{VERSION 6 0 "IBM INTEL NT" "6.0" } {USTYLETAB {CSTYLE "Maple Input" -1 0 "Courier" 0 1 255 0 0 1 0 1 0 0 1 0 0 0 0 1 }{CSTYLE "2D Math" -1 2 "Times" 0 1 0 0 0 0 0 0 2 0 0 0 0 0 0 1 }{CSTYLE "Hyperlink" -1 17 "" 0 1 0 128 128 1 2 0 1 0 0 0 0 0 0 1 }{CSTYLE "2D Comment" 2 18 "" 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 1 } {CSTYLE "2D Output" 2 20 "" 0 1 0 0 255 1 0 0 0 0 0 0 0 0 0 1 } {CSTYLE "Blue Emphasis" -1 256 "Times" 0 0 0 0 255 1 0 1 0 0 0 0 0 0 0 1 }{CSTYLE "Green Emphasis" -1 257 "Times" 1 12 0 128 0 1 0 1 0 0 0 0 0 0 0 1 }{CSTYLE "Maroon Emphasis" -1 258 "Times" 1 12 128 0 128 1 0 1 0 0 0 0 0 0 0 1 }{CSTYLE "Dark Red Emphasis" -1 259 "Times" 1 12 128 0 0 1 0 1 0 0 0 0 0 0 0 0 }{CSTYLE "Purple Emphasis" -1 260 "Times " 1 12 102 0 230 1 0 1 0 0 0 0 0 0 0 0 }{CSTYLE "Red Emphasis" -1 261 "Times" 1 12 255 0 0 1 0 1 0 0 0 0 0 0 0 0 }{CSTYLE "Grey Emphasis" -1 262 "Times" 1 12 96 52 84 1 0 1 0 0 0 0 0 0 0 0 }{CSTYLE "Magenta E mphasis" -1 263 "Times" 1 12 200 0 200 1 0 1 0 0 0 0 0 0 0 0 }{CSTYLE "Orange Emphasis" -1 264 "Times" 1 12 225 100 10 1 0 1 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 265 "" 0 1 0 0 0 0 1 0 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 266 "" 0 1 0 0 0 0 1 0 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 267 "" 0 1 0 0 0 0 1 0 0 0 0 0 0 0 0 0 }{CSTYLE "" 261 268 "" 1 18 0 0 0 0 0 0 0 0 0 0 0 0 0 0 }{PSTYLE "Normal" -1 0 1 {CSTYLE "" -1 -1 "Times" 1 12 0 0 0 1 2 2 2 2 2 2 1 1 1 1 }1 1 0 0 0 0 1 0 1 0 2 2 0 1 }{PSTYLE "Headi ng 1" -1 3 1 {CSTYLE "" -1 -1 "Times" 1 18 0 0 128 1 2 1 2 2 2 2 1 1 1 1 }1 1 0 0 8 4 3 0 3 0 2 2 0 1 }{PSTYLE "Heading 2" -1 4 1 {CSTYLE " " -1 -1 "Times" 1 14 128 0 0 1 2 1 2 2 2 2 1 1 1 1 }1 1 0 0 8 2 1 0 1 0 2 2 0 1 }{PSTYLE "Maple Output" -1 11 1 {CSTYLE "" -1 -1 "Times" 1 12 0 0 0 1 2 2 2 2 2 2 1 1 1 1 }3 3 0 0 0 0 1 0 1 0 2 2 0 1 }{PSTYLE " Maple Output" -1 12 1 {CSTYLE "" -1 -1 "Times" 1 12 0 0 0 1 2 2 2 2 2 2 1 1 1 1 }1 3 0 0 0 0 1 0 1 0 2 2 0 1 }{PSTYLE "Maple Plot" -1 13 1 {CSTYLE "" -1 -1 "Times" 1 12 0 0 0 1 2 2 2 2 2 2 1 1 1 1 }3 1 0 0 0 0 1 0 1 0 2 2 0 1 }{PSTYLE "Normal" -1 256 1 {CSTYLE "" -1 -1 "Times" 1 12 0 0 0 1 2 2 2 2 2 2 1 1 1 1 }3 1 0 0 0 0 1 0 1 0 2 2 0 1 } {PSTYLE "Normal" -1 257 1 {CSTYLE "" -1 -1 "Times" 1 12 0 0 0 1 2 2 2 2 2 2 1 1 1 1 }1 1 0 0 0 0 1 0 1 0 2 2 0 1 }{PSTYLE "Normal" -1 258 1 {CSTYLE "" -1 -1 "Times" 1 12 0 0 0 1 2 2 2 2 2 2 1 1 1 1 }2 1 0 0 0 0 1 0 1 0 2 2 0 1 }} {SECT 0 {PARA 3 "" 0 "" {TEXT -1 66 "Derivation of 13 stage, combined \+ order 7 and 8 Runge-Kutta schemes" }}{PARA 0 "" 0 "" {TEXT -1 46 "by P eter Stone, Gabriola Island, B.C., Canada " }}{PARA 0 "" 0 "" {TEXT -1 18 "Version: 7.10.2011" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 8 "restart;" }}}{EXCHG {PARA 0 "> " 0 " " {MPLTEXT 1 0 1 ";" }}}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 58 "load proc edures for constructing Runge-Kutta schemes etc. " }}{PARA 0 "" 0 "" {TEXT -1 18 "The Maple m-files " }{TEXT 262 9 "butcher.m" }{TEXT -1 2 ", " }{TEXT 262 7 "roots.m" }{TEXT -1 6 " and " }{TEXT 262 6 "intg.m " }{TEXT -1 33 " are required by this worksheet. " }}{PARA 0 "" 0 "" {TEXT -1 134 "They can be read into a Maple session by commands simila r to those that follow, where each file path gives the location of the m-file." }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 102 "read \"C:\\\\Ma ple/procdrs/butcher.m\";\nread \"C:\\\\Maple/procdrs/roots.m\";\nread \+ \"C:\\\\Maple/procdrs/intg.m\";" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}{PARA 0 " " 0 "" {TEXT -1 27 "#==========================" }}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 27 "Relations between the nodes" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 84 "The Runge-Kutta schemes c onsidered in this worksheet are such that the nodes nodes " } {XPPEDIT 18 0 "c[3]" "6#&%\"cG6#\"\"$" }{TEXT -1 3 ", " }{XPPEDIT 18 0 "c[4]" "6#&%\"cG6#\"\"%" }{TEXT -1 3 ", " }{XPPEDIT 18 0 "c[5]" "6# &%\"cG6#\"\"&" }{TEXT -1 7 " and " }{XPPEDIT 18 0 "c[6];" "6#&%\"cG6 #\"\"'" }{TEXT -1 27 " satisfy certain relations." }}{PARA 0 "" 0 "" {TEXT -1 63 "These arise from the stage order conditions for stages 4 \+ to 6. " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 395 "SO_eqs := [op(StageOrderConditions(2,4..6,'expanded' )),op(StageOrderConditions(3,4..6,'expanded')),\n op(Stage OrderConditions(4,6..6,'expanded'))]:\n\ne_1 := \{seq(a[i,2]=0,i=4..6) ,seq(a[i,3]=0,i=6..6)\}:\nnode_eqs := subs(e_1,SO_eqs):\nnops(node_eqs );\nnops(indets(node_eqs));\nsol := solve(\{op(node_eqs)\},indets(node _eqs) minus \{c[4],c[6]\}):\nnode_eqns := \{seq(c[i]=subs(sol,c[i]),i= [3,5])\};" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#\"\"(" }}{PARA 11 "" 1 " " {XPPMATH 20 "6#\"\"*" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%*node_eqns G<$/&%\"cG6#\"\"$,$*&#\"\"#F*\"\"\"&F(6#\"\"%F/F//&F(6#\"\"&,$*&#F/F.F /*(&F(6#\"\"'F/,&*&F2F/F0F/!\"\"*&F*F/F;F/F/F/,&*&F*F/F0F/F@*&F.F/F;F/ F/F@F/F/" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 13 "Thus we have " }{XPPEDIT 18 0 "c[3]=2/3" "6#/&%\"cG6#\"\"$*&\"\"# \"\"\"F'!\"\"" }{TEXT -1 1 " " }{XPPEDIT 18 0 "c[4]" "6#&%\"cG6#\"\"% " }{TEXT -1 16 " and the nodes " }{XPPEDIT 18 0 "c[4];" "6#&%\"cG6#\" \"%" }{TEXT -1 3 ", " }{XPPEDIT 18 0 "c[5];" "6#&%\"cG6#\"\"&" } {TEXT -1 3 ", " }{XPPEDIT 18 0 "c[6];" "6#&%\"cG6#\"\"'" }{TEXT -1 31 " are related by the equation: " }}{PARA 256 "" 0 "" {TEXT -1 1 " \+ " }{XPPEDIT 18 0 "3*c[6]^2-4*c[6]*c[4]-4*c[5]*c[6]+6*c[5]*c[4] = 0" "6 #/,**&\"\"$\"\"\"*$&%\"cG6#\"\"'\"\"#F'F'*(\"\"%F'&F*6#F,F'&F*6#F/F'! \"\"*(F/F'&F*6#\"\"&F'&F*6#F,F'F4*(F,F'&F*6#F8F'&F*6#F/F'F'\"\"!" } {TEXT -1 2 ". " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}}{PARA 0 "" 0 "" {TEXT 268 20 "____________ ________" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 25 "#========================" }} {SECT 1 {PARA 4 "" 0 "" {TEXT -1 22 "Enright-Verner scheme " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 216 "See: The Relat ive Efficiency of Alternative Defect Control Schemes for High-Order Co ntinuous Runge-Kutta Formulas\n W. H. Enright SIAM Journal on N umerical Analysis, Vol. 30, No. 5. (Oct., 1993), pp. 1419-1445." }} {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 15 "The nodes are: " }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "c[2] = 139/ 2500" "6#/&%\"cG6#\"\"#*&\"$R\"\"\"\"\"%+D!\"\"" }{TEXT -1 3 ", " } {XPPEDIT 18 0 "c[3] = 473499/4616000" "6#/&%\"cG6#\"\"$*&\"'*\\t%\"\" \"\"(+gh%!\"\"" }{TEXT -1 3 ", " }{XPPEDIT 18 0 "c[4] = 1420497/92320 00" "6#/&%\"cG6#\"\"%*&\"((\\?9\"\"\"\"(+?B*!\"\"" }{TEXT -1 3 ", " } {XPPEDIT 18 0 "c[5] = 1923/5000" "6#/&%\"cG6#\"\"&*&\"%B>\"\"\"\"%+]! \"\"" }{TEXT -1 3 ", " }{XPPEDIT 18 0 "c[6] = 923/2000" "6#/&%\"cG6# \"\"'*&\"$B*\"\"\"\"%+?!\"\"" }{TEXT -1 3 ", " }{XPPEDIT 18 0 "c[7] = 769/5000" "6#/&%\"cG6#\"\"(*&\"$p(\"\"\"\"%+]!\"\"" }{TEXT -1 2 ", " }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "c[8] = 8571/10000" "6#/&%\"cG6#\"\")*&\"%r&)\"\"\"\"&++\"!\"\"" }{TEXT -1 3 ", " } {XPPEDIT 18 0 "c[9] = 14280181565693441800/15023510624554266963" "6#/& %\"cG6#\"\"**&\"5+=W$pl:=!G9\"\"\"\"5jpEaXi5N-:!\"\"" }{TEXT -1 3 ", \+ " }{XPPEDIT 18 0 "c[10] = 3611/5000" "6#/&%\"cG6#\"#5*&\"%6O\"\"\"\"%+ ]!\"\"" }{TEXT -1 3 ", " }{XPPEDIT 18 0 "c[11] = 15/16" "6#/&%\"cG6# \"#6*&\"#:\"\"\"\"#;!\"\"" }{TEXT -1 3 ", " }{XPPEDIT 18 0 "c[12] = 1 " "6#/&%\"cG6#\"#7\"\"\"" }{TEXT -1 3 ", " }{XPPEDIT 18 0 "c[13]=1" " 6#/&%\"cG6#\"#8\"\"\"" }{TEXT -1 2 ". " }}{PARA 257 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}{PARA 0 "" 0 "" {TEXT -1 52 "#---------------------------------------------------" }} {SECT 1 {PARA 4 "" 0 "" {TEXT -1 28 "checking the combined scheme" }} {PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 36 "coefficients of the comb ined scheme " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 9696 "ee := \{c[2]=139/2500,\nc[3]=473499/4616000,\nc[ 4]=1420497/9232000,\nc[5]=1923/5000,\nc[6]=923/2000,\nc[7]=769/5000,\n c[8]=8571/10000,\nc[9]=14280181565693441800/15023510624554266963,\nc[1 0]=3611/5000,\nc[11]=15/16,\nc[12]=1,\nc[13]=1,\n\na[2,1]=139/2500,\na [3,1]=94226774499/11846945536000,\na[3,2]=224201303001/2369389107200, \na[4,1]=1420497/36928000,\na[4,2]=0,\na[4,3]=4261491/36928000,\na[5,1 ]=3590880889914277/9341720958375000,\na[5,2]=0,\na[5,3]=-1122077307242 443/778476746531250,\na[5,4]=1683359084698258/1167715119796875,\na[6,1 ]=3279216863/71027928000,\na[6,2]=0,\na[6,3]=0,\na[6,4]=1418466643672/ 6146756958375,\na[6,5]=725783021041/3932390759616,\na[7,1]=22628776819 090891/378192197956950000,\na[7,2]=0,\na[7,3]=0,\na[7,4]=1163268788376 70912/1047320101758291075,\na[7,5]=-21022506263989/614436056190000,\na [7,6]=182198401/10649112500,\na[8,1]=-150702609045465151/2801423688570 00000,\na[8,2]=0,\na[8,3]=0,\na[8,4]=-47048174572430533795112/67815739 75400382234375,\na[8,5]=-29670244019740727/6363654238520000,\na[8,6]=5 03350581600824913/125990342179812500,\na[8,7]=9,\na[9,1]=-160459180355 4452653108566670820097679909295434905910255443941805885523703480313807 6235914485211856830024864166368323768200/98294831577286648441985440349 2165597537825783008095189369352011632206843937744670849326949363324414 6421522730587295739613,\na[9,2]=0,\na[9,3]=0,\na[9,4]=-235853914788187 7873039445811751103289867403691705393271280994482043263320271984518766 662589229245453919623760032009543680000000/217835526181955565558022800 3414294640863851721229066080313495435838630220158871171685663384701630 90032498196064101837445053273,\na[9,5]=-991136638972626168678903371416 4561000939004055351649246830581228024297073540333828269473981586837653 24439618282500000000/7984814200884600278929892522760577519033126919472 6743910364273272231784282184770467794155269096224513726772081370189773 ,\na[9,6]=994112798212104133871493524972117858295471493586969526467810 33905129048593757052549024957474512389085654050445280000000/1021975007 1154355529199360565479179548497227516418867455143546824991883616697162 903707552409044338358355837306818391433,\na[9,7]=194327599672380134095 8982917199129613636780737930235250070813284250984315744488097793107325 32821200046895000000000/1199601148822772264965667393113649089125646329 2620473601841875115170259043531987881275965965525693289412424400177,\n a[9,8]=-47381438676841221895938162441994505404833843721635499519905253 87550768038015218275414120082248510000000000/4562738155611920982891652 8385326434273376137158228892503158792567081801761628344925618992749059 885819540261,\na[10,1]=10050976387926430682409615346304117463662936424 8095333923106653001873/\n 2294903240076446280423617562174361558 32461488260089524115475000000000,\na[10,2]=0,\na[10,3]=0,\na[10,4]=350 76261889213578261995286390053983221937920015616/\n 890366240305 2468890234321680409039895089390971875,\na[10,5]=2987705347224854522778 2869189767925950557009/\n 1044370915864536295808990574013420660 6110000,\na[10,6]=-726020251827988894428939665538442860127767700195888 38776297031451/4091843938040500792607167383471827610617886385154824480 0289187500,\na[10,7]=-153229120638643705121301459884920986054865029941 07816190/3130069400645604876183669549605539076333579290524919889,\na[1 0,8]=66085154677219418645471125072555541174985695924/31022273664823506 2495097951962638800384203417327,\na[10,9]=-334756546189656076254902666 7823123036634583052726552531003001623087575523942032472860095736887713 2012320553021/56360718148650508277541923761487301604316912513035933026 9370345097328575740250457475506596993780051575000000000,\na[11,1]=-510 1097760197841615137571256611109219669965728737004453654940399435749927 779/346025479006521849802539491211365444054798764102117496898501075318 3759952128,\na[11,2]=0,\na[11,3]=0,\na[11,4]=-739282853758412257967453 242147288028248514875000/\n 6724418287565482909832394371337420 7709631980757,\na[11,5]=-304301954438407952266341438991435702455078125 /\n 26817588459995310678515661957421952038871616,\na[11,6]=794 4893992399893116476631697520363855501654462308622798446100145004371780 765625/\n 88700516160292865925282760403576865729843588493948975 4462273147402132735951354,\na[11,7]=2714666628898351286148107969160694 88062485834784939879774912795880859375/\n 170800578682088075849 11919572395871967414618088449127584294090356526117,\na[11,8]=-54767163 9173493535187249719091874257724281347519101322819710937500/\n \+ 5545897345777551056363997101744668576293320106258184122878184564689,\n a[11,9]=27880505422945647305191478577013040105627752107138627482025385 2826023602554035252878888463451556710568650135086751734733498388408614 837563/570327897589938424716739897067258193159963418613722438239821039 8440764234613614101505591454120074773344256118618572821579031366188464 7470336,\na[11,10]=-7641031089620713267943821483116886435546875/\n \+ 1865115549729059107236140767953375513727732867,\na[12,1]=-217329 6165244568434534168496725754283210370856714048295955495704392191998074 219/\n 8263297098171974682539128149962767212977192718730471214 96939265708086637102000,\na[12,2]=0,\na[12,3]=0,\na[12,4]=-11764858893 34472345397948024769479865991267657667189808793600000/\n 12823 8274125745784857395023683055391720808636521055761250085619,\na[12,5]=- 901641526273088667293677983960434306145731566007759375000/\n 4 7006051321531726477650870675115597779536141098588395119,\na[12,6]=4461 9197709345843038609995810290222092713734357503573021289399188441075277 893825000/\n 30472268298677596678056782370835849160284566290690 14496162891378892137605913688013,\na[12,7]=948339742665210716931560767 459705347432692833888702670755668800833857753125000/\n 54100203 068523816523920547203712158257383235486719126879290708372319555115953, \na[12,8]=-46935998653078424236784116396799798528099205206247952269403 7718750000/\n 126200006879774002960489363671712464681453610603 0908513651797310900389,\na[12,9]=-664127481006260866245417353167879111 7254386081601341297887155553784655731178130017572382606930519018125943 41608922016461968723754067215934799054852630446938063/9474053136757793 2788646204486183484687947386153958347999510116829111616884404043875158 4392785913224239894029843821464363224405440877235651675827150051218506 000,\na[12,10]=1245946889494312367182787180786885712459679616415841517 13065625000/\n 24422661199951239145999264873446308735658803040 03136401896150897137,\na[12,11]=56106474231011098569963962401476898179 3940948761680464947352/\n 671379391526329055316725343694451428 305592235340194400714475,\na[13,1]=21125373654396053178422144002949396 00845864356126794907105288476157299972777/\n 979112999257634895 4381966935712572673760150144277222568673000574989723295600,\na[13,2]=0 ,\na[13,3]=0,\na[13,4]=63401732568650174882469655273037557867024014649 473659852800000/\n 75974371795646999000568266827900958765018677 94005624909986891,\na[13,5]=608676173779740463756063981922330580225886 3489403125000/\n 2784859079002788542258846760345051095481316496 802932391,\na[13,6]=-2559665601805637409198888850698428172565517218973 577146475607855715703433900000/\n 1517075781450111169555356439 342115045941218379260955323225394329122945220575603,\na[13,7]=-3562962 724414944757891560164749973139486260772587248231213362868690625000/\n \+ 40897663917896889951622204551861984944595571459523457320790174 0578867341,\na[13,8]=2727477409394889057232717284650541132862437801524 9298079900000000/\n 1115922486918625141794316350997246426030217 325603407120793811703663,\na[13,9]=15176499802732877948107114809286897 6296994135944658935524468324347086887398053746271794553089124289218801 9850453060256773054273287757818339/17931096351403549517324822951619492 6905902591095541696921117714941628050882978212428484921029234475322157 42446890559805801281057084758250800,\na[13,10]=36524642706569597276845 612368244490741747970924877885921875000/\n 6720450834551542949 1838411146746729660269255184823759889942481,\na[13,11]=0,\na[13,12]=0, \n\nb[1]=55284231195707975647126708111723/1258814250475014554517603612 114000,\nb[2]=0,\nb[3]=0,\nb[4]=0,\nb[5]=0,\nb[6]=10165204821428220551 8610445783893750000000/289586658278060310247144250081091360673509,\nb[ 7]=6219731958882093270433753490048828125000000/25268792314562899182186 819800834272764054341,\nb[8]=47152588201493258732167370792968750000000 0/523728817391198795728810839649314044495553,\nb[9]=639984196590745029 6097946702704438053351349956217971614578815398125835488217018355729426 1050806789914954901252698438375102943237148428019253408419/14067383878 2241599806769359995003242677277239475163095364640616831300634376164529 51326819824531063499969576146304645311203502485409773697867038000,\nb[ 10]=594562755257530592552224345703125000000/12380272091032768230143141 7435953442122031,\nb[11]=-1886691133979705639959153870454656/397947585 885835383951563312487675,\nb[12]=-50061468875139778913910254637881/141 186641936036184819986782313781,\nb[13]=0,\n\n`b*`[1]=36355439922940214 75202312550589/83920950031667636967840240807600,\n`b*`[2]=0,\n`b*`[3]= 0,\n`b*`[4]=0,\n`b*`[5]=0,\n`b*`[6]=9147255330833622123302075012200000 0000/270389036674192633283981559366098375979,\n`b*`[7]=312457108791068 59500854236329453125000000/125747467177230198813995913261775760852127, \n`b*`[8]=7580382785455138868782239796250000000000/3387300808997803156 4549954803189465564389,\n`b*`[9]=-126466820753477200038941690908829580 6823076185957298091511366356638145411319432807599860924173256812108555 319505561297378438513/314581336823638793763598373286113109900992592339 3634680707983641010013290201378987758381956291343572387481003787557675 0158649200,\n`b*`[10]=2363202400377740499662213515625000000/1916747498 2246118950523520271861502109,\n`b*`[11]=0,\n`b*`[12]=0,\n`b*`[13]=2965 876353908674111604434604609/47062213978678728273328927437927\}:" }}} {PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 20 "printed coe fficients" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 357 "subs(ee,matrix([seq([c[i],seq(a[i,j],j=1..i-1),``$(8 -i)],i=2..8)]));\nfor ii from 9 to 13 do\n print(``);\n print(c[ii ]=subs(ee,c[ii])); \n for jj to ii-1 do\n print(a[ii,jj]=subs(e e,a[ii,jj]));\n end do:\nend do:print(``);\nfor ii to 13 do\n prin t(b[ii]=subs(ee,b[ii]));\nend do:print(``);\nfor ii to 13 do\n print (`b*`[ii]=subs(ee,`b*`[ii]));\nend do:" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#K%'matrixG6#7)7*#\"$R\"\"%+DF(%!GF+F+F+F+F+7*#\"'*\\t%\"(+gh%#\" ,*\\unA%*\"/+g`Xp%=\"#\"-,II,UA\".+s5*QpBF+F+F+F+F+7*#\"((\\?9\"(+?B*# F8\")+!Gp$\"\"!#\"(\"\\hUF;F+F+F+F+7*#\"%B>\"%+]#\"1xU\"**)3)3f$\"1+]P e4sT$*F<#!1VCC2t2A6\"0]7`YnZy(#\"1e#)p%3fLo\"\"1voz>^rn6F+F+F+7*#\"$B* \"%+?#\"+jo@zK\",+!Gz-rFoxGE#\"3++&p&z>#>y$FcgV9'#\"*,%)>#=\",+D6\\1\"F+7*#\"%r&)\"&++\"#!3^ ^YX!4Eq]\"\"3+++d)oB9!GFSCq' H\"1++_QUljj#\"38\\#3g\"e]L]\"3+D\")z@M!*f7\"\"*Q)pprint226\"" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#%!G" }}{PARA 11 "" 1 "" {XPPMATH 20 "6 #/&%\"cG6#\"\"*#\"5+=W$pl:=!G9\"5jpEaXi5N-:" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/&%\"aG6$\"\"*\"\"\"#!er+#oPKoj;k[-Io&=@&[9fBw!QJ![.P_& )e!=%RWb-\"f!\\V&H4*zw4?3nm&3Jl_Wb.=f/;\"dr8'RdH(eIF_@k9WKj$\\pK\\3nWx $R%o?K;,_$p$*=&43IyDy`(fl@\\.W&)>W[mGx:$[H)*" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/&%\"aG6$\"\"*\"\"#\"\"!" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/&%\"aG6$\"\"*\"\"$\"\"!" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/&% \"aG6$\"\"*\"\"%#!jr+++!oV&4?.gPi>RXX#H#*eimw=X)>F?LEV?[%*4GrKR0)\\K+4j,ZQjcor6()e,AI'QeV&\\8.3m!H 7s^Q'3k%H9M+G-ebcb>=Eb$y@" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/&%\"aG6$ \"\"*\"\"&#!cr++++DG='RWKlPoe\")RZp#GQLSN2(HC!G7eIoC\\;NbS+R4+hX;9P.*y 'ohis*Qm8\"**\"brt(*=q83sns8XA'4p_:%zn/x%=#G%yJAFtUO5RuEZ>p7L!>vdgF_#* )H*y-g%)3?9[)z" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/&%\"aG6$\"\"*\"\"'# \"br+++!GX/0ac3*Q7XZd\\-\\D0dPf[!H^!R.\"yYE&ppe$\\raHey6s\\_$\\rQ8/@@) z7T**\"brL9R=oIPeNe$QV/4Cb2P!H;(p;O)=*\\#oaV^Xn)=k^Fs\\[&z\"zacg$*>HbN a62](>-\"" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/&%\"aG6$\"\"*\"\"(#\"]r+ +++]*o/+7#G`K2Jz(4)[WdJ%)4D%G832]_BIzt!yOOhH\"*>\"\\rx ,SCCT*G$pDb'f'fF\")y)>`V!f-<:^(=%=gt/i#HjkD\"*3\\O6$Rnc'\\EsF#)[6g*>\" " }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/&%\"aG6$\"\"*\"\")#!eq+++++^[A3?T Tv#=_,Q!o2b(QD0*>&*\\N;sVQ$[S0X*>Wi\"Qf*=ATonQ9QZ\"fqh-a>e))f!\\F**=c# \\MG;w,=3nDzeJ]#*)G#er8wLFMkK&QGl\"*G)4#>hb\"QFc%" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%!G" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/&%\"cG6#\"#5#\" %6O\"%+]" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/&%\"aG6$\"#5\"\"\"#\"`ot= +`m5BRL&4[UOHmju6/jM:'4CoIk#zQw405\"`o++++]Z:T_*3g#)[hC$e:Ou@cA$)R0!R'G&*>EyN@*)= Ew]$\"Rv=(4R*3&*)R!4/o@VB!*)oC0.Cm.*)" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/&%\"aG6$\"#5\"\"&#\"M4qb]f#zw*=pGyF_a[AZ`q()H\"M++61m?M,u0**3eHOX' e\"4PW5" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/&%\"aG6$\"#5\"\"'#!\\o^9.( Hw(Q))e>+xwF,'GWQbmR*GW*)))z#=D?gs\"\\o+v=*G+[C[:&Q')yh5w#=Z$Qnrg#z+0/ QR%=4%" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/&%\"aG6$\"#5\"\"(#!Y!>;y5%* H]'[0')4#\\))f9I@^qV'Q17HK:\"X*))>\\_!HzNLw!Rbg\\&pO=w[gX1Sp+8$" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#/&%\"aG6$\"#5\"\")#\"PCfp&)\\sna^3m\"QFtT.UQ+)QE'>&z4&\\i]B[mtA-J" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/&%\"aG6$\"#5\"\"*#!iq@Ib?B,Kr()ot&4gGZK?%R_vv3B;+.5`_ls_IeMm.BJ #ymE!\\Dwgl*=YlvM$\"jq++++]d^+y$*pf1bZd/DSddGt4X.Pp-Lf.8D\"pJ/;I([hP#> ax#30l[\"=2Oc" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%!G" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/&%\"cG6#\"#6#\"#:\"#;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/&%\"aG6$\"#6\"\"\"#!gozx#*\\dV*RS\\l`W+P(Gd'*p'>#46hc7dP^hTy>gx 4,^\"goG@&*fP=`2,&)*o\\<@5k()zaSWl8@\"\\RD!)\\=_1!za-Y$" }}{PARA 11 " " 1 "" {XPPMATH 20 "6#/&%\"aG6$\"#6\"\"#\"\"!" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/&%\"aG6$\"#6\"\"$\"\"!" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/&%\"aG6$\"#6\"\"%#!Q+]([^[#G!)GZ@C`u'zD7%eP&GGR(\"Pd2)>j4x?uLrV RK)4H[lvG=Ws'" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/&%\"aG6$\"#6\"\"&#!N D\"y]X-dV\"**Q9MmA&zSQW&>I/$\"M;;()Q?&>Ud>m:&y1J&**f%)e(eRs&>> \"\\e2)3#oy0!3<" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/&%\"aG6$\"#6\"\")# !]o+v$4r>GK,\">vM\"GCxDu=4>(\\s=NN\\t\"R;nZ&\"^o*okX=yG7%=ei5?LHw&oYu, r*ROc5bxdM(*ea&" }}{PARA 12 "" 1 "" {XPPMATH 20 "6#/&%\"aG6$\"#6\"\"*# \"esjv$[h3%)Q)\\LZt^n3N,lo0rc:Xj%)))yGDNSb-O-EG&QD?[F'Qr5_xi0,/8qdy9>0 tkXHU00)y#\"gsO.ZZY)=m8.z:#Gd='=hDWLxu+7a9f0:59OhMUwS%)R5#)R#QCs8'=M'* fJ>es1(*)RnrC%Q**e(*yKq&" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/&%\"aG6$ \"#6\"#5#!LvoaNk)o6$[@Q%zE82i*3J5k(\"OnGtFP^vL&zwShB2\"f!H(\\b6l=" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#%!G" }}{PARA 11 "" 1 "" {XPPMATH 20 "6 #/&%\"cG6#\"#7\"\"\"" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/&%\"aG6$\"#7 \"\"\"#!jo>U2)*>>#R/d\\bfH[Src3P5KGads'\\oT`M%oXClhHt@\"io+?5Pm33dERp \\@r/t=F>xH@nF'*\\\"G\"RDou><)4(Hj#)" }}{PARA 11 "" 1 "" {XPPMATH 20 " 6#/&%\"aG6$\"#7\"\"#\"\"!" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/&%\"aG6$ \"#7\"\"$\"\"!" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/&%\"aG6$\"#7\"\"%#! [o++g$z3)*=nwln7*f')z%pZ-[zRXBZM$*)e[w6\"jn>c3]7wb5_O'33s\"RbIoB]Rd[yX d7u#Q#G\"" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/&%\"aG6$\"#7\"\"&#!Z+]Pf x+m:tXhIM/'R)zn$Hn')3ti_T;!*\"Y>^R)e)4Th`zxf:^nq3lxksJ:K^g+Z" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/&%\"aG6$\"#7\"\"'#\"^p+]#Q*yFv5W)=*R*G@Id.v NMPr#4A-H5e**4'QI%eM4x>>Y%\"]p8!)o8fgP@*)y8*G;'\\9!p!HmXGg\"\\e$3P#yc! ymfx')HoAZI" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/&%\"aG6$\"#7\"\"(#\"io +]7`x&Q$3!)ocvqEq))Q$GpKuM0(fuwg:$pr5_mU(R$[*\"ho`f6b&>BP32Hzo7>n[NKQd #e@r.sa?R_;Q_oI?+T&" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/&%\"aG6$\"#7\" \")#!`o++v=x.%pA&zC1_?*4G&)z*z'R;TyOUUyIl)*f$p%\"ao*Q+4J(z^O^34.1h`9ok CrrOO*[gH+u(zo++i7" }}{PARA 12 "" 1 "" {XPPMATH 20 "6#/&%\"aG6$\"#7\" \"*#!jtj!QpWIE&[0*zMf@nSvB(o>Y;?#*3;M%f7=!>0$pg#Qsv,I\"y6tbYy`b:()yHT8 g\"3'Qas6\"zy;`tTXi'3E15[FTm\"jt+g]=70]r#en^cBx3W0WAjVY@Q%)HS*)RUA8fy# R%e^(Q//W)oh6\"Ho65&**zMeR:'QZzo%[$='[/ik)yKzdn8`SZ*" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/&%\"aG6$\"#7\"#5#\"]o+]ilIr^TeT;'z'fCr&)oy!=(y#=nBJ %\\*)o%fC\"\"^oPr*3:'*=SOJ+/.)ect3jWt[E**f9R7&**>hEUC" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/&%\"aG6$\"#7\"#6#\"gn_t%\\Y!oh([4%Rz\")*oZ,C'R'*p &)465Buk5c\"gnvWr+W>S`B#f0$G9X%pV`s;`0Hj_\"Rz8n" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%!G" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/&%\"cG6#\"#8\" \"\"" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/&%\"aG6$\"#8\"\"\"#\"goxF(**H dhZ)G0r!\\zEhNke%3gR\\H+W@UyJ0'RaOPD6#\"go+cHB(*)\\d+InoDAxU9],wtEd7d$ p'>Qa*[jd#**H6z*" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/&%\"aG6$\"#8\"\"# \"\"!" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/&%\"aG6$\"#8\"\"$\"\"!" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#/&%\"aG6$\"#8\"\"%#\"in++!G&)fOZ\\Y,Cq 'ybPIFb'pC)[<]'oDt,M'\"hn\"*o)*4\\i0Szn=]we4!z#oEo0+**pk&zrV(f(" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#/&%\"aG6$\"#8\"\"&#\"X+]7.%*[j)eA!eIB# >)R1cPYS(zP4uj0=gl'fD\"jo.cd?_%H7HVRDKKb4Ez$=7%f/:@MRkNb&p66]9yvq^\"" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#/&%\"aG6$\"#8\"\"(#!do+]i!poGO87B[ses2 E'[RJ(*\\Z;g:*yvW\\TCF'Hc$\"coTt')y0u,z?tXB&f9d&fW\\)>'=b/Ai^**)o*y\"R m(*3%" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/&%\"aG6$\"#8\"\")#\"\\o++++* z!)H\\_,yViG8T0l%G!)=#*GC\"*3`Xzriu`!)R()o3ZVKoW_N*eY%f8%*pHw*oG4[62\"[z(Gt-)* \\w^\"\"gs+3DeZ3d5G,e!)f0*oWUd@KvWBH5#\\[GC@yH)30G;%\\r<6#ppTb4\"f-f!p #\\>;&H#[K<&\\NS^j4Jz\"" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/&%\"aG6$\" #8\"#5#\"in+](=#f)y([#4(zuT2\\W#oBhXoF(fplqUY_O\"in\"[U**))fP#[=b#p-mH nuY6TQ=\\Ha^X$3X?n" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/&%\"aG6$\"#8\"# 6\"\"!" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/&%\"aG6$\"#8\"#7\"\"!" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#%!G" }}{PARA 11 "" 1 "" {XPPMATH 20 "6 #/&%\"bG6#\"\"\"#\"AB<63n7Zc(zq&>JUGb\"C+S67Ogt>i\"MTV0kFFM3!)>o=#=** Gc9Bzo_#" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/&%\"bG6#\"\")#\"K++++voHz qt;K(eK\\,#)e_r%\"K`b\\WSJ\\'R3\")Gdz)>\"R<)GP_" }}{PARA 12 "" 1 "" {XPPMATH 20 "6#/&%\"bG6#\"\"*#\"]t>%3MD>!G%[rBVH5v$Q%)p_7!\\&\\\"**y13 0hUHdN=q@)[Ne7)R:)yXhrz@c*\\8N`!QWq-n%z4'H]u!f'>%)*R'\"]t+!Qq'ypt(4a[- N?6`k/j9w&p**\\j5`C)>oK^HX;wVj+8$ohSYO&4j^ZRsFxEC.]**f$pn!)*fTAyQQnS\" " }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/&%\"bG6#\"#5#\"H+++DJqXVA_DfIvDbF c%f\"KJ?7UM&fV<9V,BoF.\"4s-Q7" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/&%\" bG6#\"#6#!CcYXqQ:f*RcqzR8\"p')=\"Bvw[7Lc^RQNe)eeZzR" }}{PARA 11 "" 1 " " {XPPMATH 20 "6#/&%\"bG6#\"#7#!A\")yja-\"R\"*y(R^()o91]\"B\"y8By')*>[ =Og$>k'=T\"" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/&%\"bG6#\"#8\"\"!" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#%!G" }}{PARA 11 "" 1 "" {XPPMATH 20 "6 #/&%#b*G6#\"\"\"#\"@*e]DJ-_Z@SH#*RaNO\"A+w!3CSy'pjn;.]4#R)" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/&%#b*G6#\"\"#\"\"!" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/&%#b*G6#\"\"$\"\"!" }}{PARA 11 "" 1 "" {XPPMATH 20 "6# /&%#b*G6#\"\"%\"\"!" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/&%#b*G6#\"\"& \"\"!" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/&%#b*G6#\"\"'#\"G++++?7]2-L7 AO$3Lbs9*\"HzfP)4m$f:)RGLE>um.*Qq#" }}{PARA 11 "" 1 "" {XPPMATH 20 "6# /&%#b*G6#\"\"(#\"J+++DJXHjBa3]fo5z3rX7$\"KF@&3wvIs`b3@\"oDtT#4')*f2GV>8TX\"QmNm8^\"4)Hdf=wI #o!eH)34pT*Q+?xMv?oYE\"\"[s+#\\'e,vwb(y.5[(QsNM\"Hc>Qex)*y8?!H8+,TO)zq !oMORBf#*4!*4J6'Gt$)fjPzQO#oL\"e9$" }}{PARA 11 "" 1 "" {XPPMATH 20 "6# /&%#b*G6#\"#5#\"F+++Dc^8Am*\\SxP+C?jB\"G4@]h=F?N_]*=hC#)\\Zn\">" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#/&%#b*G6#\"#6\"\"!" }}{PARA 11 "" 1 " " {XPPMATH 20 "6#/&%#b*G6#\"#7\"\"!" }}{PARA 11 "" 1 "" {XPPMATH 20 "6 #/&%#b*G6#\"#8#\"@4YgMWg6Tn3RNwe'H\"AFzVF*GLFG(y'yR@iq%" }}}{PARA 0 " " 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}} }{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 151 "RK8_13eqs := [op(RowSumConditions(13,'expanded')),op(OrderCondi tions(8,13,'expanded'))]:\n`RK7_13eqs*` := subs(b=`b*`,OrderConditions (7,13,'expanded')):" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 " " {TEXT -1 6 "Check:" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 110 "sub s(ee,RK8_13eqs):\nmap(u->lhs(u)-rhs(u),%);\nnops(%);\nsubs(ee,`RK7_13e qs*`):\nmap(u->lhs(u)-rhs(u),%);\nnops(%);" }}{PARA 12 "" 1 "" {XPPMATH 20 "6#7`x\"\"!F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F $F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F $F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F $F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F $F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F $F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F $F$F$F$F$F$F$F$F$F$F$F$F$" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#\"$7#" }} {PARA 12 "" 1 "" {XPPMATH 20 "6#7ap\"\"!F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$ F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$ F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#\"#&)" }}}{PARA 257 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 78 "Set-up stage-order condtions to check for stage-orders from 2 to 6 inclusive. " }}{EXCHG {PARA 0 "> \+ " 0 "" {MPLTEXT 1 0 87 "for ct from 2 to 6 do\n so||ct||_13 := Stage OrderConditions(ct,13,'expanded');\nend do:" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 59 "Stages 3 to 13 have the followi ng respective stage-orders. " }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 175 "[seq([seq(expand(subs(ee,so||i||_13[j])),i=2..6)],j=1..11)]:\nmap (proc(L) local i; for i to nops(L) do if not evalb(L[i]) then break en d if end do; i end proc,%):\nsimplify(%); " }}{PARA 11 "" 1 "" {XPPMATH 20 "6#7-\"\"#\"\"$F%\"\"%F&F&F&F&F&F&F&" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 27 "The simplifying condition s:" }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "Sum(b[i]*a[i,j] ,i = j+1 .. 13) = b[j]*(1-c[j]);" "6#/-%$SumG6$*&&%\"bG6#%\"iG\"\"\"&% \"aG6$F+%\"jGF,/F+;,&F0F,F,F,\"#8*&&F)6#F0F,,&F,F,&%\"cG6#F0!\"\"F," } {TEXT -1 3 ", " }{XPPEDIT 18 0 "j=1" "6#/%\"jG\"\"\"" }{TEXT -1 9 " . . 12, " }}{PARA 0 "" 0 "" {TEXT -1 8 "(where " }{XPPEDIT 18 0 "c[1]= 0" "6#/&%\"cG6#\"\"\"\"\"!" }{TEXT -1 17 " ) are satisfied." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 151 "[ Sum(b[i]*a[i,1],i=1+1..13)=b[1],seq(Sum(b[i]*a[i,j],i=j+1..13)=b[j]*(1 -c[j]),j=2..12)]:\nmap(u->lhs(u)-rhs(u),subs(ee,eval(subs(Sum=add,%))) );\nnops(%);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#7.\"\"!F$F$F$F$F$F$F$F $F$F$F$" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#\"#7" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 27 "The simplifying condition s:" }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "Sum(`b*`[i]*a[i ,j],i = j+1 .. 13) = `b*`[j]*(1-c[j]);" "6#/-%$SumG6$*&&%#b*G6#%\"iG\" \"\"&%\"aG6$F+%\"jGF,/F+;,&F0F,F,F,\"#8*&&F)6#F0F,,&F,F,&%\"cG6#F0!\" \"F," }{TEXT -1 3 ", " }{XPPEDIT 18 0 "j=1" "6#/%\"jG\"\"\"" }{TEXT -1 9 " . . 12, " }}{PARA 0 "" 0 "" {TEXT -1 8 "(where " }{XPPEDIT 18 0 "c[1]=0" "6#/&%\"cG6#\"\"\"\"\"!" }{TEXT -1 17 " ) are satisfied." } }{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 163 "[Sum(`b*`[i]*a[i,1],i=1+1..13)=`b*`[1],seq(Sum(`b*`[i]*a[i,j],i =j+1..13)=`b*`[j]*(1-c[j]),j=2..12)]:\nmap(u->lhs(u)-rhs(u),subs(ee,ev al(subs(Sum=add,%))));\nnops(%);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#7. \"\"!F$F$F$F$F$F$F$F$F$F$F$" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#\"#7" } }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 93 "We can \+ calculate the principal error norm, that is, the 2-norm of the princip al error terms. " }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 162 "errterm s8_13 := PrincipalErrorTerms(8,13,'expanded'):\nsm := 0:\nfor ct to no ps(errterms8_13) do\n sm := sm+(evalf(subs(ee,errterms8_13[ct])))^2; \nend do:\nsqrt(sm);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#$\"+4`_&H\"!#: " }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 78 "We c an also calculate the principal error norm of the order 7 embedded sch eme." }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 184 "`errterms7_13*` := \+ subs(b=`b*`,PrincipalErrorTerms(7,13,'expanded')):\nsm := 0:\nfor ct t o nops(`errterms7_13*`) do\n sm := sm+(evalf(subs(ee,`errterms7_13*` [ct])))^2;\nend do:\nsqrt(sm);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#$\"+ UuoBF!#9" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}}{PARA 0 "" 0 "" {TEXT -1 52 "#----------------- ----------------------------------" }}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 34 "construction of the order 8 scheme" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 93 "First we construct a 12 stage orde r 8 scheme starting with a consideration of stages 2 to 8. " }}{PARA 0 "" 0 "" {TEXT -1 20 "We specify the nodes" }}{PARA 256 "" 0 "" {TEXT -1 2 " " }{XPPEDIT 18 0 "c[2] = 139/2500;" "6#/&%\"cG6#\"\"#*& \"$R\"\"\"\"\"%+D!\"\"" }{TEXT -1 2 ", " }{XPPEDIT 18 0 "c[5] = 1923/5 000;" "6#/&%\"cG6#\"\"&*&\"%B>\"\"\"\"%+]!\"\"" }{TEXT -1 2 ", " } {XPPEDIT 18 0 "c[6] = 923/2000;" "6#/&%\"cG6#\"\"'*&\"$B*\"\"\"\"%+?! \"\"" }{TEXT -1 2 ", " }{XPPEDIT 18 0 "c[7] = 769/5000;" "6#/&%\"cG6# \"\"(*&\"$p(\"\"\"\"%+]!\"\"" }{TEXT -1 3 ", " }{XPPEDIT 18 0 "c[8] = 8571/10000;" "6#/&%\"cG6#\"\")*&\"%r&)\"\"\"\"&++\"!\"\"" }{TEXT -1 2 " " }}{PARA 0 "" 0 "" {TEXT -1 33 "and the zero linking coefficient s" }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "a[4,2]=0" "6#/&% \"aG6$\"\"%\"\"#\"\"!" }{TEXT -1 2 ", " }{XPPEDIT 18 0 "a[5,2]=0" "6#/ &%\"aG6$\"\"&\"\"#\"\"!" }{TEXT -1 2 ", " }{XPPEDIT 18 0 "a[6,2]=0" "6 #/&%\"aG6$\"\"'\"\"#\"\"!" }{TEXT -1 2 ", " }{XPPEDIT 18 0 "a[6,3]=0" "6#/&%\"aG6$\"\"'\"\"$\"\"!" }{TEXT -1 2 ", " }{XPPEDIT 18 0 "a[7,2]=0 " "6#/&%\"aG6$\"\"(\"\"#\"\"!" }{TEXT -1 2 ", " }{XPPEDIT 18 0 "a[7,3] =0" "6#/&%\"aG6$\"\"(\"\"$\"\"!" }{TEXT -1 2 ", " }{XPPEDIT 18 0 "a[8, 2] = 0;" "6#/&%\"aG6$\"\")\"\"#\"\"!" }{TEXT -1 2 ", " }{XPPEDIT 18 0 "a[8,3] = 0;" "6#/&%\"aG6$\"\")\"\"$\"\"!" }{TEXT -1 3 ", " } {XPPEDIT 18 0 "a[8,4]=0" "6#/&%\"aG6$\"\")\"\"%\"\"!" }{TEXT -1 2 ". \+ " }}{PARA 0 "" 0 "" {TEXT -1 48 "In addition we specify the linking co efficient " }{XPPEDIT 18 0 "a[8,7]=9" "6#/&%\"aG6$\"\")\"\"(\"\"*" } {TEXT -1 1 "." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 69 "The stage-order equations relating to these stages, such \+ that stages " }{XPPEDIT 18 0 "3,4,5,6,7,8;" "6(\"\"$\"\"%\"\"&\"\"'\" \"(\"\")" }{TEXT -1 19 " have stage orders " }{XPPEDIT 18 0 "2,3,3,4,4 ,4;" "6(\"\"#\"\"$F$\"\"%F%F%" }{TEXT -1 48 " respectively can be solv ed to obtain the nodes " }{XPPEDIT 18 0 "c[3], c[4]" "6$&%\"cG6#\"\"$& F$6#\"\"%" }{TEXT -1 67 " and the remaining non-zero linking coefficie nts for these stages. " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT 260 4 "Note" }{TEXT -1 31 ": Enright specifies the nodes \+ " }{XPPEDIT 18 0 "c[2]" "6#&%\"cG6#\"\"#" }{TEXT -1 3 ", " }{XPPEDIT 18 0 "c[5]" "6#&%\"cG6#\"\"&" }{TEXT -1 3 ", " }{XPPEDIT 18 0 "c[6]" "6#&%\"cG6#\"\"'" }{TEXT -1 3 ", " }{XPPEDIT 18 0 "c[7]" "6#&%\"cG6# \"\"(" }{TEXT -1 7 " and " }{XPPEDIT 18 0 "c[8]" "6#&%\"cG6#\"\")" } {TEXT -1 29 " in floating point form as: " }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "c[2] = 0" "6#/&%\"cG6#\"\"#\"\"!" }{TEXT -1 8 ".0556, " }{XPPEDIT 18 0 "c[5] = 0" "6#/&%\"cG6#\"\"&\"\"!" }{TEXT -1 8 ".3846, " }{XPPEDIT 18 0 "c[6] = 0" "6#/&%\"cG6#\"\"'\"\"!" } {TEXT -1 8 ".4615, " }{XPPEDIT 18 0 "c[7] = 0" "6#/&%\"cG6#\"\"(\"\"! " }{TEXT -1 8 ".1538, " }{XPPEDIT 18 0 "c[8] = 0" "6#/&%\"cG6#\"\")\" \"!" }{TEXT -1 7 ".8571. " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 458 "RSeqs := RowSumConditions(8,'expan ded'):\nSOeqs := [op(StageOrderConditions(2,8,'expanded')),op(StageOrd erConditions(3,4..8,'expanded')),\n op(StageOrderConditi ons(4,6..8,'expanded'))]:\ne1 := \{c[2]=139/2500,c[5]=1923/5000,c[6]=9 23/2000,c[7]=769/5000,c[8]=8571/10000,\n seq(a[i,2]=0,i=4..8),se q(a[i,3]=0,i=6..8),a[8,7]=9\}:\neqns := map(expand,subs(e1,[op(RSeqs), op(SOeqs)])):\nconvert(ListTools[Enumerate](%),matrix);\n``;\nindets(e qns);\nnops(%);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#K%'matrixG6#777$\" \"\"/&%\"aG6$\"\"#F(#\"$R\"\"%+D7$F-/,&&F+6$\"\"$F(F(&F+6$F6F-F(&%\"cG 6#F67$F6/,&&F+6$\"\"%F(F(&F+6$FAF6F(&F:6#FA7$FA/,(&F+6$\"\"&F(F(&F+6$F KF6F(&F+6$FKFAF(#\"%B>\"%+]7$FK/,(&F+6$\"\"'F(F(&F+6$FXFAF(&F+6$FXFKF( #\"$B*\"%+?7$FX/,*&F+6$\"\"(F(F(&F+6$F_oFAF(&F+6$F_oFKF(&F+6$F_oFXF(# \"$p(FR7$F_o/,,&F+6$\"\")F(F(\"\"*F(&F+6$F]pFAF(&F+6$F]pFKF(&F+6$F]pFX F(#\"%r&)\"&++\"7$F]p/,$*&F.F(F7F(F(,$*&#F(F-F(*$)F9F-F(F(F(7$F^p/*&FB F(F9F(,$*&F^qF(*$)FDF-F(F(F(7$\"#5/,&*&FLF(F9F(F(*&FNF(FDF(F(#\"(Hzp$ \")+++]7$\"#6/,&*&FYF(FDF(F(*&FPF(FenF(F(#\"'H>&)\"(+++)7$\"#7/,(*&F`o F(FDF(F(*&FPF(FboF(F(*&FgnF(FdoF(F(#\"'h8fF`r7$\"#8/,*#\"%@pFRF(*&F_pF (FDF(F(*&FPF(FapF(F(*&FgnF(FcpF(F(#\")T?Yt\"*++++#7$\"#9/*&FBF(F`qF(,$ *&#F(F6F(*$)FDF6F(F(F(7$\"#:/,&*&FLF(F`qF(F(*&FNF(FgqF(F(#\"+*[s.P#\"- ++++]77$\"#;/,&*&FYF(FgqF(F(*&#F_r\")+++DF(FenF(F(#\"*n/L'y\",++++S#7$ \"#/,&*&FYF(FgtF(F(*&#\"+nu66rF`uF(FenF(F(# \"-T5-$yD(\"/++++++k7$\"#?/,(*&F`oF(FgtF(F(*&FjwF(FboF(F(*&#Fju\"+++++ !)F(FdoF(F(#\"-@B$yq\\$\"1+++++++D7$\"#@/,*#\"+\"[4G4%F`uF(*&F_pF(FgtF (F(*&FjwF(FapF(F(*&FfxF(FcpF(F(#\"1\"o&)yYrmR&\"2++++++++%Q)pprint166 \"" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%!G" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#<7&%\"aG6$\"\"(\"\"\"&F%6$\"\"'\"\"%&F%6$\"\"$\"\"#&F%6 $\"\"&F,&%\"cG6#F/&F%6$F3F/&F%6$\"\")F(&F%6$F,F/&F%6$F;F,&F%6$F/F(&F%6 $F,F(&F%6$F'F3&F%6$F;F3&F%6$F0F(&F%6$F+F3&F%6$F'F,&F56#F,&F%6$F;F+&F%6 $F'F+&F%6$F3F(&F%6$F+F(" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#\"#@" }}} {PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 46 "e2 := solve(\{op(eqns)\});\ne3 := `union`(e1,e2):" }}{PARA 12 "" 1 "" {XPPMATH 20 "6#>%#e2G<7/&%\"cG6#\"\"$#\"'*\\t%\"(+gh%/&%\"aG6$\" \"&F*#!1VCC2t2A6\"0]7`YnZy(/&F06$\"\"%F*#\"(\"\\hU\")+!Gp$/&F06$\"\") \"\"'#\"38\\#3g\"e]L]\"3+D\")z@M!*f7/&F06$F2\"\"\"#\"1xU\"**)3)3f$\"1+ ]Pe4sT$*/&F06$FAFH#\"+jo@zK\",+!Gz-r/&F06$FAF2#\"-T5-$yD(\".;'f2RKR/&F 06$\"\"(F9#\"374nP)yoK;\"\"4v5He<5?t/\"/&F06$F@FH#!3^^YX!4Eq]\"\"3+++d )oB9!G/&F(6#F9#\"((\\?9\"(+?B*/&F06$F*FH#\",*\\unA%*\"/+g`Xp%=\"/&F06$ F@F2#!2F2u>SCq'H\"1++_QUljj/&F06$FenF2#!/*)RE1D-@\"0++>cgV9'/&F06$FenF H#\"2\"*34>oxGE#\"3++&p&z>#>y$/&F06$F*\"\"##\"-,II,UA\".+s5*QpB/&F06$F enFA#\"*,%)>#=\",+D6\\1\"/&F06$F@F9#!87^zL0VsX<[q%\"7vVB#Q+a(Rd\"y'/&F 06$F`qFH#\"$R\"\"%+D/&F06$FAF9#\".sOkm%=9\".v$epvYh/&F06$F2F9#\"1e#)p% 3fLo\"\"1voz>^rn6/&F06$F9FH#FcoF<" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }} {PARA 0 "" 0 "" {TEXT 260 4 "Note" }{TEXT -1 2 ": " }}{PARA 0 "" 0 "" {TEXT -1 8 "We have " }{XPPEDIT 18 0 "c[3]=2/3" "6#/&%\"cG6#\"\"$*&\" \"#\"\"\"F'!\"\"" }{TEXT -1 1 " " }{XPPEDIT 18 0 "c[4]" "6#&%\"cG6#\" \"%" }{TEXT -1 16 " and the nodes " }{XPPEDIT 18 0 "c[4];" "6#&%\"cG6 #\"\"%" }{TEXT -1 3 ", " }{XPPEDIT 18 0 "c[5];" "6#&%\"cG6#\"\"&" } {TEXT -1 3 ", " }{XPPEDIT 18 0 "c[6];" "6#&%\"cG6#\"\"'" }{TEXT -1 31 " are related by the equation: " }}{PARA 256 "" 0 "" {TEXT -1 1 " \+ " }{XPPEDIT 18 0 "3*c[6]^2-4*c[6]*c[4]-4*c[5]*c[6]+6*c[5]*c[4] = 0" "6 #/,**&\"\"$\"\"\"*$&%\"cG6#\"\"'\"\"#F'F'*(\"\"%F'&F*6#F,F'&F*6#F/F'! \"\"*(F/F'&F*6#\"\"&F'&F*6#F,F'F4*(F,F'&F*6#F8F'&F*6#F/F'F'\"\"!" } {TEXT -1 2 ". " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 86 "subs(e3,matrix([seq([c[i],seq(a[i,j],j=1..i-1),` `$(9-i)],i=2..8)]));\n``;\nevalf[8](%%);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#K%'matrixG6#7)7+#\"$R\"\"%+DF(%!GF+F+F+F+F+F+7+#\"'*\\t%\"(+gh%# \",*\\unA%*\"/+g`Xp%=\"#\"-,II,UA\".+s5*QpBF+F+F+F+F+F+7+#\"((\\?9\"(+ ?B*#F8\")+!Gp$\"\"!#\"(\"\\hUF;F+F+F+F+F+7+#\"%B>\"%+]#\"1xU\"**)3)3f$ \"1+]Pe4sT$*F<#!1VCC2t2A6\"0]7`YnZy(#\"1e#)p%3fLo\"\"1voz>^rn6F+F+F+F+ 7+#\"$B*\"%+?#\"+jo@zK\",+!Gz-rFoxGE#\"3++&p&z>#>y$FcgV9'#\"*,%)>#=\",+D6\\1\"F+F+7+#\"%r &)\"&++\"#!3^^YX!4Eq]\"\"3+++d)oB9!GFSCq'H\"1++_QUljj#\"38\\#3g\"e]L]\"3+D\")z@M!*f7\"\"*F+Q)pprin t176\"" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%!G" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#K%'matrixG6#7)7+$\")++gb!\"*F(%!GF+F+F+F+F+F+7+$\")xxD5 !\")$\")nn`z!#5$\")'4CY*F*F+F+F+F+F+F+7+$\")mmQ:F/$\")lmYQF*$\"\"!F;$ \")***R:\"F/F+F+F+F+F+7+$\")++YQF/$\")!=R%QF/F:$!)bPT9!\"($\")PeT9FEF+ F+F+F+7+$\")++:YF/$\")$*z;YF*F:F:$\")nm2BF/$\")MlX=F/F+F+F+7+$\")++Q:F /$\")mS$)fF*F:F:$\")*426\"F/$!)6V@MF*$\")d#4r\"F*F+F+7+$\")++r&)F/$!)3 ]z`F/F:F:$!)#[w$pFE$!)QXiYFE$\")@:&*RFE$\"\"*F;F+Q)pprint186\"" }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 27 "We do not need to specify " } {XPPEDIT 18 0 "c[9];" "6#&%\"cG6#\"\"*" }{TEXT -1 42 ", because accord ing to Verner, the nodes " }{XPPEDIT 18 0 "c[6];" "6#&%\"cG6#\"\"'" } {TEXT -1 2 ", " }{XPPEDIT 18 0 "c[7];" "6#&%\"cG6#\"\"(" }{TEXT -1 2 " , " }{XPPEDIT 18 0 "c[8];" "6#&%\"cG6#\"\")" }{TEXT -1 2 ", " } {XPPEDIT 18 0 "c[9];" "6#&%\"cG6#\"\"*" }{TEXT -1 30 " are related by the equation " }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "Int (p(x)*``((1-x)^3/3!),x = 0 .. 1)*Int(q(x)*(1-x),x = 0 .. 1) = Int(p(x) *``((1-x)^2/2!),x = 0 .. 1)*Int(q(x)*``((1-x)^2/2!),x = 0 .. 1);" "6#/ *&-%$IntG6$*&-%\"pG6#%\"xG\"\"\"-%!G6#*&,&F-F-F,!\"\"\"\"$-%*factorial G6#F4F3F-/F,;\"\"!F-F--F&6$*&-%\"qG6#F,F-,&F-F-F,F3F-/F,;F:F-F-*&-F&6$ *&-F*6#F,F--F/6#*&,&F-F-F,F3\"\"#-F66#FNF3F-/F,;F:F-F--F&6$*&-F?6#F,F- -F/6#*&,&F-F-F,F3FN-F66#FNF3F-/F,;F:F-F-" }{TEXT -1 2 ", " }}{PARA 0 " " 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 7 "where " }{XPPEDIT 18 0 "p(x) = x*(x-c[6])*(x-c[7])*(x-c[8]);" "6#/-%\"pG6#%\"xG**F'\"\" \",&F'F)&%\"cG6#\"\"'!\"\"F),&F'F)&F,6#\"\"(F/F),&F'F)&F,6#\"\")F/F)" }{TEXT -1 7 " and " }{XPPEDIT 18 0 "q(x) = (x-c[9])*p(x);" "6#/-%\"q G6#%\"xG*&,&F'\"\"\"&%\"cG6#\"\"*!\"\"F*-%\"pG6#F'F*" }{TEXT -1 2 ". \+ " }}{PARA 0 "" 0 "" {TEXT -1 1 " " }}{PARA 0 "" 0 "" {TEXT -1 155 "See : J.H. Verner, SIAM Journal of Numerical Analysis 1978, 772-790, \"Exp licit Runge-Kutta methods with estimates of the Local Truncation Error .\" (page 780)" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 90 "p := x -> x*(x-c[6])*(x-c[7])*(x-c[8]):\n'p(x)'= p(x);\nq := x -> (x-c[9])*p(x):\n'q(x)'=q(x);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/-%\"pG6#%\"xG**F'\"\"\",&F'F)&%\"cG6#\"\"'!\"\"F),&F'F )&F,6#\"\"(F/F),&F'F)&F,6#\"\")F/F)" }}{PARA 11 "" 1 "" {XPPMATH 20 "6 #/-%\"qG6#%\"xG*,,&F'\"\"\"&%\"cG6#\"\"*!\"\"F*F'F*,&F'F*&F,6#\"\"'F/F *,&F'F*&F,6#\"\"(F/F*,&F'F*&F,6#\"\")F/F*" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 239 "Ieq := Int('p(x)' *(1-x)^3/3!,x=0..1)*Int('q(x)'*(1-x),x=0..1)=\n Int('p(x)'*(1-x)^2/2! ,x=0..1)*Int('q(x)'*(1-x)^2/2!,x=0..1);\nvalue(Ieq):\nReq := simplify( lhs(%)-rhs(%))=0:\nL := ilcm(op(map(denom,[coeffs(lhs(Req))]))):\nCeq \+ := expand(L*Req);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%$IeqG/*&-%$IntG 6$,$*&#\"\"\"\"\"'F-*&-%\"pG6#%\"xGF-),&F-F-F3!\"\"\"\"$F-F-F-/F3;\"\" !F-F--F(6$*&-%\"qGF2F-F5F-F8F-*&-F(6$,$*&#F-\"\"#F-*&F0F-)F5FFF-F-F-F8 F--F(6$,$*&FEF-*&F>F-FHF-F-F-F8F-" }}{PARA 12 "" 1 "" {XPPMATH 20 "6#> %$CeqG/,hp*&\"\"$\"\"\"&%\"cG6#\"\"'F)!\"\"*&F(F)&F+6#\"\")F)F.*&F(F)& F+6#\"\"(F)F.*&\"#7F)&F+6#\"\"*F)F)*(\"#9F))F*\"\"#F)F4F)F.*(\"#GF)F*F )F4F)F)*(FAF)F0F)F*F)F)*(FAF)F0F)F4F)F)*(F=F)F0F)F>F)F.*(F=F))F0F?F)F* F)F.*(F=F)F*F))F4F?F)F.*(F=F)FFF)F4F)F.*(F=F)F0F)FHF)F.*(\"#UF)F9F)F*F )F.*(FLF)F4F)F9F)F.*(FLF)F0F)F9F)F.*(FLF)F9F)F>F)F)*(FLF)FHF)F9F)F)*(F LF)FFF)F9F)F)*,\"$)eF)F9F)F*F)F4F)F0F)F.*,\"$G(F)F0F)F>F)F9F)F4F)F)*,F UF)FFF)F*F)F4F)F9F)F)*,\"$!)*F)FFF)F>F)F9F)F4F)F.*,FUF)F0F)FHF)F9F)F*F )F)*,FXF)FFF)FHF)F9F)F*F)F.*,FXF)F>F)FHF)F9F)F0F)F.*,\"%q9F)F>F)FHF)FF F)F9F)F)**\"$J#F)F*F)F4F)F0F)F.**\"#)*F)F>F)FHF)F0F)F.**F[oF)FFF)FHF)F *F)F.**F[oF)FFF)F>F)F4F)F.**\"$o\"F)FFF)F*F)F4F)F)**F_oF)F0F)FHF)F*F)F )**F_oF)F0F)F>F)F4F)F)**\"$a\"F)F9F)F*F)F4F)F)**FcoF)F0F)F9F)F*F)F)**F coF)F0F)F4F)F9F)F)**F_oF)F0F)FHF)F9F)F.**F_oF)F0F)F>F)F9F)F.**F_oF)FFF )F*F)F9F)F.**\"$'>F)FFF)F>F)F9F)F)**F_oF)FFF)F4F)F9F)F.**FjoF)FFF)FHF) F9F)F)**F_oF)F9F)F>F)F4F)F.**F_oF)F*F)FHF)F9F)F.**FjoF)F>F)FHF)F9F)F) \"\"!" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 60 "The previous equation involving integrals is equivalent to: " }} {PARA 256 "" 0 "" {TEXT -1 2 " " }{XPPEDIT 18 0 "14*c[6]^2*c[7]+3*c[6 ]-28*c[6]*c[7]+3*c[7]+588*c[9]*c[6]*c[7]*c[8]+3*c[8]-28*c[8]*c[6]-28*c [8]*c[7]+14*c[8]*c[6]^2+14*c[8]^2*c[6]+231*c[6]*c[7]*c[8]+98*c[6]^2*c[ 7]^2*c[8]+14*c[6]*c[7]^2+98*c[8]^2*c[7]^2*c[6]+98*c[8]^2*c[6]^2*c[7]-1 68*c[8]^2*c[6]*c[7]-168*c[8]*c[7]^2*c[6]-168*c[8]*c[6]^2*c[7]+14*c[8]^ 2*c[7]-12*c[9]+14*c[8]*c[7]^2+42*c[9]*c[6]-154*c[9]*c[6]*c[7]+42*c[7]* c[9]+42*c[8]*c[9]-154*c[8]*c[9]*c[6]-154*c[8]*c[7]*c[9]-42*c[9]*c[6]^2 -42*c[7]^2*c[9]-42*c[8]^2*c[9]+168*c[8]*c[7]^2*c[9]-728*c[8]*c[6]^2*c[ 9]*c[7]-728*c[8]^2*c[6]*c[7]*c[9]+980*c[8]^2*c[6]^2*c[9]*c[7]-728*c[8] *c[7]^2*c[9]*c[6]+980*c[8]^2*c[7]^2*c[9]*c[6]+980*c[6]^2*c[7]^2*c[9]*c [8]-1470*c[6]^2*c[7]^2*c[8]^2*c[9]+168*c[8]*c[6]^2*c[9]+168*c[8]^2*c[6 ]*c[9]-196*c[8]^2*c[6]^2*c[9]+168*c[8]^2*c[7]*c[9]-196*c[8]^2*c[7]^2*c [9]+168*c[9]*c[6]^2*c[7]+168*c[6]*c[7]^2*c[9]-196*c[6]^2*c[7]^2*c[9]=0 " "6#/,hp*(\"#9\"\"\"*$&%\"cG6#\"\"'\"\"#F'&F*6#\"\"(F'F'*&\"\"$F'&F*6 #F,F'F'*(\"#GF'&F*6#F,F'&F*6#F0F'!\"\"*&F2F'&F*6#F0F'F'*,\"$)eF'&F*6# \"\"*F'&F*6#F,F'&F*6#F0F'&F*6#\"\")F'F'*&F2F'&F*6#FJF'F'*(F6F'&F*6#FJF '&F*6#F,F'F;*(F6F'&F*6#FJF'&F*6#F0F'F;*(F&F'&F*6#FJF'&F*6#F,F-F'*(F&F' *$&F*6#FJF-F'&F*6#F,F'F'**\"$J#F'&F*6#F,F'&F*6#F0F'&F*6#FJF'F'**\"#)*F '*$&F*6#F,F-F'&F*6#F0F-&F*6#FJF'F'*(F&F'&F*6#F,F'&F*6#F0F-F'**FfoF'*$& F*6#FJF-F'&F*6#F0F-&F*6#F,F'F'**FfoF'*$&F*6#FJF-F'&F*6#F,F-&F*6#F0F'F' **\"$o\"F'*$&F*6#FJF-F'&F*6#F,F'&F*6#F0F'F;**FdqF'&F*6#FJF'&F*6#F0F-&F *6#F,F'F;**FdqF'&F*6#FJF'&F*6#F,F-&F*6#F0F'F;*(F&F'*$&F*6#FJF-F'&F*6#F 0F'F'*&\"#7F'&F*6#FCF'F;*(F&F'&F*6#FJF'&F*6#F0F-F'*(\"#UF'&F*6#FCF'&F* 6#F,F'F'**\"$a\"F'&F*6#FCF'&F*6#F,F'&F*6#F0F'F;*(FjsF'&F*6#F0F'&F*6#FC F'F'*(FjsF'&F*6#FJF'&F*6#FCF'F'**F`tF'&F*6#FJF'&F*6#FCF'&F*6#F,F'F;**F `tF'&F*6#FJF'&F*6#F0F'&F*6#FCF'F;*(FjsF'&F*6#FCF'&F*6#F,F-F;*(FjsF'*$& F*6#F0F-F'&F*6#FCF'F;*(FjsF'*$&F*6#FJF-F'&F*6#FCF'F;**FdqF'&F*6#FJF'&F *6#F0F-&F*6#FCF'F'*,\"$G(F'&F*6#FJF'&F*6#F,F-&F*6#FCF'&F*6#F0F'F;*,Fhw F'*$&F*6#FJF-F'&F*6#F,F'&F*6#F0F'&F*6#FCF'F;*,\"$!)*F'*$&F*6#FJF-F'&F* 6#F,F-&F*6#FCF'&F*6#F0F'F'*,FhwF'&F*6#FJF'&F*6#F0F-&F*6#FCF'&F*6#F,F'F ;*,F\\yF'*$&F*6#FJF-F'&F*6#F0F-&F*6#FCF'&F*6#F,F'F'*,F\\yF'*$&F*6#F,F- F'&F*6#F0F-&F*6#FCF'&F*6#FJF'F'*,\"%q9F'*$&F*6#F,F-F'&F*6#F0F-&F*6#FJF -&F*6#FCF'F;**FdqF'&F*6#FJF'&F*6#F,F-&F*6#FCF'F'**FdqF'*$&F*6#FJF-F'&F *6#F,F'&F*6#FCF'F'**\"$'>F'*$&F*6#FJF-F'&F*6#F,F-&F*6#FCF'F;**FdqF'*$& F*6#FJF-F'&F*6#F0F'&F*6#FCF'F'**F^]lF'*$&F*6#FJF-F'&F*6#F0F-&F*6#FCF'F ;**FdqF'&F*6#FCF'&F*6#F,F-&F*6#F0F'F'**FdqF'&F*6#F,F'&F*6#F0F-&F*6#FCF 'F'**F^]lF'*$&F*6#F,F-F'&F*6#F0F-&F*6#FCF'F;\"\"!" }{TEXT -1 2 ". " }} {PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 75 "e4 := solve(subs(e3,Ceq),\{c[9]\}):\nc[9]=subs(e4,c[9]);\ne5 := `u nion`(e3,e4):" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/&%\"cG6#\"\"*#\"5+=W $pl:=!G9\"5jpEaXi5N-:" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 7 "Thus " }{XPPEDIT 18 0 "c[9] = 14280181565693441800/ 15023510624554266963" "6#/&%\"cG6#\"\"**&\"5+=W$pl:=!G9\"\"\"\"5jpEaXi 5N-:!\"\"" }{TEXT -1 1 "." }}{PARA 0 "" 0 "" {TEXT -1 84 "Now we can u se the quadrature equations to find the weights once the remaining nod es" }}{PARA 256 "" 0 "" {TEXT -1 2 " " }{XPPEDIT 18 0 "c[10] = 3611/5 000;" "6#/&%\"cG6#\"#5*&\"%6O\"\"\"\"%+]!\"\"" }{TEXT -1 3 ", " } {XPPEDIT 18 0 "c[11] = 15/16;" "6#/&%\"cG6#\"#6*&\"#:\"\"\"\"#;!\"\"" }{TEXT -1 3 ", " }{XPPEDIT 18 0 "c[12] = 1;" "6#/&%\"cG6#\"#7\"\"\"" }{TEXT -1 1 " " }}{PARA 0 "" 0 "" {TEXT -1 38 "are specified along wit h the weights " }{XPPEDIT 18 0 "b[2]=0" "6#/&%\"bG6#\"\"#\"\"!" } {TEXT -1 2 ", " }{XPPEDIT 18 0 "b[3]=0" "6#/&%\"bG6#\"\"$\"\"!" } {TEXT -1 2 ", " }{XPPEDIT 18 0 "b[4]=0" "6#/&%\"bG6#\"\"%\"\"!" } {TEXT -1 2 ", " }{XPPEDIT 18 0 "b[5]=0" "6#/&%\"bG6#\"\"&\"\"!" } {TEXT -1 2 ". " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT 260 4 "Note" }{TEXT -1 31 ": Enright specifies the nodes " } {XPPEDIT 18 0 "c[10]" "6#&%\"cG6#\"#5" }{TEXT -1 7 " and " } {XPPEDIT 18 0 "c[11]" "6#&%\"cG6#\"#6" }{TEXT -1 29 " in floating poi nt form as: " }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "c[10] =0" "6#/&%\"cG6#\"#5\"\"!" }{TEXT -1 8 ".7222, " }{XPPEDIT 18 0 "c[11 ] = 0" "6#/&%\"cG6#\"#6\"\"!" }{TEXT -1 7 ".9375. " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 292 "Qeqs := QuadratureConditions(8,12,'expanded'):\ne 6 := \{seq(b[i]=0,i=2..5),c[10]=3611/5000,c[11]=15/16,c[12]=1\}:\ne7 : = `union`(e5,e6):\nquadeqns := subs(e7,Qeqs):\nfor ct to nops(quadeqns ) do\n print(`equation `||ct); print(``);print(quadeqns[ct]);print (``);\nend do:\nindets(quadeqns);\nnops(%);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%-equation~~~1G" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%!G " }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/,2&%\"bG6#\"\"\"F(&F&6#\"\"'F(&F& 6#\"\"(F(&F&6#\"\")F(&F&6#\"\"*F(&F&6#\"#5F(&F&6#\"#6F(&F&6#\"#7F(F(" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%!G" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%-equation~~~2G" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%!G" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/,0*&#\"$B*\"%+?\"\"\"&%\"bG6#\"\"'F)F)*&#\"$p( \"%+]F)&F+6#\"\"(F)F)*&#\"%r&)\"&++\"F)&F+6#\"\")F)F)*&#\"5+=W$pl:=!G9 \"5jpEaXi5N-:F)&F+6#\"\"*F)F)*&#\"%6OF1F)&F+6#\"#5F)F)*&#\"#:\"#;F)&F+ 6#\"#6F)F)&F+6#\"#7F)#F)\"\"#" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%!G" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%-equation~~~3G" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%!G" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/,0*&#\"'H>&)\"( +++%\"\"\"&%\"bG6#\"\"'F)F)*&#\"'h8f\")+++DF)&F+6#\"\"(F)F)*&#\")T?Yt \"*++++\"F)&F+6#\"\")F)F)*&#\"H++C()*Hbp,I%))zq\"\\beBR?\"HpLCt-pTEj!f S\\4'[reqD#F)&F+6#\"\"*F)F)*&#\")@$RI\"F1F)&F+6#\"#5F)F)*&#\"$D#\"$c#F )&F+6#\"#6F)F)&F+6#\"#7F)#F)\"\"$" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#% !G" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%-equation~~~4G" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%!G" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/,0*&#\"*n/ L'y\"+++++!)\"\"\"&%\"bG6#\"\"'F)F)*&#\"*4mva%\"-++++]7F)&F+6#\"\"(F)F )*&#\"-6M:V'H'\".++++++\"F)&F+6#\"\")F)F)*&#\"en+++KEo0pf%[-d>G!Hr#)** 3Qy$pr#e17H\"enZ$=bH)\\*)RWv(p2%ewmW)e:;ti&HeX*3R$F)&F+6#\"\"*F)F)*&# \",J\"))\\3ZF1F)&F+6#\"#5F)F)*&#\"%vL\"%'4%F)&F+6#\"#6F)F)&F+6#\"#7F)# F)\"\"%" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%!G" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%-equation~~~5G" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%!G " }}{PARA 12 "" 1 "" {XPPMATH 20 "6#/,0*&#\"-T5-$yD(\"/++++++;\"\"\"&% \"bG6#\"\"'F)F)*&#\"-@B$yq\\$\"0++++++D'F)&F+6#\"\"(F)F)*&#\"1\"o&)yYr mR&\"2++++++++\"F)&F+6#\"\")F)F)*&#\"ho++++w\"Gcu+jx&[$owq4\\()z$yv'*G @3YjX\")*HKuG[eT\"hoh,Z-(e\"pbc8=o/'\\h'y*o)HkY(y_$4G#pX^g(HB/9V4&F)&F +6#\"\"*F)F)*&#\"0T59#*Q-q\"F1F)&F+6#\"#5F)F)*&#\"&D1&\"&Ob'F)&F+6#\"# 6F)F)&F+6#\"#7F)#F)\"\"&" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%!G" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#%-equation~~~6G" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%!G" }}{PARA 12 "" 1 "" {XPPMATH 20 "6#/,0*&#\"0V3UGx*) p'\"2+++++++?$\"\"\"&%\"bG6#\"\"'F)F)*&#\"0\\[0B`#*o#\"4+++++++]7$F)&F +6#\"\"(F)F)*&#\"5^=<[7:r[DY\"6++++++++++\"F)&F+6#\"\")F)F)*&#\"[q++++ +o:'\\%=lhkDA-c(**ya;jmoIi8WxQ,9JV5BgW#e:fJ[!*QQf\"[qV5fN2itpyPFifd@1Q ;)oR!4j:Xcud)y'R*G_^[(o\"[x_rvR6[Ml(F)&F+6#\"\"*F)F)*&#\"3^!*H@XFcRhF1 F)&F+6#\"#5F)F)*&#\"'v$f(\"(w&[5F)&F+6#\"#6F)F)&F+6#\"#7F)#F)F-" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#%!G" }}{PARA 11 "" 1 "" {XPPMATH 20 "6 #%-equation~~~7G" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%!G" }}{PARA 12 " " 1 "" {XPPMATH 20 "6#/,0*&#\"3*3QCL.cJ='\"5+++++++++k\"\"\"&%\"bG6#\" \"'F)F)*&#\"3\"))y\"HMd.o?\"8+++++++++Dc\"F)&F+6#\"\"(F)F)*&#\"9@\\$43 [tj+0X'R\":++++++++++++\"F)&F+6#\"\")F)F)*&#\"^r++++++CuW1RPM(4D*yN*Qn &zHHcH+.H%Rm+9xs'*oYHx.t@@7kI(=fnn@'3@#RdBhujc=zu*)ya!f;)\\6F)&F+6#\"\"*F)F)*&#\"7hJ(3T'H 2h*p@#F1F)&F+6#\"#5F)F)*&#\")D1R6\");sx;F)&F+6#\"#6F)F)&F+6#\"#7F)#F)F 4" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%!G" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%-equation~~~8G" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%!G" }}{PARA 12 "" 1 "" {XPPMATH 20 "6#/,0*&#\"6ZhNSe(=I02d\"9++++++++++!G\"\"\"\"& %\"bG6#\"\"'F)F)*&#\"6*[f&Qq'z%>.f\"\";++++++++++]7yF)&F+6#\"\"(F)F)*& #\"=\"*y?$p,/ri4CxzR$\">++++++++++++++\"F)&F+6#\"\")F)F)*&#\"bs+++++++ K#RI$\\)451EC=8F!e%yB$)3Bd_JEd`KSZ@%>6?)\\M7*)QL5wdwvQ`+%Rjv&4&=rex4@ \"\"bsnQa&zCQIOq+N12370JV](eT#QpYX(f@Ct=>U*oLt^4Ddd]B`L^L>XdvC " 0 "" {MPLTEXT 1 0 199 "e8 := solve(\{op(quade qns)\}):\ne9 := `union`(e7,e8):\nfor ii to 12 do\n wt_val := subs(e9 ,b[ii]);\n if wt_val<>0 then print(b[ii]=wt_val) end if;\nend do:\n` `;\nsubs(e9,[seq(b[i],i=1..12)]):\nevalf[8](%);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/&%\"bG6#\"\"\"#\"AB<63n7Zc(zq&>JUGb\"C+S67Ogt>i\"MTV0kFFM3!)>o=#=** Gc9Bzo_#" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/&%\"bG6#\"\")#\"K++++voHz qt;K(eK\\,#)e_r%\"K`b\\WSJ\\'R3\")Gdz)>\"R<)GP_" }}{PARA 12 "" 1 "" {XPPMATH 20 "6#/&%\"bG6#\"\"*#\"]t>%3MD>!G%[rBVH5v$Q%)p_7!\\&\\\"**y13 0hUHdN=q@)[Ne7)R:)yXhrz@c*\\8N`!QWq-n%z4'H]u!f'>%)*R'\"]t+!Qq'ypt(4a[- N?6`k/j9w&p**\\j5`C)>oK^HX;wVj+8$ohSYO&4j^ZRsFxEC.]**f$pn!)*fTAyQQnS\" " }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/&%\"bG6#\"#5#\"H+++DJqXVA_DfIvDbF c%f\"KJ?7UM&fV<9V,BoF.\"4s-Q7" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/&%\" bG6#\"#6#!CcYXqQ:f*RcqzR8\"p')=\"Bvw[7Lc^RQNe)eeZzR" }}{PARA 11 "" 1 " " {XPPMATH 20 "6#/&%\"bG6#\"#7#!A\")yja-\"R\"*y(R^()o91]\"B\"y8By')*>[ =Og$>k'=T\"" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%!G" }}{PARA 11 "" 1 " " {XPPMATH 20 "6#7.$\")/x\"R%!\"*$\"\"!F(F'F'F'$\")jC5N!\")$\")$G9Y#F+ $\")\\C.!*F+$\")(=%\\X!\"($\"):]-[!#5$!)W0TZF2$!)_wXNF+" }}}{PARA 0 " " 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}} {SECT 1 {PARA 4 "" 0 "" {TEXT -1 2 "e9" }}{PARA 0 "" 0 "" {TEXT -1 0 " " }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 2054 "e9 := \{b[4] = 0, c[12 ] = 1, a[7,2] = 0, b[2] = 0, b[3] = 0, c[3] = 473499/4616000, a[5,3] = -1122077307242443/778476746531250, a[4,3] = 4261491/36928000, a[8,6] \+ = 503350581600824913/125990342179812500, b[5] = 0, a[6,3] = 0, a[4,2] \+ = 0, a[8,3] = 0, a[8,2] = 0, a[7,3] = 0, a[6,2] = 0, a[5,2] = 0, a[5,1 ] = 3590880889914277/9341720958375000, a[6,1] = 3279216863/71027928000 , c[10] = 3611/5000, c[11] = 15/16, a[6,5] = 725783021041/393239075961 6, b[10] = 594562755257530592552224345703125000000/1238027209103276823 01431417435953442122031, b[12] = -50061468875139778913910254637881/141 186641936036184819986782313781, a[7,4] = 116326878837670912/1047320101 758291075, b[1] = 55284231195707975647126708111723/1258814250475014554 517603612114000, b[9] = 6399841965907450296097946702704438053351349956 2179716145788153981258354882170183557294261050806789914954901252698438 375102943237148428019253408419/140673838782241599806769359995003242677 2772394751630953646406168313006343761645295132681982453106349996957614 6304645311203502485409773697867038000, a[8,1] = -150702609045465151/28 0142368857000000, b[7] = 6219731958882093270433753490048828125000000/2 5268792314562899182186819800834272764054341, c[4] = 1420497/9232000, a [3,1] = 94226774499/11846945536000, b[11] = -1886691133979705639959153 870454656/397947585885835383951563312487675, c[9] = 142801815656934418 00/15023510624554266963, b[6] = 10165204821428220551861044578389375000 0000/289586658278060310247144250081091360673509, b[8] = 47152588201493 2587321673707929687500000000/52372881739119879572881083964931404449555 3, a[8,5] = -29670244019740727/6363654238520000, a[7,5] = -21022506263 989/614436056190000, c[2] = 139/2500, c[5] = 1923/5000, c[6] = 923/200 0, c[7] = 769/5000, c[8] = 8571/10000, a[7,1] = 22628776819090891/3781 92197956950000, a[3,2] = 224201303001/2369389107200, a[7,6] = 18219840 1/10649112500, a[8,4] = -47048174572430533795112/678157397540038223437 5, a[8,7] = 9, a[2,1] = 139/2500, a[6,4] = 1418466643672/6146756958375 , a[5,4] = 1683359084698258/1167715119796875, a[4,1] = 1420497/3692800 0\}:" }{TEXT -1 0 "" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}}{PARA 0 "" 0 "" {TEXT -1 0 "" }} {PARA 0 "" 0 "" {TEXT -1 55 "Now we specify the remaining zero linking coefficients:" }}{PARA 256 "" 0 "" {TEXT -1 2 " " }{XPPEDIT 18 0 "a[ 9,2] = 0,a[9,3] = 0,a[10,2] = 0,a[10,3] = 0,a[11,2] = 0,a[11,3] = 0,a[ 12,2] = 0,a[12,3] = 0;" "6*/&%\"aG6$\"\"*\"\"#\"\"!/&F%6$F'\"\"$F)/&F% 6$\"#5F(F)/&F%6$F1F-F)/&F%6$\"#6F(F)/&F%6$F8F-F)/&F%6$\"#7F(F)/&F%6$F? F-F)" }{TEXT -1 1 "." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 " " {TEXT -1 149 "We include the stage-order conditions for stages 9 to \+ 12 so that all these stages all have stage-order 4 and incorporate the simplifying conditions: " }}{PARA 256 "" 0 "" {TEXT -1 1 " " } {XPPEDIT 18 0 "Sum(b[i]*a[i,j],i = j+1 .. 12) = b[j]*(1-c[j]);" "6#/-% $SumG6$*&&%\"bG6#%\"iG\"\"\"&%\"aG6$F+%\"jGF,/F+;,&F0F,F,F,\"#7*&&F)6# F0F,,&F,F,&%\"cG6#F0!\"\"F," }{TEXT -1 1 " " }}{PARA 0 "" 0 "" {TEXT -1 5 "for " }{XPPEDIT 18 0 "j=1" "6#/%\"jG\"\"\"" }{TEXT -1 10 " ( w ith " }{XPPEDIT 18 0 "c[1]=0" "6#/&%\"cG6#\"\"\"\"\"!" }{TEXT -1 5 " \+ ), " }{XPPEDIT 18 0 "j = 5;" "6#/%\"jG\"\"&" }{TEXT -1 3 ", " } {XPPEDIT 18 0 "j=7" "6#/%\"jG\"\"(" }{TEXT -1 3 ", " }{XPPEDIT 18 0 " j=9" "6#/%\"jG\"\"*" }{TEXT -1 7 " and " }{XPPEDIT 18 0 "j=10" "6#/% \"jG\"#5" }{TEXT -1 2 ". " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 " " 0 "" {TEXT -1 195 "Then it turns out that the following additional c ollection of \"simple\" order conditions (given in abreviated form) is sufficient to determine the remaining linking coefficients in stages \+ 9 to 12." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 190 "SO8 := SimpleOrderConditions(8):\n[seq([i,SO8[i]],i= [37,61,81,87,102,106,113,117,125])]:\nlinalg[augment](linalg[delcols]( %,2..2),matrix([[` `]$(linalg[rowdim](%))]),linalg[delcols](%,1..1)); " }}{PARA 11 "" 1 "" {XPPMATH 20 "6#K%'matrixG6#7+7%\"#P%#~~G/*(%\"bG \"\"\"%\"cGF--%!G6#*&%\"aGF--F06#*&F3F--F06#*&F3F--F06#*&F3F-F.F-F-F-F -F-#F-\"$S)7%\"#hF)/*(F,F-F.F--F06#*&)F.\"\"%F-F3F-F-#F-\"#N7%\"#\")F) /*&F,F--F06#*(F3F-F.F--F06#*&F3F--F06#*&F3F--F06#*&)F.\"\"#F-F3F-F-F-F -F-#F-\"%gL7%\"#()F)/*(F,F-F.F--F06#*(F3F-F.F--F06#*(F3F-F.F-F:F-F-F-# F-\"$%Q7%\"$-\"F)/*(F,F-F.F--F06#*&F3F--F06#*&F3F--F06#*&)F.\"\"$F-F3F -F-F-F-#F-\"$g*7%\"$1\"F)/*&F,F--F06#*&F3F--F06#*&F3F-FCF-F-F-#F-\"%!o \"7%\"$8\"F)/*(F,F-FZF-F[pF-#F-\"$g\"7%\"$<\"F)/*&F,F--F06#*(F3F-F.F-F CF-F-#F-\"$!G7%\"$D\"F)/*(F,F-F.F--F06#*&)F.\"\"&F-F3F-F-#F-\"#[Q(ppri nt26\"" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 20 "The associa ted trees" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 461 "AST8 := AllSimpleTrees(8):\nwhch := [37,61,81,87,102 ,106,113,117,125]:\nm := 3: ## number of trees per row\nnn := nops(whc h): q := iquo(nn,m,'r'):\nfor i to nn do \n p||i := DrawTree(AST8[whc h[i]],height=4,width=2,show_ordercondition=true,\n scale=1.2,color= COLOR(RGB,0,.7,0),font_color=black);\nend do:\npp := plot([[1,1]],styl e=line,axes=none):\nplots[display](convert([seq([p||((k-1)*m+1..m*k)], k=1..q),\n `if`(r>0,[p||(m*q+1..nn),pp$(m*(q+1)-nn)],NULL)],array)); " }}{PARA 13 "" 1 "" {GLPLOT2D 852 1131 1131 {PLOTDATA 2 "6\\u-%%TEXTG 6&7$$\"#g\"\"!$!+5`7'*o!\")Q/b~c~(a~c~)~=~~6\"-%'COLOURG6&%$RGBGF)F)F) -%%FONTG6$%(COURIERG\"#5-F$6&7$F'$!+)o#)p\"oF,Q1~~~~~~~~~~~~~~~1F.F/F3 -F$6&7$F'$!+dZrToF,Q2~~~~~~~~~~~~~~~__F.F/F3-F$6&7$F'$!+++++qF,Q2~~~~~ ~~~~~~~~~~48F.F/F3-F$6&7$F'$!+g)*32oF,Q1~~~~~~~~~5~~~~~~F.F/F3-%'CURVE SG6&7*7$F'$!+ " 0 "" {MPLTEXT 1 0 1 ";" }}}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT 260 4 "Note" }{TEXT -1 119 ": The 5 simplifying conditions can be replaced b y the 5 extra order conditions given (in abreviated form) as follows.. " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 174 "SO8 := SimpleOrderConditions(8):\n[seq([i,SO8[i]],i= [21,27,31,36,127])]:\nlinalg[augment](linalg[delcols](%,2..2),matrix([ [` `]$(linalg[rowdim](%))]),linalg[delcols](%,1..1));" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#K%'matrixG6#7'7%\"#@%#~~G/*&%\"bG\"\"\"-%!G6#*&%\" aGF--F/6#*&F2F--F/6#*&)%\"cG\"\"#F-F2F-F-F-F-#F-\"$g$7%\"#FF)/*&F,F--F /6#*&F2F--F/6#*&)F:\"\"$F-F2F-F-F-#F-\"$?\"7%\"#JF)/*&F,F--F/6#*&)F:\" \"%F-F2F-F-#F-\"#I7%\"#OF)/*&F,F--F/6#*(F2F-F:F--F/6#*&F2F--F/6#*&F2F- -F/6#*&F2F-F:F-F-F-F-F-#F-\"%357%\"$F\"F)/*&F,F--F/6#*&)F:\"\"'F-F2F-F -#F-\"#cQ(pprint36\"" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 462 "SOeqs2 := [seq(add(a[i,j],j= 1..i-1)=c[i ],i=9..12),op(StageOrderConditions(2,9..12,'expanded')),\n op(Stage OrderConditions(3,9..12,'expanded')),op(StageOrderConditions(4,9..12,' expanded'))]:\nSO8_12 := SimpleOrderConditions(8,12,'expanded'):\nord_ cdns := [seq(SO8_12[i],i=[37,61,81,87,102,113,106,117,125])]:\nsimp_eq s := [add(b[i]*a[i,1],i=1+1..12)=b[1],seq(add(b[i]*a[i,j],i=j+1..12)=b [j]*(1-c[j]),j=[5,7,9,10])]:\ncdns := [op(SOeqs2),op(ord_cdns),op(simp _eqs)]:" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 107 "e10 :=\{seq(a[i,2]=0,i=9..12),seq(a[i,3]=0,i=9..12) \}:\ne11 := `union`(e9,e10):\neqns2 := subs(e11,\{op(cdns)\}):" }}} {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 39 "There are 30 equations and 30 unknowns." }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 36 "nops(eqns2);\nindets(eqns2);\nnops(%);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#\"#I" }}{PARA 12 "" 1 "" {XPPMATH 20 "6#<@&%\"aG6$\"#7 \"#5&F%6$\"\"*\"\"(&F%6$F'\"\"&&F%6$F'F,&F%6$\"#6F+&F%6$F+\"\"%&F%6$F' F+&F%6$F(F,&F%6$F(F/&F%6$F(\"\")&F%6$F+F@&F%6$F4F7&F%6$F(\"\"\"&F%6$F4 FG&F%6$F(\"\"'&F%6$F+FL&F%6$F(F+&F%6$F4F@&F%6$F4F(&F%6$F'FL&F%6$F'F7&F %6$F'FG&F%6$F'F@&F%6$F+FG&F%6$F'F4&F%6$F+F/&F%6$F4FL&F%6$F4F,&F%6$F(F7 &F%6$F4F/" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#\"#I" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 22 "infolevel[so lve] := 4:" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 74 "e12 := solve( \{op(eqns2)\}):\ninfolevel[solve] := 0:\ne13 := `union`(e11,e12):" }}} {PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 4 "e13 " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 7419 "e13 := \{c [7] = 769/5000, c[6] = 923/2000, c[5] = 1923/5000, a[8,3] = 0, a[8,2] \+ = 0, a[7,3] = 0, a[7,2] = 0, a[6,3] = 0, a[6,2] = 0, a[5,2] = 0, a[4,2 ] = 0, c[8] = 8571/10000, a[8,7] = 9, c[2] = 139/2500, c[9] = 14280181 565693441800/15023510624554266963, a[5,3] = -1122077307242443/77847674 6531250, a[4,3] = 4261491/36928000, a[3,1] = 94226774499/1184694553600 0, a[6,1] = 3279216863/71027928000, a[12,7] = 948339742665210716931560 767459705347432692833888702670755668800833857753125000/541002030685238 16523920547203712158257383235486719126879290708372319555115953, a[2,1] = 139/2500, b[10] = 594562755257530592552224345703125000000/123802720 910327682301431417435953442122031, b[9] = 6399841965907450296097946702 7044380533513499562179716145788153981258354882170183557294261050806789 914954901252698438375102943237148428019253408419/140673838782241599806 7693599950032426772772394751630953646406168313006343761645295132681982 4531063499969576146304645311203502485409773697867038000, b[7] = 621973 1958882093270433753490048828125000000/25268792314562899182186819800834 272764054341, c[10] = 3611/5000, c[11] = 15/16, c[12] = 1, a[10,9] = - 3347565461896560762549026667823123036634583052726552531003001623087575 5239420324728600957368877132012320553021/56360718148650508277541923761 4873016043169125130359330269370345097328575740250457475506596993780051 575000000000, a[9,8] = -4738143867684122189593816244199450540483384372 163549951990525387550768038015218275414120082248510000000000/456273815 5611920982891652838532643427337613715822889250315879256708180176162834 4925618992749059885819540261, b[5] = 0, b[2] = 0, b[3] = 0, b[4] = 0, \+ b[11] = -1886691133979705639959153870454656/39794758588583538395156331 2487675, a[12,10] = 12459468894943123671827871807868857124596796164158 4151713065625000/24422661199951239145999264873446308735658803040031364 01896150897137, b[6] = 101652048214282205518610445783893750000000/2895 86658278060310247144250081091360673509, a[11,8] = -5476716391734935351 87249719091874257724281347519101322819710937500/5545897345777551056363 997101744668576293320106258184122878184564689, a[7,1] = 22628776819090 891/378192197956950000, b[1] = 55284231195707975647126708111723/125881 4250475014554517603612114000, a[11,7] = 271466662889835128614810796916 069488062485834784939879774912795880859375/170800578682088075849119195 72395871967414618088449127584294090356526117, a[10,4] = 35076261889213 578261995286390053983221937920015616/890366240305246889023432168040903 9895089390971875, b[12] = -50061468875139778913910254637881/1411866419 36036184819986782313781, c[4] = 1420497/9232000, a[8,6] = 503350581600 824913/125990342179812500, a[7,6] = 182198401/10649112500, a[8,5] = -2 9670244019740727/6363654238520000, a[7,5] = -21022506263989/6144360561 90000, a[7,4] = 116326878837670912/1047320101758291075, a[8,4] = -4704 8174572430533795112/6781573975400382234375, a[6,4] = 1418466643672/614 6756958375, a[5,4] = 1683359084698258/1167715119796875, c[3] = 473499/ 4616000, a[9,6] = 9941127982121041338714935249721178582954714935869695 2646781033905129048593757052549024957474512389085654050445280000000/10 2197500711543555291993605654791795484972275164188674551435468249918836 16697162903707552409044338358355837306818391433, a[6,5] = 725783021041 /3932390759616, a[5,1] = 3590880889914277/9341720958375000, a[8,1] = - 150702609045465151/280142368857000000, a[9,1] = -160459180355445265310 8566670820097679909295434905910255443941805885523703480313807623591448 5211856830024864166368323768200/98294831577286648441985440349216559753 7825783008095189369352011632206843937744670849326949363324414642152273 0587295739613, a[3,2] = 224201303001/2369389107200, b[8] = 47152588201 4932587321673707929687500000000/52372881739119879572881083964931404449 5553, a[4,1] = 1420497/36928000, a[9,2] = 0, a[10,2] = 0, a[11,2] = 0, a[12,2] = 0, a[9,3] = 0, a[10,3] = 0, a[11,3] = 0, a[12,3] = 0, a[12, 8] = -4693599865307842423678411639679979852809920520624795226940377187 50000/1262000068797740029604893636717124646814536106030908513651797310 900389, a[11,10] = -7641031089620713267943821483116886435546875/186511 5549729059107236140767953375513727732867, a[10,1] = 100509763879264306 824096153463041174636629364248095333923106653001873/229490324007644628 042361756217436155832461488260089524115475000000000, a[10,5] = 2987705 3472248545227782869189767925950557009/10443709158645362958089905740134 206606110000, a[11,5] = -304301954438407952266341438991435702455078125 /26817588459995310678515661957421952038871616, a[11,4] = -739282853758 412257967453242147288028248514875000/672441828756548290983239437133742 07709631980757, a[12,1] = -2173296165244568434534168496725754283210370 856714048295955495704392191998074219/826329709817197468253912814996276 721297719271873047121496939265708086637102000, a[12,4] = -117648588933 4472345397948024769479865991267657667189808793600000/12823827412574578 4857395023683055391720808636521055761250085619, a[10,6] = -72602025182 798889442893966553844286012776770019588838776297031451/409184393804050 07926071673834718276106178863851548244800289187500, a[12,6] = 44619197 7093458430386099958102902220927137343575035730212893991884410752778938 25000/3047226829867759667805678237083584916028456629069014496162891378 892137605913688013, a[9,5] = -9911366389726261686789033714164561000939 0040553516492468305812280242970735403338282694739815868376532443961828 2500000000/79848142008846002789298925227605775190331269194726743910364 273272231784282184770467794155269096224513726772081370189773, a[11,6] \+ = 79448939923998931164766316975203638555016544623086227984461001450043 71780765625/8870051616029286592528276040357686572984358849394897544622 73147402132735951354, a[10,7] = -1532291206386437051213014598849209860 5486502994107816190/31300694006456048761836695496055390763335792905249 19889, a[9,7] = 194327599672380134095898291719912961363678073793023525 007081328425098431574448809779310732532821200046895000000000/119960114 8822772264965667393113649089125646329262047360184187511517025904353198 7881275965965525693289412424400177, a[11,1] = -51010977601978416151375 71256611109219669965728737004453654940399435749927779/3460254790065218 498025394912113654440547987641021174968985010753183759952128, a[11,9] \+ = 27880505422945647305191478577013040105627752107138627482025385282602 3602554035252878888463451556710568650135086751734733498388408614837563 /570327897589938424716739897067258193159963418613722438239821039844076 4234613614101505591454120074773344256118618572821579031366188464747033 6, a[12,5] = -90164152627308866729367798396043430614573156600775937500 0/47006051321531726477650870675115597779536141098588395119, a[12,11] = 561064742310110985699639624014768981793940948761680464947352/67137939 1526329055316725343694451428305592235340194400714475, a[12,9] = -66412 7481006260866245417353167879111725438608160134129788715555378465573117 8130017572382606930519018125943416089220164619687237540672159347990548 52630446938063/9474053136757793278864620448618348468794738615395834799 9510116829111616884404043875158439278591322423989402984382146436322440 5440877235651675827150051218506000, a[9,4] = -235853914788187787303944 5811751103289867403691705393271280994482043263320271984518766662589229 245453919623760032009543680000000/217835526181955565558022800341429464 0863851721229066080313495435838630220158871171685663384701630900324981 96064101837445053273, a[10,8] = 66085154677219418645471125072555541174 985695924/310222736648235062495097951962638800384203417327\}:" }}} {PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 20 "printed coe fficients" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 283 "subs(e13,matrix([seq([c[i],seq(a[i,j],j=1..i-1),``$( 8-i)],i=2..8)]));\nfor ii from 9 to 12 do\n print(``);\n print(c[i i]=subs(e13,c[ii])); \n for jj to ii-1 do\n print(a[ii,jj]=subs (e13,a[ii,jj]));\n end do:\nend do:\n``;\nfor ii to 12 do\n print( b[ii]=subs(e13,b[ii]));\nend do:" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#K% 'matrixG6#7)7*#\"$R\"\"%+DF(%!GF+F+F+F+F+7*#\"'*\\t%\"(+gh%#\",*\\unA% *\"/+g`Xp%=\"#\"-,II,UA\".+s5*QpBF+F+F+F+F+7*#\"((\\?9\"(+?B*#F8\")+!G p$\"\"!#\"(\"\\hUF;F+F+F+F+7*#\"%B>\"%+]#\"1xU\"**)3)3f$\"1+]Pe4sT$*F< #!1VCC2t2A6\"0]7`YnZy(#\"1e#)p%3fLo\"\"1voz>^rn6F+F+F+7*#\"$B*\"%+?#\" +jo@zK\",+!Gz-rFoxGE#\"3++&p&z>#>y$FcgV9'#\"*,%)>#=\",+D6\\1\"F+7*#\"%r&)\"&++\"#!3^^YX!4Eq] \"\"3+++d)oB9!GFSCq'H\"1++_Q Uljj#\"38\\#3g\"e]L]\"3+D\")z@M!*f7\"\"*Q)pprint196\"" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%!G" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/&%\"cG6#\" \"*#\"5+=W$pl:=!G9\"5jpEaXi5N-:" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/&% \"aG6$\"\"*\"\"\"#!er+#oPKoj;k[-Io&=@&[9fBw!QJ![.P_&)e!=%RWb-\"f!\\V&H 4*zw4?3nm&3Jl_Wb.=f/;\"dr8'RdH(eIF_@k9WKj$\\pK\\3nWx$R%o?K;,_$p$*=&43I yDy`(fl@\\.W&)>W[mGx:$[H)*" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/&%\"aG6 $\"\"*\"\"#\"\"!" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/&%\"aG6$\"\"*\"\" $\"\"!" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/&%\"aG6$\"\"*\"\"%#!jr+++!o V&4?.gPi>RXX#H#*eimw=X)>F?LEV?[%*4GrKR0)\\K+4j,ZQjcor6()e,AI'QeV&\\8.3m!H7s^Q'3k%H9M+G-ebcb>=Eb$y@ " }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/&%\"aG6$\"\"*\"\"&#!cr++++DG='RWK lPoe\")RZp#GQLSN2(HC!G7eIoC\\;NbS+R4+hX;9P.*y'ohis*Qm8\"**\"brt(*=q83s ns8XA'4p_:%zn/x%=#G%yJAFtUO5RuEZ>p7L!>vdgF_#*)H*y-g%)3?9[)z" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/&%\"aG6$\"\"*\"\"'#\"br+++!GX/0ac3*Q7XZd\\- \\D0dPf[!H^!R.\"yYE&ppe$\\raHey6s\\_$\\rQ8/@@)z7T**\"brL9R=oIPeNe$QV/4 Cb2P!H;(p;O)=*\\#oaV^Xn)=k^Fs\\[&z\"zacg$*>HbNa62](>-\"" }}{PARA 11 " " 1 "" {XPPMATH 20 "6#/&%\"aG6$\"\"*\"\"(#\"]r++++]*o/+7#G`K2Jz(4)[WdJ %)4D%G832]_BIzt!yOOhH\"*>\"\\rx,SCCT*G$pDb'f'fF\")y)>` V!f-<:^(=%=gt/i#HjkD\"*3\\O6$Rnc'\\EsF#)[6g*>\"" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/&%\"aG6$\"\"*\"\")#!eq+++++^[A3?TTv#=_,Q!o2b(QD0*>&*\\ N;sVQ$[S0X*>Wi\"Qf*=ATonQ9QZ\"fqh-a>e))f!\\F**=c#\\MG;w,=3nDzeJ]#*)G#e r8wLFMkK&QGl\"*G)4#>hb\"QFc%" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%!G" } }{PARA 11 "" 1 "" {XPPMATH 20 "6#/&%\"cG6#\"#5#\"%6O\"%+]" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/&%\"aG6$\"#5\"\"\"#\"`ot=+`m5BRL&4[UOHmju6/jM: '4CoIk#zQw405\"`o++++]Z:T_*3g#)[hC$e:Ou@cA$)R0!R'G&*>EyN@*)=Ew]$\"Rv=(4R*3&* )R!4/o@VB!*)oC0.Cm.*)" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/&%\"aG6$\"#5 \"\"&#\"M4qb]f#zw*=pGyF_a[AZ`q()H\"M++61m?M,u0**3eHOX'e\"4PW5" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#/&%\"aG6$\"#5\"\"'#!\\o^9.(Hw(Q))e>+xw F,'GWQbmR*GW*)))z#=D?gs\"\\o+v=*G+[C[:&Q')yh5w#=Z$Qnrg#z+0/QR%=4%" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#/&%\"aG6$\"#5\"\"(#!Y!>;y5%*H]'[0')4# \\))f9I@^qV'Q17HK:\"X*))>\\_!HzNLw!Rbg\\&pO=w[gX1Sp+8$" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/&%\"aG6$\"#5\"\")#\"PCfp&)\\sna^3m\" QFtT.UQ+)QE'>&z4&\\i]B[mtA-J" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/&%\"a G6$\"#5\"\"*#!iq@Ib?B,Kr()ot&4gGZK?%R_vv3B;+.5`_ls_IeMm.BJ#ymE!\\Dwgl* =YlvM$\"jq++++]d^+y$*pf1bZd/DSddGt4X.Pp-Lf.8D\"pJ/;I([hP#>ax#30l[\"=2O c" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%!G" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/&%\"cG6#\"#6#\"#:\"#;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/&%\"a G6$\"#6\"\"\"#!gozx#*\\dV*RS\\l`W+P(Gd'*p'>#46hc7dP^hTy>gx4,^\"goG@&*f P=`2,&)*o\\<@5k()zaSWl8@\"\\RD!)\\=_1!za-Y$" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/&%\"aG6$\"#6\"\"#\"\"!" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/&%\"aG6$\"#6\"\"$\"\"!" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/&%\" aG6$\"#6\"\"%#!Q+]([^[#G!)GZ@C`u'zD7%eP&GGR(\"Pd2)>j4x?uLrVRK)4H[lvG=W s'" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/&%\"aG6$\"#6\"\"&#!ND\"y]X-dV\" **Q9MmA&zSQW&>I/$\"M;;()Q?&>Ud>m:&y1J&**f%)e(eRs&>>\"\\e2 )3#oy0!3<" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/&%\"aG6$\"#6\"\")#!]o+v$ 4r>GK,\">vM\"GCxDu=4>(\\s=NN\\t\"R;nZ&\"^o*okX=yG7%=ei5?LHw&oYu,r*ROc5 bxdM(*ea&" }}{PARA 12 "" 1 "" {XPPMATH 20 "6#/&%\"aG6$\"#6\"\"*#\"esjv $[h3%)Q)\\LZt^n3N,lo0rc:Xj%)))yGDNSb-O-EG&QD?[F'Qr5_xi0,/8qdy9>0tkXHU0 0)y#\"gsO.ZZY)=m8.z:#Gd='=hDWLxu+7a9f0:59OhMUwS%)R5#)R#QCs8'=M'*fJ>es1 (*)RnrC%Q**e(*yKq&" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/&%\"aG6$\"#6\"# 5#!LvoaNk)o6$[@Q%zE82i*3J5k(\"OnGtFP^vL&zwShB2\"f!H(\\b6l=" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%!G" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/&%\" cG6#\"#7\"\"\"" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/&%\"aG6$\"#7\"\"\"# !jo>U2)*>>#R/d\\bfH[Src3P5KGads'\\oT`M%oXClhHt@\"io+?5Pm33dERp\\@r/t=F >xH@nF'*\\\"G\"RDou><)4(Hj#)" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/&%\"a G6$\"#7\"\"#\"\"!" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/&%\"aG6$\"#7\"\" $\"\"!" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/&%\"aG6$\"#7\"\"%#![o++g$z3 )*=nwln7*f')z%pZ-[zRXBZM$*)e[w6\"jn>c3]7wb5_O'33s\"RbIoB]Rd[yXd7u#Q#G \"" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/&%\"aG6$\"#7\"\"&#!Z+]Pfx+m:tXh IM/'R)zn$Hn')3ti_T;!*\"Y>^R)e)4Th`zxf:^nq3lxksJ:K^g+Z" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/&%\"aG6$\"#7\"\"'#\"^p+]#Q*yFv5W)=*R*G@Id.vNMPr#4 A-H5e**4'QI%eM4x>>Y%\"]p8!)o8fgP@*)y8*G;'\\9!p!HmXGg\"\\e$3P#yc!ymfx') HoAZI" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/&%\"aG6$\"#7\"\"(#\"io+]7`x& Q$3!)ocvqEq))Q$GpKuM0(fuwg:$pr5_mU(R$[*\"ho`f6b&>BP32Hzo7>n[NKQd#e@r.s a?R_;Q_oI?+T&" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/&%\"aG6$\"#7\"\")#!` o++v=x.%pA&zC1_?*4G&)z*z'R;TyOUUyIl)*f$p%\"ao*Q+4J(z^O^34.1h`9okCrrOO* [gH+u(zo++i7" }}{PARA 12 "" 1 "" {XPPMATH 20 "6#/&%\"aG6$\"#7\"\"*#!jt j!QpWIE&[0*zMf@nSvB(o>Y;?#*3;M%f7=!>0$pg#Qsv,I\"y6tbYy`b:()yHT8g\"3'Qa s6\"zy;`tTXi'3E15[FTm\"jt+g]=70]r#en^cBx3W0WAjVY@Q%)HS*)RUA8fy#R%e^(Q/ /W)oh6\"Ho65&**zMeR:'QZzo%[$='[/ik)yKzdn8`SZ*" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/&%\"aG6$\"#7\"#5#\"]o+]ilIr^TeT;'z'fCr&)oy!=(y#=nBJ%\\ *)o%fC\"\"^oPr*3:'*=SOJ+/.)ect3jWt[E**f9R7&**>hEUC" }}{PARA 11 "" 1 " " {XPPMATH 20 "6#/&%\"aG6$\"#7\"#6#\"gn_t%\\Y!oh([4%Rz\")*oZ,C'R'*p&)4 65Buk5c\"gnvWr+W>S`B#f0$G9X%pV`s;`0Hj_\"Rz8n" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%!G" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/&%\"bG6#\"\"\"# \"AB<63n7Zc(zq&>JUGb\"C+S67Ogt>i\"MTV0kFFM3!)>o=#=**Gc9Bzo_#" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#/&%\"bG6#\"\")#\"K++++voHzqt;K(eK\\,#) e_r%\"K`b\\WSJ\\'R3\")Gdz)>\"R<)GP_" }}{PARA 12 "" 1 "" {XPPMATH 20 "6 #/&%\"bG6#\"\"*#\"]t>%3MD>!G%[rBVH5v$Q%)p_7!\\&\\\"**y130hUHdN=q@)[Ne7 )R:)yXhrz@c*\\8N`!QWq-n%z4'H]u!f'>%)*R'\"]t+!Qq'ypt(4a[-N?6`k/j9w&p** \\j5`C)>oK^HX;wVj+8$ohSYO&4j^ZRsFxEC.]**f$pn!)*fTAyQQnS\"" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/&%\"bG6#\"#5#\"H+++DJqXVA_DfIvDbFc%f\"KJ?7UM&f V<9V,BoF.\"4s-Q7" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/&%\"bG6#\"#6#!CcY XqQ:f*RcqzR8\"p')=\"Bvw[7Lc^RQNe)eeZzR" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/&%\"bG6#\"#7#!A\")yja-\"R\"*y(R^()o91]\"B\"y8By')*>[=Og$>k'=T\" " }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}}{PARA 0 "" 0 "" {TEXT -1 52 "#------- --------------------------------------------" }}{SECT 1 {PARA 4 "" 0 " " {TEXT -1 43 "construction of the order 7 embedded scheme" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }} }{SECT 1 {PARA 4 "" 0 "" {TEXT -1 4 "e13 " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 7419 "e13 := \{c[7] = 769 /5000, c[6] = 923/2000, c[5] = 1923/5000, a[8,3] = 0, a[8,2] = 0, a[7, 3] = 0, a[7,2] = 0, a[6,3] = 0, a[6,2] = 0, a[5,2] = 0, a[4,2] = 0, c[ 8] = 8571/10000, a[8,7] = 9, c[2] = 139/2500, c[9] = 14280181565693441 800/15023510624554266963, a[5,3] = -1122077307242443/778476746531250, \+ a[4,3] = 4261491/36928000, a[3,1] = 94226774499/11846945536000, a[6,1] = 3279216863/71027928000, a[12,7] = 948339742665210716931560767459705 347432692833888702670755668800833857753125000/541002030685238165239205 47203712158257383235486719126879290708372319555115953, a[2,1] = 139/25 00, b[10] = 594562755257530592552224345703125000000/123802720910327682 301431417435953442122031, b[9] = 6399841965907450296097946702704438053 3513499562179716145788153981258354882170183557294261050806789914954901 252698438375102943237148428019253408419/140673838782241599806769359995 0032426772772394751630953646406168313006343761645295132681982453106349 9969576146304645311203502485409773697867038000, b[7] = 621973195888209 3270433753490048828125000000/25268792314562899182186819800834272764054 341, c[10] = 3611/5000, c[11] = 15/16, c[12] = 1, a[10,9] = -334756546 1896560762549026667823123036634583052726552531003001623087575523942032 4728600957368877132012320553021/56360718148650508277541923761487301604 3169125130359330269370345097328575740250457475506596993780051575000000 000, a[9,8] = -4738143867684122189593816244199450540483384372163549951 990525387550768038015218275414120082248510000000000/456273815561192098 2891652838532643427337613715822889250315879256708180176162834492561899 2749059885819540261, b[5] = 0, b[2] = 0, b[3] = 0, b[4] = 0, b[11] = - 1886691133979705639959153870454656/397947585885835383951563312487675, \+ a[12,10] = 12459468894943123671827871807868857124596796164158415171306 5625000/24422661199951239145999264873446308735658803040031364018961508 97137, b[6] = 101652048214282205518610445783893750000000/2895866582780 60310247144250081091360673509, a[11,8] = -5476716391734935351872497190 91874257724281347519101322819710937500/5545897345777551056363997101744 668576293320106258184122878184564689, a[7,1] = 22628776819090891/37819 2197956950000, b[1] = 55284231195707975647126708111723/125881425047501 4554517603612114000, a[11,7] = 271466662889835128614810796916069488062 485834784939879774912795880859375/170800578682088075849119195723958719 67414618088449127584294090356526117, a[10,4] = 35076261889213578261995 286390053983221937920015616/890366240305246889023432168040903989508939 0971875, b[12] = -50061468875139778913910254637881/1411866419360361848 19986782313781, c[4] = 1420497/9232000, a[8,6] = 503350581600824913/12 5990342179812500, a[7,6] = 182198401/10649112500, a[8,5] = -2967024401 9740727/6363654238520000, a[7,5] = -21022506263989/614436056190000, a[ 7,4] = 116326878837670912/1047320101758291075, a[8,4] = -4704817457243 0533795112/6781573975400382234375, a[6,4] = 1418466643672/614675695837 5, a[5,4] = 1683359084698258/1167715119796875, c[3] = 473499/4616000, \+ a[9,6] = 9941127982121041338714935249721178582954714935869695264678103 3905129048593757052549024957474512389085654050445280000000/10219750071 1543555291993605654791795484972275164188674551435468249918836166971629 03707552409044338358355837306818391433, a[6,5] = 725783021041/39323907 59616, a[5,1] = 3590880889914277/9341720958375000, a[8,1] = -150702609 045465151/280142368857000000, a[9,1] = -160459180355445265310856667082 0097679909295434905910255443941805885523703480313807623591448521185683 0024864166368323768200/98294831577286648441985440349216559753782578300 8095189369352011632206843937744670849326949363324414642152273058729573 9613, a[3,2] = 224201303001/2369389107200, b[8] = 47152588201493258732 1673707929687500000000/523728817391198795728810839649314044495553, a[4 ,1] = 1420497/36928000, a[9,2] = 0, a[10,2] = 0, a[11,2] = 0, a[12,2] \+ = 0, a[9,3] = 0, a[10,3] = 0, a[11,3] = 0, a[12,3] = 0, a[12,8] = -469 359986530784242367841163967997985280992052062479522694037718750000/126 2000068797740029604893636717124646814536106030908513651797310900389, a [11,10] = -7641031089620713267943821483116886435546875/186511554972905 9107236140767953375513727732867, a[10,1] = 100509763879264306824096153 463041174636629364248095333923106653001873/229490324007644628042361756 217436155832461488260089524115475000000000, a[10,5] = 2987705347224854 5227782869189767925950557009/10443709158645362958089905740134206606110 000, a[11,5] = -304301954438407952266341438991435702455078125/26817588 459995310678515661957421952038871616, a[11,4] = -739282853758412257967 453242147288028248514875000/672441828756548290983239437133742077096319 80757, a[12,1] = -2173296165244568434534168496725754283210370856714048 295955495704392191998074219/826329709817197468253912814996276721297719 271873047121496939265708086637102000, a[12,4] = -117648588933447234539 7948024769479865991267657667189808793600000/12823827412574578485739502 3683055391720808636521055761250085619, a[10,6] = -72602025182798889442 893966553844286012776770019588838776297031451/409184393804050079260716 73834718276106178863851548244800289187500, a[12,6] = 44619197709345843 038609995810290222092713734357503573021289399188441075277893825000/304 7226829867759667805678237083584916028456629069014496162891378892137605 913688013, a[9,5] = -9911366389726261686789033714164561000939004055351 6492468305812280242970735403338282694739815868376532443961828250000000 0/79848142008846002789298925227605775190331269194726743910364273272231 784282184770467794155269096224513726772081370189773, a[11,6] = 7944893 9923998931164766316975203638555016544623086227984461001450043717807656 25/8870051616029286592528276040357686572984358849394897544622731474021 32735951354, a[10,7] = -1532291206386437051213014598849209860548650299 4107816190/3130069400645604876183669549605539076333579290524919889, a[ 9,7] = 194327599672380134095898291719912961363678073793023525007081328 425098431574448809779310732532821200046895000000000/119960114882277226 4965667393113649089125646329262047360184187511517025904353198788127596 5965525693289412424400177, a[11,1] = -51010977601978416151375712566111 09219669965728737004453654940399435749927779/3460254790065218498025394 912113654440547987641021174968985010753183759952128, a[11,9] = 2788050 5422945647305191478577013040105627752107138627482025385282602360255403 5252878888463451556710568650135086751734733498388408614837563/57032789 7589938424716739897067258193159963418613722438239821039844076423461361 41015055914541200747733442561186185728215790313661884647470336, a[12,5 ] = -901641526273088667293677983960434306145731566007759375000/4700605 1321531726477650870675115597779536141098588395119, a[12,11] = 56106474 2310110985699639624014768981793940948761680464947352/67137939152632905 5316725343694451428305592235340194400714475, a[12,9] = -66412748100626 0866245417353167879111725438608160134129788715555378465573117813001757 2382606930519018125943416089220164619687237540672159347990548526304469 38063/9474053136757793278864620448618348468794738615395834799951011682 9111616884404043875158439278591322423989402984382146436322440544087723 5651675827150051218506000, a[9,4] = -235853914788187787303944581175110 3289867403691705393271280994482043263320271984518766662589229245453919 623760032009543680000000/217835526181955565558022800341429464086385172 1229066080313495435838630220158871171685663384701630900324981960641018 37445053273, a[10,8] = 66085154677219418645471125072555541174985695924 /310222736648235062495097951962638800384203417327\}:" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}} {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 61 "We can ob tain an embedded 11 stage order 7 scheme as follows." }}{PARA 0 "" 0 " " {TEXT -1 94 "We remove stages 11 and 12 from the 12 stage order 8 sc heme and introduce a new stage 11 with " }{XPPEDIT 18 0 "c[11] = 1;" " 6#/&%\"cG6#\"#6\"\"\"" }{TEXT -1 3 ", " }{XPPEDIT 18 0 "a[11,2] = 0" "6#/&%\"aG6$\"#6\"\"#\"\"!" }{TEXT -1 3 ", " }{XPPEDIT 18 0 "a[11,3] \+ = 0;" "6#/&%\"aG6$\"#6\"\"$\"\"!" }{TEXT -1 3 ", " }{XPPEDIT 18 0 "`b *`[2]=0" "6#/&%#b*G6#\"\"#\"\"!" }{TEXT -1 3 ", " }{XPPEDIT 18 0 "`b* `[3]=0" "6#/&%#b*G6#\"\"$\"\"!" }{TEXT -1 3 ", " }{XPPEDIT 18 0 "`b*` [4]=0" "6#/&%#b*G6#\"\"%\"\"!" }{TEXT -1 7 " and " }{XPPEDIT 18 0 "` b*`[5]=0" "6#/&%#b*G6#\"\"&\"\"!" }{TEXT -1 58 ", where the weights of the order 7 scheme are denoted by " }{XPPEDIT 18 0 "`b*`" "6#%#b*G" }{TEXT -1 1 "." }}{PARA 0 "" 0 "" {TEXT -1 1 " " }}{PARA 0 "" 0 "" {TEXT -1 156 "We incorporate the order 7 quadrature conditions, the ro w sum conditions for this stage and stage-order conditions so that thi s new stage has stage-order 4." }}{PARA 0 "" 0 "" {TEXT -1 48 "We also incorporate the simplifying conditions: " }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "Sum(`b*`[i]*a[i,j],i=j+1..11)=`b*`[j]*(1-c[j] )" "6#/-%$SumG6$*&&%#b*G6#%\"iG\"\"\"&%\"aG6$F+%\"jGF,/F+;,&F0F,F,F,\" #6*&&F)6#F0F,,&F,F,&%\"cG6#F0!\"\"F," }{TEXT -1 1 "," }}{PARA 0 "" 0 " " {TEXT -1 5 "for " }{XPPEDIT 18 0 "j=1" "6#/%\"jG\"\"\"" }{TEXT -1 10 " ( with " }{XPPEDIT 18 0 "c[1]=0" "6#/&%\"cG6#\"\"\"\"\"!" } {TEXT -1 5 " ), " }{XPPEDIT 18 0 "j = 8;" "6#/%\"jG\"\")" }{TEXT -1 3 ", " }{XPPEDIT 18 0 "j=9" "6#/%\"jG\"\"*" }{TEXT -1 7 " and " } {XPPEDIT 18 0 "j=10" "6#/%\"jG\"#5" }{TEXT -1 2 ". " }}{PARA 0 "" 0 " " {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 399 "`Qeqs*` : = subs(b=`b*`,QuadratureConditions(7,11,'expanded')):\nSO_eqs := [add( a[11,j],j=1..10)=c[11],add(a[11,j]*c[j],j=2..10)=1/2*c[11]^2,\n \+ add(a[11,j]*c[j]^2,j=2..10)=1/3*c[11]^3,add(a[11,j]*c[j]^3,j=2..10)=1 /4*c[11]^4]:\n`simp_eqs*` := [add(`b*`[i]*a[i,1],i=2..11)=`b*`[1],seq( add(`b*`[i]*a[i,j],i=j+1..11)=`b*`[j]*(1-c[j]),j=[8,9,10])]:\n`cdns*` \+ := [op(`simp_eqs*`),op(SO_eqs),op(`Qeqs*`)]:" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 144 "e14 := `uni on`(remove(u_->member(op(1,lhs(u_)),[11,12]) or op(0,lhs(u_))=b,e13), \n \{c[11]=1,a[11,2]=0,a[11,3]=0,seq(`b*`[i]=0,i=2..5)\}):\n" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 53 "We hav e 15 equations for the 15 unknown coefficients." }}{EXCHG {PARA 0 "> \+ " 0 "" {MPLTEXT 1 0 60 "eqns3 := subs(e14,`cdns*`):\nnops(%);\nindets( eqns3);\nnops(%);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#\"#:" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#<1&%#b*G6#\"\"*&F%6#\"#5&F%6#\"#6&F%6#\"\"\"&F% 6#\"\"'&F%6#\"\"(&F%6#\"\")&%\"aG6$F-F'&F;6$F-\"\"%&F;6$F-F0&F;6$F-F9& F;6$F-F*&F;6$F-F3&F;6$F-F6&F;6$F-\"\"&" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#\"#:" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 " " {MPLTEXT 1 0 22 "infolevel[solve] := 4:" }}}{EXCHG {PARA 0 "> " 0 " " {MPLTEXT 1 0 74 "e15 := solve(\{op(eqns3)\}):\ninfolevel[solve] := 0 :\ne16 := `union`(e14,e15):" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}{SECT 1 {PARA 4 "" 0 " " {TEXT -1 3 "e16" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 5803 "e16 := \{a[10,3] = 0, a[11,3] = 0, a[11,2] \+ = 0, a[7,2] = 0, a[9,3] = 0, c[3] = 473499/4616000, a[5,3] = -11220773 07242443/778476746531250, a[4,3] = 4261491/36928000, a[8,6] = 50335058 1600824913/125990342179812500, a[9,7] = 194327599672380134095898291719 9129613636780737930235250070813284250984315744488097793107325328212000 46895000000000/1199601148822772264965667393113649089125646329262047360 1841875115170259043531987881275965965525693289412424400177, a[6,3] = 0 , a[10,2] = 0, a[4,2] = 0, a[8,3] = 0, a[8,2] = 0, a[7,3] = 0, a[6,2] \+ = 0, a[5,2] = 0, a[9,2] = 0, a[5,1] = 3590880889914277/934172095837500 0, a[6,1] = 3279216863/71027928000, c[10] = 3611/5000, a[10,8] = 66085 154677219418645471125072555541174985695924/310222736648235062495097951 962638800384203417327, `b*`[2] = 0, `b*`[3] = 0, `b*`[4] = 0, `b*`[5] \+ = 0, a[6,5] = 725783021041/3932390759616, a[9,1] = -160459180355445265 3108566670820097679909295434905910255443941805885523703480313807623591 4485211856830024864166368323768200/98294831577286648441985440349216559 7537825783008095189369352011632206843937744670849326949363324414642152 2730587295739613, a[11,8] = 272747740939488905723271728465054113286243 78015249298079900000000/1115922486918625141794316350997246426030217325 603407120793811703663, a[11,9] = 1517649980273287794810711480928689762 9699413594465893552446832434708688739805374627179455308912428921880198 50453060256773054273287757818339/1793109635140354951732482295161949269 0590259109554169692111771494162805088297821242848492102923447532215742 446890559805801281057084758250800, a[10,6] = -726020251827988894428939 66553844286012776770019588838776297031451/4091843938040500792607167383 4718276106178863851548244800289187500, c[11] = 1, a[11,7] = -356296272 4414944757891560164749973139486260772587248231213362868690625000/40897 6639178968899516222045518619849445955714595234573207901740578867341, a [10,5] = 29877053472248545227782869189767925950557009/1044370915864536 2958089905740134206606110000, a[10,7] = -15322912063864370512130145988 492098605486502994107816190/313006940064560487618366954960553907633357 9290524919889, a[7,4] = 116326878837670912/1047320101758291075, a[9,5] = -991136638972626168678903371416456100093900405535164924683058122802 429707354033382826947398158683765324439618282500000000/798481420088460 0278929892522760577519033126919472674391036427327223178428218477046779 4155269096224513726772081370189773, a[9,6] = 9941127982121041338714935 2497211785829547149358696952646781033905129048593757052549024957474512 389085654050445280000000/102197500711543555291993605654791795484972275 1641886745514354682499188361669716290370755240904433835835583730681839 1433, a[10,4] = 35076261889213578261995286390053983221937920015616/890 3662403052468890234321680409039895089390971875, a[8,1] = -150702609045 465151/280142368857000000, `b*`[6] = 914725533083362212330207501220000 00000/270389036674192633283981559366098375979, a[9,4] = -2358539147881 8778730394458117511032898674036917053932712809944820432633202719845187 66662589229245453919623760032009543680000000/2178355261819555655580228 0034142946408638517212290660803134954358386302201588711716856633847016 3090032498196064101837445053273, a[10,9] = -33475654618965607625490266 6782312303663458305272655253100300162308757552394203247286009573688771 32012320553021/5636071814865050827754192376148730160431691251303593302 69370345097328575740250457475506596993780051575000000000, c[4] = 14204 97/9232000, a[3,1] = 94226774499/11846945536000, a[11,5] = 60867617377 97404637560639819223305802258863489403125000/2784859079002788542258846 760345051095481316496802932391, `b*`[10] = 236320240037774049966221351 5625000000/19167474982246118950523520271861502109, a[11,1] = 211253736 5439605317842214400294939600845864356126794907105288476157299972777/97 9112999257634895438196693571257267376015014427722256867300057498972329 5600, a[11,6] = -25596656018056374091988888506984281725655172189735771 46475607855715703433900000/1517075781450111169555356439342115045941218 379260955323225394329122945220575603, `b*`[8] = 7580382785455138868782 239796250000000000/33873008089978031564549954803189465564389, `b*`[9] \+ = -1264668207534772000389416909088295806823076185957298091511366356638 145411319432807599860924173256812108555319505561297378438513/314581336 8236387937635983732861131099009925923393634680707983641010013290201378 9877583819562913435723874810037875576750158649200, a[11,4] = 634017325 68650174882469655273037557867024014649473659852800000/7597437179564699 900056826682790095876501867794005624909986891, c[9] = 1428018156569344 1800/15023510624554266963, `b*`[1] = 3635543992294021475202312550589/8 3920950031667636967840240807600, a[10,1] = 100509763879264306824096153 463041174636629364248095333923106653001873/229490324007644628042361756 217436155832461488260089524115475000000000, `b*`[7] = 3124571087910685 9500854236329453125000000/125747467177230198813995913261775760852127, \+ a[8,5] = -29670244019740727/6363654238520000, a[7,5] = -21022506263989 /614436056190000, c[2] = 139/2500, c[5] = 1923/5000, c[6] = 923/2000, \+ c[7] = 769/5000, c[8] = 8571/10000, a[7,1] = 22628776819090891/3781921 97956950000, `b*`[11] = 2965876353908674111604434604609/47062213978678 728273328927437927, a[11,10] = 365246427065695972768456123682444907417 47970924877885921875000/6720450834551542949183841114674672966026925518 4823759889942481, a[3,2] = 224201303001/2369389107200, a[7,6] = 182198 401/10649112500, a[8,4] = -47048174572430533795112/6781573975400382234 375, a[9,8] = -4738143867684122189593816244199450540483384372163549951 990525387550768038015218275414120082248510000000000/456273815561192098 2891652838532643427337613715822889250315879256708180176162834492561899 2749059885819540261, a[8,7] = 9, a[2,1] = 139/2500, a[6,4] = 141846664 3672/6146756958375, a[5,4] = 1683359084698258/1167715119796875, a[4,1] = 1420497/36928000\}:" }{TEXT -1 0 "" }}}{PARA 0 "" 0 "" {TEXT -1 0 " " }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 20 "printed coefficients" }}{PARA 0 "" 0 " " {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 289 "subs(e16, matrix([seq([c[i],seq(a[i,j],j=1..i-1),``$(8-i)],i=2..8)]));\nfor ii f rom 9 to 11 do\n print(``);\n print(c[ii]=subs(e16,c[ii])); \n f or jj to ii-1 do\n print(a[ii,jj]=subs(e16,a[ii,jj]));\n end do :\nend do:\n``;\nfor ii to 11 do\n print(`b*`[ii]=subs(e16,`b*`[ii]) );\nend do:" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#K%'matrixG6#7)7*#\"$R\" \"%+DF(%!GF+F+F+F+F+7*#\"'*\\t%\"(+gh%#\",*\\unA%*\"/+g`Xp%=\"#\"-,II, UA\".+s5*QpBF+F+F+F+F+7*#\"((\\?9\"(+?B*#F8\")+!Gp$\"\"!#\"(\"\\hUF;F+ F+F+F+7*#\"%B>\"%+]#\"1xU\"**)3)3f$\"1+]Pe4sT$*F<#!1VCC2t2A6\"0]7`YnZy (#\"1e#)p%3fLo\"\"1voz>^rn6F+F+F+7*#\"$B*\"%+?#\"+jo@zK\",+!Gz-rFoxGE# \"3++&p&z>#>y$FcgV9'# \"*,%)>#=\",+D6\\1\"F+7*#\"%r&)\"&++\"#!3^^YX!4Eq]\"\"3+++d)oB9!GFSCq'H\"1++_QUljj#\"38\\#3g\"e]L] \"3+D\")z@M!*f7\"\"*Q)pprint206\"" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#% !G" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/&%\"cG6#\"\"*#\"5+=W$pl:=!G9\"5 jpEaXi5N-:" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/&%\"aG6$\"\"*\"\"\"#!er +#oPKoj;k[-Io&=@&[9fBw!QJ![.P_&)e!=%RWb-\"f!\\V&H4*zw4?3nm&3Jl_Wb.=f/; \"dr8'RdH(eIF_@k9WKj$\\pK\\3nWx$R%o?K;,_$p$*=&43IyDy`(fl@\\.W&)>W[mGx: $[H)*" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/&%\"aG6$\"\"*\"\"#\"\"!" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#/&%\"aG6$\"\"*\"\"$\"\"!" }}{PARA 11 " " 1 "" {XPPMATH 20 "6#/&%\"aG6$\"\"*\"\"%#!jr+++!oV&4?.gPi>RXX#H#*eimw =X)>F?LEV?[%*4GrKR0)\\K+4j,Z Qjcor6()e,AI'QeV&\\8.3m!H7s^Q'3k%H9M+G-ebcb>=Eb$y@" }}{PARA 11 "" 1 " " {XPPMATH 20 "6#/&%\"aG6$\"\"*\"\"&#!cr++++DG='RWKlPoe\")RZp#GQLSN2(H C!G7eIoC\\;NbS+R4+hX;9P.*y'ohis*Qm8\"**\"brt(*=q83sns8XA'4p_:%zn/x%=#G %yJAFtUO5RuEZ>p7L!>vdgF_#*)H*y-g%)3?9[)z" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/&%\"aG6$\"\"*\"\"'#\"br+++!GX/0ac3*Q7XZd\\-\\D0dPf[!H^!R.\"yYE& ppe$\\raHey6s\\_$\\rQ8/@@)z7T**\"brL9R=oIPeNe$QV/4Cb2P!H;(p;O)=*\\#oaV ^Xn)=k^Fs\\[&z\"zacg$*>HbNa62](>-\"" }}{PARA 11 "" 1 "" {XPPMATH 20 "6 #/&%\"aG6$\"\"*\"\"(#\"]r++++]*o/+7#G`K2Jz(4)[WdJ%)4D%G832]_BIzt!yOOhH \"*>\"\\rx,SCCT*G$pDb'f'fF\")y)>`V!f-<:^(=%=gt/i#HjkD \"*3\\O6$Rnc'\\EsF#)[6g*>\"" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/&%\"aG 6$\"\"*\"\")#!eq+++++^[A3?TTv#=_,Q!o2b(QD0*>&*\\N;sVQ$[S0X*>Wi\"Qf*=AT onQ9QZ\"fqh-a>e))f!\\F**=c#\\MG;w,=3nDzeJ]#*)G#er8wLFMkK&QGl\"*G)4#>hb \"QFc%" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%!G" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/&%\"cG6#\"#5#\"%6O\"%+]" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/&%\"aG6$\"#5\"\"\"#\"`ot=+`m5BRL&4[UOHmju6/jM:'4CoIk#zQw405\"`o ++++]Z:T_*3g#)[hC$e:Ou@cA$)R0!R'G&*>EyN@*)=Ew]$\"Rv=(4R*3&*)R!4/o@VB!*)oC0.Cm.*) " }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/&%\"aG6$\"#5\"\"&#\"M4qb]f#zw*=pG yF_a[AZ`q()H\"M++61m?M,u0**3eHOX'e\"4PW5" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/&%\"aG6$\"#5\"\"'#!\\o^9.(Hw(Q))e>+xwF,'GWQbmR*GW*)))z#=D?gs\" \\o+v=*G+[C[:&Q')yh5w#=Z$Qnrg#z+0/QR%=4%" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/&%\"aG6$\"#5\"\"(#!Y!>;y5%*H]'[0')4#\\))f9I@^qV'Q17HK:\"X*))>\\ _!HzNLw!Rbg\\&pO=w[gX1Sp+8$" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/&%\"aG 6$\"#5\"\")#\"PCfp&)\\sna^3m\"QFtT.UQ+)QE'>&z4&\\i]B[mtA- J" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/&%\"aG6$\"#5\"\"*#!iq@Ib?B,Kr()o t&4gGZK?%R_vv3B;+.5`_ls_IeMm.BJ#ymE!\\Dwgl*=YlvM$\"jq++++]d^+y$*pf1bZd /DSddGt4X.Pp-Lf.8D\"pJ/;I([hP#>ax#30l[\"=2Oc" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%!G" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/&%\"cG6#\"#6\" \"\"" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/&%\"aG6$\"#6\"\"\"#\"goxF(**H dhZ)G0r!\\zEhNke%3gR\\H+W@UyJ0'RaOPD6#\"go+cHB(*)\\d+InoDAxU9],wtEd7d$ p'>Qa*[jd#**H6z*" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/&%\"aG6$\"#6\"\"# \"\"!" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/&%\"aG6$\"#6\"\"$\"\"!" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#/&%\"aG6$\"#6\"\"%#\"in++!G&)fOZ\\Y,Cq 'ybPIFb'pC)[<]'oDt,M'\"hn\"*o)*4\\i0Szn=]we4!z#oEo0+**pk&zrV(f(" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#/&%\"aG6$\"#6\"\"&#\"X+]7.%*[j)eA!eIB# >)R1cPYS(zP4uj0=gl'fD\"jo.cd?_%H7HVRDKKb4Ez$=7%f/:@MRkNb&p66]9yvq^\"" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#/&%\"aG6$\"#6\"\"(#!do+]i!poGO87B[ses2 E'[RJ(*\\Z;g:*yvW\\TCF'Hc$\"coTt')y0u,z?tXB&f9d&fW\\)>'=b/Ai^**)o*y\"R m(*3%" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/&%\"aG6$\"#6\"\")#\"\\o++++* z!)H\\_,yViG8T0l%G!)=#*GC\"*3`Xzriu`!)R()o3ZVKoW_N*eY%f8%*pHw*oG4[62\"[z(Gt-)* \\w^\"\"gs+3DeZ3d5G,e!)f0*oWUd@KvWBH5#\\[GC@yH)30G;%\\r<6#ppTb4\"f-f!p #\\>;&H#[K<&\\NS^j4Jz\"" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/&%\"aG6$\" #6\"#5#\"in+](=#f)y([#4(zuT2\\W#oBhXoF(fplqUY_O\"in\"[U**))fP#[=b#p-mH nuY6TQ=\\Ha^X$3X?n" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%!G" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/&%#b*G6#\"\"\"#\"@*e]DJ-_Z@SH#*RaNO\"A+w!3CSy' pjn;.]4#R)" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/&%#b*G6#\"\"#\"\"!" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#/&%#b*G6#\"\"$\"\"!" }}{PARA 11 "" 1 " " {XPPMATH 20 "6#/&%#b*G6#\"\"%\"\"!" }}{PARA 11 "" 1 "" {XPPMATH 20 " 6#/&%#b*G6#\"\"&\"\"!" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/&%#b*G6#\"\" '#\"G++++?7]2-L7AO$3Lbs9*\"HzfP)4m$f:)RGLE>um.*Qq#" }}{PARA 11 "" 1 " " {XPPMATH 20 "6#/&%#b*G6#\"\"(#\"J+++DJXHjBa3]fo5z3rX7$\"KF@&3wvIs`b3@\"oDtT#4')*f2GV>8TX \"QmNm8^\"4)Hdf=wI#o!eH)34pT*Q+?xMv?oYE\"\"[s+#\\'e,vwb(y.5[(QsNM\"Hc> Qex)*y8?!H8+,TO)zq!oMORBf#*4!*4J6'Gt$)fjPzQO#oL\"e9$" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/&%#b*G6#\"#5#\"F+++Dc^8Am*\\SxP+C?jB\"G4@]h=F?N_]*= hC#)\\Zn\">" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/&%#b*G6#\"#6#\"@4YgMWg 6Tn3RNwe'H\"AFzVF*GLFG(y'yR@iq%" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 104 "`RK7_11eqs*` := s ubs(b=`b*`,[op(RowSumConditions(11,'expanded')),op(OrderConditions(7,1 1,'expanded'))]):" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 5 "Check" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 57 "subs(e1 6,`RK7_11eqs*`):\nmap(u->lhs(u)-rhs(u),%);\nnops(%);" }}{PARA 12 "" 1 "" {XPPMATH 20 "6#7[q\"\"!F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$ F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$ F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$ F$F$" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#\"#&*" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 93 "We can calculate the prin cipal error norm, that is, the 2-norm of the principal error terms. " }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 185 "`errterms7_11*` := subs(b =`b*`,PrincipalErrorTerms(7,11,'expanded')):\nsm := 0:\nfor ct to nops (`errterms7_11*`) do\n sm := sm+(evalf(subs(e16,`errterms7_11*`[ct]) ))^2;\nend do:\nsqrt(sm);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#$\"+UuoBF !#9" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 138 " We can include the new stage for the embedded scheme as an additional \+ 13th stage added to the order 8 scheme along with the coefficients " }{XPPEDIT 18 0 "a[13,11] = 0;" "6#/&%\"aG6$\"#8\"#6\"\"!" }{TEXT -1 7 " and " }{XPPEDIT 18 0 "a[13,12] = 0;" "6#/&%\"aG6$\"#8\"#7\"\"!" } {TEXT -1 1 "." }}{PARA 0 "" 0 "" {TEXT -1 13 "The weights " } {XPPEDIT 18 0 "`b*`[i]" "6#&%#b*G6#%\"iG" }{TEXT -1 7 " for " } {XPPEDIT 18 0 "i=1" "6#/%\"iG\"\"\"" }{TEXT -1 89 " . . 10 of the 13 s tage combined scheme are those of the 11 stage scheme and the weight \+ " }{XPPEDIT 18 0 "`b*`[11];" "6#&%#b*G6#\"#6" }{TEXT -1 34 " in the 1 1 stage scheme becomes " }{XPPEDIT 18 0 "`b*`[13];" "6#&%#b*G6#\"#8" }{TEXT -1 25 " in the 13 stage scheme." }}{PARA 0 "" 0 "" {TEXT -1 8 "We set " }{XPPEDIT 18 0 "c[13] = 1;" "6#/&%\"cG6#\"#8\"\"\"" }{TEXT -1 3 ", " }{XPPEDIT 18 0 "`b*`[11] = 0;" "6#/&%#b*G6#\"#6\"\"!" } {TEXT -1 7 " and " }{XPPEDIT 18 0 "`b*`[12] = 0;" "6#/&%#b*G6#\"#7\" \"!" }{TEXT -1 1 "." }}{PARA 0 "" 0 "" {TEXT -1 66 "We can make the or der 8 scheme into a 13 stage scheme by setting " }{XPPEDIT 18 0 "b[13 ] = 0;" "6#/&%\"bG6#\"#8\"\"!" }{TEXT -1 1 "." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 201 "e17 := \{c[ 13]=1,seq(a[13,i]=subs(e16,a[11,i]),i=1..10),a[13,11]=0,a[13,12]=0,b[1 3]=0,\nseq(`b*`[i]=subs(e16,`b*`[i]),i=1..10),`b*`[11]=0,`b*`[12]=0,`b *`[13]=subs(e16,`b*`[11])\}:\ne18 := `union`(e13,e17):" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}} {SECT 1 {PARA 4 "" 0 "" {TEXT -1 3 "e18" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 9696 "e18 := \{b[13] = 0, a [13,11] = 0, a[13,12] = 0, a[10,3] = 0, b[4] = 0, a[11,3] = 0, a[12,3] = 0, c[13] = 1, c[12] = 1, a[11,2] = 0, a[12,2] = 0, a[7,2] = 0, b[2] = 0, b[3] = 0, a[9,3] = 0, c[3] = 473499/4616000, a[5,3] = -112207730 7242443/778476746531250, a[4,3] = 4261491/36928000, a[8,6] = 503350581 600824913/125990342179812500, a[9,7] = 1943275996723801340958982917199 1296136367807379302352500708132842509843157444880977931073253282120004 6895000000000/11996011488227722649656673931136490891256463292620473601 841875115170259043531987881275965965525693289412424400177, a[13,7] = - 3562962724414944757891560164749973139486260772587248231213362868690625 000/408976639178968899516222045518619849445955714595234573207901740578 867341, a[13,6] = -255966560180563740919888885069842817256551721897357 7146475607855715703433900000/15170757814501111695553564393421150459412 18379260955323225394329122945220575603, a[13,5] = 60867617377974046375 60639819223305802258863489403125000/2784859079002788542258846760345051 095481316496802932391, a[13,4] = 6340173256865017488246965527303755786 7024014649473659852800000/75974371795646999000568266827900958765018677 94005624909986891, a[13,1] = 21125373654396053178422144002949396008458 64356126794907105288476157299972777/9791129992576348954381966935712572 673760150144277222568673000574989723295600, a[13,10] = 365246427065695 97276845612368244490741747970924877885921875000/6720450834551542949183 8411146746729660269255184823759889942481, `b*`[13] = 29658763539086741 11604434604609/47062213978678728273328927437927, a[13,9] = 15176499802 7328779481071148092868976296994135944658935524468324347086887398053746 2717945530891242892188019850453060256773054273287757818339/17931096351 4035495173248229516194926905902591095541696921117714941628050882978212 42848492102923447532215742446890559805801281057084758250800, a[13,8] = 27274774093948890572327172846505411328624378015249298079900000000/111 5922486918625141794316350997246426030217325603407120793811703663, `b*` [12] = 0, `b*`[11] = 0, b[5] = 0, a[6,3] = 0, a[10,2] = 0, a[4,2] = 0, a[8,3] = 0, a[8,2] = 0, a[7,3] = 0, a[6,2] = 0, a[5,2] = 0, a[9,2] = \+ 0, a[5,1] = 3590880889914277/9341720958375000, a[6,1] = 3279216863/710 27928000, c[10] = 3611/5000, c[11] = 15/16, a[10,8] = 6608515467721941 8645471125072555541174985695924/31022273664823506249509795196263880038 4203417327, `b*`[2] = 0, `b*`[3] = 0, `b*`[4] = 0, `b*`[5] = 0, a[6,5] = 725783021041/3932390759616, a[9,1] = -16045918035544526531085666708 2009767990929543490591025544394180588552370348031380762359144852118568 30024864166368323768200/9829483157728664844198544034921655975378257830 0809518936935201163220684393774467084932694936332441464215227305872957 39613, b[10] = 594562755257530592552224345703125000000/123802720910327 682301431417435953442122031, a[10,6] = -726020251827988894428939665538 44286012776770019588838776297031451/4091843938040500792607167383471827 6106178863851548244800289187500, b[12] = -5006146887513977891391025463 7881/141186641936036184819986782313781, a[10,5] = 29877053472248545227 782869189767925950557009/10443709158645362958089905740134206606110000, a[11,8] = -5476716391734935351872497190918742577242813475191013228197 10937500/5545897345777551056363997101744668576293320106258184122878184 564689, a[13,3] = 0, a[12,6] = 446191977093458430386099958102902220927 13734357503573021289399188441075277893825000/3047226829867759667805678 237083584916028456629069014496162891378892137605913688013, a[12,8] = - 469359986530784242367841163967997985280992052062479522694037718750000/ 1262000068797740029604893636717124646814536106030908513651797310900389 , a[10,7] = -15322912063864370512130145988492098605486502994107816190/ 3130069400645604876183669549605539076333579290524919889, a[7,4] = 1163 26878837670912/1047320101758291075, a[11,10] = -7641031089620713267943 821483116886435546875/1865115549729059107236140767953375513727732867, \+ a[13,2] = 0, a[9,5] = -99113663897262616867890337141645610009390040553 5164924683058122802429707354033382826947398158683765324439618282500000 000/798481420088460027892989252276057751903312691947267439103642732722 31784282184770467794155269096224513726772081370189773, a[9,6] = 994112 7982121041338714935249721178582954714935869695264678103390512904859375 7052549024957474512389085654050445280000000/10219750071154355529199360 5654791795484972275164188674551435468249918836166971629037075524090443 38358355837306818391433, a[10,4] = 35076261889213578261995286390053983 221937920015616/8903662403052468890234321680409039895089390971875, a[1 2,7] = 948339742665210716931560767459705347432692833888702670755668800 833857753125000/541002030685238165239205472037121582573832354867191268 79290708372319555115953, b[1] = 55284231195707975647126708111723/12588 14250475014554517603612114000, b[9] = 63998419659074502960979467027044 3805335134995621797161457881539812583548821701835572942610508067899149 54901252698438375102943237148428019253408419/1406738387822415998067693 5999500324267727723947516309536464061683130063437616452951326819824531 063499969576146304645311203502485409773697867038000, a[8,1] = -1507026 09045465151/280142368857000000, b[7] = 6219731958882093270433753490048 828125000000/25268792314562899182186819800834272764054341, `b*`[6] = 9 1472553308336221233020750122000000000/27038903667419263328398155936609 8375979, a[9,4] = -235853914788187787303944581175110328986740369170539 3271280994482043263320271984518766662589229245453919623760032009543680 000000/217835526181955565558022800341429464086385172122906608031349543 583863022015887117168566338470163090032498196064101837445053273, a[10, 9] = -3347565461896560762549026667823123036634583052726552531003001623 0875755239420324728600957368877132012320553021/56360718148650508277541 9237614873016043169125130359330269370345097328575740250457475506596993 780051575000000000, c[4] = 1420497/9232000, a[3,1] = 94226774499/11846 945536000, a[12,10] = 124594688949431236718278718078688571245967961641 584151713065625000/244226611999512391459992648734463087356588030400313 6401896150897137, `b*`[10] = 2363202400377740499662213515625000000/191 67474982246118950523520271861502109, a[12,1] = -2173296165244568434534 168496725754283210370856714048295955495704392191998074219/826329709817 197468253912814996276721297719271873047121496939265708086637102000, a[ 11,4] = -739282853758412257967453242147288028248514875000/672441828756 54829098323943713374207709631980757, a[12,5] = -9016415262730886672936 77983960434306145731566007759375000/4700605132153172647765087067511559 7779536141098588395119, `b*`[8] = 758038278545513886878223979625000000 0000/33873008089978031564549954803189465564389, b[11] = -1886691133979 705639959153870454656/397947585885835383951563312487675, `b*`[9] = -12 6466820753477200038941690908829580682307618595729809151136635663814541 1319432807599860924173256812108555319505561297378438513/31458133682363 8793763598373286113109900992592339363468070798364101001329020137898775 83819562913435723874810037875576750158649200, c[9] = 14280181565693441 800/15023510624554266963, `b*`[1] = 3635543992294021475202312550589/83 920950031667636967840240807600, a[12,4] = -117648588933447234539794802 4769479865991267657667189808793600000/12823827412574578485739502368305 5391720808636521055761250085619, b[6] = 101652048214282205518610445783 893750000000/289586658278060310247144250081091360673509, a[12,11] = 56 1064742310110985699639624014768981793940948761680464947352/67137939152 6329055316725343694451428305592235340194400714475, a[11,9] = 278805054 2294564730519147857701304010562775210713862748202538528260236025540352 52878888463451556710568650135086751734733498388408614837563/5703278975 8993842471673989706725819315996341861372243823982103984407642346136141 015055914541200747733442561186185728215790313661884647470336, a[10,1] \+ = 10050976387926430682409615346304117463662936424809533392310665300187 3/22949032400764462804236175621743615583246148826008952411547500000000 0, a[12,9] = -66412748100626086624541735316787911172543860816013412978 8715555378465573117813001757238260693051901812594341608922016461968723 754067215934799054852630446938063/947405313675779327886462044861834846 8794738615395834799951011682911161688440404387515843927859132242398940 29843821464363224405440877235651675827150051218506000, `b*`[7] = 31245 710879106859500854236329453125000000/125747467177230198813995913261775 760852127, b[8] = 471525882014932587321673707929687500000000/523728817 391198795728810839649314044495553, a[11,6] = 7944893992399893116476631 697520363855501654462308622798446100145004371780765625/887005161602928 659252827604035768657298435884939489754462273147402132735951354, a[8,5 ] = -29670244019740727/6363654238520000, a[7,5] = -21022506263989/6144 36056190000, a[11,7] = 27146666288983512861481079691606948806248583478 4939879774912795880859375/17080057868208807584911919572395871967414618 088449127584294090356526117, c[2] = 139/2500, c[5] = 1923/5000, c[6] = 923/2000, c[7] = 769/5000, c[8] = 8571/10000, a[7,1] = 22628776819090 891/378192197956950000, a[11,5] = -30430195443840795226634143899143570 2455078125/26817588459995310678515661957421952038871616, a[3,2] = 2242 01303001/2369389107200, a[7,6] = 182198401/10649112500, a[8,4] = -4704 8174572430533795112/6781573975400382234375, a[9,8] = -4738143867684122 1895938162441994505404833843721635499519905253875507680380152182754141 20082248510000000000/4562738155611920982891652838532643427337613715822 8892503158792567081801761628344925618992749059885819540261, a[8,7] = 9 , a[2,1] = 139/2500, a[6,4] = 1418466643672/6146756958375, a[5,4] = 16 83359084698258/1167715119796875, a[4,1] = 1420497/36928000, a[11,1] = \+ -510109776019784161513757125661110921966996572873700445365494039943574 9927779/34602547900652184980253949121136544405479876410211749689850107 53183759952128\}:" }{TEXT -1 0 "" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 20 "printed coefficients" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 362 "subs(e18,ma trix([seq([c[i],seq(a[i,j],j=1..i-1),``$(8-i)],i=2..8)]));\nfor ii fro m 9 to 13 do\n print(``);\n print(c[ii]=subs(e18,c[ii])); \n for jj to ii-1 do\n print(a[ii,jj]=subs(e18,a[ii,jj]));\n end do: \nend do:print(``);\nfor ii to 13 do\n print(b[ii]=subs(e18,b[ii])); \nend do:print(``);\nfor ii to 13 do\n print(`b*`[ii]=subs(e18,`b*`[ ii]));\nend do:" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#K%'matrixG6#7)7*#\" $R\"\"%+DF(%!GF+F+F+F+F+7*#\"'*\\t%\"(+gh%#\",*\\unA%*\"/+g`Xp%=\"#\"- ,II,UA\".+s5*QpBF+F+F+F+F+7*#\"((\\?9\"(+?B*#F8\")+!Gp$\"\"!#\"(\"\\hU F;F+F+F+F+7*#\"%B>\"%+]#\"1xU\"**)3)3f$\"1+]Pe4sT$*F<#!1VCC2t2A6\"0]7` YnZy(#\"1e#)p%3fLo\"\"1voz>^rn6F+F+F+7*#\"$B*\"%+?#\"+jo@zK\",+!Gz-rF< F<#\".sOkm%=9\".v$epvYh#\"-T5-$yD(\".;'f2RKRF+F+7*#\"$p(FB#\"2\"*34>ox GE#\"3++&p&z>#>y$FcgV 9'#\"*,%)>#=\",+D6\\1\"F+7*#\"%r&)\"&++\"#!3^^YX!4Eq]\"\"3+++d)oB9!GF< F<#!87^zL0VsX<[q%\"7vVB#Q+a(Rd\"y'#!2F2u>SCq'H\"1++_QUljj#\"38\\#3g\"e ]L]\"3+D\")z@M!*f7\"\"*Q)pprint216\"" }}{PARA 11 "" 1 "" {XPPMATH 20 " 6#%!G" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/&%\"cG6#\"\"*#\"5+=W$pl:=!G9 \"5jpEaXi5N-:" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/&%\"aG6$\"\"*\"\"\"# !er+#oPKoj;k[-Io&=@&[9fBw!QJ![.P_&)e!=%RWb-\"f!\\V&H4*zw4?3nm&3Jl_Wb.= f/;\"dr8'RdH(eIF_@k9WKj$\\pK\\3nWx$R%o?K;,_$p$*=&43IyDy`(fl@\\.W&)>W[m Gx:$[H)*" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/&%\"aG6$\"\"*\"\"#\"\"!" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/&%\"aG6$\"\"*\"\"$\"\"!" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/&%\"aG6$\"\"*\"\"%#!jr+++!oV&4?.gPi>RXX#H#* eimw=X)>F?LEV?[%*4GrKR0)\\K+ 4j,ZQjcor6()e,AI'QeV&\\8.3m!H7s^Q'3k%H9M+G-ebcb>=Eb$y@" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/&%\"aG6$\"\"*\"\"&#!cr++++DG='RWKlPoe\")RZp#GQLSN 2(HC!G7eIoC\\;NbS+R4+hX;9P.*y'ohis*Qm8\"**\"brt(*=q83sns8XA'4p_:%zn/x% =#G%yJAFtUO5RuEZ>p7L!>vdgF_#*)H*y-g%)3?9[)z" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/&%\"aG6$\"\"*\"\"'#\"br+++!GX/0ac3*Q7XZd\\-\\D0dPf[!H^ !R.\"yYE&ppe$\\raHey6s\\_$\\rQ8/@@)z7T**\"brL9R=oIPeNe$QV/4Cb2P!H;(p;O )=*\\#oaV^Xn)=k^Fs\\[&z\"zacg$*>HbNa62](>-\"" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/&%\"aG6$\"\"*\"\"(#\"]r++++]*o/+7#G`K2Jz(4)[WdJ%)4D%G8 32]_BIzt!yOOhH\"*>\"\\rx,SCCT*G$pDb'f'fF\")y)>`V!f-<:^ (=%=gt/i#HjkD\"*3\\O6$Rnc'\\EsF#)[6g*>\"" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/&%\"aG6$\"\"*\"\")#!eq+++++^[A3?TTv#=_,Q!o2b(QD0*>&*\\N;sVQ$[S0 X*>Wi\"Qf*=ATonQ9QZ\"fqh-a>e))f!\\F**=c#\\MG;w,=3nDzeJ]#*)G#er8wLFMkK& QGl\"*G)4#>hb\"QFc%" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%!G" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/&%\"cG6#\"#5#\"%6O\"%+]" }}{PARA 11 "" 1 " " {XPPMATH 20 "6#/&%\"aG6$\"#5\"\"\"#\"`ot=+`m5BRL&4[UOHmju6/jM:'4CoIk #zQw405\"`o++++]Z:T_*3g#)[hC$e:Ou@cA$)R0!R'G&*>EyN@*)=Ew]$\"Rv=(4R*3&*)R!4/o@VB!* )oC0.Cm.*)" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/&%\"aG6$\"#5\"\"&#\"M4q b]f#zw*=pGyF_a[AZ`q()H\"M++61m?M,u0**3eHOX'e\"4PW5" }}{PARA 11 "" 1 " " {XPPMATH 20 "6#/&%\"aG6$\"#5\"\"'#!\\o^9.(Hw(Q))e>+xwF,'GWQbmR*GW*)) )z#=D?gs\"\\o+v=*G+[C[:&Q')yh5w#=Z$Qnrg#z+0/QR%=4%" }}{PARA 11 "" 1 " " {XPPMATH 20 "6#/&%\"aG6$\"#5\"\"(#!Y!>;y5%*H]'[0')4#\\))f9I@^qV'Q17H K:\"X*))>\\_!HzNLw!Rbg\\&pO=w[gX1Sp+8$" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/&%\"aG6$\"#5\"\")#\"PCfp&)\\sna^3m\"QFtT.UQ+)QE'>& z4&\\i]B[mtA-J" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/&%\"aG6$\"#5\"\"*#! iq@Ib?B,Kr()ot&4gGZK?%R_vv3B;+.5`_ls_IeMm.BJ#ymE!\\Dwgl*=YlvM$\"jq++++ ]d^+y$*pf1bZd/DSddGt4X.Pp-Lf.8D\"pJ/;I([hP#>ax#30l[\"=2Oc" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%!G" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/&%\"cG6 #\"#6#\"#:\"#;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/&%\"aG6$\"#6\"\"\"# !gozx#*\\dV*RS\\l`W+P(Gd'*p'>#46hc7dP^hTy>gx4,^\"goG@&*fP=`2,&)*o\\<@5 k()zaSWl8@\"\\RD!)\\=_1!za-Y$" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/&%\" aG6$\"#6\"\"#\"\"!" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/&%\"aG6$\"#6\" \"$\"\"!" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/&%\"aG6$\"#6\"\"%#!Q+]([^ [#G!)GZ@C`u'zD7%eP&GGR(\"Pd2)>j4x?uLrVRK)4H[lvG=Ws'" }}{PARA 11 "" 1 " " {XPPMATH 20 "6#/&%\"aG6$\"#6\"\"&#!ND\"y]X-dV\"**Q9MmA&zSQW&>I/$\"M; ;()Q?&>Ud>m:&y1J&**f%)e(eRs&>>\"\\e2)3#oy0!3<" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/&%\"aG6$\"#6\"\")#!]o+v$4r>GK,\">vM\"GCxDu=4>(\\s=N N\\t\"R;nZ&\"^o*okX=yG7%=ei5?LHw&oYu,r*ROc5bxdM(*ea&" }}{PARA 12 "" 1 "" {XPPMATH 20 "6#/&%\"aG6$\"#6\"\"*#\"esjv$[h3%)Q)\\LZt^n3N,lo0rc:Xj% )))yGDNSb-O-EG&QD?[F'Qr5_xi0,/8qdy9>0tkXHU00)y#\"gsO.ZZY)=m8.z:#Gd='=h DWLxu+7a9f0:59OhMUwS%)R5#)R#QCs8'=M'*fJ>es1(*)RnrC%Q**e(*yKq&" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#/&%\"aG6$\"#6\"#5#!LvoaNk)o6$[@Q%zE82i *3J5k(\"OnGtFP^vL&zwShB2\"f!H(\\b6l=" }}{PARA 11 "" 1 "" {XPPMATH 20 " 6#%!G" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/&%\"cG6#\"#7\"\"\"" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/&%\"aG6$\"#7\"\"\"#!jo>U2)*>>#R/d\\bfH[Src3 P5KGads'\\oT`M%oXClhHt@\"io+?5Pm33dERp\\@r/t=F>xH@nF'*\\\"G\"RDou><)4( Hj#)" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/&%\"aG6$\"#7\"\"#\"\"!" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#/&%\"aG6$\"#7\"\"$\"\"!" }}{PARA 11 " " 1 "" {XPPMATH 20 "6#/&%\"aG6$\"#7\"\"%#![o++g$z3)*=nwln7*f')z%pZ-[zR XBZM$*)e[w6\"jn>c3]7wb5_O'33s\"RbIoB]Rd[yXd7u#Q#G\"" }}{PARA 11 "" 1 " " {XPPMATH 20 "6#/&%\"aG6$\"#7\"\"&#!Z+]Pfx+m:tXhIM/'R)zn$Hn')3ti_T;!* \"Y>^R)e)4Th`zxf:^nq3lxksJ:K^g+Z" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/& %\"aG6$\"#7\"\"'#\"^p+]#Q*yFv5W)=*R*G@Id.vNMPr#4A-H5e**4'QI%eM4x>>Y%\" ]p8!)o8fgP@*)y8*G;'\\9!p!HmXGg\"\\e$3P#yc!ymfx')HoAZI" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/&%\"aG6$\"#7\"\"(#\"io+]7`x&Q$3!)ocvqEq))Q$GpKuM0 (fuwg:$pr5_mU(R$[*\"ho`f6b&>BP32Hzo7>n[NKQd#e@r.sa?R_;Q_oI?+T&" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#/&%\"aG6$\"#7\"\")#!`o++v=x.%pA&zC1_?* 4G&)z*z'R;TyOUUyIl)*f$p%\"ao*Q+4J(z^O^34.1h`9okCrrOO*[gH+u(zo++i7" }} {PARA 12 "" 1 "" {XPPMATH 20 "6#/&%\"aG6$\"#7\"\"*#!jtj!QpWIE&[0*zMf@n SvB(o>Y;?#*3;M%f7=!>0$pg#Qsv,I\"y6tbYy`b:()yHT8g\"3'Qas6\"zy;`tTXi'3E1 5[FTm\"jt+g]=70]r#en^cBx3W0WAjVY@Q%)HS*)RUA8fy#R%e^(Q//W)oh6\"Ho65&**z MeR:'QZzo%[$='[/ik)yKzdn8`SZ*" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/&%\" aG6$\"#7\"#5#\"]o+]ilIr^TeT;'z'fCr&)oy!=(y#=nBJ%\\*)o%fC\"\"^oPr*3:'*= SOJ+/.)ect3jWt[E**f9R7&**>hEUC" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/&% \"aG6$\"#7\"#6#\"gn_t%\\Y!oh([4%Rz\")*oZ,C'R'*p&)465Buk5c\"gnvWr+W>S`B #f0$G9X%pV`s;`0Hj_\"Rz8n" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%!G" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#/&%\"cG6#\"#8\"\"\"" }}{PARA 11 "" 1 " " {XPPMATH 20 "6#/&%\"aG6$\"#8\"\"\"#\"goxF(**HdhZ)G0r!\\zEhNke%3gR\\H +W@UyJ0'RaOPD6#\"go+cHB(*)\\d+InoDAxU9],wtEd7d$p'>Qa*[jd#**H6z*" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#/&%\"aG6$\"#8\"\"#\"\"!" }}{PARA 11 " " 1 "" {XPPMATH 20 "6#/&%\"aG6$\"#8\"\"$\"\"!" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/&%\"aG6$\"#8\"\"%#\"in++!G&)fOZ\\Y,Cq'ybPIFb'pC)[<]'oD t,M'\"hn\"*o)*4\\i0Szn=]we4!z#oEo0+**pk&zrV(f(" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/&%\"aG6$\"#8\"\"&#\"X+]7.%*[j)eA!eIB#>)R1cPYS(zP4uj0=gl'fD \"jo.cd?_%H7HVRDKKb4Ez$=7%f/:@MRkNb&p66]9yvq^\"" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/&%\"aG6$\"#8\"\"(#!do+]i!poGO87B[ses2E'[RJ(*\\Z;g:*yvW \\TCF'Hc$\"coTt')y0u,z?tXB&f9d&fW\\)>'=b/Ai^**)o*y\"Rm(*3%" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/&%\"aG6$\"#8\"\")#\"\\o++++*z!)H\\_,yViG8T0 l%G!)=#* GC\"*3`Xzriu`!)R()o3ZVKoW_N*eY%f8%*pHw*oG4[62\"[z(Gt-)*\\w^\"\"gs+3DeZ 3d5G,e!)f0*oWUd@KvWBH5#\\[GC@yH)30G;%\\r<6#ppTb4\"f-f!p#\\>;&H#[K<&\\N S^j4Jz\"" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/&%\"aG6$\"#8\"#5#\"in+](= #f)y([#4(zuT2\\W#oBhXoF(fplqUY_O\"in\"[U**))fP#[=b#p-mHnuY6TQ=\\Ha^X$3 X?n" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/&%\"aG6$\"#8\"#6\"\"!" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#/&%\"aG6$\"#8\"#7\"\"!" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%!G" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/&%\"bG6#\" \"\"#\"AB<63n7Zc(zq&>JUGb\"C+S67Ogt>i\"MTV0kFFM3!)>o=#=**Gc9Bzo_#" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#/&%\"bG6#\"\")#\"K++++voHzqt;K(eK\\,#) e_r%\"K`b\\WSJ\\'R3\")Gdz)>\"R<)GP_" }}{PARA 12 "" 1 "" {XPPMATH 20 "6 #/&%\"bG6#\"\"*#\"]t>%3MD>!G%[rBVH5v$Q%)p_7!\\&\\\"**y130hUHdN=q@)[Ne7 )R:)yXhrz@c*\\8N`!QWq-n%z4'H]u!f'>%)*R'\"]t+!Qq'ypt(4a[-N?6`k/j9w&p** \\j5`C)>oK^HX;wVj+8$ohSYO&4j^ZRsFxEC.]**f$pn!)*fTAyQQnS\"" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/&%\"bG6#\"#5#\"H+++DJqXVA_DfIvDbFc%f\"KJ?7UM&f V<9V,BoF.\"4s-Q7" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/&%\"bG6#\"#6#!CcY XqQ:f*RcqzR8\"p')=\"Bvw[7Lc^RQNe)eeZzR" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/&%\"bG6#\"#7#!A\")yja-\"R\"*y(R^()o91]\"B\"y8By')*>[=Og$>k'=T\" " }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/&%\"bG6#\"#8\"\"!" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%!G" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/&%#b*G6#\" \"\"#\"@*e]DJ-_Z@SH#*RaNO\"A+w!3CSy'pjn;.]4#R)" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/&%#b*G6#\"\"#\"\"!" }}{PARA 11 "" 1 "" {XPPMATH 20 "6# /&%#b*G6#\"\"$\"\"!" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/&%#b*G6#\"\"% \"\"!" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/&%#b*G6#\"\"&\"\"!" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/&%#b*G6#\"\"'#\"G++++?7]2-L7AO$3Lbs9*\"HzfP )4m$f:)RGLE>um.*Qq#" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/&%#b*G6#\"\"(# \"J+++DJXHjBa3]fo5z3rX7$\"KF@&3wvIs`b3@\"oDtT#4')*f2GV>8TX\"QmNm8^\"4)Hdf=wI#o!eH)34pT*Q+? xMv?oYE\"\"[s+#\\'e,vwb(y.5[(QsNM\"Hc>Qex)*y8?!H8+,TO)zq!oMORBf#*4!*4J 6'Gt$)fjPzQO#oL\"e9$" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/&%#b*G6#\"#5# \"F+++Dc^8Am*\\SxP+C?jB\"G4@]h=F?N_]*=hC#)\\Zn\">" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/&%#b*G6#\"#6\"\"!" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/ &%#b*G6#\"#7\"\"!" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/&%#b*G6#\"#8#\"@ 4YgMWg6Tn3RNwe'H\"AFzVF*GLFG(y'yR@iq%" }}}{PARA 0 "" 0 "" {TEXT -1 0 " " }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 151 "RK8_13eqs : = [op(RowSumConditions(13,'expanded')),op(OrderConditions(8,13,'expand ed'))]:\n`RK7_13eqs*` := subs(b=`b*`,OrderConditions(7,13,'expanded')) :" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 6 "Chec k:" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 112 "subs(e18,RK8_13eqs): \nmap(u->lhs(u)-rhs(u),%);\nnops(%);\nsubs(e18,`RK7_13eqs*`):\nmap(u-> lhs(u)-rhs(u),%);\nnops(%);" }}{PARA 12 "" 1 "" {XPPMATH 20 "6#7`x\"\" !F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F $F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F $F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F $F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F $F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F $F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F $F$" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#\"$7#" }}{PARA 12 "" 1 "" {XPPMATH 20 "6#7ap\"\"!F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F $F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F $F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$" }}{PARA 11 "" 1 " " {XPPMATH 20 "6#\"#&)" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 26 "#-------------------------" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}}{PARA 0 "" 0 "" {TEXT -1 52 "#-------------- -------------------------------------" }}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 38 "characteristics of the embedded scheme" }}{PARA 0 "" 0 " " {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}} {SECT 1 {PARA 4 "" 0 "" {TEXT -1 36 "coefficients of the combined sche me " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 9696 "ee := \{c[2]=139/2500,\nc[3]=473499/4616000,\nc[4]= 1420497/9232000,\nc[5]=1923/5000,\nc[6]=923/2000,\nc[7]=769/5000,\nc[8 ]=8571/10000,\nc[9]=14280181565693441800/15023510624554266963,\nc[10]= 3611/5000,\nc[11]=15/16,\nc[12]=1,\nc[13]=1,\n\na[2,1]=139/2500,\na[3, 1]=94226774499/11846945536000,\na[3,2]=224201303001/2369389107200,\na[ 4,1]=1420497/36928000,\na[4,2]=0,\na[4,3]=4261491/36928000,\na[5,1]=35 90880889914277/9341720958375000,\na[5,2]=0,\na[5,3]=-1122077307242443/ 778476746531250,\na[5,4]=1683359084698258/1167715119796875,\na[6,1]=32 79216863/71027928000,\na[6,2]=0,\na[6,3]=0,\na[6,4]=1418466643672/6146 756958375,\na[6,5]=725783021041/3932390759616,\na[7,1]=226287768190908 91/378192197956950000,\na[7,2]=0,\na[7,3]=0,\na[7,4]=11632687883767091 2/1047320101758291075,\na[7,5]=-21022506263989/614436056190000,\na[7,6 ]=182198401/10649112500,\na[8,1]=-150702609045465151/28014236885700000 0,\na[8,2]=0,\na[8,3]=0,\na[8,4]=-47048174572430533795112/678157397540 0382234375,\na[8,5]=-29670244019740727/6363654238520000,\na[8,6]=50335 0581600824913/125990342179812500,\na[8,7]=9,\na[9,1]=-1604591803554452 6531085666708200976799092954349059102554439418058855237034803138076235 914485211856830024864166368323768200/982948315772866484419854403492165 5975378257830080951893693520116322068439377446708493269493633244146421 522730587295739613,\na[9,2]=0,\na[9,3]=0,\na[9,4]=-2358539147881877873 0394458117511032898674036917053932712809944820432633202719845187666625 89229245453919623760032009543680000000/2178355261819555655580228003414 2946408638517212290660803134954358386302201588711716856633847016309003 2498196064101837445053273,\na[9,5]=-9911366389726261686789033714164561 0009390040553516492468305812280242970735403338282694739815868376532443 9618282500000000/79848142008846002789298925227605775190331269194726743 910364273272231784282184770467794155269096224513726772081370189773,\na [9,6]=9941127982121041338714935249721178582954714935869695264678103390 5129048593757052549024957474512389085654050445280000000/10219750071154 3555291993605654791795484972275164188674551435468249918836166971629037 07552409044338358355837306818391433,\na[9,7]=1943275996723801340958982 9171991296136367807379302352500708132842509843157444880977931073253282 1200046895000000000/11996011488227722649656673931136490891256463292620 473601841875115170259043531987881275965965525693289412424400177,\na[9, 8]=-473814386768412218959381624419945054048338437216354995199052538755 0768038015218275414120082248510000000000/45627381556119209828916528385 3264342733761371582288925031587925670818017616283449256189927490598858 19540261,\na[10,1]=100509763879264306824096153463041174636629364248095 333923106653001873/\n 22949032400764462804236175621743615583246 1488260089524115475000000000,\na[10,2]=0,\na[10,3]=0,\na[10,4]=3507626 1889213578261995286390053983221937920015616/\n 8903662403052468 890234321680409039895089390971875,\na[10,5]=29877053472248545227782869 189767925950557009/\n 10443709158645362958089905740134206606110 000,\na[10,6]=-7260202518279888944289396655384428601277677001958883877 6297031451/40918439380405007926071673834718276106178863851548244800289 187500,\na[10,7]=-1532291206386437051213014598849209860548650299410781 6190/3130069400645604876183669549605539076333579290524919889,\na[10,8] =66085154677219418645471125072555541174985695924/310222736648235062495 097951962638800384203417327,\na[10,9]=-3347565461896560762549026667823 1230366345830527265525310030016230875755239420324728600957368877132012 320553021/563607181486505082775419237614873016043169125130359330269370 345097328575740250457475506596993780051575000000000,\na[11,1]=-5101097 760197841615137571256611109219669965728737004453654940399435749927779/ 3460254790065218498025394912113654440547987641021174968985010753183759 952128,\na[11,2]=0,\na[11,3]=0,\na[11,4]=-7392828537584122579674532421 47288028248514875000/\n 67244182875654829098323943713374207709 631980757,\na[11,5]=-304301954438407952266341438991435702455078125/\n \+ 26817588459995310678515661957421952038871616,\na[11,6]=7944893 9923998931164766316975203638555016544623086227984461001450043717807656 25/\n 887005161602928659252827604035768657298435884939489754462 273147402132735951354,\na[11,7]=27146666288983512861481079691606948806 2485834784939879774912795880859375/\n 1708005786820880758491191 9572395871967414618088449127584294090356526117,\na[11,8]=-547671639173 493535187249719091874257724281347519101322819710937500/\n 5545 897345777551056363997101744668576293320106258184122878184564689,\na[11 ,9]=278805054229456473051914785770130401056277521071386274820253852826 0236025540352528788884634515567105686501350867517347334983884086148375 63/5703278975899384247167398970672581931599634186137224382398210398440 7642346136141015055914541200747733442561186185728215790313661884647470 336,\na[11,10]=-7641031089620713267943821483116886435546875/\n \+ 1865115549729059107236140767953375513727732867,\na[12,1]=-2173296165 244568434534168496725754283210370856714048295955495704392191998074219/ \n 82632970981719746825391281499627672129771927187304712149693 9265708086637102000,\na[12,2]=0,\na[12,3]=0,\na[12,4]=-117648588933447 2345397948024769479865991267657667189808793600000/\n 128238274 125745784857395023683055391720808636521055761250085619,\na[12,5]=-9016 41526273088667293677983960434306145731566007759375000/\n 47006 051321531726477650870675115597779536141098588395119,\na[12,6]=44619197 7093458430386099958102902220927137343575035730212893991884410752778938 25000/\n 304722682986775966780567823708358491602845662906901449 6162891378892137605913688013,\na[12,7]=9483397426652107169315607674597 05347432692833888702670755668800833857753125000/\n 541002030685 23816523920547203712158257383235486719126879290708372319555115953,\na[ 12,8]=-469359986530784242367841163967997985280992052062479522694037718 750000/\n 1262000068797740029604893636717124646814536106030908 513651797310900389,\na[12,9]=-6641274810062608662454173531678791117254 3860816013412978871555537846557311781300175723826069305190181259434160 8922016461968723754067215934799054852630446938063/94740531367577932788 6462044861834846879473861539583479995101168291116168844040438751584392 785913224239894029843821464363224405440877235651675827150051218506000, \na[12,10]=12459468894943123671827871807868857124596796164158415171306 5625000/\n 244226611999512391459992648734463087356588030400313 6401896150897137,\na[12,11]=561064742310110985699639624014768981793940 948761680464947352/\n 6713793915263290553167253436944514283055 92235340194400714475,\na[13,1]=211253736543960531784221440029493960084 5864356126794907105288476157299972777/\n 9791129992576348954381 966935712572673760150144277222568673000574989723295600,\na[13,2]=0,\na [13,3]=0,\na[13,4]=634017325686501748824696552730375578670240146494736 59852800000/\n 759743717956469990005682668279009587650186779400 5624909986891,\na[13,5]=6086761737797404637560639819223305802258863489 403125000/\n 27848590790027885422588467603450510954813164968029 32391,\na[13,6]=-25596656018056374091988888506984281725655172189735771 46475607855715703433900000/\n 15170757814501111695553564393421 15045941218379260955323225394329122945220575603,\na[13,7]=-35629627244 14944757891560164749973139486260772587248231213362868690625000/\n \+ 408976639178968899516222045518619849445955714595234573207901740578 867341,\na[13,8]=27274774093948890572327172846505411328624378015249298 079900000000/\n 11159224869186251417943163509972464260302173256 03407120793811703663,\na[13,9]=151764998027328779481071148092868976296 9941359446589355244683243470868873980537462717945530891242892188019850 453060256773054273287757818339/179310963514035495173248229516194926905 9025910955416969211177149416280508829782124284849210292344753221574244 6890559805801281057084758250800,\na[13,10]=365246427065695972768456123 68244490741747970924877885921875000/\n 67204508345515429491838 411146746729660269255184823759889942481,\na[13,11]=0,\na[13,12]=0,\n\n b[1]=55284231195707975647126708111723/12588142504750145545176036121140 00,\nb[2]=0,\nb[3]=0,\nb[4]=0,\nb[5]=0,\nb[6]=101652048214282205518610 445783893750000000/289586658278060310247144250081091360673509,\nb[7]=6 219731958882093270433753490048828125000000/252687923145628991821868198 00834272764054341,\nb[8]=471525882014932587321673707929687500000000/52 3728817391198795728810839649314044495553,\nb[9]=6399841965907450296097 9467027044380533513499562179716145788153981258354882170183557294261050 806789914954901252698438375102943237148428019253408419/140673838782241 5998067693599950032426772772394751630953646406168313006343761645295132 6819824531063499969576146304645311203502485409773697867038000,\nb[10]= 594562755257530592552224345703125000000/123802720910327682301431417435 953442122031,\nb[11]=-1886691133979705639959153870454656/3979475858858 35383951563312487675,\nb[12]=-50061468875139778913910254637881/1411866 41936036184819986782313781,\nb[13]=0,\n\n`b*`[1]=363554399229402147520 2312550589/83920950031667636967840240807600,\n`b*`[2]=0,\n`b*`[3]=0,\n `b*`[4]=0,\n`b*`[5]=0,\n`b*`[6]=91472553308336221233020750122000000000 /270389036674192633283981559366098375979,\n`b*`[7]=3124571087910685950 0854236329453125000000/125747467177230198813995913261775760852127,\n`b *`[8]=7580382785455138868782239796250000000000/33873008089978031564549 954803189465564389,\n`b*`[9]=-1264668207534772000389416909088295806823 0761859572980915113663566381454113194328075998609241732568121085553195 05561297378438513/3145813368236387937635983732861131099009925923393634 6807079836410100132902013789877583819562913435723874810037875576750158 649200,\n`b*`[10]=2363202400377740499662213515625000000/19167474982246 118950523520271861502109,\n`b*`[11]=0,\n`b*`[12]=0,\n`b*`[13]=29658763 53908674111604434604609/47062213978678728273328927437927\}:" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 5 "Let \+ " }{XPPEDIT 18 0 "T[8, 13];" "6#&%\"TG6$\"\")\"#8" }{TEXT -1 128 " de note the vector whose components are the principal error terms of the \+ 13 stage, order 8 scheme (the error terms of order 9)." }}{PARA 0 "" 0 "" {TEXT -1 5 "Let " }{XPPEDIT 18 0 "`T*`[7,13];" "6#&%#T*G6$\"\"( \"#8" }{TEXT -1 146 " denote the vector whose components are the prin cipal error terms of the embedded 13 stage, order 7 scheme (the error \+ terms of order 8) and let " }{XPPEDIT 18 0 "`T*`[8,13];" "6#&%#T*G6$ \"\")\"#8" }{TEXT -1 99 " denote the vector whose components are the \+ error terms of order 9 of the embedded order 7 scheme." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 39 "Denote the 2-norm of these vectors by " }{XPPEDIT 18 0 "abs(abs(T[8,13]));" "6#-%$absG6#- F$6#&%\"TG6$\"\")\"#8" }{TEXT -1 3 ", " }{XPPEDIT 18 0 "abs(abs(`T*`[ 7, 13]));" "6#-%$absG6#-F$6#&%#T*G6$\"\"(\"#8" }{TEXT -1 7 " and " } {XPPEDIT 18 0 "abs(abs(`T*`[8, 13]));" "6#-%$absG6#-F$6#&%#T*G6$\"\") \"#8" }{TEXT -1 15 " respectively." }}{PARA 0 "" 0 "" {TEXT -1 0 "" } }{PARA 0 "" 0 "" {TEXT -1 8 "Define: " }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "A[9] = abs(abs(T[8,13]));" "6#/&%\"AG6#\"\"*-%$abs G6#-F)6#&%\"TG6$\"\")\"#8" }{TEXT -1 6 ", " }{XPPEDIT 18 0 "B[9] = abs(abs(`T*`[8,13]))/abs(abs(`T*`[7,13]));" "6#/&%\"BG6#\"\"**&-%$abs G6#-F*6#&%#T*G6$\"\")\"#8\"\"\"-F*6#-F*6#&F/6$\"\"(F2!\"\"" }{TEXT -1 6 ", " }{XPPEDIT 18 0 "C[9] = abs(abs(`T*`[8,13]-T[8,13]))/abs(abs (`T*`[7,13]));" "6#/&%\"CG6#\"\"**&-%$absG6#-F*6#,&&%#T*G6$\"\")\"#8\" \"\"&%\"TG6$F2F3!\"\"F4-F*6#-F*6#&F06$\"\"(F3F8" }{TEXT -1 1 "." }} {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 77 "Dormand a nd Prince have suggested that as well as attempting to ensure that " }{XPPEDIT 18 0 "A[9];" "6#&%\"AG6#\"\"*" }{TEXT -1 73 " is a minimum, if the embedded scheme is to be used for error control, " }{XPPEDIT 18 0 "B[9];" "6#&%\"BG6#\"\"*" }{TEXT -1 7 " and " }{XPPEDIT 18 0 "C [9];" "6#&%\"CG6#\"\"*" }{TEXT -1 27 " should be chosen so that " }} {PARA 0 "" 0 "" {TEXT -1 68 "they are \"similar in magnitude\" and als o not differ too much from 1." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 168 "errterms8_13 := PrincipalEr rorTerms(8,13,'expanded'):\n`errterms8_13*` :=subs(b=`b*`,errterms8_13 ):\n`errterms7_13*` := subs(b=`b*`,PrincipalErrorTerms(7,13,'expanded' )):" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 333 "snmB := sqrt(add(evalf(subs(ee,`errterms8_13*`[i]))^ 2,i=1..nops(`errterms8_13*`))):\nsdnB := sqrt(add(evalf(subs(ee,`errte rms7_13*`[i]))^2,i=1..nops(`errterms7_13*`))):\nsnmC := sqrt(add((eval f(subs(ee,`errterms8_13*`[i])-subs(ee,errterms8_13[i])))^2,i=1..nops(e rrterms8_13))):\n'B[9]'= evalf[8](snmB/sdnB);\n'C[9]'= evalf[8](snmC/s dnB);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/&%\"BG6#\"\"*$\")vqW>!\"(" } }{PARA 11 "" 1 "" {XPPMATH 20 "6#/&%\"CG6#\"\"*$\")3wG>!\"(" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}}{PARA 0 "" 0 "" {TEXT -1 52 "#------------------------------------ ---------------" }}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 26 "absolute stabi lity regions" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 36 "coeffici ents of the combined scheme " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 9696 "ee := \{c[2]=139/2500,\nc[ 3]=473499/4616000,\nc[4]=1420497/9232000,\nc[5]=1923/5000,\nc[6]=923/2 000,\nc[7]=769/5000,\nc[8]=8571/10000,\nc[9]=14280181565693441800/1502 3510624554266963,\nc[10]=3611/5000,\nc[11]=15/16,\nc[12]=1,\nc[13]=1, \n\na[2,1]=139/2500,\na[3,1]=94226774499/11846945536000,\na[3,2]=22420 1303001/2369389107200,\na[4,1]=1420497/36928000,\na[4,2]=0,\na[4,3]=42 61491/36928000,\na[5,1]=3590880889914277/9341720958375000,\na[5,2]=0, \na[5,3]=-1122077307242443/778476746531250,\na[5,4]=1683359084698258/1 167715119796875,\na[6,1]=3279216863/71027928000,\na[6,2]=0,\na[6,3]=0, \na[6,4]=1418466643672/6146756958375,\na[6,5]=725783021041/39323907596 16,\na[7,1]=22628776819090891/378192197956950000,\na[7,2]=0,\na[7,3]=0 ,\na[7,4]=116326878837670912/1047320101758291075,\na[7,5]=-21022506263 989/614436056190000,\na[7,6]=182198401/10649112500,\na[8,1]=-150702609 045465151/280142368857000000,\na[8,2]=0,\na[8,3]=0,\na[8,4]=-470481745 72430533795112/6781573975400382234375,\na[8,5]=-29670244019740727/6363 654238520000,\na[8,6]=503350581600824913/125990342179812500,\na[8,7]=9 ,\na[9,1]=-16045918035544526531085666708200976799092954349059102554439 418058855237034803138076235914485211856830024864166368323768200/982948 3157728664844198544034921655975378257830080951893693520116322068439377 446708493269493633244146421522730587295739613,\na[9,2]=0,\na[9,3]=0,\n a[9,4]=-23585391478818778730394458117511032898674036917053932712809944 82043263320271984518766662589229245453919623760032009543680000000/2178 3552618195556555802280034142946408638517212290660803134954358386302201 5887117168566338470163090032498196064101837445053273,\na[9,5]=-9911366 3897262616867890337141645610009390040553516492468305812280242970735403 3382826947398158683765324439618282500000000/79848142008846002789298925 2276057751903312691947267439103642732722317842821847704677941552690962 24513726772081370189773,\na[9,6]=9941127982121041338714935249721178582 9547149358696952646781033905129048593757052549024957474512389085654050 445280000000/102197500711543555291993605654791795484972275164188674551 43546824991883616697162903707552409044338358355837306818391433,\na[9,7 ]=19432759967238013409589829171991296136367807379302352500708132842509 8431574448809779310732532821200046895000000000/11996011488227722649656 6739311364908912564632926204736018418751151702590435319878812759659655 25693289412424400177,\na[9,8]=-473814386768412218959381624419945054048 3384372163549951990525387550768038015218275414120082248510000000000/45 6273815561192098289165283853264342733761371582288925031587925670818017 61628344925618992749059885819540261,\na[10,1]=100509763879264306824096 153463041174636629364248095333923106653001873/\n 22949032400764 4628042361756217436155832461488260089524115475000000000,\na[10,2]=0,\n a[10,3]=0,\na[10,4]=35076261889213578261995286390053983221937920015616 /\n 8903662403052468890234321680409039895089390971875,\na[10,5] =29877053472248545227782869189767925950557009/\n 10443709158645 362958089905740134206606110000,\na[10,6]=-7260202518279888944289396655 3844286012776770019588838776297031451/40918439380405007926071673834718 276106178863851548244800289187500,\na[10,7]=-1532291206386437051213014 5988492098605486502994107816190/31300694006456048761836695496055390763 33579290524919889,\na[10,8]=660851546772194186454711250725555411749856 95924/310222736648235062495097951962638800384203417327,\na[10,9]=-3347 5654618965607625490266678231230366345830527265525310030016230875755239 420324728600957368877132012320553021/563607181486505082775419237614873 0160431691251303593302693703450973285757402504574755065969937800515750 00000000,\na[11,1]=-51010977601978416151375712566111092196699657287370 04453654940399435749927779/3460254790065218498025394912113654440547987 641021174968985010753183759952128,\na[11,2]=0,\na[11,3]=0,\na[11,4]=-7 39282853758412257967453242147288028248514875000/\n 67244182875 654829098323943713374207709631980757,\na[11,5]=-3043019544384079522663 41438991435702455078125/\n 26817588459995310678515661957421952 038871616,\na[11,6]=79448939923998931164766316975203638555016544623086 22798446100145004371780765625/\n 887005161602928659252827604035 768657298435884939489754462273147402132735951354,\na[11,7]=27146666288 9835128614810796916069488062485834784939879774912795880859375/\n \+ 17080057868208807584911919572395871967414618088449127584294090356526 117,\na[11,8]=-5476716391734935351872497190918742577242813475191013228 19710937500/\n 55458973457775510563639971017446685762933201062 58184122878184564689,\na[11,9]=278805054229456473051914785770130401056 2775210713862748202538528260236025540352528788884634515567105686501350 86751734733498388408614837563/5703278975899384247167398970672581931599 6341861372243823982103984407642346136141015055914541200747733442561186 185728215790313661884647470336,\na[11,10]=-764103108962071326794382148 3116886435546875/\n 18651155497290591072361407679533755137277 32867,\na[12,1]=-21732961652445684345341684967257542832103708567140482 95955495704392191998074219/\n 82632970981719746825391281499627 6721297719271873047121496939265708086637102000,\na[12,2]=0,\na[12,3]=0 ,\na[12,4]=-1176485889334472345397948024769479865991267657667189808793 600000/\n 1282382741257457848573950236830553917208086365210557 61250085619,\na[12,5]=-90164152627308866729367798396043430614573156600 7759375000/\n 470060513215317264776508706751155977795361410985 88395119,\na[12,6]=446191977093458430386099958102902220927137343575035 73021289399188441075277893825000/\n 304722682986775966780567823 7083584916028456629069014496162891378892137605913688013,\na[12,7]=9483 3974266521071693156076745970534743269283388870267075566880083385775312 5000/\n 5410020306852381652392054720371215825738323548671912687 9290708372319555115953,\na[12,8]=-469359986530784242367841163967997985 280992052062479522694037718750000/\n 1262000068797740029604893 636717124646814536106030908513651797310900389,\na[12,9]=-6641274810062 6086624541735316787911172543860816013412978871555537846557311781300175 7238260693051901812594341608922016461968723754067215934799054852630446 938063/947405313675779327886462044861834846879473861539583479995101168 2911161688440404387515843927859132242398940298438214643632244054408772 35651675827150051218506000,\na[12,10]=12459468894943123671827871807868 8571245967961641584151713065625000/\n 244226611999512391459992 6487344630873565880304003136401896150897137,\na[12,11]=561064742310110 985699639624014768981793940948761680464947352/\n 6713793915263 29055316725343694451428305592235340194400714475,\na[13,1]=211253736543 9605317842214400294939600845864356126794907105288476157299972777/\n \+ 97911299925763489543819669357125726737601501442772225686730005749 89723295600,\na[13,2]=0,\na[13,3]=0,\na[13,4]=634017325686501748824696 55273037557867024014649473659852800000/\n 759743717956469990005 6826682790095876501867794005624909986891,\na[13,5]=6086761737797404637 560639819223305802258863489403125000/\n 27848590790027885422588 46760345051095481316496802932391,\na[13,6]=-25596656018056374091988888 50698428172565517218973577146475607855715703433900000/\n 15170 7578145011116955535643934211504594121837926095532322539432912294522057 5603,\na[13,7]=-356296272441494475789156016474997313948626077258724823 1213362868690625000/\n 408976639178968899516222045518619849445 955714595234573207901740578867341,\na[13,8]=27274774093948890572327172 846505411328624378015249298079900000000/\n 11159224869186251417 94316350997246426030217325603407120793811703663,\na[13,9]=151764998027 3287794810711480928689762969941359446589355244683243470868873980537462 717945530891242892188019850453060256773054273287757818339/179310963514 0354951732482295161949269059025910955416969211177149416280508829782124 2848492102923447532215742446890559805801281057084758250800,\na[13,10]= 36524642706569597276845612368244490741747970924877885921875000/\n \+ 67204508345515429491838411146746729660269255184823759889942481,\na [13,11]=0,\na[13,12]=0,\n\nb[1]=55284231195707975647126708111723/12588 14250475014554517603612114000,\nb[2]=0,\nb[3]=0,\nb[4]=0,\nb[5]=0,\nb[ 6]=101652048214282205518610445783893750000000/289586658278060310247144 250081091360673509,\nb[7]=6219731958882093270433753490048828125000000/ 25268792314562899182186819800834272764054341,\nb[8]=471525882014932587 321673707929687500000000/523728817391198795728810839649314044495553,\n b[9]=63998419659074502960979467027044380533513499562179716145788153981 2583548821701835572942610508067899149549012526984383751029432371484280 19253408419/1406738387822415998067693599950032426772772394751630953646 4061683130063437616452951326819824531063499969576146304645311203502485 409773697867038000,\nb[10]=594562755257530592552224345703125000000/123 802720910327682301431417435953442122031,\nb[11]=-188669113397970563995 9153870454656/397947585885835383951563312487675,\nb[12]=-5006146887513 9778913910254637881/141186641936036184819986782313781,\nb[13]=0,\n\n`b *`[1]=3635543992294021475202312550589/83920950031667636967840240807600 ,\n`b*`[2]=0,\n`b*`[3]=0,\n`b*`[4]=0,\n`b*`[5]=0,\n`b*`[6]=91472553308 336221233020750122000000000/270389036674192633283981559366098375979,\n `b*`[7]=31245710879106859500854236329453125000000/12574746717723019881 3995913261775760852127,\n`b*`[8]=7580382785455138868782239796250000000 000/33873008089978031564549954803189465564389,\n`b*`[9]=-1264668207534 7720003894169090882958068230761859572980915113663566381454113194328075 99860924173256812108555319505561297378438513/3145813368236387937635983 7328611310990099259233936346807079836410100132902013789877583819562913 435723874810037875576750158649200,\n`b*`[10]=2363202400377740499662213 515625000000/19167474982246118950523520271861502109,\n`b*`[11]=0,\n`b* `[12]=0,\n`b*`[13]=2965876353908674111604434604609/4706221397867872827 3328927437927\}:" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 78 "The stability function R for the 13 stage, orde r 8 scheme is given as follows." }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 76 "subs(ee,StabilityFunction(8,13,'expanded')):\nR := unapply(%,z ):\n'R(z)'=R(z);" }}{PARA 12 "" 1 "" {XPPMATH 20 "6#/-%\"RG6#%\"zG,<\" \"\"F)F'F)*&#F)\"\"#F)*$)F'F,F)F)F)*&#F)\"\"'F)*$)F'\"\"$F)F)F)*&#F)\" #CF)*$)F'\"\"%F)F)F)*&#F)\"$?\"F)*$)F'\"\"&F)F)F)*&#F)\"$?(F)*$)F'F1F) F)F)*&#F)\"%S]F)*$)F'\"\"(F)F)F)*&#F)\"&?.%F)*$)F'\"\")F)F)F)*&#\"R8ZY lWM#y'4Nl2h!=:&>8)))fuf3@\"W+++++GLCA'\\BTGzk.54F7Oz4-A*yF)*$)F'\"\"*F )F)F)*&#\"S\\[L4@S`aY![&*49.ztp'>/Ps,KX\"Z++++++S=lr-6%p=w@>64\\\\-V)o AQv\"F)*$)F'\"#5F)F)F)*&#\"TzH^5#oX%Qee#Qs1K:HC*)zEZ+)H?F)*$)F'\"#6F)F)F)*&#\"J(=P@pd%*))4O3p!HD>iUJU %>\"T++++++?br/S\"okK)3+M6Z&yf\"R/\"F)*$)F'\"#7F)F)F)" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 124 "We can find the poi nt where the boundary of the stability region intersects the negative \+ real axis by solving the equation: " }}{PARA 256 "" 0 "" {TEXT -1 1 " \+ " }{XPPEDIT 18 0 "R(z) = 1;" "6#/-%\"RG6#%\"zG\"\"\"" }{TEXT -1 2 ". \+ " }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 26 "z0 := newton(R(z)=1,z=-6 );" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%#z0G$!+ZRcUc!\"*" }}}{PARA 0 " " 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 302 "z0 : = newton(R(z)=1,z=-6):\np1 := plot([R(z),1],z=-6.69..0.49,color=[red,b lue]):\np2 := plot([[[z0,1]]$3],style=point,symbol=[circle,cross,diamo nd],color=black):\np3 := plot([[z0,0],[z0,1]],linestyle=3,color=COLOR( RGB,0,.5,0)):\nplots[display]([p1,p2,p3],view=[-6.69..0.49,-0.07..1.47 ],font=[HELVETICA,9]);" }}{PARA 13 "" 1 "" {GLPLOT2D 360 253 253 {PLOTDATA 2 "6+-%'CURVESG6$7Z7$$!3Q++++++!p'!#<$\"3G*RtJ]!z(>&F*7$$!3/ M3F3T(3l'F*$\"3kJuq)pq>\"\\F*7$$!3ym;a;#[pNbU\"Fbp7$$!3yLeHls)GX%F*$\"30g5w\"\\&*G5\"Fbp7$$!3Umm\"y=v5 H%F*$\"3K_?HjCmR#)!#>7$$!3#ommK8**R:%F*$\"3a!)pYMD(Rd'F[s7$$!3O+D'[4ZX *RF*$\"3AXKZ6K#oD&F[s7$$!3=nmwA]S`QF*$\"30$Hp[')*yDXF[s7$$!33+D')*z\\& )p$F*$\"37zH4M9;*3%F[s7$$!3'**\\(e254^NF*$\"3UYLt)o\"ebRF[s7$$!3wm;HFK B(R$F*$\"3?ALkM,N`SF[s7$$!3im\"zdvVfD$F*$\"3_%*G#yI()GK%F[s7$$!3kL$3K( Ra.JF*$\"3I%p6gU6Qy%F[s7$$!3oLeHIMCXHF*$\"3WJFfjq*pV&F[s7$$!37+vLUBW2G F*$\"3_Z!)yoUK[hF[s7$$!3EL$e@R8'eEF*$\"3355!=-v72(F[s7$$!3G++qB&e[]#F* $\"3;W=7HSa1#)F[s7$$!3?+]_\"QRWN#F*$\"3CW/T&[(o:&*F[s7$$!33+D1F6!*3AF* $\"3)eX?xei$*4\"Fbp7$$!3?+]24dIZ?F*$\"3_.=(Q9&Q\"H\"Fbp7$$!3#pmm(Q]5-> F*$\"3)fwB;#R#G\\\"Fbp7$$!3-++vIB2ZiJK;F*-%'COLOURG6&%$RGBG$\"*++++\" !\")$\"\"!Fd]lFc]l-F$6$7S7$F($\"\"\"Fd]l7$F=Fi]l7$FGFi]l7$FQFi]l7$FenF i]l7$F_oFi]l7$FdoFi]l7$FioFi]l7$F^pFi]l7$FdpFi]l7$FipFi]l7$F^qFi]l7$Fc qFi]l7$FhqFi]l7$F]rFi]l7$FbrFi]l7$FgrFi]l7$F]sFi]l7$FbsFi]l7$FgsFi]l7$ F\\tFi]l7$FatFi]l7$FftFi]l7$F[uFi]l7$F`uFi]l7$FeuFi]l7$FjuFi]l7$F_vFi] l7$FdvFi]l7$FivFi]l7$F^wFi]l7$FcwFi]l7$FhwFi]l7$F]xFi]l7$FbxFi]l7$FgxF i]l7$F\\yFi]l7$FayFi]l7$FfyFi]l7$F[zFi]l7$F`zFi]l7$FezFi]l7$FjzFi]l7$F _[lFi]l7$Fd[lFi]l7$Fi[lFi]l7$F^\\lFi]l7$Fc\\lFi]l7$Fh\\lFi]l-F]]l6&F_] lFc]lFc]lF`]l-F$6&7#7$$!3>+++ZRcUcF*Fi]l-%'SYMBOLG6#%'CIRCLEG-F]]l6&F_ ]lFd]lFd]lFd]l-%&STYLEG6#%&POINTG-F$6&F_al-Fdal6#%&CROSSGFgalFial-F$6& F_al-Fdal6#%(DIAMONDGFgalFial-F$6%7$7$FaalFc]lF`al-%&COLORG6&F_]lFc]l$ \"\"&!\"\"Fc]l-%*LINESTYLEG6#\"\"$-%+AXESLABELSG6%Q\"z6\"Q!Ficl-%%FONT G6#%(DEFAULTG-F\\dl6$%*HELVETICAG\"\"*-%%VIEWG6$;$!$p'!\"#$\"#\\Fidl;$ !\"(Fidl$\"$Z\"Fidl" 1 2 0 1 10 0 2 9 1 4 2 1.000000 45.000000 45.000000 0 0 "Curve 1" "Curve 2" "Curve 3" "Curve 4" "Curve 5" "Curve 6" }}}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 79 "T he following picture shows the stability region (excluding tiny \"sate llites\")." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 " " {MPLTEXT 1 0 1033 "R := z -> add(z^j/j!,j=0..8)+ 2108597459888 131951518061076535096782344465464713/789220209793612270910036479284123 496222433280000000000*z^9+\n453201723704196697379031409954804654534021 09334849/175382268843024949091119217618694110271651840000000000000*z^1 0+\n246498511389404768171532067238258583844568210512979/20298004726798 924291967158948812406976423116800000000000000*z^11+\n19442314262192529 069083609889457692137187/104391597854711340008832646814004715520000000 000000*z^12:\npts := []: z0 := 0:\nfor ct from 0 to 320 do\n zz := n ewton(R(z)=exp(ct*Pi/20*I),z=z0):\n z0 := zz:\n pts := [op(pts),[R e(zz),Im(zz)]]:\nend do:\np1 := plot(pts,color=COLOR(RGB,.22,0,.45)): \np2 := plots[polygonplot]([seq([pts[i-1],pts[i],[-4.6,0]],i=2..nops(p ts))],\n style=patchnogrid,color=COLOR(RGB,.45,.33,.95)):\np3 := plot([[[-6.69,0],[1.39,0]],[[0,-6.49],[0,6.49]]],color=black,lines tyle=3):\nplots[display]([p||(1..3)],view=[-6.69..1.39,-6.49..6.49],fo nt=[HELVETICA,9],\n labels=[`Re(z)`,`Im(z)`],axes=boxed,s caling=constrained);" }}{PARA 13 "" 1 "" {GLPLOT2D 524 539 539 {PLOTDATA 2 "6+-%'CURVESG6$7]_l7$$\"\"!F)F(7$F($\"3++++Fjzq:!#=7$F($\" 3)******Rl#fTJF-7$F($\"3D+++\")*)Q7ZF-7$$\"3*******f&Qj0e!#F$\"3o***** zJ&=$G'F-7$$\"3%******feat5&!#E$\"3[+++0<)R&yF-7$$\"31+++1B$\\%H!#D$\" 3E+++\"GyZU*F-7$$\"3/+++#yC[3;4$F U$\"3#******RZ(zq:FL7$$\"3S+++6Jh$y'FU$\"34+++ow(ys\"FL7$$\"32+++irtI8 !#A$\"3)*******\\v&\\)=FL7$$\"3++++(fc&\\BF_o$\"3:+++)fO?/#FL7$$\"3#)* ****f?)RNPF_o$\"3%)*****Rj9\"*>#FL7$$\"3?+++O&\\oI&F_o$\"3))*****fG$>c BFL7$$\"3i+++;zU&f'F_o$\"3:+++l\"zK^#FL7$$\"3`+++)Ro`!oF_o$\"35+++>(*Q qEFL7$$\"3s*****RB2q(\\F_o$\"34+++?HcFGFL7$$\"3!*******3aM))eFU$\"3%** ****z'=([)HFL7$$!3\")*****pTR6.&F_o$\"3!******RGXC9$FL7$$!3J+++(pYR!fF _o$\"32+++>W\\+LFL7$$\"33+++*RBbP\"!#@$\"3++++[gMfMFL7$$\"3K+++u%H(z)) Fbr$\"3)*******4&y%>OFL7$$\"3#******\\gb$**G!#?$\"3++++Owa\"y$FL7$$\"3 e******3#Q![vF]s$\"3!)*****zd^j%RFL7$$\"36+++5.$Ru\"!#>$\"3%)******)eZ X6%FL7$$\"39+++g5\"yr$Fhs$\"3G+++$)[^&G%FL7$$\"3q******)Ra9H(Fhs$\"3I+ ++H\"RUX%FL7$$\"3++++A>Rh7F-$\"3*)******RT$R\\FL7$$\"3/+++0$obC$F-$\"35+++%RI2-&FL7$$\"3F+++7@(3^ $F-$\"3-+++X_N%4&FL7$$\"39+++GCq%p$F-$\"3/+++/-&>;&FL7$$\"3>+++PQp2QF- $\"3V+++Mc`C_FL7$$\"3!)*****RiZ*eQF-$\"3c*****H2NFG&FL7$$\"3G+++MN:cQF -$\"3++++\">YpL&FL7$$\"3w*****paHe!QF-$\"37+++lWV(Q&FL7$$\"3t*****fFTN r$F-$\"3B+++@eQMaFL7$$\"3y*****>ewSe$F-$\"3#*******Ro$zZ&FL7$$\"3\")** ***>'*z:U$F-$\"3z*****fE#>=bFL7$$\"3#******>Jc'HKF-$\"3I+++O^BbbFL7$$ \"3E+++RCX6IF-$\"3m******HD8*e&FL7$$\"3#******\\XA(pFF-$\"3q*****4qQ*> cFL7$$\"32+++z6)o]#F-$\"3C+++1mpZcFL7$$\"3/+++-$[]A#F-$\"3I+++[%QCn&FL 7$$\"36+++L]3E>F-$\"3q*****pf&=%p&FL7$$\"3!******HNC;h\"F-$\"3H+++p![H r&FL7$$\"3))*****f]&4$G\"F-$\"33+++hOsGdFL7$$\"3N*****RAIuT*Fhs$\"3%)* ****Hv'\\TdFL7$$\"3++++[AS')eFhs$\"3S+++SnB^dFL7$$\"3)******p'y'pC#Fhs $\"3/+++:g*yv&FL7$$!3++++_hS$\\\"Fhs$\"3;+++atShdFL7$$!3$)*****z!G'*G` Fhs$\"3s*****4_!ohdFL7$$!33+++=/&eD*Fhs$\"3/+++1yfedFL7$$!33+++Y$[sK\" F-$\"33+++>\"3?v&FL7$$!3$******4C:!Q+++^.:LIF- $\"3s*****>)H'eo&FL7$$!3)******f=>E\\$F-$\"39+++p+ddcFL7$$!3u******pfD tRF-$\"3g*****4\"eZBcFL7$$!3')*****>amT[%F-$\"3g******o^x#e&FL7$$!3f** ***zl!=T]F-$\"3P+++%pSX`&FL7$$!3_+++]#GBn&F-$\"3E+++B_+yaFL7$$!3:+++\" yOaU'F-$\"3G+++\"Q'p8aFL7$$!35+++\\g\\jtF-$\"3)******4O[lM&FL7$$!3B+++ [ta-&)F-$\"3g*****H()Q%)G&FL7$$!3F+++*\\pAu*F-$\"3i*****R#zC\\_FL7$$!3 '******z)*)='4\"FL$\"3*******4$>-y*=9FL$\"3d*** **z(od0_FL7$$!3%******zR5R^\"FL$\"3,++++4I-_FL7$$!3$******z`ATg\"FL$\" 3y*****HBW&)>&FL7$$!3)******pv$Q!p\"FL$\"39+++htm$>&FL7$$!33+++?JJtFL$\"3+ +++U+%*o^FL7$$!3A+++v)[m+#FL$\"3i*****>IBn:&FL7$$!37+++ZMS!3#FL$\"31++ +g>NU^FL7$$!31+++@hd_@FL$\"3S++++/xD^FL7$$!3#******HVqLA#FL$\"3.+++\\* Rp5&FL7$$!3y*****\\klHH#FL$\"3e*****H9Me3&FL7$$!3++++yH_hBFL$\"3#)**** *\\%fVi]FL7$$!35+++%e!>HCFL$\"3O+++WjtO]FL7$$!39+++3v5'\\#FL$\"3$***** *\\,M(3]FL7$$!3))*****Hv1Cc#FL$\"3))*****H#eVy\\FL7$$!3.+++=v@GEFL$\"3 U+++b!ee%\\FL7$$!3))*****>amOp#FL$\"3/+++[#G5\"\\FL7$$!39+++s%y)eFFL$ \"3A+++,v)R([FL7$$!3.+++#3vR#GFL$\"3A+++jIzM[FL7$$!3%)*****R>t!*)GFL$ \"33+++#*3_$z%FL7$$!37+++!>\"GaHFL$\"3/+++txE]ZFL7$$!3-+++_Tp>IFL$\"3, +++3C:0ZFL7$$!3%******R#zQ&3$FL$\"3=+++_[JeYFL7$$!3!********o79:$FL$\" 3&)*****4s8*4YFL7$$!3*******z.(ys/` NFL$\"3U+++il7'H%FL7$$!3++++Jk(*>OFL$\"31+++$)3BTUFL7$$!3)******po:lo$ FL$\"3i*****zE,e=%FL7$$!31+++KC^_PFL$\"3k*****>sW)HTFL7$$!3,+++Xi#y\"Q FL$\"3A+++OAMtSFL7$$!3-+++=JL#)QFL$\"3=+++RXD;SFL7$$!39+++\"oFf%RFL$\" 3A+++yz_eRFL7$$!3v*****\\uB&3SFL$\"3;+++?**4+RFL7$$!3/+++fO0qSFL$\"37+ ++nL!4%QFL7$$!3&)******=pYITFL$\"3-+++p1(3y$FL7$$!3u*****zRG(*=%FL$\"3 1+++\"RO*>PFL7$$!33+++Qk\"yC%FL$\"3#******RWR!eOFL7$$!3R+++v4s/VFL$\"3 7+++VY7&f$FL7$$!3g******f+++x\\WKLFL7$$!3++++hstrXFL$\"3?+++MP)QE$FL7$$!36+++P+x@YFL$ \"3%)*****4ZVT>$FL7$$!35+++$=Q2n%FL$\"3*)*****feHK7$FL7$$!3#******HOo' =ZFL$\"3++++Gq:^IFL7$$!3s*****4J&elZFL$\"3'******R/]z(HFL7$$!3q*****pK 5:\"[FL$\"3?+++p;k.HFL7$$!3y*****H#)fk&[FL$\"3$)*****\\#HFGGFL7$$!3;++ +**RW+\\FL$\"3%)*******p\"*=v#FL7$$!3O+++PdYV\\FL$\"3%******fL^Xn#FL7$ $!3I+++>)>b)\\FL$\"3;+++Y*3jf#FL7$$!3G+++IDfE]FL$\"36+++lNA#FL7$$!3D+ ++x_?;_FL$\"3&******4x?66#FL7$$!3!)*****H4(o]_FL$\"3%******R?,\"G?FL7$ $!37+++A#4RG&FL$\"3/+++TncW>FL7$$!3i*****4MAeJ&FL$\"31+++'*3ag=FL7$$!3 #******fNzjM&FL$\"3#*******)GTgx\"FL7$$!3u*****\\gObP&FL$\"35+++0/3\"p \"FL7$$!3o*****z#[D.aFL$\"3++++'\\mcg\"FL7$$!3D+++&))*\\HaFL$\"3,+++%f /)>:FL7$$!3!)*****\\CVUX&FL$\"3#******\\c(\\L9FL7$$!3O+++sAYxaFL$\"3-+ ++druY8FL7$$!3!******HIS\"*\\&FL$\"3++++F]bf7FL7$$!3&*******ylE>bFL$\" 3#******HsB><\"FL7$$!3q*****>+Oy`&FL$\"3/+++!fdQ3\"FL7$$!3/+++w([[b&FL $\"3%*******Qij`**F-7$$!3k*****z*)4.d&FL$\"3e*****z@AX1*F-7$$!3E+++N&G Ue&FL$\"3?+++'\\y8<)F-7$$!3o*****RGK \"\\2cFL$\"3?+++>`\"QP'F-7$$!3<+++ju'oh&FL$\"3q******Q0$*paF-7$$!3!*** ***H;jZi&FL$\"3++++@%)3jXF-7$$!3i******ya>JcFL$\"3;+++Lvo`OF-7$$!30+++ g+=OcFL$\"34+++.V0 c=k&FL$\"3[+++:4(z9*Fhs7$$!3>+++ZRcUcFLF(7$F`]n$!3[+++:4(z9*Fhs7$F[]n$ !3********=p.H=F-7$Ff\\n$!34+++.V`\"QP'F-7$F][n$!3W++ +ieRusF-7$Fhjm$!3?+++'\\y8<)F-7$Fcjm$!3e*****z@AX1*F-7$F^jm$!3%******* Qij`**F-7$Fiim$!3/+++!fdQ3\"FL7$Fdim$!3#******HsB><\"FL7$F_im$!3++++F] bf7FL7$Fjhm$!3-+++druY8FL7$Fehm$!3#******\\c(\\L9FL7$F`hm$!3,+++%f/)>: FL7$F[hm$!3++++'\\mcg\"FL7$Ffgm$!35+++0/3\"p\"FL7$Fagm$!3#*******)GTgx \"FL7$F\\gm$!31+++'*3ag=FL7$Fgfm$!3/+++TncW>FL7$Fbfm$!3%******R?,\"G?F L7$F]fm$!3&******4x?66#FL7$Fhem$!39+++Jbf$>#FL7$Fcem$!39+++r*)[vAFL7$F ^em$!3!)*****49enN#FL7$Fidm$!3?+++LUNPCFL7$Fddm$!36+++lNA$FL7$Fbam$!3 ?+++MP)QE$FL7$F]am$!3>+++x\\WKLFL7$Fh`m$!3')*****HSJ)*R$FL7$Fc`m$!3#)* ****fqcgY$FL7$F^`m$!3z*****H$Q9JNFL7$Fi_m$!37+++VY7&f$FL7$Fd_m$!3#**** **RWR!eOFL7$F__m$!31+++\"RO*>PFL7$Fj^m$!3-+++p1(3y$FL7$Fe^m$!37+++nL!4 %QFL7$F`^m$!3;+++?**4+RFL7$F[^m$!3A+++yz_eRFL7$Ff]m$!3=+++RXD;SFL7$Fa] m$!3A+++OAMtSFL7$F\\]m$!3k*****>sW)HTFL7$Fg\\m$!3i*****zE,e=%FL7$Fb\\m $!31+++$)3BTUFL7$F]\\m$!3U+++il7'H%FL7$Fh[m$!3m*****f\\]/N%FL7$Fc[m$!3 x******f78/WFL7$F^[m$!3!******4(Q1dWFL7$Fijl$!3O+++GK64XFL7$Fdjl$!3A++ +.37gXFL7$F_jl$!3&)*****4s8*4YFL7$Fjil$!3=+++_[JeYFL7$Feil$!3,+++3C:0Z FL7$F`il$!3/+++txE]ZFL7$F[il$!33+++#*3_$z%FL7$Ffhl$!3A+++jIzM[FL7$Fahl $!3A+++,v)R([FL7$F\\hl$!3/+++[#G5\"\\FL7$Fggl$!3U+++b!ee%\\FL7$Fbgl$!3 ))*****H#eVy\\FL7$F]gl$!3$******\\,M(3]FL7$Fhfl$!3O+++WjtO]FL7$Fcfl$!3 #)*****\\%fVi]FL7$F^fl$!3e*****H9Me3&FL7$Fiel$!3.+++\\*Rp5&FL7$Fdel$!3 S++++/xD^FL7$F_el$!31+++g>NU^FL7$Fjdl$!3i*****>IBn:&FL7$Fedl$!3++++U+% *o^FL7$F`dl$!3C+++bB3z^FL7$F[dl$!3/+++:rE(=&FL7$Ffcl$!39+++htm$>&FL7$F acl$!3y*****HBW&)>&FL7$F\\cl$!3,++++4I-_FL7$Fgbl$!3d*****z(od0_FL7$Fbb l$!3&)*****f*)=%4_FL7$F]bl$!3n*****4L(f:_FL7$Fhal$!3*******4$>)H'eo&FL7$Fa^l$!3e*****4t***3dFL7$F\\^l$!30+++kk[FdFL7$Fg]l$!3e *****R4>\"3?v&FL7$F]]l$!3/+++1yfedFL7$Fh\\l$!3s**** *4_!ohdFL7$Fc\\l$!3;+++atShdFL7$F^\\l$!3/+++:g*yv&FL7$Fi[l$!3S+++SnB^d FL7$Fd[l$!3%)*****Hv'\\TdFL7$F_[l$!33+++hOsGdFL7$Fjz$!3H+++p![Hr&FL7$F ez$!3q*****pf&=%p&FL7$F`z$!3I+++[%QCn&FL7$F[z$!3C+++1mpZcFL7$Ffy$!3q** ***4qQ*>cFL7$Fay$!3m******HD8*e&FL7$F\\y$!3I+++O^BbbFL7$Fgx$!3z*****fE #>=bFL7$Fbx$!3#*******Ro$zZ&FL7$F]x$!3B+++@eQMaFL7$Fhw$!37+++lWV(Q&FL7 $Fcw$!3++++\">YpL&FL7$F^w$!3c*****H2NFG&FL7$Fiv$!3V+++Mc`C_FL7$Fdv$!3/ +++/-&>;&FL7$F_v$!3-+++X_N%4&FL7$Fju$!35+++%RI2-&FL7$Feu$!3:+++)>T$R\\ FL7$F`u$!3!******H*>OFL7$ F`r$!3++++[gMfMFL7$F[r$!32+++>W\\+LFL7$Ffq$!3!******RGXC9$FL7$Faq$!3%* *****z'=([)HFL7$F\\q$!34+++?HcFGFL7$Fgp$!35+++>(*QqEFL7$Fbp$!3:+++l\"z K^#FL7$F]p$!3))*****fG$>cBFL7$Fho$!3%)*****Rj9\"*>#FL7$Fco$!3:+++)fO?/ #FL7$F]o$!3)*******\\v&\\)=FL7$$\"3')*****>68Oy'FU$!34+++ow(ys\"FL7$$ \"3?+++*\\3;4$FU$!3#******RZ(zq:FL7$FS$!30+++Ctr89FL7$FN$!3-+++\\tjc7F L7$$\"37+++'yC#Ff_pFa]p7%F^bp7$$\"+O&\\oI&Fdap$\"+'G$>cBFf_pFa]p7%Fdbp7$$ \"+;zU&f'Fdap$\"+l\"zK^#Ff_pFa]p7%Fjbp7$$\"+)Ro`!oFdap$\"+>(*QqEFf_pFa ]p7%F`cp7$$\"+Ms+x\\Fdap$\"+?HcFGFf_pFa]p7%Ffcp7$$\"+4aM))eFa`p$\"+o=( [)HFf_pFa]p7%F\\dp7$$!+<%R6.&Fdap$\"+%GXC9$Ff_pFa]p7%Fbdp7$$!+(pYR!fFd ap$\"+>W\\+LFf_pFa]p7%Fhdp7$$\"+*RBbP\"!#8$\"+[gMfMFf_pFa]p7%F^ep7$$\" +u%H(z))Faep$\"+5&y%>OFf_pFa]p7%Feep7$$\"+0cN**G!#7$\"+Owa\"y$Ff_pFa]p 7%F[fp7$$\"+4#Q![vF^fp$\"+y:NYRFf_pFa]p7%Fbfp7$$\"+5.$Ru\"!#6$\"+*eZX6 %Ff_pFa]p7%Fhfp7$$\"+g5\"yr$F[gp$\"+$)[^&G%Ff_pFa]p7%F_gp7$$\"+*Ra9H(F [gp$\"+H\"RUX%Ff_pFa]p7%Fegp7$$\"+A>Rh7F`]p$\"+ST$R\\Ff_pFa]p7%F]ip7$$\"+0$obC$F`]p$\"+ %RI2-&Ff_pFa]p7%Fcip7$$\"+7@(3^$F`]p$\"+X_N%4&Ff_pFa]p7%Fiip7$$\"+GCq% p$F`]p$\"+/-&>;&Ff_pFa]p7%F_jp7$$\"+PQp2QF`]p$\"+Mc`C_Ff_pFa]p7%Fejp7$ $\"+Cw%*eQF`]p$\"+t]t#G&Ff_pFa]p7%F[[q7$$\"+MN:cQF`]p$\"+\">YpL&Ff_pFa ]p7%Fa[q7$$\"+Z&He!QF`]p$\"+lWV(Q&Ff_pFa]p7%Fg[q7$$\"+w7a8PF`]p$\"+@eQ MaFf_pFa]p7%F]\\q7$$\"+#ewSe$F`]p$\"+So$zZ&Ff_pFa]p7%Fc\\q7$$\"+i*z:U$ F`]p$\"+mA>=bFf_pFa]p7%Fi\\q7$$\"+7jlHKF`]p$\"+O^BbbFf_pFa]p7%F_]q7$$ \"+RCX6IF`]p$\"+ID8*e&Ff_pFa]p7%Fe]q7$$\"+bCspFF`]p$\"+,(Q*>cFf_pFa]p7 %F[^q7$$\"+z6)o]#F`]p$\"+1mpZcFf_pFa]p7%Fa^q7$$\"+-$[]A#F`]p$\"+[%QCn& Ff_pFa]p7%Fg^q7$$\"+L]3E>F`]p$\"+(f&=%p&Ff_pFa]p7%F]_q7$$\"+`Vi6;F`]p$ \"+p![Hr&Ff_pFa]p7%Fc_q7$$\"+1b4$G\"F`]p$\"+hOsGdFf_pFa]p7%Fi_q7$$\"+C -V<%*F[gp$\"+`n\\TdFf_pFa]p7%F_`q7$$\"+[AS')eF[gp$\"+SnB^dFf_pFa]p7%Fe `q7$$\"+ny'pC#F[gp$\"+:g*yv&Ff_pFa]p7%F[aq7$$!+_hS$\\\"F[gp$\"+atShdFf _pFa]p7%Faaq7$$!+3G'*G`F[gp$\"+@0ohdFf_pFa]p7%Fgaq7$$!+=/&eD*F[gp$\"+1 yfedFf_pFa]p7%F]bq7$$!+Y$[sK\"F`]p$\"+>\"3?v&Ff_pFa]p7%Fcbq7$$!+T_,QE\\$F`]p$\"+p+ddcFf_pFa]p7%Fadq7$$!+qfDtRF`]p$\"+6eZBcFf_pF a]p7%Fgdq7$$!+Ul;%[%F`]p$\"+p^x#e&Ff_pFa]p7%F]eq7$$!+e1=T]F`]p$\"+%pSX `&Ff_pFa]p7%Fceq7$$!+]#GBn&F`]p$\"+B_+yaFf_pFa]p7%Fieq7$$!+\"yOaU'F`]p $\"+\"Q'p8aFf_pFa]p7%F_fq7$$!+\\g\\jtF`]p$\"+h$[lM&Ff_pFa]p7%Fefq7$$!+ [ta-&)F`]p$\"+t)Q%)G&Ff_pFa]p7%F[gq7$$!+*\\pAu*F`]p$\"+CzC\\_Ff_pFa]p7 %Fagq7$$!+))*)='4\"Ff_p$\"+J>&Ff_pFa]p7%Feiq7$$!+dPQ!p\"Ff_p$\"+htm$>&Ff_pFa]p7%F[jq7 $$!+?JJtFf_p$\"+U+%*o^Ff_pFa]p7%F][r7$$!+v)[m+#Ff_p$\"+-Lsc^Ff _pFa]p7%Fc[r7$$!+ZMS!3#Ff_p$\"+g>NU^Ff_pFa]p7%Fi[r7$$!+@hd_@Ff_p$\"++/ xD^Ff_pFa]p7%F_\\r7$$!+L/PBAFf_p$\"+\\*Rp5&Ff_pFa]p7%Fe\\r7$$!+Xc'HH#F f_p$\"+VT$e3&Ff_pFa]p7%F[]r7$$!+yH_hBFf_p$\"+XfVi]Ff_pFa]p7%Fa]r7$$!+% e!>HCFf_p$\"+WjtO]Ff_pFa]p7%Fg]r7$$!+3v5'\\#Ff_p$\"+:St3]Ff_pFa]p7%F]^ r7$$!+`nSiDFf_p$\"+BeVy\\Ff_pFa]p7%Fc^r7$$!+=v@GEFf_p$\"+b!ee%\\Ff_pFa ]p7%Fi^r7$$!+Ulm$p#Ff_p$\"+[#G5\"\\Ff_pFa]p7%F__r7$$!+s%y)eFFf_p$\"+,v )R([Ff_pFa]p7%Fe_r7$$!+#3vR#GFf_p$\"+jIzM[Ff_pFa]p7%F[`r7$$!+%>t!*)GFf _p$\"+#*3_$z%Ff_pFa]p7%Fa`r7$$!+!>\"GaHFf_p$\"+txE]ZFf_pFa]p7%Fg`r7$$! +_Tp>IFf_p$\"+3C:0ZFf_pFa]p7%F]ar7$$!+CzQ&3$Ff_p$\"+_[JeYFf_pFa]p7%Fca r7$$!+!p79:$Ff_p$\"+@P\"*4YFf_pFa]p7%Fiar7$$!+Qqys/`NFf_p$\"+il7'H%Ff_pFa]p7%F]dr7$$!+Jk(*> OFf_p$\"+$)3BTUFf_pFa]p7%Fcdr7$$!+(o:lo$Ff_p$\"+o7!e=%Ff_pFa]p7%Fidr7$ $!+KC^_PFf_p$\"+AZ%)HTFf_pFa]p7%F_er7$$!+Xi#y\"QFf_p$\"+OAMtSFf_pFa]p7 %Feer7$$!+=JL#)QFf_p$\"+RXD;SFf_pFa]p7%F[fr7$$!+\"oFf%RFf_p$\"+yz_eRFf _pFa]p7%Fafr7$$!+XP_3SFf_p$\"+?**4+RFf_pFa]p7%Fgfr7$$!+fO0qSFf_p$\"+nL !4%QFf_pFa]p7%F]gr7$$!+>pYITFf_p$\"+p1(3y$Ff_pFa]p7%Fcgr7$$!+)RG(*=%Ff _p$\"+\"RO*>PFf_pFa]p7%Figr7$$!+Qk\"yC%Ff_p$\"+W%R!eOFf_pFa]p7%F_hr7$$ !+v4s/VFf_p$\"+VY7&f$Ff_pFa]p7%Fehr7$$!+g$Ff_pFa]p7%Fijr7$$!+$=Q2n%Ff_p$ \"+'eHK7$Ff_pFa]p7%F_[s7$$!+j$o'=ZFf_p$\"+Gq:^IFf_pFa]p7%Fe[s7$$!+6`el ZFf_p$\"+W+&z(HFf_pFa]p7%F[\\s7$$!+F.^6[Ff_p$\"+p;k.HFf_pFa]p7%Fa\\s7$ $!+B)fk&[Ff_p$\"+DHFGGFf_pFa]p7%Fg\\s7$$!+**RW+\\Ff_p$\"++<*=v#Ff_pFa] p7%F]]s7$$!+PdYV\\Ff_p$\"+O8buEFf_pFa]p7%Fc]s7$$!+>)>b)\\Ff_p$\"+Y*3jf #Ff_pFa]p7%Fi]s7$$!+IDfE]Ff_p$\"+lNA#Ff_pFa]p7%Fg_s7$$ !+x_?;_Ff_p$\"+r276@Ff_pFa]p7%F]`s7$$!+$4(o]_Ff_p$\"+/75G?Ff_pFa]p7%Fc `s7$$!+A#4RG&Ff_p$\"+TncW>Ff_pFa]p7%Fi`s7$$!+TB#eJ&Ff_p$\"+'*3ag=Ff_pF a]p7%F_as7$$!+c$zjM&Ff_p$\"+*GTgx\"Ff_pFa]p7%Feas7$$!+0m`v`Ff_p$\"+0/3 \"p\"Ff_pFa]p7%F[bs7$$!+G[D.aFf_p$\"+'\\mcg\"Ff_pFa]p7%Fabs7$$!+&))*\\ HaFf_p$\"+%f/)>:Ff_pFa]p7%Fgbs7$$!+XKCaaFf_p$\"+lv\\L9Ff_pFa]p7%F]cs7$ $!+sAYxaFf_p$\"+druY8Ff_pFa]p7%Fccs7$$!+..9*\\&Ff_p$\"+F]bf7Ff_pFa]p7% Fics7$$!+zlE>bFf_p$\"+BP#><\"Ff_pFa]p7%F_ds7$$!+-g$y`&Ff_p$\"+!fdQ3\"F f_pFa]p7%Feds7$$!+w([[b&Ff_p$\"+Rij`**F`]pFa]p7%F[es7$$!+)*)4.d&Ff_p$ \"+=A_k!*F`]pFa]p7%Faes7$$!+N&GUe&Ff_p$\"+'\\y8<)F`]pFa]p7%Fges7$$!+%G `\"QP'F`]pFa]p7%Fcf s7$$!+ju'oh&Ff_p$\"+R0$*paF`]pFa]p7%Fifs7$$!+jJwCcFf_p$\"+@%)3jXF`]pFa ]p7%F_gs7$$!+za>JcFf_p$\"+Lvo`OF`]pFa]p7%Fegs7$$!+g+=OcFf_p$\"+.Vp.H=F`]pFa]p7%Fahs7$$!+_g&=k&Ff_p$\"+:4 (z9*F[gpFa]p7%Fghs7$$!+ZRcUcFf_pF(Fa]p7%F]is7$Fhhs$!+:4(z9*F[gpFa]p7%F ais7$Fbhs$!+>p.H=F`]pFa]p7%Feis7$F\\hs$!+.V`\"QP'F`]pFa]p7%Fijs7$F^fs$!+ieRusF`]pFa]p7%F][t7$Fhes $!+'\\y8<)F`]pFa]p7%Fa[t7$Fbes$!+=A_k!*F`]pFa]p7%Fe[t7$F\\es$!+Rij`**F `]pFa]p7%Fi[t7$Ffds$!+!fdQ3\"Ff_pFa]p7%F]\\t7$F`ds$!+BP#><\"Ff_pFa]p7% Fa\\t7$Fjcs$!+F]bf7Ff_pFa]p7%Fe\\t7$Fdcs$!+druY8Ff_pFa]p7%Fi\\t7$F^cs$ !+lv\\L9Ff_pFa]p7%F]]t7$Fhbs$!+%f/)>:Ff_pFa]p7%Fa]t7$Fbbs$!+'\\mcg\"Ff _pFa]p7%Fe]t7$F\\bs$!+0/3\"p\"Ff_pFa]p7%Fi]t7$Ffas$!+*GTgx\"Ff_pFa]p7% F]^t7$F`as$!+'*3ag=Ff_pFa]p7%Fa^t7$Fj`s$!+TncW>Ff_pFa]p7%Fe^t7$Fd`s$!+ /75G?Ff_pFa]p7%Fi^t7$F^`s$!+r276@Ff_pFa]p7%F]_t7$Fh_s$!+Jbf$>#Ff_pFa]p 7%Fa_t7$Fb_s$!+r*)[vAFf_pFa]p7%Fe_t7$F\\_s$!+T\"enN#Ff_pFa]p7%Fi_t7$Ff ^s$!+LUNPCFf_pFa]p7%F]`t7$F`^s$!+lNA$Ff_pFa]p7%Febt7$Fdjr$!+MP)QE$Ff_pFa]p7%Fibt7$F^jr$! +x\\WKLFf_pFa]p7%F]ct7$Fhir$!+.9$)*R$Ff_pFa]p7%Fact7$Fbir$!+1n0mMFf_pF a]p7%Fect7$F\\ir$!+LQ9JNFf_pFa]p7%Fict7$Ffhr$!+VY7&f$Ff_pFa]p7%F]dt7$F `hr$!+W%R!eOFf_pFa]p7%Fadt7$Fjgr$!+\"RO*>PFf_pFa]p7%Fedt7$Fdgr$!+p1(3y $Ff_pFa]p7%Fidt7$F^gr$!+nL!4%QFf_pFa]p7%F]et7$Fhfr$!+?**4+RFf_pFa]p7%F aet7$Fbfr$!+yz_eRFf_pFa]p7%Feet7$F\\fr$!+RXD;SFf_pFa]p7%Fiet7$Ffer$!+O AMtSFf_pFa]p7%F]ft7$F`er$!+AZ%)HTFf_pFa]p7%Faft7$Fjdr$!+o7!e=%Ff_pFa]p 7%Feft7$Fddr$!+$)3BTUFf_pFa]p7%Fift7$F^dr$!+il7'H%Ff_pFa]p7%F]gt7$Fhcr $!+'\\]/N%Ff_pFa]p7%Fagt7$Fbcr$!+g78/WFf_pFa]p7%Fegt7$F\\cr$!+rQ1dWFf_ pFa]p7%Figt7$Ffbr$!+GK64XFf_pFa]p7%F]ht7$F`br$!+.37gXFf_pFa]p7%Faht7$F jar$!+@P\"*4YFf_pFa]p7%Feht7$Fdar$!+_[JeYFf_pFa]p7%Fiht7$F^ar$!+3C:0ZF f_pFa]p7%F]it7$Fh`r$!+txE]ZFf_pFa]p7%Fait7$Fb`r$!+#*3_$z%Ff_pFa]p7%Fei t7$F\\`r$!+jIzM[Ff_pFa]p7%Fiit7$Ff_r$!+,v)R([Ff_pFa]p7%F]jt7$F`_r$!+[# G5\"\\Ff_pFa]p7%Fajt7$Fj^r$!+b!ee%\\Ff_pFa]p7%Fejt7$Fd^r$!+BeVy\\Ff_pF a]p7%Fijt7$F^^r$!+:St3]Ff_pFa]p7%F][u7$Fh]r$!+WjtO]Ff_pFa]p7%Fa[u7$Fb] r$!+XfVi]Ff_pFa]p7%Fe[u7$F\\]r$!+VT$e3&Ff_pFa]p7%Fi[u7$Ff\\r$!+\\*Rp5& Ff_pFa]p7%F]\\u7$F`\\r$!++/xD^Ff_pFa]p7%Fa\\u7$Fj[r$!+g>NU^Ff_pFa]p7%F e\\u7$Fd[r$!+-Lsc^Ff_pFa]p7%Fi\\u7$F^[r$!+U+%*o^Ff_pFa]p7%F]]u7$Fhjq$! +bB3z^Ff_pFa]p7%Fa]u7$Fbjq$!+:rE(=&Ff_pFa]p7%Fe]u7$F\\jq$!+htm$>&Ff_pF a]p7%Fi]u7$Ffiq$!+LUa)>&Ff_pFa]p7%F]^u7$F`iq$!++4I-_Ff_pFa]p7%Fa^u7$Fj hq$!+yod0_Ff_pFa]p7%Fe^u7$Fdhq$!+'*)=%4_Ff_pFa]p7%Fi^u7$F^hq$!+Jtf:_Ff _pFa]p7%F]_u7$Fhgq$!+J>\"3?v&Ff_pFa]p7%Fibu7$F^bq$!+1yfedFf_pFa]p7%F]cu7$Fhaq$!+@0oh dFf_pFa]p7%Facu7$Fbaq$!+atShdFf_pFa]p7%Fecu7$F\\aq$!+:g*yv&Ff_pFa]p7%F icu7$Ff`q$!+SnB^dFf_pFa]p7%F]du7$F``q$!+`n\\TdFf_pFa]p7%Fadu7$Fj_q$!+h OsGdFf_pFa]p7%Fedu7$Fd_q$!+p![Hr&Ff_pFa]p7%Fidu7$F^_q$!+(f&=%p&Ff_pFa] p7%F]eu7$Fh^q$!+[%QCn&Ff_pFa]p7%Faeu7$Fb^q$!+1mpZcFf_pFa]p7%Feeu7$F\\^ q$!+,(Q*>cFf_pFa]p7%Fieu7$Ff]q$!+ID8*e&Ff_pFa]p7%F]fu7$F`]q$!+O^BbbFf_ pFa]p7%Fafu7$Fj\\q$!+mA>=bFf_pFa]p7%Fefu7$Fd\\q$!+So$zZ&Ff_pFa]p7%Fifu 7$F^\\q$!+@eQMaFf_pFa]p7%F]gu7$Fh[q$!+lWV(Q&Ff_pFa]p7%Fagu7$Fb[q$!+\"> YpL&Ff_pFa]p7%Fegu7$F\\[q$!+t]t#G&Ff_pFa]p7%Figu7$Ffjp$!+Mc`C_Ff_pFa]p 7%F]hu7$F`jp$!+/-&>;&Ff_pFa]p7%Fahu7$Fjip$!+X_N%4&Ff_pFa]p7%Fehu7$Fdip $!+%RI2-&Ff_pFa]p7%Fihu7$F^ip$!+)>T$R\\Ff_pFa]p7%F]iu7$Fhhp$!+$*>OFf_pFa]p7%Fa[v7$F_ep$!+[gMfMFf_pFa]p7%Fe[v7$F idp$!+>W\\+LFf_pFa]p7%Fi[v7$Fcdp$!+%GXC9$Ff_pFa]p7%F]\\v7$F]dp$!+o=([) HFf_pFa]p7%Fa\\v7$Fgcp$!+?HcFGFf_pFa]p7%Fe\\v7$Facp$!+>(*QqEFf_pFa]p7% Fi\\v7$F[cp$!+l\"zK^#Ff_pFa]p7%F]]v7$Febp$!+'G$>cBFf_pFa]p7%Fa]v7$F_bp $!+MY6*>#Ff_pFa]p7%Fe]v7$Fiap$!+)fO?/#Ff_pFa]p7%Fi]v7$Fbap$!+]v&\\)=Ff _pFa]p7%F]^v7$$\"+7Jh$y'Fa`p$!+ow(ys\"Ff_pFa]p7%Fa^v7$$\"+*\\3;4$Fa`p$ !+uuzq:Ff_pFa]p7%Fg^v7$F_`p$!+Ctr89Ff_pFa]p7%F]_v7$Fi_p$!+\\tjc7Ff_pFa ]p7%Fa_v7$$\"+'yC " 0 "" {MPLTEXT 1 0 730 "R := z -> add(z^j/j!,j=0..8)+ 2108597459888131951518061076535096782344465 464713/789220209793612270910036479284123496222433280000000000*z^9+\n45 320172370419669737903140995480465453402109334849/175382268843024949091 119217618694110271651840000000000000*z^10+\n24649851138940476817153206 7238258583844568210512979/20298004726798924291967158948812406976423116 800000000000000*z^11+\n19442314262192529069083609889457692137187/10439 1597854711340008832646814004715520000000000000*z^12:\nDigits := 18:\np ts := []: z0 := 0:\nfor ct from 0 to 240 do\n zz := newton(R(z)=exp( ct*Pi/60*I),z=z0):\n z0 := zz:\n pts := [op(pts),[surd(Re(zz),11), Im(zz)]]:\nend do:\nplot(pts,color=COLOR(RGB,.3,.2,.95),thickness=2,fo nt=[HELVETICA,9]);\nDigits := 10:" }}{PARA 13 "" 1 "" {GLPLOT2D 287 308 308 {PLOTDATA 2 "6(-%'CURVESG6#7]z7$$\"\"!F)F(7$F($\"3w))H)fv()fB& !#>7$$\"3&Q!4G8!)QiGF-$\"3,*f'>^v>Z5!#=7$$\"3()3'* z(Q*zh#F37$$\"3&))Hv.B\"obxF-$\"3SLPh`EfTJF37$$\"3AanQrv?:*)F-$\"3z,CG H9>lOF37$$\"3'Hmdcr]c+\"F3$\"3nvT40-z)=%F37$$\"3ZRJ)G1B\"=6F3$\"3JySD \")*)Q7ZF37$$\"3[bPucY0H7F3$\"3E3HF3$\"3'3bBiQ z6!*)F37$$\"3h'\\\"3?G9n?F3$\"3sLnd!GyZU*F37$$\"3ut*yPoYf;#F3$\"3o&=i \"HsP[**F37$$\"3m&Q_[@)RjAF3$\"3\\y\"\\\\i(>Z5!#<7$$\"3R-Ot`FZfBF3$\"3 m*G&>Ovb*4\"Fhq7$$\"37KDJ))39aCF3$\"3r*HH*eu\">:\"Fhq7$$\"3o:!)e(fota# F3$\"3r)>lbRxU?\"Fhq7$$\"3.!*[5)H<\"REF3$\"3@()[][tjc7Fhq7$$\"39!eqE!Q MHFF3$\"3c/X#*>t**38Fhq7$$\"3Cf:L'>+!=GF3$\"3R^N\\6tNh8Fhq7$$\"3p&e.3j L]!HF3$\"3n.l-Ctr89Fhq7$$\"3Z?48NgQ!*HF3$\"3,RJ2dt2m9Fhq7$$\"3H?1,PP*R 2$F3$\"3!Gei%3uV=:Fhq7$$\"3bg$*p&)pybJF3$\"3OAu#QZ(zq:Fhq7$$\"3H=&eMO* oNKF3$\"3o'R[hadJi\"Fhq7$$\"3z-()ynqh8LF3$\"3sqtP:w^v;Fhq7$$\"3xBe)3/y %*Q$F3$\"3WpC]tJP>Fhq7$$\"36s&*)*\\#GQt$F3$\"3R!R!)p.x'*)>Fhq7 $$\"3F%4j***ps%z$F3$\"3L=+P)fO?/#Fhq7$$\"3\"*G5l@\"eD&QF3$\"3&puwT.'R% 4#Fhq7$$\"3xj^U^q42RF3$\"3k$3CMObn9#Fhq7$$\"3g+ms#Fhq7$$\"3=tkO[OA0SF3$\"3px-'\\$RZ^AFhq7$$\"33wXBb%f\"[SF3$\"3M(Qi>TL QI#Fhq7$$\"3%*[T\"=auk3%F3$\"3IayO'G$>cBFhq7$$\"3foC[BUm>TF3$\"3oq)y%y Qb3CFhq7$$\"3JO]pC\"4r9%F3$\"3cw7ZNc\"4Y#Fhq7$$\"3\"G'f\"zHL!oTF3$\"3I Gtbk\"zK^#Fhq7$$\"32>\"**)*4S9=%F3$\"3D`S#>FXcc#Fhq7$$\"3e@f#)G%4g=%F3 $\"3/MJ)z+:!=EFhq7$$\"3%=V9_ZB*zTF3$\"3r*z\\%>(*QqEFhq7$$\"3$)\\PT3$\\ 0;%F3$\"3+,)R(36xAFFhq7$$\"3#*)4P)4O!Q7%F3$\"3X\"eN6Ih^x#Fhq7$$\"3^M9# G30F1%F3$\"3O9d7?HcFGFhq7$$\"3-NE[]NPjRF3$\"3T;eEr\"z*zGFhq7$$\"3#Q#*Q I0\\'*y$F3$\"3t:R%[$RTKHFhq7$$\"3+Umkpc9YLF3$\"3Xgjbn=([)HFhq7$$!39zx' p\"G_)f$F3$\"3(oS1L^et.$Fhq7$$!3)*y*>\")H)Q3RF3$\"3c?1&>$Fhq7$$!3j] D*y*\\a\"=%F3$\"3!*3\"3&=6uZKFhq7$$!3yQ3XAsFETF3$\"3c%3d%>W\\+LFhq7$$! 3i_j(R/\")3'QF3$\"353H*GjNLN$Fhq7$$\"3W/gTy'oS\"RF3$\"3c7#fc$Fhq7$$\"3J!3L'yiOz_F3$\"3&eVD&4&y%>OFhq7 $$\"3Z\"H&[*\\!)4\\&F3$\"3K\\B%)o)[Kn$Fhq7$$\"3`Y.\"\\Tl&*o&F3$\"3/Zb, s^)eZX6%Fhq7$$\"3&=yADW(Q'3(F3$\" 3$*4%es%GMrTFhq7$$\"3$3je%oh$3D(F3$\"3YsS^_oQGUFhq7$$\"3#R@#))o#[NT(F3 $\"3Z\"fCH)[^&G%Fhq7$$\"32]TLO$*ztvF3$\"3'*f#GeumCM%Fhq7$$\"3-9./C!)RI xF3$\"3?l/sI#p))R%Fhq7$$\"3#=8@*z@p\")yF3$\"3mIfmG\"RUX%Fhq7$$\"3gEr>u &e)F3$\"3J42qW\"o\"RZFhq7$$ \"3aqzA-NBj')F3$\"30O'H(=bLxZFhq7$$\"3t'on,!oxJ()F3$\"3B$[B**pZK\"[Fhq 7$$\"3uRvKf)R>z)F3$\"3#fnjI*>\">;eNz[F hq7$$\"3%*QBBu)47*))F3$\"3w)o0q(o,5\\Fhq7$$\"3F)p+a`n?$*)F3$\"3')o!R\" )>T$R\\Fhq7$$\"3)Hs'oI(R!o*)F3$\"39ZV;/^[n\\Fhq7$$\"3!3J%eM_r***)F3$\" 3=9pP:)yX*\\Fhq7$$\"32%p/]C\"fF!*F3$\"3?61V%RI2-&Fhq7$$\"3X/rkNv3_!*F3 $\"3'4+DtXHg/&Fhq7$$\"3rL#[f5fN2*F3$\"3\\h!>E7*F3$\"3S5uNa^,S^Fhq7$$\"3*39.xk*eM\"*F3$\"3'fhQP?]>;&Fhq7$$\"37\\u' *3-mW\"*F3$\"3WnYPdM\\)y#er\"*F3$\"3F:C1e0 I>`Fhq7$$\"3\"yR;s(eYpL&Fhq7$$\"3i7B#[JCw;*F3$\"3EUQb 2)yTN&Fhq7$$\"3KMsU*flR;*F3$\"3YYvPwf+r`Fhq7$$\"3AX-x!QJ#f\"*F3$\"3#[q b`YMuQ&Fhq7$$\"3U2/mjY\"*F3$\"3 #>)*pa9>\">aFhq7$$\"3B*[0Z19)Q\"*F3$\"3wM`;@eQMaFhq7$$\"3(>%HaOD**H\"* F3$\"3R%Q%=O\\F\\aFhq7$$\"3ur!=L0!=?\"*F3$\"30w#*pi1zjaFhq7$$\"3^i&)* \\^!Q4\"*F3$\"3'[Mm,%o$zZ&Fhq7$$\"3-nyrkPf(4*F3$\"3s9a&[,<<\\&Fhq7$$\" 3]03\"ff:[3*F3$\"3L0&yEZM^]&Fhq7$$\"3C-nU0x.r!*F3$\"3fg=PmA>=bFhq7$$\" 35()pO\"fZi0*F3$\"3C)z8zB$*3`&Fhq7$$\"3>_H!GRG//*F3$\"3]M)=p.SKa&Fhq7$ $\"3ZX0zm'eN-*F3$\"3S,.jN^BbbFhq7$$\"3)odBh47c+*F3$\"3jR[.T3)oc&Fhq7$$ \"3j!3Y*=rb')*)F3$\"3^0#RXIz\"ybFhq7$$\"3Vd\\trs%4%)*))F3$\"3.J#H9qQ*>cFhq7$$\"3k^\"))oy;J())F3$\"3g >$*eA#G&HcFhq7$$\"3.I0Z.=QY))F3$\"3G')3')f/yQcFhq7$$\"34uA\"Qv:\"=))F3 $\"3!3BLkg'pZcFhq7$$\"37'[8vS;#)y)F3$\"31x,iBxFccFhq7$$\"3TW*=Nvllv)F3 $\"3AKpuRZ_kcFhq7$$\"3hz?!z5FIs)F3$\"3.M#p%[%QCn&Fhq7$$\"3xuabu9W(o)F3 $\"3N)pul]>+o&Fhq7$$\"3%pT/$>Hi\\')F3$\"3glJIJ%oso&Fhq7$$\"30(G!HAEN4' )F3$\"3Uo7@(f&=%p&Fhq7$$\"3@^!f98rjc)F3$\"33MhfK7x+dFhq7$$\"3B$)4nC\"o ._)F3$\"3iaQ[:a-2dFhq7$$\"3X$)zn`'o4Z)F3$\"30'*)z\"p![Hr&Fhq7$$\"3;>LN 5Vr<%)F3$\"3N&yqt&*Q&=dFhq7$$\"3wo3I!)o.g$)F3$\"3mT!f(ywzBdFhq7$$\"3kr qNh0A(H)F3$\"3OZi?hOsGdFhq7$$\"3$Q[@Y>Y$G#)F3$\"3:TANbhJLdFhq7$$\"3pm% 4\")31A:)F3$\"3W=:nFUdPdFhq7$$\"3ek()[,&or1)F3$\"3*>\"4#Hv'\\TdFhq7$$ \"3wB/<>5&4(zF3$\"3Mk`#fS#3XdFhq7$$\"3DN!3y=<-'yF3$\"3Zh4k_'H$[dFhq7$$ \"3:zUub&)zHxF3$\"3QRiUSnB^dFhq7$$\"3]Tj9kJ-rvF3$\"3YhRW(o,Qv&Fhq7$$\" 3n%zmPhCvO(F3$\"3PZf5rA-cdFhq7$$\"3*)fcM^Y#=3(F3$\"3bdHY:g*yv&Fhq7$$\" 3#[F$Q46#fe'F3$\"3>M*Hu&fW4TO#oF3$\"3qw&Fhq7$$!3*zMYOR--m(F3$\"3?%f]2_!ohdFhq7$$!3 /,*F3$\"3T#4T'QO.xcFhq7$$!3N/i,[r_]!*F3$\"3r#3ilf5wm&Fhq 7$$!3*y(ev#\\**z3*F3$\"3JZ$['o+ddcFhq7$$!3\\a`xec^C\"*F3$\"3#*R*4qI')o k&Fhq7$$!3(3!ozyQ>g\"*F3$\"3%y/(R::`NcFhq7$$!3Lj)>?7\\^>*F3$\"3/0s(4\" eZBcFhq7$$!3];Cl$Q&\\H#*F3$\"3u8Bkrso5cFhq7$$!3oyTw#*zMj#*F3$\"3\\plPD @8(f&Fhq7$$!3-$*=72g#oH*F3$\"30sw>p^x#e&Fhq7$$!3%fuvIua+L*F3$\"3D`0N_0 enbFhq7$$!3`82`]%oJO*F3$\"3^)*pWHJ^^bFhq7$$!3FJ<9pIJ'R*F3$\"3N5H+%pSX` &Fhq7$$!3!\\uv/')['H%*F3$\"3\\?+q^vj;bFhq7$$!3*yGt,X_LY*F3$\"3FFUx*3!z (\\&Fhq7$$!3xdY\\dth(\\*F3$\"3Lpw-B_+yaFhq7$$!3_M*3xM^E`*F3$\"3TEv#ovA tX&Fhq7$$!3jD)H@)ymo&*F3$\"3%=yOKbKeV&Fhq7$$!3'=ST5zneg*F3$\"3xu!H9Q'p 8aFhq7$$!3'o$)fs)pSW'*F3$\"3e%3!*)4;<\"R&Fhq7$$!3nY_q;([Vo*F3$\"3H$=Kx tG'o`Fhq7$$!3\"z'ylktgD(*F3$\"3eP]Gh$[lM&Fhq7$$!3wpsx1]!zw*F3$\"39bJ$e M\"[D`Fhq7$$!3<0Jxq=x5)*F3$\"3C@qm9!ofI&Fhq7$$!3O'p]k**4O&)*F3$\"3;ZS% G()Q%)G&Fhq7$$!38o1-DG!e*)*F3$\"3Zv7^8,9t_Fhq7$$!3Gj,>gA#o$**F3$\"36-[ z0k6g_Fhq7$$!3tJ4*H)3Hw**F3$\"34))HLCzC\\_Fhq7$$!3yku)e2*R,5Fhq$\"3u6] g3CJS_Fhq7$$!3'>%=O7P)\\+\"Fhq$\"37x)pS`VIB&Fhq7$$!3Uex\"e+%Q35Fhq$\"3 !RbA9$>$Qau\"[CA&Fhq7$$!3hS5h&fiY,\"Fhq$ \"32jgz3El=_Fhq7$$!3B(=C4qiv,\"Fhq$\"3g#=.9L(f:_Fhq7$$!3<.7:J\">.-\"Fh q$\"3JB)=3@CJ@&Fhq7$$!3YfWLVM%H-\"Fhq$\"3-.s!)*H,6@&Fhq7$$!3?`0*))\\Ya -\"Fhq$\"3vX'>f*)=%4_Fhq7$$!3t5z,8$Qy-\"Fhq$\"3LYUkPe)z?&Fhq7$$!3VCt9: !G,.\"Fhq$\"3Nb3/jis1_Fhq7$$!3q9b.rQKK5Fhq$\"3!f4sz(od0_Fhq7$$!35*f5lL LW.\"Fhq$\"3k6Is+Z[/_Fhq7$$!3w\"Rp^7jk.\"Fhq$\"39$Gz-A0M?&Fhq7$$!3@:X! *z#>%Q5Fhq$\"3_v^B+4I-_Fhq7$$!3Wm8sUsIS5Fhq$\"3955Xl*R6?&Fhq7$$!3z*pF+ #>8U5Fhq$\"3V$*oP&Fhq7$$!3C`];Ux*Q/\"Fhq$\"3kf-DLUa)>&Fhq7$$!3&o @^or3c/\"Fhq$\"3%o2SGrmq>&Fhq7$$!3N'\\#pw%os/\"Fhq$\"3\"[8&Fhq7 $$!3/-<<@.))[5Fhq$\"3wJr9htm$>&Fhq7$$!3FPUi`sW]5Fhq$\"3J(3kiQ=<>&Fhq7$ $!3H&[TP,s>0\"Fhq$\"3*)*>oV7)e*=&Fhq7$$!3FoV71rX`5Fhq$\"3JGc=:rE(=&Fhq 7$$!322?PD[!\\0\"Fhq$\"3F?woBru%=&Fhq7$$!3-UL9ysJc5Fhq$\"3(\\/\\V&)R/f5Fhq$\"3Bj&4/zDf<&F hq7$$!3#=&[>x;Og5Fhq$\"36x?GOkas^Fhq7$$!3mu5Z05lh5Fhq$\"3/)Gu;/S*o^Fhq -%%FONTG6$%*HELVETICAG\"\"*-%*THICKNESSG6#\"\"#-%&COLORG6&%$RGBG$\"\"$ !\"\"$F]goFdgo$\"#&*!\"#-%+AXESLABELSG6$Q!6\"F\\ho-%%VIEWG6$%(DEFAULTG Faho" 1 2 0 1 10 2 2 6 1 4 2 1.000000 45.000000 45.000000 0 0 "Curve 1 " }}}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 108 "Th e relevant intersection points of the boundary curve with the imaginar y axis can be determined as follows." }}{PARA 0 "" 0 "" {TEXT -1 87 "F irst we look for points on the boundary curve either side of each inte rsection point. " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> \+ " 0 "" {MPLTEXT 1 0 288 "Digits := 15:\nz0 := 3*I:\nfor ct from 94 to \+ 97 do\n newton(R(z)=exp(ct*Pi/100*I),z=z0);\nend do;``;\nz0 := 3.4*I :\nfor ct from 106 to 109 do\n newton(R(z)=exp(ct*Pi/100*I),z=z0);\n end do;``;\nz0 := 5.75*I:\nfor ct from 267 to 270 do\n newton(R(z)=e xp(ct*Pi/100*I),z=z0);\nend do;\nDigits := 10:" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#^$$\"09JgQ89l\"!#>$\"0m%HQSR`H!#9" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#^$$\"0Y+(3aM))e!#?$\"0Ocv'=([)H!#9" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#^$$!0uCK>J&p`!#?$\"0#e>9+O;I!#9" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#^$$!0;$e?^Q+$\"0zj`%)fy/$!#9" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%!G" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#^$$!0lW%[ $\"0eLkk(=KL!#9" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#^$$!0z'[[91%*=!#>$ \"0Y!3fd\"RO$!#9" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#^$$\"0oi_!=$ \"0$\\Y'G#o&R$!#9" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#^$$\"0pF)[ks()o!# >$\"0vZ[-\"\\FM!#9" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%!G" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#^$$\"0rcW[)oDw!#<$\"0!*e))z#ofd!#9" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#^$$\"0\">m`>yW9!#=$\"0e(e%y&Qgd!#9" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#^$$!0v)3eIevt!#<$\"0ei\"*)3'4w&!#9" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#^$$!0c+C:1M\\\"!#;$\"0AxVN29w&!#9" }}}{PARA 0 " " 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 107 "Then we apply the bisection method to calculate the parameter value associated with the intersection point." }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 478 "Dig its := 15:\nreal_part := proc(u)\n Re(newton(R(z)=exp(u*Pi*I),z=3*I) )\nend proc:\nu0 := bisect('real_part'(u),u=0.94..0.97);\nnewton(R(z)= exp(u0*Pi*I),z=3*I);``;\nreal_part := proc(u)\n Re(newton(R(z)=exp(u *Pi*I),z=3.4*I))\nend proc:\nu0 := bisect('real_part'(u),u=1.06..1.09) ;\nnewton(R(z)=exp(u0*Pi*I),z=3.4*I);``;\nreal_part := proc(u)\n Re( newton(R(z)=exp(u*Pi*I),z=5.75*I))\nend proc:\nu0 := bisect('real_part '(u),u=2.67..2.70);\nnewton(R(z)=exp(u0*Pi*I),z=5.75*I);\nDigits := 10 :" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%#u0G$\"04vS.vGb*!#:" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#^$$\"0N)Q%H<\"yl!#H$\"0Uwn')>:+$!#9" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#%!G" }}{PARA 11 "" 1 "" {XPPMATH 20 "6 #>%#u0G$\"0s]u%fgv5!#9" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#^#$\"0X[\"R* =%#u0G$\"0gS(4E>!o#!#9" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#^#$\"0Rm=2)Rgd!#9" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 4 "The " }{TEXT 260 112 "largest interval on the nonnegati ve imaginary axis that contains the origin and lies inside the stabili ty region" }{TEXT -1 5 " is " }{XPPEDIT 18 0 "[ 0, 3.0015 ]" "6#7$\" \"!-%&FloatG6$\"&:+$!\"%" }{TEXT -1 18 " (approximately)." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 4 "The " }{TEXT 260 72 "intersection of the stability region with the nonnegative imaginar y axis" }{TEXT -1 5 " is " }{XPPEDIT 18 0 "`union`([0, 3.0015],[3.381 7, 5.7604]);" "6#-%&unionG6$7$\"\"!-%&FloatG6$\"&:+$!\"%7$-F)6$\"& " 0 "" {MPLTEXT 1 0 98 "subs(e e,subs(b=`b*`,StabilityFunction(7,13,'expanded'))):\n`R*` := unapply(% ,z):\n'`R*`(z)'=`R*`(z);" }}{PARA 12 "" 1 "" {XPPMATH 20 "6#/-%#R*G6#% \"zG,:\"\"\"F)F'F)*&#F)\"\"#F)*$)F'F,F)F)F)*&#F)\"\"'F)*$)F'\"\"$F)F)F )*&#F)\"#CF)*$)F'\"\"%F)F)F)*&#F)\"$?\"F)*$)F'\"\"&F)F)F)*&#F)\"$?(F)* $)F'F1F)F)F)*&#F)\"%S]F)*$)F'\"\"(F)F)F)*&#\"R8d;GRu0E&RW(><4RY/RE:CA& fK\"W+++++))QPq#e?S@8%R$==roAj,n`J\"F)*$)F'\"\")F)F)F)*&#\"WVYJSW$QAS0 Q@#3=_uCm_NwI98\\`=\"fn++++++![K,X*R&)yQu\"R``DH4V,P#[Mu'F)*$)F'\"\"*F )F)F)*&#\"RhI!fh$y3#F)*$)F'\"#6F)F)F)" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }} {PARA 0 "" 0 "" {TEXT -1 124 "We can find the point where the boundary of the stability region intersects the negative real axis by solving \+ the equation: " }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "`R* `(z) = 1;" "6#/-%#R*G6#%\"zG\"\"\"" }{TEXT -1 2 ". " }}{PARA 0 "" 0 " " {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 30 "z_0 := new ton(`R*`(z)=1,z=-5);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%$z_0G$!+Q/)3 q&!\"*" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 318 "z_0 := newton(`R*`(z)=1,z=-5):\np_1 := plot([`R*`(z) ,1],z=-6.39..0.49,color=[red,blue]):\np_2 := plot([[[z_0,1]]$3],style= point,symbol=[circle,cross,diamond],color=black):\np_3 := plot([[z_0,0 ],[z_0,1]],linestyle=3,color=COLOR(RGB,0,.5,0)):\nplots[display]([p_1, p_2,p_3],view=[-6.39..0.49,-0.07..1.47],font=[HELVETICA,9]);" }}{PARA 13 "" 1 "" {GLPLOT2D 369 264 264 {PLOTDATA 2 "6+-%'CURVESG6$7Z7$$!3o** ***********Q'!#<$\"3erKTOuV_NF*7$$!3-LL$[*)3DN'F*$\"3[:57]r9HLF*7$$!3Q mmm*y<]J'F*$\"3y$yBW0K&=JF*7$$!3u****\\%oEvF'F*$\"3BgC)oP_*>HF*7$$!33L LLzb.SiF*$\"3k>3A$e'zKFF*7$$!3g***\\!fRzuhF*$\"3%)=]E\"fCEV#F*7$$!3:mm wQBb4hF*$\"3Gt@?,wRi@F*7$$!3I***\\LJ\"=OgF*$\"3y-zA`0'4*=F*7$$!3MLL$zG 5G'fF*$\"3/'HE(*pW0l\"F*7$$!3-LLt6H&*))eF*$\"31*y&zKblO9F*7$$!3pKL`Nb4 :eF*$\"3w&zOG$Q/[7F*7$$!31++b!>*eTdF*$\"3'p,i`9xF3\"F*7$$!3_mmcXG3ocF* $\"3E1%e\\+G^P*!#=7$$!3zKL.jKyJbF*$\"3_(HVx2?)QrFco7$$!3%)****4SHl!R&F *$\"3m>N\"oI@OM&Fco7$$!37LLBedpW_F*$\"3(yQmGCL#HRFco7$$!3Y*****Gm1#*4& F*$\"3gL7DkfQqGFco7$$!3UmmYXKb\\\\F*$\"35^'ycOoW1#Fco7$$!37LL$RVQx\"[F *$\"3k6aZ'R4*Q:Fco7$$!3[****RwhMpYF*$\"3Vwj*)f;z06Fco7$$!35+++1YM?XF*$ \"3-uKmG*>t*z!#>7$$!3c****f$)RvwVF*$\"3ip5NnLmkfF\\r7$$!39LLBU*fjC%F*$ \"3#)\\qALCw1ZF\\r7$$!3Kmm1:)384%F*$\"3Y..#y.Elv$F\\r7$$!3UmmEs,'*fRF* $\"3kQ9%y$GO,LF\\r7$$!3_****Hv/<2QF*$\"3-%o%4>()\\tIF\\r7$$!3?mmE/d#>n $F*$\"3ACh\\J%f(zIF\\r7$$!3b****Hb2aBNF*$\"3#)H=[NUNiKF\\r7$$!3]*****Q >VAQ$F*$\"3-`Y$y/\\pd$F\\r7$$!3%fmm;,9[B$F*$\"3v^D4>l_KSF\\r7$$!3%emm* **zU*4$F*$\"39S))\\0m*yb%F\\r7$$!3)GLLj*eR`HF*$\"3#4C,TH')*Q_F\\r7$$!3 )GLLAe4m'pEF*$\"3BWudt1`HpF\\r7$$ !3oKL`Hd0FDF*$\"3c5$zj(ye))zF\\r7$$!3]****>f^szBF*$\"33\\ezYb*fD*F\\r7 $$!3w****RR4fNAF*$\"33v%3TgQ\"p5Fco7$$!3a****\\qO8'4#F*$\"3$=0g`+%>H7F co7$$!3^****>],HT>F*$\"3[C.@CW6N9Fco7$$!3PmmEgj:-=F*$\"3#)*=fP%\\Q\\;F co7$$!33*****>M,Ol\"F*$\"3T3_!eMsN\">Fco7$$!3LLL$eT))*=:F*$\"3%fK1[u?$ *=#Fco7$$!3S****fJ4#=P\"F*$\"3u4*e!)eTk`#Fco7$$!3Fmm'zzZLB\"F*$\"3KII` rY:8HFco7$$!3C****4uMg)3\"F*$\"3+VhEPG'oO$Fco7$$!3#ymmmS%*3Z*Fco$\"3sk %eht#oyQFco7$$!3e$****\\it$*)zFco$\"3sQ=>lo1)\\%Fco7$$!3-DLLHI\\ilFco$ \"3))fMcOFco $\"3Udg1m(ow$pFco7$$!3(z)***>J]kK#Fco$\"3Z%\\-C8[V#zFco7$$!33*emms_T-) F\\r$\"375qTbT$*G#*Fco7$$\"3$RrmmM)*pg&F\\r$\"3A4)y1qrw0\"F*7$$\"3a0++ Z[19?Fco$\"3Lmv5^=7B7F*7$$\"3o4++4170MFco$\"3-m0^$>ncS\"F*7$$\"3!***** **********[Fco$\"3!QoL)>iJK;F*-%'COLOURG6&%$RGBG$\"*++++\"!\")$\"\"!Fd ]lFc]l-F$6$7S7$F($\"\"\"Fd]l7$F=Fi]l7$FGFi]l7$FQFi]l7$FenFi]l7$F_oFi]l 7$FeoFi]l7$FjoFi]l7$F_pFi]l7$FdpFi]l7$FipFi]l7$F^qFi]l7$FcqFi]l7$FhqFi ]l7$F^rFi]l7$FcrFi]l7$FhrFi]l7$F]sFi]l7$FbsFi]l7$FgsFi]l7$F\\tFi]l7$Fa tFi]l7$FftFi]l7$F[uFi]l7$F`uFi]l7$FeuFi]l7$FjuFi]l7$F_vFi]l7$FdvFi]l7$ FivFi]l7$F^wFi]l7$FcwFi]l7$FhwFi]l7$F]xFi]l7$FbxFi]l7$FgxFi]l7$F\\yFi] l7$FayFi]l7$FfyFi]l7$F[zFi]l7$F`zFi]l7$FezFi]l7$FjzFi]l7$F_[lFi]l7$Fd[ lFi]l7$Fi[lFi]l7$F^\\lFi]l7$Fc\\lFi]l7$Fh\\lFi]l-F]]l6&F_]lFc]lFc]lF`] l-F$6&7#7$$!3W+++Q/)3q&F*Fi]l-%'SYMBOLG6#%'CIRCLEG-F]]l6&F_]lFd]lFd]lF d]l-%&STYLEG6#%&POINTG-F$6&F_al-Fdal6#%&CROSSGFgalFial-F$6&F_al-Fdal6# %(DIAMONDGFgalFial-F$6%7$7$FaalFc]lF`al-%&COLORG6&F_]lFc]l$\"\"&!\"\"F c]l-%*LINESTYLEG6#\"\"$-%+AXESLABELSG6%Q\"z6\"Q!Ficl-%%FONTG6#%(DEFAUL TG-F\\dl6$%*HELVETICAG\"\"*-%%VIEWG6$;$!$R'!\"#$\"#\\Fidl;$!\"(Fidl$\" $Z\"Fidl" 1 2 0 1 10 0 2 9 1 4 2 1.000000 45.000000 45.000000 0 0 "Cur ve 1" "Curve 2" "Curve 3" "Curve 4" "Curve 5" "Curve 6" }}}}{PARA 0 " " 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 83 "The following pict ure shows the stability region for the 13 stage, order 7 scheme. " }} {PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1030 "`R*` := z -> add(z^j/j!,j=0..7)+\n325952224152639044639091719744 3952605743928165713/13153670163226871181833941321402058270373888000000 0000*z^8+185349131430763552662474521808221380540223834440314643/674344 82370143092925535339174387885399450132480000000000000*z^9+956892459490 2894163178434426866771147217615903061/40472831271467295944104434835083 256216535040000000000000*z^10+1097097017546100249376713718817001764889 87/20878319570942268001766529362800943104000000000000*z^11:\npts := [] :\nz0 := 0:\nfor ct from 0 to 320 do\n zz := newton(`R*`(z)=exp(ct*P i/20*I),z=z0):\n z0 := zz:\n pts := [op(pts),[Re(zz),Im(zz)]]:\nen d do:\np_1 := plot(pts,color=COLOR(RGB,.17,0,.35)):\np_2 := plots[poly gonplot]([seq([pts[i-1],pts[i],[-4.4,0]],i=2..nops(pts))],\n \+ style=patchnogrid,color=COLOR(RGB,.38,0,.75)):\np_3 := plot([[[-6.69,0 ],[1.39,0]],[[0,-6.39],[0,6.39]]],color=black,linestyle=3):\nplots[dis play]([p_||(1..3)],view=[-6.69..1.39,-6.39..6.39],font=[HELVETICA,9], \n labels=[`Re(z)`,`Im(z)`],axes=boxed,scaling=constrained) ;" }}{PARA 13 "" 1 "" {GLPLOT2D 381 565 565 {PLOTDATA 2 "6+-%'CURVESG6 $7]_l7$$\"\"!F)F(7$F($\"3++++Fjzq:!#=7$F($\"3)******Rl#fTJF-7$F($\"3<+ ++!)*)Q7ZF-7$$\"3%******\\5Vy9\"!#F$\"3!)*****fI&=$G'F-7$$!3')*****Hg' *G;'F7$\"3++++K;)R&yF-7$$!3y******>vRO#*!#E$\"3>+++&)zxC%*F-7$$!3)**** **47&4*p&!#D$\"3++++bub*4\"!#<7$$!3y*****>c)\\UC!#C$\"33+++8sjc7FK7$$! 3A+++orO'H)FO$\"3)*******\\tr89FK7$$!31+++1$f9P#!#B$\"33+++a')zq:FK7$$ !3%*******\\z4$*eFZ$\"3#******46$)ys\"FK7$$!3++++svu&H\"!#A$\"33+++:_( \\)=FK7$$!3))*****4?xp`#F_o$\"3')*****zG%3U?FK7$$!3o*****fp:[R%F_o$\"3 #******fFG#*>#FK7$$!3y*****pRQs`'F_o$\"33+++1&RkN#FK7$$!3o*****z6E%\\v F_o$\"33+++=Gx8DFK7$$!3++++S+hfOF_o$\"3!******\\;;8n#FK7$$\"3/+++rK[_8 !#@$\"3&*******)3.#HGFK7$$\"3Y+++,Yc'3'F^q$\"3%******z(\\i()HFK7$$\"3+ +++4ek+Q)oMFK7$$\"35+++CU2o;!#>$\"36+++Jo&>j$FK7$$\"3 .+++QX;?JFir$\"3y******)*[u&z$FK7$$\"3m*****R#GF*\\&Fir$\"3')*****pHOv &RFK7$$\"3S+++)4gt'*)Fir$\"3y*****pGx<6%FK7$$\"3%*********oiC8F-$\"3e* *****HM$=D%FK7$$\"3/+++MJdn+]jd%FK7$$\"3?+++#R@eu #F-$\"3'******\\U]9m%FK7$$\"3'******H+P0#HF-$\"3.+++7;+RZFK7$$\"3-+++l `8EIF-$\"3A+++3'=.\"[FK7$$\"3E+++s_fpIF-$\"3')*****4Zei([FK7$$\"3%)*** **HIQw0$F-$\"39+++'*3RP\\FK7$$\"3!)******GeT'*HF-$\"3;+++ou5%*\\FK7$$ \"3w*****>5V9*GF-$\"3=+++()))oY]FK7$$\"39+++w-iZFF-$\"3=+++vdM&4&FK7$$ \"3')*****4x!GpDF-$\"34+++gKCS^FK7$$\"3!******HiX-O#F-$\"39+++Mc^\"=&F K7$$\"3,+++Rl(Q7#F-$\"3w*****f#\\F>_FK7$$\"3#*******o`7j=F-$\"3O+++ydh `_FK7$$\"31+++z#z0e\"F-$\"3k*****43=YG&FK7$$\"3')*****zN,&y7F-$\"3h*** **>D[BJ&FK7$$\"3Q+++M^m)e*Fir$\"3o******)pfoL&FK7$$\"3')*******yCRB'Fi r$\"3I+++dF>e`FK7$$\"3*)********ylNFFir$\"3G+++!GujP&FK7$$!3R+++lOvL*) Fiq$\"3o*****\\#pT\"R&FK7$$!3D+++(ypBk%Fir$\"34+++c\"=LS&FK7$$!3R+++Bv Q-&)Fir$\"3p******=*e?T&FK7$$!3/+++IVjY7F-$\"3!)*****f!>gaFK7$$!3))*****zBb(o?F-$\"3Q+++Z*R)=aFK7$$!3/ +++2)=T\\#F-$\"3<+++2bM9aFK7$$!3*)*****>3X#HHF-$\"3$******R&eE1aFK7$$! 35+++=jxuLF-$\"3A+++\"3BWR&FK7$$!3t*****4#\\*=$QF-$\"39+++GVfy`FK7$$!3 2+++,@b-VF-$\"35+++BP]e`FK7$$!3%)*****HLM)*y%F-$\"3!)*****R,:QL&FK7$$! 3w*******>e&)H&F-$\"3m*****f\")HTI&FK7$$!3\\*****>JSi$eF-$\"3k*****f80 !p_FK7$$!3F+++!=TYT'F-$\"3i******>m-G_FK7$$!3]+++%>j>0(F-$\"3O+++p)G5= &FK7$$!3]*****4s`Zx(F-$\"3b*****Ric'G^FK7$$!36+++\\#RYh)F-$\"3M+++o$oM 2&FK7$$!3#******HMy\"*e*F-$\"30+++gM\"4-&FK7$$!3/+++Zy:n5FK$\"3-+++0O_ x\\FK7$$!3/++++]kz6FK$\"3#)*****4F&GY\\FK7$$!3)******40\"e!H\"FK$\"3l* ****HfJb#\\FK7$$!3++++1$>rR\"FK$\"3i*****zc%)=\"\\FK7$$!3#*******fx^) \\\"FK$\"3&)*****RG*Q-\\FK7$$!3++++u<(\\f\"FK$\"3u*****>hj\\*[FK7$$!3- +++jQ)po\"FK$\"33+++%\\I#))[FK7$$!3!*******za5vFK$\"3i*****p \"4ek[FK7$$!34+++JN!4-#FK$\"3')*****pxUT&[FK7$$!31+++:9#z4#FK$\"3B+++( yM?%[FK7$$!3')*****RX\")H<#FK$\"3.+++:_7G[FK7$$!3#******Hu>jC#FK$\"3U+ ++p)\\QzXFK7$$!30+++%os8#HFK $\"3********>m\\VXFK7$$!3'******RGle)HFK$\"3e*****>;Zd]%FK7$$!39+++giB ]IFK$\"3e*****\\t:iY%FK7$$!3&*******Qxc9JFK$\"3s*****\\g(*\\U%FK7$$!3# )*****phF*yJFK$\"3)******H'Q?#Q%FK7$$!3/+++2bOVKFK$\"3=+++A)fzL%FK7$$! 39+++r(3zI$FK$\"3)******4)=S#H%FK7$$!3!)*****HFfDP$FK$\"3%)*****f#GnXU FK7$$!3()*****R`\"HPMFK$\"3u*****>\"e\"z>%FK7$$!36+++MG0-NFK$\"3++++Aw E\\TFK7$$!3%)*****4^lnc$FK$\"3=+++**>&)*4%FK7$$!3.+++C8LJOFK$\"3S+++QR x\\SFK7$$!3++++ftj&p$FK$\"37+++qc6**RFK7$$!3'******Hoi&fPFK$\"31+++IZ$ z%RFK7$$!37+++?\\)H#QFK$\"3%******4;ki*QFK7$$!3%******>.'y&)QFK$\"3=++ +$\\9T%QFK7$$!3-+++co&y%RFK$\"31+++,qZ\"z$FK7$$!3w*****>B+\"4SFK$\"38+ ++IwKQPFK7$$!3i******oFVpSFK$\"33+++(3JYo$FK7$$!3,+++wayGTFK$\"3()**** *fpW.j$FK7$$!3!*******HN5(=%FK$\"3=+++(z@ad$FK7$$!38+++_aMWUFK$\"38+++ %e9)>NFK7$$!3e*****\\%>[+VFK$\"39+++sjZjMFK7$$!3C+++[X\\bVFK$\"3#)**** **[LO1MFK7$$!3>+++`UP4WFK$\"3'******z*eV[LFK7$$!3q*****4:?@Y%FK$\"3y** ***Hdf'*G$FK7$$!3))*****p7QP^%FK$\"3?+++5e+IKFK7$$!3q*****RkRUc%FK$\"3 ')*****>W_%pJFK7$$!3W+++s0k8YFK$\"3-+++pS)z5$FK7$$!37+++q+'>m%FK$\"3%* ******)H#fXIFK7$$!3P+++([>#4ZFK$\"3)******f*eF#)HFK7$$!3w*****HFTav%FK $\"3@+++33/=HFK7$$!3)******R)yk+[FK$\"33+++(3+H&GFK7$$!30+++\"pg[%[FK$ \"39+++_O(oy#FK7$$!3Q+++.*)4))[FK$\"3=+++-z)*>FFK7$$!3,+++N&y.$\\FK$\" 35+++O^F_EFK7$$!3)******>Z6<(\\FK$\"3-+++(ysPe#FK7$$!3g*****>m/@,&FK$ \"35+++`C_9DFK7$$!37+++:&f:0&FK$\"3))*****RyoXW#FK7$$!3M+++I:2!4&FK$\" 3A+++G#eRP#FK7$$!3;+++2-jF^FK$\"3=+++@xt-BFK7$$!3G+++\\#>U;&FK$\"39+++ %R`4B#FK7$$!3M+++Vq\")*>&FK$\"3=+++c$\\'e@FK7$$!3:+++%R(RM_FK$\"3!)*** ***Qm'e3#FK7$$!3I+++d/$zE&FK$\"3$)*****pNUE,#FK7$$!3l*******z$Q+`FK$\" 34+++9\"4!R>FK7$$!38+++DNsJ`FK$\"3%******\\n%*\\'=FK7$$!3!)*****pQ:>O& FK$\"3'******4!>i!z\"FK7$$!3K+++Re#4R&FK$\"3/+++0*3fr\"FK7$$!30+++pHs= aFK$\"33+++=%p3k\"FK7$$!3!)******)Hx_W&FK$\"3%********G8bc\"FK7$$!3i** ***HUi0Z&FK$\"31+++4r%)*[\"FK7$$!3o******)[bX\\&FK$\"36+++')[(QT\"FK7$ $!3(*******GvB5E\"FK 7$$!3i******)38'ebFK$\"3/+++)QQT=\"FK7$$!3!******f;*GxbFK$\"3'******4[ bp5\"FK7$$!3a*****f7>Yf&FK$\"3%******pos%H5FK7$$!3J+++#*Qg5cFK$\"3g*** **z=Jp^*F-7$$!3A+++MxCDcFK$\"3()*****ft=it)F-7$$!3&******H'ybQcFK$\"33 +++;]m_zF-7$$!31+++ZRa]cFK$\"3/+++.SPmrF-7$$!3c*****pc<7m&FK$\"3o***** HF\"[xjF-7$$!3?+++:;fqcFK$\"3;+++kh:'e&F-7$$!3e*****>lz'ycFK$\"3++++\\ Gg#z%F-7$$!3%)******R_\\&o&FK$\"3w*****4>gq*RF-7$$!3g*****fF^5p&FK$\"3 !*******)G+)*>$F-7$$!3?+++[$f`p&FK$\"35+++oc7,CF-7$$!3E+++a\"H%)p&FK$ \"3$******\\9l8g\"F-7$$!3h*****>+o-q&FK$\"3.+++\"y%o3!)Fir7$$!3W+++Q/) 3q&FKF(7$F`]n$!3.+++\"y%o3!)Fir7$F[]n$!3$******\\9l8g\"F-7$Ff\\n$!35++ +oc7,CF-7$Fa\\n$!3!*******)G+)*>$F-7$F\\\\n$!3w*****4>gq*RF-7$Fg[n$!3+ +++\\Gg#z%F-7$Fb[n$!3;+++kh:'e&F-7$F][n$!3o*****HF\"[xjF-7$Fhjm$!3/+++ .SPmrF-7$Fcjm$!33+++;]m_zF-7$F^jm$!3()*****ft=it)F-7$Fiim$!3g*****z=Jp ^*F-7$Fdim$!3%******pos%H5FK7$F_im$!3'******4[bp5\"FK7$Fjhm$!3/+++)QQT =\"FK7$Fehm$!37+++Y(>5E\"FK7$F`hm$!3%******fw)fP8FK7$F[hm$!36+++')[(QT \"FK7$Ffgm$!31+++4r%)*[\"FK7$Fagm$!3%********G8bc\"FK7$F\\gm$!33+++=%p 3k\"FK7$Fgfm$!3/+++0*3fr\"FK7$Fbfm$!3'******4!>i!z\"FK7$F]fm$!3%****** \\n%*\\'=FK7$Fhem$!34+++9\"4!R>FK7$Fcem$!3$)*****pNUE,#FK7$F^em$!3!)** ****Qm'e3#FK7$Fidm$!3=+++c$\\'e@FK7$Fddm$!39+++%R`4B#FK7$F_dm$!3=+++@x t-BFK7$Fjcm$!3A+++G#eRP#FK7$Fecm$!3))*****RyoXW#FK7$F`cm$!35+++`C_9DFK 7$F[cm$!3-+++(ysPe#FK7$Ffbm$!35+++O^F_EFK7$Fabm$!3=+++-z)*>FFK7$F\\bm$ !39+++_O(oy#FK7$Fgam$!33+++(3+H&GFK7$Fbam$!3@+++33/=HFK7$F]am$!3)***** *f*eF#)HFK7$Fh`m$!3%*******)H#fXIFK7$Fc`m$!3-+++pS)z5$FK7$F^`m$!3')*** **>W_%pJFK7$Fi_m$!3?+++5e+IKFK7$Fd_m$!3y*****Hdf'*G$FK7$F__m$!3'****** z*eV[LFK7$Fj^m$!3#)******[LO1MFK7$Fe^m$!39+++sjZjMFK7$F`^m$!38+++%e9)> NFK7$F[^m$!3=+++(z@ad$FK7$Ff]m$!3()*****fpW.j$FK7$Fa]m$!33+++(3JYo$FK7 $F\\]m$!38+++IwKQPFK7$Fg\\m$!31+++,qZ\"z$FK7$Fb\\m$!3=+++$\\9T%QFK7$F] \\m$!3%******4;ki*QFK7$Fh[m$!31+++IZ$z%RFK7$Fc[m$!37+++qc6**RFK7$F^[m$ !3S+++QRx\\SFK7$Fijl$!3=+++**>&)*4%FK7$Fdjl$!3++++AwE\\TFK7$F_jl$!3u** ***>\"e\"z>%FK7$Fjil$!3%)*****f#GnXUFK7$Feil$!3)******4)=S#H%FK7$F`il$ !3=+++A)fzL%FK7$F[il$!3)******H'Q?#Q%FK7$Ffhl$!3s*****\\g(*\\U%FK7$Fah l$!3e*****\\t:iY%FK7$F\\hl$!3e*****>;Zd]%FK7$Fggl$!3********>m\\VXFK7$ Fbgl$!3m*****>)\\QzXFK7$F]gl$!3e*****pk[Lh%FK7$Fhfl$!3K+++s$Q`k%FK7$Fc fl$!3y*****4)yJvYFK7$F^fl$!3U+++DDE.ZFK7$Fiel$!3$)*****H@e\"HZFK7$Fdel $!3c*****px+Iv%FK7$F_el$!3u******RfzuZFK7$Fjdl$!3W+++T+c%z%FK7$Fedl$!3 U+++phj\\*[FK7$Fhal$!3&)**** *RG*Q-\\FK7$Fcal$!3i*****zc%)=\"\\FK7$F^al$!3l*****HfJb#\\FK7$Fi`l$!3# )*****4F&GY\\FK7$Fd`l$!3-+++0O_x\\FK7$F_`l$!30+++gM\"4-&FK7$Fj_l$!3M++ +o$oM2&FK7$Fe_l$!3b*****Ric'G^FK7$F`_l$!3O+++p)G5=&FK7$F[_l$!3i******> m-G_FK7$Ff^l$!3k*****f80!p_FK7$Fa^l$!3m*****f\")HTI&FK7$F\\^l$!3!)**** *R,:QL&FK7$Fg]l$!35+++BP]e`FK7$Fb]l$!39+++GVfy`FK7$F]]l$!3A+++\"3BWR&F K7$Fh\\l$!3$******R&eE1aFK7$Fc\\l$!3<+++2bM9aFK7$F^\\l$!3Q+++Z*R)=aFK7 $Fi[l$!3l*****HJ*))>aFK7$Fd[l$!3!)*****f!>ge`FK7$Ffy$!3o******)pfoL&FK7$Fay$!3h*****>D[BJ&FK7$F\\y$!3 k*****43=YG&FK7$Fgx$!3O+++ydh`_FK7$Fbx$!3w*****f#\\F>_FK7$F]x$!39+++Mc ^\"=&FK7$Fhw$!34+++gKCS^FK7$Fcw$!3=+++vdM&4&FK7$F^w$!3=+++()))oY]FK7$F iv$!3;+++ou5%*\\FK7$Fdv$!39+++'*3RP\\FK7$F_v$!3')*****4Zei([FK7$Fju$!3 A+++3'=.\"[FK7$Feu$!3.+++7;+RZFK7$F`u$!3'******\\U]9m%FK7$F[u$!3')**** *>+]jd%FK7$Fft$!3-+++zXp\"[%FK7$Fat$!3#)*****4r;YP%FK7$F\\t$!3e******H M$=D%FK7$Fgs$!3y*****pGx<6%FK7$Fbs$!3')*****pHOv&RFK7$F]s$!3y******)*[ u&z$FK7$Fgr$!36+++Jo&>j$FK7$Fbr$!3/+++!>Q)oMFK7$F]r$!31+++Pn92LFK7$Fgq $!3/+++$\\Po9$FK7$Fbq$!3%******z(\\i()HFK7$F\\q$!3&*******)3.#HGFK7$Fg p$!3!******\\;;8n#FK7$Fbp$!33+++=Gx8DFK7$F]p$!33+++1&RkN#FK7$Fho$!3#** ****fFG#*>#FK7$Fco$!3')*****zG%3U?FK7$F]o$!33+++:_(\\)=FK7$Fhn$!3#**** **46$)ys\"FK7$FX$!33+++a')zq:FK7$FS$!3)*******\\tr89FK7$FM$!33+++8sjc7 FK7$$!3u*****\\6&4*p&FH$!3++++bub*4\"FK7$$!3i*****\\w'RO#*FB$!3>+++&)z xC%*F-7$$!3!)*****zl'*G;'F7$!3++++K;)R&yF-7$$\"3-+++yI%y9\"F7$!3!)**** *fI&=$G'F-7$F($!3<+++!)*)Q7ZF-7$F($!3)******Rl#fTJF-7$F($!3++++Fjzq:F- F'-%&COLORG6&%$RGBG$\"##Fa_pF]]p7%Fjap7$$!+(RQs`'F`ap$\"+1&RkN#Fa_pF]]p7%F`bp7$$!+= hU\\vF`ap$\"+=Gx8DFa_pF]]p7%Ffbp7$$!+S+hfOF`ap$\"+lhJrEFa_pF]]p7%F\\cp 7$$\"+rK[_8!#8$\"+*3.#HGFa_pF]]p7%Fbcp7$$\"+,Yc'3'Fecp$\"+y\\i()HFa_pF ]]p7%Ficp7$$\"+4ek+Q)oMFa_pF]]p7%F\\ep7$$\"+CU2o;! #6$\"+Jo&>j$Fa_pF]]p7%Fbep7$$\"+QX;?JFeep$\"+**[u&z$Fa_pF]]p7%Fiep7$$ \"+CGF*\\&Feep$\"+(HOv&RFa_pF]]p7%F_fp7$$\"+)4gt'*)Feep$\"+(Gx<6%Fa_pF ]]p7%Fefp7$$\"++piC8F\\]p$\"+IM$=D%Fa_pF]]p7%F[gp7$$\"+MJdn_Fa_pF]]p7%F_\\q7$$\"+p` 7j=F\\]p$\"+ydh`_Fa_pF]]p7%Fe\\q7$$\"+z#z0e\"F\\]p$\"+\"3=YG&Fa_pF]]p7 %F[]q7$$\"+e8]y7F\\]p$\"+_#[BJ&Fa_pF]]p7%Fa]q7$$\"+M^m)e*Feep$\"+*pfoL &Fa_pF]]p7%Fg]q7$$\"+!zCRB'Feep$\"+dF>e`Fa_pF]]p7%F]^q7$$\"++zlNFFeep$ \"+!GujP&Fa_pF]]p7%Fc^q7$$!+lOvL*)Fbdp$\"+DpT\"R&Fa_pF]]p7%Fi^q7$$!+(y pBk%Feep$\"+c\"=LS&Fa_pF]]p7%F__q7$$!+BvQ-&)Feep$\"+>*e?T&Fa_pF]]p7%Fe _q7$$!+IVjY7F\\]p$\"+1>gaFa _pF]]p7%Fa`q7$$!+Q_vo?F\\]p$\"+Z*R)=aFa_pF]]p7%Fg`q7$$!+2)=T\\#F\\]p$ \"+2bM9aFa_pF]]p7%F]aq7$$!+#3X#HHF\\]p$\"+aeE1aFa_pF]]p7%Fcaq7$$!+=jxu LF\\]p$\"+\"3BWR&Fa_pF]]p7%Fiaq7$$!+@\\*=$QF\\]p$\"+GVfy`Fa_pF]]p7%F_b q7$$!+,@b-VF\\]p$\"+BP]e`Fa_pF]]p7%Febq7$$!+LV$)*y%F\\]p$\"+9]\"QL&Fa_ pF]]p7%F[cq7$$!++#e&)H&F\\]p$\"+;)HTI&Fa_pF]]p7%Facq7$$!+7.COeF\\]p$\" +O^+p_Fa_pF]]p7%Fgcq7$$!+!=TYT'F\\]p$\"+?m-G_Fa_pF]]p7%F]dq7$$!+%>j>0( F\\]p$\"+p)G5=&Fa_pF]]p7%Fcdq7$$!+@PvuxF\\]p$\"+CmlG^Fa_pF]]p7%Fidq7$$ !+\\#RYh)F\\]p$\"+o$oM2&Fa_pF]]p7%F_eq7$$!+V$y\"*e*F\\]p$\"+gM\"4-&Fa_ pF]]p7%Feeq7$$!+Zy:n5Fa_p$\"+0O_x\\Fa_pF]]p7%F[fq7$$!++]kz6Fa_p$\"+r_G Y\\Fa_pF]]p7%Fafq7$$!+^5e!H\"Fa_p$\"+$fJb#\\Fa_pF]]p7%Fgfq7$$!+1$>rR\" Fa_p$\"+oX)=\"\\Fa_pF]]p7%F]gq7$$!+gx^)\\\"Fa_p$\"+%G*Q-\\Fa_pF]]p7%Fc gq7$$!+u<(\\f\"Fa_p$\"+7O'\\*[Fa_pF]]p7%Figq7$$!+jQ)po\"Fa_p$\"+%\\I#) )[Fa_pF]]p7%F_hq7$$!+![0^x\"Fa_p$\"+HwG\")[Fa_pF]]p7%Fehq7$$!+Io%)f=Fa _p$\"+v!QN([Fa_pF]]p7%F[iq7$$!+))GlT>Fa_p$\"+<4ek[Fa_pF]]p7%Faiq7$$!+J N!4-#Fa_p$\"+xF9a[Fa_pF]]p7%Fgiq7$$!+:9#z4#Fa_p$\"+(yM?%[Fa_pF]]p7%F]j q7$$!+a9)H<#Fa_p$\"+:_7G[Fa_pF]]p7%Fcjq7$$!+V(>jC#Fa_p$\"+psA gEFa_p$\"+\")yJvYFa_pF]]p7%F]]r7$$!+H9?EFFa_p$\"+s$Q`k%Fa_pF]]p7%Fc]r7 $$!+DSk\"z#Fa_p$\"+Z'[Lh%Fa_pF]]p7%Fi]r7$$!+QdmcGFa_p$\"+#)\\QzXFa_pF] ]p7%F_^r7$$!+%os8#HFa_p$\"+?m\\VXFa_pF]]p7%Fe^r7$$!+%Gle)HFa_p$\"+iru0 XFa_pF]]p7%F[_r7$$!+giB]IFa_p$\"+Nd@mWFa_pF]]p7%Fa_r7$$!+Rxc9JFa_p$\"+ 0w*\\U%Fa_pF]]p7%Fg_r7$$!+%Fa_pF]]p 7%Fear7$$!+MG0-NFa_p$\"+AwE\\TFa_pF]]p7%F[br7$$!+6bwmNFa_p$\"+**>&)*4% Fa_pF]]p7%Fabr7$$!+C8LJOFa_p$\"+QRx\\SFa_pF]]p7%Fgbr7$$!+ftj&p$Fa_p$\" +qc6**RFa_pF]]p7%F]cr7$$!+$oi&fPFa_p$\"+IZ$z%RFa_pF]]p7%Fccr7$$!+?\\)H #QFa_p$\"+hTE'*QFa_pF]]p7%Ficr7$$!+Kgy&)QFa_p$\"+$\\9T%QFa_pF]]p7%F_dr 7$$!+co&y%RFa_p$\"+,qZ\"z$Fa_pF]]p7%Fedr7$$!+K-54SFa_p$\"+IwKQPFa_pF]] p7%F[er7$$!+pFVpSFa_p$\"+(3JYo$Fa_pF]]p7%Faer7$$!+wayGTFa_p$\"+'pW.j$F a_pF]]p7%Fger7$$!+IN5(=%Fa_p$\"+(z@ad$Fa_pF]]p7%F]fr7$$!+_aMWUFa_p$\"+ %e9)>NFa_pF]]p7%Fcfr7$$!+X>[+VFa_p$\"+sjZjMFa_pF]]p7%Fifr7$$!+[X\\bVFa _p$\"+\\LO1MFa_pF]]p7%F_gr7$$!+`UP4WFa_p$\"+)*eV[LFa_pF]]p7%Fegr7$$!+^ ,7iWFa_p$\"+t&f'*G$Fa_pF]]p7%F[hr7$$!+F\"QP^%Fa_p$\"+5e+IKFa_pF]]p7%Fa hr7$$!+W'RUc%Fa_p$\"+UCXpJFa_pF]]p7%Fghr7$$!+s0k8YFa_p$\"+pS)z5$Fa_pF] ]p7%F]ir7$$!+q+'>m%Fa_p$\"+*H#fXIFa_pF]]p7%Fcir7$$!+([>#4ZFa_p$\"+'*eF #)HFa_pF]]p7%Fiir7$$!+t7WbZFa_p$\"+33/=HFa_pF]]p7%F_jr7$$!+%)yk+[Fa_p$ \"+(3+H&GFa_pF]]p7%Fejr7$$!+\"pg[%[Fa_p$\"+_O(oy#Fa_pF]]p7%F[[s7$$!+.* )4))[Fa_p$\"+-z)*>FFa_pF]]p7%Fa[s7$$!+N&y.$\\Fa_p$\"+O^F_EFa_pF]]p7%Fg [s7$$!+s9rr\\Fa_p$\"+(ysPe#Fa_pF]]p7%F]\\s7$$!+iY57]Fa_p$\"+`C_9DFa_pF ]]p7%Fc\\s7$$!+:&f:0&Fa_p$\"+%yoXW#Fa_pF]]p7%Fi\\s7$$!+I:2!4&Fa_p$\"+G #eRP#Fa_pF]]p7%F_]s7$$!+2-jF^Fa_p$\"+@xt-BFa_pF]]p7%Fe]s7$$!+\\#>U;&Fa _p$\"+%R`4B#Fa_pF]]p7%F[^s7$$!+Vq\")*>&Fa_p$\"+c$\\'e@Fa_pF]]p7%Fa^s7$ $!+%R(RM_Fa_p$\"+Rm'e3#Fa_pF]]p7%Fg^s7$$!+d/$zE&Fa_p$\"+dBk7?Fa_pF]]p7 %F]_s7$$!++QQ+`Fa_p$\"+9\"4!R>Fa_pF]]p7%Fc_s7$$!+DNsJ`Fa_p$\"+vY*\\'=F a_pF]]p7%Fi_s7$$!+(Q:>O&Fa_p$\"+,>i!z\"Fa_pF]]p7%F_`s7$$!+Re#4R&Fa_p$ \"+0*3fr\"Fa_pF]]p7%Fe`s7$$!+pHs=aFa_p$\"+=%p3k\"Fa_pF]]p7%F[as7$$!+*H x_W&Fa_p$\"+!H8bc\"Fa_pF]]p7%Faas7$$!+BCcqaFa_p$\"+4r%)*[\"Fa_pF]]p7%F gas7$$!+*[bX\\&Fa_p$\"+')[(QT\"Fa_pF]]p7%F]bs7$$!+HvB5E\"Fa_pF]]p7%Fibs7$$!+*38'ebFa_p$\" +)QQT=\"Fa_pF]]p7%F_cs7$$!+m\"*GxbFa_p$\"+\"[bp5\"Fa_pF]]p7%Fecs7$$!+E \">Yf&Fa_p$\"+(os%H5Fa_pF]]p7%F[ds7$$!+#*Qg5cFa_p$\"+)=Jp^*F\\]pF]]p7% Fads7$$!+MxCDcFa_p$\"+O(=it)F\\]pF]]p7%Fgds7$$!+jybQcFa_p$\"+;]m_zF\\] pF]]p7%F]es7$$!+ZRa]cFa_p$\"+.SPmrF\\]pF]]p7%Fces7$$!+nv@hcFa_p$\"+t7[ xjF\\]pF]]p7%Fies7$$!+:;fqcFa_p$\"+kh:'e&F\\]pF]]p7%F_fs7$$!+_'z'ycFa_ p$\"+\\Gg#z%F\\]pF]]p7%Fefs7$$!+S_\\&o&Fa_p$\"+\">gq*RF\\]pF]]p7%F[gs7 $$!+w70\"p&Fa_p$\"+*G+)*>$F\\]pF]]p7%Fags7$$!+[$f`p&Fa_p$\"+oc7,CF\\]p F]]p7%Fggs7$$!+a\"H%)p&Fa_p$\"+X^O,;F\\]pF]]p7%F]hs7$$!+-!o-q&Fa_p$\"+ \"y%o3!)FeepF]]p7%Fchs7$$!+Q/)3q&Fa_pF(F]]p7%Fihs7$Fdhs$!+\"y%o3!)Feep F]]p7%F]is7$F^hs$!+X^O,;F\\]pF]]p7%Fais7$Fhgs$!+oc7,CF\\]pF]]p7%Feis7$ Fbgs$!+*G+)*>$F\\]pF]]p7%Fiis7$F\\gs$!+\">gq*RF\\]pF]]p7%F]js7$Fffs$!+ \\Gg#z%F\\]pF]]p7%Fajs7$F`fs$!+kh:'e&F\\]pF]]p7%Fejs7$Fjes$!+t7[xjF\\] pF]]p7%Fijs7$Fdes$!+.SPmrF\\]pF]]p7%F][t7$F^es$!+;]m_zF\\]pF]]p7%Fa[t7 $Fhds$!+O(=it)F\\]pF]]p7%Fe[t7$Fbds$!+)=Jp^*F\\]pF]]p7%Fi[t7$F\\ds$!+( os%H5Fa_pF]]p7%F]\\t7$Ffcs$!+\"[bp5\"Fa_pF]]p7%Fa\\t7$F`cs$!+)QQT=\"Fa _pF]]p7%Fe\\t7$Fjbs$!+Y(>5E\"Fa_pF]]p7%Fi\\t7$Fdbs$!+m()fP8Fa_pF]]p7%F ]]t7$F^bs$!+')[(QT\"Fa_pF]]p7%Fa]t7$Fhas$!+4r%)*[\"Fa_pF]]p7%Fe]t7$Fba s$!+!H8bc\"Fa_pF]]p7%Fi]t7$F\\as$!+=%p3k\"Fa_pF]]p7%F]^t7$Ff`s$!+0*3fr \"Fa_pF]]p7%Fa^t7$F``s$!+,>i!z\"Fa_pF]]p7%Fe^t7$Fj_s$!+vY*\\'=Fa_pF]]p 7%Fi^t7$Fd_s$!+9\"4!R>Fa_pF]]p7%F]_t7$F^_s$!+dBk7?Fa_pF]]p7%Fa_t7$Fh^s $!+Rm'e3#Fa_pF]]p7%Fe_t7$Fb^s$!+c$\\'e@Fa_pF]]p7%Fi_t7$F\\^s$!+%R`4B#F a_pF]]p7%F]`t7$Ff]s$!+@xt-BFa_pF]]p7%Fa`t7$F`]s$!+G#eRP#Fa_pF]]p7%Fe`t 7$Fj\\s$!+%yoXW#Fa_pF]]p7%Fi`t7$Fd\\s$!+`C_9DFa_pF]]p7%F]at7$F^\\s$!+( ysPe#Fa_pF]]p7%Faat7$Fh[s$!+O^F_EFa_pF]]p7%Feat7$Fb[s$!+-z)*>FFa_pF]]p 7%Fiat7$F\\[s$!+_O(oy#Fa_pF]]p7%F]bt7$Ffjr$!+(3+H&GFa_pF]]p7%Fabt7$F`j r$!+33/=HFa_pF]]p7%Febt7$Fjir$!+'*eF#)HFa_pF]]p7%Fibt7$Fdir$!+*H#fXIFa _pF]]p7%F]ct7$F^ir$!+pS)z5$Fa_pF]]p7%Fact7$Fhhr$!+UCXpJFa_pF]]p7%Fect7 $Fbhr$!+5e+IKFa_pF]]p7%Fict7$F\\hr$!+t&f'*G$Fa_pF]]p7%F]dt7$Ffgr$!+)*e V[LFa_pF]]p7%Fadt7$F`gr$!+\\LO1MFa_pF]]p7%Fedt7$Fjfr$!+sjZjMFa_pF]]p7% Fidt7$Fdfr$!+%e9)>NFa_pF]]p7%F]et7$F^fr$!+(z@ad$Fa_pF]]p7%Faet7$Fher$! +'pW.j$Fa_pF]]p7%Feet7$Fber$!+(3JYo$Fa_pF]]p7%Fiet7$F\\er$!+IwKQPFa_pF ]]p7%F]ft7$Ffdr$!+,qZ\"z$Fa_pF]]p7%Faft7$F`dr$!+$\\9T%QFa_pF]]p7%Feft7 $Fjcr$!+hTE'*QFa_pF]]p7%Fift7$Fdcr$!+IZ$z%RFa_pF]]p7%F]gt7$F^cr$!+qc6* *RFa_pF]]p7%Fagt7$Fhbr$!+QRx\\SFa_pF]]p7%Fegt7$Fbbr$!+**>&)*4%Fa_pF]]p 7%Figt7$F\\br$!+AwE\\TFa_pF]]p7%F]ht7$Ffar$!+7e\"z>%Fa_pF]]p7%Faht7$F` ar$!+EGnXUFa_pF]]p7%Feht7$Fj`r$!+\")=S#H%Fa_pF]]p7%Fiht7$Fd`r$!+A)fzL% Fa_pF]]p7%F]it7$F^`r$!+jQ?#Q%Fa_pF]]p7%Fait7$Fh_r$!+0w*\\U%Fa_pF]]p7%F eit7$Fb_r$!+Nd@mWFa_pF]]p7%Fiit7$F\\_r$!+iru0XFa_pF]]p7%F]jt7$Ff^r$!+? m\\VXFa_pF]]p7%Fajt7$F`^r$!+#)\\QzXFa_pF]]p7%Fejt7$Fj]r$!+Z'[Lh%Fa_pF] ]p7%Fijt7$Fd]r$!+s$Q`k%Fa_pF]]p7%F][u7$F^]r$!+\")yJvYFa_pF]]p7%Fa[u7$F h\\r$!+DDE.ZFa_pF]]p7%Fe[u7$Fb\\r$!+8#e\"HZFa_pF]]p7%Fi[u7$F\\\\r$!+x2 +`ZFa_pF]]p7%F]\\u7$Ff[r$!+SfzuZFa_pF]]p7%Fa\\u7$F`[r$!+T+c%z%Fa_pF]]p 7%Fe\\u7$Fjjq$!+paF a_pF]]p7%Ficu7$F\\`q$!+1>g*e?T&Fa_pF]]p7%Fad u7$F`_q$!+c\"=LS&Fa_pF]]p7%Fedu7$Fj^q$!+DpT\"R&Fa_pF]]p7%Fidu7$Fd^q$!+ !GujP&Fa_pF]]p7%F]eu7$F^^q$!+dF>e`Fa_pF]]p7%Faeu7$Fh]q$!+*pfoL&Fa_pF]] p7%Feeu7$Fb]q$!+_#[BJ&Fa_pF]]p7%Fieu7$F\\]q$!+\"3=YG&Fa_pF]]p7%F]fu7$F f\\q$!+ydh`_Fa_pF]]p7%Fafu7$F`\\q$!+E\\F>_Fa_pF]]p7%Fefu7$Fj[q$!+Mc^\" =&Fa_pF]]p7%Fifu7$Fd[q$!+gKCS^Fa_pF]]p7%F]gu7$F^[q$!+vdM&4&Fa_pF]]p7%F agu7$Fhjp$!+()))oY]Fa_pF]]p7%Fegu7$Fbjp$!+ou5%*\\Fa_pF]]p7%Figu7$F\\jp $!+'*3RP\\Fa_pF]]p7%F]hu7$Ffip$!+r%ei([Fa_pF]]p7%Fahu7$F`ip$!+3'=.\"[F a_pF]]p7%Fehu7$Fjhp$!+7;+RZFa_pF]]p7%Fihu7$Fdhp$!+D/XhYFa_pF]]p7%F]iu7 $F^hp$!+-+NwXFa_pF]]p7%Faiu7$Fhgp$!+zXp\"[%Fa_pF]]p7%Feiu7$Fbgp$!+6nhu VFa_pF]]p7%Fiiu7$F\\gp$!+IM$=D%Fa_pF]]p7%F]ju7$Fffp$!+(Gx<6%Fa_pF]]p7% Faju7$F`fp$!+(HOv&RFa_pF]]p7%Feju7$Fjep$!+**[u&z$Fa_pF]]p7%Fiju7$Fcep$ !+Jo&>j$Fa_pF]]p7%F][v7$F]ep$!+!>Q)oMFa_pF]]p7%Fa[v7$Fgdp$!+Pn92LFa_pF ]]p7%Fe[v7$F`dp$!+$\\Po9$Fa_pF]]p7%Fi[v7$Fjcp$!+y\\i()HFa_pF]]p7%F]\\v 7$Fccp$!+*3.#HGFa_pF]]p7%Fa\\v7$F]cp$!+lhJrEFa_pF]]p7%Fe\\v7$Fgbp$!+=G x8DFa_pF]]p7%Fi\\v7$Fabp$!+1&RkN#Fa_pF]]p7%F]]v7$F[bp$!+w#G#*>#Fa_pF]] p7%Fa]v7$Feap$!+)G%3U?Fa_pF]]p7%Fe]v7$F^ap$!+:_(\\)=Fa_pF]]p7%Fi]v7$Fh `p$!+6J)ys\"Fa_pF]]p7%F]^v7$Fa`p$!+a')zq:Fa_pF]]p7%Fa^v7$F[`p$!+]tr89F a_pF]]p7%Fe^v7$Fd_p$!+8sjc7Fa_pF]]p7%Fi^v7$$!+:^4*p&FK$!+bub*4\"Fa_pF] ]p7%F]_v7$$!+lnRO#*F-$!+&)zxC%*F\\]pF]]p7%Fc_v7$$!+em*G;'Fir$!+K;)R&yF \\]pF]]p7%Fi_v7$$\"+yI%y9\"Fir$!+1`=$G'F\\]pF]]p7%F_`v7$F($!+!)*)Q7ZF \\]pF]]p7%Fe`v7$F($!+aEfTJF\\]pF]]p7%Fi`v7$F($!+Fjzq:F\\]pF]]p7%F]avF' F]]p-F]\\p6&F_\\p$\"#QFb\\pF($\"#vFb\\p-%&STYLEG6#%,PATCHNOGRIDG-F$6%7 $7$$!3Q++++++!p'FKF(7$$\"3!**************Q\"FKF(-%'COLOURG6&F_\\pF)F)F )-%*LINESTYLEG6#\"\"$-F$6%7$7$F($!3o*************Q'FK7$F($\"3o******** *****Q'FKFdbvFgbv-%(SCALINGG6#%,CONSTRAINEDG-%+AXESLABELSG6%%&Re(z)G%& Im(z)G-%%FONTG6#%(DEFAULTG-F^dv6$%*HELVETICAG\"\"*-%*AXESSTYLEG6#%$BOX G-%%VIEWG6$;$!$p'Fb\\p$\"$R\"Fb\\p;$!$R'Fb\\p$\"$R'Fb\\p" 1 2 0 1 10 0 2 9 1 2 1 1.000000 45.000000 45.000000 0 0 "Curve 1" "Curve 2" "Curv e 3" "Curve 4" }}}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 75 "The stability interval for the embedded order 7 scheme is (approximately) " }{XPPEDIT 18 0 "[-5.7009, 0];" "6#7$,$-%&FloatG6$ \"&4q&!\"%!\"\"\"\"!" }{TEXT -1 2 ". " }}{PARA 0 "" 0 "" {TEXT -1 0 " " }}{PARA 0 "" 0 "" {TEXT -1 79 "The stability regions for the two sch emes can be displayed in the same picture." }}{EXCHG {PARA 0 "> " 0 " " {MPLTEXT 1 0 154 "plots[display]([p_1,p_2,p||(1..3)],view=[-6.49..1. 19,-6.39..6.39],font=[HELVETICA,9],\n labels=[`Re(z)`,`Im(z) `],axes=boxed,scaling=constrained);" }}{PARA 13 "" 1 "" {GLPLOT2D 362 580 580 {PLOTDATA 2 "6--%'CURVESG6$7]_l7$$\"\"!F)F(7$F($\"3++++Fjzq:!# =7$F($\"3)******Rl#fTJF-7$F($\"3<+++!)*)Q7ZF-7$$\"3%******\\5Vy9\"!#F$ \"3!)*****fI&=$G'F-7$$!3')*****Hg'*G;'F7$\"3++++K;)R&yF-7$$!3y******>v RO#*!#E$\"3>+++&)zxC%*F-7$$!3)******47&4*p&!#D$\"3++++bub*4\"!#<7$$!3y *****>c)\\UC!#C$\"33+++8sjc7FK7$$!3A+++orO'H)FO$\"3)*******\\tr89FK7$$ !31+++1$f9P#!#B$\"33+++a')zq:FK7$$!3%*******\\z4$*eFZ$\"3#******46$)ys \"FK7$$!3++++svu&H\"!#A$\"33+++:_(\\)=FK7$$!3))*****4?xp`#F_o$\"3')*** **zG%3U?FK7$$!3o*****fp:[R%F_o$\"3#******fFG#*>#FK7$$!3y*****pRQs`'F_o $\"33+++1&RkN#FK7$$!3o*****z6E%\\vF_o$\"33+++=Gx8DFK7$$!3++++S+hfOF_o$ \"3!******\\;;8n#FK7$$\"3/+++rK[_8!#@$\"3&*******)3.#HGFK7$$\"3Y+++,Yc '3'F^q$\"3%******z(\\i()HFK7$$\"3++++4ek+Q)oMFK7$$\"3 5+++CU2o;!#>$\"36+++Jo&>j$FK7$$\"3.+++QX;?JFir$\"3y******)*[u&z$FK7$$ \"3m*****R#GF*\\&Fir$\"3')*****pHOv&RFK7$$\"3S+++)4gt'*)Fir$\"3y*****p Gx<6%FK7$$\"3%*********oiC8F-$\"3e******HM$=D%FK7$$\"3/+++MJdn+]jd%FK7$$\"3?+++#R@eu#F-$\"3'******\\U]9m%FK7$$\"3'***** *H+P0#HF-$\"3.+++7;+RZFK7$$\"3-+++l`8EIF-$\"3A+++3'=.\"[FK7$$\"3E+++s_ fpIF-$\"3')*****4Zei([FK7$$\"3%)*****HIQw0$F-$\"39+++'*3RP\\FK7$$\"3!) ******GeT'*HF-$\"3;+++ou5%*\\FK7$$\"3w*****>5V9*GF-$\"3=+++()))oY]FK7$ $\"39+++w-iZFF-$\"3=+++vdM&4&FK7$$\"3')*****4x!GpDF-$\"34+++gKCS^FK7$$ \"3!******HiX-O#F-$\"39+++Mc^\"=&FK7$$\"3,+++Rl(Q7#F-$\"3w*****f#\\F>_ FK7$$\"3#*******o`7j=F-$\"3O+++ydh`_FK7$$\"31+++z#z0e\"F-$\"3k*****43= YG&FK7$$\"3')*****zN,&y7F-$\"3h*****>D[BJ&FK7$$\"3Q+++M^m)e*Fir$\"3o** ****)pfoL&FK7$$\"3')*******yCRB'Fir$\"3I+++dF>e`FK7$$\"3*)********ylNF Fir$\"3G+++!GujP&FK7$$!3R+++lOvL*)Fiq$\"3o*****\\#pT\"R&FK7$$!3D+++(yp Bk%Fir$\"34+++c\"=LS&FK7$$!3R+++BvQ-&)Fir$\"3p******=*e?T&FK7$$!3/+++I VjY7F-$\"3!)*****f!>gaFK7$$!3))* ****zBb(o?F-$\"3Q+++Z*R)=aFK7$$!3/+++2)=T\\#F-$\"3<+++2bM9aFK7$$!3*)** ***>3X#HHF-$\"3$******R&eE1aFK7$$!35+++=jxuLF-$\"3A+++\"3BWR&FK7$$!3t* ****4#\\*=$QF-$\"39+++GVfy`FK7$$!32+++,@b-VF-$\"35+++BP]e`FK7$$!3%)*** **HLM)*y%F-$\"3!)*****R,:QL&FK7$$!3w*******>e&)H&F-$\"3m*****f\")HTI&F K7$$!3\\*****>JSi$eF-$\"3k*****f80!p_FK7$$!3F+++!=TYT'F-$\"3i******>m- G_FK7$$!3]+++%>j>0(F-$\"3O+++p)G5=&FK7$$!3]*****4s`Zx(F-$\"3b*****Ric' G^FK7$$!36+++\\#RYh)F-$\"3M+++o$oM2&FK7$$!3#******HMy\"*e*F-$\"30+++gM \"4-&FK7$$!3/+++Zy:n5FK$\"3-+++0O_x\\FK7$$!3/++++]kz6FK$\"3#)*****4F&G Y\\FK7$$!3)******40\"e!H\"FK$\"3l*****HfJb#\\FK7$$!3++++1$>rR\"FK$\"3i *****zc%)=\"\\FK7$$!3#*******fx^)\\\"FK$\"3&)*****RG*Q-\\FK7$$!3++++u< (\\f\"FK$\"3u*****>hj\\*[FK7$$!3-+++jQ)po\"FK$\"33+++%\\I#))[FK7$$!3!* ******za5vFK$\"3i*****p\"4ek[FK7$$!34+++JN!4-#FK$\"3')*****p xUT&[FK7$$!31+++:9#z4#FK$\"3B+++(yM?%[FK7$$!3')*****RX\")H<#FK$\"3.+++ :_7G[FK7$$!3#******Hu>jC#FK$\"3U+++p)\\QzXFK7$$!30+++%os8#HFK$\"3********>m\\VXFK7$$!3'******RGle)H FK$\"3e*****>;Zd]%FK7$$!39+++giB]IFK$\"3e*****\\t:iY%FK7$$!3&*******Qx c9JFK$\"3s*****\\g(*\\U%FK7$$!3#)*****phF*yJFK$\"3)******H'Q?#Q%FK7$$! 3/+++2bOVKFK$\"3=+++A)fzL%FK7$$!39+++r(3zI$FK$\"3)******4)=S#H%FK7$$!3 !)*****HFfDP$FK$\"3%)*****f#GnXUFK7$$!3()*****R`\"HPMFK$\"3u*****>\"e \"z>%FK7$$!36+++MG0-NFK$\"3++++AwE\\TFK7$$!3%)*****4^lnc$FK$\"3=+++**> &)*4%FK7$$!3.+++C8LJOFK$\"3S+++QRx\\SFK7$$!3++++ftj&p$FK$\"37+++qc6**R FK7$$!3'******Hoi&fPFK$\"31+++IZ$z%RFK7$$!37+++?\\)H#QFK$\"3%******4;k i*QFK7$$!3%******>.'y&)QFK$\"3=+++$\\9T%QFK7$$!3-+++co&y%RFK$\"31+++,q Z\"z$FK7$$!3w*****>B+\"4SFK$\"38+++IwKQPFK7$$!3i******oFVpSFK$\"33+++( 3JYo$FK7$$!3,+++wayGTFK$\"3()*****fpW.j$FK7$$!3!*******HN5(=%FK$\"3=++ +(z@ad$FK7$$!38+++_aMWUFK$\"38+++%e9)>NFK7$$!3e*****\\%>[+VFK$\"39+++s jZjMFK7$$!3C+++[X\\bVFK$\"3#)******[LO1MFK7$$!3>+++`UP4WFK$\"3'******z *eV[LFK7$$!3q*****4:?@Y%FK$\"3y*****Hdf'*G$FK7$$!3))*****p7QP^%FK$\"3? +++5e+IKFK7$$!3q*****RkRUc%FK$\"3')*****>W_%pJFK7$$!3W+++s0k8YFK$\"3-+ ++pS)z5$FK7$$!37+++q+'>m%FK$\"3%*******)H#fXIFK7$$!3P+++([>#4ZFK$\"3)* *****f*eF#)HFK7$$!3w*****HFTav%FK$\"3@+++33/=HFK7$$!3)******R)yk+[FK$ \"33+++(3+H&GFK7$$!30+++\"pg[%[FK$\"39+++_O(oy#FK7$$!3Q+++.*)4))[FK$\" 3=+++-z)*>FFK7$$!3,+++N&y.$\\FK$\"35+++O^F_EFK7$$!3)******>Z6<(\\FK$\" 3-+++(ysPe#FK7$$!3g*****>m/@,&FK$\"35+++`C_9DFK7$$!37+++:&f:0&FK$\"3)) *****RyoXW#FK7$$!3M+++I:2!4&FK$\"3A+++G#eRP#FK7$$!3;+++2-jF^FK$\"3=+++ @xt-BFK7$$!3G+++\\#>U;&FK$\"39+++%R`4B#FK7$$!3M+++Vq\")*>&FK$\"3=+++c$ \\'e@FK7$$!3:+++%R(RM_FK$\"3!)******Qm'e3#FK7$$!3I+++d/$zE&FK$\"3$)*** **pNUE,#FK7$$!3l*******z$Q+`FK$\"34+++9\"4!R>FK7$$!38+++DNsJ`FK$\"3%** ****\\n%*\\'=FK7$$!3!)*****pQ:>O&FK$\"3'******4!>i!z\"FK7$$!3K+++Re#4R &FK$\"3/+++0*3fr\"FK7$$!30+++pHs=aFK$\"33+++=%p3k\"FK7$$!3!)******)Hx_ W&FK$\"3%********G8bc\"FK7$$!3i*****HUi0Z&FK$\"31+++4r%)*[\"FK7$$!3o** ****)[bX\\&FK$\"36+++')[(QT\"FK7$$!3(*******GvB5E\"FK7$$!3i******)38'ebFK$\"3/+++)QQT=\"FK 7$$!3!******f;*GxbFK$\"3'******4[bp5\"FK7$$!3a*****f7>Yf&FK$\"3%****** pos%H5FK7$$!3J+++#*Qg5cFK$\"3g*****z=Jp^*F-7$$!3A+++MxCDcFK$\"3()***** ft=it)F-7$$!3&******H'ybQcFK$\"33+++;]m_zF-7$$!31+++ZRa]cFK$\"3/+++.SP mrF-7$$!3c*****pc<7m&FK$\"3o*****HF\"[xjF-7$$!3?+++:;fqcFK$\"3;+++kh:' e&F-7$$!3e*****>lz'ycFK$\"3++++\\Gg#z%F-7$$!3%)******R_\\&o&FK$\"3w*** **4>gq*RF-7$$!3g*****fF^5p&FK$\"3!*******)G+)*>$F-7$$!3?+++[$f`p&FK$\" 35+++oc7,CF-7$$!3E+++a\"H%)p&FK$\"3$******\\9l8g\"F-7$$!3h*****>+o-q&F K$\"3.+++\"y%o3!)Fir7$$!3W+++Q/)3q&FKF(7$F`]n$!3.+++\"y%o3!)Fir7$F[]n$ !3$******\\9l8g\"F-7$Ff\\n$!35+++oc7,CF-7$Fa\\n$!3!*******)G+)*>$F-7$F \\\\n$!3w*****4>gq*RF-7$Fg[n$!3++++\\Gg#z%F-7$Fb[n$!3;+++kh:'e&F-7$F][ n$!3o*****HF\"[xjF-7$Fhjm$!3/+++.SPmrF-7$Fcjm$!33+++;]m_zF-7$F^jm$!3() *****ft=it)F-7$Fiim$!3g*****z=Jp^*F-7$Fdim$!3%******pos%H5FK7$F_im$!3' ******4[bp5\"FK7$Fjhm$!3/+++)QQT=\"FK7$Fehm$!37+++Y(>5E\"FK7$F`hm$!3%* *****fw)fP8FK7$F[hm$!36+++')[(QT\"FK7$Ffgm$!31+++4r%)*[\"FK7$Fagm$!3%* *******G8bc\"FK7$F\\gm$!33+++=%p3k\"FK7$Fgfm$!3/+++0*3fr\"FK7$Fbfm$!3' ******4!>i!z\"FK7$F]fm$!3%******\\n%*\\'=FK7$Fhem$!34+++9\"4!R>FK7$Fce m$!3$)*****pNUE,#FK7$F^em$!3!)******Qm'e3#FK7$Fidm$!3=+++c$\\'e@FK7$Fd dm$!39+++%R`4B#FK7$F_dm$!3=+++@xt-BFK7$Fjcm$!3A+++G#eRP#FK7$Fecm$!3))* ****RyoXW#FK7$F`cm$!35+++`C_9DFK7$F[cm$!3-+++(ysPe#FK7$Ffbm$!35+++O^F_ EFK7$Fabm$!3=+++-z)*>FFK7$F\\bm$!39+++_O(oy#FK7$Fgam$!33+++(3+H&GFK7$F bam$!3@+++33/=HFK7$F]am$!3)******f*eF#)HFK7$Fh`m$!3%*******)H#fXIFK7$F c`m$!3-+++pS)z5$FK7$F^`m$!3')*****>W_%pJFK7$Fi_m$!3?+++5e+IKFK7$Fd_m$! 3y*****Hdf'*G$FK7$F__m$!3'******z*eV[LFK7$Fj^m$!3#)******[LO1MFK7$Fe^m $!39+++sjZjMFK7$F`^m$!38+++%e9)>NFK7$F[^m$!3=+++(z@ad$FK7$Ff]m$!3()*** **fpW.j$FK7$Fa]m$!33+++(3JYo$FK7$F\\]m$!38+++IwKQPFK7$Fg\\m$!31+++,qZ \"z$FK7$Fb\\m$!3=+++$\\9T%QFK7$F]\\m$!3%******4;ki*QFK7$Fh[m$!31+++IZ$ z%RFK7$Fc[m$!37+++qc6**RFK7$F^[m$!3S+++QRx\\SFK7$Fijl$!3=+++**>&)*4%FK 7$Fdjl$!3++++AwE\\TFK7$F_jl$!3u*****>\"e\"z>%FK7$Fjil$!3%)*****f#GnXUF K7$Feil$!3)******4)=S#H%FK7$F`il$!3=+++A)fzL%FK7$F[il$!3)******H'Q?#Q% FK7$Ffhl$!3s*****\\g(*\\U%FK7$Fahl$!3e*****\\t:iY%FK7$F\\hl$!3e*****>; Zd]%FK7$Fggl$!3********>m\\VXFK7$Fbgl$!3m*****>)\\QzXFK7$F]gl$!3e***** pk[Lh%FK7$Fhfl$!3K+++s$Q`k%FK7$Fcfl$!3y*****4)yJvYFK7$F^fl$!3U+++DDE.Z FK7$Fiel$!3$)*****H@e\"HZFK7$Fdel$!3c*****px+Iv%FK7$F_el$!3u******Rfzu ZFK7$Fjdl$!3W+++T+c%z%FK7$Fedl$!3U+++phj\\*[FK7$Fhal$!3&)*****RG*Q-\\FK7$Fcal$!3i*****zc%)=\"\\FK7 $F^al$!3l*****HfJb#\\FK7$Fi`l$!3#)*****4F&GY\\FK7$Fd`l$!3-+++0O_x\\FK7 $F_`l$!30+++gM\"4-&FK7$Fj_l$!3M+++o$oM2&FK7$Fe_l$!3b*****Ric'G^FK7$F`_ l$!3O+++p)G5=&FK7$F[_l$!3i******>m-G_FK7$Ff^l$!3k*****f80!p_FK7$Fa^l$! 3m*****f\")HTI&FK7$F\\^l$!3!)*****R,:QL&FK7$Fg]l$!35+++BP]e`FK7$Fb]l$! 39+++GVfy`FK7$F]]l$!3A+++\"3BWR&FK7$Fh\\l$!3$******R&eE1aFK7$Fc\\l$!3< +++2bM9aFK7$F^\\l$!3Q+++Z*R)=aFK7$Fi[l$!3l*****HJ*))>aFK7$Fd[l$!3!)*** **f!>ge`FK7$Ffy$!3o******)pfoL &FK7$Fay$!3h*****>D[BJ&FK7$F\\y$!3k*****43=YG&FK7$Fgx$!3O+++ydh`_FK7$F bx$!3w*****f#\\F>_FK7$F]x$!39+++Mc^\"=&FK7$Fhw$!34+++gKCS^FK7$Fcw$!3=+ ++vdM&4&FK7$F^w$!3=+++()))oY]FK7$Fiv$!3;+++ou5%*\\FK7$Fdv$!39+++'*3RP \\FK7$F_v$!3')*****4Zei([FK7$Fju$!3A+++3'=.\"[FK7$Feu$!3.+++7;+RZFK7$F `u$!3'******\\U]9m%FK7$F[u$!3')*****>+]jd%FK7$Fft$!3-+++zXp\"[%FK7$Fat $!3#)*****4r;YP%FK7$F\\t$!3e******HM$=D%FK7$Fgs$!3y*****pGx<6%FK7$Fbs$ !3')*****pHOv&RFK7$F]s$!3y******)*[u&z$FK7$Fgr$!36+++Jo&>j$FK7$Fbr$!3/ +++!>Q)oMFK7$F]r$!31+++Pn92LFK7$Fgq$!3/+++$\\Po9$FK7$Fbq$!3%******z(\\ i()HFK7$F\\q$!3&*******)3.#HGFK7$Fgp$!3!******\\;;8n#FK7$Fbp$!33+++=Gx 8DFK7$F]p$!33+++1&RkN#FK7$Fho$!3#******fFG#*>#FK7$Fco$!3')*****zG%3U?F K7$F]o$!33+++:_(\\)=FK7$Fhn$!3#******46$)ys\"FK7$FX$!33+++a')zq:FK7$FS $!3)*******\\tr89FK7$FM$!33+++8sjc7FK7$$!3u*****\\6&4*p&FH$!3++++bub*4 \"FK7$$!3i*****\\w'RO#*FB$!3>+++&)zxC%*F-7$$!3!)*****zl'*G;'F7$!3++++K ;)R&yF-7$$\"3-+++yI%y9\"F7$!3!)*****fI&=$G'F-7$F($!3<+++!)*)Q7ZF-7$F($ !3)******Rl#fTJF-7$F($!3++++Fjzq:F-F'-%&COLORG6&%$RGBG$\"##Fa_pF]]p7%Fjap7$$!+(RQs` 'F`ap$\"+1&RkN#Fa_pF]]p7%F`bp7$$!+=hU\\vF`ap$\"+=Gx8DFa_pF]]p7%Ffbp7$$ !+S+hfOF`ap$\"+lhJrEFa_pF]]p7%F\\cp7$$\"+rK[_8!#8$\"+*3.#HGFa_pF]]p7%F bcp7$$\"+,Yc'3'Fecp$\"+y\\i()HFa_pF]]p7%Ficp7$$\"+4ek+Q)oMFa_pF]]p7%F\\ep7$$\"+CU2o;!#6$\"+Jo&>j$Fa_pF]]p7%Fbep7$$\"+QX ;?JFeep$\"+**[u&z$Fa_pF]]p7%Fiep7$$\"+CGF*\\&Feep$\"+(HOv&RFa_pF]]p7%F _fp7$$\"+)4gt'*)Feep$\"+(Gx<6%Fa_pF]]p7%Fefp7$$\"++piC8F\\]p$\"+IM$=D% Fa_pF]]p7%F[gp7$$\"+MJdn_Fa_pF]]p7%F_\\q7$$\"+p`7j=F\\]p$\"+ydh`_Fa_pF]]p7%Fe\\q7$$\"+ z#z0e\"F\\]p$\"+\"3=YG&Fa_pF]]p7%F[]q7$$\"+e8]y7F\\]p$\"+_#[BJ&Fa_pF]] p7%Fa]q7$$\"+M^m)e*Feep$\"+*pfoL&Fa_pF]]p7%Fg]q7$$\"+!zCRB'Feep$\"+dF> e`Fa_pF]]p7%F]^q7$$\"++zlNFFeep$\"+!GujP&Fa_pF]]p7%Fc^q7$$!+lOvL*)Fbdp $\"+DpT\"R&Fa_pF]]p7%Fi^q7$$!+(ypBk%Feep$\"+c\"=LS&Fa_pF]]p7%F__q7$$!+ BvQ-&)Feep$\"+>*e?T&Fa_pF]]p7%Fe_q7$$!+IVjY7F\\]p$\"+1>gaFa_pF]]p7%Fa`q7$$!+Q_vo?F\\]p$\"+Z*R)=aF a_pF]]p7%Fg`q7$$!+2)=T\\#F\\]p$\"+2bM9aFa_pF]]p7%F]aq7$$!+#3X#HHF\\]p$ \"+aeE1aFa_pF]]p7%Fcaq7$$!+=jxuLF\\]p$\"+\"3BWR&Fa_pF]]p7%Fiaq7$$!+@\\ *=$QF\\]p$\"+GVfy`Fa_pF]]p7%F_bq7$$!+,@b-VF\\]p$\"+BP]e`Fa_pF]]p7%Febq 7$$!+LV$)*y%F\\]p$\"+9]\"QL&Fa_pF]]p7%F[cq7$$!++#e&)H&F\\]p$\"+;)HTI&F a_pF]]p7%Facq7$$!+7.COeF\\]p$\"+O^+p_Fa_pF]]p7%Fgcq7$$!+!=TYT'F\\]p$\" +?m-G_Fa_pF]]p7%F]dq7$$!+%>j>0(F\\]p$\"+p)G5=&Fa_pF]]p7%Fcdq7$$!+@Pvux F\\]p$\"+CmlG^Fa_pF]]p7%Fidq7$$!+\\#RYh)F\\]p$\"+o$oM2&Fa_pF]]p7%F_eq7 $$!+V$y\"*e*F\\]p$\"+gM\"4-&Fa_pF]]p7%Feeq7$$!+Zy:n5Fa_p$\"+0O_x\\Fa_p F]]p7%F[fq7$$!++]kz6Fa_p$\"+r_GY\\Fa_pF]]p7%Fafq7$$!+^5e!H\"Fa_p$\"+$f Jb#\\Fa_pF]]p7%Fgfq7$$!+1$>rR\"Fa_p$\"+oX)=\"\\Fa_pF]]p7%F]gq7$$!+gx^) \\\"Fa_p$\"+%G*Q-\\Fa_pF]]p7%Fcgq7$$!+u<(\\f\"Fa_p$\"+7O'\\*[Fa_pF]]p7 %Figq7$$!+jQ)po\"Fa_p$\"+%\\I#))[Fa_pF]]p7%F_hq7$$!+![0^x\"Fa_p$\"+HwG \")[Fa_pF]]p7%Fehq7$$!+Io%)f=Fa_p$\"+v!QN([Fa_pF]]p7%F[iq7$$!+))GlT>Fa _p$\"+<4ek[Fa_pF]]p7%Faiq7$$!+JN!4-#Fa_p$\"+xF9a[Fa_pF]]p7%Fgiq7$$!+:9 #z4#Fa_p$\"+(yM?%[Fa_pF]]p7%F]jq7$$!+a9)H<#Fa_p$\"+:_7G[Fa_pF]]p7%Fcjq 7$$!+V(>jC#Fa_p$\"+psAgEFa_p$\"+\")yJvYFa_pF]]p7%F]]r7$$!+H9? EFFa_p$\"+s$Q`k%Fa_pF]]p7%Fc]r7$$!+DSk\"z#Fa_p$\"+Z'[Lh%Fa_pF]]p7%Fi]r 7$$!+QdmcGFa_p$\"+#)\\QzXFa_pF]]p7%F_^r7$$!+%os8#HFa_p$\"+?m\\VXFa_pF] ]p7%Fe^r7$$!+%Gle)HFa_p$\"+iru0XFa_pF]]p7%F[_r7$$!+giB]IFa_p$\"+Nd@mWF a_pF]]p7%Fa_r7$$!+Rxc9JFa_p$\"+0w*\\U%Fa_pF]]p7%Fg_r7$$!+%Fa_pF]]p7%Fear7$$!+MG0-NFa_p$\"+AwE\\TFa_pF]]p7 %F[br7$$!+6bwmNFa_p$\"+**>&)*4%Fa_pF]]p7%Fabr7$$!+C8LJOFa_p$\"+QRx\\SF a_pF]]p7%Fgbr7$$!+ftj&p$Fa_p$\"+qc6**RFa_pF]]p7%F]cr7$$!+$oi&fPFa_p$\" +IZ$z%RFa_pF]]p7%Fccr7$$!+?\\)H#QFa_p$\"+hTE'*QFa_pF]]p7%Ficr7$$!+Kgy& )QFa_p$\"+$\\9T%QFa_pF]]p7%F_dr7$$!+co&y%RFa_p$\"+,qZ\"z$Fa_pF]]p7%Fed r7$$!+K-54SFa_p$\"+IwKQPFa_pF]]p7%F[er7$$!+pFVpSFa_p$\"+(3JYo$Fa_pF]]p 7%Faer7$$!+wayGTFa_p$\"+'pW.j$Fa_pF]]p7%Fger7$$!+IN5(=%Fa_p$\"+(z@ad$F a_pF]]p7%F]fr7$$!+_aMWUFa_p$\"+%e9)>NFa_pF]]p7%Fcfr7$$!+X>[+VFa_p$\"+s jZjMFa_pF]]p7%Fifr7$$!+[X\\bVFa_p$\"+\\LO1MFa_pF]]p7%F_gr7$$!+`UP4WFa_ p$\"+)*eV[LFa_pF]]p7%Fegr7$$!+^,7iWFa_p$\"+t&f'*G$Fa_pF]]p7%F[hr7$$!+F \"QP^%Fa_p$\"+5e+IKFa_pF]]p7%Fahr7$$!+W'RUc%Fa_p$\"+UCXpJFa_pF]]p7%Fgh r7$$!+s0k8YFa_p$\"+pS)z5$Fa_pF]]p7%F]ir7$$!+q+'>m%Fa_p$\"+*H#fXIFa_pF] ]p7%Fcir7$$!+([>#4ZFa_p$\"+'*eF#)HFa_pF]]p7%Fiir7$$!+t7WbZFa_p$\"+33/= HFa_pF]]p7%F_jr7$$!+%)yk+[Fa_p$\"+(3+H&GFa_pF]]p7%Fejr7$$!+\"pg[%[Fa_p $\"+_O(oy#Fa_pF]]p7%F[[s7$$!+.*)4))[Fa_p$\"+-z)*>FFa_pF]]p7%Fa[s7$$!+N &y.$\\Fa_p$\"+O^F_EFa_pF]]p7%Fg[s7$$!+s9rr\\Fa_p$\"+(ysPe#Fa_pF]]p7%F] \\s7$$!+iY57]Fa_p$\"+`C_9DFa_pF]]p7%Fc\\s7$$!+:&f:0&Fa_p$\"+%yoXW#Fa_p F]]p7%Fi\\s7$$!+I:2!4&Fa_p$\"+G#eRP#Fa_pF]]p7%F_]s7$$!+2-jF^Fa_p$\"+@x t-BFa_pF]]p7%Fe]s7$$!+\\#>U;&Fa_p$\"+%R`4B#Fa_pF]]p7%F[^s7$$!+Vq\")*>& Fa_p$\"+c$\\'e@Fa_pF]]p7%Fa^s7$$!+%R(RM_Fa_p$\"+Rm'e3#Fa_pF]]p7%Fg^s7$ $!+d/$zE&Fa_p$\"+dBk7?Fa_pF]]p7%F]_s7$$!++QQ+`Fa_p$\"+9\"4!R>Fa_pF]]p7 %Fc_s7$$!+DNsJ`Fa_p$\"+vY*\\'=Fa_pF]]p7%Fi_s7$$!+(Q:>O&Fa_p$\"+,>i!z\" Fa_pF]]p7%F_`s7$$!+Re#4R&Fa_p$\"+0*3fr\"Fa_pF]]p7%Fe`s7$$!+pHs=aFa_p$ \"+=%p3k\"Fa_pF]]p7%F[as7$$!+*Hx_W&Fa_p$\"+!H8bc\"Fa_pF]]p7%Faas7$$!+B CcqaFa_p$\"+4r%)*[\"Fa_pF]]p7%Fgas7$$!+*[bX\\&Fa_p$\"+')[(QT\"Fa_pF]]p 7%F]bs7$$!+HvB5E\"F a_pF]]p7%Fibs7$$!+*38'ebFa_p$\"+)QQT=\"Fa_pF]]p7%F_cs7$$!+m\"*GxbFa_p$ \"+\"[bp5\"Fa_pF]]p7%Fecs7$$!+E\">Yf&Fa_p$\"+(os%H5Fa_pF]]p7%F[ds7$$!+ #*Qg5cFa_p$\"+)=Jp^*F\\]pF]]p7%Fads7$$!+MxCDcFa_p$\"+O(=it)F\\]pF]]p7% Fgds7$$!+jybQcFa_p$\"+;]m_zF\\]pF]]p7%F]es7$$!+ZRa]cFa_p$\"+.SPmrF\\]p F]]p7%Fces7$$!+nv@hcFa_p$\"+t7[xjF\\]pF]]p7%Fies7$$!+:;fqcFa_p$\"+kh:' e&F\\]pF]]p7%F_fs7$$!+_'z'ycFa_p$\"+\\Gg#z%F\\]pF]]p7%Fefs7$$!+S_\\&o& Fa_p$\"+\">gq*RF\\]pF]]p7%F[gs7$$!+w70\"p&Fa_p$\"+*G+)*>$F\\]pF]]p7%Fa gs7$$!+[$f`p&Fa_p$\"+oc7,CF\\]pF]]p7%Fggs7$$!+a\"H%)p&Fa_p$\"+X^O,;F\\ ]pF]]p7%F]hs7$$!+-!o-q&Fa_p$\"+\"y%o3!)FeepF]]p7%Fchs7$$!+Q/)3q&Fa_pF( F]]p7%Fihs7$Fdhs$!+\"y%o3!)FeepF]]p7%F]is7$F^hs$!+X^O,;F\\]pF]]p7%Fais 7$Fhgs$!+oc7,CF\\]pF]]p7%Feis7$Fbgs$!+*G+)*>$F\\]pF]]p7%Fiis7$F\\gs$!+ \">gq*RF\\]pF]]p7%F]js7$Fffs$!+\\Gg#z%F\\]pF]]p7%Fajs7$F`fs$!+kh:'e&F \\]pF]]p7%Fejs7$Fjes$!+t7[xjF\\]pF]]p7%Fijs7$Fdes$!+.SPmrF\\]pF]]p7%F] [t7$F^es$!+;]m_zF\\]pF]]p7%Fa[t7$Fhds$!+O(=it)F\\]pF]]p7%Fe[t7$Fbds$!+ )=Jp^*F\\]pF]]p7%Fi[t7$F\\ds$!+(os%H5Fa_pF]]p7%F]\\t7$Ffcs$!+\"[bp5\"F a_pF]]p7%Fa\\t7$F`cs$!+)QQT=\"Fa_pF]]p7%Fe\\t7$Fjbs$!+Y(>5E\"Fa_pF]]p7 %Fi\\t7$Fdbs$!+m()fP8Fa_pF]]p7%F]]t7$F^bs$!+')[(QT\"Fa_pF]]p7%Fa]t7$Fh as$!+4r%)*[\"Fa_pF]]p7%Fe]t7$Fbas$!+!H8bc\"Fa_pF]]p7%Fi]t7$F\\as$!+=%p 3k\"Fa_pF]]p7%F]^t7$Ff`s$!+0*3fr\"Fa_pF]]p7%Fa^t7$F``s$!+,>i!z\"Fa_pF] ]p7%Fe^t7$Fj_s$!+vY*\\'=Fa_pF]]p7%Fi^t7$Fd_s$!+9\"4!R>Fa_pF]]p7%F]_t7$ F^_s$!+dBk7?Fa_pF]]p7%Fa_t7$Fh^s$!+Rm'e3#Fa_pF]]p7%Fe_t7$Fb^s$!+c$\\'e @Fa_pF]]p7%Fi_t7$F\\^s$!+%R`4B#Fa_pF]]p7%F]`t7$Ff]s$!+@xt-BFa_pF]]p7%F a`t7$F`]s$!+G#eRP#Fa_pF]]p7%Fe`t7$Fj\\s$!+%yoXW#Fa_pF]]p7%Fi`t7$Fd\\s$ !+`C_9DFa_pF]]p7%F]at7$F^\\s$!+(ysPe#Fa_pF]]p7%Faat7$Fh[s$!+O^F_EFa_pF ]]p7%Feat7$Fb[s$!+-z)*>FFa_pF]]p7%Fiat7$F\\[s$!+_O(oy#Fa_pF]]p7%F]bt7$ Ffjr$!+(3+H&GFa_pF]]p7%Fabt7$F`jr$!+33/=HFa_pF]]p7%Febt7$Fjir$!+'*eF#) HFa_pF]]p7%Fibt7$Fdir$!+*H#fXIFa_pF]]p7%F]ct7$F^ir$!+pS)z5$Fa_pF]]p7%F act7$Fhhr$!+UCXpJFa_pF]]p7%Fect7$Fbhr$!+5e+IKFa_pF]]p7%Fict7$F\\hr$!+t &f'*G$Fa_pF]]p7%F]dt7$Ffgr$!+)*eV[LFa_pF]]p7%Fadt7$F`gr$!+\\LO1MFa_pF] ]p7%Fedt7$Fjfr$!+sjZjMFa_pF]]p7%Fidt7$Fdfr$!+%e9)>NFa_pF]]p7%F]et7$F^f r$!+(z@ad$Fa_pF]]p7%Faet7$Fher$!+'pW.j$Fa_pF]]p7%Feet7$Fber$!+(3JYo$Fa _pF]]p7%Fiet7$F\\er$!+IwKQPFa_pF]]p7%F]ft7$Ffdr$!+,qZ\"z$Fa_pF]]p7%Faf t7$F`dr$!+$\\9T%QFa_pF]]p7%Feft7$Fjcr$!+hTE'*QFa_pF]]p7%Fift7$Fdcr$!+I Z$z%RFa_pF]]p7%F]gt7$F^cr$!+qc6**RFa_pF]]p7%Fagt7$Fhbr$!+QRx\\SFa_pF]] p7%Fegt7$Fbbr$!+**>&)*4%Fa_pF]]p7%Figt7$F\\br$!+AwE\\TFa_pF]]p7%F]ht7$ Ffar$!+7e\"z>%Fa_pF]]p7%Faht7$F`ar$!+EGnXUFa_pF]]p7%Feht7$Fj`r$!+\")=S #H%Fa_pF]]p7%Fiht7$Fd`r$!+A)fzL%Fa_pF]]p7%F]it7$F^`r$!+jQ?#Q%Fa_pF]]p7 %Fait7$Fh_r$!+0w*\\U%Fa_pF]]p7%Feit7$Fb_r$!+Nd@mWFa_pF]]p7%Fiit7$F\\_r $!+iru0XFa_pF]]p7%F]jt7$Ff^r$!+?m\\VXFa_pF]]p7%Fajt7$F`^r$!+#)\\QzXFa_ pF]]p7%Fejt7$Fj]r$!+Z'[Lh%Fa_pF]]p7%Fijt7$Fd]r$!+s$Q`k%Fa_pF]]p7%F][u7 $F^]r$!+\")yJvYFa_pF]]p7%Fa[u7$Fh\\r$!+DDE.ZFa_pF]]p7%Fe[u7$Fb\\r$!+8# e\"HZFa_pF]]p7%Fi[u7$F\\\\r$!+x2+`ZFa_pF]]p7%F]\\u7$Ff[r$!+SfzuZFa_pF] ]p7%Fa\\u7$F`[r$!+T+c%z%Fa_pF]]p7%Fe\\u7$Fjjq$!+paFa_pF]]p7%Ficu7$F\\`q$!+1>g*e?T&Fa_pF]]p7%Fadu7$F`_q$!+c\"=LS&Fa_pF]]p7%Fedu7$Fj^q$ !+DpT\"R&Fa_pF]]p7%Fidu7$Fd^q$!+!GujP&Fa_pF]]p7%F]eu7$F^^q$!+dF>e`Fa_p F]]p7%Faeu7$Fh]q$!+*pfoL&Fa_pF]]p7%Feeu7$Fb]q$!+_#[BJ&Fa_pF]]p7%Fieu7$ F\\]q$!+\"3=YG&Fa_pF]]p7%F]fu7$Ff\\q$!+ydh`_Fa_pF]]p7%Fafu7$F`\\q$!+E \\F>_Fa_pF]]p7%Fefu7$Fj[q$!+Mc^\"=&Fa_pF]]p7%Fifu7$Fd[q$!+gKCS^Fa_pF]] p7%F]gu7$F^[q$!+vdM&4&Fa_pF]]p7%Fagu7$Fhjp$!+()))oY]Fa_pF]]p7%Fegu7$Fb jp$!+ou5%*\\Fa_pF]]p7%Figu7$F\\jp$!+'*3RP\\Fa_pF]]p7%F]hu7$Ffip$!+r%ei ([Fa_pF]]p7%Fahu7$F`ip$!+3'=.\"[Fa_pF]]p7%Fehu7$Fjhp$!+7;+RZFa_pF]]p7% Fihu7$Fdhp$!+D/XhYFa_pF]]p7%F]iu7$F^hp$!+-+NwXFa_pF]]p7%Faiu7$Fhgp$!+z Xp\"[%Fa_pF]]p7%Feiu7$Fbgp$!+6nhuVFa_pF]]p7%Fiiu7$F\\gp$!+IM$=D%Fa_pF] ]p7%F]ju7$Fffp$!+(Gx<6%Fa_pF]]p7%Faju7$F`fp$!+(HOv&RFa_pF]]p7%Feju7$Fj ep$!+**[u&z$Fa_pF]]p7%Fiju7$Fcep$!+Jo&>j$Fa_pF]]p7%F][v7$F]ep$!+!>Q)oM Fa_pF]]p7%Fa[v7$Fgdp$!+Pn92LFa_pF]]p7%Fe[v7$F`dp$!+$\\Po9$Fa_pF]]p7%Fi [v7$Fjcp$!+y\\i()HFa_pF]]p7%F]\\v7$Fccp$!+*3.#HGFa_pF]]p7%Fa\\v7$F]cp$ !+lhJrEFa_pF]]p7%Fe\\v7$Fgbp$!+=Gx8DFa_pF]]p7%Fi\\v7$Fabp$!+1&RkN#Fa_p F]]p7%F]]v7$F[bp$!+w#G#*>#Fa_pF]]p7%Fa]v7$Feap$!+)G%3U?Fa_pF]]p7%Fe]v7 $F^ap$!+:_(\\)=Fa_pF]]p7%Fi]v7$Fh`p$!+6J)ys\"Fa_pF]]p7%F]^v7$Fa`p$!+a' )zq:Fa_pF]]p7%Fa^v7$F[`p$!+]tr89Fa_pF]]p7%Fe^v7$Fd_p$!+8sjc7Fa_pF]]p7% Fi^v7$$!+:^4*p&FK$!+bub*4\"Fa_pF]]p7%F]_v7$$!+lnRO#*F-$!+&)zxC%*F\\]pF ]]p7%Fc_v7$$!+em*G;'Fir$!+K;)R&yF\\]pF]]p7%Fi_v7$$\"+yI%y9\"Fir$!+1`=$ G'F\\]pF]]p7%F_`v7$F($!+!)*)Q7ZF\\]pF]]p7%Fe`v7$F($!+aEfTJF\\]pF]]p7%F i`v7$F($!+Fjzq:F\\]pF]]p7%F]avF'F]]p-F]\\p6&F_\\p$\"#QFb\\pF($\"#vFb\\ p-%&STYLEG6#%,PATCHNOGRIDG-F$6$7]_lF'F*F.7$F($\"3D+++\")*)Q7ZF-7$$\"3* ******f&Qj0eF7$\"3o*****zJ&=$G'F-7$$\"3%******feat5&FB$\"3[+++0<)R&yF- 7$$\"31+++1B$\\%HFH$\"3E+++\"GyZU*F-7$$\"3/+++#yC[3;4$FZ$\"3#******RZ(zq:FK7$$\"3S+++6Jh$y'FZ$\"34+++ow(ys \"FK7$$\"32+++irtI8F_o$\"3)*******\\v&\\)=FK7$$\"3++++(fc&\\BF_o$\"3:+ ++)fO?/#FK7$$\"3#)*****f?)RNPF_o$\"3%)*****Rj9\"*>#FK7$$\"3?+++O&\\oI& F_o$\"3))*****fG$>cBFK7$$\"3i+++;zU&f'F_o$\"3:+++l\"zK^#FK7$$\"3`+++)R o`!oF_o$\"35+++>(*QqEFK7$$\"3s*****RB2q(\\F_o$\"34+++?HcFGFK7$$\"3!*** ****3aM))eFZ$\"3%******z'=([)HFK7$$!3\")*****pTR6.&F_o$\"3!******RGXC9 $FK7$$!3J+++(pYR!fF_o$\"32+++>W\\+LFK7$$\"33+++*RBbP\"F^q$\"3++++[gMfM FK7$$\"3K+++u%H(z))F^q$\"3)*******4&y%>OFK7$$\"3#******\\gb$**GFiq$\"3 ++++Owa\"y$FK7$$\"3e******3#Q![vFiq$\"3!)*****zd^j%RFK7$$\"36+++5.$Ru \"Fir$\"3%)******)eZX6%FK7$$\"39+++g5\"yr$Fir$\"3G+++$)[^&G%FK7$$\"3q* *****)Ra9H(Fir$\"3I+++H\"RUX%FK7$$\"3++++A>Rh7F-$\"3*)******R T$R\\FK7$$\"3/+++0$obC$F-$\"35+++%RI 2-&FK7$$\"3F+++7@(3^$F-$\"3-+++X_N%4&FK7$$\"39+++GCq%p$F-$\"3/+++/-&>; &FK7$$\"3>+++PQp2QF-$\"3V+++Mc`C_FK7$$\"3!)*****RiZ*eQF-$\"3c*****H2NF G&FK7$$\"3G+++MN:cQF-$\"3++++\">YpL&FK7$$\"3w*****paHe!QF-$\"37+++lWV( Q&FK7$$\"3t*****fFTNr$F-$\"3B+++@eQMaFK7$$\"3y*****>ewSe$F-$\"3#****** *Ro$zZ&FK7$$\"3\")*****>'*z:U$F-$\"3z*****fE#>=bFK7$$\"3#******>Jc'HKF -$\"3I+++O^BbbFK7$$\"3E+++RCX6IF-$\"3m******HD8*e&FK7$$\"3#******\\XA( pFF-$\"3q*****4qQ*>cFK7$$\"32+++z6)o]#F-$\"3C+++1mpZcFK7$$\"3/+++-$[]A #F-$\"3I+++[%QCn&FK7$$\"36+++L]3E>F-$\"3q*****pf&=%p&FK7$$\"3!******HN C;h\"F-$\"3H+++p![Hr&FK7$$\"3))*****f]&4$G\"F-$\"33+++hOsGdFK7$$\"3N** ***RAIuT*Fir$\"3%)*****Hv'\\TdFK7$$\"3++++[AS')eFir$\"3S+++SnB^dFK7$$ \"3)******p'y'pC#Fir$\"3/+++:g*yv&FK7$$!3++++_hS$\\\"Fir$\"3;+++atShdF K7$$!3$)*****z!G'*G`Fir$\"3s*****4_!ohdFK7$$!33+++=/&eD*Fir$\"3/+++1yf edFK7$$!33+++Y$[sK\"F-$\"33+++>\"3?v&FK7$$!3$******4C:!Q+++^.:LIF-$\"3s*****>)H'eo&FK7$$!3)******f=>E\\$F-$\"39+++p+ ddcFK7$$!3u******pfDtRF-$\"3g*****4\"eZBcFK7$$!3')*****>amT[%F-$\"3g** ****o^x#e&FK7$$!3f*****zl!=T]F-$\"3P+++%pSX`&FK7$$!3_+++]#GBn&F-$\"3E+ ++B_+yaFK7$$!3:+++\"yOaU'F-$\"3G+++\"Q'p8aFK7$$!35+++\\g\\jtF-$\"3)*** ***4O[lM&FK7$$!3B+++[ta-&)F-$\"3g*****H()Q%)G&FK7$$!3F+++*\\pAu*F-$\"3 i*****R#zC\\_FK7$$!3'******z)*)='4\"FK$\"3*******4$>-y*=9FK$\"3d*****z(od0_FK7$$!3%******zR5R^\"FK$\"3,++++4I-_FK7$$! 3$******z`ATg\"FK$\"3y*****HBW&)>&FK7$$!3)******pv$Q!p\"FK$\"39+++htm$ >&FK7$$!33+++?JJtFK$\"3++++U+%*o^FK7$$!3A+++v)[m+#FK$\"3i*****>IBn:&FK7$$! 37+++ZMS!3#FK$\"31+++g>NU^FK7$$!31+++@hd_@FK$\"3S++++/xD^FK7$$!3#***** *HVqLA#FK$\"3.+++\\*Rp5&FK7$$!3y*****\\klHH#FK$\"3e*****H9Me3&FK7$$!3+ +++yH_hBFK$\"3#)*****\\%fVi]FK7$$!35+++%e!>HCFK$\"3O+++WjtO]FK7$$!39++ +3v5'\\#FK$\"3$******\\,M(3]FK7$$!3))*****Hv1Cc#FK$\"3))*****H#eVy\\FK 7$$!3.+++=v@GEFK$\"3U+++b!ee%\\FK7$$!3))*****>amOp#FK$\"3/+++[#G5\"\\F K7$$!39+++s%y)eFFK$\"3A+++,v)R([FK7$$!3.+++#3vR#GFK$\"3A+++jIzM[FK7$$! 3%)*****R>t!*)GFK$\"33+++#*3_$z%FK7$$!37+++!>\"GaHFK$\"3/+++txE]ZFK7$$ !3-+++_Tp>IFK$\"3,+++3C:0ZFK7$$!3%******R#zQ&3$FK$\"3=+++_[JeYFK7$$!3! ********o79:$FK$\"3&)*****4s8*4YFK7$$!3*******z.(ys/`NFK$\"3U+++il7'H%FK7$$!3++++Jk(*>OFK$\"31+++$)3BTUFK 7$$!3)******po:lo$FK$\"3i*****zE,e=%FK7$$!31+++KC^_PFK$\"3k*****>sW)HT FK7$$!3,+++Xi#y\"QFK$\"3A+++OAMtSFK7$$!3-+++=JL#)QFK$\"3=+++RXD;SFK7$$ !39+++\"oFf%RFK$\"3A+++yz_eRFK7$$!3v*****\\uB&3SFK$\"3;+++?**4+RFK7$$! 3/+++fO0qSFK$\"37+++nL!4%QFK7$$!3&)******=pYITFK$\"3-+++p1(3y$FK7$$!3u *****zRG(*=%FK$\"31+++\"RO*>PFK7$$!33+++Qk\"yC%FK$\"3#******RWR!eOFK7$ $!3R+++v4s/VFK$\"37+++VY7&f$FK7$$!3g******f+++x\\WKLFK7$$!3++++hstrXFK$\"3?+++MP)QE$F K7$$!36+++P+x@YFK$\"3%)*****4ZVT>$FK7$$!35+++$=Q2n%FK$\"3*)*****feHK7$ FK7$$!3#******HOo'=ZFK$\"3++++Gq:^IFK7$$!3s*****4J&elZFK$\"3'******R/] z(HFK7$$!3q*****pK5:\"[FK$\"3?+++p;k.HFK7$$!3y*****H#)fk&[FK$\"3$)**** *\\#HFGGFK7$$!3;+++**RW+\\FK$\"3%)*******p\"*=v#FK7$$!3O+++PdYV\\FK$\" 3%******fL^Xn#FK7$$!3I+++>)>b)\\FK$\"3;+++Y*3jf#FK7$$!3G+++IDfE]FK$\"3 6+++lNA#FK7$$!3D+++x_?;_FK$\"3&******4x?66#FK7$$!3!)*****H4(o]_FK$\"3 %******R?,\"G?FK7$$!37+++A#4RG&FK$\"3/+++TncW>FK7$$!3i*****4MAeJ&FK$\" 31+++'*3ag=FK7$$!3#******fNzjM&FK$\"3#*******)GTgx\"FK7$$!3u*****\\gOb P&FK$\"35+++0/3\"p\"FK7$$!3o*****z#[D.aFK$\"3++++'\\mcg\"FK7$$!3D+++&) )*\\HaFK$\"3,+++%f/)>:FK7$$!3!)*****\\CVUX&FK$\"3#******\\c(\\L9FK7$$! 3O+++sAYxaFK$\"3-+++druY8FK7$$!3!******HIS\"*\\&FK$\"3++++F]bf7FK7$$!3 &*******ylE>bFK$\"3#******HsB><\"FK7$$!3q*****>+Oy`&FK$\"3/+++!fdQ3\"F K7$$!3/+++w([[b&FK$\"3%*******Qij`**F-7$$!3k*****z*)4.d&FK$\"3e*****z@ AX1*F-7$$!3E+++N&GUe&FK$\"3?+++'\\y8<)F-7$$!3o*****RGK\"\\2cFK$\"3?+++>`\"QP'F-7$$!3<+++ju'oh&FK$\"3q****** Q0$*paF-7$$!3!******H;jZi&FK$\"3++++@%)3jXF-7$$!3i******ya>JcFK$\"3;++ +Lvo`OF-7$$!30+++g+=OcFK$\"34+++.V0c=k&FK$\"3[+++:4(z9*Fir7$$!3>+++ZRcUcFKF(7$Fiby$!3[++ +:4(z9*Fir7$Fdby$!3********=p.H=F-7$F_by$!34+++.V`\"QP'F -7$Ff`y$!3W+++ieRusF-7$Fa`y$!3?+++'\\y8<)F-7$F\\`y$!3e*****z@AX1*F-7$F g_y$!3%*******Qij`**F-7$Fb_y$!3/+++!fdQ3\"FK7$F]_y$!3#******HsB><\"FK7 $Fh^y$!3++++F]bf7FK7$Fc^y$!3-+++druY8FK7$F^^y$!3#******\\c(\\L9FK7$Fi] y$!3,+++%f/)>:FK7$Fd]y$!3++++'\\mcg\"FK7$F_]y$!35+++0/3\"p\"FK7$Fj\\y$ !3#*******)GTgx\"FK7$Fe\\y$!31+++'*3ag=FK7$F`\\y$!3/+++TncW>FK7$F[\\y$ !3%******R?,\"G?FK7$Ff[y$!3&******4x?66#FK7$Fa[y$!39+++Jbf$>#FK7$F\\[y $!39+++r*)[vAFK7$Fgjx$!3!)*****49enN#FK7$Fbjx$!3?+++LUNPCFK7$F]jx$!36+ ++lNA$FK7$F[gx$!3?+++MP)QE$FK7$Fffx$!3>+++x\\WKLFK7$Fafx$!3')*****HSJ) *R$FK7$F\\fx$!3#)*****fqcgY$FK7$Fgex$!3z*****H$Q9JNFK7$Fbex$!37+++VY7& f$FK7$F]ex$!3#******RWR!eOFK7$Fhdx$!31+++\"RO*>PFK7$Fcdx$!3-+++p1(3y$F K7$F^dx$!37+++nL!4%QFK7$Ficx$!3;+++?**4+RFK7$Fdcx$!3A+++yz_eRFK7$F_cx$ !3=+++RXD;SFK7$Fjbx$!3A+++OAMtSFK7$Febx$!3k*****>sW)HTFK7$F`bx$!3i**** *zE,e=%FK7$F[bx$!31+++$)3BTUFK7$Ffax$!3U+++il7'H%FK7$Faax$!3m*****f\\] /N%FK7$F\\ax$!3x******f78/WFK7$Fg`x$!3!******4(Q1dWFK7$Fb`x$!3O+++GK64 XFK7$F]`x$!3A+++.37gXFK7$Fh_x$!3&)*****4s8*4YFK7$Fc_x$!3=+++_[JeYFK7$F ^_x$!3,+++3C:0ZFK7$Fi^x$!3/+++txE]ZFK7$Fd^x$!33+++#*3_$z%FK7$F_^x$!3A+ ++jIzM[FK7$Fj]x$!3A+++,v)R([FK7$Fe]x$!3/+++[#G5\"\\FK7$F`]x$!3U+++b!ee %\\FK7$F[]x$!3))*****H#eVy\\FK7$Ff\\x$!3$******\\,M(3]FK7$Fa\\x$!3O+++ WjtO]FK7$F\\\\x$!3#)*****\\%fVi]FK7$Fg[x$!3e*****H9Me3&FK7$Fb[x$!3.+++ \\*Rp5&FK7$F][x$!3S++++/xD^FK7$Fhjw$!31+++g>NU^FK7$Fcjw$!3i*****>IBn:& FK7$F^jw$!3++++U+%*o^FK7$Fiiw$!3C+++bB3z^FK7$Fdiw$!3/+++:rE(=&FK7$F_iw $!39+++htm$>&FK7$Fjhw$!3y*****HBW&)>&FK7$Fehw$!3,++++4I-_FK7$F`hw$!3d* ****z(od0_FK7$F[hw$!3&)*****f*)=%4_FK7$Ffgw$!3n*****4L(f:_FK7$Fagw$!3* ******4$>)H'eo&FK7$Fjcw$!3e*****4t***3dFK7$Fecw$!30+++ kk[FdFK7$F`cw$!3e*****R4>\"3?v&FK7$Ffbw$!3/+++1yfed FK7$Fabw$!3s*****4_!ohdFK7$F\\bw$!3;+++atShdFK7$Fgaw$!3/+++:g*yv&FK7$F baw$!3S+++SnB^dFK7$F]aw$!3%)*****Hv'\\TdFK7$Fh`w$!33+++hOsGdFK7$Fc`w$! 3H+++p![Hr&FK7$F^`w$!3q*****pf&=%p&FK7$Fi_w$!3I+++[%QCn&FK7$Fd_w$!3C++ +1mpZcFK7$F__w$!3q*****4qQ*>cFK7$Fj^w$!3m******HD8*e&FK7$Fe^w$!3I+++O^ BbbFK7$F`^w$!3z*****fE#>=bFK7$F[^w$!3#*******Ro$zZ&FK7$Ff]w$!3B+++@eQM aFK7$Fa]w$!37+++lWV(Q&FK7$F\\]w$!3++++\">YpL&FK7$Fg\\w$!3c*****H2NFG&F K7$Fb\\w$!3V+++Mc`C_FK7$F]\\w$!3/+++/-&>;&FK7$Fh[w$!3-+++X_N%4&FK7$Fc[ w$!35+++%RI2-&FK7$F^[w$!3:+++)>T$R\\FK7$Fijv$!3!******H*>OFK7$F\\hv$!3++++[gMfMFK7$Fggv$ !32+++>W\\+LFK7$Fbgv$!3!******RGXC9$FK7$F]gv$!3%******z'=([)HFK7$Fhfv$ !34+++?HcFGFK7$Fcfv$!35+++>(*QqEFK7$F^fv$!3:+++l\"zK^#FK7$Fiev$!3))*** **fG$>cBFK7$Fdev$!3%)*****Rj9\"*>#FK7$F_ev$!3:+++)fO?/#FK7$Fjdv$!3)*** ****\\v&\\)=FK7$$\"3')*****>68Oy'FZ$!34+++ow(ys\"FK7$$\"3?+++*\\3;4$FZ $!3#******RZ(zq:FK7$F[dv$!30+++Ctr89FK7$Ffcv$!3-+++\\tjc7FK7$$\"37+++' yC#Fa_pF\\b[l7%Faf[l7$$\"+O &\\oI&F`ap$\"+'G$>cBFa_pF\\b[l7%Fgf[l7$$\"+;zU&f'F`ap$\"+l\"zK^#Fa_pF \\b[l7%F]g[l7$$\"+)Ro`!oF`ap$\"+>(*QqEFa_pF\\b[l7%Fcg[l7$$\"+Ms+x\\F`a p$\"+?HcFGFa_pF\\b[l7%Fig[l7$$\"+4aM))eFc`p$\"+o=([)HFa_pF\\b[l7%F_h[l 7$$!+<%R6.&F`ap$\"+%GXC9$Fa_pF\\b[l7%Feh[l7$$!+(pYR!fF`ap$\"+>W\\+LFa_ pF\\b[l7%F[i[l7$$\"+*RBbP\"Fecp$\"+[gMfMFa_pF\\b[l7%Fai[l7$$\"+u%H(z)) Fecp$\"+5&y%>OFa_pF\\b[l7%Fgi[l7$$\"+0cN**GFbdp$\"+Owa\"y$Fa_pF\\b[l7% F]j[l7$$\"+4#Q![vFbdp$\"+y:NYRFa_pF\\b[l7%Fcj[l7$$\"+5.$Ru\"Feep$\"+*e ZX6%Fa_pF\\b[l7%Fij[l7$$\"+g5\"yr$Feep$\"+$)[^&G%Fa_pF\\b[l7%F_[\\l7$$ \"+*Ra9H(Feep$\"+H\"RUX%Fa_pF\\b[l7%Fe[\\l7$$\"+A>Rh7F\\]p$\"+ST$R\\Fa _pF\\b[l7%F]]\\l7$$\"+0$obC$F\\]p$\"+%RI2-&Fa_pF\\b[l7%Fc]\\l7$$\"+7@( 3^$F\\]p$\"+X_N%4&Fa_pF\\b[l7%Fi]\\l7$$\"+GCq%p$F\\]p$\"+/-&>;&Fa_pF\\ b[l7%F_^\\l7$$\"+PQp2QF\\]p$\"+Mc`C_Fa_pF\\b[l7%Fe^\\l7$$\"+Cw%*eQF\\] p$\"+t]t#G&Fa_pF\\b[l7%F[_\\l7$$\"+MN:cQF\\]p$\"+\">YpL&Fa_pF\\b[l7%Fa _\\l7$$\"+Z&He!QF\\]p$\"+lWV(Q&Fa_pF\\b[l7%Fg_\\l7$$\"+w7a8PF\\]p$\"+@ eQMaFa_pF\\b[l7%F]`\\l7$$\"+#ewSe$F\\]p$\"+So$zZ&Fa_pF\\b[l7%Fc`\\l7$$ \"+i*z:U$F\\]p$\"+mA>=bFa_pF\\b[l7%Fi`\\l7$$\"+7jlHKF\\]p$\"+O^BbbFa_p F\\b[l7%F_a\\l7$$\"+RCX6IF\\]p$\"+ID8*e&Fa_pF\\b[l7%Fea\\l7$$\"+bCspFF \\]p$\"+,(Q*>cFa_pF\\b[l7%F[b\\l7$$\"+z6)o]#F\\]p$\"+1mpZcFa_pF\\b[l7% Fab\\l7$$\"+-$[]A#F\\]p$\"+[%QCn&Fa_pF\\b[l7%Fgb\\l7$$\"+L]3E>F\\]p$\" +(f&=%p&Fa_pF\\b[l7%F]c\\l7$$\"+`Vi6;F\\]p$\"+p![Hr&Fa_pF\\b[l7%Fcc\\l 7$$\"+1b4$G\"F\\]p$\"+hOsGdFa_pF\\b[l7%Fic\\l7$$\"+C-V<%*Feep$\"+`n\\T dFa_pF\\b[l7%F_d\\l7$$\"+[AS')eFeep$\"+SnB^dFa_pF\\b[l7%Fed\\l7$$\"+ny 'pC#Feep$\"+:g*yv&Fa_pF\\b[l7%F[e\\l7$$!+_hS$\\\"Feep$\"+atShdFa_pF\\b [l7%Fae\\l7$$!+3G'*G`Feep$\"+@0ohdFa_pF\\b[l7%Fge\\l7$$!+=/&eD*Feep$\" +1yfedFa_pF\\b[l7%F]f\\l7$$!+Y$[sK\"F\\]p$\"+>\"3?v&Fa_pF\\b[l7%Fcf\\l 7$$!+T_,QE\\$F\\]p$\"+p+ddcFa_pF\\b[l 7%Fah\\l7$$!+qfDtRF\\]p$\"+6eZBcFa_pF\\b[l7%Fgh\\l7$$!+Ul;%[%F\\]p$\"+ p^x#e&Fa_pF\\b[l7%F]i\\l7$$!+e1=T]F\\]p$\"+%pSX`&Fa_pF\\b[l7%Fci\\l7$$ !+]#GBn&F\\]p$\"+B_+yaFa_pF\\b[l7%Fii\\l7$$!+\"yOaU'F\\]p$\"+\"Q'p8aFa _pF\\b[l7%F_j\\l7$$!+\\g\\jtF\\]p$\"+h$[lM&Fa_pF\\b[l7%Fej\\l7$$!+[ta- &)F\\]p$\"+t)Q%)G&Fa_pF\\b[l7%F[[]l7$$!+*\\pAu*F\\]p$\"+CzC\\_Fa_pF\\b [l7%Fa[]l7$$!+))*)='4\"Fa_p$\"+J>&Fa_pF\\b[l7%Fe]]l7$$!+dPQ!p\"Fa_ p$\"+htm$>&Fa_pF\\b[l7%F[^]l7$$!+?JJtFa_p$\"+U+%*o^Fa_pF \\b[l7%F]_]l7$$!+v)[m+#Fa_p$\"+-Lsc^Fa_pF\\b[l7%Fc_]l7$$!+ZMS!3#Fa_p$ \"+g>NU^Fa_pF\\b[l7%Fi_]l7$$!+@hd_@Fa_p$\"++/xD^Fa_pF\\b[l7%F_`]l7$$!+ L/PBAFa_p$\"+\\*Rp5&Fa_pF\\b[l7%Fe`]l7$$!+Xc'HH#Fa_p$\"+VT$e3&Fa_pF\\b [l7%F[a]l7$$!+yH_hBFa_p$\"+XfVi]Fa_pF\\b[l7%Faa]l7$$!+%e!>HCFa_p$\"+Wj tO]Fa_pF\\b[l7%Fga]l7$$!+3v5'\\#Fa_p$\"+:St3]Fa_pF\\b[l7%F]b]l7$$!+`nS iDFa_p$\"+BeVy\\Fa_pF\\b[l7%Fcb]l7$$!+=v@GEFa_p$\"+b!ee%\\Fa_pF\\b[l7% Fib]l7$$!+Ulm$p#Fa_p$\"+[#G5\"\\Fa_pF\\b[l7%F_c]l7$$!+s%y)eFFa_p$\"+,v )R([Fa_pF\\b[l7%Fec]l7$$!+#3vR#GFa_p$\"+jIzM[Fa_pF\\b[l7%F[d]l7$$!+%>t !*)GFa_p$\"+#*3_$z%Fa_pF\\b[l7%Fad]l7$$!+!>\"GaHFa_p$\"+txE]ZFa_pF\\b[ l7%Fgd]l7$$!+_Tp>IFa_p$\"+3C:0ZFa_pF\\b[l7%F]e]l7$$!+CzQ&3$Fa_p$\"+_[J eYFa_pF\\b[l7%Fce]l7$$!+!p79:$Fa_p$\"+@P\"*4YFa_pF\\b[l7%Fie]l7$$!+Qqy s/`NFa _p$\"+il7'H%Fa_pF\\b[l7%F]h]l7$$!+Jk(*>OFa_p$\"+$)3BTUFa_pF\\b[l7%Fch] l7$$!+(o:lo$Fa_p$\"+o7!e=%Fa_pF\\b[l7%Fih]l7$$!+KC^_PFa_p$\"+AZ%)HTFa_ pF\\b[l7%F_i]l7$$!+Xi#y\"QFa_p$\"+OAMtSFa_pF\\b[l7%Fei]l7$$!+=JL#)QFa_ p$\"+RXD;SFa_pF\\b[l7%F[j]l7$$!+\"oFf%RFa_p$\"+yz_eRFa_pF\\b[l7%Faj]l7 $$!+XP_3SFa_p$\"+?**4+RFa_pF\\b[l7%Fgj]l7$$!+fO0qSFa_p$\"+nL!4%QFa_pF \\b[l7%F][^l7$$!+>pYITFa_p$\"+p1(3y$Fa_pF\\b[l7%Fc[^l7$$!+)RG(*=%Fa_p$ \"+\"RO*>PFa_pF\\b[l7%Fi[^l7$$!+Qk\"yC%Fa_p$\"+W%R!eOFa_pF\\b[l7%F_\\^ l7$$!+v4s/VFa_p$\"+VY7&f$Fa_pF\\b[l7%Fe\\^l7$$!+g$Fa_pF\\ b[l7%Fi^^l7$$!+$=Q2n%Fa_p$\"+'eHK7$Fa_pF\\b[l7%F__^l7$$!+j$o'=ZFa_p$\" +Gq:^IFa_pF\\b[l7%Fe_^l7$$!+6`elZFa_p$\"+W+&z(HFa_pF\\b[l7%F[`^l7$$!+F .^6[Fa_p$\"+p;k.HFa_pF\\b[l7%Fa`^l7$$!+B)fk&[Fa_p$\"+DHFGGFa_pF\\b[l7% Fg`^l7$$!+**RW+\\Fa_p$\"++<*=v#Fa_pF\\b[l7%F]a^l7$$!+PdYV\\Fa_p$\"+O8b uEFa_pF\\b[l7%Fca^l7$$!+>)>b)\\Fa_p$\"+Y*3jf#Fa_pF\\b[l7%Fia^l7$$!+IDf E]Fa_p$\"+lNA#Fa_pF\\b[l7%Fgc^l7$$!+x_?;_F a_p$\"+r276@Fa_pF\\b[l7%F]d^l7$$!+$4(o]_Fa_p$\"+/75G?Fa_pF\\b[l7%Fcd^l 7$$!+A#4RG&Fa_p$\"+TncW>Fa_pF\\b[l7%Fid^l7$$!+TB#eJ&Fa_p$\"+'*3ag=Fa_p F\\b[l7%F_e^l7$$!+c$zjM&Fa_p$\"+*GTgx\"Fa_pF\\b[l7%Fee^l7$$!+0m`v`Fa_p $\"+0/3\"p\"Fa_pF\\b[l7%F[f^l7$$!+G[D.aFa_p$\"+'\\mcg\"Fa_pF\\b[l7%Faf ^l7$$!+&))*\\HaFa_p$\"+%f/)>:Fa_pF\\b[l7%Fgf^l7$$!+XKCaaFa_p$\"+lv\\L9 Fa_pF\\b[l7%F]g^l7$$!+sAYxaFa_p$\"+druY8Fa_pF\\b[l7%Fcg^l7$$!+..9*\\&F a_p$\"+F]bf7Fa_pF\\b[l7%Fig^l7$$!+zlE>bFa_p$\"+BP#><\"Fa_pF\\b[l7%F_h^ l7$$!+-g$y`&Fa_p$\"+!fdQ3\"Fa_pF\\b[l7%Feh^l7$$!+w([[b&Fa_p$\"+Rij`**F \\]pF\\b[l7%F[i^l7$$!+)*)4.d&Fa_p$\"+=A_k!*F\\]pF\\b[l7%Fai^l7$$!+N&GU e&Fa_p$\"+'\\y8<)F\\]pF\\b[l7%Fgi^l7$$!+%G`\"QP'F\\]pF\\b[l7%Fcj^l7$$!+ju'oh&Fa_p$\"+ R0$*paF\\]pF\\b[l7%Fij^l7$$!+jJwCcFa_p$\"+@%)3jXF\\]pF\\b[l7%F_[_l7$$! +za>JcFa_p$\"+Lvo`OF\\]pF\\b[l7%Fe[_l7$$!+g+=OcFa_p$\"+.Vp.H=F\\]pF\\b[l7%Fa\\_l7$$!+_g&=k&Fa_p$\"+ :4(z9*FeepF\\b[l7%Fg\\_l7$$!+ZRcUcFa_pF(F\\b[l7%F]]_l7$Fh\\_l$!+:4(z9* FeepF\\b[l7%Fa]_l7$Fb\\_l$!+>p.H=F\\]pF\\b[l7%Fe]_l7$F\\\\_l$!+.V`\"QP'F\\]p F\\b[l7%Fi^_l7$F^j^l$!+ieRusF\\]pF\\b[l7%F]__l7$Fhi^l$!+'\\y8<)F\\]pF \\b[l7%Fa__l7$Fbi^l$!+=A_k!*F\\]pF\\b[l7%Fe__l7$F\\i^l$!+Rij`**F\\]pF \\b[l7%Fi__l7$Ffh^l$!+!fdQ3\"Fa_pF\\b[l7%F]`_l7$F`h^l$!+BP#><\"Fa_pF\\ b[l7%Fa`_l7$Fjg^l$!+F]bf7Fa_pF\\b[l7%Fe`_l7$Fdg^l$!+druY8Fa_pF\\b[l7%F i`_l7$F^g^l$!+lv\\L9Fa_pF\\b[l7%F]a_l7$Fhf^l$!+%f/)>:Fa_pF\\b[l7%Faa_l 7$Fbf^l$!+'\\mcg\"Fa_pF\\b[l7%Fea_l7$F\\f^l$!+0/3\"p\"Fa_pF\\b[l7%Fia_ l7$Ffe^l$!+*GTgx\"Fa_pF\\b[l7%F]b_l7$F`e^l$!+'*3ag=Fa_pF\\b[l7%Fab_l7$ Fjd^l$!+TncW>Fa_pF\\b[l7%Feb_l7$Fdd^l$!+/75G?Fa_pF\\b[l7%Fib_l7$F^d^l$ !+r276@Fa_pF\\b[l7%F]c_l7$Fhc^l$!+Jbf$>#Fa_pF\\b[l7%Fac_l7$Fbc^l$!+r*) [vAFa_pF\\b[l7%Fec_l7$F\\c^l$!+T\"enN#Fa_pF\\b[l7%Fic_l7$Ffb^l$!+LUNPC Fa_pF\\b[l7%F]d_l7$F`b^l$!+lNA$Fa_pF\\b[l7%Fef_l7$Fd^^l $!+MP)QE$Fa_pF\\b[l7%Fif_l7$F^^^l$!+x\\WKLFa_pF\\b[l7%F]g_l7$Fh]^l$!+. 9$)*R$Fa_pF\\b[l7%Fag_l7$Fb]^l$!+1n0mMFa_pF\\b[l7%Feg_l7$F\\]^l$!+LQ9J NFa_pF\\b[l7%Fig_l7$Ff\\^l$!+VY7&f$Fa_pF\\b[l7%F]h_l7$F`\\^l$!+W%R!eOF a_pF\\b[l7%Fah_l7$Fj[^l$!+\"RO*>PFa_pF\\b[l7%Feh_l7$Fd[^l$!+p1(3y$Fa_p F\\b[l7%Fih_l7$F^[^l$!+nL!4%QFa_pF\\b[l7%F]i_l7$Fhj]l$!+?**4+RFa_pF\\b [l7%Fai_l7$Fbj]l$!+yz_eRFa_pF\\b[l7%Fei_l7$F\\j]l$!+RXD;SFa_pF\\b[l7%F ii_l7$Ffi]l$!+OAMtSFa_pF\\b[l7%F]j_l7$F`i]l$!+AZ%)HTFa_pF\\b[l7%Faj_l7 $Fjh]l$!+o7!e=%Fa_pF\\b[l7%Fej_l7$Fdh]l$!+$)3BTUFa_pF\\b[l7%Fij_l7$F^h ]l$!+il7'H%Fa_pF\\b[l7%F][`l7$Fhg]l$!+'\\]/N%Fa_pF\\b[l7%Fa[`l7$Fbg]l$ !+g78/WFa_pF\\b[l7%Fe[`l7$F\\g]l$!+rQ1dWFa_pF\\b[l7%Fi[`l7$Fff]l$!+GK6 4XFa_pF\\b[l7%F]\\`l7$F`f]l$!+.37gXFa_pF\\b[l7%Fa\\`l7$Fje]l$!+@P\"*4Y Fa_pF\\b[l7%Fe\\`l7$Fde]l$!+_[JeYFa_pF\\b[l7%Fi\\`l7$F^e]l$!+3C:0ZFa_p F\\b[l7%F]]`l7$Fhd]l$!+txE]ZFa_pF\\b[l7%Fa]`l7$Fbd]l$!+#*3_$z%Fa_pF\\b [l7%Fe]`l7$F\\d]l$!+jIzM[Fa_pF\\b[l7%Fi]`l7$Ffc]l$!+,v)R([Fa_pF\\b[l7% F]^`l7$F`c]l$!+[#G5\"\\Fa_pF\\b[l7%Fa^`l7$Fjb]l$!+b!ee%\\Fa_pF\\b[l7%F e^`l7$Fdb]l$!+BeVy\\Fa_pF\\b[l7%Fi^`l7$F^b]l$!+:St3]Fa_pF\\b[l7%F]_`l7 $Fha]l$!+WjtO]Fa_pF\\b[l7%Fa_`l7$Fba]l$!+XfVi]Fa_pF\\b[l7%Fe_`l7$F\\a] l$!+VT$e3&Fa_pF\\b[l7%Fi_`l7$Ff`]l$!+\\*Rp5&Fa_pF\\b[l7%F]``l7$F``]l$! ++/xD^Fa_pF\\b[l7%Fa``l7$Fj_]l$!+g>NU^Fa_pF\\b[l7%Fe``l7$Fd_]l$!+-Lsc^ Fa_pF\\b[l7%Fi``l7$F^_]l$!+U+%*o^Fa_pF\\b[l7%F]a`l7$Fh^]l$!+bB3z^Fa_pF \\b[l7%Faa`l7$Fb^]l$!+:rE(=&Fa_pF\\b[l7%Fea`l7$F\\^]l$!+htm$>&Fa_pF\\b [l7%Fia`l7$Ff]]l$!+LUa)>&Fa_pF\\b[l7%F]b`l7$F`]]l$!++4I-_Fa_pF\\b[l7%F ab`l7$Fj\\]l$!+yod0_Fa_pF\\b[l7%Feb`l7$Fd\\]l$!+'*)=%4_Fa_pF\\b[l7%Fib `l7$F^\\]l$!+Jtf:_Fa_pF\\b[l7%F]c`l7$Fh[]l$!+J> \"3?v&Fa_pF\\b[l7%Fif`l7$F^f\\l$!+1yfe dFa_pF\\b[l7%F]g`l7$Fhe\\l$!+@0ohdFa_pF\\b[l7%Fag`l7$Fbe\\l$!+atShdFa_ pF\\b[l7%Feg`l7$F\\e\\l$!+:g*yv&Fa_pF\\b[l7%Fig`l7$Ffd\\l$!+SnB^dFa_pF \\b[l7%F]h`l7$F`d\\l$!+`n\\TdFa_pF\\b[l7%Fah`l7$Fjc\\l$!+hOsGdFa_pF\\b [l7%Feh`l7$Fdc\\l$!+p![Hr&Fa_pF\\b[l7%Fih`l7$F^c\\l$!+(f&=%p&Fa_pF\\b[ l7%F]i`l7$Fhb\\l$!+[%QCn&Fa_pF\\b[l7%Fai`l7$Fbb\\l$!+1mpZcFa_pF\\b[l7% Fei`l7$F\\b\\l$!+,(Q*>cFa_pF\\b[l7%Fii`l7$Ffa\\l$!+ID8*e&Fa_pF\\b[l7%F ]j`l7$F`a\\l$!+O^BbbFa_pF\\b[l7%Faj`l7$Fj`\\l$!+mA>=bFa_pF\\b[l7%Fej`l 7$Fd`\\l$!+So$zZ&Fa_pF\\b[l7%Fij`l7$F^`\\l$!+@eQMaFa_pF\\b[l7%F][al7$F h_\\l$!+lWV(Q&Fa_pF\\b[l7%Fa[al7$Fb_\\l$!+\">YpL&Fa_pF\\b[l7%Fe[al7$F \\_\\l$!+t]t#G&Fa_pF\\b[l7%Fi[al7$Ff^\\l$!+Mc`C_Fa_pF\\b[l7%F]\\al7$F` ^\\l$!+/-&>;&Fa_pF\\b[l7%Fa\\al7$Fj]\\l$!+X_N%4&Fa_pF\\b[l7%Fe\\al7$Fd ]\\l$!+%RI2-&Fa_pF\\b[l7%Fi\\al7$F^]\\l$!+)>T$R\\Fa_pF\\b[l7%F]]al7$Fh \\\\l$!+$*>OFa_pF\\b[l7%Fa_al7$Fbi[l$!+[gMfMFa_pF\\b[l7%Fe_al7$F\\i[l$!+>W \\+LFa_pF\\b[l7%Fi_al7$Ffh[l$!+%GXC9$Fa_pF\\b[l7%F]`al7$F`h[l$!+o=([)H Fa_pF\\b[l7%Fa`al7$Fjg[l$!+?HcFGFa_pF\\b[l7%Fe`al7$Fdg[l$!+>(*QqEFa_pF \\b[l7%Fi`al7$F^g[l$!+l\"zK^#Fa_pF\\b[l7%F]aal7$Fhf[l$!+'G$>cBFa_pF\\b [l7%Faaal7$Fbf[l$!+MY6*>#Fa_pF\\b[l7%Feaal7$F\\f[l$!+)fO?/#Fa_pF\\b[l7 %Fiaal7$Ffe[l$!+]v&\\)=Fa_pF\\b[l7%F]bal7$$\"+7Jh$y'Fc`p$!+ow(ys\"Fa_p F\\b[l7%Fabal7$$\"+*\\3;4$Fc`p$!+uuzq:Fa_pF\\b[l7%Fgbal7$Fdd[l$!+Ctr89 Fa_pF\\b[l7%F]cal7$F^d[l$!+\\tjc7Fa_pF\\b[l7%Facal7$$\"+'yC\"Fb\\p;$!$R'Fb\\p$\"$R'Fb\\p" 1 2 0 1 10 0 2 9 1 2 1 1.000000 45.000000 45.000000 0 0 "Curve 1" "Curve 2" "Curve 3" "Curve \+ 4" "Curve 5" "Curve 6" }}}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}}{PARA 0 "" 0 "" {TEXT -1 52 " #---------------------------------------------------" }}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 35 "coefficients of the combined scheme" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }} }{SECT 1 {PARA 4 "" 0 "" {TEXT -1 36 "coefficients of the combined sch eme " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 9696 "ee := \{c[2]=139/2500,\nc[3]=473499/4616000,\nc[4]= 1420497/9232000,\nc[5]=1923/5000,\nc[6]=923/2000,\nc[7]=769/5000,\nc[8 ]=8571/10000,\nc[9]=14280181565693441800/15023510624554266963,\nc[10]= 3611/5000,\nc[11]=15/16,\nc[12]=1,\nc[13]=1,\n\na[2,1]=139/2500,\na[3, 1]=94226774499/11846945536000,\na[3,2]=224201303001/2369389107200,\na[ 4,1]=1420497/36928000,\na[4,2]=0,\na[4,3]=4261491/36928000,\na[5,1]=35 90880889914277/9341720958375000,\na[5,2]=0,\na[5,3]=-1122077307242443/ 778476746531250,\na[5,4]=1683359084698258/1167715119796875,\na[6,1]=32 79216863/71027928000,\na[6,2]=0,\na[6,3]=0,\na[6,4]=1418466643672/6146 756958375,\na[6,5]=725783021041/3932390759616,\na[7,1]=226287768190908 91/378192197956950000,\na[7,2]=0,\na[7,3]=0,\na[7,4]=11632687883767091 2/1047320101758291075,\na[7,5]=-21022506263989/614436056190000,\na[7,6 ]=182198401/10649112500,\na[8,1]=-150702609045465151/28014236885700000 0,\na[8,2]=0,\na[8,3]=0,\na[8,4]=-47048174572430533795112/678157397540 0382234375,\na[8,5]=-29670244019740727/6363654238520000,\na[8,6]=50335 0581600824913/125990342179812500,\na[8,7]=9,\na[9,1]=-1604591803554452 6531085666708200976799092954349059102554439418058855237034803138076235 914485211856830024864166368323768200/982948315772866484419854403492165 5975378257830080951893693520116322068439377446708493269493633244146421 522730587295739613,\na[9,2]=0,\na[9,3]=0,\na[9,4]=-2358539147881877873 0394458117511032898674036917053932712809944820432633202719845187666625 89229245453919623760032009543680000000/2178355261819555655580228003414 2946408638517212290660803134954358386302201588711716856633847016309003 2498196064101837445053273,\na[9,5]=-9911366389726261686789033714164561 0009390040553516492468305812280242970735403338282694739815868376532443 9618282500000000/79848142008846002789298925227605775190331269194726743 910364273272231784282184770467794155269096224513726772081370189773,\na [9,6]=9941127982121041338714935249721178582954714935869695264678103390 5129048593757052549024957474512389085654050445280000000/10219750071154 3555291993605654791795484972275164188674551435468249918836166971629037 07552409044338358355837306818391433,\na[9,7]=1943275996723801340958982 9171991296136367807379302352500708132842509843157444880977931073253282 1200046895000000000/11996011488227722649656673931136490891256463292620 473601841875115170259043531987881275965965525693289412424400177,\na[9, 8]=-473814386768412218959381624419945054048338437216354995199052538755 0768038015218275414120082248510000000000/45627381556119209828916528385 3264342733761371582288925031587925670818017616283449256189927490598858 19540261,\na[10,1]=100509763879264306824096153463041174636629364248095 333923106653001873/\n 22949032400764462804236175621743615583246 1488260089524115475000000000,\na[10,2]=0,\na[10,3]=0,\na[10,4]=3507626 1889213578261995286390053983221937920015616/\n 8903662403052468 890234321680409039895089390971875,\na[10,5]=29877053472248545227782869 189767925950557009/\n 10443709158645362958089905740134206606110 000,\na[10,6]=-7260202518279888944289396655384428601277677001958883877 6297031451/40918439380405007926071673834718276106178863851548244800289 187500,\na[10,7]=-1532291206386437051213014598849209860548650299410781 6190/3130069400645604876183669549605539076333579290524919889,\na[10,8] =66085154677219418645471125072555541174985695924/310222736648235062495 097951962638800384203417327,\na[10,9]=-3347565461896560762549026667823 1230366345830527265525310030016230875755239420324728600957368877132012 320553021/563607181486505082775419237614873016043169125130359330269370 345097328575740250457475506596993780051575000000000,\na[11,1]=-5101097 760197841615137571256611109219669965728737004453654940399435749927779/ 3460254790065218498025394912113654440547987641021174968985010753183759 952128,\na[11,2]=0,\na[11,3]=0,\na[11,4]=-7392828537584122579674532421 47288028248514875000/\n 67244182875654829098323943713374207709 631980757,\na[11,5]=-304301954438407952266341438991435702455078125/\n \+ 26817588459995310678515661957421952038871616,\na[11,6]=7944893 9923998931164766316975203638555016544623086227984461001450043717807656 25/\n 887005161602928659252827604035768657298435884939489754462 273147402132735951354,\na[11,7]=27146666288983512861481079691606948806 2485834784939879774912795880859375/\n 1708005786820880758491191 9572395871967414618088449127584294090356526117,\na[11,8]=-547671639173 493535187249719091874257724281347519101322819710937500/\n 5545 897345777551056363997101744668576293320106258184122878184564689,\na[11 ,9]=278805054229456473051914785770130401056277521071386274820253852826 0236025540352528788884634515567105686501350867517347334983884086148375 63/5703278975899384247167398970672581931599634186137224382398210398440 7642346136141015055914541200747733442561186185728215790313661884647470 336,\na[11,10]=-7641031089620713267943821483116886435546875/\n \+ 1865115549729059107236140767953375513727732867,\na[12,1]=-2173296165 244568434534168496725754283210370856714048295955495704392191998074219/ \n 82632970981719746825391281499627672129771927187304712149693 9265708086637102000,\na[12,2]=0,\na[12,3]=0,\na[12,4]=-117648588933447 2345397948024769479865991267657667189808793600000/\n 128238274 125745784857395023683055391720808636521055761250085619,\na[12,5]=-9016 41526273088667293677983960434306145731566007759375000/\n 47006 051321531726477650870675115597779536141098588395119,\na[12,6]=44619197 7093458430386099958102902220927137343575035730212893991884410752778938 25000/\n 304722682986775966780567823708358491602845662906901449 6162891378892137605913688013,\na[12,7]=9483397426652107169315607674597 05347432692833888702670755668800833857753125000/\n 541002030685 23816523920547203712158257383235486719126879290708372319555115953,\na[ 12,8]=-469359986530784242367841163967997985280992052062479522694037718 750000/\n 1262000068797740029604893636717124646814536106030908 513651797310900389,\na[12,9]=-6641274810062608662454173531678791117254 3860816013412978871555537846557311781300175723826069305190181259434160 8922016461968723754067215934799054852630446938063/94740531367577932788 6462044861834846879473861539583479995101168291116168844040438751584392 785913224239894029843821464363224405440877235651675827150051218506000, \na[12,10]=12459468894943123671827871807868857124596796164158415171306 5625000/\n 244226611999512391459992648734463087356588030400313 6401896150897137,\na[12,11]=561064742310110985699639624014768981793940 948761680464947352/\n 6713793915263290553167253436944514283055 92235340194400714475,\na[13,1]=211253736543960531784221440029493960084 5864356126794907105288476157299972777/\n 9791129992576348954381 966935712572673760150144277222568673000574989723295600,\na[13,2]=0,\na [13,3]=0,\na[13,4]=634017325686501748824696552730375578670240146494736 59852800000/\n 759743717956469990005682668279009587650186779400 5624909986891,\na[13,5]=6086761737797404637560639819223305802258863489 403125000/\n 27848590790027885422588467603450510954813164968029 32391,\na[13,6]=-25596656018056374091988888506984281725655172189735771 46475607855715703433900000/\n 15170757814501111695553564393421 15045941218379260955323225394329122945220575603,\na[13,7]=-35629627244 14944757891560164749973139486260772587248231213362868690625000/\n \+ 408976639178968899516222045518619849445955714595234573207901740578 867341,\na[13,8]=27274774093948890572327172846505411328624378015249298 079900000000/\n 11159224869186251417943163509972464260302173256 03407120793811703663,\na[13,9]=151764998027328779481071148092868976296 9941359446589355244683243470868873980537462717945530891242892188019850 453060256773054273287757818339/179310963514035495173248229516194926905 9025910955416969211177149416280508829782124284849210292344753221574244 6890559805801281057084758250800,\na[13,10]=365246427065695972768456123 68244490741747970924877885921875000/\n 67204508345515429491838 411146746729660269255184823759889942481,\na[13,11]=0,\na[13,12]=0,\n\n b[1]=55284231195707975647126708111723/12588142504750145545176036121140 00,\nb[2]=0,\nb[3]=0,\nb[4]=0,\nb[5]=0,\nb[6]=101652048214282205518610 445783893750000000/289586658278060310247144250081091360673509,\nb[7]=6 219731958882093270433753490048828125000000/252687923145628991821868198 00834272764054341,\nb[8]=471525882014932587321673707929687500000000/52 3728817391198795728810839649314044495553,\nb[9]=6399841965907450296097 9467027044380533513499562179716145788153981258354882170183557294261050 806789914954901252698438375102943237148428019253408419/140673838782241 5998067693599950032426772772394751630953646406168313006343761645295132 6819824531063499969576146304645311203502485409773697867038000,\nb[10]= 594562755257530592552224345703125000000/123802720910327682301431417435 953442122031,\nb[11]=-1886691133979705639959153870454656/3979475858858 35383951563312487675,\nb[12]=-50061468875139778913910254637881/1411866 41936036184819986782313781,\nb[13]=0,\n\n`b*`[1]=363554399229402147520 2312550589/83920950031667636967840240807600,\n`b*`[2]=0,\n`b*`[3]=0,\n `b*`[4]=0,\n`b*`[5]=0,\n`b*`[6]=91472553308336221233020750122000000000 /270389036674192633283981559366098375979,\n`b*`[7]=3124571087910685950 0854236329453125000000/125747467177230198813995913261775760852127,\n`b *`[8]=7580382785455138868782239796250000000000/33873008089978031564549 954803189465564389,\n`b*`[9]=-1264668207534772000389416909088295806823 0761859572980915113663566381454113194328075998609241732568121085553195 05561297378438513/3145813368236387937635983732861131099009925923393634 6807079836410100132902013789877583819562913435723874810037875576750158 649200,\n`b*`[10]=2363202400377740499662213515625000000/19167474982246 118950523520271861502109,\n`b*`[11]=0,\n`b*`[12]=0,\n`b*`[13]=29658763 53908674111604434604609/47062213978678728273328927437927\}:" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT 259 5 "nodes " }{TEXT -1 2 ": " }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 32 "seq(c[i ]=subs(ee,c[i]),i=2..13);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6./&%\"cG6# \"\"##\"$R\"\"%+D/&F%6#\"\"$#\"'*\\t%\"(+gh%/&F%6#\"\"%#\"((\\?9\"(+?B */&F%6#\"\"&#\"%B>\"%+]/&F%6#\"\"'#\"$B*\"%+?/&F%6#\"\"(#\"$p(F?/&F%6# \"\")#\"%r&)\"&++\"/&F%6#\"\"*#\"5+=W$pl:=!G9\"5jpEaXi5N-:/&F%6#\"#5# \"%6OF?/&F%6#\"#6#\"#:\"#;/&F%6#\"#7\"\"\"/&F%6#\"#8Ffo" }}}{PARA 0 " " 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT 259 20 "linking coefficie nts" }{TEXT -1 1 ":" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 50 "seq(s eq(a[i,j]=subs(ee,a[i,j]),j=1..i-1),i=1..13);" }}{PARA 12 "" 1 "" {XPPMATH 20 "6jo/&%\"aG6$\"\"#\"\"\"#\"$R\"\"%+D/&F%6$\"\"$F(#\",*\\un A%*\"/+g`Xp%=\"/&F%6$F/F'#\"-,II,UA\".+s5*QpB/&F%6$\"\"%F(#\"((\\?9\") +!Gp$/&F%6$F^rn6/&F%6$\"\"'F(#\"+jo@zK\",+!Gz-r/&F%6$F\\oF'FC/& F%6$F\\oF/FC/&F%6$F\\oF<#\".sOkm%=9\".v$epvYh/&F%6$F\\oFL#\"-T5-$yD(\" .;'f2RKR/&F%6$\"\"(F(#\"2\"*34>oxGE#\"3++&p&z>#>y$/&F%6$FepF'FC/&F%6$F epF/FC/&F%6$FepF<#\"374nP)yoK;\"\"4v5He<5?t/\"/&F%6$FepFL#!/*)RE1D-@\" 0++>cgV9'/&F%6$FepF\\o#\"*,%)>#=\",+D6\\1\"/&F%6$\"\")F(#!3^^YX!4Eq]\" \"3+++d)oB9!G/&F%6$FdrF'FC/&F%6$FdrF/FC/&F%6$FdrF<#!87^zL0VsX<[q%\"7vV B#Q+a(Rd\"y'/&F%6$FdrFL#!2F2u>SCq'H\"1++_QUljj/&F%6$FdrF\\o#\"38\\#3g \"e]L]\"3+D\")z@M!*f7/&F%6$FdrFep\"\"*/&F%6$FctF(#!er+#oPKoj;k[-Io&=@& [9fBw!QJ![.P_&)e!=%RWb-\"f!\\V&H4*zw4?3nm&3Jl_Wb.=f/;\"dr8'RdH(eIF_@k9 WKj$\\pK\\3nWx$R%o?K;,_$p$*=&43IyDy`(fl@\\.W&)>W[mGx:$[H)*/&F%6$FctF'F C/&F%6$FctF/FC/&F%6$FctF<#!jr+++!oV&4?.gPi>RXX#H#*eimw=X)>F?LEV?[%*4Gr KR0)\\K+4j,ZQjcor6()e,AI'QeV &\\8.3m!H7s^Q'3k%H9M+G-ebcb>=Eb$y@/&F%6$FctFL#!cr++++DG='RWKlPoe\")RZp #GQLSN2(HC!G7eIoC\\;NbS+R4+hX;9P.*y'ohis*Qm8\"**\"brt(*=q83sns8XA'4p_: %zn/x%=#G%yJAFtUO5RuEZ>p7L!>vdgF_#*)H*y-g%)3?9[)z/&F%6$FctF\\o#\"br+++ !GX/0ac3*Q7XZd\\-\\D0dPf[!H^!R.\"yYE&ppe$\\raHey6s\\_$\\rQ8/@@)z7T**\" brL9R=oIPeNe$QV/4Cb2P!H;(p;O)=*\\#oaV^Xn)=k^Fs\\[&z\"zacg$*>HbNa62](>- \"/&F%6$FctFep#\"]r++++]*o/+7#G`K2Jz(4)[WdJ%)4D%G832]_BIzt!yOOhH\"*>\"\\rx,SCCT*G$pDb'f'fF\")y)>`V!f-<:^(=%=gt/i#HjkD\"*3\\O 6$Rnc'\\EsF#)[6g*>\"/&F%6$FctFdr#!eq+++++^[A3?TTv#=_,Q!o2b(QD0*>&*\\N; sVQ$[S0X*>Wi\"Qf*=ATonQ9QZ\"fqh-a>e))f!\\F**=c#\\MG;w,=3nDzeJ]#*)G#er8 wLFMkK&QGl\"*G)4#>hb\"QFc%/&F%6$\"#5F(#\"`ot=+`m5BRL&4[UOHmju6/jM:'4Co Ik#zQw405\"`o++++]Z:T_*3g#)[hC$e:Ou@cA$)R0!R'G&*>EyN@*)=Ew]$\"Rv=(4R*3&*)R!4/ o@VB!*)oC0.Cm.*)/&F%6$FawFL#\"M4qb]f#zw*=pGyF_a[AZ`q()H\"M++61m?M,u0** 3eHOX'e\"4PW5/&F%6$FawF\\o#!\\o^9.(Hw(Q))e>+xwF,'GWQbmR*GW*)))z#=D?gs \"\\o+v=*G+[C[:&Q')yh5w#=Z$Qnrg#z+0/QR%=4%/&F%6$FawFep#!Y!>;y5%*H]'[0' )4#\\))f9I@^qV'Q17HK:\"X*))>\\_!HzNLw!Rbg\\&pO=w[gX1Sp+8$/&F%6$FawFdr# \"PCfp&)\\sna^3m\"QFtT.UQ+)QE'>&z4&\\i]B[mtA-J/&F%6$FawFc t#!iq@Ib?B,Kr()ot&4gGZK?%R_vv3B;+.5`_ls_IeMm.BJ#ymE!\\Dwgl*=YlvM$\"jq+ +++]d^+y$*pf1bZd/DSddGt4X.Pp-Lf.8D\"pJ/;I([hP#>ax#30l[\"=2Oc/&F%6$\"#6 F(#!gozx#*\\dV*RS\\l`W+P(Gd'*p'>#46hc7dP^hTy>gx4,^\"goG@&*fP=`2,&)*o\\ <@5k()zaSWl8@\"\\RD!)\\=_1!za-Y$/&F%6$FbzF'FC/&F%6$FbzF/FC/&F%6$FbzF<# !Q+]([^[#G!)GZ@C`u'zD7%eP&GGR(\"Pd2)>j4x?uLrVRK)4H[lvG=Ws'/&F%6$FbzFL# !ND\"y]X-dV\"**Q9MmA&zSQW&>I/$\"M;;()Q?&>Ud>m:&y1J&**f%)e(eRs&>>\"\\e2)3#oy0! 3GK,\">vM\"GCxDu=4>(\\s=NN\\t\"R;nZ&\"^o*okX=yG 7%=ei5?LHw&oYu,r*ROc5bxdM(*ea&/&F%6$FbzFct#\"esjv$[h3%)Q)\\LZt^n3N,lo0 rc:Xj%)))yGDNSb-O-EG&QD?[F'Qr5_xi0,/8qdy9>0tkXHU00)y#\"gsO.ZZY)=m8.z:# Gd='=hDWLxu+7a9f0:59OhMUwS%)R5#)R#QCs8'=M'*fJ>es1(*)RnrC%Q**e(*yKq&/&F %6$FbzFaw#!LvoaNk)o6$[@Q%zE82i*3J5k(\"OnGtFP^vL&zwShB2\"f!H(\\b6l=/&F% 6$\"#7F(#!jo>U2)*>>#R/d\\bfH[Src3P5KGads'\\oT`M%oXClhHt@\"io+?5Pm33dER p\\@r/t=F>xH@nF'*\\\"G\"RDou><)4(Hj#)/&F%6$Fi]lF'FC/&F%6$Fi]lF/FC/&F%6 $Fi]lF<#![o++g$z3)*=nwln7*f')z%pZ-[zRXBZM$*)e[w6\"jn>c3]7wb5_O'33s\"Rb IoB]Rd[yXd7u#Q#G\"/&F%6$Fi]lFL#!Z+]Pfx+m:tXhIM/'R)zn$Hn')3ti_T;!*\"Y>^ R)e)4Th`zxf:^nq3lxksJ:K^g+Z/&F%6$Fi]lF\\o#\"^p+]#Q*yFv5W)=*R*G@Id.vNMP r#4A-H5e**4'QI%eM4x>>Y%\"]p8!)o8fgP@*)y8*G;'\\9!p!HmXGg\"\\e$3P#yc!ymf x')HoAZI/&F%6$Fi]lFep#\"io+]7`x&Q$3!)ocvqEq))Q$GpKuM0(fuwg:$pr5_mU(R$[ *\"ho`f6b&>BP32Hzo7>n[NKQd#e@r.sa?R_;Q_oI?+T&/&F%6$Fi]lFdr#!`o++v=x.%p A&zC1_?*4G&)z*z'R;TyOUUyIl)*f$p%\"ao*Q+4J(z^O^34.1h`9okCrrOO*[gH+u(zo+ +i7/&F%6$Fi]lFct#!jtj!QpWIE&[0*zMf@nSvB(o>Y;?#*3;M%f7=!>0$pg#Qsv,I\"y6 tbYy`b:()yHT8g\"3'Qas6\"zy;`tTXi'3E15[FTm\"jt+g]=70]r#en^cBx3W0WAjVY@Q %)HS*)RUA8fy#R%e^(Q//W)oh6\"Ho65&**zMeR:'QZzo%[$='[/ik)yKzdn8`SZ*/&F%6 $Fi]lFaw#\"]o+]ilIr^TeT;'z'fCr&)oy!=(y#=nBJ%\\*)o%fC\"\"^oPr*3:'*=SOJ+ /.)ect3jWt[E**f9R7&**>hEUC/&F%6$Fi]lFbz#\"gn_t%\\Y!oh([4%Rz\")*oZ,C'R' *p&)465Buk5c\"gnvWr+W>S`B#f0$G9X%pV`s;`0Hj_\"Rz8n/&F%6$\"#8F(#\"goxF(* *HdhZ)G0r!\\zEhNke%3gR\\H+W@UyJ0'RaOPD6#\"go+cHB(*)\\d+InoDAxU9],wtEd7 d$p'>Qa*[jd#**H6z*/&F%6$FfalF'FC/&F%6$FfalF/FC/&F%6$FfalF<#\"in++!G&)f OZ\\Y,Cq'ybPIFb'pC)[<]'oDt,M'\"hn\"*o)*4\\i0Szn=]we4!z#oEo0+**pk&zrV(f (/&F%6$FfalFL#\"X+]7.%*[j)eA!eIB#>)R1cPYS(zP4 uj0=gl'fD\"jo.cd?_%H7HVRDKKb4Ez$=7%f/:@MRkNb&p66]9yvq^\"/&F%6$FfalFep# !do+]i!poGO87B[ses2E'[RJ(*\\Z;g:*yvW\\TCF'Hc$\"coTt')y0u,z?tXB&f9d&fW \\)>'=b/Ai^**)o*y\"Rm(*3%/&F%6$FfalFdr#\"\\o++++*z!)H\\_,yViG8T0l%G!)=#*GC\"*3`Xzriu`!)R()o3ZVKoW_N*eY%f8%*pHw*oG4[62\" [z(Gt-)*\\w^\"\"gs+3DeZ3d5G,e!)f0*oWUd@KvWBH5#\\[GC@yH)30G;%\\r<6#ppTb 4\"f-f!p#\\>;&H#[K<&\\NS^j4Jz\"/&F%6$FfalFaw#\"in+](=#f)y([#4(zuT2\\W# oBhXoF(fplqUY_O\"in\"[U**))fP#[=b#p-mHnuY6TQ=\\Ha^X$3X?n/&F%6$FfalFbzF C/&F%6$FfalFi]lFC" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT 259 39 "weights for the 13 stage order 8 scheme" }{TEXT -1 1 ": " }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 32 "seq(b[i]=subs(ee,b[i]),i =1..13);" }}{PARA 12 "" 1 "" {XPPMATH 20 "6//&%\"bG6#\"\"\"#\"AB<63n7Z c(zq&>JUGb\"C+S67Ogt>i\"MTV0kFFM3!)>o =#=**Gc9Bzo_#/&F%6#\"\")#\"K++++voHzqt;K(eK\\,#)e_r%\"K`b\\WSJ\\'R3\") Gdz)>\"R<)GP_/&F%6#\"\"*#\"]t>%3MD>!G%[rBVH5v$Q%)p_7!\\&\\\"**y130hUHd N=q@)[Ne7)R:)yXhrz@c*\\8N`!QWq-n%z4'H]u!f'>%)*R'\"]t+!Qq'ypt(4a[-N?6`k /j9w&p**\\j5`C)>oK^HX;wVj+8$ohSYO&4j^ZRsFxEC.]**f$pn!)*fTAyQQnS\"/&F%6 #\"#5#\"H+++DJqXVA_DfIvDbFc%f\"KJ?7UM&fV<9V,BoF.\"4s-Q7/&F%6#\"#6#!CcY XqQ:f*RcqzR8\"p')=\"Bvw[7Lc^RQNe)eeZzR/&F%6#\"#7#!A\")yja-\"R\"*y(R^() o91]\"B\"y8By')*>[=Og$>k'=T\"/&F%6#\"#8F/" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT 259 39 "weights for the 13 stage order 7 scheme" }{TEXT -1 1 ":" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 38 "seq(`b*`[i]=subs(ee,`b*`[i]),i=1..13);" }}{PARA 12 "" 1 "" {XPPMATH 20 "6//&%#b*G6#\"\"\"#\"@*e]DJ-_Z@SH#*RaNO\"A+w!3CSy'pjn;.]4#R)/&F%6# \"\"#\"\"!/&F%6#\"\"$F//&F%6#\"\"%F//&F%6#\"\"&F//&F%6#\"\"'#\"G++++?7 ]2-L7AO$3Lbs9*\"HzfP)4m$f:)RGLE>um.*Qq#/&F%6#\"\"(#\"J+++DJXHjBa3]fo5z 3rX7$\"KF@&3wvIs`b3@\"oDtT#4' )*f2GV>8TX\"QmNm8^\"4)Hdf=wI#o!eH)34pT*Q+?xMv?oYE\"\"[s+#\\'e,vwb(y.5[ (QsNM\"Hc>Qex)*y8?!H8+,TO)zq!oMORBf#*4!*4J6'Gt$)fjPzQO#oL\"e9$/&F%6#\" #5#\"F+++Dc^8Am*\\SxP+C?jB\"G4@]h=F?N_]*=hC#)\\Zn\">/&F%6#\"#6F//&F%6# \"#7F//&F%6#\"#8#\"@4YgMWg6Tn3RNwe'H\"AFzVF*GLFG(y'yR@iq%" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }} }}{PARA 0 "" 0 "" {TEXT -1 52 "#-------------------------------------- -------------" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}} {PARA 0 "" 0 "" {TEXT -1 25 "#========================" }}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 23 "Verner's \"Maple\" scheme" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 115 "This scheme is obta ined from the 13 stage scheme supplied to Maple by Jim Verner for the \+ dverk78 method in Maple's " }{TEXT 0 6 "dsolve" }{TEXT -1 2 ". " }} {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 15 "The nodes are: " }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "c[2] = 1/16 " "6#/&%\"cG6#\"\"#*&\"\"\"F)\"#;!\"\"" }{TEXT -1 3 ", " }{XPPEDIT 18 0 "c[3] = 112/1065" "6#/&%\"cG6#\"\"$*&\"$7\"\"\"\"\"%l5!\"\"" } {TEXT -1 3 ", " }{XPPEDIT 18 0 "c[4] = 56/355" "6#/&%\"cG6#\"\"%*&\"# c\"\"\"\"$b$!\"\"" }{TEXT -1 3 ", " }{XPPEDIT 18 0 "c[5] = 39/100" "6 #/&%\"cG6#\"\"&*&\"#R\"\"\"\"$+\"!\"\"" }{TEXT -1 3 ", " }{XPPEDIT 18 0 "c[6] = 7/15" "6#/&%\"cG6#\"\"'*&\"\"(\"\"\"\"#:!\"\"" }{TEXT -1 3 ", " }{XPPEDIT 18 0 "c[7] = 39/250" "6#/&%\"cG6#\"\"(*&\"#R\"\"\"\" $]#!\"\"" }{TEXT -1 3 ", " }{XPPEDIT 18 0 "c[8] = 24/25" "6#/&%\"cG6# \"\")*&\"#C\"\"\"\"#D!\"\"" }{TEXT -1 3 ", " }{XPPEDIT 18 0 "c[9] = 1 4435868/16178861" "6#/&%\"cG6#\"\"**&\")oeV9\"\"\"\")h)yh\"!\"\"" } {TEXT -1 3 ", " }{XPPEDIT 18 0 "c[10] = 11/12" "6#/&%\"cG6#\"#5*&\"#6 \"\"\"\"#7!\"\"" }{TEXT -1 3 ", " }{XPPEDIT 18 0 "c[11] = 19/20" "6#/ &%\"cG6#\"#6*&\"#>\"\"\"\"#?!\"\"" }{TEXT -1 3 ", " }{XPPEDIT 18 0 "c [12] = 1" "6#/&%\"cG6#\"#7\"\"\"" }{TEXT -1 3 ", " }{XPPEDIT 18 0 "c[ 13]=1" "6#/&%\"cG6#\"#8\"\"\"" }{TEXT -1 2 ". " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}{PARA 0 "" 0 "" {TEXT -1 52 "#---------------------------------------------- -----" }}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 28 "checking the combined sc heme" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 35 "coefficient s of the combined scheme" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 4879 "ee := \{c[2]=1/16,\nc[3]=112/1065 ,\nc[4]=56/355,\nc[5]=39/100,\nc[6]=7/15,\nc[7]=39/250,\nc[8]=24/25,\n c[9]=14435868/16178861,\nc[10]=11/12,\nc[11]=19/20,\nc[12]=1,\nc[13]=1 ,\n\na[2,1]=1/16,\na[3,1]=18928/1134225,\na[3,2]=100352/1134225,\na[4, 1]=14/355,\na[4,2]=0,\na[4,3]=42/355,\na[5,1]=94495479/250880000,\na[5 ,2]=0,\na[5,3]=-352806597/250880000,\na[5,4]=178077159/125440000,\na[6 ,1]=12089/252720,\na[6,2]=0,\na[6,3]=0,\na[6,4]=2505377/10685520,\na[6 ,5]=960400/5209191,\na[7,1]=21400899/350000000,\na[7,2]=0,\na[7,3]=0, \na[7,4]=3064329829899/27126050000000,\na[7,5]=-21643947/592609375,\na [7,6]=124391943/6756250000,\na[8,1]=-15365458811/13609565775,\na[8,2]= 0,\na[8,3]=0,\na[8,4]=-7/5,\na[8,5]=-8339128164608/939060038475,\na[8, 6]=341936800488/47951126225,\na[8,7]=1993321838240/380523459069,\na[9, 1]=-1840911252282376584438157336464708426954728061551/\n 299192 3615171151921596253813483118262195533733898,\na[9,2]=0,\na[9,3]=0,\na[ 9,4]=-14764960804048657303638372252908780219281424435/\n 298169 2102565021975611711269209606363661854518,\na[9,5]=-8753250485021304411 18613421785266742862694404520560000/\n 170212030428894418395571 677575961339495435011888324169,\na[9,6]=763205196415429092566184979837 0645637589377834346780/\n 17340872574188115830498003475818652604 79233950396659,\na[9,7]=7519834791971137517048532179652347729899303513 750000/\n 1045677303502317596597890707812349832637339039997351, \na[9,8]=1366042683489166351293315549358278750/\n 14463141822426 7718165055326464180836641,\na[10,1]=-63077736705254280154824845013881/ 78369357853786633855112190394368,\na[10,2]=0,\na[10,3]=0,\na[10,4]=-31 948346510820970247215/6956009216960026632192,\na[10,5]=-33786048053942 55292453489375/517042670569824692230499952,\na[10,6]=10015878441833259 81198091450220795/184232684207722503701669953872896,\na[10,7]=18702307 5231349900768014890274453125/25224698849808178010752575653374848,\na[1 0,8]=1908158550070998850625/117087067039189929394176,\na[10,9]=-529568 18288156668227044990077324877908565/\n 29127799594774339863498 22224412353951940608,\na[11,1]=-10116106591826909534781157993685116703 /9562819945036894030442231411871744000,\na[11,2]=0,\na[11,3]=0,\na[11, 4]=-9623541317323077848129/3864449564977792573440,\na[11,5]=-482334833 3146829406881375/576413233634141239944816,\na[11,6]=656611924651499688 4067001154977284529/970305487021846325473990863582315520,\na[11,7]=222 6455130519213549256016892506730559375/36488044315967525557743564838004 7355776,\na[11,8]=39747262782380466933662225/1756032802431424164410720 256,\na[11,9]=48175771419260955335244683805171548038966866545122229/\n 1989786420513815146528880165952064118903852843612160000,\na[11 ,10]=-2378292068163246/47768728487211875,\na[12,1]=-321802217475859983 1659045535578571/1453396753634469525663775847094384,\na[12,2]=0,\na[12 ,3]=0,\na[12,4]=26290092604284231996745/5760876126062860430544,\na[12, 5]=-697069297560926452045586710000/41107967755245430594036502319,\na[1 2,6]=1827357820434213461438077550902273440/139381013914245317709567680 839641697,\na[12,7]=643504802814241550941949227194107500000/2421246091 18836550860494007545333945331,\na[12,8]=162259938151380266113750/59091 082835244183497007,\na[12,9]=-2302825163287352381854541485685701561667 8575554130463402/\n 200131691831914445034439052404056033499784 24504151629055,\na[12,10]=7958341351371843889152/328446798844320358130 5,\na[12,11]=-507974327957860843878400/121555654819179042718967,\na[13 ,1]=-549080624436801105208519835138333/3539871090287071396871008856004 00,\na[13,2]=0,\na[13,3]=0,\na[13,4]=29116675312186033956481/533181895 7833865866320,\na[13,5]=-91153092961177216058210567600/760926765301702 8089793994539,\na[13,6]=1540775569495234383390307262972464/16475124873 3597841985831445059895,\na[13,7]=-202748825453638632121202135762230000 0/7563800784313191306927092359781550321,\na[13,8]=-7075929545777566000 25/2430671607007101253926,\na[13,9]=-315869406877370103440389763510384 832076674110731/\n 5508459010167147941145163578595317549263005 00125,\na[13,10]=4305840920849725632512/5066413598663346018375,\na[13, 11]=0,\na[13,12]=0,\n\nb[1]=4631674879841/103782082379976,\nb[2]=0,\nb [3]=0,\nb[4]=0,\nb[5]=0,\nb[6]=14327219974204125/40489566827933216,\nb [7]=2720762324010009765625000/10917367480696813922225349,\nb[8]=-49853 3005859375/95352091037424,\nb[9]=4059320304637772479267050305961754374 02459637909765779/\n 788039194363218410832018860412015372297691150 88303952,\nb[10]=-10290327637248/1082076946951,\nb[11]=863264105888000 /85814662253313,\nb[12]=-29746300739/247142463456,\nb[13]=0,\n\n`b*`[1 ]=325503096889/7282953149472,\n`b*`[2]=0,\n`b*`[3]=0,\n`b*`[4]=0,\n`b* `[5]=0,\n`b*`[6]=495437430316125/1396191959583904,\n`b*`[7]=7607410722 16796875000/3055518466469861159313,\n`b*`[8]=1186143278515625/44497642 4841312,\n`b*`[9]=13791579353894559147128282108092181066885426809/3515 333032435874239702147176443897942760320040,\n`b*`[10]=-30733299644928/ 5410384734755,\n`b*`[11]=0,\n`b*`[12]=0,\n`b*`[13]=-45884771325/823808 21152\}:" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 20 "p rinted coefficients" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 357 "subs(ee,matrix([seq([c[i],seq(a[i,j],j=1.. i-1),``$(8-i)],i=2..8)]));\nfor ii from 9 to 13 do\n print(``);\n \+ print(c[ii]=subs(ee,c[ii])); \n for jj to ii-1 do\n print(a[ii, jj]=subs(ee,a[ii,jj]));\n end do:\nend do:print(``);\nfor ii to 13 d o\n print(b[ii]=subs(ee,b[ii]));\nend do:print(``);\nfor ii to 13 do \n print(`b*`[ii]=subs(ee,`b*`[ii]));\nend do:" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#K%'matrixG6#7)7*#\"$R\"\"%+DF(%!GF+F+F+F+F+7*#\"'*\\t% \"(+gh%#\",*\\unA%*\"/+g`Xp%=\"#\"-,II,UA\".+s5*QpBF+F+F+F+F+7*#\"((\\ ?9\"(+?B*#F8\")+!Gp$\"\"!#\"(\"\\hUF;F+F+F+F+7*#\"%B>\"%+]#\"1xU\"**)3 )3f$\"1+]Pe4sT$*F<#!1VCC2t2A6\"0]7`YnZy(#\"1e#)p%3fLo\"\"1voz>^rn6F+F+ F+7*#\"$B*\"%+?#\"+jo@zK\",+!Gz-rFoxGE#\"3++&p&z>#>y$FcgV9'#\"*,%)>#=\",+D6\\1\"F+7*#\"%r&) \"&++\"#!3^^YX!4Eq]\"\"3+++d)oB9!GFSCq'H\"1++_QUljj#\"38\\#3g\"e]L]\"3+D\")z@M!*f7\"\"*Q)pprint226 \"" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%!G" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/&%\"cG6#\"\"*#\"5+=W$pl:=!G9\"5jpEaXi5N-:" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/&%\"aG6$\"\"*\"\"\"#!er+#oPKoj;k[-Io&=@&[9fBw! QJ![.P_&)e!=%RWb-\"f!\\V&H4*zw4?3nm&3Jl_Wb.=f/;\"dr8'RdH(eIF_@k9WKj$\\ pK\\3nWx$R%o?K;,_$p$*=&43IyDy`(fl@\\.W&)>W[mGx:$[H)*" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/&%\"aG6$\"\"*\"\"#\"\"!" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/&%\"aG6$\"\"*\"\"$\"\"!" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/&%\"aG6$\"\"*\"\"%#!jr+++!oV&4?.gPi>RXX#H#*eimw=X)>F?LEV?[%*4Gr KR0)\\K+4j,ZQjcor6()e,AI'QeV &\\8.3m!H7s^Q'3k%H9M+G-ebcb>=Eb$y@" }}{PARA 11 "" 1 "" {XPPMATH 20 "6# /&%\"aG6$\"\"*\"\"&#!cr++++DG='RWKlPoe\")RZp#GQLSN2(HC!G7eIoC\\;NbS+R4 +hX;9P.*y'ohis*Qm8\"**\"brt(*=q83sns8XA'4p_:%zn/x%=#G%yJAFtUO5RuEZ>p7L !>vdgF_#*)H*y-g%)3?9[)z" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/&%\"aG6$\" \"*\"\"'#\"br+++!GX/0ac3*Q7XZd\\-\\D0dPf[!H^!R.\"yYE&ppe$\\raHey6s\\_$ \\rQ8/@@)z7T**\"brL9R=oIPeNe$QV/4Cb2P!H;(p;O)=*\\#oaV^Xn)=k^Fs\\[&z\"z acg$*>HbNa62](>-\"" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/&%\"aG6$\"\"*\" \"(#\"]r++++]*o/+7#G`K2Jz(4)[WdJ%)4D%G832]_BIzt!yOOhH\"*>\"\\rx,SCCT*G$pDb'f'fF\")y)>`V!f-<:^(=%=gt/i#HjkD\"*3\\O6$Rnc'\\EsF #)[6g*>\"" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/&%\"aG6$\"\"*\"\")#!eq++ +++^[A3?TTv#=_,Q!o2b(QD0*>&*\\N;sVQ$[S0X*>Wi\"Qf*=ATonQ9QZ\"fqh-a>e))f !\\F**=c#\\MG;w,=3nDzeJ]#*)G#er8wLFMkK&QGl\"*G)4#>hb\"QFc%" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%!G" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/&%\" cG6#\"#5#\"%6O\"%+]" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/&%\"aG6$\"#5\" \"\"#\"`ot=+`m5BRL&4[UOHmju6/jM:'4CoIk#zQw405\"`o++++]Z:T_*3g#)[hC$e:O u@cA$)R0!R' G&*>EyN@*)=Ew]$\"Rv=(4R*3&*)R!4/o@VB!*)oC0.Cm.*)" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/&%\"aG6$\"#5\"\"&#\"M4qb]f#zw*=pGyF_a[AZ`q()H\"M++61m? M,u0**3eHOX'e\"4PW5" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/&%\"aG6$\"#5\" \"'#!\\o^9.(Hw(Q))e>+xwF,'GWQbmR*GW*)))z#=D?gs\"\\o+v=*G+[C[:&Q')yh5w# =Z$Qnrg#z+0/QR%=4%" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/&%\"aG6$\"#5\" \"(#!Y!>;y5%*H]'[0')4#\\))f9I@^qV'Q17HK:\"X*))>\\_!HzNLw!Rbg\\&pO=w[gX 1Sp+8$" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/&%\"aG6$\"#5\"\")#\"PCfp&) \\sna^3m\"QFtT.UQ+)QE'>&z4&\\i]B[mtA-J" }}{PARA 11 "" 1 " " {XPPMATH 20 "6#/&%\"aG6$\"#5\"\"*#!iq@Ib?B,Kr()ot&4gGZK?%R_vv3B;+.5` _ls_IeMm.BJ#ymE!\\Dwgl*=YlvM$\"jq++++]d^+y$*pf1bZd/DSddGt4X.Pp-Lf.8D\" pJ/;I([hP#>ax#30l[\"=2Oc" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%!G" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#/&%\"cG6#\"#6#\"#:\"#;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/&%\"aG6$\"#6\"\"\"#!gozx#*\\dV*RS\\l`W+P(Gd'*p'># 46hc7dP^hTy>gx4,^\"goG@&*fP=`2,&)*o\\<@5k()zaSWl8@\"\\RD!)\\=_1!za-Y$ " }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/&%\"aG6$\"#6\"\"#\"\"!" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/&%\"aG6$\"#6\"\"$\"\"!" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/&%\"aG6$\"#6\"\"%#!Q+]([^[#G!)GZ@C`u'zD7%eP&GGR(\"Pd2) >j4x?uLrVRK)4H[lvG=Ws'" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/&%\"aG6$\"# 6\"\"&#!ND\"y]X-dV\"**Q9MmA&zSQW&>I/$\"M;;()Q?&>Ud>m:&y1J&**f%)e (eRs&>>\"\\e2)3#oy0!3<" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/&%\"aG6$\"# 6\"\")#!]o+v$4r>GK,\">vM\"GCxDu=4>(\\s=NN\\t\"R;nZ&\"^o*okX=yG7%=ei5?L Hw&oYu,r*ROc5bxdM(*ea&" }}{PARA 12 "" 1 "" {XPPMATH 20 "6#/&%\"aG6$\"# 6\"\"*#\"esjv$[h3%)Q)\\LZt^n3N,lo0rc:Xj%)))yGDNSb-O-EG&QD?[F'Qr5_xi0,/ 8qdy9>0tkXHU00)y#\"gsO.ZZY)=m8.z:#Gd='=hDWLxu+7a9f0:59OhMUwS%)R5#)R#QC s8'=M'*fJ>es1(*)RnrC%Q**e(*yKq&" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/&% \"aG6$\"#6\"#5#!LvoaNk)o6$[@Q%zE82i*3J5k(\"OnGtFP^vL&zwShB2\"f!H(\\b6l =" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%!G" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/&%\"cG6#\"#7\"\"\"" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/&%\"aG6$ \"#7\"\"\"#!jo>U2)*>>#R/d\\bfH[Src3P5KGads'\\oT`M%oXClhHt@\"io+?5Pm33d ERp\\@r/t=F>xH@nF'*\\\"G\"RDou><)4(Hj#)" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/&%\"aG6$\"#7\"\"#\"\"!" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/&%\" aG6$\"#7\"\"$\"\"!" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/&%\"aG6$\"#7\" \"%#![o++g$z3)*=nwln7*f')z%pZ-[zRXBZM$*)e[w6\"jn>c3]7wb5_O'33s\"RbIoB] Rd[yXd7u#Q#G\"" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/&%\"aG6$\"#7\"\"&#! Z+]Pfx+m:tXhIM/'R)zn$Hn')3ti_T;!*\"Y>^R)e)4Th`zxf:^nq3lxksJ:K^g+Z" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#/&%\"aG6$\"#7\"\"'#\"^p+]#Q*yFv5W)=*R* G@Id.vNMPr#4A-H5e**4'QI%eM4x>>Y%\"]p8!)o8fgP@*)y8*G;'\\9!p!HmXGg\"\\e$ 3P#yc!ymfx')HoAZI" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/&%\"aG6$\"#7\"\" (#\"io+]7`x&Q$3!)ocvqEq))Q$GpKuM0(fuwg:$pr5_mU(R$[*\"ho`f6b&>BP32Hzo7> n[NKQd#e@r.sa?R_;Q_oI?+T&" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/&%\"aG6$ \"#7\"\")#!`o++v=x.%pA&zC1_?*4G&)z*z'R;TyOUUyIl)*f$p%\"ao*Q+4J(z^O^34. 1h`9okCrrOO*[gH+u(zo++i7" }}{PARA 12 "" 1 "" {XPPMATH 20 "6#/&%\"aG6$ \"#7\"\"*#!jtj!QpWIE&[0*zMf@nSvB(o>Y;?#*3;M%f7=!>0$pg#Qsv,I\"y6tbYy`b: ()yHT8g\"3'Qas6\"zy;`tTXi'3E15[FTm\"jt+g]=70]r#en^cBx3W0WAjVY@Q%)HS*)R UA8fy#R%e^(Q//W)oh6\"Ho65&**zMeR:'QZzo%[$='[/ik)yKzdn8`SZ*" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/&%\"aG6$\"#7\"#5#\"]o+]ilIr^TeT;'z'fCr&)oy! =(y#=nBJ%\\*)o%fC\"\"^oPr*3:'*=SOJ+/.)ect3jWt[E**f9R7&**>hEUC" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#/&%\"aG6$\"#7\"#6#\"gn_t%\\Y!oh([4%Rz \")*oZ,C'R'*p&)465Buk5c\"gnvWr+W>S`B#f0$G9X%pV`s;`0Hj_\"Rz8n" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%!G" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/&%\" cG6#\"#8\"\"\"" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/&%\"aG6$\"#8\"\"\"# \"goxF(**HdhZ)G0r!\\zEhNke%3gR\\H+W@UyJ0'RaOPD6#\"go+cHB(*)\\d+InoDAxU 9],wtEd7d$p'>Qa*[jd#**H6z*" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/&%\"aG6 $\"#8\"\"#\"\"!" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/&%\"aG6$\"#8\"\"$ \"\"!" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/&%\"aG6$\"#8\"\"%#\"in++!G&) fOZ\\Y,Cq'ybPIFb'pC)[<]'oDt,M'\"hn\"*o)*4\\i0Szn=]we4!z#oEo0+**pk&zrV( f(" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/&%\"aG6$\"#8\"\"&#\"X+]7.%*[j)e A!eIB#>)R1cPYS(zP4uj0=gl'fD\"jo.cd?_%H7HVRDKKb4Ez$=7%f/:@MRkNb&p66]9yvq^\" " }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/&%\"aG6$\"#8\"\"(#!do+]i!poGO87B[ ses2E'[RJ(*\\Z;g:*yvW\\TCF'Hc$\"coTt')y0u,z?tXB&f9d&fW\\)>'=b/Ai^**)o* y\"Rm(*3%" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/&%\"aG6$\"#8\"\")#\"\\o+ +++*z!)H\\_,yViG8T0l%G!)=#*GC\"*3`Xzriu`!)R()o3ZVKoW_N*eY%f8%*pHw*oG4[62\"[z(G t-)*\\w^\"\"gs+3DeZ3d5G,e!)f0*oWUd@KvWBH5#\\[GC@yH)30G;%\\r<6#ppTb4\"f -f!p#\\>;&H#[K<&\\NS^j4Jz\"" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/&%\"aG 6$\"#8\"#5#\"in+](=#f)y([#4(zuT2\\W#oBhXoF(fplqUY_O\"in\"[U**))fP#[=b# p-mHnuY6TQ=\\Ha^X$3X?n" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/&%\"aG6$\"# 8\"#6\"\"!" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/&%\"aG6$\"#8\"#7\"\"!" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%!G" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/&%\"bG6#\"\"\"#\"AB<63n7Zc(zq&>JUGb\"C+S67Ogt>i\"MTV0kFFM3 !)>o=#=**Gc9Bzo_#" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/&%\"bG6#\"\")#\" K++++voHzqt;K(eK\\,#)e_r%\"K`b\\WSJ\\'R3\")Gdz)>\"R<)GP_" }}{PARA 12 " " 1 "" {XPPMATH 20 "6#/&%\"bG6#\"\"*#\"]t>%3MD>!G%[rBVH5v$Q%)p_7!\\&\\ \"**y130hUHdN=q@)[Ne7)R:)yXhrz@c*\\8N`!QWq-n%z4'H]u!f'>%)*R'\"]t+!Qq'y pt(4a[-N?6`k/j9w&p**\\j5`C)>oK^HX;wVj+8$ohSYO&4j^ZRsFxEC.]**f$pn!)*fTA yQQnS\"" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/&%\"bG6#\"#5#\"H+++DJqXVA_ DfIvDbFc%f\"KJ?7UM&fV<9V,BoF.\"4s-Q7" }}{PARA 11 "" 1 "" {XPPMATH 20 " 6#/&%\"bG6#\"#6#!CcYXqQ:f*RcqzR8\"p')=\"Bvw[7Lc^RQNe)eeZzR" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/&%\"bG6#\"#7#!A\")yja-\"R\"*y(R^()o91]\"B\" y8By')*>[=Og$>k'=T\"" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/&%\"bG6#\"#8 \"\"!" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%!G" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/&%#b*G6#\"\"\"#\"@*e]DJ-_Z@SH#*RaNO\"A+w!3CSy'pjn;.]4# R)" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/&%#b*G6#\"\"#\"\"!" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/&%#b*G6#\"\"$\"\"!" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/&%#b*G6#\"\"%\"\"!" }}{PARA 11 "" 1 "" {XPPMATH 20 "6# /&%#b*G6#\"\"&\"\"!" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/&%#b*G6#\"\"'# \"G++++?7]2-L7AO$3Lbs9*\"HzfP)4m$f:)RGLE>um.*Qq#" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/&%#b*G6#\"\"(#\"J+++DJXHjBa3]fo5z3rX7$\"KF@&3wvIs`b3@\"oDtT#4')*f2GV>8TX \"QmNm8^\"4)Hdf=wI#o!eH)34pT*Q+?xMv?oYE\"\"[s+#\\'e,vwb(y.5[(QsNM\"Hc> Qex)*y8?!H8+,TO)zq!oMORBf#*4!*4J6'Gt$)fjPzQO#oL\"e9$" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/&%#b*G6#\"#5#\"F+++Dc^8Am*\\SxP+C?jB\"G4@]h=F?N_]*= hC#)\\Zn\">" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/&%#b*G6#\"#6\"\"!" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#/&%#b*G6#\"#7\"\"!" }}{PARA 11 "" 1 " " {XPPMATH 20 "6#/&%#b*G6#\"#8#\"@4YgMWg6Tn3RNwe'H\"AFzVF*GLFG(y'yR@iq %" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 151 "RK8_13eqs := [op(RowSumConditions(13,'expa nded')),op(OrderConditions(8,13,'expanded'))]:\n`RK7_13eqs*` := subs(b =`b*`,OrderConditions(7,13,'expanded')):" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 6 "Check:" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 110 "subs(ee,RK8_13eqs):\nmap(u->lhs(u)-rhs(u),%);\nnops( %);\nsubs(ee,`RK7_13eqs*`):\nmap(u->lhs(u)-rhs(u),%);\nnops(%);" }} {PARA 12 "" 1 "" {XPPMATH 20 "6#7`x\"\"!F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$ F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$ F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$ F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$ F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$ F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$ F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#\"$7#" }}{PARA 12 "" 1 "" {XPPMATH 20 "6#7ap\"\"!F$F$F$ F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$ F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$ F$F$F$F$F$F$F$F$F$F$F$" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#\"#&)" }}} {PARA 257 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 78 "Set-up \+ stage-order condtions to check for stage-orders from 2 to 6 inclusive. " }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 87 "for ct from 2 to 6 do\n so||ct||_13 := StageOrderConditions(ct,13,'expanded');\nend do:" }} }{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 59 "Stages 3 to 13 have the following respective stage-orders. " }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 175 "[seq([seq(expand(subs(ee,so||i||_13[j])),i =2..6)],j=1..11)]:\nmap(proc(L) local i; for i to nops(L) do if not ev alb(L[i]) then break end if end do; i end proc,%):\nsimplify(%); " }} {PARA 11 "" 1 "" {XPPMATH 20 "6#7-\"\"#\"\"$F%\"\"%F&F&F&F&F&F&F&" }}} {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 27 "The simpl ifying conditions:" }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "Sum(b[i]*a[i,j],i = j+1 .. 13) = b[j]*(1-c[j]);" "6#/-%$SumG6$*&&%\"b G6#%\"iG\"\"\"&%\"aG6$F+%\"jGF,/F+;,&F0F,F,F,\"#8*&&F)6#F0F,,&F,F,&%\" cG6#F0!\"\"F," }{TEXT -1 3 ", " }{XPPEDIT 18 0 "j=1" "6#/%\"jG\"\"\" " }{TEXT -1 9 " . . 12, " }}{PARA 0 "" 0 "" {TEXT -1 8 "(where " } {XPPEDIT 18 0 "c[1]=0" "6#/&%\"cG6#\"\"\"\"\"!" }{TEXT -1 17 " ) are s atisfied." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 151 "[Sum(b[i]*a[i,1],i=1+1..13)=b[1],seq(Sum(b[i]*a[i,j] ,i=j+1..13)=b[j]*(1-c[j]),j=2..12)]:\nmap(u->lhs(u)-rhs(u),subs(ee,eva l(subs(Sum=add,%))));\nnops(%);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#7. \"\"!F$F$F$F$F$F$F$F$F$F$F$" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#\"#7" } }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 27 "The sim plifying conditions:" }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "Sum(`b*`[i]*a[i,j],i = j+1 .. 13) = `b*`[j]*(1-c[j]);" "6#/-%$SumG6 $*&&%#b*G6#%\"iG\"\"\"&%\"aG6$F+%\"jGF,/F+;,&F0F,F,F,\"#8*&&F)6#F0F,,& F,F,&%\"cG6#F0!\"\"F," }{TEXT -1 3 ", " }{XPPEDIT 18 0 "j=1" "6#/%\"j G\"\"\"" }{TEXT -1 9 " . . 12, " }}{PARA 0 "" 0 "" {TEXT -1 8 "(where \+ " }{XPPEDIT 18 0 "c[1]=0" "6#/&%\"cG6#\"\"\"\"\"!" }{TEXT -1 17 " ) a re satisfied." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 163 "[Sum(`b*`[i]*a[i,1],i=1+1..13)=`b*`[1],seq(Sum( `b*`[i]*a[i,j],i=j+1..13)=`b*`[j]*(1-c[j]),j=2..12)]:\nmap(u->lhs(u)-r hs(u),subs(ee,eval(subs(Sum=add,%))));\nnops(%);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#7.\"\"!F$F$F$F$F$F$F$F$F$F$F$" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#\"#7" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 " " {TEXT -1 93 "We can calculate the principal error norm, that is, the 2-norm of the principal error terms. " }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 162 "errterms8_13 := PrincipalErrorTerms(8,13,'expanded') :\nsm := 0:\nfor ct to nops(errterms8_13) do\n sm := sm+(evalf(subs( ee,errterms8_13[ct])))^2;\nend do:\nsqrt(sm);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#$\"+\"H[4N)!#;" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }} {PARA 0 "" 0 "" {TEXT -1 78 "We can also calculate the principal error norm of the order 7 embedded scheme." }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 184 "`errterms7_13*` := subs(b=`b*`,PrincipalErrorTerms(7 ,13,'expanded')):\nsm := 0:\nfor ct to nops(`errterms7_13*`) do\n sm := sm+(evalf(subs(ee,`errterms7_13*`[ct])))^2;\nend do:\nsqrt(sm);" } }{PARA 11 "" 1 "" {XPPMATH 20 "6#$\"+sNs?$*!#:" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}}{PARA 0 "" 0 "" {TEXT -1 52 "#---------------------------------------------- -----" }}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 34 "construction of the orde r 8 scheme" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 93 "First we construct a 12 stage order 8 scheme starting with a co nsideration of stages 2 to 8. " }}{PARA 0 "" 0 "" {TEXT -1 20 "We spec ify the nodes" }}{PARA 256 "" 0 "" {TEXT -1 2 " " }{XPPEDIT 18 0 "c[2 ] = 1/16;" "6#/&%\"cG6#\"\"#*&\"\"\"F)\"#;!\"\"" }{TEXT -1 2 ", " } {XPPEDIT 18 0 "c[5] = 39/100;" "6#/&%\"cG6#\"\"&*&\"#R\"\"\"\"$+\"!\" \"" }{TEXT -1 2 ", " }{XPPEDIT 18 0 "c[6] = 7/15;" "6#/&%\"cG6#\"\"'*& \"\"(\"\"\"\"#:!\"\"" }{TEXT -1 2 ", " }{XPPEDIT 18 0 "c[7] = 39/250; " "6#/&%\"cG6#\"\"(*&\"#R\"\"\"\"$]#!\"\"" }{TEXT -1 3 ", " } {XPPEDIT 18 0 "c[8] = 24/25;" "6#/&%\"cG6#\"\")*&\"#C\"\"\"\"#D!\"\"" }{TEXT -1 2 " " }}{PARA 0 "" 0 "" {TEXT -1 33 "and the zero linking c oefficients" }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "a[4,2] =0" "6#/&%\"aG6$\"\"%\"\"#\"\"!" }{TEXT -1 2 ", " }{XPPEDIT 18 0 "a[5, 2]=0" "6#/&%\"aG6$\"\"&\"\"#\"\"!" }{TEXT -1 2 ", " }{XPPEDIT 18 0 "a[ 6,2]=0" "6#/&%\"aG6$\"\"'\"\"#\"\"!" }{TEXT -1 2 ", " }{XPPEDIT 18 0 " a[6,3]=0" "6#/&%\"aG6$\"\"'\"\"$\"\"!" }{TEXT -1 2 ", " }{XPPEDIT 18 0 "a[7,2]=0" "6#/&%\"aG6$\"\"(\"\"#\"\"!" }{TEXT -1 2 ", " }{XPPEDIT 18 0 "a[7,3]=0" "6#/&%\"aG6$\"\"(\"\"$\"\"!" }{TEXT -1 2 ", " } {XPPEDIT 18 0 "a[8,2] = 0;" "6#/&%\"aG6$\"\")\"\"#\"\"!" }{TEXT -1 2 " , " }{XPPEDIT 18 0 "a[8,3] = 0;" "6#/&%\"aG6$\"\")\"\"$\"\"!" }{TEXT -1 3 ", " }{XPPEDIT 18 0 "a[8,4]=0" "6#/&%\"aG6$\"\")\"\"%\"\"!" } {TEXT -1 2 ". " }}{PARA 0 "" 0 "" {TEXT -1 48 "In addition we specify \+ the linking coefficient " }{XPPEDIT 18 0 "a[8,4] = -7/5;" "6#/&%\"aG6 $\"\")\"\"%,$*&\"\"(\"\"\"\"\"&!\"\"F." }{TEXT -1 1 "." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 69 "The stage-order equa tions relating to these stages, such that stages " }{XPPEDIT 18 0 "3,4 ,5,6,7,8;" "6(\"\"$\"\"%\"\"&\"\"'\"\"(\"\")" }{TEXT -1 19 " have stag e orders " }{XPPEDIT 18 0 "2,3,3,4,4,4;" "6(\"\"#\"\"$F$\"\"%F%F%" } {TEXT -1 48 " respectively can be solved to obtain the nodes " } {XPPEDIT 18 0 "c[3], c[4]" "6$&%\"cG6#\"\"$&F$6#\"\"%" }{TEXT -1 67 " \+ and the remaining non-zero linking coefficients for these stages. " }} {PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 443 "RSeqs := RowSumConditions(8,'expanded'):\nSOeqs := [op(StageOrder Conditions(2,8,'expanded')),op(StageOrderConditions(3,4..8,'expanded') ),\n op(StageOrderConditions(4,6..8,'expanded'))]:\ne1 : = \{c[2]=1/16,c[5]=39/100,c[6]=7/15,c[7]=39/250,c[8]=24/25,\n se q(a[i,2]=0,i=4..8),seq(a[i,3]=0,i=6..8),a[8,4]=-7/5\}:\neqns := map(ex pand,subs(e1,[op(RSeqs),op(SOeqs)])):\nconvert(ListTools[Enumerate](%) ,matrix);\n``;\nindets(eqns);\nnops(%);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#K%'matrixG6#777$\"\"\"/&%\"aG6$\"\"#F(#F(\"#;7$F-/,&&F+6$\"\"$F( F(&F+6$F5F-F(&%\"cG6#F57$F5/,&&F+6$\"\"%F(F(&F+6$F@F5F(&F96#F@7$F@/,(& F+6$\"\"&F(F(&F+6$FJF5F(&F+6$FJF@F(#\"#R\"$+\"7$FJ/,(&F+6$\"\"'F(F(&F+ 6$FWF@F(&F+6$FWFJF(#\"\"(\"#:7$FW/,*&F+6$FgnF(F(&F+6$FgnF@F(&F+6$FgnFJ F(&F+6$FgnFWF(#FP\"$]#7$Fgn/,,&F+6$\"\")F(F(#FgnFJ!\"\"&F+6$F[pFJF(&F+ 6$F[pFWF(&F+6$F[pFgnF(#\"#C\"#D7$F[p/,$*&F.F(F6F(F(,$*&#F(F-F(*$)F8F-F (F(F(7$\"\"*/*&FAF(F8F(,$*&F]qF(*$)FCF-F(F(F(7$\"#5/,&*&FKF(F8F(F(*&FM F(FCF(F(#\"%@:\"&++#7$\"#6/,&*&FXF(FCF(F(*&FOF(FZF(F(#\"#\\\"$]%7$\"#7 /,(*&F^oF(FCF(F(*&FOF(F`oF(F(*&FfnF(FboF(F(#F_r\"'+]77$\"#8/,**&#FgnFJ F(FCF(F]p*&FOF(F^pF(F(*&FfnF(F`pF(F(*&FdoF(FbpF(F(#\"$)G\"$D'7$\"#9/*& FAF(F_qF(,$*&#F(F5F(*$)FCF5F(F(F(7$Fhn/,&*&FKF(F_qF(F(*&FMF(FgqF(F(#\" &t(>\"(+++\"7$F//,&*&FXF(FgqF(F(*&#F_r\"&++\"F(FZF(F(#\"$V$\"&D,\"7$\" #/,&*&FXF(FgtF(F(*&#\"&>$fF_uF(FZF(F(#\"%,C\"'+D?7$ \"#?/,(*&F^oF(FgtF(F(*&FiwF(F`oF(F(*&#Fhu\"%vLF(FboF(F(#\"(TMJ#\",+++D c\"7$\"#@/,**&#FgnFJF(FftF(F]p*&FiwF(F^pF(F(*&FexF(F`pF(F(*&#FjwFdvF(F bpF(F(#\"&WH)\"'D1RQ(pprint06\"" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%!G " }}{PARA 11 "" 1 "" {XPPMATH 20 "6#<7&%\"aG6$\"\"(\"\"\"&F%6$\"\"'\" \"%&F%6$\"\"$\"\"#&F%6$\"\"&F,&%\"cG6#F/&F%6$F3F/&F%6$\"\")F(&F%6$F,F/ &F%6$F;F'&F%6$F/F(&F%6$F,F(&F%6$F'F3&F%6$F;F3&F%6$F0F(&F%6$F+F3&F%6$F' F,&F56#F,&F%6$F;F+&F%6$F'F+&F%6$F3F(&F%6$F+F(" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#\"#@" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 46 "e2 := solve(\{op(eqns)\});\ne3 := `union` (e1,e2):" }}{PARA 12 "" 1 "" {XPPMATH 20 "6#>%#e2G<7/&%\"aG6$\"\"(\"\" \"#\")**3S@\"*+++]$/&F(6$\"\"'\"\"%#\"(x`]#\")?bo5/&F(6$\"\"$\"\"##\"' _.5\"(DU8\"/&F(6$\"\"&F3#\"*fr2y\"\"*++WD\"/&%\"cG6#F:#\"$7\"\"%l5/&F( 6$FBF:#!*(f1GN\"*++)3D/&F(6$\"\")F+#!,6)eaO:\",vdc4O\"/&F(6$F3F:#\"#U \"$b$/&F(6$FVF*#\".S#Q=K$*>\"-p!fM_!Q/&F(6$F:F+#\"&G*=F>/&F(6$F3F+#\"# 9Fin/&F(6$F*FB#!)ZRk@\"*v$4Ef/&F(6$FVFB#!.3Y;G\"R$)\"-v%Q+1R*/&F(6$F;F +#F+\"#;/&F(6$F2FB#\"'+/'*\"(\">4_/&F(6$F*F3#\".**)H)HV1$\"/+++]g7F/&F H6#F3#\"#cFin/&F(6$FVF2#\"-)[+o$>M\",Di7^z%/&F(6$F*F2#\"*V>RC\"\"+++Dc n/&F(6$FBF+#\")za\\%*FR/&F(6$F2F+#\"&*37\"'?FD" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT 260 4 "Note" }{TEXT -1 2 ": " }} {PARA 0 "" 0 "" {TEXT -1 8 "We have " }{XPPEDIT 18 0 "c[3]=2/3" "6#/&% \"cG6#\"\"$*&\"\"#\"\"\"F'!\"\"" }{TEXT -1 1 " " }{XPPEDIT 18 0 "c[4] " "6#&%\"cG6#\"\"%" }{TEXT -1 16 " and the nodes " }{XPPEDIT 18 0 "c[ 4];" "6#&%\"cG6#\"\"%" }{TEXT -1 3 ", " }{XPPEDIT 18 0 "c[5];" "6#&% \"cG6#\"\"&" }{TEXT -1 3 ", " }{XPPEDIT 18 0 "c[6];" "6#&%\"cG6#\"\"' " }{TEXT -1 31 " are related by the equation: " }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "3*c[6]^2-4*c[6]*c[4]-4*c[5]*c[6]+6*c[5] *c[4] = 0" "6#/,**&\"\"$\"\"\"*$&%\"cG6#\"\"'\"\"#F'F'*(\"\"%F'&F*6#F, F'&F*6#F/F'!\"\"*(F/F'&F*6#\"\"&F'&F*6#F,F'F4*(F,F'&F*6#F8F'&F*6#F/F'F '\"\"!" }{TEXT -1 2 ". " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 86 "subs(e3,matrix([seq([c[i],seq(a[i,j ],j=1..i-1),``$(9-i)],i=2..8)]));\n``;\nevalf[8](%%);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#K%'matrixG6#7)7+#\"\"\"\"#;F(%!GF+F+F+F+F+F+7+#\"$ 7\"\"%l5#\"&G*=\"(DU8\"#\"'_.5F2F+F+F+F+F+F+7+#\"#c\"$b$#\"#9F8\"\"!# \"#UF8F+F+F+F+F+7+#\"#R\"$+\"#\")za\\%*\"*++)3DF;#!*(f1GNFD#\"*fr2y\" \"*++WD\"F+F+F+F+7+#\"\"(\"#:#\"&*37\"'?FDF;F;#\"(x`]#\")?bo5#\"'+/'* \"(\">4_F+F+F+7+#F@\"$]##\")**3S@\"*+++]$F;F;#\".**)H)HV1$\"/+++]g7F#! )ZRk@\"*v$4Ef#\"*V>RC\"\"+++DcnF+F+7+#\"#C\"#D#!,6)eaO:\",vdc4O\"F;F;# !\"(\"\"&#!.3Y;G\"R$)\"-v%Q+1R*#\"-)[+o$>M\",Di7^z%#\".S#Q=K$*>\"-p!fM _!QF+Q(pprint96\"" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%!G" }}{PARA 11 " " 1 "" {XPPMATH 20 "6#K%'matrixG6#7)7+$\")++]i!\"*F(%!GF+F+F+F+F+F+7+$ \")Kk^5!\")$\")Z!)o;F*$\")siZ))F*F+F+F+F+F+F+7+$\")[Yx:F/$\")?mVRF*$\" \"!F:$\")')4$=\"F/F+F+F+F+F+7+$\")+++RF/$\")4cmPF/F9$!)jF19!\"($\")-i> 9FDF+F+F+F+7+$\")nmmYF/$\")\\b$y%F*F9F9$\")okWBF/$\")WmV=F/F+F+F+7+$\" )++g:F/$\")Ea9hF*F9F9$\")JmH6F/$!)EJ_OF*$\"))Q6%=F*F+F+7+$\")+++'*F/$! )!>!H6FDF9F9$!)+++9FD$!)HH!)))FD$\")T%48(FD$\")yOQ_FDF+Q)pprint106\"" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 27 "We do not need to specify " }{XPPEDIT 18 0 "c[9];" "6#&%\"cG6#\"\"*" }{TEXT -1 42 ", because a ccording to Verner, the nodes " }{XPPEDIT 18 0 "c[6];" "6#&%\"cG6#\" \"'" }{TEXT -1 2 ", " }{XPPEDIT 18 0 "c[7];" "6#&%\"cG6#\"\"(" }{TEXT -1 2 ", " }{XPPEDIT 18 0 "c[8];" "6#&%\"cG6#\"\")" }{TEXT -1 2 ", " } {XPPEDIT 18 0 "c[9];" "6#&%\"cG6#\"\"*" }{TEXT -1 30 " are related by the equation " }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "Int (p(x)*``((1-x)^3/3!),x = 0 .. 1)*Int(q(x)*(1-x),x = 0 .. 1) = Int(p(x) *``((1-x)^2/2!),x = 0 .. 1)*Int(q(x)*``((1-x)^2/2!),x = 0 .. 1);" "6#/ *&-%$IntG6$*&-%\"pG6#%\"xG\"\"\"-%!G6#*&,&F-F-F,!\"\"\"\"$-%*factorial G6#F4F3F-/F,;\"\"!F-F--F&6$*&-%\"qG6#F,F-,&F-F-F,F3F-/F,;F:F-F-*&-F&6$ *&-F*6#F,F--F/6#*&,&F-F-F,F3\"\"#-F66#FNF3F-/F,;F:F-F--F&6$*&-F?6#F,F- -F/6#*&,&F-F-F,F3FN-F66#FNF3F-/F,;F:F-F-" }{TEXT -1 2 ", " }}{PARA 0 " " 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 7 "where " }{XPPEDIT 18 0 "p(x) = x*(x-c[6])*(x-c[7])*(x-c[8]);" "6#/-%\"pG6#%\"xG**F'\"\" \",&F'F)&%\"cG6#\"\"'!\"\"F),&F'F)&F,6#\"\"(F/F),&F'F)&F,6#\"\")F/F)" }{TEXT -1 7 " and " }{XPPEDIT 18 0 "q(x) = (x-c[9])*p(x);" "6#/-%\"q G6#%\"xG*&,&F'\"\"\"&%\"cG6#\"\"*!\"\"F*-%\"pG6#F'F*" }{TEXT -1 2 ". \+ " }}{PARA 0 "" 0 "" {TEXT -1 1 " " }}{PARA 0 "" 0 "" {TEXT -1 155 "See : J.H. Verner, SIAM Journal of Numerical Analysis 1978, 772-790, \"Exp licit Runge-Kutta methods with estimates of the Local Truncation Error .\" (page 780)" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 90 "p := x -> x*(x-c[6])*(x-c[7])*(x-c[8]):\n'p(x)'= p(x);\nq := x -> (x-c[9])*p(x):\n'q(x)'=q(x);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/-%\"pG6#%\"xG**F'\"\"\",&F'F)&%\"cG6#\"\"'!\"\"F),&F'F )&F,6#\"\"(F/F),&F'F)&F,6#\"\")F/F)" }}{PARA 11 "" 1 "" {XPPMATH 20 "6 #/-%\"qG6#%\"xG*,,&F'\"\"\"&%\"cG6#\"\"*!\"\"F*F'F*,&F'F*&F,6#\"\"'F/F *,&F'F*&F,6#\"\"(F/F*,&F'F*&F,6#\"\")F/F*" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 239 "Ieq := Int('p(x)' *(1-x)^3/3!,x=0..1)*Int('q(x)'*(1-x),x=0..1)=\n Int('p(x)'*(1-x)^2/2! ,x=0..1)*Int('q(x)'*(1-x)^2/2!,x=0..1);\nvalue(Ieq):\nReq := simplify( lhs(%)-rhs(%))=0:\nL := ilcm(op(map(denom,[coeffs(lhs(Req))]))):\nCeq \+ := expand(L*Req);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%$IeqG/*&-%$IntG 6$,$*&#\"\"\"\"\"'F-*&-%\"pG6#%\"xGF-),&F-F-F3!\"\"\"\"$F-F-F-/F3;\"\" !F-F--F(6$*&-%\"qGF2F-F5F-F8F-*&-F(6$,$*&#F-\"\"#F-*&F0F-)F5FFF-F-F-F8 F--F(6$,$*&FEF-*&F>F-FHF-F-F-F8F-" }}{PARA 12 "" 1 "" {XPPMATH 20 "6#> %$CeqG/,hp*&\"\"$\"\"\"&%\"cG6#\"\"(F)!\"\"*&\"#7F)&F+6#\"\"*F)F)*&F(F )&F+6#\"\")F)F.*&F(F)&F+6#\"\"'F)F.*,\"$)eF)F1F)F9F)F*F)F5F)F.*(\"#UF) F1F)F9F)F.*,\"$!)*F)F9F))F*\"\"#F))F5FCF)F1F)F.*,FAF))F9FCF)F*F)FDF)F1 F)F.*,FAF)FFF)FBF)F5F)F1F)F.*,\"$G(F)F1F)FFF)F*F)F5F)F)*,\"%q9F)FFF)FB F)FDF)F1F)F)*,FIF)F1F)F9F)FBF)F5F)F)**\"$J#F)F9F)F*F)F5F)F.**\"$a\"F)F 1F)F9F)F*F)F)**FPF)F5F)F1F)F9F)F)**FPF)F5F)F*F)F1F)F)**\"$o\"F)F1F)F9F )FBF)F.**FTF)F5F)FBF)F1F)F.**FTF)F9F)FBF)F5F)F)**FTF)FDF)F1F)F9F)F.**F TF)FDF)F*F)F1F)F.**FTF)F9F)F*F)FDF)F)**FTF)F1F)FFF)F*F)F.**FTF)F5F)F1F )FFF)F.**FTF)FFF)F*F)F5F)F)**\"#)*F)FFF)FBF)F5F)F.**FhnF)FFF)F*F)FDF)F .**FhnF)F9F)FBF)FDF)F.**\"$'>F)FDF)FFF)F1F)F)**F\\oF)FDF)FBF)F1F)F)**F \\oF)FFF)FBF)F1F)F)*,FIF)F1F)F9F)F*F)FDF)F)*(\"#GF)F9F)F*F)F)*(FaoF)F5 F)F9F)F)*(FaoF)F5F)F*F)F)*(F?F)F*F)F1F)F.*(F?F)F5F)F1F)F.*(F?F)FBF)F1F )F)*(\"#9F)F9F)FBF)F.*(FhoF)F5F)FBF)F.*(F?F)FDF)F1F)F)*(FhoF)FDF)F9F)F .*(FhoF)FDF)F*F)F.*(F?F)F1F)FFF)F)*(FhoF)FFF)F*F)F.*(FhoF)F5F)FFF)F.\" \"!" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 60 "T he previous equation involving integrals is equivalent to: " }}{PARA 256 "" 0 "" {TEXT -1 2 " " }{XPPEDIT 18 0 "14*c[6]^2*c[7]+3*c[6]-28*c [6]*c[7]+3*c[7]+588*c[9]*c[6]*c[7]*c[8]+3*c[8]-28*c[8]*c[6]-28*c[8]*c[ 7]+14*c[8]*c[6]^2+14*c[8]^2*c[6]+231*c[6]*c[7]*c[8]+98*c[6]^2*c[7]^2*c [8]+14*c[6]*c[7]^2+98*c[8]^2*c[7]^2*c[6]+98*c[8]^2*c[6]^2*c[7]-168*c[8 ]^2*c[6]*c[7]-168*c[8]*c[7]^2*c[6]-168*c[8]*c[6]^2*c[7]+14*c[8]^2*c[7] -12*c[9]+14*c[8]*c[7]^2+42*c[9]*c[6]-154*c[9]*c[6]*c[7]+42*c[7]*c[9]+4 2*c[8]*c[9]-154*c[8]*c[9]*c[6]-154*c[8]*c[7]*c[9]-42*c[9]*c[6]^2-42*c[ 7]^2*c[9]-42*c[8]^2*c[9]+168*c[8]*c[7]^2*c[9]-728*c[8]*c[6]^2*c[9]*c[7 ]-728*c[8]^2*c[6]*c[7]*c[9]+980*c[8]^2*c[6]^2*c[9]*c[7]-728*c[8]*c[7]^ 2*c[9]*c[6]+980*c[8]^2*c[7]^2*c[9]*c[6]+980*c[6]^2*c[7]^2*c[9]*c[8]-14 70*c[6]^2*c[7]^2*c[8]^2*c[9]+168*c[8]*c[6]^2*c[9]+168*c[8]^2*c[6]*c[9] -196*c[8]^2*c[6]^2*c[9]+168*c[8]^2*c[7]*c[9]-196*c[8]^2*c[7]^2*c[9]+16 8*c[9]*c[6]^2*c[7]+168*c[6]*c[7]^2*c[9]-196*c[6]^2*c[7]^2*c[9]=0" "6#/ ,hp*(\"#9\"\"\"*$&%\"cG6#\"\"'\"\"#F'&F*6#\"\"(F'F'*&\"\"$F'&F*6#F,F'F '*(\"#GF'&F*6#F,F'&F*6#F0F'!\"\"*&F2F'&F*6#F0F'F'*,\"$)eF'&F*6#\"\"*F' &F*6#F,F'&F*6#F0F'&F*6#\"\")F'F'*&F2F'&F*6#FJF'F'*(F6F'&F*6#FJF'&F*6#F ,F'F;*(F6F'&F*6#FJF'&F*6#F0F'F;*(F&F'&F*6#FJF'&F*6#F,F-F'*(F&F'*$&F*6# FJF-F'&F*6#F,F'F'**\"$J#F'&F*6#F,F'&F*6#F0F'&F*6#FJF'F'**\"#)*F'*$&F*6 #F,F-F'&F*6#F0F-&F*6#FJF'F'*(F&F'&F*6#F,F'&F*6#F0F-F'**FfoF'*$&F*6#FJF -F'&F*6#F0F-&F*6#F,F'F'**FfoF'*$&F*6#FJF-F'&F*6#F,F-&F*6#F0F'F'**\"$o \"F'*$&F*6#FJF-F'&F*6#F,F'&F*6#F0F'F;**FdqF'&F*6#FJF'&F*6#F0F-&F*6#F,F 'F;**FdqF'&F*6#FJF'&F*6#F,F-&F*6#F0F'F;*(F&F'*$&F*6#FJF-F'&F*6#F0F'F'* &\"#7F'&F*6#FCF'F;*(F&F'&F*6#FJF'&F*6#F0F-F'*(\"#UF'&F*6#FCF'&F*6#F,F' F'**\"$a\"F'&F*6#FCF'&F*6#F,F'&F*6#F0F'F;*(FjsF'&F*6#F0F'&F*6#FCF'F'*( FjsF'&F*6#FJF'&F*6#FCF'F'**F`tF'&F*6#FJF'&F*6#FCF'&F*6#F,F'F;**F`tF'&F *6#FJF'&F*6#F0F'&F*6#FCF'F;*(FjsF'&F*6#FCF'&F*6#F,F-F;*(FjsF'*$&F*6#F0 F-F'&F*6#FCF'F;*(FjsF'*$&F*6#FJF-F'&F*6#FCF'F;**FdqF'&F*6#FJF'&F*6#F0F -&F*6#FCF'F'*,\"$G(F'&F*6#FJF'&F*6#F,F-&F*6#FCF'&F*6#F0F'F;*,FhwF'*$&F *6#FJF-F'&F*6#F,F'&F*6#F0F'&F*6#FCF'F;*,\"$!)*F'*$&F*6#FJF-F'&F*6#F,F- &F*6#FCF'&F*6#F0F'F'*,FhwF'&F*6#FJF'&F*6#F0F-&F*6#FCF'&F*6#F,F'F;*,F\\ yF'*$&F*6#FJF-F'&F*6#F0F-&F*6#FCF'&F*6#F,F'F'*,F\\yF'*$&F*6#F,F-F'&F*6 #F0F-&F*6#FCF'&F*6#FJF'F'*,\"%q9F'*$&F*6#F,F-F'&F*6#F0F-&F*6#FJF-&F*6# FCF'F;**FdqF'&F*6#FJF'&F*6#F,F-&F*6#FCF'F'**FdqF'*$&F*6#FJF-F'&F*6#F,F '&F*6#FCF'F'**\"$'>F'*$&F*6#FJF-F'&F*6#F,F-&F*6#FCF'F;**FdqF'*$&F*6#FJ F-F'&F*6#F0F'&F*6#FCF'F'**F^]lF'*$&F*6#FJF-F'&F*6#F0F-&F*6#FCF'F;**Fdq F'&F*6#FCF'&F*6#F,F-&F*6#F0F'F'**FdqF'&F*6#F,F'&F*6#F0F-&F*6#FCF'F'**F ^]lF'*$&F*6#F,F-F'&F*6#F0F-&F*6#FCF'F;\"\"!" }{TEXT -1 2 ". " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 75 "e4 := solve(subs(e3,Ceq),\{c[9]\}):\nc[9]=subs(e4,c[9]);\ne5 := `union`( e3,e4):" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/&%\"cG6#\"\"*#\")oeV9\")h) yh\"" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 7 "T hus " }{XPPEDIT 18 0 "c[9]=14435868/16178861" "6#/&%\"cG6#\"\"**&\") oeV9\"\"\"\")h)yh\"!\"\"" }{TEXT -1 1 "." }}{PARA 0 "" 0 "" {TEXT -1 84 "Now we can use the quadrature equations to find the weights once t he remaining nodes" }}{PARA 256 "" 0 "" {TEXT -1 2 " " }{XPPEDIT 18 0 "c[10] = 11/12;" "6#/&%\"cG6#\"#5*&\"#6\"\"\"\"#7!\"\"" }{TEXT -1 3 ", " }{XPPEDIT 18 0 "c[11] = 19/20;" "6#/&%\"cG6#\"#6*&\"#>\"\"\"\"#? !\"\"" }{TEXT -1 3 ", " }{XPPEDIT 18 0 "c[12] = 1;" "6#/&%\"cG6#\"#7 \"\"\"" }{TEXT -1 1 " " }}{PARA 0 "" 0 "" {TEXT -1 38 "are specified a long with the weights " }{XPPEDIT 18 0 "b[2]=0" "6#/&%\"bG6#\"\"#\"\" !" }{TEXT -1 2 ", " }{XPPEDIT 18 0 "b[3]=0" "6#/&%\"bG6#\"\"$\"\"!" } {TEXT -1 2 ", " }{XPPEDIT 18 0 "b[4]=0" "6#/&%\"bG6#\"\"%\"\"!" } {TEXT -1 2 ", " }{XPPEDIT 18 0 "b[5]=0" "6#/&%\"bG6#\"\"&\"\"!" } {TEXT -1 2 ". " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 288 "Qeqs := QuadratureConditions(8,12,'expanded'): \ne6 := \{seq(b[i]=0,i=2..5),c[10]=11/12,c[11]=19/20,c[12]=1\}:\ne7 := `union`(e5,e6):\nquadeqns := subs(e7,Qeqs):\nfor ct to nops(quadeqns) do\n print(`equation `||ct); print(``);print(quadeqns[ct]);print( ``);\nend do:\nindets(quadeqns);\nnops(%);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%-equation~~~1G" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%!G " }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/,2&%\"bG6#\"\"\"F(&F&6#\"\"'F(&F& 6#\"\"(F(&F&6#\"\")F(&F&6#\"\"*F(&F&6#\"#5F(&F&6#\"#6F(&F&6#\"#7F(F(" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%!G" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%-equation~~~2G" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%!G" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/,0*&#\"\"(\"#:\"\"\"&%\"bG6#\"\"'F)F)*&#\"#R\" $]#F)&F+6#F'F)F)*&#\"#C\"#DF)&F+6#\"\")F)F)*&#\")oeV9\")h)yh\"F)&F+6# \"\"*F)F)*&#\"#6\"#7F)&F+6#\"#5F)F)*&#\"#>\"#?F)&F+6#FDF)F)&F+6#FEF)#F )\"\"#" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%!G" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%-equation~~~3G" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%!G " }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/,0*&#\"#\\\"$D#\"\"\"&%\"bG6#\"\" 'F)F)*&#\"%@:\"&+D'F)&F+6#\"\"(F)F)*&#\"$w&\"$D'F)&F+6#\"\")F)F)*&#\"0 CM\"\\G%R3#\"0@tDVbvh#F)&F+6#\"\"*F)F)*&#\"$@\"\"$W\"F)&F+6#\"#5F)F)*& #\"$h$\"$+%F)&F+6#\"#6F)F)&F+6#\"#7F)#F)\"\"$" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%!G" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%-equation~~~4G " }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%!G" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/,0*&#\"$V$\"%vL\"\"\"&%\"bG6#\"\"'F)F)*&#\"&>$f\")+]i:F)&F+6#\" \"(F)F)*&#\"&CQ\"\"&Dc\"F)&F+6#\"\")F)F)*&#\"7K?H!ek*)Q_$3I\"7\"Q\"p$o R.b1\\B%F)&F+6#\"\"*F)F)*&#\"%J8\"%GwPSv6xPtd%)zTq4?oAX_$fUkf^oF)&F+6# \"\"*F)F)*&#\"&TY\"\"&O2#F)&F+6#\"#5F)F)*&#\"'@.8\"'++;F)&F+6#\"#6F)F) &F+6#\"#7F)#F)\"\"&" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%!G" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%-equation~~~6G" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%!G" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/,0*&#\"&2o\"\"' v$f(\"\"\"&%\"bG6#\"\"'F)F)*&#\")*>C-*\"-++]il(*F)&F+6#\"\"(F)F)*&#\"( CE'z\"(Dcw*F)&F+6#\"\")F)F)*&#\"Eov.dPL-h&Q[ke'[WBpi\"F,.&G.AejDv7^\"G ZE5&36F)&F+6#\"\"*F)F)*&#\"'^5;\"'K)[#F)&F+6#\"#5F)F)*&#\"(*4wC\"(++?$ F)&F+6#\"#6F)F)&F+6#\"#7F)#F)F-" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%!G " }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%-equation~~~7G" }}{PARA 11 "" 1 " " {XPPMATH 20 "6#%!G" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/,0*&#\"'\\w6 \")D1R6\"\"\"&%\"bG6#\"\"'F)F)*&#\"+hPu=N\"0+++D19W#F)&F+6#\"\"(F)F)*& #\"*wH5\">\"*D19W#F)&F+6#\"\")F)F)*&#\"LC!*o-%Q&3YGC[$[R(*))='4%=]!*\" Mhro`CA!H,f`I#z)f*4,\\LW$z\"F)&F+6#\"\"*F)F)*&#\"(h:x\"\"(%)f)HF)&F+6# \"#5F)F)*&#\")\")e/Z\")+++kF)&F+6#\"#6F)F)&F+6#\"#7F)#F)F4" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%!G" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%-equ ation~~~8G" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%!G" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/,0*&#\"'VN#)\"*v$f3<\"\"\"&%\"bG6#\"\"'F)F)*&#\"-zm+Js 8\"2+++]i:N5'F)&F+6#\"\"(F)F)*&#\"+C9Z'e%\"+Dc^.hF)&F+6#\"\")F)F)*&#\" TKG^&f\\T0#H.e<0%zt&eA#G))HEZ18\"T@OI:NGP(HU[-tiZ;g:RB]lqe,HF)&F+6#\" \"*F)F)*&#\")rr[>\")3=$e$F)&F+6#\"#5F)F)*&#\"*R<(Q*)\"++++!G\"F)&F+6# \"#6F)F)&F+6#\"#7F)#F)F;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%!G" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#<*&%\"bG6#\"\"(&F%6#\"#7&F%6#\"#5&F%6# \"\"*&F%6#\"#6&F%6#\"\"'&F%6#\"\"\"&F%6#\"\")" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#\"\")" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 199 "e8 := solve(\{op(quadeqns)\}):\ne9 := `u nion`(e7,e8):\nfor ii to 12 do\n wt_val := subs(e9,b[ii]);\n if wt _val<>0 then print(b[ii]=wt_val) end if;\nend do:\n``;\nsubs(e9,[seq(b [i],i=1..12)]):\nevalf[8](%);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/&%\" bG6#\"\"\"#\".T)z[nJY\"0w*zB3#y.\"" }}{PARA 11 "" 1 "" {XPPMATH 20 "6# /&%\"bG6#\"\"'#\"2DT?u*>sK9\"2;K$z#oc*[S" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/&%\"bG6#\"\"(#\":+]il(4+,CBw?F\";\\`AAR\"op![nt\"4\"" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/&%\"bG6#\"\")#!0v$fe+L&)\\\"/Cu.\"4_`*" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#/&%\"bG6#\"\"*#\"Wzdw4zjfCSPa<'fI]qEzC xPYI?$fS\"V_RI)3:\"p(Hs`,7/')=?$3T=KO%>R!)y" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/&%\"bG6#\"#5#!/[sjF.H5\".^p%p2#3\"" }}{PARA 11 "" 1 " " {XPPMATH 20 "6#/&%\"bG6#\"#6#\"0+!))e5kK')\"/8LDiY\"e)" }}{PARA 11 " " 1 "" {XPPMATH 20 "6#/&%\"bG6#\"#7#!,R2IY(H\"-cMYUrC" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%!G" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#7.$\")\\)GY %!\"*$\"\"!F(F'F'F'$\")n\\QN!\")$\")99#\\#F+$!)&Q$G_!\"($\")b;^^F0$!)H z4&*F0$\")M'f+\"!\"'$!)&4O?\"F+" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}{SECT 1 {PARA 4 "" 0 " " {TEXT -1 2 "e9" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> \+ " 0 "" {MPLTEXT 1 0 1278 "e9 := \{b[5] = 0, b[2] = 0, b[3] = 0, b[4] = 0, c[12] = 1, a[7,2] = 0, a[6,3] = 0, a[4,2] = 0, a[8,3] = 0, a[8,2] \+ = 0, a[7,3] = 0, a[6,2] = 0, a[5,2] = 0, a[6,4] = 2505377/10685520, a[ 3,2] = 100352/1134225, a[5,4] = 178077159/125440000, c[3] = 112/1065, \+ a[5,3] = -352806597/250880000, b[10] = -10290327637248/1082076946951, \+ a[8,1] = -15365458811/13609565775, a[4,3] = 42/355, c[2] = 1/16, c[5] \+ = 39/100, c[6] = 7/15, c[8] = 24/25, a[8,4] = -7/5, c[7] = 39/250, a[8 ,7] = 1993321838240/380523459069, a[3,1] = 18928/1134225, a[4,1] = 14/ 355, a[7,5] = -21643947/592609375, b[1] = 4631674879841/10378208237997 6, a[8,5] = -8339128164608/939060038475, a[7,1] = 21400899/350000000, \+ b[7] = 2720762324010009765625000/10917367480696813922225349, a[2,1] = \+ 1/16, b[12] = -29746300739/247142463456, a[6,5] = 960400/5209191, a[7, 4] = 3064329829899/27126050000000, b[11] = 863264105888000/85814662253 313, c[4] = 56/355, c[10] = 11/12, c[11] = 19/20, b[8] = -498533005859 375/95352091037424, a[8,6] = 341936800488/47951126225, a[7,6] = 124391 943/6756250000, b[9] = 40593203046377724792670503059617543740245963790 9765779/78803919436321841083201886041201537229769115088303952, b[6] = \+ 14327219974204125/40489566827933216, a[5,1] = 94495479/250880000, a[6, 1] = 12089/252720, c[9] = 14435868/16178861\}:" }{TEXT -1 0 "" }}} {PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 55 "Now we specify the remaining zero linking coefficients:" }}{PARA 256 "" 0 "" {TEXT -1 2 " " }{XPPEDIT 18 0 "a[9,2] = 0,a[9,3] = 0,a[10,2] \+ = 0,a[10,3] = 0,a[11,2] = 0,a[11,3] = 0,a[12,2] = 0,a[12,3] = 0;" "6*/ &%\"aG6$\"\"*\"\"#\"\"!/&F%6$F'\"\"$F)/&F%6$\"#5F(F)/&F%6$F1F-F)/&F%6$ \"#6F(F)/&F%6$F8F-F)/&F%6$\"#7F(F)/&F%6$F?F-F)" }{TEXT -1 1 "." }} {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 149 "We inclu de the stage-order conditions for stages 9 to 12 so that all these sta ges all have stage-order 4 and incorporate the simplifying conditions: " }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "Sum(b[i]*a[i,j], i = j+1 .. 12) = b[j]*(1-c[j]);" "6#/-%$SumG6$*&&%\"bG6#%\"iG\"\"\"&% \"aG6$F+%\"jGF,/F+;,&F0F,F,F,\"#7*&&F)6#F0F,,&F,F,&%\"cG6#F0!\"\"F," } {TEXT -1 1 " " }}{PARA 0 "" 0 "" {TEXT -1 5 "for " }{XPPEDIT 18 0 "j= 1" "6#/%\"jG\"\"\"" }{TEXT -1 10 " ( with " }{XPPEDIT 18 0 "c[1]=0" "6#/&%\"cG6#\"\"\"\"\"!" }{TEXT -1 5 " ), " }{XPPEDIT 18 0 "j = 5;" " 6#/%\"jG\"\"&" }{TEXT -1 3 ", " }{XPPEDIT 18 0 "j=7" "6#/%\"jG\"\"(" }{TEXT -1 3 ", " }{XPPEDIT 18 0 "j=9" "6#/%\"jG\"\"*" }{TEXT -1 7 " \+ and " }{XPPEDIT 18 0 "j=10" "6#/%\"jG\"#5" }{TEXT -1 2 ". " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 195 "Then it turns \+ out that the following additional collection of \"simple\" order condi tions (given in abreviated form) is sufficient to determine the remain ing linking coefficients in stages 9 to 12." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 190 "SO8 := SimpleOrde rConditions(8):\n[seq([i,SO8[i]],i=[37,61,81,87,102,106,113,117,125])] :\nlinalg[augment](linalg[delcols](%,2..2),matrix([[` `]$(linalg[rowd im](%))]),linalg[delcols](%,1..1));" }}{PARA 11 "" 1 "" {XPPMATH 20 "6 #K%'matrixG6#7+7%\"#P%#~~G/*(%\"bG\"\"\"%\"cGF--%!G6#*&%\"aGF--F06#*&F 3F--F06#*&F3F--F06#*&F3F-F.F-F-F-F-F-#F-\"$S)7%\"#hF)/*(F,F-F.F--F06#* &)F.\"\"%F-F3F-F-#F-\"#N7%\"#\")F)/*&F,F--F06#*(F3F-F.F--F06#*&F3F--F0 6#*&F3F--F06#*&)F.\"\"#F-F3F-F-F-F-F-#F-\"%gL7%\"#()F)/*(F,F-F.F--F06# *(F3F-F.F--F06#*(F3F-F.F-F:F-F-F-#F-\"$%Q7%\"$-\"F)/*(F,F-F.F--F06#*&F 3F--F06#*&F3F--F06#*&)F.\"\"$F-F3F-F-F-F-#F-\"$g*7%\"$1\"F)/*&F,F--F06 #*&F3F--F06#*&F3F-FCF-F-F-#F-\"%!o\"7%\"$8\"F)/*(F,F-FZF-F[pF-#F-\"$g \"7%\"$<\"F)/*&F,F--F06#*(F3F-F.F-FCF-F-#F-\"$!G7%\"$D\"F)/*(F,F-F.F-- F06#*&)F.\"\"&F-F3F-F-#F-\"#[Q(pprint26\"" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 20 "The associated trees" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 461 "AST8 := All SimpleTrees(8):\nwhch := [37,61,81,87,102,106,113,117,125]:\nm := 3: # # number of trees per row\nnn := nops(whch): q := iquo(nn,m,'r'):\nfor i to nn do \n p||i := DrawTree(AST8[whch[i]],height=4,width=2,show_o rdercondition=true,\n scale=1.2,color=COLOR(RGB,0,.7,0),font_color= black);\nend do:\npp := plot([[1,1]],style=line,axes=none):\nplots[dis play](convert([seq([p||((k-1)*m+1..m*k)],k=1..q),\n `if`(r>0,[p||(m*q +1..nn),pp$(m*(q+1)-nn)],NULL)],array));" }}{PARA 13 "" 1 "" {GLPLOT2D 852 1131 1131 {PLOTDATA 2 "6\\u-%%TEXTG6&7$$\"\"!F($!+!)fvQ \"*!\"*Q;b~c~(a~(a~(a~(a~c))))~=~~~6\"-%'COLOURG6&%$RGBGF(F(F(-%%FONTG 6$%(COURIERG\"#5-F$6&7$F'$!+g+d#[)F+Q=~~~~~~~~~~~~~~~~~~~~~~~~~~~1F-F. F2-F$6&7$F'$!+g\"Gwo)F+Q?~~~~~~~~~~~~~~~~~~~~~~~~~~~___F-F.F2-F$6&7$F' $!+**********F+Q?~~~~~~~~~~~~~~~~~~~~~~~~~~~840F-F.F2-%'CURVESG6&7)7$F '$!+#)\\<)3(F+7$$!+++++]F+$!+')*R0n$F+7$$\"+++++]F+FS7$FV$!*!*\\!HDF+7 $FV$\"+3+tkJF+7$FV$\"+)*\\O#e'F+7$FV$\"+)*********F+-%'SYMBOLG6#%'CIRC LEG-%&COLORG6&F1F'F'F'-%&STYLEG6#%&POINTG-FJ6&FL-F_o6#%(DIAMONDGFboFeo -FJ6&FL-F_o6#%&CROSSGFboFeo-FJ6%7$FMFP-Fco6&F1F'$\"\"(!\"\"F'-%*THICKN ESSG6#\"\"#-FJ6%7$FMFUFfpF[q-FJ6%7$FUFXFfpF[q-FJ6%7$FXFenFfpF[q-FJ6%7$ FenFhnFfpF[q-FJ6%7$FhnF[oFfpF[q-FJ6%7#7$F'F\\o-F/6&F1$F6FjpF'F'-Ffo6#% %LINEG-FJ6%7#7$$!+++++5!\")FNFbrFer-FJ6%7#7$$\"+++++5F^sFNFbrFer-F$6&7 $$\"#IF($!+5`7'*QF^sQ8b~c~(a~(a~(a~c~)))~=~~~F-F.F2-F$6&7$Fhs$!+)o#)p \"QF^sQ:~~~~~~~~~~~~~~~~~~~~~~~~1F-F.F2-F$6&7$Fhs$!+dZrTQF^sQ<~~~~~~~~ ~~~~~~~~~~~~~~~~___F-F.F2-F$6&7$Fhs$!+++++SF^sQ<~~~~~~~~~~~~~~~~~~~~~~ ~~960F-F.F2-F$6&7$Fhs$!+g)*32QF^sQ:~~~~~~~~~~~~~~~3~~~~~~~~~F-F.F2-FJ6 &7*7$Fhs$!+ " 0 "" {MPLTEXT 1 0 1 ";" }}}} {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT 260 4 "Note" } {TEXT -1 119 ": The 5 simplifying conditions can be replaced by the 5 \+ extra order conditions given (in abreviated form) as follows.. " }} {PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 174 "SO8 := SimpleOrderConditions(8):\n[seq([i,SO8[i]],i=[21,27,31,36, 127])]:\nlinalg[augment](linalg[delcols](%,2..2),matrix([[` `]$(linal g[rowdim](%))]),linalg[delcols](%,1..1));" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#K%'matrixG6#7'7%\"#@%#~~G/*&%\"bG\"\"\"-%!G6#*&%\"aGF-- F/6#*&F2F--F/6#*&)%\"cG\"\"#F-F2F-F-F-F-#F-\"$g$7%\"#FF)/*&F,F--F/6#*& F2F--F/6#*&)F:\"\"$F-F2F-F-F-#F-\"$?\"7%\"#JF)/*&F,F--F/6#*&)F:\"\"%F- F2F-F-#F-\"#I7%\"#OF)/*&F,F--F/6#*(F2F-F:F--F/6#*&F2F--F/6#*&F2F--F/6# *&F2F-F:F-F-F-F-F-#F-\"%357%\"$F\"F)/*&F,F--F/6#*&)F:\"\"'F-F2F-F-#F- \"#cQ(pprint36\"" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 462 "SOeqs2 := [seq(add(a[i,j],j= 1..i-1)=c[i],i= 9..12),op(StageOrderConditions(2,9..12,'expanded')),\n op(StageOrde rConditions(3,9..12,'expanded')),op(StageOrderConditions(4,9..12,'expa nded'))]:\nSO8_12 := SimpleOrderConditions(8,12,'expanded'):\nord_cdns := [seq(SO8_12[i],i=[37,61,81,87,102,113,106,117,125])]:\nsimp_eqs := [add(b[i]*a[i,1],i=1+1..12)=b[1],seq(add(b[i]*a[i,j],i=j+1..12)=b[j]* (1-c[j]),j=[5,7,9,10])]:\ncdns := [op(SOeqs2),op(ord_cdns),op(simp_eqs )]:" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 107 "e10 :=\{seq(a[i,2]=0,i=9..12),seq(a[i,3]=0,i=9..12) \}:\ne11 := `union`(e9,e10):\neqns2 := subs(e11,\{op(cdns)\}):" }}} {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 39 "There are 30 equations and 30 unknowns." }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 36 "nops(eqns2);\nindets(eqns2);\nnops(%);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#\"#I" }}{PARA 12 "" 1 "" {XPPMATH 20 "6#<@&%\"aG6$\"#7 \"#5&F%6$\"\"*\"\"(&F%6$F'\"\"&&F%6$F'F,&F%6$\"#6F+&F%6$F+\"\"%&F%6$F' F+&F%6$F(F,&F%6$F(F/&F%6$F(\"\")&F%6$F+F@&F%6$F4F7&F%6$F(\"\"\"&F%6$F4 FG&F%6$F(\"\"'&F%6$F+FL&F%6$F(F+&F%6$F4F@&F%6$F4F(&F%6$F'FL&F%6$F'F7&F %6$F'FG&F%6$F'F@&F%6$F+FG&F%6$F'F4&F%6$F+F/&F%6$F4FL&F%6$F4F,&F%6$F(F7 &F%6$F4F/" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#\"#I" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 22 "infolevel[so lve] := 4:" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 74 "e12 := solve( \{op(eqns2)\}):\ninfolevel[solve] := 0:\ne13 := `union`(e11,e12):" }}} {PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 4 "e13 " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 3887 "e13 := \{b [5] = 0, b[2] = 0, b[3] = 0, a[9,3] = 0, a[10,3] = 0, b[4] = 0, a[11,3 ] = 0, a[12,3] = 0, a[12,10] = 7958341351371843889152/3284467988443203 581305, c[12] = 1, a[11,2] = 0, a[12,2] = 0, a[7,2] = 0, a[6,3] = 0, a [10,2] = 0, a[4,2] = 0, a[8,3] = 0, a[8,2] = 0, a[7,3] = 0, a[6,2] = 0 , a[5,2] = 0, a[6,4] = 2505377/10685520, a[9,2] = 0, a[3,2] = 100352/1 134225, a[5,4] = 178077159/125440000, c[3] = 112/1065, a[5,3] = -35280 6597/250880000, b[10] = -10290327637248/1082076946951, a[8,1] = -15365 458811/13609565775, a[9,7] = 75198347919711375170485321796523477298993 03513750000/1045677303502317596597890707812349832637339039997351, a[12 ,5] = -697069297560926452045586710000/41107967755245430594036502319, a [4,3] = 42/355, a[12,7] = 643504802814241550941949227194107500000/2421 24609118836550860494007545333945331, a[11,9] = 48175771419260955335244 683805171548038966866545122229/198978642051381514652888016595206411890 3852843612160000, a[9,4] = -147649608040486573036383722529087802192814 24435/2981692102565021975611711269209606363661854518, c[2] = 1/16, c[5 ] = 39/100, c[6] = 7/15, a[12,9] = -2302825163287352381854541485685701 5616678575554130463402/20013169183191444503443905240405603349978424504 151629055, a[10,7] = 187023075231349900768014890274453125/252246988498 08178010752575653374848, a[10,5] = -3378604805394255292453489375/51704 2670569824692230499952, a[10,8] = 1908158550070998850625/1170870670391 89929394176, a[9,8] = 1366042683489166351293315549358278750/1446314182 24267718165055326464180836641, a[11,4] = -9623541317323077848129/38644 49564977792573440, a[10,1] = -63077736705254280154824845013881/7836935 7853786633855112190394368, a[11,1] = -10116106591826909534781157993685 116703/9562819945036894030442231411871744000, c[8] = 24/25, a[8,4] = - 7/5, c[7] = 39/250, a[10,6] = 1001587844183325981198091450220795/18423 2684207722503701669953872896, a[8,7] = 1993321838240/380523459069, a[9 ,6] = 7632051964154290925661849798370645637589377834346780/17340872574 18811583049800347581865260479233950396659, a[10,9] = -5295681828815666 8227044990077324877908565/2912779959477433986349822224412353951940608, a[11,8] = 39747262782380466933662225/1756032802431424164410720256, a[ 11,10] = -2378292068163246/47768728487211875, a[12,6] = 18273578204342 13461438077550902273440/139381013914245317709567680839641697, a[3,1] = 18928/1134225, a[4,1] = 14/355, a[7,5] = -21643947/592609375, a[12,4] = 26290092604284231996745/5760876126062860430544, b[1] = 463167487984 1/103782082379976, a[12,1] = -3218022174758599831659045535578571/14533 96753634469525663775847094384, a[8,5] = -8339128164608/939060038475, a [12,8] = 162259938151380266113750/59091082835244183497007, a[7,1] = 21 400899/350000000, b[7] = 2720762324010009765625000/1091736748069681392 2225349, a[2,1] = 1/16, a[9,1] = -184091125228237658443815733646470842 6954728061551/2991923615171151921596253813483118262195533733898, b[12] = -29746300739/247142463456, a[6,5] = 960400/5209191, a[7,4] = 306432 9829899/27126050000000, b[11] = 863264105888000/85814662253313, c[4] = 56/355, c[10] = 11/12, c[11] = 19/20, a[12,11] = -5079743279578608438 78400/121555654819179042718967, b[8] = -498533005859375/95352091037424 , a[9,5] = -875325048502130441118613421785266742862694404520560000/170 212030428894418395571677575961339495435011888324169, a[8,6] = 34193680 0488/47951126225, a[7,6] = 124391943/6756250000, b[9] = 40593203046377 7247926705030596175437402459637909765779/78803919436321841083201886041 201537229769115088303952, b[6] = 14327219974204125/40489566827933216, \+ a[11,6] = 6566119246514996884067001154977284529/9703054870218463254739 90863582315520, a[11,7] = 2226455130519213549256016892506730559375/364 880443159675255577435648380047355776, a[10,4] = -319483465108209702472 15/6956009216960026632192, a[5,1] = 94495479/250880000, a[6,1] = 12089 /252720, c[9] = 14435868/16178861, a[11,5] = -482334833314682940688137 5/576413233634141239944816\}:" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 20 "printed coefficients" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 283 "subs(e13,ma trix([seq([c[i],seq(a[i,j],j=1..i-1),``$(8-i)],i=2..8)]));\nfor ii fro m 9 to 12 do\n print(``);\n print(c[ii]=subs(e13,c[ii])); \n for jj to ii-1 do\n print(a[ii,jj]=subs(e13,a[ii,jj]));\n end do: \nend do:\n``;\nfor ii to 12 do\n print(b[ii]=subs(e13,b[ii]));\nend do:" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#K%'matrixG6#7)7*#\"\"\"\"#;F(% !GF+F+F+F+F+7*#\"$7\"\"%l5#\"&G*=\"(DU8\"#\"'_.5F2F+F+F+F+F+7*#\"#c\"$ b$#\"#9F8\"\"!#\"#UF8F+F+F+F+7*#\"#R\"$+\"#\")za\\%*\"*++)3DF;#!*(f1GN FD#\"*fr2y\"\"*++WD\"F+F+F+7*#\"\"(\"#:#\"&*37\"'?FDF;F;#\"(x`]#\")?bo 5#\"'+/'*\"(\">4_F+F+7*#F@\"$]##\")**3S@\"*+++]$F;F;#\".**)H)HV1$\"/++ +]g7F#!)ZRk@\"*v$4Ef#\"*V>RC\"\"+++DcnF+7*#\"#C\"#D#!,6)eaO:\",vdc4O\" F;F;#!\"(\"\"&#!.3Y;G\"R$)\"-v%Q+1R*#\"-)[+o$>M\",Di7^z%#\".S#Q=K$*>\" -p!fM_!QQ)pprint116\"" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%!G" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/&%\"cG6#\"\"*#\")oeV9\")h)yh\"" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/&%\"aG6$\"\"*\"\"\"#!R^:1GZ&pU3ZYOt:QWewBG_7\" 4%=\"R)*QtLb>i#=J[8QD'f@>:r^hB>*H" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/ &%\"aG6$\"\"*\"\"#\"\"!" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/&%\"aG6$\" \"*\"\"$\"\"!" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/&%\"aG6$\"\"*\"\"%#! PNWU\"G>-y3HDs$QOId'[S!3'\\w9\"O=X&=mjjg4#p7r6c(>-lD5#p\")H" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/&%\"aG6$\"\"*\"\"&#!W++c?XS%piGum_y@Mh=6WI@ ][]Kv)\"WpTK))=,Na\\R8'fdx;d&R=W*)G/.7-<" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/&%\"aG6$\"\"*\"\"'#\"U!yYV$yP*ePckq$)z\\=mD4HaT'>0Kw\"UfmR]RBz/ El=eZ.!)\\Ie6)=uD(3M<" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/&%\"aG6$\"\" *\"\"(#\"U++v8NI**)HxM_'z@`[q^P6(>zM)>v\"U^t**R!RtjK)\\B\"yq!*yf'f7^&Qj'y`yNp$y" }}{PARA 11 "" 1 "" {XPPMATH 20 "6# /&%\"aG6$\"#5\"\"#\"\"!" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/&%\"aG6$\" #5\"\"$\"\"!" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/&%\"aG6$\"#5\"\"%#!8: sCq4#3^Y$[>$\"7#>Km-gp@4g&p" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/&%\"aG 6$\"#5\"\"&#!=v$*[`CHbUR0[gyL\"<_**\\IApC)p0nUq^" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/&%\"aG6$\"#5\"\"'#\"C&z?-X\"4)>\")fK$=Wye,5\"B'*G(Q&*p ;q.Ds2UoKU=" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/&%\"aG6$\"#5\"\"(#\"ED JXu-*[,o2!*\\8BvI-(=\"D[[P`cd_2,y\"3)\\))pC_#" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/&%\"aG6$\"#5\"\")#\"7D1&))*42]&e\"3>\"9wTRH**=Rq1(3<\" " }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/&%\"aG6$\"#5\"\"*#!Jl&3z([Kx+*\\/ F#om:)G=o&H&\"L31%>&RN7WAA)\\j)RVx%f*zF\"H" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%!G" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/&%\"cG6#\"#6#\" #>\"#?" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/&%\"aG6$\"#6\"\"\"#!G.n6&o$ *z:\"yM&4p#=f1h65\"F+Sur=TJAWIS*o.X*>Gc*" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/&%\"aG6$\"#6\"\"#\"\"!" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/&%\" aG6$\"#6\"\"$\"\"!" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/&%\"aG6$\"#6\" \"%#!7H\"[y2BtJTNi*\"7SMd#zx\\c\\W'Q" }}{PARA 11 "" 1 "" {XPPMATH 20 " 6#/&%\"aG6$\"#6\"\"&#!:v8)oSHo9L$[L#[\"9;[%*R79MOB8kd" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/&%\"aG6$\"#6\"\"'#\"FHXGx\\:,q1%)o*\\^Y#>hc'\"E?b J#ej3*RZDj%=-([0.(*" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/&%\"aG6$\"#6\" \"(#\"Iv$f0t1D*o,c#\\N@>08bkA#\"HwdNZ+Q[cVxbDv'fJW!)[O" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/&%\"aG6$\"#6\"\")#\";DAmLpY!Q#yisuR\"=c-s5W;C9V-G .c<" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/&%\"aG6$\"#6\"\"*#\"VHA7Xl'o'* Q![:<0QoW_Lb4E>9xv\"[\"X++;7O%G&Q!*=T1_f;!))Gl9:Q^?ky*)>" }}{PARA 11 " " 1 "" {XPPMATH 20 "6#/&%\"aG6$\"#6\"#5#!1YK;o?HyB\"2v=@([G(ox%" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#%!G" }}{PARA 11 "" 1 "" {XPPMATH 20 "6 #/&%\"cG6#\"#7\"\"\"" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/&%\"aG6$\"#7 \"\"\"#!Cr&yb`X!f;$)*feZB%G/E4!HE\"7W0VgG1Eh(3w&" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/&%\"aG6$\"#7\"\"&#!?++r'eX?XE4c(Hpqp\">>B]OSfIaCbx'z5T " }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/&%\"aG6$\"#7\"\"'#\"FSMF-4bx!Q9Y8 UV?yNF=\"E(pT'R3on&4xJXU\"R,\"QR\"" }}{PARA 11 "" 1 "" {XPPMATH 20 "6# /&%\"aG6$\"#7\"\"(#\"H++]2T>F#\\>%4bTU\"G![]V'\"HJ`%RLXv+%\\g3bO)=\"4Y 7U#" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/&%\"aG6$\"#7\"\")#\"9]P6m-Q^\" Q*fA;\"82q\\$=W_$G3\"4f" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/&%\"aG6$\" #7\"\"*#!Y-MYITbv&ymh:q&o&[TX&=Q_tGj^#GI#\"Yb!H;:/XUy*\\Lg0/C0RW.XW\"> $=pJ,?" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/&%\"aG6$\"#7\"#5#\"7_\"*)Q% =P^8Mez\"708e.KW))zY%G$" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/&%\"aG6$\" #7\"#6#!9+%yQ%3'y&zKuz]\"9n*=F/z\">[lb:7" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%!G" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/&%\"bG6#\"\"\"#\".T)z[nJ Y\"0w*zB3#y.\"" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/&%\"bG6#\"\"#\"\"! " }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/&%\"bG6#\"\"$\"\"!" }}{PARA 11 " " 1 "" {XPPMATH 20 "6#/&%\"bG6#\"\"%\"\"!" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/&%\"bG6#\"\"&\"\"!" }}{PARA 11 "" 1 "" {XPPMATH 20 "6# /&%\"bG6#\"\"'#\"2DT?u*>sK9\"2;K$z#oc*[S" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/&%\"bG6#\"\"(#\":+]il(4+,CBw?F\";\\`AAR\"op![nt\"4\"" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/&%\"bG6#\"\")#!0v$fe+L&)\\\"/Cu.\"4_`*" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#/&%\"bG6#\"\"*#\"Wzdw4zjfCSPa<'fI]qEzC xPYI?$fS\"V_RI)3:\"p(Hs`,7/')=?$3T=KO%>R!)y" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/&%\"bG6#\"#5#!/[sjF.H5\".^p%p2#3\"" }}{PARA 11 "" 1 " " {XPPMATH 20 "6#/&%\"bG6#\"#6#\"0+!))e5kK')\"/8LDiY\"e)" }}{PARA 11 " " 1 "" {XPPMATH 20 "6#/&%\"bG6#\"#7#!,R2IY(H\"-cMYUrC" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}} {PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}}{PARA 0 "" 0 "" {TEXT -1 52 "#------------------------------ ---------------------" }}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 43 "construc tion of the order 7 embedded scheme" }}{PARA 0 "" 0 "" {TEXT -1 0 "" } }{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}{SECT 1 {PARA 4 "" 0 " " {TEXT -1 4 "e13 " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 " > " 0 "" {MPLTEXT 1 0 3887 "e13 := \{b[5] = 0, b[2] = 0, b[3] = 0, a[9 ,3] = 0, a[10,3] = 0, b[4] = 0, a[11,3] = 0, a[12,3] = 0, a[12,10] = 7 958341351371843889152/3284467988443203581305, c[12] = 1, a[11,2] = 0, \+ a[12,2] = 0, a[7,2] = 0, a[6,3] = 0, a[10,2] = 0, a[4,2] = 0, a[8,3] = 0, a[8,2] = 0, a[7,3] = 0, a[6,2] = 0, a[5,2] = 0, a[6,4] = 2505377/1 0685520, a[9,2] = 0, a[3,2] = 100352/1134225, a[5,4] = 178077159/12544 0000, c[3] = 112/1065, a[5,3] = -352806597/250880000, b[10] = -1029032 7637248/1082076946951, a[8,1] = -15365458811/13609565775, a[9,7] = 751 9834791971137517048532179652347729899303513750000/10456773035023175965 97890707812349832637339039997351, a[12,5] = -6970692975609264520455867 10000/41107967755245430594036502319, a[4,3] = 42/355, a[12,7] = 643504 802814241550941949227194107500000/242124609118836550860494007545333945 331, a[11,9] = 48175771419260955335244683805171548038966866545122229/1 989786420513815146528880165952064118903852843612160000, a[9,4] = -1476 4960804048657303638372252908780219281424435/29816921025650219756117112 69209606363661854518, c[2] = 1/16, c[5] = 39/100, c[6] = 7/15, a[12,9] = -23028251632873523818545414856857015616678575554130463402/200131691 83191444503443905240405603349978424504151629055, a[10,7] = 18702307523 1349900768014890274453125/25224698849808178010752575653374848, a[10,5] = -3378604805394255292453489375/517042670569824692230499952, a[10,8] \+ = 1908158550070998850625/117087067039189929394176, a[9,8] = 1366042683 489166351293315549358278750/144631418224267718165055326464180836641, a [11,4] = -9623541317323077848129/3864449564977792573440, a[10,1] = -63 077736705254280154824845013881/78369357853786633855112190394368, a[11, 1] = -10116106591826909534781157993685116703/9562819945036894030442231 411871744000, c[8] = 24/25, a[8,4] = -7/5, c[7] = 39/250, a[10,6] = 10 01587844183325981198091450220795/184232684207722503701669953872896, a[ 8,7] = 1993321838240/380523459069, a[9,6] = 76320519641542909256618497 98370645637589377834346780/1734087257418811583049800347581865260479233 950396659, a[10,9] = -52956818288156668227044990077324877908565/291277 9959477433986349822224412353951940608, a[11,8] = 397472627823804669336 62225/1756032802431424164410720256, a[11,10] = -2378292068163246/47768 728487211875, a[12,6] = 1827357820434213461438077550902273440/13938101 3914245317709567680839641697, a[3,1] = 18928/1134225, a[4,1] = 14/355, a[7,5] = -21643947/592609375, a[12,4] = 26290092604284231996745/57608 76126062860430544, b[1] = 4631674879841/103782082379976, a[12,1] = -32 18022174758599831659045535578571/1453396753634469525663775847094384, a [8,5] = -8339128164608/939060038475, a[12,8] = 16225993815138026611375 0/59091082835244183497007, a[7,1] = 21400899/350000000, b[7] = 2720762 324010009765625000/10917367480696813922225349, a[2,1] = 1/16, a[9,1] = -1840911252282376584438157336464708426954728061551/299192361517115192 1596253813483118262195533733898, b[12] = -29746300739/247142463456, a[ 6,5] = 960400/5209191, a[7,4] = 3064329829899/27126050000000, b[11] = \+ 863264105888000/85814662253313, c[4] = 56/355, c[10] = 11/12, c[11] = \+ 19/20, a[12,11] = -507974327957860843878400/121555654819179042718967, \+ b[8] = -498533005859375/95352091037424, a[9,5] = -87532504850213044111 8613421785266742862694404520560000/17021203042889441839557167757596133 9495435011888324169, a[8,6] = 341936800488/47951126225, a[7,6] = 12439 1943/6756250000, b[9] = 4059320304637772479267050305961754374024596379 09765779/78803919436321841083201886041201537229769115088303952, b[6] = 14327219974204125/40489566827933216, a[11,6] = 6566119246514996884067 001154977284529/970305487021846325473990863582315520, a[11,7] = 222645 5130519213549256016892506730559375/36488044315967525557743564838004735 5776, a[10,4] = -31948346510820970247215/6956009216960026632192, a[5,1 ] = 94495479/250880000, a[6,1] = 12089/252720, c[9] = 14435868/1617886 1, a[11,5] = -4823348333146829406881375/576413233634141239944816\}:" } }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 61 "We can obtain an embedded 11 stage order 7 scheme as follows." }} {PARA 0 "" 0 "" {TEXT -1 94 "We remove stages 11 and 12 from the 12 st age order 8 scheme and introduce a new stage 11 with " }{XPPEDIT 18 0 "c[11] = 1;" "6#/&%\"cG6#\"#6\"\"\"" }{TEXT -1 3 ", " }{XPPEDIT 18 0 "a[11,2] = 0" "6#/&%\"aG6$\"#6\"\"#\"\"!" }{TEXT -1 3 ", " }{XPPEDIT 18 0 "a[11,3] = 0;" "6#/&%\"aG6$\"#6\"\"$\"\"!" }{TEXT -1 3 ", " } {XPPEDIT 18 0 "`b*`[2]=0" "6#/&%#b*G6#\"\"#\"\"!" }{TEXT -1 3 ", " } {XPPEDIT 18 0 "`b*`[3]=0" "6#/&%#b*G6#\"\"$\"\"!" }{TEXT -1 3 ", " } {XPPEDIT 18 0 "`b*`[4]=0" "6#/&%#b*G6#\"\"%\"\"!" }{TEXT -1 7 " and \+ " }{XPPEDIT 18 0 "`b*`[5]=0" "6#/&%#b*G6#\"\"&\"\"!" }{TEXT -1 58 ", w here the weights of the order 7 scheme are denoted by " }{XPPEDIT 18 0 "`b*`" "6#%#b*G" }{TEXT -1 1 "." }}{PARA 0 "" 0 "" {TEXT -1 1 " " }} {PARA 0 "" 0 "" {TEXT -1 156 "We incorporate the order 7 quadrature co nditions, the row sum conditions for this stage and stage-order condit ions so that this new stage has stage-order 4." }}{PARA 0 "" 0 "" {TEXT -1 48 "We also incorporate the simplifying conditions: " }} {PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "Sum(`b*`[i]*a[i,j],i= j+1..11)=`b*`[j]*(1-c[j])" "6#/-%$SumG6$*&&%#b*G6#%\"iG\"\"\"&%\"aG6$F +%\"jGF,/F+;,&F0F,F,F,\"#6*&&F)6#F0F,,&F,F,&%\"cG6#F0!\"\"F," }{TEXT -1 1 "," }}{PARA 0 "" 0 "" {TEXT -1 5 "for " }{XPPEDIT 18 0 "j=1" "6# /%\"jG\"\"\"" }{TEXT -1 10 " ( with " }{XPPEDIT 18 0 "c[1]=0" "6#/&% \"cG6#\"\"\"\"\"!" }{TEXT -1 5 " ), " }{XPPEDIT 18 0 "j = 8;" "6#/%\" jG\"\")" }{TEXT -1 3 ", " }{XPPEDIT 18 0 "j=9" "6#/%\"jG\"\"*" } {TEXT -1 7 " and " }{XPPEDIT 18 0 "j=10" "6#/%\"jG\"#5" }{TEXT -1 2 ". " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 399 "`Qeqs*` := subs(b=`b*`,QuadratureConditions(7,11,'ex panded')):\nSO_eqs := [add(a[11,j],j=1..10)=c[11],add(a[11,j]*c[j],j=2 ..10)=1/2*c[11]^2,\n add(a[11,j]*c[j]^2,j=2..10)=1/3*c[11]^3,ad d(a[11,j]*c[j]^3,j=2..10)=1/4*c[11]^4]:\n`simp_eqs*` := [add(`b*`[i]*a [i,1],i=2..11)=`b*`[1],seq(add(`b*`[i]*a[i,j],i=j+1..11)=`b*`[j]*(1-c[ j]),j=[8,9,10])]:\n`cdns*` := [op(`simp_eqs*`),op(SO_eqs),op(`Qeqs*`)] :" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 144 "e14 := `union`(remove(u_->member(op(1,lhs(u_)),[11,1 2]) or op(0,lhs(u_))=b,e13),\n \{c[11]=1,a[11,2]=0,a[11,3]=0,s eq(`b*`[i]=0,i=2..5)\}):\n" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 53 "We have 15 equations for the 15 unknown coeffic ients." }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 60 "eqns3 := subs(e14, `cdns*`):\nnops(%);\nindets(eqns3);\nnops(%);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#\"#:" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#<1&%#b*G6#\"\"* &F%6#\"#5&F%6#\"#6&F%6#\"\"\"&F%6#\"\"'&F%6#\"\"(&F%6#\"\")&%\"aG6$F-F '&F;6$F-\"\"%&F;6$F-F0&F;6$F-F9&F;6$F-F*&F;6$F-F3&F;6$F-F6&F;6$F-\"\"& " }}{PARA 11 "" 1 "" {XPPMATH 20 "6#\"#:" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 22 "infolevel[solve] := 4 :" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 74 "e15 := solve(\{op(eqns 3)\}):\ninfolevel[solve] := 0:\ne16 := `union`(e14,e15):" }}}{PARA 0 " " 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}} {SECT 1 {PARA 4 "" 0 "" {TEXT -1 3 "e16" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 3092 "e16 := \{a[7,1] = 214 00899/350000000, `b*`[6] = 495437430316125/1396191959583904, a[10,3] = 0, a[11,3] = 0, a[11,2] = 0, a[7,2] = 0, a[9,3] = 0, a[6,3] = 0, a[10 ,2] = 0, a[4,2] = 0, a[8,3] = 0, a[8,2] = 0, a[7,3] = 0, a[6,2] = 0, a [5,2] = 0, a[6,4] = 2505377/10685520, a[9,2] = 0, a[3,2] = 100352/1134 225, a[5,4] = 178077159/125440000, c[3] = 112/1065, a[5,3] = -35280659 7/250880000, a[8,1] = -15365458811/13609565775, a[9,7] = 7519834791971 137517048532179652347729899303513750000/104567730350231759659789070781 2349832637339039997351, a[4,3] = 42/355, a[9,4] = -1476496080404865730 3638372252908780219281424435/29816921025650219756117112692096063636618 54518, c[2] = 1/16, c[5] = 39/100, c[6] = 7/15, a[10,7] = 187023075231 349900768014890274453125/25224698849808178010752575653374848, a[10,5] \+ = -3378604805394255292453489375/517042670569824692230499952, a[10,8] = 1908158550070998850625/117087067039189929394176, a[9,8] = 13660426834 89166351293315549358278750/144631418224267718165055326464180836641, a[ 10,1] = -63077736705254280154824845013881/7836935785378663385511219039 4368, c[8] = 24/25, a[8,4] = -7/5, c[7] = 39/250, a[10,6] = 1001587844 183325981198091450220795/184232684207722503701669953872896, a[8,7] = 1 993321838240/380523459069, a[9,6] = 7632051964154290925661849798370645 637589377834346780/173408725741881158304980034758186526047923395039665 9, a[10,9] = -52956818288156668227044990077324877908565/29127799594774 33986349822224412353951940608, a[3,1] = 18928/1134225, a[4,1] = 14/355 , a[7,5] = -21643947/592609375, a[8,5] = -8339128164608/939060038475, \+ a[2,1] = 1/16, a[9,1] = -184091125228237658443815733646470842695472806 1551/2991923615171151921596253813483118262195533733898, a[6,5] = 96040 0/5209191, a[7,4] = 3064329829899/27126050000000, c[4] = 56/355, c[10] = 11/12, a[9,5] = -87532504850213044111861342178526674286269440452056 0000/170212030428894418395571677575961339495435011888324169, a[8,6] = \+ 341936800488/47951126225, a[7,6] = 124391943/6756250000, a[10,4] = -31 948346510820970247215/6956009216960026632192, a[5,1] = 94495479/250880 000, a[6,1] = 12089/252720, c[9] = 14435868/16178861, a[11,1] = -54908 0624436801105208519835138333/353987109028707139687100885600400, `b*`[2 ] = 0, `b*`[3] = 0, `b*`[4] = 0, `b*`[5] = 0, a[11,8] = -7075929545777 56600025/2430671607007101253926, a[11,6] = 154077556949523438339030726 2972464/164751248733597841985831445059895, c[11] = 1, `b*`[11] = -4588 4771325/82380821152, a[11,10] = 4305840920849725632512/506641359866334 6018375, `b*`[8] = 1186143278515625/444976424841312, `b*`[10] = -30733 299644928/5410384734755, `b*`[1] = 325503096889/7282953149472, a[11,9] = -315869406877370103440389763510384832076674110731/55084590101671479 4114516357859531754926300500125, a[11,5] = -91153092961177216058210567 600/7609267653017028089793994539, `b*`[7] = 760741072216796875000/3055 518466469861159313, a[11,7] = -2027488254536386321212021357622300000/7 563800784313191306927092359781550321, a[11,4] = 2911667531218603395648 1/5331818957833865866320, `b*`[9] = 1379157935389455914712828210809218 1066885426809/3515333032435874239702147176443897942760320040\}:" } {TEXT -1 0 "" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 20 "printed coefficients" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 289 "subs(e16,matrix([seq([c[i],seq(a[i ,j],j=1..i-1),``$(8-i)],i=2..8)]));\nfor ii from 9 to 11 do\n print( ``);\n print(c[ii]=subs(e16,c[ii])); \n for jj to ii-1 do\n p rint(a[ii,jj]=subs(e16,a[ii,jj]));\n end do:\nend do:\n``;\nfor ii t o 11 do\n print(`b*`[ii]=subs(e16,`b*`[ii]));\nend do:" }}{PARA 11 " " 1 "" {XPPMATH 20 "6#K%'matrixG6#7)7*#\"\"\"\"#;F(%!GF+F+F+F+F+7*#\"$ 7\"\"%l5#\"&G*=\"(DU8\"#\"'_.5F2F+F+F+F+F+7*#\"#c\"$b$#\"#9F8\"\"!#\"# UF8F+F+F+F+7*#\"#R\"$+\"#\")za\\%*\"*++)3DF;#!*(f1GNFD#\"*fr2y\"\"*++W D\"F+F+F+7*#\"\"(\"#:#\"&*37\"'?FDF;F;#\"(x`]#\")?bo5#\"'+/'*\"(\">4_F +F+7*#F@\"$]##\")**3S@\"*+++]$F;F;#\".**)H)HV1$\"/+++]g7F#!)ZRk@\"*v$4 Ef#\"*V>RC\"\"+++DcnF+7*#\"#C\"#D#!,6)eaO:\",vdc4O\"F;F;#!\"(\"\"&#!.3 Y;G\"R$)\"-v%Q+1R*#\"-)[+o$>M\",Di7^z%#\".S#Q=K$*>\"-p!fM_!QQ)pprint13 6\"" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%!G" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/&%\"cG6#\"\"*#\")oeV9\")h)yh\"" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/&%\"aG6$\"\"*\"\"\"#!R^:1GZ&pU3ZYOt:QWewBG_7\"4%=\"R)* QtLb>i#=J[8QD'f@>:r^hB>*H" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/&%\"aG6$ \"\"*\"\"#\"\"!" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/&%\"aG6$\"\"*\"\"$ \"\"!" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/&%\"aG6$\"\"*\"\"%#!PNWU\"G> -y3HDs$QOId'[S!3'\\w9\"O=X&=mjjg4#p7r6c(>-lD5#p\")H" }}{PARA 11 "" 1 " " {XPPMATH 20 "6#/&%\"aG6$\"\"*\"\"&#!W++c?XS%piGum_y@Mh=6WI@][]Kv)\"W pTK))=,Na\\R8'fdx;d&R=W*)G/.7-<" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/&% \"aG6$\"\"*\"\"'#\"U!yYV$yP*ePckq$)z\\=mD4HaT'>0Kw\"UfmR]RBz/El=eZ.!) \\Ie6)=uD(3M<" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/&%\"aG6$\"\"*\"\"(# \"U++v8NI**)HxM_'z@`[q^P6(>zM)>v\"U^t**R!RtjK)\\B\"yq!*yf'f7^&Qj'y`yNp$y" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/&%\"aG6$ \"#5\"\"#\"\"!" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/&%\"aG6$\"#5\"\"$\" \"!" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/&%\"aG6$\"#5\"\"%#!8:sCq4#3^Y$ [>$\"7#>Km-gp@4g&p" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/&%\"aG6$\"#5\" \"&#!=v$*[`CHbUR0[gyL\"<_**\\IApC)p0nUq^" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/&%\"aG6$\"#5\"\"'#\"C&z?-X\"4)>\")fK$=Wye,5\"B'*G(Q&*p;q.Ds2UoK U=" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/&%\"aG6$\"#5\"\"(#\"EDJXu-*[,o2 !*\\8BvI-(=\"D[[P`cd_2,y\"3)\\))pC_#" }}{PARA 11 "" 1 "" {XPPMATH 20 " 6#/&%\"aG6$\"#5\"\")#\"7D1&))*42]&e\"3>\"9wTRH**=Rq1(3<\"" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/&%\"aG6$\"#5\"\"*#!Jl&3z([Kx+*\\/F#om:)G=o&H& \"L31%>&RN7WAA)\\j)RVx%f*zF\"H" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%!G " }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/&%\"cG6#\"#6\"\"\"" }}{PARA 11 " " 1 "" {XPPMATH 20 "6#/&%\"aG6$\"#6\"\"\"#!BL$Q^$)>&3_5,oVC13\\&\"B+/g &)35(oRrqG!4r)RN" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/&%\"aG6$\"#6\"\"# \"\"!" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/&%\"aG6$\"#6\"\"$\"\"!" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#/&%\"aG6$\"#6\"\"%#\"8\"[cR.'=7`n;\"H \"7?j'e'Q$y&*==L&" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/&%\"aG6$\"#6\"\" &#!>+wc5#eg@x6'H4`6*\"=RX*Rz*3Gq,`wE4w" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/&%\"aG6$\"#6\"\"'#\"CkC(HE2.R$QM_\\pbxS:\"B&*)f]WJe)>%yfL([7vk \"" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/&%\"aG6$\"#6\"\"(#!F++IAwN@?@@j QOXD)[F?\"F@.b\"yfB4FpI\">8Vy+Qc(" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/ &%\"aG6$\"#6\"\")#!6D+gcxdaHf2(\"7ERD,r+2;nIC" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/&%\"aG6$\"#6\"\"*#!QJ26um2K[Q5Nw*QSM5qt(oSpeJ\"QD,]+j# \\vJ&fyN;X6%z9n,,f%3b" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/&%\"aG6$\"#6 \"#5#\"77DjD(\\3#4%eI%\"7v$=gMj')f8k1&" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%!G" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/&%#b*G6#\"\"\"#\"-*)o4.b K\".s%\\J&HG(" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/&%#b*G6#\"\"#\"\"!" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/&%#b*G6#\"\"$\"\"!" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/&%#b*G6#\"\"%\"\"!" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/&%#b*G6#\"\"&\"\"!" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/&%#b*G6# \"\"'#\"0DhJIuV&\\\"1/Ref>>'R\"" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/&% #b*G6#\"\"(#\"6+](oz;A2T2w\"78$f6')pkY=b0$" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/&%#b*G6#\"\")#\"1Dc^yK9'=\"\"078%[Uw\\W" }}{PARA 11 " " 1 "" {XPPMATH 20 "6#/&%#b*G6#\"\"*#\"P4oU&)o1\"=#43@GGr9fX*QNz:z8\"O S+KgF%z*QWwr9-(RU(eVKIL:N" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/&%#b*G6# \"#5#!/G\\k*HL2$\".bZt%Q5a" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/&%#b*G6 #\"#6#!,D8x%)e%\",_6#3Q#)" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}}{PARA 0 "" 0 "" {TEXT -1 0 " " }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 104 "`RK7_11eqs*` := subs(b= `b*`,[op(RowSumConditions(11,'expanded')),op(OrderConditions(7,11,'exp anded'))]):" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 5 "Check" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 57 "subs(e16,`RK7 _11eqs*`):\nmap(u->lhs(u)-rhs(u),%);\nnops(%);" }}{PARA 12 "" 1 "" {XPPMATH 20 "6#7[q\"\"!F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F $F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F $F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F $" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#\"#&*" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 93 "We can calculate the principal \+ error norm, that is, the 2-norm of the principal error terms. " }} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 185 "`errterms7_11*` := subs(b=` b*`,PrincipalErrorTerms(7,11,'expanded')):\nsm := 0:\nfor ct to nops(` errterms7_11*`) do\n sm := sm+(evalf(subs(e16,`errterms7_11*`[ct]))) ^2;\nend do:\nsqrt(sm);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#$\"+sNs?$*! #:" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 138 "W e can include the new stage for the embedded scheme as an additional 1 3th stage added to the order 8 scheme along with the coefficients " } {XPPEDIT 18 0 "a[13,11] = 0;" "6#/&%\"aG6$\"#8\"#6\"\"!" }{TEXT -1 7 " and " }{XPPEDIT 18 0 "a[13,12] = 0;" "6#/&%\"aG6$\"#8\"#7\"\"!" } {TEXT -1 1 "." }}{PARA 0 "" 0 "" {TEXT -1 13 "The weights " } {XPPEDIT 18 0 "`b*`[i]" "6#&%#b*G6#%\"iG" }{TEXT -1 7 " for " } {XPPEDIT 18 0 "i=1" "6#/%\"iG\"\"\"" }{TEXT -1 89 " . . 10 of the 13 s tage combined scheme are those of the 11 stage scheme and the weight \+ " }{XPPEDIT 18 0 "`b*`[11];" "6#&%#b*G6#\"#6" }{TEXT -1 34 " in the 1 1 stage scheme becomes " }{XPPEDIT 18 0 "`b*`[13];" "6#&%#b*G6#\"#8" }{TEXT -1 25 " in the 13 stage scheme." }}{PARA 0 "" 0 "" {TEXT -1 8 "We set " }{XPPEDIT 18 0 "c[13] = 1;" "6#/&%\"cG6#\"#8\"\"\"" }{TEXT -1 3 ", " }{XPPEDIT 18 0 "`b*`[11] = 0;" "6#/&%#b*G6#\"#6\"\"!" } {TEXT -1 7 " and " }{XPPEDIT 18 0 "`b*`[12] = 0;" "6#/&%#b*G6#\"#7\" \"!" }{TEXT -1 1 "." }}{PARA 0 "" 0 "" {TEXT -1 66 "We can make the or der 8 scheme into a 13 stage scheme by setting " }{XPPEDIT 18 0 "b[13 ] = 0;" "6#/&%\"bG6#\"#8\"\"!" }{TEXT -1 1 "." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 201 "e17 := \{c[ 13]=1,seq(a[13,i]=subs(e16,a[11,i]),i=1..10),a[13,11]=0,a[13,12]=0,b[1 3]=0,\nseq(`b*`[i]=subs(e16,`b*`[i]),i=1..10),`b*`[11]=0,`b*`[12]=0,`b *`[13]=subs(e16,`b*`[11])\}:\ne18 := `union`(e13,e17):" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}} {SECT 1 {PARA 4 "" 0 "" {TEXT -1 3 "e18" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 5012 "e18 := \{b[13] = 0, a [7,1] = 21400899/350000000, `b*`[6] = 495437430316125/1396191959583904 , b[7] = 2720762324010009765625000/10917367480696813922225349, a[13,11 ] = 0, a[13,12] = 0, a[10,3] = 0, b[4] = 0, a[11,3] = 0, a[12,3] = 0, \+ c[13] = 1, a[12,10] = 7958341351371843889152/3284467988443203581305, c [12] = 1, a[11,2] = 0, a[12,2] = 0, a[7,2] = 0, b[2] = 0, b[3] = 0, a[ 9,3] = 0, `b*`[12] = 0, `b*`[11] = 0, b[5] = 0, a[6,3] = 0, a[10,2] = \+ 0, a[4,2] = 0, a[8,3] = 0, a[8,2] = 0, a[7,3] = 0, a[6,2] = 0, a[5,2] \+ = 0, a[6,4] = 2505377/10685520, a[9,2] = 0, a[3,2] = 100352/1134225, a [5,4] = 178077159/125440000, c[3] = 112/1065, a[5,3] = -352806597/2508 80000, b[10] = -10290327637248/1082076946951, a[8,1] = -15365458811/13 609565775, a[9,7] = 75198347919711375170485321796523477298993035137500 00/1045677303502317596597890707812349832637339039997351, a[12,5] = -69 7069297560926452045586710000/41107967755245430594036502319, a[4,3] = 4 2/355, a[12,7] = 643504802814241550941949227194107500000/2421246091188 36550860494007545333945331, a[11,9] = 48175771419260955335244683805171 548038966866545122229/198978642051381514652888016595206411890385284361 2160000, a[9,4] = -14764960804048657303638372252908780219281424435/298 1692102565021975611711269209606363661854518, c[2] = 1/16, c[5] = 39/10 0, c[6] = 7/15, a[12,9] = -2302825163287352381854541485685701561667857 5554130463402/20013169183191444503443905240405603349978424504151629055 , a[10,7] = 187023075231349900768014890274453125/252246988498081780107 52575653374848, a[10,5] = -3378604805394255292453489375/51704267056982 4692230499952, a[10,8] = 1908158550070998850625/1170870670391899293941 76, a[9,8] = 1366042683489166351293315549358278750/1446314182242677181 65055326464180836641, a[11,4] = -9623541317323077848129/38644495649777 92573440, a[10,1] = -63077736705254280154824845013881/7836935785378663 3855112190394368, a[11,1] = -10116106591826909534781157993685116703/95 62819945036894030442231411871744000, c[8] = 24/25, a[8,4] = -7/5, c[7] = 39/250, a[10,6] = 1001587844183325981198091450220795/18423268420772 2503701669953872896, a[8,7] = 1993321838240/380523459069, a[9,6] = 763 2051964154290925661849798370645637589377834346780/17340872574188115830 49800347581865260479233950396659, a[10,9] = -5295681828815666822704499 0077324877908565/2912779959477433986349822224412353951940608, a[11,8] \+ = 39747262782380466933662225/1756032802431424164410720256, a[11,10] = \+ -2378292068163246/47768728487211875, a[12,6] = 18273578204342134614380 77550902273440/139381013914245317709567680839641697, a[3,1] = 18928/11 34225, a[4,1] = 14/355, a[7,5] = -21643947/592609375, a[12,4] = 262900 92604284231996745/5760876126062860430544, b[1] = 4631674879841/1037820 82379976, a[12,1] = -3218022174758599831659045535578571/14533967536344 69525663775847094384, a[8,5] = -8339128164608/939060038475, a[12,8] = \+ 162259938151380266113750/59091082835244183497007, a[2,1] = 1/16, a[9,1 ] = -1840911252282376584438157336464708426954728061551/299192361517115 1921596253813483118262195533733898, b[12] = -29746300739/247142463456, a[6,5] = 960400/5209191, a[7,4] = 3064329829899/27126050000000, b[11] = 863264105888000/85814662253313, c[4] = 56/355, c[10] = 11/12, c[11] = 19/20, a[12,11] = -507974327957860843878400/12155565481917904271896 7, b[8] = -498533005859375/95352091037424, a[9,5] = -87532504850213044 1118613421785266742862694404520560000/17021203042889441839557167757596 1339495435011888324169, a[8,6] = 341936800488/47951126225, a[7,6] = 12 4391943/6756250000, b[9] = 4059320304637772479267050305961754374024596 37909765779/78803919436321841083201886041201537229769115088303952, b[6 ] = 14327219974204125/40489566827933216, a[11,6] = 6566119246514996884 067001154977284529/970305487021846325473990863582315520, a[11,7] = 222 6455130519213549256016892506730559375/36488044315967525557743564838004 7355776, a[10,4] = -31948346510820970247215/6956009216960026632192, a[ 5,1] = 94495479/250880000, a[6,1] = 12089/252720, c[9] = 14435868/1617 8861, a[11,5] = -4823348333146829406881375/576413233634141239944816, a [13,6] = 1540775569495234383390307262972464/16475124873359784198583144 5059895, `b*`[2] = 0, `b*`[3] = 0, `b*`[4] = 0, `b*`[5] = 0, a[13,9] = -315869406877370103440389763510384832076674110731/5508459010167147941 14516357859531754926300500125, a[13,5] = -9115309296117721605821056760 0/7609267653017028089793994539, a[13,10] = 4305840920849725632512/5066 413598663346018375, `b*`[13] = -45884771325/82380821152, `b*`[8] = 118 6143278515625/444976424841312, `b*`[10] = -30733299644928/541038473475 5, `b*`[1] = 325503096889/7282953149472, a[13,3] = 0, a[13,4] = 291166 75312186033956481/5331818957833865866320, `b*`[7] = 760741072216796875 000/3055518466469861159313, a[13,8] = -707592954577756600025/243067160 7007101253926, a[13,2] = 0, a[13,7] = -2027488254536386321212021357622 300000/7563800784313191306927092359781550321, `b*`[9] = 13791579353894 559147128282108092181066885426809/351533303243587423970214717644389794 2760320040, a[13,1] = -549080624436801105208519835138333/3539871090287 07139687100885600400\}:" }{TEXT -1 0 "" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 20 "printed coefficients" }}{PARA 0 "" 0 " " {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 362 "subs(e18, matrix([seq([c[i],seq(a[i,j],j=1..i-1),``$(8-i)],i=2..8)]));\nfor ii f rom 9 to 13 do\n print(``);\n print(c[ii]=subs(e18,c[ii])); \n f or jj to ii-1 do\n print(a[ii,jj]=subs(e18,a[ii,jj]));\n end do :\nend do:print(``);\nfor ii to 13 do\n print(b[ii]=subs(e18,b[ii])) ;\nend do:print(``);\nfor ii to 13 do\n print(`b*`[ii]=subs(e18,`b*` [ii]));\nend do:" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#K%'matrixG6#7)7*# \"\"\"\"#;F(%!GF+F+F+F+F+7*#\"$7\"\"%l5#\"&G*=\"(DU8\"#\"'_.5F2F+F+F+F +F+7*#\"#c\"$b$#\"#9F8\"\"!#\"#UF8F+F+F+F+7*#\"#R\"$+\"#\")za\\%*\"*++ )3DF;#!*(f1GNFD#\"*fr2y\"\"*++WD\"F+F+F+7*#\"\"(\"#:#\"&*37\"'?FDF;F;# \"(x`]#\")?bo5#\"'+/'*\"(\">4_F+F+7*#F@\"$]##\")**3S@\"*+++]$F;F;#\".* *)H)HV1$\"/+++]g7F#!)ZRk@\"*v$4Ef#\"*V>RC\"\"+++DcnF+7*#\"#C\"#D#!,6)e aO:\",vdc4O\"F;F;#!\"(\"\"&#!.3Y;G\"R$)\"-v%Q+1R*#\"-)[+o$>M\",Di7^z%# \".S#Q=K$*>\"-p!fM_!QQ)pprint126\"" }}{PARA 11 "" 1 "" {XPPMATH 20 "6# %!G" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/&%\"cG6#\"\"*#\")oeV9\")h)yh\" " }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/&%\"aG6$\"\"*\"\"\"#!R^:1GZ&pU3ZY Ot:QWewBG_7\"4%=\"R)*QtLb>i#=J[8QD'f@>:r^hB>*H" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/&%\"aG6$\"\"*\"\"#\"\"!" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/&%\"aG6$\"\"*\"\"$\"\"!" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/&% \"aG6$\"\"*\"\"%#!PNWU\"G>-y3HDs$QOId'[S!3'\\w9\"O=X&=mjjg4#p7r6c(>-lD 5#p\")H" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/&%\"aG6$\"\"*\"\"&#!W++c?X S%piGum_y@Mh=6WI@][]Kv)\"WpTK))=,Na\\R8'fdx;d&R=W*)G/.7-<" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/&%\"aG6$\"\"*\"\"'#\"U!yYV$yP*ePckq$)z\\=mD4Ha T'>0Kw\"UfmR]RBz/El=eZ.!)\\Ie6)=uD(3M<" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/&%\"aG6$\"\"*\"\"(#\"U++v8NI**)HxM_'z@`[q^P6(>zM)>v\"U^t**R!Rtj K)\\B\"yq!*yf'f7^&Qj'y`yNp$y" }}{PARA 11 "" 1 " " {XPPMATH 20 "6#/&%\"aG6$\"#5\"\"#\"\"!" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/&%\"aG6$\"#5\"\"$\"\"!" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/&%\" aG6$\"#5\"\"%#!8:sCq4#3^Y$[>$\"7#>Km-gp@4g&p" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/&%\"aG6$\"#5\"\"&#!=v$*[`CHbUR0[gyL\"<_**\\IApC)p0nUq^ " }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/&%\"aG6$\"#5\"\"'#\"C&z?-X\"4)>\" )fK$=Wye,5\"B'*G(Q&*p;q.Ds2UoKU=" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/& %\"aG6$\"#5\"\"(#\"EDJXu-*[,o2!*\\8BvI-(=\"D[[P`cd_2,y\"3)\\))pC_#" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#/&%\"aG6$\"#5\"\")#\"7D1&))*42]&e\"3> \"9wTRH**=Rq1(3<\"" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/&%\"aG6$\"#5\" \"*#!Jl&3z([Kx+*\\/F#om:)G=o&H&\"L31%>&RN7WAA)\\j)RVx%f*zF\"H" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#%!G" }}{PARA 11 "" 1 "" {XPPMATH 20 "6 #/&%\"cG6#\"#6#\"#>\"#?" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/&%\"aG6$\" #6\"\"\"#!G.n6&o$*z:\"yM&4p#=f1h65\"F+Sur=TJAWIS*o.X*>Gc*" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/&%\"aG6$\"#6\"\"#\"\"!" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/&%\"aG6$\"#6\"\"$\"\"!" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/&%\"aG6$\"#6\"\"%#!7H\"[y2BtJTNi*\"7SMd#zx\\c\\W'Q" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/&%\"aG6$\"#6\"\"&#!:v8)oSHo9L$[L#[\"9;[%*R79MO B8kd" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/&%\"aG6$\"#6\"\"'#\"FHXGx\\:, q1%)o*\\^Y#>hc'\"E?bJ#ej3*RZDj%=-([0.(*" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/&%\"aG6$\"#6\"\"(#\"Iv$f0t1D*o,c#\\N@>08bkA#\"HwdNZ+Q[cVxbDv'fJ W!)[O" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/&%\"aG6$\"#6\"\")#\";DAmLpY! Q#yisuR\"=c-s5W;C9V-G.c<" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/&%\"aG6$ \"#6\"\"*#\"VHA7Xl'o'*Q![:<0QoW_Lb4E>9xv\"[\"X++;7O%G&Q!*=T1_f;!))Gl9: Q^?ky*)>" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/&%\"aG6$\"#6\"#5#!1YK;o?H yB\"2v=@([G(ox%" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%!G" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/&%\"cG6#\"#7\"\"\"" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/&%\"aG6$\"#7\"\"\"#!Cr&yb`X!f;$)*feZB%G/E4!HE\"7W0VgG1Eh(3w&" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#/&%\"aG6$\"#7\"\"&#!?++r'eX?XE4c(Hpqp \">>B]OSfIaCbx'z5T" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/&%\"aG6$\"#7\" \"'#\"FSMF-4bx!Q9Y8UV?yNF=\"E(pT'R3on&4xJXU\"R,\"QR\"" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/&%\"aG6$\"#7\"\"(#\"H++]2T>F#\\>%4bTU\"G![]V'\"HJ `%RLXv+%\\g3bO)=\"4Y7U#" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/&%\"aG6$\" #7\"\")#\"9]P6m-Q^\"Q*fA;\"82q\\$=W_$G3\"4f" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/&%\"aG6$\"#7\"\"*#!Y-MYITbv&ymh:q&o&[TX&=Q_tGj^#GI#\"Y b!H;:/XUy*\\Lg0/C0RW.XW\">$=pJ,?" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/& %\"aG6$\"#7\"#5#\"7_\"*)Q%=P^8Mez\"708e.KW))zY%G$" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/&%\"aG6$\"#7\"#6#!9+%yQ%3'y&zKuz]\"9n*=F/z\">[lb:7" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#%!G" }}{PARA 11 "" 1 "" {XPPMATH 20 "6 #/&%\"cG6#\"#8\"\"\"" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/&%\"aG6$\"#8 \"\"\"#!BL$Q^$)>&3_5,oVC13\\&\"B+/g&)35(oRrqG!4r)RN" }}{PARA 11 "" 1 " " {XPPMATH 20 "6#/&%\"aG6$\"#8\"\"#\"\"!" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/&%\"aG6$\"#8\"\"$\"\"!" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/&%\" aG6$\"#8\"\"%#\"8\"[cR.'=7`n;\"H\"7?j'e'Q$y&*==L&" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/&%\"aG6$\"#8\"\"&#!>+wc5#eg@x6'H4`6*\"=RX*Rz*3Gq,`wE4w " }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/&%\"aG6$\"#8\"\"'#\"CkC(HE2.R$QM_ \\pbxS:\"B&*)f]WJe)>%yfL([7vk\"" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/&% \"aG6$\"#8\"\"(#!F++IAwN@?@@jQOXD)[F?\"F@.b\"yfB4FpI\">8Vy+Qc(" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#/&%\"aG6$\"#8\"\")#!6D+gcxdaHf2(\"7ERD ,r+2;nIC" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/&%\"aG6$\"#8\"\"*#!QJ26um 2K[Q5Nw*QSM5qt(oSpeJ\"QD,]+j#\\vJ&fyN;X6%z9n,,f%3b" }}{PARA 11 "" 1 " " {XPPMATH 20 "6#/&%\"aG6$\"#8\"#5#\"77DjD(\\3#4%eI%\"7v$=gMj')f8k1&" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/&%\"aG6$\"#8\"#6\"\"!" }}{PARA 11 " " 1 "" {XPPMATH 20 "6#/&%\"aG6$\"#8\"#7\"\"!" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%!G" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/&%\"bG6#\"\"\"# \".T)z[nJY\"0w*zB3#y.\"" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/&%\"bG6#\" \"#\"\"!" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/&%\"bG6#\"\"$\"\"!" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#/&%\"bG6#\"\"%\"\"!" }}{PARA 11 "" 1 " " {XPPMATH 20 "6#/&%\"bG6#\"\"&\"\"!" }}{PARA 11 "" 1 "" {XPPMATH 20 " 6#/&%\"bG6#\"\"'#\"2DT?u*>sK9\"2;K$z#oc*[S" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/&%\"bG6#\"\"(#\":+]il(4+,CBw?F\";\\`AAR\"op![nt\"4\"" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/&%\"bG6#\"\")#!0v$fe+L&)\\\"/Cu.\"4 _`*" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/&%\"bG6#\"\"*#\"Wzdw4zjfCSPa<' fI]qEzCxPYI?$fS\"V_RI)3:\"p(Hs`,7/')=?$3T=KO%>R!)y" }}{PARA 11 "" 1 " " {XPPMATH 20 "6#/&%\"bG6#\"#5#!/[sjF.H5\".^p%p2#3\"" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/&%\"bG6#\"#6#\"0+!))e5kK')\"/8LDiY\"e)" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/&%\"bG6#\"#7#!,R2IY(H\"-cMYUrC" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/&%\"bG6#\"#8\"\"!" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%!G" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/&%#b*G6#\"\"\"#\"-*)o4.b K\".s%\\J&HG(" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/&%#b*G6#\"\"#\"\"!" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/&%#b*G6#\"\"$\"\"!" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/&%#b*G6#\"\"%\"\"!" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/&%#b*G6#\"\"&\"\"!" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/&%#b*G6# \"\"'#\"0DhJIuV&\\\"1/Ref>>'R\"" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/&% #b*G6#\"\"(#\"6+](oz;A2T2w\"78$f6')pkY=b0$" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/&%#b*G6#\"\")#\"1Dc^yK9'=\"\"078%[Uw\\W" }}{PARA 11 " " 1 "" {XPPMATH 20 "6#/&%#b*G6#\"\"*#\"P4oU&)o1\"=#43@GGr9fX*QNz:z8\"O S+KgF%z*QWwr9-(RU(eVKIL:N" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/&%#b*G6# \"#5#!/G\\k*HL2$\".bZt%Q5a" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/&%#b*G6 #\"#6\"\"!" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/&%#b*G6#\"#7\"\"!" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#/&%#b*G6#\"#8#!,D8x%)e%\",_6#3Q#)" }}} {PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 151 "RK8_13eqs := [op(RowSumConditions(13,'expanded')),op (OrderConditions(8,13,'expanded'))]:\n`RK7_13eqs*` := subs(b=`b*`,Orde rConditions(7,13,'expanded')):" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }} {PARA 0 "" 0 "" {TEXT -1 6 "Check:" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 112 "subs(e18,RK8_13eqs):\nmap(u->lhs(u)-rhs(u),%);\nnops (%);\nsubs(e18,`RK7_13eqs*`):\nmap(u->lhs(u)-rhs(u),%);\nnops(%);" }} {PARA 12 "" 1 "" {XPPMATH 20 "6#7`x\"\"!F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$ F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$ F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$ F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$ F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$ F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$ F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#\"$7#" }}{PARA 12 "" 1 "" {XPPMATH 20 "6#7ap\"\"!F$F$F$ F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$ F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$ F$F$F$F$F$F$F$F$F$F$F$" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#\"#&)" }}} {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 26 "#-------- -----------------" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}} {PARA 0 "" 0 "" {TEXT -1 52 "#---------------------------------------- -----------" }}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 38 "characteristics of the embedded scheme" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 35 "coefficients of the combined scheme" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 4879 "ee := \{c[2]=1/16,\nc[3] =112/1065,\nc[4]=56/355,\nc[5]=39/100,\nc[6]=7/15,\nc[7]=39/250,\nc[8] =24/25,\nc[9]=14435868/16178861,\nc[10]=11/12,\nc[11]=19/20,\nc[12]=1, \nc[13]=1,\n\na[2,1]=1/16,\na[3,1]=18928/1134225,\na[3,2]=100352/11342 25,\na[4,1]=14/355,\na[4,2]=0,\na[4,3]=42/355,\na[5,1]=94495479/250880 000,\na[5,2]=0,\na[5,3]=-352806597/250880000,\na[5,4]=178077159/125440 000,\na[6,1]=12089/252720,\na[6,2]=0,\na[6,3]=0,\na[6,4]=2505377/10685 520,\na[6,5]=960400/5209191,\na[7,1]=21400899/350000000,\na[7,2]=0,\na [7,3]=0,\na[7,4]=3064329829899/27126050000000,\na[7,5]=-21643947/59260 9375,\na[7,6]=124391943/6756250000,\na[8,1]=-15365458811/13609565775, \na[8,2]=0,\na[8,3]=0,\na[8,4]=-7/5,\na[8,5]=-8339128164608/9390600384 75,\na[8,6]=341936800488/47951126225,\na[8,7]=1993321838240/3805234590 69,\na[9,1]=-1840911252282376584438157336464708426954728061551/\n \+ 2991923615171151921596253813483118262195533733898,\na[9,2]=0,\na[9, 3]=0,\na[9,4]=-14764960804048657303638372252908780219281424435/\n \+ 2981692102565021975611711269209606363661854518,\na[9,5]=-8753250485 02130441118613421785266742862694404520560000/\n 170212030428894 418395571677575961339495435011888324169,\na[9,6]=763205196415429092566 1849798370645637589377834346780/\n 17340872574188115830498003475 81865260479233950396659,\na[9,7]=7519834791971137517048532179652347729 899303513750000/\n 10456773035023175965978907078123498326373390 39997351,\na[9,8]=1366042683489166351293315549358278750/\n 14463 1418224267718165055326464180836641,\na[10,1]=-630777367052542801548248 45013881/78369357853786633855112190394368,\na[10,2]=0,\na[10,3]=0,\na[ 10,4]=-31948346510820970247215/6956009216960026632192,\na[10,5]=-33786 04805394255292453489375/517042670569824692230499952,\na[10,6]=10015878 44183325981198091450220795/184232684207722503701669953872896,\na[10,7] =187023075231349900768014890274453125/25224698849808178010752575653374 848,\na[10,8]=1908158550070998850625/117087067039189929394176,\na[10,9 ]=-52956818288156668227044990077324877908565/\n 29127799594774 33986349822224412353951940608,\na[11,1]=-10116106591826909534781157993 685116703/9562819945036894030442231411871744000,\na[11,2]=0,\na[11,3]= 0,\na[11,4]=-9623541317323077848129/3864449564977792573440,\na[11,5]=- 4823348333146829406881375/576413233634141239944816,\na[11,6]=656611924 6514996884067001154977284529/970305487021846325473990863582315520,\na[ 11,7]=2226455130519213549256016892506730559375/36488044315967525557743 5648380047355776,\na[11,8]=39747262782380466933662225/1756032802431424 164410720256,\na[11,9]=48175771419260955335244683805171548038966866545 122229/\n 19897864205138151465288801659520641189038528436121600 00,\na[11,10]=-2378292068163246/47768728487211875,\na[12,1]=-321802217 4758599831659045535578571/1453396753634469525663775847094384,\na[12,2] =0,\na[12,3]=0,\na[12,4]=26290092604284231996745/576087612606286043054 4,\na[12,5]=-697069297560926452045586710000/41107967755245430594036502 319,\na[12,6]=1827357820434213461438077550902273440/139381013914245317 709567680839641697,\na[12,7]=643504802814241550941949227194107500000/2 42124609118836550860494007545333945331,\na[12,8]=162259938151380266113 750/59091082835244183497007,\na[12,9]=-2302825163287352381854541485685 7015616678575554130463402/\n 200131691831914445034439052404056 03349978424504151629055,\na[12,10]=7958341351371843889152/328446798844 3203581305,\na[12,11]=-507974327957860843878400/1215556548191790427189 67,\na[13,1]=-549080624436801105208519835138333/3539871090287071396871 00885600400,\na[13,2]=0,\na[13,3]=0,\na[13,4]=29116675312186033956481/ 5331818957833865866320,\na[13,5]=-91153092961177216058210567600/760926 7653017028089793994539,\na[13,6]=1540775569495234383390307262972464/16 4751248733597841985831445059895,\na[13,7]=-202748825453638632121202135 7622300000/7563800784313191306927092359781550321,\na[13,8]=-7075929545 77756600025/2430671607007101253926,\na[13,9]=-315869406877370103440389 763510384832076674110731/\n 5508459010167147941145163578595317 54926300500125,\na[13,10]=4305840920849725632512/506641359866334601837 5,\na[13,11]=0,\na[13,12]=0,\n\nb[1]=4631674879841/103782082379976,\nb [2]=0,\nb[3]=0,\nb[4]=0,\nb[5]=0,\nb[6]=14327219974204125/404895668279 33216,\nb[7]=2720762324010009765625000/10917367480696813922225349,\nb[ 8]=-498533005859375/95352091037424,\nb[9]=4059320304637772479267050305 96175437402459637909765779/\n 788039194363218410832018860412015372 29769115088303952,\nb[10]=-10290327637248/1082076946951,\nb[11]=863264 105888000/85814662253313,\nb[12]=-29746300739/247142463456,\nb[13]=0, \n\n`b*`[1]=325503096889/7282953149472,\n`b*`[2]=0,\n`b*`[3]=0,\n`b*`[ 4]=0,\n`b*`[5]=0,\n`b*`[6]=495437430316125/1396191959583904,\n`b*`[7]= 760741072216796875000/3055518466469861159313,\n`b*`[8]=118614327851562 5/444976424841312,\n`b*`[9]=137915793538945591471282821080921810668854 26809/3515333032435874239702147176443897942760320040,\n`b*`[10]=-30733 299644928/5410384734755,\n`b*`[11]=0,\n`b*`[12]=0,\n`b*`[13]=-45884771 325/82380821152\}:" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}}{PARA 0 "" 0 "" {TEXT -1 0 "" }} {PARA 0 "" 0 "" {TEXT -1 5 "Let " }{XPPEDIT 18 0 "T[8, 13];" "6#&%\"T G6$\"\")\"#8" }{TEXT -1 128 " denote the vector whose components are \+ the principal error terms of the 13 stage, order 8 scheme (the error t erms of order 9)." }}{PARA 0 "" 0 "" {TEXT -1 5 "Let " }{XPPEDIT 18 0 "`T*`[7,13];" "6#&%#T*G6$\"\"(\"#8" }{TEXT -1 146 " denote the vect or whose components are the principal error terms of the embedded 13 s tage, order 7 scheme (the error terms of order 8) and let " } {XPPEDIT 18 0 "`T*`[8,13];" "6#&%#T*G6$\"\")\"#8" }{TEXT -1 99 " deno te the vector whose components are the error terms of order 9 of the e mbedded order 7 scheme." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 39 "Denote the 2-norm of these vectors by " }{XPPEDIT 18 0 "abs(abs(T[8,13]));" "6#-%$absG6#-F$6#&%\"TG6$\"\")\"#8" }{TEXT -1 3 ", " }{XPPEDIT 18 0 "abs(abs(`T*`[7, 13]));" "6#-%$absG6#-F$6#&% #T*G6$\"\"(\"#8" }{TEXT -1 7 " and " }{XPPEDIT 18 0 "abs(abs(`T*`[8, 13]));" "6#-%$absG6#-F$6#&%#T*G6$\"\")\"#8" }{TEXT -1 15 " respectiv ely." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 8 "De fine: " }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "A[9] = abs( abs(T[8,13]));" "6#/&%\"AG6#\"\"*-%$absG6#-F)6#&%\"TG6$\"\")\"#8" } {TEXT -1 6 ", " }{XPPEDIT 18 0 "B[9] = abs(abs(`T*`[8,13]))/abs(ab s(`T*`[7,13]));" "6#/&%\"BG6#\"\"**&-%$absG6#-F*6#&%#T*G6$\"\")\"#8\" \"\"-F*6#-F*6#&F/6$\"\"(F2!\"\"" }{TEXT -1 6 ", " }{XPPEDIT 18 0 " C[9] = abs(abs(`T*`[8,13]-T[8,13]))/abs(abs(`T*`[7,13]));" "6#/&%\"CG6 #\"\"**&-%$absG6#-F*6#,&&%#T*G6$\"\")\"#8\"\"\"&%\"TG6$F2F3!\"\"F4-F*6 #-F*6#&F06$\"\"(F3F8" }{TEXT -1 1 "." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 77 "Dormand and Prince have suggested that \+ as well as attempting to ensure that " }{XPPEDIT 18 0 "A[9];" "6#&%\" AG6#\"\"*" }{TEXT -1 73 " is a minimum, if the embedded scheme is to \+ be used for error control, " }{XPPEDIT 18 0 "B[9];" "6#&%\"BG6#\"\"* " }{TEXT -1 7 " and " }{XPPEDIT 18 0 "C[9];" "6#&%\"CG6#\"\"*" } {TEXT -1 27 " should be chosen so that " }}{PARA 0 "" 0 "" {TEXT -1 68 "they are \"similar in magnitude\" and also not differ too much fro m 1." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 168 "errterms8_13 := PrincipalErrorTerms(8,13,'expanded') :\n`errterms8_13*` :=subs(b=`b*`,errterms8_13):\n`errterms7_13*` := su bs(b=`b*`,PrincipalErrorTerms(7,13,'expanded')):" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 333 "snmB := sqr t(add(evalf(subs(ee,`errterms8_13*`[i]))^2,i=1..nops(`errterms8_13*`)) ):\nsdnB := sqrt(add(evalf(subs(ee,`errterms7_13*`[i]))^2,i=1..nops(`e rrterms7_13*`))):\nsnmC := sqrt(add((evalf(subs(ee,`errterms8_13*`[i]) -subs(ee,errterms8_13[i])))^2,i=1..nops(errterms8_13))):\n'B[9]'= eval f[8](snmB/sdnB);\n'C[9]'= evalf[8](snmC/sdnB);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/&%\"BG6#\"\"*$\")uP:?!\"(" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/&%\"CG6#\"\"*$\"))y`-#!\"(" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}}{PARA 0 "" 0 "" {TEXT -1 52 "#--------------------------------------------------- " }}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 27 "absolute stability regions " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 35 "coefficients of the co mbined scheme" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 4879 "ee := \{c[2]=1/16,\nc[3]=112/1065,\nc[4]=56/35 5,\nc[5]=39/100,\nc[6]=7/15,\nc[7]=39/250,\nc[8]=24/25,\nc[9]=14435868 /16178861,\nc[10]=11/12,\nc[11]=19/20,\nc[12]=1,\nc[13]=1,\n\na[2,1]=1 /16,\na[3,1]=18928/1134225,\na[3,2]=100352/1134225,\na[4,1]=14/355,\na [4,2]=0,\na[4,3]=42/355,\na[5,1]=94495479/250880000,\na[5,2]=0,\na[5,3 ]=-352806597/250880000,\na[5,4]=178077159/125440000,\na[6,1]=12089/252 720,\na[6,2]=0,\na[6,3]=0,\na[6,4]=2505377/10685520,\na[6,5]=960400/52 09191,\na[7,1]=21400899/350000000,\na[7,2]=0,\na[7,3]=0,\na[7,4]=30643 29829899/27126050000000,\na[7,5]=-21643947/592609375,\na[7,6]=12439194 3/6756250000,\na[8,1]=-15365458811/13609565775,\na[8,2]=0,\na[8,3]=0, \na[8,4]=-7/5,\na[8,5]=-8339128164608/939060038475,\na[8,6]=3419368004 88/47951126225,\na[8,7]=1993321838240/380523459069,\na[9,1]=-184091125 2282376584438157336464708426954728061551/\n 2991923615171151921 596253813483118262195533733898,\na[9,2]=0,\na[9,3]=0,\na[9,4]=-1476496 0804048657303638372252908780219281424435/\n 2981692102565021975 611711269209606363661854518,\na[9,5]=-87532504850213044111861342178526 6742862694404520560000/\n 1702120304288944183955716775759613394 95435011888324169,\na[9,6]=7632051964154290925661849798370645637589377 834346780/\n 173408725741881158304980034758186526047923395039665 9,\na[9,7]=7519834791971137517048532179652347729899303513750000/\n \+ 1045677303502317596597890707812349832637339039997351,\na[9,8]=1366 042683489166351293315549358278750/\n 144631418224267718165055326 464180836641,\na[10,1]=-63077736705254280154824845013881/7836935785378 6633855112190394368,\na[10,2]=0,\na[10,3]=0,\na[10,4]=-319483465108209 70247215/6956009216960026632192,\na[10,5]=-337860480539425529245348937 5/517042670569824692230499952,\na[10,6]=100158784418332598119809145022 0795/184232684207722503701669953872896,\na[10,7]=187023075231349900768 014890274453125/25224698849808178010752575653374848,\na[10,8]=19081585 50070998850625/117087067039189929394176,\na[10,9]=-5295681828815666822 7044990077324877908565/\n 291277995947743398634982222441235395 1940608,\na[11,1]=-10116106591826909534781157993685116703/956281994503 6894030442231411871744000,\na[11,2]=0,\na[11,3]=0,\na[11,4]=-962354131 7323077848129/3864449564977792573440,\na[11,5]=-4823348333146829406881 375/576413233634141239944816,\na[11,6]=6566119246514996884067001154977 284529/970305487021846325473990863582315520,\na[11,7]=2226455130519213 549256016892506730559375/364880443159675255577435648380047355776,\na[1 1,8]=39747262782380466933662225/1756032802431424164410720256,\na[11,9] =48175771419260955335244683805171548038966866545122229/\n 19897 86420513815146528880165952064118903852843612160000,\na[11,10]=-2378292 068163246/47768728487211875,\na[12,1]=-3218022174758599831659045535578 571/1453396753634469525663775847094384,\na[12,2]=0,\na[12,3]=0,\na[12, 4]=26290092604284231996745/5760876126062860430544,\na[12,5]=-697069297 560926452045586710000/41107967755245430594036502319,\na[12,6]=18273578 20434213461438077550902273440/139381013914245317709567680839641697,\na [12,7]=643504802814241550941949227194107500000/24212460911883655086049 4007545333945331,\na[12,8]=162259938151380266113750/590910828352441834 97007,\na[12,9]=-23028251632873523818545414856857015616678575554130463 402/\n 2001316918319144450344390524040560334997842450415162905 5,\na[12,10]=7958341351371843889152/3284467988443203581305,\na[12,11]= -507974327957860843878400/121555654819179042718967,\na[13,1]=-54908062 4436801105208519835138333/353987109028707139687100885600400,\na[13,2]= 0,\na[13,3]=0,\na[13,4]=29116675312186033956481/5331818957833865866320 ,\na[13,5]=-91153092961177216058210567600/7609267653017028089793994539 ,\na[13,6]=1540775569495234383390307262972464/164751248733597841985831 445059895,\na[13,7]=-2027488254536386321212021357622300000/75638007843 13191306927092359781550321,\na[13,8]=-707592954577756600025/2430671607 007101253926,\na[13,9]=-3158694068773701034403897635103848320766741107 31/\n 550845901016714794114516357859531754926300500125,\na[13, 10]=4305840920849725632512/5066413598663346018375,\na[13,11]=0,\na[13, 12]=0,\n\nb[1]=4631674879841/103782082379976,\nb[2]=0,\nb[3]=0,\nb[4]= 0,\nb[5]=0,\nb[6]=14327219974204125/40489566827933216,\nb[7]=272076232 4010009765625000/10917367480696813922225349,\nb[8]=-498533005859375/95 352091037424,\nb[9]=40593203046377724792670503059617543740245963790976 5779/\n 78803919436321841083201886041201537229769115088303952,\nb[ 10]=-10290327637248/1082076946951,\nb[11]=863264105888000/858146622533 13,\nb[12]=-29746300739/247142463456,\nb[13]=0,\n\n`b*`[1]=32550309688 9/7282953149472,\n`b*`[2]=0,\n`b*`[3]=0,\n`b*`[4]=0,\n`b*`[5]=0,\n`b*` [6]=495437430316125/1396191959583904,\n`b*`[7]=760741072216796875000/3 055518466469861159313,\n`b*`[8]=1186143278515625/444976424841312,\n`b* `[9]=13791579353894559147128282108092181066885426809/35153330324358742 39702147176443897942760320040,\n`b*`[10]=-30733299644928/5410384734755 ,\n`b*`[11]=0,\n`b*`[12]=0,\n`b*`[13]=-45884771325/82380821152\}:" }}} {PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 78 "The stability function R for the 13 stage, order 8 scheme is given as follows." }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 76 "subs(ee,Stabili tyFunction(8,13,'expanded')):\nR := unapply(%,z):\n'R(z)'=R(z);" }} {PARA 12 "" 1 "" {XPPMATH 20 "6#/-%\"RG6#%\"zG,<\"\"\"F)F'F)*&#F)\"\"# F)*$)F'F,F)F)F)*&#F)\"\"'F)*$)F'\"\"$F)F)F)*&#F)\"#CF)*$)F'\"\"%F)F)F) *&#F)\"$?\"F)*$)F'\"\"&F)F)F)*&#F)\"$?(F)*$)F'F1F)F)F)*&#F)\"%S]F)*$)F '\"\"(F)F)F)*&#F)\"&?.%F)*$)F'\"\")F)F)F)*&#\"<8N**RK\")4d2<*\\#H*\"B+ ++!eg?(=&*H&Rl+7rMF)*$)F'\"\"*F)F)F)*&#\"<6V7hsckzL$)*z)f&\"C++](y,=@R F.Ju*yXQ@F)*$)F'\"#5F)F)F)*&#\"=ZJw%3m,n\">#eA]6\"\"D++++:2s%o&48C(*eJ Qb)F)*$)F'\"#6F)F)F)*&#\"6L)y2sF26\">z&\"@++]iqD-@aB8AFe:)F)*$)F'\"#7F )F)F)" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 124 "We can find the point where the boundary of the stability region \+ intersects the negative real axis by solving the equation: " }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "R(z) = 1;" "6#/-%\"RG6#%\"z G\"\"\"" }{TEXT -1 2 ". " }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 26 " z0 := newton(R(z)=1,z=-5);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%#z0G$! +4-]&y&!\"*" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 302 "z0 := newton(R(z)=1,z=-5):\np1 := plot([R(z),1],z =-6.69..0.49,color=[red,blue]):\np2 := plot([[[z0,1]]$3],style=point,s ymbol=[circle,cross,diamond],color=black):\np3 := plot([[z0,0],[z0,1]] ,linestyle=3,color=COLOR(RGB,0,.5,0)):\nplots[display]([p1,p2,p3],view =[-6.69..0.49,-0.07..1.47],font=[HELVETICA,9]);" }}{PARA 13 "" 1 "" {GLPLOT2D 360 253 253 {PLOTDATA 2 "6+-%'CURVESG6$7X7$$!3Q++++++!p'!#<$ \"3gD5C\"*)z\"pOF*7$$!3ym;a;#[I*>F*7$$!3)QLL#G\\5nhF*$\"3#fOWd()**Gy\"F*7$$!3UL$3p-F+4'F*$\"31M )oB'p3#f\"F*7$$!3X]PRjaJ8gF*$\"3dy2&*RO\"*>9F*7$$!3fm\"z)**QgOfF*$\"3Q O%RqzuSE\"F*7$$!3)Q$ef@5O%z&F*$\"39![-JcOS,\"F*7$$!3E+D;gn2ZcF*$\"35y \"3fDfu,)!#=7$$!3gLeHm^v%\\&F*$\"3![9]WE%>ViFco7$$!3Q+Drp?#HM&F*$\"3-? z`sJeI[Fco7$$!3ymm@nIu'=&F*$\"3g$p=!*HnOo$Fco7$$!3oL$3o^!=\\]F*$\"3)pB >nF%R&)GFco7$$!37++lmwJ%*[F*$\"3a.*G_2(z!=#Fco7$$!3F++vM*=)QZF*$\"3V'Q z\\gg6k\"Fco7$$!3'****\\L4n*)e%F*$\"3![GVO1B&[7Fco7$$!3yLeHls)GX%F*$\" 3y??%yk\"o)y*!#>7$$!3Umm\"y=v5H%F*$\"34W*H!*=g&QuF\\r7$$!3#ommK8**R:%F *$\"3&p6h+xsx-'F\\r7$$!3O+D'[4ZX*RF*$\"3X/J(*4dT7\\F\\r7$$!3=nmwA]S`QF *$\"3(o0(y`&*>+VF\\r7$$!33+D')*z\\&)p$F*$\"3E@#*Rj*e(\\RF\\r7$$!3'**\\ (e254^NF*$\"3C0Mk'4$)*oQF\\r7$$!3wm;HFKB(R$F*$\"3uCAMjmn,SF\\r7$$!3im \"zdvVfD$F*$\"3'f)3I$z?8H%F\\r7$$!3kL$3K(Ra.JF*$\"3O&)R@&e]cw%F\\r7$$! 3oLeHIMCXHF*$\"3U\"*foN(=qU&F\\r7$$!37+vLUBW2GF*$\"3(zoR&Qa_UhF\\r7$$! 3EL$e@R8'eEF*$\"3/Oe*f4B\"oqF\\r7$$!3G++qB&e[]#F*$\"3.6c:/\"4\\?)F\\r7 $$!3?+]_\"QRWN#F*$\"3)\\'yIC.&[^*F\\r7$$!33+D1F6!*3AF*$\"3wqB#*H*>$*4 \"Fco7$$!3?+]24dIZ?F*$\"3EF*$\"3stgAPX\"G \\\"Fco7$$!3-++vIB2ZiJK;F*-%'COLOURG6&%$RGBG$\"*++++\"!\")$\"\"!Fj\\ lFi\\l-F$6$7S7$F($\"\"\"Fj\\l7$F3F_]l7$F=F_]l7$FGF_]l7$FQF_]l7$FenF_]l 7$FjnF_]l7$F_oF_]l7$FeoF_]l7$FjoF_]l7$F_pF_]l7$FdpF_]l7$FipF_]l7$F^qF_ ]l7$FcqF_]l7$FhqF_]l7$F^rF_]l7$FcrF_]l7$FhrF_]l7$F]sF_]l7$FbsF_]l7$Fgs F_]l7$F\\tF_]l7$FatF_]l7$FftF_]l7$F[uF_]l7$F`uF_]l7$FeuF_]l7$FjuF_]l7$ F_vF_]l7$FdvF_]l7$FivF_]l7$F^wF_]l7$FcwF_]l7$FhwF_]l7$F]xF_]l7$FbxF_]l 7$FgxF_]l7$F\\yF_]l7$FayF_]l7$FfyF_]l7$F[zF_]l7$F`zF_]l7$FezF_]l7$FjzF _]l7$F_[lF_]l7$Fd[lF_]l7$Fi[lF_]l7$F^\\lF_]l-Fc\\l6&Fe\\lFi\\lFi\\lFf \\l-F$6&7#7$$!3G+++4-]&y&F*F_]l-%'SYMBOLG6#%'CIRCLEG-Fc\\l6&Fe\\lFj\\l Fj\\lFj\\l-%&STYLEG6#%&POINTG-F$6&Fe`l-Fj`l6#%&CROSSGF]alF_al-F$6&Fe`l -Fj`l6#%(DIAMONDGF]alF_al-F$6%7$7$Fg`lFi\\lFf`l-%&COLORG6&Fe\\lFi\\l$ \"\"&!\"\"Fi\\l-%*LINESTYLEG6#\"\"$-%+AXESLABELSG6%Q\"z6\"Q!F_cl-%%FON TG6#%(DEFAULTG-Fbcl6$%*HELVETICAG\"\"*-%%VIEWG6$;$!$p'!\"#$\"#\\F_dl;$ !\"(F_dl$\"$Z\"F_dl" 1 2 0 1 10 0 2 9 1 4 2 1.000000 45.000000 45.000000 0 0 "Curve 1" "Curve 2" "Curve 3" "Curve 4" "Curve 5" "Curve 6" }}}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 50 "T he following picture shows the stability region. " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 857 "R := z -> a dd(z^j/j!,j=0..8)+929249917075709813239993513/347112006539529951872060 580000000*z^9+\n 559879983337964567261124311/21384578974310327392118 01787500000*z^10+\n 1115022582191670166084763147/8553831589724130956 8472071500000000*z^11+\n 579191110727720778833/815582722132354210225 7062500000*z^12:\npts := []:\nz0 := 0:\nfor ct from 0 to 320 do\n zz := newton(R(z)=exp(ct*Pi/20*I),z=z0):\n z0 := zz:\n pts := [op(pt s),[Re(zz),Im(zz)]]:\nend do:\np1 := plot(pts,color=COLOR(RGB,.1,.45,0 )):\np2 := plots[polygonplot]([seq([pts[i-1],pts[i],[-4.6,0]],i=2..nop s(pts))],\n style=patchnogrid,color=COLOR(RGB,.4,.75,.35)):\n p3 := plot([[[-6.69,0],[1.19,0]],[[0,-6.39],[0,6.39]]],color=black,lin estyle=3):\nplots[display]([p||(1..3)],view=[-6.69..1.19,-6.39..6.39], font=[HELVETICA,9],\n labels=[`Re(z)`,`Im(z)`],axes=boxed ,scaling=constrained);" }}{PARA 13 "" 1 "" {GLPLOT2D 381 565 565 {PLOTDATA 2 "6+-%'CURVESG6$7]_l7$$\"\"!F)F(7$F($\"3++++Fjzq:!#=7$F($\" 3)******Rl#fTJF-7$F($\"3D+++\")*)Q7ZF-7$$\"3W+++%R]ei&!#F$\"3g*****pJ& =$G'F-7$$\"3J+++3CqT\\!#E$\"3)*******)p\")R&yF-7$$\"30+++v^-VG!#D$\"3$ )******[#yZU*F-7$$\"36+++dg097!#C$\"33+++Bvb*4\"!#<7$$\"3D+++%\\C@9%FI $\"3%******\\IPmD\"FL7$$\"3-+++1sP#=\"!#B$\"3)******p>#FL7$$\"3q*****Rd``t$F_o$\"3/++ +B)yhN#FL7$$\"3C+++NK;*H$F_o$\"3'******pz`K^#FL7$$\"33+++Dk\"*[bZiK #!#@$\"35+++&=%y%)HFL7$$!37+++&)RC'y%Fcq$\"32+++.1MUJFL7$$!3%******fr6 %H!)Fcq$\"3!)*****zK./I$FL7$$!35+++'[o]6\"!#?$\"37+++6%[$fMFL7$$!3/+++ xa-h6Fcr$\"3:+++YKv>OFL7$$!3E+++H+iuNFcq$\"3=+++$=rCy$FL7$$\"3++++_LrW DFcr$\"3++++^-q[RFL7$$\"3)******py$)z,\"!#>$\"3P+++X/\"*>TFL7$$\"37+++ E0_.GFhs$\"3G+++rt$pH%FL7$$\"3c******o*=*zlFhs$\"3G+++XpkvWFL7$$\"31++ +mZL#H\"F-$\"3<+++WZ$*QYFL7$$\"3!******H'4uH?F-$\"3Q+++T)=,x%FL7$$\"3/ +++6y8'o#F-$\"3i******Q6)R([FL7$$\"3#******foT8@$F-$\"3K+++vD-h\\FL7$$ \"3F+++&))3Th$F-$\"36+++cwhP]FL7$$\"33+++(3,F\"RF-$\"3T+++ro-2^FL7$$\" 3D+++))Q_BTF-$\"3E+++(za4<&FL7$$\"3y*****4N(zfUF-$\"3))*****RTU.B&FL7$ $\"3-+++\"=o?L%F-$\"3A+++9#QdG&FL7$$\"33+++mQ')[VF-$\"3u******[(yuL&FL 7$$\"32+++;L?GOcFL7 $$\"3=+++%\\jQ(GF-$\"3S+++L:?gcFL7$$\"3%)*****pul;f#F-$\"3I+++,QB\"o&F L7$$\"35+++5w,%H#F-$\"3?+++,3Q*p&FL7$$\"35+++l0R#)>F-$\"3&)*****HPLYr& FL7$$\"33+++!)o2e;F-$\"3U+++I(pps&FL7$$\"3/+++m^?A8F-$\"3=+++'o`jt&FL7 $$\"3Y*****fu/vv*Fhs$\"3&******fTKFu&FL7$$\"3A+++5JQ&>'Fhs$\"3g*****zd Lgu&FL7$$\"3$******z'>TUDFhs$\"3G+++u:;YdFL7$$!31+++AeM'>\"Fhs$\"3>+++ 8E*Hu&FL7$$!3z*****R\"=w<]Fhs$\"3i*****\\3ojt&FL7$$!3M*****\\)\\7@*)Fh s$\"3y*****4a&3EdFL7$$!33+++fh*3H\"F-$\"36+++(>')=r&FL7$$!3/+++m?%))p \"F-$\"3O+++btV$p&FL7$$!3.+++wdO<@F-$\"3U+++9tIqcFL7$$!31+++*R2!\\DF-$ \"3p*****HFH>k&FL7$$!3#******RK?#)*HF-$\"3&******\\!*\\vg&FL7$$!37+++@ =#HZ$F-$\"3)*******z3;mbFL7$$!3'******4\\Hx)RF-$\"3d******yIV;bFL7$$!3 ++++jp-rXF-$\"3A+++'**)ycaFL7$$!3%)*****plC&z_F-$\"3C+++OiI'Q&FL7$$!3* ******z,[;@'F-$\"3l*****fa[!4`FL7$$!3z*****>(\\>EuF-$\"38+++s>1U_FL7$$ !3-+++_)4.x)F-$\"3i*****pBy>?&FL7$$!3-+++d'Hg+\"FL$\"3=+++**3(R=&FL7$$ !3%******fC&fC6FL$\"3e*****R**)\\x^FL7$$!3*******H!\\cL7FL$\"3K+++hD:w ^FL7$$!3.+++ETpM8FL$\"3S+++e5kw^FL7$$!3)******R'f_H9FL$\"3y******HkBx^ FL7$$!3!******f)RE>:FL$\"3))*****>'=)p<&FL7$$!3)******ReH[g\"FL$\"3O++ +B8Kv^FL7$$!33+++)GLpo\"FL$\"3?++++\">><&FL7$$!3#******fEJhw\"FL$\"3e* ****zaml;&FL7$$!3'******zRmG%=FL$\"3C+++\"=J\"f^FL7$$!3%******Hr(\\<>F L$\"3K+++Tv_\\^FL7$$!33+++`;K!*>FL$\"3&******p\"**pP^FL7$$!3=+++-xeh?F L$\"3!)*****>&=hB^FL7$$!3++++\"45:8#FL$\"3K+++9(Rs5&FL7$$!3#*******[nF +AFL$\"3w******R)o&)3&FL7$$!3=+++fi0oAFL$\"3;+++(R\"fn]FL7$$!3$)*****H N.]L#FL$\"3H+++qaIW]FL7$$!3$******\\Dj7S#FL$\"3'******H4:(=]FL7$$!3A++ +g](pY#FL$\"3?+++c5$3*\\FL7$$!3=+++EUFKDFL$\"3w*****zSs1'\\FL7$$!33+++ aTH(f#FL$\"3I+++]&o#G\\FL7$$!3)*******Rn;iEFL$\"3g*****R#=m$*[FL7$$!3- +++]<-FFFL$\"3!)*****pV5p&[FL7$$!31+++.[)>z#FL$\"3(******zR\"4=[FL7$$! 3-+++QQ7$FL$\"3!)*****\\10sf%FL7$$!3%)******=hP*=$FL$\"3c***** Hc)y[XFL7$$!3-+++.&HsD$FL$\"3s******Q$y$*\\%FL7$$!3++++0[VDLFL$\"3C+++ `69\\WFL7$$!3#)******)3pQR$FL$\"3U+++#o=#)R%FL7$$!3#******p3&QiMFL$\"3 7+++A*>nM%FL7$$!3&)******Q<#3`$FL$\"35+++,sr%H%FL7$$!3/+++W[,*f$FL$\"3 e*****z*fCUUFL7$$!3++++;k!om$FL$\"3!)*****R-3$*=%FL7$$!31+++6=0MPFL$\" 39+++@l(e8%FL7$$!3,+++\\Vi+QFL$\"35+++TE!>3%FL7$$!3A+++7wTmQFL$\"3y*** **R#>KFSFL7$$!3'******f(eMJRFL$\"3%******R?g?(RFL7$$!3!)*****zBV`*RFL$ \"3/+++#fQg\"RFL7$$!35+++]>OeSFL$\"3?+++RudL4Z %FL$\"3#******>.d=V$FL7$$!3e*****HS1f_%FL$\"3%)*****z#3,mLFL7$$!3;+++s !e*zXFL$\"3%)*****fht))H$FL7$$!3(******Rc'*oot%FL$\"3;+++Dbo*3$FL7$$ !3y******Rr](y%FL$\"35+++'e)R+++62(Q%HFL7$$!3 M+++aLQ')[FL$\"3y*****R4N\"pGFL7$$!3w*****p;HY$\\FL$\"3!)*****z7IKz#FL 7$$!3K+++4/2#)\\FL$\"3<+++yx>;FFL7$$!3&******zl(oG]FL$\"3*******>y\"3Q EFL7$$!3?+++&f^W2&FL$\"3%)*****HyD*eDFL7$$!3v*****zQB$>^FL$\"3++++O:xy CFL7$$!3K+++/bDj^FL$\"3)******RIdwR#FL7$$!3/+++&*G>1_FL$\"3********plh :BFL7$$!3?+++LW2[_FL$\"3:+++LrnKAFL7$$!3!)*****p]M))G&FL$\"37+++^1')[@ FL7$$!3e******RYSG`FL$\"3-+++5E=k?FL7$$!37+++V]rm`FL$\"3%******Rq_'y>F L7$$!3&)******=gp.aFL$\"3'******fbvA*=FL7$$!3C+++v\"z#RaFL$\"3(******R r^]!=FL7$$!39+++:%)RtaFL$\"3&******z$)ypr\"FL7$$!3i*****zr!*f]&FL$\"3 \"******f-`!G;FL7$$!3:+++=n*p`&FL$\"34+++%>q#Q:FL7$$!3e*****p6hjc&FL$ \"33+++%[FwW\"FL7$$!31+++_H.%f&FL$\"3)******zkChN\"FL7$$!3E+++[d'*>cFL $\"31+++%QlPE\"FL7$$!38+++'e4^km&F L$\"3%*******f&>l2\"FL7$$!3?+++qL$po&FL$\"33+++p%4n\")*F-7$$!3\")***** \\(e`0dFL$\"3s*****>8P/'))F-7$$!3L+++PUBAdFL$\"3Y+++Lyx'*yF-7$$!3Q+++A !3qt&FL$\"3b+++(4Di#pF-7$$!3=+++b/%)\\dFL$\"3E+++U'o$\\fF-7$$!3s***** \\3=2w&FL$\"3))******RK*o'\\F-7$$!3#)*****RUI'pdFL$\"31+++ZUdzRF-7$$!3 Q+++q&plx&FL$\"3))*****>kp#))HF-7$$!39+++S)H:y&FL$\"3++++?'3R*>F-7$$!3 q*****4W2Xy&FL$\"3W+++i+xu**Fhs7$$!3G+++4-]&y&FLF(7$F`]n$!3W+++i+xu**F hs7$F[]n$!3++++?'3R*>F-7$Ff\\n$!3))*****>kp#))HF-7$Fa\\n$!31+++ZUdzRF- 7$F\\\\n$!3))******RK*o'\\F-7$Fg[n$!3E+++U'o$\\fF-7$Fb[n$!3b+++(4Di#pF -7$F][n$!3Y+++Lyx'*yF-7$Fhjm$!3s*****>8P/'))F-7$Fcjm$!33+++p%4n\")*F-7 $F^jm$!3%*******f&>l2\"FL7$Fiim$!33+++(fe0<\"FL7$Fdim$!31+++%QlPE\"FL7 $F_im$!3)******zkChN\"FL7$Fjhm$!33+++%[FwW\"FL7$Fehm$!34+++%>q#Q:FL7$F `hm$!3\"******f-`!G;FL7$F[hm$!3&******z$)ypr\"FL7$Ffgm$!3(******Rr^]!= FL7$Fagm$!3'******fbvA*=FL7$F\\gm$!3%******Rq_'y>FL7$Fgfm$!3-+++5E=k?F L7$Fbfm$!37+++^1')[@FL7$F]fm$!3:+++LrnKAFL7$Fhem$!3********plh:BFL7$Fc em$!3)******RIdwR#FL7$F^em$!3++++O:xyCFL7$Fidm$!3%)*****HyD*eDFL7$Fddm $!3*******>y\"3QEFL7$F_dm$!3<+++yx>;FFL7$Fjcm$!3!)*****z7IKz#FL7$Fecm$ !3y*****R4N\"pGFL7$F`cm$!3>+++62(Q%HFL7$F[cm$!35+++'e)R.d=V$FL7$Fh`m$!3/+++ 8\"Hk\\$FL7$Fc`m$!3?+++cJvfNFL7$F^`m$!3?+++bU'=i$FL7$Fi_m$!3%******zB1 Go$FL7$Fd_m$!37+++s7jUPFL7$F__m$!3$)*****4L*R,QFL7$Fj^m$!3?+++RuKFSFL7$ Ff]m$!35+++TE!>3%FL7$Fa]m$!39+++@l(e8%FL7$F\\]m$!3!)*****R-3$*=%FL7$Fg \\m$!3e*****z*fCUUFL7$Fb\\m$!35+++,sr%H%FL7$F]\\m$!37+++A*>nM%FL7$Fh[m $!3U+++#o=#)R%FL7$Fc[m$!3C+++`69\\WFL7$F^[m$!3s******Q$y$*\\%FL7$Fijl$ !3c*****Hc)y[XFL7$Fdjl$!3!)*****\\10sf%FL7$F_jl$!3Q+++ihWWYFL7$Fjil$!3 #******pOE.p%FL7$Feil$!3\"******4tmYt%FL7$F`il$!3%)*****\\2.tx%FL7$F[i l$!3(******zR\"4=[FL7$Ffhl$!3!)*****pV5p&[FL7$Fahl$!3g*****R#=m$*[FL7$ F\\hl$!3I+++]&o#G\\FL7$Fggl$!3w*****zSs1'\\FL7$Fbgl$!3?+++c5$3*\\FL7$F ]gl$!3'******H4:(=]FL7$Fhfl$!3H+++qaIW]FL7$Fcfl$!3;+++(R\"fn]FL7$F^fl$ !3w******R)o&)3&FL7$Fiel$!3K+++9(Rs5&FL7$Fdel$!3!)*****>&=hB^FL7$F_el$ !3&******p\"**pP^FL7$Fjdl$!3K+++Tv_\\^FL7$Fedl$!3C+++\"=J\"f^FL7$F`dl$ !3e*****zaml;&FL7$F[dl$!3?++++\">><&FL7$Ffcl$!3O+++B8Kv^FL7$Facl$!3))* ****>'=)p<&FL7$F\\cl$!3y******HkBx^FL7$Fgbl$!3S+++e5kw^FL7$Fbbl$!3K+++ hD:w^FL7$F]bl$!3e*****R**)\\x^FL7$Fhal$!3=+++**3(R=&FL7$Fcal$!3i*****p By>?&FL7$F^al$!38+++s>1U_FL7$Fi`l$!3l*****fa[!4`FL7$Fd`l$!3C+++OiI'Q&F L7$F_`l$!3A+++'**)ycaFL7$Fj_l$!3d******yIV;bFL7$Fe_l$!3)*******z3;mbFL 7$F`_l$!3&******\\!*\\vg&FL7$F[_l$!3p*****HFH>k&FL7$Ff^l$!3U+++9tIqcFL 7$Fa^l$!3O+++btV$p&FL7$F\\^l$!36+++(>')=r&FL7$Fg]l$!3y*****4a&3EdFL7$F b]l$!3i*****\\3ojt&FL7$F]]l$!3>+++8E*Hu&FL7$Fh\\l$!3G+++u:;YdFL7$Fc\\l $!3g*****zdLgu&FL7$F^\\l$!3&******fTKFu&FL7$Fi[l$!3=+++'o`jt&FL7$Fd[l$ !3U+++I(pps&FL7$F_[l$!3&)*****HPLYr&FL7$Fjz$!3?+++,3Q*p&FL7$Fez$!3I+++ ,QB\"o&FL7$F`z$!3S+++L:?gcFL7$F[z$!3T+++N>GOcFL7$Ffy$!36+++*zi%4cFL7$F ay$!3=+++YDszbFL7$F\\y$!3i*****HgIqa&FL7$Fgx$!3S+++LsM6bFL7$Fbx$!3s*** **z]AEZ&FL7$F]x$!3A+++nQzIaFL7$Fhw$!3W+++J8y&Q&FL7$Fcw$!3u******[(yuL& FL7$F^w$!3A+++9#QdG&FL7$Fiv$!3))*****RTU.B&FL7$Fdv$!3E+++(za4<&FL7$F_v $!3T+++ro-2^FL7$Fju$!36+++cwhP]FL7$Feu$!3K+++vD-h\\FL7$F`u$!3i******Q6 )R([FL7$F[u$!3Q+++T)=,x%FL7$Fft$!3<+++WZ$*QYFL7$Fat$!3G+++XpkvWFL7$F\\ t$!3G+++rt$pH%FL7$Ffs$!3P+++X/\"*>TFL7$Fas$!3++++^-q[RFL7$F\\s$!3=+++$ =rCy$FL7$Fgr$!3:+++YKv>OFL7$Far$!37+++6%[$fMFL7$F\\r$!3!)*****zK./I$FL 7$$!3!)*****R)RC'y%Fcq$!32+++.1MUJFL7$Faq$!35+++&=%y%)HFL7$F\\q$!3')** ***H]*\\FGFL7$$\"33+++Fk\"*[#FL7$Fco$!3&)*****poK?/#FL7$F ]o$!35+++(pb\\)=FL7$Fhn$!35+++Zo(ys\"FL7$$\"3&)*******>`'3HFU$!3#***** *f8(zq:FL7$FS$!3)******p>#Fh_pFd]p7%F`bp7$$\"+uNNNPFfap$\"+B)yhN#Fh_pFd]p7%Ffbp7$$ \"+NK;*H$Ffap$\"+(z`K^#Fh_pFd]p7%F\\cp7$$\"+Dk\"*[OFh_pFd]p7%Fhep7$$!+H+iuNFadp$\"+$=rCy$Fh_pFd]p7%F^fp7$ $\"+_LrWDFdep$\"+^-q[RFh_pFd]p7%Fdfp7$$\"+(y$)z,\"!#6$\"+X/\"*>TFh_pFd ]p7%Fjfp7$$\"+E0_.GF]gp$\"+rt$pH%Fh_pFd]p7%Fagp7$$\"+p*=*zlF]gp$\"+Xpk vWFh_pFd]p7%Fggp7$$\"+mZL#H\"Fc]p$\"+WZ$*QYFh_pFd]p7%F]hp7$$\"+j4uH?Fc ]p$\"+T)=,x%Fh_pFd]p7%Fchp7$$\"+6y8'o#Fc]p$\"+R6)R([Fh_pFd]p7%Fihp7$$ \"+'oT8@$Fc]p$\"+vD-h\\Fh_pFd]p7%F_ip7$$\"+&))3Th$Fc]p$\"+cwhP]Fh_pFd] p7%Feip7$$\"+(3,F\"RFc]p$\"+ro-2^Fh_pFd]p7%F[jp7$$\"+))Q_BTFc]p$\"+(za 4<&Fh_pFd]p7%Fajp7$$\"+^tzfUFc]p$\"+9CMI_Fh_pFd]p7%Fgjp7$$\"+\"=o?L%Fc ]p$\"+9#QdG&Fh_pFd]p7%F][q7$$\"+mQ')[VFc]p$\"+\\(yuL&Fh_pFd]p7%Fc[q7$$ \"+;L?GOcFh_pFd]p7%Fc^q7$$\"+%\\jQ(GFc]p$\"+L:?gcFh_pFd]p7%Fi ^q7$$\"+Zdm\"f#Fc]p$\"+,QB\"o&Fh_pFd]p7%F__q7$$\"+5w,%H#Fc]p$\"+,3Q*p& Fh_pFd]p7%Fe_q7$$\"+l0R#)>Fc]p$\"+tLj9dFh_pFd]p7%F[`q7$$\"+!)o2e;Fc]p$ \"+I(pps&Fh_pFd]p7%Fa`q7$$\"+m^?A8Fc]p$\"+'o`jt&Fh_pFd]p7%Fg`q7$$\"+YZ ]d(*F]gp$\"+;CtUdFh_pFd]p7%F]aq7$$\"+5JQ&>'F]gp$\"+yN.YdFh_pFd]p7%Fcaq 7$$\"+o>TUDF]gp$\"+u:;YdFh_pFd]p7%Fiaq7$$!+AeM'>\"F]gp$\"+8E*Hu&Fh_pFd ]p7%F_bq7$$!+9=w<]F]gp$\"+&3ojt&Fh_pFd]p7%Febq7$$!+&)\\7@*)F]gp$\"+Tb3 EdFh_pFd]p7%F[cq7$$!+fh*3H\"Fc]p$\"+(>')=r&Fh_pFd]p7%Facq7$$!+m?%))p\" Fc]p$\"+btV$p&Fh_pFd]p7%Fgcq7$$!+wdO<@Fc]p$\"+9tIqcFh_pFd]p7%F]dq7$$!+ *R2!\\DFc]p$\"+t#H>k&Fh_pFd]p7%Fcdq7$$!+C.A)*HFc]p$\"+0*\\vg&Fh_pFd]p7 %Fidq7$$!+@=#HZ$Fc]p$\"+!)3;mbFh_pFd]p7%F_eq7$$!+\"\\Hx)RFc]p$\"+zIV;b Fh_pFd]p7%Feeq7$$!+jp-rXFc]p$\"+'**)ycaFh_pFd]p7%F[fq7$$!+dY_z_Fc]p$\" +OiI'Q&Fh_pFd]p7%Fafq7$$!+=![;@'Fc]p$\"+Y&[!4`Fh_pFd]p7%Fgfq7$$!+s\\>E uFc]p$\"+s>1U_Fh_pFd]p7%F]gq7$$!+_)4.x)Fc]p$\"+P#y>?&Fh_pFd]p7%Fcgq7$$ !+d'Hg+\"Fh_p$\"+**3(R=&Fh_pFd]p7%Figq7$$!+Y_fC6Fh_p$\"+%**)\\x^Fh_pFd ]p7%F_hq7$$!+.\\cL7Fh_p$\"+hD:w^Fh_pFd]p7%Fehq7$$!+ETpM8Fh_p$\"+e5kw^F h_pFd]p7%F[iq7$$!+kf_H9Fh_p$\"+IkBx^Fh_pFd]p7%Faiq7$$!+')RE>:Fh_p$\"+i =)p<&Fh_pFd]p7%Fgiq7$$!+%eH[g\"Fh_p$\"+B8Kv^Fh_pFd]p7%F]jq7$$!+)GLpo\" Fh_p$\"++\">><&Fh_pFd]p7%Fcjq7$$!+m78mFh_p$\"+Tv_\\^Fh_pFd]p 7%Fe[r7$$!+`;K!*>Fh_p$\"+<**pP^Fh_pFd]p7%F[\\r7$$!+-xeh?Fh_p$\"+_=hB^F h_pFd]p7%Fa\\r7$$!+\"45:8#Fh_p$\"+9(Rs5&Fh_pFd]p7%Fg\\r7$$!+\\nF+AFh_p $\"+S)o&)3&Fh_pFd]p7%F]]r7$$!+fi0oAFh_p$\"+(R\"fn]Fh_pFd]p7%Fc]r7$$!+` L+NBFh_p$\"+qaIW]Fh_pFd]p7%Fi]r7$$!+bKE,CFh_p$\"+$4:(=]Fh_pFd]p7%F_^r7 $$!+g](pY#Fh_p$\"+c5$3*\\Fh_pFd]p7%Fe^r7$$!+EUFKDFh_p$\"+3Cng\\Fh_pFd] p7%F[_r7$$!+aTH(f#Fh_p$\"+]&o#G\\Fh_pFd]p7%Fa_r7$$!+Sn;iEFh_p$\"+C=m$* [Fh_pFd]p7%Fg_r7$$!+]<-FFFh_p$\"+P/\"p&[Fh_pFd]p7%F]`r7$$!+.[)>z#Fh_p$ \"+)R\"4=[Fh_pFd]p7%Fc`r7$$!+QQ7$Fh_p$\"+l]?(f%Fh_pFd]p7%F abr7$$!+>hP*=$Fh_p$\"+j&)y[XFh_pFd]p7%Fgbr7$$!+.&HsD$Fh_p$\"+R$y$*\\%F h_pFd]p7%F]cr7$$!+0[VDLFh_p$\"+`69\\WFh_pFd]p7%Fccr7$$!+*3pQR$Fh_p$\"+ #o=#)R%Fh_pFd]p7%Ficr7$$!+(3&QiMFh_p$\"+A*>nM%Fh_pFd]p7%F_dr7$$!+R<#3` $Fh_p$\"+,sr%H%Fh_pFd]p7%Fedr7$$!+W[,*f$Fh_p$\"+)*fCUUFh_pFd]p7%F[er7$ $!+;k!om$Fh_p$\"+C!3$*=%Fh_pFd]p7%Faer7$$!+6=0MPFh_p$\"+@l(e8%Fh_pFd]p 7%Fger7$$!+\\Vi+QFh_p$\"+TE!>3%Fh_pFd]p7%F]fr7$$!+7wTmQFh_p$\"+C>KFSFh _pFd]p7%Fcfr7$$!+weMJRFh_p$\"+/-1sRFh_pFd]p7%Fifr7$$!+QKM&*RFh_p$\"+#f Qg\"RFh_pFd]p7%F_gr7$$!+]>OeSFh_p$\"+Ru;FFh_pFd]p7%F[^s7$$!+ewoG]Fh_p$\"+#y\"3QEFh_pFd]p7%Fa^s7$$!+&f^W2& Fh_p$\"+$yD*eDFh_pFd]p7%Fg^s7$$!+)QB$>^Fh_p$\"+O:xyCFh_pFd]p7%F]_s7$$! +/bDj^Fh_p$\"+/tl(R#Fh_pFd]p7%Fc_s7$$!+&*G>1_Fh_p$\"+qlh:BFh_pFd]p7%Fi _s7$$!+LW2[_Fh_p$\"+LrnKAFh_pFd]p7%F_`s7$$!+2X$))G&Fh_p$\"+^1')[@Fh_pF d]p7%Fe`s7$$!+SYSG`Fh_p$\"+5E=k?Fh_pFd]p7%F[as7$$!+V]rm`Fh_p$\"+/Fly>F h_pFd]p7%Faas7$$!+>gp.aFh_p$\"+cbF#*=Fh_pFd]p7%Fgas7$$!+v\"z#RaFh_p$\" +9<00=Fh_pFd]p7%F]bs7$$!+:%)RtaFh_p$\"+Q)ypr\"Fh_pFd]p7%Fcbs7$$!+=2*f] &Fh_p$\"+EI0G;Fh_pFd]p7%Fibs7$$!+=n*p`&Fh_p$\"+%>q#Q:Fh_pFd]p7%F_cs7$$ !+<6OmbFh_p$\"+%[FwW\"Fh_pFd]p7%Fecs7$$!+_H.%f&Fh_p$\"+[Y7c8Fh_pFd]p7% F[ds7$$!+[d'*>cFh_p$\"+%QlPE\"Fh_pFd]p7%Fads7$$!+'el2\"Fh_pFd]p7%F]es7$$!+qL$po&Fh_p$ \"+p%4n\")*Fc]pFd]p7%Fces7$$!+ve`0dFh_p$\"+KrVg))Fc]pFd]p7%Fies7$$!+PU BAdFh_p$\"+Lyx'*yFc]pFd]p7%F_fs7$$!+A!3qt&Fh_p$\"+(4Di#pFc]pFd]p7%Fefs 7$$!+b/%)\\dFh_p$\"+U'o$\\fFc]pFd]p7%F[gs7$$!+&3=2w&Fh_p$\"+SK*o'\\Fc] pFd]p7%Fags7$$!+C/jpdFh_p$\"+ZUdzRFc]pFd]p7%Fggs7$$!+q&plx&Fh_p$\"+U'p #))HFc]pFd]p7%F]hs7$$!+S)H:y&Fh_p$\"+?'3R*>Fc]pFd]p7%Fchs7$$!+Tu]%y&Fh _p$\"+i+xu**F]gpFd]p7%Fihs7$$!+4-]&y&Fh_pF(Fd]p7%F_is7$Fjhs$!+i+xu**F] gpFd]p7%Fcis7$Fdhs$!+?'3R*>Fc]pFd]p7%Fgis7$F^hs$!+U'p#))HFc]pFd]p7%F[j s7$Fhgs$!+ZUdzRFc]pFd]p7%F_js7$Fbgs$!+SK*o'\\Fc]pFd]p7%Fcjs7$F\\gs$!+U 'o$\\fFc]pFd]p7%Fgjs7$Fffs$!+(4Di#pFc]pFd]p7%F[[t7$F`fs$!+Lyx'*yFc]pFd ]p7%F_[t7$Fjes$!+KrVg))Fc]pFd]p7%Fc[t7$Fdes$!+p%4n\")*Fc]pFd]p7%Fg[t7$ F^es$!+g&>l2\"Fh_pFd]p7%F[\\t7$Fhds$!+(fe0<\"Fh_pFd]p7%F_\\t7$Fbds$!+% QlPE\"Fh_pFd]p7%Fc\\t7$F\\ds$!+[Y7c8Fh_pFd]p7%Fg\\t7$Ffcs$!+%[FwW\"Fh_ pFd]p7%F[]t7$F`cs$!+%>q#Q:Fh_pFd]p7%F_]t7$Fjbs$!+EI0G;Fh_pFd]p7%Fc]t7$ Fdbs$!+Q)ypr\"Fh_pFd]p7%Fg]t7$F^bs$!+9<00=Fh_pFd]p7%F[^t7$Fhas$!+cbF#* =Fh_pFd]p7%F_^t7$Fbas$!+/Fly>Fh_pFd]p7%Fc^t7$F\\as$!+5E=k?Fh_pFd]p7%Fg ^t7$Ff`s$!+^1')[@Fh_pFd]p7%F[_t7$F``s$!+LrnKAFh_pFd]p7%F__t7$Fj_s$!+ql h:BFh_pFd]p7%Fc_t7$Fd_s$!+/tl(R#Fh_pFd]p7%Fg_t7$F^_s$!+O:xyCFh_pFd]p7% F[`t7$Fh^s$!+$yD*eDFh_pFd]p7%F_`t7$Fb^s$!+#y\"3QEFh_pFd]p7%Fc`t7$F\\^s $!+yx>;FFh_pFd]p7%Fg`t7$Ff]s$!+G,B$z#Fh_pFd]p7%F[at7$F`]s$!+%4N\"pGFh_ pFd]p7%F_at7$Fj\\s$!+62(Q%HFh_pFd]p7%Fcat7$Fd\\s$!+'e)RKFSFh_pFd]p7%Fget7$F^fr$!+TE!>3%Fh_pFd] p7%F[ft7$Fher$!+@l(e8%Fh_pFd]p7%F_ft7$Fber$!+C!3$*=%Fh_pFd]p7%Fcft7$F \\er$!+)*fCUUFh_pFd]p7%Fgft7$Ffdr$!+,sr%H%Fh_pFd]p7%F[gt7$F`dr$!+A*>nM %Fh_pFd]p7%F_gt7$Fjcr$!+#o=#)R%Fh_pFd]p7%Fcgt7$Fdcr$!+`69\\WFh_pFd]p7% Fggt7$F^cr$!+R$y$*\\%Fh_pFd]p7%F[ht7$Fhbr$!+j&)y[XFh_pFd]p7%F_ht7$Fbbr $!+l]?(f%Fh_pFd]p7%Fcht7$F\\br$!+ihWWYFh_pFd]p7%Fght7$Ffar$!+njK!p%Fh_ pFd]p7%F[it7$F`ar$!+JnmMZFh_pFd]p7%F_it7$Fj`r$!+vIIxZFh_pFd]p7%Fcit7$F d`r$!+)R\"4=[Fh_pFd]p7%Fgit7$F^`r$!+P/\"p&[Fh_pFd]p7%F[jt7$Fh_r$!+C=m$ *[Fh_pFd]p7%F_jt7$Fb_r$!+]&o#G\\Fh_pFd]p7%Fcjt7$F\\_r$!+3Cng\\Fh_pFd]p 7%Fgjt7$Ff^r$!+c5$3*\\Fh_pFd]p7%F[[u7$F`^r$!+$4:(=]Fh_pFd]p7%F_[u7$Fj] r$!+qaIW]Fh_pFd]p7%Fc[u7$Fd]r$!+(R\"fn]Fh_pFd]p7%Fg[u7$F^]r$!+S)o&)3&F h_pFd]p7%F[\\u7$Fh\\r$!+9(Rs5&Fh_pFd]p7%F_\\u7$Fb\\r$!+_=hB^Fh_pFd]p7% Fc\\u7$F\\\\r$!+<**pP^Fh_pFd]p7%Fg\\u7$Ff[r$!+Tv_\\^Fh_pFd]p7%F[]u7$F` [r$!+\"=J\"f^Fh_pFd]p7%F_]u7$Fjjq$!+[lcm^Fh_pFd]p7%Fc]u7$Fdjq$!++\">>< &Fh_pFd]p7%Fg]u7$F^jq$!+B8Kv^Fh_pFd]p7%F[^u7$Fhiq$!+i=)p<&Fh_pFd]p7%F_ ^u7$Fbiq$!+IkBx^Fh_pFd]p7%Fc^u7$F\\iq$!+e5kw^Fh_pFd]p7%Fg^u7$Ffhq$!+hD :w^Fh_pFd]p7%F[_u7$F`hq$!+%**)\\x^Fh_pFd]p7%F__u7$Fjgq$!+**3(R=&Fh_pFd ]p7%Fc_u7$Fdgq$!+P#y>?&Fh_pFd]p7%Fg_u7$F^gq$!+s>1U_Fh_pFd]p7%F[`u7$Fhf q$!+Y&[!4`Fh_pFd]p7%F_`u7$Fbfq$!+OiI'Q&Fh_pFd]p7%Fc`u7$F\\fq$!+'**)yca Fh_pFd]p7%Fg`u7$Ffeq$!+zIV;bFh_pFd]p7%F[au7$F`eq$!+!)3;mbFh_pFd]p7%F_a u7$Fjdq$!+0*\\vg&Fh_pFd]p7%Fcau7$Fddq$!+t#H>k&Fh_pFd]p7%Fgau7$F^dq$!+9 tIqcFh_pFd]p7%F[bu7$Fhcq$!+btV$p&Fh_pFd]p7%F_bu7$Fbcq$!+(>')=r&Fh_pFd] p7%Fcbu7$F\\cq$!+Tb3EdFh_pFd]p7%Fgbu7$Ffbq$!+&3ojt&Fh_pFd]p7%F[cu7$F`b q$!+8E*Hu&Fh_pFd]p7%F_cu7$Fjaq$!+u:;YdFh_pFd]p7%Fccu7$Fdaq$!+yN.YdFh_p Fd]p7%Fgcu7$F^aq$!+;CtUdFh_pFd]p7%F[du7$Fh`q$!+'o`jt&Fh_pFd]p7%F_du7$F b`q$!+I(pps&Fh_pFd]p7%Fcdu7$F\\`q$!+tLj9dFh_pFd]p7%Fgdu7$Ff_q$!+,3Q*p& Fh_pFd]p7%F[eu7$F`_q$!+,QB\"o&Fh_pFd]p7%F_eu7$Fj^q$!+L:?gcFh_pFd]p7%Fc eu7$Fd^q$!+N>GOcFh_pFd]p7%Fgeu7$F^^q$!+*zi%4cFh_pFd]p7%F[fu7$Fh]q$!+YD szbFh_pFd]p7%F_fu7$Fb]q$!+.1.ZbFh_pFd]p7%Fcfu7$F\\]q$!+LsM6bFh_pFd]p7% Fgfu7$Ff\\q$!+3DisaFh_pFd]p7%F[gu7$F`\\q$!+nQzIaFh_pFd]p7%F_gu7$Fj[q$! +J8y&Q&Fh_pFd]p7%Fcgu7$Fd[q$!+\\(yuL&Fh_pFd]p7%Fggu7$F^[q$!+9#QdG&Fh_p Fd]p7%F[hu7$Fhjp$!+9CMI_Fh_pFd]p7%F_hu7$Fbjp$!+(za4<&Fh_pFd]p7%Fchu7$F \\jp$!+ro-2^Fh_pFd]p7%Fghu7$Ffip$!+cwhP]Fh_pFd]p7%F[iu7$F`ip$!+vD-h\\F h_pFd]p7%F_iu7$Fjhp$!+R6)R([Fh_pFd]p7%Fciu7$Fdhp$!+T)=,x%Fh_pFd]p7%Fgi u7$F^hp$!+WZ$*QYFh_pFd]p7%F[ju7$Fhgp$!+XpkvWFh_pFd]p7%F_ju7$Fbgp$!+rt$ pH%Fh_pFd]p7%Fcju7$F[gp$!+X/\"*>TFh_pFd]p7%Fgju7$Fefp$!+^-q[RFh_pFd]p7 %F[[v7$F_fp$!+$=rCy$Fh_pFd]p7%F_[v7$Fiep$!+YKv>OFh_pFd]p7%Fc[v7$Fbep$! +6%[$fMFh_pFd]p7%Fg[v7$F\\ep$!+GLS+LFh_pFd]p7%F[\\v7$$!+%)RC'y%Fadp$!+ .1MUJFh_pFd]p7%F_\\v7$F_dp$!+&=%y%)HFh_pFd]p7%Fe\\v7$Ficp$!+.&*\\FGFh_ pFd]p7%Fi\\v7$$\"+Fk\"*[#Fh_pFd]p7%F[^ v7$F[bp$!+(oK?/#Fh_pFd]p7%F_^v7$Fdap$!+(pb\\)=Fh_pFd]p7%Fc^v7$F^ap$!+Z o(ys\"Fh_pFd]p7%Fg^v7$$\"++Kl3HFc`p$!+Orzq:Fh_pFd]p7%F[_v7$Fa`p$!+(>

\"F[]p;$!$R'F[]p$\"$R'F[]p" 1 2 0 1 10 0 2 9 1 2 1 1.000000 45.000000 45.000000 0 0 "Curve 1" "Curve 2 " "Curve 3" "Curve 4" }}}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 4 "The " }{TEXT 260 30 "interval of absolute stability" }{TEXT -1 89 " (or stability interval) is the intersection of the sta bility region with the real line." }}{PARA 0 "" 0 "" {TEXT -1 59 "For \+ this scheme the stability interval is (approximately) " }{XPPEDIT 18 0 "[-5.7855, 0];" "6#7$,$-%&FloatG6$\"&by&!\"%!\"\"\"\"!" }{TEXT -1 2 ". " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 115 "W e can distort the boundary curve horizontally by taking the 11th root \+ of the real part of points along the curve. " }}{PARA 0 "" 0 "" {TEXT -1 150 "In this way we see that the intersection of the stability regi on with the non-negative imaginary axis consists of the union of two d isjoint intervals." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 " > " 0 "" {MPLTEXT 1 0 557 "R := z -> add(z^j/j!,j=0..8)+92924991707570 9813239993513/347112006539529951872060580000000*z^9+\n 5598799833379 64567261124311/2138457897431032739211801787500000*z^10+\n 1115022582 191670166084763147/85538315897241309568472071500000000*z^11+\n 57919 1110727720778833/8155827221323542102257062500000*z^12:\nDigits := 18: \npts := []: z0 := 0:\nfor ct from 0 to 200 do\n zz := newton(R(z)=e xp(ct*Pi/60*I),z=z0):\n z0 := zz:\n pts := [op(pts),[surd(Re(zz),1 1),Im(zz)]]:\nend do:\nplot(pts,color=COLOR(RGB,.2,.75,.18),thickness= 2,font=[HELVETICA,9]);\nDigits := 10:" }}{PARA 13 "" 1 "" {GLPLOT2D 287 308 308 {PLOTDATA 2 "6(-%'CURVESG6#7ew7$$\"\"!F)F(7$F($\"3v))H)fv( )fB&!#>7$$\"3%=K[2-SZ&GF-$\"3$*)f'>^v>Z5!#=7$$\"3=\"*f5!pae7%F-$\"3(=% \\zEjzq:F37$$\"3SkE?*)yyc`F-$\"3e#z$R-^R%4#F37$$\"3z[Xjy$>yb'F-$\"3G!* e*z(Q*zh#F37$$\"3>`AX.fnMxF-$\"32X@h`EfTJF37$$\"33,CP\"p@4*))F-$\"3DYg FH9>lOF37$$\"3K5ykFB*G+\"F3$\"3v^I20-z)=%F37$$\"3Zrh&))QJ]6\"F3$\"3NVJ >\")*)Q7ZF37$$\"3E?]tCWiD7F3$\"3#)3d,ex)fB&F37$$\"3'R9g6&HvM8F3$\"3kyc BOlefdF37$$\"3-&oH[2uCW\"F3$\"36[z.<`=$G'F37$$\"3S!*3@ak#)[:F3$\"3cdJK -Ty1oF37$$\"3Es`?AV$Ql\"F3$\"3\"Qe\")\\*GQItF37$$\"3yE&Q;F5vv\"F3$\"3) Q0.#*p\")R&yF37$$\"3mbQdF3$\" 3zhF]n$z6!*)F37$$\"3if+IC``g?F3$\"3ACj=\\#yZU*F37$$\"3WSq#f`M)e@F3$\"3 Yy/%zubAF3$\"3R#y\"z;w>Z5!#<7$$\"3T^Dg*HE7N#F3$\" 3Sd[[Bvb*4\"Fhq7$$\"3!=\\iCM]_W#F3$\"3!*Q:^Ru\">:\"Fhq7$$\"3cmYH-6xPDF 3$\"3GRrUmtF/7Fhq7$$\"3*Q>cxaR(GEF3$\"3@W()[0tjc7Fhq7$$\"3]Z9/1.5=FF3$ \"3,&yitD(**38Fhq7$$\"3\")HdP/7z0GF3$\"3*\\2[e7BL#H$F3$\"32EPN**p^v;Fhq7$$\" 3\\r$oQQPeO$F3$\"3M!)*>q%o(ys\"Fhq7$$\"3Hz<9:e!oV$F3$\"3![d/DgO-y\"Fhq 7$$\"3EYRqhE00NF3$\"31v@,KifK=Fhq7$$\"3)y+5[3j.d$F3$\"3ZJa5(pb\\)=Fhq7 $$\"3Y_qs<\\[KOF3$\"3a=(eg&\\JP>Fhq7$$\"3g,(Qpf=6p$F3$\"3?\\1>mRn*)>Fh q7$$\"3\"3lW,C.fu$F3$\"3&GCkuoK?/#Fhq7$$\"3!*p+;JpR'z$F3$\"3W=.p(3\"R% 4#Fhq7$$\"3s#o(*R,[?%QF3$\"30v#=(\\\"\\n9#Fhq7$$\"3n*y*R2>:#)QF3$\"3u( *z/\")o5*>#Fhq7$$\"3KS52dNy:RF3$\"3e;hMEVY^AFhq7$$\"3eGKFymoTRF3$\"35b !)R$e@QI#Fhq7$$\"3x%H&H=73eRF3$\"3LhH2B)yhN#Fhq7$$\"3Iw(oww-B'RF3$\"3$ 4X?RJO&3CFhq7$$\"3Z!3%>!>d+&RF3$\"3lQ\\0_W*3Y#Fhq7$$\"3=G0NB;l8RF3$\"3 5YU,(z`K^#Fhq7$$\"30Go/0.\"o$QF3$\"3SwNI9^hlDFhq7$$\"3<*>pK:`Cn$F3$\"3 &yz``Uzzh#Fhq7$$\"3/4DN)\\2l*HF3$\"3#\\$\\jl![.n#Fhq7$$!3Hw2++hG4PF3$ \"3dOLn^FsAFFhq7$$!3@e[Ip1.BSF3$\"3\"*fmsdc5vFFhq7$$!3J\"*p<%[**RB%F3$ \"3k#3yG]*\\FGFhq7$$!311>I!)\\Q-WF3$\"3nHXK\\w!*zGFhq7$$!3@)pX-#oVYXF3 $\"3\"RWeE@MB$HFhq7$$!3D2bK4%ySn%F3$\"3x$R%)\\=%y%)HFhq7$$!33m$Hy9q$*y %F3$\"3([*exrNEPIFhq7$$!3bSu$Fhq7$$!38?(R7h #Gf^F3$\"3\"**QnSyRwC$Fhq7$$!3%4on>tv7B&F3$\"3j*[@$GLS+LFhq7$$!3**\\z0 F/]%H&F3$\"3zUl%HFjKN$Fhq7$$!3lv7Tun!zM&F3$\"3!GR@L%zB1MFhq7$$!3/,U*yp -)*Q&F3$\"3oHo.6%[$fMFhq7$$!3Du4![]3vT&F3$\"3UCEg lEaF3$\"3g2U*Gmwgc$Fhq7$$!3[]1zz$G'4aF3$\"3CBZdXKv>OFhq7$$!3nph@?o[^`F 3$\"3zOgV:RotOFhq7$$!3a%oP'3%Hg@&F3$\"3*\\crBV3zs$Fhq7$$!3wY)e(G\\Bg[F 3$\"3P\"o-I=rCy$Fhq7$$\"3'*4k6>$R:w%F3$\"3w\"fnMA@u$QFhq7$$\"3^P;!3.!* [V&F3$\"3vJcqq>\"G*QFhq7$$\"3m$*R#R(oj4eF3$\"3*fdK6D+([RFhq7$$\"36R&\\ u+pC5'F3$\"3!phhz?W^+%Fhq7$$\"3s#*3hLF+djF3$\"3nmmMA#*>iSFhq7$$\"3Md^2 %z,+f'F3$\"36@q%\\W5*>TFhq7$$\"3&)HS**HMt4oF3$\"3qQTb#p*HyTFhq7$$\"3[t V$*)y*y?qF3$\"34#\\PFQVtB%Fhq7$$\"3l#Q\")G8PdA(F3$\"3+$o$eqt$pH%Fhq7$$ \"3z&zr,k#pDuF3$\"3&p>p:&e%oN%Fhq7$$\"3&*4iX!4]/i(F3$\"3Gd&RoJRmT%Fhq7 $$\"3k'G\"p`?Y3yF3$\"3=Ql*\\%pkvWFhq7$$\"3)zs?Mm%)p)zF3$\"3ukx\"Q`V)F3$\"3Oj6yQ1P'o%Fhq7$$\"32aP%=wF3b)F3$\"3bG0oFU* *HZFhq7$$\"37HVW0qZ]')F3$\"3Q//6T)=,x%Fhq7$$\"3lMA_=3=O()F3$\"3[Pm(Gd \">2[Fhq7$$\"3!p\"*pS'e&*4))F3$\"3+'[b@$RnT[Fhq7$$\"39jV;cclt))F3$\"3: T*=#R6)R([Fhq7$$\"3JU1\\C<()G*)F3$\"3_k0bf@Y/\\Fhq7$$\"3%fP$*['\\#p(*) F3$\"38^r)HV+M$\\Fhq7$$\"3mjeopZ*)=!*F3$\"3p1iMvD-h\\Fhq7$$\"3HCIs['ec 0*F3$\"3ih6Mv(3v)\\Fhq7$$\"3>-Zm$\\Iz3*F3$\"3^OLuVB+8]Fhq7$$\"3\\mt,&R $H;\"*F3$\"3$=n8flds3n3*>*F3$\"34(>F\"fx))G^Fhq7$$\"3b6;Z-%eO@*F3$\"3:=9#o\"f=]^Fhq7$$ \"3'p$p&p$GBE#*F3$\"3CvX:(za4<&Fhq7$$\"36F*4`uypB*F3$\"3!\\\"z8YAA\">& Fhq7$$\"3(GyEL%3/Y#*F3$\"3^iy7w:,6_Fhq7$$\"3Zv6!*3Qa`#*F3$\"3g#)\\09CM I_Fhq7$$\"3iNq9yaff#*F3$\"3rXt%fPJ#\\_Fhq7$$\"3F$=LxU*Gk#*F3$\"3;*z+mg #pn_Fhq7$$\"3jt@*p#*F3$\"3_?Qz> 'yLI&Fhq7$$\"3Rqkr%G))4F*F3$\"3>1iXYGi?`Fhq7$$\"39[+6Ey'4F*F3$\"35+4u[ (yuL&Fhq7$$\"37OIgK^!*p#*F3$\"3?_KY0K&RN&Fhq7$$\"3>`cb.7%yE*F3$\"3.?bH wA0q`Fhq7$$\"34;#GGd6[E*F3$\"37L95J8y&Q&Fhq7$$\"3vsZ\"y&o%3E*F3$\"3.Z& =n:X6S&Fhq7$$\"3at@.:K(fD*F3$\"3QA0CX![hT&Fhq7$$\"38WJ!or7-D*F3$\"3vev 0nQzIaFhq7$$\"3U3'41m$eV#*F3$\"3V%[!HJh3XaFhq7$$\"3345J?35O#*F3$\"32lA wN!G!faFhq7$$\"3es^[YcxF#*F3$\"3^:)f#3DisaFhq7$$\"3]=5b@kh=#*F3$\"3Igi _SA(e[&Fhq7$$\"37N2*RPG'3#*F3$\"31o><;(z()\\&Fhq7$$\"33h$Q[v8y>*F3$\"3 N^')[LsM6bFhq7$$\"3]*e)ey=<'=*F3$\"3G9X)R#pdBbFhq7$$\"3c,\"f(H\"*pt\"* F3$\"3]#3Atwqa`&Fhq7$$\"3F]CZN*)Qg\"*F3$\"3x]zA.1.ZbFhq7$$\"34AJd.Pkf'\\:pbFhq7$$\"37\"z? w.A`6*F3$\"3w:XNYDszbFhq7$$\"3swbx\\U`)4*F3$\"3@H-mNA'**e&Fhq7$$\"3`A( )[;$G33*F3$\"3I94\"fFv)*f&Fhq7$$\"3<;m4ls-*F3$\"3$=;6=Flwi&Fhq7$$ \"3n3Q.bs@+!*F3$\"3$>s;Y$>GOcFhq7$$\"3b\"e7\\BCu(*)F3$\"3%*y6U.ldWcFhq 7$$\"3uxugazZ`*)F3$\"3;WTF^&\\Dl&Fhq7$$\"3&o]H.(zJG*)F3$\"3%e([UL:?gcF hq7$$\"3Q!e-Nev=!*)F3$\"3z*>>TyKvm&Fhq7$$\"3+2gUGE2u))F3$\"3:bM'H^VXn& Fhq7$$\"3ro#H&)**>[%))F3$\"3&>ms3!QB\"o&Fhq7$$\"3w[MnVe,9))F3$\"3lG/h& f.wo&Fhq7$$\"3iO]?\\La\"y)F3$\"3?'H>pq_Op&Fhq7$$\"3$3!*ot(zEZ()F3$\"3, ^?B,3Q*p&Fhq7$$\"3Uij)F3$\"3E^fRtLj9dFhq7$$\"396m%\\h/')e)F3$\" 3=?:e()42>dFhq7$$\"3X\"Q&*\\3tBa)F3$\"3^'eu+b$=BdFhq7$$\"3X>&p7kzG\\)F 3$\"3(zcO)H(pps&Fhq7$$\"3D4nh2ioR%)F3$\"3VU/S,!G/t&Fhq7$$\"3r$)*fr5_AQ )F3$\"3?O?*\\jcNt&Fhq7$$\"3#3P:$)F3$\"3(>q0eo`jt&Fhq7$$\"3I)y*pf Bw^#)F3$\"3;yLM#)p\")QdFhq7$$\"3Lql#oB5n<)F3$\"3'\\K9P6W4u&Fhq7$$\"3nM <\"frEK4)F3$\"3\"*=(zdTKFu&Fhq7$$\"3G$)3Uy*4#**zF3$\"3\"Q\\6g&*yTu&Fhq 7$$\"3#oM/#=Hi\"*yF3$\"3kS!=p^!GXdFhq7$$\"3uES+JC$ew(F3$\"3xv-(ydLgu&F hq7$$\"3l,e/!**HUh(F3$\"3Zc(R^HMku&Fhq7$$\"3+B\")*\\%f\"HU(F3$\"3Y-b$* z%yku&Fhq7$$\"3o;`8qF!=;(F3$\"37Nu>u:;YdFhq7$$\"3YpP(fxF2u'F3$\"3zZ7(Q iyau&Fhq7$$\"3=*QovAN$)3&F3$\"3+Z8N\\UUWdFhq7$$!328L%[*pV(o'F3$\"3#fTM Fh#*Hu&Fhq7$$!3%f$)4>))f19(F3$\"3!z#)F3$\"3d@ aq!Htqr&Fhq7$$!3,_&\\Pq$z,$)F3$\"3'))[rl>')=r&Fhq7$$!3\"*>)RPLBS%y)F3$\"3#R9Z&4P,_cFhq7$$!3coXY2][J))F3$\"32 \\eos#H>k&Fhq7$$!3l^\"o,Rgn())F3$\"3+ws&36!=JcFhq7$$!3!*zFr5HX?*)F3$\" 3LBZdiEt>cFhq7$$!3FU-M*pnF'*)F3$\"3(3z\")[!*\\vg&Fhq7$$!3XC*GN]'*Q+*F3 $\"3r*y>/-\"f%f&Fhq7$$!3x6E&\\.BS/*F3$\"36_U5#463e&Fhq7$$!3cvsOy!GL3*F 3$\"36HR#)z3;mbFhq7$$!3Z7Y^aY*>7*F3$\"3vx-!ex'e]bFhq7$$!3w-j#\\]8-;*F3 $\"3n$*[O26.MbFhq7$$!3G*y\")o8*=)>*F3$\"3R=>ryIV;bFhq7$$!3vlSO%fYhB*F3 $\"3;NI*3&3t(\\&Fhq7$$!30'p$=H*QVF*F3$\"3'**oZaglyZ&Fhq7$$!3Mi&4$y[08$ *F3$\"3Q!Rjc**)ycaFhq7$$!3\\gb*4?DEN*F3$\"3-SEUJiZMaFhq7$$!3k8 " 0 "" {MPLTEXT 1 0 290 "Digits := 15:\nz0 := 2.7*I:\nfor ct from 84 to 87 do\n newton(R(z)=exp(ct*Pi/100*I),z=z0 );\nend do;``;\nz0 := 3.8*I:\nfor ct from 120 to 123 do\n newton(R(z )=exp(ct*Pi/100*I),z=z0);\nend do;``;\nz0 := 5.75*I:\nfor ct from 277 \+ to 280 do\n newton(R(z)=exp(ct*Pi/100*I),z=z0);\nend do;\nDigits := \+ 10:" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#^$$\"0H3!ol676!#>$\"0VY^FE*QE!# 9" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#^$$\"0NAiU;*[$\"0B1nf)>LF!#9" } }{PARA 11 "" 1 "" {XPPMATH 20 "6#%!G" }}{PARA 11 "" 1 "" {XPPMATH 20 " 6#^$$!0L&HH+iuN!#=$\"0o-I=rCy$!#9" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#^ $$!0,\\@%))oFF!#?$\"0$)QVS\"R:Q!#9" }}{PARA 11 "" 1 "" {XPPMATH 20 "6# ^$$\"0b@xr?ZY%!#=$\"0*z+QBY[Q!#9" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#^$ $\"036:m7r+\"!#<$\"06es.#4$!#<$\"0'*G:6lYu& !#9" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#^$$!08#3)oC\">W!#<$\"0P%G4s*Qu& !#9" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#^$$!03c<#eM'>\"!#;$\"0UMFh#*Hu& !#9" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 107 " Then we apply the bisection method to calculate the parameter value as sociated with the intersection point." }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 482 "Digits := 15:\nreal_part := proc(u)\n Re(newton(R( z)=exp(u*Pi*I),z=2.7*I))\nend proc:\nu0 := bisect('real_part'(u),u=0.8 4..0.87);\nnewton(R(z)=exp(u0*Pi*I),z=2.7*I);``;\nreal_part := proc(u) \n Re(newton(R(z)=exp(u*Pi*I),z=3.8*I))\nend proc:\nu0 := bisect('re al_part'(u),u=1.20..1.23);\nnewton(R(z)=exp(u0*Pi*I),z=3.8*I);``;\nrea l_part := proc(u)\n Re(newton(R(z)=exp(u*Pi*I),z=5.75*I))\nend proc: \nu0 := bisect('real_part'(u),u=2.77..2.80);\nnewton(R(z)=exp(u0*Pi*I) ,z=5.75*I);\nDigits := 10:" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%#u0G$ \"0yFR&*)p;&)!#:" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#^#$\"0'H%eP&fvE!#9 " }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%!G" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%#u0G$\"0(f/\\#o+@\"!#9" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#^$$! 0@\"=fBk+u!#H$\"0cpie;c\"Q!#9" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%!G" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%#u0G$\"0'>od>7%y#!#9" }}{PARA 11 " " 1 "" {XPPMATH 20 "6#^$$\"0C*yt&H:Z\"!#G$\"0'y,f]OWd!#9" }}}{PARA 0 " " 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 4 "The " }{TEXT 260 112 "largest interval on the nonnegative imaginary axis that contains \+ the origin and lies inside the stability region" }{TEXT -1 5 " is " } {XPPEDIT 18 0 "[0, 2.6756];" "6#7$\"\"!-%&FloatG6$\"&cn#!\"%" }{TEXT -1 18 " (approximately)." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 " " 0 "" {TEXT -1 4 "The " }{TEXT 260 72 "intersection of the stability \+ region with the nonnegative imaginary axis" }{TEXT -1 5 " is " } {XPPEDIT 18 0 "`union`([0, 2.6756],[3.8156, 5.7444]);" "6#-%&unionG6$7 $\"\"!-%&FloatG6$\"&cn#!\"%7$-F)6$\"&c\"QF,-F)6$\"&Wu&F," }{TEXT -1 17 " (approximately)." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 37 "#------------------------------------" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 81 "The stability function R* for the 13 stage, order 7 scheme is given as follows." }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 98 "subs(ee,subs(b=`b*`,StabilityFuncti on(7,13,'expanded'))):\n`R*` := unapply(%,z):\n'`R*`(z)'=`R*`(z);" }} {PARA 12 "" 1 "" {XPPMATH 20 "6#/-%#R*G6#%\"zG,:\"\"\"F)F'F)*&#F)\"\"# F)*$)F'F,F)F)F)*&#F)\"\"'F)*$)F'\"\"$F)F)F)*&#F)\"#CF)*$)F'\"\"%F)F)F) *&#F)\"$?\"F)*$)F'\"\"&F)F)F)*&#F)\"$?(F)*$)F'F1F)F)F)*&#F)\"%S]F)*$)F '\"\"(F)F)F)*&#\"<2c%)f1Q&[:GZ$**=(\"A+++:SB\"4C-S\"D++++VT%p8>E[%zJm2r\"F)*$)F'\"#5F)F)F)*&#\"74e3 R8&3am'*=#\"@++]79X?%3ZEWa;J;F)*$)F'\"#6F)F)F)" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 124 "We can find the point wh ere the boundary of the stability region intersects the negative real \+ axis by solving the equation: " }}{PARA 256 "" 0 "" {TEXT -1 1 " " } {XPPEDIT 18 0 "`R*`(z) = 1;" "6#/-%#R*G6#%\"zG\"\"\"" }{TEXT -1 2 ". \+ " }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 32 "z_0 := newton(`R*`(z)=1, z=-5.5);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%$z_0G$!+Q(R:`&!\"*" }}} {PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 320 "z_0 := newton(`R*`(z)=1,z=-5.5):\np_1 := plot([`R*`(z),1],z=-6.29 ..0.49,color=[red,blue]):\np_2 := plot([[[z_0,1]]$3],style=point,symbo l=[circle,cross,diamond],color=black):\np_3 := plot([[z_0,0],[z_0,1]], linestyle=3,color=COLOR(RGB,0,.5,0)):\nplots[display]([p_1,p_2,p_3],vi ew=[-6.29..0.49,-0.07..1.47],font=[HELVETICA,9]);" }}{PARA 13 "" 1 "" {GLPLOT2D 380 273 273 {PLOTDATA 2 "6+-%'CURVESG6$7Z7$$!3/++++++!H'!#<$ \"33+k<$*=P\"Q%F*7$$!3'**\\(oBQ0`iF*$\"3??TL:#Rp4%F*7$$!3))**\\PZw5;iF *$\"3(oW&Qc;AHQF*7$$!3!)*\\i5Zh\"zhF*$\"3ynRiN;MxNF*7$$!3s***\\ZH:A9'F *$\"3[.kdy@YSLF*7$$!3M]7$Q&>#z2'F*$\"3u5,2(Rg;'HF*7$$!31+D\"HhGO,'F*$ \"3qcpzSx%>i#F*7$$!3r\\(=o-C8%fF*$\"3oO$fll6@G#F*7$$!3E+]sS%>!peF*$\"3 sDu)4gTC)>F*7$$!3e++!HdNiz&F*$\"3#Q`r;cNqr\"F*7$$!3,+]20-!*yY%Fho7$$!30+D'R>oz,&F*$\"3nuG))f R??KFho7$$!3i***\\:(**[q[F*$\"3oZ'=45z;G#Fho7$$!3g**\\(H2\"fSZF*$\"3Oq Xl^g'pn\"Fho7$$!3e***\\'zcN%f%F*$\"3OrAP-K*[=\"Fho7$$!3/++vH)>vW%F*$\" 3A#=_()QSBU)!#>7$$!3t***\\.G;gI%F*$\"3I%)=?yaz#='F\\r7$$!3')*\\77]3uBEF*$\"3#ze& =@>DZsF\\r7$$!3Y**\\KvJ?$[#F*$\"3\"f=!>^Q,U$)F\\r7$$!3#)****pPS,QBF*$ \"3'[2U$[#F*$\"3hkx&)z%4A6\"Fho7$$!3'**\\7$= Xae?F*$\"3+hH))*\\6iF\"Fho7$$!3G**\\dI;&f!>F*$\"3[\"fe,T4n[\"Fho7$$!3S ++5M,%)oC+8u>Fho7$$!3 Y*\\7G+)y*[\"F*$\"3g:*HPq#>aAFho7$$!3-***\\3efZM\"F*$\"3U<'*Gd\">gg#Fh o7$$!3M+D6Q\"*H37F*$\"3S@pfQv/()HFho7$$!3W*\\ihlec1\"F*$\"3TAG%fL3]W$F ho7$$!3A'***\\F^,i#*Fho$\"3O$pd_g_0'RFho7$$!3@'*\\7y!G?!yFho$\"3U$fI;_ IJe%Fho7$$!3Q$****f-()eR'Fho$\"3qI;9HM4v_Fho7$$!3=)**\\nq/z&\\Fho$\"3? BkQ)\\#*34'Fho7$$!3<#*\\72'H=`$Fho$\"3HZb>6m[CqFho7$$!3$[****pp99A#Fho $\"3Av$G4n?!3!)Fho7$$!3=h**\\(H8`>(F\\r$\"3sJ%z/3XdI*Fho7$$\"3JW++!e5x B'F\\r$\"3G>6wYOOk5F*7$$\"3'*4]()y:,c?Fho$\"3**y3>BKEG7F*7$$\"3b1]72&[ oU$Fho$\"3Ia#zLtC(39F*7$$\"3!***************[Fho$\"3+*Q\"=?iJK;F*-%'CO LOURG6&%$RGBG$\"*++++\"!\")$\"\"!Fd]lFc]l-F$6$7S7$F($\"\"\"Fd]l7$F=Fi] l7$FGFi]l7$FQFi]l7$FenFi]l7$F_oFi]l7$FdoFi]l7$FjoFi]l7$F_pFi]l7$FdpFi] l7$FipFi]l7$F^qFi]l7$FcqFi]l7$FhqFi]l7$F^rFi]l7$FcrFi]l7$FhrFi]l7$F]sF i]l7$FbsFi]l7$FgsFi]l7$F\\tFi]l7$FatFi]l7$FftFi]l7$F[uFi]l7$F`uFi]l7$F euFi]l7$FjuFi]l7$F_vFi]l7$FdvFi]l7$FivFi]l7$F^wFi]l7$FcwFi]l7$FhwFi]l7 $F]xFi]l7$FbxFi]l7$FgxFi]l7$F\\yFi]l7$FayFi]l7$FfyFi]l7$F[zFi]l7$F`zFi ]l7$FezFi]l7$FjzFi]l7$F_[lFi]l7$Fd[lFi]l7$Fi[lFi]l7$F^\\lFi]l7$Fc\\lFi ]l7$Fh\\lFi]l-F]]l6&F_]lFc]lFc]lF`]l-F$6&7#7$$!3@+++Q(R:`&F*Fi]l-%'SYM BOLG6#%'CIRCLEG-F]]l6&F_]lFd]lFd]lFd]l-%&STYLEG6#%&POINTG-F$6&F_al-Fda l6#%&CROSSGFgalFial-F$6&F_al-Fdal6#%(DIAMONDGFgalFial-F$6%7$7$FaalFc]l F`al-%&COLORG6&F_]lFc]l$\"\"&!\"\"Fc]l-%*LINESTYLEG6#\"\"$-%+AXESLABEL SG6%Q\"z6\"Q!Ficl-%%FONTG6#%(DEFAULTG-F\\dl6$%*HELVETICAG\"\"*-%%VIEWG 6$;$!$H'!\"#$\"#\\Fidl;$!\"(Fidl$\"$Z\"Fidl" 1 2 0 1 10 0 2 9 1 4 2 1.000000 45.000000 45.000000 0 0 "Curve 1" "Curve 2" "Curve 3" "Curve \+ 4" "Curve 5" "Curve 6" }}}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 83 "The following picture shows the stability region for the 13 stage, order 7 scheme. " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 866 "`R*` := z -> add(z^j/j!,j=0 ..7)+718993472815485380659845607/28926000544960829322671715000000*z^8+ \n 112137385675416852770920712717/3991788075204594446528696670000000 0*z^9+\n 4002240912340197109791160147/171076631794482619136944143000 00000*z^10+\n 2189666540851339085809/1631165444264708420451412500000 *z^11:\npts := []:\nz0 := 0:\nfor ct from 0 to 320 do\n zz := newton (`R*`(z)=exp(ct*Pi/20*I),z=z0):\n z0 := zz:\n pts := [op(pts),[Re( zz),Im(zz)]]:\nend do:\np_1 := plot(pts,color=COLOR(RGB,0,.2,0)):\np_2 := plots[polygonplot]([seq([pts[i-1],pts[i],[-4.6,0]],i=2..nops(pts)) ],\n style=patchnogrid,color=COLOR(RGB,.2,.55,.25)):\np_3 := \+ plot([[[-6.49,0],[1.19,0]],[[0,-6.39],[0,6.39]]],color=black,linestyle =3):\nplots[display]([p_||(1..3)],view=[-6.49..1.19,-6.39..6.39],font= [HELVETICA,9],\n labels=[`Re(z)`,`Im(z)`],axes=boxed,scalin g=constrained);" }}{PARA 13 "" 1 "" {GLPLOT2D 381 565 565 {PLOTDATA 2 "6+-%'CURVESG6$7]_l7$$\"\"!F)F(7$F($\"3++++Fjzq:!#=7$F($\"3)******Rl#f TJF-7$$!3'*******e2>*o\"!#F$\"3<+++!)*)Q7ZF-7$$!3=+++$Rw_&>!#E$\"37+++ 5`=$G'F-7$$!33+++?/,i8!#D$\"3=+++h;)R&yF-7$$!3<+++7P6FoF@$\"32+++e\"yZ U*F-7$$!3>+++mI5/F!#C$\"31+++Ovb*4\"!#<7$$!3g*****RCDB%*)FK$\"3#****** \\_PmD\"FN7$$!3;+++%fVkb#!#B$\"30+++%Q=PT\"FN7$$!3C+++$*3CjkFW$\"31+++ (p,3d\"FN7$$!3'******H>0iY\"!#A$\"3'******z?\"*ys\"FN7$$!3%)*****\\)*[ \"3IF\\o$\"3'*******))\\*\\)=FN7$$!3%******H)R&ye&F\\o$\"3!******>8H@/ #FN7$$!3a******4[j=$*F\\o$\"3/+++oNK*>#FN7$$!3/+++V)zrN\"!#@$\"3%)**** *fiImN#FN7$$!3++++^+_(e\"Fap$\"30+++=n89DFN7$$!3))*****fwHP')*F\\o$\"3 A+++Xq(>n#FN7$$\"3,+++;EpK=Fap$\"3++++6?NIGFN7$$\"3]+++aML@'*Fap$\"3'* *****z>T&*)HFN7$$\"3!******papIv#!#?$\"3#******fE.*\\JFN7$$\"37+++lF3# ['F[r$\"3*)*****>ZH=J$FN7$$\"3#*******>D5v8!#>$\"33+++qGcvMFN7$$\"39++ +g:([r#Ffr$\"3\")******>rmSOFN7$$\"34+++#pui,&Ffr$\"3\"******\\*4![!QF N7$$\"3i*****zqhA`)Ffr$\"3#******fg*=iRFN7$$\"3(******4`\\EI\"F-$\"3B+ ++l0-0TFN7$$\"3))*****Rt(QvI'*[FN7$$\"3y*****>wIx)GF-$\" 3,+++O2MW\\FN7$$\"3++++EK<:FF-$\"3C+++([Q'))\\FN7$$\"3-+++*=t<^#F-$\"3 !)******ynKH]FN7$$\"3')*****f\\25G#F-$\"3q*****z(e^m]FN7$$\"30+++1R#f- #F-$\"3p*****f_*H+^FN7$$\"3-+++g3>\\=&FN7$$\"3Y**** *za&y4\")Ffr$\"3$)*****pr#y-_FN7$$\"3++++3?x\"o%Ffr$\"3))*****\\)*z/A& FN7$$\"3#******fC`t7\"Ffr$\"3y*****zMd]B&FN7$$!3(******R?f@a#Ffr$\"3%) *****HF9lC&FN7$$!3^*****>M)H5F-$ \"3++++x2)*f_FN7$$!32+++)pmbT\"F-$\"3%)*****HX**=E&FN7$$!33+++Z*G4#=F- $\"33+++;0^g_FN7$$!38+++\"Qy\\B#F-$\"3!******f72dD&FN7$$!31+++O\"\\yl# F-$\"3#******p\")\\tC&FN7$$!31+++l_FN7$$!3))*****\\03r)RF-$\"3s******[1#*)>&FN7$$! 3-+++D@ZcWF-$\"37+++Rb.u^FN7$$!39+++yO1X\\F-$\"34+++Yp7W^FN7$$!3u***** **H0*faF-$\"3f*****pE$p3^FN7$$!3b*****Rc\\A,'F-$\"3I+++(z4s1&FN7$$!3$* ******>t7?mF-$\"3O+++c/J>]FN7$$!3V+++hJN6tF-$\"3S+++\\!=`'\\FN7$$!3O++ +]DaA\")F-$\"3t*****Hgnu!\\FN7$$!3k*****RIa/3*F-$\"3%******zbK9&[FN7$$ !3#******48di,\"FN$\"3J+++lk#\\![FN7$$!3)******\\A5&H6FN$\"3c*****\\@1 >x%FN7$$!3'******f^;5C\"FN$\"3w*****>Ca0v%FN7$$!3)******Hs%eZ8FN$\"37+ ++JQ&pt%FN7$$!31+++%H0&[9FN$\"3J+++!Qzxs%FN7$$!30+++H14W:FN$\"3J+++j'o 2s%FN7$$!31+++uq%\\j\"FN$\"3w*****z[pWr%FN7$$!3-+++i:q@FN$\"3G+++R8-#o%FN7$$!3*******Rnex.#FN$\"3% ******HR9.n%FN7$$!3/+++ei\"46#FN$\"3;+++hb\"ol%FN7$$!3-+++%G!H#=#FN$\" 3F+++B(Q9k%FN7$$!35+++X>4_AFN$\"3(******R\"[7CYFN7$$!37+++Ai]?BFN$\"3% )******3U$[g%FN7$$!3!)******yxp(Q#FN$\"3w******>=a$e%FN7$$!3/+++^d\"QX #FN$\"3A+++jSBgXFN7$$!3>+++2w**=DFN$\"32+++nr!\\`%FN7$$!3')*****4AsLe# FN$\"3l*****>[mv]%FN7$$!3%******H?iqk#FN$\"3#)*****\\eE#yWFN7$$!3!)*** **>^&=5FFN$\"3y*****\\*>\"pW%FN7$$!3/+++7j&Gx#FN$\"3*******zQeOT%FN7$$ !3$******H*\\=NGFN$\"3-+++HT^yVFN7$$!33+++rsF(*GFN$\"3,+++T>aTVFN7$$!3 ))*****zHK#fHFN$\"3h******G/#GI%FN7$$!32+++W(R6-$FN$\"3E+++E^WiUFN7$$! 3'******H+wI3$FN$\"3*)*****RjG0A%FN7$$!3!)*****Rw*4XJFN$\"3)******>X*> xTFN7$$!3$*******zuC2KFN$\"3*)******\\\"*fKTFN7$$!39+++;%H&pKFN$\"3M++ +Tx(o3%FN7$$!3')*****H/F>L$FN$\"3()*****\\X(=SSFN7$$!3=+++([#R%R$FN$\" 3)*******ean#*RFN7$$!3#******fU]oX$FN$\"3!*******=kZWRFN7$$!32+++)Q-#> NFN$\"3!******4!fq&*QFN7$$!3%)*****f%HL\"e$FN$\"3=+++EbXYQFN7$$!33+++S p6VOFN$\"3%******>U!z'z$FN7$$!37+++hnU/PFN$\"39+++*3\\nu$FN7$$!3'***** **[)Q^w$FN$\"3?+++d`M'p$FN7$$!3@+++I)Q^#QFN$\"3=+++I=dXOFN7$$!33+++l]K %)QFN$\"3y******pTS%f$FN7$$!37+++l1hURFN$\"3$*******yb!Ga$FN7$$!3=+++6 U#***RFN$\"39+++Z5t!\\$FN7$$!3W+++*[4i0%FN$\"3?+++p58QMFN7$$!36+++wXU6 TFN$\"3)******\\ta\\Q$FN7$$!3G+++S0alTFN$\"3.+++o@:JLFN7$$!3k*****R'*R &=UFN$\"3)*******yjnwKFN7$$!3o*****zV:/F%FN$\"3z*****Rk%[@KFN7$$!3S+++ +\"o6K%FN$\"3')******)\\Rb;$FN7$$!3%******zE13P%FN$\"39+++[%4)3JFN7$$! 3F+++dTM>WFN$\"3/+++i%p70$FN7$$!3))*****\\p+oY%FN$\"3'*******R8!H*HFN7 $$!3#******\\Q)>8XFN$\"3++++YRpLHFN7$$!3++++7AceXFN$\"3#)*****zNVO(GFN 7$$!3e*****\\b=Hg%FN$\"3#)******HHv7GFN7$$!3-+++&4%HYYFN$\"35+++UK.^FF N7$$!3i*****Rr9()o%FN$\"3=+++D>])o#FN7$$!3,+++\"R/-t%FN$\"3,+++NL=DEFN 7$$!3S+++CTyqZFN$\"3#******z/36c#FN7$$!3j*****H\"4Z5[FN$\"3)*******[@J '\\#FN7$$!3/+++joF\\[FN$\"3\"*******Gj$3V#FN7$$!39+++]%3s)[FN$\"39+++# yCZO#FN7$$!3g*****\\.mU#\\FN$\"3++++fR-)H#FN7$$!3K+++IOWg\\FN$\"3/+++6 7yIAFN7$$!3%)******))*Gd*\\FN$\"3/+++CM/j@FN7$$!3m******))R5I]FN$\"3?+ ++'pb[4#FN7$$!3!)*****RIXN1&FN$\"3?+++c,EE?FN7$$!3I+++/`-'4&FN$\"35+++ $*\\Hd>FN7$$!3G+++FJ^F^FN$\"3/+++IP*z)=FN7$$!3S+++Pe(z:&FN$\"3,+++\"y% Q==FN7$$!3O+++1'zt=&FN$\"31+++&G\"\\[J$y;FN7$$!3$*******fr(GC&FN$\"33+++2w\"zg\"FN7$$!35+++f$3*o_FN$\"3 '*******)>fs`\"FN7$$!3g*****\\.dPH&FN$\"3++++()3Om9FN7$$!3M+++w\"*R<`F N$\"3#******>[C_R\"FN7$$!3A+++WX\")R`FN$\"3++++E%\\QK\"FN7$$!3y*****R! p)4O&FN$\"3++++CNB_7FN7$$!3%)*****f8/4Q&FN$\"3'******4mt.=\"FN7$$!3G++ +?#e&*R&FN$\"3*******R\\m#36FN7$$!3O+++@^%pT&FN$\"3%*******)44f.\"FN7$ $!3F+++&fkIV&FN$\"3L+++Xd*Hj*F-7$$!3w*****>!*>zW&FN$\"3)******H\"fP/*) F-7$$!33+++bu^haFN$\"3e*****>\"*[K<)F-7$$!3?+++)RmQZ&FN$\"3\"******f?d 'RuF-7$$!33+++?\"y\\[&FN$\"3M+++r>n.nF-7$$!3i*****frl[\\&FN$\"37+++\") fRlfF-7$$!3K+++%QVN]&FN$\"3)******Hdl\\A&F-7$$!3c******)zD5^&FN$\"3')* *****z:b#[%F-7$$!3Q+++duKn.nF-7$Fe jm$!3\"******f?d'RuF-7$F`jm$!3e*****>\"*[K<)F-7$F[jm$!3)******H\"fP/*) F-7$Ffim$!3L+++Xd*Hj*F-7$Faim$!3%*******)44f.\"FN7$F\\im$!3*******R\\m #36FN7$Fghm$!3'******4mt.=\"FN7$Fbhm$!3++++CNB_7FN7$F]hm$!3++++E%\\QK \"FN7$Fhgm$!3#******>[C_R\"FN7$Fcgm$!3++++()3Om9FN7$F^gm$!3'*******)>f s`\"FN7$Fifm$!33+++2w\"zg\"FN7$Fdfm$!3!********>J$y;FN7$F_fm$!31+++&G \"\\[FN7$F[em$!3?+++c,EE?FN7$Ffdm$!3?+++'pb[4#FN7$Fadm$!3/+++CM/j@FN7$F\\d m$!3/+++67yIAFN7$Fgcm$!3++++fR-)H#FN7$Fbcm$!39+++#yCZO#FN7$F]cm$!3\"** *****Gj$3V#FN7$Fhbm$!3)*******[@J'\\#FN7$Fcbm$!3#******z/36c#FN7$F^bm$ !3,+++NL=DEFN7$Fiam$!3=+++D>])o#FN7$Fdam$!35+++UK.^FFN7$F_am$!3#)***** *HHv7GFN7$Fj`m$!3#)*****zNVO(GFN7$Fe`m$!3++++YRpLHFN7$F``m$!3'*******R 8!H*HFN7$F[`m$!3/+++i%p70$FN7$Ff_m$!39+++[%4)3JFN7$Fa_m$!3')******)\\R b;$FN7$F\\_m$!3z*****Rk%[@KFN7$Fg^m$!3)*******yjnwKFN7$Fb^m$!3.+++o@:J LFN7$F]^m$!3)******\\ta\\Q$FN7$Fh]m$!3?+++p58QMFN7$Fc]m$!39+++Z5t!\\$F N7$F^]m$!3$*******yb!Ga$FN7$Fi\\m$!3y******pTS%f$FN7$Fd\\m$!3=+++I=dXO FN7$F_\\m$!3?+++d`M'p$FN7$Fj[m$!39+++*3\\nu$FN7$Fe[m$!3%******>U!z'z$F N7$F`[m$!3=+++EbXYQFN7$F[[m$!3!******4!fq&*QFN7$Ffjl$!3!*******=kZWRFN 7$Fajl$!3)*******ean#*RFN7$F\\jl$!3()*****\\X(=SSFN7$Fgil$!3M+++Tx(o3% FN7$Fbil$!3*)******\\\"*fKTFN7$F]il$!3)******>X*>xTFN7$Fhhl$!3*)*****R jG0A%FN7$Fchl$!3E+++E^WiUFN7$F^hl$!3h******G/#GI%FN7$Figl$!3,+++T>aTVF N7$Fdgl$!3-+++HT^yVFN7$F_gl$!3*******zQeOT%FN7$Fjfl$!3y*****\\*>\"pW%F N7$Fefl$!3#)*****\\eE#yWFN7$F`fl$!3l*****>[mv]%FN7$F[fl$!32+++nr!\\`%F N7$Ffel$!3A+++jSBgXFN7$Fael$!3w******>=a$e%FN7$F\\el$!3%)******3U$[g%F N7$Fgdl$!3(******R\"[7CYFN7$Fbdl$!3F+++B(Q9k%FN7$F]dl$!3;+++hb\"ol%FN7 $Fhcl$!3%******HR9.n%FN7$Fccl$!3G+++R8-#o%FN7$F^cl$!3%)*****4Ih?p%FN7$ Fibl$!3(******R3;1q%FN7$Fdbl$!3#*******RG&zq%FN7$F_bl$!3w*****z[pWr%FN 7$Fjal$!3J+++j'o2s%FN7$Feal$!3J+++!Qzxs%FN7$F`al$!37+++JQ&pt%FN7$F[al$ !3w*****>Ca0v%FN7$Ff`l$!3c*****\\@1>x%FN7$Fa`l$!3J+++lk#\\![FN7$F\\`l$ !3%******zbK9&[FN7$Fg_l$!3t*****Hgnu!\\FN7$Fb_l$!3S+++\\!=`'\\FN7$F]_l $!3O+++c/J>]FN7$Fh^l$!3I+++(z4s1&FN7$Fc^l$!3f*****pE$p3^FN7$F^^l$!34++ +Yp7W^FN7$Fi]l$!37+++Rb.u^FN7$Fd]l$!3s******[1#*)>&FN7$F_]l$!3/+++C9@> _FN7$Fj\\l$!3e******[*f_B&FN7$Fe\\l$!3#******p\")\\tC&FN7$F`\\l$!3!*** ***f72dD&FN7$F[\\l$!33+++;0^g_FN7$Ff[l$!3%)*****HX**=E&FN7$Fa[l$!3++++ x2)*f_FN7$F\\[l$!3!)*****p!Q$[D&FN7$Fgz$!3%)*****HF9lC&FN7$Fbz$!3y**** *zMd]B&FN7$F]z$!3))*****\\)*z/A&FN7$Fhy$!3$)*****pr#y-_FN7$Fcy$!33+++Z 5&>=&FN7$F^y$!3#*******3l&z:&FN7$Fix$!3y*****zWd28&FN7$Fdx$!3p*****f_* H+^FN7$F_x$!3q*****z(e^m]FN7$Fjw$!3!)******ynKH]FN7$Few$!3C+++([Q'))\\ FN7$F`w$!3,+++O2MW\\FN7$F[w$!3#)******G>I'*[FN7$Ffv$!3q*****4]gV%[FN7$ Fav$!3y********)4$)y%FN7$F\\v$!3\")******yH(ys%FN7$$\"3x*******QQ'GJF- $!35+++P(eEm%FN7$Fbu$!33+++U&*3#f%FN7$F]u$!37+++#Qz_^%FN7$Fht$!3!)**** **Hc$3V%FN7$Fct$!3h*****peplL%FN7$F^t$!3/+++b(R#HUFN7$Fis$!3B+++l0-0TF N7$Fds$!3#******fg*=iRFN7$F_s$!3\"******\\*4![!QFN7$Fjr$!3\")******>rm SOFN7$Fdr$!33+++qGcvMFN7$F_r$!3*)*****>ZH=J$FN7$Fiq$!3#******fE.*\\JFN 7$Fdq$!3'******z>T&*)HFN7$F_q$!3++++6?NIGFN7$Fjp$!3A+++Xq(>n#FN7$Fep$! 30+++=n89DFN7$F_p$!3%)*****fiImN#FN7$Fjo$!3/+++oNK*>#FN7$Feo$!3!****** >8H@/#FN7$F`o$!3'*******))\\*\\)=FN7$Fjn$!3'******z?\"*ys\"FN7$$!3u*** **>!4CjkFW$!31+++(p,3d\"FN7$$!31+++&fVkb#FW$!30+++%Q=PT\"FN7$FP$!3#*** ***\\_PmD\"FN7$$!3))*****p1.Tq#FK$!31+++Ovb*4\"FN7$$!3l+++8P6FoF@$!32+ ++e\"yZU*F-7$F>$!3=+++h;)R&yF-7$$!3))*****4Rw_&>F:$!37+++5`=$G'F-7$$!3 3+++Z2>*o\"F4$!3<+++!)*)Q7ZF-7$F($!3)******Rl#fTJF-7$F($!3++++Fjzq:F-F '-%&COLORG6&%$RGBGF($\"\"#!\"\"F(-%)POLYGONSG6^_l7%F'7$F($\"+Fjzq:!#57 $$!#YFj\\pF(7%F_]p7$F($\"+aEfTJFb]pFc]p7%Fg]p7$$!+f2>*o\"Ffr$\"+!)*)Q7 ZFb]pFc]p7%F[^p7$$!+$Rw_&>F-$\"+5`=$G'Fb]pFc]p7%Fa^p7$$!+?/,i8FN$\"+h; )R&yFb]pFc]p7%Fg^p7$$!+7P6FoFN$\"+e\"yZU*Fb]pFc]p7%F]_p7$$!+mI5/F!#;$ \"+Ovb*4\"!\"*Fc]p7%Fc_p7$$!+W_KU*)Ff_p$\"+Dvjc7Fi_pFc]p7%F[`p7$$!+%fV kb#!#:$\"+%Q=PT\"Fi_pFc]p7%Fa`p7$$!+$*3CjkFd`p$\"+(p,3d\"Fi_pFc]p7%Fh` p7$$!+$>0iY\"!#9$\"+37*ys\"Fi_pFc]p7%F^ap7$$!+&)*[\"3IFaap$\"+*)\\*\\) =Fi_pFc]p7%Feap7$$!+$)R&ye&Faap$\"+K\"H@/#Fi_pFc]p7%F[bp7$$!+5[j=$*Faa p$\"+oNK*>#Fi_pFc]p7%Fabp7$$!+V)zrN\"!#8$\"+E1jcBFi_pFc]p7%Fgbp7$$!+^+ _(e\"Fjbp$\"+=n89DFi_pFc]p7%F^cp7$$!+m(HP')*Faap$\"+Xq(>n#Fi_pFc]p7%Fd cp7$$\"+;EpK=Fjbp$\"+6?NIGFi_pFc]p7%Fjcp7$$\"+aML@'*Fjbp$\"+)>T&*)HFi_ pFc]p7%F`dp7$$\"+Z&pIv#!#7$\"+mK!*\\JFi_pFc]p7%Ffdp7$$\"+lF3#['Fidp$\" +s%H=J$Fi_pFc]p7%F]ep7$$\"+?D5v8!#6$\"+qGcvMFi_pFc]p7%Fcep7$$\"+g:([r# Ffep$\"+?rmSOFi_pFc]p7%Fjep7$$\"+#pui,&Ffep$\"+&*4![!QFi_pFc]p7%F`fp7$ $\"+3I'*[Fi_pFc]p7%Fhjp7$$\"+i2t() GFb]p$\"+O2MW\\Fi_pFc]p7%F^[q7$$\"+EK<:FFb]p$\"+([Q'))\\Fi_pFc]p7%Fd[q 7$$\"+*=t<^#Fb]p$\"+znKH]Fi_pFc]p7%Fj[q7$$\"+'\\25G#Fb]p$\"+ye^m]Fi_pF c]p7%F`\\q7$$\"+1R#f-#Fb]p$\"+E&*H+^Fi_pFc]p7%Ff\\q7$$\"+g3>\\=&Fi_pFc]p7%Fh]q7$$\"+[by4\")Ffep$\"+5Fb]p$\"+x2)*f_Fi_pFc]p7%F\\`q7$$!+)pmbT\"F b]p$\"+`%**=E&Fi_pFc]p7%Fb`q7$$!+Z*G4#=Fb]p$\"+;0^g_Fi_pFc]p7%Fh`q7$$! +\"Qy\\B#Fb]p$\"+Erqb_Fi_pFc]p7%F^aq7$$!+O\"\\yl#Fb]p$\"+<)\\tC&Fi_pFc ]p7%Fdaq7$$!+l _Fi_pFc]p7%F`bq7$$!+b!3r)RFb]p$\"+\\1#*)>&Fi_pFc]p7%Ffbq7$$!+D@ZcWFb]p $\"+Rb.u^Fi_pFc]p7%F\\cq7$$!+yO1X\\Fb]p$\"+Yp7W^Fi_pFc]p7%Fbcq7$$!++`! *faFb]p$\"+nKp3^Fi_pFc]p7%Fhcq7$$!+k&\\A,'Fb]p$\"+(z4s1&Fi_pFc]p7%F^dq 7$$!+?t7?mFb]p$\"+c/J>]Fi_pFc]p7%Fddq7$$!+hJN6tFb]p$\"+\\!=`'\\Fi_pFc] p7%Fjdq7$$!+]DaA\")Fb]p$\"+.wY2\\Fi_pFc]p7%F`eq7$$!+/VX!3*Fb]p$\"+eDV^ [Fi_pFc]p7%Ffeq7$$!+JrD;5Fi_p$\"+lk#\\![Fi_pFc]p7%F\\fq7$$!+D-^H6Fi_p$ \"+:i!>x%Fi_pFc]p7%Fbfq7$$!+;l,T7Fi_p$\"+UUb]ZFi_pFc]p7%Fhfq7$$!+BZeZ8 Fi_p$\"+JQ&pt%Fi_pFc]p7%F^gq7$$!+%H0&[9Fi_p$\"+!Qzxs%Fi_pFc]p7%Fdgq7$$ !+H14W:Fi_p$\"+j'o2s%Fi_pFc]p7%Fjgq7$$!+uq%\\j\"Fi_p$\"+)[pWr%Fi_pFc]p 7%F`hq7$$!+i:q@Fi_p $\"+R8-#o%Fi_pFc]p7%Fhiq7$$!+u'ex.#Fi_p$\"+$R9.n%Fi_pFc]p7%F^jq7$$!+ei \"46#Fi_p$\"+hb\"ol%Fi_pFc]p7%Fdjq7$$!+%G!H#=#Fi_p$\"+B(Q9k%Fi_pFc]p7% Fjjq7$$!+X>4_AFi_p$\"+9[7CYFi_pFc]p7%F`[r7$$!+Ai]?BFi_p$\"+4U$[g%Fi_pF c]p7%Ff[r7$$!+zxp(Q#Fi_p$\"+?=a$e%Fi_pFc]p7%F\\\\r7$$!+^d\"QX#Fi_p$\"+ jSBgXFi_pFc]p7%Fb\\r7$$!+2w**=DFi_p$\"+nr!\\`%Fi_pFc]p7%Fh\\r7$$!+@AP$ e#Fi_p$\"+#[mv]%Fi_pFc]p7%F^]r7$$!+.A1ZEFi_p$\"+&eE#yWFi_pFc]p7%Fd]r7$ $!+7b=5FFi_p$\"+&*>\"pW%Fi_pFc]p7%Fj]r7$$!+7j&Gx#Fi_p$\"+)QeOT%Fi_pFc] p7%F`^r7$$!+$*\\=NGFi_p$\"+HT^yVFi_pFc]p7%Ff^r7$$!+rsF(*GFi_p$\"+T>aTV Fi_pFc]p7%F\\_r7$$!+)HK#fHFi_p$\"+H/#GI%Fi_pFc]p7%Fb_r7$$!+W(R6-$Fi_p$ \"+E^WiUFi_pFc]p7%Fh_r7$$!+.g2$3$Fi_p$\"+M'G0A%Fi_pFc]p7%F^`r7$$!+k(*4 XJFi_p$\"+_%*>xTFi_pFc]p7%Fd`r7$$!+![Zs?$Fi_p$\"+]\"*fKTFi_pFc]p7%Fj`r 7$$!+;%H&pKFi_p$\"+Tx(o3%Fi_pFc]p7%F`ar7$$!+Vq#>L$Fi_p$\"+bu=SSFi_pFc] p7%Ffar7$$!+([#R%R$Fi_p$\"+fan#*RFi_pFc]p7%F\\br7$$!+E/&oX$Fi_p$\"+>kZ WRFi_pFc]p7%Fbbr7$$!+)Q-#>NFi_p$\"+,fq&*QFi_pFc]p7%Fhbr7$$!+YHL\"e$Fi_ p$\"+EbXYQFi_pFc]p7%F^cr7$$!+Sp6VOFi_p$\"+A/z'z$Fi_pFc]p7%Fdcr7$$!+hnU /PFi_p$\"+*3\\nu$Fi_pFc]p7%Fjcr7$$!+\\)Q^w$Fi_p$\"+d`M'p$Fi_pFc]p7%F`d r7$$!+I)Q^#QFi_p$\"+I=dXOFi_pFc]p7%Ffdr7$$!+l]K%)QFi_p$\"+qTS%f$Fi_pFc ]p7%F\\er7$$!+l1hURFi_p$\"+zb!Ga$Fi_pFc]p7%Fber7$$!+6U#***RFi_p$\"+Z5t !\\$Fi_pFc]p7%Fher7$$!+*[4i0%Fi_p$\"+p58QMFi_pFc]p7%F^fr7$$!+wXU6TFi_p $\"+NZ&\\Q$Fi_pFc]p7%Fdfr7$$!+S0alTFi_p$\"+o@:JLFi_pFc]p7%Fjfr7$$!+k*R &=UFi_p$\"+zjnwKFi_pFc]p7%F`gr7$$!+QaTqUFi_p$\"+WY[@KFi_pFc]p7%Ffgr7$$ !++\"o6K%Fi_p$\"+*\\Rb;$Fi_pFc]p7%F\\hr7$$!+oi!3P%Fi_p$\"+[%4)3JFi_pFc ]p7%Fbhr7$$!+dTM>WFi_p$\"+i%p70$Fi_pFc]p7%Fhhr7$$!+&p+oY%Fi_p$\"+S8!H* HFi_pFc]p7%F^ir7$$!+&Q)>8XFi_p$\"+YRpLHFi_pFc]p7%Fdir7$$!+7AceXFi_p$\" +eLktGFi_pFc]p7%Fjir7$$!+b&=Hg%Fi_p$\"+IHv7GFi_pFc]p7%F`jr7$$!+&4%HYYF i_p$\"+UK.^FFi_pFc]p7%Ffjr7$$!+9Zr)o%Fi_p$\"+D>])o#Fi_pFc]p7%F\\[s7$$! +\"R/-t%Fi_p$\"+NL=DEFi_pFc]p7%Fb[s7$$!+CTyqZFi_p$\"+[!36c#Fi_pFc]p7%F h[s7$$!+84Z5[Fi_p$\"+\\@J'\\#Fi_pFc]p7%F^\\s7$$!+joF\\[Fi_p$\"+Hj$3V#F i_pFc]p7%Fd\\s7$$!+]%3s)[Fi_p$\"+#yCZO#Fi_pFc]p7%Fj\\s7$$!+NgEC\\Fi_p$ \"+fR-)H#Fi_pFc]p7%F`]s7$$!+IOWg\\Fi_p$\"+67yIAFi_pFc]p7%Ff]s7$$!+*)*G d*\\Fi_p$\"+CM/j@Fi_pFc]p7%F\\^s7$$!+*)R5I]Fi_p$\"+'pb[4#Fi_pFc]p7%Fb^ s7$$!+/`aj]Fi_p$\"+c,EE?Fi_pFc]p7%Fh^s7$$!+/`-'4&Fi_p$\"+$*\\Hd>Fi_pFc ]p7%F^_s7$$!+FJ^F^Fi_p$\"+IP*z)=Fi_pFc]p7%Fd_s7$$!+Pe(z:&Fi_p$\"+\"y%Q ==Fi_pFc]p7%Fj_s7$$!+1'zt=&Fi_p$\"+&G\"\\[fs`\"Fi_pFc]p7%Fbas7$$!+Nqv$H&Fi_p$\"+()3Om9Fi_pFc]p7 %Fhas7$$!+w\"*R<`Fi_p$\"+#[C_R\"Fi_pFc]p7%F^bs7$$!+WX\")R`Fi_p$\"+E%\\ QK\"Fi_pFc]p7%Fdbs7$$!+/p)4O&Fi_p$\"+CNB_7Fi_pFc]p7%Fjbs7$$!+OT!4Q&Fi_ p$\"+hOP!=\"Fi_pFc]p7%F`cs7$$!+?#e&*R&Fi_p$\"+%\\m#36Fi_pFc]p7%Ffcs7$$ !+@^%pT&Fi_p$\"+*44f.\"Fi_pFc]p7%F\\ds7$$!+&fkIV&Fi_p$\"+Xd*Hj*Fb]pFc] p7%Fbds7$$!+-*>zW&Fi_p$\"+8fP/*)Fb]pFc]p7%Fhds7$$!+bu^haFi_p$\"+7*[K<) Fb]pFc]p7%F^es7$$!+)RmQZ&Fi_p$\"+1slRuFb]pFc]p7%Fdes7$$!+?\"y\\[&Fi_p$ \"+r>n.nFb]pFc]p7%Fjes7$$!+;d'[\\&Fi_p$\"+\")fRlfFb]pFc]p7%F`fs7$$!+%Q VN]&Fi_p$\"+tb'\\A&Fb]pFc]p7%Fffs7$$!+*zD5^&Fi_p$\"+!e^D[%Fb]pFc]p7%F \\gs7$$!+duKn.nFb]pFc]p7%Fd [t7$Fees$!+1slRuFb]pFc]p7%Fh[t7$F_es$!+7*[K<)Fb]pFc]p7%F\\\\t7$Fids$!+ 8fP/*)Fb]pFc]p7%F`\\t7$Fcds$!+Xd*Hj*Fb]pFc]p7%Fd\\t7$F]ds$!+*44f.\"Fi_ pFc]p7%Fh\\t7$Fgcs$!+%\\m#36Fi_pFc]p7%F\\]t7$Facs$!+hOP!=\"Fi_pFc]p7%F `]t7$F[cs$!+CNB_7Fi_pFc]p7%Fd]t7$Febs$!+E%\\QK\"Fi_pFc]p7%Fh]t7$F_bs$! +#[C_R\"Fi_pFc]p7%F\\^t7$Fias$!+()3Om9Fi_pFc]p7%F`^t7$Fcas$!+*>fs`\"Fi _pFc]p7%Fd^t7$F]as$!+2w\"zg\"Fi_pFc]p7%Fh^t7$Fg`s$!++7Ly;Fi_pFc]p7%F\\ _t7$Fa`s$!+&G\"\\[Fi_pFc]p7%F\\`t7$Fi^s$!+c,EE?Fi_ pFc]p7%F``t7$Fc^s$!+'pb[4#Fi_pFc]p7%Fd`t7$F]^s$!+CM/j@Fi_pFc]p7%Fh`t7$ Fg]s$!+67yIAFi_pFc]p7%F\\at7$Fa]s$!+fR-)H#Fi_pFc]p7%F`at7$F[]s$!+#yCZO #Fi_pFc]p7%Fdat7$Fe\\s$!+Hj$3V#Fi_pFc]p7%Fhat7$F_\\s$!+\\@J'\\#Fi_pFc] p7%F\\bt7$Fi[s$!+[!36c#Fi_pFc]p7%F`bt7$Fc[s$!+NL=DEFi_pFc]p7%Fdbt7$F][ s$!+D>])o#Fi_pFc]p7%Fhbt7$Fgjr$!+UK.^FFi_pFc]p7%F\\ct7$Fajr$!+IHv7GFi_ pFc]p7%F`ct7$F[jr$!+eLktGFi_pFc]p7%Fdct7$Feir$!+YRpLHFi_pFc]p7%Fhct7$F _ir$!+S8!H*HFi_pFc]p7%F\\dt7$Fihr$!+i%p70$Fi_pFc]p7%F`dt7$Fchr$!+[%4)3 JFi_pFc]p7%Fddt7$F]hr$!+*\\Rb;$Fi_pFc]p7%Fhdt7$Fggr$!+WY[@KFi_pFc]p7%F \\et7$Fagr$!+zjnwKFi_pFc]p7%F`et7$F[gr$!+o@:JLFi_pFc]p7%Fdet7$Fefr$!+N Z&\\Q$Fi_pFc]p7%Fhet7$F_fr$!+p58QMFi_pFc]p7%F\\ft7$Fier$!+Z5t!\\$Fi_pF c]p7%F`ft7$Fcer$!+zb!Ga$Fi_pFc]p7%Fdft7$F]er$!+qTS%f$Fi_pFc]p7%Fhft7$F gdr$!+I=dXOFi_pFc]p7%F\\gt7$Fadr$!+d`M'p$Fi_pFc]p7%F`gt7$F[dr$!+*3\\nu $Fi_pFc]p7%Fdgt7$Fecr$!+A/z'z$Fi_pFc]p7%Fhgt7$F_cr$!+EbXYQFi_pFc]p7%F \\ht7$Fibr$!+,fq&*QFi_pFc]p7%F`ht7$Fcbr$!+>kZWRFi_pFc]p7%Fdht7$F]br$!+ fan#*RFi_pFc]p7%Fhht7$Fgar$!+bu=SSFi_pFc]p7%F\\it7$Faar$!+Tx(o3%Fi_pFc ]p7%F`it7$F[ar$!+]\"*fKTFi_pFc]p7%Fdit7$Fe`r$!+_%*>xTFi_pFc]p7%Fhit7$F _`r$!+M'G0A%Fi_pFc]p7%F\\jt7$Fi_r$!+E^WiUFi_pFc]p7%F`jt7$Fc_r$!+H/#GI% Fi_pFc]p7%Fdjt7$F]_r$!+T>aTVFi_pFc]p7%Fhjt7$Fg^r$!+HT^yVFi_pFc]p7%F\\[ u7$Fa^r$!+)QeOT%Fi_pFc]p7%F`[u7$F[^r$!+&*>\"pW%Fi_pFc]p7%Fd[u7$Fe]r$!+ &eE#yWFi_pFc]p7%Fh[u7$F_]r$!+#[mv]%Fi_pFc]p7%F\\\\u7$Fi\\r$!+nr!\\`%Fi _pFc]p7%F`\\u7$Fc\\r$!+jSBgXFi_pFc]p7%Fd\\u7$F]\\r$!+?=a$e%Fi_pFc]p7%F h\\u7$Fg[r$!+4U$[g%Fi_pFc]p7%F\\]u7$Fa[r$!+9[7CYFi_pFc]p7%F`]u7$F[[r$! +B(Q9k%Fi_pFc]p7%Fd]u7$Fejq$!+hb\"ol%Fi_pFc]p7%Fh]u7$F_jq$!+$R9.n%Fi_p Fc]p7%F\\^u7$Fiiq$!+R8-#o%Fi_pFc]p7%F`^u7$Fciq$!+,81#p%Fi_pFc]p7%Fd^u7 $F]iq$!+%3;1q%Fi_pFc]p7%Fh^u7$Fghq$!+SG&zq%Fi_pFc]p7%F\\_u7$Fahq$!+)[p Wr%Fi_pFc]p7%F`_u7$F[hq$!+j'o2s%Fi_pFc]p7%Fd_u7$Fegq$!+!Qzxs%Fi_pFc]p7 %Fh_u7$F_gq$!+JQ&pt%Fi_pFc]p7%F\\`u7$Fifq$!+UUb]ZFi_pFc]p7%F``u7$Fcfq$ !+:i!>x%Fi_pFc]p7%Fd`u7$F]fq$!+lk#\\![Fi_pFc]p7%Fh`u7$Fgeq$!+eDV^[Fi_p Fc]p7%F\\au7$Faeq$!+.wY2\\Fi_pFc]p7%F`au7$F[eq$!+\\!=`'\\Fi_pFc]p7%Fda u7$Fedq$!+c/J>]Fi_pFc]p7%Fhau7$F_dq$!+(z4s1&Fi_pFc]p7%F\\bu7$Ficq$!+nK p3^Fi_pFc]p7%F`bu7$Fccq$!+Yp7W^Fi_pFc]p7%Fdbu7$F]cq$!+Rb.u^Fi_pFc]p7%F hbu7$Fgbq$!+\\1#*)>&Fi_pFc]p7%F\\cu7$Fabq$!+C9@>_Fi_pFc]p7%F`cu7$F[bq$ !+\\*f_B&Fi_pFc]p7%Fdcu7$Feaq$!+<)\\tC&Fi_pFc]p7%Fhcu7$F_aq$!+Erqb_Fi_ pFc]p7%F\\du7$Fi`q$!+;0^g_Fi_pFc]p7%F`du7$Fc`q$!+`%**=E&Fi_pFc]p7%Fddu 7$F]`q$!+x2)*f_Fi_pFc]p7%Fhdu7$Fg_q$!+2Q$[D&Fi_pFc]p7%F\\eu7$Fa_q$!+tU ^Y_Fi_pFc]p7%F`eu7$F[_q$!+[t0N_Fi_pFc]p7%Fdeu7$Fe^q$!+&)*z/A&Fi_pFc]p7 %Fheu7$F_^q$!+=&Fi_pFc]p7%F`fu7$Fc]q$ !+4l&z:&Fi_pFc]p7%Fdfu7$F]]q$!+[uvI^Fi_pFc]p7%Fhfu7$Fg\\q$!+E&*H+^Fi_p Fc]p7%F\\gu7$Fa\\q$!+ye^m]Fi_pFc]p7%F`gu7$F[\\q$!+znKH]Fi_pFc]p7%Fdgu7 $Fe[q$!+([Q'))\\Fi_pFc]p7%Fhgu7$F_[q$!+O2MW\\Fi_pFc]p7%F\\hu7$Fijp$!+H >I'*[Fi_pFc]p7%F`hu7$Fcjp$!+,0OW[Fi_pFc]p7%Fdhu7$F]jp$!++*4$)y%Fi_pFc] p7%Fhhu7$Fgip$!+zH(ys%Fi_pFc]p7%F\\iu7$$\"+!RQ'GJFb]p$!+P(eEm%Fi_pFc]p 7%F`iu7$F[ip$!+U&*3#f%Fi_pFc]p7%Ffiu7$Fehp$!+#Qz_^%Fi_pFc]p7%Fjiu7$F_h p$!+Ic$3V%Fi_pFc]p7%F^ju7$Figp$!+(eplL%Fi_pFc]p7%Fbju7$Fcgp$!+b(R#HUFi _pFc]p7%Ffju7$F]gp$!+l0-0TFi_pFc]p7%Fjju7$Fgfp$!+1'*=iRFi_pFc]p7%F^[v7 $Fafp$!+&*4![!QFi_pFc]p7%Fb[v7$F[fp$!+?rmSOFi_pFc]p7%Ff[v7$Fdep$!+qGcv MFi_pFc]p7%Fj[v7$F^ep$!+s%H=J$Fi_pFc]p7%F^\\v7$Fgdp$!+mK!*\\JFi_pFc]p7 %Fb\\v7$Fadp$!+)>T&*)HFi_pFc]p7%Ff\\v7$F[dp$!+6?NIGFi_pFc]p7%Fj\\v7$Fe cp$!+Xq(>n#Fi_pFc]p7%F^]v7$F_cp$!+=n89DFi_pFc]p7%Fb]v7$Fhbp$!+E1jcBFi_ pFc]p7%Ff]v7$Fbbp$!+oNK*>#Fi_pFc]p7%Fj]v7$F\\bp$!+K\"H@/#Fi_pFc]p7%F^^ v7$Ffap$!+*)\\*\\)=Fi_pFc]p7%Fb^v7$F_ap$!+37*ys\"Fi_pFc]p7%Ff^v7$$!+-4 CjkFd`p$!+(p,3d\"Fi_pFc]p7%Fj^v7$$!+&fVkb#Fd`p$!+%Q=PT\"Fi_pFc]p7%F`_v 7$F\\`p$!+Dvjc7Fi_pFc]p7%Ff_v7$$!+nI5/FFf_p$!+Ovb*4\"Fi_pFc]p7%Fj_v7$$ !+8P6FoFN$!+e\"yZU*Fb]pFc]p7%F``v7$Fh^p$!+h;)R&yFb]pFc]p7%Ff`v7$$!+\"R w_&>F-$!+5`=$G'Fb]pFc]p7%Fj`v7$$!+Z2>*o\"Ffr$!+!)*)Q7ZFb]pFc]p7%F`av7$ F($!+aEfTJFb]pFc]p7%Ffav7$F($!+Fjzq:Fb]pFc]p7%FjavF'Fc]p-Fe\\p6&Fg\\pF h\\p$\"#b!\"#$\"#DFbbv-%&STYLEG6#%,PATCHNOGRIDG-F$6%7$7$$!3A++++++!\\' FNF(7$$\"3%**************=\"FNF(-%'COLOURG6&Fg\\pF)F)F)-%*LINESTYLEG6# \"\"$-F$6%7$7$F($!3o*************Q'FN7$F($\"3o*************Q'FNFbcvFec v-%(SCALINGG6#%,CONSTRAINEDG-%+AXESLABELSG6%%&Re(z)G%&Im(z)G-%%FONTG6# %(DEFAULTG-F\\ev6$%*HELVETICAG\"\"*-%*AXESSTYLEG6#%$BOXG-%%VIEWG6$;$!$ \\'Fbbv$\"$>\"Fbbv;$!$R'Fbbv$\"$R'Fbbv" 1 2 0 1 10 0 2 9 1 2 1 1.000000 45.000000 45.000000 0 0 "Curve 1" "Curve 2" "Curve 3" "Curve \+ 4" }}}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 75 "Th e stability interval for the embedded order 7 scheme is (approximately ) " }{XPPEDIT 18 0 "[-5.5315, 0];" "6#7$,$-%&FloatG6$\"&:`&!\"%!\"\" \"\"!" }{TEXT -1 2 ". " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 79 "The stability regions for the two schemes can be dis played in the same picture." }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 154 "plots[display]([p_1,p_2,p||(1..3)],view=[-6.49..1.19,-6.39..6.39] ,font=[HELVETICA,9],\n labels=[`Re(z)`,`Im(z)`],axes=boxed,s caling=constrained);" }}{PARA 13 "" 1 "" {GLPLOT2D 362 580 580 {PLOTDATA 2 "6--%'CURVESG6$7]_l7$$\"\"!F)F(7$F($\"3++++Fjzq:!#=7$F($\" 3)******Rl#fTJF-7$$!3'*******e2>*o\"!#F$\"3<+++!)*)Q7ZF-7$$!3=+++$Rw_& >!#E$\"37+++5`=$G'F-7$$!33+++?/,i8!#D$\"3=+++h;)R&yF-7$$!3<+++7P6FoF@$ \"32+++e\"yZU*F-7$$!3>+++mI5/F!#C$\"31+++Ovb*4\"!#<7$$!3g*****RCDB%*)F K$\"3#******\\_PmD\"FN7$$!3;+++%fVkb#!#B$\"30+++%Q=PT\"FN7$$!3C+++$*3C jkFW$\"31+++(p,3d\"FN7$$!3'******H>0iY\"!#A$\"3'******z?\"*ys\"FN7$$!3 %)*****\\)*[\"3IF\\o$\"3'*******))\\*\\)=FN7$$!3%******H)R&ye&F\\o$\"3 !******>8H@/#FN7$$!3a******4[j=$*F\\o$\"3/+++oNK*>#FN7$$!3/+++V)zrN\"! #@$\"3%)*****fiImN#FN7$$!3++++^+_(e\"Fap$\"30+++=n89DFN7$$!3))*****fwH P')*F\\o$\"3A+++Xq(>n#FN7$$\"3,+++;EpK=Fap$\"3++++6?NIGFN7$$\"3]+++aML @'*Fap$\"3'******z>T&*)HFN7$$\"3!******papIv#!#?$\"3#******fE.*\\JFN7$ $\"37+++lF3#['F[r$\"3*)*****>ZH=J$FN7$$\"3#*******>D5v8!#>$\"33+++qGcv MFN7$$\"39+++g:([r#Ffr$\"3\")******>rmSOFN7$$\"34+++#pui,&Ffr$\"3\"*** ***\\*4![!QFN7$$\"3i*****zqhA`)Ffr$\"3#******fg*=iRFN7$$\"3(******4`\\ EI\"F-$\"3B+++l0-0TFN7$$\"3))*****Rt(QvI'*[FN7$$\"3y*** **>wIx)GF-$\"3,+++O2MW\\FN7$$\"3++++EK<:FF-$\"3C+++([Q'))\\FN7$$\"3-++ +*=t<^#F-$\"3!)******ynKH]FN7$$\"3')*****f\\25G#F-$\"3q*****z(e^m]FN7$ $\"30+++1R#f-#F-$\"3p*****f_*H+^FN7$$\"3-+++g3>\\=& FN7$$\"3Y*****za&y4\")Ffr$\"3$)*****pr#y-_FN7$$\"3++++3?x\"o%Ffr$\"3)) *****\\)*z/A&FN7$$\"3#******fC`t7\"Ffr$\"3y*****zMd]B&FN7$$!3(******R? f@a#Ffr$\"3%)*****HF9lC&FN7$$!3^*****>M)H5F-$\"3++++x2)*f_FN7$$!32+++)pmbT\"F-$\"3%)*****HX**=E&FN7$$! 33+++Z*G4#=F-$\"33+++;0^g_FN7$$!38+++\"Qy\\B#F-$\"3!******f72dD&FN7$$! 31+++O\"\\yl#F-$\"3#******p\")\\tC&FN7$$!31+++l_FN7$$!3))*****\\03r)RF-$\"3s****** [1#*)>&FN7$$!3-+++D@ZcWF-$\"37+++Rb.u^FN7$$!39+++yO1X\\F-$\"34+++Yp7W^ FN7$$!3u*******H0*faF-$\"3f*****pE$p3^FN7$$!3b*****Rc\\A,'F-$\"3I+++(z 4s1&FN7$$!3$*******>t7?mF-$\"3O+++c/J>]FN7$$!3V+++hJN6tF-$\"3S+++\\!=` '\\FN7$$!3O+++]DaA\")F-$\"3t*****Hgnu!\\FN7$$!3k*****RIa/3*F-$\"3%**** **zbK9&[FN7$$!3#******48di,\"FN$\"3J+++lk#\\![FN7$$!3)******\\A5&H6FN$ \"3c*****\\@1>x%FN7$$!3'******f^;5C\"FN$\"3w*****>Ca0v%FN7$$!3)******H s%eZ8FN$\"37+++JQ&pt%FN7$$!31+++%H0&[9FN$\"3J+++!Qzxs%FN7$$!30+++H14W: FN$\"3J+++j'o2s%FN7$$!31+++uq%\\j\"FN$\"3w*****z[pWr%FN7$$!3-+++i:q@FN$\"3G+++R8-#o%FN7$$!3******* Rnex.#FN$\"3%******HR9.n%FN7$$!3/+++ei\"46#FN$\"3;+++hb\"ol%FN7$$!3-++ +%G!H#=#FN$\"3F+++B(Q9k%FN7$$!35+++X>4_AFN$\"3(******R\"[7CYFN7$$!37++ +Ai]?BFN$\"3%)******3U$[g%FN7$$!3!)******yxp(Q#FN$\"3w******>=a$e%FN7$ $!3/+++^d\"QX#FN$\"3A+++jSBgXFN7$$!3>+++2w**=DFN$\"32+++nr!\\`%FN7$$!3 ')*****4AsLe#FN$\"3l*****>[mv]%FN7$$!3%******H?iqk#FN$\"3#)*****\\eE#y WFN7$$!3!)*****>^&=5FFN$\"3y*****\\*>\"pW%FN7$$!3/+++7j&Gx#FN$\"3***** **zQeOT%FN7$$!3$******H*\\=NGFN$\"3-+++HT^yVFN7$$!33+++rsF(*GFN$\"3,++ +T>aTVFN7$$!3))*****zHK#fHFN$\"3h******G/#GI%FN7$$!32+++W(R6-$FN$\"3E+ ++E^WiUFN7$$!3'******H+wI3$FN$\"3*)*****RjG0A%FN7$$!3!)*****Rw*4XJFN$ \"3)******>X*>xTFN7$$!3$*******zuC2KFN$\"3*)******\\\"*fKTFN7$$!39+++; %H&pKFN$\"3M+++Tx(o3%FN7$$!3')*****H/F>L$FN$\"3()*****\\X(=SSFN7$$!3=+ ++([#R%R$FN$\"3)*******ean#*RFN7$$!3#******fU]oX$FN$\"3!*******=kZWRFN 7$$!32+++)Q-#>NFN$\"3!******4!fq&*QFN7$$!3%)*****f%HL\"e$FN$\"3=+++EbX YQFN7$$!33+++Sp6VOFN$\"3%******>U!z'z$FN7$$!37+++hnU/PFN$\"39+++*3\\nu $FN7$$!3'*******[)Q^w$FN$\"3?+++d`M'p$FN7$$!3@+++I)Q^#QFN$\"3=+++I=dXO FN7$$!33+++l]K%)QFN$\"3y******pTS%f$FN7$$!37+++l1hURFN$\"3$*******yb!G a$FN7$$!3=+++6U#***RFN$\"39+++Z5t!\\$FN7$$!3W+++*[4i0%FN$\"3?+++p58QMF N7$$!36+++wXU6TFN$\"3)******\\ta\\Q$FN7$$!3G+++S0alTFN$\"3.+++o@:JLFN7 $$!3k*****R'*R&=UFN$\"3)*******yjnwKFN7$$!3o*****zV:/F%FN$\"3z*****Rk% [@KFN7$$!3S++++\"o6K%FN$\"3')******)\\Rb;$FN7$$!3%******zE13P%FN$\"39+ ++[%4)3JFN7$$!3F+++dTM>WFN$\"3/+++i%p70$FN7$$!3))*****\\p+oY%FN$\"3'** *****R8!H*HFN7$$!3#******\\Q)>8XFN$\"3++++YRpLHFN7$$!3++++7AceXFN$\"3# )*****zNVO(GFN7$$!3e*****\\b=Hg%FN$\"3#)******HHv7GFN7$$!3-+++&4%HYYFN $\"35+++UK.^FFN7$$!3i*****Rr9()o%FN$\"3=+++D>])o#FN7$$!3,+++\"R/-t%FN$ \"3,+++NL=DEFN7$$!3S+++CTyqZFN$\"3#******z/36c#FN7$$!3j*****H\"4Z5[FN$ \"3)*******[@J'\\#FN7$$!3/+++joF\\[FN$\"3\"*******Gj$3V#FN7$$!39+++]%3 s)[FN$\"39+++#yCZO#FN7$$!3g*****\\.mU#\\FN$\"3++++fR-)H#FN7$$!3K+++IOW g\\FN$\"3/+++67yIAFN7$$!3%)******))*Gd*\\FN$\"3/+++CM/j@FN7$$!3m****** ))R5I]FN$\"3?+++'pb[4#FN7$$!3!)*****RIXN1&FN$\"3?+++c,EE?FN7$$!3I+++/` -'4&FN$\"35+++$*\\Hd>FN7$$!3G+++FJ^F^FN$\"3/+++IP*z)=FN7$$!3S+++Pe(z:& FN$\"3,+++\"y%Q==FN7$$!3O+++1'zt=&FN$\"31+++&G\"\\[J$y;FN7$$!3$*******fr(GC&FN$\"33+++2w\"zg\"FN7$$!35+ ++f$3*o_FN$\"3'*******)>fs`\"FN7$$!3g*****\\.dPH&FN$\"3++++()3Om9FN7$$ !3M+++w\"*R<`FN$\"3#******>[C_R\"FN7$$!3A+++WX\")R`FN$\"3++++E%\\QK\"F N7$$!3y*****R!p)4O&FN$\"3++++CNB_7FN7$$!3%)*****f8/4Q&FN$\"3'******4mt .=\"FN7$$!3G+++?#e&*R&FN$\"3*******R\\m#36FN7$$!3O+++@^%pT&FN$\"3%**** ***)44f.\"FN7$$!3F+++&fkIV&FN$\"3L+++Xd*Hj*F-7$$!3w*****>!*>zW&FN$\"3) ******H\"fP/*)F-7$$!33+++bu^haFN$\"3e*****>\"*[K<)F-7$$!3?+++)RmQZ&FN$ \"3\"******f?d'RuF-7$$!33+++?\"y\\[&FN$\"3M+++r>n.nF-7$$!3i*****frl[\\ &FN$\"37+++\")fRlfF-7$$!3K+++%QVN]&FN$\"3)******Hdl\\A&F-7$$!3c******) zD5^&FN$\"3')******z:b#[%F-7$$!3Q+++duKn.nF-7$Fejm$!3\"******f?d'RuF-7$F`jm$!3e*****>\"*[K<)F-7$F[jm$!3) ******H\"fP/*)F-7$Ffim$!3L+++Xd*Hj*F-7$Faim$!3%*******)44f.\"FN7$F\\im $!3*******R\\m#36FN7$Fghm$!3'******4mt.=\"FN7$Fbhm$!3++++CNB_7FN7$F]hm $!3++++E%\\QK\"FN7$Fhgm$!3#******>[C_R\"FN7$Fcgm$!3++++()3Om9FN7$F^gm$ !3'*******)>fs`\"FN7$Fifm$!33+++2w\"zg\"FN7$Fdfm$!3!********>J$y;FN7$F _fm$!31+++&G\"\\[FN7$F[em$!3?+++c,EE?FN7$Ffdm$!3?+++'pb[4#FN7$Fadm$!3/+++C M/j@FN7$F\\dm$!3/+++67yIAFN7$Fgcm$!3++++fR-)H#FN7$Fbcm$!39+++#yCZO#FN7 $F]cm$!3\"*******Gj$3V#FN7$Fhbm$!3)*******[@J'\\#FN7$Fcbm$!3#******z/3 6c#FN7$F^bm$!3,+++NL=DEFN7$Fiam$!3=+++D>])o#FN7$Fdam$!35+++UK.^FFN7$F_ am$!3#)******HHv7GFN7$Fj`m$!3#)*****zNVO(GFN7$Fe`m$!3++++YRpLHFN7$F``m $!3'*******R8!H*HFN7$F[`m$!3/+++i%p70$FN7$Ff_m$!39+++[%4)3JFN7$Fa_m$!3 ')******)\\Rb;$FN7$F\\_m$!3z*****Rk%[@KFN7$Fg^m$!3)*******yjnwKFN7$Fb^ m$!3.+++o@:JLFN7$F]^m$!3)******\\ta\\Q$FN7$Fh]m$!3?+++p58QMFN7$Fc]m$!3 9+++Z5t!\\$FN7$F^]m$!3$*******yb!Ga$FN7$Fi\\m$!3y******pTS%f$FN7$Fd\\m $!3=+++I=dXOFN7$F_\\m$!3?+++d`M'p$FN7$Fj[m$!39+++*3\\nu$FN7$Fe[m$!3%** ****>U!z'z$FN7$F`[m$!3=+++EbXYQFN7$F[[m$!3!******4!fq&*QFN7$Ffjl$!3!** *****=kZWRFN7$Fajl$!3)*******ean#*RFN7$F\\jl$!3()*****\\X(=SSFN7$Fgil$ !3M+++Tx(o3%FN7$Fbil$!3*)******\\\"*fKTFN7$F]il$!3)******>X*>xTFN7$Fhh l$!3*)*****RjG0A%FN7$Fchl$!3E+++E^WiUFN7$F^hl$!3h******G/#GI%FN7$Figl$ !3,+++T>aTVFN7$Fdgl$!3-+++HT^yVFN7$F_gl$!3*******zQeOT%FN7$Fjfl$!3y*** **\\*>\"pW%FN7$Fefl$!3#)*****\\eE#yWFN7$F`fl$!3l*****>[mv]%FN7$F[fl$!3 2+++nr!\\`%FN7$Ffel$!3A+++jSBgXFN7$Fael$!3w******>=a$e%FN7$F\\el$!3%)* *****3U$[g%FN7$Fgdl$!3(******R\"[7CYFN7$Fbdl$!3F+++B(Q9k%FN7$F]dl$!3;+ ++hb\"ol%FN7$Fhcl$!3%******HR9.n%FN7$Fccl$!3G+++R8-#o%FN7$F^cl$!3%)*** **4Ih?p%FN7$Fibl$!3(******R3;1q%FN7$Fdbl$!3#*******RG&zq%FN7$F_bl$!3w* ****z[pWr%FN7$Fjal$!3J+++j'o2s%FN7$Feal$!3J+++!Qzxs%FN7$F`al$!37+++JQ& pt%FN7$F[al$!3w*****>Ca0v%FN7$Ff`l$!3c*****\\@1>x%FN7$Fa`l$!3J+++lk#\\ ![FN7$F\\`l$!3%******zbK9&[FN7$Fg_l$!3t*****Hgnu!\\FN7$Fb_l$!3S+++\\!= `'\\FN7$F]_l$!3O+++c/J>]FN7$Fh^l$!3I+++(z4s1&FN7$Fc^l$!3f*****pE$p3^FN 7$F^^l$!34+++Yp7W^FN7$Fi]l$!37+++Rb.u^FN7$Fd]l$!3s******[1#*)>&FN7$F_] l$!3/+++C9@>_FN7$Fj\\l$!3e******[*f_B&FN7$Fe\\l$!3#******p\")\\tC&FN7$ F`\\l$!3!******f72dD&FN7$F[\\l$!33+++;0^g_FN7$Ff[l$!3%)*****HX**=E&FN7 $Fa[l$!3++++x2)*f_FN7$F\\[l$!3!)*****p!Q$[D&FN7$Fgz$!3%)*****HF9lC&FN7 $Fbz$!3y*****zMd]B&FN7$F]z$!3))*****\\)*z/A&FN7$Fhy$!3$)*****pr#y-_FN7 $Fcy$!33+++Z5&>=&FN7$F^y$!3#*******3l&z:&FN7$Fix$!3y*****zWd28&FN7$Fdx $!3p*****f_*H+^FN7$F_x$!3q*****z(e^m]FN7$Fjw$!3!)******ynKH]FN7$Few$!3 C+++([Q'))\\FN7$F`w$!3,+++O2MW\\FN7$F[w$!3#)******G>I'*[FN7$Ffv$!3q*** **4]gV%[FN7$Fav$!3y********)4$)y%FN7$F\\v$!3\")******yH(ys%FN7$$\"3x** *****QQ'GJF-$!35+++P(eEm%FN7$Fbu$!33+++U&*3#f%FN7$F]u$!37+++#Qz_^%FN7$ Fht$!3!)******Hc$3V%FN7$Fct$!3h*****peplL%FN7$F^t$!3/+++b(R#HUFN7$Fis$ !3B+++l0-0TFN7$Fds$!3#******fg*=iRFN7$F_s$!3\"******\\*4![!QFN7$Fjr$!3 \")******>rmSOFN7$Fdr$!33+++qGcvMFN7$F_r$!3*)*****>ZH=J$FN7$Fiq$!3#*** ***fE.*\\JFN7$Fdq$!3'******z>T&*)HFN7$F_q$!3++++6?NIGFN7$Fjp$!3A+++Xq( >n#FN7$Fep$!30+++=n89DFN7$F_p$!3%)*****fiImN#FN7$Fjo$!3/+++oNK*>#FN7$F eo$!3!******>8H@/#FN7$F`o$!3'*******))\\*\\)=FN7$Fjn$!3'******z?\"*ys \"FN7$$!3u*****>!4CjkFW$!31+++(p,3d\"FN7$$!31+++&fVkb#FW$!30+++%Q=PT\" FN7$FP$!3#******\\_PmD\"FN7$$!3))*****p1.Tq#FK$!31+++Ovb*4\"FN7$$!3l++ +8P6FoF@$!32+++e\"yZU*F-7$F>$!3=+++h;)R&yF-7$$!3))*****4Rw_&>F:$!37+++ 5`=$G'F-7$$!33+++Z2>*o\"F4$!3<+++!)*)Q7ZF-7$F($!3)******Rl#fTJF-7$F($! 3++++Fjzq:F-F'-%&COLORG6&%$RGBGF($\"\"#!\"\"F(-%)POLYGONSG6^_l7%F'7$F( $\"+Fjzq:!#57$$!#YFj\\pF(7%F_]p7$F($\"+aEfTJFb]pFc]p7%Fg]p7$$!+f2>*o\" Ffr$\"+!)*)Q7ZFb]pFc]p7%F[^p7$$!+$Rw_&>F-$\"+5`=$G'Fb]pFc]p7%Fa^p7$$!+ ?/,i8FN$\"+h;)R&yFb]pFc]p7%Fg^p7$$!+7P6FoFN$\"+e\"yZU*Fb]pFc]p7%F]_p7$ $!+mI5/F!#;$\"+Ovb*4\"!\"*Fc]p7%Fc_p7$$!+W_KU*)Ff_p$\"+Dvjc7Fi_pFc]p7% F[`p7$$!+%fVkb#!#:$\"+%Q=PT\"Fi_pFc]p7%Fa`p7$$!+$*3CjkFd`p$\"+(p,3d\"F i_pFc]p7%Fh`p7$$!+$>0iY\"!#9$\"+37*ys\"Fi_pFc]p7%F^ap7$$!+&)*[\"3IFaap $\"+*)\\*\\)=Fi_pFc]p7%Feap7$$!+$)R&ye&Faap$\"+K\"H@/#Fi_pFc]p7%F[bp7$ $!+5[j=$*Faap$\"+oNK*>#Fi_pFc]p7%Fabp7$$!+V)zrN\"!#8$\"+E1jcBFi_pFc]p7 %Fgbp7$$!+^+_(e\"Fjbp$\"+=n89DFi_pFc]p7%F^cp7$$!+m(HP')*Faap$\"+Xq(>n# Fi_pFc]p7%Fdcp7$$\"+;EpK=Fjbp$\"+6?NIGFi_pFc]p7%Fjcp7$$\"+aML@'*Fjbp$ \"+)>T&*)HFi_pFc]p7%F`dp7$$\"+Z&pIv#!#7$\"+mK!*\\JFi_pFc]p7%Ffdp7$$\"+ lF3#['Fidp$\"+s%H=J$Fi_pFc]p7%F]ep7$$\"+?D5v8!#6$\"+qGcvMFi_pFc]p7%Fce p7$$\"+g:([r#Ffep$\"+?rmSOFi_pFc]p7%Fjep7$$\"+#pui,&Ffep$\"+&*4![!QFi_ pFc]p7%F`fp7$$\"+3I'*[Fi_pFc]p7%Fhj p7$$\"+i2t()GFb]p$\"+O2MW\\Fi_pFc]p7%F^[q7$$\"+EK<:FFb]p$\"+([Q'))\\Fi _pFc]p7%Fd[q7$$\"+*=t<^#Fb]p$\"+znKH]Fi_pFc]p7%Fj[q7$$\"+'\\25G#Fb]p$ \"+ye^m]Fi_pFc]p7%F`\\q7$$\"+1R#f-#Fb]p$\"+E&*H+^Fi_pFc]p7%Ff\\q7$$\"+ g3>\\=&Fi_pFc]p7%Fh]q7$$\"+[by4\")Ffep$\"+5Fb]p$\"+x2)*f_Fi_pFc]p7%F\\`q 7$$!+)pmbT\"Fb]p$\"+`%**=E&Fi_pFc]p7%Fb`q7$$!+Z*G4#=Fb]p$\"+;0^g_Fi_pF c]p7%Fh`q7$$!+\"Qy\\B#Fb]p$\"+Erqb_Fi_pFc]p7%F^aq7$$!+O\"\\yl#Fb]p$\"+ <)\\tC&Fi_pFc]p7%Fdaq7$$!+l_Fi_pFc]p7%F`bq7$$!+b!3r)RFb]p$\"+\\1#*)>&Fi_pFc]p7%Ffbq7 $$!+D@ZcWFb]p$\"+Rb.u^Fi_pFc]p7%F\\cq7$$!+yO1X\\Fb]p$\"+Yp7W^Fi_pFc]p7 %Fbcq7$$!++`!*faFb]p$\"+nKp3^Fi_pFc]p7%Fhcq7$$!+k&\\A,'Fb]p$\"+(z4s1&F i_pFc]p7%F^dq7$$!+?t7?mFb]p$\"+c/J>]Fi_pFc]p7%Fddq7$$!+hJN6tFb]p$\"+\\ !=`'\\Fi_pFc]p7%Fjdq7$$!+]DaA\")Fb]p$\"+.wY2\\Fi_pFc]p7%F`eq7$$!+/VX!3 *Fb]p$\"+eDV^[Fi_pFc]p7%Ffeq7$$!+JrD;5Fi_p$\"+lk#\\![Fi_pFc]p7%F\\fq7$ $!+D-^H6Fi_p$\"+:i!>x%Fi_pFc]p7%Fbfq7$$!+;l,T7Fi_p$\"+UUb]ZFi_pFc]p7%F hfq7$$!+BZeZ8Fi_p$\"+JQ&pt%Fi_pFc]p7%F^gq7$$!+%H0&[9Fi_p$\"+!Qzxs%Fi_p Fc]p7%Fdgq7$$!+H14W:Fi_p$\"+j'o2s%Fi_pFc]p7%Fjgq7$$!+uq%\\j\"Fi_p$\"+) [pWr%Fi_pFc]p7%F`hq7$$!+i:q@Fi_p$\"+R8-#o%Fi_pFc]p7%Fhiq7$$!+u'ex.#Fi_p$\"+$R9.n%Fi_pFc]p 7%F^jq7$$!+ei\"46#Fi_p$\"+hb\"ol%Fi_pFc]p7%Fdjq7$$!+%G!H#=#Fi_p$\"+B(Q 9k%Fi_pFc]p7%Fjjq7$$!+X>4_AFi_p$\"+9[7CYFi_pFc]p7%F`[r7$$!+Ai]?BFi_p$ \"+4U$[g%Fi_pFc]p7%Ff[r7$$!+zxp(Q#Fi_p$\"+?=a$e%Fi_pFc]p7%F\\\\r7$$!+^ d\"QX#Fi_p$\"+jSBgXFi_pFc]p7%Fb\\r7$$!+2w**=DFi_p$\"+nr!\\`%Fi_pFc]p7% Fh\\r7$$!+@AP$e#Fi_p$\"+#[mv]%Fi_pFc]p7%F^]r7$$!+.A1ZEFi_p$\"+&eE#yWFi _pFc]p7%Fd]r7$$!+7b=5FFi_p$\"+&*>\"pW%Fi_pFc]p7%Fj]r7$$!+7j&Gx#Fi_p$\" +)QeOT%Fi_pFc]p7%F`^r7$$!+$*\\=NGFi_p$\"+HT^yVFi_pFc]p7%Ff^r7$$!+rsF(* GFi_p$\"+T>aTVFi_pFc]p7%F\\_r7$$!+)HK#fHFi_p$\"+H/#GI%Fi_pFc]p7%Fb_r7$ $!+W(R6-$Fi_p$\"+E^WiUFi_pFc]p7%Fh_r7$$!+.g2$3$Fi_p$\"+M'G0A%Fi_pFc]p7 %F^`r7$$!+k(*4XJFi_p$\"+_%*>xTFi_pFc]p7%Fd`r7$$!+![Zs?$Fi_p$\"+]\"*fKT Fi_pFc]p7%Fj`r7$$!+;%H&pKFi_p$\"+Tx(o3%Fi_pFc]p7%F`ar7$$!+Vq#>L$Fi_p$ \"+bu=SSFi_pFc]p7%Ffar7$$!+([#R%R$Fi_p$\"+fan#*RFi_pFc]p7%F\\br7$$!+E/ &oX$Fi_p$\"+>kZWRFi_pFc]p7%Fbbr7$$!+)Q-#>NFi_p$\"+,fq&*QFi_pFc]p7%Fhbr 7$$!+YHL\"e$Fi_p$\"+EbXYQFi_pFc]p7%F^cr7$$!+Sp6VOFi_p$\"+A/z'z$Fi_pFc] p7%Fdcr7$$!+hnU/PFi_p$\"+*3\\nu$Fi_pFc]p7%Fjcr7$$!+\\)Q^w$Fi_p$\"+d`M' p$Fi_pFc]p7%F`dr7$$!+I)Q^#QFi_p$\"+I=dXOFi_pFc]p7%Ffdr7$$!+l]K%)QFi_p$ \"+qTS%f$Fi_pFc]p7%F\\er7$$!+l1hURFi_p$\"+zb!Ga$Fi_pFc]p7%Fber7$$!+6U# ***RFi_p$\"+Z5t!\\$Fi_pFc]p7%Fher7$$!+*[4i0%Fi_p$\"+p58QMFi_pFc]p7%F^f r7$$!+wXU6TFi_p$\"+NZ&\\Q$Fi_pFc]p7%Fdfr7$$!+S0alTFi_p$\"+o@:JLFi_pFc] p7%Fjfr7$$!+k*R&=UFi_p$\"+zjnwKFi_pFc]p7%F`gr7$$!+QaTqUFi_p$\"+WY[@KFi _pFc]p7%Ffgr7$$!++\"o6K%Fi_p$\"+*\\Rb;$Fi_pFc]p7%F\\hr7$$!+oi!3P%Fi_p$ \"+[%4)3JFi_pFc]p7%Fbhr7$$!+dTM>WFi_p$\"+i%p70$Fi_pFc]p7%Fhhr7$$!+&p+o Y%Fi_p$\"+S8!H*HFi_pFc]p7%F^ir7$$!+&Q)>8XFi_p$\"+YRpLHFi_pFc]p7%Fdir7$ $!+7AceXFi_p$\"+eLktGFi_pFc]p7%Fjir7$$!+b&=Hg%Fi_p$\"+IHv7GFi_pFc]p7%F `jr7$$!+&4%HYYFi_p$\"+UK.^FFi_pFc]p7%Ffjr7$$!+9Zr)o%Fi_p$\"+D>])o#Fi_p Fc]p7%F\\[s7$$!+\"R/-t%Fi_p$\"+NL=DEFi_pFc]p7%Fb[s7$$!+CTyqZFi_p$\"+[! 36c#Fi_pFc]p7%Fh[s7$$!+84Z5[Fi_p$\"+\\@J'\\#Fi_pFc]p7%F^\\s7$$!+joF\\[ Fi_p$\"+Hj$3V#Fi_pFc]p7%Fd\\s7$$!+]%3s)[Fi_p$\"+#yCZO#Fi_pFc]p7%Fj\\s7 $$!+NgEC\\Fi_p$\"+fR-)H#Fi_pFc]p7%F`]s7$$!+IOWg\\Fi_p$\"+67yIAFi_pFc]p 7%Ff]s7$$!+*)*Gd*\\Fi_p$\"+CM/j@Fi_pFc]p7%F\\^s7$$!+*)R5I]Fi_p$\"+'pb[ 4#Fi_pFc]p7%Fb^s7$$!+/`aj]Fi_p$\"+c,EE?Fi_pFc]p7%Fh^s7$$!+/`-'4&Fi_p$ \"+$*\\Hd>Fi_pFc]p7%F^_s7$$!+FJ^F^Fi_p$\"+IP*z)=Fi_pFc]p7%Fd_s7$$!+Pe( z:&Fi_p$\"+\"y%Q==Fi_pFc]p7%Fj_s7$$!+1'zt=&Fi_p$\"+&G\"\\[fs`\"Fi_pFc]p7%Fbas7$$!+Nqv$H&Fi_p$\" +()3Om9Fi_pFc]p7%Fhas7$$!+w\"*R<`Fi_p$\"+#[C_R\"Fi_pFc]p7%F^bs7$$!+WX \")R`Fi_p$\"+E%\\QK\"Fi_pFc]p7%Fdbs7$$!+/p)4O&Fi_p$\"+CNB_7Fi_pFc]p7%F jbs7$$!+OT!4Q&Fi_p$\"+hOP!=\"Fi_pFc]p7%F`cs7$$!+?#e&*R&Fi_p$\"+%\\m#36 Fi_pFc]p7%Ffcs7$$!+@^%pT&Fi_p$\"+*44f.\"Fi_pFc]p7%F\\ds7$$!+&fkIV&Fi_p $\"+Xd*Hj*Fb]pFc]p7%Fbds7$$!+-*>zW&Fi_p$\"+8fP/*)Fb]pFc]p7%Fhds7$$!+bu ^haFi_p$\"+7*[K<)Fb]pFc]p7%F^es7$$!+)RmQZ&Fi_p$\"+1slRuFb]pFc]p7%Fdes7 $$!+?\"y\\[&Fi_p$\"+r>n.nFb]pFc]p7%Fjes7$$!+;d'[\\&Fi_p$\"+\")fRlfFb]p Fc]p7%F`fs7$$!+%QVN]&Fi_p$\"+tb'\\A&Fb]pFc]p7%Fffs7$$!+*zD5^&Fi_p$\"+! e^D[%Fb]pFc]p7%F\\gs7$$!+duKn.nFb]pFc]p7%Fd[t7$Fees$!+1slRuFb]pFc]p7%Fh[t7$F_es$!+7*[K<)Fb]pFc]p7 %F\\\\t7$Fids$!+8fP/*)Fb]pFc]p7%F`\\t7$Fcds$!+Xd*Hj*Fb]pFc]p7%Fd\\t7$F ]ds$!+*44f.\"Fi_pFc]p7%Fh\\t7$Fgcs$!+%\\m#36Fi_pFc]p7%F\\]t7$Facs$!+hO P!=\"Fi_pFc]p7%F`]t7$F[cs$!+CNB_7Fi_pFc]p7%Fd]t7$Febs$!+E%\\QK\"Fi_pFc ]p7%Fh]t7$F_bs$!+#[C_R\"Fi_pFc]p7%F\\^t7$Fias$!+()3Om9Fi_pFc]p7%F`^t7$ Fcas$!+*>fs`\"Fi_pFc]p7%Fd^t7$F]as$!+2w\"zg\"Fi_pFc]p7%Fh^t7$Fg`s$!++7 Ly;Fi_pFc]p7%F\\_t7$Fa`s$!+&G\"\\[Fi_pFc]p7%F\\`t7 $Fi^s$!+c,EE?Fi_pFc]p7%F``t7$Fc^s$!+'pb[4#Fi_pFc]p7%Fd`t7$F]^s$!+CM/j@ Fi_pFc]p7%Fh`t7$Fg]s$!+67yIAFi_pFc]p7%F\\at7$Fa]s$!+fR-)H#Fi_pFc]p7%F` at7$F[]s$!+#yCZO#Fi_pFc]p7%Fdat7$Fe\\s$!+Hj$3V#Fi_pFc]p7%Fhat7$F_\\s$! +\\@J'\\#Fi_pFc]p7%F\\bt7$Fi[s$!+[!36c#Fi_pFc]p7%F`bt7$Fc[s$!+NL=DEFi_ pFc]p7%Fdbt7$F][s$!+D>])o#Fi_pFc]p7%Fhbt7$Fgjr$!+UK.^FFi_pFc]p7%F\\ct7 $Fajr$!+IHv7GFi_pFc]p7%F`ct7$F[jr$!+eLktGFi_pFc]p7%Fdct7$Feir$!+YRpLHF i_pFc]p7%Fhct7$F_ir$!+S8!H*HFi_pFc]p7%F\\dt7$Fihr$!+i%p70$Fi_pFc]p7%F` dt7$Fchr$!+[%4)3JFi_pFc]p7%Fddt7$F]hr$!+*\\Rb;$Fi_pFc]p7%Fhdt7$Fggr$!+ WY[@KFi_pFc]p7%F\\et7$Fagr$!+zjnwKFi_pFc]p7%F`et7$F[gr$!+o@:JLFi_pFc]p 7%Fdet7$Fefr$!+NZ&\\Q$Fi_pFc]p7%Fhet7$F_fr$!+p58QMFi_pFc]p7%F\\ft7$Fie r$!+Z5t!\\$Fi_pFc]p7%F`ft7$Fcer$!+zb!Ga$Fi_pFc]p7%Fdft7$F]er$!+qTS%f$F i_pFc]p7%Fhft7$Fgdr$!+I=dXOFi_pFc]p7%F\\gt7$Fadr$!+d`M'p$Fi_pFc]p7%F`g t7$F[dr$!+*3\\nu$Fi_pFc]p7%Fdgt7$Fecr$!+A/z'z$Fi_pFc]p7%Fhgt7$F_cr$!+E bXYQFi_pFc]p7%F\\ht7$Fibr$!+,fq&*QFi_pFc]p7%F`ht7$Fcbr$!+>kZWRFi_pFc]p 7%Fdht7$F]br$!+fan#*RFi_pFc]p7%Fhht7$Fgar$!+bu=SSFi_pFc]p7%F\\it7$Faar $!+Tx(o3%Fi_pFc]p7%F`it7$F[ar$!+]\"*fKTFi_pFc]p7%Fdit7$Fe`r$!+_%*>xTFi _pFc]p7%Fhit7$F_`r$!+M'G0A%Fi_pFc]p7%F\\jt7$Fi_r$!+E^WiUFi_pFc]p7%F`jt 7$Fc_r$!+H/#GI%Fi_pFc]p7%Fdjt7$F]_r$!+T>aTVFi_pFc]p7%Fhjt7$Fg^r$!+HT^y VFi_pFc]p7%F\\[u7$Fa^r$!+)QeOT%Fi_pFc]p7%F`[u7$F[^r$!+&*>\"pW%Fi_pFc]p 7%Fd[u7$Fe]r$!+&eE#yWFi_pFc]p7%Fh[u7$F_]r$!+#[mv]%Fi_pFc]p7%F\\\\u7$Fi \\r$!+nr!\\`%Fi_pFc]p7%F`\\u7$Fc\\r$!+jSBgXFi_pFc]p7%Fd\\u7$F]\\r$!+?= a$e%Fi_pFc]p7%Fh\\u7$Fg[r$!+4U$[g%Fi_pFc]p7%F\\]u7$Fa[r$!+9[7CYFi_pFc] p7%F`]u7$F[[r$!+B(Q9k%Fi_pFc]p7%Fd]u7$Fejq$!+hb\"ol%Fi_pFc]p7%Fh]u7$F_ jq$!+$R9.n%Fi_pFc]p7%F\\^u7$Fiiq$!+R8-#o%Fi_pFc]p7%F`^u7$Fciq$!+,81#p% Fi_pFc]p7%Fd^u7$F]iq$!+%3;1q%Fi_pFc]p7%Fh^u7$Fghq$!+SG&zq%Fi_pFc]p7%F \\_u7$Fahq$!+)[pWr%Fi_pFc]p7%F`_u7$F[hq$!+j'o2s%Fi_pFc]p7%Fd_u7$Fegq$! +!Qzxs%Fi_pFc]p7%Fh_u7$F_gq$!+JQ&pt%Fi_pFc]p7%F\\`u7$Fifq$!+UUb]ZFi_pF c]p7%F``u7$Fcfq$!+:i!>x%Fi_pFc]p7%Fd`u7$F]fq$!+lk#\\![Fi_pFc]p7%Fh`u7$ Fgeq$!+eDV^[Fi_pFc]p7%F\\au7$Faeq$!+.wY2\\Fi_pFc]p7%F`au7$F[eq$!+\\!=` '\\Fi_pFc]p7%Fdau7$Fedq$!+c/J>]Fi_pFc]p7%Fhau7$F_dq$!+(z4s1&Fi_pFc]p7% F\\bu7$Ficq$!+nKp3^Fi_pFc]p7%F`bu7$Fccq$!+Yp7W^Fi_pFc]p7%Fdbu7$F]cq$!+ Rb.u^Fi_pFc]p7%Fhbu7$Fgbq$!+\\1#*)>&Fi_pFc]p7%F\\cu7$Fabq$!+C9@>_Fi_pF c]p7%F`cu7$F[bq$!+\\*f_B&Fi_pFc]p7%Fdcu7$Feaq$!+<)\\tC&Fi_pFc]p7%Fhcu7 $F_aq$!+Erqb_Fi_pFc]p7%F\\du7$Fi`q$!+;0^g_Fi_pFc]p7%F`du7$Fc`q$!+`%**= E&Fi_pFc]p7%Fddu7$F]`q$!+x2)*f_Fi_pFc]p7%Fhdu7$Fg_q$!+2Q$[D&Fi_pFc]p7% F\\eu7$Fa_q$!+tU^Y_Fi_pFc]p7%F`eu7$F[_q$!+[t0N_Fi_pFc]p7%Fdeu7$Fe^q$!+ &)*z/A&Fi_pFc]p7%Fheu7$F_^q$!+=&Fi_pF c]p7%F`fu7$Fc]q$!+4l&z:&Fi_pFc]p7%Fdfu7$F]]q$!+[uvI^Fi_pFc]p7%Fhfu7$Fg \\q$!+E&*H+^Fi_pFc]p7%F\\gu7$Fa\\q$!+ye^m]Fi_pFc]p7%F`gu7$F[\\q$!+znKH ]Fi_pFc]p7%Fdgu7$Fe[q$!+([Q'))\\Fi_pFc]p7%Fhgu7$F_[q$!+O2MW\\Fi_pFc]p7 %F\\hu7$Fijp$!+H>I'*[Fi_pFc]p7%F`hu7$Fcjp$!+,0OW[Fi_pFc]p7%Fdhu7$F]jp$ !++*4$)y%Fi_pFc]p7%Fhhu7$Fgip$!+zH(ys%Fi_pFc]p7%F\\iu7$$\"+!RQ'GJFb]p$ !+P(eEm%Fi_pFc]p7%F`iu7$F[ip$!+U&*3#f%Fi_pFc]p7%Ffiu7$Fehp$!+#Qz_^%Fi_ pFc]p7%Fjiu7$F_hp$!+Ic$3V%Fi_pFc]p7%F^ju7$Figp$!+(eplL%Fi_pFc]p7%Fbju7 $Fcgp$!+b(R#HUFi_pFc]p7%Ffju7$F]gp$!+l0-0TFi_pFc]p7%Fjju7$Fgfp$!+1'*=i RFi_pFc]p7%F^[v7$Fafp$!+&*4![!QFi_pFc]p7%Fb[v7$F[fp$!+?rmSOFi_pFc]p7%F f[v7$Fdep$!+qGcvMFi_pFc]p7%Fj[v7$F^ep$!+s%H=J$Fi_pFc]p7%F^\\v7$Fgdp$!+ mK!*\\JFi_pFc]p7%Fb\\v7$Fadp$!+)>T&*)HFi_pFc]p7%Ff\\v7$F[dp$!+6?NIGFi_ pFc]p7%Fj\\v7$Fecp$!+Xq(>n#Fi_pFc]p7%F^]v7$F_cp$!+=n89DFi_pFc]p7%Fb]v7 $Fhbp$!+E1jcBFi_pFc]p7%Ff]v7$Fbbp$!+oNK*>#Fi_pFc]p7%Fj]v7$F\\bp$!+K\"H @/#Fi_pFc]p7%F^^v7$Ffap$!+*)\\*\\)=Fi_pFc]p7%Fb^v7$F_ap$!+37*ys\"Fi_pF c]p7%Ff^v7$$!+-4CjkFd`p$!+(p,3d\"Fi_pFc]p7%Fj^v7$$!+&fVkb#Fd`p$!+%Q=PT \"Fi_pFc]p7%F`_v7$F\\`p$!+Dvjc7Fi_pFc]p7%Ff_v7$$!+nI5/FFf_p$!+Ovb*4\"F i_pFc]p7%Fj_v7$$!+8P6FoFN$!+e\"yZU*Fb]pFc]p7%F``v7$Fh^p$!+h;)R&yFb]pFc ]p7%Ff`v7$$!+\"Rw_&>F-$!+5`=$G'Fb]pFc]p7%Fj`v7$$!+Z2>*o\"Ffr$!+!)*)Q7Z Fb]pFc]p7%F`av7$F($!+aEfTJFb]pFc]p7%Ffav7$F($!+Fjzq:Fb]pFc]p7%FjavF'Fc ]p-Fe\\p6&Fg\\pFh\\p$\"#b!\"#$\"#DFbbv-%&STYLEG6#%,PATCHNOGRIDG-F$6$7] _lF'F*F.7$F($\"3D+++\")*)Q7ZF-7$$\"3W+++%R]ei&F4$\"3g*****pJ&=$G'F-7$$ \"3J+++3CqT\\F:$\"3)*******)p\")R&yF-7$$\"30+++v^-VGF@$\"3$)******[#yZ U*F-7$$\"36+++dg097FK$\"33+++Bvb*4\"FN7$$\"3D+++%\\C@9%FK$\"3%******\\ IPmD\"FN7$$\"3-+++1sP#=\"FW$\"3)******p>#FN7$$\"3q*****Rd``t$F\\o$\"3/+++B)yhN#FN 7$$\"3C+++NK;*H$F\\o$\"3'******pz`K^#FN7$$\"33+++Dk\"*[bZiK#Fap$\" 35+++&=%y%)HFN7$$!37+++&)RC'y%Fap$\"32+++.1MUJFN7$$!3%******fr6%H!)Fap $\"3!)*****zK./I$FN7$$!35+++'[o]6\"F[r$\"37+++6%[$fMFN7$$!3/+++xa-h6F[ r$\"3:+++YKv>OFN7$$!3E+++H+iuNFap$\"3=+++$=rCy$FN7$$\"3++++_LrWDF[r$\" 3++++^-q[RFN7$$\"3)******py$)z,\"Ffr$\"3P+++X/\"*>TFN7$$\"37+++E0_.GFf r$\"3G+++rt$pH%FN7$$\"3c******o*=*zlFfr$\"3G+++XpkvWFN7$$\"31+++mZL#H \"F-$\"3<+++WZ$*QYFN7$$\"3!******H'4uH?F-$\"3Q+++T)=,x%FN7$$\"3/+++6y8 'o#F-$\"3i******Q6)R([FN7$$\"3#******foT8@$F-$\"3K+++vD-h\\FN7$$\"3F++ +&))3Th$F-$\"36+++cwhP]FN7$$\"33+++(3,F\"RF-$\"3T+++ro-2^FN7$$\"3D+++) )Q_BTF-$\"3E+++(za4<&FN7$$\"3y*****4N(zfUF-$\"3))*****RTU.B&FN7$$\"3-+ ++\"=o?L%F-$\"3A+++9#QdG&FN7$$\"33+++mQ')[VF-$\"3u******[(yuL&FN7$$\"3 2+++;L?GOcFN7$$\"3= +++%\\jQ(GF-$\"3S+++L:?gcFN7$$\"3%)*****pul;f#F-$\"3I+++,QB\"o&FN7$$\" 35+++5w,%H#F-$\"3?+++,3Q*p&FN7$$\"35+++l0R#)>F-$\"3&)*****HPLYr&FN7$$ \"33+++!)o2e;F-$\"3U+++I(pps&FN7$$\"3/+++m^?A8F-$\"3=+++'o`jt&FN7$$\"3 Y*****fu/vv*Ffr$\"3&******fTKFu&FN7$$\"3A+++5JQ&>'Ffr$\"3g*****zdLgu&F N7$$\"3$******z'>TUDFfr$\"3G+++u:;YdFN7$$!31+++AeM'>\"Ffr$\"3>+++8E*Hu &FN7$$!3z*****R\"=w<]Ffr$\"3i*****\\3ojt&FN7$$!3M*****\\)\\7@*)Ffr$\"3 y*****4a&3EdFN7$$!33+++fh*3H\"F-$\"36+++(>')=r&FN7$$!3/+++m?%))p\"F-$ \"3O+++btV$p&FN7$$!3.+++wdO<@F-$\"3U+++9tIqcFN7$$!31+++*R2!\\DF-$\"3p* ****HFH>k&FN7$$!3#******RK?#)*HF-$\"3&******\\!*\\vg&FN7$$!37+++@=#HZ$ F-$\"3)*******z3;mbFN7$$!3'******4\\Hx)RF-$\"3d******yIV;bFN7$$!3++++j p-rXF-$\"3A+++'**)ycaFN7$$!3%)*****plC&z_F-$\"3C+++OiI'Q&FN7$$!3****** *z,[;@'F-$\"3l*****fa[!4`FN7$$!3z*****>(\\>EuF-$\"38+++s>1U_FN7$$!3-++ +_)4.x)F-$\"3i*****pBy>?&FN7$$!3-+++d'Hg+\"FN$\"3=+++**3(R=&FN7$$!3%** ****fC&fC6FN$\"3e*****R**)\\x^FN7$$!3*******H!\\cL7FN$\"3K+++hD:w^FN7$ $!3.+++ETpM8FN$\"3S+++e5kw^FN7$$!3)******R'f_H9FN$\"3y******HkBx^FN7$$ !3!******f)RE>:FN$\"3))*****>'=)p<&FN7$$!3)******ReH[g\"FN$\"3O+++B8Kv ^FN7$$!33+++)GLpo\"FN$\"3?++++\">><&FN7$$!3#******fEJhw\"FN$\"3e*****z aml;&FN7$$!3'******zRmG%=FN$\"3C+++\"=J\"f^FN7$$!3%******Hr(\\<>FN$\"3 K+++Tv_\\^FN7$$!33+++`;K!*>FN$\"3&******p\"**pP^FN7$$!3=+++-xeh?FN$\"3 !)*****>&=hB^FN7$$!3++++\"45:8#FN$\"3K+++9(Rs5&FN7$$!3#*******[nF+AFN$ \"3w******R)o&)3&FN7$$!3=+++fi0oAFN$\"3;+++(R\"fn]FN7$$!3$)*****HN.]L# FN$\"3H+++qaIW]FN7$$!3$******\\Dj7S#FN$\"3'******H4:(=]FN7$$!3A+++g](p Y#FN$\"3?+++c5$3*\\FN7$$!3=+++EUFKDFN$\"3w*****zSs1'\\FN7$$!33+++aTH(f #FN$\"3I+++]&o#G\\FN7$$!3)*******Rn;iEFN$\"3g*****R#=m$*[FN7$$!3-+++]< -FFFN$\"3!)*****pV5p&[FN7$$!31+++.[)>z#FN$\"3(******zR\"4=[FN7$$!3-+++ QQ7$FN$\"3!)*****\\10sf%FN7$$!3%)******=hP*=$FN$\"3c*****Hc)y[ XFN7$$!3-+++.&HsD$FN$\"3s******Q$y$*\\%FN7$$!3++++0[VDLFN$\"3C+++`69\\ WFN7$$!3#)******)3pQR$FN$\"3U+++#o=#)R%FN7$$!3#******p3&QiMFN$\"37+++A *>nM%FN7$$!3&)******Q<#3`$FN$\"35+++,sr%H%FN7$$!3/+++W[,*f$FN$\"3e**** *z*fCUUFN7$$!3++++;k!om$FN$\"3!)*****R-3$*=%FN7$$!31+++6=0MPFN$\"39+++ @l(e8%FN7$$!3,+++\\Vi+QFN$\"35+++TE!>3%FN7$$!3A+++7wTmQFN$\"3y*****R#> KFSFN7$$!3'******f(eMJRFN$\"3%******R?g?(RFN7$$!3!)*****zBV`*RFN$\"3/+ ++#fQg\"RFN7$$!35+++]>OeSFN$\"3?+++RudL4Z%FN$ \"3#******>.d=V$FN7$$!3e*****HS1f_%FN$\"3%)*****z#3,mLFN7$$!3;+++s!e*z XFN$\"3%)*****fht))H$FN7$$!3(******Rc'*oot%FN$\"3;+++Dbo*3$FN7$$!3y* *****Rr](y%FN$\"35+++'e)R+++62(Q%HFN7$$!3M+++ aLQ')[FN$\"3y*****R4N\"pGFN7$$!3w*****p;HY$\\FN$\"3!)*****z7IKz#FN7$$! 3K+++4/2#)\\FN$\"3<+++yx>;FFN7$$!3&******zl(oG]FN$\"3*******>y\"3QEFN7 $$!3?+++&f^W2&FN$\"3%)*****HyD*eDFN7$$!3v*****zQB$>^FN$\"3++++O:xyCFN7 $$!3K+++/bDj^FN$\"3)******RIdwR#FN7$$!3/+++&*G>1_FN$\"3********plh:BFN 7$$!3?+++LW2[_FN$\"3:+++LrnKAFN7$$!3!)*****p]M))G&FN$\"37+++^1')[@FN7$ $!3e******RYSG`FN$\"3-+++5E=k?FN7$$!37+++V]rm`FN$\"3%******Rq_'y>FN7$$ !3&)******=gp.aFN$\"3'******fbvA*=FN7$$!3C+++v\"z#RaFN$\"3(******Rr^]! =FN7$$!39+++:%)RtaFN$\"3&******z$)ypr\"FN7$$!3i*****zr!*f]&FN$\"3\"*** ***f-`!G;FN7$$!3:+++=n*p`&FN$\"34+++%>q#Q:FN7$$!3e*****p6hjc&FN$\"33++ +%[FwW\"FN7$$!31+++_H.%f&FN$\"3)******zkChN\"FN7$$!3E+++[d'*>cFN$\"31+ ++%QlPE\"FN7$$!38+++'e4^km&FN$\"3% *******f&>l2\"FN7$$!3?+++qL$po&FN$\"33+++p%4n\")*F-7$$!3\")*****\\(e`0 dFN$\"3s*****>8P/'))F-7$$!3L+++PUBAdFN$\"3Y+++Lyx'*yF-7$$!3Q+++A!3qt&F N$\"3b+++(4Di#pF-7$$!3=+++b/%)\\dFN$\"3E+++U'o$\\fF-7$$!3s*****\\3=2w& FN$\"3))******RK*o'\\F-7$$!3#)*****RUI'pdFN$\"31+++ZUdzRF-7$$!3Q+++q&p lx&FN$\"3))*****>kp#))HF-7$$!39+++S)H:y&FN$\"3++++?'3R*>F-7$$!3q*****4 W2Xy&FN$\"3W+++i+xu**Ffr7$$!3G+++4-]&y&FNF(7$Fgcy$!3W+++i+xu**Ffr7$Fbc y$!3++++?'3R*>F-7$F]cy$!3))*****>kp#))HF-7$Fhby$!31+++ZUdzRF-7$Fcby$!3 ))******RK*o'\\F-7$F^by$!3E+++U'o$\\fF-7$Fiay$!3b+++(4Di#pF-7$Fday$!3Y +++Lyx'*yF-7$F_ay$!3s*****>8P/'))F-7$Fj`y$!33+++p%4n\")*F-7$Fe`y$!3%** *****f&>l2\"FN7$F``y$!33+++(fe0<\"FN7$F[`y$!31+++%QlPE\"FN7$Ff_y$!3)** ****zkChN\"FN7$Fa_y$!33+++%[FwW\"FN7$F\\_y$!34+++%>q#Q:FN7$Fg^y$!3\"** ****f-`!G;FN7$Fb^y$!3&******z$)ypr\"FN7$F]^y$!3(******Rr^]!=FN7$Fh]y$! 3'******fbvA*=FN7$Fc]y$!3%******Rq_'y>FN7$F^]y$!3-+++5E=k?FN7$Fi\\y$!3 7+++^1')[@FN7$Fd\\y$!3:+++LrnKAFN7$F_\\y$!3********plh:BFN7$Fj[y$!3)** ****RIdwR#FN7$Fe[y$!3++++O:xyCFN7$F`[y$!3%)*****HyD*eDFN7$F[[y$!3***** **>y\"3QEFN7$Ffjx$!3<+++yx>;FFN7$Fajx$!3!)*****z7IKz#FN7$F\\jx$!3y**** *R4N\"pGFN7$Fgix$!3>+++62(Q%HFN7$Fbix$!35+++'e)R.d=V$FN7$F_gx$!3/+++8\"Hk\\$ FN7$Fjfx$!3?+++cJvfNFN7$Fefx$!3?+++bU'=i$FN7$F`fx$!3%******zB1Go$FN7$F [fx$!37+++s7jUPFN7$Ffex$!3$)*****4L*R,QFN7$Faex$!3?+++RuKFSFN7$F]dx$!3 5+++TE!>3%FN7$Fhcx$!39+++@l(e8%FN7$Fccx$!3!)*****R-3$*=%FN7$F^cx$!3e** ***z*fCUUFN7$Fibx$!35+++,sr%H%FN7$Fdbx$!37+++A*>nM%FN7$F_bx$!3U+++#o=# )R%FN7$Fjax$!3C+++`69\\WFN7$Feax$!3s******Q$y$*\\%FN7$F`ax$!3c*****Hc) y[XFN7$F[ax$!3!)*****\\10sf%FN7$Ff`x$!3Q+++ihWWYFN7$Fa`x$!3#******pOE. p%FN7$F\\`x$!3\"******4tmYt%FN7$Fg_x$!3%)*****\\2.tx%FN7$Fb_x$!3(***** *zR\"4=[FN7$F]_x$!3!)*****pV5p&[FN7$Fh^x$!3g*****R#=m$*[FN7$Fc^x$!3I++ +]&o#G\\FN7$F^^x$!3w*****zSs1'\\FN7$Fi]x$!3?+++c5$3*\\FN7$Fd]x$!3'**** **H4:(=]FN7$F_]x$!3H+++qaIW]FN7$Fj\\x$!3;+++(R\"fn]FN7$Fe\\x$!3w****** R)o&)3&FN7$F`\\x$!3K+++9(Rs5&FN7$F[\\x$!3!)*****>&=hB^FN7$Ff[x$!3&**** **p\"**pP^FN7$Fa[x$!3K+++Tv_\\^FN7$F\\[x$!3C+++\"=J\"f^FN7$Fgjw$!3e*** **zaml;&FN7$Fbjw$!3?++++\">><&FN7$F]jw$!3O+++B8Kv^FN7$Fhiw$!3))*****>' =)p<&FN7$Fciw$!3y******HkBx^FN7$F^iw$!3S+++e5kw^FN7$Fihw$!3K+++hD:w^FN 7$Fdhw$!3e*****R**)\\x^FN7$F_hw$!3=+++**3(R=&FN7$Fjgw$!3i*****pBy>?&FN 7$Fegw$!38+++s>1U_FN7$F`gw$!3l*****fa[!4`FN7$F[gw$!3C+++OiI'Q&FN7$Fffw $!3A+++'**)ycaFN7$Fafw$!3d******yIV;bFN7$F\\fw$!3)*******z3;mbFN7$Fgew $!3&******\\!*\\vg&FN7$Fbew$!3p*****HFH>k&FN7$F]ew$!3U+++9tIqcFN7$Fhdw $!3O+++btV$p&FN7$Fcdw$!36+++(>')=r&FN7$F^dw$!3y*****4a&3EdFN7$Ficw$!3i *****\\3ojt&FN7$Fdcw$!3>+++8E*Hu&FN7$F_cw$!3G+++u:;YdFN7$Fjbw$!3g***** zdLgu&FN7$Febw$!3&******fTKFu&FN7$F`bw$!3=+++'o`jt&FN7$F[bw$!3U+++I(pp s&FN7$Ffaw$!3&)*****HPLYr&FN7$Faaw$!3?+++,3Q*p&FN7$F\\aw$!3I+++,QB\"o& FN7$Fg`w$!3S+++L:?gcFN7$Fb`w$!3T+++N>GOcFN7$F]`w$!36+++*zi%4cFN7$Fh_w$ !3=+++YDszbFN7$Fc_w$!3i*****HgIqa&FN7$F^_w$!3S+++LsM6bFN7$Fi^w$!3s**** *z]AEZ&FN7$Fd^w$!3A+++nQzIaFN7$F_^w$!3W+++J8y&Q&FN7$Fj]w$!3u******[(yu L&FN7$Fe]w$!3A+++9#QdG&FN7$F`]w$!3))*****RTU.B&FN7$F[]w$!3E+++(za4<&FN 7$Ff\\w$!3T+++ro-2^FN7$Fa\\w$!36+++cwhP]FN7$F\\\\w$!3K+++vD-h\\FN7$Fg[ w$!3i******Q6)R([FN7$Fb[w$!3Q+++T)=,x%FN7$F][w$!3<+++WZ$*QYFN7$Fhjv$!3 G+++XpkvWFN7$Fcjv$!3G+++rt$pH%FN7$F^jv$!3P+++X/\"*>TFN7$Fiiv$!3++++^-q [RFN7$Fdiv$!3=+++$=rCy$FN7$F_iv$!3:+++YKv>OFN7$Fjhv$!37+++6%[$fMFN7$Fe hv$!3!)*****zK./I$FN7$$!3!)*****R)RC'y%Fap$!32+++.1MUJFN7$F[hv$!35+++& =%y%)HFN7$Ffgv$!3')*****H]*\\FGFN7$$\"33+++Fk\"*[#FN7 $F]fv$!3&)*****poK?/#FN7$Fhev$!35+++(pb\\)=FN7$Fcev$!35+++Zo(ys\"FN7$$ \"3&)*******>`'3HFW$!3#******f8(zq:FN7$Fidv$!3)******p>#Fi_pFc]p7%F\\g[l7$$\"+uNNNPFaap$ \"+B)yhN#Fi_pFc]p7%Fbg[l7$$\"+NK;*H$Faap$\"+(z`K^#Fi_pFc]p7%Fhg[l7$$\" +Dk\"*[OFi_pFc]p7%Fbj[l7$$ !+H+iuNFjbp$\"+$=rCy$Fi_pFc]p7%Fhj[l7$$\"+_LrWDFidp$\"+^-q[RFi_pFc]p7% F^[\\l7$$\"+(y$)z,\"Ffep$\"+X/\"*>TFi_pFc]p7%Fd[\\l7$$\"+E0_.GFfep$\"+ rt$pH%Fi_pFc]p7%Fj[\\l7$$\"+p*=*zlFfep$\"+XpkvWFi_pFc]p7%F`\\\\l7$$\"+ mZL#H\"Fb]p$\"+WZ$*QYFi_pFc]p7%Ff\\\\l7$$\"+j4uH?Fb]p$\"+T)=,x%Fi_pFc] p7%F\\]\\l7$$\"+6y8'o#Fb]p$\"+R6)R([Fi_pFc]p7%Fb]\\l7$$\"+'oT8@$Fb]p$ \"+vD-h\\Fi_pFc]p7%Fh]\\l7$$\"+&))3Th$Fb]p$\"+cwhP]Fi_pFc]p7%F^^\\l7$$ \"+(3,F\"RFb]p$\"+ro-2^Fi_pFc]p7%Fd^\\l7$$\"+))Q_BTFb]p$\"+(za4<&Fi_pF c]p7%Fj^\\l7$$\"+^tzfUFb]p$\"+9CMI_Fi_pFc]p7%F`_\\l7$$\"+\"=o?L%Fb]p$ \"+9#QdG&Fi_pFc]p7%Ff_\\l7$$\"+mQ')[VFb]p$\"+\\(yuL&Fi_pFc]p7%F\\`\\l7 $$\"+;L?GOcFi_pFc]p7%F\\c\\l7$$\"+%\\jQ(GFb]p$\"+L :?gcFi_pFc]p7%Fbc\\l7$$\"+Zdm\"f#Fb]p$\"+,QB\"o&Fi_pFc]p7%Fhc\\l7$$\"+ 5w,%H#Fb]p$\"+,3Q*p&Fi_pFc]p7%F^d\\l7$$\"+l0R#)>Fb]p$\"+tLj9dFi_pFc]p7 %Fdd\\l7$$\"+!)o2e;Fb]p$\"+I(pps&Fi_pFc]p7%Fjd\\l7$$\"+m^?A8Fb]p$\"+'o `jt&Fi_pFc]p7%F`e\\l7$$\"+YZ]d(*Ffep$\"+;CtUdFi_pFc]p7%Ffe\\l7$$\"+5JQ &>'Ffep$\"+yN.YdFi_pFc]p7%F\\f\\l7$$\"+o>TUDFfep$\"+u:;YdFi_pFc]p7%Fbf \\l7$$!+AeM'>\"Ffep$\"+8E*Hu&Fi_pFc]p7%Fhf\\l7$$!+9=w<]Ffep$\"+&3ojt&F i_pFc]p7%F^g\\l7$$!+&)\\7@*)Ffep$\"+Tb3EdFi_pFc]p7%Fdg\\l7$$!+fh*3H\"F b]p$\"+(>')=r&Fi_pFc]p7%Fjg\\l7$$!+m?%))p\"Fb]p$\"+btV$p&Fi_pFc]p7%F`h \\l7$$!+wdO<@Fb]p$\"+9tIqcFi_pFc]p7%Ffh\\l7$$!+*R2!\\DFb]p$\"+t#H>k&Fi _pFc]p7%F\\i\\l7$$!+C.A)*HFb]p$\"+0*\\vg&Fi_pFc]p7%Fbi\\l7$$!+@=#HZ$Fb ]p$\"+!)3;mbFi_pFc]p7%Fhi\\l7$$!+\"\\Hx)RFb]p$\"+zIV;bFi_pFc]p7%F^j\\l 7$$!+jp-rXFb]p$\"+'**)ycaFi_pFc]p7%Fdj\\l7$$!+dY_z_Fb]p$\"+OiI'Q&Fi_pF c]p7%Fjj\\l7$$!+=![;@'Fb]p$\"+Y&[!4`Fi_pFc]p7%F`[]l7$$!+s\\>EuFb]p$\"+ s>1U_Fi_pFc]p7%Ff[]l7$$!+_)4.x)Fb]p$\"+P#y>?&Fi_pFc]p7%F\\\\]l7$$!+d'H g+\"Fi_p$\"+**3(R=&Fi_pFc]p7%Fb\\]l7$$!+Y_fC6Fi_p$\"+%**)\\x^Fi_pFc]p7 %Fh\\]l7$$!+.\\cL7Fi_p$\"+hD:w^Fi_pFc]p7%F^]]l7$$!+ETpM8Fi_p$\"+e5kw^F i_pFc]p7%Fd]]l7$$!+kf_H9Fi_p$\"+IkBx^Fi_pFc]p7%Fj]]l7$$!+')RE>:Fi_p$\" +i=)p<&Fi_pFc]p7%F`^]l7$$!+%eH[g\"Fi_p$\"+B8Kv^Fi_pFc]p7%Ff^]l7$$!+)GL po\"Fi_p$\"++\">><&Fi_pFc]p7%F\\_]l7$$!+m78mFi_p$\"+Tv_\\^ Fi_pFc]p7%F^`]l7$$!+`;K!*>Fi_p$\"+<**pP^Fi_pFc]p7%Fd`]l7$$!+-xeh?Fi_p$ \"+_=hB^Fi_pFc]p7%Fj`]l7$$!+\"45:8#Fi_p$\"+9(Rs5&Fi_pFc]p7%F`a]l7$$!+ \\nF+AFi_p$\"+S)o&)3&Fi_pFc]p7%Ffa]l7$$!+fi0oAFi_p$\"+(R\"fn]Fi_pFc]p7 %F\\b]l7$$!+`L+NBFi_p$\"+qaIW]Fi_pFc]p7%Fbb]l7$$!+bKE,CFi_p$\"+$4:(=]F i_pFc]p7%Fhb]l7$$!+g](pY#Fi_p$\"+c5$3*\\Fi_pFc]p7%F^c]l7$$!+EUFKDFi_p$ \"+3Cng\\Fi_pFc]p7%Fdc]l7$$!+aTH(f#Fi_p$\"+]&o#G\\Fi_pFc]p7%Fjc]l7$$!+ Sn;iEFi_p$\"+C=m$*[Fi_pFc]p7%F`d]l7$$!+]<-FFFi_p$\"+P/\"p&[Fi_pFc]p7%F fd]l7$$!+.[)>z#Fi_p$\"+)R\"4=[Fi_pFc]p7%F\\e]l7$$!+QQ7 $Fi_p$\"+l]?(f%Fi_pFc]p7%Fjf]l7$$!+>hP*=$Fi_p$\"+j&)y[XFi_pFc]p7%F`g]l 7$$!+.&HsD$Fi_p$\"+R$y$*\\%Fi_pFc]p7%Ffg]l7$$!+0[VDLFi_p$\"+`69\\WFi_p Fc]p7%F\\h]l7$$!+*3pQR$Fi_p$\"+#o=#)R%Fi_pFc]p7%Fbh]l7$$!+(3&QiMFi_p$ \"+A*>nM%Fi_pFc]p7%Fhh]l7$$!+R<#3`$Fi_p$\"+,sr%H%Fi_pFc]p7%F^i]l7$$!+W [,*f$Fi_p$\"+)*fCUUFi_pFc]p7%Fdi]l7$$!+;k!om$Fi_p$\"+C!3$*=%Fi_pFc]p7% Fji]l7$$!+6=0MPFi_p$\"+@l(e8%Fi_pFc]p7%F`j]l7$$!+\\Vi+QFi_p$\"+TE!>3%F i_pFc]p7%Ffj]l7$$!+7wTmQFi_p$\"+C>KFSFi_pFc]p7%F\\[^l7$$!+weMJRFi_p$\" +/-1sRFi_pFc]p7%Fb[^l7$$!+QKM&*RFi_p$\"+#fQg\"RFi_pFc]p7%Fh[^l7$$!+]>O eSFi_p$\"+Ru;FFi _pFc]p7%Fdb^l7$$!+ewoG]Fi_p$\"+#y\"3QEFi_pFc]p7%Fjb^l7$$!+&f^W2&Fi_p$ \"+$yD*eDFi_pFc]p7%F`c^l7$$!+)QB$>^Fi_p$\"+O:xyCFi_pFc]p7%Ffc^l7$$!+/b Dj^Fi_p$\"+/tl(R#Fi_pFc]p7%F\\d^l7$$!+&*G>1_Fi_p$\"+qlh:BFi_pFc]p7%Fbd ^l7$$!+LW2[_Fi_p$\"+LrnKAFi_pFc]p7%Fhd^l7$$!+2X$))G&Fi_p$\"+^1')[@Fi_p Fc]p7%F^e^l7$$!+SYSG`Fi_p$\"+5E=k?Fi_pFc]p7%Fde^l7$$!+V]rm`Fi_p$\"+/Fl y>Fi_pFc]p7%Fje^l7$$!+>gp.aFi_p$\"+cbF#*=Fi_pFc]p7%F`f^l7$$!+v\"z#RaFi _p$\"+9<00=Fi_pFc]p7%Fff^l7$$!+:%)RtaFi_p$\"+Q)ypr\"Fi_pFc]p7%F\\g^l7$ $!+=2*f]&Fi_p$\"+EI0G;Fi_pFc]p7%Fbg^l7$$!+=n*p`&Fi_p$\"+%>q#Q:Fi_pFc]p 7%Fhg^l7$$!+<6OmbFi_p$\"+%[FwW\"Fi_pFc]p7%F^h^l7$$!+_H.%f&Fi_p$\"+[Y7c 8Fi_pFc]p7%Fdh^l7$$!+[d'*>cFi_p$\"+%QlPE\"Fi_pFc]p7%Fjh^l7$$!+'el2\"Fi_pFc]p7%Ffi^l7 $$!+qL$po&Fi_p$\"+p%4n\")*Fb]pFc]p7%F\\j^l7$$!+ve`0dFi_p$\"+KrVg))Fb]p Fc]p7%Fbj^l7$$!+PUBAdFi_p$\"+Lyx'*yFb]pFc]p7%Fhj^l7$$!+A!3qt&Fi_p$\"+( 4Di#pFb]pFc]p7%F^[_l7$$!+b/%)\\dFi_p$\"+U'o$\\fFb]pFc]p7%Fd[_l7$$!+&3= 2w&Fi_p$\"+SK*o'\\Fb]pFc]p7%Fj[_l7$$!+C/jpdFi_p$\"+ZUdzRFb]pFc]p7%F`\\ _l7$$!+q&plx&Fi_p$\"+U'p#))HFb]pFc]p7%Ff\\_l7$$!+S)H:y&Fi_p$\"+?'3R*>F b]pFc]p7%F\\]_l7$$!+Tu]%y&Fi_p$\"+i+xu**FfepFc]p7%Fb]_l7$$!+4-]&y&Fi_p F(Fc]p7%Fh]_l7$Fc]_l$!+i+xu**FfepFc]p7%F\\^_l7$F]]_l$!+?'3R*>Fb]pFc]p7 %F`^_l7$Fg\\_l$!+U'p#))HFb]pFc]p7%Fd^_l7$Fa\\_l$!+ZUdzRFb]pFc]p7%Fh^_l 7$F[\\_l$!+SK*o'\\Fb]pFc]p7%F\\__l7$Fe[_l$!+U'o$\\fFb]pFc]p7%F`__l7$F_ [_l$!+(4Di#pFb]pFc]p7%Fd__l7$Fij^l$!+Lyx'*yFb]pFc]p7%Fh__l7$Fcj^l$!+Kr Vg))Fb]pFc]p7%F\\`_l7$F]j^l$!+p%4n\")*Fb]pFc]p7%F``_l7$Fgi^l$!+g&>l2\" Fi_pFc]p7%Fd`_l7$Fai^l$!+(fe0<\"Fi_pFc]p7%Fh`_l7$F[i^l$!+%QlPE\"Fi_pFc ]p7%F\\a_l7$Feh^l$!+[Y7c8Fi_pFc]p7%F`a_l7$F_h^l$!+%[FwW\"Fi_pFc]p7%Fda _l7$Fig^l$!+%>q#Q:Fi_pFc]p7%Fha_l7$Fcg^l$!+EI0G;Fi_pFc]p7%F\\b_l7$F]g^ l$!+Q)ypr\"Fi_pFc]p7%F`b_l7$Fgf^l$!+9<00=Fi_pFc]p7%Fdb_l7$Faf^l$!+cbF# *=Fi_pFc]p7%Fhb_l7$F[f^l$!+/Fly>Fi_pFc]p7%F\\c_l7$Fee^l$!+5E=k?Fi_pFc] p7%F`c_l7$F_e^l$!+^1')[@Fi_pFc]p7%Fdc_l7$Fid^l$!+LrnKAFi_pFc]p7%Fhc_l7 $Fcd^l$!+qlh:BFi_pFc]p7%F\\d_l7$F]d^l$!+/tl(R#Fi_pFc]p7%F`d_l7$Fgc^l$! +O:xyCFi_pFc]p7%Fdd_l7$Fac^l$!+$yD*eDFi_pFc]p7%Fhd_l7$F[c^l$!+#y\"3QEF i_pFc]p7%F\\e_l7$Feb^l$!+yx>;FFi_pFc]p7%F`e_l7$F_b^l$!+G,B$z#Fi_pFc]p7 %Fde_l7$Fia^l$!+%4N\"pGFi_pFc]p7%Fhe_l7$Fca^l$!+62(Q%HFi_pFc]p7%F\\f_l 7$F]a^l$!+'e)RKFSFi_pFc]p7%F`j_l7$Fgj]l$!+TE!>3%Fi_pFc]p7%Fd j_l7$Faj]l$!+@l(e8%Fi_pFc]p7%Fhj_l7$F[j]l$!+C!3$*=%Fi_pFc]p7%F\\[`l7$F ei]l$!+)*fCUUFi_pFc]p7%F`[`l7$F_i]l$!+,sr%H%Fi_pFc]p7%Fd[`l7$Fih]l$!+A *>nM%Fi_pFc]p7%Fh[`l7$Fch]l$!+#o=#)R%Fi_pFc]p7%F\\\\`l7$F]h]l$!+`69\\W Fi_pFc]p7%F`\\`l7$Fgg]l$!+R$y$*\\%Fi_pFc]p7%Fd\\`l7$Fag]l$!+j&)y[XFi_p Fc]p7%Fh\\`l7$F[g]l$!+l]?(f%Fi_pFc]p7%F\\]`l7$Fef]l$!+ihWWYFi_pFc]p7%F `]`l7$F_f]l$!+njK!p%Fi_pFc]p7%Fd]`l7$Fie]l$!+JnmMZFi_pFc]p7%Fh]`l7$Fce ]l$!+vIIxZFi_pFc]p7%F\\^`l7$F]e]l$!+)R\"4=[Fi_pFc]p7%F`^`l7$Fgd]l$!+P/ \"p&[Fi_pFc]p7%Fd^`l7$Fad]l$!+C=m$*[Fi_pFc]p7%Fh^`l7$F[d]l$!+]&o#G\\Fi _pFc]p7%F\\_`l7$Fec]l$!+3Cng\\Fi_pFc]p7%F`_`l7$F_c]l$!+c5$3*\\Fi_pFc]p 7%Fd_`l7$Fib]l$!+$4:(=]Fi_pFc]p7%Fh_`l7$Fcb]l$!+qaIW]Fi_pFc]p7%F\\``l7 $F]b]l$!+(R\"fn]Fi_pFc]p7%F```l7$Fga]l$!+S)o&)3&Fi_pFc]p7%Fd``l7$Faa]l $!+9(Rs5&Fi_pFc]p7%Fh``l7$F[a]l$!+_=hB^Fi_pFc]p7%F\\a`l7$Fe`]l$!+<**pP ^Fi_pFc]p7%F`a`l7$F_`]l$!+Tv_\\^Fi_pFc]p7%Fda`l7$Fi_]l$!+\"=J\"f^Fi_pF c]p7%Fha`l7$Fc_]l$!+[lcm^Fi_pFc]p7%F\\b`l7$F]_]l$!++\">><&Fi_pFc]p7%F` b`l7$Fg^]l$!+B8Kv^Fi_pFc]p7%Fdb`l7$Fa^]l$!+i=)p<&Fi_pFc]p7%Fhb`l7$F[^] l$!+IkBx^Fi_pFc]p7%F\\c`l7$Fe]]l$!+e5kw^Fi_pFc]p7%F`c`l7$F_]]l$!+hD:w^ Fi_pFc]p7%Fdc`l7$Fi\\]l$!+%**)\\x^Fi_pFc]p7%Fhc`l7$Fc\\]l$!+**3(R=&Fi_ pFc]p7%F\\d`l7$F]\\]l$!+P#y>?&Fi_pFc]p7%F`d`l7$Fg[]l$!+s>1U_Fi_pFc]p7% Fdd`l7$Fa[]l$!+Y&[!4`Fi_pFc]p7%Fhd`l7$F[[]l$!+OiI'Q&Fi_pFc]p7%F\\e`l7$ Fej\\l$!+'**)ycaFi_pFc]p7%F`e`l7$F_j\\l$!+zIV;bFi_pFc]p7%Fde`l7$Fii\\l $!+!)3;mbFi_pFc]p7%Fhe`l7$Fci\\l$!+0*\\vg&Fi_pFc]p7%F\\f`l7$F]i\\l$!+t #H>k&Fi_pFc]p7%F`f`l7$Fgh\\l$!+9tIqcFi_pFc]p7%Fdf`l7$Fah\\l$!+btV$p&Fi _pFc]p7%Fhf`l7$F[h\\l$!+(>')=r&Fi_pFc]p7%F\\g`l7$Feg\\l$!+Tb3EdFi_pFc] p7%F`g`l7$F_g\\l$!+&3ojt&Fi_pFc]p7%Fdg`l7$Fif\\l$!+8E*Hu&Fi_pFc]p7%Fhg `l7$Fcf\\l$!+u:;YdFi_pFc]p7%F\\h`l7$F]f\\l$!+yN.YdFi_pFc]p7%F`h`l7$Fge \\l$!+;CtUdFi_pFc]p7%Fdh`l7$Fae\\l$!+'o`jt&Fi_pFc]p7%Fhh`l7$F[e\\l$!+I (pps&Fi_pFc]p7%F\\i`l7$Fed\\l$!+tLj9dFi_pFc]p7%F`i`l7$F_d\\l$!+,3Q*p&F i_pFc]p7%Fdi`l7$Fic\\l$!+,QB\"o&Fi_pFc]p7%Fhi`l7$Fcc\\l$!+L:?gcFi_pFc] p7%F\\j`l7$F]c\\l$!+N>GOcFi_pFc]p7%F`j`l7$Fgb\\l$!+*zi%4cFi_pFc]p7%Fdj `l7$Fab\\l$!+YDszbFi_pFc]p7%Fhj`l7$F[b\\l$!+.1.ZbFi_pFc]p7%F\\[al7$Fea \\l$!+LsM6bFi_pFc]p7%F`[al7$F_a\\l$!+3DisaFi_pFc]p7%Fd[al7$Fi`\\l$!+nQ zIaFi_pFc]p7%Fh[al7$Fc`\\l$!+J8y&Q&Fi_pFc]p7%F\\\\al7$F]`\\l$!+\\(yuL& Fi_pFc]p7%F`\\al7$Fg_\\l$!+9#QdG&Fi_pFc]p7%Fd\\al7$Fa_\\l$!+9CMI_Fi_pF c]p7%Fh\\al7$F[_\\l$!+(za4<&Fi_pFc]p7%F\\]al7$Fe^\\l$!+ro-2^Fi_pFc]p7% F`]al7$F_^\\l$!+cwhP]Fi_pFc]p7%Fd]al7$Fi]\\l$!+vD-h\\Fi_pFc]p7%Fh]al7$ Fc]\\l$!+R6)R([Fi_pFc]p7%F\\^al7$F]]\\l$!+T)=,x%Fi_pFc]p7%F`^al7$Fg\\ \\l$!+WZ$*QYFi_pFc]p7%Fd^al7$Fa\\\\l$!+XpkvWFi_pFc]p7%Fh^al7$F[\\\\l$! +rt$pH%Fi_pFc]p7%F\\_al7$Fe[\\l$!+X/\"*>TFi_pFc]p7%F`_al7$F_[\\l$!+^-q [RFi_pFc]p7%Fd_al7$Fij[l$!+$=rCy$Fi_pFc]p7%Fh_al7$Fcj[l$!+YKv>OFi_pFc] p7%F\\`al7$F]j[l$!+6%[$fMFi_pFc]p7%F``al7$Fgi[l$!+GLS+LFi_pFc]p7%Fd`al 7$$!+%)RC'y%Fjbp$!+.1MUJFi_pFc]p7%Fh`al7$F[i[l$!+&=%y%)HFi_pFc]p7%F^aa l7$Feh[l$!+.&*\\FGFi_pFc]p7%Fbaal7$$\"+Fk\"*[#Fi_pFc]p7%Fdbal7$Fgf[l$!+(oK?/#Fi_pFc]p7%Fhbal7$Faf[l$! +(pb\\)=Fi_pFc]p7%F\\cal7$F[f[l$!+Zo(ys\"Fi_pFc]p7%F`cal7$$\"++Kl3HFd` p$!+Orzq:Fi_pFc]p7%Fdcal7$F_e[l$!+(>\"Fbbv;$!$R'Fbbv$\"$R'Fbbv" 1 2 0 1 10 0 2 9 1 2 1 1.000000 45.000000 45.000000 0 0 "Curve 1" "Curve 2" "Curve 3" " Curve 4" "Curve 5" "Curve 6" }}}}{PARA 0 "" 0 "" {TEXT -1 0 "" }} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}}{PARA 0 "" 0 "" {TEXT -1 52 "#---------------------------------------------------" }}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 36 "coefficients for the combined scheme" } }{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 35 "coefficients of the co mbined scheme" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 4879 "ee := \{c[2]=1/16,\nc[3]=112/1065,\nc[4]=56/35 5,\nc[5]=39/100,\nc[6]=7/15,\nc[7]=39/250,\nc[8]=24/25,\nc[9]=14435868 /16178861,\nc[10]=11/12,\nc[11]=19/20,\nc[12]=1,\nc[13]=1,\n\na[2,1]=1 /16,\na[3,1]=18928/1134225,\na[3,2]=100352/1134225,\na[4,1]=14/355,\na [4,2]=0,\na[4,3]=42/355,\na[5,1]=94495479/250880000,\na[5,2]=0,\na[5,3 ]=-352806597/250880000,\na[5,4]=178077159/125440000,\na[6,1]=12089/252 720,\na[6,2]=0,\na[6,3]=0,\na[6,4]=2505377/10685520,\na[6,5]=960400/52 09191,\na[7,1]=21400899/350000000,\na[7,2]=0,\na[7,3]=0,\na[7,4]=30643 29829899/27126050000000,\na[7,5]=-21643947/592609375,\na[7,6]=12439194 3/6756250000,\na[8,1]=-15365458811/13609565775,\na[8,2]=0,\na[8,3]=0, \na[8,4]=-7/5,\na[8,5]=-8339128164608/939060038475,\na[8,6]=3419368004 88/47951126225,\na[8,7]=1993321838240/380523459069,\na[9,1]=-184091125 2282376584438157336464708426954728061551/\n 2991923615171151921 596253813483118262195533733898,\na[9,2]=0,\na[9,3]=0,\na[9,4]=-1476496 0804048657303638372252908780219281424435/\n 2981692102565021975 611711269209606363661854518,\na[9,5]=-87532504850213044111861342178526 6742862694404520560000/\n 1702120304288944183955716775759613394 95435011888324169,\na[9,6]=7632051964154290925661849798370645637589377 834346780/\n 173408725741881158304980034758186526047923395039665 9,\na[9,7]=7519834791971137517048532179652347729899303513750000/\n \+ 1045677303502317596597890707812349832637339039997351,\na[9,8]=1366 042683489166351293315549358278750/\n 144631418224267718165055326 464180836641,\na[10,1]=-63077736705254280154824845013881/7836935785378 6633855112190394368,\na[10,2]=0,\na[10,3]=0,\na[10,4]=-319483465108209 70247215/6956009216960026632192,\na[10,5]=-337860480539425529245348937 5/517042670569824692230499952,\na[10,6]=100158784418332598119809145022 0795/184232684207722503701669953872896,\na[10,7]=187023075231349900768 014890274453125/25224698849808178010752575653374848,\na[10,8]=19081585 50070998850625/117087067039189929394176,\na[10,9]=-5295681828815666822 7044990077324877908565/\n 291277995947743398634982222441235395 1940608,\na[11,1]=-10116106591826909534781157993685116703/956281994503 6894030442231411871744000,\na[11,2]=0,\na[11,3]=0,\na[11,4]=-962354131 7323077848129/3864449564977792573440,\na[11,5]=-4823348333146829406881 375/576413233634141239944816,\na[11,6]=6566119246514996884067001154977 284529/970305487021846325473990863582315520,\na[11,7]=2226455130519213 549256016892506730559375/364880443159675255577435648380047355776,\na[1 1,8]=39747262782380466933662225/1756032802431424164410720256,\na[11,9] =48175771419260955335244683805171548038966866545122229/\n 19897 86420513815146528880165952064118903852843612160000,\na[11,10]=-2378292 068163246/47768728487211875,\na[12,1]=-3218022174758599831659045535578 571/1453396753634469525663775847094384,\na[12,2]=0,\na[12,3]=0,\na[12, 4]=26290092604284231996745/5760876126062860430544,\na[12,5]=-697069297 560926452045586710000/41107967755245430594036502319,\na[12,6]=18273578 20434213461438077550902273440/139381013914245317709567680839641697,\na [12,7]=643504802814241550941949227194107500000/24212460911883655086049 4007545333945331,\na[12,8]=162259938151380266113750/590910828352441834 97007,\na[12,9]=-23028251632873523818545414856857015616678575554130463 402/\n 2001316918319144450344390524040560334997842450415162905 5,\na[12,10]=7958341351371843889152/3284467988443203581305,\na[12,11]= -507974327957860843878400/121555654819179042718967,\na[13,1]=-54908062 4436801105208519835138333/353987109028707139687100885600400,\na[13,2]= 0,\na[13,3]=0,\na[13,4]=29116675312186033956481/5331818957833865866320 ,\na[13,5]=-91153092961177216058210567600/7609267653017028089793994539 ,\na[13,6]=1540775569495234383390307262972464/164751248733597841985831 445059895,\na[13,7]=-2027488254536386321212021357622300000/75638007843 13191306927092359781550321,\na[13,8]=-707592954577756600025/2430671607 007101253926,\na[13,9]=-3158694068773701034403897635103848320766741107 31/\n 550845901016714794114516357859531754926300500125,\na[13, 10]=4305840920849725632512/5066413598663346018375,\na[13,11]=0,\na[13, 12]=0,\n\nb[1]=4631674879841/103782082379976,\nb[2]=0,\nb[3]=0,\nb[4]= 0,\nb[5]=0,\nb[6]=14327219974204125/40489566827933216,\nb[7]=272076232 4010009765625000/10917367480696813922225349,\nb[8]=-498533005859375/95 352091037424,\nb[9]=40593203046377724792670503059617543740245963790976 5779/\n 78803919436321841083201886041201537229769115088303952,\nb[ 10]=-10290327637248/1082076946951,\nb[11]=863264105888000/858146622533 13,\nb[12]=-29746300739/247142463456,\nb[13]=0,\n\n`b*`[1]=32550309688 9/7282953149472,\n`b*`[2]=0,\n`b*`[3]=0,\n`b*`[4]=0,\n`b*`[5]=0,\n`b*` [6]=495437430316125/1396191959583904,\n`b*`[7]=760741072216796875000/3 055518466469861159313,\n`b*`[8]=1186143278515625/444976424841312,\n`b* `[9]=13791579353894559147128282108092181066885426809/35153330324358742 39702147176443897942760320040,\n`b*`[10]=-30733299644928/5410384734755 ,\n`b*`[11]=0,\n`b*`[12]=0,\n`b*`[13]=-45884771325/82380821152\}:" }}} {PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT 259 5 "nodes" }{TEXT -1 2 ": " }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 32 "s eq(c[i]=subs(ee,c[i]),i=2..13);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6./&% \"cG6#\"\"##\"\"\"\"#;/&F%6#\"\"$#\"$7\"\"%l5/&F%6#\"\"%#\"#c\"$b$/&F% 6#\"\"&#\"#R\"$+\"/&F%6#\"\"'#\"\"(\"#:/&F%6#FE#F>\"$]#/&F%6#\"\")#\"# C\"#D/&F%6#\"\"*#\")oeV9\")h)yh\"/&F%6#\"#5#\"#6\"#7/&F%6#Fin#\"#>\"#? /&F%6#FjnF)/&F%6#\"#8F)" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 " " 0 "" {TEXT 259 20 "linking coefficients" }{TEXT -1 1 ":" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 50 "seq(seq(a[i,j]=subs(ee,a[i,j]),j=1. .i-1),i=1..13);" }}{PARA 12 "" 1 "" {XPPMATH 20 "6jo/&%\"aG6$\"\"#\"\" \"#F(\"#;/&F%6$\"\"$F(#\"&G*=\"(DU8\"/&F%6$F.F'#\"'_.5F1/&F%6$\"\"%F(# \"#9\"$b$/&F%6$F:F'\"\"!/&F%6$F:F.#\"#UF=/&F%6$\"\"&F(#\")za\\%*\"*++) 3D/&F%6$FJF'FA/&F%6$FJF.#!*(f1GNFM/&F%6$FJF:#\"*fr2y\"\"*++WD\"/&F%6$ \"\"'F(#\"&*37\"'?FD/&F%6$FinF'FA/&F%6$FinF.FA/&F%6$FinF:#\"(x`]#\")?b o5/&F%6$FinFJ#\"'+/'*\"(\">4_/&F%6$\"\"(F(#\")**3S@\"*+++]$/&F%6$FbpF' FA/&F%6$FbpF.FA/&F%6$FbpF:#\".**)H)HV1$\"/+++]g7F/&F%6$FbpFJ#!)ZRk@\"* v$4Ef/&F%6$FbpFin#\"*V>RC\"\"+++Dcn/&F%6$\"\")F(#!,6)eaO:\",vdc4O\"/&F %6$FarF'FA/&F%6$FarF.FA/&F%6$FarF:#!\"(FJ/&F%6$FarFJ#!.3Y;G\"R$)\"-v%Q +1R*/&F%6$FarFin#\"-)[+o$>M\",Di7^z%/&F%6$FarFbp#\".S#Q=K$*>\"-p!fM_!Q /&F%6$\"\"*F(#!R^:1GZ&pU3ZYOt:QWewBG_7\"4%=\"R)*QtLb>i#=J[8QD'f@>:r^hB >*H/&F%6$FetF'FA/&F%6$FetF.FA/&F%6$FetF:#!PNWU\"G>-y3HDs$QOId'[S!3'\\w 9\"O=X&=mjjg4#p7r6c(>-lD5#p\")H/&F%6$FetFJ#!W++c?XS%piGum_y@Mh=6WI@][] Kv)\"WpTK))=,Na\\R8'fdx;d&R=W*)G/.7-0Kw\"UfmR]RBz/El=eZ.!)\\Ie6)=uD(3MzM)>v\"U^t**R!RtjK)\\B\"yq!*yf'f7^&Qj'y`yNp$y/&F%6$F`wF'FA/&F%6$F`wF.FA/&F%6$F `wF:#!8:sCq4#3^Y$[>$\"7#>Km-gp@4g&p/&F%6$F`wFJ#!=v$*[`CHbUR0[gyL\"<_** \\IApC)p0nUq^/&F%6$F`wFin#\"C&z?-X\"4)>\")fK$=Wye,5\"B'*G(Q&*p;q.Ds2Uo KU=/&F%6$F`wFbp#\"EDJXu-*[,o2!*\\8BvI-(=\"D[[P`cd_2,y\"3)\\))pC_#/&F%6 $F`wFar#\"7D1&))*42]&e\"3>\"9wTRH**=Rq1(3<\"/&F%6$F`wFet#!Jl&3z([Kx+* \\/F#om:)G=o&H&\"L31%>&RN7WAA)\\j)RVx%f*zF\"H/&F%6$\"#6F(#!G.n6&o$*z: \"yM&4p#=f1h65\"F+Sur=TJAWIS*o.X*>Gc*/&F%6$FazF'FA/&F%6$FazF.FA/&F%6$F azF:#!7H\"[y2BtJTNi*\"7SMd#zx\\c\\W'Q/&F%6$FazFJ#!:v8)oSHo9L$[L#[\"9;[ %*R79MOB8kd/&F%6$FazFin#\"FHXGx\\:,q1%)o*\\^Y#>hc'\"E?bJ#ej3*RZDj%=-([ 0.(*/&F%6$FazFbp#\"Iv$f0t1D*o,c#\\N@>08bkA#\"HwdNZ+Q[cVxbDv'fJW!)[O/&F %6$FazFar#\";DAmLpY!Q#yisuR\"=c-s5W;C9V-G.c9xv\"[\"X++;7O%G&Q!*=T1_f;!))Gl9:Q^?ky*)>/&F%6$FazF`w#!1 YK;o?HyB\"2v=@([G(ox%/&F%6$\"#7F(#!Cr&yb`X!f;$)*feZB%G/E4!HE\"7W0V gG1Eh(3w&/&F%6$Fh]lFJ#!?++r'eX?XE4c(Hpqp\">>B]OSfIaCbx'z5T/&F%6$Fh]lFi n#\"FSMF-4bx!Q9Y8UV?yNF=\"E(pT'R3on&4xJXU\"R,\"QR\"/&F%6$Fh]lFbp#\"H++ ]2T>F#\\>%4bTU\"G![]V'\"HJ`%RLXv+%\\g3bO)=\"4Y7U#/&F%6$Fh]lFar#\"9]P6m -Q^\"Q*fA;\"82q\\$=W_$G3\"4f/&F%6$Fh]lFet#!Y-MYITbv&ymh:q&o&[TX&=Q_tGj ^#GI#\"Yb!H;:/XUy*\\Lg0/C0RW.XW\">$=pJ,?/&F%6$Fh]lF`w#\"7_\"*)Q%=P^8Me z\"708e.KW))zY%G$/&F%6$Fh]lFaz#!9+%yQ%3'y&zKuz]\"9n*=F/z\">[lb:7/&F%6$ \"#8F(#!BL$Q^$)>&3_5,oVC13\\&\"B+/g&)35(oRrqG!4r)RN/&F%6$FealF'FA/&F%6 $FealF.FA/&F%6$FealF:#\"8\"[cR.'=7`n;\"H\"7?j'e'Q$y&*==L&/&F%6$FealFJ# !>+wc5#eg@x6'H4`6*\"=RX*Rz*3Gq,`wE4w/&F%6$FealFin#\"CkC(HE2.R$QM_\\pbx S:\"B&*)f]WJe)>%yfL([7vk\"/&F%6$FealFbp#!F++IAwN@?@@jQOXD)[F?\"F@.b\"y fB4FpI\">8Vy+Qc(/&F%6$FealFar#!6D+gcxdaHf2(\"7ERD,r+2;nIC/&F%6$FealFet #!QJ26um2K[Q5Nw*QSM5qt(oSpeJ\"QD,]+j#\\vJ&fyN;X6%z9n,,f%3b/&F%6$FealF` w#\"77DjD(\\3#4%eI%\"7v$=gMj')f8k1&/&F%6$FealFazFA/&F%6$FealFh]lFA" }} }{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT 259 39 "weights for the 13 stage order 8 scheme" }{TEXT -1 1 ":" }}{EXCHG {PARA 0 "> \+ " 0 "" {MPLTEXT 1 0 32 "seq(b[i]=subs(ee,b[i]),i=1..13);" }}{PARA 12 " " 1 "" {XPPMATH 20 "6//&%\"bG6#\"\"\"#\".T)z[nJY\"0w*zB3#y.\"/&F%6#\" \"#\"\"!/&F%6#\"\"$F//&F%6#\"\"%F//&F%6#\"\"&F//&F%6#\"\"'#\"2DT?u*>sK 9\"2;K$z#oc*[S/&F%6#\"\"(#\":+]il(4+,CBw?F\";\\`AAR\"op![nt\"4\"/&F%6# \"\")#!0v$fe+L&)\\\"/Cu.\"4_`*/&F%6#\"\"*#\"Wzdw4zjfCSPa<'fI]qEzCxPYI? $fS\"V_RI)3:\"p(Hs`,7/')=?$3T=KO%>R!)y/&F%6#\"#5#!/[sjF.H5\".^p%p2#3\" /&F%6#\"#6#\"0+!))e5kK')\"/8LDiY\"e)/&F%6#\"#7#!,R2IY(H\"-cMYUrC/&F%6# \"#8F/" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT 259 39 "weights for the 13 stage order 7 scheme" }{TEXT -1 1 ":" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 38 "seq(`b*`[i]=subs(ee,`b*`[i]),i=1..1 3);" }}{PARA 12 "" 1 "" {XPPMATH 20 "6//&%#b*G6#\"\"\"#\"-*)o4.bK\".s% \\J&HG(/&F%6#\"\"#\"\"!/&F%6#\"\"$F//&F%6#\"\"%F//&F%6#\"\"&F//&F%6#\" \"'#\"0DhJIuV&\\\"1/Ref>>'R\"/&F%6#\"\"(#\"6+](oz;A2T2w\"78$f6')pkY=b0 $/&F%6#\"\")#\"1Dc^yK9'=\"\"078%[Uw\\W/&F%6#\"\"*#\"P4oU&)o1\"=#43@GGr 9fX*QNz:z8\"OS+KgF%z*QWwr9-(RU(eVKIL:N/&F%6#\"#5#!/G\\k*HL2$\".bZt%Q5a /&F%6#\"#6F//&F%6#\"#7F//&F%6#\"#8#!,D8x%)e%\",_6#3Q#)" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}} {PARA 0 "" 0 "" {TEXT -1 52 "#---------------------------------------- -----------" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}}{PARA 0 "" 0 "" {TEXT -1 40 "#=======================================" }} {SECT 1 {PARA 4 "" 0 "" {TEXT -1 41 "a modification of Verner's \"Mapl e\" scheme" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 97 "This scheme is a modification of Jim Verner's \"Maple\" scheme \+ considered in the previous section. " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 15 "The nodes are: " }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "c[2] = 3/50" "6#/&%\"cG6#\"\"#*&\"\"$\" \"\"\"#]!\"\"" }{TEXT -1 3 ", " }{XPPEDIT 18 0 "c[3] = 469/4510" "6#/ &%\"cG6#\"\"$*&\"$p%\"\"\"\"%5X!\"\"" }{TEXT -1 3 ", " }{XPPEDIT 18 0 "c[4] = 1407/9020" "6#/&%\"cG6#\"\"%*&\"%29\"\"\"\"%?!*!\"\"" } {TEXT -1 3 ", " }{XPPEDIT 18 0 "c[5] = 100/257" "6#/&%\"cG6#\"\"&*&\" $+\"\"\"\"\"$d#!\"\"" }{TEXT -1 3 ", " }{XPPEDIT 18 0 "c[6] = 7/15" " 6#/&%\"cG6#\"\"'*&\"\"(\"\"\"\"#:!\"\"" }{TEXT -1 3 ", " }{XPPEDIT 18 0 "c[7] = 12/77" "6#/&%\"cG6#\"\"(*&\"#7\"\"\"\"#x!\"\"" }{TEXT -1 3 ", " }{XPPEDIT 18 0 "c[8] = 26/27" "6#/&%\"cG6#\"\")*&\"#E\"\"\"\"# F!\"\"" }{TEXT -1 3 ", " }{XPPEDIT 18 0 "c[9] = 7101051/7944748" "6#/ &%\"cG6#\"\"**&\"(^55(\"\"\"\"([Z%z!\"\"" }{TEXT -1 3 ", " }{XPPEDIT 18 0 "c[10] = 11/12" "6#/&%\"cG6#\"#5*&\"#6\"\"\"\"#7!\"\"" }{TEXT -1 3 ", " }{XPPEDIT 18 0 "c[11] = 19/20" "6#/&%\"cG6#\"#6*&\"#>\"\"\"\"# ?!\"\"" }{TEXT -1 3 ", " }{XPPEDIT 18 0 "c[12] = 1" "6#/&%\"cG6#\"#7 \"\"\"" }{TEXT -1 3 ", " }{XPPEDIT 18 0 "c[13]=1" "6#/&%\"cG6#\"#8\" \"\"" }{TEXT -1 2 ". " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}{PARA 0 "" 0 "" {TEXT -1 52 "#------ ---------------------------------------------" }}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 28 "checking the combined scheme" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 36 "coefficients of the combined scheme " }} {PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 5227 "ee := \{c[2]=3/50,\nc[3]=469/4510,\nc[4]=1407/9020,\nc[5]=100/25 7,\nc[6]=7/15,\nc[7]=12/77,\nc[8]=26/27,\nc[9]=7101051/7944748,\nc[10] =11/12,\nc[11]=19/20,\nc[12]=1,\nc[13]=1,\n\na[2,1]=3/50,\na[3,1]=1693 09/12204060,\na[3,2]=219961/2440812,\na[4,1]=1407/36080,\na[4,2]=0,\na [4,3]=4221/36080,\na[5,1]=12985296430100/33603736057857,\na[5,2]=0,\na [5,3]=-16218027650000/11201245352619,\na[5,4]=48744170200000/336037360 57857,\na[6,1]=3817807/81405000,\na[6,2]=0,\na[6,3]=0,\na[6,4]=1027419 1312/43991343405,\na[6,5]=40755997793/218862405000,\na[7,1]=1590080257 8/262289621825,\na[7,2]=0,\na[7,3]=0,\na[7,4]=6340315606886400/5627491 9101668113,\na[7,5]=-451131008276934/12909147305279225,\na[7,6]=984103 149600/56231606512081,\na[8,1]=-6560178806172785891/567285870653568000 0,\na[8,2]=0,\na[8,3]=0,\na[8,4]=-17/2,\na[8,5]=-587060588751798636931 6613/652465735085103413760000,\na[8,6]=16204960050140425727/2255297505 252764544,\na[8,7]=9658417761218465607893/776920972593666355200,\na[9, 1]=-140461899806965182831986352616636648283813334213167183799/\n \+ 231595702353396941018788535727415998878963658147758080000,\na[9,2]=0 ,\na[9,3]=0,\na[9,4]=-230388436455789139579016226863459716720858844533 129533/\n 28162599104012402938735024801544932535316065892462592 ,\na[9,5]=-97906309983749574521298472575929535517651647721497960654138 44391/\n 187912086354556872209131148539200248025949734645340979 0689280000,\na[9,6]=19704323580398550825484436390223525959900159607690 4803130443/\n 44561965459500736173318259755870730512740108168829 012803584,\na[9,7]=294620817758575088586344956781032015012776829359971 133/\n 28166985596967449488723424839046646360350769597644800,\na [9,8]=126873616630173037746837292557884226969/132172985196530229490710 50320940567220224,\na[10,1]=-41617847659972885613040591872450035471/52 813005326633659912790449394810880000,\na[10,2]=0,\na[10,3]=0,\na[10,4] =-269951087764496912154173876831/29845195411834947899362922496,\na[10, 5]=-2749986656729113102825386117572168116123/4224161445521692301226128 85222850560000,\na[10,6]=324525706880938787969090744301341259977691/60 253355178721829324099288547678254006272,\na[10,7]=41487408138830374659 637178881584696821/3493656315517022583105833038931558400,\na[10,8]=932 598433870374903885/57293130256189894463488,\na[10,9]=-3345742205378079 321732624680747620/191228884555191941725363563271035831,\na[11,1]=-152 032134715446654236835762683933540260220443/144031229933501187933392929 852012953600000000,\na[11,2]=0,\na[11,3]=0,\na[11,4]=-1992628243045045 9437356479596949/2204175587073348224489986560000,\na[11,5]=-2861670807 732347804933810531212726938261187/343166554822474341490702928140697600 000000,\na[11,6]=1334063326593468056350327316268714319601715109229/198 658539882559017093840573946438829079429120000, a[11,7]=255870600859869 06515751471801297011229903719273/2019042940187611700532950824131079569 408000000,\na[11,8]=1292083162729235236212706557/477016463963366755242 88000000,\na[11,9]=2091032384514247750201495665581688757138358851/6072 9869176412564938423797095435796334060237500,\na[11,10]=-37769438225840 3/5816420236687500,\na[12,1]=-240860055167915269684777564157507287321/ 117003858749851375574083424673098438400,\na[12,2]=0,\na[12,3]=0,\na[12 ,4]=-98894702310834163626964955900/10545476035409921643978852639,\na[1 2,5]=-252251601740617744737074278562623715189/164181778496132296852310 88852333734400,\na[12,6]=840813041585474673597744565360698902394175/70 780871835683840841986501581774346003376,\na[12,7]=24189514716339363218 9444055554841293171617/15082171924595104472807447927564153421312,\na[1 2,8]=42847753308025275666/29882754550596969299,\na[12,9]=-771733622226 4885240564054011744574965898992045440/84790334188947973255478145542071 28260520626293873,\na[12,10]=16262726151696512/8940486723127047,\na[12 ,11]=-16001553345352000000/6505707112968117383,\na[13,1]=-274402293455 9056024469622604306540529/1738222573548705964451797934060438400,\na[13 ,2]=0,\na[13,3]=0,\na[13,4]=-35144243656562246120945761400/59532618684 37390070644832157,\na[13,5]=-54637123672393421308996942594910689159/46 34295873089356958410234071688953600,\na[13,6]=612390812902422662436708 59206851281463175/6659681426411632770651421460598964953048,\na[13,7]=5 134209062671060909676995239537507099025/462964694026104436494336717002 613458688,\na[13,8]=-370363100142446513442/1535150001377392224067,\na[ 13,9]=-212020682512329076068221666486441382648974080/39026028930928949 7513964664381911193640454941,\na[13,10]=2242504532188281088/2908865886 282319623,\na[13,11]=0,\na[13,12]=0,\n\nb[1]=61903079921/1389136000824 ,\nb[2]=0,\nb[3]=0,\nb[4]=0,\nb[5]=0,\nb[6]=402653566621875/1136204715 874336,\nb[7]=1872020731216849170881/7515476193013636507650,\nb[8]=-48 961117784712267/15177413999438600,\nb[9]=18246781664777103430227808871 39376956665357164944/332726480751222257863787417240798843822259974895, \nb[10]=-126172885488/13167945725,\nb[11]=41251039800000/5264998725539 ,\nb[12]=-107501739/674957600,\nb[13]=0,\n\n`b*`[1]=5709741119/1279467 36918,\n`b*`[2]=0,\n`b*`[3]=0,\n`b*`[4]=0,\n`b*`[5]=0,\n`b*`[6]=974589 49528125/274256310728288,\n`b*`[7]=79520000370367181779/31954600330066 1568600,\n`b*`[8]=5689283182401789/2168201999919800,\n`b*`[9]=12458274 43533127167605703273831223125580576/2981022074572680330885817370627029 57958355,\n`b*`[10]=-76671637728/13167945725,\n`b*`[11]=0,\n`b*`[12]=0 ,\n`b*`[13]=-5522626587/8774448800\}:" }}}{PARA 0 "" 0 "" {TEXT -1 0 " " }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 20 "printed coefficients" }}{PARA 0 "" 0 " " {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 357 "subs(ee,m atrix([seq([c[i],seq(a[i,j],j=1..i-1),``$(7-i)],i=2..7)]));\nfor ii fr om 8 to 13 do\n print(``);\n print(c[ii]=subs(ee,c[ii])); \n for jj to ii-1 do\n print(a[ii,jj]=subs(ee,a[ii,jj]));\n end do:\n end do:print(``);\nfor ii to 13 do\n print(b[ii]=subs(ee,b[ii]));\ne nd do:print(``);\nfor ii to 13 do\n print(`b*`[ii]=subs(ee,`b*`[ii]) );\nend do:" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#K%'matrixG6#7(7)#\"\"$ \"#]F(%!GF+F+F+F+7)#\"$p%\"%5X#\"'4$p\"\")gS?7#\"'h*>#\"(73W#F+F+F+F+7 )#\"%29\"%?!*#F8\"&!3O\"\"!#\"%@UF;F+F+F+7)#\"$+\"\"$d##\"/+,V'H&)H\" \"/dy0OPgLF<#!/++lF!=i\"\"/>ENX7?6#\"/++?qTu[FEF+F+7)#\"\"(\"#:#\"(2y \"Q\")+]S\")Fu-\"\",0MM\"*R%#\",$z(*fvS\"-+]Si)=#F+7)#\"#7\" #x#\",yD!3!f\"\"-D=i*Gi#F\\Fc#!0MpF358^%\"2D #z_IZ\"4H\"#\"-+'\\J5%)*\"/\"37lgJi&Q(pprint76\"" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%!G" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/&%\"cG6#\"\")# \"#E\"#F" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/&%\"aG6$\"\")\"\"\"#!4\"* eysh!)y,c'\"4++oNlqeGn&" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/&%\"aG6$\" \")\"\"#\"\"!" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/&%\"aG6$\"\")\"\"$\" \"!" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/&%\"aG6$\"\")\"\"%#!#<\"\"#" } }{PARA 11 "" 1 "" {XPPMATH 20 "6#/&%\"aG6$\"\")\"\"&#!:8mJpj)z^()egqe \"9++w8M5&3NdY_'" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/&%\"aG6$\"\")\"\" '#\"5FdUS,0g\\?;\"4WXw__](HbA" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/&%\" aG6$\"\")\"\"(#\"7$*ygl%=7w<%e'*\"6+_NmOfs4#px" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%!G" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/&%\"cG6#\"\"*# \"(^55(\"([Z%z" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/&%\"aG6$\"\"*\"\"\" #!Z*z$=nJ@ML\"QG[mj;EN')>$G=lp!)**=YS\"\"Z++3ex9eO'*y))*fTFd`)y=5%pR`B q&fJ#" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/&%\"aG6$\"\"*\"\"#\"\"!" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#/&%\"aG6$\"\"*\"\"$\"\"!" }}{PARA 11 " " 1 "" {XPPMATH 20 "6#/&%\"aG6$\"\"*\"\"%#!WL&HJ`W)e3s;(fM'oA;!z&R\"*y bkV)QI#\"V#fiC*e1;``K\\a,[-N(QHS7S5*fi\"G" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/&%\"aG6$\"\"*\"\"&#![o\"RWQTlgz\\@xk^w^N&Hfds%)H@Xd\\P )*4j!z*\"[o++G*o!z4MXYt\\f-[-?R&[684A(obaj37z=" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/&%\"aG6$\"\"*\"\"'#\"gnV/8.[!p2'f,!*ff_B-ROW[D3b)R!eBV q>\"fn%e.G,H)o\"3,u70tqevf#=LcW" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/&%\"aG6$\"\"*\"\"(#\"WL6(*f$Hox7],K5yc\\M'e)3v&ex\"3i%H\"V+[k(f p2NgjkY!R[UB()[\\u'pf&)p;G" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/&%\"aG6 $\"\"*\"\")#\"HppA%)yb#HPouPI&)H'\\AHO**y%\\$=T&>X)H " }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/&%\"aG6$\"#5\"\"&#!IBh6o@d>bX))G7>" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%!G" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#/&%\"cG6#\"#6#\"#>\"#?" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/&%\"aG6$\"#6\"\"\"#!NV/Ag-aLRoid$oBamW:Z8K?:\"N++ ++O&H,_)HHRLz=,N$*H7.W\"" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/&%\"aG6$ \"#6\"\"#\"\"!" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/&%\"aG6$\"#6\"\"$\" \"!" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/&%\"aG6$\"#6\"\"%#!A\\pfzkNP%f /XICGE*>\"@++c')**[C#[L2(evT?#" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/&% \"aG6$\"#6\"\"&#!L(=h#Qps77`5Q$\\!yMKx!3nhG\"K++++wpS\"GHq!\\TVZA[bmJM " }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/&%\"aG6$\"#6\"\"'#\"RH#4^r,'>Vroi JF.Nc!oMfEL1M8\"Q++7H%z!H)Qk%RdSQ4" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/&%\"aG6$\"#6\"\"(#\"Pt#>P!*H7,(H,=Z^d^1p)f3gqeD\"O+++3 %p&z58C3&H`+?" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/&%\"aG6$\"#6 \"\")#\"=dlq7iBN#HF;$3#H\"\">+++)GCbnOjRY;qZ" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/&%\"aG6$\"#6\"\"*#\"O^)e$Qrv)o\"elc\\,-vZU^%QK54#\"P+v BgSL'zNa4(zB%Q\\c7k+f&\\'pijT $3J-Z*))*\">RE&)yRk@*4a.waa5" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/&%\"a G6$\"#7\"\"&#!H*=:Pii&yU2PZu<1u,;D_#\"G+WtLB&))3J_oHKh\\y<=k\"" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#/&%\"aG6$\"#7\"\"'#\"KvTR-*)pg`cWxftYZ &eTI\"3%)\"JwL+YVx\"e,l)>%3%QoN=(3yq" }}{PARA 11 "" 1 "" {XPPMATH 20 " 6#/&%\"aG6$\"#7\"\"(#\"K<;<$HT[bbSW*=KORjr9&*=C\"J78U`TcFzW2GZ/^fC><#3 :" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/&%\"aG6$\"#7\"\")#\"5mcFD!3LvZG% \"5*Hppf]Xv#))H" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/&%\"aG6$\"#7\"\"*# !RSa/#**)*e'\\dW<,aScS_)[EAiL+9wX47YAccOCW^$\"=d@$[k q+RP%o=E`f" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/&%\"aG6$\"#8\"\"&#!Gf\" *o5\\fUp**38U$RsO7PY&\"F+O&*)orSB5%epN*3teHMY" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/&%\"aG6$\"#8\"\"'#\"JvJY\"G^o?f3nViEU-H\"3R7'\"I[I&\\' *)fg9U^1xK;TE9ofm" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/&%\"aG6$\"#8\"\" (#\"ID!*42v`R_*pn441rE14U8&\"H)oeMh-qrOV\\OW5ESpkHY" }}{PARA 11 "" 1 " " {XPPMATH 20 "6#/&%\"aG6$\"#8\"\")#!6UM^YC9+JOq$\"7nSA#Rx8+]^`\"" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#/&%\"aG6$\"#8\"\"*#!N!3u*[EQTk[m;Aog2H B^#o??@\"NT\\XSO>6>QkY'R^(\\*G4$*Gg-R" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/&%\"aG6$\"#8\"#5#\"4)3\"G)=KX]UA\"4B'>BG')e')3H" }}{PARA 11 "" 1 " " {XPPMATH 20 "6#/&%\"aG6$\"#8\"#6\"\"!" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/&%\"aG6$\"#8\"#7\"\"!" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%!G" } }{PARA 11 "" 1 "" {XPPMATH 20 "6#/&%\"bG6#\"\"\"#\",@*zI!>'\".C3+O\"*Q \"" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/&%\"bG6#\"\"#\"\"!" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/&%\"bG6#\"\"$\"\"!" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/&%\"bG6#\"\"%\"\"!" }}{PARA 11 "" 1 "" {XPPMATH 20 "6# /&%\"bG6#\"\"&\"\"!" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/&%\"bG6#\"\"'# \"0v=imNl-%\"1OV(er/i8\"" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/&%\"bG6# \"\"(#\"7\")3<\\o@J2-s=\"7]w]OO,$>wa^(" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/&%\"bG6#\"\")#!2nAr%y<6'*[\"2+'Q%**RTx^\"" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/&%\"bG6#\"\"*#\"RW\\;d`mcpPRr)3yAIM5xZm\"yY#=\"Q&*[(*f A#Q%))zSsT(yjyDA7v![EFL" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/&%\"bG6#\" #5#!-)[&)G6u4d\"-=ptYz7" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/&%#b*G6#\"\"#\"\"!" }}{PARA 11 "" 1 "" {XPPMATH 20 "6# /&%#b*G6#\"\"$\"\"!" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/&%#b*G6#\"\"% \"\"!" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/&%#b*G6#\"\"&\"\"!" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/&%#b*G6#\"\"'#\"/D\"G&\\*eu*\"0)GG2JcUF" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#/&%#b*G6#\"\"(#\"5z<=n.P++_z\"6+'o:m+L +Y&>$" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/&%#b*G6#\"\")#\"1*y,C=$G*o& \"1+)>***>?o@" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/&%#b*G6#\"\"*#\"Lw0e DJAJQF.dgnr7LNWFeC\"\"Kb$ez&Hqiqt\"e)3L!osX2A5)H" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/&%#b*G6#\"#5#!,Gxjrm(\",Dd%z;8" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/&%#b*G6#\"#6\"\"!" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/ &%#b*G6#\"#7\"\"!" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/&%#b*G6#\"#8#!+( eEE_&\"++)[Wx)" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> \+ " 0 "" {MPLTEXT 1 0 1 ";" }}}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 151 "RK8_13eqs := [op(RowSumConditions( 13,'expanded')),op(OrderConditions(8,13,'expanded'))]:\n`RK7_13eqs*` : = subs(b=`b*`,OrderConditions(7,13,'expanded')):" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 6 "Check:" }}{EXCHG {PARA 0 " > " 0 "" {MPLTEXT 1 0 110 "subs(ee,RK8_13eqs):\nmap(u->lhs(u)-rhs(u),% );\nnops(%);\nsubs(ee,`RK7_13eqs*`):\nmap(u->lhs(u)-rhs(u),%);\nnops(% );" }}{PARA 12 "" 1 "" {XPPMATH 20 "6#7`x\"\"!F$F$F$F$F$F$F$F$F$F$F$F$ F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$ F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$ F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$ F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$ F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$ F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#\"$7#" }}{PARA 12 "" 1 "" {XPPMATH 20 "6#7ap\"\"!F$F$F$ F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$ F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$ F$F$F$F$F$F$F$F$F$F$F$" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#\"#&)" }}} {PARA 257 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 78 "Set-up \+ stage-order condtions to check for stage-orders from 2 to 6 inclusive. " }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 87 "for ct from 2 to 6 do\n so||ct||_13 := StageOrderConditions(ct,13,'expanded');\nend do:" }} }{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 59 "Stages 3 to 13 have the following respective stage-orders. " }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 175 "[seq([seq(expand(subs(ee,so||i||_13[j])),i =2..6)],j=1..11)]:\nmap(proc(L) local i; for i to nops(L) do if not ev alb(L[i]) then break end if end do; i end proc,%):\nsimplify(%); " }} {PARA 11 "" 1 "" {XPPMATH 20 "6#7-\"\"#\"\"$F%\"\"%F&F&F&F&F&F&F&" }}} {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 27 "The simpl ifying conditions:" }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "Sum(b[i]*a[i,j],i = j+1 .. 13) = b[j]*(1-c[j]);" "6#/-%$SumG6$*&&%\"b G6#%\"iG\"\"\"&%\"aG6$F+%\"jGF,/F+;,&F0F,F,F,\"#8*&&F)6#F0F,,&F,F,&%\" cG6#F0!\"\"F," }{TEXT -1 3 ", " }{XPPEDIT 18 0 "j=1" "6#/%\"jG\"\"\" " }{TEXT -1 9 " . . 12, " }}{PARA 0 "" 0 "" {TEXT -1 8 "(where " } {XPPEDIT 18 0 "c[1]=0" "6#/&%\"cG6#\"\"\"\"\"!" }{TEXT -1 17 " ) are s atisfied." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 151 "[Sum(b[i]*a[i,1],i=1+1..13)=b[1],seq(Sum(b[i]*a[i,j] ,i=j+1..13)=b[j]*(1-c[j]),j=2..12)]:\nmap(u->lhs(u)-rhs(u),subs(ee,eva l(subs(Sum=add,%))));\nnops(%);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#7. \"\"!F$F$F$F$F$F$F$F$F$F$F$" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#\"#7" } }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 27 "The sim plifying conditions:" }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "Sum(`b*`[i]*a[i,j],i = j+1 .. 13) = `b*`[j]*(1-c[j]);" "6#/-%$SumG6 $*&&%#b*G6#%\"iG\"\"\"&%\"aG6$F+%\"jGF,/F+;,&F0F,F,F,\"#8*&&F)6#F0F,,& F,F,&%\"cG6#F0!\"\"F," }{TEXT -1 3 ", " }{XPPEDIT 18 0 "j=1" "6#/%\"j G\"\"\"" }{TEXT -1 9 " . . 12, " }}{PARA 0 "" 0 "" {TEXT -1 8 "(where \+ " }{XPPEDIT 18 0 "c[1]=0" "6#/&%\"cG6#\"\"\"\"\"!" }{TEXT -1 17 " ) a re satisfied." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 163 "[Sum(`b*`[i]*a[i,1],i=1+1..13)=`b*`[1],seq(Sum( `b*`[i]*a[i,j],i=j+1..13)=`b*`[j]*(1-c[j]),j=2..12)]:\nmap(u->lhs(u)-r hs(u),subs(ee,eval(subs(Sum=add,%))));\nnops(%);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#7.\"\"!F$F$F$F$F$F$F$F$F$F$F$" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#\"#7" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 " " {TEXT -1 93 "We can calculate the principal error norm, that is, the 2-norm of the principal error terms. " }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 162 "errterms8_13 := PrincipalErrorTerms(8,13,'expanded') :\nsm := 0:\nfor ct to nops(errterms8_13) do\n sm := sm+(evalf(subs( ee,errterms8_13[ct])))^2;\nend do:\nsqrt(sm);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#$\"+.RzQl!#;" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 78 "We can also calculate the principal error norm \+ of the order 7 embedded scheme." }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 184 "`errterms7_13*` := subs(b=`b*`,PrincipalErrorTerms(7,13,'expa nded')):\nsm := 0:\nfor ct to nops(`errterms7_13*`) do\n sm := sm+(e valf(subs(ee,`errterms7_13*`[ct])))^2;\nend do:\nsqrt(sm);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#$\"+nsG.#*!#:" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}}{PARA 0 "" 0 " " {TEXT -1 52 "#---------------------------------------------------" } }{SECT 1 {PARA 4 "" 0 "" {TEXT -1 34 "construction of the order 8 sche me" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 93 "Fir st we construct a 12 stage order 8 scheme starting with a consideratio n of stages 2 to 8. " }}{PARA 0 "" 0 "" {TEXT -1 20 "We specify the no des" }}{PARA 256 "" 0 "" {TEXT -1 2 " " }{XPPEDIT 18 0 "c[2] = 3/50; " "6#/&%\"cG6#\"\"#*&\"\"$\"\"\"\"#]!\"\"" }{TEXT -1 2 ", " }{XPPEDIT 18 0 "c[5] = 100/257;" "6#/&%\"cG6#\"\"&*&\"$+\"\"\"\"\"$d#!\"\"" } {TEXT -1 2 ", " }{XPPEDIT 18 0 "c[6] = 7/15;" "6#/&%\"cG6#\"\"'*&\"\"( \"\"\"\"#:!\"\"" }{TEXT -1 2 ", " }{XPPEDIT 18 0 "c[7] = 12/77;" "6#/& %\"cG6#\"\"(*&\"#7\"\"\"\"#x!\"\"" }{TEXT -1 3 ", " }{XPPEDIT 18 0 "c [8] = 26/27;" "6#/&%\"cG6#\"\")*&\"#E\"\"\"\"#F!\"\"" }{TEXT -1 2 " \+ " }}{PARA 0 "" 0 "" {TEXT -1 33 "and the zero linking coefficients" }} {PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "a[4,2]=0" "6#/&%\"aG6 $\"\"%\"\"#\"\"!" }{TEXT -1 2 ", " }{XPPEDIT 18 0 "a[5,2]=0" "6#/&%\"a G6$\"\"&\"\"#\"\"!" }{TEXT -1 2 ", " }{XPPEDIT 18 0 "a[6,2]=0" "6#/&% \"aG6$\"\"'\"\"#\"\"!" }{TEXT -1 2 ", " }{XPPEDIT 18 0 "a[6,3]=0" "6#/ &%\"aG6$\"\"'\"\"$\"\"!" }{TEXT -1 2 ", " }{XPPEDIT 18 0 "a[7,2]=0" "6 #/&%\"aG6$\"\"(\"\"#\"\"!" }{TEXT -1 2 ", " }{XPPEDIT 18 0 "a[7,3]=0" "6#/&%\"aG6$\"\"(\"\"$\"\"!" }{TEXT -1 2 ", " }{XPPEDIT 18 0 "a[8,2] = 0;" "6#/&%\"aG6$\"\")\"\"#\"\"!" }{TEXT -1 2 ", " }{XPPEDIT 18 0 "a[8 ,3] = 0;" "6#/&%\"aG6$\"\")\"\"$\"\"!" }{TEXT -1 3 ", " }{XPPEDIT 18 0 "a[8,4]=0" "6#/&%\"aG6$\"\")\"\"%\"\"!" }{TEXT -1 2 ". " }}{PARA 0 " " 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 48 "In addition we spe cify the linking coefficient " }{XPPEDIT 18 0 "a[8,4] = -17/2;" "6#/& %\"aG6$\"\")\"\"%,$*&\"#<\"\"\"\"\"#!\"\"F." }{TEXT -1 1 "." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 69 "The stage-order equations relating to these stages, such that stages " }{XPPEDIT 18 0 "3,4,5,6,7,8;" "6(\"\"$\"\"%\"\"&\"\"'\"\"(\"\")" }{TEXT -1 19 " hav e stage orders " }{XPPEDIT 18 0 "2,3,3,4,4,4;" "6(\"\"#\"\"$F$\"\"%F%F %" }{TEXT -1 48 " respectively can be solved to obtain the nodes " } {XPPEDIT 18 0 "c[3], c[4]" "6$&%\"cG6#\"\"$&F$6#\"\"%" }{TEXT -1 67 " \+ and the remaining non-zero linking coefficients for these stages. " }} {PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 444 "RSeqs := RowSumConditions(8,'expanded'):\nSOeqs := [op(StageOrder Conditions(2,8,'expanded')),op(StageOrderConditions(3,4..8,'expanded') ),\n op(StageOrderConditions(4,6..8,'expanded'))]:\ne1 : = \{c[2]=3/50,c[5]=100/257,c[6]=7/15,c[7]=12/77,c[8]=26/27,\n se q(a[i,2]=0,i=4..8),seq(a[i,3]=0,i=6..8),a[8,4]=-17/2\}:\neqns := map(e xpand,subs(e1,[op(RSeqs),op(SOeqs)])):\nconvert(ListTools[Enumerate](% ),matrix);\n``;\nindets(eqns);\nnops(%);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#K%'matrixG6#777$\"\"\"/&%\"aG6$\"\"#F(#\"\"$\"#]7$F-/,&&F+6$F/F( F(&F+6$F/F-F(&%\"cG6#F/7$F//,&&F+6$\"\"%F(F(&F+6$F@F/F(&F96#F@7$F@/,(& F+6$\"\"&F(F(&F+6$FJF/F(&F+6$FJF@F(#\"$+\"\"$d#7$FJ/,(&F+6$\"\"'F(F(&F +6$FWF@F(&F+6$FWFJF(#\"\"(\"#:7$FW/,*&F+6$FgnF(F(&F+6$FgnF@F(&F+6$FgnF JF(&F+6$FgnFWF(#\"#7\"#x7$Fgn/,,&F+6$\"\")F(F(#\"#/,&*&FXF(FitF(F(*&#F`u\")$fup\"F(FZF( F(#\"%,C\"'+D?7$\"#?/,(*&F^oF(FitF(F(*&F\\xF(F`oF(F(*&#F[v\"%vLF(FboF( F(#\"%%=&\")TI:N7$\"#@/,**&#F^pF-F(FhtF(F_p*&F\\xF(F`pF(F(*&FhxF(FbpF( F(*&#\"%G " 0 "" {MPLTEXT 1 0 46 "e2 := solve(\{op(eqns)\} );\ne3 := `union`(e1,e2):" }}{PARA 12 "" 1 "" {XPPMATH 20 "6#>%#e2G<7/ &%\"aG6$\"\"&\"\"\"#\"/+,V'H&)H\"\"/dy0OPgL/&F(6$\"\"'F*#\",$z(*fvS\"- +]Si)=#/&F(6$\"\"(F+#\",yD!3!f\"\"-D=i*Gi#/&F(6$F2F+#\"(2y\"Q\")+]S\") /&F(6$F9F*#!0MpF358^%\"2D#z_IZ\"4H\"/&F(6$F9F2#\"-+'\\J5%)*\"/\"37lgJi &/&F(6$\"\"#F+#\"\"$\"#]/&F(6$F2\"\"%#\",78>u-\"\",0MM\"*R%/&F(6$F*FY# \"/++?qTu[F./&F(6$FTFR#\"'h*>#\"(73W#/&%\"cG6#FT#\"$p%\"%5X/&F(6$F*FT# !/++lF!=i\"\"/>ENX7?6/&F(6$FYFT#\"%@U\"&!3O/&F(6$\"\")F2#\"5FdUS,0g\\? ;\"4WXw__](HbA/&F(6$FhpF+#!4\"*eysh!)y,c'\"4++oNlqeGn&/&F(6$FhpF*#!:8m Jpj)z^()egqe\"9++w8M5&3NdY_'/&Fdo6#FY#\"%29\"%?!*/&F(6$FhpF9#\"7$*ygl% =7w<%e'*\"6+_NmOfs4#px/&F(6$FYF+#F\\rFdp/&F(6$FTF+#\"'4$p\"\")gS?7/&F( 6$F9FY#\"1+k)og:.M'\"28\"o;5>\\Fc" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }} {PARA 0 "" 0 "" {TEXT 260 4 "Note" }{TEXT -1 2 ": " }}{PARA 0 "" 0 "" {TEXT -1 8 "We have " }{XPPEDIT 18 0 "c[3]=2/3" "6#/&%\"cG6#\"\"$*&\" \"#\"\"\"F'!\"\"" }{TEXT -1 1 " " }{XPPEDIT 18 0 "c[4]" "6#&%\"cG6#\" \"%" }{TEXT -1 16 " and the nodes " }{XPPEDIT 18 0 "c[4];" "6#&%\"cG6 #\"\"%" }{TEXT -1 3 ", " }{XPPEDIT 18 0 "c[5];" "6#&%\"cG6#\"\"&" } {TEXT -1 3 ", " }{XPPEDIT 18 0 "c[6];" "6#&%\"cG6#\"\"'" }{TEXT -1 31 " are related by the equation: " }}{PARA 256 "" 0 "" {TEXT -1 1 " \+ " }{XPPEDIT 18 0 "3*c[6]^2-4*c[6]*c[4]-4*c[5]*c[6]+6*c[5]*c[4] = 0" "6 #/,**&\"\"$\"\"\"*$&%\"cG6#\"\"'\"\"#F'F'*(\"\"%F'&F*6#F,F'&F*6#F/F'! \"\"*(F/F'&F*6#\"\"&F'&F*6#F,F'F4*(F,F'&F*6#F8F'&F*6#F/F'F'\"\"!" } {TEXT -1 2 ". " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 86 "subs(e3,matrix([seq([c[i],seq(a[i,j],j=1..i-1),` `$(9-i)],i=2..8)]));\n``;\nevalf[8](%%);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#K%'matrixG6#7)7+#\"\"$\"#]F(%!GF+F+F+F+F+F+7+#\"$p%\"%5X#\"'4$p \"\")gS?7#\"'h*>#\"(73W#F+F+F+F+F+F+7+#\"%29\"%?!*#F8\"&!3O\"\"!#\"%@U F;F+F+F+F+F+7+#\"$+\"\"$d##\"/+,V'H&)H\"\"/dy0OPgLF<#!/++lF!=i\"\"/>EN X7?6#\"/++?qTu[FEF+F+F+F+7+#\"\"(\"#:#\"(2y\"Q\")+]S\")Fu-\" \",0MM\"*R%#\",$z(*fvS\"-+]Si)=#F+F+F+7+#\"#7\"#x#\",yD!3!f\"\"-D=i*Gi #F\\Fc#!0MpF358^%\"2D#z_IZ\"4H\"#\"-+'\\J5%) *\"/\"37lgJi&F+F+7+#\"#E\"#F#!4\"*eysh!)y,c'\"4++oNlqeGn&FCkQF/F9$!)s(yW\"!\"($\")\"e0X\"FDF+F+F+F+7+$\")nmmYF/$\")D*)*o%F*F 9F9$\")I]NBF/$\")W " 0 "" {MPLTEXT 1 0 1 ";" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 27 "We do not need to specify " }{XPPEDIT 18 0 "c[9];" "6#&% \"cG6#\"\"*" }{TEXT -1 42 ", because according to Verner, the nodes \+ " }{XPPEDIT 18 0 "c[6];" "6#&%\"cG6#\"\"'" }{TEXT -1 2 ", " }{XPPEDIT 18 0 "c[7];" "6#&%\"cG6#\"\"(" }{TEXT -1 2 ", " }{XPPEDIT 18 0 "c[8]; " "6#&%\"cG6#\"\")" }{TEXT -1 2 ", " }{XPPEDIT 18 0 "c[9];" "6#&%\"cG6 #\"\"*" }{TEXT -1 30 " are related by the equation " }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "Int(p(x)*``((1-x)^3/3!),x = 0 .. 1 )*Int(q(x)*(1-x),x = 0 .. 1) = Int(p(x)*``((1-x)^2/2!),x = 0 .. 1)*Int (q(x)*``((1-x)^2/2!),x = 0 .. 1);" "6#/*&-%$IntG6$*&-%\"pG6#%\"xG\"\" \"-%!G6#*&,&F-F-F,!\"\"\"\"$-%*factorialG6#F4F3F-/F,;\"\"!F-F--F&6$*&- %\"qG6#F,F-,&F-F-F,F3F-/F,;F:F-F-*&-F&6$*&-F*6#F,F--F/6#*&,&F-F-F,F3\" \"#-F66#FNF3F-/F,;F:F-F--F&6$*&-F?6#F,F--F/6#*&,&F-F-F,F3FN-F66#FNF3F- /F,;F:F-F-" }{TEXT -1 2 ", " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 7 "where " }{XPPEDIT 18 0 "p(x) = x*(x-c[6])*(x-c[ 7])*(x-c[8]);" "6#/-%\"pG6#%\"xG**F'\"\"\",&F'F)&%\"cG6#\"\"'!\"\"F),& F'F)&F,6#\"\"(F/F),&F'F)&F,6#\"\")F/F)" }{TEXT -1 7 " and " } {XPPEDIT 18 0 "q(x) = (x-c[9])*p(x);" "6#/-%\"qG6#%\"xG*&,&F'\"\"\"&% \"cG6#\"\"*!\"\"F*-%\"pG6#F'F*" }{TEXT -1 2 ". " }}{PARA 0 "" 0 "" {TEXT -1 1 " " }}{PARA 0 "" 0 "" {TEXT -1 155 "See: J.H. Verner, SIAM \+ Journal of Numerical Analysis 1978, 772-790, \"Explicit Runge-Kutta me thods with estimates of the Local Truncation Error.\" (page 780)" }} {PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 90 "p := x -> x*(x-c[6])*(x-c[7])*(x-c[8]):\n'p(x)'=p(x);\nq := x -> ( x-c[9])*p(x):\n'q(x)'=q(x);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/-%\"pG 6#%\"xG**F'\"\"\",&F'F)&%\"cG6#\"\"'!\"\"F),&F'F)&F,6#\"\"(F/F),&F'F)& F,6#\"\")F/F)" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/-%\"qG6#%\"xG*,,&F' \"\"\"&%\"cG6#\"\"*!\"\"F*F'F*,&F'F*&F,6#\"\"'F/F*,&F'F*&F,6#\"\"(F/F* ,&F'F*&F,6#\"\")F/F*" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 239 "Ieq := Int('p(x)'*(1-x)^3/3!,x=0..1)*Int ('q(x)'*(1-x),x=0..1)=\n Int('p(x)'*(1-x)^2/2!,x=0..1)*Int('q(x)'*(1- x)^2/2!,x=0..1);\nvalue(Ieq):\nReq := simplify(lhs(%)-rhs(%))=0:\nL := ilcm(op(map(denom,[coeffs(lhs(Req))]))):\nCeq := expand(L*Req);" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#>%$IeqG/*&-%$IntG6$,$*&#\"\"\"\"\"'F-* &-%\"pG6#%\"xGF-),&F-F-F3!\"\"\"\"$F-F-F-/F3;\"\"!F-F--F(6$*&-%\"qGF2F -F5F-F8F-*&-F(6$,$*&#F-\"\"#F-*&F0F-)F5FFF-F-F-F8F--F(6$,$*&FEF-*&F>F- FHF-F-F-F8F-" }}{PARA 12 "" 1 "" {XPPMATH 20 "6#>%$CeqG/,hp*&\"\"$\"\" \"&%\"cG6#\"\"'F)!\"\"*&F(F)&F+6#\"\")F)F.*&F(F)&F+6#\"\"(F)F.*&\"#7F) &F+6#\"\"*F)F)*(\"#9F))F*\"\"#F)F4F)F.*(\"#GF)F*F)F4F)F)*(FAF)F0F)F*F) F)*(FAF)F0F)F4F)F)*(F=F)F0F)F>F)F.*(F=F))F0F?F)F*F)F.*(F=F)F*F))F4F?F) F.*(F=F)FFF)F4F)F.*(F=F)F0F)FHF)F.*(\"#UF)F9F)F*F)F.*(FLF)F4F)F9F)F.*( FLF)F0F)F9F)F.*(FLF)F9F)F>F)F)*(FLF)FHF)F9F)F)*(FLF)FFF)F9F)F)*,\"$)eF )F9F)F*F)F4F)F0F)F.*,\"$G(F)F0F)F>F)F9F)F4F)F)*,FUF)FFF)F*F)F4F)F9F)F) *,\"$!)*F)FFF)F>F)F9F)F4F)F.*,FUF)F0F)FHF)F9F)F*F)F)*,FXF)FFF)FHF)F9F) F*F)F.*,FXF)F>F)FHF)F9F)F0F)F.*,\"%q9F)F>F)FHF)FFF)F9F)F)**\"$J#F)F*F) F4F)F0F)F.**\"#)*F)F>F)FHF)F0F)F.**F[oF)FFF)FHF)F*F)F.**F[oF)FFF)F>F)F 4F)F.**\"$o\"F)FFF)F*F)F4F)F)**F_oF)F0F)FHF)F*F)F)**F_oF)F0F)F>F)F4F)F )**\"$a\"F)F9F)F*F)F4F)F)**FcoF)F0F)F9F)F*F)F)**FcoF)F0F)F4F)F9F)F)**F _oF)F0F)FHF)F9F)F.**F_oF)F0F)F>F)F9F)F.**F_oF)FFF)F*F)F9F)F.**\"$'>F)F FF)F>F)F9F)F)**F_oF)FFF)F4F)F9F)F.**FjoF)FFF)FHF)F9F)F)**F_oF)F9F)F>F) F4F)F.**F_oF)F*F)FHF)F9F)F.**FjoF)F>F)FHF)F9F)F)\"\"!" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 60 "The previous equatio n involving integrals is equivalent to: " }}{PARA 256 "" 0 "" {TEXT -1 2 " " }{XPPEDIT 18 0 "14*c[6]^2*c[7]+3*c[6]-28*c[6]*c[7]+3*c[7]+58 8*c[9]*c[6]*c[7]*c[8]+3*c[8]-28*c[8]*c[6]-28*c[8]*c[7]+14*c[8]*c[6]^2+ 14*c[8]^2*c[6]+231*c[6]*c[7]*c[8]+98*c[6]^2*c[7]^2*c[8]+14*c[6]*c[7]^2 +98*c[8]^2*c[7]^2*c[6]+98*c[8]^2*c[6]^2*c[7]-168*c[8]^2*c[6]*c[7]-168* c[8]*c[7]^2*c[6]-168*c[8]*c[6]^2*c[7]+14*c[8]^2*c[7]-12*c[9]+14*c[8]*c [7]^2+42*c[9]*c[6]-154*c[9]*c[6]*c[7]+42*c[7]*c[9]+42*c[8]*c[9]-154*c[ 8]*c[9]*c[6]-154*c[8]*c[7]*c[9]-42*c[9]*c[6]^2-42*c[7]^2*c[9]-42*c[8]^ 2*c[9]+168*c[8]*c[7]^2*c[9]-728*c[8]*c[6]^2*c[9]*c[7]-728*c[8]^2*c[6]* c[7]*c[9]+980*c[8]^2*c[6]^2*c[9]*c[7]-728*c[8]*c[7]^2*c[9]*c[6]+980*c[ 8]^2*c[7]^2*c[9]*c[6]+980*c[6]^2*c[7]^2*c[9]*c[8]-1470*c[6]^2*c[7]^2*c [8]^2*c[9]+168*c[8]*c[6]^2*c[9]+168*c[8]^2*c[6]*c[9]-196*c[8]^2*c[6]^2 *c[9]+168*c[8]^2*c[7]*c[9]-196*c[8]^2*c[7]^2*c[9]+168*c[9]*c[6]^2*c[7] +168*c[6]*c[7]^2*c[9]-196*c[6]^2*c[7]^2*c[9]=0" "6#/,hp*(\"#9\"\"\"*$& %\"cG6#\"\"'\"\"#F'&F*6#\"\"(F'F'*&\"\"$F'&F*6#F,F'F'*(\"#GF'&F*6#F,F' &F*6#F0F'!\"\"*&F2F'&F*6#F0F'F'*,\"$)eF'&F*6#\"\"*F'&F*6#F,F'&F*6#F0F' &F*6#\"\")F'F'*&F2F'&F*6#FJF'F'*(F6F'&F*6#FJF'&F*6#F,F'F;*(F6F'&F*6#FJ F'&F*6#F0F'F;*(F&F'&F*6#FJF'&F*6#F,F-F'*(F&F'*$&F*6#FJF-F'&F*6#F,F'F'* *\"$J#F'&F*6#F,F'&F*6#F0F'&F*6#FJF'F'**\"#)*F'*$&F*6#F,F-F'&F*6#F0F-&F *6#FJF'F'*(F&F'&F*6#F,F'&F*6#F0F-F'**FfoF'*$&F*6#FJF-F'&F*6#F0F-&F*6#F ,F'F'**FfoF'*$&F*6#FJF-F'&F*6#F,F-&F*6#F0F'F'**\"$o\"F'*$&F*6#FJF-F'&F *6#F,F'&F*6#F0F'F;**FdqF'&F*6#FJF'&F*6#F0F-&F*6#F,F'F;**FdqF'&F*6#FJF' &F*6#F,F-&F*6#F0F'F;*(F&F'*$&F*6#FJF-F'&F*6#F0F'F'*&\"#7F'&F*6#FCF'F;* (F&F'&F*6#FJF'&F*6#F0F-F'*(\"#UF'&F*6#FCF'&F*6#F,F'F'**\"$a\"F'&F*6#FC F'&F*6#F,F'&F*6#F0F'F;*(FjsF'&F*6#F0F'&F*6#FCF'F'*(FjsF'&F*6#FJF'&F*6# FCF'F'**F`tF'&F*6#FJF'&F*6#FCF'&F*6#F,F'F;**F`tF'&F*6#FJF'&F*6#F0F'&F* 6#FCF'F;*(FjsF'&F*6#FCF'&F*6#F,F-F;*(FjsF'*$&F*6#F0F-F'&F*6#FCF'F;*(Fj sF'*$&F*6#FJF-F'&F*6#FCF'F;**FdqF'&F*6#FJF'&F*6#F0F-&F*6#FCF'F'*,\"$G( F'&F*6#FJF'&F*6#F,F-&F*6#FCF'&F*6#F0F'F;*,FhwF'*$&F*6#FJF-F'&F*6#F,F'& F*6#F0F'&F*6#FCF'F;*,\"$!)*F'*$&F*6#FJF-F'&F*6#F,F-&F*6#FCF'&F*6#F0F'F '*,FhwF'&F*6#FJF'&F*6#F0F-&F*6#FCF'&F*6#F,F'F;*,F\\yF'*$&F*6#FJF-F'&F* 6#F0F-&F*6#FCF'&F*6#F,F'F'*,F\\yF'*$&F*6#F,F-F'&F*6#F0F-&F*6#FCF'&F*6# FJF'F'*,\"%q9F'*$&F*6#F,F-F'&F*6#F0F-&F*6#FJF-&F*6#FCF'F;**FdqF'&F*6#F JF'&F*6#F,F-&F*6#FCF'F'**FdqF'*$&F*6#FJF-F'&F*6#F,F'&F*6#FCF'F'**\"$'> F'*$&F*6#FJF-F'&F*6#F,F-&F*6#FCF'F;**FdqF'*$&F*6#FJF-F'&F*6#F0F'&F*6#F CF'F'**F^]lF'*$&F*6#FJF-F'&F*6#F0F-&F*6#FCF'F;**FdqF'&F*6#FCF'&F*6#F,F -&F*6#F0F'F'**FdqF'&F*6#F,F'&F*6#F0F-&F*6#FCF'F'**F^]lF'*$&F*6#F,F-F'& F*6#F0F-&F*6#FCF'F;\"\"!" }{TEXT -1 2 ". " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 75 "e4 := solve(subs(e3,C eq),\{c[9]\}):\nc[9]=subs(e4,c[9]);\ne5 := `union`(e3,e4):" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/&%\"cG6#\"\"*#\"(^55(\"([Z%z" }}}{PARA 0 " " 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 7 "Thus " }{XPPEDIT 18 0 "c[9] = 7101051/7944748" "6#/&%\"cG6#\"\"**&\"(^55(\"\"\"\"([Z%z! \"\"" }{TEXT -1 1 "." }}{PARA 0 "" 0 "" {TEXT -1 84 "Now we can use th e quadrature equations to find the weights once the remaining nodes" } }{PARA 256 "" 0 "" {TEXT -1 2 " " }{XPPEDIT 18 0 "c[10] = 11/12;" "6# /&%\"cG6#\"#5*&\"#6\"\"\"\"#7!\"\"" }{TEXT -1 3 ", " }{XPPEDIT 18 0 " c[11] = 19/20;" "6#/&%\"cG6#\"#6*&\"#>\"\"\"\"#?!\"\"" }{TEXT -1 3 ", \+ " }{XPPEDIT 18 0 "c[12] = 1;" "6#/&%\"cG6#\"#7\"\"\"" }{TEXT -1 1 " \+ " }}{PARA 0 "" 0 "" {TEXT -1 38 "are specified along with the weights \+ " }{XPPEDIT 18 0 "b[2]=0" "6#/&%\"bG6#\"\"#\"\"!" }{TEXT -1 2 ", " } {XPPEDIT 18 0 "b[3]=0" "6#/&%\"bG6#\"\"$\"\"!" }{TEXT -1 2 ", " } {XPPEDIT 18 0 "b[4]=0" "6#/&%\"bG6#\"\"%\"\"!" }{TEXT -1 2 ", " } {XPPEDIT 18 0 "b[5]=0" "6#/&%\"bG6#\"\"&\"\"!" }{TEXT -1 2 ". " }} {PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 288 "Qeqs := QuadratureConditions(8,12,'expanded'):\ne6 := \{seq(b[i]= 0,i=2..5),c[10]=11/12,c[11]=19/20,c[12]=1\}:\ne7 := `union`(e5,e6):\nq uadeqns := subs(e7,Qeqs):\nfor ct to nops(quadeqns) do\n print(`equa tion `||ct); print(``);print(quadeqns[ct]);print(``);\nend do:\ninde ts(quadeqns);\nnops(%);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%-equation~ ~~1G" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%!G" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/,2&%\"bG6#\"\"\"F(&F&6#\"\"'F(&F&6#\"\"(F(&F&6#\"\")F( &F&6#\"\"*F(&F&6#\"#5F(&F&6#\"#6F(&F&6#\"#7F(F(" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%!G" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%-equation~~~2G " }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%!G" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/,0*&#\"\"(\"#:\"\"\"&%\"bG6#\"\"'F)F)*&#\"#7\"#xF)&F+6#F'F)F)*& #\"#E\"#FF)&F+6#\"\")F)F)*&#\"(^55(\"([Z%zF)&F+6#\"\"*F)F)*&#\"#6F0F)& F+6#\"#5F)F)*&#\"#>\"#?F)&F+6#FDF)F)&F+6#F0F)#F)\"\"#" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%!G" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%-equation~ ~~3G" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%!G" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/,0*&#\"#\\\"$D#\"\"\"&%\"bG6#\"\"'F)F)*&#\"$W\"\"%HfF) &F+6#\"\"(F)F)*&#\"$w'\"$H(F)&F+6#\"\")F)F)*&#\"/,YID\\U]\"//Ny?!>J'F) &F+6#\"\"*F)F)*&#\"$@\"F0F)&F+6#\"#5F)F)*&#\"$h$\"$+%F)&F+6#\"#6F)F)&F +6#\"#7F)#F)\"\"$" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%!G" }}{PARA 11 " " 1 "" {XPPMATH 20 "6#%-equation~~~4G" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%!G" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/,0*&#\"$V$\"%vL\"\"\"&%\"bG 6#\"\"'F)F)*&#\"%G<\"'LlXF)&F+6#\"\"(F)F)*&#\"&wv\"\"&$o>F)&F+6#\"\")F )F)*&#\"6^cBi\"fi'*p!e$\"6#*p$=qJTrk9]F)&F+6#\"\"*F)F)*&#\"%J8F0F)&F+6 #\"#5F)F)*&#\"%fo\"%+!)F)&F+6#\"#6F)F)&F+6#\"#7F)#F)\"\"%" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%!G" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%-equati on~~~5G" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%!G" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/,0*&#\"%,C\"&D1&\"\"\"&%\"bG6#\"\"'F)F)*&#\"&O2#\")TI: NF)&F+6#\"\"(F)F)*&#\"'wpX\"'T9`F)&F+6#\"\")F)F)*&#\"=,#p4tEa# \"=;!=&Qg!*4%oYy5S)RF)&F+6#\"\"*F)F)*&#\"&TY\"F0F)&F+6#\"#5F)F)*&#\"'@ .8\"'++;F)&F+6#\"#6F)F)&F+6#\"#7F)#F)\"\"&" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%!G" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%-equation~~~6G " }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%!G" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/,0*&#\"&2o\"\"'v$f(\"\"\"&%\"bG6#\"\"'F)F)*&#\"'K)[#\"+dTy1FF)& F+6#\"\"(F)F)*&#\")w8)=\"\")2*[V\"F)&F+6#\"\")F)F)*&#\"D^-`;t2T!)3c#RC I^c0=\"Do*z0$RqzTE!ysMrh>lJF)&F+6#\"\"*F)F)*&#\"'^5;F0F)&F+6#\"#5F)F)* &#\"(*4wC\"(++?$F)&F+6#\"#6F)F)&F+6#\"#7F)#F)F-" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%!G" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%-equation~~~7G " }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%!G" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/,0*&#\"'\\w6\")D1R6\"\"\"&%\"bG6#\"\"'F)F)*&#\"(%)f)H\"-*3!QA%3 #F)&F+6#\"\"(F)F)*&#\"*wd\"*3$\"**[?uQF)&F+6#\"\")F)F)*&#\"K,QRbWi>,=S K)=ePot+T@G\"\"Kk!3'Rt1VW<\"GeSW*=>&foY^#F)&F+6#\"\"*F)F)*&#\"(h:x\"F0 F)&F+6#\"#5F)F)*&#\")\")e/Z\")+++kF)&F+6#\"#6F)F)&F+6#\"#7F)#F)F4" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#%!G" }}{PARA 11 "" 1 "" {XPPMATH 20 "6 #%-equation~~~8G" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%!G" }}{PARA 11 " " 1 "" {XPPMATH 20 "6#/,0*&#\"'VN#)\"*v$f3<\"\"\"&%\"bG6#\"\"'F)F)*&# \")3=$e$\"/`oEB&[g\"F)&F+6#\"\"(F)F)*&#\"+w,\"=.)\",.KNg/\"F)&F+6#\"\" )F)F)*&#\"Q^[)fg)fe$R'yUck&pwC@9Do[X5*\"RsyC([;-O5U4W79.tHh8BH3%y*>F)& F+6#\"\"*F)F)*&#\")rr[>F0F)&F+6#\"#5F)F)*&#\"*R<(Q*)\"++++!G\"F)&F+6# \"#6F)F)&F+6#\"#7F)#F)F;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%!G" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#<*&%\"bG6#\"\"(&F%6#\"#5&F%6#\"\"\"&F% 6#\"#7&F%6#\"#6&F%6#\"\")&F%6#\"\"*&F%6#\"\"'" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#\"\")" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 199 "e8 := solve(\{op(quadeqns)\}):\ne9 := `u nion`(e7,e8):\nfor ii to 12 do\n wt_val := subs(e9,b[ii]);\n if wt _val<>0 then print(b[ii]=wt_val) end if;\nend do:\n``;\nsubs(e9,[seq(b [i],i=1..12)]):\nevalf[8](%);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/&%\" bG6#\"\"\"#\",@*zI!>'\".C3+O\"*Q\"" }}{PARA 11 "" 1 "" {XPPMATH 20 "6# /&%\"bG6#\"\"'#\"0v=imNl-%\"1OV(er/i8\"" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/&%\"bG6#\"\"(#\"7\")3<\\o@J2-s=\"7]w]OO,$>wa^(" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/&%\"bG6#\"\")#!2nAr%y<6'*[\"2+'Q%**RTx^\"" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#/&%\"bG6#\"\"*#\"RW\\;d`mcpPRr)3yAIM5x Zm\"yY#=\"Q&*[(*fA#Q%))zSsT(yjyDA7v![EFL" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/&%\"bG6#\"#5#!-)[&)GfA$!\"($\")&=S[&F0$!)+#=e*F 0$\")s&\\$yF0$!)%=Ff\"F+" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 2 "e9" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1420 "e9 := \{b[5] = 0, b[2] = 0, b[3] = 0, b[4] = 0, c[1 2] = 1, a[7,1] = 15900802578/262289621825, a[7,2] = 0, a[6,3] = 0, a[4 ,2] = 0, a[8,3] = 0, a[8,2] = 0, a[7,3] = 0, a[6,2] = 0, a[5,2] = 0, c [10] = 11/12, c[11] = 19/20, a[7,4] = 6340315606886400/562749191016681 13, b[9] = 1824678166477710343022780887139376956665357164944/332726480 751222257863787417240798843822259974895, a[8,1] = -6560178806172785891 /5672858706535680000, a[6,4] = 10274191312/43991343405, a[5,3] = -1621 8027650000/11201245352619, a[5,4] = 48744170200000/33603736057857, c[3 ] = 469/4510, c[4] = 1407/9020, c[6] = 7/15, a[4,3] = 4221/36080, a[6, 1] = 3817807/81405000, a[7,5] = -451131008276934/12909147305279225, a[ 8,7] = 9658417761218465607893/776920972593666355200, b[6] = 4026535666 21875/1136204715874336, c[2] = 3/50, c[5] = 100/257, c[7] = 12/77, c[8 ] = 26/27, a[8,6] = 16204960050140425727/2255297505252764544, a[8,4] = -17/2, a[3,2] = 219961/2440812, a[2,1] = 3/50, b[1] = 61903079921/138 9136000824, a[5,1] = 12985296430100/33603736057857, a[3,1] = 169309/12 204060, a[8,5] = -5870605887517986369316613/652465735085103413760000, \+ c[9] = 7101051/7944748, b[7] = 1872020731216849170881/7515476193013636 507650, a[4,1] = 1407/36080, a[6,5] = 40755997793/218862405000, a[7,6] = 984103149600/56231606512081, b[12] = -107501739/674957600, b[8] = - 48961117784712267/15177413999438600, b[11] = 41251039800000/5264998725 539, b[10] = -126172885488/13167945725\}:" }{TEXT -1 0 "" }}}{PARA 0 " " 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}} }{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 55 "Now we s pecify the remaining zero linking coefficients:" }}{PARA 256 "" 0 "" {TEXT -1 2 " " }{XPPEDIT 18 0 "a[9,2] = 0,a[9,3] = 0,a[10,2] = 0,a[10 ,3] = 0,a[11,2] = 0,a[11,3] = 0,a[12,2] = 0,a[12,3] = 0;" "6*/&%\"aG6$ \"\"*\"\"#\"\"!/&F%6$F'\"\"$F)/&F%6$\"#5F(F)/&F%6$F1F-F)/&F%6$\"#6F(F) /&F%6$F8F-F)/&F%6$\"#7F(F)/&F%6$F?F-F)" }{TEXT -1 1 "." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 149 "We include the stag e-order conditions for stages 9 to 12 so that all these stages all hav e stage-order 4 and incorporate the simplifying conditions: " }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "Sum(b[i]*a[i,j],i = j+1 .. \+ 12) = b[j]*(1-c[j]);" "6#/-%$SumG6$*&&%\"bG6#%\"iG\"\"\"&%\"aG6$F+%\"j GF,/F+;,&F0F,F,F,\"#7*&&F)6#F0F,,&F,F,&%\"cG6#F0!\"\"F," }{TEXT -1 1 " " }}{PARA 0 "" 0 "" {TEXT -1 5 "for " }{XPPEDIT 18 0 "j=1" "6#/%\"jG \"\"\"" }{TEXT -1 10 " ( with " }{XPPEDIT 18 0 "c[1]=0" "6#/&%\"cG6# \"\"\"\"\"!" }{TEXT -1 5 " ), " }{XPPEDIT 18 0 "j = 5;" "6#/%\"jG\"\" &" }{TEXT -1 3 ", " }{XPPEDIT 18 0 "j=7" "6#/%\"jG\"\"(" }{TEXT -1 3 ", " }{XPPEDIT 18 0 "j=9" "6#/%\"jG\"\"*" }{TEXT -1 7 " and " } {XPPEDIT 18 0 "j=10" "6#/%\"jG\"#5" }{TEXT -1 2 ". " }}{PARA 0 "" 0 " " {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 195 "Then it turns out that the following additional collection of \"simple\" order conditions (g iven in abreviated form) is sufficient to determine the remaining link ing coefficients in stages 9 to 12." }}{PARA 0 "" 0 "" {TEXT -1 0 "" } }{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 190 "SO8 := SimpleOrderConditio ns(8):\n[seq([i,SO8[i]],i=[37,61,81,87,102,106,113,117,125])]:\nlinalg [augment](linalg[delcols](%,2..2),matrix([[` `]$(linalg[rowdim](%))]) ,linalg[delcols](%,1..1));" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#K%'matri xG6#7+7%\"#P%#~~G/*(%\"bG\"\"\"%\"cGF--%!G6#*&%\"aGF--F06#*&F3F--F06#* &F3F--F06#*&F3F-F.F-F-F-F-F-#F-\"$S)7%\"#hF)/*(F,F-F.F--F06#*&)F.\"\"% F-F3F-F-#F-\"#N7%\"#\")F)/*&F,F--F06#*(F3F-F.F--F06#*&F3F--F06#*&F3F-- F06#*&)F.\"\"#F-F3F-F-F-F-F-#F-\"%gL7%\"#()F)/*(F,F-F.F--F06#*(F3F-F.F --F06#*(F3F-F.F-F:F-F-F-#F-\"$%Q7%\"$-\"F)/*(F,F-F.F--F06#*&F3F--F06#* &F3F--F06#*&)F.\"\"$F-F3F-F-F-F-#F-\"$g*7%\"$1\"F)/*&F,F--F06#*&F3F--F 06#*&F3F-FCF-F-F-#F-\"%!o\"7%\"$8\"F)/*(F,F-FZF-F[pF-#F-\"$g\"7%\"$<\" F)/*&F,F--F06#*(F3F-F.F-FCF-F-#F-\"$!G7%\"$D\"F)/*(F,F-F.F--F06#*&)F. \"\"&F-F3F-F-#F-\"#[Q(pprint26\"" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}{SECT 1 {PARA 4 "" 0 " " {TEXT -1 20 "The associated trees" }}{PARA 0 "" 0 "" {TEXT -1 0 "" } }{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 461 "AST8 := AllSimpleTrees(8): \nwhch := [37,61,81,87,102,106,113,117,125]:\nm := 3: ## number of tre es per row\nnn := nops(whch): q := iquo(nn,m,'r'):\nfor i to nn do \n \+ p||i := DrawTree(AST8[whch[i]],height=4,width=2,show_ordercondition=t rue,\n scale=1.2,color=COLOR(RGB,0,.7,0),font_color=black);\nend do :\npp := plot([[1,1]],style=line,axes=none):\nplots[display](convert([ seq([p||((k-1)*m+1..m*k)],k=1..q),\n `if`(r>0,[p||(m*q+1..nn),pp$(m*( q+1)-nn)],NULL)],array));" }}{PARA 13 "" 1 "" {GLPLOT2D 852 1131 1131 {PLOTDATA 2 "6\\u-%%TEXTG6&7$$\"\"!F($!+!)fvQ\"*!\"*Q;b~c~(a~(a~(a~(a~ c))))~=~~~6\"-%'COLOURG6&%$RGBGF(F(F(-%%FONTG6$%(COURIERG\"#5-F$6&7$F' $!+g+d#[)F+Q=~~~~~~~~~~~~~~~~~~~~~~~~~~~1F-F.F2-F$6&7$F'$!+g\"Gwo)F+Q? ~~~~~~~~~~~~~~~~~~~~~~~~~~~___F-F.F2-F$6&7$F'$!+**********F+Q?~~~~~~~~ ~~~~~~~~~~~~~~~~~~~840F-F.F2-%'CURVESG6&7)7$F'$!+#)\\<)3(F+7$$!+++++]F +$!+')*R0n$F+7$$\"+++++]F+FS7$FV$!*!*\\!HDF+7$FV$\"+3+tkJF+7$FV$\"+)* \\O#e'F+7$FV$\"+)*********F+-%'SYMBOLG6#%'CIRCLEG-%&COLORG6&F1F'F'F'-% &STYLEG6#%&POINTG-FJ6&FL-F_o6#%(DIAMONDGFboFeo-FJ6&FL-F_o6#%&CROSSGFbo Feo-FJ6%7$FMFP-Fco6&F1F'$\"\"(!\"\"F'-%*THICKNESSG6#\"\"#-FJ6%7$FMFUFf pF[q-FJ6%7$FUFXFfpF[q-FJ6%7$FXFenFfpF[q-FJ6%7$FenFhnFfpF[q-FJ6%7$FhnF[ oFfpF[q-FJ6%7#7$F'F\\o-F/6&F1$F6FjpF'F'-Ffo6#%%LINEG-FJ6%7#7$$!+++++5! \")FNFbrFer-FJ6%7#7$$\"+++++5F^sFNFbrFer-F$6&7$$\"#IF($!+5`7'*oF^sQ4b~ (a~c~(a~c~))~=~~~F-F.F2-F$6&7$Fhs$!+)o#)p\"oF^sQ6~~~~~~~~~~~~~~~~~~~~1 F-F.F2-F$6&7$Fhs$!+dZrToF^sQ8~~~~~~~~~~~~~~~~~~~~___F-F.F2-F$6&7$Fhs$! +++++qF^sQ8~~~~~~~~~~~~~~~~~~~~280F-F.F2-F$6&7$Fhs$!+g)*32oF^sQ6~~~~~~ ~~~~~~4~~~~~~~~F-F.F2-FJ6&7*7$Fhs$!+ " 0 "" {MPLTEXT 1 0 1 ";" }}}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT 260 4 "Note" }{TEXT -1 119 ": The 5 si mplifying conditions can be replaced by the 5 extra order conditions g iven (in abreviated form) as follows.. " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 174 "SO8 := SimpleOrderCond itions(8):\n[seq([i,SO8[i]],i=[21,27,31,36,127])]:\nlinalg[augment](li nalg[delcols](%,2..2),matrix([[` `]$(linalg[rowdim](%))]),linalg[delc ols](%,1..1));" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#K%'matrixG6#7'7%\"#@ %#~~G/*&%\"bG\"\"\"-%!G6#*&%\"aGF--F/6#*&F2F--F/6#*&)%\"cG\"\"#F-F2F-F -F-F-#F-\"$g$7%\"#FF)/*&F,F--F/6#*&F2F--F/6#*&)F:\"\"$F-F2F-F-F-#F-\"$ ?\"7%\"#JF)/*&F,F--F/6#*&)F:\"\"%F-F2F-F-#F-\"#I7%\"#OF)/*&F,F--F/6#*( F2F-F:F--F/6#*&F2F--F/6#*&F2F--F/6#*&F2F-F:F-F-F-F-F-#F-\"%357%\"$F\"F )/*&F,F--F/6#*&)F:\"\"'F-F2F-F-#F-\"#cQ(pprint36\"" }}}{PARA 0 "" 0 " " {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 462 "SOeqs2 := [seq(add(a[i,j],j= 1..i-1)=c[i],i=9..12),op(StageOrderConditions(2,9. .12,'expanded')),\n op(StageOrderConditions(3,9..12,'expanded')),op (StageOrderConditions(4,9..12,'expanded'))]:\nSO8_12 := SimpleOrderCon ditions(8,12,'expanded'):\nord_cdns := [seq(SO8_12[i],i=[37,61,81,87,1 02,113,106,117,125])]:\nsimp_eqs := [add(b[i]*a[i,1],i=1+1..12)=b[1],s eq(add(b[i]*a[i,j],i=j+1..12)=b[j]*(1-c[j]),j=[5,7,9,10])]:\ncdns := [ op(SOeqs2),op(ord_cdns),op(simp_eqs)]:" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 107 "e10 :=\{seq(a[i,2]=0,i =9..12),seq(a[i,3]=0,i=9..12)\}:\ne11 := `union`(e9,e10):\neqns2 := su bs(e11,\{op(cdns)\}):" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 39 "There are 30 equations and 30 unknowns." }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 36 "nops(eqns2);\nindets(eqns2);\nnops( %);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#\"#I" }}{PARA 12 "" 1 "" {XPPMATH 20 "6#<@&%\"aG6$\"#7\"#5&F%6$\"\"*\"\"(&F%6$F'\"\"&&F%6$F'F,& F%6$\"#6F+&F%6$F+\"\"%&F%6$F'F+&F%6$F(F,&F%6$F(F/&F%6$F(\"\")&F%6$F+F@ &F%6$F4F7&F%6$F(\"\"\"&F%6$F4FG&F%6$F(\"\"'&F%6$F+FL&F%6$F(F+&F%6$F4F@ &F%6$F4F(&F%6$F'FL&F%6$F'F7&F%6$F'FG&F%6$F'F@&F%6$F+FG&F%6$F'F4&F%6$F+ F/&F%6$F4FL&F%6$F4F,&F%6$F(F7&F%6$F4F/" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#\"#I" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 " " {MPLTEXT 1 0 22 "infolevel[solve] := 4:" }}}{EXCHG {PARA 0 "> " 0 " " {MPLTEXT 1 0 74 "e12 := solve(\{op(eqns2)\}):\ninfolevel[solve] := 0 :\ne13 := `union`(e11,e12):" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}{SECT 1 {PARA 4 "" 0 " " {TEXT -1 4 "e13 " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 " > " 0 "" {MPLTEXT 1 0 4265 "e13 := \{b[5] = 0, b[2] = 0, b[3] = 0, a[9 ,3] = 0, a[10,3] = 0, b[4] = 0, a[11,3] = 0, a[12,3] = 0, c[12] = 1, a [7,1] = 15900802578/262289621825, a[11,2] = 0, a[12,2] = 0, a[7,2] = 0 , a[6,3] = 0, a[10,2] = 0, a[4,2] = 0, a[8,3] = 0, a[8,2] = 0, a[7,3] \+ = 0, a[6,2] = 0, a[5,2] = 0, c[10] = 11/12, c[11] = 19/20, a[7,4] = 63 40315606886400/56274919101668113, a[9,2] = 0, a[10,6] = 32452570688093 8787969090744301341259977691/60253355178721829324099288547678254006272 , b[9] = 1824678166477710343022780887139376956665357164944/33272648075 1222257863787417240798843822259974895, a[8,1] = -6560178806172785891/5 672858706535680000, a[10,1] = -41617847659972885613040591872450035471/ 52813005326633659912790449394810880000, a[6,4] = 10274191312/439913434 05, a[5,3] = -16218027650000/11201245352619, a[5,4] = 48744170200000/3 3603736057857, c[3] = 469/4510, a[9,6] = 19704323580398550825484436390 2235259599001596076904803130443/44561965459500736173318259755870730512 740108168829012803584, c[4] = 1407/9020, c[6] = 7/15, a[11,8] = 129208 3162729235236212706557/47701646396336675524288000000, a[4,3] = 4221/36 080, a[6,1] = 3817807/81405000, a[7,5] = -451131008276934/129091473052 79225, a[10,4] = -269951087764496912154173876831/298451954118349478993 62922496, a[8,7] = 9658417761218465607893/776920972593666355200, a[12, 8] = 42847753308025275666/29882754550596969299, a[12,6] = 840813041585 474673597744565360698902394175/707808718356838408419865015817743460033 76, a[12,7] = 241895147163393632189444055554841293171617/1508217192459 5104472807447927564153421312, b[6] = 402653566621875/1136204715874336, a[10,5] = -2749986656729113102825386117572168116123/42241614455216923 0122612885222850560000, a[9,1] = -140461899806965182831986352616636648 283813334213167183799/231595702353396941018788535727415998878963658147 758080000, a[11,4] = -19926282430450459437356479596949/220417558707334 8224489986560000, a[10,9] = -3345742205378079321732624680747620/191228 884555191941725363563271035831, c[2] = 3/50, c[5] = 100/257, c[7] = 12 /77, c[8] = 26/27, a[8,6] = 16204960050140425727/2255297505252764544, \+ a[8,4] = -17/2, a[3,2] = 219961/2440812, a[2,1] = 3/50, b[1] = 6190307 9921/1389136000824, a[5,1] = 12985296430100/33603736057857, a[3,1] = 1 69309/12204060, a[8,5] = -5870605887517986369316613/652465735085103413 760000, c[9] = 7101051/7944748, b[7] = 1872020731216849170881/75154761 93013636507650, a[12,5] = -252251601740617744737074278562623715189/164 18177849613229685231088852333734400, a[11,7] = 25587060085986906515751 471801297011229903719273/201904294018761170053295082413107956940800000 0, a[9,7] = 294620817758575088586344956781032015012776829359971133/281 66985596967449488723424839046646360350769597644800, a[10,7] = 41487408 138830374659637178881584696821/3493656315517022583105833038931558400, \+ a[11,5] = -2861670807732347804933810531212726938261187/343166554822474 341490702928140697600000000, a[11,10] = -377694382258403/5816420236687 500, a[4,1] = 1407/36080, a[6,5] = 40755997793/218862405000, a[11,1] = -152032134715446654236835762683933540260220443/1440312299335011879333 92929852012953600000000, a[11,6] = 13340633265934680563503273162687143 19601715109229/198658539882559017093840573946438829079429120000, a[7,6 ] = 984103149600/56231606512081, a[12,10] = 16262726151696512/89404867 23127047, a[9,5] = -97906309983749574521298472575929535517651647721497 96065413844391/1879120863545568722091311485392002480259497346453409790 689280000, b[12] = -107501739/674957600, b[8] = -48961117784712267/151 77413999438600, a[12,11] = -16001553345352000000/6505707112968117383, \+ a[10,8] = 932598433870374903885/57293130256189894463488, b[11] = 41251 039800000/5264998725539, b[10] = -126172885488/13167945725, a[9,4] = - 230388436455789139579016226863459716720858844533129533/281625991040124 02938735024801544932535316065892462592, a[12,1] = -2408600551679152696 84777564157507287321/117003858749851375574083424673098438400, a[11,9] \+ = 2091032384514247750201495665581688757138358851/607298691764125649384 23797095435796334060237500, a[9,8] = 126873616630173037746837292557884 226969/13217298519653022949071050320940567220224, a[12,9] = -771733622 2264885240564054011744574965898992045440/84790334188947973255478145542 07128260520626293873, a[12,4] = -98894702310834163626964955900/1054547 6035409921643978852639\}:" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}}{PARA 0 "" 0 "" {TEXT -1 0 " " }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 20 "printed coefficients" }}{PARA 0 "" 0 "" {TEXT -1 0 " " }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 283 "subs(e13,matrix([seq([c [i],seq(a[i,j],j=1..i-1),``$(7-i)],i=2..7)]));\nfor ii from 8 to 12 do \n print(``);\n print(c[ii]=subs(e13,c[ii])); \n for jj to ii-1 \+ do\n print(a[ii,jj]=subs(e13,a[ii,jj]));\n end do:\nend do:\n`` ;\nfor ii to 12 do\n print(b[ii]=subs(e13,b[ii]));\nend do:" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#K%'matrixG6#7(7)#\"\"$\"#]F(%!GF+F+F+F +7)#\"$p%\"%5X#\"'4$p\"\")gS?7#\"'h*>#\"(73W#F+F+F+F+7)#\"%29\"%?!*#F8 \"&!3O\"\"!#\"%@UF;F+F+F+7)#\"$+\"\"$d##\"/+,V'H&)H\"\"/dy0OPgLF<#!/++ lF!=i\"\"/>ENX7?6#\"/++?qTu[FEF+F+7)#\"\"(\"#:#\"(2y\"Q\")+]S\")Fu-\"\",0MM\"*R%#\",$z(*fvS\"-+]Si)=#F+7)#\"#7\"#x#\",yD!3!f\"\"- D=i*Gi#F\\Fc#!0MpF358^%\"2D#z_IZ\"4H\"#\"-+' \\J5%)*\"/\"37lgJi&Q(pprint86\"" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%!G " }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/&%\"cG6#\"\")#\"#E\"#F" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/&%\"aG6$\"\")\"\"\"#!4\"*eysh!)y,c'\"4++oNl qeGn&" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/&%\"aG6$\"\")\"\"#\"\"!" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#/&%\"aG6$\"\")\"\"$\"\"!" }}{PARA 11 " " 1 "" {XPPMATH 20 "6#/&%\"aG6$\"\")\"\"%#!#<\"\"#" }}{PARA 11 "" 1 " " {XPPMATH 20 "6#/&%\"aG6$\"\")\"\"&#!:8mJpj)z^()egqe\"9++w8M5&3NdY_' " }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/&%\"aG6$\"\")\"\"'#\"5FdUS,0g\\?; \"4WXw__](HbA" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/&%\"aG6$\"\")\"\"(# \"7$*ygl%=7w<%e'*\"6+_NmOfs4#px" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%!G " }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/&%\"cG6#\"\"*#\"(^55(\"([Z%z" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#/&%\"aG6$\"\"*\"\"\"#!Z*z$=nJ@ML\"QG[m j;EN')>$G=lp!)**=YS\"\"Z++3ex9eO'*y))*fTFd`)y=5%pR`Bq&fJ#" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/&%\"aG6$\"\"*\"\"#\"\"!" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/&%\"aG6$\"\"*\"\"$\"\"!" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/&%\"aG6$\"\"*\"\"%#!WL&HJ`W)e3s;(fM'oA;!z&R\"*ybkV)QI#\"V#fiC*e 1;``K\\a,[-N(QHS7S5*fi\"G" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/&%\"aG6$ \"\"*\"\"&#![o\"RWQTlgz\\@xk^w^N&Hfds%)H@Xd\\P)*4j!z*\"[o++G*o!z4MXYt \\f-[-?R&[684A(obaj37z=" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/&%\"aG6$\" \"*\"\"'#\"gnV/8.[!p2'f,!*ff_B-ROW[D3b)R!eBVq>\"fn%e.G,H)o\"3,u70tqevf #=LcW" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/&%\"aG6$\"\"*\"\"(# \"WL6(*f$Hox7],K5yc\\M'e)3v&ex\"3i%H\"V+[k(fp2NgjkY!R[UB()[\\u'pf&)p;G " }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/&%\"aG6$\"\"*\"\")#\"HppA%)yb#HPo uPI&)H'\\AHO**y%\\$=T&>X)H" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/&%\"aG6$\"#5\"\"&#!IBh6o@d>bX))G7>" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#%!G" }}{PARA 11 "" 1 "" {XPPMATH 20 "6 #/&%\"cG6#\"#6#\"#>\"#?" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/&%\"aG6$\" #6\"\"\"#!NV/Ag-aLRoid$oBamW:Z8K?:\"N++++O&H,_)HHRLz=,N$*H7.W\"" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#/&%\"aG6$\"#6\"\"#\"\"!" }}{PARA 11 " " 1 "" {XPPMATH 20 "6#/&%\"aG6$\"#6\"\"$\"\"!" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/&%\"aG6$\"#6\"\"%#!A\\pfzkNP%f/XICGE*>\"@++c')**[C#[L2 (evT?#" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/&%\"aG6$\"#6\"\"&#!L(=h#Qps 77`5Q$\\!yMKx!3nhG\"K++++wpS\"GHq!\\TVZA[bmJM" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/&%\"aG6$\"#6\"\"'#\"RH#4^r,'>VroiJF.Nc!oMfEL1M8\"Q++7H %z!H)Qk%RdSQ4" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/&%\"aG6$ \"#6\"\"(#\"Pt#>P!*H7,(H,=Z^d^1p)f3gqeD\"O+++3%p&z58C3&H`+?" } }{PARA 11 "" 1 "" {XPPMATH 20 "6#/&%\"aG6$\"#6\"\")#\"=dlq7iBN#HF;$3#H \"\">+++)GCbnOjRY;qZ" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/&%\"aG6$\"#6 \"\"*#\"O^)e$Qrv)o\"elc\\,-vZU^%QK54#\"P+vBgSL'zNa4(zB%Q\\c7k+f&\\'pijT$3J-Z*))*\">RE&)y Rk@*4a.waa5" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/&%\"aG6$\"#7\"\"&#!H*= :Pii&yU2PZu<1u,;D_#\"G+WtLB&))3J_oHKh\\y<=k\"" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/&%\"aG6$\"#7\"\"'#\"KvTR-*)pg`cWxftYZ&eTI\"3%)\"JwL+YV x\"e,l)>%3%QoN=(3yq" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/&%\"aG6$\"#7\" \"(#\"K<;<$HT[bbSW*=KORjr9&*=C\"J78U`TcFzW2GZ/^fC><#3:" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/&%\"aG6$\"#7\"\")#\"5mcFD!3LvZG%\"5*Hppf]Xv#))H" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/&%\"aG6$\"#7\"\"*#!RSa/#**)*e'\\dW< ,aScS_)[EAiL'\".C3+O\"*Q\"" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/&%\"bG6#\"\"#\"\"!" }}{PARA 11 "" 1 "" {XPPMATH 20 "6# /&%\"bG6#\"\"$\"\"!" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/&%\"bG6#\"\"% \"\"!" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/&%\"bG6#\"\"&\"\"!" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/&%\"bG6#\"\"'#\"0v=imNl-%\"1OV(er/i8\"" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#/&%\"bG6#\"\"(#\"7\")3<\\o@J2-s=\"7]w] OO,$>wa^(" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/&%\"bG6#\"\")#!2nAr%y<6' *[\"2+'Q%**RTx^\"" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/&%\"bG6#\"\"*#\" RW\\;d`mcpPRr)3yAIM5xZm\"yY#=\"Q&*[(*fA#Q%))zSsT(yjyDA7v![EFL" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#/&%\"bG6#\"#5#!-)[&)G " 0 "" {MPLTEXT 1 0 1 ";" }}}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}}{PARA 0 "" 0 "" {TEXT -1 52 "#----------------- ----------------------------------" }}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 43 "construction of the order 7 embedded scheme" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 4 "e13 " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 4265 "e13 := \{b[5] = 0, b[2] = \+ 0, b[3] = 0, a[9,3] = 0, a[10,3] = 0, b[4] = 0, a[11,3] = 0, a[12,3] = 0, c[12] = 1, a[7,1] = 15900802578/262289621825, a[11,2] = 0, a[12,2] = 0, a[7,2] = 0, a[6,3] = 0, a[10,2] = 0, a[4,2] = 0, a[8,3] = 0, a[8 ,2] = 0, a[7,3] = 0, a[6,2] = 0, a[5,2] = 0, c[10] = 11/12, c[11] = 19 /20, a[7,4] = 6340315606886400/56274919101668113, a[9,2] = 0, a[10,6] \+ = 324525706880938787969090744301341259977691/6025335517872182932409928 8547678254006272, b[9] = 182467816647771034302278088713937695666535716 4944/332726480751222257863787417240798843822259974895, a[8,1] = -65601 78806172785891/5672858706535680000, a[10,1] = -41617847659972885613040 591872450035471/52813005326633659912790449394810880000, a[6,4] = 10274 191312/43991343405, a[5,3] = -16218027650000/11201245352619, a[5,4] = \+ 48744170200000/33603736057857, c[3] = 469/4510, a[9,6] = 1970432358039 85508254844363902235259599001596076904803130443/4456196545950073617331 8259755870730512740108168829012803584, c[4] = 1407/9020, c[6] = 7/15, \+ a[11,8] = 1292083162729235236212706557/47701646396336675524288000000, \+ a[4,3] = 4221/36080, a[6,1] = 3817807/81405000, a[7,5] = -451131008276 934/12909147305279225, a[10,4] = -269951087764496912154173876831/29845 195411834947899362922496, a[8,7] = 9658417761218465607893/776920972593 666355200, a[12,8] = 42847753308025275666/29882754550596969299, a[12,6 ] = 840813041585474673597744565360698902394175/70780871835683840841986 501581774346003376, a[12,7] = 2418951471633936321894440555548412931716 17/15082171924595104472807447927564153421312, b[6] = 402653566621875/1 136204715874336, a[10,5] = -2749986656729113102825386117572168116123/4 22416144552169230122612885222850560000, a[9,1] = -14046189980696518283 1986352616636648283813334213167183799/23159570235339694101878853572741 5998878963658147758080000, a[11,4] = -19926282430450459437356479596949 /2204175587073348224489986560000, a[10,9] = -3345742205378079321732624 680747620/191228884555191941725363563271035831, c[2] = 3/50, c[5] = 10 0/257, c[7] = 12/77, c[8] = 26/27, a[8,6] = 16204960050140425727/22552 97505252764544, a[8,4] = -17/2, a[3,2] = 219961/2440812, a[2,1] = 3/50 , b[1] = 61903079921/1389136000824, a[5,1] = 12985296430100/3360373605 7857, a[3,1] = 169309/12204060, a[8,5] = -5870605887517986369316613/65 2465735085103413760000, c[9] = 7101051/7944748, b[7] = 187202073121684 9170881/7515476193013636507650, a[12,5] = -252251601740617744737074278 562623715189/16418177849613229685231088852333734400, a[11,7] = 2558706 0085986906515751471801297011229903719273/20190429401876117005329508241 31079569408000000, a[9,7] = 294620817758575088586344956781032015012776 829359971133/28166985596967449488723424839046646360350769597644800, a[ 10,7] = 41487408138830374659637178881584696821/34936563155170225831058 33038931558400, a[11,5] = -2861670807732347804933810531212726938261187 /343166554822474341490702928140697600000000, a[11,10] = -3776943822584 03/5816420236687500, a[4,1] = 1407/36080, a[6,5] = 40755997793/2188624 05000, a[11,1] = -152032134715446654236835762683933540260220443/144031 229933501187933392929852012953600000000, a[11,6] = 1334063326593468056 350327316268714319601715109229/198658539882559017093840573946438829079 429120000, a[7,6] = 984103149600/56231606512081, a[12,10] = 1626272615 1696512/8940486723127047, a[9,5] = -9790630998374957452129847257592953 551765164772149796065413844391/187912086354556872209131148539200248025 9497346453409790689280000, b[12] = -107501739/674957600, b[8] = -48961 117784712267/15177413999438600, a[12,11] = -16001553345352000000/65057 07112968117383, a[10,8] = 932598433870374903885/5729313025618989446348 8, b[11] = 41251039800000/5264998725539, b[10] = -126172885488/1316794 5725, a[9,4] = -230388436455789139579016226863459716720858844533129533 /28162599104012402938735024801544932535316065892462592, a[12,1] = -240 860055167915269684777564157507287321/117003858749851375574083424673098 438400, a[11,9] = 2091032384514247750201495665581688757138358851/60729 869176412564938423797095435796334060237500, a[9,8] = 12687361663017303 7746837292557884226969/13217298519653022949071050320940567220224, a[12 ,9] = -7717336222264885240564054011744574965898992045440/8479033418894 797325547814554207128260520626293873, a[12,4] = -988947023108341636269 64955900/10545476035409921643978852639\}:" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 61 "We can obtain an emb edded 11 stage order 7 scheme as follows." }}{PARA 0 "" 0 "" {TEXT -1 94 "We remove stages 11 and 12 from the 12 stage order 8 scheme and in troduce a new stage 11 with " }{XPPEDIT 18 0 "c[11] = 1;" "6#/&%\"cG6# \"#6\"\"\"" }{TEXT -1 3 ", " }{XPPEDIT 18 0 "a[11,2] = 0" "6#/&%\"aG6 $\"#6\"\"#\"\"!" }{TEXT -1 3 ", " }{XPPEDIT 18 0 "a[11,3] = 0;" "6#/& %\"aG6$\"#6\"\"$\"\"!" }{TEXT -1 3 ", " }{XPPEDIT 18 0 "`b*`[2]=0" "6 #/&%#b*G6#\"\"#\"\"!" }{TEXT -1 3 ", " }{XPPEDIT 18 0 "`b*`[3]=0" "6# /&%#b*G6#\"\"$\"\"!" }{TEXT -1 3 ", " }{XPPEDIT 18 0 "`b*`[4]=0" "6#/ &%#b*G6#\"\"%\"\"!" }{TEXT -1 7 " and " }{XPPEDIT 18 0 "`b*`[5]=0" " 6#/&%#b*G6#\"\"&\"\"!" }{TEXT -1 58 ", where the weights of the order \+ 7 scheme are denoted by " }{XPPEDIT 18 0 "`b*`" "6#%#b*G" }{TEXT -1 1 "." }}{PARA 0 "" 0 "" {TEXT -1 1 " " }}{PARA 0 "" 0 "" {TEXT -1 156 "We incorporate the order 7 quadrature conditions, the row sum conditi ons for this stage and stage-order conditions so that this new stage h as stage-order 4." }}{PARA 0 "" 0 "" {TEXT -1 48 "We also incorporate \+ the simplifying conditions: " }}{PARA 256 "" 0 "" {TEXT -1 1 " " } {XPPEDIT 18 0 "Sum(`b*`[i]*a[i,j],i=j+1..11)=`b*`[j]*(1-c[j])" "6#/-%$ SumG6$*&&%#b*G6#%\"iG\"\"\"&%\"aG6$F+%\"jGF,/F+;,&F0F,F,F,\"#6*&&F)6#F 0F,,&F,F,&%\"cG6#F0!\"\"F," }{TEXT -1 1 "," }}{PARA 0 "" 0 "" {TEXT -1 5 "for " }{XPPEDIT 18 0 "j=1" "6#/%\"jG\"\"\"" }{TEXT -1 10 " ( w ith " }{XPPEDIT 18 0 "c[1]=0" "6#/&%\"cG6#\"\"\"\"\"!" }{TEXT -1 5 " \+ ), " }{XPPEDIT 18 0 "j = 8;" "6#/%\"jG\"\")" }{TEXT -1 3 ", " } {XPPEDIT 18 0 "j=9" "6#/%\"jG\"\"*" }{TEXT -1 7 " and " }{XPPEDIT 18 0 "j=10" "6#/%\"jG\"#5" }{TEXT -1 2 ". " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 399 "`Qeqs*` := subs(b =`b*`,QuadratureConditions(7,11,'expanded')):\nSO_eqs := [add(a[11,j], j=1..10)=c[11],add(a[11,j]*c[j],j=2..10)=1/2*c[11]^2,\n add(a[1 1,j]*c[j]^2,j=2..10)=1/3*c[11]^3,add(a[11,j]*c[j]^3,j=2..10)=1/4*c[11] ^4]:\n`simp_eqs*` := [add(`b*`[i]*a[i,1],i=2..11)=`b*`[1],seq(add(`b*` [i]*a[i,j],i=j+1..11)=`b*`[j]*(1-c[j]),j=[8,9,10])]:\n`cdns*` := [op(` simp_eqs*`),op(SO_eqs),op(`Qeqs*`)]:" }}}{PARA 0 "" 0 "" {TEXT -1 0 " " }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 144 "e14 := `union`(remove(u _->member(op(1,lhs(u_)),[11,12]) or op(0,lhs(u_))=b,e13),\n \{ c[11]=1,a[11,2]=0,a[11,3]=0,seq(`b*`[i]=0,i=2..5)\}):\n" }}}{PARA 0 " " 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 53 "We have 15 equatio ns for the 15 unknown coefficients." }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 60 "eqns3 := subs(e14,`cdns*`):\nnops(%);\nindets(eqns3); \nnops(%);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#\"#:" }}{PARA 11 "" 1 " " {XPPMATH 20 "6#<1&%#b*G6#\"\"*&F%6#\"#5&F%6#\"#6&F%6#\"\"\"&F%6#\"\" '&F%6#\"\"(&F%6#\"\")&%\"aG6$F-F'&F;6$F-\"\"%&F;6$F-F0&F;6$F-F9&F;6$F- F*&F;6$F-F3&F;6$F-F6&F;6$F-\"\"&" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#\" #:" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 22 "infolevel[solve] := 4:" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 74 "e15 := solve(\{op(eqns3)\}):\ninfolevel[solve] := 0: \ne16 := `union`(e14,e15):" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 3 "e16" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 3414 "e16 := \{a[10,3] = 0, a[11,3] = 0, a[11,2] = 0, a[5 ,1] = 12985296430100/33603736057857, a[7,2] = 0, `b*`[9] = 12458274435 33127167605703273831223125580576/2981022074572680330885817370627029579 58355, a[9,3] = 0, a[6,3] = 0, a[10,2] = 0, a[4,2] = 0, a[8,3] = 0, a[ 8,2] = 0, a[7,3] = 0, a[6,2] = 0, a[5,2] = 0, a[9,2] = 0, c[6] = 7/15, c[10] = 11/12, `b*`[6] = 97458949528125/274256310728288, `b*`[2] = 0, `b*`[3] = 0, `b*`[4] = 0, `b*`[5] = 0, a[10,7] = 41487408138830374659 637178881584696821/3493656315517022583105833038931558400, `b*`[8] = 56 89283182401789/2168201999919800, a[6,5] = 40755997793/218862405000, a[ 7,1] = 15900802578/262289621825, a[6,1] = 3817807/81405000, a[7,5] = - 451131008276934/12909147305279225, a[7,6] = 984103149600/5623160651208 1, a[11,1] = -2744022934559056024469622604306540529/173822257354870596 4451797934060438400, a[11,6] = 612390812902422662436708592068512814631 75/6659681426411632770651421460598964953048, `b*`[7] = 795200003703671 81779/319546003300661568600, a[8,4] = -17/2, c[11] = 1, c[8] = 26/27, \+ c[7] = 12/77, c[5] = 100/257, c[2] = 3/50, a[11,5] = -5463712367239342 1308996942594910689159/4634295873089356958410234071688953600, a[2,1] = 3/50, a[9,5] = -97906309983749574521298472575929535517651647721497960 65413844391/1879120863545568722091311485392002480259497346453409790689 280000, a[6,4] = 10274191312/43991343405, `b*`[1] = 5709741119/1279467 36918, a[5,4] = 48744170200000/33603736057857, a[3,2] = 219961/2440812 , a[11,7] = 5134209062671060909676995239537507099025/46296469402610443 6494336717002613458688, c[3] = 469/4510, a[5,3] = -16218027650000/1120 1245352619, a[11,4] = -35144243656562246120945761400/59532618684373900 70644832157, a[4,3] = 4221/36080, `b*`[10] = -76671637728/13167945725, a[10,4] = -269951087764496912154173876831/298451954118349478993629224 96, a[10,1] = -41617847659972885613040591872450035471/5281300532663365 9912790449394810880000, a[8,6] = 16204960050140425727/2255297505252764 544, a[10,8] = 932598433870374903885/57293130256189894463488, a[10,6] \+ = 324525706880938787969090744301341259977691/6025335517872182932409928 8547678254006272, a[11,10] = 2242504532188281088/2908865886282319623, \+ a[9,1] = -140461899806965182831986352616636648283813334213167183799/23 1595702353396941018788535727415998878963658147758080000, a[8,1] = -656 0178806172785891/5672858706535680000, a[8,5] = -5870605887517986369316 613/652465735085103413760000, a[9,7] = 2946208177585750885863449567810 32015012776829359971133/2816698559696744948872342483904664636035076959 7644800, c[4] = 1407/9020, `b*`[11] = -5522626587/8774448800, c[9] = 7 101051/7944748, a[9,6] = 197043235803985508254844363902235259599001596 076904803130443/445619654595007361733182597558707305127401081688290128 03584, a[8,7] = 9658417761218465607893/776920972593666355200, a[11,8] \+ = -370363100142446513442/1535150001377392224067, a[4,1] = 1407/36080, \+ a[9,8] = 126873616630173037746837292557884226969/132172985196530229490 71050320940567220224, a[11,9] = -2120206825123290760682216664864413826 48974080/390260289309289497513964664381911193640454941, a[10,9] = -334 5742205378079321732624680747620/191228884555191941725363563271035831, \+ a[9,4] = -230388436455789139579016226863459716720858844533129533/28162 599104012402938735024801544932535316065892462592, a[10,5] = -274998665 6729113102825386117572168116123/42241614455216923012261288522285056000 0, a[3,1] = 169309/12204060, a[7,4] = 6340315606886400/562749191016681 13\}:" }{TEXT -1 0 "" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}}{PARA 0 "" 0 "" {TEXT -1 0 "" }} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}{SECT 1 {PARA 4 "" 0 " " {TEXT -1 20 "printed coefficients" }}{PARA 0 "" 0 "" {TEXT -1 0 "" } }{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 289 "subs(e16,matrix([seq([c[i] ,seq(a[i,j],j=1..i-1),``$(7-i)],i=2..7)]));\nfor ii from 8 to 11 do\n \+ print(``);\n print(c[ii]=subs(e16,c[ii])); \n for jj to ii-1 do \n print(a[ii,jj]=subs(e16,a[ii,jj]));\n end do:\nend do:\n``; \nfor ii to 11 do\n print(`b*`[ii]=subs(e16,`b*`[ii]));\nend do:" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#K%'matrixG6#7(7)#\"\"$\"#]F(%!GF+F+F+F +7)#\"$p%\"%5X#\"'4$p\"\")gS?7#\"'h*>#\"(73W#F+F+F+F+7)#\"%29\"%?!*#F8 \"&!3O\"\"!#\"%@UF;F+F+F+7)#\"$+\"\"$d##\"/+,V'H&)H\"\"/dy0OPgLF<#!/++ lF!=i\"\"/>ENX7?6#\"/++?qTu[FEF+F+7)#\"\"(\"#:#\"(2y\"Q\")+]S\")Fu-\"\",0MM\"*R%#\",$z(*fvS\"-+]Si)=#F+7)#\"#7\"#x#\",yD!3!f\"\"- D=i*Gi#F\\Fc#!0MpF358^%\"2D#z_IZ\"4H\"#\"-+' \\J5%)*\"/\"37lgJi&Q(pprint96\"" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%!G " }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/&%\"cG6#\"\")#\"#E\"#F" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/&%\"aG6$\"\")\"\"\"#!4\"*eysh!)y,c'\"4++oNl qeGn&" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/&%\"aG6$\"\")\"\"#\"\"!" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#/&%\"aG6$\"\")\"\"$\"\"!" }}{PARA 11 " " 1 "" {XPPMATH 20 "6#/&%\"aG6$\"\")\"\"%#!#<\"\"#" }}{PARA 11 "" 1 " " {XPPMATH 20 "6#/&%\"aG6$\"\")\"\"&#!:8mJpj)z^()egqe\"9++w8M5&3NdY_' " }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/&%\"aG6$\"\")\"\"'#\"5FdUS,0g\\?; \"4WXw__](HbA" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/&%\"aG6$\"\")\"\"(# \"7$*ygl%=7w<%e'*\"6+_NmOfs4#px" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%!G " }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/&%\"cG6#\"\"*#\"(^55(\"([Z%z" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#/&%\"aG6$\"\"*\"\"\"#!Z*z$=nJ@ML\"QG[m j;EN')>$G=lp!)**=YS\"\"Z++3ex9eO'*y))*fTFd`)y=5%pR`Bq&fJ#" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/&%\"aG6$\"\"*\"\"#\"\"!" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/&%\"aG6$\"\"*\"\"$\"\"!" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/&%\"aG6$\"\"*\"\"%#!WL&HJ`W)e3s;(fM'oA;!z&R\"*ybkV)QI#\"V#fiC*e 1;``K\\a,[-N(QHS7S5*fi\"G" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/&%\"aG6$ \"\"*\"\"&#![o\"RWQTlgz\\@xk^w^N&Hfds%)H@Xd\\P)*4j!z*\"[o++G*o!z4MXYt \\f-[-?R&[684A(obaj37z=" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/&%\"aG6$\" \"*\"\"'#\"gnV/8.[!p2'f,!*ff_B-ROW[D3b)R!eBVq>\"fn%e.G,H)o\"3,u70tqevf #=LcW" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/&%\"aG6$\"\"*\"\"(# \"WL6(*f$Hox7],K5yc\\M'e)3v&ex\"3i%H\"V+[k(fp2NgjkY!R[UB()[\\u'pf&)p;G " }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/&%\"aG6$\"\"*\"\")#\"HppA%)yb#HPo uPI&)H'\\AHO**y%\\$=T&>X)H" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/&%\"aG6$\"#5\"\"&#!IBh6o@d>bX))G7>" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#%!G" }}{PARA 11 "" 1 "" {XPPMATH 20 "6 #/&%\"cG6#\"#6\"\"\"" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/&%\"aG6$\"#6 \"\"\"#!FH0a1VgA'pW-c!fX$H-WF\"F+%Q/1Mzz^W'fq[NdA#Q<" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/&%\"aG6$\"#6\"\"#\"\"!" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/&%\"aG6$\"#6\"\"$\"\"!" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/&%\"aG6$\"#6\"\"%#!>+9wX47YAccOCW^$\"=d@$[kq+RP%o=E`f" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/&%\"aG6$\"#6\"\"&#!Gf\"*o5\\fUp**38U$RsO7PY &\"F+O&*)orSB5%epN*3teHMY" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/&%\"aG6$ \"#6\"\"'#\"JvJY\"G^o?f3nViEU-H\"3R7'\"I[I&\\'*)fg9U^1xK;TE9ofm" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#/&%\"aG6$\"#6\"\"(#\"ID!*42v`R_*pn441r E14U8&\"H)oeMh-qrOV\\OW5ESpkHY" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/&% \"aG6$\"#6\"\")#!6UM^YC9+JOq$\"7nSA#Rx8+]^`\"" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/&%\"aG6$\"#6\"\"*#!N!3u*[EQTk[m;Aog2HB^#o??@\"NT\\XSO> 6>QkY'R^(\\*G4$*Gg-R" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/&%\"aG6$\"#6 \"#5#\"4)3\"G)=KX]UA\"4B'>BG')e')3H" }}{PARA 11 "" 1 "" {XPPMATH 20 "6 #%!G" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/&%#b*G6#\"\"\"#\"+>6u4d\"-=pt Yz7" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/&%#b*G6#\"\"#\"\"!" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/&%#b*G6#\"\"$\"\"!" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/&%#b*G6#\"\"%\"\"!" }}{PARA 11 "" 1 "" {XPPMATH 20 "6# /&%#b*G6#\"\"&\"\"!" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/&%#b*G6#\"\"'# \"/D\"G&\\*eu*\"0)GG2JcUF" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/&%#b*G6# \"\"(#\"5z<=n.P++_z\"6+'o:m+L+Y&>$" }}{PARA 11 "" 1 "" {XPPMATH 20 "6# /&%#b*G6#\"\")#\"1*y,C=$G*o&\"1+)>***>?o@" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/&%#b*G6#\"\"*#\"Lw0eDJAJQF.dgnr7LNWFeC\"\"Kb$ez&Hqiqt \"e)3L!osX2A5)H" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/&%#b*G6#\"#5#!,Gxj rm(\",Dd%z;8" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/&%#b*G6#\"#6#!+(eEE_& \"++)[Wx)" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 " " {MPLTEXT 1 0 1 ";" }}}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 104 "`RK7_11eqs*` := subs(b=`b*`,[op(RowSumCo nditions(11,'expanded')),op(OrderConditions(7,11,'expanded'))]):" }}} {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 5 "Check" }} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 57 "subs(e16,`RK7_11eqs*`):\nmap (u->lhs(u)-rhs(u),%);\nnops(%);" }}{PARA 12 "" 1 "" {XPPMATH 20 "6#7[q \"\"!F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F $F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F $F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#\"#&*" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 93 "We can calculate the principal error norm, that i s, the 2-norm of the principal error terms. " }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 185 "`errterms7_11*` := subs(b=`b*`,PrincipalErrorTerm s(7,11,'expanded')):\nsm := 0:\nfor ct to nops(`errterms7_11*`) do\n \+ sm := sm+(evalf(subs(e16,`errterms7_11*`[ct])))^2;\nend do:\nsqrt(sm) ;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#$\"+nsG.#*!#:" }}}{PARA 0 "" 0 " " {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 138 "We can include the new stage for the embedded scheme as an additional 13th stage added to th e order 8 scheme along with the coefficients " }{XPPEDIT 18 0 "a[13,1 1] = 0;" "6#/&%\"aG6$\"#8\"#6\"\"!" }{TEXT -1 7 " and " }{XPPEDIT 18 0 "a[13,12] = 0;" "6#/&%\"aG6$\"#8\"#7\"\"!" }{TEXT -1 1 "." }} {PARA 0 "" 0 "" {TEXT -1 13 "The weights " }{XPPEDIT 18 0 "`b*`[i]" " 6#&%#b*G6#%\"iG" }{TEXT -1 7 " for " }{XPPEDIT 18 0 "i=1" "6#/%\"iG \"\"\"" }{TEXT -1 89 " . . 10 of the 13 stage combined scheme are thos e of the 11 stage scheme and the weight " }{XPPEDIT 18 0 "`b*`[11];" "6#&%#b*G6#\"#6" }{TEXT -1 34 " in the 11 stage scheme becomes " } {XPPEDIT 18 0 "`b*`[13];" "6#&%#b*G6#\"#8" }{TEXT -1 25 " in the 13 s tage scheme." }}{PARA 0 "" 0 "" {TEXT -1 8 "We set " }{XPPEDIT 18 0 " c[13] = 1;" "6#/&%\"cG6#\"#8\"\"\"" }{TEXT -1 3 ", " }{XPPEDIT 18 0 " `b*`[11] = 0;" "6#/&%#b*G6#\"#6\"\"!" }{TEXT -1 7 " and " }{XPPEDIT 18 0 "`b*`[12] = 0;" "6#/&%#b*G6#\"#7\"\"!" }{TEXT -1 1 "." }}{PARA 0 "" 0 "" {TEXT -1 66 "We can make the order 8 scheme into a 13 stage sc heme by setting " }{XPPEDIT 18 0 "b[13] = 0;" "6#/&%\"bG6#\"#8\"\"!" }{TEXT -1 1 "." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 201 "e17 := \{c[13]=1,seq(a[13,i]=subs(e16,a[11,i]), i=1..10),a[13,11]=0,a[13,12]=0,b[13]=0,\nseq(`b*`[i]=subs(e16,`b*`[i]) ,i=1..10),`b*`[11]=0,`b*`[12]=0,`b*`[13]=subs(e16,`b*`[11])\}:\ne18 := `union`(e13,e17):" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 3 "e1 8" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 5414 "e18 := \{b[13] = 0, a[13,11] = 0, a[13,12] = 0, a[1 0,3] = 0, b[4] = 0, a[11,3] = 0, a[12,3] = 0, c[13] = 1, c[12] = 1, a[ 11,2] = 0, a[5,1] = 12985296430100/33603736057857, a[12,2] = 0, a[7,2] = 0, `b*`[9] = 1245827443533127167605703273831223125580576/2981022074 57268033088581737062702957958355, b[2] = 0, b[3] = 0, a[9,3] = 0, `b*` [12] = 0, `b*`[11] = 0, b[5] = 0, a[6,3] = 0, a[10,2] = 0, a[4,2] = 0, a[8,3] = 0, a[8,2] = 0, a[7,3] = 0, a[6,2] = 0, a[5,2] = 0, a[9,2] = \+ 0, c[6] = 7/15, c[10] = 11/12, c[11] = 19/20, `b*`[6] = 97458949528125 /274256310728288, b[11] = 41251039800000/5264998725539, `b*`[2] = 0, ` b*`[3] = 0, `b*`[4] = 0, `b*`[5] = 0, a[10,7] = 4148740813883037465963 7178881584696821/3493656315517022583105833038931558400, `b*`[8] = 5689 283182401789/2168201999919800, a[6,5] = 40755997793/218862405000, a[12 ,11] = -16001553345352000000/6505707112968117383, a[7,1] = 15900802578 /262289621825, a[6,1] = 3817807/81405000, a[7,5] = -451131008276934/12 909147305279225, b[10] = -126172885488/13167945725, a[7,6] = 984103149 600/56231606512081, `b*`[7] = 79520000370367181779/3195460033006615686 00, a[8,4] = -17/2, c[8] = 26/27, c[7] = 12/77, c[5] = 100/257, c[2] = 3/50, a[2,1] = 3/50, a[9,5] = -97906309983749574521298472575929535517 65164772149796065413844391/1879120863545568722091311485392002480259497 346453409790689280000, a[11,4] = -19926282430450459437356479596949/220 4175587073348224489986560000, b[9] = 182467816647771034302278088713937 6956665357164944/332726480751222257863787417240798843822259974895, a[1 2,7] = 241895147163393632189444055554841293171617/15082171924595104472 807447927564153421312, a[6,4] = 10274191312/43991343405, `b*`[1] = 570 9741119/127946736918, a[5,4] = 48744170200000/33603736057857, a[3,2] = 219961/2440812, b[6] = 402653566621875/1136204715874336, c[3] = 469/4 510, a[5,3] = -16218027650000/11201245352619, a[12,5] = -2522516017406 17744737074278562623715189/16418177849613229685231088852333734400, b[1 ] = 61903079921/1389136000824, a[4,3] = 4221/36080, `b*`[10] = -766716 37728/13167945725, a[13,3] = 0, b[8] = -48961117784712267/151774139994 38600, a[10,4] = -269951087764496912154173876831/298451954118349478993 62922496, a[10,1] = -41617847659972885613040591872450035471/5281300532 6633659912790449394810880000, a[11,1] = -15203213471544665423683576268 3933540260220443/144031229933501187933392929852012953600000000, a[13,4 ] = -35144243656562246120945761400/5953261868437390070644832157, a[8,6 ] = 16204960050140425727/2255297505252764544, a[13,2] = 0, b[7] = 1872 020731216849170881/7515476193013636507650, a[11,7] = 25587060085986906 515751471801297011229903719273/201904294018761170053295082413107956940 8000000, b[12] = -107501739/674957600, a[10,8] = 932598433870374903885 /57293130256189894463488, a[10,6] = 3245257068809387879690907443013412 59977691/60253355178721829324099288547678254006272, a[11,6] = 13340633 26593468056350327316268714319601715109229/1986585398825590170938405739 46438829079429120000, a[11,8] = 1292083162729235236212706557/477016463 96336675524288000000, a[12,4] = -98894702310834163626964955900/1054547 6035409921643978852639, a[9,1] = -140461899806965182831986352616636648 283813334213167183799/231595702353396941018788535727415998878963658147 758080000, a[8,1] = -6560178806172785891/5672858706535680000, a[11,9] \+ = 2091032384514247750201495665581688757138358851/607298691764125649384 23797095435796334060237500, a[8,5] = -5870605887517986369316613/652465 735085103413760000, a[9,7] = 29462081775857508858634495678103201501277 6829359971133/28166985596967449488723424839046646360350769597644800, a [12,10] = 16262726151696512/8940486723127047, a[12,9] = -7717336222264 885240564054011744574965898992045440/847903341889479732554781455420712 8260520626293873, c[4] = 1407/9020, c[9] = 7101051/7944748, a[9,6] = 1 97043235803985508254844363902235259599001596076904803130443/4456196545 9500736173318259755870730512740108168829012803584, a[12,1] = -24086005 5167915269684777564157507287321/11700385874985137557408342467309843840 0, a[8,7] = 9658417761218465607893/776920972593666355200, a[11,5] = -2 861670807732347804933810531212726938261187/343166554822474341490702928 140697600000000, a[12,6] = 840813041585474673597744565360698902394175/ 70780871835683840841986501581774346003376, a[4,1] = 1407/36080, a[12,8 ] = 42847753308025275666/29882754550596969299, a[11,10] = -37769438225 8403/5816420236687500, a[9,8] = 12687361663017303774683729255788422696 9/13217298519653022949071050320940567220224, a[10,9] = -33457422053780 79321732624680747620/191228884555191941725363563271035831, a[13,10] = \+ 2242504532188281088/2908865886282319623, `b*`[13] = -5522626587/877444 8800, a[13,9] = -212020682512329076068221666486441382648974080/3902602 89309289497513964664381911193640454941, a[13,8] = -3703631001424465134 42/1535150001377392224067, a[13,7] = 513420906267106090967699523953750 7099025/462964694026104436494336717002613458688, a[13,6] = 61239081290 242266243670859206851281463175/665968142641163277065142146059896495304 8, a[13,5] = -54637123672393421308996942594910689159/46342958730893569 58410234071688953600, a[9,4] = -23038843645578913957901622686345971672 0858844533129533/28162599104012402938735024801544932535316065892462592 , a[13,1] = -2744022934559056024469622604306540529/1738222573548705964 451797934060438400, a[10,5] = -274998665672911310282538611757216811612 3/422416144552169230122612885222850560000, a[3,1] = 169309/12204060, a [7,4] = 6340315606886400/56274919101668113\}:" }{TEXT -1 0 "" }}} {PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 20 "printed coe fficients" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 362 "subs(e18,matrix([seq([c[i],seq(a[i,j],j=1..i-1),``$( 7-i)],i=2..7)]));\nfor ii from 8 to 13 do\n print(``);\n print(c[i i]=subs(e18,c[ii])); \n for jj to ii-1 do\n print(a[ii,jj]=subs (e18,a[ii,jj]));\n end do:\nend do:print(``);\nfor ii to 13 do\n p rint(b[ii]=subs(e18,b[ii]));\nend do:print(``);\nfor ii to 13 do\n p rint(`b*`[ii]=subs(e18,`b*`[ii]));\nend do:" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#K%'matrixG6#7(7)#\"\"$\"#]F(%!GF+F+F+F+7)#\"$p%\"%5X#\" '4$p\"\")gS?7#\"'h*>#\"(73W#F+F+F+F+7)#\"%29\"%?!*#F8\"&!3O\"\"!#\"%@U F;F+F+F+7)#\"$+\"\"$d##\"/+,V'H&)H\"\"/dy0OPgLF<#!/++lF!=i\"\"/>ENX7?6 #\"/++?qTu[FEF+F+7)#\"\"(\"#:#\"(2y\"Q\")+]S\")Fu-\"\",0MM\" *R%#\",$z(*fvS\"-+]Si)=#F+7)#\"#7\"#x#\",yD!3!f\"\"-D=i*Gi#F\\Fc#!0MpF358^%\"2D#z_IZ\"4H\"#\"-+'\\J5%)*\"/\"37lgJi &Q)pprint106\"" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%!G" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/&%\"cG6#\"\")#\"#E\"#F" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/&%\"aG6$\"\")\"\"\"#!4\"*eysh!)y,c'\"4++oNlqeGn&" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#/&%\"aG6$\"\")\"\"#\"\"!" }}{PARA 11 " " 1 "" {XPPMATH 20 "6#/&%\"aG6$\"\")\"\"$\"\"!" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/&%\"aG6$\"\")\"\"%#!#<\"\"#" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/&%\"aG6$\"\")\"\"&#!:8mJpj)z^()egqe\"9++w8M5&3NdY_'" } }{PARA 11 "" 1 "" {XPPMATH 20 "6#/&%\"aG6$\"\")\"\"'#\"5FdUS,0g\\?;\"4 WXw__](HbA" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/&%\"aG6$\"\")\"\"(#\"7$ *ygl%=7w<%e'*\"6+_NmOfs4#px" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%!G" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#/&%\"cG6#\"\"*#\"(^55(\"([Z%z" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#/&%\"aG6$\"\"*\"\"\"#!Z*z$=nJ@ML\"QG[m j;EN')>$G=lp!)**=YS\"\"Z++3ex9eO'*y))*fTFd`)y=5%pR`Bq&fJ#" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/&%\"aG6$\"\"*\"\"#\"\"!" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/&%\"aG6$\"\"*\"\"$\"\"!" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/&%\"aG6$\"\"*\"\"%#!WL&HJ`W)e3s;(fM'oA;!z&R\"*ybkV)QI#\"V#fiC*e 1;``K\\a,[-N(QHS7S5*fi\"G" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/&%\"aG6$ \"\"*\"\"&#![o\"RWQTlgz\\@xk^w^N&Hfds%)H@Xd\\P)*4j!z*\"[o++G*o!z4MXYt \\f-[-?R&[684A(obaj37z=" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/&%\"aG6$\" \"*\"\"'#\"gnV/8.[!p2'f,!*ff_B-ROW[D3b)R!eBVq>\"fn%e.G,H)o\"3,u70tqevf #=LcW" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/&%\"aG6$\"\"*\"\"(# \"WL6(*f$Hox7],K5yc\\M'e)3v&ex\"3i%H\"V+[k(fp2NgjkY!R[UB()[\\u'pf&)p;G " }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/&%\"aG6$\"\"*\"\")#\"HppA%)yb#HPo uPI&)H'\\AHO**y%\\$=T&>X)H" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/&%\"aG6$\"#5\"\"&#!IBh6o@d>bX))G7>" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#%!G" }}{PARA 11 "" 1 "" {XPPMATH 20 "6 #/&%\"cG6#\"#6#\"#>\"#?" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/&%\"aG6$\" #6\"\"\"#!NV/Ag-aLRoid$oBamW:Z8K?:\"N++++O&H,_)HHRLz=,N$*H7.W\"" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#/&%\"aG6$\"#6\"\"#\"\"!" }}{PARA 11 " " 1 "" {XPPMATH 20 "6#/&%\"aG6$\"#6\"\"$\"\"!" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/&%\"aG6$\"#6\"\"%#!A\\pfzkNP%f/XICGE*>\"@++c')**[C#[L2 (evT?#" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/&%\"aG6$\"#6\"\"&#!L(=h#Qps 77`5Q$\\!yMKx!3nhG\"K++++wpS\"GHq!\\TVZA[bmJM" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/&%\"aG6$\"#6\"\"'#\"RH#4^r,'>VroiJF.Nc!oMfEL1M8\"Q++7H %z!H)Qk%RdSQ4" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/&%\"aG6$ \"#6\"\"(#\"Pt#>P!*H7,(H,=Z^d^1p)f3gqeD\"O+++3%p&z58C3&H`+?" } }{PARA 11 "" 1 "" {XPPMATH 20 "6#/&%\"aG6$\"#6\"\")#\"=dlq7iBN#HF;$3#H \"\">+++)GCbnOjRY;qZ" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/&%\"aG6$\"#6 \"\"*#\"O^)e$Qrv)o\"elc\\,-vZU^%QK54#\"P+vBgSL'zNa4(zB%Q\\c7k+f&\\'pijT$3J-Z*))*\">RE&)y Rk@*4a.waa5" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/&%\"aG6$\"#7\"\"&#!H*= :Pii&yU2PZu<1u,;D_#\"G+WtLB&))3J_oHKh\\y<=k\"" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/&%\"aG6$\"#7\"\"'#\"KvTR-*)pg`cWxftYZ&eTI\"3%)\"JwL+YV x\"e,l)>%3%QoN=(3yq" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/&%\"aG6$\"#7\" \"(#\"K<;<$HT[bbSW*=KORjr9&*=C\"J78U`TcFzW2GZ/^fC><#3:" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/&%\"aG6$\"#7\"\")#\"5mcFD!3LvZG%\"5*Hppf]Xv#))H" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/&%\"aG6$\"#7\"\"*#!RSa/#**)*e'\\dW< ,aScS_)[EAiL+9wX47YAccOCW^$\"=d@$[kq+RP%o=E`f" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/&%\"aG6$\"#8\"\"&#!Gf\"*o5\\fUp**38U$RsO7PY &\"F+O&*)orSB5%epN*3teHMY" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/&%\"aG6$ \"#8\"\"'#\"JvJY\"G^o?f3nViEU-H\"3R7'\"I[I&\\'*)fg9U^1xK;TE9ofm" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#/&%\"aG6$\"#8\"\"(#\"ID!*42v`R_*pn441r E14U8&\"H)oeMh-qrOV\\OW5ESpkHY" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/&% \"aG6$\"#8\"\")#!6UM^YC9+JOq$\"7nSA#Rx8+]^`\"" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/&%\"aG6$\"#8\"\"*#!N!3u*[EQTk[m;Aog2HB^#o??@\"NT\\XSO> 6>QkY'R^(\\*G4$*Gg-R" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/&%\"aG6$\"#8 \"#5#\"4)3\"G)=KX]UA\"4B'>BG')e')3H" }}{PARA 11 "" 1 "" {XPPMATH 20 "6 #/&%\"aG6$\"#8\"#6\"\"!" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/&%\"aG6$\" #8\"#7\"\"!" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%!G" }}{PARA 11 "" 1 " " {XPPMATH 20 "6#/&%\"bG6#\"\"\"#\",@*zI!>'\".C3+O\"*Q\"" }}{PARA 11 " " 1 "" {XPPMATH 20 "6#/&%\"bG6#\"\"#\"\"!" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/&%\"bG6#\"\"$\"\"!" }}{PARA 11 "" 1 "" {XPPMATH 20 "6# /&%\"bG6#\"\"%\"\"!" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/&%\"bG6#\"\"& \"\"!" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/&%\"bG6#\"\"'#\"0v=imNl-%\"1 OV(er/i8\"" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/&%\"bG6#\"\"(#\"7\")3< \\o@J2-s=\"7]w]OO,$>wa^(" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/&%\"bG6# \"\")#!2nAr%y<6'*[\"2+'Q%**RTx^\"" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/ &%\"bG6#\"\"*#\"RW\\;d`mcpPRr)3yAIM5xZm\"yY#=\"Q&*[(*fA#Q%))zSsT(yjyDA 7v![EFL" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/&%\"bG6#\"#5#!-)[&)G6u4d\"-=ptYz7" }}{PARA 11 "" 1 "" {XPPMATH 20 "6# /&%#b*G6#\"\"#\"\"!" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/&%#b*G6#\"\"$ \"\"!" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/&%#b*G6#\"\"%\"\"!" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/&%#b*G6#\"\"&\"\"!" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/&%#b*G6#\"\"'#\"/D\"G&\\*eu*\"0)GG2JcUF" }}{PARA 11 " " 1 "" {XPPMATH 20 "6#/&%#b*G6#\"\"(#\"5z<=n.P++_z\"6+'o:m+L+Y&>$" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#/&%#b*G6#\"\")#\"1*y,C=$G*o&\"1+)>***> ?o@" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/&%#b*G6#\"\"*#\"Lw0eDJAJQF.dgn r7LNWFeC\"\"Kb$ez&Hqiqt\"e)3L!osX2A5)H" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/&%#b*G6#\"#5#!,Gxjrm(\",Dd%z;8" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/&%#b*G6#\"#6\"\"!" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/&%#b*G6#\"#7 \"\"!" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/&%#b*G6#\"#8#!+(eEE_&\"++)[W x)" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 151 "RK8_13eqs := [op(RowSumConditions(13,'expa nded')),op(OrderConditions(8,13,'expanded'))]:\n`RK7_13eqs*` := subs(b =`b*`,OrderConditions(7,13,'expanded')):" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 6 "Check:" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 112 "subs(e18,RK8_13eqs):\nmap(u->lhs(u)-rhs(u),%);\nnops (%);\nsubs(e18,`RK7_13eqs*`):\nmap(u->lhs(u)-rhs(u),%);\nnops(%);" }} {PARA 12 "" 1 "" {XPPMATH 20 "6#7`x\"\"!F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$ F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$ F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$ F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$ F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$ F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$ F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#\"$7#" }}{PARA 12 "" 1 "" {XPPMATH 20 "6#7ap\"\"!F$F$F$ F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$ F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$ F$F$F$F$F$F$F$F$F$F$F$" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#\"#&)" }}} {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 26 "#-------- -----------------" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}} {PARA 0 "" 0 "" {TEXT -1 52 "#---------------------------------------- -----------" }}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 38 "characteristics of the embedded scheme" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 36 "coefficients of the combined scheme " }}{PARA 0 "" 0 "" {TEXT -1 0 " " }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 5227 "ee := \{c[2]=3/50,\nc[ 3]=469/4510,\nc[4]=1407/9020,\nc[5]=100/257,\nc[6]=7/15,\nc[7]=12/77, \nc[8]=26/27,\nc[9]=7101051/7944748,\nc[10]=11/12,\nc[11]=19/20,\nc[12 ]=1,\nc[13]=1,\n\na[2,1]=3/50,\na[3,1]=169309/12204060,\na[3,2]=219961 /2440812,\na[4,1]=1407/36080,\na[4,2]=0,\na[4,3]=4221/36080,\na[5,1]=1 2985296430100/33603736057857,\na[5,2]=0,\na[5,3]=-16218027650000/11201 245352619,\na[5,4]=48744170200000/33603736057857,\na[6,1]=3817807/8140 5000,\na[6,2]=0,\na[6,3]=0,\na[6,4]=10274191312/43991343405,\na[6,5]=4 0755997793/218862405000,\na[7,1]=15900802578/262289621825,\na[7,2]=0, \na[7,3]=0,\na[7,4]=6340315606886400/56274919101668113,\na[7,5]=-45113 1008276934/12909147305279225,\na[7,6]=984103149600/56231606512081,\na[ 8,1]=-6560178806172785891/5672858706535680000,\na[8,2]=0,\na[8,3]=0,\n a[8,4]=-17/2,\na[8,5]=-5870605887517986369316613/652465735085103413760 000,\na[8,6]=16204960050140425727/2255297505252764544,\na[8,7]=9658417 761218465607893/776920972593666355200,\na[9,1]=-1404618998069651828319 86352616636648283813334213167183799/\n 231595702353396941018788 535727415998878963658147758080000,\na[9,2]=0,\na[9,3]=0,\na[9,4]=-2303 88436455789139579016226863459716720858844533129533/\n 281625991 04012402938735024801544932535316065892462592,\na[9,5]=-979063099837495 7452129847257592953551765164772149796065413844391/\n 1879120863 545568722091311485392002480259497346453409790689280000,\na[9,6]=197043 235803985508254844363902235259599001596076904803130443/\n 445619 65459500736173318259755870730512740108168829012803584,\na[9,7]=2946208 17758575088586344956781032015012776829359971133/\n 2816698559696 7449488723424839046646360350769597644800,\na[9,8]=12687361663017303774 6837292557884226969/13217298519653022949071050320940567220224,\na[10,1 ]=-41617847659972885613040591872450035471/5281300532663365991279044939 4810880000,\na[10,2]=0,\na[10,3]=0,\na[10,4]=-269951087764496912154173 876831/29845195411834947899362922496,\na[10,5]=-2749986656729113102825 386117572168116123/422416144552169230122612885222850560000,\na[10,6]=3 24525706880938787969090744301341259977691/6025335517872182932409928854 7678254006272,\na[10,7]=41487408138830374659637178881584696821/3493656 315517022583105833038931558400,\na[10,8]=932598433870374903885/5729313 0256189894463488,\na[10,9]=-3345742205378079321732624680747620/1912288 84555191941725363563271035831,\na[11,1]=-15203213471544665423683576268 3933540260220443/144031229933501187933392929852012953600000000,\na[11, 2]=0,\na[11,3]=0,\na[11,4]=-19926282430450459437356479596949/220417558 7073348224489986560000,\na[11,5]=-286167080773234780493381053121272693 8261187/343166554822474341490702928140697600000000,\na[11,6]=133406332 6593468056350327316268714319601715109229/19865853988255901709384057394 6438829079429120000, a[11,7]=25587060085986906515751471801297011229903 719273/2019042940187611700532950824131079569408000000,\na[11,8]=129208 3162729235236212706557/47701646396336675524288000000,\na[11,9]=2091032 384514247750201495665581688757138358851/607298691764125649384237970954 35796334060237500,\na[11,10]=-377694382258403/5816420236687500,\na[12, 1]=-240860055167915269684777564157507287321/11700385874985137557408342 4673098438400,\na[12,2]=0,\na[12,3]=0,\na[12,4]=-988947023108341636269 64955900/10545476035409921643978852639,\na[12,5]=-25225160174061774473 7074278562623715189/16418177849613229685231088852333734400,\na[12,6]=8 40813041585474673597744565360698902394175/7078087183568384084198650158 1774346003376,\na[12,7]=241895147163393632189444055554841293171617/150 82171924595104472807447927564153421312,\na[12,8]=42847753308025275666/ 29882754550596969299,\na[12,9]=-77173362222648852405640540117445749658 98992045440/8479033418894797325547814554207128260520626293873,\na[12,1 0]=16262726151696512/8940486723127047,\na[12,11]=-16001553345352000000 /6505707112968117383,\na[13,1]=-2744022934559056024469622604306540529/ 1738222573548705964451797934060438400,\na[13,2]=0,\na[13,3]=0,\na[13,4 ]=-35144243656562246120945761400/5953261868437390070644832157,\na[13,5 ]=-54637123672393421308996942594910689159/4634295873089356958410234071 688953600,\na[13,6]=61239081290242266243670859206851281463175/66596814 26411632770651421460598964953048,\na[13,7]=513420906267106090967699523 9537507099025/462964694026104436494336717002613458688,\na[13,8]=-37036 3100142446513442/1535150001377392224067,\na[13,9]=-2120206825123290760 68221666486441382648974080/3902602893092894975139646643819111936404549 41,\na[13,10]=2242504532188281088/2908865886282319623,\na[13,11]=0,\na [13,12]=0,\n\nb[1]=61903079921/1389136000824,\nb[2]=0,\nb[3]=0,\nb[4]= 0,\nb[5]=0,\nb[6]=402653566621875/1136204715874336,\nb[7]=187202073121 6849170881/7515476193013636507650,\nb[8]=-48961117784712267/1517741399 9438600,\nb[9]=1824678166477710343022780887139376956665357164944/33272 6480751222257863787417240798843822259974895,\nb[10]=-126172885488/1316 7945725,\nb[11]=41251039800000/5264998725539,\nb[12]=-107501739/674957 600,\nb[13]=0,\n\n`b*`[1]=5709741119/127946736918,\n`b*`[2]=0,\n`b*`[3 ]=0,\n`b*`[4]=0,\n`b*`[5]=0,\n`b*`[6]=97458949528125/274256310728288, \n`b*`[7]=79520000370367181779/319546003300661568600,\n`b*`[8]=5689283 182401789/2168201999919800,\n`b*`[9]=124582744353312716760570327383122 3125580576/298102207457268033088581737062702957958355,\n`b*`[10]=-7667 1637728/13167945725,\n`b*`[11]=0,\n`b*`[12]=0,\n`b*`[13]=-5522626587/8 774448800\}:" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 " " 0 "" {TEXT -1 5 "Let " }{XPPEDIT 18 0 "T[8, 13];" "6#&%\"TG6$\"\") \"#8" }{TEXT -1 128 " denote the vector whose components are the prin cipal error terms of the 13 stage, order 8 scheme (the error terms of \+ order 9)." }}{PARA 0 "" 0 "" {TEXT -1 5 "Let " }{XPPEDIT 18 0 "`T*`[7 ,13];" "6#&%#T*G6$\"\"(\"#8" }{TEXT -1 146 " denote the vector whose \+ components are the principal error terms of the embedded 13 stage, ord er 7 scheme (the error terms of order 8) and let " }{XPPEDIT 18 0 "`T *`[8,13];" "6#&%#T*G6$\"\")\"#8" }{TEXT -1 99 " denote the vector who se components are the error terms of order 9 of the embedded order 7 s cheme." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 39 "Denote the 2-norm of these vectors by " }{XPPEDIT 18 0 "abs(abs(T[8, 13]));" "6#-%$absG6#-F$6#&%\"TG6$\"\")\"#8" }{TEXT -1 3 ", " } {XPPEDIT 18 0 "abs(abs(`T*`[7, 13]));" "6#-%$absG6#-F$6#&%#T*G6$\"\"( \"#8" }{TEXT -1 7 " and " }{XPPEDIT 18 0 "abs(abs(`T*`[8, 13]));" "6 #-%$absG6#-F$6#&%#T*G6$\"\")\"#8" }{TEXT -1 15 " respectively." }} {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 8 "Define: " }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "A[9] = abs(abs(T[8, 13]));" "6#/&%\"AG6#\"\"*-%$absG6#-F)6#&%\"TG6$\"\")\"#8" }{TEXT -1 6 ", " }{XPPEDIT 18 0 "B[9] = abs(abs(`T*`[8,13]))/abs(abs(`T*`[7,13 ]));" "6#/&%\"BG6#\"\"**&-%$absG6#-F*6#&%#T*G6$\"\")\"#8\"\"\"-F*6#-F* 6#&F/6$\"\"(F2!\"\"" }{TEXT -1 6 ", " }{XPPEDIT 18 0 "C[9] = abs(a bs(`T*`[8,13]-T[8,13]))/abs(abs(`T*`[7,13]));" "6#/&%\"CG6#\"\"**&-%$a bsG6#-F*6#,&&%#T*G6$\"\")\"#8\"\"\"&%\"TG6$F2F3!\"\"F4-F*6#-F*6#&F06$ \"\"(F3F8" }{TEXT -1 1 "." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 " " 0 "" {TEXT -1 77 "Dormand and Prince have suggested that as well as \+ attempting to ensure that " }{XPPEDIT 18 0 "A[9];" "6#&%\"AG6#\"\"*" }{TEXT -1 73 " is a minimum, if the embedded scheme is to be used for error control, " }{XPPEDIT 18 0 "B[9];" "6#&%\"BG6#\"\"*" }{TEXT -1 7 " and " }{XPPEDIT 18 0 "C[9];" "6#&%\"CG6#\"\"*" }{TEXT -1 27 " s hould be chosen so that " }}{PARA 0 "" 0 "" {TEXT -1 68 "they are \"si milar in magnitude\" and also not differ too much from 1." }}{PARA 0 " " 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 168 "errt erms8_13 := PrincipalErrorTerms(8,13,'expanded'):\n`errterms8_13*` :=s ubs(b=`b*`,errterms8_13):\n`errterms7_13*` := subs(b=`b*`,PrincipalErr orTerms(7,13,'expanded')):" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 333 "snmB := sqrt(add(evalf(subs(ee,`er rterms8_13*`[i]))^2,i=1..nops(`errterms8_13*`))):\nsdnB := sqrt(add(ev alf(subs(ee,`errterms7_13*`[i]))^2,i=1..nops(`errterms7_13*`))):\nsnmC := sqrt(add((evalf(subs(ee,`errterms8_13*`[i])-subs(ee,errterms8_13[i ])))^2,i=1..nops(errterms8_13))):\n'B[9]'= evalf[8](snmB/sdnB);\n'C[9] '= evalf[8](snmC/sdnB);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/&%\"BG6#\" \"*$\")\\)Q,#!\"(" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/&%\"CG6#\"\"*$\" )[L>?!\"(" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 " " {MPLTEXT 1 0 1 ";" }}}}{PARA 0 "" 0 "" {TEXT -1 52 "#--------------- ------------------------------------" }}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 27 "absolute stability regions " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}{SECT 1 {PARA 4 "" 0 " " {TEXT -1 36 "coefficients of the combined scheme " }}{PARA 0 "" 0 " " {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 5227 "ee := \{ c[2]=3/50,\nc[3]=469/4510,\nc[4]=1407/9020,\nc[5]=100/257,\nc[6]=7/15, \nc[7]=12/77,\nc[8]=26/27,\nc[9]=7101051/7944748,\nc[10]=11/12,\nc[11] =19/20,\nc[12]=1,\nc[13]=1,\n\na[2,1]=3/50,\na[3,1]=169309/12204060,\n a[3,2]=219961/2440812,\na[4,1]=1407/36080,\na[4,2]=0,\na[4,3]=4221/360 80,\na[5,1]=12985296430100/33603736057857,\na[5,2]=0,\na[5,3]=-1621802 7650000/11201245352619,\na[5,4]=48744170200000/33603736057857,\na[6,1] =3817807/81405000,\na[6,2]=0,\na[6,3]=0,\na[6,4]=10274191312/439913434 05,\na[6,5]=40755997793/218862405000,\na[7,1]=15900802578/262289621825 ,\na[7,2]=0,\na[7,3]=0,\na[7,4]=6340315606886400/56274919101668113,\na [7,5]=-451131008276934/12909147305279225,\na[7,6]=984103149600/5623160 6512081,\na[8,1]=-6560178806172785891/5672858706535680000,\na[8,2]=0, \na[8,3]=0,\na[8,4]=-17/2,\na[8,5]=-5870605887517986369316613/65246573 5085103413760000,\na[8,6]=16204960050140425727/2255297505252764544,\na [8,7]=9658417761218465607893/776920972593666355200,\na[9,1]=-140461899 806965182831986352616636648283813334213167183799/\n 23159570235 3396941018788535727415998878963658147758080000,\na[9,2]=0,\na[9,3]=0, \na[9,4]=-230388436455789139579016226863459716720858844533129533/\n \+ 28162599104012402938735024801544932535316065892462592,\na[9,5]=-9 790630998374957452129847257592953551765164772149796065413844391/\n \+ 1879120863545568722091311485392002480259497346453409790689280000, \na[9,6]=197043235803985508254844363902235259599001596076904803130443/ \n 44561965459500736173318259755870730512740108168829012803584, \na[9,7]=294620817758575088586344956781032015012776829359971133/\n \+ 28166985596967449488723424839046646360350769597644800,\na[9,8]=1268 73616630173037746837292557884226969/1321729851965302294907105032094056 7220224,\na[10,1]=-41617847659972885613040591872450035471/528130053266 33659912790449394810880000,\na[10,2]=0,\na[10,3]=0,\na[10,4]=-26995108 7764496912154173876831/29845195411834947899362922496,\na[10,5]=-274998 6656729113102825386117572168116123/42241614455216923012261288522285056 0000,\na[10,6]=324525706880938787969090744301341259977691/602533551787 21829324099288547678254006272,\na[10,7]=414874081388303746596371788815 84696821/3493656315517022583105833038931558400,\na[10,8]=9325984338703 74903885/57293130256189894463488,\na[10,9]=-33457422053780793217326246 80747620/191228884555191941725363563271035831,\na[11,1]=-1520321347154 46654236835762683933540260220443/1440312299335011879333929298520129536 00000000,\na[11,2]=0,\na[11,3]=0,\na[11,4]=-19926282430450459437356479 596949/2204175587073348224489986560000,\na[11,5]=-28616708077323478049 33810531212726938261187/343166554822474341490702928140697600000000,\na [11,6]=1334063326593468056350327316268714319601715109229/1986585398825 59017093840573946438829079429120000, a[11,7]=2558706008598690651575147 1801297011229903719273/2019042940187611700532950824131079569408000000, \na[11,8]=1292083162729235236212706557/47701646396336675524288000000, \na[11,9]=2091032384514247750201495665581688757138358851/6072986917641 2564938423797095435796334060237500,\na[11,10]=-377694382258403/5816420 236687500,\na[12,1]=-240860055167915269684777564157507287321/117003858 749851375574083424673098438400,\na[12,2]=0,\na[12,3]=0,\na[12,4]=-9889 4702310834163626964955900/10545476035409921643978852639,\na[12,5]=-252 251601740617744737074278562623715189/164181778496132296852310888523337 34400,\na[12,6]=840813041585474673597744565360698902394175/70780871835 683840841986501581774346003376,\na[12,7]=24189514716339363218944405555 4841293171617/15082171924595104472807447927564153421312,\na[12,8]=4284 7753308025275666/29882754550596969299,\na[12,9]=-771733622226488524056 4054011744574965898992045440/84790334188947973255478145542071282605206 26293873,\na[12,10]=16262726151696512/8940486723127047,\na[12,11]=-160 01553345352000000/6505707112968117383,\na[13,1]=-274402293455905602446 9622604306540529/1738222573548705964451797934060438400,\na[13,2]=0,\na [13,3]=0,\na[13,4]=-35144243656562246120945761400/59532618684373900706 44832157,\na[13,5]=-54637123672393421308996942594910689159/46342958730 89356958410234071688953600,\na[13,6]=612390812902422662436708592068512 81463175/6659681426411632770651421460598964953048,\na[13,7]=5134209062 671060909676995239537507099025/462964694026104436494336717002613458688 ,\na[13,8]=-370363100142446513442/1535150001377392224067,\na[13,9]=-21 2020682512329076068221666486441382648974080/39026028930928949751396466 4381911193640454941,\na[13,10]=2242504532188281088/2908865886282319623 ,\na[13,11]=0,\na[13,12]=0,\n\nb[1]=61903079921/1389136000824,\nb[2]=0 ,\nb[3]=0,\nb[4]=0,\nb[5]=0,\nb[6]=402653566621875/1136204715874336,\n b[7]=1872020731216849170881/7515476193013636507650,\nb[8]=-48961117784 712267/15177413999438600,\nb[9]=18246781664777103430227808871393769566 65357164944/332726480751222257863787417240798843822259974895,\nb[10]=- 126172885488/13167945725,\nb[11]=41251039800000/5264998725539,\nb[12]= -107501739/674957600,\nb[13]=0,\n\n`b*`[1]=5709741119/127946736918,\n` b*`[2]=0,\n`b*`[3]=0,\n`b*`[4]=0,\n`b*`[5]=0,\n`b*`[6]=97458949528125/ 274256310728288,\n`b*`[7]=79520000370367181779/319546003300661568600, \n`b*`[8]=5689283182401789/2168201999919800,\n`b*`[9]=1245827443533127 167605703273831223125580576/298102207457268033088581737062702957958355 ,\n`b*`[10]=-76671637728/13167945725,\n`b*`[11]=0,\n`b*`[12]=0,\n`b*`[ 13]=-5522626587/8774448800\}:" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 78 "The stability function R for th e 13 stage, order 8 scheme is given as follows." }}{EXCHG {PARA 0 "> \+ " 0 "" {MPLTEXT 1 0 76 "subs(ee,StabilityFunction(8,13,'expanded')):\n R := unapply(%,z):\n'R(z)'=R(z);" }}{PARA 12 "" 1 "" {XPPMATH 20 "6#/- %\"RG6#%\"zG,<\"\"\"F)F'F)*&#F)\"\"#F)*$)F'F,F)F)F)*&#F)\"\"'F)*$)F'\" \"$F)F)F)*&#F)\"#CF)*$)F'\"\"%F)F)F)*&#F)\"$?\"F)*$)F'\"\"&F)F)F)*&#F) \"$?(F)*$)F'F1F)F)F)*&#F)\"%S]F)*$)F'\"\"(F)F)F)*&#F)\"&?.%F)*$)F'\"\" )F)F)F)*&#\"Gn71Tey$3M`0#)G#4t!)G')>\"L++Sa!)fxX6C>yuV')Qg>ujR(F)*$)F' \"\"*F)F)F)*&#\"J\\:fkUM_jZ)pn4^JAB!GcF#\"P++w,(GP<(\\npkNK^J([\"H+!)Gg!=wOhZG3\\/#> " 0 "" {MPLTEXT 1 0 26 "z0 := newton(R(z)=1,z=-6);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%#z0G$!+f]$f\"f!\"*" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 302 "z0 := newton(R(z)=1,z=-6) :\np1 := plot([R(z),1],z=-6.69..0.49,color=[red,blue]):\np2 := plot([[ [z0,1]]$3],style=point,symbol=[circle,cross,diamond],color=black):\np3 := plot([[z0,0],[z0,1]],linestyle=3,color=COLOR(RGB,0,.5,0)):\nplots[ display]([p1,p2,p3],view=[-6.69..0.49,-0.07..1.47],font=[HELVETICA,9]) ;" }}{PARA 13 "" 1 "" {GLPLOT2D 360 253 253 {PLOTDATA 2 "6+-%'CURVESG6 $7X7$$!3Q++++++!p'!#<$\"37X<)3X()3*HF*7$$!3ym;a;#[w#p&[$e @#F*7$$!31n\"Hj^BtR'F*$\"3$oxbea9v,#F*7$$!3$)\\P%H<`2K'F*$\"3[(o%ewFq7 =F*7$$!3[L$e&HG=WiF*$\"3A(*\\v>k*ei\"F*7$$!3)QLL#G\\5nhF*$\"3u$=j:xXZX \"F*7$$!3UL$3p-F+4'F*$\"3SfDzCzI*H\"F*7$$!3X]PRjaJ8gF*$\"3)*)=iH%>/f6F *7$$!3fm\"z)**QgOfF*$\"3e=2#=Hs?.\"F*7$$!3)Q$ef@5O%z&F*$\"3i\"=\"e1sH$ G)!#=7$$!3E+D;gn2ZcF*$\"3S[jO1&=Kb'F^o7$$!3gLeHm^v%\\&F*$\"3Xff@p^82^F ^o7$$!3Q+Drp?#HM&F*$\"3_'=ZD-6e&RF^o7$$!3ymm@nIu'=&F*$\"37,;DM]=@IF^o7 $$!3oL$3o^!=\\]F*$\"3C'f;cC@4P#F^o7$$!37++lmwJ%*[F*$\"3sMcGCjR(z\"F^o7 $$!3F++vM*=)QZF*$\"3_qC:0]xe8F^o7$$!3'****\\L4n*)e%F*$\"39L)*z.)>//\"F ^o7$$!3yLeHls)GX%F*$\"3xsDZG(ziA)!#>7$$!3Umm\"y=v5H%F*$\"3m$GVD-=6M'F \\r7$$!3#ommK8**R:%F*$\"3=o\"oFb)=B_F\\r7$$!3O+D'[4ZX*RF*$\"3m^x1ltLfV F\\r7$$!3=nmwA]S`QF*$\"34)*p&46s%3RF\\r7$$!33+D')*z\\&)p$F*$\"3+fe>j;^ &o$F\\r7$$!3'**\\(e254^NF*$\"3iMu50S@!p$F\\r7$$!3wm;HFKB(R$F*$\"3;a/F# ov[)QF\\r7$$!3im\"zdvVfD$F*$\"3&f4thzpO@%F\\r7$$!3kL$3K(Ra.JF*$\"3M(eD gCimr%F\\r7$$!3oLeHIMCXHF*$\"3se(=yU!R(R&F\\r7$$!37+vLUBW2GF*$\"38<0Ux ,#Q7'F\\r7$$!3EL$e@R8'eEF*$\"3)[CRo4Gq0(F\\r7$$!3G++qB&e[]#F*$\"3E)F\\r7$$!3?+]_\"QRWN#F*$\"3l?OmX8Q6&*F\\r7$$!33+D1F6!*3AF*$\"3'>e #yW78*4\"F^o7$$!3?+]24dIZ?F*$\"3#yX/o2u7H\"F^o7$$!3#pmm(Q]5->F*$\"33iO !\\-pF\\\"F^o7$$!3-++vIB2ZQHu\"F^o7$$!30Me*[l*e1;F*$\"3&) pX*=d_d+#F^o7$$!3m***\\Q)\\+`9F*$\"3%*GFiOPoQBF^o7$$!3um\"Hvx$\\38F*$ \"3ws&ez/vAq#F^o7$$!3c*\\7z#zVd6F*$\"3X)zvOD1H9$F^o7$$!3[mmTCKv45F*$\" 3ogVm4%*3VOF^o7$$!3].]il-T^&)F^o$\"3eggEZBB_UF^o7$$!3#)GLLR5JiqF^o$\"3 ZmqOBl+N\\F^o7$$!3^GL30?\\RbF^o$\"3_%Q\\S8enu&F^o7$$!3ch;H\"o#GHSF^o$ \"3[VaQ./g$o'F^o7$$!30%****p:d:k#F^o$\"3#y0+VjR&ywF^o7$$!3/lmT,r1^5F^o $\"3#[aZ&RXG-!*F^o7$$\"3\"Gsmmkh[r$F\\r$\"3,a'o0DZy.\"F*7$$\"3;,]P^YA) )=F^o$\"3*eDN@]Ey?\"F*7$$\"3J,]i9p$*RLF^o$\"3QHNXLV`'R\"F*7$$\"3!***** **********[F^o$\"3&QiB)>iJK;F*-%'COLOURG6&%$RGBG$\"*++++\"!\")$\"\"!Fj \\lFi\\l-F$6$7S7$F($\"\"\"Fj\\l7$F3F_]l7$F=F_]l7$FGF_]l7$FQF_]l7$FenF_ ]l7$FjnF_]l7$F`oF_]l7$FeoF_]l7$FjoF_]l7$F_pF_]l7$FdpF_]l7$FipF_]l7$F^q F_]l7$FcqF_]l7$FhqF_]l7$F^rF_]l7$FcrF_]l7$FhrF_]l7$F]sF_]l7$FbsF_]l7$F gsF_]l7$F\\tF_]l7$FatF_]l7$FftF_]l7$F[uF_]l7$F`uF_]l7$FeuF_]l7$FjuF_]l 7$F_vF_]l7$FdvF_]l7$FivF_]l7$F^wF_]l7$FcwF_]l7$FhwF_]l7$F]xF_]l7$FbxF_ ]l7$FgxF_]l7$F\\yF_]l7$FayF_]l7$FfyF_]l7$F[zF_]l7$F`zF_]l7$FezF_]l7$Fj zF_]l7$F_[lF_]l7$Fd[lF_]l7$Fi[lF_]l7$F^\\lF_]l-Fc\\l6&Fe\\lFi\\lFi\\lF f\\l-F$6&7#7$$!3s******e]$f\"fF*F_]l-%'SYMBOLG6#%'CIRCLEG-Fc\\l6&Fe\\l Fj\\lFj\\lFj\\l-%&STYLEG6#%&POINTG-F$6&Fe`l-Fj`l6#%&CROSSGF]alF_al-F$6 &Fe`l-Fj`l6#%(DIAMONDGF]alF_al-F$6%7$7$Fg`lFi\\lFf`l-%&COLORG6&Fe\\lFi \\l$\"\"&!\"\"Fi\\l-%*LINESTYLEG6#\"\"$-%+AXESLABELSG6%Q\"z6\"Q!F_cl-% %FONTG6#%(DEFAULTG-Fbcl6$%*HELVETICAG\"\"*-%%VIEWG6$;$!$p'!\"#$\"#\\F_ dl;$!\"(F_dl$\"$Z\"F_dl" 1 2 0 1 10 0 2 9 1 4 2 1.000000 45.000000 45.000000 0 0 "Curve 1" "Curve 2" "Curve 3" "Curve 4" "Curve 5" "Curve 6" }}}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 50 "T he following picture shows the stability region. " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 964 "R := z -> a dd(z^j/j!,j=0..8)+\n 19862880730922882055334083785841061267/7396 374196038864374781924114577598054400000*z^9+\n 22756280232231510 967698476352344264591549/875760290307785697431678942862445920033177600 00*z^10+\n 142888784395492966602500449811503638307/1042571774175 9353540853320748362451428966400000*z^11+\n 487315132356469674971 73728701/306827817192044908284761367618060288000*z^12:\npts := []:\nz0 := 0:\nfor ct from 0 to 320 do\n zz := newton(R(z)=exp(ct*Pi/20*I), z=z0):\n z0 := zz:\n pts := [op(pts),[Re(zz),Im(zz)]]:\nend do:\np 1 := plot(pts,color=COLOR(RGB,.35,.1,0)):\np2 := plots[polygonplot]([s eq([pts[i-1],pts[i],[-4.6,0]],i=2..nops(pts))],\n style=patch nogrid,color=COLOR(RGB,.75,.4,.35)):\np3 := plot([[[-6.69,0],[1.19,0]] ,[[0,-6.39],[0,6.39]]],color=black,linestyle=3):\nplots[display]([p||( 1..3)],view=[-6.69..1.19,-6.39..6.39],font=[HELVETICA,9],\n \+ labels=[`Re(z)`,`Im(z)`],axes=boxed,scaling=constrained);" }}{PARA 13 "" 1 "" {GLPLOT2D 381 565 565 {PLOTDATA 2 "6+-%'CURVESG6$7]_l7$$\" \"!F)F(7$F($\"3++++Fjzq:!#=7$F($\"3)******Rl#fTJF-7$F($\"3D+++\")*)Q7Z F-7$$\"3y******eekSZ!#F$\"3_*****fJ&=$G'F-7$$\"3-+++bK\"=<%!#E$\"3e*** **Rp\")R&yF-7$$\"30+++m(\\gS#!#D$\"3X*****4ByZU*F-7$$\"31+++b/&3.\"!#C $\"31+++?vb*4\"!#<7$$\"3-+++sJeKNFI$\"35+++2tjc7FL7$$\"3#*******f9R95! #B$\"3********Rsr89FL7$$\"3:+++jY*e^#FU$\"3/+++Otzq:FL7$$\"3k*****>JBh \\&FU$\"34+++2v(ys\"FL7$$\"3)*******)G&yp5!#A$\"31+++%[d\\)=FL7$$\"39+ ++;e(H'=F_o$\"36+++#)o.U?FL7$$\"3++++<\\#)))GF_o$\"3/+++^c6*>#FL7$$\"3 (******Rr>5\"RF_o$\"33+++(Q&>cBFL7$$\"3-+++U!QBP%F_o$\"3>+++)>#G8DFL7$ $\"3#)*****fyUHI$F_o$\"3')*****H3#RqEFL7$$!3)******f_a'fiFU$\"3(****** z)*fv#GFL7$$!3b*****zS/gq)F_o$\"37+++JQ&[)HFL7$$!3*)*****HRKN6#!#@$\"3 7+++qZRUJFL7$$!3G+++(z0+[$Fhq$\"30+++\"*\\Q+LFL7$$!3#)******3g)f!RFhq$ \"3))******HT9fMFL7$$!3c*****\\_#f5!)F_o$\"3&)*****>ya\">OFL7$$\"3)*** ***H6,(Q6!#?$\"3))*****R986y$FL7$$\"3e*****4dDNR%F]s$\"3=+++n(ff%RFL7$ $\"31+++rRA$>\"!#>$\"3z*****\\ZtZ6%FL7$$\"3<+++-_;8GFhs$\"3g*****f,V!) G%FL7$$\"37+++376OgFhs$\"3u*****>dnFY%FL7$$\"31+++\\$[R9\"F-$\"3v***** z(H(oi%FL7$$\"34+++=mu:=F-$\"3))*****>q!HaCF-$\" 37+++-*3i([FL7$$\"3))*****p\\`S)HF-$\"3Q+++o:2p\\FL7$$\"3'******H\\)R) R$F-$\"3!)*****\\(*4,0&FL7$$\"3++++Q:!*4PF-$\"3++++%)f/B^FL7$$\"3***** ***e&QI$RF-$\"33+++TV!**=&FL7$$\"3A+++4wS!3%F-$\"3w*****HlB=D&FL7$$\"3 ++++fmRiTF-$\"3B+++_UZ4`FL7$$\"3')*****\\&=a(=%F-$\"3U+++^'pKO&FL7$$\" 3%******4mHH;%F-$\"3A+++NsZ8aFL7$$\"3-+++>%=X4%F-$\"3'******\\Zy-Y&FL7 $$\"3w*****fJwt)RF-$\"3H+++>F!Q]&FL7$$\"3A+++[_&e%QF-$\"3w*****f\\YTa& FL7$$\"3C+++A%>Pn$F-$\"3')*******e%Q\"e&FL7$$\"3'******\\lUUZ$F-$\"3-+ ++-jd:cFL7$$\"3?+++4RG]KF-$\"3'*******\\)onk&FL7$$\"3&)*****pKZV+$F-$ \"3m*****>.**\\n&FL7$$\"3/+++V3jQFF-$\"3L+++QQH+dFL7$$\"3'******p&f1bC F-$\"3i*****>PqEs&FL7$$\"3%******4N]`:#F-$\"3@+++A`8UdFL7$$\"3++++jV(4 %=F-$\"3%******z2%oedFL7$$\"3,+++()=C8:F-$\"3i*****p]*HsdFL7$$\"3++++# *eGt6F-$\"3?+++$Q]Hy&FL7$$\"3w*****>Z$y?#)Fhs$\"3w*****\\O*e!z&FL7$$\" 3>+++J*QVg%Fhs$\"33+++9/:&z&FL7$$\"3M*****4L\"f+*)F]s$\"3!)*****pVXlz& FL7$$!3!*******eMF/y&FL7$$!3#******p!fM)[\"F-$ \"3'******f'*Rww&FL7$$!3-+++\\[_1>F-$\"3m*****\\o/2v&FL7$$!3()*****\\` qiL#F-$\"3v*****HGC#HdFL7$$!3!******>.Z,y#F-$\"3m******)3&p-dFL7$$!3C+ ++O2_UKF-$\"31+++o9YqcFL7$$!3*)*****pZ-4t$F-$\"3z*****H?r;j&FL7$$!3C++ ++nfeUF-$\"3-+++S\"[_e&FL7$$!3#)*****\\Pe+&[F-$\"3@+++3))**HbFL7$$!3c+ ++q$o3b&F-$\"3y******>*Q`Y&FL7$$!3G+++s:QG&FL7 $$!3/+++svC85FL$\"3q*****>JG1E&FL7$$!3'*******yK5L6FL$\"37+++C#*f]_FL7 $$!3#******4s\"3W7FL$\"3)******Rv)3Z_FL7$$!31+++b&ztM\"FL$\"3!******>+ QjC&FL7$$!3)*******GdNW9FL$\"3A+++\">#HY_FL7$$!3!******fHfh`\"FL$\"3i* ****\\y!zX_FL7$$!3&******p5-Pi\"FL$\"3!*******f.:W_FL7$$!32++++0q2+@()y\"FL$\"3&)*****znOfB&FL7$$!3%******pQ FL$\"39+++*)\\#)>_FL7$$!3; +++=)\\!=?FL$\"3$******HhW&3_FL7$$!37+++fu%44#FL$\"3k*****R?S]>&FL7$$! 3#******4ZrC;#FL$\"3))*****R_y#z^FL7$$!3#******>49GB#FL$\"3'*******fyB h^FL7$$!36+++^p9-BFL$\"3\")*****z:049&FL7$$!3'*******4miqBFL$\"3M+++dX F=^FL7$$!39+++f*)RQCFL$\"3e*****z(pM$4&FL7$$!3')*****\\i-c]#FL$\"3)*** ***\\bIh1&FL7$$!3')*****R^rBd#FL$\"3%******pvTm.&FL7$$!39+++`k$)QEFL$ \"3u*****H72\\+&FL7$$!3)******z)e70FFL$\"3w*****HYl4(\\FL7$$!3A+++#[l8 x#FL$\"3s*****>Qq[$\\FL7$$!31+++sknPGFL$\"33+++iGp'*[FL7$$!3@+++jE^Bl&[FL7$$!3/+++Zg&4(HFL$\"3$******>*RZ9[FL7$$!3))*****zC 6\"QIFL$\"3Y+++%*pnqZFL7$$!3%)*****zG*p0JFL$\"3*)*****z<%GDZFL7$$!3/++ +3:vtJFL$\"3q*****fkh%yYFL7$$!3\"******HskAC$FL$\"38+++9HQIYFL7$$!30++ +^()>6LFL$\"39+++s)>7e%FL7$$!3;+++prZ!Q$FL$\"3U+++FG8JXFL7$$!35+++^:** \\MFL$\"3i*****H/h-[%FL7$$!3?+++X(3'>NFL$\"33+++V_rGWFL7$$!3%)******p+ =*e$FL$\"37+++.JdwVFL7$$!3?+++G5beOFL$\"3m*****z6yQK%FL7$$!3\"******H@ qvs$FL$\"3e*****z`T1F%FL7$$!37+++kk4'z$FL$\"3z*****ReYo@%FL7$$!3'***** **fQ+kQFL$\"3K+++dRXiTFL7$$!3-+++`X=JRFL$\"3E+++IxS2TFL7$$!3!******4!) \\v*RFL$\"35+++Z4k^SFL7$$!3')******G'HI1%FL$\"37+++403&*RFL7$$!31+++1; dFTFL$\"3&)*****R>^w$RFL7$$!3y******=\"R6>%FL$\"3#*******z(y#zQFL7$$!3 %******4U4PD%FL$\"38+++1C*)>QFL7$$!3t*****>nr_J%FL$\"3')******)GE%fPFL 7$$!3y*****>/DeP%FL$\"3=+++^4#yp$FL7$$!3H+++^pPNWFL$\"3&******45C]j$FL 7$$!3))*****HUTR\\%FL$\"3%******RD\"*4d$FL7$$!3w*****zaP:b%FL$\"3:+++' ='o0NFL7$$!3#******p*z=3YFL$\"3\")*****>D\"3RMFL7$$!31+++uv\"Rm%FL$\"3 1+++*ed6P$FL7$$!3')*****zu^(=ZFL$\"3$******>?0>I$FL7$$!3%)*****><:Fx%F L$\"3<+++)*GKJKFL7$$!3%)*****4;Ie#[FL$\"3y*****\\/=%fJFL7$$!33+++\"=: \"y[FL$\"31++++h?'3$FL7$$!3\"******RP$eH\\FL$\"3')*****p))4<,$FL7$$!3= +++\">T-)\\FL$\"3y*****\\fef$HFL7$$!3T+++?s3I]FL$\"3=+++4l)*eGFL7$$!3o *****RG6\"z]FL$\"3#)*****p]J3y#FL7$$!3s******))QHF^FL$\"3)******>RL:q# FL7$$!3I+++\"31Y<&FL$\"3@+++m@8@EFL7$$!3>+++;(45A&FL$\"3A+++TjmRDFL7$$ !3'******z8ekE&FL$\"3!******4VrrX#FL7$$!3C+++*>(*3J&FL$\"3#)*****pqyOP #FL7$$!3++++PlEa`FL$\"3!)*****fJ9#*G#FL7$$!37+++:5]'R&FL$\"3<+++#z)z.A FL7$$!3z*****4?KvV&FL$\"3:+++#)pW<@FL7$$!3))*****f\")*GxaFL$\"3()***** 4Oo,.#FL7$$!3%*******\\Iq:bFL$\"3++++zw'>%>FL7$$!3O+++D6qTr&FL$\"33+++#*>M$R\"FL7$$!3A+++D7MTdFL$ \"3!******RXe')H\"FL7$$!3K+++\"*GomdFL$\"30+++v;1.7FL7$$!37+++@3:!z&FL $\"32+++A#pl5\"FL7$$!3%)*****f![q6eFL$\"3\"******zY1#45FL7$$!3H+++-(38 $eFL$\"3k*****R@v+6*F-7$$!3#)*****>QI*[eFL$\"3!)*****Hdf,7)F-7$$!3G+++ p;akeFL$\"3v*****f.`G7(F-7$$!3O+++L#=\"yeFL$\"3T+++`Cz=hF-7$$!3s****** f%R'*)eFL$\"3C+++R]r3^F-7$$!3o******3$)3**eFL$\"35+++HOX$4%F-7$$!3d*** ***[6X1fFL$\"3%******\\CGR2$F-7$$!38+++)e<<\"fFL$\"3#*******GS8^?F-7$$ !3;+++L.)[\"fFL$\"3$*******He7E5F-7$$!3s******e]$f\"fFLF(7$F`]n$!3$*** ****He7E5F-7$F[]n$!3#*******GS8^?F-7$Ff\\n$!3%******\\CGR2$F-7$Fa\\n$! 35+++HOX$4%F-7$F\\\\n$!3C+++R]r3^F-7$Fg[n$!3T+++`Cz=hF-7$Fb[n$!3v***** f.`G7(F-7$F][n$!3!)*****Hdf,7)F-7$Fhjm$!3k*****R@v+6*F-7$Fcjm$!3\"**** **zY1#45FL7$F^jm$!32+++A#pl5\"FL7$Fiim$!30+++v;1.7FL7$Fdim$!3!******RX e')H\"FL7$F_im$!33+++#*>M$R\"FL7$Fjhm$!31+++Q95([\"FL7$Fehm$!3/+++T9$* z:FL7$F`hm$!3%******H%3$=n\"FL7$F[hm$!3!******RI,Gw\"FL7$Ffgm$!3-+++$* e%G&=FL7$Fagm$!3++++zw'>%>FL7$F\\gm$!3()*****4Oo,.#FL7$Fgfm$!3:+++#)pW <@FL7$Fbfm$!3<+++#z)z.AFL7$F]fm$!3!)*****fJ9#*G#FL7$Fhem$!3#)*****pqyO P#FL7$Fcem$!3!******4VrrX#FL7$F^em$!3A+++TjmRDFL7$Fidm$!3@+++m@8@EFL7$ Fddm$!3)******>RL:q#FL7$F_dm$!3#)*****p]J3y#FL7$Fjcm$!3=+++4l)*eGFL7$F ecm$!3y*****\\fef$HFL7$F`cm$!3')*****p))4<,$FL7$F[cm$!31++++h?'3$FL7$F fbm$!3y*****\\/=%fJFL7$Fabm$!3<+++)*GKJKFL7$F\\bm$!3$******>?0>I$FL7$F gam$!31+++*ed6P$FL7$Fbam$!3\")*****>D\"3RMFL7$F]am$!3:+++'='o0NFL7$Fh` m$!3%******RD\"*4d$FL7$Fc`m$!3&******45C]j$FL7$F^`m$!3=+++^4#yp$FL7$Fi _m$!3')******)GE%fPFL7$Fd_m$!38+++1C*)>QFL7$F__m$!3#*******z(y#zQFL7$F j^m$!3&)*****R>^w$RFL7$Fe^m$!37+++403&*RFL7$F`^m$!35+++Z4k^SFL7$F[^m$! 3E+++IxS2TFL7$Ff]m$!3K+++dRXiTFL7$Fa]m$!3z*****ReYo@%FL7$F\\]m$!3e**** *z`T1F%FL7$Fg\\m$!3m*****z6yQK%FL7$Fb\\m$!37+++.JdwVFL7$F]\\m$!33+++V_ rGWFL7$Fh[m$!3i*****H/h-[%FL7$Fc[m$!3U+++FG8JXFL7$F^[m$!39+++s)>7e%FL7 $Fijl$!38+++9HQIYFL7$Fdjl$!3q*****fkh%yYFL7$F_jl$!3*)*****z<%GDZFL7$Fj il$!3Y+++%*pnqZFL7$Feil$!3$******>*RZ9[FL7$F`il$!3))*****>^Bl&[FL7$F[i l$!33+++iGp'*[FL7$Ffhl$!3s*****>Qq[$\\FL7$Fahl$!3w*****HYl4(\\FL7$F\\h l$!3u*****H72\\+&FL7$Fggl$!3%******pvTm.&FL7$Fbgl$!3)******\\bIh1&FL7$ F]gl$!3e*****z(pM$4&FL7$Fhfl$!3M+++dXF=^FL7$Fcfl$!3\")*****z:049&FL7$F ^fl$!3'*******fyBh^FL7$Fiel$!3))*****R_y#z^FL7$Fdel$!3k*****R?S]>&FL7$ F_el$!3$******HhW&3_FL7$Fjdl$!39+++*)\\#)>_FL7$Fedl$!3*)*****f*4$*G_FL 7$F`dl$!3&)*****znOfB&FL7$F[dl$!3S+++.L&4C&FL7$Ffcl$!3!*******f.:W_FL7 $Facl$!3i*****\\y!zX_FL7$F\\cl$!3A+++\">#HY_FL7$Fgbl$!3!******>+QjC&FL 7$Fbbl$!3)******Rv)3Z_FL7$F]bl$!37+++C#*f]_FL7$Fhal$!3q*****>JG1E&FL7$ Fcal$!3%******>s:QG&FL7$F^al$!3C+++UobG`FL7$Fi`l$!3;+++(\\*=%R&FL7$Fd` l$!3y******>*Q`Y&FL7$F_`l$!3@+++3))**HbFL7$Fj_l$!3-+++S\"[_e&FL7$Fe_l$ !3z*****H?r;j&FL7$F`_l$!31+++o9YqcFL7$F[_l$!3m******)3&p-dFL7$Ff^l$!3v *****HGC#HdFL7$Fa^l$!3m*****\\o/2v&FL7$F\\^l$!3'******f'*Rww&FL7$Fg]l$ !3G+++q&>/y&FL7$$!3J+++#H1\\\"oFhs$!3y******HpM*y&FL7$F]]l$!31+++*))fY z&FL7$Fh\\l$!3!)*****pVXlz&FL7$Fc\\l$!33+++9/:&z&FL7$F^\\l$!3w*****\\O *e!z&FL7$Fi[l$!3?+++$Q]Hy&FL7$Fd[l$!3i*****p]*HsdFL7$F_[l$!3%******z2% oedFL7$Fjz$!3@+++A`8UdFL7$Fez$!3i*****>PqEs&FL7$F`z$!3L+++QQH+dFL7$F[z $!3m*****>.**\\n&FL7$Ffy$!3'*******\\)onk&FL7$Fay$!3-+++-jd:cFL7$F\\y$ !3')*******e%Q\"e&FL7$Fgx$!3w*****f\\YTa&FL7$Fbx$!3H+++>F!Q]&FL7$F]x$! 3'******\\Zy-Y&FL7$Fhw$!3A+++NsZ8aFL7$Fcw$!3U+++^'pKO&FL7$F^w$!3B+++_U Z4`FL7$Fiv$!3w*****HlB=D&FL7$Fdv$!33+++TV!**=&FL7$F_v$!3++++%)f/B^FL7$ Fju$!3!)*****\\(*4,0&FL7$Feu$!3Q+++o:2p\\FL7$F`u$!37+++-*3i([FL7$F[u$! 3))*****>dnFY%FL7$F\\t$! 3g*****f,V!)G%FL7$Ffs$!3z*****\\ZtZ6%FL7$Fas$!3=+++n(ff%RFL7$F[s$!3))* ****R986y$FL7$Ffr$!3&)*****>ya\">OFL7$Far$!3))******HT9fMFL7$F\\r$!30+ ++\"*\\Q+LFL7$$!36+++#RKN6#Fhq$!37+++qZRUJFL7$Faq$!37+++JQ&[)HFL7$$!3m *****\\_a'fiFU$!3(******z)*fv#GFL7$Fgp$!3')*****H3#RqEFL7$Fbp$!3>+++)> #G8DFL7$F]p$!33+++(Q&>cBFL7$Fho$!3/+++^c6*>#FL7$Fco$!36+++#)o.U?FL7$F] o$!31+++%[d\\)=FL7$Fhn$!34+++2v(ys\"FL7$$\"3-+++cY*e^#FU$!3/+++Otzq:FL 7$FS$!3********Rsr89FL7$FN$!35+++2tjc7FL7$FG$!31+++?vb*4\"FL7$$\"3()** ***pw\\gS#FC$!3X*****4ByZU*F-7$$\"3++++mK\"=<%F=$!3e*****Rp\")R&yF-7$$ \"3w*****z)ekSZF7$!3_*****fJ&=$G'F-7$F($!3D+++\")*)Q7ZF-7$F($!3)****** Rl#fTJF-7$F($!3++++Fjzq:F-F'-%&COLORG6&%$RGBG$\"#N!\"#$\"\"\"!\"\"F(-% )POLYGONSG6^_l7%F'7$F($\"+Fjzq:!#57$$!#YF[]pF(7%F`]p7$F($\"+aEfTJFc]pF d]p7%Fh]p7$F($\"+\")*)Q7ZFc]pFd]p7%F\\^p7$$\"+fekSZFhs$\"+;`=$G'Fc]pFd ]p7%F`^p7$$\"+bK\"=<%F-$\"+%p\")R&yFc]pFd]p7%Ff^p7$$\"+m(\\gS#FL$\"+J# yZU*Fc]pFd]p7%F\\_p7$$\"+b/&3.\"!#;$\"+?vb*4\"!\"*Fd]p7%Fb_p7$$\"+sJeK NFe_p$\"+2tjc7Fh_pFd]p7%Fj_p7$$\"+g9R95!#:$\"+Ssr89Fh_pFd]p7%F``p7$$\" +jY*e^#Fc`p$\"+Otzq:Fh_pFd]p7%Fg`p7$$\"+7L7'\\&Fc`p$\"+2v(ys\"Fh_pFd]p 7%F]ap7$$\"+*G&yp5!#9$\"+%[d\\)=Fh_pFd]p7%Fcap7$$\"+;e(H'=Ffap$\"+#)o. U?Fh_pFd]p7%Fjap7$$\"+<\\#)))GFfap$\"+^c6*>#Fh_pFd]p7%F`bp7$$\"+9(>5\" RFfap$\"+(Q&>cBFh_pFd]p7%Ffbp7$$\"+U!QBP%Ffap$\"+)>#G8DFh_pFd]p7%F\\cp 7$$\"+'yUHI$Ffap$\"+$3#RqEFh_pFd]p7%Fbcp7$$!+EXlfiFc`p$\"+))*fv#GFh_pF d]p7%Fhcp7$$!+3W+1()Ffap$\"+JQ&[)HFh_pFd]p7%F^dp7$$!+$RKN6#!#8$\"+qZRU JFh_pFd]p7%Fddp7$$!+(z0+[$Fgdp$\"+\"*\\Q+LFh_pFd]p7%F[ep7$$!+4g)f!RFgd p$\"+IT9fMFh_pFd]p7%Faep7$$!+DDf5!)Ffap$\"+#ya\">OFh_pFd]p7%Fgep7$$\"+ 86qQ6!#7$\"+WJ6\"y$Fh_pFd]p7%F]fp7$$\"+rb_$R%F`fp$\"+n(ff%RFh_pFd]p7%F dfp7$$\"+rRA$>\"!#6$\"+vMx9TFh_pFd]p7%Fjfp7$$\"+-_;8GF]gp$\"+;I/)G%Fh_ pFd]p7%Fagp7$$\"+376OgF]gp$\"+svwiWFh_pFd]p7%Fggp7$$\"+\\$[R9\"Fc]p$\" +yH(oi%Fh_pFd]p7%F]hp7$$\"+=mu:=Fc]p$\"+s^#\\w%Fh_pFd]p7%Fchp7$$\"+-2H aCFc]p$\"+-*3i([Fh_pFd]p7%Fihp7$$\"+(\\`S)HFc]p$\"+o:2p\\Fh_pFd]p7%F_i p7$$\"+$\\)R)R$Fc]p$\"+v*4,0&Fh_pFd]p7%Feip7$$\"+Q:!*4PFc]p$\"+%)f/B^F h_pFd]p7%F[jp7$$\"+f&QI$RFc]p$\"+TV!**=&Fh_pFd]p7%Fajp7$$\"+4wS!3%Fc]p $\"+`O#=D&Fh_pFd]p7%Fgjp7$$\"+fmRiTFc]p$\"+_UZ4`Fh_pFd]p7%F][q7$$\"+b= a(=%Fc]p$\"+^'pKO&Fh_pFd]p7%Fc[q7$$\"+h'HH;%Fc]p$\"+NsZ8aFh_pFd]p7%Fi[ q7$$\"+>%=X4%Fc]p$\"+v%y-Y&Fh_pFd]p7%F_\\q7$$\"+;jP()RFc]p$\"+>F!Q]&Fh _pFd]p7%Fe\\q7$$\"+[_&e%QFc]p$\"+'\\YTa&Fh_pFd]p7%F[]q7$$\"+A%>Pn$Fc]p $\"+!f%Q\"e&Fh_pFd]p7%Fa]q7$$\"+bECuMFc]p$\"+-jd:cFh_pFd]p7%Fg]q7$$\"+ 4RG]KFc]p$\"+])onk&Fh_pFd]p7%F]^q7$$\"+FtM/IFc]p$\"+K!**\\n&Fh_pFd]p7% Fc^q7$$\"+V3jQFFc]p$\"+QQH+dFh_pFd]p7%Fi^q7$$\"+df1bCFc]p$\"+s.nAdFh_p Fd]p7%F__q7$$\"+^.Nb@Fc]p$\"+A`8UdFh_pFd]p7%Fe_q7$$\"+jV(4%=Fc]p$\"+yS oedFh_pFd]p7%F[`q7$$\"+()=C8:Fc]p$\"+2&*HsdFh_pFd]p7%Fa`q7$$\"+#*eGt6F c]p$\"+$Q]Hy&Fh_pFd]p7%Fg`q7$$\"+sMy?#)F]gp$\"+l$*e!z&Fh_pFd]p7%F]aq7$ $\"+J*QVg%F]gp$\"+9/:&z&Fh_pFd]p7%Fcaq7$$\"+J8f+*)F`fp$\"+Paa'z&Fh_pFd ]p7%Fiaq7$$!+fMF/y&Fh_pFd]p7%F[cq7$$!+2fM)[ \"Fc]p$\"+m*Rww&Fh_pFd]p7%Facq7$$!+\\[_1>Fc]p$\"+&o/2v&Fh_pFd]p7%Fgcq7 $$!+N0FOBFc]p$\"+$GC#HdFh_pFd]p7%F]dq7$$!+Kq9!y#Fc]p$\"+*3&p-dFh_pFd]p 7%Fcdq7$$!+O2_UKFc]p$\"+o9YqcFh_pFd]p7%Fidq7$$!+xC!4t$Fc]p$\"+.7nJcFh_ pFd]p7%F_eq7$$!++nfeUFc]p$\"+S\"[_e&Fh_pFd]p7%Feeq7$$!+v$e+&[Fc]p$\"+3 ))**HbFh_pFd]p7%F[fq7$$!+q$o3b&Fc]p$\"+?*Q`Y&Fh_pFd]p7%Fafq7$$!+#HY_Fh_pFd]p7%Faiq7$$!+'Hfh`\"Fh_p$\"+&y!zX_Fh_pFd]p7%Fgiq7$$!+2@qB ;Fh_p$\"+g.:W_Fh_pFd]p7%F]jq7$$!++0q2Fh_p$\"+*)\\#)>_Fh_pFd]p7%Fe[r7$$!+=)\\!=?Fh_p$\"+8Ya 3_Fh_pFd]p7%F[\\r7$$!+fu%44#Fh_p$\"+/-/&>&Fh_pFd]p7%Fa\\r7$$!+r9Zi@Fh_ p$\"+C&y#z^Fh_pFd]p7%Fg\\r7$$!+#49GB#Fh_p$\"+gyBh^Fh_pFd]p7%F]]r7$$!+^ p9-BFh_p$\"+e^!49&Fh_pFd]p7%Fc]r7$$!+5miqBFh_p$\"+dXF=^Fh_pFd]p7%Fi]r7 $$!+f*)RQCFh_p$\"+ypM$4&Fh_pFd]p7%F_^r7$$!+DEg0DFh_p$\"+b08m]Fh_pFd]p7 %Fe^r7$$!+9:PsDFh_p$\"+d6LFh_p$\"+s)>7e%Fh_pFd]p7%F]cr7$$!+prZ!Q$Fh_p$\"+FG8JXF h_pFd]p7%Fccr7$$!+^:**\\MFh_p$\"+V5E![%Fh_pFd]p7%Ficr7$$!+X(3'>NFh_p$ \"+V_rGWFh_pFd]p7%F_dr7$$!+q+=*e$Fh_p$\"+.JdwVFh_pFd]p7%Fedr7$$!+G5beO Fh_p$\"+=\"yQK%Fh_pFd]p7%F[er7$$!+8-dFPFh_p$\"+Q:kqUFh_pFd]p7%Faer7$$! +kk4'z$Fh_p$\"+%eYo@%Fh_pFd]p7%Fger7$$!+gQ+kQFh_p$\"+dRXiTFh_pFd]p7%F] fr7$$!+`X=JRFh_p$\"+IxS2TFh_pFd]p7%Fcfr7$$!+,)\\v*RFh_p$\"+Z4k^SFh_pFd ]p7%Fifr7$$!+H'HI1%Fh_p$\"+403&*RFh_pFd]p7%F_gr7$$!+1;dFTFh_p$\"+%>^w$ RFh_pFd]p7%Fegr7$$!+>\"R6>%Fh_p$\"+!yy#zQFh_pFd]p7%F[hr7$$!+@%4PD%Fh_p $\"+1C*)>QFh_pFd]p7%Fahr7$$!+s;F:VFh_p$\"+*GE%fPFh_pFd]p7%Fghr7$$!+U]# eP%Fh_p$\"+^4#yp$Fh_pFd]p7%F]ir7$$!+^pPNWFh_p$\"+,T-NOFh_pFd]p7%Fcir7$ $!+B9%R\\%Fh_p$\"+a7*4d$Fh_pFd]p7%Fiir7$$!+[v`^XFh_p$\"+'='o0NFh_pFd]p 7%F_jr7$$!+(*z=3YFh_p$\"+_73RMFh_pFd]p7%Fejr7$$!+uv\"Rm%Fh_p$\"+*ed6P$ Fh_pFd]p7%F[[s7$$!+[I$Fh_pFd]p7%Fa[s7$$!+s^rsZFh_p$\"+ )*GKJKFh_pFd]p7%Fg[s7$$!+h,$e#[Fh_p$\"+X!=%fJFh_pFd]p7%F]\\s7$$!+\"=: \"y[Fh_p$\"++h?'3$Fh_pFd]p7%Fc\\s7$$!+uLeH\\Fh_p$\"+())4<,$Fh_pFd]p7%F i\\s7$$!+\">T-)\\Fh_p$\"+&fef$HFh_pFd]p7%F_]s7$$!+?s3I]Fh_p$\"+4l)*eGF h_pFd]p7%Fe]s7$$!+%G6\"z]Fh_p$\"+2:$3y#Fh_pFd]p7%F[^s7$$!+*)QHF^Fh_p$ \"+#RL:q#Fh_pFd]p7%Fa^s7$$!+\"31Y<&Fh_p$\"+m@8@EFh_pFd]p7%Fg^s7$$!+;(4 5A&Fh_p$\"+TjmRDFh_pFd]p7%F]_s7$$!+Q\"ekE&Fh_p$\"+J9(*3J&Fh_p$\"+2(yOP#Fh_pFd]p7%Fi_s7$$!+PlEa`Fh_p$\"+;V@*G#Fh_pFd] p7%F_`s7$$!+:5]'R&Fh_p$\"+#z)z.AFh_pFd]p7%Fe`s7$$!+,A`PaFh_p$\"+#)pW<@ Fh_pFd]p7%F[as7$$!+;)*GxaFh_p$\"+h$o,.#Fh_pFd]p7%Faas7$$!+]Iq:bFh_p$\" +zw'>%>Fh_pFd]p7%Fgas7$$!+DM$R\"Fh_pFd]p7%F[ds7$$!+D7MTdFh_p$\"+a%e' )H\"Fh_pFd]p7%Fads7$$!+\"*GomdFh_p$\"+v;1.7Fh_pFd]p7%Fgds7$$!+@3:!z&Fh _p$\"+A#pl5\"Fh_pFd]p7%F]es7$$!+1[q6eFh_p$\"+ok?45Fh_pFd]p7%Fces7$$!+- (38$eFh_p$\"+9_25\"*Fc]pFd]p7%Fies7$$!+#QI*[eFh_p$\"+t&f,7)Fc]pFd]p7%F _fs7$$!+p;akeFh_p$\"+OI&G7(Fc]pFd]p7%Fefs7$$!+L#=\"yeFh_p$\"+`Cz=hFc]p Fd]p7%F[gs7$$!+g%R'*)eFh_p$\"+R]r3^Fc]pFd]p7%Fags7$$!+4$)3**eFh_p$\"+H OX$4%Fc]pFd]p7%Fggs7$$!+\\6X1fFh_p$\"+X#GR2$Fc]pFd]p7%F]hs7$$!+)e<<\"f Fh_p$\"+HS8^?Fc]pFd]p7%Fchs7$$!+L.)[\"fFh_p$\"+Ie7E5Fc]pFd]p7%Fihs7$$! +f]$f\"fFh_pF(Fd]p7%F_is7$Fjhs$!+Ie7E5Fc]pFd]p7%Fcis7$Fdhs$!+HS8^?Fc]p Fd]p7%Fgis7$F^hs$!+X#GR2$Fc]pFd]p7%F[js7$Fhgs$!+HOX$4%Fc]pFd]p7%F_js7$ Fbgs$!+R]r3^Fc]pFd]p7%Fcjs7$F\\gs$!+`Cz=hFc]pFd]p7%Fgjs7$Fffs$!+OI&G7( Fc]pFd]p7%F[[t7$F`fs$!+t&f,7)Fc]pFd]p7%F_[t7$Fjes$!+9_25\"*Fc]pFd]p7%F c[t7$Fdes$!+ok?45Fh_pFd]p7%Fg[t7$F^es$!+A#pl5\"Fh_pFd]p7%F[\\t7$Fhds$! +v;1.7Fh_pFd]p7%F_\\t7$Fbds$!+a%e')H\"Fh_pFd]p7%Fc\\t7$F\\ds$!+#*>M$R \"Fh_pFd]p7%Fg\\t7$Ffcs$!+Q95([\"Fh_pFd]p7%F[]t7$F`cs$!+T9$*z:Fh_pFd]p 7%F_]t7$Fjbs$!+V3$=n\"Fh_pFd]p7%Fc]t7$Fdbs$!+/8!Gw\"Fh_pFd]p7%Fg]t7$F^ bs$!+$*e%G&=Fh_pFd]p7%F[^t7$Fhas$!+zw'>%>Fh_pFd]p7%F_^t7$Fbas$!+h$o,.# Fh_pFd]p7%Fc^t7$F\\as$!+#)pW<@Fh_pFd]p7%Fg^t7$Ff`s$!+#z)z.AFh_pFd]p7%F [_t7$F``s$!+;V@*G#Fh_pFd]p7%F__t7$Fj_s$!+2(yOP#Fh_pFd]p7%Fc_t7$Fd_s$!+ J9I$Fh_pFd]p7%Fcbt7$F\\[s$! +*ed6P$Fh_pFd]p7%Fgbt7$Ffjr$!+_73RMFh_pFd]p7%F[ct7$F`jr$!+'='o0NFh_pFd ]p7%F_ct7$Fjir$!+a7*4d$Fh_pFd]p7%Fcct7$Fdir$!+,T-NOFh_pFd]p7%Fgct7$F^i r$!+^4#yp$Fh_pFd]p7%F[dt7$Fhhr$!+*GE%fPFh_pFd]p7%F_dt7$Fbhr$!+1C*)>QFh _pFd]p7%Fcdt7$F\\hr$!+!yy#zQFh_pFd]p7%Fgdt7$Ffgr$!+%>^w$RFh_pFd]p7%F[e t7$F`gr$!+403&*RFh_pFd]p7%F_et7$Fjfr$!+Z4k^SFh_pFd]p7%Fcet7$Fdfr$!+IxS 2TFh_pFd]p7%Fget7$F^fr$!+dRXiTFh_pFd]p7%F[ft7$Fher$!+%eYo@%Fh_pFd]p7%F _ft7$Fber$!+Q:kqUFh_pFd]p7%Fcft7$F\\er$!+=\"yQK%Fh_pFd]p7%Fgft7$Ffdr$! +.JdwVFh_pFd]p7%F[gt7$F`dr$!+V_rGWFh_pFd]p7%F_gt7$Fjcr$!+V5E![%Fh_pFd] p7%Fcgt7$Fdcr$!+FG8JXFh_pFd]p7%Fggt7$F^cr$!+s)>7e%Fh_pFd]p7%F[ht7$Fhbr $!+9HQIYFh_pFd]p7%F_ht7$Fbbr$!+Y;YyYFh_pFd]p7%Fcht7$F\\br$!+yTGDZFh_pF d]p7%Fght7$Ffar$!+%*pnqZFh_pFd]p7%F[it7$F`ar$!+#*RZ9[Fh_pFd]p7%F_it7$F j`r$!+7N_c[Fh_pFd]p7%Fcit7$Fd`r$!+iGp'*[Fh_pFd]p7%Fgit7$F^`r$!+#Qq[$\\ Fh_pFd]p7%F[jt7$Fh_r$!+ja'4(\\Fh_pFd]p7%F_jt7$Fb_r$!+Br!\\+&Fh_pFd]p7% Fcjt7$F\\_r$!+d&Fh_pFd]p7%Fc\\u7$F\\\\r$!+8Ya3_Fh_pFd]p7%Fg\\u7$Ff[r$!+*) \\#)>_Fh_pFd]p7%F[]u7$F`[r$!+'*4$*G_Fh_pFd]p7%F_]u7$Fjjq$!+ym$fB&Fh_pF d]p7%Fc]u7$Fdjq$!+.L&4C&Fh_pFd]p7%Fg]u7$F^jq$!+g.:W_Fh_pFd]p7%F[^u7$Fh iq$!+&y!zX_Fh_pFd]p7%F_^u7$Fbiq$!+\">#HY_Fh_pFd]p7%Fc^u7$F\\iq$!+-!QjC &Fh_pFd]p7%Fg^u7$Ffhq$!+a()3Z_Fh_pFd]p7%F[_u7$F`hq$!+C#*f]_Fh_pFd]p7%F __u7$Fjgq$!+7$G1E&Fh_pFd]p7%Fc_u7$Fdgq$!+Ad\"QG&Fh_pFd]p7%Fg_u7$F^gq$! +UobG`Fh_pFd]p7%F[`u7$Fhfq$!+(\\*=%R&Fh_pFd]p7%F_`u7$Fbfq$!+?*Q`Y&Fh_p Fd]p7%Fc`u7$F\\fq$!+3))**HbFh_pFd]p7%Fg`u7$Ffeq$!+S\"[_e&Fh_pFd]p7%F[a u7$F`eq$!+.7nJcFh_pFd]p7%F_au7$Fjdq$!+o9YqcFh_pFd]p7%Fcau7$Fddq$!+*3&p -dFh_pFd]p7%Fgau7$F^dq$!+$GC#HdFh_pFd]p7%F[bu7$Fhcq$!+&o/2v&Fh_pFd]p7% F_bu7$Fbcq$!+m*Rww&Fh_pFd]p7%Fcbu7$F\\cq$!+q&>/y&Fh_pFd]p7%Fgbu7$$!+#H 1\\\"oF]gp$!+IpM*y&Fh_pFd]p7%F[cu7$F`bq$!+*))fYz&Fh_pFd]p7%Facu7$Fjaq$ !+Paa'z&Fh_pFd]p7%Fecu7$Fdaq$!+9/:&z&Fh_pFd]p7%Ficu7$F^aq$!+l$*e!z&Fh_ pFd]p7%F]du7$Fh`q$!+$Q]Hy&Fh_pFd]p7%Fadu7$Fb`q$!+2&*HsdFh_pFd]p7%Fedu7 $F\\`q$!+ySoedFh_pFd]p7%Fidu7$Ff_q$!+A`8UdFh_pFd]p7%F]eu7$F`_q$!+s.nAd Fh_pFd]p7%Faeu7$Fj^q$!+QQH+dFh_pFd]p7%Feeu7$Fd^q$!+K!**\\n&Fh_pFd]p7%F ieu7$F^^q$!+])onk&Fh_pFd]p7%F]fu7$Fh]q$!+-jd:cFh_pFd]p7%Fafu7$Fb]q$!+! f%Q\"e&Fh_pFd]p7%Fefu7$F\\]q$!+'\\YTa&Fh_pFd]p7%Fifu7$Ff\\q$!+>F!Q]&Fh _pFd]p7%F]gu7$F`\\q$!+v%y-Y&Fh_pFd]p7%Fagu7$Fj[q$!+NsZ8aFh_pFd]p7%Fegu 7$Fd[q$!+^'pKO&Fh_pFd]p7%Figu7$F^[q$!+_UZ4`Fh_pFd]p7%F]hu7$Fhjp$!+`O#= D&Fh_pFd]p7%Fahu7$Fbjp$!+TV!**=&Fh_pFd]p7%Fehu7$F\\jp$!+%)f/B^Fh_pFd]p 7%Fihu7$Ffip$!+v*4,0&Fh_pFd]p7%F]iu7$F`ip$!+o:2p\\Fh_pFd]p7%Faiu7$Fjhp $!+-*3i([Fh_pFd]p7%Feiu7$Fdhp$!+s^#\\w%Fh_pFd]p7%Fiiu7$F^hp$!+yH(oi%Fh _pFd]p7%F]ju7$Fhgp$!+svwiWFh_pFd]p7%Faju7$Fbgp$!+;I/)G%Fh_pFd]p7%Feju7 $F[gp$!+vMx9TFh_pFd]p7%Fiju7$Fefp$!+n(ff%RFh_pFd]p7%F][v7$F^fp$!+WJ6\" y$Fh_pFd]p7%Fa[v7$Fhep$!+#ya\">OFh_pFd]p7%Fe[v7$Fbep$!+IT9fMFh_pFd]p7% Fi[v7$F\\ep$!+\"*\\Q+LFh_pFd]p7%F]\\v7$$!+#RKN6#Fgdp$!+qZRUJFh_pFd]p7% Fa\\v7$F_dp$!+JQ&[)HFh_pFd]p7%Fg\\v7$$!+DXlfiFc`p$!+))*fv#GFh_pFd]p7%F []v7$Fccp$!+$3#RqEFh_pFd]p7%Fa]v7$F]cp$!+)>#G8DFh_pFd]p7%Fe]v7$Fgbp$!+ (Q&>cBFh_pFd]p7%Fi]v7$Fabp$!+^c6*>#Fh_pFd]p7%F]^v7$F[bp$!+#)o.U?Fh_pFd ]p7%Fa^v7$Fdap$!+%[d\\)=Fh_pFd]p7%Fe^v7$F^ap$!+2v(ys\"Fh_pFd]p7%Fi^v7$ $\"+cY*e^#Fc`p$!+Otzq:Fh_pFd]p7%F]_v7$Fa`p$!+Ssr89Fh_pFd]p7%Fc_v7$F[`p $!+2tjc7Fh_pFd]p7%Fg_v7$Fc_p$!+?vb*4\"Fh_pFd]p7%F[`v7$$\"+n(\\gS#FL$!+ J#yZU*Fc]pFd]p7%F_`v7$$\"+mK\"=<%F-$!+%p\")R&yFc]pFd]p7%Fe`v7$$\"+))ek SZFhs$!+;`=$G'Fc]pFd]p7%F[av7$F($!+\")*)Q7ZFc]pFd]p7%Faav7$F($!+aEfTJF c]pFd]p7%Feav7$F($!+Fjzq:Fc]pFd]p7%FiavF'Fd]p-Fc\\p6&Fe\\p$\"#vFh\\p$ \"\"%F[]pFf\\p-%&STYLEG6#%,PATCHNOGRIDG-F$6%7$7$$!3Q++++++!p'FLF(7$$\" 3%**************=\"FLF(-%'COLOURG6&Fe\\pF)F)F)-%*LINESTYLEG6#\"\"$-F$6 %7$7$F($!3o*************Q'FL7$F($\"3o*************Q'FLF`cvFccv-%(SCALI NGG6#%,CONSTRAINEDG-%+AXESLABELSG6%%&Re(z)G%&Im(z)G-%%FONTG6#%(DEFAULT G-Fjdv6$%*HELVETICAG\"\"*-%*AXESSTYLEG6#%$BOXG-%%VIEWG6$;$!$p'Fh\\p$\" $>\"Fh\\p;$!$R'Fh\\p$\"$R'Fh\\p" 1 2 0 1 10 0 2 9 1 2 1 1.000000 45.000000 45.000000 0 0 "Curve 1" "Curve 2" "Curve 3" "Curve 4" }}}} {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 4 "The " } {TEXT 260 30 "interval of absolute stability" }{TEXT -1 89 " (or stab ility interval) is the intersection of the stability region with the r eal line." }}{PARA 0 "" 0 "" {TEXT -1 59 "For this scheme the stabilit y interval is (approximately) " }{XPPEDIT 18 0 "[-5.9159, 0];" "6#7$, $-%&FloatG6$\"&f\"f!\"%!\"\"\"\"!" }{TEXT -1 2 ". " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 116 "We can distort the bound ary curve horizontally by taking the 11th root of the real part of poi nts along the curve. " }}{PARA 0 "" 0 "" {TEXT -1 150 "In this way we see that the intersection of the stability region with the non-negati ve imaginary axis consists of the union of two disjoint intervals." }} {PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 664 "R := z -> add(z^j/j!,j=0..8)+\n 198628807309228820553340837 85841061267/7396374196038864374781924114577598054400000*z^9+\n 2 2756280232231510967698476352344264591549/87576029030778569743167894286 244592003317760000*z^10+\n 1428887843954929666025004498115036383 07/10425717741759353540853320748362451428966400000*z^11+\n 48731 513235646967497173728701/306827817192044908284761367618060288000*z^12: \nDigits := 18:\npts := []: z0 := 0:\nfor ct from 0 to 200 do\n zz : = newton(R(z)=exp(ct*Pi/60*I),z=z0):\n z0 := zz:\n pts := [op(pts) ,[surd(Re(zz),11),Im(zz)]]:\nend do:\nplot(pts,color=COLOR(RGB,.75,.2, .18),thickness=2,font=[HELVETICA,9]);\nDigits := 10:" }}{PARA 13 "" 1 "" {GLPLOT2D 287 308 308 {PLOTDATA 2 "6(-%'CURVESG6#7ew7$$\"\"!F)F(7$F ($\"3v))H)fv()fB&!#>7$$\"3SNU=Y%H*4GF-$\"3!))f'>^v>Z5!#=7$$\"3z0KYnl7h SF-$\"33P\\zEjzq:F37$$\"3wry>x@\"GF&F-$\"3xHPR-^R%4#F37$$\"3(zblm#p6bk F-$\"3'=V&*z(Q*zh#F37$$\"35$4/?/!p8wF-$\"3\")R)4Ol#fTJF37$$\"3=k$eIa_? v)F-$\"3D(4n#H9>lOF37$$\"3ROaHwqas)*F-$\"3=dV/0-z)=%F37$$\"3hPo5<&zw4 \"F3$\"3%[x86)*)Q7ZF37$$\"3T#z`hU%f17F3$\"3v&o?yv()fB&F37$$\"3Ow#\\1D \"398F3$\"3*)y5!e`'efdF37$$\"3%[%*)z,z>?9F3$\"3PMk9;`=$G'F37$$\"38H+Mv U)\\_\"F3$\"3>N8i+Ty1oF37$$\"3Z,'pdBm%G;F3$\"3BQZ$>*GQItF37$$\"3`h`O'Q e1t\"F3$\"3&>FfSp\")R&yF37$$\"3&*oIHLgcJ=F3$\"3YOWe70ex$)F37$$\"3Tjx<& f'=J>F3$\"32=$z]Nz6!*)F37$$\"3pj,5=1^H?F3$\"3n4)G9ByZU*F37$$\"3(R\"3Z+ D_E@F3$\"3N/]4arP[**F37$$\"3qZ;C(*4?AAF3$\"3pJV#Qh(>Z5!#<7$$\"3p`:0]&> lJ#F3$\"3t:\"Fhq7$$\"3qBO/C ]%4]#F3$\"3OQ=![OxU?\"Fhq7$$\"37(RIXMt4f#F3$\"3ZPRW2tjc7Fhq7$$\"3_l(*o @V[zEF3$\"3!4iAkE(**38Fhq7$$\"3+IX.D`UmFF3$\"3eL\\VVsNh8Fhq7$$\"3'>Df7 %yt^GF3$\"3tow`Rsr89Fhq7$$\"3bPjn()pNNHF3$\"3!yu:[DxgY\"Fhq7$$\"31@`t* )3@7!)GP%=:Fhq7$$\"3EP;'***)>s4$F3$\"3X[>9Otzq:Fhq7$$\"3d+ &eri&HvJF3$\"3L!*R7%RdJi\"Fhq7$$\"3,t`(fK=Fhq7$$\"30raAD/zKNF3$\"3lb#\\U[d\\)=Fhq7$ $\"3#=dt81Dnf$F3$\"3dv&y[OFhq7$$\"3QE4_uGrdOF3$\"3m$)eHmrn*)>Fhq7$ $\"37L7N^M^:PF3$\"3%yRL<)o.U?Fhq7$$\"3E7CT4U%)pPF3$\"3W`gT:lR%4#Fhq7$$ \"3ok9+O(o.#QF3$\"3$oj1\"*3cn9#Fhq7$$\"3584;T0omQF3$\"3zH!32l:\"*>#Fhq 7$$\"308ZWI)z#3RF3$\"3g+YK&Gv9D#Fhq7$$\"3nNMN&GRX%RF3$\"37[YAH^$QI#Fhq 7$$\"3;)Rr(G=luRF3$\"3rdN=(Q&>cBFhq7$$\"3S0mfGua(*RF3$\"3f^=s`jb3CFhq7 $$\"3YX#fM_`<,%F3$\"3oHDyQ%=4Y#Fhq7$$\"3bpuEN^9:SF3$\"3JHqT)>#G8DFhq7$ $\"3uQ)*>S#eW+%F3$\"3(f,o5P[cc#Fhq7$$\"3!o%39F!GU(RF3$\"35*e\\+#z,=EFh q7$$\"3GW>Z.!fS\"RF3$\"3^'GJG3#RqEFhq7$$\"35=c*y\"[Z*z$F3$\"3-C>tFCxAF Fhq7$$\"3N&*o&4%ftQNF3$\"3UtPi=4;vFFhq7$$!350B-$)*)zkLF3$\"3jT$Q#))*fv #GFhq7$$!3qi0z_'fq&QF3$\"3#z&Rl?E(*zGFhq7$$!3Wv0&pS#p*4%F3$\"3=`X`VCSK HFhq7$$!3T!p'H#*=duUF3$\"3&yZF3$\"3!4%Q%zG&*\\>$Fhq7$$!3BV-ak'\\7z%F3$\"3-E3]6 [lZKFhq7$$!3M]\\5cqR[[F3$\"3S_pe!*\\Q+LFhq7$$!3P28a\"eO)))[F3$\"3%Q!4$ =F*>`LFhq7$$!3g$z+sb*\\3\\F3$\"3+Zg))*48hS$Fhq7$$!3Pz_^qGc**[F3$\"3WWZ ,IT9fMFhq7$$!3'GVZ11eb%[F3$\"3Tr\\s[CJ7NFhq7$$!37:=F>)yEq%F3$\"3>o-Z'z Scc$Fhq7$$!3!GFe@&RMUUF3$\"3fV\"\\OFhq7$$\"3\\'e*p&HSyd%F3$\"3O* Rv#oI)Gn$Fhq7$$\"3sI!p%ocLx]F3$\"3@.?T/v&os$Fhq7$$\"3to%fig)3+aF3$\"3! RPKQ986y$Fhq7$$\"3_%*ewKT!>m&F3$\"3%=r!ot!)oNQFhq7$$\"3pk)yq&=\"G*eF3$ \"3[O)RQ\"Hi!*QFhq7$$\"3G+f)Q3Y`5'F3$\"3Lx>am(ff%RFhq7$$\"3q&>*4\"o?gI 'F3$\"3M]]!=ES<+%Fhq7$$\"3r^g.bVp)\\'F3$\"30M\"zk8-!eSFhq7$$\"3>3oXu'[ eo'F3$\"3n)*='[ZtZ6%Fhq7$$\"3OH--wL9poF3$\"3VHZq7L1sTFhq7$$\"3h+v(p\"= m\\qF3$\"3-j7!4#o%)HUFhq7$$\"3T**zICM*zA(F3$\"3/q%ph,V!)G%Fhq7$$\"3hCf '='>@/uF3$\"3USNc#p$[YVFhq7$$\"3)\\_gG4&yxvF3$\"3%GhG5uv[S%Fhq7$$\"3H9 x>QsYZxF3$\"3a[8[svwiWFhq7$$\"3,s(ephu7\"zF3$\"3TJhK8Qa>XFhq7$$\"3u&4% 3/#ym1)F3$\"3eVI+!\\![uXFhq7$$\"3\"\\mS'oc36#)F3$\"3v01PyH(oi%Fhq7$$\" 3mCz)\\[tCM)F3$\"3&R!pN,U>wYFhq7$$\"3>v!*o,w%)f%)F3$\"3B\"3stK*>AZFhq7 $$\"3)=WF:Z/Lc)F3$\"3S`=Js^#\\w%Fhq7$$\"3yx&*)*f`v`')F3$\"3m,3>b%3Y![F hq7$$\"3FtAG7UaK()F3$\"3k%HC([WeT[Fhq7$$\"3,D?(z!f8,))F3$\"3L(ehv]Fhq7$$\"3F-t Fv$eo6*F3$\"3%)p3b\"Q&[*4&Fhq7$$\"3MhD%=T**z8*F3$\"3;$fsV)f/B^Fhq7$$\" 3Y![)yjV^c\"*F3$\"3]E\"*Gs\"Qf9&Fhq7$$\"3<\"ys,\\eE<*F3$\"3(*[3Un2@o^F hq7$$\"3CV!3oq[m=*F3$\"3n2&*zSV!**=&Fhq7$$\"3NE47U-n)>*F3$\"3#3!\\H#y_ 5@&Fhq7$$\"3wv34#)>))3#*F3$\"3+CTrZXoJ_Fhq7$$\"3h]r'fx?u@*F3$\"3Eb4G`O #=D&Fhq7$$\"3\">5!R=\\SC#*F3$\"3KA*4AZ!\\r_Fhq7$$\"3xA@^xp$*H#*F3$\"3' 30sdP-2H&Fhq7$$\"30#*[*>51TB*F3$\"3E!>c;Du%4`Fhq7$$\"3$R0:4\"**)pB*F3$ \"38=!Q]#*=yK&Fhq7$$\"3V`n7[glQ#*F3$\"3I:25-vuX`Fhq7$$\"35S!e^Yj\"R#*F 3$\"3rI,T^'pKO&Fhq7$$\"3z@G9$[j&Q#*F3$\"3R@&=_$QR!Q&Fhq7$$\"3bM7st1!pB *F3$\"3O#e6-]FrR&Fhq7$$\"3\\:Ze!f8UB*F3$\"3l0#Q_BxMT&Fhq7$$\"3On>'oPN0 B*F3$\"3l]5.-*[%HaFhq7$$\"35BsUoU*eA*F3$\"3SbGtVx/XaFhq7$$\"3,6x94SJ?# *F3$\"3g*z3^Zy-Y&Fhq7$$\"3Gl7$3>9Q@*F3$\"3dN!*ed`9vaFhq7$$\"3!HX,!G0T1 #*F3$\"3g#G-*F3$\"3ffm**=F!Q]&Fhq7$$\"3\">w -\\RO*)=*F3$\"3%QZr^'**fG*) ))F3$\"3:G\"\\\"QQH+dFhq7$$\"3F,O[,,_h))F3$\"3f&RULQw!3dFhq7$$\"33(zuM idA$))F3$\"3H)4\"*R9Nbr&Fhq7$$\"3p)GLjg)Q,))F3$\"3/uS`r.nAdFhq7$$\"3xw vjL7zo()F3$\"3#**4rD=#[HdFhq7$$\"38%\\E#3[KM()F3$\"3r*RbZbqft&Fhq7$$\" 3o)>^.,Eyp)F3$\"3w+5DA`8UdFhq7$$\"3bJE\"*RT5f')F3$\"3>p![2dZht&)3 odFhq7$$\"3c%*GaP2gA%)F3$\"3QO%=t]*HsdFhq7$$\"3j4uVu:#RO)F3$\"3;T7;y6= wdFhq7$$\"3o0p')y)R+I)F3$\"3a(o340K(zdFhq7$$\"3?79&Gv4+B)F3$\"3G7\\x#Q ]Hy&Fhq7$$\"3C$)*zfR&e_\")F3$\"3'p&*=H@Mey&Fhq7$$\"3P(GmXTyg1)F3$\"3q& p75N\"Q)y&Fhq7$$\"3#)R@00y6ozF3$\"396Hyk$*e!z&Fhq7$$\"3#=,'3*f>_&yF3$ \"3J$ys(F3$\"3!\\N9x*p(Rz&Fhq7$$\"3BN=_DT4fvF3 $\"3&z7GQT]^z&Fhq7$$\"3Q#40R/+!\\tF3$\"3'3boaEsfz&Fhq7$$\"3vl[&z]*4]qF 3$\"3peWOz'Qkz&Fhq7$$\"3L&H0z8Q+^'F3$\"3(==,oVXlz&Fhq7$$!3afWnUq44gF3$ \"3=Vc+]zG'z&Fhq7$$!3<`,Y6kA\")oF3$\"3+\")o7O7m&z&Fhq7$$!3I'y91.6>D(F3 $\"3L\"eE&)))fYz&Fhq7$$!3*HXW*>@Ly'y&Fhq7$$!3X-m\\!4H22)F3$\"3``S7^/\"Qy&Fhq7$$!3n.D[6 nVo\")F3$\"3V4mMq&>/y&Fhq7$$!38p?h)pHjD)F3$\"3`kC<'>,mx&Fhq7$$!3:U6$\\ M[jL)F3$\"3YF3WY]MsdFhq7$$!3+OgpB$3*4%)F3$\"32Pfpl*Rww&Fhq7$$!3m3#Q`Qw !y%)F3$\"3#eA*yUQZidFhq7$$!3Iwv&o:y;a)F3$\"314I!*=N$ov&Fhq7$$!3(*>YYut O,')F3$\"3OQ!>\\o/2v&Fhq7$$!39YxJy?nd')F3$\"3q1v,n<2WdFhq7$$!3HkUZhk-6 ()F3$\"3/:=T*zhbvc))F3$\"3klS-'*f87dFhq7$$!3aY:)z&*[9!* )F3$\"3_r0#*)3&p-dFhq7$$!3f&f+1WoX%*)F3$\"3+?6,#)>i#p&Fhq7$$!3-.gRE8I' )*)F3$\"3YFL')3z)=o&Fhq7$$!3Zjvu#H?o-*F3$\"3f#3V$o9YqcFhq7$$!3R12vTxGm !*F3$\"3VN(*e^%3$ecFhq7$$!33cG[l(f[5*F3$\"3'[PyDs\"RXcFhq7$$!3/#[$z>** oU\"*F3$\"3DXEK.7nJcFhq7$$!3fZ%yJ)H$*z\"*F3$\"3]\"Rxv//rh&Fhq7$$!35%G \">])[n@*F3$\"3J9umR^k,cFhq7$$!3>El.smI`#*F3$\"3kt>`R\"[_e&Fhq7$$!32NW p'G*y*G*F3$\"3_3va`(Q\"4S*F3$\"3]m\"[3%pY4bFhq7$$ !3zPpD>jMR%*F3$\"3W_@6Iw)y[&Fhq-%%FONTG6$%*HELVETICAG\"\"*-%*THICKNESS G6#\"\"#-%&COLORG6&%$RGBG$\"#v!\"#$Fejn!\"\"$\"#=F\\[o-%+AXESLABELSG6$ Q!6\"Fd[o-%%VIEWG6$%(DEFAULTGFi[o" 1 2 0 1 10 2 2 6 1 4 2 1.000000 45.000000 45.000000 0 0 "Curve 1" }}}}{PARA 0 "" 0 "" {TEXT -1 0 "" }} {PARA 0 "" 0 "" {TEXT -1 108 "The relevant intersection points of the \+ boundary curve with the imaginary axis can be determined as follows." }}{PARA 0 "" 0 "" {TEXT -1 87 "First we look for points on the boundar y curve either side of each intersection point. " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 289 "Digits := 1 5:\nz0 := 2.8*I:\nfor ct from 88 to 91 do\n newton(R(z)=exp(ct*Pi/10 0*I),z=z0);\nend do;``;\nz0 := 3.6*I:\nfor ct from 114 to 117 do\n n ewton(R(z)=exp(ct*Pi/100*I),z=z0);\nend do;``;\nz0 := 5.8*I:\nfor ct f rom 275 to 278 do\n newton(R(z)=exp(ct*Pi/100*I),z=z0);\nend do;\nDi gits := 10:" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#^$$\"0zP+;'\\!Q\"!#>$\" 0o$QfCokF!#9" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#^$$\"0b&e]t?qX!#?$\"0q 4K7>hz#!#9" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#^$$!09(zDXlfi!#?$\"0MQ#) )*fv#G!#9" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#^$$!02@e>D!y=!#>$\"0aUNs0 !fG!#9" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%!G" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#^$$!0M!#=$\"0M(4zD-(e$!#9" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#^$$!0E`X_#f5!)!#>$\"09\\O!#9" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#^$$\"0VunIaeb'!#>$\"0ki92k8l$!#9" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#^$$\"0!\\#o$HzAD!#=$\"0I'f(=dOo$!#9" }}{PARA 11 "" 1 " " {XPPMATH 20 "6#%!G" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#^$$\"04s4L\"f+ *)!#<$\"0=,oVXlz&!#9" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#^$$\"0c&fn<i !#<$\"0#[67B>'z&!#9" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#^$$!0feB?+MQ\"! #;$\"00R-P;ez&!#9" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 107 "Then we apply the bisection method to calculate the para meter value associated with the intersection point." }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 480 "Digits := 15:\nreal_part := proc(u)\n Re (newton(R(z)=exp(u*Pi*I),z=2.8*I))\nend proc:\nu0 := bisect('real_part '(u),u=0.88..0.91);\nnewton(R(z)=exp(u0*Pi*I),z=2.8*I);``;\nreal_part \+ := proc(u)\n Re(newton(R(z)=exp(u*Pi*I),z=3.6*I))\nend proc:\nu0 := \+ bisect('real_part'(u),u=1.14..1.17);\nnewton(R(z)=exp(u0*Pi*I),z=3.6*I );``;\nreal_part := proc(u)\n Re(newton(R(z)=exp(u*Pi*I),z=5.8*I))\n end proc:\nu0 := bisect('real_part'(u),u=2.75..2.78);\nnewton(R(z)=exp (u0*Pi*I),z=5.8*I);\nDigits := 10:" }}{PARA 11 "" 1 "" {XPPMATH 20 "6# >%#u0G$\"0s8F9qS%*)!#:" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#^$$\"0)z#3SO TR$!#H$\"0in%#u0G$\"0&pH.a\"e:\"!#9" }}{PARA 11 "" 1 " " {XPPMATH 20 "6#^$$!0eGrt8$R#)!#H$\"0')*H!)f(yj$!#9" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%!G" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%#u0G$\"0\"* QN3(zhF!#9" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#^$$\"0;&=hHF3\"*!#H$\"0g x68,kz&!#9" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 4 "The " }{TEXT 260 112 "largest interval on the nonnegative imagin ary axis that contains the origin and lies inside the stability region " }{TEXT -1 5 " is " }{XPPEDIT 18 0 "[0, 2.80997];" "6#7$\"\"!-%&Floa tG6$\"'(*4G!\"&" }{TEXT -1 18 " (approximately)." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 4 "The " }{TEXT 260 72 "inter section of the stability region with the nonnegative imaginary axis" } {TEXT -1 5 " is " }{XPPEDIT 18 0 "`union`([0, 2.80997],[3.6379, 5.796 4]);" "6#-%&unionG6$7$\"\"!-%&FloatG6$\"'(*4G!\"&7$-F)6$\"&zj$!\"%-F)6 $\"&kz&F1" }{TEXT -1 17 " (approximately)." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 37 "#------------------------------ ------" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 81 "The stability function R* for the 13 stage, order 7 scheme is given as follows." }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 98 "subs(ee,subs (b=`b*`,StabilityFunction(7,13,'expanded'))):\n`R*` := unapply(%,z):\n '`R*`(z)'=`R*`(z);" }}{PARA 12 "" 1 "" {XPPMATH 20 "6#/-%#R*G6#%\"zG,: \"\"\"F)F'F)*&#F)\"\"#F)*$)F'F,F)F)F)*&#F)\"\"'F)*$)F'\"\"$F)F)F)*&#F) \"#CF)*$)F'\"\"%F)F)F)*&#F)\"$?\"F)*$)F'\"\"&F)F)F)*&#F)\"$?(F)*$)F'F1 F)F)F)*&#F)\"%S]F)*$)F'\"\"(F)F)F)*&#\"F29$e(3j#\\/uD>(Ri9B%)e$\"K++!3 E,>Osyk*f3n96;@&3W\"F)*$)F'\"\")F)F)F)*&#\"H(*)*\\_yOEP:\"z4UrJgb**QC \"M++SeR&pTswU*eU " 0 "" {MPLTEXT 1 0 30 "z_0 := newton(`R*`(z)=1, z=-5);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%$z_0G$!+nQ2Ed!\"*" }}} {PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 318 "z_0 := newton(`R*`(z)=1,z=-5):\np_1 := plot([`R*`(z),1],z=-6.39.. 0.49,color=[red,blue]):\np_2 := plot([[[z_0,1]]$3],style=point,symbol= [circle,cross,diamond],color=black):\np_3 := plot([[z_0,0],[z_0,1]],li nestyle=3,color=COLOR(RGB,0,.5,0)):\nplots[display]([p_1,p_2,p_3],view =[-6.39..0.49,-0.07..1.47],font=[HELVETICA,9]);" }}{PARA 13 "" 1 "" {GLPLOT2D 381 277 277 {PLOTDATA 2 "6+-%'CURVESG6$7Z7$$!3o************* Q'!#<$\"3IQ'R@NOHl$F*7$$!3-LL$[*)3DN'F*$\"3HyjKCVs5MF*7$$!3Qmmm*y<]J'F *$\"3'o+UvptH=$F*7$$!3u****\\%oEvF'F*$\"3e!4/t8<*oHF*7$$!33LLLzb.SiF*$ \"3oU#4P*H#yw#F*7$$!3g***\\!fRzuhF*$\"3'zb\")HdJnW#F*7$$!3:mmwQBb4hF*$ \"3'HOtz\\d$f@F*7$$!3I***\\LJ\"=OgF*$\"3%yF*)>*eTdF*$\"3$=)R&Qk>H.\"F*7$$!3_mmcXG3ocF*$\"3 Yk$Q-_J6&))!#=7$$!3zKL.jKyJbF*$\"3\\*=K_V!y-mFco7$$!3%)****4SHl!R&F*$ \"3%fB0m#)4)G[Fco7$$!37LLBedpW_F*$\"3M(Ra**f(>eMFco7$$!3Y*****Gm1#*4&F *$\"3!fX7pzPWX#Fco7$$!3UmmYXKb\\\\F*$\"3/!y8N%eF47$$!35+++1YM?X F*$\"3tY\"3g++C-'Fgq7$$!3c****f$)RvwVF*$\"3qP&*H+qL.WFgq7$$!39LLBU*fjC %F*$\"3[any`.6lMFgq7$$!3Kmm1:)384%F*$\"3;./#p4R!HGFgq7$$!3UmmEs,'*fRF* $\"3sdD%*[o&))e#Fgq7$$!3_****Hv/<2QF*$\"3IDeb6hQfDFgq7$$!3?mmE/d#>n$F* $\"3>Ph*)Hlg,FFgq7$$!3b****Hb2aBNF*$\"3)f%fw]t$z*HFgq7$$!3]*****Q>VAQ$ F*$\"3qM]E$*\\(GR$Fgq7$$!3%fmm;,9[B$F*$\"3+jWVgiQ4RFgq7$$!3%emm***zU*4 $F*$\"3Qgy!>W`ZZ%Fgq7$$!3)GLLj*eR`HF*$\"3W6qy1I0'=&Fgq7$$!3)GLLAe4K&4n/'Fgq7$$!3a*****)*>m'pEF*$\"33iv\\Uc`4pFgq7$$!3oKL`Hd0FD F*$\"3)[NbQm8q(zFgq7$$!3]****>f^szBF*$\"3]np_Jbt\\#*Fgq7$$!3w****RR4fN AF*$\"3G=*Q)z`\")o5Fco7$$!3a****\\qO8'4#F*$\"3E\"o:UGN!H7Fco7$$!3^**** >],HT>F*$\"3pkeW@,0N9Fco7$$!3PmmEgj:-=F*$\"3'QCY4hg$\\;Fco7$$!33*****> M,Ol\"F*$\"3`L@:'HmN\">Fco7$$!3LLL$eT))*=:F*$\"3[:Lu'f?$*=#Fco7$$!3S** **fJ4#=P\"F*$\"3#*y2&*oHWODFco7$$!3Fmm'zzZLB\"F*$\"3rB*>R&e:8HFco7$$!3 C****4uMg)3\"F*$\"3!f6%*[]joO$Fco7$$!3#ymmmS%*3Z*Fco$\"39q`uIIoyQFco7$ $!3e$****\\it$*)zFco$\"3!)Rz=hp1)\\%Fco7$$!3-DLLHI\\ilFco$\"3/tZ<(GNz= &Fco7$$!34CLLJ:I.^Fco$\"3XT***HctH+'Fco7$$!3'Glmc(G>cOFco$\"3dVnOm(ow$ pFco7$$!3(z)***>J]kK#Fco$\"3OG:TK\"[V#zFco7$$!33*emms_T-)Fgq$\"3B6qTbT $*G#*Fco7$$\"3$RrmmM)*pg&Fgq$\"3A4)y1qrw0\"F*7$$\"3a0++Z[19?Fco$\"3!y% z5^=7B7F*7$$\"3o4++4170MFco$\"3$*Q!QN>ncS\"F*7$$\"3!***************[Fc o$\"3HqvP?iJK;F*-%'COLOURG6&%$RGBG$\"*++++\"!\")$\"\"!Fd]lFc]l-F$6$7S7 $F($\"\"\"Fd]l7$F=Fi]l7$FGFi]l7$FQFi]l7$FenFi]l7$F_oFi]l7$FeoFi]l7$Fjo Fi]l7$F_pFi]l7$FdpFi]l7$FipFi]l7$F^qFi]l7$FcqFi]l7$FiqFi]l7$F^rFi]l7$F crFi]l7$FhrFi]l7$F]sFi]l7$FbsFi]l7$FgsFi]l7$F\\tFi]l7$FatFi]l7$FftFi]l 7$F[uFi]l7$F`uFi]l7$FeuFi]l7$FjuFi]l7$F_vFi]l7$FdvFi]l7$FivFi]l7$F^wFi ]l7$FcwFi]l7$FhwFi]l7$F]xFi]l7$FbxFi]l7$FgxFi]l7$F\\yFi]l7$FayFi]l7$Ff yFi]l7$F[zFi]l7$F`zFi]l7$FezFi]l7$FjzFi]l7$F_[lFi]l7$Fd[lFi]l7$Fi[lFi] l7$F^\\lFi]l7$Fc\\lFi]l7$Fh\\lFi]l-F]]l6&F_]lFc]lFc]lF`]l-F$6&7#7$$!3@ +++nQ2EdF*Fi]l-%'SYMBOLG6#%'CIRCLEG-F]]l6&F_]lFd]lFd]lFd]l-%&STYLEG6#% &POINTG-F$6&F_al-Fdal6#%&CROSSGFgalFial-F$6&F_al-Fdal6#%(DIAMONDGFgalF ial-F$6%7$7$FaalFc]lF`al-%&COLORG6&F_]lFc]l$\"\"&!\"\"Fc]l-%*LINESTYLE G6#\"\"$-%+AXESLABELSG6%Q\"z6\"Q!Ficl-%%FONTG6#%(DEFAULTG-F\\dl6$%*HEL VETICAG\"\"*-%%VIEWG6$;$!$R'!\"#$\"#\\Fidl;$!\"(Fidl$\"$Z\"Fidl" 1 2 0 1 10 0 2 9 1 4 2 1.000000 45.000000 45.000000 0 0 "Curve 1" "Curve 2 " "Curve 3" "Curve 4" "Curve 5" "Curve 6" }}}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 83 "The following picture shows the stability region for the 12 stage, order 7 scheme. " }}{PARA 0 "" 0 " " {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 947 "`R*` := z -> add(z^j/j!,j=0..7)+3588423146239719257404492630875831407/144085211 611146708599647872361901260800000*z^8+\n 24389955603171420979115 3726367852499897/86162956543465731742589427672416953958400000*z^9+\n \+ 342077050237920691600610257810140139/1468824702981030366420586186 018942156800000*z^10+\n 154918302898061856320802469891/511379695 32007484714126894603010048000*z^11:\npts := []:\nz0 := 0:\nfor ct from 0 to 320 do\n zz := newton(`R*`(z)=exp(ct*Pi/20*I),z=z0):\n z0 := zz:\n pts := [op(pts),[Re(zz),Im(zz)]]:\nend do:\np_1 := plot(pts,c olor=COLOR(RGB,.2,0,0)):\np_2 := plots[polygonplot]([seq([pts[i-1],pts [i],[-4.4,0]],i=2..nops(pts))],\n style=patchnogrid,color=COL OR(RGB,.65,.3,.25)):\np_3 := plot([[[-6.49,0],[1.19,0]],[[0,-6.39],[0, 6.39]]],color=black,linestyle=3):\nplots[display]([p_||(1..3)],view=[- 6.49..1.19,-6.39..6.39],font=[HELVETICA,9],\n labels=[`Re(z )`,`Im(z)`],axes=boxed,scaling=constrained);" }}{PARA 13 "" 1 "" {GLPLOT2D 381 565 565 {PLOTDATA 2 "6+-%'CURVESG6$7]_l7$$\"\"!F)F(7$F($ \"3++++Fjzq:!#=7$F($\"3)******Rl#fTJF-7$$!3!******47(GcG!#F$\"3D+++\") *)Q7ZF-7$$!39+++eFZ/J!#E$\"3Y+++9`=$G'F-7$$!39+++A6:I?!#D$\"3m*****\\p \")R&yF-7$$!3C+++8nk'e*F@$\"3`+++Q$yZU*F-7$$!3w*****f;1Wf$!#C$\"3'**** ***4wb*4\"!#<7$$!3/+++NobI6!#B$\"3/+++wxjc7FN7$$!3++++JDF'3$FR$\"3%*** ***z6>PT\"FN7$$!35++++*[9Z(FR$\"3%******Hf.3d\"FN7$$!3;+++A`$\\i\"!#A$ \"35+++>c*ys\"FN7$$!3%*******)p2K>$F\\o$\"3)******zN/])=FN7$$!3o****** )[[il&F\\o$\"3(******>YZ@/#FN7$$!3i+++3ue!*))F\\o$\"3=+++$zc$*>#FN7$$! 3%******zEiI=\"!#@$\"37+++^locBFN7$$!3********RJ&z7\"Fap$\"3')*****>)R A9DFN7$$\"3#)*****H5HQC#FR$\"3!******RD-@n#FN7$$\"30+++bv(f!QFap$\"3/+ ++6P^IGFN7$$\"30+++_$egJ\"!#?$\"3?+++k'=(*)HFN7$$\"3<+++[a4ULFfq$\"3%) *****z&p.]JFN7$$\"3%******z(p*4R(Ffq$\"3-+++U!*y6LFN7$$\"30+++C>m-:!#> $\"3')*****f\">5vMFN7$$\"3++++\"zz!oGFfr$\"3=+++$p!RROFN7$$\"3#******z X8i9&Ffr$\"3#)*****fYpA!QFN7$$\"3$******H$elU&)Ffr$\"3/+++f%p$eRFN7$$ \"3#*******)))fKG\"F-$\"3e*****fq>25%FN7$$\"39+++>vUN^)Hy]FN 7$$\"3)******H*>'z#>F-$\"3P+++cH%H6&FN7$$\"3)******\\%4?Z;F-$\"38+++'[ FU9&FN7$$\"3)******fV'oY8F-$\"3c*****Rt>A<&FN7$$\"3&*******R2XG5F-$\"3 E+++EY(p>&FN7$$\"3P+++2(HE%pFfr$\"3%*******)GM&=_FN7$$\"3G+++!fflX$Ffr $\"3G+++(eFpB&FN7$$!3!*******)*z?0;Ffq$\"3q******y$p@D&FN7$$!3E+++FYT( *QFfr$\"3'******H\\fUE&FN7$$!3Y*****R>6[u(Ffr$\"3C+++)Q\"=t_FN7$$!3'** ****fzD&p6F-$\"3-+++Ku:F-$\"3A+++)eh8G&FN7$$!3)**** **foC&))>F-$\"3W+++%o'[!G&FN7$$!3++++v-.7CF-$\"37+++a0n4)G$F-$\"3u******['HmD&FN7$ $!30+++s>KUPF-$\"3G+++.4,T_FN7$$!3s*****H*H`4UF-$\"3=+++]r9@_FN7$$!3w* ****p#=k#p%F-$\"35+++cNq'>&FN7$$!3++++yQ@'>&F-$\"3u*****\\Ozs;&FN7$$!3 ?+++'f,us&F-$\"3J+++9,UK^FN7$$!3%*******pPX(H'F-$\"3q*****R#)y;4&FN7$$ !3G+++K*[R#pF-$\"3<+++g0\"[/&FN7$$!3Z*****fmhIj(F-$\"3C+++.)*G#*\\FN7$ $!3/+++'ycvX)F-$\"3.+++zuRO\\FN7$$!3X+++L%*F-$\"3!)*****HX+C)[FN7$$ !3%******p]B(\\5FN$\"3f*****4)*or$[FN7$$!3/+++#3cD;\"FN$\"3T+++j)[V![F N7$$!35+++5L?u7FN$\"3i*****zekDy%FN7$$!3-+++^M[\"Q\"FN$\"3=+++F&e$oZFN 7$$!3++++r=_$[\"FN$\"3#******p<>'eZFN7$$!3/+++*Q'[!e\"FN$\"3i*******)p 9^ZFN7$$!35+++7&))Gn\"FN$\"3n*****fg#\\WZFN7$$!3'******HZ(HhFN$\"3&)*** **HA*[@ZFN7$$!3y*****pK.v+#FN$\"3V+++29N6ZFN7$$!3!*******Hla%3#FN$\"3o *****46v&*p%FN7$$!3A+++%G\"ff@FN$\"3E+++2i-'o%FN7$$!3++++>+)GB#FN$\"3) )*****\\R61n%FN7$$!3!******f$Hi/BFN$\"3S+++^hE`YFN7$$!37+++dZ+vBFN$\"3 z*****f$o%Rj%FN7$$!3-+++\"G!>WCFN$\"39+++(eDEh%FN7$$!35+++B*GB^#FN$\"3 #)******>rG*e%FN7$$!3#)*****zOe&zDFN$\"3o*****R!p#Rc%FN7$$!31+++du+YEF N$\"3R+++e.bOXFN7$$!3#******zL)z6FFN$\"3x*****p#G<2XFN7$$!3#******4\"z /xFFN$\"3s*****>&>N2WFN7$$!3)******ppZ1(HFN$\"3g*****48^.P%FN7$$!3') *****zd-[.$FN$\"3q*****>A2;L%FN7$$!3;+++Sj\"*)4$FN$\"3k*****H;;7H%FN7$ $!3?+++*))eI;$FN$\"3S+++!o*G\\UFN7$$!3<+++B@GFKFN$\"3%)*****4y`f?%FN7$ $!3-+++vfh\"H$FN$\"3M+++DdMhTFN7$$!3'******>7lgN$FN$\"3)*******Q%4c6%F N7$$!3%)*****p/21U$FN$\"3o*****Hg*))oSFN7$$!3'******R+#>&[$FN$\"3i**** *>&\\K@SFN7$$!3?+++n^u\\NFN$\"3?+++6;/tRFN7$$!3))*****f&3<9OFN$\"3')** ****3t9CRFN7$$!3-+++T$e$yOFN$\"31+++UssuQFN7$$!3')******e%)=UPFN$\"32+ ++3?%[#QFN7$$!3?+++5-a0QFN$\"3y*****zlFXx$FN7$$!3')*******f'HoQFN$\"33 +++$e(zBPFN7$$!39+++p#\\.$RFN$\"37+++#pXEn$FN7$$!31+++20%FN$\"37+++c$y*oNFN7$$!3++++A!*Q6TFN$\"34+++ j!*Q;NFN7$$!3K+++B7!)pTFN$\"3?+++%fPKY$FN7$$!3u******Go;FUFN$\"34+++Sy Z4MFN7$$!3))*****f:dMG%FN$\"3=+++PZ1bLFN7$$!3*******RNa'QVFN$\"35+++^g &**H$FN7$$!3w*****>)*\\FR%FN$\"3()*****>r8TC$FN7$$!35+++@OuXWFN$\"35++ +*y/v=$FN7$$!39+++T:k(\\%FN$\"3<+++'>-,8$FN7$$!3%******HWb%[XFN$\"3!)* ****\\<&)=2$FN7$$!3%)*****>F,#)f%FN$\"3#)*****fjRG,$FN7$$!3s******z!)* ok%FN$\"31+++e$eH&HFN7$$!3g******Gpc%p%FN$\"3'******>3TA*GFN7$$!3%**** **z!)H7u%FN$\"3<+++)\\%pIGFN7$$!3e******p&3py%FN$\"3)******p5K$oFFN7$$ !3E+++yQiJ[FN$\"3*******>&R<0FFN7$$!3'******>=%Rv[FN$\"3;+++)>Y7k#FN7$ $!3l*****zvM#=\\FN$\"3=+++N0ewDFN7$$!3E+++No:g\\FN$\"3\")******>M@6DFN 7$$!3e******eo;,]FN$\"3')*****R9&=XCFN7$$!3u*****4&fET]FN$\"3%******Hv Q&yBFN7$$!3))*****4e\\/3&FN$\"3;+++x)=8J#FN7$$!3W+++.vq=^FN$\"3++++$\\ qNC#FN7$$!3'******>&R-c^FN$\"3;+++BxLv@FN7$$!3S+++0\"yB>&FN$\"3%****** >qim5#FN7$$!3d*****HwWxA&FN$\"3-+++)f%eP?FN7$$!3?+++7_4i_FN$\"3#****** p$)Q\"o>FN7$$!3y*****f?)R&H&FN$\"3++++VlN)*=FN7$$!3))*****>+@wK&FN$\"3 2+++TUEG=FN7$$!3K+++&QI(e`FN$\"31+++yP)yv\"FN7$$!3w*****4j$p)Q&FN$\"3) ******fYKso\"FN7$$!3w*****4PzuT&FN$\"3-+++1MK;;FN7$$!3#)*****\\Je]W&FN $\"3/+++Tf;X:FN7$$!31+++VQSraFN$\"3%******\\@mPZ\"FN7$$!3++++TC\\'\\&F N$\"30+++!yF@S\"FN7$$!3K++++SI?bFN$\"3++++RADI8FN7$$!3G+++\"*>#Ga&FN$ \"33+++&))R\"e7FN7$$!3Q+++!zPS5FN7$$!3-+++%Q `(>cFN$\"3a*****HOeJn*F-7$$!35+++&ezcj&FN$\"3q*****zR!>S*)F-7$$!3++++E ^G]cFN$\"3&******z+:\\?)F-7$$!3u*****zTvNm&FN$\"3e******3IRnuF-7$$!3m* ****Rwdbn&FN$\"3J+++A)3xs'F-7$$!3U+++54C'o&FN$\"3W+++UC(f)fF-7$$!3m*** **RcMcp&FN$\"3=+++c1KU_F-7$$!3l*****p*)[Pq&FN$\"3/+++?u\"p\\%F-7$$!3l* ****R.%f5dFN$\"3=+++aL&*\\PF-7$$!3$******Rczhr&FN$\"3'******\\HW;+$F-7 $$!3.+++pS^?dFN$\"3%******\\#*GAD#F-7$$!3z*****pq/Os&FN$\"3'******z[l> ]\"F-7$$!3V+++goXDdFN$\"3s******=qF6vFfr7$$!3@+++nQ2EdFNF(7$Fb]n$!3s** ****=qF6vFfr7$F]]n$!3'******z[l>]\"F-7$Fh\\n$!3%******\\#*GAD#F-7$Fc\\ n$!3'******\\HW;+$F-7$F^\\n$!3=+++aL&*\\PF-7$Fi[n$!3/+++?u\"p\\%F-7$Fd [n$!3=+++c1KU_F-7$F_[n$!3W+++UC(f)fF-7$Fjjm$!3J+++A)3xs'F-7$Fejm$!3e** ****3IRnuF-7$F`jm$!3&******z+:\\?)F-7$F[jm$!3q*****zR!>S*)F-7$Ffim$!3a *****HOeJn*F-7$Faim$!3\"******>!zPS5FN7$F\\im$!3'******4,.K6\"FN7$Fghm $!3/+++8.z&=\"FN7$Fbhm$!33+++&))R\"e7FN7$F]hm$!3++++RADI8FN7$Fhgm$!30+ ++!yF@S\"FN7$Fcgm$!3%******\\@mPZ\"FN7$F^gm$!3/+++Tf;X:FN7$Fifm$!3-+++ 1MK;;FN7$Fdfm$!3)******fYKso\"FN7$F_fm$!31+++yP)yv\"FN7$Fjem$!32+++TUE G=FN7$Feem$!3++++VlN)*=FN7$F`em$!3#******p$)Q\"o>FN7$F[em$!3-+++)f%eP? FN7$Ffdm$!3%******>qim5#FN7$Fadm$!3;+++BxLv@FN7$F\\dm$!3++++$\\qNC#FN7 $Fgcm$!3;+++x)=8J#FN7$Fbcm$!3%******HvQ&yBFN7$F]cm$!3')*****R9&=XCFN7$ Fhbm$!3\")******>M@6DFN7$Fcbm$!3=+++N0ewDFN7$F^bm$!3;+++)>Y7k#FN7$Fiam $!3*******>&R<0FFN7$Fdam$!3)******p5K$oFFN7$F_am$!3<+++)\\%pIGFN7$Fj`m $!3'******>3TA*GFN7$Fe`m$!31+++e$eH&HFN7$F``m$!3#)*****fjRG,$FN7$F[`m$ !3!)*****\\<&)=2$FN7$Ff_m$!3<+++'>-,8$FN7$Fa_m$!35+++*y/v=$FN7$F\\_m$! 3()*****>r8TC$FN7$Fg^m$!35+++^g&**H$FN7$Fb^m$!3=+++PZ1bLFN7$F]^m$!34++ +SyZ4MFN7$Fh]m$!3?+++%fPKY$FN7$Fc]m$!34+++j!*Q;NFN7$F^]m$!37+++c$y*oNF N7$Fi\\m$!3$******zK]5i$FN7$Fd\\m$!37+++#pXEn$FN7$F_\\m$!33+++$e(zBPFN 7$Fj[m$!3y*****zlFXx$FN7$Fe[m$!32+++3?%[#QFN7$F`[m$!31+++UssuQFN7$F[[m $!3')******3t9CRFN7$Ffjl$!3?+++6;/tRFN7$Fajl$!3i*****>&\\K@SFN7$F\\jl$ !3o*****Hg*))oSFN7$Fgil$!3)*******Q%4c6%FN7$Fbil$!3M+++DdMhTFN7$F]il$! 3%)*****4y`f?%FN7$Fhhl$!3S+++!o*G\\UFN7$Fchl$!3k*****H;;7H%FN7$F^hl$!3 q*****>A2;L%FN7$Figl$!3g*****48^.P%FN7$Fdgl$!3'******>&>N2WFN7$F_gl$!3 X+++&zHDW%FN7$Fjfl$!3s*****>rG*e%FN7$Fael$!39+++ (eDEh%FN7$F\\el$!3z*****f$o%Rj%FN7$Fgdl$!3S+++^hE`YFN7$Fbdl$!3))***** \\R61n%FN7$F]dl$!3E+++2i-'o%FN7$Fhcl$!3o*****46v&*p%FN7$Fccl$!3V+++29N 6ZFN7$F^cl$!3&)*****HA*[@ZFN7$Fibl$!3U+++O?=IZFN7$Fdbl$!3L+++W9rPZFN7$ F_bl$!3n*****fg#\\WZFN7$Fjal$!3i*******)p9^ZFN7$Feal$!3#******p<>'eZFN 7$F`al$!3=+++F&e$oZFN7$F[al$!3i*****zekDy%FN7$Ff`l$!3T+++j)[V![FN7$Fa` l$!3f*****4)*or$[FN7$F\\`l$!3!)*****HX+C)[FN7$Fg_l$!3.+++zuRO\\FN7$Fb_ l$!3C+++.)*G#*\\FN7$F]_l$!3<+++g0\"[/&FN7$Fh^l$!3q*****R#)y;4&FN7$Fc^l $!3J+++9,UK^FN7$F^^l$!3u*****\\Ozs;&FN7$Fi]l$!35+++cNq'>&FN7$Fd]l$!3=+ ++]r9@_FN7$F_]l$!3G+++.4,T_FN7$Fj\\l$!3u******['HmD&FN7$Fe\\l$!3s***** *pjFo_FN7$F`\\l$!37+++a0&FN7$F^y$!3c*****Rt>A<&F N7$Fix$!38+++'[FU9&FN7$Fdx$!3P+++cH%H6&FN7$F_x$!3%******>^)Hy]FN7$Fjw$ !3C+++%R8-/&FN7$Few$!3#******RU#f)*\\FN7$F`w$!3S+++$HBL&\\FN7$F[w$!3Q+ ++r9F/\\FN7$Ffv$!3!)*****R=r7&[FN7$Fav$!3%******\\\"*4Tz%FN7$F\\v$!3)) *****zJ-Dt%FN7$Fgu$!3U+++rn/mYFN7$Fbu$!3M+++!f`Tf%FN7$F]u$!3@+++?z#f^% FN7$Fht$!3D+++Qk)*HWFN7$Fct$!34+++d\"4UL%FN7$F^t$!3n*****p9ubA%FN7$Fis $!3e*****fq>25%FN7$Fds$!3/+++f%p$eRFN7$F_s$!3#)*****fYpA!QFN7$Fjr$!3=+ ++$p!RROFN7$Fdr$!3')*****f\">5vMFN7$F_r$!3-+++U!*y6LFN7$Fjq$!3%)*****z &p.]JFN7$Fdq$!3?+++k'=(*)HFN7$F_q$!3/+++6P^IGFN7$Fjp$!3!******RD-@n#FN 7$Fep$!3')*****>)RA9DFN7$F_p$!37+++^locBFN7$Fjo$!3=+++$zc$*>#FN7$Feo$! 3(******>YZ@/#FN7$F`o$!3)******zN/])=FN7$Fjn$!35+++>c*ys\"FN7$$!3y**** **)*)[9Z(FR$!3%******Hf.3d\"FN7$FV$!3%******z6>PT\"FN7$FP$!3/+++wxjc7F N7$FI$!3'*******4wb*4\"FN7$$!3w*****>rYme*F@$!3`+++Q$yZU*F-7$$!3'***** *H7^,.#F@$!3m*****\\p\")R&yF-7$$!3-+++gFZ/JF:$!3Y+++9`=$G'F-7$$!3;+++e rGcGF4$!3D+++\")*)Q7ZF-7$F($!3)******Rl#fTJF-7$F($!3++++Fjzq:F-F'-%&CO LORG6&%$RGBG$\"\"#!\"\"F(F(-%)POLYGONSG6^_l7%F'7$F($\"+Fjzq:!#57$$!#WF j\\pF(7%F_]p7$F($\"+aEfTJFb]pFc]p7%Fg]p7$$!+@rGcGFfr$\"+\")*)Q7ZFb]pFc ]p7%F[^p7$$!+eFZ/JF-$\"+9`=$G'Fb]pFc]p7%Fa^p7$$!+A6:I?FN$\"+&p\")R&yFb ]pFc]p7%Fg^p7$$!+8nk'e*FN$\"+Q$yZU*Fb]pFc]p7%F]_p7$$!+mhS%f$!#;$\"+5wb *4\"!\"*Fc]p7%Fc_p7$$!+NobI6!#:$\"+wxjc7Fi_pFc]p7%F[`p7$$!+JDF'3$F^`p$ \"+=\">PT\"Fi_pFc]p7%Fb`p7$$!++*[9Z(F^`p$\"+$f.3d\"Fi_pFc]p7%Fh`p7$$!+ A`$\\i\"!#9$\"+>c*ys\"Fi_pFc]p7%F^ap7$$!+*p2K>$Faap$\"+eV+&)=Fi_pFc]p7 %Feap7$$!+*[[il&Faap$\"+iu9U?Fi_pFc]p7%F[bp7$$!+3ue!*))Faap$\"+$zc$*># Fi_pFc]p7%Fabp7$$!+oA1$=\"!#8$\"+^locBFi_pFc]p7%Fgbp7$$!+SJ&z7\"Fjbp$ \"+#)RA9DFi_pFc]p7%F^cp7$$\"+.\"HQC#F^`p$\"+aA5sEFi_pFc]p7%Fdcp7$$\"+b v(f!QFjbp$\"+6P^IGFi_pFc]p7%Fjcp7$$\"+_$egJ\"!#7$\"+k'=(*)HFi_pFc]p7%F `dp7$$\"+[a4ULFcdp$\"+ep.]JFi_pFc]p7%Fgdp7$$\"+yp*4R(Fcdp$\"+U!*y6LFi_ pFc]p7%F]ep7$$\"+C>m-:!#6$\"+;>5vMFi_pFc]p7%Fcep7$$\"+\"zz!oGFfep$\"+$ p!RROFi_pFc]p7%Fjep7$$\"+eM@Y^Ffep$\"+m%pA!QFi_pFc]p7%F`fp7$$\"+LelU&) Ffep$\"+f%p$eRFi_pFc]p7%Fffp7$$\"+*))fKG\"Fb]p$\"+1(>25%Fi_pFc]p7%F\\g p7$$\"+>vUN'z#>Fb]p$\"+cH%H6&Fi_pFc]p7%Ff\\q7$$\"+X4?Z;Fb]p$\"+'[FU9&Fi_ pFc]p7%F\\]q7$$\"+OkoY8Fb]p$\"+M(>A<&Fi_pFc]p7%Fb]q7$$\"+S2XG5Fb]p$\"+ EY(p>&Fi_pFc]p7%Fh]q7$$\"+2(HE%pFfep$\"+*GM&=_Fi_pFc]p7%F^^q7$$\"+!ffl X$Ffep$\"+(eFpB&Fi_pFc]p7%Fd^q7$$!+**z?0;Fcdp$\"+z$p@D&Fi_pFc]p7%Fj^q7 $$!+FYT(*QFfep$\"+$\\fUE&Fi_pFc]p7%F`_q7$$!+%>6[u(Ffep$\"+)Q\"=t_Fi_pF c]p7%Ff_q7$$!+'zD&p6Fb]p$\"+Ku:Fb]p$\"+)eh 8G&Fi_pFc]p7%Fb`q7$$!+'oC&))>Fb]p$\"+%o'[!G&Fi_pFc]p7%Fh`q7$$!+v-.7CFb ]p$\"+a0KUPFb]p$\"+.4,T_Fi_pFc]p7%F`bq 7$$!+$*H`4UFb]p$\"+]r9@_Fi_pFc]p7%Ffbq7$$!+F=k#p%Fb]p$\"+cNq'>&Fi_pFc] p7%F\\cq7$$!+yQ@'>&Fb]p$\"+l$zs;&Fi_pFc]p7%Fbcq7$$!+'f,us&Fb]p$\"+9,UK ^Fi_pFc]p7%Fhcq7$$!+qPX(H'Fb]p$\"+C)y;4&Fi_pFc]p7%F^dq7$$!+K*[R#pFb]p$ \"+g0\"[/&Fi_pFc]p7%Fddq7$$!+m;1LwFb]p$\"+.)*G#*\\Fi_pFc]p7%Fjdq7$$!+' ycvX)Fb]p$\"+zuRO\\Fi_pFc]p7%F`eq7$$!+L%*Fb]p$\"+`/S#)[Fi_pFc]p7%Ff eq7$$!+2Ns\\5Fi_p$\"+\")*or$[Fi_pFc]p7%F\\fq7$$!+#3cD;\"Fi_p$\"+j)[V![ Fi_pFc]p7%Fbfq7$$!+5L?u7Fi_p$\"+)ekDy%Fi_pFc]p7%Fhfq7$$!+^M[\"Q\"Fi_p$ \"+F&e$oZFi_pFc]p7%F^gq7$$!+r=_$[\"Fi_p$\"+x\">'eZFi_pFc]p7%Fdgq7$$!+* Q'[!e\"Fi_p$\"+!*p9^ZFi_pFc]p7%Fjgq7$$!+7&))Gn\"Fi_p$\"+1E\\WZFi_pFc]p 7%F`hq7$$!+tuHhFi_p$\"+B#*[@ZFi_pFc]p7%Fbiq7$$!+FL]2?Fi_p$\"+ 29N6ZFi_pFc]p7%Fhiq7$$!+Ila%3#Fi_p$\"+6^d*p%Fi_pFc]p7%F^jq7$$!+%G\"ff@ Fi_p$\"+2i-'o%Fi_pFc]p7%Fdjq7$$!+>+)GB#Fi_p$\"+&R61n%Fi_pFc]p7%Fjjq7$$ !+OHi/BFi_p$\"+^hE`YFi_pFc]p7%F`[r7$$!+dZ+vBFi_p$\"+Oo%Rj%Fi_pFc]p7%Ff [r7$$!+\"G!>WCFi_p$\"+(eDEh%Fi_pFc]p7%F\\\\r7$$!+B*GB^#Fi_p$\"+?rG*e%F i_pFc]p7%Fb\\r7$$!+o$e&zDFi_p$\"+/p#Rc%Fi_pFc]p7%Fh\\r7$$!+du+YEFi_p$ \"+e.bOXFi_pFc]p7%F^]r7$$!+Q$)z6FFi_p$\"+FG<2XFi_pFc]p7%Fd]r7$$!+6z/xF Fi_p$\"+s,#eZ%Fi_pFc]p7%Fj]r7$$!+7&o=%GFi_p$\"+&zHDW%Fi_pFc]p7%F`^r7$$ !+)ynj!HFi_p$\"+_>N2WFi_pFc]p7%Ff^r7$$!+(pZ1(HFi_p$\"+J6NqVFi_pFc]p7%F \\_r7$$!+yD![.$Fi_p$\"+AsgJVFi_pFc]p7%Fb_r7$$!+Sj\"*)4$Fi_p$\"+jh@\"H% Fi_pFc]p7%Fh_r7$$!+*))eI;$Fi_p$\"+!o*G\\UFi_pFc]p7%F^`r7$$!+B@GFKFi_p$ \"+\"y`f?%Fi_pFc]p7%Fd`r7$$!+vfh\"H$Fi_p$\"+DdMhTFi_pFc]p7%Fj`r7$$!+A^ 1cLFi_p$\"+R%4c6%Fi_pFc]p7%F`ar7$$!+Zqg?MFi_p$\"+.'*))oSFi_pFc]p7%Ffar 7$$!+/?>&[$Fi_p$\"+_\\K@SFi_pFc]p7%F\\br7$$!+n^u\\NFi_p$\"+6;/tRFi_pFc ]p7%Fbbr7$$!+c3<9OFi_p$\"+4t9CRFi_pFc]p7%Fhbr7$$!+T$e$yOFi_p$\"+UssuQF i_pFc]p7%F^cr7$$!+f%)=UPFi_p$\"+3?%[#QFi_pFc]p7%Fdcr7$$!+5-a0QFi_p$\"+ ew_uPFi_pFc]p7%Fjcr7$$!++mHoQFi_p$\"+$e(zBPFi_pFc]p7%F`dr7$$!+p#\\.$RF i_p$\"+#pXEn$Fi_pFc]p7%Ffdr7$$!+20%Fi_p$\"+c$y*oNFi_pFc]p7%Fber7$$!+A!*Q6TFi_p$\"+j!*Q;NFi_pFc]p7 %Fher7$$!+B7!)pTFi_p$\"+%fPKY$Fi_pFc]p7%F^fr7$$!+Ho;FUFi_p$\"+SyZ4MFi_ pFc]p7%Fdfr7$$!+crX$G%Fi_p$\"+PZ1bLFi_pFc]p7%Fjfr7$$!+aVlQVFi_p$\"+^g& **H$Fi_pFc]p7%F`gr7$$!+#)*\\FR%Fi_p$\"+7P6WKFi_pFc]p7%Ffgr7$$!+@OuXWFi _p$\"+*y/v=$Fi_pFc]p7%F\\hr7$$!+T:k(\\%Fi_p$\"+'>-,8$Fi_pFc]p7%Fbhr7$$ !+VaX[XFi_p$\"+v^)=2$Fi_pFc]p7%Fhhr7$$!+s7?)f%Fi_p$\"+O'RG,$Fi_pFc]p7% F^ir7$$!+!3)*ok%Fi_p$\"+e$eH&HFi_pFc]p7%Fdir7$$!+Hpc%p%Fi_p$\"+#3TA*GF i_pFc]p7%Fjir7$$!+3)H7u%Fi_p$\"+)\\%pIGFi_pFc]p7%F`jr7$$!+q&3py%Fi_p$ \"+2@LoFFi_pFc]p7%Ffjr7$$!+yQiJ[Fi_p$\"+_R<0FFi_pFc]p7%F\\[s7$$!+#=%Rv [Fi_p$\"+)>Y7k#Fi_pFc]p7%Fb[s7$$!+eZB=\\Fi_p$\"+N0ewDFi_pFc]p7%Fh[s7$$ !+No:g\\Fi_p$\"+?M@6DFi_pFc]p7%F^\\s7$$!+fo;,]Fi_p$\"+W^=XCFi_pFc]p7%F d\\s7$$!+^fET]Fi_p$\"+`(Q&yBFi_pFc]p7%Fj\\s7$$!+\"e\\/3&Fi_p$\"+x)=8J# Fi_pFc]p7%F`]s7$$!+.vq=^Fi_p$\"+$\\qNC#Fi_pFc]p7%Ff]s7$$!+_R-c^Fi_p$\" +BxLv@Fi_pFc]p7%F\\^s7$$!+0\"yB>&Fi_p$\"+-Fm1@Fi_pFc]p7%Fb^s7$$!+jZuF_ Fi_p$\"+)f%eP?Fi_pFc]p7%Fh^s7$$!+7_4i_Fi_p$\"+P)Q\"o>Fi_pFc]p7%F^_s7$$ !+1#)R&H&Fi_p$\"+VlN)*=Fi_pFc]p7%Fd_s7$$!+-5iF`Fi_p$\"+TUEG=Fi_pFc]p7% Fj_s7$$!+&QI(e`Fi_p$\"+yP)yv\"Fi_pFc]p7%F``s7$$!+JOp)Q&Fi_p$\"+mCB(o\" Fi_pFc]p7%Ff`s7$$!+r$zuT&Fi_p$\"+1MK;;Fi_pFc]p7%F\\as7$$!+:$e]W&Fi_p$ \"+Tf;X:Fi_pFc]p7%Fbas7$$!+VQSraFi_p$\"+:iwt9Fi_pFc]p7%Fhas7$$!+TC\\' \\&Fi_p$\"+!yF@S\"Fi_pFc]p7%F^bs7$$!++SI?bFi_p$\"+RADI8Fi_pFc]p7%Fdbs7 $$!+\"*>#Ga&Fi_p$\"+&))R\"e7Fi_pFc]p7%Fjbs7$$!+cFi_p$\"+j$eJn*Fb]pFc]p7%Fbds7$$!+&ezcj&Fi_ p$\"+)R!>S*)Fb]pFc]p7%Fhds7$$!+E^G]cFi_p$\"+3]\"\\?)Fb]pFc]p7%F^es7$$! +=adjcFi_p$\"+4IRnuFb]pFc]p7%Fdes7$$!+kxbvcFi_p$\"+A)3xs'Fb]pFc]p7%Fje s7$$!+54C'o&Fi_p$\"+UC(f)fFb]pFc]p7%F`fs7$$!+kXj&p&Fi_p$\"+c1KU_Fb]pFc ]p7%Fffs7$$!+(*)[Pq&Fi_p$\"+?u\"p\\%Fb]pFc]p7%F\\gs7$$!+MSf5dFi_p$\"+a L&*\\PFb]pFc]p7%Fbgs7$$!+k&zhr&Fi_p$\"+&HW;+$Fb]pFc]p7%Fhgs7$$!+pS^?dF i_p$\"+D*GAD#Fb]pFc]p7%F^hs7$$!+2ZgBdFi_p$\"+)[l>]\"Fb]pFc]p7%Fdhs7$$! +goXDdFi_p$\"+>qF6vFfepFc]p7%Fjhs7$$!+nQ2EdFi_pF(Fc]p7%F`is7$F[is$!+>q F6vFfepFc]p7%Fdis7$Fehs$!+)[l>]\"Fb]pFc]p7%Fhis7$F_hs$!+D*GAD#Fb]pFc]p 7%F\\js7$Figs$!+&HW;+$Fb]pFc]p7%F`js7$Fcgs$!+aL&*\\PFb]pFc]p7%Fdjs7$F] gs$!+?u\"p\\%Fb]pFc]p7%Fhjs7$Fgfs$!+c1KU_Fb]pFc]p7%F\\[t7$Fafs$!+UC(f) fFb]pFc]p7%F`[t7$F[fs$!+A)3xs'Fb]pFc]p7%Fd[t7$Fees$!+4IRnuFb]pFc]p7%Fh [t7$F_es$!+3]\"\\?)Fb]pFc]p7%F\\\\t7$Fids$!+)R!>S*)Fb]pFc]p7%F`\\t7$Fc ds$!+j$eJn*Fb]pFc]p7%Fd\\t7$F]ds$!+-zPS5Fi_pFc]p7%Fh\\t7$Fgcs$!+6I?86F i_pFc]p7%F\\]t7$Facs$!+8.z&=\"Fi_pFc]p7%F`]t7$F[cs$!+&))R\"e7Fi_pFc]p7 %Fd]t7$Febs$!+RADI8Fi_pFc]p7%Fh]t7$F_bs$!+!yF@S\"Fi_pFc]p7%F\\^t7$Fias $!+:iwt9Fi_pFc]p7%F`^t7$Fcas$!+Tf;X:Fi_pFc]p7%Fd^t7$F]as$!+1MK;;Fi_pFc ]p7%Fh^t7$Fg`s$!+mCB(o\"Fi_pFc]p7%F\\_t7$Fa`s$!+yP)yv\"Fi_pFc]p7%F`_t7 $F[`s$!+TUEG=Fi_pFc]p7%Fd_t7$Fe_s$!+VlN)*=Fi_pFc]p7%Fh_t7$F__s$!+P)Q\" o>Fi_pFc]p7%F\\`t7$Fi^s$!+)f%eP?Fi_pFc]p7%F``t7$Fc^s$!+-Fm1@Fi_pFc]p7% Fd`t7$F]^s$!+BxLv@Fi_pFc]p7%Fh`t7$Fg]s$!+$\\qNC#Fi_pFc]p7%F\\at7$Fa]s$ !+x)=8J#Fi_pFc]p7%F`at7$F[]s$!+`(Q&yBFi_pFc]p7%Fdat7$Fe\\s$!+W^=XCFi_p Fc]p7%Fhat7$F_\\s$!+?M@6DFi_pFc]p7%F\\bt7$Fi[s$!+N0ewDFi_pFc]p7%F`bt7$ Fc[s$!+)>Y7k#Fi_pFc]p7%Fdbt7$F][s$!+_R<0FFi_pFc]p7%Fhbt7$Fgjr$!+2@LoFF i_pFc]p7%F\\ct7$Fajr$!+)\\%pIGFi_pFc]p7%F`ct7$F[jr$!+#3TA*GFi_pFc]p7%F dct7$Feir$!+e$eH&HFi_pFc]p7%Fhct7$F_ir$!+O'RG,$Fi_pFc]p7%F\\dt7$Fihr$! +v^)=2$Fi_pFc]p7%F`dt7$Fchr$!+'>-,8$Fi_pFc]p7%Fddt7$F]hr$!+*y/v=$Fi_pF c]p7%Fhdt7$Fggr$!+7P6WKFi_pFc]p7%F\\et7$Fagr$!+^g&**H$Fi_pFc]p7%F`et7$ F[gr$!+PZ1bLFi_pFc]p7%Fdet7$Fefr$!+SyZ4MFi_pFc]p7%Fhet7$F_fr$!+%fPKY$F i_pFc]p7%F\\ft7$Fier$!+j!*Q;NFi_pFc]p7%F`ft7$Fcer$!+c$y*oNFi_pFc]p7%Fd ft7$F]er$!+G.0@OFi_pFc]p7%Fhft7$Fgdr$!+#pXEn$Fi_pFc]p7%F\\gt7$Fadr$!+$ e(zBPFi_pFc]p7%F`gt7$F[dr$!+ew_uPFi_pFc]p7%Fdgt7$Fecr$!+3?%[#QFi_pFc]p 7%Fhgt7$F_cr$!+UssuQFi_pFc]p7%F\\ht7$Fibr$!+4t9CRFi_pFc]p7%F`ht7$Fcbr$ !+6;/tRFi_pFc]p7%Fdht7$F]br$!+_\\K@SFi_pFc]p7%Fhht7$Fgar$!+.'*))oSFi_p Fc]p7%F\\it7$Faar$!+R%4c6%Fi_pFc]p7%F`it7$F[ar$!+DdMhTFi_pFc]p7%Fdit7$ Fe`r$!+\"y`f?%Fi_pFc]p7%Fhit7$F_`r$!+!o*G\\UFi_pFc]p7%F\\jt7$Fi_r$!+jh @\"H%Fi_pFc]p7%F`jt7$Fc_r$!+AsgJVFi_pFc]p7%Fdjt7$F]_r$!+J6NqVFi_pFc]p7 %Fhjt7$Fg^r$!+_>N2WFi_pFc]p7%F\\[u7$Fa^r$!+&zHDW%Fi_pFc]p7%F`[u7$F[^r$ !+s,#eZ%Fi_pFc]p7%Fd[u7$Fe]r$!+FG<2XFi_pFc]p7%Fh[u7$F_]r$!+e.bOXFi_pFc ]p7%F\\\\u7$Fi\\r$!+/p#Rc%Fi_pFc]p7%F`\\u7$Fc\\r$!+?rG*e%Fi_pFc]p7%Fd \\u7$F]\\r$!+(eDEh%Fi_pFc]p7%Fh\\u7$Fg[r$!+Oo%Rj%Fi_pFc]p7%F\\]u7$Fa[r $!+^hE`YFi_pFc]p7%F`]u7$F[[r$!+&R61n%Fi_pFc]p7%Fd]u7$Fejq$!+2i-'o%Fi_p Fc]p7%Fh]u7$F_jq$!+6^d*p%Fi_pFc]p7%F\\^u7$Fiiq$!+29N6ZFi_pFc]p7%F`^u7$ Fciq$!+B#*[@ZFi_pFc]p7%Fd^u7$F]iq$!+O?=IZFi_pFc]p7%Fh^u7$Fghq$!+W9rPZF i_pFc]p7%F\\_u7$Fahq$!+1E\\WZFi_pFc]p7%F`_u7$F[hq$!+!*p9^ZFi_pFc]p7%Fd _u7$Fegq$!+x\">'eZFi_pFc]p7%Fh_u7$F_gq$!+F&e$oZFi_pFc]p7%F\\`u7$Fifq$! +)ekDy%Fi_pFc]p7%F``u7$Fcfq$!+j)[V![Fi_pFc]p7%Fd`u7$F]fq$!+\")*or$[Fi_ pFc]p7%Fh`u7$Fgeq$!+`/S#)[Fi_pFc]p7%F\\au7$Faeq$!+zuRO\\Fi_pFc]p7%F`au 7$F[eq$!+.)*G#*\\Fi_pFc]p7%Fdau7$Fedq$!+g0\"[/&Fi_pFc]p7%Fhau7$F_dq$!+ C)y;4&Fi_pFc]p7%F\\bu7$Ficq$!+9,UK^Fi_pFc]p7%F`bu7$Fccq$!+l$zs;&Fi_pFc ]p7%Fdbu7$F]cq$!+cNq'>&Fi_pFc]p7%Fhbu7$Fgbq$!+]r9@_Fi_pFc]p7%F\\cu7$Fa bq$!+.4,T_Fi_pFc]p7%F`cu7$F[bq$!+\\'HmD&Fi_pFc]p7%Fdcu7$Feaq$!+qjFo_Fi _pFc]p7%Fhcu7$F_aq$!+a0&Fi_pFc]p7%Fdfu7$Fc]q$!+M(>A<&Fi_pFc]p7% Fhfu7$F]]q$!+'[FU9&Fi_pFc]p7%F\\gu7$Fg\\q$!+cH%H6&Fi_pFc]p7%F`gu7$Fa\\ q$!+7&)Hy]Fi_pFc]p7%Fdgu7$F[\\q$!+%R8-/&Fi_pFc]p7%Fhgu7$Fe[q$!+CCf)*\\ Fi_pFc]p7%F\\hu7$F_[q$!+$HBL&\\Fi_pFc]p7%F`hu7$Fijp$!+r9F/\\Fi_pFc]p7% Fdhu7$Fcjp$!+%=r7&[Fi_pFc]p7%Fhhu7$F]jp$!+:*4Tz%Fi_pFc]p7%F\\iu7$Fgip$ !+=B]KZFi_pFc]p7%F`iu7$Faip$!+rn/mYFi_pFc]p7%Fdiu7$F[ip$!+!f`Tf%Fi_pFc ]p7%Fhiu7$Fehp$!+?z#f^%Fi_pFc]p7%F\\ju7$F_hp$!+Qk)*HWFi_pFc]p7%F`ju7$F igp$!+d\"4UL%Fi_pFc]p7%Fdju7$Fcgp$!+ZTdDUFi_pFc]p7%Fhju7$F]gp$!+1(>25% Fi_pFc]p7%F\\[v7$Fgfp$!+f%p$eRFi_pFc]p7%F`[v7$Fafp$!+m%pA!QFi_pFc]p7%F d[v7$F[fp$!+$p!RROFi_pFc]p7%Fh[v7$Fdep$!+;>5vMFi_pFc]p7%F\\\\v7$F^ep$! +U!*y6LFi_pFc]p7%F`\\v7$Fhdp$!+ep.]JFi_pFc]p7%Fd\\v7$Fadp$!+k'=(*)HFi_ pFc]p7%Fh\\v7$F[dp$!+6P^IGFi_pFc]p7%F\\]v7$Fecp$!+aA5sEFi_pFc]p7%F`]v7 $F_cp$!+#)RA9DFi_pFc]p7%Fd]v7$Fhbp$!+^locBFi_pFc]p7%Fh]v7$Fbbp$!+$zc$* >#Fi_pFc]p7%F\\^v7$F\\bp$!+iu9U?Fi_pFc]p7%F`^v7$Ffap$!+eV+&)=Fi_pFc]p7 %Fd^v7$F_ap$!+>c*ys\"Fi_pFc]p7%Fh^v7$$!+**)[9Z(F^`p$!+$f.3d\"Fi_pFc]p7 %F\\_v7$Fc`p$!+=\">PT\"Fi_pFc]p7%Fb_v7$F\\`p$!+wxjc7Fi_pFc]p7%Ff_v7$Fd _p$!+5wb*4\"Fi_pFc]p7%Fj_v7$$!+7nk'e*FN$!+Q$yZU*Fb]pFc]p7%F^`v7$$!+B6: I?FN$!+&p\")R&yFb]pFc]p7%Fd`v7$$!+gFZ/JF-$!+9`=$G'Fb]pFc]p7%Fj`v7$$!+e rGcGFfr$!+\")*)Q7ZFb]pFc]p7%F`av7$F($!+aEfTJFb]pFc]p7%Ffav7$F($!+Fjzq: Fb]pFc]p7%FjavF'Fc]p-Fe\\p6&Fg\\p$\"#l!\"#$\"\"$Fj\\p$\"#DFbbv-%&STYLE G6#%,PATCHNOGRIDG-F$6%7$7$$!3A++++++!\\'FNF(7$$\"3%**************=\"FN F(-%'COLOURG6&Fg\\pF)F)F)-%*LINESTYLEG6#Fdbv-F$6%7$7$F($!3o*********** **Q'FN7$F($\"3o*************Q'FNFdcvFgcv-%(SCALINGG6#%,CONSTRAINEDG-%+ AXESLABELSG6%%&Re(z)G%&Im(z)G-%%FONTG6#%(DEFAULTG-F]ev6$%*HELVETICAG\" \"*-%*AXESSTYLEG6#%$BOXG-%%VIEWG6$;$!$\\'Fbbv$\"$>\"Fbbv;$!$R'Fbbv$\"$ R'Fbbv" 1 2 0 1 10 0 2 9 1 2 1 1.000000 45.000000 45.000000 0 0 "Curve 1" "Curve 2" "Curve 3" "Curve 4" }}}}{PARA 0 "" 0 "" {TEXT -1 0 "" }} {PARA 0 "" 0 "" {TEXT -1 75 "The stability interval for the embedded o rder 7 scheme is (approximately) " }{XPPEDIT 18 0 "[-5.7261, 0];" "6# 7$,$-%&FloatG6$\"&hs&!\"%!\"\"\"\"!" }{TEXT -1 2 ". " }}{PARA 0 "" 0 " " {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 79 "The stability regions f or the two schemes can be displayed in the same picture." }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 154 "plots[display]([p_1,p_2,p||(1..3)] ,view=[-6.49..1.19,-6.39..6.39],font=[HELVETICA,9],\n labels =[`Re(z)`,`Im(z)`],axes=boxed,scaling=constrained);" }}{PARA 13 "" 1 " " {GLPLOT2D 362 580 580 {PLOTDATA 2 "6--%'CURVESG6$7]_l7$$\"\"!F)F(7$F ($\"3++++Fjzq:!#=7$F($\"3)******Rl#fTJF-7$$!3!******47(GcG!#F$\"3D+++ \")*)Q7ZF-7$$!39+++eFZ/J!#E$\"3Y+++9`=$G'F-7$$!39+++A6:I?!#D$\"3m***** \\p\")R&yF-7$$!3C+++8nk'e*F@$\"3`+++Q$yZU*F-7$$!3w*****f;1Wf$!#C$\"3'* ******4wb*4\"!#<7$$!3/+++NobI6!#B$\"3/+++wxjc7FN7$$!3++++JDF'3$FR$\"3% ******z6>PT\"FN7$$!35++++*[9Z(FR$\"3%******Hf.3d\"FN7$$!3;+++A`$\\i\"! #A$\"35+++>c*ys\"FN7$$!3%*******)p2K>$F\\o$\"3)******zN/])=FN7$$!3o*** ***)[[il&F\\o$\"3(******>YZ@/#FN7$$!3i+++3ue!*))F\\o$\"3=+++$zc$*>#FN7 $$!3%******zEiI=\"!#@$\"37+++^locBFN7$$!3********RJ&z7\"Fap$\"3')***** >)RA9DFN7$$\"3#)*****H5HQC#FR$\"3!******RD-@n#FN7$$\"30+++bv(f!QFap$\" 3/+++6P^IGFN7$$\"30+++_$egJ\"!#?$\"3?+++k'=(*)HFN7$$\"3<+++[a4ULFfq$\" 3%)*****z&p.]JFN7$$\"3%******z(p*4R(Ffq$\"3-+++U!*y6LFN7$$\"30+++C>m-: !#>$\"3')*****f\">5vMFN7$$\"3++++\"zz!oGFfr$\"3=+++$p!RROFN7$$\"3#**** **zX8i9&Ffr$\"3#)*****fYpA!QFN7$$\"3$******H$elU&)Ffr$\"3/+++f%p$eRFN7 $$\"3#*******)))fKG\"F-$\"3e*****fq>25%FN7$$\"39+++>vUN^)Hy] FN7$$\"3)******H*>'z#>F-$\"3P+++cH%H6&FN7$$\"3)******\\%4?Z;F-$\"38+++ '[FU9&FN7$$\"3)******fV'oY8F-$\"3c*****Rt>A<&FN7$$\"3&*******R2XG5F-$ \"3E+++EY(p>&FN7$$\"3P+++2(HE%pFfr$\"3%*******)GM&=_FN7$$\"3G+++!fflX$ Ffr$\"3G+++(eFpB&FN7$$!3!*******)*z?0;Ffq$\"3q******y$p@D&FN7$$!3E+++F YT(*QFfr$\"3'******H\\fUE&FN7$$!3Y*****R>6[u(Ffr$\"3C+++)Q\"=t_FN7$$!3 '******fzD&p6F-$\"3-+++Ku:F-$\"3A+++)eh8G&FN7$$!3)* *****foC&))>F-$\"3W+++%o'[!G&FN7$$!3++++v-.7CF-$\"37+++a0n4)G$F-$\"3u******['HmD&F N7$$!30+++s>KUPF-$\"3G+++.4,T_FN7$$!3s*****H*H`4UF-$\"3=+++]r9@_FN7$$! 3w*****p#=k#p%F-$\"35+++cNq'>&FN7$$!3++++yQ@'>&F-$\"3u*****\\Ozs;&FN7$ $!3?+++'f,us&F-$\"3J+++9,UK^FN7$$!3%*******pPX(H'F-$\"3q*****R#)y;4&FN 7$$!3G+++K*[R#pF-$\"3<+++g0\"[/&FN7$$!3Z*****fmhIj(F-$\"3C+++.)*G#*\\F N7$$!3/+++'ycvX)F-$\"3.+++zuRO\\FN7$$!3X+++L%*F-$\"3!)*****HX+C)[FN 7$$!3%******p]B(\\5FN$\"3f*****4)*or$[FN7$$!3/+++#3cD;\"FN$\"3T+++j)[V ![FN7$$!35+++5L?u7FN$\"3i*****zekDy%FN7$$!3-+++^M[\"Q\"FN$\"3=+++F&e$o ZFN7$$!3++++r=_$[\"FN$\"3#******p<>'eZFN7$$!3/+++*Q'[!e\"FN$\"3i****** *)p9^ZFN7$$!35+++7&))Gn\"FN$\"3n*****fg#\\WZFN7$$!3'******HZ(HhFN$\"3&) *****HA*[@ZFN7$$!3y*****pK.v+#FN$\"3V+++29N6ZFN7$$!3!*******Hla%3#FN$ \"3o*****46v&*p%FN7$$!3A+++%G\"ff@FN$\"3E+++2i-'o%FN7$$!3++++>+)GB#FN$ \"3))*****\\R61n%FN7$$!3!******f$Hi/BFN$\"3S+++^hE`YFN7$$!37+++dZ+vBFN $\"3z*****f$o%Rj%FN7$$!3-+++\"G!>WCFN$\"39+++(eDEh%FN7$$!35+++B*GB^#FN $\"3#)******>rG*e%FN7$$!3#)*****zOe&zDFN$\"3o*****R!p#Rc%FN7$$!31+++du +YEFN$\"3R+++e.bOXFN7$$!3#******zL)z6FFN$\"3x*****p#G<2XFN7$$!3#****** 4\"z/xFFN$\"3s*****>&>N2WFN7$$!3)******ppZ1(HFN$\"3g*****48^.P%FN7$$ !3')*****zd-[.$FN$\"3q*****>A2;L%FN7$$!3;+++Sj\"*)4$FN$\"3k*****H;;7H% FN7$$!3?+++*))eI;$FN$\"3S+++!o*G\\UFN7$$!3<+++B@GFKFN$\"3%)*****4y`f?% FN7$$!3-+++vfh\"H$FN$\"3M+++DdMhTFN7$$!3'******>7lgN$FN$\"3)*******Q%4 c6%FN7$$!3%)*****p/21U$FN$\"3o*****Hg*))oSFN7$$!3'******R+#>&[$FN$\"3i *****>&\\K@SFN7$$!3?+++n^u\\NFN$\"3?+++6;/tRFN7$$!3))*****f&3<9OFN$\"3 ')******3t9CRFN7$$!3-+++T$e$yOFN$\"31+++UssuQFN7$$!3')******e%)=UPFN$ \"32+++3?%[#QFN7$$!3?+++5-a0QFN$\"3y*****zlFXx$FN7$$!3')*******f'HoQFN $\"33+++$e(zBPFN7$$!39+++p#\\.$RFN$\"37+++#pXEn$FN7$$!31+++20%FN$\"37+++c$y*oNFN7$$!3++++A!*Q6TFN$ \"34+++j!*Q;NFN7$$!3K+++B7!)pTFN$\"3?+++%fPKY$FN7$$!3u******Go;FUFN$\" 34+++SyZ4MFN7$$!3))*****f:dMG%FN$\"3=+++PZ1bLFN7$$!3*******RNa'QVFN$\" 35+++^g&**H$FN7$$!3w*****>)*\\FR%FN$\"3()*****>r8TC$FN7$$!35+++@OuXWFN $\"35+++*y/v=$FN7$$!39+++T:k(\\%FN$\"3<+++'>-,8$FN7$$!3%******HWb%[XFN $\"3!)*****\\<&)=2$FN7$$!3%)*****>F,#)f%FN$\"3#)*****fjRG,$FN7$$!3s*** ***z!)*ok%FN$\"31+++e$eH&HFN7$$!3g******Gpc%p%FN$\"3'******>3TA*GFN7$$ !3%******z!)H7u%FN$\"3<+++)\\%pIGFN7$$!3e******p&3py%FN$\"3)******p5K$ oFFN7$$!3E+++yQiJ[FN$\"3*******>&R<0FFN7$$!3'******>=%Rv[FN$\"3;+++)>Y 7k#FN7$$!3l*****zvM#=\\FN$\"3=+++N0ewDFN7$$!3E+++No:g\\FN$\"3\")****** >M@6DFN7$$!3e******eo;,]FN$\"3')*****R9&=XCFN7$$!3u*****4&fET]FN$\"3%* *****HvQ&yBFN7$$!3))*****4e\\/3&FN$\"3;+++x)=8J#FN7$$!3W+++.vq=^FN$\"3 ++++$\\qNC#FN7$$!3'******>&R-c^FN$\"3;+++BxLv@FN7$$!3S+++0\"yB>&FN$\"3 %******>qim5#FN7$$!3d*****HwWxA&FN$\"3-+++)f%eP?FN7$$!3?+++7_4i_FN$\"3 #******p$)Q\"o>FN7$$!3y*****f?)R&H&FN$\"3++++VlN)*=FN7$$!3))*****>+@wK &FN$\"32+++TUEG=FN7$$!3K+++&QI(e`FN$\"31+++yP)yv\"FN7$$!3w*****4j$p)Q& FN$\"3)******fYKso\"FN7$$!3w*****4PzuT&FN$\"3-+++1MK;;FN7$$!3#)*****\\ Je]W&FN$\"3/+++Tf;X:FN7$$!31+++VQSraFN$\"3%******\\@mPZ\"FN7$$!3++++TC \\'\\&FN$\"30+++!yF@S\"FN7$$!3K++++SI?bFN$\"3++++RADI8FN7$$!3G+++\"*># Ga&FN$\"33+++&))R\"e7FN7$$!3Q+++!zPS5FN7$$!3 -+++%Q`(>cFN$\"3a*****HOeJn*F-7$$!35+++&ezcj&FN$\"3q*****zR!>S*)F-7$$! 3++++E^G]cFN$\"3&******z+:\\?)F-7$$!3u*****zTvNm&FN$\"3e******3IRnuF-7 $$!3m*****Rwdbn&FN$\"3J+++A)3xs'F-7$$!3U+++54C'o&FN$\"3W+++UC(f)fF-7$$ !3m*****RcMcp&FN$\"3=+++c1KU_F-7$$!3l*****p*)[Pq&FN$\"3/+++?u\"p\\%F-7 $$!3l*****R.%f5dFN$\"3=+++aL&*\\PF-7$$!3$******Rczhr&FN$\"3'******\\HW ;+$F-7$$!3.+++pS^?dFN$\"3%******\\#*GAD#F-7$$!3z*****pq/Os&FN$\"3'**** **z[l>]\"F-7$$!3V+++goXDdFN$\"3s******=qF6vFfr7$$!3@+++nQ2EdFNF(7$Fb]n $!3s******=qF6vFfr7$F]]n$!3'******z[l>]\"F-7$Fh\\n$!3%******\\#*GAD#F- 7$Fc\\n$!3'******\\HW;+$F-7$F^\\n$!3=+++aL&*\\PF-7$Fi[n$!3/+++?u\"p\\% F-7$Fd[n$!3=+++c1KU_F-7$F_[n$!3W+++UC(f)fF-7$Fjjm$!3J+++A)3xs'F-7$Fejm $!3e******3IRnuF-7$F`jm$!3&******z+:\\?)F-7$F[jm$!3q*****zR!>S*)F-7$Ff im$!3a*****HOeJn*F-7$Faim$!3\"******>!zPS5FN7$F\\im$!3'******4,.K6\"FN 7$Fghm$!3/+++8.z&=\"FN7$Fbhm$!33+++&))R\"e7FN7$F]hm$!3++++RADI8FN7$Fhg m$!30+++!yF@S\"FN7$Fcgm$!3%******\\@mPZ\"FN7$F^gm$!3/+++Tf;X:FN7$Fifm$ !3-+++1MK;;FN7$Fdfm$!3)******fYKso\"FN7$F_fm$!31+++yP)yv\"FN7$Fjem$!32 +++TUEG=FN7$Feem$!3++++VlN)*=FN7$F`em$!3#******p$)Q\"o>FN7$F[em$!3-+++ )f%eP?FN7$Ffdm$!3%******>qim5#FN7$Fadm$!3;+++BxLv@FN7$F\\dm$!3++++$\\q NC#FN7$Fgcm$!3;+++x)=8J#FN7$Fbcm$!3%******HvQ&yBFN7$F]cm$!3')*****R9&= XCFN7$Fhbm$!3\")******>M@6DFN7$Fcbm$!3=+++N0ewDFN7$F^bm$!3;+++)>Y7k#FN 7$Fiam$!3*******>&R<0FFN7$Fdam$!3)******p5K$oFFN7$F_am$!3<+++)\\%pIGFN 7$Fj`m$!3'******>3TA*GFN7$Fe`m$!31+++e$eH&HFN7$F``m$!3#)*****fjRG,$FN7 $F[`m$!3!)*****\\<&)=2$FN7$Ff_m$!3<+++'>-,8$FN7$Fa_m$!35+++*y/v=$FN7$F \\_m$!3()*****>r8TC$FN7$Fg^m$!35+++^g&**H$FN7$Fb^m$!3=+++PZ1bLFN7$F]^m $!34+++SyZ4MFN7$Fh]m$!3?+++%fPKY$FN7$Fc]m$!34+++j!*Q;NFN7$F^]m$!37+++c $y*oNFN7$Fi\\m$!3$******zK]5i$FN7$Fd\\m$!37+++#pXEn$FN7$F_\\m$!33+++$e (zBPFN7$Fj[m$!3y*****zlFXx$FN7$Fe[m$!32+++3?%[#QFN7$F`[m$!31+++UssuQFN 7$F[[m$!3')******3t9CRFN7$Ffjl$!3?+++6;/tRFN7$Fajl$!3i*****>&\\K@SFN7$ F\\jl$!3o*****Hg*))oSFN7$Fgil$!3)*******Q%4c6%FN7$Fbil$!3M+++DdMhTFN7$ F]il$!3%)*****4y`f?%FN7$Fhhl$!3S+++!o*G\\UFN7$Fchl$!3k*****H;;7H%FN7$F ^hl$!3q*****>A2;L%FN7$Figl$!3g*****48^.P%FN7$Fdgl$!3'******>&>N2WFN7$F _gl$!3X+++&zHDW%FN7$Fjfl$!3s*****>rG*e%FN7$Fael$ !39+++(eDEh%FN7$F\\el$!3z*****f$o%Rj%FN7$Fgdl$!3S+++^hE`YFN7$Fbdl$!3)) *****\\R61n%FN7$F]dl$!3E+++2i-'o%FN7$Fhcl$!3o*****46v&*p%FN7$Fccl$!3V+ ++29N6ZFN7$F^cl$!3&)*****HA*[@ZFN7$Fibl$!3U+++O?=IZFN7$Fdbl$!3L+++W9rP ZFN7$F_bl$!3n*****fg#\\WZFN7$Fjal$!3i*******)p9^ZFN7$Feal$!3#******p<> 'eZFN7$F`al$!3=+++F&e$oZFN7$F[al$!3i*****zekDy%FN7$Ff`l$!3T+++j)[V![FN 7$Fa`l$!3f*****4)*or$[FN7$F\\`l$!3!)*****HX+C)[FN7$Fg_l$!3.+++zuRO\\FN 7$Fb_l$!3C+++.)*G#*\\FN7$F]_l$!3<+++g0\"[/&FN7$Fh^l$!3q*****R#)y;4&FN7 $Fc^l$!3J+++9,UK^FN7$F^^l$!3u*****\\Ozs;&FN7$Fi]l$!35+++cNq'>&FN7$Fd]l $!3=+++]r9@_FN7$F_]l$!3G+++.4,T_FN7$Fj\\l$!3u******['HmD&FN7$Fe\\l$!3s ******pjFo_FN7$F`\\l$!37+++a0&FN7$F^y$!3c*****Rt >A<&FN7$Fix$!38+++'[FU9&FN7$Fdx$!3P+++cH%H6&FN7$F_x$!3%******>^)Hy]FN7 $Fjw$!3C+++%R8-/&FN7$Few$!3#******RU#f)*\\FN7$F`w$!3S+++$HBL&\\FN7$F[w $!3Q+++r9F/\\FN7$Ffv$!3!)*****R=r7&[FN7$Fav$!3%******\\\"*4Tz%FN7$F\\v $!3))*****zJ-Dt%FN7$Fgu$!3U+++rn/mYFN7$Fbu$!3M+++!f`Tf%FN7$F]u$!3@+++? z#f^%FN7$Fht$!3D+++Qk)*HWFN7$Fct$!34+++d\"4UL%FN7$F^t$!3n*****p9ubA%FN 7$Fis$!3e*****fq>25%FN7$Fds$!3/+++f%p$eRFN7$F_s$!3#)*****fYpA!QFN7$Fjr $!3=+++$p!RROFN7$Fdr$!3')*****f\">5vMFN7$F_r$!3-+++U!*y6LFN7$Fjq$!3%)* ****z&p.]JFN7$Fdq$!3?+++k'=(*)HFN7$F_q$!3/+++6P^IGFN7$Fjp$!3!******RD- @n#FN7$Fep$!3')*****>)RA9DFN7$F_p$!37+++^locBFN7$Fjo$!3=+++$zc$*>#FN7$ Feo$!3(******>YZ@/#FN7$F`o$!3)******zN/])=FN7$Fjn$!35+++>c*ys\"FN7$$!3 y******)*)[9Z(FR$!3%******Hf.3d\"FN7$FV$!3%******z6>PT\"FN7$FP$!3/+++w xjc7FN7$FI$!3'*******4wb*4\"FN7$$!3w*****>rYme*F@$!3`+++Q$yZU*F-7$$!3' ******H7^,.#F@$!3m*****\\p\")R&yF-7$$!3-+++gFZ/JF:$!3Y+++9`=$G'F-7$$!3 ;+++erGcGF4$!3D+++\")*)Q7ZF-7$F($!3)******Rl#fTJF-7$F($!3++++Fjzq:F-F' -%&COLORG6&%$RGBG$\"\"#!\"\"F(F(-%)POLYGONSG6^_l7%F'7$F($\"+Fjzq:!#57$ $!#WFj\\pF(7%F_]p7$F($\"+aEfTJFb]pFc]p7%Fg]p7$$!+@rGcGFfr$\"+\")*)Q7ZF b]pFc]p7%F[^p7$$!+eFZ/JF-$\"+9`=$G'Fb]pFc]p7%Fa^p7$$!+A6:I?FN$\"+&p\") R&yFb]pFc]p7%Fg^p7$$!+8nk'e*FN$\"+Q$yZU*Fb]pFc]p7%F]_p7$$!+mhS%f$!#;$ \"+5wb*4\"!\"*Fc]p7%Fc_p7$$!+NobI6!#:$\"+wxjc7Fi_pFc]p7%F[`p7$$!+JDF'3 $F^`p$\"+=\">PT\"Fi_pFc]p7%Fb`p7$$!++*[9Z(F^`p$\"+$f.3d\"Fi_pFc]p7%Fh` p7$$!+A`$\\i\"!#9$\"+>c*ys\"Fi_pFc]p7%F^ap7$$!+*p2K>$Faap$\"+eV+&)=Fi_ pFc]p7%Feap7$$!+*[[il&Faap$\"+iu9U?Fi_pFc]p7%F[bp7$$!+3ue!*))Faap$\"+$ zc$*>#Fi_pFc]p7%Fabp7$$!+oA1$=\"!#8$\"+^locBFi_pFc]p7%Fgbp7$$!+SJ&z7\" Fjbp$\"+#)RA9DFi_pFc]p7%F^cp7$$\"+.\"HQC#F^`p$\"+aA5sEFi_pFc]p7%Fdcp7$ $\"+bv(f!QFjbp$\"+6P^IGFi_pFc]p7%Fjcp7$$\"+_$egJ\"!#7$\"+k'=(*)HFi_pFc ]p7%F`dp7$$\"+[a4ULFcdp$\"+ep.]JFi_pFc]p7%Fgdp7$$\"+yp*4R(Fcdp$\"+U!*y 6LFi_pFc]p7%F]ep7$$\"+C>m-:!#6$\"+;>5vMFi_pFc]p7%Fcep7$$\"+\"zz!oGFfep $\"+$p!RROFi_pFc]p7%Fjep7$$\"+eM@Y^Ffep$\"+m%pA!QFi_pFc]p7%F`fp7$$\"+L elU&)Ffep$\"+f%p$eRFi_pFc]p7%Fffp7$$\"+*))fKG\"Fb]p$\"+1(>25%Fi_pFc]p7 %F\\gp7$$\"+>vUN'z#>Fb]p$\"+cH%H6&Fi_pFc]p7%Ff\\q7$$\"+X4?Z;Fb]p$\"+'[FU9 &Fi_pFc]p7%F\\]q7$$\"+OkoY8Fb]p$\"+M(>A<&Fi_pFc]p7%Fb]q7$$\"+S2XG5Fb]p $\"+EY(p>&Fi_pFc]p7%Fh]q7$$\"+2(HE%pFfep$\"+*GM&=_Fi_pFc]p7%F^^q7$$\"+ !fflX$Ffep$\"+(eFpB&Fi_pFc]p7%Fd^q7$$!+**z?0;Fcdp$\"+z$p@D&Fi_pFc]p7%F j^q7$$!+FYT(*QFfep$\"+$\\fUE&Fi_pFc]p7%F`_q7$$!+%>6[u(Ffep$\"+)Q\"=t_F i_pFc]p7%Ff_q7$$!+'zD&p6Fb]p$\"+Ku:Fb]p$\" +)eh8G&Fi_pFc]p7%Fb`q7$$!+'oC&))>Fb]p$\"+%o'[!G&Fi_pFc]p7%Fh`q7$$!+v-. 7CFb]p$\"+a0KUPFb]p$\"+.4,T_Fi_pFc]p7% F`bq7$$!+$*H`4UFb]p$\"+]r9@_Fi_pFc]p7%Ffbq7$$!+F=k#p%Fb]p$\"+cNq'>&Fi_ pFc]p7%F\\cq7$$!+yQ@'>&Fb]p$\"+l$zs;&Fi_pFc]p7%Fbcq7$$!+'f,us&Fb]p$\"+ 9,UK^Fi_pFc]p7%Fhcq7$$!+qPX(H'Fb]p$\"+C)y;4&Fi_pFc]p7%F^dq7$$!+K*[R#pF b]p$\"+g0\"[/&Fi_pFc]p7%Fddq7$$!+m;1LwFb]p$\"+.)*G#*\\Fi_pFc]p7%Fjdq7$ $!+'ycvX)Fb]p$\"+zuRO\\Fi_pFc]p7%F`eq7$$!+L%*Fb]p$\"+`/S#)[Fi_pFc]p 7%Ffeq7$$!+2Ns\\5Fi_p$\"+\")*or$[Fi_pFc]p7%F\\fq7$$!+#3cD;\"Fi_p$\"+j) [V![Fi_pFc]p7%Fbfq7$$!+5L?u7Fi_p$\"+)ekDy%Fi_pFc]p7%Fhfq7$$!+^M[\"Q\"F i_p$\"+F&e$oZFi_pFc]p7%F^gq7$$!+r=_$[\"Fi_p$\"+x\">'eZFi_pFc]p7%Fdgq7$ $!+*Q'[!e\"Fi_p$\"+!*p9^ZFi_pFc]p7%Fjgq7$$!+7&))Gn\"Fi_p$\"+1E\\WZFi_p Fc]p7%F`hq7$$!+tuHhFi_p$\"+B#*[@ZFi_pFc]p7%Fbiq7$$!+FL]2?Fi_p $\"+29N6ZFi_pFc]p7%Fhiq7$$!+Ila%3#Fi_p$\"+6^d*p%Fi_pFc]p7%F^jq7$$!+%G \"ff@Fi_p$\"+2i-'o%Fi_pFc]p7%Fdjq7$$!+>+)GB#Fi_p$\"+&R61n%Fi_pFc]p7%Fj jq7$$!+OHi/BFi_p$\"+^hE`YFi_pFc]p7%F`[r7$$!+dZ+vBFi_p$\"+Oo%Rj%Fi_pFc] p7%Ff[r7$$!+\"G!>WCFi_p$\"+(eDEh%Fi_pFc]p7%F\\\\r7$$!+B*GB^#Fi_p$\"+?r G*e%Fi_pFc]p7%Fb\\r7$$!+o$e&zDFi_p$\"+/p#Rc%Fi_pFc]p7%Fh\\r7$$!+du+YEF i_p$\"+e.bOXFi_pFc]p7%F^]r7$$!+Q$)z6FFi_p$\"+FG<2XFi_pFc]p7%Fd]r7$$!+6 z/xFFi_p$\"+s,#eZ%Fi_pFc]p7%Fj]r7$$!+7&o=%GFi_p$\"+&zHDW%Fi_pFc]p7%F`^ r7$$!+)ynj!HFi_p$\"+_>N2WFi_pFc]p7%Ff^r7$$!+(pZ1(HFi_p$\"+J6NqVFi_pFc] p7%F\\_r7$$!+yD![.$Fi_p$\"+AsgJVFi_pFc]p7%Fb_r7$$!+Sj\"*)4$Fi_p$\"+jh@ \"H%Fi_pFc]p7%Fh_r7$$!+*))eI;$Fi_p$\"+!o*G\\UFi_pFc]p7%F^`r7$$!+B@GFKF i_p$\"+\"y`f?%Fi_pFc]p7%Fd`r7$$!+vfh\"H$Fi_p$\"+DdMhTFi_pFc]p7%Fj`r7$$ !+A^1cLFi_p$\"+R%4c6%Fi_pFc]p7%F`ar7$$!+Zqg?MFi_p$\"+.'*))oSFi_pFc]p7% Ffar7$$!+/?>&[$Fi_p$\"+_\\K@SFi_pFc]p7%F\\br7$$!+n^u\\NFi_p$\"+6;/tRFi _pFc]p7%Fbbr7$$!+c3<9OFi_p$\"+4t9CRFi_pFc]p7%Fhbr7$$!+T$e$yOFi_p$\"+Us suQFi_pFc]p7%F^cr7$$!+f%)=UPFi_p$\"+3?%[#QFi_pFc]p7%Fdcr7$$!+5-a0QFi_p $\"+ew_uPFi_pFc]p7%Fjcr7$$!++mHoQFi_p$\"+$e(zBPFi_pFc]p7%F`dr7$$!+p#\\ .$RFi_p$\"+#pXEn$Fi_pFc]p7%Ffdr7$$!+20%Fi_p$\"+c$y*oNFi_pFc]p7%Fber7$$!+A!*Q6TFi_p$\"+j!*Q;NFi_pF c]p7%Fher7$$!+B7!)pTFi_p$\"+%fPKY$Fi_pFc]p7%F^fr7$$!+Ho;FUFi_p$\"+SyZ4 MFi_pFc]p7%Fdfr7$$!+crX$G%Fi_p$\"+PZ1bLFi_pFc]p7%Fjfr7$$!+aVlQVFi_p$\" +^g&**H$Fi_pFc]p7%F`gr7$$!+#)*\\FR%Fi_p$\"+7P6WKFi_pFc]p7%Ffgr7$$!+@Ou XWFi_p$\"+*y/v=$Fi_pFc]p7%F\\hr7$$!+T:k(\\%Fi_p$\"+'>-,8$Fi_pFc]p7%Fbh r7$$!+VaX[XFi_p$\"+v^)=2$Fi_pFc]p7%Fhhr7$$!+s7?)f%Fi_p$\"+O'RG,$Fi_pFc ]p7%F^ir7$$!+!3)*ok%Fi_p$\"+e$eH&HFi_pFc]p7%Fdir7$$!+Hpc%p%Fi_p$\"+#3T A*GFi_pFc]p7%Fjir7$$!+3)H7u%Fi_p$\"+)\\%pIGFi_pFc]p7%F`jr7$$!+q&3py%Fi _p$\"+2@LoFFi_pFc]p7%Ffjr7$$!+yQiJ[Fi_p$\"+_R<0FFi_pFc]p7%F\\[s7$$!+#= %Rv[Fi_p$\"+)>Y7k#Fi_pFc]p7%Fb[s7$$!+eZB=\\Fi_p$\"+N0ewDFi_pFc]p7%Fh[s 7$$!+No:g\\Fi_p$\"+?M@6DFi_pFc]p7%F^\\s7$$!+fo;,]Fi_p$\"+W^=XCFi_pFc]p 7%Fd\\s7$$!+^fET]Fi_p$\"+`(Q&yBFi_pFc]p7%Fj\\s7$$!+\"e\\/3&Fi_p$\"+x)= 8J#Fi_pFc]p7%F`]s7$$!+.vq=^Fi_p$\"+$\\qNC#Fi_pFc]p7%Ff]s7$$!+_R-c^Fi_p $\"+BxLv@Fi_pFc]p7%F\\^s7$$!+0\"yB>&Fi_p$\"+-Fm1@Fi_pFc]p7%Fb^s7$$!+jZ uF_Fi_p$\"+)f%eP?Fi_pFc]p7%Fh^s7$$!+7_4i_Fi_p$\"+P)Q\"o>Fi_pFc]p7%F^_s 7$$!+1#)R&H&Fi_p$\"+VlN)*=Fi_pFc]p7%Fd_s7$$!+-5iF`Fi_p$\"+TUEG=Fi_pFc] p7%Fj_s7$$!+&QI(e`Fi_p$\"+yP)yv\"Fi_pFc]p7%F``s7$$!+JOp)Q&Fi_p$\"+mCB( o\"Fi_pFc]p7%Ff`s7$$!+r$zuT&Fi_p$\"+1MK;;Fi_pFc]p7%F\\as7$$!+:$e]W&Fi_ p$\"+Tf;X:Fi_pFc]p7%Fbas7$$!+VQSraFi_p$\"+:iwt9Fi_pFc]p7%Fhas7$$!+TC\\ '\\&Fi_p$\"+!yF@S\"Fi_pFc]p7%F^bs7$$!++SI?bFi_p$\"+RADI8Fi_pFc]p7%Fdbs 7$$!+\"*>#Ga&Fi_p$\"+&))R\"e7Fi_pFc]p7%Fjbs7$$!+cFi_p$\"+j$eJn*Fb]pFc]p7%Fbds7$$!+&ezcj&Fi _p$\"+)R!>S*)Fb]pFc]p7%Fhds7$$!+E^G]cFi_p$\"+3]\"\\?)Fb]pFc]p7%F^es7$$ !+=adjcFi_p$\"+4IRnuFb]pFc]p7%Fdes7$$!+kxbvcFi_p$\"+A)3xs'Fb]pFc]p7%Fj es7$$!+54C'o&Fi_p$\"+UC(f)fFb]pFc]p7%F`fs7$$!+kXj&p&Fi_p$\"+c1KU_Fb]pF c]p7%Fffs7$$!+(*)[Pq&Fi_p$\"+?u\"p\\%Fb]pFc]p7%F\\gs7$$!+MSf5dFi_p$\"+ aL&*\\PFb]pFc]p7%Fbgs7$$!+k&zhr&Fi_p$\"+&HW;+$Fb]pFc]p7%Fhgs7$$!+pS^?d Fi_p$\"+D*GAD#Fb]pFc]p7%F^hs7$$!+2ZgBdFi_p$\"+)[l>]\"Fb]pFc]p7%Fdhs7$$ !+goXDdFi_p$\"+>qF6vFfepFc]p7%Fjhs7$$!+nQ2EdFi_pF(Fc]p7%F`is7$F[is$!+> qF6vFfepFc]p7%Fdis7$Fehs$!+)[l>]\"Fb]pFc]p7%Fhis7$F_hs$!+D*GAD#Fb]pFc] p7%F\\js7$Figs$!+&HW;+$Fb]pFc]p7%F`js7$Fcgs$!+aL&*\\PFb]pFc]p7%Fdjs7$F ]gs$!+?u\"p\\%Fb]pFc]p7%Fhjs7$Fgfs$!+c1KU_Fb]pFc]p7%F\\[t7$Fafs$!+UC(f )fFb]pFc]p7%F`[t7$F[fs$!+A)3xs'Fb]pFc]p7%Fd[t7$Fees$!+4IRnuFb]pFc]p7%F h[t7$F_es$!+3]\"\\?)Fb]pFc]p7%F\\\\t7$Fids$!+)R!>S*)Fb]pFc]p7%F`\\t7$F cds$!+j$eJn*Fb]pFc]p7%Fd\\t7$F]ds$!+-zPS5Fi_pFc]p7%Fh\\t7$Fgcs$!+6I?86 Fi_pFc]p7%F\\]t7$Facs$!+8.z&=\"Fi_pFc]p7%F`]t7$F[cs$!+&))R\"e7Fi_pFc]p 7%Fd]t7$Febs$!+RADI8Fi_pFc]p7%Fh]t7$F_bs$!+!yF@S\"Fi_pFc]p7%F\\^t7$Fia s$!+:iwt9Fi_pFc]p7%F`^t7$Fcas$!+Tf;X:Fi_pFc]p7%Fd^t7$F]as$!+1MK;;Fi_pF c]p7%Fh^t7$Fg`s$!+mCB(o\"Fi_pFc]p7%F\\_t7$Fa`s$!+yP)yv\"Fi_pFc]p7%F`_t 7$F[`s$!+TUEG=Fi_pFc]p7%Fd_t7$Fe_s$!+VlN)*=Fi_pFc]p7%Fh_t7$F__s$!+P)Q \"o>Fi_pFc]p7%F\\`t7$Fi^s$!+)f%eP?Fi_pFc]p7%F``t7$Fc^s$!+-Fm1@Fi_pFc]p 7%Fd`t7$F]^s$!+BxLv@Fi_pFc]p7%Fh`t7$Fg]s$!+$\\qNC#Fi_pFc]p7%F\\at7$Fa] s$!+x)=8J#Fi_pFc]p7%F`at7$F[]s$!+`(Q&yBFi_pFc]p7%Fdat7$Fe\\s$!+W^=XCFi _pFc]p7%Fhat7$F_\\s$!+?M@6DFi_pFc]p7%F\\bt7$Fi[s$!+N0ewDFi_pFc]p7%F`bt 7$Fc[s$!+)>Y7k#Fi_pFc]p7%Fdbt7$F][s$!+_R<0FFi_pFc]p7%Fhbt7$Fgjr$!+2@Lo FFi_pFc]p7%F\\ct7$Fajr$!+)\\%pIGFi_pFc]p7%F`ct7$F[jr$!+#3TA*GFi_pFc]p7 %Fdct7$Feir$!+e$eH&HFi_pFc]p7%Fhct7$F_ir$!+O'RG,$Fi_pFc]p7%F\\dt7$Fihr $!+v^)=2$Fi_pFc]p7%F`dt7$Fchr$!+'>-,8$Fi_pFc]p7%Fddt7$F]hr$!+*y/v=$Fi_ pFc]p7%Fhdt7$Fggr$!+7P6WKFi_pFc]p7%F\\et7$Fagr$!+^g&**H$Fi_pFc]p7%F`et 7$F[gr$!+PZ1bLFi_pFc]p7%Fdet7$Fefr$!+SyZ4MFi_pFc]p7%Fhet7$F_fr$!+%fPKY $Fi_pFc]p7%F\\ft7$Fier$!+j!*Q;NFi_pFc]p7%F`ft7$Fcer$!+c$y*oNFi_pFc]p7% Fdft7$F]er$!+G.0@OFi_pFc]p7%Fhft7$Fgdr$!+#pXEn$Fi_pFc]p7%F\\gt7$Fadr$! +$e(zBPFi_pFc]p7%F`gt7$F[dr$!+ew_uPFi_pFc]p7%Fdgt7$Fecr$!+3?%[#QFi_pFc ]p7%Fhgt7$F_cr$!+UssuQFi_pFc]p7%F\\ht7$Fibr$!+4t9CRFi_pFc]p7%F`ht7$Fcb r$!+6;/tRFi_pFc]p7%Fdht7$F]br$!+_\\K@SFi_pFc]p7%Fhht7$Fgar$!+.'*))oSFi _pFc]p7%F\\it7$Faar$!+R%4c6%Fi_pFc]p7%F`it7$F[ar$!+DdMhTFi_pFc]p7%Fdit 7$Fe`r$!+\"y`f?%Fi_pFc]p7%Fhit7$F_`r$!+!o*G\\UFi_pFc]p7%F\\jt7$Fi_r$!+ jh@\"H%Fi_pFc]p7%F`jt7$Fc_r$!+AsgJVFi_pFc]p7%Fdjt7$F]_r$!+J6NqVFi_pFc] p7%Fhjt7$Fg^r$!+_>N2WFi_pFc]p7%F\\[u7$Fa^r$!+&zHDW%Fi_pFc]p7%F`[u7$F[^ r$!+s,#eZ%Fi_pFc]p7%Fd[u7$Fe]r$!+FG<2XFi_pFc]p7%Fh[u7$F_]r$!+e.bOXFi_p Fc]p7%F\\\\u7$Fi\\r$!+/p#Rc%Fi_pFc]p7%F`\\u7$Fc\\r$!+?rG*e%Fi_pFc]p7%F d\\u7$F]\\r$!+(eDEh%Fi_pFc]p7%Fh\\u7$Fg[r$!+Oo%Rj%Fi_pFc]p7%F\\]u7$Fa[ r$!+^hE`YFi_pFc]p7%F`]u7$F[[r$!+&R61n%Fi_pFc]p7%Fd]u7$Fejq$!+2i-'o%Fi_ pFc]p7%Fh]u7$F_jq$!+6^d*p%Fi_pFc]p7%F\\^u7$Fiiq$!+29N6ZFi_pFc]p7%F`^u7 $Fciq$!+B#*[@ZFi_pFc]p7%Fd^u7$F]iq$!+O?=IZFi_pFc]p7%Fh^u7$Fghq$!+W9rPZ Fi_pFc]p7%F\\_u7$Fahq$!+1E\\WZFi_pFc]p7%F`_u7$F[hq$!+!*p9^ZFi_pFc]p7%F d_u7$Fegq$!+x\">'eZFi_pFc]p7%Fh_u7$F_gq$!+F&e$oZFi_pFc]p7%F\\`u7$Fifq$ !+)ekDy%Fi_pFc]p7%F``u7$Fcfq$!+j)[V![Fi_pFc]p7%Fd`u7$F]fq$!+\")*or$[Fi _pFc]p7%Fh`u7$Fgeq$!+`/S#)[Fi_pFc]p7%F\\au7$Faeq$!+zuRO\\Fi_pFc]p7%F`a u7$F[eq$!+.)*G#*\\Fi_pFc]p7%Fdau7$Fedq$!+g0\"[/&Fi_pFc]p7%Fhau7$F_dq$! +C)y;4&Fi_pFc]p7%F\\bu7$Ficq$!+9,UK^Fi_pFc]p7%F`bu7$Fccq$!+l$zs;&Fi_pF c]p7%Fdbu7$F]cq$!+cNq'>&Fi_pFc]p7%Fhbu7$Fgbq$!+]r9@_Fi_pFc]p7%F\\cu7$F abq$!+.4,T_Fi_pFc]p7%F`cu7$F[bq$!+\\'HmD&Fi_pFc]p7%Fdcu7$Feaq$!+qjFo_F i_pFc]p7%Fhcu7$F_aq$!+a0&Fi_pFc]p7%Fdfu7$Fc]q$!+M(>A<&Fi_pFc]p7 %Fhfu7$F]]q$!+'[FU9&Fi_pFc]p7%F\\gu7$Fg\\q$!+cH%H6&Fi_pFc]p7%F`gu7$Fa \\q$!+7&)Hy]Fi_pFc]p7%Fdgu7$F[\\q$!+%R8-/&Fi_pFc]p7%Fhgu7$Fe[q$!+CCf)* \\Fi_pFc]p7%F\\hu7$F_[q$!+$HBL&\\Fi_pFc]p7%F`hu7$Fijp$!+r9F/\\Fi_pFc]p 7%Fdhu7$Fcjp$!+%=r7&[Fi_pFc]p7%Fhhu7$F]jp$!+:*4Tz%Fi_pFc]p7%F\\iu7$Fgi p$!+=B]KZFi_pFc]p7%F`iu7$Faip$!+rn/mYFi_pFc]p7%Fdiu7$F[ip$!+!f`Tf%Fi_p Fc]p7%Fhiu7$Fehp$!+?z#f^%Fi_pFc]p7%F\\ju7$F_hp$!+Qk)*HWFi_pFc]p7%F`ju7 $Figp$!+d\"4UL%Fi_pFc]p7%Fdju7$Fcgp$!+ZTdDUFi_pFc]p7%Fhju7$F]gp$!+1(>2 5%Fi_pFc]p7%F\\[v7$Fgfp$!+f%p$eRFi_pFc]p7%F`[v7$Fafp$!+m%pA!QFi_pFc]p7 %Fd[v7$F[fp$!+$p!RROFi_pFc]p7%Fh[v7$Fdep$!+;>5vMFi_pFc]p7%F\\\\v7$F^ep $!+U!*y6LFi_pFc]p7%F`\\v7$Fhdp$!+ep.]JFi_pFc]p7%Fd\\v7$Fadp$!+k'=(*)HF i_pFc]p7%Fh\\v7$F[dp$!+6P^IGFi_pFc]p7%F\\]v7$Fecp$!+aA5sEFi_pFc]p7%F`] v7$F_cp$!+#)RA9DFi_pFc]p7%Fd]v7$Fhbp$!+^locBFi_pFc]p7%Fh]v7$Fbbp$!+$zc $*>#Fi_pFc]p7%F\\^v7$F\\bp$!+iu9U?Fi_pFc]p7%F`^v7$Ffap$!+eV+&)=Fi_pFc] p7%Fd^v7$F_ap$!+>c*ys\"Fi_pFc]p7%Fh^v7$$!+**)[9Z(F^`p$!+$f.3d\"Fi_pFc] p7%F\\_v7$Fc`p$!+=\">PT\"Fi_pFc]p7%Fb_v7$F\\`p$!+wxjc7Fi_pFc]p7%Ff_v7$ Fd_p$!+5wb*4\"Fi_pFc]p7%Fj_v7$$!+7nk'e*FN$!+Q$yZU*Fb]pFc]p7%F^`v7$$!+B 6:I?FN$!+&p\")R&yFb]pFc]p7%Fd`v7$$!+gFZ/JF-$!+9`=$G'Fb]pFc]p7%Fj`v7$$! +erGcGFfr$!+\")*)Q7ZFb]pFc]p7%F`av7$F($!+aEfTJFb]pFc]p7%Ffav7$F($!+Fjz q:Fb]pFc]p7%FjavF'Fc]p-Fe\\p6&Fg\\p$\"#l!\"#$\"\"$Fj\\p$\"#DFbbv-%&STY LEG6#%,PATCHNOGRIDG-F$6$7]_lF'F*F.7$F(F57$$\"3y******eekSZF4$\"3_***** fJ&=$G'F-7$$\"3-+++bK\"=<%F:$\"3e*****Rp\")R&yF-7$$\"30+++m(\\gS#F@$\" 3X*****4ByZU*F-7$$\"31+++b/&3.\"FK$\"31+++?vb*4\"FN7$$\"3-+++sJeKNFK$ \"35+++2tjc7FN7$$\"3#*******f9R95FR$\"3********Rsr89FN7$$\"3:+++jY*e^# FR$\"3/+++Otzq:FN7$$\"3k*****>JBh\\&FR$\"34+++2v(ys\"FN7$$\"3)*******) G&yp5F\\o$\"31+++%[d\\)=FN7$$\"39+++;e(H'=F\\o$\"36+++#)o.U?FN7$$\"3++ ++<\\#)))GF\\o$\"3/+++^c6*>#FN7$$\"3(******Rr>5\"RF\\o$\"33+++(Q&>cBFN 7$$\"3-+++U!QBP%F\\o$\"3>+++)>#G8DFN7$$\"3#)*****fyUHI$F\\o$\"3')***** H3#RqEFN7$$!3)******f_a'fiFR$\"3(******z)*fv#GFN7$$!3b*****zS/gq)F\\o$ \"37+++JQ&[)HFN7$$!3*)*****HRKN6#Fap$\"37+++qZRUJFN7$$!3G+++(z0+[$Fap$ \"30+++\"*\\Q+LFN7$$!3#)******3g)f!RFap$\"3))******HT9fMFN7$$!3c***** \\_#f5!)F\\o$\"3&)*****>ya\">OFN7$$\"3)******H6,(Q6Ffq$\"3))*****R986y $FN7$$\"3e*****4dDNR%Ffq$\"3=+++n(ff%RFN7$$\"31+++rRA$>\"Ffr$\"3z***** \\ZtZ6%FN7$$\"3<+++-_;8GFfr$\"3g*****f,V!)G%FN7$$\"37+++376OgFfr$\"3u* ****>dnFY%FN7$$\"31+++\\$[R9\"F-$\"3v*****z(H(oi%FN7$$\"34+++=mu:=F-$ \"3))*****>q!HaCF-$\"37+++-*3i([FN7$$\"3))*****p \\`S)HF-$\"3Q+++o:2p\\FN7$$\"3'******H\\)R)R$F-$\"3!)*****\\(*4,0&FN7$ $\"3++++Q:!*4PF-$\"3++++%)f/B^FN7$$\"3********e&QI$RF-$\"33+++TV!**=&F N7$$\"3A+++4wS!3%F-$\"3w*****HlB=D&FN7$$\"3++++fmRiTF-$\"3B+++_UZ4`FN7 $$\"3')*****\\&=a(=%F-$\"3U+++^'pKO&FN7$$\"3%******4mHH;%F-$\"3A+++NsZ 8aFN7$$\"3-+++>%=X4%F-$\"3'******\\Zy-Y&FN7$$\"3w*****fJwt)RF-$\"3H+++ >F!Q]&FN7$$\"3A+++[_&e%QF-$\"3w*****f\\YTa&FN7$$\"3C+++A%>Pn$F-$\"3')* ******e%Q\"e&FN7$$\"3'******\\lUUZ$F-$\"3-+++-jd:cFN7$$\"3?+++4RG]KF-$ \"3'*******\\)onk&FN7$$\"3&)*****pKZV+$F-$\"3m*****>.**\\n&FN7$$\"3/++ +V3jQFF-$\"3L+++QQH+dFN7$$\"3'******p&f1bCF-$\"3i*****>PqEs&FN7$$\"3%* *****4N]`:#F-$\"3@+++A`8UdFN7$$\"3++++jV(4%=F-$\"3%******z2%oedFN7$$\" 3,+++()=C8:F-$\"3i*****p]*HsdFN7$$\"3++++#*eGt6F-$\"3?+++$Q]Hy&FN7$$\" 3w*****>Z$y?#)Ffr$\"3w*****\\O*e!z&FN7$$\"3>+++J*QVg%Ffr$\"33+++9/:&z& FN7$$\"3M*****4L\"f+*)Ffq$\"3!)*****pVXlz&FN7$$!3!*******eMF/y&FN7$$!3#******p!fM)[\"F-$\"3'******f'*Rww&FN7$$!3-++ +\\[_1>F-$\"3m*****\\o/2v&FN7$$!3()*****\\`qiL#F-$\"3v*****HGC#HdFN7$$ !3!******>.Z,y#F-$\"3m******)3&p-dFN7$$!3C+++O2_UKF-$\"31+++o9YqcFN7$$ !3*)*****pZ-4t$F-$\"3z*****H?r;j&FN7$$!3C++++nfeUF-$\"3-+++S\"[_e&FN7$ $!3#)*****\\Pe+&[F-$\"3@+++3))**HbFN7$$!3c+++q$o3b&F-$\"3y******>*Q`Y& FN7$$!3G+++s:QG&FN7$$!3/+++svC85FN$\"3q*****>J G1E&FN7$$!3'*******yK5L6FN$\"37+++C#*f]_FN7$$!3#******4s\"3W7FN$\"3)** ****Rv)3Z_FN7$$!31+++b&ztM\"FN$\"3!******>+QjC&FN7$$!3)*******GdNW9FN$ \"3A+++\">#HY_FN7$$!3!******fHfh`\"FN$\"3i*****\\y!zX_FN7$$!3&******p5 -Pi\"FN$\"3!*******f.:W_FN7$$!32++++0q2+@()y\"FN$\"3&)*****znOfB&FN7$$!3%******pQFN$\"39+++*)\\#)>_FN7$$!3;+++=)\\!=?FN$\"3$******HhW& 3_FN7$$!37+++fu%44#FN$\"3k*****R?S]>&FN7$$!3#******4ZrC;#FN$\"3))***** R_y#z^FN7$$!3#******>49GB#FN$\"3'*******fyBh^FN7$$!36+++^p9-BFN$\"3\") *****z:049&FN7$$!3'*******4miqBFN$\"3M+++dXF=^FN7$$!39+++f*)RQCFN$\"3e *****z(pM$4&FN7$$!3')*****\\i-c]#FN$\"3)******\\bIh1&FN7$$!3')*****R^r Bd#FN$\"3%******pvTm.&FN7$$!39+++`k$)QEFN$\"3u*****H72\\+&FN7$$!3)**** **z)e70FFN$\"3w*****HYl4(\\FN7$$!3A+++#[l8x#FN$\"3s*****>Qq[$\\FN7$$!3 1+++sknPGFN$\"33+++iGp'*[FN7$$!3@+++jE^Bl&[FN7$$!3/++ +Zg&4(HFN$\"3$******>*RZ9[FN7$$!3))*****zC6\"QIFN$\"3Y+++%*pnqZFN7$$!3 %)*****zG*p0JFN$\"3*)*****z<%GDZFN7$$!3/+++3:vtJFN$\"3q*****fkh%yYFN7$ $!3\"******HskAC$FN$\"38+++9HQIYFN7$$!30+++^()>6LFN$\"39+++s)>7e%FN7$$ !3;+++prZ!Q$FN$\"3U+++FG8JXFN7$$!35+++^:**\\MFN$\"3i*****H/h-[%FN7$$!3 ?+++X(3'>NFN$\"33+++V_rGWFN7$$!3%)******p+=*e$FN$\"37+++.JdwVFN7$$!3?+ ++G5beOFN$\"3m*****z6yQK%FN7$$!3\"******H@qvs$FN$\"3e*****z`T1F%FN7$$! 37+++kk4'z$FN$\"3z*****ReYo@%FN7$$!3'*******fQ+kQFN$\"3K+++dRXiTFN7$$! 3-+++`X=JRFN$\"3E+++IxS2TFN7$$!3!******4!)\\v*RFN$\"35+++Z4k^SFN7$$!3' )******G'HI1%FN$\"37+++403&*RFN7$$!31+++1;dFTFN$\"3&)*****R>^w$RFN7$$! 3y******=\"R6>%FN$\"3#*******z(y#zQFN7$$!3%******4U4PD%FN$\"38+++1C*)> QFN7$$!3t*****>nr_J%FN$\"3')******)GE%fPFN7$$!3y*****>/DeP%FN$\"3=+++^ 4#yp$FN7$$!3H+++^pPNWFN$\"3&******45C]j$FN7$$!3))*****HUTR\\%FN$\"3%** ****RD\"*4d$FN7$$!3w*****zaP:b%FN$\"3:+++'='o0NFN7$$!3#******p*z=3YFN$ \"3\")*****>D\"3RMFN7$$!31+++uv\"Rm%FN$\"31+++*ed6P$FN7$$!3')*****zu^( =ZFN$\"3$******>?0>I$FN7$$!3%)*****><:Fx%FN$\"3<+++)*GKJKFN7$$!3%)**** *4;Ie#[FN$\"3y*****\\/=%fJFN7$$!33+++\"=:\"y[FN$\"31++++h?'3$FN7$$!3\" ******RP$eH\\FN$\"3')*****p))4<,$FN7$$!3=+++\">T-)\\FN$\"3y*****\\fef$ HFN7$$!3T+++?s3I]FN$\"3=+++4l)*eGFN7$$!3o*****RG6\"z]FN$\"3#)*****p]J3 y#FN7$$!3s******))QHF^FN$\"3)******>RL:q#FN7$$!3I+++\"31Y<&FN$\"3@+++m @8@EFN7$$!3>+++;(45A&FN$\"3A+++TjmRDFN7$$!3'******z8ekE&FN$\"3!******4 VrrX#FN7$$!3C+++*>(*3J&FN$\"3#)*****pqyOP#FN7$$!3++++PlEa`FN$\"3!)**** *fJ9#*G#FN7$$!37+++:5]'R&FN$\"3<+++#z)z.AFN7$$!3z*****4?KvV&FN$\"3:+++ #)pW<@FN7$$!3))*****f\")*GxaFN$\"3()*****4Oo,.#FN7$$!3%*******\\Iq:bFN $\"3++++zw'>%>FN7$$!3O+++D6 qTr&FN$\"33+++#*>M$R\"FN7$$!3A+++D7MTdFN$\"3!******RXe')H\"FN7$$!3K+++ \"*GomdFN$\"30+++v;1.7FN7$$!37+++@3:!z&FN$\"32+++A#pl5\"FN7$$!3%)***** f![q6eFN$\"3\"******zY1#45FN7$$!3H+++-(38$eFN$\"3k*****R@v+6*F-7$$!3#) *****>QI*[eFN$\"3!)*****Hdf,7)F-7$$!3G+++p;akeFN$\"3v*****f.`G7(F-7$$! 3O+++L#=\"yeFN$\"3T+++`Cz=hF-7$$!3s******f%R'*)eFN$\"3C+++R]r3^F-7$$!3 o******3$)3**eFN$\"35+++HOX$4%F-7$$!3d******[6X1fFN$\"3%******\\CGR2$F -7$$!38+++)e<<\"fFN$\"3#*******GS8^?F-7$$!3;+++L.)[\"fFN$\"3$*******He 7E5F-7$$!3s******e]$f\"fFNF(7$Fgcy$!3$*******He7E5F-7$Fbcy$!3#*******G S8^?F-7$F]cy$!3%******\\CGR2$F-7$Fhby$!35+++HOX$4%F-7$Fcby$!3C+++R]r3^ F-7$F^by$!3T+++`Cz=hF-7$Fiay$!3v*****f.`G7(F-7$Fday$!3!)*****Hdf,7)F-7 $F_ay$!3k*****R@v+6*F-7$Fj`y$!3\"******zY1#45FN7$Fe`y$!32+++A#pl5\"FN7 $F``y$!30+++v;1.7FN7$F[`y$!3!******RXe')H\"FN7$Ff_y$!33+++#*>M$R\"FN7$ Fa_y$!31+++Q95([\"FN7$F\\_y$!3/+++T9$*z:FN7$Fg^y$!3%******H%3$=n\"FN7$ Fb^y$!3!******RI,Gw\"FN7$F]^y$!3-+++$*e%G&=FN7$Fh]y$!3++++zw'>%>FN7$Fc ]y$!3()*****4Oo,.#FN7$F^]y$!3:+++#)pW<@FN7$Fi\\y$!3<+++#z)z.AFN7$Fd\\y $!3!)*****fJ9#*G#FN7$F_\\y$!3#)*****pqyOP#FN7$Fj[y$!3!******4VrrX#FN7$ Fe[y$!3A+++TjmRDFN7$F`[y$!3@+++m@8@EFN7$F[[y$!3)******>RL:q#FN7$Ffjx$! 3#)*****p]J3y#FN7$Fajx$!3=+++4l)*eGFN7$F\\jx$!3y*****\\fef$HFN7$Fgix$! 3')*****p))4<,$FN7$Fbix$!31++++h?'3$FN7$F]ix$!3y*****\\/=%fJFN7$Fhhx$! 3<+++)*GKJKFN7$Fchx$!3$******>?0>I$FN7$F^hx$!31+++*ed6P$FN7$Figx$!3\") *****>D\"3RMFN7$Fdgx$!3:+++'='o0NFN7$F_gx$!3%******RD\"*4d$FN7$Fjfx$!3 &******45C]j$FN7$Fefx$!3=+++^4#yp$FN7$F`fx$!3')******)GE%fPFN7$F[fx$!3 8+++1C*)>QFN7$Ffex$!3#*******z(y#zQFN7$Faex$!3&)*****R>^w$RFN7$F\\ex$! 37+++403&*RFN7$Fgdx$!35+++Z4k^SFN7$Fbdx$!3E+++IxS2TFN7$F]dx$!3K+++dRXi TFN7$Fhcx$!3z*****ReYo@%FN7$Fccx$!3e*****z`T1F%FN7$F^cx$!3m*****z6yQK% FN7$Fibx$!37+++.JdwVFN7$Fdbx$!33+++V_rGWFN7$F_bx$!3i*****H/h-[%FN7$Fja x$!3U+++FG8JXFN7$Feax$!39+++s)>7e%FN7$F`ax$!38+++9HQIYFN7$F[ax$!3q**** *fkh%yYFN7$Ff`x$!3*)*****z<%GDZFN7$Fa`x$!3Y+++%*pnqZFN7$F\\`x$!3$***** *>*RZ9[FN7$Fg_x$!3))*****>^Bl&[FN7$Fb_x$!33+++iGp'*[FN7$F]_x$!3s*****> Qq[$\\FN7$Fh^x$!3w*****HYl4(\\FN7$Fc^x$!3u*****H72\\+&FN7$F^^x$!3%**** **pvTm.&FN7$Fi]x$!3)******\\bIh1&FN7$Fd]x$!3e*****z(pM$4&FN7$F_]x$!3M+ ++dXF=^FN7$Fj\\x$!3\")*****z:049&FN7$Fe\\x$!3'*******fyBh^FN7$F`\\x$!3 ))*****R_y#z^FN7$F[\\x$!3k*****R?S]>&FN7$Ff[x$!3$******HhW&3_FN7$Fa[x$ !39+++*)\\#)>_FN7$F\\[x$!3*)*****f*4$*G_FN7$Fgjw$!3&)*****znOfB&FN7$Fb jw$!3S+++.L&4C&FN7$F]jw$!3!*******f.:W_FN7$Fhiw$!3i*****\\y!zX_FN7$Fci w$!3A+++\">#HY_FN7$F^iw$!3!******>+QjC&FN7$Fihw$!3)******Rv)3Z_FN7$Fdh w$!37+++C#*f]_FN7$F_hw$!3q*****>JG1E&FN7$Fjgw$!3%******>s:QG&FN7$Fegw$ !3C+++UobG`FN7$F`gw$!3;+++(\\*=%R&FN7$F[gw$!3y******>*Q`Y&FN7$Fffw$!3@ +++3))**HbFN7$Fafw$!3-+++S\"[_e&FN7$F\\fw$!3z*****H?r;j&FN7$Fgew$!31++ +o9YqcFN7$Fbew$!3m******)3&p-dFN7$F]ew$!3v*****HGC#HdFN7$Fhdw$!3m***** \\o/2v&FN7$Fcdw$!3'******f'*Rww&FN7$F^dw$!3G+++q&>/y&FN7$$!3J+++#H1\\ \"oFfr$!3y******HpM*y&FN7$Fdcw$!31+++*))fYz&FN7$F_cw$!3!)*****pVXlz&FN 7$Fjbw$!33+++9/:&z&FN7$Febw$!3w*****\\O*e!z&FN7$F`bw$!3?+++$Q]Hy&FN7$F [bw$!3i*****p]*HsdFN7$Ffaw$!3%******z2%oedFN7$Faaw$!3@+++A`8UdFN7$F\\a w$!3i*****>PqEs&FN7$Fg`w$!3L+++QQH+dFN7$Fb`w$!3m*****>.**\\n&FN7$F]`w$ !3'*******\\)onk&FN7$Fh_w$!3-+++-jd:cFN7$Fc_w$!3')*******e%Q\"e&FN7$F^ _w$!3w*****f\\YTa&FN7$Fi^w$!3H+++>F!Q]&FN7$Fd^w$!3'******\\Zy-Y&FN7$F_ ^w$!3A+++NsZ8aFN7$Fj]w$!3U+++^'pKO&FN7$Fe]w$!3B+++_UZ4`FN7$F`]w$!3w*** **HlB=D&FN7$F[]w$!33+++TV!**=&FN7$Ff\\w$!3++++%)f/B^FN7$Fa\\w$!3!)**** *\\(*4,0&FN7$F\\\\w$!3Q+++o:2p\\FN7$Fg[w$!37+++-*3i([FN7$Fb[w$!3))**** *>dnFY%FN7$Fcjv$!3g*** **f,V!)G%FN7$F^jv$!3z*****\\ZtZ6%FN7$Fiiv$!3=+++n(ff%RFN7$Fdiv$!3))*** **R986y$FN7$F_iv$!3&)*****>ya\">OFN7$Fjhv$!3))******HT9fMFN7$Fehv$!30+ ++\"*\\Q+LFN7$$!36+++#RKN6#Fap$!37+++qZRUJFN7$F[hv$!37+++JQ&[)HFN7$$!3 m*****\\_a'fiFR$!3(******z)*fv#GFN7$Fagv$!3')*****H3#RqEFN7$F\\gv$!3>+ ++)>#G8DFN7$Fgfv$!33+++(Q&>cBFN7$Fbfv$!3/+++^c6*>#FN7$F]fv$!36+++#)o.U ?FN7$Fhev$!31+++%[d\\)=FN7$Fcev$!34+++2v(ys\"FN7$$\"3-+++cY*e^#FR$!3/+ ++Otzq:FN7$Fidv$!3********Rsr89FN7$Fddv$!35+++2tjc7FN7$F_dv$!31+++?vb* 4\"FN7$$\"3()*****pw\\gS#F@$!3X*****4ByZU*F-7$$\"3++++mK\"=<%F:$!3e*** **Rp\")R&yF-7$$\"3w*****z)ekSZF4$!3_*****fJ&=$G'F-7$F(F\\\\pF^\\pFa\\p F'-Fe\\p6&Fg\\p$\"#NFbbv$\"\"\"Fj\\pF(-F\\]p6^_l7%F'F_]p7$$!#YFj\\pF(7 %F_]pFg]pFjb[l7%Fg]p7$F(F^^pFjb[l7%F_c[l7$$\"+fekSZFfr$\"+;`=$G'Fb]pFj b[l7%Fac[l7$$\"+bK\"=<%F-$\"+%p\")R&yFb]pFjb[l7%Fgc[l7$$\"+m(\\gS#FN$ \"+J#yZU*Fb]pFjb[l7%F]d[l7$$\"+b/&3.\"Ff_p$\"+?vb*4\"Fi_pFjb[l7%Fcd[l7 $$\"+sJeKNFf_p$\"+2tjc7Fi_pFjb[l7%Fid[l7$$\"+g9R95F^`p$\"+Ssr89Fi_pFjb [l7%F_e[l7$$\"+jY*e^#F^`p$\"+Otzq:Fi_pFjb[l7%Fee[l7$$\"+7L7'\\&F^`p$\" +2v(ys\"Fi_pFjb[l7%F[f[l7$$\"+*G&yp5Faap$\"+%[d\\)=Fi_pFjb[l7%Faf[l7$$ \"+;e(H'=Faap$\"+#)o.U?Fi_pFjb[l7%Fgf[l7$$\"+<\\#)))GFaap$\"+^c6*>#Fi_ pFjb[l7%F]g[l7$$\"+9(>5\"RFaap$\"+(Q&>cBFi_pFjb[l7%Fcg[l7$$\"+U!QBP%Fa ap$\"+)>#G8DFi_pFjb[l7%Fig[l7$$\"+'yUHI$Faap$\"+$3#RqEFi_pFjb[l7%F_h[l 7$$!+EXlfiF^`p$\"+))*fv#GFi_pFjb[l7%Feh[l7$$!+3W+1()Faap$\"+JQ&[)HFi_p Fjb[l7%F[i[l7$$!+$RKN6#Fjbp$\"+qZRUJFi_pFjb[l7%Fai[l7$$!+(z0+[$Fjbp$\" +\"*\\Q+LFi_pFjb[l7%Fgi[l7$$!+4g)f!RFjbp$\"+IT9fMFi_pFjb[l7%F]j[l7$$!+ DDf5!)Faap$\"+#ya\">OFi_pFjb[l7%Fcj[l7$$\"+86qQ6Fcdp$\"+WJ6\"y$Fi_pFjb [l7%Fij[l7$$\"+rb_$R%Fcdp$\"+n(ff%RFi_pFjb[l7%F_[\\l7$$\"+rRA$>\"Ffep$ \"+vMx9TFi_pFjb[l7%Fe[\\l7$$\"+-_;8GFfep$\"+;I/)G%Fi_pFjb[l7%F[\\\\l7$ $\"+376OgFfep$\"+svwiWFi_pFjb[l7%Fa\\\\l7$$\"+\\$[R9\"Fb]p$\"+yH(oi%Fi _pFjb[l7%Fg\\\\l7$$\"+=mu:=Fb]p$\"+s^#\\w%Fi_pFjb[l7%F]]\\l7$$\"+-2HaC Fb]p$\"+-*3i([Fi_pFjb[l7%Fc]\\l7$$\"+(\\`S)HFb]p$\"+o:2p\\Fi_pFjb[l7%F i]\\l7$$\"+$\\)R)R$Fb]p$\"+v*4,0&Fi_pFjb[l7%F_^\\l7$$\"+Q:!*4PFb]p$\"+ %)f/B^Fi_pFjb[l7%Fe^\\l7$$\"+f&QI$RFb]p$\"+TV!**=&Fi_pFjb[l7%F[_\\l7$$ \"+4wS!3%Fb]p$\"+`O#=D&Fi_pFjb[l7%Fa_\\l7$$\"+fmRiTFb]p$\"+_UZ4`Fi_pFj b[l7%Fg_\\l7$$\"+b=a(=%Fb]p$\"+^'pKO&Fi_pFjb[l7%F]`\\l7$$\"+h'HH;%Fb]p $\"+NsZ8aFi_pFjb[l7%Fc`\\l7$$\"+>%=X4%Fb]p$\"+v%y-Y&Fi_pFjb[l7%Fi`\\l7 $$\"+;jP()RFb]p$\"+>F!Q]&Fi_pFjb[l7%F_a\\l7$$\"+[_&e%QFb]p$\"+'\\YTa&F i_pFjb[l7%Fea\\l7$$\"+A%>Pn$Fb]p$\"+!f%Q\"e&Fi_pFjb[l7%F[b\\l7$$\"+bEC uMFb]p$\"+-jd:cFi_pFjb[l7%Fab\\l7$$\"+4RG]KFb]p$\"+])onk&Fi_pFjb[l7%Fg b\\l7$$\"+FtM/IFb]p$\"+K!**\\n&Fi_pFjb[l7%F]c\\l7$$\"+V3jQFFb]p$\"+QQH +dFi_pFjb[l7%Fcc\\l7$$\"+df1bCFb]p$\"+s.nAdFi_pFjb[l7%Fic\\l7$$\"+^.Nb @Fb]p$\"+A`8UdFi_pFjb[l7%F_d\\l7$$\"+jV(4%=Fb]p$\"+ySoedFi_pFjb[l7%Fed \\l7$$\"+()=C8:Fb]p$\"+2&*HsdFi_pFjb[l7%F[e\\l7$$\"+#*eGt6Fb]p$\"+$Q]H y&Fi_pFjb[l7%Fae\\l7$$\"+sMy?#)Ffep$\"+l$*e!z&Fi_pFjb[l7%Fge\\l7$$\"+J *QVg%Ffep$\"+9/:&z&Fi_pFjb[l7%F]f\\l7$$\"+J8f+*)Fcdp$\"+Paa'z&Fi_pFjb[ l7%Fcf\\l7$$!+fMF/y&Fi_pFjb[l7%Feg\\l7 $$!+2fM)[\"Fb]p$\"+m*Rww&Fi_pFjb[l7%F[h\\l7$$!+\\[_1>Fb]p$\"+&o/2v&Fi_ pFjb[l7%Fah\\l7$$!+N0FOBFb]p$\"+$GC#HdFi_pFjb[l7%Fgh\\l7$$!+Kq9!y#Fb]p $\"+*3&p-dFi_pFjb[l7%F]i\\l7$$!+O2_UKFb]p$\"+o9YqcFi_pFjb[l7%Fci\\l7$$ !+xC!4t$Fb]p$\"+.7nJcFi_pFjb[l7%Fii\\l7$$!++nfeUFb]p$\"+S\"[_e&Fi_pFjb [l7%F_j\\l7$$!+v$e+&[Fb]p$\"+3))**HbFi_pFjb[l7%Fej\\l7$$!+q$o3b&Fb]p$ \"+?*Q`Y&Fi_pFjb[l7%F[[]l7$$!+#HY_Fi_pFjb[l7 %F[^]l7$$!+'Hfh`\"Fi_p$\"+&y!zX_Fi_pFjb[l7%Fa^]l7$$!+2@qB;Fi_p$\"+g.:W _Fi_pFjb[l7%Fg^]l7$$!++0q2Fi_p$\"+*)\\#)>_Fi_pFjb[l7%F_`]l7$$!+=)\\!=?Fi_p$\"+8Ya3_F i_pFjb[l7%Fe`]l7$$!+fu%44#Fi_p$\"+/-/&>&Fi_pFjb[l7%F[a]l7$$!+r9Zi@Fi_p $\"+C&y#z^Fi_pFjb[l7%Faa]l7$$!+#49GB#Fi_p$\"+gyBh^Fi_pFjb[l7%Fga]l7$$! +^p9-BFi_p$\"+e^!49&Fi_pFjb[l7%F]b]l7$$!+5miqBFi_p$\"+dXF=^Fi_pFjb[l7% Fcb]l7$$!+f*)RQCFi_p$\"+ypM$4&Fi_pFjb[l7%Fib]l7$$!+DEg0DFi_p$\"+b08m]F i_pFjb[l7%F_c]l7$$!+9:PsDFi_p$\"+d6LFi_p$\"+s)>7e%Fi_pFjb [l7%Fgg]l7$$!+prZ!Q$Fi_p$\"+FG8JXFi_pFjb[l7%F]h]l7$$!+^:**\\MFi_p$\"+V 5E![%Fi_pFjb[l7%Fch]l7$$!+X(3'>NFi_p$\"+V_rGWFi_pFjb[l7%Fih]l7$$!+q+=* e$Fi_p$\"+.JdwVFi_pFjb[l7%F_i]l7$$!+G5beOFi_p$\"+=\"yQK%Fi_pFjb[l7%Fei ]l7$$!+8-dFPFi_p$\"+Q:kqUFi_pFjb[l7%F[j]l7$$!+kk4'z$Fi_p$\"+%eYo@%Fi_p Fjb[l7%Faj]l7$$!+gQ+kQFi_p$\"+dRXiTFi_pFjb[l7%Fgj]l7$$!+`X=JRFi_p$\"+I xS2TFi_pFjb[l7%F][^l7$$!+,)\\v*RFi_p$\"+Z4k^SFi_pFjb[l7%Fc[^l7$$!+H'HI 1%Fi_p$\"+403&*RFi_pFjb[l7%Fi[^l7$$!+1;dFTFi_p$\"+%>^w$RFi_pFjb[l7%F_ \\^l7$$!+>\"R6>%Fi_p$\"+!yy#zQFi_pFjb[l7%Fe\\^l7$$!+@%4PD%Fi_p$\"+1C*) >QFi_pFjb[l7%F[]^l7$$!+s;F:VFi_p$\"+*GE%fPFi_pFjb[l7%Fa]^l7$$!+U]#eP%F i_p$\"+^4#yp$Fi_pFjb[l7%Fg]^l7$$!+^pPNWFi_p$\"+,T-NOFi_pFjb[l7%F]^^l7$ $!+B9%R\\%Fi_p$\"+a7*4d$Fi_pFjb[l7%Fc^^l7$$!+[v`^XFi_p$\"+'='o0NFi_pFj b[l7%Fi^^l7$$!+(*z=3YFi_p$\"+_73RMFi_pFjb[l7%F__^l7$$!+uv\"Rm%Fi_p$\"+ *ed6P$Fi_pFjb[l7%Fe_^l7$$!+[I$Fi_pFjb[l7%F[`^l7$$!+s^r sZFi_p$\"+)*GKJKFi_pFjb[l7%Fa`^l7$$!+h,$e#[Fi_p$\"+X!=%fJFi_pFjb[l7%Fg `^l7$$!+\"=:\"y[Fi_p$\"++h?'3$Fi_pFjb[l7%F]a^l7$$!+uLeH\\Fi_p$\"+())4< ,$Fi_pFjb[l7%Fca^l7$$!+\">T-)\\Fi_p$\"+&fef$HFi_pFjb[l7%Fia^l7$$!+?s3I ]Fi_p$\"+4l)*eGFi_pFjb[l7%F_b^l7$$!+%G6\"z]Fi_p$\"+2:$3y#Fi_pFjb[l7%Fe b^l7$$!+*)QHF^Fi_p$\"+#RL:q#Fi_pFjb[l7%F[c^l7$$!+\"31Y<&Fi_p$\"+m@8@EF i_pFjb[l7%Fac^l7$$!+;(45A&Fi_p$\"+TjmRDFi_pFjb[l7%Fgc^l7$$!+Q\"ekE&Fi_ p$\"+J9(*3J&Fi_p$\"+2(yOP#Fi_pFjb[l7%Fcd^l7$ $!+PlEa`Fi_p$\"+;V@*G#Fi_pFjb[l7%Fid^l7$$!+:5]'R&Fi_p$\"+#z)z.AFi_pFjb [l7%F_e^l7$$!+,A`PaFi_p$\"+#)pW<@Fi_pFjb[l7%Fee^l7$$!+;)*GxaFi_p$\"+h$ o,.#Fi_pFjb[l7%F[f^l7$$!+]Iq:bFi_p$\"+zw'>%>Fi_pFjb[l7%Faf^l7$$!+DM$R\"Fi_pFjb[l7%Feh^l7$$!+D7MTdFi_p$\"+a%e')H\"Fi_pFjb[l7%F[i^ l7$$!+\"*GomdFi_p$\"+v;1.7Fi_pFjb[l7%Fai^l7$$!+@3:!z&Fi_p$\"+A#pl5\"Fi _pFjb[l7%Fgi^l7$$!+1[q6eFi_p$\"+ok?45Fi_pFjb[l7%F]j^l7$$!+-(38$eFi_p$ \"+9_25\"*Fb]pFjb[l7%Fcj^l7$$!+#QI*[eFi_p$\"+t&f,7)Fb]pFjb[l7%Fij^l7$$ !+p;akeFi_p$\"+OI&G7(Fb]pFjb[l7%F_[_l7$$!+L#=\"yeFi_p$\"+`Cz=hFb]pFjb[ l7%Fe[_l7$$!+g%R'*)eFi_p$\"+R]r3^Fb]pFjb[l7%F[\\_l7$$!+4$)3**eFi_p$\"+ HOX$4%Fb]pFjb[l7%Fa\\_l7$$!+\\6X1fFi_p$\"+X#GR2$Fb]pFjb[l7%Fg\\_l7$$!+ )e<<\"fFi_p$\"+HS8^?Fb]pFjb[l7%F]]_l7$$!+L.)[\"fFi_p$\"+Ie7E5Fb]pFjb[l 7%Fc]_l7$$!+f]$f\"fFi_pF(Fjb[l7%Fi]_l7$Fd]_l$!+Ie7E5Fb]pFjb[l7%F]^_l7$ F^]_l$!+HS8^?Fb]pFjb[l7%Fa^_l7$Fh\\_l$!+X#GR2$Fb]pFjb[l7%Fe^_l7$Fb\\_l $!+HOX$4%Fb]pFjb[l7%Fi^_l7$F\\\\_l$!+R]r3^Fb]pFjb[l7%F]__l7$Ff[_l$!+`C z=hFb]pFjb[l7%Fa__l7$F`[_l$!+OI&G7(Fb]pFjb[l7%Fe__l7$Fjj^l$!+t&f,7)Fb] pFjb[l7%Fi__l7$Fdj^l$!+9_25\"*Fb]pFjb[l7%F]`_l7$F^j^l$!+ok?45Fi_pFjb[l 7%Fa`_l7$Fhi^l$!+A#pl5\"Fi_pFjb[l7%Fe`_l7$Fbi^l$!+v;1.7Fi_pFjb[l7%Fi`_ l7$F\\i^l$!+a%e')H\"Fi_pFjb[l7%F]a_l7$Ffh^l$!+#*>M$R\"Fi_pFjb[l7%Faa_l 7$F`h^l$!+Q95([\"Fi_pFjb[l7%Fea_l7$Fjg^l$!+T9$*z:Fi_pFjb[l7%Fia_l7$Fdg ^l$!+V3$=n\"Fi_pFjb[l7%F]b_l7$F^g^l$!+/8!Gw\"Fi_pFjb[l7%Fab_l7$Fhf^l$! +$*e%G&=Fi_pFjb[l7%Feb_l7$Fbf^l$!+zw'>%>Fi_pFjb[l7%Fib_l7$F\\f^l$!+h$o ,.#Fi_pFjb[l7%F]c_l7$Ffe^l$!+#)pW<@Fi_pFjb[l7%Fac_l7$F`e^l$!+#z)z.AFi_ pFjb[l7%Fec_l7$Fjd^l$!+;V@*G#Fi_pFjb[l7%Fic_l7$Fdd^l$!+2(yOP#Fi_pFjb[l 7%F]d_l7$F^d^l$!+J9I$Fi_pFjb[l7%F]g_l7$Ff_^l$!+*ed6P$Fi_pFjb[l7%Fa g_l7$F`_^l$!+_73RMFi_pFjb[l7%Feg_l7$Fj^^l$!+'='o0NFi_pFjb[l7%Fig_l7$Fd ^^l$!+a7*4d$Fi_pFjb[l7%F]h_l7$F^^^l$!+,T-NOFi_pFjb[l7%Fah_l7$Fh]^l$!+^ 4#yp$Fi_pFjb[l7%Feh_l7$Fb]^l$!+*GE%fPFi_pFjb[l7%Fih_l7$F\\]^l$!+1C*)>Q Fi_pFjb[l7%F]i_l7$Ff\\^l$!+!yy#zQFi_pFjb[l7%Fai_l7$F`\\^l$!+%>^w$RFi_p Fjb[l7%Fei_l7$Fj[^l$!+403&*RFi_pFjb[l7%Fii_l7$Fd[^l$!+Z4k^SFi_pFjb[l7% F]j_l7$F^[^l$!+IxS2TFi_pFjb[l7%Faj_l7$Fhj]l$!+dRXiTFi_pFjb[l7%Fej_l7$F bj]l$!+%eYo@%Fi_pFjb[l7%Fij_l7$F\\j]l$!+Q:kqUFi_pFjb[l7%F][`l7$Ffi]l$! +=\"yQK%Fi_pFjb[l7%Fa[`l7$F`i]l$!+.JdwVFi_pFjb[l7%Fe[`l7$Fjh]l$!+V_rGW Fi_pFjb[l7%Fi[`l7$Fdh]l$!+V5E![%Fi_pFjb[l7%F]\\`l7$F^h]l$!+FG8JXFi_pFj b[l7%Fa\\`l7$Fhg]l$!+s)>7e%Fi_pFjb[l7%Fe\\`l7$Fbg]l$!+9HQIYFi_pFjb[l7% Fi\\`l7$F\\g]l$!+Y;YyYFi_pFjb[l7%F]]`l7$Fff]l$!+yTGDZFi_pFjb[l7%Fa]`l7 $F`f]l$!+%*pnqZFi_pFjb[l7%Fe]`l7$Fje]l$!+#*RZ9[Fi_pFjb[l7%Fi]`l7$Fde]l $!+7N_c[Fi_pFjb[l7%F]^`l7$F^e]l$!+iGp'*[Fi_pFjb[l7%Fa^`l7$Fhd]l$!+#Qq[ $\\Fi_pFjb[l7%Fe^`l7$Fbd]l$!+ja'4(\\Fi_pFjb[l7%Fi^`l7$F\\d]l$!+Br!\\+& Fi_pFjb[l7%F]_`l7$Ffc]l$!+d&Fi_pFjb[l7%F]a`l7$Ff`]l$!+8 Ya3_Fi_pFjb[l7%Faa`l7$F``]l$!+*)\\#)>_Fi_pFjb[l7%Fea`l7$Fj_]l$!+'*4$*G _Fi_pFjb[l7%Fia`l7$Fd_]l$!+ym$fB&Fi_pFjb[l7%F]b`l7$F^_]l$!+.L&4C&Fi_pF jb[l7%Fab`l7$Fh^]l$!+g.:W_Fi_pFjb[l7%Feb`l7$Fb^]l$!+&y!zX_Fi_pFjb[l7%F ib`l7$F\\^]l$!+\">#HY_Fi_pFjb[l7%F]c`l7$Ff]]l$!+-!QjC&Fi_pFjb[l7%Fac`l 7$F`]]l$!+a()3Z_Fi_pFjb[l7%Fec`l7$Fj\\]l$!+C#*f]_Fi_pFjb[l7%Fic`l7$Fd \\]l$!+7$G1E&Fi_pFjb[l7%F]d`l7$F^\\]l$!+Ad\"QG&Fi_pFjb[l7%Fad`l7$Fh[]l $!+UobG`Fi_pFjb[l7%Fed`l7$Fb[]l$!+(\\*=%R&Fi_pFjb[l7%Fid`l7$F\\[]l$!+? *Q`Y&Fi_pFjb[l7%F]e`l7$Ffj\\l$!+3))**HbFi_pFjb[l7%Fae`l7$F`j\\l$!+S\"[ _e&Fi_pFjb[l7%Fee`l7$Fji\\l$!+.7nJcFi_pFjb[l7%Fie`l7$Fdi\\l$!+o9YqcFi_ pFjb[l7%F]f`l7$F^i\\l$!+*3&p-dFi_pFjb[l7%Faf`l7$Fhh\\l$!+$GC#HdFi_pFjb [l7%Fef`l7$Fbh\\l$!+&o/2v&Fi_pFjb[l7%Fif`l7$F\\h\\l$!+m*Rww&Fi_pFjb[l7 %F]g`l7$Ffg\\l$!+q&>/y&Fi_pFjb[l7%Fag`l7$$!+#H1\\\"oFfep$!+IpM*y&Fi_pF jb[l7%Feg`l7$Fjf\\l$!+*))fYz&Fi_pFjb[l7%F[h`l7$Fdf\\l$!+Paa'z&Fi_pFjb[ l7%F_h`l7$F^f\\l$!+9/:&z&Fi_pFjb[l7%Fch`l7$Fhe\\l$!+l$*e!z&Fi_pFjb[l7% Fgh`l7$Fbe\\l$!+$Q]Hy&Fi_pFjb[l7%F[i`l7$F\\e\\l$!+2&*HsdFi_pFjb[l7%F_i `l7$Ffd\\l$!+ySoedFi_pFjb[l7%Fci`l7$F`d\\l$!+A`8UdFi_pFjb[l7%Fgi`l7$Fj c\\l$!+s.nAdFi_pFjb[l7%F[j`l7$Fdc\\l$!+QQH+dFi_pFjb[l7%F_j`l7$F^c\\l$! +K!**\\n&Fi_pFjb[l7%Fcj`l7$Fhb\\l$!+])onk&Fi_pFjb[l7%Fgj`l7$Fbb\\l$!+- jd:cFi_pFjb[l7%F[[al7$F\\b\\l$!+!f%Q\"e&Fi_pFjb[l7%F_[al7$Ffa\\l$!+'\\ YTa&Fi_pFjb[l7%Fc[al7$F`a\\l$!+>F!Q]&Fi_pFjb[l7%Fg[al7$Fj`\\l$!+v%y-Y& Fi_pFjb[l7%F[\\al7$Fd`\\l$!+NsZ8aFi_pFjb[l7%F_\\al7$F^`\\l$!+^'pKO&Fi_ pFjb[l7%Fc\\al7$Fh_\\l$!+_UZ4`Fi_pFjb[l7%Fg\\al7$Fb_\\l$!+`O#=D&Fi_pFj b[l7%F[]al7$F\\_\\l$!+TV!**=&Fi_pFjb[l7%F_]al7$Ff^\\l$!+%)f/B^Fi_pFjb[ l7%Fc]al7$F`^\\l$!+v*4,0&Fi_pFjb[l7%Fg]al7$Fj]\\l$!+o:2p\\Fi_pFjb[l7%F [^al7$Fd]\\l$!+-*3i([Fi_pFjb[l7%F_^al7$F^]\\l$!+s^#\\w%Fi_pFjb[l7%Fc^a l7$Fh\\\\l$!+yH(oi%Fi_pFjb[l7%Fg^al7$Fb\\\\l$!+svwiWFi_pFjb[l7%F[_al7$ F\\\\\\l$!+;I/)G%Fi_pFjb[l7%F__al7$Ff[\\l$!+vMx9TFi_pFjb[l7%Fc_al7$F`[ \\l$!+n(ff%RFi_pFjb[l7%Fg_al7$Fjj[l$!+WJ6\"y$Fi_pFjb[l7%F[`al7$Fdj[l$! +#ya\">OFi_pFjb[l7%F_`al7$F^j[l$!+IT9fMFi_pFjb[l7%Fc`al7$Fhi[l$!+\"*\\ Q+LFi_pFjb[l7%Fg`al7$$!+#RKN6#Fjbp$!+qZRUJFi_pFjb[l7%F[aal7$F\\i[l$!+J Q&[)HFi_pFjb[l7%Faaal7$$!+DXlfiF^`p$!+))*fv#GFi_pFjb[l7%Feaal7$F`h[l$! +$3#RqEFi_pFjb[l7%F[bal7$Fjg[l$!+)>#G8DFi_pFjb[l7%F_bal7$Fdg[l$!+(Q&>c BFi_pFjb[l7%Fcbal7$F^g[l$!+^c6*>#Fi_pFjb[l7%Fgbal7$Fhf[l$!+#)o.U?Fi_pF jb[l7%F[cal7$Fbf[l$!+%[d\\)=Fi_pFjb[l7%F_cal7$F\\f[l$!+2v(ys\"Fi_pFjb[ l7%Fccal7$$\"+cY*e^#F^`p$!+Otzq:Fi_pFjb[l7%Fgcal7$F`e[l$!+Ssr89Fi_pFjb [l7%F]dal7$Fjd[l$!+2tjc7Fi_pFjb[l7%Fadal7$Fdd[l$!+?vb*4\"Fi_pFjb[l7%Fe dal7$$\"+n(\\gS#FN$!+J#yZU*Fb]pFjb[l7%Fidal7$$\"+mK\"=<%F-$!+%p\")R&yF b]pFjb[l7%F_eal7$$\"+))ekSZFfr$!+;`=$G'Fb]pFjb[l7%Feeal7$F(FcavFjb[l7% F[falFfavFjb[l7%FfavFjavFjb[l7%FjavF'Fjb[l-Fe\\p6&Fg\\p$\"#vFbbv$\"\"% Fj\\pFcb[lFgbv-F$6%7$7$$!3Q++++++!p'FNF(7$$\"3%**************=\"FNF(-% 'COLOURG6&Fg\\pF)F)F)-%*LINESTYLEG6#Fdbv-F$6%7$7$F($!3o*************Q' FN7$F($\"3o*************Q'FNF^galFagal-%(SCALINGG6#%,CONSTRAINEDG-%+AX ESLABELSG6%%&Re(z)G%&Im(z)G-%%FONTG6#%(DEFAULTG-Fghal6$%*HELVETICAG\" \"*-%*AXESSTYLEG6#%$BOXG-%%VIEWG6$;$!$\\'Fbbv$\"$>\"Fbbv;$!$R'Fbbv$\"$ R'Fbbv" 1 2 0 1 10 0 2 9 1 2 1 1.000000 45.000000 45.000000 0 0 "Curve 1" "Curve 2" "Curve 3" "Curve 4" "Curve 5" "Curve 6" }}}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}} {PARA 0 "" 0 "" {TEXT -1 52 "#---------------------------------------- -----------" }}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 36 "coefficients for t he combined scheme" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 " > " 0 "" {MPLTEXT 1 0 1 ";" }}}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 36 "co efficients of the combined scheme " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 5227 "ee := \{c[2]=3/50,\nc[3]=4 69/4510,\nc[4]=1407/9020,\nc[5]=100/257,\nc[6]=7/15,\nc[7]=12/77,\nc[8 ]=26/27,\nc[9]=7101051/7944748,\nc[10]=11/12,\nc[11]=19/20,\nc[12]=1, \nc[13]=1,\n\na[2,1]=3/50,\na[3,1]=169309/12204060,\na[3,2]=219961/244 0812,\na[4,1]=1407/36080,\na[4,2]=0,\na[4,3]=4221/36080,\na[5,1]=12985 296430100/33603736057857,\na[5,2]=0,\na[5,3]=-16218027650000/112012453 52619,\na[5,4]=48744170200000/33603736057857,\na[6,1]=3817807/81405000 ,\na[6,2]=0,\na[6,3]=0,\na[6,4]=10274191312/43991343405,\na[6,5]=40755 997793/218862405000,\na[7,1]=15900802578/262289621825,\na[7,2]=0,\na[7 ,3]=0,\na[7,4]=6340315606886400/56274919101668113,\na[7,5]=-4511310082 76934/12909147305279225,\na[7,6]=984103149600/56231606512081,\na[8,1]= -6560178806172785891/5672858706535680000,\na[8,2]=0,\na[8,3]=0,\na[8,4 ]=-17/2,\na[8,5]=-5870605887517986369316613/652465735085103413760000, \na[8,6]=16204960050140425727/2255297505252764544,\na[8,7]=96584177612 18465607893/776920972593666355200,\na[9,1]=-14046189980696518283198635 2616636648283813334213167183799/\n 2315957023533969410187885357 27415998878963658147758080000,\na[9,2]=0,\na[9,3]=0,\na[9,4]=-23038843 6455789139579016226863459716720858844533129533/\n 2816259910401 2402938735024801544932535316065892462592,\na[9,5]=-9790630998374957452 129847257592953551765164772149796065413844391/\n 18791208635455 68722091311485392002480259497346453409790689280000,\na[9,6]=1970432358 03985508254844363902235259599001596076904803130443/\n 4456196545 9500736173318259755870730512740108168829012803584,\na[9,7]=29462081775 8575088586344956781032015012776829359971133/\n 28166985596967449 488723424839046646360350769597644800,\na[9,8]=126873616630173037746837 292557884226969/13217298519653022949071050320940567220224,\na[10,1]=-4 1617847659972885613040591872450035471/52813005326633659912790449394810 880000,\na[10,2]=0,\na[10,3]=0,\na[10,4]=-2699510877644969121541738768 31/29845195411834947899362922496,\na[10,5]=-27499866567291131028253861 17572168116123/422416144552169230122612885222850560000,\na[10,6]=32452 5706880938787969090744301341259977691/60253355178721829324099288547678 254006272,\na[10,7]=41487408138830374659637178881584696821/34936563155 17022583105833038931558400,\na[10,8]=932598433870374903885/57293130256 189894463488,\na[10,9]=-3345742205378079321732624680747620/19122888455 5191941725363563271035831,\na[11,1]=-152032134715446654236835762683933 540260220443/144031229933501187933392929852012953600000000,\na[11,2]=0 ,\na[11,3]=0,\na[11,4]=-19926282430450459437356479596949/2204175587073 348224489986560000,\na[11,5]=-2861670807732347804933810531212726938261 187/343166554822474341490702928140697600000000,\na[11,6]=1334063326593 468056350327316268714319601715109229/198658539882559017093840573946438 829079429120000, a[11,7]=255870600859869065157514718012970112299037192 73/2019042940187611700532950824131079569408000000,\na[11,8]=1292083162 729235236212706557/47701646396336675524288000000,\na[11,9]=20910323845 14247750201495665581688757138358851/6072986917641256493842379709543579 6334060237500,\na[11,10]=-377694382258403/5816420236687500,\na[12,1]=- 240860055167915269684777564157507287321/117003858749851375574083424673 098438400,\na[12,2]=0,\na[12,3]=0,\na[12,4]=-9889470231083416362696495 5900/10545476035409921643978852639,\na[12,5]=-252251601740617744737074 278562623715189/16418177849613229685231088852333734400,\na[12,6]=84081 3041585474673597744565360698902394175/70780871835683840841986501581774 346003376,\na[12,7]=241895147163393632189444055554841293171617/1508217 1924595104472807447927564153421312,\na[12,8]=42847753308025275666/2988 2754550596969299,\na[12,9]=-771733622226488524056405401174457496589899 2045440/8479033418894797325547814554207128260520626293873,\na[12,10]=1 6262726151696512/8940486723127047,\na[12,11]=-16001553345352000000/650 5707112968117383,\na[13,1]=-2744022934559056024469622604306540529/1738 222573548705964451797934060438400,\na[13,2]=0,\na[13,3]=0,\na[13,4]=-3 5144243656562246120945761400/5953261868437390070644832157,\na[13,5]=-5 4637123672393421308996942594910689159/46342958730893569584102340716889 53600,\na[13,6]=61239081290242266243670859206851281463175/665968142641 1632770651421460598964953048,\na[13,7]=5134209062671060909676995239537 507099025/462964694026104436494336717002613458688,\na[13,8]=-370363100 142446513442/1535150001377392224067,\na[13,9]=-21202068251232907606822 1666486441382648974080/390260289309289497513964664381911193640454941, \na[13,10]=2242504532188281088/2908865886282319623,\na[13,11]=0,\na[13 ,12]=0,\n\nb[1]=61903079921/1389136000824,\nb[2]=0,\nb[3]=0,\nb[4]=0, \nb[5]=0,\nb[6]=402653566621875/1136204715874336,\nb[7]=18720207312168 49170881/7515476193013636507650,\nb[8]=-48961117784712267/151774139994 38600,\nb[9]=1824678166477710343022780887139376956665357164944/3327264 80751222257863787417240798843822259974895,\nb[10]=-126172885488/131679 45725,\nb[11]=41251039800000/5264998725539,\nb[12]=-107501739/67495760 0,\nb[13]=0,\n\n`b*`[1]=5709741119/127946736918,\n`b*`[2]=0,\n`b*`[3]= 0,\n`b*`[4]=0,\n`b*`[5]=0,\n`b*`[6]=97458949528125/274256310728288,\n` b*`[7]=79520000370367181779/319546003300661568600,\n`b*`[8]=5689283182 401789/2168201999919800,\n`b*`[9]=124582744353312716760570327383122312 5580576/298102207457268033088581737062702957958355,\n`b*`[10]=-7667163 7728/13167945725,\n`b*`[11]=0,\n`b*`[12]=0,\n`b*`[13]=-5522626587/8774 448800\}:" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 " " {MPLTEXT 1 0 1 ";" }}}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT 259 5 "nodes" }{TEXT -1 2 ": " }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 32 "seq(c[i]=subs(ee,c[i]),i=2..13);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6./&%\"cG6#\"\"##\"\"$\"#]/&F%6#F)#\"$p%\"%5X/&F%6#\"\"%# \"%29\"%?!*/&F%6#\"\"&#\"$+\"\"$d#/&F%6#\"\"'#\"\"(\"#:/&F%6#FD#\"#7\" #x/&F%6#\"\")#\"#E\"#F/&F%6#\"\"*#\"(^55(\"([Z%z/&F%6#\"#5#\"#6FJ/&F%6 #Fin#\"#>\"#?/&F%6#FJ\"\"\"/&F%6#\"#8Fco" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT 259 20 "linking coefficients" }{TEXT -1 1 ":" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 50 "seq(seq(a[i,j]=subs( ee,a[i,j]),j=1..i-1),i=1..13);" }}{PARA 12 "" 1 "" {XPPMATH 20 "6jo/&% \"aG6$\"\"#\"\"\"#\"\"$\"#]/&F%6$F*F(#\"'4$p\"\")gS?7/&F%6$F*F'#\"'h*> #\"(73W#/&F%6$\"\"%F(#\"%29\"&!3O/&F%6$F;F'\"\"!/&F%6$F;F*#\"%@UF>/&F% 6$\"\"&F(#\"/+,V'H&)H\"\"/dy0OPgL/&F%6$FKF'FB/&F%6$FKF*#!/++lF!=i\"\"/ >ENX7?6/&F%6$FKF;#\"/++?qTu[FN/&F%6$\"\"'F(#\"(2y\"Q\")+]S\")/&F%6$Fjn F'FB/&F%6$FjnF*FB/&F%6$FjnF;#\",78>u-\"\",0MM\"*R%/&F%6$FjnFK#\",$z(*f vS\"-+]Si)=#/&F%6$\"\"(F(#\",yD!3!f\"\"-D=i*Gi#/&F%6$FcpF'FB/&F%6$FcpF *FB/&F%6$FcpF;#\"1+k)og:.M'\"28\"o;5>\\Fc/&F%6$FcpFK#!0MpF358^%\"2D#z_ IZ\"4H\"/&F%6$FcpFjn#\"-+'\\J5%)*\"/\"37lgJi&/&F%6$\"\")F(#!4\"*eysh!) y,c'\"4++oNlqeGn&/&F%6$FbrF'FB/&F%6$FbrF*FB/&F%6$FbrF;#!#$G=lp!)**=YS\"\"Z++3ex9eO'*y))*fTFd`)y=5%pR`Bq&fJ#/&F% 6$FftF'FB/&F%6$FftF*FB/&F%6$FftF;#!WL&HJ`W)e3s;(fM'oA;!z&R\"*ybkV)QI# \"V#fiC*e1;``K\\a,[-N(QHS7S5*fi\"G/&F%6$FftFK#![o\"RWQTlgz\\@xk^w^N&Hf ds%)H@Xd\\P)*4j!z*\"[o++G*o!z4MXYt\\f-[-?R&[684A(obaj37z=/&F%6$FftFjn# \"gnV/8.[!p2'f,!*ff_B-ROW[D3b)R!eBVq>\"fn%e.G,H)o\"3,u70tqevf#=LcW/&F%6$FftFcp#\"WL6(*f$Hox7],K5yc\\M'e)3v&ex\"3i%H\"V+[k(fp2NgjkY!R [UB()[\\u'pf&)p;G/&F%6$FftFbr#\"HppA%)yb#HPouPI&)H'\\ AHO**y%\\$=T&>X)H/&F%6$FawFK#!IBh6o@d>bX))G7>/&F%6$\"#6F(#!NV/Ag-aLRoid$oBamW:Z8K ?:\"N++++O&H,_)HHRLz=,N$*H7.W\"/&F%6$FbzF'FB/&F%6$FbzF*FB/&F%6$FbzF;#! A\\pfzkNP%f/XICGE*>\"@++c')**[C#[L2(evT?#/&F%6$FbzFK#!L(=h#Qps77`5Q$\\ !yMKx!3nhG\"K++++wpS\"GHq!\\TVZA[bmJM/&F%6$FbzFjn#\"RH#4^r,'>VroiJF.Nc !oMfEL1M8\"Q++7H%z!H)Qk%RdSQ4/&F%6$FbzFcp#\"Pt#>P!*H7,(H,= Z^d^1p)f3gqeD\"O+++3%p&z58C3&H`+?/&F%6$FbzFbr#\"=dlq7iBN#HF;$3 #H\"\">+++)GCbnOjRY;qZ/&F%6$FbzFft#\"O^)e$Qrv)o\"elc\\,-vZU^%QK54#\"P+ vBgSL'zNa4(zB%Q\\c7k+f&\\'pijT$3J-Z*))*\">RE&)yRk@*4a.waa5/&F %6$Fi]lFK#!H*=:Pii&yU2PZu<1u,;D_#\"G+WtLB&))3J_oHKh\\y<=k\"/&F%6$Fi]lF jn#\"KvTR-*)pg`cWxftYZ&eTI\"3%)\"JwL+YVx\"e,l)>%3%QoN=(3yq/&F%6$Fi]lFc p#\"K<;<$HT[bbSW*=KORjr9&*=C\"J78U`TcFzW2GZ/^fC><#3:/&F%6$Fi]lFbr#\"5m cFD!3LvZG%\"5*Hppf]Xv#))H/&F%6$Fi]lFft#!RSa/#**)*e'\\dW<,aScS_)[EAiL+9wX47YAccOCW^$\"=d@$[kq+RP%o=E`f/&F%6$FfalFK#!Gf\"*o5\\fUp**38U$Rs O7PY&\"F+O&*)orSB5%epN*3teHMY/&F%6$FfalFjn#\"JvJY\"G^o?f3nViEU-H\"3R7' \"I[I&\\'*)fg9U^1xK;TE9ofm/&F%6$FfalFcp#\"ID!*42v`R_*pn441rE14U8&\"H)o eMh-qrOV\\OW5ESpkHY/&F%6$FfalFbr#!6UM^YC9+JOq$\"7nSA#Rx8+]^`\"/&F%6$Ff alFft#!N!3u*[EQTk[m;Aog2HB^#o??@\"NT\\XSO>6>QkY'R^(\\*G4$*Gg-R/&F%6$Ff alFaw#\"4)3\"G)=KX]UA\"4B'>BG')e')3H/&F%6$FfalFbzFB/&F%6$FfalFi]lFB" } }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT 259 39 "weight s for the 13 stage order 8 scheme" }{TEXT -1 1 ":" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 32 "seq(b[i]=subs(ee,b[i]),i=1..13);" }}{PARA 12 "" 1 "" {XPPMATH 20 "6//&%\"bG6#\"\"\"#\",@*zI!>'\".C3+O\"*Q\"/&F%6#\" \"#\"\"!/&F%6#\"\"$F//&F%6#\"\"%F//&F%6#\"\"&F//&F%6#\"\"'#\"0v=imNl-% \"1OV(er/i8\"/&F%6#\"\"(#\"7\")3<\\o@J2-s=\"7]w]OO,$>wa^(/&F%6#\"\")#! 2nAr%y<6'*[\"2+'Q%**RTx^\"/&F%6#\"\"*#\"RW\\;d`mcpPRr)3yAIM5xZm\"yY#= \"Q&*[(*fA#Q%))zSsT(yjyDA7v![EFL/&F%6#\"#5#!-)[&)G " 0 "" {MPLTEXT 1 0 38 "seq(`b*`[i]=subs(ee,`b*`[i]),i=1..13);" }} {PARA 12 "" 1 "" {XPPMATH 20 "6//&%#b*G6#\"\"\"#\"+>6u4d\"-=ptYz7/&F%6 #\"\"#\"\"!/&F%6#\"\"$F//&F%6#\"\"%F//&F%6#\"\"&F//&F%6#\"\"'#\"/D\"G& \\*eu*\"0)GG2JcUF/&F%6#\"\"(#\"5z<=n.P++_z\"6+'o:m+L+Y&>$/&F%6#\"\")# \"1*y,C=$G*o&\"1+)>***>?o@/&F%6#\"\"*#\"Lw0eDJAJQF.dgnr7LNWFeC\"\"Kb$e z&Hqiqt\"e)3L!osX2A5)H/&F%6#\"#5#!,Gxjrm(\",Dd%z;8/&F%6#\"#6F//&F%6#\" #7F//&F%6#\"#8#!+(eEE_&\"++)[Wx)" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}}{PARA 0 "" 0 "" {TEXT -1 52 "#---------------------------------------------------" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}}{PARA 0 "" 0 "" {TEXT -1 40 " #=======================================" }}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 21 "an \"efficient\" scheme" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 86 "This scheme could be regarded as anothe r modification of Jim Verner's \"Maple\" scheme. " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 15 "The nodes are: " }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "c[2] = 27/400" "6#/&%\"cG6# \"\"#*&\"#F\"\"\"\"$+%!\"\"" }{TEXT -1 3 ", " }{XPPEDIT 18 0 "c[3] = \+ 469/4510" "6#/&%\"cG6#\"\"$*&\"$p%\"\"\"\"%5X!\"\"" }{TEXT -1 3 ", " }{XPPEDIT 18 0 "c[4] = 1407/9020" "6#/&%\"cG6#\"\"%*&\"%29\"\"\"\"%?!* !\"\"" }{TEXT -1 3 ", " }{XPPEDIT 18 0 "c[5] = 100/257" "6#/&%\"cG6# \"\"&*&\"$+\"\"\"\"\"$d#!\"\"" }{TEXT -1 3 ", " }{XPPEDIT 18 0 "c[6] \+ = 7/15" "6#/&%\"cG6#\"\"'*&\"\"(\"\"\"\"#:!\"\"" }{TEXT -1 3 ", " } {XPPEDIT 18 0 "c[7] = 7/45" "6#/&%\"cG6#\"\"(*&F'\"\"\"\"#X!\"\"" } {TEXT -1 3 ", " }{XPPEDIT 18 0 "c[8] = 58/59" "6#/&%\"cG6#\"\")*&\"#e \"\"\"\"#f!\"\"" }{TEXT -1 3 ", " }{XPPEDIT 18 0 "c[9] = 49117478/551 44743" "6#/&%\"cG6#\"\"**&\")yu6\\\"\"\"\")VZ9b!\"\"" }{TEXT -1 3 ", \+ " }{XPPEDIT 18 0 "c[10] = 13/14" "6#/&%\"cG6#\"#5*&\"#8\"\"\"\"#9!\"\" " }{TEXT -1 3 ", " }{XPPEDIT 18 0 "c[11] = 62/65" "6#/&%\"cG6#\"#6*& \"#i\"\"\"\"#l!\"\"" }{TEXT -1 3 ", " }{XPPEDIT 18 0 "c[12] = 1" "6#/ &%\"cG6#\"#7\"\"\"" }{TEXT -1 3 ", " }{XPPEDIT 18 0 "c[13]=1" "6#/&% \"cG6#\"#8\"\"\"" }{TEXT -1 2 ". " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}{PARA 0 "" 0 "" {TEXT -1 52 "#---------------------------------------------------" }}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 28 "checking the combined scheme" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 36 "coefficients of the combined s cheme " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 5959 "ee := \{c[2]=27/400,\nc[3]=469/4510,\nc[4]=1407/902 0,\nc[5]=100/257,\nc[6]=7/15,\nc[7]=7/45,\nc[8]=58/59,\nc[9]=49117478/ 55144743,\nc[10]=13/14,\nc[11]=62/65,\nc[12]=1,\nc[13]=1,\n\na[2,1]=27 /400,\na[3,1]=1311793/54918270,\na[3,2]=439922/5491827,\na[4,1]=1407/3 6080,\na[4,2]=0,\na[4,3]=4221/36080,\na[5,1]=12985296430100/3360373605 7857,\na[5,2]=0,\na[5,3]=-16218027650000/11201245352619,\na[5,4]=48744 170200000/33603736057857,\na[6,1]=3817807/81405000,\na[6,2]=0,\na[6,3] =0,\na[6,4]=10274191312/43991343405,\na[6,5]=40755997793/218862405000, \na[7,1]=8883007/146529000,\na[7,2]=0,\na[7,3]=0,\na[7,4]=534360690137 12/475502430864645,\na[7,5]=-4116355777093/117791746371000,\na[7,6]=76 35614/436305285,\na[8,1]=-25750155116216595121447/19172397107027415328 000,\na[8,2]=0,\na[8,3]=0,\na[8,4]=-743/55,\na[8,5]=-31934383116845286 58341050903/315992014923229256968928000,\na[8,6]=183334983592060082563 257/22930186940004788732288,\na[8,7]=18586487074230223824062061/103569 2891721620976018560,\na[9,1]=-9221221718947165522178079732779342046723 48766351979550680715901/\n 157401190198671640845526912912949987 6539846630029827642536608000,\na[9,2]=0,\na[9,3]=0,\na[9,4]=-249392411 2965609110422434487743812058912776576445331741068763/\n 2485761 88041998999363340039748303974885433202092103843801085,\na[9,5]=-347410 454252950206893609060802736441487435576445275533884950354547756949/\n \+ 697473627660918834815922125361784386305095155212074816816271884 67808000,\na[9,6]=5327733707939584478612584810530320112314665632405586 015822099459/\n 124980895520514732019873673632122559682826398009 3048262260372352,\na[9,7]=63805647563182428528344562389882620890969933 364805025903916023/\n 522253688012544809193253721243016911311851 3294896584316063360,\na[9,8]=16756974250619222216361657668160522447776 912/1649765930508280715886033484777254665540228007,\na[10,1]=-37029278 24534109782314804940532859074364487/4131257703459996302316696251055190 381568000,\na[10,2]=0,\na[10,3]=0,\na[10,4]=-4854249491928577458940897 7236397/3795156546073735065277428293920,\na[10,5]=-2291942554267427743 1989520078944496953957193/3194620077548089472408777624402038658048000, \na[10,6]=2484540315797359406715529397919751486619672787/4216486903888 02974367036270632096098190348288,\na[10,7]=170546976143576425414211056 33253210641175183343/1072180740487803620784937862662845301329326080,\n a[10,8]=242464961680690545467502075/12396488665185571302473970944,\na[ 10,9]=-276720658930244250241932341390181638290085115/96637936155048616 17193707372304668805996412672,\na[11,1]=-88826526149223807070829373027 93930626864630973971/\n 80180429743401295397733653580724007953 35265440000,\na[11,2]=0,\na[11,3]=0,\na[11,4]=-91300774029817320031962 039591895693/6641776260885597741930170995083525,\na[11,5]=-11220125505 3474401906816716367850973725245563029/\n 130451942058366052280 92670613416996666049440000,\na[11,6]=183334809040500814662036381242837 807618576330592954213/\n 26459846979744846747689472891117696020 043251727064960,\na[11,7]=59085344439790277683397952622868924535489307 74924082527/\n 337801676952094167280622617772889760985970195055 625600,\na[11,8]=6612449503769637840690016046967566/225556868777060284 586258384373942375,\na[11,9]=-1029378427823652249730886986351378997098 7031746931000834/\n 878782636339557288924677338637495764024419 7745505586187675,\na[11,10]=-4215629637956404624565504/108123612591292 234376500125,\na[12,1]=-535308736842973754888311604833674190949559277/ \n 507665470526061438136955303900248854246636000,\na[12,2]=0, \na[12,3]=0,\na[12,4]=-6581385500218652420284298344406312/842464028888 378018027275808653215,\na[12,5]=-2298109906288490803293215283725071869 06863097/\n 29548105906232803976194189596626971177604000,\na[1 2,6]=333621743628604464651535937000728865436063541833/\n 540424 31430277972080199874664118662613066091504,\na[12,7]=281436223028901228 360767240757054593496984174199893/\n 24784655195828222920388514 383970641394482546714960,\na[12,8]=-195816087241661023276342391/399896 648012674685277875106,\na[12,9]=14464527008223808452459454468398839957 5644348254551267965/\n 3148564668287108076358894831813590230011 87603321870639973,\na[12,10]=-113365555877618519272588672/930915042822 32088010468779,\na[12,11]=9008463031864887557094375/660686234780068058 4331678,\na[13,1]=-35556891109301815175584262675897743594621/233372257 66986990952929827252863229628000,\na[13,2]=0,\na[13,3]=0,\na[13,4]=-15 763198989262628277502495250456/1200562192841616547810297566045,\na[13, 5]=-474499229690075043966853347490522346181911/\n 421078379665 79029834778387168434144652000,\na[13,6]=679545987701237138790035764835 967076741310679/\n 77013733239195245746002612595738122195430352 ,\na[13,7]=1368817102190976218912189847996057845647169577/\n 75 630904954724828436404430778019911819343440,\na[13,8]=-3292496872165727 322673258/59253630028479896512047753,\na[13,9]=-2684522309490576529735 302727427307676030962009390/\n 3462833663065024895730647437447 9340566374128519867, a[13,10]=64631913714031794733568/4864237536243029 98802649,\na[13,11]=0,\na[13,12]=0,\n\nb[1]=1049982578583103/236273203 35287760,\nb[2]=0,\nb[3]=0,\nb[4]=0,\nb[5]=0,\nb[6]=768239620641637875 /2161676880469378496,\nb[7]=7884214609987698755625/3168197534294333398 0208,\nb[8]=-19726902451409487557291/9259475236790114044800,\nb[9]=229 8544897725758629882107777744216351971045149850043139161/\n 7323526 61995952143676542507482891823353467854709785979600,\nb[10]=-2307497857 833742448/278745570101106225,\nb[11]=148963664643228251375/20207195606 693950848,\nb[12]=576091109999/2308683585600,\nb[13]=0,\n\n`b*`[1]=485 0006989829/108881660531280,\n`b*`[2]=0,\n`b*`[3]=0,\n`b*`[4]=0,\n`b*`[ 5]=0,\n`b*`[6]=5801298400186875/16253209627589312,\n`b*`[7]=2408095926 060530625/9691641279578872432,\n`b*`[8]=55006229857804792033/157474068 65289309600,\n`b*`[9]=363484892569724108095352508636899702948448557463 /\n 160778106318887934237641558033288694994268792800,\n`b*`[10] =-14197733827415152/4039790871030525,\n`b*`[11]=0,\n`b*`[12]=0,\n`b*`[ 13]=-9030611407/4779883200\}:" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 20 "printed coefficients" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 357 "subs(ee,mat rix([seq([c[i],seq(a[i,j],j=1..i-1),``$(7-i)],i=2..7)]));\nfor ii from 8 to 13 do\n print(``);\n print(c[ii]=subs(ee,c[ii])); \n for j j to ii-1 do\n print(a[ii,jj]=subs(ee,a[ii,jj]));\n end do:\nen d do:print(``);\nfor ii to 13 do\n print(b[ii]=subs(ee,b[ii]));\nend do:print(``);\nfor ii to 13 do\n print(`b*`[ii]=subs(ee,`b*`[ii])); \nend do:" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#K%'matrixG6#7(7)#\"#F\"$+ %F(%!GF+F+F+F+7)#\"$p%\"%5X#\"($z68\")q#=\\&#\"'A*R%\"(F=\\&F+F+F+F+7) #\"%29\"%?!*#F8\"&!3O\"\"!#\"%@UF;F+F+F+7)#\"$+\"\"$d##\"/+,V'H&)H\"\" /dy0OPgLF<#!/++lF!=i\"\"/>ENX7?6#\"/++?qTu[FEF+F+7)#\"\"(\"#:#\"(2y\"Q \")+]S\")Fu-\"\",0MM\"*R%#\",$z(*fvS\"-+]Si)=#F+7)#FM\"#X#\" (2I)))\"*+!Hl9F" } }{PARA 11 "" 1 "" {XPPMATH 20 "6#/&%\"aG6$\"\")\"\"#\"\"!" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/&%\"aG6$\"\")\"\"$\"\"!" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/&%\"aG6$\"\")\"\"%#!$V(\"#b" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/&%\"aG6$\"\")\"\"&#!=.40T$e'GXo6$QM>$\"<+!G*opDHK#\\,# *fJ" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/&%\"aG6$\"\")\"\"'#\"9dKc#3g?f $)\\L$=\"8)GK()y/+%p=IH#" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/&%\"aG6$ \"\")\"\"(#\";h?1CQAIU2(['e=\":g&=g(4i@<*GpN5" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%!G" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/&%\"cG6#\"\"*# \")yu6\\\")VZ9b" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/&%\"aG6$\"\"*\"\" \"#!jn,fr!o]&z>Nm([Bn/U$zFtz!y@_lr%*=!>,u:" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/&%\"aG6$\"\"*\" \"#\"\"!" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/&%\"aG6$\"\"*\"\"$\"\"!" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/&%\"aG6$\"\"*\"\"%#!hnj(o5uJ`Wwlx7* e?\"Qu([MCU5\"4c'H6CR\\#\"gn&3,Q%Q5#4-KV&)[(RI[(R+Mj$***)*>/)=w&[#" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#/&%\"aG6$\"\"*\"\"&#!co\\pvZXN]\\)Q`v_ WwbV([Tkt-314O*o?]HDa/TZ$\"bo+!3yY)=F;o\"[27_:&40jQ%yh`7Af\"[$)=4mFOZ( p" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/&%\"aG6$\"\"*\"\"'#\"[of%*4Ae,'e 0CjlYJ7,KI0\"[e7'yWeRzqLxK&\"[o_BPgAE[I4!)REGofD7KOnt)>?t90_&*3)\\7" } }{PARA 11 "" 1 "" {XPPMATH 20 "6#/&%\"aG6$\"\"*\"\"(#\"inBg\"R!f-0[OL* p4*3i#))*QiXMG&GC=jvk0Q'\"hngL1;Ve'*[H8&=J6p,V7s`K>4[a7!)o`A_" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#/&%\"aG6$\"\"*\"\")#\"M7pxZC_g\"owlhj@ A#>1Dupv;\"O2!G-alYDxZ[Lg)er!G30$fw\\;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%!G" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/&%\"cG6#\"#5#\"#8\"#9" } }{PARA 11 "" 1 "" {XPPMATH 20 "6#/&%\"aG6$\"#5\"\"\"#!L([kV2fG`S\\![J# y4T`Cy#Hq$\"L+!o:Q!>b5D'p;BI'**fMqd78%" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/&%\"aG6$\"#5\"\"#\"\"!" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/&%\" aG6$\"#5\"\"$\"\"!" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/&%\"aG6$\"#5\" \"%#!A(ROs(*3%*eudG>\\\\U&[\"@?RHGuFl]ttgac^z$" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/&%\"aG6$\"#5\"\"&#!M$>dR&p\\W*y+_*)>VxUnUbU>H#\"L+![!e 'Q?SCwx3CZ*3[v2?Y>$" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/&%\"aG6$\"#5\" \"'#\"O(ys'>m[^(>zRHbr1%ftz:.a%[#\"N)G[.>)4'4K1FOqOuH!))Q!p[;U" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#/&%\"aG6$\"#5\"\"(#\"PVL=v6k5KDLc5@9aU wN9wpaq\"\"O!3E$H8IXGmiy$\\y?O!y[S2=s5" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/&%\"aG6$\"#5\"\")#\"W4(RZ-8d&=l')['R7" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#/&%\"aG6$\"#5\"\"*#!N:^3!HQ;=!RTB$>C]U CI*e1sw#\"OsET'*f!)oYIstq$><;'[]:Ozj'*" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%!G" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/&%\"cG6#\"#6#\"#i\"#l" } }{PARA 11 "" 1 "" {XPPMATH 20 "6#/&%\"aG6$\"#6\"\"\"#!RrR(4jkoiIRz-t$H 322QA\\h_E)))\"R++Wl_L&z+C2e`Ot(R&H,MuH/=!)" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/&%\"aG6$\"#6\"\"#\"\"!" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/&%\"aG6$\"#6\"\"$\"\"!" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/&%\" aG6$\"#6\"\"%#!D$p&*=fR?'>.?t\")HSx+8*\"CDN3&*4u(f&)3EwSuM0b7?7\"\"P++W\\gm'*pT81n#4G_gOe?%>XI\"" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/&%\"aG6$\"#6\"\"'#\"W8U&HfIjd=w!y$GC\"QO?m93]S!4[L$=\" Vg\\1Fqf)4w*)Gx< Ei!GnT4_pn,yL" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/&%\"aG6$\"#6\"\")#\" Cmv'p/;+pSyjpP]\\Ch'\"EvB%RP%Qeie%GgqxoobD#" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/&%\"aG6$\"#6\"\"*#!YM3+JpuJq)4(**y8N')p)3t\\AlByUy$H5 \"envw='e0bu(>W-kd\\P'QtnC*)Gd&Rjj#yy)" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/&%\"aG6$\"#6\"#5#!:/bcCYSczjHc@%\"uO$[g6$))[vtH%ot3``\"N+gjYU&)[-!RIbp8Q91E0Zlw]" }}{PARA 11 " " 1 "" {XPPMATH 20 "6#/&%\"aG6$\"#7\"\"#\"\"!" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/&%\"aG6$\"#7\"\"$\"\"!" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/&%\"aG6$\"#7\"\"%#!C7jSW$)H%G?Cl=-]&Q\"e'\"B:Kl3eFF!=!y$)))GSYU )" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/&%\"aG6$\"#7\"\"&#!N(4jo!p=2DPG: KH.3\\)G1*4\")H#\"M+Sgx6(pi'f*=%>wR!GB1f5[&H" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/&%\"aG6$\"#7\"\"'#\"QL=ajgVl)G2+Pf`^YY/'GOu@OL\"P/:4mI hi'=Tmu)*>!3szFI9VUS&" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/&%\"aG6$\"#7 \"\"(#\"T$*)*>uT)p\\$faqvSswg$G7!*GIAO9G\"Sg\\rYD[%RT1(RQ9&)Q?HAGe>bYy C" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/&%\"aG6$\"#7\"\")#!<\"RUjFB5mTs3 ;e>\"<1^(yF&ouE,[m*)*R" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/&%\"aG6$\"# 7\"\"*#\"ZlzE^XD[Vkv&*R))RoWXfCX3QA3q_kW\"\"Zt*R1(=K.w=,I-f8=$[*)ej23r GoYc[J" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/&%\"aG6$\"#7\"#5#!& =w(eblL6\";z(o/,)3KAG/:4$*" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/&%\"aG6 $\"#7\"#6#\":vV4dv)['=.j%3!*\":y;L%e!o+yMiog'" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%!G" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/&%\"cG6#\"#8\" \"\"" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/&%\"aG6$\"#8\"\"\"#!J@YfVx*en iUev^\"=I46*obN\"J+!G'HK'GDF)HH&4*p)pwDsLB" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/&%\"aG6$\"#8\"\"#\"\"!" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/&%\"aG6$\"#8\"\"$\"\"!" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/&%\" aG6$\"#8\"\"%#!Ac/D&\\-vFGEE*)*)>jd\"\"@Xgc(H5ya;;%G>i0?\"" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/&%\"aG6$\"#8\"\"&#!K6>=YB_!\\ZL&o'R/v+pH#* \\u%\"J+?lWTVorQyZ$)H!zl'z$y5U" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/&% \"aG6$\"#8\"\"'#\"Nz1JTn2nf$[wN+zQrB,x)faz'\"M_.V&>A\"Qdf7E+YdC&>RKt8q (" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/&%\"aG6$\"#8\"\"(#\"Ox&prkXy0'*z %)*=7*=i(4>-r\")o8\"MSMM>=\"*>!y2V/kVG[sa\\!4jv" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/&%\"aG6$\"#8\"\")#!:eKnAtsl@(o\\#H$\";`x/7l*)z%G+j`#f " }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/&%\"aG6$\"#8\"\"*#!R!R4?'4.wwIFus -`tHld!\\4B_%o#\"Sn)>>Pm0MzWPukId*[-lImLGY$" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/&%\"aG6$\"#8\"#5#\"8oNt%zJSr8>jk\"9\\E!))*HICOvBk[" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#/&%\"aG6$\"#8\"#6\"\"!" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/&%\"aG6$\"#8\"#7\"\"!" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%!G" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/&%\"bG6#\"\"\"# \"1.JeyD)*\\5\"2gxGN.KFO#" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/&%\"bG6# \"\"#\"\"!" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/&%\"bG6#\"\"$\"\"!" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#/&%\"bG6#\"\"%\"\"!" }}{PARA 11 "" 1 " " {XPPMATH 20 "6#/&%\"bG6#\"\"&\"\"!" }}{PARA 11 "" 1 "" {XPPMATH 20 " 6#/&%\"bG6#\"\"'#\"3vyjT1iR#o(\"4'\\y$p/)onh@" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/&%\"bG6#\"\"(#\"7Dcv)p()*4Y@%)y\"83-)RLVHMv>oJ" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#/&%\"bG6#\"\")#!8\"Hdv[49X-ps>\"7+[/9, zO_Zf#*" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/&%\"bG6#\"\"*#\"enh\"RJ/]) \\^/r>N;Uuxx5#))H'eds(*[a)H#\"Z+'zfy4Z&yY`L#=*G[2DawO9_f*>m_Bt" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#/&%\"bG6#\"#5#!4[CuLy&y\\2B\"3Di5,,dX( y#" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/&%\"bG6#\"#6#\"6v8DGKkkO'*[\"\" 5[3&Rp1c>2-#" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/&%\"bG6#\"#7#\"-***46 4w&\".+ce$o3B" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/&%\"bG6#\"#8\"\"!" } }{PARA 11 "" 1 "" {XPPMATH 20 "6#%!G" }}{PARA 11 "" 1 "" {XPPMATH 20 " 6#/&%#b*G6#\"\"\"#\".H)*)p+][\"0!GJ0m\"))3\"" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/&%#b*G6#\"\"#\"\"!" }}{PARA 11 "" 1 "" {XPPMATH 20 "6# /&%#b*G6#\"\"$\"\"!" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/&%#b*G6#\"\"% \"\"!" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/&%#b*G6#\"\"&\"\"!" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/&%#b*G6#\"\"'#\"1vo=+%)H,e\"27$*eF'4KD;" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#/&%#b*G6#\"\"(#\"4D1`gg#f43C\"4KC()y&z 7k\"p*" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/&%#b*G6#\"\")#\"5L?z/y&)Hi+ b\"5+'4$*GloSZd\"" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/&%#b*G6#\"\"*#\" Qjub[%[Hq**oj3DN&43TspD*[[j$\"Q+GzoU*\\p)GL!e:kPU$z))=j5y2;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/&%#b*G6#\"#5#!2_^TFQt(>9\"1D0.r3zRS" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#/&%#b*G6#\"#6\"\"!" }}{PARA 11 "" 1 " " {XPPMATH 20 "6#/&%#b*G6#\"#7\"\"!" }}{PARA 11 "" 1 "" {XPPMATH 20 "6 #/&%#b*G6#\"#8#!+29hI!*\"++K))zZ" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 151 "RK8_13eqs := [op( RowSumConditions(13,'expanded')),op(OrderConditions(8,13,'expanded'))] :\n`RK7_13eqs*` := subs(b=`b*`,OrderConditions(7,13,'expanded')):" }}} {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 6 "Check:" }} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 110 "subs(ee,RK8_13eqs):\nmap(u- >lhs(u)-rhs(u),%);\nnops(%);\nsubs(ee,`RK7_13eqs*`):\nmap(u->lhs(u)-rh s(u),%);\nnops(%);" }}{PARA 12 "" 1 "" {XPPMATH 20 "6#7`x\"\"!F$F$F$F$ F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$ F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$ F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$ F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$ F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$ F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#\"$7#" }}{PARA 12 "" 1 "" {XPPMATH 20 "6#7ap\"\"!F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F $F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F $F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#\"#&)" }}}{PARA 257 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 78 "Set-up stage-order condtions to check for stage-orders fr om 2 to 6 inclusive. " }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 87 "for ct from 2 to 6 do\n so||ct||_13 := StageOrderConditions(ct,13,'expa nded');\nend do:" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 59 "Stages 3 to 13 have the following respective stage-orders . " }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 175 "[seq([seq(expand(subs (ee,so||i||_13[j])),i=2..6)],j=1..11)]:\nmap(proc(L) local i; for i to nops(L) do if not evalb(L[i]) then break end if end do; i end proc,%) :\nsimplify(%); " }}{PARA 11 "" 1 "" {XPPMATH 20 "6#7-\"\"#\"\"$F%\"\" %F&F&F&F&F&F&F&" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 27 "The simplifying conditions:" }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "Sum(b[i]*a[i,j],i = j+1 .. 13) = b[j]*(1-c[j] );" "6#/-%$SumG6$*&&%\"bG6#%\"iG\"\"\"&%\"aG6$F+%\"jGF,/F+;,&F0F,F,F, \"#8*&&F)6#F0F,,&F,F,&%\"cG6#F0!\"\"F," }{TEXT -1 3 ", " }{XPPEDIT 18 0 "j=1" "6#/%\"jG\"\"\"" }{TEXT -1 9 " . . 12, " }}{PARA 0 "" 0 "" {TEXT -1 8 "(where " }{XPPEDIT 18 0 "c[1]=0" "6#/&%\"cG6#\"\"\"\"\"! " }{TEXT -1 17 " ) are satisfied." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 151 "[Sum(b[i]*a[i,1],i=1+1..13) =b[1],seq(Sum(b[i]*a[i,j],i=j+1..13)=b[j]*(1-c[j]),j=2..12)]:\nmap(u-> lhs(u)-rhs(u),subs(ee,eval(subs(Sum=add,%))));\nnops(%);" }}{PARA 11 " " 1 "" {XPPMATH 20 "6#7.\"\"!F$F$F$F$F$F$F$F$F$F$F$" }}{PARA 11 "" 1 " " {XPPMATH 20 "6#\"#7" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 27 "The simplifying conditions:" }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "Sum(`b*`[i]*a[i,j],i = j+1 .. 13) = `b* `[j]*(1-c[j]);" "6#/-%$SumG6$*&&%#b*G6#%\"iG\"\"\"&%\"aG6$F+%\"jGF,/F+ ;,&F0F,F,F,\"#8*&&F)6#F0F,,&F,F,&%\"cG6#F0!\"\"F," }{TEXT -1 3 ", " } {XPPEDIT 18 0 "j=1" "6#/%\"jG\"\"\"" }{TEXT -1 9 " . . 12, " }}{PARA 0 "" 0 "" {TEXT -1 8 "(where " }{XPPEDIT 18 0 "c[1]=0" "6#/&%\"cG6#\" \"\"\"\"!" }{TEXT -1 17 " ) are satisfied." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 163 "[Sum(`b*`[i]*a[i, 1],i=1+1..13)=`b*`[1],seq(Sum(`b*`[i]*a[i,j],i=j+1..13)=`b*`[j]*(1-c[j ]),j=2..12)]:\nmap(u->lhs(u)-rhs(u),subs(ee,eval(subs(Sum=add,%))));\n nops(%);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#7.\"\"!F$F$F$F$F$F$F$F$F$F $F$" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#\"#7" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 93 "We can calculate the principal \+ error norm, that is, the 2-norm of the principal error terms. " }} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 162 "errterms8_13 := PrincipalEr rorTerms(8,13,'expanded'):\nsm := 0:\nfor ct to nops(errterms8_13) do \n sm := sm+(evalf(subs(ee,errterms8_13[ct])))^2;\nend do:\nsqrt(sm) ;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#$\"+US&Rt&!#;" }}}{PARA 0 "" 0 " " {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 78 "We can also calculate t he principal error norm of the order 7 embedded scheme." }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 184 "`errterms7_13*` := subs(b=`b*`,Pri ncipalErrorTerms(7,13,'expanded')):\nsm := 0:\nfor ct to nops(`errterm s7_13*`) do\n sm := sm+(evalf(subs(ee,`errterms7_13*`[ct])))^2;\nend do:\nsqrt(sm);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#$\"+!oeQ+\"!#9" }}} {PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}}{PARA 0 "" 0 "" {TEXT -1 52 "#------------------------------ ---------------------" }}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 34 "construc tion of the order 8 scheme" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 93 "First we construct a 12 stage order 8 scheme star ting with a consideration of stages 2 to 8. " }}{PARA 0 "" 0 "" {TEXT -1 20 "We specify the nodes" }}{PARA 256 "" 0 "" {TEXT -1 2 " " } {XPPEDIT 18 0 "c[2] = 27/400;" "6#/&%\"cG6#\"\"#*&\"#F\"\"\"\"$+%!\"\" " }{TEXT -1 2 ", " }{XPPEDIT 18 0 "c[5] = 100/257;" "6#/&%\"cG6#\"\"&* &\"$+\"\"\"\"\"$d#!\"\"" }{TEXT -1 2 ", " }{XPPEDIT 18 0 "c[6] = 7/15; " "6#/&%\"cG6#\"\"'*&\"\"(\"\"\"\"#:!\"\"" }{TEXT -1 2 ", " }{XPPEDIT 18 0 "c[7] = 7/45;" "6#/&%\"cG6#\"\"(*&F'\"\"\"\"#X!\"\"" }{TEXT -1 3 ", " }{XPPEDIT 18 0 "c[8] = 58/59;" "6#/&%\"cG6#\"\")*&\"#e\"\"\"\"#f !\"\"" }{TEXT -1 2 " " }}{PARA 0 "" 0 "" {TEXT -1 33 "and the zero li nking coefficients" }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "a[4,2]=0" "6#/&%\"aG6$\"\"%\"\"#\"\"!" }{TEXT -1 2 ", " }{XPPEDIT 18 0 "a[5,2]=0" "6#/&%\"aG6$\"\"&\"\"#\"\"!" }{TEXT -1 2 ", " }{XPPEDIT 18 0 "a[6,2]=0" "6#/&%\"aG6$\"\"'\"\"#\"\"!" }{TEXT -1 2 ", " } {XPPEDIT 18 0 "a[6,3]=0" "6#/&%\"aG6$\"\"'\"\"$\"\"!" }{TEXT -1 2 ", \+ " }{XPPEDIT 18 0 "a[7,2]=0" "6#/&%\"aG6$\"\"(\"\"#\"\"!" }{TEXT -1 2 " , " }{XPPEDIT 18 0 "a[7,3]=0" "6#/&%\"aG6$\"\"(\"\"$\"\"!" }{TEXT -1 2 ", " }{XPPEDIT 18 0 "a[8,2] = 0;" "6#/&%\"aG6$\"\")\"\"#\"\"!" } {TEXT -1 2 ", " }{XPPEDIT 18 0 "a[8,3] = 0;" "6#/&%\"aG6$\"\")\"\"$\" \"!" }{TEXT -1 3 ", " }{XPPEDIT 18 0 "a[8,4]=0" "6#/&%\"aG6$\"\")\"\" %\"\"!" }{TEXT -1 2 ". " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 48 "In addition we specify the linking coefficient " } {XPPEDIT 18 0 "a[8,4] = -743/55;" "6#/&%\"aG6$\"\")\"\"%,$*&\"$V(\"\" \"\"#b!\"\"F." }{TEXT -1 1 "." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }} {PARA 0 "" 0 "" {TEXT -1 69 "The stage-order equations relating to the se stages, such that stages " }{XPPEDIT 18 0 "3,4,5,6,7,8;" "6(\"\"$\" \"%\"\"&\"\"'\"\"(\"\")" }{TEXT -1 19 " have stage orders " }{XPPEDIT 18 0 "2,3,3,4,4,4;" "6(\"\"#\"\"$F$\"\"%F%F%" }{TEXT -1 48 " respectiv ely can be solved to obtain the nodes " }{XPPEDIT 18 0 "c[3], c[4]" "6 $&%\"cG6#\"\"$&F$6#\"\"%" }{TEXT -1 67 " and the remaining non-zero li nking coefficients for these stages. " }}{PARA 0 "" 0 "" {TEXT -1 0 " " }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 447 "RSeqs := RowSumConditio ns(8,'expanded'):\nSOeqs := [op(StageOrderConditions(2,8,'expanded')), op(StageOrderConditions(3,4..8,'expanded')),\n op(StageO rderConditions(4,6..8,'expanded'))]:\ne1 := \{c[2]=27/400,c[5]=100/257 ,c[6]=7/15,c[7]=7/45,c[8]=58/59,\n seq(a[i,2]=0,i=4..8),seq(a[i, 3]=0,i=6..8),a[8,4]=-743/55\}:\neqns := map(expand,subs(e1,[op(RSeqs), op(SOeqs)])):\nconvert(ListTools[Enumerate](%),matrix);\n``;\nindets(e qns);\nnops(%);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#K%'matrixG6#777$\" \"\"/&%\"aG6$\"\"#F(#\"#F\"$+%7$F-/,&&F+6$\"\"$F(F(&F+6$F6F-F(&%\"cG6# F67$F6/,&&F+6$\"\"%F(F(&F+6$FAF6F(&F:6#FA7$FA/,(&F+6$\"\"&F(F(&F+6$FKF 6F(&F+6$FKFAF(#\"$+\"\"$d#7$FK/,(&F+6$\"\"'F(F(&F+6$FXFAF(&F+6$FXFKF(# \"\"(\"#:7$FX/,*&F+6$FhnF(F(&F+6$FhnFAF(&F+6$FhnFKF(&F+6$FhnFXF(#Fhn\" #X7$Fhn/,,&F+6$\"\")F(F(#\"$V(\"#b!\"\"&F+6$F\\pFKF(&F+6$F\\pFXF(&F+6$ F\\pFhnF(#\"#e\"#f7$F\\p/,$*&F.F(F7F(F(,$*&#F(F-F(*$)F9F-F(F(F(7$\"\"* /*&FBF(F9F(,$*&F`qF(*$)FDF-F(F(F(7$\"#5/,&*&FLF(F9F(F(*&FNF(FDF(F(#\"% +]\"&\\g'7$\"#6/,&*&FYF(FDF(F(*&FPF(FenF(F(#\"#\\\"$]%7$\"#7/,(*&F_oF( FDF(F(*&FPF(FaoF(F(*&FgnF(FcoF(F(#F[s\"%]S7$\"#8/,**&#F^pF_pF(FDF(F`p* &FPF(FapF(F(*&FgnF(FcpF(F(*&FeoF(FepF(F(#\"%#o\"\"%\"[$7$\"#9/*&FBF(Fb qF(,$*&#F(F6F(*$)FDF6F(F(F(7$Fin/,&*&FLF(FbqF(F(*&FNF(FjqF(F(#\"(+++\" \")zP#4&7$\"#;/,&*&FYF(FjqF(F(*&#\"&++\"FcrF(FenF(F(#\"$V$\"&D,\"7$\"# \"'Phh7$\"#>/,&*&FYF(FjtF(F(*&#Fau\")$fup\"F(FenF(F(#\"%,C\"'+D? 7$\"#?/,(*&F_oF(FjtF(F(*&F]xF(FaoF(F(*&#F\\v\"%vLF(FcoF(F(#F`x\")+DS;7 $\"#@/,**&#F^pF_pF(FitF(F`p*&F]xF(FapF(F(*&FixF(FcpF(F(*&#F\\v\"&D6*F( FepF(F(#\"(C\"HG\")ht67Q)pprint306\"" }}{PARA 11 "" 1 "" {XPPMATH 20 " 6#%!G" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#<7&%\"aG6$\"\"(\"\"\"&F%6$\" \"'\"\"%&F%6$\"\"$\"\"#&F%6$\"\"&F,&%\"cG6#F/&F%6$F3F/&F%6$\"\")F(&F%6 $F,F/&F%6$F;F'&F%6$F/F(&F%6$F,F(&F%6$F'F3&F%6$F;F3&F%6$F0F(&F%6$F+F3&F %6$F'F,&F56#F,&F%6$F;F+&F%6$F'F+&F%6$F3F(&F%6$F+F(" }}{PARA 11 "" 1 " " {XPPMATH 20 "6#\"#@" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 46 "e2 := solve(\{op(eqns)\});\ne3 := ` union`(e1,e2):" }}{PARA 12 "" 1 "" {XPPMATH 20 "6#>%#e2G<7/&%\"aG6$\" \")\"\"\"#!8Z97&f;i6b,vD\"8+!G`TFq5(Rs\">/&F(6$\"\"&F+#\"/+,V'H&)H\"\" /dy0OPgL/&F(6$F*\"\"(#\";h?1CQAIU2(['e=\":g&=g(4i@<*GpN5/&F(6$F9F2#!.$ 4xdN;T\"0+5PY$\"<+!G*opDHK#\\,#*fJ/&F(6$FF\"\"%#\",78>u- \"\",0MM\"*R%/&F(6$F2Fao#\"/++?qTu[F5/&%\"cG6#FM#\"$p%\"%5X/&F(6$F2FM# !/++lF!=i\"\"/>ENX7?6/&F(6$FaoFM#\"%@U\"&!3O/&F(6$FMF+#\"($z68\")q#=\\ &/&F(6$FNF+#\"#F\"$+%/&F\\p6#Fao#\"%29\"%?!*/&F(6$F9Fao#\"/7P,pgV`\"0X Y'3V-bZ/&F(6$FaoF+#F]rF\\q/&F(6$F*FF#\"9dKc#3g?f$)\\L$=\"8)GK()y/+%p=I H#/&F(6$F9F+#\"(2I)))\"*+!Hl9" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }} {PARA 0 "" 0 "" {TEXT 260 4 "Note" }{TEXT -1 2 ": " }}{PARA 0 "" 0 "" {TEXT -1 8 "We have " }{XPPEDIT 18 0 "c[3]=2/3" "6#/&%\"cG6#\"\"$*&\" \"#\"\"\"F'!\"\"" }{TEXT -1 1 " " }{XPPEDIT 18 0 "c[4]" "6#&%\"cG6#\" \"%" }{TEXT -1 16 " and the nodes " }{XPPEDIT 18 0 "c[4];" "6#&%\"cG6 #\"\"%" }{TEXT -1 3 ", " }{XPPEDIT 18 0 "c[5];" "6#&%\"cG6#\"\"&" } {TEXT -1 3 ", " }{XPPEDIT 18 0 "c[6];" "6#&%\"cG6#\"\"'" }{TEXT -1 31 " are related by the equation: " }}{PARA 256 "" 0 "" {TEXT -1 1 " \+ " }{XPPEDIT 18 0 "3*c[6]^2-4*c[6]*c[4]-4*c[5]*c[6]+6*c[5]*c[4] = 0" "6 #/,**&\"\"$\"\"\"*$&%\"cG6#\"\"'\"\"#F'F'*(\"\"%F'&F*6#F,F'&F*6#F/F'! \"\"*(F/F'&F*6#\"\"&F'&F*6#F,F'F4*(F,F'&F*6#F8F'&F*6#F/F'F'\"\"!" } {TEXT -1 2 ". " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 86 "subs(e3,matrix([seq([c[i],seq(a[i,j],j=1..i-1),` `$(9-i)],i=2..8)]));\n``;\nevalf[8](%%);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#K%'matrixG6#7)7+#\"#F\"$+%F(%!GF+F+F+F+F+F+7+#\"$p%\"%5X#\"($z68 \")q#=\\&#\"'A*R%\"(F=\\&F+F+F+F+F+F+7+#\"%29\"%?!*#F8\"&!3O\"\"!#\"%@ UF;F+F+F+F+F+7+#\"$+\"\"$d##\"/+,V'H&)H\"\"/dy0OPgLF<#!/++lF!=i\"\"/>E NX7?6#\"/++?qTu[FEF+F+F+F+7+#\"\"(\"#:#\"(2y\"Q\")+]S\")Fu- \"\",0MM\"*R%#\",$z(*fvS\"-+]Si)=#F+F+F+7+#FM\"#X#\"(2I)))\"*+!Hl9FF$ \"<+!G*opDHK#\\,#*fJ#\"9dKc#3g?f$)\\L$=\"8)GK()y/+%p=IH##\";h?1CQAIU2( ['e=\":g&=g(4i@<*GpN5F+Q)pprint316\"" }}{PARA 11 "" 1 "" {XPPMATH 20 " 6#%!G" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#K%'matrixG6#7)7+$\")++]n!\"*F (%!GF+F+F+F+F+F+7+$\")8\"*R5!\")$\")xi)Q#F*$\")a[5!)F*F+F+F+F+F+F+7+$ \")q')f:F/$\")um**QF*$\"\"!F:$\")-!*p6F/F+F+F+F+F+7+$\")10\"*QF/$\")>C kQF/F9$!)s(yW\"!\"($\")\"e0X\"FDF+F+F+F+7+$\")nmmYF/$\")D*)*o%F*F9F9$ \")I]NBF/$\")W " 0 "" {MPLTEXT 1 0 1 ";" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 27 "We do not need to specify " }{XPPEDIT 18 0 "c[9];" "6#&% \"cG6#\"\"*" }{TEXT -1 42 ", because according to Verner, the nodes \+ " }{XPPEDIT 18 0 "c[6];" "6#&%\"cG6#\"\"'" }{TEXT -1 2 ", " }{XPPEDIT 18 0 "c[7];" "6#&%\"cG6#\"\"(" }{TEXT -1 2 ", " }{XPPEDIT 18 0 "c[8]; " "6#&%\"cG6#\"\")" }{TEXT -1 2 ", " }{XPPEDIT 18 0 "c[9];" "6#&%\"cG6 #\"\"*" }{TEXT -1 30 " are related by the equation " }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "Int(p(x)*``((1-x)^3/3!),x = 0 .. 1 )*Int(q(x)*(1-x),x = 0 .. 1) = Int(p(x)*``((1-x)^2/2!),x = 0 .. 1)*Int (q(x)*``((1-x)^2/2!),x = 0 .. 1);" "6#/*&-%$IntG6$*&-%\"pG6#%\"xG\"\" \"-%!G6#*&,&F-F-F,!\"\"\"\"$-%*factorialG6#F4F3F-/F,;\"\"!F-F--F&6$*&- %\"qG6#F,F-,&F-F-F,F3F-/F,;F:F-F-*&-F&6$*&-F*6#F,F--F/6#*&,&F-F-F,F3\" \"#-F66#FNF3F-/F,;F:F-F--F&6$*&-F?6#F,F--F/6#*&,&F-F-F,F3FN-F66#FNF3F- /F,;F:F-F-" }{TEXT -1 2 ", " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 7 "where " }{XPPEDIT 18 0 "p(x) = x*(x-c[6])*(x-c[ 7])*(x-c[8]);" "6#/-%\"pG6#%\"xG**F'\"\"\",&F'F)&%\"cG6#\"\"'!\"\"F),& F'F)&F,6#\"\"(F/F),&F'F)&F,6#\"\")F/F)" }{TEXT -1 7 " and " } {XPPEDIT 18 0 "q(x) = (x-c[9])*p(x);" "6#/-%\"qG6#%\"xG*&,&F'\"\"\"&% \"cG6#\"\"*!\"\"F*-%\"pG6#F'F*" }{TEXT -1 2 ". " }}{PARA 0 "" 0 "" {TEXT -1 1 " " }}{PARA 0 "" 0 "" {TEXT -1 155 "See: J.H. Verner, SIAM \+ Journal of Numerical Analysis 1978, 772-790, \"Explicit Runge-Kutta me thods with estimates of the Local Truncation Error.\" (page 780)" }} {PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 90 "p := x -> x*(x-c[6])*(x-c[7])*(x-c[8]):\n'p(x)'=p(x);\nq := x -> ( x-c[9])*p(x):\n'q(x)'=q(x);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/-%\"pG 6#%\"xG**F'\"\"\",&F'F)&%\"cG6#\"\"'!\"\"F),&F'F)&F,6#\"\"(F/F),&F'F)& F,6#\"\")F/F)" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/-%\"qG6#%\"xG*,,&F' \"\"\"&%\"cG6#\"\"*!\"\"F*F'F*,&F'F*&F,6#\"\"'F/F*,&F'F*&F,6#\"\"(F/F* ,&F'F*&F,6#\"\")F/F*" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 239 "Ieq := Int('p(x)'*(1-x)^3/3!,x=0..1)*Int ('q(x)'*(1-x),x=0..1)=\n Int('p(x)'*(1-x)^2/2!,x=0..1)*Int('q(x)'*(1- x)^2/2!,x=0..1);\nvalue(Ieq):\nReq := simplify(lhs(%)-rhs(%))=0:\nL := ilcm(op(map(denom,[coeffs(lhs(Req))]))):\nCeq := expand(L*Req);" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#>%$IeqG/*&-%$IntG6$,$*&#\"\"\"\"\"'F-* &-%\"pG6#%\"xGF-),&F-F-F3!\"\"\"\"$F-F-F-/F3;\"\"!F-F--F(6$*&-%\"qGF2F -F5F-F8F-*&-F(6$,$*&#F-\"\"#F-*&F0F-)F5FFF-F-F-F8F--F(6$,$*&FEF-*&F>F- FHF-F-F-F8F-" }}{PARA 12 "" 1 "" {XPPMATH 20 "6#>%$CeqG/,hp*&\"\"$\"\" \"&%\"cG6#\"\"'F)!\"\"*&F(F)&F+6#\"\")F)F.*&F(F)&F+6#\"\"(F)F.*&\"#7F) &F+6#\"\"*F)F)*(\"#9F))F*\"\"#F)F4F)F.*(\"#GF)F*F)F4F)F)*(FAF)F0F)F*F) F)*(FAF)F0F)F4F)F)*(F=F)F0F)F>F)F.*(F=F))F0F?F)F*F)F.*(F=F)F*F))F4F?F) F.*(F=F)FFF)F4F)F.*(F=F)F0F)FHF)F.*(\"#UF)F9F)F*F)F.*(FLF)F4F)F9F)F.*( FLF)F0F)F9F)F.*(FLF)F9F)F>F)F)*(FLF)FHF)F9F)F)*(FLF)FFF)F9F)F)*,\"$)eF )F9F)F*F)F4F)F0F)F.*,\"$G(F)F0F)F>F)F9F)F4F)F)*,FUF)FFF)F*F)F4F)F9F)F) *,\"$!)*F)FFF)F>F)F9F)F4F)F.*,FUF)F0F)FHF)F9F)F*F)F)*,FXF)FFF)FHF)F9F) F*F)F.*,FXF)F>F)FHF)F9F)F0F)F.*,\"%q9F)F>F)FHF)FFF)F9F)F)**\"$J#F)F*F) F4F)F0F)F.**\"#)*F)F>F)FHF)F0F)F.**F[oF)FFF)FHF)F*F)F.**F[oF)FFF)F>F)F 4F)F.**\"$o\"F)FFF)F*F)F4F)F)**F_oF)F0F)FHF)F*F)F)**F_oF)F0F)F>F)F4F)F )**\"$a\"F)F9F)F*F)F4F)F)**FcoF)F0F)F9F)F*F)F)**FcoF)F0F)F4F)F9F)F)**F _oF)F0F)FHF)F9F)F.**F_oF)F0F)F>F)F9F)F.**F_oF)FFF)F*F)F9F)F.**\"$'>F)F FF)F>F)F9F)F)**F_oF)FFF)F4F)F9F)F.**FjoF)FFF)FHF)F9F)F)**F_oF)F9F)F>F) F4F)F.**F_oF)F*F)FHF)F9F)F.**FjoF)F>F)FHF)F9F)F)\"\"!" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 60 "The previous equatio n involving integrals is equivalent to: " }}{PARA 256 "" 0 "" {TEXT -1 2 " " }{XPPEDIT 18 0 "14*c[6]^2*c[7]+3*c[6]-28*c[6]*c[7]+3*c[7]+58 8*c[9]*c[6]*c[7]*c[8]+3*c[8]-28*c[8]*c[6]-28*c[8]*c[7]+14*c[8]*c[6]^2+ 14*c[8]^2*c[6]+231*c[6]*c[7]*c[8]+98*c[6]^2*c[7]^2*c[8]+14*c[6]*c[7]^2 +98*c[8]^2*c[7]^2*c[6]+98*c[8]^2*c[6]^2*c[7]-168*c[8]^2*c[6]*c[7]-168* c[8]*c[7]^2*c[6]-168*c[8]*c[6]^2*c[7]+14*c[8]^2*c[7]-12*c[9]+14*c[8]*c [7]^2+42*c[9]*c[6]-154*c[9]*c[6]*c[7]+42*c[7]*c[9]+42*c[8]*c[9]-154*c[ 8]*c[9]*c[6]-154*c[8]*c[7]*c[9]-42*c[9]*c[6]^2-42*c[7]^2*c[9]-42*c[8]^ 2*c[9]+168*c[8]*c[7]^2*c[9]-728*c[8]*c[6]^2*c[9]*c[7]-728*c[8]^2*c[6]* c[7]*c[9]+980*c[8]^2*c[6]^2*c[9]*c[7]-728*c[8]*c[7]^2*c[9]*c[6]+980*c[ 8]^2*c[7]^2*c[9]*c[6]+980*c[6]^2*c[7]^2*c[9]*c[8]-1470*c[6]^2*c[7]^2*c [8]^2*c[9]+168*c[8]*c[6]^2*c[9]+168*c[8]^2*c[6]*c[9]-196*c[8]^2*c[6]^2 *c[9]+168*c[8]^2*c[7]*c[9]-196*c[8]^2*c[7]^2*c[9]+168*c[9]*c[6]^2*c[7] +168*c[6]*c[7]^2*c[9]-196*c[6]^2*c[7]^2*c[9]=0" "6#/,hp*(\"#9\"\"\"*$& %\"cG6#\"\"'\"\"#F'&F*6#\"\"(F'F'*&\"\"$F'&F*6#F,F'F'*(\"#GF'&F*6#F,F' &F*6#F0F'!\"\"*&F2F'&F*6#F0F'F'*,\"$)eF'&F*6#\"\"*F'&F*6#F,F'&F*6#F0F' &F*6#\"\")F'F'*&F2F'&F*6#FJF'F'*(F6F'&F*6#FJF'&F*6#F,F'F;*(F6F'&F*6#FJ F'&F*6#F0F'F;*(F&F'&F*6#FJF'&F*6#F,F-F'*(F&F'*$&F*6#FJF-F'&F*6#F,F'F'* *\"$J#F'&F*6#F,F'&F*6#F0F'&F*6#FJF'F'**\"#)*F'*$&F*6#F,F-F'&F*6#F0F-&F *6#FJF'F'*(F&F'&F*6#F,F'&F*6#F0F-F'**FfoF'*$&F*6#FJF-F'&F*6#F0F-&F*6#F ,F'F'**FfoF'*$&F*6#FJF-F'&F*6#F,F-&F*6#F0F'F'**\"$o\"F'*$&F*6#FJF-F'&F *6#F,F'&F*6#F0F'F;**FdqF'&F*6#FJF'&F*6#F0F-&F*6#F,F'F;**FdqF'&F*6#FJF' &F*6#F,F-&F*6#F0F'F;*(F&F'*$&F*6#FJF-F'&F*6#F0F'F'*&\"#7F'&F*6#FCF'F;* (F&F'&F*6#FJF'&F*6#F0F-F'*(\"#UF'&F*6#FCF'&F*6#F,F'F'**\"$a\"F'&F*6#FC F'&F*6#F,F'&F*6#F0F'F;*(FjsF'&F*6#F0F'&F*6#FCF'F'*(FjsF'&F*6#FJF'&F*6# FCF'F'**F`tF'&F*6#FJF'&F*6#FCF'&F*6#F,F'F;**F`tF'&F*6#FJF'&F*6#F0F'&F* 6#FCF'F;*(FjsF'&F*6#FCF'&F*6#F,F-F;*(FjsF'*$&F*6#F0F-F'&F*6#FCF'F;*(Fj sF'*$&F*6#FJF-F'&F*6#FCF'F;**FdqF'&F*6#FJF'&F*6#F0F-&F*6#FCF'F'*,\"$G( F'&F*6#FJF'&F*6#F,F-&F*6#FCF'&F*6#F0F'F;*,FhwF'*$&F*6#FJF-F'&F*6#F,F'& F*6#F0F'&F*6#FCF'F;*,\"$!)*F'*$&F*6#FJF-F'&F*6#F,F-&F*6#FCF'&F*6#F0F'F '*,FhwF'&F*6#FJF'&F*6#F0F-&F*6#FCF'&F*6#F,F'F;*,F\\yF'*$&F*6#FJF-F'&F* 6#F0F-&F*6#FCF'&F*6#F,F'F'*,F\\yF'*$&F*6#F,F-F'&F*6#F0F-&F*6#FCF'&F*6# FJF'F'*,\"%q9F'*$&F*6#F,F-F'&F*6#F0F-&F*6#FJF-&F*6#FCF'F;**FdqF'&F*6#F JF'&F*6#F,F-&F*6#FCF'F'**FdqF'*$&F*6#FJF-F'&F*6#F,F'&F*6#FCF'F'**\"$'> F'*$&F*6#FJF-F'&F*6#F,F-&F*6#FCF'F;**FdqF'*$&F*6#FJF-F'&F*6#F0F'&F*6#F CF'F'**F^]lF'*$&F*6#FJF-F'&F*6#F0F-&F*6#FCF'F;**FdqF'&F*6#FCF'&F*6#F,F -&F*6#F0F'F'**FdqF'&F*6#F,F'&F*6#F0F-&F*6#FCF'F'**F^]lF'*$&F*6#F,F-F'& F*6#F0F-&F*6#FCF'F;\"\"!" }{TEXT -1 2 ". " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 75 "e4 := solve(subs(e3,C eq),\{c[9]\}):\nc[9]=subs(e4,c[9]);\ne5 := `union`(e3,e4):" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/&%\"cG6#\"\"*#\")yu6\\\")VZ9b" }}}{PARA 0 " " 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 7 "Thus " }{XPPEDIT 18 0 "c[9] = 49117478/55144743" "6#/&%\"cG6#\"\"**&\")yu6\\\"\"\"\")VZ 9b!\"\"" }{TEXT -1 2 ". " }}{PARA 0 "" 0 "" {TEXT -1 84 "Now we can us e the quadrature equations to find the weights once the remaining node s" }}{PARA 256 "" 0 "" {TEXT -1 2 " " }{XPPEDIT 18 0 "c[10] = 13/14; " "6#/&%\"cG6#\"#5*&\"#8\"\"\"\"#9!\"\"" }{TEXT -1 3 ", " }{XPPEDIT 18 0 "c[11] = 62/65;" "6#/&%\"cG6#\"#6*&\"#i\"\"\"\"#l!\"\"" }{TEXT -1 3 ", " }{XPPEDIT 18 0 "c[12] = 1;" "6#/&%\"cG6#\"#7\"\"\"" }{TEXT -1 1 " " }}{PARA 0 "" 0 "" {TEXT -1 38 "are specified along with the w eights " }{XPPEDIT 18 0 "b[2]=0" "6#/&%\"bG6#\"\"#\"\"!" }{TEXT -1 2 ", " }{XPPEDIT 18 0 "b[3]=0" "6#/&%\"bG6#\"\"$\"\"!" }{TEXT -1 2 ", " }{XPPEDIT 18 0 "b[4]=0" "6#/&%\"bG6#\"\"%\"\"!" }{TEXT -1 2 ", " } {XPPEDIT 18 0 "b[5]=0" "6#/&%\"bG6#\"\"&\"\"!" }{TEXT -1 2 ". " }} {PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 288 "Qeqs := QuadratureConditions(8,12,'expanded'):\ne6 := \{seq(b[i]= 0,i=2..5),c[10]=13/14,c[11]=62/65,c[12]=1\}:\ne7 := `union`(e5,e6):\nq uadeqns := subs(e7,Qeqs):\nfor ct to nops(quadeqns) do\n print(`equa tion `||ct); print(``);print(quadeqns[ct]);print(``);\nend do:\ninde ts(quadeqns);\nnops(%);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%-equation~ ~~1G" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%!G" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/,2&%\"bG6#\"\"\"F(&F&6#\"\"'F(&F&6#\"\"(F(&F&6#\"\")F( &F&6#\"\"*F(&F&6#\"#5F(&F&6#\"#6F(&F&6#\"#7F(F(" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%!G" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%-equation~~~2G " }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%!G" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/,0*&#\"\"(\"#:\"\"\"&%\"bG6#\"\"'F)F)*&#F'\"#XF)&F+6#F'F)F)*&# \"#e\"#fF)&F+6#\"\")F)F)*&#\")yu6\\\")VZ9bF)&F+6#\"\"*F)F)*&#\"#8\"#9F )&F+6#\"#5F)F)*&#\"#i\"#lF)&F+6#\"#6F)F)&F+6#\"#7F)#F)\"\"#" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%!G" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%-equ ation~~~3G" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%!G" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/,0*&#\"#\\\"$D#\"\"\"&%\"bG6#\"\"'F)F)*&#F'\"%D?F)&F+6 #\"\"(F)F)*&#\"%kL\"%\"[$F)&F+6#\"\")F)F)*&#\"1%[!3Xm_7C\"1\\g`!oU4/$F )&F+6#\"\"*F)F)*&#\"$p\"\"$'>F)&F+6#\"#5F)F)*&#\"%WQ\"%DUF)&F+6#\"#6F) F)&F+6#\"#7F)#F)\"\"$" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%!G" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%-equation~~~4G" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%!G" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/,0*&#\"$V$\"%vL \"\"\"&%\"bG6#\"\"'F)F)*&#F'\"&D6*F)&F+6#\"\"(F)F)*&#\"'7^>\"'z`?F)&F+ 6#\"\")F)F)*&#\"9_$*4\"[aTTCs\\=\"\"92/MC:*ef-?pn\"F)&F+6#\"\"*F)F)*&# \"%(>#\"%WFF)&F+6#\"#5F)F)*&#\"'G$Q#\"'DYFF)&F+6#\"#6F)F)&F+6#\"#7F)#F )\"\"%" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%!G" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%-equation~~~5G" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%!G " }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/,0*&#\"%,C\"&D1&\"\"\"&%\"bG6#\" \"'F)F)*&#F'\"(D15%F)&F+6#\"\"(F)F)*&#\")'\\;8\"\")ht67F)&F+6#\"\")F)F )*&#\"@cUnP))*Rh&HBK\"[G?e\"@,/`))*Hc'4x0jQKtC*F)&F+6#\"\"*F)F)*&#\"&h &G\"&;%QF)&F+6#\"#5F)F)*&#\")Ojx9\")D1&y\"F)&F+6#\"#6F)F)&F+6#\"#7F)#F )\"\"&" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%!G" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%-equation~~~6G" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%!G " }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/,0*&#\"&2o\"\"'v$f(\"\"\"&%\"bG6# \"\"'F)F)*&#F'\"*D\"GX=F)&F+6#\"\"(F)F)*&#\"*onNc'\"**HC\\rF)&F+6#\"\" )F)F)*&#\"HojCS3\\,W339LHsE6x(eG\"HV>$3\">g]v_;$f%3%yywT*4&F)&F+6#\"\" *F)F)*&#\"'$Hr$\"'Cy`F)&F+6#\"#5F)F)*&#\"*KG8;*\"+D1Hg6F)&F+6#\"#6F)F) &F+6#\"#7F)#F)F-" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%!G" }}{PARA 11 " " 1 "" {XPPMATH 20 "6#%-equation~~~7G" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%!G" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/,0*&#\"'\\w6\")D1R6\"\"\"&% \"bG6#\"\"'F)F)*&#F'\"+Dcw.$)F)&F+6#\"\"(F)F)*&#\",WDpo!Q\",TO`!=UF)&F +6#\"\")F)F)*&#\"P/*>[\\-I^pBh%H\"R1))[eQ>#fTS\"\"P\\c#HTO\"o=OIu'e.Q( *[?Yt217GF)&F+6#\"\"*F)F)*&#\"(4o#[\"(O&HvF)&F+6#\"#5F)F)*&#\",%eN-!o& \",D1*)=a(F)&F+6#\"#6F)F)&F+6#\"#7F)#F)F3" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%!G" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%-equation~~~8G " }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%!G" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/,0*&#\"'VN#)\"*v$f3<\"\"\"&%\"bG6#\"\"'F)F)*&#F'\"-DJXpOPF)&F+6 #\"\"(F)F)*&#\"._v;%)z?#\".>[[^')[#F)&F+6#\"\")F)F)*&#\"W7@o[ " 0 "" {MPLTEXT 1 0 199 "e8 := solve(\{op(quadeqns)\}):\ne9 := `u nion`(e7,e8):\nfor ii to 12 do\n wt_val := subs(e9,b[ii]);\n if wt _val<>0 then print(b[ii]=wt_val) end if;\nend do:\n``;\nsubs(e9,[seq(b [i],i=1..12)]):\nevalf[8](%);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/&%\" bG6#\"\"\"#\"1.JeyD)*\\5\"2gxGN.KFO#" }}{PARA 11 "" 1 "" {XPPMATH 20 " 6#/&%\"bG6#\"\"'#\"3vyjT1iR#o(\"4'\\y$p/)onh@" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/&%\"bG6#\"\"(#\"7Dcv)p()*4Y@%)y\"83-)RLVHMv>oJ" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#/&%\"bG6#\"\")#!8\"Hdv[49X-ps>\"7+[/9, zO_Zf#*" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/&%\"bG6#\"\"*#\"enh\"RJ/]) \\^/r>N;Uuxx5#))H'eds(*[a)H#\"Z+'zfy4Z&yY`L#=*G[2DawO9_f*>m_Bt" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#/&%\"bG6#\"#5#!4[CuLy&y\\2B\"3Di5,,dX( y#" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/&%\"bG6#\"#6#\"6v8DGKkkO'*[\"\" 5[3&Rp1c>2-#" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/&%\"bG6#\"#7#\"-***46 4w&\".+ce$o3B" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%!G" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#7.$\")U$RW%!\"*$\"\"!F(F'F'F'$\")f!Rb$!\")$\")*[&)[# F+$!)eXI@!\"($\")mdQJF0$!)3:y#)F0$\")H\"=P(F0$\")HK&\\#F+" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }} }{SECT 1 {PARA 4 "" 0 "" {TEXT -1 2 "e9" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1512 "e9 := \{b[4] = 0, a[8 ,1] = -25750155116216595121447/19172397107027415328000, c[12] = 1, a[5 ,1] = 12985296430100/33603736057857, a[7,2] = 0, b[2] = 0, b[3] = 0, a [8,7] = 18586487074230223824062061/1035692891721620976018560, a[7,5] = -4116355777093/117791746371000, a[7,6] = 7635614/436305285, b[5] = 0, a[6,3] = 0, a[4,2] = 0, a[8,3] = 0, a[8,2] = 0, a[7,3] = 0, a[6,2] = \+ 0, a[5,2] = 0, c[6] = 7/15, a[3,2] = 439922/5491827, a[6,5] = 40755997 793/218862405000, b[11] = 148963664643228251375/20207195606693950848, \+ a[6,1] = 3817807/81405000, c[5] = 100/257, c[10] = 13/14, c[11] = 62/6 5, b[1] = 1049982578583103/23627320335287760, a[8,5] = -31934383116845 28658341050903/315992014923229256968928000, a[6,4] = 10274191312/43991 343405, a[5,4] = 48744170200000/33603736057857, b[7] = 788421460998769 8755625/31681975342943333980208, c[3] = 469/4510, a[5,3] = -1621802765 0000/11201245352619, a[4,3] = 4221/36080, a[3,1] = 1311793/54918270, b [12] = 576091109999/2308683585600, a[2,1] = 27/400, a[8,4] = -743/55, \+ c[8] = 58/59, c[7] = 7/45, c[2] = 27/400, b[8] = -19726902451409487557 291/9259475236790114044800, c[4] = 1407/9020, a[7,4] = 53436069013712/ 475502430864645, a[4,1] = 1407/36080, b[10] = -2307497857833742448/278 745570101106225, c[9] = 49117478/55144743, b[9] = 22985448977257586298 82107777744216351971045149850043139161/7323526619959521436765425074828 91823353467854709785979600, b[6] = 768239620641637875/2161676880469378 496, a[8,6] = 183334983592060082563257/22930186940004788732288, a[7,1] = 8883007/146529000\}:" }{TEXT -1 0 "" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 55 "Now we specify the remain ing zero linking coefficients:" }}{PARA 256 "" 0 "" {TEXT -1 2 " " } {XPPEDIT 18 0 "a[9,2] = 0,a[9,3] = 0,a[10,2] = 0,a[10,3] = 0,a[11,2] = 0,a[11,3] = 0,a[12,2] = 0,a[12,3] = 0;" "6*/&%\"aG6$\"\"*\"\"#\"\"!/& F%6$F'\"\"$F)/&F%6$\"#5F(F)/&F%6$F1F-F)/&F%6$\"#6F(F)/&F%6$F8F-F)/&F%6 $\"#7F(F)/&F%6$F?F-F)" }{TEXT -1 1 "." }}{PARA 0 "" 0 "" {TEXT -1 0 " " }}{PARA 0 "" 0 "" {TEXT -1 149 "We include the stage-order condition s for stages 9 to 12 so that all these stages all have stage-order 4 a nd incorporate the simplifying conditions: " }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "Sum(b[i]*a[i,j],i = j+1 .. 12) = b[j]*( 1-c[j]);" "6#/-%$SumG6$*&&%\"bG6#%\"iG\"\"\"&%\"aG6$F+%\"jGF,/F+;,&F0F ,F,F,\"#7*&&F)6#F0F,,&F,F,&%\"cG6#F0!\"\"F," }{TEXT -1 1 " " }}{PARA 0 "" 0 "" {TEXT -1 5 "for " }{XPPEDIT 18 0 "j=1" "6#/%\"jG\"\"\"" } {TEXT -1 10 " ( with " }{XPPEDIT 18 0 "c[1]=0" "6#/&%\"cG6#\"\"\"\" \"!" }{TEXT -1 5 " ), " }{XPPEDIT 18 0 "j = 5;" "6#/%\"jG\"\"&" } {TEXT -1 3 ", " }{XPPEDIT 18 0 "j=7" "6#/%\"jG\"\"(" }{TEXT -1 3 ", \+ " }{XPPEDIT 18 0 "j=9" "6#/%\"jG\"\"*" }{TEXT -1 7 " and " } {XPPEDIT 18 0 "j=10" "6#/%\"jG\"#5" }{TEXT -1 2 ". " }}{PARA 0 "" 0 " " {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 195 "Then it turns out that the following additional collection of \"simple\" order conditions (g iven in abreviated form) is sufficient to determine the remaining link ing coefficients in stages 9 to 12." }}{PARA 0 "" 0 "" {TEXT -1 0 "" } }{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 190 "SO8 := SimpleOrderConditio ns(8):\n[seq([i,SO8[i]],i=[37,61,81,87,102,106,113,117,125])]:\nlinalg [augment](linalg[delcols](%,2..2),matrix([[` `]$(linalg[rowdim](%))]) ,linalg[delcols](%,1..1));" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#K%'matri xG6#7+7%\"#P%#~~G/*(%\"bG\"\"\"%\"cGF--%!G6#*&%\"aGF--F06#*&F3F--F06#* &F3F--F06#*&F3F-F.F-F-F-F-F-#F-\"$S)7%\"#hF)/*(F,F-F.F--F06#*&)F.\"\"% F-F3F-F-#F-\"#N7%\"#\")F)/*&F,F--F06#*(F3F-F.F--F06#*&F3F--F06#*&F3F-- F06#*&)F.\"\"#F-F3F-F-F-F-F-#F-\"%gL7%\"#()F)/*(F,F-F.F--F06#*(F3F-F.F --F06#*(F3F-F.F-F:F-F-F-#F-\"$%Q7%\"$-\"F)/*(F,F-F.F--F06#*&F3F--F06#* &F3F--F06#*&)F.\"\"$F-F3F-F-F-F-#F-\"$g*7%\"$1\"F)/*&F,F--F06#*&F3F--F 06#*&F3F-FCF-F-F-#F-\"%!o\"7%\"$8\"F)/*(F,F-FZF-F[pF-#F-\"$g\"7%\"$<\" F)/*&F,F--F06#*(F3F-F.F-FCF-F-#F-\"$!G7%\"$D\"F)/*(F,F-F.F--F06#*&)F. \"\"&F-F3F-F-#F-\"#[Q(pprint26\"" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}{SECT 1 {PARA 4 "" 0 " " {TEXT -1 20 "The associated trees" }}{PARA 0 "" 0 "" {TEXT -1 0 "" } }{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 461 "AST8 := AllSimpleTrees(8): \nwhch := [37,61,81,87,102,106,113,117,125]:\nm := 3: ## number of tre es per row\nnn := nops(whch): q := iquo(nn,m,'r'):\nfor i to nn do \n \+ p||i := DrawTree(AST8[whch[i]],height=4,width=2,show_ordercondition=t rue,\n scale=1.2,color=COLOR(RGB,0,.7,0),font_color=black);\nend do :\npp := plot([[1,1]],style=line,axes=none):\nplots[display](convert([ seq([p||((k-1)*m+1..m*k)],k=1..q),\n `if`(r>0,[p||(m*q+1..nn),pp$(m*( q+1)-nn)],NULL)],array));" }}{PARA 13 "" 1 "" {GLPLOT2D 852 1131 1131 {PLOTDATA 2 "6\\u-%'CURVESG6#-%'LEGENDG6#QB__never_display_this_legend _entry6\"-%%TEXTG6&7$$\"#I\"\"!$!+#4`7'*)!\"*Q/b~c~(a~c~)~=~~F*-%'COLO URG6&%$RGBGF1F1F1-%%FONTG6$%(COURIERG\"#5-F,6&7$F/$!+wo#)p\")F4Q1~~~~~ ~~~~~~~~~~1F*F6F:-F,6&7$F/$!+pv9<%)F4Q2~~~~~~~~~~~~~~~__F*F6F:-F,6&7$F /$!+++++5!\")Q2~~~~~~~~~~~~~~~35F*F6F:-F,6&7$F/$!+*f)*32)F4Q1~~~~~~~~~ 4~~~~~~F*F6F:-F$6&7)7$F/$!+nh/)['F47$$\"+++++DFP$!+DY.mBF47$$\"+++++NF PF[o7$$\"+nmmmJFP$\"+F[rFepFep-Fgp6#%%LINEG-F$6% 7#7$$\"+++++?FPFfnFesFhs-F$6%7#7$$\"+++++SFPFfnFesFhs-F,6&7$F/$!+5`7'* QFPQ8b~c~(a~(a~(a~c~)))~=~~~F*F6F:-F,6&7$F/$!+)o#)p\"QFPQ:~~~~~~~~~~~~ ~~~~~~~~~~~~1F*F6F:-F,6&7$F/$!+dZrTQFPQ<~~~~~~~~~~~~~~~~~~~~~~~~___F*F 6F:-F,6&7$F/$!+++++SFPQ<~~~~~~~~~~~~~~~~~~~~~~~~960F*F6F:-F,6&7$F/$!+g )*32QFPQ:~~~~~~~~~~~~~~~3~~~~~~~~~F*F6F:-F$6&7*7$F/$!+ " 0 "" {MPLTEXT 1 0 1 ";" }}}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 " " {TEXT 260 4 "Note" }{TEXT -1 119 ": The 5 simplifying conditions can be replaced by the 5 extra order conditions given (in abreviated form ) as follows.. " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> \+ " 0 "" {MPLTEXT 1 0 174 "SO8 := SimpleOrderConditions(8):\n[seq([i,SO8 [i]],i=[21,27,31,36,127])]:\nlinalg[augment](linalg[delcols](%,2..2),m atrix([[` `]$(linalg[rowdim](%))]),linalg[delcols](%,1..1));" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#K%'matrixG6#7'7%\"#@%#~~G/*&%\"bG\"\" \"-%!G6#*&%\"aGF--F/6#*&F2F--F/6#*&)%\"cG\"\"#F-F2F-F-F-F-#F-\"$g$7%\" #FF)/*&F,F--F/6#*&F2F--F/6#*&)F:\"\"$F-F2F-F-F-#F-\"$?\"7%\"#JF)/*&F,F --F/6#*&)F:\"\"%F-F2F-F-#F-\"#I7%\"#OF)/*&F,F--F/6#*(F2F-F:F--F/6#*&F2 F--F/6#*&F2F--F/6#*&F2F-F:F-F-F-F-F-#F-\"%357%\"$F\"F)/*&F,F--F/6#*&)F :\"\"'F-F2F-F-#F-\"#cQ(pprint36\"" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 462 "SOeqs2 := [seq(add(a[i,j],j = 1..i-1)=c[i],i=9..12),op(StageOrderConditions(2,9..12,'expanded')), \n op(StageOrderConditions(3,9..12,'expanded')),op(StageOrderCondit ions(4,9..12,'expanded'))]:\nSO8_12 := SimpleOrderConditions(8,12,'exp anded'):\nord_cdns := [seq(SO8_12[i],i=[37,61,81,87,102,113,106,117,12 5])]:\nsimp_eqs := [add(b[i]*a[i,1],i=1+1..12)=b[1],seq(add(b[i]*a[i,j ],i=j+1..12)=b[j]*(1-c[j]),j=[5,7,9,10])]:\ncdns := [op(SOeqs2),op(ord _cdns),op(simp_eqs)]:" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 107 "e10 :=\{seq(a[i,2]=0,i=9..12),seq( a[i,3]=0,i=9..12)\}:\ne11 := `union`(e9,e10):\neqns2 := subs(e11,\{op( cdns)\}):" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 39 "There are 30 equations and 30 unknowns." }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 36 "nops(eqns2);\nindets(eqns2);\nnops(%);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#\"#I" }}{PARA 12 "" 1 "" {XPPMATH 20 "6#<@&% \"aG6$\"#7\"#5&F%6$\"\"*\"\"(&F%6$F'\"\"&&F%6$F'F,&F%6$\"#6F+&F%6$F+\" \"%&F%6$F'F+&F%6$F(F,&F%6$F(F/&F%6$F(\"\")&F%6$F+F@&F%6$F4F7&F%6$F(\" \"\"&F%6$F4FG&F%6$F(\"\"'&F%6$F+FL&F%6$F(F+&F%6$F4F@&F%6$F4F(&F%6$F'FL &F%6$F'F7&F%6$F'FG&F%6$F'F@&F%6$F+FG&F%6$F'F4&F%6$F+F/&F%6$F4FL&F%6$F4 F,&F%6$F(F7&F%6$F4F/" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#\"#I" }}} {PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 22 "infolevel[solve] := 4:" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 74 "e12 := solve(\{op(eqns2)\}):\ninfolevel[solve] := 0:\ne13 := `unio n`(e11,e12):" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 4 "e13 " } }{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 4745 "e13 := \{a[11,4] = -91300774029817320031962039591895693/664177 6260885597741930170995083525, a[10,3] = 0, b[4] = 0, a[11,3] = 0, a[12 ,3] = 0, a[8,1] = -25750155116216595121447/19172397107027415328000, c[ 12] = 1, a[11,2] = 0, a[5,1] = 12985296430100/33603736057857, a[12,2] \+ = 0, a[7,2] = 0, b[2] = 0, b[3] = 0, a[9,3] = 0, a[10,4] = -4854249491 9285774589408977236397/3795156546073735065277428293920, a[11,1] = -888 2652614922380707082937302793930626864630973971/80180429743401295397733 65358072400795335265440000, a[8,7] = 18586487074230223824062061/103569 2891721620976018560, a[12,4] = -6581385500218652420284298344406312/842 464028888378018027275808653215, a[7,5] = -4116355777093/11779174637100 0, a[7,6] = 7635614/436305285, b[5] = 0, a[6,3] = 0, a[10,2] = 0, a[4, 2] = 0, a[8,3] = 0, a[8,2] = 0, a[7,3] = 0, a[6,2] = 0, a[5,2] = 0, a[ 9,2] = 0, c[6] = 7/15, a[10,6] = 2484540315797359406715529397919751486 619672787/421648690388802974367036270632096098190348288, a[3,2] = 4399 22/5491827, a[6,5] = 40755997793/218862405000, b[11] = 148963664643228 251375/20207195606693950848, a[6,1] = 3817807/81405000, c[5] = 100/257 , c[10] = 13/14, c[11] = 62/65, b[1] = 1049982578583103/23627320335287 760, a[8,5] = -3193438311684528658341050903/31599201492322925696892800 0, a[6,4] = 10274191312/43991343405, a[5,4] = 48744170200000/336037360 57857, b[7] = 7884214609987698755625/31681975342943333980208, c[3] = 4 69/4510, a[12,5] = -229810990628849080329321528372507186906863097/2954 8105906232803976194189596626971177604000, a[5,3] = -16218027650000/112 01245352619, a[4,3] = 4221/36080, a[3,1] = 1311793/54918270, a[11,8] = 6612449503769637840690016046967566/2255568687770602845862583843739423 75, a[9,8] = 16756974250619222216361657668160522447776912/164976593050 8280715886033484777254665540228007, a[11,5] = -11220125505347440190681 6716367850973725245563029/13045194205836605228092670613416996666049440 000, a[12,11] = 9008463031864887557094375/6606862347800680584331678, b [12] = 576091109999/2308683585600, a[9,6] = 53277337079395844786125848 10530320112314665632405586015822099459/1249808955205147320198736736321 225596828263980093048262260372352, a[9,4] = -2493924112965609110422434 487743812058912776576445331741068763/248576188041998999363340039748303 974885433202092103843801085, a[10,1] = -370292782453410978231480494053 2859074364487/4131257703459996302316696251055190381568000, a[2,1] = 27 /400, a[11,10] = -4215629637956404624565504/10812361259129223437650012 5, a[12,1] = -535308736842973754888311604833674190949559277/5076654705 26061438136955303900248854246636000, a[8,4] = -743/55, c[8] = 58/59, c [7] = 7/45, c[2] = 27/400, a[12,10] = -113365555877618519272588672/930 91504282232088010468779, a[12,6] = 33362174362860446465153593700072886 5436063541833/54042431430277972080199874664118662613066091504, b[8] = \+ -19726902451409487557291/9259475236790114044800, a[11,6] = 18333480904 0500814662036381242837807618576330592954213/26459846979744846747689472 891117696020043251727064960, c[4] = 1407/9020, a[10,8] = 2424649616806 90545467502075/12396488665185571302473970944, a[9,5] = -34741045425295 0206893609060802736441487435576445275533884950354547756949/69747362766 091883481592212536178438630509515521207481681627188467808000, a[12,9] \+ = 144645270082238084524594544683988399575644348254551267965/3148564668 28710807635889483181359023001187603321870639973, a[12,7] = 28143622302 8901228360767240757054593496984174199893/24784655195828222920388514383 970641394482546714960, a[10,7] = 1705469761435764254142110563325321064 1175183343/1072180740487803620784937862662845301329326080, a[12,8] = - 195816087241661023276342391/399896648012674685277875106, a[9,7] = 6380 5647563182428528344562389882620890969933364805025903916023/52225368801 25448091932537212430169113118513294896584316063360, a[11,7] = 59085344 43979027768339795262286892453548930774924082527/3378016769520941672806 22617772889760985970195055625600, a[10,5] = -2291942554267427743198952 0078944496953957193/3194620077548089472408777624402038658048000, a[7,4 ] = 53436069013712/475502430864645, a[4,1] = 1407/36080, a[10,9] = -27 6720658930244250241932341390181638290085115/96637936155048616171937073 72304668805996412672, b[10] = -2307497857833742448/278745570101106225, c[9] = 49117478/55144743, a[11,9] = -10293784278236522497308869863513 789970987031746931000834/878782636339557288924677338637495764024419774 5505586187675, a[9,1] = -922122171894716552217807973277934204672348766 351979550680715901/157401190198671640845526912912949987653984663002982 7642536608000, b[9] = 229854489772575862988210777774421635197104514985 0043139161/732352661995952143676542507482891823353467854709785979600, \+ b[6] = 768239620641637875/2161676880469378496, a[8,6] = 18333498359206 0082563257/22930186940004788732288, a[7,1] = 8883007/146529000\}:" } {TEXT -1 0 "" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 20 "printed coefficients" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 283 "subs(e13,matrix([seq([c[i],seq(a[i ,j],j=1..i-1),``$(7-i)],i=2..7)]));\nfor ii from 8 to 12 do\n print( ``);\n print(c[ii]=subs(e13,c[ii])); \n for jj to ii-1 do\n p rint(a[ii,jj]=subs(e13,a[ii,jj]));\n end do:\nend do:\n``;\nfor ii t o 12 do\n print(b[ii]=subs(e13,b[ii]));\nend do:" }}{PARA 11 "" 1 " " {XPPMATH 20 "6#K%'matrixG6#7(7)#\"#F\"$+%F(%!GF+F+F+F+7)#\"$p%\"%5X# \"($z68\")q#=\\&#\"'A*R%\"(F=\\&F+F+F+F+7)#\"%29\"%?!*#F8\"&!3O\"\"!# \"%@UF;F+F+F+7)#\"$+\"\"$d##\"/+,V'H&)H\"\"/dy0OPgLF<#!/++lF!=i\"\"/>E NX7?6#\"/++?qTu[FEF+F+7)#\"\"(\"#:#\"(2y\"Q\")+]S\")Fu-\"\", 0MM\"*R%#\",$z(*fvS\"-+]Si)=#F+7)#FM\"#X#\"(2I)))\"*+!Hl9F" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/&%\"aG6$\"\")\"\"#\"\"!" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/&%\"aG6$\"\")\"\"$\"\"!" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/&% \"aG6$\"\")\"\"%#!$V(\"#b" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/&%\"aG6$ \"\")\"\"&#!=.40T$e'GXo6$QM>$\"<+!G*opDHK#\\,#*fJ" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/&%\"aG6$\"\")\"\"'#\"9dKc#3g?f$)\\L$=\"8)GK()y/+%p=IH# " }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/&%\"aG6$\"\")\"\"(#\";h?1CQAIU2([ 'e=\":g&=g(4i@<*GpN5" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%!G" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/&%\"cG6#\"\"*#\")yu6\\\")VZ9b" }}{PARA 11 " " 1 "" {XPPMATH 20 "6#/&%\"aG6$\"\"*\"\"\"#!jn,fr!o]&z>Nm([Bn/U$zFtz!y @_lr%*=!>,u:" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/&%\"aG6$\"\"*\"\"#\"\"!" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/&%\"aG6$\"\"*\"\"$\"\"!" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/&%\"aG6$\"\"*\"\"%#!hnj(o5uJ`Wwlx7*e?\"Qu([MCU5\"4c'H6CR\\#\"gn &3,Q%Q5#4-KV&)[(RI[(R+Mj$***)*>/)=w&[#" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/&%\"aG6$\"\"*\"\"&#!co\\pvZXN]\\)Q`v_WwbV([Tkt-314O*o?]HDa/TZ$ \"bo+!3yY)=F;o\"[27_:&40jQ%yh`7Af\"[$)=4mFOZ(p" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/&%\"aG6$\"\"*\"\"'#\"[of%*4Ae,'e0CjlYJ7,KI0\"[e7'yWeRz qLxK&\"[o_BPgAE[I4!)REGofD7KOnt)>?t90_&*3)\\7" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/&%\"aG6$\"\"*\"\"(#\"inBg\"R!f-0[OL*p4*3i#))*QiXMG&GC= jvk0Q'\"hngL1;Ve'*[H8&=J6p,V7s`K>4[a7!)o`A_" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/&%\"aG6$\"\"*\"\")#\"M7pxZC_g\"owlhj@A#>1Dupv;\"O2!G-a lYDxZ[Lg)er!G30$fw\\;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%!G" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/&%\"cG6#\"#5#\"#8\"#9" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/&%\"aG6$\"#5\"\"\"#!L([kV2fG`S\\![J#y4T`Cy#Hq$\"L+!o:Q !>b5D'p;BI'**fMqd78%" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/&%\"aG6$\"#5 \"\"#\"\"!" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/&%\"aG6$\"#5\"\"$\"\"! " }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/&%\"aG6$\"#5\"\"%#!A(ROs(*3%*eudG >\\\\U&[\"@?RHGuFl]ttgac^z$" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/&%\"aG 6$\"#5\"\"&#!M$>dR&p\\W*y+_*)>VxUnUbU>H#\"L+![!e'Q?SCwx3CZ*3[v2?Y>$" } }{PARA 11 "" 1 "" {XPPMATH 20 "6#/&%\"aG6$\"#5\"\"'#\"O(ys'>m[^(>zRHbr 1%ftz:.a%[#\"N)G[.>)4'4K1FOqOuH!))Q!p[;U" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/&%\"aG6$\"#5\"\"(#\"PVL=v6k5KDLc5@9aUwN9wpaq\"\"O!3E$H8IXGmiy$ \\y?O!y[S2=s5" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/&%\"aG6$\"#5\"\")#\" W4(RZ-8d&=l')['R7" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/&%\"aG6$\"#5\"\"*#!N:^3!HQ;=!RTB$>C]UCI*e1sw#\"OsET'*f!)oYIstq$><; '[]:Ozj'*" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%!G" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/&%\"cG6#\"#6#\"#i\"#l" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/&%\"aG6$\"#6\"\"\"#!RrR(4jkoiIRz-t$H322QA\\h_E)))\"R++Wl_L&z+C2e`O t(R&H,MuH/=!)" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/&%\"aG6$\"#6\"\"#\" \"!" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/&%\"aG6$\"#6\"\"$\"\"!" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#/&%\"aG6$\"#6\"\"%#!D$p&*=fR?'>.?t\")H Sx+8*\"CDN3&*4u(f&)3EwSuM0b7?7\"\"P++W\\gm'*pT81n#4G_gOe?%> XI\"" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/&%\"aG6$\"#6\"\"'#\"W8U&HfIjd =w!y$GC\"QO?m93]S!4[L$=\"Vg\\1Fqf)4w*)GxuO$[g6$))[vtH%ot3``\"N+gjYU&)[-!RIbp8Q91 E0Zlw]" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/&%\"aG6$\"#7\"\"#\"\"!" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#/&%\"aG6$\"#7\"\"$\"\"!" }}{PARA 11 " " 1 "" {XPPMATH 20 "6#/&%\"aG6$\"#7\"\"%#!C7jSW$)H%G?Cl=-]&Q\"e'\"B:Kl 3eFF!=!y$)))GSYU)" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/&%\"aG6$\"#7\"\" &#!N(4jo!p=2DPG:KH.3\\)G1*4\")H#\"M+Sgx6(pi'f*=%>wR!GB1f5[&H" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/&%\"aG6$\"#7\"\"'#\"QL=ajgVl)G2+Pf`^YY/'GOu @OL\"P/:4mIhi'=Tmu)*>!3szFI9VUS&" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/& %\"aG6$\"#7\"\"(#\"T$*)*>uT)p\\$faqvSswg$G7!*GIAO9G\"Sg\\rYD[%RT1(RQ9& )Q?HAGe>bYyC" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/&%\"aG6$\"#7\"\")#!< \"RUjFB5mTs3;e>\"<1^(yF&ouE,[m*)*R" }}{PARA 11 "" 1 "" {XPPMATH 20 "6# /&%\"aG6$\"#7\"\"*#\"ZlzE^XD[Vkv&*R))RoWXfCX3QA3q_kW\"\"Zt*R1(=K.w=,I- f8=$[*)ej23rGoYc[J" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/&%\"aG6$\"#7\"# 5#!&=w(eblL6\";z(o/,)3KAG/:4$*" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/&%\"aG6$\"#7\"#6#\":vV4dv)['=.j%3!*\":y;L%e!o+yMiog'" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%!G" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/&%\" bG6#\"\"\"#\"1.JeyD)*\\5\"2gxGN.KFO#" }}{PARA 11 "" 1 "" {XPPMATH 20 " 6#/&%\"bG6#\"\"#\"\"!" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/&%\"bG6#\"\" $\"\"!" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/&%\"bG6#\"\"%\"\"!" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#/&%\"bG6#\"\"&\"\"!" }}{PARA 11 "" 1 " " {XPPMATH 20 "6#/&%\"bG6#\"\"'#\"3vyjT1iR#o(\"4'\\y$p/)onh@" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/&%\"bG6#\"\"(#\"7Dcv)p()*4Y@%)y\"83-)RLVHMv >oJ" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/&%\"bG6#\"\")#!8\"Hdv[49X-ps> \"7+[/9,zO_Zf#*" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/&%\"bG6#\"\"*#\"en h\"RJ/])\\^/r>N;Uuxx5#))H'eds(*[a)H#\"Z+'zfy4Z&yY`L#=*G[2DawO9_f*>m_Bt " }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/&%\"bG6#\"#5#!4[CuLy&y\\2B\"3Di5, ,dX(y#" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/&%\"bG6#\"#6#\"6v8DGKkkO'*[ \"\"5[3&Rp1c>2-#" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/&%\"bG6#\"#7#\"-* **464w&\".+ce$o3B" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 " > " 0 "" {MPLTEXT 1 0 1 ";" }}}}{PARA 0 "" 0 "" {TEXT -1 0 "" }} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}}{PARA 0 "" 0 "" {TEXT -1 52 "#---------------------------------------------------" }}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 43 "construction of the order 7 embedded sc heme" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 4 "e13 " }} {PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 4745 "e13 := \{a[11,4] = -91300774029817320031962039591895693/66417762 60885597741930170995083525, a[10,3] = 0, b[4] = 0, a[11,3] = 0, a[12,3 ] = 0, a[8,1] = -25750155116216595121447/19172397107027415328000, c[12 ] = 1, a[11,2] = 0, a[5,1] = 12985296430100/33603736057857, a[12,2] = \+ 0, a[7,2] = 0, b[2] = 0, b[3] = 0, a[9,3] = 0, a[10,4] = -485424949192 85774589408977236397/3795156546073735065277428293920, a[11,1] = -88826 52614922380707082937302793930626864630973971/8018042974340129539773365 358072400795335265440000, a[8,7] = 18586487074230223824062061/10356928 91721620976018560, a[12,4] = -6581385500218652420284298344406312/84246 4028888378018027275808653215, a[7,5] = -4116355777093/117791746371000, a[7,6] = 7635614/436305285, b[5] = 0, a[6,3] = 0, a[10,2] = 0, a[4,2] = 0, a[8,3] = 0, a[8,2] = 0, a[7,3] = 0, a[6,2] = 0, a[5,2] = 0, a[9, 2] = 0, c[6] = 7/15, a[10,6] = 248454031579735940671552939791975148661 9672787/421648690388802974367036270632096098190348288, a[3,2] = 439922 /5491827, a[6,5] = 40755997793/218862405000, b[11] = 14896366464322825 1375/20207195606693950848, a[6,1] = 3817807/81405000, c[5] = 100/257, \+ c[10] = 13/14, c[11] = 62/65, b[1] = 1049982578583103/2362732033528776 0, a[8,5] = -3193438311684528658341050903/315992014923229256968928000, a[6,4] = 10274191312/43991343405, a[5,4] = 48744170200000/33603736057 857, b[7] = 7884214609987698755625/31681975342943333980208, c[3] = 469 /4510, a[12,5] = -229810990628849080329321528372507186906863097/295481 05906232803976194189596626971177604000, a[5,3] = -16218027650000/11201 245352619, a[4,3] = 4221/36080, a[3,1] = 1311793/54918270, a[11,8] = 6 612449503769637840690016046967566/225556868777060284586258384373942375 , a[9,8] = 16756974250619222216361657668160522447776912/16497659305082 80715886033484777254665540228007, a[11,5] = -1122012550534744019068167 16367850973725245563029/1304519420583660522809267061341699666604944000 0, a[12,11] = 9008463031864887557094375/6606862347800680584331678, b[1 2] = 576091109999/2308683585600, a[9,6] = 5327733707939584478612584810 530320112314665632405586015822099459/124980895520514732019873673632122 5596828263980093048262260372352, a[9,4] = -249392411296560911042243448 7743812058912776576445331741068763/24857618804199899936334003974830397 4885433202092103843801085, a[10,1] = -37029278245341097823148049405328 59074364487/4131257703459996302316696251055190381568000, a[2,1] = 27/4 00, a[11,10] = -4215629637956404624565504/108123612591292234376500125, a[12,1] = -535308736842973754888311604833674190949559277/507665470526 061438136955303900248854246636000, a[8,4] = -743/55, c[8] = 58/59, c[7 ] = 7/45, c[2] = 27/400, a[12,10] = -113365555877618519272588672/93091 504282232088010468779, a[12,6] = 3336217436286044646515359370007288654 36063541833/54042431430277972080199874664118662613066091504, b[8] = -1 9726902451409487557291/9259475236790114044800, a[11,6] = 1833348090405 00814662036381242837807618576330592954213/2645984697974484674768947289 1117696020043251727064960, c[4] = 1407/9020, a[10,8] = 242464961680690 545467502075/12396488665185571302473970944, a[9,5] = -3474104542529502 06893609060802736441487435576445275533884950354547756949/6974736276609 1883481592212536178438630509515521207481681627188467808000, a[12,9] = \+ 144645270082238084524594544683988399575644348254551267965/314856466828 710807635889483181359023001187603321870639973, a[12,7] = 2814362230289 01228360767240757054593496984174199893/2478465519582822292038851438397 0641394482546714960, a[10,7] = 170546976143576425414211056332532106411 75183343/1072180740487803620784937862662845301329326080, a[12,8] = -19 5816087241661023276342391/399896648012674685277875106, a[9,7] = 638056 47563182428528344562389882620890969933364805025903916023/5222536880125 448091932537212430169113118513294896584316063360, a[11,7] = 5908534443 979027768339795262286892453548930774924082527/337801676952094167280622 617772889760985970195055625600, a[10,5] = -229194255426742774319895200 78944496953957193/3194620077548089472408777624402038658048000, a[7,4] \+ = 53436069013712/475502430864645, a[4,1] = 1407/36080, a[10,9] = -2767 20658930244250241932341390181638290085115/9663793615504861617193707372 304668805996412672, b[10] = -2307497857833742448/278745570101106225, c [9] = 49117478/55144743, a[11,9] = -1029378427823652249730886986351378 9970987031746931000834/87878263633955728892467733863749576402441977455 05586187675, a[9,1] = -92212217189471655221780797327793420467234876635 1979550680715901/15740119019867164084552691291294998765398466300298276 42536608000, b[9] = 22985448977257586298821077777442163519710451498500 43139161/732352661995952143676542507482891823353467854709785979600, b[ 6] = 768239620641637875/2161676880469378496, a[8,6] = 1833349835920600 82563257/22930186940004788732288, a[7,1] = 8883007/146529000\}:" } {TEXT -1 0 "" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 " " 0 "" {TEXT -1 61 "We can obtain an embedded 11 stage order 7 scheme \+ as follows." }}{PARA 0 "" 0 "" {TEXT -1 94 "We remove stages 11 and 12 from the 12 stage order 8 scheme and introduce a new stage 11 with " }{XPPEDIT 18 0 "c[11] = 1;" "6#/&%\"cG6#\"#6\"\"\"" }{TEXT -1 3 ", " }{XPPEDIT 18 0 "a[11,2] = 0" "6#/&%\"aG6$\"#6\"\"#\"\"!" }{TEXT -1 3 " , " }{XPPEDIT 18 0 "a[11,3] = 0;" "6#/&%\"aG6$\"#6\"\"$\"\"!" }{TEXT -1 3 ", " }{XPPEDIT 18 0 "`b*`[2]=0" "6#/&%#b*G6#\"\"#\"\"!" }{TEXT -1 3 ", " }{XPPEDIT 18 0 "`b*`[3]=0" "6#/&%#b*G6#\"\"$\"\"!" }{TEXT -1 3 ", " }{XPPEDIT 18 0 "`b*`[4]=0" "6#/&%#b*G6#\"\"%\"\"!" }{TEXT -1 7 " and " }{XPPEDIT 18 0 "`b*`[5]=0" "6#/&%#b*G6#\"\"&\"\"!" } {TEXT -1 58 ", where the weights of the order 7 scheme are denoted by \+ " }{XPPEDIT 18 0 "`b*`" "6#%#b*G" }{TEXT -1 1 "." }}{PARA 0 "" 0 "" {TEXT -1 1 " " }}{PARA 0 "" 0 "" {TEXT -1 156 "We incorporate the orde r 7 quadrature conditions, the row sum conditions for this stage and s tage-order conditions so that this new stage has stage-order 4." }} {PARA 0 "" 0 "" {TEXT -1 48 "We also incorporate the simplifying condi tions: " }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "Sum(`b*`[i ]*a[i,j],i=j+1..11)=`b*`[j]*(1-c[j])" "6#/-%$SumG6$*&&%#b*G6#%\"iG\"\" \"&%\"aG6$F+%\"jGF,/F+;,&F0F,F,F,\"#6*&&F)6#F0F,,&F,F,&%\"cG6#F0!\"\"F ," }{TEXT -1 1 "," }}{PARA 0 "" 0 "" {TEXT -1 5 "for " }{XPPEDIT 18 0 "j=1" "6#/%\"jG\"\"\"" }{TEXT -1 10 " ( with " }{XPPEDIT 18 0 "c[1 ]=0" "6#/&%\"cG6#\"\"\"\"\"!" }{TEXT -1 5 " ), " }{XPPEDIT 18 0 "j = \+ 8;" "6#/%\"jG\"\")" }{TEXT -1 3 ", " }{XPPEDIT 18 0 "j=9" "6#/%\"jG\" \"*" }{TEXT -1 7 " and " }{XPPEDIT 18 0 "j=10" "6#/%\"jG\"#5" } {TEXT -1 2 ". " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 399 "`Qeqs*` := subs(b=`b*`,QuadratureConditions(7,1 1,'expanded')):\nSO_eqs := [add(a[11,j],j=1..10)=c[11],add(a[11,j]*c[j ],j=2..10)=1/2*c[11]^2,\n add(a[11,j]*c[j]^2,j=2..10)=1/3*c[11] ^3,add(a[11,j]*c[j]^3,j=2..10)=1/4*c[11]^4]:\n`simp_eqs*` := [add(`b*` [i]*a[i,1],i=2..11)=`b*`[1],seq(add(`b*`[i]*a[i,j],i=j+1..11)=`b*`[j]* (1-c[j]),j=[8,9,10])]:\n`cdns*` := [op(`simp_eqs*`),op(SO_eqs),op(`Qeq s*`)]:" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 144 "e14 := `union`(remove(u_->member(op(1,lhs(u_)),[11,1 2]) or op(0,lhs(u_))=b,e13),\n \{c[11]=1,a[11,2]=0,a[11,3]=0,s eq(`b*`[i]=0,i=2..5)\}):\n" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 53 "We have 15 equations for the 15 unknown coeffic ients." }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 60 "eqns3 := subs(e14, `cdns*`):\nnops(%);\nindets(eqns3);\nnops(%);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#\"#:" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#<1&%#b*G6#\"\"* &F%6#\"#5&F%6#\"#6&F%6#\"\"\"&F%6#\"\"'&F%6#\"\"(&F%6#\"\")&%\"aG6$F-F '&F;6$F-\"\"%&F;6$F-F0&F;6$F-F9&F;6$F-F*&F;6$F-F3&F;6$F-F6&F;6$F-\"\"& " }}{PARA 11 "" 1 "" {XPPMATH 20 "6#\"#:" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 22 "infolevel[solve] := 4 :" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 74 "e15 := solve(\{op(eqns 3)\}):\ninfolevel[solve] := 0:\ne16 := `union`(e14,e15):" }}}{PARA 0 " " 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}} {SECT 1 {PARA 4 "" 0 "" {TEXT -1 3 "e16" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 3689 "e16 := \{a[10,3] = 0, a[11,3] = 0, a[8,1] = -25750155116216595121447/1917239710702741532800 0, a[11,2] = 0, a[5,1] = 12985296430100/33603736057857, a[7,2] = 0, a[ 9,3] = 0, a[10,4] = -48542494919285774589408977236397/3795156546073735 065277428293920, a[8,7] = 18586487074230223824062061/10356928917216209 76018560, a[7,5] = -4116355777093/117791746371000, a[7,6] = 7635614/43 6305285, a[6,3] = 0, a[10,2] = 0, a[4,2] = 0, a[8,3] = 0, a[8,2] = 0, \+ a[7,3] = 0, a[6,2] = 0, a[5,2] = 0, a[9,2] = 0, c[6] = 7/15, `b*`[9] = 363484892569724108095352508636899702948448557463/16077810631888793423 7641558033288694994268792800, a[10,6] = 248454031579735940671552939791 9751486619672787/421648690388802974367036270632096098190348288, `b*`[2 ] = 0, `b*`[3] = 0, `b*`[4] = 0, `b*`[5] = 0, a[3,2] = 439922/5491827, `b*`[1] = 4850006989829/108881660531280, a[6,5] = 40755997793/2188624 05000, `b*`[6] = 5801298400186875/16253209627589312, a[6,1] = 3817807/ 81405000, c[11] = 1, c[5] = 100/257, c[10] = 13/14, a[8,5] = -31934383 11684528658341050903/315992014923229256968928000, a[6,4] = 10274191312 /43991343405, a[5,4] = 48744170200000/33603736057857, c[3] = 469/4510, a[5,3] = -16218027650000/11201245352619, a[4,3] = 4221/36080, a[3,1] \+ = 1311793/54918270, a[9,8] = 16756974250619222216361657668160522447776 912/1649765930508280715886033484777254665540228007, a[11,7] = 13688171 02190976218912189847996057845647169577/7563090495472482843640443077801 9911819343440, a[9,6] = 5327733707939584478612584810530320112314665632 405586015822099459/124980895520514732019873673632122559682826398009304 8262260372352, a[9,4] = -249392411296560911042243448774381205891277657 6445331741068763/24857618804199899936334003974830397488543320209210384 3801085, `b*`[7] = 2408095926060530625/9691641279578872432, `b*`[11] = -9030611407/4779883200, a[10,1] = -3702927824534109782314804940532859 074364487/4131257703459996302316696251055190381568000, a[11,9] = -2684 522309490576529735302727427307676030962009390/346283366306502489573064 74374479340566374128519867, a[2,1] = 27/400, `b*`[8] = 550062298578047 92033/15747406865289309600, a[11,1] = -3555689110930181517558426267589 7743594621/23337225766986990952929827252863229628000, a[8,4] = -743/55 , c[8] = 58/59, c[7] = 7/45, c[2] = 27/400, a[11,5] = -474499229690075 043966853347490522346181911/42107837966579029834778387168434144652000, `b*`[10] = -14197733827415152/4039790871030525, c[4] = 1407/9020, a[1 1,6] = 679545987701237138790035764835967076741310679/77013733239195245 746002612595738122195430352, a[10,8] = 242464961680690545467502075/123 96488665185571302473970944, a[9,5] = -34741045425295020689360906080273 6441487435576445275533884950354547756949/69747362766091883481592212536 178438630509515521207481681627188467808000, a[10,7] = 1705469761435764 2541421105633253210641175183343/10721807404878036207849378626628453013 29326080, a[9,7] = 638056475631824285283445623898826208909699333648050 25903916023/5222536880125448091932537212430169113118513294896584316063 360, a[11,10] = 64631913714031794733568/486423753624302998802649, a[11 ,4] = -15763198989262628277502495250456/120056219284161654781029756604 5, a[10,5] = -22919425542674277431989520078944496953957193/31946200775 48089472408777624402038658048000, a[7,4] = 53436069013712/475502430864 645, a[4,1] = 1407/36080, a[10,9] = -276720658930244250241932341390181 638290085115/9663793615504861617193707372304668805996412672, a[11,8] = -3292496872165727322673258/59253630028479896512047753, c[9] = 4911747 8/55144743, a[9,1] = -922122171894716552217807973277934204672348766351 979550680715901/157401190198671640845526912912949987653984663002982764 2536608000, a[8,6] = 183334983592060082563257/22930186940004788732288, a[7,1] = 8883007/146529000\}:" }{TEXT -1 0 "" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 20 "printed coefficients" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 289 "s ubs(e16,matrix([seq([c[i],seq(a[i,j],j=1..i-1),``$(7-i)],i=2..7)]));\n for ii from 8 to 11 do\n print(``);\n print(c[ii]=subs(e16,c[ii])) ; \n for jj to ii-1 do\n print(a[ii,jj]=subs(e16,a[ii,jj]));\n \+ end do:\nend do:\n``;\nfor ii to 11 do\n print(`b*`[ii]=subs(e16,` b*`[ii]));\nend do:" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#K%'matrixG6#7(7 )#\"#F\"$+%F(%!GF+F+F+F+7)#\"$p%\"%5X#\"($z68\")q#=\\&#\"'A*R%\"(F=\\& F+F+F+F+7)#\"%29\"%?!*#F8\"&!3O\"\"!#\"%@UF;F+F+F+7)#\"$+\"\"$d##\"/+, V'H&)H\"\"/dy0OPgLF<#!/++lF!=i\"\"/>ENX7?6#\"/++?qTu[FEF+F+7)#\"\"(\"# :#\"(2y\"Q\")+]S\")Fu-\"\",0MM\"*R%#\",$z(*fvS\"-+]Si)=#F+7) #FM\"#X#\"(2I)))\"*+!Hl9F" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/&%\"aG6$\"\")\"\"#\"\"!" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/&%\"aG6$\"\")\"\"$\"\"!" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/&%\"aG6$\"\")\"\"%#!$V(\"#b" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/&%\"aG6$\"\")\"\"&#!=.40T$e'GXo6$QM>$\"<+!G*opDHK #\\,#*fJ" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/&%\"aG6$\"\")\"\"'#\"9dKc #3g?f$)\\L$=\"8)GK()y/+%p=IH#" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/&%\" aG6$\"\")\"\"(#\";h?1CQAIU2(['e=\":g&=g(4i@<*GpN5" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%!G" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/&%\"cG6#\"\"*# \")yu6\\\")VZ9b" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/&%\"aG6$\"\"*\"\" \"#!jn,fr!o]&z>Nm([Bn/U$zFtz!y@_lr%*=!>,u:" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/&%\"aG6$\"\"*\" \"#\"\"!" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/&%\"aG6$\"\"*\"\"$\"\"!" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/&%\"aG6$\"\"*\"\"%#!hnj(o5uJ`Wwlx7* e?\"Qu([MCU5\"4c'H6CR\\#\"gn&3,Q%Q5#4-KV&)[(RI[(R+Mj$***)*>/)=w&[#" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#/&%\"aG6$\"\"*\"\"&#!co\\pvZXN]\\)Q`v_ WwbV([Tkt-314O*o?]HDa/TZ$\"bo+!3yY)=F;o\"[27_:&40jQ%yh`7Af\"[$)=4mFOZ( p" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/&%\"aG6$\"\"*\"\"'#\"[of%*4Ae,'e 0CjlYJ7,KI0\"[e7'yWeRzqLxK&\"[o_BPgAE[I4!)REGofD7KOnt)>?t90_&*3)\\7" } }{PARA 11 "" 1 "" {XPPMATH 20 "6#/&%\"aG6$\"\"*\"\"(#\"inBg\"R!f-0[OL* p4*3i#))*QiXMG&GC=jvk0Q'\"hngL1;Ve'*[H8&=J6p,V7s`K>4[a7!)o`A_" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#/&%\"aG6$\"\"*\"\")#\"M7pxZC_g\"owlhj@ A#>1Dupv;\"O2!G-alYDxZ[Lg)er!G30$fw\\;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%!G" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/&%\"cG6#\"#5#\"#8\"#9" } }{PARA 11 "" 1 "" {XPPMATH 20 "6#/&%\"aG6$\"#5\"\"\"#!L([kV2fG`S\\![J# y4T`Cy#Hq$\"L+!o:Q!>b5D'p;BI'**fMqd78%" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/&%\"aG6$\"#5\"\"#\"\"!" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/&%\" aG6$\"#5\"\"$\"\"!" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/&%\"aG6$\"#5\" \"%#!A(ROs(*3%*eudG>\\\\U&[\"@?RHGuFl]ttgac^z$" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/&%\"aG6$\"#5\"\"&#!M$>dR&p\\W*y+_*)>VxUnUbU>H#\"L+![!e 'Q?SCwx3CZ*3[v2?Y>$" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/&%\"aG6$\"#5\" \"'#\"O(ys'>m[^(>zRHbr1%ftz:.a%[#\"N)G[.>)4'4K1FOqOuH!))Q!p[;U" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#/&%\"aG6$\"#5\"\"(#\"PVL=v6k5KDLc5@9aU wN9wpaq\"\"O!3E$H8IXGmiy$\\y?O!y[S2=s5" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/&%\"aG6$\"#5\"\")#\"W4(RZ-8d&=l')['R7" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#/&%\"aG6$\"#5\"\"*#!N:^3!HQ;=!RTB$>C]U CI*e1sw#\"OsET'*f!)oYIstq$><;'[]:Ozj'*" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%!G" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/&%\"cG6#\"#6\"\"\"" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#/&%\"aG6$\"#6\"\"\"#!J@YfVx*eniUev^\"= I46*obN\"J+!G'HK'GDF)HH&4*p)pwDsLB" }}{PARA 11 "" 1 "" {XPPMATH 20 "6# /&%\"aG6$\"#6\"\"#\"\"!" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/&%\"aG6$\" #6\"\"$\"\"!" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/&%\"aG6$\"#6\"\"%#!Ac /D&\\-vFGEE*)*)>jd\"\"@Xgc(H5ya;;%G>i0?\"" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/&%\"aG6$\"#6\"\"&#!K6>=YB_!\\ZL&o'R/v+pH#*\\u%\"J+?lWT VorQyZ$)H!zl'z$y5U" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/&%\"aG6$\"#6\" \"'#\"Nz1JTn2nf$[wN+zQrB,x)faz'\"M_.V&>A\"Qdf7E+YdC&>RKt8q(" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/&%\"aG6$\"#6\"\"(#\"Ox&prkXy0'*z%)*=7*=i(4> -r\")o8\"MSMM>=\"*>!y2V/kVG[sa\\!4jv" }}{PARA 11 "" 1 "" {XPPMATH 20 " 6#/&%\"aG6$\"#6\"\")#!:eKnAtsl@(o\\#H$\";`x/7l*)z%G+j`#f" }}{PARA 11 " " 1 "" {XPPMATH 20 "6#/&%\"aG6$\"#6\"\"*#!R!R4?'4.wwIFus-`tHld!\\4B_%o #\"Sn)>>Pm0MzWPukId*[-lImLGY$" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/&% \"aG6$\"#6\"#5#\"8oNt%zJSr8>jk\"9\\E!))*HICOvBk[" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%!G" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/&%#b*G6#\"\"\"# \".H)*)p+][\"0!GJ0m\"))3\"" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/&%#b*G6 #\"\"#\"\"!" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/&%#b*G6#\"\"$\"\"!" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#/&%#b*G6#\"\"%\"\"!" }}{PARA 11 "" 1 " " {XPPMATH 20 "6#/&%#b*G6#\"\"&\"\"!" }}{PARA 11 "" 1 "" {XPPMATH 20 " 6#/&%#b*G6#\"\"'#\"1vo=+%)H,e\"27$*eF'4KD;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/&%#b*G6#\"\"(#\"4D1`gg#f43C\"4KC()y&z7k\"p*" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/&%#b*G6#\"\")#\"5L?z/y&)Hi+b\"5+'4$*GloSZd \"" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/&%#b*G6#\"\"*#\"Qjub[%[Hq**oj3D N&43TspD*[[j$\"Q+GzoU*\\p)GL!e:kPU$z))=j5y2;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/&%#b*G6#\"#5#!2_^TFQt(>9\"1D0.r3zRS" }}{PARA 11 "" 1 " " {XPPMATH 20 "6#/&%#b*G6#\"#6#!+29hI!*\"++K))zZ" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 104 "` RK7_11eqs*` := subs(b=`b*`,[op(RowSumConditions(11,'expanded')),op(Ord erConditions(7,11,'expanded'))]):" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }} {PARA 0 "" 0 "" {TEXT -1 5 "Check" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 57 "subs(e16,`RK7_11eqs*`):\nmap(u->lhs(u)-rhs(u),%);\nno ps(%);" }}{PARA 12 "" 1 "" {XPPMATH 20 "6#7[q\"\"!F$F$F$F$F$F$F$F$F$F$ F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$ F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$ F$F$F$F$F$F$F$F$F$F$F$F$F$F$" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#\"#&* " }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 93 "We c an calculate the principal error norm, that is, the 2-norm of the prin cipal error terms. " }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 185 "`err terms7_11*` := subs(b=`b*`,PrincipalErrorTerms(7,11,'expanded')):\nsm \+ := 0:\nfor ct to nops(`errterms7_11*`) do\n sm := sm+(evalf(subs(e16 ,`errterms7_11*`[ct])))^2;\nend do:\nsqrt(sm);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#$\"+!oeQ+\"!#9" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }} {PARA 0 "" 0 "" {TEXT -1 138 "We can include the new stage for the emb edded scheme as an additional 13th stage added to the order 8 scheme a long with the coefficients " }{XPPEDIT 18 0 "a[13,11] = 0;" "6#/&%\"a G6$\"#8\"#6\"\"!" }{TEXT -1 7 " and " }{XPPEDIT 18 0 "a[13,12] = 0; " "6#/&%\"aG6$\"#8\"#7\"\"!" }{TEXT -1 1 "." }}{PARA 0 "" 0 "" {TEXT -1 13 "The weights " }{XPPEDIT 18 0 "`b*`[i]" "6#&%#b*G6#%\"iG" } {TEXT -1 7 " for " }{XPPEDIT 18 0 "i=1" "6#/%\"iG\"\"\"" }{TEXT -1 89 " . . 10 of the 13 stage combined scheme are those of the 11 stage \+ scheme and the weight " }{XPPEDIT 18 0 "`b*`[11];" "6#&%#b*G6#\"#6" } {TEXT -1 34 " in the 11 stage scheme becomes " }{XPPEDIT 18 0 "`b*`[ 13];" "6#&%#b*G6#\"#8" }{TEXT -1 25 " in the 13 stage scheme." }} {PARA 0 "" 0 "" {TEXT -1 8 "We set " }{XPPEDIT 18 0 "c[13] = 1;" "6#/ &%\"cG6#\"#8\"\"\"" }{TEXT -1 3 ", " }{XPPEDIT 18 0 "`b*`[11] = 0;" " 6#/&%#b*G6#\"#6\"\"!" }{TEXT -1 7 " and " }{XPPEDIT 18 0 "`b*`[12] = 0;" "6#/&%#b*G6#\"#7\"\"!" }{TEXT -1 1 "." }}{PARA 0 "" 0 "" {TEXT -1 66 "We can make the order 8 scheme into a 13 stage scheme by settin g " }{XPPEDIT 18 0 "b[13] = 0;" "6#/&%\"bG6#\"#8\"\"!" }{TEXT -1 1 ". " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 201 "e17 := \{c[13]=1,seq(a[13,i]=subs(e16,a[11,i]),i=1..10),a[13, 11]=0,a[13,12]=0,b[13]=0,\nseq(`b*`[i]=subs(e16,`b*`[i]),i=1..10),`b*` [11]=0,`b*`[12]=0,`b*`[13]=subs(e16,`b*`[11])\}:\ne18 := `union`(e13,e 17):" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 3 "e18" }} {PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 5998 "e18 := \{b[13] = 0, a[13,11] = 0, a[13,12] = 0, a[11,4] = -91300 774029817320031962039591895693/6641776260885597741930170995083525, a[1 0,3] = 0, b[4] = 0, a[11,3] = 0, a[12,3] = 0, c[13] = 1, a[8,1] = -257 50155116216595121447/19172397107027415328000, c[12] = 1, a[11,2] = 0, \+ a[5,1] = 12985296430100/33603736057857, a[12,2] = 0, a[7,2] = 0, b[2] \+ = 0, b[3] = 0, a[9,3] = 0, a[10,4] = -48542494919285774589408977236397 /3795156546073735065277428293920, a[11,1] = -8882652614922380707082937 302793930626864630973971/801804297434012953977336535807240079533526544 0000, a[8,7] = 18586487074230223824062061/1035692891721620976018560, a [12,4] = -6581385500218652420284298344406312/8424640288883780180272758 08653215, a[7,5] = -4116355777093/117791746371000, a[7,6] = 7635614/43 6305285, `b*`[12] = 0, `b*`[11] = 0, b[5] = 0, a[6,3] = 0, a[10,2] = 0 , a[4,2] = 0, a[8,3] = 0, a[8,2] = 0, a[7,3] = 0, a[6,2] = 0, a[5,2] = 0, a[9,2] = 0, c[6] = 7/15, `b*`[9] = 3634848925697241080953525086368 99702948448557463/160778106318887934237641558033288694994268792800, a[ 10,6] = 2484540315797359406715529397919751486619672787/421648690388802 974367036270632096098190348288, `b*`[2] = 0, `b*`[3] = 0, `b*`[4] = 0, `b*`[5] = 0, a[3,2] = 439922/5491827, `b*`[1] = 4850006989829/1088816 60531280, a[6,5] = 40755997793/218862405000, `b*`[6] = 580129840018687 5/16253209627589312, b[11] = 148963664643228251375/2020719560669395084 8, a[6,1] = 3817807/81405000, c[5] = 100/257, c[10] = 13/14, c[11] = 6 2/65, b[1] = 1049982578583103/23627320335287760, a[8,5] = -31934383116 84528658341050903/315992014923229256968928000, a[6,4] = 10274191312/43 991343405, a[5,4] = 48744170200000/33603736057857, b[7] = 788421460998 7698755625/31681975342943333980208, c[3] = 469/4510, a[12,5] = -229810 990628849080329321528372507186906863097/295481059062328039761941895966 26971177604000, a[5,3] = -16218027650000/11201245352619, a[4,3] = 4221 /36080, a[3,1] = 1311793/54918270, a[13,3] = 0, a[11,8] = 661244950376 9637840690016046967566/225556868777060284586258384373942375, a[9,8] = \+ 16756974250619222216361657668160522447776912/1649765930508280715886033 484777254665540228007, a[11,5] = -112201255053474401906816716367850973 725245563029/13045194205836605228092670613416996666049440000, a[12,11] = 9008463031864887557094375/6606862347800680584331678, b[12] = 576091 109999/2308683585600, a[9,6] = 532773370793958447861258481053032011231 4665632405586015822099459/12498089552051473201987367363212255968282639 80093048262260372352, a[13,2] = 0, a[9,4] = -2493924112965609110422434 487743812058912776576445331741068763/248576188041998999363340039748303 974885433202092103843801085, `b*`[7] = 2408095926060530625/96916412795 78872432, a[10,1] = -3702927824534109782314804940532859074364487/41312 57703459996302316696251055190381568000, a[2,1] = 27/400, a[11,10] = -4 215629637956404624565504/108123612591292234376500125, a[12,1] = -53530 8736842973754888311604833674190949559277/50766547052606143813695530390 0248854246636000, `b*`[8] = 55006229857804792033/15747406865289309600, a[8,4] = -743/55, c[8] = 58/59, c[7] = 7/45, c[2] = 27/400, a[12,10] \+ = -113365555877618519272588672/93091504282232088010468779, a[12,6] = 3 33621743628604464651535937000728865436063541833/5404243143027797208019 9874664118662613066091504, b[8] = -19726902451409487557291/92594752367 90114044800, a[11,6] = 18333480904050081466203638124283780761857633059 2954213/26459846979744846747689472891117696020043251727064960, `b*`[10 ] = -14197733827415152/4039790871030525, c[4] = 1407/9020, a[10,8] = 2 42464961680690545467502075/12396488665185571302473970944, a[9,5] = -34 7410454252950206893609060802736441487435576445275533884950354547756949 /697473627660918834815922125361784386305095155212074816816271884678080 00, a[12,9] = 14464527008223808452459454468398839957564434825455126796 5/314856466828710807635889483181359023001187603321870639973, a[12,7] = 281436223028901228360767240757054593496984174199893/24784655195828222 920388514383970641394482546714960, a[10,7] = 1705469761435764254142110 5633253210641175183343/1072180740487803620784937862662845301329326080, a[12,8] = -195816087241661023276342391/399896648012674685277875106, a [9,7] = 63805647563182428528344562389882620890969933364805025903916023 /5222536880125448091932537212430169113118513294896584316063360, a[11,7 ] = 5908534443979027768339795262286892453548930774924082527/3378016769 52094167280622617772889760985970195055625600, a[10,5] = -2291942554267 4277431989520078944496953957193/31946200775480894724087776244020386580 48000, a[7,4] = 53436069013712/475502430864645, a[4,1] = 1407/36080, a [10,9] = -276720658930244250241932341390181638290085115/96637936155048 61617193707372304668805996412672, b[10] = -2307497857833742448/2787455 70101106225, c[9] = 49117478/55144743, a[11,9] = -10293784278236522497 308869863513789970987031746931000834/878782636339557288924677338637495 7640244197745505586187675, a[9,1] = -922122171894716552217807973277934 204672348766351979550680715901/157401190198671640845526912912949987653 9846630029827642536608000, b[9] = 229854489772575862988210777774421635 1971045149850043139161/73235266199595214367654250748289182335346785470 9785979600, a[13,1] = -35556891109301815175584262675897743594621/23337 225766986990952929827252863229628000, a[13,5] = -474499229690075043966 853347490522346181911/42107837966579029834778387168434144652000, a[13, 4] = -15763198989262628277502495250456/1200562192841616547810297566045 , a[13,8] = -3292496872165727322673258/59253630028479896512047753, a[1 3,7] = 1368817102190976218912189847996057845647169577/7563090495472482 8436404430778019911819343440, a[13,6] = 679545987701237138790035764835 967076741310679/77013733239195245746002612595738122195430352, `b*`[13] = -9030611407/4779883200, a[13,10] = 64631913714031794733568/48642375 3624302998802649, b[6] = 768239620641637875/2161676880469378496, a[13, 9] = -2684522309490576529735302727427307676030962009390/34628336630650 248957306474374479340566374128519867, a[8,6] = 18333498359206008256325 7/22930186940004788732288, a[7,1] = 8883007/146529000\}:" }{TEXT -1 0 "" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 20 "p rinted coefficients" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 362 "subs(e18,matrix([seq([c[i],seq(a[i,j],j=1. .i-1),``$(7-i)],i=2..7)]));\nfor ii from 8 to 13 do\n print(``);\n \+ print(c[ii]=subs(e18,c[ii])); \n for jj to ii-1 do\n print(a[i i,jj]=subs(e18,a[ii,jj]));\n end do:\nend do:print(``);\nfor ii to 1 3 do\n print(b[ii]=subs(e18,b[ii]));\nend do:print(``);\nfor ii to 1 3 do\n print(`b*`[ii]=subs(e18,`b*`[ii]));\nend do:" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#K%'matrixG6#7(7)#\"#F\"$+%F(%!GF+F+F+F+7)#\"$p%\"% 5X#\"($z68\")q#=\\&#\"'A*R%\"(F=\\&F+F+F+F+7)#\"%29\"%?!*#F8\"&!3O\"\" !#\"%@UF;F+F+F+7)#\"$+\"\"$d##\"/+,V'H&)H\"\"/dy0OPgLF<#!/++lF!=i\"\"/ >ENX7?6#\"/++?qTu[FEF+F+7)#\"\"(\"#:#\"(2y\"Q\")+]S\")Fu-\" \",0MM\"*R%#\",$z(*fvS\"-+]Si)=#F+7)#FM\"#X#\"(2I)))\"*+!Hl9F" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/&%\"aG6$\"\")\"\"#\"\"!" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/&%\"aG6$\"\")\"\"$\"\"!" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/&% \"aG6$\"\")\"\"%#!$V(\"#b" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/&%\"aG6$ \"\")\"\"&#!=.40T$e'GXo6$QM>$\"<+!G*opDHK#\\,#*fJ" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/&%\"aG6$\"\")\"\"'#\"9dKc#3g?f$)\\L$=\"8)GK()y/+%p=IH# " }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/&%\"aG6$\"\")\"\"(#\";h?1CQAIU2([ 'e=\":g&=g(4i@<*GpN5" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%!G" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/&%\"cG6#\"\"*#\")yu6\\\")VZ9b" }}{PARA 11 " " 1 "" {XPPMATH 20 "6#/&%\"aG6$\"\"*\"\"\"#!jn,fr!o]&z>Nm([Bn/U$zFtz!y @_lr%*=!>,u:" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/&%\"aG6$\"\"*\"\"#\"\"!" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/&%\"aG6$\"\"*\"\"$\"\"!" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/&%\"aG6$\"\"*\"\"%#!hnj(o5uJ`Wwlx7*e?\"Qu([MCU5\"4c'H6CR\\#\"gn &3,Q%Q5#4-KV&)[(RI[(R+Mj$***)*>/)=w&[#" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/&%\"aG6$\"\"*\"\"&#!co\\pvZXN]\\)Q`v_WwbV([Tkt-314O*o?]HDa/TZ$ \"bo+!3yY)=F;o\"[27_:&40jQ%yh`7Af\"[$)=4mFOZ(p" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/&%\"aG6$\"\"*\"\"'#\"[of%*4Ae,'e0CjlYJ7,KI0\"[e7'yWeRz qLxK&\"[o_BPgAE[I4!)REGofD7KOnt)>?t90_&*3)\\7" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/&%\"aG6$\"\"*\"\"(#\"inBg\"R!f-0[OL*p4*3i#))*QiXMG&GC= jvk0Q'\"hngL1;Ve'*[H8&=J6p,V7s`K>4[a7!)o`A_" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/&%\"aG6$\"\"*\"\")#\"M7pxZC_g\"owlhj@A#>1Dupv;\"O2!G-a lYDxZ[Lg)er!G30$fw\\;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%!G" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/&%\"cG6#\"#5#\"#8\"#9" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/&%\"aG6$\"#5\"\"\"#!L([kV2fG`S\\![J#y4T`Cy#Hq$\"L+!o:Q !>b5D'p;BI'**fMqd78%" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/&%\"aG6$\"#5 \"\"#\"\"!" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/&%\"aG6$\"#5\"\"$\"\"! " }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/&%\"aG6$\"#5\"\"%#!A(ROs(*3%*eudG >\\\\U&[\"@?RHGuFl]ttgac^z$" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/&%\"aG 6$\"#5\"\"&#!M$>dR&p\\W*y+_*)>VxUnUbU>H#\"L+![!e'Q?SCwx3CZ*3[v2?Y>$" } }{PARA 11 "" 1 "" {XPPMATH 20 "6#/&%\"aG6$\"#5\"\"'#\"O(ys'>m[^(>zRHbr 1%ftz:.a%[#\"N)G[.>)4'4K1FOqOuH!))Q!p[;U" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/&%\"aG6$\"#5\"\"(#\"PVL=v6k5KDLc5@9aUwN9wpaq\"\"O!3E$H8IXGmiy$ \\y?O!y[S2=s5" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/&%\"aG6$\"#5\"\")#\" W4(RZ-8d&=l')['R7" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/&%\"aG6$\"#5\"\"*#!N:^3!HQ;=!RTB$>C]UCI*e1sw#\"OsET'*f!)oYIstq$><; '[]:Ozj'*" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%!G" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/&%\"cG6#\"#6#\"#i\"#l" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/&%\"aG6$\"#6\"\"\"#!RrR(4jkoiIRz-t$H322QA\\h_E)))\"R++Wl_L&z+C2e`O t(R&H,MuH/=!)" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/&%\"aG6$\"#6\"\"#\" \"!" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/&%\"aG6$\"#6\"\"$\"\"!" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#/&%\"aG6$\"#6\"\"%#!D$p&*=fR?'>.?t\")H Sx+8*\"CDN3&*4u(f&)3EwSuM0b7?7\"\"P++W\\gm'*pT81n#4G_gOe?%> XI\"" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/&%\"aG6$\"#6\"\"'#\"W8U&HfIjd =w!y$GC\"QO?m93]S!4[L$=\"Vg\\1Fqf)4w*)GxuO$[g6$))[vtH%ot3``\"N+gjYU&)[-!RIbp8Q91 E0Zlw]" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/&%\"aG6$\"#7\"\"#\"\"!" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#/&%\"aG6$\"#7\"\"$\"\"!" }}{PARA 11 " " 1 "" {XPPMATH 20 "6#/&%\"aG6$\"#7\"\"%#!C7jSW$)H%G?Cl=-]&Q\"e'\"B:Kl 3eFF!=!y$)))GSYU)" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/&%\"aG6$\"#7\"\" &#!N(4jo!p=2DPG:KH.3\\)G1*4\")H#\"M+Sgx6(pi'f*=%>wR!GB1f5[&H" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/&%\"aG6$\"#7\"\"'#\"QL=ajgVl)G2+Pf`^YY/'GOu @OL\"P/:4mIhi'=Tmu)*>!3szFI9VUS&" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/& %\"aG6$\"#7\"\"(#\"T$*)*>uT)p\\$faqvSswg$G7!*GIAO9G\"Sg\\rYD[%RT1(RQ9& )Q?HAGe>bYyC" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/&%\"aG6$\"#7\"\")#!< \"RUjFB5mTs3;e>\"<1^(yF&ouE,[m*)*R" }}{PARA 11 "" 1 "" {XPPMATH 20 "6# /&%\"aG6$\"#7\"\"*#\"ZlzE^XD[Vkv&*R))RoWXfCX3QA3q_kW\"\"Zt*R1(=K.w=,I- f8=$[*)ej23rGoYc[J" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/&%\"aG6$\"#7\"# 5#!&=w(eblL6\";z(o/,)3KAG/:4$*" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/&%\"aG6$\"#7\"#6#\":vV4dv)['=.j%3!*\":y;L%e!o+yMiog'" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%!G" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/&%\" cG6#\"#8\"\"\"" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/&%\"aG6$\"#8\"\"\"# !J@YfVx*eniUev^\"=I46*obN\"J+!G'HK'GDF)HH&4*p)pwDsLB" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/&%\"aG6$\"#8\"\"#\"\"!" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/&%\"aG6$\"#8\"\"$\"\"!" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/&%\"aG6$\"#8\"\"%#!Ac/D&\\-vFGEE*)*)>jd\"\"@Xgc(H5ya;;%G>i0?\" " }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/&%\"aG6$\"#8\"\"&#!K6>=YB_!\\ZL&o 'R/v+pH#*\\u%\"J+?lWTVorQyZ$)H!zl'z$y5U" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/&%\"aG6$\"#8\"\"'#\"Nz1JTn2nf$[wN+zQrB,x)faz'\"M_.V&>A\"Qdf7E+Y dC&>RKt8q(" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/&%\"aG6$\"#8\"\"(#\"Ox& prkXy0'*z%)*=7*=i(4>-r\")o8\"MSMM>=\"*>!y2V/kVG[sa\\!4jv" }}{PARA 11 " " 1 "" {XPPMATH 20 "6#/&%\"aG6$\"#8\"\")#!:eKnAtsl@(o\\#H$\";`x/7l*)z% G+j`#f" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/&%\"aG6$\"#8\"\"*#!R!R4?'4. wwIFus-`tHld!\\4B_%o#\"Sn)>>Pm0MzWPukId*[-lImLGY$" }}{PARA 11 "" 1 " " {XPPMATH 20 "6#/&%\"aG6$\"#8\"#5#\"8oNt%zJSr8>jk\"9\\E!))*HICOvBk[" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/&%\"aG6$\"#8\"#6\"\"!" }}{PARA 11 " " 1 "" {XPPMATH 20 "6#/&%\"aG6$\"#8\"#7\"\"!" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%!G" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/&%\"bG6#\"\"\"# \"1.JeyD)*\\5\"2gxGN.KFO#" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/&%\"bG6# \"\"#\"\"!" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/&%\"bG6#\"\"$\"\"!" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#/&%\"bG6#\"\"%\"\"!" }}{PARA 11 "" 1 " " {XPPMATH 20 "6#/&%\"bG6#\"\"&\"\"!" }}{PARA 11 "" 1 "" {XPPMATH 20 " 6#/&%\"bG6#\"\"'#\"3vyjT1iR#o(\"4'\\y$p/)onh@" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/&%\"bG6#\"\"(#\"7Dcv)p()*4Y@%)y\"83-)RLVHMv>oJ" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#/&%\"bG6#\"\")#!8\"Hdv[49X-ps>\"7+[/9, zO_Zf#*" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/&%\"bG6#\"\"*#\"enh\"RJ/]) \\^/r>N;Uuxx5#))H'eds(*[a)H#\"Z+'zfy4Z&yY`L#=*G[2DawO9_f*>m_Bt" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#/&%\"bG6#\"#5#!4[CuLy&y\\2B\"3Di5,,dX( y#" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/&%\"bG6#\"#6#\"6v8DGKkkO'*[\"\" 5[3&Rp1c>2-#" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/&%\"bG6#\"#7#\"-***46 4w&\".+ce$o3B" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/&%\"bG6#\"#8\"\"!" } }{PARA 11 "" 1 "" {XPPMATH 20 "6#%!G" }}{PARA 11 "" 1 "" {XPPMATH 20 " 6#/&%#b*G6#\"\"\"#\".H)*)p+][\"0!GJ0m\"))3\"" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/&%#b*G6#\"\"#\"\"!" }}{PARA 11 "" 1 "" {XPPMATH 20 "6# /&%#b*G6#\"\"$\"\"!" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/&%#b*G6#\"\"% \"\"!" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/&%#b*G6#\"\"&\"\"!" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/&%#b*G6#\"\"'#\"1vo=+%)H,e\"27$*eF'4KD;" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#/&%#b*G6#\"\"(#\"4D1`gg#f43C\"4KC()y&z 7k\"p*" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/&%#b*G6#\"\")#\"5L?z/y&)Hi+ b\"5+'4$*GloSZd\"" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/&%#b*G6#\"\"*#\" Qjub[%[Hq**oj3DN&43TspD*[[j$\"Q+GzoU*\\p)GL!e:kPU$z))=j5y2;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/&%#b*G6#\"#5#!2_^TFQt(>9\"1D0.r3zRS" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#/&%#b*G6#\"#6\"\"!" }}{PARA 11 "" 1 " " {XPPMATH 20 "6#/&%#b*G6#\"#7\"\"!" }}{PARA 11 "" 1 "" {XPPMATH 20 "6 #/&%#b*G6#\"#8#!+29hI!*\"++K))zZ" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 151 "RK8_13eqs := [op( RowSumConditions(13,'expanded')),op(OrderConditions(8,13,'expanded'))] :\n`RK7_13eqs*` := subs(b=`b*`,OrderConditions(7,13,'expanded')):" }}} {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 6 "Check:" }} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 112 "subs(e18,RK8_13eqs):\nmap(u ->lhs(u)-rhs(u),%);\nnops(%);\nsubs(e18,`RK7_13eqs*`):\nmap(u->lhs(u)- rhs(u),%);\nnops(%);" }}{PARA 12 "" 1 "" {XPPMATH 20 "6#7`x\"\"!F$F$F$ F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$ F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$ F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$ F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$ F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$ F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#\"$7#" }}{PARA 12 "" 1 "" {XPPMATH 20 "6#7ap\"\"!F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F $F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F $F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#\"#&)" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 26 "#-------------------------" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}}{PARA 0 "" 0 "" {TEXT -1 52 "#----------------- ----------------------------------" }}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 38 "characteristics of the embedded scheme" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 36 "coefficients of the combined scheme " } }{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 5959 "ee := \{c[2]=27/400,\nc[3]=469/4510,\nc[4]=1407/9020,\nc[5]=10 0/257,\nc[6]=7/15,\nc[7]=7/45,\nc[8]=58/59,\nc[9]=49117478/55144743,\n c[10]=13/14,\nc[11]=62/65,\nc[12]=1,\nc[13]=1,\n\na[2,1]=27/400,\na[3, 1]=1311793/54918270,\na[3,2]=439922/5491827,\na[4,1]=1407/36080,\na[4, 2]=0,\na[4,3]=4221/36080,\na[5,1]=12985296430100/33603736057857,\na[5, 2]=0,\na[5,3]=-16218027650000/11201245352619,\na[5,4]=48744170200000/3 3603736057857,\na[6,1]=3817807/81405000,\na[6,2]=0,\na[6,3]=0,\na[6,4] =10274191312/43991343405,\na[6,5]=40755997793/218862405000,\na[7,1]=88 83007/146529000,\na[7,2]=0,\na[7,3]=0,\na[7,4]=53436069013712/47550243 0864645,\na[7,5]=-4116355777093/117791746371000,\na[7,6]=7635614/43630 5285,\na[8,1]=-25750155116216595121447/19172397107027415328000,\na[8,2 ]=0,\na[8,3]=0,\na[8,4]=-743/55,\na[8,5]=-3193438311684528658341050903 /315992014923229256968928000,\na[8,6]=183334983592060082563257/2293018 6940004788732288,\na[8,7]=18586487074230223824062061/10356928917216209 76018560,\na[9,1]=-922122171894716552217807973277934204672348766351979 550680715901/\n 15740119019867164084552691291294998765398466300 29827642536608000,\na[9,2]=0,\na[9,3]=0,\na[9,4]=-24939241129656091104 22434487743812058912776576445331741068763/\n 248576188041998999 363340039748303974885433202092103843801085,\na[9,5]=-34741045425295020 6893609060802736441487435576445275533884950354547756949/\n 6974 7362766091883481592212536178438630509515521207481681627188467808000,\n a[9,6]=532773370793958447861258481053032011231466563240558601582209945 9/\n 12498089552051473201987367363212255968282639800930482622603 72352,\na[9,7]=6380564756318242852834456238988262089096993336480502590 3916023/\n 52225368801254480919325372124301691131185132948965843 16063360,\na[9,8]=16756974250619222216361657668160522447776912/1649765 930508280715886033484777254665540228007,\na[10,1]=-3702927824534109782 314804940532859074364487/4131257703459996302316696251055190381568000, \na[10,2]=0,\na[10,3]=0,\na[10,4]=-48542494919285774589408977236397/37 95156546073735065277428293920,\na[10,5]=-22919425542674277431989520078 944496953957193/3194620077548089472408777624402038658048000,\na[10,6]= 2484540315797359406715529397919751486619672787/42164869038880297436703 6270632096098190348288,\na[10,7]=1705469761435764254142110563325321064 1175183343/1072180740487803620784937862662845301329326080,\na[10,8]=24 2464961680690545467502075/12396488665185571302473970944,\na[10,9]=-276 720658930244250241932341390181638290085115/966379361550486161719370737 2304668805996412672,\na[11,1]=-888265261492238070708293730279393062686 4630973971/\n 801804297434012953977336535807240079533526544000 0,\na[11,2]=0,\na[11,3]=0,\na[11,4]=-913007740298173200319620395918956 93/6641776260885597741930170995083525,\na[11,5]=-112201255053474401906 816716367850973725245563029/\n 1304519420583660522809267061341 6996666049440000,\na[11,6]=1833348090405008146620363812428378076185763 30592954213/\n 264598469797448467476894728911176960200432517270 64960,\na[11,7]=590853444397902776833979526228689245354893077492408252 7/\n 337801676952094167280622617772889760985970195055625600,\na [11,8]=6612449503769637840690016046967566/2255568687770602845862583843 73942375,\na[11,9]=-10293784278236522497308869863513789970987031746931 000834/\n 8787826363395572889246773386374957640244197745505586 187675,\na[11,10]=-4215629637956404624565504/1081236125912922343765001 25,\na[12,1]=-535308736842973754888311604833674190949559277/\n \+ 507665470526061438136955303900248854246636000,\na[12,2]=0,\na[12,3]=0 ,\na[12,4]=-6581385500218652420284298344406312/84246402888837801802727 5808653215,\na[12,5]=-229810990628849080329321528372507186906863097/\n 29548105906232803976194189596626971177604000,\na[12,6]=333621 743628604464651535937000728865436063541833/\n 54042431430277972 080199874664118662613066091504,\na[12,7]=28143622302890122836076724075 7054593496984174199893/\n 2478465519582822292038851438397064139 4482546714960,\na[12,8]=-195816087241661023276342391/39989664801267468 5277875106,\na[12,9]=1446452700822380845245945446839883995756443482545 51267965/\n 314856466828710807635889483181359023001187603321870 639973,\na[12,10]=-113365555877618519272588672/93091504282232088010468 779,\na[12,11]=9008463031864887557094375/6606862347800680584331678,\na [13,1]=-35556891109301815175584262675897743594621/23337225766986990952 929827252863229628000,\na[13,2]=0,\na[13,3]=0,\na[13,4]=-1576319898926 2628277502495250456/1200562192841616547810297566045,\na[13,5]=-4744992 29690075043966853347490522346181911/\n 42107837966579029834778 387168434144652000,\na[13,6]=67954598770123713879003576483596707674131 0679/\n 77013733239195245746002612595738122195430352,\na[13,7]= 1368817102190976218912189847996057845647169577/\n 7563090495472 4828436404430778019911819343440,\na[13,8]=-3292496872165727322673258/5 9253630028479896512047753,\na[13,9]=-268452230949057652973530272742730 7676030962009390/\n 346283366306502489573064743744793405663741 28519867, a[13,10]=64631913714031794733568/486423753624302998802649,\n a[13,11]=0,\na[13,12]=0,\n\nb[1]=1049982578583103/23627320335287760,\n b[2]=0,\nb[3]=0,\nb[4]=0,\nb[5]=0,\nb[6]=768239620641637875/2161676880 469378496,\nb[7]=7884214609987698755625/31681975342943333980208,\nb[8] =-19726902451409487557291/9259475236790114044800,\nb[9]=22985448977257 58629882107777744216351971045149850043139161/\n 732352661995952143 676542507482891823353467854709785979600,\nb[10]=-2307497857833742448/2 78745570101106225,\nb[11]=148963664643228251375/20207195606693950848, \nb[12]=576091109999/2308683585600,\nb[13]=0,\n\n`b*`[1]=4850006989829 /108881660531280,\n`b*`[2]=0,\n`b*`[3]=0,\n`b*`[4]=0,\n`b*`[5]=0,\n`b* `[6]=5801298400186875/16253209627589312,\n`b*`[7]=2408095926060530625/ 9691641279578872432,\n`b*`[8]=55006229857804792033/1574740686528930960 0,\n`b*`[9]=363484892569724108095352508636899702948448557463/\n \+ 160778106318887934237641558033288694994268792800,\n`b*`[10]=-14197733 827415152/4039790871030525,\n`b*`[11]=0,\n`b*`[12]=0,\n`b*`[13]=-90306 11407/4779883200\}:" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}}{PARA 0 "" 0 "" {TEXT -1 0 "" }} {PARA 0 "" 0 "" {TEXT -1 5 "Let " }{XPPEDIT 18 0 "T[8, 13];" "6#&%\"T G6$\"\")\"#8" }{TEXT -1 128 " denote the vector whose components are \+ the principal error terms of the 13 stage, order 8 scheme (the error t erms of order 9)." }}{PARA 0 "" 0 "" {TEXT -1 5 "Let " }{XPPEDIT 18 0 "`T*`[7,13];" "6#&%#T*G6$\"\"(\"#8" }{TEXT -1 146 " denote the vect or whose components are the principal error terms of the embedded 13 s tage, order 7 scheme (the error terms of order 8) and let " } {XPPEDIT 18 0 "`T*`[8,13];" "6#&%#T*G6$\"\")\"#8" }{TEXT -1 99 " deno te the vector whose components are the error terms of order 9 of the e mbedded order 7 scheme." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 39 "Denote the 2-norm of these vectors by " }{XPPEDIT 18 0 "abs(abs(T[8,13]));" "6#-%$absG6#-F$6#&%\"TG6$\"\")\"#8" }{TEXT -1 3 ", " }{XPPEDIT 18 0 "abs(abs(`T*`[7, 13]));" "6#-%$absG6#-F$6#&% #T*G6$\"\"(\"#8" }{TEXT -1 7 " and " }{XPPEDIT 18 0 "abs(abs(`T*`[8, 13]));" "6#-%$absG6#-F$6#&%#T*G6$\"\")\"#8" }{TEXT -1 15 " respectiv ely." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 8 "De fine: " }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "A[9] = abs( abs(T[8,13]));" "6#/&%\"AG6#\"\"*-%$absG6#-F)6#&%\"TG6$\"\")\"#8" } {TEXT -1 6 ", " }{XPPEDIT 18 0 "B[9] = abs(abs(`T*`[8,13]))/abs(ab s(`T*`[7,13]));" "6#/&%\"BG6#\"\"**&-%$absG6#-F*6#&%#T*G6$\"\")\"#8\" \"\"-F*6#-F*6#&F/6$\"\"(F2!\"\"" }{TEXT -1 6 ", " }{XPPEDIT 18 0 " C[9] = abs(abs(`T*`[8,13]-T[8,13]))/abs(abs(`T*`[7,13]));" "6#/&%\"CG6 #\"\"**&-%$absG6#-F*6#,&&%#T*G6$\"\")\"#8\"\"\"&%\"TG6$F2F3!\"\"F4-F*6 #-F*6#&F06$\"\"(F3F8" }{TEXT -1 1 "." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 77 "Dormand and Prince have suggested that \+ as well as attempting to ensure that " }{XPPEDIT 18 0 "A[9];" "6#&%\" AG6#\"\"*" }{TEXT -1 73 " is a minimum, if the embedded scheme is to \+ be used for error control, " }{XPPEDIT 18 0 "B[9];" "6#&%\"BG6#\"\"* " }{TEXT -1 7 " and " }{XPPEDIT 18 0 "C[9];" "6#&%\"CG6#\"\"*" } {TEXT -1 27 " should be chosen so that " }}{PARA 0 "" 0 "" {TEXT -1 68 "they are \"similar in magnitude\" and also not differ too much fro m 1." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 168 "errterms8_13 := PrincipalErrorTerms(8,13,'expanded') :\n`errterms8_13*` :=subs(b=`b*`,errterms8_13):\n`errterms7_13*` := su bs(b=`b*`,PrincipalErrorTerms(7,13,'expanded')):" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 333 "snmB := sqr t(add(evalf(subs(ee,`errterms8_13*`[i]))^2,i=1..nops(`errterms8_13*`)) ):\nsdnB := sqrt(add(evalf(subs(ee,`errterms7_13*`[i]))^2,i=1..nops(`e rrterms7_13*`))):\nsnmC := sqrt(add((evalf(subs(ee,`errterms8_13*`[i]) -subs(ee,errterms8_13[i])))^2,i=1..nops(errterms8_13))):\n'B[9]'= eval f[8](snmB/sdnB);\n'C[9]'= evalf[8](snmC/sdnB);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/&%\"BG6#\"\"*$\")3#43#!\"(" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/&%\"CG6#\"\"*$\") " 0 "" {MPLTEXT 1 0 1 ";" }}}}{PARA 0 "" 0 "" {TEXT -1 52 "#--------------------------------------------------- " }}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 27 "absolute stability regions " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 36 "coefficients of the co mbined scheme " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 5959 "ee := \{c[2]=27/400,\nc[3]=469/4510,\nc[4]=140 7/9020,\nc[5]=100/257,\nc[6]=7/15,\nc[7]=7/45,\nc[8]=58/59,\nc[9]=4911 7478/55144743,\nc[10]=13/14,\nc[11]=62/65,\nc[12]=1,\nc[13]=1,\n\na[2, 1]=27/400,\na[3,1]=1311793/54918270,\na[3,2]=439922/5491827,\na[4,1]=1 407/36080,\na[4,2]=0,\na[4,3]=4221/36080,\na[5,1]=12985296430100/33603 736057857,\na[5,2]=0,\na[5,3]=-16218027650000/11201245352619,\na[5,4]= 48744170200000/33603736057857,\na[6,1]=3817807/81405000,\na[6,2]=0,\na [6,3]=0,\na[6,4]=10274191312/43991343405,\na[6,5]=40755997793/21886240 5000,\na[7,1]=8883007/146529000,\na[7,2]=0,\na[7,3]=0,\na[7,4]=5343606 9013712/475502430864645,\na[7,5]=-4116355777093/117791746371000,\na[7, 6]=7635614/436305285,\na[8,1]=-25750155116216595121447/191723971070274 15328000,\na[8,2]=0,\na[8,3]=0,\na[8,4]=-743/55,\na[8,5]=-319343831168 4528658341050903/315992014923229256968928000,\na[8,6]=1833349835920600 82563257/22930186940004788732288,\na[8,7]=18586487074230223824062061/1 035692891721620976018560,\na[9,1]=-92212217189471655221780797327793420 4672348766351979550680715901/\n 1574011901986716408455269129129 499876539846630029827642536608000,\na[9,2]=0,\na[9,3]=0,\na[9,4]=-2493 924112965609110422434487743812058912776576445331741068763/\n 24 8576188041998999363340039748303974885433202092103843801085,\na[9,5]=-3 4741045425295020689360906080273644148743557644527553388495035454775694 9/\n 6974736276609188348159221253617843863050951552120748168162 7188467808000,\na[9,6]=53277337079395844786125848105303201123146656324 05586015822099459/\n 1249808955205147320198736736321225596828263 980093048262260372352,\na[9,7]=638056475631824285283445623898826208909 69933364805025903916023/\n 5222536880125448091932537212430169113 118513294896584316063360,\na[9,8]=167569742506192222163616576681605224 47776912/1649765930508280715886033484777254665540228007,\na[10,1]=-370 2927824534109782314804940532859074364487/41312577034599963023166962510 55190381568000,\na[10,2]=0,\na[10,3]=0,\na[10,4]=-48542494919285774589 408977236397/3795156546073735065277428293920,\na[10,5]=-22919425542674 277431989520078944496953957193/319462007754808947240877762440203865804 8000,\na[10,6]=2484540315797359406715529397919751486619672787/42164869 0388802974367036270632096098190348288,\na[10,7]=1705469761435764254142 1105633253210641175183343/10721807404878036207849378626628453013293260 80,\na[10,8]=242464961680690545467502075/12396488665185571302473970944 ,\na[10,9]=-276720658930244250241932341390181638290085115/966379361550 4861617193707372304668805996412672,\na[11,1]=-888265261492238070708293 7302793930626864630973971/\n 801804297434012953977336535807240 0795335265440000,\na[11,2]=0,\na[11,3]=0,\na[11,4]=-913007740298173200 31962039591895693/6641776260885597741930170995083525,\na[11,5]=-112201 255053474401906816716367850973725245563029/\n 1304519420583660 5228092670613416996666049440000,\na[11,6]=1833348090405008146620363812 42837807618576330592954213/\n 264598469797448467476894728911176 96020043251727064960,\na[11,7]=590853444397902776833979526228689245354 8930774924082527/\n 3378016769520941672806226177728897609859701 95055625600,\na[11,8]=6612449503769637840690016046967566/2255568687770 60284586258384373942375,\na[11,9]=-10293784278236522497308869863513789 970987031746931000834/\n 8787826363395572889246773386374957640 244197745505586187675,\na[11,10]=-4215629637956404624565504/1081236125 91292234376500125,\na[12,1]=-53530873684297375488831160483367419094955 9277/\n 507665470526061438136955303900248854246636000,\na[12,2 ]=0,\na[12,3]=0,\na[12,4]=-6581385500218652420284298344406312/84246402 8888378018027275808653215,\na[12,5]=-229810990628849080329321528372507 186906863097/\n 29548105906232803976194189596626971177604000, \na[12,6]=333621743628604464651535937000728865436063541833/\n 5 4042431430277972080199874664118662613066091504,\na[12,7]=2814362230289 01228360767240757054593496984174199893/\n 247846551958282229203 88514383970641394482546714960,\na[12,8]=-195816087241661023276342391/3 99896648012674685277875106,\na[12,9]=144645270082238084524594544683988 399575644348254551267965/\n 31485646682871080763588948318135902 3001187603321870639973,\na[12,10]=-113365555877618519272588672/9309150 4282232088010468779,\na[12,11]=9008463031864887557094375/6606862347800 680584331678,\na[13,1]=-35556891109301815175584262675897743594621/2333 7225766986990952929827252863229628000,\na[13,2]=0,\na[13,3]=0,\na[13,4 ]=-15763198989262628277502495250456/1200562192841616547810297566045,\n a[13,5]=-474499229690075043966853347490522346181911/\n 4210783 7966579029834778387168434144652000,\na[13,6]=6795459877012371387900357 64835967076741310679/\n 770137332391952457460026125957381221954 30352,\na[13,7]=1368817102190976218912189847996057845647169577/\n \+ 75630904954724828436404430778019911819343440,\na[13,8]=-32924968721 65727322673258/59253630028479896512047753,\na[13,9]=-26845223094905765 29735302727427307676030962009390/\n 34628336630650248957306474 374479340566374128519867, a[13,10]=64631913714031794733568/48642375362 4302998802649,\na[13,11]=0,\na[13,12]=0,\n\nb[1]=1049982578583103/2362 7320335287760,\nb[2]=0,\nb[3]=0,\nb[4]=0,\nb[5]=0,\nb[6]=7682396206416 37875/2161676880469378496,\nb[7]=7884214609987698755625/31681975342943 333980208,\nb[8]=-19726902451409487557291/9259475236790114044800,\nb[9 ]=2298544897725758629882107777744216351971045149850043139161/\n 73 2352661995952143676542507482891823353467854709785979600,\nb[10]=-23074 97857833742448/278745570101106225,\nb[11]=148963664643228251375/202071 95606693950848,\nb[12]=576091109999/2308683585600,\nb[13]=0,\n\n`b*`[1 ]=4850006989829/108881660531280,\n`b*`[2]=0,\n`b*`[3]=0,\n`b*`[4]=0,\n `b*`[5]=0,\n`b*`[6]=5801298400186875/16253209627589312,\n`b*`[7]=24080 95926060530625/9691641279578872432,\n`b*`[8]=55006229857804792033/1574 7406865289309600,\n`b*`[9]=3634848925697241080953525086368997029484485 57463/\n 160778106318887934237641558033288694994268792800,\n`b* `[10]=-14197733827415152/4039790871030525,\n`b*`[11]=0,\n`b*`[12]=0,\n `b*`[13]=-9030611407/4779883200\}:" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" } }{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 78 "The stability function R \+ for the 13 stage, order 8 scheme is given as follows." }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 76 "subs(ee,StabilityFunction(8,13,'expanded' )):\nR := unapply(%,z):\n'R(z)'=R(z);" }}{PARA 12 "" 1 "" {XPPMATH 20 "6#/-%\"RG6#%\"zG,<\"\"\"F)F'F)*&#F)\"\"#F)*$)F'F,F)F)F)*&#F)\"\"'F)*$ )F'\"\"$F)F)F)*&#F)\"#CF)*$)F'\"\"%F)F)F)*&#F)\"$?\"F)*$)F'\"\"&F)F)F) *&#F)\"$?(F)*$)F'F1F)F)F)*&#F)\"%S]F)*$)F'\"\"(F)F)F)*&#F)\"&?.%F)*$)F '\"\")F)F)F)*&#\"M(GO'R6l1yclB?YCE(4!foTF6\"R+++s5@5#*znxM(=1$yCEEO)Ql *=%F)*$)F'\"\"*F)F)F)*&#\"N2\"GBysE)\\(pZ,#=Hu'z:T67*\\$\"U+++!Gh(*pK& )*)4RI*3VR%>%)=xa&=U8F)*$)F'\"#5F)F)F)*&#\"L\"=')>S[(yo,T0/U9ZSs(yG!> \"\"S+++[oWeev$*f&[EO!e)))[x0P.'f#)F)*$)F'\"#6F)F)F)*&#\"G$GKME'>rhj>O :;_DT/AK\"Q++'pp]\\o*3 " 0 "" {MPLTEXT 1 0 26 "z0 := newton(R(z)=1,z=- 6);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%#z0G$!+l&)Q7g!\"*" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 302 "z 0 := newton(R(z)=1,z=-6):\np1 := plot([R(z),1],z=-6.69..0.49,color=[re d,blue]):\np2 := plot([[[z0,1]]$3],style=point,symbol=[circle,cross,di amond],color=black):\np3 := plot([[z0,0],[z0,1]],linestyle=3,color=COL OR(RGB,0,.5,0)):\nplots[display]([p1,p2,p3],view=[-6.69..0.49,-0.07..1 .47],font=[HELVETICA,9]);" }}{PARA 13 "" 1 "" {GLPLOT2D 360 253 253 {PLOTDATA 2 "6+-%'CURVESG6$7X7$$!3Q++++++!p'!#<$\"3$QWe:())4QDF*7$$!3y m;a;#[ZZ/(Rt'R\"F*7$$!3)QLL#G \\5nhF*$\"3;ad-we8_7F*7$$!3UL$3p-F+4'F*$\"3C^v9)fu/7\"F*7$$!3X]PRjaJ8g F*$\"3ar<:i*p8+\"F*7$$!3fm\"z)**QgOfF*$\"3V9/W*p&)H$*)!#=7$$!3)Q$ef@5O %z&F*$\"3%e*p\"Re_K>(Fin7$$!3E+D;gn2ZcF*$\"3+&[2Vnk*4dFin7$$!3gLeHm^v% \\&F*$\"3k'G]C>>`Y%Fin7$$!3Q+Drp?#HM&F*$\"3A*R0@5X2Z$Fin7$$!3ymm@nIu'= &F*$\"3iK>_*f]1m#Fin7$$!3oL$3o^!=\\]F*$\"3a1#ezK8a4#Fin7$$!37++lmwJ%*[ F*$\"3#*[02a(*o&f\"Fin7$$!3F++vM*=)QZF*$\"3-&z!*fosF@\"Fin7$$!3'****\\ L4n*)e%F*$\"3+k;?3>oX$*!#>7$$!3yLeHls)GX%F*$\"30F*H&fi*HW(Fgq7$$!3Umm \"y=v5H%F*$\"3))\\S<&y%3+eFgq7$$!3#ommK8**R:%F*$\"3wv3z/z'>$[Fgq7$$!3O +D'[4ZX*RF*$\"3+yAbm]b%4%Fgq7$$!3=nmwA]S`QF*$\"3()y)y>H/Ms$Fgq7$$!33+D ')*z\\&)p$F*$\"3)4u8!G')RiNFgq7$$!3'**\\(e254^NF*$\"3>AwO%ppzg$Fgq7$$! 3wm;HFKB(R$F*$\"3KR!fAH-=$QFgq7$$!3im\"zdvVfD$F*$\"3!3OY`qd(yTFgq7$$!3 kL$3K(Ra.JF*$\"352YXIi'[p%Fgq7$$!3oLeHIMCXHF*$\"3M\")[BtZL%Q&Fgq7$$!37 +vLUBW2GF*$\"3Ce8Sosj:hFgq7$$!3EL$e@R8'eEF*$\"3!*e<&3g0A0(Fgq7$$!3G++q B&e[]#F*$\"35LE1P`#f>)Fgq7$$!3?+]_\"QRWN#F*$\"3=**Ga0`))4&*Fgq7$$!33+D 1F6!*3AF*$\"3GC7rJ*\\!*4\"Fin7$$!3?+]24dIZ?F*$\"3i/iPdXB\"H\"Fin7$$!3# pmm(Q]5->F*$\"3/0`O(G\\F\\\"Fin7$$!3-++vIB2ZL8Xag!H9$Fin7$$! 3[mmTCKv45F*$\"3_S^:_$*3VOFin7$$!3].]il-T^&)Fin$\"3_+^]MBB_UFin7$$!3#) GLLR5JiqFin$\"3/Jl4@l+N\\Fin7$$!3^GL30?\\RbFin$\"3oiJK;F*-%'COLOURG6&%$RGBG$\"*+++ +\"!\")$\"\"!Fj\\lFi\\l-F$6$7S7$F($\"\"\"Fj\\l7$F3F_]l7$F=F_]l7$FGF_]l 7$FQF_]l7$FenF_]l7$F[oF_]l7$F`oF_]l7$FeoF_]l7$FjoF_]l7$F_pF_]l7$FdpF_] l7$FipF_]l7$F^qF_]l7$FcqF_]l7$FiqF_]l7$F^rF_]l7$FcrF_]l7$FhrF_]l7$F]sF _]l7$FbsF_]l7$FgsF_]l7$F\\tF_]l7$FatF_]l7$FftF_]l7$F[uF_]l7$F`uF_]l7$F euF_]l7$FjuF_]l7$F_vF_]l7$FdvF_]l7$FivF_]l7$F^wF_]l7$FcwF_]l7$FhwF_]l7 $F]xF_]l7$FbxF_]l7$FgxF_]l7$F\\yF_]l7$FayF_]l7$FfyF_]l7$F[zF_]l7$F`zF_ ]l7$FezF_]l7$FjzF_]l7$F_[lF_]l7$Fd[lF_]l7$Fi[lF_]l7$F^\\lF_]l-Fc\\l6&F e\\lFi\\lFi\\lFf\\l-F$6&7#7$$!3=+++l&)Q7gF*F_]l-%'SYMBOLG6#%'CIRCLEG-F c\\l6&Fe\\lFj\\lFj\\lFj\\l-%&STYLEG6#%&POINTG-F$6&Fe`l-Fj`l6#%&CROSSGF ]alF_al-F$6&Fe`l-Fj`l6#%(DIAMONDGF]alF_al-F$6%7$7$Fg`lFi\\lFf`l-%&COLO RG6&Fe\\lFi\\l$\"\"&!\"\"Fi\\l-%*LINESTYLEG6#\"\"$-%+AXESLABELSG6%Q\"z 6\"Q!F_cl-%%FONTG6#%(DEFAULTG-Fbcl6$%*HELVETICAG\"\"*-%%VIEWG6$;$!$p'! \"#$\"#\\F_dl;$!\"(F_dl$\"$Z\"F_dl" 1 2 0 1 10 0 2 9 1 4 2 1.000000 45.000000 45.000000 0 0 "Curve 1" "Curve 2" "Curve 3" "Curve 4" "Curve 5" "Curve 6" }}}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 50 "The following picture shows the stability region. " }} {PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 992 "R := z -> add(z^j/j!,j=0..8)+\n 1127416859009726244620236556780 6651139636287/4189653883626262478306187347767799210211072000000*z^9+\n 349912114115796742918201476974982672782328107/134218554771884194394 3089303909898532699761280000000*z^10+\n 1190287877240471442040541016 878748401986181/82596033705774888858036264855993755858446848000000*z^1 1+\n 32220441255216153619636171196263432283/155615715677546892051372 672917089684950696960000*z^12:\npts := []: z0 := 0:\nfor ct from 0 to \+ 320 do\n zz := newton(R(z)=exp(ct*Pi/20*I),z=z0):\n z0 := zz:\n \+ pts := [op(pts),[Re(zz),Im(zz)]]:\nend do:\np1 := plot(pts,color=COLOR (RGB,.45,0,0)):\np2 := plots[polygonplot]([seq([pts[i-1],pts[i],[-4.6, 0]],i=2..nops(pts))],\n style=patchnogrid,color=COLOR(RGB,.95 ,.3,.2)):\np3 := plot([[[-6.89,0],[1.19,0]],[[0,-6.39],[0,6.39]]],colo r=black,linestyle=3):\nplots[display]([p||(1..3)],view=[-6.89..1.19,-6 .39..6.39],font=[HELVETICA,9],\n labels=[`Re(z)`,`Im(z)`] ,axes=boxed,scaling=constrained);" }}{PARA 13 "" 1 "" {GLPLOT2D 381 565 565 {PLOTDATA 2 "6+-%'CURVESG6$7]_l7$$\"\"!F)F(7$F($\"3++++Fjzq:!# =7$F($\"3)******Rl#fTJF-7$F($\"3D+++\")*)Q7ZF-7$$\"3y*****pqK.M%!#F$\" 3a+++:`=$G'F-7$$\"3w*****>hz&>Q!#E$\"3E+++*o\")R&yF-7$$\"3-+++z@#G?#!# D$\"3-+++6#yZU*F-7$$\"3h+++VMfO%*FC$\"3#******H^d&*4\"!#<7$$\"3=+++E$z GB$!#C$\"3-+++!HPmD\"FK7$$\"3i******zsKy#*FO$\"34+++,sr89FK7$$\"3))*** **f'Q-*H#!#B$\"31+++pszq:FK7$$\"3/+++r+:9]FZ$\"3++++7u(ys\"FK7$$\"3#3+ ++HQCt*FZ$\"31+++!Rd\\)=FK7$$\"37+++Eee'o\"!#A$\"3&)*****>&o.U?FK7$$\" 3;+++&HP>f#Fdo$\"3'******zw:\"*>#FK7$$\"3E+++]`/YMFdo$\"3y*****4k&>cBF K7$$\"37+++TiM&o$Fdo$\"3$)******)>#G8DFK7$$\"3)******\\jmHK#Fdo$\"3'** ****zX!RqEFK7$$!3&)*****4ha'H?Fdo$\"3;+++#H`v#GFK7$$!3'******\\IrO3\"! #@$\"3/+++(yM[)HFK7$$!3)******pOH;Z#Fcq$\"34+++!p\\B9$FK7$$!37++++\"=j 9%Fcq$\"3z*****zG!H+LFK7$$!3%*******Qs)eA&Fcq$\"3%******R3i*eMFK7$$!3% ******zqc&pMFcq$\"3(******4OH)=OFK7$$\"3\\*****>(zT:gFcq$\"3#*******)* )o0y$FK7$$\"3,+++1!QiK$!#?$\"3;+++:[6XRFK7$$\"3d******)GTI%)*Fbs$\"3n* *****4pg8TFK7$$\"30+++)fu[T#!#>$\"3-+++(pboG%FK7$$\"3(******z]@8M&F]t$ \"3/+++'eEHY%FK7$$\"3)******\\NB,0\"F-$\"3k******GH?JYFK7$$\"3%******* 4`\")H,XXx%FK7$$\"3++++\"[+sR#F-$\"3I+++PcS*)[FK7$$\"3% )*****RAWu&HF-$\"3s******>J@%)\\FK7$$\"3u*****Hlb$)R$F-$\"3e*****\\SVj 1&FK7$$\"3F+++,yPKPF-$\"3#)*****\\Zu*R^FK7$$\"3%******z\"HyuRF-$\"39++ +*[[t?&FK7$$\"3')*****fi'*)QTF-$\"3I+++Sxqp_FK7$$\"3w*****>uTcB%F-$\"3 k*****RktxK&FK7$$\"3C+++')>$RF%F-$\"35+++u`)>Q&FK7$$\"3s*****R/A6E%F-$ \"3!)*****R,BEV&FK7$$\"3u*****fUlL?%F-$\"3?+++XQ()zaFK7$$\"3++++')o(e5 %F-$\"3y*****>FpQ_&FK7$$\"33+++UA7tRF-$\"3')*****p&eqkbFK7$$\"3F+++^s& *3QF-$\"3%******>PdCg&FK7$$\"3$)*****\\aJnh$F-$\"3;+++Y;=PcFK7$$\"35++ +t%o$*R$F-$\"3K+++4V#*ocFK7$$\"3')*****>)zUfJF-$\"3!)******p1s(p&FK7$$ \"3%******R:d\"**GF-$\"3n******RkfBdFK7$$\"3v*****zOJ0i#F-$\"3E+++OxcY dFK7$$\"32+++jzGDBF-$\"3e*****zeSmw&FK7$$\"3()*****pf`\\,#F-$\"3#)**** *\\35Qy&FK7$$\"3'*******of'3p\"F-$\"3#******za8F- $\"37+++PrN4eFK7$$\"3++++/'Hf+\"F-$\"3I+++xvl+++Pf*G#eFK7$$\"3)******H>3)yFF]t$\"3%******R@')\\#eFK7$$!30+++[*Rq+ \"F]t$\"3w*****pq;Q#eFK7$$!3m*****4IL])[F]t$\"3t*****zNW#>eFK7$$!3Y*** **>8mV&))F]t$\"3n*****4;(36eFK7$$!3/+++()pt\"H\"F-$\"3Q+++HQ6*z&FK7$$! 3%******R=u!3Chd8#F-$\"3y*****44cCw &FK7$$!3?+++W<6xDF-$\"3z*****zH/pt&FK7$$!3;+++q53OIF-$\"3))*****>=Odq& FK7$$!3G+++m;Z>NF-$\"33+++5v6ocFK7$$!3!)*****pC7$RSF-$\"3$******RswHi& FK7$$!3&******>XDxh%F-$\"3!******zbd!pbFK7$$!39+++i\"*\\'H&F-$\"3C+++= DX0bFK7$$!35+++0J\"*[hF-$\"3T+++ob,MaFK7$$!3s******oi**\\sF-$\"3I+++S` Hl`FK7$$!3q*****pL:5a)F-$\"3k*****zmsjJ&FK7$$!3_*****Hc&\\Y)*F-$\"3)** ****>WU2H&FK7$$!3(******4dvr5\"FK$\"3q******4#H*z_FK7$$!3*******p!)R/A \"FK$\"37+++yZ^w_FK7$$!3!******>JLcK\"FK$\"3o*****fN%Gw_FK7$$!32+++j?? C9FK$\"35+++8a'pF&FK7$$!3&******Rsut^\"FK$\"3g*****pw#Hx_FK7$$!36+++OF 71;FK$\"31+++`.`w_FK7$$!3!******zR07p\"FK$\"3U+++eGBu_FK7$$!3!******4H ?Kx\"FK$\"3M+++aA7q_FK7$$!3/+++]Rk_=FK$\"3w******o1-k_FK7$$!3\"******R Ki)H>FK$\"3))*****H;7eD&FK7$$!3*******RC$>0?FK$\"35+++\\,UX_FK7$$!35++ +$4.*y?FK$\"3l*****4m$zK_FK7$$!35+++E$>7:#FK$\"3u*****z())*y@&FK7$$!3@ +++N+MAAFK$\"35+++?Rr+_FK7$$!33+++k8W#H#FK$\"3#******pwD7=&FK7$$!3;+++ FLohBFK$\"39+++b'G%f^FK7$$!36++++W@ICFK$\"35+++`NKN^FK7$$!3!******p3v \")\\#FK$\"3U+++)\\=*3^FK7$$!39+++\"f+dc#FK$\"3A+++C'H-3&FK7$$!3!***** *>fAHj#FK$\"3#)*****\\!GG\\]FK7$$!3)*******p+(**p#FK$\"3.+++sd6;]FK7$$ !3%)*****H2ppw#FK$\"3/+++c0y!)\\FK7$$!3')*****4ETS$GFK$\"3\")******=gM V\\FK7$$!3:+++H6I,HFK$\"3V++++,!R!\\FK7$$!3?+++t=&)oHFK$\"39+++:8bi[FK 7$$!35+++%3!yOIFK$\"38++++*G%>[FK7$$!3%)*****>q\\^5$FK$\"3%*******R5ou ZFK7$$!3#******>,'*R<$FK$\"3F+++:2ZGZFK7$$!3A+++92KVKFK$\"3C+++(3p4o%F K7$$!3++++v\"*38LFK$\"3g*****H&oMKYFK7$$!3))*****ptIKQ$FK$\"3G+++=Xw#e %FK7$$!3/+++UCk`MFK$\"3x*****R(HOKXFK7$$!3()*****>\"e>CNFK$\"3;+++2eD \"[%FK7$$!3!******\\oXZf$FK$\"3@+++8W_HWFK7$$!3%)*****>xR^m$FK$\"3%)** ***zR;sP%FK7$$!35+++'fF_t$FK$\"3')*****z0ZVK%FK7$$!3.+++Ny'[!QFK$\"3)) *****zA.4F%FK7$$!39+++'eLR(QFK$\"3++++^&[o@%FK7$$!3%)******)[:B%RFK$\" 3e*****4EH@;%FK7$$!3P+++>I#*4SFK$\"3++++w(zm5%FK7$$!35+++`VowSFK$\"33+ ++XwU]SFK7$$!3;+++!HXD9%FK$\"3%******Hl(H$*RFK7$$!3I+++7wY2UFK$\"3'*** ***\\1:_$RFK7$$!3A+++%=F9F%FK$\"3!*******p!3h(QFK7$$!3H+++**>TMVFK$\"3 \"******Re4f\"QFK7$$!3)*******y-U'R%FK$\"3?+++T&eXv$FK7$$!3%******zpeu X%FK$\"3;+++03+#p$FK7$$!3@+++22a)*=GOFK7$$!3m*****4)\\owX FK$\"3A+++jq3jNFK7$$!3O+++;R\"\\j%FK$\"3;+++lBm'\\$FK7$$!3O+++n@D#p%FK $\"3\")*****H4%*)GMFK7$$!3W+++%=D([ZFK$\"3#*******=#p(fLFK7$$!3-+++DxN /[FK$\"3y*****>C$G*G$FK7$$!3y*****4Ns\"f[FK$\"36+++G*Ru@$FK7$$!3!***** *\\#z=8\\FK$\"3!)*****\\$3DWJFK7$$!37+++T\"=k'\\FK$\"3=+++$fM(pIFK7$$! 38+++m,()=]FK$\"3!******z\"f\"R*HFK7$$!3s******\\Maq]FK$\"3%)*****>OCo \"HFK7$$!3q*****H!)G97&FK$\"37+++WG\\QGFK7$$!3s*******y2:<&FK$\"3;+++8 g&*eFFK7$$!37+++;Dv?_FK$\"3!)*****4b[#yEFK7$$!3e*****>!e7p_FK$\"35+++6 NS'f#FK7$$!36+++]\"3X8DFK7$$!3%******RjmIO&FK$\"3))* ****4+;%HCFK7$$!3R+++X+_3aFK$\"3@+++^$>VM#FK7$$!3))*****H=wGX&FK$\"3?+ ++c`j#*4<#FK7$$!3')******=X,QbFK$ \"3'******z@uF3#FK7$$!37+++%)*\\'ybFK$\"3#******HA=N*>FK7$$!3*******H# p*yh&FK$\"3++++!y=K!>FK7$$!35+++k4obcFK$\"3'******pBn=\"=FK7$$!3')**** *poG>p&FK$\"3*)*****R``%>+++&4ols&FK$\"3$******Hqnfi\"FK7$$!3 e*****\\%*G&fdFK$\"3'******4Y,9`\"FK7$$!3#)*****f'Hu!z&FK$\"3%*******R )\\dV\"FK7$$!3q*****z,W,#eFK$\"34+++SG,R8FK7$$!3)******z?ow%eFK$\"3#** ****f/(>T7FK7$$!3g*****\\-aK(eFK$\"3*)*****\\?4?N*fFK$\"38+++M9,RUF-7$$!3.+++hgv,gFK$\"3%)*****H$e+%=$F-7$$!37+++Ep l2gFK$\"3()******p+'\\7#F-7$$!3\"******pt/7,'FK$\"3++++!zsJ1\"F-7$$!3= +++l&)Q7gFKF(7$F`]n$!3++++!zsJ1\"F-7$F[]n$!3()******p+'\\7#F-7$Ff\\n$! 3%)*****H$e+%=$F-7$Fa\\n$!38+++M9,RUF-7$F\\\\n$!3W*****RNg()G&F-7$Fg[n $!3=+++%pQ@L'F-7$Fb[n$!3)******Rk]\"otF-7$F][n$!3;+++*eGfR)F-7$Fhjm$!3 \\******4zt9%*F-7$Fcjm$!32+++&o(RU5FK7$F^jm$!3*)*****\\?T7FK7$Fdim$!34+++SG,R8FK7$F_im$!3%*******R)\\dV\"FK7$Fjhm $!3'******4Y,9`\"FK7$Fehm$!3$******Hqnfi\"FK7$F`hm$!3*)*****R``%>FK7$Fagm$!3#******HA=N*>FK7$ F\\gm$!3'******z@uF3#FK7$Fgfm$!3#******>j#*4<#FK7$Fbfm$!3?+++c`VM#FK7$Fhem$!3))*****4+;%HCFK7$Fcem$!3()*****>\"3X8DFK7$ F^em$!35+++6NS'f#FK7$Fidm$!3!)*****4b[#yEFK7$Fddm$!3;+++8g&*eFFK7$F_dm $!37+++WG\\QGFK7$Fjcm$!3%)*****>OCo\"HFK7$Fecm$!3!******z\"f\"R*HFK7$F `cm$!3=+++$fM(pIFK7$F[cm$!3!)*****\\$3DWJFK7$Ffbm$!36+++G*Ru@$FK7$Fabm $!3y*****>C$G*G$FK7$F\\bm$!3#*******=#p(fLFK7$Fgam$!3\")*****H4%*)GMFK 7$Fbam$!3;+++lBm'\\$FK7$F]am$!3A+++jq3jNFK7$Fh`m$!3:+++>)*=GOFK7$Fc`m$ !3;+++03+#p$FK7$F^`m$!3?+++T&eXv$FK7$Fi_m$!3\"******Re4f\"QFK7$Fd_m$!3 !*******p!3h(QFK7$F__m$!3'******\\1:_$RFK7$Fj^m$!3%******Hl(H$*RFK7$Fe ^m$!33+++XwU]SFK7$F`^m$!3++++w(zm5%FK7$F[^m$!3e*****4EH@;%FK7$Ff]m$!3+ +++^&[o@%FK7$Fa]m$!3))*****zA.4F%FK7$F\\]m$!3')*****z0ZVK%FK7$Fg\\m$!3 %)*****zR;sP%FK7$Fb\\m$!3@+++8W_HWFK7$F]\\m$!3;+++2eD\"[%FK7$Fh[m$!3x* ****R(HOKXFK7$Fc[m$!3G+++=Xw#e%FK7$F^[m$!3g*****H&oMKYFK7$Fijl$!3C+++( 3p4o%FK7$Fdjl$!3F+++:2ZGZFK7$F_jl$!3%*******R5ouZFK7$Fjil$!38++++*G%>[ FK7$Feil$!39+++:8bi[FK7$F`il$!3V++++,!R!\\FK7$F[il$!3\")******=gMV\\FK 7$Ffhl$!3/+++c0y!)\\FK7$Fahl$!3.+++sd6;]FK7$F\\hl$!3#)*****\\!GG\\]FK7 $Fggl$!3A+++C'H-3&FK7$Fbgl$!3U+++)\\=*3^FK7$F]gl$!35+++`NKN^FK7$Fhfl$! 39+++b'G%f^FK7$Fcfl$!3#******pwD7=&FK7$F^fl$!35+++?Rr+_FK7$Fiel$!3u*** **z())*y@&FK7$Fdel$!3l*****4m$zK_FK7$F_el$!35+++\\,UX_FK7$Fjdl$!3))*** **H;7eD&FK7$Fedl$!3w******o1-k_FK7$F`dl$!3M+++aA7q_FK7$F[dl$!3U+++eGBu _FK7$Ffcl$!31+++`.`w_FK7$Facl$!3g*****pw#Hx_FK7$F\\cl$!35+++8a'pF&FK7$ Fgbl$!3o*****fN%Gw_FK7$Fbbl$!37+++yZ^w_FK7$F]bl$!3q******4#H*z_FK7$Fha l$!3)******>WU2H&FK7$$!3i*****fL:5a)F-$!3k*****zmsjJ&FK7$F^al$!3I+++S` Hl`FK7$Fi`l$!3T+++ob,MaFK7$Fd`l$!3C+++=DX0bFK7$F_`l$!3!******zbd!pbFK7 $Fj_l$!3$******RswHi&FK7$Fe_l$!33+++5v6ocFK7$F`_l$!3))*****>=Odq&FK7$F [_l$!3z*****zH/pt&FK7$Ff^l$!3y*****44cCw&FK7$Fa^l$!3#******f`HIy&FK7$F \\^l$!3Q+++HQ6*z&FK7$Fg]l$!3n*****4;(36eFK7$Fb]l$!3t*****zNW#>eFK7$F]] l$!3w*****pq;Q#eFK7$Fh\\l$!3%******R@')\\#eFK7$Fc\\l$!3>+++Pf*G#eFK7$F ^\\l$!3I+++xvlPdCg&FK7$Fgx$!3')*****p&eqkbFK7$Fbx$!3y*****>FpQ_ &FK7$F]x$!3?+++XQ()zaFK7$Fhw$!3!)*****R,BEV&FK7$Fcw$!35+++u`)>Q&FK7$F^ w$!3k*****RktxK&FK7$Fiv$!3I+++Sxqp_FK7$Fdv$!39+++*[[t?&FK7$F_v$!3#)*** **\\Zu*R^FK7$Fju$!3e*****\\SVj1&FK7$Feu$!3s******>J@%)\\FK7$F`u$!3I+++ PcS*)[FK7$F[u$!3#******>,XXx%FK7$Fft$!3k******GH?JYFK7$Fat$!3/+++'eEHY %FK7$F[t$!3-+++(pboG%FK7$Ffs$!3n******4pg8TFK7$F`s$!3;+++:[6XRFK7$F[s$ !3#*******)*)o0y$FK7$Ffr$!3(******4OH)=OFK7$Far$!3%******R3i*eMFK7$F\\ r$!3z*****zG!H+LFK7$Fgq$!34+++!p\\B9$FK7$Faq$!3/+++(yM[)HFK7$F\\q$!3;+ ++#H`v#GFK7$$\"3++++Mm'HK#Fdo$!3'******zX!RqEFK7$$\"3w*****>CY`o$Fdo$! 3$)******)>#G8DFK7$F]p$!3y*****4k&>cBFK7$Fho$!3'******zw:\"*>#FK7$Fbo$ !3&)*****>&o.U?FK7$$\"3l+++U#QCt*FZ$!31+++!Rd\\)=FK7$Fhn$!3++++7u(ys\" FK7$$\"3!*******pQ-*H#FZ$!31+++pszq:FK7$$\"3#******p4F$y#*FO$!34+++,sr 89FK7$FM$!3-+++!HPmD\"FK7$$\"3a*****zV$fO%*FC$!3#******H^d&*4\"FK7$$\" 30+++x@#G?#FC$!3-+++6#yZU*F-7$$\"3A+++/'z&>QF=$!3E+++*o\")R&yF-7$$\"3* )*****4nK.M%F7$!3a+++:`=$G'F-7$F($!3D+++\")*)Q7ZF-7$F($!3)******Rl#fTJ F-7$F($!3++++Fjzq:F-F'-%&COLORG6&%$RGBG$\"#X!\"#F(F(-%)POLYGONSG6^_l7% F'7$F($\"+Fjzq:!#57$$!#Y!\"\"F(7%Fc]p7$F($\"+aEfTJFf]pFg]p7%F\\^p7$F($ \"+\")*)Q7ZFf]pFg]p7%F`^p7$$\"+2FLSVF]t$\"+:`=$G'Ff]pFg]p7%Fd^p7$$\"+7 'z&>QF-$\"+*o\")R&yFf]pFg]p7%Fj^p7$$\"+z@#G?#FK$\"+6#yZU*Ff]pFg]p7%F`_ p7$$\"+VMfO%*FK$\"+8vb*4\"!\"*Fg]p7%Ff_p7$$\"+E$zGB$!#;$\"+!HPmD\"F[`p Fg]p7%F]`p7$$\"+!GF$y#*F``p$\"+,sr89F[`pFg]p7%Fd`p7$$\"+mQ-*H#!#:$\"+p szq:F[`pFg]p7%Fj`p7$$\"+r+:9]F]ap$\"+7u(ys\"F[`pFg]p7%Faap7$$\"+!HQCt* F]ap$\"+!Rd\\)=F[`pFg]p7%Fgap7$$\"+Eee'o\"!#9$\"+_o.U?F[`pFg]p7%F]bp7$ $\"+&HP>f#F`bp$\"+od6*>#F[`pFg]p7%Fdbp7$$\"+]`/YMF`bp$\"+Tc>cBF[`pFg]p 7%Fjbp7$$\"+TiM&o$F`bp$\"+*>#G8DF[`pFg]p7%F`cp7$$\"+Nm'HK#F`bp$\"+e/Rq EF[`pFg]p7%Ffcp7$$!+6YlH?F`bp$\"+#H`v#GF[`pFg]p7%F\\dp7$$!+08n$3\"!#8$ \"+(yM[)HF[`pFg]p7%Fbdp7$$!+n$H;Z#Fedp$\"+!p\\B9$F[`pFg]p7%Fidp7$$!++ \"=j9%Fedp$\"+)G!H+LF[`pFg]p7%F_ep7$$!+Rs)eA&Fedp$\"+%3i*eMF[`pFg]p7%F eep7$$!+3nbpMFedp$\"+h$H)=OF[`pFg]p7%F[fp7$$\"+szT:gFedp$\"+**)o0y$F[` pFg]p7%Fafp7$$\"+1!QiK$!#7$\"+:[6XRF[`pFg]p7%Fgfp7$$\"+*GTI%)*Fjfp$\"+ 5pg8TF[`pFg]p7%F^gp7$$\"+)fu[T#!#6$\"+(pboG%F[`pFg]p7%Fdgp7$$\"+3:KT`F ggp$\"+'eEHY%F[`pFg]p7%F[hp7$$\"+bL7]5Ff]p$\"+HH?JYF[`pFg]p7%Fahp7$$\" +5`\")H$RF%Ff]p$\"+u`)>Q&F[`pFg]p7%Fg[ q7$$\"+W?7hUFf]p$\"+9IiKaF[`pFg]p7%F]\\q7$$\"+EaO.UFf]p$\"+XQ()zaF[`pF g]p7%Fc\\q7$$\"+')o(e5%Ff]p$\"+s#pQ_&F[`pFg]p7%Fi\\q7$$\"+UA7tRFf]p$\" +deqkbF[`pFg]p7%F_]q7$$\"+^s&*3QFf]p$\"+stX-cF[`pFg]p7%Fe]q7$$\"+X:t;O Ff]p$\"+Y;=PcF[`pFg]p7%F[^q7$$\"+t%o$*R$Ff]p$\"+4V#*ocF[`pFg]p7%Fa^q7$ $\"+#)zUfJFf]p$\"+q1s(p&F[`pFg]p7%Fg^q7$$\"+ar:**GFf]p$\"+SkfBdF[`pFg] p7%F]_q7$$\"+o8`?EFf]p$\"+OxcYdF[`pFg]p7%Fc_q7$$\"+jzGDBFf]p$\"+)eSmw& F[`pFg]p7%Fi_q7$$\"+(f`\\,#Ff]p$\"+&35Qy&F[`pFg]p7%F_`q7$$\"+pf'3p\"Ff ]p$\"+y\"f!)z&F[`pFg]p7%Fe`q7$$\"+y,>a8Ff]p$\"+PrN4eF[`pFg]p7%F[aq7$$ \"+/'Hf+\"Ff]p$\"+xvl3)yFFggp$\"+9i)\\#eF[`pFg]p7%F]bq7$$!+[*Rq+\"Fggp$\"+2n \"Q#eF[`pFg]p7%Fcbq7$$!+,L.&)[Fggp$\"+eVC>eF[`pFg]p7%Fibq7$$!+KhOa))Fg gp$\"+hr36eF[`pFg]p7%F_cq7$$!+()pt\"H\"Ff]p$\"+HQ6*z&F[`pFg]p7%Fecq7$$ !+%=u!3NFf]p$\"+5v6ocF[`pFg]p7%Fceq7$$!+ZAJRSFf]p$\"+Cn(H i&F[`pFg]p7%Fieq7$$!+_asF[`p$\"+j@\"eD&F[`pFg]p7%Fi [r7$$!+WK>0?F[`p$\"+\\,UX_F[`pFg]p7%F_\\r7$$!+$4.*y?F[`p$\"+hOzK_F[`pF g]p7%Fe\\r7$$!+E$>7:#F[`p$\"+y))*y@&F[`pFg]p7%F[]r7$$!+N+MAAF[`p$\"+?R r+_F[`pFg]p7%Fa]r7$$!+k8W#H#F[`p$\"+ndA\"=&F[`pFg]p7%Fg]r7$$!+FLohBF[` p$\"+b'G%f^F[`pFg]p7%F]^r7$$!++W@ICF[`p$\"+`NKN^F[`pFg]p7%Fc^r7$$!+(3v \")\\#F[`p$\"+)\\=*3^F[`pFg]p7%Fi^r7$$!+\"f+dc#F[`p$\"+C'H-3&F[`pFg]p7 %F__r7$$!+#fAHj#F[`p$\"+0GG\\]F[`pFg]p7%Fe_r7$$!+q+(**p#F[`p$\"+sd6;]F [`pFg]p7%F[`r7$$!+t!ppw#F[`p$\"+c0y!)\\F[`pFg]p7%Fa`r7$$!+h7/MGF[`p$\" +>gMV\\F[`pFg]p7%Fg`r7$$!+H6I,HF[`p$\"++,!R!\\F[`pFg]p7%F]ar7$$!+t=&)o HF[`p$\"+:8bi[F[`pFg]p7%Fcar7$$!+%3!yOIF[`p$\"++*G%>[F[`pFg]p7%Fiar7$$ !+-(\\^5$F[`p$\"+S5ouZF[`pFg]p7%F_br7$$!+7g*R<$F[`p$\"+:2ZGZF[`pFg]p7% Febr7$$!+92KVKF[`p$\"+(3p4o%F[`pFg]p7%F[cr7$$!+v\"*38LF[`p$\"+`oMKYF[` pFg]p7%Facr7$$!+P2B$Q$F[`p$\"+=Xw#e%F[`pFg]p7%Fgcr7$$!+UCk`MF[`p$\"+uH OKXF[`pFg]p7%F]dr7$$!+7e>CNF[`p$\"+2eD\"[%F[`pFg]p7%Fcdr7$$!+&oXZf$F[` p$\"+8W_HWF[`pFg]p7%Fidr7$$!+s(R^m$F[`p$\"+)R;sP%F[`pFg]p7%F_er7$$!+'f F_t$F[`p$\"+eqMCVF[`pFg]p7%Feer7$$!+Ny'[!QF[`p$\"+GK!4F%F[`pFg]p7%F[fr 7$$!+'eLR(QF[`p$\"+^&[o@%F[`pFg]p7%Fafr7$$!+*[:B%RF[`p$\"+h#H@;%F[`pFg ]p7%Fgfr7$$!+>I#*4SF[`p$\"+w(zm5%F[`pFg]p7%F]gr7$$!+`VowSF[`p$\"+XwU]S F[`pFg]p7%Fcgr7$$!+!HXD9%F[`p$\"+`wH$*RF[`pFg]p7%Figr7$$!+7wY2UF[`p$\" +l]@NRF[`pFg]p7%F_hr7$$!+%=F9F%F[`p$\"+q!3h(QF[`pFg]p7%Fehr7$$!+**>TMV F[`p$\"+%e4f\"QF[`pFg]p7%F[ir7$$!+z-U'R%F[`p$\"+T&eXv$F[`pFg]p7%Fair7$ $!+)peuX%F[`p$\"+03+#p$F[`pFg]p7%Fgir7$$!+22a)*=GOF[`pFg]p7 %F]jr7$$!+\")\\owXF[`p$\"+jq3jNF[`pFg]p7%Fcjr7$$!+;R\"\\j%F[`p$\"+lBm' \\$F[`pFg]p7%Fijr7$$!+n@D#p%F[`p$\"+$4%*)GMF[`pFg]p7%F_[s7$$!+%=D([ZF[ `p$\"+>#p(fLF[`pFg]p7%Fe[s7$$!+DxN/[F[`p$\"+UKG*G$F[`pFg]p7%F[\\s7$$!+ ^BVM#F[`pFg]p7%Fc`s7$$!+$=wGX&F[`p$\"+c`X,QbF[`p$\"+=Ux# 3#F[`pFg]p7%Feas7$$!+%)*\\'ybF[`p$\"+B#=N*>F[`pFg]p7%F[bs7$$!+Bp*yh&F[ `p$\"+!y=K!>F[`pFg]p7%Fabs7$$!+k4obcF[`p$\"+Ps'=\"=F[`pFg]p7%Fgbs7$$!+ (oG>p&F[`p$\"+MNX>T7F[`pFg]p7%F[es7$$!+DSDteF[`p$\"+0sJU6F[`pFg]p7%Faes7$$!+*\\Uo*eF[`p $\"+&o(RU5F[`pFg]p7%Fges7$$!+HuP=fF[`p$\"+5zt9%*Ff]pFg]p7%F]fs7$$!+zc! y$fF[`p$\"+*eGfR)Ff]pFg]p7%Fcfs7$$!+mv2bfF[`p$\"+W1:otFf]pFg]p7%Fifs7$ $!+=t9qfF[`p$\"+%pQ@L'Ff]pFg]p7%F_gs7$$!+,O(H)fF[`p$\"+a.w)G&Ff]pFg]p7 %Fegs7$$!+#4?N*fF[`p$\"+M9,RUFf]pFg]p7%F[hs7$$!+hgv,gF[`p$\"+Le+%=$Ff] pFg]p7%Fahs7$$!+Epl2gF[`p$\"+q+'\\7#Ff]pFg]p7%Fghs7$$!+PZ?6gF[`p$\"+!z sJ1\"Ff]pFg]p7%F]is7$$!+l&)Q7gF[`pF(Fg]p7%Fcis7$F^is$!+!zsJ1\"Ff]pFg]p 7%Fgis7$Fhhs$!+q+'\\7#Ff]pFg]p7%F[js7$Fbhs$!+Le+%=$Ff]pFg]p7%F_js7$F\\ hs$!+M9,RUFf]pFg]p7%Fcjs7$Ffgs$!+a.w)G&Ff]pFg]p7%Fgjs7$F`gs$!+%pQ@L'Ff ]pFg]p7%F[[t7$Fjfs$!+W1:otFf]pFg]p7%F_[t7$Fdfs$!+*eGfR)Ff]pFg]p7%Fc[t7 $F^fs$!+5zt9%*Ff]pFg]p7%Fg[t7$Fhes$!+&o(RU5F[`pFg]p7%F[\\t7$Fbes$!+0sJ U6F[`pFg]p7%F_\\t7$F\\es$!+Yq>T7F[`pFg]p7%Fc\\t7$Ffds$!+SG,R8F[`pFg]p7 %Fg\\t7$F`ds$!+S)\\dV\"F[`pFg]p7%F[]t7$Fjcs$!+h9SJ:F[`pFg]p7%F_]t7$Fdc s$!+.x'fi\"F[`pFg]p7%Fc]t7$F^cs$!+MNX>F[`pFg]p7%F_^t7$F\\bs$!+B#=N*>F[`pFg]p7%Fc ^t7$Ffas$!+=Ux#3#F[`pFg]p7%Fg^t7$F`as$!+KE*4<#F[`pFg]p7%F[_t7$Fj`s$!+c `VM#F[`pFg]p7%Fc_t7$F^`s$!+,gTHCF[`pFg]p7 %Fg_t7$Fh_s$!+73X8DF[`pFg]p7%F[`t7$Fb_s$!+6NS'f#F[`pFg]p7%F_`t7$F\\_s$ !+^&[#yEF[`pFg]p7%Fc`t7$Ff^s$!+8g&*eFF[`pFg]p7%Fg`t7$F`^s$!+WG\\QGF[`p Fg]p7%F[at7$Fj]s$!+iV#o\"HF[`pFg]p7%F_at7$Fd]s$!+=f\"R*HF[`pFg]p7%Fcat 7$F^]s$!+$fM(pIF[`pFg]p7%Fgat7$Fh\\s$!+N3DWJF[`pFg]p7%F[bt7$Fb\\s$!+G* Ru@$F[`pFg]p7%F_bt7$F\\\\s$!+UKG*G$F[`pFg]p7%Fcbt7$Ff[s$!+>#p(fLF[`pFg ]p7%Fgbt7$F`[s$!+$4%*)GMF[`pFg]p7%F[ct7$Fjjr$!+lBm'\\$F[`pFg]p7%F_ct7$ Fdjr$!+jq3jNF[`pFg]p7%Fcct7$F^jr$!+>)*=GOF[`pFg]p7%Fgct7$Fhir$!+03+#p$ F[`pFg]p7%F[dt7$Fbir$!+T&eXv$F[`pFg]p7%F_dt7$F\\ir$!+%e4f\"QF[`pFg]p7% Fcdt7$Ffhr$!+q!3h(QF[`pFg]p7%Fgdt7$F`hr$!+l]@NRF[`pFg]p7%F[et7$Fjgr$!+ `wH$*RF[`pFg]p7%F_et7$Fdgr$!+XwU]SF[`pFg]p7%Fcet7$F^gr$!+w(zm5%F[`pFg] p7%Fget7$Fhfr$!+h#H@;%F[`pFg]p7%F[ft7$Fbfr$!+^&[o@%F[`pFg]p7%F_ft7$F\\ fr$!+GK!4F%F[`pFg]p7%Fcft7$Ffer$!+eqMCVF[`pFg]p7%Fgft7$F`er$!+)R;sP%F[ `pFg]p7%F[gt7$Fjdr$!+8W_HWF[`pFg]p7%F_gt7$Fddr$!+2eD\"[%F[`pFg]p7%Fcgt 7$F^dr$!+uHOKXF[`pFg]p7%Fggt7$Fhcr$!+=Xw#e%F[`pFg]p7%F[ht7$Fbcr$!+`oMK YF[`pFg]p7%F_ht7$F\\cr$!+(3p4o%F[`pFg]p7%Fcht7$Ffbr$!+:2ZGZF[`pFg]p7%F ght7$F`br$!+S5ouZF[`pFg]p7%F[it7$Fjar$!++*G%>[F[`pFg]p7%F_it7$Fdar$!+: 8bi[F[`pFg]p7%Fcit7$F^ar$!++,!R!\\F[`pFg]p7%Fgit7$Fh`r$!+>gMV\\F[`pFg] p7%F[jt7$Fb`r$!+c0y!)\\F[`pFg]p7%F_jt7$F\\`r$!+sd6;]F[`pFg]p7%Fcjt7$Ff _r$!+0GG\\]F[`pFg]p7%Fgjt7$F`_r$!+C'H-3&F[`pFg]p7%F[[u7$Fj^r$!+)\\=*3^ F[`pFg]p7%F_[u7$Fd^r$!+`NKN^F[`pFg]p7%Fc[u7$F^^r$!+b'G%f^F[`pFg]p7%Fg[ u7$Fh]r$!+ndA\"=&F[`pFg]p7%F[\\u7$Fb]r$!+?Rr+_F[`pFg]p7%F_\\u7$F\\]r$! +y))*y@&F[`pFg]p7%Fc\\u7$Ff\\r$!+hOzK_F[`pFg]p7%Fg\\u7$F`\\r$!+\\,UX_F [`pFg]p7%F[]u7$Fj[r$!+j@\"eD&F[`pFg]p7%F_]u7$Fd[r$!+p1-k_F[`pFg]p7%Fc] u7$F^[r$!+aA7q_F[`pFg]p7%Fg]u7$Fhjq$!+eGBu_F[`pFg]p7%F[^u7$Fbjq$!+`.`w _F[`pFg]p7%F_^u7$F\\jq$!+nFHx_F[`pFg]p7%Fc^u7$Ffiq$!+8a'pF&F[`pFg]p7%F g^u7$F`iq$!+cVGw_F[`pFg]p7%F[_u7$Fjhq$!+yZ^w_F[`pFg]p7%F__u7$Fdhq$!+5# H*z_F[`pFg]p7%Fc_u7$F^hq$!+UCu!H&F[`pFg]p7%Fg_u7$$!+O`,T&)Ff]p$!+oEP;` F[`pFg]p7%F[`u7$Fbgq$!+S`Hl`F[`pFg]p7%Fa`u7$F\\gq$!+ob,MaF[`pFg]p7%Fe` u7$Fffq$!+=DX0bF[`pFg]p7%Fi`u7$F`fq$!+ev0pbF[`pFg]p7%F]au7$Fjeq$!+Cn(H i&F[`pFg]p7%Faau7$Fdeq$!+5v6ocF[`pFg]p7%Feau7$F^eq$!+#=Odq&F[`pFg]p7%F iau7$Fhdq$!+)H/pt&F[`pFg]p7%F]bu7$Fbdq$!+\"4cCw&F[`pFg]p7%Fabu7$F\\dq$ !+O&HIy&F[`pFg]p7%Febu7$Ffcq$!+HQ6*z&F[`pFg]p7%Fibu7$F`cq$!+hr36eF[`pF g]p7%F]cu7$Fjbq$!+eVC>eF[`pFg]p7%Facu7$Fdbq$!+2n\"Q#eF[`pFg]p7%Fecu7$F ^bq$!+9i)\\#eF[`pFg]p7%Ficu7$Fhaq$!+Pf*G#eF[`pFg]p7%F]du7$Fbaq$!+xvlQ&F[`pFg]p7%F]hu7$Fb[q$!+WOxF`F[`pFg]p7%Fahu7$F\\[q$!+Sxq p_F[`pFg]p7%Fehu7$Ffjp$!+*[[t?&F[`pFg]p7%Fihu7$F`jp$!+vW(*R^F[`pFg]p7% F]iu7$Fjip$!+0MMm]F[`pFg]p7%Faiu7$Fdip$!+?J@%)\\F[`pFg]p7%Feiu7$F^ip$! +PcS*)[F[`pFg]p7%Fiiu7$Fhhp$!+7]auZF[`pFg]p7%F]ju7$Fbhp$!+HH?JYF[`pFg] p7%Faju7$F\\hp$!+'eEHY%F[`pFg]p7%Feju7$Fegp$!+(pboG%F[`pFg]p7%Fiju7$F_ gp$!+5pg8TF[`pFg]p7%F][v7$Fhfp$!+:[6XRF[`pFg]p7%Fa[v7$Fbfp$!+**)o0y$F[ `pFg]p7%Fe[v7$F\\fp$!+h$H)=OF[`pFg]p7%Fi[v7$Ffep$!+%3i*eMF[`pFg]p7%F] \\v7$F`ep$!+)G!H+LF[`pFg]p7%Fa\\v7$Fjdp$!+!p\\B9$F[`pFg]p7%Fe\\v7$Fcdp $!+(yM[)HF[`pFg]p7%Fi\\v7$F]dp$!+#H`v#GF[`pFg]p7%F]]v7$$\"+Mm'HK#F`bp$ !+e/RqEF[`pFg]p7%Fa]v7$$\"+UiM&o$F`bp$!+*>#G8DF[`pFg]p7%Fg]v7$F[cp$!+T c>cBF[`pFg]p7%F]^v7$Febp$!+od6*>#F[`pFg]p7%Fa^v7$F^bp$!+_o.U?F[`pFg]p7 %Fe^v7$$\"+U#QCt*F]ap$!+!Rd\\)=F[`pFg]p7%Fi^v7$Fbap$!+7u(ys\"F[`pFg]p7 %F__v7$$\"+qQ-*H#F]ap$!+pszq:F[`pFg]p7%Fc_v7$$\"+(4F$y#*F``p$!+,sr89F[ `pFg]p7%Fi_v7$F^`p$!+!HPmD\"F[`pFg]p7%F_`v7$$\"+QMfO%*FK$!+8vb*4\"F[`p Fg]p7%Fc`v7$$\"+x@#G?#FK$!+6#yZU*Ff]pFg]p7%Fi`v7$$\"+/'z&>QF-$!+*o\")R &yFf]pFg]p7%F_av7$$\"+rELSVF]t$!+:`=$G'Ff]pFg]p7%Feav7$F($!+\")*)Q7ZFf ]pFg]p7%F[bv7$F($!+aEfTJFf]pFg]p7%F_bv7$F($!+Fjzq:Ff]pFg]p7%FcbvF'Fg]p -Fi\\p6&F[]p$\"#&*F^]p$\"\"$Fj]p$\"\"#Fj]p-%&STYLEG6#%,PATCHNOGRIDG-F$ 6%7$7$$!3o*************)oFKF(7$$\"3%**************=\"FKF(-%'COLOURG6&F []pF)F)F)-%*LINESTYLEG6#F\\cv-F$6%7$7$F($!3o*************Q'FK7$F($\"3o *************Q'FKF\\dvF_dv-%*AXESSTYLEG6#%$BOXG-%(SCALINGG6#%,CONSTRAI NEDG-%+AXESLABELSG6%%&Re(z)G%&Im(z)G-%%FONTG6#%(DEFAULTG-Fiev6$%*HELVE TICAG\"\"*-%%VIEWG6$;$!$*oF^]p$\"$>\"F^]p;$!$R'F^]p$\"$R'F^]p" 1 2 0 1 10 0 2 9 1 2 1 1.000000 45.000000 45.000000 0 0 "Curve 1" "Curve 2" "Curve 3" "Curve 4" }}}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 " " {TEXT -1 4 "The " }{TEXT 260 30 "interval of absolute stability" } {TEXT -1 89 " (or stability interval) is the intersection of the stab ility region with the real line." }}{PARA 0 "" 0 "" {TEXT -1 59 "For t his scheme the stability interval is (approximately) " }{XPPEDIT 18 0 "[-6.0124, 0];" "6#7$,$-%&FloatG6$\"&C,'!\"%!\"\"\"\"!" }{TEXT -1 2 ". " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 114 "W e can distort the boundary curve horizontally by taking the 11th root \+ of the real part of points along the curve." }}{PARA 0 "" 0 "" {TEXT -1 150 "In this way we see that the intersection of the stability regi on with the non-negative imaginary axis consists of the union of two d isjoint intervals." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 " > " 0 "" {MPLTEXT 1 0 694 "R := z -> add(z^j/j!,j=0..8)+\n 112741685 90097262446202365567806651139636287/4189653883626262478306187347767799 210211072000000*z^9+\n 349912114115796742918201476974982672782328107 /1342185547718841943943089303909898532699761280000000*z^10+\n 119028 7877240471442040541016878748401986181/82596033705774888858036264855993 755858446848000000*z^11+\n 32220441255216153619636171196263432283/15 5615715677546892051372672917089684950696960000*z^12:\nDigits := 18:\np ts := []: z0 := 0:\nfor ct from 0 to 200 do\n zz := newton(R(z)=exp( ct*Pi/60*I),z=z0):\n z0 := zz:\n pts := [op(pts),[surd(Re(zz),11), Im(zz)]]:\nend do:\nplot(pts,color=COLOR(RGB,.95,.15,.1),thickness=2,f ont=[HELVETICA,9]);\nDigits := 10:" }}{PARA 13 "" 1 "" {GLPLOT2D 287 308 308 {PLOTDATA 2 "6(-%'CURVESG6#7ew7$$\"\"!F)F(7$F($\"3v))H)fv()fB& !#>7$$\"3T&e!e8)euy#F-$\"3s)f'>^v>Z5!#=7$$\"3zjZzu>lGSF-$\"3$R$\\zEjzq :F37$$\"3hjW8G;lI_F-$\"3E)o$R-^R%4#F37$$\"35(\\98w2NS'F-$\"3@E^*z(Q*zh #F37$$\"3tqu%RtCGb(F-$\"3M&G3Ol#fTJF37$$\"3zQwUlb4#o)F-$\"3sy4EH9>lOF3 7$$\"3ch^tUTk$z*F-$\"3m3W-0-z)=%F37$$\"3@lPwyy!*)3\"F3$\"3c!\\d5)*)Q7Z F37$$\"3+fyrEP&p>\"F3$\"3m\"4zwv()fB&F37$$\"3*pv/:/$e.8F3$\"3n[mZNlefd F37$$\"34plJfJ&)39F3$\"3(**peaJ&=$G'F37$$\"3-9s51N!G^\"F3$\"3z^kD*4%y1 oF37$$\"3VeY/W&fah\"F3$\"3/m`P*)GQItF37$$\"3q@Ad7b$or\"F3$\"3q(y\"\\*o \")R&yF37$$\"3d/r#*oj$p\"=F3$\"3$Q$=y/0ex$)F37$$\"33eDEx\"fd\">F3$\"3. E.EU$z6!*)F37$$\"3l%f&y/TH8?F3$\"3eOo46#yZU*F37$$\"34pS>G^_4@F3$\"3HLX 'G7x$[**F37$$\"3T$R.*=0V/AF3$\"3KGm;4w>Z5!#<7$$\"3tV'pF:$)zH#F3$\"3Y]s S8vb*4\"Fhq7$$\"3.F'4Xn],R#F3$\"3V*yzoU<>:\"Fhq7$$\"3(eY))f`&*3[#F3$\" 3(G5/;NxU?\"Fhq7$$\"34o=E>\\iW]7$e6HF3$\"3)))y>v?xgY\"Fhq7$$\"3) z>[;p)o#*HF3$\"3LP#\\3BP%=:Fhq7$$\"3RG]*QAU>2$F3$\"3Vt>joszq:Fhq7$$\"3 b_)ere^#\\JF3$\"3%)[CL;t:B;Fhq7$$\"3oEPWVQ^CKF3$\"3R')QAnt^v;Fhq7$$\"3 )*))\\xNNh(H$F3$\"3.HjB7u(ys\"Fhq7$$\"3ss$of`?%oLF3$\"3J<=,SuB!y\"Fhq7 $$\"3!*4,UBtyOMF3$\"32w=FPufK=Fhq7$$\"30jd)*4ba-NF3$\"3Sk%G'*Qd\\)=Fhq 7$$\"3Z3)\\%[8]lNF3$\"3V)e)*HGFhq7$$\"3g)y?^%)Hai$F3$\"3dUZ\"e5x'* )>Fhq7$$\"3f8'Rk\"p1#o$F3$\"33pzD_o.U?Fhq7$$\"3nl))*Hu)4NPF3$\"3b=sxEl R%4#Fhq7$$\"3'3G(G1q9%y$F3$\"3COt>]hvY@Fhq7$$\"3`P_N-zuGQF3$\"3&ptdxw: \"*>#Fhq7$$\"36]ZXO9KoQF3$\"3,%)QYfaZ^AFhq7$$\"3$G?Co%\\7-RF3$\"3*R;tJ NNQI#Fhq7$$\"3`YPl[)z\"HRF3$\"3`sY(3k&>cBFhq7$$\"3I6M\"yL]\"[RF3$\"33# yd')fc&3CFhq7$$\"36Y-od)Hr&RF3$\"3!3:'*3h=4Y#Fhq7$$\"3^EQ*)fWB`RF3$\"3 gV6?*>#G8DFhq7$$\"3!\\mpGrU<$RF3$\"33U2sc![cc#Fhq7$$\"3[J2P:X)R)QF3$\" 37kkP)3`;(G'*zGFhq7$$!3inV^k9[3UF3$\" 3\"eNTol)QKHFhq7$$!3o,ML;&)\\gVF3$\"3**Hl8(yM[)HFhq7$$!30ixt]\">#*[%F3 $\"3R=8!GA1t.$Fhq7$$!3*H8h$QOE,YF3$\"3'f(QHl)3)*3$Fhq7$$!3$eTraT3**p%F 3$\"3b?#H.p\\B9$Fhq7$$!3:\"=G&=ut'y%F3$\"3-Btm$*o$\\>$Fhq7$$!3\\?2#46z A'[F3$\"3$*[WG]+eZKFhq7$$!3X=(p)>6BE\\F3$\"3WrT`(G!H+LFhq7$$!3#*\\\"G \"p_Wx\\F3$\"3pc*fFZ!3`LFhq7$$!3n&o2Q>yO,&F3$\"3Ux?]:a'fS$Fhq7$$!3&=)o ]a0'4.&F3$\"3Etq\"Q3i*eMFhq7$$!3Us 0#Hps\\F3$\"39TjpV1PlNFhq7$$!3^T%=s!=2Z[F3$\"3')[wbg$H)=OFhq7$$!3@9H$> a$[(\\%F3$\"3;%*ew@R\\sOFhq7$$\"3=!*>$z;Yq\\%F3$\"3@tVEabREPFhq7$$\"3G bo!>+Dd4&F3$\"3_H)Q*)*)o0y$Fhq7$$\"3l$*yp0FGUaF3$\"3u0>G-?0NQFhq7$$\"3 ,'Ru#zXC:dF3$\"3K'zQ02'))*)QFhq7$$\"35UIZ=Q#G&fF3$\"3ti:B:[6XRFhq7$$\" 3#ykAoly*phF3$\"3)fe826$y+SFhq7$$\"3!oZ^*y\\@ujF3$\"3b2Et6X$p0%Fhq7$$ \"3%[)e8fl()plF3$\"3>[nH5pg8TFhq7$$\"3&RF>$3YqfnF3$\"3B]2=8_#3<%Fhq7$$ \"3kR(Hm-%\\XpF3$\"3(yj1+R*eGUFhq7$$\"3O-dQ,IPGrF3$\"3wRjL(pboG%Fhq7$$ \"3F$p_s#y))3tF3$\"3g$RC)3)3bM%Fhq7$$\"3UPQ;9p'p[(F3$\"3f.EUlUK/WFhq7$ $\"35gqu7d\"=m(F3$\"3pi.?'eEHY%Fhq7$$\"32n4\\WT#=$yF3$\"3fLk^%*4w?XFhq 7$$\"30y#eb[DY*zF3$\"3'pZ*G\\A6xXFhq7$$\"333a)[o_u9)F3$\"3()3f$)GH?JYF hq7$$\"32'*G[1Gy(G)F3$\"3+(eY^+gBo%Fhq7$$\"3Wd*=,))*)RT)F3$\"3^s!>Lir, t%Fhq7$$\"3I&3VH2Xc_)F3$\"39;G*=,XXx%Fhq7$$\"3jr>&ydlLi)F3$\"3S;:JY1l: [Fhq7$$\"3ujCG6uT3()F3$\"3F9\"RU57Q&[Fhq7$$\"3xS?zTYK#y)F3$\"3r>)ynj0% *)[Fhq7$$\"3vV*o*GegY))F3$\"3$*)=YSF'zA\\Fhq7$$\"3G?L#GhRE!*)F3$\"3Z\\ ]hIwIa\\Fhq7$$\"308Xpush^*)F3$\"3s7)>,78U)\\Fhq7$$\"3\\m/I&yTX**)F3$\" 3i*zUc$zt7]Fhq7$$\"3,v.S?qCK!*F3$\"3!=kUBEl+/&Fhq7$$\"3>MkuPIUl!*F3$\" 3U_*4]SVj1&Fhq7$$\"3#yaeT)3k%4*F3$\"3!3ef;S#p\"4&Fhq7$$\"3AE^K;WP?\"*F 3$\"3HWu(ok4i6&Fhq7$$\"3?F6tE&=I9*F3$\"3&RBE^Zu*R^Fhq7$$\"3#\\t,P(R!H; *F3$\"3%*[\\?==0j^Fhq7$$\"3?MJW$H4.=*F3$\"3ARwF&4&\\&=&Fhq7$$\"3w\\'yi Jqa>*F3$\"3H:Jv)[[t?&Fhq7$$\"33r0.hxe3#*F3$\"3v`9FP(['G_Fhq7$$\"39\"H= aIL)>#*F3$\"3C'*\\0_lU\\_Fhq7$$\"3yN?y`WNH#*F3$\"3JJ7BSxqp_Fhq7$$\"3AU pTc%ysB*F3$\"3wq'p**49&*G&Fhq7$$\"38VlqzarV#*F3$\"3$>D#RPT')3`Fhq7$$\" 3%f;%R#=h([#*F3$\"3#>8zOktxK&Fhq7$$\"3clA->))\\_#*F3$\"3gL6&>;ciM&Fhq7 $$\"3!*Hx&\\'4+b#*F3$\"3$fprnOBVO&Fhq7$$\"3g4@&o$4Lc#*F3$\"3\\C>)RP&)> Q&Fhq7$$\"3xGA\\mRac#*F3$\"3f^Ln%*4D*R&Fhq7$$\"3tKLpt\")ob#*F3$\"3!pc: T%z7;aFhq7$$\"35m5Lu`!QD*F3$\"3ybP;9IiKaFhq7$$\"3[$e`@vJ4D*F3$\"3vVV!f @U([aFhq7$$\"3FUy;<%)4Z#*F3$\"3=wz#z*3\\kaFhq7$$\"3VE#z(o=LU#*F3$\"30I DbWQ()zaFhq7$$\"3n/^>yVlO#*F3$\"3Y%=$[d`*[\\&Fhq7$$\"3:u@m,V3I#*F3$\"3 %>&ztB$f&4bFhq7$$\"321msLjjA#*F3$\"3-)*[9s#pQ_&Fhq7$$\"3x1_v2 *F3$\"3THxCdeqkbFhq7$$\"3dGg%=ATU=*F3$\"3$RxRMMHwd&Fhq7$$\"30zN`iv]s\" *F3$\"3Kt!RFZ7-f&Fhq7$$\"3&3EP\">a\"*f\"*F3$\"3jb'[APdCg&Fhq7$$\"3w%f) [jqXY\"*F3$\"3ptq35gO9cFhq7$$\"3&\\6+1b@@8*F3$\"3P?#eq?Sfi&Fhq7$$\"34m $e3g%*o6*F3$\"3J0#**ek\"=PcFhq7$$\"3;Hf))R%e25*F3$\"3WbIZy=4[cFhq7$$\" 3[(H6?f\"p$3*F3$\"31jT@JBnecFhq7$$\"3itA#>ioc1*F3$\"38SZ34V#*ocFhq7$$ \"3Jp/Y<)fm/*F3$\"3._%e#)**[)ycFhq7$$\"3&[6_X!3jE!*F3$\"3&eg6xYZ%)o&Fh q7$$\"3\\5w%G3Ub+*F3$\"340<()p1s(p&Fhq7$$\"3N:agn%[L)*)F3$\"3H))pWS%pm q&Fhq7$$\"3hd%)\\>%)**f*)F3$\"3)*H!Qv\\%H:dFhq7$$\"3!)=>1RKVN*)F3$\"3+ .I6SkfBdFhq7$$\"3'ys^g4'e4*)F3$\"3%HG6fuv:t&Fhq7$$\"3mV2]b3Q#)))F3$\"3 X(HD)oFBRdFhq7$$\"3:Bh(4lIP&))F3$\"3=EvzNxcYdFhq7$$\"3*pIa.9ON#))F3$\" 3oL!eKu!e`dFhq7$$\"3aLZteLo\"z)F3$\"3d*f[NdCs)F3$\"3sKd'z#posdFhq7$$\"3cs$3*[bv%o)F3 $\"3Ka3&>I5%ydFhq7$$\"3d;S@(pGZk)F3$\"3xT#y^35Qy&Fhq7$$\"3(e4x2lK@g)F3 $\"3*HpR7\\&)))y&Fhq7$$\"34vL(\\)onc&)F3$\"3et7elbj$z&Fhq7$$\"3Oo%)eqA ,3&)F3$\"3v.k4y\"f!)z&Fhq7$$\"37a(GPY9dX)F3$\"3r>55:]:-eFhq7$$\"3p_fIz 3E*R)F3$\"3Nn^jq:#f!eFhq7$$\"3Wt8w6!)*zL)F3$\"3&e4Fq8d$4eFhq7$$\"3iFYz xP4r#)F3$\"3'[*pk%zfC\"eFhq7$$\"3]/J%Gwlu>)F3$\"3>&zn2SF_\"eFhq7$$\"3# *Qfq2!pc6)F3$\"3K\\(Ruddw\"eFhq7$$\"3S9%H!oArB!)F3$\"3,GOF)pZ(>eFhq7$$ \"3L)4L^bR(=zF3$\"37#49S([\\@eFhq7$$\"3$**eh:!oV'z(F3$\"3\\z)pn$f*G#eF hq7$$\"3cDV*>t:)\\wF3$\"3d31vAu%R#eFhq7$$\"3zDq\"e^njY(F3$\"3ZI4M`bkCe Fhq7$$\"3#f,')H_B*>sF3$\"3!\\5@V@')\\#eFhq7$$\"3([G\\C'pbPoF3$\"3#GV=I $\\'\\#eFhq7$$\"3\\%Gs<-7;$eF3$\"3*o*e9`odCeFhq7$$!3%4\")[QdHNe'F3$\"3 ,KB02n\"Q#eFhq7$$!3CjXxrW#R4(F3$\"3R3>2&yyE#eFhq7$$!3hdc*HOa&)Q(F3$\"3 !z.>;!p:@eFhq7$$!34XKqq&p)*f(F3$\"3aXtidVC>eFhq7$$!3Be4US`7mxF3$\"39*Q x))*Q$p\"eFhq7$$!3,q%G?MOR!zF3$\"3'*ymppw@9eFhq7$$!3z#>a#453A!)F3$\"3; SmWhr36eFhq7$$!3ip'HHw$yD\")F3$\"3R`S8bJ`2eFhq7$$!3x7Sn>fT=#)F3$\"3O[ \"*Rdca.eFhq7$$!3+jiVR^G-$)F3$\"3:4&R!HQ6*z&Fhq7$$!3bM$*ySR/z$)F3$\"3l !**H^!fA%z&Fhq7$$!3neqI8*=*\\%)F3$\"3%H\\(p0\"p))y&Fhq7$$!3H+Pjtw%e^)F 3$\"3aLA\"e`HIy&Fhq7$$!3$=hN+%\\cx&)F3$\"3wKa(Q2#pwdFhq7$$!3\"33[p$*ec j)F3$\"3L!3OoCS)pdFhq7$$!3)oF\"f\"p41p)F3$\"3Bpy3\"4cCw&Fhq7$$!3)3%3]a e\"Gu)F3$\"3T[?$z**>Xv&Fhq7$$!3)*G8c;Sh#z)F3$\"3f\\V#Ga5gu&Fhq7$$!30%R TT*HHS))F3$\"3[c^G)H/pt&Fhq7$$!332hrtY5')))F3$\"3:$y^:jvrs&Fhq7$$!3[Zx ^uCFI*)F3$\"3lxx7!\\'z;dFhq7$$!3X3g*o:)*H(*)F3$\"3r92)==Odq&Fhq7$$!3=Z &=[\\nW,*F3$\"3j2n8i6'Rp&Fhq7$$!3eO0B(>b[0*F3$\"3ngXU`$*QcFhq7$$!3*\\j!*3gU*3#*F3$\"3^Z3!RswHi&Fh q7$$!3%4D3x5ikC*F3$\"3bGG=\"[Tgg&Fhq7$$!3OAFf7>'RG*F3$\"3Rx%G%\\A3)e&F hq7$$!3&*Q@_AWm@$*F3$\"3*)G%=#ev0pbFhq7$$!3qgDN)*y\")f$*F3$\"3&Rd0/9R* [bFhq7$$!3.mR&**4*p)R*F3$\"3S]&[!pQsFbFhq-%%FONTG6$%*HELVETICAG\"\"*-% *THICKNESSG6#\"\"#-%&COLORG6&%$RGBG$\"#&*!\"#$\"#:F\\[o$\"\"\"!\"\"-%+ AXESLABELSG6$Q!6\"Fe[o-%%VIEWG6$%(DEFAULTGFj[o" 1 2 0 1 10 2 2 6 1 4 2 1.000000 45.000000 45.000000 0 0 "Curve 1" }}}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 108 "The relevant intersectio n points of the boundary curve with the imaginary axis can be determin ed as follows." }}{PARA 0 "" 0 "" {TEXT -1 87 "First we look for point s on the boundary curve either side of each intersection point. " }} {PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 289 "Digits := 15:\nz0 := 2.8*I:\nfor ct from 87 to 90 do\n newton(R (z)=exp(ct*Pi/100*I),z=z0);\nend do;``;\nz0 := 3.7*I:\nfor ct from 116 to 119 do\n newton(R(z)=exp(ct*Pi/100*I),z=z0);\nend do;``;\nz0 := \+ 5.8*I:\nfor ct from 277 to 280 do\n newton(R(z)=exp(ct*Pi/100*I),z=z 0);\nend do;\nDigits := 10:" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#^$$\"0 \"z.1QNF5!#>$\"0IF,LYKt#!#9" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#^$$\"0d (>gH5$p\"!#?$\"0#)f9SyYw#!#9" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#^$$!0Z '[Dw[c%)!#?$\"0nq'oQ6'z#!#9" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#^$$!0[Y 1ha'H?!#>$\"0I5:H`v#G!#9" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%!G" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#^$$!0e!y43;;C!#=$\"0-6IZ,5l$!#9" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#^$$!0l4H*4'>,\"!#=$\"0tkOAaKo$!#9" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#^$$\"0(fo9Y2*3)!#>$\"0c?*[Yf:P!#9" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#^$$\"0NL9P$GAJ!#=$\"0,@6BI![P!#9" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#%!G" }}{PARA 11 "" 1 "" {XPPMATH 20 "6 #^$$\"0jm_)Qov7!#;$\"0`&QBo\"\\#e!#9" }}{PARA 11 "" 1 "" {XPPMATH 20 " 6#^$$\"0**=7Z#)[=&!#<$\"0c\"pfSoCe!#9" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#^$$!0)>*yEsUU#!#<$\"0jY@v " 0 "" {MPLTEXT 1 0 480 "Digits := 15:\nreal_part \+ := proc(u)\n Re(newton(R(z)=exp(u*Pi*I),z=2.8*I))\nend proc:\nu0 := \+ bisect('real_part'(u),u=0.87..0.90);\nnewton(R(z)=exp(u0*Pi*I),z=2.8*I );``;\nreal_part := proc(u)\n Re(newton(R(z)=exp(u*Pi*I),z=3.7*I))\n end proc:\nu0 := bisect('real_part'(u),u=1.16..1.19);\nnewton(R(z)=exp (u0*Pi*I),z=3.7*I);``;\nreal_part := proc(u)\n Re(newton(R(z)=exp(u* Pi*I),z=5.8*I))\nend proc:\nu0 := bisect('real_part'(u),u=2.77..2.80); \nnewton(R(z)=exp(u0*Pi*I),z=5.8*I);\nDigits := 10:" }}{PARA 11 "" 1 " " {XPPMATH 20 "6#>%#u0G$\"0'[()Q,&y\"))!#:" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#^#$\"0G.=T*GqF!#9" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%! G" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%#u0G$\"0#*R#=0'e<\"!#9" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#^$$!0v$Hd!**3!**!#H$\"0\\$[%G'>-P!#9" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%!G" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%#u0G$\"0A9')G>oy#!#9" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#^#$\"0/%Q =)[W#e!#9" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 4 "The " }{TEXT 260 112 "largest interval on the nonnegative imagin ary axis that contains the origin and lies inside the stability region " }{TEXT -1 5 " is " }{XPPEDIT 18 0 "[0, 2.7703];" "6#7$\"\"!-%&Float G6$\"&.x#!\"%" }{TEXT -1 18 " (approximately)." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 4 "The " }{TEXT 260 72 "inter section of the stability region with the nonnegative imaginary axis" } {TEXT -1 5 " is " }{XPPEDIT 18 0 "`union`([0, 2.7703],[3.7022, 5.8244 ]);" "6#-%&unionG6$7$\"\"!-%&FloatG6$\"&.x#!\"%7$-F)6$\"&Aq$F,-F)6$\"& W#eF," }{TEXT -1 17 " (approximately)." }}{PARA 0 "" 0 "" {TEXT -1 0 " " }}{PARA 0 "" 0 "" {TEXT -1 37 "#------------------------------------ " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 81 "The s tability function R* for the 13 stage, order 7 scheme is given as fo llows." }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 98 "subs(ee,subs(b=`b* `,StabilityFunction(7,13,'expanded'))):\n`R*` := unapply(%,z):\n'`R*`( z)'=`R*`(z);" }}{PARA 12 "" 1 "" {XPPMATH 20 "6#/-%#R*G6#%\"zG,:\"\"\" F)F'F)*&#F)\"\"#F)*$)F'F,F)F)F)*&#F)\"\"'F)*$)F'\"\"$F)F)F)*&#F)\"#CF) *$)F'\"\"%F)F)F)*&#F)\"$?\"F)*$)F'\"\"&F)F)F)*&#F)\"$?(F)*$)F'F1F)F)F) *&#F)\"%S]F)*$)F'\"\"(F)F)F)*&#\"K**[]Xe`$*f<]t(eJ@*zzLM#[$\"P++CqtO(* fAf6H(oF\\(3UXH^lR\"F)*$)F'\"\")F)F)F)*&#\"LTF)3FBSp(yu,9OgxKeavUw#\"Q +++c=%4`5=t0#z5Kv!oY)HA)fs*F)*$)F'\"\"*F)F)F)*&#\"NpHnjQ3pZgowMG3%\\Gt )o1(f5\"T+++gPDLU%GjOYV'pZYJZHd#=&RZ%F)*$)F'\"#5F)F)F)*&#\"I " 0 "" {MPLTEXT 1 0 30 "z_0 := ne wton(`R*`(z)=1,z=-6);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%$z_0G$!+0=* yw&!\"*" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 318 "z_0 := newton(`R*`(z)=1,z=-6):\np_1 := plot([`R*`(z) ,1],z=-6.49..0.49,color=[red,blue]):\np_2 := plot([[[z_0,1]]$3],style= point,symbol=[circle,cross,diamond],color=black):\np_3 := plot([[z_0,0 ],[z_0,1]],linestyle=3,color=COLOR(RGB,0,.5,0)):\nplots[display]([p_1, p_2,p_3],view=[-6.49..0.49,-0.07..1.47],font=[HELVETICA,9]);" }}{PARA 13 "" 1 "" {GLPLOT2D 387 266 266 {PLOTDATA 2 "6+-%'CURVESG6$7Z7$$!3A++ ++++!\\'!#<$\"3yWHy10*))*RF*7$$!3+n\"zf'R'>X'F*$\"3olzZ)Rmbt$F*7$$!3xL $e>$z#RT'F*$\"3QeFzpZ%y[$F*7$$!3a+v$z*=*eP'F*$\"3!fXsP)[!\\D$F*7$$!3Ln m\"R'e&yL'F*$\"3(\\jNVwjf.$F*7$$!3o](oU'fmriF*$\"3O6%3ghSho#F*7$$!3,M3 ikgZ0iF*$\"3;/(=#4*yFP#F*7$$!3m]7))*fQ58'F*$\"3oR]()GQsf?F*7$$!3Kn;9N6 gcgF*$\"37_kye**4%y\"F*7$$!3Cnmc]-n\")fF*$\"3+%>K`h4/a\"F*7$$!3;n;*fOR n!fF*$\"3ufYY\")*3pK\"F*7$$!3[]7$[hk@$eF*$\"3Z2Xq_)*3T6F*7$$!3#Q$3nj)* eddF*$\"3%oCTK\"Q**)y*!#=7$$!3ymTb#=4$>cF*$\"3P*>?UbtxJ(Fco7$$!3e+vyYv 7waF*$\"3u6w)p2WLO&Fco7$$!3hmTD%*)[!G`F*$\"3%R1Us]X#\\QFco7$$!3w*\\P=8 X/=&F*$\"3Qbma>@)pt#Fco7$$!3@LLQ>lhG]F*$\"3zH%=fr^#3>Fco7$$!3km;*[z&)[ *[F*$\"3e)GHiL-,Q\"Fco7$$!3I++:tmLWZF*$\"3cvh\\#[s#f&*!#>7$$!39++D#QpJ f%F*$\"3!4IP'H3mcmFgq7$$!3I++&oo\"\\ZWF*$\"3XkhV***z?![Fgq7$$!3FnTD$Q- _J%F*$\"30(*G>YW72PFgq7$$!3-LL)fg(*y:%F*$\"36v9kG#*GOHFgq7$$!3#HLLf#)R Y-%F*$\"36]d3<'3Oh#Fgq7$$!3S+v[[$H'pQF*$\"3f,nG3!zM^#Fgq7$$!3zLLV5)=Ct $F*$\"3P^n.@po3EFgq7$$!3))*\\([.r(=e$F*$\"3C#ymBLjU'GFgq7$$!3&**\\i%)z D&QMF*$\"3:TKYP_%oA$Fgq7$$!3[L$3-v`*)G$F*$\"3O6)e8J\">8PFgq7$$!3AL3d=* *f^JF*$\"3#e)z-yI=_UFgq7$$!31n;Hb_W.IF*$\"313&*pji-N\\Fgq7$$!3ymTD)>a& \\GF*$\"3IcUS(H^]w&Fgq7$$!3<+Dr!e\"f:FF*$\"3)H$\\z^Mu*f'Fgq7$$!3!omTPG 34d#F*$\"3.W71RZ#\\j(Fgq7$$!30++q!GO9U#F*$\"33CD\\V'o>())Fgq7$$!3]+]x' 32_F#F*$\"3s#4i8`6t-\"Fco7$$!3-+voAGsL@F*$\"3')zg*\\m/P=\"Fco7$$!3k+]# )p'Gm(>F*$\"3ePWtE?B&Q\"Fco7$$!36MLV'esa$=F*$\"3-@M)4UA`f\"Fco7$$!3)** **\\#Q$eZo\"F*$\"3-#3n(H%p[&=Fco7$$!3?nT&)G))=[:F*$\"3'fCKN4>j7#Fco7$$ !3y***\\BG#)))R\"F*$\"3Ws:lNcsoCFco7$$!3)R$3#yX'Re7F*$\"3U())p*)[#4TGF co7$$!3%**\\P?H[:6\"F*$\"3]*ROXR\"\\!H$Fco7$$!3URL$eot(z'*Fco$\"3a%y!* R$e])z$Fco7$$!3u3](=<>n<)Fco$\"3Q=D@*>#e9WFco7$$!3almmK!*4HnFco$\"3JMe tBs@-^Fco7$$!3ynm\"fN)p[_Fco$\"3J3b'HpBj\"fFco7$$!3IJ$3U9c0y$Fco$\"3E= VoIR#>&oFco7$$!3)))****p#f[JCFco$\"3,7**3a)\\:%yFco7$$!3!eILe:#*H&))Fg q$\"3a(3.&Qsv_\"*Fco7$$\"3u%HLL6'Gw\\Fgq$\"31ljpG=-^5F*7$$\"37,]7:\"=@ (>Fco$\"3t$yJ-+-!=7F*7$$\"3/&*\\(3r#R$Q$Fco$\"3(>ar)*G;ES\"F*7$$\"3!** *************[Fco$\"3RJ4]?iJK;F*-%'COLOURG6&%$RGBG$\"*++++\"!\")$\"\"! Fd]lFc]l-F$6$7S7$F($\"\"\"Fd]l7$F=Fi]l7$FGFi]l7$FQFi]l7$FenFi]l7$F_oFi ]l7$FeoFi]l7$FjoFi]l7$F_pFi]l7$FdpFi]l7$FipFi]l7$F^qFi]l7$FcqFi]l7$Fiq Fi]l7$F^rFi]l7$FcrFi]l7$FhrFi]l7$F]sFi]l7$FbsFi]l7$FgsFi]l7$F\\tFi]l7$ FatFi]l7$FftFi]l7$F[uFi]l7$F`uFi]l7$FeuFi]l7$FjuFi]l7$F_vFi]l7$FdvFi]l 7$FivFi]l7$F^wFi]l7$FcwFi]l7$FhwFi]l7$F]xFi]l7$FbxFi]l7$FgxFi]l7$F\\yF i]l7$FayFi]l7$FfyFi]l7$F[zFi]l7$F`zFi]l7$FezFi]l7$FjzFi]l7$F_[lFi]l7$F d[lFi]l7$Fi[lFi]l7$F^\\lFi]l7$Fc\\lFi]l7$Fh\\lFi]l-F]]l6&F_]lFc]lFc]lF `]l-F$6&7#7$$!3m*****\\!=*yw&F*Fi]l-%'SYMBOLG6#%'CIRCLEG-F]]l6&F_]lFd] lFd]lFd]l-%&STYLEG6#%&POINTG-F$6&F_al-Fdal6#%&CROSSGFgalFial-F$6&F_al- Fdal6#%(DIAMONDGFgalFial-F$6%7$7$FaalFc]lF`al-%&COLORG6&F_]lFc]l$\"\"& !\"\"Fc]l-%*LINESTYLEG6#\"\"$-%+AXESLABELSG6%Q\"z6\"Q!Ficl-%%FONTG6#%( DEFAULTG-F\\dl6$%*HELVETICAG\"\"*-%%VIEWG6$;$!$\\'!\"#$\"#\\Fidl;$!\"( Fidl$\"$Z\"Fidl" 1 2 0 1 10 0 2 9 1 4 2 1.000000 45.000000 45.000000 0 0 "Curve 1" "Curve 2" "Curve 3" "Curve 4" "Curve 5" "Curve 6" }}}} {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 83 "The follo wing picture shows the stability region for the 12 stage, order 7 sche me. " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 999 "`R*` := z -> add(z^j/j!,j=0..7)+\n 3482343379799 21315877350175993535845504899/1396551294542087492768729115922599736737 0240000*z^8+\n 2764275545832776036140174787694023270882741/9725982 22984668075321079205731810530941856000000*z^9+\n 10597066887328494 0828347668604769083863672969/44739518257294731464769643463663284423325 3760000000*z^10+\n 5843271787637141271137547988122362454617/149131 7275243157715492321448788776147444179200000*z^11:\npts := []: z0 := 0: \nfor ct from 0 to 320 do\n zz := newton(`R*`(z)=exp(ct*Pi/20*I),z=z 0):\n z0 := zz:\n pts := [op(pts),[Re(zz),Im(zz)]]:\nend do:\np_1 \+ := plot(pts,color=COLOR(RGB,.2,0,0)):\np_2 := plots[polygonplot]([seq( [pts[i-1],pts[i],[-4.4,0]],i=2..nops(pts))],\n style=patchnog rid,color=COLOR(RGB,.7,0,0)):\np_3 := plot([[[-6.49,0],[1.19,0]],[[0,- 6.39],[0,6.39]]],color=black,linestyle=3):\nplots[display]([p_||(1..3) ],view=[-6.49..1.19,-6.39..6.39],font=[HELVETICA,9],\n labe ls=[`Re(z)`,`Im(z)`],axes=boxed,scaling=constrained);" }}{PARA 13 "" 1 "" {GLPLOT2D 381 565 565 {PLOTDATA 2 "6+-%'CURVESG6$7]_l7$$\"\"!F)F( 7$F($\"3++++Fjzq:!#=7$F($\"3)******Rl#fTJF-7$$!3#******R()e`b$!#F$\"3D +++\")*)Q7ZF-7$$!3E+++dIlrP!#E$\"3g*****pJ&=$G'F-7$$!3&******zR#e/C!#D $\"3?+++:<)R&yF-7$$!3*******f\\xy5\"!#C$\"3%)*****pVyZU*F-7$$!3/+++qa* 31%FF$\"3!******pkd&*4\"!#<7$$!3!******R]v=D\"!#B$\"3!******\\*yjc7FN7 $$!3?+++***)feLFR$\"33+++R%>PT\"FN7$$!3^+++*=R@,)FR$\"3/+++fV!3d\"FN7$ $!3&)*****\\LL:s\"!#A$\"3!******pE(*ys\"FN7$$!3M+++*eS1N$F\\o$\"3#**** **pg2])=FN7$$!3y*****H\"pp$*eF\\o$\"3@+++&Q`@/#FN7$$!3O+++R]@H#*F\\o$ \"3-+++goO*>#FN7$$!3))******eE^I7!#@$\"3?++++EqcBFN7$$!3'******z$4Y(> \"Fap$\"3)*******>\"[U^#FN7$$!3G*****R!y\")>\"*FR$\"3%******HsO@n#FN7$ $\"3)******f)*p]f$Fap$\"3()******H4cIGFN7$$\"3++++xsft7!#?$\"3++++j>y* )HFN7$$\"39+++4&4[D$Ffq$\"3!******>IB,:$FN7$$\"3e*****4a'G:sFfq$\"34++ +vo\">J$FN7$$\"3)******>NV*o9!#>$\"37+++q5KvMFN7$$\"32+++I%o#3GFfr$\"3 7+++Xf#)ROFN7$$\"3M+++&3_P0&Ffr$\"31+++EW=.QFN7$$\"3g*****HyW5V)Ffr$\" 3'*******\\::gRFN7$$\"3)******R.CXF\"F-$\"3@+++mpi.TFN7$$\"38+++3q%Qt \"F-$\"35+++UZ^HUFN7$$\"3#*******=%4h:#F-$\"3(*******=o!*QVFN7$$\"3>++ +v&*f2DF-$\"3&******\\N*>NWFN7$$\"3++++)*f?zFF-$\"3C+++g^]@XFN7$$\"3C+ ++l4CtHF-$\"3\")*****>$f,+YFN7$$\"35+++Od*f4$F-$\"3'*******H,;sYFN7$$ \"3/+++mWkaJF-$\"38+++0!f)QZFN7$$\"3w******eC7cJF-$\"3.+++:qr+[FN7$$\" 3:+++&*[x1JF-$\"3#)*****>^V\"e[FN7$$\"3()******QjG7IF-$\"3,+++#GF9\"\\ FN7$$\"3E+++wcqxGF-$\"3y*****\\n\"yg\\FN7$$\"3,+++i!*\\2FF-$\"3/+++X

h2c]#F-$\"3c*****\\VI$[]FN7$$\"37+++aD]vAF-$\"3S+++N !on3&FN7$$\"32+++CfB??F-$\"3\")******ovx@^FN7$$\"3\"******\\]'[UXsj$F-$\"3#******R$z(eD&FN7$$!3()*****>2P` 5%F-$\"3n*****REMkB&FN7$$!31+++4iN*e%F-$\"3()*****\\(fT7_FN7$$!35+++Ef w$4&F-$\"3%)*****zYAM=&FN7$$!3X+++*4hbi&F-$\"3#******Hb'**[^FN7$$!3]++ +j8v&>'F-$\"3v******p0o3^FN7$$!3q*****4L(e@oF-$\"3)******H**)>i]FN7$$! 3'******H$G&*GvF-$\"3y*****\\G_*4]FN7$$!3a+++eM$4N)F-$\"3')******>P3a \\FN7$$!37+++nY]6$*F-$\"3g*****R&yt**[FN7$$!32+++%)oAR5FN$\"37+++`h\"R &[FN7$$!3#******\\`;G:\"FN$\"3h*****fL;1#[FN7$$!3-+++?gXl7FN$\"3%***** *fT\\')z%FN7$$!33+++,nwt8FN$\"3!*******3__%y%FN7$$!35+++(y!zw9FN$\"31+ ++&zU]x%FN7$$!3!******H+xYd\"FN$\"3K+++C<$zw%FN7$$!33+++]6%zm\"FN$\"3Q +++6vphZFN7$$!3)******\\#G;dFN$\"3=+++(='3SZFN7$$!32+++M*3c+#FN$\"3\"** ******4?/t%FN7$$!3/+++RyN$3#FN$\"30+++%e5\">ZFN7$$!3;+++um4f@FN$\"3?++ +i4-1ZFN7$$!3++++#erIB#FN$\"3W+++5j0\"p%FN7$$!32+++(o&\\0BFN$\"3E+++(3 ^Tn%FN7$$!3#*******yfbwBFN$\"3K+++V5EbYFN7$$!3=+++!**=kW#FN$\"3'****** 4)zNMYFN7$$!3:+++.aB:DFN$\"3#******HYE9h%FN7$$!3')******oP9$e#FN$\"3/+ ++D>Y'e%FN7$$!3=+++LMF]EFN$\"3;+++2)p%fXFN7$$!3++++=nu;FFN$\"3S+++$[l/ `%FN7$$!3%******fR!o#y#FN$\"3++++#zu%*\\%FN7$$!3y*****RS'=[GFN$\"33+++ =]`mWFN7$$!3*)******e#yO$FN$\"39+++BE#>9%F N7$$!3<+++:/.LMFN$\"3=+++UyS&4%FN7$$!3/+++U,G)\\$FN$\"3y******4S-[SFN7 $$!3!)*****zk(\\jNFN$\"3-+++#QFN$\"3:+++efu,QFN7$$!3%)*****RO9a)QFN$\"3'*******y_/ ^PFN7$$!3!******fWb\"[RFN$\"3A+++_.!**p$FN7$$!3.+++&*=55SFN$\"3/+++l2H [OFN7$$!35+++s.c%='f$FN7$$!3#)*****>l\"HJTFN$\"3/+++w ?aVNFN7$$!3c******)[5/>%FN$\"3))*****zM>.\\$FN7$$!3++++l[[[UFN$\"3++++ %*4ZOMFN7$$!3k*****f&[[0VFN$\"3=+++AD&>Q$FN7$$!37+++58RhVFN$\"34+++<@s ELFN7$$!38+++`W>;WFN$\"3=+++**>uqKFN7$$!3!******p^#*)pWFN$\"3/+++7%zR@ $FN7$$!3@+++)\\+)4$FN7$$!3A+++&RJWi%FN$\"3>+++T\"e(QIFN7$$!3g*****H#o!Qn%FN$\"3y ******y$f'yHFN7$$!3P+++sX9AZFN$\"39+++R\"3x\"HFN7$$!3c******)Rl%pZFN$ \"3))*****4s5f&GFN7$$!3L+++&)**y:[FN$\"3#)*****4:!G$z#FN7$$!3c*****R!z 8h[FN$\"31+++8f$)HFFN7$$!3D+++Dm_0\\FN$\"3A+++SNglEFN7$$!3o*****Rcq*[ \\FN$\"3*******\\.91g#FN7$$!3?+++P-[\"*\\FN$\"37+++EJ!\\`#FN7$$!3?+++R :1L]FN$\"3-+++o.^oCFN7$$!3m*****4@:P2&FN$\"37+++)4y9S#FN7$$!3#)*****>^ OM6&FN$\"3$)*****zG]QL#FN7$$!3!)*****z9:A:&FN$\"33+++e8nlAFN7$$!37+++` a.!>&FN$\"3?+++&*\\)p>#FN7$$!3G+++@o(oA&FN$\"3*)*****R0Ly7#FN7$$!3!)** ***HN9FE&FN$\"35+++%fa#e?FN7$$!3%)*****>p>vH&FN$\"3))*****H6&G))>FN7$$ !3H+++k>EJ`FN$\"3.+++()f&z\">FN7$$!3Q+++%y3RO&FN$\"3!*******fTHZ=FN7$$ !3;+++_sU&R&FN$\"3)******R0Ajx\"FN7$$!3-++++\\yDaFN$\"32+++!pd]q\"FN7$ $!3?+++P0&\\X&FN$\"3.+++\")\\^L;FN7$$!3q*****R(\\*G[&FN$\"3#******4e&FN$\" 3)*******Ru)=F\"FN7$$!3/+++z&zMg&FN$\"33+++&)4!))>\"FN7$$!3C+++0BlBcFN $\"3\"******R;ja7\"FN7$$!3U+++*z([UcFN$\"3%******R!\\(=0\"FN7$$!3;+++Z J)*fcFN$\"3%******HV!Q!y*F-7$$!3U+++D%Qhn&FN$\"3))*****Htd&R!*F-7$$!3# ******fPc4p&FN$\"3++++u-L'H)F-7$$!3q*****z0UWq&FN$\"3E+++)4o2b(F-7$$!3 o*****fU-mr&FN$\"3x*****\\ikH!oF-7$$!3V+++#*eWFdFN$\"3O+++G'QI0'F-7$$! 3')*****\\%=)pt&FN$\"3<+++Va8,`F-7$$!3q******f,AXdFN$\"37+++dtUZXF-7$$ !3I+++'pq@v&FN$\"3;+++1J6#z$F-7$$!3/+++/G%yv&FN$\"3x*****RW;a.$F-7$$!3 W+++l[CidFN$\"3'******\\y$exAF-7$$!3e*****f(QQldFN$\"32+++W4))=:F-7$$! 3O++++^EndFN$\"3k*****f&3*ef(Ffr7$$!3m*****\\!=*yw&FNF(7$Fb]n$!3k***** f&3*ef(Ffr7$F]]n$!32+++W4))=:F-7$Fh\\n$!3'******\\y$exAF-7$Fc\\n$!3x** ***RW;a.$F-7$F^\\n$!3;+++1J6#z$F-7$Fi[n$!37+++dtUZXF-7$Fd[n$!3<+++Va8, `F-7$F_[n$!3O+++G'QI0'F-7$Fjjm$!3x*****\\ikH!oF-7$Fejm$!3E+++)4o2b(F-7 $F`jm$!3++++u-L'H)F-7$F[jm$!3))*****Htd&R!*F-7$Ffim$!3%******HV!Q!y*F- 7$Faim$!3%******R!\\(=0\"FN7$F\\im$!3\"******R;ja7\"FN7$Fghm$!33+++&)4 !))>\"FN7$Fbhm$!3)*******Ru)=F\"FN7$F]hm$!3$******4-AZM\"FN7$Fhgm$!3/+ ++,TI<9FN7$Fcgm$!3))******fBj*[\"FN7$F^gm$!3#******4FN7$F`em$!3))*****H6&G))>FN7$F[em$! 35+++%fa#e?FN7$Ffdm$!3*)*****R0Ly7#FN7$Fadm$!3?+++&*\\)p>#FN7$F\\dm$!3 3+++e8nlAFN7$Fgcm$!3$)*****zG]QL#FN7$Fbcm$!37+++)4y9S#FN7$F]cm$!3-+++o .^oCFN7$Fhbm$!37+++EJ!\\`#FN7$Fcbm$!3*******\\.91g#FN7$F^bm$!3A+++SNgl EFN7$Fiam$!31+++8f$)HFFN7$Fdam$!3#)*****4:!G$z#FN7$F_am$!3))*****4s5f& GFN7$Fj`m$!39+++R\"3x\"HFN7$Fe`m$!3y******y$f'yHFN7$F``m$!3>+++T\"e(QI FN7$F[`m$!3'******>)\\+)4$FN7$Ff_m$!3#******\\L2k:$FN7$Fa_m$!3/+++7%zR @$FN7$F\\_m$!3=+++**>uqKFN7$Fg^m$!34+++<@sELFN7$Fb^m$!3=+++AD&>Q$FN7$F ]^m$!3++++%*4ZOMFN7$Fh]m$!3))*****zM>.\\$FN7$Fc]m$!3/+++w?aVNFN7$F^]m$ !3#******>c%='f$FN7$Fi\\m$!3/+++l2H[OFN7$Fd\\m$!3A+++_.!**p$FN7$F_\\m$ !3'*******y_/^PFN7$Fj[m$!3:+++efu,QFN7$Fe[m$!3;+++Ws+_QFN7$F`[m$!3')** ****[Y\"=!RFN7$F[[m$!3A+++&GJ6&RFN7$Ffjl$!3-+++9%FN7$Fbil$!39+++s6V(= %FN7$F]il$!3L+++Y1zJUFN7$Fhhl$!3!)******[!f[F%FN7$Fchl$!3w*******)4];V FN7$F^hl$!37+++I@fcVFN7$Figl$!3K+++L@-&R%FN7$Fdgl$!3B+++OhpJWFN7$F_gl$ !33+++=]`mWFN7$Fjfl$!3++++#zu%*\\%FN7$Fefl$!3S+++$[l/`%FN7$F`fl$!3;+++ 2)p%fXFN7$F[fl$!3/+++D>Y'e%FN7$Ffel$!3#******HYE9h%FN7$Fael$!3'******4 )zNMYFN7$F\\el$!3K+++V5EbYFN7$Fgdl$!3E+++(3^Tn%FN7$Fbdl$!3W+++5j0\"p%F N7$F]dl$!3?+++i4-1ZFN7$Fhcl$!30+++%e5\">ZFN7$Fccl$!3\"********4?/t%FN7 $F^cl$!3=+++(='3SZFN7$Fibl$!3w*****RI1$[ZFN7$Fdbl$!3=+++P$o`v%FN7$F_bl $!3Q+++6vphZFN7$Fjal$!3K+++C<$zw%FN7$Feal$!31+++&zU]x%FN7$F`al$!3!**** ***3__%y%FN7$F[al$!3%******fT\\')z%FN7$Ff`l$!3h*****fL;1#[FN7$Fa`l$!37 +++`h\"R&[FN7$F\\`l$!3g*****R&yt**[FN7$Fg_l$!3')******>P3a\\FN7$Fb_l$! 3y*****\\G_*4]FN7$F]_l$!3)******H**)>i]FN7$Fh^l$!3v******p0o3^FN7$Fc^l $!3#******Hb'**[^FN7$F^^l$!3%)*****zYAM=&FN7$Fi]l$!3()*****\\(fT7_FN7$ Fd]l$!3n*****REMkB&FN7$F_]l$!3#******R$z(eD&FN7$Fj\\l$!3++++*y!3r_FN7$ Fe\\l$!3I+++vYJ#G&FN7$F`\\l$!3'******\\6)z*G&FN7$F[\\l$!3U+++v[q$H&FN7 $Ff[l$!3K+++@8<%H&FN7$Fa[l$!3#)*****\\Q-8H&FN7$F\\[l$!3A+++%Hw^G&FN7$F gz$!3!******zD[eF&FN7$Fbz$!3A+++_LNj_FN7$F]z$!3o*****\\v3xC&FN7$Fhy$!3 D+++*[:*G_FN7$Fcy$!3o*****\\wfp?&FN7$F^y$!3M+++CS\"==&FN7$Fix$!3g***** *ywV`^FN7$Fdx$!3\")******ovx@^FN7$F_x$!3S+++N!on3&FN7$Fjw$!3c*****\\VI $[]FN7$Few$!3/+++X^V\"e[FN7$Fav$!3.+++:qr+[FN7$F\\v$!38+++0!f)QZFN7$Fgu $!3'*******H,;sYFN7$Fbu$!3\")*****>$f,+YFN7$F]u$!3C+++g^]@XFN7$Fht$!3& ******\\N*>NWFN7$Fct$!3(*******=o!*QVFN7$F^t$!35+++UZ^HUFN7$Fis$!3@+++ mpi.TFN7$Fds$!3'*******\\::gRFN7$F_s$!31+++EW=.QFN7$Fjr$!37+++Xf#)ROFN 7$Fdr$!37+++q5KvMFN7$F_r$!34+++vo\">J$FN7$$\"39+++5&4[D$Ffq$!3!******> IB,:$FN7$Fdq$!3++++j>y*)HFN7$F_q$!3()******H4cIGFN7$$!3g*****\\!y\")> \"*FR$!3%******HsO@n#FN7$Fep$!3)*******>\"[U^#FN7$F_p$!3?++++EqcBFN7$F jo$!3-+++goO*>#FN7$Feo$!3@+++&Q`@/#FN7$F`o$!3#******pg2])=FN7$Fjn$!3!* *****pE(*ys\"FN7$$!3y*****>>R@,)FR$!3/+++fV!3d\"FN7$FV$!33+++R%>PT\"FN 7$FP$!3!******\\*yjc7FN7$FJ$!3!******pkd&*4\"FN7$FD$!3%)*****pVyZU*F-7 $$!38+++(R#e/CF@$!3?+++:<)R&yF-7$$!37+++Z@lrPF:$!3g*****pJ&=$G'F-7$$!3 w*****fan`b$F4$!3D+++\")*)Q7ZF-7$F($!3)******Rl#fTJF-7$F($!3++++Fjzq:F -F'-%&COLORG6&%$RGBG$\"\"#!\"\"F(F(-%)POLYGONSG6^_l7%F'7$F($\"+Fjzq:!# 57$$!#WFh\\pF(7%F]]p7$F($\"+aEfTJF`]pFa]p7%Fe]p7$$!+u)e`b$Ffr$\"+\")*) Q7ZF`]pFa]p7%Fi]p7$$!+dIlrPF-$\"+<`=$G'F`]pFa]p7%F_^p7$$!+)R#e/CFN$\"+ :<)R&yF`]pFa]p7%Fe^p7$$!+'\\xy5\"!#;$\"+P%yZU*F`]pFa]p7%F[_p7$$!+qa*31 %F^_p$\"+Zwb*4\"!\"*Fa]p7%Fb_p7$$!+/b(=D\"!#:$\"+&*yjc7Fg_pFa]p7%Fi_p7 $$!+***)feLF\\`p$\"+R%>PT\"Fg_pFa]p7%F``p7$$!+*=R@,)F\\`p$\"+fV!3d\"Fg _pFa]p7%Ff`p7$$!+NL`@#Fg_pFa]p7%F_bp7$$!+fE^I7!#8$\"++EqcBFg_pFa]p7%Febp7$$!+Q 4Y(>\"Fhbp$\"+?\"[U^#Fg_pFa]p7%F\\cp7$$!+/y\")>\"*F\\`p$\"+Bn8sEFg_pFa ]p7%Fbcp7$$\"+')*p]f$Fhbp$\"+I4cIGFg_pFa]p7%Fhcp7$$\"+xsft7!#7$\"+j>y* )HFg_pFa]p7%F^dp7$$\"+4&4[D$Fadp$\"+-L7]JFg_pFa]p7%Fedp7$$\"+TlG:sFadp $\"+vo\">J$Fg_pFa]p7%F[ep7$$\"+_L%*o9!#6$\"+q5KvMFg_pFa]p7%Faep7$$\"+I %o#3GFdep$\"+Xf#)ROFg_pFa]p7%Fhep7$$\"+&3_P0&Fdep$\"+EW=.QFg_pFa]p7%F^ fp7$$\"+$yW5V)Fdep$\"+]::gRFg_pFa]p7%Fdfp7$$\"+MS_u7F`]p$\"+mpi.TFg_pF a]p7%Fjfp7$$\"+3q%Qt\"F`]p$\"+UZ^HUFg_pFa]p7%F`gp7$$\"+>%4h:#F`]p$\"+> o!*QVFg_pFa]p7%Ffgp7$$\"+v&*f2DF`]p$\"+b$*>NWFg_pFa]p7%F\\hp7$$\"+)*f? zFF`]p$\"+g^]@XFg_pFa]p7%Fbhp7$$\"+l4CtHF`]p$\"+Kf,+YFg_pFa]p7%Fhhp7$$ \"+Od*f4$F`]p$\"+I,;sYFg_pFa]p7%F^ip7$$\"+mWkaJF`]p$\"+0!f)QZFg_pFa]p7 %Fdip7$$\"+fC7cJF`]p$\"+:qr+[Fg_pFa]p7%Fjip7$$\"+&*[x1JF`]p$\"+7N9e[Fg _pFa]p7%F`jp7$$\"+RjG7IF`]p$\"+#GF9\"\\Fg_pFa]p7%Ffjp7$$\"+wcqxGF`]p$ \"+v;yg\\Fg_pFa]p7%F\\[q7$$\"+i!*\\2FF`]p$\"+XXsj$F`]p$\"+Mz(eD&Fg_pFa]p7% F^bq7$$!+sqL0TF`]p$\"+kUVO_Fg_pFa]p7%Fdbq7$$!+4iN*e%F`]p$\"+vfT7_Fg_pF a]p7%Fjbq7$$!+Efw$4&F`]p$\"+oCU$=&Fg_pFa]p7%F`cq7$$!+*4hbi&F`]p$\"+`l* *[^Fg_pFa]p7%Ffcq7$$!+j8v&>'F`]p$\"+q0o3^Fg_pFa]p7%F\\dq7$$!+Jte@oF`]p $\"+$**)>i]Fg_pFa]p7%Fbdq7$$!+LG&*GvF`]p$\"+&G_*4]Fg_pFa]p7%Fhdq7$$!+e M$4N)F`]p$\"+?P3a\\Fg_pFa]p7%F^eq7$$!+nY]6$*F`]p$\"+ayt**[Fg_pFa]p7%Fd eq7$$!+%)oAR5Fg_p$\"+`h\"R&[Fg_pFa]p7%Fjeq7$$!+Nl\"G:\"Fg_p$\"+Ojh?[Fg _pFa]p7%F`fq7$$!+?gXl7Fg_p$\"+;%\\')z%Fg_pFa]p7%Fffq7$$!+,nwt8Fg_p$\"+ 4__%y%Fg_pFa]p7%F\\gq7$$!+(y!zw9Fg_p$\"+&zU]x%Fg_pFa]p7%Fbgq7$$!+.qnu: Fg_p$\"+C<$zw%Fg_pFa]p7%Fhgq7$$!+]6%zm\"Fg_p$\"+6vphZFg_pFa]p7%F^hq7$$ !+DG;dFg_p$\"+(='3SZFg_pFa]p7%F`iq7$$!+M*3c+#Fg_p$\"++,UIZFg_p Fa]p7%Ffiq7$$!+RyN$3#Fg_p$\"+%e5\">ZFg_pFa]p7%F\\jq7$$!+um4f@Fg_p$\"+i 4-1ZFg_pFa]p7%Fbjq7$$!+#erIB#Fg_p$\"+5j0\"p%Fg_pFa]p7%Fhjq7$$!+(o&\\0B Fg_p$\"+(3^Tn%Fg_pFa]p7%F^[r7$$!+zfbwBFg_p$\"+V5EbYFg_pFa]p7%Fd[r7$$!+ !**=kW#Fg_p$\"+\")zNMYFg_pFa]p7%Fj[r7$$!+.aB:DFg_p$\"+jkU6YFg_pFa]p7%F `\\r7$$!+pP9$e#Fg_p$\"+D>Y'e%Fg_pFa]p7%Ff\\r7$$!+LMF]EFg_p$\"+2)p%fXFg _pFa]p7%F\\]r7$$!+=nu;FFg_p$\"+$[l/`%Fg_pFa]p7%Fb]r7$$!+'R!o#y#Fg_p$\" +#zu%*\\%Fg_pFa]p7%Fh]r7$$!+/k=[GFg_p$\"+=]`mWFg_pFa]p7%F^^r7$$!+f#yO$Fg_p$\"+BE#>9 %Fg_pFa]p7%F^ar7$$!+:/.LMFg_p$\"+UyS&4%Fg_pFa]p7%Fdar7$$!+U,G)\\$Fg_p$ \"+5S-[SFg_pFa]p7%Fjar7$$!+[w\\jNFg_p$\"+#QFg_p$\"+efu,QFg_p Fa]p7%Fhcr7$$!+kVT&)QFg_p$\"+z_/^PFg_pFa]p7%F^dr7$$!+Ya:[RFg_p$\"+_.!* *p$Fg_pFa]p7%Fddr7$$!+&*=55SFg_p$\"+l2H[OFg_pFa]p7%Fjdr7$$!+s.&Fg_p$\"+&*\\)p>#F g_pFa]p7%Fj]s7$$!+@o(oA&Fg_p$\"+aI$y7#Fg_pFa]p7%F`^s7$$!+`Vri_Fg_p$\"+ %fa#e?Fg_pFa]p7%Ff^s7$$!+#p>vH&Fg_p$\"+8^G))>Fg_pFa]p7%F\\_s7$$!+k>EJ` Fg_p$\"+()f&z\">Fg_pFa]p7%Fb_s7$$!+%y3RO&Fg_p$\"+gTHZ=Fg_pFa]p7%Fh_s7$ $!+_sU&R&Fg_p$\"+a?Kwe&Fg_p$\"+Su)=F\"Fg_pFa]p7%Fhbs7$$!+z&zMg&Fg_p$\"+&)4!))>\"Fg_pFa]p 7%F^cs7$$!+0BlBcFg_p$\"+kJYD6Fg_pFa]p7%Fdcs7$$!+*z([UcFg_p$\"+/\\(=0\" Fg_pFa]p7%Fjcs7$$!+ZJ)*fcFg_p$\"+L/Q!y*F`]pFa]p7%F`ds7$$!+D%Qhn&Fg_p$ \"+LxbR!*F`]pFa]p7%Ffds7$$!+wj&4p&Fg_p$\"+u-L'H)F`]pFa]p7%F\\es7$$!+e? W/dFg_p$\"+)4o2b(F`]pFa]p7%Fbes7$$!+ECg;dFg_p$\"+DY'H!oF`]pFa]p7%Fhes7 $$!+#*eWFdFg_p$\"+G'QI0'F`]pFa]p7%F^fs7$$!+X=)pt&Fg_p$\"+Va8,`F`]pFa]p 7%Fdfs7$$!+g,AXdFg_p$\"+dtUZXF`]pFa]p7%Fjfs7$$!+'pq@v&Fg_p$\"+1J6#z$F` ]pFa]p7%F`gs7$$!+/G%yv&Fg_p$\"+WkTNIF`]pFa]p7%Ffgs7$$!+l[CidFg_p$\"+&y $exAF`]pFa]p7%F\\hs7$$!+wQQldFg_p$\"+W4))=:F`]pFa]p7%Fbhs7$$!++^EndFg_ p$\"+c3*ef(FdepFa]p7%Fhhs7$$!+0=*yw&Fg_pF(Fa]p7%F^is7$Fihs$!+c3*ef(Fde pFa]p7%Fbis7$Fchs$!+W4))=:F`]pFa]p7%Ffis7$F]hs$!+&y$exAF`]pFa]p7%Fjis7 $Fggs$!+WkTNIF`]pFa]p7%F^js7$Fags$!+1J6#z$F`]pFa]p7%Fbjs7$F[gs$!+dtUZX F`]pFa]p7%Ffjs7$Fefs$!+Va8,`F`]pFa]p7%Fjjs7$F_fs$!+G'QI0'F`]pFa]p7%F^[ t7$Fies$!+DY'H!oF`]pFa]p7%Fb[t7$Fces$!+)4o2b(F`]pFa]p7%Ff[t7$F]es$!+u- L'H)F`]pFa]p7%Fj[t7$Fgds$!+LxbR!*F`]pFa]p7%F^\\t7$Fads$!+L/Q!y*F`]pFa] p7%Fb\\t7$F[ds$!+/\\(=0\"Fg_pFa]p7%Ff\\t7$Fecs$!+kJYD6Fg_pFa]p7%Fj\\t7 $F_cs$!+&)4!))>\"Fg_pFa]p7%F^]t7$Fibs$!+Su)=F\"Fg_pFa]p7%Fb]t7$Fcbs$!+ @?sW8Fg_pFa]p7%Ff]t7$F]bs$!+,TI<9Fg_pFa]p7%Fj]t7$Fgas$!+gBj*[\"Fg_pFa] p7%F^^t7$Faas$!+rTqh:Fg_pFa]p7%Fb^t7$F[as$!+\")\\^L;Fg_pFa]p7%Ff^t7$Fe `s$!+!pd]q\"Fg_pFa]p7%Fj^t7$F_`s$!+a?KwFg_pFa]p7%Ff_t7$F]_s$!+8^G))>Fg_pFa]p7%Fj _t7$Fg^s$!+%fa#e?Fg_pFa]p7%F^`t7$Fa^s$!+aI$y7#Fg_pFa]p7%Fb`t7$F[^s$!+& *\\)p>#Fg_pFa]p7%Ff`t7$Fe]s$!+e8nlAFg_pFa]p7%Fj`t7$F_]s$!+)G]QL#Fg_pFa ]p7%F^at7$Fi\\s$!+)4y9S#Fg_pFa]p7%Fbat7$Fc\\s$!+o.^oCFg_pFa]p7%Ffat7$F ]\\s$!+EJ!\\`#Fg_pFa]p7%Fjat7$Fg[s$!+NSh+EFg_pFa]p7%F^bt7$Fa[s$!+SNglE Fg_pFa]p7%Fbbt7$F[[s$!+8f$)HFFg_pFa]p7%Ffbt7$Fejr$!+^,G$z#Fg_pFa]p7%Fj bt7$F_jr$!+@2\"f&GFg_pFa]p7%F^ct7$Fiir$!+R\"3x\"HFg_pFa]p7%Fbct7$Fcir$ !+z$f'yHFg_pFa]p7%Ffct7$F]ir$!+T\"e(QIFg_pFa]p7%Fjct7$Fghr$!+#)\\+)4$F g_pFa]p7%F^dt7$Fahr$!+NtScJFg_pFa]p7%Fbdt7$F[hr$!+7%zR@$Fg_pFa]p7%Ffdt 7$Fegr$!+**>uqKFg_pFa]p7%Fjdt7$F_gr$!+<@sELFg_pFa]p7%F^et7$Fifr$!+AD&> Q$Fg_pFa]p7%Fbet7$Fcfr$!+%*4ZOMFg_pFa]p7%Ffet7$F]fr$!+[$>.\\$Fg_pFa]p7 %Fjet7$Fger$!+w?aVNFg_pFa]p7%F^ft7$Faer$!+iX='f$Fg_pFa]p7%Fbft7$F[er$! +l2H[OFg_pFa]p7%Ffft7$Fedr$!+_.!**p$Fg_pFa]p7%Fjft7$F_dr$!+z_/^PFg_pFa ]p7%F^gt7$Ficr$!+efu,QFg_pFa]p7%Fbgt7$Fccr$!+Ws+_QFg_pFa]p7%Ffgt7$F]cr $!+\\Y\"=!RFg_pFa]p7%Fjgt7$Fgbr$!+&GJ6&RFg_pFa]p7%F^ht7$Fabr$!+9%Fg_pFa]p7%F^it7$Fi`r$!+s6V(=%Fg_pFa]p7%Fbit7$Fc`r$!+Y1 zJUFg_pFa]p7%Ffit7$F]`r$!+\\!f[F%Fg_pFa]p7%Fjit7$Fg_r$!+!*4];VFg_pFa]p 7%F^jt7$Fa_r$!+I@fcVFg_pFa]p7%Fbjt7$F[_r$!+L@-&R%Fg_pFa]p7%Ffjt7$Fe^r$ !+OhpJWFg_pFa]p7%Fjjt7$F_^r$!+=]`mWFg_pFa]p7%F^[u7$Fi]r$!+#zu%*\\%Fg_p Fa]p7%Fb[u7$Fc]r$!+$[l/`%Fg_pFa]p7%Ff[u7$F]]r$!+2)p%fXFg_pFa]p7%Fj[u7$ Fg\\r$!+D>Y'e%Fg_pFa]p7%F^\\u7$Fa\\r$!+jkU6YFg_pFa]p7%Fb\\u7$F[\\r$!+ \")zNMYFg_pFa]p7%Ff\\u7$Fe[r$!+V5EbYFg_pFa]p7%Fj\\u7$F_[r$!+(3^Tn%Fg_p Fa]p7%F^]u7$Fijq$!+5j0\"p%Fg_pFa]p7%Fb]u7$Fcjq$!+i4-1ZFg_pFa]p7%Ff]u7$ F]jq$!+%e5\">ZFg_pFa]p7%Fj]u7$Fgiq$!++,UIZFg_pFa]p7%F^^u7$Faiq$!+(='3S ZFg_pFa]p7%Fb^u7$F[iq$!+/jI[ZFg_pFa]p7%Ff^u7$Fehq$!+P$o`v%Fg_pFa]p7%Fj ^u7$F_hq$!+6vphZFg_pFa]p7%F^_u7$Figq$!+C<$zw%Fg_pFa]p7%Fb_u7$Fcgq$!+&z U]x%Fg_pFa]p7%Ff_u7$F]gq$!+4__%y%Fg_pFa]p7%Fj_u7$Fgfq$!+;%\\')z%Fg_pFa ]p7%F^`u7$Fafq$!+Ojh?[Fg_pFa]p7%Fb`u7$F[fq$!+`h\"R&[Fg_pFa]p7%Ff`u7$Fe eq$!+ayt**[Fg_pFa]p7%Fj`u7$F_eq$!+?P3a\\Fg_pFa]p7%F^au7$Fidq$!+&G_*4]F g_pFa]p7%Fbau7$Fcdq$!+$**)>i]Fg_pFa]p7%Ffau7$F]dq$!+q0o3^Fg_pFa]p7%Fja u7$Fgcq$!+`l**[^Fg_pFa]p7%F^bu7$Facq$!+oCU$=&Fg_pFa]p7%Fbbu7$F[cq$!+vf T7_Fg_pFa]p7%Ffbu7$Febq$!+kUVO_Fg_pFa]p7%Fjbu7$F_bq$!+Mz(eD&Fg_pFa]p7% F^cu7$Fiaq$!+*y!3r_Fg_pFa]p7%Fbcu7$Fcaq$!+vYJ#G&Fg_pFa]p7%Ffcu7$F]aq$! +:\")z*G&Fg_pFa]p7%Fjcu7$Fg`q$!+v[q$H&Fg_pFa]p7%F^du7$Fa`q$!+@8<%H&Fg_ pFa]p7%Fbdu7$F[`q$!+&Q-8H&Fg_pFa]p7%Ffdu7$Fe_q$!+%Hw^G&Fg_pFa]p7%Fjdu7 $F__q$!+e#[eF&Fg_pFa]p7%F^eu7$Fi^q$!+_LNj_Fg_pFa]p7%Fbeu7$Fc^q$!+b(3xC &Fg_pFa]p7%Ffeu7$F]^q$!+*[:*G_Fg_pFa]p7%Fjeu7$Fg]q$!+l(fp?&Fg_pFa]p7%F ^fu7$Fa]q$!+CS\"==&Fg_pFa]p7%Fbfu7$F[]q$!+zwV`^Fg_pFa]p7%Fffu7$Fe\\q$! +pvx@^Fg_pFa]p7%Fjfu7$F_\\q$!+N!on3&Fg_pFa]p7%F^gu7$Fi[q$!+N/L[]Fg_pFa ]p7%Fbgu7$Fc[q$!+XNW Fg_pFa]p7%Fjiu7$Fggp$!+>o!*QVFg_pFa]p7%F^ju7$Fagp$!+UZ^HUFg_pFa]p7%Fbj u7$F[gp$!+mpi.TFg_pFa]p7%Ffju7$Fefp$!+]::gRFg_pFa]p7%Fjju7$F_fp$!+EW=. QFg_pFa]p7%F^[v7$Fiep$!+Xf#)ROFg_pFa]p7%Fb[v7$Fbep$!+q5KvMFg_pFa]p7%Ff [v7$F\\ep$!+vo\">J$Fg_pFa]p7%Fj[v7$$\"+5&4[D$Fadp$!+-L7]JFg_pFa]p7%F^ \\v7$F_dp$!+j>y*)HFg_pFa]p7%Fd\\v7$Ficp$!+I4cIGFg_pFa]p7%Fh\\v7$$!+0y \")>\"*F\\`p$!+Bn8sEFg_pFa]p7%F\\]v7$F]cp$!+?\"[U^#Fg_pFa]p7%Fb]v7$Ffb p$!++EqcBFg_pFa]p7%Ff]v7$F`bp$!+goO*>#Fg_pFa]p7%Fj]v7$Fjap$!+&Q`@/#Fg_ pFa]p7%F^^v7$Fdap$!+2w+&)=Fg_pFa]p7%Fb^v7$F]ap$!+ns*ys\"Fg_pFa]p7%Ff^v 7$$!+#>R@,)F\\`p$!+fV!3d\"Fg_pFa]p7%Fj^v7$Fa`p$!+R%>PT\"Fg_pFa]p7%F`_v 7$Fj_p$!+&*yjc7Fg_pFa]p7%Fd_v7$Fc_p$!+Zwb*4\"Fg_pFa]p7%Fh_v7$F\\_p$!+P %yZU*F`]pFa]p7%F\\`v7$$!+(R#e/CFN$!+:<)R&yF`]pFa]p7%F``v7$$!+Z@lrPF-$! +<`=$G'F`]pFa]p7%Ff`v7$$!+YvObNFfr$!+\")*)Q7ZF`]pFa]p7%F\\av7$F($!+aEf TJF`]pFa]p7%Fbav7$F($!+Fjzq:F`]pFa]p7%FfavF'Fa]p-Fc\\p6&Fe\\p$\"\"(Fh \\pF(F(-%&STYLEG6#%,PATCHNOGRIDG-F$6%7$7$$!3A++++++!\\'FNF(7$$\"3%**** **********=\"FNF(-%'COLOURG6&Fe\\pF)F)F)-%*LINESTYLEG6#\"\"$-F$6%7$7$F ($!3o*************Q'FN7$F($\"3o*************Q'FNF[cvF^cv-%*AXESSTYLEG6 #%$BOXG-%(SCALINGG6#%,CONSTRAINEDG-%+AXESLABELSG6%%&Re(z)G%&Im(z)G-%%F ONTG6#%(DEFAULTG-Fidv6$%*HELVETICAG\"\"*-%%VIEWG6$;$!$\\'!\"#$\"$>\"Ff ev;$!$R'Ffev$\"$R'Ffev" 1 2 0 1 10 0 2 9 1 2 1 1.000000 45.000000 45.000000 0 0 "Curve 1" "Curve 2" "Curve 3" "Curve 4" }}}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 75 "The stability interv al for the embedded order 7 scheme is (approximately) " }{XPPEDIT 18 0 "[-5.7679, 0];" "6#7$,$-%&FloatG6$\"&zw&!\"%!\"\"\"\"!" }{TEXT -1 2 ". " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 79 "Th e stability regions for the two schemes can be displayed in the same p icture." }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 154 "plots[display]([ p_1,p_2,p||(1..3)],view=[-6.89..1.19,-6.39..6.39],font=[HELVETICA,9], \n labels=[`Re(z)`,`Im(z)`],axes=boxed,scaling=constrained); " }}{PARA 13 "" 1 "" {GLPLOT2D 362 580 580 {PLOTDATA 2 "6--%'CURVESG6$ 7]_l7$$\"\"!F)F(7$F($\"3++++Fjzq:!#=7$F($\"3)******Rl#fTJF-7$$!3#***** *R()e`b$!#F$\"3D+++\")*)Q7ZF-7$$!3E+++dIlrP!#E$\"3g*****pJ&=$G'F-7$$!3 &******zR#e/C!#D$\"3?+++:<)R&yF-7$$!3*******f\\xy5\"!#C$\"3%)*****pVyZ U*F-7$$!3/+++qa*31%FF$\"3!******pkd&*4\"!#<7$$!3!******R]v=D\"!#B$\"3! ******\\*yjc7FN7$$!3?+++***)feLFR$\"33+++R%>PT\"FN7$$!3^+++*=R@,)FR$\" 3/+++fV!3d\"FN7$$!3&)*****\\LL:s\"!#A$\"3!******pE(*ys\"FN7$$!3M+++*eS 1N$F\\o$\"3#******pg2])=FN7$$!3y*****H\"pp$*eF\\o$\"3@+++&Q`@/#FN7$$!3 O+++R]@H#*F\\o$\"3-+++goO*>#FN7$$!3))******eE^I7!#@$\"3?++++EqcBFN7$$! 3'******z$4Y(>\"Fap$\"3)*******>\"[U^#FN7$$!3G*****R!y\")>\"*FR$\"3%** ****HsO@n#FN7$$\"3)******f)*p]f$Fap$\"3()******H4cIGFN7$$\"3++++xsft7! #?$\"3++++j>y*)HFN7$$\"39+++4&4[D$Ffq$\"3!******>IB,:$FN7$$\"3e*****4a 'G:sFfq$\"34+++vo\">J$FN7$$\"3)******>NV*o9!#>$\"37+++q5KvMFN7$$\"32++ +I%o#3GFfr$\"37+++Xf#)ROFN7$$\"3M+++&3_P0&Ffr$\"31+++EW=.QFN7$$\"3g*** **HyW5V)Ffr$\"3'*******\\::gRFN7$$\"3)******R.CXF\"F-$\"3@+++mpi.TFN7$ $\"38+++3q%Qt\"F-$\"35+++UZ^HUFN7$$\"3#*******=%4h:#F-$\"3(*******=o!* QVFN7$$\"3>+++v&*f2DF-$\"3&******\\N*>NWFN7$$\"3++++)*f?zFF-$\"3C+++g^ ]@XFN7$$\"3C+++l4CtHF-$\"3\")*****>$f,+YFN7$$\"35+++Od*f4$F-$\"3'***** **H,;sYFN7$$\"3/+++mWkaJF-$\"38+++0!f)QZFN7$$\"3w******eC7cJF-$\"3.+++ :qr+[FN7$$\"3:+++&*[x1JF-$\"3#)*****>^V\"e[FN7$$\"3()******QjG7IF-$\"3 ,+++#GF9\"\\FN7$$\"3E+++wcqxGF-$\"3y*****\\n\"yg\\FN7$$\"3,+++i!*\\2FF -$\"3/+++Xh2c]#F-$\"3c*****\\VI$[]FN7$$\"37+++aD]v AF-$\"3S+++N!on3&FN7$$\"32+++CfB??F-$\"3\")******ovx@^FN7$$\"3\"****** \\]'[UXsj$F-$\"3#******R$z(eD&FN7$ $!3()*****>2P`5%F-$\"3n*****REMkB&FN7$$!31+++4iN*e%F-$\"3()*****\\(fT7 _FN7$$!35+++Efw$4&F-$\"3%)*****zYAM=&FN7$$!3X+++*4hbi&F-$\"3#******Hb' **[^FN7$$!3]+++j8v&>'F-$\"3v******p0o3^FN7$$!3q*****4L(e@oF-$\"3)***** *H**)>i]FN7$$!3'******H$G&*GvF-$\"3y*****\\G_*4]FN7$$!3a+++eM$4N)F-$\" 3')******>P3a\\FN7$$!37+++nY]6$*F-$\"3g*****R&yt**[FN7$$!32+++%)oAR5FN $\"37+++`h\"R&[FN7$$!3#******\\`;G:\"FN$\"3h*****fL;1#[FN7$$!3-+++?gXl 7FN$\"3%******fT\\')z%FN7$$!33+++,nwt8FN$\"3!*******3__%y%FN7$$!35+++( y!zw9FN$\"31+++&zU]x%FN7$$!3!******H+xYd\"FN$\"3K+++C<$zw%FN7$$!33+++] 6%zm\"FN$\"3Q+++6vphZFN7$$!3)******\\#G;dFN$\"3=+++(='3SZFN7$$!32+++M*3 c+#FN$\"3\"********4?/t%FN7$$!3/+++RyN$3#FN$\"30+++%e5\">ZFN7$$!3;+++u m4f@FN$\"3?+++i4-1ZFN7$$!3++++#erIB#FN$\"3W+++5j0\"p%FN7$$!32+++(o&\\0 BFN$\"3E+++(3^Tn%FN7$$!3#*******yfbwBFN$\"3K+++V5EbYFN7$$!3=+++!**=kW# FN$\"3'******4)zNMYFN7$$!3:+++.aB:DFN$\"3#******HYE9h%FN7$$!3')******o P9$e#FN$\"3/+++D>Y'e%FN7$$!3=+++LMF]EFN$\"3;+++2)p%fXFN7$$!3++++=nu;FF N$\"3S+++$[l/`%FN7$$!3%******fR!o#y#FN$\"3++++#zu%*\\%FN7$$!3y*****RS' =[GFN$\"33+++=]`mWFN7$$!3*)******e#yO$FN$ \"39+++BE#>9%FN7$$!3<+++:/.LMFN$\"3=+++UyS&4%FN7$$!3/+++U,G)\\$FN$\"3y ******4S-[SFN7$$!3!)*****zk(\\jNFN$\"3-+++#QFN$\"3:+++efu,QFN7$$!3%)*****RO9a)QFN$ \"3'*******y_/^PFN7$$!3!******fWb\"[RFN$\"3A+++_.!**p$FN7$$!3.+++&*=55 SFN$\"3/+++l2H[OFN7$$!35+++s.c%='f$FN7$$!3#)*****>l\" HJTFN$\"3/+++w?aVNFN7$$!3c******)[5/>%FN$\"3))*****zM>.\\$FN7$$!3++++l [[[UFN$\"3++++%*4ZOMFN7$$!3k*****f&[[0VFN$\"3=+++AD&>Q$FN7$$!37+++58Rh VFN$\"34+++<@sELFN7$$!38+++`W>;WFN$\"3=+++**>uqKFN7$$!3!******p^#*)pWF N$\"3/+++7%zR@$FN7$$!3@+++)\\+)4$FN7$$!3A+++&RJWi%FN$\"3>+++T\"e(QIFN7$$!3g***** H#o!Qn%FN$\"3y******y$f'yHFN7$$!3P+++sX9AZFN$\"39+++R\"3x\"HFN7$$!3c** ****)Rl%pZFN$\"3))*****4s5f&GFN7$$!3L+++&)**y:[FN$\"3#)*****4:!G$z#FN7 $$!3c*****R!z8h[FN$\"31+++8f$)HFFN7$$!3D+++Dm_0\\FN$\"3A+++SNglEFN7$$! 3o*****Rcq*[\\FN$\"3*******\\.91g#FN7$$!3?+++P-[\"*\\FN$\"37+++EJ!\\`# FN7$$!3?+++R:1L]FN$\"3-+++o.^oCFN7$$!3m*****4@:P2&FN$\"37+++)4y9S#FN7$ $!3#)*****>^OM6&FN$\"3$)*****zG]QL#FN7$$!3!)*****z9:A:&FN$\"33+++e8nlA FN7$$!37+++`a.!>&FN$\"3?+++&*\\)p>#FN7$$!3G+++@o(oA&FN$\"3*)*****R0Ly7 #FN7$$!3!)*****HN9FE&FN$\"35+++%fa#e?FN7$$!3%)*****>p>vH&FN$\"3))***** H6&G))>FN7$$!3H+++k>EJ`FN$\"3.+++()f&z\">FN7$$!3Q+++%y3RO&FN$\"3!***** **fTHZ=FN7$$!3;+++_sU&R&FN$\"3)******R0Ajx\"FN7$$!3-++++\\yDaFN$\"32++ +!pd]q\"FN7$$!3?+++P0&\\X&FN$\"3.+++\")\\^L;FN7$$!3q*****R(\\*G[&FN$\" 3#******4e&FN$\"3)*******Ru)=F\"FN7$$!3/+++z&zMg&FN$\"33+++&)4!))>\"FN7$$! 3C+++0BlBcFN$\"3\"******R;ja7\"FN7$$!3U+++*z([UcFN$\"3%******R!\\(=0\" FN7$$!3;+++ZJ)*fcFN$\"3%******HV!Q!y*F-7$$!3U+++D%Qhn&FN$\"3))*****Htd &R!*F-7$$!3#******fPc4p&FN$\"3++++u-L'H)F-7$$!3q*****z0UWq&FN$\"3E+++) 4o2b(F-7$$!3o*****fU-mr&FN$\"3x*****\\ikH!oF-7$$!3V+++#*eWFdFN$\"3O+++ G'QI0'F-7$$!3')*****\\%=)pt&FN$\"3<+++Va8,`F-7$$!3q******f,AXdFN$\"37+ ++dtUZXF-7$$!3I+++'pq@v&FN$\"3;+++1J6#z$F-7$$!3/+++/G%yv&FN$\"3x*****R W;a.$F-7$$!3W+++l[CidFN$\"3'******\\y$exAF-7$$!3e*****f(QQldFN$\"32+++ W4))=:F-7$$!3O++++^EndFN$\"3k*****f&3*ef(Ffr7$$!3m*****\\!=*yw&FNF(7$F b]n$!3k*****f&3*ef(Ffr7$F]]n$!32+++W4))=:F-7$Fh\\n$!3'******\\y$exAF-7 $Fc\\n$!3x*****RW;a.$F-7$F^\\n$!3;+++1J6#z$F-7$Fi[n$!37+++dtUZXF-7$Fd[ n$!3<+++Va8,`F-7$F_[n$!3O+++G'QI0'F-7$Fjjm$!3x*****\\ikH!oF-7$Fejm$!3E +++)4o2b(F-7$F`jm$!3++++u-L'H)F-7$F[jm$!3))*****Htd&R!*F-7$Ffim$!3%*** ***HV!Q!y*F-7$Faim$!3%******R!\\(=0\"FN7$F\\im$!3\"******R;ja7\"FN7$Fg hm$!33+++&)4!))>\"FN7$Fbhm$!3)*******Ru)=F\"FN7$F]hm$!3$******4-AZM\"F N7$Fhgm$!3/+++,TI<9FN7$Fcgm$!3))******fBj*[\"FN7$F^gm$!3#******4FN7$F`em$!3))*****H6&G) )>FN7$F[em$!35+++%fa#e?FN7$Ffdm$!3*)*****R0Ly7#FN7$Fadm$!3?+++&*\\)p># FN7$F\\dm$!33+++e8nlAFN7$Fgcm$!3$)*****zG]QL#FN7$Fbcm$!37+++)4y9S#FN7$ F]cm$!3-+++o.^oCFN7$Fhbm$!37+++EJ!\\`#FN7$Fcbm$!3*******\\.91g#FN7$F^b m$!3A+++SNglEFN7$Fiam$!31+++8f$)HFFN7$Fdam$!3#)*****4:!G$z#FN7$F_am$!3 ))*****4s5f&GFN7$Fj`m$!39+++R\"3x\"HFN7$Fe`m$!3y******y$f'yHFN7$F``m$! 3>+++T\"e(QIFN7$F[`m$!3'******>)\\+)4$FN7$Ff_m$!3#******\\L2k:$FN7$Fa_ m$!3/+++7%zR@$FN7$F\\_m$!3=+++**>uqKFN7$Fg^m$!34+++<@sELFN7$Fb^m$!3=++ +AD&>Q$FN7$F]^m$!3++++%*4ZOMFN7$Fh]m$!3))*****zM>.\\$FN7$Fc]m$!3/+++w? aVNFN7$F^]m$!3#******>c%='f$FN7$Fi\\m$!3/+++l2H[OFN7$Fd\\m$!3A+++_.!** p$FN7$F_\\m$!3'*******y_/^PFN7$Fj[m$!3:+++efu,QFN7$Fe[m$!3;+++Ws+_QFN7 $F`[m$!3')******[Y\"=!RFN7$F[[m$!3A+++&GJ6&RFN7$Ffjl$!3-+++9%FN7$Fbil $!39+++s6V(=%FN7$F]il$!3L+++Y1zJUFN7$Fhhl$!3!)******[!f[F%FN7$Fchl$!3w *******)4];VFN7$F^hl$!37+++I@fcVFN7$Figl$!3K+++L@-&R%FN7$Fdgl$!3B+++Oh pJWFN7$F_gl$!33+++=]`mWFN7$Fjfl$!3++++#zu%*\\%FN7$Fefl$!3S+++$[l/`%FN7 $F`fl$!3;+++2)p%fXFN7$F[fl$!3/+++D>Y'e%FN7$Ffel$!3#******HYE9h%FN7$Fae l$!3'******4)zNMYFN7$F\\el$!3K+++V5EbYFN7$Fgdl$!3E+++(3^Tn%FN7$Fbdl$!3 W+++5j0\"p%FN7$F]dl$!3?+++i4-1ZFN7$Fhcl$!30+++%e5\">ZFN7$Fccl$!3\"**** ****4?/t%FN7$F^cl$!3=+++(='3SZFN7$Fibl$!3w*****RI1$[ZFN7$Fdbl$!3=+++P$ o`v%FN7$F_bl$!3Q+++6vphZFN7$Fjal$!3K+++C<$zw%FN7$Feal$!31+++&zU]x%FN7$ F`al$!3!*******3__%y%FN7$F[al$!3%******fT\\')z%FN7$Ff`l$!3h*****fL;1#[ FN7$Fa`l$!37+++`h\"R&[FN7$F\\`l$!3g*****R&yt**[FN7$Fg_l$!3')******>P3a \\FN7$Fb_l$!3y*****\\G_*4]FN7$F]_l$!3)******H**)>i]FN7$Fh^l$!3v******p 0o3^FN7$Fc^l$!3#******Hb'**[^FN7$F^^l$!3%)*****zYAM=&FN7$Fi]l$!3()**** *\\(fT7_FN7$Fd]l$!3n*****REMkB&FN7$F_]l$!3#******R$z(eD&FN7$Fj\\l$!3++ ++*y!3r_FN7$Fe\\l$!3I+++vYJ#G&FN7$F`\\l$!3'******\\6)z*G&FN7$F[\\l$!3U +++v[q$H&FN7$Ff[l$!3K+++@8<%H&FN7$Fa[l$!3#)*****\\Q-8H&FN7$F\\[l$!3A++ +%Hw^G&FN7$Fgz$!3!******zD[eF&FN7$Fbz$!3A+++_LNj_FN7$F]z$!3o*****\\v3x C&FN7$Fhy$!3D+++*[:*G_FN7$Fcy$!3o*****\\wfp?&FN7$F^y$!3M+++CS\"==&FN7$ Fix$!3g******ywV`^FN7$Fdx$!3\")******ovx@^FN7$F_x$!3S+++N!on3&FN7$Fjw$ !3c*****\\VI$[]FN7$Few$!3/+++X^V\"e[FN7$Fav$!3.+++:qr+[FN7$F\\v$!38+++0 !f)QZFN7$Fgu$!3'*******H,;sYFN7$Fbu$!3\")*****>$f,+YFN7$F]u$!3C+++g^]@ XFN7$Fht$!3&******\\N*>NWFN7$Fct$!3(*******=o!*QVFN7$F^t$!35+++UZ^HUFN 7$Fis$!3@+++mpi.TFN7$Fds$!3'*******\\::gRFN7$F_s$!31+++EW=.QFN7$Fjr$!3 7+++Xf#)ROFN7$Fdr$!37+++q5KvMFN7$F_r$!34+++vo\">J$FN7$$\"39+++5&4[D$Ff q$!3!******>IB,:$FN7$Fdq$!3++++j>y*)HFN7$F_q$!3()******H4cIGFN7$$!3g** ***\\!y\")>\"*FR$!3%******HsO@n#FN7$Fep$!3)*******>\"[U^#FN7$F_p$!3?++ ++EqcBFN7$Fjo$!3-+++goO*>#FN7$Feo$!3@+++&Q`@/#FN7$F`o$!3#******pg2])=F N7$Fjn$!3!******pE(*ys\"FN7$$!3y*****>>R@,)FR$!3/+++fV!3d\"FN7$FV$!33+ ++R%>PT\"FN7$FP$!3!******\\*yjc7FN7$FJ$!3!******pkd&*4\"FN7$FD$!3%)*** **pVyZU*F-7$$!38+++(R#e/CF@$!3?+++:<)R&yF-7$$!37+++Z@lrPF:$!3g*****pJ& =$G'F-7$$!3w*****fan`b$F4$!3D+++\")*)Q7ZF-7$F($!3)******Rl#fTJF-7$F($! 3++++Fjzq:F-F'-%&COLORG6&%$RGBG$\"\"#!\"\"F(F(-%)POLYGONSG6^_l7%F'7$F( $\"+Fjzq:!#57$$!#WFh\\pF(7%F]]p7$F($\"+aEfTJF`]pFa]p7%Fe]p7$$!+u)e`b$F fr$\"+\")*)Q7ZF`]pFa]p7%Fi]p7$$!+dIlrPF-$\"+<`=$G'F`]pFa]p7%F_^p7$$!+) R#e/CFN$\"+:<)R&yF`]pFa]p7%Fe^p7$$!+'\\xy5\"!#;$\"+P%yZU*F`]pFa]p7%F[_ p7$$!+qa*31%F^_p$\"+Zwb*4\"!\"*Fa]p7%Fb_p7$$!+/b(=D\"!#:$\"+&*yjc7Fg_p Fa]p7%Fi_p7$$!+***)feLF\\`p$\"+R%>PT\"Fg_pFa]p7%F``p7$$!+*=R@,)F\\`p$ \"+fV!3d\"Fg_pFa]p7%Ff`p7$$!+NL`@#Fg_pFa]p7%F_bp7$$!+fE^I7!#8$\"++EqcBFg_pFa]p 7%Febp7$$!+Q4Y(>\"Fhbp$\"+?\"[U^#Fg_pFa]p7%F\\cp7$$!+/y\")>\"*F\\`p$\" +Bn8sEFg_pFa]p7%Fbcp7$$\"+')*p]f$Fhbp$\"+I4cIGFg_pFa]p7%Fhcp7$$\"+xsft 7!#7$\"+j>y*)HFg_pFa]p7%F^dp7$$\"+4&4[D$Fadp$\"+-L7]JFg_pFa]p7%Fedp7$$ \"+TlG:sFadp$\"+vo\">J$Fg_pFa]p7%F[ep7$$\"+_L%*o9!#6$\"+q5KvMFg_pFa]p7 %Faep7$$\"+I%o#3GFdep$\"+Xf#)ROFg_pFa]p7%Fhep7$$\"+&3_P0&Fdep$\"+EW=.Q Fg_pFa]p7%F^fp7$$\"+$yW5V)Fdep$\"+]::gRFg_pFa]p7%Fdfp7$$\"+MS_u7F`]p$ \"+mpi.TFg_pFa]p7%Fjfp7$$\"+3q%Qt\"F`]p$\"+UZ^HUFg_pFa]p7%F`gp7$$\"+>% 4h:#F`]p$\"+>o!*QVFg_pFa]p7%Ffgp7$$\"+v&*f2DF`]p$\"+b$*>NWFg_pFa]p7%F \\hp7$$\"+)*f?zFF`]p$\"+g^]@XFg_pFa]p7%Fbhp7$$\"+l4CtHF`]p$\"+Kf,+YFg_ pFa]p7%Fhhp7$$\"+Od*f4$F`]p$\"+I,;sYFg_pFa]p7%F^ip7$$\"+mWkaJF`]p$\"+0 !f)QZFg_pFa]p7%Fdip7$$\"+fC7cJF`]p$\"+:qr+[Fg_pFa]p7%Fjip7$$\"+&*[x1JF `]p$\"+7N9e[Fg_pFa]p7%F`jp7$$\"+RjG7IF`]p$\"+#GF9\"\\Fg_pFa]p7%Ffjp7$$ \"+wcqxGF`]p$\"+v;yg\\Fg_pFa]p7%F\\[q7$$\"+i!*\\2FF`]p$\"+XXsj$F`]p$\"+Mz (eD&Fg_pFa]p7%F^bq7$$!+sqL0TF`]p$\"+kUVO_Fg_pFa]p7%Fdbq7$$!+4iN*e%F`]p $\"+vfT7_Fg_pFa]p7%Fjbq7$$!+Efw$4&F`]p$\"+oCU$=&Fg_pFa]p7%F`cq7$$!+*4h bi&F`]p$\"+`l**[^Fg_pFa]p7%Ffcq7$$!+j8v&>'F`]p$\"+q0o3^Fg_pFa]p7%F\\dq 7$$!+Jte@oF`]p$\"+$**)>i]Fg_pFa]p7%Fbdq7$$!+LG&*GvF`]p$\"+&G_*4]Fg_pFa ]p7%Fhdq7$$!+eM$4N)F`]p$\"+?P3a\\Fg_pFa]p7%F^eq7$$!+nY]6$*F`]p$\"+ayt* *[Fg_pFa]p7%Fdeq7$$!+%)oAR5Fg_p$\"+`h\"R&[Fg_pFa]p7%Fjeq7$$!+Nl\"G:\"F g_p$\"+Ojh?[Fg_pFa]p7%F`fq7$$!+?gXl7Fg_p$\"+;%\\')z%Fg_pFa]p7%Fffq7$$! +,nwt8Fg_p$\"+4__%y%Fg_pFa]p7%F\\gq7$$!+(y!zw9Fg_p$\"+&zU]x%Fg_pFa]p7% Fbgq7$$!+.qnu:Fg_p$\"+C<$zw%Fg_pFa]p7%Fhgq7$$!+]6%zm\"Fg_p$\"+6vphZFg_ pFa]p7%F^hq7$$!+DG;dFg_p$\"+(='3SZFg_pFa]p7%F`iq7$$!+M*3c+#Fg_ p$\"++,UIZFg_pFa]p7%Ffiq7$$!+RyN$3#Fg_p$\"+%e5\">ZFg_pFa]p7%F\\jq7$$!+ um4f@Fg_p$\"+i4-1ZFg_pFa]p7%Fbjq7$$!+#erIB#Fg_p$\"+5j0\"p%Fg_pFa]p7%Fh jq7$$!+(o&\\0BFg_p$\"+(3^Tn%Fg_pFa]p7%F^[r7$$!+zfbwBFg_p$\"+V5EbYFg_pF a]p7%Fd[r7$$!+!**=kW#Fg_p$\"+\")zNMYFg_pFa]p7%Fj[r7$$!+.aB:DFg_p$\"+jk U6YFg_pFa]p7%F`\\r7$$!+pP9$e#Fg_p$\"+D>Y'e%Fg_pFa]p7%Ff\\r7$$!+LMF]EFg _p$\"+2)p%fXFg_pFa]p7%F\\]r7$$!+=nu;FFg_p$\"+$[l/`%Fg_pFa]p7%Fb]r7$$!+ 'R!o#y#Fg_p$\"+#zu%*\\%Fg_pFa]p7%Fh]r7$$!+/k=[GFg_p$\"+=]`mWFg_pFa]p7% F^^r7$$!+f#yO $Fg_p$\"+BE#>9%Fg_pFa]p7%F^ar7$$!+:/.LMFg_p$\"+UyS&4%Fg_pFa]p7%Fdar7$$ !+U,G)\\$Fg_p$\"+5S-[SFg_pFa]p7%Fjar7$$!+[w\\jNFg_p$\"+#QFg_p $\"+efu,QFg_pFa]p7%Fhcr7$$!+kVT&)QFg_p$\"+z_/^PFg_pFa]p7%F^dr7$$!+Ya:[ RFg_p$\"+_.!**p$Fg_pFa]p7%Fddr7$$!+&*=55SFg_p$\"+l2H[OFg_pFa]p7%Fjdr7$ $!+s.%Fg_p$\"+[$>.\\$Fg_pFa]p7%F\\fr7$$!+l[[[UFg_p$\"+%*4ZOMFg _pFa]p7%Fbfr7$$!+c[[0VFg_p$\"+AD&>Q$Fg_pFa]p7%Fhfr7$$!+58RhVFg_p$\"+<@ sELFg_pFa]p7%F^gr7$$!+`W>;WFg_p$\"+**>uqKFg_pFa]p7%Fdgr7$$!+&Fg_p $\"+&*\\)p>#Fg_pFa]p7%Fj]s7$$!+@o(oA&Fg_p$\"+aI$y7#Fg_pFa]p7%F`^s7$$!+ `Vri_Fg_p$\"+%fa#e?Fg_pFa]p7%Ff^s7$$!+#p>vH&Fg_p$\"+8^G))>Fg_pFa]p7%F \\_s7$$!+k>EJ`Fg_p$\"+()f&z\">Fg_pFa]p7%Fb_s7$$!+%y3RO&Fg_p$\"+gTHZ=Fg _pFa]p7%Fh_s7$$!+_sU&R&Fg_p$\"+a?Kwe&Fg_p$\"+Su)=F\"Fg_pFa]p7%Fhbs7$$!+z&zMg&Fg_p$\"+&)4 !))>\"Fg_pFa]p7%F^cs7$$!+0BlBcFg_p$\"+kJYD6Fg_pFa]p7%Fdcs7$$!+*z([UcFg _p$\"+/\\(=0\"Fg_pFa]p7%Fjcs7$$!+ZJ)*fcFg_p$\"+L/Q!y*F`]pFa]p7%F`ds7$$ !+D%Qhn&Fg_p$\"+LxbR!*F`]pFa]p7%Ffds7$$!+wj&4p&Fg_p$\"+u-L'H)F`]pFa]p7 %F\\es7$$!+e?W/dFg_p$\"+)4o2b(F`]pFa]p7%Fbes7$$!+ECg;dFg_p$\"+DY'H!oF` ]pFa]p7%Fhes7$$!+#*eWFdFg_p$\"+G'QI0'F`]pFa]p7%F^fs7$$!+X=)pt&Fg_p$\"+ Va8,`F`]pFa]p7%Fdfs7$$!+g,AXdFg_p$\"+dtUZXF`]pFa]p7%Fjfs7$$!+'pq@v&Fg_ p$\"+1J6#z$F`]pFa]p7%F`gs7$$!+/G%yv&Fg_p$\"+WkTNIF`]pFa]p7%Ffgs7$$!+l[ CidFg_p$\"+&y$exAF`]pFa]p7%F\\hs7$$!+wQQldFg_p$\"+W4))=:F`]pFa]p7%Fbhs 7$$!++^EndFg_p$\"+c3*ef(FdepFa]p7%Fhhs7$$!+0=*yw&Fg_pF(Fa]p7%F^is7$Fih s$!+c3*ef(FdepFa]p7%Fbis7$Fchs$!+W4))=:F`]pFa]p7%Ffis7$F]hs$!+&y$exAF` ]pFa]p7%Fjis7$Fggs$!+WkTNIF`]pFa]p7%F^js7$Fags$!+1J6#z$F`]pFa]p7%Fbjs7 $F[gs$!+dtUZXF`]pFa]p7%Ffjs7$Fefs$!+Va8,`F`]pFa]p7%Fjjs7$F_fs$!+G'QI0' F`]pFa]p7%F^[t7$Fies$!+DY'H!oF`]pFa]p7%Fb[t7$Fces$!+)4o2b(F`]pFa]p7%Ff [t7$F]es$!+u-L'H)F`]pFa]p7%Fj[t7$Fgds$!+LxbR!*F`]pFa]p7%F^\\t7$Fads$!+ L/Q!y*F`]pFa]p7%Fb\\t7$F[ds$!+/\\(=0\"Fg_pFa]p7%Ff\\t7$Fecs$!+kJYD6Fg_ pFa]p7%Fj\\t7$F_cs$!+&)4!))>\"Fg_pFa]p7%F^]t7$Fibs$!+Su)=F\"Fg_pFa]p7% Fb]t7$Fcbs$!+@?sW8Fg_pFa]p7%Ff]t7$F]bs$!+,TI<9Fg_pFa]p7%Fj]t7$Fgas$!+g Bj*[\"Fg_pFa]p7%F^^t7$Faas$!+rTqh:Fg_pFa]p7%Fb^t7$F[as$!+\")\\^L;Fg_pF a]p7%Ff^t7$Fe`s$!+!pd]q\"Fg_pFa]p7%Fj^t7$F_`s$!+a?KwFg_pFa]p7%Ff_t7$F]_s$!+8^G)) >Fg_pFa]p7%Fj_t7$Fg^s$!+%fa#e?Fg_pFa]p7%F^`t7$Fa^s$!+aI$y7#Fg_pFa]p7%F b`t7$F[^s$!+&*\\)p>#Fg_pFa]p7%Ff`t7$Fe]s$!+e8nlAFg_pFa]p7%Fj`t7$F_]s$! +)G]QL#Fg_pFa]p7%F^at7$Fi\\s$!+)4y9S#Fg_pFa]p7%Fbat7$Fc\\s$!+o.^oCFg_p Fa]p7%Ffat7$F]\\s$!+EJ!\\`#Fg_pFa]p7%Fjat7$Fg[s$!+NSh+EFg_pFa]p7%F^bt7 $Fa[s$!+SNglEFg_pFa]p7%Fbbt7$F[[s$!+8f$)HFFg_pFa]p7%Ffbt7$Fejr$!+^,G$z #Fg_pFa]p7%Fjbt7$F_jr$!+@2\"f&GFg_pFa]p7%F^ct7$Fiir$!+R\"3x\"HFg_pFa]p 7%Fbct7$Fcir$!+z$f'yHFg_pFa]p7%Ffct7$F]ir$!+T\"e(QIFg_pFa]p7%Fjct7$Fgh r$!+#)\\+)4$Fg_pFa]p7%F^dt7$Fahr$!+NtScJFg_pFa]p7%Fbdt7$F[hr$!+7%zR@$F g_pFa]p7%Ffdt7$Fegr$!+**>uqKFg_pFa]p7%Fjdt7$F_gr$!+<@sELFg_pFa]p7%F^et 7$Fifr$!+AD&>Q$Fg_pFa]p7%Fbet7$Fcfr$!+%*4ZOMFg_pFa]p7%Ffet7$F]fr$!+[$> .\\$Fg_pFa]p7%Fjet7$Fger$!+w?aVNFg_pFa]p7%F^ft7$Faer$!+iX='f$Fg_pFa]p7 %Fbft7$F[er$!+l2H[OFg_pFa]p7%Ffft7$Fedr$!+_.!**p$Fg_pFa]p7%Fjft7$F_dr$ !+z_/^PFg_pFa]p7%F^gt7$Ficr$!+efu,QFg_pFa]p7%Fbgt7$Fccr$!+Ws+_QFg_pFa] p7%Ffgt7$F]cr$!+\\Y\"=!RFg_pFa]p7%Fjgt7$Fgbr$!+&GJ6&RFg_pFa]p7%F^ht7$F abr$!+9%Fg_pFa]p7%F^it7$Fi`r$!+s6V(=%Fg_pFa]p7%Fb it7$Fc`r$!+Y1zJUFg_pFa]p7%Ffit7$F]`r$!+\\!f[F%Fg_pFa]p7%Fjit7$Fg_r$!+! *4];VFg_pFa]p7%F^jt7$Fa_r$!+I@fcVFg_pFa]p7%Fbjt7$F[_r$!+L@-&R%Fg_pFa]p 7%Ffjt7$Fe^r$!+OhpJWFg_pFa]p7%Fjjt7$F_^r$!+=]`mWFg_pFa]p7%F^[u7$Fi]r$! +#zu%*\\%Fg_pFa]p7%Fb[u7$Fc]r$!+$[l/`%Fg_pFa]p7%Ff[u7$F]]r$!+2)p%fXFg_ pFa]p7%Fj[u7$Fg\\r$!+D>Y'e%Fg_pFa]p7%F^\\u7$Fa\\r$!+jkU6YFg_pFa]p7%Fb \\u7$F[\\r$!+\")zNMYFg_pFa]p7%Ff\\u7$Fe[r$!+V5EbYFg_pFa]p7%Fj\\u7$F_[r $!+(3^Tn%Fg_pFa]p7%F^]u7$Fijq$!+5j0\"p%Fg_pFa]p7%Fb]u7$Fcjq$!+i4-1ZFg_ pFa]p7%Ff]u7$F]jq$!+%e5\">ZFg_pFa]p7%Fj]u7$Fgiq$!++,UIZFg_pFa]p7%F^^u7 $Faiq$!+(='3SZFg_pFa]p7%Fb^u7$F[iq$!+/jI[ZFg_pFa]p7%Ff^u7$Fehq$!+P$o`v %Fg_pFa]p7%Fj^u7$F_hq$!+6vphZFg_pFa]p7%F^_u7$Figq$!+C<$zw%Fg_pFa]p7%Fb _u7$Fcgq$!+&zU]x%Fg_pFa]p7%Ff_u7$F]gq$!+4__%y%Fg_pFa]p7%Fj_u7$Fgfq$!+; %\\')z%Fg_pFa]p7%F^`u7$Fafq$!+Ojh?[Fg_pFa]p7%Fb`u7$F[fq$!+`h\"R&[Fg_pF a]p7%Ff`u7$Feeq$!+ayt**[Fg_pFa]p7%Fj`u7$F_eq$!+?P3a\\Fg_pFa]p7%F^au7$F idq$!+&G_*4]Fg_pFa]p7%Fbau7$Fcdq$!+$**)>i]Fg_pFa]p7%Ffau7$F]dq$!+q0o3^ Fg_pFa]p7%Fjau7$Fgcq$!+`l**[^Fg_pFa]p7%F^bu7$Facq$!+oCU$=&Fg_pFa]p7%Fb bu7$F[cq$!+vfT7_Fg_pFa]p7%Ffbu7$Febq$!+kUVO_Fg_pFa]p7%Fjbu7$F_bq$!+Mz( eD&Fg_pFa]p7%F^cu7$Fiaq$!+*y!3r_Fg_pFa]p7%Fbcu7$Fcaq$!+vYJ#G&Fg_pFa]p7 %Ffcu7$F]aq$!+:\")z*G&Fg_pFa]p7%Fjcu7$Fg`q$!+v[q$H&Fg_pFa]p7%F^du7$Fa` q$!+@8<%H&Fg_pFa]p7%Fbdu7$F[`q$!+&Q-8H&Fg_pFa]p7%Ffdu7$Fe_q$!+%Hw^G&Fg _pFa]p7%Fjdu7$F__q$!+e#[eF&Fg_pFa]p7%F^eu7$Fi^q$!+_LNj_Fg_pFa]p7%Fbeu7 $Fc^q$!+b(3xC&Fg_pFa]p7%Ffeu7$F]^q$!+*[:*G_Fg_pFa]p7%Fjeu7$Fg]q$!+l(fp ?&Fg_pFa]p7%F^fu7$Fa]q$!+CS\"==&Fg_pFa]p7%Fbfu7$F[]q$!+zwV`^Fg_pFa]p7% Fffu7$Fe\\q$!+pvx@^Fg_pFa]p7%Fjfu7$F_\\q$!+N!on3&Fg_pFa]p7%F^gu7$Fi[q$ !+N/L[]Fg_pFa]p7%Fbgu7$Fc[q$!+XNWFg_pFa]p7%Fjiu7$Fggp$!+>o!*QVFg_pFa]p7%F^ju7$Fagp$!+UZ^HU Fg_pFa]p7%Fbju7$F[gp$!+mpi.TFg_pFa]p7%Ffju7$Fefp$!+]::gRFg_pFa]p7%Fjju 7$F_fp$!+EW=.QFg_pFa]p7%F^[v7$Fiep$!+Xf#)ROFg_pFa]p7%Fb[v7$Fbep$!+q5Kv MFg_pFa]p7%Ff[v7$F\\ep$!+vo\">J$Fg_pFa]p7%Fj[v7$$\"+5&4[D$Fadp$!+-L7]J Fg_pFa]p7%F^\\v7$F_dp$!+j>y*)HFg_pFa]p7%Fd\\v7$Ficp$!+I4cIGFg_pFa]p7%F h\\v7$$!+0y\")>\"*F\\`p$!+Bn8sEFg_pFa]p7%F\\]v7$F]cp$!+?\"[U^#Fg_pFa]p 7%Fb]v7$Ffbp$!++EqcBFg_pFa]p7%Ff]v7$F`bp$!+goO*>#Fg_pFa]p7%Fj]v7$Fjap$ !+&Q`@/#Fg_pFa]p7%F^^v7$Fdap$!+2w+&)=Fg_pFa]p7%Fb^v7$F]ap$!+ns*ys\"Fg_ pFa]p7%Ff^v7$$!+#>R@,)F\\`p$!+fV!3d\"Fg_pFa]p7%Fj^v7$Fa`p$!+R%>PT\"Fg_ pFa]p7%F`_v7$Fj_p$!+&*yjc7Fg_pFa]p7%Fd_v7$Fc_p$!+Zwb*4\"Fg_pFa]p7%Fh_v 7$F\\_p$!+P%yZU*F`]pFa]p7%F\\`v7$$!+(R#e/CFN$!+:<)R&yF`]pFa]p7%F``v7$$ !+Z@lrPF-$!+<`=$G'F`]pFa]p7%Ff`v7$$!+YvObNFfr$!+\")*)Q7ZF`]pFa]p7%F\\a v7$F($!+aEfTJF`]pFa]p7%Fbav7$F($!+Fjzq:F`]pFa]p7%FfavF'Fa]p-Fc\\p6&Fe \\p$\"\"(Fh\\pF(F(-%&STYLEG6#%,PATCHNOGRIDG-F$6$7]_lF'F*F.7$F(F57$$\"3 y*****pqK.M%F4$\"3a+++:`=$G'F-7$$\"3w*****>hz&>QF:$\"3E+++*o\")R&yF-7$ $\"3-+++z@#G?#F@$\"3-+++6#yZU*F-7$$\"3h+++VMfO%*F@$\"3#******H^d&*4\"F N7$$\"3=+++E$zGB$FF$\"3-+++!HPmD\"FN7$$\"3i******zsKy#*FF$\"34+++,sr89 FN7$$\"3))*****f'Q-*H#FR$\"31+++pszq:FN7$$\"3/+++r+:9]FR$\"3++++7u(ys \"FN7$$\"3#3+++HQCt*FR$\"31+++!Rd\\)=FN7$$\"37+++Eee'o\"F\\o$\"3&)**** *>&o.U?FN7$$\"3;+++&HP>f#F\\o$\"3'******zw:\"*>#FN7$$\"3E+++]`/YMF\\o$ \"3y*****4k&>cBFN7$$\"37+++TiM&o$F\\o$\"3$)******)>#G8DFN7$$\"3)****** \\jmHK#F\\o$\"3'******zX!RqEFN7$$!3&)*****4ha'H?F\\o$\"3;+++#H`v#GFN7$ $!3'******\\IrO3\"Fap$\"3/+++(yM[)HFN7$$!3)******pOH;Z#Fap$\"34+++!p\\ B9$FN7$$!37++++\"=j9%Fap$\"3z*****zG!H+LFN7$$!3%*******Qs)eA&Fap$\"3%* *****R3i*eMFN7$$!3%******zqc&pMFap$\"3(******4OH)=OFN7$$\"3\\*****>(zT :gFap$\"3#*******)*)o0y$FN7$$\"3,+++1!QiK$Ffq$\"3;+++:[6XRFN7$$\"3d*** ***)GTI%)*Ffq$\"3n******4pg8TFN7$$\"30+++)fu[T#Ffr$\"3-+++(pboG%FN7$$ \"3(******z]@8M&Ffr$\"3/+++'eEHY%FN7$$\"3)******\\NB,0\"F-$\"3k******G H?JYFN7$$\"3%*******4`\")H,XXx%FN7$$\"3++++\"[+sR#F-$\" 3I+++PcS*)[FN7$$\"3%)*****RAWu&HF-$\"3s******>J@%)\\FN7$$\"3u*****Hlb$ )R$F-$\"3e*****\\SVj1&FN7$$\"3F+++,yPKPF-$\"3#)*****\\Zu*R^FN7$$\"3%** ****z\"HyuRF-$\"39+++*[[t?&FN7$$\"3')*****fi'*)QTF-$\"3I+++Sxqp_FN7$$ \"3w*****>uTcB%F-$\"3k*****RktxK&FN7$$\"3C+++')>$RF%F-$\"35+++u`)>Q&FN 7$$\"3s*****R/A6E%F-$\"3!)*****R,BEV&FN7$$\"3u*****fUlL?%F-$\"3?+++XQ( )zaFN7$$\"3++++')o(e5%F-$\"3y*****>FpQ_&FN7$$\"33+++UA7tRF-$\"3')***** p&eqkbFN7$$\"3F+++^s&*3QF-$\"3%******>PdCg&FN7$$\"3$)*****\\aJnh$F-$\" 3;+++Y;=PcFN7$$\"35+++t%o$*R$F-$\"3K+++4V#*ocFN7$$\"3')*****>)zUfJF-$ \"3!)******p1s(p&FN7$$\"3%******R:d\"**GF-$\"3n******RkfBdFN7$$\"3v*** **zOJ0i#F-$\"3E+++OxcYdFN7$$\"32+++jzGDBF-$\"3e*****zeSmw&FN7$$\"3()** ***pf`\\,#F-$\"3#)*****\\35Qy&FN7$$\"3'*******of'3p\"F-$\"3#******za8F-$\"37+++PrN4eFN7$$\"3++++/'Hf+\"F-$\"3I+++xvl+++Pf*G#eFN7$$\"3)******H>3)yFFfr$\"3%******R@ ')\\#eFN7$$!30+++[*Rq+\"Ffr$\"3w*****pq;Q#eFN7$$!3m*****4IL])[Ffr$\"3t *****zNW#>eFN7$$!3Y*****>8mV&))Ffr$\"3n*****4;(36eFN7$$!3/+++()pt\"H\" F-$\"3Q+++HQ6*z&FN7$$!3%******R=u!3 Chd8#F-$\"3y*****44cCw&FN7$$!3?+++W<6xDF-$\"3z*****zH/pt&FN7$$!3;+++q5 3OIF-$\"3))*****>=Odq&FN7$$!3G+++m;Z>NF-$\"33+++5v6ocFN7$$!3!)*****pC7 $RSF-$\"3$******RswHi&FN7$$!3&******>XDxh%F-$\"3!******zbd!pbFN7$$!39+ ++i\"*\\'H&F-$\"3C+++=DX0bFN7$$!35+++0J\"*[hF-$\"3T+++ob,MaFN7$$!3s*** ***oi**\\sF-$\"3I+++S`Hl`FN7$$!3q*****pL:5a)F-$\"3k*****zmsjJ&FN7$$!3_ *****Hc&\\Y)*F-$\"3)******>WU2H&FN7$$!3(******4dvr5\"FN$\"3q******4#H* z_FN7$$!3*******p!)R/A\"FN$\"37+++yZ^w_FN7$$!3!******>JLcK\"FN$\"3o*** **fN%Gw_FN7$$!32+++j??C9FN$\"35+++8a'pF&FN7$$!3&******Rsut^\"FN$\"3g** ***pw#Hx_FN7$$!36+++OF71;FN$\"31+++`.`w_FN7$$!3!******zR07p\"FN$\"3U++ +eGBu_FN7$$!3!******4H?Kx\"FN$\"3M+++aA7q_FN7$$!3/+++]Rk_=FN$\"3w***** *o1-k_FN7$$!3\"******RKi)H>FN$\"3))*****H;7eD&FN7$$!3*******RC$>0?FN$ \"35+++\\,UX_FN7$$!35+++$4.*y?FN$\"3l*****4m$zK_FN7$$!35+++E$>7:#FN$\" 3u*****z())*y@&FN7$$!3@+++N+MAAFN$\"35+++?Rr+_FN7$$!33+++k8W#H#FN$\"3# ******pwD7=&FN7$$!3;+++FLohBFN$\"39+++b'G%f^FN7$$!36++++W@ICFN$\"35+++ `NKN^FN7$$!3!******p3v\")\\#FN$\"3U+++)\\=*3^FN7$$!39+++\"f+dc#FN$\"3A +++C'H-3&FN7$$!3!******>fAHj#FN$\"3#)*****\\!GG\\]FN7$$!3)*******p+(** p#FN$\"3.+++sd6;]FN7$$!3%)*****H2ppw#FN$\"3/+++c0y!)\\FN7$$!3')*****4E TS$GFN$\"3\")******=gMV\\FN7$$!3:+++H6I,HFN$\"3V++++,!R!\\FN7$$!3?+++t =&)oHFN$\"39+++:8bi[FN7$$!35+++%3!yOIFN$\"38++++*G%>[FN7$$!3%)*****>q \\^5$FN$\"3%*******R5ouZFN7$$!3#******>,'*R<$FN$\"3F+++:2ZGZFN7$$!3A++ +92KVKFN$\"3C+++(3p4o%FN7$$!3++++v\"*38LFN$\"3g*****H&oMKYFN7$$!3))*** **ptIKQ$FN$\"3G+++=Xw#e%FN7$$!3/+++UCk`MFN$\"3x*****R(HOKXFN7$$!3()*** **>\"e>CNFN$\"3;+++2eD\"[%FN7$$!3!******\\oXZf$FN$\"3@+++8W_HWFN7$$!3% )*****>xR^m$FN$\"3%)*****zR;sP%FN7$$!35+++'fF_t$FN$\"3')*****z0ZVK%FN7 $$!3.+++Ny'[!QFN$\"3))*****zA.4F%FN7$$!39+++'eLR(QFN$\"3++++^&[o@%FN7$ $!3%)******)[:B%RFN$\"3e*****4EH@;%FN7$$!3P+++>I#*4SFN$\"3++++w(zm5%FN 7$$!35+++`VowSFN$\"33+++XwU]SFN7$$!3;+++!HXD9%FN$\"3%******Hl(H$*RFN7$ $!3I+++7wY2UFN$\"3'******\\1:_$RFN7$$!3A+++%=F9F%FN$\"3!*******p!3h(QF N7$$!3H+++**>TMVFN$\"3\"******Re4f\"QFN7$$!3)*******y-U'R%FN$\"3?+++T& eXv$FN7$$!3%******zpeuX%FN$\"3;+++03+#p$FN7$$!3@+++22a)*= GOFN7$$!3m*****4)\\owXFN$\"3A+++jq3jNFN7$$!3O+++;R\"\\j%FN$\"3;+++lBm' \\$FN7$$!3O+++n@D#p%FN$\"3\")*****H4%*)GMFN7$$!3W+++%=D([ZFN$\"3#***** **=#p(fLFN7$$!3-+++DxN/[FN$\"3y*****>C$G*G$FN7$$!3y*****4Ns\"f[FN$\"36 +++G*Ru@$FN7$$!3!******\\#z=8\\FN$\"3!)*****\\$3DWJFN7$$!37+++T\"=k'\\ FN$\"3=+++$fM(pIFN7$$!38+++m,()=]FN$\"3!******z\"f\"R*HFN7$$!3s****** \\Maq]FN$\"3%)*****>OCo\"HFN7$$!3q*****H!)G97&FN$\"37+++WG\\QGFN7$$!3s *******y2:<&FN$\"3;+++8g&*eFFN7$$!37+++;Dv?_FN$\"3!)*****4b[#yEFN7$$!3 e*****>!e7p_FN$\"35+++6NS'f#FN7$$!36+++]\"3X8DFN7$$! 3%******RjmIO&FN$\"3))*****4+;%HCFN7$$!3R+++X+_3aFN$\"3@+++^$>VM#FN7$$ !3))*****H=wGX&FN$\"3?+++c`j#*4<# FN7$$!3')******=X,QbFN$\"3'******z@uF3#FN7$$!37+++%)*\\'ybFN$\"3#***** *HA=N*>FN7$$!3*******H#p*yh&FN$\"3++++!y=K!>FN7$$!35+++k4obcFN$\"3'*** ***pBn=\"=FN7$$!3')*****poG>p&FN$\"3*)*****R``%>+++&4ols&FN$ \"3$******Hqnfi\"FN7$$!3e*****\\%*G&fdFN$\"3'******4Y,9`\"FN7$$!3#)*** **f'Hu!z&FN$\"3%*******R)\\dV\"FN7$$!3q*****z,W,#eFN$\"34+++SG,R8FN7$$ !3)******z?ow%eFN$\"3#******f/(>T7FN7$$!3g*****\\-aK(eFN$\"3*)*****\\? 4?N*fFN$\"38+++M9,RUF-7$$!3.+++hgv,gFN$\"3%)* ****H$e+%=$F-7$$!37+++Epl2gFN$\"3()******p+'\\7#F-7$$!3\"******pt/7,'F N$\"3++++!zsJ1\"F-7$$!3=+++l&)Q7gFNF(7$F^cy$!3++++!zsJ1\"F-7$Fiby$!3() ******p+'\\7#F-7$Fdby$!3%)*****H$e+%=$F-7$F_by$!38+++M9,RUF-7$Fjay$!3W *****RNg()G&F-7$Feay$!3=+++%pQ@L'F-7$F`ay$!3)******Rk]\"otF-7$F[ay$!3; +++*eGfR)F-7$Ff`y$!3\\******4zt9%*F-7$Fa`y$!32+++&o(RU5FN7$F\\`y$!3*)* ****\\?T7FN7$Fb_y$!34+++SG,R8FN7$F]_y$!3%** *****R)\\dV\"FN7$Fh^y$!3'******4Y,9`\"FN7$Fc^y$!3$******Hqnfi\"FN7$F^^ y$!3*)*****R``%>FN7$F_] y$!3#******HA=N*>FN7$Fj\\y$!3'******z@uF3#FN7$Fe\\y$!3#******>j#*4<#FN 7$F`\\y$!3?+++c`VM#FN7$Ff[y$!3))*****4+;%HCFN7$F a[y$!3()*****>\"3X8DFN7$F\\[y$!35+++6NS'f#FN7$Fgjx$!3!)*****4b[#yEFN7$ Fbjx$!3;+++8g&*eFFN7$F]jx$!37+++WG\\QGFN7$Fhix$!3%)*****>OCo\"HFN7$Fci x$!3!******z\"f\"R*HFN7$F^ix$!3=+++$fM(pIFN7$Fihx$!3!)*****\\$3DWJFN7$ Fdhx$!36+++G*Ru@$FN7$F_hx$!3y*****>C$G*G$FN7$Fjgx$!3#*******=#p(fLFN7$ Fegx$!3\")*****H4%*)GMFN7$F`gx$!3;+++lBm'\\$FN7$F[gx$!3A+++jq3jNFN7$Ff fx$!3:+++>)*=GOFN7$Fafx$!3;+++03+#p$FN7$F\\fx$!3?+++T&eXv$FN7$Fgex$!3 \"******Re4f\"QFN7$Fbex$!3!*******p!3h(QFN7$F]ex$!3'******\\1:_$RFN7$F hdx$!3%******Hl(H$*RFN7$Fcdx$!33+++XwU]SFN7$F^dx$!3++++w(zm5%FN7$Ficx$ !3e*****4EH@;%FN7$Fdcx$!3++++^&[o@%FN7$F_cx$!3))*****zA.4F%FN7$Fjbx$!3 ')*****z0ZVK%FN7$Febx$!3%)*****zR;sP%FN7$F`bx$!3@+++8W_HWFN7$F[bx$!3;+ ++2eD\"[%FN7$Ffax$!3x*****R(HOKXFN7$Faax$!3G+++=Xw#e%FN7$F\\ax$!3g**** *H&oMKYFN7$Fg`x$!3C+++(3p4o%FN7$Fb`x$!3F+++:2ZGZFN7$F]`x$!3%*******R5o uZFN7$Fh_x$!38++++*G%>[FN7$Fc_x$!39+++:8bi[FN7$F^_x$!3V++++,!R!\\FN7$F i^x$!3\")******=gMV\\FN7$Fd^x$!3/+++c0y!)\\FN7$F_^x$!3.+++sd6;]FN7$Fj] x$!3#)*****\\!GG\\]FN7$Fe]x$!3A+++C'H-3&FN7$F`]x$!3U+++)\\=*3^FN7$F[]x $!35+++`NKN^FN7$Ff\\x$!39+++b'G%f^FN7$Fa\\x$!3#******pwD7=&FN7$F\\\\x$ !35+++?Rr+_FN7$Fg[x$!3u*****z())*y@&FN7$Fb[x$!3l*****4m$zK_FN7$F][x$!3 5+++\\,UX_FN7$Fhjw$!3))*****H;7eD&FN7$Fcjw$!3w******o1-k_FN7$F^jw$!3M+ ++aA7q_FN7$Fiiw$!3U+++eGBu_FN7$Fdiw$!31+++`.`w_FN7$F_iw$!3g*****pw#Hx_ FN7$Fjhw$!35+++8a'pF&FN7$Fehw$!3o*****fN%Gw_FN7$F`hw$!37+++yZ^w_FN7$F[ hw$!3q******4#H*z_FN7$Ffgw$!3)******>WU2H&FN7$$!3i*****fL:5a)F-$!3k*** **zmsjJ&FN7$F\\gw$!3I+++S`Hl`FN7$Fgfw$!3T+++ob,MaFN7$Fbfw$!3C+++=DX0bF N7$F]fw$!3!******zbd!pbFN7$Fhew$!3$******RswHi&FN7$Fcew$!33+++5v6ocFN7 $F^ew$!3))*****>=Odq&FN7$Fidw$!3z*****zH/pt&FN7$Fddw$!3y*****44cCw&FN7 $F_dw$!3#******f`HIy&FN7$Fjcw$!3Q+++HQ6*z&FN7$Fecw$!3n*****4;(36eFN7$F `cw$!3t*****zNW#>eFN7$F[cw$!3w*****pq;Q#eFN7$Ffbw$!3%******R@')\\#eFN7 $Fabw$!3>+++Pf*G#eFN7$F\\bw$!3I+++xvlPdCg&FN7$Fe^w$!3')** ***p&eqkbFN7$F`^w$!3y*****>FpQ_&FN7$F[^w$!3?+++XQ()zaFN7$Ff]w$!3!)**** *R,BEV&FN7$Fa]w$!35+++u`)>Q&FN7$F\\]w$!3k*****RktxK&FN7$Fg\\w$!3I+++Sx qp_FN7$Fb\\w$!39+++*[[t?&FN7$F]\\w$!3#)*****\\Zu*R^FN7$Fh[w$!3e*****\\ SVj1&FN7$Fc[w$!3s******>J@%)\\FN7$F^[w$!3I+++PcS*)[FN7$Fijv$!3#******> ,XXx%FN7$Fdjv$!3k******GH?JYFN7$F_jv$!3/+++'eEHY%FN7$Fjiv$!3-+++(pboG% FN7$Feiv$!3n******4pg8TFN7$F`iv$!3;+++:[6XRFN7$F[iv$!3#*******)*)o0y$F N7$Ffhv$!3(******4OH)=OFN7$Fahv$!3%******R3i*eMFN7$F\\hv$!3z*****zG!H+ LFN7$Fggv$!34+++!p\\B9$FN7$Fbgv$!3/+++(yM[)HFN7$F]gv$!3;+++#H`v#GFN7$$ \"3++++Mm'HK#F\\o$!3'******zX!RqEFN7$$\"3w*****>CY`o$F\\o$!3$)******)> #G8DFN7$F^fv$!3y*****4k&>cBFN7$Fiev$!3'******zw:\"*>#FN7$Fdev$!3&)**** *>&o.U?FN7$$\"3l+++U#QCt*FR$!31+++!Rd\\)=FN7$Fjdv$!3++++7u(ys\"FN7$$\" 3!*******pQ-*H#FR$!31+++pszq:FN7$$\"3#******p4F$y#*FF$!34+++,sr89FN7$F [dv$!3-+++!HPmD\"FN7$$\"3a*****zV$fO%*F@$!3#******H^d&*4\"FN7$$\"30+++ x@#G?#F@$!3-+++6#yZU*F-7$$\"3A+++/'z&>QF:$!3E+++*o\")R&yF-7$$\"3*)**** *4nK.M%F4$!3a+++:`=$G'F-7$F(Fj[pF\\\\pF_\\pF'-Fc\\p6&Fe\\p$\"#X!\"#F(F (-Fj\\p6^_l7%F'F]]p7$$!#YFh\\pF(7%F]]pFe]pFfb[l7%Fe]p7$F(F\\^pFfb[l7%F [c[l7$$\"+2FLSVFfr$\"+:`=$G'F`]pFfb[l7%F]c[l7$$\"+7'z&>QF-$\"+*o\")R&y F`]pFfb[l7%Fcc[l7$$\"+z@#G?#FN$\"+6#yZU*F`]pFfb[l7%Fic[l7$$\"+VMfO%*FN $\"+8vb*4\"Fg_pFfb[l7%F_d[l7$$\"+E$zGB$F^_p$\"+!HPmD\"Fg_pFfb[l7%Fed[l 7$$\"+!GF$y#*F^_p$\"+,sr89Fg_pFfb[l7%F[e[l7$$\"+mQ-*H#F\\`p$\"+pszq:Fg _pFfb[l7%Fae[l7$$\"+r+:9]F\\`p$\"+7u(ys\"Fg_pFfb[l7%Fge[l7$$\"+!HQCt*F \\`p$\"+!Rd\\)=Fg_pFfb[l7%F]f[l7$$\"+Eee'o\"F_ap$\"+_o.U?Fg_pFfb[l7%Fc f[l7$$\"+&HP>f#F_ap$\"+od6*>#Fg_pFfb[l7%Fif[l7$$\"+]`/YMF_ap$\"+Tc>cBF g_pFfb[l7%F_g[l7$$\"+TiM&o$F_ap$\"+*>#G8DFg_pFfb[l7%Feg[l7$$\"+Nm'HK#F _ap$\"+e/RqEFg_pFfb[l7%F[h[l7$$!+6YlH?F_ap$\"+#H`v#GFg_pFfb[l7%Fah[l7$ $!+08n$3\"Fhbp$\"+(yM[)HFg_pFfb[l7%Fgh[l7$$!+n$H;Z#Fhbp$\"+!p\\B9$Fg_p Ffb[l7%F]i[l7$$!++\"=j9%Fhbp$\"+)G!H+LFg_pFfb[l7%Fci[l7$$!+Rs)eA&Fhbp$ \"+%3i*eMFg_pFfb[l7%Fii[l7$$!+3nbpMFhbp$\"+h$H)=OFg_pFfb[l7%F_j[l7$$\" +szT:gFhbp$\"+**)o0y$Fg_pFfb[l7%Fej[l7$$\"+1!QiK$Fadp$\"+:[6XRFg_pFfb[ l7%F[[\\l7$$\"+*GTI%)*Fadp$\"+5pg8TFg_pFfb[l7%Fa[\\l7$$\"+)fu[T#Fdep$ \"+(pboG%Fg_pFfb[l7%Fg[\\l7$$\"+3:KT`Fdep$\"+'eEHY%Fg_pFfb[l7%F]\\\\l7 $$\"+bL7]5F`]p$\"+HH?JYFg_pFfb[l7%Fc\\\\l7$$\"+5`\")H$RF%F`]p$\"+u`)>Q&Fg_pFfb[l7%F i_\\l7$$\"+W?7hUF`]p$\"+9IiKaFg_pFfb[l7%F_`\\l7$$\"+EaO.UF`]p$\"+XQ()z aFg_pFfb[l7%Fe`\\l7$$\"+')o(e5%F`]p$\"+s#pQ_&Fg_pFfb[l7%F[a\\l7$$\"+UA 7tRF`]p$\"+deqkbFg_pFfb[l7%Faa\\l7$$\"+^s&*3QF`]p$\"+stX-cFg_pFfb[l7%F ga\\l7$$\"+X:t;OF`]p$\"+Y;=PcFg_pFfb[l7%F]b\\l7$$\"+t%o$*R$F`]p$\"+4V# *ocFg_pFfb[l7%Fcb\\l7$$\"+#)zUfJF`]p$\"+q1s(p&Fg_pFfb[l7%Fib\\l7$$\"+a r:**GF`]p$\"+SkfBdFg_pFfb[l7%F_c\\l7$$\"+o8`?EF`]p$\"+OxcYdFg_pFfb[l7% Fec\\l7$$\"+jzGDBF`]p$\"+)eSmw&Fg_pFfb[l7%F[d\\l7$$\"+(f`\\,#F`]p$\"+& 35Qy&Fg_pFfb[l7%Fad\\l7$$\"+pf'3p\"F`]p$\"+y\"f!)z&Fg_pFfb[l7%Fgd\\l7$ $\"+y,>a8F`]p$\"+PrN4eFg_pFfb[l7%F]e\\l7$$\"+/'Hf+\"F`]p$\"+xvl3)yFFdep $\"+9i)\\#eFg_pFfb[l7%F_f\\l7$$!+[*Rq+\"Fdep$\"+2n\"Q#eFg_pFfb[l7%Fef \\l7$$!+,L.&)[Fdep$\"+eVC>eFg_pFfb[l7%F[g\\l7$$!+KhOa))Fdep$\"+hr36eFg _pFfb[l7%Fag\\l7$$!+()pt\"H\"F`]p$\"+HQ6*z&Fg_pFfb[l7%Fgg\\l7$$!+%=u!3 NF`]p$\"+5v6ocFg_pFfb[l7%Fei\\l7$$!+ZAJRSF`]p $\"+Cn(Hi&Fg_pFfb[l7%F[j\\l7$$!+_asFg_p$\"+j@\"eD&Fg_pFfb[l7%F[`]l7$$!+WK>0?Fg_p$\"+\\ ,UX_Fg_pFfb[l7%Fa`]l7$$!+$4.*y?Fg_p$\"+hOzK_Fg_pFfb[l7%Fg`]l7$$!+E$>7: #Fg_p$\"+y))*y@&Fg_pFfb[l7%F]a]l7$$!+N+MAAFg_p$\"+?Rr+_Fg_pFfb[l7%Fca] l7$$!+k8W#H#Fg_p$\"+ndA\"=&Fg_pFfb[l7%Fia]l7$$!+FLohBFg_p$\"+b'G%f^Fg_ pFfb[l7%F_b]l7$$!++W@ICFg_p$\"+`NKN^Fg_pFfb[l7%Feb]l7$$!+(3v\")\\#Fg_p $\"+)\\=*3^Fg_pFfb[l7%F[c]l7$$!+\"f+dc#Fg_p$\"+C'H-3&Fg_pFfb[l7%Fac]l7 $$!+#fAHj#Fg_p$\"+0GG\\]Fg_pFfb[l7%Fgc]l7$$!+q+(**p#Fg_p$\"+sd6;]Fg_pF fb[l7%F]d]l7$$!+t!ppw#Fg_p$\"+c0y!)\\Fg_pFfb[l7%Fcd]l7$$!+h7/MGFg_p$\" +>gMV\\Fg_pFfb[l7%Fid]l7$$!+H6I,HFg_p$\"++,!R!\\Fg_pFfb[l7%F_e]l7$$!+t =&)oHFg_p$\"+:8bi[Fg_pFfb[l7%Fee]l7$$!+%3!yOIFg_p$\"++*G%>[Fg_pFfb[l7% F[f]l7$$!+-(\\^5$Fg_p$\"+S5ouZFg_pFfb[l7%Faf]l7$$!+7g*R<$Fg_p$\"+:2ZGZ Fg_pFfb[l7%Fgf]l7$$!+92KVKFg_p$\"+(3p4o%Fg_pFfb[l7%F]g]l7$$!+v\"*38LFg _p$\"+`oMKYFg_pFfb[l7%Fcg]l7$$!+P2B$Q$Fg_p$\"+=Xw#e%Fg_pFfb[l7%Fig]l7$ $!+UCk`MFg_p$\"+uHOKXFg_pFfb[l7%F_h]l7$$!+7e>CNFg_p$\"+2eD\"[%Fg_pFfb[ l7%Feh]l7$$!+&oXZf$Fg_p$\"+8W_HWFg_pFfb[l7%F[i]l7$$!+s(R^m$Fg_p$\"+)R; sP%Fg_pFfb[l7%Fai]l7$$!+'fF_t$Fg_p$\"+eqMCVFg_pFfb[l7%Fgi]l7$$!+Ny'[!Q Fg_p$\"+GK!4F%Fg_pFfb[l7%F]j]l7$$!+'eLR(QFg_p$\"+^&[o@%Fg_pFfb[l7%Fcj] l7$$!+*[:B%RFg_p$\"+h#H@;%Fg_pFfb[l7%Fij]l7$$!+>I#*4SFg_p$\"+w(zm5%Fg_ pFfb[l7%F_[^l7$$!+`VowSFg_p$\"+XwU]SFg_pFfb[l7%Fe[^l7$$!+!HXD9%Fg_p$\" +`wH$*RFg_pFfb[l7%F[\\^l7$$!+7wY2UFg_p$\"+l]@NRFg_pFfb[l7%Fa\\^l7$$!+% =F9F%Fg_p$\"+q!3h(QFg_pFfb[l7%Fg\\^l7$$!+**>TMVFg_p$\"+%e4f\"QFg_pFfb[ l7%F]]^l7$$!+z-U'R%Fg_p$\"+T&eXv$Fg_pFfb[l7%Fc]^l7$$!+)peuX%Fg_p$\"+03 +#p$Fg_pFfb[l7%Fi]^l7$$!+22a)*=GOFg_pFfb[l7%F_^^l7$$!+\")\\ owXFg_p$\"+jq3jNFg_pFfb[l7%Fe^^l7$$!+;R\"\\j%Fg_p$\"+lBm'\\$Fg_pFfb[l7 %F[_^l7$$!+n@D#p%Fg_p$\"+$4%*)GMFg_pFfb[l7%Fa_^l7$$!+%=D([ZFg_p$\"+>#p (fLFg_pFfb[l7%Fg_^l7$$!+DxN/[Fg_p$\"+UKG*G$Fg_pFfb[l7%F]`^l7$$!+^BVM#Fg_pFfb[l7%Fed^l7$$!+$=wGX&Fg_p $\"+c`X,QbFg_p$\"+=Ux#3#Fg_pFfb[l7%Fge^l7$$!+%)*\\'ybFg_p$\"+B#=N*>Fg_pFf b[l7%F]f^l7$$!+Bp*yh&Fg_p$\"+!y=K!>Fg_pFfb[l7%Fcf^l7$$!+k4obcFg_p$\"+P s'=\"=Fg_pFfb[l7%Fif^l7$$!+(oG>p&Fg_p$\"+MNX>T7Fg_pFfb[l7%F]i^l7$$!+DSDteFg_p $\"+0sJU6Fg_pFfb[l7%Fci^l7$$!+*\\Uo*eFg_p$\"+&o(RU5Fg_pFfb[l7%Fii^l7$$ !+HuP=fFg_p$\"+5zt9%*F`]pFfb[l7%F_j^l7$$!+zc!y$fFg_p$\"+*eGfR)F`]pFfb[ l7%Fej^l7$$!+mv2bfFg_p$\"+W1:otF`]pFfb[l7%F[[_l7$$!+=t9qfFg_p$\"+%pQ@L 'F`]pFfb[l7%Fa[_l7$$!+,O(H)fFg_p$\"+a.w)G&F`]pFfb[l7%Fg[_l7$$!+#4?N*fF g_p$\"+M9,RUF`]pFfb[l7%F]\\_l7$$!+hgv,gFg_p$\"+Le+%=$F`]pFfb[l7%Fc\\_l 7$$!+Epl2gFg_p$\"+q+'\\7#F`]pFfb[l7%Fi\\_l7$$!+PZ?6gFg_p$\"+!zsJ1\"F`] pFfb[l7%F_]_l7$$!+l&)Q7gFg_pF(Ffb[l7%Fe]_l7$F`]_l$!+!zsJ1\"F`]pFfb[l7% Fi]_l7$Fj\\_l$!+q+'\\7#F`]pFfb[l7%F]^_l7$Fd\\_l$!+Le+%=$F`]pFfb[l7%Fa^ _l7$F^\\_l$!+M9,RUF`]pFfb[l7%Fe^_l7$Fh[_l$!+a.w)G&F`]pFfb[l7%Fi^_l7$Fb [_l$!+%pQ@L'F`]pFfb[l7%F]__l7$F\\[_l$!+W1:otF`]pFfb[l7%Fa__l7$Ffj^l$!+ *eGfR)F`]pFfb[l7%Fe__l7$F`j^l$!+5zt9%*F`]pFfb[l7%Fi__l7$Fji^l$!+&o(RU5 Fg_pFfb[l7%F]`_l7$Fdi^l$!+0sJU6Fg_pFfb[l7%Fa`_l7$F^i^l$!+Yq>T7Fg_pFfb[ l7%Fe`_l7$Fhh^l$!+SG,R8Fg_pFfb[l7%Fi`_l7$Fbh^l$!+S)\\dV\"Fg_pFfb[l7%F] a_l7$F\\h^l$!+h9SJ:Fg_pFfb[l7%Faa_l7$Ffg^l$!+.x'fi\"Fg_pFfb[l7%Fea_l7$ F`g^l$!+MNX>Fg_pFfb[l7%Fab_l7$F^f^l$!+B#=N*>Fg_pFfb[l7%Feb_l7$Fhe^l$!+=Ux# 3#Fg_pFfb[l7%Fib_l7$Fbe^l$!+KE*4<#Fg_pFfb[l7%F]c_l7$F\\e^l$!+c`VM#Fg_pFfb[l7%Fec_l7$F`d^l$!+,gTHCFg_pFfb[l7% Fic_l7$Fjc^l$!+73X8DFg_pFfb[l7%F]d_l7$Fdc^l$!+6NS'f#Fg_pFfb[l7%Fad_l7$ F^c^l$!+^&[#yEFg_pFfb[l7%Fed_l7$Fhb^l$!+8g&*eFFg_pFfb[l7%Fid_l7$Fbb^l$ !+WG\\QGFg_pFfb[l7%F]e_l7$F\\b^l$!+iV#o\"HFg_pFfb[l7%Fae_l7$Ffa^l$!+=f \"R*HFg_pFfb[l7%Fee_l7$F`a^l$!+$fM(pIFg_pFfb[l7%Fie_l7$Fj`^l$!+N3DWJFg _pFfb[l7%F]f_l7$Fd`^l$!+G*Ru@$Fg_pFfb[l7%Faf_l7$F^`^l$!+UKG*G$Fg_pFfb[ l7%Fef_l7$Fh_^l$!+>#p(fLFg_pFfb[l7%Fif_l7$Fb_^l$!+$4%*)GMFg_pFfb[l7%F] g_l7$F\\_^l$!+lBm'\\$Fg_pFfb[l7%Fag_l7$Ff^^l$!+jq3jNFg_pFfb[l7%Feg_l7$ F`^^l$!+>)*=GOFg_pFfb[l7%Fig_l7$Fj]^l$!+03+#p$Fg_pFfb[l7%F]h_l7$Fd]^l$ !+T&eXv$Fg_pFfb[l7%Fah_l7$F^]^l$!+%e4f\"QFg_pFfb[l7%Feh_l7$Fh\\^l$!+q! 3h(QFg_pFfb[l7%Fih_l7$Fb\\^l$!+l]@NRFg_pFfb[l7%F]i_l7$F\\\\^l$!+`wH$*R Fg_pFfb[l7%Fai_l7$Ff[^l$!+XwU]SFg_pFfb[l7%Fei_l7$F`[^l$!+w(zm5%Fg_pFfb [l7%Fii_l7$Fjj]l$!+h#H@;%Fg_pFfb[l7%F]j_l7$Fdj]l$!+^&[o@%Fg_pFfb[l7%Fa j_l7$F^j]l$!+GK!4F%Fg_pFfb[l7%Fej_l7$Fhi]l$!+eqMCVFg_pFfb[l7%Fij_l7$Fb i]l$!+)R;sP%Fg_pFfb[l7%F][`l7$F\\i]l$!+8W_HWFg_pFfb[l7%Fa[`l7$Ffh]l$!+ 2eD\"[%Fg_pFfb[l7%Fe[`l7$F`h]l$!+uHOKXFg_pFfb[l7%Fi[`l7$Fjg]l$!+=Xw#e% Fg_pFfb[l7%F]\\`l7$Fdg]l$!+`oMKYFg_pFfb[l7%Fa\\`l7$F^g]l$!+(3p4o%Fg_pF fb[l7%Fe\\`l7$Fhf]l$!+:2ZGZFg_pFfb[l7%Fi\\`l7$Fbf]l$!+S5ouZFg_pFfb[l7% F]]`l7$F\\f]l$!++*G%>[Fg_pFfb[l7%Fa]`l7$Ffe]l$!+:8bi[Fg_pFfb[l7%Fe]`l7 $F`e]l$!++,!R!\\Fg_pFfb[l7%Fi]`l7$Fjd]l$!+>gMV\\Fg_pFfb[l7%F]^`l7$Fdd] l$!+c0y!)\\Fg_pFfb[l7%Fa^`l7$F^d]l$!+sd6;]Fg_pFfb[l7%Fe^`l7$Fhc]l$!+0G G\\]Fg_pFfb[l7%Fi^`l7$Fbc]l$!+C'H-3&Fg_pFfb[l7%F]_`l7$F\\c]l$!+)\\=*3^ Fg_pFfb[l7%Fa_`l7$Ffb]l$!+`NKN^Fg_pFfb[l7%Fe_`l7$F`b]l$!+b'G%f^Fg_pFfb [l7%Fi_`l7$Fja]l$!+ndA\"=&Fg_pFfb[l7%F]``l7$Fda]l$!+?Rr+_Fg_pFfb[l7%Fa ``l7$F^a]l$!+y))*y@&Fg_pFfb[l7%Fe``l7$Fh`]l$!+hOzK_Fg_pFfb[l7%Fi``l7$F b`]l$!+\\,UX_Fg_pFfb[l7%F]a`l7$F\\`]l$!+j@\"eD&Fg_pFfb[l7%Faa`l7$Ff_]l $!+p1-k_Fg_pFfb[l7%Fea`l7$F`_]l$!+aA7q_Fg_pFfb[l7%Fia`l7$Fj^]l$!+eGBu_ Fg_pFfb[l7%F]b`l7$Fd^]l$!+`.`w_Fg_pFfb[l7%Fab`l7$F^^]l$!+nFHx_Fg_pFfb[ l7%Feb`l7$Fh]]l$!+8a'pF&Fg_pFfb[l7%Fib`l7$Fb]]l$!+cVGw_Fg_pFfb[l7%F]c` l7$F\\]]l$!+yZ^w_Fg_pFfb[l7%Fac`l7$Ff\\]l$!+5#H*z_Fg_pFfb[l7%Fec`l7$F` \\]l$!+UCu!H&Fg_pFfb[l7%Fic`l7$$!+O`,T&)F`]p$!+oEP;`Fg_pFfb[l7%F]d`l7$ Fd[]l$!+S`Hl`Fg_pFfb[l7%Fcd`l7$F^[]l$!+ob,MaFg_pFfb[l7%Fgd`l7$Fhj\\l$! +=DX0bFg_pFfb[l7%F[e`l7$Fbj\\l$!+ev0pbFg_pFfb[l7%F_e`l7$F\\j\\l$!+Cn(H i&Fg_pFfb[l7%Fce`l7$Ffi\\l$!+5v6ocFg_pFfb[l7%Fge`l7$F`i\\l$!+#=Odq&Fg_ pFfb[l7%F[f`l7$Fjh\\l$!+)H/pt&Fg_pFfb[l7%F_f`l7$Fdh\\l$!+\"4cCw&Fg_pFf b[l7%Fcf`l7$F^h\\l$!+O&HIy&Fg_pFfb[l7%Fgf`l7$Fhg\\l$!+HQ6*z&Fg_pFfb[l7 %F[g`l7$Fbg\\l$!+hr36eFg_pFfb[l7%F_g`l7$F\\g\\l$!+eVC>eFg_pFfb[l7%Fcg` l7$Fff\\l$!+2n\"Q#eFg_pFfb[l7%Fgg`l7$F`f\\l$!+9i)\\#eFg_pFfb[l7%F[h`l7 $Fje\\l$!+Pf*G#eFg_pFfb[l7%F_h`l7$Fde\\l$!+xvlQ&Fg_pFfb[l7%F_\\al7 $Fd_\\l$!+WOxF`Fg_pFfb[l7%Fc\\al7$F^_\\l$!+Sxqp_Fg_pFfb[l7%Fg\\al7$Fh^ \\l$!+*[[t?&Fg_pFfb[l7%F[]al7$Fb^\\l$!+vW(*R^Fg_pFfb[l7%F_]al7$F\\^\\l $!+0MMm]Fg_pFfb[l7%Fc]al7$Ff]\\l$!+?J@%)\\Fg_pFfb[l7%Fg]al7$F`]\\l$!+P cS*)[Fg_pFfb[l7%F[^al7$Fj\\\\l$!+7]auZFg_pFfb[l7%F_^al7$Fd\\\\l$!+HH?J YFg_pFfb[l7%Fc^al7$F^\\\\l$!+'eEHY%Fg_pFfb[l7%Fg^al7$Fh[\\l$!+(pboG%Fg _pFfb[l7%F[_al7$Fb[\\l$!+5pg8TFg_pFfb[l7%F__al7$F\\[\\l$!+:[6XRFg_pFfb [l7%Fc_al7$Ffj[l$!+**)o0y$Fg_pFfb[l7%Fg_al7$F`j[l$!+h$H)=OFg_pFfb[l7%F [`al7$Fji[l$!+%3i*eMFg_pFfb[l7%F_`al7$Fdi[l$!+)G!H+LFg_pFfb[l7%Fc`al7$ F^i[l$!+!p\\B9$Fg_pFfb[l7%Fg`al7$Fhh[l$!+(yM[)HFg_pFfb[l7%F[aal7$Fbh[l $!+#H`v#GFg_pFfb[l7%F_aal7$$\"+Mm'HK#F_ap$!+e/RqEFg_pFfb[l7%Fcaal7$$\" +UiM&o$F_ap$!+*>#G8DFg_pFfb[l7%Fiaal7$F`g[l$!+Tc>cBFg_pFfb[l7%F_bal7$F jf[l$!+od6*>#Fg_pFfb[l7%Fcbal7$Fdf[l$!+_o.U?Fg_pFfb[l7%Fgbal7$$\"+U#QC t*F\\`p$!+!Rd\\)=Fg_pFfb[l7%F[cal7$Fhe[l$!+7u(ys\"Fg_pFfb[l7%Facal7$$ \"+qQ-*H#F\\`p$!+pszq:Fg_pFfb[l7%Fecal7$$\"+(4F$y#*F^_p$!+,sr89Fg_pFfb [l7%F[dal7$Ffd[l$!+!HPmD\"Fg_pFfb[l7%Fadal7$$\"+QMfO%*FN$!+8vb*4\"Fg_p Ffb[l7%Fedal7$$\"+x@#G?#FN$!+6#yZU*F`]pFfb[l7%F[eal7$$\"+/'z&>QF-$!+*o \")R&yF`]pFfb[l7%Faeal7$$\"+rELSVFfr$!+:`=$G'F`]pFfb[l7%Fgeal7$F(F_avF fb[l7%F]falFbavFfb[l7%FbavFfavFfb[l7%FfavF'Ffb[l-Fc\\p6&Fe\\p$\"#&*Fbb [l$\"\"$Fh\\pFf\\pF^bv-F$6%7$7$$!3o*************)oFNF(7$$\"3%********* *****=\"FNF(-%'COLOURG6&Fe\\pF)F)F)-%*LINESTYLEG6#Fffal-F$6%7$7$F($!3o *************Q'FN7$F($\"3o*************Q'FNF`galFcgal-%*AXESSTYLEG6#%$ BOXG-%(SCALINGG6#%,CONSTRAINEDG-%+AXESLABELSG6%%&Re(z)G%&Im(z)G-%%FONT G6#%(DEFAULTG-F]ial6$%*HELVETICAG\"\"*-%%VIEWG6$;$!$*oFbb[l$\"$>\"Fbb[ l;$!$R'Fbb[l$\"$R'Fbb[l" 1 2 0 1 10 0 2 9 1 2 1 1.000000 45.000000 45.000000 0 0 "Curve 1" "Curve 2" "Curve 3" "Curve 4" "Curve 5" "Curve 6" }}}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}}{PARA 0 "" 0 "" {TEXT -1 52 "#----------------- ----------------------------------" }}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 35 "coefficients of the combined scheme" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 36 "coefficients of the combined scheme " }} {PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 5959 "ee := \{c[2]=27/400,\nc[3]=469/4510,\nc[4]=1407/9020,\nc[5]=100/ 257,\nc[6]=7/15,\nc[7]=7/45,\nc[8]=58/59,\nc[9]=49117478/55144743,\nc[ 10]=13/14,\nc[11]=62/65,\nc[12]=1,\nc[13]=1,\n\na[2,1]=27/400,\na[3,1] =1311793/54918270,\na[3,2]=439922/5491827,\na[4,1]=1407/36080,\na[4,2] =0,\na[4,3]=4221/36080,\na[5,1]=12985296430100/33603736057857,\na[5,2] =0,\na[5,3]=-16218027650000/11201245352619,\na[5,4]=48744170200000/336 03736057857,\na[6,1]=3817807/81405000,\na[6,2]=0,\na[6,3]=0,\na[6,4]=1 0274191312/43991343405,\na[6,5]=40755997793/218862405000,\na[7,1]=8883 007/146529000,\na[7,2]=0,\na[7,3]=0,\na[7,4]=53436069013712/4755024308 64645,\na[7,5]=-4116355777093/117791746371000,\na[7,6]=7635614/4363052 85,\na[8,1]=-25750155116216595121447/19172397107027415328000,\na[8,2]= 0,\na[8,3]=0,\na[8,4]=-743/55,\na[8,5]=-3193438311684528658341050903/3 15992014923229256968928000,\na[8,6]=183334983592060082563257/229301869 40004788732288,\na[8,7]=18586487074230223824062061/1035692891721620976 018560,\na[9,1]=-92212217189471655221780797327793420467234876635197955 0680715901/\n 1574011901986716408455269129129499876539846630029 827642536608000,\na[9,2]=0,\na[9,3]=0,\na[9,4]=-2493924112965609110422 434487743812058912776576445331741068763/\n 24857618804199899936 3340039748303974885433202092103843801085,\na[9,5]=-3474104542529502068 93609060802736441487435576445275533884950354547756949/\n 697473 62766091883481592212536178438630509515521207481681627188467808000,\na[ 9,6]=5327733707939584478612584810530320112314665632405586015822099459/ \n 1249808955205147320198736736321225596828263980093048262260372 352,\na[9,7]=638056475631824285283445623898826208909699333648050259039 16023/\n 5222536880125448091932537212430169113118513294896584316 063360,\na[9,8]=16756974250619222216361657668160522447776912/164976593 0508280715886033484777254665540228007,\na[10,1]=-370292782453410978231 4804940532859074364487/4131257703459996302316696251055190381568000,\na [10,2]=0,\na[10,3]=0,\na[10,4]=-48542494919285774589408977236397/37951 56546073735065277428293920,\na[10,5]=-22919425542674277431989520078944 496953957193/3194620077548089472408777624402038658048000,\na[10,6]=248 4540315797359406715529397919751486619672787/42164869038880297436703627 0632096098190348288,\na[10,7]=1705469761435764254142110563325321064117 5183343/1072180740487803620784937862662845301329326080,\na[10,8]=24246 4961680690545467502075/12396488665185571302473970944,\na[10,9]=-276720 658930244250241932341390181638290085115/966379361550486161719370737230 4668805996412672,\na[11,1]=-888265261492238070708293730279393062686463 0973971/\n 8018042974340129539773365358072400795335265440000, \na[11,2]=0,\na[11,3]=0,\na[11,4]=-91300774029817320031962039591895693 /6641776260885597741930170995083525,\na[11,5]=-11220125505347440190681 6716367850973725245563029/\n 130451942058366052280926706134169 96666049440000,\na[11,6]=183334809040500814662036381242837807618576330 592954213/\n 26459846979744846747689472891117696020043251727064 960,\na[11,7]=5908534443979027768339795262286892453548930774924082527/ \n 337801676952094167280622617772889760985970195055625600,\na[1 1,8]=6612449503769637840690016046967566/225556868777060284586258384373 942375,\na[11,9]=-1029378427823652249730886986351378997098703174693100 0834/\n 878782636339557288924677338637495764024419774550558618 7675,\na[11,10]=-4215629637956404624565504/108123612591292234376500125 ,\na[12,1]=-535308736842973754888311604833674190949559277/\n 5 07665470526061438136955303900248854246636000,\na[12,2]=0,\na[12,3]=0, \na[12,4]=-6581385500218652420284298344406312/842464028888378018027275 808653215,\na[12,5]=-229810990628849080329321528372507186906863097/\n \+ 29548105906232803976194189596626971177604000,\na[12,6]=3336217 43628604464651535937000728865436063541833/\n 540424314302779720 80199874664118662613066091504,\na[12,7]=281436223028901228360767240757 054593496984174199893/\n 24784655195828222920388514383970641394 482546714960,\na[12,8]=-195816087241661023276342391/399896648012674685 277875106,\na[12,9]=14464527008223808452459454468398839957564434825455 1267965/\n 3148564668287108076358894831813590230011876033218706 39973,\na[12,10]=-113365555877618519272588672/930915042822320880104687 79,\na[12,11]=9008463031864887557094375/6606862347800680584331678,\na[ 13,1]=-35556891109301815175584262675897743594621/233372257669869909529 29827252863229628000,\na[13,2]=0,\na[13,3]=0,\na[13,4]=-15763198989262 628277502495250456/1200562192841616547810297566045,\na[13,5]=-47449922 9690075043966853347490522346181911/\n 421078379665790298347783 87168434144652000,\na[13,6]=679545987701237138790035764835967076741310 679/\n 77013733239195245746002612595738122195430352,\na[13,7]=1 368817102190976218912189847996057845647169577/\n 75630904954724 828436404430778019911819343440,\na[13,8]=-3292496872165727322673258/59 253630028479896512047753,\na[13,9]=-2684522309490576529735302727427307 676030962009390/\n 3462833663065024895730647437447934056637412 8519867, a[13,10]=64631913714031794733568/486423753624302998802649,\na [13,11]=0,\na[13,12]=0,\n\nb[1]=1049982578583103/23627320335287760,\nb [2]=0,\nb[3]=0,\nb[4]=0,\nb[5]=0,\nb[6]=768239620641637875/21616768804 69378496,\nb[7]=7884214609987698755625/31681975342943333980208,\nb[8]= -19726902451409487557291/9259475236790114044800,\nb[9]=229854489772575 8629882107777744216351971045149850043139161/\n 7323526619959521436 76542507482891823353467854709785979600,\nb[10]=-2307497857833742448/27 8745570101106225,\nb[11]=148963664643228251375/20207195606693950848,\n b[12]=576091109999/2308683585600,\nb[13]=0,\n\n`b*`[1]=4850006989829/1 08881660531280,\n`b*`[2]=0,\n`b*`[3]=0,\n`b*`[4]=0,\n`b*`[5]=0,\n`b*`[ 6]=5801298400186875/16253209627589312,\n`b*`[7]=2408095926060530625/96 91641279578872432,\n`b*`[8]=55006229857804792033/15747406865289309600, \n`b*`[9]=363484892569724108095352508636899702948448557463/\n 1 60778106318887934237641558033288694994268792800,\n`b*`[10]=-1419773382 7415152/4039790871030525,\n`b*`[11]=0,\n`b*`[12]=0,\n`b*`[13]=-9030611 407/4779883200\}:" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 " > " 0 "" {MPLTEXT 1 0 1 ";" }}}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT 259 5 "nodes" }{TEXT -1 2 ": " }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 32 "seq(c[i]=subs(ee,c[i]),i=2..13);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6./&%\"cG6#\"\"##\"#F\"$+%/&F%6#\"\"$#\"$p%\"%5X/&F% 6#\"\"%#\"%29\"%?!*/&F%6#\"\"&#\"$+\"\"$d#/&F%6#\"\"'#\"\"(\"#:/&F%6#F E#FE\"#X/&F%6#\"\")#\"#e\"#f/&F%6#\"\"*#\")yu6\\\")VZ9b/&F%6#\"#5#\"#8 \"#9/&F%6#\"#6#\"#i\"#l/&F%6#\"#7\"\"\"/&F%6#FinFfo" }}}{PARA 0 "" 0 " " {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT 259 20 "linking coefficients" }{TEXT -1 1 ":" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 50 "seq(seq(a[ i,j]=subs(ee,a[i,j]),j=1..i-1),i=1..13);" }}{PARA 12 "" 1 "" {XPPMATH 20 "6jo/&%\"aG6$\"\"#\"\"\"#\"#F\"$+%/&F%6$\"\"$F(#\"($z68\")q#=\\&/&F %6$F/F'#\"'A*R%\"(F=\\&/&F%6$\"\"%F(#\"%29\"&!3O/&F%6$FENX7?6/&F%6$FLF<#\"/++?qTu[FO/&F%6$\"\"'F(#\"(2y\"Q\") +]S\")/&F%6$F[oF'FC/&F%6$F[oF/FC/&F%6$F[oF<#\",78>u-\"\",0MM\"*R%/&F%6 $F[oFL#\",$z(*fvS\"-+]Si)=#/&F%6$\"\"(F(#\"(2I)))\"*+!Hl9/&F%6$FdpF'FC /&F%6$FdpF/FC/&F%6$FdpF<#\"/7P,pgV`\"0XY'3V-bZ/&F%6$FdpFL#!.$4xdN;T\"0 +5PY/&F%6$FcrF'FC/&F%6$FcrF/FC/&F%6$FcrF<#!$V(\"#b/&F%6$FcrFL# !=.40T$e'GXo6$QM>$\"<+!G*opDHK#\\,#*fJ/&F%6$FcrF[o#\"9dKc#3g?f$)\\L$= \"8)GK()y/+%p=IH#/&F%6$FcrFdp#\";h?1CQAIU2(['e=\":g&=g(4i@<*GpN5/&F%6$ \"\"*F(#!jn,fr!o]&z>Nm([Bn/U$zFtz!y@_lr%*=!>,u:/&F%6$FhtF'FC/&F%6$FhtF/FC/&F%6$FhtF<#!hnj(o5uJ` Wwlx7*e?\"Qu([MCU5\"4c'H6CR\\#\"gn&3,Q%Q5#4-KV&)[(RI[(R+Mj$***)*>/)=w& [#/&F%6$FhtFL#!co\\pvZXN]\\)Q`v_WwbV([Tkt-314O*o?]HDa/TZ$\"bo+!3yY)=F; o\"[27_:&40jQ%yh`7Af\"[$)=4mFOZ(p/&F%6$FhtF[o#\"[of%*4Ae,'e0CjlYJ7,KI0 \"[e7'yWeRzqLxK&\"[o_BPgAE[I4!)REGofD7KOnt)>?t90_&*3)\\7/&F%6$FhtFdp# \"inBg\"R!f-0[OL*p4*3i#))*QiXMG&GC=jvk0Q'\"hngL1;Ve'*[H8&=J6p,V7s`K>4[ a7!)o`A_/&F%6$FhtFcr#\"M7pxZC_g\"owlhj@A#>1Dupv;\"O2!G-alYDxZ[Lg)er!G3 0$fw\\;/&F%6$\"#5F(#!L([kV2fG`S\\![J#y4T`Cy#Hq$\"L+!o:Q!>b5D'p;BI'**fM qd78%/&F%6$FcwF'FC/&F%6$FcwF/FC/&F%6$FcwF<#!A(ROs(*3%*eudG>\\\\U&[\"@? RHGuFl]ttgac^z$/&F%6$FcwFL#!M$>dR&p\\W*y+_*)>VxUnUbU>H#\"L+![!e'Q?SCwx 3CZ*3[v2?Y>$/&F%6$FcwF[o#\"O(ys'>m[^(>zRHbr1%ftz:.a%[#\"N)G[.>)4'4K1FO qOuH!))Q!p[;U/&F%6$FcwFdp#\"PVL=v6k5KDLc5@9aUwN9wpaq\"\"O!3E$H8IXGmiy$ \\y?O!y[S2=s5/&F%6$FcwFcr#\"W4(RZ-8d&=l')['R7/&F%6 $FcwFht#!N:^3!HQ;=!RTB$>C]UCI*e1sw#\"OsET'*f!)oYIstq$><;'[]:Ozj'*/&F%6 $\"#6F(#!RrR(4jkoiIRz-t$H322QA\\h_E)))\"R++Wl_L&z+C2e`Ot(R&H,MuH/=!)/& F%6$FdzF'FC/&F%6$FdzF/FC/&F%6$FdzF<#!D$p&*=fR?'>.?t\")HSx+8*\"CDN3&*4< I>u(f&)3EwSuM0b7?7\"\"P++W\\gm'*pT8 1n#4G_gOe?%>XI\"/&F%6$FdzF[o#\"W8U&HfIjd=w!y$GC\"QO?m93]S!4[L$=\"Vg\\1 Fqf)4w*)GxuO$[g6$))[vtH%ot3``\"N+gjYU&) [-!RIbp8Q91E0Zlw]/&F%6$F[^lF'FC/&F%6$F[^lF/FC/&F%6$F[^lF<#!C7jSW$)H%G? Cl=-]&Q\"e'\"B:Kl3eFF!=!y$)))GSYU)/&F%6$F[^lFL#!N(4jo!p=2DPG:KH.3\\)G1 *4\")H#\"M+Sgx6(pi'f*=%>wR!GB1f5[&H/&F%6$F[^lF[o#\"QL=ajgVl)G2+Pf`^YY/ 'GOu@OL\"P/:4mIhi'=Tmu)*>!3szFI9VUS&/&F%6$F[^lFdp#\"T$*)*>uT)p\\$faqvS swg$G7!*GIAO9G\"Sg\\rYD[%RT1(RQ9&)Q?HAGe>bYyC/&F%6$F[^lFcr#!<\"RUjFB5m Ts3;e>\"<1^(yF&ouE,[m*)*R/&F%6$F[^lFht#\"ZlzE^XD[Vkv&*R))RoWXfCX3QA3q_ kW\"\"Zt*R1(=K.w=,I-f8=$[*)ej23rGoYc[J/&F%6$F[^lFcw#!&=w(eblL6 \";z(o/,)3KAG/:4$*/&F%6$F[^lFdz#\":vV4dv)['=.j%3!*\":y;L%e!o+yMiog'/&F %6$\"#8F(#!J@YfVx*eniUev^\"=I46*obN\"J+!G'HK'GDF)HH&4*p)pwDsLB/&F%6$Fh alF'FC/&F%6$FhalF/FC/&F%6$FhalF<#!Ac/D&\\-vFGEE*)*)>jd\"\"@Xgc(H5ya;;% G>i0?\"/&F%6$FhalFL#!K6>=YB_!\\ZL&o'R/v+pH#*\\u%\"J+?lWTVorQyZ$)H!zl'z $y5U/&F%6$FhalF[o#\"Nz1JTn2nf$[wN+zQrB,x)faz'\"M_.V&>A\"Qdf7E+YdC&>RKt 8q(/&F%6$FhalFdp#\"Ox&prkXy0'*z%)*=7*=i(4>-r\")o8\"MSMM>=\"*>!y2V/kVG[ sa\\!4jv/&F%6$FhalFcr#!:eKnAtsl@(o\\#H$\";`x/7l*)z%G+j`#f/&F%6$FhalFht #!R!R4?'4.wwIFus-`tHld!\\4B_%o#\"Sn)>>Pm0MzWPukId*[-lImLGY$/&F%6$Fha lFcw#\"8oNt%zJSr8>jk\"9\\E!))*HICOvBk[/&F%6$FhalFdzFC/&F%6$FhalF[^lFC " }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT 259 39 "wei ghts for the 13 stage order 8 scheme" }{TEXT -1 1 ":" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 32 "seq(b[i]=subs(ee,b[i]),i=1..13);" }} {PARA 12 "" 1 "" {XPPMATH 20 "6//&%\"bG6#\"\"\"#\"1.JeyD)*\\5\"2gxGN.K FO#/&F%6#\"\"#\"\"!/&F%6#\"\"$F//&F%6#\"\"%F//&F%6#\"\"&F//&F%6#\"\"'# \"3vyjT1iR#o(\"4'\\y$p/)onh@/&F%6#\"\"(#\"7Dcv)p()*4Y@%)y\"83-)RLVHMv> oJ/&F%6#\"\")#!8\"Hdv[49X-ps>\"7+[/9,zO_Zf#*/&F%6#\"\"*#\"enh\"RJ/])\\ ^/r>N;Uuxx5#))H'eds(*[a)H#\"Z+'zfy4Z&yY`L#=*G[2DawO9_f*>m_Bt/&F%6#\"#5 #!4[CuLy&y\\2B\"3Di5,,dX(y#/&F%6#\"#6#\"6v8DGKkkO'*[\"\"5[3&Rp1c>2-#/& F%6#\"#7#\"-***464w&\".+ce$o3B/&F%6#\"#8F/" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT 259 39 "weights for the 13 stage order 7 scheme" }{TEXT -1 1 ":" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 38 "seq(`b*`[i]=subs(ee,`b*`[i]),i=1..13);" }}{PARA 12 "" 1 "" {XPPMATH 20 "6//&%#b*G6#\"\"\"#\".H)*)p+][\"0!GJ0m\"))3\"/&F%6#\"\"#\"\"!/&F%6# \"\"$F//&F%6#\"\"%F//&F%6#\"\"&F//&F%6#\"\"'#\"1vo=+%)H,e\"27$*eF'4KD; /&F%6#\"\"(#\"4D1`gg#f43C\"4KC()y&z7k\"p*/&F%6#\"\")#\"5L?z/y&)Hi+b\"5 +'4$*GloSZd\"/&F%6#\"\"*#\"Qjub[%[Hq**oj3DN&43TspD*[[j$\"Q+GzoU*\\p)GL !e:kPU$z))=j5y2;/&F%6#\"#5#!2_^TFQt(>9\"1D0.r3zRS/&F%6#\"#6F//&F%6#\"# 7F//&F%6#\"#8#!+29hI!*\"++K))zZ" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}}{PARA 0 "" 0 "" {TEXT -1 52 "#---------------------------------------------------" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}}{PARA 0 "" 0 "" {TEXT -1 34 " #=================================" }}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 36 "a scheme of Tsitouras and Papakostas" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 91 "See: Cheap Error Estimation for Runge-Kutta methods, by Ch. Tsitouras and S.N. Papakostas, " }}{PARA 0 "" 0 "" {TEXT -1 73 " Siam Journal on Scientific Computing, V ol. 20, Issue 6, Nov 1999." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 29 "The nodes of the scheme are: " }}{PARA 256 "" 0 " " {TEXT -1 1 " " }{XPPEDIT 18 0 "c[2] = 9/142;" "6#/&%\"cG6#\"\"#*&\" \"*\"\"\"\"$U\"!\"\"" }{TEXT -1 3 ", " }{XPPEDIT 18 0 "c[3] = 24514/2 38491;" "6#/&%\"cG6#\"\"$*&\"&9X#\"\"\"\"'\"\\Q#!\"\"" }{TEXT -1 3 ", \+ " }{XPPEDIT 18 0 "c[4] = 12257/79497;" "6#/&%\"cG6#\"\"%*&\"&dA\"\"\" \"\"&(\\z!\"\"" }{TEXT -1 3 ", " }{XPPEDIT 18 0 "c[5] = 50/129;" "6#/ &%\"cG6#\"\"&*&\"#]\"\"\"\"$H\"!\"\"" }{TEXT -1 3 ", " }{XPPEDIT 18 0 "c[6] = 34/73;" "6#/&%\"cG6#\"\"'*&\"#M\"\"\"\"#t!\"\"" }{TEXT -1 3 ", " }{XPPEDIT 18 0 "c[7] = 23/148;" "6#/&%\"cG6#\"\"(*&\"#B\"\"\"\"$ [\"!\"\"" }{TEXT -1 3 ", " }{XPPEDIT 18 0 "c[8] = 142/141;" "6#/&%\"c G6#\"\")*&\"$U\"\"\"\"\"$T\"!\"\"" }{TEXT -1 3 ", " }{XPPEDIT 18 0 "c [9] = 29104120198/33218531707" "6#/&%\"cG6#\"\"**&\",)>?T5H\"\"\"\",2< `=K$!\"\"" }{TEXT -1 3 ", " }{XPPEDIT 18 0 "c[10] = 83/91;" "6#/&%\"c G6#\"#5*&\"#$)\"\"\"\"#\"*!\"\"" }{TEXT -1 3 ", " }{XPPEDIT 18 0 "c[1 1] = 143/149;" "6#/&%\"cG6#\"#6*&\"$V\"\"\"\"\"$\\\"!\"\"" }{TEXT -1 3 ", " }{XPPEDIT 18 0 "c[12] = 1" "6#/&%\"cG6#\"#7\"\"\"" }{TEXT -1 3 ", " }{XPPEDIT 18 0 "c[13]=1" "6#/&%\"cG6#\"#8\"\"\"" }{TEXT -1 2 " . " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT 260 4 "Not e" }{TEXT -1 12 ": The node " }{XPPEDIT 18 0 "c[8]=142/141" "6#/&%\"c G6#\"\")*&\"$U\"\"\"\"\"$T\"!\"\"" }{TEXT -1 44 " of this scheme lies outside the interval " }{XPPEDIT 18 0 "[0,1]" "6#7$\"\"!\"\"\"" } {TEXT -1 176 ". This means that the scheme cannot be used in a situati on where a desired numerical solution has a right hand boundary caused , for example, by the slope field being undefined." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}{PARA 0 "" 0 "" {TEXT -1 52 "#---------------------------------------------- -----" }}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 28 "checking the combined sc heme" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 36 "coefficient s of the combined scheme " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 7144 "ee := \{c[2]=9/142,\nc[3]=24514/2 38491,\nc[4]=12257/79497,\nc[5]=50/129,\nc[6]=34/73,\nc[7]=23/148,\nc[ 8]=142/141,\nc[9]=29104120198/33218531707,\nc[10]=83/91,\nc[11]=143/14 9,\nc[12]=1,\nc[13]=1,\n\na[2,1]=9/142,\na[3,1]=9950845450/51190161372 9,\na[3,2]=42666469916/511901613729,\na[4,1]=12257/317988,\na[4,2]=0, \na[4,3]=12257/105996,\na[5,1]=14131686489425/35833975601529,\na[5,2]= 0,\na[5,3]=-17700454220625/11944658533843,\na[5,4]=17619604667500/1194 4658533843,\na[6,1]=4418011/96055225,\na[6,2]=0,\na[6,3]=0,\na[6,4]=97 57757988/41995818067,\na[6,5]=19921512441/106300094275,\na[7,1]=480285 555889619/7997789253816320,\na[7,2]=0,\na[7,3]=0,\na[7,4]=521621065566 96460538643/464887033284472899365888,\na[7,5]=-16450769949384489/49000 9784398315520,\na[7,6]=864813381633887/51718297665732608,\na[8,1]=-700 1048352088587137143/4449830351030385148800,\na[8,2]=0,\na[8,3]=0,\na[8 ,4]=-2522765548294599044197935933/1915963195493758447677901792,\na[8,5 ]=-318556182235222634647116091/27172411532484037214054400,\na[8,6]=120 5563885850790193966807/132365727505912221222912,\na[8,7]=254/39,\na[9, 1]=-206293963997166891228011792644285393944628558742266044635547670708 45753369/\n 428817596627701559566575134700121144880760179811281 05307375594963152353200,\na[9,2]=0,\na[9,3]=0,\na[9,4]=-33154430743436 59404779422149387397712986453181141168247590906370819301077749322753/ \n 498517112641608872807821838566847987729514312398278633788185 835348997643218553476,\na[9,5]=-27374940941165406028694814116482845210 9898379203526945684314474186724062841643/\n 6042758395137755250 3967840653825589117167443021628367691773162913607984889600,\na[9,6]=16 6563725188745127382680605043099249004376722636092450288092298657383277 31797537/\n 4276990138533930522782076385771016009097627930550149 628282203007913538394549504,\na[9,7]=420080800333543055908043229440848 05264441066760038302359736803632/\n 4865302423216534910074823287 811605599629170295030631799935804001,\na[9,8]=668459780930716338000066 627236927417191947396177093524377824/\n 711004529488846437797990 87713322002499799983434685642170251833,\na[10,1]=-17936039463222609008 28212460706877142477132870159527/\n 23130975685119907537816497 19556084665131024900300800,\na[10,2]=0,\na[10,3]=0,\na[10,4]=-14776874 123722838192315406145167687512425345723701/\n 1847893530366076 701102014146927696206329050105856, a[10,5]=-19587020919884661714856757 105130246995757906603/\n 2911893296942532038566182868170393629 789388800,\na[10,6]=63643808632590716771122592364555064774176997806133 00364807/\n 115042817458494213340657909154944343881498834918839 4057728,\na[10,7]=27725164402569748756040320433848245155581006369/\n \+ 2544159473655547770881695354241256106302348256,\na[10,8]=1074424 7163960019876833255044784609639/53476180473990134882549194776830450329 6,\na[10,9]=-509777379307928082322044174972489793998782172800111031978 62899/\n 13009156945649136756132803140818373586449643931913379 94183389184,\na[11,1]=-35876257170689524872144934419668970487370507558 12600710793/\n 30157331640332296247720060864294676850466399837 06974412800,\na[11,2]=0,\na[11,3]=0,\na[11,4]=-54530117112678047312115 01837262816944661201619903/\n 76497332089971639707244864471073 3101329290196992,\na[11,5]=-884348836774584715070440485633026464653487 653/\n 92725983515963799584249219499787659014963200,\na[11,6]= 26823469063654084387375587616552322383082061411417182757389742951/\n \+ 3541299744763681675473620647087123057228744296123642301654630400 , a[11,7]=142363419491686507162007051071007722765323162710521029/\n \+ 12634887202368261565807449771335728082273587905775840,\na[11,8]=6 4747617454909275289531520412519442831235890581/\n 1269317188117 975960996670628974453443777854830080,\na[11,9]=11263380825327272097987 4303367503891597499261046700689572459050065039333987335667/\n 1 4045142912450345328121813771190345010140398305182184650645792916115633 74608650240,\na[11,10]=-10612202518573994431153697720606405883/6708254 6658259846778754594976831647575,\na[12,1]=-753408116554498247829620233 5922049210803875045423/\n 192195756654408487560745980516584163 87002479820800,\na[12,2]=0,\na[12,3]=0,\na[12,4]=237696087452786717802 270375283034262859273455/\n 60688480889936839261131818793519097 177034752,\na[12,5]=-20610578209826329263318986584876108069323/7356333 776438834884293014575840932659200,\na[12,6]=51260471529841028040709654 458903254781320136131844164563/\n 20998023776318546907382302106 788168158765514131585630208,\na[12,7]=-3077214437173472971196810795615 384000211457151011/\n 1272435592582280820597059432060116893200 684680384,\na[12,8]=-1539218116260541896259682954580256454049/45346708 30750360983422999946409310393344,\na[12,9]=241886539350268429372116296 787271276553970618941104594460614948326132797451456131/\n 12406 6963266246589252891612004112305105402628318832478010155534534366231609 0597376,\na[12,10]=-80556486832245966191717452425924975/41444540951867 6597565032008051106461,\na[12,11]=294478168087450034759414279281446335 0/4965161383073676983610218096030654529,\na[13,1]=-7757739937862944832 927743694336116203639371542761/\n 5225100678421794325654850845 451340473577260032000,\na[13,2]=0,\na[13,3]=0,\na[13,4]=-4338895460095 21405913741133329446636837810749/1814887961156430016213106911693222333 46938880,\na[13,5]=-246044720162308748108107126829066792329071/2199910 3311417807170100413255475150848000,\na[13,6]=2140331235425829844389060 818616719848637810765257179167/\n 24542818203601189763802825281 5369258646052549878743040,\na[13,7]=1573990926219809229258666534611598 771063240529/214535514317495538101650508596881057760818080,\na[13,8]=6 2408280667309375445301959066100433563/48383200462519838495123555593274 51645440,\na[13,9]=114560982224949367761811372535950664299815320560322 6883141207089968379/\n 2690280216682276847636742784083574313955 9294105398677975568538466119680,\na[13,10]=-40895035687587468313908967 8053832/7674292714443204070455739595109785,\na[13,11]=0,\na[13,12]=0, \n\nb[1]=55038446513529253801/1239280570055853383520,\nb[2]=0,\nb[3]=0 ,\nb[4]=0,\nb[5]=0,\nb[6]=2335496795323782464411846394611/659836878329 4020109895379936256,\nb[7]=7636073376527143565375240869888/30725949199 261296642046754748645,\nb[8]=-4237087214169934312729607487/12735791394 214625116604076160,\nb[9]=40850529129113324176099551412198433591436392 7884426780078325258228227984174126699/\n 2126248746121936974666551 59405635202123821166475348052734199905546455860773575680,\nb[10]=-1108 225296327029096435947/405679075893979729103310,\nb[11]=246098829120621 3825688467985/1756342789520947764222671739,\nb[12]=4808707937311/50545 545388065,\nb[13]=0,\n\n`b*`[1]=715953338020208413/1609455285786822576 0,\n`b*`[2]=0,\n`b*`[3]=0,\n`b*`[4]=0,\n`b*`[5]=0,\n`b*`[6]=6228433516 2928966987066121/175437206755843240272669696,\n`b*`[7]=184309146777472 302831695872/742417767522160213324368465,\n`b*`[8]=-509771598215811385 123057257/210282269969085698264101760,\n`b*`[9]=2701602489646143640362 891402924962500766379885231830470264478480621389/\n 16885752692 94196295712420798364925687791337062814161130291144634516480,\n`b*`[10] =-7218534073012286740367561/3986456306153224954098780,\n`b*`[11]=0,\n` b*`[12]=0,\n`b*`[13]=5752173075461/1925544586212\}:" }}}{PARA 0 "" 0 " " {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}} {PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 20 "printed coefficients" }} {PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 357 "subs(ee,matrix([seq([c[i],seq(a[i,j],j=1..i-1),``$(7-i)],i=2..7)] ));\nfor ii from 8 to 13 do\n print(``);\n print(c[ii]=subs(ee,c[i i])); \n for jj to ii-1 do\n print(a[ii,jj]=subs(ee,a[ii,jj])); \n end do:\nend do:print(``);\nfor ii to 13 do\n print(b[ii]=subs( ee,b[ii]));\nend do:print(``);\nfor ii to 13 do\n print(`b*`[ii]=sub s(ee,`b*`[ii]));\nend do:" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#K%'matrix G6#7(7)#\"\"*\"$U\"F(%!GF+F+F+F+7)#\"&9X#\"'\"\\Q##\"+]a%3&**\"-HPh,>^ #\",;*pkmUF2F+F+F+F+7)#\"&dA\"\"&(\\z#F7\"'))zJ\"\"!#F7\"''*f5F+F+F+7) #\"#]\"$H\"#\"/D%*['oJT\"\"/H:gvR$e$F;#!/D1Aa/q<\"/VQ`eY%>\"#\"/+vm/'> w\"FGF+F+7)#\"#M\"#t#\"(6!=W\")D_0'*F;F;#\"+))zvd(*\",n!=e*>%#\",TC^@* >\"-vU4+j5F+7)#\"#B\"$[\"#\"0>'*)eb&G![\"1?j\"QD*y(*zF;F;#\"8V'Q0Y'pcl 5i@&\"9))eO**GZ%GLq)[Y#!2*[%Q\\*p2X;\"3?bJ)R%y4+\\#\"0()Qj\"Q8[')\"23E tlwH=<&Q)pprint366\"" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%!G" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/&%\"cG6#\"\")#\"$U\"\"$T\"" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/&%\"aG6$\"\")\"\"\"#!7Vr8(e)3_$[5+(\"7+)[^QI5NI) \\W" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/&%\"aG6$\"\")\"\"#\"\"!" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#/&%\"aG6$\"\")\"\"$\"\"!" }}{PARA 11 " " 1 "" {XPPMATH 20 "6#/&%\"aG6$\"\")\"\"%#!=Lf$z>W!*f%H[bwAD\"=#z,znZ% eP\\&>jf\">" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/&%\"aG6$\"\")\"\"&#!< \"4;rkMEANA=c&=$\";+W09s.%[K:Tsr#" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/ &%\"aG6$\"\")\"\"'#\":2o'R>!z]e)Qc07\"97HA@A\"f]FdOK\"" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/&%\"aG6$\"\")\"\"(#\"$a#\"#R" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%!G" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/&%\"cG6#\"\"*# \",)>?T5H\",2<`=K$" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/&%\"aG6$\"\"*\" \"\"#!eopLvX32nZbjWgEU(e&GY%RR&GWEz6!G7*o;(*R'RH1#\"eo+KN_J'\\fvtI0\"G 6)z,w!)[9@,qM^dm&f:qFmf<)G%" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/&%\"aG 6$\"\"*\"\"#\"\"!" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/&%\"aG6$\"\"*\" \"$\"\"!" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/&%\"aG6$\"\"*\"\"%#!]p`FK \\x2,$>3P14fZ#o69\"=`k)Hr(R(Q\\@UzZSfOMuIW:L\"\\pwMb=Kk(**[`$e=)yL'y#) R7V^Hx)z%ocQ=#y!G()3;k7r^)\\" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/&%\"a G6$\"\"*\"\"&#!ioV;%G1Cn=uWJ%oXp_.#z$)*)4@XG[;T\"[pGgSl6%4%\\PF\"ho+'* )[)zg8H;t>(\\;y`2*>^ov4 8B" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/&%\"aG6$\"#5\"\"#\"\"!" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#/&%\"aG6$\"#5\"\"$\"\"!" }}{PARA 11 " " 1 "" {XPPMATH 20 "6#/&%\"aG6$\"#5\"\"%#!S,PsX`U7von^91aJ#>QGsBT(ox9 \"Rce5]!Hj?'pFp99?5,n2m.`$*y%=" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/&% \"aG6$\"#5\"\"&#!P.m!zv&*pCI^5dn&[rhY))>4-(e>\"O+))Q*yHORq\"oG=m&Q?`Up H$*=\"H" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/&%\"aG6$\"#5\"\"'#\"en2[O+ Lh!y*p+'R;ZUu5\"H'H.XIox%> \\D)[8!*RZ!=wM&" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/&%\"aG6$\"#5\"\"*# !in**G'y>.6,!G<#y)*Rz*[s\\$Rk\\ket$=39. G8cn8\\c%p:4I\"" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%!G" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/&%\"cG6#\"#6#\"$V\"\"$\\\"" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/&%\"aG6$\"#6\"\"\"#!en$z52g7ev]qt[q*o'>W$\\9s[_*oqrDwe $\"en+GTupq$)*Rm/&on%Hk31?xC'HK.kJt:I" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/&%\"aG6$\"#6\"\"#\"\"!" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/&%\"aG6 $\"#6\"\"$\"\"!" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/&%\"aG6$\"#6\"\"%# !R.*>;?hY%p\"GEP=]67t/yE6<,`a\"Q#*p>!HH85L2rW'[C2(R;(**3Kt\\w" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#/&%\"aG6$\"#6\"\"&#!N`w[`YYEIj&[S/2:Ze un$)[V))\"M+K'\\,fwy*\\>#\\Ue*zjf^$)fs#*" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/&%\"aG6$\"#6\"\"'#\"\\o^Hu*QdF=<9Th?3$QABb;wevtQ%3aO1pM#o#\"[o+ /ja;IUO7'HW(Gs0Br3Z1itan\"ojZu*HTN" }}{PARA 11 "" 1 "" {XPPMATH 20 "6# /&%\"aG6$\"#6\"\"(#\"WH5_5F;B`wAx+r502?;2lo\"\\>MOU\"\"VSex0zetA3GdLr( \\u!ech#oB?()[j7" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/&%\"aG6$\"#6\"\") #\"P\"e!*eBJGW>DT?:`*Gv#4\\Xr P\"=7G`M]C\"H9XS\"" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/&%\"aG6$\"#6\"# 5#!G$)eS11s(p`6V%*Rd=D?71\"\"GvvkJo(\\fa(yn%)f#ema#3n" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%!G" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/&%\"cG6#\" #7\"\"\"" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/&%\"aG6$\"#7\"\"\"#!RBa/v Q!3@\\?#fL-iHyC)\\al63Mv\"S+3#)zC+(Q;%e;0)fugv[3Wlcd>#>" }}{PARA 11 " " 1 "" {XPPMATH 20 "6#/&%\"aG6$\"#7\"\"#\"\"!" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/&%\"aG6$\"#7\"\"$\"\"!" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/&%\"aG6$\"#7\"\"%#\"NbMFfGEMIGv.F-yr'y_u3'pP#\"M_Z.xr4>Nz==8h#R o$**)3[)og" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/&%\"aG6$\"#7\"\"&#!JB$p !3h([e')*=LEHj#)4#y0h?\"I+#fE$4%ed9IH%)[$)QkxLjN(" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/&%\"aG6$\"#7\"\"'#\"YjX;W=8O,K\"yaK!*eWl42/G5%)H:Zg7& \"Y3-j&eJT^l(e\"o\")y1@I#Q2pa=jxB!)*4#" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/&%\"aG6$\"#7\"\"(#!R65:d9@+SQ:cz5o>rHZtrV9s2$\"R%Q!o%o+K*o6g?Vf qf?3G#e#fNCF\"" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/&%\"aG6$\"#7\"\")#! I\\SXc-eaHofi*=agi6=#R:\"IWLR5$4k%***HU$)4O]2$3nMX" }}{PARA 11 "" 1 " " {XPPMATH 20 "6#/&%\"aG6$\"#7\"\"*#\"\\pJhX^uzKhK[\\hgWf/6%*=1(Rbw7F( y'H;@PH%o-NRl)=C\"]pwtf!4;BmV`Mb:5!yC$)=$GES0^I7T+7;*GD*eYiEjp1C\"" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#/&%\"aG6$\"#7\"#5#!Dv\\#fU_ur\">mfCKo[ c0)\"Ehk5^!3?.lvfw'=&4aW9%" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/&%\"aG6 $\"#7\"#6#\"F]LY9GzUTfZ.]u3o\"yWH\"FHXlIg4=-h$)pntIQh^'\\" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%!G" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/&%\"cG6 #\"#8\"\"\"" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/&%\"aG6$\"#8\"\"\"#!Rh Far$RO?;hL%pVx#H$[%H'y$*Rxv(\"R+?.gsdt/M^a%3&[lDVz@%y15D_" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/&%\"aG6$\"#8\"\"#\"\"!" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/&%\"aG6$\"#8\"\"$\"\"!" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/&%\"aG6$\"#8\"\"%#!N\\2\"y$ojY%HL8TP\"fS@&4ga*)QV\"N!))QpMLAKp6 p58i,Ik:hz)[\"=" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/&%\"aG6$\"#8\"\"&# !Kr!HBzm!Ho72\"3\"[(3B;?Z/Y#\"J+![3:vaD8/5qr!yT6L5**>#" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/&%\"aG6$\"#8\"\"'#\"Xn\"zrDl2\"yj[)>nh=31*QW)HeUN 7LS@\"WSIuy)\\D0Y'e#p`\"GDG!Qw*=,O?=GaC" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/&%\"aG6$\"#8\"\"(#\"OH0Cj5x)f6Y`m'e#H#4)>i#4*Rd\"\"N!3=3wd5)of3 0l,\"Qb\\IXaP4tm!G3C'\"ISak^uKfbN7&\\Q)>DY+KQ[" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/&%\"aG6$\"#8\"\"*#\"aoz$o**3279$)oA.c?`\")*Hk1&f`s8\"= wn$\\\\A#)4c9\"\"bo!o>hYQ&ob(zn)R0THf&RJuN3%yUnjZoF#o;-G!p#" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/&%\"aG6$\"#8\"#5#!BKQ0y'*3RJoue(oN]*3%\"C&y 4^fRdXqS?VWr#Huw" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/&%\"aG6$\"#8\"#6 \"\"!" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/&%\"aG6$\"#8\"#7\"\"!" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#%!G" }}{PARA 11 "" 1 "" {XPPMATH 20 "6 #/&%\"bG6#\"\"\"#\"5,QDHN^Y%Q]&\"7?NQ`e0q0GR7" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/&%\"bG6#\"\"#\"\"!" }}{PARA 11 "" 1 "" {XPPMATH 20 "6# /&%\"bG6#\"\"$\"\"!" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/&%\"bG6#\"\"% \"\"!" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/&%\"bG6#\"\"&\"\"!" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/&%\"bG6#\"\"'#\"@6YRY=TkCyB`z'\\NB\"@ci$*z` *)4,-%H$yo$)f'" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/&%\"bG6#\"\"(#\"@)) )p3Cv`cVr_wL2Ow\"AX'[ZvY?k'Hh#*>\\fsI" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/&%\"bG6#\"\")#!=([2'HFJM*pT@(3PU\">gh2/m6DY@%R\"zNF\"" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/&%\"bG6#\"\"*#\"\\p*pET<%)zAG#e_Ky+yEW)y#RO9fL %)>79b*4wTK8\"H\"H0&3%\"\\p!ovNxgeXYb!**>MF0[`Zm6#Q7-_j0%f^lmup$>7Y([i 7#" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/&%\"bG6#\"#5#!:ZfV'4HqK'HD#36\" 95L5H(zR*e2zcS" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/&%\"bG6#\"#6#\"=&)z Y)oDQ@17H))4Y#\"=R%Hp_d)o\" " }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/&%#b*G6#\"#5#!:hvOSnG7I2M&=s\":!y )4a\\A`hIck)R" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/&%#b*G6#\"#6\"\"!" } }{PARA 11 "" 1 "" {XPPMATH 20 "6#/&%#b*G6#\"#7\"\"!" }}{PARA 11 "" 1 " " {XPPMATH 20 "6#/&%#b*G6#\"#8#\".ha2t@v&\".7ieWb#>" }}}{PARA 0 "" 0 " " {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}} {PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 151 "RK8_13eqs := [op(RowSumConditions(13,'expanded')),op(OrderConditi ons(8,13,'expanded'))]:\n`RK7_13eqs*` := subs(b=`b*`,OrderConditions(7 ,13,'expanded')):" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 6 "Check:" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 110 "subs( ee,RK8_13eqs):\nmap(u->lhs(u)-rhs(u),%);\nnops(%);\nsubs(ee,`RK7_13eqs *`):\nmap(u->lhs(u)-rhs(u),%);\nnops(%);" }}{PARA 12 "" 1 "" {XPPMATH 20 "6#7`x\"\"!F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$ F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$ F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$ F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$ F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$ F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$ F$F$F$F$F$F$F$F$" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#\"$7#" }}{PARA 12 "" 1 "" {XPPMATH 20 "6#7ap\"\"!F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F $F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F $F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#\"#&)" }}}{PARA 257 "" 0 "" {TEXT -1 0 "" }} {PARA 0 "" 0 "" {TEXT -1 78 "Set-up stage-order condtions to check for stage-orders from 2 to 6 inclusive. " }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 87 "for ct from 2 to 6 do\n so||ct||_13 := StageOrderCo nditions(ct,13,'expanded');\nend do:" }}}{PARA 0 "" 0 "" {TEXT -1 0 " " }}{PARA 0 "" 0 "" {TEXT -1 59 "Stages 3 to 13 have the following res pective stage-orders. " }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 175 "[ seq([seq(expand(subs(ee,so||i||_13[j])),i=2..6)],j=1..11)]:\nmap(proc( L) local i; for i to nops(L) do if not evalb(L[i]) then break end if e nd do; i end proc,%):\nsimplify(%); " }}{PARA 11 "" 1 "" {XPPMATH 20 " 6#7-\"\"#\"\"$F%\"\"%F&F&F&F&F&F&F&" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 27 "The simplifying conditions:" }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "Sum(b[i]*a[i,j],i = j+1 .. \+ 13) = b[j]*(1-c[j]);" "6#/-%$SumG6$*&&%\"bG6#%\"iG\"\"\"&%\"aG6$F+%\"j GF,/F+;,&F0F,F,F,\"#8*&&F)6#F0F,,&F,F,&%\"cG6#F0!\"\"F," }{TEXT -1 3 " , " }{XPPEDIT 18 0 "j=1" "6#/%\"jG\"\"\"" }{TEXT -1 9 " . . 12, " }} {PARA 0 "" 0 "" {TEXT -1 8 "(where " }{XPPEDIT 18 0 "c[1]=0" "6#/&%\" cG6#\"\"\"\"\"!" }{TEXT -1 17 " ) are satisfied." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 151 "[Sum(b[i]*a [i,1],i=1+1..13)=b[1],seq(Sum(b[i]*a[i,j],i=j+1..13)=b[j]*(1-c[j]),j=2 ..12)]:\nmap(u->lhs(u)-rhs(u),subs(ee,eval(subs(Sum=add,%))));\nnops(% );" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#7.\"\"!F$F$F$F$F$F$F$F$F$F$F$" } }{PARA 11 "" 1 "" {XPPMATH 20 "6#\"#7" }}}{PARA 0 "" 0 "" {TEXT -1 0 " " }}{PARA 0 "" 0 "" {TEXT -1 27 "The simplifying conditions:" }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "Sum(`b*`[i]*a[i,j],i = j+1 \+ .. 13) = `b*`[j]*(1-c[j]);" "6#/-%$SumG6$*&&%#b*G6#%\"iG\"\"\"&%\"aG6$ F+%\"jGF,/F+;,&F0F,F,F,\"#8*&&F)6#F0F,,&F,F,&%\"cG6#F0!\"\"F," }{TEXT -1 3 ", " }{XPPEDIT 18 0 "j=1" "6#/%\"jG\"\"\"" }{TEXT -1 9 " . . 12, " }}{PARA 0 "" 0 "" {TEXT -1 8 "(where " }{XPPEDIT 18 0 "c[1]=0" "6# /&%\"cG6#\"\"\"\"\"!" }{TEXT -1 17 " ) are satisfied." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 163 "[Sum(`b* `[i]*a[i,1],i=1+1..13)=`b*`[1],seq(Sum(`b*`[i]*a[i,j],i=j+1..13)=`b*`[ j]*(1-c[j]),j=2..12)]:\nmap(u->lhs(u)-rhs(u),subs(ee,eval(subs(Sum=add ,%))));\nnops(%);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#7.\"\"!F$F$F$F$F$ F$F$F$F$F$F$" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#\"#7" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 93 "We can calculate the p rincipal error norm, that is, the 2-norm of the principal error terms. " }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 162 "errterms8_13 := Princi palErrorTerms(8,13,'expanded'):\nsm := 0:\nfor ct to nops(errterms8_13 ) do\n sm := sm+(evalf(subs(ee,errterms8_13[ct])))^2;\nend do:\nsqrt (sm);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#$\"+&Qb'[t!#;" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 78 "We can also calculat e the principal error norm of the order 7 embedded scheme." }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 184 "`errterms7_13*` := subs(b=`b*`,Pri ncipalErrorTerms(7,13,'expanded')):\nsm := 0:\nfor ct to nops(`errterm s7_13*`) do\n sm := sm+(evalf(subs(ee,`errterms7_13*`[ct])))^2;\nend do:\nsqrt(sm);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#$\"+Gw`c5!#9" }}} {PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}}{PARA 0 "" 0 "" {TEXT -1 52 "#------------------------------ ---------------------" }}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 34 "construc tion of the order 8 scheme" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 93 "First we construct a 12 stage order 8 scheme star ting with a consideration of stages 2 to 8. " }}{PARA 0 "" 0 "" {TEXT -1 20 "We specify the nodes" }}{PARA 256 "" 0 "" {TEXT -1 2 " " } {XPPEDIT 18 0 "c[2] = 9/142;" "6#/&%\"cG6#\"\"#*&\"\"*\"\"\"\"$U\"!\" \"" }{TEXT -1 2 ", " }{XPPEDIT 18 0 "c[5] = 50/129;" "6#/&%\"cG6#\"\"& *&\"#]\"\"\"\"$H\"!\"\"" }{TEXT -1 2 ", " }{XPPEDIT 18 0 "c[6] = 34/73 ;" "6#/&%\"cG6#\"\"'*&\"#M\"\"\"\"#t!\"\"" }{TEXT -1 2 ", " }{XPPEDIT 18 0 "c[7] = 23/148;" "6#/&%\"cG6#\"\"(*&\"#B\"\"\"\"$[\"!\"\"" } {TEXT -1 3 ", " }{XPPEDIT 18 0 "c[8] = 142/141;" "6#/&%\"cG6#\"\")*& \"$U\"\"\"\"\"$T\"!\"\"" }{TEXT -1 2 " " }}{PARA 0 "" 0 "" {TEXT -1 33 "and the zero linking coefficients" }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "a[4,2]=0" "6#/&%\"aG6$\"\"%\"\"#\"\"!" }{TEXT -1 2 ", " }{XPPEDIT 18 0 "a[5,2]=0" "6#/&%\"aG6$\"\"&\"\"#\"\"!" }{TEXT -1 2 ", " }{XPPEDIT 18 0 "a[6,2]=0" "6#/&%\"aG6$\"\"'\"\"#\"\"!" } {TEXT -1 2 ", " }{XPPEDIT 18 0 "a[6,3]=0" "6#/&%\"aG6$\"\"'\"\"$\"\"! " }{TEXT -1 2 ", " }{XPPEDIT 18 0 "a[7,2]=0" "6#/&%\"aG6$\"\"(\"\"#\" \"!" }{TEXT -1 2 ", " }{XPPEDIT 18 0 "a[7,3]=0" "6#/&%\"aG6$\"\"(\"\"$ \"\"!" }{TEXT -1 2 ", " }{XPPEDIT 18 0 "a[8,2] = 0;" "6#/&%\"aG6$\"\") \"\"#\"\"!" }{TEXT -1 2 ", " }{XPPEDIT 18 0 "a[8,3] = 0;" "6#/&%\"aG6$ \"\")\"\"$\"\"!" }{TEXT -1 3 ", " }{XPPEDIT 18 0 "a[8,4]=0" "6#/&%\"a G6$\"\")\"\"%\"\"!" }{TEXT -1 2 ". " }}{PARA 0 "" 0 "" {TEXT -1 48 "In addition we specify the linking coefficient " }{XPPEDIT 18 0 "a[8,7] = 254/39;" "6#/&%\"aG6$\"\")\"\"(*&\"$a#\"\"\"\"#R!\"\"" }{TEXT -1 1 "." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 69 "The stage-order equations relating to these stages, such that stages " } {XPPEDIT 18 0 "3,4,5,6,7,8;" "6(\"\"$\"\"%\"\"&\"\"'\"\"(\"\")" } {TEXT -1 19 " have stage orders " }{XPPEDIT 18 0 "2,3,3,4,4,4;" "6(\" \"#\"\"$F$\"\"%F%F%" }{TEXT -1 48 " respectively can be solved to obta in the nodes " }{XPPEDIT 18 0 "c[3], c[4]" "6$&%\"cG6#\"\"$&F$6#\"\"% " }{TEXT -1 67 " and the remaining non-zero linking coefficients for t hese stages. " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 449 "RSeqs := RowSumConditions(8,'expanded'):\nSOeqs := [op(StageOrderConditions(2,8,'expanded')),op(StageOrderConditions( 3,4..8,'expanded')),\n op(StageOrderConditions(4,6..8,'e xpanded'))]:\ne1 := \{c[2]=9/142,c[5]=50/129,c[6]=34/73,c[7]=23/148,c[ 8]=142/141,\n seq(a[i,2]=0,i=4..8),seq(a[i,3]=0,i=6..8),a[8,7]=2 54/39\}:\neqns := map(expand,subs(e1,[op(RSeqs),op(SOeqs)])):\nconvert (ListTools[Enumerate](%),matrix);\n``;\nindets(eqns);\nnops(%);" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#K%'matrixG6#777$\"\"\"/&%\"aG6$\"\"#F( #\"\"*\"$U\"7$F-/,&&F+6$\"\"$F(F(&F+6$F6F-F(&%\"cG6#F67$F6/,&&F+6$\"\" %F(F(&F+6$FAF6F(&F:6#FA7$FA/,(&F+6$\"\"&F(F(&F+6$FKF6F(&F+6$FKFAF(#\"# ]\"$H\"7$FK/,(&F+6$\"\"'F(F(&F+6$FXFAF(&F+6$FXFKF(#\"#M\"#t7$FX/,*&F+6 $\"\"(F(F(&F+6$F_oFAF(&F+6$F_oFKF(&F+6$F_oFXF(#\"#B\"$[\"7$F_o/,,&F+6$ \"\")F(F(#\"$a#\"#RF(&F+6$F^pFAF(&F+6$F^pFKF(&F+6$F^pFXF(#F0\"$T\"7$F^ p/,$*&F.F(F7F(F(,$*&#F(F-F(*$)F9F-F(F(F(7$F//*&FBF(F9F(,$*&F`qF(*$)FDF -F(F(F(7$\"#5/,&*&FLF(F9F(F(*&FNF(FDF(F(#\"%]7\"&Tm\"7$\"#6/,&*&FYF(FD F(F(*&FPF(FenF(F(#\"$y&\"%H`7$\"#7/,(*&F`oF(FDF(F(*&FPF(FboF(F(*&FgnF( FdoF(F(#\"$H&\"&3Q%7$\"#8/,*#\"%@H\"%')GF(*&FbpF(FDF(F(*&FPF(FdpF(F(*& FgnF(FfpF(F(#\"#\"&\"))>7$\"#9/*&FBF(FbqF(,$*&#F(F6F(*$)FDF6F(F(F(7 $\"#:/,&*&FLF(FbqF(F(*&FNF(FiqF(F(#\"'+]7\"(n+W'7$\"#;/,&*&FYF(FiqF(F( *&#\"%+DFbrF(FenF(F(#\"&/$R\"(^q;\"7$\"#/,&*&FYF(F[uF(F (*&#Fcu\"(*oY@F(FenF(F(#\"'%3M$\")T#)RG7$\"#?/,(*&F`oF(F[uF(F(*&F_xF(F boF(F(*&#F^v\"'>7$\"#@/,*#\"(4_a\"\")W\\@j F(*&FbpF(F[uF(F(*&F_xF(FdpF(F(*&F[yF(FfpF(F(#\"*Cnk,\"\"*hTD&RQ)pprint 306\"" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%!G" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#<7&%\"aG6$\"\"(\"\"'&F%6$\"\")\"\"\"&F%6$F+\"\"&&F%6$F+ F(&F%6$F'\"\"%&F%6$F'F/&F%6$F(F,&F%6$F/F4&F%6$F/\"\"$&F%6$F+F4&F%6$F'F ,&F%6$F(F4&F%6$F(F/&F%6$F/F,&F%6$F4F=&%\"cG6#F4&F%6$F=F,&F%6$F=\"\"#&F K6#F=&F%6$F4F,&F%6$FQF," }}{PARA 11 "" 1 "" {XPPMATH 20 "6#\"#@" }}} {PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 46 "e2 := solve(\{op(eqns)\});\ne3 := `union`(e1,e2):" }}{PARA 12 "" 1 "" {XPPMATH 20 "6#>%#e2G<7/&%\"aG6$\"\"%\"\"\"#\"&dA\"\"'))zJ/&F(6$ \"\"(F+#\"0>'*)eb&G![\"1?j\"QD*y(*z/&F(6$F2\"\"'#\"0()Qj\"Q8[')\"23Etl wH=<&/&%\"cG6#F*#F-\"&(\\z/&F(6$F2\"\"&#!2*[%Q\\*p2X;\"3?bJ)R%y4+\\/&F (6$FFF+#\"/D%*['oJT\"\"/H:gvR$e$/&F(6$F9F*#\"+))zvd(*\",n!=e*>%/&F(6$F 9F+#\"(6!=W\")D_0'*/&F(6$FFF*#\"/+vm/'>w\"\"/VQ`eY%>\"/&F?6#\"\"$#\"&9 X#\"'\"\\Q#/&F(6$FFF_o#!/D1Aa/q\"-vU4+j5/&F(6$\"\")F9#\":2o'R>!z]e)Qc07\"97HA@A\"f]FdOK\"/&F(6$ FfpFF#!<\"4;rkMEANA=c&=$\";+W09s.%[K:Tsr#/&F(6$F2F*#\"8V'Q0Y'pcl5i@&\" 9))eO**GZ%GLq)[Y/&F(6$F_oF+#\"+]a%3&**\"-HPh,>^/&F(6$FfpF+#!7Vr8(e)3_$ [5+(\"7+)[^QI5NI)\\W/&F(6$F_o\"\"##\",;*pkmUF[r/&F(6$FerF+#\"\"*\"$U\" /&F(6$FfpF*#!=Lf$z>W!*f%H[bwAD\"=#z,znZ%eP\\&>jf\">" }}}{PARA 0 "" 0 " " {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT 260 4 "Note" }{TEXT -1 2 ": " }}{PARA 0 "" 0 "" {TEXT -1 8 "We have " }{XPPEDIT 18 0 "c[3]=2/3" "6#/ &%\"cG6#\"\"$*&\"\"#\"\"\"F'!\"\"" }{TEXT -1 1 " " }{XPPEDIT 18 0 "c[4 ]" "6#&%\"cG6#\"\"%" }{TEXT -1 16 " and the nodes " }{XPPEDIT 18 0 "c [4];" "6#&%\"cG6#\"\"%" }{TEXT -1 3 ", " }{XPPEDIT 18 0 "c[5];" "6#&% \"cG6#\"\"&" }{TEXT -1 3 ", " }{XPPEDIT 18 0 "c[6];" "6#&%\"cG6#\"\"' " }{TEXT -1 31 " are related by the equation: " }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "3*c[6]^2-4*c[6]*c[4]-4*c[5]*c[6]+6*c[5] *c[4] = 0" "6#/,**&\"\"$\"\"\"*$&%\"cG6#\"\"'\"\"#F'F'*(\"\"%F'&F*6#F, F'&F*6#F/F'!\"\"*(F/F'&F*6#\"\"&F'&F*6#F,F'F4*(F,F'&F*6#F8F'&F*6#F/F'F '\"\"!" }{TEXT -1 2 ". " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 146 "subs(e3,matrix([seq([c[i],seq(a[i, j],j=1..i-1),``$(8-i)],i=2..7)]));``;\nc[8]=subs(e3,c[8]);\nfor jj to \+ 7 do print(a[8,jj]=subs(e3,a[8,jj])) end do;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#K%'matrixG6#7(7*#\"\"*\"$U\"F(%!GF+F+F+F+F+7*#\"&9X#\"' \"\\Q##\"+]a%3&**\"-HPh,>^#\",;*pkmUF2F+F+F+F+F+7*#\"&dA\"\"&(\\z#F7\" '))zJ\"\"!#F7\"''*f5F+F+F+F+7*#\"#]\"$H\"#\"/D%*['oJT\"\"/H:gvR$e$F;#! /D1Aa/q<\"/VQ`eY%>\"#\"/+vm/'>w\"FGF+F+F+7*#\"#M\"#t#\"(6!=W\")D_0'*F; F;#\"+))zvd(*\",n!=e*>%#\",TC^@*>\"-vU4+j5F+F+7*#\"#B\"$[\"#\"0>'*)eb& G![\"1?j\"QD*y(*zF;F;#\"8V'Q0Y'pcl5i@&\"9))eO**GZ%GLq)[Y#!2*[%Q\\*p2X; \"3?bJ)R%y4+\\#\"0()Qj\"Q8[')\"23EtlwH=<&F+Q)pprint316\"" }}{PARA 11 " " 1 "" {XPPMATH 20 "6#%!G" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/&%\"cG6# \"\")#\"$U\"\"$T\"" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/&%\"aG6$\"\")\" \"\"#!7Vr8(e)3_$[5+(\"7+)[^QI5NI)\\W" }}{PARA 11 "" 1 "" {XPPMATH 20 " 6#/&%\"aG6$\"\")\"\"#\"\"!" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/&%\"aG6 $\"\")\"\"$\"\"!" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/&%\"aG6$\"\")\"\" %#!=Lf$z>W!*f%H[bwAD\"=#z,znZ%eP\\&>jf\">" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/&%\"aG6$\"\")\"\"&#!<\"4;rkMEANA=c&=$\";+W09s.%[K:Tsr# " }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/&%\"aG6$\"\")\"\"'#\":2o'R>!z]e)Q c07\"97HA@A\"f]FdOK\"" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/&%\"aG6$\"\" )\"\"(#\"$a#\"#R" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 81 "subs(e3,matrix([seq([c[i],seq(a[i,j],j=1..i-1 ),``$(9-i)],i=2..8)])):\nevalf[8](%);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#K%'matrixG6#7)7+$\")#G!Qj!\"*F(%!GF+F+F+F+F+F+7+$\")&zy-\"!\")$\")! )*Q%>F*$\")l*[L)F*F+F+F+F+F+F+7+$\")#>=a\"F/$\")![X&QF*$\"\"!F:$\")WOc 6F/F+F+F+F+F+7+$\")!pf(QF/$\")elVRF/F9$!)>(=[\"!\"($\")L5v9FDF+F+F+F+7 +$\")U`dYF/$\")*[%*f%F*F9F9$\")r]BBF/$\")B3u=F/F+F+F+7+$\")T0a:F/$\")! H_+'F*F9F9$\")$Q?7\"F/$!)IBdLF*$\")8;s;F*F+F+7+$\")A425FD$!)$HLd\"FDF9 F9$!))3nJ\"FD$!):Ns6!\"'$\")]#y5*FD$\")0#G^'FDF+Q)pprint326\"" }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 27 "We do not need to specify " } {XPPEDIT 18 0 "c[9];" "6#&%\"cG6#\"\"*" }{TEXT -1 42 ", because accord ing to Verner, the nodes " }{XPPEDIT 18 0 "c[6];" "6#&%\"cG6#\"\"'" } {TEXT -1 2 ", " }{XPPEDIT 18 0 "c[7];" "6#&%\"cG6#\"\"(" }{TEXT -1 2 " , " }{XPPEDIT 18 0 "c[8];" "6#&%\"cG6#\"\")" }{TEXT -1 2 ", " } {XPPEDIT 18 0 "c[9];" "6#&%\"cG6#\"\"*" }{TEXT -1 30 " are related by the equation " }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "Int (p(x)*``((1-x)^3/3!),x = 0 .. 1)*Int(q(x)*(1-x),x = 0 .. 1) = Int(p(x) *``((1-x)^2/2!),x = 0 .. 1)*Int(q(x)*``((1-x)^2/2!),x = 0 .. 1);" "6#/ *&-%$IntG6$*&-%\"pG6#%\"xG\"\"\"-%!G6#*&,&F-F-F,!\"\"\"\"$-%*factorial G6#F4F3F-/F,;\"\"!F-F--F&6$*&-%\"qG6#F,F-,&F-F-F,F3F-/F,;F:F-F-*&-F&6$ *&-F*6#F,F--F/6#*&,&F-F-F,F3\"\"#-F66#FNF3F-/F,;F:F-F--F&6$*&-F?6#F,F- -F/6#*&,&F-F-F,F3FN-F66#FNF3F-/F,;F:F-F-" }{TEXT -1 2 ", " }}{PARA 0 " " 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 7 "where " }{XPPEDIT 18 0 "p(x) = x*(x-c[6])*(x-c[7])*(x-c[8]);" "6#/-%\"pG6#%\"xG**F'\"\" \",&F'F)&%\"cG6#\"\"'!\"\"F),&F'F)&F,6#\"\"(F/F),&F'F)&F,6#\"\")F/F)" }{TEXT -1 7 " and " }{XPPEDIT 18 0 "q(x) = (x-c[9])*p(x);" "6#/-%\"q G6#%\"xG*&,&F'\"\"\"&%\"cG6#\"\"*!\"\"F*-%\"pG6#F'F*" }{TEXT -1 2 ". \+ " }}{PARA 0 "" 0 "" {TEXT -1 1 " " }}{PARA 0 "" 0 "" {TEXT -1 155 "See : J.H. Verner, SIAM Journal of Numerical Analysis 1978, 772-790, \"Exp licit Runge-Kutta methods with estimates of the Local Truncation Error .\" (page 780)" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 90 "p := x -> x*(x-c[6])*(x-c[7])*(x-c[8]):\n'p(x)'= p(x);\nq := x -> (x-c[9])*p(x):\n'q(x)'=q(x);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/-%\"pG6#%\"xG**F'\"\"\",&F'F)&%\"cG6#\"\"'!\"\"F),&F'F )&F,6#\"\"(F/F),&F'F)&F,6#\"\")F/F)" }}{PARA 11 "" 1 "" {XPPMATH 20 "6 #/-%\"qG6#%\"xG*,,&F'\"\"\"&%\"cG6#\"\"*!\"\"F*F'F*,&F'F*&F,6#\"\"'F/F *,&F'F*&F,6#\"\"(F/F*,&F'F*&F,6#\"\")F/F*" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 239 "Ieq := Int('p(x)' *(1-x)^3/3!,x=0..1)*Int('q(x)'*(1-x),x=0..1)=\n Int('p(x)'*(1-x)^2/2! ,x=0..1)*Int('q(x)'*(1-x)^2/2!,x=0..1);\nvalue(Ieq):\nReq := simplify( lhs(%)-rhs(%))=0:\nL := ilcm(op(map(denom,[coeffs(lhs(Req))]))):\nCeq \+ := expand(L*Req);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%$IeqG/*&-%$IntG 6$,$*&#\"\"\"\"\"'F-*&-%\"pG6#%\"xGF-),&F-F-F3!\"\"\"\"$F-F-F-/F3;\"\" !F-F--F(6$*&-%\"qGF2F-F5F-F8F-*&-F(6$,$*&#F-\"\"#F-*&F0F-)F5FFF-F-F-F8 F--F(6$,$*&FEF-*&F>F-FHF-F-F-F8F-" }}{PARA 12 "" 1 "" {XPPMATH 20 "6#> %$CeqG/,hp*&\"\"$\"\"\"&%\"cG6#\"\"(F)!\"\"*&F(F)&F+6#\"\")F)F.*(\"#UF ))F0\"\"#F)&F+6#\"\"*F)F)**\"#)*F)F5F))F*F6F)&F+6#\"\"'F)F.**F;F)F5F)) F=F6F)F*F)F.**\"$o\"F)F5F)F=F)F*F)F)*(\"#9F)F0F)FF)F5F)FAF)F7F)F)*,\"%q9 F)FAF)FF'*$&F*6#FJF-F'&F*6#F,F-&F*6#FCF'F;**FdqF'*$&F*6#FJF-F'&F*6#F 0F'&F*6#FCF'F'**F^]lF'*$&F*6#FJF-F'&F*6#F0F-&F*6#FCF'F;**FdqF'&F*6#FCF '&F*6#F,F-&F*6#F0F'F'**FdqF'&F*6#F,F'&F*6#F0F-&F*6#FCF'F'**F^]lF'*$&F* 6#F,F-F'&F*6#F0F-&F*6#FCF'F;\"\"!" }{TEXT -1 2 ". " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 75 "e4 := solve( subs(e3,Ceq),\{c[9]\}):\nc[9]=subs(e4,c[9]);\ne5 := `union`(e3,e4):" } }{PARA 11 "" 1 "" {XPPMATH 20 "6#/&%\"cG6#\"\"*#\",)>?T5H\",2<`=K$" }} }{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 6 "Thus " } {XPPEDIT 18 0 "c[9] = 29104120198/33218531707" "6#/&%\"cG6#\"\"**&\",) >?T5H\"\"\"\",2<`=K$!\"\"" }{TEXT -1 2 ". " }}{PARA 0 "" 0 "" {TEXT -1 85 "Now we can use the quadrature equations to find the weights onc e the remaining nodes " }}{PARA 256 "" 0 "" {TEXT -1 2 " " }{XPPEDIT 18 0 "c[10] = 83/91;" "6#/&%\"cG6#\"#5*&\"#$)\"\"\"\"#\"*!\"\"" } {TEXT -1 3 ", " }{XPPEDIT 18 0 "c[11] = 143/149;" "6#/&%\"cG6#\"#6*& \"$V\"\"\"\"\"$\\\"!\"\"" }{TEXT -1 3 ", " }{XPPEDIT 18 0 "c[12] = 1; " "6#/&%\"cG6#\"#7\"\"\"" }{TEXT -1 1 " " }}{PARA 0 "" 0 "" {TEXT -1 38 "are specified along with the weights " }{XPPEDIT 18 0 "b[2]=0" "6 #/&%\"bG6#\"\"#\"\"!" }{TEXT -1 2 ", " }{XPPEDIT 18 0 "b[3]=0" "6#/&% \"bG6#\"\"$\"\"!" }{TEXT -1 2 ", " }{XPPEDIT 18 0 "b[4]=0" "6#/&%\"bG6 #\"\"%\"\"!" }{TEXT -1 2 ", " }{XPPEDIT 18 0 "b[5]=0" "6#/&%\"bG6#\"\" &\"\"!" }{TEXT -1 2 ". " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 290 "Qeqs := QuadratureConditions(8,12, 'expanded'):\ne6 := \{seq(b[i]=0,i=2..5),c[10]=83/91,c[11]=143/149,c[1 2]=1\}:\ne7 := `union`(e5,e6):\nquadeqns := subs(e7,Qeqs):\nfor ct to \+ nops(quadeqns) do\n print(`equation `||ct); print(``);print(quadeq ns[ct]);print(``);\nend do:\nindets(quadeqns);\nnops(%);" }}{PARA 11 " " 1 "" {XPPMATH 20 "6#%-equation~~~1G" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%!G" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/,2&%\"bG6#\"\"\"F(&F&6#\"\" 'F(&F&6#\"\"(F(&F&6#\"\")F(&F&6#\"\"*F(&F&6#\"#5F(&F&6#\"#6F(&F&6#\"#7 F(F(" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%!G" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%-equation~~~2G" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%!G " }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/,0*&#\"#M\"#t\"\"\"&%\"bG6#\"\"'F )F)*&#\"#B\"$[\"F)&F+6#\"\"(F)F)*&#\"$U\"\"$T\"F)&F+6#\"\")F)F)*&#\",) >?T5H\",2<`=K$F)&F+6#\"\"*F)F)*&#\"#$)\"#\"*F)&F+6#\"#5F)F)*&#\"$V\"\" $\\\"F)&F+6#\"#6F)F)&F+6#\"#7F)#F)\"\"#" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%!G" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%-equation~~~3G" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%!G" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/,0*& #\"%c6\"%H`\"\"\"&%\"bG6#\"\"'F)F)*&#\"$H&\"&/>#F)&F+6#\"\"(F)F)*&#\"& k,#\"&\"))>F)&F+6#\"\")F)F)*&#\"6/#f:j*\\7)\\q%)\"7\\QLk*o([3Z.6F)&F+6 #\"\"*F)F)*&#\"%*)o\"%\"G)F)&F+6#\"#5F)F)*&#\"&\\/#\"&,A#F)&F+6#\"#6F) F)&F+6#\"#7F)#F)\"\"$" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%!G" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%-equation~~~4G" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%!G" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/,0*&#\"&/$R\"'< !*Q\"\"\"&%\"bG6#\"\"'F)F)*&#\"&n@\"\"(#zTKF)&F+6#\"\"(F)F)*&#\"()GjG \"(@K!GF)&F+6#\"\")F)F)*&#\"A#R-#ptyH(REoc&RElC\"AV-&)R^^TO/#ex8obm$F) &F+6#\"\"*F)F)*&#\"'(yr&\"'rNvF)&F+6#\"#5F)F)*&#\"(2U#H\"(\\zI$F)&F+6# \"#6F)F)&F+6#\"#7F)#F)\"\"%" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%!G" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#%-equation~~~5G" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%!G" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/,0*&#\"(OjL\"\" )T#)RG\"\"\"&%\"bG6#\"\"'F)F)*&#\"'T)z#\"*;_yz%F)&F+6#\"\"(F)F)*&#\"*' *oe1%\"*hTD&RF)&F+6#\"\")F)F)*&#\"K;O6d,i[U$=mN3)4mb[Q$\\<(\"L,[:Fj$ej 9%G&*pb&)*G39zk<7F)&F+6#\"\"*F)F)*&#\")@$eu%\")h\\doF)&F+6#\"#5F)F)*&# \"*,;;=%\"*,W)G\\F)&F+6#\"#6F)F)&F+6#\"#7F)#F)\"\"&" }}{PARA 11 "" 1 " " {XPPMATH 20 "6#%!G" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%-equation~~~6 G" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%!G" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/,0*&#\")CaVX\"+$frI2#\"\"\"&%\"bG6#\"\"'F)F)*&#\"(VjV'\",o>@35( F)&F+6#\"\"(F)F)*&#\",K#R`td\",,n$3tbF)&F+6#\"\")F)F)*&#\"VofT1[K!GoS* H&)3lnzp1I!4Tr8?)3#\"V2`x:@=a?*=(z_1-Hp*)\\x^#>%eZ[/%F)&F+6#\"\"*F)F)* &#\"+V1/RR\"+^9KSiF)&F+6#\"#5F)F)*&#\",V*3rzf\",\\dxRM(F)&F+6#\"#6F)F) &F+6#\"#7F)#F)F-" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%!G" }}{PARA 11 " " 1 "" {XPPMATH 20 "6#%-equation~~~7G" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%!G" }}{PARA 12 "" 1 "" {XPPMATH 20 "6#/,0*&#\"+;W![a\"\"-*GEUL^\" \"\"\"&%\"bG6#\"\"'F)F)*&#\"**)e.[\"\"/k7P:#40\"F)&F+6#\"\"(F)F)*&#\". W4<=%)>)\".T[(z/eyF)&F+6#\"\")F)F)*&#\"jnk;_eUT6WK#f6aZ]X\"o$f\\sla>Or 6rj_xg\"[o\\!frY!QM[(*HmgyE,M[aEV6r'G]\"[Dx*QOM\"F)&F+6#\"\"*F)F)*&#\" -pLPSpK\"-T?DpycF)&F+6#\"#5F)F)*&#\".\\)yl)4b)\"/,meED%4\"F)&F+6#\"#6F )F)&F+6#\"#7F)#F)F4" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%!G" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%-equation~~~8G" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%!G" }}{PARA 12 "" 1 "" {XPPMATH 20 "6#/,0*&#\",W,NBD& \"/(4>&)RZ5\"\"\"\"&%\"bG6#\"\"'F)F)*&#\"+Za#[S$\"1sq%\\(QOb:F)&F+6#\" \"(F)F)*&#\"1[SF!QvT;\"\"1\"e_Ww%)z5\"F)&F+6#\"\")F)F)*&#\"eos%pH%[b&= $=5SCKl-<5w;&F)&F+6#\"#5F)F)*&#\"12 ax!3\"zA7\"1\\NShkVI;F)&F+6#\"#6F)F)&F+6#\"#7F)#F)F;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%!G" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#<*&%\"bG6#\"# 6&F%6#\"#7&F%6#\"\")&F%6#\"\"'&F%6#\"\"(&F%6#\"\"*&F%6#\"#5&F%6#\"\"\" " }}{PARA 11 "" 1 "" {XPPMATH 20 "6#\"\")" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 199 "e8 := solve(\{op( quadeqns)\}):\ne9 := `union`(e7,e8):\nfor ii to 12 do\n wt_val := su bs(e9,b[ii]);\n if wt_val<>0 then print(b[ii]=wt_val) end if;\nend d o:\n``;\nsubs(e9,[seq(b[i],i=1..12)]):\nevalf[8](%);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/&%\"bG6#\"\"\"#\"5,QDHN^Y%Q]&\"7?NQ`e0q0GR7" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#/&%\"bG6#\"\"'#\"@6YRY=TkCyB`z'\\NB\"@ ci$*z`*)4,-%H$yo$)f'" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/&%\"bG6#\"\"( #\"@)))p3Cv`cVr_wL2Ow\"AX'[ZvY?k'Hh#*>\\fsI" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/&%\"bG6#\"\")#!=([2'HFJM*pT@(3PU\">gh2/m6DY@%R\"zNF\" " }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/&%\"bG6#\"\"*#\"\\p*pET<%)zAG#e_K y+yEW)y#RO9fL%)>79b*4wTK8\"H\"H0&3%\"\\p!ovNxgeXYb!**>MF0[`Zm6#Q7-_j0% f^lmup$>7Y([i7#" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/&%\"bG6#\"#5#!:ZfV '4HqK'HD#36\"95L5H(zR*e2zcS" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/&%\"bG 6#\"#6#\"=&)zY)oDQ@17H))4Y#\"=R_[#F+$!)K\"pK$F+$\"))[7#>!\"($!)$y " 0 "" {MPLTEXT 1 0 1 ";" }}}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 2 "e9" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1747 "e9 := \{a[6,2] = 0, a[5,2] = 0, a[7,2] = 0, b[10] = -1108225296327029096435947/405679075893979729103310, a[4,2] = 0, a[8, 3] = 0, a[7,3] = 0, a[6,3] = 0, c[7] = 23/148, a[4,1] = 12257/317988, \+ a[7,1] = 480285555889619/7997789253816320, a[7,6] = 864813381633887/51 718297665732608, c[4] = 12257/79497, a[7,5] = -16450769949384489/49000 9784398315520, a[5,1] = 14131686489425/35833975601529, a[6,4] = 975775 7988/41995818067, a[6,1] = 4418011/96055225, b[8] = -42370872141699343 12729607487/12735791394214625116604076160, a[5,4] = 17619604667500/119 44658533843, c[3] = 24514/238491, a[5,3] = -17700454220625/11944658533 843, b[5] = 0, a[4,3] = 12257/105996, b[2] = 0, b[3] = 0, b[4] = 0, b[ 6] = 2335496795323782464411846394611/6598368783294020109895379936256, \+ a[6,5] = 19921512441/106300094275, b[7] = 7636073376527143565375240869 888/30725949199261296642046754748645, b[1] = 55038446513529253801/1239 280570055853383520, c[9] = 29104120198/33218531707, c[11] = 143/149, c [10] = 83/91, c[12] = 1, a[8,6] = 1205563885850790193966807/1323657275 05912221222912, a[8,5] = -318556182235222634647116091/2717241153248403 7214054400, a[7,4] = 52162106556696460538643/464887033284472899365888, a[3,1] = 9950845450/511901613729, c[6] = 34/73, c[5] = 50/129, a[8,1] = -7001048352088587137143/4449830351030385148800, a[3,2] = 4266646991 6/511901613729, a[8,2] = 0, b[11] = 2460988291206213825688467985/17563 42789520947764222671739, b[9] = 40850529129113324176099551412198433591 4363927884426780078325258228227984174126699/21262487461219369746665515 9405635202123821166475348052734199905546455860773575680, c[8] = 142/14 1, c[2] = 9/142, a[8,7] = 254/39, a[2,1] = 9/142, a[8,4] = -2522765548 294599044197935933/1915963195493758447677901792, b[12] = 4808707937311 /50545545388065\}:" }{TEXT -1 0 "" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 55 "Now we specify the remaining ze ro linking coefficients:" }}{PARA 256 "" 0 "" {TEXT -1 2 " " } {XPPEDIT 18 0 "a[9,2] = 0,a[9,3] = 0,a[10,2] = 0,a[10,3] = 0,a[11,2] = 0,a[11,3] = 0,a[12,2] = 0,a[12,3] = 0;" "6*/&%\"aG6$\"\"*\"\"#\"\"!/& F%6$F'\"\"$F)/&F%6$\"#5F(F)/&F%6$F1F-F)/&F%6$\"#6F(F)/&F%6$F8F-F)/&F%6 $\"#7F(F)/&F%6$F?F-F)" }{TEXT -1 1 "." }}{PARA 0 "" 0 "" {TEXT -1 0 " " }}{PARA 0 "" 0 "" {TEXT -1 149 "We include the stage-order condition s for stages 9 to 12 so that all these stages all have stage-order 4 a nd incorporate the simplifying conditions: " }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "Sum(b[i]*a[i,j],i = j+1 .. 12) = b[j]*( 1-c[j]);" "6#/-%$SumG6$*&&%\"bG6#%\"iG\"\"\"&%\"aG6$F+%\"jGF,/F+;,&F0F ,F,F,\"#7*&&F)6#F0F,,&F,F,&%\"cG6#F0!\"\"F," }{TEXT -1 1 " " }}{PARA 0 "" 0 "" {TEXT -1 5 "for " }{XPPEDIT 18 0 "j=1" "6#/%\"jG\"\"\"" } {TEXT -1 10 " ( with " }{XPPEDIT 18 0 "c[1]=0" "6#/&%\"cG6#\"\"\"\" \"!" }{TEXT -1 5 " ), " }{XPPEDIT 18 0 "j = 5;" "6#/%\"jG\"\"&" } {TEXT -1 3 ", " }{XPPEDIT 18 0 "j=7" "6#/%\"jG\"\"(" }{TEXT -1 3 ", \+ " }{XPPEDIT 18 0 "j=9" "6#/%\"jG\"\"*" }{TEXT -1 7 " and " } {XPPEDIT 18 0 "j=10" "6#/%\"jG\"#5" }{TEXT -1 2 ". " }}{PARA 0 "" 0 " " {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 195 "Then it turns out that the following additional collection of \"simple\" order conditions (g iven in abreviated form) is sufficient to determine the remaining link ing coefficients in stages 9 to 12." }}{PARA 0 "" 0 "" {TEXT -1 0 "" } }{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 190 "SO8 := SimpleOrderConditio ns(8):\n[seq([i,SO8[i]],i=[37,61,81,87,102,106,113,117,125])]:\nlinalg [augment](linalg[delcols](%,2..2),matrix([[` `]$(linalg[rowdim](%))]) ,linalg[delcols](%,1..1));" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#K%'matri xG6#7+7%\"#P%#~~G/*(%\"bG\"\"\"%\"cGF--%!G6#*&%\"aGF--F06#*&F3F--F06#* &F3F--F06#*&F3F-F.F-F-F-F-F-#F-\"$S)7%\"#hF)/*(F,F-F.F--F06#*&)F.\"\"% F-F3F-F-#F-\"#N7%\"#\")F)/*&F,F--F06#*(F3F-F.F--F06#*&F3F--F06#*&F3F-- F06#*&)F.\"\"#F-F3F-F-F-F-F-#F-\"%gL7%\"#()F)/*(F,F-F.F--F06#*(F3F-F.F --F06#*(F3F-F.F-F:F-F-F-#F-\"$%Q7%\"$-\"F)/*(F,F-F.F--F06#*&F3F--F06#* &F3F--F06#*&)F.\"\"$F-F3F-F-F-F-#F-\"$g*7%\"$1\"F)/*&F,F--F06#*&F3F--F 06#*&F3F-FCF-F-F-#F-\"%!o\"7%\"$8\"F)/*(F,F-FZF-F[pF-#F-\"$g\"7%\"$<\" F)/*&F,F--F06#*(F3F-F.F-FCF-F-#F-\"$!G7%\"$D\"F)/*(F,F-F.F--F06#*&)F. \"\"&F-F3F-F-#F-\"#[Q(pprint26\"" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}{SECT 1 {PARA 4 "" 0 " " {TEXT -1 20 "The associated trees" }}{PARA 0 "" 0 "" {TEXT -1 0 "" } }{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 461 "AST8 := AllSimpleTrees(8): \nwhch := [37,61,81,87,102,106,113,117,125]:\nm := 3: ## number of tre es per row\nnn := nops(whch): q := iquo(nn,m,'r'):\nfor i to nn do \n \+ p||i := DrawTree(AST8[whch[i]],height=4,width=2,show_ordercondition=t rue,\n scale=1.2,color=COLOR(RGB,0,.7,0),font_color=black);\nend do :\npp := plot([[1,1]],style=line,axes=none):\nplots[display](convert([ seq([p||((k-1)*m+1..m*k)],k=1..q),\n `if`(r>0,[p||(m*q+1..nn),pp$(m*( q+1)-nn)],NULL)],array));" }}{PARA 13 "" 1 "" {GLPLOT2D 852 1131 1131 {PLOTDATA 2 "6\\u-%%TEXTG6&7$$\"#g\"\"!$!+5`7'*o!\")Q/b~c~(a~c~)~=~~6 \"-%'COLOURG6&%$RGBGF)F)F)-%%FONTG6$%(COURIERG\"#5-F$6&7$F'$!+)o#)p\"o F,Q1~~~~~~~~~~~~~~~1F.F/F3-F$6&7$F'$!+dZrToF,Q2~~~~~~~~~~~~~~~__F.F/F3 -F$6&7$F'$!+++++qF,Q2~~~~~~~~~~~~~~~48F.F/F3-F$6&7$F'$!+g)*32oF,Q1~~~~ ~~~~~5~~~~~~F.F/F3-%'CURVESG6&7*7$F'$!+ " 0 "" {MPLTEXT 1 0 1 ";" }}}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT 260 4 "Note" }{TEXT -1 119 ": The 5 simplifying condit ions can be replaced by the 5 extra order conditions given (in abrevia ted form) as follows.. " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 174 "SO8 := SimpleOrderConditions(8):\n [seq([i,SO8[i]],i=[21,27,31,36,127])]:\nlinalg[augment](linalg[delcols ](%,2..2),matrix([[` `]$(linalg[rowdim](%))]),linalg[delcols](%,1..1) );" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#K%'matrixG6#7'7%\"#@%#~~G/*&%\"b G\"\"\"-%!G6#*&%\"aGF--F/6#*&F2F--F/6#*&)%\"cG\"\"#F-F2F-F-F-F-#F-\"$g $7%\"#FF)/*&F,F--F/6#*&F2F--F/6#*&)F:\"\"$F-F2F-F-F-#F-\"$?\"7%\"#JF)/ *&F,F--F/6#*&)F:\"\"%F-F2F-F-#F-\"#I7%\"#OF)/*&F,F--F/6#*(F2F-F:F--F/6 #*&F2F--F/6#*&F2F--F/6#*&F2F-F:F-F-F-F-F-#F-\"%357%\"$F\"F)/*&F,F--F/6 #*&)F:\"\"'F-F2F-F-#F-\"#cQ(pprint36\"" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 462 "SOeqs2 := [seq(add(a[i ,j],j= 1..i-1)=c[i],i=9..12),op(StageOrderConditions(2,9..12,'expanded ')),\n op(StageOrderConditions(3,9..12,'expanded')),op(StageOrderCo nditions(4,9..12,'expanded'))]:\nSO8_12 := SimpleOrderConditions(8,12, 'expanded'):\nord_cdns := [seq(SO8_12[i],i=[37,61,81,87,102,113,106,11 7,125])]:\nsimp_eqs := [add(b[i]*a[i,1],i=1+1..12)=b[1],seq(add(b[i]*a [i,j],i=j+1..12)=b[j]*(1-c[j]),j=[5,7,9,10])]:\ncdns := [op(SOeqs2),op (ord_cdns),op(simp_eqs)]:" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 107 "e10 :=\{seq(a[i,2]=0,i=9..12),seq( a[i,3]=0,i=9..12)\}:\ne11 := `union`(e9,e10):\neqns2 := subs(e11,\{op( cdns)\}):" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 39 "There are 30 equations and 30 unknowns." }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 36 "nops(eqns2);\nindets(eqns2);\nnops(%);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#\"#I" }}{PARA 12 "" 1 "" {XPPMATH 20 "6#<@&% \"aG6$\"#5\"\"'&F%6$F'\"\"(&F%6$F'\"\")&F%6$F'\"\"*&F%6$\"#6\"\"\"&F%6 $F1F+&F%6$F1F.&F%6$F'\"\"%&F%6$F'\"\"&&F%6$\"#7F'&F%6$FBF4&F%6$FBF(&F% 6$FBF+&F%6$FBF.&F%6$FBF1&F%6$FBF<&F%6$FBF?&F%6$F4F'&F%6$FBF5&F%6$F4F+& F%6$F4F.&F%6$F4F1&F%6$F4F<&F%6$F4F?&F%6$F4F(&F%6$F'F5&F%6$F1F(&F%6$F1F ?&F%6$F1F5&F%6$F1F<" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#\"#I" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 22 "in folevel[solve] := 4:" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 74 "e12 := solve(\{op(eqns2)\}):\ninfolevel[solve] := 0:\ne13 := `union`(e11, e12):" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 4 "e13 " }} {PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 5577 "e13 := \{a[9,7] = 4200808003335430559080432294408480526444106676 0038302359736803632/48653024232165349100748232878116055996291702950306 31799935804001, a[6,2] = 0, a[5,2] = 0, a[7,2] = 0, b[10] = -110822529 6327029096435947/405679075893979729103310, a[4,2] = 0, a[9,3] = 0, a[1 0,3] = 0, a[8,3] = 0, a[12,3] = 0, a[9,5] = -2737494094116540602869481 41164828452109898379203526945684314474186724062841643/6042758395137755 2503967840653825589117167443021628367691773162913607984889600, a[7,3] \+ = 0, a[6,3] = 0, c[7] = 23/148, a[12,2] = 0, a[4,1] = 12257/317988, a[ 10,9] = -5097773793079280823220441749724897939987821728001110319786289 9/1300915694564913675613280314081837358644964393191337994183389184, a[ 10,6] = 6364380863259071677112259236455506477417699780613300364807/115 0428174584942133406579091549443438814988349188394057728, a[7,1] = 4802 85555889619/7997789253816320, a[7,6] = 864813381633887/517182976657326 08, c[4] = 12257/79497, a[7,5] = -16450769949384489/490009784398315520 , a[12,6] = 51260471529841028040709654458903254781320136131844164563/2 0998023776318546907382302106788168158765514131585630208, a[5,1] = 1413 1686489425/35833975601529, a[6,4] = 9757757988/41995818067, a[6,1] = 4 418011/96055225, a[10,1] = -179360394632226090082821246070687714247713 2870159527/2313097568511990753781649719556084665131024900300800, b[8] \+ = -4237087214169934312729607487/12735791394214625116604076160, a[5,4] \+ = 17619604667500/11944658533843, c[3] = 24514/238491, a[5,3] = -177004 54220625/11944658533843, b[5] = 0, a[4,3] = 12257/105996, b[2] = 0, b[ 3] = 0, b[4] = 0, b[6] = 2335496795323782464411846394611/6598368783294 020109895379936256, a[12,1] = -753408116554498247829620233592204921080 3875045423/19219575665440848756074598051658416387002479820800, a[6,5] \+ = 19921512441/106300094275, a[11,3] = 0, b[7] = 7636073376527143565375 240869888/30725949199261296642046754748645, b[1] = 5503844651352925380 1/1239280570055853383520, a[10,8] = 1074424716396001987683325504478460 9639/534761804739901348825491947768304503296, c[9] = 29104120198/33218 531707, a[9,8] = 66845978093071633800006662723692741719194739617709352 4377824/71100452948884643779799087713322002499799983434685642170251833 , a[9,6] = 16656372518874512738268060504309924900437672263609245028809 229865738327731797537/427699013853393052278207638577101600909762793055 0149628282203007913538394549504, c[11] = 143/149, c[10] = 83/91, a[10, 2] = 0, a[9,2] = 0, c[12] = 1, a[11,2] = 0, a[8,6] = 12055638858507901 93966807/132365727505912221222912, a[8,5] = -3185561822352226346471160 91/27172411532484037214054400, a[7,4] = 52162106556696460538643/464887 033284472899365888, a[3,1] = 9950845450/511901613729, c[6] = 34/73, a[ 12,8] = -1539218116260541896259682954580256454049/45346708307503609834 22999946409310393344, c[5] = 50/129, a[8,1] = -7001048352088587137143/ 4449830351030385148800, a[12,9] = 241886539350268429372116296787271276 553970618941104594460614948326132797451456131/124066963266246589252891 6120041123051054026283188324780101555345343662316090597376, a[3,2] = 4 2666469916/511901613729, a[8,2] = 0, b[11] = 2460988291206213825688467 985/1756342789520947764222671739, b[9] = 40850529129113324176099551412 1984335914363927884426780078325258228227984174126699/21262487461219369 7466655159405635202123821166475348052734199905546455860773575680, a[9, 4] = -3315443074343659404779422149387397712986453181141168247590906370 819301077749322753/498517112641608872807821838566847987729514312398278 633788185835348997643218553476, c[8] = 142/141, c[2] = 9/142, a[8,7] = 254/39, a[2,1] = 9/142, a[11,5] = -8843488367745847150704404856330264 64653487653/92725983515963799584249219499787659014963200, a[11,1] = -3 587625717068952487214493441966897048737050755812600710793/301573316403 3229624772006086429467685046639983706974412800, a[10,7] = 277251644025 69748756040320433848245155581006369/2544159473655547770881695354241256 106302348256, a[11,9] = 1126338082532727209798743033675038915974992610 46700689572459050065039333987335667/1404514291245034532812181377119034 501014039830518218465064579291611563374608650240, a[10,4] = -147768741 23722838192315406145167687512425345723701/1847893530366076701102014146 927696206329050105856, a[11,7] = 1423634194916865071620070510710077227 65323162710521029/1263488720236826156580744977133572808227358790577584 0, a[12,10] = -80556486832245966191717452425924975/4144454095186765975 65032008051106461, a[9,1] = -20629396399716689122801179264428539394462 855874226604463554767070845753369/428817596627701559566575134700121144 88076017981128105307375594963152353200, a[12,5] = -2061057820982632926 3318986584876108069323/7356333776438834884293014575840932659200, a[8,4 ] = -2522765548294599044197935933/1915963195493758447677901792, a[11,6 ] = 26823469063654084387375587616552322383082061411417182757389742951/ 3541299744763681675473620647087123057228744296123642301654630400, b[12 ] = 4808707937311/50545545388065, a[11,8] = 64747617454909275289531520 412519442831235890581/126931718811797596099667062897445344377785483008 0, a[12,11] = 2944781680874500347594142792814463350/496516138307367698 3610218096030654529, a[10,5] = -19587020919884661714856757105130246995 757906603/2911893296942532038566182868170393629789388800, a[12,7] = -3 077214437173472971196810795615384000211457151011/127243559258228082059 7059432060116893200684680384, a[11,4] = -54530117112678047312115018372 62816944661201619903/764973320899716397072448644710733101329290196992, a[11,10] = -10612202518573994431153697720606405883/670825466582598467 78754594976831647575, a[12,4] = 23769608745278671780227037528303426285 9273455/60688480889936839261131818793519097177034752\}:" }{TEXT -1 0 " " }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 20 "p rinted coefficients" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 283 "subs(e13,matrix([seq([c[i],seq(a[i,j],j=1. .i-1),``$(7-i)],i=2..7)]));\nfor ii from 8 to 12 do\n print(``);\n \+ print(c[ii]=subs(e13,c[ii])); \n for jj to ii-1 do\n print(a[i i,jj]=subs(e13,a[ii,jj]));\n end do:\nend do:\n``;\nfor ii to 12 do \n print(b[ii]=subs(e13,b[ii]));\nend do:" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#K%'matrixG6#7(7)#\"\"*\"$U\"F(%!GF+F+F+F+7)#\"&9X#\"'\" \\Q##\"+]a%3&**\"-HPh,>^#\",;*pkmUF2F+F+F+F+7)#\"&dA\"\"&(\\z#F7\"'))z J\"\"!#F7\"''*f5F+F+F+7)#\"#]\"$H\"#\"/D%*['oJT\"\"/H:gvR$e$F;#!/D1Aa/ q<\"/VQ`eY%>\"#\"/+vm/'>w\"FGF+F+7)#\"#M\"#t#\"(6!=W\")D_0'*F;F;#\"+)) zvd(*\",n!=e*>%#\",TC^@*>\"-vU4+j5F+7)#\"#B\"$[\"#\"0>'*)eb&G![\"1?j\" QD*y(*zF;F;#\"8V'Q0Y'pcl5i@&\"9))eO**GZ%GLq)[Y#!2*[%Q\\*p2X;\"3?bJ)R%y 4+\\#\"0()Qj\"Q8[')\"23EtlwH=<&Q)pprint336\"" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%!G" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/&%\"cG6#\"\")# \"$U\"\"$T\"" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/&%\"aG6$\"\")\"\"\"#! 7Vr8(e)3_$[5+(\"7+)[^QI5NI)\\W" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/&% \"aG6$\"\")\"\"#\"\"!" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/&%\"aG6$\"\" )\"\"$\"\"!" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/&%\"aG6$\"\")\"\"%#!=L f$z>W!*f%H[bwAD\"=#z,znZ%eP\\&>jf\">" }}{PARA 11 "" 1 "" {XPPMATH 20 " 6#/&%\"aG6$\"\")\"\"&#!<\"4;rkMEANA=c&=$\";+W09s.%[K:Tsr#" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/&%\"aG6$\"\")\"\"'#\":2o'R>!z]e)Qc07\"97HA@A\" f]FdOK\"" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/&%\"aG6$\"\")\"\"(#\"$a# \"#R" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%!G" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/&%\"cG6#\"\"*#\",)>?T5H\",2<`=K$" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/&%\"aG6$\"\"*\"\"\"#!eopLvX32nZbjWgEU(e&GY%RR&GWEz6!G7 *o;(*R'RH1#\"eo+KN_J'\\fvtI0\"G6)z,w!)[9@,qM^dm&f:qFmf<)G%" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/&%\"aG6$\"\"*\"\"#\"\"!" }}{PARA 11 "" 1 " " {XPPMATH 20 "6#/&%\"aG6$\"\"*\"\"$\"\"!" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/&%\"aG6$\"\"*\"\"%#!]p`FK\\x2,$>3P14fZ#o69\"=`k)Hr(R(Q \\@UzZSfOMuIW:L\"\\pwMb=Kk(**[`$e=)yL'y#)R7V^Hx)z%ocQ=#y!G()3;k7r^)\\ " }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/&%\"aG6$\"\"*\"\"&#!ioV;%G1Cn=uWJ %oXp_.#z$)*)4@XG[;T\"[pGgSl6%4%\\PF\"ho+'*)[)zg8H;t>(\\;y`2*>^ov48B" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/&%\"aG6$\"#5\"\"#\"\"!" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/&%\" aG6$\"#5\"\"$\"\"!" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/&%\"aG6$\"#5\" \"%#!S,PsX`U7von^91aJ#>QGsBT(ox9\"Rce5]!Hj?'pFp99?5,n2m.`$*y%=" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#/&%\"aG6$\"#5\"\"&#!P.m!zv&*pCI^5dn&[r hY))>4-(e>\"O+))Q*yHORq\"oG=m&Q?`UpH$*=\"H" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/&%\"aG6$\"#5\"\"'#\"en2[O+Lh!y*p+'R;ZUu5\"H'H.XIox%>\\D)[8!*RZ!=wM&" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/&%\"aG6$\"#5\"\"*#!in**G'y>.6,!G<#y)*Rz*[s \\$Rk\\ket$=39.G8cn8\\c%p:4I\"" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#%!G" }}{PARA 11 "" 1 "" {XPPMATH 20 "6 #/&%\"cG6#\"#6#\"$V\"\"$\\\"" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/&%\"a G6$\"#6\"\"\"#!en$z52g7ev]qt[q*o'>W$\\9s[_*oqrDwe$\"en+GTupq$)*Rm/&on% Hk31?xC'HK.kJt:I" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/&%\"aG6$\"#6\"\"# \"\"!" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/&%\"aG6$\"#6\"\"$\"\"!" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#/&%\"aG6$\"#6\"\"%#!R.*>;?hY%p\"GEP=]6 7t/yE6<,`a\"Q#*p>!HH85L2rW'[C2(R;(**3Kt\\w" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/&%\"aG6$\"#6\"\"&#!N`w[`YYEIj&[S/2:Zeun$)[V))\"M+K'\\, fwy*\\>#\\Ue*zjf^$)fs#*" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/&%\"aG6$\" #6\"\"'#\"\\o^Hu*QdF=<9Th?3$QABb;wevtQ%3aO1pM#o#\"[o+/ja;IUO7'HW(Gs0Br 3Z1itan\"ojZu*HTN" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/&%\"aG6$\"#6\"\" (#\"WH5_5F;B`wAx+r502?;2lo\"\\>MOU\"\"VSex0zetA3GdLr(\\u!ech#oB?()[j7 " }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/&%\"aG6$\"#6\"\")#\"P\"e!*eBJGW>D T?:`*Gv#4\\XrP\"=7G`M]C\"H9XS \"" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/&%\"aG6$\"#6\"#5#!G$)eS11s(p`6V %*Rd=D?71\"\"GvvkJo(\\fa(yn%)f#ema#3n" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%!G" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/&%\"cG6#\"#7\"\"\"" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#/&%\"aG6$\"#7\"\"\"#!RBa/vQ!3@\\?#fL-i HyC)\\al63Mv\"S+3#)zC+(Q;%e;0)fugv[3Wlcd>#>" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/&%\"aG6$\"#7\"\"#\"\"!" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/&%\"aG6$\"#7\"\"$\"\"!" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/&%\" aG6$\"#7\"\"%#\"NbMFfGEMIGv.F-yr'y_u3'pP#\"M_Z.xr4>Nz==8h#Ro$**)3[)og " }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/&%\"aG6$\"#7\"\"&#!JB$p!3h([e')*= LEHj#)4#y0h?\"I+#fE$4%ed9IH%)[$)QkxLjN(" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/&%\"aG6$\"#7\"\"'#\"YjX;W=8O,K\"yaK!*eWl42/G5%)H:Zg7&\"Y3-j&eJT ^l(e\"o\")y1@I#Q2pa=jxB!)*4#" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/&%\"a G6$\"#7\"\"(#!R65:d9@+SQ:cz5o>rHZtrV9s2$\"R%Q!o%o+K*o6g?Vfqf?3G#e#fNCF \"" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/&%\"aG6$\"#7\"\")#!I\\SXc-eaHof i*=agi6=#R:\"IWLR5$4k%***HU$)4O]2$3nMX" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/&%\"aG6$\"#7\"\"*#\"\\pJhX^uzKhK[\\hgWf/6%*=1(Rbw7F(y'H;@PH%o-N Rl)=C\"]pwtf!4;BmV`Mb:5!yC$)=$GES0^I7T+7;*GD*eYiEjp1C\"" }}{PARA 11 " " 1 "" {XPPMATH 20 "6#/&%\"aG6$\"#7\"#5#!Dv\\#fU_ur\">mfCKo[c0)\"Ehk5^ !3?.lvfw'=&4aW9%" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/&%\"aG6$\"#7\"#6# \"F]LY9GzUTfZ.]u3o\"yWH\"FHXlIg4=-h$)pntIQh^'\\" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%!G" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/&%\"bG6#\"\"\"# \"5,QDHN^Y%Q]&\"7?NQ`e0q0GR7" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/&%\"b G6#\"\"#\"\"!" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/&%\"bG6#\"\"$\"\"!" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/&%\"bG6#\"\"%\"\"!" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/&%\"bG6#\"\"&\"\"!" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/&%\"bG6#\"\"'#\"@6YRY=TkCyB`z'\\NB\"@ci$*z`*)4,-%H$yo$)f'" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#/&%\"bG6#\"\"(#\"@)))p3Cv`cVr_wL2Ow\"A X'[ZvY?k'Hh#*>\\fsI" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/&%\"bG6#\"\")# !=([2'HFJM*pT@(3PU\">gh2/m6DY@%R\"zNF\"" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/&%\"bG6#\"\"*#\"\\p*pET<%)zAG#e_Ky+yEW)y#RO9fL%)>79b*4wTK8\"H\" H0&3%\"\\p!ovNxgeXYb!**>MF0[`Zm6#Q7-_j0%f^lmup$>7Y([i7#" }}{PARA 11 " " 1 "" {XPPMATH 20 "6#/&%\"bG6#\"#5#!:ZfV'4HqK'HD#36\"95L5H(zR*e2zcS" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/&%\"bG6#\"#6#\"=&)zY)oDQ@17H))4Y#\" =R " 0 "" {MPLTEXT 1 0 1 ";" }}}}{PARA 0 "" 0 "" {TEXT -1 0 "" }} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}}{PARA 0 "" 0 "" {TEXT -1 52 "#---------------------------------------------------" }}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 43 "construction of the order 7 embedded sc heme" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 4 "e13 " }} {PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 5577 "e13 := \{a[9,7] = 4200808003335430559080432294408480526444106676 0038302359736803632/48653024232165349100748232878116055996291702950306 31799935804001, a[6,2] = 0, a[5,2] = 0, a[7,2] = 0, b[10] = -110822529 6327029096435947/405679075893979729103310, a[4,2] = 0, a[9,3] = 0, a[1 0,3] = 0, a[8,3] = 0, a[12,3] = 0, a[9,5] = -2737494094116540602869481 41164828452109898379203526945684314474186724062841643/6042758395137755 2503967840653825589117167443021628367691773162913607984889600, a[7,3] \+ = 0, a[6,3] = 0, c[7] = 23/148, a[12,2] = 0, a[4,1] = 12257/317988, a[ 10,9] = -5097773793079280823220441749724897939987821728001110319786289 9/1300915694564913675613280314081837358644964393191337994183389184, a[ 10,6] = 6364380863259071677112259236455506477417699780613300364807/115 0428174584942133406579091549443438814988349188394057728, a[7,1] = 4802 85555889619/7997789253816320, a[7,6] = 864813381633887/517182976657326 08, c[4] = 12257/79497, a[7,5] = -16450769949384489/490009784398315520 , a[12,6] = 51260471529841028040709654458903254781320136131844164563/2 0998023776318546907382302106788168158765514131585630208, a[5,1] = 1413 1686489425/35833975601529, a[6,4] = 9757757988/41995818067, a[6,1] = 4 418011/96055225, a[10,1] = -179360394632226090082821246070687714247713 2870159527/2313097568511990753781649719556084665131024900300800, b[8] \+ = -4237087214169934312729607487/12735791394214625116604076160, a[5,4] \+ = 17619604667500/11944658533843, c[3] = 24514/238491, a[5,3] = -177004 54220625/11944658533843, b[5] = 0, a[4,3] = 12257/105996, b[2] = 0, b[ 3] = 0, b[4] = 0, b[6] = 2335496795323782464411846394611/6598368783294 020109895379936256, a[12,1] = -753408116554498247829620233592204921080 3875045423/19219575665440848756074598051658416387002479820800, a[6,5] \+ = 19921512441/106300094275, a[11,3] = 0, b[7] = 7636073376527143565375 240869888/30725949199261296642046754748645, b[1] = 5503844651352925380 1/1239280570055853383520, a[10,8] = 1074424716396001987683325504478460 9639/534761804739901348825491947768304503296, c[9] = 29104120198/33218 531707, a[9,8] = 66845978093071633800006662723692741719194739617709352 4377824/71100452948884643779799087713322002499799983434685642170251833 , a[9,6] = 16656372518874512738268060504309924900437672263609245028809 229865738327731797537/427699013853393052278207638577101600909762793055 0149628282203007913538394549504, c[11] = 143/149, c[10] = 83/91, a[10, 2] = 0, a[9,2] = 0, c[12] = 1, a[11,2] = 0, a[8,6] = 12055638858507901 93966807/132365727505912221222912, a[8,5] = -3185561822352226346471160 91/27172411532484037214054400, a[7,4] = 52162106556696460538643/464887 033284472899365888, a[3,1] = 9950845450/511901613729, c[6] = 34/73, a[ 12,8] = -1539218116260541896259682954580256454049/45346708307503609834 22999946409310393344, c[5] = 50/129, a[8,1] = -7001048352088587137143/ 4449830351030385148800, a[12,9] = 241886539350268429372116296787271276 553970618941104594460614948326132797451456131/124066963266246589252891 6120041123051054026283188324780101555345343662316090597376, a[3,2] = 4 2666469916/511901613729, a[8,2] = 0, b[11] = 2460988291206213825688467 985/1756342789520947764222671739, b[9] = 40850529129113324176099551412 1984335914363927884426780078325258228227984174126699/21262487461219369 7466655159405635202123821166475348052734199905546455860773575680, a[9, 4] = -3315443074343659404779422149387397712986453181141168247590906370 819301077749322753/498517112641608872807821838566847987729514312398278 633788185835348997643218553476, c[8] = 142/141, c[2] = 9/142, a[8,7] = 254/39, a[2,1] = 9/142, a[11,5] = -8843488367745847150704404856330264 64653487653/92725983515963799584249219499787659014963200, a[11,1] = -3 587625717068952487214493441966897048737050755812600710793/301573316403 3229624772006086429467685046639983706974412800, a[10,7] = 277251644025 69748756040320433848245155581006369/2544159473655547770881695354241256 106302348256, a[11,9] = 1126338082532727209798743033675038915974992610 46700689572459050065039333987335667/1404514291245034532812181377119034 501014039830518218465064579291611563374608650240, a[10,4] = -147768741 23722838192315406145167687512425345723701/1847893530366076701102014146 927696206329050105856, a[11,7] = 1423634194916865071620070510710077227 65323162710521029/1263488720236826156580744977133572808227358790577584 0, a[12,10] = -80556486832245966191717452425924975/4144454095186765975 65032008051106461, a[9,1] = -20629396399716689122801179264428539394462 855874226604463554767070845753369/428817596627701559566575134700121144 88076017981128105307375594963152353200, a[12,5] = -2061057820982632926 3318986584876108069323/7356333776438834884293014575840932659200, a[8,4 ] = -2522765548294599044197935933/1915963195493758447677901792, a[11,6 ] = 26823469063654084387375587616552322383082061411417182757389742951/ 3541299744763681675473620647087123057228744296123642301654630400, b[12 ] = 4808707937311/50545545388065, a[11,8] = 64747617454909275289531520 412519442831235890581/126931718811797596099667062897445344377785483008 0, a[12,11] = 2944781680874500347594142792814463350/496516138307367698 3610218096030654529, a[10,5] = -19587020919884661714856757105130246995 757906603/2911893296942532038566182868170393629789388800, a[12,7] = -3 077214437173472971196810795615384000211457151011/127243559258228082059 7059432060116893200684680384, a[11,4] = -54530117112678047312115018372 62816944661201619903/764973320899716397072448644710733101329290196992, a[11,10] = -10612202518573994431153697720606405883/670825466582598467 78754594976831647575, a[12,4] = 23769608745278671780227037528303426285 9273455/60688480889936839261131818793519097177034752\}:" }{TEXT -1 0 " " }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 " " {TEXT -1 61 "We can obtain an embedded 11 stage order 7 scheme as fo llows." }}{PARA 0 "" 0 "" {TEXT -1 94 "We remove stages 11 and 12 from the 12 stage order 8 scheme and introduce a new stage 11 with " } {XPPEDIT 18 0 "c[11] = 1;" "6#/&%\"cG6#\"#6\"\"\"" }{TEXT -1 3 ", " } {XPPEDIT 18 0 "a[11,2] = 0" "6#/&%\"aG6$\"#6\"\"#\"\"!" }{TEXT -1 3 ", " }{XPPEDIT 18 0 "a[11,3] = 0;" "6#/&%\"aG6$\"#6\"\"$\"\"!" }{TEXT -1 3 ", " }{XPPEDIT 18 0 "`b*`[2]=0" "6#/&%#b*G6#\"\"#\"\"!" }{TEXT -1 3 ", " }{XPPEDIT 18 0 "`b*`[3]=0" "6#/&%#b*G6#\"\"$\"\"!" }{TEXT -1 3 ", " }{XPPEDIT 18 0 "`b*`[4]=0" "6#/&%#b*G6#\"\"%\"\"!" }{TEXT -1 7 " and " }{XPPEDIT 18 0 "`b*`[5]=0" "6#/&%#b*G6#\"\"&\"\"!" } {TEXT -1 58 ", where the weights of the order 7 scheme are denoted by \+ " }{XPPEDIT 18 0 "`b*`" "6#%#b*G" }{TEXT -1 1 "." }}{PARA 0 "" 0 "" {TEXT -1 1 " " }}{PARA 0 "" 0 "" {TEXT -1 156 "We incorporate the orde r 7 quadrature conditions, the row sum conditions for this stage and s tage-order conditions so that this new stage has stage-order 4." }} {PARA 0 "" 0 "" {TEXT -1 48 "We also incorporate the simplifying condi tions: " }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "Sum(`b*`[i ]*a[i,j],i=j+1..11)=`b*`[j]*(1-c[j])" "6#/-%$SumG6$*&&%#b*G6#%\"iG\"\" \"&%\"aG6$F+%\"jGF,/F+;,&F0F,F,F,\"#6*&&F)6#F0F,,&F,F,&%\"cG6#F0!\"\"F ," }{TEXT -1 1 "," }}{PARA 0 "" 0 "" {TEXT -1 5 "for " }{XPPEDIT 18 0 "j=1" "6#/%\"jG\"\"\"" }{TEXT -1 10 " ( with " }{XPPEDIT 18 0 "c[1 ]=0" "6#/&%\"cG6#\"\"\"\"\"!" }{TEXT -1 5 " ), " }{XPPEDIT 18 0 "j = \+ 8;" "6#/%\"jG\"\")" }{TEXT -1 3 ", " }{XPPEDIT 18 0 "j=9" "6#/%\"jG\" \"*" }{TEXT -1 7 " and " }{XPPEDIT 18 0 "j=10" "6#/%\"jG\"#5" } {TEXT -1 2 ". " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 399 "`Qeqs*` := subs(b=`b*`,QuadratureConditions(7,1 1,'expanded')):\nSO_eqs := [add(a[11,j],j=1..10)=c[11],add(a[11,j]*c[j ],j=2..10)=1/2*c[11]^2,\n add(a[11,j]*c[j]^2,j=2..10)=1/3*c[11] ^3,add(a[11,j]*c[j]^3,j=2..10)=1/4*c[11]^4]:\n`simp_eqs*` := [add(`b*` [i]*a[i,1],i=2..11)=`b*`[1],seq(add(`b*`[i]*a[i,j],i=j+1..11)=`b*`[j]* (1-c[j]),j=[8,9,10])]:\n`cdns*` := [op(`simp_eqs*`),op(SO_eqs),op(`Qeq s*`)]:" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 144 "e14 := `union`(remove(u_->member(op(1,lhs(u_)),[11,1 2]) or op(0,lhs(u_))=b,e13),\n \{c[11]=1,a[11,2]=0,a[11,3]=0,s eq(`b*`[i]=0,i=2..5)\}):\n" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 53 "We have 15 equations for the 15 unknown coeffic ients." }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 60 "eqns3 := subs(e14, `cdns*`):\nnops(%);\nindets(eqns3);\nnops(%);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#\"#:" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#<1&%\"aG6$\"#6 \"\"\"&F%6$F'\"#5&F%6$F'\"\"(&F%6$F'\"\")&F%6$F'\"\"*&F%6$F'\"\"%&F%6$ F'\"\"&&F%6$F'\"\"'&%#b*G6#F=&F?6#F.&F?6#F1&F?6#F4&F?6#F+&F?6#F'&F?6#F (" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#\"#:" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 22 "infolevel[solve] : = 4:" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 74 "e15 := solve(\{op(e qns3)\}):\ninfolevel[solve] := 0:\ne16 := `union`(e14,e15):" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 3 "e16" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 4346 "e16 := \{a[9,7] \+ = 42008080033354305590804322944084805264441066760038302359736803632/48 65302423216534910074823287811605599629170295030631799935804001, a[6,2] = 0, a[5,2] = 0, a[7,2] = 0, a[4,2] = 0, a[9,3] = 0, a[10,3] = 0, a[8 ,3] = 0, a[9,5] = -273749409411654060286948141164828452109898379203526 945684314474186724062841643/604275839513775525039678406538255891171674 43021628367691773162913607984889600, a[7,3] = 0, a[6,3] = 0, c[7] = 23 /148, `b*`[8] = -509771598215811385123057257/2102822699690856982641017 60, a[4,1] = 12257/317988, a[10,9] = -50977737930792808232204417497248 979399878217280011103197862899/130091569456491367561328031408183735864 4964393191337994183389184, a[10,6] = 636438086325907167711225923645550 6477417699780613300364807/11504281745849421334065790915494434388149883 49188394057728, a[7,1] = 480285555889619/7997789253816320, a[7,6] = 86 4813381633887/51718297665732608, c[4] = 12257/79497, a[7,5] = -1645076 9949384489/490009784398315520, a[5,1] = 14131686489425/35833975601529, a[6,4] = 9757757988/41995818067, a[6,1] = 4418011/96055225, a[10,1] = -1793603946322260900828212460706877142477132870159527/231309756851199 0753781649719556084665131024900300800, a[5,4] = 17619604667500/1194465 8533843, c[3] = 24514/238491, a[5,3] = -17700454220625/11944658533843, a[4,3] = 12257/105996, a[6,5] = 19921512441/106300094275, a[11,3] = 0 , a[10,8] = 10744247163960019876833255044784609639/5347618047399013488 25491947768304503296, c[9] = 29104120198/33218531707, a[9,8] = 6684597 80930716338000066627236927417191947396177093524377824/7110045294888464 3779799087713322002499799983434685642170251833, a[9,6] = 1665637251887 4512738268060504309924900437672263609245028809229865738327731797537/42 7699013853393052278207638577101600909762793055014962828220300791353839 4549504, c[10] = 83/91, a[10,2] = 0, a[9,2] = 0, a[11,2] = 0, c[11] = \+ 1, a[8,6] = 1205563885850790193966807/132365727505912221222912, `b*`[5 ] = 0, `b*`[3] = 0, `b*`[4] = 0, `b*`[2] = 0, a[8,5] = -31855618223522 2634647116091/27172411532484037214054400, a[7,4] = 5216210655669646053 8643/464887033284472899365888, a[3,1] = 9950845450/511901613729, c[6] \+ = 34/73, c[5] = 50/129, a[8,1] = -7001048352088587137143/4449830351030 385148800, a[3,2] = 42666469916/511901613729, a[8,2] = 0, `b*`[11] = 5 752173075461/1925544586212, a[9,4] = -33154430743436594047794221493873 97712986453181141168247590906370819301077749322753/4985171126416088728 07821838566847987729514312398278633788185835348997643218553476, a[11,8 ] = 62408280667309375445301959066100433563/483832004625198384951235555 9327451645440, a[11,9] = 114560982224949367761811372535950664299815320 5603226883141207089968379/26902802166822768476367427840835743139559294 105398677975568538466119680, c[8] = 142/141, a[11,6] = 214033123542582 9844389060818616719848637810765257179167/24542818203601189763802825281 5369258646052549878743040, a[11,10] = -4089503568758746831390896780538 32/7674292714443204070455739595109785, c[2] = 9/142, `b*`[6] = 6228433 5162928966987066121/175437206755843240272669696, a[8,7] = 254/39, a[2, 1] = 9/142, `b*`[10] = -7218534073012286740367561/39864563061532249540 98780, a[10,7] = 27725164402569748756040320433848245155581006369/25441 59473655547770881695354241256106302348256, `b*`[9] = 27016024896461436 40362891402924962500766379885231830470264478480621389/1688575269294196 295712420798364925687791337062814161130291144634516480, a[10,4] = -147 76874123722838192315406145167687512425345723701/1847893530366076701102 014146927696206329050105856, a[11,5] = -246044720162308748108107126829 066792329071/21999103311417807170100413255475150848000, a[9,1] = -2062 9396399716689122801179264428539394462855874226604463554767070845753369 /428817596627701559566575134700121144880760179811281053073755949631523 53200, a[11,7] = 1573990926219809229258666534611598771063240529/214535 514317495538101650508596881057760818080, a[11,4] = -433889546009521405 913741133329446636837810749/181488796115643001621310691169322233346938 880, a[8,4] = -2522765548294599044197935933/19159631954937584476779017 92, `b*`[1] = 715953338020208413/16094552857868225760, `b*`[7] = 18430 9146777472302831695872/742417767522160213324368465, a[10,5] = -1958702 0919884661714856757105130246995757906603/29118932969425320385661828681 70393629789388800, a[11,1] = -7757739937862944832927743694336116203639 371542761/5225100678421794325654850845451340473577260032000\}:" } {TEXT -1 0 "" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 20 "printed coefficients" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 289 "subs(e16,matrix([seq([c[i],seq(a[i ,j],j=1..i-1),``$(7-i)],i=2..7)]));\nfor ii from 8 to 11 do\n print( ``);\n print(c[ii]=subs(e16,c[ii])); \n for jj to ii-1 do\n p rint(a[ii,jj]=subs(e16,a[ii,jj]));\n end do:\nend do:\n``;\nfor ii t o 11 do\n print(`b*`[ii]=subs(e16,`b*`[ii]));\nend do:" }}{PARA 11 " " 1 "" {XPPMATH 20 "6#K%'matrixG6#7(7)#\"\"*\"$U\"F(%!GF+F+F+F+7)#\"&9 X#\"'\"\\Q##\"+]a%3&**\"-HPh,>^#\",;*pkmUF2F+F+F+F+7)#\"&dA\"\"&(\\z#F 7\"'))zJ\"\"!#F7\"''*f5F+F+F+7)#\"#]\"$H\"#\"/D%*['oJT\"\"/H:gvR$e$F;# !/D1Aa/q<\"/VQ`eY%>\"#\"/+vm/'>w\"FGF+F+7)#\"#M\"#t#\"(6!=W\")D_0'*F;F ;#\"+))zvd(*\",n!=e*>%#\",TC^@*>\"-vU4+j5F+7)#\"#B\"$[\"#\"0>'*)eb&G![ \"1?j\"QD*y(*zF;F;#\"8V'Q0Y'pcl5i@&\"9))eO**GZ%GLq)[Y#!2*[%Q\\*p2X;\"3 ?bJ)R%y4+\\#\"0()Qj\"Q8[')\"23EtlwH=<&Q)pprint346\"" }}{PARA 11 "" 1 " " {XPPMATH 20 "6#%!G" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/&%\"cG6#\"\") #\"$U\"\"$T\"" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/&%\"aG6$\"\")\"\"\"# !7Vr8(e)3_$[5+(\"7+)[^QI5NI)\\W" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/&% \"aG6$\"\")\"\"#\"\"!" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/&%\"aG6$\"\" )\"\"$\"\"!" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/&%\"aG6$\"\")\"\"%#!=L f$z>W!*f%H[bwAD\"=#z,znZ%eP\\&>jf\">" }}{PARA 11 "" 1 "" {XPPMATH 20 " 6#/&%\"aG6$\"\")\"\"&#!<\"4;rkMEANA=c&=$\";+W09s.%[K:Tsr#" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/&%\"aG6$\"\")\"\"'#\":2o'R>!z]e)Qc07\"97HA@A\" f]FdOK\"" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/&%\"aG6$\"\")\"\"(#\"$a# \"#R" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%!G" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/&%\"cG6#\"\"*#\",)>?T5H\",2<`=K$" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/&%\"aG6$\"\"*\"\"\"#!eopLvX32nZbjWgEU(e&GY%RR&GWEz6!G7 *o;(*R'RH1#\"eo+KN_J'\\fvtI0\"G6)z,w!)[9@,qM^dm&f:qFmf<)G%" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/&%\"aG6$\"\"*\"\"#\"\"!" }}{PARA 11 "" 1 " " {XPPMATH 20 "6#/&%\"aG6$\"\"*\"\"$\"\"!" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/&%\"aG6$\"\"*\"\"%#!]p`FK\\x2,$>3P14fZ#o69\"=`k)Hr(R(Q \\@UzZSfOMuIW:L\"\\pwMb=Kk(**[`$e=)yL'y#)R7V^Hx)z%ocQ=#y!G()3;k7r^)\\ " }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/&%\"aG6$\"\"*\"\"&#!ioV;%G1Cn=uWJ %oXp_.#z$)*)4@XG[;T\"[pGgSl6%4%\\PF\"ho+'*)[)zg8H;t>(\\;y`2*>^ov48B" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/&%\"aG6$\"#5\"\"#\"\"!" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/&%\" aG6$\"#5\"\"$\"\"!" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/&%\"aG6$\"#5\" \"%#!S,PsX`U7von^91aJ#>QGsBT(ox9\"Rce5]!Hj?'pFp99?5,n2m.`$*y%=" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#/&%\"aG6$\"#5\"\"&#!P.m!zv&*pCI^5dn&[r hY))>4-(e>\"O+))Q*yHORq\"oG=m&Q?`UpH$*=\"H" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/&%\"aG6$\"#5\"\"'#\"en2[O+Lh!y*p+'R;ZUu5\"H'H.XIox%>\\D)[8!*RZ!=wM&" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/&%\"aG6$\"#5\"\"*#!in**G'y>.6,!G<#y)*Rz*[s \\$Rk\\ket$=39.G8cn8\\c%p:4I\"" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#%!G" }}{PARA 11 "" 1 "" {XPPMATH 20 "6 #/&%\"cG6#\"#6\"\"\"" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/&%\"aG6$\"#6 \"\"\"#!RhFar$RO?;hL%pVx#H$[%H'y$*Rxv(\"R+?.gsdt/M^a%3&[lDVz@%y15D_" } }{PARA 11 "" 1 "" {XPPMATH 20 "6#/&%\"aG6$\"#6\"\"#\"\"!" }}{PARA 11 " " 1 "" {XPPMATH 20 "6#/&%\"aG6$\"#6\"\"$\"\"!" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/&%\"aG6$\"#6\"\"%#!N\\2\"y$ojY%HL8TP\"fS@&4ga*)QV\"N!) )QpMLAKp6p58i,Ik:hz)[\"=" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/&%\"aG6$ \"#6\"\"&#!Kr!HBzm!Ho72\"3\"[(3B;?Z/Y#\"J+![3:vaD8/5qr!yT6L5**>#" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#/&%\"aG6$\"#6\"\"'#\"Xn\"zrDl2\"yj[)>n h=31*QW)HeUN7LS@\"WSIuy)\\D0Y'e#p`\"GDG!Qw*=,O?=GaC" }}{PARA 11 "" 1 " " {XPPMATH 20 "6#/&%\"aG6$\"#6\"\"(#\"OH0Cj5x)f6Y`m'e#H#4)>i#4*Rd\"\"N !3=3wd5)of30l,\"Qb\\IXaP4tm!G3C'\"ISak^uKfbN7&\\Q)>DY+KQ[" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/&%\"aG6$\"#6\"\"*#\"aoz$o**3279$)oA.c?`\")* Hk1&f`s8\"=wn$\\\\A#)4c9\"\"bo!o>hYQ&ob(zn)R0THf&RJuN3%yUnjZoF#o;-G!p# " }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/&%\"aG6$\"#6\"#5#!BKQ0y'*3RJoue(o N]*3%\"C&y4^fRdXqS?VWr#Huw" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%!G" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#/&%#b*G6#\"\"\"#\"38%3--QL&fr\"5gdAoy& Gb%4;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/&%#b*G6#\"\"#\"\"!" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/&%#b*G6#\"\"$\"\"!" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/&%#b*G6#\"\"%\"\"!" }}{PARA 11 "" 1 "" {XPPMATH 20 "6# /&%#b*G6#\"\"&\"\"!" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/&%#b*G6#\"\"'# \";@h1()p'*GH;NVGi\"<'ppEFSK%ev1sVv\"" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/&%#b*G6#\"\"(#\"%Hp_d )o\"" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/&%#b*G6#\"#5#!:hvOSnG7I2M&=s \":!y)4a\\A`hIck)R" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/&%#b*G6#\"#6#\" .ha2t@v&\".7ieWb#>" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}}{PARA 0 "" 0 "" {TEXT -1 0 "" }} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 104 "`RK7_11eqs*` := subs(b=`b*` ,[op(RowSumConditions(11,'expanded')),op(OrderConditions(7,11,'expande d'))]):" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 5 "Check" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 57 "subs(e16,`RK7_11 eqs*`):\nmap(u->lhs(u)-rhs(u),%);\nnops(%);" }}{PARA 12 "" 1 "" {XPPMATH 20 "6#7[q\"\"!F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F $F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F $F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F $" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#\"#&*" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 93 "We can calculate the principal \+ error norm, that is, the 2-norm of the principal error terms. " }} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 185 "`errterms7_11*` := subs(b=` b*`,PrincipalErrorTerms(7,11,'expanded')):\nsm := 0:\nfor ct to nops(` errterms7_11*`) do\n sm := sm+(evalf(subs(e16,`errterms7_11*`[ct]))) ^2;\nend do:\nsqrt(sm);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#$\"+Gw`c5!# 9" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 138 "We can include the new stage for the embedded scheme as an additional 13 th stage added to the order 8 scheme along with the coefficients " } {XPPEDIT 18 0 "a[13,11] = 0;" "6#/&%\"aG6$\"#8\"#6\"\"!" }{TEXT -1 7 " and " }{XPPEDIT 18 0 "a[13,12] = 0;" "6#/&%\"aG6$\"#8\"#7\"\"!" } {TEXT -1 1 "." }}{PARA 0 "" 0 "" {TEXT -1 13 "The weights " } {XPPEDIT 18 0 "`b*`[i]" "6#&%#b*G6#%\"iG" }{TEXT -1 7 " for " } {XPPEDIT 18 0 "i=1" "6#/%\"iG\"\"\"" }{TEXT -1 89 " . . 10 of the 13 s tage combined scheme are those of the 11 stage scheme and the weight \+ " }{XPPEDIT 18 0 "`b*`[11];" "6#&%#b*G6#\"#6" }{TEXT -1 34 " in the 1 1 stage scheme becomes " }{XPPEDIT 18 0 "`b*`[13];" "6#&%#b*G6#\"#8" }{TEXT -1 25 " in the 13 stage scheme." }}{PARA 0 "" 0 "" {TEXT -1 8 "We set " }{XPPEDIT 18 0 "c[13] = 1;" "6#/&%\"cG6#\"#8\"\"\"" }{TEXT -1 3 ", " }{XPPEDIT 18 0 "`b*`[11] = 0;" "6#/&%#b*G6#\"#6\"\"!" } {TEXT -1 7 " and " }{XPPEDIT 18 0 "`b*`[12] = 0;" "6#/&%#b*G6#\"#7\" \"!" }{TEXT -1 1 "." }}{PARA 0 "" 0 "" {TEXT -1 66 "We can make the or der 8 scheme into a 13 stage scheme by setting " }{XPPEDIT 18 0 "b[13 ] = 0;" "6#/&%\"bG6#\"#8\"\"!" }{TEXT -1 1 "." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 201 "e17 := \{c[ 13]=1,seq(a[13,i]=subs(e16,a[11,i]),i=1..10),a[13,11]=0,a[13,12]=0,b[1 3]=0,\nseq(`b*`[i]=subs(e16,`b*`[i]),i=1..10),`b*`[11]=0,`b*`[12]=0,`b *`[13]=subs(e16,`b*`[11])\}:\ne18 := `union`(e13,e17):" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}} {SECT 1 {PARA 4 "" 0 "" {TEXT -1 3 "e18" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 7109 "e18 := \{a[9,7] = 420 08080033354305590804322944084805264441066760038302359736803632/4865302 423216534910074823287811605599629170295030631799935804001, a[6,2] = 0, a[5,2] = 0, a[7,2] = 0, b[10] = -1108225296327029096435947/4056790758 93979729103310, a[4,2] = 0, a[9,3] = 0, a[10,3] = 0, a[8,3] = 0, a[12, 3] = 0, a[9,5] = -2737494094116540602869481411648284521098983792035269 45684314474186724062841643/6042758395137755250396784065382558911716744 3021628367691773162913607984889600, a[7,3] = 0, a[6,3] = 0, c[7] = 23/ 148, a[12,2] = 0, `b*`[8] = -509771598215811385123057257/2102822699690 85698264101760, a[4,1] = 12257/317988, a[10,9] = -50977737930792808232 204417497248979399878217280011103197862899/130091569456491367561328031 4081837358644964393191337994183389184, a[10,6] = 636438086325907167711 2259236455506477417699780613300364807/11504281745849421334065790915494 43438814988349188394057728, a[7,1] = 480285555889619/7997789253816320, a[7,6] = 864813381633887/51718297665732608, c[4] = 12257/79497, a[7,5 ] = -16450769949384489/490009784398315520, c[13] = 1, a[12,6] = 512604 71529841028040709654458903254781320136131844164563/2099802377631854690 7382302106788168158765514131585630208, a[5,1] = 14131686489425/3583397 5601529, a[6,4] = 9757757988/41995818067, a[6,1] = 4418011/96055225, a [10,1] = -1793603946322260900828212460706877142477132870159527/2313097 568511990753781649719556084665131024900300800, b[8] = -423708721416993 4312729607487/12735791394214625116604076160, a[5,4] = 17619604667500/1 1944658533843, c[3] = 24514/238491, a[5,3] = -17700454220625/119446585 33843, b[5] = 0, a[4,3] = 12257/105996, b[2] = 0, b[3] = 0, b[4] = 0, \+ b[6] = 2335496795323782464411846394611/6598368783294020109895379936256 , a[13,8] = 62408280667309375445301959066100433563/4838320046251983849 512355559327451645440, a[12,1] = -753408116554498247829620233592204921 0803875045423/19219575665440848756074598051658416387002479820800, a[6, 5] = 19921512441/106300094275, a[11,3] = 0, a[13,1] = -775773993786294 4832927743694336116203639371542761/52251006784217943256548508454513404 73577260032000, a[13,2] = 0, a[13,3] = 0, b[7] = 763607337652714356537 5240869888/30725949199261296642046754748645, b[1] = 550384465135292538 01/1239280570055853383520, a[10,8] = 107442471639600198768332550447846 09639/534761804739901348825491947768304503296, c[9] = 29104120198/3321 8531707, a[9,8] = 6684597809307163380000666272369274171919473961770935 24377824/7110045294888464377979908771332200249979998343468564217025183 3, a[9,6] = 1665637251887451273826806050430992490043767226360924502880 9229865738327731797537/42769901385339305227820763857710160090976279305 50149628282203007913538394549504, `b*`[12] = 0, c[11] = 143/149, c[10] = 83/91, `b*`[11] = 0, a[13,11] = 0, a[10,2] = 0, a[13,12] = 0, b[13] = 0, a[9,2] = 0, c[12] = 1, a[11,2] = 0, a[13,10] = -4089503568758746 83139089678053832/7674292714443204070455739595109785, a[8,6] = 1205563 885850790193966807/132365727505912221222912, `b*`[5] = 0, `b*`[3] = 0, `b*`[4] = 0, `b*`[2] = 0, a[8,5] = -318556182235222634647116091/27172 411532484037214054400, a[7,4] = 52162106556696460538643/46488703328447 2899365888, a[3,1] = 9950845450/511901613729, a[13,9] = 11456098222494 93677618113725359506642998153205603226883141207089968379/2690280216682 2768476367427840835743139559294105398677975568538466119680, c[6] = 34/ 73, a[12,8] = -1539218116260541896259682954580256454049/45346708307503 60983422999946409310393344, c[5] = 50/129, a[8,1] = -70010483520885871 37143/4449830351030385148800, a[12,9] = 241886539350268429372116296787 271276553970618941104594460614948326132797451456131/124066963266246589 2528916120041123051054026283188324780101555345343662316090597376, a[3, 2] = 42666469916/511901613729, a[8,2] = 0, b[11] = 2460988291206213825 688467985/1756342789520947764222671739, b[9] = 40850529129113324176099 5514121984335914363927884426780078325258228227984174126699/21262487461 2193697466655159405635202123821166475348052734199905546455860773575680 , a[9,4] = -3315443074343659404779422149387397712986453181141168247590 906370819301077749322753/498517112641608872807821838566847987729514312 398278633788185835348997643218553476, c[8] = 142/141, c[2] = 9/142, `b *`[6] = 62284335162928966987066121/175437206755843240272669696, a[8,7] = 254/39, a[2,1] = 9/142, a[11,5] = -88434883677458471507044048563302 6464653487653/92725983515963799584249219499787659014963200, a[11,1] = \+ -3587625717068952487214493441966897048737050755812600710793/3015733164 033229624772006086429467685046639983706974412800, `b*`[10] = -72185340 73012286740367561/3986456306153224954098780, a[10,7] = 277251644025697 48756040320433848245155581006369/2544159473655547770881695354241256106 302348256, `b*`[9] = 2701602489646143640362891402924962500766379885231 830470264478480621389/168857526929419629571242079836492568779133706281 4161130291144634516480, a[11,9] = 112633808253272720979874303367503891 597499261046700689572459050065039333987335667/140451429124503453281218 1377119034501014039830518218465064579291611563374608650240, a[10,4] = \+ -14776874123722838192315406145167687512425345723701/184789353036607670 1102014146927696206329050105856, a[11,7] = 142363419491686507162007051 071007722765323162710521029/126348872023682615658074497713357280822735 87905775840, a[12,10] = -80556486832245966191717452425924975/414445409 518676597565032008051106461, a[9,1] = -2062939639971668912280117926442 8539394462855874226604463554767070845753369/42881759662770155956657513 470012114488076017981128105307375594963152353200, a[12,5] = -206105782 09826329263318986584876108069323/7356333776438834884293014575840932659 200, a[8,4] = -2522765548294599044197935933/19159631954937584476779017 92, a[11,6] = 26823469063654084387375587616552322383082061411417182757 389742951/354129974476368167547362064708712305722874429612364230165463 0400, `b*`[1] = 715953338020208413/16094552857868225760, `b*`[7] = 184 309146777472302831695872/742417767522160213324368465, b[12] = 48087079 37311/50545545388065, a[11,8] = 64747617454909275289531520412519442831 235890581/1269317188117975960996670628974453443777854830080, a[12,11] \+ = 2944781680874500347594142792814463350/496516138307367698361021809603 0654529, a[10,5] = -19587020919884661714856757105130246995757906603/29 11893296942532038566182868170393629789388800, a[12,7] = -3077214437173 472971196810795615384000211457151011/127243559258228082059705943206011 6893200684680384, a[11,4] = -54530117112678047312115018372628169446612 01619903/764973320899716397072448644710733101329290196992, a[11,10] = \+ -10612202518573994431153697720606405883/670825466582598467787545949768 31647575, `b*`[13] = 5752173075461/1925544586212, a[13,6] = 2140331235 425829844389060818616719848637810765257179167/245428182036011897638028 252815369258646052549878743040, a[13,7] = 1573990926219809229258666534 611598771063240529/214535514317495538101650508596881057760818080, a[13 ,5] = -246044720162308748108107126829066792329071/21999103311417807170 100413255475150848000, a[12,4] = 2376960874527867178022703752830342628 59273455/60688480889936839261131818793519097177034752, a[13,4] = -4338 89546009521405913741133329446636837810749/1814887961156430016213106911 69322233346938880\}:" }{TEXT -1 0 "" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 20 "printed coefficients" }}{PARA 0 "" 0 " " {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 362 "subs(e18, matrix([seq([c[i],seq(a[i,j],j=1..i-1),``$(7-i)],i=2..7)]));\nfor ii f rom 8 to 13 do\n print(``);\n print(c[ii]=subs(e18,c[ii])); \n f or jj to ii-1 do\n print(a[ii,jj]=subs(e18,a[ii,jj]));\n end do :\nend do:print(``);\nfor ii to 13 do\n print(b[ii]=subs(e18,b[ii])) ;\nend do:print(``);\nfor ii to 13 do\n print(`b*`[ii]=subs(e18,`b*` [ii]));\nend do:" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#K%'matrixG6#7(7)# \"\"*\"$U\"F(%!GF+F+F+F+7)#\"&9X#\"'\"\\Q##\"+]a%3&**\"-HPh,>^#\",;*pk mUF2F+F+F+F+7)#\"&dA\"\"&(\\z#F7\"'))zJ\"\"!#F7\"''*f5F+F+F+7)#\"#]\"$ H\"#\"/D%*['oJT\"\"/H:gvR$e$F;#!/D1Aa/q<\"/VQ`eY%>\"#\"/+vm/'>w\"FGF+F +7)#\"#M\"#t#\"(6!=W\")D_0'*F;F;#\"+))zvd(*\",n!=e*>%#\",TC^@*>\"-vU4+ j5F+7)#\"#B\"$[\"#\"0>'*)eb&G![\"1?j\"QD*y(*zF;F;#\"8V'Q0Y'pcl5i@&\"9) )eO**GZ%GLq)[Y#!2*[%Q\\*p2X;\"3?bJ)R%y4+\\#\"0()Qj\"Q8[')\"23EtlwH=<&Q )pprint356\"" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%!G" }}{PARA 11 "" 1 " " {XPPMATH 20 "6#/&%\"cG6#\"\")#\"$U\"\"$T\"" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/&%\"aG6$\"\")\"\"\"#!7Vr8(e)3_$[5+(\"7+)[^QI5NI)\\W" } }{PARA 11 "" 1 "" {XPPMATH 20 "6#/&%\"aG6$\"\")\"\"#\"\"!" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/&%\"aG6$\"\")\"\"$\"\"!" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/&%\"aG6$\"\")\"\"%#!=Lf$z>W!*f%H[bwAD\"=#z,znZ%eP\\&>j f\">" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/&%\"aG6$\"\")\"\"&#!<\"4;rkME ANA=c&=$\";+W09s.%[K:Tsr#" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/&%\"aG6$ \"\")\"\"'#\":2o'R>!z]e)Qc07\"97HA@A\"f]FdOK\"" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/&%\"aG6$\"\")\"\"(#\"$a#\"#R" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%!G" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/&%\"cG6#\"\"*# \",)>?T5H\",2<`=K$" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/&%\"aG6$\"\"*\" \"\"#!eopLvX32nZbjWgEU(e&GY%RR&GWEz6!G7*o;(*R'RH1#\"eo+KN_J'\\fvtI0\"G 6)z,w!)[9@,qM^dm&f:qFmf<)G%" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/&%\"aG 6$\"\"*\"\"#\"\"!" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/&%\"aG6$\"\"*\" \"$\"\"!" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/&%\"aG6$\"\"*\"\"%#!]p`FK \\x2,$>3P14fZ#o69\"=`k)Hr(R(Q\\@UzZSfOMuIW:L\"\\pwMb=Kk(**[`$e=)yL'y#) R7V^Hx)z%ocQ=#y!G()3;k7r^)\\" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/&%\"a G6$\"\"*\"\"&#!ioV;%G1Cn=uWJ%oXp_.#z$)*)4@XG[;T\"[pGgSl6%4%\\PF\"ho+'* )[)zg8H;t>(\\;y`2*>^ov4 8B" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/&%\"aG6$\"#5\"\"#\"\"!" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#/&%\"aG6$\"#5\"\"$\"\"!" }}{PARA 11 " " 1 "" {XPPMATH 20 "6#/&%\"aG6$\"#5\"\"%#!S,PsX`U7von^91aJ#>QGsBT(ox9 \"Rce5]!Hj?'pFp99?5,n2m.`$*y%=" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/&% \"aG6$\"#5\"\"&#!P.m!zv&*pCI^5dn&[rhY))>4-(e>\"O+))Q*yHORq\"oG=m&Q?`Up H$*=\"H" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/&%\"aG6$\"#5\"\"'#\"en2[O+ Lh!y*p+'R;ZUu5\"H'H.XIox%> \\D)[8!*RZ!=wM&" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/&%\"aG6$\"#5\"\"*# !in**G'y>.6,!G<#y)*Rz*[s\\$Rk\\ket$=39. G8cn8\\c%p:4I\"" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%!G" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/&%\"cG6#\"#6#\"$V\"\"$\\\"" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/&%\"aG6$\"#6\"\"\"#!en$z52g7ev]qt[q*o'>W$\\9s[_*oqrDwe $\"en+GTupq$)*Rm/&on%Hk31?xC'HK.kJt:I" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/&%\"aG6$\"#6\"\"#\"\"!" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/&%\"aG6 $\"#6\"\"$\"\"!" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/&%\"aG6$\"#6\"\"%# !R.*>;?hY%p\"GEP=]67t/yE6<,`a\"Q#*p>!HH85L2rW'[C2(R;(**3Kt\\w" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#/&%\"aG6$\"#6\"\"&#!N`w[`YYEIj&[S/2:Ze un$)[V))\"M+K'\\,fwy*\\>#\\Ue*zjf^$)fs#*" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/&%\"aG6$\"#6\"\"'#\"\\o^Hu*QdF=<9Th?3$QABb;wevtQ%3aO1pM#o#\"[o+ /ja;IUO7'HW(Gs0Br3Z1itan\"ojZu*HTN" }}{PARA 11 "" 1 "" {XPPMATH 20 "6# /&%\"aG6$\"#6\"\"(#\"WH5_5F;B`wAx+r502?;2lo\"\\>MOU\"\"VSex0zetA3GdLr( \\u!ech#oB?()[j7" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/&%\"aG6$\"#6\"\") #\"P\"e!*eBJGW>DT?:`*Gv#4\\Xr P\"=7G`M]C\"H9XS\"" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/&%\"aG6$\"#6\"# 5#!G$)eS11s(p`6V%*Rd=D?71\"\"GvvkJo(\\fa(yn%)f#ema#3n" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%!G" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/&%\"cG6#\" #7\"\"\"" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/&%\"aG6$\"#7\"\"\"#!RBa/v Q!3@\\?#fL-iHyC)\\al63Mv\"S+3#)zC+(Q;%e;0)fugv[3Wlcd>#>" }}{PARA 11 " " 1 "" {XPPMATH 20 "6#/&%\"aG6$\"#7\"\"#\"\"!" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/&%\"aG6$\"#7\"\"$\"\"!" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/&%\"aG6$\"#7\"\"%#\"NbMFfGEMIGv.F-yr'y_u3'pP#\"M_Z.xr4>Nz==8h#R o$**)3[)og" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/&%\"aG6$\"#7\"\"&#!JB$p !3h([e')*=LEHj#)4#y0h?\"I+#fE$4%ed9IH%)[$)QkxLjN(" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/&%\"aG6$\"#7\"\"'#\"YjX;W=8O,K\"yaK!*eWl42/G5%)H:Zg7& \"Y3-j&eJT^l(e\"o\")y1@I#Q2pa=jxB!)*4#" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/&%\"aG6$\"#7\"\"(#!R65:d9@+SQ:cz5o>rHZtrV9s2$\"R%Q!o%o+K*o6g?Vf qf?3G#e#fNCF\"" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/&%\"aG6$\"#7\"\")#! I\\SXc-eaHofi*=agi6=#R:\"IWLR5$4k%***HU$)4O]2$3nMX" }}{PARA 11 "" 1 " " {XPPMATH 20 "6#/&%\"aG6$\"#7\"\"*#\"\\pJhX^uzKhK[\\hgWf/6%*=1(Rbw7F( y'H;@PH%o-NRl)=C\"]pwtf!4;BmV`Mb:5!yC$)=$GES0^I7T+7;*GD*eYiEjp1C\"" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#/&%\"aG6$\"#7\"#5#!Dv\\#fU_ur\">mfCKo[ c0)\"Ehk5^!3?.lvfw'=&4aW9%" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/&%\"aG6 $\"#7\"#6#\"F]LY9GzUTfZ.]u3o\"yWH\"FHXlIg4=-h$)pntIQh^'\\" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%!G" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/&%\"cG6 #\"#8\"\"\"" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/&%\"aG6$\"#8\"\"\"#!Rh Far$RO?;hL%pVx#H$[%H'y$*Rxv(\"R+?.gsdt/M^a%3&[lDVz@%y15D_" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/&%\"aG6$\"#8\"\"#\"\"!" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/&%\"aG6$\"#8\"\"$\"\"!" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/&%\"aG6$\"#8\"\"%#!N\\2\"y$ojY%HL8TP\"fS@&4ga*)QV\"N!))QpMLAKp6 p58i,Ik:hz)[\"=" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/&%\"aG6$\"#8\"\"&# !Kr!HBzm!Ho72\"3\"[(3B;?Z/Y#\"J+![3:vaD8/5qr!yT6L5**>#" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/&%\"aG6$\"#8\"\"'#\"Xn\"zrDl2\"yj[)>nh=31*QW)HeUN 7LS@\"WSIuy)\\D0Y'e#p`\"GDG!Qw*=,O?=GaC" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/&%\"aG6$\"#8\"\"(#\"OH0Cj5x)f6Y`m'e#H#4)>i#4*Rd\"\"N!3=3wd5)of3 0l,\"Qb\\IXaP4tm!G3C'\"ISak^uKfbN7&\\Q)>DY+KQ[" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/&%\"aG6$\"#8\"\"*#\"aoz$o**3279$)oA.c?`\")*Hk1&f`s8\"= wn$\\\\A#)4c9\"\"bo!o>hYQ&ob(zn)R0THf&RJuN3%yUnjZoF#o;-G!p#" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/&%\"aG6$\"#8\"#5#!BKQ0y'*3RJoue(oN]*3%\"C&y 4^fRdXqS?VWr#Huw" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/&%\"aG6$\"#8\"#6 \"\"!" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/&%\"aG6$\"#8\"#7\"\"!" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#%!G" }}{PARA 11 "" 1 "" {XPPMATH 20 "6 #/&%\"bG6#\"\"\"#\"5,QDHN^Y%Q]&\"7?NQ`e0q0GR7" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/&%\"bG6#\"\"#\"\"!" }}{PARA 11 "" 1 "" {XPPMATH 20 "6# /&%\"bG6#\"\"$\"\"!" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/&%\"bG6#\"\"% \"\"!" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/&%\"bG6#\"\"&\"\"!" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/&%\"bG6#\"\"'#\"@6YRY=TkCyB`z'\\NB\"@ci$*z` *)4,-%H$yo$)f'" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/&%\"bG6#\"\"(#\"@)) )p3Cv`cVr_wL2Ow\"AX'[ZvY?k'Hh#*>\\fsI" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/&%\"bG6#\"\")#!=([2'HFJM*pT@(3PU\">gh2/m6DY@%R\"zNF\"" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/&%\"bG6#\"\"*#\"\\p*pET<%)zAG#e_Ky+yEW)y#RO9fL %)>79b*4wTK8\"H\"H0&3%\"\\p!ovNxgeXYb!**>MF0[`Zm6#Q7-_j0%f^lmup$>7Y([i 7#" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/&%\"bG6#\"#5#!:ZfV'4HqK'HD#36\" 95L5H(zR*e2zcS" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/&%\"bG6#\"#6#\"=&)z Y)oDQ@17H))4Y#\"=R%Hp_d)o\" " }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/&%#b*G6#\"#5#!:hvOSnG7I2M&=s\":!y )4a\\A`hIck)R" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/&%#b*G6#\"#6\"\"!" } }{PARA 11 "" 1 "" {XPPMATH 20 "6#/&%#b*G6#\"#7\"\"!" }}{PARA 11 "" 1 " " {XPPMATH 20 "6#/&%#b*G6#\"#8#\".ha2t@v&\".7ieWb#>" }}}{PARA 0 "" 0 " " {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}} {PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 151 "RK8_13eqs := [op(RowSumConditions(13,'expanded')),op(OrderConditi ons(8,13,'expanded'))]:\n`RK7_13eqs*` := subs(b=`b*`,OrderConditions(7 ,13,'expanded')):" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 6 "Check:" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 112 "subs( e18,RK8_13eqs):\nmap(u->lhs(u)-rhs(u),%);\nnops(%);\nsubs(e18,`RK7_13e qs*`):\nmap(u->lhs(u)-rhs(u),%);\nnops(%);" }}{PARA 12 "" 1 "" {XPPMATH 20 "6#7`x\"\"!F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F $F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F $F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F $F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F $F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F $F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F $F$F$F$F$F$F$F$F$F$F$F$F$" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#\"$7#" }} {PARA 12 "" 1 "" {XPPMATH 20 "6#7ap\"\"!F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$ F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$ F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#\"#&)" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 26 "#-------------------------" }} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}}{PARA 0 "" 0 "" {TEXT -1 52 "#---------------------------------------------------" }}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 38 "characteristics of the embedded scheme " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 36 "coefficients of the \+ combined scheme " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> \+ " 0 "" {MPLTEXT 1 0 7144 "ee := \{c[2]=9/142,\nc[3]=24514/238491,\nc[4 ]=12257/79497,\nc[5]=50/129,\nc[6]=34/73,\nc[7]=23/148,\nc[8]=142/141, \nc[9]=29104120198/33218531707,\nc[10]=83/91,\nc[11]=143/149,\nc[12]=1 ,\nc[13]=1,\n\na[2,1]=9/142,\na[3,1]=9950845450/511901613729,\na[3,2]= 42666469916/511901613729,\na[4,1]=12257/317988,\na[4,2]=0,\na[4,3]=122 57/105996,\na[5,1]=14131686489425/35833975601529,\na[5,2]=0,\na[5,3]=- 17700454220625/11944658533843,\na[5,4]=17619604667500/11944658533843, \na[6,1]=4418011/96055225,\na[6,2]=0,\na[6,3]=0,\na[6,4]=9757757988/41 995818067,\na[6,5]=19921512441/106300094275,\na[7,1]=480285555889619/7 997789253816320,\na[7,2]=0,\na[7,3]=0,\na[7,4]=52162106556696460538643 /464887033284472899365888,\na[7,5]=-16450769949384489/4900097843983155 20,\na[7,6]=864813381633887/51718297665732608,\na[8,1]=-70010483520885 87137143/4449830351030385148800,\na[8,2]=0,\na[8,3]=0,\na[8,4]=-252276 5548294599044197935933/1915963195493758447677901792,\na[8,5]=-31855618 2235222634647116091/27172411532484037214054400,\na[8,6]=12055638858507 90193966807/132365727505912221222912,\na[8,7]=254/39,\na[9,1]=-2062939 6399716689122801179264428539394462855874226604463554767070845753369/\n 42881759662770155956657513470012114488076017981128105307375594 963152353200,\na[9,2]=0,\na[9,3]=0,\na[9,4]=-3315443074343659404779422 149387397712986453181141168247590906370819301077749322753/\n 49 8517112641608872807821838566847987729514312398278633788185835348997643 218553476,\na[9,5]=-27374940941165406028694814116482845210989837920352 6945684314474186724062841643/\n 6042758395137755250396784065382 5589117167443021628367691773162913607984889600,\na[9,6]=16656372518874 512738268060504309924900437672263609245028809229865738327731797537/\n \+ 4276990138533930522782076385771016009097627930550149628282203007 913538394549504,\na[9,7]=420080800333543055908043229440848052644410667 60038302359736803632/\n 4865302423216534910074823287811605599629 170295030631799935804001,\na[9,8]=668459780930716338000066627236927417 191947396177093524377824/\n 711004529488846437797990877133220024 99799983434685642170251833,\na[10,1]=-17936039463222609008282124607068 77142477132870159527/\n 23130975685119907537816497195560846651 31024900300800,\na[10,2]=0,\na[10,3]=0,\na[10,4]=-14776874123722838192 315406145167687512425345723701/\n 1847893530366076701102014146 927696206329050105856, a[10,5]=-19587020919884661714856757105130246995 757906603/\n 2911893296942532038566182868170393629789388800,\n a[10,6]=6364380863259071677112259236455506477417699780613300364807/\n \+ 1150428174584942133406579091549443438814988349188394057728,\na[ 10,7]=27725164402569748756040320433848245155581006369/\n 254415 9473655547770881695354241256106302348256,\na[10,8]=1074424716396001987 6833255044784609639/534761804739901348825491947768304503296,\na[10,9]= -50977737930792808232204417497248979399878217280011103197862899/\n \+ 1300915694564913675613280314081837358644964393191337994183389184, \na[11,1]=-3587625717068952487214493441966897048737050755812600710793/ \n 3015733164033229624772006086429467685046639983706974412800, \na[11,2]=0,\na[11,3]=0,\na[11,4]=-54530117112678047312115018372628169 44661201619903/\n 76497332089971639707244864471073310132929019 6992,\na[11,5]=-884348836774584715070440485633026464653487653/\n \+ 92725983515963799584249219499787659014963200,\na[11,6]=268234690636 54084387375587616552322383082061411417182757389742951/\n 354129 9744763681675473620647087123057228744296123642301654630400, a[11,7]=14 2363419491686507162007051071007722765323162710521029/\n 1263488 7202368261565807449771335728082273587905775840,\na[11,8]=6474761745490 9275289531520412519442831235890581/\n 1269317188117975960996670 628974453443777854830080,\na[11,9]=11263380825327272097987430336750389 1597499261046700689572459050065039333987335667/\n 1404514291245 034532812181377119034501014039830518218465064579291611563374608650240, \na[11,10]=-10612202518573994431153697720606405883/6708254665825984677 8754594976831647575,\na[12,1]=-753408116554498247829620233592204921080 3875045423/\n 192195756654408487560745980516584163870024798208 00,\na[12,2]=0,\na[12,3]=0,\na[12,4]=237696087452786717802270375283034 262859273455/\n 60688480889936839261131818793519097177034752,\n a[12,5]=-20610578209826329263318986584876108069323/7356333776438834884 293014575840932659200,\na[12,6]=51260471529841028040709654458903254781 320136131844164563/\n 20998023776318546907382302106788168158765 514131585630208,\na[12,7]=-3077214437173472971196810795615384000211457 151011/\n 1272435592582280820597059432060116893200684680384,\n a[12,8]=-1539218116260541896259682954580256454049/45346708307503609834 22999946409310393344,\na[12,9]=241886539350268429372116296787271276553 970618941104594460614948326132797451456131/\n 12406696326624658 92528916120041123051054026283188324780101555345343662316090597376,\na[ 12,10]=-80556486832245966191717452425924975/41444540951867659756503200 8051106461,\na[12,11]=2944781680874500347594142792814463350/4965161383 073676983610218096030654529,\na[13,1]=-7757739937862944832927743694336 116203639371542761/\n 5225100678421794325654850845451340473577 260032000,\na[13,2]=0,\na[13,3]=0,\na[13,4]=-4338895460095214059137411 33329446636837810749/181488796115643001621310691169322233346938880,\na [13,5]=-246044720162308748108107126829066792329071/2199910331141780717 0100413255475150848000,\na[13,6]=2140331235425829844389060818616719848 637810765257179167/\n 24542818203601189763802825281536925864605 2549878743040,\na[13,7]=1573990926219809229258666534611598771063240529 /214535514317495538101650508596881057760818080,\na[13,8]=6240828066730 9375445301959066100433563/4838320046251983849512355559327451645440,\na [13,9]=114560982224949367761811372535950664299815320560322688314120708 9968379/\n 2690280216682276847636742784083574313955929410539867 7975568538466119680,\na[13,10]=-408950356875874683139089678053832/7674 292714443204070455739595109785,\na[13,11]=0,\na[13,12]=0,\n\nb[1]=5503 8446513529253801/1239280570055853383520,\nb[2]=0,\nb[3]=0,\nb[4]=0,\nb [5]=0,\nb[6]=2335496795323782464411846394611/6598368783294020109895379 936256,\nb[7]=7636073376527143565375240869888/307259491992612966420467 54748645,\nb[8]=-4237087214169934312729607487/127357913942146251166040 76160,\nb[9]=408505291291133241760995514121984335914363927884426780078 325258228227984174126699/\n 21262487461219369746665515940563520212 3821166475348052734199905546455860773575680,\nb[10]=-11082252963270290 96435947/405679075893979729103310,\nb[11]=2460988291206213825688467985 /1756342789520947764222671739,\nb[12]=4808707937311/50545545388065,\nb [13]=0,\n\n`b*`[1]=715953338020208413/16094552857868225760,\n`b*`[2]=0 ,\n`b*`[3]=0,\n`b*`[4]=0,\n`b*`[5]=0,\n`b*`[6]=62284335162928966987066 121/175437206755843240272669696,\n`b*`[7]=184309146777472302831695872/ 742417767522160213324368465,\n`b*`[8]=-509771598215811385123057257/210 282269969085698264101760,\n`b*`[9]=27016024896461436403628914029249625 00766379885231830470264478480621389/\n 168857526929419629571242 0798364925687791337062814161130291144634516480,\n`b*`[10]=-72185340730 12286740367561/3986456306153224954098780,\n`b*`[11]=0,\n`b*`[12]=0,\n` b*`[13]=5752173075461/1925544586212\}:" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 5 "Let " }{XPPEDIT 18 0 "T[8 , 13];" "6#&%\"TG6$\"\")\"#8" }{TEXT -1 128 " denote the vector whose components are the principal error terms of the 13 stage, order 8 sch eme (the error terms of order 9)." }}{PARA 0 "" 0 "" {TEXT -1 5 "Let \+ " }{XPPEDIT 18 0 "`T*`[7,13];" "6#&%#T*G6$\"\"(\"#8" }{TEXT -1 146 " \+ denote the vector whose components are the principal error terms of th e embedded 13 stage, order 7 scheme (the error terms of order 8) and l et " }{XPPEDIT 18 0 "`T*`[8,13];" "6#&%#T*G6$\"\")\"#8" }{TEXT -1 99 " denote the vector whose components are the error terms of order 9 o f the embedded order 7 scheme." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }} {PARA 0 "" 0 "" {TEXT -1 39 "Denote the 2-norm of these vectors by " }{XPPEDIT 18 0 "abs(abs(T[8,13]));" "6#-%$absG6#-F$6#&%\"TG6$\"\")\"#8 " }{TEXT -1 3 ", " }{XPPEDIT 18 0 "abs(abs(`T*`[7, 13]));" "6#-%$absG 6#-F$6#&%#T*G6$\"\"(\"#8" }{TEXT -1 7 " and " }{XPPEDIT 18 0 "abs(ab s(`T*`[8, 13]));" "6#-%$absG6#-F$6#&%#T*G6$\"\")\"#8" }{TEXT -1 15 " \+ respectively." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 8 "Define: " }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "A[9] = abs(abs(T[8,13]));" "6#/&%\"AG6#\"\"*-%$absG6#-F)6#&%\"TG6$ \"\")\"#8" }{TEXT -1 6 ", " }{XPPEDIT 18 0 "B[9] = abs(abs(`T*`[8, 13]))/abs(abs(`T*`[7,13]));" "6#/&%\"BG6#\"\"**&-%$absG6#-F*6#&%#T*G6$ \"\")\"#8\"\"\"-F*6#-F*6#&F/6$\"\"(F2!\"\"" }{TEXT -1 6 ", " } {XPPEDIT 18 0 "C[9] = abs(abs(`T*`[8,13]-T[8,13]))/abs(abs(`T*`[7,13]) );" "6#/&%\"CG6#\"\"**&-%$absG6#-F*6#,&&%#T*G6$\"\")\"#8\"\"\"&%\"TG6$ F2F3!\"\"F4-F*6#-F*6#&F06$\"\"(F3F8" }{TEXT -1 1 "." }}{PARA 0 "" 0 " " {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 77 "Dormand and Prince have suggested that as well as attempting to ensure that " }{XPPEDIT 18 0 "A[9];" "6#&%\"AG6#\"\"*" }{TEXT -1 73 " is a minimum, if the embed ded scheme is to be used for error control, " }{XPPEDIT 18 0 "B[9];" "6#&%\"BG6#\"\"*" }{TEXT -1 7 " and " }{XPPEDIT 18 0 "C[9];" "6#&%\" CG6#\"\"*" }{TEXT -1 27 " should be chosen so that " }}{PARA 0 "" 0 " " {TEXT -1 68 "they are \"similar in magnitude\" and also not differ t oo much from 1." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> \+ " 0 "" {MPLTEXT 1 0 168 "errterms8_13 := PrincipalErrorTerms(8,13,'exp anded'):\n`errterms8_13*` :=subs(b=`b*`,errterms8_13):\n`errterms7_13* ` := subs(b=`b*`,PrincipalErrorTerms(7,13,'expanded')):" }}}{PARA 0 " " 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 333 "snmB := sqrt(add(evalf(subs(ee,`errterms8_13*`[i]))^2,i=1..nops(`errterms8 _13*`))):\nsdnB := sqrt(add(evalf(subs(ee,`errterms7_13*`[i]))^2,i=1.. nops(`errterms7_13*`))):\nsnmC := sqrt(add((evalf(subs(ee,`errterms8_1 3*`[i])-subs(ee,errterms8_13[i])))^2,i=1..nops(errterms8_13))):\n'B[9] '= evalf[8](snmB/sdnB);\n'C[9]'= evalf[8](snmC/sdnB);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/&%\"BG6#\"\"*$\"):)4.#!\"(" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/&%\"CG6#\"\"*$\")o'4.#!\"(" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}}{PARA 0 "" 0 "" {TEXT -1 52 "#--------------------------------------------------- " }}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 26 "absolute stability regions" } }{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 36 "coefficients of the co mbined scheme " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 7144 "ee := \{c[2]=9/142,\nc[3]=24514/238491,\nc[4]= 12257/79497,\nc[5]=50/129,\nc[6]=34/73,\nc[7]=23/148,\nc[8]=142/141,\n c[9]=29104120198/33218531707,\nc[10]=83/91,\nc[11]=143/149,\nc[12]=1, \nc[13]=1,\n\na[2,1]=9/142,\na[3,1]=9950845450/511901613729,\na[3,2]=4 2666469916/511901613729,\na[4,1]=12257/317988,\na[4,2]=0,\na[4,3]=1225 7/105996,\na[5,1]=14131686489425/35833975601529,\na[5,2]=0,\na[5,3]=-1 7700454220625/11944658533843,\na[5,4]=17619604667500/11944658533843,\n a[6,1]=4418011/96055225,\na[6,2]=0,\na[6,3]=0,\na[6,4]=9757757988/4199 5818067,\na[6,5]=19921512441/106300094275,\na[7,1]=480285555889619/799 7789253816320,\na[7,2]=0,\na[7,3]=0,\na[7,4]=52162106556696460538643/4 64887033284472899365888,\na[7,5]=-16450769949384489/490009784398315520 ,\na[7,6]=864813381633887/51718297665732608,\na[8,1]=-7001048352088587 137143/4449830351030385148800,\na[8,2]=0,\na[8,3]=0,\na[8,4]=-25227655 48294599044197935933/1915963195493758447677901792,\na[8,5]=-3185561822 35222634647116091/27172411532484037214054400,\na[8,6]=1205563885850790 193966807/132365727505912221222912,\na[8,7]=254/39,\na[9,1]=-206293963 99716689122801179264428539394462855874226604463554767070845753369/\n \+ 4288175966277015595665751347001211448807601798112810530737559496 3152353200,\na[9,2]=0,\na[9,3]=0,\na[9,4]=-331544307434365940477942214 9387397712986453181141168247590906370819301077749322753/\n 4985 1711264160887280782183856684798772951431239827863378818583534899764321 8553476,\na[9,5]=-2737494094116540602869481411648284521098983792035269 45684314474186724062841643/\n 604275839513775525039678406538255 89117167443021628367691773162913607984889600,\na[9,6]=1665637251887451 2738268060504309924900437672263609245028809229865738327731797537/\n \+ 427699013853393052278207638577101600909762793055014962828220300791 3538394549504,\na[9,7]=42008080033354305590804322944084805264441066760 038302359736803632/\n 486530242321653491007482328781160559962917 0295030631799935804001,\na[9,8]=66845978093071633800006662723692741719 1947396177093524377824/\n 71100452948884643779799087713322002499 799983434685642170251833,\na[10,1]=-1793603946322260900828212460706877 142477132870159527/\n 2313097568511990753781649719556084665131 024900300800,\na[10,2]=0,\na[10,3]=0,\na[10,4]=-1477687412372283819231 5406145167687512425345723701/\n 184789353036607670110201414692 7696206329050105856, a[10,5]=-1958702091988466171485675710513024699575 7906603/\n 2911893296942532038566182868170393629789388800,\na[ 10,6]=6364380863259071677112259236455506477417699780613300364807/\n \+ 1150428174584942133406579091549443438814988349188394057728,\na[10 ,7]=27725164402569748756040320433848245155581006369/\n 25441594 73655547770881695354241256106302348256,\na[10,8]=107442471639600198768 33255044784609639/534761804739901348825491947768304503296,\na[10,9]=-5 0977737930792808232204417497248979399878217280011103197862899/\n \+ 1300915694564913675613280314081837358644964393191337994183389184,\n a[11,1]=-3587625717068952487214493441966897048737050755812600710793/\n 3015733164033229624772006086429467685046639983706974412800,\n a[11,2]=0,\na[11,3]=0,\na[11,4]=-5453011711267804731211501837262816944 661201619903/\n 7649733208997163970724486447107331013292901969 92,\na[11,5]=-884348836774584715070440485633026464653487653/\n \+ 92725983515963799584249219499787659014963200,\na[11,6]=26823469063654 084387375587616552322383082061411417182757389742951/\n 35412997 44763681675473620647087123057228744296123642301654630400, a[11,7]=1423 63419491686507162007051071007722765323162710521029/\n 126348872 02368261565807449771335728082273587905775840,\na[11,8]=647476174549092 75289531520412519442831235890581/\n 126931718811797596099667062 8974453443777854830080,\na[11,9]=1126338082532727209798743033675038915 97499261046700689572459050065039333987335667/\n 140451429124503 4532812181377119034501014039830518218465064579291611563374608650240,\n a[11,10]=-10612202518573994431153697720606405883/670825466582598467787 54594976831647575,\na[12,1]=-75340811655449824782962023359220492108038 75045423/\n 19219575665440848756074598051658416387002479820800 ,\na[12,2]=0,\na[12,3]=0,\na[12,4]=23769608745278671780227037528303426 2859273455/\n 60688480889936839261131818793519097177034752,\na[ 12,5]=-20610578209826329263318986584876108069323/735633377643883488429 3014575840932659200,\na[12,6]=5126047152984102804070965445890325478132 0136131844164563/\n 2099802377631854690738230210678816815876551 4131585630208,\na[12,7]=-307721443717347297119681079561538400021145715 1011/\n 1272435592582280820597059432060116893200684680384,\na[ 12,8]=-1539218116260541896259682954580256454049/4534670830750360983422 999946409310393344,\na[12,9]=24188653935026842937211629678727127655397 0618941104594460614948326132797451456131/\n 1240669632662465892 528916120041123051054026283188324780101555345343662316090597376,\na[12 ,10]=-80556486832245966191717452425924975/4144454095186765975650320080 51106461,\na[12,11]=2944781680874500347594142792814463350/496516138307 3676983610218096030654529,\na[13,1]=-775773993786294483292774369433611 6203639371542761/\n 522510067842179432565485084545134047357726 0032000,\na[13,2]=0,\na[13,3]=0,\na[13,4]=-433889546009521405913741133 329446636837810749/181488796115643001621310691169322233346938880,\na[1 3,5]=-246044720162308748108107126829066792329071/219991033114178071701 00413255475150848000,\na[13,6]=214033123542582984438906081861671984863 7810765257179167/\n 2454281820360118976380282528153692586460525 49878743040,\na[13,7]=1573990926219809229258666534611598771063240529/2 14535514317495538101650508596881057760818080,\na[13,8]=624082806673093 75445301959066100433563/4838320046251983849512355559327451645440,\na[1 3,9]=11456098222494936776181137253595066429981532056032268831412070899 68379/\n 269028021668227684763674278408357431395592941053986779 75568538466119680,\na[13,10]=-408950356875874683139089678053832/767429 2714443204070455739595109785,\na[13,11]=0,\na[13,12]=0,\n\nb[1]=550384 46513529253801/1239280570055853383520,\nb[2]=0,\nb[3]=0,\nb[4]=0,\nb[5 ]=0,\nb[6]=2335496795323782464411846394611/659836878329402010989537993 6256,\nb[7]=7636073376527143565375240869888/30725949199261296642046754 748645,\nb[8]=-4237087214169934312729607487/12735791394214625116604076 160,\nb[9]=40850529129113324176099551412198433591436392788442678007832 5258228227984174126699/\n 2126248746121936974666551594056352021238 21166475348052734199905546455860773575680,\nb[10]=-1108225296327029096 435947/405679075893979729103310,\nb[11]=2460988291206213825688467985/1 756342789520947764222671739,\nb[12]=4808707937311/50545545388065,\nb[1 3]=0,\n\n`b*`[1]=715953338020208413/16094552857868225760,\n`b*`[2]=0, \n`b*`[3]=0,\n`b*`[4]=0,\n`b*`[5]=0,\n`b*`[6]=622843351629289669870661 21/175437206755843240272669696,\n`b*`[7]=184309146777472302831695872/7 42417767522160213324368465,\n`b*`[8]=-509771598215811385123057257/2102 82269969085698264101760,\n`b*`[9]=270160248964614364036289140292496250 0766379885231830470264478480621389/\n 1688575269294196295712420 798364925687791337062814161130291144634516480,\n`b*`[10]=-721853407301 2286740367561/3986456306153224954098780,\n`b*`[11]=0,\n`b*`[12]=0,\n`b *`[13]=5752173075461/1925544586212\}:" }}}{PARA 0 "" 0 "" {TEXT -1 0 " " }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 78 "The stability function R \+ for the 13 stage, order 8 scheme is given as follows." }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 76 "subs(ee,StabilityFunction(8,13,'expanded' )):\nR := unapply(%,z):\n'R(z)'=R(z);" }}{PARA 12 "" 1 "" {XPPMATH 20 "6#/-%\"RG6#%\"zG,<\"\"\"F)F'F)*&#F)\"\"#F)*$)F'F,F)F)F)*&#F)\"\"'F)*$ )F'\"\"$F)F)F)*&#F)\"#CF)*$)F'\"\"%F)F)F)*&#F)\"$?\"F)*$)F'\"\"&F)F)F) *&#F)\"$?(F)*$)F'F1F)F)F)*&#F)\"%S]F)*$)F'\"\"(F)F)F)*&#F)\"&?.%F)*$)F '\"\")F)F)F)*&#\"ItSWIV))f#=Q+\"Q,r9\"o-)zV\"O?bND$[(oOX4Ozyv='o%4z\\O K;F)*$)F'\"\"*F)F)F)*&#\"L8'HrA8fk>Glkcl[EK6>Ef?\"R![#p;f/)>BNr#*)4&[H jn[eKg&p*zF)*$)F'\"#5F)F)F)*&#\"Kx1DE],()y=R.ej28#Gc?'Qa\"Sgl-*)[CcqX0 yU>(e!42'fyt@V]E%F)*$)F'\"#6F)F)F)*&#\"=lNcgALWLBvk&f\"R\"G_$3nFIb#[L[ ZjnC/l#\\Z'F)*$)F'\"#7F)F)F)" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }} {PARA 0 "" 0 "" {TEXT -1 124 "We can find the point where the boundary of the stability region intersects the negative real axis by solving \+ the equation: " }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "R(z ) = 1;" "6#/-%\"RG6#%\"zG\"\"\"" }{TEXT -1 2 ". " }}{EXCHG {PARA 0 "> \+ " 0 "" {MPLTEXT 1 0 26 "z0 := newton(R(z)=1,z=-6);" }}{PARA 11 "" 1 " " {XPPMATH 20 "6#>%#z0G$!+%[$)o!f!\"*" }}}{PARA 0 "" 0 "" {TEXT -1 0 " " }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 302 "z0 := newton(R(z)=1,z=- 6):\np1 := plot([R(z),1],z=-6.69..0.49,color=[red,blue]):\np2 := plot( [[[z0,1]]$3],style=point,symbol=[circle,cross,diamond],color=black):\n p3 := plot([[z0,0],[z0,1]],linestyle=3,color=COLOR(RGB,0,.5,0)):\nplot s[display]([p1,p2,p3],view=[-6.69..0.49,-0.07..1.47],font=[HELVETICA,9 ]);" }}{PARA 13 "" 1 "" {GLPLOT2D 360 253 253 {PLOTDATA 2 "6+-%'CURVES G6$7X7$$!3Q++++++!p'!#<$\"31o8o@08!*HF*7$$!3ym;a;#[#z#QQ[sQCF*7$$!3-^iqu*4aY'F*$\"3Tc@D0OiEAF*7$$ !31n\"Hj^BtR'F*$\"3]O?bzI5I?F*7$$!3$)\\P%H<`2K'F*$\"3u(H;!*fPm#=F*7$$! 3[L$e&HG=WiF*$\"38A_*[S%fS;F*7$$!3)QLL#G\\5nhF*$\"3/.w*[eV(p9F*7$$!3UL $3p-F+4'F*$\"3c(=eD-QUJ\"F*7$$!3X]PRjaJ8gF*$\"3!p[VA9DO<\"F*7$$!3fm\"z )**QgOfF*$\"3%=e]Wr+h/\"F*7$$!3)Q$ef@5O%z&F*$\"3F+#)>A4r4%)!#=7$$!3E+D ;gn2ZcF*$\"3qlRV,]!Gm'F^o7$$!3gLeHm^v%\\&F*$\"3%Gd!f'fn))>&F^o7$$!3Q+D rp?#HM&F*$\"3`%HA4h!pISF^o7$$!3ymm@nIu'=&F*$\"3WPai%QO03$F^o7$$!3oL$3o ^!=\\]F*$\"3\\$R&)yE;%=CF^o7$$!37++lmwJ%*[F*$\"3#G^AHf_O$=F^o7$$!3F++v M*=)QZF*$\"3!=t$QOO)eQ\"F^o7$$!3'****\\L4n*)e%F*$\"3_I+d=L`g5F^o7$$!3y LeHls)GX%F*$\"33RrEU$4tP)!#>7$$!3Umm\"y=v5H%F*$\"3=%eypj&[YkF\\r7$$!3# ommK8**R:%F*$\"3->ZbC!o&*H&F\\r7$$!3O+D'[4ZX*RF*$\"3;'=H;ZN3T%F\\r7$$! 3=nmwA]S`QF*$\"3Zk8]Nh8WRF\\r7$$!33+D')*z\\&)p$F*$\"3oo`@pB&)3PF\\r7$$ !3'**\\(e254^NF*$\"3Lb&R;suaq$F\\r7$$!3wm;HFKB(R$F*$\"3iJb3GkW%*QF\\r7 $$!3im\"zdvVfD$F*$\"3S;Ll>(p(>UF\\r7$$!3kL$3K(Ra.JF*$\"3\\0mF?EK?ZF\\r 7$$!3oLeHIMCXHF*$\"3MAu%[$G[*R&F\\r7$$!37+vLUBW2GF*$\"3bAw53V2DhF\\r7$ $!3EL$e@R8'eEF*$\"3/$*Gpu,tdqF\\r7$$!3G++qB&e[]#F*$\"3qjYuF;,*>)F\\r7$ $!3?+]_\"QRWN#F*$\"3;%3=u)pd6&*F\\r7$$!33+D1F6!*3AF*$\"3/:C([RT\"*4\"F ^o7$$!3?+]24dIZ?F*$\"3o4IH4)y7H\"F^o7$$!3#pmm(Q]5->F*$\"3G)H#QN8x#\\\" F^o7$$!3-++vIB2ZNz!QRoQBF^o7$$!3um\"Hvx$\\38F*$\"3Ug*338vAq #F^o7$$!3c*\\7z#zVd6F*$\"3aNog$G1H9$F^o7$$!3[mmTCKv45F*$\"3W&RD%>%*3VO F^o7$$!3].]il-T^&)F^o$\"3;EGw\\BB_UF^o7$$!3#)GLLR5JiqF^o$\"3%>U!)Q_1]$ \\F^o7$$!3^GL30?\\RbF^o$\"3'>Z;T8enu&F^o7$$!3ch;H\"o#GHSF^o$\"3#***)*Q ./g$o'F^o7$$!30%****p:d:k#F^o$\"3;s,IM'R&ywF^o7$$!3/lmT,r1^5F^o$\"3#[a Z&RXG-!*F^o7$$\"3\"Gsmmkh[r$F\\r$\"3,a'o0DZy.\"F*7$$\"3;,]P^YA))=F^o$ \"3Ab_8-l#y?\"F*7$$\"3J,]i9p$*RLF^o$\"3'*fLXLV`'R\"F*7$$\"3!********** *****[F^o$\"3xyu\")>iJK;F*-%'COLOURG6&%$RGBG$\"*++++\"!\")$\"\"!Fj\\lF i\\l-F$6$7S7$F($\"\"\"Fj\\l7$F3F_]l7$F=F_]l7$FGF_]l7$FQF_]l7$FenF_]l7$ FjnF_]l7$F`oF_]l7$FeoF_]l7$FjoF_]l7$F_pF_]l7$FdpF_]l7$FipF_]l7$F^qF_]l 7$FcqF_]l7$FhqF_]l7$F^rF_]l7$FcrF_]l7$FhrF_]l7$F]sF_]l7$FbsF_]l7$FgsF_ ]l7$F\\tF_]l7$FatF_]l7$FftF_]l7$F[uF_]l7$F`uF_]l7$FeuF_]l7$FjuF_]l7$F_ vF_]l7$FdvF_]l7$FivF_]l7$F^wF_]l7$FcwF_]l7$FhwF_]l7$F]xF_]l7$FbxF_]l7$ FgxF_]l7$F\\yF_]l7$FayF_]l7$FfyF_]l7$F[zF_]l7$F`zF_]l7$FezF_]l7$FjzF_] l7$F_[lF_]l7$Fd[lF_]l7$Fi[lF_]l7$F^\\lF_]l-Fc\\l6&Fe\\lFi\\lFi\\lFf\\l -F$6&7#7$$!3u*****R[$)o!fF*F_]l-%'SYMBOLG6#%'CIRCLEG-Fc\\l6&Fe\\lFj\\l Fj\\lFj\\l-%&STYLEG6#%&POINTG-F$6&Fe`l-Fj`l6#%&CROSSGF]alF_al-F$6&Fe`l -Fj`l6#%(DIAMONDGF]alF_al-F$6%7$7$Fg`lFi\\lFf`l-%&COLORG6&Fe\\lFi\\l$ \"\"&!\"\"Fi\\l-%*LINESTYLEG6#\"\"$-%+AXESLABELSG6%Q\"z6\"Q!F_cl-%%FON TG6#%(DEFAULTG-Fbcl6$%*HELVETICAG\"\"*-%%VIEWG6$;$!$p'!\"#$\"#\\F_dl;$ !\"(F_dl$\"$Z\"F_dl" 1 2 0 1 10 0 2 9 1 4 2 1.000000 45.000000 45.000000 0 0 "Curve 1" "Curve 2" "Curve 3" "Curve 4" "Curve 5" "Curve 6" }}}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 50 "T he following picture shows the stability region. " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 961 "R := z -> a dd(z^j/j!,j=0..8)+\n 4379802681147101381003818259884330444073/163236 4979094686187578793609453668748325355520*z^9+\n 20592619113226486556 64652819645913227129613/7996956032584867632948509892713523198045916692 480*z^10+\n 543862056282130763580339187887015026250677/4265043217378 5960709058719427805457056244889026560*z^11+\n 3915956475233344332260 563565/64749265042467634748334825530276708352*z^12:\npts := []: z0 := \+ 0:\nfor ct from 0 to 320 do\n zz := newton(R(z)=exp(ct*Pi/20*I),z=z0 ):\n z0 := zz:\n pts := [op(pts),[Re(zz),Im(zz)]]:\nend do:\np1 := plot(pts,color=COLOR(RGB,0,.38,.48)):\np2 := plots[polygonplot]([seq( [pts[i-1],pts[i],[-3.2,0]],i=2..nops(pts))],\n style=patchnog rid,color=COLOR(RGB,0,.75,.95)):\np3 := plot([[[-6.89,0],[1.19,0]],[[0 ,-6.39],[0,6.39]]],color=black,linestyle=3):\nplots[display]([p||(1..3 )],view=[-6.89..1.19,-6.39..6.39],font=[HELVETICA,9],\n l abels=[`Re(z)`,`Im(z)`],axes=boxed,scaling=constrained);" }}{PARA 13 " " 1 "" {GLPLOT2D 381 565 565 {PLOTDATA 2 "6+-%'CURVESG6$7]_l7$$\"\"!F) F(7$F($\"3++++Fjzq:!#=7$F($\"3)******Rl#fTJF-7$F($\"3D+++\")*)Q7ZF-7$$ \"3D+++'*>RUZ!#F$\"3g*****pJ&=$G'F-7$$\"3#)*****\\Q@C<%!#E$\"3#)*****p p\")R&yF-7$$\"3,+++#=zeS#!#D$\"3m*****pCyZU*F-7$$\"3'******HL71.\"!#C$ \"3)******p_d&*4\"!#<7$$\"3<+++uDpJNFI$\"3++++Itjc7FL7$$\"3********oOR 95!#B$\"3#*******3tr89FL7$$\"3!)*****\\`;y^#FU$\"3%******z_(zq:FL7$$\" 3R+++Z&[&4bFU$\"3++++(*z(ys\"FL7$$\"3'*******3!*)f2\"!#A$\"35+++\\'e\\ )=FL7$$\"3)******48wf)=F_o$\"37+++'\\R?/#FL7$$\"3!******>1;A'HF_o$\"3) ******f@@\"*>#FL7$$\"3>+++'fQ+7%F_o$\"3#)*****>o1iN#FL7$$\"3w******[vs :\\F_o$\"3!)******RTI8DFL7$$\"3\"******>NZCh%F_o$\"3%******z)HVqEFL7$$ \"3%******RkxGL#F_o$\"3(******pLLw#GFL7$$!31+++'o*z!Q#F_o$\"35+++d0)\\ )HFL7$$!3_*****z(4MW#)F_o$\"3y*****f,1E9$FL7$$!3U*****4bf#y&*F_o$\"3,+ ++7]s+LFL7$$\"3M+++w8uN')F_o$\"37+++6CnfMFL7$$\"3#)*****RnHn(z!#@$\"3. +++DY%*>OFL7$$\"33+++Le,AF!#?$\"3/+++r$QAy$FL7$$\"3=+++s1t5sF]s$\"3?++ +\\GWZRFL7$$\"31+++X%G')o\"!#>$\"3C+++2#4k6%FL7$$\"3)******fd0$fOFhs$ \"3k*****Hac))G%FL7$$\"3S*******zfTL(Fhs$\"3K+++e0#*fWFL7$$\"3-+++:zp& H\"F-$\"3h*****RtBkh%FL7$$\"3()*****>$48S>F-$\"39+++!eIou%FL7$$\"33+++ )*H&)HDF-$\"3/+++nmW`[FL7$$\"33+++CI$G,$F-$\"3>+++Gr$Q%\\FL7$$\"3w**** *\\9iuQ$F-$\"3J+++hI[B]FL7$$\"3u*****HDOcm$F-$\"3E+++')R^&4&FL7$$\"3++ ++'4N/'QF-$\"3g*****pyy;;&FL7$$\"3')*****pu@L)RF-$\"3.+++fm)HA&FL7$$\" 3;+++ZG(Q/%F-$\"3#)*****\\)e/!G&FL7$$\"3'******HLa+0%F-$\"3d******f.CL `FL7$$\"3v*******)\\a3SF-$\"39+++8U#GQ&FL7$$\"34+++4E+DRF-$\"31+++\"=u *GaFL7$$\"3t*****4olU!QF-$\"3?+++Y$>=Z&FL7$$\"3E+++PS]]OF-$\"3e*****z@ e9^&FL7$$\"3-+++AELnMF-$\"3')*****pcoza&FL7$$\"37+++u#**yD$F-$\"3U+++o HT\"e&FL7$$\"3D+++nS&\\-$F-$\"3G+++'yT=h&FL7$$\"3/+++$>04x#F-$\"3E+++? YHRcFL7$$\"3#******4nky\\#F-$\"3O+++'y+Qm&FL7$$\"31+++cfo2AF-$\"3u**** *z=z`o&FL7$$\"3$******>![*>!>F-$\"3#)*****\\wPSq&FL7$$\"3'*******oT@#e \"F-$\"3\"******4fs(>dFL7$$\"3%******>f$e\\7F-$\"3/+++_mcKdFL7$$\"3c** ***4dI<0*Fhs$\"3P+++S#)QUdFL7$$\"3!******\\G5*)\\&Fhs$\"3Q+++/*)=\\dFL 7$$\"3&)*****fvd[%=Fhs$\"3;+++#z+Hv&FL7$$!3&)*****>C;Y!>Fhs$\"3S+++&>L Mv&FL7$$!3K+++()QbXdFhs$\"3?+++,!o1v&FL7$$!3k+++\"f*=w'*Fhs$\"3s*****4 l`Wu&FL7$$!3)*******yFvp8F-$\"38+++8qfMdFL7$$!37+++J)H9y\"F-$\"3(***** *fN_3s&FL7$$!36+++TPl.AF-$\"3C+++(G1Hq&FL7$$!3F+++'>D#QEF-$\"3?+++lrN! o&FL7$$!3E+++uX?)3$F-$\"3y******)\\(o_cFL7$$!37+++YkueNF-$\"3/++++%H#> cFL7$$!3-+++K$p'eSF-$\"3#******\\UJ\"zbFL7$$!3!)******RP^.YF-$\"3G+++# Qp8`&FL7$$!3M+++\\'e9A&F-$\"3S+++Br(\\Z&FL7$$!3k*****Ri=F'fF-$\"3%)*** ***y&Q,T&FL7$$!3w*****pt%*))*oF-$\"3,+++oejT`FL7$$!3W+++u!>b0)F-$\"3w* ****RhvDG&FL7$$!3(********e;2K*F-$\"3k*****RwIWC&FL7$$!3'******4u*zb5F L$\"3U++++(R[A&FL7$$!3!******>wZH@&FL7$$!3%******zaD&z8FL$\"3Q+++Qa&>@&FL7$$!3))*******)z6 u9FL$\"3K+++FGV6_FL7$$!33+++l=(Qc\"FL$\"3:+++%z^.@&FL7$$!37+++h9h\\;FL $\"32+++&3&43_FL7$$!3*)*****p;\"*>t\"FL$\"3:+++$pzU?&FL7$$!32+++dL`6=F L$\"3s*****RYh')>&FL7$$!3!******>')f'))=FL$\"3M+++193\">&FL7$$!3'***** *p=;P'>FL$\"3o******\\UV\"=&FL7$$!3\")*****p\")*)p.#FL$\"39+++c%\\'p^F L7$$!3$)*****z2C(3@FL$\"3=+++t%zc:&FL7$$!3y*****RuG\"z@FL$\"3u*****zS# \\R^FL7$$!3!)*****zE)Q[AFL$\"3*******R]n57&FL7$$!3&)*****4DomJ#FL$\"3k *****p\\#R+^FL7$$!33+++n07%Q#FL$\"3!******HIiu2&FL7$$!37+++Lu)3X#FL$\" 3#)*****HqyA0&FL7$$!32+++nX5&*\\FL7$$!3!******p18%[EFL$\"3i*****4QVL'\\FL7$$!3.+++[)fPr# FL$\"31+++JALH\\FL7$$!3')******ew1zFFL$\"3G+++K%=K*[FL7$$!36+++0dXWGFL $\"3;+++c\\2b[FL7$$!32+++Ba.5HFL$\"3M+++'R%*\\\"[FL7$$!3%******>*f!f(H FL$\"3w*****H?!4tZFL7$$!3;+++J([@/$FL$\"3o*****HT'\\HZFL7$$!3!******4$ 4#)3JFL$\"37+++1`O%o%FL7$$!31+++g*\\f<$FL$\"3v*****pgiyj%FL7$$!3@+++\\ (GNC$FL$\"3m*****REg,f%FL7$$!35+++CP^6LFL$\"3I+++itUTXFL7$$!31+++!*f#) zLFL$\"3'******HH?=\\%FL7$$!35+++@fN[MFL$\"3w*****pht9W%FL7$$!3z****** G1(p^$FL$\"3i*****H*H\\!R%FL7$$!3-+++`J_&e$FL$\"31+++V6&*QVFL7$$!3%)** ***f*>'Ql$FL$\"3/+++Gr)oG%FL7$$!3%******prR=s$FL$\"3#******Hc3VB%FL7$$ !3%******>v>$*y$FL$\"3K+++if>\"=%FL7$$!3=+++j5=cQFL$\"3$)*****zC3v7%FL 7$$!3@+++;2KARFL$\"3I+++S&)=tSFL7$$!3%******zraw)RFL$\"3:+++*fp\"=SFL7 $$!3!******zc<@0%FL$\"3)******\\ayB'RFL7$$!3))*****\\)4m:TFL$\"3()**** **)pSd!RFL7$$!3#)*****R<_#yTFL$\"3')*****zd#=[QFL7$$!3*)*****4!>()RUFL $\"3'*******GTj*y$FL7$$!3<+++nD^+VFL$\"3;+++a..IPFL7$$!3!)*****>Ow,O%F L$\"39+++yCJpOFL7$$!3@+++WN()=WFL$\"3\")*****\\xGug$FL7$$!39+++L3iwWFL $\"3#)*****f:NVa$FL7$$!3$)*****\\!*RM`%FL$\"3?+++'f&**zMFL7$$!3#)***** *QfN*e%FL$\"3*******pV#Q9MFL7$$!3))*****f<'RWYFL$\"3!*******olZZLFL7$$ !3?+++v%)e)p%FL$\"3)******\\]n#zKFL7$$!3A+++E)f>v%FL$\"3&*******)H`(4K FL7$$!3c******Q[`/[FL$\"3++++\"GS*QJFL7$$!33+++AULc[FL$\"3A+++oD%o1$FL 7$$!3))*****>Itt!\\FL$\"3++++;8[$*HFL7$$!31+++[1md\\FL$\"3%)******)o$) )=HFL7$$!3,+++')o>2]FL$\"3\")******e:3VGFL7$$!3G+++*ytf0&FL$\"37+++r*4 hw#FL7$$!3++++5P(R5&FL$\"3))*****\\X0!)o#FL7$$!3/+++:%p6:&FL$\"3z***** *eU!)3EFL7$$!3++++QV_(>&FL$\"3.+++b1aGDFL7$$!3V+++3BvG&FL$\"3?+++8X%\\O#FL7$$!3$******>Za5L&FL$\"37+ ++[$e;G#FL7$$!3U+++1G_t`FL$\"3')*****H;,u>#FL7$$!3!******Ryf[T&FL$\"3/ +++P5=7@FL7$$!3?+++JZ*\\X&FL$\"3&)*****fA+g-#FL7$$!3E+++4e&Q\\&FL$\"3- +++^e&)Q>FL7$$!3P+++Z5PJbFL$\"3)********)3u]=FL7$$!3'******>=puc&FL$\" 3)******zRX;w\"FL7$$!3'********G!3-cFL$\"3!******H*ybr;FL7$$!3\"****** RCO^j&FL$\"31+++joY!e\"FL7$$!3G+++A5dmcFL$\"31+++UBO)[\"FL7$$!3A+++/4K 'p&FL$\"3-+++BvB&R\"FL7$$!3&)*****pfCVs&FL$\"3++++s.4,8FL7$$!3G+++AL_] dFL$\"32+++A_#f?\"FL7$$!3k*****R!4'[x&FL$\"3*)*****\\Ha(46FL7$$!3;+++! )QG(z&FL$\"3*******48*f75FL7$$!3V+++i;uFL7$F\\gm$!3&)***** fA+g-#FL7$Fgfm$!3/+++P5=7@FL7$Fbfm$!3')*****H;,u>#FL7$F]fm$!37+++[$e;G #FL7$Fhem$!3?+++8X%\\O#FL7$Fcem$!3!)*****RSXsW#FL7$F^em$!3.+++b1aGDFL7 $Fidm$!3z******eU!)3EFL7$Fddm$!3))*****\\X0!)o#FL7$F_dm$!37+++r*4hw#FL 7$Fjcm$!3\")******e:3VGFL7$Fecm$!3%)******)o$))=HFL7$F`cm$!3++++;8[$*H FL7$F[cm$!3A+++oD%o1$FL7$Ffbm$!3++++\"GS*QJFL7$Fabm$!3&*******)H`(4KFL 7$F\\bm$!3)******\\]n#zKFL7$Fgam$!3!*******olZZLFL7$Fbam$!3*******pV#Q 9MFL7$F]am$!3?+++'f&**zMFL7$Fh`m$!3#)*****f:NVa$FL7$Fc`m$!3\")*****\\x Gug$FL7$F^`m$!39+++yCJpOFL7$Fi_m$!3;+++a..IPFL7$Fd_m$!3'*******GTj*y$F L7$F__m$!3')*****zd#=[QFL7$Fj^m$!3()******)pSd!RFL7$Fe^m$!3)******\\ay B'RFL7$F`^m$!3:+++*fp\"=SFL7$F[^m$!3I+++S&)=tSFL7$Ff]m$!3$)*****zC3v7% FL7$Fa]m$!3K+++if>\"=%FL7$F\\]m$!3#******Hc3VB%FL7$Fg\\m$!3/+++Gr)oG%F L7$Fb\\m$!31+++V6&*QVFL7$F]\\m$!3i*****H*H\\!R%FL7$Fh[m$!3w*****pht9W% FL7$Fc[m$!3'******HH?=\\%FL7$F^[m$!3I+++itUTXFL7$Fijl$!3m*****REg,f%FL 7$Fdjl$!3v*****pgiyj%FL7$F_jl$!37+++1`O%o%FL7$Fjil$!3o*****HT'\\HZFL7$ Feil$!3w*****H?!4tZFL7$F`il$!3M+++'R%*\\\"[FL7$F[il$!3;+++c\\2b[FL7$Ff hl$!3G+++K%=K*[FL7$Fahl$!31+++JALH\\FL7$F\\hl$!3i*****4QVL'\\FL7$Fggl$ !3I+++7k>&*\\FL7$Fbgl$!3d*****\\%3&[-&FL7$F]gl$!3#)*****HqyA0&FL7$Fhfl $!3!******HIiu2&FL7$Fcfl$!3k*****p\\#R+^FL7$F^fl$!3*******R]n57&FL7$Fi el$!3u*****zS#\\R^FL7$Fdel$!3=+++t%zc:&FL7$F_el$!39+++c%\\'p^FL7$Fjdl$ !3o******\\UV\"=&FL7$Fedl$!3M+++193\">&FL7$F`dl$!3s*****RYh')>&FL7$F[d l$!3:+++$pzU?&FL7$Ffcl$!32+++&3&43_FL7$Facl$!3:+++%z^.@&FL7$F\\cl$!3K+ ++FGV6_FL7$Fgbl$!3Q+++Qa&>@&FL7$Fbbl$!3w*****>wZH@&FL7$F]bl$!3J+++sq=; _FL7$Fhal$!3U++++(R[A&FL7$Fcal$!3k*****RwIWC&FL7$F^al$!3w*****RhvDG&FL 7$Fi`l$!3,+++oejT`FL7$Fd`l$!3%)******y&Q,T&FL7$F_`l$!3S+++Br(\\Z&FL7$F j_l$!3G+++#Qp8`&FL7$Fe_l$!3#******\\UJ\"zbFL7$F`_l$!3/++++%H#>cFL7$F[_ l$!3y******)\\(o_cFL7$Ff^l$!3?+++lrN!o&FL7$Fa^l$!3C+++(G1Hq&FL7$F\\^l$ !3(******fN_3s&FL7$Fg]l$!38+++8qfMdFL7$Fb]l$!3s*****4l`Wu&FL7$F]]l$!3? +++,!o1v&FL7$Fh\\l$!3S+++&>LMv&FL7$Fc\\l$!3;+++#z+Hv&FL7$F^\\l$!3Q+++/ *)=\\dFL7$Fi[l$!3P+++S#)QUdFL7$Fd[l$!3/+++_mcKdFL7$F_[l$!3\"******4fs( >dFL7$Fjz$!3#)*****\\wPSq&FL7$Fez$!3u*****z=z`o&FL7$F`z$!3O+++'y+Qm&FL 7$F[z$!3E+++?YHRcFL7$Ffy$!3G+++'yT=h&FL7$Fay$!3U+++oHT\"e&FL7$F\\y$!3' )*****pcoza&FL7$Fgx$!3e*****z@e9^&FL7$Fbx$!3?+++Y$>=Z&FL7$F]x$!31+++\" =u*GaFL7$Fhw$!39+++8U#GQ&FL7$Fcw$!3d******f.CL`FL7$F^w$!3#)*****\\)e/! G&FL7$Fiv$!3.+++fm)HA&FL7$Fdv$!3g*****pyy;;&FL7$F_v$!3E+++')R^&4&FL7$F ju$!3J+++hI[B]FL7$Feu$!3>+++Gr$Q%\\FL7$F`u$!3/+++nmW`[FL7$F[u$!39+++!e Iou%FL7$Fft$!3h*****RtBkh%FL7$Fat$!3K+++e0#*fWFL7$F\\t$!3k*****Hac))G% FL7$Ffs$!3C+++2#4k6%FL7$Fas$!3?+++\\GWZRFL7$F[s$!3/+++r$QAy$FL7$Fer$!3 .+++DY%*>OFL7$F`r$!37+++6CnfMFL7$F[r$!3,+++7]s+LFL7$$!3X*****>)4MW#)F_ o$!3y*****f,1E9$FL7$Faq$!35+++d0)\\)HFL7$$\"3#******\\kxGL#F_o$!3(**** **pLLw#GFL7$Fgp$!3%******z)HVqEFL7$Fbp$!3!)******RTI8DFL7$F]p$!3#)**** *>o1iN#FL7$Fho$!3)******f@@\"*>#FL7$Fco$!37+++'\\R?/#FL7$F]o$!35+++\\' e\\)=FL7$Fhn$!3++++(*z(ys\"FL7$$\"37+++Ol\"y^#FU$!3%******z_(zq:FL7$FS $!3#*******3tr89FL7$FN$!3++++Itjc7FL7$$\"3,+++KBhI5FI$!3)******p_d&*4 \"FL7$FA$!3m*****pCyZU*F-7$$\"31+++\"Q@C<%F=$!3#)*****pp\")R&yF-7$$\"3 c*****p3#RUZF7$!3g*****pJ&=$G'F-7$F($!3D+++\")*)Q7ZF-7$F($!3)******Rl# fTJF-7$F($!3++++Fjzq:F-F'-%&COLORG6&%$RGBGF($\"#Q!\"#$\"#[Ff\\p-%)POLY GONSG6^_l7%F'7$F($\"+Fjzq:!#57$$!#K!\"\"F(7%F]]p7$F($\"+aEfTJF`]pFa]p7 %Ff]p7$F($\"+\")*)Q7ZF`]pFa]p7%Fj]p7$$\"+'*>RUZFhs$\"+<`=$G'F`]pFa]p7% F^^p7$$\"+&Q@C<%F-$\"+(p\")R&yF`]pFa]p7%Fd^p7$$\"+#=zeS#FL$\"+Z#yZU*F` ]pFa]p7%Fj^p7$$\"+LBhI5!#;$\"+Fvb*4\"!\"*Fa]p7%F`_p7$$\"+uDpJNFc_p$\"+ Itjc7Ff_pFa]p7%Fh_p7$$\"+pOR95!#:$\"+4tr89Ff_pFa]p7%F^`p7$$\"+Nl\"y^#F a`p$\"+Gvzq:Ff_pFa]p7%Fe`p7$$\"+Z&[&4bFa`p$\"+(*z(ys\"Ff_pFa]p7%F[ap7$ $\"+4!*)f2\"!#9$\"+\\'e\\)=Ff_pFa]p7%Faap7$$\"+Jh(f)=Fdap$\"+'\\R?/#Ff _pFa]p7%Fhap7$$\"+ig@iHFdap$\"+;77*>#Ff_pFa]p7%F^bp7$$\"+'fQ+7%Fdap$\" +#o1iN#Ff_pFa]p7%Fdbp7$$\"+\\vs:\\Fdap$\"+STI8DFf_pFa]p7%Fjbp7$$\"+_tW 7YFdap$\"+))HVqEFf_pFa]p7%F`cp7$$\"+Ww(GL#Fdap$\"+PLjFGFf_pFa]p7%Ffcp7 $$!+'o*z!Q#Fdap$\"+d0)\\)HFf_pFa]p7%F\\dp7$$!+y4MW#)Fdap$\"+;ggUJFf_pF a]p7%Fbdp7$$!+^&f#y&*Fdap$\"+7]s+LFf_pFa]p7%Fhdp7$$\"+w8uN')Fdap$\"+6C nfMFf_pFa]p7%F^ep7$$\"+u'Hn(z!#8$\"+DY%*>OFf_pFa]p7%Fdep7$$\"+Le,AF!#7 $\"+r$QAy$Ff_pFa]p7%F[fp7$$\"+s1t5sF^fp$\"+\\GWZRFf_pFa]p7%Fbfp7$$\"+X %G')o\"!#6$\"+2#4k6%Ff_pFa]p7%Fhfp7$$\"+wbIfOF[gp$\"+Vl&))G%Ff_pFa]p7% F_gp7$$\"++)fTL(F[gp$\"+e0#*fWFf_pFa]p7%Fegp7$$\"+:zp&H\"F`]p$\"+MPU;Y Ff_pFa]p7%F[hp7$$\"+K48S>F`]p$\"+!eIou%Ff_pFa]p7%Fahp7$$\"+)*H&)HDF`]p $\"+nmW`[Ff_pFa]p7%Fghp7$$\"+CI$G,$F`]p$\"+Gr$Q%\\Ff_pFa]p7%F]ip7$$\"+ X@Y(Q$F`]p$\"+hI[B]Ff_pFa]p7%Fcip7$$\"+`ijlOF`]p$\"+')R^&4&Ff_pFa]p7%F iip7$$\"+'4N/'QF`]p$\"+(yy;;&Ff_pFa]p7%F_jp7$$\"+Z=Z&Ff_pFa]p7 %Fc\\q7$$\"+PS]]OF`]p$\"+=#e9^&Ff_pFa]p7%Fi\\q7$$\"+AELnMF`]p$\"+n&oza &Ff_pFa]p7%F_]q7$$\"+u#**yD$F`]p$\"+oHT\"e&Ff_pFa]p7%Fe]q7$$\"+nS&\\-$ F`]p$\"+'yT=h&Ff_pFa]p7%F[^q7$$\"+$>04x#F`]p$\"+?YHRcFf_pFa]p7%Fa^q7$$ \"+rY'y\\#F`]p$\"+'y+Qm&Ff_pFa]p7%Fg^q7$$\"+cfo2AF`]p$\"+)=z`o&Ff_pFa] p7%F]_q7$$\"+-[*>!>F`]p$\"+lx./dFf_pFa]p7%Fc_q7$$\"+pT@#e\"F`]p$\"+\"f s(>dFf_pFa]p7%Fi_q7$$\"+#f$e\\7F`]p$\"+_mcKdFf_pFa]p7%F_`q7$$\"+r0t^!* F[gp$\"+S#)QUdFf_pFa]p7%Fe`q7$$\"+&G5*)\\&F[gp$\"+/*)=\\dFf_pFa]p7%F[a q7$$\"+cx&[%=F[gp$\"+#z+Hv&Ff_pFa]p7%Faaq7$$!+Uih/>F[gp$\"+&>LMv&Ff_pF a]p7%Fgaq7$$!+()QbXdF[gp$\"+,!o1v&Ff_pFa]p7%F]bq7$$!+\"f*=w'*F[gp$\"+^ OXWdFf_pFa]p7%Fcbq7$$!+zFvp8F`]p$\"+8qfMdFf_pFa]p7%Fibq7$$!+J)H9y\"F`] p$\"+cB&3s&Ff_pFa]p7%F_cq7$$!+TPl.AF`]p$\"+(G1Hq&Ff_pFa]p7%Fecq7$$!+'> D#QEF`]p$\"+lrN!o&Ff_pFa]p7%F[dq7$$!+uX?)3$F`]p$\"+*\\(o_cFf_pFa]p7%Fa dq7$$!+YkueNF`]p$\"++%H#>cFf_pFa]p7%Fgdq7$$!+K$p'eSF`]p$\"+D98zbFf_pFa ]p7%F]eq7$$!+SP^.YF`]p$\"+#Qp8`&Ff_pFa]p7%Fceq7$$!+\\'e9A&F`]p$\"+Br( \\Z&Ff_pFa]p7%Fieq7$$!+C'=F'fF`]p$\"+z&Q,T&Ff_pFa]p7%F_fq7$$!+PZ*))*oF `]p$\"+oejT`Ff_pFa]p7%Fefq7$$!+u!>b0)F`]p$\"+9cd#G&Ff_pFa]p7%F[gq7$$!+ !f;2K*F`]p$\"+k2VW_Ff_pFa]p7%Fagq7$$!+T(*zb5Ff_p$\"++(R[A&Ff_pFa]p7%Fg gq7$$!+s6`r6Ff_p$\"+sq=;_Ff_pFa]p7%F]hq7$$!+in1z7Ff_p$\"+ix%H@&Ff_pFa] p7%Fchq7$$!+[b_z8Ff_p$\"+Qa&>@&Ff_pFa]p7%Fihq7$$!+!*z6u9Ff_p$\"+FGV6_F f_pFa]p7%F_iq7$$!+l=(Qc\"Ff_p$\"+%z^.@&Ff_pFa]p7%Feiq7$$!+h9h\\;Ff_p$ \"+&3&43_Ff_pFa]p7%F[jq7$$!+n6*>t\"Ff_p$\"+$pzU?&Ff_pFa]p7%Fajq7$$!+dL `6=Ff_p$\"+k9m)>&Ff_pFa]p7%Fgjq7$$!+i)f'))=Ff_p$\"+193\">&Ff_pFa]p7%F] [r7$$!+(=;P'>Ff_p$\"+]UV\"=&Ff_pFa]p7%Fc[r7$$!+<)*)p.#Ff_p$\"+c%\\'p^F f_pFa]p7%Fi[r7$$!+ySs3@Ff_p$\"+t%zc:&Ff_pFa]p7%F_\\r7$$!+W(G\"z@Ff_p$ \"+3C\\R^Ff_pFa]p7%Fe\\r7$$!+o#)Q[AFf_p$\"+/v1@^Ff_pFa]p7%F[]r7$$!+^#o mJ#Ff_p$\"+(\\#R+^Ff_pFa]p7%Fa]r7$$!+n07%Q#Ff_p$\"+.BYx]Ff_pFa]p7%Fg]r 7$$!+Lu)3X#Ff_p$\"+.(yA0&Ff_pFa]p7%F]^r7$$!+nX5&*\\Ff_pFa]p7%Fi^r7$$!+nIT[EFf_p$\"+\"QVL '\\Ff_pFa]p7%F__r7$$!+[)fPr#Ff_p$\"+JALH\\Ff_pFa]p7%Fe_r7$$!+fw1zFFf_p $\"+K%=K*[Ff_pFa]p7%F[`r7$$!+0dXWGFf_p$\"+c\\2b[Ff_pFa]p7%Fa`r7$$!+Ba. 5HFf_p$\"+'R%*\\\"[Ff_pFa]p7%Fg`r7$$!+#*f!f(HFf_p$\"+.-4tZFf_pFa]p7%F] ar7$$!+J([@/$Ff_p$\"+8k\\HZFf_pFa]p7%Fcar7$$!+J4#)3JFf_p$\"+1`O%o%Ff_p Fa]p7%Fiar7$$!+g*\\f<$Ff_p$\"+2E'yj%Ff_pFa]p7%F_br7$$!+\\(GNC$Ff_p$\"+ k-;!f%Ff_pFa]p7%Febr7$$!+CP^6LFf_p$\"+itUTXFf_pFa]p7%F[cr7$$!+!*f#)zLF f_p$\"+$H?=\\%Ff_pFa]p7%Facr7$$!+@fN[MFf_p$\"+'Ql$Ff_p$\"+Gr)oG%Ff_pFa]p7%Fidr7$$!+<(R=s$Ff_p$\"+j&3VB% Ff_pFa]p7%F_er7$$!+_(>$*y$Ff_p$\"+if>\"=%Ff_pFa]p7%Feer7$$!+j5=cQFf_p$ \"+[#3v7%Ff_pFa]p7%F[fr7$$!+;2KARFf_p$\"+S&)=tSFf_pFa]p7%Fafr7$$!+=Zl( )RFf_p$\"+*fp\"=SFf_pFa]p7%Fgfr7$$!+ov6_SFf_p$\"+X&yB'RFf_pFa]p7%F]gr7 $$!+&)4m:TFf_p$\"+*pSd!RFf_pFa]p7%Fcgr7$$!+u@DyTFf_p$\"+yD=[QFf_pFa]p7 %Figr7$$!+,>()RUFf_p$\"+HTj*y$Ff_pFa]p7%F_hr7$$!+nD^+VFf_p$\"+a..IPFf_ pFa]p7%Fehr7$$!+ijv%Ff_p$\"+*H`(4KFf_pFa]p7%Fe[s7$$!+R[`/[Ff_p$\"+\"GS*QJFf_pFa]p 7%F[\\s7$$!+AULc[Ff_p$\"+oD%o1$Ff_pFa]p7%Fa\\s7$$!+-LP2\\Ff_p$\"+;8[$* HFf_pFa]p7%Fg\\s7$$!+[1md\\Ff_p$\"+*o$))=HFf_pFa]p7%F]]s7$$!+')o>2]Ff_ p$\"+f:3VGFf_pFa]p7%Fc]s7$$!+*ytf0&Ff_p$\"+r*4hw#Ff_pFa]p7%Fi]s7$$!+5P (R5&Ff_p$\"+ba+)o#Ff_pFa]p7%F_^s7$$!+:%p6:&Ff_p$\"+fU!)3EFf_pFa]p7%Fe^ s7$$!+QV_(>&Ff_p$\"+b1aGDFf_pFa]p7%F[_s7$$!+#Ff_pFa]p7%Fc`s7$$!+%yf[T&Ff_p$ \"+P5=7@Ff_pFa]p7%Fi`s7$$!+JZ*\\X&Ff_p$\"+E-+E?Ff_pFa]p7%F_as7$$!+4e&Q \\&Ff_p$\"+^e&)Q>Ff_pFa]p7%Feas7$$!+Z5PJbFf_p$\"+!*3u]=Ff_pFa]p7%F[bs7 $$!+#=puc&Ff_p$\"+)RX;w\"Ff_pFa]p7%Fabs7$$!+!H!3-cFf_p$\"+$*ybr;Ff_pFa ]p7%Fgbs7$$!+Wi8NcFf_p$\"+joY!e\"Ff_pFa]p7%F]cs7$$!+A5dmcFf_p$\"+UBO)[ \"Ff_pFa]p7%Fccs7$$!+/4K'p&Ff_p$\"+BvB&R\"Ff_pFa]p7%Fics7$$!+(fCVs&Ff_ p$\"+s.4,8Ff_pFa]p7%F_ds7$$!+AL_]dFf_p$\"+A_#f?\"Ff_pFa]p7%Feds7$$!+/4 '[x&Ff_p$\"+&Ha(46Ff_pFa]p7%F[es7$$!+!)QG(z&Ff_p$\"+J\"*f75Ff_pFa]p7%F aes7$$!+i;uFf_pFa]p7%F]^t7$F`as$!+ E-+E?Ff_pFa]p7%Fa^t7$Fj`s$!+P5=7@Ff_pFa]p7%Fe^t7$Fd`s$!+j6S(>#Ff_pFa]p 7%Fi^t7$F^`s$!+[$e;G#Ff_pFa]p7%F]_t7$Fh_s$!+8X%\\O#Ff_pFa]p7%Fa_t7$Fb_ s$!+/aCZCFf_pFa]p7%Fe_t7$F\\_s$!+b1aGDFf_pFa]p7%Fi_t7$Ff^s$!+fU!)3EFf_ pFa]p7%F]`t7$F`^s$!+ba+)o#Ff_pFa]p7%Fa`t7$Fj]s$!+r*4hw#Ff_pFa]p7%Fe`t7 $Fd]s$!+f:3VGFf_pFa]p7%Fi`t7$F^]s$!+*o$))=HFf_pFa]p7%F]at7$Fh\\s$!+;8[ $*HFf_pFa]p7%Faat7$Fb\\s$!+oD%o1$Ff_pFa]p7%Feat7$F\\\\s$!+\"GS*QJFf_pF a]p7%Fiat7$Ff[s$!+*H`(4KFf_pFa]p7%F]bt7$F`[s$!+0vEzKFf_pFa]p7%Fabt7$Fj jr$!+plZZLFf_pFa]p7%Febt7$Fdjr$!+PCQ9MFf_pFa]p7%Fibt7$F^jr$!+'f&**zMFf _pFa]p7%F]ct7$Fhir$!+c^LWNFf_pFa]p7%Fact7$Fbir$!+v(Gug$Ff_pFa]p7%Fect7 $F\\ir$!+yCJpOFf_pFa]p7%Fict7$Ffhr$!+a..IPFf_pFa]p7%F]dt7$F`hr$!+HTj*y $Ff_pFa]p7%Fadt7$Fjgr$!+yD=[QFf_pFa]p7%Fedt7$Fdgr$!+*pSd!RFf_pFa]p7%Fi dt7$F^gr$!+X&yB'RFf_pFa]p7%F]et7$Fhfr$!+*fp\"=SFf_pFa]p7%Faet7$Fbfr$!+ S&)=tSFf_pFa]p7%Feet7$F\\fr$!+[#3v7%Ff_pFa]p7%Fiet7$Ffer$!+if>\"=%Ff_p Fa]p7%F]ft7$F`er$!+j&3VB%Ff_pFa]p7%Faft7$Fjdr$!+Gr)oG%Ff_pFa]p7%Feft7$ Fddr$!+V6&*QVFf_pFa]p7%Fift7$F^dr$!+$*H\\!R%Ff_pFa]p7%F]gt7$Fhcr$!+&*\\Ff_pFa]p7%Fejt7$Fd^r$!+X3&[-&Ff_pF a]p7%Fijt7$F^^r$!+.(yA0&Ff_pFa]p7%F][u7$Fh]r$!+.BYx]Ff_pFa]p7%Fa[u7$Fb ]r$!+(\\#R+^Ff_pFa]p7%Fe[u7$F\\]r$!+/v1@^Ff_pFa]p7%Fi[u7$Ff\\r$!+3C\\R ^Ff_pFa]p7%F]\\u7$F`\\r$!+t%zc:&Ff_pFa]p7%Fa\\u7$Fj[r$!+c%\\'p^Ff_pFa] p7%Fe\\u7$Fd[r$!+]UV\"=&Ff_pFa]p7%Fi\\u7$F^[r$!+193\">&Ff_pFa]p7%F]]u7 $Fhjq$!+k9m)>&Ff_pFa]p7%Fa]u7$Fbjq$!+$pzU?&Ff_pFa]p7%Fe]u7$F\\jq$!+&3& 43_Ff_pFa]p7%Fi]u7$Ffiq$!+%z^.@&Ff_pFa]p7%F]^u7$F`iq$!+FGV6_Ff_pFa]p7% Fa^u7$Fjhq$!+Qa&>@&Ff_pFa]p7%Fe^u7$Fdhq$!+ix%H@&Ff_pFa]p7%Fi^u7$F^hq$! +sq=;_Ff_pFa]p7%F]_u7$Fhgq$!++(R[A&Ff_pFa]p7%Fa_u7$Fbgq$!+k2VW_Ff_pFa] p7%Fe_u7$F\\gq$!+9cd#G&Ff_pFa]p7%Fi_u7$Fffq$!+oejT`Ff_pFa]p7%F]`u7$F`f q$!+z&Q,T&Ff_pFa]p7%Fa`u7$Fjeq$!+Br(\\Z&Ff_pFa]p7%Fe`u7$Fdeq$!+#Qp8`&F f_pFa]p7%Fi`u7$F^eq$!+D98zbFf_pFa]p7%F]au7$Fhdq$!++%H#>cFf_pFa]p7%Faau 7$Fbdq$!+*\\(o_cFf_pFa]p7%Feau7$F\\dq$!+lrN!o&Ff_pFa]p7%Fiau7$Ffcq$!+( G1Hq&Ff_pFa]p7%F]bu7$F`cq$!+cB&3s&Ff_pFa]p7%Fabu7$Fjbq$!+8qfMdFf_pFa]p 7%Febu7$Fdbq$!+^OXWdFf_pFa]p7%Fibu7$F^bq$!+,!o1v&Ff_pFa]p7%F]cu7$Fhaq$ !+&>LMv&Ff_pFa]p7%Facu7$Fbaq$!+#z+Hv&Ff_pFa]p7%Fecu7$F\\aq$!+/*)=\\dFf _pFa]p7%Ficu7$Ff`q$!+S#)QUdFf_pFa]p7%F]du7$F``q$!+_mcKdFf_pFa]p7%Fadu7 $Fj_q$!+\"fs(>dFf_pFa]p7%Fedu7$Fd_q$!+lx./dFf_pFa]p7%Fidu7$F^_q$!+)=z` o&Ff_pFa]p7%F]eu7$Fh^q$!+'y+Qm&Ff_pFa]p7%Faeu7$Fb^q$!+?YHRcFf_pFa]p7%F eeu7$F\\^q$!+'yT=h&Ff_pFa]p7%Fieu7$Ff]q$!+oHT\"e&Ff_pFa]p7%F]fu7$F`]q$ !+n&oza&Ff_pFa]p7%Fafu7$Fj\\q$!+=#e9^&Ff_pFa]p7%Fefu7$Fd\\q$!+Y$>=Z&Ff _pFa]p7%Fifu7$F^\\q$!+\"=u*GaFf_pFa]p7%F]gu7$Fh[q$!+8U#GQ&Ff_pFa]p7%Fa gu7$Fb[q$!+g.CL`Ff_pFa]p7%Fegu7$F\\[q$!+&)e/!G&Ff_pFa]p7%Figu7$Ffjp$!+ fm)HA&Ff_pFa]p7%F]hu7$F`jp$!+(yy;;&Ff_pFa]p7%Fahu7$Fjip$!+')R^&4&Ff_pF a]p7%Fehu7$Fdip$!+hI[B]Ff_pFa]p7%Fihu7$F^ip$!+Gr$Q%\\Ff_pFa]p7%F]iu7$F hhp$!+nmW`[Ff_pFa]p7%Faiu7$Fbhp$!+!eIou%Ff_pFa]p7%Feiu7$F\\hp$!+MPU;YF f_pFa]p7%Fiiu7$Ffgp$!+e0#*fWFf_pFa]p7%F]ju7$F`gp$!+Vl&))G%Ff_pFa]p7%Fa ju7$Fifp$!+2#4k6%Ff_pFa]p7%Feju7$Fcfp$!+\\GWZRFf_pFa]p7%Fiju7$F\\fp$!+ r$QAy$Ff_pFa]p7%F][v7$Feep$!+DY%*>OFf_pFa]p7%Fa[v7$F_ep$!+6CnfMFf_pFa] p7%Fe[v7$Fidp$!+7]s+LFf_pFa]p7%Fi[v7$$!+#)4MW#)Fdap$!+;ggUJFf_pFa]p7%F ]\\v7$F]dp$!+d0)\\)HFf_pFa]p7%Fc\\v7$$\"+Xw(GL#Fdap$!+PLjFGFf_pFa]p7%F g\\v7$Facp$!+))HVqEFf_pFa]p7%F]]v7$F[cp$!+STI8DFf_pFa]p7%Fa]v7$Febp$!+ #o1iN#Ff_pFa]p7%Fe]v7$F_bp$!+;77*>#Ff_pFa]p7%Fi]v7$Fiap$!+'\\R?/#Ff_pF a]p7%F]^v7$Fbap$!+\\'e\\)=Ff_pFa]p7%Fa^v7$F\\ap$!+(*z(ys\"Ff_pFa]p7%Fe ^v7$$\"+Ol\"y^#Fa`p$!+Gvzq:Ff_pFa]p7%Fi^v7$F_`p$!+4tr89Ff_pFa]p7%F__v7 $Fi_p$!+Itjc7Ff_pFa]p7%Fc_v7$$\"+KBhI5Fc_p$!+Fvb*4\"Ff_pFa]p7%Fg_v7$F[ _p$!+Z#yZU*F`]pFa]p7%F]`v7$$\"+\"Q@C<%F-$!+(p\")R&yF`]pFa]p7%Fa`v7$$\" +(3#RUZFhs$!+<`=$G'F`]pFa]p7%Fg`v7$F($!+\")*)Q7ZF`]pFa]p7%F]av7$F($!+a EfTJF`]pFa]p7%Faav7$F($!+Fjzq:F`]pFa]p7%FeavF'Fa]p-Fa\\p6&Fc\\pF($\"#v Ff\\p$\"#&*Ff\\p-%&STYLEG6#%,PATCHNOGRIDG-F$6%7$7$$!3o*************)oF LF(7$$\"3%**************=\"FLF(-%'COLOURG6&Fc\\pF)F)F)-%*LINESTYLEG6# \"\"$-F$6%7$7$F($!3o*************Q'FL7$F($\"3o*************Q'FLF\\cvF_ cv-%+AXESLABELSG6%%&Re(z)G%&Im(z)G-%%FONTG6#%(DEFAULTG-Fbdv6$%*HELVETI CAG\"\"*-%*AXESSTYLEG6#%$BOXG-%(SCALINGG6#%,CONSTRAINEDG-%%VIEWG6$;$!$ *oFf\\p$\"$>\"Ff\\p;$!$R'Ff\\p$\"$R'Ff\\p" 1 2 0 1 10 0 2 9 1 2 1 1.000000 45.000000 45.000000 0 0 "Curve 1" "Curve 2" "Curve 3" "Curve \+ 4" }}}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 4 "The " }{TEXT 260 30 "interval of absolute stability" }{TEXT -1 89 " (or \+ stability interval) is the intersection of the stability region with t he real line." }}{PARA 0 "" 0 "" {TEXT -1 59 "For this scheme the stab ility interval is (approximately) " }{XPPEDIT 18 0 "[-5.9069, 0];" "6 #7$,$-%&FloatG6$\"&p!f!\"%!\"\"\"\"!" }{TEXT -1 2 ". " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 114 "We can distort the bo undary curve horizontally by taking the 11th root of the real part of \+ points along the curve." }}{PARA 0 "" 0 "" {TEXT -1 150 "In this way w e see that the intersection of the stability region with the non-negat ive imaginary axis consists of the union of two disjoint intervals." } }{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 660 "R := z -> add(z^j/j!,j=0..8)+\n 43798026811471013810038182598 84330444073/1632364979094686187578793609453668748325355520*z^9+\n 20 59261911322648655664652819645913227129613/7996956032584867632948509892 713523198045916692480*z^10+\n 54386205628213076358033918788701502625 0677/42650432173785960709058719427805457056244889026560*z^11+\n 3915 956475233344332260563565/64749265042467634748334825530276708352*z^12: \nDigits := 18:\npts := []: z0 := 0:\nfor ct from 0 to 200 do\n zz : = newton(R(z)=exp(ct*Pi/60*I),z=z0):\n z0 := zz:\n pts := [op(pts) ,[surd(Re(zz),11),Im(zz)]]:\nend do:\nplot(pts,color=COLOR(RGB,0,.65,. 85),thickness=2,font=[HELVETICA,9]);\nDigits := 10:" }}{PARA 13 "" 1 " " {GLPLOT2D 287 308 308 {PLOTDATA 2 "6(-%'CURVESG6#7ew7$$\"\"!F)F(7$F( $\"3v))H)fv()fB&!#>7$$\"3x3N$*)\\T,\"GF-$\"3%))f'>^v>Z5!#=7$$\"3xsdD+j UhSF-$\"3ZQ\\zEjzq:F37$$\"3CEyj)Q*=t_F-$\"3M[PR-^R%4#F37$$\"3e+)\\e2gb X'F-$\"3_qb*z(Q*zh#F37$$\"3/H?meh=9wF-$\"34d0h`EfTJF37$$\"3dDTa_se_()F -$\"3]y*p#H9>lOF37$$\"3[>x]DZ5t)*F-$\"3EwR00-z)=%F37$$\"3-')fS-ft(4\"F 3$\"3%=mT6)*)Q7ZF37$$\"3a*3]bn\\m?\"F3$\"30II*yv()fB&F37$$\"3iHDzcN898 F3$\"3IXE(f`'efdF37$$\"37oNPWaC?9F3$\"3#GJClJ&=$G'F37$$\"3!)**eYm_-D:F 3$\"3l`OS,Ty1oF37$$\"37_a!Q(*)\\G;F3$\"3@O9Z$*GQItF37$$\"3zEI_>8oIF3 $\"3bj%oTOz6!*)F37$$\"3Z'fDo`(\\H?F3$\"3l\\c\"oCyZU*F37$$\"3&e7d4j'\\E @F3$\"3#*3^XzrP[**F37$$\"3,,$Q:)H;AAF3$\"37^a!zh(>Z5!#<7$$\"3&=)=q)*3Z ;BF3$\"3^\"f%fEvb*4\"Fhq7$$\"3u;_?x(*Q4CF3$\"3'4jrjW<>:\"Fhq7$$\"3O\"Q vC?%)3]#F3$\"3!z`!))ztF/7Fhq7$$\"3w)z3^&R\"4f#F3$\"3zD](*Htjc7Fhq7$$\" 38BYy@RVzEF3$\"3k?Lg*H(**38Fhq7$$\"33uKw:RRmFF3$\"3wG;l\"Hd8O\"Fhq7$$ \"3[N#efSQ<&GF3$\"39&**[(3tr89Fhq7$$\"3]I0g\\hSNHF3$\"3ctA/`t2m9Fhq7$$ \"3KVbU>(Ht,$F3$\"3j))z&fUP%=:Fhq7$$\"38r@x6\\V(4$F3$\"37N#yz_(zq:Fhq7 $$\"3?0%Hm,Sc<$F3$\"3h*oq%ew:B;Fhq7$$\"3cPp/C\\&=D$F3$\"3\"\\`Gc\"y^v; Fhq7$$\"3:SK&G.!)fK$F3$\"3O4[f'*z(ys\"Fhq7$$\"3?6&zk\"\\!zR$F3$\"3fCv' y>Q-y\"Fhq7$$\"3-$H-do1vY$F3$\"3r5p5;%)fK=Fhq7$$\"306ldnzkMNF3$\"3YCV \\\\'e\\)=Fhq7$$\"3$*y.;\"Ru\"*f$F3$\"37oN#)**)=t$>Fhq7$$\"3!3>())H9\" 4m$F3$\"3ns$RbFhq7$$\"3z<\"H9Rg'>PF3$\"3eCU(f\\R?/#Fhq7$$\"3]r[p eK>vPF3$\"30[I2(*)*R%4#Fhq7$$\"3zs&eQ&fCFQF3$\"3FM[$*Q/wY@Fhq7$$\"3%)Q )[LR4b(QF3$\"3ZK.a:77*>#Fhq7$$\"3'3DlrP<'>RF3$\"3\"yWuqO#[^AFhq7$$\"3X jB=B)H\"fRF3$\"3S!)**G&4WQI#Fhq7$$\"32S6I^*3N*RF3$\"3viYR#o1iN#Fhq7$$ \"3vgi/=W3ASF3$\"3bd\\(R^q&3CFhq7$$\"3o;YbA/+WSF3$\"3QYm\\1h$4Y#Fhq7$$ \"3$)Q([.&>8eSF3$\"3tlG%*RTI8DFhq7$$\"3=s?;6]%H1%F3$\"3I;T;'\\vcc#Fhq7 $$\"3V\\O\"zeZi0%F3$\"3)\\M@NI^!=EFhq7$$\"380UH&\\1Z.%F3$\"3!GG-%))HVq EFhq7$$\"3_7&))=>'z#*RF3$\"3(4+$=MB#Gs#Fhq7$$\"3;V\")Hf.5?RF3$\"3'*o;> [:AvFFhq7$$\"3!>lHS;qAz$F3$\"3')e0@PLjFGFhq7$$\"3$*R#\\w=:%=NF3$\"3gOi -#*41!)GFhq7$$!3!fLRYb<8L$F3$\"3Tfp]\"[3D$HFhq7$$!3_\"yGBz'G*z$F3$\"3^ A(\\mb!)\\)HFhq7$$!3'HDWOW>V,%F3$\"3*>*R*\\'G[PIFhq7$$!3OsFDGp]bTF3$\" 3Kpwtv?-!4$Fhq7$$!3'R%*4B:^MD%F3$\"3f\"H*z:ggUJFhq7$$!3&Q]JX6#)eJ%F3$ \"3\"\\DCp\"QC&>$Fhq7$$!3qpKunH^$Fhq7$$\"3L*e%Gt&p?)\\F3$\"3AdeFkZLmNFhq7$$\"3***=%[0h9G _F3$\"3#3,y`iW*>OFhq7$$\"3*>J*R>ZE[aF3$\"3\")*[8CMxPn$Fhq7$$\"3M@xTS(G Al&F3$\"3$>96m1kys$Fhq7$$\"3e!***>;$>`%eF3$\"3U=a*3PQAy$Fhq7$$\"36#3fJ /d2.'F3$\"3T@BByc$p$QFhq7$$\"3`-\"H5$Ro5iF3$\"3kTa-o@*>*QFhq7$$\"3@8iv llh'Q'F3$\"3;T()**[GWZRFhq7$$\"3CcJN8QnflF3$\"3'z/W\"=$=L+%Fhq7$$\"3Au -(H!HpInF3$\"3%HopH[R'fSFhq7$$\"30n<\\d.G+pF3$\"3]YA\\2#4k6%Fhq7$$\"3n Fmf>j\")oqF3$\"3B_u56&*ftTFhq7$$\"3rOSx3#=kB(F3$\"3@1*\\1$G8JUFhq7$$\" 3*GYXa&e'GS(F3$\"3J47*Hac))G%Fhq7$$\"3cc6Y:!H+*F3$\"3.$o^b&oQr\\Fhq7$$\"3H ;arR]\"=*F3$\"3]tbO)G=E=&Fhq7$$\"3r,2cOX=!>* F3$\"3lBr\"H'p/._Fhq7$$\"3zc35*Rks>*F3$\"3-$G5!fm)HA&Fhq7$$\"3b$RR9!\\6#*F3$\"3hM-&[?#>)H&Fhq7$$\"3 @]#o%f*)*=@*F3$\"3$GE(eF1#fJ&Fhq7$$\"3\\]R\"*)Gm6@*F3$\"35Pq[g.CL`Fhq7 $$\"3$)3h&>jP$4#*F3$\"3J>LAF&f,N&Fhq7$$\"3p\"Ry$GAX1#*F3$\"36ws)[L&om` Fhq7$$\"3*f][$zOa-#*F3$\"3Lm+t7U#GQ&Fhq7$$\"3#*o?'4YSw>*F3$\"3L$*yYF>e )R&Fhq7$$\"3CqKIYjw\">*F3$\"34[fAzO'RT&Fhq7$$\"3E\")Q&*[2%\\=*F3$\"3Aw :V\"=u*GaFhq7$$\"3gay<)**yr<*F3$\"3c)*zNFxhVaFhq7$$\"3#)=(yPb#\\o\"*F3 $\"3/%>LbC)*yX&Fhq7$$\"3w6!en;*))e\"*F3$\"3DRu$eM>=Z&Fhq7$$\"3D)4/.*HP [\"*F3$\"37$*pzdVQ&[&Fhq7$$\"3j&3PUkWp8*F3$\"3b1ZKijf)\\&Fhq7$$\"3U?Qw M7gC\"*F3$\"3Z'o[z@e9^&Fhq7$$\"3[!>,-LO86*F3$\"3nP!)R#esR_&Fhq7$$\"34x T:#*)Rr4*F3$\"3^8FFI>9ObFhq7$$\"3upfVg\")*>3*F3$\"3B+oSn&oza&Fhq7$$\"3 [#)pt)[$*e1*F3$\"35WlVUYXfbFhq7$$\"3spuiET!)[!*F3$\"3$QY-g:-1d&Fhq7$$ \"3R0O%=%RqI!*F3$\"3Nrg%z'HT\"e&Fhq7$$\"333@([2i:,*F3$\"3&os.G!)))=f&F hq7$$\"3*>(H]:DM\"**)F3$\"3*e5^QDJ?g&Fhq7$$\"3N#G>\")e.+(*)F3$\"3-P,\" fyT=h&Fhq7$$\"3;Nc%)Gt\\Z*)F3$\"3%*R[4PBZIcFhq7$$\"3aXKYX[v)*))F3$\"3@+'[*>YHRcFhq7$$\"3AQD1eOQs))F3$\" 3Tm*3zf*yZcFhq7$$\"3A&4L*oFdW))F3$\"3g)pQc3efl&Fhq7$$\"3:lIBVxA:))F3$ \"3^(3Rcy+Qm&Fhq7$$\"3q3Q\"p4SUy)F3$\"34JXXo#=8n&Fhq7$$\"3Yh4_)z%[^()F 3$\"3KKg&)p4^ycFhq7$$\"3Mtl\"y9p&Fhq7$$\"3;(o5#QH/T')F3$\"373<.]E9)p&Fhq7$$\"3\"=G=/T2 &*f)F3$\"3YeN`kx./dFhq7$$\"3ue$*eSm=b&)F3$\"3e*>&GD\"3'4dFhq7$$\"336!f 6=]x])F3$\"3Hwz]vK&[r&Fhq7$$\"3yAt!>a(zc%)F3$\"34S/[\"fs(>dFhq7$$\"3S^ 72Gs$=S)F3$\"3b$3IgFlVs&Fhq7$$\"3&eXw@>dAM)F3$\"3\"QQe9NI'GdFhq7$$\"3P u8_#3#Gx#)F3$\"3QC>&=lmDt&Fhq7$$\"3zP73u%4f?)F3$\"3^=\\s9GFZ%RdFhq7$$\"3Li<37VwpF3$\"3Me3n%>LMv&Fhq7$$!3]2P)o;=zI(F3$\"3a*4.;f%)Gv &Fhq7$$!3'RYOGc1l`(F3$\"3%[wh0sk>v&Fhq7$$!3R[pu&Fh q7$$!3hvl^O>2(3)F3$\"3?Np=^OXWdFhq7$$!3`i4J\"4GB=)F3$\"3!>Zp:t$eTdFhq7 $$!33bo(yZe#o#)F3$\"34i8t)>,$QdFhq7$$!3m\\uv4RmY$)F3$\"3X8Yl7qfMdFhq7$ $!3\"ec/E8o)=%)F3$\"3W%='fv8YIdFhq7$$!3Il=lwX(e[)F3$\"3NlB[kO)es&Fhq7$ $!392BNbsg)F3$\"3V![9A%\\N:dFhq7$ $!3+QWe_Bvi')F3$\"3&3;O+%yP4dFhq7$$!3y?C3,'o`r)F3$\"3NlF&oG1Hq&Fhq7$$! 3BVE\"*G1Xl()F3$\"3i*[<&GU#fp&Fhq7$$!3*)H77tFH8))F3$\"3nZO1&>9%)o&Fhq7 $$!3[z!eQi[\"f))F3$\"3Ba)>a;d.o&Fhq7$$!3-dlM*yRK!*)F3$\"3IA&3:UKcFhq7$$!30q]x`1gS\"*F3$\"3IXr[T)[m g&Fhq7$$!3og*f)*=Lq<*F3$\"3\\F:a2dH$f&Fhq7$$!38^Z'RyXH@*F3$\"30$4LZUJ \"zbFhq7$$!39C&[Ns#[[#*F3$\"3`dThBe6kbFhq7$$!3]YY&ev'z$G*F3$\"3seiCH#3 #[bFhq7$$!3k;(**yp_!>$*F3$\"3DHpC#Qp8`&Fhq7$$!3c.M@/6Va$*F3$\"3T6$G5nk N^&Fhq7$$!3*z%Qx+;8!R*F3$\"3.GpX)RpZ\\&Fhq7$$!3o[3*G+vjU*F3$\"3d+t)H7x \\Z&Fhq7$$!316(4zY-MY*F3$\"3M*>RcW7UX&Fhq7$$!3y.FfZ&o9]*F3$\"3TP%3It\\ DV&Fhq-%&COLORG6&%$RGBGF($\"#l!\"#$\"#&)Fcjn-%+AXESLABELSG6$Q!6\"Fijn- %%FONTG6$%*HELVETICAG\"\"*-%*THICKNESSG6#\"\"#-%%VIEWG6$%(DEFAULTGFg[o " 1 2 0 1 10 2 2 6 1 4 2 1.000000 45.000000 45.000000 0 0 "Curve 1" }} }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 108 "The re levant intersection points of the boundary curve with the imaginary ax is can be determined as follows." }}{PARA 0 "" 0 "" {TEXT -1 87 "First we look for points on the boundary curve either side of each intersec tion point. " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 290 "Digits := 15:\nz0 := 2.9*I:\nfor ct from 91 to 94 do\n newton(R(z)=exp(ct*Pi/100*I),z=z0);\nend do;``;\nz0 := 3.4*I: \nfor ct from 107 to 110 do\n newton(R(z)=exp(ct*Pi/100*I),z=z0);\ne nd do;``;\nz0 := 5.75*I:\nfor ct from 271 to 274 do\n newton(R(z)=ex p(ct*Pi/100*I),z=z0);\nend do;\nDigits := 10:" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#^$$\"0_V*e\"p2e\"!#>$\"0I]&)y(3fG!#9" }}{PARA 11 "" 1 " " {XPPMATH 20 "6#^$$\"04h4@0HF(!#?$\"0qtIw[0*G!#9" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#^$$!0^ThQ4YB#!#?$\"0!Q%4=ccwij7!#>$\"0o*RmS\\`H!#9" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%!G" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#^$$!0?nP=DS'f!#> $\"0X=e&3=kL!#9" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#^$$!0GG2ygTd#!#>$\" 0fpJSmfR$!#9" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#^$$\"0iT+q**[>#!#>$\"0 2!36bzFM!#9" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#^$$\"021dPTdj)!#>$\"0WP 3Ts'fM!#9" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%!G" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#^$$\"0C#*4uTC5\"!#;$\"0iNF)\\E`d!#9" }}{PARA 11 "" 1 " " {XPPMATH 20 "6#^$$\"0kJ!p\"GDc$!#<$\"0C:XB,Nv&!#9" }}{PARA 11 "" 1 " " {XPPMATH 20 "6#^$$!0%HZ\"RRn$R!#<$\"0**)oa'3Ov&!#9" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#^$$!0T!)*\\lIZ6!#;$\"0e#3/je`d!#9" }}}{PARA 0 "" 0 " " {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 107 "Then we apply the bise ction method to calculate the parameter value associated with the inte rsection point." }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 482 "Digits : = 15:\nreal_part := proc(u)\n Re(newton(R(z)=exp(u*Pi*I),z=2.9*I))\n end proc:\nu0 := bisect('real_part'(u),u=0.91..0.94);\nnewton(R(z)=exp (u0*Pi*I),z=2.9*I);``;\nreal_part := proc(u)\n Re(newton(R(z)=exp(u* Pi*I),z=3.4*I))\nend proc:\nu0 := bisect('real_part'(u),u=1.07..1.10); \nnewton(R(z)=exp(u0*Pi*I),z=3.4*I);``;\nreal_part := proc(u)\n Re(n ewton(R(z)=exp(u*Pi*I),z=5.75*I))\nend proc:\nu0 := bisect('real_part' (u),u=2.71..2.74);\nnewton(R(z)=exp(u0*Pi*I),z=5.75*I);\nDigits := 10: " }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%#u0G$\"0@pYictF*!#:" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#^#$\"0/yd'4*[\"H!#9" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%!G" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%#u0G$\"0[y2W(y &3\"!#9" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#^#$\"0]kkt\"Q9M!#9" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#%!G" }}{PARA 11 "" 1 "" {XPPMATH 20 "6 #>%#u0G$\"0gB**ocZs#!#9" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#^$$\"0@6=W* y8u!#H$\"0IoKXoNv&!#9" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 4 "The " }{TEXT 260 112 "largest interval on the nonnega tive imaginary axis that contains the origin and lies inside the stabi lity region" }{TEXT -1 5 " is " }{XPPEDIT 18 0 "[0, 2.9149];" "6#7$\" \"!-%&FloatG6$\"&\\\"H!\"%" }{TEXT -1 18 " (approximately)." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 4 "The " }{TEXT 260 72 "intersection of the stability region with the nonnegative imag inary axis" }{TEXT -1 5 " is " }{XPPEDIT 18 0 "`union`([0, 2.9149],[3 .4144, 5.7536]);" "6#-%&unionG6$7$\"\"!-%&FloatG6$\"&\\\"H!\"%7$-F)6$ \"&WT$F,-F)6$\"&Ov&F," }{TEXT -1 17 " (approximately)." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 37 "#------------------- -----------------" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 81 "The stability function R* for the 13 stage, order 7 sch eme is given as follows." }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 98 " subs(ee,subs(b=`b*`,StabilityFunction(7,13,'expanded'))):\n`R*` := una pply(%,z):\n'`R*`(z)'=`R*`(z);" }}{PARA 12 "" 1 "" {XPPMATH 20 "6#/-%# R*G6#%\"zG,:\"\"\"F)F'F)*&#F)\"\"#F)*$)F'F,F)F)F)*&#F)\"\"'F)*$)F'\"\" $F)F)F)*&#F)\"#CF)*$)F'\"\"%F)F)F)*&#F)\"$?\"F)*$)F'\"\"&F)F)F)*&#F)\" $?(F)*$)F'F1F)F)F)*&#F)\"%S]F)*$)F'\"\"(F)F)F)*&#\"I:\"QKG6hI?8#ov%G*RBj)3XKs@SIJ`F)*$)F'\"\"*F)F)F)*&#\"M(3-! fP)QjX`)*Qj#z]*ylaw\"*R\"\"SS)RNLnVe&=3<9z!)ej5%*y1E[c(R'F)*$)F'\"#5F) F)F)*&#\">N4HI+KPj(*Q " 0 "" {MPLTEXT 1 0 30 "z_0 := newton(`R*`(z)=1,z=-6);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%$z_0G $!+mtFWe!\"*" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 318 "z_0 := newton(`R*`(z)=1,z=-6):\np_1 := plot([`R *`(z),1],z=-6.49..0.49,color=[red,blue]):\np_2 := plot([[[z_0,1]]$3],s tyle=point,symbol=[circle,cross,diamond],color=black):\np_3 := plot([[ z_0,0],[z_0,1]],linestyle=3,color=COLOR(RGB,0,.5,0)):\nplots[display]( [p_1,p_2,p_3],view=[-6.49..0.49,-0.07..1.47],font=[HELVETICA,9]);" }} {PARA 13 "" 1 "" {GLPLOT2D 387 266 266 {PLOTDATA 2 "6+-%'CURVESG6$7Z7$ $!3A++++++!\\'!#<$\"316@;lK0'z$F*7$$!3+n\"zf'R'>X'F*$\"3K:@')R?'*GNF*7 $$!3xL$e>$z#RT'F*$\"3c1**Ro1hyKF*7$$!3a+v$z*=*eP'F*$\"39>?**=F2WIF*7$$ !3Lnm\"R'e&yL'F*$\"3(\\1J4amW#GF*7$$!3o](oU'fmriF*$\"3kfJ*R[SaZ#F*7$$! 3,M3ikgZ0iF*$\"3;&>XiP&)\\;#F*7$$!3m]7))*fQ58'F*$\"3_`3(fLQs&=F*7$$!3K n;9N6gcgF*$\"3UDjin0i)e\"F*7$$!3Cnmc]-n\")fF*$\"3Y_6?E,G`8F*7$$!3;n;*f ORn!fF*$\"3')oBu[F2\\6F*7$$!3[]7$[hk@$eF*$\"3oD<6\"\\x8t*!#=7$$!3#Q$3n j)*eddF*$\"3aN1)eZC@@)F^o7$$!3ymTb#=4$>cF*$\"3\"=!\\Q)zgW$fF^o7$$!3e+v yYv7waF*$\"3+)*y1YS2wTF^o7$$!3hmTD%*)[!G`F*$\"3INw'HPNA&GF^o7$$!3w*\\P =8X/=&F*$\"3GQ&=z(o:7>F^o7$$!3@LLQ>lhG]F*$\"3g@/&Rb?)R7F^o7$$!3km;*[z& )[*[F*$\"3Fe9%\\G\"f9$)!#>7$$!3I++:tmLWZF*$\"3\\'exX&3PD_Fbq7$$!39++D# QpJf%F*$\"39qtoky*[G$Fbq7$$!3I++&oo\"\\ZWF*$\"3)or&p$e'\\!>#Fbq7$$!3Fn TD$Q-_J%F*$\"3k'oYP)oMg;Fbq7$$!3-LL)fg(*y:%F*$\"3$))H!)*HK%pU\"Fbq7$$! 3#HLLf#)RY-%F*$\"3a21;A[ci9Fbq7$$!3S+v[[$H'pQF*$\"3;0>;.fk'o\"Fbq7$$!3 zLLV5)=Ct$F*$\"335(H(\\>p+?Fbq7$$!3))*\\([.r(=e$F*$\"3G$GVoT%\\PCFbq7$ $!3&**\\i%)zD&QMF*$\"3a&\\GF*$\"3@!H(*eFryq&Fbq7$$!3<+Dr!e\"f:FF*$\"3c!Q.si%[qxie))Fbq7$$!3 ]+]x'32_F#F*$\"3\\o[rL([l-\"F^o7$$!3-+voAGsL@F*$\"35un;7hF$=\"F^o7$$!3 k+]#)p'Gm(>F*$\"3%)Q\"e$fj,&Q\"F^o7$$!36MLV'esa$=F*$\"3%e#3)y:(oCF^o7$$!3)R$3#yX'Re7F*$\"3 C9UgA')3TGF^o7$$!3%**\\P?H[:6\"F*$\"3%4BND4!\\!H$F^o7$$!3URL$eot(z'*F^ o$\"35'))pGW0&)z$F^o7$$!3u3](=<>n<)F^o$\"3NZLm2@e9WF^o7$$!3almmK!*4HnF ^o$\"3A\\&pi?&oF^o7$$!3)))****p#f[JCF^o$\"3/%)f3a)\\:%yF^o7$$!3!eI Le:#*H&))Fbq$\"3U'3.&Qsv_\"*F^o7$$\"3u%HLL6'Gw\\Fbq$\"31ljpG=-^5F*7$$ \"37,]7:\"=@(>F^o$\"3,= " 0 "" {MPLTEXT 1 0 967 "`R*` := z -> add(z^j/j!,j=0..7)+\n 981818561817 1843015125133517316585453517/39421059018953305890508281044629415350714 3680*z^8+\n 1519063132162087060889934788678388865254037/533130402172 324508863233992847568213203061112832*z^9+\n 139917654657895079263389 85345633883759002087/6397564826067894106358807914170818558436733353984 0*z^10+\n 16116291738976337320030290935/1494213808672330032653880589 1602317312*z^11:\npts := []: z0 := 0:\nfor ct from 0 to 320 do\n zz \+ := newton(`R*`(z)=exp(ct*Pi/20*I),z=z0):\n z0 := zz:\n pts := [op( pts),[Re(zz),Im(zz)]]:\nend do:\np_1 := plot(pts,color=COLOR(RGB,0,.3, .4)):\np_2 := plots[polygonplot]([seq([pts[i-1],pts[i],[-4.2,0]],i=2.. nops(pts))],\n style=patchnogrid,color=COLOR(RGB,0,.55,.75)): \np_3 := plot([[[-6.49,0],[1.19,0]],[[0,-6.39],[0,6.39]]],color=black, linestyle=3):\nplots[display]([p_||(1..3)],view=[-6.49..1.19,-6.39..6. 39],font=[HELVETICA,9],\n labels=[`Re(z)`,`Im(z)`],axes=box ed,scaling=constrained);" }}{PARA 13 "" 1 "" {GLPLOT2D 381 565 565 {PLOTDATA 2 "6+-%'CURVESG6$7]_l7$$\"\"!F)F(7$F($\"3++++Fjzq:!#=7$F($\" 3)******Rl#fTJF-7$$!3-+++!)4'p/$!#F$\"3D+++\")*)Q7ZF-7$$!39+++Q*R)4M!# E$\"3Q+++8`=$G'F-7$$!39+++'3k.H#!#D$\"3E+++*o\")R&yF-7$$!3(******fFca5 \"!#C$\"3X+++P$yZU*F-7$$!39+++mk)4@%FF$\"31+++Awb*4\"!#<7$$!3#******zB qrL\"!#B$\"3#******H'yjc7FN7$$!3)******f]2@m$FR$\"3/+++.&>PT\"FN7$$!3i +++Z9xP))FR$\"33+++J\\!3d\"FN7$$!3'******4vZH!>!#A$\"34+++`&**ys\"FN7$ $!3))*****pYJ6n$F\\o$\"3-+++SX,&)=FN7$$!3[*****\\RV[I'F\\o$\"3!******> Ar@/#FN7$$!3S*****4e^uQ*F\\o$\"33+++4wS*>#FN7$$!3++++#=^JFN7$$\" 3*******4([v75!#>$\"37+++]M@8LFN7$$\"3++++Uvx))>Far$\"3/+++k$\\kZ$FN7$ $\"33+++:q^qOFar$\"3&)*****>%HhROFN7$$\"3^+++MUnDjFar$\"3=+++:(z\"*z$F N7$$\"37+++^IQ')**Far$\"3!******\\T#3\\RFN7$$\"3()*****>[CIU\"F-$\"3A+ ++(4!o$3%FN7$$\"35+++*)3pS=F-$\"3e******Qof,UFN7$$\"3$******Ra[Y?#F-$ \"3G++++++.i:FYFN7$$\"37+++mmpRHF-$\"3)******>$)H@p%FN7$$\"31++ +:Lf,HF-$\"3!*******4hL_ZFN7$$\"3.+++Y!\\i\"GF-$\"3i*****\\ZC\"3[FN7$$ \"3!)*******>()*)o#F-$\"3m*****>S^(f[FN7$$\"3:+++S:bCDF-$\"30+++&Q9u! \\FN7$$\"3*******p%f9FBF-$\"35+++(Gs7&\\FN7$$\"3/+++;6[+@F-$\"3C+++(Gd 9*\\FN7$$\"35+++()>#y%=F-$\"3f*****>s\"3G]FN7$$\"3-+++9M.s:F-$\"3w**** *\\#=Ch]FN7$$\"3#******f'Giv7F-$\"3!)*****zh>54&FN7$$\"3E+++aUz2'*Far$ \"3\"******zr$[<^FN7$$\"3;+++'H&3%H'Far$\"3v*******[*oS^FN7$$\"3$***** *>`?;$GFar$\"3S+++!zy1;&FN7$$!3g*****zVlOl(Ffq$\"3'*******=%zu<&FN7$$! 3'******4]JZ[%Far$\"37+++6V5\">&FN7$$!36+++??G;$)Far$\"3m*****fX]:?&FN 7$$!3*******Hxq^A\"F-$\"31+++xuz3_FN7$$!3*)*****f))e%G;F-$\"3C+++$z0G@ &FN7$$!3/+++3.0T?F-$\"3Q+++TU^8_FN7$$!3-+++nhsiCF-$\"3#)*****>,P3@&FN7 $$!3%)*****4^KN*GF-$\"3l*****>tfY?&FN7$$!35+++Bz%QL$F-$\"3)******H^M[> &FN7$$!33++++0\\%y$F-$\"3#******R%Q<\"=&FN7$$!3')*****HA')oC%F-$\"3W++ +CMWj^FN7$$!37+++\"zNLs%F-$\"3#*******)oa89&FN7$$!3))*****\\LUu@&F-$\" 3W+++w&fX6&FN7$$!3D+++KL![t&F-$\"3!******\\yaE3&FN7$$!3[*****pkBTG'F-$ \"3y*****p&\\@X]FN7$$!3s*******\\=*yoF-$\"39+++5X!>+&FN7$$!3X*****pra' RvF-$\"3G+++B\\y_\\FN7$$!3K+++%)3Z%H)F-$\"3'*********\\-**[FN7$$!35+++ T(e<<*F-$\"3*******f$Q/W[FN7$$!3,+++g^g<5FN$\"3;+++pP!Qz%FN7$$!3)***** *zW1m7\"FN$\"3/+++O)RQv%FN7$$!3-+++33tP7FN$\"3i*****z:@as%FN7$$!3-+++w 2HY8FN$\"3p*****pnmhq%FN7$$!3)******p7w-X\"FN$\"3s*****>y\")Gp%FN7$$!3 5+++6lH\\:FN$\"39+++:*))Ho%FN7$$!3*)*****p()QOk\"FN$\"3k*****4i,Zn%FN7 $$!32+++S!*zL,'eY FN7$$!3)******f%Rf.>FN$\"3=+++A'y%\\YFN7$$!3'******467T)>FN$\"3f*****z E5\"RYFN7$$!3)******>^+A1#FN$\"3i*****RI^si%FN7$$!32+++n/;Q@FN$\"3++++ 89t8YFN7$$!3:+++(f]A@#FN$\"3E+++K7V)f%FN7$$!3#******p%[p%G#FN$\"3?+++3 qE\"e%FN7$$!3!******fX*obBFN$\"3Q+++80=iXFN7$$!3\"*******>)3aU#FN$\"3T +++VC8TXFN7$$!3!******\\N5S\\#FN$\"3-+++q\")4=XFN7$$!3%)*****>ZQ;c#FN$ \"3k*****4;lI\\%FN7$$!3(*******oxUGEFN$\"35+++7;.mWFN7$$!31+++,b]%p#FN $\"33+++4g+PWFN7$$!3=+++WL**fFFN$\"3V+++0u+1WFN7$$!3#)*****HV3]#GFN$\" 3C+++Vi1tVFN7$$!3%*******4Om*)GFN$\"3q******fbAQVFN7$$!35+++;o1aHFN$\" 3*)******zBa,VFN7$$!3$******Hc>$=IFN$\"33+++X!*3jUFN7$$!3))*****HY9D3$ FN$\"3y*****R=aHA%FN7$$!31+++.>tYJFN$\"3m******QHC\"=%FN7$$!3*)*****f5 O5@$FN$\"3N+++\"3w!QTFN7$$!3!)*****\\'4ZvKFN$\"31+++%o(e$4%FN7$$!33+++ di0SLFN$\"3G+++!>@z/%FN7$$!33+++@\\y/MFN$\"3C+++kRA,SFN7$$!37+++J?ipMF N$\"37+++Z2k`RFN7$$!30+++@f]MNFN$\"3.+++JnI0RFN7$$!3')*****4a^$*f$FN$ \"3%)*****>8Tj&QFN7$$!3A+++(pbSm$FN$\"35+++C@%o!QFN7$$!36+++2N]GPFN$\" 3)******>s$)ov$FN7$$!3%******pAvDz$FN$\"3*******4([^1PFN7$$!3)******Rh _h&QFN$\"3))*****zbgdl$FN7$$!3\")*****RCC\">RFN$\"33+++HXi/OFN7$$!3)** *****Q%*Q\")RFN$\"3y*****>+$4`NFN7$$!3p*****pvgG/%FN$\"35+++;)Q6]$FN7$ $!3J+++)>lM5%FN$\"3=+++Lbs[MFN7$$!3i*****4OWJ;%FN$\"33+++#=6eR$FN7$$!3 4+++nQ&=A%FN$\"3z*****pZ^BM$FN7$$!33+++(Gi&zUFN$\"3))*****fW-$)G$FN7$$ !3l*****f!)\\iL%FN$\"3********)\\AOB$FN7$$!3?+++pn!>R%FN$\"3%)******=S FyJFN7$$!3m*****f?KlW%FN$\"3$)*****z_CA7$FN7$$!3()*****zRK,]%FN$\"3;++ +dvWlIFN7$$!35+++4&>Fb%FN$\"3!)*****4DBz+$FN7$$!3()*****zE5Vg%FN$\"3/+ ++(yQ'\\HFN7$$!3y*****HrC\\l%FN$\"31+++V')e!*GFN7$$!3c*****\\'\\e/ZFN$ \"30+++@ZxIGFN7$$!3s*****H-9Lv%FN$\"3!)*****pM1-x#FN7$$!3i*****HeM6![F N$\"38+++;,!*3FFN7$$!3%)*****R(y1[[FN$\"3#******Hgzok#FN7$$!3U+++KD8%* [FN$\"3')******y[<%e#FN7$$!33+++[NMR\\FN$\"3%******4'=#3_#FN7$$!3G+++? 8r$)\\FN$\"3A+++a9'oX#FN7$$!39+++s3CF]FN$\"37+++K&QBR#FN7$$!3#)*****>F J*p]FN$\"3.+++]2IFBFN7$$!3-+++/`x6^FN$\"3%******pE(zhAFN7$$!3o******=% fF:&FN$\"35+++6u(e>#FN7$$!3++++xR'G>&FN$\"3,+++z$*eH@FN7$$!3E+++`P1K_F N$\"35+++S!zH1#FN7$$!3#)*****RpG.F&FN$\"3#******p*))3'*>FN7$$!3%)***** *\\[i2`FN$\"3-+++zs&*G>FN7$$!3&)*****zU:RM&FN$\"33+++txhh=FN7$$!3++++# )=;z`FN$\"33+++I*)4%z\"FN7$$!3P+++*)\\K8aFN$\"32+++;UUEG@'Q\"FN7$$!3/+++aT\\$f&FN$\"3%****** *o7&yJ\"FN7$$!37+++%4$4>cFN$\"35+++NrQ\\7FN7$$!3f*****z`gLk&FN$\"3#*** ***\\)*=3=\"FN7$$!39+++4!zim&FN$\"30+++N[976FN7$$!3e*****p5Myo&FN$\"3 \"******>hiL/\"FN7$$!3.+++=V,3dFN$\"3l******\\aqW(*F-7$$!3++++o2\"os&F N$\"3m*****\\-vY0*F-7$$!3K+++ip@WdFN$\"3o*****4hKNO)F-7$$!3?++++&G-w&F N$\"3++++:sGrwF-7$$!3M+++#yU[x&FN$\"3S+++z1'z(pF-7$$!3U+++4(e!)y&FN$\" 3K+++c()e$G'F-7$$!3u*****4Rw)*z&FN$\"3i*****zq@#)e&F-7$$!3t*****R&oH5e FN$\"3/+++SW#>*[F-7$$!3o*****Rw@$>eFN$\"39+++*=wZ>%F-7$$!3%******\\;`p #eFN$\"3u*****f!)po\\$F-7$$!3h*****fB$>LeFN$\"3-+++X1J)z#F-7$$!3\"**** **R2W!QeFN$\"3))*****z3:#*4#F-7$$!3)******\\^2:%eFN$\"3'******pw3(*R\" F-7$$!3:+++!)\\eVeFN$\"3s******[\\C**pFar7$$!3.+++mtFWeFNF(7$Fb]n$!3s* *****[\\C**pFar7$F]]n$!3'******pw3(*R\"F-7$Fh\\n$!3))*****z3:#*4#F-7$F c\\n$!3-+++X1J)z#F-7$F^\\n$!3u*****f!)po\\$F-7$Fi[n$!39+++*=wZ>%F-7$Fd [n$!3/+++SW#>*[F-7$F_[n$!3i*****zq@#)e&F-7$Fjjm$!3K+++c()e$G'F-7$Fejm$ !3S+++z1'z(pF-7$F`jm$!3++++:sGrwF-7$F[jm$!3o*****4hKNO)F-7$Ffim$!3m*** **\\-vY0*F-7$Faim$!3l******\\aqW(*F-7$F\\im$!3\"******>hiL/\"FN7$Fghm$ !30+++N[976FN7$Fbhm$!3#******\\)*=3=\"FN7$F]hm$!35+++NrQ\\7FN7$Fhgm$!3 %*******o7&yJ\"FN7$Fcgm$!3!*******>G@'Q\"FN7$F^gm$!3'*******\\@Za9FN7$ Fifm$!3'******\\)ziA:FN7$Fdfm$!3))*****fzw1f\"FN7$F_fm$!35+++w@he;FN7$ Fjem$!32+++;UUEFN7$Ffdm$!3#******p*))3'*>FN7$Fadm$!35+++S!zH1#FN7$F\\dm$!3, +++z$*eH@FN7$Fgcm$!35+++6u(e>#FN7$Fbcm$!3%******pE(zhAFN7$F]cm$!3.+++] 2IFBFN7$Fhbm$!37+++K&QBR#FN7$Fcbm$!3A+++a9'oX#FN7$F^bm$!3%******4'=#3_ #FN7$Fiam$!3')******y[<%e#FN7$Fdam$!3#******Hgzok#FN7$F_am$!38+++;,!*3 FFN7$Fj`m$!3!)*****pM1-x#FN7$Fe`m$!30+++@ZxIGFN7$F``m$!31+++V')e!*GFN7 $F[`m$!3/+++(yQ'\\HFN7$Ff_m$!3!)*****4DBz+$FN7$Fa_m$!3;+++dvWlIFN7$F\\ _m$!3$)*****z_CA7$FN7$Fg^m$!3%)******=SFyJFN7$Fb^m$!3********)\\AOB$FN 7$F]^m$!3))*****fW-$)G$FN7$Fh]m$!3z*****pZ^BM$FN7$Fc]m$!33+++#=6eR$FN7 $F^]m$!3=+++Lbs[MFN7$Fi\\m$!35+++;)Q6]$FN7$Fd\\m$!3y*****>+$4`NFN7$F_ \\m$!33+++HXi/OFN7$Fj[m$!3))*****zbgdl$FN7$Fe[m$!3*******4([^1PFN7$F`[ m$!3)******>s$)ov$FN7$F[[m$!35+++C@%o!QFN7$Ffjl$!3%)*****>8Tj&QFN7$Faj l$!3.+++JnI0RFN7$F\\jl$!37+++Z2k`RFN7$Fgil$!3C+++kRA,SFN7$Fbil$!3G+++! >@z/%FN7$F]il$!31+++%o(e$4%FN7$Fhhl$!3N+++\"3w!QTFN7$Fchl$!3m******QHC \"=%FN7$F^hl$!3y*****R=aHA%FN7$Figl$!33+++X!*3jUFN7$Fdgl$!3*)******zBa ,VFN7$F_gl$!3q******fbAQVFN7$Fjfl$!3C+++Vi1tVFN7$Fefl$!3V+++0u+1WFN7$F `fl$!33+++4g+PWFN7$F[fl$!35+++7;.mWFN7$Ffel$!3k*****4;lI\\%FN7$Fael$!3 -+++q\")4=XFN7$F\\el$!3T+++VC8TXFN7$Fgdl$!3Q+++80=iXFN7$Fbdl$!3?+++3qE \"e%FN7$F]dl$!3E+++K7V)f%FN7$Fhcl$!3++++89t8YFN7$Fccl$!3i*****RI^si%FN 7$F^cl$!3f*****zE5\"RYFN7$Fibl$!3=+++A'y%\\YFN7$Fdbl$!39+++&>,'eYFN7$F _bl$!3%)*****z%R$om%FN7$Fjal$!3k*****4i,Zn%FN7$Feal$!39+++:*))Ho%FN7$F `al$!3s*****>y\")Gp%FN7$F[al$!3p*****pnmhq%FN7$Ff`l$!3i*****z:@as%FN7$ Fa`l$!3/+++O)RQv%FN7$F\\`l$!3;+++pP!Qz%FN7$Fg_l$!3*******f$Q/W[FN7$Fb_ l$!3'*********\\-**[FN7$F]_l$!3G+++B\\y_\\FN7$Fh^l$!39+++5X!>+&FN7$Fc^ l$!3y*****p&\\@X]FN7$F^^l$!3!******\\yaE3&FN7$Fi]l$!3W+++w&fX6&FN7$Fd] l$!3#*******)oa89&FN7$F_]l$!3W+++CMWj^FN7$Fj\\l$!3#******R%Q<\"=&FN7$F e\\l$!3)******H^M[>&FN7$F`\\l$!3l*****>tfY?&FN7$F[\\l$!3#)*****>,P3@&F N7$Ff[l$!3Q+++TU^8_FN7$Fa[l$!3C+++$z0G@&FN7$F\\[l$!31+++xuz3_FN7$Fgz$! 3m*****fX]:?&FN7$Fbz$!37+++6V5\">&FN7$$!3/+++Pam`wFfq$!3'*******=%zu<& FN7$Fhy$!3S+++!zy1;&FN7$Fcy$!3v*******[*oS^FN7$F^y$!3\"******zr$[<^FN7 $Fix$!3!)*****zh>54&FN7$Fdx$!3w*****\\#=Ch]FN7$F_x$!3f*****>s\"3G]FN7$ Fjw$!3C+++(Gd9*\\FN7$Few$!35+++(Gs7&\\FN7$F`w$!30+++&Q9u!\\FN7$F[w$!3m *****>S^(f[FN7$Ffv$!3i*****\\ZC\"3[FN7$Fav$!3!*******4hL_ZFN7$F\\v$!3) ******>$)H@p%FN7$Fgu$!3>+++.i:FYFN7$Fbu$!37+++:o\"pb%FN7$F]u$!3k****** 4Mn![%FN7$Fht$!3m*****zN0tR%FN7$Fct$!3G+++%HhROFN7$Fer$!3/+++k$\\kZ$FN7$F_r$!37+++]M@8LFN7$Fjq$!3 '*******)>#=^JFN7$Fdq$!3%)*****\\qI0*HFN7$F_q$!30+++$QW5$GFN7$Fjp$!3'* *****f&oUsEFN7$Fep$!3%)*****Hi5W^#FN7$F_p$!3y*****4I(ycBFN7$Fjo$!33+++ 4wS*>#FN7$Feo$!3!******>Ar@/#FN7$F`o$!3-+++SX,&)=FN7$$!3\"******RvZH!> F\\o$!34+++`&**ys\"FN7$$!3[*****RXrx$))FR$!33+++J\\!3d\"FN7$FV$!3/+++. &>PT\"FN7$FP$!3#******H'yjc7FN7$FJ$!31+++Awb*4\"FN7$FD$!3X+++P$yZU*F-7 $$!3#*******)3k.H#F@$!3E+++*o\")R&yF-7$$!32+++K*R)4MF:$!3Q+++8`=$G'F-7 $$!3y*****>N!3ZIF4$!3D+++\")*)Q7ZF-7$F($!3)******Rl#fTJF-7$F($!3++++Fj zq:F-F'-%&COLORG6&%$RGBGF($\"\"$!\"\"$\"\"%Fh\\p-%)POLYGONSG6^_l7%F'7$ F($\"+Fjzq:!#57$$!#UFh\\pF(7%F_]p7$F($\"+aEfTJFb]pFc]p7%Fg]p7$$!+!)4'p /$Far$\"+\")*)Q7ZFb]pFc]p7%F[^p7$$!+Q*R)4MF-$\"+8`=$G'Fb]pFc]p7%Fa^p7$ $!+'3k.H#FN$\"+*o\")R&yFb]pFc]p7%Fg^p7$$!+wiX06!#;$\"+P$yZU*Fb]pFc]p7% F]_p7$$!+mk)4@%F`_p$\"+Awb*4\"!\"*Fc]p7%Fd_p7$$!+Q-PT\"Fi_pFc]p7%Fb`p7$$!+Z9xP))F^`p$\"+J \\!3d\"Fi_pFc]p7%Fh`p7$$!+^x%H!>!#9$\"+`&**ys\"Fi_pFc]p7%F^ap7$$!+n98r OFaap$\"+SX,&)=Fi_pFc]p7%Feap7$$!+&RV[I'Faap$\"+A7#Fi_pFc]p7%Fabp7$$!+#=^JFi_pFc]p7%Fgdp7$$\"+r[v 75!#6$\"+]M@8LFi_pFc]p7%F]ep7$$\"+Uvx))>F`ep$\"+k$\\kZ$Fi_pFc]p7%Fdep7 $$\"+:q^qOF`ep$\"+UHhROFi_pFc]p7%Fjep7$$\"+MUnDjF`ep$\"+:(z\"*z$Fi_pFc ]p7%F`fp7$$\"+^IQ')**F`ep$\"+:C3\\RFi_pFc]p7%Fffp7$$\"+#[CIU\"Fb]p$\"+ (4!o$3%Fi_pFc]p7%F\\gp7$$\"+*)3pS=Fb]p$\"+Rof,UFi_pFc]p7%Fbgp7$$\"+W&[ Y?#Fb]p$\"+#y%=Fb]p$ \"+A<3G]Fi_pFc]p7%F`\\q7$$\"+9M.s:Fb]p$\"+D=Ch]Fi_pFc]p7%Ff\\q7$$\"+mG iv7Fb]p$\"+='>54&Fi_pFc]p7%F\\]q7$$\"+aUz2'*F`ep$\"+=P[<^Fi_pFc]p7%Fb] q7$$\"+'H&3%H'F`ep$\"+!\\*oS^Fi_pFc]p7%Fh]q7$$\"+K0iJGF`ep$\"+!zy1;&Fi _pFc]p7%F^^q7$$!+Qam`wFcdp$\"+>%zu<&Fi_pFc]p7%Fd^q7$$!+,:t%[%F`ep$\"+6 V5\">&Fi_pFc]p7%Fj^q7$$!+??G;$)F`ep$\"+c/b,_Fi_pFc]p7%F`_q7$$!+t2&Fi_pF c]p7%Fdaq7$$!++0\\%y$Fb]p$\"+WQ<\"=&Fi_pFc]p7%Fjaq7$$!+Bi)oC%Fb]p$\"+C MWj^Fi_pFc]p7%F`bq7$$!+\"zNLs%Fb]p$\"+*oa89&Fi_pFc]p7%Ffbq7$$!+NBW<_Fb ]p$\"+w&fX6&Fi_pFc]p7%F\\cq7$$!+KL![t&Fb]p$\"+&yaE3&Fi_pFc]p7%Fbcq7$$! +ZO7%G'Fb]p$\"+d\\@X]Fi_pFc]p7%Fhcq7$$!++&=*yoFb]p$\"+5X!>+&Fi_pFc]p7% F^dq7$$!+,'eYFi_pFc]p7%Ffhq7$$!+YRf.>Fi_p$\"+A'y %\\YFi_pFc]p7%F\\iq7$$!+6@6%)>Fi_p$\"+o-6RYFi_pFc]p7%Fbiq7$$!+70?i?Fi_ p$\"+/8DFYFi_pFc]p7%Fhiq7$$!+n/;Q@Fi_p$\"+89t8YFi_pFc]p7%F^jq7$$!+(f]A @#Fi_p$\"+K7V)f%Fi_pFc]p7%Fdjq7$$!+Z[p%G#Fi_p$\"+3qE\"e%Fi_pFc]p7%Fjjq 7$$!+c%*obBFi_p$\"+80=iXFi_pFc]p7%F`[r7$$!+?)3aU#Fi_p$\"+VC8TXFi_pFc]p 7%Ff[r7$$!+b.,%\\#Fi_p$\"+q\")4=XFi_pFc]p7%F\\\\r7$$!+s%Q;c#Fi_p$\"+h^ 1$\\%Fi_pFc]p7%Fb\\r7$$!+pxUGEFi_p$\"+7;.mWFi_pFc]p7%Fh\\r7$$!+,b]%p#F i_p$\"+4g+PWFi_pFc]p7%F^]r7$$!+WL**fFFi_p$\"+0u+1WFi_pFc]p7%Fd]r7$$!+L %3]#GFi_p$\"+Vi1tVFi_pFc]p7%Fj]r7$$!+5Om*)GFi_p$\"+gbAQVFi_pFc]p7%F`^r 7$$!+;o1aHFi_p$\"+!QU:I%Fi_pFc]p7%Ff^r7$$!+j&>$=IFi_p$\"+X!*3jUFi_pFc] p7%F\\_r7$$!+jW^#3$Fi_p$\"+%=aHA%Fi_pFc]p7%Fb_r7$$!+.>tYJFi_p$\"+RHC\" =%Fi_pFc]p7%Fh_r7$$!+1h.6KFi_p$\"+\"3w!QTFi_pFc]p7%F^`r7$$!+l4ZvKFi_p$ \"+%o(e$4%Fi_pFc]p7%Fd`r7$$!+di0SLFi_p$\"+!>@z/%Fi_pFc]p7%Fj`r7$$!+@\\ y/MFi_p$\"+kRA,SFi_pFc]p7%F`ar7$$!+J?ipMFi_p$\"+Z2k`RFi_pFc]p7%Ffar7$$ !+@f]MNFi_p$\"+JnI0RFi_pFc]p7%F\\br7$$!+T:N*f$Fi_p$\"+K6McQFi_pFc]p7%F bbr7$$!+(pbSm$Fi_p$\"+C@%o!QFi_pFc]p7%Fhbr7$$!+2N]GPFi_p$\"+AP)ov$Fi_p Fc]p7%F^cr7$$!+F_d#z$Fi_p$\"+r[^1PFi_pFc]p7%Fdcr7$$!+9E:cQFi_p$\"+e0wb OFi_pFc]p7%Fjcr7$$!+WU7>RFi_p$\"+HXi/OFi_pFc]p7%F`dr7$$!+R%*Q\")RFi_p$ \"+-I4`NFi_pFc]p7%Ffdr7$$!+d2'G/%Fi_p$\"+;)Q6]$Fi_pFc]p7%F\\er7$$!+)>l M5%Fi_p$\"+Lbs[MFi_pFc]p7%Fber7$$!+hV9jTFi_p$\"+#=6eR$Fi_pFc]p7%Fher7$ $!+nQ&=A%Fi_p$\"+x9NULFi_pFc]p7%F^fr7$$!+(Gi&zUFi_p$\"+YCI)G$Fi_pFc]p7 %Fdfr7$$!+1)\\iL%Fi_p$\"+*\\AOB$Fi_pFc]p7%Fjfr7$$!+pn!>R%Fi_p$\"+>SFyJ Fi_pFc]p7%F`gr7$$!+1A`YWFi_p$\"+GXAAJFi_pFc]p7%Ffgr7$$!+)RK,]%Fi_p$\"+ dvWlIFi_pFc]p7%F\\hr7$$!+4&>Fb%Fi_p$\"+^K#z+$Fi_pFc]p7%Fbhr7$$!+o-J/YF i_p$\"+(yQ'\\HFi_pFc]p7%Fhhr7$$!+8Z#\\l%Fi_p$\"+V')e!*GFi_pFc]p7%F^ir7 $$!+l\\e/ZFi_p$\"+@ZxIGFi_pFc]p7%Fdir7$$!+BSJ`ZFi_p$\"+Zj?qFFi_pFc]p7% Fjir7$$!+$eM6![Fi_p$\"+;,!*3FFi_pFc]p7%F`jr7$$!+uy1[[Fi_p$\"+.'zok#Fi_ pFc]p7%Ffjr7$$!+KD8%*[Fi_p$\"+z[<%e#Fi_pFc]p7%F\\[s7$$!+[NMR\\Fi_p$\"+ h=#3_#Fi_pFc]p7%Fb[s7$$!+?8r$)\\Fi_p$\"+a9'oX#Fi_pFc]p7%Fh[s7$$!+s3CF] Fi_p$\"+K&QBR#Fi_pFc]p7%F^\\s7$$!+s7$*p]Fi_p$\"+]2IFBFi_pFc]p7%Fd\\s7$ $!+/`x6^Fi_p$\"+nszhAFi_pFc]p7%Fj\\s7$$!+>%fF:&Fi_p$\"+6u(e>#Fi_pFc]p7 %F`]s7$$!+xR'G>&Fi_p$\"+z$*eH@Fi_pFc]p7%Ff]s7$$!+`P1K_Fi_p$\"+S!zH1#Fi _pFc]p7%F\\^s7$$!+%pG.F&Fi_p$\"+(*))3'*>Fi_pFc]p7%Fb^s7$$!+][i2`Fi_p$ \"+zs&*G>Fi_pFc]p7%Fh^s7$$!+Ga\"RM&Fi_p$\"+txhh=Fi_pFc]p7%F^_s7$$!+#)= ;z`Fi_p$\"+I*)4%z\"Fi_pFc]p7%Fd_s7$$!+*)\\K8aFi_p$\"+;UUEcFi_p$\"+NrQ\\7Fi_pFc]p7%Fdbs7$$!+ Q0OVcFi_p$\"+&)*=3=\"Fi_pFc]p7%Fjbs7$$!+4!zim&Fi_p$\"+N[976Fi_pFc]p7%F `cs7$$!+2T$yo&Fi_p$\"+7EOV5Fi_pFc]p7%Ffcs7$$!+=V,3dFi_p$\"+]aqW(*Fb]pF c]p7%F\\ds7$$!+o2\"os&Fi_p$\"+D]na!*Fb]pFc]p7%Fbds7$$!+ip@WdFi_p$\"+6E `j$)Fb]pFc]p7%Fhds7$$!++&G-w&Fi_p$\"+:sGrwFb]pFc]p7%F^es7$$!+#yU[x&Fi_ p$\"+z1'z(pFb]pFc]p7%Fdes7$$!+4(e!)y&Fi_p$\"+c()e$G'Fb]pFc]p7%Fjes7$$! +\"Rw)*z&Fi_p$\"+3*[Fb]pFc]p7 %Fffs7$$!+keFi_p$\"+*=wZ>%Fb]pFc]p7%F\\gs7$$!+lJ&p#eFi_p$\"+1)po\\$ Fb]pFc]p7%Fbgs7$$!+OK>LeFi_p$\"+X1J)z#Fb]pFc]p7%Fhgs7$$!+uS/QeFi_p$\"+ )3:#*4#Fb]pFc]p7%F^hs7$$!+:v]TeFi_p$\"+n(3(*R\"Fb]pFc]p7%Fdhs7$$!+!)\\ eVeFi_p$\"+\\\\C**pF`epFc]p7%Fjhs7$$!+mtFWeFi_pF(Fc]p7%F`is7$F[is$!+\\ \\C**pF`epFc]p7%Fdis7$Fehs$!+n(3(*R\"Fb]pFc]p7%Fhis7$F_hs$!+)3:#*4#Fb] pFc]p7%F\\js7$Figs$!+X1J)z#Fb]pFc]p7%F`js7$Fcgs$!+1)po\\$Fb]pFc]p7%Fdj s7$F]gs$!+*=wZ>%Fb]pFc]p7%Fhjs7$Fgfs$!+SW#>*[Fb]pFc]p7%F\\[t7$Fafs$!+3 Fi_pFc]p7%F``t7$Fc^s$!+(*))3'*>Fi_p Fc]p7%Fd`t7$F]^s$!+S!zH1#Fi_pFc]p7%Fh`t7$Fg]s$!+z$*eH@Fi_pFc]p7%F\\at7 $Fa]s$!+6u(e>#Fi_pFc]p7%F`at7$F[]s$!+nszhAFi_pFc]p7%Fdat7$Fe\\s$!+]2IF BFi_pFc]p7%Fhat7$F_\\s$!+K&QBR#Fi_pFc]p7%F\\bt7$Fi[s$!+a9'oX#Fi_pFc]p7 %F`bt7$Fc[s$!+h=#3_#Fi_pFc]p7%Fdbt7$F][s$!+z[<%e#Fi_pFc]p7%Fhbt7$Fgjr$ !+.'zok#Fi_pFc]p7%F\\ct7$Fajr$!+;,!*3FFi_pFc]p7%F`ct7$F[jr$!+Zj?qFFi_p Fc]p7%Fdct7$Feir$!+@ZxIGFi_pFc]p7%Fhct7$F_ir$!+V')e!*GFi_pFc]p7%F\\dt7 $Fihr$!+(yQ'\\HFi_pFc]p7%F`dt7$Fchr$!+^K#z+$Fi_pFc]p7%Fddt7$F]hr$!+dvW lIFi_pFc]p7%Fhdt7$Fggr$!+GXAAJFi_pFc]p7%F\\et7$Fagr$!+>SFyJFi_pFc]p7%F `et7$F[gr$!+*\\AOB$Fi_pFc]p7%Fdet7$Fefr$!+YCI)G$Fi_pFc]p7%Fhet7$F_fr$! +x9NULFi_pFc]p7%F\\ft7$Fier$!+#=6eR$Fi_pFc]p7%F`ft7$Fcer$!+Lbs[MFi_pFc ]p7%Fdft7$F]er$!+;)Q6]$Fi_pFc]p7%Fhft7$Fgdr$!+-I4`NFi_pFc]p7%F\\gt7$Fa dr$!+HXi/OFi_pFc]p7%F`gt7$F[dr$!+e0wbOFi_pFc]p7%Fdgt7$Fecr$!+r[^1PFi_p Fc]p7%Fhgt7$F_cr$!+AP)ov$Fi_pFc]p7%F\\ht7$Fibr$!+C@%o!QFi_pFc]p7%F`ht7 $Fcbr$!+K6McQFi_pFc]p7%Fdht7$F]br$!+JnI0RFi_pFc]p7%Fhht7$Fgar$!+Z2k`RF i_pFc]p7%F\\it7$Faar$!+kRA,SFi_pFc]p7%F`it7$F[ar$!+!>@z/%Fi_pFc]p7%Fdi t7$Fe`r$!+%o(e$4%Fi_pFc]p7%Fhit7$F_`r$!+\"3w!QTFi_pFc]p7%F\\jt7$Fi_r$! +RHC\"=%Fi_pFc]p7%F`jt7$Fc_r$!+%=aHA%Fi_pFc]p7%Fdjt7$F]_r$!+X!*3jUFi_p Fc]p7%Fhjt7$Fg^r$!+!QU:I%Fi_pFc]p7%F\\[u7$Fa^r$!+gbAQVFi_pFc]p7%F`[u7$ F[^r$!+Vi1tVFi_pFc]p7%Fd[u7$Fe]r$!+0u+1WFi_pFc]p7%Fh[u7$F_]r$!+4g+PWFi _pFc]p7%F\\\\u7$Fi\\r$!+7;.mWFi_pFc]p7%F`\\u7$Fc\\r$!+h^1$\\%Fi_pFc]p7 %Fd\\u7$F]\\r$!+q\")4=XFi_pFc]p7%Fh\\u7$Fg[r$!+VC8TXFi_pFc]p7%F\\]u7$F a[r$!+80=iXFi_pFc]p7%F`]u7$F[[r$!+3qE\"e%Fi_pFc]p7%Fd]u7$Fejq$!+K7V)f% Fi_pFc]p7%Fh]u7$F_jq$!+89t8YFi_pFc]p7%F\\^u7$Fiiq$!+/8DFYFi_pFc]p7%F`^ u7$Fciq$!+o-6RYFi_pFc]p7%Fd^u7$F]iq$!+A'y%\\YFi_pFc]p7%Fh^u7$Fghq$!+&> ,'eYFi_pFc]p7%F\\_u7$Fahq$!+[R$om%Fi_pFc]p7%F`_u7$F[hq$!+@;quYFi_pFc]p 7%Fd_u7$Fegq$!+:*))Ho%Fi_pFc]p7%Fh_u7$F_gq$!+#y\")Gp%Fi_pFc]p7%F\\`u7$ Fifq$!+xm;1ZFi_pFc]p7%F``u7$Fcfq$!+e6UDZFi_pFc]p7%Fd`u7$F]fq$!+O)RQv%F i_pFc]p7%Fh`u7$Fgeq$!+pP!Qz%Fi_pFc]p7%F\\au7$Faeq$!+OQ/W[Fi_pFc]p7%F`a u7$F[eq$!++]-**[Fi_pFc]p7%Fdau7$Fedq$!+B\\y_\\Fi_pFc]p7%Fhau7$F_dq$!+5 X!>+&Fi_pFc]p7%F\\bu7$Ficq$!+d\\@X]Fi_pFc]p7%F`bu7$Fccq$!+&yaE3&Fi_pFc ]p7%Fdbu7$F]cq$!+w&fX6&Fi_pFc]p7%Fhbu7$Fgbq$!+*oa89&Fi_pFc]p7%F\\cu7$F abq$!+CMWj^Fi_pFc]p7%F`cu7$F[bq$!+WQ<\"=&Fi_pFc]p7%Fdcu7$Feaq$!+8X$[>& Fi_pFc]p7%Fhcu7$F_aq$!+K(fY?&Fi_pFc]p7%F\\du7$Fi`q$!+7q$3@&Fi_pFc]p7%F `du7$Fc`q$!+TU^8_Fi_pFc]p7%Fddu7$F]`q$!+$z0G@&Fi_pFc]p7%Fhdu7$Fg_q$!+x uz3_Fi_pFc]p7%F\\eu7$Fa_q$!+c/b,_Fi_pFc]p7%F`eu7$F[_q$!+6V5\">&Fi_pFc] p7%Fdeu7$$!+Pam`wFcdp$!+>%zu<&Fi_pFc]p7%Fheu7$F_^q$!+!zy1;&Fi_pFc]p7%F ^fu7$Fi]q$!+!\\*oS^Fi_pFc]p7%Fbfu7$Fc]q$!+=P[<^Fi_pFc]p7%Fffu7$F]]q$!+ ='>54&Fi_pFc]p7%Fjfu7$Fg\\q$!+D=Ch]Fi_pFc]p7%F^gu7$Fa\\q$!+A<3G]Fi_pFc ]p7%Fbgu7$F[\\q$!+(Gd9*\\Fi_pFc]p7%Ffgu7$Fe[q$!+(Gs7&\\Fi_pFc]p7%Fjgu7 $F_[q$!+&Q9u!\\Fi_pFc]p7%F^hu7$Fijp$!+-9vf[Fi_pFc]p7%Fbhu7$Fcjp$!+vW73 [Fi_pFc]p7%Ffhu7$F]jp$!+5hL_ZFi_pFc]p7%Fjhu7$Fgip$!+K)H@p%Fi_pFc]p7%F^ iu7$Faip$!+.i:FYFi_pFc]p7%Fbiu7$F[ip$!+:o\"pb%Fi_pFc]p7%Ffiu7$Fehp$!+5 Mn![%Fi_pFc]p7%Fjiu7$F_hp$!+e`I(R%Fi_pFc]p7%F^ju7$Figp$!+#=^JFi_pFc]p7%Fb\\v7$Fadp$!+02`!*HFi_pFc]p7%Ff\\v7$F[dp$!+ $QW5$GFi_pFc]p7%Fj\\v7$Fecp$!+coUsEFi_pFc]p7%F^]v7$F_cp$!+B1T9DFi_pFc] p7%Fb]v7$Fhbp$!+,tycBFi_pFc]p7%Ff]v7$Fbbp$!+4wS*>#Fi_pFc]p7%Fj]v7$F\\b p$!+A7Faap$!+ `&**ys\"Fi_pFc]p7%Ff^v7$$!+a9xP))F^`p$!+J\\!3d\"Fi_pFc]p7%F\\_v7$Fc`p$ !+.&>PT\"Fi_pFc]p7%Fb_v7$F\\`p$!+jyjc7Fi_pFc]p7%Ff_v7$Fe_p$!+Awb*4\"Fi _pFc]p7%Fj_v7$F^_p$!+P$yZU*Fb]pFc]p7%F^`v7$$!+*3k.H#FN$!+*o\")R&yFb]pF c]p7%Fb`v7$$!+K*R)4MF-$!+8`=$G'Fb]pFc]p7%Fh`v7$$!+_.3ZIFar$!+\")*)Q7ZF b]pFc]p7%F^av7$F($!+aEfTJFb]pFc]p7%Fdav7$F($!+Fjzq:Fb]pFc]p7%FhavF'Fc] p-Fc\\p6&Fe\\pF($\"#b!\"#$\"#vF`bv-%&STYLEG6#%,PATCHNOGRIDG-F$6%7$7$$! 3A++++++!\\'FNF(7$$\"3%**************=\"FNF(-%'COLOURG6&Fe\\pF)F)F)-%* LINESTYLEG6#Fg\\p-F$6%7$7$F($!3o*************Q'FN7$F($\"3o************ *Q'FNF`cvFccv-%+AXESLABELSG6%%&Re(z)G%&Im(z)G-%%FONTG6#%(DEFAULTG-Fedv 6$%*HELVETICAG\"\"*-%*AXESSTYLEG6#%$BOXG-%(SCALINGG6#%,CONSTRAINEDG-%% VIEWG6$;$!$\\'F`bv$\"$>\"F`bv;$!$R'F`bv$\"$R'F`bv" 1 2 0 1 10 0 2 9 1 2 1 1.000000 45.000000 45.000000 0 0 "Curve 1" "Curve 2" "Curve 3" "Cu rve 4" }}}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 75 "The stability interval for the embedded order 7 scheme is (approxi mately) " }{XPPEDIT 18 0 "[-5.8443, 0];" "6#7$,$-%&FloatG6$\"&V%e!\"% !\"\"\"\"!" }{TEXT -1 2 ". " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 79 "The stability regions for the two schemes can b e displayed in the same picture." }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 154 "plots[display]([p_1,p_2,p||(1..3)],view=[-6.89..1.19,-6.39..6 .39],font=[HELVETICA,9],\n labels=[`Re(z)`,`Im(z)`],axes=box ed,scaling=constrained);" }}{PARA 13 "" 1 "" {GLPLOT2D 362 580 580 {PLOTDATA 2 "6--%'CURVESG6$7]_l7$$\"\"!F)F(7$F($\"3++++Fjzq:!#=7$F($\" 3)******Rl#fTJF-7$$!3-+++!)4'p/$!#F$\"3D+++\")*)Q7ZF-7$$!39+++Q*R)4M!# E$\"3Q+++8`=$G'F-7$$!39+++'3k.H#!#D$\"3E+++*o\")R&yF-7$$!3(******fFca5 \"!#C$\"3X+++P$yZU*F-7$$!39+++mk)4@%FF$\"31+++Awb*4\"!#<7$$!3#******zB qrL\"!#B$\"3#******H'yjc7FN7$$!3)******f]2@m$FR$\"3/+++.&>PT\"FN7$$!3i +++Z9xP))FR$\"33+++J\\!3d\"FN7$$!3'******4vZH!>!#A$\"34+++`&**ys\"FN7$ $!3))*****pYJ6n$F\\o$\"3-+++SX,&)=FN7$$!3[*****\\RV[I'F\\o$\"3!******> Ar@/#FN7$$!3S*****4e^uQ*F\\o$\"33+++4wS*>#FN7$$!3++++#=^JFN7$$\" 3*******4([v75!#>$\"37+++]M@8LFN7$$\"3++++Uvx))>Far$\"3/+++k$\\kZ$FN7$ $\"33+++:q^qOFar$\"3&)*****>%HhROFN7$$\"3^+++MUnDjFar$\"3=+++:(z\"*z$F N7$$\"37+++^IQ')**Far$\"3!******\\T#3\\RFN7$$\"3()*****>[CIU\"F-$\"3A+ ++(4!o$3%FN7$$\"35+++*)3pS=F-$\"3e******Qof,UFN7$$\"3$******Ra[Y?#F-$ \"3G++++++.i:FYFN7$$\"37+++mmpRHF-$\"3)******>$)H@p%FN7$$\"31++ +:Lf,HF-$\"3!*******4hL_ZFN7$$\"3.+++Y!\\i\"GF-$\"3i*****\\ZC\"3[FN7$$ \"3!)*******>()*)o#F-$\"3m*****>S^(f[FN7$$\"3:+++S:bCDF-$\"30+++&Q9u! \\FN7$$\"3*******p%f9FBF-$\"35+++(Gs7&\\FN7$$\"3/+++;6[+@F-$\"3C+++(Gd 9*\\FN7$$\"35+++()>#y%=F-$\"3f*****>s\"3G]FN7$$\"3-+++9M.s:F-$\"3w**** *\\#=Ch]FN7$$\"3#******f'Giv7F-$\"3!)*****zh>54&FN7$$\"3E+++aUz2'*Far$ \"3\"******zr$[<^FN7$$\"3;+++'H&3%H'Far$\"3v*******[*oS^FN7$$\"3$***** *>`?;$GFar$\"3S+++!zy1;&FN7$$!3g*****zVlOl(Ffq$\"3'*******=%zu<&FN7$$! 3'******4]JZ[%Far$\"37+++6V5\">&FN7$$!36+++??G;$)Far$\"3m*****fX]:?&FN 7$$!3*******Hxq^A\"F-$\"31+++xuz3_FN7$$!3*)*****f))e%G;F-$\"3C+++$z0G@ &FN7$$!3/+++3.0T?F-$\"3Q+++TU^8_FN7$$!3-+++nhsiCF-$\"3#)*****>,P3@&FN7 $$!3%)*****4^KN*GF-$\"3l*****>tfY?&FN7$$!35+++Bz%QL$F-$\"3)******H^M[> &FN7$$!33++++0\\%y$F-$\"3#******R%Q<\"=&FN7$$!3')*****HA')oC%F-$\"3W++ +CMWj^FN7$$!37+++\"zNLs%F-$\"3#*******)oa89&FN7$$!3))*****\\LUu@&F-$\" 3W+++w&fX6&FN7$$!3D+++KL![t&F-$\"3!******\\yaE3&FN7$$!3[*****pkBTG'F-$ \"3y*****p&\\@X]FN7$$!3s*******\\=*yoF-$\"39+++5X!>+&FN7$$!3X*****pra' RvF-$\"3G+++B\\y_\\FN7$$!3K+++%)3Z%H)F-$\"3'*********\\-**[FN7$$!35+++ T(e<<*F-$\"3*******f$Q/W[FN7$$!3,+++g^g<5FN$\"3;+++pP!Qz%FN7$$!3)***** *zW1m7\"FN$\"3/+++O)RQv%FN7$$!3-+++33tP7FN$\"3i*****z:@as%FN7$$!3-+++w 2HY8FN$\"3p*****pnmhq%FN7$$!3)******p7w-X\"FN$\"3s*****>y\")Gp%FN7$$!3 5+++6lH\\:FN$\"39+++:*))Ho%FN7$$!3*)*****p()QOk\"FN$\"3k*****4i,Zn%FN7 $$!32+++S!*zL,'eY FN7$$!3)******f%Rf.>FN$\"3=+++A'y%\\YFN7$$!3'******467T)>FN$\"3f*****z E5\"RYFN7$$!3)******>^+A1#FN$\"3i*****RI^si%FN7$$!32+++n/;Q@FN$\"3++++ 89t8YFN7$$!3:+++(f]A@#FN$\"3E+++K7V)f%FN7$$!3#******p%[p%G#FN$\"3?+++3 qE\"e%FN7$$!3!******fX*obBFN$\"3Q+++80=iXFN7$$!3\"*******>)3aU#FN$\"3T +++VC8TXFN7$$!3!******\\N5S\\#FN$\"3-+++q\")4=XFN7$$!3%)*****>ZQ;c#FN$ \"3k*****4;lI\\%FN7$$!3(*******oxUGEFN$\"35+++7;.mWFN7$$!31+++,b]%p#FN $\"33+++4g+PWFN7$$!3=+++WL**fFFN$\"3V+++0u+1WFN7$$!3#)*****HV3]#GFN$\" 3C+++Vi1tVFN7$$!3%*******4Om*)GFN$\"3q******fbAQVFN7$$!35+++;o1aHFN$\" 3*)******zBa,VFN7$$!3$******Hc>$=IFN$\"33+++X!*3jUFN7$$!3))*****HY9D3$ FN$\"3y*****R=aHA%FN7$$!31+++.>tYJFN$\"3m******QHC\"=%FN7$$!3*)*****f5 O5@$FN$\"3N+++\"3w!QTFN7$$!3!)*****\\'4ZvKFN$\"31+++%o(e$4%FN7$$!33+++ di0SLFN$\"3G+++!>@z/%FN7$$!33+++@\\y/MFN$\"3C+++kRA,SFN7$$!37+++J?ipMF N$\"37+++Z2k`RFN7$$!30+++@f]MNFN$\"3.+++JnI0RFN7$$!3')*****4a^$*f$FN$ \"3%)*****>8Tj&QFN7$$!3A+++(pbSm$FN$\"35+++C@%o!QFN7$$!36+++2N]GPFN$\" 3)******>s$)ov$FN7$$!3%******pAvDz$FN$\"3*******4([^1PFN7$$!3)******Rh _h&QFN$\"3))*****zbgdl$FN7$$!3\")*****RCC\">RFN$\"33+++HXi/OFN7$$!3)** *****Q%*Q\")RFN$\"3y*****>+$4`NFN7$$!3p*****pvgG/%FN$\"35+++;)Q6]$FN7$ $!3J+++)>lM5%FN$\"3=+++Lbs[MFN7$$!3i*****4OWJ;%FN$\"33+++#=6eR$FN7$$!3 4+++nQ&=A%FN$\"3z*****pZ^BM$FN7$$!33+++(Gi&zUFN$\"3))*****fW-$)G$FN7$$ !3l*****f!)\\iL%FN$\"3********)\\AOB$FN7$$!3?+++pn!>R%FN$\"3%)******=S FyJFN7$$!3m*****f?KlW%FN$\"3$)*****z_CA7$FN7$$!3()*****zRK,]%FN$\"3;++ +dvWlIFN7$$!35+++4&>Fb%FN$\"3!)*****4DBz+$FN7$$!3()*****zE5Vg%FN$\"3/+ ++(yQ'\\HFN7$$!3y*****HrC\\l%FN$\"31+++V')e!*GFN7$$!3c*****\\'\\e/ZFN$ \"30+++@ZxIGFN7$$!3s*****H-9Lv%FN$\"3!)*****pM1-x#FN7$$!3i*****HeM6![F N$\"38+++;,!*3FFN7$$!3%)*****R(y1[[FN$\"3#******Hgzok#FN7$$!3U+++KD8%* [FN$\"3')******y[<%e#FN7$$!33+++[NMR\\FN$\"3%******4'=#3_#FN7$$!3G+++? 8r$)\\FN$\"3A+++a9'oX#FN7$$!39+++s3CF]FN$\"37+++K&QBR#FN7$$!3#)*****>F J*p]FN$\"3.+++]2IFBFN7$$!3-+++/`x6^FN$\"3%******pE(zhAFN7$$!3o******=% fF:&FN$\"35+++6u(e>#FN7$$!3++++xR'G>&FN$\"3,+++z$*eH@FN7$$!3E+++`P1K_F N$\"35+++S!zH1#FN7$$!3#)*****RpG.F&FN$\"3#******p*))3'*>FN7$$!3%)***** *\\[i2`FN$\"3-+++zs&*G>FN7$$!3&)*****zU:RM&FN$\"33+++txhh=FN7$$!3++++# )=;z`FN$\"33+++I*)4%z\"FN7$$!3P+++*)\\K8aFN$\"32+++;UUEG@'Q\"FN7$$!3/+++aT\\$f&FN$\"3%****** *o7&yJ\"FN7$$!37+++%4$4>cFN$\"35+++NrQ\\7FN7$$!3f*****z`gLk&FN$\"3#*** ***\\)*=3=\"FN7$$!39+++4!zim&FN$\"30+++N[976FN7$$!3e*****p5Myo&FN$\"3 \"******>hiL/\"FN7$$!3.+++=V,3dFN$\"3l******\\aqW(*F-7$$!3++++o2\"os&F N$\"3m*****\\-vY0*F-7$$!3K+++ip@WdFN$\"3o*****4hKNO)F-7$$!3?++++&G-w&F N$\"3++++:sGrwF-7$$!3M+++#yU[x&FN$\"3S+++z1'z(pF-7$$!3U+++4(e!)y&FN$\" 3K+++c()e$G'F-7$$!3u*****4Rw)*z&FN$\"3i*****zq@#)e&F-7$$!3t*****R&oH5e FN$\"3/+++SW#>*[F-7$$!3o*****Rw@$>eFN$\"39+++*=wZ>%F-7$$!3%******\\;`p #eFN$\"3u*****f!)po\\$F-7$$!3h*****fB$>LeFN$\"3-+++X1J)z#F-7$$!3\"**** **R2W!QeFN$\"3))*****z3:#*4#F-7$$!3)******\\^2:%eFN$\"3'******pw3(*R\" F-7$$!3:+++!)\\eVeFN$\"3s******[\\C**pFar7$$!3.+++mtFWeFNF(7$Fb]n$!3s* *****[\\C**pFar7$F]]n$!3'******pw3(*R\"F-7$Fh\\n$!3))*****z3:#*4#F-7$F c\\n$!3-+++X1J)z#F-7$F^\\n$!3u*****f!)po\\$F-7$Fi[n$!39+++*=wZ>%F-7$Fd [n$!3/+++SW#>*[F-7$F_[n$!3i*****zq@#)e&F-7$Fjjm$!3K+++c()e$G'F-7$Fejm$ !3S+++z1'z(pF-7$F`jm$!3++++:sGrwF-7$F[jm$!3o*****4hKNO)F-7$Ffim$!3m*** **\\-vY0*F-7$Faim$!3l******\\aqW(*F-7$F\\im$!3\"******>hiL/\"FN7$Fghm$ !30+++N[976FN7$Fbhm$!3#******\\)*=3=\"FN7$F]hm$!35+++NrQ\\7FN7$Fhgm$!3 %*******o7&yJ\"FN7$Fcgm$!3!*******>G@'Q\"FN7$F^gm$!3'*******\\@Za9FN7$ Fifm$!3'******\\)ziA:FN7$Fdfm$!3))*****fzw1f\"FN7$F_fm$!35+++w@he;FN7$ Fjem$!32+++;UUEFN7$Ffdm$!3#******p*))3'*>FN7$Fadm$!35+++S!zH1#FN7$F\\dm$!3, +++z$*eH@FN7$Fgcm$!35+++6u(e>#FN7$Fbcm$!3%******pE(zhAFN7$F]cm$!3.+++] 2IFBFN7$Fhbm$!37+++K&QBR#FN7$Fcbm$!3A+++a9'oX#FN7$F^bm$!3%******4'=#3_ #FN7$Fiam$!3')******y[<%e#FN7$Fdam$!3#******Hgzok#FN7$F_am$!38+++;,!*3 FFN7$Fj`m$!3!)*****pM1-x#FN7$Fe`m$!30+++@ZxIGFN7$F``m$!31+++V')e!*GFN7 $F[`m$!3/+++(yQ'\\HFN7$Ff_m$!3!)*****4DBz+$FN7$Fa_m$!3;+++dvWlIFN7$F\\ _m$!3$)*****z_CA7$FN7$Fg^m$!3%)******=SFyJFN7$Fb^m$!3********)\\AOB$FN 7$F]^m$!3))*****fW-$)G$FN7$Fh]m$!3z*****pZ^BM$FN7$Fc]m$!33+++#=6eR$FN7 $F^]m$!3=+++Lbs[MFN7$Fi\\m$!35+++;)Q6]$FN7$Fd\\m$!3y*****>+$4`NFN7$F_ \\m$!33+++HXi/OFN7$Fj[m$!3))*****zbgdl$FN7$Fe[m$!3*******4([^1PFN7$F`[ m$!3)******>s$)ov$FN7$F[[m$!35+++C@%o!QFN7$Ffjl$!3%)*****>8Tj&QFN7$Faj l$!3.+++JnI0RFN7$F\\jl$!37+++Z2k`RFN7$Fgil$!3C+++kRA,SFN7$Fbil$!3G+++! >@z/%FN7$F]il$!31+++%o(e$4%FN7$Fhhl$!3N+++\"3w!QTFN7$Fchl$!3m******QHC \"=%FN7$F^hl$!3y*****R=aHA%FN7$Figl$!33+++X!*3jUFN7$Fdgl$!3*)******zBa ,VFN7$F_gl$!3q******fbAQVFN7$Fjfl$!3C+++Vi1tVFN7$Fefl$!3V+++0u+1WFN7$F `fl$!33+++4g+PWFN7$F[fl$!35+++7;.mWFN7$Ffel$!3k*****4;lI\\%FN7$Fael$!3 -+++q\")4=XFN7$F\\el$!3T+++VC8TXFN7$Fgdl$!3Q+++80=iXFN7$Fbdl$!3?+++3qE \"e%FN7$F]dl$!3E+++K7V)f%FN7$Fhcl$!3++++89t8YFN7$Fccl$!3i*****RI^si%FN 7$F^cl$!3f*****zE5\"RYFN7$Fibl$!3=+++A'y%\\YFN7$Fdbl$!39+++&>,'eYFN7$F _bl$!3%)*****z%R$om%FN7$Fjal$!3k*****4i,Zn%FN7$Feal$!39+++:*))Ho%FN7$F `al$!3s*****>y\")Gp%FN7$F[al$!3p*****pnmhq%FN7$Ff`l$!3i*****z:@as%FN7$ Fa`l$!3/+++O)RQv%FN7$F\\`l$!3;+++pP!Qz%FN7$Fg_l$!3*******f$Q/W[FN7$Fb_ l$!3'*********\\-**[FN7$F]_l$!3G+++B\\y_\\FN7$Fh^l$!39+++5X!>+&FN7$Fc^ l$!3y*****p&\\@X]FN7$F^^l$!3!******\\yaE3&FN7$Fi]l$!3W+++w&fX6&FN7$Fd] l$!3#*******)oa89&FN7$F_]l$!3W+++CMWj^FN7$Fj\\l$!3#******R%Q<\"=&FN7$F e\\l$!3)******H^M[>&FN7$F`\\l$!3l*****>tfY?&FN7$F[\\l$!3#)*****>,P3@&F N7$Ff[l$!3Q+++TU^8_FN7$Fa[l$!3C+++$z0G@&FN7$F\\[l$!31+++xuz3_FN7$Fgz$! 3m*****fX]:?&FN7$Fbz$!37+++6V5\">&FN7$$!3/+++Pam`wFfq$!3'*******=%zu<& FN7$Fhy$!3S+++!zy1;&FN7$Fcy$!3v*******[*oS^FN7$F^y$!3\"******zr$[<^FN7 $Fix$!3!)*****zh>54&FN7$Fdx$!3w*****\\#=Ch]FN7$F_x$!3f*****>s\"3G]FN7$ Fjw$!3C+++(Gd9*\\FN7$Few$!35+++(Gs7&\\FN7$F`w$!30+++&Q9u!\\FN7$F[w$!3m *****>S^(f[FN7$Ffv$!3i*****\\ZC\"3[FN7$Fav$!3!*******4hL_ZFN7$F\\v$!3) ******>$)H@p%FN7$Fgu$!3>+++.i:FYFN7$Fbu$!37+++:o\"pb%FN7$F]u$!3k****** 4Mn![%FN7$Fht$!3m*****zN0tR%FN7$Fct$!3G+++%HhROFN7$Fer$!3/+++k$\\kZ$FN7$F_r$!37+++]M@8LFN7$Fjq$!3 '*******)>#=^JFN7$Fdq$!3%)*****\\qI0*HFN7$F_q$!30+++$QW5$GFN7$Fjp$!3'* *****f&oUsEFN7$Fep$!3%)*****Hi5W^#FN7$F_p$!3y*****4I(ycBFN7$Fjo$!33+++ 4wS*>#FN7$Feo$!3!******>Ar@/#FN7$F`o$!3-+++SX,&)=FN7$$!3\"******RvZH!> F\\o$!34+++`&**ys\"FN7$$!3[*****RXrx$))FR$!33+++J\\!3d\"FN7$FV$!3/+++. &>PT\"FN7$FP$!3#******H'yjc7FN7$FJ$!31+++Awb*4\"FN7$FD$!3X+++P$yZU*F-7 $$!3#*******)3k.H#F@$!3E+++*o\")R&yF-7$$!32+++K*R)4MF:$!3Q+++8`=$G'F-7 $$!3y*****>N!3ZIF4$!3D+++\")*)Q7ZF-7$F($!3)******Rl#fTJF-7$F($!3++++Fj zq:F-F'-%&COLORG6&%$RGBGF($\"\"$!\"\"$\"\"%Fh\\p-%)POLYGONSG6^_l7%F'7$ F($\"+Fjzq:!#57$$!#UFh\\pF(7%F_]p7$F($\"+aEfTJFb]pFc]p7%Fg]p7$$!+!)4'p /$Far$\"+\")*)Q7ZFb]pFc]p7%F[^p7$$!+Q*R)4MF-$\"+8`=$G'Fb]pFc]p7%Fa^p7$ $!+'3k.H#FN$\"+*o\")R&yFb]pFc]p7%Fg^p7$$!+wiX06!#;$\"+P$yZU*Fb]pFc]p7% F]_p7$$!+mk)4@%F`_p$\"+Awb*4\"!\"*Fc]p7%Fd_p7$$!+Q-PT\"Fi_pFc]p7%Fb`p7$$!+Z9xP))F^`p$\"+J \\!3d\"Fi_pFc]p7%Fh`p7$$!+^x%H!>!#9$\"+`&**ys\"Fi_pFc]p7%F^ap7$$!+n98r OFaap$\"+SX,&)=Fi_pFc]p7%Feap7$$!+&RV[I'Faap$\"+A7#Fi_pFc]p7%Fabp7$$!+#=^JFi_pFc]p7%Fgdp7$$\"+r[v 75!#6$\"+]M@8LFi_pFc]p7%F]ep7$$\"+Uvx))>F`ep$\"+k$\\kZ$Fi_pFc]p7%Fdep7 $$\"+:q^qOF`ep$\"+UHhROFi_pFc]p7%Fjep7$$\"+MUnDjF`ep$\"+:(z\"*z$Fi_pFc ]p7%F`fp7$$\"+^IQ')**F`ep$\"+:C3\\RFi_pFc]p7%Fffp7$$\"+#[CIU\"Fb]p$\"+ (4!o$3%Fi_pFc]p7%F\\gp7$$\"+*)3pS=Fb]p$\"+Rof,UFi_pFc]p7%Fbgp7$$\"+W&[ Y?#Fb]p$\"+#y%=Fb]p$ \"+A<3G]Fi_pFc]p7%F`\\q7$$\"+9M.s:Fb]p$\"+D=Ch]Fi_pFc]p7%Ff\\q7$$\"+mG iv7Fb]p$\"+='>54&Fi_pFc]p7%F\\]q7$$\"+aUz2'*F`ep$\"+=P[<^Fi_pFc]p7%Fb] q7$$\"+'H&3%H'F`ep$\"+!\\*oS^Fi_pFc]p7%Fh]q7$$\"+K0iJGF`ep$\"+!zy1;&Fi _pFc]p7%F^^q7$$!+Qam`wFcdp$\"+>%zu<&Fi_pFc]p7%Fd^q7$$!+,:t%[%F`ep$\"+6 V5\">&Fi_pFc]p7%Fj^q7$$!+??G;$)F`ep$\"+c/b,_Fi_pFc]p7%F`_q7$$!+t2&Fi_pF c]p7%Fdaq7$$!++0\\%y$Fb]p$\"+WQ<\"=&Fi_pFc]p7%Fjaq7$$!+Bi)oC%Fb]p$\"+C MWj^Fi_pFc]p7%F`bq7$$!+\"zNLs%Fb]p$\"+*oa89&Fi_pFc]p7%Ffbq7$$!+NBW<_Fb ]p$\"+w&fX6&Fi_pFc]p7%F\\cq7$$!+KL![t&Fb]p$\"+&yaE3&Fi_pFc]p7%Fbcq7$$! +ZO7%G'Fb]p$\"+d\\@X]Fi_pFc]p7%Fhcq7$$!++&=*yoFb]p$\"+5X!>+&Fi_pFc]p7% F^dq7$$!+,'eYFi_pFc]p7%Ffhq7$$!+YRf.>Fi_p$\"+A'y %\\YFi_pFc]p7%F\\iq7$$!+6@6%)>Fi_p$\"+o-6RYFi_pFc]p7%Fbiq7$$!+70?i?Fi_ p$\"+/8DFYFi_pFc]p7%Fhiq7$$!+n/;Q@Fi_p$\"+89t8YFi_pFc]p7%F^jq7$$!+(f]A @#Fi_p$\"+K7V)f%Fi_pFc]p7%Fdjq7$$!+Z[p%G#Fi_p$\"+3qE\"e%Fi_pFc]p7%Fjjq 7$$!+c%*obBFi_p$\"+80=iXFi_pFc]p7%F`[r7$$!+?)3aU#Fi_p$\"+VC8TXFi_pFc]p 7%Ff[r7$$!+b.,%\\#Fi_p$\"+q\")4=XFi_pFc]p7%F\\\\r7$$!+s%Q;c#Fi_p$\"+h^ 1$\\%Fi_pFc]p7%Fb\\r7$$!+pxUGEFi_p$\"+7;.mWFi_pFc]p7%Fh\\r7$$!+,b]%p#F i_p$\"+4g+PWFi_pFc]p7%F^]r7$$!+WL**fFFi_p$\"+0u+1WFi_pFc]p7%Fd]r7$$!+L %3]#GFi_p$\"+Vi1tVFi_pFc]p7%Fj]r7$$!+5Om*)GFi_p$\"+gbAQVFi_pFc]p7%F`^r 7$$!+;o1aHFi_p$\"+!QU:I%Fi_pFc]p7%Ff^r7$$!+j&>$=IFi_p$\"+X!*3jUFi_pFc] p7%F\\_r7$$!+jW^#3$Fi_p$\"+%=aHA%Fi_pFc]p7%Fb_r7$$!+.>tYJFi_p$\"+RHC\" =%Fi_pFc]p7%Fh_r7$$!+1h.6KFi_p$\"+\"3w!QTFi_pFc]p7%F^`r7$$!+l4ZvKFi_p$ \"+%o(e$4%Fi_pFc]p7%Fd`r7$$!+di0SLFi_p$\"+!>@z/%Fi_pFc]p7%Fj`r7$$!+@\\ y/MFi_p$\"+kRA,SFi_pFc]p7%F`ar7$$!+J?ipMFi_p$\"+Z2k`RFi_pFc]p7%Ffar7$$ !+@f]MNFi_p$\"+JnI0RFi_pFc]p7%F\\br7$$!+T:N*f$Fi_p$\"+K6McQFi_pFc]p7%F bbr7$$!+(pbSm$Fi_p$\"+C@%o!QFi_pFc]p7%Fhbr7$$!+2N]GPFi_p$\"+AP)ov$Fi_p Fc]p7%F^cr7$$!+F_d#z$Fi_p$\"+r[^1PFi_pFc]p7%Fdcr7$$!+9E:cQFi_p$\"+e0wb OFi_pFc]p7%Fjcr7$$!+WU7>RFi_p$\"+HXi/OFi_pFc]p7%F`dr7$$!+R%*Q\")RFi_p$ \"+-I4`NFi_pFc]p7%Ffdr7$$!+d2'G/%Fi_p$\"+;)Q6]$Fi_pFc]p7%F\\er7$$!+)>l M5%Fi_p$\"+Lbs[MFi_pFc]p7%Fber7$$!+hV9jTFi_p$\"+#=6eR$Fi_pFc]p7%Fher7$ $!+nQ&=A%Fi_p$\"+x9NULFi_pFc]p7%F^fr7$$!+(Gi&zUFi_p$\"+YCI)G$Fi_pFc]p7 %Fdfr7$$!+1)\\iL%Fi_p$\"+*\\AOB$Fi_pFc]p7%Fjfr7$$!+pn!>R%Fi_p$\"+>SFyJ Fi_pFc]p7%F`gr7$$!+1A`YWFi_p$\"+GXAAJFi_pFc]p7%Ffgr7$$!+)RK,]%Fi_p$\"+ dvWlIFi_pFc]p7%F\\hr7$$!+4&>Fb%Fi_p$\"+^K#z+$Fi_pFc]p7%Fbhr7$$!+o-J/YF i_p$\"+(yQ'\\HFi_pFc]p7%Fhhr7$$!+8Z#\\l%Fi_p$\"+V')e!*GFi_pFc]p7%F^ir7 $$!+l\\e/ZFi_p$\"+@ZxIGFi_pFc]p7%Fdir7$$!+BSJ`ZFi_p$\"+Zj?qFFi_pFc]p7% Fjir7$$!+$eM6![Fi_p$\"+;,!*3FFi_pFc]p7%F`jr7$$!+uy1[[Fi_p$\"+.'zok#Fi_ pFc]p7%Ffjr7$$!+KD8%*[Fi_p$\"+z[<%e#Fi_pFc]p7%F\\[s7$$!+[NMR\\Fi_p$\"+ h=#3_#Fi_pFc]p7%Fb[s7$$!+?8r$)\\Fi_p$\"+a9'oX#Fi_pFc]p7%Fh[s7$$!+s3CF] Fi_p$\"+K&QBR#Fi_pFc]p7%F^\\s7$$!+s7$*p]Fi_p$\"+]2IFBFi_pFc]p7%Fd\\s7$ $!+/`x6^Fi_p$\"+nszhAFi_pFc]p7%Fj\\s7$$!+>%fF:&Fi_p$\"+6u(e>#Fi_pFc]p7 %F`]s7$$!+xR'G>&Fi_p$\"+z$*eH@Fi_pFc]p7%Ff]s7$$!+`P1K_Fi_p$\"+S!zH1#Fi _pFc]p7%F\\^s7$$!+%pG.F&Fi_p$\"+(*))3'*>Fi_pFc]p7%Fb^s7$$!+][i2`Fi_p$ \"+zs&*G>Fi_pFc]p7%Fh^s7$$!+Ga\"RM&Fi_p$\"+txhh=Fi_pFc]p7%F^_s7$$!+#)= ;z`Fi_p$\"+I*)4%z\"Fi_pFc]p7%Fd_s7$$!+*)\\K8aFi_p$\"+;UUEcFi_p$\"+NrQ\\7Fi_pFc]p7%Fdbs7$$!+ Q0OVcFi_p$\"+&)*=3=\"Fi_pFc]p7%Fjbs7$$!+4!zim&Fi_p$\"+N[976Fi_pFc]p7%F `cs7$$!+2T$yo&Fi_p$\"+7EOV5Fi_pFc]p7%Ffcs7$$!+=V,3dFi_p$\"+]aqW(*Fb]pF c]p7%F\\ds7$$!+o2\"os&Fi_p$\"+D]na!*Fb]pFc]p7%Fbds7$$!+ip@WdFi_p$\"+6E `j$)Fb]pFc]p7%Fhds7$$!++&G-w&Fi_p$\"+:sGrwFb]pFc]p7%F^es7$$!+#yU[x&Fi_ p$\"+z1'z(pFb]pFc]p7%Fdes7$$!+4(e!)y&Fi_p$\"+c()e$G'Fb]pFc]p7%Fjes7$$! +\"Rw)*z&Fi_p$\"+3*[Fb]pFc]p7 %Fffs7$$!+keFi_p$\"+*=wZ>%Fb]pFc]p7%F\\gs7$$!+lJ&p#eFi_p$\"+1)po\\$ Fb]pFc]p7%Fbgs7$$!+OK>LeFi_p$\"+X1J)z#Fb]pFc]p7%Fhgs7$$!+uS/QeFi_p$\"+ )3:#*4#Fb]pFc]p7%F^hs7$$!+:v]TeFi_p$\"+n(3(*R\"Fb]pFc]p7%Fdhs7$$!+!)\\ eVeFi_p$\"+\\\\C**pF`epFc]p7%Fjhs7$$!+mtFWeFi_pF(Fc]p7%F`is7$F[is$!+\\ \\C**pF`epFc]p7%Fdis7$Fehs$!+n(3(*R\"Fb]pFc]p7%Fhis7$F_hs$!+)3:#*4#Fb] pFc]p7%F\\js7$Figs$!+X1J)z#Fb]pFc]p7%F`js7$Fcgs$!+1)po\\$Fb]pFc]p7%Fdj s7$F]gs$!+*=wZ>%Fb]pFc]p7%Fhjs7$Fgfs$!+SW#>*[Fb]pFc]p7%F\\[t7$Fafs$!+3 Fi_pFc]p7%F``t7$Fc^s$!+(*))3'*>Fi_p Fc]p7%Fd`t7$F]^s$!+S!zH1#Fi_pFc]p7%Fh`t7$Fg]s$!+z$*eH@Fi_pFc]p7%F\\at7 $Fa]s$!+6u(e>#Fi_pFc]p7%F`at7$F[]s$!+nszhAFi_pFc]p7%Fdat7$Fe\\s$!+]2IF BFi_pFc]p7%Fhat7$F_\\s$!+K&QBR#Fi_pFc]p7%F\\bt7$Fi[s$!+a9'oX#Fi_pFc]p7 %F`bt7$Fc[s$!+h=#3_#Fi_pFc]p7%Fdbt7$F][s$!+z[<%e#Fi_pFc]p7%Fhbt7$Fgjr$ !+.'zok#Fi_pFc]p7%F\\ct7$Fajr$!+;,!*3FFi_pFc]p7%F`ct7$F[jr$!+Zj?qFFi_p Fc]p7%Fdct7$Feir$!+@ZxIGFi_pFc]p7%Fhct7$F_ir$!+V')e!*GFi_pFc]p7%F\\dt7 $Fihr$!+(yQ'\\HFi_pFc]p7%F`dt7$Fchr$!+^K#z+$Fi_pFc]p7%Fddt7$F]hr$!+dvW lIFi_pFc]p7%Fhdt7$Fggr$!+GXAAJFi_pFc]p7%F\\et7$Fagr$!+>SFyJFi_pFc]p7%F `et7$F[gr$!+*\\AOB$Fi_pFc]p7%Fdet7$Fefr$!+YCI)G$Fi_pFc]p7%Fhet7$F_fr$! +x9NULFi_pFc]p7%F\\ft7$Fier$!+#=6eR$Fi_pFc]p7%F`ft7$Fcer$!+Lbs[MFi_pFc ]p7%Fdft7$F]er$!+;)Q6]$Fi_pFc]p7%Fhft7$Fgdr$!+-I4`NFi_pFc]p7%F\\gt7$Fa dr$!+HXi/OFi_pFc]p7%F`gt7$F[dr$!+e0wbOFi_pFc]p7%Fdgt7$Fecr$!+r[^1PFi_p Fc]p7%Fhgt7$F_cr$!+AP)ov$Fi_pFc]p7%F\\ht7$Fibr$!+C@%o!QFi_pFc]p7%F`ht7 $Fcbr$!+K6McQFi_pFc]p7%Fdht7$F]br$!+JnI0RFi_pFc]p7%Fhht7$Fgar$!+Z2k`RF i_pFc]p7%F\\it7$Faar$!+kRA,SFi_pFc]p7%F`it7$F[ar$!+!>@z/%Fi_pFc]p7%Fdi t7$Fe`r$!+%o(e$4%Fi_pFc]p7%Fhit7$F_`r$!+\"3w!QTFi_pFc]p7%F\\jt7$Fi_r$! +RHC\"=%Fi_pFc]p7%F`jt7$Fc_r$!+%=aHA%Fi_pFc]p7%Fdjt7$F]_r$!+X!*3jUFi_p Fc]p7%Fhjt7$Fg^r$!+!QU:I%Fi_pFc]p7%F\\[u7$Fa^r$!+gbAQVFi_pFc]p7%F`[u7$ F[^r$!+Vi1tVFi_pFc]p7%Fd[u7$Fe]r$!+0u+1WFi_pFc]p7%Fh[u7$F_]r$!+4g+PWFi _pFc]p7%F\\\\u7$Fi\\r$!+7;.mWFi_pFc]p7%F`\\u7$Fc\\r$!+h^1$\\%Fi_pFc]p7 %Fd\\u7$F]\\r$!+q\")4=XFi_pFc]p7%Fh\\u7$Fg[r$!+VC8TXFi_pFc]p7%F\\]u7$F a[r$!+80=iXFi_pFc]p7%F`]u7$F[[r$!+3qE\"e%Fi_pFc]p7%Fd]u7$Fejq$!+K7V)f% Fi_pFc]p7%Fh]u7$F_jq$!+89t8YFi_pFc]p7%F\\^u7$Fiiq$!+/8DFYFi_pFc]p7%F`^ u7$Fciq$!+o-6RYFi_pFc]p7%Fd^u7$F]iq$!+A'y%\\YFi_pFc]p7%Fh^u7$Fghq$!+&> ,'eYFi_pFc]p7%F\\_u7$Fahq$!+[R$om%Fi_pFc]p7%F`_u7$F[hq$!+@;quYFi_pFc]p 7%Fd_u7$Fegq$!+:*))Ho%Fi_pFc]p7%Fh_u7$F_gq$!+#y\")Gp%Fi_pFc]p7%F\\`u7$ Fifq$!+xm;1ZFi_pFc]p7%F``u7$Fcfq$!+e6UDZFi_pFc]p7%Fd`u7$F]fq$!+O)RQv%F i_pFc]p7%Fh`u7$Fgeq$!+pP!Qz%Fi_pFc]p7%F\\au7$Faeq$!+OQ/W[Fi_pFc]p7%F`a u7$F[eq$!++]-**[Fi_pFc]p7%Fdau7$Fedq$!+B\\y_\\Fi_pFc]p7%Fhau7$F_dq$!+5 X!>+&Fi_pFc]p7%F\\bu7$Ficq$!+d\\@X]Fi_pFc]p7%F`bu7$Fccq$!+&yaE3&Fi_pFc ]p7%Fdbu7$F]cq$!+w&fX6&Fi_pFc]p7%Fhbu7$Fgbq$!+*oa89&Fi_pFc]p7%F\\cu7$F abq$!+CMWj^Fi_pFc]p7%F`cu7$F[bq$!+WQ<\"=&Fi_pFc]p7%Fdcu7$Feaq$!+8X$[>& Fi_pFc]p7%Fhcu7$F_aq$!+K(fY?&Fi_pFc]p7%F\\du7$Fi`q$!+7q$3@&Fi_pFc]p7%F `du7$Fc`q$!+TU^8_Fi_pFc]p7%Fddu7$F]`q$!+$z0G@&Fi_pFc]p7%Fhdu7$Fg_q$!+x uz3_Fi_pFc]p7%F\\eu7$Fa_q$!+c/b,_Fi_pFc]p7%F`eu7$F[_q$!+6V5\">&Fi_pFc] p7%Fdeu7$$!+Pam`wFcdp$!+>%zu<&Fi_pFc]p7%Fheu7$F_^q$!+!zy1;&Fi_pFc]p7%F ^fu7$Fi]q$!+!\\*oS^Fi_pFc]p7%Fbfu7$Fc]q$!+=P[<^Fi_pFc]p7%Fffu7$F]]q$!+ ='>54&Fi_pFc]p7%Fjfu7$Fg\\q$!+D=Ch]Fi_pFc]p7%F^gu7$Fa\\q$!+A<3G]Fi_pFc ]p7%Fbgu7$F[\\q$!+(Gd9*\\Fi_pFc]p7%Ffgu7$Fe[q$!+(Gs7&\\Fi_pFc]p7%Fjgu7 $F_[q$!+&Q9u!\\Fi_pFc]p7%F^hu7$Fijp$!+-9vf[Fi_pFc]p7%Fbhu7$Fcjp$!+vW73 [Fi_pFc]p7%Ffhu7$F]jp$!+5hL_ZFi_pFc]p7%Fjhu7$Fgip$!+K)H@p%Fi_pFc]p7%F^ iu7$Faip$!+.i:FYFi_pFc]p7%Fbiu7$F[ip$!+:o\"pb%Fi_pFc]p7%Ffiu7$Fehp$!+5 Mn![%Fi_pFc]p7%Fjiu7$F_hp$!+e`I(R%Fi_pFc]p7%F^ju7$Figp$!+#=^JFi_pFc]p7%Fb\\v7$Fadp$!+02`!*HFi_pFc]p7%Ff\\v7$F[dp$!+ $QW5$GFi_pFc]p7%Fj\\v7$Fecp$!+coUsEFi_pFc]p7%F^]v7$F_cp$!+B1T9DFi_pFc] p7%Fb]v7$Fhbp$!+,tycBFi_pFc]p7%Ff]v7$Fbbp$!+4wS*>#Fi_pFc]p7%Fj]v7$F\\b p$!+A7Faap$!+ `&**ys\"Fi_pFc]p7%Ff^v7$$!+a9xP))F^`p$!+J\\!3d\"Fi_pFc]p7%F\\_v7$Fc`p$ !+.&>PT\"Fi_pFc]p7%Fb_v7$F\\`p$!+jyjc7Fi_pFc]p7%Ff_v7$Fe_p$!+Awb*4\"Fi _pFc]p7%Fj_v7$F^_p$!+P$yZU*Fb]pFc]p7%F^`v7$$!+*3k.H#FN$!+*o\")R&yFb]pF c]p7%Fb`v7$$!+K*R)4MF-$!+8`=$G'Fb]pFc]p7%Fh`v7$$!+_.3ZIFar$!+\")*)Q7ZF b]pFc]p7%F^av7$F($!+aEfTJFb]pFc]p7%Fdav7$F($!+Fjzq:Fb]pFc]p7%FhavF'Fc] p-Fc\\p6&Fe\\pF($\"#b!\"#$\"#vF`bv-%&STYLEG6#%,PATCHNOGRIDG-F$6$7]_lF' F*F.7$F(F57$$\"3D+++'*>RUZF4$\"3g*****pJ&=$G'F-7$$\"3#)*****\\Q@C<%F:$ \"3#)*****pp\")R&yF-7$$\"3,+++#=zeS#F@$\"3m*****pCyZU*F-7$$\"3'******H L71.\"FF$\"3)******p_d&*4\"FN7$$\"3<+++uDpJNFF$\"3++++Itjc7FN7$$\"3*** *****oOR95FR$\"3#*******3tr89FN7$$\"3!)*****\\`;y^#FR$\"3%******z_(zq: FN7$$\"3R+++Z&[&4bFR$\"3++++(*z(ys\"FN7$$\"3'*******3!*)f2\"F\\o$\"35+ ++\\'e\\)=FN7$$\"3)******48wf)=F\\o$\"37+++'\\R?/#FN7$$\"3!******>1;A' HF\\o$\"3)******f@@\"*>#FN7$$\"3>+++'fQ+7%F\\o$\"3#)*****>o1iN#FN7$$\" 3w******[vs:\\F\\o$\"3!)******RTI8DFN7$$\"3\"******>NZCh%F\\o$\"3%**** **z)HVqEFN7$$\"3%******RkxGL#F\\o$\"3(******pLLw#GFN7$$!31+++'o*z!Q#F \\o$\"35+++d0)\\)HFN7$$!3_*****z(4MW#)F\\o$\"3y*****f,1E9$FN7$$!3U**** *4bf#y&*F\\o$\"3,+++7]s+LFN7$$\"3M+++w8uN')F\\o$\"37+++6CnfMFN7$$\"3#) *****RnHn(zFap$\"3.+++DY%*>OFN7$$\"33+++Le,AFFfq$\"3/+++r$QAy$FN7$$\"3 =+++s1t5sFfq$\"3?+++\\GWZRFN7$$\"31+++X%G')o\"Far$\"3C+++2#4k6%FN7$$\" 3)******fd0$fOFar$\"3k*****Hac))G%FN7$$\"3S*******zfTL(Far$\"3K+++e0#* fWFN7$$\"3-+++:zp&H\"F-$\"3h*****RtBkh%FN7$$\"3()*****>$48S>F-$\"39+++ !eIou%FN7$$\"33+++)*H&)HDF-$\"3/+++nmW`[FN7$$\"33+++CI$G,$F-$\"3>+++Gr $Q%\\FN7$$\"3w*****\\9iuQ$F-$\"3J+++hI[B]FN7$$\"3u*****HDOcm$F-$\"3E++ +')R^&4&FN7$$\"3++++'4N/'QF-$\"3g*****pyy;;&FN7$$\"3')*****pu@L)RF-$\" 3.+++fm)HA&FN7$$\"3;+++ZG(Q/%F-$\"3#)*****\\)e/!G&FN7$$\"3'******HLa+0 %F-$\"3d******f.CL`FN7$$\"3v*******)\\a3SF-$\"39+++8U#GQ&FN7$$\"34+++4 E+DRF-$\"31+++\"=u*GaFN7$$\"3t*****4olU!QF-$\"3?+++Y$>=Z&FN7$$\"3E+++P S]]OF-$\"3e*****z@e9^&FN7$$\"3-+++AELnMF-$\"3')*****pcoza&FN7$$\"37+++ u#**yD$F-$\"3U+++oHT\"e&FN7$$\"3D+++nS&\\-$F-$\"3G+++'yT=h&FN7$$\"3/++ +$>04x#F-$\"3E+++?YHRcFN7$$\"3#******4nky\\#F-$\"3O+++'y+Qm&FN7$$\"31+ ++cfo2AF-$\"3u*****z=z`o&FN7$$\"3$******>![*>!>F-$\"3#)*****\\wPSq&FN7 $$\"3'*******oT@#e\"F-$\"3\"******4fs(>dFN7$$\"3%******>f$e\\7F-$\"3/+ ++_mcKdFN7$$\"3c*****4dI<0*Far$\"3P+++S#)QUdFN7$$\"3!******\\G5*)\\&Fa r$\"3Q+++/*)=\\dFN7$$\"3&)*****fvd[%=Far$\"3;+++#z+Hv&FN7$$!3&)*****>C ;Y!>Far$\"3S+++&>LMv&FN7$$!3K+++()QbXdFar$\"3?+++,!o1v&FN7$$!3k+++\"f* =w'*Far$\"3s*****4l`Wu&FN7$$!3)*******yFvp8F-$\"38+++8qfMdFN7$$!37+++J )H9y\"F-$\"3(******fN_3s&FN7$$!36+++TPl.AF-$\"3C+++(G1Hq&FN7$$!3F+++'> D#QEF-$\"3?+++lrN!o&FN7$$!3E+++uX?)3$F-$\"3y******)\\(o_cFN7$$!37+++Yk ueNF-$\"3/++++%H#>cFN7$$!3-+++K$p'eSF-$\"3#******\\UJ\"zbFN7$$!3!)**** **RP^.YF-$\"3G+++#Qp8`&FN7$$!3M+++\\'e9A&F-$\"3S+++Br(\\Z&FN7$$!3k**** *Ri=F'fF-$\"3%)******y&Q,T&FN7$$!3w*****pt%*))*oF-$\"3,+++oejT`FN7$$!3 W+++u!>b0)F-$\"3w*****RhvDG&FN7$$!3(********e;2K*F-$\"3k*****RwIWC&FN7 $$!3'******4u*zb5FN$\"3U++++(R[A&FN7$$!3!******>wZH@&FN7$$!3%******zaD&z8FN$\"3Q+++Qa&>@&F N7$$!3))*******)z6u9FN$\"3K+++FGV6_FN7$$!33+++l=(Qc\"FN$\"3:+++%z^.@&F N7$$!37+++h9h\\;FN$\"32+++&3&43_FN7$$!3*)*****p;\"*>t\"FN$\"3:+++$pzU? &FN7$$!32+++dL`6=FN$\"3s*****RYh')>&FN7$$!3!******>')f'))=FN$\"3M+++19 3\">&FN7$$!3'******p=;P'>FN$\"3o******\\UV\"=&FN7$$!3\")*****p\")*)p.# FN$\"39+++c%\\'p^FN7$$!3$)*****z2C(3@FN$\"3=+++t%zc:&FN7$$!3y*****RuG \"z@FN$\"3u*****zS#\\R^FN7$$!3!)*****zE)Q[AFN$\"3*******R]n57&FN7$$!3& )*****4DomJ#FN$\"3k*****p\\#R+^FN7$$!33+++n07%Q#FN$\"3!******HIiu2&FN7 $$!37+++Lu)3X#FN$\"3#)*****HqyA0&FN7$$!32+++nX5&*\\FN7$$!3!******p18%[EFN$\"3i*****4QVL' \\FN7$$!3.+++[)fPr#FN$\"31+++JALH\\FN7$$!3')******ew1zFFN$\"3G+++K%=K* [FN7$$!36+++0dXWGFN$\"3;+++c\\2b[FN7$$!32+++Ba.5HFN$\"3M+++'R%*\\\"[FN 7$$!3%******>*f!f(HFN$\"3w*****H?!4tZFN7$$!3;+++J([@/$FN$\"3o*****HT' \\HZFN7$$!3!******4$4#)3JFN$\"37+++1`O%o%FN7$$!31+++g*\\f<$FN$\"3v**** *pgiyj%FN7$$!3@+++\\(GNC$FN$\"3m*****REg,f%FN7$$!35+++CP^6LFN$\"3I+++i tUTXFN7$$!31+++!*f#)zLFN$\"3'******HH?=\\%FN7$$!35+++@fN[MFN$\"3w***** pht9W%FN7$$!3z******G1(p^$FN$\"3i*****H*H\\!R%FN7$$!3-+++`J_&e$FN$\"31 +++V6&*QVFN7$$!3%)*****f*>'Ql$FN$\"3/+++Gr)oG%FN7$$!3%******prR=s$FN$ \"3#******Hc3VB%FN7$$!3%******>v>$*y$FN$\"3K+++if>\"=%FN7$$!3=+++j5=cQ FN$\"3$)*****zC3v7%FN7$$!3@+++;2KARFN$\"3I+++S&)=tSFN7$$!3%******zraw) RFN$\"3:+++*fp\"=SFN7$$!3!******zc<@0%FN$\"3)******\\ayB'RFN7$$!3))*** **\\)4m:TFN$\"3()******)pSd!RFN7$$!3#)*****R<_#yTFN$\"3')*****zd#=[QFN 7$$!3*)*****4!>()RUFN$\"3'*******GTj*y$FN7$$!3<+++nD^+VFN$\"3;+++a..IP FN7$$!3!)*****>Ow,O%FN$\"39+++yCJpOFN7$$!3@+++WN()=WFN$\"3\")*****\\xG ug$FN7$$!39+++L3iwWFN$\"3#)*****f:NVa$FN7$$!3$)*****\\!*RM`%FN$\"3?+++ 'f&**zMFN7$$!3#)******QfN*e%FN$\"3*******pV#Q9MFN7$$!3))*****f<'RWYFN$ \"3!*******olZZLFN7$$!3?+++v%)e)p%FN$\"3)******\\]n#zKFN7$$!3A+++E)f>v %FN$\"3&*******)H`(4KFN7$$!3c******Q[`/[FN$\"3++++\"GS*QJFN7$$!33+++AU Lc[FN$\"3A+++oD%o1$FN7$$!3))*****>Itt!\\FN$\"3++++;8[$*HFN7$$!31+++[1m d\\FN$\"3%)******)o$))=HFN7$$!3,+++')o>2]FN$\"3\")******e:3VGFN7$$!3G+ ++*ytf0&FN$\"37+++r*4hw#FN7$$!3++++5P(R5&FN$\"3))*****\\X0!)o#FN7$$!3/ +++:%p6:&FN$\"3z******eU!)3EFN7$$!3++++QV_(>&FN$\"3.+++b1aGDFN7$$!3V++ +3BvG&FN$\"3?+++8X%\\O#FN7$$!3 $******>Za5L&FN$\"37+++[$e;G#FN7$$!3U+++1G_t`FN$\"3')*****H;,u>#FN7$$! 3!******Ryf[T&FN$\"3/+++P5=7@FN7$$!3?+++JZ*\\X&FN$\"3&)*****fA+g-#FN7$ $!3E+++4e&Q\\&FN$\"3-+++^e&)Q>FN7$$!3P+++Z5PJbFN$\"3)********)3u]=FN7$ $!3'******>=puc&FN$\"3)******zRX;w\"FN7$$!3'********G!3-cFN$\"3!****** H*ybr;FN7$$!3\"******RCO^j&FN$\"31+++joY!e\"FN7$$!3G+++A5dmcFN$\"31+++ UBO)[\"FN7$$!3A+++/4K'p&FN$\"3-+++BvB&R\"FN7$$!3&)*****pfCVs&FN$\"3+++ +s.4,8FN7$$!3G+++AL_]dFN$\"32+++A_#f?\"FN7$$!3k*****R!4'[x&FN$\"3*)*** **\\Ha(46FN7$$!3;+++!)QG(z&FN$\"3*******48*f75FN7$$!3V+++i;uF N7$F_]y$!3&)*****fA+g-#FN7$Fj\\y$!3/+++P5=7@FN7$Fe\\y$!3')*****H;,u>#F N7$F`\\y$!37+++[$e;G#FN7$F[\\y$!3?+++8X%\\O#FN7$Ff[y$!3!)*****RSXsW#FN 7$Fa[y$!3.+++b1aGDFN7$F\\[y$!3z******eU!)3EFN7$Fgjx$!3))*****\\X0!)o#F N7$Fbjx$!37+++r*4hw#FN7$F]jx$!3\")******e:3VGFN7$Fhix$!3%)******)o$))= HFN7$Fcix$!3++++;8[$*HFN7$F^ix$!3A+++oD%o1$FN7$Fihx$!3++++\"GS*QJFN7$F dhx$!3&*******)H`(4KFN7$F_hx$!3)******\\]n#zKFN7$Fjgx$!3!*******olZZLF N7$Fegx$!3*******pV#Q9MFN7$F`gx$!3?+++'f&**zMFN7$F[gx$!3#)*****f:NVa$F N7$Fffx$!3\")*****\\xGug$FN7$Fafx$!39+++yCJpOFN7$F\\fx$!3;+++a..IPFN7$ Fgex$!3'*******GTj*y$FN7$Fbex$!3')*****zd#=[QFN7$F]ex$!3()******)pSd!R FN7$Fhdx$!3)******\\ayB'RFN7$Fcdx$!3:+++*fp\"=SFN7$F^dx$!3I+++S&)=tSFN 7$Ficx$!3$)*****zC3v7%FN7$Fdcx$!3K+++if>\"=%FN7$F_cx$!3#******Hc3VB%FN 7$Fjbx$!3/+++Gr)oG%FN7$Febx$!31+++V6&*QVFN7$F`bx$!3i*****H*H\\!R%FN7$F [bx$!3w*****pht9W%FN7$Ffax$!3'******HH?=\\%FN7$Faax$!3I+++itUTXFN7$F\\ ax$!3m*****REg,f%FN7$Fg`x$!3v*****pgiyj%FN7$Fb`x$!37+++1`O%o%FN7$F]`x$ !3o*****HT'\\HZFN7$Fh_x$!3w*****H?!4tZFN7$Fc_x$!3M+++'R%*\\\"[FN7$F^_x $!3;+++c\\2b[FN7$Fi^x$!3G+++K%=K*[FN7$Fd^x$!31+++JALH\\FN7$F_^x$!3i*** **4QVL'\\FN7$Fj]x$!3I+++7k>&*\\FN7$Fe]x$!3d*****\\%3&[-&FN7$F`]x$!3#)* ****HqyA0&FN7$F[]x$!3!******HIiu2&FN7$Ff\\x$!3k*****p\\#R+^FN7$Fa\\x$! 3*******R]n57&FN7$F\\\\x$!3u*****zS#\\R^FN7$Fg[x$!3=+++t%zc:&FN7$Fb[x$ !39+++c%\\'p^FN7$F][x$!3o******\\UV\"=&FN7$Fhjw$!3M+++193\">&FN7$Fcjw$ !3s*****RYh')>&FN7$F^jw$!3:+++$pzU?&FN7$Fiiw$!32+++&3&43_FN7$Fdiw$!3:+ ++%z^.@&FN7$F_iw$!3K+++FGV6_FN7$Fjhw$!3Q+++Qa&>@&FN7$Fehw$!3w*****>wZH @&FN7$F`hw$!3J+++sq=;_FN7$F[hw$!3U++++(R[A&FN7$Ffgw$!3k*****RwIWC&FN7$ Fagw$!3w*****RhvDG&FN7$F\\gw$!3,+++oejT`FN7$Fgfw$!3%)******y&Q,T&FN7$F bfw$!3S+++Br(\\Z&FN7$F]fw$!3G+++#Qp8`&FN7$Fhew$!3#******\\UJ\"zbFN7$Fc ew$!3/++++%H#>cFN7$F^ew$!3y******)\\(o_cFN7$Fidw$!3?+++lrN!o&FN7$Fddw$ !3C+++(G1Hq&FN7$F_dw$!3(******fN_3s&FN7$Fjcw$!38+++8qfMdFN7$Fecw$!3s** ***4l`Wu&FN7$F`cw$!3?+++,!o1v&FN7$F[cw$!3S+++&>LMv&FN7$Ffbw$!3;+++#z+H v&FN7$Fabw$!3Q+++/*)=\\dFN7$F\\bw$!3P+++S#)QUdFN7$Fgaw$!3/+++_mcKdFN7$ Fbaw$!3\"******4fs(>dFN7$F]aw$!3#)*****\\wPSq&FN7$Fh`w$!3u*****z=z`o&F N7$Fc`w$!3O+++'y+Qm&FN7$F^`w$!3E+++?YHRcFN7$Fi_w$!3G+++'yT=h&FN7$Fd_w$ !3U+++oHT\"e&FN7$F__w$!3')*****pcoza&FN7$Fj^w$!3e*****z@e9^&FN7$Fe^w$! 3?+++Y$>=Z&FN7$F`^w$!31+++\"=u*GaFN7$F[^w$!39+++8U#GQ&FN7$Ff]w$!3d**** **f.CL`FN7$Fa]w$!3#)*****\\)e/!G&FN7$F\\]w$!3.+++fm)HA&FN7$Fg\\w$!3g** ***pyy;;&FN7$Fb\\w$!3E+++')R^&4&FN7$F]\\w$!3J+++hI[B]FN7$Fh[w$!3>+++Gr $Q%\\FN7$Fc[w$!3/+++nmW`[FN7$F^[w$!39+++!eIou%FN7$Fijv$!3h*****RtBkh%F N7$Fdjv$!3K+++e0#*fWFN7$F_jv$!3k*****Hac))G%FN7$Fjiv$!3C+++2#4k6%FN7$F eiv$!3?+++\\GWZRFN7$F`iv$!3/+++r$QAy$FN7$F[iv$!3.+++DY%*>OFN7$Ffhv$!37 +++6CnfMFN7$Fahv$!3,+++7]s+LFN7$$!3X*****>)4MW#)F\\o$!3y*****f,1E9$FN7 $Fggv$!35+++d0)\\)HFN7$$\"3#******\\kxGL#F\\o$!3(******pLLw#GFN7$F]gv$ !3%******z)HVqEFN7$Fhfv$!3!)******RTI8DFN7$Fcfv$!3#)*****>o1iN#FN7$F^f v$!3)******f@@\"*>#FN7$Fiev$!37+++'\\R?/#FN7$Fdev$!35+++\\'e\\)=FN7$F_ ev$!3++++(*z(ys\"FN7$$\"37+++Ol\"y^#FR$!3%******z_(zq:FN7$Fedv$!3#**** ***3tr89FN7$F`dv$!3++++Itjc7FN7$$\"3,+++KBhI5FF$!3)******p_d&*4\"FN7$F fcv$!3m*****pCyZU*F-7$$\"31+++\"Q@C<%F:$!3#)*****pp\")R&yF-7$$\"3c**** *p3#RUZF4$!3g*****pJ&=$G'F-7$F(Fj[pF\\\\pF_\\pF'-Fc\\p6&Fe\\pF($\"#QF` bv$\"#[F`bv-F\\]p6^_l7%F'F_]p7$$!#KFh\\pF(7%F_]pFg]pFdb[l7%Fg]p7$F(F^^ pFdb[l7%Fib[l7$$\"+'*>RUZFar$\"+<`=$G'Fb]pFdb[l7%F[c[l7$$\"+&Q@C<%F-$ \"+(p\")R&yFb]pFdb[l7%Fac[l7$$\"+#=zeS#FN$\"+Z#yZU*Fb]pFdb[l7%Fgc[l7$$ \"+LBhI5F`_p$\"+Fvb*4\"Fi_pFdb[l7%F]d[l7$$\"+uDpJNF`_p$\"+Itjc7Fi_pFdb [l7%Fcd[l7$$\"+pOR95F^`p$\"+4tr89Fi_pFdb[l7%Fid[l7$$\"+Nl\"y^#F^`p$\"+ Gvzq:Fi_pFdb[l7%F_e[l7$$\"+Z&[&4bF^`p$\"+(*z(ys\"Fi_pFdb[l7%Fee[l7$$\" +4!*)f2\"Faap$\"+\\'e\\)=Fi_pFdb[l7%F[f[l7$$\"+Jh(f)=Faap$\"+'\\R?/#Fi _pFdb[l7%Faf[l7$$\"+ig@iHFaap$\"+;77*>#Fi_pFdb[l7%Fgf[l7$$\"+'fQ+7%Faa p$\"+#o1iN#Fi_pFdb[l7%F]g[l7$$\"+\\vs:\\Faap$\"+STI8DFi_pFdb[l7%Fcg[l7 $$\"+_tW7YFaap$\"+))HVqEFi_pFdb[l7%Fig[l7$$\"+Ww(GL#Faap$\"+PLjFGFi_pF db[l7%F_h[l7$$!+'o*z!Q#Faap$\"+d0)\\)HFi_pFdb[l7%Feh[l7$$!+y4MW#)Faap$ \"+;ggUJFi_pFdb[l7%F[i[l7$$!+^&f#y&*Faap$\"+7]s+LFi_pFdb[l7%Fai[l7$$\" +w8uN')Faap$\"+6CnfMFi_pFdb[l7%Fgi[l7$$\"+u'Hn(zFjbp$\"+DY%*>OFi_pFdb[ l7%F]j[l7$$\"+Le,AFFcdp$\"+r$QAy$Fi_pFdb[l7%Fcj[l7$$\"+s1t5sFcdp$\"+\\ GWZRFi_pFdb[l7%Fij[l7$$\"+X%G')o\"F`ep$\"+2#4k6%Fi_pFdb[l7%F_[\\l7$$\" +wbIfOF`ep$\"+Vl&))G%Fi_pFdb[l7%Fe[\\l7$$\"++)fTL(F`ep$\"+e0#*fWFi_pFd b[l7%F[\\\\l7$$\"+:zp&H\"Fb]p$\"+MPU;YFi_pFdb[l7%Fa\\\\l7$$\"+K48S>Fb] p$\"+!eIou%Fi_pFdb[l7%Fg\\\\l7$$\"+)*H&)HDFb]p$\"+nmW`[Fi_pFdb[l7%F]] \\l7$$\"+CI$G,$Fb]p$\"+Gr$Q%\\Fi_pFdb[l7%Fc]\\l7$$\"+X@Y(Q$Fb]p$\"+hI[ B]Fi_pFdb[l7%Fi]\\l7$$\"+`ijlOFb]p$\"+')R^&4&Fi_pFdb[l7%F_^\\l7$$\"+'4 N/'QFb]p$\"+(yy;;&Fi_pFdb[l7%Fe^\\l7$$\"+Z=Z&Fi_ pFdb[l7%Fi`\\l7$$\"+PS]]OFb]p$\"+=#e9^&Fi_pFdb[l7%F_a\\l7$$\"+AELnMFb] p$\"+n&oza&Fi_pFdb[l7%Fea\\l7$$\"+u#**yD$Fb]p$\"+oHT\"e&Fi_pFdb[l7%F[b \\l7$$\"+nS&\\-$Fb]p$\"+'yT=h&Fi_pFdb[l7%Fab\\l7$$\"+$>04x#Fb]p$\"+?YH RcFi_pFdb[l7%Fgb\\l7$$\"+rY'y\\#Fb]p$\"+'y+Qm&Fi_pFdb[l7%F]c\\l7$$\"+c fo2AFb]p$\"+)=z`o&Fi_pFdb[l7%Fcc\\l7$$\"+-[*>!>Fb]p$\"+lx./dFi_pFdb[l7 %Fic\\l7$$\"+pT@#e\"Fb]p$\"+\"fs(>dFi_pFdb[l7%F_d\\l7$$\"+#f$e\\7Fb]p$ \"+_mcKdFi_pFdb[l7%Fed\\l7$$\"+r0t^!*F`ep$\"+S#)QUdFi_pFdb[l7%F[e\\l7$ $\"+&G5*)\\&F`ep$\"+/*)=\\dFi_pFdb[l7%Fae\\l7$$\"+cx&[%=F`ep$\"+#z+Hv& Fi_pFdb[l7%Fge\\l7$$!+Uih/>F`ep$\"+&>LMv&Fi_pFdb[l7%F]f\\l7$$!+()QbXdF `ep$\"+,!o1v&Fi_pFdb[l7%Fcf\\l7$$!+\"f*=w'*F`ep$\"+^OXWdFi_pFdb[l7%Fif \\l7$$!+zFvp8Fb]p$\"+8qfMdFi_pFdb[l7%F_g\\l7$$!+J)H9y\"Fb]p$\"+cB&3s&F i_pFdb[l7%Feg\\l7$$!+TPl.AFb]p$\"+(G1Hq&Fi_pFdb[l7%F[h\\l7$$!+'>D#QEFb ]p$\"+lrN!o&Fi_pFdb[l7%Fah\\l7$$!+uX?)3$Fb]p$\"+*\\(o_cFi_pFdb[l7%Fgh \\l7$$!+YkueNFb]p$\"++%H#>cFi_pFdb[l7%F]i\\l7$$!+K$p'eSFb]p$\"+D98zbFi _pFdb[l7%Fci\\l7$$!+SP^.YFb]p$\"+#Qp8`&Fi_pFdb[l7%Fii\\l7$$!+\\'e9A&Fb ]p$\"+Br(\\Z&Fi_pFdb[l7%F_j\\l7$$!+C'=F'fFb]p$\"+z&Q,T&Fi_pFdb[l7%Fej \\l7$$!+PZ*))*oFb]p$\"+oejT`Fi_pFdb[l7%F[[]l7$$!+u!>b0)Fb]p$\"+9cd#G&F i_pFdb[l7%Fa[]l7$$!+!f;2K*Fb]p$\"+k2VW_Fi_pFdb[l7%Fg[]l7$$!+T(*zb5Fi_p $\"++(R[A&Fi_pFdb[l7%F]\\]l7$$!+s6`r6Fi_p$\"+sq=;_Fi_pFdb[l7%Fc\\]l7$$ !+in1z7Fi_p$\"+ix%H@&Fi_pFdb[l7%Fi\\]l7$$!+[b_z8Fi_p$\"+Qa&>@&Fi_pFdb[ l7%F_]]l7$$!+!*z6u9Fi_p$\"+FGV6_Fi_pFdb[l7%Fe]]l7$$!+l=(Qc\"Fi_p$\"+%z ^.@&Fi_pFdb[l7%F[^]l7$$!+h9h\\;Fi_p$\"+&3&43_Fi_pFdb[l7%Fa^]l7$$!+n6*> t\"Fi_p$\"+$pzU?&Fi_pFdb[l7%Fg^]l7$$!+dL`6=Fi_p$\"+k9m)>&Fi_pFdb[l7%F] _]l7$$!+i)f'))=Fi_p$\"+193\">&Fi_pFdb[l7%Fc_]l7$$!+(=;P'>Fi_p$\"+]UV\" =&Fi_pFdb[l7%Fi_]l7$$!+<)*)p.#Fi_p$\"+c%\\'p^Fi_pFdb[l7%F_`]l7$$!+ySs3 @Fi_p$\"+t%zc:&Fi_pFdb[l7%Fe`]l7$$!+W(G\"z@Fi_p$\"+3C\\R^Fi_pFdb[l7%F[ a]l7$$!+o#)Q[AFi_p$\"+/v1@^Fi_pFdb[l7%Faa]l7$$!+^#omJ#Fi_p$\"+(\\#R+^F i_pFdb[l7%Fga]l7$$!+n07%Q#Fi_p$\"+.BYx]Fi_pFdb[l7%F]b]l7$$!+Lu)3X#Fi_p $\"+.(yA0&Fi_pFdb[l7%Fcb]l7$$!+nX5&*\\Fi_pFdb[l7%F_c]l7$$!+nIT[EFi_p$\"+\"QVL'\\Fi_pFd b[l7%Fec]l7$$!+[)fPr#Fi_p$\"+JALH\\Fi_pFdb[l7%F[d]l7$$!+fw1zFFi_p$\"+K %=K*[Fi_pFdb[l7%Fad]l7$$!+0dXWGFi_p$\"+c\\2b[Fi_pFdb[l7%Fgd]l7$$!+Ba.5 HFi_p$\"+'R%*\\\"[Fi_pFdb[l7%F]e]l7$$!+#*f!f(HFi_p$\"+.-4tZFi_pFdb[l7% Fce]l7$$!+J([@/$Fi_p$\"+8k\\HZFi_pFdb[l7%Fie]l7$$!+J4#)3JFi_p$\"+1`O%o %Fi_pFdb[l7%F_f]l7$$!+g*\\f<$Fi_p$\"+2E'yj%Fi_pFdb[l7%Fef]l7$$!+\\(GNC $Fi_p$\"+k-;!f%Fi_pFdb[l7%F[g]l7$$!+CP^6LFi_p$\"+itUTXFi_pFdb[l7%Fag]l 7$$!+!*f#)zLFi_p$\"+$H?=\\%Fi_pFdb[l7%Fgg]l7$$!+@fN[MFi_p$\"+'Ql$Fi_p$\"+Gr)oG%Fi_pFdb[l7%F_i]l7$ $!+<(R=s$Fi_p$\"+j&3VB%Fi_pFdb[l7%Fei]l7$$!+_(>$*y$Fi_p$\"+if>\"=%Fi_p Fdb[l7%F[j]l7$$!+j5=cQFi_p$\"+[#3v7%Fi_pFdb[l7%Faj]l7$$!+;2KARFi_p$\"+ S&)=tSFi_pFdb[l7%Fgj]l7$$!+=Zl()RFi_p$\"+*fp\"=SFi_pFdb[l7%F][^l7$$!+o v6_SFi_p$\"+X&yB'RFi_pFdb[l7%Fc[^l7$$!+&)4m:TFi_p$\"+*pSd!RFi_pFdb[l7% Fi[^l7$$!+u@DyTFi_p$\"+yD=[QFi_pFdb[l7%F_\\^l7$$!+,>()RUFi_p$\"+HTj*y$ Fi_pFdb[l7%Fe\\^l7$$!+nD^+VFi_p$\"+a..IPFi_pFdb[l7%F[]^l7$$!+ijv%Fi _p$\"+*H`(4KFi_pFdb[l7%F[`^l7$$!+R[`/[Fi_p$\"+\"GS*QJFi_pFdb[l7%Fa`^l7 $$!+AULc[Fi_p$\"+oD%o1$Fi_pFdb[l7%Fg`^l7$$!+-LP2\\Fi_p$\"+;8[$*HFi_pFd b[l7%F]a^l7$$!+[1md\\Fi_p$\"+*o$))=HFi_pFdb[l7%Fca^l7$$!+')o>2]Fi_p$\" +f:3VGFi_pFdb[l7%Fia^l7$$!+*ytf0&Fi_p$\"+r*4hw#Fi_pFdb[l7%F_b^l7$$!+5P (R5&Fi_p$\"+ba+)o#Fi_pFdb[l7%Feb^l7$$!+:%p6:&Fi_p$\"+fU!)3EFi_pFdb[l7% F[c^l7$$!+QV_(>&Fi_p$\"+b1aGDFi_pFdb[l7%Fac^l7$$!+#Fi_pFdb[l7%Fid^l7$$ !+%yf[T&Fi_p$\"+P5=7@Fi_pFdb[l7%F_e^l7$$!+JZ*\\X&Fi_p$\"+E-+E?Fi_pFdb[ l7%Fee^l7$$!+4e&Q\\&Fi_p$\"+^e&)Q>Fi_pFdb[l7%F[f^l7$$!+Z5PJbFi_p$\"+!* 3u]=Fi_pFdb[l7%Faf^l7$$!+#=puc&Fi_p$\"+)RX;w\"Fi_pFdb[l7%Fgf^l7$$!+!H! 3-cFi_p$\"+$*ybr;Fi_pFdb[l7%F]g^l7$$!+Wi8NcFi_p$\"+joY!e\"Fi_pFdb[l7%F cg^l7$$!+A5dmcFi_p$\"+UBO)[\"Fi_pFdb[l7%Fig^l7$$!+/4K'p&Fi_p$\"+BvB&R \"Fi_pFdb[l7%F_h^l7$$!+(fCVs&Fi_p$\"+s.4,8Fi_pFdb[l7%Feh^l7$$!+AL_]dFi _p$\"+A_#f?\"Fi_pFdb[l7%F[i^l7$$!+/4'[x&Fi_p$\"+&Ha(46Fi_pFdb[l7%Fai^l 7$$!+!)QG(z&Fi_p$\"+J\"*f75Fi_pFdb[l7%Fgi^l7$$!+i;uFi_pFdb[l7%Fcb_l7$Ffe^l$!+E-+E?Fi_pFdb[l7% Fgb_l7$F`e^l$!+P5=7@Fi_pFdb[l7%F[c_l7$Fjd^l$!+j6S(>#Fi_pFdb[l7%F_c_l7$ Fdd^l$!+[$e;G#Fi_pFdb[l7%Fcc_l7$F^d^l$!+8X%\\O#Fi_pFdb[l7%Fgc_l7$Fhc^l $!+/aCZCFi_pFdb[l7%F[d_l7$Fbc^l$!+b1aGDFi_pFdb[l7%F_d_l7$F\\c^l$!+fU!) 3EFi_pFdb[l7%Fcd_l7$Ffb^l$!+ba+)o#Fi_pFdb[l7%Fgd_l7$F`b^l$!+r*4hw#Fi_p Fdb[l7%F[e_l7$Fja^l$!+f:3VGFi_pFdb[l7%F_e_l7$Fda^l$!+*o$))=HFi_pFdb[l7 %Fce_l7$F^a^l$!+;8[$*HFi_pFdb[l7%Fge_l7$Fh`^l$!+oD%o1$Fi_pFdb[l7%F[f_l 7$Fb`^l$!+\"GS*QJFi_pFdb[l7%F_f_l7$F\\`^l$!+*H`(4KFi_pFdb[l7%Fcf_l7$Ff _^l$!+0vEzKFi_pFdb[l7%Fgf_l7$F`_^l$!+plZZLFi_pFdb[l7%F[g_l7$Fj^^l$!+PC Q9MFi_pFdb[l7%F_g_l7$Fd^^l$!+'f&**zMFi_pFdb[l7%Fcg_l7$F^^^l$!+c^LWNFi_ pFdb[l7%Fgg_l7$Fh]^l$!+v(Gug$Fi_pFdb[l7%F[h_l7$Fb]^l$!+yCJpOFi_pFdb[l7 %F_h_l7$F\\]^l$!+a..IPFi_pFdb[l7%Fch_l7$Ff\\^l$!+HTj*y$Fi_pFdb[l7%Fgh_ l7$F`\\^l$!+yD=[QFi_pFdb[l7%F[i_l7$Fj[^l$!+*pSd!RFi_pFdb[l7%F_i_l7$Fd[ ^l$!+X&yB'RFi_pFdb[l7%Fci_l7$F^[^l$!+*fp\"=SFi_pFdb[l7%Fgi_l7$Fhj]l$!+ S&)=tSFi_pFdb[l7%F[j_l7$Fbj]l$!+[#3v7%Fi_pFdb[l7%F_j_l7$F\\j]l$!+if>\" =%Fi_pFdb[l7%Fcj_l7$Ffi]l$!+j&3VB%Fi_pFdb[l7%Fgj_l7$F`i]l$!+Gr)oG%Fi_p Fdb[l7%F[[`l7$Fjh]l$!+V6&*QVFi_pFdb[l7%F_[`l7$Fdh]l$!+$*H\\!R%Fi_pFdb[ l7%Fc[`l7$F^h]l$!+&*\\Fi_pFdb[l7%F[_`l7$Fjb]l$!+X3&[-&Fi_pFdb [l7%F__`l7$Fdb]l$!+.(yA0&Fi_pFdb[l7%Fc_`l7$F^b]l$!+.BYx]Fi_pFdb[l7%Fg_ `l7$Fha]l$!+(\\#R+^Fi_pFdb[l7%F[``l7$Fba]l$!+/v1@^Fi_pFdb[l7%F_``l7$F \\a]l$!+3C\\R^Fi_pFdb[l7%Fc``l7$Ff`]l$!+t%zc:&Fi_pFdb[l7%Fg``l7$F``]l$ !+c%\\'p^Fi_pFdb[l7%F[a`l7$Fj_]l$!+]UV\"=&Fi_pFdb[l7%F_a`l7$Fd_]l$!+19 3\">&Fi_pFdb[l7%Fca`l7$F^_]l$!+k9m)>&Fi_pFdb[l7%Fga`l7$Fh^]l$!+$pzU?&F i_pFdb[l7%F[b`l7$Fb^]l$!+&3&43_Fi_pFdb[l7%F_b`l7$F\\^]l$!+%z^.@&Fi_pFd b[l7%Fcb`l7$Ff]]l$!+FGV6_Fi_pFdb[l7%Fgb`l7$F`]]l$!+Qa&>@&Fi_pFdb[l7%F[ c`l7$Fj\\]l$!+ix%H@&Fi_pFdb[l7%F_c`l7$Fd\\]l$!+sq=;_Fi_pFdb[l7%Fcc`l7$ F^\\]l$!++(R[A&Fi_pFdb[l7%Fgc`l7$Fh[]l$!+k2VW_Fi_pFdb[l7%F[d`l7$Fb[]l$ !+9cd#G&Fi_pFdb[l7%F_d`l7$F\\[]l$!+oejT`Fi_pFdb[l7%Fcd`l7$Ffj\\l$!+z&Q ,T&Fi_pFdb[l7%Fgd`l7$F`j\\l$!+Br(\\Z&Fi_pFdb[l7%F[e`l7$Fji\\l$!+#Qp8`& Fi_pFdb[l7%F_e`l7$Fdi\\l$!+D98zbFi_pFdb[l7%Fce`l7$F^i\\l$!++%H#>cFi_pF db[l7%Fge`l7$Fhh\\l$!+*\\(o_cFi_pFdb[l7%F[f`l7$Fbh\\l$!+lrN!o&Fi_pFdb[ l7%F_f`l7$F\\h\\l$!+(G1Hq&Fi_pFdb[l7%Fcf`l7$Ffg\\l$!+cB&3s&Fi_pFdb[l7% Fgf`l7$F`g\\l$!+8qfMdFi_pFdb[l7%F[g`l7$Fjf\\l$!+^OXWdFi_pFdb[l7%F_g`l7 $Fdf\\l$!+,!o1v&Fi_pFdb[l7%Fcg`l7$F^f\\l$!+&>LMv&Fi_pFdb[l7%Fgg`l7$Fhe \\l$!+#z+Hv&Fi_pFdb[l7%F[h`l7$Fbe\\l$!+/*)=\\dFi_pFdb[l7%F_h`l7$F\\e\\ l$!+S#)QUdFi_pFdb[l7%Fch`l7$Ffd\\l$!+_mcKdFi_pFdb[l7%Fgh`l7$F`d\\l$!+ \"fs(>dFi_pFdb[l7%F[i`l7$Fjc\\l$!+lx./dFi_pFdb[l7%F_i`l7$Fdc\\l$!+)=z` o&Fi_pFdb[l7%Fci`l7$F^c\\l$!+'y+Qm&Fi_pFdb[l7%Fgi`l7$Fhb\\l$!+?YHRcFi_ pFdb[l7%F[j`l7$Fbb\\l$!+'yT=h&Fi_pFdb[l7%F_j`l7$F\\b\\l$!+oHT\"e&Fi_pF db[l7%Fcj`l7$Ffa\\l$!+n&oza&Fi_pFdb[l7%Fgj`l7$F`a\\l$!+=#e9^&Fi_pFdb[l 7%F[[al7$Fj`\\l$!+Y$>=Z&Fi_pFdb[l7%F_[al7$Fd`\\l$!+\"=u*GaFi_pFdb[l7%F c[al7$F^`\\l$!+8U#GQ&Fi_pFdb[l7%Fg[al7$Fh_\\l$!+g.CL`Fi_pFdb[l7%F[\\al 7$Fb_\\l$!+&)e/!G&Fi_pFdb[l7%F_\\al7$F\\_\\l$!+fm)HA&Fi_pFdb[l7%Fc\\al 7$Ff^\\l$!+(yy;;&Fi_pFdb[l7%Fg\\al7$F`^\\l$!+')R^&4&Fi_pFdb[l7%F[]al7$ Fj]\\l$!+hI[B]Fi_pFdb[l7%F_]al7$Fd]\\l$!+Gr$Q%\\Fi_pFdb[l7%Fc]al7$F^] \\l$!+nmW`[Fi_pFdb[l7%Fg]al7$Fh\\\\l$!+!eIou%Fi_pFdb[l7%F[^al7$Fb\\\\l $!+MPU;YFi_pFdb[l7%F_^al7$F\\\\\\l$!+e0#*fWFi_pFdb[l7%Fc^al7$Ff[\\l$!+ Vl&))G%Fi_pFdb[l7%Fg^al7$F`[\\l$!+2#4k6%Fi_pFdb[l7%F[_al7$Fjj[l$!+\\GW ZRFi_pFdb[l7%F__al7$Fdj[l$!+r$QAy$Fi_pFdb[l7%Fc_al7$F^j[l$!+DY%*>OFi_p Fdb[l7%Fg_al7$Fhi[l$!+6CnfMFi_pFdb[l7%F[`al7$Fbi[l$!+7]s+LFi_pFdb[l7%F _`al7$$!+#)4MW#)Faap$!+;ggUJFi_pFdb[l7%Fc`al7$Ffh[l$!+d0)\\)HFi_pFdb[l 7%Fi`al7$$\"+Xw(GL#Faap$!+PLjFGFi_pFdb[l7%F]aal7$Fjg[l$!+))HVqEFi_pFdb [l7%Fcaal7$Fdg[l$!+STI8DFi_pFdb[l7%Fgaal7$F^g[l$!+#o1iN#Fi_pFdb[l7%F[b al7$Fhf[l$!+;77*>#Fi_pFdb[l7%F_bal7$Fbf[l$!+'\\R?/#Fi_pFdb[l7%Fcbal7$F \\f[l$!+\\'e\\)=Fi_pFdb[l7%Fgbal7$Ffe[l$!+(*z(ys\"Fi_pFdb[l7%F[cal7$$ \"+Ol\"y^#F^`p$!+Gvzq:Fi_pFdb[l7%F_cal7$Fjd[l$!+4tr89Fi_pFdb[l7%Fecal7 $Fdd[l$!+Itjc7Fi_pFdb[l7%Fical7$$\"+KBhI5F`_p$!+Fvb*4\"Fi_pFdb[l7%F]da l7$Fhc[l$!+Z#yZU*Fb]pFdb[l7%Fcdal7$$\"+\"Q@C<%F-$!+(p\")R&yFb]pFdb[l7% Fgdal7$$\"+(3#RUZFar$!+<`=$G'Fb]pFdb[l7%F]eal7$F(FaavFdb[l7%FcealFdavF db[l7%FdavFhavFdb[l7%FhavF'Fdb[l-Fc\\p6&Fe\\pF(Fabv$\"#&*F`bvFcbv-F$6% 7$7$$!3o*************)oFNF(7$$\"3%**************=\"FNF(-%'COLOURG6&Fe \\pF)F)F)-%*LINESTYLEG6#Fg\\p-F$6%7$7$F($!3o*************Q'FN7$F($\"3o *************Q'FNFdfalFgfal-%+AXESLABELSG6%%&Re(z)G%&Im(z)G-%%FONTG6#% (DEFAULTG-Figal6$%*HELVETICAG\"\"*-%*AXESSTYLEG6#%$BOXG-%(SCALINGG6#%, CONSTRAINEDG-%%VIEWG6$;$!$*oF`bv$\"$>\"F`bv;$!$R'F`bv$\"$R'F`bv" 1 2 0 1 10 0 2 9 1 2 1 1.000000 45.000000 45.000000 0 0 "Curve 1" "Curve 2 " "Curve 3" "Curve 4" "Curve 5" "Curve 6" }}}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}}{PARA 0 "" 0 "" {TEXT -1 52 "#--------------------------------------------------- " }}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 35 "coefficients of the combined \+ scheme" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 36 "coefficient s of the combined scheme " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 7144 "ee := \{c[2]=9/142,\nc[3]=24514/2 38491,\nc[4]=12257/79497,\nc[5]=50/129,\nc[6]=34/73,\nc[7]=23/148,\nc[ 8]=142/141,\nc[9]=29104120198/33218531707,\nc[10]=83/91,\nc[11]=143/14 9,\nc[12]=1,\nc[13]=1,\n\na[2,1]=9/142,\na[3,1]=9950845450/51190161372 9,\na[3,2]=42666469916/511901613729,\na[4,1]=12257/317988,\na[4,2]=0, \na[4,3]=12257/105996,\na[5,1]=14131686489425/35833975601529,\na[5,2]= 0,\na[5,3]=-17700454220625/11944658533843,\na[5,4]=17619604667500/1194 4658533843,\na[6,1]=4418011/96055225,\na[6,2]=0,\na[6,3]=0,\na[6,4]=97 57757988/41995818067,\na[6,5]=19921512441/106300094275,\na[7,1]=480285 555889619/7997789253816320,\na[7,2]=0,\na[7,3]=0,\na[7,4]=521621065566 96460538643/464887033284472899365888,\na[7,5]=-16450769949384489/49000 9784398315520,\na[7,6]=864813381633887/51718297665732608,\na[8,1]=-700 1048352088587137143/4449830351030385148800,\na[8,2]=0,\na[8,3]=0,\na[8 ,4]=-2522765548294599044197935933/1915963195493758447677901792,\na[8,5 ]=-318556182235222634647116091/27172411532484037214054400,\na[8,6]=120 5563885850790193966807/132365727505912221222912,\na[8,7]=254/39,\na[9, 1]=-206293963997166891228011792644285393944628558742266044635547670708 45753369/\n 428817596627701559566575134700121144880760179811281 05307375594963152353200,\na[9,2]=0,\na[9,3]=0,\na[9,4]=-33154430743436 59404779422149387397712986453181141168247590906370819301077749322753/ \n 498517112641608872807821838566847987729514312398278633788185 835348997643218553476,\na[9,5]=-27374940941165406028694814116482845210 9898379203526945684314474186724062841643/\n 6042758395137755250 3967840653825589117167443021628367691773162913607984889600,\na[9,6]=16 6563725188745127382680605043099249004376722636092450288092298657383277 31797537/\n 4276990138533930522782076385771016009097627930550149 628282203007913538394549504,\na[9,7]=420080800333543055908043229440848 05264441066760038302359736803632/\n 4865302423216534910074823287 811605599629170295030631799935804001,\na[9,8]=668459780930716338000066 627236927417191947396177093524377824/\n 711004529488846437797990 87713322002499799983434685642170251833,\na[10,1]=-17936039463222609008 28212460706877142477132870159527/\n 23130975685119907537816497 19556084665131024900300800,\na[10,2]=0,\na[10,3]=0,\na[10,4]=-14776874 123722838192315406145167687512425345723701/\n 1847893530366076 701102014146927696206329050105856, a[10,5]=-19587020919884661714856757 105130246995757906603/\n 2911893296942532038566182868170393629 789388800,\na[10,6]=63643808632590716771122592364555064774176997806133 00364807/\n 115042817458494213340657909154944343881498834918839 4057728,\na[10,7]=27725164402569748756040320433848245155581006369/\n \+ 2544159473655547770881695354241256106302348256,\na[10,8]=1074424 7163960019876833255044784609639/53476180473990134882549194776830450329 6,\na[10,9]=-509777379307928082322044174972489793998782172800111031978 62899/\n 13009156945649136756132803140818373586449643931913379 94183389184,\na[11,1]=-35876257170689524872144934419668970487370507558 12600710793/\n 30157331640332296247720060864294676850466399837 06974412800,\na[11,2]=0,\na[11,3]=0,\na[11,4]=-54530117112678047312115 01837262816944661201619903/\n 76497332089971639707244864471073 3101329290196992,\na[11,5]=-884348836774584715070440485633026464653487 653/\n 92725983515963799584249219499787659014963200,\na[11,6]= 26823469063654084387375587616552322383082061411417182757389742951/\n \+ 3541299744763681675473620647087123057228744296123642301654630400 , a[11,7]=142363419491686507162007051071007722765323162710521029/\n \+ 12634887202368261565807449771335728082273587905775840,\na[11,8]=6 4747617454909275289531520412519442831235890581/\n 1269317188117 975960996670628974453443777854830080,\na[11,9]=11263380825327272097987 4303367503891597499261046700689572459050065039333987335667/\n 1 4045142912450345328121813771190345010140398305182184650645792916115633 74608650240,\na[11,10]=-10612202518573994431153697720606405883/6708254 6658259846778754594976831647575,\na[12,1]=-753408116554498247829620233 5922049210803875045423/\n 192195756654408487560745980516584163 87002479820800,\na[12,2]=0,\na[12,3]=0,\na[12,4]=237696087452786717802 270375283034262859273455/\n 60688480889936839261131818793519097 177034752,\na[12,5]=-20610578209826329263318986584876108069323/7356333 776438834884293014575840932659200,\na[12,6]=51260471529841028040709654 458903254781320136131844164563/\n 20998023776318546907382302106 788168158765514131585630208,\na[12,7]=-3077214437173472971196810795615 384000211457151011/\n 1272435592582280820597059432060116893200 684680384,\na[12,8]=-1539218116260541896259682954580256454049/45346708 30750360983422999946409310393344,\na[12,9]=241886539350268429372116296 787271276553970618941104594460614948326132797451456131/\n 12406 6963266246589252891612004112305105402628318832478010155534534366231609 0597376,\na[12,10]=-80556486832245966191717452425924975/41444540951867 6597565032008051106461,\na[12,11]=294478168087450034759414279281446335 0/4965161383073676983610218096030654529,\na[13,1]=-7757739937862944832 927743694336116203639371542761/\n 5225100678421794325654850845 451340473577260032000,\na[13,2]=0,\na[13,3]=0,\na[13,4]=-4338895460095 21405913741133329446636837810749/1814887961156430016213106911693222333 46938880,\na[13,5]=-246044720162308748108107126829066792329071/2199910 3311417807170100413255475150848000,\na[13,6]=2140331235425829844389060 818616719848637810765257179167/\n 24542818203601189763802825281 5369258646052549878743040,\na[13,7]=1573990926219809229258666534611598 771063240529/214535514317495538101650508596881057760818080,\na[13,8]=6 2408280667309375445301959066100433563/48383200462519838495123555593274 51645440,\na[13,9]=114560982224949367761811372535950664299815320560322 6883141207089968379/\n 2690280216682276847636742784083574313955 9294105398677975568538466119680,\na[13,10]=-40895035687587468313908967 8053832/7674292714443204070455739595109785,\na[13,11]=0,\na[13,12]=0, \n\nb[1]=55038446513529253801/1239280570055853383520,\nb[2]=0,\nb[3]=0 ,\nb[4]=0,\nb[5]=0,\nb[6]=2335496795323782464411846394611/659836878329 4020109895379936256,\nb[7]=7636073376527143565375240869888/30725949199 261296642046754748645,\nb[8]=-4237087214169934312729607487/12735791394 214625116604076160,\nb[9]=40850529129113324176099551412198433591436392 7884426780078325258228227984174126699/\n 2126248746121936974666551 59405635202123821166475348052734199905546455860773575680,\nb[10]=-1108 225296327029096435947/405679075893979729103310,\nb[11]=246098829120621 3825688467985/1756342789520947764222671739,\nb[12]=4808707937311/50545 545388065,\nb[13]=0,\n\n`b*`[1]=715953338020208413/1609455285786822576 0,\n`b*`[2]=0,\n`b*`[3]=0,\n`b*`[4]=0,\n`b*`[5]=0,\n`b*`[6]=6228433516 2928966987066121/175437206755843240272669696,\n`b*`[7]=184309146777472 302831695872/742417767522160213324368465,\n`b*`[8]=-509771598215811385 123057257/210282269969085698264101760,\n`b*`[9]=2701602489646143640362 891402924962500766379885231830470264478480621389/\n 16885752692 94196295712420798364925687791337062814161130291144634516480,\n`b*`[10] =-7218534073012286740367561/3986456306153224954098780,\n`b*`[11]=0,\n` b*`[12]=0,\n`b*`[13]=5752173075461/1925544586212\}:" }}}{PARA 0 "" 0 " " {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}} {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT 259 5 "nodes" } {TEXT -1 2 ": " }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 32 "seq(c[i]=s ubs(ee,c[i]),i=2..13);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6./&%\"cG6#\" \"##\"\"*\"$U\"/&F%6#\"\"$#\"&9X#\"'\"\\Q#/&F%6#\"\"%#\"&dA\"\"&(\\z/& F%6#\"\"&#\"#]\"$H\"/&F%6#\"\"'#\"#M\"#t/&F%6#\"\"(#\"#B\"$[\"/&F%6#\" \")#F*\"$T\"/&F%6#F)#\",)>?T5H\",2<`=K$/&F%6#\"#5#\"#$)\"#\"*/&F%6#\"# 6#\"$V\"\"$\\\"/&F%6#\"#7\"\"\"/&F%6#\"#8Ffo" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT 259 20 "linking coefficients" } {TEXT -1 1 ":" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 50 "seq(seq(a[i ,j]=subs(ee,a[i,j]),j=1..i-1),i=1..13);" }}{PARA 12 "" 1 "" {XPPMATH 20 "6jo/&%\"aG6$\"\"#\"\"\"#\"\"*\"$U\"/&F%6$\"\"$F(#\"+]a%3&**\"-HPh, >^/&F%6$F/F'#\",;*pkmUF2/&F%6$\"\"%F(#\"&dA\"\"'))zJ/&F%6$F;F'\"\"!/&F %6$F;F/#F=\"''*f5/&F%6$\"\"&F(#\"/D%*['oJT\"\"/H:gvR$e$/&F%6$FKF'FB/&F %6$FKF/#!/D1Aa/q<\"/VQ`eY%>\"/&F%6$FKF;#\"/+vm/'>w\"FW/&F%6$\"\"'F(#\" (6!=W\")D_0'*/&F%6$FjnF'FB/&F%6$FjnF/FB/&F%6$FjnF;#\"+))zvd(*\",n!=e*> %/&F%6$FjnFK#\",TC^@*>\"-vU4+j5/&F%6$\"\"(F(#\"0>'*)eb&G![\"1?j\"QD*y( *z/&F%6$FcpF'FB/&F%6$FcpF/FB/&F%6$FcpF;#\"8V'Q0Y'pcl5i@&\"9))eO**GZ%GL q)[Y/&F%6$FcpFK#!2*[%Q\\*p2X;\"3?bJ)R%y4+\\/&F%6$FcpFjn#\"0()Qj\"Q8[') \"23EtlwH=<&/&F%6$\"\")F(#!7Vr8(e)3_$[5+(\"7+)[^QI5NI)\\W/&F%6$FbrF'FB /&F%6$FbrF/FB/&F%6$FbrF;#!=Lf$z>W!*f%H[bwAD\"=#z,znZ%eP\\&>jf\">/&F%6$ FbrFK#!<\"4;rkMEANA=c&=$\";+W09s.%[K:Tsr#/&F%6$FbrFjn#\":2o'R>!z]e)Qc0 7\"97HA@A\"f]FdOK\"/&F%6$FbrFcp#\"$a#\"#R/&F%6$F*F(#!eopLvX32nZbjWgEU( e&GY%RR&GWEz6!G7*o;(*R'RH1#\"eo+KN_J'\\fvtI0\"G6)z,w!)[9@,qM^dm&f:qFmf <)G%/&F%6$F*F'FB/&F%6$F*F/FB/&F%6$F*F;#!]p`FK\\x2,$>3P14fZ#o69\"=`k)Hr (R(Q\\@UzZSfOMuIW:L\"\\pwMb=Kk(**[`$e=)yL'y#)R7V^Hx)z%ocQ=#y!G()3;k7r^ )\\/&F%6$F*FK#!ioV;%G1Cn=uWJ%oXp_.#z$)*)4@XG[;T\"[pGgSl6%4%\\PF\"ho+'* )[)zg8H;t>(\\;y`2*>^ov48B/&F% 6$FawF'FB/&F%6$FawF/FB/&F%6$FawF;#!S,PsX`U7von^91aJ#>QGsBT(ox9\"Rce5]! Hj?'pFp99?5,n2m.`$*y%=/&F%6$FawFK#!P.m!zv&*pCI^5dn&[rhY))>4-(e>\"O+))Q *yHORq\"oG=m&Q?`UpH$*=\"H/&F%6$FawFjn#\"en2[O+Lh!y*p+'R;ZUu5\"H'H.XIox%>\\D)[8!*RZ!=wM&/&F%6$FawF*#!in**G'y>.6,!G <#y)*Rz*[s\\$Rk\\ket$=39.G8cn8\\c%p:4I \"/&F%6$\"#6F(#!en$z52g7ev]qt[q*o'>W$\\9s[_*oqrDwe$\"en+GTupq$)*Rm/&on %Hk31?xC'HK.kJt:I/&F%6$FbzF'FB/&F%6$FbzF/FB/&F%6$FbzF;#!R.*>;?hY%p\"GE P=]67t/yE6<,`a\"Q#*p>!HH85L2rW'[C2(R;(**3Kt\\w/&F%6$FbzFK#!N`w[`YYEIj& [S/2:Zeun$)[V))\"M+K'\\,fwy*\\>#\\Ue*zjf^$)fs#*/&F%6$FbzFjn#\"\\o^Hu*Q dF=<9Th?3$QABb;wevtQ%3aO1pM#o#\"[o+/ja;IUO7'HW(Gs0Br3Z1itan\"ojZu*HTN/ &F%6$FbzFcp#\"WH5_5F;B`wAx+r502?;2lo\"\\>MOU\"\"VSex0zetA3GdLr(\\u!ech #oB?()[j7/&F%6$FbzFbr#\"P\"e!*eBJGW>DT?:`*Gv#4\\XrP\"=7G`M]C\"H9XS\"/&F%6$ FbzFaw#!G$)eS11s(p`6V%*Rd=D?71\"\"GvvkJo(\\fa(yn%)f#ema#3n/&F%6$\"#7F( #!RBa/vQ!3@\\?#fL-iHyC)\\al63Mv\"S+3#)zC+(Q;%e;0)fugv[3Wlcd>#>/&F%6$Fi ]lF'FB/&F%6$Fi]lF/FB/&F%6$Fi]lF;#\"NbMFfGEMIGv.F-yr'y_u3'pP#\"M_Z.xr4> Nz==8h#Ro$**)3[)og/&F%6$Fi]lFK#!JB$p!3h([e')*=LEHj#)4#y0h?\"I+#fE$4%ed 9IH%)[$)QkxLjN(/&F%6$Fi]lFjn#\"YjX;W=8O,K\"yaK!*eWl42/G5%)H:Zg7&\"Y3-j &eJT^l(e\"o\")y1@I#Q2pa=jxB!)*4#/&F%6$Fi]lFcp#!R65:d9@+SQ:cz5o>rHZtrV9 s2$\"R%Q!o%o+K*o6g?Vfqf?3G#e#fNCF\"/&F%6$Fi]lFbr#!I\\SXc-eaHofi*=agi6= #R:\"IWLR5$4k%***HU$)4O]2$3nMX/&F%6$Fi]lF*#\"\\pJhX^uzKhK[\\hgWf/6%*=1 (Rbw7F(y'H;@PH%o-NRl)=C\"]pwtf!4;BmV`Mb:5!yC$)=$GES0^I7T+7;*GD*eYiEjp1 C\"/&F%6$Fi]lFaw#!Dv\\#fU_ur\">mfCKo[c0)\"Ehk5^!3?.lvfw'=&4aW9%/&F%6$F i]lFbz#\"F]LY9GzUTfZ.]u3o\"yWH\"FHXlIg4=-h$)pntIQh^'\\/&F%6$\"#8F(#!Rh Far$RO?;hL%pVx#H$[%H'y$*Rxv(\"R+?.gsdt/M^a%3&[lDVz@%y15D_/&F%6$FfalF'F B/&F%6$FfalF/FB/&F%6$FfalF;#!N\\2\"y$ojY%HL8TP\"fS@&4ga*)QV\"N!))QpMLA Kp6p58i,Ik:hz)[\"=/&F%6$FfalFK#!Kr!HBzm!Ho72\"3\"[(3B;?Z/Y#\"J+![3:vaD 8/5qr!yT6L5**>#/&F%6$FfalFjn#\"Xn\"zrDl2\"yj[)>nh=31*QW)HeUN7LS@\"WSIu y)\\D0Y'e#p`\"GDG!Qw*=,O?=GaC/&F%6$FfalFcp#\"OH0Cj5x)f6Y`m'e#H#4)>i#4* Rd\"\"N!3=3wd5)of30l,\"Qb\\IXaP4tm!G3C' \"ISak^uKfbN7&\\Q)>DY+KQ[/&F%6$FfalF*#\"aoz$o**3279$)oA.c?`\")*Hk1&f`s 8\"=wn$\\\\A#)4c9\"\"bo!o>hYQ&ob(zn)R0THf&RJuN3%yUnjZoF#o;-G!p#/&F%6$F falFaw#!BKQ0y'*3RJoue(oN]*3%\"C&y4^fRdXqS?VWr#Huw/&F%6$FfalFbzFB/&F%6$ FfalFi]lFB" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT 259 39 "weights for the 13 stage order 8 scheme" }{TEXT -1 1 ":" }} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 32 "seq(b[i]=subs(ee,b[i]),i=1.. 13);" }}{PARA 12 "" 1 "" {XPPMATH 20 "6//&%\"bG6#\"\"\"#\"5,QDHN^Y%Q]& \"7?NQ`e0q0GR7/&F%6#\"\"#\"\"!/&F%6#\"\"$F//&F%6#\"\"%F//&F%6#\"\"&F// &F%6#\"\"'#\"@6YRY=TkCyB`z'\\NB\"@ci$*z`*)4,-%H$yo$)f'/&F%6#\"\"(#\"@) ))p3Cv`cVr_wL2Ow\"AX'[ZvY?k'Hh#*>\\fsI/&F%6#\"\")#!=([2'HFJM*pT@(3PU\" >gh2/m6DY@%R\"zNF\"/&F%6#\"\"*#\"\\p*pET<%)zAG#e_Ky+yEW)y#RO9fL%)>79b* 4wTK8\"H\"H0&3%\"\\p!ovNxgeXYb!**>MF0[`Zm6#Q7-_j0%f^lmup$>7Y([i7#/&F%6 #\"#5#!:ZfV'4HqK'HD#36\"95L5H(zR*e2zcS/&F%6#\"#6#\"=&)zY)oDQ@17H))4Y# \"=R \+ " 0 "" {MPLTEXT 1 0 38 "seq(`b*`[i]=subs(ee,`b*`[i]),i=1..13);" }} {PARA 12 "" 1 "" {XPPMATH 20 "6//&%#b*G6#\"\"\"#\"38%3--QL&fr\"5gdAoy& Gb%4;/&F%6#\"\"#\"\"!/&F%6#\"\"$F//&F%6#\"\"%F//&F%6#\"\"&F//&F%6#\"\" '#\";@h1()p'*GH;NVGi\"<'ppEFSK%ev1sVv\"/&F%6#\"\"(#\"%Hp_d)o\"/&F%6#\"#5#!:hvOSnG7I2M&=s\":!y)4a\\A`hIc k)R/&F%6#\"#6F//&F%6#\"#7F//&F%6#\"#8#\".ha2t@v&\".7ieWb#>" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}}{PARA 0 "" 0 "" {TEXT -1 52 "#------------------------------------ ---------------" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}} {PARA 0 "" 0 "" {TEXT -1 46 "#======================================== =====" }}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 49 "a modification of the Ts itouras-Papakostas scheme" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 " " 0 "" {TEXT -1 107 "This scheme is a modification of the scheme of Ts itouras and Papakostas considered in the previous section." }}{PARA 0 "" 0 "" {TEXT -1 114 "It has similar charactersitics to those of the T sitouras-Papakostas scheme but all the nodes are in the interval " } {XPPEDIT 18 0 "[0,1]" "6#7$\"\"!\"\"\"" }{TEXT -1 2 ". " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 15 "The nodes are: " }} {PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "c[2] = 2/25;" "6#/&% \"cG6#\"\"#*&F'\"\"\"\"#D!\"\"" }{TEXT -1 3 ", " }{XPPEDIT 18 0 "c[3] = 24514/238491;" "6#/&%\"cG6#\"\"$*&\"&9X#\"\"\"\"'\"\\Q#!\"\"" } {TEXT -1 3 ", " }{XPPEDIT 18 0 "c[4] = 12257/79497;" "6#/&%\"cG6#\"\" %*&\"&dA\"\"\"\"\"&(\\z!\"\"" }{TEXT -1 3 ", " }{XPPEDIT 18 0 "c[5] = 50/129;" "6#/&%\"cG6#\"\"&*&\"#]\"\"\"\"$H\"!\"\"" }{TEXT -1 3 ", " }{XPPEDIT 18 0 "c[6] = 34/73;" "6#/&%\"cG6#\"\"'*&\"#M\"\"\"\"#t!\"\" " }{TEXT -1 3 ", " }{XPPEDIT 18 0 "c[7] = 23/148;" "6#/&%\"cG6#\"\"(* &\"#B\"\"\"\"$[\"!\"\"" }{TEXT -1 3 ", " }{XPPEDIT 18 0 "c[8] = 196/1 97;" "6#/&%\"cG6#\"\")*&\"$'>\"\"\"\"$(>!\"\"" }{TEXT -1 3 ", " } {XPPEDIT 18 0 "c[9] = 14118145156/16038718811" "6#/&%\"cG6#\"\"**&\",c ^9=T\"\"\"\"\",6)=(Qg\"!\"\"" }{TEXT -1 3 ", " }{XPPEDIT 18 0 "c[10] \+ = 39/43;" "6#/&%\"cG6#\"#5*&\"#R\"\"\"\"#V!\"\"" }{TEXT -1 3 ", " } {XPPEDIT 18 0 "c[11] = 20/21;" "6#/&%\"cG6#\"#6*&\"#?\"\"\"\"#@!\"\"" }{TEXT -1 3 ", " }{XPPEDIT 18 0 "c[12] = 1" "6#/&%\"cG6#\"#7\"\"\"" } {TEXT -1 3 ", " }{XPPEDIT 18 0 "c[13]=1" "6#/&%\"cG6#\"#8\"\"\"" } {TEXT -1 2 ". " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}{PARA 0 "" 0 "" {TEXT -1 52 "#------------- --------------------------------------" }}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 28 "checking the combined scheme" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 36 "coefficients of the combined scheme " }}{PARA 0 " " 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 6866 "ee \+ := \{c[2]=2/25,\nc[3]=24514/238491,\nc[4]=12257/79497,\nc[5]=50/129,\n c[6]=34/73,\nc[7]=23/148,\nc[8]=196/197,\nc[9]=14118145156/16038718811 ,\nc[10]=39/43,\nc[11]=20/21,\nc[12]=1,\nc[13]=1,\n\na[2,1]=2/25,\na[3 ,1]=2090517149/56877957081,\na[3,2]=3755851225/56877957081,\na[4,1]=12 257/317988,\na[4,2]=0,\na[4,3]=12257/105996,\na[5,1]=14131686489425/35 833975601529,\na[5,2]=0,\na[5,3]=-17700454220625/11944658533843,\na[5, 4]=17619604667500/11944658533843,\na[6,1]=4418011/96055225,\na[6,2]=0, \na[6,3]=0,\na[6,4]=9757757988/41995818067,\na[6,5]=19921512441/106300 094275,\na[7,1]=480285555889619/7997789253816320,\na[7,2]=0,\na[7,3]=0 ,\na[7,4]=52162106556696460538643/464887033284472899365888,\na[7,5]=-1 6450769949384489/490009784398315520,\na[7,6]=864813381633887/517182976 65732608,\na[8,1]=-41924855046865190259777/29067997485765047436800,\na [8,2]=0,\na[8,3]=0,\na[8,4]=-1573638249639795538408923368457/337926761 654282072170175876224,\na[8,5]=-3144752379896908085076884133/286731750 405790169489612800,\na[8,6]=51820007951705750121680295/605264567764177 0902708224,\na[8,7]=19/2,\na[9,1]=-14415222609369643343604689319116750 509914398656839842108465299109803139261/\n 28666460834013981370 272729820284862049438129018451533980749016707010746880,\na[9,2]=0,\na[ 9,3]=0,\na[9,4]=-52241674673322800280629263356533673805112659910840603 9793785642181694795272896367/\n 7244755263234658466960959547766 4264037757415889555156446859013382512921014002304,\na[9,5]=-8249597526 146667555031510474907199042876873605462323913470411629578679723249/\n \+ 175634113965120890123649800888811453867971790089262494990757048 7472295239680,\na[9,6]=24956600980553216370236458050032746816720224485 74640720517109843115480217671807/\n 6215583383437936975613964725 36427684004319023110349698478188147786343185391616,\na[9,7]=6592677222 2165868913654906182786964314250556615077953597292687341/\n 71147 45707172215165239946022376277735416754491161939984260041034,\na[9,8]=3 142544284003094691478527859401247056878071594901520381912/\n 322 898507178552018028498957174197954770661454622943631853501,\na[10,1]=-9 867878858058255039217746719915871073167304993320293/\n 1377100 6297995217364330134342641992580302241127219200,\na[10,2]=0,\na[10,3]=0 ,\na[10,4]=-502106193116073366202094897549565186098556680379/\n \+ 59760431150000217012599575938528786496649056256,\na[10,5]=-331923915 046080335870716315310401046907/52747587227799808210222170585805619200, \na[10,6]=1643147679523140016971844191613351711631376872651547306145/ \n 315813057849945223645118819513361571639686784746580082688,\n a[10,7]=2224286847389514767028684777161157168782986309/199937022215481 792634238811967979191208436288,\na[10,8]=49853649680049123712136863581 080955445/2729660625860504418029771633420026504704,\na[10,9]=-12037479 16407015470833526610982521566345739350276392522818995/\n 43146 504717622211052116150230647115135945640260504121035832832,\na[11,1]=-1 677150560435164751643947644188299249566198916438951/\n 1524268 586148523692143823046819588641870992602157568,\na[11,2]=0,\na[11,3]=0, \na[11,4]=-2302011554101675211827716537011959670051291/274508714084612 705166461135767825539832704,\na[11,5]=-1688914466951040738709921153124 3899609/1891391571300777047888087779210764288,\na[11,6]=39709960976242 773089741835270931984854511347767956576333171/\n 55777102945292 87682275315773022359477180471284918417776640,\na[11,7]=956248096794157 656802748368725840687390347399/779613242452844952548748442912924295256 71778,\na[11,8]=10788313637659292115861359463369872242886/189380589527 306698462795692309376152568545,\na[11,9]=89599398445595686856355698434 884216540038527287573830949657578490242/\n 57938562133849581348 3588345965887373464382184677844152053023899833767,\na[11,10]=-17863359 07413925929868932792832/7814458367137598885469908059185,\na[12,1]=-494 10934155054900182882239365042643979034016237/3173162398390380199808869 7641133618410239344640,\na[12,2]=0,\na[12,3]=0,\na[12,4]=-107461164912 8981113804182104383729704094059/24431005425533714040243336977409209758 1056,\na[12,5]=-222258155907862440804259443447899304774477/18911475471 642929222051825301977449103360,\na[12,6]=11410876327678156749715579835 467111092952376241624115293181/\n 12490816871574728439882863958 34110562049959611482227605504,\na[12,7]=898131843514608291105404022971 2538210777263716775/936984062290713822722287218869196809455428475968, \na[12,8]=465544633218086513326922426838859475/27467662576695046776938 324444449900032, a[12,9]=-28081309843646368614165858697384177345304907 2566975573877036454598695025876625/\n 105067490432026768588795 40117292890849606027959545703832394231856364095276862976, a[12,10]=509 08388575977059406815779042800/1299277084579288698410581464578279,\na[1 2,11]=-5373869958976816279363488081975/1282498194951127765005523505962 24,\na[13,1]=-68348459007984933111036701115865624594922076006017/\n \+ 45345007463014711631562120393860177858153484042240,\na[13,2]=0, \na[13,3]=0,\na[13,4]=-1421462835415589956778218218848342421532216119/ 349123046432081302269174752451415423519748096,\na[13,5]=-2984312771192 069486473441797268836080579319/262376761616550996396340471077385680977 920,\na[13,6]=41138701015420147024849320550595044614732541444419842765 9/\n 46460295006072909591428170750088268098474825959918534656, \na[13,7]=15921139337350831535408442010420239146408066675/174938297710 5047216940523536513333290389079616,\na[13,8]=-644840962682924479669958 56630949967425/5208159300615981523251328952205675621888,\na[13,9]=-842 304890904016106357610472171364471743518622807702364576842637748625/\n \+ 16001107104536584370686873062421769553685916890578695272031985 694176768, a[13,10]=1124804893366359619795687870063500/180260981966042 12076234696000197113,\na[13,11]=0,\na[13,12]=0,\n\nb[1]=87454519138996 0761/19691638733588931200,\nb[2]=0,\nb[3]=0,\nb[4]=0,\nb[5]=0,\nb[6]=4 20364804728976000977525051149/1187588452078929614685280764000,\nb[7]=2 5102415824721153301532049778688/101007568454396515184681192806875,\nb[ 8]=11291146602515386733913667333/8277895500290250100346304000,\nb[9]=5 1522714764404063835592355866713645307758690796273878782033726047743318 218807/\n 19282221074716176043598190917468876903092208069910099389 016386439619421788800,\nb[10]=-582592205611226468655591/16553820700475 4039774500,\nb[11]=43046993917969354593/35912185355757478400,\nb[12]=- 357120621384161/262158303907500,\nb[13]=0,\n\n`b*`[1]=1347079303976143 1/302948288209060480,\n`b*`[2]=0,\n`b*`[3]=0,\n`b*`[4]=0,\n`b*`[5]=0, \n`b*`[6]=225910982205515703574046329/636776649908273251841973600,\n`b *`[7]=50630668490428660369928685568/203890933496965109375618071875,\n` b*`[8]=3655081058088732259074359101/827789550029025010034630400,\n`b*` [9]=143021962768113394257793894874175054076879315632471495031519549972 079/\n 63497997867281994630609975410189529208101511709161157616 376634016000,\n`b*`[10]=-6457286835032584968329/2546741646226985227300 ,\n`b*`[11]=0,\n`b*`[12]=0,\n`b*`[13]=-4954671690517567/13107915195375 00\}:" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 20 "p rinted coefficients" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 357 "subs(ee,matrix([seq([c[i],seq(a[i,j],j=1.. i-1),``$(7-i)],i=2..7)]));\nfor ii from 8 to 13 do\n print(``);\n \+ print(c[ii]=subs(ee,c[ii])); \n for jj to ii-1 do\n print(a[ii, jj]=subs(ee,a[ii,jj]));\n end do:\nend do:print(``);\nfor ii to 13 d o\n print(b[ii]=subs(ee,b[ii]));\nend do:print(``);\nfor ii to 13 do \n print(`b*`[ii]=subs(ee,`b*`[ii]));\nend do:" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#K%'matrixG6#7(7)#\"\"#\"#DF(%!GF+F+F+F+7)#\"&9X#\"'\"\\ Q##\"+\\r^!4#\",\"3dz(o&#\"+D7&ev$F2F+F+F+F+7)#\"&dA\"\"&(\\z#F7\"'))z J\"\"!#F7\"''*f5F+F+F+7)#\"#]\"$H\"#\"/D%*['oJT\"\"/H:gvR$e$F;#!/D1Aa/ q<\"/VQ`eY%>\"#\"/+vm/'>w\"FGF+F+7)#\"#M\"#t#\"(6!=W\")D_0'*F;F;#\"+)) zvd(*\",n!=e*>%#\",TC^@*>\"-vU4+j5F+7)#\"#B\"$[\"#\"0>'*)eb&G![\"1?j\" QD*y(*zF;F;#\"8V'Q0Y'pcl5i@&\"9))eO**GZ%GLq)[Y#!2*[%Q\\*p2X;\"3?bJ)R%y 4+\\#\"0()Qj\"Q8[')\"23EtlwH=<&Q)pprint296\"" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%!G" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/&%\"cG6#\"\")# \"$'>\"$(>" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/&%\"aG6$\"\")\"\"\"#!8x (f->lo/b[#>%\"8+oVZ]w&[(*z1H" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/&%\"a G6$\"\")\"\"#\"\"!" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/&%\"aG6$\"\")\" \"$\"\"!" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/&%\"aG6$\"\")\"\"%#!@d%oL #*3%QbzR'\\#QOd\"\"?Ci(e\" \"#" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%!G" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/&%\"cG6#\"\"*#\",c^9=T\"\",6)=(Qg\"" }}{PARA 11 "" 1 " " {XPPMATH 20 "6#/&%\"aG6$\"\"*\"\"\"#!eoh#RJ!)4\"*Hl%3@%)Rol)R9*40v; \">$*o/OMV'p$4EA:W\"\"eo!)ou5qq;!\\2)R`^%=!H\"Q%\\?'[G?)HFFq8)R,M3Ym'G " }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/&%\"aG6$\"\"*\"\"#\"\"!" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/&%\"aG6$\"\"*\"\"$\"\"!" }}{PARA 11 "" 1 " " {XPPMATH 20 "6#/&%\"aG6$\"\"*\"\"%#!\\pnj*GF&z%p\"=Ucy$zRgS3\"*fE60Q nLlNj#H1G+GKtYnTA&\"[p/B+95#H^#Q8!foWc^b*)eTdx.kUmxaf4'pYeYBj_vWs" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#/&%\"aG6$\"\"*\"\"&#!go\\Ksz'y&H;TqM\" RKiagto(G/*>2\\Z5:.bvmYh_(f\\#)\"go!oR_Hsu[qv!*\\\\i#*3!zrz'QX6)))3!) \\O7!*37lR6Mc<" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/&%\"aG6$\"\"*\"\"'# \"jo2=n<-[:J%)4r^?2ku&[C-s;ouK+0ekBqj@`0)4gc\\#\"io;;R&=VjyZ\")=y%)p\\ .6B!>V+%oFk`skRhvp$zV$Q$e:i" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/&%\"aG 6$\"\"*\"\"(#\"\\oTto#H(f`z2:mb]UJkpy#=1\\l8*oe;AAxEf'\"[oM5/gU)*R>;\" \\anTNxFwB-Y*R_;:A<2du9r" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/&%\"aG6$ \"\"*\"\")#\"en7>Q?:!\\fr!yo0Z7Sfy_y9p%4.SGWD9$\"gn,N&=jVHia9mqZ&z>ur& *)\\G!=?byr])*GK" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%!G" }}{PARA 11 " " 1 "" {XPPMATH 20 "6#/&%\"cG6#\"#5#\"#R\"#V" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/&%\"aG6$\"#5\"\"\"#!U$H?L*\\InJ2re\"*>nu<#R]De!e)yy')* \"V+#>s7TAI!e#*>kUV8IVO<_*zH15x8" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/& %\"aG6$\"#5\"\"#\"\"!" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/&%\"aG6$\"#5 \"\"$\"\"!" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/&%\"aG6$\"#5\"\"%#!Qz.o c&)4'=l&\\v*[4-iOtg6$>1@]\"Pci0\\m\\'yG&Qfd*f7q@++:J/wf" }}{PARA 11 " " 1 "" {XPPMATH 20 "6#/&%\"aG6$\"#5\"\"&#!H2p/,/J:jrqeL!3Y]\"R#>L\"G+# >c!eeq@A5#3)*zFseZF&" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/&%\"aG6$\"#5 \"\"'#\"enXhIZ:lsoPJ;r^Lh\">W=(p,SJ_zw9V;\"Z)o#3!eYZy'oR;dhL^>)=^kB_%* \\y08eJ" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/&%\"aG6$\"#5\"\"(#\"O4j)Hy or:hrx%oGqw9&*QZoGCA\"N)GO%37>zz'>\")QUj#z\"[:A-P**>" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/&%\"aG6$\"#5\"\")#\"GXa&43\"ejo87P7\\+o\\O&)\\\"I/Z ]E+UL;xH!=W]geig'HF" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/&%\"aG6$\"#5\" \"*#!hn&**=G_#Rw-NRdMm:_#)4hEN$3Z:qS;zu.7\"inKG$e.@T]g-kXf8:rkI-:;@06A im&\\#*H)=Ww%Rk^Z;N/c]rn\"\"Uov:-E*4( =k)e>o/BQ9#pB&[heoU_\"" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/&%\"aG6$\"# 6\"\"#\"\"!" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/&%\"aG6$\"#6\"\"$\"\"! " }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/&%\"aG6$\"#6\"\"%#!L\"H^+nf>,PlrF =@v;5a:,-B\"K/F$)Rb#ywN6Ym^q7Y39(3XF" }}{PARA 11 "" 1 "" {XPPMATH 20 " 6#/&%\"aG6$\"#6\"\"&#!G4'**QCJ:@*4(Q2/^pY9*)o\"\"F)Gk2@zx3))y/x2Ir:R\" *=" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/&%\"aG6$\"#6\"\"'#\"fnrJLwl&zwZ 8^a[)>$4FN=u*3tFCw4'*4(R\"enSmx<%=\\Gr/=x%fB-tdJvAo(GHXH5xd&" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/&%\"aG6$\"#6\"\"(#\"N*RZ.R(oSeso$[F!oldTz'4 [i&*\"My!Q!R)RiJ<$" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/&%\"aG6$\"#7\"\"#\"\"!" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/&%\"aG6$\"#7\"\"$\"\"!" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/&%\" aG6$\"#7\"\"%#!LfS4/(HPQ/@=/Q6\")*G\"\\;hu5\"Kc5e(4#4u(pLV-/9P`Da+JW# " }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/&%\"aG6$\"#7\"\"&#!KxWx/$**yWV%fU !3Wiy!f:eAA\"JgL5\\u(>ID=0A#HHkraZ6*=" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/&%\"aG6$\"#7\"\"'#\"fn\"=$H:TiTiP_H46rYN)zbr\\n:ywKw3T6\"en/bgFA[6 'f*\\?c5T$eR'G))R%GZdro\"3\\7" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/&%\" aG6$\"#7\"\"(#\"Rvnrjsx5#QDrH-/a5\"H3Y^V=8)*)\"QofZGaX4o>p)=sGAF#Qr!Hi S)p$*" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/&%\"aG6$\"#7\"\")#\"Ev%f)QoU ApK8l3=KjWbY\"GK+!*\\WWC$QpxY]pwDmnu#" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/&%\"aG6$\"#7\"\"*#!ioDm(e-&p)fak.xQdvpcs!\\IXt<%Q(pee;9'ojkV)483G \"[pwH'oF&4kj&=B%RKQqX&fz-1'\\3*GH<,az)eon-K/\\n]5" }}{PARA 11 "" 1 " " {XPPMATH 20 "6#/&%\"aG6$\"#7\"#5#\"A+G/zd\"oSfq(fd)Q34&\"Cz#yXY\"e5% )p)GzX3x#*H\"" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/&%\"aG6$\"#7\"#6#!@v >3)[j$zi\"o(*e*pQP&\"BCif]Bb+lx7^\\>)\\#G\"" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%!G" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/&%\"cG6#\"#8\" \"\"" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/&%\"aG6$\"#8\"\"\"#!Sh@K:UU$[)=#=#yn&**e:a$GY@9\"N'4[(>NU:9X_Z< pAI\"3Kk/B\"\\$" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/&%\"aG6$\"#8\"\"&# !L>$z03O)oszTMZ'[p?>rFJ%)H\"K?z(4o&Qx5ZSjR'*4b;;wwBE" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/&%\"aG6$\"#8\"\"'#\"ZfwU)>WWTDt9Y/&f]0K\\[-Z,U:5qQ6 %\"YcY`=*ff#[Z)4o#)3]2.s_py0*o\"fo`&p'fjO$*[![7\"\"D8r>+gpMi27 Ug'>)4E!=" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/&%\"aG6$\"#8\"#6\"\"!" } }{PARA 11 "" 1 "" {XPPMATH 20 "6#/&%\"aG6$\"#8\"#7\"\"!" }}{PARA 11 " " 1 "" {XPPMATH 20 "6#%!G" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/&%\"bG6# \"\"\"#\"3h2'**Q\">XX()\"5+7$*)eL(Q;p>" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/&%\"bG6#\"\"#\"\"!" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/&%\"bG6# \"\"$\"\"!" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/&%\"bG6#\"\"%\"\"!" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#/&%\"bG6#\"\"&\"\"!" }}{PARA 11 "" 1 " " {XPPMATH 20 "6#/&%\"bG6#\"\"'#\"?\\60Dv(4+w*GZ![O?%\"@+Sw!G&o9'H*y?X )e(=\"" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/&%\"bG6#\"\"(#\"A)oy(\\?`,L :@Z#eT-^#\"Bvo!G>\"o%=:lRa%ov+,\"" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/ &%\"bG6#\"\")#\">Ltm8Rt'Q:DgY6H6\"=+SIY.5]-H+b*yF)" }}{PARA 11 "" 1 " " {XPPMATH 20 "6#/&%\"bG6#\"\"*#\"ho2)=#=LuZgsL?yyQF'z!pexIXOrmeN#fNQ1 /Ww9F_^\"ho+))y@%>'RkQ;!*Q*45*p!3A4.p()ou\"4>)fVg<;Z2@AG>" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/&%\"bG6#\"#5#!9\"fb'okA6c?#f#e\"9+XxRSv/q?Qb; " }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/&%\"bG6#\"#6#\"5$fa$pz\"R*p/V\"5+ %yuvb`=7f$" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/&%\"bG6#\"#7#!0hTQ@17d$ \"0+v!RIe@E" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/&%\"bG6#\"#8\"\"!" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#%!G" }}{PARA 11 "" 1 "" {XPPMATH 20 "6 #/&%#b*G6#\"\"\"#\"2J9wRIzqM\"\"3![g!4#)G[HI" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/&%#b*G6#\"\"#\"\"!" }}{PARA 11 "" 1 "" {XPPMATH 20 "6# /&%#b*G6#\"\"$\"\"!" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/&%#b*G6#\"\"% \"\"!" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/&%#b*G6#\"\"&\"\"!" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/&%#b*G6#\"\"'#\"%=D t#3*\\mxO'" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/&%#b*G6#\"\"(#\">oboG*p .mG/\\o1j]\"?v=2=cP4^'p\\L4*Q?" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/&%# b*G6#\"\")#\"=,\"fV2fAt)3e53bO\"<+/jM+,D!H+b*yF)" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/&%#b*G6#\"\"*#\"`oz?(*\\&>:.&\\rCj:$zo2a]-V\"\"_o+g,MmP;w:h\"4<^,\"3#H&*=5a(*41j%*>Gny*z\\j" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/&%#b*G6#\"#5#!7H$o\\eK]$oGdk\"7+tA&)pAY;uYD" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#/&%#b*G6#\"#6\"\"!" }}{PARA 11 "" 1 " " {XPPMATH 20 "6#/&%#b*G6#\"#7\"\"!" }}{PARA 11 "" 1 "" {XPPMATH 20 "6 #/&%#b*G6#\"#8#!1nv^!prY&\\\"1+v`>:z58" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 151 "RK8_13eqs : = [op(RowSumConditions(13,'expanded')),op(OrderConditions(8,13,'expand ed'))]:\n`RK7_13eqs*` := subs(b=`b*`,OrderConditions(7,13,'expanded')) :" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 6 "Chec k:" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 110 "subs(ee,RK8_13eqs):\n map(u->lhs(u)-rhs(u),%);\nnops(%);\nsubs(ee,`RK7_13eqs*`):\nmap(u->lhs (u)-rhs(u),%);\nnops(%);" }}{PARA 12 "" 1 "" {XPPMATH 20 "6#7`x\"\"!F$ F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$ F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$ F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$ F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$ F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$ F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$ " }}{PARA 11 "" 1 "" {XPPMATH 20 "6#\"$7#" }}{PARA 12 "" 1 "" {XPPMATH 20 "6#7ap\"\"!F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F $F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F $F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$" }}{PARA 11 "" 1 " " {XPPMATH 20 "6#\"#&)" }}}{PARA 257 "" 0 "" {TEXT -1 0 "" }}{PARA 0 " " 0 "" {TEXT -1 78 "Set-up stage-order condtions to check for stage-or ders from 2 to 6 inclusive. " }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 87 "for ct from 2 to 6 do\n so||ct||_13 := StageOrderConditions(ct,1 3,'expanded');\nend do:" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 " " 0 "" {TEXT -1 59 "Stages 3 to 13 have the following respective stage -orders. " }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 175 "[seq([seq(expa nd(subs(ee,so||i||_13[j])),i=2..6)],j=1..11)]:\nmap(proc(L) local i; f or i to nops(L) do if not evalb(L[i]) then break end if end do; i end \+ proc,%):\nsimplify(%); " }}{PARA 11 "" 1 "" {XPPMATH 20 "6#7-\"\"#\"\" $F%\"\"%F&F&F&F&F&F&F&" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 27 "The simplifying conditions:" }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "Sum(b[i]*a[i,j],i = j+1 .. 13) = b[j]*( 1-c[j]);" "6#/-%$SumG6$*&&%\"bG6#%\"iG\"\"\"&%\"aG6$F+%\"jGF,/F+;,&F0F ,F,F,\"#8*&&F)6#F0F,,&F,F,&%\"cG6#F0!\"\"F," }{TEXT -1 3 ", " } {XPPEDIT 18 0 "j=1" "6#/%\"jG\"\"\"" }{TEXT -1 9 " . . 12, " }}{PARA 0 "" 0 "" {TEXT -1 8 "(where " }{XPPEDIT 18 0 "c[1]=0" "6#/&%\"cG6#\" \"\"\"\"!" }{TEXT -1 17 " ) are satisfied." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 151 "[Sum(b[i]*a[i,1], i=1+1..13)=b[1],seq(Sum(b[i]*a[i,j],i=j+1..13)=b[j]*(1-c[j]),j=2..12)] :\nmap(u->lhs(u)-rhs(u),subs(ee,eval(subs(Sum=add,%))));\nnops(%);" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#7.\"\"!F$F$F$F$F$F$F$F$F$F$F$" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#\"#7" }}}{PARA 0 "" 0 "" {TEXT -1 0 " " }}{PARA 0 "" 0 "" {TEXT -1 27 "The simplifying conditions:" }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "Sum(`b*`[i]*a[i,j],i = j+1 \+ .. 13) = `b*`[j]*(1-c[j]);" "6#/-%$SumG6$*&&%#b*G6#%\"iG\"\"\"&%\"aG6$ F+%\"jGF,/F+;,&F0F,F,F,\"#8*&&F)6#F0F,,&F,F,&%\"cG6#F0!\"\"F," }{TEXT -1 3 ", " }{XPPEDIT 18 0 "j=1" "6#/%\"jG\"\"\"" }{TEXT -1 9 " . . 12, " }}{PARA 0 "" 0 "" {TEXT -1 8 "(where " }{XPPEDIT 18 0 "c[1]=0" "6# /&%\"cG6#\"\"\"\"\"!" }{TEXT -1 17 " ) are satisfied." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 163 "[Sum(`b* `[i]*a[i,1],i=1+1..13)=`b*`[1],seq(Sum(`b*`[i]*a[i,j],i=j+1..13)=`b*`[ j]*(1-c[j]),j=2..12)]:\nmap(u->lhs(u)-rhs(u),subs(ee,eval(subs(Sum=add ,%))));\nnops(%);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#7.\"\"!F$F$F$F$F$ F$F$F$F$F$F$" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#\"#7" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 93 "We can calculate the p rincipal error norm, that is, the 2-norm of the principal error terms. " }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 162 "errterms8_13 := Princi palErrorTerms(8,13,'expanded'):\nsm := 0:\nfor ct to nops(errterms8_13 ) do\n sm := sm+(evalf(subs(ee,errterms8_13[ct])))^2;\nend do:\nsqrt (sm);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#$\"+I*4OJ(!#;" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 78 "We can also calculat e the principal error norm of the order 7 embedded scheme." }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 184 "`errterms7_13*` := subs(b=`b*`,Pri ncipalErrorTerms(7,13,'expanded')):\nsm := 0:\nfor ct to nops(`errterm s7_13*`) do\n sm := sm+(evalf(subs(ee,`errterms7_13*`[ct])))^2;\nend do:\nsqrt(sm);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#$\"+g8875!#9" }}} {PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}}{PARA 0 "" 0 "" {TEXT -1 52 "#------------------------------ ---------------------" }}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 34 "construc tion of the order 8 scheme" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 93 "First we construct a 12 stage order 8 scheme star ting with a consideration of stages 2 to 8. " }}{PARA 0 "" 0 "" {TEXT -1 20 "We specify the nodes" }}{PARA 256 "" 0 "" {TEXT -1 2 " " } {XPPEDIT 18 0 "c[2] = 2/25;" "6#/&%\"cG6#\"\"#*&F'\"\"\"\"#D!\"\"" } {TEXT -1 2 ", " }{XPPEDIT 18 0 "c[5] = 50/129;" "6#/&%\"cG6#\"\"&*&\"# ]\"\"\"\"$H\"!\"\"" }{TEXT -1 2 ", " }{XPPEDIT 18 0 "c[6] = 34/73;" "6 #/&%\"cG6#\"\"'*&\"#M\"\"\"\"#t!\"\"" }{TEXT -1 2 ", " }{XPPEDIT 18 0 "c[7] = 23/148;" "6#/&%\"cG6#\"\"(*&\"#B\"\"\"\"$[\"!\"\"" }{TEXT -1 3 ", " }{XPPEDIT 18 0 "c[8] = 196/197;" "6#/&%\"cG6#\"\")*&\"$'>\"\" \"\"$(>!\"\"" }{TEXT -1 2 " " }}{PARA 0 "" 0 "" {TEXT -1 33 "and the \+ zero linking coefficients" }}{PARA 256 "" 0 "" {TEXT -1 1 " " } {XPPEDIT 18 0 "a[4,2]=0" "6#/&%\"aG6$\"\"%\"\"#\"\"!" }{TEXT -1 2 ", \+ " }{XPPEDIT 18 0 "a[5,2]=0" "6#/&%\"aG6$\"\"&\"\"#\"\"!" }{TEXT -1 2 " , " }{XPPEDIT 18 0 "a[6,2]=0" "6#/&%\"aG6$\"\"'\"\"#\"\"!" }{TEXT -1 2 ", " }{XPPEDIT 18 0 "a[6,3]=0" "6#/&%\"aG6$\"\"'\"\"$\"\"!" }{TEXT -1 2 ", " }{XPPEDIT 18 0 "a[7,2]=0" "6#/&%\"aG6$\"\"(\"\"#\"\"!" } {TEXT -1 2 ", " }{XPPEDIT 18 0 "a[7,3]=0" "6#/&%\"aG6$\"\"(\"\"$\"\"! " }{TEXT -1 2 ", " }{XPPEDIT 18 0 "a[8,2] = 0;" "6#/&%\"aG6$\"\")\"\"# \"\"!" }{TEXT -1 2 ", " }{XPPEDIT 18 0 "a[8,3] = 0;" "6#/&%\"aG6$\"\") \"\"$\"\"!" }{TEXT -1 3 ", " }{XPPEDIT 18 0 "a[8,4]=0" "6#/&%\"aG6$\" \")\"\"%\"\"!" }{TEXT -1 2 ". " }}{PARA 0 "" 0 "" {TEXT -1 48 "In addi tion we specify the linking coefficient " }{XPPEDIT 18 0 "a[8,7] = 19 /2;" "6#/&%\"aG6$\"\")\"\"(*&\"#>\"\"\"\"\"#!\"\"" }{TEXT -1 1 "." }} {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 69 "The stage -order equations relating to these stages, such that stages " } {XPPEDIT 18 0 "3,4,5,6,7,8;" "6(\"\"$\"\"%\"\"&\"\"'\"\"(\"\")" } {TEXT -1 19 " have stage orders " }{XPPEDIT 18 0 "2,3,3,4,4,4;" "6(\" \"#\"\"$F$\"\"%F%F%" }{TEXT -1 48 " respectively can be solved to obta in the nodes " }{XPPEDIT 18 0 "c[3], c[4]" "6$&%\"cG6#\"\"$&F$6#\"\"% " }{TEXT -1 67 " and the remaining non-zero linking coefficients for t hese stages. " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 446 "RSeqs := RowSumConditions(8,'expanded'):\nSOeqs := [op(StageOrderConditions(2,8,'expanded')),op(StageOrderConditions( 3,4..8,'expanded')),\n op(StageOrderConditions(4,6..8,'e xpanded'))]:\ne1 := \{c[2]=2/25,c[5]=50/129,c[6]=34/73,c[7]=23/148,c[8 ]=196/197,\n seq(a[i,2]=0,i=4..8),seq(a[i,3]=0,i=6..8),a[8,7]=19 /2\}:\neqns := map(expand,subs(e1,[op(RSeqs),op(SOeqs)])):\nconvert(Li stTools[Enumerate](%),matrix);\n``;\nindets(eqns);\nnops(%);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#K%'matrixG6#777$\"\"\"/&%\"aG6$\"\"#F(#F-\"# D7$F-/,&&F+6$\"\"$F(F(&F+6$F5F-F(&%\"cG6#F57$F5/,&&F+6$\"\"%F(F(&F+6$F @F5F(&F96#F@7$F@/,(&F+6$\"\"&F(F(&F+6$FJF5F(&F+6$FJF@F(#\"#]\"$H\"7$FJ /,(&F+6$\"\"'F(F(&F+6$FWF@F(&F+6$FWFJF(#\"#M\"#t7$FW/,*&F+6$\"\"(F(F(& F+6$F^oF@F(&F+6$F^oFJF(&F+6$F^oFWF(#\"#B\"$[\"7$F^o/,,&F+6$\"\")F(F(# \"#>F-F(&F+6$F]pF@F(&F+6$F]pFJF(&F+6$F]pFWF(#\"$'>\"$(>7$F]p/,$*&F.F(F 6F(F(,$*&#F(F-F(*$)F8F-F(F(F(7$\"\"*/*&FAF(F8F(,$*&F_qF(*$)FCF-F(F(F(7 $\"#5/,&*&FKF(F8F(F(*&FMF(FCF(F(#\"%]7\"&Tm\"7$\"#6/,&*&FXF(FCF(F(*&FO F(FZF(F(#\"$y&\"%H`7$\"#7/,(*&F_oF(FCF(F(*&FOF(FaoF(F(*&FfnF(FcoF(F(# \"$H&\"&3Q%7$\"#8/,*#\"$P%\"$'HF(*&F`pF(FCF(F(*&FOF(FbpF(F(*&FfnF(FdpF (F(#\"&3#>\"&4)Q7$\"#9/*&FAF(FaqF(,$*&#F(F5F(*$)FCF5F(F(F(7$\"#:/,&*&F KF(FaqF(F(*&FMF(FiqF(F(#\"'+]7\"(n+W'7$\"#;/,&*&FXF(FiqF(F(*&#\"%+DFbr F(FZF(F(#\"&/$R\"(^q;\"7$\"#h$H#7$F_p/,&*&FXF(F[uF(F(*&#Fcu\"(*oY@F (FZF(F(#\"'%3M$\")T#)RG7$\"#?/,(*&F_oF(F[uF(F(*&F]xF(FaoF(F(*&#F^v\"'< !*QF(FcoF(F(#\"'T)z#\"+k39>>7$\"#@/,*#\"'t6B\"(%e$['F(*&F`pF(F[uF(F(*& F]xF(FbpF(F(*&FixF(FdpF(F(#\"*ks%*o$\"+\"[Qh]\"Q)pprint166\"" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%!G" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#<7&% \"aG6$\"\"(\"\"'&F%6$\"\")\"\"\"&F%6$F+\"\"&&F%6$F+F(&F%6$F'\"\"%&F%6$ F'F/&F%6$F(F,&F%6$F/F4&F%6$F/\"\"$&F%6$F+F4&F%6$F'F,&F%6$F(F4&F%6$F(F/ &F%6$F/F,&F%6$F4F=&%\"cG6#F4&F%6$F=F,&F%6$F=\"\"#&FK6#F=&F%6$F4F,&F%6$ FQF," }}{PARA 11 "" 1 "" {XPPMATH 20 "6#\"#@" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 46 "e2 := solve( \{op(eqns)\});\ne3 := `union`(e1,e2):" }}{PARA 12 "" 1 "" {XPPMATH 20 "6#>%#e2G<7/&%\"aG6$\"\"%\"\"\"#\"&dA\"\"'))zJ/&F(6$\"\"(F+#\"0>'*)eb& G![\"1?j\"QD*y(*z/&F(6$F2\"\"'#\"0()Qj\"Q8[')\"23EtlwH=<&/&%\"cG6#F*#F -\"&(\\z/&F(6$F2\"\"&#!2*[%Q\\*p2X;\"3?bJ)R%y4+\\/&F(6$\"\"$F+#\"+\\r^ !4#\",\"3dz(o&/&F(6$\"\"#F+#FT\"#D/&F(6$FFF+#\"/D%*['oJT\"\"/H:gvR$e$/ &F(6$F9F*#\"+))zvd(*\",n!=e*>%/&F(6$F9F+#\"(6!=W\")D_0'*/&F(6$FFF*#\"/ +vm/'>w\"\"/VQ`eY%>\"/&F?6#FM#\"&9X#\"'\"\\Q#/&F(6$FFFM#!/D1Aa/q\"-vU4+j5/&F(6$\"\")F9#\";&H!o@, v0<&z+?=&\":C#3F!4xTwnXE0'/&F(6$F2F*#\"8V'Q0Y'pcl5i@&\"9))eO**GZ%GLq)[ Y/&F(6$FbqFF#!=LT)o2&33p*)zBvWJ\"<+Gh*[p,z0/vJnG/&F(6$FbqF+#!8x(f->lo/ b[#>%\"8+oVZ]w&[(*z1H/&F(6$FMFT#\"+D7&ev$FP/&F(6$FbqF*#!@d%oL#*3%QbzR' \\#QOd\"\"?Ci(e " 0 "" {MPLTEXT 1 0 146 "subs(e3,matrix([seq([c[i],seq(a[i,j],j=1..i-1),``$(8 -i)],i=2..7)]));``;\nc[8]=subs(e3,c[8]);\nfor jj to 7 do print(a[8,jj] =subs(e3,a[8,jj])) end do;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#K%'matri xG6#7(7*#\"\"#\"#DF(%!GF+F+F+F+F+7*#\"&9X#\"'\"\\Q##\"+\\r^!4#\",\"3dz (o&#\"+D7&ev$F2F+F+F+F+F+7*#\"&dA\"\"&(\\z#F7\"'))zJ\"\"!#F7\"''*f5F+F +F+F+7*#\"#]\"$H\"#\"/D%*['oJT\"\"/H:gvR$e$F;#!/D1Aa/q<\"/VQ`eY%>\"#\" /+vm/'>w\"FGF+F+F+7*#\"#M\"#t#\"(6!=W\")D_0'*F;F;#\"+))zvd(*\",n!=e*>% #\",TC^@*>\"-vU4+j5F+F+7*#\"#B\"$[\"#\"0>'*)eb&G![\"1?j\"QD*y(*zF;F;# \"8V'Q0Y'pcl5i@&\"9))eO**GZ%GLq)[Y#!2*[%Q\\*p2X;\"3?bJ)R%y4+\\#\"0()Qj \"Q8[')\"23EtlwH=<&F+Q)pprint256\"" }}{PARA 11 "" 1 "" {XPPMATH 20 "6# %!G" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/&%\"cG6#\"\")#\"$'>\"$(>" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#/&%\"aG6$\"\")\"\"\"#!8x(f->lo/b[#>%\" 8+oVZ]w&[(*z1H" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/&%\"aG6$\"\")\"\"# \"\"!" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/&%\"aG6$\"\")\"\"$\"\"!" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#/&%\"aG6$\"\")\"\"%#!@d%oL#*3%QbzR'\\# QOd\"\"?Ci(e\"\"#" }}} {PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 81 "subs(e3,matrix([seq([c[i],seq(a[i,j],j=1..i-1),``$(9-i)],i=2..8)]) ):\nevalf[8](%);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#K%'matrixG6#7)7+$ \")+++!)!\"*F(%!GF+F+F+F+F+F+7+$\")&zy-\"!\")$\")NWvOF*$\")6N.mF*F+F+F +F+F+F+7+$\")#>=a\"F/$\")![X&QF*$\"\"!F:$\")WOc6F/F+F+F+F+F+7+$\")!pf( QF/$\")elVRF/F9$!)>(=[\"!\"($\")L5v9FDF+F+F+F+7+$\")U`dYF/$\")*[%*f%F* F9F9$\")r]BBF/$\")B3u=F/F+F+F+7+$\")T0a:F/$\")!H_+'F*F9F9$\")$Q?7\"F/$ !)IBdLF*$\")8;s;F*F+F+7+$\")'Q#\\**F/$!)GIU9FDF9F9$!)NucYFD$!)wv'4\"! \"'$\")mah&)FD$\")+++&*FDF+Q)pprint206\"" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 27 "We do not need to specify " }{XPPEDIT 18 0 "c[9];" "6#&% \"cG6#\"\"*" }{TEXT -1 42 ", because according to Verner, the nodes \+ " }{XPPEDIT 18 0 "c[6];" "6#&%\"cG6#\"\"'" }{TEXT -1 2 ", " }{XPPEDIT 18 0 "c[7];" "6#&%\"cG6#\"\"(" }{TEXT -1 2 ", " }{XPPEDIT 18 0 "c[8]; " "6#&%\"cG6#\"\")" }{TEXT -1 2 ", " }{XPPEDIT 18 0 "c[9];" "6#&%\"cG6 #\"\"*" }{TEXT -1 30 " are related by the equation " }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "Int(p(x)*``((1-x)^3/3!),x = 0 .. 1 )*Int(q(x)*(1-x),x = 0 .. 1) = Int(p(x)*``((1-x)^2/2!),x = 0 .. 1)*Int (q(x)*``((1-x)^2/2!),x = 0 .. 1);" "6#/*&-%$IntG6$*&-%\"pG6#%\"xG\"\" \"-%!G6#*&,&F-F-F,!\"\"\"\"$-%*factorialG6#F4F3F-/F,;\"\"!F-F--F&6$*&- %\"qG6#F,F-,&F-F-F,F3F-/F,;F:F-F-*&-F&6$*&-F*6#F,F--F/6#*&,&F-F-F,F3\" \"#-F66#FNF3F-/F,;F:F-F--F&6$*&-F?6#F,F--F/6#*&,&F-F-F,F3FN-F66#FNF3F- /F,;F:F-F-" }{TEXT -1 2 ", " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 7 "where " }{XPPEDIT 18 0 "p(x) = x*(x-c[6])*(x-c[ 7])*(x-c[8]);" "6#/-%\"pG6#%\"xG**F'\"\"\",&F'F)&%\"cG6#\"\"'!\"\"F),& F'F)&F,6#\"\"(F/F),&F'F)&F,6#\"\")F/F)" }{TEXT -1 7 " and " } {XPPEDIT 18 0 "q(x) = (x-c[9])*p(x);" "6#/-%\"qG6#%\"xG*&,&F'\"\"\"&% \"cG6#\"\"*!\"\"F*-%\"pG6#F'F*" }{TEXT -1 2 ". " }}{PARA 0 "" 0 "" {TEXT -1 1 " " }}{PARA 0 "" 0 "" {TEXT -1 155 "See: J.H. Verner, SIAM \+ Journal of Numerical Analysis 1978, 772-790, \"Explicit Runge-Kutta me thods with estimates of the Local Truncation Error.\" (page 780)" }} {PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 90 "p := x -> x*(x-c[6])*(x-c[7])*(x-c[8]):\n'p(x)'=p(x);\nq := x -> ( x-c[9])*p(x):\n'q(x)'=q(x);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/-%\"pG 6#%\"xG**F'\"\"\",&F'F)&%\"cG6#\"\"'!\"\"F),&F'F)&F,6#\"\"(F/F),&F'F)& F,6#\"\")F/F)" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/-%\"qG6#%\"xG*,,&F' \"\"\"&%\"cG6#\"\"*!\"\"F*F'F*,&F'F*&F,6#\"\"'F/F*,&F'F*&F,6#\"\"(F/F* ,&F'F*&F,6#\"\")F/F*" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 239 "Ieq := Int('p(x)'*(1-x)^3/3!,x=0..1)*Int ('q(x)'*(1-x),x=0..1)=\n Int('p(x)'*(1-x)^2/2!,x=0..1)*Int('q(x)'*(1- x)^2/2!,x=0..1);\nvalue(Ieq):\nReq := simplify(lhs(%)-rhs(%))=0:\nL := ilcm(op(map(denom,[coeffs(lhs(Req))]))):\nCeq := expand(L*Req);" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#>%$IeqG/*&-%$IntG6$,$*&#\"\"\"\"\"'F-* &-%\"pG6#%\"xGF-),&F-F-F3!\"\"\"\"$F-F-F-/F3;\"\"!F-F--F(6$*&-%\"qGF2F -F5F-F8F-*&-F(6$,$*&#F-\"\"#F-*&F0F-)F5FFF-F-F-F8F--F(6$,$*&FEF-*&F>F- FHF-F-F-F8F-" }}{PARA 12 "" 1 "" {XPPMATH 20 "6#>%$CeqG/,hp*(\"#G\"\" \"&%\"cG6#\"\")F)&F+6#\"\"'F)F)**\"$o\"F)F*F))&F+6#\"\"(\"\"#F)&F+6#\" \"*F)!\"\"**F2F)F*F))F.F7F)F8F)F;**F2F))F*F7F)F.F)F8F)F;**F2F)F8F)F=F) F4F)F;**F2F)F.F)F3F)F8F)F;**\"$'>F)F=F)F3F)F8F)F)**F2F)F?F)F.F)F4F)F)* *F2F)F*F)F3F)F.F)F)**F2F)F*F)F=F)F4F)F)*(\"#9F)F?F)F4F)F;*(FHF)F?F)F.F )F;*(\"#UF)F8F)F=F)F)**FCF)F?F)F=F)F8F)F)**F2F)F?F)F4F)F8F)F;**FCF)F?F )F3F)F8F)F)*(F(F)F.F)F4F)F)*(FKF)F8F)F.F)F;*,\"$)eF)F8F)F.F)F4F)F*F)F; *,\"$!)*F)F?F)F=F)F8F)F4F)F;*,\"$G(F)F*F)F3F)F8F)F.F)F)*,FTF)F?F)F3F)F 8F)F.F)F;*,FTF)F=F)F3F)F8F)F*F)F;*,\"%q9F)F=F)F3F)F?F)F8F)F)**\"$a\"F) F8F)F.F)F4F)F)**FfnF)F*F)F8F)F.F)F)**FfnF)F*F)F4F)F8F)F)*,FVF)F*F)F=F) F8F)F4F)F)*(FHF)F*F)F3F)F;**\"#)*F)F=F)F3F)F*F)F;**F\\oF)F?F)F3F)F.F)F ;**F\\oF)F?F)F=F)F4F)F;*,FVF)F?F)F.F)F4F)F8F)F)*(FHF)F=F)F4F)F;*(F(F)F *F)F4F)F)*(FHF)F*F)F=F)F;*(FHF)F.F)F3F)F;*(FKF)F4F)F8F)F;*(FKF)F*F)F8F )F;*(FKF)F?F)F8F)F)**\"$J#F)F.F)F4F)F*F)F;*(FKF)F3F)F8F)F)*&\"#7F)F8F) F)*&\"\"$F)F*F)F;*&F]pF)F4F)F;*&F]pF)F.F)F;\"\"!" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 60 "The previous equation inv olving integrals is equivalent to: " }}{PARA 256 "" 0 "" {TEXT -1 2 " \+ " }{XPPEDIT 18 0 "14*c[6]^2*c[7]+3*c[6]-28*c[6]*c[7]+3*c[7]+588*c[9]* c[6]*c[7]*c[8]+3*c[8]-28*c[8]*c[6]-28*c[8]*c[7]+14*c[8]*c[6]^2+14*c[8] ^2*c[6]+231*c[6]*c[7]*c[8]+98*c[6]^2*c[7]^2*c[8]+14*c[6]*c[7]^2+98*c[8 ]^2*c[7]^2*c[6]+98*c[8]^2*c[6]^2*c[7]-168*c[8]^2*c[6]*c[7]-168*c[8]*c[ 7]^2*c[6]-168*c[8]*c[6]^2*c[7]+14*c[8]^2*c[7]-12*c[9]+14*c[8]*c[7]^2+4 2*c[9]*c[6]-154*c[9]*c[6]*c[7]+42*c[7]*c[9]+42*c[8]*c[9]-154*c[8]*c[9] *c[6]-154*c[8]*c[7]*c[9]-42*c[9]*c[6]^2-42*c[7]^2*c[9]-42*c[8]^2*c[9]+ 168*c[8]*c[7]^2*c[9]-728*c[8]*c[6]^2*c[9]*c[7]-728*c[8]^2*c[6]*c[7]*c[ 9]+980*c[8]^2*c[6]^2*c[9]*c[7]-728*c[8]*c[7]^2*c[9]*c[6]+980*c[8]^2*c[ 7]^2*c[9]*c[6]+980*c[6]^2*c[7]^2*c[9]*c[8]-1470*c[6]^2*c[7]^2*c[8]^2*c [9]+168*c[8]*c[6]^2*c[9]+168*c[8]^2*c[6]*c[9]-196*c[8]^2*c[6]^2*c[9]+1 68*c[8]^2*c[7]*c[9]-196*c[8]^2*c[7]^2*c[9]+168*c[9]*c[6]^2*c[7]+168*c[ 6]*c[7]^2*c[9]-196*c[6]^2*c[7]^2*c[9]=0" "6#/,hp*(\"#9\"\"\"*$&%\"cG6# \"\"'\"\"#F'&F*6#\"\"(F'F'*&\"\"$F'&F*6#F,F'F'*(\"#GF'&F*6#F,F'&F*6#F0 F'!\"\"*&F2F'&F*6#F0F'F'*,\"$)eF'&F*6#\"\"*F'&F*6#F,F'&F*6#F0F'&F*6#\" \")F'F'*&F2F'&F*6#FJF'F'*(F6F'&F*6#FJF'&F*6#F,F'F;*(F6F'&F*6#FJF'&F*6# F0F'F;*(F&F'&F*6#FJF'&F*6#F,F-F'*(F&F'*$&F*6#FJF-F'&F*6#F,F'F'**\"$J#F '&F*6#F,F'&F*6#F0F'&F*6#FJF'F'**\"#)*F'*$&F*6#F,F-F'&F*6#F0F-&F*6#FJF' F'*(F&F'&F*6#F,F'&F*6#F0F-F'**FfoF'*$&F*6#FJF-F'&F*6#F0F-&F*6#F,F'F'** FfoF'*$&F*6#FJF-F'&F*6#F,F-&F*6#F0F'F'**\"$o\"F'*$&F*6#FJF-F'&F*6#F,F' &F*6#F0F'F;**FdqF'&F*6#FJF'&F*6#F0F-&F*6#F,F'F;**FdqF'&F*6#FJF'&F*6#F, F-&F*6#F0F'F;*(F&F'*$&F*6#FJF-F'&F*6#F0F'F'*&\"#7F'&F*6#FCF'F;*(F&F'&F *6#FJF'&F*6#F0F-F'*(\"#UF'&F*6#FCF'&F*6#F,F'F'**\"$a\"F'&F*6#FCF'&F*6# F,F'&F*6#F0F'F;*(FjsF'&F*6#F0F'&F*6#FCF'F'*(FjsF'&F*6#FJF'&F*6#FCF'F'* *F`tF'&F*6#FJF'&F*6#FCF'&F*6#F,F'F;**F`tF'&F*6#FJF'&F*6#F0F'&F*6#FCF'F ;*(FjsF'&F*6#FCF'&F*6#F,F-F;*(FjsF'*$&F*6#F0F-F'&F*6#FCF'F;*(FjsF'*$&F *6#FJF-F'&F*6#FCF'F;**FdqF'&F*6#FJF'&F*6#F0F-&F*6#FCF'F'*,\"$G(F'&F*6# FJF'&F*6#F,F-&F*6#FCF'&F*6#F0F'F;*,FhwF'*$&F*6#FJF-F'&F*6#F,F'&F*6#F0F '&F*6#FCF'F;*,\"$!)*F'*$&F*6#FJF-F'&F*6#F,F-&F*6#FCF'&F*6#F0F'F'*,FhwF '&F*6#FJF'&F*6#F0F-&F*6#FCF'&F*6#F,F'F;*,F\\yF'*$&F*6#FJF-F'&F*6#F0F-& F*6#FCF'&F*6#F,F'F'*,F\\yF'*$&F*6#F,F-F'&F*6#F0F-&F*6#FCF'&F*6#FJF'F'* ,\"%q9F'*$&F*6#F,F-F'&F*6#F0F-&F*6#FJF-&F*6#FCF'F;**FdqF'&F*6#FJF'&F*6 #F,F-&F*6#FCF'F'**FdqF'*$&F*6#FJF-F'&F*6#F,F'&F*6#FCF'F'**\"$'>F'*$&F* 6#FJF-F'&F*6#F,F-&F*6#FCF'F;**FdqF'*$&F*6#FJF-F'&F*6#F0F'&F*6#FCF'F'** F^]lF'*$&F*6#FJF-F'&F*6#F0F-&F*6#FCF'F;**FdqF'&F*6#FCF'&F*6#F,F-&F*6#F 0F'F'**FdqF'&F*6#F,F'&F*6#F0F-&F*6#FCF'F'**F^]lF'*$&F*6#F,F-F'&F*6#F0F -&F*6#FCF'F;\"\"!" }{TEXT -1 2 ". " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 75 "e4 := solve(subs(e3,Ceq),\{c [9]\}):\nc[9]=subs(e4,c[9]);\ne5 := `union`(e3,e4):" }}{PARA 11 "" 1 " " {XPPMATH 20 "6#/&%\"cG6#\"\"*#\",c^9=T\"\",6)=(Qg\"" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 6 "Thus " }{XPPEDIT 18 0 "c[9] = 14118145156/16038718811" "6#/&%\"cG6#\"\"**&\",c^9=T\"\"\"\" \",6)=(Qg\"!\"\"" }{TEXT -1 2 ". " }}{PARA 0 "" 0 "" {TEXT -1 85 "Now \+ we can use the quadrature equations to find the weights once the remai ning nodes " }}{PARA 256 "" 0 "" {TEXT -1 2 " " }{XPPEDIT 18 0 "c[10] = 39/43;" "6#/&%\"cG6#\"#5*&\"#R\"\"\"\"#V!\"\"" }{TEXT -1 3 ", " } {XPPEDIT 18 0 "c[11] = 20/21;" "6#/&%\"cG6#\"#6*&\"#?\"\"\"\"#@!\"\"" }{TEXT -1 3 ", " }{XPPEDIT 18 0 "c[12] = 1;" "6#/&%\"cG6#\"#7\"\"\"" }{TEXT -1 1 " " }}{PARA 0 "" 0 "" {TEXT -1 38 "are specified along wit h the weights " }{XPPEDIT 18 0 "b[2]=0" "6#/&%\"bG6#\"\"#\"\"!" } {TEXT -1 2 ", " }{XPPEDIT 18 0 "b[3]=0" "6#/&%\"bG6#\"\"$\"\"!" } {TEXT -1 2 ", " }{XPPEDIT 18 0 "b[4]=0" "6#/&%\"bG6#\"\"%\"\"!" } {TEXT -1 2 ", " }{XPPEDIT 18 0 "b[5]=0" "6#/&%\"bG6#\"\"&\"\"!" } {TEXT -1 2 ". " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 288 "Qeqs := QuadratureConditions(8,12,'expanded'): \ne6 := \{seq(b[i]=0,i=2..5),c[10]=39/43,c[11]=20/21,c[12]=1\}:\ne7 := `union`(e5,e6):\nquadeqns := subs(e7,Qeqs):\nfor ct to nops(quadeqns) do\n print(`equation `||ct); print(``);print(quadeqns[ct]);print( ``);\nend do:\nindets(quadeqns);\nnops(%);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%-equation~~~1G" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%!G " }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/,2&%\"bG6#\"\"\"F(&F&6#\"\"'F(&F& 6#\"\"(F(&F&6#\"\")F(&F&6#\"\"*F(&F&6#\"#5F(&F&6#\"#6F(&F&6#\"#7F(F(" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%!G" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%-equation~~~2G" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%!G" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/,0*&#\"#M\"#t\"\"\"&%\"bG6#\"\"'F)F)*&#\"#B\"$ [\"F)&F+6#\"\"(F)F)*&#\"$'>\"$(>F)&F+6#\"\")F)F)*&#\",c^9=T\"\",6)=(Qg \"F)&F+6#\"\"*F)F)*&#\"#R\"#VF)&F+6#\"#5F)F)*&#\"#?\"#@F)&F+6#\"#6F)F) &F+6#\"#7F)#F)\"\"#" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%!G" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%-equation~~~3G" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%!G" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/,0*&#\"%c6\"%H` \"\"\"&%\"bG6#\"\"'F)F)*&#\"$H&\"&/>#F)&F+6#\"\"(F)F)*&#\"&;%Q\"&4)QF) &F+6#\"\")F)F)*&#\"6OVE')ekA?K*>\"6@PDD$)4,0Cd#F)&F+6#\"\"*F)F)*&#\"%@ :\"%\\=F)&F+6#\"#5F)F)*&#\"$+%\"$T%F)&F+6#\"#6F)F)&F+6#\"#7F)#F)\"\"$ " }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%!G" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%-equation~~~4G" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%!G" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/,0*&#\"&/$R\"'$f\"&2&zF)&F +6#\"#5F)F)*&#\"%+!)\"%h#*F)&F+6#\"#6F)F)&F+6#\"#7F)#F)\"\"%" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%!G" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%-equ ation~~~5G" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%!G" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/,0*&#\"(OjL\"\")T#)RG\"\"\"&%\"bG6#\"\"'F)F)*&#\"'T)z# \"*;_yz%F)&F+6#\"\"(F)F)*&#\"+c!*yv9\"+\"[Qh]\"F)&F+6#\"\")F)F)*&#\"J' *3_lK\">C\"3Dhz'>Z;ro#H(R\"JTeMC!*HhR;_f1wuJ0ansh'F)&F+6#\"\"*F)F)*&# \"(TMJ#\"(,)=MF)&F+6#\"#5F)F)*&#\"'++;\"'\"[%>F)&F+6#\"#6F)F)&F+6#\"#7 F)#F)\"\"&" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%!G" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%-equation~~~6G" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%!G " }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/,0*&#\")CaVX\"+$frI2#\"\"\"&%\"bG 6#\"\"'F)F)*&#\"(VjV'\",o>@35(F)&F+6#\"\"(F)F)*&#\"-w\\la#*G\"-d2G4nHF )&F+6#\"\")F)F)*&#\"Tw(z\"z'zfyLUi0`.#>GABkFh#e.4c\"U^]J;qerx$HC_([#Hg !GXiuzL\\Kh5F)&F+6#\"\"*F)F)*&#\")*>C-*\"*V%3q9F)&F+6#\"#5F)F)*&#\"(++ ?$\"(,T3%F)&F+6#\"#6F)F)&F+6#\"#7F)#F)F-" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%!G" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%-equation~~~7G" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%!G" }}{PARA 12 "" 1 "" {XPPMATH 20 "6#/,0*& #\"+;W![a\"\"-*GEUL^\"\"\"\"&%\"bG6#\"\"'F)F)*&#\"**)e.[\"\"/k7P:#40\" F)&F+6#\"\"(F)F)*&#\"/'HvB\"Rpc\"/H\"4$G=*=z\"inhV7wBd&*)*4b!zW3(zyh/hi)eq!p!G!=#H Aq\"F)&F+6#\"\"*F)F)*&#\"+hPu=N\"+\\IO@jF)&F+6#\"#5F)F)*&#\")+++k\")@h w&)F)&F+6#\"#6F)F)&F+6#\"#7F)#F)F4" }}{PARA 11 "" 1 "" {XPPMATH 20 "6# %!G" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%-equation~~~8G" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%!G" }}{PARA 12 "" 1 "" {XPPMATH 20 "6#/,0*&#\",W, NBD&\"/(4>&)RZ5\"\"\"\"&%\"bG6#\"\"'F)F)*&#\"+Za#[S$\"1sq%\\(QOb:F)&F+ 6#\"\"(F)F)*&#\"2;!eb#o+76\"\"28%)*oZ!*\\^6F)&F+6#\"\")F)F)*&#\"coO(o7 ?x_]Wp)Hg:LW+sVTl*fIa!QMSH,q&fO/!=6\"corZ0YgvdX!3g+i;xh=oCT/yYV3d%G " 0 "" {MPLTEXT 1 0 199 "e8 := solve(\{op(quade qns)\}):\ne9 := `union`(e7,e8):\nfor ii to 12 do\n wt_val := subs(e9 ,b[ii]);\n if wt_val<>0 then print(b[ii]=wt_val) end if;\nend do:\n` `;\nsubs(e9,[seq(b[i],i=1..12)]):\nevalf[8](%);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/&%\"bG6#\"\"\"#\"3h2'**Q\">XX()\"5+7$*)eL(Q;p>" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#/&%\"bG6#\"\"'#\"?\\60Dv(4+w*GZ![O?%\" @+Sw!G&o9'H*y?X)e(=\"" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/&%\"bG6#\"\" (#\"A)oy(\\?`,L:@Z#eT-^#\"Bvo!G>\"o%=:lRa%ov+,\"" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/&%\"bG6#\"\")#\">Ltm8Rt'Q:DgY6H6\"=+SIY.5]-H+b*yF)" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#/&%\"bG6#\"\"*#\"ho2)=#=LuZgsL?yyQF'z! pexIXOrmeN#fNQ1/Ww9F_^\"ho+))y@%>'RkQ;!*Q*45*p!3A4.p()ou\"4>)fVg<;Z2@A G>" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/&%\"bG6#\"#5#!9\"fb'okA6c?#f#e \"9+XxRSv/q?Qb;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/&%\"bG6#\"#6#\"5$f a$pz\"R*p/V\"5+%yuvb`=7f$" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/&%\"bG6# \"#7#!0hTQ@17d$\"0+v!RIe@E" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%!G" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#7.$\")2?TW!\"*$\"\"!F(F'F'F'$\")0lRN! \")$\"):?&[#F+$\")<,k8!\"($\")A.sEF0$!)@Q>NF0$\")Qn)>\"F0$!)FBi8F0" }} }{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 2 "e9" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1724 "e9 := \{a[ 6,2] = 0, a[5,2] = 0, a[7,2] = 0, a[4,2] = 0, a[8,3] = 0, a[7,3] = 0, \+ a[6,3] = 0, c[7] = 23/148, b[8] = 11291146602515386733913667333/827789 5500290250100346304000, a[4,1] = 12257/317988, a[7,1] = 48028555588961 9/7997789253816320, a[7,6] = 864813381633887/51718297665732608, c[4] = 12257/79497, a[7,5] = -16450769949384489/490009784398315520, a[3,1] = 2090517149/56877957081, a[2,1] = 2/25, a[5,1] = 14131686489425/358339 75601529, a[6,4] = 9757757988/41995818067, a[6,1] = 4418011/96055225, \+ b[9] = 515227147644040638355923558667136453077586907962738787820337260 47743318218807/1928222107471617604359819091746887690309220806991009938 9016386439619421788800, a[5,4] = 17619604667500/11944658533843, c[3] = 24514/238491, a[5,3] = -17700454220625/11944658533843, b[5] = 0, a[4, 3] = 12257/105996, b[2] = 0, b[3] = 0, b[4] = 0, a[8,7] = 19/2, a[6,5] = 19921512441/106300094275, a[8,6] = 51820007951705750121680295/60526 45677641770902708224, b[10] = -582592205611226468655591/16553820700475 4039774500, c[8] = 196/197, c[9] = 14118145156/16038718811, c[12] = 1, c[10] = 39/43, a[7,4] = 52162106556696460538643/464887033284472899365 888, a[8,5] = -3144752379896908085076884133/28673175040579016948961280 0, b[12] = -357120621384161/262158303907500, b[1] = 874545191389960761 /19691638733588931200, a[8,1] = -41924855046865190259777/2906799748576 5047436800, c[11] = 20/21, c[6] = 34/73, c[5] = 50/129, a[8,2] = 0, a[ 3,2] = 3755851225/56877957081, b[11] = 43046993917969354593/3591218535 5757478400, b[6] = 420364804728976000977525051149/11875884520789296146 85280764000, b[7] = 25102415824721153301532049778688/10100756845439651 5184681192806875, a[8,4] = -1573638249639795538408923368457/3379267616 54282072170175876224, c[2] = 2/25\}:" }{TEXT -1 0 "" }}}{PARA 0 "" 0 " " {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}} {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 55 "Now we sp ecify the remaining zero linking coefficients:" }}{PARA 256 "" 0 "" {TEXT -1 2 " " }{XPPEDIT 18 0 "a[9,2] = 0,a[9,3] = 0,a[10,2] = 0,a[10 ,3] = 0,a[11,2] = 0,a[11,3] = 0,a[12,2] = 0,a[12,3] = 0;" "6*/&%\"aG6$ \"\"*\"\"#\"\"!/&F%6$F'\"\"$F)/&F%6$\"#5F(F)/&F%6$F1F-F)/&F%6$\"#6F(F) /&F%6$F8F-F)/&F%6$\"#7F(F)/&F%6$F?F-F)" }{TEXT -1 1 "." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 149 "We include the stag e-order conditions for stages 9 to 12 so that all these stages all hav e stage-order 4 and incorporate the simplifying conditions: " }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "Sum(b[i]*a[i,j],i = j+1 .. \+ 12) = b[j]*(1-c[j]);" "6#/-%$SumG6$*&&%\"bG6#%\"iG\"\"\"&%\"aG6$F+%\"j GF,/F+;,&F0F,F,F,\"#7*&&F)6#F0F,,&F,F,&%\"cG6#F0!\"\"F," }{TEXT -1 1 " " }}{PARA 0 "" 0 "" {TEXT -1 5 "for " }{XPPEDIT 18 0 "j=1" "6#/%\"jG \"\"\"" }{TEXT -1 10 " ( with " }{XPPEDIT 18 0 "c[1]=0" "6#/&%\"cG6# \"\"\"\"\"!" }{TEXT -1 5 " ), " }{XPPEDIT 18 0 "j = 5;" "6#/%\"jG\"\" &" }{TEXT -1 3 ", " }{XPPEDIT 18 0 "j=7" "6#/%\"jG\"\"(" }{TEXT -1 3 ", " }{XPPEDIT 18 0 "j=9" "6#/%\"jG\"\"*" }{TEXT -1 7 " and " } {XPPEDIT 18 0 "j=10" "6#/%\"jG\"#5" }{TEXT -1 2 ". " }}{PARA 0 "" 0 " " {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 195 "Then it turns out that the following additional collection of \"simple\" order conditions (g iven in abreviated form) is sufficient to determine the remaining link ing coefficients in stages 9 to 12." }}{PARA 0 "" 0 "" {TEXT -1 0 "" } }{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 190 "SO8 := SimpleOrderConditio ns(8):\n[seq([i,SO8[i]],i=[37,61,81,87,102,106,113,117,125])]:\nlinalg [augment](linalg[delcols](%,2..2),matrix([[` `]$(linalg[rowdim](%))]) ,linalg[delcols](%,1..1));" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#K%'matri xG6#7+7%\"#P%#~~G/*(%\"bG\"\"\"%\"cGF--%!G6#*&%\"aGF--F06#*&F3F--F06#* &F3F--F06#*&F3F-F.F-F-F-F-F-#F-\"$S)7%\"#hF)/*(F,F-F.F--F06#*&)F.\"\"% F-F3F-F-#F-\"#N7%\"#\")F)/*&F,F--F06#*(F3F-F.F--F06#*&F3F--F06#*&F3F-- F06#*&)F.\"\"#F-F3F-F-F-F-F-#F-\"%gL7%\"#()F)/*(F,F-F.F--F06#*(F3F-F.F --F06#*(F3F-F.F-F:F-F-F-#F-\"$%Q7%\"$-\"F)/*(F,F-F.F--F06#*&F3F--F06#* &F3F--F06#*&)F.\"\"$F-F3F-F-F-F-#F-\"$g*7%\"$1\"F)/*&F,F--F06#*&F3F--F 06#*&F3F-FCF-F-F-#F-\"%!o\"7%\"$8\"F)/*(F,F-FZF-F[pF-#F-\"$g\"7%\"$<\" F)/*&F,F--F06#*(F3F-F.F-FCF-F-#F-\"$!G7%\"$D\"F)/*(F,F-F.F--F06#*&)F. \"\"&F-F3F-F-#F-\"#[Q(pprint26\"" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}{SECT 1 {PARA 4 "" 0 " " {TEXT -1 20 "The associated trees" }}{PARA 0 "" 0 "" {TEXT -1 0 "" } }{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 461 "AST8 := AllSimpleTrees(8): \nwhch := [37,61,81,87,102,106,113,117,125]:\nm := 3: ## number of tre es per row\nnn := nops(whch): q := iquo(nn,m,'r'):\nfor i to nn do \n \+ p||i := DrawTree(AST8[whch[i]],height=4,width=2,show_ordercondition=t rue,\n scale=1.2,color=COLOR(RGB,0,.7,0),font_color=black);\nend do :\npp := plot([[1,1]],style=line,axes=none):\nplots[display](convert([ seq([p||((k-1)*m+1..m*k)],k=1..q),\n `if`(r>0,[p||(m*q+1..nn),pp$(m*( q+1)-nn)],NULL)],array));" }}{PARA 13 "" 1 "" {GLPLOT2D 852 1131 1131 {PLOTDATA 2 "6\\u-%%TEXTG6&7$$\"#g\"\"!$!+5`7'*o!\")Q/b~c~(a~c~)~=~~6 \"-%'COLOURG6&%$RGBGF)F)F)-%%FONTG6$%(COURIERG\"#5-F$6&7$F'$!+)o#)p\"o F,Q1~~~~~~~~~~~~~~~1F.F/F3-F$6&7$F'$!+dZrToF,Q2~~~~~~~~~~~~~~~__F.F/F3 -F$6&7$F'$!+++++qF,Q2~~~~~~~~~~~~~~~48F.F/F3-F$6&7$F'$!+g)*32oF,Q1~~~~ ~~~~~5~~~~~~F.F/F3-%'CURVESG6&7*7$F'$!+ " 0 "" {MPLTEXT 1 0 1 ";" }}}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT 260 4 "Note" }{TEXT -1 119 ": The 5 simplifying condit ions can be replaced by the 5 extra order conditions given (in abrevia ted form) as follows.. " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 174 "SO8 := SimpleOrderConditions(8):\n [seq([i,SO8[i]],i=[21,27,31,36,127])]:\nlinalg[augment](linalg[delcols ](%,2..2),matrix([[` `]$(linalg[rowdim](%))]),linalg[delcols](%,1..1) );" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#K%'matrixG6#7'7%\"#@%#~~G/*&%\"b G\"\"\"-%!G6#*&%\"aGF--F/6#*&F2F--F/6#*&)%\"cG\"\"#F-F2F-F-F-F-#F-\"$g $7%\"#FF)/*&F,F--F/6#*&F2F--F/6#*&)F:\"\"$F-F2F-F-F-#F-\"$?\"7%\"#JF)/ *&F,F--F/6#*&)F:\"\"%F-F2F-F-#F-\"#I7%\"#OF)/*&F,F--F/6#*(F2F-F:F--F/6 #*&F2F--F/6#*&F2F--F/6#*&F2F-F:F-F-F-F-F-#F-\"%357%\"$F\"F)/*&F,F--F/6 #*&)F:\"\"'F-F2F-F-#F-\"#cQ(pprint36\"" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 462 "SOeqs2 := [seq(add(a[i ,j],j= 1..i-1)=c[i],i=9..12),op(StageOrderConditions(2,9..12,'expanded ')),\n op(StageOrderConditions(3,9..12,'expanded')),op(StageOrderCo nditions(4,9..12,'expanded'))]:\nSO8_12 := SimpleOrderConditions(8,12, 'expanded'):\nord_cdns := [seq(SO8_12[i],i=[37,61,81,87,102,113,106,11 7,125])]:\nsimp_eqs := [add(b[i]*a[i,1],i=1+1..12)=b[1],seq(add(b[i]*a [i,j],i=j+1..12)=b[j]*(1-c[j]),j=[5,7,9,10])]:\ncdns := [op(SOeqs2),op (ord_cdns),op(simp_eqs)]:" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 107 "e10 :=\{seq(a[i,2]=0,i=9..12),seq( a[i,3]=0,i=9..12)\}:\ne11 := `union`(e9,e10):\neqns2 := subs(e11,\{op( cdns)\}):" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 39 "There are 30 equations and 30 unknowns." }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 36 "nops(eqns2);\nindets(eqns2);\nnops(%);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#\"#I" }}{PARA 12 "" 1 "" {XPPMATH 20 "6#<@&% \"aG6$\"#5\"\"'&F%6$F'\"\"(&F%6$F'\"\")&F%6$F'\"\"*&F%6$\"#6\"\"\"&F%6 $F1F+&F%6$F1F.&F%6$F'\"\"%&F%6$F'\"\"&&F%6$\"#7F'&F%6$FBF4&F%6$FBF(&F% 6$FBF+&F%6$FBF.&F%6$FBF1&F%6$FBF<&F%6$FBF?&F%6$F4F'&F%6$FBF5&F%6$F4F+& F%6$F4F.&F%6$F4F1&F%6$F4F<&F%6$F4F?&F%6$F4F(&F%6$F'F5&F%6$F1F(&F%6$F1F ?&F%6$F1F5&F%6$F1F<" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#\"#I" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 22 "in folevel[solve] := 4:" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 74 "e12 := solve(\{op(eqns2)\}):\ninfolevel[solve] := 0:\ne13 := `union`(e11, e12):" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 4 "e13 " }} {PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 5369 "e13 := \{a[6,2] = 0, a[5,2] = 0, a[7,2] = 0, a[4,2] = 0, a[9,3] \+ = 0, a[10,3] = 0, a[8,3] = 0, a[12,3] = 0, a[7,3] = 0, a[6,3] = 0, c[7 ] = 23/148, a[12,2] = 0, b[8] = 11291146602515386733913667333/82778955 00290250100346304000, a[4,1] = 12257/317988, a[7,1] = 480285555889619/ 7997789253816320, a[7,6] = 864813381633887/51718297665732608, c[4] = 1 2257/79497, a[7,5] = -16450769949384489/490009784398315520, a[3,1] = 2 090517149/56877957081, a[2,1] = 2/25, a[5,1] = 14131686489425/35833975 601529, a[6,4] = 9757757988/41995818067, a[6,1] = 4418011/96055225, b[ 9] = 51522714764404063835592355866713645307758690796273878782033726047 743318218807/192822210747161760435981909174688769030922080699100993890 16386439619421788800, a[11,9] = 89599398445595686856355698434884216540 038527287573830949657578490242/579385621338495813483588345965887373464 382184677844152053023899833767, a[5,4] = 17619604667500/11944658533843 , a[10,7] = 2224286847389514767028684777161157168782986309/19993702221 5481792634238811967979191208436288, c[3] = 24514/238491, a[5,3] = -177 00454220625/11944658533843, a[9,1] = -14415222609369643343604689319116 750509914398656839842108465299109803139261/286664608340139813702727298 20284862049438129018451533980749016707010746880, b[5] = 0, a[4,3] = 12 257/105996, b[2] = 0, b[3] = 0, b[4] = 0, a[8,7] = 19/2, a[6,5] = 1992 1512441/106300094275, a[11,3] = 0, a[10,5] = -331923915046080335870716 315310401046907/52747587227799808210222170585805619200, a[12,5] = -222 258155907862440804259443447899304774477/189114754716429292220518253019 77449103360, a[10,1] = -9867878858058255039217746719915871073167304993 320293/13771006297995217364330134342641992580302241127219200, a[8,6] = 51820007951705750121680295/6052645677641770902708224, b[10] = -582592 205611226468655591/165538207004754039774500, c[8] = 196/197, a[12,6] = 11410876327678156749715579835467111092952376241624115293181/124908168 7157472843988286395834110562049959611482227605504, a[10,2] = 0, c[9] = 14118145156/16038718811, a[9,2] = 0, a[11,5] = -168891446695104073870 99211531243899609/1891391571300777047888087779210764288, a[12,11] = -5 373869958976816279363488081975/128249819495112776500552350596224, a[12 ,9] = -280813098436463686141658586973841773453049072566975573877036454 598695025876625/105067490432026768588795401172928908496060279595457038 32394231856364095276862976, c[12] = 1, a[11,2] = 0, a[11,8] = 10788313 637659292115861359463369872242886/189380589527306698462795692309376152 568545, a[11,10] = -1786335907413925929868932792832/781445836713759888 5469908059185, c[10] = 39/43, a[10,8] = 498536496800491237121368635810 80955445/2729660625860504418029771633420026504704, a[9,4] = -522416746 7332280028062926335653367380511265991084060397937856421816947952728963 67/7244755263234658466960959547766426403775741588955515644685901338251 2921014002304, a[12,1] = -49410934155054900182882239365042643979034016 237/31731623983903801998088697641133618410239344640, a[7,4] = 52162106 556696460538643/464887033284472899365888, a[8,5] = -314475237989690808 5076884133/286731750405790169489612800, a[12,8] = 46554463321808651332 6922426838859475/27467662576695046776938324444449900032, b[12] = -3571 20621384161/262158303907500, b[1] = 874545191389960761/196916387335889 31200, a[8,1] = -41924855046865190259777/29067997485765047436800, c[11 ] = 20/21, c[6] = 34/73, c[5] = 50/129, a[12,4] = -1074611649128981113 804182104383729704094059/244310054255337140402433369774092097581056, a [12,10] = 50908388575977059406815779042800/129927708457928869841058146 4578279, a[8,2] = 0, a[11,6] = 397099609762427730897418352709319848545 11347767956576333171/5577710294529287682275315773022359477180471284918 417776640, a[3,2] = 3755851225/56877957081, b[11] = 430469939179693545 93/35912185355757478400, a[9,7] = 659267722221658689136549061827869643 14250556615077953597292687341/7114745707172215165239946022376277735416 754491161939984260041034, a[9,8] = 31425442840030946914785278594012470 56878071594901520381912/3228985071785520180284989571741979547706614546 22943631853501, a[11,7] = 95624809679415765680274836872584068739034739 9/77961324245284495254874844291292429525671778, a[9,5] = -824959752614 6667555031510474907199042876873605462323913470411629578679723249/17563 4113965120890123649800888811453867971790089262494990757048747229523968 0, a[9,6] = 2495660098055321637023645805003274681672022448574640720517 109843115480217671807/621558338343793697561396472536427684004319023110 349698478188147786343185391616, a[10,9] = -120374791640701547083352661 0982521566345739350276392522818995/43146504717622211052116150230647115 135945640260504121035832832, b[6] = 420364804728976000977525051149/118 7588452078929614685280764000, a[10,4] = -50210619311607336620209489754 9565186098556680379/59760431150000217012599575938528786496649056256, a [12,7] = 8981318435146082911054040229712538210777263716775/93698406229 0713822722287218869196809455428475968, a[11,4] = -23020115541016752118 27716537011959670051291/274508714084612705166461135767825539832704, a[ 10,6] = 1643147679523140016971844191613351711631376872651547306145/315 813057849945223645118819513361571639686784746580082688, b[7] = 2510241 5824721153301532049778688/101007568454396515184681192806875, a[8,4] = \+ -1573638249639795538408923368457/337926761654282072170175876224, a[11, 1] = -1677150560435164751643947644188299249566198916438951/15242685861 48523692143823046819588641870992602157568, c[2] = 2/25\}:" }{TEXT -1 0 "" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 20 "p rinted coefficients" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 283 "subs(e13,matrix([seq([c[i],seq(a[i,j],j=1. .i-1),``$(7-i)],i=2..7)]));\nfor ii from 8 to 12 do\n print(``);\n \+ print(c[ii]=subs(e13,c[ii])); \n for jj to ii-1 do\n print(a[i i,jj]=subs(e13,a[ii,jj]));\n end do:\nend do:\n``;\nfor ii to 12 do \n print(b[ii]=subs(e13,b[ii]));\nend do:" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#K%'matrixG6#7(7)#\"\"#\"#DF(%!GF+F+F+F+7)#\"&9X#\"'\"\\ Q##\"+\\r^!4#\",\"3dz(o&#\"+D7&ev$F2F+F+F+F+7)#\"&dA\"\"&(\\z#F7\"'))z J\"\"!#F7\"''*f5F+F+F+7)#\"#]\"$H\"#\"/D%*['oJT\"\"/H:gvR$e$F;#!/D1Aa/ q<\"/VQ`eY%>\"#\"/+vm/'>w\"FGF+F+7)#\"#M\"#t#\"(6!=W\")D_0'*F;F;#\"+)) zvd(*\",n!=e*>%#\",TC^@*>\"-vU4+j5F+7)#\"#B\"$[\"#\"0>'*)eb&G![\"1?j\" QD*y(*zF;F;#\"8V'Q0Y'pcl5i@&\"9))eO**GZ%GLq)[Y#!2*[%Q\\*p2X;\"3?bJ)R%y 4+\\#\"0()Qj\"Q8[')\"23EtlwH=<&Q)pprint266\"" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%!G" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/&%\"cG6#\"\")# \"$'>\"$(>" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/&%\"aG6$\"\")\"\"\"#!8x (f->lo/b[#>%\"8+oVZ]w&[(*z1H" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/&%\"a G6$\"\")\"\"#\"\"!" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/&%\"aG6$\"\")\" \"$\"\"!" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/&%\"aG6$\"\")\"\"%#!@d%oL #*3%QbzR'\\#QOd\"\"?Ci(e\" \"#" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%!G" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/&%\"cG6#\"\"*#\",c^9=T\"\",6)=(Qg\"" }}{PARA 11 "" 1 " " {XPPMATH 20 "6#/&%\"aG6$\"\"*\"\"\"#!eoh#RJ!)4\"*Hl%3@%)Rol)R9*40v; \">$*o/OMV'p$4EA:W\"\"eo!)ou5qq;!\\2)R`^%=!H\"Q%\\?'[G?)HFFq8)R,M3Ym'G " }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/&%\"aG6$\"\"*\"\"#\"\"!" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/&%\"aG6$\"\"*\"\"$\"\"!" }}{PARA 11 "" 1 " " {XPPMATH 20 "6#/&%\"aG6$\"\"*\"\"%#!\\pnj*GF&z%p\"=Ucy$zRgS3\"*fE60Q nLlNj#H1G+GKtYnTA&\"[p/B+95#H^#Q8!foWc^b*)eTdx.kUmxaf4'pYeYBj_vWs" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#/&%\"aG6$\"\"*\"\"&#!go\\Ksz'y&H;TqM\" RKiagto(G/*>2\\Z5:.bvmYh_(f\\#)\"go!oR_Hsu[qv!*\\\\i#*3!zrz'QX6)))3!) \\O7!*37lR6Mc<" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/&%\"aG6$\"\"*\"\"'# \"jo2=n<-[:J%)4r^?2ku&[C-s;ouK+0ekBqj@`0)4gc\\#\"io;;R&=VjyZ\")=y%)p\\ .6B!>V+%oFk`skRhvp$zV$Q$e:i" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/&%\"aG 6$\"\"*\"\"(#\"\\oTto#H(f`z2:mb]UJkpy#=1\\l8*oe;AAxEf'\"[oM5/gU)*R>;\" \\anTNxFwB-Y*R_;:A<2du9r" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/&%\"aG6$ \"\"*\"\")#\"en7>Q?:!\\fr!yo0Z7Sfy_y9p%4.SGWD9$\"gn,N&=jVHia9mqZ&z>ur& *)\\G!=?byr])*GK" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%!G" }}{PARA 11 " " 1 "" {XPPMATH 20 "6#/&%\"cG6#\"#5#\"#R\"#V" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/&%\"aG6$\"#5\"\"\"#!U$H?L*\\InJ2re\"*>nu<#R]De!e)yy')* \"V+#>s7TAI!e#*>kUV8IVO<_*zH15x8" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/& %\"aG6$\"#5\"\"#\"\"!" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/&%\"aG6$\"#5 \"\"$\"\"!" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/&%\"aG6$\"#5\"\"%#!Qz.o c&)4'=l&\\v*[4-iOtg6$>1@]\"Pci0\\m\\'yG&Qfd*f7q@++:J/wf" }}{PARA 11 " " 1 "" {XPPMATH 20 "6#/&%\"aG6$\"#5\"\"&#!H2p/,/J:jrqeL!3Y]\"R#>L\"G+# >c!eeq@A5#3)*zFseZF&" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/&%\"aG6$\"#5 \"\"'#\"enXhIZ:lsoPJ;r^Lh\">W=(p,SJ_zw9V;\"Z)o#3!eYZy'oR;dhL^>)=^kB_%* \\y08eJ" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/&%\"aG6$\"#5\"\"(#\"O4j)Hy or:hrx%oGqw9&*QZoGCA\"N)GO%37>zz'>\")QUj#z\"[:A-P**>" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/&%\"aG6$\"#5\"\")#\"GXa&43\"ejo87P7\\+o\\O&)\\\"I/Z ]E+UL;xH!=W]geig'HF" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/&%\"aG6$\"#5\" \"*#!hn&**=G_#Rw-NRdMm:_#)4hEN$3Z:qS;zu.7\"inKG$e.@T]g-kXf8:rkI-:;@06A im&\\#*H)=Ww%Rk^Z;N/c]rn\"\"Uov:-E*4( =k)e>o/BQ9#pB&[heoU_\"" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/&%\"aG6$\"# 6\"\"#\"\"!" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/&%\"aG6$\"#6\"\"$\"\"! " }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/&%\"aG6$\"#6\"\"%#!L\"H^+nf>,PlrF =@v;5a:,-B\"K/F$)Rb#ywN6Ym^q7Y39(3XF" }}{PARA 11 "" 1 "" {XPPMATH 20 " 6#/&%\"aG6$\"#6\"\"&#!G4'**QCJ:@*4(Q2/^pY9*)o\"\"F)Gk2@zx3))y/x2Ir:R\" *=" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/&%\"aG6$\"#6\"\"'#\"fnrJLwl&zwZ 8^a[)>$4FN=u*3tFCw4'*4(R\"enSmx<%=\\Gr/=x%fB-tdJvAo(GHXH5xd&" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/&%\"aG6$\"#6\"\"(#\"N*RZ.R(oSeso$[F!oldTz'4 [i&*\"My!Q!R)RiJ<$" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/&%\"aG6$\"#7\"\"#\"\"!" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/&%\"aG6$\"#7\"\"$\"\"!" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/&%\" aG6$\"#7\"\"%#!LfS4/(HPQ/@=/Q6\")*G\"\\;hu5\"Kc5e(4#4u(pLV-/9P`Da+JW# " }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/&%\"aG6$\"#7\"\"&#!KxWx/$**yWV%fU !3Wiy!f:eAA\"JgL5\\u(>ID=0A#HHkraZ6*=" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/&%\"aG6$\"#7\"\"'#\"fn\"=$H:TiTiP_H46rYN)zbr\\n:ywKw3T6\"en/bgFA[6 'f*\\?c5T$eR'G))R%GZdro\"3\\7" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/&%\" aG6$\"#7\"\"(#\"Rvnrjsx5#QDrH-/a5\"H3Y^V=8)*)\"QofZGaX4o>p)=sGAF#Qr!Hi S)p$*" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/&%\"aG6$\"#7\"\")#\"Ev%f)QoU ApK8l3=KjWbY\"GK+!*\\WWC$QpxY]pwDmnu#" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/&%\"aG6$\"#7\"\"*#!ioDm(e-&p)fak.xQdvpcs!\\IXt<%Q(pee;9'ojkV)483G \"[pwH'oF&4kj&=B%RKQqX&fz-1'\\3*GH<,az)eon-K/\\n]5" }}{PARA 11 "" 1 " " {XPPMATH 20 "6#/&%\"aG6$\"#7\"#5#\"A+G/zd\"oSfq(fd)Q34&\"Cz#yXY\"e5% )p)GzX3x#*H\"" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/&%\"aG6$\"#7\"#6#!@v >3)[j$zi\"o(*e*pQP&\"BCif]Bb+lx7^\\>)\\#G\"" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%!G" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/&%\"bG6#\"\"\"# \"3h2'**Q\">XX()\"5+7$*)eL(Q;p>" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/&% \"bG6#\"\"#\"\"!" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/&%\"bG6#\"\"$\"\" !" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/&%\"bG6#\"\"%\"\"!" }}{PARA 11 " " 1 "" {XPPMATH 20 "6#/&%\"bG6#\"\"&\"\"!" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/&%\"bG6#\"\"'#\"?\\60Dv(4+w*GZ![O?%\"@+Sw!G&o9'H*y?X)e (=\"" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/&%\"bG6#\"\"(#\"A)oy(\\?`,L:@ Z#eT-^#\"Bvo!G>\"o%=:lRa%ov+,\"" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/&% \"bG6#\"\")#\">Ltm8Rt'Q:DgY6H6\"=+SIY.5]-H+b*yF)" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/&%\"bG6#\"\"*#\"ho2)=#=LuZgsL?yyQF'z!pexIXOrmeN#fNQ1/W w9F_^\"ho+))y@%>'RkQ;!*Q*45*p!3A4.p()ou\"4>)fVg<;Z2@AG>" }}{PARA 11 " " 1 "" {XPPMATH 20 "6#/&%\"bG6#\"#5#!9\"fb'okA6c?#f#e\"9+XxRSv/q?Qb;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/&%\"bG6#\"#6#\"5$fa$pz\"R*p/V\"5+%y uvb`=7f$" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/&%\"bG6#\"#7#!0hTQ@17d$\" 0+v!RIe@E" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 " " {MPLTEXT 1 0 1 ";" }}}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}}{PARA 0 "" 0 "" {TEXT -1 52 "#----- ----------------------------------------------" }}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 43 "construction of the order 7 embedded scheme" }} {PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 4 "e13 " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 5369 "e13 := \{a [6,2] = 0, a[5,2] = 0, a[7,2] = 0, a[4,2] = 0, a[9,3] = 0, a[10,3] = 0 , a[8,3] = 0, a[12,3] = 0, a[7,3] = 0, a[6,3] = 0, c[7] = 23/148, a[12 ,2] = 0, b[8] = 11291146602515386733913667333/827789550029025010034630 4000, a[4,1] = 12257/317988, a[7,1] = 480285555889619/7997789253816320 , a[7,6] = 864813381633887/51718297665732608, c[4] = 12257/79497, a[7, 5] = -16450769949384489/490009784398315520, a[3,1] = 2090517149/568779 57081, a[2,1] = 2/25, a[5,1] = 14131686489425/35833975601529, a[6,4] = 9757757988/41995818067, a[6,1] = 4418011/96055225, b[9] = 51522714764 404063835592355866713645307758690796273878782033726047743318218807/192 8222107471617604359819091746887690309220806991009938901638643961942178 8800, a[11,9] = 895993984455956868563556984348842165400385272875738309 49657578490242/5793856213384958134835883459658873734643821846778441520 53023899833767, a[5,4] = 17619604667500/11944658533843, a[10,7] = 2224 286847389514767028684777161157168782986309/199937022215481792634238811 967979191208436288, c[3] = 24514/238491, a[5,3] = -17700454220625/1194 4658533843, a[9,1] = -144152226093696433436046893191167505099143986568 39842108465299109803139261/2866646083401398137027272982028486204943812 9018451533980749016707010746880, b[5] = 0, a[4,3] = 12257/105996, b[2] = 0, b[3] = 0, b[4] = 0, a[8,7] = 19/2, a[6,5] = 19921512441/10630009 4275, a[11,3] = 0, a[10,5] = -331923915046080335870716315310401046907/ 52747587227799808210222170585805619200, a[12,5] = -2222581559078624408 04259443447899304774477/18911475471642929222051825301977449103360, a[1 0,1] = -9867878858058255039217746719915871073167304993320293/137710062 97995217364330134342641992580302241127219200, a[8,6] = 518200079517057 50121680295/6052645677641770902708224, b[10] = -5825922056112264686555 91/165538207004754039774500, c[8] = 196/197, a[12,6] = 114108763276781 56749715579835467111092952376241624115293181/1249081687157472843988286 395834110562049959611482227605504, a[10,2] = 0, c[9] = 14118145156/160 38718811, a[9,2] = 0, a[11,5] = -1688914466951040738709921153124389960 9/1891391571300777047888087779210764288, a[12,11] = -53738699589768162 79363488081975/128249819495112776500552350596224, a[12,9] = -280813098 436463686141658586973841773453049072566975573877036454598695025876625/ 1050674904320267685887954011729289084960602795954570383239423185636409 5276862976, c[12] = 1, a[11,2] = 0, a[11,8] = 107883136376592921158613 59463369872242886/189380589527306698462795692309376152568545, a[11,10] = -1786335907413925929868932792832/7814458367137598885469908059185, c [10] = 39/43, a[10,8] = 49853649680049123712136863581080955445/2729660 625860504418029771633420026504704, a[9,4] = -5224167467332280028062926 33565336738051126599108406039793785642181694795272896367/7244755263234 6584669609595477664264037757415889555156446859013382512921014002304, a [12,1] = -49410934155054900182882239365042643979034016237/317316239839 03801998088697641133618410239344640, a[7,4] = 52162106556696460538643/ 464887033284472899365888, a[8,5] = -3144752379896908085076884133/28673 1750405790169489612800, a[12,8] = 465544633218086513326922426838859475 /27467662576695046776938324444449900032, b[12] = -357120621384161/2621 58303907500, b[1] = 874545191389960761/19691638733588931200, a[8,1] = \+ -41924855046865190259777/29067997485765047436800, c[11] = 20/21, c[6] \+ = 34/73, c[5] = 50/129, a[12,4] = -10746116491289811138041821043837297 04094059/244310054255337140402433369774092097581056, a[12,10] = 509083 88575977059406815779042800/1299277084579288698410581464578279, a[8,2] \+ = 0, a[11,6] = 3970996097624277308974183527093198485451134776795657633 3171/5577710294529287682275315773022359477180471284918417776640, a[3,2 ] = 3755851225/56877957081, b[11] = 43046993917969354593/3591218535575 7478400, a[9,7] = 6592677222216586891365490618278696431425055661507795 3597292687341/71147457071722151652399460223762777354167544911619399842 60041034, a[9,8] = 314254428400309469147852785940124705687807159490152 0381912/322898507178552018028498957174197954770661454622943631853501, \+ a[11,7] = 956248096794157656802748368725840687390347399/77961324245284 495254874844291292429525671778, a[9,5] = -8249597526146667555031510474 907199042876873605462323913470411629578679723249/175634113965120890123 6498008888114538679717900892624949907570487472295239680, a[9,6] = 2495 6600980553216370236458050032746816720224485746407205171098431154802176 71807/6215583383437936975613964725364276840043190231103496984781881477 86343185391616, a[10,9] = -1203747916407015470833526610982521566345739 350276392522818995/431465047176222110521161502306471151359456402605041 21035832832, b[6] = 420364804728976000977525051149/1187588452078929614 685280764000, a[10,4] = -502106193116073366202094897549565186098556680 379/59760431150000217012599575938528786496649056256, a[12,7] = 8981318 435146082911054040229712538210777263716775/936984062290713822722287218 869196809455428475968, a[11,4] = -230201155410167521182771653701195967 0051291/274508714084612705166461135767825539832704, a[10,6] = 16431476 79523140016971844191613351711631376872651547306145/3158130578499452236 45118819513361571639686784746580082688, b[7] = 25102415824721153301532 049778688/101007568454396515184681192806875, a[8,4] = -157363824963979 5538408923368457/337926761654282072170175876224, a[11,1] = -1677150560 435164751643947644188299249566198916438951/152426858614852369214382304 6819588641870992602157568, c[2] = 2/25\}:" }{TEXT -1 0 "" }}}{PARA 0 " " 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}} }{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 61 "We can o btain an embedded 11 stage order 7 scheme as follows." }}{PARA 0 "" 0 "" {TEXT -1 94 "We remove stages 11 and 12 from the 12 stage order 8 s cheme and introduce a new stage 11 with " }{XPPEDIT 18 0 "c[11] = 1;" "6#/&%\"cG6#\"#6\"\"\"" }{TEXT -1 3 ", " }{XPPEDIT 18 0 "a[11,2] = 0 " "6#/&%\"aG6$\"#6\"\"#\"\"!" }{TEXT -1 3 ", " }{XPPEDIT 18 0 "a[11,3 ] = 0;" "6#/&%\"aG6$\"#6\"\"$\"\"!" }{TEXT -1 3 ", " }{XPPEDIT 18 0 " `b*`[2]=0" "6#/&%#b*G6#\"\"#\"\"!" }{TEXT -1 3 ", " }{XPPEDIT 18 0 "` b*`[3]=0" "6#/&%#b*G6#\"\"$\"\"!" }{TEXT -1 3 ", " }{XPPEDIT 18 0 "`b *`[4]=0" "6#/&%#b*G6#\"\"%\"\"!" }{TEXT -1 7 " and " }{XPPEDIT 18 0 "`b*`[5]=0" "6#/&%#b*G6#\"\"&\"\"!" }{TEXT -1 58 ", where the weights \+ of the order 7 scheme are denoted by " }{XPPEDIT 18 0 "`b*`" "6#%#b*G " }{TEXT -1 1 "." }}{PARA 0 "" 0 "" {TEXT -1 1 " " }}{PARA 0 "" 0 "" {TEXT -1 156 "We incorporate the order 7 quadrature conditions, the ro w sum conditions for this stage and stage-order conditions so that thi s new stage has stage-order 4." }}{PARA 0 "" 0 "" {TEXT -1 48 "We also incorporate the simplifying conditions: " }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "Sum(`b*`[i]*a[i,j],i=j+1..11)=`b*`[j]*(1-c[j] )" "6#/-%$SumG6$*&&%#b*G6#%\"iG\"\"\"&%\"aG6$F+%\"jGF,/F+;,&F0F,F,F,\" #6*&&F)6#F0F,,&F,F,&%\"cG6#F0!\"\"F," }{TEXT -1 1 "," }}{PARA 0 "" 0 " " {TEXT -1 5 "for " }{XPPEDIT 18 0 "j=1" "6#/%\"jG\"\"\"" }{TEXT -1 10 " ( with " }{XPPEDIT 18 0 "c[1]=0" "6#/&%\"cG6#\"\"\"\"\"!" } {TEXT -1 5 " ), " }{XPPEDIT 18 0 "j = 8;" "6#/%\"jG\"\")" }{TEXT -1 3 ", " }{XPPEDIT 18 0 "j=9" "6#/%\"jG\"\"*" }{TEXT -1 7 " and " } {XPPEDIT 18 0 "j=10" "6#/%\"jG\"#5" }{TEXT -1 2 ". " }}{PARA 0 "" 0 " " {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 399 "`Qeqs*` : = subs(b=`b*`,QuadratureConditions(7,11,'expanded')):\nSO_eqs := [add( a[11,j],j=1..10)=c[11],add(a[11,j]*c[j],j=2..10)=1/2*c[11]^2,\n \+ add(a[11,j]*c[j]^2,j=2..10)=1/3*c[11]^3,add(a[11,j]*c[j]^3,j=2..10)=1 /4*c[11]^4]:\n`simp_eqs*` := [add(`b*`[i]*a[i,1],i=2..11)=`b*`[1],seq( add(`b*`[i]*a[i,j],i=j+1..11)=`b*`[j]*(1-c[j]),j=[8,9,10])]:\n`cdns*` \+ := [op(`simp_eqs*`),op(SO_eqs),op(`Qeqs*`)]:" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 144 "e14 := `uni on`(remove(u_->member(op(1,lhs(u_)),[11,12]) or op(0,lhs(u_))=b,e13), \n \{c[11]=1,a[11,2]=0,a[11,3]=0,seq(`b*`[i]=0,i=2..5)\}):\n" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 53 "We hav e 15 equations for the 15 unknown coefficients." }}{EXCHG {PARA 0 "> \+ " 0 "" {MPLTEXT 1 0 60 "eqns3 := subs(e14,`cdns*`):\nnops(%);\nindets( eqns3);\nnops(%);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#\"#:" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#<1&%\"aG6$\"#6\"\"\"&F%6$F'\"#5&F%6$F'\"\"(&F%6 $F'\"\")&F%6$F'\"\"*&F%6$F'\"\"%&F%6$F'\"\"&&F%6$F'\"\"'&%#b*G6#F=&F?6 #F.&F?6#F1&F?6#F4&F?6#F+&F?6#F'&F?6#F(" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#\"#:" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 " " {MPLTEXT 1 0 22 "infolevel[solve] := 4:" }}}{EXCHG {PARA 0 "> " 0 " " {MPLTEXT 1 0 74 "e15 := solve(\{op(eqns3)\}):\ninfolevel[solve] := 0 :\ne16 := `union`(e14,e15):" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}{SECT 1 {PARA 4 "" 0 " " {TEXT -1 3 "e16" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 4329 "e16 := \{a[6,2] = 0, a[5,2] = 0, a[7,2] = 0 , a[4,2] = 0, a[9,3] = 0, a[10,3] = 0, a[11,4] = -14214628354155899567 78218218848342421532216119/3491230464320813022691747524514154235197480 96, a[8,3] = 0, a[7,3] = 0, a[6,3] = 0, c[7] = 23/148, a[4,1] = 12257/ 317988, `b*`[10] = -6457286835032584968329/2546741646226985227300, a[7 ,1] = 480285555889619/7997789253816320, a[7,6] = 864813381633887/51718 297665732608, `b*`[6] = 225910982205515703574046329/636776649908273251 841973600, c[4] = 12257/79497, a[7,5] = -16450769949384489/49000978439 8315520, a[3,1] = 2090517149/56877957081, a[2,1] = 2/25, a[5,1] = 1413 1686489425/35833975601529, a[6,4] = 9757757988/41995818067, a[6,1] = 4 418011/96055225, a[5,4] = 17619604667500/11944658533843, a[10,7] = 222 4286847389514767028684777161157168782986309/19993702221548179263423881 1967979191208436288, c[3] = 24514/238491, a[5,3] = -17700454220625/119 44658533843, a[9,1] = -14415222609369643343604689319116750509914398656 839842108465299109803139261/286664608340139813702727298202848620494381 29018451533980749016707010746880, a[4,3] = 12257/105996, `b*`[9] = 143 021962768113394257793894874175054076879315632471495031519549972079/634 97997867281994630609975410189529208101511709161157616376634016000, a[8 ,7] = 19/2, a[6,5] = 19921512441/106300094275, a[11,3] = 0, a[10,5] = \+ -331923915046080335870716315310401046907/52747587227799808210222170585 805619200, `b*`[7] = 50630668490428660369928685568/2038909334969651093 75618071875, a[10,1] = -9867878858058255039217746719915871073167304993 320293/13771006297995217364330134342641992580302241127219200, a[8,6] = 51820007951705750121680295/6052645677641770902708224, a[11,7] = 15921 139337350831535408442010420239146408066675/174938297710504721694052353 6513333290389079616, c[8] = 196/197, a[10,2] = 0, c[9] = 14118145156/1 6038718811, a[9,2] = 0, a[11,2] = 0, c[11] = 1, c[10] = 39/43, a[10,8] = 49853649680049123712136863581080955445/2729660625860504418029771633 420026504704, a[9,4] = -5224167467332280028062926335653367380511265991 08406039793785642181694795272896367/7244755263234658466960959547766426 4037757415889555156446859013382512921014002304, `b*`[5] = 0, `b*`[3] = 0, `b*`[4] = 0, `b*`[2] = 0, a[7,4] = 52162106556696460538643/4648870 33284472899365888, a[11,9] = -8423048909040161063576104721713644717435 18622807702364576842637748625/1600110710453658437068687306242176955368 5916890578695272031985694176768, a[8,5] = -314475237989690808507688413 3/286731750405790169489612800, a[11,8] = -6448409626829244796699585663 0949967425/5208159300615981523251328952205675621888, a[8,1] = -4192485 5046865190259777/29067997485765047436800, a[11,6] = 411387010154201470 248493205505950446147325414444198427659/464602950060729095914281707500 88268098474825959918534656, `b*`[11] = -4954671690517567/1310791519537 500, c[6] = 34/73, a[11,10] = 1124804893366359619795687870063500/18026 098196604212076234696000197113, c[5] = 50/129, a[8,2] = 0, a[3,2] = 37 55851225/56877957081, a[11,5] = -2984312771192069486473441797268836080 579319/262376761616550996396340471077385680977920, a[9,7] = 6592677222 2165868913654906182786964314250556615077953597292687341/71147457071722 15165239946022376277735416754491161939984260041034, a[9,8] = 314254428 4003094691478527859401247056878071594901520381912/32289850717855201802 8498957174197954770661454622943631853501, a[9,5] = -824959752614666755 5031510474907199042876873605462323913470411629578679723249/17563411396 51208901236498008888114538679717900892624949907570487472295239680, a[9 ,6] = 2495660098055321637023645805003274681672022448574640720517109843 115480217671807/621558338343793697561396472536427684004319023110349698 478188147786343185391616, a[11,1] = -683484590079849331110367011158656 24594922076006017/45345007463014711631562120393860177858153484042240, \+ a[10,9] = -12037479164070154708335266109825215663457393502763925228189 95/43146504717622211052116150230647115135945640260504121035832832, a[1 0,4] = -502106193116073366202094897549565186098556680379/5976043115000 0217012599575938528786496649056256, `b*`[8] = 365508105808873225907435 9101/827789550029025010034630400, a[10,6] = 16431476795231400169718441 91613351711631376872651547306145/3158130578499452236451188195133615716 39686784746580082688, a[8,4] = -1573638249639795538408923368457/337926 761654282072170175876224, `b*`[1] = 13470793039761431/3029482882090604 80, c[2] = 2/25\}:" }{TEXT -1 0 "" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 20 "printed coefficients" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 289 "subs(e16,ma trix([seq([c[i],seq(a[i,j],j=1..i-1),``$(7-i)],i=2..7)]));\nfor ii fro m 8 to 11 do\n print(``);\n print(c[ii]=subs(e16,c[ii])); \n for jj to ii-1 do\n print(a[ii,jj]=subs(e16,a[ii,jj]));\n end do: \nend do:\n``;\nfor ii to 11 do\n print(`b*`[ii]=subs(e16,`b*`[ii])) ;\nend do:" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#K%'matrixG6#7(7)#\"\"#\" #DF(%!GF+F+F+F+7)#\"&9X#\"'\"\\Q##\"+\\r^!4#\",\"3dz(o&#\"+D7&ev$F2F+F +F+F+7)#\"&dA\"\"&(\\z#F7\"'))zJ\"\"!#F7\"''*f5F+F+F+7)#\"#]\"$H\"#\"/ D%*['oJT\"\"/H:gvR$e$F;#!/D1Aa/q<\"/VQ`eY%>\"#\"/+vm/'>w\"FGF+F+7)#\"# M\"#t#\"(6!=W\")D_0'*F;F;#\"+))zvd(*\",n!=e*>%#\",TC^@*>\"-vU4+j5F+7)# \"#B\"$[\"#\"0>'*)eb&G![\"1?j\"QD*y(*zF;F;#\"8V'Q0Y'pcl5i@&\"9))eO**GZ %GLq)[Y#!2*[%Q\\*p2X;\"3?bJ)R%y4+\\#\"0()Qj\"Q8[')\"23EtlwH=<&Q)pprint 276\"" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%!G" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/&%\"cG6#\"\")#\"$'>\"$(>" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/&%\"aG6$\"\")\"\"\"#!8x(f->lo/b[#>%\"8+oVZ]w&[(*z1H" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/&%\"aG6$\"\")\"\"#\"\"!" }}{PARA 11 "" 1 " " {XPPMATH 20 "6#/&%\"aG6$\"\")\"\"$\"\"!" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/&%\"aG6$\"\")\"\"%#!@d%oL#*3%QbzR'\\#QOd\"\"?Ci(e\"\"#" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%!G" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/&%\"cG6#\"\"*# \",c^9=T\"\",6)=(Qg\"" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/&%\"aG6$\"\" *\"\"\"#!eoh#RJ!)4\"*Hl%3@%)Rol)R9*40v;\">$*o/OMV'p$4EA:W\"\"eo!)ou5qq ;!\\2)R`^%=!H\"Q%\\?'[G?)HFFq8)R,M3Ym'G" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/&%\"aG6$\"\"*\"\"#\"\"!" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/&% \"aG6$\"\"*\"\"$\"\"!" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/&%\"aG6$\"\" *\"\"%#!\\pnj*GF&z%p\"=Ucy$zRgS3\"*fE60QnLlNj#H1G+GKtYnTA&\"[p/B+95#H^ #Q8!foWc^b*)eTdx.kUmxaf4'pYeYBj_vWs" }}{PARA 11 "" 1 "" {XPPMATH 20 "6 #/&%\"aG6$\"\"*\"\"&#!go\\Ksz'y&H;TqM\"RKiagto(G/*>2\\Z5:.bvmYh_(f\\#) \"go!oR_Hsu[qv!*\\\\i#*3!zrz'QX6)))3!)\\O7!*37lR6Mc<" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/&%\"aG6$\"\"*\"\"'#\"jo2=n<-[:J%)4r^?2ku&[C-s;ouK+0 ekBqj@`0)4gc\\#\"io;;R&=VjyZ\")=y%)p\\.6B!>V+%oFk`skRhvp$zV$Q$e:i" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#/&%\"aG6$\"\"*\"\"(#\"\\oTto#H(f`z2:mb ]UJkpy#=1\\l8*oe;AAxEf'\"[oM5/gU)*R>;\"\\anTNxFwB-Y*R_;:A<2du9r" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#/&%\"aG6$\"\"*\"\")#\"en7>Q?:!\\fr!yo0 Z7Sfy_y9p%4.SGWD9$\"gn,N&=jVHia9mqZ&z>ur&*)\\G!=?byr])*GK" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%!G" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/&%\"cG6 #\"#5#\"#R\"#V" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/&%\"aG6$\"#5\"\"\"# !U$H?L*\\InJ2re\"*>nu<#R]De!e)yy')*\"V+#>s7TAI!e#*>kUV8IVO<_*zH15x8" } }{PARA 11 "" 1 "" {XPPMATH 20 "6#/&%\"aG6$\"#5\"\"#\"\"!" }}{PARA 11 " " 1 "" {XPPMATH 20 "6#/&%\"aG6$\"#5\"\"$\"\"!" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/&%\"aG6$\"#5\"\"%#!Qz.oc&)4'=l&\\v*[4-iOtg6$>1@]\"Pci0 \\m\\'yG&Qfd*f7q@++:J/wf" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/&%\"aG6$ \"#5\"\"&#!H2p/,/J:jrqeL!3Y]\"R#>L\"G+#>c!eeq@A5#3)*zFseZF&" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/&%\"aG6$\"#5\"\"'#\"enXhIZ:lsoPJ;r^Lh\">W=( p,SJ_zw9V;\"Z)o#3!eYZy'oR;dhL^>)=^kB_%*\\y08eJ" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/&%\"aG6$\"#5\"\"(#\"O4j)Hyor:hrx%oGqw9&*QZoGCA\"N)GO%3 7>zz'>\")QUj#z\"[:A-P**>" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/&%\"aG6$ \"#5\"\")#\"GXa&43\"ejo87P7\\+o\\O&)\\\"I/Z]E+UL;xH!=W]geig'HF" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#/&%\"aG6$\"#5\"\"*#!hn&**=G_#Rw-NRdMm: _#)4hEN$3Z:qS;zu.7\"inKG$e.@T]g-kXf8:rkI-:;@06Aih@K:UU$[)=#=#yn&**e:a$GY@9\"N'4[(>NU:9X_Z< pAI\"3Kk/B\"\\$" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/&%\"aG6$\"#6\"\"&# !L>$z03O)oszTMZ'[p?>rFJ%)H\"K?z(4o&Qx5ZSjR'*4b;;wwBE" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/&%\"aG6$\"#6\"\"'#\"ZfwU)>WWTDt9Y/&f]0K\\[-Z,U:5qQ6 %\"YcY`=*ff#[Z)4o#)3]2.s_py0*o\"fo`&p'fjO$*[![7\"\"D8r>+gpMi27 Ug'>)4E!=" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%!G" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/&%#b*G6#\"\"\"#\"2J9wRIzqM\"\"3![g!4#)G[HI" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/&%#b*G6#\"\"#\"\"!" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/&%#b*G6#\"\"$\"\"!" }}{PARA 11 "" 1 "" {XPPMATH 20 "6# /&%#b*G6#\"\"%\"\"!" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/&%#b*G6#\"\"& \"\"!" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/&%#b*G6#\"\"'#\"%=Dt#3*\\mxO'" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/&%#b*G6# \"\"(#\">oboG*p.mG/\\o1j]\"?v=2=cP4^'p\\L4*Q?" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/&%#b*G6#\"\")#\"=,\"fV2fAt)3e53bO\"<+/jM+,D!H+b*yF)" } }{PARA 11 "" 1 "" {XPPMATH 20 "6#/&%#b*G6#\"\"*#\"`oz?(*\\&>:.&\\rCj:$ zo2a]-V\"\"_o+g,MmP;w:h\"4<^,\"3#H&*=5a(*41j%*>Gny*z\\ j" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/&%#b*G6#\"#5#!7H$o\\eK]$oGdk\"7+ tA&)pAY;uYD" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/&%#b*G6#\"#6#!1nv^!prY &\\\"1+v`>:z58" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> \+ " 0 "" {MPLTEXT 1 0 1 ";" }}}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 104 "`RK7_11eqs*` := subs(b=`b*`,[op(Ro wSumConditions(11,'expanded')),op(OrderConditions(7,11,'expanded'))]): " }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 5 "Check " }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 57 "subs(e16,`RK7_11eqs*`): \nmap(u->lhs(u)-rhs(u),%);\nnops(%);" }}{PARA 12 "" 1 "" {XPPMATH 20 " 6#7[q\"\"!F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$ F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$ F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#\"#&*" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }} {PARA 0 "" 0 "" {TEXT -1 93 "We can calculate the principal error norm , that is, the 2-norm of the principal error terms. " }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 185 "`errterms7_11*` := subs(b=`b*`,Principal ErrorTerms(7,11,'expanded')):\nsm := 0:\nfor ct to nops(`errterms7_11* `) do\n sm := sm+(evalf(subs(e16,`errterms7_11*`[ct])))^2;\nend do: \nsqrt(sm);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#$\"+g8875!#9" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 138 "We can include the new stage for the embedded scheme as an additional 13th stage add ed to the order 8 scheme along with the coefficients " }{XPPEDIT 18 0 "a[13,11] = 0;" "6#/&%\"aG6$\"#8\"#6\"\"!" }{TEXT -1 7 " and " } {XPPEDIT 18 0 "a[13,12] = 0;" "6#/&%\"aG6$\"#8\"#7\"\"!" }{TEXT -1 1 " ." }}{PARA 0 "" 0 "" {TEXT -1 13 "The weights " }{XPPEDIT 18 0 "`b*`[ i]" "6#&%#b*G6#%\"iG" }{TEXT -1 7 " for " }{XPPEDIT 18 0 "i=1" "6#/% \"iG\"\"\"" }{TEXT -1 89 " . . 10 of the 13 stage combined scheme are \+ those of the 11 stage scheme and the weight " }{XPPEDIT 18 0 "`b*`[11 ];" "6#&%#b*G6#\"#6" }{TEXT -1 34 " in the 11 stage scheme becomes \+ " }{XPPEDIT 18 0 "`b*`[13];" "6#&%#b*G6#\"#8" }{TEXT -1 25 " in the 1 3 stage scheme." }}{PARA 0 "" 0 "" {TEXT -1 8 "We set " }{XPPEDIT 18 0 "c[13] = 1;" "6#/&%\"cG6#\"#8\"\"\"" }{TEXT -1 3 ", " }{XPPEDIT 18 0 "`b*`[11] = 0;" "6#/&%#b*G6#\"#6\"\"!" }{TEXT -1 7 " and " } {XPPEDIT 18 0 "`b*`[12] = 0;" "6#/&%#b*G6#\"#7\"\"!" }{TEXT -1 1 "." } }{PARA 0 "" 0 "" {TEXT -1 66 "We can make the order 8 scheme into a 13 stage scheme by setting " }{XPPEDIT 18 0 "b[13] = 0;" "6#/&%\"bG6#\" #8\"\"!" }{TEXT -1 1 "." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 201 "e17 := \{c[13]=1,seq(a[13,i]=subs( e16,a[11,i]),i=1..10),a[13,11]=0,a[13,12]=0,b[13]=0,\nseq(`b*`[i]=subs (e16,`b*`[i]),i=1..10),`b*`[11]=0,`b*`[12]=0,`b*`[13]=subs(e16,`b*`[11 ])\}:\ne18 := `union`(e13,e17):" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}{SECT 1 {PARA 4 "" 0 " " {TEXT -1 3 "e18" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 6915 "e18 := \{a[6,2] = 0, a[5,2] = 0, a[7,2] = 0 , a[4,2] = 0, a[9,3] = 0, a[10,3] = 0, a[8,3] = 0, a[12,3] = 0, a[7,3] = 0, a[6,3] = 0, c[7] = 23/148, a[12,2] = 0, b[8] = 11291146602515386 733913667333/8277895500290250100346304000, a[4,1] = 12257/317988, `b*` [10] = -6457286835032584968329/2546741646226985227300, a[7,1] = 480285 555889619/7997789253816320, a[7,6] = 864813381633887/51718297665732608 , `b*`[6] = 225910982205515703574046329/636776649908273251841973600, c [4] = 12257/79497, a[7,5] = -16450769949384489/490009784398315520, a[3 ,1] = 2090517149/56877957081, a[2,1] = 2/25, c[13] = 1, a[5,1] = 14131 686489425/35833975601529, a[6,4] = 9757757988/41995818067, a[6,1] = 44 18011/96055225, b[9] = 51522714764404063835592355866713645307758690796 273878782033726047743318218807/192822210747161760435981909174688769030 92208069910099389016386439619421788800, a[11,9] = 89599398445595686856 355698434884216540038527287573830949657578490242/579385621338495813483 588345965887373464382184677844152053023899833767, a[5,4] = 17619604667 500/11944658533843, a[10,7] = 2224286847389514767028684777161157168782 986309/199937022215481792634238811967979191208436288, c[3] = 24514/238 491, a[5,3] = -17700454220625/11944658533843, a[9,1] = -14415222609369 643343604689319116750509914398656839842108465299109803139261/286664608 34013981370272729820284862049438129018451533980749016707010746880, b[5 ] = 0, a[4,3] = 12257/105996, `b*`[9] = 143021962768113394257793894874 175054076879315632471495031519549972079/634979978672819946306099754101 89529208101511709161157616376634016000, b[2] = 0, b[3] = 0, b[4] = 0, \+ a[13,10] = 1124804893366359619795687870063500/180260981966042120762346 96000197113, a[13,9] = -8423048909040161063576104721713644717435186228 07702364576842637748625/1600110710453658437068687306242176955368591689 0578695272031985694176768, a[13,8] = -64484096268292447966995856630949 967425/5208159300615981523251328952205675621888, `b*`[13] = -495467169 0517567/1310791519537500, a[8,7] = 19/2, a[6,5] = 19921512441/10630009 4275, a[11,3] = 0, a[10,5] = -331923915046080335870716315310401046907/ 52747587227799808210222170585805619200, a[12,5] = -2222581559078624408 04259443447899304774477/18911475471642929222051825301977449103360, a[1 3,2] = 0, a[13,3] = 0, `b*`[7] = 50630668490428660369928685568/2038909 33496965109375618071875, a[10,1] = -9867878858058255039217746719915871 073167304993320293/137710062979952173643301343426419925803022411272192 00, a[8,6] = 51820007951705750121680295/6052645677641770902708224, b[1 0] = -582592205611226468655591/165538207004754039774500, c[8] = 196/19 7, `b*`[12] = 0, `b*`[11] = 0, a[13,11] = 0, a[12,6] = 114108763276781 56749715579835467111092952376241624115293181/1249081687157472843988286 395834110562049959611482227605504, a[10,2] = 0, a[13,12] = 0, b[13] = \+ 0, c[9] = 14118145156/16038718811, a[9,2] = 0, a[11,5] = -168891446695 10407387099211531243899609/1891391571300777047888087779210764288, a[12 ,11] = -5373869958976816279363488081975/128249819495112776500552350596 224, a[12,9] = -280813098436463686141658586973841773453049072566975573 877036454598695025876625/105067490432026768588795401172928908496060279 59545703832394231856364095276862976, c[12] = 1, a[11,2] = 0, a[11,8] = 10788313637659292115861359463369872242886/189380589527306698462795692 309376152568545, a[11,10] = -1786335907413925929868932792832/781445836 7137598885469908059185, c[10] = 39/43, a[10,8] = 498536496800491237121 36863581080955445/2729660625860504418029771633420026504704, a[9,4] = - 5224167467332280028062926335653367380511265991084060397937856421816947 95272896367/7244755263234658466960959547766426403775741588955515644685 9013382512921014002304, a[12,1] = -49410934155054900182882239365042643 979034016237/31731623983903801998088697641133618410239344640, `b*`[5] \+ = 0, `b*`[3] = 0, `b*`[4] = 0, `b*`[2] = 0, a[7,4] = 52162106556696460 538643/464887033284472899365888, a[8,5] = -314475237989690808507688413 3/286731750405790169489612800, a[12,8] = 46554463321808651332692242683 8859475/27467662576695046776938324444449900032, b[12] = -3571206213841 61/262158303907500, b[1] = 874545191389960761/19691638733588931200, a[ 8,1] = -41924855046865190259777/29067997485765047436800, c[11] = 20/21 , c[6] = 34/73, c[5] = 50/129, a[12,4] = -1074611649128981113804182104 383729704094059/244310054255337140402433369774092097581056, a[12,10] = 50908388575977059406815779042800/1299277084579288698410581464578279, \+ a[8,2] = 0, a[11,6] = 397099609762427730897418352709319848545113477679 56576333171/5577710294529287682275315773022359477180471284918417776640 , a[13,1] = -68348459007984933111036701115865624594922076006017/453450 07463014711631562120393860177858153484042240, a[3,2] = 3755851225/5687 7957081, a[13,4] = -1421462835415589956778218218848342421532216119/349 123046432081302269174752451415423519748096, a[13,5] = -298431277119206 9486473441797268836080579319/26237676161655099639634047107738568097792 0, b[11] = 43046993917969354593/35912185355757478400, a[13,7] = 159211 39337350831535408442010420239146408066675/1749382977105047216940523536 513333290389079616, a[13,6] = 4113870101542014702484932055059504461473 25414444198427659/4646029500607290959142817075008826809847482595991853 4656, a[9,7] = 6592677222216586891365490618278696431425055661507795359 7292687341/71147457071722151652399460223762777354167544911619399842600 41034, a[9,8] = 314254428400309469147852785940124705687807159490152038 1912/322898507178552018028498957174197954770661454622943631853501, a[1 1,7] = 956248096794157656802748368725840687390347399/77961324245284495 254874844291292429525671778, a[9,5] = -8249597526146667555031510474907 199042876873605462323913470411629578679723249/175634113965120890123649 8008888114538679717900892624949907570487472295239680, a[9,6] = 2495660 0980553216370236458050032746816720224485746407205171098431154802176718 07/6215583383437936975613964725364276840043190231103496984781881477863 43185391616, a[10,9] = -1203747916407015470833526610982521566345739350 276392522818995/431465047176222110521161502306471151359456402605041210 35832832, b[6] = 420364804728976000977525051149/1187588452078929614685 280764000, a[10,4] = -502106193116073366202094897549565186098556680379 /59760431150000217012599575938528786496649056256, a[12,7] = 8981318435 146082911054040229712538210777263716775/936984062290713822722287218869 196809455428475968, a[11,4] = -230201155410167521182771653701195967005 1291/274508714084612705166461135767825539832704, `b*`[8] = 36550810580 88732259074359101/827789550029025010034630400, a[10,6] = 1643147679523 140016971844191613351711631376872651547306145/315813057849945223645118 819513361571639686784746580082688, b[7] = 2510241582472115330153204977 8688/101007568454396515184681192806875, a[8,4] = -15736382496397955384 08923368457/337926761654282072170175876224, a[11,1] = -167715056043516 4751643947644188299249566198916438951/15242685861485236921438230468195 88641870992602157568, `b*`[1] = 13470793039761431/302948288209060480, \+ c[2] = 2/25\}:" }{TEXT -1 0 "" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 20 "printed coefficients" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 362 "subs(e18,ma trix([seq([c[i],seq(a[i,j],j=1..i-1),``$(7-i)],i=2..7)]));\nfor ii fro m 8 to 13 do\n print(``);\n print(c[ii]=subs(e18,c[ii])); \n for jj to ii-1 do\n print(a[ii,jj]=subs(e18,a[ii,jj]));\n end do: \nend do:print(``);\nfor ii to 13 do\n print(b[ii]=subs(e18,b[ii])); \nend do:print(``);\nfor ii to 13 do\n print(`b*`[ii]=subs(e18,`b*`[ ii]));\nend do:" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#K%'matrixG6#7(7)#\" \"#\"#DF(%!GF+F+F+F+7)#\"&9X#\"'\"\\Q##\"+\\r^!4#\",\"3dz(o&#\"+D7&ev$ F2F+F+F+F+7)#\"&dA\"\"&(\\z#F7\"'))zJ\"\"!#F7\"''*f5F+F+F+7)#\"#]\"$H \"#\"/D%*['oJT\"\"/H:gvR$e$F;#!/D1Aa/q<\"/VQ`eY%>\"#\"/+vm/'>w\"FGF+F+ 7)#\"#M\"#t#\"(6!=W\")D_0'*F;F;#\"+))zvd(*\",n!=e*>%#\",TC^@*>\"-vU4+j 5F+7)#\"#B\"$[\"#\"0>'*)eb&G![\"1?j\"QD*y(*zF;F;#\"8V'Q0Y'pcl5i@&\"9)) eO**GZ%GLq)[Y#!2*[%Q\\*p2X;\"3?bJ)R%y4+\\#\"0()Qj\"Q8[')\"23EtlwH=<&Q) pprint286\"" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%!G" }}{PARA 11 "" 1 " " {XPPMATH 20 "6#/&%\"cG6#\"\")#\"$'>\"$(>" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/&%\"aG6$\"\")\"\"\"#!8x(f->lo/b[#>%\"8+oVZ]w&[(*z1H" } }{PARA 11 "" 1 "" {XPPMATH 20 "6#/&%\"aG6$\"\")\"\"#\"\"!" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/&%\"aG6$\"\")\"\"$\"\"!" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/&%\"aG6$\"\")\"\"%#!@d%oL#*3%QbzR'\\#QOd\"\"?Ci(e\"\"#" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%!G" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/&%\"cG6#\"\"*# \",c^9=T\"\",6)=(Qg\"" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/&%\"aG6$\"\" *\"\"\"#!eoh#RJ!)4\"*Hl%3@%)Rol)R9*40v;\">$*o/OMV'p$4EA:W\"\"eo!)ou5qq ;!\\2)R`^%=!H\"Q%\\?'[G?)HFFq8)R,M3Ym'G" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/&%\"aG6$\"\"*\"\"#\"\"!" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/&% \"aG6$\"\"*\"\"$\"\"!" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/&%\"aG6$\"\" *\"\"%#!\\pnj*GF&z%p\"=Ucy$zRgS3\"*fE60QnLlNj#H1G+GKtYnTA&\"[p/B+95#H^ #Q8!foWc^b*)eTdx.kUmxaf4'pYeYBj_vWs" }}{PARA 11 "" 1 "" {XPPMATH 20 "6 #/&%\"aG6$\"\"*\"\"&#!go\\Ksz'y&H;TqM\"RKiagto(G/*>2\\Z5:.bvmYh_(f\\#) \"go!oR_Hsu[qv!*\\\\i#*3!zrz'QX6)))3!)\\O7!*37lR6Mc<" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/&%\"aG6$\"\"*\"\"'#\"jo2=n<-[:J%)4r^?2ku&[C-s;ouK+0 ekBqj@`0)4gc\\#\"io;;R&=VjyZ\")=y%)p\\.6B!>V+%oFk`skRhvp$zV$Q$e:i" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#/&%\"aG6$\"\"*\"\"(#\"\\oTto#H(f`z2:mb ]UJkpy#=1\\l8*oe;AAxEf'\"[oM5/gU)*R>;\"\\anTNxFwB-Y*R_;:A<2du9r" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#/&%\"aG6$\"\"*\"\")#\"en7>Q?:!\\fr!yo0 Z7Sfy_y9p%4.SGWD9$\"gn,N&=jVHia9mqZ&z>ur&*)\\G!=?byr])*GK" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%!G" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/&%\"cG6 #\"#5#\"#R\"#V" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/&%\"aG6$\"#5\"\"\"# !U$H?L*\\InJ2re\"*>nu<#R]De!e)yy')*\"V+#>s7TAI!e#*>kUV8IVO<_*zH15x8" } }{PARA 11 "" 1 "" {XPPMATH 20 "6#/&%\"aG6$\"#5\"\"#\"\"!" }}{PARA 11 " " 1 "" {XPPMATH 20 "6#/&%\"aG6$\"#5\"\"$\"\"!" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/&%\"aG6$\"#5\"\"%#!Qz.oc&)4'=l&\\v*[4-iOtg6$>1@]\"Pci0 \\m\\'yG&Qfd*f7q@++:J/wf" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/&%\"aG6$ \"#5\"\"&#!H2p/,/J:jrqeL!3Y]\"R#>L\"G+#>c!eeq@A5#3)*zFseZF&" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/&%\"aG6$\"#5\"\"'#\"enXhIZ:lsoPJ;r^Lh\">W=( p,SJ_zw9V;\"Z)o#3!eYZy'oR;dhL^>)=^kB_%*\\y08eJ" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/&%\"aG6$\"#5\"\"(#\"O4j)Hyor:hrx%oGqw9&*QZoGCA\"N)GO%3 7>zz'>\")QUj#z\"[:A-P**>" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/&%\"aG6$ \"#5\"\")#\"GXa&43\"ejo87P7\\+o\\O&)\\\"I/Z]E+UL;xH!=W]geig'HF" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#/&%\"aG6$\"#5\"\"*#!hn&**=G_#Rw-NRdMm: _#)4hEN$3Z:qS;zu.7\"inKG$e.@T]g-kXf8:rkI-:;@06Aim&\\#*H)=Ww%Rk^Z;N/c]rn\"\"Uov:-E*4(=k)e>o/BQ9#pB&[heoU_\"" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#/&%\"aG6$\"#6\"\"#\"\"!" }}{PARA 11 " " 1 "" {XPPMATH 20 "6#/&%\"aG6$\"#6\"\"$\"\"!" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/&%\"aG6$\"#6\"\"%#!L\"H^+nf>,PlrF=@v;5a:,-B\"K/F$)Rb#y wN6Ym^q7Y39(3XF" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/&%\"aG6$\"#6\"\"&# !G4'**QCJ:@*4(Q2/^pY9*)o\"\"F)Gk2@zx3))y/x2Ir:R\"*=" }}{PARA 11 "" 1 " " {XPPMATH 20 "6#/&%\"aG6$\"#6\"\"'#\"fnrJLwl&zwZ8^a[)>$4FN=u*3tFCw4'* 4(R\"enSmx<%=\\Gr/=x%fB-tdJvAo(GHXH5xd&" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/&%\"aG6$\"#6\"\"(#\"N*RZ.R(oSeso$[F!oldTz'4[i&*\"My!Q!R)RiJ<$" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/&%\"aG6 $\"#7\"\"#\"\"!" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/&%\"aG6$\"#7\"\"$ \"\"!" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/&%\"aG6$\"#7\"\"%#!LfS4/(HPQ /@=/Q6\")*G\"\\;hu5\"Kc5e(4#4u(pLV-/9P`Da+JW#" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/&%\"aG6$\"#7\"\"&#!KxWx/$**yWV%fU!3Wiy!f:eAA\"JgL5\\u( >ID=0A#HHkraZ6*=" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/&%\"aG6$\"#7\"\"' #\"fn\"=$H:TiTiP_H46rYN)zbr\\n:ywKw3T6\"en/bgFA[6'f*\\?c5T$eR'G))R%GZd ro\"3\\7" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/&%\"aG6$\"#7\"\"(#\"Rvnrj sx5#QDrH-/a5\"H3Y^V=8)*)\"QofZGaX4o>p)=sGAF#Qr!HiS)p$*" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/&%\"aG6$\"#7\"\")#\"Ev%f)QoUApK8l3=KjWbY\"GK+!*\\ WWC$QpxY]pwDmnu#" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/&%\"aG6$\"#7\"\"* #!ioDm(e-&p)fak.xQdvpcs!\\IXt<%Q(pee;9'ojkV)483G\"[pwH'oF&4kj&=B%RKQqX &fz-1'\\3*GH<,az)eon-K/\\n]5" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/&%\"a G6$\"#7\"#5#\"A+G/zd\"oSfq(fd)Q34&\"Cz#yXY\"e5%)p)GzX3x#*H\"" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/&%\"aG6$\"#7\"#6#!@v>3)[j$zi\"o(*e*pQP&\"BC if]Bb+lx7^\\>)\\#G\"" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%!G" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/&%\"cG6#\"#8\"\"\"" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/&%\"aG6$\"#8\"\"\"#!Sh@K:UU$ [)=#=#yn&**e:a$GY@9\"N'4[(>NU:9X_Z$z03O)oszTMZ'[p?>rFJ%)H\"K?z(4o &Qx5ZSjR'*4b;;wwBE" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/&%\"aG6$\"#8\" \"'#\"ZfwU)>WWTDt9Y/&f]0K\\[-Z,U:5qQ6%\"YcY`=*ff#[Z)4o#)3]2.s_py0*o\"fo`&p'fjO$*[![7\"\"D8r>+gpMi27Ug'>)4E!=" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/&%\"aG6$\"#8\"#6\"\"!" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/&%\"aG6$\"#8\"#7\"\"!" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%!G" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#/&%\"bG6#\"\"\"#\"3h2'**Q\">XX()\"5+7$ *)eL(Q;p>" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/&%\"bG6#\"\"#\"\"!" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#/&%\"bG6#\"\"$\"\"!" }}{PARA 11 "" 1 " " {XPPMATH 20 "6#/&%\"bG6#\"\"%\"\"!" }}{PARA 11 "" 1 "" {XPPMATH 20 " 6#/&%\"bG6#\"\"&\"\"!" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/&%\"bG6#\"\" '#\"?\\60Dv(4+w*GZ![O?%\"@+Sw!G&o9'H*y?X)e(=\"" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/&%\"bG6#\"\"(#\"A)oy(\\?`,L:@Z#eT-^#\"Bvo!G>\"o%=:lRa% ov+,\"" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/&%\"bG6#\"\")#\">Ltm8Rt'Q:D gY6H6\"=+SIY.5]-H+b*yF)" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/&%\"bG6#\" \"*#\"ho2)=#=LuZgsL?yyQF'z!pexIXOrmeN#fNQ1/Ww9F_^\"ho+))y@%>'RkQ;!*Q*4 5*p!3A4.p()ou\"4>)fVg<;Z2@AG>" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/&%\" bG6#\"#5#!9\"fb'okA6c?#f#e\"9+XxRSv/q?Qb;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/&%\"bG6#\"#6#\"5$fa$pz\"R*p/V\"5+%yuvb`=7f$" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/&%\"bG6#\"#7#!0hTQ@17d$\"0+v!RIe@E" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#/&%\"bG6#\"#8\"\"!" }}{PARA 11 "" 1 " " {XPPMATH 20 "6#%!G" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/&%#b*G6#\"\" \"#\"2J9wRIzqM\"\"3![g!4#)G[HI" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/&%# b*G6#\"\"#\"\"!" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/&%#b*G6#\"\"$\"\"! " }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/&%#b*G6#\"\"%\"\"!" }}{PARA 11 " " 1 "" {XPPMATH 20 "6#/&%#b*G6#\"\"&\"\"!" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/&%#b*G6#\"\"'#\"%=Dt#3*\\mxO' " }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/&%#b*G6#\"\"(#\">oboG*p.mG/\\o1j] \"?v=2=cP4^'p\\L4*Q?" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/&%#b*G6#\"\") #\"=,\"fV2fAt)3e53bO\"<+/jM+,D!H+b*yF)" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/&%#b*G6#\"\"*#\"`oz?(*\\&>:.&\\rCj:$zo2a]-V\"\" _o+g,MmP;w:h\"4<^,\"3#H&*=5a(*41j%*>Gny*z\\j" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/&%#b*G6#\"#5#!7H$o\\eK]$oGdk\"7+tA&)pAY;uYD" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/&%#b*G6#\"#6\"\"!" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/&%#b*G6#\"#7\"\"!" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/ &%#b*G6#\"#8#!1nv^!prY&\\\"1+v`>:z58" }}}{PARA 0 "" 0 "" {TEXT -1 0 " " }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 151 "RK8_13eqs : = [op(RowSumConditions(13,'expanded')),op(OrderConditions(8,13,'expand ed'))]:\n`RK7_13eqs*` := subs(b=`b*`,OrderConditions(7,13,'expanded')) :" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 6 "Chec k:" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 112 "subs(e18,RK8_13eqs): \nmap(u->lhs(u)-rhs(u),%);\nnops(%);\nsubs(e18,`RK7_13eqs*`):\nmap(u-> lhs(u)-rhs(u),%);\nnops(%);" }}{PARA 12 "" 1 "" {XPPMATH 20 "6#7`x\"\" !F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F $F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F $F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F $F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F $F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F $F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F $F$" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#\"$7#" }}{PARA 12 "" 1 "" {XPPMATH 20 "6#7ap\"\"!F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F $F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F $F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$" }}{PARA 11 "" 1 " " {XPPMATH 20 "6#\"#&)" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 26 "#-------------------------" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}}{PARA 0 "" 0 "" {TEXT -1 52 "#-------------- -------------------------------------" }}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 38 "characteristics of the embedded scheme" }}{PARA 0 "" 0 " " {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}} {SECT 1 {PARA 4 "" 0 "" {TEXT -1 36 "coefficients of the combined sche me " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 6866 "ee := \{c[2]=2/25,\nc[3]=24514/238491,\nc[4]=12257/ 79497,\nc[5]=50/129,\nc[6]=34/73,\nc[7]=23/148,\nc[8]=196/197,\nc[9]=1 4118145156/16038718811,\nc[10]=39/43,\nc[11]=20/21,\nc[12]=1,\nc[13]=1 ,\n\na[2,1]=2/25,\na[3,1]=2090517149/56877957081,\na[3,2]=3755851225/5 6877957081,\na[4,1]=12257/317988,\na[4,2]=0,\na[4,3]=12257/105996,\na[ 5,1]=14131686489425/35833975601529,\na[5,2]=0,\na[5,3]=-17700454220625 /11944658533843,\na[5,4]=17619604667500/11944658533843,\na[6,1]=441801 1/96055225,\na[6,2]=0,\na[6,3]=0,\na[6,4]=9757757988/41995818067,\na[6 ,5]=19921512441/106300094275,\na[7,1]=480285555889619/7997789253816320 ,\na[7,2]=0,\na[7,3]=0,\na[7,4]=52162106556696460538643/46488703328447 2899365888,\na[7,5]=-16450769949384489/490009784398315520,\na[7,6]=864 813381633887/51718297665732608,\na[8,1]=-41924855046865190259777/29067 997485765047436800,\na[8,2]=0,\na[8,3]=0,\na[8,4]=-1573638249639795538 408923368457/337926761654282072170175876224,\na[8,5]=-3144752379896908 085076884133/286731750405790169489612800,\na[8,6]=51820007951705750121 680295/6052645677641770902708224,\na[8,7]=19/2,\na[9,1]=-1441522260936 9643343604689319116750509914398656839842108465299109803139261/\n \+ 28666460834013981370272729820284862049438129018451533980749016707010 746880,\na[9,2]=0,\na[9,3]=0,\na[9,4]=-5224167467332280028062926335653 36738051126599108406039793785642181694795272896367/\n 724475526 3234658466960959547766426403775741588955515644685901338251292101400230 4,\na[9,5]=-8249597526146667555031510474907199042876873605462323913470 411629578679723249/\n 17563411396512089012364980088881145386797 17900892624949907570487472295239680,\na[9,6]=2495660098055321637023645 805003274681672022448574640720517109843115480217671807/\n 621558 3383437936975613964725364276840043190231103496984781881477863431853916 16,\na[9,7]=6592677222216586891365490618278696431425055661507795359729 2687341/\n 71147457071722151652399460223762777354167544911619399 84260041034,\na[9,8]=3142544284003094691478527859401247056878071594901 520381912/\n 322898507178552018028498957174197954770661454622943 631853501,\na[10,1]=-9867878858058255039217746719915871073167304993320 293/\n 13771006297995217364330134342641992580302241127219200, \na[10,2]=0,\na[10,3]=0,\na[10,4]=-50210619311607336620209489754956518 6098556680379/\n 597604311500002170125995759385287864966490562 56,\na[10,5]=-331923915046080335870716315310401046907/5274758722779980 8210222170585805619200,\na[10,6]=1643147679523140016971844191613351711 631376872651547306145/\n 31581305784994522364511881951336157163 9686784746580082688,\na[10,7]=2224286847389514767028684777161157168782 986309/199937022215481792634238811967979191208436288,\na[10,8]=4985364 9680049123712136863581080955445/27296606258605044180297716334200265047 04,\na[10,9]=-12037479164070154708335266109825215663457393502763925228 18995/\n 43146504717622211052116150230647115135945640260504121 035832832,\na[11,1]=-1677150560435164751643947644188299249566198916438 951/\n 1524268586148523692143823046819588641870992602157568,\n a[11,2]=0,\na[11,3]=0,\na[11,4]=-2302011554101675211827716537011959670 051291/274508714084612705166461135767825539832704,\na[11,5]=-168891446 69510407387099211531243899609/1891391571300777047888087779210764288,\n a[11,6]=39709960976242773089741835270931984854511347767956576333171/\n 5577710294529287682275315773022359477180471284918417776640,\na [11,7]=956248096794157656802748368725840687390347399/77961324245284495 254874844291292429525671778,\na[11,8]=10788313637659292115861359463369 872242886/189380589527306698462795692309376152568545,\na[11,9]=8959939 8445595686856355698434884216540038527287573830949657578490242/\n \+ 57938562133849581348358834596588737346438218467784415205302389983376 7,\na[11,10]=-1786335907413925929868932792832/781445836713759888546990 8059185,\na[12,1]=-49410934155054900182882239365042643979034016237/317 31623983903801998088697641133618410239344640,\na[12,2]=0,\na[12,3]=0, \na[12,4]=-1074611649128981113804182104383729704094059/244310054255337 140402433369774092097581056,\na[12,5]=-2222581559078624408042594434478 99304774477/18911475471642929222051825301977449103360,\na[12,6]=114108 76327678156749715579835467111092952376241624115293181/\n 124908 1687157472843988286395834110562049959611482227605504,\na[12,7]=8981318 435146082911054040229712538210777263716775/936984062290713822722287218 869196809455428475968,\na[12,8]=465544633218086513326922426838859475/2 7467662576695046776938324444449900032, a[12,9]=-2808130984364636861416 58586973841773453049072566975573877036454598695025876625/\n 10 5067490432026768588795401172928908496060279595457038323942318563640952 76862976, a[12,10]=50908388575977059406815779042800/129927708457928869 8410581464578279,\na[12,11]=-5373869958976816279363488081975/128249819 495112776500552350596224,\na[13,1]=-6834845900798493311103670111586562 4594922076006017/\n 453450074630147116315621203938601778581534 84042240,\na[13,2]=0,\na[13,3]=0,\na[13,4]=-14214628354155899567782182 18848342421532216119/349123046432081302269174752451415423519748096,\na [13,5]=-2984312771192069486473441797268836080579319/262376761616550996 396340471077385680977920,\na[13,6]=41138701015420147024849320550595044 6147325414444198427659/\n 4646029500607290959142817075008826809 8474825959918534656,\na[13,7]=1592113933735083153540844201042023914640 8066675/1749382977105047216940523536513333290389079616,\na[13,8]=-6448 4096268292447966995856630949967425/52081593006159815232513289522056756 21888,\na[13,9]=-84230489090401610635761047217136447174351862280770236 4576842637748625/\n 160011071045365843706868730624217695536859 16890578695272031985694176768, a[13,10]=112480489336635961979568787006 3500/18026098196604212076234696000197113,\na[13,11]=0,\na[13,12]=0,\n \nb[1]=874545191389960761/19691638733588931200,\nb[2]=0,\nb[3]=0,\nb[4 ]=0,\nb[5]=0,\nb[6]=420364804728976000977525051149/1187588452078929614 685280764000,\nb[7]=25102415824721153301532049778688/10100756845439651 5184681192806875,\nb[8]=11291146602515386733913667333/8277895500290250 100346304000,\nb[9]=51522714764404063835592355866713645307758690796273 878782033726047743318218807/\n 19282221074716176043598190917468876 903092208069910099389016386439619421788800,\nb[10]=-582592205611226468 655591/165538207004754039774500,\nb[11]=43046993917969354593/359121853 55757478400,\nb[12]=-357120621384161/262158303907500,\nb[13]=0,\n\n`b* `[1]=13470793039761431/302948288209060480,\n`b*`[2]=0,\n`b*`[3]=0,\n`b *`[4]=0,\n`b*`[5]=0,\n`b*`[6]=225910982205515703574046329/636776649908 273251841973600,\n`b*`[7]=50630668490428660369928685568/20389093349696 5109375618071875,\n`b*`[8]=3655081058088732259074359101/82778955002902 5010034630400,\n`b*`[9]=1430219627681133942577938948741750540768793156 32471495031519549972079/\n 634979978672819946306099754101895292 08101511709161157616376634016000,\n`b*`[10]=-6457286835032584968329/25 46741646226985227300,\n`b*`[11]=0,\n`b*`[12]=0,\n`b*`[13]=-49546716905 17567/1310791519537500\}:" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}}{PARA 0 "" 0 "" {TEXT -1 0 " " }}{PARA 0 "" 0 "" {TEXT -1 5 "Let " }{XPPEDIT 18 0 "T[8, 13];" "6#& %\"TG6$\"\")\"#8" }{TEXT -1 128 " denote the vector whose components \+ are the principal error terms of the 13 stage, order 8 scheme (the err or terms of order 9)." }}{PARA 0 "" 0 "" {TEXT -1 5 "Let " }{XPPEDIT 18 0 "`T*`[7,13];" "6#&%#T*G6$\"\"(\"#8" }{TEXT -1 146 " denote the v ector whose components are the principal error terms of the embedded 1 3 stage, order 7 scheme (the error terms of order 8) and let " } {XPPEDIT 18 0 "`T*`[8,13];" "6#&%#T*G6$\"\")\"#8" }{TEXT -1 99 " deno te the vector whose components are the error terms of order 9 of the e mbedded order 7 scheme." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 39 "Denote the 2-norm of these vectors by " }{XPPEDIT 18 0 "abs(abs(T[8,13]));" "6#-%$absG6#-F$6#&%\"TG6$\"\")\"#8" }{TEXT -1 3 ", " }{XPPEDIT 18 0 "abs(abs(`T*`[7, 13]));" "6#-%$absG6#-F$6#&% #T*G6$\"\"(\"#8" }{TEXT -1 7 " and " }{XPPEDIT 18 0 "abs(abs(`T*`[8, 13]));" "6#-%$absG6#-F$6#&%#T*G6$\"\")\"#8" }{TEXT -1 15 " respectiv ely." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 8 "De fine: " }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "A[9] = abs( abs(T[8,13]));" "6#/&%\"AG6#\"\"*-%$absG6#-F)6#&%\"TG6$\"\")\"#8" } {TEXT -1 6 ", " }{XPPEDIT 18 0 "B[9] = abs(abs(`T*`[8,13]))/abs(ab s(`T*`[7,13]));" "6#/&%\"BG6#\"\"**&-%$absG6#-F*6#&%#T*G6$\"\")\"#8\" \"\"-F*6#-F*6#&F/6$\"\"(F2!\"\"" }{TEXT -1 6 ", " }{XPPEDIT 18 0 " C[9] = abs(abs(`T*`[8,13]-T[8,13]))/abs(abs(`T*`[7,13]));" "6#/&%\"CG6 #\"\"**&-%$absG6#-F*6#,&&%#T*G6$\"\")\"#8\"\"\"&%\"TG6$F2F3!\"\"F4-F*6 #-F*6#&F06$\"\"(F3F8" }{TEXT -1 1 "." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 77 "Dormand and Prince have suggested that \+ as well as attempting to ensure that " }{XPPEDIT 18 0 "A[9];" "6#&%\" AG6#\"\"*" }{TEXT -1 73 " is a minimum, if the embedded scheme is to \+ be used for error control, " }{XPPEDIT 18 0 "B[9];" "6#&%\"BG6#\"\"* " }{TEXT -1 7 " and " }{XPPEDIT 18 0 "C[9];" "6#&%\"CG6#\"\"*" } {TEXT -1 27 " should be chosen so that " }}{PARA 0 "" 0 "" {TEXT -1 68 "they are \"similar in magnitude\" and also not differ too much fro m 1." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 168 "errterms8_13 := PrincipalErrorTerms(8,13,'expanded') :\n`errterms8_13*` :=subs(b=`b*`,errterms8_13):\n`errterms7_13*` := su bs(b=`b*`,PrincipalErrorTerms(7,13,'expanded')):" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 333 "snmB := sqr t(add(evalf(subs(ee,`errterms8_13*`[i]))^2,i=1..nops(`errterms8_13*`)) ):\nsdnB := sqrt(add(evalf(subs(ee,`errterms7_13*`[i]))^2,i=1..nops(`e rrterms7_13*`))):\nsnmC := sqrt(add((evalf(subs(ee,`errterms8_13*`[i]) -subs(ee,errterms8_13[i])))^2,i=1..nops(errterms8_13))):\n'B[9]'= eval f[8](snmB/sdnB);\n'C[9]'= evalf[8](snmC/sdnB);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/&%\"BG6#\"\"*$\")#4]*>!\"(" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/&%\"CG6#\"\"*$\")w(>*>!\"(" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}}{PARA 0 "" 0 "" {TEXT -1 52 "#--------------------------------------------------- " }}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 26 "absolute stability regions" } }{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 36 "coefficients of the co mbined scheme " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 6866 "ee := \{c[2]=2/25,\nc[3]=24514/238491,\nc[4]=1 2257/79497,\nc[5]=50/129,\nc[6]=34/73,\nc[7]=23/148,\nc[8]=196/197,\nc [9]=14118145156/16038718811,\nc[10]=39/43,\nc[11]=20/21,\nc[12]=1,\nc[ 13]=1,\n\na[2,1]=2/25,\na[3,1]=2090517149/56877957081,\na[3,2]=3755851 225/56877957081,\na[4,1]=12257/317988,\na[4,2]=0,\na[4,3]=12257/105996 ,\na[5,1]=14131686489425/35833975601529,\na[5,2]=0,\na[5,3]=-177004542 20625/11944658533843,\na[5,4]=17619604667500/11944658533843,\na[6,1]=4 418011/96055225,\na[6,2]=0,\na[6,3]=0,\na[6,4]=9757757988/41995818067, \na[6,5]=19921512441/106300094275,\na[7,1]=480285555889619/79977892538 16320,\na[7,2]=0,\na[7,3]=0,\na[7,4]=52162106556696460538643/464887033 284472899365888,\na[7,5]=-16450769949384489/490009784398315520,\na[7,6 ]=864813381633887/51718297665732608,\na[8,1]=-41924855046865190259777/ 29067997485765047436800,\na[8,2]=0,\na[8,3]=0,\na[8,4]=-15736382496397 95538408923368457/337926761654282072170175876224,\na[8,5]=-31447523798 96908085076884133/286731750405790169489612800,\na[8,6]=518200079517057 50121680295/6052645677641770902708224,\na[8,7]=19/2,\na[9,1]=-14415222 609369643343604689319116750509914398656839842108465299109803139261/\n \+ 286664608340139813702727298202848620494381290184515339807490167 07010746880,\na[9,2]=0,\na[9,3]=0,\na[9,4]=-52241674673322800280629263 3565336738051126599108406039793785642181694795272896367/\n 7244 7552632346584669609595477664264037757415889555156446859013382512921014 002304,\na[9,5]=-82495975261466675550315104749071990428768736054623239 13470411629578679723249/\n 175634113965120890123649800888811453 8679717900892624949907570487472295239680,\na[9,6]=24956600980553216370 23645805003274681672022448574640720517109843115480217671807/\n 6 2155833834379369756139647253642768400431902311034969847818814778634318 5391616,\na[9,7]=65926772222165868913654906182786964314250556615077953 597292687341/\n 711474570717221516523994602237627773541675449116 1939984260041034,\na[9,8]=31425442840030946914785278594012470568780715 94901520381912/\n 3228985071785520180284989571741979547706614546 22943631853501,\na[10,1]=-98678788580582550392177467199158710731673049 93320293/\n 13771006297995217364330134342641992580302241127219 200,\na[10,2]=0,\na[10,3]=0,\na[10,4]=-5021061931160733662020948975495 65186098556680379/\n 59760431150000217012599575938528786496649 056256,\na[10,5]=-331923915046080335870716315310401046907/527475872277 99808210222170585805619200,\na[10,6]=164314767952314001697184419161335 1711631376872651547306145/\n 3158130578499452236451188195133615 71639686784746580082688,\na[10,7]=222428684738951476702868477716115716 8782986309/199937022215481792634238811967979191208436288,\na[10,8]=498 53649680049123712136863581080955445/2729660625860504418029771633420026 504704,\na[10,9]=-1203747916407015470833526610982521566345739350276392 522818995/\n 4314650471762221105211615023064711513594564026050 4121035832832,\na[11,1]=-167715056043516475164394764418829924956619891 6438951/\n 152426858614852369214382304681958864187099260215756 8,\na[11,2]=0,\na[11,3]=0,\na[11,4]=-230201155410167521182771653701195 9670051291/274508714084612705166461135767825539832704,\na[11,5]=-16889 144669510407387099211531243899609/189139157130077704788808777921076428 8,\na[11,6]=3970996097624277308974183527093198485451134776795657633317 1/\n 5577710294529287682275315773022359477180471284918417776640 ,\na[11,7]=956248096794157656802748368725840687390347399/7796132424528 4495254874844291292429525671778,\na[11,8]=1078831363765929211586135946 3369872242886/189380589527306698462795692309376152568545,\na[11,9]=895 99398445595686856355698434884216540038527287573830949657578490242/\n \+ 5793856213384958134835883459658873734643821846778441520530238998 33767,\na[11,10]=-1786335907413925929868932792832/78144583671375988854 69908059185,\na[12,1]=-49410934155054900182882239365042643979034016237 /31731623983903801998088697641133618410239344640,\na[12,2]=0,\na[12,3] =0,\na[12,4]=-1074611649128981113804182104383729704094059/244310054255 337140402433369774092097581056,\na[12,5]=-2222581559078624408042594434 47899304774477/18911475471642929222051825301977449103360,\na[12,6]=114 10876327678156749715579835467111092952376241624115293181/\n 124 9081687157472843988286395834110562049959611482227605504,\na[12,7]=8981 318435146082911054040229712538210777263716775/936984062290713822722287 218869196809455428475968,\na[12,8]=46554463321808651332692242683885947 5/27467662576695046776938324444449900032, a[12,9]=-2808130984364636861 41658586973841773453049072566975573877036454598695025876625/\n \+ 105067490432026768588795401172928908496060279595457038323942318563640 95276862976, a[12,10]=50908388575977059406815779042800/129927708457928 8698410581464578279,\na[12,11]=-5373869958976816279363488081975/128249 819495112776500552350596224,\na[13,1]=-6834845900798493311103670111586 5624594922076006017/\n 453450074630147116315621203938601778581 53484042240,\na[13,2]=0,\na[13,3]=0,\na[13,4]=-14214628354155899567782 18218848342421532216119/349123046432081302269174752451415423519748096, \na[13,5]=-2984312771192069486473441797268836080579319/262376761616550 996396340471077385680977920,\na[13,6]=41138701015420147024849320550595 0446147325414444198427659/\n 4646029500607290959142817075008826 8098474825959918534656,\na[13,7]=1592113933735083153540844201042023914 6408066675/1749382977105047216940523536513333290389079616,\na[13,8]=-6 4484096268292447966995856630949967425/52081593006159815232513289522056 75621888,\na[13,9]=-84230489090401610635761047217136447174351862280770 2364576842637748625/\n 160011071045365843706868730624217695536 85916890578695272031985694176768, a[13,10]=112480489336635961979568787 0063500/18026098196604212076234696000197113,\na[13,11]=0,\na[13,12]=0, \n\nb[1]=874545191389960761/19691638733588931200,\nb[2]=0,\nb[3]=0,\nb [4]=0,\nb[5]=0,\nb[6]=420364804728976000977525051149/11875884520789296 14685280764000,\nb[7]=25102415824721153301532049778688/101007568454396 515184681192806875,\nb[8]=11291146602515386733913667333/82778955002902 50100346304000,\nb[9]=515227147644040638355923558667136453077586907962 73878782033726047743318218807/\n 192822210747161760435981909174688 76903092208069910099389016386439619421788800,\nb[10]=-5825922056112264 68655591/165538207004754039774500,\nb[11]=43046993917969354593/3591218 5355757478400,\nb[12]=-357120621384161/262158303907500,\nb[13]=0,\n\n` b*`[1]=13470793039761431/302948288209060480,\n`b*`[2]=0,\n`b*`[3]=0,\n `b*`[4]=0,\n`b*`[5]=0,\n`b*`[6]=225910982205515703574046329/6367766499 08273251841973600,\n`b*`[7]=50630668490428660369928685568/203890933496 965109375618071875,\n`b*`[8]=3655081058088732259074359101/827789550029 025010034630400,\n`b*`[9]=14302196276811339425779389487417505407687931 5632471495031519549972079/\n 6349799786728199463060997541018952 9208101511709161157616376634016000,\n`b*`[10]=-6457286835032584968329/ 2546741646226985227300,\n`b*`[11]=0,\n`b*`[12]=0,\n`b*`[13]=-495467169 0517567/1310791519537500\}:" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 78 "The stability function R for th e 13 stage, order 8 scheme is given as follows." }}{EXCHG {PARA 0 "> \+ " 0 "" {MPLTEXT 1 0 76 "subs(ee,StabilityFunction(8,13,'expanded')):\n R := unapply(%,z):\n'R(z)'=R(z);" }}{PARA 12 "" 1 "" {XPPMATH 20 "6#/- %\"RG6#%\"zG,<\"\"\"F)F'F)*&#F)\"\"#F)*$)F'F,F)F)F)*&#F)\"\"'F)*$)F'\" \"$F)F)F)*&#F)\"#CF)*$)F'\"\"%F)F)F)*&#F)\"$?\"F)*$)F'\"\"&F)F)F)*&#F) \"$?(F)*$)F'F1F)F)F)*&#F)\"%S]F)*$)F'\"\"(F)F)F)*&#F)\"&?.%F)*$)F'\"\" )F)F)F)*&#\"KPTWN))G=\\.nZhfNq'3S0,?\"\"P+o^tPH5*eK6^;,M%QXs?pI,Z%F)*$ )F'\"\"*F)F)F)*&#\"JfeGvi\"zyHZo]fMT%*[9z*H&\"Q!Gx\"e8N()4*47&f`kzwGLN =,Ec?F)*$)F'\"#5F)F)F)*&#\"JtCDfLb[XKz;H?W_H\"\\B78\"R+G4U6B(*y0:n]iU1 s\"=528y925F)*$)F'\"#6F)F)F)*&#\"?@5G;d(*Ry!>t55 " 0 "" {MPLTEXT 1 0 26 "z0 := newton(R(z)=1,z=-6);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%#z0G$!+Fn " 0 "" {MPLTEXT 1 0 302 "z0 := newton(R(z)=1,z=-6): \np1 := plot([R(z),1],z=-6.69..0.49,color=[red,blue]):\np2 := plot([[[ z0,1]]$3],style=point,symbol=[circle,cross,diamond],color=black):\np3 \+ := plot([[z0,0],[z0,1]],linestyle=3,color=COLOR(RGB,0,.5,0)):\nplots[d isplay]([p1,p2,p3],view=[-6.69..0.49,-0.07..1.47],font=[HELVETICA,9]); " }}{PARA 13 "" 1 "" {GLPLOT2D 360 253 253 {PLOTDATA 2 "6+-%'CURVESG6$ 7X7$$!3Q++++++!p'!#<$\"3i,dF_b#H$HF*7$$!3ym;a;#[F*7$$!3$)\\P%H<`2K'F*$\"31tEv[xI'y\"F *7$$!3[L$e&HG=WiF*$\"3-z?Y@$GNg\"F*7$$!3)QLL#G\\5nhF*$\"3EFf,f1#eV\"F* 7$$!3UL$3p-F+4'F*$\"3u^(yN:,LG\"F*7$$!3X]PRjaJ8gF*$\"37c*pWc%\\X6F*7$$ !3fm\"z)**QgOfF*$\"3*>%HqPcg?5F*7$$!3)Q$ef@5O%z&F*$\"3-+-ee,0*>)!#=7$$ !3E+D;gn2ZcF*$\"3o+Q4(zO=\\'F^o7$$!3gLeHm^v%\\&F*$\"3ya*z4\"4qi]F^o7$$ !3Q+Drp?#HM&F*$\"3FpT5_;UBRF^o7$$!3ymm@nIu'=&F*$\"3[^(f(G!>w*HF^o7$$!3 oL$3o^!=\\]F*$\"3c7sz?')*HN#F^o7$$!37++lmwJ%*[F*$\"3%yu=/lPTy\"F^o7$$! 3F++vM*=)QZF*$\"3a7F_Q-#*[8F^o7$$!3'****\\L4n*)e%F*$\"3nl#GV<))H.\"F^o 7$$!3yLeHls)GX%F*$\"3oDmk(**)po\")!#>7$$!3Umm\"y=v5H%F*$\"3+]r1:#p')H' F\\r7$$!3#ommK8**R:%F*$\"35LsDvj_!>&F\\r7$$!3O+D'[4ZX*RF*$\"3x&)yFVqVN VF\\r7$$!3=nmwA]S`QF*$\"3ed/%eq60*QF\\r7$$!33+D')*z\\&)p$F*$\"3g5'[s-^ Dn$F\\r7$$!3'**\\(e254^NF*$\"3%*pR<;_&3o$F\\r7$$!3wm;HFKB(R$F*$\"3Chju cBLyQF\\r7$$!3im\"zdvVfD$F*$\"3(>Lq9'f/4UF\\r7$$!3kL$3K(Ra.JF*$\"3cmW< \\Mb8ZF\\r7$$!3oLeHIMCXHF*$\"3uv#3\\B)Q&R&F\\r7$$!37+vLUBW2GF*$\"3:,\\ MB&*[AhF\\r7$$!3EL$e@R8'eEF*$\"3!)H(**[H(>cqF\\r7$$!3G++qB&e[]#F*$\"3E ^i/Z_9)>)F\\r7$$!3?+]_\"QRWN#F*$\"3\"*Gs&*>w46&*F\\r7$$!33+D1F6!*3AF*$ \"3-RFW*H:\"*4\"F^o7$$!3?+]24dIZ?F*$\"3'GJhj7m7H\"F^o7$$!3#pmm(Q]5->F* $\"3Eu*)f5]w#\\\"F^o7$$!3-++vIB2Zw^d+#F^o7$$!3m***\\Q)\\+`9F*$\"3K(z'GIMoQBF^o7$$!3um\"Hvx$ \\38F*$\"3H0QXQ\\F-FF^o7$$!3c*\\7z#zVd6F*$\"3YAT[@i!H9$F^o7$$!3[mmTCKv 45F*$\"3BYrj,%*3VOF^o7$$!3].]il-T^&)F^o$\"39x+&eMKAD%F^o7$$!3#)GLLR5Ji qF^o$\"3A****=Bl+N\\F^o7$$!3^GL30?\\RbF^o$\"3ES%RS8enu&F^o7$$!3ch;H\"o #GHSF^o$\"3K7bQ./g$o'F^o7$$!30%****p:d:k#F^o$\"3Ot+IM'R&ywF^o7$$!3/lmT ,r1^5F^o$\"3#[aZ&RXG-!*F^o7$$\"3\"Gsmmkh[r$F\\r$\"3,a'o0DZy.\"F*7$$\"3 ;,]P^YA))=F^o$\"3mb_8-l#y?\"F*7$$\"3J,]i9p$*RLF^o$\"3Y\\MXLV`'R\"F*7$$ \"3!***************[F^o$\"3(fT?)>iJK;F*-%'COLOURG6&%$RGBG$\"*++++\"!\" )$\"\"!Fj\\lFi\\l-F$6$7S7$F($\"\"\"Fj\\l7$F3F_]l7$F=F_]l7$FGF_]l7$FQF_ ]l7$FenF_]l7$FjnF_]l7$F`oF_]l7$FeoF_]l7$FjoF_]l7$F_pF_]l7$FdpF_]l7$Fip F_]l7$F^qF_]l7$FcqF_]l7$FhqF_]l7$F^rF_]l7$FcrF_]l7$FhrF_]l7$F]sF_]l7$F bsF_]l7$FgsF_]l7$F\\tF_]l7$FatF_]l7$FftF_]l7$F[uF_]l7$F`uF_]l7$FeuF_]l 7$FjuF_]l7$F_vF_]l7$FdvF_]l7$FivF_]l7$F^wF_]l7$FcwF_]l7$FhwF_]l7$F]xF_ ]l7$FbxF_]l7$FgxF_]l7$F\\yF_]l7$FayF_]l7$FfyF_]l7$F[zF_]l7$F`zF_]l7$Fe zF_]l7$FjzF_]l7$F_[lF_]l7$Fd[lF_]l7$Fi[lF_]l7$F^\\lF_]l-Fc\\l6&Fe\\lFi \\lFi\\lFf\\l-F$6&7#7$$!3U+++Fn " 0 "" {MPLTEXT 1 0 965 "R := z -> add(z^j/j!,j=0..8)+\n 1200105400867035596147670349182 88835444137/44701306920724538434011651113258910293773516800*z^9+\n 5 2997914489441345950684729787916275285859/20562601183533287679645359512 0990987351358177280*z^10+\n 1312234912952442029167932454855335925247 3/1007147813071018172064262506715057897231142092800*z^11+\n 25817101 0731907839975716281021/2716403483879029825452896180756614742016*z^12: \npts := []: z0 := 0:\nfor ct from 0 to 320 do\n zz := newton(R(z)=e xp(ct*Pi/20*I),z=z0):\n z0 := zz:\n pts := [op(pts),[Re(zz),Im(zz) ]]:\nend do:\np1 := plot(pts,color=COLOR(RGB,.48,.1,.48)):\np2 := plot s[polygonplot]([seq([pts[i-1],pts[i],[-4.1,0]],i=2..nops(pts))],\n \+ style=patchnogrid,color=COLOR(RGB,.95,.2,.95)):\np3 := plot([[[- 6.89,0],[1.19,0]],[[0,-6.39],[0,6.39]]],color=black,linestyle=3):\nplo ts[display]([p||(1..3)],view=[-6.89..1.19,-6.39..6.39],font=[HELVETICA ,9],\n labels=[`Re(z)`,`Im(z)`],axes=boxed,scaling=constr ained);" }}{PARA 13 "" 1 "" {GLPLOT2D 381 565 565 {PLOTDATA 2 "6+-%'CU RVESG6$7]_l7$$\"\"!F)F(7$F($\"3++++Fjzq:!#=7$F($\"3)******Rl#fTJF-7$F( $\"3D+++\")*)Q7ZF-7$$\"3(******>L(4CY!#F$\"3_*****fJ&=$G'F-7$$\"3!**** **4Qg%pS!#E$\"3u*****fp\")R&yF-7$$\"3%******R6atM#!#D$\"3G+++T#yZU*F-7 $$\"3'*******Hy+15!#C$\"3-+++Dvb*4\"!#<7$$\"3#)*****fH@%\\MFI$\"3/+++D tjc7FL7$$\"3^+++%eu^\"**FI$\"32++++tr89FL7$$\"3-+++\"[kMY#!#B$\"37+++9 vzq:FL7$$\"3-+++\")*HwR&FZ$\"3&******H)z(ys\"FL7$$\"3#******RR))f0\"!# A$\"35+++\\'e\\)=FL7$$\"3-+++xo\\b=F_o$\"3&)*****\\_R?/#FL7$$\"35+++/( >]#HF_o$\"3%)*****zE@\"*>#FL7$$\"3=+++\"y@A4%F_o$\"3?+++\\m?cBFL7$$\"3 &******zH1]$\\F_o$\"33+++7OI8DFL7$$\"3A+++*Q*yYZF_o$\"3%)*****>(3VqEFL 7$$\"37+++EOFL7$$\"3)*******H%**[h#!#?$\"3A+++g<$=y$FL7$$\"3$)*****p&>42pF]s$ \"3#******R/wn%RFL7$$\"3++++%y+Uh\"!#>$\"3<+++q(*Q:TFL7$$\"3o******Rf! H\\$Fhs$\"3#******>_[S\\ZFL7$$\"3++++ 8'>P[#F-$\"35+++5t!y&[FL7$$\"39+++:!>Y(HF-$\"3G+++8DY\\\\FL7$$\"3?+++# *G=dLF-$\"3#******z.E+.&FL7$$\"3!******>4tDk$F-$\"37+++&)*pF5&FL7$$\"3 y*****\\`fP%QF-$\"3i*****HKF&p^FL7$$\"3%******4kfA(RF-$\"3T+++hZNJ_FL7 $$\"3$)*****44Nx.%F-$\"3G+++*z()))G&FL7$$\"3;+++x@B[SF-$\"3I+++oi_U`FL 7$$\"3E+++]#)\\5SF-$\"3?+++%zLDR&FL7$$\"3')******>%[-$RF-$\"3S+++ZG4Ra FL7$$\"3E+++g$pB\"QF-$\"3C+++\"*fL#[&FL7$$\"3;+++R>2hOF-$\"3w*****H(QO AbFL7$$\"3y*****\\`1+[$F-$\"3')*****ffb#fbFL7$$\"3)******>\">NsKF-$\"3 k*****4guIf&FL7$$\"36+++K`)3/$F-$\"3#******\\'=(Qi&FL7$$\"3=+++.!R!)y# F-$\"3/+++uuo^cFL7$$\"3F+++6i%f^#F-$\"3O+++)=^ln&FL7$$\"3-+++k#zkA#F-$ \"37+++uB[)p&FL7$$\"3#******Rl!G@>F-$\"3f*****H[*[B) y,;F-$\"3V+++'=pNt&FL7$$\"3))*****z%[Dp7F-$\"3%******p<0nu&FL7$$\"3I++ +v\\iZ#*Fhs$\"3%)*****Rgmov&FL7$$\"3')*****pk1Bp&Fhs$\"3u******eg+kdFL 7$$\"33+++B-=M?Fhs$\"3O+++vp0odFL7$$!3-+++W2%3s\"Fhs$\"3?+++[-$*odFL7$ $!3I+++t**oobFhs$\"3')*****>!)4lw&FL7$$!3V*****zDxu]*Fhs$\"3K+++(pY1w& FL7$$!3))*****H%\\!QN\"F-$\"3=+++E7:^dFL7$$!3++++T<\\m@Y.`FL7$$!3o*****p,VQE*F-$\"3++++;Rlj_FL7$$!35+++%*op]5FL$\"3 !)*****42*yU_FL7$$!3/+++#G*Gn6FL$\"35+++?BRL_FL7$$!3-+++;apv7FL$\"3f** ***\\(GwH_FL7$$!3/+++cS'pP\"FL$\"3I+++f/fG_FL7$$!33+++AvHs9FL$\"3M+++P @,G_FL7$$!3/+++x9ti:FL$\"3v*****p]]pA&FL7$$!3!*******)p+\"\\;FL$\"3U++ +M#fZA&FL7$$!3++++,\"p?t\"FL$\"3u******pu.@_FL7$$!3*******H\"p;7=FL$\" 3N+++,(Hb@&FL7$$!3/+++QA#)*)=FL$\"3m*****fWp!3_FL7$$!3#******4`&Ql>FL$ \"3K+++lja)>&FL7$$!3!******4<[\"R?FL$\"39+++&R')o=&FL7$$!3%)*****>`c86 #FL$\"3!*******z%RI<&FL7$$!3')*****>wAA=#FL$\"3++++=>(p:&FL7$$!3!***** *f8L>D#FL$\"3&)*****4hh'Q^FL7$$!3#*******=Yl?BFL$\"36+++x_4=^FL7$$!3!* ******p+a)Q#FL$\"39+++EqE&4&FL7$$!3/+++dBtbCFL$\"3)******R'zm$G/&FL7$$!33+++^td)e#FL$\"3F+++f+E8]FL7$$!3=+++U/ \\aEFL$\"3K+++xbZ\")\\FL7$$!3!)*****>>L-s#FL$\"3))*****\\&G_Z\\FL7$$!3 ?+++j!Hfy#FL$\"3++++%Rc9\"\\FL7$$!3%)******))op^GFL$\"3e*****R)zMt[FL7 $$!3y*****>\"yk@LFL$\"3m******ziYfXFL7$$!3#******f7N)*Q$FL$\"3G+++d\" R(4XFL7$$!3'******\\n&peMFL$\"3g*****>1\\#fWFL7$$!3>+++4QkFNFL$\"30+++ 'e-\"3WFL7$$!3)******H*R`'f$FL$\"3y*****H*QPcVFL7$$!3!)*******f:_m$FL$ \"3*)******3Q5/VFL7$$!3-+++'[TNt$FL$\"3y*****z%=I^UFL7$$!31+++V[P,QFL$ \"31+++\"Q]z>%FL7$$!3%)*****f%RfoQFL$\"3V+++>+,WTFL7$$!3\")*****>%[4NR FL$\"3U+++M`U*3%FL7$$!3M+++(G#z+SFL$\"3N+++q-8MSFL7$$!3%******zR>c1%FL $\"3'*******pG0yRFL7$$!3=+++]k_HTFL$\"3.+++X\">6#RFL7$$!3)******p=zC>% FL$\"3/+++0hDjQFL7$$!3g*****p$pXaUFL$\"3\")*****>.%R/QFL7$$!3<+++(p]aJ %FL$\"3!)******[\"oWu$FL7$$!35+++'HhaP%FL$\"35+++4)>Mo$FL7$$!3L+++ww\\ MWFL$\"3()*****zN(>@OFL7$$!3y*****>GvD\\%FL$\"39+++QnvdNFL7$$!3?+++_Xr \\XFL$\"3=+++J>1$\\$FL7$$!3,+++f%Rfg%FL$\"3#)*****pE&3FMFL7$$!3/+++wgF hYFL$\"3#)*****fi2)fLFL7$$!3z*****f7^dr%FL$\"3'******p]=7H$FL7$$!3E+++ -0RpZFL$\"3%)*****p&fJ@KFL7$$!3r*****pw5.,<&FL$\"3')******3M(ph#FL7$$!3h*****>Kbl@&FL$\"38+++nY DODFL7$$!3e*****p\"**4i_FL$\"37+++)y*\\aCFL7$$!3$)*****>s$o1`FL$\"33++ +NgtrBFL7$$!3%******f;Z-N&FL$\"3))*****pD&)zG#FL7$$!3S+++Lcs#R&FL$\"39 +++=ME.AFL7$$!3h*****\\(30MaFL$\"30+++H0e<@FL7$$!3u*****4P_TZ&FL$\"3*) *****R)4%4.#FL7$$!3B+++e&eH^&FL$\"3)******4BWL%>FL7$$!3-+++**yR]bFL$\" 3!******zt&ya=FL7$$!33+++^(*R'e&FL$\"3)******\\=e_w\"FL7$$!3B+++Y]*3i& FL$\"3))*****p\"Gvu;FL7$$!3C+++(y;Ql&FL$\"3))******)*3E$e\"FL7$$!3=+++ 2/5&o&FL$\"33+++^_x!\\\"FL7$$!3E+++-So9dFL$\"31+++*y\"H(R\"FL7$$!3!)** ***>\\3Du&FL$\"35+++p5\"GI\"FL7$$!3c*****Rt<&odFL$\"36+++\"zRt?\"FL7$$ !3w*****fheEz&FL$\"3$******>A#*36\"FL7$$!3Q+++e6)[\"eFL$\"3#******z^\" \\85FL7$$!3x*****RjQ^$eFL$\"3C+++(y3<:*F-7$$!3S+++GxQ`eFL$\"33+++#)\\u f\")F-7$$!39+++=()epeFL$\"3')******GCefrF-7$$!3K+++-dq$)eFL$\"3y*****> )***=:'F-7$$!3/+++Uoq&*eFL$\"3'******4D'\\P^F-7$$!3!******4ikb!fFL$\"3 %******\\![G+++]J w=fFL$\"3!******4X1O1#F-7$$!3H+++pE2AfFL$\"31+++WMVK5F-7$$!3U+++Fn)***=:'F-7$Fb[n$!3')******GCefrF-7$F][n$!33+++#)\\uf\")F-7$Fhjm$!3C +++(y3<:*F-7$Fcjm$!3#******z^\"\\85FL7$F^jm$!3$******>A#*36\"FL7$Fiim$ !36+++\"zRt?\"FL7$Fdim$!35+++p5\"GI\"FL7$F_im$!31+++*y\"H(R\"FL7$Fjhm$ !33+++^_x!\\\"FL7$Fehm$!3))******)*3E$e\"FL7$F`hm$!3))*****p\"Gvu;FL7$ F[hm$!3)******\\=e_w\"FL7$Ffgm$!3!******zt&ya=FL7$Fagm$!3)******4BWL%> FL7$F\\gm$!3*)*****R)4%4.#FL7$Fgfm$!30+++H0e<@FL7$Fbfm$!39+++=ME.AFL7$ F]fm$!3))*****pD&)zG#FL7$Fhem$!33+++NgtrBFL7$Fcem$!37+++)y*\\aCFL7$F^e m$!38+++nYDODFL7$Fidm$!3')******3M(ph#FL7$Fddm$!3=+++=Ui'p#FL7$F_dm$!3 1+++,?1$\\$FL7$Fh`m$!39+++QnvdNFL7$Fc`m$!3()*****z N(>@OFL7$F^`m$!35+++4)>Mo$FL7$Fi_m$!3!)******[\"oWu$FL7$Fd_m$!3\")**** *>.%R/QFL7$F__m$!3/+++0hDjQFL7$Fj^m$!3.+++X\">6#RFL7$Fe^m$!3'*******pG 0yRFL7$F`^m$!3N+++q-8MSFL7$F[^m$!3U+++M`U*3%FL7$Ff]m$!3V+++>+,WTFL7$Fa ]m$!31+++\"Q]z>%FL7$F\\]m$!3y*****z%=I^UFL7$Fg\\m$!3*)******3Q5/VFL7$F b\\m$!3y*****H*QPcVFL7$F]\\m$!30+++'e-\"3WFL7$Fh[m$!3g*****>1\\#fWFL7$ Fc[m$!3G+++d\"R(4XFL7$F^[m$!3m******ziYfXFL7$Fijl$!3(*******ybH3YFL7$F djl$!3O+++\\52cYFL7$F_jl$!32+++6Yi-ZFL7$Fjil$!37+++-eyZZFL7$Feil$!3=++ +q1R\"z%FL7$F`il$!3m*****f!))GL[FL7$F[il$!3e*****R)zMt[FL7$Ffhl$!3++++ %Rc9\"\\FL7$Fahl$!3))*****\\&G_Z\\FL7$F\\hl$!3K+++xbZ\")\\FL7$Fggl$!3F +++f+E8]FL7$Fbgl$!3-+++>m$G/&FL7$F]gl$!3)******R'z(p:&F L7$Fdel$!3!*******z%RI<&FL7$F_el$!39+++&R')o=&FL7$Fjdl$!3K+++lja)>&FL7 $Fedl$!3m*****fWp!3_FL7$F`dl$!3N+++,(Hb@&FL7$F[dl$!3u******pu.@_FL7$Ff cl$!3U+++M#fZA&FL7$Facl$!3v*****p]]pA&FL7$F\\cl$!3M+++P@,G_FL7$Fgbl$!3 I+++f/fG_FL7$Fbbl$!3f*****\\(GwH_FL7$F]bl$!35+++?BRL_FL7$Fhal$!3!)**** *42*yU_FL7$Fcal$!3++++;Rlj_FL7$F^al$!3%)******>@Y.`FL7$Fi`l$!3Q+++l\\` j`FL7$Fd`l$!3%*******)G%zJaFL7$F_`l$!3G+++g9%e\\&FL7$Fj_l$!3%)*****4a@ 9b&FL7$Fe_l$!3M+++()oZ)f&FL7$F`_l$!3U+++\\:(zj&FL7$F[_l$!39+++&o04n&FL 7$Ff^l$!39+++X)3\")p&FL7$Fa^l$!3M+++uCB?dFL7$F\\^l$!3'******HA#yPdFL7$ Fg]l$!3=+++E7:^dFL7$Fb]l$!3K+++(pY1w&FL7$F]]l$!3')*****>!)4lw&FL7$Fh\\ l$!3?+++[-$*odFL7$Fc\\l$!3O+++vp0odFL7$F^\\l$!3u******eg+kdFL7$Fi[l$!3 %)*****Rgmov&FL7$Fd[l$!3%******p<0nu&FL7$F_[l$!3V+++'=pNt&FL7$Fjz$!3f* ****H[*[[S\\ZFL7$Fft $!3/+++k4$oh%FL7$Fat$!3?+++j8#)eWFL7$F\\t$!3#******>_O FL7$F`r$!3;+++3(3V qEFL7$Fbp$!33+++7OI8DFL7$F]p$!3?+++\\m?cBFL7$$\"34+++0(>]#HF_o$!3%)*** **zE@\"*>#FL7$Fco$!3&)*****\\_R?/#FL7$F]o$!35+++\\'e\\)=FL7$Fhn$!3&*** ***H)z(ys\"FL7$$\"3')*****fYkMY#FZ$!37+++9vzq:FL7$$\"39*****\\eu^\"**F I$!32++++tr89FL7$FN$!3/+++Dtjc7FL7$$\"3.+++$\\2g+\"FI$!3-+++Dvb*4\"FL7 $$\"3%******fs`tM#FC$!3G+++T#yZU*F-7$$\"3)******pQg%pSF=$!3u*****fp\") R&yF-7$$\"3*)******4t4CYF7$!3_*****fJ&=$G'F-7$F($!3D+++\")*)Q7ZF-7$F($ !3)******Rl#fTJF-7$F($!3++++Fjzq:F-F'-%&COLORG6&%$RGBG$\"#[!\"#$\"\"\" !\"\"Fh\\p-%)POLYGONSG6^_l7%F'7$F($\"+Fjzq:!#57$$!#TF]]pF(7%Fb]p7$F($ \"+aEfTJFe]pFf]p7%Fj]p7$F($\"+\")*)Q7ZFe]pFf]p7%F^^p7$$\"+Kt4CYFhs$\"+ ;`=$G'Fe]pFf]p7%Fb^p7$$\"+\"Qg%pSF-$\"+'p\")R&yFe]pFf]p7%Fh^p7$$\"+9TN ZBFL$\"+T#yZU*Fe]pFf]p7%F^_p7$$\"+Iy+15!#;$\"+Dvb*4\"!\"*Ff]p7%Fd_p7$$ \"+'H@%\\MFg_p$\"+Dtjc7Fj_pFf]p7%F\\`p7$$\"+%eu^\"**Fg_p$\"++tr89Fj_pF f]p7%Fb`p7$$\"+\"[kMY#!#:$\"+9vzq:Fj_pFf]p7%Fh`p7$$\"+\")*HwR&F[ap$\"+ $)z(ys\"Fj_pFf]p7%F_ap7$$\"+%R))f0\"!#9$\"+\\'e\\)=Fj_pFf]p7%Feap7$$\" +xo\\b=Fhap$\"+D&R?/#Fj_pFf]p7%F\\bp7$$\"+/(>]#HFhap$\"+o77*>#Fj_pFf]p 7%Fbbp7$$\"+\"y@A4%Fhap$\"+\\m?cBFj_pFf]p7%Fhbp7$$\"+)H1]$\\Fhap$\"+7O I8DFj_pFf]p7%F^cp7$$\"+*Q*yYZFhap$\"+s3VqEFj_pFf]p7%Fdcp7$$\"+EOFj_p Ff]p7%Fhep7$$\"+I%**[h#!#7$\"+g<$=y$Fj_pFf]p7%F_fp7$$\"+d>42pFbfp$\"+W gxYRFj_pFf]p7%Fffp7$$\"+%y+Uh\"!#6$\"+q(*Q:TFj_pFf]p7%F\\gp7$$\"+Sf!H \\$F_gp$\"+Av^(G%Fj_pFf]p7%Fcgp7$$\"+eO+6qF_gp$\"+j8#)eWFj_pFf]p7%Figp 7$$\"+eH7[7Fe]p$\"+k4$oh%Fj_pFf]p7%F_hp7$$\"+6Zz))=Fe]p$\"+?[S\\ZFj_pF f]p7%Fehp7$$\"+8'>P[#Fe]p$\"+5t!y&[Fj_pFf]p7%F[ip7$$\"+:!>Y(HFe]p$\"+8 DY\\\\Fj_pFf]p7%Faip7$$\"+#*G=dLFe]p$\"+Qg-I]Fj_pFf]p7%Fgip7$$\"+#4tDk $Fe]p$\"+&)*pF5&Fj_pFf]p7%F]jp7$$\"+N&fP%QFe]p$\"+Bt_p^Fj_pFf]p7%Fcjp7 $$\"+T'fA(RFe]p$\"+hZNJ_Fj_pFf]p7%Fijp7$$\"+\"4Nx.%Fe]p$\"+*z()))G&Fj_ pFf]p7%F_[q7$$\"+x@B[SFe]p$\"+oi_U`Fj_pFf]p7%Fe[q7$$\"+]#)\\5SFe]p$\"+ %zLDR&Fj_pFf]p7%F[\\q7$$\"+?%[-$RFe]p$\"+ZG4RaFj_pFf]p7%Fa\\q7$$\"+g$p B\"QFe]p$\"+\"*fL#[&Fj_pFf]p7%Fg\\q7$$\"+R>2hOFe]p$\"+tQOAbFj_pFf]p7%F ]]q7$$\"+Nl+![$Fe]p$\"+'fb#fbFj_pFf]p7%Fc]q7$$\"+7>NsKFe]p$\"+,Y2$f&Fj _pFf]p7%Fi]q7$$\"+K`)3/$Fe]p$\"+l=(Qi&Fj_pFf]p7%F_^q7$$\"+.!R!)y#Fe]p$ \"+uuo^cFj_pFf]p7%Fe^q7$$\"+6i%f^#Fe]p$\"+)=^ln&Fj_pFf]p7%F[_q7$$\"+k# zkA#Fe]p$\"+uB[)p&Fj_pFf]p7%Fa_q7$$\"+a1G@>Fe]p$\"+$[*[Fj_p$\"+lja)>&Fj_pFf]p7%Fg[r7$$!+r\"[\"R? Fj_p$\"+&R')o=&Fj_pFf]p7%F]\\r7$$!+KlN6@Fj_p$\"+![RI<&Fj_pFf]p7%Fc\\r7 $$!+iFA#=#Fj_p$\"+=>(p:&Fj_pFf]p7%Fi\\r7$$!+OJ$>D#Fj_p$\"+6;mQ^Fj_pFf] p7%F_]r7$$!+>Yl?BFj_p$\"+x_4=^Fj_pFf]p7%Fe]r7$$!+q+a)Q#Fj_p$\"+EqE&4&F j_pFf]p7%F[^r7$$!+dBtbCFj_p$\"+kzm $G/&Fj_pFf]p7%Fg^r7$$!+^td)e#Fj_p$\"+f+E8]Fj_pFf]p7%F]_r7$$!+U/\\aEFj_ p$\"+xbZ\")\\Fj_pFf]p7%Fc_r7$$!+#>L-s#Fj_p$\"+bG_Z\\Fj_pFf]p7%Fi_r7$$! +j!Hfy#Fj_p$\"+%Rc9\"\\Fj_pFf]p7%F_`r7$$!+*)op^GFj_p$\"+%)zMt[Fj_pFf]p 7%Fe`r7$$!+7yk@LFj_p$\"+!Gm%fXFj_pFf]p7%F_cr7$$! +E^$)*Q$Fj_p$\"+d\"R(4XFj_pFf]p7%Fecr7$$!+vcpeMFj_p$\"+i!\\#fWFj_pFf]p 7%F[dr7$$!+4QkFNFj_p$\"+'e-\"3WFj_pFf]p7%Fadr7$$!+$*R`'f$Fj_p$\"+$*QPc VFj_pFf]p7%Fgdr7$$!++c@lOFj_p$\"+4Q5/VFj_pFf]p7%F]er7$$!+'[TNt$Fj_p$\" +[=I^UFj_pFf]p7%Fcer7$$!+V[P,QFj_p$\"+\"Q]z>%Fj_pFf]p7%Fier7$$!+YRfoQF j_p$\"+>+,WTFj_pFf]p7%F_fr7$$!+U[4NRFj_p$\"+M`U*3%Fj_pFf]p7%Fefr7$$!+( G#z+SFj_p$\"+q-8MSFj_pFf]p7%F[gr7$$!+)R>c1%Fj_p$\"+qG0yRFj_pFf]p7%Fagr 7$$!+]k_HTFj_p$\"+X\">6#RFj_pFf]p7%Fggr7$$!+(=zC>%Fj_p$\"+0hDjQFj_pFf] p7%F]hr7$$!+PpXaUFj_p$\"+KSR/QFj_pFf]p7%Fchr7$$!+(p]aJ%Fj_p$\"+\\\"oWu $Fj_pFf]p7%Fihr7$$!+'HhaP%Fj_p$\"+4)>Mo$Fj_pFf]p7%F_ir7$$!+ww\\MWFj_p$ \"+et>@OFj_pFf]p7%Feir7$$!+#GvD\\%Fj_p$\"+QnvdNFj_pFf]p7%F[jr7$$!+_Xr \\XFj_p$\"+J>1$\\$Fj_pFf]p7%Fajr7$$!+f%Rfg%Fj_p$\"+n_3FMFj_pFf]p7%Fgjr 7$$!+wgFhYFj_p$\"+Ew!)fLFj_pFf]p7%F][s7$$!+E6v:ZFj_p$\"+2&=7H$Fj_pFf]p 7%Fc[s7$$!+-0RpZFj_p$\"+dfJ@KFj_pFf]p7%Fi[s7$$!+nx@A[Fj_p$\"+>j5]JFj_p Ff]p7%F_\\s7$$!+PEDu[Fj_p$\"+\\PgxIFj_pFf]p7%Fe\\s7$$!+;&4b#\\Fj_p$\"+ #[HQ+$Fj_pFf]p7%F[]s7$$!+Jh*f(\\Fj_p$\"+B3\")GHFj_pFf]p7%Fa]s7$$!+cBrD ]Fj_p$\"+p)zD&GFj_pFf]p7%Fg]s7$$!+4$\\Y2&Fj_p$\"+,?Fj_pFf]p7%Fias7$$!+**yR]bFj_p$\"+Qdya=Fj_pFf]p7%F_bs7$$ !+^(*R'e&Fj_p$\"+&=e_w\"Fj_pFf]p7%Febs7$$!+Y]*3i&Fj_p$\"+Fj_pFf]p7% Fa^t7$Fdas$!+%)4%4.#Fj_pFf]p7%Fe^t7$F^as$!+H0e<@Fj_pFf]p7%Fi^t7$Fh`s$! +=ME.AFj_pFf]p7%F]_t7$Fb`s$!+d_)zG#Fj_pFf]p7%Fa_t7$F\\`s$!+NgtrBFj_pFf ]p7%Fe_t7$Ff_s$!+)y*\\aCFj_pFf]p7%Fi_t7$F`_s$!+nYDODFj_pFf]p7%F]`t7$Fj ^s$!+4M(ph#Fj_pFf]p7%Fa`t7$Fd^s$!+=Ui'p#Fj_pFf]p7%Fe`t7$F^^s$!+,?j5]JFj_pFf]p7%F]bt7$Fj[s$!+dfJ@KFj_pFf]p7%Fabt7$Fd[s$!+2&=7H$Fj_pFf] p7%Febt7$F^[s$!+Ew!)fLFj_pFf]p7%Fibt7$Fhjr$!+n_3FMFj_pFf]p7%F]ct7$Fbjr $!+J>1$\\$Fj_pFf]p7%Fact7$F\\jr$!+QnvdNFj_pFf]p7%Fect7$Ffir$!+et>@OFj_ pFf]p7%Fict7$F`ir$!+4)>Mo$Fj_pFf]p7%F]dt7$Fjhr$!+\\\"oWu$Fj_pFf]p7%Fad t7$Fdhr$!+KSR/QFj_pFf]p7%Fedt7$F^hr$!+0hDjQFj_pFf]p7%Fidt7$Fhgr$!+X\"> 6#RFj_pFf]p7%F]et7$Fbgr$!+qG0yRFj_pFf]p7%Faet7$F\\gr$!+q-8MSFj_pFf]p7% Feet7$Fffr$!+M`U*3%Fj_pFf]p7%Fiet7$F`fr$!+>+,WTFj_pFf]p7%F]ft7$Fjer$!+ \"Q]z>%Fj_pFf]p7%Faft7$Fder$!+[=I^UFj_pFf]p7%Feft7$F^er$!+4Q5/VFj_pFf] p7%Fift7$Fhdr$!+$*QPcVFj_pFf]p7%F]gt7$Fbdr$!+'e-\"3WFj_pFf]p7%Fagt7$F \\dr$!+i!\\#fWFj_pFf]p7%Fegt7$Ffcr$!+d\"R(4XFj_pFf]p7%Figt7$F`cr$!+!Gm %fXFj_pFf]p7%F]ht7$Fjbr$!+zbH3YFj_pFf]p7%Faht7$Fdbr$!+\\52cYFj_pFf]p7% Feht7$F^br$!+6Yi-ZFj_pFf]p7%Fiht7$Fhar$!+-eyZZFj_pFf]p7%F]it7$Fbar$!+q 1R\"z%Fj_pFf]p7%Fait7$F\\ar$!+1))GL[Fj_pFf]p7%Feit7$Ff`r$!+%)zMt[Fj_pF f]p7%Fiit7$F``r$!+%Rc9\"\\Fj_pFf]p7%F]jt7$Fj_r$!+bG_Z\\Fj_pFf]p7%Fajt7 $Fd_r$!+xbZ\")\\Fj_pFf]p7%Fejt7$F^_r$!+f+E8]Fj_pFf]p7%Fijt7$Fh^r$!+>m$ G/&Fj_pFf]p7%F][u7$Fb^r$!+kz(p:&Fj_pFf]p7%Fa\\u7$Fd\\r$!+![RI<&Fj_pFf]p7%Fe\\u7$F^\\r$!+&R')o= &Fj_pFf]p7%Fi\\u7$Fh[r$!+lja)>&Fj_pFf]p7%F]]u7$Fb[r$!+Y%p!3_Fj_pFf]p7% Fa]u7$F\\[r$!+,(Hb@&Fj_pFf]p7%Fe]u7$Ffjq$!+qu.@_Fj_pFf]p7%Fi]u7$F`jq$! +M#fZA&Fj_pFf]p7%F]^u7$Fjiq$!+20&pA&Fj_pFf]p7%Fa^u7$Fdiq$!+P@,G_Fj_pFf ]p7%Fe^u7$F^iq$!+f/fG_Fj_pFf]p7%Fi^u7$Fhhq$!+vGwH_Fj_pFf]p7%F]_u7$Fbhq $!+?BRL_Fj_pFf]p7%Fa_u7$F\\hq$!+r!*yU_Fj_pFf]p7%Fe_u7$Ffgq$!+;Rlj_Fj_p Ff]p7%Fi_u7$F`gq$!+?@Y.`Fj_pFf]p7%F]`u7$Fjfq$!+l\\`j`Fj_pFf]p7%Fa`u7$F dfq$!+*G%zJaFj_pFf]p7%Fe`u7$F^fq$!+g9%e\\&Fj_pFf]p7%Fi`u7$Fheq$!+T:U^b Fj_pFf]p7%F]au7$Fbeq$!+()oZ)f&Fj_pFf]p7%Faau7$F\\eq$!+\\:(zj&Fj_pFf]p7 %Feau7$Ffdq$!+&o04n&Fj_pFf]p7%Fiau7$F`dq$!+X)3\")p&Fj_pFf]p7%F]bu7$Fjc q$!+uCB?dFj_pFf]p7%Fabu7$Fdcq$!+BAyPdFj_pFf]p7%Febu7$F^cq$!+E7:^dFj_pF f]p7%Fibu7$Fhbq$!+(pY1w&Fj_pFf]p7%F]cu7$Fbbq$!+-)4lw&Fj_pFf]p7%Facu7$F \\bq$!+[-$*odFj_pFf]p7%Fecu7$Ffaq$!+vp0odFj_pFf]p7%Ficu7$F`aq$!+fg+kdF j_pFf]p7%F]du7$Fj`q$!+/m'ov&Fj_pFf]p7%Fadu7$Fd`q$!+x^qYdFj_pFf]p7%Fedu 7$F^`q$!+'=pNt&Fj_pFf]p7%Fidu7$Fh_q$!+$[*[OFj_pFf]p7%Fe[v7$Fcep$!+3]#HFhap$!+o77*>#Fj_pFf]p7%F[^v7$F]bp$!+D&R?/#Fj_pFf]p7%F a^v7$Ffap$!+\\'e\\)=Fj_pFf]p7%Fe^v7$F`ap$!+$)z(ys\"Fj_pFf]p7%Fi^v7$$\" +mWYjCF[ap$!+9vzq:Fj_pFf]p7%F]_v7$$\"+&eu^\"**Fg_p$!++tr89Fj_pFf]p7%Fc _v7$F]`p$!+Dtjc7Fj_pFf]p7%Fi_v7$$\"+$\\2g+\"Fg_p$!+Dvb*4\"Fj_pFf]p7%F] `v7$$\"+EPNZBFL$!+T#yZU*Fe]pFf]p7%Fc`v7$$\"+(Qg%pSF-$!+'p\")R&yFe]pFf] p7%Fi`v7$$\"+5t4CYFhs$!+;`=$G'Fe]pFf]p7%F_av7$F($!+\")*)Q7ZFe]pFf]p7%F eav7$F($!+aEfTJFe]pFf]p7%Fiav7$F($!+Fjzq:Fe]pFf]p7%F]bvF'Ff]p-Fe\\p6&F g\\p$\"#&*Fj\\p$\"\"#F]]pFcbv-%&STYLEG6#%,PATCHNOGRIDG-F$6%7$7$$!3o*** **********)oFLF(7$$\"3%**************=\"FLF(-%'COLOURG6&Fg\\pF)F)F)-%* LINESTYLEG6#\"\"$-F$6%7$7$F($!3o*************Q'FL7$F($\"3o************ *Q'FLFdcvFgcv-%+AXESLABELSG6%%&Re(z)G%&Im(z)G-%%FONTG6#%(DEFAULTG-Fjdv 6$%*HELVETICAG\"\"*-%*AXESSTYLEG6#%$BOXG-%(SCALINGG6#%,CONSTRAINEDG-%% VIEWG6$;$!$*oFj\\p$\"$>\"Fj\\p;$!$R'Fj\\p$\"$R'Fj\\p" 1 2 0 1 10 0 2 9 1 2 1 1.000000 45.000000 45.000000 0 0 "Curve 1" "Curve 2" "Curve 3 " "Curve 4" }}}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 4 "The " }{TEXT 260 30 "interval of absolute stability" }{TEXT -1 89 " (or stability interval) is the intersection of the stability reg ion with the real line." }}{PARA 0 "" 0 "" {TEXT -1 59 "For this schem e the stability interval is (approximately) " }{XPPEDIT 18 0 "[-5.925 2, 0];" "6#7$,$-%&FloatG6$\"&_#f!\"%!\"\"\"\"!" }{TEXT -1 2 ". " }} {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 114 "We can d istort the boundary curve horizontally by taking the 11th root of the \+ real part of points along the curve." }}{PARA 0 "" 0 "" {TEXT -1 150 " In this way we see that the intersection of the stability region with \+ the non-negative imaginary axis consists of the union of two disjoint \+ intervals." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 " " {MPLTEXT 1 0 662 "R := z -> add(z^j/j!,j=0..8)+\n 1200105400867035 59614767034918288835444137/4470130692072453843401165111325891029377351 6800*z^9+\n 52997914489441345950684729787916275285859/20562601183533 2876796453595120990987351358177280*z^10+\n 1312234912952442029167932 4548553359252473/1007147813071018172064262506715057897231142092800*z^1 1+\n 258171010731907839975716281021/27164034838790298254528961807566 14742016*z^12:\nDigits := 18:\npts := []: z0 := 0:\nfor ct from 0 to 2 00 do\n zz := newton(R(z)=exp(ct*Pi/60*I),z=z0):\n z0 := zz:\n p ts := [op(pts),[surd(Re(zz),11),Im(zz)]]:\nend do:\nplot(pts,color=COL OR(RGB,.85,0,.85),thickness=2,font=[HELVETICA,9]);\nDigits := 10:" }} {PARA 13 "" 1 "" {GLPLOT2D 287 308 308 {PLOTDATA 2 "6(-%'CURVESG6#7ew7 $$\"\"!F)F(7$F($\"3v))H)fv()fB&!#>7$$\"3*))f.OP%e.GF-$\"3#))f'>^v>Z5!# =7$$\"3W^c*o\"\\&>0%F-$\"3aP\\zEjzq:F37$$\"3c#fDx5-4E&F-$\"32OPR-^R%4# F37$$\"3'\\wrb\\L0W'F-$\"3O![&*z(Q*zh#F37$$\"3NL^V#e&['f(F-$\"3O*45Ol# fTJF37$$\"3@E3(QorAt)F-$\"3g!=o#H9>lOF37$$\"3r)4mNmH-&)*F-$\"3-H\"[]?! z)=%F37$$\"37mg*3#y>&4\"F3$\"3k6_7\")*)Q7ZF37$$\"3!GZ\"RNj'Q?\"F3$\"31 %y^yv()fB&F37$$\"3p!))=QI56J\"F3$\"3E*\\ye`'efdF37$$\"3J5'))3s()pT\"F3 $\"3'QgDjJ&=$G'F37$$\"3_*f*zm'Q:_\"F3$\"3vG7,,Ty1oF37$$\"348&fFH*yC;F3 $\"34e'RF*GQItF37$$\"3&*[sx'fans\"F3$\"3.1ok&p\")R&yF37$$\"37t3Y(RSu#= F3$\"3#ee#f:0ex$)F37$$\"35i/2dZ%o#>F3$\"3!Ru!eg$z6!*)F37$$\"3nUc'R,f\\ -#F3$\"3r&[w6CyZU*F37$$\"3V4+LH&o<7#F3$\"3<3:)3Z5!#<7$$\"3X*QAI\"zQ6BF3$\"3S*p%yCvb*4\"Fhq7$$\"3l,+&**e UTS#F3$\"3F^-&QW<>:\"Fhq7$$\"3pCC8$[#[&\\#F3$\"37H*fkPxU?\"Fhq7$$\"3fq %ev4o`e#F3$\"36'>`aKPmD\"Fhq7$$\"3$4#H\"38bPn#F3$\"3'>fxPH(**38Fhq7$$ \"3:$4,mI%fgFF3$\"3zh)QVGd8O\"Fhq7$$\"3a*3L'>6$e%GF3$\"3f%pJF3$\"3.%RISkdJi \"Fhq7$$\"3'o_Aa<`cC$F3$\"3%)482,y^v;Fhq7$$\"3w76SI0y>LF3$\"32)R)*H)z( ys\"Fhq7$$\"3cMNHZNt\"R$F3$\"3fM)\\m=Q-y\"Fhq7$$\"3!*GR&*RDRhMF3$\"3Bf $4!4%)fK=Fhq7$$\"3RofaWRiGNF3$\"3&>,3&['e\\)=Fhq7$$\"31%=@8'zF$f$F3$\" 3y`:.2*=t$>Fhq7$$\"3#)=Fg/c=bOF3$\"3&**=TG>z'*)>Fhq7$$\"3Z(HZH(\\:9PF3 $\"3!Rb]Y_R?/#Fhq7$$\"38Rr$**R%4#Fhq7$$\"30xq$*HmOAQ F3$\"3Mw2A)[gn9#Fhq7$$\"3hg//\"*)f5(QF3$\"3j(*[Uo77*>#Fhq7$$\"3.Lw:^wp :RF3$\"3jOky7C[^AFhq7$$\"3#))>8G:je&RF3$\"3Q'yxc6WQI#Fhq7$$\"3fvxkc-0 \"*RF3$\"3sg.0\\m?cBFhq7$$\"3&R%QD1Fj?SF3$\"3[rVe%Qq&3CFhq7$$\"3)4W2B6 =Q/%F3$\"3sv9F?e$4Y#Fhq7$$\"3)[:;4Bw&fSF3$\"3'eGv>h.L^#Fhq7$$\"3!3$ePq :_mSF3$\"3!3ZG0hucc#Fhq7$$\"38lNc#>4F1%F3$\"3%*R<$\\!*\\!=EFhq7$$\"3wj .:)y]_/%F3$\"3+R\">A(3VqEFhq7$$\"3p#z;\\?9&4SF3$\"3o)*eaM#>Gs#Fhq7$$\" 3ur`M\"G$>ZRF3$\"3F3+/Cr@vFFhq7$$\"3#)4e;b][SQF3$\"3zAu$f:Fw#GFhq7$$\" 3X>IXRf^NOF3$\"3ZaS\\5D0!)GFhq7$$!3xi+)HYu(>BF3$\"3))[d8Cq\\KHFhq7$$!3 $Q+s$Fhq7$$!3s\"p`zK*G$H%F3$\"3Dv7W4:*yC$Fhq7$$ !36QMsG!)zhUF3$\"37dL5Fkl+LFhq7$$!3Vm%HN0(*y6%F3$\"3w>3ku!4NN$Fhq7$$!3 Jo:*HfA/;$F3$\"3`)HT;'\\Y1MFhq7$$\"3:duM.1y!H%F3$\"3'R$f(zqT&fMFhq7$$ \"3\"z)Q9JkT%o%F3$\"3ddc4@#fF^$Fhq7$$\"3y/'HK_k3(\\F3$\"3$)f)\\'e)Rhc$ Fhq7$$\"3!*4SlyG)G@&F3$\"31Jjse&3(>OFhq7$$\"3$)G\\NWhRIaF3$\"3cHK?4H\\ tOFhq7$$\"3t2%3b/eCj&F3$\"3$Hve16Bvs$Fhq7$$\"3ah/xMV-CeF3$\"3n#eu/wJ=y $Fhq7$$\"3]u&QzH]\"3gF3$\"3*HDbPM`k$QFhq7$$\"3&>#Qc.X)o='F3$\"3sQpE^KU \"*QFhq7$$\"3K`-g^pohjF3$\"34Ur@`UD +%Fhq7$$\"3x(RBN;mNq'F3$\"3e5AO>]ueSFhq7$$\"3)3hfSqh?(oF3$\"3#pJN)p(*Q :TFhq7$$\"3ogEJ=#>&RqF3$\"35XVfTaXsTFhq7$$\"3nzyqS')31sF3$\"3Bkzfag()H UFhq7$$\"3\"e2\"H(f6;P(F3$\"3jW(\\C_#3ZFhq7$$\"3y&3&y4Y1%f)F3$ \"3r?9m>[S\\ZFhq7$$\"3J#z^a![2w')F3$\"3bCy1BF'yy%Fhq7$$\"3k*RP$GT'yu)F 3$\"3]!4MG#[*Q#[Fhq7$$\"3eaHA#*yn5))F3$\"36(*zK5t!y&[Fhq7$$\"3iFhR!f)o l))F3$\"3nL1!=.#))*)[Fhq7$$\"3A4mGD'RR\"*)F3$\"3)4;(p!fk.#\\Fhq7$$\"3( ))\\%zO2Lc*)F3$\"3%f_nE^i%\\\\Fhq7$$\"3QqS#*)pCO**)F3$\"333[%)G#[t(\\F hq7$$\"3m?9%Q&GYE!*F3$\"3**e](y#Q;/]Fhq7$$\"3x))=^))GQb!*F3$\"3$))e4!Q g-I]Fhq7$$\"3%R7xrwN33*F3$\"3)\\E*Rq2.b]Fhq7$$\"34zHM<1?.\"*F3$\"30Q+/ -qDz]Fhq7$$\"3F'R%y#H(zA\"*F3$\"3`)y6Z)*pF5&Fhq7$$\"3(p_iru'*)R\"*F3$ \"36P,I]QiD^Fhq7$$\"3D>()e-&HZ:*F3$\"3='))fuqjy9&Fhq7$$\"3!G3)QED\\n\" *F3$\"3$>]pHKF&p^Fhq7$$\"30gCR1[Ny\"*F3$\"3/s'>'pkk!>&Fhq7$$\"3iDuDu=Y (=*F3$\"3V;Gd3![7@&Fhq7$$\"3>wJ-$[R\\>*F3$\"3\"oy#)4wa8B&Fhq7$$\"3%GN! y0m*3?*F3$\"3$eP!Q7i)4D&Fhq7$$\"3/;<*)))yU0#*F3$\"3Wr'*4%3f,F&Fhq7$$\" 3Ckb'em:'3#*F3$\"3=&>i')z()))G&Fhq7$$\"3wQ1-;;`5#*F3$\"3'z/`-'[=2`Fhq7 $$\"3od\"\\Y8Q7@*F3$\"3-*47p;h]K&Fhq7$$\"3,hehn%*y5#*F3$\"3GW]bni_U`Fh q7$$\"3z?YZ_EB4#*F3$\"3:w>@u&)ef`Fhq7$$\"3&[;<'*H3m?*F3$\"3lazuPbDw`Fh q7$$\"3YD'*=S7&H?*F3$\"3&>isOzLDR&Fhq7$$\"3dhb7i5H)>*F3$\"3q*HZLGH%3aF hq7$$\"3&R.Q8b_E>*F3$\"3`7i([NZRU&Fhq7$$\"3S=ak^g0'=*F3$\"3$Hpgr%G4RaF hq7$$\"3y)3'=ax^y\"*F3$\"3F5M&*f,(QX&Fhq7$$\"3y@ovC*\\+<*F3$\"3:+W**p7*F3$\"3 P)eWE(QOAbFhq7$$\"3tTyO<)RR6*F3$\"3(eeq+;1]`&Fhq7$$\"3(Rp!zY![**4*F3$ \"3H[q+AEIZbFhq7$$\"3,VBw^5,&3*F3$\"3CF4d&fb#fbFhq7$$\"3SAQa')=6p!*F3$ \"3k9_%pDn3d&Fhq7$$\"3z$y!\\F&HA0*F3$\"3fW/HJ'R@e&Fhq7$$\"3Dg&>bgQV.*F 3$\"3&=9\\0guIf&Fhq7$$\"3<=*p?24a,*F3$\"3#*=2Y4Rn.cFhq7$$\"3Ecyi)y0a** )F3$\"3LMJ0q\"RRh&Fhq7$$\"35`<1F!)Gu*)F3$\"3B5U$['=(Qi&Fhq7$$\"3_B2]k) 3?&*)F3$\"3xl&o)[LZLcFhq7$$\"34ku*>X9&G*)F3$\"3(*)e%*3&[uUcFhq7$$\"3Ry +VIJu.*)F3$\"3*R')3OZ(o^cFhq7$$\"3Hay`fVix))F3$\"3QjAA$>-.m&Fhq7$$\"3G BH*fTx+&))F3$\"3c&*=Pe)*eocFhq7$$\"3\"[fRI!)45#))F3$\"3)\\vb%)=^ln&Fhq 7$$\"30X97'H:.z)F3$\"3/?=Wrn=%o&Fhq7$$\"3?C,1C:(yv)F3$\"3j8,>hq\\\"p&F hq7$$\"3Tl@w]p`B()F3$\"3LjXJuB[)p&Fhq7$$\"3\"*Q!yN2Zro)F3$\"3$y4,m)G90 dFhq7$$\"3dlHJ%e4&[')F3$\"3G_!*)*G'y9r&Fhq7$$\"3tJ8VO\")R2')F3$\"3i9W6 $[*[umwKa&>T)F3$\"3b; w&Fhq7$$\"3\"p<'*Q*H F1xF3$\"3+%=>!fg+kdFhq7$$\"3%>&=/C3#4a(F3$\"3q))4@MMqldFhq7$$\"3))Rr\\ aiWEtF3$\"3NhXSL[0ndFhq7$$\"3@B;aCB1=qF3$\"3yp'=X(p0odFhq7$$\"3Tuv@^U3 UkF3$\"3>OCykiqodFhq7$$!30zf`mHIHhF3$\"3r1$>5y)**odFhq7$$!3BPJt*=W@\"p F3$\"3Dtw)zCI*odFhq7$$!33o$eJ`)))osF3$\"3Mt\"eX,'\\odFhq7$$!3?n!4^t8%3 vF3$\"3Jo/[F5pndFhq7$$!3G&HV,\"o!4p(F3$\"3%4nhC!)4lw&Fhq7$$!3![eZj41$R yF3$\"3WGUC#RY\\w&Fhq7$$!3\"el(ey.\"\\'zF3$\"3EnEL^V*Hw&Fhq7$$!3^1&>\" )e]T2)F3$\"3dcf(ppY1w&Fhq7$$!3J\\I`w!\\5<)F3$\"3T#Glf'e*yv&Fhq7$$!38RW 9=eHe#)F3$\"3UUfvmOtadFhq7$$!3V,m3^ByP$)F3$\"3Z-vOE7:^dFhq7$$!3)4&=eq> *3T)F3$\"30H'f9$*Qru&Fhq7$$!3Z3(Q&=)o'y%)F3$\"3TXX#HO'oUdFhq7$$!3L$f:K *>#>a)F3$\"3`#)\\GBAyPdFhq7$$!3q#4bhf#H,')F3$\"3^pb9cUTKdFhq7$$!3\"*o3 kh&)Hd')F3$\"3;am#p:pls&Fhq7$$!3T'4$p2cO5()F3$\"3.C'HTZK-s&Fhq7$$!3I(* >\\8#\\3w)F3$\"3bo&o/])Q8dFhq7$$!3J?3?C104))F3$\"3Ra?eDcFhq7$$ !3!)RgB;>ou\"*F3$\"3)zxS]qHCh&Fhq7$$!3*[g^Lfu1@*F3$\"3>t\"yq)oZ)f&Fhq7 $$!3*\\7`e&=EY#*F3$\"3W6y@[Vo$e&Fhq7$$!3x\\v%GoJ*F3$\"3&HgJ0a@9b&Fhq7$$!3P))[b.H9_$*F3$\"3$*Q'>7exQ`&F hq7$$!3Exg1%>HxQ*F3$\"3w??NyTN:bFhq7$$!3b6j*=;+QU*F3$\"3#[h8+YTe\\&Fhq 7$$!3Ua#Hzk)eg%*F3$\"3+!\\#4etNvaFhq7$$!3A$oIE2U$)\\*F3$\"3!H;7aikRX&F hq-%&COLORG6&%$RGBG$\"#&)!\"#F(Fajn-%+AXESLABELSG6$Q!6\"Fgjn-%%FONTG6$ %*HELVETICAG\"\"*-%*THICKNESSG6#\"\"#-%%VIEWG6$%(DEFAULTGFe[o" 1 2 0 1 10 2 2 6 1 4 2 1.000000 45.000000 45.000000 0 0 "Curve 1" }}}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 108 "The relevant i ntersection points of the boundary curve with the imaginary axis can b e determined as follows." }}{PARA 0 "" 0 "" {TEXT -1 87 "First we look for points on the boundary curve either side of each intersection poi nt. " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 289 "Digits := 15:\nz0 := 2.9*I:\nfor ct from 92 to 95 do \n newton(R(z)=exp(ct*Pi/100*I),z=z0);\nend do;``;\nz0 := 3.4*I:\nfo r ct from 107 to 110 do\n newton(R(z)=exp(ct*Pi/100*I),z=z0);\nend d o;``;\nz0 := 5.8*I:\nfor ct from 271 to 274 do\n newton(R(z)=exp(ct* Pi/100*I),z=z0);\nend do;\nDigits := 10:" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#^$$\"0]Av4i7>\"!#>$\"0Gz]uR0*G!#9" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#^$$\"0qp?e-l/$!#?$\"0Dh*zj+AH!#9" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#^$$!07(=#*o3zm!#?$\"0ja^>\"[`H!#9" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#^$$!0>nT$pm9$\"0_p-Hl\\)H!#9" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%!G" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#^$$!0#G;+:9Q\\!#>$\"0-#eg 94kL!#9" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#^$$!0eW3g&[!o\"!#>$\"0Rt6rk eR$!#9" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#^$$\"0^i'zD[(*G!#>$\"0)zK&4! oFM!#9" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#^$$\"0G&Q\"e-h2*!#>$\"0$f(zq T&fM!#9" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%!G" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#^$$\"0^&QC$p2H\"!#;$\"0X=U8*[od!#9" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#^$$\"0.ZS5u_V&!#<$\"0tX:W$zod!#9" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#^$$!0#o#)yf4v?!#<$\"0h]'>!p*od!#9" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#^$$!0OALl)3B'*!#<$\"0+#\\X\\,pd!#9" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 107 "Then we apply the bisect ion method to calculate the parameter value associated with the inters ection point." }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 480 "Digits := \+ 15:\nreal_part := proc(u)\n Re(newton(R(z)=exp(u*Pi*I),z=2.9*I))\nen d proc:\nu0 := bisect('real_part'(u),u=0.92..0.95);\nnewton(R(z)=exp(u 0*Pi*I),z=2.9*I);``;\nreal_part := proc(u)\n Re(newton(R(z)=exp(u*Pi *I),z=3.4*I))\nend proc:\nu0 := bisect('real_part'(u),u=1.07..1.10);\n newton(R(z)=exp(u0*Pi*I),z=3.4*I);``;\nreal_part := proc(u)\n Re(new ton(R(z)=exp(u*Pi*I),z=5.8*I))\nend proc:\nu0 := bisect('real_part'(u) ,u=2.71..2.74);\nnewton(R(z)=exp(u0*Pi*I),z=5.8*I);\nDigits := 10:" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#>%#u0G$\"0k%R\"fTAL*!#:" }}{PARA 11 " " 1 "" {XPPMATH 20 "6#^$$\"0-A)z=^Ek!#H$\"0,iqT`@$H!#9" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%!G" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%#u0G$\"0v ?j&30%3\"!#9" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#^#$\"0#\\)[OZ(3M!#9" } }{PARA 11 "" 1 "" {XPPMATH 20 "6#%!G" }}{PARA 11 "" 1 "" {XPPMATH 20 " 6#>%#u0G$\"0#4=Y?CFF!#9" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#^#$\"0*p#Q] L*od!#9" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 4 "The " }{TEXT 260 112 "largest interval on the nonnegative imaginary axis that contains the origin and lies inside the stability region" } {TEXT -1 5 " is " }{XPPEDIT 18 0 "[0, 2.9322];" "6#7$\"\"!-%&FloatG6$ \"&A$H!\"%" }{TEXT -1 18 " (approximately)." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 4 "The " }{TEXT 260 72 "intersectio n of the stability region with the nonnegative imaginary axis" }{TEXT -1 5 " is " }{XPPEDIT 18 0 "`union`([0, 2.9322],[3.4087, 5.7689]);" " 6#-%&unionG6$7$\"\"!-%&FloatG6$\"&A$H!\"%7$-F)6$\"&(3MF,-F)6$\"&*odF, " }{TEXT -1 17 " (approximately)." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }} {PARA 0 "" 0 "" {TEXT -1 37 "#------------------------------------" }} {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 81 "The stabi lity function R* for the 13 stage, order 7 scheme is given as follow s." }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 98 "subs(ee,subs(b=`b*`,St abilityFunction(7,13,'expanded'))):\n`R*` := unapply(%,z):\n'`R*`(z)'= `R*`(z);" }}{PARA 12 "" 1 "" {XPPMATH 20 "6#/-%#R*G6#%\"zG,:\"\"\"F)F' F)*&#F)\"\"#F)*$)F'F,F)F)F)*&#F)\"\"'F)*$)F'\"\"$F)F)F)*&#F)\"#CF)*$)F '\"\"%F)F)F)*&#F)\"$?\"F)*$)F'\"\"&F)F)F)*&#F)\"$?(F)*$)F'F1F)F)F)*&#F )\"%S]F)*$)F'\"\"(F)F)F)*&#\"JjO8dI$[c5jT2Ys&p:[o7l#\"O+/v^)\\@@DT2Wpm VB#eWJoJk5F)*$)F'\"\")F)F)F)*&#\"K46zfX2m@$)fspWt\")okkWj(*\"Q+)GOE_AJ =$o=*fvgYzabs>+rU$F)*$)F'\"\"*F)F)F)*&#\"K465s:jZlXR.LU&[Ja,D$QO\"RS#= a'3\"))*y#z'4wG;P9Im#o%43]k\"F)*$)F'\"#5F)F)F)*&#\"A:MT`&G;W]!of]w\"*) 3\"\"I/Z1,@V)3\"4!Rfit " 0 "" {MPLTEXT 1 0 30 "z_0 := newton(`R*`(z)=1, z=-6);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%$z_0G$!+5J#p'e!\"*" }}} {PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 318 "z_0 := newton(`R*`(z)=1,z=-6):\np_1 := plot([`R*`(z),1],z=-6.49.. 0.49,color=[red,blue]):\np_2 := plot([[[z_0,1]]$3],style=point,symbol= [circle,cross,diamond],color=black):\np_3 := plot([[z_0,0],[z_0,1]],li nestyle=3,color=COLOR(RGB,0,.5,0)):\nplots[display]([p_1,p_2,p_3],view =[-6.49..0.49,-0.07..1.47],font=[HELVETICA,9]);" }}{PARA 13 "" 1 "" {GLPLOT2D 387 266 266 {PLOTDATA 2 "6+-%'CURVESG6$7Z7$$!3A++++++!\\'!#< $\"3-\\dm!3(\\'f$F*7$$!3+n\"zf'R'>X'F*$\"3%=shUmZSM$F*7$$!3xL$e>$z#RT' F*$\"3Ckep6&zt5$F*7$$!3a+v$z*=*eP'F*$\"3'4sMf)>i&)GF*7$$!3Lnm\"R'e&yL' F*$\"3eX[Knj%zn#F*7$$!3o](oU'fmriF*$\"3].v$*RK!yM#F*7$$!3,M3ikgZ0iF*$ \"3kk`X)3aS0#F*7$$!3m]7))*fQ58'F*$\"35HgY?;xicF*$\"3auT,nsK^cF^o7$$!3e+vyYv7waF*$\"3-IVFFS!> )RF^o7$$!3hmTD%*)[!G`F*$\"3n+\"\\2OpSs#F^o7$$!3w*\\P=8X/=&F*$\"3QTPsL6 9I=F^o7$$!3@LLQ>lhG]F*$\"3i+K(R2(G!>\"F^o7$$!3km;*[z&)[*[F*$\"3^W1E4we 8!)!#>7$$!3I++:tmLWZF*$\"3mY.u2u[p]Fbq7$$!39++D#QpJf%F*$\"3X/EW7oZ>KFb q7$$!3I++&oo\"\\ZWF*$\"39,;()yihw@Fbq7$$!3FnTD$Q-_J%F*$\"37F?&p=;Hn\"F bq7$$!3-LL)fg(*y:%F*$\"3KBg\"f$4ma9Fbq7$$!3#HLLf#)RY-%F*$\"3e\"o)*=s\\ W\\\"Fbq7$$!3S+v[[$H'pQF*$\"3EnL*3W*p</%[kb $G?Fbq7$$!3))*\\([.r(=e$F*$\"3IlL:!f=.Y#Fbq7$$!3&**\\i%)zD&QMF*$\"3Q`? &fi\\a%HFbq7$$!3[L$3-v`*)G$F*$\"3*zGp:uNP_$Fbq7$$!3AL3d=**f^JF*$\"3DcR >8@wATFbq7$$!31n;Hb_W.IF*$\"3;'zZ;0z4&[Fbq7$$!3ymTD)>a&\\GF*$\"3\"fA$* QH#o7dFbq7$$!3<+Dr!e\"f:FF*$\"3W&*=m]v\"ec'Fbq7$$!3!omTPG34d#F*$\"3'HZ %z[RB9wFbq7$$!30++q!GO9U#F*$\"3Y;^!e(4$*f))Fbq7$$!3]+]x'32_F#F*$\"3PQK &)*QEm-\"F^o7$$!3-+voAGsL@F*$\"3)*y)*ej8K$=\"F^o7$$!3k+]#)p'Gm(>F*$\"3 a?T2'**R]Q\"F^o7$$!36MLV'esa$=F*$\"317XD+SA&f\"F^o7$$!3)****\\#Q$eZo\" F*$\"3U6Ag)*Q#[&=F^o7$$!3?nT&)G))=[:F*$\"3n<)F^o$\"3=r&>,7#e9WF^o7$$!3almmK!*4HnF^o$\"3e0\\q3s@-^F^o7$$! 3ynm\"fN)p[_F^o$\"35\"\\.6pBj\"fF^o7$$!3IJ$3U9c0y$F^o$\"39#Rh0$R#>&oF^ o7$$!3)))****p#f[JCF^o$\"3Ysl3a)\\:%yF^o7$$!3!eILe:#*H&))Fbq$\"3a(3.&Q sv_\"*F^o7$$\"3u%HLL6'Gw\\Fbq$\"31ljpG=-^5F*7$$\"37,]7:\"=@(>F^o$\"3WG " 0 "" {MPLTEXT 1 0 972 "`R* ` := z -> add(z^j/j!,j=0..7)+\n 265126848156957246074163105648330571 33663/1064316831445822343666944074125212149851750400*z^8+\n 97634464 6468817344697259832166074559791109/34271001972555479466075599186831831 2252263628800*z^9+\n 363832501543148542330339456547631572101109/1645 008094682663014371628760967927898810865418240*z^10+\n 10889176505968 050441628553413415/6338274795717736259390091088432101064704*z^11:\npts := []: z0 := 0:\nfor ct from 0 to 320 do\n zz := newton(`R*`(z)=exp (ct*Pi/20*I),z=z0):\n z0 := zz:\n pts := [op(pts),[Re(zz),Im(zz)]] :\nend do:\np_1 := plot(pts,color=COLOR(RGB,.32,0,.32)):\np_2 := plots [polygonplot]([seq([pts[i-1],pts[i],[-4.2,0]],i=2..nops(pts))],\n \+ style=patchnogrid,color=COLOR(RGB,.64,0,.64)):\np_3 := plot([[[-6 .49,0],[1.19,0]],[[0,-6.39],[0,6.39]]],color=black,linestyle=3):\nplot s[display]([p_||(1..3)],view=[-6.49..1.19,-6.39..6.39],font=[HELVETICA ,9],\n labels=[`Re(z)`,`Im(z)`],axes=boxed,scaling=constrai ned);" }}{PARA 13 "" 1 "" {GLPLOT2D 381 565 565 {PLOTDATA 2 "6+-%'CURV ESG6$7]_l7$$\"\"!F)F(7$F($\"3++++Fjzq:!#=7$F($\"3)******Rl#fTJF-7$$!3: +++u%498$!#F$\"3D+++\")*)Q7ZF-7$$!3!******>j0XZ$!#E$\"3Q+++8`=$G'F-7$$ !3$******H()))\\J#!#D$\"3]*****Hp\")R&yF-7$$!3)******z\"=V46!#C$\"3Z** ***>NyZU*F-7$$!31+++U(*=,UFF$\"3%******fid&*4\"!#<7$$!3,+++I@vF8!#B$\" 3/+++qyjc7FN7$$!39+++t`4BOFR$\"3/+++.&>PT\"FN7$$!3))*****fcv1s)FR$\"35 +++v[!3d\"FN7$$!3))*****pOZY(=!#A$\"3/+++%G**ys\"FN7$$!3&******RjNWh$F \\o$\"3!******Hl8])=FN7$$!3c+++Jmc7iF\\o$\"3<+++5)o@/#FN7$$!37+++rkU!G *F\\o$\"3'******H%=S*>#FN7$$!3.+++n2z66!#@$\"3;++++[xcBFN7$$!3&******H P?j@(F\\o$\"3\"******pg&Q9DFN7$$\"3++++:3rN7Fap$\"3++++c,QsEFN7$$\"3&) *****>`!#?$\"39+++kiR!*HFN7$ $\"3i*****pqJxe%Ffq$\"3))*****pqw4:$FN7$$\"3h+++)p!*=r*Ffq$\"3()*****f pHHJ$FN7$$\"3*)*****pU([5>!#>$\"3#******R4DhZ$FN7$$\"3E+++ct(R`$Ffr$\" 3@+++_cSROFN7$$\"3m*****zTIV6'Ffr$\"3)******zzi%*z$FN7$$\"30+++XAu7(*F fr$\"3-+++*\\,0FF-$\"3M+++5**G([%FN7$$\"3!******\\lVI &GF-$\"3c******=$>Tc%FN7$$\"3++++7:8MHF-$\"3$)*****z&*y[j%FN7$$\"3?+++ gs,bHF-$\"3y*****\\!HL+ZFN7$$\"3B+++-j\"y'Ffr$ \"3))*****H8FN:&FN7$$\"3s*****>@(eGLFfr$\"3#******>+8R<&FN7$$!34+++goP 4EFfq$\"3Q+++iq5\">&FN7$$!3#******phj](RFfr$\"3H+++4F70_FN7$$!3(****** \\8LM!yFfr$\"3(******Hbdf@&FN7$$!3+++++yvt6F-$\"3c*****f,#fB_FN7$$!31+ ++`A4x:F-$\"3@+++Kv)zA&FN7$$!3/+++I$)))*)>F-$\"3/+++,U3H_FN7$$!3-+++K& 4>T#F-$\"3T+++&z(zE_FN7$$!3-++++8=VGF-$\"3&******p!f,@_FN7$$!3-+++Mk0% G$F-$\"3f*****f5$f6_FN7$$!32+++:'=`t$F-$\"3#)*****p%\\M)>&FN7$$!37+++B _M)>%F-$\"3W+++94/\"=&FN7$$!3y*****z'\\OvYF-$\"3#******p+(Rf^FN7$$!3_+ ++d,()p^F-$\"3t*****pHqI8&FN7$$!3Y+++hEG(o&F-$\"3N+++GAm,^FN7$$!3r**** *\\ODgB'F-$\"3))*****zo[Z1&FN7$$!3K+++U'p\"HoF-$\"39+++nJ)>-&FN7$$!3S+ ++geb'[(F-$\"3w******eCQt\\FN7$$!3_*****R@adB)F-$\"3')*****f+'**>\\FN7 $$!32+++&Gec5*F-$\"3I+++yv-l[FN7$$!3,+++u@P55FN$\"3C+++,NG9[FN7$$!3*** ****fGB#>6FN$\"3')*****HI*[tZFN7$$!3++++@()pI7FN$\"3********)y(HWZFN7$ $!31+++:W*)R8FN$\"3z******G+_CZFN7$$!3,+++@OhW9FN$\"3y******\\f&4r%FN7 $$!3%******>6tVa\"FN$\"3*)*****4ej4q%FN7$$!35+++6\\UR;FN$\"3q*****zV(p #p%FN7$$!33+++HeDIFN$\"3'******p\"z!zm%FN7$$!3'******R4tB)>FN$\"3%**** **z#GudYFN7$$!3))******G\"451#FN$\"3m*****49*4YYFN7$$!33+++?q\\P@FN$\" 3p*****H!4!Gj%FN7$$!3++++]()47AFN$\"3l*****H#[s>e%FN7$$!3!****** *Q&=nU#FN$\"3U+++_*)3hXFN7$$!3'*******o')y&\\#FN$\"3s******HvEQXFN7$$! 3=+++$*z(Qc#FN$\"3*)*****RqTM^%FN7$$!3)******Rw@6j#FN$\"3W+++***3m[%FN 7$$!3%)*****>iZwp#FN$\"3m*****fIxxX%FN7$$!3')*****RSxNw#FN$\"32+++(4lp U%FN7$$!3'******H@G!HGFN$\"3Q+++)=-UR%FN7$$!3')******>F6%*GFN$\"3!)*** **\\*3`fVFN7$$!35+++:'Q*eHFN$\"3z*****zP2IK%FN7$$!3;+++XrgBIFN$\"3i*** **z%Hq%G%FN7$$!3#******Ht5#)3$FN$\"3#******H,0ZC%FN7$$!3')*****HuHG:$F N$\"3i*****4LATwZ$FN$\"3!)*******z\")e(RFN7$$! 38+++'y+Ha$FN$\"3))*****\\Ljv#RFN7$$!39+++)3B\"3OFN$\"3%)******oLfyQFN 7$$!3=+++2l?tOFN$\"33+++$zq!HQFN7$$!3)******\\TP!QPFN$\"3/+++e22zPFN7$ $!34+++2p\\-QFN$\"3#)*****>jV'GPFN7$$!3?+++IrYmQFN$\"3/+++uf\"yn$FN7$$ !3*)*****>hO)HRFN$\"32+++1JfEOFN7$$!37+++XU]#*RFN$\"3y*****\\si\\d$FN7 $$!3=+++e=QaSFN$\"38+++s*)*G_$FN7$$!3l*****f\\&R:TFN$\"3#********\\m.Z $FN7$$!3Q+++ec[vTFN$\"31+++2UK*>$FN7$$!3'******He84Y%FN$\"3))*****p\"H)H9$FN 7$$!3U+++E0\"\\^%FN$\"3'******\\3Lg3$FN7$$!3))*****>/))yc%FN$\"3#)**** *zQG$GIFN7$$!3I+++\\=')>YFN$\"31+++@d&)pHFN7$$!3w*****4.^3n%FN$\"3/+++ *>41\"HFN7$$!3*)******=o(3s%FN$\"3;+++(G!f]GFN7$$!3;+++&Rh*pZFN$\"33++ +*z2)*y#FN7$$!3D+++=n7=[FN$\"36+++gxFGFFN7$$!3E+++yLRl[FN$\"3!)******4 J-mEFN7$$!3M+++v%z<\"\\FN$\"3:+++TK2.EFN7$$!3y******)e*Hd\\FN$\"3#**** **HNj%RDFN7$$!3\"******p$Q'>+&FN$\"3++++DOBvCFN7$$!3I+++)4xd/&FN$\"3:+ ++6#G/T#FN7$$!3q*****RVQ()3&FN$\"3'*******zS4XBFN7$$!3\")******e2%38&F N$\"33+++l(z#zAFN7$$!3s*****Hxq?<&FN$\"3\")*****>0MI@#FN7$$!3s******)= 4C@&FN$\"31+++/ZSY@FN7$$!3E+++;7$=D&FN$\"3!)*****pFP%z?FN7$$!33+++(G2. H&FN$\"30+++$4u@,#FN7$$!3G+++mR!yK&FN$\"3$*******GMlW>FN7$$!3=+++t\\Gk `FN$\"3'******>&*3p(=FN7$$!3#******fD7(*R&FN$\"31+++3%p*3=FN7$$!38+++ \\q/MaFN$\"3*)*****Rbe3u\"FN7$$!3c*****R&3DnaFN$\"3/+++9`fs;FN7$$!3/++ +0jG*\\&FN$\"3++++9T>/;FN7$$!3B+++(*y6IbFN$\"3*******zMlc`\"FN7$$!3y** ***Hb7(fbFN$\"3-+++Sf,n9FN7$$!3%)*****p.S!)e&FN$\"3#******4&*\\#)R\"FN 7$$!3c*****HAt]h&FN$\"3.+++'>p$H8FN7$$!3S+++M#)yScFN$\"3++++^QPg7FN7$$ !3%)*****>Zk^m&FN$\"3))*****z,j7>\"FN7$$!3s*****za%=)o&FN$\"3-+++V_.A6 FN7$$!3v******[T$)4dFN$\"3%******z(*)o_5FN7$$!3h******\\=5IdFN$\"32+++ ^(HA$)*F-7$$!3O+++s)y*[dFN$\"3b*****pCmj8*F-7$$!37+++F)ekw&FN$\"3_**** *RL-$R%)F-7$$!3)******RVPDy&FN$\"3a*****zT_5u(F-7$$!3#)*****HA7sz&FN$ \"3-+++!*QkTqF-7$$!3s******QA[5eFN$\"3G+++e#=6M'F-7$$!3\"******HuZB#eF N$\"3m*****fxJ&RcF-7$$!3i*****p\"*4G$eFN$\"30+++5c&p$\\F-7$$!3n*****\\ eq=%eFN$\"36+++`bZLUF-7$$!3K+++S>`\\eFN$\"3?+++18>HNF-7$$!3!******>I'z beFN$\"3v*****pO:U#GF-7$$!3E+++%*emgeFN$\"3++++L:n=@F-7$$!3/+++`E9keFN $\"3))*****R3$p79F-7$$!3q*****p2Gi'eFN$\"3!*******Gn?kqFfr7$$!35+++5J# p'eFNF(7$Fb]n$!3!*******Gn?kqFfr7$F]]n$!3))*****R3$p79F-7$Fh\\n$!3++++ L:n=@F-7$Fc\\n$!3v*****pO:U#GF-7$F^\\n$!3?+++18>HNF-7$Fi[n$!36+++`bZLU F-7$Fd[n$!30+++5c&p$\\F-7$F_[n$!3m*****fxJ&RcF-7$Fjjm$!3G+++e#=6M'F-7$ Fejm$!3-+++!*QkTqF-7$F`jm$!3a*****zT_5u(F-7$F[jm$!3_*****RL-$R%)F-7$Ff im$!3b*****pCmj8*F-7$Faim$!32+++^(HA$)*F-7$F\\im$!3%******z(*)o_5FN7$F ghm$!3-+++V_.A6FN7$Fbhm$!3))*****z,j7>\"FN7$F]hm$!3++++^QPg7FN7$Fhgm$! 3.+++'>p$H8FN7$Fcgm$!3#******4&*\\#)R\"FN7$F^gm$!3-+++Sf,n9FN7$Fifm$!3 *******zMlc`\"FN7$Fdfm$!3++++9T>/;FN7$F_fm$!3/+++9`fs;FN7$Fjem$!3*)*** **Rbe3u\"FN7$Feem$!31+++3%p*3=FN7$F`em$!3'******>&*3p(=FN7$F[em$!3$*** ****GMlW>FN7$Ffdm$!30+++$4u@,#FN7$Fadm$!3!)*****pFP%z?FN7$F\\dm$!31+++ /ZSY@FN7$Fgcm$!3\")*****>0MI@#FN7$Fbcm$!33+++l(z#zAFN7$F]cm$!3'******* zS4XBFN7$Fhbm$!3:+++6#G/T#FN7$Fcbm$!3++++DOBvCFN7$F^bm$!3#******HNj%RD FN7$Fiam$!3:+++TK2.EFN7$Fdam$!3!)******4J-mEFN7$F_am$!36+++gxFGFFN7$Fj `m$!33+++*z2)*y#FN7$Fe`m$!3;+++(G!f]GFN7$F``m$!3/+++*>41\"HFN7$F[`m$!3 1+++@d&)pHFN7$Ff_m$!3#)*****zQG$GIFN7$Fa_m$!3'******\\3Lg3$FN7$F\\_m$! 3))*****p\"H)H9$FN7$Fg^m$!33+++Pz>*>$FN7$Fb^m$!3;+++=ZqaKFN7$F]^m$!3#) ******Gd`4LFN7$Fh]m$!3/+++c%GPO$FN7$Fc]m$!31+++2UKjV'GPFN7$F`[m$!3/+++e 22zPFN7$F[[m$!33+++$zq!HQFN7$Ffjl$!3%)******oLfyQFN7$Fajl$!3))*****\\L jv#RFN7$F\\jl$!3!)*******z\")e(RFN7$Fgil$!3p*****zLHM-%FN7$Fbil$!3q*** **p2r+2%FN7$F]il$!36+++\"3jc6%FN7$Fhhl$!3M+++r\"f+;%FN7$Fchl$!3i*****4 L>e %FN7$Fbdl$!3g*****4P%y+YFN7$F]dl$!3l*****H#[s\\FN7$F]_l$!3w******eCQ t\\FN7$Fh^l$!39+++nJ)>-&FN7$Fc^l$!3))*****zo[Z1&FN7$F^^l$!3N+++GAm,^FN 7$Fi]l$!3t*****pHqI8&FN7$Fd]l$!3#******p+(Rf^FN7$F_]l$!3W+++94/\"=&FN7 $Fj\\l$!3#)*****p%\\M)>&FN7$Fe\\l$!3f*****f5$f6_FN7$F`\\l$!3&******p!f ,@_FN7$F[\\l$!3T+++&z(zE_FN7$Ff[l$!3/+++,U3H_FN7$Fa[l$!3@+++Kv)zA&FN7$ F\\[l$!3c*****f,#fB_FN7$Fgz$!3(******Hbdf@&FN7$Fbz$!3H+++4F70_FN7$F]z$ !3Q+++iq5\">&FN7$Fhy$!3#******>+8R<&FN7$Fcy$!3))*****H8FN:&FN7$F^y$!3F +++U8#*H^FN7$Fix$!3&*******)e`I5&FN7$Fdx$!3q******R%oG2&FN7$F_x$!3)*** ***fH(HR]FN7$Fjw$!3s*****f=eA+&FN7$Few$!3))*****f([lh\\FN7$F`w$!3G+++B [P<\\FN7$F[w$!3C+++z^Gp[FN7$Ffv$!3))*****4cDs\"[FN7$Fav$!35+++fa*4w%FN 7$F\\v$!3y*****\\!HL+ZFN7$Fgu$!3$)*****z&*y[j%FN7$Fbu$!3c******=$>Tc%F N7$F]u$!3M+++5**G([%FN7$Fht$!36+++$pHKS%FN7$Fct$!3F+++dc<5VFN7$F^t$!3T +++(3!e0UFN7$Fis$!3C+++#FN7$Feo$!3<+++5)o@/#FN7$F`o$!3!******Hl8]) =FN7$Fjn$!3/+++%G**ys\"FN7$$!3K******ebn?()FR$!35+++v[!3d\"FN7$FV$!3/+ ++.&>PT\"FN7$FP$!3/+++qyjc7FN7$FJ$!3%******fid&*4\"FN7$FD$!3Z*****>NyZ U*F-7$$!36+++s)))\\J#F@$!3]*****Hp\")R&yF-7$$!3;+++Jc]uMF:$!3Q+++8`=$G 'F-7$$!3;+++1&498$F4$!3D+++\")*)Q7ZF-7$F($!3)******Rl#fTJF-7$F($!3++++ Fjzq:F-F'-%&COLORG6&%$RGBG$\"#K!\"#F(Fd\\p-%)POLYGONSG6^_l7%F'7$F($\"+ Fjzq:!#57$$!#U!\"\"F(7%F[]p7$F($\"+aEfTJF^]pF_]p7%Fd]p7$$!+u%498$Ffr$ \"+\")*)Q7ZF^]pF_]p7%Fh]p7$$!+Kc]uMF-$\"+8`=$G'F^]pF_]p7%F^^p7$$!+t))) \\J#FN$\"+$p\")R&yF^]pF_]p7%Fd^p7$$!+==V46!#;$\"+_$yZU*F^]pF_]p7%Fj^p7 $$!+U(*=,UF]_p$\"+Ewb*4\"!\"*F_]p7%Fa_p7$$!+I@vF8!#:$\"+qyjc7Ff_pF_]p7 %Fh_p7$$!+t`4BOF[`p$\"+.&>PT\"Ff_pF_]p7%F_`p7$$!+mbn?()F[`p$\"+v[!3d\" Ff_pF_]p7%Fe`p7$$!+ntku=!#9$\"+%G**ys\"Ff_pF_]p7%F[ap7$$!+McV9OF^ap$\" +`O,&)=Ff_pF_]p7%Fbap7$$!+Jmc7iF^ap$\"+5)o@/#Ff_pF_]p7%Fhap7$$!+rkU!G* F^ap$\"+V=S*>#Ff_pF_]p7%F^bp7$$!+n2z66!#8$\"++[xcBFf_pF_]p7%Fdbp7$$!+t .K;sF^ap$\"+2cQ9DFf_pF_]p7%F[cp7$$\"+:3rN7Fgbp$\"+c,QsEFf_pF_]p7%Facp7 $$\"+Kv@MnFgbp$\"+&ei4$GFf_pF_]p7%Fgcp7$$\"+8OXW>!#7$\"+kiR!*HFf_pF_]p 7%F]dp7$$\"+2!#6$\"+%4DhZ$Ff_pF_]p7%F`ep7$$\"+ct(R`$Fce p$\"+_cSROFf_pF_]p7%Fgep7$$\"+=/L9hFcep$\"+)zi%*z$Ff_pF_]p7%F]fp7$$\"+ XAu7(*Fcep$\"+$>Tc%Ff_pF_]p7%Fghp7$$\"+ 7:8MHF^]p$\"+e*y[j%Ff_pF_]p7%F]ip7$$\"+gs,bHF^]p$\"+0HL+ZFf_pF_]p7%Fci p7$$\"+-j\"y'Fcep$\" +Lr_`^Ff_pF_]p7%Fe]q7$$\"+7seGLFcep$\"+-I\"R<&Ff_pF_]p7%F[^q7$$!+goP4E F`dp$\"+iq5\">&Ff_pF_]p7%Fa^q7$$!+F^]p$\"+,U3H_Ff_pF _]p7%F_`q7$$!+K&4>T#F^]p$\"+&z(zE_Ff_pF_]p7%Fe`q7$$!++8=VGF^]p$\"+2f,@ _Ff_pF_]p7%F[aq7$$!+Mk0%G$F^]p$\"+1Jf6_Ff_pF_]p7%Faaq7$$!+:'=`t$F^]p$ \"+Z\\M)>&Ff_pF_]p7%Fgaq7$$!+B_M)>%F^]p$\"+94/\"=&Ff_pF_]p7%F]bq7$$!+o \\OvYF^]p$\"+2qRf^Ff_pF_]p7%Fcbq7$$!+d,()p^F^]p$\"+(HqI8&Ff_pF_]p7%Fib q7$$!+hEG(o&F^]p$\"+GAm,^Ff_pF_]p7%F_cq7$$!+l`-OiF^]p$\"+)o[Z1&Ff_pF_] p7%Fecq7$$!+U'p\"HoF^]p$\"+nJ)>-&Ff_pF_]p7%F[dq7$$!+geb'[(F^]p$\"+fCQt \\Ff_pF_]p7%Fadq7$$!+9UvN#)F^]p$\"+1g**>\\Ff_pF_]p7%Fgdq7$$!+&Gec5*F^] p$\"+yv-l[Ff_pF_]p7%F]eq7$$!+u@P55Ff_p$\"+,NG9[Ff_pF_]p7%Fceq7$$!+'GB# >6Ff_p$\"+.$*[tZFf_pF_]p7%Fieq7$$!+@()pI7Ff_p$\"+*y(HWZFf_pF_]p7%F_fq7 $$!+:W*)R8Ff_p$\"+H+_CZFf_pF_]p7%Fefq7$$!+@OhW9Ff_p$\"+]f&4r%Ff_pF_]p7 %F[gq7$$!+7JPW:Ff_p$\"+\"ej4q%Ff_pF_]p7%Fagq7$$!+6\\UR;Ff_p$\"+Qup#p%F f_pF_]p7%Fggq7$$!+HeDIFf_p$\"+F f_p$\"+GGudYFf_pF_]p7%F_iq7$$!+H\"451#Ff_p$\"+T\"*4YYFf_pF_]p7%Feiq7$$ !+?q\\P@Ff_p$\"+.4!Gj%Ff_pF_]p7%F[jq7$$!+]()47AFf_p$\"+B[s>e%Ff_p F_]p7%F][r7$$!+R&=nU#Ff_p$\"+_*)3hXFf_pF_]p7%Fc[r7$$!+p')y&\\#Ff_p$\"+ IvEQXFf_pF_]p7%Fi[r7$$!+$*z(Qc#Ff_p$\"+/*>$Ff_pF_]p7%F]gr7$$!+$e84Y%Ff_ p$\"++&Ff_p$\"+DOBvCFf_pF_]p7%Fe[s7$$!+)4xd/&Ff_p$\"+6#G/T#Ff_pF_]p7 %F[\\s7$$!+M%Q()3&Ff_p$\"+!3%4XBFf_pF_]p7%Fa\\s7$$!+f2%38&Ff_p$\"+l(z# zAFf_pF_]p7%Fg\\s7$$!+t22s^Ff_p$\"+_S.8AFf_pF_]p7%F]]s7$$!+*=4C@&Ff_p$ \"+/ZSY@Ff_pF_]p7%Fc]s7$$!+;7$=D&Ff_p$\"+xsVz?Ff_pF_]p7%Fi]s7$$!+(G2.H &Ff_p$\"+$4u@,#Ff_pF_]p7%F_^s7$$!+mR!yK&Ff_p$\"+HMlW>Ff_pF_]p7%Fe^s7$$ !+t\\Gk`Ff_p$\"+_*3p(=Ff_pF_]p7%F[_s7$$!+cAr*R&Ff_p$\"+3%p*3=Ff_pF_]p7 %Fa_s7$$!+\\q/MaFf_p$\"+a&e3u\"Ff_pF_]p7%Fg_s7$$!+a3DnaFf_p$\"+9`fs;Ff _pF_]p7%F]`s7$$!+0jG*\\&Ff_p$\"+9T>/;Ff_pF_]p7%Fc`s7$$!+(*y6IbFf_p$\"+ [`mN:Ff_pF_]p7%Fi`s7$$!+`DrfbFf_p$\"+Sf,n9Ff_pF_]p7%F_as7$$!+P+/)e&Ff_ p$\"+^*\\#)R\"Ff_pF_]p7%Feas7$$!+BK2:cFf_p$\"+'>p$H8Ff_pF_]p7%F[bs7$$! +M#)yScFf_p$\"+^QPg7Ff_pF_]p7%Fabs7$$!+sW;lcFf_p$\"+=IE\">\"Ff_pF_]p7% Fgbs7$$!+[X=)o&Ff_p$\"+V_.A6Ff_pF_]p7%F]cs7$$!+\\T$)4dFf_p$\"+y*)o_5Ff _pF_]p7%Fccs7$$!+]=5IdFf_p$\"+^(HA$)*F^]pF_]p7%Fics7$$!+s)y*[dFf_p$\"+ ZiOO\"*F^]pF_]p7%F_ds7$$!+F)ekw&Ff_p$\"+MBIR%)F^]pF_]p7%Feds7$$!+Mu`#y &Ff_p$\"+=C0TxF^]pF_]p7%F[es7$$!+BA@(z&Ff_p$\"+!*QkTqF^]pF_]p7%Faes7$$ !+RA[5eFf_p$\"+e#=6M'F^]pF_]p7%Fges7$$!+VxMAeFf_p$\"+w<`RcF^]pF_]p7%F] fs7$$!+<*4G$eFf_p$\"+5c&p$\\F^]pF_]p7%Fcfs7$$!+&eq=%eFf_p$\"+`bZLUF^]p F_]p7%Fifs7$$!+S>`\\eFf_p$\"+18>HNF^]pF_]p7%F_gs7$$!+-jzbeFf_p$\"+n`@C GF^]pF_]p7%Fegs7$$!+%*emgeFf_p$\"+L:n=@F^]pF_]p7%F[hs7$$!+`E9keFf_p$\" +%3$p79F^]pF_]p7%Fahs7$$!+x!Gi'eFf_p$\"+Hn?kqFcepF_]p7%Fghs7$$!+5J#p'e Ff_pF(F_]p7%F]is7$Fhhs$!+Hn?kqFcepF_]p7%Fais7$Fbhs$!+%3$p79F^]pF_]p7%F eis7$F\\hs$!+L:n=@F^]pF_]p7%Fiis7$Ffgs$!+n`@CGF^]pF_]p7%F]js7$F`gs$!+1 8>HNF^]pF_]p7%Fajs7$Fjfs$!+`bZLUF^]pF_]p7%Fejs7$Fdfs$!+5c&p$\\F^]pF_]p 7%Fijs7$F^fs$!+w<`RcF^]pF_]p7%F][t7$Fhes$!+e#=6M'F^]pF_]p7%Fa[t7$Fbes$ !+!*QkTqF^]pF_]p7%Fe[t7$F\\es$!+=C0TxF^]pF_]p7%Fi[t7$Ffds$!+MBIR%)F^]p F_]p7%F]\\t7$F`ds$!+ZiOO\"*F^]pF_]p7%Fa\\t7$Fjcs$!+^(HA$)*F^]pF_]p7%Fe \\t7$Fdcs$!+y*)o_5Ff_pF_]p7%Fi\\t7$F^cs$!+V_.A6Ff_pF_]p7%F]]t7$Fhbs$!+ =IE\">\"Ff_pF_]p7%Fa]t7$Fbbs$!+^QPg7Ff_pF_]p7%Fe]t7$F\\bs$!+'>p$H8Ff_p F_]p7%Fi]t7$Ffas$!+^*\\#)R\"Ff_pF_]p7%F]^t7$F`as$!+Sf,n9Ff_pF_]p7%Fa^t 7$Fj`s$!+[`mN:Ff_pF_]p7%Fe^t7$Fd`s$!+9T>/;Ff_pF_]p7%Fi^t7$F^`s$!+9`fs; Ff_pF_]p7%F]_t7$Fh_s$!+a&e3u\"Ff_pF_]p7%Fa_t7$Fb_s$!+3%p*3=Ff_pF_]p7%F e_t7$F\\_s$!+_*3p(=Ff_pF_]p7%Fi_t7$Ff^s$!+HMlW>Ff_pF_]p7%F]`t7$F`^s$!+ $4u@,#Ff_pF_]p7%Fa`t7$Fj]s$!+xsVz?Ff_pF_]p7%Fe`t7$Fd]s$!+/ZSY@Ff_pF_]p 7%Fi`t7$F^]s$!+_S.8AFf_pF_]p7%F]at7$Fh\\s$!+l(z#zAFf_pF_]p7%Faat7$Fb\\ s$!+!3%4XBFf_pF_]p7%Feat7$F\\\\s$!+6#G/T#Ff_pF_]p7%Fiat7$Ff[s$!+DOBvCF f_pF_]p7%F]bt7$F`[s$!+`LYRDFf_pF_]p7%Fabt7$Fjjr$!+TK2.EFf_pF_]p7%Febt7 $Fdjr$!+5J-mEFf_pF_]p7%Fibt7$F^jr$!+gxFGFFf_pF_]p7%F]ct7$Fhir$!+*z2)*y #Ff_pF_]p7%Fact7$Fbir$!+(G!f]GFf_pF_]p7%Fect7$F\\ir$!+*>41\"HFf_pF_]p7 %Fict7$Ffhr$!+@d&)pHFf_pF_]p7%F]dt7$F`hr$!+)QG$GIFf_pF_]p7%Fadt7$Fjgr$ !+&3Lg3$Ff_pF_]p7%Fedt7$Fdgr$!+*>$Ff_p F_]p7%F]et7$Fhfr$!+=ZqaKFf_pF_]p7%Faet7$Fbfr$!+Hd`4LFf_pF_]p7%Feet7$F \\fr$!+c%GPO$Ff_pF_]p7%Fiet7$Ffer$!+2UK>e%Ff_pF_]p7%F]]u7$Fhjq$!+rVy+YFf_pF_]p7%Fa]u7$Fbjq$!+B[s< YFf_pF_]p7%Fe]u7$F\\jq$!+.4!Gj%Ff_pF_]p7%Fi]u7$Ffiq$!+T\"*4YYFf_pF_]p7 %F]^u7$F`iq$!+GGudYFf_pF_]p7%Fa^u7$Fjhq$!+\\Ff_pF_]p7%Faau7$Fbdq$!+fCQt\\Ff_pF_]p7%Feau7$F\\dq$! +nJ)>-&Ff_pF_]p7%Fiau7$Ffcq$!+)o[Z1&Ff_pF_]p7%F]bu7$F`cq$!+GAm,^Ff_pF_ ]p7%Fabu7$Fjbq$!+(HqI8&Ff_pF_]p7%Febu7$Fdbq$!+2qRf^Ff_pF_]p7%Fibu7$F^b q$!+94/\"=&Ff_pF_]p7%F]cu7$Fhaq$!+Z\\M)>&Ff_pF_]p7%Facu7$Fbaq$!+1Jf6_F f_pF_]p7%Fecu7$F\\aq$!+2f,@_Ff_pF_]p7%Ficu7$Ff`q$!+&z(zE_Ff_pF_]p7%F]d u7$F``q$!+,U3H_Ff_pF_]p7%Fadu7$Fj_q$!+Kv)zA&Ff_pF_]p7%Fedu7$Fd_q$!+;?f B_Ff_pF_]p7%Fidu7$F^_q$!+`v&f@&Ff_pF_]p7%F]eu7$Fh^q$!+4F70_Ff_pF_]p7%F aeu7$Fb^q$!+iq5\">&Ff_pF_]p7%Feeu7$F\\^q$!+-I\"R<&Ff_pF_]p7%Fieu7$Ff]q $!+Lr_`^Ff_pF_]p7%F]fu7$F`]q$!+U8#*H^Ff_pF_]p7%Fafu7$Fj\\q$!+*e`I5&Ff_ pF_]p7%Fefu7$Fd\\q$!+S%oG2&Ff_pF_]p7%Fifu7$F^\\q$!+'H(HR]Ff_pF_]p7%F]g u7$Fh[q$!+'=eA+&Ff_pF_]p7%Fagu7$Fb[q$!+w[lh\\Ff_pF_]p7%Fegu7$F\\[q$!+B [P<\\Ff_pF_]p7%Figu7$Ffjp$!+z^Gp[Ff_pF_]p7%F]hu7$F`jp$!+hbA<[Ff_pF_]p7 %Fahu7$Fjip$!+fa*4w%Ff_pF_]p7%Fehu7$Fdip$!+0HL+ZFf_pF_]p7%Fihu7$F^ip$! +e*y[j%Ff_pF_]p7%F]iu7$Fhhp$!+>$>Tc%Ff_pF_]p7%Faiu7$Fbhp$!+5**G([%Ff_p F_]p7%Feiu7$F\\hp$!+$pHKS%Ff_pF_]p7%Fiiu7$Ffgp$!+dc<5VFf_pF_]p7%F]ju7$ F`gp$!+(3!e0UFf_pF_]p7%Faju7$Fjfp$!+#Ff_pF_]p7%Fe]v7$Fiap $!+5)o@/#Ff_pF_]p7%F[^v7$Fcap$!+`O,&)=Ff_pF_]p7%F_^v7$F\\ap$!+%G**ys\" Ff_pF_]p7%Fc^v7$$!+fbn?()F[`p$!+v[!3d\"Ff_pF_]p7%Fg^v7$F``p$!+.&>PT\"F f_pF_]p7%F]_v7$Fi_p$!+qyjc7Ff_pF_]p7%Fa_v7$Fb_p$!+Ewb*4\"Ff_pF_]p7%Fe_ v7$F[_p$!+_$yZU*F^]pF_]p7%Fi_v7$$!+s)))\\J#FN$!+$p\")R&yF^]pF_]p7%F]`v 7$$!+Jc]uMF-$!+8`=$G'F^]pF_]p7%Fc`v7$$!+1&498$Ffr$!+\")*)Q7ZF^]pF_]p7% Fi`v7$F($!+aEfTJF^]pF_]p7%F_av7$F($!+Fjzq:F^]pF_]p7%FcavF'F_]p-Fa\\p6& Fc\\p$\"#kFf\\pF(Fiav-%&STYLEG6#%,PATCHNOGRIDG-F$6%7$7$$!3A++++++!\\'F NF(7$$\"3%**************=\"FNF(-%'COLOURG6&Fc\\pF)F)F)-%*LINESTYLEG6# \"\"$-F$6%7$7$F($!3o*************Q'FN7$F($\"3o*************Q'FNFhbvF[c v-%+AXESLABELSG6%%&Re(z)G%&Im(z)G-%%FONTG6#%(DEFAULTG-F^dv6$%*HELVETIC AG\"\"*-%*AXESSTYLEG6#%$BOXG-%(SCALINGG6#%,CONSTRAINEDG-%%VIEWG6$;$!$ \\'Ff\\p$\"$>\"Ff\\p;$!$R'Ff\\p$\"$R'Ff\\p" 1 2 0 1 10 0 2 9 1 2 1 1.000000 45.000000 45.000000 0 0 "Curve 1" "Curve 2" "Curve 3" "Curve \+ 4" }}}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 75 "Th e stability interval for the embedded order 7 scheme is (approximately ) " }{XPPEDIT 18 0 "[-5.8669, 0];" "6#7$,$-%&FloatG6$\"&p'e!\"%!\"\" \"\"!" }{TEXT -1 2 ". " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 79 "The stability regions for the two schemes can be dis played in the same picture." }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 154 "plots[display]([p_1,p_2,p||(1..3)],view=[-6.89..1.19,-6.39..6.39] ,font=[HELVETICA,9],\n labels=[`Re(z)`,`Im(z)`],axes=boxed,s caling=constrained);" }}{PARA 13 "" 1 "" {GLPLOT2D 362 580 580 {PLOTDATA 2 "6--%'CURVESG6$7]_l7$$\"\"!F)F(7$F($\"3++++Fjzq:!#=7$F($\" 3)******Rl#fTJF-7$$!3:+++u%498$!#F$\"3D+++\")*)Q7ZF-7$$!3!******>j0XZ$ !#E$\"3Q+++8`=$G'F-7$$!3$******H()))\\J#!#D$\"3]*****Hp\")R&yF-7$$!3)* *****z\"=V46!#C$\"3Z*****>NyZU*F-7$$!31+++U(*=,UFF$\"3%******fid&*4\"! #<7$$!3,+++I@vF8!#B$\"3/+++qyjc7FN7$$!39+++t`4BOFR$\"3/+++.&>PT\"FN7$$ !3))*****fcv1s)FR$\"35+++v[!3d\"FN7$$!3))*****pOZY(=!#A$\"3/+++%G**ys \"FN7$$!3&******RjNWh$F\\o$\"3!******Hl8])=FN7$$!3c+++Jmc7iF\\o$\"3<++ +5)o@/#FN7$$!37+++rkU!G*F\\o$\"3'******H%=S*>#FN7$$!3.+++n2z66!#@$\"3; ++++[xcBFN7$$!3&******HP?j@(F\\o$\"3\"******pg&Q9DFN7$$\"3++++:3rN7Fap $\"3++++c,QsEFN7$$\"3&)*****>`!#?$\"39+++kiR!*HFN7$$\"3i*****pqJxe%Ffq$\"3))*****pqw4:$FN7$$\"3h++ +)p!*=r*Ffq$\"3()*****fpHHJ$FN7$$\"3*)*****pU([5>!#>$\"3#******R4DhZ$F N7$$\"3E+++ct(R`$Ffr$\"3@+++_cSROFN7$$\"3m*****zTIV6'Ffr$\"3)******zzi %*z$FN7$$\"30+++XAu7(*Ffr$\"3-+++*\\,0FF-$\"3M+++5**G ([%FN7$$\"3!******\\lVI&GF-$\"3c******=$>Tc%FN7$$\"3++++7:8MHF-$\"3$)* ****z&*y[j%FN7$$\"3?+++gs,bHF-$\"3y*****\\!HL+ZFN7$$\"3B+++-j\"y'Ffr$\"3))*****H8FN:&FN7$$\"3s*****>@(eGLFfr$\"3#*** ***>+8R<&FN7$$!34+++goP4EFfq$\"3Q+++iq5\">&FN7$$!3#******phj](RFfr$\"3 H+++4F70_FN7$$!3(******\\8LM!yFfr$\"3(******Hbdf@&FN7$$!3+++++yvt6F-$ \"3c*****f,#fB_FN7$$!31+++`A4x:F-$\"3@+++Kv)zA&FN7$$!3/+++I$)))*)>F-$ \"3/+++,U3H_FN7$$!3-+++K&4>T#F-$\"3T+++&z(zE_FN7$$!3-++++8=VGF-$\"3&** ****p!f,@_FN7$$!3-+++Mk0%G$F-$\"3f*****f5$f6_FN7$$!32+++:'=`t$F-$\"3#) *****p%\\M)>&FN7$$!37+++B_M)>%F-$\"3W+++94/\"=&FN7$$!3y*****z'\\OvYF-$ \"3#******p+(Rf^FN7$$!3_+++d,()p^F-$\"3t*****pHqI8&FN7$$!3Y+++hEG(o&F- $\"3N+++GAm,^FN7$$!3r*****\\ODgB'F-$\"3))*****zo[Z1&FN7$$!3K+++U'p\"Ho F-$\"39+++nJ)>-&FN7$$!3S+++geb'[(F-$\"3w******eCQt\\FN7$$!3_*****R@adB )F-$\"3')*****f+'**>\\FN7$$!32+++&Gec5*F-$\"3I+++yv-l[FN7$$!3,+++u@P55 FN$\"3C+++,NG9[FN7$$!3*******fGB#>6FN$\"3')*****HI*[tZFN7$$!3++++@()pI 7FN$\"3********)y(HWZFN7$$!31+++:W*)R8FN$\"3z******G+_CZFN7$$!3,+++@Oh W9FN$\"3y******\\f&4r%FN7$$!3%******>6tVa\"FN$\"3*)*****4ej4q%FN7$$!35 +++6\\UR;FN$\"3q*****zV(p#p%FN7$$!33+++HeDIFN$\"3'******p\"z!zm%FN7$$! 3'******R4tB)>FN$\"3%******z#GudYFN7$$!3))******G\"451#FN$\"3m*****49* 4YYFN7$$!33+++?q\\P@FN$\"3p*****H!4!Gj%FN7$$!3++++]()47AFN$\"3l*****H# [s>e%FN7$$!3!*******Q&=nU#FN$\"3U+++_*)3hXFN7$$!3'*******o')y&\\ #FN$\"3s******HvEQXFN7$$!3=+++$*z(Qc#FN$\"3*)*****RqTM^%FN7$$!3)****** Rw@6j#FN$\"3W+++***3m[%FN7$$!3%)*****>iZwp#FN$\"3m*****fIxxX%FN7$$!3') *****RSxNw#FN$\"32+++(4lpU%FN7$$!3'******H@G!HGFN$\"3Q+++)=-UR%FN7$$!3 ')******>F6%*GFN$\"3!)*****\\*3`fVFN7$$!35+++:'Q*eHFN$\"3z*****zP2IK%F N7$$!3;+++XrgBIFN$\"3i*****z%Hq%G%FN7$$!3#******Ht5#)3$FN$\"3#******H, 0ZC%FN7$$!3')*****HuHG:$FN$\"3i*****4LATwZ$FN$ \"3!)*******z\")e(RFN7$$!38+++'y+Ha$FN$\"3))*****\\Ljv#RFN7$$!39+++)3B \"3OFN$\"3%)******oLfyQFN7$$!3=+++2l?tOFN$\"33+++$zq!HQFN7$$!3)****** \\TP!QPFN$\"3/+++e22zPFN7$$!34+++2p\\-QFN$\"3#)*****>jV'GPFN7$$!3?+++I rYmQFN$\"3/+++uf\"yn$FN7$$!3*)*****>hO)HRFN$\"32+++1JfEOFN7$$!37+++XU] #*RFN$\"3y*****\\si\\d$FN7$$!3=+++e=QaSFN$\"38+++s*)*G_$FN7$$!3l*****f \\&R:TFN$\"3#********\\m.Z$FN7$$!3Q+++ec[vTFN$\"31+++2UK*>$FN7$$!3'******He8 4Y%FN$\"3))*****p\"H)H9$FN7$$!3U+++E0\"\\^%FN$\"3'******\\3Lg3$FN7$$!3 ))*****>/))yc%FN$\"3#)*****zQG$GIFN7$$!3I+++\\=')>YFN$\"31+++@d&)pHFN7 $$!3w*****4.^3n%FN$\"3/+++*>41\"HFN7$$!3*)******=o(3s%FN$\"3;+++(G!f]G FN7$$!3;+++&Rh*pZFN$\"33+++*z2)*y#FN7$$!3D+++=n7=[FN$\"36+++gxFGFFN7$$ !3E+++yLRl[FN$\"3!)******4J-mEFN7$$!3M+++v%z<\"\\FN$\"3:+++TK2.EFN7$$! 3y******)e*Hd\\FN$\"3#******HNj%RDFN7$$!3\"******p$Q'>+&FN$\"3++++DOBv CFN7$$!3I+++)4xd/&FN$\"3:+++6#G/T#FN7$$!3q*****RVQ()3&FN$\"3'*******zS 4XBFN7$$!3\")******e2%38&FN$\"33+++l(z#zAFN7$$!3s*****Hxq?<&FN$\"3\")* ****>0MI@#FN7$$!3s******)=4C@&FN$\"31+++/ZSY@FN7$$!3E+++;7$=D&FN$\"3!) *****pFP%z?FN7$$!33+++(G2.H&FN$\"30+++$4u@,#FN7$$!3G+++mR!yK&FN$\"3$** *****GMlW>FN7$$!3=+++t\\Gk`FN$\"3'******>&*3p(=FN7$$!3#******fD7(*R&FN $\"31+++3%p*3=FN7$$!38+++\\q/MaFN$\"3*)*****Rbe3u\"FN7$$!3c*****R&3Dna FN$\"3/+++9`fs;FN7$$!3/+++0jG*\\&FN$\"3++++9T>/;FN7$$!3B+++(*y6IbFN$\" 3*******zMlc`\"FN7$$!3y*****Hb7(fbFN$\"3-+++Sf,n9FN7$$!3%)*****p.S!)e& FN$\"3#******4&*\\#)R\"FN7$$!3c*****HAt]h&FN$\"3.+++'>p$H8FN7$$!3S+++M #)yScFN$\"3++++^QPg7FN7$$!3%)*****>Zk^m&FN$\"3))*****z,j7>\"FN7$$!3s** ***za%=)o&FN$\"3-+++V_.A6FN7$$!3v******[T$)4dFN$\"3%******z(*)o_5FN7$$ !3h******\\=5IdFN$\"32+++^(HA$)*F-7$$!3O+++s)y*[dFN$\"3b*****pCmj8*F-7 $$!37+++F)ekw&FN$\"3_*****RL-$R%)F-7$$!3)******RVPDy&FN$\"3a*****zT_5u (F-7$$!3#)*****HA7sz&FN$\"3-+++!*QkTqF-7$$!3s******QA[5eFN$\"3G+++e#=6 M'F-7$$!3\"******HuZB#eFN$\"3m*****fxJ&RcF-7$$!3i*****p\"*4G$eFN$\"30+ ++5c&p$\\F-7$$!3n*****\\eq=%eFN$\"36+++`bZLUF-7$$!3K+++S>`\\eFN$\"3?++ +18>HNF-7$$!3!******>I'zbeFN$\"3v*****pO:U#GF-7$$!3E+++%*emgeFN$\"3+++ +L:n=@F-7$$!3/+++`E9keFN$\"3))*****R3$p79F-7$$!3q*****p2Gi'eFN$\"3!*** ****Gn?kqFfr7$$!35+++5J#p'eFNF(7$Fb]n$!3!*******Gn?kqFfr7$F]]n$!3))*** **R3$p79F-7$Fh\\n$!3++++L:n=@F-7$Fc\\n$!3v*****pO:U#GF-7$F^\\n$!3?+++1 8>HNF-7$Fi[n$!36+++`bZLUF-7$Fd[n$!30+++5c&p$\\F-7$F_[n$!3m*****fxJ&RcF -7$Fjjm$!3G+++e#=6M'F-7$Fejm$!3-+++!*QkTqF-7$F`jm$!3a*****zT_5u(F-7$F[ jm$!3_*****RL-$R%)F-7$Ffim$!3b*****pCmj8*F-7$Faim$!32+++^(HA$)*F-7$F\\ im$!3%******z(*)o_5FN7$Fghm$!3-+++V_.A6FN7$Fbhm$!3))*****z,j7>\"FN7$F] hm$!3++++^QPg7FN7$Fhgm$!3.+++'>p$H8FN7$Fcgm$!3#******4&*\\#)R\"FN7$F^g m$!3-+++Sf,n9FN7$Fifm$!3*******zMlc`\"FN7$Fdfm$!3++++9T>/;FN7$F_fm$!3/ +++9`fs;FN7$Fjem$!3*)*****Rbe3u\"FN7$Feem$!31+++3%p*3=FN7$F`em$!3'**** **>&*3p(=FN7$F[em$!3$*******GMlW>FN7$Ffdm$!30+++$4u@,#FN7$Fadm$!3!)*** **pFP%z?FN7$F\\dm$!31+++/ZSY@FN7$Fgcm$!3\")*****>0MI@#FN7$Fbcm$!33+++l (z#zAFN7$F]cm$!3'*******zS4XBFN7$Fhbm$!3:+++6#G/T#FN7$Fcbm$!3++++DOBvC FN7$F^bm$!3#******HNj%RDFN7$Fiam$!3:+++TK2.EFN7$Fdam$!3!)******4J-mEFN 7$F_am$!36+++gxFGFFN7$Fj`m$!33+++*z2)*y#FN7$Fe`m$!3;+++(G!f]GFN7$F``m$ !3/+++*>41\"HFN7$F[`m$!31+++@d&)pHFN7$Ff_m$!3#)*****zQG$GIFN7$Fa_m$!3' ******\\3Lg3$FN7$F\\_m$!3))*****p\"H)H9$FN7$Fg^m$!33+++Pz>*>$FN7$Fb^m$ !3;+++=ZqaKFN7$F]^m$!3#)******Gd`4LFN7$Fh]m$!3/+++c%GPO$FN7$Fc]m$!31++ +2UKjV'GPFN7$F`[m$!3/+++e22zPFN7$F[[m$!33+++$zq!HQFN7$Ffjl$!3%)******oL fyQFN7$Fajl$!3))*****\\Ljv#RFN7$F\\jl$!3!)*******z\")e(RFN7$Fgil$!3p** ***zLHM-%FN7$Fbil$!3q*****p2r+2%FN7$F]il$!36+++\"3jc6%FN7$Fhhl$!3M+++r \"f+;%FN7$Fchl$!3i*****4L>e%FN7$Fbdl$!3g*****4P%y+YFN7$F]dl$!3l*****H#[s< YFN7$Fhcl$!3p*****H!4!Gj%FN7$Fccl$!3m*****49*4YYFN7$F^cl$!3%******z#Gu dYFN7$Fibl$!3'******p\"z!zm%FN7$Fdbl$!37+++eu%on%FN7$F_bl$!3;+++*fH\\o %FN7$Fjal$!3q*****zV(p#p%FN7$Feal$!3*)*****4ej4q%FN7$F`al$!3y******\\f &4r%FN7$F[al$!3z******G+_CZFN7$Ff`l$!3********)y(HWZFN7$Fa`l$!3')***** HI*[tZFN7$F\\`l$!3C+++,NG9[FN7$Fg_l$!3I+++yv-l[FN7$Fb_l$!3')*****f+'** >\\FN7$F]_l$!3w******eCQt\\FN7$Fh^l$!39+++nJ)>-&FN7$Fc^l$!3))*****zo[Z 1&FN7$F^^l$!3N+++GAm,^FN7$Fi]l$!3t*****pHqI8&FN7$Fd]l$!3#******p+(Rf^F N7$F_]l$!3W+++94/\"=&FN7$Fj\\l$!3#)*****p%\\M)>&FN7$Fe\\l$!3f*****f5$f 6_FN7$F`\\l$!3&******p!f,@_FN7$F[\\l$!3T+++&z(zE_FN7$Ff[l$!3/+++,U3H_F N7$Fa[l$!3@+++Kv)zA&FN7$F\\[l$!3c*****f,#fB_FN7$Fgz$!3(******Hbdf@&FN7 $Fbz$!3H+++4F70_FN7$F]z$!3Q+++iq5\">&FN7$Fhy$!3#******>+8R<&FN7$Fcy$!3 ))*****H8FN:&FN7$F^y$!3F+++U8#*H^FN7$Fix$!3&*******)e`I5&FN7$Fdx$!3q** ****R%oG2&FN7$F_x$!3)******fH(HR]FN7$Fjw$!3s*****f=eA+&FN7$Few$!3))*** **f([lh\\FN7$F`w$!3G+++B[P<\\FN7$F[w$!3C+++z^Gp[FN7$Ffv$!3))*****4cDs \"[FN7$Fav$!35+++fa*4w%FN7$F\\v$!3y*****\\!HL+ZFN7$Fgu$!3$)*****z&*y[j %FN7$Fbu$!3c******=$>Tc%FN7$F]u$!3M+++5**G([%FN7$Fht$!36+++$pHKS%FN7$F ct$!3F+++dc<5VFN7$F^t$!3T+++(3!e0UFN7$Fis$!3C+++#FN7$Feo$!3<+++5)o @/#FN7$F`o$!3!******Hl8])=FN7$Fjn$!3/+++%G**ys\"FN7$$!3K******ebn?()FR $!35+++v[!3d\"FN7$FV$!3/+++.&>PT\"FN7$FP$!3/+++qyjc7FN7$FJ$!3%******fi d&*4\"FN7$FD$!3Z*****>NyZU*F-7$$!36+++s)))\\J#F@$!3]*****Hp\")R&yF-7$$ !3;+++Jc]uMF:$!3Q+++8`=$G'F-7$$!3;+++1&498$F4$!3D+++\")*)Q7ZF-7$F($!3) ******Rl#fTJF-7$F($!3++++Fjzq:F-F'-%&COLORG6&%$RGBG$\"#K!\"#F(Fd\\p-%) POLYGONSG6^_l7%F'7$F($\"+Fjzq:!#57$$!#U!\"\"F(7%F[]p7$F($\"+aEfTJF^]pF _]p7%Fd]p7$$!+u%498$Ffr$\"+\")*)Q7ZF^]pF_]p7%Fh]p7$$!+Kc]uMF-$\"+8`=$G 'F^]pF_]p7%F^^p7$$!+t)))\\J#FN$\"+$p\")R&yF^]pF_]p7%Fd^p7$$!+==V46!#;$ \"+_$yZU*F^]pF_]p7%Fj^p7$$!+U(*=,UF]_p$\"+Ewb*4\"!\"*F_]p7%Fa_p7$$!+I@ vF8!#:$\"+qyjc7Ff_pF_]p7%Fh_p7$$!+t`4BOF[`p$\"+.&>PT\"Ff_pF_]p7%F_`p7$ $!+mbn?()F[`p$\"+v[!3d\"Ff_pF_]p7%Fe`p7$$!+ntku=!#9$\"+%G**ys\"Ff_pF_] p7%F[ap7$$!+McV9OF^ap$\"+`O,&)=Ff_pF_]p7%Fbap7$$!+Jmc7iF^ap$\"+5)o@/#F f_pF_]p7%Fhap7$$!+rkU!G*F^ap$\"+V=S*>#Ff_pF_]p7%F^bp7$$!+n2z66!#8$\"++ [xcBFf_pF_]p7%Fdbp7$$!+t.K;sF^ap$\"+2cQ9DFf_pF_]p7%F[cp7$$\"+:3rN7Fgbp $\"+c,QsEFf_pF_]p7%Facp7$$\"+Kv@MnFgbp$\"+&ei4$GFf_pF_]p7%Fgcp7$$\"+8O XW>!#7$\"+kiR!*HFf_pF_]p7%F]dp7$$\"+2!#6$\"+%4DhZ$Ff_pF _]p7%F`ep7$$\"+ct(R`$Fcep$\"+_cSROFf_pF_]p7%Fgep7$$\"+=/L9hFcep$\"+)zi %*z$Ff_pF_]p7%F]fp7$$\"+XAu7(*Fcep$\"+$ >Tc%Ff_pF_]p7%Fghp7$$\"+7:8MHF^]p$\"+e*y[j%Ff_pF_]p7%F]ip7$$\"+gs,bHF^ ]p$\"+0HL+ZFf_pF_]p7%Fcip7$$\"+-j\"y'Fcep$\"+Lr_`^Ff_pF_]p7%Fe]q7$$\"+7seGLFcep$\"+-I\"R <&Ff_pF_]p7%F[^q7$$!+goP4EF`dp$\"+iq5\">&Ff_pF_]p7%Fa^q7$$!+F^]p$\"+,U3H_Ff_pF_]p7%F_`q7$$!+K&4>T#F^]p$\"+&z(zE_Ff_pF_]p7% Fe`q7$$!++8=VGF^]p$\"+2f,@_Ff_pF_]p7%F[aq7$$!+Mk0%G$F^]p$\"+1Jf6_Ff_pF _]p7%Faaq7$$!+:'=`t$F^]p$\"+Z\\M)>&Ff_pF_]p7%Fgaq7$$!+B_M)>%F^]p$\"+94 /\"=&Ff_pF_]p7%F]bq7$$!+o\\OvYF^]p$\"+2qRf^Ff_pF_]p7%Fcbq7$$!+d,()p^F^ ]p$\"+(HqI8&Ff_pF_]p7%Fibq7$$!+hEG(o&F^]p$\"+GAm,^Ff_pF_]p7%F_cq7$$!+l `-OiF^]p$\"+)o[Z1&Ff_pF_]p7%Fecq7$$!+U'p\"HoF^]p$\"+nJ)>-&Ff_pF_]p7%F[ dq7$$!+geb'[(F^]p$\"+fCQt\\Ff_pF_]p7%Fadq7$$!+9UvN#)F^]p$\"+1g**>\\Ff_ pF_]p7%Fgdq7$$!+&Gec5*F^]p$\"+yv-l[Ff_pF_]p7%F]eq7$$!+u@P55Ff_p$\"+,NG 9[Ff_pF_]p7%Fceq7$$!+'GB#>6Ff_p$\"+.$*[tZFf_pF_]p7%Fieq7$$!+@()pI7Ff_p $\"+*y(HWZFf_pF_]p7%F_fq7$$!+:W*)R8Ff_p$\"+H+_CZFf_pF_]p7%Fefq7$$!+@Oh W9Ff_p$\"+]f&4r%Ff_pF_]p7%F[gq7$$!+7JPW:Ff_p$\"+\"ej4q%Ff_pF_]p7%Fagq7 $$!+6\\UR;Ff_p$\"+Qup#p%Ff_pF_]p7%Fggq7$$!+HeDIFf_p$\"+Ff_p$\"+GGudYFf_pF_]p7%F_iq7$$!+H\"451#Ff_p$\" +T\"*4YYFf_pF_]p7%Feiq7$$!+?q\\P@Ff_p$\"+.4!Gj%Ff_pF_]p7%F[jq7$$!+]()4 7AFf_p$\"+B[s>e%Ff_pF_]p7%F][r7$$!+R&=nU#Ff_p$\"+_*)3hXFf_pF_]p7% Fc[r7$$!+p')y&\\#Ff_p$\"+IvEQXFf_pF_]p7%Fi[r7$$!+$*z(Qc#Ff_p$\"+/*>$Ff _pF_]p7%F]gr7$$!+$e84Y%Ff_p$\"++&Ff_p$\"+DOBvCFf_pF_]p7%Fe[s7$$!+)4 xd/&Ff_p$\"+6#G/T#Ff_pF_]p7%F[\\s7$$!+M%Q()3&Ff_p$\"+!3%4XBFf_pF_]p7%F a\\s7$$!+f2%38&Ff_p$\"+l(z#zAFf_pF_]p7%Fg\\s7$$!+t22s^Ff_p$\"+_S.8AFf_ pF_]p7%F]]s7$$!+*=4C@&Ff_p$\"+/ZSY@Ff_pF_]p7%Fc]s7$$!+;7$=D&Ff_p$\"+xs Vz?Ff_pF_]p7%Fi]s7$$!+(G2.H&Ff_p$\"+$4u@,#Ff_pF_]p7%F_^s7$$!+mR!yK&Ff_ p$\"+HMlW>Ff_pF_]p7%Fe^s7$$!+t\\Gk`Ff_p$\"+_*3p(=Ff_pF_]p7%F[_s7$$!+cA r*R&Ff_p$\"+3%p*3=Ff_pF_]p7%Fa_s7$$!+\\q/MaFf_p$\"+a&e3u\"Ff_pF_]p7%Fg _s7$$!+a3DnaFf_p$\"+9`fs;Ff_pF_]p7%F]`s7$$!+0jG*\\&Ff_p$\"+9T>/;Ff_pF_ ]p7%Fc`s7$$!+(*y6IbFf_p$\"+[`mN:Ff_pF_]p7%Fi`s7$$!+`DrfbFf_p$\"+Sf,n9F f_pF_]p7%F_as7$$!+P+/)e&Ff_p$\"+^*\\#)R\"Ff_pF_]p7%Feas7$$!+BK2:cFf_p$ \"+'>p$H8Ff_pF_]p7%F[bs7$$!+M#)yScFf_p$\"+^QPg7Ff_pF_]p7%Fabs7$$!+sW;l cFf_p$\"+=IE\">\"Ff_pF_]p7%Fgbs7$$!+[X=)o&Ff_p$\"+V_.A6Ff_pF_]p7%F]cs7 $$!+\\T$)4dFf_p$\"+y*)o_5Ff_pF_]p7%Fccs7$$!+]=5IdFf_p$\"+^(HA$)*F^]pF_ ]p7%Fics7$$!+s)y*[dFf_p$\"+ZiOO\"*F^]pF_]p7%F_ds7$$!+F)ekw&Ff_p$\"+MBI R%)F^]pF_]p7%Feds7$$!+Mu`#y&Ff_p$\"+=C0TxF^]pF_]p7%F[es7$$!+BA@(z&Ff_p $\"+!*QkTqF^]pF_]p7%Faes7$$!+RA[5eFf_p$\"+e#=6M'F^]pF_]p7%Fges7$$!+VxM AeFf_p$\"+w<`RcF^]pF_]p7%F]fs7$$!+<*4G$eFf_p$\"+5c&p$\\F^]pF_]p7%Fcfs7 $$!+&eq=%eFf_p$\"+`bZLUF^]pF_]p7%Fifs7$$!+S>`\\eFf_p$\"+18>HNF^]pF_]p7 %F_gs7$$!+-jzbeFf_p$\"+n`@CGF^]pF_]p7%Fegs7$$!+%*emgeFf_p$\"+L:n=@F^]p F_]p7%F[hs7$$!+`E9keFf_p$\"+%3$p79F^]pF_]p7%Fahs7$$!+x!Gi'eFf_p$\"+Hn? kqFcepF_]p7%Fghs7$$!+5J#p'eFf_pF(F_]p7%F]is7$Fhhs$!+Hn?kqFcepF_]p7%Fai s7$Fbhs$!+%3$p79F^]pF_]p7%Feis7$F\\hs$!+L:n=@F^]pF_]p7%Fiis7$Ffgs$!+n` @CGF^]pF_]p7%F]js7$F`gs$!+18>HNF^]pF_]p7%Fajs7$Fjfs$!+`bZLUF^]pF_]p7%F ejs7$Fdfs$!+5c&p$\\F^]pF_]p7%Fijs7$F^fs$!+w<`RcF^]pF_]p7%F][t7$Fhes$!+ e#=6M'F^]pF_]p7%Fa[t7$Fbes$!+!*QkTqF^]pF_]p7%Fe[t7$F\\es$!+=C0TxF^]pF_ ]p7%Fi[t7$Ffds$!+MBIR%)F^]pF_]p7%F]\\t7$F`ds$!+ZiOO\"*F^]pF_]p7%Fa\\t7 $Fjcs$!+^(HA$)*F^]pF_]p7%Fe\\t7$Fdcs$!+y*)o_5Ff_pF_]p7%Fi\\t7$F^cs$!+V _.A6Ff_pF_]p7%F]]t7$Fhbs$!+=IE\">\"Ff_pF_]p7%Fa]t7$Fbbs$!+^QPg7Ff_pF_] p7%Fe]t7$F\\bs$!+'>p$H8Ff_pF_]p7%Fi]t7$Ffas$!+^*\\#)R\"Ff_pF_]p7%F]^t7 $F`as$!+Sf,n9Ff_pF_]p7%Fa^t7$Fj`s$!+[`mN:Ff_pF_]p7%Fe^t7$Fd`s$!+9T>/;F f_pF_]p7%Fi^t7$F^`s$!+9`fs;Ff_pF_]p7%F]_t7$Fh_s$!+a&e3u\"Ff_pF_]p7%Fa_ t7$Fb_s$!+3%p*3=Ff_pF_]p7%Fe_t7$F\\_s$!+_*3p(=Ff_pF_]p7%Fi_t7$Ff^s$!+H MlW>Ff_pF_]p7%F]`t7$F`^s$!+$4u@,#Ff_pF_]p7%Fa`t7$Fj]s$!+xsVz?Ff_pF_]p7 %Fe`t7$Fd]s$!+/ZSY@Ff_pF_]p7%Fi`t7$F^]s$!+_S.8AFf_pF_]p7%F]at7$Fh\\s$! +l(z#zAFf_pF_]p7%Faat7$Fb\\s$!+!3%4XBFf_pF_]p7%Feat7$F\\\\s$!+6#G/T#Ff _pF_]p7%Fiat7$Ff[s$!+DOBvCFf_pF_]p7%F]bt7$F`[s$!+`LYRDFf_pF_]p7%Fabt7$ Fjjr$!+TK2.EFf_pF_]p7%Febt7$Fdjr$!+5J-mEFf_pF_]p7%Fibt7$F^jr$!+gxFGFFf _pF_]p7%F]ct7$Fhir$!+*z2)*y#Ff_pF_]p7%Fact7$Fbir$!+(G!f]GFf_pF_]p7%Fec t7$F\\ir$!+*>41\"HFf_pF_]p7%Fict7$Ffhr$!+@d&)pHFf_pF_]p7%F]dt7$F`hr$!+ )QG$GIFf_pF_]p7%Fadt7$Fjgr$!+&3Lg3$Ff_pF_]p7%Fedt7$Fdgr$!+*>$Ff_pF_]p7%F]et7$Fhfr$!+=ZqaKFf_pF_]p7%Faet7$Fbf r$!+Hd`4LFf_pF_]p7%Feet7$F\\fr$!+c%GPO$Ff_pF_]p7%Fiet7$Ffer$!+2UK>e%Ff_pF_]p7%F]]u7$Fhjq$!+rVy+YF f_pF_]p7%Fa]u7$Fbjq$!+B[s\\Ff_pF_]p7%Faau7$Fbdq$!+fC Qt\\Ff_pF_]p7%Feau7$F\\dq$!+nJ)>-&Ff_pF_]p7%Fiau7$Ffcq$!+)o[Z1&Ff_pF_] p7%F]bu7$F`cq$!+GAm,^Ff_pF_]p7%Fabu7$Fjbq$!+(HqI8&Ff_pF_]p7%Febu7$Fdbq $!+2qRf^Ff_pF_]p7%Fibu7$F^bq$!+94/\"=&Ff_pF_]p7%F]cu7$Fhaq$!+Z\\M)>&Ff _pF_]p7%Facu7$Fbaq$!+1Jf6_Ff_pF_]p7%Fecu7$F\\aq$!+2f,@_Ff_pF_]p7%Ficu7 $Ff`q$!+&z(zE_Ff_pF_]p7%F]du7$F``q$!+,U3H_Ff_pF_]p7%Fadu7$Fj_q$!+Kv)zA &Ff_pF_]p7%Fedu7$Fd_q$!+;?fB_Ff_pF_]p7%Fidu7$F^_q$!+`v&f@&Ff_pF_]p7%F] eu7$Fh^q$!+4F70_Ff_pF_]p7%Faeu7$Fb^q$!+iq5\">&Ff_pF_]p7%Feeu7$F\\^q$!+ -I\"R<&Ff_pF_]p7%Fieu7$Ff]q$!+Lr_`^Ff_pF_]p7%F]fu7$F`]q$!+U8#*H^Ff_pF_ ]p7%Fafu7$Fj\\q$!+*e`I5&Ff_pF_]p7%Fefu7$Fd\\q$!+S%oG2&Ff_pF_]p7%Fifu7$ F^\\q$!+'H(HR]Ff_pF_]p7%F]gu7$Fh[q$!+'=eA+&Ff_pF_]p7%Fagu7$Fb[q$!+w[lh \\Ff_pF_]p7%Fegu7$F\\[q$!+B[P<\\Ff_pF_]p7%Figu7$Ffjp$!+z^Gp[Ff_pF_]p7% F]hu7$F`jp$!+hbA<[Ff_pF_]p7%Fahu7$Fjip$!+fa*4w%Ff_pF_]p7%Fehu7$Fdip$!+ 0HL+ZFf_pF_]p7%Fihu7$F^ip$!+e*y[j%Ff_pF_]p7%F]iu7$Fhhp$!+>$>Tc%Ff_pF_] p7%Faiu7$Fbhp$!+5**G([%Ff_pF_]p7%Feiu7$F\\hp$!+$pHKS%Ff_pF_]p7%Fiiu7$F fgp$!+dc<5VFf_pF_]p7%F]ju7$F`gp$!+(3!e0UFf_pF_]p7%Faju7$Fjfp$!+#Ff_pF_]p7%Fe]v7$Fiap$!+5)o@/#Ff_pF_]p7%F[^v7$Fcap$!+`O,&)=Ff_p F_]p7%F_^v7$F\\ap$!+%G**ys\"Ff_pF_]p7%Fc^v7$$!+fbn?()F[`p$!+v[!3d\"Ff_ pF_]p7%Fg^v7$F``p$!+.&>PT\"Ff_pF_]p7%F]_v7$Fi_p$!+qyjc7Ff_pF_]p7%Fa_v7 $Fb_p$!+Ewb*4\"Ff_pF_]p7%Fe_v7$F[_p$!+_$yZU*F^]pF_]p7%Fi_v7$$!+s)))\\J #FN$!+$p\")R&yF^]pF_]p7%F]`v7$$!+Jc]uMF-$!+8`=$G'F^]pF_]p7%Fc`v7$$!+1& 498$Ffr$!+\")*)Q7ZF^]pF_]p7%Fi`v7$F($!+aEfTJF^]pF_]p7%F_av7$F($!+Fjzq: F^]pF_]p7%FcavF'F_]p-Fa\\p6&Fc\\p$\"#kFf\\pF(Fiav-%&STYLEG6#%,PATCHNOG RIDG-F$6$7]_lF'F*F.7$F(F57$$\"3(******>L(4CYF4$\"3_*****fJ&=$G'F-7$$\" 3!******4Qg%pSF:$\"3u*****fp\")R&yF-7$$\"3%******R6atM#F@$\"3G+++T#yZU *F-7$$\"3'*******Hy+15FF$\"3-+++Dvb*4\"FN7$$\"3#)*****fH@%\\MFF$\"3/++ +Dtjc7FN7$$\"3^+++%eu^\"**FF$\"32++++tr89FN7$$\"3-+++\"[kMY#FR$\"37+++ 9vzq:FN7$$\"3-+++\")*HwR&FR$\"3&******H)z(ys\"FN7$$\"3#******RR))f0\"F \\o$\"35+++\\'e\\)=FN7$$\"3-+++xo\\b=F\\o$\"3&)*****\\_R?/#FN7$$\"35++ +/(>]#HF\\o$\"3%)*****zE@\"*>#FN7$$\"3=+++\"y@A4%F\\o$\"3?+++\\m?cBFN7 $$\"3&******zH1]$\\F\\o$\"33+++7OI8DFN7$$\"3A+++*Q*yYZF\\o$\"3%)*****> (3VqEFN7$$\"37+++EOFN7$$\"3)*******H%**[h#Ffq$\"3A+++g<$=y$FN7$$\"3$)*** **p&>42pFfq$\"3#******R/wn%RFN7$$\"3++++%y+Uh\"Ffr$\"3<+++q(*Q:TFN7$$ \"3o******Rf!H\\$Ffr$\"3#******>_[S\\ ZFN7$$\"3++++8'>P[#F-$\"35+++5t!y&[FN7$$\"39+++:!>Y(HF-$\"3G+++8DY\\\\ FN7$$\"3?+++#*G=dLF-$\"3#******z.E+.&FN7$$\"3!******>4tDk$F-$\"37+++&) *pF5&FN7$$\"3y*****\\`fP%QF-$\"3i*****HKF&p^FN7$$\"3%******4kfA(RF-$\" 3T+++hZNJ_FN7$$\"3$)*****44Nx.%F-$\"3G+++*z()))G&FN7$$\"3;+++x@B[SF-$ \"3I+++oi_U`FN7$$\"3E+++]#)\\5SF-$\"3?+++%zLDR&FN7$$\"3')******>%[-$RF -$\"3S+++ZG4RaFN7$$\"3E+++g$pB\"QF-$\"3C+++\"*fL#[&FN7$$\"3;+++R>2hOF- $\"3w*****H(QOAbFN7$$\"3y*****\\`1+[$F-$\"3')*****ffb#fbFN7$$\"3)***** *>\">NsKF-$\"3k*****4guIf&FN7$$\"36+++K`)3/$F-$\"3#******\\'=(Qi&FN7$$ \"3=+++.!R!)y#F-$\"3/+++uuo^cFN7$$\"3F+++6i%f^#F-$\"3O+++)=^ln&FN7$$\" 3-+++k#zkA#F-$\"37+++uB[)p&FN7$$\"3#******Rl!G@>F-$\"3f*****H[*[B)y,;F-$\"3V+++'=pNt&FN7$$\"3))*****z%[Dp7F-$\"3%******p<0 nu&FN7$$\"3I+++v\\iZ#*Ffr$\"3%)*****Rgmov&FN7$$\"3')*****pk1Bp&Ffr$\"3 u******eg+kdFN7$$\"33+++B-=M?Ffr$\"3O+++vp0odFN7$$!3-+++W2%3s\"Ffr$\"3 ?+++[-$*odFN7$$!3I+++t**oobFfr$\"3')*****>!)4lw&FN7$$!3V*****zDxu]*Ffr $\"3K+++(pY1w&FN7$$!3))*****H%\\!QN\"F-$\"3=+++E7:^dFN7$$!3++++T<\\m@Y.`FN7$$!3o*****p,VQE*F-$\"3++++;Rlj_FN7$$!35++ +%*op]5FN$\"3!)*****42*yU_FN7$$!3/+++#G*Gn6FN$\"35+++?BRL_FN7$$!3-+++; apv7FN$\"3f*****\\(GwH_FN7$$!3/+++cS'pP\"FN$\"3I+++f/fG_FN7$$!33+++AvH s9FN$\"3M+++P@,G_FN7$$!3/+++x9ti:FN$\"3v*****p]]pA&FN7$$!3!*******)p+ \"\\;FN$\"3U+++M#fZA&FN7$$!3++++,\"p?t\"FN$\"3u******pu.@_FN7$$!3***** **H\"p;7=FN$\"3N+++,(Hb@&FN7$$!3/+++QA#)*)=FN$\"3m*****fWp!3_FN7$$!3#* *****4`&Ql>FN$\"3K+++lja)>&FN7$$!3!******4<[\"R?FN$\"39+++&R')o=&FN7$$ !3%)*****>`c86#FN$\"3!*******z%RI<&FN7$$!3')*****>wAA=#FN$\"3++++=>(p: &FN7$$!3!******f8L>D#FN$\"3&)*****4hh'Q^FN7$$!3#*******=Yl?BFN$\"36+++ x_4=^FN7$$!3!*******p+a)Q#FN$\"39+++EqE&4&FN7$$!3/+++dBtbCFN$\"3)***** *R'zm$G/&FN7$$!33+++^td)e#FN$\"3F+++f+E8 ]FN7$$!3=+++U/\\aEFN$\"3K+++xbZ\")\\FN7$$!3!)*****>>L-s#FN$\"3))***** \\&G_Z\\FN7$$!3?+++j!Hfy#FN$\"3++++%Rc9\"\\FN7$$!3%)******))op^GFN$\"3 e*****R)zMt[FN7$$!3y*****>\"yk@LFN$\"3m******ziYfXFN7$$!3#******f7N)* Q$FN$\"3G+++d\"R(4XFN7$$!3'******\\n&peMFN$\"3g*****>1\\#fWFN7$$!3>+++ 4QkFNFN$\"30+++'e-\"3WFN7$$!3)******H*R`'f$FN$\"3y*****H*QPcVFN7$$!3!) *******f:_m$FN$\"3*)******3Q5/VFN7$$!3-+++'[TNt$FN$\"3y*****z%=I^UFN7$ $!31+++V[P,QFN$\"31+++\"Q]z>%FN7$$!3%)*****f%RfoQFN$\"3V+++>+,WTFN7$$! 3\")*****>%[4NRFN$\"3U+++M`U*3%FN7$$!3M+++(G#z+SFN$\"3N+++q-8MSFN7$$!3 %******zR>c1%FN$\"3'*******pG0yRFN7$$!3=+++]k_HTFN$\"3.+++X\">6#RFN7$$ !3)******p=zC>%FN$\"3/+++0hDjQFN7$$!3g*****p$pXaUFN$\"3\")*****>.%R/QF N7$$!3<+++(p]aJ%FN$\"3!)******[\"oWu$FN7$$!35+++'HhaP%FN$\"35+++4)>Mo$ FN7$$!3L+++ww\\MWFN$\"3()*****zN(>@OFN7$$!3y*****>GvD\\%FN$\"39+++Qnvd NFN7$$!3?+++_Xr\\XFN$\"3=+++J>1$\\$FN7$$!3,+++f%Rfg%FN$\"3#)*****pE&3F MFN7$$!3/+++wgFhYFN$\"3#)*****fi2)fLFN7$$!3z*****f7^dr%FN$\"3'******p] =7H$FN7$$!3E+++-0RpZFN$\"3%)*****p&fJ@KFN7$$!3r*****pw5.,<&FN$\"3')******3M(ph#FN7$$!3h*****>K bl@&FN$\"38+++nYDODFN7$$!3e*****p\"**4i_FN$\"37+++)y*\\aCFN7$$!3$)**** *>s$o1`FN$\"33+++NgtrBFN7$$!3%******f;Z-N&FN$\"3))*****pD&)zG#FN7$$!3S +++Lcs#R&FN$\"39+++=ME.AFN7$$!3h*****\\(30MaFN$\"30+++H0e<@FN7$$!3u*** **4P_TZ&FN$\"3*)*****R)4%4.#FN7$$!3B+++e&eH^&FN$\"3)******4BWL%>FN7$$! 3-+++**yR]bFN$\"3!******zt&ya=FN7$$!33+++^(*R'e&FN$\"3)******\\=e_w\"F N7$$!3B+++Y]*3i&FN$\"3))*****p\"Gvu;FN7$$!3C+++(y;Ql&FN$\"3))******)*3 E$e\"FN7$$!3=+++2/5&o&FN$\"33+++^_x!\\\"FN7$$!3E+++-So9dFN$\"31+++*y\" H(R\"FN7$$!3!)*****>\\3Du&FN$\"35+++p5\"GI\"FN7$$!3c*****Rt<&odFN$\"36 +++\"zRt?\"FN7$$!3w*****fheEz&FN$\"3$******>A#*36\"FN7$$!3Q+++e6)[\"eF N$\"3#******z^\"\\85FN7$$!3x*****RjQ^$eFN$\"3C+++(y3<:*F-7$$!3S+++GxQ` eFN$\"33+++#)\\uf\")F-7$$!39+++=()epeFN$\"3')******GCefrF-7$$!3K+++-dq $)eFN$\"3y*****>)***=:'F-7$$!3/+++Uoq&*eFN$\"3'******4D'\\P^F-7$$!3!** ****4ikb!fFN$\"3%******\\![G+++]Jw=fFN$\"3!******4X1O1#F-7$$!3H+++pE2AfFN$\"31+++WMVK5F -7$$!3U+++Fn)***=:'F-7$F]ay$!3')******GCefrF-7$Fh`y$!33+++#)\\uf\" )F-7$Fc`y$!3C+++(y3<:*F-7$F^`y$!3#******z^\"\\85FN7$Fi_y$!3$******>A#* 36\"FN7$Fd_y$!36+++\"zRt?\"FN7$F__y$!35+++p5\"GI\"FN7$Fj^y$!31+++*y\"H (R\"FN7$Fe^y$!33+++^_x!\\\"FN7$F`^y$!3))******)*3E$e\"FN7$F[^y$!3))*** **p\"Gvu;FN7$Ff]y$!3)******\\=e_w\"FN7$Fa]y$!3!******zt&ya=FN7$F\\]y$! 3)******4BWL%>FN7$Fg\\y$!3*)*****R)4%4.#FN7$Fb\\y$!30+++H0e<@FN7$F]\\y $!39+++=ME.AFN7$Fh[y$!3))*****pD&)zG#FN7$Fc[y$!33+++NgtrBFN7$F^[y$!37+ ++)y*\\aCFN7$Fijx$!38+++nYDODFN7$Fdjx$!3')******3M(ph#FN7$F_jx$!3=+++= Ui'p#FN7$Fjix$!31+++,?1$\\$FN7$Fcfx$!39+++QnvdNFN7 $F^fx$!3()*****zN(>@OFN7$Fiex$!35+++4)>Mo$FN7$Fdex$!3!)******[\"oWu$FN 7$F_ex$!3\")*****>.%R/QFN7$Fjdx$!3/+++0hDjQFN7$Fedx$!3.+++X\">6#RFN7$F `dx$!3'*******pG0yRFN7$F[dx$!3N+++q-8MSFN7$Ffcx$!3U+++M`U*3%FN7$Facx$! 3V+++>+,WTFN7$F\\cx$!31+++\"Q]z>%FN7$Fgbx$!3y*****z%=I^UFN7$Fbbx$!3*)* *****3Q5/VFN7$F]bx$!3y*****H*QPcVFN7$Fhax$!30+++'e-\"3WFN7$Fcax$!3g*** **>1\\#fWFN7$F^ax$!3G+++d\"R(4XFN7$Fi`x$!3m******ziYfXFN7$Fd`x$!3(**** ***ybH3YFN7$F_`x$!3O+++\\52cYFN7$Fj_x$!32+++6Yi-ZFN7$Fe_x$!37+++-eyZZF N7$F`_x$!3=+++q1R\"z%FN7$F[_x$!3m*****f!))GL[FN7$Ff^x$!3e*****R)zMt[FN 7$Fa^x$!3++++%Rc9\"\\FN7$F\\^x$!3))*****\\&G_Z\\FN7$Fg]x$!3K+++xbZ\") \\FN7$Fb]x$!3F+++f+E8]FN7$F]]x$!3-+++>m$G/&FN7$Fh\\x$!3)******R'z(p:&FN7$F_[x$!3!*******z%RI<&FN7$Fjjw$!39+++&R')o=&FN7$Fej w$!3K+++lja)>&FN7$F`jw$!3m*****fWp!3_FN7$F[jw$!3N+++,(Hb@&FN7$Ffiw$!3u ******pu.@_FN7$Faiw$!3U+++M#fZA&FN7$F\\iw$!3v*****p]]pA&FN7$Fghw$!3M++ +P@,G_FN7$Fbhw$!3I+++f/fG_FN7$F]hw$!3f*****\\(GwH_FN7$Fhgw$!35+++?BRL_ FN7$Fcgw$!3!)*****42*yU_FN7$F^gw$!3++++;Rlj_FN7$Fifw$!3%)******>@Y.`FN 7$Fdfw$!3Q+++l\\`j`FN7$F_fw$!3%*******)G%zJaFN7$Fjew$!3G+++g9%e\\&FN7$ Feew$!3%)*****4a@9b&FN7$F`ew$!3M+++()oZ)f&FN7$F[ew$!3U+++\\:(zj&FN7$Ff dw$!39+++&o04n&FN7$Fadw$!39+++X)3\")p&FN7$F\\dw$!3M+++uCB?dFN7$Fgcw$!3 '******HA#yPdFN7$Fbcw$!3=+++E7:^dFN7$F]cw$!3K+++(pY1w&FN7$Fhbw$!3')*** **>!)4lw&FN7$Fcbw$!3?+++[-$*odFN7$F^bw$!3O+++vp0odFN7$Fiaw$!3u******eg +kdFN7$Fdaw$!3%)*****Rgmov&FN7$F_aw$!3%******p<0nu&FN7$Fj`w$!3V+++'=pN t&FN7$Fe`w$!3f*****H[*[[S\\ZFN7$Fajv$!3/+++k4$oh%FN7$F\\jv$!3?+++j8# )eWFN7$Fgiv$!3#******>_OFN7$F^hv$!3;+++3(3VqEFN7$F`fv$!33+++7OI8 DFN7$F[fv$!3?+++\\m?cBFN7$$\"34+++0(>]#HF\\o$!3%)*****zE@\"*>#FN7$Faev $!3&)*****\\_R?/#FN7$F\\ev$!35+++\\'e\\)=FN7$Fgdv$!3&******H)z(ys\"FN7 $$\"3')*****fYkMY#FR$!37+++9vzq:FN7$$\"39*****\\eu^\"**FF$!32++++tr89F N7$Fhcv$!3/+++Dtjc7FN7$$\"3.+++$\\2g+\"FF$!3-+++Dvb*4\"FN7$$\"3%****** fs`tM#F@$!3G+++T#yZU*F-7$$\"3)******pQg%pSF:$!3u*****fp\")R&yF-7$$\"3* )******4t4CYF4$!3_*****fJ&=$G'F-7$F(Fh[pFj[pF]\\pF'-Fa\\p6&Fc\\p$\"#[F f\\p$\"\"\"Fb]pFia[l-Fh\\p6^_l7%F'F[]p7$$!#TFb]pF(7%F[]pFd]pF`b[l7%Fd] p7$F(F[^pF`b[l7%Feb[l7$$\"+Kt4CYFfr$\"+;`=$G'F^]pF`b[l7%Fgb[l7$$\"+\"Q g%pSF-$\"+'p\")R&yF^]pF`b[l7%F]c[l7$$\"+9TNZBFN$\"+T#yZU*F^]pF`b[l7%Fc c[l7$$\"+Iy+15F]_p$\"+Dvb*4\"Ff_pF`b[l7%Fic[l7$$\"+'H@%\\MF]_p$\"+Dtjc 7Ff_pF`b[l7%F_d[l7$$\"+%eu^\"**F]_p$\"++tr89Ff_pF`b[l7%Fed[l7$$\"+\"[k MY#F[`p$\"+9vzq:Ff_pF`b[l7%F[e[l7$$\"+\")*HwR&F[`p$\"+$)z(ys\"Ff_pF`b[ l7%Fae[l7$$\"+%R))f0\"F^ap$\"+\\'e\\)=Ff_pF`b[l7%Fge[l7$$\"+xo\\b=F^ap $\"+D&R?/#Ff_pF`b[l7%F]f[l7$$\"+/(>]#HF^ap$\"+o77*>#Ff_pF`b[l7%Fcf[l7$ $\"+\"y@A4%F^ap$\"+\\m?cBFf_pF`b[l7%Fif[l7$$\"+)H1]$\\F^ap$\"+7OI8DFf_ pF`b[l7%F_g[l7$$\"+*Q*yYZF^ap$\"+s3VqEFf_pF`b[l7%Feg[l7$$\"+EOFf_pF`b[l7%Fii[l7$$\"+I%**[h#F`dp$\"+g<$=y$Ff_pF`b[l7%F_j[l7$$\"+d >42pF`dp$\"+WgxYRFf_pF`b[l7%Fej[l7$$\"+%y+Uh\"Fcep$\"+q(*Q:TFf_pF`b[l7 %F[[\\l7$$\"+Sf!H\\$Fcep$\"+Av^(G%Ff_pF`b[l7%Fa[\\l7$$\"+eO+6qFcep$\"+ j8#)eWFf_pF`b[l7%Fg[\\l7$$\"+eH7[7F^]p$\"+k4$oh%Ff_pF`b[l7%F]\\\\l7$$ \"+6Zz))=F^]p$\"+?[S\\ZFf_pF`b[l7%Fc\\\\l7$$\"+8'>P[#F^]p$\"+5t!y&[Ff_ pF`b[l7%Fi\\\\l7$$\"+:!>Y(HF^]p$\"+8DY\\\\Ff_pF`b[l7%F_]\\l7$$\"+#*G=d LF^]p$\"+Qg-I]Ff_pF`b[l7%Fe]\\l7$$\"+#4tDk$F^]p$\"+&)*pF5&Ff_pF`b[l7%F [^\\l7$$\"+N&fP%QF^]p$\"+Bt_p^Ff_pF`b[l7%Fa^\\l7$$\"+T'fA(RF^]p$\"+hZN J_Ff_pF`b[l7%Fg^\\l7$$\"+\"4Nx.%F^]p$\"+*z()))G&Ff_pF`b[l7%F]_\\l7$$\" +x@B[SF^]p$\"+oi_U`Ff_pF`b[l7%Fc_\\l7$$\"+]#)\\5SF^]p$\"+%zLDR&Ff_pF`b [l7%Fi_\\l7$$\"+?%[-$RF^]p$\"+ZG4RaFf_pF`b[l7%F_`\\l7$$\"+g$pB\"QF^]p$ \"+\"*fL#[&Ff_pF`b[l7%Fe`\\l7$$\"+R>2hOF^]p$\"+tQOAbFf_pF`b[l7%F[a\\l7 $$\"+Nl+![$F^]p$\"+'fb#fbFf_pF`b[l7%Faa\\l7$$\"+7>NsKF^]p$\"+,Y2$f&Ff_ pF`b[l7%Fga\\l7$$\"+K`)3/$F^]p$\"+l=(Qi&Ff_pF`b[l7%F]b\\l7$$\"+.!R!)y# F^]p$\"+uuo^cFf_pF`b[l7%Fcb\\l7$$\"+6i%f^#F^]p$\"+)=^ln&Ff_pF`b[l7%Fib \\l7$$\"+k#zkA#F^]p$\"+uB[)p&Ff_pF`b[l7%F_c\\l7$$\"+a1G@>F^]p$\"+$[*[< dFf_pF`b[l7%Fec\\l7$$\"+K#)y,;F^]p$\"+'=pNt&Ff_pF`b[l7%F[d\\l7$$\"+[[D p7F^]p$\"+x^qYdFf_pF`b[l7%Fad\\l7$$\"+v\\iZ#*Fcep$\"+/m'ov&Ff_pF`b[l7% Fgd\\l7$$\"+ZmI#p&Fcep$\"+fg+kdFf_pF`b[l7%F]e\\l7$$\"+B-=M?Fcep$\"+vp0 odFf_pF`b[l7%Fce\\l7$$!+W2%3s\"Fcep$\"+[-$*odFf_pF`b[l7%Fie\\l7$$!+t** oobFcep$\"+-)4lw&Ff_pF`b[l7%F_f\\l7$$!+esZ2&*Fcep$\"+(pY1w&Ff_pF`b[l7% Fef\\l7$$!+V\\!QN\"F^]p$\"+E7:^dFf_pF`b[l7%F[g\\l7$$!+T<\\mFf_p$\"+lja )>&Ff_pF`b[l7%Fe_]l7$$!+r\"[\"R?Ff_p$\"+&R')o=&Ff_pF`b[l7%F[`]l7$$!+Kl N6@Ff_p$\"+![RI<&Ff_pF`b[l7%Fa`]l7$$!+iFA#=#Ff_p$\"+=>(p:&Ff_pF`b[l7%F g`]l7$$!+OJ$>D#Ff_p$\"+6;mQ^Ff_pF`b[l7%F]a]l7$$!+>Yl?BFf_p$\"+x_4=^Ff_ pF`b[l7%Fca]l7$$!+q+a)Q#Ff_p$\"+EqE&4&Ff_pF`b[l7%Fia]l7$$!+dBtbCFf_p$ \"+kzm$G/&Ff_pF`b[l7%Feb]l7$$!+^ td)e#Ff_p$\"+f+E8]Ff_pF`b[l7%F[c]l7$$!+U/\\aEFf_p$\"+xbZ\")\\Ff_pF`b[l 7%Fac]l7$$!+#>L-s#Ff_p$\"+bG_Z\\Ff_pF`b[l7%Fgc]l7$$!+j!Hfy#Ff_p$\"+%Rc 9\"\\Ff_pF`b[l7%F]d]l7$$!+*)op^GFf_p$\"+%)zMt[Ff_pF`b[l7%Fcd]l7$$!+7yk @LFf_p$\"+!Gm%fXFf_pF`b[l7%F]g]l7$$! +E^$)*Q$Ff_p$\"+d\"R(4XFf_pF`b[l7%Fcg]l7$$!+vcpeMFf_p$\"+i!\\#fWFf_pF` b[l7%Fig]l7$$!+4QkFNFf_p$\"+'e-\"3WFf_pF`b[l7%F_h]l7$$!+$*R`'f$Ff_p$\" +$*QPcVFf_pF`b[l7%Feh]l7$$!++c@lOFf_p$\"+4Q5/VFf_pF`b[l7%F[i]l7$$!+'[T Nt$Ff_p$\"+[=I^UFf_pF`b[l7%Fai]l7$$!+V[P,QFf_p$\"+\"Q]z>%Ff_pF`b[l7%Fg i]l7$$!+YRfoQFf_p$\"+>+,WTFf_pF`b[l7%F]j]l7$$!+U[4NRFf_p$\"+M`U*3%Ff_p F`b[l7%Fcj]l7$$!+(G#z+SFf_p$\"+q-8MSFf_pF`b[l7%Fij]l7$$!+)R>c1%Ff_p$\" +qG0yRFf_pF`b[l7%F_[^l7$$!+]k_HTFf_p$\"+X\">6#RFf_pF`b[l7%Fe[^l7$$!+(= zC>%Ff_p$\"+0hDjQFf_pF`b[l7%F[\\^l7$$!+PpXaUFf_p$\"+KSR/QFf_pF`b[l7%Fa \\^l7$$!+(p]aJ%Ff_p$\"+\\\"oWu$Ff_pF`b[l7%Fg\\^l7$$!+'HhaP%Ff_p$\"+4)> Mo$Ff_pF`b[l7%F]]^l7$$!+ww\\MWFf_p$\"+et>@OFf_pF`b[l7%Fc]^l7$$!+#GvD\\ %Ff_p$\"+QnvdNFf_pF`b[l7%Fi]^l7$$!+_Xr\\XFf_p$\"+J>1$\\$Ff_pF`b[l7%F_^ ^l7$$!+f%Rfg%Ff_p$\"+n_3FMFf_pF`b[l7%Fe^^l7$$!+wgFhYFf_p$\"+Ew!)fLFf_p F`b[l7%F[_^l7$$!+E6v:ZFf_p$\"+2&=7H$Ff_pF`b[l7%Fa_^l7$$!+-0RpZFf_p$\"+ dfJ@KFf_pF`b[l7%Fg_^l7$$!+nx@A[Ff_p$\"+>j5]JFf_pF`b[l7%F]`^l7$$!+PEDu[ Ff_p$\"+\\PgxIFf_pF`b[l7%Fc`^l7$$!+;&4b#\\Ff_p$\"+#[HQ+$Ff_pF`b[l7%Fi` ^l7$$!+Jh*f(\\Ff_p$\"+B3\")GHFf_pF`b[l7%F_a^l7$$!+cBrD]Ff_p$\"+p)zD&GF f_pF`b[l7%Fea^l7$$!+4$\\Y2&Ff_p$\"+,?Ff_pF`b[l7%Fge^l7$$!+**yR]bFf_p$\"+Qdya=Ff_pF`b[l 7%F]f^l7$$!+^(*R'e&Ff_p$\"+&=e_w\"Ff_pF`b[l7%Fcf^l7$$!+Y]*3i&Ff_p$\"+< Gvu;Ff_pF`b[l7%Fif^l7$$!+(y;Ql&Ff_p$\"+**3E$e\"Ff_pF`b[l7%F_g^l7$$!+2/ 5&o&Ff_p$\"+^_x!\\\"Ff_pF`b[l7%Feg^l7$$!+-So9dFf_p$\"+*y\"H(R\"Ff_pF`b [l7%F[h^l7$$!+#\\3Du&Ff_p$\"+p5\"GI\"Ff_pF`b[l7%Fah^l7$$!+Mx^odFf_p$\" +\"zRt?\"Ff_pF`b[l7%Fgh^l7$$!+;'eEz&Ff_p$\"+AA*36\"Ff_pF`b[l7%F]i^l7$$ !+e6)[\"eFf_p$\"+=:\\85Ff_pF`b[l7%Fci^l7$$!+M'Q^$eFf_p$\"+(y3<:*F^]pF` b[l7%Fii^l7$$!+GxQ`eFf_p$\"+#)\\uf\")F^]pF`b[l7%F_j^l7$$!+=()epeFf_p$ \"+HCefrF^]pF`b[l7%Fej^l7$$!+-dq$)eFf_p$\"+#)***=:'F^]pF`b[l7%F[[_l7$$ !+Uoq&*eFf_p$\"+^i\\P^F^]pF`b[l7%Fa[_l7$$!+@Yc0fFf_p$\"+0[GFf_pF`b[l7%F_b_l7$Fbe^l$!+%)4%4.#Ff_pF`b[l 7%Fcb_l7$F\\e^l$!+H0e<@Ff_pF`b[l7%Fgb_l7$Ffd^l$!+=ME.AFf_pF`b[l7%F[c_l 7$F`d^l$!+d_)zG#Ff_pF`b[l7%F_c_l7$Fjc^l$!+NgtrBFf_pF`b[l7%Fcc_l7$Fdc^l $!+)y*\\aCFf_pF`b[l7%Fgc_l7$F^c^l$!+nYDODFf_pF`b[l7%F[d_l7$Fhb^l$!+4M( ph#Ff_pF`b[l7%F_d_l7$Fbb^l$!+=Ui'p#Ff_pF`b[l7%Fcd_l7$F\\b^l$!+,?j5]JFf_pF`b[l7%F[f_l7$Fh_^l$!+dfJ@KFf_pF`b[l7%F_f_l7$Fb_^ l$!+2&=7H$Ff_pF`b[l7%Fcf_l7$F\\_^l$!+Ew!)fLFf_pF`b[l7%Fgf_l7$Ff^^l$!+n _3FMFf_pF`b[l7%F[g_l7$F`^^l$!+J>1$\\$Ff_pF`b[l7%F_g_l7$Fj]^l$!+QnvdNFf _pF`b[l7%Fcg_l7$Fd]^l$!+et>@OFf_pF`b[l7%Fgg_l7$F^]^l$!+4)>Mo$Ff_pF`b[l 7%F[h_l7$Fh\\^l$!+\\\"oWu$Ff_pF`b[l7%F_h_l7$Fb\\^l$!+KSR/QFf_pF`b[l7%F ch_l7$F\\\\^l$!+0hDjQFf_pF`b[l7%Fgh_l7$Ff[^l$!+X\">6#RFf_pF`b[l7%F[i_l 7$F`[^l$!+qG0yRFf_pF`b[l7%F_i_l7$Fjj]l$!+q-8MSFf_pF`b[l7%Fci_l7$Fdj]l$ !+M`U*3%Ff_pF`b[l7%Fgi_l7$F^j]l$!+>+,WTFf_pF`b[l7%F[j_l7$Fhi]l$!+\"Q]z >%Ff_pF`b[l7%F_j_l7$Fbi]l$!+[=I^UFf_pF`b[l7%Fcj_l7$F\\i]l$!+4Q5/VFf_pF `b[l7%Fgj_l7$Ffh]l$!+$*QPcVFf_pF`b[l7%F[[`l7$F`h]l$!+'e-\"3WFf_pF`b[l7 %F_[`l7$Fjg]l$!+i!\\#fWFf_pF`b[l7%Fc[`l7$Fdg]l$!+d\"R(4XFf_pF`b[l7%Fg[ `l7$F^g]l$!+!Gm%fXFf_pF`b[l7%F[\\`l7$Fhf]l$!+zbH3YFf_pF`b[l7%F_\\`l7$F bf]l$!+\\52cYFf_pF`b[l7%Fc\\`l7$F\\f]l$!+6Yi-ZFf_pF`b[l7%Fg\\`l7$Ffe]l $!+-eyZZFf_pF`b[l7%F[]`l7$F`e]l$!+q1R\"z%Ff_pF`b[l7%F_]`l7$Fjd]l$!+1)) GL[Ff_pF`b[l7%Fc]`l7$Fdd]l$!+%)zMt[Ff_pF`b[l7%Fg]`l7$F^d]l$!+%Rc9\"\\F f_pF`b[l7%F[^`l7$Fhc]l$!+bG_Z\\Ff_pF`b[l7%F_^`l7$Fbc]l$!+xbZ\")\\Ff_pF `b[l7%Fc^`l7$F\\c]l$!+f+E8]Ff_pF`b[l7%Fg^`l7$Ffb]l$!+>m$G/&Ff_pF`b[l7% F[_`l7$F`b]l$!+kz(p:&Ff_pF`b[l7%F_``l7$Fb`]l$!+![RI<&Ff_pF`b[l7%Fc``l7$F\\`]l$!+&R')o =&Ff_pF`b[l7%Fg``l7$Ff_]l$!+lja)>&Ff_pF`b[l7%F[a`l7$F`_]l$!+Y%p!3_Ff_p F`b[l7%F_a`l7$Fj^]l$!+,(Hb@&Ff_pF`b[l7%Fca`l7$Fd^]l$!+qu.@_Ff_pF`b[l7% Fga`l7$F^^]l$!+M#fZA&Ff_pF`b[l7%F[b`l7$Fh]]l$!+20&pA&Ff_pF`b[l7%F_b`l7 $Fb]]l$!+P@,G_Ff_pF`b[l7%Fcb`l7$F\\]]l$!+f/fG_Ff_pF`b[l7%Fgb`l7$Ff\\]l $!+vGwH_Ff_pF`b[l7%F[c`l7$F`\\]l$!+?BRL_Ff_pF`b[l7%F_c`l7$Fj[]l$!+r!*y U_Ff_pF`b[l7%Fcc`l7$Fd[]l$!+;Rlj_Ff_pF`b[l7%Fgc`l7$F^[]l$!+?@Y.`Ff_pF` b[l7%F[d`l7$Fhj\\l$!+l\\`j`Ff_pF`b[l7%F_d`l7$Fbj\\l$!+*G%zJaFf_pF`b[l7 %Fcd`l7$F\\j\\l$!+g9%e\\&Ff_pF`b[l7%Fgd`l7$Ffi\\l$!+T:U^bFf_pF`b[l7%F[ e`l7$F`i\\l$!+()oZ)f&Ff_pF`b[l7%F_e`l7$Fjh\\l$!+\\:(zj&Ff_pF`b[l7%Fce` l7$Fdh\\l$!+&o04n&Ff_pF`b[l7%Fge`l7$F^h\\l$!+X)3\")p&Ff_pF`b[l7%F[f`l7 $Fhg\\l$!+uCB?dFf_pF`b[l7%F_f`l7$Fbg\\l$!+BAyPdFf_pF`b[l7%Fcf`l7$F\\g \\l$!+E7:^dFf_pF`b[l7%Fgf`l7$Fff\\l$!+(pY1w&Ff_pF`b[l7%F[g`l7$F`f\\l$! +-)4lw&Ff_pF`b[l7%F_g`l7$Fje\\l$!+[-$*odFf_pF`b[l7%Fcg`l7$Fde\\l$!+vp0 odFf_pF`b[l7%Fgg`l7$F^e\\l$!+fg+kdFf_pF`b[l7%F[h`l7$Fhd\\l$!+/m'ov&Ff_ pF`b[l7%F_h`l7$Fbd\\l$!+x^qYdFf_pF`b[l7%Fch`l7$F\\d\\l$!+'=pNt&Ff_pF`b [l7%Fgh`l7$Ffc\\l$!+$[*[OFf_pF`b[l7%Fc_al7$Fdi [l$!+3]#HF^ap$!+o77*># Ff_pF`b[l7%Fiaal7$F^f[l$!+D&R?/#Ff_pF`b[l7%F_bal7$Fhe[l$!+\\'e\\)=Ff_p F`b[l7%Fcbal7$Fbe[l$!+$)z(ys\"Ff_pF`b[l7%Fgbal7$$\"+mWYjCF[`p$!+9vzq:F f_pF`b[l7%F[cal7$$\"+&eu^\"**F]_p$!++tr89Ff_pF`b[l7%Facal7$F`d[l$!+Dtj c7Ff_pF`b[l7%Fgcal7$$\"+$\\2g+\"F]_p$!+Dvb*4\"Ff_pF`b[l7%F[dal7$$\"+EP NZBFN$!+T#yZU*F^]pF`b[l7%Fadal7$$\"+(Qg%pSF-$!+'p\")R&yF^]pF`b[l7%Fgda l7$$\"+5t4CYFfr$!+;`=$G'F^]pF`b[l7%F]eal7$F(F\\avF`b[l7%FcealF_avF`b[l 7%F_avFcavF`b[l7%FcavF'F`b[l-Fa\\p6&Fc\\p$\"#&*Ff\\p$\"\"#Fb]pFiealF[b v-F$6%7$7$$!3o*************)oFNF(7$$\"3%**************=\"FNF(-%'COLOUR G6&Fc\\pF)F)F)-%*LINESTYLEG6#\"\"$-F$6%7$7$F($!3o*************Q'FN7$F( $\"3o*************Q'FNFffalFifal-%+AXESLABELSG6%%&Re(z)G%&Im(z)G-%%FON TG6#%(DEFAULTG-F\\hal6$%*HELVETICAG\"\"*-%*AXESSTYLEG6#%$BOXG-%(SCALIN GG6#%,CONSTRAINEDG-%%VIEWG6$;$!$*oFf\\p$\"$>\"Ff\\p;$!$R'Ff\\p$\"$R'Ff \\p" 1 2 0 1 10 0 2 9 1 2 1 1.000000 45.000000 45.000000 0 0 "Curve 1 " "Curve 2" "Curve 3" "Curve 4" "Curve 5" "Curve 6" }}}}{PARA 0 "" 0 " " {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}} {PARA 0 "" 0 "" {TEXT -1 52 "#---------------------------------------- -----------" }}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 35 "coefficients of th e combined scheme" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 36 "coe fficients of the combined scheme " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 6866 "ee := \{c[2]=2/25,\nc[3]=2 4514/238491,\nc[4]=12257/79497,\nc[5]=50/129,\nc[6]=34/73,\nc[7]=23/14 8,\nc[8]=196/197,\nc[9]=14118145156/16038718811,\nc[10]=39/43,\nc[11]= 20/21,\nc[12]=1,\nc[13]=1,\n\na[2,1]=2/25,\na[3,1]=2090517149/56877957 081,\na[3,2]=3755851225/56877957081,\na[4,1]=12257/317988,\na[4,2]=0, \na[4,3]=12257/105996,\na[5,1]=14131686489425/35833975601529,\na[5,2]= 0,\na[5,3]=-17700454220625/11944658533843,\na[5,4]=17619604667500/1194 4658533843,\na[6,1]=4418011/96055225,\na[6,2]=0,\na[6,3]=0,\na[6,4]=97 57757988/41995818067,\na[6,5]=19921512441/106300094275,\na[7,1]=480285 555889619/7997789253816320,\na[7,2]=0,\na[7,3]=0,\na[7,4]=521621065566 96460538643/464887033284472899365888,\na[7,5]=-16450769949384489/49000 9784398315520,\na[7,6]=864813381633887/51718297665732608,\na[8,1]=-419 24855046865190259777/29067997485765047436800,\na[8,2]=0,\na[8,3]=0,\na [8,4]=-1573638249639795538408923368457/337926761654282072170175876224, \na[8,5]=-3144752379896908085076884133/286731750405790169489612800,\na [8,6]=51820007951705750121680295/6052645677641770902708224,\na[8,7]=19 /2,\na[9,1]=-144152226093696433436046893191167505099143986568398421084 65299109803139261/\n 286664608340139813702727298202848620494381 29018451533980749016707010746880,\na[9,2]=0,\na[9,3]=0,\na[9,4]=-52241 6746733228002806292633565336738051126599108406039793785642181694795272 896367/\n 72447552632346584669609595477664264037757415889555156 446859013382512921014002304,\na[9,5]=-82495975261466675550315104749071 99042876873605462323913470411629578679723249/\n 175634113965120 8901236498008888114538679717900892624949907570487472295239680,\na[9,6] =249566009805532163702364580500327468167202244857464072051710984311548 0217671807/\n 62155833834379369756139647253642768400431902311034 9698478188147786343185391616,\na[9,7]=65926772222165868913654906182786 964314250556615077953597292687341/\n 711474570717221516523994602 2376277735416754491161939984260041034,\na[9,8]=31425442840030946914785 27859401247056878071594901520381912/\n 3228985071785520180284989 57174197954770661454622943631853501,\na[10,1]=-98678788580582550392177 46719915871073167304993320293/\n 13771006297995217364330134342 641992580302241127219200,\na[10,2]=0,\na[10,3]=0,\na[10,4]=-5021061931 16073366202094897549565186098556680379/\n 59760431150000217012 599575938528786496649056256,\na[10,5]=-3319239150460803358707163153104 01046907/52747587227799808210222170585805619200,\na[10,6]=164314767952 3140016971844191613351711631376872651547306145/\n 3158130578499 45223645118819513361571639686784746580082688,\na[10,7]=222428684738951 4767028684777161157168782986309/19993702221548179263423881196797919120 8436288,\na[10,8]=49853649680049123712136863581080955445/2729660625860 504418029771633420026504704,\na[10,9]=-1203747916407015470833526610982 521566345739350276392522818995/\n 4314650471762221105211615023 0647115135945640260504121035832832,\na[11,1]=-167715056043516475164394 7644188299249566198916438951/\n 152426858614852369214382304681 9588641870992602157568,\na[11,2]=0,\na[11,3]=0,\na[11,4]=-230201155410 1675211827716537011959670051291/27450871408461270516646113576782553983 2704,\na[11,5]=-16889144669510407387099211531243899609/189139157130077 7047888087779210764288,\na[11,6]=3970996097624277308974183527093198485 4511347767956576333171/\n 5577710294529287682275315773022359477 180471284918417776640,\na[11,7]=95624809679415765680274836872584068739 0347399/77961324245284495254874844291292429525671778,\na[11,8]=1078831 3637659292115861359463369872242886/18938058952730669846279569230937615 2568545,\na[11,9]=8959939844559568685635569843488421654003852728757383 0949657578490242/\n 5793856213384958134835883459658873734643821 84677844152053023899833767,\na[11,10]=-1786335907413925929868932792832 /7814458367137598885469908059185,\na[12,1]=-49410934155054900182882239 365042643979034016237/31731623983903801998088697641133618410239344640, \na[12,2]=0,\na[12,3]=0,\na[12,4]=-10746116491289811138041821043837297 04094059/244310054255337140402433369774092097581056,\na[12,5]=-2222581 55907862440804259443447899304774477/1891147547164292922205182530197744 9103360,\na[12,6]=1141087632767815674971557983546711109295237624162411 5293181/\n 1249081687157472843988286395834110562049959611482227 605504,\na[12,7]=8981318435146082911054040229712538210777263716775/936 984062290713822722287218869196809455428475968,\na[12,8]=46554463321808 6513326922426838859475/27467662576695046776938324444449900032, a[12,9] =-28081309843646368614165858697384177345304907256697557387703645459869 5025876625/\n 105067490432026768588795401172928908496060279595 45703832394231856364095276862976, a[12,10]=509083885759770594068157790 42800/1299277084579288698410581464578279,\na[12,11]=-53738699589768162 79363488081975/128249819495112776500552350596224,\na[13,1]=-6834845900 7984933111036701115865624594922076006017/\n 453450074630147116 31562120393860177858153484042240,\na[13,2]=0,\na[13,3]=0,\na[13,4]=-14 21462835415589956778218218848342421532216119/3491230464320813022691747 52451415423519748096,\na[13,5]=-29843127711920694864734417972688360805 79319/262376761616550996396340471077385680977920,\na[13,6]=41138701015 4201470248493205505950446147325414444198427659/\n 4646029500607 2909591428170750088268098474825959918534656,\na[13,7]=1592113933735083 1535408442010420239146408066675/17493829771050472169405235365133332903 89079616,\na[13,8]=-64484096268292447966995856630949967425/52081593006 15981523251328952205675621888,\na[13,9]=-84230489090401610635761047217 1364471743518622807702364576842637748625/\n 160011071045365843 70686873062421769553685916890578695272031985694176768, a[13,10]=112480 4893366359619795687870063500/18026098196604212076234696000197113,\na[1 3,11]=0,\na[13,12]=0,\n\nb[1]=874545191389960761/19691638733588931200, \nb[2]=0,\nb[3]=0,\nb[4]=0,\nb[5]=0,\nb[6]=420364804728976000977525051 149/1187588452078929614685280764000,\nb[7]=251024158247211533015320497 78688/101007568454396515184681192806875,\nb[8]=11291146602515386733913 667333/8277895500290250100346304000,\nb[9]=515227147644040638355923558 66713645307758690796273878782033726047743318218807/\n 192822210747 16176043598190917468876903092208069910099389016386439619421788800,\nb[ 10]=-582592205611226468655591/165538207004754039774500,\nb[11]=4304699 3917969354593/35912185355757478400,\nb[12]=-357120621384161/2621583039 07500,\nb[13]=0,\n\n`b*`[1]=13470793039761431/302948288209060480,\n`b* `[2]=0,\n`b*`[3]=0,\n`b*`[4]=0,\n`b*`[5]=0,\n`b*`[6]=22591098220551570 3574046329/636776649908273251841973600,\n`b*`[7]=506306684904286603699 28685568/203890933496965109375618071875,\n`b*`[8]=36550810580887322590 74359101/827789550029025010034630400,\n`b*`[9]=14302196276811339425779 3894874175054076879315632471495031519549972079/\n 6349799786728 1994630609975410189529208101511709161157616376634016000,\n`b*`[10]=-64 57286835032584968329/2546741646226985227300,\n`b*`[11]=0,\n`b*`[12]=0, \n`b*`[13]=-4954671690517567/1310791519537500\}:" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT 259 5 "nodes" }{TEXT -1 2 ": " }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 32 "seq(c[i]=subs(ee ,c[i]),i=2..13);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6./&%\"cG6#\"\"##F' \"#D/&F%6#\"\"$#\"&9X#\"'\"\\Q#/&F%6#\"\"%#\"&dA\"\"&(\\z/&F%6#\"\"&# \"#]\"$H\"/&F%6#\"\"'#\"#M\"#t/&F%6#\"\"(#\"#B\"$[\"/&F%6#\"\")#\"$'> \"$(>/&F%6#\"\"*#\",c^9=T\"\",6)=(Qg\"/&F%6#\"#5#\"#R\"#V/&F%6#\"#6#\" #?\"#@/&F%6#\"#7\"\"\"/&F%6#\"#8Fgo" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT 259 20 "linking coefficients" }{TEXT -1 1 ":" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 50 "seq(seq(a[i,j]=subs(ee,a[i ,j]),j=1..i-1),i=1..13);" }}{PARA 12 "" 1 "" {XPPMATH 20 "6jo/&%\"aG6$ \"\"#\"\"\"#F'\"#D/&F%6$\"\"$F(#\"+\\r^!4#\",\"3dz(o&/&F%6$F.F'#\"+D7& ev$F1/&F%6$\"\"%F(#\"&dA\"\"'))zJ/&F%6$F:F'\"\"!/&F%6$F:F.#F<\"''*f5/& F%6$\"\"&F(#\"/D%*['oJT\"\"/H:gvR$e$/&F%6$FJF'FA/&F%6$FJF.#!/D1Aa/q<\" /VQ`eY%>\"/&F%6$FJF:#\"/+vm/'>w\"FV/&F%6$\"\"'F(#\"(6!=W\")D_0'*/&F%6$ FinF'FA/&F%6$FinF.FA/&F%6$FinF:#\"+))zvd(*\",n!=e*>%/&F%6$FinFJ#\",TC^ @*>\"-vU4+j5/&F%6$\"\"(F(#\"0>'*)eb&G![\"1?j\"QD*y(*z/&F%6$FbpF'FA/&F% 6$FbpF.FA/&F%6$FbpF:#\"8V'Q0Y'pcl5i@&\"9))eO**GZ%GLq)[Y/&F%6$FbpFJ#!2* [%Q\\*p2X;\"3?bJ)R%y4+\\/&F%6$FbpFin#\"0()Qj\"Q8[')\"23EtlwH=<&/&F%6$ \"\")F(#!8x(f->lo/b[#>%\"8+oVZ]w&[(*z1H/&F%6$FarF'FA/&F%6$FarF.FA/&F%6 $FarF:#!@d%oL#*3%QbzR'\\#QOd\"\"?Ci(eF'/&F%6$\"\"*F(#!eoh#RJ!)4\"*Hl%3@%)Rol)R9*40v;\" >$*o/OMV'p$4EA:W\"\"eo!)ou5qq;!\\2)R`^%=!H\"Q%\\?'[G?)HFFq8)R,M3Ym'G/& F%6$FetF'FA/&F%6$FetF.FA/&F%6$FetF:#!\\pnj*GF&z%p\"=Ucy$zRgS3\"*fE60Qn LlNj#H1G+GKtYnTA&\"[p/B+95#H^#Q8!foWc^b*)eTdx.kUmxaf4'pYeYBj_vWs/&F%6$ FetFJ#!go\\Ksz'y&H;TqM\"RKiagto(G/*>2\\Z5:.bvmYh_(f\\#)\"go!oR_Hsu[qv! *\\\\i#*3!zrz'QX6)))3!)\\O7!*37lR6McV+%oFk`skRhvp$z V$Q$e:i/&F%6$FetFbp#\"\\oTto#H(f`z2:mb]UJkpy#=1\\l8*oe;AAxEf'\"[oM5/gU )*R>;\"\\anTNxFwB-Y*R_;:A<2du9r/&F%6$FetFar#\"en7>Q?:!\\fr!yo0Z7Sfy_y9 p%4.SGWD9$\"gn,N&=jVHia9mqZ&z>ur&*)\\G!=?byr])*GK/&F%6$\"#5F(#!U$H?L* \\InJ2re\"*>nu<#R]De!e)yy')*\"V+#>s7TAI!e#*>kUV8IVO<_*zH15x8/&F%6$F`wF 'FA/&F%6$F`wF.FA/&F%6$F`wF:#!Qz.oc&)4'=l&\\v*[4-iOtg6$>1@]\"Pci0\\m\\' yG&Qfd*f7q@++:J/wf/&F%6$F`wFJ#!H2p/,/J:jrqeL!3Y]\"R#>L\"G+#>c!eeq@A5#3 )*zFseZF&/&F%6$F`wFin#\"enXhIZ:lsoPJ;r^Lh\">W=(p,SJ_zw9V;\"Z)o#3!eYZy' oR;dhL^>)=^kB_%*\\y08eJ/&F%6$F`wFbp#\"O4j)Hyor:hrx%oGqw9&*QZoGCA\"N)GO %37>zz'>\")QUj#z\"[:A-P**>/&F%6$F`wFar#\"GXa&43\"ejo87P7\\+o\\O&)\\\"I /Z]E+UL;xH!=W]geig'HF/&F%6$F`wFet#!hn&**=G_#Rw-NRdMm:_#)4hEN$3Z:qS;zu. 7\"inKG$e.@T]g-kXf8:rkI-:;@06Aim&\\#*H)= Ww%Rk^Z;N/c]rn\"\"Uov:-E*4(=k)e>o/BQ9#pB&[heoU_\"/&F%6$FazF'FA/&F%6$Fa zF.FA/&F%6$FazF:#!L\"H^+nf>,PlrF=@v;5a:,-B\"K/F$)Rb#ywN6Ym^q7Y39(3XF/& F%6$FazFJ#!G4'**QCJ:@*4(Q2/^pY9*)o\"\"F)Gk2@zx3))y/x2Ir:R\"*=/&F%6$Faz Fin#\"fnrJLwl&zwZ8^a[)>$4FN=u*3tFCw4'*4(R\"enSmx<%=\\Gr/=x%fB-tdJvAo(G HXH5xd&/&F%6$FazFbp#\"N*RZ.R(oSeso$[F!oldTz'4[i&*\"My!Q!R)RiJ<$/&F%6$Fh]lF'FA/&F%6$Fh]lF.FA/&F%6$Fh]l F:#!LfS4/(HPQ/@=/Q6\")*G\"\\;hu5\"Kc5e(4#4u(pLV-/9P`Da+JW#/&F%6$Fh]lFJ #!KxWx/$**yWV%fU!3Wiy!f:eAA\"JgL5\\u(>ID=0A#HHkraZ6*=/&F%6$Fh]lFin#\"f n\"=$H:TiTiP_H46rYN)zbr\\n:ywKw3T6\"en/bgFA[6'f*\\?c5T$eR'G))R%GZdro\" 3\\7/&F%6$Fh]lFbp#\"Rvnrjsx5#QDrH-/a5\"H3Y^V=8)*)\"QofZGaX4o>p)=sGAF#Q r!HiS)p$*/&F%6$Fh]lFar#\"Ev%f)QoUApK8l3=KjWbY\"GK+!*\\WWC$QpxY]pwDmnu# /&F%6$Fh]lFet#!ioDm(e-&p)fak.xQdvpcs!\\IXt<%Q(pee;9'ojkV)483G\"[pwH'oF &4kj&=B%RKQqX&fz-1'\\3*GH<,az)eon-K/\\n]5/&F%6$Fh]lF`w#\"A+G/zd\"oSfq( fd)Q34&\"Cz#yXY\"e5%)p)GzX3x#*H\"/&F%6$Fh]lFaz#!@v>3)[j$zi\"o(*e*pQP& \"BCif]Bb+lx7^\\>)\\#G\"/&F%6$\"#8F(#!Sh@K:UU$[)=#=#yn&**e:a$GY@9\"N'4[(>NU:9X_Z$z03O)oszTMZ'[p?>rFJ%)H\"K?z(4o&Qx5ZSjR'*4b;;wwBE/&F%6$FealFin# \"ZfwU)>WWTDt9Y/&f]0K\\[-Z,U:5qQ6%\"YcY`=*ff#[Z)4o#)3]2.s_py0*o \"fo`&p'fjO$*[![7\"\"D8r>+g pMi27Ug'>)4E!=/&F%6$FealFazFA/&F%6$FealFh]lFA" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT 259 39 "weights for the 13 stage order 8 scheme" }{TEXT -1 1 ":" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 32 "seq(b[i]=subs(ee,b[i]),i=1..13);" }}{PARA 12 "" 1 "" {XPPMATH 20 "6//&%\"bG6#\"\"\"#\"3h2'**Q\">XX()\"5+7$*)eL(Q;p>/&F%6#\"\"#\"\"!/ &F%6#\"\"$F//&F%6#\"\"%F//&F%6#\"\"&F//&F%6#\"\"'#\"?\\60Dv(4+w*GZ![O? %\"@+Sw!G&o9'H*y?X)e(=\"/&F%6#\"\"(#\"A)oy(\\?`,L:@Z#eT-^#\"Bvo!G>\"o% =:lRa%ov+,\"/&F%6#\"\")#\">Ltm8Rt'Q:DgY6H6\"=+SIY.5]-H+b*yF)/&F%6#\"\" *#\"ho2)=#=LuZgsL?yyQF'z!pexIXOrmeN#fNQ1/Ww9F_^\"ho+))y@%>'RkQ;!*Q*45* p!3A4.p()ou\"4>)fVg<;Z2@AG>/&F%6#\"#5#!9\"fb'okA6c?#f#e\"9+XxRSv/q?Qb; /&F%6#\"#6#\"5$fa$pz\"R*p/V\"5+%yuvb`=7f$/&F%6#\"#7#!0hTQ@17d$\"0+v!RI e@E/&F%6#\"#8F/" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT 259 39 "weights for the 13 stage order 7 scheme" }{TEXT -1 1 ": " }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 38 "seq(`b*`[i]=subs(ee,`b*` [i]),i=1..13);" }}{PARA 12 "" 1 "" {XPPMATH 20 "6//&%#b*G6#\"\"\"#\"2J 9wRIzqM\"\"3![g!4#)G[HI/&F%6#\"\"#\"\"!/&F%6#\"\"$F//&F%6#\"\"%F//&F%6 #\"\"&F//&F%6#\"\"'#\"%=Dt#3*\\mxO'/&F%6#\"\"( #\">oboG*p.mG/\\o1j]\"?v=2=cP4^'p\\L4*Q?/&F%6#\"\")#\"=,\"fV2fAt)3e53b O\"<+/jM+,D!H+b*yF)/&F%6#\"\"*#\"`oz?(*\\&>:.&\\rCj:$zo2a]-V\"\"_o+g,MmP;w:h\"4<^,\"3#H&*=5a(*41j%*>Gny*z\\j/&F%6#\"#5#!7H$o \\eK]$oGdk\"7+tA&)pAY;uYD/&F%6#\"#6F//&F%6#\"#7F//&F%6#\"#8#!1nv^!prY& \\\"1+v`>:z58" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}}{PARA 0 "" 0 "" {TEXT -1 52 "#------------ ---------------------------------------" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}}{PARA 0 "" 0 "" {TEXT -1 46 "#================= ============================" }}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 32 "V erner's \"most efficient\" scheme" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }} {PARA 0 "" 0 "" {TEXT -1 66 "The following ( \"most efficient\" ) sche me comes from his website: " }{URLLINK 17 "http://www.math.sfu.ca/~jve rner/" 4 "http://www.math.sfu.ca/~jverner/" "" }{TEXT -1 1 "." }} {PARA 0 "" 0 "" {TEXT -1 144 "See: J.H. Verner, SIAM Journal of Numeri cal Analysis 1978, 772-790, \"Explicit Runge-Kutta methods with estima tes of the Local Truncation Error.\"" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 16 "The nodes are: " }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "c[2] = 1/20" "6#/&%\"cG6#\"\"#*&\"\"\"F )\"#?!\"\"" }{TEXT -1 3 ", " }{XPPEDIT 18 0 "c[3] = 341/3200" "6#/&% \"cG6#\"\"$*&\"$T$\"\"\"\"%+K!\"\"" }{TEXT -1 3 ", " }{XPPEDIT 18 0 " c[4] = 1023/6400" "6#/&%\"cG6#\"\"%*&\"%B5\"\"\"\"%+k!\"\"" }{TEXT -1 3 ", " }{XPPEDIT 18 0 "c[5] = 39/100" "6#/&%\"cG6#\"\"&*&\"#R\"\"\"\" $+\"!\"\"" }{TEXT -1 3 ", " }{XPPEDIT 18 0 "c[6] = 93/200" "6#/&%\"cG 6#\"\"'*&\"#$*\"\"\"\"$+#!\"\"" }{TEXT -1 3 ", " }{XPPEDIT 18 0 "c[7] = 31/200" "6#/&%\"cG6#\"\"(*&\"#J\"\"\"\"$+#!\"\"" }{TEXT -1 3 ", " }{XPPEDIT 18 0 "c[8] = 943/1000" "6#/&%\"cG6#\"\")*&\"$V*\"\"\"\"%+5! \"\"" }{TEXT -1 3 ", " }{XPPEDIT 18 0 "c[9] = 7067558016280/783715016 0667" "6#/&%\"cG6#\"\"**&\".!G;!ev1(\"\"\"\".n1;]r$y!\"\"" }{TEXT -1 3 ", " }{XPPEDIT 18 0 "c[10] = 909/1000" "6#/&%\"cG6#\"#5*&\"$4*\"\" \"\"%+5!\"\"" }{TEXT -1 3 ", " }{XPPEDIT 18 0 "c[11] = 47/50" "6#/&% \"cG6#\"#6*&\"#Z\"\"\"\"#]!\"\"" }{TEXT -1 3 ", " }{XPPEDIT 18 0 "c[1 2] = 1" "6#/&%\"cG6#\"#7\"\"\"" }{TEXT -1 1 "." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}{PARA 0 "" 0 "" {TEXT -1 52 "#---------------------------------------------- -----" }}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 28 "checking the combined sc heme" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 36 "coefficient s of the combined scheme " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 7041 "ee := \{c[2]=1/20,\nc[3]=341/3200 ,\nc[4]=1023/6400,\nc[5]=39/100,\nc[6]=93/200,\nc[7]=31/200,\nc[8]=943 /1000,\nc[9]=7067558016280/7837150160667,\nc[10]=909/1000,\nc[11]=47/5 0,\nc[12]=1,\nc[13]=1,\n\na[2,1]=1/20,\na[3,1]=-7161/1024000,\na[3,2]= 116281/1024000,\na[4,1]=1023/25600,\na[4,2]=0,\na[4,3]=3069/25600,\na[ 5,1]=4202367/11628100,\na[5,2]=0,\na[5,3]=-3899844/2907025,\na[5,4]=39 82992/2907025,\na[6,1]=5611/114400,\na[6,2]=0,\na[6,3]=0,\na[6,4]=3174 4/135025,\na[6,5]=923521/5106400,\na[7,1]=21173/343200,\na[7,2]=0,\na[ 7,3]=0,\na[7,4]=8602624/76559175,\na[7,5]=-26782109/689364000,\na[7,6] =5611/283500,\na[8,1]=-1221101821869329/690812928000000,\na[8,2]=0,\na [8,3]=0,\na[8,4]=-125/2,\na[8,5]=-1024030607959889/168929280000000,\na [8,6]=1501408353528689/265697280000000,\na[8,7]=6070139212132283/92502 016000000,\na[9,1]=-14725142644862158038813847088772642463460444333070 94207829051978044531801133057155/\n 1246894801620032001157059 621643986024803301558393487900440453636168046069686436608,\na[9,2]=0, \na[9,3]=0,\na[9,4]=-5172294311085668458375175655246981230039025336933 699114138315270772319372469280000/\n 124619381004809145897278 630571215298365257079410236252921850936749076487132995191,\na[9,5]=-12 0706792584692548079789364417331879494845715161204699665345142964068916 52614970375/\n 2722031154761657221710478184531100699497284085 048389015085076961673446140398628096,\na[9,6]=780125155843893641323090 552530431036567795592568497182701460674803126770111481625/\n \+ 1831104254127319721978898745071587868592261029808618595052414430736291 43100805376,\na[9,7]=6641131229599116421347821358391064699281403281605 77035357155340392950009492511875/\n 1517846559858624813633302 3107295349175279765150089078301139943253016877823170816,\na[9,8]=10332 848184452015604056836767286656859124007796970668046446015775000000/\n \+ 1312703550036033648073834248740727914537972028638950165249582 733679393783,\na[10,1]=-2905557336033741508853861844223103644131406051 1/\n 22674759891089577691327962602370597632000000000,\na[10,2 ]=0,\na[10,3]=0,\na[10,4]=-20462749524591049105403365239069/4542519134 99893469596231268750,\na[10,5]=-180269259803172281163724663224981097/3 8100922558256871086579832832000000,\na[10,6]=2112767021417280287012828 6992003940810655221489/\n 4679473877997892906145822697976708 633673728000,\na[10,7]=31860723517364931240515126584966086992765341442 5413/\n 6714716715558965303132938072935465423910912000000,\n a[10,8]=212083202434519082281842245535894/2002242604477567256382286537 1173879,\na[10,9]=-269840492940084251872116648508712979856226984822951 7793703413951226714583/\n 4695456749139343150770004420808711 41884676035902717550325616728175875000000,\na[11,1]=-23426598458140868 36951207140065609179073838476242943917/\n 135848096135105677 7022231400139158760857532162795520000,\na[11,2]=0,\na[11,3]=0,\na[11,4 ]=-996286030132538159613930889652/16353068885996164905464325675,\na[11 ,5]=-26053085959256534152588089363841/4377552804565683061011299942400, \na[11,6]=20980822345096760292224086794978105312644533925634933539/\n \+ 3775889992007550803878727839115494641972212962174156800,\na[ 11,7]=890722993756379186418929622095833835264322635782294899/\n \+ 13921242001395112657501941955594013822830119803764736,\na[11,8]=16 1021426143124178389075121929246710833125/10997207722131034650667041364 346422894371443,\na[11,9]=30076066976810251783423249756545243494667226 6195876496371874262392684852243925359864884962513/\n 4655443 3375013464555850653366045056037608247796155212857518928103156804923641 06674524398280000,\na[11,10]=-31155237437111730665923206875/3928621415 94230515010338956291,\na[12,1]=-28665569918256639717782953291010338875 34912787724034363/\n 868226711619262703011213925016143612030 669233795338240,\na[12,2]=0,\na[12,3]=0,\na[12,4]=-1695708871417146867 6387054358954754000/143690415119654683326368228101570221,\na[12,5]=-45 83493974484572912949314673356033540575/4519577036552507471573130342703 35135744,\na[12,6]=234630538855340425865625847344618441915474017251994 9575/\n 2567267164078954028927449783011514862541831852896624 64,\na[12,7]=165712155931984680217128369091361069858625657348480866262 5/\n 1343148041125514647725915510495609350536164443208810905 6,\na[12,8]=345685379554677052215495825476969226377187500/747711674369 30077221667203179551347546362089,\na[12,9]=\n -3205890962717072542791 4343121527275340081027740232102405713615707572490561670152301603520870 48674542196011/\n 947569549683965814783015124451273604984657747127257 615372449205973192657306017239103491074738324033259120,\na[12,10]=4027 9545832706233433100438588458933210937500/88964608427994828469169721263 77338947215101,\na[12,11]=-6122933601070769591613093993993358877250/10 50517001510235513198246721302027675953,\na[13,1]=-61867590553548250067 2800859344538410358660153899637/\n 2035442821182140471001194 75340667684874292102389760,\na[13,2]=0,\na[13,3]=0,\na[13,4]=-44111949 16804718600478400319122931000/40373053902469967450761491269633019,\na[ 13,5]=-16734711409449292534539422531728520225/180124371529008866930720 3927210237952,\na[13,6]=1351375197570546790980421841527496777612547518 65630525/\n 1602958779448628959777132636191189511270371659398 3488,\na[13,7]=38937568367409876012548551903492196137929710431584875/ \n 340956454090191606099548798001469306974758443147264,\na[13 ,8]=-6748865855011993037732355335815350667265625/\n 70028803 95717424621213565406715087764770357,\na[13,9]=-17560055203074509281954 22767042525091954178296002788308926563193523662404739779789732685671/ \n 348767814578469983605688098046186480904607278021030540735 333862087061574934154942830062320,\na[13,10]=5338102458923561108401389 7674181629296875/8959357584795694524874969598508592944141,\na[13,11]=0 ,\na[13,12]=0,\n\nb[1]=44901867737754616851973/10140464099802310133806 80,\nb[2]=0,\nb[3]=0,\nb[4]=0,\nb[5]=0,\nb[6]=791638675191615279648100 000/2235604725089973126411512319,\nb[7]=3847749490868980348119500000/1 5517045062138271618141237517,\nb[8]=-13734512432397741476562500000/875 132892924995907746928783,\nb[9]=12274765470313196878428812037740635050 319234276006986398294443554969616342274215316330684448207141/\n \+ 4893451474937155176503858341435109348888292806866096544828965267965233 53052166757299452852166040,\nb[10]=-9798363684577739445312500000/30872 2986341456031822630699,\nb[11]=282035543183190840068750/12295407629873 040425991,\nb[12]=-306814272936976936753/1299331183183744997286,\nb[13 ]=0,\n\n`b*`[1]=10835401739407019406577/244521829356935137978320,\n`b* `[2]=0,\n`b*`[3]=0,\n`b*`[4]=0,\n`b*`[5]=0,\n`b*`[6]=13908189778321895 491375000/39221135527894265375640567,\n`b*`[7]=73487947527027243487625 000/296504045773342769773399443,\n`b*`[8]=68293140641257649609375000/1 5353208647806945749946119,\n`b*`[9]=2206064794899667861101771137997457 8860522018208949721559448560203338437626022142776381/\n 11115420 0926232587451295918579572721575901057756573607964137662138157723668092 9558640,\n`b*`[10]=-547971229495642458203125000/2323721402570099164256 3601,\n`b*`[11]=0,\n`b*`[12]=0,\n`b*`[13]=-28735456870978964189/797834 93704265043693\}:" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 " > " 0 "" {MPLTEXT 1 0 1 ";" }}}}{PARA 0 "" 0 "" {TEXT -1 0 "" }} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}{SECT 1 {PARA 4 "" 0 " " {TEXT -1 20 "printed coefficients" }}{PARA 0 "" 0 "" {TEXT -1 0 "" } }{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 357 "subs(ee,matrix([seq([c[i], seq(a[i,j],j=1..i-1),``$(8-i)],i=2..8)]));\nfor ii from 9 to 13 do\n \+ print(``);\n print(c[ii]=subs(ee,c[ii])); \n for jj to ii-1 do\n \+ print(a[ii,jj]=subs(ee,a[ii,jj]));\n end do:\nend do:print(``); \nfor ii to 13 do\n print(b[ii]=subs(ee,b[ii]));\nend do:print(``); \nfor ii to 13 do\n print(`b*`[ii]=subs(ee,`b*`[ii]));\nend do:" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#K%'matrixG6#7)7*#\"\"\"\"#?F(%!GF+F+F+ F+F+7*#\"$T$\"%+K#!%hr\"(+S-\"#\"'\"G;\"F2F+F+F+F+F+7*#\"%B5\"%+k#F7\" &+c#\"\"!#\"%pIF:F+F+F+F+7*#\"#R\"$+\"#\"(nB?%\")+\"G;\"F;#!(W)**Q\"(D q!H#\"(#*H)RFGF+F+F+7*#\"#$*\"$+##\"%6c\"'+W6F;F;#\"&W<$\"'D]8#\"'@N#* \"(+k5&F+F+7*#\"#JFM#\"&t6#\"'+KMF;F;#\"(CEg)\")v\"fl(#!)4@yE\"*+SO*o# FO\"'+NGF+7*#\"$V*\"%+5#!1H$p=#=5@7\"0+++GH\"3pF;F;#!$D\"\"\"##!1*))fz gIS-\"\"0+++!GH*o\"#\"1*oGNN39]\"\"0+++!G(pl##\"1$GK@@R,2'\"/+++;?]#*Q )pprint466\"" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%!G" }}{PARA 11 "" 1 " " {XPPMATH 20 "6#/&%\"cG6#\"\"*#\".!G;!ev1(\".n1;]r$y" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/&%\"aG6$\"\"*\"\"\"#!]pbr0L6!=`W!y>0Hy?%42LVWgMYU Ex)3ZQ\")Q!e@'[kU^s9\"]p3mV'opg/ohj`/W+z[$Re:I.[-')Rk@'fq:,?.?;![*oC\" " }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/&%\"aG6$\"\"*\"\"#\"\"!" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/&%\"aG6$\"\"*\"\"$\"\"!" }}{PARA 11 "" 1 " " {XPPMATH 20 "6#/&%\"aG6$\"\"*\"\"%#!]p++GpCP>Bxq_JQT6*pLpLD!R+B\")pC bc&*H8([w!\\n$4&=#HDO-TzqDl$)H:7dI'ys*e94[+\"Q>Y7 " }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/&%\"aG6$\"\"*\"\"&#!^pv.(\\h_;*oS 'H9X`m*p/7;:d%[\\z=LDa%" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/&%\"aG6$\"#5\"\"&#!E(4\") \\AjYsj6GsJ!)f#p-=\"D+++KG$)zl3roDeD#45Q" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/&%\"aG6$\"#5\"\"'#\"P*[@_l53%R+#*pGG,(G!G<9-nF6#\"O+!GPnL'3n(zp Ae91H*y*z(QZzY" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/&%\"aG6$\"#5\"\"(# \"T8aU9MlF*p3m\\eE^^S7$\\OXV-K 37#\"DzQvcKka!\\;'*f))oIN;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/&%\"aG6$\"#6\"\"&#!ATQO*3)e_T`c#ff3`g#\"@+C%** H651$olX!GbxV" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/&%\"aG6$\"#6\"\"'#\" YRN$\\jDR`WEJ0\"y\\z'3CAHgn4XB#3)4#\"X+o:u@'H@s>k%\\:\"RysyQ!3b2?***)e x$" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/&%\"aG6$\"#6\"\"(#\"W**[H#yNEKk _$Q$e4A'H*=k=zjv$*Hs!*)\"VOZw.)>,$G#Q,%fb>%>]dE6&R,?C@R\"" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/&%\"aG6$\"#6\"\")#\"KDJ$3rY#H>7v!*QyT7VhU@5;\" MV9P%*GUYVOTqm]Y.J@s2s*4\"" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/&%\"aG6 $\"#6\"\"*#\"hp8D'\\)[')f`#RC_[o#RiU(=P'\\we>mAnY\\V_ac(\\KU$y^-\"o(p1 w+$\"ip++G)RCXn1TO#\\!o:.\"G*=v&G@bhzZ#3w.c]/mLl]ebkM,vLVal%" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/&%\"aG6$\"#6\"#5#!>vo?BfmI<6PuBb6$\"?\"Hc*Q .,:0B%fT@'GR" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%!G" }}{PARA 11 "" 1 " " {XPPMATH 20 "6#/&%\"cG6#\"#7\"\"\"" }}{PARA 11 "" 1 "" {XPPMATH 20 " 6#/&%\"aG6$\"#7\"\"\"#!XjV.Cxy7\\`()Q.,\"H`Hy<(RmD=*pbmG\"WS#Q`zL#p1.7 O9;]#R@6Iqi#>;rE#o)" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/&%\"aG6$\"#7\" \"#\"\"!" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/&%\"aG6$\"#7\"\"$\"\"!" } }{PARA 11 "" 1 "" {XPPMATH 20 "6#/&%\"aG6$\"#7\"\"%#!G+Sva*eV0(Qw'o9<9 ()3dp\"\"E@-d,\"G#ojK$oa'>^T!pV\"" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/ &%\"aG6$\"#7\"\"&#!Iv0aLgNtYJ\\H\"Hd%[uR\\$e%\"HWd8N.FMIJdru]_l.x&>X" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/&%\"aG6$\"#7\"\"'#\"Xv&\\*>DW= YMZeileUS`&)Q0jM#\"WkCm*G&=$=ai[^6Iy\\u#*GS&*yS;nsc#" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/&%\"aG6$\"#7\"\"(#\"enDEm3[[tlD'e)p5O\"4p$Gr@!o%)>$ f:7d;\"Yc!4\")3KWkh`]$4c\\5b\"fsZY^D6/[JM\"" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/&%\"aG6$\"#7\"\")#\"N+v=xjAppZDe\\:A0xYbz`oX$\"M*3ijaZ 8bzJ?n;Ax+$pVn6xu" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/&%\"aG6$\"#7\"\" *#!eq6g>UXn[q3_.;I_,nh0\\svq:Or0C5K-uF53S`FF:7VV\"zUD2tf?\\CP:wDFrudY)\\gt7XC^,$y9e'Ro\\&pv%*" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#/&%\"aG6$\"#7\"#5#\"M+v$4@L*e%)eQ/5LMB 1F$eaz-%\"L,^@Z*QtPE@(p\"p%G[*zU3Y'*))" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/&%\"aG6$\"#7\"#6#!I]s()eL*R*R48;fp22,O$H7'\"I`fnF?I@nC)>8bB5:+< 00\"" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%!G" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/&%\"cG6#\"#8\"\"\"" }}{PARA 11 "" 1 "" {XPPMATH 20 "6# /&%\"aG6$\"#8\"\"\"#!TP'**Q:g'e.TQXMf3!Gn+D[Nb!fn='\"Tg(*Q-@Hu[on1Mv%> ,5ZS@=@GWN?" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/&%\"aG6$\"#8\"\"#\"\"! " }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/&%\"aG6$\"#8\"\"$\"\"!" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/&%\"aG6$\"#8\"\"%#!F+5$H7>.Sy/g=Z!o\"\\>6W \"D>Ijp7\\h2Xn*pC!R0t.%" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/&%\"aG6$\" #8\"\"&#!GD-_G<`A%RX`#H\\%49rMn\"\"F_zB5s#R?2$p')3!H:PC,=" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/&%\"aG6$\"#8\"\"'#\"WD0jl=va7wx'\\F:%=U!)4zY0d (>v8N\"\"V)[$)Rf;Pq7^*=\">OE8x(f*G'[%z(eHg\"" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/&%\"aG6$\"#8\"\"(#\"Vv[eJ/rHz8'>#\\.>b[D,w)4uOov$*Q\"T ks9V%eZ(pIp9+)z[&*41;>!4ak&4M" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/&%\" aG6$\"#8\"\")#!LDcEn1N:eLbBtPI*>,be')[n\"Ld.xkx3:nSlN@@YUjl#*3$)y-gHyT&>4DD/nFU&>G4X2._0gv\"\"ep?B1IG%\\:M\\dhq3iQLN2aI5-ysg/4 ['=Y!)4)o0O)*p%yX\"yw[$" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/&%\"aG6$\" #8\"#5#\"JvoHH;=uw*Q,%36cB*eC5Q`\"ITT%Hf3&)fp\\([_%p&z%ed$f*)" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#/&%\"aG6$\"#8\"#6\"\"!" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/&%\"aG6$\"#8\"#7\"\"!" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%!G" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/&%\"bG6#\"\"\"# \"8t>&ohaxtn=!\\%\":!o!Q85B!)*4k/95" }}{PARA 11 "" 1 "" {XPPMATH 20 "6 #/&%\"bG6#\"\"#\"\"!" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/&%\"bG6#\"\"$ \"\"!" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/&%\"bG6#\"\"%\"\"!" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/&%\"bG6#\"\"&\"\"!" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/&%\"bG6#\"\"'#\"<++5['z_h\">v'Q;z\"=>B^6k7t**3DZgNA" } }{PARA 11 "" 1 "" {XPPMATH 20 "6#/&%\"bG6#\"\"(#\"=++]>\"[.)*o3\\\\x%Q \">++]ilZTxRKC^MP\"\"<$yGpu2f*\\#H*G8v)" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/&%\"bG6#\"\"*#\"]qTr?[WoIjJ:UFUjhp\\bVWH)R')p+wUB>.0N1uP?\")G%y o>8.ZlZF7\"[qSg;_GX*Hdn;_INBlzEl*G[a'4mo!GH)))[$4^VT$eQ]w^:P\\Z^M*[" } }{PARA 11 "" 1 "" {XPPMATH 20 "6#/&%\"bG6#\"#5#!=++]7`WRxd%oj$)z*\"<*p IE#=.c9M')Hs3$" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/&%\"bG6#\"#6#\"9](o +%3>$=Vb.#G\"8\"*fUSI()HwS&H7" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/&%\" bG6#\"#7#!6`n$p(p$HF9oI\"7'G(*\\u$=$=J$*H\"" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/&%\"bG6#\"#8\"\"!" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#% !G" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/&%#b*G6#\"\"\"#\"8xlS>qSRh%*\\d%p!yk3KN:" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/&%#b*G6#\"\"*# \"ap\"QwF9AgiP%QL?g&[%f:s\\*3#=?_g)yX(*z8r<5hym**[zkg?#\"`pS'e&H4oOsd \"Q@mPT'zgtlvd5!fd@Fdz&=fH^ueKi#4?a66" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/&%#b*G6#\"#5#!<+]7.#eCk&\\H7(za\";,OcU;*4qDS@PK#" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/&%#b*G6#\"#6\"\"!" }}{PARA 11 "" 1 "" {XPPMATH 20 " 6#/&%#b*G6#\"#7\"\"!" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/&%#b*G6#\"#8# !5*=k*y4(oXN(G\"5$pV]E/P\\$yz" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 151 "RK8_13eqs := [op( RowSumConditions(13,'expanded')),op(OrderConditions(8,13,'expanded'))] :\n`RK7_13eqs*` := subs(b=`b*`,OrderConditions(7,13,'expanded')):" }}} {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 6 "Check:" }} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 110 "subs(ee,RK8_13eqs):\nmap(u- >lhs(u)-rhs(u),%);\nnops(%);\nsubs(ee,`RK7_13eqs*`):\nmap(u->lhs(u)-rh s(u),%);\nnops(%);" }}{PARA 12 "" 1 "" {XPPMATH 20 "6#7`x\"\"!F$F$F$F$ F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$ F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$ F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$ F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$ F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$ F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#\"$7#" }}{PARA 12 "" 1 "" {XPPMATH 20 "6#7ap\"\"!F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F $F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F $F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#\"#&)" }}}{PARA 257 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 78 "Set-up stage-order condtions to check for stage-orders fr om 2 to 6 inclusive. " }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 87 "for ct from 2 to 6 do\n so||ct||_13 := StageOrderConditions(ct,13,'expa nded');\nend do:" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 59 "Stages 3 to 13 have the following respective stage-orders . " }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 175 "[seq([seq(expand(subs (ee,so||i||_13[j])),i=2..6)],j=1..11)]:\nmap(proc(L) local i; for i to nops(L) do if not evalb(L[i]) then break end if end do; i end proc,%) :\nsimplify(%); " }}{PARA 11 "" 1 "" {XPPMATH 20 "6#7-\"\"#\"\"$F%\"\" %F&F&F&F&F&F&F&" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 27 "The simplifying conditions:" }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "Sum(b[i]*a[i,j],i = j+1 .. 13) = b[j]*(1-c[j] );" "6#/-%$SumG6$*&&%\"bG6#%\"iG\"\"\"&%\"aG6$F+%\"jGF,/F+;,&F0F,F,F, \"#8*&&F)6#F0F,,&F,F,&%\"cG6#F0!\"\"F," }{TEXT -1 3 ", " }{XPPEDIT 18 0 "j=1" "6#/%\"jG\"\"\"" }{TEXT -1 9 " . . 12, " }}{PARA 0 "" 0 "" {TEXT -1 8 "(where " }{XPPEDIT 18 0 "c[1]=0" "6#/&%\"cG6#\"\"\"\"\"! " }{TEXT -1 17 " ) are satisfied." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 151 "[Sum(b[i]*a[i,1],i=1+1..13) =b[1],seq(Sum(b[i]*a[i,j],i=j+1..13)=b[j]*(1-c[j]),j=2..12)]:\nmap(u-> lhs(u)-rhs(u),subs(ee,eval(subs(Sum=add,%))));\nnops(%);" }}{PARA 11 " " 1 "" {XPPMATH 20 "6#7.\"\"!F$F$F$F$F$F$F$F$F$F$F$" }}{PARA 11 "" 1 " " {XPPMATH 20 "6#\"#7" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 27 "The simplifying conditions:" }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "Sum(`b*`[i]*a[i,j],i = j+1 .. 13) = `b* `[j]*(1-c[j]);" "6#/-%$SumG6$*&&%#b*G6#%\"iG\"\"\"&%\"aG6$F+%\"jGF,/F+ ;,&F0F,F,F,\"#8*&&F)6#F0F,,&F,F,&%\"cG6#F0!\"\"F," }{TEXT -1 3 ", " } {XPPEDIT 18 0 "j=1" "6#/%\"jG\"\"\"" }{TEXT -1 9 " . . 12, " }}{PARA 0 "" 0 "" {TEXT -1 8 "(where " }{XPPEDIT 18 0 "c[1]=0" "6#/&%\"cG6#\" \"\"\"\"!" }{TEXT -1 17 " ) are satisfied." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 163 "[Sum(`b*`[i]*a[i, 1],i=1+1..13)=`b*`[1],seq(Sum(`b*`[i]*a[i,j],i=j+1..13)=`b*`[j]*(1-c[j ]),j=2..12)]:\nmap(u->lhs(u)-rhs(u),subs(ee,eval(subs(Sum=add,%))));\n nops(%);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#7.\"\"!F$F$F$F$F$F$F$F$F$F $F$" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#\"#7" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 93 "We can calculate the principal \+ error norm, that is, the 2-norm of the principal error terms. " }} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 162 "errterms8_13 := PrincipalEr rorTerms(8,13,'expanded'):\nsm := 0:\nfor ct to nops(errterms8_13) do \n sm := sm+(evalf(subs(ee,errterms8_13[ct])))^2;\nend do:\nsqrt(sm) ;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#$\"+Pg'y#G!#;" }}}{PARA 0 "" 0 " " {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 78 "We can also calculate t he principal error norm of the order 7 embedded scheme." }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 184 "`errterms7_13*` := subs(b=`b*`,Pri ncipalErrorTerms(7,13,'expanded')):\nsm := 0:\nfor ct to nops(`errterm s7_13*`) do\n sm := sm+(evalf(subs(ee,`errterms7_13*`[ct])))^2;\nend do:\nsqrt(sm);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#$\"+PY3Y$)!#:" }}} {PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}}{PARA 0 "" 0 "" {TEXT -1 52 "#------------------------------ ---------------------" }}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 34 "construc tion of the order 8 scheme" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 93 "First we construct a 12 stage order 8 scheme star ting with a consideration of stages 2 to 8. " }}{PARA 0 "" 0 "" {TEXT -1 20 "We specify the nodes" }}{PARA 256 "" 0 "" {TEXT -1 2 " " } {XPPEDIT 18 0 "c[2] = 1/20;" "6#/&%\"cG6#\"\"#*&\"\"\"F)\"#?!\"\"" } {TEXT -1 2 ", " }{XPPEDIT 18 0 "c[5] = 39/100;" "6#/&%\"cG6#\"\"&*&\"# R\"\"\"\"$+\"!\"\"" }{TEXT -1 2 ", " }{XPPEDIT 18 0 "c[6] = 93/200;" " 6#/&%\"cG6#\"\"'*&\"#$*\"\"\"\"$+#!\"\"" }{TEXT -1 2 ", " }{XPPEDIT 18 0 "c[7] = 31/200;" "6#/&%\"cG6#\"\"(*&\"#J\"\"\"\"$+#!\"\"" }{TEXT -1 3 ", " }{XPPEDIT 18 0 "c[8] = 943/1000;" "6#/&%\"cG6#\"\")*&\"$V* \"\"\"\"%+5!\"\"" }{TEXT -1 2 " " }}{PARA 0 "" 0 "" {TEXT -1 33 "and \+ the zero linking coefficients" }}{PARA 256 "" 0 "" {TEXT -1 1 " " } {XPPEDIT 18 0 "a[4,2]=0" "6#/&%\"aG6$\"\"%\"\"#\"\"!" }{TEXT -1 2 ", \+ " }{XPPEDIT 18 0 "a[5,2]=0" "6#/&%\"aG6$\"\"&\"\"#\"\"!" }{TEXT -1 2 " , " }{XPPEDIT 18 0 "a[6,2]=0" "6#/&%\"aG6$\"\"'\"\"#\"\"!" }{TEXT -1 2 ", " }{XPPEDIT 18 0 "a[6,3]=0" "6#/&%\"aG6$\"\"'\"\"$\"\"!" }{TEXT -1 2 ", " }{XPPEDIT 18 0 "a[7,2]=0" "6#/&%\"aG6$\"\"(\"\"#\"\"!" } {TEXT -1 2 ", " }{XPPEDIT 18 0 "a[7,3]=0" "6#/&%\"aG6$\"\"(\"\"$\"\"! " }{TEXT -1 2 ", " }{XPPEDIT 18 0 "a[8,2] = 0;" "6#/&%\"aG6$\"\")\"\"# \"\"!" }{TEXT -1 2 ", " }{XPPEDIT 18 0 "a[8,3] = 0;" "6#/&%\"aG6$\"\") \"\"$\"\"!" }{TEXT -1 3 ", " }{XPPEDIT 18 0 "a[8,4]=0" "6#/&%\"aG6$\" \")\"\"%\"\"!" }{TEXT -1 2 ". " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }} {PARA 0 "" 0 "" {TEXT -1 48 "In addition we specify the linking coeffi cient " }{XPPEDIT 18 0 "a[8,4] = -125/2;" "6#/&%\"aG6$\"\")\"\"%,$*& \"$D\"\"\"\"\"\"#!\"\"F." }{TEXT -1 1 "." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 69 "The stage-order equations relating to these stages, such that stages " }{XPPEDIT 18 0 "3,4,5,6,7,8;" "6( \"\"$\"\"%\"\"&\"\"'\"\"(\"\")" }{TEXT -1 19 " have stage orders " } {XPPEDIT 18 0 "2,3,3,4,4,4;" "6(\"\"#\"\"$F$\"\"%F%F%" }{TEXT -1 48 " \+ respectively can be solved to obtain the nodes " }{XPPEDIT 18 0 "c[3], c[4]" "6$&%\"cG6#\"\"$&F$6#\"\"%" }{TEXT -1 67 " and the remaining no n-zero linking coefficients for these stages. " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 453 "RSeqs := Ro wSumConditions(8,'expanded'):\nSOeqs := [op(StageOrderConditions(2,8,' expanded')),op(StageOrderConditions(3,4..8,'expanded')),\n \+ op(StageOrderConditions(4,6..8,'expanded'))]:\ne1 := \{c[2]=1/20,c[ 5]=39/100,c[6]=93/200,c[7]=31/200,c[8]=943/1000,\n seq(a[i,2] =0,i=4..8),seq(a[i,3]=0,i=6..8),a[8,4]=-125/2\}:\neqns := map(expand,s ubs(e1,[op(RSeqs),op(SOeqs)])):\nconvert(ListTools[Enumerate](%),matri x);\n``;\nindets(eqns);\nnops(%);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#K %'matrixG6#777$\"\"\"/&%\"aG6$\"\"#F(#F(\"#?7$F-/,&&F+6$\"\"$F(F(&F+6$ F5F-F(&%\"cG6#F57$F5/,&&F+6$\"\"%F(F(&F+6$F@F5F(&F96#F@7$F@/,(&F+6$\" \"&F(F(&F+6$FJF5F(&F+6$FJF@F(#\"#R\"$+\"7$FJ/,(&F+6$\"\"'F(F(&F+6$FWF@ F(&F+6$FWFJF(#\"#$*\"$+#7$FW/,*&F+6$\"\"(F(F(&F+6$F^oF@F(&F+6$F^oFJF(& F+6$F^oFWF(#\"#JFhn7$F^o/,,&F+6$\"\")F(F(#\"$D\"F-!\"\"&F+6$F\\pFJF(&F +6$F\\pFWF(&F+6$F\\pF^oF(#\"$V*\"%+57$F\\p/,$*&F.F(F6F(F(,$*&#F(F-F(*$ )F8F-F(F(F(7$\"\"*/*&FAF(F8F(,$*&F_qF(*$)FCF-F(F(F(7$\"#5/,&*&FKF(F8F( F(*&FMF(FCF(F(#\"%@:\"&++#7$\"#6/,&*&FXF(FCF(F(*&FOF(FZF(F(#\"%\\')\"& ++)7$\"#7/,(*&F_oF(FCF(F(*&FOF(FaoF(F(*&FfnF(FcoF(F(#\"$h*F[s7$\"#8/,* *&#F^pF-F(FCF(F_p*&FOF(F`pF(F(*&FfnF(FbpF(F(*&FeoF(FdpF(F(#\"'\\#*))\" (+++#7$\"#9/*&FAF(FaqF(,$*&#F(F5F(*$)FCF5F(F(F(7$\"#:/,&*&FKF(FaqF(F(* &FMF(FiqF(F(#\"&t(>\"(+++\"7$\"#;/,&*&FXF(FiqF(F(*&#Far\"&++\"F(FZF(F( #\"'>\"o#\"(+++)7$\"#/,&*&FXF(FitF(F(*&#\"&>$f FbuF(FZF(F(#\"),_![(\"+++++k7$F//,(*&F_oF(FitF(F(*&F]xF(FaoF(F(*&#\"'d V!)F]vF(FcoF(F(#\"'@N#*Fax7$\"#@/,**&#F^pF-F(FhtF(F_p*&F]xF(F`pF(F(*&F hxF(FbpF(F(*&#FhvF]vF(FdpF(F(#\"-,Syj2z\".++++++%Q(pprint06\"" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#%!G" }}{PARA 11 "" 1 "" {XPPMATH 20 "6 #<7&%\"aG6$\"\"%\"\"\"&%\"cG6#F'&F%6$\"\"&F'&F%6$F.F(&F%6$F.\"\"$&F*6# F3&F%6$F'F3&F%6$\"\"#F(&F%6$F3F(&F%6$F3F:&F%6$\"\"'F(&F%6$FAF'&F%6$FAF .&F%6$\"\"(F(&F%6$FHF'&F%6$FHF.&F%6$FHFA&F%6$\"\")F(&F%6$FQF.&F%6$FQFA &F%6$FQFH" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#\"#@" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 46 "e2 := solve( \{op(eqns)\});\ne3 := `union`(e1,e2):" }}{PARA 12 "" 1 "" {XPPMATH 20 "6#>%#e2G<7/&%\"aG6$\"\"'\"\"\"#\"%6c\"'+W6/&F(6$\"\"&F+#\"(nB?%\")+\" G;\"/&F(6$\"\"#F+#F+\"#?/&F(6$\"\"(\"\"%#\"(CEg)\")v\"fl(/&F(6$\"\"$F9 #\"'\"G;\"\"(+S-\"/&F(6$FGF+#!%hrFJ/&F(6$F@F+#\"%B5\"&+c#/&F(6$\"\")F2 #!1*))fzgIS-\"\"0+++!GH*o\"/&F(6$FYF?#\"1$GK@@R,2'\"/+++;?]#*/&F(6$F?F *#F-\"'+NG/&F(6$F?F2#!)4@yE\"*+SO*o/&%\"cG6#F@#FT\"%+k/&F(6$F*F@#\"&W< $\"'D]8/&F(6$F2F@#\"(#*H)R\"(Dq!H/&Fjo6#FG#\"$T$\"%+K/&F(6$F2FG#!(W)** QFip/&F(6$F@FG#\"%pIFU/&F(6$FYF*#\"1*oGNN39]\"\"0+++!G(pl#/&F(6$F*F2# \"'@N#*\"(+k5&/&F(6$FYF+#!1H$p=#=5@7\"0+++GH\"3p/&F(6$F?F+#\"&t6#\"'+K M" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT 260 4 "Not e" }{TEXT -1 2 ": " }}{PARA 0 "" 0 "" {TEXT -1 8 "We have " }{XPPEDIT 18 0 "c[3]=2/3" "6#/&%\"cG6#\"\"$*&\"\"#\"\"\"F'!\"\"" }{TEXT -1 1 " \+ " }{XPPEDIT 18 0 "c[4]" "6#&%\"cG6#\"\"%" }{TEXT -1 16 " and the nodes " }{XPPEDIT 18 0 "c[4];" "6#&%\"cG6#\"\"%" }{TEXT -1 3 ", " } {XPPEDIT 18 0 "c[5];" "6#&%\"cG6#\"\"&" }{TEXT -1 3 ", " }{XPPEDIT 18 0 "c[6];" "6#&%\"cG6#\"\"'" }{TEXT -1 31 " are related by the equa tion: " }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "3*c[6]^2-4* c[6]*c[4]-4*c[5]*c[6]+6*c[5]*c[4] = 0" "6#/,**&\"\"$\"\"\"*$&%\"cG6#\" \"'\"\"#F'F'*(\"\"%F'&F*6#F,F'&F*6#F/F'!\"\"*(F/F'&F*6#\"\"&F'&F*6#F,F 'F4*(F,F'&F*6#F8F'&F*6#F/F'F'\"\"!" }{TEXT -1 2 ". " }}{PARA 0 "" 0 " " {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 86 "subs(e3,ma trix([seq([c[i],seq(a[i,j],j=1..i-1),``$(9-i)],i=2..8)]));\n``;\nevalf [8](%%);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#K%'matrixG6#7)7+#\"\"\"\"# ?F(%!GF+F+F+F+F+F+7+#\"$T$\"%+K#!%hr\"(+S-\"#\"'\"G;\"F2F+F+F+F+F+F+7+ #\"%B5\"%+k#F7\"&+c#\"\"!#\"%pIF:F+F+F+F+F+7+#\"#R\"$+\"#\"(nB?%\")+\" G;\"F;#!(W)**Q\"(Dq!H#\"(#*H)RFGF+F+F+F+7+#\"#$*\"$+##\"%6c\"'+W6F;F;# \"&W<$\"'D]8#\"'@N#*\"(+k5&F+F+F+7+#\"#JFM#\"&t6#\"'+KMF;F;#\"(CEg)\") v\"fl(#!)4@yE\"*+SO*o#FO\"'+NGF+F+7+#\"$V*\"%+5#!1H$p=#=5@7\"0+++GH\"3 pF;F;#!$D\"\"\"##!1*))fzgIS-\"\"0+++!GH*o\"#\"1*oGNN39]\"\"0+++!G(pl## \"1$GK@@R,2'\"/+++;?]#*F+Q(pprint16\"" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%!G" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#K%'matrixG6#7)7+$\")+++]!\"* F(%!GF+F+F+F+F+F+7+$\")]il5!\")$!)T;$*p!#5$\")mbN6F/F+F+F+F+F+F+7+$\") vV)f\"F/$\")Q4'*RF*$\"\"!F;$\")\"G))>\"F/F+F+F+F+F+7+$\")+++RF/$\")c(R h$F/F:$!)T_T8!\"($\")l7q8FEF+F+F+F+7+$\")++]YF/$\").s/\\F*F:F:$\")?(4N #F/$\")fb3=F/F+F+F+7+$\")++]:F/$\")!*GphF*F:F:$\")olB6F/$!)h/&)QF*$\") ()=z>F*F+F+7+$\")++I%*F/$!)-jn " 0 "" {MPLTEXT 1 0 1 ";" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 27 " We do not need to specify " }{XPPEDIT 18 0 "c[9];" "6#&%\"cG6#\"\"*" }{TEXT -1 42 ", because according to Verner, the nodes " }{XPPEDIT 18 0 "c[6];" "6#&%\"cG6#\"\"'" }{TEXT -1 2 ", " }{XPPEDIT 18 0 "c[7]; " "6#&%\"cG6#\"\"(" }{TEXT -1 2 ", " }{XPPEDIT 18 0 "c[8];" "6#&%\"cG6 #\"\")" }{TEXT -1 2 ", " }{XPPEDIT 18 0 "c[9];" "6#&%\"cG6#\"\"*" } {TEXT -1 30 " are related by the equation " }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "Int(p(x)*``((1-x)^3/3!),x = 0 .. 1)*Int (q(x)*(1-x),x = 0 .. 1) = Int(p(x)*``((1-x)^2/2!),x = 0 .. 1)*Int(q(x) *``((1-x)^2/2!),x = 0 .. 1);" "6#/*&-%$IntG6$*&-%\"pG6#%\"xG\"\"\"-%!G 6#*&,&F-F-F,!\"\"\"\"$-%*factorialG6#F4F3F-/F,;\"\"!F-F--F&6$*&-%\"qG6 #F,F-,&F-F-F,F3F-/F,;F:F-F-*&-F&6$*&-F*6#F,F--F/6#*&,&F-F-F,F3\"\"#-F6 6#FNF3F-/F,;F:F-F--F&6$*&-F?6#F,F--F/6#*&,&F-F-F,F3FN-F66#FNF3F-/F,;F: F-F-" }{TEXT -1 2 ", " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 7 "where " }{XPPEDIT 18 0 "p(x) = x*(x-c[6])*(x-c[7])*(x- c[8]);" "6#/-%\"pG6#%\"xG**F'\"\"\",&F'F)&%\"cG6#\"\"'!\"\"F),&F'F)&F, 6#\"\"(F/F),&F'F)&F,6#\"\")F/F)" }{TEXT -1 7 " and " }{XPPEDIT 18 0 "q(x) = (x-c[9])*p(x);" "6#/-%\"qG6#%\"xG*&,&F'\"\"\"&%\"cG6#\"\"*!\" \"F*-%\"pG6#F'F*" }{TEXT -1 2 ". " }}{PARA 0 "" 0 "" {TEXT -1 1 " " }} {PARA 0 "" 0 "" {TEXT -1 155 "See: J.H. Verner, SIAM Journal of Numeri cal Analysis 1978, 772-790, \"Explicit Runge-Kutta methods with estima tes of the Local Truncation Error.\" (page 780)" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 90 "p := x -> x* (x-c[6])*(x-c[7])*(x-c[8]):\n'p(x)'=p(x);\nq := x -> (x-c[9])*p(x):\n' q(x)'=q(x);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/-%\"pG6#%\"xG**F'\"\" \",&F'F)&%\"cG6#\"\"'!\"\"F),&F'F)&F,6#\"\"(F/F),&F'F)&F,6#\"\")F/F)" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/-%\"qG6#%\"xG*,,&F'\"\"\"&%\"cG6#\" \"*!\"\"F*F'F*,&F'F*&F,6#\"\"'F/F*,&F'F*&F,6#\"\"(F/F*,&F'F*&F,6#\"\") F/F*" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 239 "Ieq := Int('p(x)'*(1-x)^3/3!,x=0..1)*Int('q(x)'*(1-x ),x=0..1)=\n Int('p(x)'*(1-x)^2/2!,x=0..1)*Int('q(x)'*(1-x)^2/2!,x=0. .1);\nvalue(Ieq):\nReq := simplify(lhs(%)-rhs(%))=0:\nL := ilcm(op(map (denom,[coeffs(lhs(Req))]))):\nCeq := expand(L*Req);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%$IeqG/*&-%$IntG6$,$*&#\"\"\"\"\"'F-*&-%\"pG6#%\"xG F-),&F-F-F3!\"\"\"\"$F-F-F-/F3;\"\"!F-F--F(6$*&-%\"qGF2F-F5F-F8F-*&-F( 6$,$*&#F-\"\"#F-*&F0F-)F5FFF-F-F-F8F--F(6$,$*&FEF-*&F>F-FHF-F-F-F8F-" }}{PARA 12 "" 1 "" {XPPMATH 20 "6#>%$CeqG/,hp*&\"#7\"\"\"&%\"cG6#\"\"* F)F)*(\"#UF)F*F)&F+6#\"\"'F)!\"\"*(F/F))&F+6#\"\")\"\"#F)F*F)F)*(F/F)) &F+6#\"\"(F9F)F*F)F)*(\"#GF)F0F)FF)FEF)F;F)F*F)F)** \"#)*F)FEF)F;F)F6F)F3**FPF)F5F)F0F)F*F)F3**FPF)F5F)F0F)FF'*$&F*6#FJF-F'&F*6#F,F -&F*6#FCF'F;**FdqF'*$&F*6#FJF-F'&F*6#F0F'&F*6#FCF'F'**F^]lF'*$&F*6#FJF -F'&F*6#F0F-&F*6#FCF'F;**FdqF'&F*6#FCF'&F*6#F,F-&F*6#F0F'F'**FdqF'&F*6 #F,F'&F*6#F0F-&F*6#FCF'F'**F^]lF'*$&F*6#F,F-F'&F*6#F0F-&F*6#FCF'F;\"\" !" }{TEXT -1 2 ". " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 " > " 0 "" {MPLTEXT 1 0 75 "e4 := solve(subs(e3,Ceq),\{c[9]\}):\nc[9]=su bs(e4,c[9]);\ne5 := `union`(e3,e4):" }}{PARA 11 "" 1 "" {XPPMATH 20 "6 #/&%\"cG6#\"\"*#\".!G;!ev1(\".n1;]r$y" }}}{PARA 0 "" 0 "" {TEXT -1 0 " " }}{PARA 0 "" 0 "" {TEXT -1 7 "Thus " }{XPPEDIT 18 0 "c[9] = 706755 8016280/7837150160667" "6#/&%\"cG6#\"\"**&\".!G;!ev1(\"\"\"\".n1;]r$y! \"\"" }{TEXT -1 2 ". " }}{PARA 0 "" 0 "" {TEXT -1 84 "Now we can use t he quadrature equations to find the weights once the remaining nodes" }}{PARA 256 "" 0 "" {XPPEDIT 18 0 "c[10] = 909/1000;" "6#/&%\"cG6#\"#5 *&\"$4*\"\"\"\"%+5!\"\"" }{TEXT -1 3 ", " }{XPPEDIT 18 0 "c[11] = 47/ 50;" "6#/&%\"cG6#\"#6*&\"#Z\"\"\"\"#]!\"\"" }{TEXT -1 3 ", " } {XPPEDIT 18 0 "c[12] = 1;" "6#/&%\"cG6#\"#7\"\"\"" }{TEXT -1 1 " " }} {PARA 0 "" 0 "" {TEXT -1 38 "are specified along with the weights " } {XPPEDIT 18 0 "b[2]=0" "6#/&%\"bG6#\"\"#\"\"!" }{TEXT -1 2 ", " } {XPPEDIT 18 0 "b[3]=0" "6#/&%\"bG6#\"\"$\"\"!" }{TEXT -1 2 ", " } {XPPEDIT 18 0 "b[4]=0" "6#/&%\"bG6#\"\"%\"\"!" }{TEXT -1 2 ", " } {XPPEDIT 18 0 "b[5]=0" "6#/&%\"bG6#\"\"&\"\"!" }{TEXT -1 2 ". " }} {PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 291 "Qeqs := QuadratureConditions(8,12,'expanded'):\ne6 := \{seq(b[i]= 0,i=2..5),c[10]=909/1000,c[11]=47/50,c[12]=1\}:\ne7 := `union`(e5,e6): \nquadeqns := subs(e7,Qeqs):\nfor ct to nops(quadeqns) do\n print(`e quation `||ct); print(``);print(quadeqns[ct]);print(``);\nend do:\ni ndets(quadeqns);\nnops(%);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%-equati on~~~1G" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%!G" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/,2&%\"bG6#\"\"\"F(&F&6#\"\"'F(&F&6#\"\"(F(&F&6#\"\")F( &F&6#\"\"*F(&F&6#\"#5F(&F&6#\"#6F(&F&6#\"#7F(F(" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%!G" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%-equation~~~2G " }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%!G" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/,0*&#\"#$*\"$+#\"\"\"&%\"bG6#\"\"'F)F)*&#\"#JF(F)&F+6#\"\"(F)F) *&#\"$V*\"%+5F)&F+6#\"\")F)F)*&#\".!G;!ev1(\".n1;]r$yF)&F+6#\"\"*F)F)* &#\"$4*F7F)&F+6#\"#5F)F)*&#\"#Z\"#]F)&F+6#\"#6F)F)&F+6#\"#7F)#F)\"\"# " }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%!G" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%-equation~~~3G" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%!G" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/,0*&#\"%\\')\"&++%\"\"\"&%\"bG6#\"\"'F)F)*& #\"$h*F(F)&F+6#\"\"(F)F)*&#\"'\\#*))\"(+++\"F)&F+6#\"\")F)F)*&#\";+%Q] u)o$[8jP]*\\\";*)[)Q\"RyU3kA4UhF)&F+6#\"\"*F)F)*&#\"'\"GE)F7F)&F+6#\"# 5F)F)*&#\"%4A\"%+DF)&F+6#\"#6F)F)&F+6#\"#7F)#F)\"\"$" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%!G" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%-equation~~~ 4G" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%!G" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/,0*&#\"'dV!)\"(+++)\"\"\"&%\"bG6#\"\"'F)F)*&#\"&\"zHF( F)&F+6#\"\"(F)F)*&#\"*2=cQ)\"+++++5F)&F+6#\"\")F)F)*&#\"H+?:Dk*pb?IkyU cID=FIN\"Hj4Y$\\=4_!*p)=S'*Hu$*\\O\"[F)&F+6#\"\"*F)F)*&#\"*H%*3^(F7F)& F+6#\"#5F)F)*&#\"'BQ5\"'+]7F)&F+6#\"#6F)F)&F+6#\"#7F)#F)\"\"%" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#%!G" }}{PARA 11 "" 1 "" {XPPMATH 20 "6 #%-equation~~~5G" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%!G" }}{PARA 11 " " 1 "" {XPPMATH 20 "6#/,0*&#\"),_![(\"+++++;\"\"\"&%\"bG6#\"\"'F)F)*&# \"'@N#*F(F)&F+6#\"\"(F)F)*&#\"-,Syj2z\".++++++\"F)&F+6#\"\")F)F)*&#\"U ++cuuh?BA:H2QbHNOMK'eQ4S]\\#\"U@BaUVw(*>0xM*HmIYv+PR_!Q(HDx$F)&F+6#\" \"*F)F)*&#\"-h4HSFoF7F)&F+6#\"#5F)F)*&#\"(\"oz[\"(++D'F)&F+6#\"#6F)F)& F+6#\"#7F)#F)\"\"&" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%!G" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%-equation~~~6G" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%!G" }}{PARA 12 "" 1 "" {XPPMATH 20 "6#/,0*&#\"+$p$)o&p\"-+++++K \"\"\"&%\"bG6#\"\"'F)F)*&#\")^\"H'GF(F)&F+6#\"\"(F)F)*&#\"0VHJ[-pX(\"1 +++++++5F)&F+6#\"\")F)F)*&#\"\\o++!o$e'p(fk;%=4mAz9aoh;pL<1e!H;1%Qj<\" \\o2\")3(R[YN![IjDP#4MdB'>C4Lbr_$)pU?)ecHF)&F+6#\"\"*F)F)*&#\"0\\N[C4h ?'F7F)&F+6#\"#5F)F)*&#\"*2]MH#\"*++]7$F)&F+6#\"#6F)F)&F+6#\"#7F)#F)F- " }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%!G" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%-equation~~~7G" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%!G" }}{PARA 12 "" 1 "" {XPPMATH 20 "6#/,0*&#\"-\\M=!*pk\"/++++++k\"\"\"&%\"bG6#\" \"'F)F)*&#\"*\"o.v))F(F)&F+6#\"\"(F)F)*&#\"3\\_5fT!f=.(\"4+++++++++\"F )&F+6#\"\")F)F)*&#\"io+++/JAQv57d\\fL$QU$3rj]qa-\"=;^\"**f[$oag\">GY7 \"iopt)[+!z^$zB(>GujM#RKW'y7[#3I+6kvQ^'\\>,sD7$H:!*z7&F7F)&F+6#\"#5F)F)*&#\"-j/7 Bm]\"-+++]7yF)&F+6#\"#6F)F)&F+6#\"#7F)#F)F:" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%!G" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#<*&%\"bG6#\"\"\" &F%6#\"\"'&F%6#\"\"(&F%6#\"\")&F%6#\"\"*&F%6#\"#5&F%6#\"#6&F%6#\"#7" } }{PARA 11 "" 1 "" {XPPMATH 20 "6#\"\")" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 199 "e8 := solve(\{op(quade qns)\}):\ne9 := `union`(e7,e8):\nfor ii to 12 do\n wt_val := subs(e9 ,b[ii]);\n if wt_val<>0 then print(b[ii]=wt_val) end if;\nend do:\n` `;\nsubs(e9,[seq(b[i],i=1..12)]):\nevalf[8](%);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/&%\"bG6#\"\"\"#\"8t>&ohaxtn=!\\%\":!o!Q85B!)*4k/95" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#/&%\"bG6#\"\"'#\"<++5['z_h\">v'Q;z\"=> B^6k7t**3DZgNA" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/&%\"bG6#\"\"(#\"=++ ]>\"[.)*o3\\\\x%Q\">++]ilZTxRKC^MP\"\"<$yGpu2f*\\#H*G8v)" }}{PARA 11 " " 1 "" {XPPMATH 20 "6#/&%\"bG6#\"\"*#\"]qTr?[WoIjJ:UFUjhp\\bVWH)R')p+w UB>.0N1uP?\")G%yo>8.ZlZF7\"[qSg;_GX*Hdn;_INBlzEl*G[a'4mo!GH)))[$4^VT$e Q]w^:P\\Z^M*[" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/&%\"bG6#\"#5#!=++]7` WRxd%oj$)z*\"<*pIE#=.c9M')Hs3$" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/&% \"bG6#\"#6#\"9](o+%3>$=Vb.#G\"8\"*fUSI()HwS&H7" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/&%\"bG6#\"#7#!6`n$p(p$HF9oI\"7'G(*\\u$=$=J$*H\"" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#%!G" }}{PARA 11 "" 1 "" {XPPMATH 20 "6 #7.$\")%*)zU%!\"*$\"\"!F(F'F'F'$\")%\\5a$!\")$\")ApzCF+$!)-Up:!\"'$\") lS3DF0$!)o$Q<$F0$\")$GQH#F0$!)YKhBF+" }}}{PARA 0 "" 0 "" {TEXT -1 0 " " }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 2 "e9" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1522 "e9 := \{a[6,1] = 5611/114400, b[5] = 0, b [2] = 0, b[3] = 0, b[4] = 0, b[8] = -13734512432397741476562500000/875 132892924995907746928783, c[12] = 1, a[6,4] = 31744/135025, b[12] = -3 06814272936976936753/1299331183183744997286, a[7,2] = 0, a[6,3] = 0, a [4,2] = 0, a[8,3] = 0, a[8,2] = 0, a[7,3] = 0, a[6,2] = 0, a[5,2] = 0, a[4,1] = 1023/25600, c[9] = 7067558016280/7837150160667, a[5,4] = 398 2992/2907025, b[9] = 1227476547031319687842881203774063505031923427600 6986398294443554969616342274215316330684448207141/48934514749371551765 0385834143510934888829280686609654482896526796523353052166757299452852 166040, b[7] = 3847749490868980348119500000/15517045062138271618141237 517, c[3] = 341/3200, a[5,3] = -3899844/2907025, b[11] = 2820355431831 90840068750/12295407629873040425991, a[4,3] = 3069/25600, a[7,1] = 211 73/343200, a[2,1] = 1/20, c[5] = 39/100, a[8,4] = -125/2, c[6] = 93/20 0, c[7] = 31/200, c[8] = 943/1000, c[2] = 1/20, a[8,6] = 1501408353528 689/265697280000000, a[6,5] = 923521/5106400, b[1] = 44901867737754616 851973/1014046409980231013380680, c[4] = 1023/6400, b[6] = 79163867519 1615279648100000/2235604725089973126411512319, a[8,1] = -1221101821869 329/690812928000000, a[7,6] = 5611/283500, a[7,5] = -26782109/68936400 0, b[10] = -9798363684577739445312500000/308722986341456031822630699, \+ a[5,1] = 4202367/11628100, c[10] = 909/1000, c[11] = 47/50, a[3,1] = - 7161/1024000, a[7,4] = 8602624/76559175, a[8,5] = -1024030607959889/16 8929280000000, a[8,7] = 6070139212132283/92502016000000, a[3,2] = 1162 81/1024000\}:" }{TEXT -1 0 "" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 55 "Now we specify the remaining ze ro linking coefficients:" }}{PARA 256 "" 0 "" {TEXT -1 2 " " } {XPPEDIT 18 0 "a[9,2] = 0,a[9,3] = 0,a[10,2] = 0,a[10,3] = 0,a[11,2] = 0,a[11,3] = 0,a[12,2] = 0,a[12,3] = 0;" "6*/&%\"aG6$\"\"*\"\"#\"\"!/& F%6$F'\"\"$F)/&F%6$\"#5F(F)/&F%6$F1F-F)/&F%6$\"#6F(F)/&F%6$F8F-F)/&F%6 $\"#7F(F)/&F%6$F?F-F)" }{TEXT -1 1 "." }}{PARA 0 "" 0 "" {TEXT -1 0 " " }}{PARA 0 "" 0 "" {TEXT -1 149 "We include the stage-order condition s for stages 9 to 12 so that all these stages all have stage-order 4 a nd incorporate the simplifying conditions: " }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "Sum(b[i]*a[i,j],i = j+1 .. 12) = b[j]*( 1-c[j]);" "6#/-%$SumG6$*&&%\"bG6#%\"iG\"\"\"&%\"aG6$F+%\"jGF,/F+;,&F0F ,F,F,\"#7*&&F)6#F0F,,&F,F,&%\"cG6#F0!\"\"F," }{TEXT -1 1 " " }}{PARA 0 "" 0 "" {TEXT -1 5 "for " }{XPPEDIT 18 0 "j=1" "6#/%\"jG\"\"\"" } {TEXT -1 10 " ( with " }{XPPEDIT 18 0 "c[1]=0" "6#/&%\"cG6#\"\"\"\" \"!" }{TEXT -1 5 " ), " }{XPPEDIT 18 0 "j = 5;" "6#/%\"jG\"\"&" } {TEXT -1 3 ", " }{XPPEDIT 18 0 "j=7" "6#/%\"jG\"\"(" }{TEXT -1 3 ", \+ " }{XPPEDIT 18 0 "j=9" "6#/%\"jG\"\"*" }{TEXT -1 7 " and " } {XPPEDIT 18 0 "j=10" "6#/%\"jG\"#5" }{TEXT -1 2 ". " }}{PARA 0 "" 0 " " {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 195 "Then it turns out that the following additional collection of \"simple\" order conditions (g iven in abreviated form) is sufficient to determine the remaining link ing coefficients in stages 9 to 12." }}{PARA 0 "" 0 "" {TEXT -1 0 "" } }{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 190 "SO8 := SimpleOrderConditio ns(8):\n[seq([i,SO8[i]],i=[37,61,81,87,102,106,113,117,125])]:\nlinalg [augment](linalg[delcols](%,2..2),matrix([[` `]$(linalg[rowdim](%))]) ,linalg[delcols](%,1..1));" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#K%'matri xG6#7+7%\"#P%#~~G/*(%\"bG\"\"\"%\"cGF--%!G6#*&%\"aGF--F06#*&F3F--F06#* &F3F--F06#*&F3F-F.F-F-F-F-F-#F-\"$S)7%\"#hF)/*(F,F-F.F--F06#*&)F.\"\"% F-F3F-F-#F-\"#N7%\"#\")F)/*&F,F--F06#*(F3F-F.F--F06#*&F3F--F06#*&F3F-- F06#*&)F.\"\"#F-F3F-F-F-F-F-#F-\"%gL7%\"#()F)/*(F,F-F.F--F06#*(F3F-F.F --F06#*(F3F-F.F-F:F-F-F-#F-\"$%Q7%\"$-\"F)/*(F,F-F.F--F06#*&F3F--F06#* &F3F--F06#*&)F.\"\"$F-F3F-F-F-F-#F-\"$g*7%\"$1\"F)/*&F,F--F06#*&F3F--F 06#*&F3F-FCF-F-F-#F-\"%!o\"7%\"$8\"F)/*(F,F-FZF-F[pF-#F-\"$g\"7%\"$<\" F)/*&F,F--F06#*(F3F-F.F-FCF-F-#F-\"$!G7%\"$D\"F)/*(F,F-F.F--F06#*&)F. \"\"&F-F3F-F-#F-\"#[Q(pprint26\"" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}{SECT 1 {PARA 4 "" 0 " " {TEXT -1 20 "The associated trees" }}{PARA 0 "" 0 "" {TEXT -1 0 "" } }{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 461 "AST8 := AllSimpleTrees(8): \nwhch := [37,61,81,87,102,106,113,117,125]:\nm := 3: ## number of tre es per row\nnn := nops(whch): q := iquo(nn,m,'r'):\nfor i to nn do \n \+ p||i := DrawTree(AST8[whch[i]],height=4,width=2,show_ordercondition=t rue,\n scale=1.2,color=COLOR(RGB,0,.7,0),font_color=black);\nend do :\npp := plot([[1,1]],style=line,axes=none):\nplots[display](convert([ seq([p||((k-1)*m+1..m*k)],k=1..q),\n `if`(r>0,[p||(m*q+1..nn),pp$(m*( q+1)-nn)],NULL)],array));" }}{PARA 13 "" 1 "" {GLPLOT2D 852 1131 1131 {PLOTDATA 2 "6\\u-%%TEXTG6&7$$\"#g\"\"!$!+5`7'*o!\")Q/b~c~(a~c~)~=~~6 \"-%'COLOURG6&%$RGBGF)F)F)-%%FONTG6$%(COURIERG\"#5-F$6&7$F'$!+)o#)p\"o F,Q1~~~~~~~~~~~~~~~1F.F/F3-F$6&7$F'$!+dZrToF,Q2~~~~~~~~~~~~~~~__F.F/F3 -F$6&7$F'$!+++++qF,Q2~~~~~~~~~~~~~~~48F.F/F3-F$6&7$F'$!+g)*32oF,Q1~~~~ ~~~~~5~~~~~~F.F/F3-%'CURVESG6&7*7$F'$!+ " 0 "" {MPLTEXT 1 0 1 ";" }}}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT 260 4 "Note" }{TEXT -1 119 ": Th e 5 simplifying conditions can be replaced by the 5 extra order condit ions given (in abreviated form) as follows.. " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 174 "SO8 := Simp leOrderConditions(8):\n[seq([i,SO8[i]],i=[21,27,31,36,127])]:\nlinalg[ augment](linalg[delcols](%,2..2),matrix([[` `]$(linalg[rowdim](%))]), linalg[delcols](%,1..1));" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#K%'matrix G6#7'7%\"#@%#~~G/*&%\"bG\"\"\"-%!G6#*&%\"aGF--F/6#*&F2F--F/6#*&)%\"cG \"\"#F-F2F-F-F-F-#F-\"$g$7%\"#FF)/*&F,F--F/6#*&F2F--F/6#*&)F:\"\"$F-F2 F-F-F-#F-\"$?\"7%\"#JF)/*&F,F--F/6#*&)F:\"\"%F-F2F-F-#F-\"#I7%\"#OF)/* &F,F--F/6#*(F2F-F:F--F/6#*&F2F--F/6#*&F2F--F/6#*&F2F-F:F-F-F-F-F-#F-\" %357%\"$F\"F)/*&F,F--F/6#*&)F:\"\"'F-F2F-F-#F-\"#cQ(pprint36\"" }}} {PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 462 "SOeqs2 := [seq(add(a[i,j],j= 1..i-1)=c[i],i=9..12),op(StageOrderC onditions(2,9..12,'expanded')),\n op(StageOrderConditions(3,9..12,' expanded')),op(StageOrderConditions(4,9..12,'expanded'))]:\nSO8_12 := \+ SimpleOrderConditions(8,12,'expanded'):\nord_cdns := [seq(SO8_12[i],i= [37,61,81,87,102,113,106,117,125])]:\nsimp_eqs := [add(b[i]*a[i,1],i=1 +1..12)=b[1],seq(add(b[i]*a[i,j],i=j+1..12)=b[j]*(1-c[j]),j=[5,7,9,10] )]:\ncdns := [op(SOeqs2),op(ord_cdns),op(simp_eqs)]:" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 107 "e10 :=\{ seq(a[i,2]=0,i=9..12),seq(a[i,3]=0,i=9..12)\}:\ne11 := `union`(e9,e10) :\neqns2 := subs(e11,\{op(cdns)\}):" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 39 "There are 30 equations and 30 unknowns. " }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 36 "nops(eqns2);\nindets(eqn s2);\nnops(%);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#\"#I" }}{PARA 12 "" 1 "" {XPPMATH 20 "6#<@&%\"aG6$\"#5\"\"%&F%6$F'\"\"&&F%6$F'\"\"'&F%6$F' \"\"(&F%6$F'\"\")&F%6$F'\"\"*&F%6$\"#6\"\"\"&F%6$F:F(&F%6$F:F+&F%6$F:F .&F%6$F:F1&F%6$F:F4&F%6$F:F7&F%6$F:F'&F%6$\"#7F;&F%6$FLF(&F%6$FLF+&F%6 $FLF.&F%6$FLF1&F%6$FLF4&F%6$FLF7&F%6$FLF'&F%6$FLF:&F%6$F7F;&F%6$F7F(&F %6$F7F+&F%6$F7F.&F%6$F7F1&F%6$F7F4&F%6$F'F;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#\"#I" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 22 "infolevel[solve] := 4:" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 74 "e12 := solve(\{op(eqns2)\}):\ninfolevel[s olve] := 0:\ne13 := `union`(e11,e12):" }}}{PARA 0 "" 0 "" {TEXT -1 0 " " }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 4 "e13 " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 5357 "e13 := \{a[12,8] = 34568537955467705221 5495825476969226377187500/74771167436930077221667203179551347546362089 , a[6,1] = 5611/114400, c[9] = 7067558016280/7837150160667, a[5,1] = 4 202367/11628100, a[9,1] = -1472514264486215803881384708877264246346044 433307094207829051978044531801133057155/124689480162003200115705962164 3986024803301558393487900440453636168046069686436608, a[11,1] = -23426 59845814086836951207140065609179073838476242943917/1358480961351056777 022231400139158760857532162795520000, a[2,1] = 1/20, a[7,4] = 8602624/ 76559175, a[8,3] = 0, c[2] = 1/20, c[5] = 39/100, c[6] = 93/200, c[7] \+ = 31/200, c[8] = 943/1000, a[8,4] = -125/2, a[4,2] = 0, a[5,2] = 0, a[ 6,2] = 0, a[7,2] = 0, a[8,2] = 0, a[6,3] = 0, a[7,3] = 0, a[3,2] = 116 281/1024000, b[9] = 12274765470313196878428812037740635050319234276006 986398294443554969616342274215316330684448207141/489345147493715517650 3858341435109348888292806866096544828965267965233530521667572994528521 66040, a[9,3] = 0, a[10,3] = 0, a[11,3] = 0, a[12,3] = 0, a[9,2] = 0, \+ a[10,2] = 0, a[11,2] = 0, a[12,2] = 0, a[10,9] = -26984049294008425187 21166485087129798562269848229517793703413951226714583/4695456749139343 15077000442080871141884676035902717550325616728175875000000, a[9,8] = \+ 1033284818445201560405683676728665685912400779697066804644601577500000 0/13127035500360336480738342487407279145379720286389501652495827336793 93783, a[12,6] = 23463053885534042586562584734461844191547401725199495 75/256726716407895402892744978301151486254183185289662464, a[10,8] = 2 12083202434519082281842245535894/20022426044775672563822865371173879, \+ b[7] = 3847749490868980348119500000/15517045062138271618141237517, b[6 ] = 791638675191615279648100000/2235604725089973126411512319, a[9,5] = -12070679258469254807978936441733187949484571516120469966534514296406 891652614970375/272203115476165722171047818453110069949728408504838901 5085076961673446140398628096, b[1] = 44901867737754616851973/101404640 9980231013380680, c[11] = 47/50, c[12] = 1, b[2] = 0, b[3] = 0, b[4] = 0, b[5] = 0, a[12,4] = -16957088714171468676387054358954754000/143690 415119654683326368228101570221, b[10] = -9798363684577739445312500000/ 308722986341456031822630699, a[9,6] = 78012515584389364132309055253043 1036567795592568497182701460674803126770111481625/18311042541273197219 7889874507158786859226102980861859505241443073629143100805376, a[11,10 ] = -31155237437111730665923206875/392862141594230515010338956291, c[1 0] = 909/1000, a[3,1] = -7161/1024000, b[12] = -306814272936976936753/ 1299331183183744997286, a[4,1] = 1023/25600, a[11,5] = -26053085959256 534152588089363841/4377552804565683061011299942400, b[8] = -1373451243 2397741476562500000/875132892924995907746928783, a[8,5] = -10240306079 59889/168929280000000, a[8,7] = 6070139212132283/92502016000000, a[7,6 ] = 5611/283500, a[7,5] = -26782109/689364000, c[4] = 1023/6400, a[6,4 ] = 31744/135025, a[5,4] = 3982992/2907025, c[3] = 341/3200, a[5,3] = \+ -3899844/2907025, a[4,3] = 3069/25600, a[8,6] = 1501408353528689/26569 7280000000, a[6,5] = 923521/5106400, b[11] = 282035543183190840068750/ 12295407629873040425991, a[8,1] = -1221101821869329/690812928000000, a [7,1] = 21173/343200, a[12,1] = -2866556991825663971778295329101033887 534912787724034363/868226711619262703011213925016143612030669233795338 240, a[11,8] = 161021426143124178389075121929246710833125/109972077221 31034650667041364346422894371443, a[11,6] = 20980822345096760292224086 794978105312644533925634933539/377588999200755080387872783911549464197 2212962174156800, a[9,4] = -517229431108566845837517565524698123003902 5336933699114138315270772319372469280000/12461938100480914589727863057 1215298365257079410236252921850936749076487132995191, a[10,5] = -18026 9259803172281163724663224981097/38100922558256871086579832832000000, a [12,11] = -6122933601070769591613093993993358877250/105051700151023551 3198246721302027675953, a[10,7] = 318607235173649312405151265849660869 927653414425413/6714716715558965303132938072935465423910912000000, a[1 2,10] = 40279545832706233433100438588458933210937500/88964608427994828 46916972126377338947215101, a[10,4] = -2046274952459104910540336523906 9/454251913499893469596231268750, a[11,9] = 30076066976810251783423249 7565452434946672266195876496371874262392684852243925359864884962513/46 5544333750134645558506533660450560376082477961552128575189281031568049 2364106674524398280000, a[10,6] = 211276702141728028701282869920039408 10655221489/4679473877997892906145822697976708633673728000, a[11,4] = \+ -996286030132538159613930889652/16353068885996164905464325675, a[11,7] = 890722993756379186418929622095833835264322635782294899/139212420013 95112657501941955594013822830119803764736, a[9,7] = 664113122959911642 134782135839106469928140328160577035357155340392950009492511875/151784 6559858624813633302310729534917527976515008907830113994325301687782317 0816, a[12,9] = -32058909627170725427914343121527275340081027740232102 40571361570757249056167015230160352087048674542196011/9475695496839658 1478301512445127360498465774712725761537244920597319265730601723910349 1074738324033259120, a[10,1] = -29055573360337415088538618442231036441 314060511/22674759891089577691327962602370597632000000000, a[12,5] = - 4583493974484572912949314673356033540575/45195770365525074715731303427 0335135744, a[12,7] = 165712155931984680217128369091361069858625657348 4808662625/13431480411255146477259155104956093505361644432088109056\}: " }{TEXT -1 0 "" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> \+ " 0 "" {MPLTEXT 1 0 1 ";" }}}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 20 "printed coefficients" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 283 "subs(e13,matrix([seq([c[i],seq(a[i ,j],j=1..i-1),``$(8-i)],i=2..8)]));\nfor ii from 9 to 12 do\n print( ``);\n print(c[ii]=subs(e13,c[ii])); \n for jj to ii-1 do\n p rint(a[ii,jj]=subs(e13,a[ii,jj]));\n end do:\nend do:\n``;\nfor ii t o 12 do\n print(b[ii]=subs(e13,b[ii]));\nend do:" }}{PARA 11 "" 1 " " {XPPMATH 20 "6#K%'matrixG6#7)7*#\"\"\"\"#?F(%!GF+F+F+F+F+7*#\"$T$\"% +K#!%hr\"(+S-\"#\"'\"G;\"F2F+F+F+F+F+7*#\"%B5\"%+k#F7\"&+c#\"\"!#\"%pI F:F+F+F+F+7*#\"#R\"$+\"#\"(nB?%\")+\"G;\"F;#!(W)**Q\"(Dq!H#\"(#*H)RFGF +F+F+7*#\"#$*\"$+##\"%6c\"'+W6F;F;#\"&W<$\"'D]8#\"'@N#*\"(+k5&F+F+7*# \"#JFM#\"&t6#\"'+KMF;F;#\"(CEg)\")v\"fl(#!)4@yE\"*+SO*o#FO\"'+NGF+7*# \"$V*\"%+5#!1H$p=#=5@7\"0+++GH\"3pF;F;#!$D\"\"\"##!1*))fzgIS-\"\"0+++! GH*o\"#\"1*oGNN39]\"\"0+++!G(pl##\"1$GK@@R,2'\"/+++;?]#*Q(pprint46\"" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%!G" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/&%\"cG6#\"\"*#\".!G;!ev1(\".n1;]r$y" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/&%\"aG6$\"\"*\"\"\"#!]pbr0L6!=`W!y>0Hy?%42LVWgMYUEx)3ZQ\")Q!e@' [kU^s9\"]p3mV'opg/ohj`/W+z[$Re:I.[-')Rk@'fq:,?.?;![*oC\"" }}{PARA 11 " " 1 "" {XPPMATH 20 "6#/&%\"aG6$\"\"*\"\"#\"\"!" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/&%\"aG6$\"\"*\"\"$\"\"!" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/&%\"aG6$\"\"*\"\"%#!]p++GpCP>Bxq_JQT6*pLpLD!R+B\")pCbc&*H8([w!\\n$4&=#HDO-TzqDl$)H:7dI'ys*e94[+\"Q>Y7" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/&%\"aG6$\"\"*\"\"&#!^pv.(\\h_;*oS'H9X`m*p/7 ;:d%[\\z=Ls>t7aU5J=" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/&%\"aG6 $\"\"*\"\"(#\"\\pv=^#\\4+&HRS`:d`.x0;G.9G*pk5Re8#yM@k6*fH78Tm\"[p;3Da%" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/&%\"aG6$\"#5\"\"&#!E(4\")\\AjYsj6Gs J!)f#p-=\"D+++KG$)zl3roDeD#45Q" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/&% \"aG6$\"#5\"\"'#\"P*[@_l53%R+#*pGG,(G!G<9-nF6#\"O+!GPnL'3n(zpAe91H*y*z (QZzY" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/&%\"aG6$\"#5\"\"(#\"T8aU9MlF *p3m\\eE^^S7$\\OXV-K37#\"DzQvcKka!\\;'*f))oIN;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/&%\"aG6$\"#6\"\"&#!ATQO*3)e_T`c#ff3`g#\"@+C%**H651$olX!GbxV" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#/&%\"aG6$\"#6\"\"'#\"YRN$\\jDR`WEJ0\"y \\z'3CAHgn4XB#3)4#\"X+o:u@'H@s>k%\\:\"RysyQ!3b2?***)ex$" }}{PARA 11 " " 1 "" {XPPMATH 20 "6#/&%\"aG6$\"#6\"\"(#\"W**[H#yNEKk_$Q$e4A'H*=k=zjv $*Hs!*)\"VOZw.)>,$G#Q,%fb>%>]dE6&R,?C@R\"" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/&%\"aG6$\"#6\"\")#\"KDJ$3rY#H>7v!*QyT7VhU@5;\"MV9P%*GU YVOTqm]Y.J@s2s*4\"" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/&%\"aG6$\"#6\" \"*#\"hp8D'\\)[')f`#RC_[o#RiU(=P'\\we>mAnY\\V_ac(\\KU$y^-\"o(p1w+$\"ip ++G)RCXn1TO#\\!o:.\"G*=v&G@bhzZ#3w.c]/mLl]ebkM,vLVal%" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/&%\"aG6$\"#6\"#5#!>vo?BfmI<6PuBb6$\"?\"Hc*Q.,:0B% fT@'GR" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%!G" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/&%\"cG6#\"#7\"\"\"" }}{PARA 11 "" 1 "" {XPPMATH 20 "6# /&%\"aG6$\"#7\"\"\"#!XjV.Cxy7\\`()Q.,\"H`Hy<(RmD=*pbmG\"WS#Q`zL#p1.7O9 ;]#R@6Iqi#>;rE#o)" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/&%\"aG6$\"#7\"\" #\"\"!" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/&%\"aG6$\"#7\"\"$\"\"!" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#/&%\"aG6$\"#7\"\"%#!G+Sva*eV0(Qw'o9<9( )3dp\"\"E@-d,\"G#ojK$oa'>^T!pV\"" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/& %\"aG6$\"#7\"\"&#!Iv0aLgNtYJ\\H\"Hd%[uR\\$e%\"HWd8N.FMIJdru]_l.x&>X" } }{PARA 11 "" 1 "" {XPPMATH 20 "6#/&%\"aG6$\"#7\"\"'#\"Xv&\\*>DW=Y MZeileUS`&)Q0jM#\"WkCm*G&=$=ai[^6Iy\\u#*GS&*yS;nsc#" }}{PARA 11 "" 1 " " {XPPMATH 20 "6#/&%\"aG6$\"#7\"\"(#\"enDEm3[[tlD'e)p5O\"4p$Gr@!o%)>$f :7d;\"Yc!4\")3KWkh`]$4c\\5b\"fsZY^D6/[JM\"" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/&%\"aG6$\"#7\"\")#\"N+v=xjAppZDe\\:A0xYbz`oX$\"M*3ijaZ 8bzJ?n;Ax+$pVn6xu" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/&%\"aG6$\"#7\"\" *#!eq6g>UXn[q3_.;I_,nh0\\svq:Or0C5K-uF53S`FF:7VV\"zUD2tf?\\CP:wDFrudY)\\gt7XC^,$y9e'Ro\\&pv%*" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#/&%\"aG6$\"#7\"#5#\"M+v$4@L*e%)eQ/5LMB 1F$eaz-%\"L,^@Z*QtPE@(p\"p%G[*zU3Y'*))" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/&%\"aG6$\"#7\"#6#!I]s()eL*R*R48;fp22,O$H7'\"I`fnF?I@nC)>8bB5:+< 00\"" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%!G" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/&%\"bG6#\"\"\"#\"8t>&ohaxtn=!\\%\":!o!Q85B!)*4k/95" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#/&%\"bG6#\"\"#\"\"!" }}{PARA 11 "" 1 " " {XPPMATH 20 "6#/&%\"bG6#\"\"$\"\"!" }}{PARA 11 "" 1 "" {XPPMATH 20 " 6#/&%\"bG6#\"\"%\"\"!" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/&%\"bG6#\"\" &\"\"!" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/&%\"bG6#\"\"'#\"<++5['z_h\" >v'Q;z\"=>B^6k7t**3DZgNA" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/&%\"bG6# \"\"(#\"=++]>\"[.)*o3\\\\x%Q\">++]ilZTxRKC^MP\"\"<$yGpu2f*\\#H*G8v)" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/&%\"bG6#\"\"*#\"]qTr?[WoIjJ:UFUjhp \\bVWH)R')p+wUB>.0N1uP?\")G%yo>8.ZlZF7\"[qSg;_GX*Hdn;_INBlzEl*G[a'4mo! GH)))[$4^VT$eQ]w^:P\\Z^M*[" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/&%\"bG6 #\"#5#!=++]7`WRxd%oj$)z*\"<*pIE#=.c9M')Hs3$" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/&%\"bG6#\"#6#\"9](o+%3>$=Vb.#G\"8\"*fUSI()HwS&H7" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#/&%\"bG6#\"#7#!6`n$p(p$HF9oI\"7'G(*\\u $=$=J$*H\"" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 " " {MPLTEXT 1 0 1 ";" }}}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}}{PARA 0 "" 0 "" {TEXT -1 52 "#----- ----------------------------------------------" }}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 43 "construction of the order 7 embedded scheme" }} {PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 4 "e13 " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 5357 "e13 := \{a [12,8] = 345685379554677052215495825476969226377187500/747711674369300 77221667203179551347546362089, a[6,1] = 5611/114400, c[9] = 7067558016 280/7837150160667, a[5,1] = 4202367/11628100, a[9,1] = -14725142644862 15803881384708877264246346044433307094207829051978044531801133057155/1 2468948016200320011570596216439860248033015583934879004404536361680460 69686436608, a[11,1] = -2342659845814086836951207140065609179073838476 242943917/1358480961351056777022231400139158760857532162795520000, a[2 ,1] = 1/20, a[7,4] = 8602624/76559175, a[8,3] = 0, c[2] = 1/20, c[5] = 39/100, c[6] = 93/200, c[7] = 31/200, c[8] = 943/1000, a[8,4] = -125/ 2, a[4,2] = 0, a[5,2] = 0, a[6,2] = 0, a[7,2] = 0, a[8,2] = 0, a[6,3] \+ = 0, a[7,3] = 0, a[3,2] = 116281/1024000, b[9] = 122747654703131968784 2881203774063505031923427600698639829444355496961634227421531633068444 8207141/48934514749371551765038583414351093488882928068660965448289652 6796523353052166757299452852166040, a[9,3] = 0, a[10,3] = 0, a[11,3] = 0, a[12,3] = 0, a[9,2] = 0, a[10,2] = 0, a[11,2] = 0, a[12,2] = 0, a[ 10,9] = -2698404929400842518721166485087129798562269848229517793703413 951226714583/469545674913934315077000442080871141884676035902717550325 616728175875000000, a[9,8] = 10332848184452015604056836767286656859124 007796970668046446015775000000/131270355003603364807383424874072791453 7972028638950165249582733679393783, a[12,6] = 234630538855340425865625 8473446184419154740172519949575/25672671640789540289274497830115148625 4183185289662464, a[10,8] = 212083202434519082281842245535894/20022426 044775672563822865371173879, b[7] = 3847749490868980348119500000/15517 045062138271618141237517, b[6] = 791638675191615279648100000/223560472 5089973126411512319, a[9,5] = -120706792584692548079789364417331879494 84571516120469966534514296406891652614970375/2722031154761657221710478 184531100699497284085048389015085076961673446140398628096, b[1] = 4490 1867737754616851973/1014046409980231013380680, c[11] = 47/50, c[12] = \+ 1, b[2] = 0, b[3] = 0, b[4] = 0, b[5] = 0, a[12,4] = -1695708871417146 8676387054358954754000/143690415119654683326368228101570221, b[10] = - 9798363684577739445312500000/308722986341456031822630699, a[9,6] = 780 1251558438936413230905525304310365677955925684971827014606748031267701 11481625/1831104254127319721978898745071587868592261029808618595052414 43073629143100805376, a[11,10] = -31155237437111730665923206875/392862 141594230515010338956291, c[10] = 909/1000, a[3,1] = -7161/1024000, b[ 12] = -306814272936976936753/1299331183183744997286, a[4,1] = 1023/256 00, a[11,5] = -26053085959256534152588089363841/4377552804565683061011 299942400, b[8] = -13734512432397741476562500000/875132892924995907746 928783, a[8,5] = -1024030607959889/168929280000000, a[8,7] = 607013921 2132283/92502016000000, a[7,6] = 5611/283500, a[7,5] = -26782109/68936 4000, c[4] = 1023/6400, a[6,4] = 31744/135025, a[5,4] = 3982992/290702 5, c[3] = 341/3200, a[5,3] = -3899844/2907025, a[4,3] = 3069/25600, a[ 8,6] = 1501408353528689/265697280000000, a[6,5] = 923521/5106400, b[11 ] = 282035543183190840068750/12295407629873040425991, a[8,1] = -122110 1821869329/690812928000000, a[7,1] = 21173/343200, a[12,1] = -28665569 91825663971778295329101033887534912787724034363/8682267116192627030112 13925016143612030669233795338240, a[11,8] = 16102142614312417838907512 1929246710833125/10997207722131034650667041364346422894371443, a[11,6] = 20980822345096760292224086794978105312644533925634933539/3775889992 007550803878727839115494641972212962174156800, a[9,4] = -5172294311085 668458375175655246981230039025336933699114138315270772319372469280000/ 1246193810048091458972786305712152983652570794102362529218509367490764 87132995191, a[10,5] = -180269259803172281163724663224981097/381009225 58256871086579832832000000, a[12,11] = -612293360107076959161309399399 3358877250/1050517001510235513198246721302027675953, a[10,7] = 3186072 35173649312405151265849660869927653414425413/6714716715558965303132938 072935465423910912000000, a[12,10] = 402795458327062334331004385884589 33210937500/8896460842799482846916972126377338947215101, a[10,4] = -20 462749524591049105403365239069/454251913499893469596231268750, a[11,9] = 3007606697681025178342324975654524349466722661958764963718742623926 84852243925359864884962513/4655443337501346455585065336604505603760824 779615521285751892810315680492364106674524398280000, a[10,6] = 2112767 0214172802870128286992003940810655221489/46794738779978929061458226979 76708633673728000, a[11,4] = -996286030132538159613930889652/163530688 85996164905464325675, a[11,7] = 89072299375637918641892962209583383526 4322635782294899/13921242001395112657501941955594013822830119803764736 , a[9,7] = 66411312295991164213478213583910646992814032816057703535715 5340392950009492511875/15178465598586248136333023107295349175279765150 089078301139943253016877823170816, a[12,9] = -320589096271707254279143 4312152727534008102774023210240571361570757249056167015230160352087048 674542196011/947569549683965814783015124451273604984657747127257615372 449205973192657306017239103491074738324033259120, a[10,1] = -290555733 60337415088538618442231036441314060511/2267475989108957769132796260237 0597632000000000, a[12,5] = -4583493974484572912949314673356033540575/ 451957703655250747157313034270335135744, a[12,7] = 1657121559319846802 171283690913610698586256573484808662625/134314804112551464772591551049 56093505361644432088109056\}:" }{TEXT -1 0 "" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 61 "We can obtain a n embedded 11 stage order 7 scheme as follows." }}{PARA 0 "" 0 "" {TEXT -1 94 "We remove stages 11 and 12 from the 12 stage order 8 sche me and introduce a new stage 11 with " }{XPPEDIT 18 0 "c[11] = 1;" "6# /&%\"cG6#\"#6\"\"\"" }{TEXT -1 3 ", " }{XPPEDIT 18 0 "a[11,2] = 0" "6 #/&%\"aG6$\"#6\"\"#\"\"!" }{TEXT -1 3 ", " }{XPPEDIT 18 0 "a[11,3] = \+ 0;" "6#/&%\"aG6$\"#6\"\"$\"\"!" }{TEXT -1 3 ", " }{XPPEDIT 18 0 "`b*` [2]=0" "6#/&%#b*G6#\"\"#\"\"!" }{TEXT -1 3 ", " }{XPPEDIT 18 0 "`b*`[ 3]=0" "6#/&%#b*G6#\"\"$\"\"!" }{TEXT -1 3 ", " }{XPPEDIT 18 0 "`b*`[4 ]=0" "6#/&%#b*G6#\"\"%\"\"!" }{TEXT -1 7 " and " }{XPPEDIT 18 0 "`b* `[5]=0" "6#/&%#b*G6#\"\"&\"\"!" }{TEXT -1 58 ", where the weights of t he order 7 scheme are denoted by " }{XPPEDIT 18 0 "`b*`" "6#%#b*G" } {TEXT -1 1 "." }}{PARA 0 "" 0 "" {TEXT -1 1 " " }}{PARA 0 "" 0 "" {TEXT -1 156 "We incorporate the order 7 quadrature conditions, the ro w sum conditions for this stage and stage-order conditions so that thi s new stage has stage-order 4." }}{PARA 0 "" 0 "" {TEXT -1 48 "We also incorporate the simplifying conditions: " }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "Sum(`b*`[i]*a[i,j],i=j+1..11)=`b*`[j]*(1-c[j] )" "6#/-%$SumG6$*&&%#b*G6#%\"iG\"\"\"&%\"aG6$F+%\"jGF,/F+;,&F0F,F,F,\" #6*&&F)6#F0F,,&F,F,&%\"cG6#F0!\"\"F," }{TEXT -1 1 "," }}{PARA 0 "" 0 " " {TEXT -1 5 "for " }{XPPEDIT 18 0 "j=1" "6#/%\"jG\"\"\"" }{TEXT -1 10 " ( with " }{XPPEDIT 18 0 "c[1]=0" "6#/&%\"cG6#\"\"\"\"\"!" } {TEXT -1 5 " ), " }{XPPEDIT 18 0 "j = 8;" "6#/%\"jG\"\")" }{TEXT -1 3 ", " }{XPPEDIT 18 0 "j=9" "6#/%\"jG\"\"*" }{TEXT -1 7 " and " } {XPPEDIT 18 0 "j=10" "6#/%\"jG\"#5" }{TEXT -1 2 ". " }}{PARA 0 "" 0 " " {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 399 "`Qeqs*` : = subs(b=`b*`,QuadratureConditions(7,11,'expanded')):\nSO_eqs := [add( a[11,j],j=1..10)=c[11],add(a[11,j]*c[j],j=2..10)=1/2*c[11]^2,\n \+ add(a[11,j]*c[j]^2,j=2..10)=1/3*c[11]^3,add(a[11,j]*c[j]^3,j=2..10)=1 /4*c[11]^4]:\n`simp_eqs*` := [add(`b*`[i]*a[i,1],i=2..11)=`b*`[1],seq( add(`b*`[i]*a[i,j],i=j+1..11)=`b*`[j]*(1-c[j]),j=[8,9,10])]:\n`cdns*` \+ := [op(`simp_eqs*`),op(SO_eqs),op(`Qeqs*`)]:" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 144 "e14 := `uni on`(remove(u_->member(op(1,lhs(u_)),[11,12]) or op(0,lhs(u_))=b,e13), \n \{c[11]=1,a[11,2]=0,a[11,3]=0,seq(`b*`[i]=0,i=2..5)\}):\n" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 53 "We hav e 15 equations for the 15 unknown coefficients." }}{EXCHG {PARA 0 "> \+ " 0 "" {MPLTEXT 1 0 60 "eqns3 := subs(e14,`cdns*`):\nnops(%);\nindets( eqns3);\nnops(%);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#\"#:" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#<1&%#b*G6#\"#6&F%6#\"\")&F%6#\"\"*&F%6#\"#5&F%6 #\"\"'&F%6#\"\"(&F%6#\"\"\"&%\"aG6$F'F9&F;6$F'\"\"%&F;6$F'\"\"&&F;6$F' F3&F;6$F'F6&F;6$F'F*&F;6$F'F-&F;6$F'F0" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#\"#:" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 " " {MPLTEXT 1 0 22 "infolevel[solve] := 4:" }}}{EXCHG {PARA 0 "> " 0 " " {MPLTEXT 1 0 74 "e15 := solve(\{op(eqns3)\}):\ninfolevel[solve] := 0 :\ne16 := `union`(e14,e15):" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}{SECT 1 {PARA 4 "" 0 " " {TEXT -1 3 "e16" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 4201 "e16 := \{a[6,1] = 5611/114400, c[9] = 70675 58016280/7837150160667, a[5,1] = 4202367/11628100, a[9,1] = -147251426 4486215803881384708877264246346044433307094207829051978044531801133057 155/124689480162003200115705962164398602480330155839348790044045363616 8046069686436608, c[11] = 1, a[2,1] = 1/20, a[7,4] = 8602624/76559175, `b*`[2] = 0, `b*`[4] = 0, `b*`[5] = 0, `b*`[3] = 0, a[8,3] = 0, c[2] \+ = 1/20, c[5] = 39/100, c[6] = 93/200, c[7] = 31/200, c[8] = 943/1000, \+ a[8,4] = -125/2, a[4,2] = 0, a[5,2] = 0, a[6,2] = 0, a[7,2] = 0, a[8,2 ] = 0, a[6,3] = 0, a[7,3] = 0, a[3,2] = 116281/1024000, a[11,9] = -175 6005520307450928195422767042525091954178296002788308926563193523662404 739779789732685671/348767814578469983605688098046186480904607278021030 540735333862087061574934154942830062320, a[9,3] = 0, a[10,3] = 0, a[11 ,3] = 0, a[9,2] = 0, a[10,2] = 0, a[11,2] = 0, a[10,9] = -269840492940 0842518721166485087129798562269848229517793703413951226714583/46954567 4913934315077000442080871141884676035902717550325616728175875000000, a [9,8] = 10332848184452015604056836767286656859124007796970668046446015 775000000/131270355003603364807383424874072791453797202863895016524958 2733679393783, a[10,8] = 212083202434519082281842245535894/20022426044 775672563822865371173879, a[9,5] = -1207067925846925480797893644173318 7949484571516120469966534514296406891652614970375/27220311547616572217 10478184531100699497284085048389015085076961673446140398628096, `b*`[1 1] = -28735456870978964189/79783493704265043693, a[11,4] = -4411194916 804718600478400319122931000/40373053902469967450761491269633019, a[9,6 ] = 780125155843893641323090552530431036567795592568497182701460674803 126770111481625/183110425412731972197889874507158786859226102980861859 505241443073629143100805376, c[10] = 909/1000, a[11,6] = 1351375197570 54679098042184152749677761254751865630525/1602958779448628959777132636 1911895112703716593983488, a[3,1] = -7161/1024000, a[4,1] = 1023/25600 , a[11,8] = -6748865855011993037732355335815350667265625/7002880395717 424621213565406715087764770357, `b*`[7] = 73487947527027243487625000/2 96504045773342769773399443, a[8,5] = -1024030607959889/168929280000000 , a[8,7] = 6070139212132283/92502016000000, a[7,6] = 5611/283500, a[7, 5] = -26782109/689364000, c[4] = 1023/6400, a[6,4] = 31744/135025, a[5 ,4] = 3982992/2907025, c[3] = 341/3200, a[5,3] = -3899844/2907025, a[4 ,3] = 3069/25600, a[8,6] = 1501408353528689/265697280000000, a[6,5] = \+ 923521/5106400, a[8,1] = -1221101821869329/690812928000000, a[7,1] = 2 1173/343200, a[11,5] = -16734711409449292534539422531728520225/1801243 715290088669307203927210237952, `b*`[1] = 10835401739407019406577/2445 21829356935137978320, `b*`[9] = 22060647948996678611017711379974578860 522018208949721559448560203338437626022142776381/111154200926232587451 2959185795727215759010577565736079641376621381577236680929558640, `b*` [6] = 13908189778321895491375000/39221135527894265375640567, a[11,10] \+ = 53381024589235611084013897674181629296875/89593575847956945248749695 98508592944141, `b*`[10] = -547971229495642458203125000/23237214025700 991642563601, a[11,7] = 3893756836740987601254855190349219613792971043 1584875/340956454090191606099548798001469306974758443147264, a[9,4] = \+ -517229431108566845837517565524698123003902533693369911413831527077231 9372469280000/12461938100480914589727863057121529836525707941023625292 1850936749076487132995191, `b*`[8] = 68293140641257649609375000/153532 08647806945749946119, a[10,5] = -180269259803172281163724663224981097/ 38100922558256871086579832832000000, a[11,1] = -6186759055354825006728 00859344538410358660153899637/2035442821182140471001194753406676848742 92102389760, a[10,7] = 31860723517364931240515126584966086992765341442 5413/6714716715558965303132938072935465423910912000000, a[10,4] = -204 62749524591049105403365239069/454251913499893469596231268750, a[10,6] \+ = 21127670214172802870128286992003940810655221489/46794738779978929061 45822697976708633673728000, a[9,7] = 664113122959911642134782135839106 469928140328160577035357155340392950009492511875/151784655985862481363 33023107295349175279765150089078301139943253016877823170816, a[10,1] = -29055573360337415088538618442231036441314060511/22674759891089577691 327962602370597632000000000\}:" }{TEXT -1 0 "" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 20 "printed coefficients" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 289 "s ubs(e16,matrix([seq([c[i],seq(a[i,j],j=1..i-1),``$(8-i)],i=2..8)]));\n for ii from 9 to 11 do\n print(``);\n print(c[ii]=subs(e16,c[ii])) ; \n for jj to ii-1 do\n print(a[ii,jj]=subs(e16,a[ii,jj]));\n \+ end do:\nend do:\n``;\nfor ii to 11 do\n print(`b*`[ii]=subs(e16,` b*`[ii]));\nend do:" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#K%'matrixG6#7)7 *#\"\"\"\"#?F(%!GF+F+F+F+F+7*#\"$T$\"%+K#!%hr\"(+S-\"#\"'\"G;\"F2F+F+F +F+F+7*#\"%B5\"%+k#F7\"&+c#\"\"!#\"%pIF:F+F+F+F+7*#\"#R\"$+\"#\"(nB?% \")+\"G;\"F;#!(W)**Q\"(Dq!H#\"(#*H)RFGF+F+F+7*#\"#$*\"$+##\"%6c\"'+W6F ;F;#\"&W<$\"'D]8#\"'@N#*\"(+k5&F+F+7*#\"#JFM#\"&t6#\"'+KMF;F;#\"(CEg) \")v\"fl(#!)4@yE\"*+SO*o#FO\"'+NGF+7*#\"$V*\"%+5#!1H$p=#=5@7\"0+++GH\" 3pF;F;#!$D\"\"\"##!1*))fzgIS-\"\"0+++!GH*o\"#\"1*oGNN39]\"\"0+++!G(pl# #\"1$GK@@R,2'\"/+++;?]#*Q(pprint56\"" }}{PARA 11 "" 1 "" {XPPMATH 20 " 6#%!G" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/&%\"cG6#\"\"*#\".!G;!ev1(\". n1;]r$y" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/&%\"aG6$\"\"*\"\"\"#!]pbr0 L6!=`W!y>0Hy?%42LVWgMYUEx)3ZQ\")Q!e@'[kU^s9\"]p3mV'opg/ohj`/W+z[$Re:I. [-')Rk@'fq:,?.?;![*oC\"" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/&%\"aG6$\" \"*\"\"#\"\"!" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/&%\"aG6$\"\"*\"\"$\" \"!" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/&%\"aG6$\"\"*\"\"%#!]p++GpCP>B xq_JQT6*pLpLD!R+B\")pCbc&*H8([w!\\n$4&=#HDO-TzqDl $)H:7dI'ys*e94[+\"Q>Y7" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/&%\"aG6$\" \"*\"\"&#!^pv.(\\h_;*oS'H9X`m*p/7;:d%[\\z=LDa%" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/&% \"aG6$\"#5\"\"&#!E(4\")\\AjYsj6GsJ!)f#p-=\"D+++KG$)zl3roDeD#45Q" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#/&%\"aG6$\"#5\"\"'#\"P*[@_l53%R+#*pGG, (G!G<9-nF6#\"O+!GPnL'3n(zpAe91H*y*z(QZzY" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/&%\"aG6$\"#5\"\"(#\"T8aU9MlF*p3m\\eE^^S7$\\OXV-K37#\"DzQv8N\"\"V)[$)Rf;Pq7^ *=\">OE8x(f*G'[%z(eHg\"" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/&%\"aG6$\" #6\"\"(#\"Vv[eJ/rHz8'>#\\.>b[D,w)4uOov$*Q\"Tks9V%eZ(pIp9+)z[&*41;>!4ak &4M" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/&%\"aG6$\"#6\"\")#!LDcEn1N:eLb BtPI*>,be')[n\"Ld.xkx3:nSlN@@YUjl#*3$)y-gHyT&>4DD/nFU&>G 4X2._0gv\"\"ep?B1IG%\\:M\\dhq3iQLN2aI5-ysg/4['=Y!)4)o0O)*p%yX\"yw[$" } }{PARA 11 "" 1 "" {XPPMATH 20 "6#/&%\"aG6$\"#6\"#5#\"JvoHH;=uw*Q,%36cB *eC5Q`\"ITT%Hf3&)fp\\([_%p&z%ed$f*)" }}{PARA 11 "" 1 "" {XPPMATH 20 "6 #%!G" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/&%#b*G6#\"\"\"#\"8xlS>qSRh%*\\d%p!yk3KN:" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/&%#b*G6#\"\"*# \"ap\"QwF9AgiP%QL?g&[%f:s\\*3#=?_g)yX(*z8r<5hym**[zkg?#\"`pS'e&H4oOsd \"Q@mPT'zgtlvd5!fd@Fdz&=fH^ueKi#4?a66" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/&%#b*G6#\"#5#!<+]7.#eCk&\\H7(za\";,OcU;*4qDS@PK#" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/&%#b*G6#\"#6#!5*=k*y4(oXN(G\"5$pV]E/P\\$yz" }}} {PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 104 "`RK7_11eqs*` := subs(b=`b*`,[op(RowSumConditions(11, 'expanded')),op(OrderConditions(7,11,'expanded'))]):" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 5 "Check" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 57 "subs(e16,`RK7_11eqs*`):\nmap(u->lhs(u)-rh s(u),%);\nnops(%);" }}{PARA 12 "" 1 "" {XPPMATH 20 "6#7[q\"\"!F$F$F$F$ F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$ F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$ F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#\"#&*" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 93 "We can calculate the principal error norm, that is, the 2-norm \+ of the principal error terms. " }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 185 "`errterms7_11*` := subs(b=`b*`,PrincipalErrorTerms(7,11,'expand ed')):\nsm := 0:\nfor ct to nops(`errterms7_11*`) do\n sm := sm+(eva lf(subs(e16,`errterms7_11*`[ct])))^2;\nend do:\nsqrt(sm);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#$\"+PY3Y$)!#:" }}}{PARA 0 "" 0 "" {TEXT -1 0 " " }}{PARA 0 "" 0 "" {TEXT -1 138 "We can include the new stage for the embedded scheme as an additional 13th stage added to the order 8 sche me along with the coefficients " }{XPPEDIT 18 0 "a[13,11] = 0;" "6#/& %\"aG6$\"#8\"#6\"\"!" }{TEXT -1 7 " and " }{XPPEDIT 18 0 "a[13,12] = 0;" "6#/&%\"aG6$\"#8\"#7\"\"!" }{TEXT -1 1 "." }}{PARA 0 "" 0 "" {TEXT -1 13 "The weights " }{XPPEDIT 18 0 "`b*`[i]" "6#&%#b*G6#%\"iG " }{TEXT -1 7 " for " }{XPPEDIT 18 0 "i=1" "6#/%\"iG\"\"\"" }{TEXT -1 89 " . . 10 of the 13 stage combined scheme are those of the 11 sta ge scheme and the weight " }{XPPEDIT 18 0 "`b*`[11];" "6#&%#b*G6#\"#6 " }{TEXT -1 34 " in the 11 stage scheme becomes " }{XPPEDIT 18 0 "`b *`[13];" "6#&%#b*G6#\"#8" }{TEXT -1 25 " in the 13 stage scheme." }} {PARA 0 "" 0 "" {TEXT -1 8 "We set " }{XPPEDIT 18 0 "c[13] = 1;" "6#/ &%\"cG6#\"#8\"\"\"" }{TEXT -1 3 ", " }{XPPEDIT 18 0 "`b*`[11] = 0;" " 6#/&%#b*G6#\"#6\"\"!" }{TEXT -1 7 " and " }{XPPEDIT 18 0 "`b*`[12] = 0;" "6#/&%#b*G6#\"#7\"\"!" }{TEXT -1 1 "." }}{PARA 0 "" 0 "" {TEXT -1 66 "We can make the order 8 scheme into a 13 stage scheme by settin g " }{XPPEDIT 18 0 "b[13] = 0;" "6#/&%\"bG6#\"#8\"\"!" }{TEXT -1 1 ". " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 201 "e17 := \{c[13]=1,seq(a[13,i]=subs(e16,a[11,i]),i=1..10),a[13, 11]=0,a[13,12]=0,b[13]=0,\nseq(`b*`[i]=subs(e16,`b*`[i]),i=1..10),`b*` [11]=0,`b*`[12]=0,`b*`[13]=subs(e16,`b*`[11])\}:\ne18 := `union`(e13,e 17):" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 3 "e18" }} {PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 6994 "e18 := \{a[12,8] = 345685379554677052215495825476969226377187500 /74771167436930077221667203179551347546362089, a[6,1] = 5611/114400, c [9] = 7067558016280/7837150160667, a[5,1] = 4202367/11628100, a[9,1] = -14725142644862158038813847088772642463460444333070942078290519780445 31801133057155/1246894801620032001157059621643986024803301558393487900 440453636168046069686436608, a[11,1] = -234265984581408683695120714006 5609179073838476242943917/13584809613510567770222314001391587608575321 62795520000, a[2,1] = 1/20, a[7,4] = 8602624/76559175, `b*`[2] = 0, `b *`[4] = 0, `b*`[5] = 0, `b*`[3] = 0, a[8,3] = 0, a[13,1] = -6186759055 35482500672800859344538410358660153899637/2035442821182140471001194753 40667684874292102389760, c[2] = 1/20, c[5] = 39/100, c[6] = 93/200, c[ 7] = 31/200, c[8] = 943/1000, a[8,4] = -125/2, a[4,2] = 0, a[5,2] = 0, a[6,2] = 0, a[7,2] = 0, a[8,2] = 0, a[6,3] = 0, a[7,3] = 0, a[3,2] = \+ 116281/1024000, b[9] = 12274765470313196878428812037740635050319234276 006986398294443554969616342274215316330684448207141/489345147493715517 6503858341435109348888292806866096544828965267965233530521667572994528 52166040, a[9,3] = 0, a[10,3] = 0, a[11,3] = 0, a[12,3] = 0, a[9,2] = \+ 0, a[10,2] = 0, a[11,2] = 0, a[12,2] = 0, a[10,9] = -26984049294008425 18721166485087129798562269848229517793703413951226714583/4695456749139 34315077000442080871141884676035902717550325616728175875000000, a[9,8] = 1033284818445201560405683676728665685912400779697066804644601577500 0000/13127035500360336480738342487407279145379720286389501652495827336 79393783, a[12,6] = 23463053885534042586562584734461844191547401725199 49575/256726716407895402892744978301151486254183185289662464, a[10,8] \+ = 212083202434519082281842245535894/2002242604477567256382286537117387 9, b[7] = 3847749490868980348119500000/15517045062138271618141237517, \+ b[6] = 791638675191615279648100000/2235604725089973126411512319, a[9,5 ] = -12070679258469254807978936441733187949484571516120469966534514296 406891652614970375/272203115476165722171047818453110069949728408504838 9015085076961673446140398628096, `b*`[11] = 0, a[13,5] = -167347114094 49292534539422531728520225/1801243715290088669307203927210237952, b[1] = 44901867737754616851973/1014046409980231013380680, c[11] = 47/50, c [12] = 1, b[2] = 0, b[3] = 0, b[4] = 0, b[5] = 0, a[12,4] = -169570887 14171468676387054358954754000/143690415119654683326368228101570221, b[ 10] = -9798363684577739445312500000/308722986341456031822630699, a[9,6 ] = 780125155843893641323090552530431036567795592568497182701460674803 126770111481625/183110425412731972197889874507158786859226102980861859 505241443073629143100805376, a[11,10] = -31155237437111730665923206875 /392862141594230515010338956291, `b*`[12] = 0, c[10] = 909/1000, a[13, 12] = 0, a[3,1] = -7161/1024000, a[13,11] = 0, a[13,10] = 533810245892 35611084013897674181629296875/8959357584795694524874969598508592944141 , b[12] = -306814272936976936753/1299331183183744997286, a[4,1] = 1023 /25600, a[11,5] = -26053085959256534152588089363841/437755280456568306 1011299942400, b[8] = -13734512432397741476562500000/87513289292499590 7746928783, `b*`[7] = 73487947527027243487625000/296504045773342769773 399443, a[13,8] = -6748865855011993037732355335815350667265625/7002880 395717424621213565406715087764770357, a[8,5] = -1024030607959889/16892 9280000000, a[8,7] = 6070139212132283/92502016000000, a[7,6] = 5611/28 3500, a[7,5] = -26782109/689364000, c[4] = 1023/6400, a[6,4] = 31744/1 35025, a[5,4] = 3982992/2907025, c[3] = 341/3200, a[5,3] = -3899844/29 07025, a[4,3] = 3069/25600, a[13,9] = -1756005520307450928195422767042 525091954178296002788308926563193523662404739779789732685671/348767814 5784699836056880980461864809046072780210305407353338620870615749341549 42830062320, a[8,6] = 1501408353528689/265697280000000, a[6,5] = 92352 1/5106400, b[11] = 282035543183190840068750/12295407629873040425991, a [8,1] = -1221101821869329/690812928000000, a[7,1] = 21173/343200, a[13 ,3] = 0, a[13,2] = 0, `b*`[1] = 10835401739407019406577/24452182935693 5137978320, `b*`[9] = 220606479489966786110177113799745788605220182089 49721559448560203338437626022142776381/1111542009262325874512959185795 727215759010577565736079641376621381577236680929558640, a[12,1] = -286 6556991825663971778295329101033887534912787724034363/86822671161926270 3011213925016143612030669233795338240, a[11,8] = 161021426143124178389 075121929246710833125/10997207722131034650667041364346422894371443, a[ 13,7] = 38937568367409876012548551903492196137929710431584875/34095645 4090191606099548798001469306974758443147264, a[13,6] = 135137519757054 679098042184152749677761254751865630525/160295877944862895977713263619 11895112703716593983488, b[13] = 0, c[13] = 1, `b*`[6] = 1390818977832 1895491375000/39221135527894265375640567, `b*`[10] = -5479712294956424 58203125000/23237214025700991642563601, a[11,6] = 20980822345096760292 224086794978105312644533925634933539/377588999200755080387872783911549 4641972212962174156800, a[9,4] = -517229431108566845837517565524698123 0039025336933699114138315270772319372469280000/12461938100480914589727 8630571215298365257079410236252921850936749076487132995191, a[13,4] = \+ -4411194916804718600478400319122931000/4037305390246996745076149126963 3019, `b*`[8] = 68293140641257649609375000/15353208647806945749946119, `b*`[13] = -28735456870978964189/79783493704265043693, a[10,5] = -180 269259803172281163724663224981097/38100922558256871086579832832000000, a[12,11] = -6122933601070769591613093993993358877250/1050517001510235 513198246721302027675953, a[10,7] = 3186072351736493124051512658496608 69927653414425413/6714716715558965303132938072935465423910912000000, a [12,10] = 40279545832706233433100438588458933210937500/889646084279948 2846916972126377338947215101, a[10,4] = -20462749524591049105403365239 069/454251913499893469596231268750, a[11,9] = 300760669768102517834232 497565452434946672266195876496371874262392684852243925359864884962513/ 4655443337501346455585065336604505603760824779615521285751892810315680 492364106674524398280000, a[10,6] = 2112767021417280287012828699200394 0810655221489/4679473877997892906145822697976708633673728000, a[11,4] \+ = -996286030132538159613930889652/16353068885996164905464325675, a[11, 7] = 890722993756379186418929622095833835264322635782294899/1392124200 1395112657501941955594013822830119803764736, a[9,7] = 6641131229599116 42134782135839106469928140328160577035357155340392950009492511875/1517 8465598586248136333023107295349175279765150089078301139943253016877823 170816, a[12,9] = -320589096271707254279143431215272753400810277402321 0240571361570757249056167015230160352087048674542196011/94756954968396 5814783015124451273604984657747127257615372449205973192657306017239103 491074738324033259120, a[10,1] = -290555733603374150885386184422310364 41314060511/22674759891089577691327962602370597632000000000, a[12,5] = -4583493974484572912949314673356033540575/451957703655250747157313034 270335135744, a[12,7] = 1657121559319846802171283690913610698586256573 484808662625/13431480411255146477259155104956093505361644432088109056 \}:" }{TEXT -1 0 "" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}}{PARA 0 "" 0 "" {TEXT -1 0 "" }} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}{SECT 1 {PARA 4 "" 0 " " {TEXT -1 20 "printed coefficients" }}{PARA 0 "" 0 "" {TEXT -1 0 "" } }{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 362 "subs(e18,matrix([seq([c[i] ,seq(a[i,j],j=1..i-1),``$(8-i)],i=2..8)]));\nfor ii from 9 to 13 do\n \+ print(``);\n print(c[ii]=subs(e18,c[ii])); \n for jj to ii-1 do \n print(a[ii,jj]=subs(e18,a[ii,jj]));\n end do:\nend do:print( ``);\nfor ii to 13 do\n print(b[ii]=subs(e18,b[ii]));\nend do:print( ``);\nfor ii to 13 do\n print(`b*`[ii]=subs(e18,`b*`[ii]));\nend do: " }}{PARA 11 "" 1 "" {XPPMATH 20 "6#K%'matrixG6#7)7*#\"\"\"\"#?F(%!GF+ F+F+F+F+7*#\"$T$\"%+K#!%hr\"(+S-\"#\"'\"G;\"F2F+F+F+F+F+7*#\"%B5\"%+k# F7\"&+c#\"\"!#\"%pIF:F+F+F+F+7*#\"#R\"$+\"#\"(nB?%\")+\"G;\"F;#!(W)**Q \"(Dq!H#\"(#*H)RFGF+F+F+7*#\"#$*\"$+##\"%6c\"'+W6F;F;#\"&W<$\"'D]8#\"' @N#*\"(+k5&F+F+7*#\"#JFM#\"&t6#\"'+KMF;F;#\"(CEg)\")v\"fl(#!)4@yE\"*+S O*o#FO\"'+NGF+7*#\"$V*\"%+5#!1H$p=#=5@7\"0+++GH\"3pF;F;#!$D\"\"\"##!1* ))fzgIS-\"\"0+++!GH*o\"#\"1*oGNN39]\"\"0+++!G(pl##\"1$GK@@R,2'\"/+++;? ]#*Q(pprint66\"" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%!G" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/&%\"cG6#\"\"*#\".!G;!ev1(\".n1;]r$y" }}{PARA 11 " " 1 "" {XPPMATH 20 "6#/&%\"aG6$\"\"*\"\"\"#!]pbr0L6!=`W!y>0Hy?%42LVWgM YUEx)3ZQ\")Q!e@'[kU^s9\"]p3mV'opg/ohj`/W+z[$Re:I.[-')Rk@'fq:,?.?;![*oC \"" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/&%\"aG6$\"\"*\"\"#\"\"!" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#/&%\"aG6$\"\"*\"\"$\"\"!" }}{PARA 11 " " 1 "" {XPPMATH 20 "6#/&%\"aG6$\"\"*\"\"%#!]p++GpCP>Bxq_JQT6*pLpLD!R+B \")pCbc&*H8([w!\\n$4&=#HDO-TzqDl$)H:7dI'ys*e94[+ \"Q>Y7" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/&%\"aG6$\"\"*\"\"&#!^pv.(\\ h_;*oS'H9X`m*p/7;:d%[\\z=LDa%" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/&%\"aG6$\"#5\"\"&#! E(4\")\\AjYsj6GsJ!)f#p-=\"D+++KG$)zl3roDeD#45Q" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/&%\"aG6$\"#5\"\"'#\"P*[@_l53%R+#*pGG,(G!G<9-nF6#\"O+!G PnL'3n(zpAe91H*y*z(QZzY" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/&%\"aG6$\" #5\"\"(#\"T8aU9MlF*p3m\\eE^^S7$\\OXV-K37#\"DzQvcKka!\\;'*f))oIN;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/&%\"aG6$\"#6\"\"&#!ATQO*3)e_T`c#ff3`g#\"@+C%** H651$olX!GbxV" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/&%\"aG6$\"#6\"\"'#\" YRN$\\jDR`WEJ0\"y\\z'3CAHgn4XB#3)4#\"X+o:u@'H@s>k%\\:\"RysyQ!3b2?***)e x$" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/&%\"aG6$\"#6\"\"(#\"W**[H#yNEKk _$Q$e4A'H*=k=zjv$*Hs!*)\"VOZw.)>,$G#Q,%fb>%>]dE6&R,?C@R\"" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/&%\"aG6$\"#6\"\")#\"KDJ$3rY#H>7v!*QyT7VhU@5;\" MV9P%*GUYVOTqm]Y.J@s2s*4\"" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/&%\"aG6 $\"#6\"\"*#\"hp8D'\\)[')f`#RC_[o#RiU(=P'\\we>mAnY\\V_ac(\\KU$y^-\"o(p1 w+$\"ip++G)RCXn1TO#\\!o:.\"G*=v&G@bhzZ#3w.c]/mLl]ebkM,vLVal%" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/&%\"aG6$\"#6\"#5#!>vo?BfmI<6PuBb6$\"?\"Hc*Q .,:0B%fT@'GR" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%!G" }}{PARA 11 "" 1 " " {XPPMATH 20 "6#/&%\"cG6#\"#7\"\"\"" }}{PARA 11 "" 1 "" {XPPMATH 20 " 6#/&%\"aG6$\"#7\"\"\"#!XjV.Cxy7\\`()Q.,\"H`Hy<(RmD=*pbmG\"WS#Q`zL#p1.7 O9;]#R@6Iqi#>;rE#o)" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/&%\"aG6$\"#7\" \"#\"\"!" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/&%\"aG6$\"#7\"\"$\"\"!" } }{PARA 11 "" 1 "" {XPPMATH 20 "6#/&%\"aG6$\"#7\"\"%#!G+Sva*eV0(Qw'o9<9 ()3dp\"\"E@-d,\"G#ojK$oa'>^T!pV\"" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/ &%\"aG6$\"#7\"\"&#!Iv0aLgNtYJ\\H\"Hd%[uR\\$e%\"HWd8N.FMIJdru]_l.x&>X" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/&%\"aG6$\"#7\"\"'#\"Xv&\\*>DW= YMZeileUS`&)Q0jM#\"WkCm*G&=$=ai[^6Iy\\u#*GS&*yS;nsc#" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/&%\"aG6$\"#7\"\"(#\"enDEm3[[tlD'e)p5O\"4p$Gr@!o%)>$ f:7d;\"Yc!4\")3KWkh`]$4c\\5b\"fsZY^D6/[JM\"" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/&%\"aG6$\"#7\"\")#\"N+v=xjAppZDe\\:A0xYbz`oX$\"M*3ijaZ 8bzJ?n;Ax+$pVn6xu" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/&%\"aG6$\"#7\"\" *#!eq6g>UXn[q3_.;I_,nh0\\svq:Or0C5K-uF53S`FF:7VV\"zUD2tf?\\CP:wDFrudY)\\gt7XC^,$y9e'Ro\\&pv%*" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#/&%\"aG6$\"#7\"#5#\"M+v$4@L*e%)eQ/5LMB 1F$eaz-%\"L,^@Z*QtPE@(p\"p%G[*zU3Y'*))" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/&%\"aG6$\"#7\"#6#!I]s()eL*R*R48;fp22,O$H7'\"I`fnF?I@nC)>8bB5:+< 00\"" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%!G" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/&%\"cG6#\"#8\"\"\"" }}{PARA 11 "" 1 "" {XPPMATH 20 "6# /&%\"aG6$\"#8\"\"\"#!TP'**Q:g'e.TQXMf3!Gn+D[Nb!fn='\"Tg(*Q-@Hu[on1Mv%> ,5ZS@=@GWN?" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/&%\"aG6$\"#8\"\"#\"\"! " }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/&%\"aG6$\"#8\"\"$\"\"!" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/&%\"aG6$\"#8\"\"%#!F+5$H7>.Sy/g=Z!o\"\\>6W \"D>Ijp7\\h2Xn*pC!R0t.%" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/&%\"aG6$\" #8\"\"&#!GD-_G<`A%RX`#H\\%49rMn\"\"F_zB5s#R?2$p')3!H:PC,=" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/&%\"aG6$\"#8\"\"'#\"WD0jl=va7wx'\\F:%=U!)4zY0d (>v8N\"\"V)[$)Rf;Pq7^*=\">OE8x(f*G'[%z(eHg\"" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/&%\"aG6$\"#8\"\"(#\"Vv[eJ/rHz8'>#\\.>b[D,w)4uOov$*Q\"T ks9V%eZ(pIp9+)z[&*41;>!4ak&4M" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/&%\" aG6$\"#8\"\")#!LDcEn1N:eLbBtPI*>,be')[n\"Ld.xkx3:nSlN@@YUjl#*3$)y-gHyT&>4DD/nFU&>G4X2._0gv\"\"ep?B1IG%\\:M\\dhq3iQLN2aI5-ysg/4 ['=Y!)4)o0O)*p%yX\"yw[$" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/&%\"aG6$\" #8\"#5#\"JvoHH;=uw*Q,%36cB*eC5Q`\"ITT%Hf3&)fp\\([_%p&z%ed$f*)" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#/&%\"aG6$\"#8\"#6\"\"!" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/&%\"aG6$\"#8\"#7\"\"!" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%!G" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/&%\"bG6#\"\"\"# \"8t>&ohaxtn=!\\%\":!o!Q85B!)*4k/95" }}{PARA 11 "" 1 "" {XPPMATH 20 "6 #/&%\"bG6#\"\"#\"\"!" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/&%\"bG6#\"\"$ \"\"!" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/&%\"bG6#\"\"%\"\"!" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/&%\"bG6#\"\"&\"\"!" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/&%\"bG6#\"\"'#\"<++5['z_h\">v'Q;z\"=>B^6k7t**3DZgNA" } }{PARA 11 "" 1 "" {XPPMATH 20 "6#/&%\"bG6#\"\"(#\"=++]>\"[.)*o3\\\\x%Q \">++]ilZTxRKC^MP\"\"<$yGpu2f*\\#H*G8v)" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/&%\"bG6#\"\"*#\"]qTr?[WoIjJ:UFUjhp\\bVWH)R')p+wUB>.0N1uP?\")G%y o>8.ZlZF7\"[qSg;_GX*Hdn;_INBlzEl*G[a'4mo!GH)))[$4^VT$eQ]w^:P\\Z^M*[" } }{PARA 11 "" 1 "" {XPPMATH 20 "6#/&%\"bG6#\"#5#!=++]7`WRxd%oj$)z*\"<*p IE#=.c9M')Hs3$" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/&%\"bG6#\"#6#\"9](o +%3>$=Vb.#G\"8\"*fUSI()HwS&H7" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/&%\" bG6#\"#7#!6`n$p(p$HF9oI\"7'G(*\\u$=$=J$*H\"" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/&%\"bG6#\"#8\"\"!" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#% !G" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/&%#b*G6#\"\"\"#\"8xlS>qSRh%*\\d%p!yk3KN:" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/&%#b*G6#\"\"*# \"ap\"QwF9AgiP%QL?g&[%f:s\\*3#=?_g)yX(*z8r<5hym**[zkg?#\"`pS'e&H4oOsd \"Q@mPT'zgtlvd5!fd@Fdz&=fH^ueKi#4?a66" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/&%#b*G6#\"#5#!<+]7.#eCk&\\H7(za\";,OcU;*4qDS@PK#" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/&%#b*G6#\"#6\"\"!" }}{PARA 11 "" 1 "" {XPPMATH 20 " 6#/&%#b*G6#\"#7\"\"!" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/&%#b*G6#\"#8# !5*=k*y4(oXN(G\"5$pV]E/P\\$yz" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 151 "RK8_13eqs := [op( RowSumConditions(13,'expanded')),op(OrderConditions(8,13,'expanded'))] :\n`RK7_13eqs*` := subs(b=`b*`,OrderConditions(7,13,'expanded')):" }}} {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 6 "Check:" }} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 112 "subs(e18,RK8_13eqs):\nmap(u ->lhs(u)-rhs(u),%);\nnops(%);\nsubs(e18,`RK7_13eqs*`):\nmap(u->lhs(u)- rhs(u),%);\nnops(%);" }}{PARA 12 "" 1 "" {XPPMATH 20 "6#7`x\"\"!F$F$F$ F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$ F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$ F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$ F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$ F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$ F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#\"$7#" }}{PARA 12 "" 1 "" {XPPMATH 20 "6#7ap\"\"!F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F $F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F $F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#\"#&)" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 26 "#-------------------------" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}}{PARA 0 "" 0 "" {TEXT -1 52 "#----------------- ----------------------------------" }}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 38 "characteristics of the embedded scheme" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 36 "coefficients of the combined scheme " } }{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 7041 "ee := \{c[2]=1/20,\nc[3]=341/3200,\nc[4]=1023/6400,\nc[5]=39/1 00,\nc[6]=93/200,\nc[7]=31/200,\nc[8]=943/1000,\nc[9]=7067558016280/78 37150160667,\nc[10]=909/1000,\nc[11]=47/50,\nc[12]=1,\nc[13]=1,\n\na[2 ,1]=1/20,\na[3,1]=-7161/1024000,\na[3,2]=116281/1024000,\na[4,1]=1023/ 25600,\na[4,2]=0,\na[4,3]=3069/25600,\na[5,1]=4202367/11628100,\na[5,2 ]=0,\na[5,3]=-3899844/2907025,\na[5,4]=3982992/2907025,\na[6,1]=5611/1 14400,\na[6,2]=0,\na[6,3]=0,\na[6,4]=31744/135025,\na[6,5]=923521/5106 400,\na[7,1]=21173/343200,\na[7,2]=0,\na[7,3]=0,\na[7,4]=8602624/76559 175,\na[7,5]=-26782109/689364000,\na[7,6]=5611/283500,\na[8,1]=-122110 1821869329/690812928000000,\na[8,2]=0,\na[8,3]=0,\na[8,4]=-125/2,\na[8 ,5]=-1024030607959889/168929280000000,\na[8,6]=1501408353528689/265697 280000000,\na[8,7]=6070139212132283/92502016000000,\na[9,1]=-147251426 4486215803881384708877264246346044433307094207829051978044531801133057 155/\n 124689480162003200115705962164398602480330155839348790 0440453636168046069686436608,\na[9,2]=0,\na[9,3]=0,\na[9,4]=-517229431 1085668458375175655246981230039025336933699114138315270772319372469280 000/\n 124619381004809145897278630571215298365257079410236252 921850936749076487132995191,\na[9,5]=-12070679258469254807978936441733 187949484571516120469966534514296406891652614970375/\n 272203 1154761657221710478184531100699497284085048389015085076961673446140398 628096,\na[9,6]=780125155843893641323090552530431036567795592568497182 701460674803126770111481625/\n 183110425412731972197889874507 158786859226102980861859505241443073629143100805376,\na[9,7]=664113122 9599116421347821358391064699281403281605770353571553403929500094925118 75/\n 1517846559858624813633302310729534917527976515008907830 1139943253016877823170816,\na[9,8]=10332848184452015604056836767286656 859124007796970668046446015775000000/\n 131270355003603364807 3834248740727914537972028638950165249582733679393783,\na[10,1]=-290555 73360337415088538618442231036441314060511/\n 2267475989108957 7691327962602370597632000000000,\na[10,2]=0,\na[10,3]=0,\na[10,4]=-204 62749524591049105403365239069/454251913499893469596231268750,\na[10,5] =-180269259803172281163724663224981097/3810092255825687108657983283200 0000,\na[10,6]=21127670214172802870128286992003940810655221489/\n \+ 4679473877997892906145822697976708633673728000,\na[10,7]=3186072 35173649312405151265849660869927653414425413/\n 671471671555 8965303132938072935465423910912000000,\na[10,8]=2120832024345190822818 42245535894/20022426044775672563822865371173879,\na[10,9]=-26984049294 00842518721166485087129798562269848229517793703413951226714583/\n \+ 4695456749139343150770004420808711418846760359027175503256167281 75875000000,\na[11,1]=-23426598458140868369512071400656091790738384762 42943917/\n 135848096135105677702223140013915876085753216279 5520000,\na[11,2]=0,\na[11,3]=0,\na[11,4]=-996286030132538159613930889 652/16353068885996164905464325675,\na[11,5]=-2605308595925653415258808 9363841/4377552804565683061011299942400,\na[11,6]=20980822345096760292 224086794978105312644533925634933539/\n 37758899920075508038 78727839115494641972212962174156800,\na[11,7]=890722993756379186418929 622095833835264322635782294899/\n 13921242001395112657501941 955594013822830119803764736,\na[11,8]=16102142614312417838907512192924 6710833125/10997207722131034650667041364346422894371443,\na[11,9]=3007 6066976810251783423249756545243494667226619587649637187426239268485224 3925359864884962513/\n 4655443337501346455585065336604505603 760824779615521285751892810315680492364106674524398280000,\na[11,10]=- 31155237437111730665923206875/392862141594230515010338956291,\na[12,1] =-2866556991825663971778295329101033887534912787724034363/\n \+ 868226711619262703011213925016143612030669233795338240,\na[12,2]=0,\n a[12,3]=0,\na[12,4]=-16957088714171468676387054358954754000/1436904151 19654683326368228101570221,\na[12,5]=-45834939744845729129493146733560 33540575/451957703655250747157313034270335135744,\na[12,6]=23463053885 53404258656258473446184419154740172519949575/\n 256726716407 895402892744978301151486254183185289662464,\na[12,7]=16571215593198468 02171283690913610698586256573484808662625/\n 134314804112551 46477259155104956093505361644432088109056,\na[12,8]=345685379554677052 215495825476969226377187500/747711674369300772216672031795513475463620 89,\na[12,9]=\n -3205890962717072542791434312152727534008102774023210 240571361570757249056167015230160352087048674542196011/\n 94756954968 3965814783015124451273604984657747127257615372449205973192657306017239 103491074738324033259120,\na[12,10]=4027954583270623343310043858845893 3210937500/8896460842799482846916972126377338947215101,\na[12,11]=-612 2933601070769591613093993993358877250/10505170015102355131982467213020 27675953,\na[13,1]=-61867590553548250067280085934453841035866015389963 7/\n 203544282118214047100119475340667684874292102389760,\na [13,2]=0,\na[13,3]=0,\na[13,4]=-4411194916804718600478400319122931000/ 40373053902469967450761491269633019,\na[13,5]=-16734711409449292534539 422531728520225/1801243715290088669307203927210237952,\na[13,6]=135137 519757054679098042184152749677761254751865630525/\n 160295877 94486289597771326361911895112703716593983488,\na[13,7]=389375683674098 76012548551903492196137929710431584875/\n 3409564540901916060 99548798001469306974758443147264,\na[13,8]=-67488658550119930377323553 35815350667265625/\n 700288039571742462121356540671508776477 0357,\na[13,9]=-175600552030745092819542276704252509195417829600278830 8926563193523662404739779789732685671/\n 3487678145784699836 0568809804618648090460727802103054073533386208706157493415494283006232 0,\na[13,10]=53381024589235611084013897674181629296875/895935758479569 4524874969598508592944141,\na[13,11]=0,\na[13,12]=0,\n\nb[1]=449018677 37754616851973/1014046409980231013380680,\nb[2]=0,\nb[3]=0,\nb[4]=0,\n b[5]=0,\nb[6]=791638675191615279648100000/2235604725089973126411512319 ,\nb[7]=3847749490868980348119500000/15517045062138271618141237517,\nb [8]=-13734512432397741476562500000/875132892924995907746928783,\nb[9]= 1227476547031319687842881203774063505031923427600698639829444355496961 6342274215316330684448207141/\n 48934514749371551765038583414351 0934888829280686609654482896526796523353052166757299452852166040,\nb[1 0]=-9798363684577739445312500000/308722986341456031822630699,\nb[11]=2 82035543183190840068750/12295407629873040425991,\nb[12]=-3068142729369 76936753/1299331183183744997286,\nb[13]=0,\n\n`b*`[1]=1083540173940701 9406577/244521829356935137978320,\n`b*`[2]=0,\n`b*`[3]=0,\n`b*`[4]=0, \n`b*`[5]=0,\n`b*`[6]=13908189778321895491375000/392211355278942653756 40567,\n`b*`[7]=73487947527027243487625000/296504045773342769773399443 ,\n`b*`[8]=68293140641257649609375000/15353208647806945749946119,\n`b* `[9]=22060647948996678611017711379974578860522018208949721559448560203 338437626022142776381/\n 111154200926232587451295918579572721575 9010577565736079641376621381577236680929558640,\n`b*`[10]=-54797122949 5642458203125000/23237214025700991642563601,\n`b*`[11]=0,\n`b*`[12]=0, \n`b*`[13]=-28735456870978964189/79783493704265043693\}:" }}}{PARA 0 " " 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}} }{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 5 "Let " } {XPPEDIT 18 0 "T[8, 13];" "6#&%\"TG6$\"\")\"#8" }{TEXT -1 128 " denot e the vector whose components are the principal error terms of the 13 \+ stage, order 8 scheme (the error terms of order 9)." }}{PARA 0 "" 0 " " {TEXT -1 5 "Let " }{XPPEDIT 18 0 "`T*`[7,13];" "6#&%#T*G6$\"\"(\"#8 " }{TEXT -1 146 " denote the vector whose components are the principa l error terms of the embedded 13 stage, order 7 scheme (the error term s of order 8) and let " }{XPPEDIT 18 0 "`T*`[8,13];" "6#&%#T*G6$\"\") \"#8" }{TEXT -1 99 " denote the vector whose components are the error terms of order 9 of the embedded order 7 scheme." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 39 "Denote the 2-norm of thes e vectors by " }{XPPEDIT 18 0 "abs(abs(T[8,13]));" "6#-%$absG6#-F$6#& %\"TG6$\"\")\"#8" }{TEXT -1 3 ", " }{XPPEDIT 18 0 "abs(abs(`T*`[7, 13 ]));" "6#-%$absG6#-F$6#&%#T*G6$\"\"(\"#8" }{TEXT -1 7 " and " } {XPPEDIT 18 0 "abs(abs(`T*`[8, 13]));" "6#-%$absG6#-F$6#&%#T*G6$\"\") \"#8" }{TEXT -1 15 " respectively." }}{PARA 0 "" 0 "" {TEXT -1 0 "" } }{PARA 0 "" 0 "" {TEXT -1 8 "Define: " }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "A[9] = abs(abs(T[8,13]));" "6#/&%\"AG6#\"\"*-%$abs G6#-F)6#&%\"TG6$\"\")\"#8" }{TEXT -1 6 ", " }{XPPEDIT 18 0 "B[9] = abs(abs(`T*`[8,13]))/abs(abs(`T*`[7,13]));" "6#/&%\"BG6#\"\"**&-%$abs G6#-F*6#&%#T*G6$\"\")\"#8\"\"\"-F*6#-F*6#&F/6$\"\"(F2!\"\"" }{TEXT -1 6 ", " }{XPPEDIT 18 0 "C[9] = abs(abs(`T*`[8,13]-T[8,13]))/abs(abs (`T*`[7,13]));" "6#/&%\"CG6#\"\"**&-%$absG6#-F*6#,&&%#T*G6$\"\")\"#8\" \"\"&%\"TG6$F2F3!\"\"F4-F*6#-F*6#&F06$\"\"(F3F8" }{TEXT -1 1 "." }} {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 77 "Dormand a nd Prince have suggested that as well as attempting to ensure that " }{XPPEDIT 18 0 "A[9];" "6#&%\"AG6#\"\"*" }{TEXT -1 73 " is a minimum, if the embedded scheme is to be used for error control, " }{XPPEDIT 18 0 "B[9];" "6#&%\"BG6#\"\"*" }{TEXT -1 7 " and " }{XPPEDIT 18 0 "C [9];" "6#&%\"CG6#\"\"*" }{TEXT -1 27 " should be chosen so that " }} {PARA 0 "" 0 "" {TEXT -1 68 "they are \"similar in magnitude\" and als o not differ too much from 1." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 168 "errterms8_13 := PrincipalEr rorTerms(8,13,'expanded'):\n`errterms8_13*` :=subs(b=`b*`,errterms8_13 ):\n`errterms7_13*` := subs(b=`b*`,PrincipalErrorTerms(7,13,'expanded' )):" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 333 "snmB := sqrt(add(evalf(subs(ee,`errterms8_13*`[i]))^ 2,i=1..nops(`errterms8_13*`))):\nsdnB := sqrt(add(evalf(subs(ee,`errte rms7_13*`[i]))^2,i=1..nops(`errterms7_13*`))):\nsnmC := sqrt(add((eval f(subs(ee,`errterms8_13*`[i])-subs(ee,errterms8_13[i])))^2,i=1..nops(e rrterms8_13))):\n'B[9]'= evalf[8](snmB/sdnB);\n'C[9]'= evalf[8](snmC/s dnB);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/&%\"BG6#\"\"*$\")!\\R(>!\"( " }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/&%\"CG6#\"\"*$\"):Q$)>!\"(" }}} {PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}}{PARA 0 "" 0 "" {TEXT -1 52 "#------------------------------ ---------------------" }}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 26 "absolute stability regions" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 " > " 0 "" {MPLTEXT 1 0 1 ";" }}}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 36 "co efficients of the combined scheme " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 7041 "ee := \{c[2]=1/20,\nc[3]=3 41/3200,\nc[4]=1023/6400,\nc[5]=39/100,\nc[6]=93/200,\nc[7]=31/200,\nc [8]=943/1000,\nc[9]=7067558016280/7837150160667,\nc[10]=909/1000,\nc[1 1]=47/50,\nc[12]=1,\nc[13]=1,\n\na[2,1]=1/20,\na[3,1]=-7161/1024000,\n a[3,2]=116281/1024000,\na[4,1]=1023/25600,\na[4,2]=0,\na[4,3]=3069/256 00,\na[5,1]=4202367/11628100,\na[5,2]=0,\na[5,3]=-3899844/2907025,\na[ 5,4]=3982992/2907025,\na[6,1]=5611/114400,\na[6,2]=0,\na[6,3]=0,\na[6, 4]=31744/135025,\na[6,5]=923521/5106400,\na[7,1]=21173/343200,\na[7,2] =0,\na[7,3]=0,\na[7,4]=8602624/76559175,\na[7,5]=-26782109/689364000, \na[7,6]=5611/283500,\na[8,1]=-1221101821869329/690812928000000,\na[8, 2]=0,\na[8,3]=0,\na[8,4]=-125/2,\na[8,5]=-1024030607959889/16892928000 0000,\na[8,6]=1501408353528689/265697280000000,\na[8,7]=60701392121322 83/92502016000000,\na[9,1]=-147251426448621580388138470887726424634604 4433307094207829051978044531801133057155/\n 12468948016200320 01157059621643986024803301558393487900440453636168046069686436608,\na[ 9,2]=0,\na[9,3]=0,\na[9,4]=-517229431108566845837517565524698123003902 5336933699114138315270772319372469280000/\n 12461938100480914 5897278630571215298365257079410236252921850936749076487132995191,\na[9 ,5]=-12070679258469254807978936441733187949484571516120469966534514296 406891652614970375/\n 272203115476165722171047818453110069949 7284085048389015085076961673446140398628096,\na[9,6]=78012515584389364 1323090552530431036567795592568497182701460674803126770111481625/\n \+ 183110425412731972197889874507158786859226102980861859505241443 073629143100805376,\na[9,7]=664113122959911642134782135839106469928140 328160577035357155340392950009492511875/\n 151784655985862481 36333023107295349175279765150089078301139943253016877823170816,\na[9,8 ]=10332848184452015604056836767286656859124007796970668046446015775000 000/\n 131270355003603364807383424874072791453797202863895016 5249582733679393783,\na[10,1]=-290555733603374150885386184422310364413 14060511/\n 22674759891089577691327962602370597632000000000, \na[10,2]=0,\na[10,3]=0,\na[10,4]=-20462749524591049105403365239069/45 4251913499893469596231268750,\na[10,5]=-180269259803172281163724663224 981097/38100922558256871086579832832000000,\na[10,6]=21127670214172802 870128286992003940810655221489/\n 46794738779978929061458226 97976708633673728000,\na[10,7]=318607235173649312405151265849660869927 653414425413/\n 67147167155589653031329380729354654239109120 00000,\na[10,8]=212083202434519082281842245535894/20022426044775672563 822865371173879,\na[10,9]=-2698404929400842518721166485087129798562269 848229517793703413951226714583/\n 46954567491393431507700044 2080871141884676035902717550325616728175875000000,\na[11,1]=-234265984 5814086836951207140065609179073838476242943917/\n 1358480961 351056777022231400139158760857532162795520000,\na[11,2]=0,\na[11,3]=0, \na[11,4]=-996286030132538159613930889652/1635306888599616490546432567 5,\na[11,5]=-26053085959256534152588089363841/437755280456568306101129 9942400,\na[11,6]=2098082234509676029222408679497810531264453392563493 3539/\n 3775889992007550803878727839115494641972212962174156 800,\na[11,7]=890722993756379186418929622095833835264322635782294899/ \n 13921242001395112657501941955594013822830119803764736,\na [11,8]=161021426143124178389075121929246710833125/10997207722131034650 667041364346422894371443,\na[11,9]=30076066976810251783423249756545243 4946672266195876496371874262392684852243925359864884962513/\n \+ 46554433375013464555850653366045056037608247796155212857518928103156 80492364106674524398280000,\na[11,10]=-31155237437111730665923206875/3 92862141594230515010338956291,\na[12,1]=-28665569918256639717782953291 01033887534912787724034363/\n 868226711619262703011213925016 143612030669233795338240,\na[12,2]=0,\na[12,3]=0,\na[12,4]=-1695708871 4171468676387054358954754000/143690415119654683326368228101570221,\na[ 12,5]=-4583493974484572912949314673356033540575/4519577036552507471573 13034270335135744,\na[12,6]=234630538855340425865625847344618441915474 0172519949575/\n 2567267164078954028927449783011514862541831 85289662464,\na[12,7]=165712155931984680217128369091361069858625657348 4808662625/\n 1343148041125514647725915510495609350536164443 2088109056,\na[12,8]=345685379554677052215495825476969226377187500/747 71167436930077221667203179551347546362089,\na[12,9]=\n -3205890962717 0725427914343121527275340081027740232102405713615707572490561670152301 60352087048674542196011/\n 947569549683965814783015124451273604984657 747127257615372449205973192657306017239103491074738324033259120,\na[12 ,10]=40279545832706233433100438588458933210937500/88964608427994828469 16972126377338947215101,\na[12,11]=-6122933601070769591613093993993358 877250/1050517001510235513198246721302027675953,\na[13,1]=-61867590553 5482500672800859344538410358660153899637/\n 2035442821182140 47100119475340667684874292102389760,\na[13,2]=0,\na[13,3]=0,\na[13,4]= -4411194916804718600478400319122931000/4037305390246996745076149126963 3019,\na[13,5]=-16734711409449292534539422531728520225/180124371529008 8669307203927210237952,\na[13,6]=1351375197570546790980421841527496777 61254751865630525/\n 1602958779448628959777132636191189511270 3716593983488,\na[13,7]=3893756836740987601254855190349219613792971043 1584875/\n 34095645409019160609954879800146930697475844314726 4,\na[13,8]=-6748865855011993037732355335815350667265625/\n \+ 7002880395717424621213565406715087764770357,\na[13,9]=-175600552030745 0928195422767042525091954178296002788308926563193523662404739779789732 685671/\n 34876781457846998360568809804618648090460727802103 0540735333862087061574934154942830062320,\na[13,10]=533810245892356110 84013897674181629296875/8959357584795694524874969598508592944141,\na[1 3,11]=0,\na[13,12]=0,\n\nb[1]=44901867737754616851973/1014046409980231 013380680,\nb[2]=0,\nb[3]=0,\nb[4]=0,\nb[5]=0,\nb[6]=79163867519161527 9648100000/2235604725089973126411512319,\nb[7]=38477494908689803481195 00000/15517045062138271618141237517,\nb[8]=-13734512432397741476562500 000/875132892924995907746928783,\nb[9]=1227476547031319687842881203774 0635050319234276006986398294443554969616342274215316330684448207141/\n 489345147493715517650385834143510934888829280686609654482896526 796523353052166757299452852166040,\nb[10]=-979836368457773944531250000 0/308722986341456031822630699,\nb[11]=282035543183190840068750/1229540 7629873040425991,\nb[12]=-306814272936976936753/1299331183183744997286 ,\nb[13]=0,\n\n`b*`[1]=10835401739407019406577/24452182935693513797832 0,\n`b*`[2]=0,\n`b*`[3]=0,\n`b*`[4]=0,\n`b*`[5]=0,\n`b*`[6]=1390818977 8321895491375000/39221135527894265375640567,\n`b*`[7]=7348794752702724 3487625000/296504045773342769773399443,\n`b*`[8]=682931406412576496093 75000/15353208647806945749946119,\n`b*`[9]=220606479489966786110177113 79974578860522018208949721559448560203338437626022142776381/\n 1 1115420092623258745129591857957272157590105775657360796413766213815772 36680929558640,\n`b*`[10]=-547971229495642458203125000/232372140257009 91642563601,\n`b*`[11]=0,\n`b*`[12]=0,\n`b*`[13]=-28735456870978964189 /79783493704265043693\}:" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}}{PARA 0 "" 0 "" {TEXT -1 0 " " }}{PARA 0 "" 0 "" {TEXT -1 78 "The stability function R for the 13 s tage, order 8 scheme is given as follows." }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 76 "subs(ee,StabilityFunction(8,13,'expanded')):\nR := un apply(%,z):\n'R(z)'=R(z);" }}{PARA 12 "" 1 "" {XPPMATH 20 "6#/-%\"RG6# %\"zG,<\"\"\"F)F'F)*&#F)\"\"#F)*$)F'F,F)F)F)*&#F)\"\"'F)*$)F'\"\"$F)F) F)*&#F)\"#CF)*$)F'\"\"%F)F)F)*&#F)\"$?\"F)*$)F'\"\"&F)F)F)*&#F)\"$?(F) *$)F'F1F)F)F)*&#F)\"%S]F)*$)F'\"\"(F)F)F)*&#F)\"&?.%F)*$)F'\"\")F)F)F) *&#\"F*e+**3-$f&G'o(4)pbN\"z$G\"\"K+++S#eO7+Kl6,\"*\\\"3vIXYYF)*$)F'\" \"*F)F)F)*&#\"GF)eBZpbp=6+'*z^R'eYfJ\"N+++++c94.+L\"z_xu.xoKhh6F)*$)F' \"#5F)F)F)*&#\"I\")3*o!3O#4B8Y)yj:zo&poy%\"Q++++++'Hj%\\+GhY/k*fK+B\"e e=F)*$)F'\"#6F)F)F)*&#\"G*>4YlV%H6Tx*>r;3c79X#\"P++++++!G0JK2@Umv6&\\c ZilE#F)*$)F'\"#7F)F)F)" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 124 "We can find the point where the boundary of the sta bility region intersects the negative real axis by solving the equatio n: " }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "R(z) = -1;" "6 #/-%\"RG6#%\"zG,$\"\"\"!\"\"" }{TEXT -1 2 ". " }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 27 "z0 := newton(R(z)=-1,z=-6);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%#z0G$!+RK6ke!\"*" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 306 "z0 := newton(R(z)=-1,z=-5): \np1 := plot([R(z),-1],z=-6.49..0.49,color=[red,blue]):\np2 := plot([[ [z0,-1]]$3],style=point,symbol=[circle,cross,diamond],color=black):\np 3 := plot([[z0,0],[z0,-1]],linestyle=3,color=COLOR(RGB,0,.5,0)):\nplot s[display]([p1,p2,p3],view=[-6.49..0.49,-1.47..1.47],font=[HELVETICA,9 ]);" }}{PARA 13 "" 1 "" {GLPLOT2D 391 285 285 {PLOTDATA 2 "6+-%'CURVES G6$7V7$$!3A++++++!\\'!#<$!3yTbO]+E'*GF*7$$!3xL$e>$z#RT'F*$!39uC/Y8lhDF *7$$!3Lnm\"R'e&yL'F*$!3#y'H\"*zv%=E#F*7$$!3o](oU'fmriF*$!3S#oo%p]\"o-# F*7$$!3,M3ikgZ0iF*$!3Rt8`K6w8=F*7$$!3m]7))*fQ58'F*$!3[5$fiii\")f\"F*7$ $!3Kn;9N6gcgF*$!3j'oQ:axcS\"F*7$$!3;n;*fORn!fF*$!3u%yA4#4_z5F*7$$!3#Q$ 3nj)*eddF*$!3RAALMIqN#)!#=7$$!3ymTb#=4$>cF*$!3G=!4>JV+O'FU7$$!3e+vyYv7 waF*$!3?vRT1MTE[FU7$$!3hmTD%*)[!G`F*$!3+\\/lg8S#f$FU7$$!3w*\\P=8X/=&F* $!3y\"o9$4GwXEFU7$$!3@LLQ>lhG]F*$!3Q\">?\"oG!R!>FU7$$!3km;*[z&)[*[F*$! 3Y1Y@DWC.9FU7$$!3I++:tmLWZF*$!35gz@&Rw[s*!#>7$$!39++D#QpJf%F*$!3!H]S&p g0)['Fcp7$$!3I++&oo\"\\ZWF*$!3Oy?w65#e9%Fcp7$$!3FnTD$Q-_J%F*$!3!oU8*Gm e9DFcp7$$!3-LL)fg(*y:%F*$!3.2LhXc'e-\"Fcp7$$!3#HLLf#)RY-%F*$!37;vvIZXz T!#@7$$!3S+v[[$H'pQF*$\"3[AO(G^^M%))!#?7$$!3zLLV5)=Ct$F*$\"3CBa&GOfcd \"Fcp7$$!3))*\\([.r(=e$F*$\"33*4T&QC5bAFcp7$$!3&**\\i%)zD&QMF*$\"3=B2t W!H*pGFcp7$$!3[L$3-v`*)G$F*$\"3na&\\GF*$\"3uPm%yQ $QSdFcp7$$!3<+Dr!e\"f:FF*$\"3%y#[BPQ[)e'Fcp7$$!3!omTPG34d#F*$\"3h/ZHFz xIwFcp7$$!30++q!GO9U#F*$\"3-AIBDo#4())Fcp7$$!3]+]x'32_F#F*$\"3s&f-NT7t -\"FU7$$!3-+voAGsL@F*$\"3%*R-P!HGP=\"FU7$$!3k+]#)p'Gm(>F*$\"3Kx#HXc]_Q \"FU7$$!36MLV'esa$=F*$\"3MK5`fC(oCFU7$$!3)R$3#yX'Re7F*$\"3.Jp?!G\"4TGFU7$$!3%**\\P?H[:6\"F*$\"37 s*46o!\\!H$FU7$$!3URL$eot(z'*FU$\"3_4Vj-b])z$FU7$$!3u3](=<>n<)FU$\"3J7 AZ&3#e9WFU7$$!3almmK!*4HnFU$\"3j43\"R>&oFU7$$!3)))****p#f[JCFU$\"3-V h2a)\\:%yFU7$$!3!eILe:#*H&))Fcp$\"3?%3.&Qsv_\"*FU7$$\"3u%HLL6'Gw\\Fcp$ \"31ljpG=-^5F*7$$\"37,]7:\"=@(>FU$\"3cX9B+?+=7F*7$$\"3/&*\\(3r#R$Q$FU$ \"3B;[%)*G;ES\"F*7$$\"3!***************[FU$\"3&)QV'*>iJK;F*-%'COLOURG6 &%$RGBG$\"*++++\"!\")$\"\"!Fb\\lFa\\l-F$6$7S7$F($!\"\"Fb\\l7$F3Fg\\l7$ F=Fg\\l7$FGFg\\l7$FLFg\\l7$FQFg\\l7$FWFg\\l7$FfnFg\\l7$F[oFg\\l7$F`oFg \\l7$FeoFg\\l7$FjoFg\\l7$F_pFg\\l7$FepFg\\l7$FjpFg\\l7$F_qFg\\l7$FdqFg \\l7$FiqFg\\l7$F_rFg\\l7$FerFg\\l7$FjrFg\\l7$F_sFg\\l7$FdsFg\\l7$FisFg \\l7$F^tFg\\l7$FctFg\\l7$FhtFg\\l7$F]uFg\\l7$FbuFg\\l7$FguFg\\l7$F\\vF g\\l7$FavFg\\l7$FfvFg\\l7$F[wFg\\l7$F`wFg\\l7$FewFg\\l7$FjwFg\\l7$F_xF g\\l7$FdxFg\\l7$FixFg\\l7$F^yFg\\l7$FcyFg\\l7$FhyFg\\l7$F]zFg\\l7$FbzF g\\l7$FgzFg\\l7$F\\[lFg\\l7$Fa[lFg\\l7$Ff[lFg\\l-F[\\l6&F]\\lFa\\lFa\\ lF^\\l-F$6&7#7$$!3#*******QK6keF*Fg\\l-%'SYMBOLG6#%'CIRCLEG-F[\\l6&F] \\lFb\\lFb\\lFb\\l-%&STYLEG6#%&POINTG-F$6&F]`l-Fb`l6#%&CROSSGFe`lFg`l- F$6&F]`l-Fb`l6#%(DIAMONDGFe`lFg`l-F$6%7$7$F_`lFa\\lF^`l-%&COLORG6&F]\\ lFa\\l$\"\"&Fh\\lFa\\l-%*LINESTYLEG6#\"\"$-%+AXESLABELSG6%Q\"z6\"Q!Ffb l-%%FONTG6#%(DEFAULTG-Fibl6$%*HELVETICAG\"\"*-%%VIEWG6$;$!$\\'!\"#$\"# \\Ffcl;$!$Z\"Ffcl$\"$Z\"Ffcl" 1 2 0 1 10 0 2 9 1 4 2 1.000000 45.000000 45.000000 0 0 "Curve 1" "Curve 2" "Curve 3" "Curve 4" "Curve 5" "Curve 6" }}}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 50 "The following picture shows the stability region. " }} {PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1640 "R := z -> add(z^j/j!,j=0..8)+\n 12837913555698097686285593 02089900589/464645307508149910111653200123658240000000*z^9+\n 31 594658639517996001118695569472358827/116161326877037477527913300030914 560000000000*z^10+\n 4786869568791563788461323092360806890881/18 5858123003259964044661280049463296000000000000*z^11+\n 245141256 08167119977411129443654609199/2266562475649511756642210732310528000000 0000000*z^12:\npts := []: z0 := 0:\nfor ct from 0 to 280 do\n zz := \+ newton(R(z)=exp(ct*Pi/20*I),z=z0):\n z0 := zz:\n pts := [op(pts),[ Re(zz),Im(zz)]]:\nend do:\np1 := plot(pts,color=COLOR(RGB,.4,.2,.05)): \np2 := plots[polygonplot]([seq([pts[i-1],pts[i],[-4.6,0]],i=2..nops(p ts))],\n style=patchnogrid,color=COLOR(RGB,.95,.45,.1)):\npts := []: z0 := .9+5.7*I:\nfor ct from 0 to 60 do\n zz := newton(R(z)= exp(ct*Pi/30*I),z=z0):\n z0 := zz:\n pts := [op(pts),[Re(zz),Im(zz )]]:\nend do:\np3 := plot(pts,color=COLOR(RGB,.4,.2,.05)):\np4 := plot s[polygonplot]([seq([pts[i-1],pts[i],[0.64,5.8]],i=2..nops(pts))],\n \+ style=patchnogrid,color=COLOR(RGB,.95,.45,.1)):\npts := []: z0 := .9-5.7*I:\nfor ct from 0 to 60 do\n zz := newton(R(z)=exp(ct*Pi/ 30*I),z=z0):\n z0 := zz:\n pts := [op(pts),[Re(zz),Im(zz)]]:\nend \+ do:\np5 := plot(pts,color=COLOR(RGB,.4,.2,.05)):\np6 := plots[polygonp lot]([seq([pts[i-1],pts[i],[0.64,-5.8]],i=2..nops(pts))],\n s tyle=patchnogrid,color=COLOR(RGB,.95,.45,.1)):\np7 := plot([[[-6.69,0] ,[1.39,0]],[[0,-6.39],[0,6.39]]],color=black,linestyle=3):\nplots[disp lay]([p||(1..7)],view=[-6.69..1.39,-6.39..6.39],font=[HELVETICA,9],\n \+ labels=[`Re(z)`,`Im(z)`],axes=boxed,scaling=constrained); " }}{PARA 13 "" 1 "" {GLPLOT2D 381 565 565 {PLOTDATA 2 "6/-%'CURVESG6$ 7e\\l7$$\"\"!F)F(7$F($\"3++++Fjzq:!#=7$F($\"3)******Rl#fTJF-7$F($\"3<+ ++!)*)Q7ZF-7$$!3#******4()=ln)!#G$\"3))*****pI&=$G'F-7$$!3;+++pDFzs!#F $\"3#)******H;)R&yF-7$$!3U+++vdlYR!#E$\"3g*****4&zxC%*F-7$$!3++++P-`j: !#D$\"3++++Gub*4\"!#<7$$!3o*****\\h'zc[FI$\"3++++nqjc7FL7$$!3!******>B cQB\"!#C$\"3&******ps;PT\"FL7$$!3))*****\\s())GEFU$\"3-+++Mkzq:FL7$$!3 a+++\"[VAx%FU$\"3++++*>wys\"FL7$$!3*)*****zHv)=vFU$\"3-+++Ff&\\)=FL7$$ !3-+++k2f)3\"!#B$\"33+++\">N?/#FL7$$!3-+++j'y^s\"Fdo$\"3++++zF6*>#FL7$ $!39+++PN-)y$Fdo$\"3?+++Ef=cBFL7$$!31+++*G\\93\"!#A$\"3++++I#\\K^#FL7$ $!3!******4i@8=$Fdp$\"3=+++EMHqEFL7$$!33+++!R'ow')Fdp$\"37+++NXIFGFL7$ $!3-+++A.]c@!#@$\"3')*****R_kU)HFL7$$!3(******fn+.#\\Fdq$\"3$)*****4Kb 69$FL7$$!3&******f%z1U5!#?$\"3-+++h*ozH$FL7$$!35+++qUMp?F_r$\"3#****** HHFZX$FL7$$!3=+++Ge%f)QF_r$\"30+++i]^6OFL7$$!3g*****H+)*>&pF_r$\"3!*** ***H,I&oPFL7$$!35+++&R&>$>\"!#>$\"37+++vO:ERFL7$$!33+++#Hw\"z>Fds$\"3C +++kt/&3%FL7$$!3o******[-(**>$Fds$\"3m*****RrxiC%FL7$$!3C+++<_R*4&Fds$ \"3C+++Y@X6WFL7$$!3b******)=8\\8)Fds$\"3G+++\\-l#e%FL7$$!3*)*****fW5^K \"F-$\"3!)*******p&ogZFL7$$!36+++&p!*>?#F-$\"30+++i#Ra$\\FL7$$!3-+++k7 ZJMF-$\"3s*****\\=Dk2&FL7$$!3z*******>qPq%F-$\"3g*****Rl-uyF-$\"3F+++$Q8,M&FL7$$!35+++H^Vj()F-$\"3S+++R-;v`FL7$$!3/+++ @Nd)f*F-$\"3%)*****p1Y&F L7$$!3++++-0Ie7FL$\"3A+++H(3LZ&FL7$$!36+++f/#pK\"FL$\"3)******\\6W7[&F L7$$!33+++I1!RR\"FL$\"3/+++.;$f[&FL7$$!3/+++!o)\\f9FL$\"3/+++=!Hv[&FL7 $$!3(******fWJR_\"FL$\"3?+++n;;'[&FL7$$!3/+++*y&Q(e\"FL$\"31+++[)H>[&F L7$$!3)******R(p-];FL$\"3y*****fU9\\Z&FL7$$!3$******>L0?r\"FL$\"39+++X 7=laFL7$$!3/+++'phMx\"FL$\"39+++*)[y_aFL7$$!32+++_>`M=FL$\"3!)******\\ @xPaFL7$$!31+++!)3N&*=FL$\"3,+++:b=?aFL7$$!3++++``0c>FL$\"3/+++Ep1+aFL 7$$!39+++Uoy;?FL$\"3%)*****f.iuP&FL7$$!3%******f-$px?FL$\"3W+++`^U_`FL 7$$!3%******4;G*Q@FL$\"3,+++X\\-D`FL7$$!3#*******y;l+AFL$\"3g*****R&3N &H&FL7$$!33+++2W-jAFL$\"35+++V)>NE&FL7$$!3++++L&FL7$$!3'******R))*\\bCFL$\"3s***** \\e&yc^FL7$$!3#******Ha(z@DFL$\"3e*****>)o@=^FL7$$!35+++6QA*e#FL$\"3I+ ++H#3'y]FL7$$!3))*****H[z%FL7$$!35+++sK`]JFL$ \"3y*****H%3\"4v%FL7$$!30+++*e.+A$FL$\"3W+++$pL(4ZFL7$$!35+++Y^n)G$FL$ \"39+++h*3$oYFL7$$!39+++&\\YkN$FL$\"3W+++^%4li%FL7$$!35+++,$[KU$FL$\"3 -+++!>3Ue%FL7$$!31+++i$Q!*[$FL$\"3&)*****f$QGTXFL7$$!31+++\"p'z`NFL$\" 3Q+++OSi(\\%FL7$$!3&)*****\\4@vh$FL$\"37+++)GFJX%FL7$$!39+++7MA!o$FL$ \"3o*****Rz.xS%FL7$$!3()*****zSE>u$FL$\"3q*****Rpv7O%FL7$$!34+++u7m-QF L$\"3A+++gpx8VFL7$$!3?+++&flC'QFL$\"3?+++XJ:lUFL7$$!37+++v;Q@RFL$\"3I+ ++/6O:UFL7$$!3++++0^XzRFL$\"3?+++P(oV;%FL7$$!3#******4XLn.%FL$\"3k**** *pza@6%FL7$$!35+++%*\\E$4%FL$\"3/+++G(3(eSFL7$$!3o*****HV(4\\TFL$\"3m* ****z\\IS+%FL7$$!3@+++FlF/UFL$\"3')*****\\PI\"[RFL7$$!3E+++\\Y%)eUFL$ \"3))*****HcG5*QFL7$$!3t*****>FRGJ%FL$\"3A+++?YvKQFL7$$!3m*****HX\"HmV FL$\"3-+++&fYLx$FL7$$!3y*****fQC#>WFL$\"3-+++M)\\Gr$FL7$$!3?+++X@lrWFL $\"3%)******GaJ^OFL7$$!3C+++P(yN_%FL$\"3-+++e$)z)e$FL7$$!3'******>e(* \\d%FL$\"3$******>Pb`_$FL7$$!3I+++&\\\"*ei%FL$\"3-+++zF/hMFL7$$!3))*** **\\@Lin%FL$\"3-+++aU\"fR$FL7$$!31+++ak)fs%FL$\"3%*******e)=+L$FL7$$!3 q*****4R2^x%FL$\"37+++-%*RjKFL7$$!3++++zkaB[FL$\"39+++a74'>$FL7$$!3))* *****=/Dr[FL$\"3@+++x;7GJFL7$$!3/+++^T;=\\FL$\"3-+++o'4&fIFL7$$!3c**** *RrKU'\\FL$\"31+++_iE!*HFL7$$!3x*****>*GS4]FL$\"3y******y^R?HFL7$$!3'* *****pcCO0&FL$\"35+++%)Q*)\\GFL7$$!39+++`<&o4&FL$\"3++++UYvyFFL7$$!3s* *****GL/R^FL$\"3A+++:e'pq#FL7$$!3/+++![j,=&FL$\"3!)******3J^MEFL7$$!3g *****R)==?_FL$\"35+++43QhDFL7$$!3e*****Rpt!f_FL$\"3++++MHb([#FL7$$!3)* *****>N>oH&FL$\"3.+++,V,8CFL7$$!3J+++`USL`FL$\"3,+++&\\^xL#FL7$$!3!*** ****3$=)o`FL$\"3%)*****4s`+++LqHH?FL7$$! 3E+++!3!p)\\&FL$\"3-+++6zK]>FL7$$!3;+++P[@GbFL$\"35+++z:jq=FL7$$!3?+++ ()zccbFL$\"3)******RsC-z\"FL7$$!3/+++QBv$e&FL$\"3'******\\NH\"4s1?Y\"FL7$$!3M+++Mkz!o&FL$\"3-+++oKZy8FL7$ $!31+++x$3@q&FL$\"3#********GGWH\"FL7$$!3o*****4==As&FL$\"32+++ce\"*47 FL7$$!3$)*****\\/36u&FL$\"3++++r3)\\7\"FL7$$!3s*****Hnc(edFL$\"3!***** *R:o'R5FL7$$!3U+++K(R^x&FL$\"3++++Ud@S&*F-7$$!3#)******)[I-z&FL$\"3))* ****\\)H$3o)F-7$$!3W+++r0+/eFL$\"3,+++e=$*=yF-7$$!3))*****f#3U;eFL$\"3 ]+++bM)[&pF-7$$!3\")*****f@iu#eFL$\"3Q+++?(G!*3'F-7$$!38+++7n4PeFL$\"3 b*****H]x;A&F-7$$!3G+++Z\")HXeFL$\"3))*****H>4JN%F-7$$!3C+++\"*H/_eFL$ \"3++++SZd$[$F-7$$!3e*****f.6t&eFL$\"3++++=)*H8EF-7$$!3=+++nf3heFL$\"3 1+++-*)[U4JN%F-7$Fdem$!3b*****H]x;A&F-7$F_em$ !3Q+++?(G!*3'F-7$Fjdm$!3]+++bM)[&pF-7$Fedm$!3,+++e=$*=yF-7$F`dm$!3))** ***\\)H$3o)F-7$F[dm$!3++++Ud@S&*F-7$Ffcm$!3!******R:o'R5FL7$Facm$!3+++ +r3)\\7\"FL7$F\\cm$!32+++ce\"*47FL7$Fgbm$!3#********GGWH\"FL7$Fbbm$!3- +++oKZy8FL7$F]bm$!3#******>s1?Y\"FL7$Fham$!3++++oj)\\a\"FL7$Fcam$!3.++ +4EPF;FL7$F^am$!3'******\\NH\"4FL7$Fj_m$!3>+++LqHH?FL7$Fe_m$!38+++$FL7$Fejl$!37+++-%*RjKFL7 $F`jl$!3%*******e)=+L$FL7$F[jl$!3-+++aU\"fR$FL7$Ffil$!3-+++zF/hMFL7$Fa il$!3$******>Pb`_$FL7$F\\il$!3-+++e$)z)e$FL7$Fghl$!3%)******GaJ^OFL7$F bhl$!3-+++M)\\Gr$FL7$F]hl$!3-+++&fYLx$FL7$Fhgl$!3A+++?YvKQFL7$Fcgl$!3) )*****HcG5*QFL7$F^gl$!3')*****\\PI\"[RFL7$Fifl$!3m*****z\\IS+%FL7$Fdfl $!3/+++G(3(eSFL7$F_fl$!3k*****pza@6%FL7$Fjel$!3?+++P(oV;%FL7$Feel$!3I+ ++/6O:UFL7$F`el$!3?+++XJ:lUFL7$F[el$!3A+++gpx8VFL7$Ffdl$!3q*****Rpv7O% FL7$Fadl$!3o*****Rz.xS%FL7$F\\dl$!37+++)GFJX%FL7$Fgcl$!3Q+++OSi(\\%FL7 $Fbcl$!3&)*****f$QGTXFL7$F]cl$!3-+++!>3Ue%FL7$Fhbl$!3W+++^%4li%FL7$Fcb l$!39+++h*3$oYFL7$F^bl$!3W+++$pL(4ZFL7$Fial$!3y*****H%3\"4v%FL7$Fdal$! 3e******\\)f>z%FL7$F_al$!3v*****zi\")H$[FL7$Fj`l$!3E+++R`/u[FL7$Fe`l$! 3U+++AS<:\\FL7$F``l$!3=+++L:Lc\\FL7$F[`l$!3c*****R89u*\\FL7$Ff_l$!3V++ +x*\\#Q]FL7$Fa_l$!3I+++H#3'y]FL7$F\\_l$!3e*****>)o@=^FL7$Fg^l$!3s***** \\e&yc^FL7$Fb^l$!3)********[GS>&FL7$F]^l$!3e*****zo#oH_FL7$Fh]l$!35+++ V)>NE&FL7$Fc]l$!3g*****R&3N&H&FL7$F^]l$!3,+++X\\-D`FL7$Fi\\l$!3W+++`^U _`FL7$Fd\\l$!3%)*****f.iuP&FL7$F_\\l$!3/+++Ep1+aFL7$Fj[l$!3,+++:b=?aFL 7$Fe[l$!3!)******\\@xPaFL7$F`[l$!39+++*)[y_aFL7$F[[l$!39+++X7=laFL7$Ff z$!3y*****fU9\\Z&FL7$Faz$!31+++[)H>[&FL7$F\\z$!3?+++n;;'[&FL7$Fgy$!3/+ ++=!Hv[&FL7$Fby$!3/+++.;$f[&FL7$F]y$!3)******\\6W7[&FL7$Fhx$!3A+++H(3L Z&FL7$Fcx$!3'******HXC>Y&FL7$F^x$!3/+++`7$oW&FL7$Fiw$!3:+++f^oFaFL7$Fd w$!3%)*****p1#FL7$Fbo$!3 3+++\">N?/#FL7$F]o$!3-+++Ff&\\)=FL7$$!3y******zMCsZFU$!3++++*>wys\"FL7 $$!3v*****pr())GEFU$!3-+++Mkzq:FL7$FS$!3&******ps;PT\"FL7$$!37+++4mzc[ FI$!3++++nqjc7FL7$$!39+++Q-`j:FI$!3++++Gub*4\"FL7$$!3'******>xbm%RFC$! 3g*****4&zxC%*F-7$$!3m*****\\es#zsF=$!3#)******H;)R&yF-7$$!3?+++!Q:ln) F7$!3))*****pI&=$G'F-7$F($!3<+++!)*)Q7ZF-7$F($!3)******Rl#fTJF-7$F($!3 ++++Fjzq:F-F'-%&COLORG6&%$RGBG$\"\"%!\"\"$\"\"#F]co$\"\"&!\"#-%)POLYGO NSG6f\\l7%F'7$F($\"+Fjzq:!#57$$!#YF]coF(7%Fgco7$F($\"+aEfTJFjcoF[do7%F _do7$F($\"+!)*)Q7ZFjcoF[do7%Fcdo7$$!+r)=ln)F_r$\"+2`=$G'FjcoF[do7%Fgdo 7$$!+pDFzsFds$\"+I;)R&yFjcoF[do7%F]eo7$$!+vdlYRF-$\"+^zxC%*FjcoF[do7%F ceo7$$!+P-`j:FL$\"+Gub*4\"!\"*F[do7%Fieo7$$!+:mzc[FL$\"+nqjc7F^foF[do7 %F`fo7$$!+Ki&QB\"!#;$\"+Fnr89F^foF[do7%Fffo7$$!+Dx))GEFifo$\"+Mkzq:F^f oF[do7%F]go7$$!+\"[VAx%Fifo$\"+*>wys\"F^foF[do7%Fcgo7$$!+)Hv)=vFifo$\" +Ff&\\)=F^foF[do7%Figo7$$!+k2f)3\"!#:$\"+\">N?/#F^foF[do7%F_ho7$$!+j'y ^s\"Fbho$\"+zF6*>#F^foF[do7%Ffho7$$!+PN-)y$Fbho$\"+Ef=cBF^foF[do7%F\\i o7$$!+*G\\93\"!#9$\"+I#\\K^#F^foF[do7%Fbio7$$!+@;K\"=$Feio$\"+EMHqEF^f oF[do7%Fiio7$$!+!R'ow')Feio$\"+NXIFGF^foF[do7%F_jo7$$!+A.]c@!#8$\"+CXE %)HF^foF[do7%Fejo7$$!+w1I?\\Fhjo$\"+@`:TJF^foF[do7%F\\[p7$$!+Yz1U5!#7$ \"+h*ozH$F^foF[do7%Fb[p7$$!+qUMp?Fe[p$\"+$HFZX$F^foF[do7%Fi[p7$$!+Ge%f )QFe[p$\"+i]^6OF^foF[do7%F_\\p7$$!+.!)*>&pFe[p$\"+8+`oPF^foF[do7%Fe\\p 7$$!+&R&>$>\"!#6$\"+vO:ERF^foF[do7%F[]p7$$!+#Hw\"z>F^]p$\"+kt/&3%F^foF [do7%Fb]p7$$!+\\-(**>$F^]p$\"+9xFYUF^foF[do7%Fh]p7$$!+<_R*4&F^]p$\"+Y@ X6WF^foF[do7%F^^p7$$!+*=8\\8)F^]p$\"+\\-l#e%F^foF[do7%Fd^p7$$!+Y/6D8Fj co$\"++dogZF^foF[do7%Fj^p7$$!+&p!*>?#Fjco$\"+i#Ra$\\F^foF[do7%F`_p7$$! +k7ZJMFjco$\"+&=Dk2&F^foF[do7%Ff_p7$$!++-x.ZFjco$\"+u^&R<&F^foF[do7%F \\`p7$$!+=0&f'eFjco$\"+H;uV_F^foF[do7%Fb`p7$$!+pp-:pFjco$\"+w,E(H&F^fo F[do7%Fh`p7$$!+>l-uyFjco$\"+$Q8,M&F^foF[do7%F^ap7$$!+H^Vj()Fjco$\"+R-; v`F^foF[do7%Fdap7$$!+@Nd)f*Fjco$\"+nq,/aF^foF[do7%Fjap7$$!+d)o!R5F^fo$ \"+f^oFaF^foF[do7%F`bp7$$!+rU\"[6\"F^fo$\"+`7$oW&F^foF[do7%Ffbp7$$!+*z Fx=\"F^fo$\"+`W#>Y&F^foF[do7%F\\cp7$$!+-0Ie7F^fo$\"+H(3LZ&F^foF[do7%Fb cp7$$!+f/#pK\"F^fo$\"+:TC\"[&F^foF[do7%Fhcp7$$!+I1!RR\"F^fo$\"+.;$f[&F ^foF[do7%F^dp7$$!+!o)\\f9F^fo$\"+=!Hv[&F^foF[do7%Fddp7$$!+Y9$R_\"F^fo$ \"+n;;'[&F^foF[do7%Fjdp7$$!+*y&Q(e\"F^fo$\"+[)H>[&F^foF[do7%F`ep7$$!+u p-];F^fo$\"+EW\"\\Z&F^foF[do7%Ffep7$$!+K`+7`M=F^fo$\"+]@xPaF^foF [do7%Fhfp7$$!+!)3N&*=F^fo$\"+:b=?aF^foF[do7%F^gp7$$!+``0c>F^fo$\"+Ep1+ aF^foF[do7%Fdgp7$$!+Uoy;?F^fo$\"+O?Yx`F^foF[do7%Fjgp7$$!+EIpx?F^fo$\"+ `^U_`F^foF[do7%F`hp7$$!+h\"G*Q@F^fo$\"+X\\-D`F^foF[do7%Ffhp7$$!+z;l+AF ^fo$\"+a3N&H&F^foF[do7%F\\ip7$$!+2W-jAF^fo$\"+V)>NE&F^foF[do7%Fbip7$$! +L&F^foF[do7% F^jp7$$!+%))*\\bCF^fo$\"+&e&yc^F^foF[do7%Fdjp7$$!+Vvz@DF^fo$\"+#)o@=^F ^foF[do7%Fjjp7$$!+6QA*e#F^fo$\"+H#3'y]F^foF[do7%F`[q7$$!+$[1!*QHF^fo$\"+R`/u[F^foF[do7%F^]q7$$!+Zky4IF^fo$\"+G;)H$[F^ foF[do7%Fd]q7$$!+M?S!3$F^fo$\"+])f>z%F^foF[do7%Fj]q7$$!+sK`]JF^fo$\"+V 3\"4v%F^foF[do7%F`^q7$$!+*e.+A$F^fo$\"+$pL(4ZF^foF[do7%Ff^q7$$!+Y^n)G$ F^fo$\"+h*3$oYF^foF[do7%F\\_q7$$!+&\\YkN$F^fo$\"+^%4li%F^foF[do7%Fb_q7 $$!+,$[KU$F^fo$\"+!>3Ue%F^foF[do7%Fh_q7$$!+i$Q!*[$F^fo$\"+OQGTXF^foF[d o7%F^`q7$$!+\"p'z`NF^fo$\"+OSi(\\%F^foF[do7%Fd`q7$$!+&4@vh$F^fo$\"+)GF JX%F^foF[do7%Fj`q7$$!+7MA!o$F^fo$\"+%z.xS%F^foF[do7%F`aq7$$!+3k#>u$F^f o$\"+%pv7O%F^foF[do7%Ffaq7$$!+u7m-QF^fo$\"+gpx8VF^foF[do7%F\\bq7$$!+&f lC'QF^fo$\"+XJ:lUF^foF[do7%Fbbq7$$!+v;Q@RF^fo$\"+/6O:UF^foF[do7%Fhbq7$ $!+0^XzRF^fo$\"+P(oV;%F^foF[do7%F^cq7$$!+^MtOSF^fo$\"+(za@6%F^foF[do7% Fdcq7$$!+%*\\E$4%F^fo$\"+G(3(eSF^foF[do7%Fjcq7$$!+Lu4\\TF^fo$\"+)\\IS+ %F^foF[do7%F`dq7$$!+FlF/UF^fo$\"+v.8[RF^foF[do7%Ffdq7$$!+\\Y%)eUF^fo$ \"+j&G5*QF^foF[do7%F\\eq7$$!+s#RGJ%F^fo$\"+?YvKQF^foF[do7%Fbeq7$$!+`9H mVF^fo$\"+&fYLx$F^foF[do7%Fheq7$$!+'QC#>WF^fo$\"+M)\\Gr$F^foF[do7%F^fq 7$$!+X@lrWF^fo$\"+HaJ^OF^foF[do7%Fdfq7$$!+P(yN_%F^fo$\"+e$)z)e$F^foF[d o7%Fjfq7$$!+#e(*\\d%F^fo$\"+s`NDNF^foF[do7%F`gq7$$!+&\\\"*ei%F^fo$\"+z F/hMF^foF[do7%Ffgq7$$!+:KBwYF^fo$\"+aU\"fR$F^foF[do7%F\\hq7$$!+ak)fs%F ^fo$\"+f)=+L$F^foF[do7%Fbhq7$$!+\"R2^x%F^fo$\"+-%*RjKF^foF[do7%Fhhq7$$ !+zkaB[F^fo$\"+a74'>$F^foF[do7%F^iq7$$!+>/Dr[F^fo$\"+x;7GJF^foF[do7%Fd iq7$$!+^T;=\\F^fo$\"+o'4&fIF^foF[do7%Fjiq7$$!+9FBk\\F^fo$\"+_iE!*HF^fo F[do7%F`jq7$$!+#*GS4]F^fo$\"+z^R?HF^foF[do7%Ffjq7$$!+nXi`]F^fo$\"+%)Q* )\\GF^foF[do7%F\\[r7$$!+`<&o4&F^fo$\"+UYvyFF^foF[do7%Fb[r7$$!+HL/R^F^f o$\"+:e'pq#F^foF[do7%Fh[r7$$!+![j,=&F^fo$\"+4J^MEF^foF[do7%F^\\r7$$!+% )==?_F^fo$\"+43QhDF^foF[do7%Fd\\r7$$!+%pt!f_F^fo$\"+MHb([#F^foF[do7%Fj \\r7$$!+_$>oH&F^fo$\"+,V,8CF^foF[do7%F`]r7$$!+`USL`F^fo$\"+&\\^xL#F^fo F[do7%Ff]r7$$!+4$=)o`F^fo$\"+@PvhAF^foF[do7%F\\^r7$$!+1a0.aF^fo$\"+hN, &=#F^foF[do7%Fb^r7$$!+%z7hV&F^fo$\"+F^foF[do7%Fd_r7$$!+P [@GbF^fo$\"+z:jq=F^foF[do7%Fj_r7$$!+()zccbF^fo$\"+CZA!z\"F^foF[do7%F`` r7$$!+QBv$e&F^fo$\"+b$H\"44JN%FjcoF[do7%F`fr 7$$!+\"*H/_eF^fo$\"+SZd$[$FjcoF[do7%Fffr7$$!+O5JdeF^fo$\"+=)*H8EFjcoF[ do7%F\\gr7$$!+nf3heF^fo$\"+-*)[U4JN%FjcoF[do7%Fbir7$F[fr$!+.vn@_FjcoF[do7% Ffir7$Feer$!+?(G!*3'FjcoF[do7%Fjir7$F_er$!+bM)[&pFjcoF[do7%F^jr7$Fidr$ !+e=$*=yFjcoF[do7%Fbjr7$Fcdr$!+&)H$3o)FjcoF[do7%Ffjr7$F]dr$!+Ud@S&*Fjc oF[do7%Fjjr7$Fgcr$!+a\"o'R5F^foF[do7%F^[s7$Facr$!+r3)\\7\"F^foF[do7%Fb [s7$F[cr$!+ce\"*47F^foF[do7%Ff[s7$Febr$!+!HGWH\"F^foF[do7%Fj[s7$F_br$! +oKZy8F^foF[do7%F^\\s7$Fiar$!+An+i9F^foF[do7%Fb\\s7$Fcar$!+oj)\\a\"F^f oF[do7%Ff\\s7$F]ar$!+4EPF;F^foF[do7%Fj\\s7$Fg`r$!+b$H\"4F^foF[do7%Fj]s7$F__r$!+LqHH?F^foF[do7%F^^s7$Fi^r$!+$F^foF[do7%F^bs7$Fihq$!+-%*RjKF^foF[do7%Fbbs7$Fchq$!+f)=+L$F^foF[do7% Ffbs7$F]hq$!+aU\"fR$F^foF[do7%Fjbs7$Fggq$!+zF/hMF^foF[do7%F^cs7$Fagq$! +s`NDNF^foF[do7%Fbcs7$F[gq$!+e$)z)e$F^foF[do7%Ffcs7$Fefq$!+HaJ^OF^foF[ do7%Fjcs7$F_fq$!+M)\\Gr$F^foF[do7%F^ds7$Fieq$!+&fYLx$F^foF[do7%Fbds7$F ceq$!+?YvKQF^foF[do7%Ffds7$F]eq$!+j&G5*QF^foF[do7%Fjds7$Fgdq$!+v.8[RF^ foF[do7%F^es7$Fadq$!+)\\IS+%F^foF[do7%Fbes7$F[dq$!+G(3(eSF^foF[do7%Ffe s7$Fecq$!+(za@6%F^foF[do7%Fjes7$F_cq$!+P(oV;%F^foF[do7%F^fs7$Fibq$!+/6 O:UF^foF[do7%Fbfs7$Fcbq$!+XJ:lUF^foF[do7%Fffs7$F]bq$!+gpx8VF^foF[do7%F jfs7$Fgaq$!+%pv7O%F^foF[do7%F^gs7$Faaq$!+%z.xS%F^foF[do7%Fbgs7$F[aq$!+ )GFJX%F^foF[do7%Ffgs7$Fe`q$!+OSi(\\%F^foF[do7%Fjgs7$F_`q$!+OQGTXF^foF[ do7%F^hs7$Fi_q$!+!>3Ue%F^foF[do7%Fbhs7$Fc_q$!+^%4li%F^foF[do7%Ffhs7$F] _q$!+h*3$oYF^foF[do7%Fjhs7$Fg^q$!+$pL(4ZF^foF[do7%F^is7$Fa^q$!+V3\"4v% F^foF[do7%Fbis7$F[^q$!+])f>z%F^foF[do7%Ffis7$Fe]q$!+G;)H$[F^foF[do7%Fj is7$F_]q$!+R`/u[F^foF[do7%F^js7$Fi\\q$!+AS<:\\F^foF[do7%Fbjs7$Fc\\q$!+ L:Lc\\F^foF[do7%Ffjs7$F]\\q$!+MTT(*\\F^foF[do7%Fjjs7$Fg[q$!+x*\\#Q]F^f oF[do7%F^[t7$Fa[q$!+H#3'y]F^foF[do7%Fb[t7$F[[q$!+#)o@=^F^foF[do7%Ff[t7 $Fejp$!+&e&yc^F^foF[do7%Fj[t7$F_jp$!+!\\GS>&F^foF[do7%F^\\t7$Fiip$!+)o #oH_F^foF[do7%Fb\\t7$Fcip$!+V)>NE&F^foF[do7%Ff\\t7$F]ip$!+a3N&H&F^foF[ do7%Fj\\t7$Fghp$!+X\\-D`F^foF[do7%F^]t7$Fahp$!+`^U_`F^foF[do7%Fb]t7$F[ hp$!+O?Yx`F^foF[do7%Ff]t7$Fegp$!+Ep1+aF^foF[do7%Fj]t7$F_gp$!+:b=?aF^fo F[do7%F^^t7$Fifp$!+]@xPaF^foF[do7%Fb^t7$Fcfp$!+*)[y_aF^foF[do7%Ff^t7$F ]fp$!+X7=laF^foF[do7%Fj^t7$Fgep$!+EW\"\\Z&F^foF[do7%F^_t7$Faep$!+[)H>[ &F^foF[do7%Fb_t7$F[ep$!+n;;'[&F^foF[do7%Ff_t7$Fedp$!+=!Hv[&F^foF[do7%F j_t7$F_dp$!+.;$f[&F^foF[do7%F^`t7$Ficp$!+:TC\"[&F^foF[do7%Fb`t7$Fccp$! +H(3LZ&F^foF[do7%Ff`t7$F]cp$!+`W#>Y&F^foF[do7%Fj`t7$Fgbp$!+`7$oW&F^foF [do7%F^at7$Fabp$!+f^oFaF^foF[do7%Fbat7$F[bp$!+nq,/aF^foF[do7%Ffat7$Fea p$!+R-;v`F^foF[do7%Fjat7$F_ap$!+$Q8,M&F^foF[do7%F^bt7$Fi`p$!+w,E(H&F^f oF[do7%Fbbt7$Fc`p$!+H;uV_F^foF[do7%Ffbt7$F]`p$!+u^&R<&F^foF[do7%Fjbt7$ Fg_p$!+&=Dk2&F^foF[do7%F^ct7$Fa_p$!+i#Ra$\\F^foF[do7%Fbct7$F[_p$!++dog ZF^foF[do7%Ffct7$Fe^p$!+\\-l#e%F^foF[do7%Fjct7$F_^p$!+Y@X6WF^foF[do7%F ^dt7$Fi]p$!+9xFYUF^foF[do7%Fbdt7$Fc]p$!+kt/&3%F^foF[do7%Ffdt7$F\\]p$!+ vO:ERF^foF[do7%Fjdt7$Ff\\p$!+8+`oPF^foF[do7%F^et7$F`\\p$!+i]^6OF^foF[d o7%Fbet7$Fj[p$!+$HFZX$F^foF[do7%Ffet7$Fc[p$!+h*ozH$F^foF[do7%Fjet7$F][ p$!+@`:TJF^foF[do7%F^ft7$Ffjo$!+CXE%)HF^foF[do7%Fbft7$$!+#Q'ow')Feio$! +NXIFGF^foF[do7%Ffft7$Fjio$!+EMHqEF^foF[do7%F\\gt7$Fcio$!+I#\\K^#F^foF [do7%F`gt7$$!+QN-)y$Fbho$!+Ef=cBF^foF[do7%Fdgt7$Fgho$!+zF6*>#F^foF[do7 %Fjgt7$F`ho$!+\">N?/#F^foF[do7%F^ht7$Fjgo$!+Ff&\\)=F^foF[do7%Fbht7$$!+ ![VAx%Fifo$!+*>wys\"F^foF[do7%Ffht7$$!+!yUP))F-$\"34+++*eV\"GdFL7$$\"3M+++ k$>M!*)F-$\"3y*****H$4\"*\\dFL7$$\"3Y******z\"H\"\\*)F-$\"3))******HPF rdFL7$$\"3%)******Hb;w*)F-$\"3y*****f?^@z&FL7$$\"3]*****\\abf)*)F-$\"3 C+++=eZ7eFL7$$\"3g*****f*exz*)F-$\"3A+++\\.>KeFL7$$\"3!)*****R%pxe*)F- $\"3))*****\\vX7&eFL7$$\"3K+++`>+C*)F-$\"3Q+++4&*fpeFL7$$\"3))******\\ DSw))F-$\"3#*******=V@()eFL7$$\"32+++B8&o\"))F-$\"3++++)4dS!fFL7$$\"3- +++(>_hu)F-$\"3C+++G\")4?fFL7$$\"3W+++5\"\\]m)F-$\"3B+++k.JNfFL7$$\"3' )*****RLLUd)F-$\"3%)*****f$)o'\\fFL7$$\"3]******f(\\VZ)F-$\"3K+++O,:jf FL7$$\"3k*****fN-gO)F-$\"3S+++k>tvfFL7$$\"3e******p*e(\\#)F-$\"3q***** pq#R()fFL7$$\"3^*****fkbh7)F-$\"35+++e56)*fFL7$$\"3[+++H0q&*zF-$\"3k** ***Hlly+'FL7$$\"3!)*****z\\x)eyF-$\"3%*******4Zj;gFL7$$\"3^+++%e\\hr(F -$\"3f******zcRCgFL7$$\"3-+++lA'zc(F-$\"3A+++\")[7JgFL7$$\"3i*****>sYZ T(F-$\"3Q+++BrzOgFL7$$\"3w*****H7BpD(F-$\"3'********H&QTgFL7$$\"3Y**** *z./\\4(F-$\"3V+++U*f[/'FL7$$\"3K+++#3)4HpF-$\"3e*****>s)=ZgFL7$$\"3a* ****\\#R\"*fnF-$\"35+++%)eL[gFL7$$\"33+++G[w(e'F-$\"3m*****\\eh#[gFL7$ $\"3o******=R28kF-$\"3#)*****R7@p/'FL7$$\"3y*****>^!GOiF-$\"3>+++7SEWg FL7$$\"3!)*****>'y%y0'F-$\"3i******eGBSgFL7$$\"3G+++IHFyeF-$\"3@+++&*> wMgFL7$$\"3i*****4(*)4)p&F-$\"3X+++xdxFgFL7$$\"3%)*****4;Kz^&F-$\"3m** ***HX'=>gFL7$$\"3x*****4.k%Q`F-$\"3e*****RX\"*)3gFL7$$\"3O+++ED]g^F-$ \"3s*****\\#*pn*fFL7$$\"3A+++/g,&)\\F-$\"3!)*****H@yE)fFL7$$\"30+++(G( >8[F-$\"3g*****>'RWmfFL7$$\"3)******>aflk%F-$\"3++++@$ey%fFL7$$\"3)*** ***zQ%3([%F-$\"3o*****>klm#fFL7$$\"3\")******R'euL%F-$\"3i******>+b-fF L7$$\"3*)******osY,UF-$\"37+++9%=^(eFL7$$\"3))*******GsY3%F-$\"3#***** *49#)Q%eFL7$$\"3))*****HHKc*RF-$\"3L+++ciC3eFL7$$\"31+++\"o_\"[RF-$\"3 #)******RPandFL7$$\"3)******R#[MlRF-$\"3'******p\\X7s&FL7$$\"3;+++#4Xa 3%F-$\"3U+++Q]wpcFL7$$\"3!******z%R\\gVF-$\"33+++y(=nh&FL7$$\"3$)***** f[$R9[F-$\"3'*******\\bWrbFL7$$\"3_+++&oStP&F-$\"3U+++kFXVbFL7$$\"3Y** ***fNy>%fF-$\"3()*****fIDI`&FL7$$\"3!******pnIGX'F-$\"3w*****ft\")\\`& FL7$$\"3'******H@wz*oF-$\"35+++E&GZa&FL7$$\"3k*****\\CB6G(F-$\"3g***** >MS$fbFL7$$\"3p*****4\\.%4wF-$\"3O+++M>/xbFL7$$\"3C+++%om(*)yF-$\"3!** *****)>Enf&FL7$$\"3A+++^OM!*)Fjco$\"+L4\"*\\dF^ fo7$$\"#kFbco$\"#eF]co7%Fc`v7$$\"+!=H\"\\*)Fjco$\"+IPFrdF^foFh`v7%F^av 7$$\"+Ib;w*)Fjco$\"+17:#z&F^foFh`v7%Fdav7$$\"+Xb&f)*)Fjco$\"+=eZ7eF^fo Fh`v7%Fjav7$$\"+'*exz*)Fjco$\"+\\.>KeF^foFh`v7%F`bv7$$\"+Wpxe*)Fjco$\" +bdC^eF^foFh`v7%Ffbv7$$\"+`>+C*)Fjco$\"+4&*fpeF^foFh`v7%F\\cv7$$\"+]DS w))Fjco$\"+>V@()eF^foFh`v7%Fbcv7$$\"+B8&o\"))Fjco$\"+)4dS!fF^foFh`v7%F hcv7$$\"+(>_hu)Fjco$\"+G\")4?fF^foFh`v7%F^dv7$$\"+5\"\\]m)Fjco$\"+k.JN fF^foFh`v7%Fddv7$$\"+MLBu&)Fjco$\"+O)o'\\fF^foFh`v7%Fjdv7$$\"+g(\\VZ)F jco$\"+O,:jfF^foFh`v7%F`ev7$$\"+cB+m$)Fjco$\"+k>tvfF^foFh`v7%Ffev7$$\" +q*e(\\#)Fjco$\"+2FR()fF^foFh`v7%F\\fv7$$\"+Yc:E\")Fjco$\"+e56)*fF^foF h`v7%Fbfv7$$\"+H0q&*zFjco$\"+`c'y+'F^foFh`v7%Fhfv7$$\"+)\\x)eyFjco$\"+ 5Zj;gF^foFh`v7%F^gv7$$\"+%e\\hr(Fjco$\"+!o&RCgF^foFh`v7%Fdgv7$$\"+lA'z c(Fjco$\"+\")[7JgF^foFh`v7%Fjgv7$$\"+Anu9uFjco$\"+BrzOgF^foFh`v7%F`hv7 $$\"+BJ#pD(Fjco$\"++`QTgF^foFh`v7%Ffhv7$$\"+QS!\\4(Fjco$\"+U*f[/'F^foF h`v7%F\\iv7$$\"+#3)4HpFjco$\"+A()=ZgF^foFh`v7%Fbiv7$$\"+DR\"*fnFjco$\" +%)eL[gF^foFh`v7%Fhiv7$$\"+G[w(e'Fjco$\"+&eh#[gF^foFh`v7%F^jv7$$\"+>R2 8kFjco$\"+C6#p/'F^foFh`v7%Fdjv7$$\"+70GOiFjco$\"+7SEWgF^foFh`v7%Fjjv7$ $\"+iy%y0'Fjco$\"+fGBSgF^foFh`v7%F`[w7$$\"+IHFyeFjco$\"+&*>wMgF^foFh`v 7%Ff[w7$$\"+r*)4)p&Fjco$\"+xdxFgF^foFh`v7%F\\\\w7$$\"+h@$z^&Fjco$\"+`k =>gF^foFh`v7%Fb\\w7$$\"+JSYQ`Fjco$\"+a9*)3gF^foFh`v7%Fh\\w7$$\"+ED]g^F jco$\"+D*pn*fF^foFh`v7%F^]w7$$\"+/g,&)\\Fjco$\"+8#yE)fF^foFh`v7%Fd]w7$ $\"+(G(>8[Fjco$\"+iRWmfF^foFh`v7%Fj]w7$$\"+U&flk%Fjco$\"+@$ey%fF^foFh` v7%F`^w7$$\"+)Q%3([%Fjco$\"+UcmEfF^foFh`v7%Ff^w7$$\"+S'euL%Fjco$\"+?+b -fF^foFh`v7%F\\_w7$$\"+psY,UFjco$\"+9%=^(eF^foFh`v7%Fb_w7$$\"+!HsY3%Fj co$\"+T@)Q%eF^foFh`v7%Fh_w7$$\"+$HKc*RFjco$\"+ciC3eF^foFh`v7%F^`w7$$\" +\"o_\"[RFjco$\"+SPandF^foFh`v7%Fd`w7$$\"+C[MlRFjco$\"+(\\X7s&F^foFh`v 7%Fj`w7$$\"+#4Xa3%Fjco$\"+Q]wpcF^foFh`v7%F`aw7$$\"+[R\\gVFjco$\"+y(=nh &F^foFh`v7%Ffaw7$$\"+'[$R9[Fjco$\"+]bWrbF^foFh`v7%F\\bw7$$\"+&oStP&Fjc o$\"+kFXVbF^foFh`v7%Fbbw7$$\"+c$y>%fFjco$\"+1`-LbF^foFh`v7%Fhbw7$$\"+x 1$GX'Fjco$\"+O<)\\`&F^foFh`v7%F^cw7$$\"+8i(z*oFjco$\"+E&GZa&F^foFh`v7% Fdcw7$$\"+XK7\"G(Fjco$\"+U.MfbF^foFh`v7%Fjcw7$$\"+\"\\.%4wFjco$\"+M>/x bF^foFh`v7%F`dw7$$\"+%om(*)yFjco$\"+*>Enf&F^foFh`v7%Ffdw7$$\"+^OC,P')Fjco$\"+,,\"Qo&F^f oFh`v7%F^fw7$$\"+bcK\\()Fjco$\"+&ppgq&F^foFh`v7%FdfwF^`vFh`vF`\\uFh\\u -F$6$7in7$F`]u$!34+++*eV\"GdFL7$Fg_v$!30+++&ppgq&FL7$Fb_v$!3s*****455Q o&FL7$F]_v$!3O+++%p;:m&FL7$Fh^v$!3=+++GYQRcFL7$Fc^v$!3L+++B7nEnf&FL7$Fi]v$!3O+++M>/xbFL7$Fd]v$!3g*****>MS$fbFL7$F_]v$ !35+++E&GZa&FL7$Fj\\v$!3w*****ft\")\\`&FL7$Fe\\v$!3()*****fIDI`&FL7$F` \\v$!3U+++kFXVbFL7$F[\\v$!3'*******\\bWrbFL7$Ff[v$!33+++y(=nh&FL7$Fa[v $!3U+++Q]wpcFL7$F\\[v$!3'******p\\X7s&FL7$Fgju$!3#)******RPandFL7$Fbju $!3L+++ciC3eFL7$F]ju$!3#******49#)Q%eFL7$Fhiu$!37+++9%=^(eFL7$Fciu$!3i ******>+b-fFL7$F^iu$!3o*****>klm#fFL7$Fihu$!3++++@$ey%fFL7$Fdhu$!3g*** **>'RWmfFL7$F_hu$!3!)*****H@yE)fFL7$Fjgu$!3s*****\\#*pn*fFL7$Fegu$!3e* ****RX\"*)3gFL7$F`gu$!3m*****HX'=>gFL7$F[gu$!3X+++xdxFgFL7$Fffu$!3@+++ &*>wMgFL7$Fafu$!3i******eGBSgFL7$F\\fu$!3>+++7SEWgFL7$Fgeu$!3#)*****R7 @p/'FL7$Fbeu$!3m*****\\eh#[gFL7$F]eu$!35+++%)eL[gFL7$Fhdu$!3e*****>s)= ZgFL7$Fcdu$!3V+++U*f[/'FL7$F^du$!3'********H&QTgFL7$Ficu$!3Q+++BrzOgFL 7$Fdcu$!3A+++\")[7JgFL7$F_cu$!3f******zcRCgFL7$Fjbu$!3%*******4Zj;gFL7 $Febu$!3k*****Hlly+'FL7$F`bu$!35+++e56)*fFL7$F[bu$!3q*****pq#R()fFL7$F fau$!3S+++k>tvfFL7$Faau$!3K+++O,:jfFL7$F\\au$!3%)*****f$)o'\\fFL7$Fg`u $!3B+++k.JNfFL7$Fb`u$!3C+++G\")4?fFL7$F]`u$!3++++)4dS!fFL7$Fh_u$!3#*** ****=V@()eFL7$Fc_u$!3Q+++4&*fpeFL7$F^_u$!3))*****\\vX7&eFL7$Fi^u$!3A++ +\\.>KeFL7$Fd^u$!3C+++=eZ7eFL7$F_^u$!3y*****f?^@z&FL7$Fj]u$!3))******H PFrdFL7$Fe]u$!3y*****H$4\"*\\dFLF]gwFgbo-Fdco6jn7%7$F_`v$!+*eV\"GdF^fo 7$Fefw$!+&ppgq&F^fo7$Fi`v$!#eF]co7%Fgbx7$F_fw$!+,,\"Qo&F^foFjbx7%F^cx7 $Fiew$!+%p;:m&F^foFjbx7%Fbcx7$Fcew$!+GYQRcF^foFjbx7%Ffcx7$F]ew$!+B7nEnf&F^foFjbx7%F^dx7$Fadw$!+M>/xbF^foFjbx7%Fbd x7$F[dw$!+U.MfbF^foFjbx7%Ffdx7$Fecw$!+E&GZa&F^foFjbx7%Fjdx7$F_cw$!+O<) \\`&F^foFjbx7%F^ex7$Fibw$!+1`-LbF^foFjbx7%Fbex7$Fcbw$!+kFXVbF^foFjbx7% Ffex7$F]bw$!+]bWrbF^foFjbx7%Fjex7$Fgaw$!+y(=nh&F^foFjbx7%F^fx7$Faaw$!+ Q]wpcF^foFjbx7%Fbfx7$F[aw$!+(\\X7s&F^foFjbx7%Fffx7$Fe`w$!+SPandF^foFjb x7%Fjfx7$F_`w$!+ciC3eF^foFjbx7%F^gx7$Fi_w$!+T@)Q%eF^foFjbx7%Fbgx7$Fc_w $!+9%=^(eF^foFjbx7%Ffgx7$F]_w$!+?+b-fF^foFjbx7%Fjgx7$Fg^w$!+UcmEfF^foF jbx7%F^hx7$Fa^w$!+@$ey%fF^foFjbx7%Fbhx7$F[^w$!+iRWmfF^foFjbx7%Ffhx7$Fe ]w$!+8#yE)fF^foFjbx7%Fjhx7$F_]w$!+D*pn*fF^foFjbx7%F^ix7$Fi\\w$!+a9*)3g F^foFjbx7%Fbix7$Fc\\w$!+`k=>gF^foFjbx7%Ffix7$F]\\w$!+xdxFgF^foFjbx7%Fj ix7$Fg[w$!+&*>wMgF^foFjbx7%F^jx7$Fa[w$!+fGBSgF^foFjbx7%Fbjx7$F[[w$!+7S EWgF^foFjbx7%Ffjx7$Fejv$!+C6#p/'F^foFjbx7%Fjjx7$F_jv$!+&eh#[gF^foFjbx7 %F^[y7$Fiiv$!+%)eL[gF^foFjbx7%Fb[y7$Fciv$!+A()=ZgF^foFjbx7%Ff[y7$F]iv$ !+U*f[/'F^foFjbx7%Fj[y7$Fghv$!++`QTgF^foFjbx7%F^\\y7$Fahv$!+BrzOgF^foF jbx7%Fb\\y7$F[hv$!+\")[7JgF^foFjbx7%Ff\\y7$Fegv$!+!o&RCgF^foFjbx7%Fj\\ y7$F_gv$!+5Zj;gF^foFjbx7%F^]y7$Fifv$!+`c'y+'F^foFjbx7%Fb]y7$Fcfv$!+e56 )*fF^foFjbx7%Ff]y7$F]fv$!+2FR()fF^foFjbx7%Fj]y7$Fgev$!+k>tvfF^foFjbx7% F^^y7$Faev$!+O,:jfF^foFjbx7%Fb^y7$F[ev$!+O)o'\\fF^foFjbx7%Ff^y7$Fedv$! +k.JNfF^foFjbx7%Fj^y7$F_dv$!+G\")4?fF^foFjbx7%F^_y7$Ficv$!+)4dS!fF^foF jbx7%Fb_y7$Fccv$!+>V@()eF^foFjbx7%Ff_y7$F]cv$!+4&*fpeF^foFjbx7%Fj_y7$F gbv$!+bdC^eF^foFjbx7%F^`y7$Fabv$!+\\.>KeF^foFjbx7%Fb`y7$F[bv$!+=eZ7eF^ foFjbx7%Ff`y7$Feav$!+17:#z&F^foFjbx7%Fj`y7$F_av$!+IPFrdF^foFjbx7%F^ay7 $Fd`v$!+L4\"*\\dF^foFjbx7%FbayFdbxFjbxF`\\uFh\\u-F$6%7$7$$!3Q++++++!p' FLF(7$$\"3!**************Q\"FLF(-%'COLOURG6&FjboF)F)F)-%*LINESTYLEG6# \"\"$-F$6%7$7$F($!3o*************Q'FL7$F($\"3o*************Q'FLF_byFbb y-%*AXESSTYLEG6#%$BOXG-%(SCALINGG6#%,CONSTRAINEDG-%+AXESLABELSG6%%&Re( z)G%&Im(z)G-%%FONTG6#%(DEFAULTG-F]dy6$%*HELVETICAG\"\"*-%%VIEWG6$;$!$p 'Fbco$\"$R\"Fbco;$!$R'Fbco$\"$R'Fbco" 1 2 0 1 10 0 2 9 1 2 1 1.000000 45.000000 45.000000 0 0 "Curve 1" "Curve 2" "Curve 3" "Curve 4" "Curve 5" "Curve 6" "Curve 7" "Curve 8" }}}}{PARA 0 "" 0 "" {TEXT -1 0 "" }} {PARA 0 "" 0 "" {TEXT -1 4 "The " }{TEXT 260 30 "interval of absolute \+ stability" }{TEXT -1 89 " (or stability interval) is the intersection of the stability region with the real line." }}{PARA 0 "" 0 "" {TEXT -1 59 "For this scheme the stability interval is (approximately) " } {XPPEDIT 18 0 "[-5.8641, 0];" "6#7$,$-%&FloatG6$\"&T'e!\"%!\"\"\"\"!" }{TEXT -1 2 ". " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 115 "We can distort the boundary curve horizontally by taking the 11th root of the real part of points along the curve. " }}{PARA 0 "" 0 "" {TEXT -1 33 "In this way we see that there is " }{TEXT 260 11 "no interval" }{TEXT -1 99 " on the nonnegative imaginary axis that contains the origin and lies inside the stability region. " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 673 "R := z -> add(z^j/j!,j=0..8)+\n 128379135556980976862855930208990 0589/464645307508149910111653200123658240000000*z^9+\n 315946586 39517996001118695569472358827/1161613268770374775279133000309145600000 00000*z^10+\n 4786869568791563788461323092360806890881/185858123 003259964044661280049463296000000000000*z^11+\n 2451412560816711 9977411129443654609199/22665624756495117566422107323105280000000000000 *z^12:\nDigits := 18:\npts := []: z0 := 0:\nfor ct from 0 to 200 do\n \+ zz := newton(R(z)=exp(ct*Pi/60*I),z=z0):\n z0 := zz:\n pts := [o p(pts),[surd(Re(zz),11),Im(zz)]]:\nend do:\nplot(pts,color=COLOR(RGB,. 9,.4,0),thickness=2,font=[HELVETICA,9]);\nDigits := 10:" }}{PARA 13 " " 1 "" {GLPLOT2D 228 258 258 {PLOTDATA 2 "6(-%'CURVESG6#7ew7$$\"\"!F)F (7$F($\"3t))H)fv()fB&!#>7$$!3[`nrk+#\\U#F-$\"3k(f'>^v>Z5!#=7$$!3\"*zt[ H$QQ]$F-$\"3`#*[zEjzq:F37$$!3wxD\")zJrZXF-$\"3=UJR-^R%4#F37$$!3\\Uyl!Q ,]c&F-$\"3636*z(Q*zh#F37$$!3#R%z=6IHglF-$\"36oye`EfTJF37$$!3[')*GL*)Rj `(F-$\"3#))p!=H9>lOF37$$!3%\\!)3^Bt[\\)F-$\"3o&)Hw/-z)=%F37$$!3KomAEj* pV*F-$\"3,84K!)*)Q7ZF37$$!3Dq7E'eRj.\"F3$\"3S!)*Gev()fB&F37$$!3lp=K2lW F6F3$\"3([$\\CJlefdF37$$!3\"=L#[pz.<7F3$\"3))Gb]1`=$G'F37$$!3V@y6$o9^I \"F3$\"3s(fD:3%y1oF37$$!3X)o'\\FPm\"R\"F3$\"3`KU?cGQItF37$$!3Oiu]B2mw9 F3$\"3g!=V/j\")R&yF37$$!3grV4)Hr+c\"F3$\"3BNQ=//ex$)F37$$!3n9^3d?&=k\" F3$\"3n3+Zx\"z6!*)F37$$!3giOb07&>s\"F3$\"3YhPa]zxC%*F37$$!39xE+4*3.!=F 3$\"3&*e/)Rsw$[**F37$$!30&y!fhu&o(=F3$\"3[Wm))\\v>Z5!#<7$$!3%H2LFJ@:&> F3$\"3xq:\"Fhq7$$!3iqQ4=I&[ 4#F3$\"3YKSb&=xU?\"Fhq7$$!3p4^.N(HL;#F3$\"3'G?Yl1PmD\"Fhq7$$!3\"H>)HZ$ R&HAF3$\"3d'49)\\p**38Fhq7$$!3xRK1!3mLH#F3$\"3p[\\BOoNh8Fhq7$$!3^,&=$f noaBF3$\"3rK5#os;PT\"Fhq7$$!36d.;_?P8CF3$\"3EM0lAm2m9Fhq7$$!3)>rD%3))G pCF3$\"3FR&QZ_O%=:Fhq7$$!3Zds'p=/B_#F3$\"3A>a#QV'zq:Fhq7$$!3T(o2\"QEHs DF3$\"3>q_0]j:B;Fhq7$$!3fJ*>_TZ\">EF3$\"3(\\J7DF;bn\"Fhq7$$!3d9'4')>*z iEF3$\"3!G`y&)>wys\"Fhq7$$!3Jj\\\"3tXKq#F3$\"3!>`rI7O-y\"Fhq7$$!3zT]]# 4)fSFF3$\"3G!>=r.'fK=Fhq7$$!3h)pkV/^^x#F3$\"3*\\IN&[2GF3$\"30QR%4x:t$>Fhq7$$!3*f9*GdagQGF3$\"3_X*\\(Rbn*)>Fhq7$$!3\\ Z%*o&R)4qGF3$\"37J+K\">N?/#Fhq7$$!3F=c,H%eU!HF3$\"3=t8&*oYR%4#Fhq7$$!3 29(fv+cS%HF3$\"3!HJQu*QvY@Fhq7$$!32#o(Q9ry#*HF3$\"3()=F'*yF6*>#Fhq7$$! 3P(**)=\"4SL0$F3$\"3guk^)=r9D#Fhq7$$!3'34>'3(Gt7$F3$\"3]Gh\"*o*GQI#Fhq 7$$!3u8@(\\)4h9KF3$\"3Yl@^Df=cBFhq7$$!3*p#)o-'*>NJ$F3$\"3]Pwu?=a3CFhq7 $$!3](>=(z!)e@MF3$\"3(G(Ryoj*3Y#Fhq7$$!35uicn[FONF3$\"3\"HX'[I#\\K^#Fh q7$$!3c*[:@m,al$F3$\"3\\X'[\"4+glDFhq7$$!3,;M$)z2HxPF3$\"3V0UVZ#[zh#Fh q7$$!3PY]vrAt+RF3$\"3FAc7EMHqEFhq7$$!3?X[!\\p()[-%F3$\"3(*\\=Rl\\jAFFh q7$$!3Nf&[N)**=\\TF3$\"3O.(=%HA(\\x#Fhq7$$!3FK\"3&e7EtUF3$\"3\\I]QNXIF GFhq7$$!3-@265?&oR%F3$\"3m@1\"z;J'zGFhq7$$!3%\\vvdT)z>XF3$\"3n.VC+9&>$ HFhq7$$!3M*))*GFK*>k%F3$\"3O4vfBXE%)HFhq7$$!3t)Qs1BmLw%F3$\"3*[#R:'))p l.$Fhq7$$!3Cr:\\m7(Q)[F3$\"3*oM$RVp'))3$Fhq7$$!31()f&))oxM+&F3$\"3\\N: X@`:TJFhq7$$!3T:h'y\"[;A^F3$\"3n+CH&*[V$>$Fhq7$$!3A>+c<%=*R_F3$\"3-f3 \\$)eqXKFhq7$$!3$*e31c&GnN&F3$\"3x-NTh*ozH$Fhq7$$!3&p#yi@%)esaF3$\"3:$ 3VXRD-N$Fhq7$$!3T1>D7Q\\(e&F3$\"32_)HO>xCS$Fhq7$$!3fUl#=6V9q&F3$\"3l7< G$HFZX$Fhq7$$!3w*>I%HwV9eF3$\"3bm_ab(zp]$Fhq7$$!37)RQ5A#[EfF3$\"3y,b++ +CfNFhq7$$!3a!R'[UhePgF3$\"3)y>q>1::h$Fhq7$$!32`%\\P5kx9'F3$\"3)*yAV!) Q\"Qm$Fhq7$$!3))*=Y5^PqD'F3$\"3V#*yt#)4z/rF3$\"3cu9paSYQTFhq7$$!3^]&4'R7!*3sF3$\"3d'> \"**Qx=#>%Fhq7$$!3Mlgdl[98tF3$\"3Q=\"3WrxiC%Fhq7$$!3!ppT$)='y<2lM9BvF3$\"3C=4fo@$eN%Fhq7$$!3p?JfK()fHwF3$\"3g 0@)f9_9T%Fhq7$$!3uxD$z#ygPxF3$\"3qVmJf\\unWFhq7$$!36`X>@yqZyF3$\"3TI^z d0zCXFhq7$$!3I][5z^^gzF3$\"38,%H)[-l#e%Fhq7$$!3%R.G=$)*pw!)F3$\"3$3j7* *oQ8k%Fhq7$$!3?I>=,]!p>)F3$\"39x%3qys2q%Fhq7$$!3'\\d%*4sf:K)F3$\"3$4#Q ')*p&ogZFhq7$$!3r#ok[r<0X)F3$\"3zYzFJM]?[Fhq7$$!3cCe]*GBDe)F3$\"3-l^k) HD#z[Fhq7$$!37$=X5YqZr)F3$\"3!R-qCERa$\\Fhq7$$!3d_Vngk;V))F3$\"3>\"[mg QMw)\\Fhq7$$!3!RW%G2tgj*)F3$\"3<\"=3wVKZ.&Fhq7$$!3ZG+I&\\=M2*F3$\"3H_c Q&=Dk2&Fhq7$$!3&>,f*)G6=<*F3$\"3\"e93TmmI6&Fhq7$$!3cS(GTQq$f#*F3$\"3-3 fCW[LX^Fhq7$$!3bM-J'yAtL*F3$\"3%[F!Gu^&R<&Fhq7$$!3G#fN=BOqS*F3$\"3F(4J C,o&*>&Fhq7$$!3ECFZU%z(p%*F3$\"3&4u*)=(QpA_Fhq7$$!3c/#yV>Um_*F3$\"3[@X \"*G;uV_Fhq7$$!3K*[sSVH&y&*F3$\"3mRQeR\"HIE&Fhq7$$!3uX)4ho#=E'*F3$\"31 me98R!3G&Fhq7$$!3[UjS6y?q'*F3$\"3,jpVw,E(H&Fhq7$$!39`Rd!3,6r*F3$\"3&*Q lmp>b7`Fhq7$$!30Pz#3;r#\\(*F3$\"3rq&\\r!H!oK&Fhq7$$!3DQFIDo0&y*F3$\"3 \\>6U$Q8,M&Fhq7$$!3P;dqL4u=)*F3$\"3[Bl4&*ec_`Fhq7$$!3Kq5q(Qh0&)*F3$\"3 EwHo#**GUO&Fhq7$$!3*41:%4(>2))*F3$\"3gsX)*Q-;v`Fhq7$$!3=ZkoyxQ4**F3$\" 3RQz#[W3aQ&Fhq7$$!3Vr\"Q3:8n$**F3$\"3ijRgv`,&R&Fhq7$$!3(y?u(zK#G'**F3$ \"3v&*y4nq,/aFhq7$$!3gC'GT*)Gy)**F3$\"3yh\"3-&[W7aFhq7$$!3v)p)[rE=,5Fh q$\"3Qu/rchK?aFhq7$$!3:Qizl,\\.5Fhq$\"3?Epxe^oFaFhq7$$!3(HJPJ'Gr05Fhq$ \"3iU]4\"GVXV&Fhq7$$!3?#RA6Sdy+\"Fhq$\"3`#fsWh>4W&Fhq7$$!3$*Q?:'oH*45F hq$\"3m18g_7$oW&Fhq7$$!3y_[$[*\\$>,\"Fhq$\"3eJWQoNH_aFhq7$$!3$3D$ze!yQ ,\"Fhq$\"3E1>NU/KdaFhq7$$!3[Tl'f8jd,\"Fhq$\"3+,L!GXC>Y&Fhq7$$!3$Gkt32% f<5Fhq$\"3Gz#e^.5Fhq$\"3TPFO;'3*paFhq7$$!3%H:jW726 -\"Fhq$\"3([\"p&*G(3LZ&Fhq7$$!3EZ%z%*G&zA5Fhq$\"31r%f%3hKwaFhq7$$!3(H! *oXZTW-\"Fhq$\"3=Rg)pxo*yaFhq7$$!3hWrU#4[g-\"Fhq$\"3&[%)[a6W7[&Fhq7$$! 30.k*)ethF5Fhq$\"3]VjBE*eJ[&Fhq7$$!3z^\">SJ^\"H5Fhq$\"3Z6$>z[>Z[&Fhq7$ $!3fc&R/%=lI5Fhq$\"3v'e]KgJf[&Fhq7$$!3`P:az17K5Fhq$\"3h\\K]T1!o[&Fhq7$ $!397![cWfN.\"Fhq$\"3,$)R,w:L([&Fhq7$$!3))*R`&Q'p\\.\"Fhq$\"3HO^l[&Fhq7$$!3%HMCAZZT/\"Fhq$\"3;9CiSr*)zaFhq7$$!3-`ZV$[s`/\"F hq$\"355mceybxaFhq7$$!3+eu^^(yl/\"Fhq$\"3ym#=bU9\\Z&Fhq7$$!3`(pZ.3nx/ \"Fhq$\"3$)[jw6\"p>Z&Fhq7$$!32#zJMBQ*[5Fhq$\"3g/@0\\SsoaFhq7$$!3_akMPH 4]5Fhq$\"3P$3g[C\"=laFhq7$$!3+D%p[)=B^5Fhq$\"3:;5['fU8Y&Fhq7$$!3w%)H$y tbB0\"Fhq$\"33.(z\\!*4sX&Fhq7$$!3zNwsH^Y`5Fhq$\"3fCX>*)[y_aFhq7$$!3#zZ z!o1ca5Fhq$\"3?])*))*>p![aFhq7$$!3?!\\@m$Hkb5Fhq$\"3O>\"yTVkIW&Fhq7$$! 3ucfF(\\7n0\"Fhq$\"32xVK]@xPaFhq7$$!3F4E'>*)px0\"Fhq$\"3-AA/$)Q>KaFhq7 $$!3tSYCWc\")e5Fhq$\"3.0hQf6LEaFhq7$$!3dLD#3E])f5Fhq$\"3b6.M:b=?aFhq7$ $!3c5O)GBu31\"Fhq$\"3=X$*>8&ePT&Fhq7$$!3&)GLJP!))=1\"Fhq$\"3KEt#)f<02a Fhq7$$!3gyDxP@*G1\"Fhq$\"3D-\\(e#p1+aFhq7$$!320Ok&)p)Q1\"Fhq$\"3-[/-md !GR&Fhq7$$!37Sc%3-t[1\"Fhq$\"3%oP(GS,F&Q&Fhq7$$!3))3*>Dn]e1\"Fhq$\"34v /ZO?Yx`Fhq7$$!3vxBef.#o1\"Fhq$\"3#\\e!o$f$Qp`Fhq7$$!3Y^99\"\\#yn5Fhq$ \"39.z)p7P5O&Fhq7$$!3MMoxmuto5Fhq$\"3Gt)3J:DCN&Fhq7$$!3p%)\\nwcop5Fhq$ \"3T&QRqT]NM&Fhq7$$!3,cgh,viq5Fhq$\"3JDr[>fTM`Fhq7$$!3?&*p!GJj:2\"Fhq$ \"3Wkz*[%\\-D`Fhq7$$!3#=ib:Z$\\s5Fhq$\"3B\\]!)*3\"Q:`Fhq7$$!31@6yG$=M2 \"Fhq$\"3F/\\:\"H)[0`Fhq7$$!3!o_dVAQV2\"Fhq$\"3.!4$=a3N&H&Fhq7$$!3pM&) G'[`_2\"Fhq$\"3*f3C$zM(\\G&Fhq7$$!3)[W9(HW;w5Fhq$\"3\\Y#[vGhVF&Fhq7$$! 3se_vb82x5Fhq$\"3%4*fUV)>NE&Fhq7$$!3<&)z>]X(z2\"Fhq$\"3&[+I^!3xz5Fhq$\"3#)fBF)o#oH_Fhq7 $$!3W)4+/Mk13\"Fhq$\"3a#G\"Qj$*)z@&Fhq7$$!3Zn[G1^b\"3\"Fhq$\"3Bu+`0=51 _Fhq7$$!3?]Ry%GVC3\"Fhq$\"3*G]w)*[GS>&Fhq7$$!32y6yN!HL3\"Fhq$\"3RaT0f$ y<=&Fhq7$$!3Kaz8#\\7U3\"Fhq$\"3$HUEdzg$p^Fhq7$$!3X6&G\"eP4&3\"Fhq$\"3C *)))4&e&yc^Fhq7$$!3onG13H(f3\"Fhq$\"3/Nf0nG1W^Fhq7$$!3Uq0;&)*\\o3\"Fhq $\"3n8-*y2.78&Fhq7$$!3Vftt+]s(3\"Fhq$\"3\"\\!f%>)o@=^Fhq7$$!3(y'GwLzf) 3\"Fhq$\"3*3W&)Q4:^5&Fhq7$$!3Nl,%3to%*3\"Fhq$\"3S@ip#f3>4&Fhq7$$!3!=JZ mIP.4\"Fhq$\"3]Ie&*G#3'y]Fhq7$$!3]zt'[a.74\"Fhq$\"3c:aCFZAl]Fhq7$$!3NR rh*Hn?4\"Fhq$\"3xvT\"feo<0&Fhq7$$!3&)*emtRGH4\"Fhq$\"31Lyiw*\\#Q]Fhq7$ $!39]FKRmy$4\"Fhq$\"3*)e#fikyY-&Fhq7$$!3[)zKU!=k%4\"Fhq$\"3\"4f&oBQ16] Fhq7$$!3R2\\a^O\\&4\"Fhq$\"3vxt$R89u*\\Fhq7$$!3wA6&[#>M'4\"Fhq$\"3Sta! >_PP)\\Fhq7$$!3)oFEbN'=(4\"Fhq$\"3)>?ku=T+(\\Fhq7$$!3/a3Zmm-)4\"Fhq$\" 3+voVL:Lc\\Fhq7$$!3KLU'edi))4\"Fhq$\"3AV0MFThU\\Fhq7$$!3[Xx$3!Qp*4\"Fh q$\"3kOGLxO*)G\\Fhq-%%FONTG6$%*HELVETICAG\"\"*-%*THICKNESSG6#\"\"#-%&C OLORG6&%$RGBG$Fajn!\"\"$\"\"%F[[oF(-%+AXESLABELSG6$Q!6\"Fa[o-%%VIEWG6$ %(DEFAULTGFf[o" 1 2 0 1 10 2 2 6 1 4 2 1.000000 45.000000 45.000000 0 0 "Curve 1" }}}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 37 "#------------------------------------" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 81 "The stability function R* for the 11 stage, order 7 scheme is given as follows." }}{EXCHG {PARA 0 " > " 0 "" {MPLTEXT 1 0 98 "subs(ee,subs(b=`b*`,StabilityFunction(7,13,' expanded'))):\n`R*` := unapply(%,z):\n'`R*`(z)'=`R*`(z);" }}{PARA 12 " " 1 "" {XPPMATH 20 "6#/-%#R*G6#%\"zG,:\"\"\"F)F'F)*&#F)\"\"#F)*$)F'F,F )F)F)*&#F)\"\"'F)*$)F'\"\"$F)F)F)*&#F)\"#CF)*$)F'\"\"%F)F)F)*&#F)\"$? \"F)*$)F'\"\"&F)F)F)*&#F)\"$?(F)*$)F'F1F)F)F)*&#F)\"%S]F)*$)F'\"\"(F)F )F)*&#\"A^>3G*H^D)Hhh0[)\\W$\"F+++g8&fL,+!o)>)en*op8F)*$)F'\"\")F)F)F) *&#\"GhG=NmSM=h)>R8;?E7+#G\"L+++++)GCp;Wf1Yck3t06!o*F)*$)F'\"\"*F)F)F) *&#\"H.,DfW1vhV1F&QY%zxG-\">\"N++++++G(p*o$o5)\\_)en\"e82f'F)*$)F'\"#5 F)F)F)*&#\"ILlcRKdO6kGdHbt\"3G5uJ\"\"P++++++KW:$oE/A[8K`xm2F&>'F)*$)F' \"#6F)F)F)" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 124 "We can find the point where the boundary of the stability regi on intersects the negative real axis by solving the equation: " }} {PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "`R*`(z) = -1;" "6#/-% #R*G6#%\"zG,$\"\"\"!\"\"" }{TEXT -1 2 ". " }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 31 "z_0 := newton(`R*`(z)=-1,z=-5);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%$z_0G$!+)HL\")p&!\"*" }}}{PARA 0 "" 0 "" {TEXT -1 0 " " }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 324 "z_0 := newton(`R*`(z)=- 1,z=-5.5):\np_1 := plot([`R*`(z),-1],z=-6.39..0.49,color=[red,blue]): \np_2 := plot([[[z_0,-1]]$3],style=point,symbol=[circle,cross,diamond] ,color=black):\np_3 := plot([[z_0,0],[z_0,-1]],linestyle=3,color=COLOR (RGB,0,.5,0)):\nplots[display]([p_1,p_2,p_3],view=[-6.39..0.49,-1.47.. 1.47],font=[HELVETICA,9]);" }}{PARA 13 "" 1 "" {GLPLOT2D 372 265 265 {PLOTDATA 2 "6+-%'CURVESG6$7W7$$!3o*************Q'!#<$!3?X%R1_#e*)QF*7 $$!3Qmmm*y<]J'F*$!3U'\\5ePxFQ$F*7$$!33LLLzb.SiF*$!3lQ&z@hwp$HF*7$$!3g* **\\!fRzuhF*$!39'34RFbMf#F*7$$!3:mmwQBb4hF*$!3;s,\"ye#)pG#F*7$$!3I*** \\LJ\"=OgF*$!3M1p#z`N?)>F*7$$!3MLL$zG5G'fF*$!3BXJ*)G-g9p#Gi)*y9F*7$$!3pKL`Nb4:eF*$!3#y@Hs11KF\"F*7$$!3_mmcXG3ocF* $!3w!p0$>.)**Q*!#=7$$!3zKL.jKyJbF*$!3pn3z2N&R-(FZ7$$!3%)****4SHl!R&F*$ !3eTF]d0e`^FZ7$$!37LLBedpW_F*$!3y7$oZ)Q^+PFZ7$$!3Y*****Gm1#*4&F*$!3$o+ -A,x^i#FZ7$$!3UmmYXKb\\\\F*$!3_qxtwzx7=FZ7$$!37LL$RVQx\"[F*$!33*GV]5HO G\"FZ7$$!3[****RwhMpYF*$!3\\T*46&\\dS%)!#>7$$!35+++1YM?XF*$!3!3%oAopOe _Fhp7$$!3c****f$)RvwVF*$!3X$zR?o-m.$Fhp7$$!39LLBU*fjC%F*$!3**=*\\S\\lq `\"Fhp7$$!3Kmm1:)384%F*$!33KPSRT7d?!#?7$$!3UmmEs,'*fRF*$\"3Id*G$**Hepl F]r7$$!3_****Hv/<2QF*$\"31U-Y'ysIY\"Fhp7$$!3?mmE/d#>n$F*$\"3MxRZm*\\#p ?Fhp7$$!3b****Hb2aBNF*$\"3#[*\\)>1Own#Fhp7$$!3]*****Q>VAQ$F*$\"3m]q?Xj mWKFhp7$$!3%fmm;,9[B$F*$\"3Cpx)p`v%fQFhp7$$!3%emm***zU*4$F*$\"3iD@'*=c ]qWFhp7$$!3)GLLj*eR`HF*$\"3kJ.`l&pL?&Fhp7$$!3)GLLAe4m'pEF*$\"3oOhR/!R:$pFhp7$$!3oKL`Hd0FDF*$\"3q$zat)p*[ *zFhp7$$!3]****>f^szBF*$\"3_4!Q&GH!GE*Fhp7$$!3w****RR4fNAF*$\"3U%3n\\- 3(p5FZ7$$!3a****\\qO8'4#F*$\"3XmIs)f;'H7FZ7$$!3^****>],HT>F*$\"3:R%\\2 P)QN9FZ7$$!3PmmEgj:-=F*$\"3e(onPkc&\\;FZ7$$!33*****>M,Ol\"F*$\"3*z&3Dz )oO\">FZ7$$!3LLL$eT))*=:F*$\"37`c5:TP*=#FZ7$$!3S****fJ4#=P\"F*$\"3+Fqm [sYODFZ7$$!3Fmm'zzZLB\"F*$\"3y,%>#>k;8HFZ7$$!3C****4uMg)3\"F*$\"3Y`$Hc YnoO$FZ7$$!3#ymmmS%*3Z*FZ$\"3yx@;_VoyQFZ7$$!3e$****\\it$*)zFZ$\"3?GX;1 t1)\\%FZ7$$!3-DLLHI\\ilFZ$\"37PI2g`$z=&FZ7$$!34CLLJ:I.^FZ$\"3aUH)HdtH+ 'FZ7$$!3'Glmc(G>cOFZ$\"3JV'zqwow$pFZ7$$!3(z)***>J]kK#FZ$\"3)eHJC8[V#zF Z7$$!33*emms_T-)Fhp$\"3c9qTbT$*G#*FZ7$$\"3$RrmmM)*pg&Fhp$\"3A4)y1qrw0 \"F*7$$\"3a0++Z[19?FZ$\"3Xn'36&=7B7F*7$$\"3o4++4170MFZ$\"3DJ!*e$>ncS\" F*7$$\"3!***************[FZ$\"3f\"R%Q@iJK;F*-%'COLOURG6&%$RGBG$\"*++++ \"!\")$\"\"!Ff\\lFe\\l-F$6$7S7$F($!\"\"Ff\\l7$F3F[]l7$F=F[]l7$FGF[]l7$ FQF[]l7$FVF[]l7$FfnF[]l7$F[oF[]l7$F`oF[]l7$FeoF[]l7$FjoF[]l7$F_pF[]l7$ FdpF[]l7$FjpF[]l7$F_qF[]l7$FdqF[]l7$FiqF[]l7$F_rF[]l7$FdrF[]l7$FirF[]l 7$F^sF[]l7$FcsF[]l7$FhsF[]l7$F]tF[]l7$FbtF[]l7$FgtF[]l7$F\\uF[]l7$FauF []l7$FfuF[]l7$F[vF[]l7$F`vF[]l7$FevF[]l7$FjvF[]l7$F_wF[]l7$FdwF[]l7$Fi wF[]l7$F^xF[]l7$FcxF[]l7$FhxF[]l7$F]yF[]l7$FbyF[]l7$FgyF[]l7$F\\zF[]l7 $FazF[]l7$FfzF[]l7$F[[lF[]l7$F`[lF[]l7$Fe[lF[]l7$Fj[lF[]l-F_\\l6&Fa\\l Fe\\lFe\\lFb\\l-F$6&7#7$$!37+++)HL\")p&F*F[]l-%'SYMBOLG6#%'CIRCLEG-F_ \\l6&Fa\\lFf\\lFf\\lFf\\l-%&STYLEG6#%&POINTG-F$6&Fa`l-Ff`l6#%&CROSSGFi `lF[al-F$6&Fa`l-Ff`l6#%(DIAMONDGFi`lF[al-F$6%7$7$Fc`lFe\\lFb`l-%&COLOR G6&Fa\\lFe\\l$\"\"&F\\]lFe\\l-%*LINESTYLEG6#\"\"$-%+AXESLABELSG6%Q\"z6 \"Q!Fjbl-%%FONTG6#%(DEFAULTG-F]cl6$%*HELVETICAG\"\"*-%%VIEWG6$;$!$R'! \"#$\"#\\Fjcl;$!$Z\"Fjcl$\"$Z\"Fjcl" 1 2 0 1 10 0 2 9 1 4 2 1.000000 45.000000 45.000000 0 0 "Curve 1" "Curve 2" "Curve 3" "Curve 4" "Curve 5" "Curve 6" }}}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 83 "The following picture shows the stability region for the \+ 13 stage, order 7 scheme. " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 968 "`R*` := z -> add(z^j/j!,j=0..7)+\n 34449848056161298255129928081951/136968967588198680000133595136 0000000*z^8+\n 28200122620161339198611834406635182861/9680110573 086456460659441669242880000000000*z^9+\n 19102287779446385270643 6175064459250103/659071358167588524981068368969728000000000000*z^10+\n 1317410280817355295728641136573239566533/6195270766775332134822 0426683154432000000000000*z^11:\npts := []: z0 := 0:\nfor ct from 0 to 360 do\n zz := newton(`R*`(z)=exp(ct*Pi/20*I),z=z0):\n z0 := zz: \n pts := [op(pts),[Re(zz),Im(zz)]]:\nend do:\np_1 := plot(pts,color =COLOR(RGB,.3,.15,0)):\np_2 := plots[polygonplot]([seq([pts[i-1],pts[i ],[-4.6,0]],i=2..nops(pts))],\n style=patchnogrid,color=COLOR (RGB,.75,.32,0)):\np_3 := plot([[[-6.49,0],[1.19,0]],[[0,-6.39],[0,6.3 9]]],color=black,linestyle=3):\nplots[display]([p_||(1..3)],view=[-6.4 9..1.19,-6.39..6.39],font=[HELVETICA,9],\n labels=[`Re(z)`, `Im(z)`],axes=boxed,scaling=constrained);" }}{PARA 13 "" 1 "" {GLPLOT2D 407 520 520 {PLOTDATA 2 "6+-%'CURVESG6$7eal7$$\"\"!F)F(7$F($ \"3++++Fjzq:!#=7$$!3#)*****Rv93H$!#G$\"3)******Rl#fTJF-7$$!3A+++BN.O$) !#F$\"3')*****H)*)Q7ZF-7$$!3G*****fbVl=)!#E$\"39+++P`=$G'F-7$$!3?+++r/ MpZ!#D$\"3[*****R&=)R&yF-7$$!33+++D(e6*>!#C$\"30+++(4zZU*F-7$$!3k***** \\rJde'FI$\"3)******R)yb*4\"!#<7$$!3\"******\\R^6$=!#B$\"3!******RdQmD \"FQ7$$!3'*******H8nWWFU$\"3#*******\\5s89FQ7$$!3O+++(f@ql*FU$\"3'**** ***Hv!3d\"FQ7$$!3-+++h,_5>!#A$\"3,+++TA!zs\"FQ7$$!3C+++wb_\"[$F_o$\"33 +++KF,&)=FQ7$$!39+++oA)e)eF_o$\"3#******fw^@/#FQ7$$!3!******p9++E*F_o$ \"3))*****>-S$*>#FQ7$$!37+++B87`8!#@$\"3=+++v)4mN#FQ7$$!3&******4,u#== Fdp$\"3y******G2,9DFQ7$$!39+++H4a'=#Fdp$\"3=+++!f;;n#FQ7$$!3'******f@& \\v@Fdp$\"3')*****f*p`HGFQ7$$!31+++OgrY7Fdp$\"3-+++uH$z)HFQ7$$\"33+++$ e4De\"Fdp$\"3++++_0/ZJFQ7$$\"3k*****pBMn2)Fdp$\"3!******4d0sI$FQ7$$\"3 !)*****4Wy59#!#?$\"3')******)4N*oMFQ7$$\"3)******HG,(RZFhr$\"3>+++/;(H j$FQ7$$\"3`+++uZK.(*Fhr$\"3=+++&[*Q+QFQ7$$\"3&******f%)y<#>!#>$\"35+++ (3hE(RFQ7$$\"33+++GY^(y$Fhs$\"3'******R0D8:%FQ7$$\"3Z+++dAA?vFhs$\"3q* ****R[R_L%FQ7$$\"30+++h\"\\KV\"F-$\"3*)*****f'*e+^%FQ7$$\"3'******>.qu J#F-$\"3s*****zEt$\\YFQ7$$\"3.+++pUIHJF-$\"3!)*****>cXLv%FQ7$$\"3C+++ \\I'*zPF-$\"3i*****Hkwq$[FQ7$$\"37+++Mq8%G%F-$\"3#*******eSh4\\FQ7$$\" 3!)*****4Mr(oYF-$\"3A+++QX7v\\FQ7$$\"35+++Ws%e&\\F-$\"39+++d&)fN]FQ7$$ \"3-+++yl%=;&F-$\"3%******H\"*f?4&FQ7$$\"3-+++j*4#*H&F-$\"3?+++Z,2X^FQ 7$$\"3g*****z[mvP&F-$\"3U+++9M&\\>&FQ7$$\"3k******4;d/aF-$\"3)*******[ [!>C&FQ7$$\"3i*****p#*ekQ&F-$\"3o*****>MYgG&FQ7$$\"3I+++EjSG`F-$\"3T++ +&pcuK&FQ7$$\"3,+++LTyM_F-$\"3e*****\\Y)=m`FQ7$$\"3B+++&oF$4^F-$\"3N++ +!\\xAS&FQ7$$\"3!)*****fqg_&\\F-$\"3f******Q#[dV&FQ7$$\"3!******4%fQvZ F-$\"3'******Rn;mY&FQ7$$\"3'*******\\M:sXF-$\"3c*****pg\"*[\\&FQ7$$\"3 !)******3KrZVF-$\"3t*****R5v0_&FQ7$$\"3'******zFcR5%F-$\"3O+++pAmVbFQ7 $$\"3*)*******HZD%QF-$\"31+++!fSTc&FQ7$$\"3w*****p2]\\c$F-$\"3E+++F*)) >e&FQ7$$\"3A+++rsWsKF-$\"3%*******zh<(f&FQ7$$\"3?+++dZ:mHF-$\"36+++&ff '4cFQ7$$\"3++++v1.ZEF-$\"3E+++QEQ>cFQ7$$\"3(******HtyeJ#F-$\"3()*****p Hsii&FQ7$$\"3)******\\pWL(>F-$\"3-+++NcBIcFQ7$$\"3*******f//*>;F-$\"3, +++$Gb6j&FQ7$$\"39+++0u$eD\"F-$\"3c*****fY$))GcFQ7$$\"3:+++Tx(=\"))Fhs $\"3;+++ERBBcFQ7$$\"3?+++()R*o&\\Fhs$\"3E+++'zqRh&FQ7$$\"3W+++eG*\\')* Fhr$\"3)******R(Hz+cFQ7$$!3********e;[7JFhs$\"3#******zt6Le&FQ7$$!3))* ****p_LHO(Fhs$\"3q*****p6=5c&FQ7$$!3-+++v_P!=\"F-$\"3l*****>&[BLbFQ7$$ !3'******\\X--l\"F-$\"3M+++9n/*\\&FQ7$$!37+++F^#y:#F-$\"3)*******Q#4sX &FQ7$$!3#)*****fvnis#F-$\"3z*****f>)31aFQ7$$!3z******y?n-MF-$\"3f***** \\C'*RM&FQ7$$!3u*****pM8NG%F-$\"3?+++HP%>F&FQ7$$!3/+++W4<\"\\&F-$\"3r* ****>:kT?&FQ7$$!34+++2%\\;#pF-$\"3#******>[wp;&FQ7$$!3!******R+F.I)F-$ \"3m*****phY,;&FQ7$$!3r*****z,H*Q&*F-$\"3=+++8>_o^FQ7$$!35+++&zR_1\"FQ $\"3!)*****\\C?I=&FQ7$$!3%******4$pwm6FQ$\"39+++xbX*>&FQ7$$!3/+++qpkg7 FQ$\"3\"******zOJe@&FQ7$$!3-+++94[[8FQ$\"3,+++&4H6B&FQ7$$!30+++uuWJ9FQ $\"3k******Q?![C&FQ7$$!33+++[*H/^\"FQ$\"3U+++>([lD&FQ7$$!3#*******pR5' e\"FQ$\"3e*****z)G?m_FQ7$$!3++++Tk**e;FQ$\"3i*****f:wOF&FQ7$$!3'****** fcD&HEE*y_FQ7$$!3!******zjG!)z\"FQ$\"3y*****>pQ>G&FQ7$$! 31+++6=yk=FQ$\"39+++scr#G&FQ7$$!3\"******>]9+$>FQ$\"3))*****zUp7G&FQ7$ $!3'******R7?R*>FQ$\"3')*****f-=wF&FQ7$$!3'******\\Bkm0#FQ$\"3')*****f $Gyr_FQ7$$!3@+++k3R=@FQ$\"3K+++rqyj_FQ7$$!3!)*****\\ZF#z@FQ$\"3m*****z 0bOD&FQ7$$!3z*****R#*)GRAFQ$\"3-+++\\?TT_FQ7$$!3=+++N1o)H#FQ$\"36+++YW 3F_FQ7$$!3;+++I6]dBFQ$\"3I+++y.q5_FQ7$$!3!)******)*R%eT#FQ$\"3y******y 0H#>&FQ7$$!3!******>N*ztCFQ$\"3!)*****Ha*)=<&FQ7$$!3/+++SZXJDFQ$\"3m** ***\\\"p`\\^FQ7$$!3#******z[&*))e#FQ$\"3\"******\\2z_7&FQ7$$!3'******> S/ik#FQ$\"3K+++?3<*4&FQ7$$!3y*****4ugMq#FQ$\"3q*****R+x72&FQ7$$!3')*** **RVQ2w#FQ$\"3$******4&RnT]FQ7$$!3y*****\\^.\"=GFQ$\"35+++g.X5]FQ7$$!3 =+++/4hvGFQ$\"37+++#R2x(\\FQ7$$!3%******>x+L$HFQ$\"3%)******GwbV\\FQ7$ $!3)******f\"\\>\"*HFQ$\"3>+++sE73\\FQ7$$!3!*******zOH\\IFQ$\"35+++$GH :([FQ7$$!34+++aSd2JFQ$\"3V+++!H/R$[FQ7$$!3%*******e%*)f;$FQ$\"39+++7(o `z%FQ7$$!34+++=:ZCKFQ$\"3Q+++>=.cZFQ7$$!31+++KP$HG$FQ$\"3))*****R$f)fr %FQ7$$!3*******pjw7M$FQ$\"3'******Ho-`n%FQ7$$!3#)*****fv$R*R$FQ$\"3P++ +46.MYFQ7$$!3%)*****\\cxrX$FQ$\"3q******>x>#f%FQ7$$!3$******4zCX^$FQ$ \"3Q+++8#3)\\XFQ7$$!3-+++E1MrNFQ$\"34+++)R]o]%FQ7$$!3'******RjTvi$FQ$ \"3G+++yxHjWFQ7$$!3%******4VdIo$FQ$\"3T+++-L6>WFQ7$$!3;+++48$yt$FQ$\"3 1+++@HDuVFQ7$$!3%******R1?=z$FQ$\"33+++E(o'GVFQ7$$!34+++&H$*\\%QFQ$\"3 5+++\"e6BG%FQ7$$!3')*******QKt*QFQ$\"33+++MK8NUFQ7$$!3=+++o$H)[RFQ$\"3 %)*****\\(z3(=%FQ7$$!3?+++!p&[**RFQ$\"3e******>R8QTFQ7$$!3'******H56$ \\SFQ$\"3y******3SB)3%FQ7$$!3C+++MDK)4%FQ$\"3$)*****Hzct.%FQ7$$!3#)*** **p.Vl9%FQ$\"39+++*4xa)RFQ7$$!3?+++w2+%>%FQ$\"3/+++!fwD$RFQ7$$!3\")*** ***))zsSUFQ$\"3A+++mUkyQFQ7$$!3!)*****4#*fnG%FQ$\"3#)*****>\"pnBQFQ7$$ !3C+++rO8KVFQ$\"35+++g%zww$FQ7$$!3U+++Tp)oP%FQ$\"3++++^_m5PFQ7$$!3#)** ***Hnc5U%FQ$\"3?+++Fhl_OFQ7$$!3Q+++%exYY%FQ$\"3%)*****pJ#o$f$FQ7$$!3q* ****pk!y2XFQ$\"3!)******H?yLNFQ7$$!3K+++!e\"R]XFQ$\"3!******\\s+IZ$FQ7 $$!3-+++U%HDf%FQ$\"35+++H+R6MFQ7$$!3%)*****>W0Uj%FQ$\"3'******HH1!\\LF Q7$$!3M+++x?UvYFQ$\"33+++f)3fG$FQ7$$!3.+++&pshr%FQ$\"3$)*****p5e@A$FQ7 $$!3e*****faTkv%FQ$\"31+++DM\"y:$FQ7$$!3#)*****pL/iz%FQ$\"37+++U7$H4$F Q7$$!3P+++d#Ha$[FQ$\"3!)*****RFjv-$FQ7$$!3M+++\"QyS([FQ$\"3y*****H5b<' HFQ7$$!33+++L$4@\"\\FQ$\"3=+++(=Xb*GFQ7$$!3A+++zqZ\\\\FQ$\"3.+++AV'*GG FQ7$$!3o*****RpNh)\\FQ$\"33+++fb.iFFQ7$$!30+++;+/A]FQ$\"3%******z_uZp# FQ7$$!3=+++#3Zr0&FQ$\"3!******H3!>FEFQ7$$!3O+++'H<94&FQ$\"33+++$=&GfDF Q7$$!3f******G`\"[7&FQ$\"3/+++Gz0\"\\#FQ7$$!3K+++D2Jd^FQ$\"3))*****Hn- DU#FQ7$$!3C+++i#y))=&FQ$\"3')******Q6h`BFQ7$$!3,+++*3)\\>_FQ$\"3-+++0N P%G#FQ7$$!3c******fc:\\_FQ$\"38+++D%zZ@#FQ7$$!3s*****z_TyF&FQ$\"3>+++f *=[9#FQ7$$!3L+++*)4b0`FQ$\"33+++;M[u?FQ7$$!30+++IOGK`FQ$\"3#******H3mP +#FQ7$$!39+++$yU!e`FQ$\"31+++hGmK>FQ7$$!3)******>*[$GQ&FQ$\"3'******zy s6'=FQ7$$!3U+++M)omS&FQ$\"3-+++P%)H*y\"FQ7$$!3V+++q^bHaFQ$\"35+++2i/<< FQ7$$!3Q+++p`]^aFQ$\"3.+++okUW;FQ7$$!3!*******35`saFQ$\"33+++qNXr:FQ7$ $!3m*****R*Hk#\\&FQ$\"3++++(pX\")\\\"FQ7$$!3!)*****Hz]=^&FQ$\"3'****** \\dCXU\"FQ7$$!3d*****4wh,`&FQ$\"3++++?]h]8FQ7$$!3U+++`0eZbFQ$\"33+++fV Ww7FQ7$$!3?+++y'3Tc&FQ$\"3.+++E@W(zbFQ$\"3$******fGP u7\"FQ7$$!3O+++\"p\"[%f&FQ$\"3++++29m_5FQ7$$!3O+++!>7$3cFQ$\"3i*****\\ OQux*F-7$$!3A+++wNA@cFQ$\"3s*****pGHr-*F-7$$!3%******z#4?LcFQ$\"3e**** *>]\\fF)F-7$$!3l*****H3FUk&FQ$\"3#******\\7NT_(F-7$$!3A+++xKGacFQ$\"3? +++g9*=x'F-7$$!35+++Z)\\Lm&FQ$\"3L+++*H!R>gF-7$$!3G+++YpSrcFQ$\"3w**** *>yonE&F-7$$!3=+++X_VycFQ$\"3#******HuHT^%F-7$$!3c******)e;Wo&FQ$\"37+ ++\"RV:w$F-7$$!3s*****4fM$*o&FQ$\"3%******f6]!4IF-7$$!3/+++R^<$p&FQ$\" 37+++JUmcAF-7$$!3Q+++?o#fp&FQ$\"3-+++D%yV]\"F-7$$!3k*****zC\"e(p&FQ$\" 39+++\\vo@vFhs7$$!37+++)HL\")p&FQF(7$Ficn$!39+++\\vo@vFhs7$Fdcn$!3-+++ D%yV]\"F-7$F_cn$!37+++JUmcAF-7$Fjbn$!3%******f6]!4IF-7$Febn$!37+++\"RV :w$F-7$F`bn$!3#******HuHT^%F-7$F[bn$!3w*****>yonE&F-7$Ffan$!3L+++*H!R> gF-7$Faan$!3?+++g9*=x'F-7$F\\an$!3#******\\7NT_(F-7$Fg`n$!3e*****>]\\f F)F-7$Fb`n$!3s*****pGHr-*F-7$F]`n$!3i*****\\OQux*F-7$Fh_n$!3++++29m_5F Q7$Fc_n$!3$******fGPu7\"FQ7$F^_n$!3.+++EFQ7$Fg[n$!3#** ****H3mP+#FQ7$Fb[n$!33+++;M[u?FQ7$F][n$!3>+++f*=[9#FQ7$Fhjm$!38+++D%zZ @#FQ7$Fcjm$!3-+++0NP%G#FQ7$F^jm$!3')******Q6h`BFQ7$Fiim$!3))*****Hn-DU #FQ7$Fdim$!3/+++Gz0\"\\#FQ7$F_im$!33+++$=&GfDFQ7$Fjhm$!3!******H3!>FEF Q7$Fehm$!3%******z_uZp#FQ7$F`hm$!33+++fb.iFFQ7$F[hm$!3.+++AV'*GGFQ7$Ff gm$!3=+++(=Xb*GFQ7$Fagm$!3y*****H5b<'HFQ7$F\\gm$!3!)*****RFjv-$FQ7$Fgf m$!37+++U7$H4$FQ7$Fbfm$!31+++DM\"y:$FQ7$F]fm$!3$)*****p5e@A$FQ7$Fhem$! 33+++f)3fG$FQ7$Fcem$!3'******HH1!\\LFQ7$F^em$!35+++H+R6MFQ7$Fidm$!3!** ****\\s+IZ$FQ7$Fddm$!3!)******H?yLNFQ7$F_dm$!3%)*****pJ#o$f$FQ7$Fjcm$! 3?+++Fhl_OFQ7$Fecm$!3++++^_m5PFQ7$F`cm$!35+++g%zww$FQ7$F[cm$!3#)*****> \"pnBQFQ7$Ffbm$!3A+++mUkyQFQ7$Fabm$!3/+++!fwD$RFQ7$F\\bm$!39+++*4xa)RF Q7$Fgam$!3$)*****Hzct.%FQ7$Fbam$!3y******3SB)3%FQ7$F]am$!3e******>R8QT FQ7$Fh`m$!3%)*****\\(z3(=%FQ7$Fc`m$!33+++MK8NUFQ7$F^`m$!35+++\"e6BG%FQ 7$Fi_m$!33+++E(o'GVFQ7$Fd_m$!31+++@HDuVFQ7$F__m$!3T+++-L6>WFQ7$Fj^m$!3 G+++yxHjWFQ7$Fe^m$!34+++)R]o]%FQ7$F`^m$!3Q+++8#3)\\XFQ7$F[^m$!3q****** >x>#f%FQ7$Ff]m$!3P+++46.MYFQ7$Fa]m$!3'******Ho-`n%FQ7$F\\]m$!3))*****R $f)fr%FQ7$Fg\\m$!3Q+++>=.cZFQ7$Fb\\m$!39+++7(o`z%FQ7$F]\\m$!3V+++!H/R$ [FQ7$Fh[m$!35+++$GH:([FQ7$Fc[m$!3>+++sE73\\FQ7$F^[m$!3%)******GwbV\\FQ 7$Fijl$!37+++#R2x(\\FQ7$Fdjl$!35+++g.X5]FQ7$F_jl$!3$******4&RnT]FQ7$Fj il$!3q*****R+x72&FQ7$Feil$!3K+++?3<*4&FQ7$F`il$!3\"******\\2z_7&FQ7$F[ il$!3m*****\\\"p`\\^FQ7$Ffhl$!3!)*****Ha*)=<&FQ7$Fahl$!3y******y0H#>&F Q7$F\\hl$!3I+++y.q5_FQ7$Fggl$!36+++YW3F_FQ7$Fbgl$!3-+++\\?TT_FQ7$F]gl$ !3m*****z0bOD&FQ7$Fhfl$!3K+++rqyj_FQ7$Fcfl$!3')*****f$Gyr_FQ7$F^fl$!3' )*****f-=wF&FQ7$Fiel$!3))*****zUp7G&FQ7$Fdel$!39+++scr#G&FQ7$F_el$!3y* ****>pQ>G&FQ7$Fjdl$!3q*****>EE*y_FQ7$Fedl$!3i*****f:wOF&FQ7$F`dl$!3e** ***z)G?m_FQ7$F[dl$!3U+++>([lD&FQ7$Ffcl$!3k******Q?![C&FQ7$Facl$!3,+++& 4H6B&FQ7$F\\cl$!3\"******zOJe@&FQ7$Fgbl$!39+++xbX*>&FQ7$Fbbl$!3!)***** \\C?I=&FQ7$F]bl$!3=+++8>_o^FQ7$Fhal$!3m*****phY,;&FQ7$Fcal$!3#******>[ wp;&FQ7$F^al$!3r*****>:kT?&FQ7$Fi`l$!3?+++HP%>F&FQ7$Fd`l$!3f*****\\C'* RM&FQ7$F_`l$!3z*****f>)31aFQ7$Fj_l$!3)*******Q#4sX&FQ7$Fe_l$!3M+++9n/* \\&FQ7$F`_l$!3l*****>&[BLbFQ7$F[_l$!3q*****p6=5c&FQ7$Ff^l$!3#******zt6 Le&FQ7$Fa^l$!3)******R(Hz+cFQ7$$\"3G+++))R*o&\\Fhs$!3E+++'zqRh&FQ7$Fg] l$!3;+++ERBBcFQ7$Fb]l$!3c*****fY$))GcFQ7$F]]l$!3,+++$Gb6j&FQ7$Fh\\l$!3 -+++NcBIcFQ7$Fc\\l$!3()*****pHsii&FQ7$F^\\l$!3E+++QEQ>cFQ7$Fi[l$!36+++ &ff'4cFQ7$Fd[l$!3%*******zh<(f&FQ7$F_[l$!3E+++F*))>e&FQ7$Fjz$!31+++!fS Tc&FQ7$Fez$!3O+++pAmVbFQ7$F`z$!3t*****R5v0_&FQ7$F[z$!3c*****pg\"*[\\&F Q7$Ffy$!3'******Rn;mY&FQ7$Fay$!3f******Q#[dV&FQ7$F\\y$!3N+++!\\xAS&FQ7 $Fgx$!3e*****\\Y)=m`FQ7$Fbx$!3T+++&pcuK&FQ7$F]x$!3o*****>MYgG&FQ7$Fhw$ !3)*******[[!>C&FQ7$Fcw$!3U+++9M&\\>&FQ7$F^w$!3?+++Z,2X^FQ7$Fiv$!3%*** ***H\"*f?4&FQ7$Fdv$!39+++d&)fN]FQ7$F_v$!3A+++QX7v\\FQ7$Fju$!3#*******e Sh4\\FQ7$Feu$!3i*****Hkwq$[FQ7$F`u$!3!)*****>cXLv%FQ7$F[u$!3s*****zEt$ \\YFQ7$Fft$!3*)*****f'*e+^%FQ7$Fat$!3q*****R[R_L%FQ7$F\\t$!3'******R0D 8:%FQ7$Ffs$!35+++(3hE(RFQ7$Fas$!3=+++&[*Q+QFQ7$F\\s$!3>+++/;(Hj$FQ7$Ff r$!3')******)4N*oMFQ7$Far$!3!******4d0sI$FQ7$$\"3-+++#e4De\"Fdp$!3++++ _0/ZJFQ7$Fgq$!3-+++uH$z)HFQ7$Fbq$!3')*****f*p`HGFQ7$F]q$!3=+++!f;;n#FQ 7$Fhp$!3y******G2,9DFQ7$Fbp$!3=+++v)4mN#FQ7$F]p$!3))*****>-S$*>#FQ7$Fh o$!3#******fw^@/#FQ7$Fco$!33+++KF,&)=FQ7$F]o$!3,+++TA!zs\"FQ7$$!3Q**** *Rf@ql*FU$!3'*******Hv!3d\"FQ7$FY$!3#*******\\5s89FQ7$FS$!3!******RdQm D\"FQ7$FM$!3)******R)yb*4\"FQ7$FG$!30+++(4zZU*F-7$FA$!3[*****R&=)R&yF- 7$$!3))******fNa'=)F=$!39+++P`=$G'F-7$$!3G+++'eQhL)F7$!3')*****H)*)Q7Z F-7$$!3;+++Z\\\"3H$F1$!3)******Rl#fTJF-7$F($!3++++Fjzq:F-F'-%&COLORG6& %$RGBG$\"\"$!\"\"$\"#:!\"#F(-%)POLYGONSG6fal7%F'7$F($\"+Fjzq:!#57$$!#Y F[gpF(7%Fcgp7$$!+aZ\"3H$Fhr$\"+aEfTJFfgpFggp7%F[hp7$$!+BN.O$)Fhs$\"+$) *)Q7ZFfgpFggp7%Fahp7$$!+cNa'=)F-$\"+P`=$G'FfgpFggp7%Fghp7$$!+r/MpZFQ$ \"+a=)R&yFfgpFggp7%F]ip7$$!+D(e6*>!#;$\"+(4zZU*FfgpFggp7%Fcip7$$!+:!#9$\"+TA!zs\"F_jpFggp7%Fd[q7$$!+wb_\"[$Fg[q$\"+KF,& )=F_jpFggp7%F[\\q7$$!+oA)e)eFg[q$\"+m<:U?F_jpFggp7%Fa\\q7$$!+Z,+g#*Fg[ q$\"+A+M*>#F_jpFggp7%Fg\\q7$$!+B87`8!#8$\"+v)4mN#F_jpFggp7%F]]q7$$!+6S F==F`]q$\"+H2,9DF_jpFggp7%Fd]q7$$!+H4a'=#F`]q$\"+!f;;n#F_jpFggp7%Fj]q7 $$!+;_\\v@F`]q$\"+'*p`HGF_jpFggp7%F`^q7$$!+OgrY7F`]q$\"+uH$z)HF_jpFggp 7%Ff^q7$$\"+$e4De\"F`]q$\"+_0/ZJF_jpFggp7%F\\_q7$$\"+PUtw!)F`]q$\"+rb? 2LF_jpFggp7%Fb_q7$$\"+T%y59#!#7$\"+*4N*oMF_jpFggp7%Fh_q7$$\"+$G,(RZF[` q$\"+/;(Hj$F_jpFggp7%F_`q7$$\"+uZK.(*F[`q$\"+&[*Q+QF_jpFggp7%Fe`q7$$\" +Y)y<#>!#6$\"+(3hE(RF_jpFggp7%F[aq7$$\"+GY^(y$F^aq$\"+a]K^TF_jpFggp7%F baq7$$\"+dAA?vF^aq$\"+%[R_L%F_jpFggp7%Fhaq7$$\"+h\"\\KV\"Ffgp$\"+m*e+^ %F_jpFggp7%F^bq7$$\"+K+Z&F_jpFggp7%Fdeq7$$\"+5;d/aFfgp$\"+\\[!>C&F_jpFggp7%Fjeq7$$\"+ F*ekQ&Ffgp$\"+Uj/'G&F_jpFggp7%F`fq7$$\"+EjSG`Ffgp$\"+&pcuK&F_jpFggp7%F ffq7$$\"+LTyM_Ffgp$\"+l%)=m`F_jpFggp7%F\\gq7$$\"+&oF$4^Ffgp$\"+!\\xAS& F_jpFggp7%Fbgq7$$\"+12Eb\\Ffgp$\"+R#[dV&F_jpFggp7%Fhgq7$$\"+TfQvZFfgp$ \"+umhmaF_jpFggp7%F^hq7$$\"+]M:sXFfgp$\"+2;*[\\&F_jpFggp7%Fdhq7$$\"+4K rZVFfgp$\"+/^d?bF_jpFggp7%Fjhq7$$\"+yi&R5%Ffgp$\"+pAmVbF_jpFggp7%F`iq7 $$\"++taUQFfgp$\"+!fSTc&F_jpFggp7%Ffiq7$$\"+x+&\\c$Ffgp$\"+F*))>e&F_jp Fggp7%F\\jq7$$\"+rsWsKFfgp$\"+!=wrf&F_jpFggp7%Fbjq7$$\"+dZ:mHFfgp$\"+& ff'4cF_jpFggp7%Fhjq7$$\"+v1.ZEFfgp$\"+QEQ>cF_jpFggp7%F^[r7$$\"+L(yeJ#F fgp$\"+(Hsii&F_jpFggp7%Fd[r7$$\"+&pWL(>Ffgp$\"+NcBIcF_jpFggp7%Fj[r7$$ \"+YS!*>;Ffgp$\"+$Gb6j&F_jpFggp7%F`\\r7$$\"+0u$eD\"Ffgp$\"+mM))GcF_jpF ggp7%Ff\\r7$$\"+Tx(=\"))F^aq$\"+ERBBcF_jpFggp7%F\\]r7$$\"+()R*o&\\F^aq $\"+'zqRh&F_jpFggp7%Fb]r7$$\"+eG*\\')*F[`q$\"+uHz+cF_jpFggp7%Fh]r7$$!+ f;[7JF^aq$\"+Q )31aF_jpFggp7%F\\`r7$$!+z?n-MFfgp$\"+Xi*RM&F_jpFggp7%Fb`r7$$!+ZL^$G%Ff gp$\"+HP%>F&F_jpFggp7%Fh`r7$$!+W4<\"\\&Ffgp$\"+_T;/_F_jpFggp7%F^ar7$$! +2%\\;#pFfgp$\"+#[wp;&F_jpFggp7%Fdar7$$!+/qK+$)Ffgp$\"+_o^F_jpFggp7%F`br7$$!+&zR_1\"F_jp$\"+X--$=&F _jpFggp7%Ffbr7$$!+Jpwm6F_jp$\"+xbX*>&F_jpFggp7%F\\cr7$$!+qpkg7F_jp$\"+ o8$e@&F_jpFggp7%Fbcr7$$!+94[[8F_jp$\"+&4H6B&F_jpFggp7%Fhcr7$$!+uuWJ9F_ jp$\"+R?![C&F_jpFggp7%F^dr7$$!+[*H/^\"F_jp$\"+>([lD&F_jpFggp7%Fddr7$$! +qR5'e\"F_jp$\"+))G?m_F_jpFggp7%Fjdr7$$!+Tk**e;F_jp$\"+chnt_F_jpFggp7% F`er7$$!+mb_HG&F _jpFggp7%F\\fr7$$!+6=yk=F_jp$\"+scr#G&F_jpFggp7%Fbfr7$$!+-X,I>F_jp$\"+ G%p7G&F_jpFggp7%Fhfr7$$!+C,#R*>F_jp$\"+E!=wF&F_jpFggp7%F^gr7$$!+NUmc?F _jp$\"+OGyr_F_jpFggp7%Fdgr7$$!+k3R=@F_jp$\"+rqyj_F_jpFggp7%Fjgr7$$!+vu Az@F_jp$\"+e]l`_F_jpFggp7%F`hr7$$!+C*)GRAF_jp$\"+\\?TT_F_jpFggp7%Ffhr7 $$!+N1o)H#F_jp$\"+YW3F_F_jpFggp7%F\\ir7$$!+I6]dBF_jp$\"+y.q5_F_jpFggp7 %Fbir7$$!+**R%eT#F_jp$\"+z0H#>&F_jpFggp7%Fhir7$$!+_$*ztCF_jp$\"+V&*)=< &F_jpFggp7%F^jr7$$!+SZXJDF_jp$\"+:p`\\^F_jpFggp7%Fdjr7$$!+)[&*))e#F_jp $\"+v!z_7&F_jpFggp7%Fjjr7$$!+-W?YEF_jp$\"+?3<*4&F_jpFggp7%F`[s7$$!+T2Y .FF_jp$\"+/qFr]F_jpFggp7%Ff[s7$$!+M%Q2w#F_jp$\"+^RnT]F_jpFggp7%F\\\\s7 $$!+:N5=GF_jp$\"+g.X5]F_jpFggp7%Fb\\s7$$!+/4hvGF_jp$\"+#R2x(\\F_jpFggp 7%Fh\\s7$$!+s2ILHF_jp$\"+HwbV\\F_jpFggp7%F^]s7$$!+;\\>\"*HF_jp$\"+sE73 \\F_jpFggp7%Fd]s7$$!+!o$H\\IF_jp$\"+$GH:([F_jpFggp7%Fj]s7$$!+aSd2JF_jp $\"+!H/R$[F_jpFggp7%F`^s7$$!+f%*)f;$F_jp$\"+7(o`z%F_jpFggp7%Ff^s7$$!+= :ZCKF_jp$\"+>=.cZF_jpFggp7%F\\_s7$$!+KP$HG$F_jp$\"+Mf)fr%F_jpFggp7%Fb_ s7$$!+PmFTLF_jp$\"+$o-`n%F_jpFggp7%Fh_s7$$!+cPR*R$F_jp$\"+46.MYF_jpFgg p7%F^`s7$$!+lv#f%F_jpFggp7%Fd`s7$$!+\"zCX^$F_jp$\"+8#3) \\XF_jpFggp7%Fj`s7$$!+E1MrNF_jp$\"+)R]o]%F_jpFggp7%F`as7$$!+M;aFOF_jp$ \"+yxHjWF_jpFggp7%Ffas7$$!+Ju0$o$F_jp$\"+-L6>WF_jpFggp7%F\\bs7$$!+48$y t$F_jp$\"+@HDuVF_jpFggp7%Fbbs7$$!+k+#=z$F_jp$\"+E(o'GVF_jpFggp7%Fhbs7$ $!+&H$*\\%QF_jp$\"+\"e6BG%F_jpFggp7%F^cs7$$!+!RKt*QF_jp$\"+MK8NUF_jpFg gp7%Fdcs7$$!+o$H)[RF_jp$\"+vz3(=%F_jpFggp7%Fjcs7$$!+!p&[**RF_jp$\"+?R8 QTF_jpFggp7%F`ds7$$!+.6J\\SF_jp$\"+4SB)3%F_jpFggp7%Ffds7$$!+MDK)4%F_jp $\"+$zct.%F_jpFggp7%F\\es7$$!+PIaYTF_jp$\"+*4xa)RF_jpFggp7%Fbes7$$!+w2 +%>%F_jp$\"+!fwD$RF_jpFggp7%Fhes7$$!+*)zsSUF_jp$\"+mUkyQF_jpFggp7%F^fs 7$$!+@*fnG%F_jp$\"+7pnBQF_jpFggp7%Fdfs7$$!+rO8KVF_jp$\"+g%zww$F_jpFggp 7%Fjfs7$$!+Tp)oP%F_jp$\"+^_m5PF_jpFggp7%F`gs7$$!+tm0@WF_jp$\"+Fhl_OF_j pFggp7%Ffgs7$$!+%exYY%F_jp$\"+FEF_jpFg gp7%Ff]t7$$!+'H<94&F_jp$\"+$=&GfDF_jpFggp7%F\\^t7$$!+H`\"[7&F_jp$\"+Gz 0\"\\#F_jpFggp7%Fb^t7$$!+D2Jd^F_jp$\"+tE]ACF_jpFggp7%Fh^t7$$!+i#y))=&F _jp$\"+R6h`BF_jpFggp7%F^_t7$$!+*3)\\>_F_jp$\"+0NP%G#F_jpFggp7%Fd_t7$$! +gc:\\_F_jp$\"+D%zZ@#F_jpFggp7%Fj_t7$$!+G:%yF&F_jp$\"+f*=[9#F_jpFggp7% F``t7$$!+*)4b0`F_jp$\"+;M[u?F_jpFggp7%Ff`t7$$!+IOGK`F_jp$\"+$3mP+#F_jp Fggp7%F\\at7$$!+$yU!e`F_jp$\"+hGmK>F_jpFggp7%Fbat7$$!+#*[$GQ&F_jp$\"+) ys6'=F_jpFggp7%Fhat7$$!+M)omS&F_jp$\"+P%)H*y\"F_jpFggp7%F^bt7$$!+q^bHa F_jp$\"+2i/<7$3cF_jp$\"+l$Qux*FfgpFggp7%F`ft7$$! +wNA@cF_jp$\"+(GHr-*FfgpFggp7%Ffft7$$!+G4?LcF_jp$\"+-&\\fF)FfgpFggp7%F \\gt7$$!+$3FUk&F_jp$\"+D^8CvFfgpFggp7%Fbgt7$$!+xKGacF_jp$\"+g9*=x'Ffgp Fggp7%Fhgt7$$!+Z)\\Lm&F_jp$\"+*H!R>gFfgpFggp7%F^ht7$$!+YpSrcF_jp$\"+#y onE&FfgpFggp7%Fdht7$$!+X_VycF_jp$\"+V(HT^%FfgpFggp7%Fjht7$$!+*e;Wo&F_j p$\"+\"RV:w$FfgpFggp7%F`it7$$!+\"fM$*o&F_jp$\"+;,04IFfgpFggp7%Ffit7$$! +R^<$p&F_jp$\"+JUmcAFfgpFggp7%F\\jt7$$!+?o#fp&F_jp$\"+D%yV]\"FfgpFggp7 %Fbjt7$$!+[7e(p&F_jp$\"+\\vo@vF^aqFggp7%Fhjt7$$!+)HL\")p&F_jpF(Fggp7%F ^[u7$Fijt$!+\\vo@vF^aqFggp7%Fb[u7$Fcjt$!+D%yV]\"FfgpFggp7%Ff[u7$F]jt$! +JUmcAFfgpFggp7%Fj[u7$Fgit$!+;,04IFfgpFggp7%F^\\u7$Fait$!+\"RV:w$FfgpF ggp7%Fb\\u7$F[it$!+V(HT^%FfgpFggp7%Ff\\u7$Feht$!+#yonE&FfgpFggp7%Fj\\u 7$F_ht$!+*H!R>gFfgpFggp7%F^]u7$Figt$!+g9*=x'FfgpFggp7%Fb]u7$Fcgt$!+D^8 CvFfgpFggp7%Ff]u7$F]gt$!+-&\\fF)FfgpFggp7%Fj]u7$Fgft$!+(GHr-*FfgpFggp7 %F^^u7$Faft$!+l$Qux*FfgpFggp7%Fb^u7$F[ft$!+29m_5F_jpFggp7%Ff^u7$Feet$! +'GPu7\"F_jpFggp7%Fj^u7$F_et$!+EF_jpFggp7%Ffau7$F]at$!+$3 mP+#F_jpFggp7%Fjau7$Fg`t$!+;M[u?F_jpFggp7%F^bu7$Fa`t$!+f*=[9#F_jpFggp7 %Fbbu7$F[`t$!+D%zZ@#F_jpFggp7%Ffbu7$Fe_t$!+0NP%G#F_jpFggp7%Fjbu7$F__t$ !+R6h`BF_jpFggp7%F^cu7$Fi^t$!+tE]ACF_jpFggp7%Fbcu7$Fc^t$!+Gz0\"\\#F_jp Fggp7%Ffcu7$F]^t$!+$=&GfDF_jpFggp7%Fjcu7$Fg]t$!+$3!>FEF_jpFggp7%F^du7$ Fa]t$!+GXx%p#F_jpFggp7%Fbdu7$F[]t$!+fb.iFF_jpFggp7%Ffdu7$Fe\\t$!+AV'*G GF_jpFggp7%Fjdu7$F_\\t$!+(=Xb*GF_jpFggp7%F^eu7$Fi[t$!+.^vhHF_jpFggp7%F beu7$Fc[t$!+uKcFIF_jpFggp7%Ffeu7$F][t$!+U7$H4$F_jpFggp7%Fjeu7$Fgjs$!+D M\"y:$F_jpFggp7%F^fu7$Fajs$!+2\"e@A$F_jpFggp7%Fbfu7$F[js$!+f)3fG$F_jpF ggp7%Fffu7$Feis$!+$H1!\\LF_jpFggp7%Fjfu7$F_is$!+H+R6MF_jpFggp7%F^gu7$F ihs$!+D2+tMF_jpFggp7%Fbgu7$Fchs$!+I?yLNF_jpFggp7%Ffgu7$F]hs$!+WF_jpFggp7%Fj[v7$Fgas$!+yxHjWF_jpFgg p7%F^\\v7$Faas$!+)R]o]%F_jpFggp7%Fb\\v7$F[as$!+8#3)\\XF_jpFggp7%Ff\\v7 $Fe`s$!+?x>#f%F_jpFggp7%Fj\\v7$F_`s$!+46.MYF_jpFggp7%F^]v7$Fi_s$!+$o-` n%F_jpFggp7%Fb]v7$Fc_s$!+Mf)fr%F_jpFggp7%Ff]v7$F]_s$!+>=.cZF_jpFggp7%F j]v7$Fg^s$!+7(o`z%F_jpFggp7%F^^v7$Fa^s$!+!H/R$[F_jpFggp7%Fb^v7$F[^s$!+ $GH:([F_jpFggp7%Ff^v7$Fe]s$!+sE73\\F_jpFggp7%Fj^v7$F_]s$!+HwbV\\F_jpFg gp7%F^_v7$Fi\\s$!+#R2x(\\F_jpFggp7%Fb_v7$Fc\\s$!+g.X5]F_jpFggp7%Ff_v7$ F]\\s$!+^RnT]F_jpFggp7%Fj_v7$Fg[s$!+/qFr]F_jpFggp7%F^`v7$Fa[s$!+?3<*4& F_jpFggp7%Fb`v7$F[[s$!+v!z_7&F_jpFggp7%Ff`v7$Fejr$!+:p`\\^F_jpFggp7%Fj `v7$F_jr$!+V&*)=<&F_jpFggp7%F^av7$Fiir$!+z0H#>&F_jpFggp7%Fbav7$Fcir$!+ y.q5_F_jpFggp7%Ffav7$F]ir$!+YW3F_F_jpFggp7%Fjav7$Fghr$!+\\?TT_F_jpFggp 7%F^bv7$Fahr$!+e]l`_F_jpFggp7%Fbbv7$F[hr$!+rqyj_F_jpFggp7%Ffbv7$Fegr$! +OGyr_F_jpFggp7%Fjbv7$F_gr$!+E!=wF&F_jpFggp7%F^cv7$Fifr$!+G%p7G&F_jpFg gp7%Fbcv7$Fcfr$!+scr#G&F_jpFggp7%Ffcv7$F]fr$!+#pQ>G&F_jpFggp7%Fjcv7$Fg er$!+ii#*y_F_jpFggp7%F^dv7$Faer$!+chnt_F_jpFggp7%Fbdv7$F[er$!+))G?m_F_ jpFggp7%Ffdv7$Fedr$!+>([lD&F_jpFggp7%Fjdv7$F_dr$!+R?![C&F_jpFggp7%F^ev 7$Ficr$!+&4H6B&F_jpFggp7%Fbev7$Fccr$!+o8$e@&F_jpFggp7%Ffev7$F]cr$!+xbX *>&F_jpFggp7%Fjev7$Fgbr$!+X--$=&F_jpFggp7%F^fv7$Fabr$!+8>_o^F_jpFggp7% Fbfv7$F[br$!+F&F_jpFggp7%Fbgv7$Fc`r$!+Xi*RM&F_jpFgg p7%Ffgv7$F]`r$!+'>)31aF_jpFggp7%Fjgv7$Fg_r$!+R#4sX&F_jpFggp7%F^hv7$Fa_ r$!+9n/*\\&F_jpFggp7%Fbhv7$F[_r$!+_[BLbF_jpFggp7%Ffhv7$Fe^r$!+<\"=5c&F _jpFggp7%Fjhv7$F_^r$!+QcF_jpFgg p7%F`[w7$Fijq$!+&ff'4cF_jpFggp7%Fd[w7$Fcjq$!+!=wrf&F_jpFggp7%Fh[w7$F]j q$!+F*))>e&F_jpFggp7%F\\\\w7$Fgiq$!+!fSTc&F_jpFggp7%F`\\w7$Faiq$!+pAmV bF_jpFggp7%Fd\\w7$F[iq$!+/^d?bF_jpFggp7%Fh\\w7$Fehq$!+2;*[\\&F_jpFggp7 %F\\]w7$F_hq$!+umhmaF_jpFggp7%F`]w7$Figq$!+R#[dV&F_jpFggp7%Fd]w7$Fcgq$ !+!\\xAS&F_jpFggp7%Fh]w7$F]gq$!+l%)=m`F_jpFggp7%F\\^w7$Fgfq$!+&pcuK&F_ jpFggp7%F`^w7$Fafq$!+Uj/'G&F_jpFggp7%Fd^w7$F[fq$!+\\[!>C&F_jpFggp7%Fh^ w7$Feeq$!+9M&\\>&F_jpFggp7%F\\_w7$F_eq$!+Z,2X^F_jpFggp7%F`_w7$Fidq$!+8 *f?4&F_jpFggp7%Fd_w7$Fcdq$!+d&)fN]F_jpFggp7%Fh_w7$F]dq$!+QX7v\\F_jpFgg p7%F\\`w7$Fgcq$!+fSh4\\F_jpFggp7%F``w7$Facq$!+Vm2P[F_jpFggp7%Fd`w7$F[c q$!+ibM`ZF_jpFggp7%Fh`w7$Febq$!+oKP\\YF_jpFggp7%F\\aw7$F_bq$!+m*e+^%F_ jpFggp7%F`aw7$Fiaq$!+%[R_L%F_jpFggp7%Fdaw7$Fcaq$!+a]K^TF_jpFggp7%Fhaw7 $F\\aq$!+(3hE(RF_jpFggp7%F\\bw7$Ff`q$!+&[*Q+QF_jpFggp7%F`bw7$F``q$!+/; (Hj$F_jpFggp7%Fdbw7$Fi_q$!+*4N*oMF_jpFggp7%Fhbw7$Fc_q$!+rb?2LF_jpFggp7 %F\\cw7$$\"+#e4De\"F`]q$!+_0/ZJF_jpFggp7%F`cw7$Fg^q$!+uH$z)HF_jpFggp7% Ffcw7$Fa^q$!+'*p`HGF_jpFggp7%Fjcw7$F[^q$!+!f;;n#F_jpFggp7%F^dw7$Fe]q$! +H2,9DF_jpFggp7%Fbdw7$F^]q$!+v)4mN#F_jpFggp7%Ffdw7$Fh\\q$!+A+M*>#F_jpF ggp7%Fjdw7$Fb\\q$!+m<:U?F_jpFggp7%F^ew7$F\\\\q$!+KF,&)=F_jpFggp7%Fbew7 $Fe[q$!+TA!zs\"F_jpFggp7%Ffew7$$!+%f@ql*Fdjp$!+Iv!3d\"F_jpFggp7%Fjew7$ Fijp$!+]5s89F_jpFggp7%F`fw7$Fbjp$!+u&QmD\"F_jpFggp7%Fdfw7$F[jp$!+%)yb* 4\"F_jpFggp7%Fhfw7$Fdip$!+(4zZU*FfgpFggp7%F\\gw7$F^ip$!+a=)R&yFfgpFggp 7%F`gw7$$!+gNa'=)F-$!+P`=$G'FfgpFggp7%Fdgw7$$!+'eQhL)Fhs$!+$)*)Q7ZFfgp Fggp7%Fjgw7$$!+Z\\\"3H$Fhr$!+aEfTJFfgpFggp7%F`hw7$F($!+Fjzq:FfgpFggp7% FfhwF'Fggp-Fffp6&Fhfp$\"#vF^gp$\"#KF^gpF(-%&STYLEG6#%,PATCHNOGRIDG-F$6 %7$7$$!3A++++++!\\'FQF(7$$\"3%**************=\"FQF(-%'COLOURG6&FhfpF)F )F)-%*LINESTYLEG6#Fjfp-F$6%7$7$F($!3o*************Q'FQ7$F($\"3o******* ******Q'FQF]jwF`jw-%*AXESSTYLEG6#%$BOXG-%(SCALINGG6#%,CONSTRAINEDG-%+A XESLABELSG6%%&Re(z)G%&Im(z)G-%%FONTG6#%(DEFAULTG-Fj[x6$%*HELVETICAG\" \"*-%%VIEWG6$;$!$\\'F^gp$\"$>\"F^gp;$!$R'F^gp$\"$R'F^gp" 1 2 0 1 10 0 2 9 1 2 1 1.000000 45.000000 45.000000 0 0 "Curve 1" "Curve 2" "Curve \+ 3" "Curve 4" }}}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 75 "The stability interval for the embedded order 7 scheme is (approximately) " }{XPPEDIT 18 0 "[-5.6981, 0];" "6#7$,$-%&FloatG6$ \"&\")p&!\"%!\"\"\"\"!" }{TEXT -1 2 ". " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 79 "The stability regions for the two sc hemes can be displayed in the same picture." }}{EXCHG {PARA 0 "> " 0 " " {MPLTEXT 1 0 154 "plots[display]([p_1,p_2,p||(1..7)],view=[-6.49..1. 19,-6.39..6.39],font=[HELVETICA,9],\n labels=[`Re(z)`,`Im(z) `],axes=boxed,scaling=constrained);" }}{PARA 13 "" 1 "" {GLPLOT2D 483 531 531 {PLOTDATA 2 "61-%'CURVESG6$7eal7$$\"\"!F)F(7$F($\"3++++Fjzq:!# =7$$!3#)*****Rv93H$!#G$\"3)******Rl#fTJF-7$$!3A+++BN.O$)!#F$\"3')***** H)*)Q7ZF-7$$!3G*****fbVl=)!#E$\"39+++P`=$G'F-7$$!3?+++r/MpZ!#D$\"3[*** **R&=)R&yF-7$$!33+++D(e6*>!#C$\"30+++(4zZU*F-7$$!3k*****\\rJde'FI$\"3) ******R)yb*4\"!#<7$$!3\"******\\R^6$=!#B$\"3!******RdQmD\"FQ7$$!3'**** ***H8nWWFU$\"3#*******\\5s89FQ7$$!3O+++(f@ql*FU$\"3'*******Hv!3d\"FQ7$ $!3-+++h,_5>!#A$\"3,+++TA!zs\"FQ7$$!3C+++wb_\"[$F_o$\"33+++KF,&)=FQ7$$ !39+++oA)e)eF_o$\"3#******fw^@/#FQ7$$!3!******p9++E*F_o$\"3))*****>-S$ *>#FQ7$$!37+++B87`8!#@$\"3=+++v)4mN#FQ7$$!3&******4,u#==Fdp$\"3y****** G2,9DFQ7$$!39+++H4a'=#Fdp$\"3=+++!f;;n#FQ7$$!3'******f@&\\v@Fdp$\"3')* ****f*p`HGFQ7$$!31+++OgrY7Fdp$\"3-+++uH$z)HFQ7$$\"33+++$e4De\"Fdp$\"3+ +++_0/ZJFQ7$$\"3k*****pBMn2)Fdp$\"3!******4d0sI$FQ7$$\"3!)*****4Wy59#! #?$\"3')******)4N*oMFQ7$$\"3)******HG,(RZFhr$\"3>+++/;(Hj$FQ7$$\"3`+++ uZK.(*Fhr$\"3=+++&[*Q+QFQ7$$\"3&******f%)y<#>!#>$\"35+++(3hE(RFQ7$$\"3 3+++GY^(y$Fhs$\"3'******R0D8:%FQ7$$\"3Z+++dAA?vFhs$\"3q*****R[R_L%FQ7$ $\"30+++h\"\\KV\"F-$\"3*)*****f'*e+^%FQ7$$\"3'******>.quJ#F-$\"3s***** zEt$\\YFQ7$$\"3.+++pUIHJF-$\"3!)*****>cXLv%FQ7$$\"3C+++\\I'*zPF-$\"3i* ****Hkwq$[FQ7$$\"37+++Mq8%G%F-$\"3#*******eSh4\\FQ7$$\"3!)*****4Mr(oYF -$\"3A+++QX7v\\FQ7$$\"35+++Ws%e&\\F-$\"39+++d&)fN]FQ7$$\"3-+++yl%=;&F- $\"3%******H\"*f?4&FQ7$$\"3-+++j*4#*H&F-$\"3?+++Z,2X^FQ7$$\"3g*****z[m vP&F-$\"3U+++9M&\\>&FQ7$$\"3k******4;d/aF-$\"3)*******[[!>C&FQ7$$\"3i* ****p#*ekQ&F-$\"3o*****>MYgG&FQ7$$\"3I+++EjSG`F-$\"3T+++&pcuK&FQ7$$\"3 ,+++LTyM_F-$\"3e*****\\Y)=m`FQ7$$\"3B+++&oF$4^F-$\"3N+++!\\xAS&FQ7$$\" 3!)*****fqg_&\\F-$\"3f******Q#[dV&FQ7$$\"3!******4%fQvZF-$\"3'******Rn ;mY&FQ7$$\"3'*******\\M:sXF-$\"3c*****pg\"*[\\&FQ7$$\"3!)******3KrZVF- $\"3t*****R5v0_&FQ7$$\"3'******zFcR5%F-$\"3O+++pAmVbFQ7$$\"3*)*******H ZD%QF-$\"31+++!fSTc&FQ7$$\"3w*****p2]\\c$F-$\"3E+++F*))>e&FQ7$$\"3A+++ rsWsKF-$\"3%*******zh<(f&FQ7$$\"3?+++dZ:mHF-$\"36+++&ff'4cFQ7$$\"3++++ v1.ZEF-$\"3E+++QEQ>cFQ7$$\"3(******HtyeJ#F-$\"3()*****pHsii&FQ7$$\"3)* *****\\pWL(>F-$\"3-+++NcBIcFQ7$$\"3*******f//*>;F-$\"3,+++$Gb6j&FQ7$$ \"39+++0u$eD\"F-$\"3c*****fY$))GcFQ7$$\"3:+++Tx(=\"))Fhs$\"3;+++ERBBcF Q7$$\"3?+++()R*o&\\Fhs$\"3E+++'zqRh&FQ7$$\"3W+++eG*\\')*Fhr$\"3)****** R(Hz+cFQ7$$!3********e;[7JFhs$\"3#******zt6Le&FQ7$$!3))*****p_LHO(Fhs$ \"3q*****p6=5c&FQ7$$!3-+++v_P!=\"F-$\"3l*****>&[BLbFQ7$$!3'******\\X-- l\"F-$\"3M+++9n/*\\&FQ7$$!37+++F^#y:#F-$\"3)*******Q#4sX&FQ7$$!3#)**** *fvnis#F-$\"3z*****f>)31aFQ7$$!3z******y?n-MF-$\"3f*****\\C'*RM&FQ7$$! 3u*****pM8NG%F-$\"3?+++HP%>F&FQ7$$!3/+++W4<\"\\&F-$\"3r*****>:kT?&FQ7$ $!34+++2%\\;#pF-$\"3#******>[wp;&FQ7$$!3!******R+F.I)F-$\"3m*****phY,; &FQ7$$!3r*****z,H*Q&*F-$\"3=+++8>_o^FQ7$$!35+++&zR_1\"FQ$\"3!)*****\\C ?I=&FQ7$$!3%******4$pwm6FQ$\"39+++xbX*>&FQ7$$!3/+++qpkg7FQ$\"3\"****** zOJe@&FQ7$$!3-+++94[[8FQ$\"3,+++&4H6B&FQ7$$!30+++uuWJ9FQ$\"3k******Q?! [C&FQ7$$!33+++[*H/^\"FQ$\"3U+++>([lD&FQ7$$!3#*******pR5'e\"FQ$\"3e**** *z)G?m_FQ7$$!3++++Tk**e;FQ$\"3i*****f:wOF&FQ7$$!3'******fcD&HEE*y_FQ7$$!3!******zjG!)z\"FQ$\"3y*****>pQ>G&FQ7$$!31+++6=yk=FQ$ \"39+++scr#G&FQ7$$!3\"******>]9+$>FQ$\"3))*****zUp7G&FQ7$$!3'******R7? R*>FQ$\"3')*****f-=wF&FQ7$$!3'******\\Bkm0#FQ$\"3')*****f$Gyr_FQ7$$!3@ +++k3R=@FQ$\"3K+++rqyj_FQ7$$!3!)*****\\ZF#z@FQ$\"3m*****z0bOD&FQ7$$!3z *****R#*)GRAFQ$\"3-+++\\?TT_FQ7$$!3=+++N1o)H#FQ$\"36+++YW3F_FQ7$$!3;++ +I6]dBFQ$\"3I+++y.q5_FQ7$$!3!)******)*R%eT#FQ$\"3y******y0H#>&FQ7$$!3! ******>N*ztCFQ$\"3!)*****Ha*)=<&FQ7$$!3/+++SZXJDFQ$\"3m*****\\\"p`\\^F Q7$$!3#******z[&*))e#FQ$\"3\"******\\2z_7&FQ7$$!3'******>S/ik#FQ$\"3K+ ++?3<*4&FQ7$$!3y*****4ugMq#FQ$\"3q*****R+x72&FQ7$$!3')*****RVQ2w#FQ$\" 3$******4&RnT]FQ7$$!3y*****\\^.\"=GFQ$\"35+++g.X5]FQ7$$!3=+++/4hvGFQ$ \"37+++#R2x(\\FQ7$$!3%******>x+L$HFQ$\"3%)******GwbV\\FQ7$$!3)******f \"\\>\"*HFQ$\"3>+++sE73\\FQ7$$!3!*******zOH\\IFQ$\"35+++$GH:([FQ7$$!34 +++aSd2JFQ$\"3V+++!H/R$[FQ7$$!3%*******e%*)f;$FQ$\"39+++7(o`z%FQ7$$!34 +++=:ZCKFQ$\"3Q+++>=.cZFQ7$$!31+++KP$HG$FQ$\"3))*****R$f)fr%FQ7$$!3*** ****pjw7M$FQ$\"3'******Ho-`n%FQ7$$!3#)*****fv$R*R$FQ$\"3P+++46.MYFQ7$$ !3%)*****\\cxrX$FQ$\"3q******>x>#f%FQ7$$!3$******4zCX^$FQ$\"3Q+++8#3) \\XFQ7$$!3-+++E1MrNFQ$\"34+++)R]o]%FQ7$$!3'******RjTvi$FQ$\"3G+++yxHjW FQ7$$!3%******4VdIo$FQ$\"3T+++-L6>WFQ7$$!3;+++48$yt$FQ$\"31+++@HDuVFQ7 $$!3%******R1?=z$FQ$\"33+++E(o'GVFQ7$$!34+++&H$*\\%QFQ$\"35+++\"e6BG%F Q7$$!3')*******QKt*QFQ$\"33+++MK8NUFQ7$$!3=+++o$H)[RFQ$\"3%)*****\\(z3 (=%FQ7$$!3?+++!p&[**RFQ$\"3e******>R8QTFQ7$$!3'******H56$\\SFQ$\"3y*** ***3SB)3%FQ7$$!3C+++MDK)4%FQ$\"3$)*****Hzct.%FQ7$$!3#)*****p.Vl9%FQ$\" 39+++*4xa)RFQ7$$!3?+++w2+%>%FQ$\"3/+++!fwD$RFQ7$$!3\")******))zsSUFQ$ \"3A+++mUkyQFQ7$$!3!)*****4#*fnG%FQ$\"3#)*****>\"pnBQFQ7$$!3C+++rO8KVF Q$\"35+++g%zww$FQ7$$!3U+++Tp)oP%FQ$\"3++++^_m5PFQ7$$!3#)*****Hnc5U%FQ$ \"3?+++Fhl_OFQ7$$!3Q+++%exYY%FQ$\"3%)*****pJ#o$f$FQ7$$!3q*****pk!y2XFQ $\"3!)******H?yLNFQ7$$!3K+++!e\"R]XFQ$\"3!******\\s+IZ$FQ7$$!3-+++U%HD f%FQ$\"35+++H+R6MFQ7$$!3%)*****>W0Uj%FQ$\"3'******HH1!\\LFQ7$$!3M+++x? UvYFQ$\"33+++f)3fG$FQ7$$!3.+++&pshr%FQ$\"3$)*****p5e@A$FQ7$$!3e*****fa Tkv%FQ$\"31+++DM\"y:$FQ7$$!3#)*****pL/iz%FQ$\"37+++U7$H4$FQ7$$!3P+++d# Ha$[FQ$\"3!)*****RFjv-$FQ7$$!3M+++\"QyS([FQ$\"3y*****H5b<'HFQ7$$!33+++ L$4@\"\\FQ$\"3=+++(=Xb*GFQ7$$!3A+++zqZ\\\\FQ$\"3.+++AV'*GGFQ7$$!3o**** *RpNh)\\FQ$\"33+++fb.iFFQ7$$!30+++;+/A]FQ$\"3%******z_uZp#FQ7$$!3=+++# 3Zr0&FQ$\"3!******H3!>FEFQ7$$!3O+++'H<94&FQ$\"33+++$=&GfDFQ7$$!3f***** *G`\"[7&FQ$\"3/+++Gz0\"\\#FQ7$$!3K+++D2Jd^FQ$\"3))*****Hn-DU#FQ7$$!3C+ ++i#y))=&FQ$\"3')******Q6h`BFQ7$$!3,+++*3)\\>_FQ$\"3-+++0NP%G#FQ7$$!3c ******fc:\\_FQ$\"38+++D%zZ@#FQ7$$!3s*****z_TyF&FQ$\"3>+++f*=[9#FQ7$$!3 L+++*)4b0`FQ$\"33+++;M[u?FQ7$$!30+++IOGK`FQ$\"3#******H3mP+#FQ7$$!39++ +$yU!e`FQ$\"31+++hGmK>FQ7$$!3)******>*[$GQ&FQ$\"3'******zys6'=FQ7$$!3U +++M)omS&FQ$\"3-+++P%)H*y\"FQ7$$!3V+++q^bHaFQ$\"35+++2i/<@W(zbFQ$\"3$******fGPu7\"FQ7$$!3O +++\"p\"[%f&FQ$\"3++++29m_5FQ7$$!3O+++!>7$3cFQ$\"3i*****\\OQux*F-7$$!3 A+++wNA@cFQ$\"3s*****pGHr-*F-7$$!3%******z#4?LcFQ$\"3e*****>]\\fF)F-7$ $!3l*****H3FUk&FQ$\"3#******\\7NT_(F-7$$!3A+++xKGacFQ$\"3?+++g9*=x'F-7 $$!35+++Z)\\Lm&FQ$\"3L+++*H!R>gF-7$$!3G+++YpSrcFQ$\"3w*****>yonE&F-7$$ !3=+++X_VycFQ$\"3#******HuHT^%F-7$$!3c******)e;Wo&FQ$\"37+++\"RV:w$F-7 $$!3s*****4fM$*o&FQ$\"3%******f6]!4IF-7$$!3/+++R^<$p&FQ$\"37+++JUmcAF- 7$$!3Q+++?o#fp&FQ$\"3-+++D%yV]\"F-7$$!3k*****zC\"e(p&FQ$\"39+++\\vo@vF hs7$$!37+++)HL\")p&FQF(7$Ficn$!39+++\\vo@vFhs7$Fdcn$!3-+++D%yV]\"F-7$F _cn$!37+++JUmcAF-7$Fjbn$!3%******f6]!4IF-7$Febn$!37+++\"RV:w$F-7$F`bn$ !3#******HuHT^%F-7$F[bn$!3w*****>yonE&F-7$Ffan$!3L+++*H!R>gF-7$Faan$!3 ?+++g9*=x'F-7$F\\an$!3#******\\7NT_(F-7$Fg`n$!3e*****>]\\fF)F-7$Fb`n$! 3s*****pGHr-*F-7$F]`n$!3i*****\\OQux*F-7$Fh_n$!3++++29m_5FQ7$Fc_n$!3$* *****fGPu7\"FQ7$F^_n$!3.+++EFQ7$Fg[n$!3#******H3mP+#FQ 7$Fb[n$!33+++;M[u?FQ7$F][n$!3>+++f*=[9#FQ7$Fhjm$!38+++D%zZ@#FQ7$Fcjm$! 3-+++0NP%G#FQ7$F^jm$!3')******Q6h`BFQ7$Fiim$!3))*****Hn-DU#FQ7$Fdim$!3 /+++Gz0\"\\#FQ7$F_im$!33+++$=&GfDFQ7$Fjhm$!3!******H3!>FEFQ7$Fehm$!3%* *****z_uZp#FQ7$F`hm$!33+++fb.iFFQ7$F[hm$!3.+++AV'*GGFQ7$Ffgm$!3=+++(=X b*GFQ7$Fagm$!3y*****H5b<'HFQ7$F\\gm$!3!)*****RFjv-$FQ7$Fgfm$!37+++U7$H 4$FQ7$Fbfm$!31+++DM\"y:$FQ7$F]fm$!3$)*****p5e@A$FQ7$Fhem$!33+++f)3fG$F Q7$Fcem$!3'******HH1!\\LFQ7$F^em$!35+++H+R6MFQ7$Fidm$!3!******\\s+IZ$F Q7$Fddm$!3!)******H?yLNFQ7$F_dm$!3%)*****pJ#o$f$FQ7$Fjcm$!3?+++Fhl_OFQ 7$Fecm$!3++++^_m5PFQ7$F`cm$!35+++g%zww$FQ7$F[cm$!3#)*****>\"pnBQFQ7$Ff bm$!3A+++mUkyQFQ7$Fabm$!3/+++!fwD$RFQ7$F\\bm$!39+++*4xa)RFQ7$Fgam$!3$) *****Hzct.%FQ7$Fbam$!3y******3SB)3%FQ7$F]am$!3e******>R8QTFQ7$Fh`m$!3% )*****\\(z3(=%FQ7$Fc`m$!33+++MK8NUFQ7$F^`m$!35+++\"e6BG%FQ7$Fi_m$!33++ +E(o'GVFQ7$Fd_m$!31+++@HDuVFQ7$F__m$!3T+++-L6>WFQ7$Fj^m$!3G+++yxHjWFQ7 $Fe^m$!34+++)R]o]%FQ7$F`^m$!3Q+++8#3)\\XFQ7$F[^m$!3q******>x>#f%FQ7$Ff ]m$!3P+++46.MYFQ7$Fa]m$!3'******Ho-`n%FQ7$F\\]m$!3))*****R$f)fr%FQ7$Fg \\m$!3Q+++>=.cZFQ7$Fb\\m$!39+++7(o`z%FQ7$F]\\m$!3V+++!H/R$[FQ7$Fh[m$!3 5+++$GH:([FQ7$Fc[m$!3>+++sE73\\FQ7$F^[m$!3%)******GwbV\\FQ7$Fijl$!37++ +#R2x(\\FQ7$Fdjl$!35+++g.X5]FQ7$F_jl$!3$******4&RnT]FQ7$Fjil$!3q*****R +x72&FQ7$Feil$!3K+++?3<*4&FQ7$F`il$!3\"******\\2z_7&FQ7$F[il$!3m***** \\\"p`\\^FQ7$Ffhl$!3!)*****Ha*)=<&FQ7$Fahl$!3y******y0H#>&FQ7$F\\hl$!3 I+++y.q5_FQ7$Fggl$!36+++YW3F_FQ7$Fbgl$!3-+++\\?TT_FQ7$F]gl$!3m*****z0b OD&FQ7$Fhfl$!3K+++rqyj_FQ7$Fcfl$!3')*****f$Gyr_FQ7$F^fl$!3')*****f-=wF &FQ7$Fiel$!3))*****zUp7G&FQ7$Fdel$!39+++scr#G&FQ7$F_el$!3y*****>pQ>G&F Q7$Fjdl$!3q*****>EE*y_FQ7$Fedl$!3i*****f:wOF&FQ7$F`dl$!3e*****z)G?m_FQ 7$F[dl$!3U+++>([lD&FQ7$Ffcl$!3k******Q?![C&FQ7$Facl$!3,+++&4H6B&FQ7$F \\cl$!3\"******zOJe@&FQ7$Fgbl$!39+++xbX*>&FQ7$Fbbl$!3!)*****\\C?I=&FQ7 $F]bl$!3=+++8>_o^FQ7$Fhal$!3m*****phY,;&FQ7$Fcal$!3#******>[wp;&FQ7$F^ al$!3r*****>:kT?&FQ7$Fi`l$!3?+++HP%>F&FQ7$Fd`l$!3f*****\\C'*RM&FQ7$F_` l$!3z*****f>)31aFQ7$Fj_l$!3)*******Q#4sX&FQ7$Fe_l$!3M+++9n/*\\&FQ7$F`_ l$!3l*****>&[BLbFQ7$F[_l$!3q*****p6=5c&FQ7$Ff^l$!3#******zt6Le&FQ7$Fa^ l$!3)******R(Hz+cFQ7$$\"3G+++))R*o&\\Fhs$!3E+++'zqRh&FQ7$Fg]l$!3;+++ER BBcFQ7$Fb]l$!3c*****fY$))GcFQ7$F]]l$!3,+++$Gb6j&FQ7$Fh\\l$!3-+++NcBIcF Q7$Fc\\l$!3()*****pHsii&FQ7$F^\\l$!3E+++QEQ>cFQ7$Fi[l$!36+++&ff'4cFQ7$ Fd[l$!3%*******zh<(f&FQ7$F_[l$!3E+++F*))>e&FQ7$Fjz$!31+++!fSTc&FQ7$Fez $!3O+++pAmVbFQ7$F`z$!3t*****R5v0_&FQ7$F[z$!3c*****pg\"*[\\&FQ7$Ffy$!3' ******Rn;mY&FQ7$Fay$!3f******Q#[dV&FQ7$F\\y$!3N+++!\\xAS&FQ7$Fgx$!3e** ***\\Y)=m`FQ7$Fbx$!3T+++&pcuK&FQ7$F]x$!3o*****>MYgG&FQ7$Fhw$!3)******* [[!>C&FQ7$Fcw$!3U+++9M&\\>&FQ7$F^w$!3?+++Z,2X^FQ7$Fiv$!3%******H\"*f?4 &FQ7$Fdv$!39+++d&)fN]FQ7$F_v$!3A+++QX7v\\FQ7$Fju$!3#*******eSh4\\FQ7$F eu$!3i*****Hkwq$[FQ7$F`u$!3!)*****>cXLv%FQ7$F[u$!3s*****zEt$\\YFQ7$Fft $!3*)*****f'*e+^%FQ7$Fat$!3q*****R[R_L%FQ7$F\\t$!3'******R0D8:%FQ7$Ffs $!35+++(3hE(RFQ7$Fas$!3=+++&[*Q+QFQ7$F\\s$!3>+++/;(Hj$FQ7$Ffr$!3')**** **)4N*oMFQ7$Far$!3!******4d0sI$FQ7$$\"3-+++#e4De\"Fdp$!3++++_0/ZJFQ7$F gq$!3-+++uH$z)HFQ7$Fbq$!3')*****f*p`HGFQ7$F]q$!3=+++!f;;n#FQ7$Fhp$!3y* *****G2,9DFQ7$Fbp$!3=+++v)4mN#FQ7$F]p$!3))*****>-S$*>#FQ7$Fho$!3#***** *fw^@/#FQ7$Fco$!33+++KF,&)=FQ7$F]o$!3,+++TA!zs\"FQ7$$!3Q*****Rf@ql*FU$ !3'*******Hv!3d\"FQ7$FY$!3#*******\\5s89FQ7$FS$!3!******RdQmD\"FQ7$FM$ !3)******R)yb*4\"FQ7$FG$!30+++(4zZU*F-7$FA$!3[*****R&=)R&yF-7$$!3))*** ***fNa'=)F=$!39+++P`=$G'F-7$$!3G+++'eQhL)F7$!3')*****H)*)Q7ZF-7$$!3;++ +Z\\\"3H$F1$!3)******Rl#fTJF-7$F($!3++++Fjzq:F-F'-%&COLORG6&%$RGBG$\" \"$!\"\"$\"#:!\"#F(-%)POLYGONSG6fal7%F'7$F($\"+Fjzq:!#57$$!#YF[gpF(7%F cgp7$$!+aZ\"3H$Fhr$\"+aEfTJFfgpFggp7%F[hp7$$!+BN.O$)Fhs$\"+$)*)Q7ZFfgp Fggp7%Fahp7$$!+cNa'=)F-$\"+P`=$G'FfgpFggp7%Fghp7$$!+r/MpZFQ$\"+a=)R&yF fgpFggp7%F]ip7$$!+D(e6*>!#;$\"+(4zZU*FfgpFggp7%Fcip7$$!+:!#9$\"+TA!zs\"F_jpFggp7%Fd[q7$$!+wb_\"[$Fg[q$\"+KF,&)=F_jpFggp 7%F[\\q7$$!+oA)e)eFg[q$\"+m<:U?F_jpFggp7%Fa\\q7$$!+Z,+g#*Fg[q$\"+A+M*> #F_jpFggp7%Fg\\q7$$!+B87`8!#8$\"+v)4mN#F_jpFggp7%F]]q7$$!+6SF==F`]q$\" +H2,9DF_jpFggp7%Fd]q7$$!+H4a'=#F`]q$\"+!f;;n#F_jpFggp7%Fj]q7$$!+;_\\v@ F`]q$\"+'*p`HGF_jpFggp7%F`^q7$$!+OgrY7F`]q$\"+uH$z)HF_jpFggp7%Ff^q7$$ \"+$e4De\"F`]q$\"+_0/ZJF_jpFggp7%F\\_q7$$\"+PUtw!)F`]q$\"+rb?2LF_jpFgg p7%Fb_q7$$\"+T%y59#!#7$\"+*4N*oMF_jpFggp7%Fh_q7$$\"+$G,(RZF[`q$\"+/;(H j$F_jpFggp7%F_`q7$$\"+uZK.(*F[`q$\"+&[*Q+QF_jpFggp7%Fe`q7$$\"+Y)y<#>!# 6$\"+(3hE(RF_jpFggp7%F[aq7$$\"+GY^(y$F^aq$\"+a]K^TF_jpFggp7%Fbaq7$$\"+ dAA?vF^aq$\"+%[R_L%F_jpFggp7%Fhaq7$$\"+h\"\\KV\"Ffgp$\"+m*e+^%F_jpFggp 7%F^bq7$$\"+K+Z &F_jpFggp7%Fdeq7$$\"+5;d/aFfgp$\"+\\[!>C&F_jpFggp7%Fjeq7$$\"+F*ekQ&Ffg p$\"+Uj/'G&F_jpFggp7%F`fq7$$\"+EjSG`Ffgp$\"+&pcuK&F_jpFggp7%Fffq7$$\"+ LTyM_Ffgp$\"+l%)=m`F_jpFggp7%F\\gq7$$\"+&oF$4^Ffgp$\"+!\\xAS&F_jpFggp7 %Fbgq7$$\"+12Eb\\Ffgp$\"+R#[dV&F_jpFggp7%Fhgq7$$\"+TfQvZFfgp$\"+umhmaF _jpFggp7%F^hq7$$\"+]M:sXFfgp$\"+2;*[\\&F_jpFggp7%Fdhq7$$\"+4KrZVFfgp$ \"+/^d?bF_jpFggp7%Fjhq7$$\"+yi&R5%Ffgp$\"+pAmVbF_jpFggp7%F`iq7$$\"++ta UQFfgp$\"+!fSTc&F_jpFggp7%Ffiq7$$\"+x+&\\c$Ffgp$\"+F*))>e&F_jpFggp7%F \\jq7$$\"+rsWsKFfgp$\"+!=wrf&F_jpFggp7%Fbjq7$$\"+dZ:mHFfgp$\"+&ff'4cF_ jpFggp7%Fhjq7$$\"+v1.ZEFfgp$\"+QEQ>cF_jpFggp7%F^[r7$$\"+L(yeJ#Ffgp$\"+ (Hsii&F_jpFggp7%Fd[r7$$\"+&pWL(>Ffgp$\"+NcBIcF_jpFggp7%Fj[r7$$\"+YS!*> ;Ffgp$\"+$Gb6j&F_jpFggp7%F`\\r7$$\"+0u$eD\"Ffgp$\"+mM))GcF_jpFggp7%Ff \\r7$$\"+Tx(=\"))F^aq$\"+ERBBcF_jpFggp7%F\\]r7$$\"+()R*o&\\F^aq$\"+'zq Rh&F_jpFggp7%Fb]r7$$\"+eG*\\')*F[`q$\"+uHz+cF_jpFggp7%Fh]r7$$!+f;[7JF^ aq$\"+Q)31aF_j pFggp7%F\\`r7$$!+z?n-MFfgp$\"+Xi*RM&F_jpFggp7%Fb`r7$$!+ZL^$G%Ffgp$\"+H P%>F&F_jpFggp7%Fh`r7$$!+W4<\"\\&Ffgp$\"+_T;/_F_jpFggp7%F^ar7$$!+2%\\;# pFfgp$\"+#[wp;&F_jpFggp7%Fdar7$$!+/qK+$)Ffgp$\"+_o^F_jpFggp7%F`br7$$!+&zR_1\"F_jp$\"+X--$=&F_jpFggp 7%Ffbr7$$!+Jpwm6F_jp$\"+xbX*>&F_jpFggp7%F\\cr7$$!+qpkg7F_jp$\"+o8$e@&F _jpFggp7%Fbcr7$$!+94[[8F_jp$\"+&4H6B&F_jpFggp7%Fhcr7$$!+uuWJ9F_jp$\"+R ?![C&F_jpFggp7%F^dr7$$!+[*H/^\"F_jp$\"+>([lD&F_jpFggp7%Fddr7$$!+qR5'e \"F_jp$\"+))G?m_F_jpFggp7%Fjdr7$$!+Tk**e;F_jp$\"+chnt_F_jpFggp7%F`er7$ $!+mb_HG&F_jpFgg p7%F\\fr7$$!+6=yk=F_jp$\"+scr#G&F_jpFggp7%Fbfr7$$!+-X,I>F_jp$\"+G%p7G& F_jpFggp7%Fhfr7$$!+C,#R*>F_jp$\"+E!=wF&F_jpFggp7%F^gr7$$!+NUmc?F_jp$\" +OGyr_F_jpFggp7%Fdgr7$$!+k3R=@F_jp$\"+rqyj_F_jpFggp7%Fjgr7$$!+vuAz@F_j p$\"+e]l`_F_jpFggp7%F`hr7$$!+C*)GRAF_jp$\"+\\?TT_F_jpFggp7%Ffhr7$$!+N1 o)H#F_jp$\"+YW3F_F_jpFggp7%F\\ir7$$!+I6]dBF_jp$\"+y.q5_F_jpFggp7%Fbir7 $$!+**R%eT#F_jp$\"+z0H#>&F_jpFggp7%Fhir7$$!+_$*ztCF_jp$\"+V&*)=<&F_jpF ggp7%F^jr7$$!+SZXJDF_jp$\"+:p`\\^F_jpFggp7%Fdjr7$$!+)[&*))e#F_jp$\"+v! z_7&F_jpFggp7%Fjjr7$$!+-W?YEF_jp$\"+?3<*4&F_jpFggp7%F`[s7$$!+T2Y.FF_jp $\"+/qFr]F_jpFggp7%Ff[s7$$!+M%Q2w#F_jp$\"+^RnT]F_jpFggp7%F\\\\s7$$!+:N 5=GF_jp$\"+g.X5]F_jpFggp7%Fb\\s7$$!+/4hvGF_jp$\"+#R2x(\\F_jpFggp7%Fh\\ s7$$!+s2ILHF_jp$\"+HwbV\\F_jpFggp7%F^]s7$$!+;\\>\"*HF_jp$\"+sE73\\F_jp Fggp7%Fd]s7$$!+!o$H\\IF_jp$\"+$GH:([F_jpFggp7%Fj]s7$$!+aSd2JF_jp$\"+!H /R$[F_jpFggp7%F`^s7$$!+f%*)f;$F_jp$\"+7(o`z%F_jpFggp7%Ff^s7$$!+=:ZCKF_ jp$\"+>=.cZF_jpFggp7%F\\_s7$$!+KP$HG$F_jp$\"+Mf)fr%F_jpFggp7%Fb_s7$$!+ PmFTLF_jp$\"+$o-`n%F_jpFggp7%Fh_s7$$!+cPR*R$F_jp$\"+46.MYF_jpFggp7%F^` s7$$!+lv#f%F_jpFggp7%Fd`s7$$!+\"zCX^$F_jp$\"+8#3)\\XF_jp Fggp7%Fj`s7$$!+E1MrNF_jp$\"+)R]o]%F_jpFggp7%F`as7$$!+M;aFOF_jp$\"+yxHj WF_jpFggp7%Ffas7$$!+Ju0$o$F_jp$\"+-L6>WF_jpFggp7%F\\bs7$$!+48$yt$F_jp$ \"+@HDuVF_jpFggp7%Fbbs7$$!+k+#=z$F_jp$\"+E(o'GVF_jpFggp7%Fhbs7$$!+&H$* \\%QF_jp$\"+\"e6BG%F_jpFggp7%F^cs7$$!+!RKt*QF_jp$\"+MK8NUF_jpFggp7%Fdc s7$$!+o$H)[RF_jp$\"+vz3(=%F_jpFggp7%Fjcs7$$!+!p&[**RF_jp$\"+?R8QTF_jpF ggp7%F`ds7$$!+.6J\\SF_jp$\"+4SB)3%F_jpFggp7%Ffds7$$!+MDK)4%F_jp$\"+$zc t.%F_jpFggp7%F\\es7$$!+PIaYTF_jp$\"+*4xa)RF_jpFggp7%Fbes7$$!+w2+%>%F_j p$\"+!fwD$RF_jpFggp7%Fhes7$$!+*)zsSUF_jp$\"+mUkyQF_jpFggp7%F^fs7$$!+@* fnG%F_jp$\"+7pnBQF_jpFggp7%Fdfs7$$!+rO8KVF_jp$\"+g%zww$F_jpFggp7%Fjfs7 $$!+Tp)oP%F_jp$\"+^_m5PF_jpFggp7%F`gs7$$!+tm0@WF_jp$\"+Fhl_OF_jpFggp7% Ffgs7$$!+%exYY%F_jp$\"+FEF_jpFggp7%Ff] t7$$!+'H<94&F_jp$\"+$=&GfDF_jpFggp7%F\\^t7$$!+H`\"[7&F_jp$\"+Gz0\"\\#F _jpFggp7%Fb^t7$$!+D2Jd^F_jp$\"+tE]ACF_jpFggp7%Fh^t7$$!+i#y))=&F_jp$\"+ R6h`BF_jpFggp7%F^_t7$$!+*3)\\>_F_jp$\"+0NP%G#F_jpFggp7%Fd_t7$$!+gc:\\_ F_jp$\"+D%zZ@#F_jpFggp7%Fj_t7$$!+G:%yF&F_jp$\"+f*=[9#F_jpFggp7%F``t7$$ !+*)4b0`F_jp$\"+;M[u?F_jpFggp7%Ff`t7$$!+IOGK`F_jp$\"+$3mP+#F_jpFggp7%F \\at7$$!+$yU!e`F_jp$\"+hGmK>F_jpFggp7%Fbat7$$!+#*[$GQ&F_jp$\"+)ys6'=F_ jpFggp7%Fhat7$$!+M)omS&F_jp$\"+P%)H*y\"F_jpFggp7%F^bt7$$!+q^bHaF_jp$\" +2i/<7$3cF_jp$\"+l$Qux*FfgpFggp7%F`ft7$$!+wNA@cF _jp$\"+(GHr-*FfgpFggp7%Ffft7$$!+G4?LcF_jp$\"+-&\\fF)FfgpFggp7%F\\gt7$$ !+$3FUk&F_jp$\"+D^8CvFfgpFggp7%Fbgt7$$!+xKGacF_jp$\"+g9*=x'FfgpFggp7%F hgt7$$!+Z)\\Lm&F_jp$\"+*H!R>gFfgpFggp7%F^ht7$$!+YpSrcF_jp$\"+#yonE&Ffg pFggp7%Fdht7$$!+X_VycF_jp$\"+V(HT^%FfgpFggp7%Fjht7$$!+*e;Wo&F_jp$\"+\" RV:w$FfgpFggp7%F`it7$$!+\"fM$*o&F_jp$\"+;,04IFfgpFggp7%Ffit7$$!+R^<$p& F_jp$\"+JUmcAFfgpFggp7%F\\jt7$$!+?o#fp&F_jp$\"+D%yV]\"FfgpFggp7%Fbjt7$ $!+[7e(p&F_jp$\"+\\vo@vF^aqFggp7%Fhjt7$$!+)HL\")p&F_jpF(Fggp7%F^[u7$Fi jt$!+\\vo@vF^aqFggp7%Fb[u7$Fcjt$!+D%yV]\"FfgpFggp7%Ff[u7$F]jt$!+JUmcAF fgpFggp7%Fj[u7$Fgit$!+;,04IFfgpFggp7%F^\\u7$Fait$!+\"RV:w$FfgpFggp7%Fb \\u7$F[it$!+V(HT^%FfgpFggp7%Ff\\u7$Feht$!+#yonE&FfgpFggp7%Fj\\u7$F_ht$ !+*H!R>gFfgpFggp7%F^]u7$Figt$!+g9*=x'FfgpFggp7%Fb]u7$Fcgt$!+D^8CvFfgpF ggp7%Ff]u7$F]gt$!+-&\\fF)FfgpFggp7%Fj]u7$Fgft$!+(GHr-*FfgpFggp7%F^^u7$ Faft$!+l$Qux*FfgpFggp7%Fb^u7$F[ft$!+29m_5F_jpFggp7%Ff^u7$Feet$!+'GPu7 \"F_jpFggp7%Fj^u7$F_et$!+EF_jpFggp7%Ffau7$F]at$!+$3mP+#F_ jpFggp7%Fjau7$Fg`t$!+;M[u?F_jpFggp7%F^bu7$Fa`t$!+f*=[9#F_jpFggp7%Fbbu7 $F[`t$!+D%zZ@#F_jpFggp7%Ffbu7$Fe_t$!+0NP%G#F_jpFggp7%Fjbu7$F__t$!+R6h` BF_jpFggp7%F^cu7$Fi^t$!+tE]ACF_jpFggp7%Fbcu7$Fc^t$!+Gz0\"\\#F_jpFggp7% Ffcu7$F]^t$!+$=&GfDF_jpFggp7%Fjcu7$Fg]t$!+$3!>FEF_jpFggp7%F^du7$Fa]t$! +GXx%p#F_jpFggp7%Fbdu7$F[]t$!+fb.iFF_jpFggp7%Ffdu7$Fe\\t$!+AV'*GGF_jpF ggp7%Fjdu7$F_\\t$!+(=Xb*GF_jpFggp7%F^eu7$Fi[t$!+.^vhHF_jpFggp7%Fbeu7$F c[t$!+uKcFIF_jpFggp7%Ffeu7$F][t$!+U7$H4$F_jpFggp7%Fjeu7$Fgjs$!+DM\"y:$ F_jpFggp7%F^fu7$Fajs$!+2\"e@A$F_jpFggp7%Fbfu7$F[js$!+f)3fG$F_jpFggp7%F ffu7$Feis$!+$H1!\\LF_jpFggp7%Fjfu7$F_is$!+H+R6MF_jpFggp7%F^gu7$Fihs$!+ D2+tMF_jpFggp7%Fbgu7$Fchs$!+I?yLNF_jpFggp7%Ffgu7$F]hs$!+WF_jpFggp7%Fj[v7$Fgas$!+yxHjWF_jpFggp7%F^ \\v7$Faas$!+)R]o]%F_jpFggp7%Fb\\v7$F[as$!+8#3)\\XF_jpFggp7%Ff\\v7$Fe`s $!+?x>#f%F_jpFggp7%Fj\\v7$F_`s$!+46.MYF_jpFggp7%F^]v7$Fi_s$!+$o-`n%F_j pFggp7%Fb]v7$Fc_s$!+Mf)fr%F_jpFggp7%Ff]v7$F]_s$!+>=.cZF_jpFggp7%Fj]v7$ Fg^s$!+7(o`z%F_jpFggp7%F^^v7$Fa^s$!+!H/R$[F_jpFggp7%Fb^v7$F[^s$!+$GH:( [F_jpFggp7%Ff^v7$Fe]s$!+sE73\\F_jpFggp7%Fj^v7$F_]s$!+HwbV\\F_jpFggp7%F ^_v7$Fi\\s$!+#R2x(\\F_jpFggp7%Fb_v7$Fc\\s$!+g.X5]F_jpFggp7%Ff_v7$F]\\s $!+^RnT]F_jpFggp7%Fj_v7$Fg[s$!+/qFr]F_jpFggp7%F^`v7$Fa[s$!+?3<*4&F_jpF ggp7%Fb`v7$F[[s$!+v!z_7&F_jpFggp7%Ff`v7$Fejr$!+:p`\\^F_jpFggp7%Fj`v7$F _jr$!+V&*)=<&F_jpFggp7%F^av7$Fiir$!+z0H#>&F_jpFggp7%Fbav7$Fcir$!+y.q5_ F_jpFggp7%Ffav7$F]ir$!+YW3F_F_jpFggp7%Fjav7$Fghr$!+\\?TT_F_jpFggp7%F^b v7$Fahr$!+e]l`_F_jpFggp7%Fbbv7$F[hr$!+rqyj_F_jpFggp7%Ffbv7$Fegr$!+OGyr _F_jpFggp7%Fjbv7$F_gr$!+E!=wF&F_jpFggp7%F^cv7$Fifr$!+G%p7G&F_jpFggp7%F bcv7$Fcfr$!+scr#G&F_jpFggp7%Ffcv7$F]fr$!+#pQ>G&F_jpFggp7%Fjcv7$Fger$!+ ii#*y_F_jpFggp7%F^dv7$Faer$!+chnt_F_jpFggp7%Fbdv7$F[er$!+))G?m_F_jpFgg p7%Ffdv7$Fedr$!+>([lD&F_jpFggp7%Fjdv7$F_dr$!+R?![C&F_jpFggp7%F^ev7$Fic r$!+&4H6B&F_jpFggp7%Fbev7$Fccr$!+o8$e@&F_jpFggp7%Ffev7$F]cr$!+xbX*>&F_ jpFggp7%Fjev7$Fgbr$!+X--$=&F_jpFggp7%F^fv7$Fabr$!+8>_o^F_jpFggp7%Fbfv7 $F[br$!+F&F_jpFggp7%Fbgv7$Fc`r$!+Xi*RM&F_jpFggp7%Ff gv7$F]`r$!+'>)31aF_jpFggp7%Fjgv7$Fg_r$!+R#4sX&F_jpFggp7%F^hv7$Fa_r$!+9 n/*\\&F_jpFggp7%Fbhv7$F[_r$!+_[BLbF_jpFggp7%Ffhv7$Fe^r$!+<\"=5c&F_jpFg gp7%Fjhv7$F_^r$!+QcF_jpFggp7%F `[w7$Fijq$!+&ff'4cF_jpFggp7%Fd[w7$Fcjq$!+!=wrf&F_jpFggp7%Fh[w7$F]jq$!+ F*))>e&F_jpFggp7%F\\\\w7$Fgiq$!+!fSTc&F_jpFggp7%F`\\w7$Faiq$!+pAmVbF_j pFggp7%Fd\\w7$F[iq$!+/^d?bF_jpFggp7%Fh\\w7$Fehq$!+2;*[\\&F_jpFggp7%F\\ ]w7$F_hq$!+umhmaF_jpFggp7%F`]w7$Figq$!+R#[dV&F_jpFggp7%Fd]w7$Fcgq$!+! \\xAS&F_jpFggp7%Fh]w7$F]gq$!+l%)=m`F_jpFggp7%F\\^w7$Fgfq$!+&pcuK&F_jpF ggp7%F`^w7$Fafq$!+Uj/'G&F_jpFggp7%Fd^w7$F[fq$!+\\[!>C&F_jpFggp7%Fh^w7$ Feeq$!+9M&\\>&F_jpFggp7%F\\_w7$F_eq$!+Z,2X^F_jpFggp7%F`_w7$Fidq$!+8*f? 4&F_jpFggp7%Fd_w7$Fcdq$!+d&)fN]F_jpFggp7%Fh_w7$F]dq$!+QX7v\\F_jpFggp7% F\\`w7$Fgcq$!+fSh4\\F_jpFggp7%F``w7$Facq$!+Vm2P[F_jpFggp7%Fd`w7$F[cq$! +ibM`ZF_jpFggp7%Fh`w7$Febq$!+oKP\\YF_jpFggp7%F\\aw7$F_bq$!+m*e+^%F_jpF ggp7%F`aw7$Fiaq$!+%[R_L%F_jpFggp7%Fdaw7$Fcaq$!+a]K^TF_jpFggp7%Fhaw7$F \\aq$!+(3hE(RF_jpFggp7%F\\bw7$Ff`q$!+&[*Q+QF_jpFggp7%F`bw7$F``q$!+/;(H j$F_jpFggp7%Fdbw7$Fi_q$!+*4N*oMF_jpFggp7%Fhbw7$Fc_q$!+rb?2LF_jpFggp7%F \\cw7$$\"+#e4De\"F`]q$!+_0/ZJF_jpFggp7%F`cw7$Fg^q$!+uH$z)HF_jpFggp7%Ff cw7$Fa^q$!+'*p`HGF_jpFggp7%Fjcw7$F[^q$!+!f;;n#F_jpFggp7%F^dw7$Fe]q$!+H 2,9DF_jpFggp7%Fbdw7$F^]q$!+v)4mN#F_jpFggp7%Ffdw7$Fh\\q$!+A+M*>#F_jpFgg p7%Fjdw7$Fb\\q$!+m<:U?F_jpFggp7%F^ew7$F\\\\q$!+KF,&)=F_jpFggp7%Fbew7$F e[q$!+TA!zs\"F_jpFggp7%Ffew7$$!+%f@ql*Fdjp$!+Iv!3d\"F_jpFggp7%Fjew7$Fi jp$!+]5s89F_jpFggp7%F`fw7$Fbjp$!+u&QmD\"F_jpFggp7%Fdfw7$F[jp$!+%)yb*4 \"F_jpFggp7%Fhfw7$Fdip$!+(4zZU*FfgpFggp7%F\\gw7$F^ip$!+a=)R&yFfgpFggp7 %F`gw7$$!+gNa'=)F-$!+P`=$G'FfgpFggp7%Fdgw7$$!+'eQhL)Fhs$!+$)*)Q7ZFfgpF ggp7%Fjgw7$$!+Z\\\"3H$Fhr$!+aEfTJFfgpFggp7%F`hw7$F($!+Fjzq:FfgpFggp7%F fhwF'Fggp-Fffp6&Fhfp$\"#vF^gp$\"#KF^gpF(-%&STYLEG6#%,PATCHNOGRIDG-F$6$ 7e\\lF'F*7$F(F27$F($\"3<+++!)*)Q7ZF-7$$!3#******4()=ln)F1$\"3))*****pI &=$G'F-7$$!3;+++pDFzsF7$\"3#)******H;)R&yF-7$$!3U+++vdlYRF=$\"3g*****4 &zxC%*F-7$$!3++++P-`j:FC$\"3++++Gub*4\"FQ7$$!3o*****\\h'zc[FC$\"3++++n qjc7FQ7$$!3!******>BcQB\"FI$\"3&******ps;PT\"FQ7$$!3))*****\\s())GEFI$ \"3-+++Mkzq:FQ7$$!3a+++\"[VAx%FI$\"3++++*>wys\"FQ7$$!3*)*****zHv)=vFI$ \"3-+++Ff&\\)=FQ7$$!3-+++k2f)3\"FU$\"33+++\">N?/#FQ7$$!3-+++j'y^s\"FU$ \"3++++zF6*>#FQ7$$!39+++PN-)y$FU$\"3?+++Ef=cBFQ7$$!31+++*G\\93\"F_o$\" 3++++I#\\K^#FQ7$$!3!******4i@8=$F_o$\"3=+++EMHqEFQ7$$!33+++!R'ow')F_o$ \"37+++NXIFGFQ7$$!3-+++A.]c@Fdp$\"3')*****R_kU)HFQ7$$!3(******fn+.#\\F dp$\"3$)*****4Kb69$FQ7$$!3&******f%z1U5Fhr$\"3-+++h*ozH$FQ7$$!35+++qUM p?Fhr$\"3#******HHFZX$FQ7$$!3=+++Ge%f)QFhr$\"30+++i]^6OFQ7$$!3g*****H+ )*>&pFhr$\"3!******H,I&oPFQ7$$!35+++&R&>$>\"Fhs$\"37+++vO:ERFQ7$$!33++ +#Hw\"z>Fhs$\"3C+++kt/&3%FQ7$$!3o******[-(**>$Fhs$\"3m*****RrxiC%FQ7$$ !3C+++<_R*4&Fhs$\"3C+++Y@X6WFQ7$$!3b******)=8\\8)Fhs$\"3G+++\\-l#e%FQ7 $$!3*)*****fW5^K\"F-$\"3!)*******p&ogZFQ7$$!36+++&p!*>?#F-$\"30+++i#Ra $\\FQ7$$!3-+++k7ZJMF-$\"3s*****\\=Dk2&FQ7$$!3z*******>qPq%F-$\"3g***** Rl-uyF-$\"3F+++$Q8,M&FQ7$$!35+++H^Vj()F-$\"3S+++ R-;v`FQ7$$!3/+++@Nd)f*F-$\"3%)*****p1Y&FQ7$$!3++++-0Ie7FQ$\"3A+++H(3LZ&FQ7$$!36+++f/#pK\"FQ$ \"3)******\\6W7[&FQ7$$!33+++I1!RR\"FQ$\"3/+++.;$f[&FQ7$$!3/+++!o)\\f9F Q$\"3/+++=!Hv[&FQ7$$!3(******fWJR_\"FQ$\"3?+++n;;'[&FQ7$$!3/+++*y&Q(e \"FQ$\"31+++[)H>[&FQ7$$!3)******R(p-];FQ$\"3y*****fU9\\Z&FQ7$$!3$***** *>L0?r\"FQ$\"39+++X7=laFQ7$$!3/+++'phMx\"FQ$\"39+++*)[y_aFQ7$$!32+++_> `M=FQ$\"3!)******\\@xPaFQ7$$!31+++!)3N&*=FQ$\"3,+++:b=?aFQ7$$!3++++``0 c>FQ$\"3/+++Ep1+aFQ7$$!39+++Uoy;?FQ$\"3%)*****f.iuP&FQ7$$!3%******f-$p x?FQ$\"3W+++`^U_`FQ7$$!3%******4;G*Q@FQ$\"3,+++X\\-D`FQ7$$!3#*******y; l+AFQ$\"3g*****R&3N&H&FQ7$$!33+++2W-jAFQ$\"35+++V)>NE&FQ7$$!3++++L&FQ7$$!3'******R ))*\\bCFQ$\"3s*****\\e&yc^FQ7$$!3#******Ha(z@DFQ$\"3e*****>)o@=^FQ7$$! 35+++6QA*e#FQ$\"3I+++H#3'y]FQ7$$!3))*****H[z%F Q7$$!35+++sK`]JFQ$\"3y*****H%3\"4v%FQ7$$!30+++*e.+A$FQ$\"3W+++$pL(4ZFQ 7$$!35+++Y^n)G$FQ$\"39+++h*3$oYFQ7$$!39+++&\\YkN$FQ$\"3W+++^%4li%FQ7$$ !35+++,$[KU$FQ$\"3-+++!>3Ue%FQ7$$!31+++i$Q!*[$FQ$\"3&)*****f$QGTXFQ7$$ !31+++\"p'z`NFQ$\"3Q+++OSi(\\%FQ7$$!3&)*****\\4@vh$FQ$\"37+++)GFJX%FQ7 $$!39+++7MA!o$FQ$\"3o*****Rz.xS%FQ7$$!3()*****zSE>u$FQ$\"3q*****Rpv7O% FQ7$$!34+++u7m-QFQ$\"3A+++gpx8VFQ7$$!3?+++&flC'QFQ$\"3?+++XJ:lUFQ7$$!3 7+++v;Q@RFQ$\"3I+++/6O:UFQ7$$!3++++0^XzRFQ$\"3?+++P(oV;%FQ7$$!3#****** 4XLn.%FQ$\"3k*****pza@6%FQ7$$!35+++%*\\E$4%FQ$\"3/+++G(3(eSFQ7$$!3o*** **HV(4\\TFQ$\"3m*****z\\IS+%FQ7$$!3@+++FlF/UFQ$\"3')*****\\PI\"[RFQ7$$ !3E+++\\Y%)eUFQ$\"3))*****HcG5*QFQ7$$!3t*****>FRGJ%FQ$\"3A+++?YvKQFQ7$ $!3m*****HX\"HmVFQ$\"3-+++&fYLx$FQ7$$!3y*****fQC#>WFQ$\"3-+++M)\\Gr$FQ 7$$!3?+++X@lrWFQ$\"3%)******GaJ^OFQ7$$!3C+++P(yN_%FQ$\"3-+++e$)z)e$FQ7 $$!3'******>e(*\\d%FQ$\"3$******>Pb`_$FQ7$$!3I+++&\\\"*ei%FQ$\"3-+++zF /hMFQ7$$!3))*****\\@Lin%FQ$\"3-+++aU\"fR$FQ7$$!31+++ak)fs%FQ$\"3%***** **e)=+L$FQ7$$!3q*****4R2^x%FQ$\"37+++-%*RjKFQ7$$!3++++zkaB[FQ$\"39+++a 74'>$FQ7$$!3))******=/Dr[FQ$\"3@+++x;7GJFQ7$$!3/+++^T;=\\FQ$\"3-+++o'4 &fIFQ7$$!3c*****RrKU'\\FQ$\"31+++_iE!*HFQ7$$!3x*****>*GS4]FQ$\"3y***** *y^R?HFQ7$$!3'******pcCO0&FQ$\"35+++%)Q*)\\GFQ7$$!39+++`<&o4&FQ$\"3+++ +UYvyFFQ7$$!3s******GL/R^FQ$\"3A+++:e'pq#FQ7$$!3/+++![j,=&FQ$\"3!)**** **3J^MEFQ7$$!3g*****R)==?_FQ$\"35+++43QhDFQ7$$!3e*****Rpt!f_FQ$\"3++++ MHb([#FQ7$$!3)******>N>oH&FQ$\"3.+++,V,8CFQ7$$!3J+++`USL`FQ$\"3,+++&\\ ^xL#FQ7$$!3!*******3$=)o`FQ$\"3%)*****4s`+++LqHH?FQ7$$!3E+++!3!p)\\&FQ$\"3-+++6zK]>FQ7$$!3;+++P[@GbFQ$\"35+++z :jq=FQ7$$!3?+++()zccbFQ$\"3)******RsC-z\"FQ7$$!3/+++QBv$e&FQ$\"3'***** *\\NH\"4s1?Y\"FQ7$$!3M+++Mkz!o&FQ$ \"3-+++oKZy8FQ7$$!31+++x$3@q&FQ$\"3#********GGWH\"FQ7$$!3o*****4==As&F Q$\"32+++ce\"*47FQ7$$!3$)*****\\/36u&FQ$\"3++++r3)\\7\"FQ7$$!3s*****Hn c(edFQ$\"3!******R:o'R5FQ7$$!3U+++K(R^x&FQ$\"3++++Ud@S&*F-7$$!3#)***** *)[I-z&FQ$\"3))*****\\)H$3o)F-7$$!3W+++r0+/eFQ$\"3,+++e=$*=yF-7$$!3))* ****f#3U;eFQ$\"3]+++bM)[&pF-7$$!3\")*****f@iu#eFQ$\"3Q+++?(G!*3'F-7$$! 38+++7n4PeFQ$\"3b*****H]x;A&F-7$$!3G+++Z\")HXeFQ$\"3))*****H>4JN%F-7$$ !3C+++\"*H/_eFQ$\"3++++SZd$[$F-7$$!3e*****f.6t&eFQ$\"3++++=)*H8EF-7$$! 3=+++nf3heFQ$\"31+++-*)[U4JN%F-7$Ffbz$!3b**** *H]x;A&F-7$Fabz$!3Q+++?(G!*3'F-7$F\\bz$!3]+++bM)[&pF-7$Fgaz$!3,+++e=$* =yF-7$Fbaz$!3))*****\\)H$3o)F-7$F]az$!3++++Ud@S&*F-7$Fh`z$!3!******R:o 'R5FQ7$Fc`z$!3++++r3)\\7\"FQ7$F^`z$!32+++ce\"*47FQ7$Fi_z$!3#********GG WH\"FQ7$Fd_z$!3-+++oKZy8FQ7$F__z$!3#******>s1?Y\"FQ7$Fj^z$!3++++oj)\\a \"FQ7$Fe^z$!3.+++4EPF;FQ7$F`^z$!3'******\\NH\"4FQ7$F\\]z$!3>+++LqHH?FQ7$Fg \\z$!38+++$FQ7$Fgg y$!37+++-%*RjKFQ7$Fbgy$!3%*******e)=+L$FQ7$F]gy$!3-+++aU\"fR$FQ7$Fhfy$ !3-+++zF/hMFQ7$Fcfy$!3$******>Pb`_$FQ7$F^fy$!3-+++e$)z)e$FQ7$Fiey$!3%) ******GaJ^OFQ7$Fdey$!3-+++M)\\Gr$FQ7$F_ey$!3-+++&fYLx$FQ7$Fjdy$!3A+++? YvKQFQ7$Fedy$!3))*****HcG5*QFQ7$F`dy$!3')*****\\PI\"[RFQ7$F[dy$!3m**** *z\\IS+%FQ7$Ffcy$!3/+++G(3(eSFQ7$Facy$!3k*****pza@6%FQ7$F\\cy$!3?+++P( oV;%FQ7$Fgby$!3I+++/6O:UFQ7$Fbby$!3?+++XJ:lUFQ7$F]by$!3A+++gpx8VFQ7$Fh ay$!3q*****Rpv7O%FQ7$Fcay$!3o*****Rz.xS%FQ7$F^ay$!37+++)GFJX%FQ7$Fi`y$ !3Q+++OSi(\\%FQ7$Fd`y$!3&)*****f$QGTXFQ7$F_`y$!3-+++!>3Ue%FQ7$Fj_y$!3W +++^%4li%FQ7$Fe_y$!39+++h*3$oYFQ7$F`_y$!3W+++$pL(4ZFQ7$F[_y$!3y*****H% 3\"4v%FQ7$Ff^y$!3e******\\)f>z%FQ7$Fa^y$!3v*****zi\")H$[FQ7$F\\^y$!3E+ ++R`/u[FQ7$Fg]y$!3U+++AS<:\\FQ7$Fb]y$!3=+++L:Lc\\FQ7$F]]y$!3c*****R89u *\\FQ7$Fh\\y$!3V+++x*\\#Q]FQ7$Fc\\y$!3I+++H#3'y]FQ7$F^\\y$!3e*****>)o@ =^FQ7$Fi[y$!3s*****\\e&yc^FQ7$Fd[y$!3)********[GS>&FQ7$F_[y$!3e*****zo #oH_FQ7$Fjjx$!35+++V)>NE&FQ7$Fejx$!3g*****R&3N&H&FQ7$F`jx$!3,+++X\\-D` FQ7$F[jx$!3W+++`^U_`FQ7$Ffix$!3%)*****f.iuP&FQ7$Faix$!3/+++Ep1+aFQ7$F \\ix$!3,+++:b=?aFQ7$Fghx$!3!)******\\@xPaFQ7$Fbhx$!39+++*)[y_aFQ7$F]hx $!39+++X7=laFQ7$Fhgx$!3y*****fU9\\Z&FQ7$Fcgx$!31+++[)H>[&FQ7$F^gx$!3?+ ++n;;'[&FQ7$Fifx$!3/+++=!Hv[&FQ7$Fdfx$!3/+++.;$f[&FQ7$F_fx$!3)******\\ 6W7[&FQ7$Fjex$!3A+++H(3LZ&FQ7$Feex$!3'******HXC>Y&FQ7$F`ex$!3/+++`7$oW &FQ7$F[ex$!3:+++f^oFaFQ7$Ffdx$!3%)*****p1#FQ7$Fi\\x$!33+++\">N?/#FQ7$Fd\\ x$!3-+++Ff&\\)=FQ7$$!3y******zMCsZFI$!3++++*>wys\"FQ7$$!3v*****pr())GE FI$!3-+++Mkzq:FQ7$Fe[x$!3&******ps;PT\"FQ7$$!37+++4mzc[FC$!3++++nqjc7F Q7$$!39+++Q-`j:FC$!3++++Gub*4\"FQ7$$!3'******>xbm%RF=$!3g*****4&zxC%*F -7$$!3m*****\\es#zsF7$!3#)******H;)R&yF-7$$!3?+++!Q:ln)F1$!3))*****pI& =$G'F-7$F($!3<+++!)*)Q7ZF-7$F(F`fpFbfpF'-Fffp6&Fhfp$\"\"%F[gp$\"\"#F[g p$\"\"&F^gp-F`gp6f\\lFbgp7%Fcgp7$F(F^hpFggp7%F_`\\l7$F($\"+!)*)Q7ZFfgp Fggp7%Fa`\\l7$$!+r)=ln)Fhr$\"+2`=$G'FfgpFggp7%Fe`\\l7$$!+pDFzsFhs$\"+I ;)R&yFfgpFggp7%F[a\\l7$$!+vdlYRF-$\"+^zxC%*FfgpFggp7%Faa\\l7$$!+P-`j:F Q$\"+Gub*4\"F_jpFggp7%Fga\\l7$$!+:mzc[FQ$\"+nqjc7F_jpFggp7%F]b\\l7$$!+ Ki&QB\"Ffip$\"+Fnr89F_jpFggp7%Fcb\\l7$$!+Dx))GEFfip$\"+Mkzq:F_jpFggp7% Fib\\l7$$!+\"[VAx%Ffip$\"+*>wys\"F_jpFggp7%F_c\\l7$$!+)Hv)=vFfip$\"+Ff &\\)=F_jpFggp7%Fec\\l7$$!+k2f)3\"Fdjp$\"+\">N?/#F_jpFggp7%F[d\\l7$$!+j 'y^s\"Fdjp$\"+zF6*>#F_jpFggp7%Fad\\l7$$!+PN-)y$Fdjp$\"+Ef=cBF_jpFggp7% Fgd\\l7$$!+*G\\93\"Fg[q$\"+I#\\K^#F_jpFggp7%F]e\\l7$$!+@;K\"=$Fg[q$\"+ EMHqEF_jpFggp7%Fce\\l7$$!+!R'ow')Fg[q$\"+NXIFGF_jpFggp7%Fie\\l7$$!+A.] c@F`]q$\"+CXE%)HF_jpFggp7%F_f\\l7$$!+w1I?\\F`]q$\"+@`:TJF_jpFggp7%Fef \\l7$$!+Yz1U5F[`q$\"+h*ozH$F_jpFggp7%F[g\\l7$$!+qUMp?F[`q$\"+$HFZX$F_j pFggp7%Fag\\l7$$!+Ge%f)QF[`q$\"+i]^6OF_jpFggp7%Fgg\\l7$$!+.!)*>&pF[`q$ \"+8+`oPF_jpFggp7%F]h\\l7$$!+&R&>$>\"F^aq$\"+vO:ERF_jpFggp7%Fch\\l7$$! +#Hw\"z>F^aq$\"+kt/&3%F_jpFggp7%Fih\\l7$$!+\\-(**>$F^aq$\"+9xFYUF_jpFg gp7%F_i\\l7$$!+<_R*4&F^aq$\"+Y@X6WF_jpFggp7%Fei\\l7$$!+*=8\\8)F^aq$\"+ \\-l#e%F_jpFggp7%F[j\\l7$$!+Y/6D8Ffgp$\"++dogZF_jpFggp7%Faj\\l7$$!+&p! *>?#Ffgp$\"+i#Ra$\\F_jpFggp7%Fgj\\l7$$!+k7ZJMFfgp$\"+&=Dk2&F_jpFggp7%F ][]l7$$!++-x.ZFfgp$\"+u^&R<&F_jpFggp7%Fc[]l7$$!+=0&f'eFfgp$\"+H;uV_F_j pFggp7%Fi[]l7$$!+pp-:pFfgp$\"+w,E(H&F_jpFggp7%F_\\]l7$$!+>l-uyFfgp$\"+ $Q8,M&F_jpFggp7%Fe\\]l7$$!+H^Vj()Ffgp$\"+R-;v`F_jpFggp7%F[]]l7$$!+@Nd) f*Ffgp$\"+nq,/aF_jpFggp7%Fa]]l7$$!+d)o!R5F_jp$\"+f^oFaF_jpFggp7%Fg]]l7 $$!+rU\"[6\"F_jp$\"+`7$oW&F_jpFggp7%F]^]l7$$!+*zFx=\"F_jp$\"+`W#>Y&F_j pFggp7%Fc^]l7$$!+-0Ie7F_jp$\"+H(3LZ&F_jpFggp7%Fi^]l7$$!+f/#pK\"F_jp$\" +:TC\"[&F_jpFggp7%F__]l7$$!+I1!RR\"F_jp$\"+.;$f[&F_jpFggp7%Fe_]l7$$!+! o)\\f9F_jp$\"+=!Hv[&F_jpFggp7%F[`]l7$$!+Y9$R_\"F_jp$\"+n;;'[&F_jpFggp7 %Fa`]l7$$!+*y&Q(e\"F_jp$\"+[)H>[&F_jpFggp7%Fg`]l7$$!+up-];F_jp$\"+EW\" \\Z&F_jpFggp7%F]a]l7$$!+K`+7`M=F_jp$\"+]@xPaF_jpFggp7%F_b]l7$$! +!)3N&*=F_jp$\"+:b=?aF_jpFggp7%Feb]l7$$!+``0c>F_jp$\"+Ep1+aF_jpFggp7%F [c]l7$$!+Uoy;?F_jp$\"+O?Yx`F_jpFggp7%Fac]l7$$!+EIpx?F_jp$\"+`^U_`F_jpF ggp7%Fgc]l7$$!+h\"G*Q@F_jp$\"+X\\-D`F_jpFggp7%F]d]l7$$!+z;l+AF_jp$\"+a 3N&H&F_jpFggp7%Fcd]l7$$!+2W-jAF_jp$\"+V)>NE&F_jpFggp7%Fid]l7$$!+L&F_jpFggp7%Fee]l7 $$!+%))*\\bCF_jp$\"+&e&yc^F_jpFggp7%F[f]l7$$!+Vvz@DF_jp$\"+#)o@=^F_jpF ggp7%Faf]l7$$!+6QA*e#F_jp$\"+H#3'y]F_jpFggp7%Fgf]l7$$!+$[1!*QHF_jp$\"+R`/u[F_jpFggp7%Feh]l7$$!+Zky4IF_jp$\"+G;)H$[F_jp Fggp7%F[i]l7$$!+M?S!3$F_jp$\"+])f>z%F_jpFggp7%Fai]l7$$!+sK`]JF_jp$\"+V 3\"4v%F_jpFggp7%Fgi]l7$$!+*e.+A$F_jp$\"+$pL(4ZF_jpFggp7%F]j]l7$$!+Y^n) G$F_jp$\"+h*3$oYF_jpFggp7%Fcj]l7$$!+&\\YkN$F_jp$\"+^%4li%F_jpFggp7%Fij ]l7$$!+,$[KU$F_jp$\"+!>3Ue%F_jpFggp7%F_[^l7$$!+i$Q!*[$F_jp$\"+OQGTXF_j pFggp7%Fe[^l7$$!+\"p'z`NF_jp$\"+OSi(\\%F_jpFggp7%F[\\^l7$$!+&4@vh$F_jp $\"+)GFJX%F_jpFggp7%Fa\\^l7$$!+7MA!o$F_jp$\"+%z.xS%F_jpFggp7%Fg\\^l7$$ !+3k#>u$F_jp$\"+%pv7O%F_jpFggp7%F]]^l7$$!+u7m-QF_jp$\"+gpx8VF_jpFggp7% Fc]^l7$$!+&flC'QF_jp$\"+XJ:lUF_jpFggp7%Fi]^l7$$!+v;Q@RF_jp$\"+/6O:UF_j pFggp7%F_^^l7$$!+0^XzRF_jp$\"+P(oV;%F_jpFggp7%Fe^^l7$$!+^MtOSF_jp$\"+( za@6%F_jpFggp7%F[_^l7$$!+%*\\E$4%F_jp$\"+G(3(eSF_jpFggp7%Fa_^l7$$!+Lu4 \\TF_jp$\"+)\\IS+%F_jpFggp7%Fg_^l7$$!+FlF/UF_jp$\"+v.8[RF_jpFggp7%F]`^ l7$$!+\\Y%)eUF_jp$\"+j&G5*QF_jpFggp7%Fc`^l7$$!+s#RGJ%F_jp$\"+?YvKQF_jp Fggp7%Fi`^l7$$!+`9HmVF_jp$\"+&fYLx$F_jpFggp7%F_a^l7$$!+'QC#>WF_jp$\"+M )\\Gr$F_jpFggp7%Fea^l7$$!+X@lrWF_jp$\"+HaJ^OF_jpFggp7%F[b^l7$$!+P(yN_% F_jp$\"+e$)z)e$F_jpFggp7%Fab^l7$$!+#e(*\\d%F_jp$\"+s`NDNF_jpFggp7%Fgb^ l7$$!+&\\\"*ei%F_jp$\"+zF/hMF_jpFggp7%F]c^l7$$!+:KBwYF_jp$\"+aU\"fR$F_ jpFggp7%Fcc^l7$$!+ak)fs%F_jp$\"+f)=+L$F_jpFggp7%Fic^l7$$!+\"R2^x%F_jp$ \"+-%*RjKF_jpFggp7%F_d^l7$$!+zkaB[F_jp$\"+a74'>$F_jpFggp7%Fed^l7$$!+>/ Dr[F_jp$\"+x;7GJF_jpFggp7%F[e^l7$$!+^T;=\\F_jp$\"+o'4&fIF_jpFggp7%Fae^ l7$$!+9FBk\\F_jp$\"+_iE!*HF_jpFggp7%Fge^l7$$!+#*GS4]F_jp$\"+z^R?HF_jpF ggp7%F]f^l7$$!+nXi`]F_jp$\"+%)Q*)\\GF_jpFggp7%Fcf^l7$$!+`<&o4&F_jp$\"+ UYvyFF_jpFggp7%Fif^l7$$!+HL/R^F_jp$\"+:e'pq#F_jpFggp7%F_g^l7$$!+![j,=& F_jp$\"+4J^MEF_jpFggp7%Feg^l7$$!+%)==?_F_jp$\"+43QhDF_jpFggp7%F[h^l7$$ !+%pt!f_F_jp$\"+MHb([#F_jpFggp7%Fah^l7$$!+_$>oH&F_jp$\"+,V,8CF_jpFggp7 %Fgh^l7$$!+`USL`F_jp$\"+&\\^xL#F_jpFggp7%F]i^l7$$!+4$=)o`F_jp$\"+@PvhA F_jpFggp7%Fci^l7$$!+1a0.aF_jp$\"+hN,&=#F_jpFggp7%Fii^l7$$!+%z7hV&F_jp$ \"+F_jpFggp7%F[[_l7$$!+P[@GbF_jp$\"+z:jq=F_jpFggp7%Fa[ _l7$$!+()zccbF_jp$\"+CZA!z\"F_jpFggp7%Fg[_l7$$!+QBv$e&F_jp$\"+b$H\"44JN%FfgpFggp7%Fga_l7$$!+\"*H/_ eF_jp$\"+SZd$[$FfgpFggp7%F]b_l7$$!+O5JdeF_jp$\"+=)*H8EFfgpFggp7%Fcb_l7 $$!+nf3heF_jp$\"+-*)[U4JN%FfgpFggp7%Fid_l7$Fba_l$!+.vn@_FfgpFg gp7%F]e_l7$F\\a_l$!+?(G!*3'FfgpFggp7%Fae_l7$Ff`_l$!+bM)[&pFfgpFggp7%Fe e_l7$F``_l$!+e=$*=yFfgpFggp7%Fie_l7$Fj__l$!+&)H$3o)FfgpFggp7%F]f_l7$Fd __l$!+Ud@S&*FfgpFggp7%Faf_l7$F^__l$!+a\"o'R5F_jpFggp7%Fef_l7$Fh^_l$!+r 3)\\7\"F_jpFggp7%Fif_l7$Fb^_l$!+ce\"*47F_jpFggp7%F]g_l7$F\\^_l$!+!HGWH \"F_jpFggp7%Fag_l7$Ff]_l$!+oKZy8F_jpFggp7%Feg_l7$F`]_l$!+An+i9F_jpFggp 7%Fig_l7$Fj\\_l$!+oj)\\a\"F_jpFggp7%F]h_l7$Fd\\_l$!+4EPF;F_jpFggp7%Fah _l7$F^\\_l$!+b$H\"4F_jpFggp7%Fai_l7$Ffj^l$!+Lq HH?F_jpFggp7%Fei_l7$F`j^l$!+$F_jpFggp7%Fe]`l7$F`d^l$!+-%*RjKF_jpFggp7%Fi]`l7$Fjc^l$!+f)=+ L$F_jpFggp7%F]^`l7$Fdc^l$!+aU\"fR$F_jpFggp7%Fa^`l7$F^c^l$!+zF/hMF_jpFg gp7%Fe^`l7$Fhb^l$!+s`NDNF_jpFggp7%Fi^`l7$Fbb^l$!+e$)z)e$F_jpFggp7%F]_` l7$F\\b^l$!+HaJ^OF_jpFggp7%Fa_`l7$Ffa^l$!+M)\\Gr$F_jpFggp7%Fe_`l7$F`a^ l$!+&fYLx$F_jpFggp7%Fi_`l7$Fj`^l$!+?YvKQF_jpFggp7%F]``l7$Fd`^l$!+j&G5* QF_jpFggp7%Fa``l7$F^`^l$!+v.8[RF_jpFggp7%Fe``l7$Fh_^l$!+)\\IS+%F_jpFgg p7%Fi``l7$Fb_^l$!+G(3(eSF_jpFggp7%F]a`l7$F\\_^l$!+(za@6%F_jpFggp7%Faa` l7$Ff^^l$!+P(oV;%F_jpFggp7%Fea`l7$F`^^l$!+/6O:UF_jpFggp7%Fia`l7$Fj]^l$ !+XJ:lUF_jpFggp7%F]b`l7$Fd]^l$!+gpx8VF_jpFggp7%Fab`l7$F^]^l$!+%pv7O%F_ jpFggp7%Feb`l7$Fh\\^l$!+%z.xS%F_jpFggp7%Fib`l7$Fb\\^l$!+)GFJX%F_jpFggp 7%F]c`l7$F\\\\^l$!+OSi(\\%F_jpFggp7%Fac`l7$Ff[^l$!+OQGTXF_jpFggp7%Fec` l7$F`[^l$!+!>3Ue%F_jpFggp7%Fic`l7$Fjj]l$!+^%4li%F_jpFggp7%F]d`l7$Fdj]l $!+h*3$oYF_jpFggp7%Fad`l7$F^j]l$!+$pL(4ZF_jpFggp7%Fed`l7$Fhi]l$!+V3\"4 v%F_jpFggp7%Fid`l7$Fbi]l$!+])f>z%F_jpFggp7%F]e`l7$F\\i]l$!+G;)H$[F_jpF ggp7%Fae`l7$Ffh]l$!+R`/u[F_jpFggp7%Fee`l7$F`h]l$!+AS<:\\F_jpFggp7%Fie` l7$Fjg]l$!+L:Lc\\F_jpFggp7%F]f`l7$Fdg]l$!+MTT(*\\F_jpFggp7%Faf`l7$F^g] l$!+x*\\#Q]F_jpFggp7%Fef`l7$Fhf]l$!+H#3'y]F_jpFggp7%Fif`l7$Fbf]l$!+#)o @=^F_jpFggp7%F]g`l7$F\\f]l$!+&e&yc^F_jpFggp7%Fag`l7$Ffe]l$!+!\\GS>&F_j pFggp7%Feg`l7$F`e]l$!+)o#oH_F_jpFggp7%Fig`l7$Fjd]l$!+V)>NE&F_jpFggp7%F ]h`l7$Fdd]l$!+a3N&H&F_jpFggp7%Fah`l7$F^d]l$!+X\\-D`F_jpFggp7%Feh`l7$Fh c]l$!+`^U_`F_jpFggp7%Fih`l7$Fbc]l$!+O?Yx`F_jpFggp7%F]i`l7$F\\c]l$!+Ep1 +aF_jpFggp7%Fai`l7$Ffb]l$!+:b=?aF_jpFggp7%Fei`l7$F`b]l$!+]@xPaF_jpFggp 7%Fii`l7$Fja]l$!+*)[y_aF_jpFggp7%F]j`l7$Fda]l$!+X7=laF_jpFggp7%Faj`l7$ F^a]l$!+EW\"\\Z&F_jpFggp7%Fej`l7$Fh`]l$!+[)H>[&F_jpFggp7%Fij`l7$Fb`]l$ !+n;;'[&F_jpFggp7%F][al7$F\\`]l$!+=!Hv[&F_jpFggp7%Fa[al7$Ff_]l$!+.;$f[ &F_jpFggp7%Fe[al7$F`_]l$!+:TC\"[&F_jpFggp7%Fi[al7$Fj^]l$!+H(3LZ&F_jpFg gp7%F]\\al7$Fd^]l$!+`W#>Y&F_jpFggp7%Fa\\al7$F^^]l$!+`7$oW&F_jpFggp7%Fe \\al7$Fh]]l$!+f^oFaF_jpFggp7%Fi\\al7$Fb]]l$!+nq,/aF_jpFggp7%F]]al7$F\\ ]]l$!+R-;v`F_jpFggp7%Fa]al7$Ff\\]l$!+$Q8,M&F_jpFggp7%Fe]al7$F`\\]l$!+w ,E(H&F_jpFggp7%Fi]al7$Fj[]l$!+H;uV_F_jpFggp7%F]^al7$Fd[]l$!+u^&R<&F_jp Fggp7%Fa^al7$F^[]l$!+&=Dk2&F_jpFggp7%Fe^al7$Fhj\\l$!+i#Ra$\\F_jpFggp7% Fi^al7$Fbj\\l$!++dogZF_jpFggp7%F]_al7$F\\j\\l$!+\\-l#e%F_jpFggp7%Fa_al 7$Ffi\\l$!+Y@X6WF_jpFggp7%Fe_al7$F`i\\l$!+9xFYUF_jpFggp7%Fi_al7$Fjh\\l $!+kt/&3%F_jpFggp7%F]`al7$Fdh\\l$!+vO:ERF_jpFggp7%Fa`al7$F^h\\l$!+8+`o PF_jpFggp7%Fe`al7$Fhg\\l$!+i]^6OF_jpFggp7%Fi`al7$Fbg\\l$!+$HFZX$F_jpFg gp7%F]aal7$F\\g\\l$!+h*ozH$F_jpFggp7%Faaal7$Fff\\l$!+@`:TJF_jpFggp7%Fe aal7$F`f\\l$!+CXE%)HF_jpFggp7%Fiaal7$$!+#Q'ow')Fg[q$!+NXIFGF_jpFggp7%F ]bal7$Fde\\l$!+EMHqEF_jpFggp7%Fcbal7$F^e\\l$!+I#\\K^#F_jpFggp7%Fgbal7$ $!+QN-)y$Fdjp$!+Ef=cBF_jpFggp7%F[cal7$Fbd\\l$!+zF6*>#F_jpFggp7%Facal7$ F\\d\\l$!+\">N?/#F_jpFggp7%Fecal7$Ffc\\l$!+Ff&\\)=F_jpFggp7%Fical7$$!+ ![VAx%Ffip$!+*>wys\"F_jpFggp7%F]dal7$$!+!yUP)) F-$\"34+++*eV\"GdFQ7$$\"3M+++k$>M!*)F-$\"3y*****H$4\"*\\dFQ7$$\"3Y**** **z\"H\"\\*)F-$\"3))******HPFrdFQ7$$\"3%)******Hb;w*)F-$\"3y*****f?^@z &FQ7$$\"3]*****\\abf)*)F-$\"3C+++=eZ7eFQ7$$\"3g*****f*exz*)F-$\"3A+++ \\.>KeFQ7$$\"3!)*****R%pxe*)F-$\"3))*****\\vX7&eFQ7$$\"3K+++`>+C*)F-$ \"3Q+++4&*fpeFQ7$$\"3))******\\DSw))F-$\"3#*******=V@()eFQ7$$\"32+++B8 &o\"))F-$\"3++++)4dS!fFQ7$$\"3-+++(>_hu)F-$\"3C+++G\")4?fFQ7$$\"3W+++5 \"\\]m)F-$\"3B+++k.JNfFQ7$$\"3')*****RLLUd)F-$\"3%)*****f$)o'\\fFQ7$$ \"3]******f(\\VZ)F-$\"3K+++O,:jfFQ7$$\"3k*****fN-gO)F-$\"3S+++k>tvfFQ7 $$\"3e******p*e(\\#)F-$\"3q*****pq#R()fFQ7$$\"3^*****fkbh7)F-$\"35+++e 56)*fFQ7$$\"3[+++H0q&*zF-$\"3k*****Hlly+'FQ7$$\"3!)*****z\\x)eyF-$\"3% *******4Zj;gFQ7$$\"3^+++%e\\hr(F-$\"3f******zcRCgFQ7$$\"3-+++lA'zc(F-$ \"3A+++\")[7JgFQ7$$\"3i*****>sYZT(F-$\"3Q+++BrzOgFQ7$$\"3w*****H7BpD(F -$\"3'********H&QTgFQ7$$\"3Y*****z./\\4(F-$\"3V+++U*f[/'FQ7$$\"3K+++#3 )4HpF-$\"3e*****>s)=ZgFQ7$$\"3a*****\\#R\"*fnF-$\"35+++%)eL[gFQ7$$\"33 +++G[w(e'F-$\"3m*****\\eh#[gFQ7$$\"3o******=R28kF-$\"3#)*****R7@p/'FQ7 $$\"3y*****>^!GOiF-$\"3>+++7SEWgFQ7$$\"3!)*****>'y%y0'F-$\"3i******eGB SgFQ7$$\"3G+++IHFyeF-$\"3@+++&*>wMgFQ7$$\"3i*****4(*)4)p&F-$\"3X+++xdx FgFQ7$$\"3%)*****4;Kz^&F-$\"3m*****HX'=>gFQ7$$\"3x*****4.k%Q`F-$\"3e** ***RX\"*)3gFQ7$$\"3O+++ED]g^F-$\"3s*****\\#*pn*fFQ7$$\"3A+++/g,&)\\F-$ \"3!)*****H@yE)fFQ7$$\"30+++(G(>8[F-$\"3g*****>'RWmfFQ7$$\"3)******>af lk%F-$\"3++++@$ey%fFQ7$$\"3)******zQ%3([%F-$\"3o*****>klm#fFQ7$$\"3\") ******R'euL%F-$\"3i******>+b-fFQ7$$\"3*)******osY,UF-$\"37+++9%=^(eFQ7 $$\"3))*******GsY3%F-$\"3#******49#)Q%eFQ7$$\"3))*****HHKc*RF-$\"3L+++ ciC3eFQ7$$\"31+++\"o_\"[RF-$\"3#)******RPandFQ7$$\"3)******R#[MlRF-$\" 3'******p\\X7s&FQ7$$\"3;+++#4Xa3%F-$\"3U+++Q]wpcFQ7$$\"3!******z%R\\gV F-$\"33+++y(=nh&FQ7$$\"3$)*****f[$R9[F-$\"3'*******\\bWrbFQ7$$\"3_+++& oStP&F-$\"3U+++kFXVbFQ7$$\"3Y*****fNy>%fF-$\"3()*****fIDI`&FQ7$$\"3!** ****pnIGX'F-$\"3w*****ft\")\\`&FQ7$$\"3'******H@wz*oF-$\"35+++E&GZa&FQ 7$$\"3k*****\\CB6G(F-$\"3g*****>MS$fbFQ7$$\"3p*****4\\.%4wF-$\"3O+++M> /xbFQ7$$\"3C+++%om(*)yF-$\"3!*******)>Enf&FQ7$$\"3A+++^OM!*)Ffgp$\"+L4\"*\\dF_jp7$$\"#kF^gp$\"#eF[gp7%F`[cl7$$\" +!=H\"\\*)Ffgp$\"+IPFrdF_jpFe[cl7%F[\\cl7$$\"+Ib;w*)Ffgp$\"+17:#z&F_jp Fe[cl7%Fa\\cl7$$\"+Xb&f)*)Ffgp$\"+=eZ7eF_jpFe[cl7%Fg\\cl7$$\"+'*exz*)F fgp$\"+\\.>KeF_jpFe[cl7%F]]cl7$$\"+Wpxe*)Ffgp$\"+bdC^eF_jpFe[cl7%Fc]cl 7$$\"+`>+C*)Ffgp$\"+4&*fpeF_jpFe[cl7%Fi]cl7$$\"+]DSw))Ffgp$\"+>V@()eF_ jpFe[cl7%F_^cl7$$\"+B8&o\"))Ffgp$\"+)4dS!fF_jpFe[cl7%Fe^cl7$$\"+(>_hu) Ffgp$\"+G\")4?fF_jpFe[cl7%F[_cl7$$\"+5\"\\]m)Ffgp$\"+k.JNfF_jpFe[cl7%F a_cl7$$\"+MLBu&)Ffgp$\"+O)o'\\fF_jpFe[cl7%Fg_cl7$$\"+g(\\VZ)Ffgp$\"+O, :jfF_jpFe[cl7%F]`cl7$$\"+cB+m$)Ffgp$\"+k>tvfF_jpFe[cl7%Fc`cl7$$\"+q*e( \\#)Ffgp$\"+2FR()fF_jpFe[cl7%Fi`cl7$$\"+Yc:E\")Ffgp$\"+e56)*fF_jpFe[cl 7%F_acl7$$\"+H0q&*zFfgp$\"+`c'y+'F_jpFe[cl7%Feacl7$$\"+)\\x)eyFfgp$\"+ 5Zj;gF_jpFe[cl7%F[bcl7$$\"+%e\\hr(Ffgp$\"+!o&RCgF_jpFe[cl7%Fabcl7$$\"+ lA'zc(Ffgp$\"+\")[7JgF_jpFe[cl7%Fgbcl7$$\"+Anu9uFfgp$\"+BrzOgF_jpFe[cl 7%F]ccl7$$\"+BJ#pD(Ffgp$\"++`QTgF_jpFe[cl7%Fcccl7$$\"+QS!\\4(Ffgp$\"+U *f[/'F_jpFe[cl7%Ficcl7$$\"+#3)4HpFfgp$\"+A()=ZgF_jpFe[cl7%F_dcl7$$\"+D R\"*fnFfgp$\"+%)eL[gF_jpFe[cl7%Fedcl7$$\"+G[w(e'Ffgp$\"+&eh#[gF_jpFe[c l7%F[ecl7$$\"+>R28kFfgp$\"+C6#p/'F_jpFe[cl7%Faecl7$$\"+70GOiFfgp$\"+7S EWgF_jpFe[cl7%Fgecl7$$\"+iy%y0'Ffgp$\"+fGBSgF_jpFe[cl7%F]fcl7$$\"+IHFy eFfgp$\"+&*>wMgF_jpFe[cl7%Fcfcl7$$\"+r*)4)p&Ffgp$\"+xdxFgF_jpFe[cl7%Fi fcl7$$\"+h@$z^&Ffgp$\"+`k=>gF_jpFe[cl7%F_gcl7$$\"+JSYQ`Ffgp$\"+a9*)3gF _jpFe[cl7%Fegcl7$$\"+ED]g^Ffgp$\"+D*pn*fF_jpFe[cl7%F[hcl7$$\"+/g,&)\\F fgp$\"+8#yE)fF_jpFe[cl7%Fahcl7$$\"+(G(>8[Ffgp$\"+iRWmfF_jpFe[cl7%Fghcl 7$$\"+U&flk%Ffgp$\"+@$ey%fF_jpFe[cl7%F]icl7$$\"+)Q%3([%Ffgp$\"+UcmEfF_ jpFe[cl7%Fcicl7$$\"+S'euL%Ffgp$\"+?+b-fF_jpFe[cl7%Fiicl7$$\"+psY,UFfgp $\"+9%=^(eF_jpFe[cl7%F_jcl7$$\"+!HsY3%Ffgp$\"+T@)Q%eF_jpFe[cl7%Fejcl7$ $\"+$HKc*RFfgp$\"+ciC3eF_jpFe[cl7%F[[dl7$$\"+\"o_\"[RFfgp$\"+SPandF_jp Fe[cl7%Fa[dl7$$\"+C[MlRFfgp$\"+(\\X7s&F_jpFe[cl7%Fg[dl7$$\"+#4Xa3%Ffgp $\"+Q]wpcF_jpFe[cl7%F]\\dl7$$\"+[R\\gVFfgp$\"+y(=nh&F_jpFe[cl7%Fc\\dl7 $$\"+'[$R9[Ffgp$\"+]bWrbF_jpFe[cl7%Fi\\dl7$$\"+&oStP&Ffgp$\"+kFXVbF_jp Fe[cl7%F_]dl7$$\"+c$y>%fFfgp$\"+1`-LbF_jpFe[cl7%Fe]dl7$$\"+x1$GX'Ffgp$ \"+O<)\\`&F_jpFe[cl7%F[^dl7$$\"+8i(z*oFfgp$\"+E&GZa&F_jpFe[cl7%Fa^dl7$ $\"+XK7\"G(Ffgp$\"+U.MfbF_jpFe[cl7%Fg^dl7$$\"+\"\\.%4wFfgp$\"+M>/xbF_j pFe[cl7%F]_dl7$$\"+%om(*)yFfgp$\"+*>Enf&F_jpFe[cl7%Fc_dl7$$\"+^OC,P')Ffgp$\"+,,\"Qo &F_jpFe[cl7%F[adl7$$\"+bcK\\()Ffgp$\"+&ppgq&F_jpFe[cl7%FaadlF[[clFe[cl FagalF`iw-F$6$7in7$F]hal$!34+++*eV\"GdFQ7$Fdjbl$!30+++&ppgq&FQ7$F_jbl$ !3s*****455Qo&FQ7$Fjibl$!3O+++%p;:m&FQ7$Feibl$!3=+++GYQRcFQ7$F`ibl$!3L +++B7nEnf&FQ7$Ffhbl$!3O+++M>/xbFQ7$Fahbl$!3g** ***>MS$fbFQ7$F\\hbl$!35+++E&GZa&FQ7$Fggbl$!3w*****ft\")\\`&FQ7$Fbgbl$! 3()*****fIDI`&FQ7$F]gbl$!3U+++kFXVbFQ7$Fhfbl$!3'*******\\bWrbFQ7$Fcfbl $!33+++y(=nh&FQ7$F^fbl$!3U+++Q]wpcFQ7$Fiebl$!3'******p\\X7s&FQ7$Fdebl$ !3#)******RPandFQ7$F_ebl$!3L+++ciC3eFQ7$Fjdbl$!3#******49#)Q%eFQ7$Fedb l$!37+++9%=^(eFQ7$F`dbl$!3i******>+b-fFQ7$F[dbl$!3o*****>klm#fFQ7$Ffcb l$!3++++@$ey%fFQ7$Facbl$!3g*****>'RWmfFQ7$F\\cbl$!3!)*****H@yE)fFQ7$Fg bbl$!3s*****\\#*pn*fFQ7$Fbbbl$!3e*****RX\"*)3gFQ7$F]bbl$!3m*****HX'=>g FQ7$Fhabl$!3X+++xdxFgFQ7$Fcabl$!3@+++&*>wMgFQ7$F^abl$!3i******eGBSgFQ7 $Fi`bl$!3>+++7SEWgFQ7$Fd`bl$!3#)*****R7@p/'FQ7$F_`bl$!3m*****\\eh#[gFQ 7$Fj_bl$!35+++%)eL[gFQ7$Fe_bl$!3e*****>s)=ZgFQ7$F`_bl$!3V+++U*f[/'FQ7$ F[_bl$!3'********H&QTgFQ7$Ff^bl$!3Q+++BrzOgFQ7$Fa^bl$!3A+++\")[7JgFQ7$ F\\^bl$!3f******zcRCgFQ7$Fg]bl$!3%*******4Zj;gFQ7$Fb]bl$!3k*****Hlly+' FQ7$F]]bl$!35+++e56)*fFQ7$Fh\\bl$!3q*****pq#R()fFQ7$Fc\\bl$!3S+++k>tvf FQ7$F^\\bl$!3K+++O,:jfFQ7$Fi[bl$!3%)*****f$)o'\\fFQ7$Fd[bl$!3B+++k.JNf FQ7$F_[bl$!3C+++G\")4?fFQ7$Fjjal$!3++++)4dS!fFQ7$Fejal$!3#*******=V@() eFQ7$F`jal$!3Q+++4&*fpeFQ7$F[jal$!3))*****\\vX7&eFQ7$Ffial$!3A+++\\.>K eFQ7$Faial$!3C+++=eZ7eFQ7$F\\ial$!3y*****f?^@z&FQ7$Fghal$!3))******HPF rdFQ7$Fbhal$!3y*****H$4\"*\\dFQFjadlFd_\\l-F`gp6jn7%7$F\\[cl$!+*eV\"Gd F_jp7$Fbadl$!+&ppgq&F_jp7$Ff[cl$!#eF[gp7%Fd]el7$F\\adl$!+,,\"Qo&F_jpFg ]el7%F[^el7$Ff`dl$!+%p;:m&F_jpFg]el7%F_^el7$F``dl$!+GYQRcF_jpFg]el7%Fc ^el7$Fj_dl$!+B7nEnf&F_jpFg]el7%F[_el7$F^ _dl$!+M>/xbF_jpFg]el7%F__el7$Fh^dl$!+U.MfbF_jpFg]el7%Fc_el7$Fb^dl$!+E& GZa&F_jpFg]el7%Fg_el7$F\\^dl$!+O<)\\`&F_jpFg]el7%F[`el7$Ff]dl$!+1`-LbF _jpFg]el7%F_`el7$F`]dl$!+kFXVbF_jpFg]el7%Fc`el7$Fj\\dl$!+]bWrbF_jpFg]e l7%Fg`el7$Fd\\dl$!+y(=nh&F_jpFg]el7%F[ael7$F^\\dl$!+Q]wpcF_jpFg]el7%F_ ael7$Fh[dl$!+(\\X7s&F_jpFg]el7%Fcael7$Fb[dl$!+SPandF_jpFg]el7%Fgael7$F \\[dl$!+ciC3eF_jpFg]el7%F[bel7$Ffjcl$!+T@)Q%eF_jpFg]el7%F_bel7$F`jcl$! +9%=^(eF_jpFg]el7%Fcbel7$Fjicl$!+?+b-fF_jpFg]el7%Fgbel7$Fdicl$!+UcmEfF _jpFg]el7%F[cel7$F^icl$!+@$ey%fF_jpFg]el7%F_cel7$Fhhcl$!+iRWmfF_jpFg]e l7%Fccel7$Fbhcl$!+8#yE)fF_jpFg]el7%Fgcel7$F\\hcl$!+D*pn*fF_jpFg]el7%F[ del7$Ffgcl$!+a9*)3gF_jpFg]el7%F_del7$F`gcl$!+`k=>gF_jpFg]el7%Fcdel7$Fj fcl$!+xdxFgF_jpFg]el7%Fgdel7$Fdfcl$!+&*>wMgF_jpFg]el7%F[eel7$F^fcl$!+f GBSgF_jpFg]el7%F_eel7$Fhecl$!+7SEWgF_jpFg]el7%Fceel7$Fbecl$!+C6#p/'F_j pFg]el7%Fgeel7$F\\ecl$!+&eh#[gF_jpFg]el7%F[fel7$Ffdcl$!+%)eL[gF_jpFg]e l7%F_fel7$F`dcl$!+A()=ZgF_jpFg]el7%Fcfel7$Fjccl$!+U*f[/'F_jpFg]el7%Fgf el7$Fdccl$!++`QTgF_jpFg]el7%F[gel7$F^ccl$!+BrzOgF_jpFg]el7%F_gel7$Fhbc l$!+\")[7JgF_jpFg]el7%Fcgel7$Fbbcl$!+!o&RCgF_jpFg]el7%Fggel7$F\\bcl$!+ 5Zj;gF_jpFg]el7%F[hel7$Ffacl$!+`c'y+'F_jpFg]el7%F_hel7$F`acl$!+e56)*fF _jpFg]el7%Fchel7$Fj`cl$!+2FR()fF_jpFg]el7%Fghel7$Fd`cl$!+k>tvfF_jpFg]e l7%F[iel7$F^`cl$!+O,:jfF_jpFg]el7%F_iel7$Fh_cl$!+O)o'\\fF_jpFg]el7%Fci el7$Fb_cl$!+k.JNfF_jpFg]el7%Fgiel7$F\\_cl$!+G\")4?fF_jpFg]el7%F[jel7$F f^cl$!+)4dS!fF_jpFg]el7%F_jel7$F`^cl$!+>V@()eF_jpFg]el7%Fcjel7$Fj]cl$! +4&*fpeF_jpFg]el7%Fgjel7$Fd]cl$!+bdC^eF_jpFg]el7%F[[fl7$F^]cl$!+\\.>Ke F_jpFg]el7%F_[fl7$Fh\\cl$!+=eZ7eF_jpFg]el7%Fc[fl7$Fb\\cl$!+17:#z&F_jpF g]el7%Fg[fl7$F\\\\cl$!+IPFrdF_jpFg]el7%F[\\fl7$Fa[cl$!+L4\"*\\dF_jpFg] el7%F_\\flFa]elFg]elFagalF`iw-F$6%7$7$$!3Q++++++!p'FQF(7$$\"3!******** ******Q\"FQF(-%'COLOURG6&FhfpF)F)F)-%*LINESTYLEG6#Fjfp-F$6%7$7$F($!3o* ************Q'FQ7$F($\"3o*************Q'FQF\\]flF_]fl-%*AXESSTYLEG6#%$ BOXG-%(SCALINGG6#%,CONSTRAINEDG-%+AXESLABELSG6%%&Re(z)G%&Im(z)G-%%FONT G6#%(DEFAULTG-Fi^fl6$%*HELVETICAG\"\"*-%%VIEWG6$;$!$\\'F^gp$\"$>\"F^gp ;$!$R'F^gp$\"$R'F^gp" 1 2 0 1 10 0 2 9 1 2 1 1.000000 45.000000 45.000000 0 0 "Curve 1" "Curve 2" "Curve 3" "Curve 4" "Curve 5" "Curve 6" "Curve 7" "Curve 8" "Curve 9" "Curve 10" }}}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}}{PARA 0 "" 0 "" {TEXT -1 52 "#---------------------------------------------- -----" }}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 36 "coefficients for the com bined scheme" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 36 "coeffici ents of the combined scheme " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 7041 "ee := \{c[2]=1/20,\nc[3]=3 41/3200,\nc[4]=1023/6400,\nc[5]=39/100,\nc[6]=93/200,\nc[7]=31/200,\nc [8]=943/1000,\nc[9]=7067558016280/7837150160667,\nc[10]=909/1000,\nc[1 1]=47/50,\nc[12]=1,\nc[13]=1,\n\na[2,1]=1/20,\na[3,1]=-7161/1024000,\n a[3,2]=116281/1024000,\na[4,1]=1023/25600,\na[4,2]=0,\na[4,3]=3069/256 00,\na[5,1]=4202367/11628100,\na[5,2]=0,\na[5,3]=-3899844/2907025,\na[ 5,4]=3982992/2907025,\na[6,1]=5611/114400,\na[6,2]=0,\na[6,3]=0,\na[6, 4]=31744/135025,\na[6,5]=923521/5106400,\na[7,1]=21173/343200,\na[7,2] =0,\na[7,3]=0,\na[7,4]=8602624/76559175,\na[7,5]=-26782109/689364000, \na[7,6]=5611/283500,\na[8,1]=-1221101821869329/690812928000000,\na[8, 2]=0,\na[8,3]=0,\na[8,4]=-125/2,\na[8,5]=-1024030607959889/16892928000 0000,\na[8,6]=1501408353528689/265697280000000,\na[8,7]=60701392121322 83/92502016000000,\na[9,1]=-147251426448621580388138470887726424634604 4433307094207829051978044531801133057155/\n 12468948016200320 01157059621643986024803301558393487900440453636168046069686436608,\na[ 9,2]=0,\na[9,3]=0,\na[9,4]=-517229431108566845837517565524698123003902 5336933699114138315270772319372469280000/\n 12461938100480914 5897278630571215298365257079410236252921850936749076487132995191,\na[9 ,5]=-12070679258469254807978936441733187949484571516120469966534514296 406891652614970375/\n 272203115476165722171047818453110069949 7284085048389015085076961673446140398628096,\na[9,6]=78012515584389364 1323090552530431036567795592568497182701460674803126770111481625/\n \+ 183110425412731972197889874507158786859226102980861859505241443 073629143100805376,\na[9,7]=664113122959911642134782135839106469928140 328160577035357155340392950009492511875/\n 151784655985862481 36333023107295349175279765150089078301139943253016877823170816,\na[9,8 ]=10332848184452015604056836767286656859124007796970668046446015775000 000/\n 131270355003603364807383424874072791453797202863895016 5249582733679393783,\na[10,1]=-290555733603374150885386184422310364413 14060511/\n 22674759891089577691327962602370597632000000000, \na[10,2]=0,\na[10,3]=0,\na[10,4]=-20462749524591049105403365239069/45 4251913499893469596231268750,\na[10,5]=-180269259803172281163724663224 981097/38100922558256871086579832832000000,\na[10,6]=21127670214172802 870128286992003940810655221489/\n 46794738779978929061458226 97976708633673728000,\na[10,7]=318607235173649312405151265849660869927 653414425413/\n 67147167155589653031329380729354654239109120 00000,\na[10,8]=212083202434519082281842245535894/20022426044775672563 822865371173879,\na[10,9]=-2698404929400842518721166485087129798562269 848229517793703413951226714583/\n 46954567491393431507700044 2080871141884676035902717550325616728175875000000,\na[11,1]=-234265984 5814086836951207140065609179073838476242943917/\n 1358480961 351056777022231400139158760857532162795520000,\na[11,2]=0,\na[11,3]=0, \na[11,4]=-996286030132538159613930889652/1635306888599616490546432567 5,\na[11,5]=-26053085959256534152588089363841/437755280456568306101129 9942400,\na[11,6]=2098082234509676029222408679497810531264453392563493 3539/\n 3775889992007550803878727839115494641972212962174156 800,\na[11,7]=890722993756379186418929622095833835264322635782294899/ \n 13921242001395112657501941955594013822830119803764736,\na [11,8]=161021426143124178389075121929246710833125/10997207722131034650 667041364346422894371443,\na[11,9]=30076066976810251783423249756545243 4946672266195876496371874262392684852243925359864884962513/\n \+ 46554433375013464555850653366045056037608247796155212857518928103156 80492364106674524398280000,\na[11,10]=-31155237437111730665923206875/3 92862141594230515010338956291,\na[12,1]=-28665569918256639717782953291 01033887534912787724034363/\n 868226711619262703011213925016 143612030669233795338240,\na[12,2]=0,\na[12,3]=0,\na[12,4]=-1695708871 4171468676387054358954754000/143690415119654683326368228101570221,\na[ 12,5]=-4583493974484572912949314673356033540575/4519577036552507471573 13034270335135744,\na[12,6]=234630538855340425865625847344618441915474 0172519949575/\n 2567267164078954028927449783011514862541831 85289662464,\na[12,7]=165712155931984680217128369091361069858625657348 4808662625/\n 1343148041125514647725915510495609350536164443 2088109056,\na[12,8]=345685379554677052215495825476969226377187500/747 71167436930077221667203179551347546362089,\na[12,9]=\n -3205890962717 0725427914343121527275340081027740232102405713615707572490561670152301 60352087048674542196011/\n 947569549683965814783015124451273604984657 747127257615372449205973192657306017239103491074738324033259120,\na[12 ,10]=40279545832706233433100438588458933210937500/88964608427994828469 16972126377338947215101,\na[12,11]=-6122933601070769591613093993993358 877250/1050517001510235513198246721302027675953,\na[13,1]=-61867590553 5482500672800859344538410358660153899637/\n 2035442821182140 47100119475340667684874292102389760,\na[13,2]=0,\na[13,3]=0,\na[13,4]= -4411194916804718600478400319122931000/4037305390246996745076149126963 3019,\na[13,5]=-16734711409449292534539422531728520225/180124371529008 8669307203927210237952,\na[13,6]=1351375197570546790980421841527496777 61254751865630525/\n 1602958779448628959777132636191189511270 3716593983488,\na[13,7]=3893756836740987601254855190349219613792971043 1584875/\n 34095645409019160609954879800146930697475844314726 4,\na[13,8]=-6748865855011993037732355335815350667265625/\n \+ 7002880395717424621213565406715087764770357,\na[13,9]=-175600552030745 0928195422767042525091954178296002788308926563193523662404739779789732 685671/\n 34876781457846998360568809804618648090460727802103 0540735333862087061574934154942830062320,\na[13,10]=533810245892356110 84013897674181629296875/8959357584795694524874969598508592944141,\na[1 3,11]=0,\na[13,12]=0,\n\nb[1]=44901867737754616851973/1014046409980231 013380680,\nb[2]=0,\nb[3]=0,\nb[4]=0,\nb[5]=0,\nb[6]=79163867519161527 9648100000/2235604725089973126411512319,\nb[7]=38477494908689803481195 00000/15517045062138271618141237517,\nb[8]=-13734512432397741476562500 000/875132892924995907746928783,\nb[9]=1227476547031319687842881203774 0635050319234276006986398294443554969616342274215316330684448207141/\n 489345147493715517650385834143510934888829280686609654482896526 796523353052166757299452852166040,\nb[10]=-979836368457773944531250000 0/308722986341456031822630699,\nb[11]=282035543183190840068750/1229540 7629873040425991,\nb[12]=-306814272936976936753/1299331183183744997286 ,\nb[13]=0,\n\n`b*`[1]=10835401739407019406577/24452182935693513797832 0,\n`b*`[2]=0,\n`b*`[3]=0,\n`b*`[4]=0,\n`b*`[5]=0,\n`b*`[6]=1390818977 8321895491375000/39221135527894265375640567,\n`b*`[7]=7348794752702724 3487625000/296504045773342769773399443,\n`b*`[8]=682931406412576496093 75000/15353208647806945749946119,\n`b*`[9]=220606479489966786110177113 79974578860522018208949721559448560203338437626022142776381/\n 1 1115420092623258745129591857957272157590105775657360796413766213815772 36680929558640,\n`b*`[10]=-547971229495642458203125000/232372140257009 91642563601,\n`b*`[11]=0,\n`b*`[12]=0,\n`b*`[13]=-28735456870978964189 /79783493704265043693\}:" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}}{PARA 0 "" 0 "" {TEXT -1 0 " " }}{PARA 0 "" 0 "" {TEXT 259 5 "nodes" }{TEXT -1 2 ": " }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 32 "seq(c[i]=subs(ee,c[i]),i=2..13);" } }{PARA 11 "" 1 "" {XPPMATH 20 "6./&%\"cG6#\"\"##\"\"\"\"#?/&F%6#\"\"$# \"$T$\"%+K/&F%6#\"\"%#\"%B5\"%+k/&F%6#\"\"&#\"#R\"$+\"/&F%6#\"\"'#\"#$ *\"$+#/&F%6#\"\"(#\"#JFF/&F%6#\"\")#\"$V*\"%+5/&F%6#\"\"*#\".!G;!ev1( \".n1;]r$y/&F%6#\"#5#\"$4*FS/&F%6#\"#6#\"#Z\"#]/&F%6#\"#7F)/&F%6#\"#8F )" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT 259 20 "li nking coefficients" }{TEXT -1 1 ":" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 50 "seq(seq(a[i,j]=subs(ee,a[i,j]),j=1..i-1),i=1..13);" } }{PARA 12 "" 1 "" {XPPMATH 20 "6jo/&%\"aG6$\"\"#\"\"\"#F(\"#?/&F%6$\" \"$F(#!%hr\"(+S-\"/&F%6$F.F'#\"'\"G;\"F1/&F%6$\"\"%F(#\"%B5\"&+c#/&F%6 $F:F'\"\"!/&F%6$F:F.#\"%pIF=/&F%6$\"\"&F(#\"(nB?%\")+\"G;\"/&F%6$FJF'F A/&F%6$FJF.#!(W)**Q\"(Dq!H/&F%6$FJF:#\"(#*H)RFV/&F%6$\"\"'F(#\"%6c\"'+ W6/&F%6$FinF'FA/&F%6$FinF.FA/&F%6$FinF:#\"&W<$\"'D]8/&F%6$FinFJ#\"'@N# *\"(+k5&/&F%6$\"\"(F(#\"&t6#\"'+KM/&F%6$FbpF'FA/&F%6$FbpF.FA/&F%6$FbpF :#\"(CEg)\")v\"fl(/&F%6$FbpFJ#!)4@yE\"*+SO*o/&F%6$FbpFin#F[o\"'+NG/&F% 6$\"\")F(#!1H$p=#=5@7\"0+++GH\"3p/&F%6$F`rF'FA/&F%6$F`rF.FA/&F%6$F`rF: #!$D\"F'/&F%6$F`rFJ#!1*))fzgIS-\"\"0+++!GH*o\"/&F%6$F`rFin#\"1*oGNN39] \"\"0+++!G(pl#/&F%6$F`rFbp#\"1$GK@@R,2'\"/+++;?]#*/&F%6$\"\"*F(#!]pbr0 L6!=`W!y>0Hy?%42LVWgMYUEx)3ZQ\")Q!e@'[kU^s9\"]p3mV'opg/ohj`/W+z[$Re:I. [-')Rk@'fq:,?.?;![*oC\"/&F%6$FdtF'FA/&F%6$FdtF.FA/&F%6$FdtF:#!]p++GpCP >Bxq_JQT6*pLpLD!R+B\")pCbc&*H8([w!\\n$4&=#HDO-Tzq Dl$)H:7dI'ys*e94[+\"Q>Y7/&F%6$FdtFJ#!^pv.(\\h_;*oS'H9X`m*p/7;:d%[\\z=L Da%/&F%6$F_w FJ#!E(4\")\\AjYsj6GsJ!)f#p-=\"D+++KG$)zl3roDeD#45Q/&F%6$F_wFin#\"P*[@_ l53%R+#*pGG,(G!G<9-nF6#\"O+!GPnL'3n(zpAe91H*y*z(QZzY/&F%6$F_wFbp#\"T8a U9MlF*p3m\\eE^^S7$\\OXV-K37#\"DzQvcKka!\\;'*f))oIN;/&F%6$F`zFJ#!ATQO*3)e_T`c#ff3`g#\"@+C%** H651$olX!GbxV/&F%6$F`zFin#\"YRN$\\jDR`WEJ0\"y\\z'3CAHgn4XB#3)4#\"X+o:u @'H@s>k%\\:\"RysyQ!3b2?***)ex$/&F%6$F`zFbp#\"W**[H#yNEKk_$Q$e4A'H*=k=z jv$*Hs!*)\"VOZw.)>,$G#Q,%fb>%>]dE6&R,?C@R\"/&F%6$F`zF`r#\"KDJ$3rY#H>7v !*QyT7VhU@5;\"MV9P%*GUYVOTqm]Y.J@s2s*4\"/&F%6$F`zFdt#\"hp8D'\\)[')f`#R C_[o#RiU(=P'\\we>mAnY\\V_ac(\\KU$y^-\"o(p1w+$\"ip++G)RCXn1TO#\\!o:.\"G *=v&G@bhzZ#3w.c]/mLl]ebkM,vLVal%/&F%6$F`zF_w#!>vo?BfmI<6PuBb6$\"?\"Hc* Q.,:0B%fT@'GR/&F%6$\"#7F(#!XjV.Cxy7\\`()Q.,\"H`Hy<(RmD=*pbmG\"WS#Q`zL# p1.7O9;]#R@6Iqi#>;rE#o)/&F%6$Fg]lF'FA/&F%6$Fg]lF.FA/&F%6$Fg]lF:#!G+Sva *eV0(Qw'o9<9()3dp\"\"E@-d,\"G#ojK$oa'>^T!pV\"/&F%6$Fg]lFJ#!Iv0aLgNtYJ \\H\"Hd%[uR\\$e%\"HWd8N.FMIJdru]_l.x&>X/&F%6$Fg]lFin#\"Xv&\\*>DW= YMZeileUS`&)Q0jM#\"WkCm*G&=$=ai[^6Iy\\u#*GS&*yS;nsc#/&F%6$Fg]lFbp#\"en DEm3[[tlD'e)p5O\"4p$Gr@!o%)>$f:7d;\"Yc!4\")3KWkh`]$4c\\5b\"fsZY^D6/[JM \"/&F%6$Fg]lF`r#\"N+v=xjAppZDe\\:A0xYbz`oX$\"M*3ijaZ8bzJ?n;Ax+$pVn6xu/ &F%6$Fg]lFdt#!eq6g>UXn[q3_.;I_,nh0\\svq:Or0C5K-uF53S`FF:7VV\"zUD2tf?\\CP:wDFrudY)\\gt7XC^,$y9e'Ro\\&pv %*/&F%6$Fg]lF_w#\"M+v$4@L*e%)eQ/5LMB1F$eaz-%\"L,^@Z*QtPE@(p\"p%G[*zU3Y '*))/&F%6$Fg]lF`z#!I]s()eL*R*R48;fp22,O$H7'\"I`fnF?I@nC)>8bB5:+<00\"/& F%6$\"#8F(#!TP'**Q:g'e.TQXMf3!Gn+D[Nb!fn='\"Tg(*Q-@Hu[on1Mv%>,5ZS@=@GW N?/&F%6$FdalF'FA/&F%6$FdalF.FA/&F%6$FdalF:#!F+5$H7>.Sy/g=Z!o\"\\>6W\"D >Ijp7\\h2Xn*pC!R0t.%/&F%6$FdalFJ#!GD-_G<`A%RX`#H\\%49rMn\"\"F_zB5s#R?2 $p')3!H:PC,=/&F%6$FdalFin#\"WD0jl=va7wx'\\F:%=U!)4zY0d(>v8N\"\"V)[$)Rf ;Pq7^*=\">OE8x(f*G'[%z(eHg\"/&F%6$FdalFbp#\"Vv[eJ/rHz8'>#\\.>b[D,w)4uO ov$*Q\"Tks9V%eZ(pIp9+)z[&*41;>!4ak&4M/&F%6$FdalF`r#!LDcEn1N:eLbBtPI*>, be')[n\"Ld.xkx3:nSlN@@YUjl#* 3$)y-gHyT&>4DD/nFU&>G4X2._0gv\"\"ep?B1IG%\\:M\\dhq3iQLN2aI5-ysg/4['=Y! )4)o0O)*p%yX\"yw[$/&F%6$FdalF_w#\"JvoHH;=uw*Q,%36cB*eC5Q`\"ITT%Hf3&)fp \\([_%p&z%ed$f*)/&F%6$FdalF`zFA/&F%6$FdalFg]lFA" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT 259 39 "weights for the 13 stage order 8 scheme" }{TEXT -1 1 ":" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 32 "seq(b[i]=subs(ee,b[i]),i=1..13);" }}{PARA 12 "" 1 "" {XPPMATH 20 "6//&%\"bG6#\"\"\"#\"8t>&ohaxtn=!\\%\":!o!Q85B!)*4k/95/&F%6#\"\"#\" \"!/&F%6#\"\"$F//&F%6#\"\"%F//&F%6#\"\"&F//&F%6#\"\"'#\"<++5['z_h\">v' Q;z\"=>B^6k7t**3DZgNA/&F%6#\"\"(#\"=++]>\"[.)*o3\\\\x%Q\">++]ilZTxRKC^MP\"\"<$yGpu2f*\\#H*G8v)/&F%6#\"\"*#\"]q Tr?[WoIjJ:UFUjhp\\bVWH)R')p+wUB>.0N1uP?\")G%yo>8.ZlZF7\"[qSg;_GX*Hdn;_ INBlzEl*G[a'4mo!GH)))[$4^VT$eQ]w^:P\\Z^M*[/&F%6#\"#5#!=++]7`WRxd%oj$)z *\"<*pIE#=.c9M')Hs3$/&F%6#\"#6#\"9](o+%3>$=Vb.#G\"8\"*fUSI()HwS&H7/&F% 6#\"#7#!6`n$p(p$HF9oI\"7'G(*\\u$=$=J$*H\"/&F%6#\"#8F/" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT 259 39 "weights for the 13 \+ stage order 7 scheme" }{TEXT -1 1 ":" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 38 "seq(`b*`[i]=subs(ee,`b*`[i]),i=1..13);" }}{PARA 12 " " 1 "" {XPPMATH 20 "6//&%#b*G6#\"\"\"#\"8xlS>qSRh%*\\d%p!yk3KN:/&F%6#\" \"*#\"ap\"QwF9AgiP%QL?g&[%f:s\\*3#=?_g)yX(*z8r<5hym**[zkg?#\"`pS'e&H4o Osd\"Q@mPT'zgtlvd5!fd@Fdz&=fH^ueKi#4?a66/&F%6#\"#5#!<+]7.#eCk&\\H7(za \";,OcU;*4qDS@PK#/&F%6#\"#6F//&F%6#\"#7F//&F%6#\"#8#!5*=k*y4(oXN(G\"5$ pV]E/P\\$yz" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}}{PARA 0 "" 0 "" {TEXT -1 52 "#-------------- -------------------------------------" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}}{PARA 0 "" 0 "" {TEXT -1 31 "#================= =============" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 23 "#======================" }}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 23 "Abreviated calculations" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}{SECT 1 {PARA 4 "" 0 " " {TEXT -1 28 "Set up order conditions etc." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 2185 "RSeqs := RowSumC onditions(8,'expanded'):\nSOeqs := [op(StageOrderConditions(2,8,'expan ded')),\n op(StageOrderConditions(3,4..8,'expanded')),\n op(StageOrderConditions(4,6..8,'expanded'))]:\nCeq := 1 4*c[6]^2*c[7]+3*c[6]-28*c[6]*c[7]+3*c[7]+588*c[9]*c[6]*c[7]*c[8]+3*c[8 ]-28*c[8]*c[6]-28*c[8]*c[7]+14*c[8]*c[6]^2+14*c[8]^2*c[6]+231*c[6]*c[7 ]*c[8]+98*c[6]^2*c[7]^2*c[8]+14*c[6]*c[7]^2+98*c[8]^2*c[7]^2*c[6]+98*c [8]^2*c[6]^2*c[7]-168*c[8]^2*c[6]*c[7]-168*c[8]*c[7]^2*c[6]-168*c[8]*c [6]^2*c[7]+14*c[8]^2*c[7]-12*c[9]+14*c[8]*c[7]^2+42*c[9]*c[6]-154*c[9] *c[6]*c[7]+42*c[7]*c[9]+42*c[8]*c[9]-154*c[8]*c[9]*c[6]-154*c[8]*c[7]* c[9]-42*c[9]*c[6]^2-42*c[7]^2*c[9]-42*c[8]^2*c[9]+168*c[8]*c[7]^2*c[9] -728*c[8]*c[6]^2*c[9]*c[7]-728*c[8]^2*c[6]*c[7]*c[9]+980*c[8]^2*c[6]^2 *c[9]*c[7]-728*c[8]*c[7]^2*c[9]*c[6]+980*c[8]^2*c[7]^2*c[9]*c[6]+980*c [6]^2*c[7]^2*c[9]*c[8]-1470*c[6]^2*c[7]^2*c[8]^2*c[9]+168*c[8]*c[6]^2* c[9]+168*c[8]^2*c[6]*c[9]-196*c[8]^2*c[6]^2*c[9]+168*c[8]^2*c[7]*c[9]- 196*c[8]^2*c[7]^2*c[9]+168*c[9]*c[6]^2*c[7]+168*c[6]*c[7]^2*c[9]-196*c [6]^2*c[7]^2*c[9]=0:\nQeqs := QuadratureConditions(8,12,'expanded'):\n SOeqs2 := [op(RowSumConditions(12,'expanded')),op(StageOrderConditions (2,9..12,'expanded')),\n op(StageOrderConditions(3,9..12,'expanded' )),op(StageOrderConditions(4,9..12,'expanded'))]:\nSO8_12 := SimpleOrd erConditions(8,12,'expanded'):\nord_cdns := [seq(SO8_12[i],i=[37,61,81 ,87,102,113,106,117,125])]:\nsimp_eqs := [add(b[i]*a[i,1],i=1+1..12)=b [1],seq(add(b[i]*a[i,j],i=j+1..12)=b[j]*(1-c[j]),j=[5,7,9,10])]:\ncdns := [op(SOeqs2),op(ord_cdns),op(simp_eqs)]:\n`Qeqs*` := subs(b=`b*`,Qu adratureConditions(7,11,'expanded')):\nSO_eqs3 := [add(a[11,j],j=1..10 )=c[11],add(a[11,j]*c[j],j=2..10)=1/2*c[11]^2,\n add(a[11,j]*c[ j]^2,j=2..10)=1/3*c[11]^3,add(a[11,j]*c[j]^3,j=2..10)=1/4*c[11]^4]:\n` simp_eqs*` := [add(`b*`[i]*a[i,1],i=2..11)=`b*`[1],seq(add(`b*`[i]*a[i ,j],i=j+1..11)=`b*`[j]*(1-c[j]),j=[8,9,10])]:\n`cdns*` := [op(`simp_eq s*`),op(SO_eqs3),op(`Qeqs*`)]:\n\nerrterms8_13 := PrincipalErrorTerms( 8,13,'expanded'):\n`errterms7_13*` := subs(b=`b*`,PrincipalErrorTerms( 7,13,'expanded')):\n`errterms8_13*` := subs(b=`b*`,PrincipalErrorTerms (8,13,'expanded')):" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}}{PARA 0 "" 0 "" {TEXT -1 0 "" }} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}{SECT 1 {PARA 4 "" 0 " " {TEXT 0 13 "calc_RKcoeffs" }{TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 3008 "calc_RKcoeffs := proc()\n local eqns,quadeqns,eqns2,eqns3,sm,ct,Rz,stb8,stb7,nmB,snm B,\n dnB,sdnB,nmC,snmC,B_9,C_9;\n global e1,e2,e3,e4,e5,e6,e 7,e8,e9,e10,e11,e12,e13,e14,e15,e16,e17,e18;\n\n e1 := \{c[2]=c_2,c[ 5]=c_5,c[6]=c_6,c[7]=c_7,c[8]=c_8,\n seq(a[i,2]=0,i=4..8),seq(a[i ,3]=0,i=6..8),a[8,a_8[1]]=a_8[2]\}:\n eqns := map(expand,subs(e1,[op (RSeqs),op(SOeqs)])):\n e2 := solve(\{op(eqns)\});\n e3 := `union` (e1,e2):\n e4 := solve(subs(e3,Ceq),\{c[9]\}):\n e5 := `union`(e3, e4):\n e6 := \{seq(b[i]=0,i=2..5),c[10]=c_10,c[11]=c_11,c[12]=1\}:\n e7 := `union`(e5,e6):\n quadeqns := subs(e7,Qeqs):\n e8 := solv e(\{op(quadeqns)\}):\n e9 := `union`(e7,e8):\n e10 := \{seq(a[i,2] =0,i=9..12),seq(a[i,3]=0,i=9..12)\}:\n e11 := `union`(e9,e10):\n e qns2 := remove(_U->evalb(lhs(_U)-rhs(_U)=0),subs(e11,\{op(cdns),op(SOe qs2)\})):\n e12 := solve(\{op(eqns2)\}):\n e13 := `union`(e11,e12) :\n e14 := `union`(remove(u_->member(op(1,lhs(u_)),[11,12]) or op(0, lhs(u_))=b,e13),\n \{c[11]=1,a[11,2]=0,a[11,3]=0,seq(`b*`[i]=0 ,i=2..5)\}):\n eqns3 := subs(e14,`cdns*`):\n e15 := solve(\{op(eqn s3)\}):\n e16 := `union`(e14,e15):\n e17 := \{c[13]=1,seq(a[13,i]= subs(e16,a[11,i]),i=1..10),a[13,11]=0,a[13,12]=0,b[13]=0,\n seq(`b *`[i]=subs(e16,`b*`[i]),i=1..10),`b*`[11]=0,`b*`[12]=0,`b*`[13]=subs(e 16,`b*`[11])\}:\n e18 := `union`(e13,e17):\n sm := 0:\n for ct t o nops(errterms8_13) do\n sm := sm+(evalf(subs(e18,errterms8_13[c t])))^2;\n end do:\n Rz := subs(e18,StabilityFunction(8,13,'expand ed'));\n stb8 := evalf[8](max(fsolve(Rz=1,z=-8..-1e-7),fsolve(Rz=-1, z=-8..-1e-7)));\n Rz := subs(e18,subs(b=`b*`,StabilityFunction(7,13, 'expanded')));\n stb7 := evalf[8](max(fsolve(Rz=1,z=-8..-1e-7),fsolv e(Rz=-1,z=-8..-1e-7)));\n nmB := 0; \n for ct to nops(`errterms8_ 13*`) do\n nmB := nmB+evalf(subs(e18,`errterms8_13*`[ct]))^2;\n \+ end do:\n snmB := sqrt(simplify(nmB));\n dnB := 0;\n for ct to \+ nops(`errterms7_13*`) do\n dnB := dnB+evalf(subs(e18,`errterms7_1 3*`[ct]))^2;\n end do;\n sdnB := sqrt(simplify(dnB));\n nmC := 0 ;\n for ct to nops(errterms8_13) do\n nmC := nmC+(evalf(subs(e1 8,`errterms8_13*`[ct]))-evalf(subs(e18,errterms8_13[ct])))^2;\n end \+ do;\n snmC := sqrt(simplify(nmC));\n B_9 := evalf[8](snmB/sdnB):\n C_9 := evalf[8](snmC/sdnB):\n print(`nodes:`,c[2]=c_2,c[3]=subs(e 18,c[3]),c[4]=subs(e18,c[4]),c[5]=c_5,c[6]=c_6,\n \+ c[7]=c_7,c[8]=c_8,c[9]=subs(e18,c[9]),c[10]=c_10,c[11]=c_11 );\n print(`order 8 weights:`,seq(b[i]=evalf[6](subs(e18,b[i])),i=[1 ,$6..12]));\n print(`order 7 weights:`,seq(`b*`[i]=evalf[6](subs(e18 ,`b*`[i])),i=[1,$6..10,13]));\n print(infinity*`-norm of linking coe ffs`=evalf(max(seq(seq(subs(e18,abs(a[i,j])),j=1..i-1),i=2..13))));\n \+ print(`2-norm of principal error of order 8 scheme` = sqrt(sm));\n \+ print(`2-norm of principal error of order 7 scheme` = sdnB);\n prin t(`order 8 stability interval` = [stb8,0]);\n print(`order 7 stabili ty interval` = [stb7,0]);\n print('B[9]'=B_9,'C[9]'=C_9)\nend proc: " }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 " " {TEXT -1 57 "#------------------------------------------------------ --" }}{PARA 0 "" 0 "" {TEXT -1 29 "Verner's \"most robust\" scheme" }} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 132 "c_2 := 1/4: c_5 := 53/125: \+ c_6 := 509/1000: c_7 := 867/1000: c_8 := 3/20:\nc_10 := 8/25: c_11 := \+ 9/20: a_8 := [4,0]:\ncalc_RKcoeffs();" }}{PARA 11 "" 1 "" {XPPMATH 20 "6-%'nodes:G/&%\"cG6#\"\"##\"\"\"\"\"%/&F&6#\"\"$#\"&@g)\"'+?w/&F&6#F+ #F1\"'+!3&/&F&6#\"\"&#\"#`\"$D\"/&F&6#\"\"'#\"$4&\"%+5/&F&6#\"\"(#\"$n )FE/&F&6#\"\")#F/\"#?/&F&6#\"\"*#\"-kd%)f^H\"-\\;\\kiT/&F&6#\"#5#FO\"# D/&F&6#\"#6#FUFQ" }}{PARA 11 "" 1 "" {XPPMATH 20 "6+%1order~8~weights: G/&%\"bG6#\"\"\"$\"''HZ%!\"(/&F&6#\"\"'$\"'5p:!\"'/&F&6#\"\"($\"'5Y=F2 /&F&6#\"\")$\"'k^AF2/&F&6#\"\"*$\"'Yz9F2/&F&6#\"#5$\"'b0wF+/&F&6#\"#6$ \"'tF7F2/&F&6#\"#7$\"'?\"=%F+" }}{PARA 11 "" 1 "" {XPPMATH 20 "6*%1ord er~7~weights:G/&%#b*G6#\"\"\"$\"'r%e%!\"(/&F&6#\"\"'$\"'>BE!\"'/&F&6# \"\"($\"'%p\">F2/&F&6#\"\")$\"'#4<#F2/&F&6#\"\"*$\"'#QF\"F2/&F&6#\"#5$ \"'0^6F2/&F&6#\"#8$\"'8cSF+" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/*&%)in finityG\"\"\"%8-norm~of~linking~coeffsGF&$\"+^ug:f!\"*" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/%L2-norm~of~principal~error~of~order~8~schemeG$\" +y-xYv!#:" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/%L2-norm~of~principal~er ror~of~order~7~schemeG$\"+qQT6@!#9" }}{PARA 11 "" 1 "" {XPPMATH 20 "6# /%;order~8~stability~intervalG7$$!)Wv>[!\"(\"\"!" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/%;order~7~stability~intervalG7$$!)gY0]!\"(\"\"!" }} {PARA 11 "" 1 "" {XPPMATH 20 "6$/&%\"BG6#\"\"*$\")wHcI!\"(/&%\"CGF&$\" )6:6IF*" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 57 "#--------------------------------------------------------" }} {PARA 0 "" 0 "" {TEXT -1 21 "Enright-Verner scheme" }}{EXCHG {PARA 0 " > " 0 "" {MPLTEXT 1 0 152 "c_2 := 139/2500: c_5 := 1923/5000: c_6 := 9 23/2000: c_7 := 769/5000: c_8 := 8571/10000:\nc_10 := 3611/5000: c_11 \+ := 15/16: a_8 := [7,9]:\ncalc_RKcoeffs();" }}{PARA 11 "" 1 "" {XPPMATH 20 "6-%'nodes:G/&%\"cG6#\"\"##\"$R\"\"%+D/&F&6#\"\"$#\"'*\\t% \"(+gh%/&F&6#\"\"%#\"((\\?9\"(+?B*/&F&6#\"\"&#\"%B>\"%+]/&F&6#\"\"'#\" $B*\"%+?/&F&6#\"\"(#\"$p(F@/&F&6#\"\")#\"%r&)\"&++\"/&F&6#\"\"*#\"5+=W $pl:=!G9\"5jpEaXi5N-:/&F&6#\"#5#\"%6OF@/&F&6#\"#6#\"#:\"#;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6+%1order~8~weights:G/&%\"bG6#\"\"\"$\"'x\"R%! \"(/&F&6#\"\"'$\"'D5N!\"'/&F&6#\"\"($\"'VhCF2/&F&6#\"\")$\"'C.!*F2/&F& 6#\"\"*$\"'U\\X!\"&/&F&6#\"#5$\"']-[!\")/&F&6#\"#6$!'0TZFE/&F&6#\"#7$! 'xXNF2" }}{PARA 11 "" 1 "" {XPPMATH 20 "6*%1order~7~weights:G/&%#b*G6# \"\"\"$\"'6KV!\"(/&F&6#\"\"'$\"'+$Q$!\"'/&F&6#\"\"($\"'![[#F2/&F&6#\" \")$\"')yB#F2/&F&6#\"\"*$!';?SF+/&F&6#\"#5$\"'#HB\"F2/&F&6#\"#8$\"'.-j F+" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/*&%)infinityG\"\"\"%8-norm~of~l inking~coeffsGF&$\"+j#R\"=>!\")" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/%L 2-norm~of~principal~error~of~order~8~schemeG$\"+4`_&H\"!#:" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/%L2-norm~of~principal~error~of~order~7~sche meG$\"+UuoBF!#9" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/%;order~8~stabilit y~intervalG7$$!)RcUc!\"(\"\"!" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/%;or der~7~stability~intervalG7$$!)/)3q&!\"(\"\"!" }}{PARA 11 "" 1 "" {XPPMATH 20 "6$/&%\"BG6#\"\"*$\")vqW>!\"(/&%\"CGF&$\")3wG>F*" }}} {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 57 "#-------- ------------------------------------------------" }}{PARA 0 "" 0 "" {TEXT -1 23 "Verner's \"Maple\" scheme" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 133 "c_2 := 1/16: c_5 := 39/100: c_6 := 7/15: c_7 := 39/2 50: c_8 := 24/25:\nc_10 := 11/12: c_11 := 19/20: a_8 := [4,-7/5]:\ncal c_RKcoeffs();" }}{PARA 11 "" 1 "" {XPPMATH 20 "6-%'nodes:G/&%\"cG6#\" \"##\"\"\"\"#;/&F&6#\"\"$#\"$7\"\"%l5/&F&6#\"\"%#\"#c\"$b$/&F&6#\"\"&# \"#R\"$+\"/&F&6#\"\"'#\"\"(\"#:/&F&6#FF#F?\"$]#/&F&6#\"\")#\"#C\"#D/&F &6#\"\"*#\")oeV9\")h)yh\"/&F&6#\"#5#\"#6\"#7/&F&6#Fjn#\"#>\"#?" }} {PARA 11 "" 1 "" {XPPMATH 20 "6+%1order~8~weights:G/&%\"bG6#\"\"\"$\"' )GY%!\"(/&F&6#\"\"'$\"']QN!\"'/&F&6#\"\"($\"'9#\\#F2/&F&6#\"\")$!'MG_! \"&/&F&6#\"\"*$\"'<^^F?/&F&6#\"#5$!'z4&*F?/&F&6#\"#6$\"''f+\"!\"%/&F&6 #\"#7$!'h.7F2" }}{PARA 11 "" 1 "" {XPPMATH 20 "6*%1order~7~weights:G/& %#b*G6#\"\"\"$\"'QpW!\"(/&F&6#\"\"'$\"'\\[N!\"'/&F&6#\"\"($\"'t*[#F2/& F&6#\"\")$\"'jlE!\"&/&F&6#\"\"*$\"'EBRF?/&F&6#\"#5$!'V!o&F?/&F&6#\"#8$ !'%)pbF2" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/*&%)infinityG\"\"\"%8-nor m~of~linking~coeffsGF&$\"+@Oq&p\"!\")" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/%L2-norm~of~principal~error~of~order~8~schemeG$\"+\"H[4N)!#;" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#/%L2-norm~of~principal~error~of~order~ 7~schemeG$\"+sNs?$*!#:" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/%;order~8~s tability~intervalG7$$!)-]&y&!\"(\"\"!" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/%;order~7~stability~intervalG7$$!)(R:`&!\"(\"\"!" }}{PARA 11 "" 1 "" {XPPMATH 20 "6$/&%\"BG6#\"\"*$\")uP:?!\"(/&%\"CGF&$\"))y`-#F*" }}} {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 57 "#-------- ------------------------------------------------" }}{PARA 0 "" 0 "" {TEXT -1 41 "a modification of Verner's \"Maple\" scheme" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 134 "c_2 := 3/50: c_5 := 100/257: c_6 : = 7/15: c_7 := 12/77: c_8 := 26/27:\nc_10 := 11/12: c_11 := 19/20: a_8 := [4,-17/2]:\ncalc_RKcoeffs();" }}{PARA 11 "" 1 "" {XPPMATH 20 "6-%' nodes:G/&%\"cG6#\"\"##\"\"$\"#]/&F&6#F*#\"$p%\"%5X/&F&6#\"\"%#\"%29\"% ?!*/&F&6#\"\"&#\"$+\"\"$d#/&F&6#\"\"'#\"\"(\"#:/&F&6#FE#\"#7\"#x/&F&6# \"\")#\"#E\"#F/&F&6#\"\"*#\"(^55(\"([Z%z/&F&6#\"#5#\"#6FK/&F&6#Fjn#\"# >\"#?" }}{PARA 11 "" 1 "" {XPPMATH 20 "6+%1order~8~weights:G/&%\"bG6# \"\"\"$\"'BcW!\"(/&F&6#\"\"'$\"'&Qa$!\"'/&F&6#\"\"($\"'*3\\#F2/&F&6#\" \")$!'#fA$!\"&/&F&6#\"\"*$\"'-%[&F?/&F&6#\"#5$!'#=e*F?/&F&6#\"#6$\"'' \\$yF?/&F&6#\"#7$!'s#f\"F2" }}{PARA 11 "" 1 "" {XPPMATH 20 "6*%1order~ 7~weights:G/&%#b*G6#\"\"\"$\"'fiW!\"(/&F&6#\"\"'$\"'d`N!\"'/&F&6#\"\"( $\"'`)[#F2/&F&6#\"\")$\"''Ri#!\"&/&F&6#\"\"*$\"'?zTF?/&F&6#\"#5$!'gAeF ?/&F&6#\"#8$!'*RH'F2" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/*&%)infinityG \"\"\"%8-norm~of~linking~coeffsGF&$\"+G#[Qg\"!\")" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/%L2-norm~of~principal~error~of~order~8~schemeG$\"+.RzQ l!#;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/%L2-norm~of~principal~error~o f~order~7~schemeG$\"+nsG.#*!#:" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/%;o rder~8~stability~intervalG7$$!)^$f\"f!\"(\"\"!" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/%;order~7~stability~intervalG7$$!)R2Ed!\"(\"\"!" }} {PARA 11 "" 1 "" {XPPMATH 20 "6$/&%\"BG6#\"\"*$\")\\)Q,#!\"(/&%\"CGF&$ \")[L>?F*" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 57 "#--------------------------------------------------------" }} {PARA 0 "" 0 "" {TEXT -1 32 "Verner's \"most efficient\" scheme" }} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 143 "c_2 := 1/20: c_5 := 39/100: c_6 := 93/200: c_7 := 31/200: c_8 := 943/1000:\nc_10 := 909/1000: c_1 1 := 47/50: a_8 := [4,-125/2]:\ncalc_RKcoeffs();" }}{PARA 11 "" 1 "" {XPPMATH 20 "6-%'nodes:G/&%\"cG6#\"\"##\"\"\"\"#?/&F&6#\"\"$#\"$T$\"%+ K/&F&6#\"\"%#\"%B5\"%+k/&F&6#\"\"&#\"#R\"$+\"/&F&6#\"\"'#\"#$*\"$+#/&F &6#\"\"(#\"#JFG/&F&6#\"\")#\"$V*\"%+5/&F&6#\"\"*#\".!G;!ev1(\".n1;]r$y /&F&6#\"#5#\"$4*FT/&F&6#\"#6#\"#Z\"#]" }}{PARA 11 "" 1 "" {XPPMATH 20 "6+%1order~8~weights:G/&%\"bG6#\"\"\"$\"'*zU%!\"(/&F&6#\"\"'$\"'0TN!\" '/&F&6#\"\"($\"'pzCF2/&F&6#\"\")$!'Up:!\"%/&F&6#\"\"*$\"'T3DF?/&F&6#\" #5$!'%Q<$F?/&F&6#\"#6$\"'$QH#F?/&F&6#\"#7$!'KhBF2" }}{PARA 11 "" 1 "" {XPPMATH 20 "6*%1order~7~weights:G/&%#b*G6#\"\"\"$\"'EJW!\"(/&F&6#\"\" '$\"'5YN!\"'/&F&6#\"\"($\"'[yCF2/&F&6#\"\")$\"'8[W!\"&/&F&6#\"\"*$\"'p %)>!\"%/&F&6#\"#5$!';eBFF/&F&6#\"#8$!'o,OF2" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/*&%)infinityG\"\"\"%8-norm~of~linking~coeffsGF&$\"+G%f PB\"!\"(" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/%L2-norm~of~principal~err or~of~order~8~schemeG$\"+Pg'y#G!#;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6# /%L2-norm~of~principal~error~of~order~7~schemeG$\"+PY3Y$)!#:" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/%;order~8~stability~intervalG7$$!)K6ke!\"( \"\"!" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/%;order~7~stability~interval G7$$!)L8)p&!\"(\"\"!" }}{PARA 11 "" 1 "" {XPPMATH 20 "6$/&%\"BG6#\"\"* $\")!\\R(>!\"(/&%\"CGF&$\"):Q$)>F*" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" } }{PARA 0 "" 0 "" {TEXT -1 57 "#--------------------------------------- -----------------" }}{PARA 0 "" 0 "" {TEXT -1 21 "An \"efficient\" sch eme" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 137 "c_2 := 27/400: c_5 : = 100/257: c_6 := 7/15: c_7 := 7/45: c_8 := 58/59:\nc_10 := 13/14: c_1 1 := 62/65: a_8 := [4,-743/55]:\ncalc_RKcoeffs();" }}{PARA 11 "" 1 "" {XPPMATH 20 "6-%'nodes:G/&%\"cG6#\"\"##\"#F\"$+%/&F&6#\"\"$#\"$p%\"%5X /&F&6#\"\"%#\"%29\"%?!*/&F&6#\"\"&#\"$+\"\"$d#/&F&6#\"\"'#\"\"(\"#:/&F &6#FF#FF\"#X/&F&6#\"\")#\"#e\"#f/&F&6#\"\"*#\")yu6\\\")VZ9b/&F&6#\"#5# \"#8\"#9/&F&6#\"#6#\"#i\"#l" }}{PARA 11 "" 1 "" {XPPMATH 20 "6+%1order ~8~weights:G/&%\"bG6#\"\"\"$\"'$RW%!\"(/&F&6#\"\"'$\"'\"Rb$!\"'/&F&6# \"\"($\"'b)[#F2/&F&6#\"\")$!'YI@!\"&/&F&6#\"\"*$\"'eQJF?/&F&6#\"#5$!': y#)F?/&F&6#\"#6$\"'\"=P(F?/&F&6#\"#7$\"'K&\\#F2" }}{PARA 11 "" 1 "" {XPPMATH 20 "6*%1order~7~weights:G/&%#b*G6#\"\"\"$\"'QaW!\"(/&F&6#\"\" '$\"'KpN!\"'/&F&6#\"\"($\"'r%[#F2/&F&6#\"\")$\"'.$\\$!\"&/&F&6#\"\"*$ \"'zgAF?/&F&6#\"#5$!'Z9NF?/&F&6#\"#8$!'I*)=F?" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/*&%)infinityG\"\"\"%8-norm~of~linking~coeffsGF&$\"+oZ' )4=!\")" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/%L2-norm~of~principal~erro r~of~order~8~schemeG$\"+US&Rt&!#;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/ %L2-norm~of~principal~error~of~order~7~schemeG$\"+!oeQ+\"!#9" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/%;order~8~stability~intervalG7$$!)')Q7g!\"( \"\"!" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/%;order~7~stability~interval G7$$!)=*yw&!\"(\"\"!" }}{PARA 11 "" 1 "" {XPPMATH 20 "6$/&%\"BG6#\"\"* $\")3#43#!\"(/&%\"CGF&$\") " 0 "" {MPLTEXT 1 0 127 "c_2 := 1/20: c_5 := 21/50: \+ c_6 := 55/108: c_7 := 13/15: c_8 := 3/20:\nc_10 := 13/34: c_11 := 7/8: a_8 := [4,0]:\ncalc_RKcoeffs();" }}{PARA 11 "" 1 "" {XPPMATH 20 "6-%' nodes:G/&%\"cG6#\"\"##\"\"\"\"#?/&F&6#\"\"$#\"%Nv\"&;/(/&F&6#\"\"%#F1 \"&Wp%/&F&6#\"\"&#\"#@\"#]/&F&6#\"\"'#\"#b\"$3\"/&F&6#\"\"(#\"#8\"#:/& F&6#\"\")#F/F+/&F&6#\"\"*#\")&>i:\"\")nST;/&F&6#\"#5#FL\"#M/&F&6#\"#6# FJFQ" }}{PARA 11 "" 1 "" {XPPMATH 20 "6+%1order~8~weights:G/&%\"bG6#\" \"\"$\"'@?W!\"(/&F&6#\"\"'$\"'w(z\"!\"'/&F&6#\"\"($\"'V8iF+/&F&6#\"\") $\"'92BF2/&F&6#\"\"*$\"'YE;F2/&F&6#\"#5$\"''[i\"F2/&F&6#\"#6$\"'yz6F2/ &F&6#\"#7$\"'O1SF+" }}{PARA 11 "" 1 "" {XPPMATH 20 "6*%1order~7~weight s:G/&%#b*G6#\"\"\"$\"'&[S%!\"(/&F&6#\"\"'$\"'?!)=!\"'/&F&6#\"\"($\"'jd =F2/&F&6#\"\")$\"''\\J#F2/&F&6#\"\"*$\"'82:F2/&F&6#\"#5$\"'3!e\"F2/&F& 6#\"#8$\"';&>%F+" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/*&%)infinityG\"\" \"%8-norm~of~linking~coeffsGF&$\"+k/Vwe!\"*" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/%L2-norm~of~principal~error~of~order~8~schemeG$\"+`twM k!#:" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/%L2-norm~of~principal~error~o f~order~7~schemeG$\"+\")30\"=#!#9" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/ %;order~8~stability~intervalG7$$!))*fB\\!\"(\"\"!" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/%;order~7~stability~intervalG7$$!)^-G^!\"(\"\"!" }} {PARA 11 "" 1 "" {XPPMATH 20 "6$/&%\"BG6#\"\"*$\")(G_w#!\"(/&%\"CGF&$ \")b>FGF*" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 57 "#--------------------------------------------------------" }} {PARA 0 "" 0 "" {TEXT -1 27 "Tsitouras-Papakostas scheme" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 141 "c_2 := 9/142: c_5 := 50/129: c_6 : = 34/73: c_7 := 23/148: c_8 := 142/141:\nc_10 := 83/91: c_11 := 143/14 9: a_8 := [7,254/39]:\ncalc_RKcoeffs();" }}{PARA 11 "" 1 "" {XPPMATH 20 "6-%'nodes:G/&%\"cG6#\"\"##\"\"*\"$U\"/&F&6#\"\"$#\"&9X#\"'\"\\Q#/& F&6#\"\"%#\"&dA\"\"&(\\z/&F&6#\"\"&#\"#]\"$H\"/&F&6#\"\"'#\"#M\"#t/&F& 6#\"\"(#\"#B\"$[\"/&F&6#\"\")#F+\"$T\"/&F&6#F*#\",)>?T5H\",2<`=K$/&F&6 #\"#5#\"#$)\"#\"*/&F&6#\"#6#\"$V\"\"$\\\"" }}{PARA 11 "" 1 "" {XPPMATH 20 "6+%1order~8~weights:G/&%\"bG6#\"\"\"$\"';TW!\"(/&F&6#\"\" '$\"'^RN!\"'/&F&6#\"\"($\"'A&[#F2/&F&6#\"\")$!'\"pK$F2/&F&6#\"\"*$\"'D @>!\"&/&F&6#\"#5$!'yJFFE/&F&6#\"#6$\"'?,9FE/&F&6#\"#7$\"'h8&*F+" }} {PARA 11 "" 1 "" {XPPMATH 20 "6*%1order~7~weights:G/&%#b*G6#\"\"\"$\"' U[W!\"(/&F&6#\"\"'$\"'C]N!\"'/&F&6#\"\"($\"'b#[#F2/&F&6#\"\")$!'BCC!\" &/&F&6#\"\"*$\"'$**f\"F?/&F&6#\"#5$!'w5=F?/&F&6#\"#8$\"'I()HF?" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#/*&%)infinityG\"\"\"%8-norm~of~linking ~coeffsGF&$\"+I:Ns6!\")" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/%L2-norm~o f~principal~error~of~order~8~schemeG$\"+&Qb'[t!#;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/%L2-norm~of~principal~error~of~order~7~schemeG$\"+Gw`c 5!#9" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/%;order~8~stability~intervalG 7$$!)N)o!f!\"(\"\"!" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/%;order~7~stab ility~intervalG7$$!)uFWe!\"(\"\"!" }}{PARA 11 "" 1 "" {XPPMATH 20 "6$/ &%\"BG6#\"\"*$\")9)4.#!\"(/&%\"CGF&$\")n'4.#F*" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 57 "#------------------------ --------------------------------" }}{PARA 0 "" 0 "" {TEXT -1 49 "a mod ification of the Tsitouras-Papakostas scheme" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 136 "c_2 := 2/25: c_5 := 50/129: c_6 := 34/73: c_7 := \+ 23/148: c_8 := 196/197:\nc_10 := 39/43: c_11 := 20/21: a_8 := [7,19/2] :\ncalc_RKcoeffs();" }}{PARA 11 "" 1 "" {XPPMATH 20 "6-%'nodes:G/&%\"c G6#\"\"##F(\"#D/&F&6#\"\"$#\"&9X#\"'\"\\Q#/&F&6#\"\"%#\"&dA\"\"&(\\z/& F&6#\"\"&#\"#]\"$H\"/&F&6#\"\"'#\"#M\"#t/&F&6#\"\"(#\"#B\"$[\"/&F&6#\" \")#\"$'>\"$(>/&F&6#\"\"*#\",c^9=T\"\",6)=(Qg\"/&F&6#\"#5#\"#R\"#V/&F& 6#\"#6#\"#?\"#@" }}{PARA 11 "" 1 "" {XPPMATH 20 "6+%1order~8~weights:G /&%\"bG6#\"\"\"$\"'?TW!\"(/&F&6#\"\"'$\"'lRN!\"'/&F&6#\"\"($\"'?&[#F2/ &F&6#\"\")$\"',k8!\"&/&F&6#\"\"*$\"'.sEF?/&F&6#\"#5$!'Q>NF?/&F&6#\"#6$ \"'n)>\"F?/&F&6#\"#7$!'Bi8F?" }}{PARA 11 "" 1 "" {XPPMATH 20 "6*%1orde r~7~weights:G/&%#b*G6#\"\"\"$\"'dYW!\"(/&F&6#\"\"'$\"'tZN!\"'/&F&6#\" \"($\"'A$[#F2/&F&6#\"\")$\"'Z:W!\"&/&F&6#\"\"*$\"'R_AF?/&F&6#\"#5$!'^N DF?/&F&6#\"#8$!'\"*zPF?" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/*&%)infini tyG\"\"\"%8-norm~of~linking~coeffsGF&$\"+$GnlA\"!\")" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/%L2-norm~of~principal~error~of~order~8~schemeG$\"+I *4OJ(!#;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/%L2-norm~of~principal~err or~of~order~7~schemeG$\"+g8875!#9" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/ %;order~8~stability~intervalG7$$!)n!\"(/&%\"CGF&$ \")w(>*>F*" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 57 "#--------------------------------------------------------" }} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}}{PARA 0 "" 0 "" {TEXT -1 23 "#======================" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }} {PARA 0 "" 0 "" {TEXT -1 29 "#============================" }}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 36 "Test-bed procedures for the examples" }} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}{SECT 1 {PARA 4 "" 0 " " {TEXT -1 10 "RK8_12step" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 4512 "rk8step := proc(x_rk8step::realco ns)\n local c2,c3,c4,c5,c6,c7,c8,c9,cA,cB,cC,a21,a31,a32,a41,a42,a43 ,a51,a52,\n a53,a54,a61,a62,a63,a64,a65,a71,a72,a73,a74,a75,a76,a81, a82,a83,a84,a85,\n a86,a87,a91,a92,a93,a94,a95,a96,a97,a98,aA1,aA2,a A3,aA4,aA5,aA6,aA7,aA8,\n aA9,aB1,aB2,aB3,aB4,aB5,aB6,aB7,aB8,aB9,aB A,aC1,aC2,aC3,aC4,aC5,aC6,aC7,\n aC8,aC9,aCA,aCB,f1,f2,f3,f4,f5,f6,f 7,f8,f9,fA,fB,fC,b1,b2,b3,b4,b5,b6,b7,\n b8,b9,bA,bB,bC,xk,yk,t,jF,j M,jS,n,h,data,fn,xx,ys,saveDigits;\n options `Copyright 2004 by Pete r Stone`;\n \n data := SOLN_;\n\n saveDigits := Digits;\n Digi ts := max(trunc(evalhf(Digits)),Digits+5);\n\n # procedure to evalua te the slope field\n fn := proc(X_,Y_)\n local val; \n val := traperror(evalf(FXY_));\n if val=lasterror or not type(val,nu meric) then\n error \"evaluation of slope field failed at %1\" ,evalf([X_,Y_],saveDigits);\n end if;\n val;\n end proc ;\n\n xx := evalf(x_rk8step);\n n := nops(data);\n\n if (data[1, 1]data[n,1] or xxdata[1,1])) then\n error \"inde pendent variable is outside the interpolation interval: %1\",evalf(dat a[1,1])..evalf(data[n,1]);\n end if;\n\n c2 := c2_; c3 := c3_; c4 \+ := c4_; c5 := c5_; c6 := c6_; c7 := c7_; c8 := c8_;\n c9 := c9_; cA \+ := cA_; cB := cB_; cC := cC_;\n a21 := c2; a31 := a31_; a32 := a32_; a41 := a41_; a42 := a42_; a43 := a43_;\n a51 := a51_; a52 := a52_; \+ a53 := a53_; a54 := a54_;\n a61 := a61_; a62 := a62_; a63 := a63_; a 64 := a64_; a65 := a65_;\n a71 := a71_; a72 := a72_; a73 := a73_; a7 4 := a74_; a75 := a75_; a76 := a76_;\n a81 := a81_; a82 := a82_; a83 := a83_; a84 := a84_; a85 := a85_; a86 := a86_;\n a87 := a87_;\n \+ a91 := a91_; a92 := a92_; a93 := a93_; a94 := a94_; a95 := a95_; a96 : = a96_; \n a97 := a97_; a98 := a98_; \n aA1 := aA1_; aA2 := aA2_ ; aA3 := aA3_; aA4 := aA4_; aA5 := aA5_; aA6 := aA6_; \n aA7 := aA7_ ; aA8 := aA8_; aA9 := aA9_;\n aB1 := aB1_; aB2 := aB2_; aB3 := aB3_; aB4 := aB4_; aB5 := aB5_; aB6 := aB6_; \n aB7 := aB7_; aB8 := aB8_; aB9 := aB9_; aBA := aBA_;\n aC1 := aC1_; aC2 := aC2_; aC3 := aC3_; \+ aC4 := aC4_; aC5 := aC5_; aC6 := aC6_; \n aC7 := aC7_; aC8 := aC8_; \+ aC9 := aC9_; aCA := aCA_; aCB := aCB_;\n b1 := b1_; b2 := b2_; b3 := b3_; b4 := b4_; b5 := b5_; b6 := b6_; b7 := b7_; \n b8 := b8_; b9 : = b9_; bA := bA_; bB := bB_; bC := bC_;\n # Perform a binary search \+ for the interval containing x.\n n := nops(data);\n jF := 0;\n j S := n+1;\n\n if data[1,1]1 do\n \+ jM := trunc((jF+jS)/2);\n if xx>=data[jM,1] then jF := j M else jS := jM end if;\n end do;\n if jM = n then jF := n-1 ; jS := n end if;\n else\n while jS-jF> 1 do\n jM := tru nc((jF+jS)/2);\n if xx<=data[jM,1] then jF := jM else jS := jM \+ end if;\n end do;\n if jM = n then jF := n-1; jS := n end if ;\n end if;\n \n # Get the data needed from the list.\n xk := d ata[jF,1];\n yk := data[jF,2];\n\n # Do one step with step-size .. \n h := xx-xk;\n f1 := fn(xk,yk);\n t := a21*f1;\n f2 := fn(xk + c2*h,yk + t*h);\n t := a31*f1 + a32*f2;\n f3 := fn(xk + c3*h,yk + t*h);\n t := a41*f1 + a42*f2 + a43*f3;\n f4 := fn(xk + c4*h,yk \+ + t*h);\n t := a51*f1 + a52*f2 + a53*f3 + a54*f4;\n f5 := fn(xk + \+ c5*h,yk + t*h);\n t := a61*f1 + a62*f2 + a63*f3 + a64*f4 + a65*f5;\n f6 := fn(xk + c6*h,yk + t*h);\n t := a71*f1 + a72*f2 + a73*f3 + a 74*f4 + a75*f5 + a76*f6;\n f7 := fn(xk + c7*h,yk + t*h);\n t := a8 1*f1 + a82*f2 + a83*f3 + a84*f4 + a85*f5 + a86*f6 + a87*f7;\n f8 := \+ fn(xk + c8*h,yk + t*h);\n t := a91*f1 + a92*f2 + a93*f3 + a94*f4 + a 95*f5 + a96*f6 + a97*f7 + a98*f8;\n f9 := fn(xk + c9*h,yk + t*h);\n \+ t := aA1*f1 + aA2*f2 + aA3*f3 + aA4*f4 + aA5*f5 + aA6*f6 + aA7*f7 + \+ aA8*f8 + aA9*f9;\n fA := fn(xk + cA*h,yk + t*h);\n t := aB1*f1 + a B2*f2 + aB3*f3 + aB4*f4 + aB5*f5 + aB6*f6 + aB7*f7 + aB8*f8 + aB9*f9\n \+ + aBA*fA;\n fB := fn(xk + cB*h,yk + t*h);\n t := aC1*f1 + \+ aC2*f2 + aC3*f3 + aC4*f4 + aC5*f5 + aC6*f6 + aC7*f7 + aC8*f8 + aC9*f9 \n \+ + aCA*fA + aCB*fB;\n fC := fn(xk + cC*h,yk + t*h);\n \n t := b1*f1 + b2*f2 + b3*f3 + b4*f4 + b5*f5 + b6*f6 + b7*f7 + b8*f8 + b9*f9 + bA*fA\n \+ + bB*fB + bC*fC;\n \n ys := yk + t*h;\n\n evalf [saveDigits](ys);\nend proc: # of rk8step" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}}{PARA 0 "" 0 "" {TEXT -1 27 "#--------------------------" }}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 27 "RK8_1 Enright-Verner scheme" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 12203 "RK8_1 := proc(f xy,x,y,xx,yy,h,stps,bb)\n local c2,c3,c4,c5,c6,c7,c8,c9,cA,cB,cC,a21 ,a31,a32,a41,a42,a43,a51,a52,\n a53,a54,a61,a62,a63,a64,a65,a71,a72, a73,a74,a75,a76,a81,a82,a83,a84,a85,\n a86,a87,a91,a92,a93,a94,a95,a 96,a97,a98,aA1,aA2,aA3,aA4,aA5,aA6,aA7,aA8,\n aA9,aB1,aB2,aB3,aB4,aB 5,aB6,aB7,aB8,aB9,aBA,aC1,aC2,aC3,aC4,aC5,aC6,aC7,\n aC8,aC9,aCA,aCB ,f1,f2,f3,f4,f5,f6,f7,f8,f9,fA,fB,fC,b1,b2,b3,b4,b5,b6,b7,\n b8,b9,b A,bB,bC,t,k,fn,xk,yk,soln,eqns,A,saveDigits;\n\n saveDigits := Digit s;\n Digits := max(trunc(evalhf(Digits)),Digits+5);\n\n fn := unap ply(fxy,x,y);\n\n A := matrix([[139/2500,139/2500,0,0,0,0,0,0,0,0,0, 0,0],\n[473499/4616000,94226774499/11846945536000,224201303001/2369389 107200,0,0,0,0,0,0,0,0,0,0],\n[1420497/9232000,1420497/36928000,0,4261 491/36928000,0,0,0,0,0,0,0,0,0],\n[1923/5000,3590880889914277/93417209 58375000,0,-1122077307242443/778476746531250,1683359084698258/11677151 19796875,0,0,0,0,0,0,0,0],\n[923/2000,3279216863/71027928000,0,0,14184 66643672/6146756958375,725783021041/3932390759616,0,0,0,0,0,0,0],\n[76 9/5000,22628776819090891/378192197956950000,0,0,116326878837670912/104 7320101758291075,-21022506263989/614436056190000,182198401/10649112500 ,0,0,0,0,0,0],\n[8571/10000,-150702609045465151/280142368857000000,0,0 ,-47048174572430533795112/6781573975400382234375,-29670244019740727/63 63654238520000,503350581600824913/125990342179812500,9,0,0,0,0,0],\n[1 4280181565693441800/15023510624554266963,-1604591803554452653108566670 8200976799092954349059102554439418058855237034803138076235914485211856 830024864166368323768200/982948315772866484419854403492165597537825783 0080951893693520116322068439377446708493269493633244146421522730587295 739613,0,0,-2358539147881877873039445811751103289867403691705393271280 994482043263320271984518766662589229245453919623760032009543680000000/ 2178355261819555655580228003414294640863851721229066080313495435838630 22015887117168566338470163090032498196064101837445053273,-991136638972 6261686789033714164561000939004055351649246830581228024297073540333828 26947398158683765324439618282500000000/7984814200884600278929892522760 5775190331269194726743910364273272231784282184770467794155269096224513 726772081370189773,994112798212104133871493524972117858295471493586969 52646781033905129048593757052549024957474512389085654050445280000000/1 0219750071154355529199360565479179548497227516418867455143546824991883 616697162903707552409044338358355837306818391433,194327599672380134095 8982917199129613636780737930235250070813284250984315744488097793107325 32821200046895000000000/1199601148822772264965667393113649089125646329 2620473601841875115170259043531987881275965965525693289412424400177,-4 7381438676841221895938162441994505404833843721635499519905253875507680 38015218275414120082248510000000000/4562738155611920982891652838532643 4273376137158228892503158792567081801761628344925618992749059885819540 261,0,0,0,0],\n[3611/5000,10050976387926430682409615346304117463662936 4248095333923106653001873/22949032400764462804236175621743615583246148 8260089524115475000000000,0,0,3507626188921357826199528639005398322193 7920015616/8903662403052468890234321680409039895089390971875,298770534 72248545227782869189767925950557009/1044370915864536295808990574013420 6606110000,-7260202518279888944289396655384428601277677001958883877629 7031451/40918439380405007926071673834718276106178863851548244800289187 500,-15322912063864370512130145988492098605486502994107816190/31300694 00645604876183669549605539076333579290524919889,6608515467721941864547 1125072555541174985695924/31022273664823506249509795196263880038420341 7327,-3347565461896560762549026667823123036634583052726552531003001623 0875755239420324728600957368877132012320553021/56360718148650508277541 9237614873016043169125130359330269370345097328575740250457475506596993 780051575000000000,0,0,0],\n[15/16,-5101097760197841615137571256611109 219669965728737004453654940399435749927779/346025479006521849802539491 2113654440547987641021174968985010753183759952128,0,0,-739282853758412 257967453242147288028248514875000/672441828756548290983239437133742077 09631980757,-304301954438407952266341438991435702455078125/26817588459 995310678515661957421952038871616,794489399239989311647663169752036385 5501654462308622798446100145004371780765625/88700516160292865925282760 4035768657298435884939489754462273147402132735951354,27146666288983512 8614810796916069488062485834784939879774912795880859375/17080057868208 807584911919572395871967414618088449127584294090356526117,-54767163917 3493535187249719091874257724281347519101322819710937500/55458973457775 51056363997101744668576293320106258184122878184564689,2788050542294564 7305191478577013040105627752107138627482025385282602360255403525287888 8463451556710568650135086751734733498388408614837563/57032789758993842 4716739897067258193159963418613722438239821039844076423461361410150559 14541200747733442561186185728215790313661884647470336,-764103108962071 3267943821483116886435546875/18651155497290591072361407679533755137277 32867,0,0],\n[1,-21732961652445684345341684967257542832103708567140482 95955495704392191998074219/8263297098171974682539128149962767212977192 71873047121496939265708086637102000,0,0,-11764858893344723453979480247 69479865991267657667189808793600000/1282382741257457848573950236830553 91720808636521055761250085619,-901641526273088667293677983960434306145 731566007759375000/470060513215317264776508706751155977795361410985883 95119,4461919770934584303860999581029022209271373435750357302128939918 8441075277893825000/30472268298677596678056782370835849160284566290690 14496162891378892137605913688013,9483397426652107169315607674597053474 32692833888702670755668800833857753125000/5410020306852381652392054720 3712158257383235486719126879290708372319555115953,-4693599865307842423 67841163967997985280992052062479522694037718750000/1262000068797740029 604893636717124646814536106030908513651797310900389,-66412748100626086 6245417353167879111725438608160134129788715555378465573117813001757238 2606930519018125943416089220164619687237540672159347990548526304469380 63/9474053136757793278864620448618348468794738615395834799951011682911 1616884404043875158439278591322423989402984382146436322440544087723565 1675827150051218506000,12459468894943123671827871807868857124596796164 1584151713065625000/24422661199951239145999264873446308735658803040031 36401896150897137,5610647423101109856996396240147689817939409487616804 64947352/671379391526329055316725343694451428305592235340194400714475, 0],\n[0,55284231195707975647126708111723/12588142504750145545176036121 14000,0,0,0,0,101652048214282205518610445783893750000000/2895866582780 60310247144250081091360673509,6219731958882093270433753490048828125000 000/25268792314562899182186819800834272764054341,471525882014932587321 673707929687500000000/523728817391198795728810839649314044495553,63998 4196590745029609794670270443805335134995621797161457881539812583548821 7018355729426105080678991495490125269843837510294323714842801925340841 9/14067383878224159980676935999500324267727723947516309536464061683130 0634376164529513268198245310634999695761463046453112035024854097736978 67038000,594562755257530592552224345703125000000/123802720910327682301 431417435953442122031,-1886691133979705639959153870454656/397947585885 835383951563312487675,-50061468875139778913910254637881/14118664193603 6184819986782313781]]);\n\n c2 := evalf(A[1,1]);\n c3 := evalf(A[2 ,1]);\n c4 := evalf(A[3,1]);\n c5 := evalf(A[4,1]);\n c6 := eval f(A[5,1]);\n c7 := evalf(A[6,1]);\n c8 := evalf(A[7,1]);\n c9 := evalf(A[8,1]);\n cA := evalf(A[9,1]);\n cB := evalf(A[10,1]);\n \+ cC := evalf(A[11,1]);\n a21 := c2;\n a31 := evalf(A[2,2]);\n a3 2 := evalf(A[2,3]);\n a41 := evalf(A[3,2]);\n a42 := evalf(A[3,3]) ;\n a43 := evalf(A[3,4]);\n a51 := evalf(A[4,2]);\n a52 := evalf (A[4,3]);\n a53 := evalf(A[4,4]);\n a54 := evalf(A[4,5]);\n a61 \+ := evalf(A[5,2]);\n a62 := evalf(A[5,3]);\n a63 := evalf(A[5,4]); \n a64 := evalf(A[5,5]);\n a65 := evalf(A[5,6]);\n a71 := evalf( A[6,2]);\n a72 := evalf(A[6,3]);\n a73 := evalf(A[6,4]);\n a74 : = evalf(A[6,5]);\n a75 := evalf(A[6,6]);\n a76 := evalf(A[6,7]);\n a81 := evalf(A[7,2]);\n a82 := evalf(A[7,3]);\n a83 := evalf(A[ 7,4]);\n a84 := evalf(A[7,5]);\n a85 := evalf(A[7,6]);\n a86 := \+ evalf(A[7,7]);\n a87 := evalf(A[7,8]);\n a91 := evalf(A[8,2]);\n \+ a92 := evalf(A[8,3]);\n a93 := evalf(A[8,4]);\n a94 := evalf(A[8, 5]);\n a95 := evalf(A[8,6]);\n a96 := evalf(A[8,7]);\n a97 := ev alf(A[8,8]);\n a98 := evalf(A[8,9]);\n aA1 := evalf(A[9,2]);\n a A2 := evalf(A[9,3]);\n aA3 := evalf(A[9,4]);\n aA4 := evalf(A[9,5] );\n aA5 := evalf(A[9,6]);\n aA6 := evalf(A[9,7]);\n aA7 := eval f(A[9,8]);\n aA8 := evalf(A[9,9]);\n aA9 := evalf(A[9,10]);\n aB 1 := evalf(A[10,2]);\n aB2 := evalf(A[10,3]);\n aB3 := evalf(A[10, 4]);\n aB4 := evalf(A[10,5]);\n aB5 := evalf(A[10,6]);\n aB6 := \+ evalf(A[10,7]);\n aB7 := evalf(A[10,8]);\n aB8 := evalf(A[10,9]); \n aB9 := evalf(A[10,10]);\n aBA := evalf(A[10,11]);\n aC1 := ev alf(A[11,2]);\n aC2 := evalf(A[11,3]);\n aC3 := evalf(A[11,4]);\n \+ aC4 := evalf(A[11,5]);\n aC5 := evalf(A[11,6]);\n aC6 := evalf(A [11,7]);\n aC7 := evalf(A[11,8]);\n aC8 := evalf(A[11,9]);\n aC9 := evalf(A[11,10]);\n aCA := evalf(A[11,11]);\n aCB := evalf(A[11 ,12]);\n b1 := evalf(A[12,2]);\n b2 := evalf(A[12,3]);\n b3 := e valf(A[12,4]);\n b4 := evalf(A[12,5]);\n b5 := evalf(A[12,6]);\n \+ b6 := evalf(A[12,7]);\n b7 := evalf(A[12,8]);\n b8 := evalf(A[12, 9]);\n b9 := evalf(A[12,10]);\n bA := evalf(A[12,11]);\n bB := e valf(A[12,12]);\n bC := evalf(A[12,13]);\n xk := evalf(xx);\n yk := evalf(yy);\n soln := [xk,yk]; \n for k from 1 to stps do\n \+ f1 := fn(xk,yk);\n t := a21*f1;\n f2 := fn(xk + c2*h,yk + \+ t*h);\n t := a31*f1 + a32*f2;\n f3 := fn(xk + c3*h,yk + t*h) ;\n t := a41*f1 + a42*f2 + a43*f3;\n f4 := fn(xk + c4*h,yk + t*h);\n t := a51*f1 + a52*f2 + a53*f3 + a54*f4;\n f5 := fn( xk + c5*h,yk + t*h);\n t := a61*f1 + a62*f2 + a63*f3 + a64*f4 + a 65*f5;\n f6 := fn(xk + c6*h,yk + t*h);\n t := a71*f1 + a72*f 2 + a73*f3 + a74*f4 + a75*f5 + a76*f6;\n f7 := fn(xk + c7*h,yk + \+ t*h);\n t := a81*f1 + a82*f2 + a83*f3 + a84*f4 + a85*f5 + a86*f6 \+ + a87*f7;\n f8 := fn(xk + c8*h,yk + t*h);\n t := a91*f1 + a9 2*f2 + a93*f3 + a94*f4 + a95*f5 + a96*f6 + a97*f7 + a98*f8;\n f9 \+ := fn(xk + c9*h,yk + t*h);\n t := aA1*f1 + aA2*f2 + aA3*f3 + aA4* f4 + aA5*f5 + aA6*f6 + aA7*f7 + aA8*f8 + aA9*f9;\n fA := fn(xk + \+ cA*h,yk + t*h);\n t := aB1*f1 + aB2*f2 + aB3*f3 + aB4*f4 + aB5*f5 + aB6*f6 + aB7*f7 + aB8*f8 + aB9*f9\n \+ + aBA*fA;\n fB : = fn(xk + cB*h,yk + t*h);\n t := aC1*f1 + aC2*f2 + aC3*f3 + aC4*f 4 + aC5*f5 + aC6*f6 + aC7*f7 + aC8*f8 + aC9*f9\n \+ + aCA*fA + aCB*fB;\n fC := fn(xk + cC*h,yk + t*h);\n \n t := b1*f1 + b2*f2 + b 3*f3 + b4*f4 + b5*f5 + b6*f6 + b7*f7 + b8*f8 + b9*f9 + bA*fA\n \+ + bB *fB + bC*fC;\n yk := yk + t*h;\n xk := xk + h:\n soln : = soln,[xk,yk];\n end do;\n if bb=true then\n eqns := \{SOLN_ =[soln],FXY_=fxy,X_=x,Y_=y,c2_=c2,c3_=c3,\n c4_=c4,c5_=c5,c6_= c6,c7_=c7,c8_=c8,c9_=c9,cA_=cA,cB_=cB,cC_=cC,\n a31_=a31,a32_= a32,a41_=a41,a42_=a42,a43_=a43,a51_=a51,a52_=a52,\n a53_=a53,a 54_=a54,a61_=a61,a62_=a62,a63_=a63,a64_=a64,a65_=a65,\n a71_=a 71,a72_=a72,a73_=a73,a74_=a74,a75_=a75,a76_=a76,a81_=a81,\n a8 2_=a82,a83_=a83,a84_=a84,a85_=a85,a86_=a86,a87_=a87,a91_=a91,\n \+ a92_=a92,a93_=a93,a94_=a94,a95_=a95,a96_=a96,a97_=a97,a98_=a98,\n \+ aA1_=aA1,aA2_=aA2,aA3_=aA3,aA4_=aA4,aA5_=aA5,aA6_=aA6,aA7_=aA7, \n aA8_=aA8,aA9_=aA9,aB1_=aB1,aB2_=aB2,aB3_=aB3,aB4_=aB4,aB5_= aB5,\n aB6_=aB6,aB7_=aB7,aB8_=aB8,aB9_=aB9,aBA_=aBA,aC1_=aC1,a C2_=aC2,\n aC3_=aC3,aC4_=aC4,aC5_=aC5,aC6_=aC6,aC7_=aC7,aC8_=a C8,aC9_=aC9,\n aCA_=aCA,aCB_=aCB,b1_=b1,b2_=b2,b3_=b3,b4_=b4,b 5_=b5,b6_=b6,\n b7_=b7,b8_=b8,b9_=b9,bA_=bA,bB_=bB,bC_=bC\};\n return subs(eqns,eval(rk8step)); \n else\n return evalf[s aveDigits]([soln]);\n end if;\nend proc:" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 36 "RK8_2 Verner's Maple dverk78 scheme " }} {PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 9332 "RK8_2 := proc(fxy,x,y,xx,yy,h,stps,bb)\n local c2,c3,c4,c5,c6, c7,c8,c9,cA,cB,cC,a21,a31,a32,a41,a42,a43,a51,a52,\n a53,a54,a61,a62 ,a63,a64,a65,a71,a72,a73,a74,a75,a76,a81,a82,a83,a84,a85,\n a86,a87, a91,a92,a93,a94,a95,a96,a97,a98,aA1,aA2,aA3,aA4,aA5,aA6,aA7,aA8,\n a A9,aB1,aB2,aB3,aB4,aB5,aB6,aB7,aB8,aB9,aBA,aC1,aC2,aC3,aC4,aC5,aC6,aC7 ,\n aC8,aC9,aCA,aCB,f1,f2,f3,f4,f5,f6,f7,f8,f9,fA,fB,fC,b1,b2,b3,b4, b5,b6,b7,\n b8,b9,bA,bB,bC,t,k,fn,xk,yk,soln,eqns,A,saveDigits;\n\n \+ saveDigits := Digits;\n Digits := max(trunc(evalhf(Digits)),Digits +5);\n\n fn := unapply(fxy,x,y);\n\n A := matrix([[1/16,1/16,0,0,0 ,0,0,0,0,0,0,0,0],\n [112/1065,18928/1134225,100352/1134225,0,0,0,0, 0,0,0,0,0,0],\n [56/355,14/355,0,42/355,0,0,0,0,0,0,0,0,0],\n [39/ 100,94495479/250880000,0,-352806597/250880000,178077159/125440000,0,0, 0,0,0,0,0,0],\n [7/15,12089/252720,0,0,2505377/10685520,960400/52091 91,0,0,0,0,0,0,0],\n [39/250,21400899/350000000,0,0,3064329829899/27 126050000000,-21643947/592609375,\n \+ 124391943/6756250000,0,0,0,0,0,0],\n [24/25,-15365458 811/13609565775,0,0,-7/5,-8339128164608/939060038475,\n \+ 341936800488/47951126225,1993321838240/380523459069,0,0,0,0,0], \n [14435868/16178861,-184091125228237658443815733646470842695472806 1551/\n 29919236151711519215962538134831182 62195533733898,0,0,\n -14764960804048657303638372252908780219281424 435/\n 29816921025650219756117112 69209606363661854518,\n -875325048502130441118613421785266742862694 404520560000/\n 170212030428894418395571 677575961339495435011888324169,\n 763205196415429092566184979837064 5637589377834346780/\n 173408725741881 1583049800347581865260479233950396659,\n 75198347919711375170485321 79652347729899303513750000/\n 10456773 03502317596597890707812349832637339039997351, 1366042683489166 351293315549358278750/144631418224267718165055326464180836641,0,0,0,0] ,\n [11/12,-63077736705254280154824845013881/78369357853786633855112 190394368,0,0,\n -31948346510820970247215/6956009216960026632192, \n -3378604805394255292453489375/517042670569824692230499952,\n \+ 1001587844183325981198091450220795/1842326842077225037016699538728 96,\n 187023075231349900768014890274453125/2522469884980817801075 2575653374848,\n 1908158550070998850625/117087067039189929394176, \n -52956818288156668227044990077324877908565/\n \+ 2912779959477433986349822224412353951940608,0,0,0],\n [19/20,-10 116106591826909534781157993685116703/956281994503689403044223141187174 4000,\n 0,0,-9623541317323077848129/3864449564977 792573440,\n -4823348333146829406881375/57641323363 4141239944816,\n 6566119246514996884067001154977284529/9703054870 21846325473990863582315520,\n 22264551305192135492560168925067305 59375/364880443159675255577435648380047355776,\n 3974726278238046 6933662225/1756032802431424164410720256,\n 4817577141926095533524 4683805171548038966866545122229/\n 19897864205138151 46528880165952064118903852843612160000,\n \+ -2378292068163246/47768728487211875,0,0],\n [1,-321802217 4758599831659045535578571/1453396753634469525663775847094384,0,0,\n \+ 26290092604284231996745/5760876126062860430544,\n -69706929756 0926452045586710000/41107967755245430594036502319,\n 182735782043 4213461438077550902273440/139381013914245317709567680839641697,\n \+ 643504802814241550941949227194107500000/24212460911883655086049400754 5333945331,\n 162259938151380266113750/59091082835244183497007,\n -23028251632873523818545414856857015616678575554130463402/\n \+ 200131691831914445034439052404056033499784245041516290 55,\n 7958341351371843889152/3284467988443203581305,\n \+ -507974327957860843878400/1215556548191790427189 67,0],\n [0,4631674879841/103782082379976,0,0,0,0,14327219974204125/ 40489566827933216,\n 2720762324010009765625000/109173674806968139 22225349,-498533005859375/95352091037424,\n 405932030463777247926 705030596175437402459637909765779/\n 7880391943632184108320188604 1201537229769115088303952,\n -10290327637248/1082076946951,863264 105888000/85814662253313,\n -29746300739/24 7142463456]]);\n\n c2 := evalf(A[1,1]);\n c3 := evalf(A[2,1]);\n \+ c4 := evalf(A[3,1]);\n c5 := evalf(A[4,1]);\n c6 := evalf(A[5,1]) ;\n c7 := evalf(A[6,1]);\n c8 := evalf(A[7,1]);\n c9 := evalf(A[ 8,1]);\n cA := evalf(A[9,1]);\n cB := evalf(A[10,1]);\n cC := ev alf(A[11,1]);\n a21 := c2;\n a31 := evalf(A[2,2]);\n a32 := eval f(A[2,3]);\n a41 := evalf(A[3,2]);\n a42 := evalf(A[3,3]);\n a43 := evalf(A[3,4]);\n a51 := evalf(A[4,2]);\n a52 := evalf(A[4,3]); \n a53 := evalf(A[4,4]);\n a54 := evalf(A[4,5]);\n a61 := evalf( A[5,2]);\n a62 := evalf(A[5,3]);\n a63 := evalf(A[5,4]);\n a64 : = evalf(A[5,5]);\n a65 := evalf(A[5,6]);\n a71 := evalf(A[6,2]);\n a72 := evalf(A[6,3]);\n a73 := evalf(A[6,4]);\n a74 := evalf(A[ 6,5]);\n a75 := evalf(A[6,6]);\n a76 := evalf(A[6,7]);\n a81 := \+ evalf(A[7,2]);\n a82 := evalf(A[7,3]);\n a83 := evalf(A[7,4]);\n \+ a84 := evalf(A[7,5]);\n a85 := evalf(A[7,6]);\n a86 := evalf(A[7, 7]);\n a87 := evalf(A[7,8]);\n a91 := evalf(A[8,2]);\n a92 := ev alf(A[8,3]);\n a93 := evalf(A[8,4]);\n a94 := evalf(A[8,5]);\n a 95 := evalf(A[8,6]);\n a96 := evalf(A[8,7]);\n a97 := evalf(A[8,8] );\n a98 := evalf(A[8,9]);\n aA1 := evalf(A[9,2]);\n aA2 := eval f(A[9,3]);\n aA3 := evalf(A[9,4]);\n aA4 := evalf(A[9,5]);\n aA5 := evalf(A[9,6]);\n aA6 := evalf(A[9,7]);\n aA7 := evalf(A[9,8]); \n aA8 := evalf(A[9,9]);\n aA9 := evalf(A[9,10]);\n aB1 := evalf (A[10,2]);\n aB2 := evalf(A[10,3]);\n aB3 := evalf(A[10,4]);\n a B4 := evalf(A[10,5]);\n aB5 := evalf(A[10,6]);\n aB6 := evalf(A[10 ,7]);\n aB7 := evalf(A[10,8]);\n aB8 := evalf(A[10,9]);\n aB9 := evalf(A[10,10]);\n aBA := evalf(A[10,11]);\n aC1 := evalf(A[11,2] );\n aC2 := evalf(A[11,3]);\n aC3 := evalf(A[11,4]);\n aC4 := ev alf(A[11,5]);\n aC5 := evalf(A[11,6]);\n aC6 := evalf(A[11,7]);\n \+ aC7 := evalf(A[11,8]);\n aC8 := evalf(A[11,9]);\n aC9 := evalf(A [11,10]);\n aCA := evalf(A[11,11]);\n aCB := evalf(A[11,12]);\n \+ b1 := evalf(A[12,2]);\n b2 := evalf(A[12,3]);\n b3 := evalf(A[12,4 ]);\n b4 := evalf(A[12,5]);\n b5 := evalf(A[12,6]);\n b6 := eval f(A[12,7]);\n b7 := evalf(A[12,8]);\n b8 := evalf(A[12,9]);\n b9 := evalf(A[12,10]);\n bA := evalf(A[12,11]);\n bB := evalf(A[12,1 2]);\n bC := evalf(A[12,13]);\n xk := evalf(xx);\n yk := evalf(y y);\n soln := [xk,yk]; \n for k from 1 to stps do\n f1 := fn( xk,yk);\n t := a21*f1;\n f2 := fn(xk + c2*h,yk + t*h);\n \+ t := a31*f1 + a32*f2;\n f3 := fn(xk + c3*h,yk + t*h);\n t \+ := a41*f1 + a42*f2 + a43*f3;\n f4 := fn(xk + c4*h,yk + t*h);\n \+ t := a51*f1 + a52*f2 + a53*f3 + a54*f4;\n f5 := fn(xk + c5*h,y k + t*h);\n t := a61*f1 + a62*f2 + a63*f3 + a64*f4 + a65*f5;\n \+ f6 := fn(xk + c6*h,yk + t*h);\n t := a71*f1 + a72*f2 + a73*f3 \+ + a74*f4 + a75*f5 + a76*f6;\n f7 := fn(xk + c7*h,yk + t*h);\n \+ t := a81*f1 + a82*f2 + a83*f3 + a84*f4 + a85*f5 + a86*f6 + a87*f7;\n f8 := fn(xk + c8*h,yk + t*h);\n t := a91*f1 + a92*f2 + a93* f3 + a94*f4 + a95*f5 + a96*f6 + a97*f7 + a98*f8;\n f9 := fn(xk + \+ c9*h,yk + t*h);\n t := aA1*f1 + aA2*f2 + aA3*f3 + aA4*f4 + aA5*f5 + aA6*f6 + aA7*f7 + aA8*f8 + aA9*f9;\n fA := fn(xk + cA*h,yk + t *h);\n t := aB1*f1 + aB2*f2 + aB3*f3 + aB4*f4 + aB5*f5 + aB6*f6 + aB7*f7 + aB8*f8 + aB9*f9\n \+ + aBA*fA;\n fB := fn(xk + c B*h,yk + t*h);\n t := aC1*f1 + aC2*f2 + aC3*f3 + aC4*f4 + aC5*f5 \+ + aC6*f6 + aC7*f7 + aC8*f8 + aC9*f9\n \+ + aCA*fA + aCB*fB;\n fC := fn(xk + cC*h,yk + t*h);\n \n t := b1*f1 + b2*f2 + b3*f3 + b4*f 4 + b5*f5 + b6*f6 + b7*f7 + b8*f8 + b9*f9 + bA*fA\n \+ + bB*fB + bC*fC ;\n yk := yk + t*h;\n xk := xk + h:\n soln := soln,[xk, yk];\n end do;\n if bb=true then\n eqns := \{SOLN_=[soln],FXY _=fxy,X_=x,Y_=y,c2_=c2,c3_=c3,\n c4_=c4,c5_=c5,c6_=c6,c7_=c7,c 8_=c8,c9_=c9,cA_=cA,cB_=cB,cC_=cC,\n a31_=a31,a32_=a32,a41_=a4 1,a42_=a42,a43_=a43,a51_=a51,a52_=a52,\n a53_=a53,a54_=a54,a61 _=a61,a62_=a62,a63_=a63,a64_=a64,a65_=a65,\n a71_=a71,a72_=a72 ,a73_=a73,a74_=a74,a75_=a75,a76_=a76,a81_=a81,\n a82_=a82,a83_ =a83,a84_=a84,a85_=a85,a86_=a86,a87_=a87,a91_=a91,\n a92_=a92, a93_=a93,a94_=a94,a95_=a95,a96_=a96,a97_=a97,a98_=a98,\n aA1_= aA1,aA2_=aA2,aA3_=aA3,aA4_=aA4,aA5_=aA5,aA6_=aA6,aA7_=aA7,\n a A8_=aA8,aA9_=aA9,aB1_=aB1,aB2_=aB2,aB3_=aB3,aB4_=aB4,aB5_=aB5,\n \+ aB6_=aB6,aB7_=aB7,aB8_=aB8,aB9_=aB9,aBA_=aBA,aC1_=aC1,aC2_=aC2,\n \+ aC3_=aC3,aC4_=aC4,aC5_=aC5,aC6_=aC6,aC7_=aC7,aC8_=aC8,aC9_=aC9, \n aCA_=aCA,aCB_=aCB,b1_=b1,b2_=b2,b3_=b3,b4_=b4,b5_=b5,b6_=b6 ,\n b7_=b7,b8_=b8,b9_=b9,bA_=bA,bB_=bB,bC_=bC\};\n return subs(eqns,eval(rk8step)); \n else\n return evalf[saveDigits]([ soln]);\n end if;\nend proc:" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}}{SECT 1 {PARA 4 "" 0 " " {TEXT -1 52 "RK8_3 modification of Verner's Maple dverk78 scheme " } }{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 9049 "RK8_3 := proc(fxy,x,y,xx,yy,h,stps,bb)\n local c2,c3,c4,c5,c 6,c7,c8,c9,cA,cB,cC,a21,a31,a32,a41,a42,a43,a51,a52,\n a53,a54,a61,a 62,a63,a64,a65,a71,a72,a73,a74,a75,a76,a81,a82,a83,a84,a85,\n a86,a8 7,a91,a92,a93,a94,a95,a96,a97,a98,aA1,aA2,aA3,aA4,aA5,aA6,aA7,aA8,\n \+ aA9,aB1,aB2,aB3,aB4,aB5,aB6,aB7,aB8,aB9,aBA,aC1,aC2,aC3,aC4,aC5,aC6,a C7,\n aC8,aC9,aCA,aCB,f1,f2,f3,f4,f5,f6,f7,f8,f9,fA,fB,fC,b1,b2,b3,b 4,b5,b6,b7,\n b8,b9,bA,bB,bC,t,k,fn,xk,yk,soln,eqns,A,saveDigits;\n \n saveDigits := Digits;\n Digits := max(trunc(evalhf(Digits)),Dig its+5);\n\n fn := unapply(fxy,x,y);\n\n A := matrix([[3/50,3/50,0, 0,0,0,0,0,0,0,0,0,0],\n[469/4510,169309/12204060,219961/2440812,0,0,0, 0,0,0,0,0,0,0],\n[1407/9020,1407/36080,0,4221/36080,0,0,0,0,0,0,0,0,0] ,\n[100/257,12985296430100/33603736057857,0,-16218027650000/1120124535 2619,48744170200000/33603736057857,0,0,0,0,0,0,0,0],\n[7/15,3817807/81 405000,0,0,10274191312/43991343405,40755997793/218862405000,0,0,0,0,0, 0,0],\n[12/77,15900802578/262289621825,0,0,6340315606886400/5627491910 1668113,-451131008276934/12909147305279225,984103149600/56231606512081 ,0,0,0,0,0,0],\n[26/27,-6560178806172785891/5672858706535680000,0,0,-1 7/2,-5870605887517986369316613/652465735085103413760000,16204960050140 425727/2255297505252764544,9658417761218465607893/77692097259366635520 0,0,0,0,0,0],\n[7101051/7944748,-1404618998069651828319863526166366482 83813334213167183799/2315957023533969410187885357274159988789636581477 58080000,0,0,-230388436455789139579016226863459716720858844533129533/2 8162599104012402938735024801544932535316065892462592,-9790630998374957 452129847257592953551765164772149796065413844391/187912086354556872209 1311485392002480259497346453409790689280000,19704323580398550825484436 3902235259599001596076904803130443/44561965459500736173318259755870730 512740108168829012803584,294620817758575088586344956781032015012776829 359971133/28166985596967449488723424839046646360350769597644800,126873 616630173037746837292557884226969/132172985196530229490710503209405672 20224,0,0,0,0],\n[11/12,-41617847659972885613040591872450035471/528130 05326633659912790449394810880000,0,0,-269951087764496912154173876831/2 9845195411834947899362922496,-2749986656729113102825386117572168116123 /422416144552169230122612885222850560000,32452570688093878796909074430 1341259977691/60253355178721829324099288547678254006272,41487408138830 374659637178881584696821/3493656315517022583105833038931558400,9325984 33870374903885/57293130256189894463488,-334574220537807932173262468074 7620/191228884555191941725363563271035831,0,0,0],\n[19/20,-15203213471 5446654236835762683933540260220443/14403122993350118793339292985201295 3600000000,0,0,-19926282430450459437356479596949/220417558707334822448 9986560000,-2861670807732347804933810531212726938261187/34316655482247 4341490702928140697600000000,13340633265934680563503273162687143196017 15109229/198658539882559017093840573946438829079429120000,255870600859 86906515751471801297011229903719273/2019042940187611700532950824131079 569408000000,1292083162729235236212706557/4770164639633667552428800000 0,2091032384514247750201495665581688757138358851/607298691764125649384 23797095435796334060237500,-377694382258403/5816420236687500,0,0],\n[1 ,-240860055167915269684777564157507287321/1170038587498513755740834246 73098438400,0,0,-98894702310834163626964955900/10545476035409921643978 852639,-252251601740617744737074278562623715189/1641817784961322968523 1088852333734400,840813041585474673597744565360698902394175/7078087183 5683840841986501581774346003376,24189514716339363218944405555484129317 1617/15082171924595104472807447927564153421312,42847753308025275666/29 882754550596969299,-7717336222264885240564054011744574965898992045440/ 8479033418894797325547814554207128260520626293873,16262726151696512/89 40486723127047,-16001553345352000000/6505707112968117383,0],\n[0,61903 079921/1389136000824,0,0,0,0,402653566621875/1136204715874336,18720207 31216849170881/7515476193013636507650,-48961117784712267/1517741399943 8600,1824678166477710343022780887139376956665357164944/332726480751222 257863787417240798843822259974895,-126172885488/13167945725,4125103980 0000/5264998725539,-107501739/674957600]]);\n\n c2 := evalf(A[1,1]); \n c3 := evalf(A[2,1]);\n c4 := evalf(A[3,1]);\n c5 := evalf(A[4 ,1]);\n c6 := evalf(A[5,1]);\n c7 := evalf(A[6,1]);\n c8 := eval f(A[7,1]);\n c9 := evalf(A[8,1]);\n cA := evalf(A[9,1]);\n cB := evalf(A[10,1]);\n cC := evalf(A[11,1]);\n a21 := c2;\n a31 := e valf(A[2,2]);\n a32 := evalf(A[2,3]);\n a41 := evalf(A[3,2]);\n \+ a42 := evalf(A[3,3]);\n a43 := evalf(A[3,4]);\n a51 := evalf(A[4,2 ]);\n a52 := evalf(A[4,3]);\n a53 := evalf(A[4,4]);\n a54 := eva lf(A[4,5]);\n a61 := evalf(A[5,2]);\n a62 := evalf(A[5,3]);\n a6 3 := evalf(A[5,4]);\n a64 := evalf(A[5,5]);\n a65 := evalf(A[5,6]) ;\n a71 := evalf(A[6,2]);\n a72 := evalf(A[6,3]);\n a73 := evalf (A[6,4]);\n a74 := evalf(A[6,5]);\n a75 := evalf(A[6,6]);\n a76 \+ := evalf(A[6,7]);\n a81 := evalf(A[7,2]);\n a82 := evalf(A[7,3]); \n a83 := evalf(A[7,4]);\n a84 := evalf(A[7,5]);\n a85 := evalf( A[7,6]);\n a86 := evalf(A[7,7]);\n a87 := evalf(A[7,8]);\n a91 : = evalf(A[8,2]);\n a92 := evalf(A[8,3]);\n a93 := evalf(A[8,4]);\n a94 := evalf(A[8,5]);\n a95 := evalf(A[8,6]);\n a96 := evalf(A[ 8,7]);\n a97 := evalf(A[8,8]);\n a98 := evalf(A[8,9]);\n aA1 := \+ evalf(A[9,2]);\n aA2 := evalf(A[9,3]);\n aA3 := evalf(A[9,4]);\n \+ aA4 := evalf(A[9,5]);\n aA5 := evalf(A[9,6]);\n aA6 := evalf(A[9, 7]);\n aA7 := evalf(A[9,8]);\n aA8 := evalf(A[9,9]);\n aA9 := ev alf(A[9,10]);\n aB1 := evalf(A[10,2]);\n aB2 := evalf(A[10,3]);\n \+ aB3 := evalf(A[10,4]);\n aB4 := evalf(A[10,5]);\n aB5 := evalf(A [10,6]);\n aB6 := evalf(A[10,7]);\n aB7 := evalf(A[10,8]);\n aB8 := evalf(A[10,9]);\n aB9 := evalf(A[10,10]);\n aBA := evalf(A[10, 11]);\n aC1 := evalf(A[11,2]);\n aC2 := evalf(A[11,3]);\n aC3 := evalf(A[11,4]);\n aC4 := evalf(A[11,5]);\n aC5 := evalf(A[11,6]); \n aC6 := evalf(A[11,7]);\n aC7 := evalf(A[11,8]);\n aC8 := eval f(A[11,9]);\n aC9 := evalf(A[11,10]);\n aCA := evalf(A[11,11]);\n \+ aCB := evalf(A[11,12]);\n b1 := evalf(A[12,2]);\n b2 := evalf(A[ 12,3]);\n b3 := evalf(A[12,4]);\n b4 := evalf(A[12,5]);\n b5 := \+ evalf(A[12,6]);\n b6 := evalf(A[12,7]);\n b7 := evalf(A[12,8]);\n \+ b8 := evalf(A[12,9]);\n b9 := evalf(A[12,10]);\n bA := evalf(A[1 2,11]);\n bB := evalf(A[12,12]);\n bC := evalf(A[12,13]);\n xk : = evalf(xx);\n yk := evalf(yy);\n soln := [xk,yk]; \n for k from 1 to stps do\n f1 := fn(xk,yk);\n t := a21*f1;\n f2 := fn(xk + c2*h,yk + t*h);\n t := a31*f1 + a32*f2;\n f3 := fn( xk + c3*h,yk + t*h);\n t := a41*f1 + a42*f2 + a43*f3;\n f4 : = fn(xk + c4*h,yk + t*h);\n t := a51*f1 + a52*f2 + a53*f3 + a54*f 4;\n f5 := fn(xk + c5*h,yk + t*h);\n t := a61*f1 + a62*f2 + \+ a63*f3 + a64*f4 + a65*f5;\n f6 := fn(xk + c6*h,yk + t*h);\n \+ t := a71*f1 + a72*f2 + a73*f3 + a74*f4 + a75*f5 + a76*f6;\n f7 := fn(xk + c7*h,yk + t*h);\n t := a81*f1 + a82*f2 + a83*f3 + a84*f4 + a85*f5 + a86*f6 + a87*f7;\n f8 := fn(xk + c8*h,yk + t*h);\n \+ t := a91*f1 + a92*f2 + a93*f3 + a94*f4 + a95*f5 + a96*f6 + a97*f7 + a98*f8;\n f9 := fn(xk + c9*h,yk + t*h);\n t := aA1*f1 + aA2 *f2 + aA3*f3 + aA4*f4 + aA5*f5 + aA6*f6 + aA7*f7 + aA8*f8 + aA9*f9;\n \+ fA := fn(xk + cA*h,yk + t*h);\n t := aB1*f1 + aB2*f2 + aB3*f 3 + aB4*f4 + aB5*f5 + aB6*f6 + aB7*f7 + aB8*f8 + aB9*f9\n \+ + \+ aBA*fA;\n fB := fn(xk + cB*h,yk + t*h);\n t := aC1*f1 + aC2* f2 + aC3*f3 + aC4*f4 + aC5*f5 + aC6*f6 + aC7*f7 + aC8*f8 + aC9*f9\n \+ + aCA*fA + aCB*fB;\n fC := fn(xk + cC*h,yk + t*h);\n \n t : = b1*f1 + b2*f2 + b3*f3 + b4*f4 + b5*f5 + b6*f6 + b7*f7 + b8*f8 + b9*f 9 + bA*fA\n \+ + bB*fB + bC*fC;\n yk := yk + t*h;\n xk := xk + h:\n soln := soln,[xk,yk];\n end do;\n if bb=true then\n \+ eqns := \{SOLN_=[soln],FXY_=fxy,X_=x,Y_=y,c2_=c2,c3_=c3,\n \+ c4_=c4,c5_=c5,c6_=c6,c7_=c7,c8_=c8,c9_=c9,cA_=cA,cB_=cB,cC_=cC,\n \+ a31_=a31,a32_=a32,a41_=a41,a42_=a42,a43_=a43,a51_=a51,a52_=a52,\n a53_=a53,a54_=a54,a61_=a61,a62_=a62,a63_=a63,a64_=a64,a65_=a6 5,\n a71_=a71,a72_=a72,a73_=a73,a74_=a74,a75_=a75,a76_=a76,a81 _=a81,\n a82_=a82,a83_=a83,a84_=a84,a85_=a85,a86_=a86,a87_=a87 ,a91_=a91,\n a92_=a92,a93_=a93,a94_=a94,a95_=a95,a96_=a96,a97_ =a97,a98_=a98,\n aA1_=aA1,aA2_=aA2,aA3_=aA3,aA4_=aA4,aA5_=aA5, aA6_=aA6,aA7_=aA7,\n aA8_=aA8,aA9_=aA9,aB1_=aB1,aB2_=aB2,aB3_= aB3,aB4_=aB4,aB5_=aB5,\n aB6_=aB6,aB7_=aB7,aB8_=aB8,aB9_=aB9,a BA_=aBA,aC1_=aC1,aC2_=aC2,\n aC3_=aC3,aC4_=aC4,aC5_=aC5,aC6_=a C6,aC7_=aC7,aC8_=aC8,aC9_=aC9,\n aCA_=aCA,aCB_=aCB,b1_=b1,b2_= b2,b3_=b3,b4_=b4,b5_=b5,b6_=b6,\n b7_=b7,b8_=b8,b9_=b9,bA_=bA, bB_=bB,bC_=bC\};\n return subs(eqns,eval(rk8step)); \n else\n \+ return evalf[saveDigits]([soln]);\n end if;\nend proc:" }}} {PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 28 "RK8_4 an \"efficient\" \+ scheme " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 9529 "RK8_4 := proc(fxy,x,y,xx,yy,h,stps,bb)\n local c2 ,c3,c4,c5,c6,c7,c8,c9,cA,cB,cC,a21,a31,a32,a41,a42,a43,a51,a52,\n a5 3,a54,a61,a62,a63,a64,a65,a71,a72,a73,a74,a75,a76,a81,a82,a83,a84,a85, \n a86,a87,a91,a92,a93,a94,a95,a96,a97,a98,aA1,aA2,aA3,aA4,aA5,aA6,a A7,aA8,\n aA9,aB1,aB2,aB3,aB4,aB5,aB6,aB7,aB8,aB9,aBA,aC1,aC2,aC3,aC 4,aC5,aC6,aC7,\n aC8,aC9,aCA,aCB,f1,f2,f3,f4,f5,f6,f7,f8,f9,fA,fB,fC ,b1,b2,b3,b4,b5,b6,b7,\n b8,b9,bA,bB,bC,t,k,fn,xk,yk,soln,eqns,A,sav eDigits;\n\n saveDigits := Digits;\n Digits := max(trunc(evalhf(Di gits)),Digits+5);\n\n fn := unapply(fxy,x,y);\n\n A := matrix([[27 /400,27/400,0,0,0,0,0,0,0,0,0,0,0],\n[469/4510,1311793/54918270,439922 /5491827,0,0,0,0,0,0,0,0,0,0],\n[1407/9020,1407/36080,0,4221/36080,0,0 ,0,0,0,0,0,0,0],\n[100/257,12985296430100/33603736057857,0,-1621802765 0000/11201245352619,48744170200000/33603736057857,0,0,0,0,0,0,0,0],\n[ 7/15,3817807/81405000,0,0,10274191312/43991343405,40755997793/21886240 5000,0,0,0,0,0,0,0],\n[7/45,8883007/146529000,0,0,53436069013712/47550 2430864645,-4116355777093/117791746371000,7635614/436305285,0,0,0,0,0, 0],\n[58/59,-25750155116216595121447/19172397107027415328000,0,0,-743/ 55,-3193438311684528658341050903/315992014923229256968928000,183334983 592060082563257/22930186940004788732288,18586487074230223824062061/103 5692891721620976018560,0,0,0,0,0],\n[49117478/55144743,-92212217189471 6552217807973277934204672348766351979550680715901/15740119019867164084 55269129129499876539846630029827642536608000,0,0,-24939241129656091104 22434487743812058912776576445331741068763/2485761880419989993633400397 48303974885433202092103843801085,-347410454252950206893609060802736441 487435576445275533884950354547756949/697473627660918834815922125361784 38630509515521207481681627188467808000,5327733707939584478612584810530 320112314665632405586015822099459/124980895520514732019873673632122559 6828263980093048262260372352,63805647563182428528344562389882620890969 933364805025903916023/522253688012544809193253721243016911311851329489 6584316063360,16756974250619222216361657668160522447776912/16497659305 08280715886033484777254665540228007,0,0,0,0],\n[13/14,-370292782453410 9782314804940532859074364487/41312577034599963023166962510551903815680 00,0,0,-48542494919285774589408977236397/37951565460737350652774282939 20,-22919425542674277431989520078944496953957193/319462007754808947240 8777624402038658048000,2484540315797359406715529397919751486619672787/ 421648690388802974367036270632096098190348288,170546976143576425414211 05633253210641175183343/1072180740487803620784937862662845301329326080 ,242464961680690545467502075/12396488665185571302473970944,-2767206589 30244250241932341390181638290085115/9663793615504861617193707372304668 805996412672,0,0,0],\n[62/65,-8882652614922380707082937302793930626864 630973971/8018042974340129539773365358072400795335265440000,0,0,-91300 774029817320031962039591895693/6641776260885597741930170995083525,-112 201255053474401906816716367850973725245563029/130451942058366052280926 70613416996666049440000,1833348090405008146620363812428378076185763305 92954213/26459846979744846747689472891117696020043251727064960,5908534 443979027768339795262286892453548930774924082527/337801676952094167280 622617772889760985970195055625600,6612449503769637840690016046967566/2 25556868777060284586258384373942375,-102937842782365224973088698635137 89970987031746931000834/8787826363395572889246773386374957640244197745 505586187675,-4215629637956404624565504/108123612591292234376500125,0, 0],\n[1,-535308736842973754888311604833674190949559277/507665470526061 438136955303900248854246636000,0,0,-6581385500218652420284298344406312 /842464028888378018027275808653215,-2298109906288490803293215283725071 86906863097/29548105906232803976194189596626971177604000,3336217436286 04464651535937000728865436063541833/5404243143027797208019987466411866 2613066091504,281436223028901228360767240757054593496984174199893/2478 4655195828222920388514383970641394482546714960,-1958160872416610232763 42391/399896648012674685277875106,144645270082238084524594544683988399 575644348254551267965/314856466828710807635889483181359023001187603321 870639973,-113365555877618519272588672/93091504282232088010468779,9008 463031864887557094375/6606862347800680584331678,0],\n[0,10499825785831 03/23627320335287760,0,0,0,0,768239620641637875/2161676880469378496,78 84214609987698755625/31681975342943333980208,-19726902451409487557291/ 9259475236790114044800,22985448977257586298821077777442163519710451498 50043139161/732352661995952143676542507482891823353467854709785979600, -2307497857833742448/278745570101106225,148963664643228251375/20207195 606693950848,576091109999/2308683585600]]);\n\n c2 := evalf(A[1,1]); \n c3 := evalf(A[2,1]);\n c4 := evalf(A[3,1]);\n c5 := evalf(A[4 ,1]);\n c6 := evalf(A[5,1]);\n c7 := evalf(A[6,1]);\n c8 := eval f(A[7,1]);\n c9 := evalf(A[8,1]);\n cA := evalf(A[9,1]);\n cB := evalf(A[10,1]);\n cC := evalf(A[11,1]);\n a21 := c2;\n a31 := e valf(A[2,2]);\n a32 := evalf(A[2,3]);\n a41 := evalf(A[3,2]);\n \+ a42 := evalf(A[3,3]);\n a43 := evalf(A[3,4]);\n a51 := evalf(A[4,2 ]);\n a52 := evalf(A[4,3]);\n a53 := evalf(A[4,4]);\n a54 := eva lf(A[4,5]);\n a61 := evalf(A[5,2]);\n a62 := evalf(A[5,3]);\n a6 3 := evalf(A[5,4]);\n a64 := evalf(A[5,5]);\n a65 := evalf(A[5,6]) ;\n a71 := evalf(A[6,2]);\n a72 := evalf(A[6,3]);\n a73 := evalf (A[6,4]);\n a74 := evalf(A[6,5]);\n a75 := evalf(A[6,6]);\n a76 \+ := evalf(A[6,7]);\n a81 := evalf(A[7,2]);\n a82 := evalf(A[7,3]); \n a83 := evalf(A[7,4]);\n a84 := evalf(A[7,5]);\n a85 := evalf( A[7,6]);\n a86 := evalf(A[7,7]);\n a87 := evalf(A[7,8]);\n a91 : = evalf(A[8,2]);\n a92 := evalf(A[8,3]);\n a93 := evalf(A[8,4]);\n a94 := evalf(A[8,5]);\n a95 := evalf(A[8,6]);\n a96 := evalf(A[ 8,7]);\n a97 := evalf(A[8,8]);\n a98 := evalf(A[8,9]);\n aA1 := \+ evalf(A[9,2]);\n aA2 := evalf(A[9,3]);\n aA3 := evalf(A[9,4]);\n \+ aA4 := evalf(A[9,5]);\n aA5 := evalf(A[9,6]);\n aA6 := evalf(A[9, 7]);\n aA7 := evalf(A[9,8]);\n aA8 := evalf(A[9,9]);\n aA9 := ev alf(A[9,10]);\n aB1 := evalf(A[10,2]);\n aB2 := evalf(A[10,3]);\n \+ aB3 := evalf(A[10,4]);\n aB4 := evalf(A[10,5]);\n aB5 := evalf(A [10,6]);\n aB6 := evalf(A[10,7]);\n aB7 := evalf(A[10,8]);\n aB8 := evalf(A[10,9]);\n aB9 := evalf(A[10,10]);\n aBA := evalf(A[10, 11]);\n aC1 := evalf(A[11,2]);\n aC2 := evalf(A[11,3]);\n aC3 := evalf(A[11,4]);\n aC4 := evalf(A[11,5]);\n aC5 := evalf(A[11,6]); \n aC6 := evalf(A[11,7]);\n aC7 := evalf(A[11,8]);\n aC8 := eval f(A[11,9]);\n aC9 := evalf(A[11,10]);\n aCA := evalf(A[11,11]);\n \+ aCB := evalf(A[11,12]);\n b1 := evalf(A[12,2]);\n b2 := evalf(A[ 12,3]);\n b3 := evalf(A[12,4]);\n b4 := evalf(A[12,5]);\n b5 := \+ evalf(A[12,6]);\n b6 := evalf(A[12,7]);\n b7 := evalf(A[12,8]);\n \+ b8 := evalf(A[12,9]);\n b9 := evalf(A[12,10]);\n bA := evalf(A[1 2,11]);\n bB := evalf(A[12,12]);\n bC := evalf(A[12,13]);\n xk : = evalf(xx);\n yk := evalf(yy);\n soln := [xk,yk]; \n for k from 1 to stps do\n f1 := fn(xk,yk);\n t := a21*f1;\n f2 := fn(xk + c2*h,yk + t*h);\n t := a31*f1 + a32*f2;\n f3 := fn( xk + c3*h,yk + t*h);\n t := a41*f1 + a42*f2 + a43*f3;\n f4 : = fn(xk + c4*h,yk + t*h);\n t := a51*f1 + a52*f2 + a53*f3 + a54*f 4;\n f5 := fn(xk + c5*h,yk + t*h);\n t := a61*f1 + a62*f2 + \+ a63*f3 + a64*f4 + a65*f5;\n f6 := fn(xk + c6*h,yk + t*h);\n \+ t := a71*f1 + a72*f2 + a73*f3 + a74*f4 + a75*f5 + a76*f6;\n f7 := fn(xk + c7*h,yk + t*h);\n t := a81*f1 + a82*f2 + a83*f3 + a84*f4 + a85*f5 + a86*f6 + a87*f7;\n f8 := fn(xk + c8*h,yk + t*h);\n \+ t := a91*f1 + a92*f2 + a93*f3 + a94*f4 + a95*f5 + a96*f6 + a97*f7 + a98*f8;\n f9 := fn(xk + c9*h,yk + t*h);\n t := aA1*f1 + aA2 *f2 + aA3*f3 + aA4*f4 + aA5*f5 + aA6*f6 + aA7*f7 + aA8*f8 + aA9*f9;\n \+ fA := fn(xk + cA*h,yk + t*h);\n t := aB1*f1 + aB2*f2 + aB3*f 3 + aB4*f4 + aB5*f5 + aB6*f6 + aB7*f7 + aB8*f8 + aB9*f9\n \+ + \+ aBA*fA;\n fB := fn(xk + cB*h,yk + t*h);\n t := aC1*f1 + aC2* f2 + aC3*f3 + aC4*f4 + aC5*f5 + aC6*f6 + aC7*f7 + aC8*f8 + aC9*f9\n \+ + aCA*fA + aCB*fB;\n fC := fn(xk + cC*h,yk + t*h);\n \n t : = b1*f1 + b2*f2 + b3*f3 + b4*f4 + b5*f5 + b6*f6 + b7*f7 + b8*f8 + b9*f 9 + bA*fA\n \+ + bB*fB + bC*fC;\n yk := yk + t*h;\n xk := xk + h:\n soln := soln,[xk,yk];\n end do;\n if bb=true then\n \+ eqns := \{SOLN_=[soln],FXY_=fxy,X_=x,Y_=y,c2_=c2,c3_=c3,\n \+ c4_=c4,c5_=c5,c6_=c6,c7_=c7,c8_=c8,c9_=c9,cA_=cA,cB_=cB,cC_=cC,\n \+ a31_=a31,a32_=a32,a41_=a41,a42_=a42,a43_=a43,a51_=a51,a52_=a52,\n a53_=a53,a54_=a54,a61_=a61,a62_=a62,a63_=a63,a64_=a64,a65_=a6 5,\n a71_=a71,a72_=a72,a73_=a73,a74_=a74,a75_=a75,a76_=a76,a81 _=a81,\n a82_=a82,a83_=a83,a84_=a84,a85_=a85,a86_=a86,a87_=a87 ,a91_=a91,\n a92_=a92,a93_=a93,a94_=a94,a95_=a95,a96_=a96,a97_ =a97,a98_=a98,\n aA1_=aA1,aA2_=aA2,aA3_=aA3,aA4_=aA4,aA5_=aA5, aA6_=aA6,aA7_=aA7,\n aA8_=aA8,aA9_=aA9,aB1_=aB1,aB2_=aB2,aB3_= aB3,aB4_=aB4,aB5_=aB5,\n aB6_=aB6,aB7_=aB7,aB8_=aB8,aB9_=aB9,a BA_=aBA,aC1_=aC1,aC2_=aC2,\n aC3_=aC3,aC4_=aC4,aC5_=aC5,aC6_=a C6,aC7_=aC7,aC8_=aC8,aC9_=aC9,\n aCA_=aCA,aCB_=aCB,b1_=b1,b2_= b2,b3_=b3,b4_=b4,b5_=b5,b6_=b6,\n b7_=b7,b8_=b8,b9_=b9,bA_=bA, bB_=bB,bC_=bC\};\n return subs(eqns,eval(rk8step)); \n else\n \+ return evalf[saveDigits]([soln]);\n end if;\nend proc:" }}} {PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 42 "RK8_5 a scheme of Tsito uras and Papakostas" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 10361 "RK8_5 := proc(fxy,x,y,xx,yy,h,stps,bb)\n local c2,c3,c4,c5,c6,c7,c8,c9,cA,cB,cC,a21,a31,a32,a41,a42,a43,a51, a52,\n a53,a54,a61,a62,a63,a64,a65,a71,a72,a73,a74,a75,a76,a81,a82,a 83,a84,a85,\n a86,a87,a91,a92,a93,a94,a95,a96,a97,a98,aA1,aA2,aA3,aA 4,aA5,aA6,aA7,aA8,\n aA9,aB1,aB2,aB3,aB4,aB5,aB6,aB7,aB8,aB9,aBA,aC1 ,aC2,aC3,aC4,aC5,aC6,aC7,\n aC8,aC9,aCA,aCB,f1,f2,f3,f4,f5,f6,f7,f8, f9,fA,fB,fC,b1,b2,b3,b4,b5,b6,b7,\n b8,b9,bA,bB,bC,t,k,fn,xk,yk,soln ,eqns,A,saveDigits;\n\n saveDigits := Digits;\n Digits := max(trun c(evalhf(Digits)),Digits+5);\n\n fn := unapply(fxy,x,y);\n\n A := \+ matrix([[9/142,9/142,0,0,0,0,0,0,0,0,0,0,0],\n[24514/238491,9950845450 /511901613729,42666469916/511901613729,0,0,0,0,0,0,0,0,0,0],\n[12257/7 9497,12257/317988,0,12257/105996,0,0,0,0,0,0,0,0,0],\n[50/129,14131686 489425/35833975601529,0,-17700454220625/11944658533843,17619604667500/ 11944658533843,0,0,0,0,0,0,0,0],\n[34/73,4418011/96055225,0,0,97577579 88/41995818067,19921512441/106300094275,0,0,0,0,0,0,0],\n[23/148,48028 5555889619/7997789253816320,0,0,52162106556696460538643/46488703328447 2899365888,-16450769949384489/490009784398315520,864813381633887/51718 297665732608,0,0,0,0,0,0],\n[142/141,-7001048352088587137143/444983035 1030385148800,0,0,-2522765548294599044197935933/1915963195493758447677 901792,-318556182235222634647116091/27172411532484037214054400,1205563 885850790193966807/132365727505912221222912,254/39,0,0,0,0,0],\n[29104 120198/33218531707,-20629396399716689122801179264428539394462855874226 604463554767070845753369/428817596627701559566575134700121144880760179 81128105307375594963152353200,0,0,-33154430743436594047794221493873977 12986453181141168247590906370819301077749322753/4985171126416088728078 21838566847987729514312398278633788185835348997643218553476,-273749409 411654060286948141164828452109898379203526945684314474186724062841643/ 6042758395137755250396784065382558911716744302162836769177316291360798 4889600,16656372518874512738268060504309924900437672263609245028809229 865738327731797537/427699013853393052278207638577101600909762793055014 9628282203007913538394549504,42008080033354305590804322944084805264441 066760038302359736803632/486530242321653491007482328781160559962917029 5030631799935804001,66845978093071633800006662723692741719194739617709 3524377824/71100452948884643779799087713322002499799983434685642170251 833,0,0,0,0],\n[83/91,-17936039463222609008282124607068771424771328701 59527/2313097568511990753781649719556084665131024900300800,0,0,-147768 74123722838192315406145167687512425345723701/1847893530366076701102014 146927696206329050105856,-19587020919884661714856757105130246995757906 603/2911893296942532038566182868170393629789388800,6364380863259071677 112259236455506477417699780613300364807/115042817458494213340657909154 9443438814988349188394057728,27725164402569748756040320433848245155581 006369/2544159473655547770881695354241256106302348256,1074424716396001 9876833255044784609639/534761804739901348825491947768304503296,-509777 37930792808232204417497248979399878217280011103197862899/1300915694564 913675613280314081837358644964393191337994183389184,0,0,0],\n[143/149, -3587625717068952487214493441966897048737050755812600710793/3015733164 033229624772006086429467685046639983706974412800,0,0,-5453011711267804 731211501837262816944661201619903/764973320899716397072448644710733101 329290196992,-884348836774584715070440485633026464653487653/9272598351 5963799584249219499787659014963200,26823469063654084387375587616552322 383082061411417182757389742951/354129974476368167547362064708712305722 8744296123642301654630400,14236341949168650716200705107100772276532316 2710521029/12634887202368261565807449771335728082273587905775840,64747 617454909275289531520412519442831235890581/126931718811797596099667062 8974453443777854830080,11263380825327272097987430336750389159749926104 6700689572459050065039333987335667/14045142912450345328121813771190345 01014039830518218465064579291611563374608650240,-106122025185739944311 53697720606405883/67082546658259846778754594976831647575,0,0],\n[1,-75 34081165544982478296202335922049210803875045423/1921957566544084875607 4598051658416387002479820800,0,0,2376960874527867178022703752830342628 59273455/60688480889936839261131818793519097177034752,-206105782098263 29263318986584876108069323/7356333776438834884293014575840932659200,51 260471529841028040709654458903254781320136131844164563/209980237763185 46907382302106788168158765514131585630208,-307721443717347297119681079 5615384000211457151011/12724355925822808205970594320601168932006846803 84,-1539218116260541896259682954580256454049/4534670830750360983422999 946409310393344,241886539350268429372116296787271276553970618941104594 460614948326132797451456131/124066963266246589252891612004112305105402 6283188324780101555345343662316090597376,-8055648683224596619171745242 5924975/414445409518676597565032008051106461,2944781680874500347594142 792814463350/4965161383073676983610218096030654529,0],\n[0,55038446513 529253801/1239280570055853383520,0,0,0,0,23354967953237824644118463946 11/6598368783294020109895379936256,7636073376527143565375240869888/307 25949199261296642046754748645,-4237087214169934312729607487/1273579139 4214625116604076160,40850529129113324176099551412198433591436392788442 6780078325258228227984174126699/21262487461219369746665515940563520212 3821166475348052734199905546455860773575680,-1108225296327029096435947 /405679075893979729103310,2460988291206213825688467985/175634278952094 7764222671739,4808707937311/50545545388065]]);\n\n c2 := evalf(A[1,1 ]);\n c3 := evalf(A[2,1]);\n c4 := evalf(A[3,1]);\n c5 := evalf( A[4,1]);\n c6 := evalf(A[5,1]);\n c7 := evalf(A[6,1]);\n c8 := e valf(A[7,1]);\n c9 := evalf(A[8,1]);\n cA := evalf(A[9,1]);\n cB := evalf(A[10,1]);\n cC := evalf(A[11,1]);\n a21 := c2;\n a31 : = evalf(A[2,2]);\n a32 := evalf(A[2,3]);\n a41 := evalf(A[3,2]);\n a42 := evalf(A[3,3]);\n a43 := evalf(A[3,4]);\n a51 := evalf(A[ 4,2]);\n a52 := evalf(A[4,3]);\n a53 := evalf(A[4,4]);\n a54 := \+ evalf(A[4,5]);\n a61 := evalf(A[5,2]);\n a62 := evalf(A[5,3]);\n \+ a63 := evalf(A[5,4]);\n a64 := evalf(A[5,5]);\n a65 := evalf(A[5, 6]);\n a71 := evalf(A[6,2]);\n a72 := evalf(A[6,3]);\n a73 := ev alf(A[6,4]);\n a74 := evalf(A[6,5]);\n a75 := evalf(A[6,6]);\n a 76 := evalf(A[6,7]);\n a81 := evalf(A[7,2]);\n a82 := evalf(A[7,3] );\n a83 := evalf(A[7,4]);\n a84 := evalf(A[7,5]);\n a85 := eval f(A[7,6]);\n a86 := evalf(A[7,7]);\n a87 := evalf(A[7,8]);\n a91 := evalf(A[8,2]);\n a92 := evalf(A[8,3]);\n a93 := evalf(A[8,4]); \n a94 := evalf(A[8,5]);\n a95 := evalf(A[8,6]);\n a96 := evalf( A[8,7]);\n a97 := evalf(A[8,8]);\n a98 := evalf(A[8,9]);\n aA1 : = evalf(A[9,2]);\n aA2 := evalf(A[9,3]);\n aA3 := evalf(A[9,4]);\n aA4 := evalf(A[9,5]);\n aA5 := evalf(A[9,6]);\n aA6 := evalf(A[ 9,7]);\n aA7 := evalf(A[9,8]);\n aA8 := evalf(A[9,9]);\n aA9 := \+ evalf(A[9,10]);\n aB1 := evalf(A[10,2]);\n aB2 := evalf(A[10,3]); \n aB3 := evalf(A[10,4]);\n aB4 := evalf(A[10,5]);\n aB5 := eval f(A[10,6]);\n aB6 := evalf(A[10,7]);\n aB7 := evalf(A[10,8]);\n \+ aB8 := evalf(A[10,9]);\n aB9 := evalf(A[10,10]);\n aBA := evalf(A[ 10,11]);\n aC1 := evalf(A[11,2]);\n aC2 := evalf(A[11,3]);\n aC3 := evalf(A[11,4]);\n aC4 := evalf(A[11,5]);\n aC5 := evalf(A[11,6 ]);\n aC6 := evalf(A[11,7]);\n aC7 := evalf(A[11,8]);\n aC8 := e valf(A[11,9]);\n aC9 := evalf(A[11,10]);\n aCA := evalf(A[11,11]); \n aCB := evalf(A[11,12]);\n b1 := evalf(A[12,2]);\n b2 := evalf (A[12,3]);\n b3 := evalf(A[12,4]);\n b4 := evalf(A[12,5]);\n b5 \+ := evalf(A[12,6]);\n b6 := evalf(A[12,7]);\n b7 := evalf(A[12,8]); \n b8 := evalf(A[12,9]);\n b9 := evalf(A[12,10]);\n bA := evalf( A[12,11]);\n bB := evalf(A[12,12]);\n bC := evalf(A[12,13]);\n x k := evalf(xx);\n yk := evalf(yy);\n soln := [xk,yk]; \n for k f rom 1 to stps do\n f1 := fn(xk,yk);\n t := a21*f1;\n f2 := fn(xk + c2*h,yk + t*h);\n t := a31*f1 + a32*f2;\n f3 := \+ fn(xk + c3*h,yk + t*h);\n t := a41*f1 + a42*f2 + a43*f3;\n f 4 := fn(xk + c4*h,yk + t*h);\n t := a51*f1 + a52*f2 + a53*f3 + a5 4*f4;\n f5 := fn(xk + c5*h,yk + t*h);\n t := a61*f1 + a62*f2 + a63*f3 + a64*f4 + a65*f5;\n f6 := fn(xk + c6*h,yk + t*h);\n \+ t := a71*f1 + a72*f2 + a73*f3 + a74*f4 + a75*f5 + a76*f6;\n f7 := fn(xk + c7*h,yk + t*h);\n t := a81*f1 + a82*f2 + a83*f3 + a84 *f4 + a85*f5 + a86*f6 + a87*f7;\n f8 := fn(xk + c8*h,yk + t*h);\n t := a91*f1 + a92*f2 + a93*f3 + a94*f4 + a95*f5 + a96*f6 + a97*f 7 + a98*f8;\n f9 := fn(xk + c9*h,yk + t*h);\n t := aA1*f1 + \+ aA2*f2 + aA3*f3 + aA4*f4 + aA5*f5 + aA6*f6 + aA7*f7 + aA8*f8 + aA9*f9; \n fA := fn(xk + cA*h,yk + t*h);\n t := aB1*f1 + aB2*f2 + aB 3*f3 + aB4*f4 + aB5*f5 + aB6*f6 + aB7*f7 + aB8*f8 + aB9*f9\n \+ \+ + aBA*fA;\n fB := fn(xk + cB*h,yk + t*h);\n t := aC1*f1 + a C2*f2 + aC3*f3 + aC4*f4 + aC5*f5 + aC6*f6 + aC7*f7 + aC8*f8 + aC9*f9\n \+ + aCA*fA + aCB*fB;\n fC := fn(xk + cC*h,yk + t*h);\n \n \+ t := b1*f1 + b2*f2 + b3*f3 + b4*f4 + b5*f5 + b6*f6 + b7*f7 + b8*f8 + b 9*f9 + bA*fA\n \+ + bB*fB + bC*fC;\n yk := yk + t*h;\n xk := xk + h:\n soln := soln,[xk,yk];\n end do;\n if bb=true then \n eqns := \{SOLN_=[soln],FXY_=fxy,X_=x,Y_=y,c2_=c2,c3_=c3,\n \+ c4_=c4,c5_=c5,c6_=c6,c7_=c7,c8_=c8,c9_=c9,cA_=cA,cB_=cB,cC_=cC,\n a31_=a31,a32_=a32,a41_=a41,a42_=a42,a43_=a43,a51_=a51,a52_=a5 2,\n a53_=a53,a54_=a54,a61_=a61,a62_=a62,a63_=a63,a64_=a64,a65 _=a65,\n a71_=a71,a72_=a72,a73_=a73,a74_=a74,a75_=a75,a76_=a76 ,a81_=a81,\n a82_=a82,a83_=a83,a84_=a84,a85_=a85,a86_=a86,a87_ =a87,a91_=a91,\n a92_=a92,a93_=a93,a94_=a94,a95_=a95,a96_=a96, a97_=a97,a98_=a98,\n aA1_=aA1,aA2_=aA2,aA3_=aA3,aA4_=aA4,aA5_= aA5,aA6_=aA6,aA7_=aA7,\n aA8_=aA8,aA9_=aA9,aB1_=aB1,aB2_=aB2,a B3_=aB3,aB4_=aB4,aB5_=aB5,\n aB6_=aB6,aB7_=aB7,aB8_=aB8,aB9_=a B9,aBA_=aBA,aC1_=aC1,aC2_=aC2,\n aC3_=aC3,aC4_=aC4,aC5_=aC5,aC 6_=aC6,aC7_=aC7,aC8_=aC8,aC9_=aC9,\n aCA_=aCA,aCB_=aCB,b1_=b1, b2_=b2,b3_=b3,b4_=b4,b5_=b5,b6_=b6,\n b7_=b7,b8_=b8,b9_=b9,bA_ =bA,bB_=bB,bC_=bC\};\n return subs(eqns,eval(rk8step)); \n else \n return evalf[saveDigits]([soln]);\n end if;\nend proc:" }}} {PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 56 "RK8_6 a modification of the Tsitouras-Papakostas scheme " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 10153 "RK8_6 := proc(fxy,x,y,xx, yy,h,stps,bb)\n local c2,c3,c4,c5,c6,c7,c8,c9,cA,cB,cC,a21,a31,a32,a 41,a42,a43,a51,a52,\n a53,a54,a61,a62,a63,a64,a65,a71,a72,a73,a74,a7 5,a76,a81,a82,a83,a84,a85,\n a86,a87,a91,a92,a93,a94,a95,a96,a97,a98 ,aA1,aA2,aA3,aA4,aA5,aA6,aA7,aA8,\n aA9,aB1,aB2,aB3,aB4,aB5,aB6,aB7, aB8,aB9,aBA,aC1,aC2,aC3,aC4,aC5,aC6,aC7,\n aC8,aC9,aCA,aCB,f1,f2,f3, f4,f5,f6,f7,f8,f9,fA,fB,fC,b1,b2,b3,b4,b5,b6,b7,\n b8,b9,bA,bB,bC,t, k,fn,xk,yk,soln,eqns,A,saveDigits;\n\n saveDigits := Digits;\n Dig its := max(trunc(evalhf(Digits)),Digits+5);\n\n fn := unapply(fxy,x, y);\n\n A := matrix([[2/25,2/25,0,0,0,0,0,0,0,0,0,0,0],\n[24514/2384 91,2090517149/56877957081,3755851225/56877957081,0,0,0,0,0,0,0,0,0,0], \n[12257/79497,12257/317988,0,12257/105996,0,0,0,0,0,0,0,0,0],\n[50/12 9,14131686489425/35833975601529,0,-17700454220625/11944658533843,17619 604667500/11944658533843,0,0,0,0,0,0,0,0],\n[34/73,4418011/96055225,0, 0,9757757988/41995818067,19921512441/106300094275,0,0,0,0,0,0,0],\n[23 /148,480285555889619/7997789253816320,0,0,52162106556696460538643/4648 87033284472899365888,-16450769949384489/490009784398315520,86481338163 3887/51718297665732608,0,0,0,0,0,0],\n[196/197,-4192485504686519025977 7/29067997485765047436800,0,0,-1573638249639795538408923368457/3379267 61654282072170175876224,-3144752379896908085076884133/2867317504057901 69489612800,51820007951705750121680295/6052645677641770902708224,19/2, 0,0,0,0,0],\n[14118145156/16038718811,-1441522260936964334360468931911 6750509914398656839842108465299109803139261/28666460834013981370272729 820284862049438129018451533980749016707010746880,0,0,-5224167467332280 02806292633565336738051126599108406039793785642181694795272896367/7244 7552632346584669609595477664264037757415889555156446859013382512921014 002304,-82495975261466675550315104749071990428768736054623239134704116 29578679723249/1756341139651208901236498008888114538679717900892624949 907570487472295239680,249566009805532163702364580500327468167202244857 4640720517109843115480217671807/62155833834379369756139647253642768400 4319023110349698478188147786343185391616,65926772222165868913654906182 786964314250556615077953597292687341/711474570717221516523994602237627 7735416754491161939984260041034,31425442840030946914785278594012470568 78071594901520381912/3228985071785520180284989571741979547706614546229 43631853501,0,0,0,0],\n[39/43,-986787885805825503921774671991587107316 7304993320293/13771006297995217364330134342641992580302241127219200,0, 0,-502106193116073366202094897549565186098556680379/597604311500002170 12599575938528786496649056256,-331923915046080335870716315310401046907 /52747587227799808210222170585805619200,164314767952314001697184419161 3351711631376872651547306145/31581305784994522364511881951336157163968 6784746580082688,2224286847389514767028684777161157168782986309/199937 022215481792634238811967979191208436288,498536496800491237121368635810 80955445/2729660625860504418029771633420026504704,-1203747916407015470 833526610982521566345739350276392522818995/431465047176222110521161502 30647115135945640260504121035832832,0,0,0],\n[20/21,-16771505604351647 51643947644188299249566198916438951/1524268586148523692143823046819588 641870992602157568,0,0,-2302011554101675211827716537011959670051291/27 4508714084612705166461135767825539832704,-1688914466951040738709921153 1243899609/1891391571300777047888087779210764288,397099609762427730897 41835270931984854511347767956576333171/5577710294529287682275315773022 359477180471284918417776640,956248096794157656802748368725840687390347 399/77961324245284495254874844291292429525671778,107883136376592921158 61359463369872242886/189380589527306698462795692309376152568545,895993 98445595686856355698434884216540038527287573830949657578490242/5793856 21338495813483588345965887373464382184677844152053023899833767,-178633 5907413925929868932792832/7814458367137598885469908059185,0,0],\n[1,-4 9410934155054900182882239365042643979034016237/31731623983903801998088 697641133618410239344640,0,0,-1074611649128981113804182104383729704094 059/244310054255337140402433369774092097581056,-2222581559078624408042 59443447899304774477/18911475471642929222051825301977449103360,1141087 6327678156749715579835467111092952376241624115293181/12490816871574728 43988286395834110562049959611482227605504,8981318435146082911054040229 712538210777263716775/936984062290713822722287218869196809455428475968 ,465544633218086513326922426838859475/27467662576695046776938324444449 900032,-28081309843646368614165858697384177345304907256697557387703645 4598695025876625/10506749043202676858879540117292890849606027959545703 832394231856364095276862976,50908388575977059406815779042800/129927708 4579288698410581464578279,-5373869958976816279363488081975/12824981949 5112776500552350596224,0],\n[0,874545191389960761/19691638733588931200 ,0,0,0,0,420364804728976000977525051149/118758845207892961468528076400 0,25102415824721153301532049778688/101007568454396515184681192806875,1 1291146602515386733913667333/8277895500290250100346304000,515227147644 04063835592355866713645307758690796273878782033726047743318218807/1928 2221074716176043598190917468876903092208069910099389016386439619421788 800,-582592205611226468655591/165538207004754039774500,430469939179693 54593/35912185355757478400,-357120621384161/262158303907500]]);\n\n \+ c2 := evalf(A[1,1]);\n c3 := evalf(A[2,1]);\n c4 := evalf(A[3,1]); \n c5 := evalf(A[4,1]);\n c6 := evalf(A[5,1]);\n c7 := evalf(A[6 ,1]);\n c8 := evalf(A[7,1]);\n c9 := evalf(A[8,1]);\n cA := eval f(A[9,1]);\n cB := evalf(A[10,1]);\n cC := evalf(A[11,1]);\n a21 := c2;\n a31 := evalf(A[2,2]);\n a32 := evalf(A[2,3]);\n a41 := evalf(A[3,2]);\n a42 := evalf(A[3,3]);\n a43 := evalf(A[3,4]);\n \+ a51 := evalf(A[4,2]);\n a52 := evalf(A[4,3]);\n a53 := evalf(A[4 ,4]);\n a54 := evalf(A[4,5]);\n a61 := evalf(A[5,2]);\n a62 := e valf(A[5,3]);\n a63 := evalf(A[5,4]);\n a64 := evalf(A[5,5]);\n \+ a65 := evalf(A[5,6]);\n a71 := evalf(A[6,2]);\n a72 := evalf(A[6,3 ]);\n a73 := evalf(A[6,4]);\n a74 := evalf(A[6,5]);\n a75 := eva lf(A[6,6]);\n a76 := evalf(A[6,7]);\n a81 := evalf(A[7,2]);\n a8 2 := evalf(A[7,3]);\n a83 := evalf(A[7,4]);\n a84 := evalf(A[7,5]) ;\n a85 := evalf(A[7,6]);\n a86 := evalf(A[7,7]);\n a87 := evalf (A[7,8]);\n a91 := evalf(A[8,2]);\n a92 := evalf(A[8,3]);\n a93 \+ := evalf(A[8,4]);\n a94 := evalf(A[8,5]);\n a95 := evalf(A[8,6]); \n a96 := evalf(A[8,7]);\n a97 := evalf(A[8,8]);\n a98 := evalf( A[8,9]);\n aA1 := evalf(A[9,2]);\n aA2 := evalf(A[9,3]);\n aA3 : = evalf(A[9,4]);\n aA4 := evalf(A[9,5]);\n aA5 := evalf(A[9,6]);\n aA6 := evalf(A[9,7]);\n aA7 := evalf(A[9,8]);\n aA8 := evalf(A[ 9,9]);\n aA9 := evalf(A[9,10]);\n aB1 := evalf(A[10,2]);\n aB2 : = evalf(A[10,3]);\n aB3 := evalf(A[10,4]);\n aB4 := evalf(A[10,5]) ;\n aB5 := evalf(A[10,6]);\n aB6 := evalf(A[10,7]);\n aB7 := eva lf(A[10,8]);\n aB8 := evalf(A[10,9]);\n aB9 := evalf(A[10,10]);\n \+ aBA := evalf(A[10,11]);\n aC1 := evalf(A[11,2]);\n aC2 := evalf( A[11,3]);\n aC3 := evalf(A[11,4]);\n aC4 := evalf(A[11,5]);\n aC 5 := evalf(A[11,6]);\n aC6 := evalf(A[11,7]);\n aC7 := evalf(A[11, 8]);\n aC8 := evalf(A[11,9]);\n aC9 := evalf(A[11,10]);\n aCA := evalf(A[11,11]);\n aCB := evalf(A[11,12]);\n b1 := evalf(A[12,2]) ;\n b2 := evalf(A[12,3]);\n b3 := evalf(A[12,4]);\n b4 := evalf( A[12,5]);\n b5 := evalf(A[12,6]);\n b6 := evalf(A[12,7]);\n b7 : = evalf(A[12,8]);\n b8 := evalf(A[12,9]);\n b9 := evalf(A[12,10]); \n bA := evalf(A[12,11]);\n bB := evalf(A[12,12]);\n bC := evalf (A[12,13]);\n xk := evalf(xx);\n yk := evalf(yy);\n soln := [xk, yk]; \n for k from 1 to stps do\n f1 := fn(xk,yk);\n t := \+ a21*f1;\n f2 := fn(xk + c2*h,yk + t*h);\n t := a31*f1 + a32* f2;\n f3 := fn(xk + c3*h,yk + t*h);\n t := a41*f1 + a42*f2 + a43*f3;\n f4 := fn(xk + c4*h,yk + t*h);\n t := a51*f1 + a52 *f2 + a53*f3 + a54*f4;\n f5 := fn(xk + c5*h,yk + t*h);\n t : = a61*f1 + a62*f2 + a63*f3 + a64*f4 + a65*f5;\n f6 := fn(xk + c6* h,yk + t*h);\n t := a71*f1 + a72*f2 + a73*f3 + a74*f4 + a75*f5 + \+ a76*f6;\n f7 := fn(xk + c7*h,yk + t*h);\n t := a81*f1 + a82* f2 + a83*f3 + a84*f4 + a85*f5 + a86*f6 + a87*f7;\n f8 := fn(xk + \+ c8*h,yk + t*h);\n t := a91*f1 + a92*f2 + a93*f3 + a94*f4 + a95*f5 + a96*f6 + a97*f7 + a98*f8;\n f9 := fn(xk + c9*h,yk + t*h);\n \+ t := aA1*f1 + aA2*f2 + aA3*f3 + aA4*f4 + aA5*f5 + aA6*f6 + aA7*f7 + aA8*f8 + aA9*f9;\n fA := fn(xk + cA*h,yk + t*h);\n t := aB1 *f1 + aB2*f2 + aB3*f3 + aB4*f4 + aB5*f5 + aB6*f6 + aB7*f7 + aB8*f8 + a B9*f9\n \+ + aBA*fA;\n fB := fn(xk + cB*h,yk + t*h);\n \+ t := aC1*f1 + aC2*f2 + aC3*f3 + aC4*f4 + aC5*f5 + aC6*f6 + aC7*f7 + \+ aC8*f8 + aC9*f9\n \+ + aCA*fA + aCB*fB;\n fC := fn(xk + cC*h,yk + t *h);\n \n t := b1*f1 + b2*f2 + b3*f3 + b4*f4 + b5*f5 + b6*f6 + \+ b7*f7 + b8*f8 + b9*f9 + bA*fA\n \+ + bB*fB + bC*fC;\n yk := yk + \+ t*h;\n xk := xk + h:\n soln := soln,[xk,yk];\n end do;\n \+ if bb=true then\n eqns := \{SOLN_=[soln],FXY_=fxy,X_=x,Y_=y,c2_= c2,c3_=c3,\n c4_=c4,c5_=c5,c6_=c6,c7_=c7,c8_=c8,c9_=c9,cA_=cA, cB_=cB,cC_=cC,\n a31_=a31,a32_=a32,a41_=a41,a42_=a42,a43_=a43, a51_=a51,a52_=a52,\n a53_=a53,a54_=a54,a61_=a61,a62_=a62,a63_= a63,a64_=a64,a65_=a65,\n a71_=a71,a72_=a72,a73_=a73,a74_=a74,a 75_=a75,a76_=a76,a81_=a81,\n a82_=a82,a83_=a83,a84_=a84,a85_=a 85,a86_=a86,a87_=a87,a91_=a91,\n a92_=a92,a93_=a93,a94_=a94,a9 5_=a95,a96_=a96,a97_=a97,a98_=a98,\n aA1_=aA1,aA2_=aA2,aA3_=aA 3,aA4_=aA4,aA5_=aA5,aA6_=aA6,aA7_=aA7,\n aA8_=aA8,aA9_=aA9,aB1 _=aB1,aB2_=aB2,aB3_=aB3,aB4_=aB4,aB5_=aB5,\n aB6_=aB6,aB7_=aB7 ,aB8_=aB8,aB9_=aB9,aBA_=aBA,aC1_=aC1,aC2_=aC2,\n aC3_=aC3,aC4_ =aC4,aC5_=aC5,aC6_=aC6,aC7_=aC7,aC8_=aC8,aC9_=aC9,\n aCA_=aCA, aCB_=aCB,b1_=b1,b2_=b2,b3_=b3,b4_=b4,b5_=b5,b6_=b6,\n b7_=b7,b 8_=b8,b9_=b9,bA_=bA,bB_=bB,bC_=bC\};\n return subs(eqns,eval(rk8s tep)); \n else\n return evalf[saveDigits]([soln]);\n end if; \nend proc:" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 36 "RK8_7 V erner \"most efficient\" scheme" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 10141 "RK8_7 := proc(fxy,x,y,xx, yy,h,stps,bb)\n local c2,c3,c4,c5,c6,c7,c8,c9,cA,cB,cC,a21,a31,a32,a 41,a42,a43,a51,a52,\n a53,a54,a61,a62,a63,a64,a65,a71,a72,a73,a74,a7 5,a76,a81,a82,a83,a84,a85,\n a86,a87,a91,a92,a93,a94,a95,a96,a97,a98 ,aA1,aA2,aA3,aA4,aA5,aA6,aA7,aA8,\n aA9,aB1,aB2,aB3,aB4,aB5,aB6,aB7, aB8,aB9,aBA,aC1,aC2,aC3,aC4,aC5,aC6,aC7,\n aC8,aC9,aCA,aCB,f1,f2,f3, f4,f5,f6,f7,f8,f9,fA,fB,fC,b1,b2,b3,b4,b5,b6,b7,\n b8,b9,bA,bB,bC,t, k,fn,xk,yk,soln,eqns,A,saveDigits;\n\n saveDigits := Digits;\n Dig its := max(trunc(evalhf(Digits)),Digits+5);\n\n fn := unapply(fxy,x, y);\n\n A := matrix([[1/20,1/20,0,0,0,0,0,0,0,0,0,0,0],\n[341/3200,- 7161/1024000,116281/1024000,0,0,0,0,0,0,0,0,0,0],\n[1023/6400,1023/256 00,0,3069/25600,0,0,0,0,0,0,0,0,0],\n[39/100,4202367/11628100,0,-38998 44/2907025,3982992/2907025,0,0,0,0,0,0,0,0],\n[93/200,5611/114400,0,0, 31744/135025,923521/5106400,0,0,0,0,0,0,0],\n[31/200,21173/343200,0,0, 8602624/76559175,-26782109/689364000,5611/283500,0,0,0,0,0,0],\n[943/1 000,-1221101821869329/690812928000000,0,0,-125/2,-1024030607959889/168 929280000000,1501408353528689/265697280000000,6070139212132283/9250201 6000000,0,0,0,0,0],\n[7067558016280/7837150160667,-1472514264486215803 881384708877264246346044433307094207829051978044531801133057155/124689 4801620032001157059621643986024803301558393487900440453636168046069686 436608,0,0,-5172294311085668458375175655246981230039025336933699114138 315270772319372469280000/124619381004809145897278630571215298365257079 410236252921850936749076487132995191,-12070679258469254807978936441733 187949484571516120469966534514296406891652614970375/272203115476165722 1710478184531100699497284085048389015085076961673446140398628096,78012 5155843893641323090552530431036567795592568497182701460674803126770111 481625/183110425412731972197889874507158786859226102980861859505241443 073629143100805376,664113122959911642134782135839106469928140328160577 035357155340392950009492511875/151784655985862481363330231072953491752 79765150089078301139943253016877823170816,1033284818445201560405683676 7286656859124007796970668046446015775000000/13127035500360336480738342 48740727914537972028638950165249582733679393783,0,0,0,0],\n[909/1000,- 29055573360337415088538618442231036441314060511/2267475989108957769132 7962602370597632000000000,0,0,-20462749524591049105403365239069/454251 913499893469596231268750,-180269259803172281163724663224981097/3810092 2558256871086579832832000000,21127670214172802870128286992003940810655 221489/4679473877997892906145822697976708633673728000,3186072351736493 12405151265849660869927653414425413/6714716715558965303132938072935465 423910912000000,212083202434519082281842245535894/20022426044775672563 822865371173879,-26984049294008425187211664850871297985622698482295177 93703413951226714583/4695456749139343150770004420808711418846760359027 17550325616728175875000000,0,0,0],\n[47/50,-23426598458140868369512071 40065609179073838476242943917/1358480961351056777022231400139158760857 532162795520000,0,0,-996286030132538159613930889652/163530688859961649 05464325675,-26053085959256534152588089363841/437755280456568306101129 9942400,20980822345096760292224086794978105312644533925634933539/37758 89992007550803878727839115494641972212962174156800,8907229937563791864 18929622095833835264322635782294899/1392124200139511265750194195559401 3822830119803764736,161021426143124178389075121929246710833125/1099720 7722131034650667041364346422894371443,30076066976810251783423249756545 2434946672266195876496371874262392684852243925359864884962513/46554433 3750134645558506533660450560376082477961552128575189281031568049236410 6674524398280000,-31155237437111730665923206875/3928621415942305150103 38956291,0,0],\n[1,-28665569918256639717782953291010338875349127877240 34363/868226711619262703011213925016143612030669233795338240,0,0,-1695 7088714171468676387054358954754000/14369041511965468332636822810157022 1,-4583493974484572912949314673356033540575/45195770365525074715731303 4270335135744,2346305388553404258656258473446184419154740172519949575/ 256726716407895402892744978301151486254183185289662464,165712155931984 6802171283690913610698586256573484808662625/13431480411255146477259155 104956093505361644432088109056,345685379554677052215495825476969226377 187500/74771167436930077221667203179551347546362089,-32058909627170725 4279143431215272753400810277402321024057136157075724905616701523016035 2087048674542196011/94756954968396581478301512445127360498465774712725 7615372449205973192657306017239103491074738324033259120,40279545832706 233433100438588458933210937500/889646084279948284691697212637733894721 5101,-6122933601070769591613093993993358877250/10505170015102355131982 46721302027675953,0],\n[0,44901867737754616851973/10140464099802310133 80680,0,0,0,0,791638675191615279648100000/2235604725089973126411512319 ,3847749490868980348119500000/15517045062138271618141237517,-137345124 32397741476562500000/875132892924995907746928783,122747654703131968784 2881203774063505031923427600698639829444355496961634227421531633068444 8207141/48934514749371551765038583414351093488882928068660965448289652 6796523353052166757299452852166040,-9798363684577739445312500000/30872 2986341456031822630699,282035543183190840068750/1229540762987304042599 1,-306814272936976936753/1299331183183744997286]]);\n\n c2 := evalf( A[1,1]);\n c3 := evalf(A[2,1]);\n c4 := evalf(A[3,1]);\n c5 := e valf(A[4,1]);\n c6 := evalf(A[5,1]);\n c7 := evalf(A[6,1]);\n c8 := evalf(A[7,1]);\n c9 := evalf(A[8,1]);\n cA := evalf(A[9,1]);\n cB := evalf(A[10,1]);\n cC := evalf(A[11,1]);\n a21 := c2;\n \+ a31 := evalf(A[2,2]);\n a32 := evalf(A[2,3]);\n a41 := evalf(A[3,2 ]);\n a42 := evalf(A[3,3]);\n a43 := evalf(A[3,4]);\n a51 := eva lf(A[4,2]);\n a52 := evalf(A[4,3]);\n a53 := evalf(A[4,4]);\n a5 4 := evalf(A[4,5]);\n a61 := evalf(A[5,2]);\n a62 := evalf(A[5,3]) ;\n a63 := evalf(A[5,4]);\n a64 := evalf(A[5,5]);\n a65 := evalf (A[5,6]);\n a71 := evalf(A[6,2]);\n a72 := evalf(A[6,3]);\n a73 \+ := evalf(A[6,4]);\n a74 := evalf(A[6,5]);\n a75 := evalf(A[6,6]); \n a76 := evalf(A[6,7]);\n a81 := evalf(A[7,2]);\n a82 := evalf( A[7,3]);\n a83 := evalf(A[7,4]);\n a84 := evalf(A[7,5]);\n a85 : = evalf(A[7,6]);\n a86 := evalf(A[7,7]);\n a87 := evalf(A[7,8]);\n a91 := evalf(A[8,2]);\n a92 := evalf(A[8,3]);\n a93 := evalf(A[ 8,4]);\n a94 := evalf(A[8,5]);\n a95 := evalf(A[8,6]);\n a96 := \+ evalf(A[8,7]);\n a97 := evalf(A[8,8]);\n a98 := evalf(A[8,9]);\n \+ aA1 := evalf(A[9,2]);\n aA2 := evalf(A[9,3]);\n aA3 := evalf(A[9, 4]);\n aA4 := evalf(A[9,5]);\n aA5 := evalf(A[9,6]);\n aA6 := ev alf(A[9,7]);\n aA7 := evalf(A[9,8]);\n aA8 := evalf(A[9,9]);\n a A9 := evalf(A[9,10]);\n aB1 := evalf(A[10,2]);\n aB2 := evalf(A[10 ,3]);\n aB3 := evalf(A[10,4]);\n aB4 := evalf(A[10,5]);\n aB5 := evalf(A[10,6]);\n aB6 := evalf(A[10,7]);\n aB7 := evalf(A[10,8]); \n aB8 := evalf(A[10,9]);\n aB9 := evalf(A[10,10]);\n aBA := eva lf(A[10,11]);\n aC1 := evalf(A[11,2]);\n aC2 := evalf(A[11,3]);\n \+ aC3 := evalf(A[11,4]);\n aC4 := evalf(A[11,5]);\n aC5 := evalf(A [11,6]);\n aC6 := evalf(A[11,7]);\n aC7 := evalf(A[11,8]);\n aC8 := evalf(A[11,9]);\n aC9 := evalf(A[11,10]);\n aCA := evalf(A[11, 11]);\n aCB := evalf(A[11,12]);\n b1 := evalf(A[12,2]);\n b2 := \+ evalf(A[12,3]);\n b3 := evalf(A[12,4]);\n b4 := evalf(A[12,5]);\n \+ b5 := evalf(A[12,6]);\n b6 := evalf(A[12,7]);\n b7 := evalf(A[12 ,8]);\n b8 := evalf(A[12,9]);\n b9 := evalf(A[12,10]);\n bA := e valf(A[12,11]);\n bB := evalf(A[12,12]);\n bC := evalf(A[12,13]); \n xk := evalf(xx);\n yk := evalf(yy);\n soln := [xk,yk]; \n f or k from 1 to stps do\n f1 := fn(xk,yk);\n t := a21*f1;\n \+ f2 := fn(xk + c2*h,yk + t*h);\n t := a31*f1 + a32*f2;\n \+ f3 := fn(xk + c3*h,yk + t*h);\n t := a41*f1 + a42*f2 + a43*f3;\n \+ f4 := fn(xk + c4*h,yk + t*h);\n t := a51*f1 + a52*f2 + a53*f 3 + a54*f4;\n f5 := fn(xk + c5*h,yk + t*h);\n t := a61*f1 + \+ a62*f2 + a63*f3 + a64*f4 + a65*f5;\n f6 := fn(xk + c6*h,yk + t*h) ;\n t := a71*f1 + a72*f2 + a73*f3 + a74*f4 + a75*f5 + a76*f6;\n \+ f7 := fn(xk + c7*h,yk + t*h);\n t := a81*f1 + a82*f2 + a83*f3 + a84*f4 + a85*f5 + a86*f6 + a87*f7;\n f8 := fn(xk + c8*h,yk + t *h);\n t := a91*f1 + a92*f2 + a93*f3 + a94*f4 + a95*f5 + a96*f6 + a97*f7 + a98*f8;\n f9 := fn(xk + c9*h,yk + t*h);\n t := aA1 *f1 + aA2*f2 + aA3*f3 + aA4*f4 + aA5*f5 + aA6*f6 + aA7*f7 + aA8*f8 + a A9*f9;\n fA := fn(xk + cA*h,yk + t*h);\n t := aB1*f1 + aB2*f 2 + aB3*f3 + aB4*f4 + aB5*f5 + aB6*f6 + aB7*f7 + aB8*f8 + aB9*f9\n \+ \+ + aBA*fA;\n fB := fn(xk + cB*h,yk + t*h);\n t := aC1* f1 + aC2*f2 + aC3*f3 + aC4*f4 + aC5*f5 + aC6*f6 + aC7*f7 + aC8*f8 + aC 9*f9\n \+ + aCA*fA + aCB*fB;\n fC := fn(xk + cC*h,yk + t*h);\n \n t := b1*f1 + b2*f2 + b3*f3 + b4*f4 + b5*f5 + b6*f6 + b7*f7 + b8* f8 + b9*f9 + bA*fA\n \+ + bB*fB + bC*fC;\n yk := yk + t*h;\n \+ xk := xk + h:\n soln := soln,[xk,yk];\n end do;\n if bb=true then\n eqns := \{SOLN_=[soln],FXY_=fxy,X_=x,Y_=y,c2_=c2,c3_=c3, \n c4_=c4,c5_=c5,c6_=c6,c7_=c7,c8_=c8,c9_=c9,cA_=cA,cB_=cB,cC_ =cC,\n a31_=a31,a32_=a32,a41_=a41,a42_=a42,a43_=a43,a51_=a51,a 52_=a52,\n a53_=a53,a54_=a54,a61_=a61,a62_=a62,a63_=a63,a64_=a 64,a65_=a65,\n a71_=a71,a72_=a72,a73_=a73,a74_=a74,a75_=a75,a7 6_=a76,a81_=a81,\n a82_=a82,a83_=a83,a84_=a84,a85_=a85,a86_=a8 6,a87_=a87,a91_=a91,\n a92_=a92,a93_=a93,a94_=a94,a95_=a95,a96 _=a96,a97_=a97,a98_=a98,\n aA1_=aA1,aA2_=aA2,aA3_=aA3,aA4_=aA4 ,aA5_=aA5,aA6_=aA6,aA7_=aA7,\n aA8_=aA8,aA9_=aA9,aB1_=aB1,aB2_ =aB2,aB3_=aB3,aB4_=aB4,aB5_=aB5,\n aB6_=aB6,aB7_=aB7,aB8_=aB8, aB9_=aB9,aBA_=aBA,aC1_=aC1,aC2_=aC2,\n aC3_=aC3,aC4_=aC4,aC5_= aC5,aC6_=aC6,aC7_=aC7,aC8_=aC8,aC9_=aC9,\n aCA_=aCA,aCB_=aCB,b 1_=b1,b2_=b2,b3_=b3,b4_=b4,b5_=b5,b6_=b6,\n b7_=b7,b8_=b8,b9_= b9,bA_=bA,bB_=bB,bC_=bC\};\n return subs(eqns,eval(rk8step)); \n \+ else\n return evalf[saveDigits]([soln]);\n end if;\nend proc: " }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}}{PARA 0 "" 0 "" {TEXT -1 26 "#----------------- --------" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 20 "Testing the examples" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}{PARA 0 "" 0 "" {TEXT -1 80 "These tests do not make use of the embedded order 7 method for error correction. " }}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 48 "Test 1 of 12 stage, order 8 R unge-Kutta methods " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "dy/dx=12*x*cos(4*x)*exp(-x)*y" "6#/* &%#dyG\"\"\"%#dxG!\"\"*,\"#7F&%\"xGF&-%$cosG6#*&\"\"%F&F+F&F&-%$expG6# ,$F+F(F&%\"yGF&" }{TEXT -1 6 ", " }{XPPEDIT 18 0 "y(0)=1" "6#/-%\" yG6#\"\"!\"\"\"" }{TEXT -1 1 " " }}{PARA 0 "" 0 "" {TEXT -1 10 "Soluti on: " }}{PARA 256 "" 0 "" {TEXT -1 2 " " }{XPPEDIT 18 0 "y=exp(-12/17 *x*cos(4*x)*exp(-x)+180/289*exp(-x)*cos(4*x)+48/17*exp(-x)*sin(4*x)*x+ 96/289*exp(-x)*sin(4*x)-180/289)" "6#/%\"yG-%$expG6#,,*,\"#7\"\"\"\"#< !\"\"%\"xGF+-%$cosG6#*&\"\"%F+F.F+F+-F&6#,$F.F-F+F-**\"$!=F+\"$*GF--F& 6#,$F.F-F+-F06#*&F3F+F.F+F+F+*,\"#[F+F,F--F&6#,$F.F-F+-%$sinG6#*&F3F+F .F+F+F.F+F+**\"#'*F+F9F--F&6#,$F.F-F+-FF6#*&F3F+F.F+F+F+*&F8F+F9F-F-" }{TEXT -1 2 ". " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> \+ " 0 "" {MPLTEXT 1 0 229 "de := diff(y(x),x)=12*x*cos(4*x)*exp(-x)*y(x) ;\nic := y(0)=1;\ndsolve(\{de,ic\},y(x)):\ny(x)=simplify(numer(rhs(%)) /convert(denom(rhs(%)),exp));\nf := unapply(rhs(%),x):\nplot(f(x),x=0. .5,0..1.45,font=[HELVETICA,9],labels=[`x`,`y(x)`]);" }}{PARA 11 "" 1 " " {XPPMATH 20 "6#>%#deG/-%%diffG6$-%\"yG6#%\"xGF,,$*,\"#7\"\"\"F,F0-%$ cosG6#,$*&\"\"%F0F,F0F0F0-%$expG6#,$F,!\"\"F0F)F0F0" }}{PARA 11 "" 1 " " {XPPMATH 20 "6#>%#icG/-%\"yG6#\"\"!\"\"\"" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/-%\"yG6#%\"xG-%$expG6#,,*&#\"#7\"#<\"\"\"*(F'F0-%$cosG 6#,$*&\"\"%F0F'F0F0F0-F)6#,$F'!\"\"F0F0F;*&#\"$!=\"$*GF0*&F8F0F2F0F0F0 *&#\"#[F/F0*(F8F0-%$sinGF4F0F'F0F0F0*&#\"#'*F?F0*&F8F0FEF0F0F0#F>F?F; " }}{PARA 13 "" 1 "" {GLPLOT2D 577 286 286 {PLOTDATA 2 "6&-%'CURVESG6$ 7er7$$\"\"!F)$\"\"\"F)7$$\"3gmmTN@Ki8!#>$\"3Fk>e\"G.6+\"!#<7$$\"3ALL$3 FWYs#F/$\"3!H*fm:2P/5F27$$\"3%)***\\iSmp3%F/$\"3Qn()\\Dat45F27$$\"3Wmm mT&)G\\aF/$\"34$Q7t`Dr,\"F27$$\"3m****\\7G$R<)F/$\"3S2-*\\9jw.\"F27$$ \"3GLLL3x&)*3\"!#=$\"3U([#>C\\El5F27$$\"3))**\\i!R(*Rc\"FJ$\"3>&=^@[0u 7\"F27$$\"3umm\"H2P\"Q?FJ$\"3k\\#o#G?)=?\"F27$$\"3!***\\PMnNrDFJ$\"3s_ j<)f!R*G\"F27$$\"3MLL$eRwX5$FJ$\"37'\\4u:c`O\"F27$$\"3_LLe*[`HP$FJ$\"3 [!\\'y0#yNR\"F27$$\"3rLLL$eI8k$FJ$\"3N\"Ha_9o@T\"F27$$\"3_L$3-8>bx$FJ$ \"3@))>@pAD<9F27$$\"3*QL$3xwq4RFJ$\"3a@g!fsi#>9F27$$\"3EM$eRA'*Q/%FJ$ \"3^DvP/8/=9F27$$\"33ML$3x%3yTFJ$\"3bF0p:\"oMT\"F27$$\"3h+]PfyG7ZFJ$\" 3e=U+Y19h8F27$$\"3emm\"z%4\\Y_FJ$\"3Yii#4W6uD\"F27$$\"3'QLL3FGT\\&FJ$ \"3c!QStI8]>\"F27$$\"32++v$flWv*FJ7$$\"3I++vVVX$\\'FJ$ \"3w/21T*\\F&*)FJ7$$\"31nm\"zWo)\\nFJ$\"3E>3;k'H:;)FJ7$$\"3%QL$3_DG1qF J$\"31le1yn9(R(FJ7$$\"3]***\\il'pisFJ$\"3E!)4GzFfsmFJ7$$\"3+MLe*[!)y_( FJ$\"3CJpN=**=vfFJ7$$\"3Qnm\"HKkIz(FJ$\"3'oU:>LtrL&FJ7$$\"3!3+]i:[#e!) FJ$\"31b0R&QB=w%FJ7$$\"3>MLe*)>VB$)FJ$\"3qsV#=-7'\\UFJ7$$\"3wmmTg()4_) )FJ$\"3,.mLb#*p3MFJ7$$\"3Y++DJbw!Q*FJ$\"3)=h%pn^Z#y#FJ7$$\"3+N$ekGkX#* *FJ$\"3i0nI\\:'RK#FJ7$$\"3%ommTIOo/\"F2$\"3\"GyFJ7$$\"3E+]7GTt%4\"F2$\"3YFEp[WPZ=FJ7$$\"3(p;/,/$o=6F2 $\"3;\"e:UqpMz\"FJ7$$\"3YLL3_>jU6F2$\"3EC/vOKMe(4+7ES\"F2$\"3U)GYYI=FJ7$$\"35+++v\"=YI\"F2$\"3/ >xDBH;*)>FJ7$$\"33++](=h(e8F2$\"3'>M4q'>VoAFJ7$$\"3&*****\\7!Q4T\"F2$ \"3Ig`=6c[gEFJ7$$\"3/++]P[6j9F2$\"31r>cjB_'>$FJ7$$\"3%o;HKR'\\5:F2$\"3 XwZW,h_FQFJ7$$\"3UL$e*[z(yb\"F2$\"3CY!yD$)***3YFJ7$$\"3w;/Ev&[ge\"F2$ \"3#G<.XQ`j9&FJ7$$\"34+Dc,#>Uh\"F2$\"3t/<(f[0bt&FJ7$$\"3V$eky#)*QU;F2$ \"3y9nBQhKrjFJ7$$\"3wmm;a/cq;F2$\"3A`yB*3;b/(FJ7$$\"3\"pm;a)))G=F2$\"3 )3HOInxF>\"F27$$\"3KLe9;0?E>F2$\"3`!yI!pI]77F27$$\"3pTg-gl[Q>F2$\"3>kb 3F2$\"3OF\\_#G6DA\"F27$$\"3WekyZ'eI'>F2$\"3Te$z >cCQA\"F27$$\"3gmmm\"pW`(>F2$\"3C(*f_UYpA7F27$$\"3dLe9TOEH?F2$\"3mC!>8 `I->\"F27$$\"3K+]i!f#=$3#F2$\"3w9E]:+C>6F27$$\"3/++D\"=EX8#F2$\"3f+))G PMfE5F27$$\"3?+](=xpe=#F2$\"3ES3-I\\16#*FJ7$$\"3mLeRA9WRAF2$\"3IMhv&[? ^3)FJ7$$\"37nm\"H28IH#F2$\"3H\\m$Q)R4@qFJ7$$\"3$p;a8d3AM#F2$\"39j2HRJ+ ZhFJ7$$\"3um;zpSS\"R#F2$\"3#>07(=j$QR&FJ7$$\"3-+v$41oWW#F2$\"3QVRl9U0B ZFJ7$$\"3GLL3_?`(\\#F2$\"3/\\cKWs=$>%FJ7$$\"3AL3_D1l_DF2$\"3o6E$fFc$yP FJ7$$\"3fL$e*)>pxg#F2$\"3ym)*p(*f`&[$FJ7$$\"3%omm\"z+vbEF2$\"3jG&[,$f< =LFJ7$$\"33+]Pf4t.FF2$\"3%R>3YHT'HFF2$\"3/C%)f*f*e+K FJ7$$\"3om\"zWi^bv#F2$\"3-Gu$[oUh>$FJ7$$\"3)*\\7.d>Y\"y#F2$\"3#p*R$)o? n4KFJ7$$\"3uLLe*Gst!GF2$\"3>.X!=mk1C$FJ7$$\"3)om\"H2\"34'GF2$\"3'[>IF2$\"3a-&\\&*p%H, TFJ7$$\"3F+]i!RU07$F2$\"3'fkDHe#=P[FJ7$$\"3+++v=S2LKF2$\"3K%\\5FaXpw&F J7$$\"3Jmmm\"p)=MLF2$\"3))zmB`6`OlFJ7$$\"3GLLeR%p\")Q$F2$\"3#o,C;(=8fo FJ7$$\"3B++](=]@W$F2$\"3#G%=QV$\\;4(FJ7$$\"3C$ekyZ2mY$F2$\"3u,muc\"4C(FJ7$$\"3hTgx.2vFNF2$\"3/^M\"Q[;l C(FJ7$$\"35L$e*[$z*RNF2$\"3=wJ%fi2nC(FJ7$$\"3)*\\PMFwrmNF2$\"3R[i&\\xl (GsFJ7$$\"3%o;Hd!fX$f$F2$\"3IEKi0hy'=(FJ7$$\"3r$e9T=%>?OF2$\"3(>gS`&3d ArFJ7$$\"3e++]iC$pk$F2$\"3ma\\oRiHQqFJ7$$\"3ILe*[t\\sp$F2$\"3'e9/wG(3M oFJ7$$\"3[m;H2qcZPF2$\"3CYQ8S*3be'FJ7$$\"3O+]7.\"fF&QF2$\"3**Q8E[N&3+' FJ7$$\"3Ymm;/OgbRF2$\"3kN#z0%oN^aFJ7$$\"3w**\\ilAFjSF2$\"3[i8#)*p//*\\ FJ7$$\"3ym\"zW7@^6%F2$\"3>C%QCunR#[FJ7$$\"3yLLL$)*pp;%F2$\"3g*yCm#3E'p %FJ7$$\"3)QL3-$H**>UF2$\"3$*o:W?mr0YFJ7$$\"3)RL$3xe,tUF2$\"3!\\Bp&*))o Xb%FJ7$$\"3h+v=n(*fDVF2$\"3kIpK$)H$3a%FJ7$$\"3Cn;HdO=yVF2$\"3u&G6!oNOh XFJ7$$\"3MMe9\"z-lU%F2$\"3kC\">#=Lu2YFJ7$$\"3a+++D>#[Z%F2$\"3w_(eqj7vn %FJ7$$\"3SnmT&G!e&e%F2$\"3W>T$>g**p!\\FJ7$$\"3#RLLL)Qk%o%F2$\"3'yDBP_q :;&FJ7$$\"37+]iSjE!z%F2$\"3J;fP@m(pV&FJ7$$\"3a+]P40O\"*[F2$\"3!>+$=fU- gcFJ7$$\"\"&F)$\"3h(Q0fOqh\"eFJ-%'COLOURG6&%$RGBG$\"#5!\"\"F(F(-%%FONT G6$%*HELVETICAG\"\"*-%+AXESLABELSG6$%\"xG%%y(x)G-%%VIEWG6$;F(F]am;F($ \"$X\"!\"#" 1 2 0 1 10 0 2 9 1 4 2 1.000000 45.000000 45.000000 0 0 "C urve 1" }}}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 32 "The following code constructs a " }{TEXT 260 17 "discrete solution " }{TEXT -1 44 " based on each of the methods and gives the " }{TEXT 260 22 "root mean square error" }{TEXT -1 18 " of each solution." }} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 801 "F := (x,y) -> 12*x*cos(4*x) *exp(-x)*y: hh := 0.01: numsteps := 500: x0 := 0: y0 := 1:\nmatrix([[` slope field: `,F(x,y)],[`initial point: `,``(x0,y0)],[`step width: \+ `,hh],\n[`no. of steps: `,numsteps]]);``;\nmthds := [`Enright-Verne r scheme`,`Verner's Maple scheme`,`a modification of Verner's Maple sc heme`,`an \"efficient\" scheme`,`a scheme of Tsitouras and Papakostas` ,`a modification of the Tsitouras-Papakostas scheme`,`Verner's \"most \+ efficient\" scheme`]: errs := []:\nDigits := 25:\nfor ct to 7 do\n F n_RK8_||ct := RK8_||ct(F(x,y),x,y,x0,y0,hh,numsteps,false);\n sm := \+ 0: numpts := nops(Fn_RK8_||ct):\n for ii to numpts do\n sm := s m+(Fn_RK8_||ct[ii,2]-f(Fn_RK8_||ct[ii,1]))^2;\n end do:\n errs := \+ [op(errs),sqrt(sm/numpts)];\nend do:\nDigits := 10:\nlinalg[transpose] ([mthds,evalf(errs)]);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#K%'matrixG6# 7&7$%0slope~field:~~~G,$*,\"#7\"\"\"%\"xGF,-%$cosG6#,$*&\"\"%F,F-F,F,F ,-%$expG6#,$F-!\"\"F,%\"yGF,F,7$%0initial~point:~G-%!G6$\"\"!F,7$%/ste p~width:~~~G$F,!\"#7$%1no.~of~steps:~~~G\"$+&Q(pprint06\"" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%!G" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#K%'matri xG6#7)7$%6Enright-Verner~schemeG$\"+#\\96$\\!#G7$%6Verner's~Maple~sche meG$\"+.')[S5!#F7$%Ha~modification~of~Verner's~Maple~schemeG$\"+K#*yTI F+7$%6an~\"efficient\"~schemeG$\"+\"R^=o#F+7$%Ea~scheme~of~Tsitouras~a nd~PapakostasG$\"+j3E)*RF+7$%Ra~modification~of~the~Tsitouras-Papakost as~schemeG$\"+=c/qRF+7$%AVerner's~\"most~efficient\"~schemeG$\"+MhgY_F +Q(pprint16\"" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 30 "The following code constructs " }{TEXT 260 20 "numerical \+ procedures" }{TEXT -1 56 " for solutions based on each of the Runge-Ku tta schemes." }}{PARA 0 "" 0 "" {TEXT -1 75 "The error in the value ob tained by each of the methods at the point where " }{XPPEDIT 18 0 "x \+ = 4.999;" "6#/%\"xG-%&FloatG6$\"%**\\!\"$" }{TEXT -1 16 " is also giv en." }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 732 "F := (x,y) -> 12*x*c os(4*x)*exp(-x)*y: hh := 0.01: numsteps := 500: x0 := 0: y0 := 1:\nmat rix([[`slope field: `,F(x,y)],[`initial point: `,``(x0,y0)],[`step w idth: `,hh],\n[`no. of steps: `,numsteps]]);``;\nmthds := [`Enrigh t-Verner scheme`,`Verner's Maple scheme`,`a modification of Verner's M aple scheme`,`an \"efficient\" scheme`,`a scheme of Tsitouras and Papa kostas`,`a modification of the Tsitouras-Papakostas scheme`,`Verner's \+ \"most efficient\" scheme`]: errs := []:\nDigits := 30:\nfor ct to 7 d o\n fn_RK8_||ct := RK8_||ct(F(x,y),x,y,x0,y0,hh,numsteps,true);\nend do:\nxx := 4.999: fxx := evalf(f(xx)):\nfor ct to 7 do\n errs := [o p(errs),abs(fn_RK8_||ct(xx)-fxx)];\nend do:\nDigits := 10:\nlinalg[tra nspose]([mthds,evalf(errs)]);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#K%'ma trixG6#7&7$%0slope~field:~~~G,$*,\"#7\"\"\"%\"xGF,-%$cosG6#,$*&\"\"%F, F-F,F,F,-%$expG6#,$F-!\"\"F,%\"yGF,F,7$%0initial~point:~G-%!G6$\"\"!F, 7$%/step~width:~~~G$F,!\"#7$%1no.~of~steps:~~~G\"$+&Q(pprint26\"" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#%!G" }}{PARA 11 "" 1 "" {XPPMATH 20 "6 #K%'matrixG6#7)7$%6Enright-Verner~schemeG$\"+YyK!3#F+7$%Ea~scheme~of~Tsi touras~and~PapakostasG$\"+)p)Q[OF+7$%Ra~modification~of~the~Tsitouras- Papakostas~schemeG$\"+uXC5NF+7$%AVerner's~\"most~efficient\"~schemeG$ \"+-(z%e_F+Q(pprint36\"" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 " " 0 "" {TEXT -1 4 "The " }{TEXT 260 22 "root mean square error" } {TEXT -1 20 " over the interval " }{XPPEDIT 18 0 "[0, 5];" "6#7$\"\"! \"\"&" }{TEXT -1 82 " of each Runge-Kutta method is estimated as foll ows using the special procedure " }{TEXT 0 5 "NCint" }{TEXT -1 98 " \+ to perform numerical integration by the 7 point Newton-Cotes method ov er 200 equal subintervals." }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 476 "mthds := [`Enright-Verner scheme`,`Verner's Maple scheme`,`a modi fication of Verner's Maple scheme`,`an \"efficient\" scheme`,`a scheme of Tsitouras and Papakostas`,`a modification of the Tsitouras-Papakos tas scheme`,`Verner's \"most efficient\" scheme`]: errs := []:\nDigits := 20:\nfor ct to 7 do\n sm := NCint((f(x)-'fn_RK8_||ct'(x))^2,x=0. .5,adaptive=false,numpoints=7,factor=200);\n errs := [op(errs),sqrt( sm/5)];\nend do:\nDigits := 10:\nlinalg[transpose]([mthds,evalf(errs)] );" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#K%'matrixG6#7)7$%6Enright-Verner ~schemeG$\"+\\-%G#\\!#G7$%6Verner's~Maple~schemeG$\"+Ze^S5!#F7$%Ha~mod ification~of~Verner's~Maple~schemeG$\"+['HH.$F+7$%6an~\"efficient\"~sc hemeG$\"+n\"H#pEF+7$%Ea~scheme~of~Tsitouras~and~PapakostasG$\"+')4'y)R F+7$%Ra~modification~of~the~Tsitouras-Papakostas~schemeG$\"+9g1fRF+7$% AVerner's~\"most~efficient\"~schemeG$\"+YFZW_F+Q(pprint46\"" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 92 "The following e rror graphs are constructed using the numerical procedures for the sol utions." }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 616 "evalf[25](plot([ f(x)-'fn_RK8_1'(x),f(x)-'fn_RK8_2'(x),f(x)-'fn_RK8_3'(x),f(x)-'fn_RK8_ 4'(x),\nf(x)-'fn_RK8_5'(x),f(x)-'fn_RK8_6'(x),f(x)-'fn_RK8_7'(x)],x=0. .5,-2.2e-18..2.4e-18,font=[HELVETICA,9],\ncolor=[COLOR(RGB,.4,0,.9),CO LOR(RGB,0,.65,0),brown,red,COLOR(RGB,0,.75,.95),magenta,COLOR(RGB,.95, .55,0)],\nlegend=[`Enright-Verner scheme`,`Verner's Maple scheme`,`a m odification of Verner's Maple scheme`,`an \"efficient\" scheme`,`a sch eme of Tsitouras and Papakostas`,`a modification of the Tsitouras-Papa kostas scheme`,`Verner's \"most efficient\" scheme`],title=`error curv es for 12 stage order 8 Runge-Kutta methods`));" }}{PARA 13 "" 1 "" {GLPLOT2D 1006 670 670 {PLOTDATA 2 "6--%'CURVESG6%7es7$$\"\"!F)F(7$$\" :qmmmmmm;a)G\\a!#E$!'M%G\"!#C7$$\":MLLLLLL$3x&)*3\"!#D$!'Su?F07$$\":++ ++++D1R(*Rc\"F4$!'T'*=F07$$\":nmmmmm;H2P\"Q?F4$!'n$o\"F07$$\":MLLLL$ek .pu/BF4$!'VVAF07$$\":++++++vVtc8d#F4$!''><$F07$$\":MLLLLeR(\\;m/FF4$!' 7\\WF07$$\":nmmmm;/^cmz$GF4$!'>,`F07$$\":+++++vo/[r7(HF4$!']0jF07$$\": MLLLLLLeRwX5$F4$!'rc$)F07$$\":MLLLLL3F%\\wQKF4$!'\"[b*F07$$\":MLLLLL$e *[`HP$F4$!(ak3\"F07$$\":MLLLLLek.Ur]$F4$!(,QK\"F07$$\":MLLLLLLLeI8k$F4 $!(u'[9F07$$\":MLLLLL3-8>bx$F4$!(Iud\"F07$$\":MLLLLL$3xwq4RF4$!(I`y\"F 07$$\":MLLLLLeRA'*Q/%F4$!(A2)=F07$$\":NLLLLLL3x%3yTF4$!()>r>F07$$\":-+ +++voHaN;J%F4$!(Kl4#F07$$\":ommmm;/^J'=XWF4$!(cV8#F07$$\":NLLL3-Q\"3Sd yWF4$!(]S8#F07$$\":-+++](=<,<'>^%F4$!(p`;#F07$$\":ommm\"Hd?%R\\`a%F4$! ([%f@F07$$\":NLLLLeRs3P(yXF4$!(6l:#F07$$\":ommm;H2LZ7bk%F4$!(-s;#F07$$ \":-+++++v$fyG7ZF4$!(o\\;#F07$$\":NLLLL$ek.%*Qz\\F4$!(*>m?F07$$\":ommm mm;z%4\\Y_F4$!([o#=F07$$\":++++++v$4'4.P&F4$!(Aap\"F07$$\":NLLLLLL3FGT \\&F4$!(n6\\\"F07$$\":qmmmmm\"HKp%zh&F4$!(1fF\"F07$$\":++++++]Pfl*F07$$\":qmmmmmmm\"HS*)fF4$!'%>)oF 07$$\":NLLLL$eRZAJ^gF4$!'K*y&F07$$\":++++++D\"y:A8hF4$!':eTF07$$\":qmm mm;a)348vhF4$!(P]'RF47$$\":NLLLLL$eR-/PiF4$!(fm^#F47$$\":qmmmmmm;HZ_O' F4$!'$Hw*F47$$\":++++++]PMaM\\'F4$\"(F)=5F47$$\":NLLLLLLeRh;i'F4$\"(c, h#F47$$\":qmmmmm;zWo)\\nF4$\"(zKV$F47$$\":++++++++]v!yoF4$\"()fATF47$$ \":NLLLLL$3_DG1qF4$\"(X#p\\F47$$\":qmmmmmmTg*[MrF4$\"(>y1&F47$$\":-+++ ++]il'pisF4$\"(7n-&F47$$\":qmmmmm;HKkIz(F4$\"(=>u$F47$$\":OLLLLL$e*)>V B$)F4$\"(2\"4DF47$$\":0++++]PfL:&*Q)F4$\"(82W#F47$$\":qmmmmm\"H#o)fb%) F4$\"(!G$Q#F47$$\":SLLLL$ekG?o@&)F4$\"(\"ojBF47$$\":0++++++]Plxe)F4$\" (A(>BF47$$\":SLLLLL3x1K*>()F4$\"(H\"[BF47$$\":qmmmmm;/w)4_))F4$\"(P(QB F47$$\":++++++DJXlU)*)F4$\"('\\jBF47$$\":NLLLLLLe9Kk6*F4$\"(a)fCF47$$ \":qmmmmmT&Q))f[#*F4$\"(BeW#F47$$\":.+++++]7`l2Q*F4$\"(N)GCF47$$\":NLL LLLekGkX#**F4$\"(ex5#F47$$\":nmmmmmmTIOo/\"F0$\"(+'*[\"F47$$\":++++++D \"GTt%4\"F0$\"'ji'*F47$$\":MLLLLL$3_>jU6F0$\"'P-mF47$$\":nmmmmT&Q`B6c6 F0$\"'5tiF47$$\":+++++](oaFfp6F0$\"'\\qfF47$$\":MLLLLe*)f:tI=\"F0$\"'` #*eF47$$\":nmmmmm\"HdNb'>\"F0$\"'MYfF47$$\":+++++v$feR.57F0$\"'ZliF47$ $\":MLLLL$e*)fV^B7F0$\"'\"ec'F47$$\":nmmmm\"z>hZ*pB\"F0$\"'wmpF47$$\": +++++++D;v/D\"F0$\"'*R%yF47$$\":+++++++]<=YI\"F0$\"(6E9\"F47$$\":+++++ ++v=h(e8F0$\"(cNm\"F47$$\":+++++++v$[6j9F0$\"(c9#HF47$$\":MLLLLLe*[z(y b\"F0$\"(b=*[F47$$\":nmmmmTg_d[ge\"F0$\"(i;o&F47$$\":+++++]i:?>Uh\"F0$ \"((oxjF47$$\":nmmm;a8Z^/$G;F0$\"(u0l'F47$$\":MLLLLeky#)*QU;F0$\"(gsw' F47$$\":nmmmT5SW[K%\\;F0$\"(HE#oF47$$\":++++]i:59vkl\"F0$\"(3_!pF47$$ \":MLLLe9\"f(zF0$!''pb#F07$$\":,+++]ilAZ9R\">F0$!'r>EF07$$\" :MLLLL$e9;0?E>F0$!'6EEF07$$\":+++++D1Rgs2&>F0$!'nFBF07$$\":nmmmmmm;pW` (>F0$!'Q'*=F07$$\":MLLLL$e9TOEH?F0$\"%5PF07$$\":,+++++D1f#=$3#F0$\"'`x CF07$$\":,+++++]7=EX8#F0$\"'2jVF07$$\":,+++++v=xpe=#F0$\"(b@N&F47$$\": ommmT5!>Gic#>#F0$\"(guY&F47$$\":MLLL$3_]%oi#*>#F0$\"(!elaF47$$\":,+++D J?39ff?#F0$\"(#pgaF47$$\":ommmmTNrfbE@#F0$\"(j9]&F47$$\":,+++]il(4&[gA #F0$\"(#4?aF47$$\":MLLLL$eRA9WRAF0$\"(I/L&F47$$\":+++++DcwCFiE#F0$\"(h &)*\\F47$$\":nmmmmm;H28IH#F0$\"(F;b%F47$$\":nmmmmm\"zpSS\"R#F0$\"(>G*H F47$$\":+++++]P41oWW#F0$\"(\"R%[#F47$$\":MLLLLL$3_?`(\\#F0$\"(6z9#F47$ $\":MLLLL$3_D1l_DF0$\"(\"41>F47$$\":MLLLLLe*)>pxg#F0$\"(Tjt\"F47$$\":, +++++v$f4t.FF0$\"(jXb\"F47$$\":ommmm;zWi^bv#F0$\"(Vd^\"F47$$\":MLLLLL$ e*Gst!GF0$\"(\\y^\"F47$$\":ommmmm\"H2\"34'GF0$\"(moc\"F47$$\":,++++++] #RW9HF0$\"(6sm\"F47$$\":,+++++]7j#>>IF0$\"(9\")*>F47$$\":,+++++D1RU07$ F0$\"(<2Q#F47$$\":,+++++](=S2LKF0$\"(&\\yEF47$$\":ommmmmm;p)=MLF0$\"(e F\"GF47$$\":,++++++v=]@W$F0$\"(pQ$HF47$$\":MLLLLLe*[$z*RNF0$\"(B]/$F47 $$\":,++++++DYKpk$F0$\"(pB1$F47$$\":ommmmm\"H2qcZPF0$\"(0\"RHF47$$\":, +++++DJ5fF&QF0$\"(k))p#F47$$\":ommmmmmTg.c&RF0$\"(b5X#F47$$\":,+++++Dc EsK1%F0$\"(I-D#F47$$\":NLLLLLLL)*pp;%F0$\"([j7#F47$$\":NLLLLL$3xe,tUF0 $\"(ub1#F47$$\":ommmmm\"HdO=yVF0$\"(%*p1#F47$$\":,++++++]#>#[Z%F0$\"(k \"=@F47$$\":ommmmm;aG!e&e%F0$\"(@'o^7\\6F4$\"'c)=#F07$$\":+++++D1*GER37F4$\"'j$G#F07$$\":MLLL$3F>*3g wE\"F4$\"'D+BF07$$\":nmmmm;z%\\v#pK\"F4$\"'xaBF07$$\":MLLLL3_+ZiaW\"F4 $\"'_'Q#F07$F8$\"')zP#F07$$\":MLLLL$3xJs1,=F4$\"'dD@F07$F=$\"'(Q)=F07$ $\":+++++D\"G))>Wr@F4$\"')pv\"F07$FB$\"'\"o`\"F07$$\":nmmmmT5!>=0QCF4$ \"'(pZ\"F07$FG$\"'w^9F07$FL$\"'7.:F07$FQ$\"'E1;F07$FV$\"'&*f)F4$\"&V\"[F47$Fcz$\"(A=u\"F47$Fg[l$\"( L#4SF47$Fa\\l$\"(Yp9'F47$Ff\\l$\"(%z1oF47$F[]l$\"(B_m(F47$F`]l$\"(?-#y F47$Fe]l$\"($yDzF47$$\":qmmm;/^cPS([%*F4$\"(xb-)F47$$\":NLLLL3_+A:n^*F 4$\"(>a4)F47$$\":qmmmTg_Ak-2b*F4$\"([9+)F47$$\":++++]7`W1!p%e*F4$\"(#3 ^zF47$$\":NLLLek`m[x'='*F4$\"(YS+)F47$$\":qmmmm;a)3\\m_'*F4$\"(oT\"zF4 7$$\":+++++Dcwf9')y*F4$\"(T(=xF47$Fj]l$\"(XD[(F47$$\":+++++D1kOY'>5F0$ \"(&H/pF47$F_^l$\"(Z\")R'F47$$\":MLLLL$e9;_yq5F0$\"($RUgF47$Fd^l$\"(8N \"eF47$$\":MLLLLe9Te3n5\"F0$\"(WQt&F47$$\":nmmmm;/,/$o=6F0$\"(U#zcF47$ $\":+++++v$4'\\d18\"F0$\"(]Rl&F47$Fi^l$\"(.xi&F47$Fc_l$\"(e#=cF47$F]`l $\"(79l&F47$Fg`l$\"(/0t&F47$Faal$\"(P)oeF47$Ffal$\"(skV'F47$F[bl$\"(wD T(F47$$\":++++++++g\\[Q\"F0$\"(#pKzF47$$\":+++++++D,Q4T\"F0$\"(yBJ)F47 $$\":++++++](=A)RU\"F0$\"(Y$3&)F47$$\":+++++++]UEqV\"F0$\"()[n')F47$$ \":++++++D\"G&[NW\"F0$\"(MKh)F47$$\":++++++]7jq+X\"F0$\"(;`])F47$$\":+ ++++]7GoJLX\"F0$\"(Vkg)F47$$\":++++++vVt#fc9F0$\"(-4q)F47$$\":+++++]Pf y`)f9F0$\"(u,])F47$F`bl$\"(c\"e&)F47$$\":nmmmm;HKR'\\5:F0$\"(v)GxF47$F ebl$\"(;%eqF47$$\":nmmmT&Q`01#\\c\"F0$\"(hX(oF47$$\":++++]P4@Ej>d\"F0$ \"(VNp'F47$$\":MLLLe*[o=f+z:F0$\"(U\"fnF47$Fjbl$\"(tQv'F47$$\":++++v$f $=B\"4$f\"F0$\"(;Vq'F47$$\":MLLL$e9T))Q8+;F0$\"(#*)HnF47$$\":nmmm\"zp) \\awrg\"F0$\"(W*4pF47$F_cl$\"(5K-(F47$Ficl$\"(#eh!)F47$F]el$\"(+*H)*F4 7$$\":nmmmmm\"zpYU%p\"F0$\")GB]6F47$Fbel$\")k9U8F47$$\":nmmmmmT55`@u\" F0$\")Vvi:F47$Fgel$\")huPF07$Fafl$\"(Ij1#F07$Fffl$\" ((=f@F07$F[gl$\"(c!=AF07$Fegl$\"(2yB#F07$F_hl$\"(#H_AF07$Fdhl$\"(#fXAF 07$Fihl$\"(3_B#F07$F^il$\"(y>;#F07$Fcil$\"(&*4,#F07$Fhil$\"(u0!=F07$F] jl$\")_yk:F47$F[\\m$\"):GX8F47$Fe\\m$\")3a'=\"F47$$\":nmmmm;a8d3AM#F0$ \")(zB3\"F47$Fj\\m$\"(+V#**F47$Fd]m$\"([q)zF47$Fi]m$\"(XC9(F47$F^^m$\" (-'RlF47$$\":,++++]i!R'f0uzEF0$\"(#=1hF47$Fc^m$\"($GRgF47$$\":MLLLL3F>HT'HFF0$ \"(#R0gF47$Fh^m$\"(Og+'F47$$\":,++++DJq&>Y\"y#F0$\"(#4WgF47$F]_m$\"(r9 6'F47$Fb_m$\"(T#fjF47$Fg_m$\"(-.t'F47$F\\`m$\"(Hfw(F47$Fa`m$\"(MF4*F47 $Ff`m$\")h&e3\"F47$F[am$\")<>Q7F47$$\":MLLLLL$eR%p\")Q$F0$\")#R?O F0$\")6]\\8F47$Fjam$\")>1L8F47$$\":MLLLL$e*[t\\sp$F0$\")#HMH\"F47$F_bm $\")'\\eC\"F47$Fdbm$\")$[b8\"F47$Fibm$\")p)))F47$$\":NLLLLL3-$H**>UF0$\"(%Q3 ()F47$Fhcm$\"(D6h)F47$$\":-++++](=n(*fDVF0$\"(qbe)F47$F]dm$\"(2`i)F47$ $\":MLLLL$e9\"z-lU%F0$\"(jQr)F47$Fbdm$\"(Nj%))F47$Fgdm$\"(\"**z#*F47$F \\em$\"(u2w*F47$Faem$\")uDG5F47$Ffem$\")Yiq5F47$F[fm$\")(p-5\"F4-F`fm6 &FbfmF($\"#l!\"#F(-Fifm6#%6Verner's~Maple~schemeG-F$6%7ctF'7$F+$\"&+=& F07$F2$\"'O*>\"F07$F[hm$\"'_T8F07$Fehm$\"'949F07$Fjhm$\"'tc9F07$F8$\"' 5$[\"F07$Fbim$\"'&RU\"F07$F=$\"'LG8F07$Fjim$\"'vp7F07$FB$\"'MZ6F07$Fbj m$\"'&y5\"F07$FG$\"'r\"3\"F07$FL$\"'nu5F07$FQ$\"'h96F07$FV$\"'#Q=\"F07 $Fen$\"'>u8F07$Fio$\"']pAF07$F]q$\"'SMMF07$Fgq$\"'HZSF07$Fes$\"'P]XF07 $$\":ommmmT5:jQe%[F4$\"'oWYF07$Fjs$\"'z7ZF07$$\":qmmm;a8(*ykh/&F4$\"'z gZF07$$\":+++++D\"yv,%H6&F4$\"'N\"y%F07$$\":NLLL$e*[=c:(z^F4$\"'uHZF07 $F_t$\"'?'p%F07$Fit$\"';kUF07$Fcu$\"'TAMF07$Fhu$\"']0HF07$F]v$\"'?@@F0 7$Fgv$\"'c35F07$Faw$\"'>^@F47$Ffw$!'&>P'F47$F[x$!(%p$*>F47$F`x$!(OdJ$F 47$Fex$!(?w:%F47$$\":+++++]P4@?>y'F4$!(K@F%F47$$\":NLLLLLeR(>(R\"oF4$! ('Rb]F47$$\":qmmmm;zptBg%oF4$!(J^*\\F47$Fjx$!(By.&F47$$\":qmmmmmTg-z@% pF4$!(A8x&F47$F_y$!(zpa'F47$Fdy$!(Oi0(F47$Fiy$!(,/W(F47$F__n$!()*>)zF4 7$Fi_n$!(eW4)F47$F^`n$!(!)H)yF47$F^z$!(7r^(F47$Ff`n$!(Z'3pF47$F[an$!(= 0C'F47$F`an$!(\"HD`F47$Fcz$!(6qB%F47$F][l$!(ggV$F47$Fg[l$!(w`[#F47$F\\ \\l$!(i%f8F47$Fa\\l$!'2-tF47$F[]l$\"'e\"3(F47$Fe]l$\"(0)>8F47$Fjbn$\"( x!)\\\"F47$F_cn$\"(%[c;F47$Fdcn$\"((\\P;F47$Ficn$\"(N6l\"F47$F^dn$\"(l Vu\"F47$Fcdn$\"(>]s\"F47$Fhdn$\"(k8x\"F47$Fj]l$\"(RMx\"F47$F`en$\"(d9j \"F47$F_^l$\"(MiY\"F47$Fhen$\"(l$=8F47$Fd^l$\"(maB\"F47$F`fn$\"(ao?\"F 47$Fefn$\"()*e=\"F47$Fjfn$\"(\"F47$F_cl$!(!oY6F47$Ficl$!'0+fF47$F]el$\"'#)zSF47$F f^o$\"(rnE\"F47$Fbel$\"(XvH#F47$F^_o$\"(8?e$F47$Fgel$\"(;r\\%F47$F\\fl $\"'xxbF07$Fafl$\"'OsjF07$Fffl$\"'fkoF07$F[gl$\"'nrrF07$F`gl$\"'1WsF07 $Fegl$\"',$G(F07$Fjgl$\"'L6tF07$F_hl$\"'b>tF07$Fdhl$\"'vPsF07$Fihl$\"' -WrF07$F^il$\"'].nF07$Fcil$\"']$)fF07$Fhil$\"'!z1&F07$F]jl$\"(fr1%F47$ F[\\m$\"(pi:$F47$Fe\\m$\"(bFg#F47$F`bo$\"(ktN#F47$Fj\\m$\"(R_@#F47$Fd] m$\"(xv#=F47$Fi]m$\"(gpg\"F47$F^^m$\"(sfW\"F47$Ffco$\"(0\"f8F47$Fc^m$ \"(9%=8F47$Fh^m$\"(yMJ\"F47$F]_m$\"(0>M\"F47$Fb_m$\"(7]S\"F47$Fg_m$\"( .Z\\\"F47$F\\`m$\"(GAt\"F47$Fa`m$\"($pF?F47$Ff`m$\"(TMW#F47$F[am$\"(cm \"GF47$Fefo$\"(T3(HF47$F`am$\"(Lv2$F47$Fbgo$\"(Ix7$F47$Feam$\"(ab8$F47 $F^io$\"(o85$F47$Fjam$\"(R&HIF47$F_bm$\"(=f#GF47$Fdbm$\"(^jd#F47$Fibm$ \"(fPM#F47$F^cm$\"(]W9#F47$Fccm$\"(O],#F47$Fhcm$\"(QI&>F47$F]dm$\"('zc >F47$Fbdm$\"(lu+#F47$Fgdm$\"(Rd5#F47$F\\em$\"(%R9AF47$Faem$\"(.FL#F47$ Ffem$\"(Z\"HCF47$F[fm$\"([m\\#F4-%'COLOURG6&Fbfm$\")#)eqk!\")$\"))eqk \"Fd\\rFe\\r-Fifm6#%Ha~modification~of~Verner's~Maple~schemeG-F$6%7[tF '7$F2$\"&:\\#F07$F=$\"&)zdF07$Fen$\"'!*H6F07$F]q$\"'.CCF07$Fgq$\"'F\\F F07$Fes$\"')H+$F07$Fbbp$\"'CSIF07$Fjs$\"'=cIF07$Fjbp$\"',kIF07$F_cp$\" '.YIF07$Fdcp$\"')y+$F07$F_t$\"'m`HF07$Fit$\"'HDDF07$Fcu$\"''zu\"F07$Fh u$\"'Ez7F07$F]v$\"&Z[&F07$Fgv$!&+0&F07$Faw$!(*fQ7F47$Ffw$!(7\"H?F47$F[ x$!(V]K$F47$F`x$!(j$*e%F47$Fex$!(HfP&F47$Fjep$!(woZ&F47$F_fp$!(%\\\\iF 47$Fdfp$!(#zuhF47$Fjx$!(6R?'F47$F\\gp$!(*=9pF47$F_y$!(6)owF47$Fdy$!(w- 8)F47$Fiy$!($eo%)F47$Fe^n$!(?<())F47$F__n$!(***p*)F47$$\":NLLLLL$3_lVG uF4$!(Z&G!*F47$Fd_n$!(!>4*)F47$$\":qmmmmm;/^KZ\\(F4$!(-&>!*F47$Fi_n$!( U5/*F47$F^`n$!(Apy)F47$F^z$!(;]Q)F47$Ff`n$!(Q;u(F47$F[an$!(yk.(F47$F`a n$!(t,4'F47$Fcz$!(jO(\\F47$F][l$!(c59%F47$Fg[l$!(vB;$F47$F\\\\l$!(G2,# F47$Fa\\l$!(=:N\"F47$Ff\\l$!'yAnF47$F[]l$\"'gy9F47$F`]l$\"'6b]F47$Fe]l $\"'uQ#)F47$F_cn$\"(@??\"F47$Fcdn$\"(:MI\"F47$Fhdn$\"(3iQ\"F47$Fj]l$\" (#\\G9F47$F`en$\"(D+N\"F47$F_^l$\"(M!H7F47$Fhen$\"(cL6\"F47$Fd^l$\"(`R /\"F47$Fefn$\"'f&***F47$Fi^l$\"'9\"z*F47$F]`l$\"'D6)*F47$Faal$\"(^k-\" F47$Ffal$\"(c'[6F47$F[bl$\"(1F47$Fc]o$!($)y'>F47$F_cl$!(=<&>F47$Ficl$!(NcY\"F47 $F]el$!'[HaF47$Ff^o$\"'dsCF47$Fbel$\"(y/@\"F47$F^_o$\"(AXV#F47$Fgel$\" (>**G$F47$F\\fl$\"'8@VF07$Fafl$\"'#**3&F07$Fffl$\"'asbF07$F[gl$\"'E#*e F07$F`gl$\"'W')fF07$Fegl$\"'M\\gF07$Fjgl$\"'1#4'F07$F_hl$\"'-F0$\"'q?hF07$Fdhl$\"'!Q4'F07$$\":MLLL$ekyZ'eI'>F0$\"'hsg F07$Fihl$\"'IQgF07$F^il$\"'&zp&F07$Fcil$\"'Ap]F07$Fhil$\"'JJUF07$F]jl$ \"(+qH$F47$F[\\m$\"(/YW#F47$Fe\\m$\"(Fb&>F47$$\":nmmm;/Ev%>J0BF0$\"(z6 *=F47$$\":nmmmmTN@#3h0\"zBF0$\"(RJs\"F47$Fj\\m$\"(\\@s\"F47$Fd]m$ \"(Ia]\"F47$Fi]m$\"(#Q`8F47$F^^m$\"(A7B\"F47$Ffco$\"(u5;\"F47$Fc^m$\"( _K7\"F47$Fh^m$\"(H=6\"F47$F]_m$\"(Fr7\"F47$Fb_m$\"(\\5<\"F47$Fg_m$\"(T 5C\"F47$F\\`m$\"(b*Q9F47$Fa`m$\"(VPp\"F47$Ff`m$\"(iDF47$Fbgo$\"(/Yg#F47$Feam$\"(rJh#F47$F^ io$\"(^pe#F47$Fjam$\"(Y$GDF47$F_bm$\"(SlN#F47$Fdbm$\"(QQ9#F47$Fibm$\"( c#\\>F47$F^cm$\"(Ney\"F47$Fccm$\"(S,o\"F47$Fhcm$\"()oG;F47$F]dm$\"(!*3 j\"F47$Fbdm$\"(CDn\"F47$Fgdm$\"(tYv\"F47$F\\em$\"(]d%=F47$Faem$\"($\\W >F47$Ffem$\"($eC?F47$F[fm$\"(T03#F4-F`\\r6&Fbfm$\"*++++\"Fd\\rF(F(-Fif m6#%6an~\"efficient\"~schemeG-F$6%7duF'7$F2$\"&$4SF07$Fehm$\"&6Q&F07$F 8$\"&r<'F07$$\":nmmmm\"z>6B`#o\"F4$\"&hY'F07$Fbim$\"&ul'F07$$\":+++++v VB:-'>>F4$\"&&F07$Fen$\"&Pb&F07$Fio$\"&7A*F07$F]q$\"'`<;F07$ Fgq$\"'CC?F07$Fes$\"'5CBF07$Fbbp$\"')=O#F07$Fjs$\"'JkBF07$F_cp$\"'M%H# F07$F_t$\"''\\;#F07$Fit$\"'_0;F07$Fcu$\"&V3(F07$Fhu$\"&&>?F07$F]v$!&!3 eF07$Fgv$!')zp\"F07$Faw$!(1*\\CF47$$\":qmmmm;/E+#4piF4$!(5pX#F47$$\":+ +++++DcwV6I'F4$!()y1LF47$$\":NLLLL$ekGb>LjF4$!(A@F$F47$Ffw$!(f\"eKF47$ $\":NLLLLL3x\"3NHkF4$!(6.9%F47$F[x$!(\\th%F47$$\":qmmmmm\"zpybdlF4$!() R#*\\F47$F`x$!(,4'fF47$$\":++++++v=#\\w&o'F4$!(@j5'F47$Fex$!(.9!oF47$F jep$!(i!3pF47$F_fp$!(??y(F47$Fdfp$!(C*)o(F47$Fjx$!(zqr(F47$F\\gp$!(%zQ &)F47$F_y$!(6'\\%*F47$Fdy$!)*QY+\"F47$Fiy$!)BAa5F47$F__n$!)ieS6F47$Fi_ n$!)h=y6F47$$\":++++++](o%G5c(F4$!)h^i6F47$$\":qmmmmm;zWwTf(F4$!)kf!= \"F47$$\":NLLLLL$3FWKFwF4$!)Ux)=\"F47$F^`n$!)iks6F47$$\":qmmmmm;aQ?Op( F4$!)?y\"=\"F47$$\":NLLLLL$ek$ons(F4$!)0)e=\"F47$$\":++++++]PM;*fxF4$! )S`p6F47$F^z$!)=lq6F47$Ff`n$!)'H>9\"F47$F[an$!)*Qt3\"F47$F`an$!)K;?5F4 7$Fcz$!(rmO*F47$F][l$!(Uve)F47$Fg[l$!(1eq(F47$F\\\\l$!(%Q(o'F47$Fa\\l$ !(su'fF47$Ff\\l$!()\\N_F47$F[]l$!(1/S%F47$F`]l$!(D:\"RF47$Fe]l$!(IdY$F 47$Fcdn$!(!pLFF47$Fj]l$!(%QhBF47$$\":nmmm;/EvR^g+\"F0$!(5*pAF47$F`en$! (YiA#F47$$\":MLLL$ekGN8CL5F0$!(8!*>#F47$F_^l$!(\"ew@F47$Fd^l$!(C5?#F47 $Fi^l$!(7U?#F47$F]`l$!(L*)>#F47$Faal$!(n=A#F47$$\":++++++](omax7F0$!(z lE#F47$Ffal$!(]3J#F47$$\":++++++]7o*oJ8F0$!(pfQ#F47$F[bl$!(9E`#F47$Fih n$!($)z8$F47$F`bl$!(YBJ%F47$$\":MLLLLek`h0o[\"F0$!(&\\%)\\F47$Fd[o$!(` v!eF47$$\":+++++v$4rr=M:F0$!(\\U]'F47$Febl$!(tC8(F47$Fa\\o$!(5nP(F47$F jbl$!(Pyj(F47$F^]o$!(QCl(F47$Fc]o$!(.0i(F47$$\":+++](=#*p;_l.;F0$!(8Vs (F47$Fh]o$!(tZ\"yF47$$\":MLL$eRZFtyp5;F0$!(6xk(F47$F_cl$!(B2v(F47$Ficl $!(kCZ(F47$F]el$!(9R*oF47$Fbel$!()=8gF47$Fgel$!(M8!\\F47$F\\fl$!'O#R%F 07$Fafl$!'&42%F07$$\":ommm\"HKknnZG=F0$!')G4%F07$$\":,+++]iSmq;a$=F0$! 'YNSF07$$\":omm;HKRhn'))Q=F0$!'Y?SF07$$\":MLLL3-QckcB%=F0$!'VoRF07$$\" :,++](=n8:m#e%=F0$!'&**)RF07$Fffl$!'KkRF07$$\":MLLL$3-jikSF07$Fihl$!'geTF07$$\":MLLL$ek.HW#)))>F0$!'u' =%F07$$\":+++++D1k;/B+#F0$!'H?UF07$$\":nmmm;/wP!Ry:?F0$!'5CUF07$F^il$! ''3C%F07$$\":ommmmT&)e6Bi0#F0$!';SUF07$Fcil$!'YfUF07$Fhil$!'G\"G%F07$F ]jl$!(!4JVF47$Fgjl$!(Z*fVF47$Fa[m$!($>#Q%F47$Ff[m$!(+QM%F47$F[\\m$!(WD N%F47$F`\\m$!(22E%F47$Fe\\m$!(QA:%F47$F`bo$!(6Sy$F47$Fj\\m$!(!)*RLF47$ Fd]m$!(I.f#F47$Fi]m$!(&*HP#F47$F^^m$!(#fDAF47$Ffco$!(BD9#F47$Fc^m$!(mH 4#F47$Fh^m$!(&px?F47$F]_m$!(UP5#F47$Fb_m$!('=s@F47$Fg_m$!(fZG#F47$$\": ,+++++D\"y#=o'HF0$!(afV#F47$F\\`m$!(![DEF47$Fa`m$!(:x2$F47$Ff`m$!(5`j$ F47$F[am$!(\\U4%F47$Fefo$!(u/H%F47$F`am$!(y]V%F47$F]go$!(Y2[%F47$Fbgo$ !(FP^%F47$F\\ho$!(lD`%F47$Feam$!(['QXF47$Fiho$!(J\"HXF47$F^io$!(9_]%F4 7$Fcio$!(ArY%F47$Fjam$!(YbT%F47$F[jo$!(l)*G%F47$F_bm$!(/_8%F47$Fdbm$!( 1xw$F47$Fibm$!()G?MF47$F^cm$!(3'HJF47$Fccm$!(^`%HF47$Fd[p$!(W)))GF47$F hcm$!(+p&GF47$F\\\\p$!(4$[GF47$F]dm$!(G6'GF47$Fbdm$!(tQ$HF47$Fgdm$!(F! yIF47$F\\em$!(&4QKF47$Faem$!(&36MF47$Ffem$!(+5b$F47$F[fm$!(,!\\OF4-F`f m6&FbfmF($\"#vF^^p$\"#&*F^^p-Fifm6#%Ea~scheme~of~Tsitouras~and~Papakos tasG-F$6%7guF'7$F2$\"&YQ%F07$Fehm$\"&%*y&F07$F8$\"&)[mF07$F_js$\"&t+(F 07$Fbim$\"&qO(F07$Fgjs$\"&2`(F07$F=$\"&)\\wF07$FB$\"&Dv(F07$FG$\"&3\"z F07$FL$\"&y1)F07$FQ$\"&`H)F07$$\":MLLL$ekyA!>Y!HF4$\"&d`)F07$FV$\"&ug) F07$$\":nmmm;/^\"QR#z.$F4$\"&)=*)F07$Fen$\"&\"Q$*F07$Fio$\"'t48F07$F]q $\"'mz=F07$Fgq$\"'R'>#F07$Fes$\"'dECF07$Fbbp$\"'#*\\CF07$Fjs$\"')GW#F0 7$F_cp$\"'urBF07$F_t$\"'9ZAF07$Fit$\"'u6bxF47$F_]u$!(S3'yF47$Fh]o$!(BI&zF47$Fg]u$!(V\\y(F47$F_cl$!( /)*)yF47$Ficl$!(/*)f(F47$F]el$!(TW)pF47$Fbel$!(@%RgF47$Fgel$!(2h([F47$ F\\fl$!'\"GO%F07$Fafl$!'MaSF07$Fa_u$!'cxSF07$Ff_u$!'>DSF07$F``u$!'$\\' RF07$Fffl$!'BkRF07$F]au$!'fMRF07$F[gl$!'J^RF07$F_hl$!'=QSF07$Fihl$!'=T TF07$F[bu$!'@bTF07$F`bu$!'!*pTF07$Febu$!'RiTF07$F^il$!'yiTF07$$\":,+++ ]i:&yLuU?F0$!'(*fTF07$F]cu$!'aQTF07$$\":MLLL$3_D`Gqp?F0$!'G\\TF07$Fcil $!'))QTF07$$\":,++++D\"G)[=g4#F0$!':@TF07$$\":,++++]PfQa)3@F0$!'tFTF07 $$\":,++++v$f$G!p@@F0$!'ynTF07$Fhil$!'TeTF07$$\":,+++]7y+8W49#F0$!'$G? %F07$$\":,++++D1*y?OZ@F0$!'$e:%F07$$\":,+++]PMx-!y`@F0$!(%F47$$\":, ++++]il(z>g@F0$!((HZUF47$$\":,++++v=U(Q.t@F0$!(e3C%F47$F]jl$!(!fKUF47$ Fgjl$!(')*pUF47$Fa[m$!(S/I%F47$Ff[m$!('*yE%F47$F[\\m$!(mIG%F47$F`\\m$! (-))>%F47$Fe\\m$!(SB4%F47$F`bo$!(0kr$F47$Fj\\m$!(x&fKF47$F_]m$!(g4#GF4 7$Fd]m$!(.!*\\#F47$Fi]m$!(FQG#F47$F^^m$!(!)>9#F47$Ffco$!(fK1#F47$Fc^m$ !(9f,#F47$Fh^m$!(v***>F47$F]_m$!(jG-#F47$Fb_m$!(Zn3#F47$Fg_m$!()z%>#F4 7$F^gu$!(Z7M#F47$F\\`m$!(H__#F47$Fa`m$!(;;'HF47$Ff`m$!(4e\\$F47$F[am$! (Ah$RF47$Fefo$!(b\\7%F47$F`am$!(MSE%F47$F]go$!(QyI%F47$Fbgo$!(!GRVF47$ F\\ho$!(brN%F47$Feam$!(DEO%F47$Fiho$!(QKN%F47$F^io$!(%**HVF47$Fcio$!(& H$H%F47$Fjam$!(#yVUF47$F[jo$!(;N7%F47$F_bm$!(od(RF47$Fdbm$!(+Vi$F47$Fi bm$!(,2H$F47$F^cm$!(\"46IF47$Fccm$!(oR$GF47$Fhcm$!(d!\\FF47$F]dm$!(xJv #F47$Fbdm$!(fJ#GF47$Fgdm$!((zhHF47$F\\em$!($o:JF47$Faem$!(&)>G$F47$Ffe m$!(YlT$F47$F[fm$!(M3^$F4-F`\\r6&FbfmF]isF(F]is-Fifm6#%Ra~modification ~of~the~Tsitouras-Papakostas~schemeG-F$6%7huF'7$$\":ommmmm;a8ABO\"F-$! %7UF07$$\":NLLLLLL3FWYs#F-$!%#f(F07$$\":-+++++]iSmp3%F-$!&32\"F07$F+$! &#o5F07$$\":0++++++D\"G$R<)F-$!%&>%F07$F2$\"%ufF07$F8$\"&%)*HF07$F=$\" &Jq%F07$Fen$\"&3&zF07$Fio$\"'fi5F07$F]q$\"'c>8F07$Fgq$\"'(GY\"F07$Fes$ \"'X&f\"F07$Fbbp$\"'_:;F07$Fjs$\"'t>;F07$Fjbp$\"'+:;F07$F_cp$\"':&e\"F 07$Fdcp$\"'/h:F07$F_t$\"'F-:F07$Fit$\"''f4\"F07$Fcu$\"&^y$F07$Fhu$!%pS F07$F]v$!&U)pF07$Fgv$!'8R;F07$Faw$!(wEE#F47$F[_t$!(5iE#F47$F`_t$!(;$oH F47$Fe_t$!(&>PHF47$Ffw$!(NI#HF47$F]`t$!(#[OOF47$F[x$!(X6,%F47$F`x$!(Y> 0&F47$Fex$!((=rcF47$Fjep$!()\\VdF47$F_fp$!(bxQ'F47$Fdfp$!(Q7J'F47$Fjx$ !(,FK'F47$F\\gp$!(bC!pF47$F_y$!(Wk_(F47$Fdy$!(z*))yF47$Fiy$!(=k;)F47$F __n$!(4Nm)F47$Fi_n$!(%3L))F47$F^`n$!(\\RF47$Fj]l$!(Lc?$F47$F_^l$!(c#\\#o\"F0$!(!p')))F47$Ff^o$!([V! *)F47$$\":nmmm;zWnp2up\"F0$!(#[o*)F47$$\":nmmmm\"zpB2R+(*)z)F07$F\\fl$!'# )e#)F07$$\":nmm;z%\\:UqC(z\"F0$!'&HJ)F07$$\":MLLLek`;,<2!=F0$!'X0#)F07 $$\":+++]PM_6)p=/=F0$!'fm#)F07$$\":nmmm;/^1&pl2=F0$!'L8$)F07$$\":++++v V['*)of9=F0$!'9#H)F07$Fafl$!'b!G)F07$Fffl$!'X\"[)F07$F[gl$!'#Qs)F07$Fe gl$!'pV*)F07$F_hl$!'M>\"*F07$Ff^s$!'#\\=*F07$Fdhl$!'bQ#*F07$F^_s$!'Ng# *F07$Fihl$!')>E*F07$F^il$!'1n!*F07$Fcil$!'w!o)F07$Fhil$!'-t#)F07$F]jl$ !(S3$yF47$F[\\m$!(BlE(F47$Fe\\m$!(]y]'F47$F`bo$!(Jqs&F47$Fj\\m$!(**G) \\F47$F_]m$!(GtJ%F47$Fd]m$!(b2\"QF47$Fi]m$!(LCV$F47$F^^m$!(kB<$F47$Ffc o$!(L`-$F47$Fc^m$!(!QVHF47$Fcdo$!(\"QCHF47$Fh^m$!(v=#HF47$F[eo$!([e$HF 47$F]_m$!(d['HF47$Fb_m$!(m82$F47$Fg_m$!(]$QKF47$F\\`m$!(W\\t$F47$Fa`m$ !(Z2S%F47$Ff`m$!(_,B&F47$F[am$!(3)3fF47$Fefo$!(Qe>'F47$F`am$!(pWS'F47$ F]go$!(u#pkF47$Fbgo$!(%R9lF47$Fggo$!(N%HlF47$F\\ho$!(+&RlF47$Faho$!(BY a'F47$Feam$!(+\\a'F47$Fiho$!(c)GlF47$F^io$!(>8\\'F47$Fcio$!(*)RV'F47$F jam$!(:&ejF47$F[jo$!(pi<'F47$F_bm$!(\\U&fF47$Fdbm$!(()*GaF47$Fibm$!(*) 4$\\F47$F^cm$!(nA^%F47$F\\[p$!(@9O%F47$Fccm$!(6fC%F47$Fd[p$!(]U;%F47$F hcm$!(d#=TF47$F\\\\p$!(lg5%F47$F]dm$!(tZ7%F47$Fd\\p$!(\\n;%F47$Fbdm$!( ,)HUF47$Fgdm$!(yrV%F47$F\\em$!(mtm%F47$Faem$!(bk\"\\F47$Ffem$!(g\"=^F4 7$F[fm$!(^$f_F4-F`fm6&FbfmF`]v$\"#bF^^pF(-Fifm6#%AVerner's~\"most~effi cient\"~schemeG-%%FONTG6$%*HELVETICAGFgfm-%&TITLEG6#%Verror~curves~for ~12~stage~order~8~Runge-Kutta~methodsG-%+AXESLABELSG6$Q\"x6\"Q!Fbdz-%% VIEWG6$;F(F[fm;$!#A!#>$\"#CF[ez" 1 2 0 1 10 0 2 9 1 4 2 1.000000 45.000000 45.000000 0 1 "Enright-Verner scheme" "Verner's Maple scheme " "a modification of Verner's Maple scheme" "an \"efficient\" scheme" "a scheme of Tsitouras and Papakostas" "a modification of the Tsitoura s-Papakostas scheme" "Verner's \"most efficient\" scheme" }}}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }} }}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 48 "Test 2 of 12 stage, order 8 Run ge-Kutta methods " }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 " dy/dx=x/y" "6#/*&%#dyG\"\"\"%#dxG!\"\"*&%\"xGF&%\"yGF(" }{TEXT -1 10 " , " }{XPPEDIT 18 0 "y(0)=1" "6#/-%\"yG6#\"\"!\"\"\"" }{TEXT -1 1 " " }}{PARA 0 "" 0 "" {TEXT -1 10 "Solution: " }}{PARA 256 "" 0 " " {TEXT -1 1 " " }{XPPEDIT 18 0 "y=sqrt(1+x^2)" "6#/%\"yG-%%sqrtG6#,& \"\"\"F)*$%\"xG\"\"#F)" }{TEXT -1 2 ". " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 32 "The following code constructs a " } {TEXT 260 17 "discrete solution" }{TEXT -1 44 " based on each of the m ethods and gives the " }{TEXT 260 22 "root mean square error" }{TEXT -1 18 " of each solution." }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 804 "G := (x,y) -> x/y: hh := 0.05: numsteps := 200: x0 := 0: y0 := 1: \nmatrix([[`slope field: `,G(x,y)],[`initial point: `,``(x0,y0)],[`s tep width: `,hh],\n[`no. of steps: `,numsteps]]);``;\nmthds := [`E nright-Verner scheme`,`Verner's Maple scheme`,`a modification of Verne r's Maple scheme`,`an \"efficient\" scheme`,`a scheme of Tsitouras and Papakostas`,`a modification of the Tsitouras-Papakostas scheme`,`Vern er's \"most efficient\" scheme`]: errs := []:\nDigits := 20:\ng := x - > sqrt(1+x^2):\nfor ct to 7 do\n Gn_RK8_||ct := RK8_||ct(G(x,y),x,y, x0,y0,hh,numsteps,false);\n sm := 0: numpts := nops(Gn_RK8_||ct):\n \+ for ii to numpts do\n sm := sm+(Gn_RK8_||ct[ii,2]-g(Gn_RK8_||ct [ii,1]))^2;\n end do:\n errs := [op(errs),sqrt(sm/numpts)];\nend d o:\nDigits := 10:\nlinalg[transpose]([mthds,evalf(errs)]);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#K%'matrixG6#7&7$%0slope~field:~~~G*&%\"xG\" \"\"%\"yG!\"\"7$%0initial~point:~G-%!G6$\"\"!F+7$%/step~width:~~~G$\" \"&!\"#7$%1no.~of~steps:~~~G\"$+#Q(pprint56\"" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%!G" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#K%'matrixG6#7)7$ %6Enright-Verner~schemeG$\"+.EwdC!#F7$%6Verner's~Maple~schemeG$\"+SX`; [!#G7$%Ha~modification~of~Verner's~Maple~schemeG$\"+)R3i;$F07$%6an~\"e fficient\"~schemeG$\"+Y\"=`g%F07$%Ea~scheme~of~Tsitouras~and~Papakosta sG$\"+KN/?>F07$%Ra~modification~of~the~Tsitouras-Papakostas~schemeG$\" +t9F%H#F07$%AVerner's~\"most~efficient\"~schemeG$\"+p$Qno$F0Q(pprint66 \"" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 30 "Th e following code constructs " }{TEXT 260 20 "numerical procedures" } {TEXT -1 56 " for solutions based on each of the Runge-Kutta schemes. " }}{PARA 0 "" 0 "" {TEXT -1 75 "The error in the value obtained by ea ch of the methods at the point where " }{XPPEDIT 18 0 "9.99;" "6#-%&F loatG6$\"$***!\"#" }{TEXT -1 16 " is also given." }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 734 "G := (x,y) -> x/y: hh := 0.05: numsteps := 2 00: x0 := 0: y0 := 1:\nmatrix([[`slope field: `,G(x,y)],[`initial po int: `,``(x0,y0)],[`step width: `,hh],\n[`no. of steps: `,numsteps ]]);``;\nmthds := [`Enright-Verner scheme`,`Verner's Maple scheme`,`a \+ modification of Verner's Maple scheme`,`an \"efficient\" scheme`,`a sc heme of Tsitouras and Papakostas`,`a modification of the Tsitouras-Pap akostas scheme`,`Verner's \"most efficient\" scheme`]: errs := []:\nDi gits := 30:\nfor ct to 7 do\n gn_RK8_||ct := RK8_||ct(G(x,y),x,y,x0, y0,hh,numsteps,true);\nend do:\ng := x -> sqrt(1+x^2):\nxx := 9.99: gx x := evalf(g(xx)):\nfor ct to 7 do\n errs := [op(errs),abs(gn_RK8_|| ct(xx)-gxx)];\nend do:\nDigits := 10:\nlinalg[transpose]([mthds,evalf( errs)]);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#K%'matrixG6#7&7$%0slope~fi eld:~~~G*&%\"xG\"\"\"%\"yG!\"\"7$%0initial~point:~G-%!G6$\"\"!F+7$%/st ep~width:~~~G$\"\"&!\"#7$%1no.~of~steps:~~~G\"$+#Q(pprint76\"" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#%!G" }}{PARA 11 "" 1 "" {XPPMATH 20 "6 #K%'matrixG6#7)7$%6Enright-Verner~schemeG$\"+vz\"*)3(!#G7$%6Verner's~M aple~schemeG$\"+O#yUS\"F+7$%Ha~modification~of~Verner's~Maple~schemeG$ \"*;Y&f))F+7$%6an~\"efficient\"~schemeG$\"+F\"zZK\"F+7$%Ea~scheme~of~T sitouras~and~PapakostasG$\"*4Yg4%F+7$%Ra~modification~of~the~Tsitouras -Papakostas~schemeG$\"*!p:XdF+7$%AVerner's~\"most~efficient\"~schemeG$ \"+)e%eP6F+Q(pprint86\"" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 " " 0 "" {TEXT -1 4 "The " }{TEXT 260 22 "root mean square error" } {TEXT -1 20 " over the interval " }{XPPEDIT 18 0 "[0, 10]" "6#7$\"\"! \"#5" }{TEXT -1 82 " of each Runge-Kutta method is estimated as follo ws using the special procedure " }{TEXT 0 5 "NCint" }{TEXT -1 98 " t o perform numerical integration by the 7 point Newton-Cotes method ove r 100 equal subintervals." }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 501 "mthds := [`Enright-Verner scheme`,`Verner's Maple scheme`,`a modi fication of Verner's Maple scheme`,`an \"efficient\" scheme`,`a scheme of Tsitouras and Papakostas`,`a modification of the Tsitouras-Papakos tas scheme`,`Verner's \"most efficient\" scheme`]: errs := []:\nDigits := 20:\ng := x -> sqrt(1+x^2):\nfor ct to 7 do\n sm := NCint((g(x)- 'gn_RK8_||ct'(x))^2,x=0..10,adaptive=false,numpoints=7,factor=100);\n \+ errs := [op(errs),sqrt(sm/10)];\nend do:\nDigits := 10:\nlinalg[tran spose]([mthds,evalf(errs)]);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#K%'mat rixG6#7)7$%6Enright-Verner~schemeG$\"+bs4aC!#F7$%6Verner's~Maple~schem eG$\"+4q@#y%!#G7$%Ha~modification~of~Verner's~Maple~schemeG$\"+l<*H7$F 07$%6an~\"efficient\"~schemeG$\"+G3huXF07$%Ea~scheme~of~Tsitouras~and~ PapakostasG$\"+*pJw$=F07$%Ra~modification~of~the~Tsitouras-Papakostas~ schemeG$\"+SD%QB#F07$%AVerner's~\"most~efficient\"~schemeG$\"+e#*zhOF0 Q(pprint96\"" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 92 "The following error graphs are constructed using the nume rical procedures for the solutions." }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 617 "evalf[25](plot(['gn_RK8_1'(x)-g(x),'gn_RK8_2'(x)-g(x ),'gn_RK8_3'(x)-g(x),'gn_RK8_4'(x)-g(x),\n'gn_RK8_5'(x)-g(x),'gn_RK8_6 '(x)-g(x),'gn_RK8_7'(x)-g(x)],x=0..10,-1.5e-18..6.7e-18,font=[HELVETIC A,9],\ncolor=[COLOR(RGB,.4,0,.9),COLOR(RGB,0,.65,0),brown,red,COLOR(RG B,0,.75,.95),magenta,COLOR(RGB,.95,.55,0)],\nlegend=[`Enright-Verner s cheme`,`Verner's Maple scheme`,`a modification of Verner's Maple schem e`,`an \"efficient\" scheme`,`a scheme of Tsitouras and Papakostas`,`a modification of the Tsitouras-Papakostas scheme`,`Verner's \"most eff icient\" scheme`],title=`error curves for 12 stage order 8 Runge-Kutta methods`));" }}{PARA 13 "" 1 "" {GLPLOT2D 1052 682 682 {PLOTDATA 2 "6 --%'CURVESG6%7[r7$$\"\"!F)F(7$$\":mmmmmm;a8ABO\"!#E$\"\"\"!#C7$$\":KLL LLLL3FWYs#F-$\"$([F07$$\":)***********\\iSmp3%F-$\"&wy#F07$$\":lmmmmmm ;a)G\\aF-$\"'wx?F07$$\":ILLLLL$3x1h6oF-$\"'lw?F07$$\":+++++++D\"G$R<)F -$\"'cw@F07$$\":lmmmmm;z%\\DO&*F-$\"'J)y%F07$$\":LLLLLLL$3x&)*3\"!#D$ \"'A4()F07$$\":mmmmm;/^\"Q(z:\"FR$\"'%Hq)F07$$\":++++++v=#**3E7FR$\"'I .()F07$$\":LLLLL$ekGg?%H\"FR$\"'aq()F07$$F,FR$\"'??#*F07$$\":+++++](=U #Q/V\"FR$\"(')H7\"F07$$\":LLLLLLe*[Vb)\\\"FR$\"(c%H=F07$$\":mmmmm;HdXq mc\"FR$\"(tI&=F07$$\":+++++++Dc'yM;FR$\"(86&=F07$$\":LLLLL$3FpE!Hq\"FR $\"(6%\\=F07$$\":mmmmmmTgx=5x\"FR$\"(!p^=F07$$\":+++++]7G)[8R=FR$\"(e- )=F07$$\":LLLLLL$e*)4D2>FR$\"(i`-#F07$$\":mmmmm;aj4n`(>FR$\"(H%oDF07$$ \":++++++DJ?$[V?FR$\"(a'zHF07$$\":LLLLL$e*)4$*f6@FR$\"(;c(HF07$$\":mmm mmmmmT:(z@FR$\"(%erHF07$$\":++++]7`p(G)*QAFR$\"(X$pHF07$$\":LLLLLeRsL] #)H#FR$\"('[vHF07$$\":mmmm;/Evz+RsFFR$\"(b81%F07$$\":LLL L$3x\")zulJGFR$\"(]HFR$\"(xfV%F07$$\":LLLLL3x1')f%4IFR$\"(8(H]F07$$\":mmmm;aj4KF(oIF R$\"(F9-&F07$$\":++++++]7y%*z7$FR$\"(EI,&F07$$\":LLLL$ek`TAE(=$FR$\"(O Y+&F07$$\":mmmmm\"H#=qHlC$FR$\"(au*\\F07$$\":++++]P4@;(z0LFR$\"(Qy*\\F 07$$\":LLLLL$eRAY1lLFR$\"(y#H]F07$$\":+++++voHa*f$[$FR$\"(hRb&F07$$\": mmmmmmTNYM@g$FR$\"(ch!z&4UFR$\"('oxhF07$$\":+++++]il(R)GM%FR$\"(J@ ;'F07$$\":LLLLL3F>*))=wWFR$\"(\\hN'F07$$\":mmmmmm\"H2Q\\4YFR$\"(P_W'F0 7$$\":+++++DcEs)zUZFR$\"(tFT'F07$$\":LLLLL$3-QO5w[FR$\"(brR'F07$$\":** ******\\7.d4cF%\\FR$\"(tMV'F07$$\":lmmmmT&Q`&3%4]FR$\"(z!RlF07$$\":ILL L$3x1651w]FR$\"(!e@lF07$$\":++++++](oMrU^FR$\"(%*R]'F07$$\":NLLLLL3-8L fn&FR$\"(\\`V'F07$$\":mmmmmmm;z_\"4iFR$\"()>(G'F07$$\":lmmmmmmm;hEG(FR $\"(V-*eF07$$\":lmmmmmm;aphN)FR$\"(`+\\&F07$$\":++++++](=ddC%*FR$\"(wE 9&F07$$\":LLLLLL$e*=)H\\5F0$\"(0*\\[F07$$\":++++++](=JN[6F0$\"(&oF07$$\":************\\7o7Tv$F0$\"(`>$=F07$$ \":LLLLLLLLQ*o]RF0$\"(Gku\"F07$$\":************\\7=lj;%F0$\"(v5m\"F07$ $\":************\\PaRY2aF0$\"(SUH\"F07$$ \":mmmmmmm\"zXu9cF0$\"(dzC\"F07$$\":***************\\y))GeF0$\"(VM?\"F 07$$\":*************\\i_QQgF0$\"(DG;\"F07$$\":************\\7y%3TiF0$ \"(>g7\"F07$$\":*************\\P![hY'F0$\"(ex3\"F07$$\":KLLLLLLLQx$omF 0$\"(2b0\"F07$$\":**************\\P+V)oF0$\"('4B5F07$$\":mmmmmm;zpe*zq F0$\"'%Q&**F07$$\":**************\\#\\'QH(F0$\"'Qn'*F07$$\":KLLLLL$e9S 8&\\(F0$\"'Q7%*F07$$\":************\\i?=bq(F0$\"'sf\"*F07$$\":KLLLLLL$ 3s?6zF0$\"'OD*)F07$$\":************\\7`Wl7)F0$\"'U#p)F07$$\":lmmmmmmm' *RRL)F0$\"'Az%)F07$$\":lmmmmmmTvJga)F0$\"'mr#)F07$$\":KLLLLL$e9tOc()F0 $\"'cv!)F07$$\":***************\\Qk\\*)F0$\"'L.zF07$$\":KLLLLLL3dg6<*F 0$\"'r9xF07$$\":lmmmmmmmw(Gp$*F0$\"'V`vF07$$\":)***********\\7oK0e*F0$ \"'q)Q(F07$$\":)***********\\(=5s#y*F0$\"'ePsF07$$\"#5F)$\"&>3(!#B-%&C OLORG6&%$RGBG$\"\"%!\"\"F($\"\"*Fc^m-%'LEGENDG6#%6Enright-Verner~schem eG-F$6%7eqF'7$F<$!&A\"GF07$FP$!'X'G\"F07$FV$!'_&G\"F07$Fen$!'d&G\"F07$ Fjn$!'u&H\"F07$F_o$!'7l8F07$Fco$!'U$o\"F07$Fho$!'uNGF07$F]p$!'(\\(GF07 $Fbp$!'$>(GF07$Fgp$!'GpGF07$F\\q$!'xsGF07$Faq$!'S=HF07$Ffq$!':eJF07$F[ r$!'i!4%F07$F`r$!'#\\\"[F07$Fer$!'S3[F07$Fjr$!'(=![F07$F_s$!'>)z%F07$F ds$!':3[F07$Fis$!',!)[F07$F^t$!'!*o^F07$Fct$!'6%4'F07$Fht$!'nQoF07$F]u $!'#*GoF07$Fbu$!'/>oF07$Fgu$!'s4oF07$F\\v$!'I1oF07$Fav$!':MoF07$Ffv$!' A$)pF07$F[w$!'29vF07$F`w$!'@6()F07$Few$!'&op)F07$Fjw$!'H#o)F07$F_x$!'v n')F07$Fdx$!'Ab')F07$Fix$!'9c')F07$F^y$!'$fr)F07$Fcy$!'7>)*F07$Fhy$!(7 A-\"F07$F]z$!(#Q=5F07$Fbz$!(X!=5F07$Fgz$!(Yy1\"F07$F\\[l$!(fa8\"F07$F` \\l$!(7Z?\"F07$F^^l$!(grB\"F07$$\":qmmm;HKk#fO4_FR$!(4QB\"F07$$\":NLLL Le9TQ=gF&FR$!(_0B\"F07$$\":++++](ozT3nU`FR$!(xzA\"F07$$\":qmmmm;z%*HB$ 4aFR$!(%\\G7F07$$\":NLLL$e9;dd(fZ&FR$!(()*R7F07$$\":+++++vV[@GEa&FR$!( d![7F07$$\":qmmm;/EDn!G4cFR$!(CXC\"F07$Fc^l$!(!)4C\"F07$$\":+++++]P4'H aUfFR$!(i1B\"F07$Fh^l$!(%)[A\"F07$F]_l$!(o:;\"F07$Fb_l$!(ry3\"F07$Fg_l $!(e--\"F07$F\\`l$!'%*>'*F07$Fa`l$!'Xc\"*F07$Ff`l$!'hK()F07$F[al$!'x\\ zF07$F`al$!'NQsF07$Feal$!'W@mF07$Fjal$!'*>2'F07$F_bl$!'z\\cF07$Fdbl$!' aL_F07$Fibl$!'[o[F07$F^cl$!')*eXF07$Fccl$!'U3VF07$Fhcl$!'bUSF07$F]dl$! 'qSQF07$Fbdl$!',HOF07$Fgdl$!'ffMF07$F\\el$!'^!H$F07$Fael$!'!Q9$F07$Ffe l$!'s.IF07$F[fl$!'U&)GF07$F`fl$!'jnFF07$Fefl$!')[l#F07$Fjfl$!'#Qc#F07$ F_gl$!'8sCF07$Fdgl$!''RQ#F07$Figl$!'].BF07$F^hl$!'fIAF07$Fchl$!'za@F07 $Fhhl$!'\"44#F07$F]il$!'qE?F07$Fbil$!'!=(>F07$Fgil$!'1:>F07$F\\jl$!'ak =F07$Fajl$!'[9=F07$Ffjl$!'2os\"F07$F`[m$!'pz;F07$Fe[m$!' eQ;F07$Fj[m$!'t*f\"F07$F_\\m$!'hl:F07$Fd\\m$!'DG:F07$Fi\\m$!'I'\\\"F07 $F^]m$!'mj9F07$Fc]m$!'tL9F07$Fh]m$!&GS\"F\\^m-F^^m6&F`^mF($\"#l!\"#F(- Fg^m6#%6Verner's~Maple~schemeG-F$6%7[rF'7$F<$!&tz\"F07$FP$!&3M(F07$FV$ !&bL(F07$Fen$!&^L(F07$Fjn$!&\")Q(F07$F_o$!&>w(F07$Fco$!&J_*F07$Fho$!'( og\"F07$F]p$!'UH;F07$Fbp$!'qF;F07$Fgp$!'=E;F07$F\\q$!'+G;F07$Faq$!'W`; F07$Ffq$!'5\"z\"F07$F[r$!'oTBF07$F`r$!'gwFF07$Fer$!'&Gx#F07$Fjr$!'3pFF 07$F_s$!'#pw#F07$Fds$!'tsFF07$Fis$!'l:GF07$F^t$!'T#*HF07$Fct$!'rsNF07$ Fht$!'JZSF07$F]u$!'bTSF07$Fbu$!'qNSF07$Fgu$!'>ISF07$F\\v$!'IGSF07$Fav$ !'>YSF07$Ffv$!'$H9%F07$F[w$!'?&\\%F07$F`w$!'\"pI&F07$Few$!'<)H&F07$Fjw $!'I*G&F07$F_x$!'W!G&F07$Fdx$!'\"HF&F07$Fix$!'Vu_F07$F^y$!'n;`F07$Fcy$ !'+7hF07$Fhy$!''4T'F07$F]z$!'*pQ'F07$Fbz$!'#))Q'F07$Fgz$!'_ynF07$F\\[l $!'))=tF07$F`\\l$!'\\dzF07$F^^l$!'1U$)F07$Fhgm$!']>$)F07$F]hm$!',)H)F0 7$Fbhm$!'-%G)F07$Fghm$!'I/$)F07$F\\im$!'EZ%)F07$Faim$!'t_&)F07$Ffim$!' _G&)F07$Fc^l$!'C/&)F07$$\":++++]i!*ye&eUdFR$!'3![)F07$$\":qmmmm\"HdX!Q #4eFR$!'Yd%)F07$$\":NLLL$3_D.0*e(eFR$!']V%)F07$F^jm$!'bk%)F07$$\":qmmm ;z>'=a>4gFR$!'@m&)F07$$\":NLLLL3-j(y%e2'FR$!'(4a)F07$$\":++++]P%)RL+D9 'FR$!'p:&)F07$Fh^l$!'P!\\)F07$F]_l$!'-C\")F07$Fb_l$!'UgvF07$Fg_l$!'eup F07$F\\`l$!'gEkF07$Fa`l$!'sIgF07$Ff`l$!':&o&F07$F[al$!'I(3&F07$F`al$!' h(f%F07$Feal$!''4>%F07$Fjal$!'cOQF07$F_bl$!'CnNF07$Fdbl$!'(HI$F07$Fibl $!'3sIF07$F^cl$!'awGF07$Fccl$!'K=FF07$Fhcl$!'_]DF07$F]dl$!'9BCF07$Fbdl $!'b*G#F07$Fgdl$!'m#=#F07$F\\el$!')f2#F07$Fael$!'U$)>F07$Ffel$!'/&*=F0 7$F[fl$!'T?=F07$F`fl$!'4Y 8F07$F]il$!'ky7F07$Fbil$!'+W7F07$Fgil$!'@37F07$F\\jl$!'Lw6F07$Fajl$!'v W6F07$Ffjl$!'Z:6F07$F[[m$!'O'3\"F07$F`[m$!'rf5F07$Fe[m$!'xL5F07$Fj[m$! 'F45F07$F_\\m$!&u()*F07$Fd\\m$!&XF07$Fjr$!'(G^%F07$F_s$!'O4XF07$Fds$!'`=XF07$Fis$!'[&e%F07$F^ t$!'%f&[F07$Fct$!'&\\s&F07$Fht$!'FDkF07$F]u$!'7;kF07$Fbu$!'%oS'F07$Fgu $!'3)R'F07$F\\v$!'\"\\R'F07$Fav$!'\\@kF07$Ffv$!'sjlF07$F[w$!'cqqF07$F` w$!'T8#)F07$Few$!'()*>)F07$Fjw$!':'=)F07$F_x$!'Ws\")F07$Fdx$!'qg\")F07 $Fix$!'5i\")F07$F^y$!'4@#)F07$Fcy$!'C)H*F07$Fhy$!'I\"p*F07$F]z$!'.b'*F 07$Fbz$!'>a'*F07$Fgz$!(Zd,\"F07$F\\[l$!(7N3\"F07$F`\\l$!(#Qd6F07$F^^l$ !(1i>\"F07$Fhgm$!(pH>\"F07$F]hm$!(e)*=\"F07$Fbhm$!(.w=\"F07$Fghm$!(5\" *=\"F07$F\\im$!(#[.7F07$Faim$!(8N@\"F07$Ffim$!(y+@\"F07$Fc^l$!(Lm?\"F0 7$Fc^o$!(*>.7F07$Fh^o$!(H**>\"F07$F]_o$!(Wv>\"F07$F^jm$!(_&)>\"F07$Fe_ o$!(Bp?\"F07$Fj_o$!(nL?\"F07$F_`o$!(/)*>\"F07$Fh^l$!(Oi>\"F07$F]_l$!(W %R6F07$Fb_l$!('zl5F07$Fg_l$!'_R**F07$F\\`l$!'u!H*F07$Fa`l$!'d$z)F07$Ff `l$!'@Z$)F07$F[al$!'/XvF07$F`al$!'E[oF07$Feal$!'?biF07$Fjal$!'#>t&F07$ F_bl$!'tJ`F07$Fdbl$!'*z$\\F07$Fibl$!'A$f%F07$F^cl$!'2,VF07$Fccl$!'hkSF 07$Fhcl$!'v8QF07$F]dl$!'IBOF07$Fbdl$!'eBMF07$Fgdl$!'ujKF07$F\\el$!'B/J F07$Fael$!'#e'HF07$Ffel$!'nLGF07$F[fl$!'3AFF07$F`fl$!'&4h#F07$Fefl$!'e /DF07$Fjfl$!'o=CF07$F_gl$!'=KBF07$Fdgl$!'+\\AF07$Figl$!'4t@F07$F^hl$!' J/@F07$Fchl$!'\"G.#F07$Fhhl$!'as>F07$F]il$!'(>\">F07$Fbil$!'$z$F07$Ffv$!'1oQF07$F[w$!'ySTF07$F `w$!'TkZF07$Few$!'ccZF07$Fjw$!'f[ZF07$Fdx$!'fLZF07$F^y$!'yhZF07$Fcy$!' y=`F07$Fhy$!'CBbF07$F]z$!'Z-bF07$Fbz$!'*z\\&F07$Fgz$!'YMdF07$F\\[l$!'1 fgF07$Fa[l$!'qIgF07$Ff[l$!'e8gF07$F[\\l$!'X@iF07$F`\\l$!'OKjF07$Fe\\l$ !'F+jF07$Fj\\l$!')*ziF07$F_]l$!'k3jF07$Fd]l$!'#QT'F07$Fi]l$!'m'R'F07$F ^^l$!'TzjF07$Fc^l$!'bYiF07$Fh^l$!'7()fF07$$\":lmmmmmm\"zp!fu'FR$!'Q]cF 07$F]_l$!'Yy_F07$$\":lmmmmmmTN:%>yFR$!'C,\\F07$Fb_l$!']PXF07$$\":ILLLL L3-js.*))FR$!'c)>%F07$Fg_l$!''p)QF07$$\":lmmmmm\"H2)y(e**FR$!'%=g$F07$ F\\`l$!'+SLF07$Fa`l$!'SEIF07$Ff`l$!'j!y#F07$F[al$!'/1CF07$F`al$!'&p9#F 07$Feal$!'OY>F07$Fjal$!'Jx=%F0 7$Fh]m$!%\"4%F\\^m-F^^m6&F`^mF($\"#vFicn$\"#&*Ficn-Fg^m6#%Ea~scheme~of ~Tsitouras~and~PapakostasG-F$6%7eqF'7$F<$!&n?#F07$FP$!&FT(F07$FV$!&sS( F07$Fen$!&fS(F07$Fjn$!&-X(F07$F_o$!&Gx(F07$Fco$!&hJ*F07$Fho$!''=^\"F07 $F]p$!''=`\"F07$Fbp$!'CI:F07$Fgp$!'yG:F07$F\\q$!'3I:F07$Faq$!'E^:F07$F fq$!')zm\"F07$F[r$!'%)R@F07$F`r$!'V9DF07$Fer$!'.6DF07$Fjr$!'h2DF07$F_s $!'a0DF07$Fds$!')*4DF07$Fis$!'9XDF07$F^t$!'2#p#F07$Fct$!'UyJF07$Fht$!' rxNF07$F]u$!'isNF07$Fbu$!'XnNF07$Fgu$!'`iNF07$F\\v$!']gNF07$Fav$!'TuNF 07$Ffv$!'Y_OF07$F[w$!'))RRF07$F`w$!'K1YF07$Few$!'t)f%F07$Fjw$!'.\"f%F0 7$F_x$!'L$e%F07$Fdx$!'kwXF07$Fix$!'*pd%F07$F^y$!'R4YF07$Fcy$!'cS_F07$F hy$!';yaF07$F]z$!'hdaF07$Fbz$!'IcaF07$Fgz$!'L]dF07$F\\[l$!'MihF07$F`\\ l$!'e,mF07$F^^l$!'2:oF07$Fhgm$!'f'z'F07$F]hm$!'`ynF07$Fbhm$!'zjnF07$Fg hm$!'OlnF07$F\\im$!'kGoF07$Faim$!'7voF07$Ffim$!'mboF07$Fc^l$!'9OoF07$F ^jm$!':lnF07$Fh^l$!'-5nF07$F]_l$!':&>'F07$Fb_l$!']gbF07$Fg_l$!'ra\\F07 $F\\`l$!'/;WF07$Fa`l$!'YuSF07$Ff`l$!'3$z$F07$F[al$!',QLF07$F`al$!'>(*H F07$Feal$!'ICFF07$Fjal$!'g!\\#F07$F_bl$!'a9BF07$Fdbl$!'SU@F07$Fibl$!'T #*>F07$F^cl$!'[l=F07$Fccl$!'#Gw\"F07$Fhcl$!')Rl\"F07$F]dl$!'Nr:F07$Fbd l$!'s%[\"F07$Fgdl$!'R:9F07$F\\el$!'AY8F07$Fael$!'>'G\"F07$Ffel$!'()G7F 07$F[fl$!'[!=\"F07$F`fl$!'HK6F07$Fefl$!';'3\"F07$Fjfl$!'\"*[5F07$F_gl$ !'R65F07$Fdgl$!&Lv*F07$Figl$!&SU*F07$F^hl$!&d7*F07$Fchl$!&d\"))F07$Fhh l$!&Vb)F07$F]il$!&;H)F07$Fbil$!&p1)F07$Fgil$!&\\$yF07$F\\jl$!&#GwF07$F ajl$!&LU(F07$Ffjl$!&NB(F07$F[[m$!&Z/(F07$F`[m$!&>(oF07$Fe[m$!&Pq'F07$F j[m$!&[a'F07$F_\\m$!&_S'F07$Fd\\m$!&CD'F07$Fi\\m$!&;7'F07$F^]m$!&\"))f F07$Fc]m$!&c'eF07$Fh]m$!%RdF\\^m-Fgio6&F`^mF`_qF(F`_q-Fg^m6#%Ra~modifi cation~of~the~Tsitouras-Papakostas~schemeG-F$6%7jqF'7$F+F(7$F2$!#8F07$ $\":lmmmmmT&Q`!eS$F-$!$<\"F07$F7$!$n(F07$$\":KLLLLLeRZF\"oZF-$!%>QF07$ F<$!%\"G'F07$FA$!%yiF07$FF$!%GnF07$FK$!&$z>F07$FP$!&y2%F07$Fen$!&_2%F0 7$F_o$!&&pVF07$Fco$!&Ni&F07$Fho$!'&R/\"F07$F]p$!'$41\"F07$Fbp$!'\")f5F 07$Ffq$!'3(=\"F07$Fjr$!'d()>F07$F_s$!'3')>F07$Fds$!'(4*>F07$Fis$!'yE?F 07$F^t$!'Vy@F07$Fct$!'K#p#F07$Fht$!'_@JF07$F]u$!'32JF07$Fav$!'fBJF07$Ffv$!'n7KF07$F[w$!'2YNF07$F`w$! 'PNVF07$Few$!'BGVF07$Fjw$!')4K%F07$F_x$!'w8VF07$Fdx$!'u2VF07$Fix$!'H5V F07$F^y$!'(GN%F07$Fcy$!'Kw^F07$Fhy$!'S'\\&F07$F]z$!'!fZ&F07$Fbz$!'^#[& F07$Fgz$!'h?fF07$F\\[l$!'[VlF07$F`\\l$!'o&Q(F07$F^^l$!'U6!)F07$Fc^l$!' lH%)F07$Fh^l$!'lo')F07$$\":lmmmm;/^1ZiF'FR$!'wV')F07$$\":lmmmmmT&Q8MVj FR$!'*ei)F07$$\":lmmmm;z>hN/T'FR$!'iR')F07$$\":lmmmmm;a))HvZ'FR$!'$[w) F07$$\":lmmmm;a)eTiWlFR$!')\\%))F07$$\":lmmmmm\"HK%=EZ[rFR$!'p2))F07$F]_l$!'f_( )F07$Fcjq$!'jn')F07$Fb_l$!'8M&)F07$Fg_l$!'F#>)F07$F\\`l$!'s:yF07$Ff`l$ !'O9rF07$F[al$!'tkkF07$F`al$!'SweF07$Feal$!'6q`F07$Fjal$!'n@\\F07$F_bl $!'EyXF07$Fdbl$!'CSUF07$Fibl$!'?WRF07$F^cl$!'L$p$F07$Fccl$!'F!\\$F07$F hcl$!''[F$F07$F]dl$!'K6JF07$Fbdl$!'#)RHF07$Fgdl$!'d-GF07$F\\el$!'glEF0 7$Fael$!'vYDF07$Ffel$!'FLCF07$F[fl$!'WPBF07$F`fl$!'-UAF07$Fefl$!'o]@F0 7$Fjfl$!'#p2#F07$F_gl$!'j-?F07$Fdgl$!'@J>F07$Figl$!'.m=F07$F^hl$!''p!= F07$Fchl$!'dX$\"#nFg`u" 1 2 0 1 10 0 2 9 1 4 2 1.000000 45.000000 45.000000 0 1 "Enright-Verner scheme" "Verner's Maple scheme" "a modification of V erner's Maple scheme" "an \"efficient\" scheme" "a scheme of Tsitouras and Papakostas" "a modification of the Tsitouras-Papakostas scheme" " Verner's \"most efficient\" scheme" }}}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 549 "evalf[25](plot([g(x)-'gn_ RK8_2'(x),g(x)-'gn_RK8_3'(x),g(x)-'gn_RK8_4'(x),\ng(x)-'gn_RK8_5'(x),g (x)-'gn_RK8_6'(x),g(x)-'gn_RK8_7'(x)],x=0..10,0..1.27e-18,font=[HELVET ICA,9],\ncolor=[COLOR(RGB,0,.65,0),brown,red,COLOR(RGB,0,.75,.95),mage nta,COLOR(RGB,.95,.55,0)],\nlegend=[`Verner's Maple scheme`,`a modific ation of Verner's Maple scheme`,`an \"efficient\" scheme`,`a scheme of Tsitouras and Papakostas`,`a modification of the Tsitouras-Papakostas scheme`,`Verner's \"most efficient\" scheme`],title=`error curves for 12 stage order 8 Runge-Kutta methods`));" }}{PARA 13 "" 1 "" {GLPLOT2D 1018 615 615 {PLOTDATA 2 "6,-%'CURVESG6%7eq7$$\"\"!F)F(7$$\" :lmmmmmm;a)G\\a!#E$\"&A\"G!#C7$$\":LLLLLLL$3x&)*3\"!#D$\"'X'G\"F07$$\" :mmmmm;/^\"Q(z:\"F4$\"'_&G\"F07$$\":++++++v=#**3E7F4$\"'d&G\"F07$$\":L LLLL$ekGg?%H\"F4$\"'u&H\"F07$$\":mmmmmm;a8ABO\"F4$\"'7l8F07$$\":+++++] (=U#Q/V\"F4$\"'U$o\"F07$$\":LLLLLLe*[Vb)\\\"F4$\"'uNGF07$$\":mmmmm;HdX qmc\"F4$\"'(\\(GF07$$\":+++++++Dc'yM;F4$\"'$>(GF07$$\":LLLLL$3FpE!Hq\" F4$\"'GpGF07$$\":mmmmmmTgx=5x\"F4$\"'xsGF07$$\":+++++]7G)[8R=F4$\"'S=H F07$$\":LLLLLL$e*)4D2>F4$\"':eJF07$$\":mmmmm;aj4n`(>F4$\"'i!4%F07$$\": ++++++DJ?$[V?F4$\"'#\\\"[F07$$\":LLLLL$e*)4$*f6@F4$\"'S3[F07$$\":mmmmm mmmT:(z@F4$\"'(=![F07$$\":++++]7`p(G)*QAF4$\"'>)z%F07$$\":LLLLLeRsL]#) H#F4$\"':3[F07$$\":mmmm;/EvzoF07$$\":mmmm ;zW#fD78FF4$\"'s4oF07$$\":+++++DJ&>+RsFF4$\"'I1oF07$$\":LLLL$3x\")zulJ GF4$\"':MoF07$$\":mmmmm;/,%\\#4*GF4$\"'A$)pF07$$\":++++]i!R+C>]HF4$\"' 29vF07$$\":LLLLL3x1')f%4IF4$\"'@6()F07$$\":mmmm;aj4KF(oIF4$\"'&op)F07$ $\":++++++]7y%*z7$F4$\"'H#o)F07$$\":LLLL$ek`TAE(=$F4$\"'vn')F07$$\":mm mmm\"H#=qHlC$F4$\"'Ab')F07$$\":++++]P4@;(z0LF4$\"'9c')F07$$\":LLLLL$eR AY1lLF4$\"'$fr)F07$$\":+++++voHa*f$[$F4$\"'7>)*F07$$\":mmmmmmTNYM@g$F4 $\"(7A-\"F07$$\":LLLLLe9TQp1s$F4$\"(#Q=5F07$$\":+++++](o/V?RQF4$\"(X!= 5F07$$\":mmmmmTg_ARx&RF4$\"(Yy1\"F07$$\":LLLLLLLe9ui2%F4$\"(fa8\"F07$$ \":mmmmmm\"H2Q\\4YF4$\"(7Z?\"F07$$\":++++++](oMrU^F4$\"(grB\"F07$$\":q mmm;HKk#fO4_F4$\"(4QB\"F07$$\":NLLLLe9TQ=gF&F4$\"(_0B\"F07$$\":++++](o zT3nU`F4$\"(xzA\"F07$$\":qmmmm;z%*HB$4aF4$\"(%\\G7F07$$\":NLLL$e9;dd(f Z&F4$\"(()*R7F07$$\":+++++vV[@GEa&F4$\"(d![7F07$$\":qmmm;/EDn!G4cF4$\" (CXC\"F07$$\":NLLLLL3-8Lfn&F4$\"(!)4C\"F07$$\":+++++]P4'HaUfF4$\"(i1B \"F07$$\":mmmmmmm;z_\"4iF4$\"(%)[A\"F07$$\":lmmmmmmm;hEG(F4$\"(o:;\"F0 7$$\":lmmmmmm;aphN)F4$\"(ry3\"F07$$\":++++++](=ddC%*F4$\"(e--\"F07$$\" :LLLLLL$e*=)H\\5F0$\"'%*>'*F07$$\":++++++](=JN[6F0$\"'Xc\"*F07$$\":mmm mmm;z/3uC\"F0$\"'hK()F07$$\":++++++]7LRDX\"F0$\"'x\\zF07$$\":mmmmmm;zR 'ok;F0$\"'NQsF07$$\":++++++]i5`h(=F0$\"'W@mF07$$\":LLLLLLL$3En$4#F0$\" '*>2'F07$$\":mmmmmmmT!RE&G#F0$\"'z\\cF07$$\":+++++++]K]4]#F0$\"'aL_F07 $$\":+++++++]PAvr#F0$\"'[o[F07$$\":+++++++]nHi#HF0$\"')*eXF07$$\":mmmm mm;z*ev:JF0$\"'U3VF07$$\":LLLLLLL$347TLF0$\"'bUSF07$$\":LLLLLLLLjM?`$F 0$\"'qSQF07$$\":************\\7o7Tv$F0$\"',HOF07$$\":LLLLLLLLQ*o]RF0$ \"'ffMF07$$\":************\\7=lj;%F0$\"'^!H$F07$$\":************\\PaR< P%F0$\"'!Q9$F07$$\":LLLLLLLe9Ege%F0$\"'s.IF07$$\":LLLLLL$eR\"3Gy%F0$\" 'U&)GF07$$\":mmmmmmmT5k]*\\F0$\"'jnFF07$$\":mmmmmm;zRQb@&F0$\"')[l#F07 $$\":************\\(=>Y2aF0$\"'#Qc#F07$$\":mmmmmmm\"zXu9cF0$\"'8sCF07$ $\":***************\\y))GeF0$\"''RQ#F07$$\":*************\\i_QQgF0$\"' ].BF07$$\":************\\7y%3TiF0$\"'fIAF07$$\":*************\\P![hY'F 0$\"'za@F07$$\":KLLLLLLLQx$omF0$\"'\"44#F07$$\":**************\\P+V)oF 0$\"'qE?F07$$\":mmmmmm;zpe*zqF0$\"'!=(>F07$$\":**************\\#\\'QH( F0$\"'1:>F07$$\":KLLLLL$e9S8&\\(F0$\"'ak=F07$$\":************\\i?=bq(F 0$\"'[9=F07$$\":KLLLLLL$3s?6zF0$\"'2os\"F07$$\":lmmmmmmm'*RRL)F0$\"'pz;F07$$\":lmmmmmmTvJga)F0$\"'eQ;F 07$$\":KLLLLL$e9tOc()F0$\"'t*f\"F07$$\":***************\\Qk\\*)F0$\"'h l:F07$$\":KLLLLLL3dg6<*F0$\"'DG:F07$$\":lmmmmmmmw(Gp$*F0$\"'I'\\\"F07$ $\":)***********\\7oK0e*F0$\"'mj9F07$$\":)***********\\(=5s#y*F0$\"'tL 9F07$$\"#5F)$\"&GS\"!#B-%&COLORG6&%$RGBGF($\"#l!\"#F(-%'LEGENDG6#%6Ver ner's~Maple~schemeG-F$6%7[rF'7$F+$\"&tz\"F07$F2$\"&3M(F07$F8$\"&bL(F07 $F=$\"&^L(F07$FB$\"&\")Q(F07$FG$\"&>w(F07$FL$\"&J_*F07$FQ$\"'(og\"F07$ FV$\"'UH;F07$Fen$\"'qF;F07$Fjn$\"'=E;F07$F_o$\"'+G;F07$Fdo$\"'W`;F07$F io$\"'5\"z\"F07$F^p$\"'oTBF07$Fcp$\"'gwFF07$Fhp$\"'&Gx#F07$F]q$\"'3pFF 07$Fbq$\"'#pw#F07$Fgq$\"'tsFF07$F\\r$\"'l:GF07$Far$\"'T#*HF07$Ffr$\"'r sNF07$F[s$\"'JZSF07$F`s$\"'bTSF07$Fes$\"'qNSF07$Fjs$\"'>ISF07$F_t$\"'I GSF07$Fdt$\"'>YSF07$Fit$\"'$H9%F07$F^u$\"'?&\\%F07$Fcu$\"'\"pI&F07$Fhu $\"'<)H&F07$F]v$\"'I*G&F07$Fbv$\"'W!G&F07$Fgv$\"'\"HF&F07$F\\w$\"'Vu_F 07$Faw$\"'n;`F07$Ffw$\"'+7hF07$F[x$\"''4T'F07$F`x$\"'*pQ'F07$Fex$\"'#) )Q'F07$Fjx$\"'_ynF07$F_y$\"'))=tF07$Fdy$\"'\\dzF07$Fiy$\"'1U$)F07$F^z$ \"']>$)F07$Fcz$\"',)H)F07$Fhz$\"'-%G)F07$F][l$\"'I/$)F07$Fb[l$\"'EZ%)F 07$Fg[l$\"'t_&)F07$F\\\\l$\"'_G&)F07$Fa\\l$\"'C/&)F07$$\":++++]i!*ye&e UdF4$\"'3![)F07$$\":qmmmm\"HdX!Q#4eF4$\"'Yd%)F07$$\":NLLL$3_D.0*e(eF4$ \"']V%)F07$Ff\\l$\"'bk%)F07$$\":qmmm;z>'=a>4gF4$\"'@m&)F07$$\":NLLLL3- j(y%e2'F4$\"'(4a)F07$$\":++++]P%)RL+D9'F4$\"'p:&)F07$F[]l$\"'P!\\)F07$ F`]l$\"'-C\")F07$Fe]l$\"'UgvF07$Fj]l$\"'eupF07$F_^l$\"'gEkF07$Fd^l$\"' sIgF07$Fi^l$\"':&o&F07$F^_l$\"'I(3&F07$Fc_l$\"'h(f%F07$Fh_l$\"''4>%F07 $F]`l$\"'cOQF07$Fb`l$\"'CnNF07$Fg`l$\"'(HI$F07$F\\al$\"'3sIF07$Faal$\" 'awGF07$Ffal$\"'K=FF07$F[bl$\"'_]DF07$F`bl$\"'9BCF07$Febl$\"'b*G#F07$F jbl$\"'m#=#F07$F_cl$\"')f2#F07$Fdcl$\"'U$)>F07$Ficl$\"'/&*=F07$F^dl$\" 'T?=F07$Fcdl$\"'4Y8F07$F`gl$\"'ky7F07$Fegl$\"'+W7F07$Fjgl$\"'@37F07$F_hl$\"'Lw6F07$Fdh l$\"'vW6F07$Fihl$\"'Z:6F07$F^il$\"'O'3\"F07$Fcil$\"'rf5F07$Fhil$\"'xL5 F07$F]jl$\"'F45F07$Fbjl$\"&u()*F07$Fgjl$\"&XF07$F]q$\"'(G^%F07$Fbq$\"'O4XF07$ Fgq$\"'`=XF07$F\\r$\"'[&e%F07$Far$\"'%f&[F07$Ffr$\"'&\\s&F07$F[s$\"'FD kF07$F`s$\"'7;kF07$Fes$\"'%oS'F07$Fjs$\"'3)R'F07$F_t$\"'\"\\R'F07$Fdt$ \"'\\@kF07$Fit$\"'sjlF07$F^u$\"'cqqF07$Fcu$\"'T8#)F07$Fhu$\"'()*>)F07$ F]v$\"':'=)F07$Fbv$\"'Ws\")F07$Fgv$\"'qg\")F07$F\\w$\"'5i\")F07$Faw$\" '4@#)F07$Ffw$\"'C)H*F07$F[x$\"'I\"p*F07$F`x$\"'.b'*F07$Fex$\"'>a'*F07$ Fjx$\"(Zd,\"F07$F_y$\"(7N3\"F07$Fdy$\"(#Qd6F07$Fiy$\"(1i>\"F07$F^z$\"( pH>\"F07$Fcz$\"(e)*=\"F07$Fhz$\"(.w=\"F07$F][l$\"(5\"*=\"F07$Fb[l$\"(# [.7F07$Fg[l$\"(8N@\"F07$F\\\\l$\"(y+@\"F07$Fa\\l$\"(Lm?\"F07$Fagm$\"(* >.7F07$Ffgm$\"(H**>\"F07$F[hm$\"(Wv>\"F07$Ff\\l$\"(_&)>\"F07$Fchm$\"(B p?\"F07$Fhhm$\"(nL?\"F07$F]im$\"(/)*>\"F07$F[]l$\"(Oi>\"F07$F`]l$\"(W% R6F07$Fe]l$\"('zl5F07$Fj]l$\"'_R**F07$F_^l$\"'u!H*F07$Fd^l$\"'d$z)F07$ Fi^l$\"'@Z$)F07$F^_l$\"'/XvF07$Fc_l$\"'E[oF07$Fh_l$\"'?biF07$F]`l$\"'# >t&F07$Fb`l$\"'tJ`F07$Fg`l$\"'*z$\\F07$F\\al$\"'A$f%F07$Faal$\"'2,VF07 $Ffal$\"'hkSF07$F[bl$\"'v8QF07$F`bl$\"'IBOF07$Febl$\"'eBMF07$Fjbl$\"'u jKF07$F_cl$\"'B/JF07$Fdcl$\"'#e'HF07$Ficl$\"'nLGF07$F^dl$\"'3AFF07$Fcd l$\"'&4h#F07$Fhdl$\"'e/DF07$F]el$\"'o=CF07$Fbel$\"'=KBF07$Fgel$\"'+\\A F07$F\\fl$\"'4t@F07$Fafl$\"'J/@F07$Fffl$\"'\"G.#F07$F[gl$\"'as>F07$F`g l$\"'(>\">F07$Fegl$\"'$z$F07$Fit$\"'1oQF07$F^u$\"'ySTF07$F cu$\"'TkZF07$Fhu$\"'ccZF07$F]v$\"'f[ZF07$Fgv$\"'fLZF07$Faw$\"'yhZF07$F fw$\"'y=`F07$F[x$\"'CBbF07$F`x$\"'Z-bF07$Fex$\"'*z\\&F07$Fjx$\"'YMdF07 $F_y$\"'1fgF07$$\":mmmmm\"z>h!z&4UF4$\"'qIgF07$$\":+++++]il(R)GM%F4$\" 'e8gF07$$\":LLLLL3F>*))=wWF4$\"'X@iF07$Fdy$\"'OKjF07$$\":+++++DcEs)zUZ F4$\"'F+jF07$$\":LLLLL$3-QO5w[F4$\"')*ziF07$$\":********\\7.d4cF%\\F4$ \"'k3jF07$$\":lmmmmT&Q`&3%4]F4$\"'#QT'F07$$\":ILLL$3x1651w]F4$\"'m'R'F 07$Fiy$\"'TzjF07$Fa\\l$\"'bYiF07$F[]l$\"'7()fF07$$\":lmmmmmm\"zp!fu'F4 $\"'Q]cF07$F`]l$\"'Yy_F07$$\":lmmmmmmTN:%>yF4$\"'C,\\F07$Fe]l$\"']PXF0 7$$\":ILLLLL3-js.*))F4$\"'c)>%F07$Fj]l$\"''p)QF07$$\":lmmmmm\"H2)y(e** F4$\"'%=g$F07$F_^l$\"'+SLF07$Fd^l$\"'SEIF07$Fi^l$\"'j!y#F07$F^_l$\"'/1 CF07$Fc_l$\"'&p9#F07$Fh_l$\"'OY>F07$F]`l$\"'Jx=%F07$F[\\m$\"%\"4% F_\\m-Fa\\m6&Fc\\mF($\"#vFf\\m$\"#&*Ff\\m-Fh\\m6#%Ea~scheme~of~Tsitour as~and~PapakostasG-F$6%7eqF'7$F+$\"&n?#F07$F2$\"&FT(F07$F8$\"&sS(F07$F =$\"&fS(F07$FB$\"&-X(F07$FG$\"&Gx(F07$FL$\"&hJ*F07$FQ$\"''=^\"F07$FV$ \"''=`\"F07$Fen$\"'CI:F07$Fjn$\"'yG:F07$F_o$\"'3I:F07$Fdo$\"'E^:F07$Fi o$\"')zm\"F07$F^p$\"'%)R@F07$Fcp$\"'V9DF07$Fhp$\"'.6DF07$F]q$\"'h2DF07 $Fbq$\"'a0DF07$Fgq$\"')*4DF07$F\\r$\"'9XDF07$Far$\"'2#p#F07$Ffr$\"'UyJ F07$F[s$\"'rxNF07$F`s$\"'isNF07$Fes$\"'XnNF07$Fjs$\"'`iNF07$F_t$\"']gN F07$Fdt$\"'TuNF07$Fit$\"'Y_OF07$F^u$\"'))RRF07$Fcu$\"'K1YF07$Fhu$\"'t) f%F07$F]v$\"'.\"f%F07$Fbv$\"'L$e%F07$Fgv$\"'kwXF07$F\\w$\"'*pd%F07$Faw $\"'R4YF07$Ffw$\"'cS_F07$F[x$\"';yaF07$F`x$\"'hdaF07$Fex$\"'IcaF07$Fjx $\"'L]dF07$F_y$\"'MihF07$Fdy$\"'e,mF07$Fiy$\"'2:oF07$F^z$\"'f'z'F07$Fc z$\"'`ynF07$Fhz$\"'zjnF07$F][l$\"'OlnF07$Fb[l$\"'kGoF07$Fg[l$\"'7voF07 $F\\\\l$\"'mboF07$Fa\\l$\"'9OoF07$Ff\\l$\"':lnF07$F[]l$\"'-5nF07$F`]l$ \"':&>'F07$Fe]l$\"']gbF07$Fj]l$\"'ra\\F07$F_^l$\"'/;WF07$Fd^l$\"'YuSF0 7$Fi^l$\"'3$z$F07$F^_l$\"',QLF07$Fc_l$\"'>(*HF07$Fh_l$\"'ICFF07$F]`l$ \"'g!\\#F07$Fb`l$\"'a9BF07$Fg`l$\"'SU@F07$F\\al$\"'T#*>F07$Faal$\"'[l= F07$Ffal$\"'#Gw\"F07$F[bl$\"')Rl\"F07$F`bl$\"'Nr:F07$Febl$\"'s%[\"F07$ Fjbl$\"'R:9F07$F_cl$\"'AY8F07$Fdcl$\"'>'G\"F07$Ficl$\"'()G7F07$F^dl$\" '[!=\"F07$Fcdl$\"'HK6F07$Fhdl$\"';'3\"F07$F]el$\"'\"*[5F07$Fbel$\"'R65 F07$Fgel$\"&Lv*F07$F\\fl$\"&SU*F07$Fafl$\"&d7*F07$Fffl$\"&d\"))F07$F[g l$\"&Vb)F07$F`gl$\"&;H)F07$Fegl$\"&p1)F07$Fjgl$\"&\\$yF07$F_hl$\"&#GwF 07$Fdhl$\"&LU(F07$Fihl$\"&NB(F07$F^il$\"&Z/(F07$Fcil$\"&>(oF07$Fhil$\" &Pq'F07$F]jl$\"&[a'F07$Fbjl$\"&_S'F07$Fgjl$\"&CD'F07$F\\[m$\"&;7'F07$F a[m$\"&\"))fF07$Ff[m$\"&c'eF07$F[\\m$\"%RdF_\\m-Febn6&Fc\\mF^hoF(F^ho- Fh\\m6#%Ra~modification~of~the~Tsitouras-Papakostas~schemeG-F$6%7jqF'7 $$FHF-F(7$$\":KLLLLLL3FWYs#F-$\"#8F07$$\":lmmmmmT&Q`!eS$F-$\"$<\"F07$$ \":)***********\\iSmp3%F-$\"$n(F07$$\":KLLLLLeRZF\"oZF-$\"%>QF07$F+$\" %\"G'F07$$\":ILLLLL$3x1h6oF-$\"%yiF07$$\":+++++++D\"G$R<)F-$\"%GnF07$$ \":lmmmmm;z%\\DO&*F-$\"&$z>F07$F2$\"&y2%F07$F=$\"&_2%F07$FG$\"&&pVF07$ FL$\"&Ni&F07$FQ$\"'&R/\"F07$FV$\"'$41\"F07$Fen$\"'\")f5F07$Fio$\"'3(= \"F07$F]q$\"'d()>F07$Fbq$\"'3')>F07$Fgq$\"'(4*>F07$F\\r$\"'yE?F07$Far$ \"'Vy@F07$Ffr$\"'K#p#F07$F[s$\"'_@JF07$F`s$\"'32JF07$Fdt$\"'fBJF07$Fit$\"'n7KF07$F^u$\"'2YNF07$Fcu $\"'PNVF07$Fhu$\"'BGVF07$F]v$\"')4K%F07$Fbv$\"'w8VF07$Fgv$\"'u2VF07$F \\w$\"'H5VF07$Faw$\"'(GN%F07$Ffw$\"'Kw^F07$F[x$\"'S'\\&F07$F`x$\"'!fZ& F07$Fex$\"'^#[&F07$Fjx$\"'h?fF07$F_y$\"'[VlF07$Fdy$\"'o&Q(F07$Fiy$\"'U 6!)F07$Fa\\l$\"'lH%)F07$F[]l$\"'lo')F07$$\":lmmmm;/^1ZiF'F4$\"'wV')F07 $$\":lmmmmmT&Q8MVjF4$\"'*ei)F07$$\":lmmmm;z>hN/T'F4$\"'iR')F07$$\":lmm mmm;a))HvZ'F4$\"'$[w)F07$$\":lmmmm;a)eTiWlF4$\"')\\%))F07$$\":lmmmmm\" HK%=EZ[rF4$\"'p2))F07$F`]l$\"'f_()F07$Fadp$\"'jn')F07$Fe]l$\"'8M&)F07$Fj] l$\"'F#>)F07$F_^l$\"'s:yF07$Fi^l$\"'O9rF07$F^_l$\"'tkkF07$Fc_l$\"'SweF 07$Fh_l$\"'6q`F07$F]`l$\"'n@\\F07$Fb`l$\"'EyXF07$Fg`l$\"'CSUF07$F\\al$ \"'?WRF07$Faal$\"'L$p$F07$Ffal$\"'F!\\$F07$F[bl$\"''[F$F07$F`bl$\"'K6J F07$Febl$\"'#)RHF07$Fjbl$\"'d-GF07$F_cl$\"'glEF07$Fdcl$\"'vYDF07$Ficl$ \"'FLCF07$F^dl$\"'WPBF07$Fcdl$\"'-UAF07$Fhdl$\"'o]@F07$F]el$\"'#p2#F07 $Fbel$\"'j-?F07$Fgel$\"'@J>F07$F\\fl$\"'.m=F07$Fafl$\"''p!=F07$Fffl$\" 'dX " 0 "" {MPLTEXT 1 0 1 ";" }}}}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 48 "Test 3 of 12 stage, order 8 Runge-Kutta methods " }} {PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "dy/dx = -x*y;" "6#/*& %#dyG\"\"\"%#dxG!\"\",$*&%\"xGF&%\"yGF&F(" }{TEXT -1 11 ", " }{XPPEDIT 18 0 "y(0)=1" "6#/-%\"yG6#\"\"!\"\"\"" }{TEXT -1 1 " " }} {PARA 0 "" 0 "" {TEXT -1 10 "Solution: " }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "y = exp(-x^2/2);" "6#/%\"yG-%$expG6#,$*&%\"xG\" \"#F+!\"\"F," }{TEXT -1 2 ". " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }} {PARA 0 "" 0 "" {TEXT -1 32 "The following code constructs a " }{TEXT 260 17 "discrete solution" }{TEXT -1 44 " based on each of the methods and gives the " }{TEXT 260 22 "root mean square error" }{TEXT -1 18 " of each solution." }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 804 "H := \+ (x,y) -> -x*y: hh := 0.1: numsteps := 100: x0 := 0: y0 := 1:\nmatrix([ [`slope field: `,H(x,y)],[`initial point: `,``(x0,y0)],[`step width: `,hh],\n[`no. of steps: `,numsteps]]);``;\nmthds := [`Enright-Ver ner scheme`,`Verner's Maple scheme`,`a modification of Verner's Maple \+ scheme`,`an \"efficient\" scheme`,`a scheme of Tsitouras and Papakosta s`,`a modification of the Tsitouras-Papakostas scheme`,`Verner's \"mos t efficient\" scheme`]: errs := []:\nDigits := 20:\nh := x -> exp(-x^2 /2):\nfor ct to 7 do\n Hn_RK8_||ct := RK8_||ct(H(x,y),x,y,x0,y0,hh,n umsteps,false);\n sm := 0: numpts := nops(Hn_RK8_||ct):\n for ii t o numpts do\n sm := sm+(Hn_RK8_||ct[ii,2]-h(Hn_RK8_||ct[ii,1]))^2 ;\n end do:\n errs := [op(errs),sqrt(sm/numpts)];\nend do:\nDigits := 10:\nlinalg[transpose]([mthds,evalf(errs)]);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#K%'matrixG6#7&7$%0slope~field:~~~G,$*&%\"xG\"\"\"%\"yGF ,!\"\"7$%0initial~point:~G-%!G6$\"\"!F,7$%/step~width:~~~G$F,F.7$%1no. ~of~steps:~~~G\"$+\"Q)pprint106\"" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#% !G" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#K%'matrixG6#7)7$%6Enright-Verner ~schemeG$\"++zM?R!#B7$%6Verner's~Maple~schemeG$\"+uP\\&y%F+7$%Ha~modif ication~of~Verner's~Maple~schemeG$\"+!>6x#QF+7$%6an~\"efficient\"~sche meG$\"+O&)=*y$F+7$%Ea~scheme~of~Tsitouras~and~PapakostasG$\"+=.T;QF+7$ %Ra~modification~of~the~Tsitouras-Papakostas~schemeG$\"+l!GA.%F+7$%AVe rner's~\"most~efficient\"~schemeG$\"+'QU]+#!#CQ)pprint116\"" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 30 "The following c ode constructs " }{TEXT 260 20 "numerical procedures" }{TEXT -1 56 " f or solutions based on each of the Runge-Kutta schemes." }}{PARA 0 "" 0 "" {TEXT -1 75 "The error in the value obtained by each of the metho ds at the point where " }{XPPEDIT 18 0 "9.99;" "6#-%&FloatG6$\"$***! \"#" }{TEXT -1 16 " is also given." }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 734 "H := (x,y) -> -x*y: hh := 0.1: numsteps := 100: x0 : = 0: y0 := 1:\nmatrix([[`slope field: `,H(x,y)],[`initial point: `,` `(x0,y0)],[`step width: `,hh],\n[`no. of steps: `,numsteps]]);``; \nmthds := [`Enright-Verner scheme`,`Verner's Maple scheme`,`a modific ation of Verner's Maple scheme`,`an \"efficient\" scheme`,`a scheme of Tsitouras and Papakostas`,`a modification of the Tsitouras-Papakostas scheme`,`Verner's \"most efficient\" scheme`]: errs := []:\nDigits := 20:\nfor ct to 7 do\n hn_RK8_||ct := RK8_||ct(H(x,y),x,y,x0,y0,hh,n umsteps,true);\nend do:\nh := x -> exp(-x^2/2):\nxx := 9.99: hxx := ev alf(h(xx)):\nfor ct to 7 do\n errs := [op(errs),abs(hn_RK8_||ct(xx)- hxx)];\nend do:\nDigits := 10:\nlinalg[transpose]([mthds,evalf(errs)]) ;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#K%'matrixG6#7&7$%0slope~field:~~~ G,$*&%\"xG\"\"\"%\"yGF,!\"\"7$%0initial~point:~G-%!G6$\"\"!F,7$%/step~ width:~~~G$F,F.7$%1no.~of~steps:~~~G\"$+\"Q(pprint66\"" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%!G" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#K%'matrixG6 #7)7$%6Enright-Verner~schemeG$\"+GaoOT!#P7$%6Verner's~Maple~schemeG$\" +'\\oa/%F+7$%Ha~modification~of~Verner's~Maple~schemeG$\"+&\\EjW$F+7$% 6an~\"efficient\"~schemeG$\"+IKl)=$F+7$%Ea~scheme~of~Tsitouras~and~Pap akostasG$\"+lo+,NF+7$%Ra~modification~of~the~Tsitouras-Papakostas~sche meG$\"+-'4!\\MF+7$%AVerner's~\"most~efficient\"~schemeG$\"+3<>@h!#QQ(p print76\"" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 4 "The " }{TEXT 260 22 "root mean square error" }{TEXT -1 110 " ove r the interval [0, 0.5] of each Runge-Kutta method is estimated as f ollows using the special procedure " }{TEXT 0 5 "NCint" }{TEXT -1 97 " to perform numerical integration by the 7 point Newton-Cotes method over 50 equal subintervals." }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 500 "mthds := [`Enright-Verner scheme`,`Verner's Maple scheme`,`a modi fication of Verner's Maple scheme`,`an \"efficient\" scheme`,`a scheme of Tsitouras and Papakostas`,`a modification of the Tsitouras-Papakos tas scheme`,`Verner's \"most efficient\" scheme`]: errs := []:\nDigits := 20:\nh := x -> exp(-x^2/2):\nfor ct to 7 do\n sm := NCint((h(x)- 'hn_RK8_||ct'(x))^2,x=0..10,adaptive=false,numpoints=7,factor=50);\n \+ errs := [op(errs),sqrt(sm/10)];\nend do:\nDigits := 10:\nlinalg[trans pose]([mthds,evalf(errs)]);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#K%'matr ixG6#7)7$%6Enright-Verner~schemeG$\"+q8qFO!#B7$%6Verner's~Maple~scheme G$\"+oz!yV%F+7$%Ha~modification~of~Verner's~Maple~schemeG$\"+XK.ZNF+7$ %6an~\"efficient\"~schemeG$\"+2?T8NF+7$%Ea~scheme~of~Tsitouras~and~Pap akostasG$\"+h:qNNF+7$%Ra~modification~of~the~Tsitouras-Papakostas~sche meG$\"+A')yPPF+7$%AVerner's~\"most~efficient\"~schemeG$\"+W@9u=!#CQ)pp rint146\"" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 92 "The following error graphs are constructed using the numerical \+ procedures for the solutions." }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 612 "evalf[20](plot(['hn_RK8_1'(x)-h(x),'hn_RK8_2'(x)-h(x),'hn_RK8_3 '(x)-h(x),'hn_RK8_4'(x)-h(x),\n'hn_RK8_5'(x)-h(x),'hn_RK8_6'(x)-h(x),' hn_RK8_7'(x)-h(x)],x=0..5,numpoints=100,font=[HELVETICA,9],\ncolor=[CO LOR(RGB,.4,0,.9),COLOR(RGB,0,.65,0),brown,red,COLOR(RGB,0,.75,.95),mag enta,COLOR(RGB,.95,.55,0)],\nlegend=[`Enright-Verner scheme`,`Verner's Maple scheme`,`a modification of Verner's Maple scheme`,`an \"efficie nt\" scheme`,`a scheme of Tsitouras and Papakostas`,`a modification of the Tsitouras-Papakostas scheme`,`Verner's \"most efficient\" scheme` ],title=`error curves for 12 stage order 8 Runge-Kutta methods`));" }} {PARA 13 "" 1 "" {GLPLOT2D 1067 728 728 {PLOTDATA 2 "6--%'CURVESG6%7_^ l7$$\"\"!F)F(7$$\"5&eeeeeeeTG&!#@$\"\"'!#?7$$\"5;<<<<S-$F0$\"&#R$\"&)=dF07$$\"5(pppppk:)f5Fhq$\"&UV&F07$$\"5999999Rw66Fhq $\"&?r&F07$$\"5!**)*)*)*)R'o%f6Fhq$\"&4U&F07$$\"5JJJJJJY#4@\"Fhq$\"&mp 'F07$$\"588888)y[wB\"Fhq$\"&:['F07$$\"5&\\\\\\\\\\%HPk7Fhq$\"&!HjF07$$ \"5[[[[[tY+w7Fhq$\"&&okF07$$\"5------kj(G\"Fhq$\"&_@(F07$$\"5cbbbbI\"o #*H\"Fhq$\"&*3(*F07$$\"544444f)**3J\"Fhq$\"&H$)*F07$$\"500000I]-O8Fhq$ \"&>^*F07$$\"5,,,,,,-:h8Fhq$\"&oF*F07$$\"511111\"eGTP\"Fhq$\"&t_*F07$$ \"566666hp5(Q\"Fhq$\"'m66F07$$\"5kjjj8^hf$R\"Fhq$\"'l?8F07$$\"5;;;;;T` 3+9Fhq$\"'b&p\"F07$$\"5oooo=JXd19Fhq$\"'F07$$\"5FFFF- @Ic%\\\"Fhq$\"'QnBF07$$\"5====oIaq+:Fhq$\"'9kHF07$$\"54444MSy%o]\"Fhq$ \"''o$HF07$$\"5+++++]-*H^\"Fhq$\"'w4HF07$$\"5qppppW1FS:Fhq$\"'j\"z#F07 $$\"5RRRRRR5bn:Fhq$\"'LSFF07$$\"5PPPPP(oj?f\"Fhq$\"'+APF07$$\"5NNNNNNj d;;Fhq$\"'w\"*[F07$$\"5>>>>>>)[Fk\"Fhq$\"'/*o%F07$$\"5......8#*o;Fhq$ \"'P5YF07$$\"5QQQQQjuj#p\"Fhq$\"'_(='F07$$\"5uttttBON;\"F07$$\"5\\\\\\\\\\*H.D%=F hq$\"(y'Q6F07$$\"5jjjjj8W+o=Fhq$\"(:v5\"F07$$\"5)))))))))QTOH*=Fhq$\"( !)>U\"F07$$\"5999999%oy\">Fhq$\"(`Ds\"F07$$\"5#GGGGy!*pR%>Fhq$\"(C%Q;F 07$$\"5^^^^^,92q>Fhq$\"(RUf\"F07$$\"5<<<<Fhq$\"(%Q<@F07$$\"5$GG GGG3\\.-#Fhq$\"(h'pBF07$$\"5cbbbbIJ?L?Fhq$\"(u(3BF07$$\"5GGGGGyr0Y?Fhq $\"(D+D#F07$$\"5kkkk9-U[_?Fhq$\"(&*HA#F07$$\"5,,,,,E7\"*e?Fhq$\"(..?#F 07$$\"5QPPP()\\#Q`1#Fhq$\"(cq=#F07$$\"5uttttt_wr?Fhq$\"((>$>#F07$$\"5' ffffM)G^%3#Fhq$\"($=ZBF07$$\"5=====$\\gs4#Fhq$\"(8p+$F07$$\"5'ff4(RWR& ))4#Fhq$\"(j$fJF07$$\"5uttBh&RZ/5#Fhq$\"(\"yzKF07$$\"5_^^w#o%3/-@Fhq$ \"(=)oKF07$$\"5HHHH/)HMO5#Fhq$\"(%)yD$F07$$\"5%[[[t/?@o5#Fhq$\"(*4OKF0 7$$\"5SSSS!H53+6#Fhq$\"(FW@$F07$$\"5^^^^w2>Q;@Fhq$\"(@9<$F07$$\"5iiiii 7dvA@Fhq$\"(o)GJF07$$\"5YYYYYrJ=Y@Fhq$\")&yv(HF-7$$\"5IIIIII1hp@Fhq$\" )Rc&)GF-7$$\"5#>>>>>9hk>#Fhq$\")b\"zy$F-7$$\"5``````;JBAFhq$\"))Hl*RF- 7$$\"5@@@@@rqKZAFhq$\")Ei*y$F-7$$\"5*))))))))))[U8F#Fhq$\")VLrOF-7$$\" 5[[[[)4@XTG#Fhq$\")FGkQF-7$$\"533333Lz%pH#Fhq$\")f\"*yZF-7$$\"5...GMt# [&)H#Fhq$\")KL$*\\F-7$$\"5)zzz/Oh[,I#Fhq$\")w,9_F-7$$\"5$HHzmQ&*[&F-7$$\"5)yyyGTH\\LI#Fhq$\")muv^F-7$$\"5yxxFlu*\\lI#Fhq$\")Mq P^F-7$$\"5onnnR#Fhq$\")uO)>&F-7$$\"5$GGGGyN\"y)R#Fhq$\")W!> 0'F-7$$\"5999R&p9#[+CFhq$\")/GhiF-7$$\"5YXX&zg$H=-CFhq$\")1wNiF-7$$\"5 xww^?DP)QS#Fhq$\")qK5iF-7$$\"53333L9Xe0CFhq$\")*y\\='F-7$$\"5rqq?e#4') *3CFhq$\")'RX8'F-7$$\"5MLLL$3n(Q7CFhq$\")2W%3'F-7$$\"5feeeLF3>>CFhq$\" )9E&)fF-7$$\"5%QQQQQ)R*fU#Fhq$\")_V()eF-7$$\"5WVVVV=\"e&\\CFhq$\")'=Fc &F-7$$\"5.....`A7tCFhq$\")Fx#Q&F-7$$\"5UTTT\"*G'3)zCFhq$\")l$[Y&F-7$$ \"5!)zzzz/]\\'[#Fhq$\")'oCr&F-7$$\"5**)*)*)RF>Q)*[#Fhq$\")*4c$fF-7$$\" 5====o!Q\"=$\\#Fhq$\")AS`iF-7$$\"5PPPPioX_'\\#Fhq$\")3D%p'F-7$$\"5cccc ccx')*\\#Fhq$\")9o$H(F-7$$\"5LLLLL30C8DFhq$\")>C#3(F-7$$\"555555gKhEDF hq$\")RdZoF-7$$\"533333$))))4b#Fhq$\")[IUkF-7$$\"5111111XOvDFhq$\")#ee C'F-7$$\"5++++D\"GI?e#Fhq$\")1ovjF-7$$\"5%RRRRk0'p)e#Fhq$\")%o^s'F-7$$ \"5)yyyG;$=O&f#Fhq$\")UTduF-7$$\"5#====ogF?g#Fhq$\")bW\"G)F-7$$\"5qppp >d\"f`h#Fhq$\")[Q)*zF-7$$\"5ddddd22pGEFhq$\")$fOs(F-7$$\"5HHHHHakp^EFh q$\")+PvsF-7$$\"5,,,,,,AquEFhq$\")i!R.(F-7$$\"5a```.;CD\"o#Fhq$\")i&H9 (F-7$$\"511111JE!yo#Fhq$\")9skuF-7$$\"5KKKKdQx2\"p#Fhq$\")t$4v(F-7$$\" 5eeee3YGN%p#Fhq$\")T/b\")F-7$$\"5%[[[)f`zi(p#Fhq$\")+*4r)F-7$$\"566666 hI!4q#Fhq$\")$)[4#*F-7$$\"5;;;;;\"\\.Sr#Fhq$\")Qc))))F-7$$\"5@@@@@@R5F FFhq$\"):Px&)F-7$$\"5eeeeeeET_FFhq$\")o*4,)F-7$$\"5'fffffR@xx#Fhq$\")[ tfxF-7$$\"5mlll!pWWSy#Fhq$\")lASzF-7$$\"5NNNN&y\\n.z#Fhq$\")7e%Q)F-7$$ \"5/000!)[0p'z#Fhq$\"))[/F*F-7$$\"5uuuuu*f8I!GFhq$\")[%H\"**F-7$$\"599 99k,(fc\"GFhq$\"))[pc*F-7$$\"5`````.eIGGFhq$\")WeJ#*F-7$$\"5'oooo='\\] `GFhq$\"):./')F-7$$\"5??????TqyGFhq$\")$Q'Q$)F-7$$\"5----xXiv%)GFhq$\" )QZK&)F-7$$\"5%QQQQ8P33*GFhq$\").c*)*)F-7$$\"5mlll!p\\go*GFhq$\")f%f() *F-7$$\"5ZZZZZAE\"H!HFhq$\"*3T%\\5F-7$$\"55666hto,:HFhq$\"*QKJ,\"F-7$$ \"5uuuuuC67FHFhq$\")vQz(*F-7$$\"5#====o:\"H`HFhq$\")04l!*F-7$$\"5*)))) ))))))=h%zHFhq$\")))oc()F-7$$\"5=<<*HFhq$\")oL-&*F-7$$\"5gff4A+*\\]*HFhq$\")[)o$**F-7$$\"5uttt[Uw;)*HF hq$\"*AGG0\"F-7$$\"5\"33e?O^E(**HFhq$\"*s=&*3\"F-7$$\"5)yyy`ZQ&G,IFhq$ \"*Z&G#4\"F-7$$\"5&\\\\*p)eDWG+$Fhq$\"*p&=(3\"F-7$$\"5-----FJS/IFhq$\" *62@3\"F-7$$\"5eeee3'4uo,$Fhq$\"*0KA/\"F-7$$\"5:::::l]MHIFhq$\"*luO+\" F-7$$\"5cbbbb!G&fbIFhq$\"*8'z!G*!#A7$$\"5'fffff\\X=3$Fhq$\"*5(4Q!*Fd`n 7$$\"5/////zos$4$Fhq$\"*\")[c#**Fd`n7$$\"577777i#3c5$Fhq$\"+oac$4\"Fd` n7$$\"5?????X'*[$Fhq$\"+*3.?+\"Fd`n7$$\"5DDDDD].02KFhq$\"+%*QD#3\"Fd`n7$$\" 5oooo=1xi>KFhq$\"+lkQR5Fd`n7$$\"577777i]?KKFhq$\"*V%o!)**Fd`n7$$\"5srr rrY&)HdKFhq$\"*G4y@*Fd`n7$$\"5JJJJJJ?R#G$Fhq$\"*WA[&*)Fd`n7$$\"5FFFFF_ r\\&H$Fhq$\"*D#=f**Fd`n7$$\"5BBBBBtAg3LFhq$\"+50'*\\5Fd`n7$$\"5>>>>>%R 2>>>pr5.[$Fhq$\"**HXv\")Fd`n7$$\"5KKKK#[*Gs$ [$Fhq$\"*g`HB)Fd`n7$$\"5ssssss]8([$Fhq$\"*4i-N)Fd`n7$$\"5#HHHHz'RJ$\\$ Fhq$\"*uB3y)Fd`n7$$\"578888jG\\*\\$Fhq$\"*$Q(oj*Fd`n7$$\"5KLLLLewO$*Fd`n7$$\"5%RRRRRW3U_$Fhq$ \"*R[&R*)Fd`n7$$\"5MMMMMMicONFhq$\"*p^\"e&)Fd`n7$$\"5FFFFx*Q/#\\NFhq$ \"*.Ym=)Fd`n7$$\"5?????XD%=c$Fhq$\"*p+6&yFd`n7$$\"5mmmm\"Hih\"oNFhq$\" *'ez:xFd`n7$$\"58888j+2[uNFhq$\"*)fJFwFd`n7$$\"5OOO')[R-kxNFhq$\"*;([6 wFd`n7$$\"5gfffMy(*z!e$Fhq$\"*-HJi(Fd`n7$$\"5$GGG.sJfRe$Fhq$\"*U[4n(Fd `n7$$\"511111c)=re$Fhq$\"*%4,mxFd`n7$$\"5$GGGG`>XNf$Fhq$\"*$>qm\")Fd`n 7$$\"5gffffM:(**f$Fhq$\"*+]2**)Fd`n7$$\"5OOOO'Q(yR1OFhq$\"*\"en!z)Fd`n 7$$\"5888888U#Gh$Fhq$\"*44\"*e)Fd`n7$$\"5mmmmm\"*onDOFhq$\"*&ys)>)Fd`n 7$$\"5?????q&H&QOFhq$\"*-=^#yFd`n7$$\"5XXXXX&zd@m$Fhq$\"*;#y.sFd`n7$$ \"5qqqqq?gy&o$Fhq$\"*j,)eqFd`n7$$\"5????X^F3#p$Fhq$\"*Pa)RtFd`n7$$\"5p ppp>#[z$)p$Fhq$\"*/'zVzFd`n7$$\"5122K)[m`**p$Fhq$\"*V:f;)Fd`n7$$\"5WWW %pv%y_,PFhq$\"*c!*o7)Fd`n7$$\"5\"==ob-.-Jq$Fhq$\"*!Qmz!)Fd`n7$$\"5=>>> %H@wYq$Fhq$\"*i\"pK!)Fd`n7$$\"5$RRR9$yX#yq$Fhq$\"*&e]RzFd`n7$$\"5ooooo VH(4r$Fhq$\"*LBt%yFd`n7$$\"5nnnn<0kcBPFhq$\"+%yL%)[(!#B7$$\"5mmmmmm)fh t$Fhq$\"+'[%)\\9(F[do7$$\"59::::l]:hPFhq$\"+9=6DlF[do7$$\"5jjjjjj-:'y$ Fhq$\"+&\\tSN'F[do7$$\"5'ooooo`MGz$Fhq$\"+5`LGmF[do7$$\"5555555)=&*z$F hq$\"+*fM+C(F[do7$$\"5LLLLL$3.i!QFhq$\"+4%Q>8(F[do7$$\"5cccccct)G\"QFh q$\"+zJi_pF[do7$$\"5-.....fDEQFhq$\"+jq_1mF[do7$$\"5\\\\\\\\\\\\WiRQFh q$\"+))4&oF'F[do7$$\"5wvvvvv(=K'QFhq$\"+')p#Qv&F[do7$$\"5------J\"o)QF hq$\"+!oG4j&F[do7$$\"5mllllS)=O*QFhq$\"+[#3#*)eF[do7$$\"5HHHHHzXU+RFhq $\"+$[kvS'F[do7$$\"5#HHHHzJIs!RFhq$\"+krcRiF[do7$$\"5ccccccg.9RFhq$\"+ qIpvgF[do7$$\"5%QQQQQ`Zw#RFhq$\"+YS&*fdF[do7$$\"5666666!f7%RFhq$\"+`J* *faF[do7$$\"5%\\\\\\\\*>Ba`RFhq$\"+K*Ha?&F[do7$$\"5yyyyyGc#e'RFhq$\"+p j!=)\\F[do7$$\"5qqqq?$Gn>(RFhq$\"+'\\\"z'*[F[do7$$\"5iiiiiP*3\"yRFhq$ \"+2@L[[F[do7$$\"5eeee$[wz6)RFhq$\"+=:]X[F[do7$$\"5aaaa/#f]U)RFhq$\"+< C\"G'[F[do7$$\"5]]]]D>9K()RFhq$\"+VC81\\F[do7$$\"5YYYYYYAR!*RFhq$\"+$e 'z#)\\F[do7$$\"5kkkkk9q+RSFhq$\"+4I,YZF[do7$$\"5333333NG\"4%Fhq$\"+4&o SG%F[do7$$\"5qqqqqq9xVTFhq$\"+X$))o%RF[do7$$\"5&\\\\\\\\\\uF:>%Fhq$\"+ >z_5OF[do7$$\"5'oooooyUAC%Fhq$\"+J)zeJ$F[do7$$\"5666666dt\"H%Fhq$\"+\" )*zb*HF[do7$$\"588888jlQXVFhq$\",PzR(*o#!#C7$$\"5mmmmmm%4=R%Fhq$\",]]6 fW#F\\^p7$$\"544444fy7XWFhq$\",uHQ1=#F\\^p7$$\"5,,,,,^*\\_\\%Fhq$\",XQ 3x,#F\\^p7$$\"5======z)[a%Fhq$\",]J\"*>u\"F\\^p7$$\"5ssssss`c%f%Fhq$\" ,&z7*ye\"F\\^p7$$\"5aaa/#R6Kxf%Fhq$\",RT8vj\"F\\^p7$$\"5OOOO6b))*3g%Fh q$\",(oy!)y;F\\^p7$$\"5===oI'flSg%Fhq$\",*4p^a;F\\^p7$$\"5********\\PB B2YFhq$\",!p5cI;F\\^p7$$\"5ijjj))>ec8YFhq$\",FIPOe\"F\\^p7$$\"5EFFFF-$ **)>YFhq$\",pI-!Q:F\\^p7$$\"5aaaa/nicKYFhq$\",Ilu/X\"F\\^p7$$\"5\"==== =BL_k%Fhq$\",$RN)yO\"F\\^p7$$\"5EFFFF-+meYFhq$\",bq,pG\"F\\^p7$$\"5sss sssn3sYFhq$\",/-E*>7F\\^p7$$\"5XXXX&z:+)yYFhq$\",j*4$))>\"F\\^p7$$\"5= ====VN^&o%Fhq$\",eYnZ>\"F\\^p7$$\"5!4444%GpA#p%Fhq$\",Cx&4?7F\\^p7$$\" 5jjjjj8.%*)p%Fhq$\",_tKZH\"F\\^p7$$\"511111\"QiNs%Fhq$\",P\"=\"\\<\"F \\^p7$$\"5[[[[[[W=[ZFhq$\",MT`e/\"F\\^p7$$\"51222d>qLgZFhq$\"+!\\'p))) *F\\^p7$$\"5lllll!f*[sZFhq$\"+pVJ<%*F\\^p7$$\"5%\\\\\\*>wecyZFhq$\"+Xh Ai#*F\\^p7$$\"5CCCCuh@k%y%Fhq$\"+WHT5#*F\\^p7$$\"5````GZ%=2z%Fhq$\"+We *yK*F\\^p7$$\"5#GGGGGt%z'z%Fhq$\"+\\;%or*F\\^p7$$\"5zzzzzHlxA[Fhq$\",$ yJ2 **\\(Fifp7$$\"5jjjjjjNou[Fhq$\",sUJb7(Fifp7$$\"5MNNN&G([;\")[Fhq$\",'* [V2-(Fifp7$$\"512222#=Yw)[Fhq$\",TIk\\-(Fifp7$$\"5yyyyG\"\\FT*[Fhq$\", AXk:@(Fifp7$$\"5]]]]]+)31!\\Fhq$\",?rG0h(Fifp7$$\"59::::S#yQ#\\Fhq$\", dY&[)y'Fifp7$$\"5zzzzzzw9Z\\Fhq$\",why_Fifp7$$\"5&\\\\\\\\*>py')\\Fhq$\",j[`,E&Fifp7$$\"5[ZZZ(*fMR $*\\Fhq$\",rzvEP&Fifp7$$\"\"&F)$\",rnt0q&Fifp-%&COLORG6&%$RGBG$\"\"%! \"\"F($\"\"*F[\\q-%'LEGENDG6#%6Enright-Verner~schemeG-F$6%7_^lF'7$F+$ \"\"\"F07$F2$\"$W%F07$F7$\"$\"\\F07$F<$!%+DF07$FA$!%*\\#F07$FF$!%L*)F0 7$FK$!%=))F07$FP$!&C'=F07$FU$!&!G=F07$FZ$!&B6$F07$Fin$!&I/$F07$F^o$!&e c%F07$Fco$!&@V%F07$Fho$!&j-'F07$F]p$!&`$eF07$Fbp$!&%*>(F07$Fgp$!&O$pF0 7$F\\q$!&6d(F07$Faq$!&.C(F07$Ffq$!&^P'F07$F\\r$!&(GgF07$Far$!&0a#F07$$ \"5-----xihN6Fhq$!&IZ#F07$Ffr$!&WM#F07$$\"5DDDDvPELs6Fhq$!&<*>F07$$\"5 gggggNm>&=\"Fhq$!%)4'F07$$\"5'ffffMjg!)>\"Fhq$\"&!ySF07$F[s$\"&,H&F07$ F`s$\"&<7&F07$Fes$\"&\")>&F07$Fjs$\"&W,'F07$F_t$\"&4.*F07$$\"5zyyyGmAX $H\"Fhq$\"'RU7F07$Fdt$\"'Q7=F07$$\"5KKKK#[*R308Fhq$\"'r$*=F07$Fit$\"'O z=F07$F^u$\"'5==F07$Fcu$\"';$y\"F07$Fhu$\"'?x=F07$F]v$\"'4hBF07$Fbv$\" 'ZrHF07$Fgv$\"'lWSF07$F\\w$\"')z+%F07$Faw$\"'[rRF07$Ffw$\"'(3$QF07$F[x $\"'\"4v$F07$Fex$\"'&\\t%F07$Fiy$\"'c8qF07$F^z$\"'wGnF07$Fcz$\"'\\#f'F 07$Fhz$\"'St')F07$F][l$\"(_:6\"F07$Fb[l$\"(\\a1\"F07$Fg[l$\"(#*R/\"F07 $F\\\\l$\"(\"zT8F07$Fa\\l$\"(FAl\"F07$Ff\\l$\"(e,e\"F07$F[]l$\"('fQ:F0 7$F`]l$\"(IW#>F07$Fe]l$\"(FXK#F07$Fj]l$\"(d!>AF07$F_^l$\"(CC:#F07$Fd^l $\"(kUl#F07$Fi^l$\"(['GJF07$F^_l$\"(Fd(HF07$Fc_l$\"((H')GF07$Fh_l$\"(j ]n$F07$F]`l$\"(!*>/%F07$Fg`l$\"(@x$QF07$F[bl$\"(/!GPF07$F`bl$\"(%yURF0 7$Febl$\"(+x*[F07$Fjbl$\"($**=^F07$F_cl$\"(+LH&F07$Fdcl$\"(2cF&F07$Fic l$\"(fzD&F07$F^dl$\"(,GA&F07$Fcdl$\"(Cy=&F07$Fhdl$\"(<%=^F07$F]el$\"(Q (\\]F07$Fbel$\")=M0[F-7$Fgel$\"),dYYF-7$$\"566666')e.$=#Fhq$\")\\HN[F- 7$F\\fl$\"))R[)eF-7$$\"5---_*))HR\")>#Fhq$\")#*QRhF-7$$\"57777(eX<)*># Fhq$\")>iMkF-7$$\"5AAAs%Gh&\\,AFhq$\")$\\\"[kF-7$$\"5KKKK#)pP<.AFhq$\" )>OCkF-7$$\"5____x$3Il?#Fhq$\")i*pP'F-7$$\"5sssss(R'))4AFhq$\")z!*HjF- 7$$\"57888jD!*f;AFhq$\")@cOiF-7$Fafl$\"),KWhF-7$Fffl$\")s\"f#eF-7$F[gl $\")w_IcF-7$F`gl$\")ImveF-7$Fegl$\")!Rx4(F-7$Fjgl$\")?Z&Q(F-7$F_hl$\") C[\"o(F-7$Fdhl$\")'\\Kl(F-7$Fihl$\")+5DwF-7$F^il$\")M0pvF-7$Fcil$\")BM 8vF-7$Fhil$\")&=HS(F-7$F]jl$\")O#QH(F-7$Fbjl$\")$36*oF-7$Fgjl$\")R&Qj' F-7$F\\[m$\")&>2q'F-7$Fa[m$\")RxVpF-7$Ff[m$\")YG/vF-7$F[\\m$\")W68')F- 7$F`\\m$\")Fm%)))F-7$Fe\\m$\")/X[))F-7$Fj\\m$\")/O7))F-7$F_]m$\")>Rw() F-7$Fd]m$\")#>[q)F-7$Fi]m$\")*HPj)F-7$F^^m$\")d*H\\)F-7$Fc^m$\")3=a$)F -7$Fh^m$\")u$H*yF-7$F]_m$\")3G?wF-7$Fb_m$\"))>Tr(F-7$Fg_m$\")v%)>!)F-7 $F\\`m$\")h`*H)F-7$Fa`m$\")5()*p)F-7$Ff`m$\")*\\oD*F-7$F[am$\"*5V:+\"F -7$F`am$\")qBA(*F-7$Feam$\")@4+%*F-7$Fjam$\")X9V))F-7$F_bm$\")`%4b)F-7 $Fdbm$\")r6,()F-7$Fibm$\")=WD\"*F-7$F^cm$\"*)pi-5F-7$Fccm$\"*b%)R5\"F- 7$Fhcm$\"*B]i1\"F-7$F]dm$\"*xE'H5F-7$Fbdm$\")Q*zp*F-7$Fgdm$\"):Fc$*F-7 $F\\em$\")$[rZ*F-7$Faem$\")U)z&)*F-7$Ffem$\"*BH,-\"F-7$F[fm$\"*8@)o5F- 7$F`fm$\"*%**)f8\"F-7$Fefm$\"*h.h>\"F-7$Fjfm$\"*!GUa6F-7$F_gm$\"*!e+96 F-7$Fdgm$\"*Tu./\"F-7$Figm$\"*D1]+\"F-7$F^hm$\"*n@a-\"F-7$Fchm$\"*:^u2 \"F-7$Fhhm$\"*WwA=\"F-7$F]im$\"*1lyD\"F-7$Fbim$\"*BhR@\"F-7$Fgim$\"*A1 9<\"F-7$F\\jm$\"*!Qp\"4\"F-7$Fajm$\"**Q;b5F-7$Ffjm$\"*LIo2\"F-7$F[[n$ \"*a1&H6F-7$F`[n$\"*YmEB\"F-7$Fe[n$\"*]/UI\"F-7$Fj[n$\"*Y\"3f7F-7$F_\\ n$\"*0W`@\"F-7$Fd\\n$\"*s,l7\"F-7$Fi\\n$\"*x4`3\"F-7$F^]n$\"*$4r46F-7$ Fc]n$\"*NV%p6F-7$F]^n$\"*)y=(G\"F-7$Fa_n$\"*>z)>8F-7$Ff_n$\"*OU7F\"F-7 $F[`n$\"*W7UA\"F-7$F``n$\"+=.\">8\"Fd`n7$Ff`n$\"+`'=*)4\"Fd`n7$F[an$\" +R**H)>\"Fd`n7$F`an$\"+J%f:J\"Fd`n7$Fean$\"+Cr&RE\"Fd`n7$Fjan$\"+ba\"z @\"Fd`n7$F_bn$\"+'3YI7\"Fd`n7$Fdbn$\"+G!\\f3\"Fd`n7$Fibn$\"+xR&**=\"Fd `n7$F^cn$\"+KaOy7Fd`n7$Fccn$\"+P,tF7Fd`n7$Fhcn$\"+8X#*y6Fd`n7$F]dn$\"+ JOr)3\"Fd`n7$Fbdn$\"+\\L([0\"Fd`n7$Fgdn$\"+Hs+l6Fd`n7$F\\en$\"+\"=2KA \"Fd`n7$Faen$\"+!\\-7<\"Fd`n7$Ffen$\"+!)eB@6Fd`n7$F[fn$\"+z-dO5Fd`n7$F `fn$\"+OX425Fd`n7$Fefn$\"+![OT5\"Fd`n7$Fjfn$\"+L%3\"f6Fd`n7$F_gn$\"+\\ !476\"Fd`n7$Fdgn$\"+tg8l5Fd`n7$Fign$\"+G%>l,\"Fd`n7$F^hn$\"*1#z>(*Fd`n 7$Fchn$\"*IQY`*Fd`n7$Fhhn$\"*w\"42%*Fd`n7$F]in$\"*83)z$*Fd`n7$Fbin$\"* Iy*)Q*Fd`n7$Fgin$\"*K>tW*Fd`n7$F\\jn$\"*;D8d*Fd`n7$Fajn$\"+4#=N+\"Fd`n 7$Ffjn$\"+\"HSl4\"Fd`n7$F[[o$\"++k@&3\"Fd`n7$F`[o$\"+A=%>1\"Fd`n7$Fe[o $\"+UOw;5Fd`n7$Fj[o$\"*`MQt*Fd`n7$F_\\o$\"*S[6J*Fd`n7$Fd\\o$\"*X2%G*)F d`n7$Fi\\o$\"*R-Fx)Fd`n7$F^]o$\"*'R>o')Fd`n7$Fc]o$\"*iZpk)Fd`n7$Fh]o$ \"*y]cl)Fd`n7$F]^o$\"*.pPq)Fd`n7$Fb^o$\"*VFL!))Fd`n7$Fg^o$\"*r:9B*Fd`n 7$F\\_o$\"+I\\(>,\"Fd`n7$Fa_o$\"*kKV*)*Fd`n7$Ff_o$\"*]fum*Fd`n7$F[`o$ \"*6m!G#*Fd`n7$F``o$\"*hOv!))Fd`n7$Fe`o$\"*\"o62\")Fd`n7$Fj`o$\"*qQ5*Fd`n7$F^bo$ \"*8j+1*Fd`n7$Fcbo$\"*b8u+*Fd`n7$Fhbo$\"*xZ]&*)Fd`n7$F]co$\"*#>;^))Fd` n7$Fbco$\"*^%R[()Fd`n7$Fgco$\"+S_H[$)F[do7$F]do$\"+/ISlzF[do7$Fbdo$\"+ o3ftsF[do7$Fgdo$\"+y\\wlqF[do7$F\\eo$\"+4$*o`tF[do7$Faeo$\"+X))G.!)F[d o7$Ffeo$\"+dPK\")yF[do7$F[fo$\"+eq;$o(F[do7$F`fo$\"+dZq+tF[do7$Fefo$\" +XkPOpF[do7$Fjfo$\"+PlcdjF[do7$F_go$\"+#QXr?'F[do7$Fdgo$\"+Y(3qZ'F[do7 $Figo$\"+'>lQ-(F[do7$F^ho$\"+R$4(RoF[do7$Fcho$\"+&Ht+m'F[do7$Fhho$\"+! 3lRJ'F[do7$F]io$\"+%\\R^)fF[do7$Fbio$\"+*ydfq&F[do7$Fgio$\"+\"3.+Y&F[d o7$F\\jo$\"+:))el`F[do7$Fajo$\"+@(=+J&F[do7$Ffjo$\"+\"QX\\I&F[do7$F[[p $\"+lQB@`F[do7$F`[p$\"+e\\4l`F[do7$Fe[p$\"+(y!HWaF[do7$Fj[p$\"+VU\"R;& F[do7$F_\\p$\"+Pr4YYF[do7$Fd\\p$\"+,6,lUF[do7$Fi\\p$\"+cA`*)QF[do7$F^] p$\"+v^XgNF[do7$Fc]p$\"+x3>2KF[do7$Fh]p$\",::#=rGF\\^p7$F^^p$\",(ey$Qg #F\\^p7$Fc^p$\",c+o]J#F\\^p7$Fh^p$\",e!QvM@F\\^p7$F]_p$\",_.E+%=F\\^p7 $Fb_p$\",w'*>An\"F\\^p7$Fg_p$\",wqtGs\"F\\^p7$F\\`p$\",IXY\\w\"F\\^p7$ Fa`p$\",.94%Rkf7F\\^p7$F^cp$\",z(H 7F\\^p7$Fbdp$\",+!yo%4\"F\\^p7$Fgdp$\",,M.].\"F\\^p7$F\\ep$\"+*QAZ&)*F \\^p7$Ffep$\"+C1'4j*F\\^p7$F`fp$\",a(p%Q,\"F\\^p7$Fefp$\",oP.yT*Fifp7$ F[gp$\",y*e.3$)Fifp7$F`gp$\",V0Lw\"yFifp7$Fegp$\",1AubU(Fifp7$Fjgp$\", V<(*QJ(Fifp7$F_hp$\",t'pn8tFifp7$Fdhp$\",g-Y)*\\(Fifp7$Fihp$\",r`LI!zF ifp7$F^ip$\",@j&R\\qFifp7$Fcip$\",G7RdG'Fifp7$Fhip$\",YM6u*eFifp7$F]jp $\",X%pcybFifp7$Fgjp$\",<\"))RcaFifp7$Fa[q$\",$H9_)*eFifp-Ff[q6&Fh[qF( $\"#l!\"#F(-F_\\q6#%6Verner's~Maple~schemeG-F$6%7_^lF'7$F+$\"\"$F07$F2 $\"%&Q\"F07$F7$\"%^:F07$F<$\"%aBF07$FA$\"%ABF07$FF$\"%[CF07$FK$\"%/CF0 7$FP$\"%T=F07$FU$\"%!z\"F07$FZ$\"$z%F07$Fin$\"$b%F07$F^o$!%<;F07$Fco$! %(f\"F07$Fho$!%)4%F07$F]p$!%!)RF07$Fbp$!%!)eF07$Fgp$!%jcF07$F\\q$!%$e% F07$Faq$!%VVF07$Ffq$\"%*3%F07$F\\r$\"%tSF07$Far$\"&ao#F07$Fh`q$\"&Yh#F 07$Ffr$\"&Re#F07$F`aq$\"&;v#F07$Feaq$\"&@f$F07$Fjaq$\"&fg'F07$F[s$\"&C M(F07$F`s$\"&t5(F07$Fes$\"&,.(F07$Fjs$\"&2](F07$F_t$\"&`R*F07$F^cq$\"' pc6F07$Fdt$\"')QF07$Ficl$\"(u*oQF07$F^dl$\"(.J%QF07$Fcdl$\"(mt\"QF07 $Fhdl$\"(%HmPF07$F]el$\"(edr$F07$Fbel$\")*pf`$F-7$Fgel$\")<=@MF-7$F\\f l$\")ZhuVF-7$Fafl$\")&f.e%F-7$Fffl$\")!pIM%F-7$F[gl$\")EK+UF-7$F`gl$\" )>#RR%F-7$Fegl$\")b8V`F-7$Fjgl$\")_EmbF-7$F_hl$\")(Gez&F-7$Fdhl$\")n_u dF-7$Fihl$\")sG`dF-7$F^il$\")*)*4r&F-7$Fcil$\")Q'*ocF-7$Fhil$\")ok&e&F -7$F]jl$\")GL.bF-7$Fbjl$\")la*>&F-7$Fgjl$\")#=(3]F-7$F\\[m$\")?ej]F-7$ Fa[m$\")k,c_F-7$Ff[m$\")'zkp&F-7$F[\\m$\")wellF-7$F`\\m$\")d\\ynF-7$Fe \\m$\")y'3v'F-7$Fj\\m$\")LLBnF-7$F_]m$\")8*ep'F-7$Fd]m$\")aGTmF-7$Fi]m $\")\"[qe'F-7$F^^m$\")gnzkF-7$Fc^m$\")$oPP'F-7$Fh^m$\")k)>-'F-7$F]_m$ \")h:=eF-7$Fb_m$\")(3^*eF-7$Fg_m$\")[/RhF-7$F\\`m$\")()*4O'F-7$Fa`m$\" )X4ymF-7$Ff`m$\")hu=rF-7$F[am$\")yW=xF-7$F`am$\")3;$\\(F-7$Feam$\")k([ C(F-7$Fjam$\")my:oF-7$F_bm$\")/F'f'F-7$Fdbm$\")5#*=nF-7$Fibm$\")RTfqF- 7$F^cm$\"))o'yxF-7$Fccm$\")cl(e)F-7$Fhcm$\")>H)F-7$F]dm$\")NC4!)F-7$ Fbdm$\")j/WvF-7$Fgdm$\")2X$G(F-7$F\\em$\")8v$Q(F-7$Faem$\")V5#p(F-7$Ff em$\")zmozF-7$F[fm$\")j?g$)F-7$F`fm$\")^o**))F-7$Fefm$\")sp#Q*F-7$Fjfm $\")ptb!*F-7$F_gm$\")4pQ()F-7$Fdgm$\")*38;)F-7$Figm$\")5F\"*yF-7$F^hm$ \")#[$f!)F-7$Fchm$\")'*4#[)F-7$Fhhm$\")B@I$*F-7$F]im$\")trU**F-7$Fbim$ \")@o&f*F-7$Fgim$\")%4$f#*F-7$F\\jm$\")XZH')F-7$Fajm$\")hm[$)F-7$Ffjm$ \")oyF&)F-7$F[[n$\")a:e*)F-7$F`[n$\")sc(z*F-7$Fe[n$\"*Yj!Q5F-7$Fj[n$\" *S[@+\"F-7$F_\\n$\")FOt'*F-7$Fd\\n$\")?Ym*)F-7$Fi\\n$\")!pnk)F-7$F^]n$ \")0@\\))F-7$Fc]n$\"))*QR$*F-7$F]^n$\"*S*>I5F-7$Fa_n$\"*Qmr0\"F-7$Ff_n $\"*P5#=5F-7$F[`n$\")`T0)*F-7$F``n$\"*oEk1*Fd`n7$Ff`n$\"*Gw>\"))Fd`n7$ F[an$\"*-s?j*Fd`n7$F`an$\"+e?_c5Fd`n7$Fean$\"+\">w\"=5Fd`n7$Fjan$\"*\" e(3\")*Fd`n7$F_bn$\"*]Zp/*Fd`n7$Fdbn$\"*PMqv)Fd`n7$Fibn$\"*u]'='*Fd`n7 $F^cn$\"+So9N5Fd`n7$Fccn$\"*#HXT**Fd`n7$Fhcn$\"*&\\DY&*Fd`n7$F]dn$\"*F =h\"))Fd`n7$Fbdn$\"*B;/b)Fd`n7$Fgdn$\"*NfcY*Fd`n7$F\\en$\"*&3x^**Fd`n7 $Faen$\"*2r'G&*Fd`n7$Ffen$\"*#)Fd`n7$Ffjn$\"*y#y$**) Fd`n7$F[[o$\"*!z4-*)Fd`n7$F`[o$\"*1v6r)Fd`n7$Fe[o$\"*vv0M)Fd`n7$Fj[o$ \"*dHZ)zFd`n7$F_\\o$\"*P`!QwFd`n7$Fd\\o$\"*A![CtFd`n7$Fi\\o$\"*0kt>(Fd `n7$F^]o$\"*6!)G6(Fd`n7$Fc]o$\"*Wgk4(Fd`n7$Fh]o$\"*vy\\5(Fd`n7$F]^o$\" *f6j9(Fd`n7$Fb^o$\"*2i/B(Fd`n7$Fg^o$\"*6R$*e(Fd`n7$F\\_o$\"*bM5L)Fd`n7 $Fa_o$\"*`?tFd`n7$Fiao$\"*%4E>vFd`n7$F^bo$\"*8bJ[(Fd`n7$Fcbo$\"*Bp'RuFd`n7 $Fhbo$\"*u\\P&F[do7$Figo$\"+<\"*oMeF[do7$F^ho$\"+%y6< o&F[do7$Fcho$\"+K!*[KbF[do7$Fhho$\"+())y\\C&F[do7$F]io$\"+*RJ=(\\F[do7 $Fbio$\"+qF(*RZF[do7$Fgio$\"+Z$pf`%F[do7$F\\jo$\"+Vi'zX%F[do7$Fajo$\"+ ^sg7WF[do7$Ffjo$\"+&e4!4WF[do7$F[[p$\"+WRNBWF[do7$F`[p$\"+Q+&3Y%F[do7$ Fe[p$\"+dg+GXF[do7$Fj[p$\"+0NS+VF[do7$F_\\p$\"+KJ#Q(QF[do7$Fd\\p$\"+gI )*fNF[do7$Fi\\p$\"+$3W-D$F[do7$F^]p$\"+$y_\"yHF[do7$Fc]p$\"+$e=bo#F[do 7$Fh]p$\",Gs]iS#F\\^p7$F^^p$\",$\\xN%=#F\\^p7$Fc^p$\",F\\^p7$Fh^ p$\",B#>P%z\"F\\^p7$F]_p$\",*o[JZ:F\\^p7$Fb_p$\",)[Un29F\\^p7$Fg_p$\", `wx1X\"F\\^p7$F\\`p$\",)y8R'[\"F\\^p7$Fa`p$\",r_%)[Y\"F\\^p7$Ff`p$\",P QuOW\"F\\^p7$F[ap$\",HyG@S\"F\\^p7$F`ap$\",'RUsh8F\\^p7$Feap$\",&4&GUG \"F\\^p7$Fjap$\",#pA567F\\^p7$F_bp$\",L,'QR6F\\^p7$Fdbp$\",$*op*z5F\\^ p7$Fibp$\",!o65h5F\\^p7$F^cp$\",9!Q4d5F\\^p7$F`ct$\",&fgzj5F\\^p7$Fccp $\",%>7py5F\\^p7$Fhct$\",uRhT5\"F\\^p7$Fhcp$\",qs!=V6F\\^p7$F]dp$\",K$ o5P5F\\^p7$Fbdp$\"+Wn\"*\\Fifp-%'COLOURG6&Fh[q$\")#)eqk!\")$\"))eqk\"Fe bxFfbx-F_\\q6#%Ha~modification~of~Verner's~Maple~schemeG-F$6%7]^lF'7$F +$Fj[qF07$F2$\"%%*=F07$F7$\"%B@F07$F<$\"%\"Fhq$\"'nO6F07$Fjaq$\"'M>9F07$$\"5kjjjQKE\\/7Fhq$\"'cK:F 07$F[s$\"'o?:F07$F`s$\"'!>Z\"F07$Fes$\"'[W9F07$Fjs$\"'p*\\\"F07$F_t$\" '8Vk+$F07$F[bl$\"(\"))>HF07$F`bl$\"(\"G'3$F07$Febl$\"('p GQF07$Fjbl$\"(k3+%F07$F_cl$\"(xk8%F07$Fdcl$\"(]E7%F07$Ficl$\"(f)3TF07$ F^dl$\"(&Q\"3%F07$Fcdl$\"(_S0%F07$Fhdl$\"(8)**RF07$F]el$\"(Wh%RF07$Fbe l$\")O:bPF-7$Fgel$\")L#4j$F-7$F\\fl$\")k$pf%F-7$Fafl$\")HP*z%F-7$Fffl$ \")vm]XF-7$F[gl$\")C<)R%F-7$F`gl$\")ED!f%F-7$Fegl$\")\"Qta&F-7$Fjgl$\" )ppsdF-7$F_hl$\"),b/gF-7$Fdhl$\")4[#)fF-7$Fihl$\")lZgfF-7$F^il$\")am;f F-7$Fcil$\")l6teF-7$Fhil$\")!*z'y&F-7$F]jl$\")1_,dF-7$Fbjl$\")&HnQ&F-7 $Fgjl$\")%Rg=&F-7$F\\[m$\")#o)Q_F-7$Fa[m$\")E.IaF-7$Ff[m$\")A\\qeF-7$F [\\m$\")3hTnF-7$F`\\m$\")O'\\&pF-7$Fe\\m$\")khEpF-7$Fj\\m$\")]O)*oF-7$ F_]m$\")(3-(oF-7$Fd]m$\")7=9oF-7$Fi]m$\")>`enF-7$F^^m$\")XO[mF-7$Fc^m$ \")%*pRlF-7$Fh^m$\")ulyhF-7$F]_m$\")V(f'fF-7$Fb_m$\")PRSgF-7$Fg_m$\")L j\"G'F-7$F\\`m$\")*3@]'F-7$Fa`m$\")(zv\"oF-7$Ff`m$\")YRcsF-7$F[am$\")% fR&yF-7$F`am$\")'fTi(F-7$Feam$\")U`rtF-7$Fjam$\")X\"[$pF-7$F_bm$\")T#p q'F-7$Fdbm$\")&zh#oF-7$Fibm$\")G!=;(F-7$F^cm$\")-\\tyF-7$Fccm$\")Q8u') F-7$Fhcm$\")4lx$)F-7$F]dm$\")k*)*3)F-7$Fbdm$\")c()>wF-7$Fgdm$\")xo_tF- 7$F\\em$\")S=\\uF-7$Faem$\")8H^xF-7$Ffem$\")JEB!)F-7$F[fm$\")r!)3%)F-7 $F`fm$\")aYS*)F-7$Fefm$\")[H;%*F-7$Fjfm$\")Q;)3*F-7$F_gm$\")A)*p()F-7$ Fdgm$\")MQ!>)F-7$Figm$\")!\\R\"zF-7$F^hm$\")9sw!)F-7$Fchm$\")e+!\\)F-7 $Fhhm$\")%y:K*F-7$F]im$\")*[8#**F-7$Fbim$\")'f]d*F-7$Fgim$\")$4%R#*F-7 $F\\jm$\")/w5')F-7$Fajm$\")#**[K)F-7$Ffjm$\"))=z\\)F-7$F[[n$\")c1<*)F- 7$F`[n$\")T\"ot*F-7$Fe[n$\"*.T0.\"F-7$Fj[n$\")C')[**F-7$F_\\n$\")LE.'* F-7$Fd\\n$\")oL,*)F-7$Fi\\n$\")^Fy&)F-7$F^]n$\")zOt()F-7$Fc]n$\")>G\\# *F-7$F]^n$\"*\"[h=5F-7$Fa_n$\"*&*zY/\"F-7$Ff_n$\"*1%=15F-7$F[`n$\")3g* o*F-7$F``n$\"*^U\"f*)Fd`n7$Ff`n$\"*[l4q)Fd`n7$F[an$\"*@aS\\*Fd`n7$F`an $\"+#e*pR5Fd`n7$Fean$\"+xU'>+\"Fd`n7$Fjan$\"*ehYl*Fd`n7$F_bn$\"*6>F!*) Fd`n7$Fdbn$\"*k$Q6')Fd`n7$Fibn$\"*LDAW*Fd`n7$F^cn$\"+#z6[,\"Fd`n7$Fccn $\"*-dhu*Fd`n7$Fhcn$\"*\\@(e$*Fd`n7$F]dn$\"*_BFk)Fd`n7$Fbdn$\"*DkmP)Fd `n7$Fgdn$\"*\"=4d#*Fd`n7$F\\en$\"*(>eA(*Fd`n7$Faen$\"*8E#4$*Fd`n7$Ffen $\"*Qr?\"*)Fd`n7$F[fn$\"*eh#R#)Fd`n7$F`fn$\"*'*)Q2!)Fd`n7$Fefn$\"*`QPy )Fd`n7$Fjfn$\"*-yNA*Fd`n7$F_gn$\"*$)>C%))Fd`n7$Fdgn$\"*/'zv%)Fd`n7$Fig n$\"*bY*)3)Fd`n7$F^hn$\"*ubYt(Fd`n7$Fchn$\"*&Qa(e(Fd`n7$Fhhn$\"*N3l[(F d`n7$F]in$\"*:m^Y(Fd`n7$Fbin$\"*;gHZ(Fd`n7$Fgin$\"*FU+_(Fd`n7$F\\jn$\" *'Rd>wFd`n7$Fajn$\"*Lm3*zFd`n7$Ffjn$\"*L%HM()Fd`n7$F[[o$\"*`DVk)Fd`n7$ F`[o$\"*7J*e%)Fd`n7$Fe[o$\"*(H1*4)Fd`n7$Fj[o$\"*r=Nv(Fd`n7$F_\\o$\"*5^ oT(Fd`n7$Fd\\o$\"*kI@6(Fd`n7$Fi\\o$\"*d@$))pFd`n7$F^]o$\"*)p[0pFd`n7$F c]o$\"*cw)))oFd`n7$Fh]o$\"*1-(Fd`n7$Fe`o$\"*#3pjkFd `n7$Fj`o$\"*'RW>jFd`n7$F_ao$\"*:Ntb'Fd`n7$Fdao$\"*!)zN2(Fd`n7$Fiao$\"* N&*QE(Fd`n7$F^bo$\"*e`*GsFd`n7$Fcbo$\"*)[%p=(Fd`n7$Fhbo$\"*ki^9(Fd`n7$ F]co$\"*$HFiqFd`n7$Fbco$\"*iv-)pFd`n7$Fgco$\"+t&R5m'F[do7$F]do$\"+)GMb N'F[do7$Fbdo$\"+7Rm.eF[do7$Fgdo$\"+x#=%RcF[do7$F\\eo$\"+$Hk-(eF[do7$Fa eo$\"+6:0!R'F[do7$Ffeo$\"+o5v#H'F[do7$F[fo$\"+h^`MhF[do7$F`fo$\"+sE;He F[do7$Fefo$\"+.;VY&F[do7$Fcho$\"+cI!3K&F [do7$Fhho$\"+\\OHW]F[do7$F]io$\"+jYf\"y%F[do7$Fbio$\"+=(z&eXF[do7$Fgio $\"+7$4AO%F[do7$F\\jo$\"+sC$pG%F[do7$Fajo$\"+z*)yUUF[do7$Ffjo$\"+0%4*Q UF[do7$F[[p$\"+m:8_UF[do7$F`[p$\"+:!=uG%F[do7$Fe[p$\"+ew'4N%F[do7$Fj[p $\"+$>&fFTF[do7$F_\\p$\"+IO)[r$F[do7$Fd\\p$\"+3!\\0T$F[do7$Fi\\p$\"+\\ ')>6JF[do7$F^]p$\"+S(e\"[GF[do7$Fc]p$\"+]UCmDF[do7$Fh]p$\",>`\\uH#F\\^ p7$F^^p$\",&*e>S3#F\\^p7$Fc^p$\",:`))G&=F\\^p7$Fh^p$\",9\"G%*3IT\"F\\^p7$Fa`p$\",:YuDR\"F\\^p7$Ff`p$\",&*H6CP\"F\\^p7$F[ap$\" ,][;HL\"F\\^p7$F`ap$\",BS1XH\"F\\^p7$Feap$\",S*f$3A\"F\\^p7$Fjap$\",]b =8:\"F\\^p7$F_bp$\",^DLJ3\"F\\^p7$Fdbp$\",3b#fE5F\\^p7$Fibp$\",r+p&35F \\^p7$F^cp$\",QQ%e/5F\\^p7$F`ct$\",q!R\"3,\"F\\^p7$Fccp$\",jDzZ-\"F\\^ p7$Fhct$\",#>$G([5F\\^p7$Fhcp$\",;Nna3\"F\\^p7$F]dp$\"+k%Qk%)*F\\^p7$F bdp$\"+\"HE[w)F\\^p7$Fgdp$\"+].0(G)F\\^p7$F\\ep$\"+\"\\&y!*yF\\^p7$Ffe p$\"+@.?7xF\\^p7$F`fp$\"+R+W=\")F\\^p7$Fefp$\",%Ru#3a(Fifp7$F[gp$\",qI aAl'Fifp7$F`gp$\",*ponfiFifp7$Fegp$\",l*\\.YfFifp7$Fjgp$\",DGNo&eFifp7 $F_hp$\",D^Yo&eFifp7$Fdhp$\",[\"*Qe+'Fifp7$Fihp$\",'ze'zK'Fifp7$F^ip$ \",1HdWk&Fifp7$Fcip$\",QM1I.&Fifp7$Fhip$\",QyF@s%Fifp7$F]jp$\",s@=qY%F ifp7$Fgjp$\",0p$[pVFifp7$Fa[q$\",O7)zAZFifp-Fabx6&Fh[q$\"*++++\"FebxF( F(-F_\\q6#%6an~\"efficient\"~schemeG-F$6%7_^lF'F^cx7$F2$\"%*)=F07$F7$ \"%=@F07$F<$\"%(G&F07$FA$\"%P_F07$FF$\"%]&*F07$FK$\"%2%*F07$FP$\"&H[\" F07$FU$\"&HX\"F07$FZ$\"&G3#F07$Fin$\"&X.#F07$F^o$\"&es#F07$Fco$\"&Bk#F 07$Fho$\"&SP$F07$F]p$\"&gE$F07$Fbp$\"&$oSF07$Fgp$\"&+#RF07$F\\q$\"&y% \\F07$Faq$\"&Mu%F07$Ffq$\"&eL'F07$F\\r$\"&Q/'F07$Far$\"&sy)F07$Fh`q$\" &^b)F07$Ffr$\"&tO)F07$F`aq$\"&oY)F07$Feaq$\"&zH*F07$Fjaq$\"'K\\7F07$F[ s$\"'F@8F07$F`s$\"'')y7F07$Fes$\"'E_7F07$Fjs$\"'y!H\"F07$F_t$\"'Ar9F07 $F^cq$\"'z$o\"F07$Fdt$\"'%p/#F07$Ffcq$\"'E$4#F07$Fit$\"'Rx?F07$F^u$\"' d4?F07$Fcu$\"'Ue>F07$Fhu$\"'2-?F07$F]v$\"'w\"H#F07$Fbv$\"''3n#F07$Fgv$ \"'5YLF07$F\\w$\"'w:LF07$Faw$\"'c&G$F07$Ffw$\"'7pJF07$F[x$\"'1\"4$F07$ Faau$\"'y7JF07$F`x$\"'R'=$F07$Fiau$\"'1bLF07$Fex$\"'X!p$F07$Fjx$\"'G2V F07$F_y$\"'VB_F07$Fdy$\"'Pv^F07$Fiy$\"'gF^F07$F^z$\"'D>\\F07$Fcz$\"'s1 [F07$Fhz$\"'\"39'F07$F][l$\"'>CxF07$Fb[l$\"'p.uF07$Fg[l$\"'2YsF07$F\\ \\l$\"'>C#*F07$Fa\\l$\"(0.8\"F07$Ff\\l$\"((*43\"F07$F[]l$\"(iB0\"F07$F `]l$\"(poJ\"F07$Fe]l$\"(BJf\"F07$Fj]l$\"(R3_\"F07$F_^l$\"(5bZ\"F07$Fd^ l$\"(g'H=F07$Fi^l$\"(lm;#F07$F^_l$\"(v21#F07$Fc_l$\"())***>F07$Fh_l$\" (I2d#F07$F]`l$\"(y'RGF07$Fg`l$\"(*>'p#F07$F[bl$\"(Z8i#F07$F`bl$\"(`7y# F07$Febl$\"(0g[$F07$Fjbl$\"(d$\\OF07$F_cl$\"(u\"yPF07$Fdcl$\"(Ybw$F07$ Ficl$\"(]Hv$F07$F^dl$\"(bys$F07$Fcdl$\"(*)Gq$F07$Fhdl$\"(\\Ll$F07$F]el $\"(GVg$F07$Fbel$\"))H*HMF-7$Fgel$\")Zw=LF-7$F\\fl$\")0x_UF-7$Fafl$\") uMcWF-7$Fffl$\")S[DUF-7$F[gl$\").1(3%F-7$F`gl$\")N(yF%F-7$Fegl$\")l:7_ F-7$Fjgl$\")n*=V&F-7$F_hl$\")Q.ecF-7$Fdhl$\")$Qsj&F-7$Fihl$\")P];cF-7$ F^il$\")4AvbF-7$Fcil$\")_=MbF-7$Fhil$\")!\\GX&F-7$F]jl$\")?\\s`F-7$Fbj l$\")U$f2&F-7$Fgjl$\")$*G!*[F-7$F\\[m$\")%H\\%\\F-7$Fa[m$\"),FN^F-7$Ff [m$\")bdqbF-7$F[\\m$\")-vHkF-7$F`\\m$\")nOSmF-7$Fe\\m$\")=I8mF-7$Fj\\m $\")$Gje'F-7$F_]m$\")cWflF-7$Fd]m$\")C&f]'F-7$Fi]m$\").#GX'F-7$F^^m$\" )ijZjF-7$Fc^m$\")m)QC'F-7$Fh^m$\")))G**eF-7$F]_m$\")(y0q&F-7$Fb_m$\")% Rux&F-7$Fg_m$\")Wk>gF-7$F\\`m$\")$R)RiF-7$Fa`m$\")@RalF-7$Ff`m$\")*\\: *pF-7$F[am$\")\">me(F-7$F`am$\")HVltF-7$Feam$\")3Q@rF-7$Fjam$\")4j*p'F -7$F_bm$\")&=`['F-7$Fdbm$\")!ezg'F-7$Fibm$\")'fq%pF-7$F^cm$\").!Hm(F-7 $Fccm$\")Tbo%)F-7$Fhcm$\")z4z\")F-7$F]dm$\")T;)*yF-7$Fbdm$\");XRuF-7$F gdm$\")K'Q=(F-7$F\\em$\")5r%G(F-7$Faem$\")s%Gf(F-7$Ffem$\")-%*oyF-7$F[ fm$\")Gtf#)F-7$F`fm$\")k<)z)F-7$Fefm$\")y]!G*F-7$Fjfm$\")'3r&*)F-7$F_g m$\")d^V')F-7$Fdgm$\")>Ys!)F-7$Figm$\")mZ2yF-7$F^hm$\")IKwzF-7$Fchm$\" )!G'*R)F-7$Fhhm$\")^E[#*F-7$F]im$\")!p=')*F-7$Fbim$\")el<&*F-7$Fgim$\" )$=S=*F-7$F\\jm$\")WNf&)F-7$Fajm$\")=;$G)F-7$Ffjm$\")OYj%)F-7$F[[n$\") wO&*))F-7$F`[n$\")p;P(*F-7$Fe[n$\"*GeA.\"F-7$Fj[n$\")2Wl**F-7$F_\\n$\" )iE>'*F-7$Fd\\n$\")_O;*)F-7$Fi\\n$\");!4g)F-7$F^]n$\")1-0))F-7$Fc]n$\" )x'yH*F-7$F]^n$\"*=1l-\"F-7$Fa_n$\"*b6P0\"F-7$Ff_n$\"*&G)[,\"F-7$F[`n$ \")7Pt(*F-7$F``n$\"*uko.*Fd`n7$Ff`n$\"*)=F'y)Fd`n7$F[an$\"*'3`7'*Fd`n7 $F`an$\"+>'4`0\"Fd`n7$Fean$\"+dx+<5Fd`n7$Fjan$\"*A<'*z*Fd`n7$F_bn$\"*K Gm.*Fd`n7$Fdbn$\"*)H!)\\()Fd`n7$Fibn$\"*UC#>'*Fd`n7$F^cn$\"+%pcf.\"Fd` n7$Fccn$\"*oI#\\**Fd`n7$Fhcn$\"*#Qs`&*Fd`n7$F]dn$\"*#=4B))Fd`n7$Fbdn$ \"*qg(f&)Fd`n7$Fgdn$\"*6ZW[*Fd`n7$F\\en$\"*M4q(**Fd`n7$Faen$\"*bOGb*Fd `n7$Ffen$\"*F\"HX\"*Fd`n7$F[fn$\"*\\*>b%)Fd`n7$F`fn$\"*W7]A)Fd`n7$Fefn $\"*j[M/*Fd`n7$Fjfn$\"*7t+^*Fd`n7$F_gn$\"*svq6*Fd`n7$Fdgn$\"*Ml!R()Fd` n7$Fign$\"*GG-M)Fd`n7$F^hn$\"*tF_(zFd`n7$Fchn$\"*Q2T#yFd`n7$Fhhn$\"*BX 7s(Fd`n7$F]in$\"*!4S+xFd`n7$Fbin$\"*]?,r(Fd`n7$Fgin$\"*/q5w(Fd`n7$F\\j n$\"*k\"3nyFd`n7$Fajn$\"*)Hrf#)Fd`n7$Ffjn$\"*N#QV!*Fd`n7$F[[o$\"*J)p^* )Fd`n7$F`[o$\"*q6(f()Fd`n7$Fe[o$\"*^ZqQ)Fd`n7$Fj[o$\"*P>#H!)Fd`n7$F_\\ o$\"*+B1o(Fd`n7$Fd\\o$\"*-(RltFd`n7$Fi\\o$\"*wNxB(Fd`n7$F^]o$\"*SUJ:(F d`n7$Fc]o$\"*6Tp8(Fd`n7$Fh]o$\"*tZf9(Fd`n7$F]^o$\"*$*H\")=(Fd`n7$Fb^o$ \"*<3OF(Fd`n7$Fg^o$\"*:Etj(Fd`n7$F\\_o$\"*.L$)Q)Fd`n7$Fa_o$\"*&Rc,#)Fd `n7$Ff_o$\"*M0N,)Fd`n7$F[`o$\"*g&G\\wFd`n7$F``o$\"*842I(Fd`n7$Fe`o$\"* [71s'Fd`n7$Fj`o$\"*3+pd'Fd`n7$F_ao$\"*2N/$oFd`n7$Fdao$\"*!>=ytFd`n7$Fi ao$\"*D%))zvFd`n7$F^bo$\"*l>Na(Fd`n7$Fcbo$\"*&Ho*\\(Fd`n7$Fhbo$\"*d#3c uFd`n7$F]co$\"*$feptFd`n7$Fbco$\"*`?SG(Fd`n7$Fgco$\"+YI*3&pF[do7$F]do$ \"+Lc4KmF[do7$Fbdo$\"+7NWcgF[do7$Fgdo$\"+kkW!*eF[do7$F\\eo$\"+MQ-PhF[d o7$Faeo$\"+:C#)*o'F[do7$Ffeo$\"+gPw)e'F[do7$F[fo$\"+M`5BkF[do7$F`fo$\" +G\"oL5'F[do7$Fefo$\"+zqz)z&F[do7$Fjfo$\"+%*)3`J&F[do7$F_go$\"+'zhb>&F [do7$Fdgo$\"+@*)3FaF[do7$Figo$\"+&R-P*eF[do7$F^ho$\"+e%F[do7$F\\jo$\"+]&fK]%F[do7$Fajo$\"+2ZndWF[do7$Ffjo $\"+i'HUX%F[do7$F[[p$\"+V&y*oWF[do7$F`[p$\"+5i?2XF[do7$Fe[p$\"+'>:bd%F [do7$Fj[p$\"+jwoZVF[do7$F_\\p$\"++!py\"RF[do7$Fd\\p$\"+r^'>g$F[do7$Fi \\p$\"+Fwq*G$F[do7$F^]p$\"+>4U:IF[do7$Fc]p$\"+*3&**>FF[do7$Fh]p$\",BkJ zV#F\\^p7$F^^p$\",g,mP@#F\\^p7$Fc^p$\",F\\^p7$Fh^p$\",H(*3(>=F\\ ^p7$F]_p$\",k.@%p:F\\^p7$Fb_p$\",k>E#G9F\\^p7$Fg_p$\",HV*)>Z\"F\\^p7$F \\`p$\",K8R$3:F\\^p7$Fa`p$\",dr9l[\"F\\^p7$Ff`p$\",wQ\"*\\Y\"F\\^p7$F[ ap$\",HLKGU\"F\\^p7$F`ap$\",Z=J=Q\"F\\^p7$Feap$\",2?\">.8F\\^p7$Fjap$ \",Zt&)*G7F\\^p7$F_bp$\",5N:i:\"F\\^p7$Fdbp$\",/jZf4\"F\\^p7$Fibp$\",P (e$o2\"F\\^p7$F^cp$\",]!f$G2\"F\\^p7$F`ct$\",5T)oz5F\\^p7$Fccp$\",.Vr[ 4\"F\\^p7$Fhct$\",%en!37\"F\\^p7$Fhcp$\",f'z^g6F\\^p7$F]dp$\",@MoG0\"F \\^p7$Fbdp$\"+j![@P*F\\^p7$Fgdp$\"+?')Rh))F\\^p7$F\\ep$\"+\")3JQ%)F\\^ p7$Faep$\"+s%*H)H)F\\^p7$Ffep$\"+;4u\\#)F\\^p7$F[fp$\"+Ig'3N)F\\^p7$F` fp$\"+R:m\"p)F\\^p7$Fefp$\",#*3Fi2)Fifp7$F[gp$\",*48eCrFifp7$F`gp$\",R [ZUq'Fifp7$Fegp$\",DBO*ojFifp7$Fjgp$\",K!o@uiFifp7$F_hp$\",?XpdF'Fifp7 $Fdhp$\",g1J\"QkFifp7$Fihp$\",nO*Q(y'Fifp7$F^ip$\",ClcU0'Fifp7$Fcip$\" ,=j@%)R&Fifp7$Fhip$\",[IT]1&Fifp7$F]jp$\",\"yUy\"z%Fifp7$Fgjp$\",4E'p) o%Fifp7$Fa[q$\",)e0xs]Fifp-Ff[q6&Fh[qF($\"#vF_ht$\"#&*F_ht-F_\\q6#%Ea~ scheme~of~Tsitouras~and~PapakostasG-F$6%7c^lF'F^cx7$F2$\"%\\>F07$F7$\" %'=#F07$F<$\"%!R&F07$FA$\"%R`F07$FF$\"%!o*F07$FK$\"%O&*F07$FP$\"&c]\"F 07$FU$\"&`Z\"F07$FZ$\"&'Q@F07$Fin$\"&#*3#F07$F^o$\"&q'GF07$Fco$\"&.y#F 07$Fho$\"&op$F07$F]p$\"&'zNF07$Fbp$\"&' F07$Faq$\"&)RfF07$Ffq$\"&e\\)F07$Fjfx$\"&DG)F07$F\\r$\"&(4\")F07$Fbgx$ \"&i?)F07$Fggx$\"&J)*)F07$F\\hx$\"'b&>\"F07$Far$\"'wH7F07$Fh`q$\"'G(> \"F07$Ffr$\"'Cr6F07$F`aq$\"'7'=\"F07$Feaq$\"'c08F07$F`ix$\"'8m9F07$Fja q$\"'XgUF07$Fcdl$\"(Q<>%F07$Fhdl $\"(dc8%F07$F]el$\"(l,3%F07$Fbel$\")))o#)QF-7$Fgel$\")veaPF-7$F\\fl$\" )pPnZF-7$Fafl$\")Wd#)\\F-7$Fffl$\")*yVs%F-7$F[gl$\")XmmXF-7$F`gl$\")LH pZF-7$Fegl$\")>,xdF-7$Fjgl$\")_W9gF-7$F_hl$\")2!)eiF-7$Fdhl$\")qzNiF-7 $Fihl$\")4'G@'F-7$F^il$\")Z>nhF-7$Fcil$\")=!=7'F-7$Fhil$\")%H=.'F-7$F] jl$\")+%H%fF-7$Fbjl$\")[#[h&F-7$Fgjl$\")nO1aF-7$F\\[m$\")qpiaF-7$Fa[m$ \")!3\\m&F-7$Ff[m$\")8UIhF-7$F[\\m$\")**[^qF-7$F`\\m$\")oExsF-7$Fe\\m$ \")ggZsF-7$Fj\\m$\")a/=sF-7$F_]m$\")Ue)=(F-7$Fd]m$\").'*HrF-7$Fi]m$\") AtrqF-7$F^^m$\")&fk&pF-7$Fc^m$\")(eF%oF-7$Fh^m$\")w*\\Y'F-7$F]_m$\")lY ViF-7$Fb_m$\")e&HK'F-7$Fg_m$\")*>!zlF-7$F\\`m$\")m'G\"oF-7$Fa`m$\")%ou 9(F-7$Ff`m$\")\\)Hh(F-7$F[am$\")J6Z#)F-7$F`am$\")661!)F-7$Feam$\"))H3u (F-7$Fjam$\")/D#G(F-7$F_bm$\")=SWqF-7$Fdbm$\")^$=<(F-7$Fibm$\")43HvF-7 $F^cm$\")iG'G)F-7$Fccm$\")K*)Q\"*F-7$Fhcm$\")Z_E))F-7$F]dm$\")CNB&)F-7 $Fbdm$\")4/'*F-7$F_gm$\")b%zE*F-7$Fdgm$\")MYb')F-7$Figm$\")gJl$)F-7$F^h m$\")m&*R&)F-7$Fchm$\")Q4#)*)F-7$Fhhm$\")pOr)*F-7$F]im$\"*dp80\"F-7$Fb im$\"*9tY,\"F-7$Fgim$\")9/\"z*F-7$F\\jm$\")9*[7*F-7$Fajm$\")!3T#))F-7$ Ffjm$\")&y+,*F-7$F[[n$\")3Vf%*F-7$F`[n$\"*r)yL5F-7$Fe[n$\"*))4[4\"F-7$ Fj[n$\"*^Jp0\"F-7$F_\\n$\"*J;--\"F-7$Fd\\n$\")(*[c%*F-7$Fi\\n$\")3[:\" *F-7$F^]n$\")t\\D$*F-7$Fc]n$\")#3m$)*F-7$F]^n$\"*JTU3\"F-7$Fa_n$\"*1^B 6\"F-7$Ff_n$\"*`h82\"F-7$F[`n$\"*HE<.\"F-7$F``n$\"**)G&R&*Fd`n7$Ff`n$ \"*0\\tE*Fd`n7$F[an$\"+)Qq?,\"Fd`n7$F`an$\"+#e=$46Fd`n7$Fean$\"+%\\c!p 5Fd`n7$Fjan$\"+OW6I5Fd`n7$F_bn$\"*z()))\\*Fd`n7$Fdbn$\"*xp/>*Fd`n7$Fib n$\"+2 #))Fd`n7$F`fn$\"*7mdd)Fd`n7$Fefn$\"**3>9%*Fd`n7$Fjfn$\"*'*\\1*)*Fd`n7$ F_gn$\"*YD>[*Fd`n7$Fdgn$\"*a'y)3*Fd`n7$Fign$\"*thRn)Fd`n7$F^hn$\"*W&4% H)Fd`n7$Fchn$\"*#*ek8)Fd`n7$Fhhn$\"*>F%G!)Fd`n7$F]in$\"*%>%e+)Fd`n7$Fb in$\"**3m9!)Fd`n7$Fgin$\"*\\Ue1)Fd`n7$F\\jn$\"*r*ft\")Fd`n7$Fajn$\"*>= \\d)Fd`n7$Ffjn$\"*>z!y$*Fd`n7$F[[o$\"*;@?G*Fd`n7$F`[o$\"*8]H3*Fd`n7$Fe [o$\"*MMlp)Fd`n7$Fj[o$\"*C*\\D$)Fd`n7$F_\\o$\"**)**R'zFd`n7$Fd\\o$\"*4 boj(Fd`n7$Fi\\o$\"*4ES](Fd`n7$F^]o$\"*Sg`T(Fd`n7$Fc]o$\"*G!y(R(Fd`n7$F h]o$\"*6TgS(Fd`n7$F]^o$\"*F5$[uFd`n7$Fb^o$\"*(G'\\`(Fd`n7$Fg^o$\"*xZe! zFd`n7$F\\_o$\"*:0Qn)Fd`n7$Fa_o$\"*tK1[)Fd`n7$Ff_o$\"*=vhG)Fd`n7$F[`o$ \"*Ii&4zFd`n7$F``o$\"*:@\"\\vFd`n7$Fe`o$\"*(****[pFd`n7$Fj`o$\"*WVbz'F d`n7$F_ao$\"*;yJ0(Fd`n7$Fdao$\"*,,=h(Fd`n7$Fiao$\"*\\>x\"yFd`n7$F^bo$ \"*=[,y(Fd`n7$Fcbo$\"*SO\\t(Fd`n7$Fhbo$\"*Mo**o(Fd`n7$F]co$\"*We2g(Fd` n7$Fbco$\"*)*3D^(Fd`n7$Fgco$\"+j<$*orF[do7$F]do$\"+8B8SoF[do7$Fbdo$\"+ Ym@YiF[do7$Fgdo$\"+%)e!32'F[do7$F\\eo$\"+l#\\4K'F[do7$Faeo$\"+*eF[do7$Fcho$\"+9.4\"F\\^p7$F^cp$\",tK%o(3\"F \\^p7$F`ct$\",JXpW4\"F\\^p7$Fccp$\",i;Z'46F\\^p7$Fhct$\",3hdc8\"F\\^p7 $Fhcp$\",I^Zb<\"F\\^p7$F]dp$\",.K#Rm5F\\^p7$Fbdp$\"+#z8D\\*F\\^p7$Fgdp $\"+&Q'3v*)F\\^p7$F\\ep$\"+1:,Y&)F\\^p7$Ffep$\"+6j+`$)F\\^p7$F`fp$\"+@ Z]%z)F\\^p7$Fefp$\",EUi%p\")Fifp7$F[gp$\",.r:o?(Fifp7$F`gp$\",7/D:y'Fi fp7$Fegp$\",TX?=W'Fifp7$Fjgp$\",R.8`M'Fifp7$F_hp$\",Nn\"fXjFifp7$Fdhp$ \",@.Lv]'Fifp7$Fihp$\",/%4RdoFifp7$F^ip$\",<1(p;hFifp7$Fcip$\",`O!4aaF ifp7$Fhip$\",^H5s6&Fifp7$F]jp$\",w#o!3%[Fifp7$Fgjp$\",*H0NNZFifp7$Fa[q $\",YRq\">^Fifp-Fabx6&Fh[qF[]\\lF(F[]\\l-F_\\q6#%Ra~modification~of~th e~Tsitouras-Papakostas~schemeG-F$6%7fclF'F^cx7$F2$\"%g@F07$F7$\"%ACF07 $$\"5)))))))))))Qr`j\"F0$\"%VCF07$$\"5;;;;;;O\\lQm#F0F]^\\l7$$\"5[[[[[[j)Qy#F0$\"%$[&F07$$\"5HHHHHHH&R!HF0$ \"%,lF07$FF$\"%))))F07$$\"5!**)*)*)*)*[jEF$F0$\"%?))F07$FK$\"%`()F07$$ \"5[\\\\\\\\\\9))\\OF0$\"%w()F07$$\"5GHHHHHaXyPF0$\"%\"4*F07$$\"534444 4%Hq!RF0$\"&'f5F07$FP$\"&FQ\"F07$$\"5rrrrrrm#>H%F0$\"&!o8F07$FU$\"&YN \"F07$$\"5!3333e&*z+o%F0$\"&3O\"F07$$\"512222d*4>\"[F0$\"&.V\"F07$$\"5 KLLLLe*RP%\\F0$\"&du\"F07$FZ$\"&%f?F07$$\"5NLLLLL=!yI&F0$\"&\\.#F07$Fi n$\"&?,#F07$$\"5+++++]FvqcF0$\"&e,#F07$$\"5!HHHHHzr9!eF0$\"&#*4#F07$$ \"5&eeeee$3>KfF0$\"&!)[#F07$F^o$\"&#))HF07$$\"5baaaaa9UDjF0$\"&+%HF07$ Fco$\"&#**GF07$$\"5+.....BU9nF0$\"&B#HF07$$\"5vvvvvv:\"4%oF0$\"&-6$F07 $$\"5][[[[[3SnpF0$\"&*eQF07$Fho$\"&(>UF07$$\"5DEEEEw)=OK(F0$\"&0:%F07$ F]p$\"&r3%F07$$\"556666'[K**o(F0$\"&U4%F07$$\"5!44444M-J'zF0$\"&]I&F07$Fbp$\"&P$eF07$$\"5DEEEEEw4J$)F0$\"&Rs&F07 $Fgp$\"&Xi&F07$$\"5NLLLLe45(p)F0$\"&Mj&F07$$\"5lkkkk9PpJ))F0$\"&:$fF07 $$\"5&ffff4Z'Gm*)F0$\"&$psF07$F\\q$\"&@*yF07$$\"5lnnnna?F07$ Fcu$\"'U%*>F07$F]v$\"'\"p7#F07$Faw$\"'cjDF07$Ffw$\"'fsCF07$F[x$\"';)R# F07$Fex$\"'^wDF07$Fiy$\"'!4.$F07$F^z$\"'b2HF07$Fcz$\"'Y:GF07$Fhz$\"'sE JF07$F][l$\"'hvMF07$Fb[l$\"'1JLF07$Fg[l$\"'cFKF07$F\\\\l$\"'>vNF07$Fa \\l$\"'72RF07$Ff\\l$\"'ROPF07$F[]l$\"'P2OF07$F`]l$\"'!>&RF07$Fe]l$\"'( )oUF07$Fj]l$\"'([2%F07$F_^l$\"'ZCRF07$Fd^l$\"''[E%F07$Fi^l$\"'MIXF07$F ^_l$\"'c3VF07$Fc_l$\"']VTF07$$\"5AAAsfcP@t>Fhq$\"'1TTF07$$\"5#HHHz;6cj (>Fhq$\"'DZTF07$$\"5jjj8wm%)\\z>Fhq$\"'mkTF07$$\"5MMMM%=#3k#)>Fhq$\"'Q '>%F07$$\"5wvvv+Kb#*))>Fhq$\"'i=VF07$Fh_l$\"'Z`XF07$$\"5)yyy`sf_$)*>Fh q$\"'JHZF07$$\"5eeeeL_\\\\,?Fhq$\"'GE[F07$$\"5HHHzT2tj/?Fhq$\"'*fz%F07 $$\"5++++]i'zx+#Fhq$\"'&ew%F07$$\"5UTTTmsV19?Fhq$\"'-1ZF07$F]`l$\"'tYY F07$Fg`l$\"'T6WF07$F[bl$\"'gUUF07$F`bl$\"'W0VF07$Febl$\"'v#o%F07$Fjbl$ \"'%ow%F07$F_cl$\"'NH[F07$Fdcl$\"'@8[F07$Ficl$\"'6(z%F07$F^dl$\"'.lZF0 7$Fcdl$\"'6LZF07$Fhdl$\"'zpYF07$F]el$\"'92YF07$Fbel$\"(`PQ%F-7$Fgel$\" (K'4UF-7$$\"5qqqq?eKKw@Fhq$\"(!*z>%F-7$Fh\\r$\"(rsA%F-7$$\"5_^^^,9&[(* =#Fhq$\"(p)>VF-7$F\\fl$\"(%F-7$Fjgl$\"(fkB%F-7$F_hl$\"(^IF%F-7$Fdhl$\"(_tD%F-7$Fihl$\"(!pT UF-7$F^il$\"(60@%F-7$Fcil$\"(B&zTF-7$Fhil$\"(\"4=TF-7$F]jl$\"(7u0%F-7$ Fbjl$\"()3LQF-7$Fgjl$\"(0gl$F-7$Fc^m$\"(2K\\$F-7$F]_m$\"(Gk9$F-7$Feam$ \"(vl!GF-7$F_bm$\"(l<]#F-7$F]dm$\"(WT*>F-7$Fgdm$\"(N+x\"F-7$Faem$\"(&R W;F-7$Fefm$\"(.L?\"F-7$Fjfm$\"(s8;\"F-7$F_gm$\"(;27\"F-7$Fdgm$\"(Jk/\" F-7$Figm$\"'M3&*F-7$Fchm$\"'PEuF-7$F]im$\"'yj@F-7$Fbim$\"'E)3#F-7$Fgim $\"'6:?F-7$$\"5????q#Q04%GFhq$\"'@X>F-7$F\\jm$\"'rw=F-7$$\"5????X^Z!)f GFhq$\"'nK=F-7$$\"5````.TX5mGFhq$\"'RbUcyIFhq$!)1nJ 9Fd`n7$Ff`n$!)\\\">Z\"Fd`n7$F[an$!)n!o\">Fd`n7$F`an$!)z#p[#Fd`n7$Fean$ !)^m'R#Fd`n7$Fjan$!)[N4BFd`n7$$\"5!33333=-D9$Fhq$!)BL;AFd`n7$F_bn$!)&= #H@Fd`n7$$\"5gffff4*)>iJFhq$!)[2#4#Fd`n7$$\"5'eeeee[k(oJFhq$!)#\\q1#Fd `n7$$\"5**)*)*)*)RFZ?<$Fhq$!)?Fj?Fd`n7$$\"577777i+LvJFhq$!)m4p?Fd`n7$$ \"5DDDDD]GhyJFhq$!)m\"))3#Fd`n7$Fdbn$!)!\\$G@Fd`n7$Fibn$!)`=ZEFd`n7$F^ cn$!)\"[T;$Fd`n7$Fccn$!)y\")QIFd`n7$Fhcn$!)>+=HFd`n7$$\"5#>>>>W!=vWKFh q$!)r9-GFd`n7$F]dn$!)-S%p#Fd`n7$$\"5ihhh'y\">djKFhq$!)#e)[EFd`n7$$\"5_ ^^^,*GX)pKFhq$!)Cq)HF$Fhq$!)4%>h#Fd`n7$$\"5UTTT;g' =hF$Fhq$!);b;EFd`n7$$\"5OOO'QdMb#zKFhq$!)hvNEFd`n7$Fbdn$!)=CvEFd`n7$Fg dn$!)I[wKFd`n7$F\\en$!)vG$o$Fd`n7$Faen$!)6pENFd`n7$Ffen$!)Y?wLFd`n7$$ \"5[ZZZ(*4G)oM$Fhq$!)SRVKFd`n7$F[fn$!)y*37$Fd`n7$$\"5'fff4AD))\\O$Fhq$ !)a%)pIFd`n7$$\"57777i*RB5P$Fhq$!)oFNIFd`n7$$\"5???qKt4/uLFhq$!)^xGIFd `n7$$\"5GGGG.Z&eqP$Fhq$!)%=L.$Fd`n7$$\"5OOO'Q27w+Q$Fhq$!)]0`IFd`n7$F`f n$!)FZ$4$Fd`n7$Fefn$!)5cmOFd`n7$Fjfn$!)P)z1%Fd`n7$F_gn$!)g()**QFd`n7$F dgn$!)C;QPFd`n7$Fign$!)ULnNFd`n7$F^hn$!)Iu5MFd`n7$Fchn$!),NZLFd`n7$Fhh n$!)*))*4LFd`n7$F]in$!)X;4LFd`n7$Fbin$!)8GFLFd`n7$Fgin$!))3@P$Fd`n7$F \\jn$!)c:aMFd`n7$Fajn$!)TtXPFd`n7$Ffjn$!)sBLVFd`n7$$\"5<===$peP5]$Fhq$ !)$RTQ%Fd`n7$$\"5ABBBt5Be-NFhq$!)&)[gVFd`n7$$\"5FGGG`Mq7/NFhq$!)X&pL%F d`n7$F[[o$!)m`8VFd`n7$$\"5UVVV$f?h(3NFhq$!)20nUFd`n7$F`[o$!)`-@UFd`n7$ Fe[o$!)'[9/%Fd`n7$Fj[o$!)G)*oQFd`n7$F_\\o$!)Zp+PFd`n7$Fd\\o$!)sP[NFd`n 7$Fi\\o$!)?@)[$Fd`n7$F^]o$!)sj`MFd`n7$Fc]o$!)S#HX$Fd`n7$Fh]o$!)(e!pMFd `n7$F]^o$!)]A3NFd`n7$Fb^o$!))o&yNFd`n7$Fg^o$!)wAlQFd`n7$F\\_o$!)(e@Y%F d`n7$Fa_o$!)e+kVFd`n7$Ff_o$!).%RE%Fd`n7$F[`o$!)!R,2%Fd`n7$F``o$!)*3Y)Q Fd`n7$$\"5#GGGGGoV.l$Fhq$!)oEAPFd`n7$Fe`o$!)\\7vNFd`n7$$\"5wwwww^[1oOF hq$!)#Gc^$Fd`n7$$\"5333333>(Rn$Fhq$!)K6xMFd`n7$$\"5utttBOa#pn$Fhq$!)K! 4Z$Fd`n7$$\"5RRRRRk*y)zOFhq$!)1dxMFd`n7$$\"5/000b#\\KGo$Fhq$!)%y9]$Fd` n7$Fj`o$!)[F[NFd`n7$F_ao$!)%z.w$Fd`n7$Fdao$!)KQ;UFd`n7$Fiao$!)BR'Q%Fd` n7$F^bo$!)<0nVFd`n7$Fcbo$!)VnTVFd`n7$Fhbo$!)JV;VFd`n7$F]co$!)*ejE%Fd`n 7$Fbco$!)Q#o@%Fd`n7$Fgco$!*zqR-%F[do7$F]do$!*!QQRQF[do7$$\"5!4444fYd'[ PFhq$!*c'*[m$F[do7$Fbdo$!*v^_]$F[do7$$\"5EFFFxkQSnPFhq$!*.G)QMF[do7$$ \"5QRRRRkEltPFhq$!*QmGR$F[do7$$\"5WXXX?kqxwPFhq$!*l)*GQ$F[do7$$\"5]^^^ ,k9!*zPFhq$!*N8gQ$F[do7$$\"5cddd#Q'e-$y$Fhq$!*<\"*oS$F[do7$Fgdo$!*ct;X $F[do7$F\\eo$!*+kxm$F[do7$Faeo$!*#=D^TF[do7$Ffeo$!*7BI5%F[do7$F[fo$!*j i)**RF[do7$F`fo$!*s]2!QF[do7$Fefo$!*HH5h$F[do7$$\"5iiiii7;U^QFhq$!*NSB X$F[do7$Fjfo$!*?g!4LF[do7$$\"5KKKKKdt6pQFhq$!*5.6D$F[do7$$\"5*)))))))) )Qf,vQFhq$!*$*GH@$F[do7$$\"5=<<E&)HF[do7$F`[p$!*,mQ-$F[do7$Fe[p$!*QT%*3$F[do7$$\"5tttt[#4pk*RFhq$!* /#)HL$F[do7$$\"5+,,,^Qfa-SFhq$!*hRr`$F[do7$$\"5GGGG`%yA'3SFhq$!*N!3_MF [do7$$\"5bbbbbI'*p9SFhq$!*5U*oLF[do7$$\"55555gAL&o-%Fhq$!*fr#3KF[do7$F j[p$!*D)*[0$F[do7$$\"5]]]]+Qh2_SFhq$!*?E*)*GF[do7$$\"5OOOOOh_9lSFhq$!* Z7-w#F[do7$$\"5HHHH/B)z;2%Fhq$!*uAqq#F[do7$$\"5AAAAs%Q9#ySFhq$!*ah%yEF [do7$$\"5ooo=cl;[\"3%Fhq$!*_9(zEF[do7$$\"5::::SY*[Z3%Fhq$!*uhkp#F[do7$ $\"5ihh6CFi,)3%Fhq$!*T7Tt#F[do7$F_\\p$!*P:(*z#F[do7$Fd\\p$!*7tGm#F[do7 $Fi\\p$!*08*pCF[do7$F^]p$!*w**RL#F[do7$Fc]p$!*]8a8#F[do7$Fh]p$!+AxVn>F \\^p7$F^^p$!+;\\n4=F\\^p7$Fc^p$!+w:/_;F\\^p7$Fh^p$!+*>e`b\"F\\^p7$F]_p $!+c(pHO\"F\\^p7$Fb_p$!+-sIg7F\\^p7$Fg_p$!+t!p,J\"F\\^p7$F\\`p$!+%yfKN \"F\\^p7$Fa`p$!+Z#zOL\"F\\^p7$Ff`p$!+[)oVJ\"F\\^p7$F[ap$!+WVaw7F\\^p7$ F`ap$!+&oe(R7F\\^p7$Feap$!+n:?p6F\\^p7$Fjap$!+TFf-6F\\^p7$F_bp$!++77P5 F\\^p7$Fdbp$!*u\\%G)*F\\^p7$Fibp$!*+wGm*F\\^p7$F^cp$!*\"GL_'*F\\^p7$Fc cp$!*\"feB**F\\^p7$Fhcp$!+>y;p5F\\^p7$F]dp$!*2aet*F\\^p7$Fbdp$!*Qofm)F \\^p7$Fgdp$!*'\\4#>)F\\^p7$F\\ep$!*\"zN*z(F\\^p7$Faep$!*dRNn(F\\^p7$Ff ep$!*+\\Mk(F\\^p7$F[fp$!*'Fifp7$Fihp$!+_]5GmFifp7$ F^ip$!+ " 0 "" {MPLTEXT 1 0 639 "evalf[20](p lot(['hn_RK8_1'(x)/h(x)-1,'hn_RK8_2'(x)/h(x)-1,'hn_RK8_3'(x)/h(x)-1,'h n_RK8_4'(x)/h(x)-1,\n'hn_RK8_5'(x)/h(x)-1,'hn_RK8_6'(x)/h(x)-1,'hn_RK8 _7'(x)/h(x)-1],x=0..10,-3.9e-7..2.27e-6,font=[HELVETICA,9],\ncolor=[CO LOR(RGB,.4,0,.9),COLOR(RGB,0,.65,0),brown,red,COLOR(RGB,0,.75,.95),mag enta,COLOR(RGB,.95,.55,0)],\nlegend=[`Enright-Verner scheme`,`Verner's Maple scheme`,`a modification of Verner's Maple scheme`,`an \"efficie nt\" scheme`,`a scheme of Tsitouras and Papakostas`,`a modification of the Tsitouras-Papakostas scheme`,`Verner's \"most efficient\" scheme` ],title=`relative error curves for 12 stage order 8 Runge-Kutta method s`));" }}{PARA 13 "" 1 "" {GLPLOT2D 953 630 630 {PLOTDATA 2 "6--%'CURV ESG6%7[o7$$\"\"!F)F(7$$\"5mmmmm;arz@!#?$\"$l*!#>7$$\"5LLLL$e9ui2%F-$\" %GHF07$$\"5mmmmm\"z_\"4iF-$\"%ecF07$$\"5lmmmmT&phN)F-$\"%`\")F07$$\"5L LLLe*=)H\\5F0$\"%D&*F07$$\"5mmmm\"z/3uC\"F0$\"&[R\"F07$$\"5++++DJ$RDX \"F0$\"&#HXF07$$\"5mmmm\"zR'ok;F0$\"'0M=F07$$\"5++++D1J:w=F0$\"'!Gd'F0 7$$\"5LLLLL3En$4#F0$\"(SqW#F07$$\"5mmmm;/RE&G#F0$\"(cfJ&F07$$\"5+++++D .&4]#F0$\")gIm;F07$$\"5+++++vB_Y2aF0$\",Nz)\\ELF07$$\"5mmmm;zXu9cF0$\",B r\"36[F07$$\"5**********\\y))GeF0$\",r@]o(oF07$$\"5********\\i_QQgF0$ \",^i&RA(*F07$$\"5*******\\7y%3TiF0$\"-#p3r/O\"F07$$\"5********\\P![hY 'F0$\"-J#\\)f\"*=F07$$\"5KLLLL$Qx$omF0$\"-F`Rc+EF07$$\"5*********\\P+V )oF0$\"-(4?Xej$F07$$\"5mmmm\"zpe*zqF0$\"-g!)z'*[[F07$$\"5*********\\# \\'QH(F0$\"-jttd:pF07$$\"5KLLLe9S8&\\(F0$\"-GTo(3G*F07$$\"5*******\\i? =bq(F0$\".x)HK$yG\"F07$$\"5KLLLL3s?6zF0$\".^#F07$$\"5lmmmmm*RRL)F0$\".\"p:g`QGF07$$\"5lmmm;a<.Y&)F 0$\".rs!o'Gl$F07$$\"5KLLLe9tOc()F0$\".kgk5!yYF07$$\"5**********\\Qk\\* )F0$\".Kf$*ok&fF07$$\"5KLLL$3dg6<*F0$\".O;YHHe(F07$$\"5++++voTAq#*F0$ \".t!\\]9=&)F07$$\"5lmmmmmxGp$*F0$\".'3k^qe&*F07$$\"5ILL$eRA5\\Z*F0$\" /;>s%)*e2\"F07$$\"5)******\\7oK0e*F0$\"/B5H5Dj'=F07$$\"5++]7.#Q?&=**F0$\"/8\"o7Dj'=F07$$\"5++v$f$=. 5K**F0$\"/X0%)*Gj'=F07$$\"5+++voa-oX**F0$\"/>gW3Vm=F07$$\"5++Dc,\">g#f **F0$\"/`)yS$en=F07$$\"5++]PMF,%G(**F0$\"/Py\"yzb(=F07$$\"5++v=nj+U')* *F0$\"/#[#)eNg\">F07$$\"#5F)$\"/)Qo_f.3#F0-%&COLORG6&%$RGBG$\"\"%!\"\" F($\"\"*Fb_l-%'LEGENDG6#%6Enright-Verner~schemeG-F$6%7[oF'7$F+$!%_DF-7 $F2$!&/-#F-7$F7$!&r[&F-7$F<$!&Y***F-7$FA$!'&41\"F-7$FF$\"&X5\"F07$FK$ \"'n!3\"F07$FP$\"'mfTF07$FU$\"(*yq7F07$FZ$\"(]#HSF07$Fin$\"(b?2)F07$F^ o$\")FW(G#F07$Fco$\")o>!f%F07$Fho$\")40:))F07$F]p$\"*%[yH;F07$Fbp$\"*H O[\"HF07$Fgp$\"*^'fg]F07$F\\q$\"*?4^c)F07$Faq$\"+57b89F07$Ffq$\"+)4*Q& H#F07$F[r$\"+o#Qal$F07$F`r$\"+*oiw%fF07$Fer$\"+'eZ`$*)F07$Fjr$\",!*4qH Z\"F07$F_s$\",GJ>[L#F07$Fds$\",Ak)G+MF07$Fis$\",\"*R%4$*[F07$F^t$\",+b ZH'pF07$Fct$\",08)=0)*F07$Fht$\"-_iFAn8F07$F]u$\"-f)[F\\*=F07$Fbu$\"-? P9y(f#F07$Fgu$\"-V3%3>i$F07$F\\v$\"-'))o?+#[F07$Fav$\"-*=DEu&oF07$Ffv$ \"-:zF*e=*F07$F[w$\".Q7z9BF\"F07$F`w$\".G*4XDj;F07$Few$\".#)fco:;#F07$ Fjw$\".@ERzMz#F07$F_x$\".nyN)3\"f$F07$Fdx$\".Z5gxWf%F07$Fix$\".DmVb]%e F07$F^y$\".4!Rw+NuF07$Fcy$\".=fMf*[$)F07$Fhy$\".(on6il$*F07$F]z$\"/!fy 6%z`5F07$Fbz$\"/-TH$=$)=\"F07$Fgz$\"/b5V=\"=L\"F07$F\\[l$\"/lJRx7,i``D=F07$Fi]l$\"/Mx!\\_m#=F07$F^^l$\"/goCaUM=F07$Fc ^l$\"/w\\*\\YQ(=F07$Fh^l$\"/9J89FM?F0-F]_l6&F__lF($\"#l!\"#F(-Ff_l6#%6 Verner's~Maple~schemeG-F$6%7[oF'7$F+$\"$S#F07$F2$\"$+#F07$F7$!%W>F-7$F <$!%i\")F-7$FA$\"$(oF07$FF$\"&1`\"F07$FK$\"&Z(yF07$FP$\"';()GF07$FU$\" '7%)*)F07$FZ$\"(.+&HF07$Fin$\"(C(QgF07$F^o$\")o)Hw\"F07$Fco$\")rs+OF07 $Fho$\")w@;qF07$F]p$\"*TnGJ\"F07$Fbp$\"*F07$F[r$\"+c/#R1$F07$F`r$\"+ h\"QT+&F07$Fer$\"+V!R!QvF07$Fjr$\",Bf*=Y7F07$F_s$\",i9T&z>F07$Fds$\",P *oQ()GF07$Fis$\",T[Q/;%F07$F^t$\",ut!yEfF07$Fct$\",/84NN)F07$Fht$\"-M= \"[c;\"F07$F]u$\"-Vr;a;;F07$Fbu$\"-4=J;.@rF07$Fhy$\".`VQao)zF07$F]z$\".([[>E&)*)F07$Fbz$\"/c=jf285F07$Fg z$\"/s>S3@N6F07$F\\[l$\"/z6u:.r7F07$Fa[l$\"/.@<&*H%R\"F07$Ff[l$\"/2(3L o$f9F07$F[\\l$\"/\\')on]F:F07$F`\\l$\"/?O%4*)\\b\"F07$Fe\\l$\"/zn%4*) \\b\"F07$Fj\\l$\"/R#[6*)\\b\"F07$F_]l$\"/OseB*\\b\"F07$Fd]l$\"/^JF07$Fin$\"($G1jF07$F^o$\")\"3Qz \"F07$Fco$\")1i8OF07$Fho$\")PPlpF07$F]p$\"*Gj>H\"F07$Fbp$\"*KSoJ#F07$F gp$\"*YM5.%F07$F\\q$\"*L&3MoF07$Faq$\"+_cIH6F07$Ffq$\"+LzeN=F07$F[r$\" +M*e^#HF07$F`r$\"+-M'=w%F07$Fer$\"+w14brF07$Fjr$\",,J7&z6F07$F_s$\",g5 Q\"p=F07$Fds$\",!RWQ@FF07$Fis$\",9`hZ\"RF07$F^t$\",`S)RobF07$Fct$\",K3 vv$yF07$Fht$\"->kDF#4\"F07$F]u$\"-*)Ru+8:F07$Fbu$\"-f/'GH2#F07$Fgu$\"- z$ff$))GF07$F\\v$\"-g!f67%QF07$Fav$\"-wu%*QgaF07$Ffv$\"-hfx#*3tF07$F[w $\".GA+S8,\"F07$F`w$\".1w4/6K\"F07$Few$\".K()38cr\"F07$Fjw$\".CZ?la@#F 07$F_x$\".Uy=ce%GF07$Fdx$\".1VKL#QOF07$Fix$\".([hK$\\i%F07$F^y$\".:1p: 'yeF07$Fcy$\".J\\$Hn)f'F07$Fhy$\".XYh9$*R(F07$F]z$\".QXh0CK)F07$Fbz$\" .by%>H\"Q*F07$Fgz$\"/9>'G050\"F07$F\\[l$\"/'Hew1l<\"F07$Fa[l$\"/H#Q'RP !H\"F07$Ff[l$\"/[f%[-0N\"F07$F[\\l$\"/=oO*GMT\"F07$F`\\l$\"/V'Q\"Q!)Q9 F07$Fe\\l$\"/s:9Q!)Q9F07$Fj\\l$\"/\"HG$Q!)Q9F07$F_]l$\"/^jPo!)Q9F07$Fd ]l$\"/!pZ#f))Q9F07$Fi]l$\"/@tjxxR9F07$F^^l$\"/iE,O%fW\"F07$Fc^l$\"/Jn, ]-x9F07$Fh^l$\"/A@AFz-;F0-F\\hm6&F__l$\"*++++\"F`hmF(F(-Ff_l6#%6an~\"e fficient\"~schemeG-F$6%7[oF'7$F+$\"$S&F07$F2$\"%4;F07$F7$\"%wKF07$F<$ \"%[cF07$FA$\"&g0\"F07$FF$\"&Dv#F07$FK$\"&W$*)F07$FP$\"'a))GF07$FU$\"' Q;()F07$FZ$\"(\"feGF07$Fin$\"(=)zeF07$F^o$\")^$Ht\"F07$Fco$\")2^hNF07$ Fho$\")3)p(pF07$F]p$\"*!3O68F07$Fbp$\"*7suP#F07$Fgp$\"*WpV<%F07$F\\q$ \"*F07$F[r$\"+G7O/JF07$F`r$\"+= *[j2&F07$Fer$\"+A7F`wF07$Fjr$\",DcAjE\"F07$F_s$\",8CWF,#F07$Fds$\",wL9 q$HF07$Fis$\",*)RxKB%F07$F^t$\",0&\\'>.'F07$Fct$\",=a(>.&)F07$Fht$\"-v )yom=\"F07$F]u$\"-,H:\"ek\"F07$Fbu$\"-^KOOdAF07$Fgu$\"-Xgut[JF07$F\\v$ \"-/HT(4>%F07$Fav$\"-:NGDjfF07$Ffv$\"--(38x)zF07$F[w$\".lf%e/16F07$F`w $\".T?#3hX9F07$Few$\".A'G:By=F07$Fjw$\".,q2_lU#F07$F_x$\".lVau#=JF07$F dx$\".\"GI\"zz)RF07$Fix$\".^\\>@72&F07$F^y$\".*Q8[#zW'F07$Fcy$\".(zs0t QsF07$Fhy$\".Is%37=\")F07$F]z$\".5BmR@8*F07$Fbz$\"/0nKXaH5F07$Fgz$\"/` /!QeN:\"F07$F\\[l$\"/ru%Qa9H\"F07$Fa[l$\"/U.Z\"F07$FP$\"'h?OF07$FU$\"(*GO5F07 $FZ$\"(dED$F07$Fin$\"(BIb'F07$F^o$\")Nn$)=F07$Fco$\")N!)=QF07$Fho$\")O w*R(F07$F]p$\"*.u%y8F07$Fbp$\"*mx1[#F07$Fgp$\"*j0%GVF07$F\\q$\"*&==btF 07$Faq$\"+-vn<7F07$Ffq$\"+NA@#)>F07$F[r$\"+M'QE;$F07$F`r$\"+c[x`^F07$F er$\"+Q3^\\xF07$Fjr$\",*)p*Qy7F07$F_s$\",A[4o-#F07$Fds$\",:c.=&HF07$Fi s$\",:!e3ZUF07$F^t$\",Vh&)=/'F07$Fct$\",4^)f/&)F07$Fht$\"-=(*>D&=\"F07 $F]u$\"-*H;k\\AF07$Fgu$\"-^3)yO8$F07$F\\v$\"-P77(p; %F07$Fav$\"-#*RGXAfF07$Ffv$\"-Q%=tg#zF07$F[w$\".xh>(['4\"F07$F`w$\".5Z af?V\"F07$Few$\".c\"*=L$f=F07$Fjw$\".&HPec+CF07$F_x$\".MMbwH3$F07$Fdx$ \".#=2K`SRF07$Fix$\".**fAD\"3]F07$F^y$\".Xm^7VO'F07$Fcy$\".#*z;vI9(F07 $Fhy$\".Qc$))))3!)F07$F]z$\".0I'p/2!*F07$Fbz$\"/&[&o:>:5F07$Fgz$\"/[.H V@P6F07$F\\[l$\"/c5h'pGF\"F07$Fa[l$\"/K9\\$3fR\"F07$Ff[l$\"/;wE1#4Y\"F 07$F[\\l$\"/[Ss8!*G:F07$F`\\l$\"/xqI*3jb\"F07$Fe\\l$\"/v-J*3jb\"F07$Fj \\l$\"/()Q^*3jb\"F07$F_]l$\"/lG@AJc:F07$Fd]l$\"/E3<\")Rc:F07$Fi]l$\"/ \\S;]Od:F07$F^^l$\"/\\9H'RSc\"F07$Fc^l$\"/yE@Zk(f\"F07$Fh^l$\"/\\KKO]L #z%F-7$F^t$!,)\\WViqF-7$Fct$!-r`([p-\"F-7$Fht$!-^2O&\\Z\"F-7$F]u$!-B$p >/5#F-7$Fbu$!-n,A$R&HF-7$Fgu$!-$QRQ\">UF-7$F\\v$!-fL(>3u&F-7$Fav$!-%yF GaP)F-7$Ffv$!.$Rc\"ed9\"F-7$F[w$!.6BdDmi\"F-7$F`w$!.&R^\"e^;#F-7$Few$! .8=B;I'GF-7$Fjw$!.$\\<>?iPF-7$F_x$!.K#=cy9\\F-7$Fdx$!.t?p6iQ'F-7$Fix$! .afz%\\[#)F-7$F^y$!/%fbudU1\"F-7$Fcy$!/eipqt.7F-7$Fhy$!/f$obn*f8F-7$F] z$!/,1w_hS:F-7$Fbz$!/(>*\\G1\\F-7$F\\[l$!/(4gO4_A#F -7$Fa[l$!/6_U!R)fCF-7$Ff[l$!/Y)H(RcvDF-7$F[\\l$!/LK&[-uq#F-7$F`\\l$!/B zLF.iFF-7$Fe\\l$!/i5MF.iFF-7$Fj\\l$!/y9bF.iFF-7$F_]l$!/[;Ul.iFF-7$Fd]l $!/SNJx9iFF-7$Fi]l$!/sRt@ajFF-7$F^^l$!/&Gi]DUx#F-7$Fc^l$!/Ko\\anLGF-7$ Fh^l$!/bg]+?)4$F--F]_l6&F__lF[ao$\"#bFj[mF(-Ff_l6#%AVerner's~\"most~ef ficient\"~schemeG-%%FONTG6$%*HELVETICAGFd_l-%&TITLEG6#%inrelative~erro r~curves~for~12~stage~order~8~Runge-Kutta~methodsG-%+AXESLABELSG6$Q\"x 6\"Q!Fbjp-%%VIEWG6$;F(Fh^l;$!#RF`hm$\"$F#F`hm" 1 2 0 1 10 0 2 9 1 4 2 1.000000 45.000000 45.000000 0 1 "Enright-Verner scheme" "Verner's Map le scheme" "a modification of Verner's Maple scheme" "an \"efficient\" scheme" "a scheme of Tsitouras and Papakostas" "a modification of the Tsitouras-Papakostas scheme" "Verner's \"most efficient\" scheme" }}} }{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 48 "Test 4 of 12 stage, o rder 8 Runge-Kutta methods " }}{PARA 0 "" 0 "" {TEXT -1 81 "F. G. Leth er: Mathematics of Computation, Vol. 20, no. 95, (July 1966) page 381. " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "dy/dx = -32*x*y*ln(2);" "6#/*&%#dyG\"\"\"%#dxG!\"\",$* *\"#KF&%\"xGF&%\"yGF&-%#lnG6#\"\"#F&F(" }{TEXT -1 6 ", " } {XPPEDIT 18 0 "y(-1) = 1/8;" "6#/-%\"yG6#,$\"\"\"!\"\"*&F(F(\"\")F)" } {TEXT -1 1 " " }}{PARA 0 "" 0 "" {TEXT -1 10 "Solution: " }}{PARA 256 "" 0 "" {TEXT -1 2 " " }{XPPEDIT 18 0 "y = 2^(13-6*x^2);" "6#/%\"yG) \"\"#,&\"#8\"\"\"*&\"\"'F)*$%\"xGF&F)!\"\"" }{TEXT -1 2 ". " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 191 "d e := diff(y(x),x)=-32*x*y(x)*ln(2);\nic := y(-1)=1/8;\ndsolve(\{de,ic \},y(x)):\ny(x)=2^simplify(log[2](rhs(%)));\nk := unapply(rhs(%),x):\n plot(k(x),x=-1..1,font=[HELVETICA,9],labels=[`x`,`y(x)`]);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%#deG/-%%diffG6$-%\"yG6#%\"xGF,,$**\"#K\"\" \"F,F0F)F0-%#lnG6#\"\"#F0!\"\"" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%#i cG/-%\"yG6#!\"\"#\"\"\"\"\")" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/-%\"y G6#%\"xG)\"\"#,&\"#8\"\"\"*&\"#;F,)F'F)F,!\"\"" }}{PARA 13 "" 1 "" {GLPLOT2D 577 286 286 {PLOTDATA 2 "6&-%'CURVESG6$7eo7$$!\"\"\"\"!$\"3+ ++++++]7!#=7$$!3ommm;p0k&*F-$\"3!Hg[[\"[$)=KF-7$$!3wKL$3$3(F-7$$!3mmmmT%p\"e()F-$\"3!=E-TWD`l\"!#<7$$!3:mmm\"4m(G$)F-$\"3 M\"fONp()[t$F=7$$!3\"QLL3i.9!zF-$\"3A!e'4A%*Rg!)F=7$$!3\"ommT!R=0vF-$ \"3%z2Mbncie\"!#;7$$!3u****\\P8#\\4(F-$\"3C>dT>$)H#3$FM7$$!3+nm;/siqmF -$\"3gp%*z`g)4*eFM7$$!3[++](y$pZiF-$\"3%R6L-Y$zz5!#:7$$!33LLL$yaE\"eF- $\"3xvp\"p)==K>Fgn7$$!3hmmm\">s%HaF-$\"3dBW_P%Gb6$Fgn7$$!3Q+++]$*4)*\\ F-$\"3e;N4:OFap7$$!3]++]PYx\" \\#F-$\"37]-4,Tp9TFap7$$!3QnmTNz>&H#F-$\"3y(*QMk^JnXFap7$$!3EMLLL7i)4# F-$\"3yCPsPtXE]Fap7$$!3#pm;aVXH)=F-$\"3_cYryDpGbFap7$$!3c****\\P'psm\" F-$\"38i9x*[!p=gFap7$$!3s*****\\F&*=Y\"F-$\"3K`b3X@JjkFap7$$!3')****\\ 74_c7F-$\"3co2Qfx9woFap7$$!3ZmmT5VBU5F-$\"3E!>K?nVAE(Fap7$$!3)3LLL3x%z #)!#>$\"3C'Q/NU&H#f(Fap7$$!3gKL$e9d;J'Fft$\"3@erx_1&z$yFap7$$!3KMLL3s$ QM%Fft$\"3<%pUH&HNA!)Fap7$$!3'ym;aQdDG$Fft$\"3%eWuwq(o%4)Fap7$$!3T,+]i vF@AFft$\"3[vW[$G&HZ\")Fap7$$!3=o;/^wj!p\"Fft$\"3]3j^OK2m\")Fap7$$!3' \\L$eRx**f6Fft$\"355#oX4%yz\")Fap7$$!3S<+D\"GyNH'!#?$\"3#QE)R9AS)=)Fap 7$$!3]^omm;zr)*!#@$\"3;#)*eHY6>>)Fap7$$\"3o'H$3x\"yY_%Fjv$\"3Q>*>/AS,> )Fap7$$\"3&yK$3_Nl.5Fft$\"3eqLS$Q`G=)Fap7$$\"3/E$ekGR[b\"Fft$\"3-l$f@n l+<)Fap7$$\"3@CL$3-Dg5#Fft$\"3kJX?)*G!=:)Fap7$$\"3e?Le*['R3KFft$\"3E'y GoI5!*4)Fap7$$\"3%pJL$ezw5VFft$\"3U_-I6(**[-)Fap7$$\"3L`mmmJ+IiFft$\"3 %pB(\\hv&o%yFap7$$\"3s*)***\\PQ#\\\")Fft$\"3!QM&=wHL5wFap7$$\"3ilm\"z \\1A-\"F-$\"3#*[#H(\\2i&H(Fap7$$\"3GKLLe\"*[H7F-$\"3))\\\\;@heFpFap7$$ \"3ylm;HCjV9F-$\"3)e+$\\9-Y,lFap7$$\"3I*******pvxl\"F-$\"3S%z:5s)zRgFa p7$$\"3g)***\\7JFn=F-$\"31))p(30[[c&Fap7$$\"3#z****\\_qn2#F-$\"3ae5F\" zuv2&Fap7$$\"3=)**\\P/q%zAF-$\"3ZUhzOe!Rg%Fap7$$\"3U)***\\i&p@[#F-$\"3 r&f%4uLbOTFap7$$\"3L)**\\(=GB2FF-$\"3WV]5@%**Rj$Fap7$$\"3B)****\\2'HKH F-$\"3ul]=$GLo:$Fap7$$\"3uJL$3UDX8$F-$\"3sKZjodBbFFap7$$\"3ElmmmZvOLF- $\"3!>\\-t_7IQ#Fap7$$\"3i******\\2goPF-$\"3Q>G9F7l&p\"Fap7$$\"3UKL$eR< *fTF-$\"3?\"Fap7$$\"3m******\\)Hxe%F-$\"3V-?C_;$p$zFgn7$$\"3 ckm;H!o-*\\F-$\"31MiF2c]v^Fgn7$$\"3y)***\\7k.6aF-$\"3#pB[/J``=$Fgn7$$ \"3#emmmT9C#eF-$\"3&*=.D]9+3>Fgn7$$\"33****\\i!*3`iF-$\"3%HX+j$our5Fgn 7$$\"3%QLLL$*zym'F-$\"3!o4*yfd(\\\"fFM7$$\"3wKLL3N1#4(F-$\"3!\\\\K5**) ='4$FM7$$\"3Nmm;HYt7vF-$\"3%o[)olFVm:FM7$$\"3Y*******p(G**yF-$\"3)3H-p cT.4)F=7$$\"3]mmmT6KU$)F-$\"35omE\\#[Ck$F=7$$\"3fKLLLbdQ()F-$\"3TxwT%Q u%>ei< " 0 "" {MPLTEXT 1 0 811 "K := (x,y) -> -32*x*y(x)*ln(2): hh := 0.01 : numsteps := 200: x0 := -1: y0 := 1/8:\nmatrix([[`slope field: `,K( x,y)],[`initial point: `,``(x0,y0)],[`step width: `,hh],\n[`no. of s teps: `,numsteps]]);``;\nmthds := [`Enright-Verner scheme`,`Verner's Maple scheme`,`a modification of Verner's Maple scheme`,`an \"efficie nt\" scheme`,`a scheme of Tsitouras and Papakostas`,`a modification of the Tsitouras-Papakostas scheme`,`Verner's \"most efficient\" scheme` ]: errs := []:\nDigits := 20:\nfor ct to 7 do\n Kn_RK8_||ct := RK8_| |ct(evalf(K(x,y)),x,y,x0,evalf(y0),hh,numsteps,false);\n sm := 0: nu mpts := nops(Kn_RK8_||ct):\n for ii to numpts do\n sm := sm+(Kn _RK8_||ct[ii,2]-k(Kn_RK8_||ct[ii,1]))^2;\n end do:\n errs := [op(e rrs),sqrt(sm/numpts)];\nend do:\nDigits := 10:\nlinalg[transpose]([mth ds,evalf(errs)]);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#K%'matrixG6#7&7$% 0slope~field:~~~G,$**\"#K\"\"\"%\"xGF,-%\"yG6#F-F,-%#lnG6#\"\"#F,!\"\" 7$%0initial~point:~G-%!G6$F5#F,\"\")7$%/step~width:~~~G$F,!\"#7$%1no.~ of~steps:~~~G\"$+#Q)pprint156\"" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%!G " }}{PARA 11 "" 1 "" {XPPMATH 20 "6#K%'matrixG6#7)7$%6Enright-Verner~s chemeG$\"+d$>I.%!#=7$%6Verner's~Maple~schemeG$\"+S@j1UF+7$%Ha~modifica tion~of~Verner's~Maple~schemeG$\"+s?%fi$F+7$%6an~\"efficient\"~schemeG $\"+Q$=\"[MF+7$%Ea~scheme~of~Tsitouras~and~PapakostasG$\"+PF+7$%AVer ner's~\"most~efficient\"~schemeG$\"+f-@)>#!#?Q)pprint166\"" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 30 "The following c ode constructs " }{TEXT 260 20 "numerical procedures" }{TEXT -1 56 " f or solutions based on each of the Runge-Kutta schemes." }}{PARA 0 "" 0 "" {TEXT -1 75 "The error in the value obtained by each of the metho ds at the point where " }{XPPEDIT 18 0 "x = 0;" "6#/%\"xG\"\"!" } {TEXT -1 20 ".995 is also given." }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 742 "K := (x,y) -> -32*x*y(x)*ln(2): hh := 0.01: numsteps := 200: x0 := -1: y0 := 1/8:\nmatrix([[`slope field: `,K(x,y)],[`in itial point: `,``(x0,y0)],[`step width: `,hh],\n[`no. of steps: `, numsteps]]);``;\nmthds := [`Enright-Verner scheme`,`Verner's Maple sch eme`,`a modification of Verner's Maple scheme`,`an \"efficient\" schem e`,`a scheme of Tsitouras and Papakostas`,`a modification of the Tsito uras-Papakostas scheme`,`Verner's \"most efficient\" scheme`]: errs := []:\nDigits := 25:\nfor ct to 7 do\n kn_RK8_||ct := RK8_||ct(evalf( K(x,y)),x,y,x0,evalf(y0),hh,numsteps,true);\nend do:\nxx := 0.995: kxx := evalf(k(xx)):\nfor ct to 7 do\n errs := [op(errs),abs(kn_RK8_||c t(xx)-kxx)];\nend do:\nDigits := 10:\nlinalg[transpose]([mthds,evalf(e rrs)]);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#K%'matrixG6#7&7$%0slope~fie ld:~~~G,$**\"#K\"\"\"%\"xGF,-%\"yG6#F-F,-%#lnG6#\"\"#F,!\"\"7$%0initia l~point:~G-%!G6$F5#F,\"\")7$%/step~width:~~~G$F,!\"#7$%1no.~of~steps:~ ~~G\"$+#Q)pprint176\"" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%!G" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#K%'matrixG6#7)7$%6Enright-Verner~schemeG$\"+ z/x@[!#B7$%6Verner's~Maple~schemeG$\"+aV5V^F+7$%Ha~modification~of~Ver ner's~Maple~schemeG$\"+6=$z9%F+7$%6an~\"efficient\"~schemeG$\"+_A-PRF+ 7$%Ea~scheme~of~Tsitouras~and~PapakostasG$\"+Rp4yTF+7$%Ra~modification ~of~the~Tsitouras-Papakostas~schemeG$\"+&p.;G%F+7$%AVerner's~\"most~ef ficient\"~schemeG$\"+\\BJ55F+Q)pprint186\"" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 4 "The " }{TEXT 260 22 "root mean s quare error" }{TEXT -1 20 " over the interval " }{XPPEDIT 18 0 "[-1,1 ]" "6#7$,$\"\"\"!\"\"F%" }{TEXT -1 82 " of each Runge-Kutta method is estimated as follows using the special procedure " }{TEXT 0 5 "NCint " }{TEXT -1 98 " to perform numerical integration by the 7 point Newt on-Cotes method over 100 equal subintervals." }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 477 "mthds := [`Enright-Verner scheme`,`Verner's Maple scheme`,`a modification of Verner's Maple scheme`,`an \"efficient\" s cheme`,`a scheme of Tsitouras and Papakostas`,`a modification of the T sitouras-Papakostas scheme`,`Verner's \"most efficient\" scheme`]: err s := []:\nDigits := 20:\nfor ct to 7 do\n sm := NCint((k(x)-'kn_RK8_ ||ct'(x))^2,x=-1..1,adaptive=false,numpoints=7,factor=100);\n errs : = [op(errs),sqrt(sm/2)];\nend do:\nDigits := 10:\nlinalg[transpose]([m thds,evalf(errs)]);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#K%'matrixG6#7)7 $%6Enright-Verner~schemeG$\"+k))3VS!#=7$%6Verner's~Maple~schemeG$\"+A^ 8 " 0 "" {MPLTEXT 1 0 599 "evalf[20](plot([k(x)-'kn_RK8_1'(x),k(x)-'kn_RK8_2'(x),k(x)-'kn_RK 8_3'(x),k(x)-'kn_RK8_4'(x),\nk(x)-'kn_RK8_5'(x),k(x)-'kn_RK8_6'(x),k(x )-'kn_RK8_7'(x)],x=-1..1,font=[HELVETICA,9],\ncolor=[COLOR(RGB,.4,0,.9 ),COLOR(RGB,0,.65,0),brown,red,COLOR(RGB,0,.75,.95),magenta,COLOR(RGB, .95,.55,0)],\nlegend=[`Enright-Verner scheme`,`Verner's Maple scheme`, `a modification of Verner's Maple scheme`,`an \"efficient\" scheme`,`a scheme of Tsitouras and Papakostas`,`a modification of the Tsitouras- Papakostas scheme`,`Verner's \"most efficient\" scheme`],title=`error \+ curves for 12 stage order 8 Runge-Kutta methods`));" }}{PARA 13 "" 1 " " {GLPLOT2D 720 752 752 {PLOTDATA 2 "6--%'CURVESG6%7eo7$$!\"\"\"\"!$F* F*7$$!5nmmmm;p0k&*!#?$\")1\"z<\"F/7$$!5LLLL$3QK\"!#>7$$!5nmmmm\"4m(G$)F/$\")06TMF<7$$!5LLLL$3i.9! zF/$\")tru\")F<7$$!5mmmm;/R=0vF/$\")&Q1o\"!#=7$$!5++++]P8#\\4(F/$\")\\ $*pLFL7$$!5mmmm;/siqmF/$\")@.^lFL7$$!5++++](y$pZiF/$\")z,87!#<7$$!5LLL LL$yaE\"eF/$\")#yS=#Ffn7$$!5mmmmm\">s%HaF/$\")#4C`$Ffn7$$!5+++++]$*4)* \\F/$\")'z$HeFfn7$$!5+++++]_&\\c%F/$\")NmN#*Ffn7$$!5+++++]1aZTF/$\")N! HQ\"!#;7$$!5mmmm;/#)[oPF/$\")\"Q#H>F`p7$$!5LLLLL$=exJ$F/$\")1\\\\FF`p7 $$!5LLLLLeW%o7$F/$\")T]^JF`p7$$!5LLLLLL2$f$HF/$\")S>$e$F`p7$$!5mmmmT&o _Qr#F/$\")u;=TF`p7$$!5********\\PYx\"\\#F/$\")0^\"o%F`p7$$!5mmmmTNz>&H #F/$\")W['>&F`p7$$!5LLLLLL7i)4#F/$\")K()=dF`p7$$!5mmmmTNa%H)=F/$\")ZH! H'F`p7$$!5********\\P'psm\"F/$\"))*yZoF`p7$$!5*********\\F&*=Y\"F/$\") (fON(F`p7$$!5********\\74_c7F/$\")bOByF`p7$$!5mmmmT5VBU5F/$\")oki#)F`p 7$$!4LLLLL3x%z#)F/$\")$p\"Q')F`p7$$!5ILLL$e9d;J'!#@$\")\\m<*)F`p7$$!4L LLL$3s$QM%F/$\")pYF\"*F`p7$$!5immmT&QdDG$Fjt$\")%p(4#*F`p7$$!5&******* \\ivF@AFjt$\")Oip#*F`p7$$!5imm;/^wj!p\"Fjt$\")g)4H*F`p7$$!5GLLLeRx**f6 Fjt$\")Qe1$*F`p7$$!5S*****\\7GyNH'!#A$\")MR;$*F`p7$$!2mmmmm\"zr)*F/$\" )^Q?$*F`p7$$\"5]LLL3x\"yY_%Fiv$\")kO=$*F`p7$$\"5ILLL3_Nl.5Fjt$\")m25$* F`p7$$\"5DLL$ekGR[b\"Fjt$\")q_&H*F`p7$$\"5?LLL$3-Dg5#Fjt$\")Cvu#*F`p7$ $\"55LLLe*['R3KFjt$\")fo9#*F`p7$$\"3LLLLezw5VF<$\")`OI\"*F`p7$$\"5]mmm mmJ+IiFjt$\")hzF*)F`p7$$\"3++++v$Q#\\\")F<$\")uoe')F`p7$$\"5lmmm\"z\\1 A-\"F/$\"))=1I)F`p7$$\"4LLLL$e\"*[H7F<$\")y)=)yF`p7$$\"5lmmm;HCjV9F/$ \")(eqR(F`p7$$\"4+++++qvxl\"F<$\")0!=(oF`p7$$\"5++++]7JFn=F/$\")MUJjF` p7$$\"4++++]_qn2#F<$\")l-xdF`p7$$\"5++++vV+ZzAF/$\")(3\"Q_F`p7$$\"4+++ +Dcp@[#F<$\")^P1ZF`p7$$\"5++++v=GB2FF/$\")teMTF`p7$$\"4++++]2'HKHF<$\" )#z;f$F`p7$$\"5NLLL$3UDX8$F/$\")9uMJF`p7$$\"4nmmmmwanL$F<$\")`@6FF`p7$ $\"4+++++v+'oPF<$\")%o!H>F`p7$$\"4LLLLeR<*fTF<$\")wBn8F`p7$$\"4+++++&) Hxe%F<$\")vCC!*Ffn7$$\"4nmmm\"H!o-*\\F<$\")(o#zeFfn7$$\"4++++DTO5T&F<$ \"))e2h$Ffn7$$\"4nmmmmT9C#eF<$\")q%[:#Ffn7$$\"4++++D1*3`iF<$\")GI-7Ffn 7$$\"4LLLLL$*zym'F<$\")E-dlFL7$$\"4LLLL$3N1#4(F<$\");;`LFL7$$\"4nmmm\" HYt7vF<$\");-E;FL7$$\"4+++++q(G**yF<$\")=]\")yF<7$$\"4nmmm;9@BM)F<$\") V\\vJF<7$$\"4LLLLL`v&Q()F<$\")P+M7F<7$$\"4++++DOl5;*F<$\")8w8OF/7$$\"4 ++++v.Uac*F<$\"(l,m%F/7$$\"\"\"F*$!(6BB'F/-%&COLORG6&%$RGBG$\"\"%F)F+$ \"\"*F)-%'LEGENDG6#%6Enright-Verner~schemeG-F$6%7eoF'7$F-$\")6?-7F/7$F 3$\")Ax:XF/7$F8$\")s:g8F<7$F>$\")!)*fa$F<7$FC$\")#4(\\%)F<7$FH$\")imS< FL7$FN$\")E6(\\$FL7$FS$\")$Hz!oFL7$FX$\")^)>E\"Ffn7$Fhn$\");EuAFfn7$F] o$\")kL!o$Ffn7$Fbo$\")TcwgFfn7$Fgo$\")`xH'*Ffn7$F\\p$\")I:U9F`p7$Fbp$ \")R47?F`p7$Fgp$\")'Qx'GF`p7$F\\q$\")M4(G$F`p7$Faq$\")#*RPPF`p7$Ffq$\" )GV&H%F`p7$F[r$\")Y0$)[F`p7$F`r$\")5@?aF`p7$Fer$\")r4lfF`p7$Fjr$\")K7h lF`p7$F_s$\").iUrF`p7$Fds$\")vEqwF`p7$Fis$\")A>g\")F`p7$F^t$\")?Q=')F` p7$Fct$\")x15!*F`p7$Fht$\")If,$*F`p7$F^u$\")[U?&*F`p7$Fcu$\")%pig*F`p7 $Fhu$\")(*po'*F`p7$F]v$\")8)4p*F`p7$Fbv$\")1D2(*F`p7$Fgv$\")@[<(*F`p7$ F]w$\")dk@(*F`p7$Fbw$\"),a>(*F`p7$Fgw$\")N*3r*F`p7$F\\x$\")wr&p*F`p7$F ax$\")([Sn*F`p7$Ffx$\")qR6'*F`p7$F[y$\")rWB&*F`p7$F`y$\")%f@J*F`p7$Fey $\")$o9.*F`p7$Fjy$\")#))zl)F`p7$F_z$\")SB@#)F`p7$Fdz$\")g`:xF`p7$Fiz$ \")tmnrF`p7$F^[l$\")r-/mF`p7$Fc[l$\")5wDgF`p7$Fh[l$\")njjaF`p7$F]\\l$ \")9+4\\F`p7$Fb\\l$\")Wd7VF`p7$Fg\\l$\").FYPF`p7$F\\]l$\"),#F`p7$F[^l$\")m$eU\"F`p7$F`^l$\")NI4%*Ffn7$ Fe^l$\")VPGhFfn7$Fj^l$\")W\"=w$Ffn7$F__l$\"),[VAFfn7$Fd_l$\")+c]7Ffn7$ Fi_l$\")g?6oFL7$F^`l$\")cDwMFL7$Fc`l$\")87\"o\"FL7$Fh`l$\")@*H7)F<7$F] al$\")\"o&eKF<7$Fbal$\").Ie7F<7$Fgal$\"))f,k$F/7$F\\bl$\"(/QP%F/7$Fabl $!(5pc'F/-Ffbl6&FhblF+$\"#l!\"#F+-F^cl6#%6Verner's~Maple~schemeG-F$6%7 eoF'7$F-$\")[%f/\"F/7$F3$\")o*G#RF/7$F8$\")a$*z6F<7$F>$\")NBsIF<7$FC$ \")]=6tF<7$FH$\")f$[]\"FL7$FN$\")k\"3-$FL7$FS$\")tFfn7$F]o$\")SwtJFfn7$Fbo$\")p1R_Ffn7$Fgo$\")6n,$)Ffn7$ F\\p$\")BZ#F`p7$F\\q$\")/OLGF`p7$Faq $\")T\\@KF`p7$Ffq$\")`[-PF`p7$F[r$\")>)*3UF`p7$F`r$\")T)>n%F`p7$Fer$\" )/lT^F`p7$Fjr$\")))RbcF`p7$F_s$\")XichF`p7$Fds$\")[V6mF`p7$Fis$\")/tLq F`p7$F^t$\")=nGuF`p7$Fct$\")-HmxF`p7$Fht$\")Qd1#)F` p7$Fcu$\")L>!G)F`p7$Fhu$\")e+M$)F`p7$F]v$\"):@`$)F`p7$Fbv$\")[Bn$)F`p7 $Fgv$\")Y0w$)F`p7$F]w$\")Lkz$)F`p7$Fbw$\")z#yP)F`p7$Fgw$\")]Pq$)F`p7$F \\x$\")SHd$)F`p7$Fax$\")nhQ$)F`p7$Ffx$\")Kh%G)F`p7$F[y$\")Q!)3#)F`p7$F `y$\")=oE!)F`p7$Fey$\")ot%y(F`p7$Fjy$\")6\"GY(F`p7$F_z$\")jM'3(F`p7$Fd z$\")VX]mF`p7$Fiz$\")N@yhF`p7$F^[l$\"))yBp&F`p7$Fc[l$\")!QR>&F`p7$Fh[l $\")AT4ZF`p7$F]\\l$\")IMJUF`p7$Fb\\l$\")cDFfn7$Fd_ l$\")IIz5Ffn7$Fi_l$\"):P#)eFL7$F^`l$\")3V0IFL7$Fc`l$\")J\"eX\"FL7$Fh`l $\")qS\\qF<7$F]al$\")$Gx$GF<7$Fbal$\")=a-6F<7$Fgal$\")<4PKF/7$F\\bl$\" ('\\2VF/7$Fabl$!(HJR&F/-%'COLOURG6&Fhbl$\")#)eqk!\")$\"))eqk\"Fd_nFe_n -F^cl6#%Ha~modification~of~Verner's~Maple~schemeG-F$6%7eoF'7$F-$\"(Pf* )*F/7$F3$\")Bd9PF/7$F8$\")%>\"=6F<7$F>$\")gH8HF<7$FC$\")q#z$pF<7$FH$\" ),oG9FL7$FN$\")gCpGFL7$FS$\")W8%e&FL7$FX$\")M\"\\.\"Ffn7$Fhn$\")\"QZ'= Ffn7$F]o$\")oIXZzF` p7$Fax$\")3pHzF`p7$Ffx$\")aLyyF`p7$F[y$\")NC1yF`p7$F`y$\")D0LwF`p7$Fey $\")>(HS(F`p7$Fjy$\")F$o4(F`p7$F_z$\")#H)QnF`p7$Fdz$\")DJCjF`p7$Fiz$\" )#H_(eF`p7$F^[l$\")%=KT&F`p7$Fc[l$\")%>#R\\F`p7$Fh[l$\")DXyWF`p7$F]\\l $\")Z#Q-%F`p7$Fb\\l$\")B%\\`$F`p7$Fg\\l$\")JvqIF`p7$F\\]l$\")E1!o#F`p7 $Fa]l$\")`$zJ#F`p7$Ff]l$\")(o\"\\;F`p7$F[^l$\")Dwo6F`p7$F`^l$\")738xFf n7$Fe^l$\")%zQ-&Ffn7$Fj^l$\")!eT3$Ffn7$F__l$\")/hR=Ffn7$Fd_l$\"),nD5Ff n7$Fi_l$\")hI)e&FL7$F^`l$\")y&Q&GFL7$Fc`l$\")w_\"Q\"FL7$Fh`l$\");2&o'F <7$F]al$\")'f')o#F<7$Fbal$\")XUV5F<7$Fgal$\")40eIF/7$F\\bl$\"(&HSSF/7$ Fabl$!(/76&F/-F`_n6&Fhbl$\"*++++\"Fd_nF+F+-F^cl6#%6an~\"efficient\"~sc hemeG-F$6%7eoF'7$F-$\")-%*o5F/7$F3$\")PG1SF/7$F8$\")LA/7F<7$F>$\")$)eL JF<7$FC$\")qc_uF<7$FH$\")YJL:FL7$FN$\")'yn2$FL7$FS$\")TN%)fFL7$FX$\")i b36Ffn7$Fhn$\")*om*>Ffn7$F]o$\")Q+IKFfn7$Fbo$\").QJ`Ffn7$Fgo$\")\"HvW) Ffn7$F\\p$\")a(\\E\"F`p7$Fbp$\")Ezkn\")F`p7$Fey$\")V,@zF`p7$Fjy$\")Ffn7$Fd_l$\")4n)4\"Ffn7$Fi_l$\")!*=*)fFL7$F^`l$\")f)41$FL7$Fc `l$\")IR$[\"FL7$Fh`l$\")>&o=(F<7$F]al$\")YO&*GF<7$Fbal$\")*fj7\"F<7$Fg al$\")U0;LF/7$F\\bl$\"(0`[%F/7$Fabl$!((*4X&F/-Ffbl6&FhblF+$\"#vF`am$\" #&*F`am-F^cl6#%Ea~scheme~of~Tsitouras~and~PapakostasG-F$6%7eoF'7$F-$\" )J#>2\"F/7$F3$\"))H4-%F/7$F8$\")-f47F<7$F>$\")/&)\\JF<7$FC$\")V)o\\(F< 7$FH$\")1>V:FL7$FN$\")E1)4$FL7$FS$\")!>y-'FL7$FX$\")Z!p6\"Ffn7$Fhn$\") /87?Ffn7$F]o$\")\"[aD$Ffn7$Fbo$\")W,u`Ffn7$Fgo$\")jf:&)Ffn7$F\\p$\")r@ v7F`p7$Fbp$\")A7zy'F`p7$Fis$\")?;:sF`p7$F^t$\" )GH?wF`p7$Fct$\");imzF`p7$Fht$\")%)QC#)F`p7$F^u$\")$zyT)F`p7$Fcu$\")Oy $\\)F`p7$Fhu$\")X)*[&)F`p7$F]v$\")foo&)F`p7$Fbv$\")42$e)F`p7$Fgv$\")#= @f)F`p7$F]w$\")&*z&f)F`p7$Fbw$\")v$Rf)F`p7$Fgw$\")AH'e)F`p7$F\\x$\")Q( Gd)F`p7$Fax$\")Yr`&)F`p7$Ffx$\")zJ)\\)F`p7$F[y$\")Fb?%)F`p7$F`y$\")?tL #)F`p7$Fey$\")\\a&)zF`p7$Fjy$\")OJbwF`p7$F_z$\")h8psF`p7$Fdz$\")$)*>#o F`p7$Fiz$\")QdPjF`p7$F^[l$\")Y?ReF`p7$Fc[l$\")S!zK&F`p7$Fh[l$\")t(3$[F `p7$F]\\l$\")PZSVF`p7$Fb\\l$\")778QF`p7$Fg\\l$\")dS7LF`p7$F\\]l$\")P(4 *GF`p7$Fa]l$\")NN+DF`p7$Ff]l$\")$p*yaFfn7$Fj^l$\")bRFLFfn7$F__l$\")_'[)>Ffn7$Fd_l$\")P y16Ffn7$Fi_l$\")*Q6.'FL7$F^`l$\")/i!3$FL7$Fc`l$\")![;\\\"FL7$Fh`l$\")) \\%>sF<7$F]al$\")F6/HF<7$Fbal$\")M4F6F<7$Fgal$\")Fw,LF/7$F\\bl$\"(*4QV F/7$Fabl$!(_Db&F/-F`_n6&FhblFg]oF+Fg]o-F^cl6#%Ra~modification~of~the~T sitouras-Papakostas~schemeG-F$6%7eoF'7$F-$!'/')=F/7$F3$!'yBjF/7$F8$!'M #p\"F<7$F>$!'E3RF<7$FC$!'zf!)F<7$FH$!'r#\\\"FL7$FN$!'))pEFL7$FS$!'(pt% FL7$FX$!&'=\")Ffn7$Fhn$!'Uo8Ffn7$F]o$!'M<@Ffn7$Fbo$!'>`LFfn7$Fgo$!'e)> &Ffn7$F\\p$!&an(F`p7$Fbp$!'eh5F`p7$Fgp$!'60:F`p7$F\\q$!'wAF`p7$Ffq$!'tYAF`p7$F[r$!'A_DF`p7$F`r$!'(>$GF`p7$Fer$!'B;JF`p7$Fjr$! '&pU$F`p7$F_s$!'cIPF`p7$Fds$!'*f+%F`p7$Fis$!'yhUF`p7$F^t$!'?,XF`p7$Fct $!'P0ZF`p7$Fht$!'od[F`p7$F^u$!'Js\\F`p7$Fcu$!'%o,&F`p7$Fhu$!'^\\]F`p7$ F]v$!'Rh]F`p7$Fbv$!'2q]F`p7$Fgv$!'*\\2&F`p7$F]w$!'Ex]F`p7$Fbw$!'_w]F`p 7$Fgw$!'\">2&F`p7$F\\x$!'-k]F`p7$Fax$!'I_]F`p7$Ffx$!'q>]F`p7$F[y$!'$Q( \\F`p7$F`y$!']j[F`p7$Fey$!'+AXF`p7$F_z$!'&QH%F`p7$Fdz$!' ,ISF`p7$Fiz$!'5WPF`p7$F^[l$!'**\\MF`p7$Fc[l$!'N[JF`p7$Fh[l$!'F`p7$F\\]l$!'/8rFfn7$Fi_l$!'^jQFL7$F^`l$! '4o=FL7$Fc`l$!&+P(FL7$Fh`l$!'j$*=F<7$F]al$\"&#ewF<7$Fbal$\"'=e;F<7$Fga l$\"(')=k\"F/7$F\\bl$\"(2]M\"F/7$Fabl$\"(Cv-\"F/-Ffbl6&FhblF[\\p$\"#bF `amF+-F^cl6#%AVerner's~\"most~efficient\"~schemeG-%%FONTG6$%*HELVETICA GF\\cl-%&TITLEG6#%Verror~curves~for~12~stage~order~8~Runge-Kutta~metho dsG-%+AXESLABELSG6$Q\"x6\"Q!F^iq-%%VIEWG6$;F(Fabl%(DEFAULTG" 1 2 0 1 10 0 2 9 1 4 2 1.000000 45.000000 45.000000 0 1 "Enright-Verner scheme " "Verner's Maple scheme" "a modification of Verner's Maple scheme" "a n \"efficient\" scheme" "a scheme of Tsitouras and Papakostas" "a modi fication of the Tsitouras-Papakostas scheme" "Verner's \"most efficien t\" scheme" }}}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 48 "Test 5 \+ of 12 stage, order 8 Runge-Kutta methods " }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "dy/dx=16/((16*x+1)*y)" "6#/*&%#dyG\"\"\"%#dxG !\"\"*&\"#;F&*&,&*&F*F&%\"xGF&F&F&F&F&%\"yGF&F(" }{TEXT -1 10 ", \+ " }{XPPEDIT 18 0 "y(0)=1" "6#/-%\"yG6#\"\"!\"\"\"" }{TEXT -1 1 " " }}{PARA 0 "" 0 "" {TEXT -1 10 "Solution: " }}{PARA 256 "" 0 "" {TEXT -1 2 " " }{XPPEDIT 18 0 "y=sqrt (2*ln(16*x+1)+1)" "6#/%\"yG-%%sqrtG6# ,&*&\"\"#\"\"\"-%#lnG6#,&*&\"#;F+%\"xGF+F+F+F+F+F+F+F+" }{TEXT -1 2 ". " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 165 "de := diff(y(x),x)=16/((16*x+1)*y(x));\nic := y(0)=1 ;\ndsolve(\{de,ic\},y(x));\ns := unapply(rhs(%),x):\nplot(s(x),x=0..0. 5,0..2.6,font=[HELVETICA,9],labels=[`x`,`y(x)`]);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%#deG/-%%diffG6$-%\"yG6#%\"xGF,,$*(\"#;\"\"\",&*&F/F0F ,F0F0F0F0!\"\"F)F3F0" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%#icG/-%\"yG6 #\"\"!\"\"\"" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/-%\"yG6#%\"xG*$,&*&\" \"#\"\"\"-%#lnG6#,&*&\"#;F,F'F,F,F,F,F,F,F,F,#F,F+" }}{PARA 13 "" 1 " " {GLPLOT2D 503 318 318 {PLOTDATA 2 "6&-%'CURVESG6$7U7$$\"\"!F)$\"\"\" F)7$$\"3WmmmT&)G\\a!#?$\"3EP,(Qyl.3\"!#<7$$\"3ILLL3x&)*3\"!#>$\"3?25A! pa&\\6F27$$\"3-+]i!R(*Rc\"F6$\"3oz*p77wF?\"F27$$\"3umm\"H2P\"Q?F6$\"3] _vibZz]7F27$$\"3MLL$eRwX5$F6$\"3!Qb+fY'3W8F27$$\"3CLL$3x%3yTF6$\"31#\\ \\E7=EU\"F27$$\"3=mm\"z%4\\Y_F6$\"3s(e$4OUg*[\"F27$$\"3)HL$eR-/PiF6$\" 3.fPtw=4W:F27$$\"3A***\\il'pisF6$\"3/07@a`R%f\"F27$$\"3`KLe*)>VB$)F6$ \"3K!\\`od36k\"F27$$\"3!))**\\7`l2Q*F6$\"3#HUv\"fmC$o\"F27$$\"3smm;/j$ o/\"!#=$\"3:'H!f>cuAjU6Fco$\"3K$o8QC!za=F27$ $\"3)*****\\P[6j9Fco$\"39iuo+OIZ=F27$$\"3KL$e*[z(yb\"Fco$\"3Q:]fA\\>F 27$$\"3))**\\iSj0x=Fco$\"3-5Hbh&QF%>F27$$\"3Wmmm\"pW`(>Fco$\"3So#znsrC '>F27$$\"35+]i!f#=$3#Fco$\"3w)>Y)R!pI)>F27$$\"3/+](=xpe=#Fco$\"3?*eB@. [<+#F27$$\"3smm\"H28IH#Fco$\"3/Fyh^(\\.-#F27$$\"3km;zpSS\"R#Fco$\"3)4U S+%ypO?F27$$\"3GLL3_?`(\\#Fco$\"3#4Cj+a0O0#F27$$\"3#HLe*)>pxg#Fco$\"3a b\\mG7Vq?F27$$\"3u**\\Pf4t.FFco$\"3Cx7m@=^%3#F27$$\"32LLe*Gst!GFco$\"3 Q>I Fco$\"3&ocGC'[]F@F27$$\"3h**\\i!RU07$Fco$\"3HCH$Q\")f.9#F27$$\"3b***\\ (=S2LKFco$\"3C`wrWc9a@F27$$\"3Kmmm\"p)=MLFco$\"3;=S,IA7m@F27$$\"3!**** *\\(=]@W$Fco$\"3w4%eC\"p]y@F27$$\"35L$e*[$z*RNFco$\"3UyOr,.R*=#F27$$\" 3#*****\\iC$pk$Fco$\"3wIdFs1%4?#F27$$\"39m;H2qcZPFco$\"3Qbx\"QY%\\6AF2 7$$\"3q**\\7.\"fF&QFco$\"3f+!e(oz@AAF27$$\"3Ymm;/OgbRFco$\"36qG(yA8CB# F27$$\"3y**\\ilAFjSFco$\"3v.zLgjzUAF27$$\"3YLLL$)*pp;%Fco$\"3IImU*yHDD #F27$$\"3?LL3xe,tUFco$\"3I%R!fhiAiAF27$$\"3em;HdO=yVFco$\"3?ogo1xfrAF2 7$$\"3))*****\\#>#[Z%Fco$\"3EO%fx0/+G#F27$$\"3immT&G!e&e%Fco$\"3)zsS%e \"3%*G#F27$$\"3;LLL$)Qk%o%Fco$\"35e8d5)>wH#F27$$\"37+]iSjE!z%Fco$\"3e% 4h.zwhI#F27$$\"35+]P40O\"*[Fco$\"3Nwd2,K=9BF27$$\"3++++++++]Fco$\"3m'> ())[`fABF2-%'COLOURG6&%$RGBG$\"#5!\"\"F(F(-%%FONTG6$%*HELVETICAG\"\"*- %+AXESLABELSG6$%\"xG%%y(x)G-%%VIEWG6$;F($\"\"&Fj[l;F($\"#EFj[l" 1 2 0 1 10 0 2 9 1 4 2 1.000000 45.000000 45.000000 0 0 "Curve 1" }}}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 32 "The following c ode constructs a " }{TEXT 260 17 "discrete solution" }{TEXT -1 44 " ba sed on each of the methods and gives the " }{TEXT 260 22 "root mean sq uare error" }{TEXT -1 18 " of each solution." }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 794 "S := (x,y) -> 16/((16*x+1)*y): hh := 0.005: numst eps := 100: x0 := 0: y0 := 1:\nmatrix([[`slope field: `,S(x,y)],[`in itial point: `,``(x0,y0)],[`step width: `,hh],\n[`no. of steps: `, numsteps]]);``;\nmthds := [`Enright-Verner scheme`,`Verner's Maple sch eme`,`a modification of Verner's Maple scheme`,`an \"efficient\" schem e`,`a scheme of Tsitouras and Papakostas`,`a modification of the Tsito uras-Papakostas scheme`,`Verner's \"most efficient\" scheme`]: errs := []:\nDigits := 20:\nfor ct to 7 do\n Sn_RK8_||ct := RK8_||ct(S(x,y) ,x,y,x0,y0,hh,numsteps,false);\n sm := 0: numpts := nops(Sn_RK8_||ct ):\n for ii to numpts do\n sm := sm+(Sn_RK8_||ct[ii,2]-s(Sn_RK8 _||ct[ii,1]))^2;\n end do:\n errs := [op(errs),sqrt(sm/numpts)];\n end do:\nDigits := 10:\nlinalg[transpose]([mthds,evalf(errs)]);" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#K%'matrixG6#7&7$%0slope~field:~~~G,$*( \"#;\"\"\",&*&F+F,%\"xGF,F,F,F,!\"\"%\"yGF0F,7$%0initial~point:~G-%!G6 $\"\"!F,7$%/step~width:~~~G$\"\"&!\"$7$%1no.~of~steps:~~~G\"$+\"Q)ppri nt206\"" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%!G" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#K%'matrixG6#7)7$%6Enright-Verner~schemeG$\"+%[%*)3z!#C7 $%6Verner's~Maple~schemeG$\"+;\\w46!#B7$%Ha~modification~of~Verner's~M aple~schemeG$\"+:(>xG(F+7$%6an~\"efficient\"~schemeG$\"+*)[RcsF+7$%Ea~ scheme~of~Tsitouras~and~PapakostasG$\"+S24;NF+7$%Ra~modification~of~th e~Tsitouras-Papakostas~schemeG$\"+d9IddF+7$%AVerner's~\"most~efficient \"~schemeG$\"+)4)3(3\"F0Q)pprint216\"" }}}{PARA 0 "" 0 "" {TEXT -1 0 " " }}{PARA 0 "" 0 "" {TEXT -1 30 "The following code constructs " } {TEXT 260 20 "numerical procedures" }{TEXT -1 56 " for solutions based on each of the Runge-Kutta schemes." }}{PARA 0 "" 0 "" {TEXT -1 75 "T he error in the value obtained by each of the methods at the point whe re " }{XPPEDIT 18 0 "x = 0;" "6#/%\"xG\"\"!" }{TEXT -1 21 ".4995 is \+ also given." }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 726 "S := (x,y) - > 16/((16*x+1)*y): hh := 0.005: numsteps := 100: x0 := 0: y0 := 1:\nma trix([[`slope field: `,S(x,y)],[`initial point: `,``(x0,y0)],[`step \+ width: `,hh],\n[`no. of steps: `,numsteps]]);``;\nmthds := [`Enrig ht-Verner scheme`,`Verner's Maple scheme`,`a modification of Verner's \+ Maple scheme`,`an \"efficient\" scheme`,`a scheme of Tsitouras and Pap akostas`,`a modification of the Tsitouras-Papakostas scheme`,`Verner's \"most efficient\" scheme`]: errs := []:\nDigits := 25:\nfor ct to 7 \+ do\n sn_RK8_||ct := RK8_||ct(S(x,y),x,y,x0,y0,hh,numsteps,true);\nen d do:\nxx := 0.4995: sxx := evalf(s(xx)):\nfor ct to 7 do\n errs := \+ [op(errs),abs(sn_RK8_||ct(xx)-sxx)];\nend do:\nDigits := 10:\nlinalg[t ranspose]([mthds,evalf(errs)]);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#K%' matrixG6#7&7$%0slope~field:~~~G,$*(\"#;\"\"\",&*&F+F,%\"xGF,F,F,F,!\" \"%\"yGF0F,7$%0initial~point:~G-%!G6$\"\"!F,7$%/step~width:~~~G$\"\"&! \"$7$%1no.~of~steps:~~~G\"$+\"Q)pprint226\"" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%!G" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#K%'matrixG6#7)7$ %6Enright-Verner~schemeG$\"+\"*4_;l!#C7$%6Verner's~Maple~schemeG$\"+v% *\\o!*F+7$%Ha~modification~of~Verner's~Maple~schemeG$\"+z4()\\fF+7$%6a n~\"efficient\"~schemeG$\"+#yyF$fF+7$%Ea~scheme~of~Tsitouras~and~Papak ostasG$\"+jl/vGF+7$%Ra~modification~of~the~Tsitouras-Papakostas~scheme G$\"+^^r-ZF+7$%AVerner's~\"most~efficient\"~schemeG$\"+q)*4u))F+Q)ppri nt236\"" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 4 "The " }{TEXT 260 22 "root mean square error" }{TEXT -1 110 " over t he interval [0, 0.5] of each Runge-Kutta method is estimated as foll ows using the special procedure " }{TEXT 0 5 "NCint" }{TEXT -1 97 " \+ to perform numerical integration by the 7 point Newton-Cotes method ov er 50 equal subintervals." }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 479 "mthds := [`Enright-Verner scheme`,`Verner's Maple scheme`,`a modi fication of Verner's Maple scheme`,`an \"efficient\" scheme`,`a scheme of Tsitouras and Papakostas`,`a modification of the Tsitouras-Papakos tas scheme`,`Verner's \"most efficient\" scheme`]: errs := []:\nDigits := 20:\nfor ct to 7 do\n sm := NCint((s(x)-'sn_RK8_||ct'(x))^2,x=0. .0.5,adaptive=false,numpoints=7,factor=50);\n errs := [op(errs),sqrt (sm/0.5)];\nend do:\nDigits := 10:\nlinalg[transpose]([mthds,evalf(err s)]);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#K%'matrixG6#7)7$%6Enright-Ver ner~schemeG$\"+y*p^!z!#C7$%6Verner's~Maple~schemeG$\"+HhY46!#B7$%Ha~mo dification~of~Verner's~Maple~schemeG$\"+oK$fG(F+7$%6an~\"efficient\"~s chemeG$\"+C'QVD(F+7$%Ea~scheme~of~Tsitouras~and~PapakostasG$\"+U`0:NF+ 7$%Ra~modification~of~the~Tsitouras-Papakostas~schemeG$\"+*oEev&F+7$%A Verner's~\"most~efficient\"~schemeG$\"+\\t$o3\"F0Q)pprint246\"" }}} {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 92 "The follo wing error graphs are constructed using the numerical procedures for t he solutions." }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 600 "evalf[20]( plot([s(x)-'sn_RK8_1'(x),s(x)-'sn_RK8_2'(x),s(x)-'sn_RK8_3'(x),s(x)-'s n_RK8_4'(x),\ns(x)-'sn_RK8_5'(x),s(x)-'sn_RK8_6'(x),s(x)-'sn_RK8_7'(x) ],x=0..0.5,font=[HELVETICA,9],\ncolor=[COLOR(RGB,.4,0,.9),COLOR(RGB,0, .65,0),brown,red,COLOR(RGB,0,.75,.95),magenta,COLOR(RGB,.95,.55,0)],\n legend=[`Enright-Verner scheme`,`Verner's Maple scheme`,`a modificatio n of Verner's Maple scheme`,`an \"efficient\" scheme`,`a scheme of Tsi touras and Papakostas`,`a modification of the Tsitouras-Papakostas sch eme`,`Verner's \"most efficient\" scheme`],title=`error curves for 12 \+ stage order 8 Runge-Kutta methods`));" }}{PARA 13 "" 1 "" {GLPLOT2D 849 587 587 {PLOTDATA 2 "6--%'CURVESG6%7^p7$$\"\"!F)F(7$$\"5SLLL3x1h6o !#BF(7$$\"5ommmTN@Ki8!#A$!\"$!#>7$$\"5NLL3FpE!Hq\"F1$!#EF47$$\"5-++]7. K[V?F1$!$=\"F47$$\"5omm\"zptjSQ#F1$!$>%F47$$\"5NLLL$3FWYs#F1$!%G7F47$$ \"5-++vo/[AlIF1$!%)3$F47$$\"5omm;aQ`!eS$F1$!%toF47$$\"5NLLeRseQYPF1$!& )y8F47$$\"5-+++D1k'p3%F1$!&J_#F47$$\"5pmmT5SpaFWF1$!&'RUF47$$\"5OLL$eR ZF\"oZF1$!&\\a'F47$$\"50++D\"y+3(3^F1$!&%R$)F47$$\"5qmmmmT&)G\\aF1$!&M I)F47$$\"50++]P4'\\/8'F1$!&LB)F47$$F,F1$!&n;)F47$$\"5qmm;zW6B`#o\"F`r$ !'2a6F47$$\"5MLL3xJs1,=F`r$!'VV6F47$$\"5nm\"Hd?pM.'=F`r$!'eS6F47$$\"5+ +]PM_@g>>F`r$!'hV6F47$$\"5ML3-j7'p)y>F`r$!'?g6F47$$\"5nmmm\"H2P\"Q?F`r $!'Eo6F47$$\"5+++]PMnNrDF`r$!'OW6F47$$\"5MLLL$eRwX5$F`r$!'r:6F47$$\"5N LLL$3x%3yTF`r$!'og5F47$$\"5ommm\"z%4\\Y_F`r$!'%[,\"F47$$\"5NLLLeR-/PiF `r$!&iz*F47$$\"5-+++DcmpisF`r$!&%*[*F47$$\"5OLLLe*)>VB$)F`r$!&-A*F47$$ \"5.+++DJbw!Q*F`r$!&)*)*)F47$$\"5nmmm;/j$o/\"!#?$!&Ry)F47$$\"5MLLL3_>j U6Fjx$!&Oi)F47$$\"5++++]i^Z]7Fjx$!&CY)F47$$\"5++++](=h(e8Fjx$!&xJ)F47$ $\"5++++]P[6j9Fjx$!&=>)F47$$\"5MLL$e*[z(yb\"Fjx$!&t3)F47$$\"5nmmm;a/cq ;Fjx$!&M(zF47$$\"5nmmmm;t,mFjx$!&5r(F47$$\"5,++]i!f#=$3#Fjx$!&5j(F47$$\"5,++](=xpe =#Fjx$!&)fvF47$$\"5nmmm\"H28IH#Fjx$!&,\\(F47$$\"5nmm;zpSS\"R#Fjx$!&,V( F47$$\"5MLLL3_?`(\\#Fjx$!&)otF47$$\"5MLL$e*)>pxg#Fjx$!&*3tF47$$\"5,++] Pf4t.FFjx$!&'fsF47$$\"5MLLLe*Gst!GFjx$!&*3sF47$$\"5,++++DRW9HFjx$!&#fr F47$$\"5,+++DJE>>IFjx$!&H6(F47$$\"5,++]i!RU07$Fjx$!&-2(F47$$\"5,+++v=S 2LKFjx$!&]-(F47$$\"5ommmm\"p)=MLFjx$!&h)pF47$$\"5,+++](=]@W$Fjx$!&k%pF 47$$\"5MLL$e*[$z*RNFjx$!&=\"pF47$$\"5,+++]iC$pk$Fjx$!&c(oF47$$\"5omm;H 2qcZPFjx$!&F%oF47$$\"5,++]7.\"fF&QFjx$!&)4oF47$$\"5ommm;/OgbRFjx$!&(yn F47$$\"5,++]ilAFjSFjx$!&tu'F47$$\"5NLLLL$)*pp;%Fjx$!&\"=nF47$$\"5NLLL3 xe,tUFjx$!&$*o'F47$$\"5omm;HdO=yVFjx$!&#[Z%Fjx$!&rj' F47$$\"5ommmT&G!e&e%Fjx$!&*4mF47$$\"5NLLLL$)Qk%o%Fjx$!&je'F47$$\"5-++] iSjE!z%Fjx$!&=c'F47$$\"5-++]P40O\"*[Fjx$!&#RlF47$$\"\"&!\"\"$!&a^'F4-% &COLORG6&%$RGBG$\"\"%FidlF($\"\"*Fidl-%'LEGENDG6#%6Enright-Verner~sche meG-F$6%7joF'7$F/$\"\"\"F47$FE$\"$i&F47$FJ$\"%i;F47$FO$\"%zVF47$FT$\"& A0\"F47$FY$\"&DM#F47$Fhn$\"&*))[F47$F]o$\"&il*F47$Fbo$\"'w!\\\"F47$Fgo $\"'M%[\"F47$Fap$\"'uf9F47$Fjp$\"']T9F47$Fdq$\"'7U:F47$Fir$\"'f- 'Q\"F`r$\"'Sh;F47$$\"5n;a)3*R(GeT\"F`r$\"'Bk;F47$Fhs$\"'&Hn\"F47$$\"5+ ]7`>+i4v9F`r$\"'G\"p\"F47$$\"5nmT&Q.$*HZ]\"F`r$\"'T<1\"F47$F`\\l$\"'._5F 47$Fe\\l$\"'MU5F47$Fj\\l$\"'(R.\"F47$F_]l$\"'ZD5F47$Fd]l$\"'9<5F47$Fi] l$\"'E55F47$F^^l$\"'@.5F47$Fc^l$\"&G'**F47$Fh^l$\"&&)*)*F47$F]_l$\"&!R )*F47$Fb_l$\"&gx*F47$Fg_l$\"&?s*F47$F\\`l$\"&nm*F47$Fa`l$\"&(='*F47$Ff `l$\"&#o&*F47$F[al$\"&D_*F47$F`al$\"&lZ*F47$Feal$\"&LV*F47$Fjal$\"&'*Q *F47$F_bl$\"&!\\$*F47$Fdbl$\"&!4$*F47$Fibl$\"&0F*F47$F^cl$\"&kB*F47$Fc cl$\"&&)>*F47$Fhcl$\"&c;*F47$F]dl$\"&;8*F47$Fbdl$\"&**4*F47$Fgdl$\"&q1 *F4-F]el6&F_elF($\"#l!\"#F(-Feel6#%6Verner's~Maple~schemeG-F$6%7ioF'F[ fl7$FE$\"$*GF47$FJ$\"$2*F47$FO$\"%6DF47$FT$\"%)H'F47$FY$\"&ZX\"F47$Fhn $\"&]8$F47$F]o$\"&!ojF47$Fbo$\"'4+5F47$Fgo$\"&y&**F47$Fap$\"&Fz*F47$Fj p$\"&Gm*F47$Fdq$\"'(e-\"F47$Fir$\"'yG6F47$F^s$\"'O:6F47$Fcs$\"'P.6F47$ F\\il$\"'J,6F47$Fail$\"'R+6F47$Ffil$\"'],6F47$Fhs$\"';16F47$F^jl$\"'x; 6F47$Fcjl$\"'YK6F47$Fhjl$\"'XH6F47$F]t$\"'[E6F47$Fbt$\"'&\\6\"F47$Fgt$ \"',/6F47$F\\u$\"';*4\"F47$Fau$\"'#f4\"F47$Ffu$\"'U(4\"F47$F[v$\"'7(4 \"F47$Fev$\"'(p-\"F47$Fjv$\"&6r*F47$F_w$\"&cF*F47$Fdw$\"&&[*)F47$Fiw$ \"&hm)F47$F^x$\"&%>%)F47$Fcx$\"&&3#)F47$Fhx$\"&.-)F47$F^y$\"&Q(yF47$Fc y$\"&ms(F47$Fhy$\"&Wf(F47$F]z$\"&&zuF47$Fbz$\"&TQ(F47$Fgz$\"&-G(F47$F \\[l$\"&#*>(F47$Fa[l$\"&?6(F47$Ff[l$\"&1/(F47$F[\\l$\"&u'pF47$F`\\l$\" &C!pF47$Fe\\l$\"&*QoF47$Fj\\l$\"&Ry'F47$F_]l$\"&#GnF47$Fd]l$\"&Nn'F47$ Fi]l$\"&%GmF47$F^^l$\"&@e'F47$Fc^l$\"&m`'F47$Fh^l$\"&W\\'F47$F]_l$\"&a X'F47$Fb_l$\"&TT'F47$Fg_l$\"&(yjF47$F\\`l$\"&BM'F47$Fa`l$\"&4J'F47$Ff` l$\"&xF'F47$F[al$\"&yC'F47$F`al$\"&w@'F47$Feal$\"&#*='F47$Fjal$\"&0;'F 47$F_bl$\"&R8'F47$Fdbl$\"&x5'F47$Fibl$\"&C3'F47$F^cl$\"&+1'F47$Fccl$\" &_.'F47$Fhcl$\"&N,'F47$F]dl$\"&7*fF47$Fbdl$\"&0(fF47$Fgdl$\"&*[fF4-%'C OLOURG6&F_el$\")#)eqk!\")$\"))eqk\"F_dnF`dn-Feel6#%Ha~modification~of~ Verner's~Maple~schemeG-F$6%7doF'F[fl7$FE$\"$%QF47$FJ$\"%G6F47$FO$\"%`H F47$FT$\"%SqF47$FY$\"&Ob\"F47$Fhn$\"&A@$F47$F]o$\"&FG'F47$Fbo$\"&Rj*F4 7$Fgo$\"&Cf*F47$Fap$\"&MV*F47$Fjp$\"&yJ*F47$Fdq$\"'O+5F47$Fir$\"'626F4 7$F^s$\"'&R4\"F47$Fcs$\"'e#3\"F47$Fhs$\"'<*3\"F47$F]t$\"'096F47$Fbt$\" 'k-6F47$Fgt$\"'!>4\"F47$F\\u$\"'O(3\"F47$Fau$\"''\\3\"F47$Ffu$\"'U)3\" F47$F[v$\"'E*3\"F47$F`v$\"'#[0\"F47$Fev$\"'hA5F47$Fjv$\"&#y'*F47$F_w$ \"&pC*F47$Fdw$\"&>#*)F47$Fiw$\"&3k)F47$F^x$\"&]R)F47$Fcx$\"&[=)F47$Fhx $\"&s*zF47$F^y$\"&6&yF47$Fcy$\"&Wq(F47$Fhy$\"&Ed(F47$F]z$\"&!euF47$Fbz $\"&HO(F47$Fgz$\"&$fsF47$F\\[l$\"&&yrF47$Fa[l$\"&;4(F47$Ff[l$\"&.-(F47 $F[\\l$\"&u%pF47$F`\\l$\"&D)oF47$Fe\\l$\"&#>oF47$Fj\\l$\"&Ww'F47$F_]l$ \"&)3nF47$Fd]l$\"&Vl'F47$Fi]l$\"&$4mF47$F^^l$\"&Kc'F47$Fc^l$\"&y^'F47$ Fh^l$\"&eZ'F47$F]_l$\"&oV'F47$Fb_l$\"&cR'F47$Fg_l$\"&.O'F47$F\\`l$\"&T K'F47$Fa`l$\"&FH'F47$Ff`l$\"&(fiF47$F[al$\"&)HiF47$F`al$\"&(*>'F47$Fea l$\"&9<'F47$Fjal$\"&G9'F47$F_bl$\"&j6'F47$Fdbl$\"&,4'F47$Fibl$\"&\\1'F 47$F^cl$\"&E/'F47$Fccl$\"&y,'F47$Fhcl$\"&j*fF47$F]dl$\"&S(fF47$Fbdl$\" &L&fF47$Fgdl$\"&=$fF4-F[dn6&F_el$\"*++++\"F_dnF(F(-Feel6#%6an~\"effici ent\"~schemeG-F$6%7doF'F[fl7$FE$\"$'=F47$FJ$\"$^&F47$FO$\"%Z9F47$FT$\" %]MF47$FY$\"%(f(F47$Fhn$\"&Wc\"F47$F]o$\"&F/$F47$Fbo$\"&Xk%F47$Fgo$\"& Xi%F47$Fap$\"&ya%F47$Fjp$\"&D\\%F47$Fdq$\"&M$[F47$Fir$\"&uN&F47$F^s$\" &QH&F47$Fcs$\"&*Q_F47$Fhs$\"&MF&F47$F]t$\"&vR&F47$Fbt$\"&BM&F47$Fgt$\" &.H&F47$F\\u$\"&%o_F47$Fau$\"&rD&F47$Ffu$\"&ZF&F47$F[v$\"&$z_F47$F`v$ \"&E6&F47$Fev$\"&k&\\F47$Fjv$\"&1p%F47$F_w$\"&8[%F47$Fdw$\"&PK%F47$Fiw $\"&u=%F47$F^x$\"&$oSF47$Fcx$\"&k'RF47$Fhx$\"&b(QF47$F^y$\"&Z!QF47$Fcy $\"&Ot$F47$Fhy$\"&(pOF47$F]z$\"&Th$F47$Fbz$\"&\"oNF47$Fgz$\"&z^$F47$F \\[l$\"&(yMF47$Fa[l$\"&mV$F47$Ff[l$\"&@S$F47$F[\\l$\"&nO$F47$F`\\l$\"& `L$F47$Fe\\l$\"&YI$F47$Fj\\l$\"&!yKF47$F_]l$\"&6D$F47$Fd]l$\"&ZA$F47$F i]l$\"&H?$F47$F^^l$\"&0=$F47$Fc^l$\"&'eJF47$Fh^l$\"&#QJF47$F]_l$\"&$>J F47$Fb_l$\"&$*4$F47$Fg_l$\"&B3$F47$F\\`l$\"&Z1$F47$Fa`l$\"&&\\IF47$Ff` l$\"&N.$F47$F[al$\"&!>IF47$F`al$\"&W+$F47$Feal$\"&2*HF47$Fjal$\"&o(HF4 7$F_bl$\"&S'HF47$Fdbl$\"&8&HF47$Fibl$\"&\"RHF47$F^cl$\"&$GHF47$Fccl$\" &j\"HF47$Fhcl$\"&e!HF47$F]dl$\"&]*GF47$Fbdl$\"&])GF47$Fgdl$\"&Y(GF4-F] el6&F_elF($\"#vFbem$\"#&*Fbem-Feel6#%Ea~scheme~of~Tsitouras~and~Papako stasG-F$6%7joF'F[fl7$FE$\"$O#F47$FJ$\"$M(F47$FO$\"%??F47$FT$\"%K]F47$F Y$\"&_:\"F47$Fhn$\"&[Z#F47$F]o$\"&))*\\F47$Fbo$\"&=#yF47$Fgo$\"&\")y(F 47$Fap$\"&!fwF47$Fjp$\"&(evF47$Fdq$\"&m/)F47$Fir$\"&F())F47$F^s$\"&tw) F47$Fcs$\"&Pn)F47$F\\il$\"&$e')F47$Fail$\"&Dl)F47$Ffil$\"&Om)F47$Fhs$ \"&Tq)F47$F^jl$\"&Oz)F47$Fcjl$\"&U#*)F47$Fhjl$\"&0!*)F47$F]t$\"&r())F4 7$Fbt$\"&iy)F47$Fgt$\"&-q)F47$F\\u$\"&Dm)F47$Fau$\"&*Q')F47$Ffu$\"&gl) F47$F[v$\"&pl)F47$F`v$\"&GP)F47$Fev$\"&@6)F47$Fjv$\"&On(F47$F_w$\"&0L( F47$Fdw$\"&C2(F47$Fiw$\"&%\\oF47$F^x$\"&Xl'F47$Fcx$\"&z['F47$Fhx$\"&\" RjF47$F^y$\"&LA'F47$Fcy$\"&q5'F47$Fhy$\"&D+'F47$F]z$\"&<\"fF47$Fbz$\"& j$eF47$Fgz$\"&Uv&F47$F\\[l$\"&-p&F47$Fa[l$\"&8i&F47$Ff[l$\"&[c&F47$F[ \\l$\"&p]&F47$F`\\l$\"&cX&F47$Fe\\l$\"&aS&F47$Fj\\l$\"&>O&F47$F_]l$\"& zJ&F47$Fd]lFdfo7$Fi]l$\"&!R_F47$F^^l$\"&C?&F47$Fc^l$\"&l;&F47$Fh^l$\"& J8&F47$F]_l$\"&B5&F47$Fb_l$\"&'p]F47$Fg_l$\"&;/&F47$F\\`l$\"&H,&F47$Fa `l$\"&!))\\F47$Ff`l$\"&='\\F47$F[al$\"&#Q\\F47$F`al$\"&V\"\\F47$Feal$ \"&>*[F47$Fjal$\"&#p[F47$F_bl$\"&#[[F47$Fdbl$\"&u#[F47$Fibl$\"&v![F47$ F^cl$\"&)*y%F47$Fccl$\"&,x%F47$Fhcl$\"&Iv%F47$F]dl$\"&at%F47$Fbdl$\"&! >ZF47$Fgdl$\"&>q%F4-F[dn6&F_elF\\boF(F\\bo-Feel6#%Ra~modification~of~t he~Tsitouras-Papakostas~schemeG-F$6%7goF'F[fl7$FE$\"$\"RF47$FJ$\"%^7F4 7$FO$\"%GNF47$FT$\"%#**)F47$FY$\"@F47$Fhn$\"&ng%F47$F]o$\"&2[*F47$F bo$\"'s,:F47$Fgo$\"'D&\\\"F47$Fap$\"'Xq9F47$Fjp$\"'d]9F47$Fdq$\"'@N:F4 7$Fir$\"'O'o\"F47$F^s$\"'Im;F47$Fcs$\"'F[;F47$Fail$\"']V;F47$Fhs$\"'F^ ;F47$F^jl$\"'Vm;F47$Fcjl$\"';*o\"F47$Fhjl$\"'n%o\"F47$F]t$\"'C!o\"F47$ Fbt$\"'.j;F47$Fgt$\"'rY;F47$F\\u$\"'VR;F47$Fau$\"'WM;F47$Ffu$\"'PO;F47 $F[v$\"'xN;F47$Fev$\"'@J:F47$Fjv$\"'7[9F47$F_w$\"'I$Q\"F47$Fdw$\"'dM8F 47$Fiw$\"'\\#H\"F47$F^x$\"'rb7F47$Fcx$\"'FC7F47$Fhx$\"'?'>\"F47$F^y$\" 'Nu6F47$Fcy$\"'S_6F47$Fhy$\"'pK6F47$F]z$\"'b:6F47$Fbz$\"'K,6F47$Fgz$\" '#e3\"F47$F\\[l$\"'ut5F47$Fa[l$\"'ug5F47$Ff[l$\"'3]5F47$F[\\l$\"' " 0 "" {MPLTEXT 1 0 2 " ;" }}}}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 48 "Test 6 of 12 stage, \+ order 8 Runge-Kutta methods " }}{PARA 256 "" 0 "" {TEXT -1 2 " " } {XPPEDIT 18 0 "dy/dx = (1+2*(x+1)*sin(3*x))*exp(-y);" "6#/*&%#dyG\"\" \"%#dxG!\"\"*&,&F&F&*(\"\"#F&,&%\"xGF&F&F&F&-%$sinG6#*&\"\"$F&F.F&F&F& F&-%$expG6#,$%\"yGF(F&" }{TEXT -1 6 ", " }{XPPEDIT 18 0 "y(0) = 0; " "6#/-%\"yG6#\"\"!F'" }{TEXT -1 1 " " }}{PARA 0 "" 0 "" {TEXT -1 10 " Solution: " }}{PARA 256 "" 0 "" {TEXT -1 2 " " }{XPPEDIT 18 0 "y=ln(x +2/9*sin(3*x)-2/3*x*cos(3*x)-2/3*cos(3*x)+5/3)" "6#/%\"yG-%#lnG6#,,%\" xG\"\"\"*(\"\"#F*\"\"*!\"\"-%$sinG6#*&\"\"$F*F)F*F*F***F,F*F3F.F)F*-%$ cosG6#*&F3F*F)F*F*F.*(F,F*F3F.-F66#*&F3F*F)F*F*F.*&\"\"&F*F3F.F*" } {TEXT -1 2 ". " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 169 "de := diff(y(x),x)=(1+2*(x+1)*sin(3*x))*exp(-y( x));\nic := y(0)=0;\ndsolve(\{de,ic\},y(x));\nu := unapply(rhs(%),x): \nplot(u(x),x=0..5,font=[HELVETICA,9],labels=[`x`,`y(x)`]);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%#deG/-%%diffG6$-%\"yG6#%\"xGF,*&,&\"\"\"F/ *(\"\"#F/,&F,F/F/F/F/-%$sinG6#,$*&\"\"$F/F,F/F/F/F/F/-%$expG6#,$F)!\" \"F/" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%#icG/-%\"yG6#\"\"!F)" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#/-%\"yG6#%\"xG-%#lnG6#,,F'\"\"\"*&#\" \"#\"\"*F,-%$sinG6#,$*&\"\"$F,F'F,F,F,F,*&#F/F6F,*&F'F,-%$cosGF3F,F,! \"\"*&#F/F6F,F:F,F<#\"\"&F6F," }}{PARA 13 "" 1 "" {GLPLOT2D 503 318 318 {PLOTDATA 2 "6&-%'CURVESG6$7bp7$$\"\"!F)F(7$$\"3GLLL3x&)*3\"!#=$\" 3QWK+t!=.P\"F-7$$\"3umm\"H2P\"Q?F-$\"3pUCE&GmM$HF-7$$\"3MLL$eRwX5$F-$ \"3l!G\"yWq,6\\F-7$$\"33ML$3x%3yTF-$\"3dz%)zauhMpF-7$$\"3emm\"z%4\\Y_F -$\"3,G5kQO>C))F-7$$\"3`LLeR-/PiF-$\"36YrjIBvP5!#<7$$\"3]***\\il'pisF- $\"3wGtPF*HL<\"FI7$$\"3>MLe*)>VB$)F-$\"3kc:o\"[JMG\"FI7$$\"3Y++DJbw!Q* F-$\"3/$y^LV#)3O\"FI7$$\"3+N$ekGkX#**F-$\"3'R4-T\"[w(Q\"FI7$$\"3%ommTI Oo/\"FI$\"3i$3slO!o09FI7$$\"3E+]7GTt%4\"FI$\"3qC'f&H6%RT\"FI7$$\"3YLL3 _>jU6FI$\"3#[%e$e^!3:9FI7$$\"3ym;HdNb'>\"FI$\"3CNMY.Kv29FI7$$\"37++]i^ Z]7FI$\"3=<^j^\"=7R\"FI7$$\"35+++v\"=YI\"FI$\"3![4*)z8y_O\"FI7$$\"33++ ](=h(e8FI$\"3yj8C4s%*H8FI7$$\"3/++]P[6j9FI$\"3![\"=po,dN7FI7$$\"3UL$e* [z(yb\"FI$\"3%eEi/R377\"FI7$$\"3wmm;a/cq;FI$\"3!frK?-E\\b*F-7$$\"3%omm mJFI$\"3yKBC%\\'\\`dF-7$$\"3gmmm\"pW`(>FI$\"3$)>nU%[*3T`F-7$$\"3_ek.HW #)))>FI$\"3kKE$Q%*GSE&F-7$$\"3?]iSmTI-?FI$\"3NxPe(47T?&F-7$$\"3*=/wP!R y:?FI$\"3MId=-(\\?;&F-7$$\"3dLe9TOEH?FI$\"3+V!)\\%**R%Q^F-7$$\"3EDc^yL uU?FI$\"3=k(\\#*\\cP8&F-7$$\"3'pT&)e6Bi0#FI$\"3[P(R22Q$[^F-7$$\"3k3_D` Gqp?FI$\"3m1F&*)f\"Q#=&F-7$$\"3K+]i!f#=$3#FI$\"3z7KO0x$fB&F-7$$\"3/++D \"=EX8#FI$\"31#)Gask8:cF-7$$\"3?+](=xpe=#FI$\"3qCq#fO$Q]iF-7$$\"37nm\" H28IH#FI$\"3sUF;\"\\Mi=)F-7$$\"3$p;a8d3AM#FI$\"3NM9xIK\")Q#*F-7$$\"3um ;zpSS\"R#FI$\"3r*)ek&)o0L5FI7$$\"3-+v$41oWW#FI$\"3if$>8HM6:\"FI7$$\"3G LL3_?`(\\#FI$\"31,>EjcGm7FI7$$\"3AL3_D1l_DFI$\"3+C:%e=o-Q\"FI7$$\"3fL$ e*)>pxg#FI$\"3%R*>K#Rqn[\"FI7$$\"33+]Pf4t.FFI$\"3A89rRoB^;FI7$$\"3uLLe *Gst!GFI$\"3eR;lLLE'z\"FI7$$\"30+++DRW9HFI$\"36)ejqPI#4>FI7$$\"3K+]7y# =o'HFI$\"3?c()=o%e2&>FI7$$\"3:++DJE>>IFI$\"3s.\"e&4:F$)>FI7$$\"3A+v$4^ n)pIFI$\"39;D$z^_h+#FI7$$\"3F+]i!RU07$FI$\"3vuj8s:f??FI7$$\"3?]il(Hv'[ JFI$\"3&\\3XiGc\\-#FI7$$\"39+vo/#3o<$FI$\"3(>zS&>rqE?FI7$$\"32](=<6T\\ ?$FI$\"3X%zI>hOe-#FI7$$\"3+++v=S2LKFI$\"3'[xpLqNB-#FI7$$\"3;L$3_NJOG$F I$\"3:>!e))R4?FI7$$\"3Jmmm\"p)=MLFI$\"3OrPOD8'y)>FI7$$\"3GLLeR%p\")Q $FI$\"3S(z'\\,LGb>FI7$$\"3B++](=]@W$FI$\"3K:aBe2s7>FI7$$\"35L$e*[$z*RN FI$\"3!>**fc?u*4=FI7$$\"3e++]iC$pk$FI$\"3'e\\^`F#Qg;FI7$$\"3[m;H2qcZPF I$\"351&p,mkr[\"FI7$$\"3O+]7.\"fF&QFI$\"3o;2PnR[#G\"FI7$$\"3Ymm;/OgbRF I$\"3/lp%G(QG#3\"FI7$$\"3*G$e*[$zV4SFI$\"3P9Az61+7**F-7$$\"3w**\\ilAFj SFI$\"3!p>ERLl'*>*F-7$$\"3#G3_]p'>*3%FI$\"3O9*z2=!e_*)F-7$$\"3ym\"zW7@ ^6%FI$\"3w4XE;/Rz()F-7$$\"3w3F>RL3GTFI$\"3JeP:9JjA()F-7$$\"3t]i!RbX59% FI$\"3mH1#H$\\k'o)F-7$$\"3#=z>'ox+aTFI$\"3oLr_-o*=n)F-7$$\"3yLLL$)*pp; %FI$\"3A7j1wipy')F-7$$\"3!Q3_+sD-=%FI$\"32pcM,k23()F-7$$\"3#Q$3xc9[$>% FI$\"3Gri,**=4g()F-7$$\"3'Qe*[$>Pn?%FI$\"3se,X+?^M))F-7$$\"3)QL3-$H**> UFI$\"3Z**e,OD#4$*)F-7$$\"3#R$ek.W]YUFI$\"3i#fiyx0s=*F-7$$\"3)RL$3xe,t UFI$\"3[2R[)*eVA&*F-7$$\"3Cn;HdO=yVFI$\"3#)>Y<=$f\\9\"FI7$$\"3MMe9\"z- lU%FI$\"3)4DVDmlMD\"FI7$$\"3a+++D>#[Z%FI$\"3qZKS'GmoO\"FI7$$\"3TM$3_5, -`%FI$\"3CFB-Gn\\(\\\"FI7$$\"3SnmT&G!e&e%FI$\"3t\\(p9r/Xi\"FI7$$\"3m+] P%37^j%FI$\"3_eaMDR_K " 0 " " {MPLTEXT 1 0 806 "U := (x,y) -> (1+2*(x+1)*sin(3*x))*exp(-y): hh := \+ 0.01: numsteps := 500: x0 := 0: y0 := 0:\nmatrix([[`slope field: `,U (x,y)],[`initial point: `,``(x0,y0)],[`step width: `,hh],\n[`no. of \+ steps: `,numsteps]]);``;\nmthds := [`Enright-Verner scheme`,`Verner' s Maple scheme`,`a modification of Verner's Maple scheme`,`an \"effici ent\" scheme`,`a scheme of Tsitouras and Papakostas`,`a modification o f the Tsitouras-Papakostas scheme`,`Verner's \"most efficient\" scheme `]: errs := []:\nDigits := 25:\nfor ct to 7 do\n Un_RK8_||ct := RK8_ ||ct(U(x,y),x,y,x0,y0,hh,numsteps,false);\n sm := 0: numpts := nops( Un_RK8_||ct):\n for ii to numpts do\n sm := sm+(Un_RK8_||ct[ii, 2]-u(Un_RK8_||ct[ii,1]))^2;\n end do:\n errs := [op(errs),sqrt(sm/ numpts)];\nend do:\nDigits := 10:\nlinalg[transpose]([mthds,evalf(errs )]);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#K%'matrixG6#7&7$%0slope~field: ~~~G*&,&\"\"\"F+*(\"\"#F+,&%\"xGF+F+F+F+-%$sinG6#,$*&\"\"$F+F/F+F+F+F+ F+-%$expG6#,$%\"yG!\"\"F+7$%0initial~point:~G-%!G6$\"\"!FA7$%/step~wid th:~~~G$F+!\"#7$%1no.~of~steps:~~~G\"$+&Q)pprint256\"" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%!G" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#K%'matrixG6 #7)7$%6Enright-Verner~schemeG$\"+C4vg7!#G7$%6Verner's~Maple~schemeG$\" +)f:@Q#!#H7$%Ha~modification~of~Verner's~Maple~schemeG$\"+-U$)R=F07$%6 an~\"efficient\"~schemeG$\"+tB)[!>F07$%Ea~scheme~of~Tsitouras~and~Papa kostasG$\"+zf8Q7F07$%Ra~modification~of~the~Tsitouras-Papakostas~schem eG$\"+K*yCP\"F07$%AVerner's~\"most~efficient\"~schemeG$\"+QQ4I9F0Q)ppr int266\"" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 30 "The following code constructs " }{TEXT 260 20 "numerical procedure s" }{TEXT -1 56 " for solutions based on each of the Runge-Kutta schem es." }}{PARA 0 "" 0 "" {TEXT -1 75 "The error in the value obtained by each of the methods at the point where " }{XPPEDIT 18 0 "x = 4.999; " "6#/%\"xG-%&FloatG6$\"%**\\!\"$" }{TEXT -1 16 " is also given." }} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 737 "U := (x,y) -> (1+2*(x+1)*si n(3*x))*exp(-y): hh := 0.01: numsteps := 500: x0 := 0: y0 := 0:\nmatri x([[`slope field: `,U(x,y)],[`initial point: `,``(x0,y0)],[`step wid th: `,hh],\n[`no. of steps: `,numsteps]]);``;\nmthds := [`Enright- Verner scheme`,`Verner's Maple scheme`,`a modification of Verner's Map le scheme`,`an \"efficient\" scheme`,`a scheme of Tsitouras and Papako stas`,`a modification of the Tsitouras-Papakostas scheme`,`Verner's \" most efficient\" scheme`]: errs := []:\nDigits := 30:\nfor ct to 7 do \n un_RK8_||ct := RK8_||ct(U(x,y),x,y,x0,y0,hh,numsteps,true);\nend \+ do:\nxx := 4.999: uxx := evalf(u(xx)):\nfor ct to 7 do\n errs := [op (errs),abs(un_RK8_||ct(xx)-uxx)];\nend do:\nDigits := 10:\nlinalg[tran spose]([mthds,evalf(errs)]);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#K%'mat rixG6#7&7$%0slope~field:~~~G*&,&\"\"\"F+*(\"\"#F+,&%\"xGF+F+F+F+-%$sin G6#,$*&\"\"$F+F/F+F+F+F+F+-%$expG6#,$%\"yG!\"\"F+7$%0initial~point:~G- %!G6$\"\"!FA7$%/step~width:~~~G$F+!\"#7$%1no.~of~steps:~~~G\"$+&Q)ppri nt276\"" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%!G" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#K%'matrixG6#7)7$%6Enright-Verner~schemeG$\"+rUJ>5!#H7$% 6Verner's~Maple~schemeG$\"++g%3\\\"F+7$%Ha~modification~of~Verner's~Ma ple~schemeG$\"+=+h?9F+7$%6an~\"efficient\"~schemeG$\"+5>T^5F+7$%Ea~sch eme~of~Tsitouras~and~PapakostasG$\"+*4\")oC\"F+7$%Ra~modification~of~t he~Tsitouras-Papakostas~schemeG$\"+&oAvI\"F+7$%AVerner's~\"most~effici ent\"~schemeG$\"+Jh$eA\"F+Q)pprint286\"" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 4 "The " }{TEXT 260 22 "root mean squa re error" }{TEXT -1 20 " over the interval " }{XPPEDIT 18 0 "[0, 5]; " "6#7$\"\"!\"\"&" }{TEXT -1 82 " of each Runge-Kutta method is estim ated as follows using the special procedure " }{TEXT 0 5 "NCint" } {TEXT -1 98 " to perform numerical integration by the 7 point Newton- Cotes method over 200 equal subintervals." }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 476 "mthds := [`Enright-Verner scheme`,`Verner's Maple sc heme`,`a modification of Verner's Maple scheme`,`an \"efficient\" sche me`,`a scheme of Tsitouras and Papakostas`,`a modification of the Tsit ouras-Papakostas scheme`,`Verner's \"most efficient\" scheme`]: errs : = []:\nDigits := 20:\nfor ct to 7 do\n sm := NCint((u(x)-'un_RK8_||c t'(x))^2,x=0..5,adaptive=false,numpoints=7,factor=200);\n errs := [o p(errs),sqrt(sm/5)];\nend do:\nDigits := 10:\nlinalg[transpose]([mthds ,evalf(errs)]);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#K%'matrixG6#7)7$%6E nright-Verner~schemeG$\"+#*)p.E\"!#G7$%6Verner's~Maple~schemeG$\"+.'pf P#!#H7$%Ha~modification~of~Verner's~Maple~schemeG$\"+\"f#GM=F07$%6an~ \"efficient\"~schemeG$\"+Y_)3!>F07$%Ea~scheme~of~Tsitouras~and~Papakos tasG$\"+'45FB\"F07$%Ra~modification~of~the~Tsitouras-Papakostas~scheme G$\"+N<%oO\"F07$%AVerner's~\"most~efficient\"~schemeG$\"+>$y\\U\"F0Q)p print296\"" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 92 "The following error graphs are constructed using the numerical \+ procedures for the solutions." }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 616 "evalf[30](plot([u(x)-'un_RK8_1'(x),u(x)-'un_RK8_2'(x),u(x)-'un_ RK8_3'(x),u(x)-'un_RK8_4'(x),\nu(x)-'un_RK8_5'(x),u(x)-'un_RK8_6'(x),u (x)-'un_RK8_7'(x)],x=0..5,-1.5e-19..6.9e-19,font=[HELVETICA,9],\ncolor =[COLOR(RGB,.4,0,.9),COLOR(RGB,0,.65,0),brown,red,COLOR(RGB,0,.75,.95) ,magenta,COLOR(RGB,.95,.55,0)],\nlegend=[`Enright-Verner scheme`,`Vern er's Maple scheme`,`a modification of Verner's Maple scheme`,`an \"eff icient\" scheme`,`a scheme of Tsitouras and Papakostas`,`a modificatio n of the Tsitouras-Papakostas scheme`,`Verner's \"most efficient\" sch eme`],title=`error curves for 12 stage order 8 Runge-Kutta methods`)); " }}{PARA 13 "" 1 "" {GLPLOT2D 874 688 688 {PLOTDATA 2 "6--%'CURVESG6% 7gs7$$\"\"!F)F(7$$\"?MLLLLLLLLL3x&)*3\"!#I$!,4l0]I\"F-7$$\"?++++++++]i !R(*Rc\"F-$!,]T!4%)>F-7$$\"?nmmmmmmmm\"H2P\"Q?F-$!,vQh[R#F-7$$\"?+++++ +++]PMnNrDF-$!,DOi6W#F-7$$\"?MLLLLLLLL$eRwX5$F-$!,v\\!\\vAF-7$$\"?NLLL LLLLL$3x%3yTF-$!,^t*\\'z\"F-7$$\"?ommmmmmmm\"z%4\\Y_F-$!,([\\F]9F-7$$ \"?NLLLLLLLLeR-/PiF-$!+qa[R7!#H7$$\"?-++++++++DcmpisF-$!+qYo$3\"FS7$$ \"?OLLLLLLLLe*)>VB$)F-$!*hV(*o*FS7$$\"?.++++++++DJbw!Q*F-$!*PHS%*)FS7$ $\"?nmmmmmmmm;/j$o/\"FS$!*$feZ&)FS7$$\"?MLLLLLLLL3_>jU6FS$!*'e3'\\)FS7 $$\"?+++++++++]i^Z]7FS$!*g\\?x)FS7$$\"?+++++++++](=h(e8FS$!*WAPR*FS7$$ \"?+++++++++]P[6j9FS$!+DXC?5FS7$$\"?MLLLLLLL$e*[z(yb\"FS$!+%*[F#3\"FS7 $$\"?++++++++Dc,#>Uh\"FS$!+Zku-6FS7$$\"?nmmmmmmmm;a/cq;FS$!,U2d:;\"F-7 $$\"?nmmmmmmmmT&)))G=*4[p\" F-7$$\"?+++++++]ilsqx$z\"FS$!,2tl4,#F-7$$\"?MLLLLLLLekGo`@=FS$!,@e:ID# F-7$$\"?,++++++D1k1nTN=FS$!,a/HXG#F-7$$\"?ommmmmm;aj%e'H\\=FS$!,zQb%z@ F-7$$\"?MLLLLLL3-jikFS$\"+A\\tgbF-7$$\"?MLLLLLLLe9;0?E>FS$\",1c,A6$F -7$$\"?+++++++]i!Rgs2&>FS$\",1V#R4))F-7$$\"?nmmmmmmmmm\"pW`(>FS$\"-)fP IbQ\"F-7$$\"?MLLLLLLek.HW#)))>FS$\"-[$4+&[z[;F$F-7$$\"?,+++++++]i! f#=$3#FS$\"-zyxl5$F-7$$\"?,++++++ ++D\"=EX8#FS$\"-Vq>sZDF-7$$\"?,++++++]i!*y?OZ@FS$\"-(R$*>xE#F-7$$\"?,+ ++++++Dcwz>g@FS$\"-O-o,hJ0BFS$!,')=wLS\"F-7$$ \"?nmmmmmm;a8A3hB-l9F-7 $$\"?nmmmmmmmTNr&3AM#FS$!,5/q6Q\"F-7$$\"?nmmmmmm;Hd?j!oO#FS$!,^URd:\"F -7$$\"?nmmmmmmm;zpSS\"R#FS$!*Kq%G#)FS7$$\"?++++++++v$41oWW#FS$!*>@J!QF S7$$\"?MLLLLLLLL3_?`(\\#FS$!*'>l^;FS7$$\"?MLLLLLLL3_D1l_DFS$!)0-!p)FS7 $$\"?MLLLLLLL$e*)>pxg#FS$!)=rHtFS7$$\"?,+++++++]Pf4t.FFS$!):&Gv'FS7$$ \"?MLLLLLLLLe*Gst!GFS$!)pO>^FS7$$\"?,+++++++++DRW9HFS$!)8cBKFS7$$\"?,+ +++++++DJE>>IFS$!)$z&=?FS7$$\"?,+++++++]i!RU07$FS$!)B\"*H:FS7$$\"?,+++ +++++v=S2LKFS$!)p\"ov\"FS7$$\"?ommmmmmmmm\"p)=MLFS$!)8l*p#FS7$$\"?,+++ +++++](=]@W$FS$!)lK6YFS7$$\"?MLLLLLLL$e*[$z*RNFS$!)L-\"G'FS7$$\"?,++++ ++++]iC$pk$FS$!)ADuTFS7$$\"?MLLLLLLLe*[t\\sp$FS$\"(#>b8FS7$$\"?ommmmmm m;H2qcZPFS$\")Oz-_FS7$$\"?,++++++++DJ]'Qx$FS$\")NykcFS7$$\"?MLLLLLLL$3 _0j,!QFS$\"(1J\"=FS7$$\"?,+++++++v=n?J8QFS$!)T%=)QFS7$$\"?ommmmmmmm;z5 YEQFS$!)B#4j*FS7$$\"?MLLLLLLLe9\"45'RQFS$!*Y7)3BFS7$$\"?,+++++++]7.\"f F&QFS$!*#f:LPFS7$$\"?MLLLLLLLek`8=/RFS$!+2#RoC\"FS7$$\"?ommmmmmmm;/Ogb RFS$!+(oDf#=FS7$$\"?,++++++]7`p2_#)RFS$!*$yF\"R'FS7$$\"?MLLLLLLLe*[$zV 4SFS$\",97E!*o#F-7$$\"?,++++++D\"yv^'*G-%FS$\",]PDyr&F-7$$\"?ommmmmm;/ E+^NOSFS$\",BwdDm)F-7$$\"?MLLLLLL3F%Ho8)\\SFS$\"-+7kyP:F-7$$\"?,++++++ +]ilAFjSFS$\"-uBKDm?F-7$$\"?MLLLLL$3x\")HP`(pSFS$\"-wfeGqCF-7$$\"?ommm mmmT&Q.[Mi2%FS$\"-J=m:$f#F-7$$\"?,+++++]7`p(e:F3%FS$\"-z\"H:/8$F-7$$\" ?MLLLLLL$3_]p'>*3%FS$\"-YpeV2MF-7$$\"?ommmmm;a)3C!yn&4%FS$\"-'*HYFBPF- 7$$\"?,++++++Dcw4*e@5%FS$\"-T!Q%G,VF-7$$\"?MLLLLL$eRAr,S'3TFS$\"-y7ugn WF-7$$\"?ommmmmmm\"zW7@^6%FS$\"-$e8g<*[F-7$$\"?-+++++]Pf$=B-;7%FS$\"-1 4QuIaF-7$$\"?NLLLLLL3F>RL3GTFS$\"-D4SJ:bF-7$$\"?ommmmm;z%\\lWkX8%FS$\" -S@5)4$fF-7$$\"?-++++++]i!RbX59%FS$\"-J`vvUjF-7$$\"?ommmmmm\"z>'ox+aTF S$\"-5ktQmmF-7$$\"?NLLLLLLLLL$)*pp;%FS$\"-B$o>v(oF-7$$\"?NLLLLL$3-8v\" RGqTFS$\"-P,]`opF-7$$\"?NLLLLLL3Fp^yftTFS$\"-l@Y\"R'pF-7$$\"?NLLLLL$eR sey6p<%FS$\"-J8!G%epF-7$$\"?NLLLLLL$3_+sD-=%FS$\"-$eGO_$pF-7$$\"?NLLLL L$3xJUlRN=%FS$\"-%Qs\"oFpF-7$$\"?NLLLLLLe9T)e`o=%FS$\"-QS7[:pF-7$$\"?N LLLLL$e9\"fAv;!>%FS$\"-MbzwznF-7$$\"?NLLLLLLL3xc9[$>%FS$\"-%*\\?^pnF-7 $$\"?NLLLLLL3-8D$4,?%FS$\"-^R/t5lF-7$$\"?NLLLLLL$e*[$>Pn?%FS$\"-Jt!fdZ 'F-7$$\"?NLLLLLLe*[=1lL@%FS$\"-](G$oFhF-7$$\"?NLLLLLLL$3-$H**>UFS$\"-v ^\\`%p&F-7$$\"?NLLLLLL$3Fpm[KB%FS$\"-7#pP\\;&F-7$$\"?NLLLLLLLek.W]YUFS $\"-#HBjze%F-7$$\"?NLLLLLL3_+sA8`UFS$\"-f.W/]SF-7$$\"?NLLLLLL$ek.9g(fU FS$\"-gR6a.OF-7$$\"?NLLLLLLeRs3!)QmUFS$\"-\\-X+kMF-7$$\"?NLLLLLLLL3xe, tUFS$\"-\"=jYv%HF-7$$\"?-++++++v$4'\\=;'G%FS$\"-G;JWCCF-7$$\"?ommmmmm; a8AyI*H%FS$\"-\"o\\K6w\"F-7$$\"?NLLLLLLe9m%z`CJ%FS$\",IL*o@7FS7$$\"?-+ ++++++v=n(*fDVFS$\"+Y@ov\"*FS7$$\"?NLLLLLL$eRAr\"*=N%FS$\"+i7imLFS7$$ \"?ommmmmmm;HdO=yVFS$\"+&\\S>O\"FS7$$\"?,+++++]PM-\\NA%Q%FS$\"*G#*[X*F S7$$\"?MLLLLLL3_vSME!R%FS$\"*%fIzkFS7$$\"?ommmmm;zp[KLI'R%FS$\"*nviO'F S7$$\"?,++++++](=UAVBS%FS$\"*0Gs.&FS7$$\"?ommmmmm\"H#o2IU9WFS$\"*D&z1[ FS7$$\"?MLLLLLLLe9\"z-lU%FS$\"*!=&zV&FS7$$\"?ommmmmm;H2eBm]WFS$\"*WWG# )*FS7$$\"?,+++++++++D>#[Z%FS$\"+%\\,3H\"FS7$$\"?ommmmmmmT5::^-XFS$\"+q ^Tk;FS7$$\"?MLLLLLLL$3_5,-`%FS$\"+1pBo=FS7$$\"?,+++++++DJ&p!*yb%FS$\"+ /Fy()=FS7$$\"?ommmmmmmmT&G!e&e%FS$\"+&y#*y(=FS7$$\"?-+++++++]P%37^j%FS $\"+E7*4u\"FS7$$\"?NLLLLLLLLL$)Qk%o%FS$\"+o+Jq:FS7$$\"?-+++++++]iSjE!z %FS$\"+tA)fG\"FS7$$\"?-+++++++]P40O\"*[FS$\"+l1QC6FS7$$\"\"&F)$\"+tvU> 5FS-%&COLORG6&%$RGBG$\"\"%!\"\"F($\"\"*F^gm-%'LEGENDG6#%6Enright-Verne r~schemeG-F$6%7btF'7$F+$\"+ei%Q5%F-7$F1$\"+yghkqF-7$F6$\"+Hgsl%*F-7$$ \"?+++++++]7G))>Wr@F-$\"+;m5Z(*F-7$$\"?MLLLLLLLek.pu/BF-$\",z\\X>.\"F- 7$$\"?nmmmmmm;/,>=0QCF-$\",CnS8/\"F-7$F;$\",G6Sn/\"F-7$$\"?MLLLLLL$eR( \\;m/FF-$\",$)4yj1\"F-7$$\"?nmmmmmmmT5ll'z$GF-$\",Di,v0\"F-7$$\"?+++++ ++](o/[r7(HF-$\",&ezlX5F-7$F@$\",*Hk\"y.\"F-7$$\"?MLLLLLLLLL$eI8k$F-$ \"+1eJ[%*F-7$FE$\"+dtbS$)F-7$FJ$\"+&fh#QkF-7$FO$\"*#[ur`FS7$FU$\"*&3`T ZFS7$FZ$\"*z`zRlF-7$Fds$\"+R Ta[nF-7$Fis$\"+Qc,1nF-7$F^t$\"+p!4)Q\\F-7$Fct$\"+hFz8>F-7$Fht$!+p[?ciF -7$F]u$!,@**\\yV\"F-7$Fbu$!,vST=2#F-7$Fgu$!,Lafr$HF-7$F\\v$!,(H(fxZ$F- 7$Fav$!,wqTDC%F-7$Ffv$!,m*zB6`F-7$F[w$!,lhw&zeF-7$F`w$!,0(*ySg&F-7$Few $!,F;[jh%F-7$Fjw$!,1qu6/%F-7$F_x$!,Ls/j*GF-7$Fdx$!,9B(*[C#F-7$Fix$!,f< ^Bf\"F-7$$\"?MLLLLLL3_]%oi#*>#FS$!++rW\"z'F-7$F^y$\"+nAr%p\"F-7$$\"?,+ +++++Dcw4&[gA#FS$\"+y]2ZnF-7$Fcy$\",Gkq_N\"F-7$Fhy$\",d!*p[6#F-7$F]z$ \",1!\\8]EF-7$Fbz$\",\">hFEFF-7$Fgz$\",%HgltFF-7$F\\[l$\",4meN$GF-7$Fa [l$\",csFZ!GF-7$Ff[l$\",ZW=Rp#F-7$F[\\l$\"+,a&o`#FS7$F`\\l$\"+\"G'Go@F S7$Fe\\l$\"+,9G`=FS7$F_]l$\"+vC5L9FS7$Fd]l$\"+d!eRB\"FS7$Fi]l$\"+OLp$4 \"FS7$F^^l$\"*$3LY**FS7$Fc^l$\"*m*4/$*FS7$Fh^l$\"*!=4\")*)FS7$F]_l$\"* aF*p*)FS7$Fb_l$\"*zu)o#*FS7$Fg_l$\"*hU5')*FS7$F\\`l$\"+DNU^5FS7$Fa`l$ \"+ComA6FS7$Ff`l$\"+@*>!\\6FS7$F[al$\"+aiT\"=\"FS7$F`al$\"+0p=07FS7$Fe al$\"+%Q.fC\"FS7$F_bl$\"+Y+t58FS7$Fibl$\"+mo809FS7$F^cl$\"+!3Q\"o;FS7$ Fccl$\"+`&HE!>FS7$$\"?,+++++]7y]&*GLiRFS$\"+^@Y%*=FS7$$\"?MLLLLLLe*[o= i!pRFS$\"+\"[5P*=FS7$$\"?,+++++DJ&>D$oUsRFS$\"+DAAo=FS7$$\"?ommmmm;/,> y9zvRFS$\"+TIdy=FS7$$\"?MLLLLL3x1'Q7c\"zRFS$\"+E+MZ=FS7$Fhcl$\"+>mO.=F S7$$\"?ommmmmmTN@_$zf*RFS$\"+JMH(p\"FS7$F]dl$\",/1&>v8F-7$Fbdl$\"+t')e q%*F-7$Fgdl$\"+.BQB_F-7$F\\el$!+MYso^F-7$Fael$!,'[w,^8F-7$F[fl$!,#p;'* 4AF-7$Fefl$!,Za(41OF-7$Fjfl$!,c$eWbTF-7$F_gl$!,*3s`4_F-7$Fdgl$!,s4mc^& F-7$Figl$!,ZA#>?jF-7$F^hl$!,(=UN%R(F-7$Fchl$!,T(GygvF-7$Fhhl$!,zKv2W)F -7$F]il$!,,^.iP*F-7$Fbil$!-]ZFk=5F-7$Fgil$!-s/Hm#3\"F-7$Fajl$!-?K$GL7 \"F-7$F[[m$!-%G'Q&H9\"F-7$Fe[m$!-EW\"y,9\"F-7$F_\\m$!-?W*y_8\"F-7$Fi\\ m$!-C3pA+6F-7$Fc]m$!,I!>-](*F-7$Fh]m$!,-iAsw)F-7$F]^m$!,#\\Sy?wF-7$Fb^ m$!,d)\\3vkF-7$Fg^m$!,@Uy(yaF-7$F\\_m$!,!=v()z^F-7$Fa_m$!,FcP!fRF-7$$ \"?ommmmm;ajMj))ezUFS$!,$)*Gl7JF-7$Ff_m$!,1cXXp#F-7$$\"?NLLLLL$eRse$[t #H%FS$!,%\\kBX:F-7$F[`m$!+EJq<(*F-7$$\"?-+++++]P%)R33)eI%FS$!*XuKW%FS7 $F``m$\"*e#H-`FS7$$\"?ommmmm;zW#4yE!>VFS$\"*Mx$4()FS7$Fe`m$\"+c\\\\$R \"FS7$$\"?ommmmmmTNrRduQVFS$\"+aq=8BFS7$Fj`m$\"+ANjrKFS7$$\"?-++++++Dc w%oP]O%FS$\"+$>GQk$FS7$F_am$\"+rQ*)eRFS7$Fdam$\"+$p\">_TFS7$Fiam$\"+=0 \\BVFS7$$\"?,+++++v$4@mQ$G$R%FS$\"+vDo%H%FS7$F^bm$\"+i1foUFS7$$\"?MLLL LLekGNyKK*R%FS$\"+R%GmK%FS7$Fcbm$\"+T%>`P%FS7$Fhbm$\"+U/uuVFS7$F]cm$\" +*3\"yLVFS7$Fbcm$\"+%*>/mTFS7$Fgcm$\"+**)fj!RFS7$Fadm$\"+]J^hKFS7$F[em $\"+4UltFFS7$Feem$\"+t+46AFS7$Fjem$\"+To$4(=FS7$F_fm$\"+'*)\\Hl\"FS7$F dfm$\"+\\%[-\\\"FS-Fifm6&F[gmF($\"#l!\"#F(-Fbgm6#%6Verner's~Maple~sche meG-F$6%7jsF'7$$\"?ommmmmmmmTN@Ki8!#J$!+!R>Gr\"Fggo7$$\"?NLLLLLLLL$3FW Ys#Fggo$!+e(fC!GFggo7$$\"?-++++++++D1k'p3%Fggo$!+?tw')HFggo7$$\"?qmmmm mmmmmT&)G\\aFggo$!+^6S\"4#Fggo7$$\"?0++++++++]7G$R<)Fggo$\"+7-5oSFggo7 $F+$\"+hKC)=\"F-7$F1$\"+aECLIF-7$F6$\"+W<*R'[F-7$Fbhm$\"+>-GT^F-7$Fghm $\"+9%p(zcF-7$F\\im$\"+ok4VeF-7$F;$\"+%oFV)fF-7$Fdim$\"+W3(=H'F-7$Fiim $\"+NA)>L'F-7$F^jm$\"+)4yDN'F-7$F@$\"+q3;kkF-7$Ffjm$\"+$Rr#4iF-7$FE$\" +![&f>dF-7$FJ$\"+'e64n%F-7$FO$\"*dD5(RFS7$FU$\"**y?5NFS7$FZ$\"*(et1KFS 7$Fin$\"*3Ze+$FS7$F^o$\"*(Q:%*GFS7$Fco$\"*\"[nyGFS7$Fho$\"*RuK&HFS7$F] p$\"*>o]4$FS7$Fbp$\"*%GM\">$FS7$Fgp$\"*@U&eJFS7$F\\q$\"*QaA0$FS7$Faq$ \"+Gsr!*HF-7$Ffq$\"+W13MJF-7$F[r$\"+=#3i^$F-7$F`r$\"+%*zxkQF-7$Fer$\"+ P_@)=%F-7$Fjr$\"+x7;\"G%F-7$F_s$\"+oAE#G%F-7$Fds$\"+7pxQTF-7$Fis$\"+\" )[mARF-7$Fct$\"*-\")3/$F-7$F]u$!+=8vj)*F-7$Fgu$!,YmWV$>F-7$Fav$!,5$fb3 GF-7$Ffv$!,J'Q\\'f$F-7$F[w$!,tz6=;%F-7$$\"?,++++++DJXR0g*3#FS$!,9e,9?% F-7$$\"?,++++++]7G)[=g4#FS$!,'\\rA6UF-7$$\"?,++++++v$4rVOC5#FS$!,F(=%= ?%F-7$F`w$!,0bT.;%F-7$$\"?,++++++]Pf$G!p@@FS$!,P:58%RF-7$Few$!,4%pa$p$ F-7$F_x$!,>e2En#F-7$Fix$!,Mqm_!=F-7$Fgbn$!,_ix^;\"F-7$F^y$!+v$>!=aF-7$ F_cn$!+-Un&[\"F-7$Fcy$\"+WmR/SF-7$Fhy$\",]LKd2\"F-7$F]z$\",ngxoj\"F-7$ Fgz$\",IJkz%=F-7$Fa[l$\",;^wr,#F-7$$\"?nmmmmmmTN'fW2XN#FS$\",)\\q,D?F- 7$Ff[l$\",&[!H6-#F-7$$\"?nmmmmmm\"H#=&>0\"zBFS$\"+,ky%)FS7$Fg_l$\"*(p>:!*FS7$F \\`l$\"*%>\")p(*FS7$Fa`l$\"+dCr!3\"FS7$F[al$\"+,.-(>\"FS7$Feal$\"+*[HB F\"FS7$Fibl$\"+hdhr8FS7$$\"?ommmmmm;aQG-ZyQFS$\"+#Q?3U\"FS7$F^cl$\"+>b TY9FS7$$\"?ommmmmmTgF;p.ds'F-7$Fbdl$\"+&42eD$F-7$Fgdl$\"*eWe/\"F-7$F\\el$ !+%eNV+(F-7$Fael$!,ubG![7F-7$F[fl$!,@ke/!=F-7$Fefl$!,7Y9mo#F-7$F_gl$!, *pYB/PF-7$Figl$!,._Q!>WF-7$F^hl$!,0fq67&F-7$Fchl$!,(=.5K_F-7$Fhhl$!,ri ZE#eF-7$F]il$!,Az55Z'F-7$Fbil$!,DO;'fqF-7$Fgil$!,$Rm*4c(F-7$Fajl$!,K)) fF#zF-7$F[[m$!,3:aZ;)F-7$F`[m$!,'yI(e:)F-7$Fe[m$!,(H:Y[\")F-7$Fj[m$!,1 m_sC)F-7$F_\\m$!,An(yM#)F-7$Fi\\m$!,y!H[G\")F-7$Fc]m$!,d7Pj_(F-7$Fh]m$ !,VYJ5'pF-7$F]^m$!,&f/qhiF-7$Fb^m$!,^9-6a&F-7$Fg^m$!,jA<$))[F-7$F\\_m$ !,P\"=*Go%F-7$Fa_m$!,@`HY&QF-7$Ff`o$!,4pX&fKF-7$Ff_m$!,`,Jw&HF-7$F^ao$ !,/I(QC@F-7$F[`m$!,95O)>9FS-% 'COLOURG6&F[gm$\")#)eqk!\")$\"))eqk\"FcdqFddq-Fbgm6#%Ha~modification~o f~Verner's~Maple~schemeG-F$6%7isF'7$Fego$!+vQ&*R@Fggo7$F[ho$!+B:O%z$Fg go7$F`ho$!+uV![F&Fggo7$Feho$!+3(z/6&Fggo7$Fjho$!+==z\\7Fggo7$F+$\"*;0( )*\\F-7$F1$\"+TI``@F-7$F6$\"+b`q^RF-7$Fbhm$\"+,DaVUF-7$Fghm$\"+;-'yz%F -7$F\\im$\"+b*z>)\\F-7$F;$\"+32TW^F-7$Fdim$\"+&3>DZ&F-7$Fiim$\"+@/'=`& F-7$F^jm$\"+rt0qbF-7$F@$\"+,*oFp&F-7$Ffjm$\"+>&ymY&F-7$FE$\"+SXHu\\F-7 $FJ$\"+kh)e*QF-7$FO$\"*]Z'3KFS7$FU$\"*E<,z#FS7$FZ$\"*^@.`#FS7$Fin$\"*v 0GO#FS7$F^o$\"*:.uE#FS7$Fco$\"*L^AD#FS7$Fho$\"*%3N;BFS7$F]p$\"*:7sW#FS 7$Fbp$\"*w38e#FS7$Fgp$\"*z.yq#FS7$Faq$\"+S+x&4$F-7$Ffq$\"+U$)*pS$F-7$F [r$\"+gwW6OF-7$$\"?MLLLLLLe9m%>(*)zF-7$F ^y$!,;M-CL\"F-7$F_cn$!+YD(*y\"*F-7$Fcy$!+]+$)eKF-7$Fhy$\"+.'G&yWF-7$F] z$\",,t'RK6F-7$Fgz$\",=l\\GU\"F-7$Fa[l$\",BL)ek;F-7$F_dp$\",+K^Ap\"F-7 $Ff[l$\",aP\"Q/EjnFS7$F\\`l$\"*0UuN(FS7$Fa`l$\"*Kp7F) FS7$F[al$\"*f?+q*FS7$Feal$\"+1Vg*3\"FS7$Fibl$\"+%R$)p?\"FS7$$\"?MLLLLL L3_vlYhlQFS$\"+zZbF7FS7$F_hp$\"+!QCsB\"FS7$$\"?,++++++Dc,\"zD8*QFS$\"+ <(>I@\"FS7$F^cl$\"+Ek4'>\"FS7$F\\ip$\"+n2!=-\"FS7$Fccl$\"*?z7*yFS7$Fhc l$\"*\"GAF@FS7$F]dl$!+n8SY`F-7$Fbdl$!,&R[9J5F-7$Fgdl$!,:P$pT9F-7$F\\el $!,tR5)RAF-7$Fael$!,]'pV-GF-7$F[fl$!,WcT8L$F-7$Fefl$!,,RhT6%F-7$F_gl$! ,`KA]'\\F-7$Figl$!,z&o_\\bF-7$Fchl$!,\"o`//iF-7$F]il$!,_zTn?(F-7$Fbil$ !,1g18q(F-7$Fgil$!,*eamS\")F-7$F[[m$!,Z`_]u)F-7$F_\\m$!,q!G?u))F-7$Fd \\m$!,IE^D!*)F-7$Fi\\m$!,!e^Nk))F-7$F^]m$!,$3nyg()F-7$Fc]m$!,s%>cA&)F- 7$Fh]m$!,PDx15)F-7$F]^m$!,R#QTSvF-7$Fb^m$!,&znK[pF-7$Fg^m$!,b[ACQ'F-7$ F\\_m$!,**Q>**='F-7$Fa_m$!,$H?dVaF-7$Ff`o$!,09F-7$F+$!+Ij:K7F-7$F1$!*'RMS6F-7$F6$ \"+5:d(R\"F-7$F;$\"+kR%fi#F-7$F@$\"+[wyLMF-7$$\"?MLLLLLLLLe*[`HP$F-$\" +][I(\\$F-7$Ffjm$\"+i5![b$F-7$$\"?MLLLLLLLL3xwq4RF-$\"+%y%*)=NF-7$FE$ \"+A%pDS$F-7$$\"?-+++++++]PfyG7ZF-$\"+HSFS7$Fin$\"*Lum$=FS7$F^o$\"*>z#oFS7$Fgp$ \"*=%3**>FS7$F\\q$\"*-Zm0#FS7$Faq$\"+f.Z$G#F-7$Ffq$\"+&))[fm#F-7$F[r$ \"+!\\')fH$F-7$F`r$\"+C#\\Xu$F-7$Fer$\"+Go;gTF-7$Fjr$\"+bZT.VF-7$F_s$ \"+?bdJWF-7$Fds$\"+)yl1]%F-7$Fis$\"+)33<_%F-7$F^t$\"+kX,&>%F-7$Fct$\"+ 2)f3j$F-7$Fht$\"+Z\"*ot=F-7$F]u$!**>%*Q7F-7$Fgu$!+4AJzYF-7$Fav$!+A4_m) *F-7$Ffv$!,9%3Gr:F-7$F[w$!,,1nP>#F-7$F\\ap$!,d&>DVBF-7$F`w$!,iTjOZ#F-7 $$\"?,++++++DcwMBF:@FS$!,4$3n%*FS7$Fi]l$\"*.n.J)FS7$F^^l$\"* )Q#)GvFS7$Fc^l$\"*LKL.(FS7$Fh^l$\"*xXvy'FS7$F]_l$\"*!3PznFS7$Fb_l$\"*, D:,(FS7$Fg_l$\"*pe(*\\(FS7$F\\`l$\"*Tk'G\")FS7$Fa`l$\"*Y&f]!*FS7$F[al$ \"+\"G'oH5FS7$Feal$\"+Tj\\D6FS7$Fibl$\"+$)*\\pB\"FS7$Feer$\"+WKvi7FS7$ F_hp$\"+jF@%G\"FS7$F]fr$\"+E)4'*G\"FS7$F^cl$\"+NV,+8FS7$F\\ip$\"+0rfe7 FS7$Fccl$\"+%e+X>\"FS7$Fhcl$\"*`+H))*FS7$F]dl$\"+sr=orF-7$Fgdl$\"+GrG. QF-7$Fael$!+Q2^p3lF-7$Fchl$!,=wAw.#F-7$Fhhl$!,4o#\\ tBF-7$F]il$!,JTZex#F-7$Fbil$!,4;\")3=$F-7$Fgil$!,-_i&yNF-7$Fajl$!,rc96 #RF-7$F[[m$!,onQnA%F-7$F`[m$!,*>F:AUF-7$Fe[m$!,Xe\\PA%F-7$Fj[m$!,_hK.Y %F-7$F_\\m$!,)=?g`WF-7$$\"?NLLLLL$3_]4R&z'>%FS$!,(\\tM]WF-7$Fd\\m$!,g \"f@4YF-7$$\"?NLLLLL$e*)4$fKU.UFS$!,teG.g%F-7$Fi\\m$!,R6KBf%F-7$$\"?NL LLLL$3Fpw7^+@%FS$!,]%)4=m%F-7$F^]m$!,0!z*3l%F-7$$\"?NLLLLL$ekGg**ym@%F S$!,#y%G&QYF-7$Fc]m$!,]#*o8h%F-7$Fh]m$!,;)[DUWF-7$F]^m$!,ws#zkTF-7$Fb^ m$!,jvn(QQF-7$Fg^m$!,Ny:\\]$F-7$F\\_m$!,\"3=:#R$F-7$Fa_m$!,T+V'GHF-7$F f`o$!,c$3PpDF-7$Ff_m$!,3#QS#Q#F-7$F^ao$!,BVpw%=F-7$F[`m$!,\"eNx_:F-7$F fao$!+x0zw7FS7$F``m$!*IjjP(FS7$F^bo$!*I!Qk_FS7$Fe`m$!*\"[2i?FS7$$\"?NL LLLL$3_]Mvs@L%FS$\"*LW+%GFS7$Ffbo$\"*U&ziTFS7$$\"?-+++++]il(fs=`M%FS$ \"*wT)HtFS7$Fj`m$\"+Z#G>9\"FS7$F^co$\"+M^Z)[\"FS7$F_am$\"+HD)z#=FS7$Fc bm$\"+$e-eV#FS7$F]cm$\"+p-!zj#FS7$F]bq$\"+,&QUr#FS7$Fbcm$\"+=;G(z#FS7$ Febq$\"+].a&y#FS7$Fgcm$\"+>GufFFS7$Fadm$\"+x#QNb#FS7$F[em$\"+UmXsAFS7$ Feem$\"+$p\"Rn=FS7$Fjem$\"+p^vx:FS7$F_fm$\"+)y9lQ\"FS7$Fdfm$\"+)\\qiC \"FS-Fifm6&F[gmF($\"#vF]go$\"#&*F]go-Fbgm6#%Ea~scheme~of~Tsitouras~and ~PapakostasG-F$6%7^sF'7$F[ho$!+K\")e@tFggo7$Feho$!,l?o5T\"Fggo7$Faas$! ,kSQ%H:Fggo7$Fjho$!,/Ah&z:Fggo7$Fias$!+_u&*4:F-7$F+$!+'>$[O8F-7$F1$!*[ frG#F-7$F6$\"+<1'QL\"F-7$F;$\"+'4#QSEF-7$F@$\"+68F:NF-7$F]cs$\"+SZJ)e$ F-7$Ffjm$\"+U@l]OF-7$Fecs$\"+Rj)*3OF-7$FE$\"+-sj#[$F-7$F]ds$\"+4;vxJF- 7$FJ$\"+!yiz&GF-7$FO$\"*314R#FS7$FU$\"**RM+@FS7$FZ$\"*.u[#>FS7$Fin$\"* ]t9\"=FS7$F^o$\"*6`lu\"FS7$Fco$\"*Wjbt\"FS7$Fho$\"*s*zvFS7$Fgp$\"*/Y)Q>FS7$Faq$\"+0D'o@#F-7$Ffq$\"+7qA1DF- 7$F[r$\"+bB`6GF-7$F][r$\"+:-)\\%GF-7$F`r$\"+e^EdGF-7$Fj[r$\"+vAqQGF-7$ Fer$\"+aG[hEF-7$F_s$\"+lrdTAF-7$Fis$\"+=8V?:F-7$Fct$!*%3K(*))F-7$F]u$! +()G2*=&F-7$Fgu$!+AAG!Q*F-7$Fav$!,o1EZS\"F-7$Ffv$!,9&yxS>F-7$F[w$!,5HP !RDF-7$F\\ap$!,A_c)*o#F-7$F`w$!,vl]&HGF-7$F_js$!,T*=&*zGF-7$Fiap$!,`h+ Q\"HF-7$Fgjs$!,(H\"fp*GF-7$Few$!,qz^)zGF-7$F_x$!,)3#o\"4DF-7$Fix$!,c)> h??F-7$Fgbn$!,^@%>-;F-7$F^y$!,$fR(*\\6F-7$F_cn$!+gt$RL)F-7$Fcy$!+%[Y/j $F-7$Fhy$\"+`Y=&*GF-7$F]z$\"+247n\"*F-7$Fgz$\",b94N@\"F-7$Fa[l$\",80m[ \\\"F-7$Ff[l$\",Jd4qd\"F-7$F[\\l$\"+6(G%R@(FS7$Fg_l$\"*OLqq(FS7$F\\`l$\"*HJ=N)FS7$Fa`l$\"*.4)Q$*FS7 $F[al$\"+Z:.v5FS7$Feal$\"+pmJx6FS7$Fibl$\"+XE')p7FS7$Feer$\"+2'HcG\"FS 7$F_hp$\"+'e)G!H\"FS7$F]fr$\"+we'=E\"FS7$F^cl$\"+$3lnC\"FS7$F\\ip$\"+G R>.6FS7$Fccl$\"*Q&QR$*FS7$Fhcl$\"*:3\\W&FS7$F]dl$\"*`m%>(*F-7$Fgdl$!+a (y6\"RF-7$Fael$!,LG9z0\"F-7$F[fl$!,u;-!38F-7$Fefl$!,!e>@'o\"F-7$F_gl$! ,+dU`6#F-7$Figl$!,f*>#3V#F-7$Fchl$!,B;bG\"GF-7$F]il$!,AmeZZ$F-7$Fbil$! ,')*)Rh%QF-7$Fgil$!,cC7x@%F-7$F[[m$!,Mnz.&[F-7$F_\\m$!,t=(H&3&F-7$Fi\\ m$!,q;4?C&F-7$Fc]m$!,#zAA=`F-7$Fh]m$!,#QUlw^F-7$F]^m$!,\\moD#\\F-7$Fg^ m$!,zhP/n6PF-7$Ff`o$!,#)*)eFM$F-7$Ff_m$!,IbUb9$F-7$F^a o$!,1SMfe#F-7$F[`m$!,v(yepAF-7$Ffao$!+]gOq>FS7$F``m$!+gGE%Q\"FS7$F^bo$ !+:C'y9\"FS7$Fe`m$!*>8e*yFS7$F[jt$!*/\\VP#FS7$Ffbo$!)'GC>)FS7$Fcjt$\"* +lE%GFS7$Fj`m$\"*zDRh(FS7$F^co$\"+HJ1\"=\"FS7$F_am$\"+G.#)*f\"FS7$Fiam $\"+-K#G;#FS7$Fcbm$\"+/RSvBFS7$Fhbm$\"+\\$z:a#FS7$F]cm$\"+%H<%pEFS7$F] bq$\"+B%\\*)y#FS7$Fbcm$\"+:^9@HFS7$Febq$\"+JH\")GHFS7$Fgcm$\"+O2p5ZQFFS7$F[em$\"+EW[LCFS7$Feem$\"+a4jx>FS7$Fjem$\"+r%G#e;FS7$ F_fm$\"+aWFS7$F^o$\"**Rtq=F S7$Fco$\"*P0h&=FS7$Fho$\"*p')o!>FS7$F]p$\"*B@4-#FS7$Fbp$\"*+,!o@FS7$Fg p$\"*aU$[BFS7$F\\q$\"*'*[C^#FS7$Faq$\"+KZCHFF-7$Fe]n$\"+l!oy!GF-7$Ffq$ \"++N8SGF-7$F]^n$\"+j@i1FF-7$F[r$\"+b(>OZ#F-7$Fer$\"*Z.ZO#F-7$Fis$!+Z& fD#RF-7$Fct$!+4=zr!*F-7$F]u$!,7CDWK\"F-7$Fav$!,,*)3=w\"F-7$F[w$!,E'Rzc CF-7$F\\ap$!,&\\:jqDF-7$F`w$!,eiA_o#F-7$F_js$!,2\"f3IFF-7$Fiap$!,EnHaw #F-7$Fgjs$!,$[!f1v#F-7$Few$!,?oYF- 7$Fgbn$!,*4RnQ:F-7$F^y$!,,-%fz5F-7$F_cn$!+OoFQvF-7$Fcy$!+=@dEEF-7$Fhy$ \"+E*p)\\UF-7$F]z$\",&)*yM&3\"F-7$Fgz$\",8m>')Q\"F-7$Fa[l$\",x1TEm\"F- 7$Ff[l$\",$ou8E*3i\"FS7$Fe\\l$\"+Vb3 K9FS7$F_]l$\"+Yz+;6FS7$Fd]l$\"*W()QX*FS7$Fi]l$\"*#zrK#)FS7$F^^l$\"*;$= !R(FS7$Fc^l$\"*z4D(oFS7$Fh^l$\"*D#3@mFS7$F]_l$\"*,$G6mFS7$Fb_l$\"*(GUX oFS7$Fg_l$\"*/W;O(FS7$F\\`l$\"*A\\P3)FS7$Fa`l$\"*#**Q_#*FS7$F[al$\"+z( 3D5\"FS7$Feal$\"+epwG7FS7$Fibl$\"+C%[fJ\"FS7$Feer$\"+VOnA8FS7$F_hp$\"+ B!z'48FS7$F]fr$\"+\"FS7$F\\ip$\"*%*>%*4*FS7$Fccl $\"*71Y&eFS7$Fhcl$!)'[P$))FS7$F]dl$!+_F'f6)F-7$Fbdl$!,*z#oDA\"F-7$Fgdl $!,.M7E_\"F-7$F\\el$!,-9[[+#F-7$Fael$!,`*fw*G#F-7$Fefl$!,UCV&)z#F-7$Fi gl$!,3&41VKF-7$F]il$!,!QKpEQF-7$Fgil$!,&Q7V9VF-7$F_\\m$!,6l<^-&F-7$Fc] m$!,\"[RV^`F-7$Fh]m$!,_%>Qr_F-7$F]^m$!,sdq73&F-7$Fg^m$!,&=E*fa%F-7$Fa_ m$!,x@<]+%F-7$Ff`o$!,'G(\\ok$F-7$Ff_m$!,Hpu6X$F-7$F^ao$!,LqU'))GF-7$F[ `m$!,\\pFCc#F-7$Ffao$!+rih]AFS7$F``m$!+]'G9j\"FS7$F^bo$!+12Qx8FS7$Fe`m $!*7h%**)*FS7$F[jt$!*oa!pQFS7$Ffbo$!*2!**[@FS7$Fcjt$\"*9?/\">FS7$Fj`m$ \"*lCpA(FS7$F^co$\"+T$RX>\"FS7$F_am$\"+4Zvl;FS7$Fiam$\"+!>'f&H#FS7$Fcb m$\"+_L]JDFS7$Fhbm$\"+!)\\&Gr#FS7$F]cm$\"+Q&e%[GFS7$F]bq$\"+Qo+pHFS7$F bcm$\"+3%pB4$FS7$Febq$\"+IJC(3$FS7$Fgcm$\"+\"e(ygIFS7$Fadm$\"+x_I!y#FS 7$F[em$\"+t*)Q7CFS7$Feem$\"+&f*z2>FS7$Fjem$\"+DPXy:FS7$F_fm$\"+bVKs8FS 7$Fdfm$\"+t$))[A\"FS-Fifm6&F[gmFh]u$\"#bF]goF(-Fbgm6#%AVerner's~\"most ~efficient\"~schemeG-%%FONTG6$%*HELVETICAGF`gm-%&TITLEG6#%Verror~curve s~for~12~stage~order~8~Runge-Kutta~methodsG-%+AXESLABELSG6$Q\"x6\"Q!Fc ^x-%%VIEWG6$;F(Fdfm;$!#:!#?$\"#pF\\_x" 1 2 0 1 10 0 2 9 1 4 2 1.000000 45.000000 45.000000 0 1 "Enright-Verner scheme" "Verner's Map le scheme" "a modification of Verner's Maple scheme" "an \"efficient\" scheme" "a scheme of Tsitouras and Papakostas" "a modification of the Tsitouras-Papakostas scheme" "Verner's \"most efficient\" scheme" }}} }{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 555 "evalf[30](plot([u(x)-'un_RK8_2'(x),u(x)-'un_RK8_3'(x),u(x)-'un_ RK8_4'(x),\nu(x)-'un_RK8_5'(x),u(x)-'un_RK8_6'(x),u(x)-'un_RK8_7'(x)], x=0..5,-1.25e-19..5.5e-20,font=[HELVETICA,9],\ncolor=[COLOR(RGB,0,.65, 0),brown,red,COLOR(RGB,0,.75,.95),magenta,COLOR(RGB,.95,.55,0)],\nlege nd=[`Verner's Maple scheme`,`a modification of Verner's Maple scheme`, `an \"efficient\" scheme`,`a scheme of Tsitouras and Papakostas`,`a mo dification of the Tsitouras-Papakostas scheme`,`Verner's \"most effici ent\" scheme`],title=`error curves for 12 stage order 8 Runge-Kutta me thods`));" }}{PARA 13 "" 1 "" {GLPLOT2D 1093 628 628 {PLOTDATA 2 "6,-% 'CURVESG6%7bt7$$\"\"!F)F(7$$\"?MLLLLLLLLL3x&)*3\"!#I$\"+ei%Q5%F-7$$\"? ++++++++]i!R(*Rc\"F-$\"+yghkqF-7$$\"?nmmmmmmmm\"H2P\"Q?F-$\"+Hgsl%*F-7 $$\"?+++++++]7G))>Wr@F-$\"+;m5Z(*F-7$$\"?MLLLLLLLek.pu/BF-$\",z\\X>.\" F-7$$\"?nmmmmmm;/,>=0QCF-$\",CnS8/\"F-7$$\"?++++++++]PMnNrDF-$\",G6Sn/ \"F-7$$\"?MLLLLLL$eR(\\;m/FF-$\",$)4yj1\"F-7$$\"?nmmmmmmmT5ll'z$GF-$\" ,Di,v0\"F-7$$\"?+++++++](o/[r7(HF-$\",&ezlX5F-7$$\"?MLLLLLLLL$eRwX5$F- $\",*Hk\"y.\"F-7$$\"?MLLLLLLLLL$eI8k$F-$\"+1eJ[%*F-7$$\"?NLLLLLLLL$3x% 3yTF-$\"+dtbS$)F-7$$\"?ommmmmmmm\"z%4\\Y_F-$\"+&fh#QkF-7$$\"?NLLLLLLLL eR-/PiF-$\"*#[ur`!#H7$$\"?-++++++++DcmpisF-$\"*&3`TZF`p7$$\"?OLLLLLLLL e*)>VB$)F-$\"*z`jU6F`p$\"*S(3IRF`p7$$\"?+ ++++++++]i^Z]7F`p$\"*I.n+%F`p7$$\"?+++++++++](=h(e8F`p$\"*rpo7%F`p7$$ \"?+++++++++]P[6j9F`p$\"*!=YjSF`p7$$\"?MLLLLLLL$e*[z(yb\"F`p$\"*RMhm$F `p7$$\"?++++++++Dc,#>Uh\"F`p$\"*qq&*=$F`p7$$\"?nmmmmmmmm;a/cq;F`p$\"+Z *)HnFF-7$$\"?nmmmmmmm;zpYU%p\"F`p$\"+pA)3v#F-7$$\"?nmmmmmmmmT&)))G=zRlF-7$$\"?MLLLLLL3-jikF`p$\"+p!4)Q\\F-7$$\"?MLLL LLLLe9;0?E>F`p$\"+hFz8>F-7$$\"?+++++++]i!Rgs2&>F`p$!+p[?ciF-7$$\"?nmmm mmmmmm\"pW`(>F`p$!,@**\\yV\"F-7$$\"?MLLLLLLek.HW#)))>F`p$!,vST=2#F-7$$ \"?+++++++]iSmTI-?F`p$!,Lafr$HF-7$$\"?nmmmmmmTgx.Ry:?F`p$!,(H(fxZ$F-7$ $\"?MLLLLLLLe9TOEH?F`p$!,wqTDC%F-7$$\"?ommmmmm;a)e6Bi0#F`p$!,m*zB6`F-7 $$\"?,+++++++]i!f#=$3#F`p$!,lhw&zeF-7$$\"?,+++++++v$fQa)3@F`p$!,0(*ySg &F-7$$\"?,++++++++D\"=EX8#F`p$!,F;[jh%F-7$$\"?,++++++]i!*y?OZ@F`p$!,1q u6/%F-7$$\"?,+++++++Dcwz>g@F`p$!,Ls/j*GF-7$$\"?,++++++](=U(Q.t@F`p$!,9 B(*[C#F-7$$\"?,+++++++](=xpe=#F`p$!,f<^Bf\"F-7$$\"?MLLLLLL3_]%oi#*>#F` p$!++rW\"z'F-7$$\"?ommmmmm;a8(fbE@#F`p$\"+nAr%p\"F-7$$\"?,++++++Dcw4&[ gA#F`p$\"+y]2ZnF-7$$\"?MLLLLLLLeRA9WRAF`p$\",Gkq_N\"F-7$$\"?+++++++]il ZsAmAF`p$\",d!*p[6#F-7$$\"?nmmmmmmmm\"H28IH#F`p$\",1!\\8]EF-7$$\"?nmmm mmmTg_Z>J0BF`p$\",\">hFEFF-7$$\"?nmmmmmm;a8A3hpxg#F`p$\"+vC5L9F`p7$$\"?,+++++++]Pf4t.F F`p$\"+d!eRB\"F`p7$$\"?MLLLLLLLLe*Gst!GF`p$\"+OLp$4\"F`p7$$\"?,+++++++ ++DRW9HF`p$\"*$3LY**F`p7$$\"?,++++++++DJE>>IF`p$\"*m*4/$*F`p7$$\"?,+++ ++++]i!RU07$F`p$\"*!=4\")*)F`p7$$\"?,++++++++v=S2LKF`p$\"*aF*p*)F`p7$$ \"?ommmmmmmmm\"p)=MLF`p$\"*zu)o#*F`p7$$\"?,++++++++](=]@W$F`p$\"*hU5') *F`p7$$\"?MLLLLLLL$e*[$z*RNF`p$\"+DNU^5F`p7$$\"?,++++++++]iC$pk$F`p$\" +ComA6F`p7$$\"?MLLLLLLLe*[t\\sp$F`p$\"+@*>!\\6F`p7$$\"?ommmmmmm;H2qcZP F`p$\"+aiT\"=\"F`p7$$\"?,++++++++DJ]'Qx$F`p$\"+0p=07F`p7$$\"?MLLLLLLL$ 3_0j,!QF`p$\"+%Q.fC\"F`p7$$\"?ommmmmmmm;z5YEQF`p$\"+Y+t58F`p7$$\"?,+++ ++++]7.\"fF&QF`p$\"+mo809F`p7$$\"?MLLLLLLLek`8=/RF`p$\"+!3Q\"o;F`p7$$ \"?ommmmmmmm;/OgbRF`p$\"+`&HE!>F`p7$$\"?,+++++]7y]&*GLiRF`p$\"+^@Y%*=F `p7$$\"?MLLLLLLe*[o=i!pRF`p$\"+\"[5P*=F`p7$$\"?,+++++DJ&>D$oUsRF`p$\"+ DAAo=F`p7$$\"?ommmmm;/,>y9zvRF`p$\"+TIdy=F`p7$$\"?MLLLLL3x1'Q7c\"zRF`p $\"+E+MZ=F`p7$$\"?,++++++]7`p2_#)RF`p$\"+>mO.=F`p7$$\"?ommmmmmTN@_$zf* RF`p$\"+JMH(p\"F`p7$$\"?MLLLLLLLe*[$zV4SF`p$\",/1&>v8F-7$$\"?,++++++D \"yv^'*G-%F`p$\"+t')eq%*F-7$$\"?ommmmmm;/E+^NOSF`p$\"+.BQB_F-7$$\"?MLL LLLL3F%Ho8)\\SF`p$!+MYso^F-7$$\"?,+++++++]ilAFjSF`p$!,'[w,^8F-7$$\"?om mmmmmT&Q.[Mi2%F`p$!,#p;'*4AF-7$$\"?MLLLLLL$3_]p'>*3%F`p$!,Za(41OF-7$$ \"?ommmmm;a)3C!yn&4%F`p$!,c$eWbTF-7$$\"?,++++++Dcw4*e@5%F`p$!,*3s`4_F- 7$$\"?MLLLLL$eRAr,S'3TF`p$!,s4mc^&F-7$$\"?ommmmmmm\"zW7@^6%F`p$!,ZA#>? jF-7$$\"?-+++++]Pf$=B-;7%F`p$!,(=UN%R(F-7$$\"?NLLLLLL3F>RL3GTF`p$!,T(G ygvF-7$$\"?ommmmm;z%\\lWkX8%F`p$!,zKv2W)F-7$$\"?-++++++]i!RbX59%F`p$!, ,^.iP*F-7$$\"?ommmmmm\"z>'ox+aTF`p$!-]ZFk=5F-7$$\"?NLLLLLLLLL$)*pp;%F` p$!-s/Hm#3\"F-7$$\"?NLLLLLL3Fp^yftTF`p$!-?K$GL7\"F-7$$\"?NLLLLLL$3_+sD -=%F`p$!-%G'Q&H9\"F-7$$\"?NLLLLLLe9T)e`o=%F`p$!-EW\"y,9\"F-7$$\"?NLLLL LLL3xc9[$>%F`p$!-?W*y_8\"F-7$$\"?NLLLLLL$e*[$>Pn?%F`p$!-C3pA+6F-7$$\"? NLLLLLLL$3-$H**>UF`p$!,I!>-](*F-7$$\"?NLLLLLL$3Fpm[KB%F`p$!,-iAsw)F-7$ $\"?NLLLLLLLek.W]YUF`p$!,#\\Sy?wF-7$$\"?NLLLLLL3_+sA8`UF`p$!,d)\\3vkF- 7$$\"?NLLLLLL$ek.9g(fUF`p$!,@Uy(yaF-7$$\"?NLLLLLLeRs3!)QmUF`p$!,!=v()z ^F-7$$\"?NLLLLLLLL3xe,tUF`p$!,FcP!fRF-7$$\"?ommmmm;ajMj))ezUF`p$!,$)*G l7JF-7$$\"?-++++++v$4'\\=;'G%F`p$!,1cXXp#F-7$$\"?NLLLLL$eRse$[t#H%F`p$ !,%\\kBX:F-7$$\"?ommmmmm;a8AyI*H%F`p$!+EJq<(*F-7$$\"?-+++++]P%)R33)eI% F`p$!*XuKW%F`p7$$\"?NLLLLLLe9m%z`CJ%F`p$\"*e#H-`F`p7$$\"?ommmmm;zW#4yE !>VF`p$\"*Mx$4()F`p7$$\"?-+++++++v=n(*fDVF`p$\"+c\\\\$R\"F`p7$$\"?ommm mmmTNrRduQVF`p$\"+aq=8BF`p7$$\"?NLLLLLL$eRAr\"*=N%F`p$\"+ANjrKF`p7$$\" ?-++++++Dcw%oP]O%F`p$\"+$>GQk$F`p7$$\"?ommmmmmm;HdO=yVF`p$\"+rQ*)eRF`p 7$$\"?,+++++]PM-\\NA%Q%F`p$\"+$p\">_TF`p7$$\"?MLLLLLL3_vSME!R%F`p$\"+= 0\\BVF`p7$$\"?,+++++v$4@mQ$G$R%F`p$\"+vDo%H%F`p7$$\"?ommmmm;zp[KLI'R%F `p$\"+i1foUF`p7$$\"?MLLLLLekGNyKK*R%F`p$\"+R%GmK%F`p7$$\"?,++++++](=UA VBS%F`p$\"+T%>`P%F`p7$$\"?ommmmmm\"H#o2IU9WF`p$\"+U/uuVF`p7$$\"?MLLLLL LLe9\"z-lU%F`p$\"+*3\"yLVF`p7$$\"?ommmmmm;H2eBm]WF`p$\"+%*>/mTF`p7$$\" ?,+++++++++D>#[Z%F`p$\"+**)fj!RF`p7$$\"?MLLLLLLL$3_5,-`%F`p$\"+]J^hKF` p7$$\"?ommmmmmmmT&G!e&e%F`p$\"+4UltFF`p7$$\"?NLLLLLLLLL$)Qk%o%F`p$\"+t +46AF`p7$$\"?-+++++++]iSjE!z%F`p$\"+To$4(=F`p7$$\"?-+++++++]P40O\"*[F` p$\"+'*)\\Hl\"F`p7$$\"\"&F)$\"+\\%[-\\\"F`p-%&COLORG6&%$RGBGF($\"#l!\" #F(-%'LEGENDG6#%6Verner's~Maple~schemeG-F$6%7jsF'7$$\"?ommmmmmmmTN@Ki8 !#J$!+!R>Gr\"F`[n7$$\"?NLLLLLLLL$3FWYs#F`[n$!+e(fC!GF`[n7$$\"?-+++++++ +D1k'p3%F`[n$!+?tw')HF`[n7$$\"?qmmmmmmmmmT&)G\\aF`[n$!+^6S\"4#F`[n7$$ \"?0++++++++]7G$R<)F`[n$\"+7-5oSF`[n7$F+$\"+hKC)=\"F-7$F1$\"+aECLIF-7$ F6$\"+W<*R'[F-7$F;$\"+>-GT^F-7$F@$\"+9%p(zcF-7$FE$\"+ok4VeF-7$FJ$\"+%o FV)fF-7$FO$\"+W3(=H'F-7$FT$\"+NA)>L'F-7$FY$\"+)4yDN'F-7$Fhn$\"+q3;kkF- 7$F]o$\"+$Rr#4iF-7$Fbo$\"+![&f>dF-7$Fgo$\"+'e64n%F-7$F\\p$\"*dD5(RF`p7 $Fbp$\"**y?5NF`p7$Fgp$\"*(et1KF`p7$F\\q$\"*3Ze+$F`p7$Faq$\"*(Q:%*GF`p7 $Ffq$\"*\"[nyGF`p7$F[r$\"*RuK&HF`p7$F`r$\"*>o]4$F`p7$Fer$\"*%GM\">$F`p 7$Fjr$\"*@U&eJF`p7$F_s$\"*QaA0$F`p7$Fds$\"+Gsr!*HF-7$F^t$\"+W13MJF-7$F ht$\"+=#3i^$F-7$F]u$\"+%*zxkQF-7$Fbu$\"+P_@)=%F-7$Fgu$\"+x7;\"G%F-7$F \\v$\"+oAE#G%F-7$Fav$\"+7pxQTF-7$Ffv$\"+\")[mARF-7$F`w$\"*-\")3/$F-7$F jw$!+=8vj)*F-7$Fdx$!,YmWV$>F-7$F^y$!,5$fb3GF-7$Fcy$!,J'Q\\'f$F-7$Fhy$! ,tz6=;%F-7$$\"?,++++++DJXR0g*3#F`p$!,9e,9?%F-7$$\"?,++++++]7G)[=g4#F`p $!,'\\rA6UF-7$$\"?,++++++v$4rVOC5#F`p$!,F(=%=?%F-7$F]z$!,0bT.;%F-7$$\" ?,++++++]Pf$G!p@@F`p$!,P:58%RF-7$Fbz$!,4%pa$p$F-7$F\\[l$!,>e2En#F-7$Ff [l$!,Mqm_!=F-7$F[\\l$!,_ix^;\"F-7$F`\\l$!+v$>!=aF-7$Fe\\l$!+-Un&[\"F-7 $Fj\\l$\"+WmR/SF-7$F_]l$\",]LKd2\"F-7$Fd]l$\",ngxoj\"F-7$F^^l$\",IJkz% =F-7$Fh^l$\",;^wr,#F-7$$\"?nmmmmmmTN'fW2XN#F`p$\",)\\q,D?F-7$F]_l$\",& [!H6-#F-7$$\"?nmmmmmm\"H#=&>0\"zBF`p$\"+,ky%)F`p7$Fibl$\"*(p>:!*F`p7$ F^cl$\"*%>\")p(*F`p7$Fccl$\"+dCr!3\"F`p7$F]dl$\"+,.-(>\"F`p7$Fgdl$\"+* [HBF\"F`p7$Fael$\"+hdhr8F`p7$$\"?ommmmmm;aQG-ZyQF`p$\"+#Q?3U\"F`p7$Ffe l$\"+>bTY9F`p7$$\"?ommmmmmTgF;p.ds'F-7$Fhhl$\"+&42eD$F-7$F]il$\"* eWe/\"F-7$Fbil$!+%eNV+(F-7$Fgil$!,ubG![7F-7$F\\jl$!,@ke/!=F-7$Fajl$!,7 Y9mo#F-7$F[[m$!,*pYB/PF-7$Fe[m$!,._Q!>WF-7$Fj[m$!,0fq67&F-7$F_\\m$!,(= .5K_F-7$Fd\\m$!,riZE#eF-7$Fi\\m$!,Az55Z'F-7$F^]m$!,DO;'fqF-7$Fc]m$!,$R m*4c(F-7$Fh]m$!,K))fF#zF-7$F]^m$!,3:aZ;)F-7$$\"?NLLLLL$3xJUlRN=%F`p$!, 'yI(e:)F-7$Fb^m$!,(H:Y[\")F-7$$\"?NLLLLL$e9\"fAv;!>%F`p$!,1m_sC)F-7$Fg ^m$!,An(yM#)F-7$F\\_m$!,y!H[G\")F-7$Fa_m$!,d7Pj_(F-7$Ff_m$!,VYJ5'pF-7$ F[`m$!,&f/qhiF-7$F``m$!,^9-6a&F-7$Fe`m$!,jA<$))[F-7$Fj`m$!,P\"=*Go%F-7 $F_am$!,@`HY&QF-7$Fdam$!,4pX&fKF-7$Fiam$!,`,Jw&HF-7$F^bm$!,/I(QC@F-7$F cbm$!,95O)>9F`p-%'COLOURG6&Fbjm$\")#)eqk!\")$\"))eqk\"F`hoFaho-Fgjm6#%Ha~modi fication~of~Verner's~Maple~schemeG-F$6%7isF'7$F^[n$!+vQ&*R@F`[n7$Fd[n$ !+B:O%z$F`[n7$Fi[n$!+uV![F&F`[n7$F^\\n$!+3(z/6&F`[n7$Fc\\n$!+==z\\7F`[ n7$F+$\"*;0()*\\F-7$F1$\"+TI``@F-7$F6$\"+b`q^RF-7$F;$\"+,DaVUF-7$F@$\" +;-'yz%F-7$FE$\"+b*z>)\\F-7$FJ$\"+32TW^F-7$FO$\"+&3>DZ&F-7$FT$\"+@/'=` &F-7$FY$\"+rt0qbF-7$Fhn$\"+,*oFp&F-7$F]o$\"+>&ymY&F-7$Fbo$\"+SXHu\\F-7 $Fgo$\"+kh)e*QF-7$F\\p$\"*]Z'3KF`p7$Fbp$\"*E<,z#F`p7$Fgp$\"*^@.`#F`p7$ F\\q$\"*v0GO#F`p7$Faq$\"*:.uE#F`p7$Ffq$\"*L^AD#F`p7$F[r$\"*%3N;BF`p7$F `r$\"*:7sW#F`p7$Fer$\"*w38e#F`p7$Fjr$\"*z.yq#F`p7$Fds$\"+S+x&4$F-7$F^t $\"+U$)*pS$F-7$Fht$\"+gwW6OF-7$$\"?MLLLLLLe9m%>(*)zF-7$F`\\l$!,;M-CL\"F-7$Fe\\l$!+YD(*y\"*F-7$Fj\\l$!+ ]+$)eKF-7$F_]l$\"+.'G&yWF-7$Fd]l$\",,t'RK6F-7$F^^l$\",=l\\GU\"F-7$Fh^l $\",BL)ek;F-7$Fhgn$\",+K^Ap\"F-7$F]_l$\",aP\"Q/EjnF`p7$F^cl$\"*0UuN(F`p7$Fccl$\"*Kp7F)F`p7$F]dl$\"*f?+q*F`p7$Fgd l$\"+1Vg*3\"F`p7$Fael$\"+%R$)p?\"F`p7$$\"?MLLLLLL3_vlYhlQF`p$\"+zZbF7F `p7$Fh[o$\"+!QCsB\"F`p7$$\"?,++++++Dc,\"zD8*QF`p$\"+<(>I@\"F`p7$Ffel$ \"+Ek4'>\"F`p7$Fe\\o$\"+n2!=-\"F`p7$F[fl$\"*?z7*yF`p7$Figl$\"*\"GAF@F` p7$Fchl$!+n8SY`F-7$Fhhl$!,&R[9J5F-7$F]il$!,:P$pT9F-7$Fbil$!,tR5)RAF-7$ Fgil$!,]'pV-GF-7$F\\jl$!,WcT8L$F-7$Fajl$!,,RhT6%F-7$F[[m$!,`KA]'\\F-7$ Fe[m$!,z&o_\\bF-7$F_\\m$!,\"o`//iF-7$Fi\\m$!,_zTn?(F-7$F^]m$!,1g18q(F- 7$Fc]m$!,*eamS\")F-7$F]^m$!,Z`_]u)F-7$Fg^m$!,q!G?u))F-7$$\"?NLLLLLL3-8 D$4,?%F`p$!,IE^D!*)F-7$F\\_m$!,!e^Nk))F-7$$\"?NLLLLLLe*[=1lL@%F`p$!,$3 nyg()F-7$Fa_m$!,s%>cA&)F-7$Ff_m$!,PDx15)F-7$F[`m$!,R#QTSvF-7$F``m$!,&z nK[pF-7$Fe`m$!,b[ACQ'F-7$Fj`m$!,**Q>**='F-7$F_am$!,$H?dVaF-7$Fdam$!,0< x2)[F-7$Fiam$!,.1]`e%F-7$F^bm$!,6[[Zx$F-7$Fcbm$!,uj\"=ILF-7$Fhbm$!+4O` :HF`p7$F]cm$!+prUF@F`p7$Fbcm$!+j&HY\"=F`p7$Fgcm$!+4^k[8F`p7$F\\dm$!*JR ?a%F`p7$Fadm$\"*LW9m&F`p7$Ffdm$\"+suxd5F`p7$F[em$\"+&f$=J:F`p7$Fifm$\" +5d:[BF`p7$Fcgm$\"+@(H%=EF`p7$Fjeo$\"+zD36FF`p7$Fhgm$\"+.re'z#F`p7$Fbf o$\"+\\b6xFF`p7$F]hm$\"+***f#RFF`p7$Fbhm$\"+?-#eU#F`p7$Fghm$\"+$4:Q2#F `p7$F\\im$\"+#z2dh\"F`p7$Faim$\"+TgyP8F`p7$Ffim$\"+W(R2<\"F`p7$F[jm$\" +hf#30\"F`p-F\\ho6&Fbjm$\"*++++\"F`hoF(F(-Fgjm6#%6an~\"efficient\"~sch emeG-F$6%7]tF'7$Fd[n$!+PAmqsF`[n7$F^\\n$!,r$\\9F-7$F+$!+Ij:K7F-7$F1$!*'RMS6F-7$F6$\"+5:d(R\"F-7$FJ$\"+kR %fi#F-7$Fhn$\"+[wyLMF-7$$\"?MLLLLLLLLe*[`HP$F-$\"+][I(\\$F-7$F]o$\"+i5 ![b$F-7$$\"?MLLLLLLLL3xwq4RF-$\"+%y%*)=NF-7$Fbo$\"+A%pDS$F-7$$\"?-++++ +++]PfyG7ZF-$\"+HSF`p7$F\\q$\"*Lum$=F`p7$Faq$\"*>z#oF`p7$Fjr$\"*=%3* *>F`p7$F_s$\"*-Zm0#F`p7$Fds$\"+f.Z$G#F-7$F^t$\"+&))[fm#F-7$Fht$\"+!\\' )fH$F-7$F]u$\"+C#\\Xu$F-7$Fbu$\"+Go;gTF-7$Fgu$\"+bZT.VF-7$F\\v$\"+?bdJ WF-7$Fav$\"+)yl1]%F-7$Ffv$\"+)33<_%F-7$F[w$\"+kX,&>%F-7$F`w$\"+2)f3j$F -7$Few$\"+Z\"*ot=F-7$Fjw$!**>%*Q7F-7$Fdx$!+4AJzYF-7$F^y$!+A4_m)*F-7$Fc y$!,9%3Gr:F-7$Fhy$!,,1nP>#F-7$Fedn$!,d&>DVBF-7$F]z$!,iTjOZ#F-7$$\"?,++ ++++DcwMBF:@F`p$!,4$3n%*F`p7$F[al$\"*.n.J)F`p 7$F`al$\"*)Q#)GvF`p7$Feal$\"*LKL.(F`p7$Fjal$\"*xXvy'F`p7$F_bl$\"*!3Pzn F`p7$Fdbl$\"*,D:,(F`p7$Fibl$\"*pe(*\\(F`p7$F^cl$\"*Tk'G\")F`p7$Fccl$\" *Y&f]!*F`p7$F]dl$\"+\"G'oH5F`p7$Fgdl$\"+Tj\\D6F`p7$Fael$\"+$)*\\pB\"F` p7$Fbip$\"+WKvi7F`p7$Fh[o$\"+jF@%G\"F`p7$Fjip$\"+E)4'*G\"F`p7$Ffel$\"+ NV,+8F`p7$Fe\\o$\"+0rfe7F`p7$F[fl$\"+%e+X>\"F`p7$Figl$\"*`+H))*F`p7$Fc hl$\"+sr=orF-7$F]il$\"+GrG.QF-7$Fgil$!+Q2^p3lF-7$F_ \\m$!,=wAw.#F-7$Fd\\m$!,4o#\\tBF-7$Fi\\m$!,JTZex#F-7$F^]m$!,4;\")3=$F- 7$Fc]m$!,-_i&yNF-7$Fh]m$!,rc96#RF-7$F]^m$!,onQnA%F-7$Fh`o$!,*>F:AUF-7$ Fb^m$!,Xe\\PA%F-7$F`ao$!,_hK.Y%F-7$Fg^m$!,)=?g`WF-7$$\"?NLLLLL$3_]4R&z '>%F`p$!,(\\tM]WF-7$Fh]q$!,g\"f@4YF-7$$\"?NLLLLL$e*)4$fKU.UF`p$!,teG.g %F-7$F\\_m$!,R6KBf%F-7$$\"?NLLLLL$3Fpw7^+@%F`p$!,]%)4=m%F-7$F`^q$!,0!z *3l%F-7$$\"?NLLLLL$ekGg**ym@%F`p$!,#y%G&QYF-7$Fa_m$!,]#*o8h%F-7$Ff_m$! ,;)[DUWF-7$F[`m$!,ws#zkTF-7$F``m$!,jvn(QQF-7$Fe`m$!,Ny:\\]$F-7$Fj`m$!, \"3=:#R$F-7$F_am$!,T+V'GHF-7$Fdam$!,c$3PpDF-7$Fiam$!,3#QS#Q#F-7$F^bm$! ,BVpw%=F-7$Fcbm$!,\"eNx_:F-7$Fhbm$!+x0zw7F`p7$F]cm$!*IjjP(F`p7$Fbcm$!* I!Qk_F`p7$Fgcm$!*\"[2i?F`p7$$\"?NLLLLL$3_]Mvs@L%F`p$\"*LW+%GF`p7$F\\dm $\"*U&ziTF`p7$$\"?-+++++]il(fs=`M%F`p$\"*wT)HtF`p7$Fadm$\"+Z#G>9\"F`p7 $Ffdm$\"+M^Z)[\"F`p7$F[em$\"+HD)z#=F`p7$Fifm$\"+$e-eV#F`p7$Fcgm$\"+p-! zj#F`p7$Fjeo$\"+,&QUr#F`p7$Fhgm$\"+=;G(z#F`p7$Fbfo$\"+].a&y#F`p7$F]hm$ \"+>GufFF`p7$Fbhm$\"+x#QNb#F`p7$Fghm$\"+UmXsAF`p7$F\\im$\"+$p\"Rn=F`p7 $Faim$\"+p^vx:F`p7$Ffim$\"+)y9lQ\"F`p7$F[jm$\"+)\\qiC\"F`p-F`jm6&FbjmF ($\"#vFejm$\"#&*Fejm-Fgjm6#%Ea~scheme~of~Tsitouras~and~PapakostasG-F$6 %7^sF'7$Fd[n$!+K\")e@tF`[n7$F^\\n$!,l?o5T\"F`[n7$Fbeq$!,kSQ%H:F`[n7$Fc \\n$!,/Ah&z:F`[n7$Fjeq$!+_u&*4:F-7$F+$!+'>$[O8F-7$F1$!*[frG#F-7$F6$\"+ <1'QL\"F-7$FJ$\"+'4#QSEF-7$Fhn$\"+68F:NF-7$F^gq$\"+SZJ)e$F-7$F]o$\"+U@ l]OF-7$Ffgq$\"+Rj)*3OF-7$Fbo$\"+-sj#[$F-7$F^hq$\"+4;vxJF-7$Fgo$\"+!yiz &GF-7$F\\p$\"*314R#F`p7$Fbp$\"**RM+@F`p7$Fgp$\"*.u[#>F`p7$F\\q$\"*]t9 \"=F`p7$Faq$\"*6`lu\"F`p7$Ffq$\"*Wjbt\"F`p7$F[r$\"*s*zvF`p7$Fjr$\"*/Y)Q>F`p7$Fds$\"+0D'o@#F-7$F^t$\"+7qA 1DF-7$Fht$\"+bB`6GF-7$Fj^p$\"+:-)\\%GF-7$F]u$\"+e^EdGF-7$Fg_p$\"+vAqQG F-7$Fbu$\"+aG[hEF-7$F\\v$\"+lrdTAF-7$Ffv$\"+=8V?:F-7$F`w$!*%3K(*))F-7$ Fjw$!+()G2*=&F-7$Fdx$!+AAG!Q*F-7$F^y$!,o1EZS\"F-7$Fcy$!,9&yxS>F-7$Fhy$ !,5HP!RDF-7$Fedn$!,A_c)*o#F-7$F]z$!,vl]&HGF-7$F`^r$!,T*=&*zGF-7$Fben$! ,`h+Q\"HF-7$Fh^r$!,(H\"fp*GF-7$Fbz$!,qz^)zGF-7$F\\[l$!,)3#o\"4DF-7$Ff[ l$!,c)>h??F-7$F[\\l$!,^@%>-;F-7$F`\\l$!,$fR(*\\6F-7$Fe\\l$!+gt$RL)F-7$ Fj\\l$!+%[Y/j$F-7$F_]l$\"+`Y=&*GF-7$Fd]l$\"+247n\"*F-7$F^^l$\",b94N@\" F-7$Fh^l$\",80m[\\\"F-7$F]_l$\",Jd4qd\"F-7$Fb_l$\"+6(G%R@(F`p7$Fibl$\"*OLqq(F`p7$F^cl$\"*H J=N)F`p7$Fccl$\"*.4)Q$*F`p7$F]dl$\"+Z:.v5F`p7$Fgdl$\"+pmJx6F`p7$Fael$ \"+XE')p7F`p7$Fbip$\"+2'HcG\"F`p7$Fh[o$\"+'e)G!H\"F`p7$Fjip$\"+we'=E\" F`p7$Ffel$\"+$3lnC\"F`p7$Fe\\o$\"+GR>.6F`p7$F[fl$\"*Q&QR$*F`p7$Figl$\" *:3\\W&F`p7$Fchl$\"*`m%>(*F-7$F]il$!+a(y6\"RF-7$Fgil$!,LG9z0\"F-7$F\\j l$!,u;-!38F-7$Fajl$!,!e>@'o\"F-7$F[[m$!,+dU`6#F-7$Fe[m$!,f*>#3V#F-7$F_ \\m$!,B;bG\"GF-7$Fi\\m$!,AmeZZ$F-7$F^]m$!,')*)Rh%QF-7$Fc]m$!,cC7x@%F-7 $F]^m$!,Mnz.&[F-7$Fg^m$!,t=(H&3&F-7$F\\_m$!,q;4?C&F-7$Fa_m$!,#zAA=`F-7 $Ff_m$!,#QUlw^F-7$F[`m$!,\\moD#\\F-7$Fe`m$!,zhP/n6PF-7 $Fdam$!,#)*)eFM$F-7$Fiam$!,IbUb9$F-7$F^bm$!,1SMfe#F-7$Fcbm$!,v(yepAF-7 $Fhbm$!+]gOq>F`p7$F]cm$!+gGE%Q\"F`p7$Fbcm$!+:C'y9\"F`p7$Fgcm$!*>8e*yF` p7$F\\^s$!*/\\VP#F`p7$F\\dm$!)'GC>)F`p7$Fd^s$\"*+lE%GF`p7$Fadm$\"*zDRh (F`p7$Ffdm$\"+HJ1\"=\"F`p7$F[em$\"+G.#)*f\"F`p7$Feem$\"+-K#G;#F`p7$Fif m$\"+/RSvBF`p7$F^gm$\"+\\$z:a#F`p7$Fcgm$\"+%H<%pEF`p7$Fjeo$\"+B%\\*)y# F`p7$Fhgm$\"+:^9@HF`p7$Fbfo$\"+JH\")GHF`p7$F]hm$\"+O2p 5ZQFF`p7$Fghm$\"+EW[LCF`p7$F\\im$\"+a4jx>F`p7$Faim$\"+r%G#e;F`p7$Ffim$ \"+aWF`p7$Fa q$\"**Rtq=F`p7$Ffq$\"*P0h&=F`p7$F[r$\"*p')o!>F`p7$F`r$\"*B@4-#F`p7$Fer $\"*+,!o@F`p7$Fjr$\"*aU$[BF`p7$F_s$\"*'*[C^#F`p7$Fds$\"+KZCHFF-7$Fis$ \"+l!oy!GF-7$F^t$\"++N8SGF-7$Fct$\"+j@i1FF-7$Fht$\"+b(>OZ#F-7$Fbu$\"*Z .ZO#F-7$Ffv$!+Z&fD#RF-7$F`w$!+4=zr!*F-7$Fjw$!,7CDWK\"F-7$F^y$!,,*)3=w \"F-7$Fhy$!,E'RzcCF-7$Fedn$!,&\\:jqDF-7$F]z$!,eiA_o#F-7$F`^r$!,2\"f3IF F-7$Fben$!,EnHaw#F-7$Fh^r$!,$[!f1v#F-7$Fbz$!,?oYF-7$F[\\l$!,*4RnQ:F-7$F`\\l$!,,-%fz5F-7$Fe\\l$!+OoFQv F-7$Fj\\l$!+=@dEEF-7$F_]l$\"+E*p)\\UF-7$Fd]l$\",&)*yM&3\"F-7$F^^l$\",8 m>')Q\"F-7$Fh^l$\",x1TEm\"F-7$F]_l$\",$ou8E*3i\"F`p7$F\\`l$\"+Vb3K9F`p7$Fa`l$\"+Yz+;6F`p7$Ff`l$\"*W()QX*F`p 7$F[al$\"*#zrK#)F`p7$F`al$\"*;$=!R(F`p7$Feal$\"*z4D(oF`p7$Fjal$\"*D#3@ mF`p7$F_bl$\"*,$G6mF`p7$Fdbl$\"*(GUXoF`p7$Fibl$\"*/W;O(F`p7$F^cl$\"*A \\P3)F`p7$Fccl$\"*#**Q_#*F`p7$F]dl$\"+z(3D5\"F`p7$Fgdl$\"+epwG7F`p7$Fa el$\"+C%[fJ\"F`p7$Fbip$\"+VOnA8F`p7$Fh[o$\"+B!z'48F`p7$Fjip$\"+\"F`p7$Fe\\o$\"*%*>%*4*F`p7$F[fl$\"*71Y&eF`p7$Figl$!)' [P$))F`p7$Fchl$!+_F'f6)F-7$Fhhl$!,*z#oDA\"F-7$F]il$!,.M7E_\"F-7$Fbil$! ,-9[[+#F-7$Fgil$!,`*fw*G#F-7$Fajl$!,UCV&)z#F-7$Fe[m$!,3&41VKF-7$Fi\\m$ !,!QKpEQF-7$Fc]m$!,&Q7V9VF-7$Fg^m$!,6l<^-&F-7$Fa_m$!,\"[RV^`F-7$Ff_m$! ,_%>Qr_F-7$F[`m$!,sdq73&F-7$Fe`m$!,&=E*fa%F-7$F_am$!,x@<]+%F-7$Fdam$!, 'G(\\ok$F-7$Fiam$!,Hpu6X$F-7$F^bm$!,LqU'))GF-7$Fcbm$!,\\pFCc#F-7$Fhbm$ !+rih]AF`p7$F]cm$!+]'G9j\"F`p7$Fbcm$!+12Qx8F`p7$Fgcm$!*7h%**)*F`p7$F\\ ^s$!*oa!pQF`p7$F\\dm$!*2!**[@F`p7$Fd^s$\"*9?/\">F`p7$Fadm$\"*lCpA(F`p7 $Ffdm$\"+T$RX>\"F`p7$F[em$\"+4Zvl;F`p7$Feem$\"+!>'f&H#F`p7$Fifm$\"+_L] JDF`p7$F^gm$\"+!)\\&Gr#F`p7$Fcgm$\"+Q&e%[GF`p7$Fjeo$\"+Qo+pHF`p7$Fhgm$ \"+3%pB4$F`p7$Fbfo$\"+IJC(3$F`p7$F]hm$\"+\"e(ygIF`p7$Fbhm$\"+x_I!y#F`p 7$Fghm$\"+t*)Q7CF`p7$F\\im$\"+&f*z2>F`p7$Faim$\"+DPXy:F`p7$Ffim$\"+bVK s8F`p7$F[jm$\"+t$))[A\"F`p-F`jm6&FbjmFias$\"#bFejmF(-Fgjm6#%AVerner's~ \"most~efficient\"~schemeG-%%FONTG6$%*HELVETICAG\"\"*-%&TITLEG6#%Verro r~curves~for~12~stage~order~8~Runge-Kutta~methodsG-%+AXESLABELSG6$Q\"x 6\"Q!Febv-%%VIEWG6$;F(F[jm;$!$D\"!#@$FdavF^cv" 1 2 0 1 10 0 2 9 1 4 2 1.000000 45.000000 45.000000 0 1 "Verner's Maple scheme" "a modificati on of Verner's Maple scheme" "an \"efficient\" scheme" "a scheme of Ts itouras and Papakostas" "a modification of the Tsitouras-Papakostas sc heme" "Verner's \"most efficient\" scheme" }}}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 49 "Test 7 of 12 stage, order 8 Runge-Kutta m ethods " }}{PARA 256 "" 0 "" {TEXT -1 2 " " }{XPPEDIT 18 0 "dy/dx=-( 1+4*cos(3*x))*(y-1/3)" "6#/*&%#dyG\"\"\"%#dxG!\"\",$*&,&F&F&*&\"\"%F&- %$cosG6#*&\"\"$F&%\"xGF&F&F&F&,&%\"yGF&*&F&F&F2F(F(F&F(" }{TEXT -1 5 " , " }{XPPEDIT 18 0 "y(0)=1" "6#/-%\"yG6#\"\"!\"\"\"" }{TEXT -1 2 ". " }}{PARA 0 "" 0 "" {TEXT -1 10 "Solution: " }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "y = 1/3" "6#/%\"yG*&\"\"\"F&\"\"$!\"\" " }{TEXT -1 1 " " }{XPPEDIT 18 0 "exp(-4/3*sin(3*x)+8/3*sin(3/2*x)*cos (3/2*x))+2/3" "6#,&-%$expG6#,&*(\"\"%\"\"\"\"\"$!\"\"-%$sinG6#*&F+F*% \"xGF*F*F,**\"\")F*F+F,-F.6#*(F+F*\"\"#F,F1F*F*-%$cosG6#*(F+F*F7F,F1F* F*F*F**&F7F*F+F,F*" }{TEXT -1 1 " " }{XPPEDIT 18 0 "exp(-4/3*sin(3*x)- x)" "6#-%$expG6#,&*(\"\"%\"\"\"\"\"$!\"\"-%$sinG6#*&F*F)%\"xGF)F)F+F0F +" }{TEXT -1 2 ". " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 " > " 0 "" {MPLTEXT 1 0 181 "de := diff(y(x),x)=-(1+4*cos(3*x))*(y(x)-1/ 3);\nic := y(0)=1;\nsimplify(dsolve(\{de,ic\},y(x)));\nv := unapply(rh s(%),x):\nplot(v(x),x=0..5,0..1.1,font=[HELVETICA,9],labels=[`x`,`y(x) `]);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%#deG/-%%diffG6$-%\"yG6#%\"xG F,,$*&,&\"\"\"F0*&\"\"%F0-%$cosG6#,$*&\"\"$F0F,F0F0F0F0F0,&F)F0#F0F8! \"\"F0F;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%#icG/-%\"yG6#\"\"!\"\"\" " }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/-%\"yG6#%\"xG,&*&#\"\"\"\"\"$F+-% $expG6#,&*&#\"\"%F,F+-%$sinG6#,$*&F,F+F'F+F+F+!\"\"*&#\"\")F,F+*&-F56# ,$*(F,F+\"\"#F9F'F+F+F+-%$cosGF?F+F+F+F+F+*&#FBF,F+-F.6#,&*&#F3F,F+F4F +F9F'F9F+F+" }}{PARA 13 "" 1 "" {GLPLOT2D 503 318 318 {PLOTDATA 2 "6&- %'CURVESG6$7ap7$$\"\"!F)$\"\"\"F)7$$\"3gmmTN@Ki8!#>$\"3W+7cSy5h&*!#=7$ $\"3ALL$3FWYs#F/$\"3KtP[t*Q;:*F27$$\"3%)***\\iSmp3%F/$\"3g.\"H>f!3q()F 27$$\"3WmmmT&)G\\aF/$\"36p*p.:G\\T)F27$$\"3m****\\7G$R<)F/$\"3a?glh]$z x(F27$$\"3GLLL3x&)*3\"F2$\"3IM[S(o-#HsF27$$\"3em\"z%\\v#pK\"F2$\"3i=)H '*Q$=:oF27$$\"3))**\\i!R(*Rc\"F2$\"3w,'pRB0LX'F27$$\"3&edVF27$$\"3%QL$3_DG1qF2$\" 3'fN^hMLe*)>VB$)F2$\"3DB(Rfp)*\\j%F27$$\"3Y++DJbw!Q*F2$ \"3%GsCu$*)zK]F27$$\"3+N$ekGkX#**F2$\"3u>+\\,YW?`F27$$\"3%ommTIOo/\"!# <$\"3q]2x8ZEqcF27$$\"3E+]7GTt%4\"Fgt$\"39b$=$pWlHgF27$$\"3YLL3_>jU6Fgt $\"3nwYdkc=KkF27$$\"3ym;HdNb'>\"Fgt$\"3l[hQOW]BpF27$$\"37++]i^Z]7Fgt$ \"3IVnF)*yXIuF27$$\"35+++v\"=YI\"Fgt$\"3ahS!3L%e=zF27$$\"33++](=h(e8Fg t$\"3l&QV-<82M)F27$$\"3&*****\\7!Q4T\"Fgt$\"3^]H\"3wS2k)F27$$\"3/++]P[ 6j9Fgt$\"3ur)[IAj$)z)F27$$\"3'=HKkAg\\Z\"Fgt$\"33z^;ogY6))F27$$\"3W$ek `h0o[\"Fgt$\"3h=q?g>u:))F27$$\"3/voH/5l)\\\"Fgt$\"3p\\\\U!)G36))F27$$ \"3%o;HKR'\\5:Fgt$\"3%G4&GMdV(z)F27$$\"3-]P4rr=M:Fgt$\"3Erd.MaCV()F27$ $\"3UL$e*[z(yb\"Fgt$\"3m)))[\\1qQl)F27$$\"34+Dc,#>Uh\"Fgt$\"3(fTb\\\\y 3J)F27$$\"3wmm;a/cq;Fgt$\"3-!y\"yF27$$\"3\"pm;a)))G=BtF27$$\"3%ommmJ>IFgt$ \"3N!o4Joz]`$F27$$\"3F+]i!RU07$Fgt$\"3=,?;D0\"Qg$F27$$\"3+++v=S2LKFgt$ \"3wRH=fZn5PF27$$\"3Jmmm\"p)=MLFgt$\"3RsXuk([b#QF27$$\"3B++](=]@W$Fgt$ \"3%4[=*QOMSRF27$$\"3mm\"H#oZ1\"\\$Fgt$\"3QK??D+QyRF27$$\"35L$e*[$z*RN Fgt$\"3UAxt;S)>+%F27$$\"3%o;Hd!fX$f$Fgt$\"3+h91z&\\y+%F27$$\"3e++]iC$p k$Fgt$\"3eIRs#H!Q\"*RF27$$\"3ILe*[t\\sp$Fgt$\"3m\"Rx)H&*[cRF27$$\"3[m; H2qcZPFgt$\"3w)))[$RF!f!RF27$$\"3O+]7.\"fF&QFgt$\"3+Efp,iIqPF27$$\"3Ym m;/OgbRFgt$\"3W-Tml[`MOF27$$\"3w**\\ilAFjSFgt$\"3&zNMj#[Z%Fgt$\"3ADU\\K%G5O$F27$$\"3S nmT&G!e&e%Fgt$\"35gRzc#\\LF2 7$$\"\"&F)$\"3Ii#4)y!3AN$F2-%'COLOURG6&%$RGBG$\"#5!\"\"F(F(-%%FONTG6$% *HELVETICAG\"\"*-%+AXESLABELSG6$%\"xG%%y(x)G-%%VIEWG6$;F(Fiel;F($\"#6F cfl" 1 2 0 1 10 0 2 9 1 4 2 1.000000 45.000000 45.000000 0 0 "Curve 1 " }}}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 32 "The following code constructs a " }{TEXT 260 17 "discrete solution" } {TEXT -1 44 " based on each of the methods and gives the " }{TEXT 260 22 "root mean square error" }{TEXT -1 18 " of each solution." }} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 801 "V := (x,y) -> -(1+4*cos(3*x ))*(y-1/3): hh := 0.02: numsteps := 250: x0 := 0: y0 := 1:\nmatrix([[` slope field: `,V(x,y)],[`initial point: `,``(x0,y0)],[`step width: \+ `,hh],\n[`no. of steps: `,numsteps]]);``;\nmthds := [`Enright-Verne r scheme`,`Verner's Maple scheme`,`a modification of Verner's Maple sc heme`,`an \"efficient\" scheme`,`a scheme of Tsitouras and Papakostas` ,`a modification of the Tsitouras-Papakostas scheme`,`Verner's \"most \+ efficient\" scheme`]: errs := []:\nDigits := 30:\nfor ct to 7 do\n V n_RK8_||ct := RK8_||ct(V(x,y),x,y,x0,y0,hh,numsteps,false);\n sm := \+ 0: numpts := nops(Vn_RK8_||ct):\n for ii to numpts do\n sm := s m+(Vn_RK8_||ct[ii,2]-v(Vn_RK8_||ct[ii,1]))^2;\n end do:\n errs := \+ [op(errs),sqrt(sm/numpts)];\nend do:\nDigits := 10:\nlinalg[transpose] ([mthds,evalf(errs)]);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#K%'matrixG6# 7&7$%0slope~field:~~~G,$*&,&\"\"\"F,*&\"\"%F,-%$cosG6#,$*&\"\"$F,%\"xG F,F,F,F,F,,&%\"yGF,#F,F4!\"\"F,F97$%0initial~point:~G-%!G6$\"\"!F,7$%/ step~width:~~~G$\"\"#!\"#7$%1no.~of~steps:~~~G\"$]#Q)pprint306\"" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#%!G" }}{PARA 11 "" 1 "" {XPPMATH 20 "6 #K%'matrixG6#7)7$%6Enright-Verner~schemeG$\"+6;p,;!#D7$%6Verner's~Mapl e~schemeG$\"+&G@u#=F+7$%Ha~modification~of~Verner's~Maple~schemeG$\"+* QwJm\"F+7$%6an~\"efficient\"~schemeG$\"+fx\"Q]\"F+7$%Ea~scheme~of~Tsit ouras~and~PapakostasG$\"+7XMT;F+7$%Ra~modification~of~the~Tsitouras-Pa pakostas~schemeG$\"+p2#Hf\"F+7$%AVerner's~\"most~efficient\"~schemeG$ \"+JRSNP!#EQ)pprint316\"" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 " " 0 "" {TEXT -1 30 "The following code constructs " }{TEXT 260 20 "num erical procedures" }{TEXT -1 56 " for solutions based on each of the R unge-Kutta schemes." }}{PARA 0 "" 0 "" {TEXT -1 75 "The error in the v alue obtained by each of the methods at the point where " }{XPPEDIT 18 0 "x = 4.999;" "6#/%\"xG-%&FloatG6$\"%**\\!\"$" }{TEXT -1 16 " is \+ also given." }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 732 "V := (x,y) - > -(1+4*cos(3*x))*(y-1/3): hh := 0.02: numsteps := 250: x0 := 0: y0 := 1:\nmatrix([[`slope field: `,V(x,y)],[`initial point: `,``(x0,y0)], [`step width: `,hh],\n[`no. of steps: `,numsteps]]);``;\nmthds := \+ [`Enright-Verner scheme`,`Verner's Maple scheme`,`a modification of Ve rner's Maple scheme`,`an \"efficient\" scheme`,`a scheme of Tsitouras \+ and Papakostas`,`a modification of the Tsitouras-Papakostas scheme`,`V erner's \"most efficient\" scheme`]: errs := []:\nDigits := 30:\nfor c t to 7 do\n vn_RK8_||ct := RK8_||ct(V(x,y),x,y,x0,y0,hh,numsteps,tru e);\nend do:\nxx := 4.999: vxx := evalf(v(xx)):\nfor ct to 7 do\n er rs := [op(errs),abs(vn_RK8_||ct(xx)-vxx)];\nend do:\nDigits := 10:\nli nalg[transpose]([mthds,evalf(errs)]);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#K%'matrixG6#7&7$%0slope~field:~~~G,$*&,&\"\"\"F,*&\"\"%F,-%$cosG6#, $*&\"\"$F,%\"xGF,F,F,F,F,,&%\"yGF,#F,F4!\"\"F,F97$%0initial~point:~G-% !G6$\"\"!F,7$%/step~width:~~~G$\"\"#!\"#7$%1no.~of~steps:~~~G\"$]#Q)pp rint326\"" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%!G" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#K%'matrixG6#7)7$%6Enright-Verner~schemeG$\"+e`OhO!#F7$% 6Verner's~Maple~schemeG$\"+))o?==F+7$%Ha~modification~of~Verner's~Mapl e~schemeG$\"+n%e\"\\>F+7$%6an~\"efficient\"~schemeG$\"+32Ke6F+7$%Ea~sc heme~of~Tsitouras~and~PapakostasG$\"+f*z4c\"F+7$%Ra~modification~of~th e~Tsitouras-Papakostas~schemeG$\"+)pK*>'*!#G7$%AVerner's~\"most~effici ent\"~schemeG$\"+GgFxTF@Q)pprint336\"" }}}{PARA 0 "" 0 "" {TEXT -1 0 " " }}{PARA 0 "" 0 "" {TEXT -1 4 "The " }{TEXT 260 22 "root mean square \+ error" }{TEXT -1 20 " over the interval " }{XPPEDIT 18 0 "[0, 5];" "6 #7$\"\"!\"\"&" }{TEXT -1 82 " of each Runge-Kutta method is estimated as follows using the special procedure " }{TEXT 0 5 "NCint" }{TEXT -1 98 " to perform numerical integration by the 7 point Newton-Cotes \+ method over 100 equal subintervals." }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 476 "mthds := [`Enright-Verner scheme`,`Verner's Maple sc heme`,`a modification of Verner's Maple scheme`,`an \"efficient\" sche me`,`a scheme of Tsitouras and Papakostas`,`a modification of the Tsit ouras-Papakostas scheme`,`Verner's \"most efficient\" scheme`]: errs : = []:\nDigits := 20:\nfor ct to 7 do\n sm := NCint((v(x)-'vn_RK8_||c t'(x))^2,x=0..5,adaptive=false,numpoints=7,factor=100);\n errs := [o p(errs),sqrt(sm/5)];\nend do:\nDigits := 10:\nlinalg[transpose]([mthds ,evalf(errs)]);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#K%'matrixG6#7)7$%6E nright-Verner~schemeG$\"+.;\\.;!#D7$%6Verner's~Maple~schemeG$\"+kswG=F +7$%Ha~modification~of~Verner's~Maple~schemeG$\"+,x\"\\m\"F+7$%6an~\"e fficient\"~schemeG$\"+G!oc]\"F+7$%Ea~scheme~of~Tsitouras~and~Papakosta sG$\"+bElV;F+7$%Ra~modification~of~the~Tsitouras-Papakostas~schemeG$\" +gUQ&f\"F+7$%AVerner's~\"most~efficient\"~schemeG$\"+*3a*QP!#EQ)pprint 346\"" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 92 "The following error graphs are constructed using the numerical proced ures for the solutions." }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 614 " evalf[25](plot([v(x)-'vn_RK8_1'(x),v(x)-'vn_RK8_2'(x),v(x)-'vn_RK8_3'( x),v(x)-'vn_RK8_4'(x),\nv(x)-'vn_RK8_5'(x),v(x)-'vn_RK8_6'(x),v(x)-'vn _RK8_7'(x)],x=0..5,-5e-17..5.7e-16,font=[HELVETICA,9],\ncolor=[COLOR(R GB,.4,0,.9),COLOR(RGB,0,.65,0),brown,red,COLOR(RGB,0,.75,.95),magenta, COLOR(RGB,.95,.55,0)],\nlegend=[`Enright-Verner scheme`,`Verner's Mapl e scheme`,`a modification of Verner's Maple scheme`,`an \"efficient\" \+ scheme`,`a scheme of Tsitouras and Papakostas`,`a modification of the \+ Tsitouras-Papakostas scheme`,`Verner's \"most efficient\" scheme`],tit le=`error curves for 12 stage order 8 Runge-Kutta methods`));" }} {PARA 13 "" 1 "" {GLPLOT2D 993 675 675 {PLOTDATA 2 "6--%'CURVESG6%7fr7 $$\"\"!F)F(7$$\":NLLLLLL3FWYs#!#E$\")A!G![!#D7$$\":qmmmmmm;a)G\\aF-$\" *Q5sz\"F07$$\":0++++++D\"G$R<)F-$\"*C$GXeF07$$\":MLLLLLL$3x&)*3\"F0$\" *O;^#yF07$$\":++++++D1R(*Rc\"F0$\"+JgYy6F07$$\":nmmmmm;H2P\"Q?F0$\"+vt NX:F07$$\":MLLLL$ek.pu/BF0$\"+f_#3a\"F07$$\":++++++vVtc8d#F0$\"+%\\*GK :F07$$\":MLLLLeR(\\;m/FF0$\"+?PNF:F07$$\":nmmmm;/^cmz$GF0$\"+cgFB:F07$ $\":+++++vo/[r7(HF0$\"+Ofik9F07$$\":MLLLLLLeRwX5$F0$\"+yM(GV\"F07$$\": MLLLLLLLeI8k$F0$\"+@&phD\"F07$$\":NLLLLLL3x%3yTF0$\"+!Rmi3\"F07$$\":-+ ++++v$fyG7ZF0$\"*/nGf*F07$$\":ommmmm;z%4\\Y_F0$\"*Xk(z()F07$$\":NLLLLL L3FGT\\&F0$\"*G!Qn&)F07$$\":++++++]PflVB$)F0$\"+OtYA6F07$$\":qmmmmm;/w)4_))F0$\"+gb&QG \"F07$$\":.+++++]7`l2Q*F0$\"+>p!*)[\"F07$$\":NLLLLLekGkX#**F0$\"+*4$zb jU6Fdu$\"+\"Q!R!z#F07$$\":nmmmmm\"HdNb'>\"Fdu$\"+e uD:KF07$$\":+++++++D;v/D\"Fdu$\"+vOK4OF07$$\":+++++++]<=YI\"Fdu$\"+)4a n$RF07$$\":+++++++v=h(e8Fdu$\"+q$[$*>%F07$$\":++++++++g\\[Q\"Fdu$\"++4 R&G%F07$$\":+++++++D,Q4T\"Fdu$\"+0_MiVF07$$\":++++++Dc6guT\"Fdu$\"+NlM wVF07$$\":++++++](=A)RU\"Fdu$\"+hdbvVF07$$\":++++++v=K/0V\"Fdu$\"+&QDY R%F07$$\":+++++++]UEqV\"Fdu$\"+^Uh\"Fdu$\"+<.!R $RF07$$\":nmmmmmmTXg0n\"Fdu$\"+IMZwNF07$$\":nmmmmm;a)))G=Fdu$\"++0^kFdu$\"++i(*Q:F07$$\":+++++D1Rgs2&>Fdu$\"+F8UB8F07$$\" :nmmmmmm;pW`(>Fdu$\"+LC'z6\"F07$$\":MLLLL$e9TOEH?Fdu$\"*(=vprF07$$\":, +++++D1f#=$3#Fdu$\"*ia\"RZF07$$\":,++++]PfQa)3@Fdu$\"*y\\I9%F07$$\":,+ ++++]7=EX8#Fdu$\"*kw>u$F07$$\":,++++D1*y?OZ@Fdu$\"*7'R*o$F07$$\":,++++ ]il(z>g@Fdu$\"*X(*\\t$F07$$\":,+++]i!RDfhm@Fdu$\"*tV!>OF07$$\":,++++v= U(Q.t@Fdu$\"*f^N^$F07$$\":,+++](o/B=Xz@Fdu$\"*,>3l$F07$$\":,+++++v=xpe =#Fdu$\"*Uv-h$F07$$\":MLLL$3_]%oi#*>#Fdu$\"*e;Xj$F07$$\":ommmmTNrfbE@# Fdu$\"*NA=^$F07$$\":,+++]il(4&[gA#Fdu$\"*+F&QOF07$$\":MLLLL$eRA9WRAFdu $\"*7GTo$F07$$\":nmmm;/E]LMGD#Fdu$\"*99Ra$F07$$\":+++++DcwCFiE#Fdu$\"* H(yOOF07$$\":MLLL$ekGg,izAFdu$\"**)R/m$F07$$\":nmmmmm;H28IH#Fdu$\"*=() )*\\$F07$$\":nmmmmTN@#3h7d<$F07$$\":nmmmmm\"zpSS\"R#Fdu$\"*Pp ,)HF07$$\":MLLLLL$3_?`(\\#Fdu$\"*8)eWAF07$$\":MLLLLLe*)>pxg#Fdu$\"*dI, !=F07$$\":,+++++v$f4t.FFdu$\"*Vikp\"F07$$\":MLLLLL$e*Gst!GFdu$\"*5Hi\" =F07$$\":,++++++]#RW9HFdu$\"*\"H:*=#F07$$\":,+++++]7j#>>IFdu$\"*hH$eGF 07$$\":,+++++D1RU07$Fdu$\"*q\"puQF07$$\":,+++++](=S2LKFdu$\"*&[UVaF07$ $\":ommmmmm;p)=MLFdu$\"*G%3CqF07$$\":MLLLLL$eR%p\")Q$Fdu$\"*Q6mx(F07$$ \":,++++++v=]@W$Fdu$\"*r\">#R)F07$$\":MLLLLekyZ2mY$Fdu$\"*V*zI')F07$$ \":ommmm;H#oZ1\"\\$Fdu$\"*frj#))F07$$\":,++++v$fe?_:NFdu$\"*B)\\q*)F07 $$\":MLLLLLe*[$z*RNFdu$\"*%HCO!*F07$$\":,++++vVti?OFdu$\"*![=W*)F07$$ \":,++++++DYKpk$Fdu$\"*,F?y)F07$$\":MLLLL$e*[t\\sp$Fdu$\"*M7:I)F07$$\" :ommmmm\"H2qcZPFdu$\"*M&R^wF07$$\":,+++++DJ5fF&QFdu$\"*+Pc,'F07$$\":om mmmmmTg.c&RFdu$\"*I96G%F07$$\":,+++++DcEsK1%Fdu$\"*&\\!)=CF07$$\":ommm m;zW7@^6%Fdu$\"*Bv2v\"F07$$\":NLLLLLLL)*pp;%Fdu$\"*N<#R7F07$$\":NLLLLL 3-$H**>UFdu$\")L*)[&*F07$$\":NLLLLL$3xe,tUFdu$\")+]AzF07$$\":ommmmm\"H dO=yVFdu$\")v$3f'F07$$\":,++++++]#>#[Z%Fdu$\")7\\c_F07$$\":ommmmm;aG!e &e%Fdu$\")G\"f(QF07$$\":NLLLLLLL)Qk%o%Fdu$\")eCaJF07$$\":-+++++D1Mm-z% Fdu$\")ey4HF07$$\":-+++++v$40O\"*[Fdu$\")Z6rIF07$$\"\"&F)$\")AsqOF0-%& COLORG6&%$RGBG$\"\"%!\"\"F($\"\"*Fjam-%'LEGENDG6#%6Enright-Verner~sche meG-F$6%7_rF'7$F+$\"*E\"y(\\\"F07$F2$\"*(e1@RF07$$\":SLLLLL$3x1h6oF-$ \"**)*\\/nF07$F7$\"*Rd'[**F07$$\":qmmmm;ajM8X^)F-$\"*#y%[y*F07$$\":SLL LLL3-)Q4b))F-$\"*Wldi*F07$$\":0++++]iSTuc>*F-$\"*9W?]*F07$$\":qmmmmm;z %\\DO&*F-$\"*(H4e'*F07$$\":NLLLL$3x\"[No()*F-$\"+vz:H6F07$$\":+++++]i: gT<-\"F0$\"+5'znF\"F07$$\":nmmmm\"z%\\l*zb5F0$\"+V*[gD\"F07$F<$\"+mc'f B\"F07$FA$\"+J@pb;F07$FF$\"+%HqI&>F07$FP$\"+lJJ@=F07$F^o$\"+2cy?;F07$F ho$\"+j#*[;7F07$F]p$\"+4]-,6F07$Fbp$\"+*y#GP5F07$Fgp$\"+RVZ>5F07$F\\q$ \"+6S**35F07$Ffq$\"+=i915F07$F`r$\"+l%G*35F07$Fer$\"+OF07$Fbu$\"+ 665PAF07$Fhu$\"+ueT\\DF07$F]v$\"+d#R6\"HF07$Fbv$\"+CVL&Q$F07$Fgv$\"+eY >/RF07$F\\w$\"+!3irT%F07$Faw$\"+aA[Z[F07$F[x$\"+^KOU^F07$Fiy$\"+WL5)G& F07$F^z$\"+01t+`F07$Fcz$\"+Y'zUI&F07$Fhz$\"+Lo(3I&F07$F][l$\"+R$f')G&F 07$Fb[l$\"+*pWBC&F07$Fg[l$\"+E\"[0<&F07$F\\\\l$\"+Zcg()[F07$Fa\\l$\"+& Ga:Z%F07$$\":nmmmmm\"zpYU%p\"Fdu$\"+[a][UF07$Ff\\l$\"+)f=,+%F07$$\":nm mmmmT55`@u\"Fdu$\"+\"*)QSr$F07$F[]l$\"+1jH4MF07$F`]l$\"+T**pUIF07$Fe]l $\"+SU-jDF07$Fj]l$\"+.&>T<#F07$F_^l$\"+K6F07$Faal$\")BVs#)F07$Febl$\"*I1)z6 F07$F_cl$\"*_wA[\"F07$Ficl$\"*Ab2,#F07$F^dl$\"*YZA)>F07$Fcdl$\"*#4b#># F07$Fhdl$\"*DmQI#F07$F]el$\"*B[1A#F07$Fbel$\"*n]#fAF07$Fgel$\"*$4\"=F# F07$F\\fl$\"*oA\"[@F07$Fafl$\"*@xg,#F07$Fffl$\"*gTN^\"F07$F[gl$\"*(eFf 7F07$F`gl$\"*U48@\"F07$Fegl$\"*6L0H\"F07$Fjgl$\"*wiIa\"F07$F_hl$\"*:,8 (>F07$Fdhl$\"*RT1a#F07$Fihl$\"*AQ%eMF07$F^il$\"*65Nc%F07$Fhil$\"*QdWt& F07$Fbjl$\"*@j65'F07$F\\[m$\"*&pBF07$F_^l$\"+FtuH;F07$Fd^l$\"+]- :b7F07$Fi^l$\"*=E&*)**F07$F^_l$\"*?t*3xF07$Fc_l$\"*yeql&F07$Fh_l$\"*k( \\r?F07$F]`l$\")z?'z'F07$Ff_n$\")s.ljF07$Fb`l$\")_)[/'F07$Fc`n$\")([^6 (F07$Fg`l$\"),c2oF07$F\\al$\")*\\-q)F07$Faal$\"*:KF9\"F07$Ffal$\"*!yE2 6F07$F[bl$\"*X/03\"F07$$\":,+++D\"Gj[GCw@Fdu$\"*`y<6\"F07$F`bl$\"*7LfK \"F07$$\":,+++v$4Y(zgE=#Fdu$\"*'pX(Q\"F07$Febl$\"*@7gO\"F07$Fjbl$\"*0E Ab\"F07$F_cl$\"*yAnc\"F07$Fdcl$\"*_%3F=F07$Ficl$\"*:,R)>F07$F^dl$\"*r6 _%>F07$Fcdl$\"*I1L7#F07$Fhdl$\"*)*>-A#F07$F]el$\"*zF#Q@F07$Fbel$\"*o(G v@F07$Fgel$\"*yt'*>#F07$F\\fl$\"*px/4#F07$Fafl$\"*MOD(>F07$Fffl$\"*Z\\ K]\"F07$F[gl$\"*bAEB\"F07$F`gl$\"*)H\"Q<\"F07$Fegl$\"*ko.D\"F07$Fjgl$ \"*DY`\\\"F07$F_hl$\"*B'4@>F07$Fdhl$\"*482`#F07$Fihl$\"*(p/=NF07$F^il$ \"*\"H?jYF07$Fhil$\"*#*f)feF07$Fbjl$\"*/)GZiF07$F\\[m$\"*Z0H\\'F07$Fa[ m$\"*KX=b'F07$Ff[m$\"*$e5dlF07$F[\\m$\"*j29^'F07$F`\\m$\"*oF&3kF07$Fe \\m$\"*l*pzgF07$Fj\\m$\"*H]Wf&F07$F_]m$\"*G\"*y=%F07$Fd]m$\"*[mw\\#F07 $F`gn$\"*i:of\"F07$Fi]m$\")&\\2N)F07$F^^m$\")s1aVF07$Fc^m$\"))pf(=F07$ Fh^m$\")>M-;F07$F]_m$\")#)RF>F07$Fb_m$\")3YaHF07$Fg_m$\")(R3p#F07$F\\` m$\")Y)=.#F07$Fa`m$\")b'en\"F07$Ff`m$\")Grh:F07$F[am$\")VaW;F07$F`am$ \")fM`>F0-%'COLOURG6&Fgam$\")#)eqk!\")$\"))eqk\"F\\`pF]`p-F^bm6#%Ha~mo dification~of~Verner's~Maple~schemeG-F$6%7hqF'7$F+$\")#R!otF07$F2$\"*< 5SN#F07$F7$\"*\"y$o#oF07$F<$\"*jXLz)F07$FA$\"+)y.*R7F07$FF$\"+QcyC:F07 $FP$\"+#=>^X\"F07$F^o$\"+*[$Q>8F07$Fho$\"*$z[V**F07$F]p$\"*!=P6*)F07$F bp$\"*&Rj#H)F07$Fgp$\"*jVM7)F07$F\\q$\"*Mqx,)F07$Ffq$\"*wFH(zF07$F`r$ \"*fwS)zF07$Fer$\"*OqH0)F07$Fjr$\"*=&p\"=)F07$F_s$\"*.7BP)F07$Fds$\"*I \"zD')F07$Fis$\"*gi_O*F07$F^t$\"+wp=T5F07$Fct$\"+MI*o<\"F07$Fht$\"+pq9 Y8F07$F]u$\"+qwhd:F07$Fbu$\"+$zU=\"=F07$Fhu$\"+$e2P3#F07$F]v$\"+J>9/CF 07$Fbv$\"+oN%)3GF07$Fgv$\"+D2o\\KF07$F\\w$\"+(Rdgo$F07$Faw$\"+IUseSF07 $F[x$\"+gyaBVF07$Fiy$\"+**[LlWF07$F^z$\"+kN@wWF07$Fcz$\"+:+p\"[%F07$Fh z$\"+3^lyWF07$F][l$\"+H_3oWF07$Fb[l$\"+(>JXU%F07$Fg[l$\"+'y1;N%F07$F\\ \\l$\"+M)3$oSF07$Fa\\l$\"+v\"4+l$F07$Fh\\n$\"+!)4dSMF07$Ff\\l$\"+?QZ0K F07$F`]n$\"+I+3VHF07$F[]l$\"+!zkSo#F07$F`]l$\"+')G*4Q#F07$Fe]l$\"+w4(f )>F07$Fj]l$\"+N6,x;F07$F_^l$\"+'e[+N\"F07$Fd^l$\"*(y)\\#)*F07$Fi^l$\"* Xs$QuF07$F^_l$\"*%*HxL&F07$Fc_l$\"*D`GY$F07$Fh_l$\")<`*\\#F07$F]`l$!)% \\dy(F07$Ff_n$!)#yOL(F07$Fb`l$!)(3)RpF07$Fc`n$!)gH@_F07$Fg`l$!)_NuZF07 $Faal$\"(*)Q5*F07$Febl$\")H\"))>%F07$F_cl$\")*e]:(F07$Ficl$\"*CK%)>\"F 07$$\":++++v$4ryy8YAFdu$\"*$pqL7F07$F^dl$\"*cvtDF07$F^il$\"*jMdV$F07$Fhil$\"*F07$Fh^m$!)c&*e8F07$F]_m$!(+'3RF07$Fb_ m$\")7tF9F07$Fg_m$\")Cn\"f\"F07$F\\`m$\")g%HA\"F07$Fa`m$\"(i;***F07$Ff `m$\"(OqG*F07$F[am$\"(6dx*F07$F`am$\")hsg6F0-Fh_p6&Fgam$\"*++++\"F\\`p F(F(-F^bm6#%6an~\"efficient\"~schemeG-F$6%7grF'7$$F\\cm!#F$!\"$F07$$\" :ommmmm;a8ABO\"F-$\"'e?^F07$$\":NLLLL$3FpE!Hq\"F-$\"(q*QeF07$$\":-++++ +DJ?$[V?F-$\")%Q2D$F07$$\":ommmm;zptjSQ#F-$\"))\\f>$F07$F+$\")eAVJF07$ $\":-++++](o/[AlIF-$\")*Hx7$F07$$\":ommmmmT&Q`!eS$F-$\")!H*QNF07$$\":N LLLL$eRseQYPF-$\")-8mnF07$$\":-+++++]iSmp3%F-$\"*'3uJ;F07$$\":OLLLLLeR ZF\"oZF-$\"*k;yd\"F07$F2$\"*'[_M;F07$F7$\"*Z55\"eF07$F<$\"*#3'H%yF07$F A$\"+'3ew<\"F07$FF$\"+%)H5@:F07$FK$\"+.>L3:F07$FP$\"+k<$4\\\"F07$FU$\" +a^X\"[\"F07$FZ$\"+'4FEZ\"F07$Fin$\"+Cf3:9F07$F^o$\"+([*4#Q\"F07$Fco$ \"+2p(=@\"F07$Fho$\"+g\"Q]0\"F07$F]p$\"*l5/U*F07$Fbp$\"*qmqr)F07$Fgp$ \"*gWo_)F07$F\\q$\"*&o]1%)F07$Ffq$\"*,:@N)F07$F`r$\"*%4ni$)F07$Fer$\"* )z8P%)F07$Fjr$\"*W$ew&)F07$F_s$\"*zZyy)F07$Fds$\"*hL11*F07$Fis$\"*3Le& )*F07$F^t$\"+=]G(4\"F07$Fct$\"+bnkV7F07$Fht$\"+FpZG9F07$F]u$\"+;w&[m\" F07$Fbu$\"+Wvcd>F07$Fhu$\"+kVpnAF07$F]v$\"+n(Hej#F07$Fbv$\"+lS-(3$F07$ Fgv$\"+]6\">d$F07$F\\w$\"+AVSXSF07$Faw$\"+C)p*\\WF07$F[x$\"+0pMRZF07$F iy$\"++/Q(*[F07$F^z$\"+'*eP4\\F07$Fcz$\"+3[G;\\F07$Fhz$\"+04C8\\F07$F] [l$\"+&f@>!\\F07$Fb[l$\"+/(QR&[F07$Fg[l$\"+UI*>x%F07$F\\\\l$\"+@88^WF0 7$Fa\\l$\"+G+$\\(RF07$Fh\\n$\"+^GpSPF07$Ff\\l$\"+oj@![$F07$F`]n$\"+0`* e>$F07$F[]l$\"+ae>?HF07$F`]l$\"+QAu+EF07$Fe]l$\"+![:;?#F07$Fj]l$\"+maT #)=F07$F_^l$\"+E\")z[:F07$Fd^l$\"+u#4B=\"F07$Fi^l$\"*/v*G$*F07$F^_l$\" *$He+rF07$Fc_l$\"*h543&F07$Fh_l$\"*i?WZ\"F07$F]`l$\"(F++\"F07$Ff_n$!'N g:F07$Fb`l$!(z$RcF07$Fc`n$\"(u_Z#F07$Fg`l$\"(fzK$F07$Faal$\"):jjZF07$F ebl$\")kePuF07$F_cl$\")wfh**F07$Ficl$\"*HvAY\"F07$Fc\\q$\"*qrE\\\"F07$ F^dl$\"*;*H`9F07$F[]q$\"*Bw:l\"F07$Fcdl$\"*qf-m\"F07$Fhdl$\"*r1zy\"F07 $F]el$\"*h#GJ5#*=F07$F\\fl$\"*(3P>=F07 $Fafl$\"*h@>t\"F07$Fe^q$\"*Y&>P:F07$Fffl$\"*)H5O8F07$F[gl$\"*ZX(y5F07$ F`gl$\"*l+'>5F07$Fegl$\"*&4.)3\"F07$Fjgl$\"*vVYI\"F07$F_hl$\"*SGFo\"F0 7$Fdhl$\"*BAdC#F07$Fihl$\"*eGL<$F07$F^il$\"*A5NC%F07$Fhil$\"*8z)[`F07$ Fbjl$\"*ac$3dF07$F\\[m$\"*5\"pTfF07$Fa[m$\"*&=Y(*fF07$Ff[m$\"*'Rz-gF07 $F[\\m$\"*\\nl&fF07$F`\\m$\"*1qn&eF07$Fe\\m$\"*,RQ`&F07$Fj\\m$\"*,<-0& F07$F_]m$\"*-y()p$F07$Fd]m$\"*+/'Q@F07$F`gn$\"*Z(=68F07$Fi]m$\")@gRgF0 7$F^^m$\")#)4ACF07$Fc^m$\"(V$[8F07$Fh^m$\"'raDF07$F]_m$\"(o&\\gF07$Fb_ m$\")Yh)3#F07$Fg_m$\")4PW@F07$F\\`m$\")^jd;F07$Fa`m$\")VF[8F07$Ff`m$\" )ZDX7F07$F[am$\")Iu78F07$F`am$\")ack:F0-Feam6&FgamF($\"#vF\\jn$\"#&*F \\jn-F^bm6#%Ea~scheme~of~Tsitouras~and~PapakostasG-F$6%7crF'7$F\\fq$\" &kY\"F07$F+$\")ytf;F07$Fcgq$\")1>m;F07$Fhgq$\")Pcn?F07$F]hq$\")xF&4&F0 7$Fbhq$\"*v2eS\"F07$Fghq$\"*:)Qf8F07$F2$\"*;))3U\"F07$F7$\"*\\jec&F07$ F<$\"*4nzh(F07$FA$\"++a/d6F07$FF$\"+>m(p\\\"F07$FK$\"+\"zfL[\"F07$FP$ \"+$)yYk9F07$FU$\"+5_)RX\"F07$FZ$\"+!f/QW\"F07$Fin$\"+)[nqQ\"F07$F^o$ \"+#*eu`8F07$Fco$\"+Lpx%=\"F07$Fho$\"+Pz-J5F07$F]p$\"*BNs?*F07$Fbp$\"* HzD_)F07$Fgp$\"*x`wL)F07$F\\q$\"*Kr7A)F07$Ffq$\"*CL0<)F07$F`r$\"*W&[$= )F07$Fer$\"*lG%e#)F07$Fjr$\"*q7rR)F07$F_s$\"*\\Trg)F07$Fds$\"*Ub_())F0 7$Fis$\"*+^Jl*F07$F^t$\"+GZ=u5F07$Fct$\"+/.;;7F07$Fht$\"+6WR&R\"F07$F] u$\"+CIGC;F07$Fbu$\"+L&Rw!>F07$Fhu$\"+?Z**3AF07$F]v$\"+Bj0oDF07$Fbv$\" +2%*H4IF07$Fgv$\"+Y`q%[$F07$F\\w$\"+*pv$[RF07$Faw$\"+^RvUVF07$F[x$\"+3 c9F07$Fd^l$\"+dm6L5F07$Fi^l$\"*fZz!yF07$F^_l $\"*0?:d&F07$Fc_l$\"*ee^b$F07$Fh_l$!'s5>F07$F]`l$!*h`BF\"F07$Ff_n$!*C! *y?\"F07$Fb`l$!*m;*47F07$Fc`n$!*:wL2\"F07$Fg`l$!)cwt**F07$Faal$!)M:NWF 07$Febl$!(E1'pF07$F_cl$\")*G%)z#F07$Ficl$\")x0x#)F07$Fc\\q$\")8(*o()F0 7$F^dl$\")(z0c)F07$F[]q$\"*1())p5F07$Fcdl$\"*%)[_4\"F07$Fhdl$\"*=-CD\" F07$F]el$\"*\\baA\"F07$Fbel$\"*IWvN\"F07$Fgel$\"*;VxY\"F07$F\\fl$\"*^6 'H9F07$Fafl$\"*4rBP\"F07$Fe^q$\"*t'oH7F07$Fffl$\"*Dj(o5F07$F[gl$\")1>( f)F07$F`gl$\")wCF\")F07$Fegl$\")@N!p)F07$Fjgl$\"*'yoU5F07$F_hl$\"*p92M \"F07$Fdhl$\"*'*p@y\"F07$Fihl$\"*hxQ_#F07$F^il$\"*ts*)R$F07$Fhil$\"*5i tI%F07$Fbjl$\"*@I#*f%F07$F\\[m$\"*J=sy%F07$Fa[m$\"*Q/:$[F07$Ff[m$\"*! \\$\\$[F07$F[\\m$\"*uxcz%F07$F`\\m$\"*;'=9ZF07$Fe\\m$\"*ye7X%F07$Fj\\m $\"*)RqbSF07$F_]m$\"*#yRBHF07$Fd]m$\"*lda`\"F07$F`gn$\")m4;xF07$Fi]m$ \")2\"\\D\"F07$F^^m$!)-5(e\"F07$Fc^m$!)')3)=$F07$Fh^m$!)$\\Tg#F07$F]_m $!)$e7V\"F07$Fb_m$\"(AaG)F07$Fg_m$\")TVq7F07$F\\`m$\")v6F5F07$Fa`m$\"( ^fF)F07$Ff`m$\"(1yi(F07$F[am$\"(U\"o!)F07$F`am$\"(E@k*F0-Fh_p6&FgamF^e qF(F^eq-F^bm6#%Ra~modification~of~the~Tsitouras-Papakostas~schemeG-F$6 %7drF'7$F+$\")^IonF07$F2$\"*)=!)[9F07$F[cm$\"*(Qr(>#F07$F7$\"*D,b(HF07 $Fccm$\"*.3l#HF07$Fhcm$\"*x*))yGF07$F]dm$\"*v\"eSGF07$Fbdm$\"*kb1(GF07 $Fgdm$\"*_WyB$F07$F\\em$\"*'3FbNF07$Faem$\"*TWv\\$F07$F<$\"*oT:W$F07$F A$\"*2cG6%F07$FF$\"*'fJuVF07$FP$\"*d:e%QF07$F^o$\"*%=GQKF07$Fco$\"*+_< m#F07$Fho$\"*Cw&eAF07$F]p$\"*afW*>F07$Fbp$\"*4t([=F07$Fgp$\"*D)*=\"=F0 7$F\\q$\"*6O+z\"F07$Ffq$\"*6OKy\"F07$F`r$\"*NO')y\"F07$Fjr$\"*\"*Hc$=F 07$Fds$\"*njr$>F07$Fis$\"*??(3@F07$F^t$\"*ZBtN#F07$Fht$\"*9/L5$F07$F]u $\"*.]!)f$F07$Fbu$\"*>$4lTF07$Fhu$\"*Ed)zZF07$F]v$\"*V]Lc&F07$Fbv$\"*@ jGi'F07$Fgv$\"*9qC#zF07$$\":++++++](omax7Fdu$\"*U\\Cf)F07$F\\w$\"*h]8I *F07$$\":++++++]7o*oJ8Fdu$\"*CH'y)*F07$Faw$\"+!yr\\/\"F07$F[x$\"+9LaC6 F07$Fiy$\"+O:9l6F07$F^z$\"+em(z;\"F07$Fcz$\"+$pj\"p6F07$Fhz$\"+le0o6F0 7$F][l$\"+V&Q\\;\"F07$Fb[l$\"+'y$\\_6F07$Fg[l$\"+WrKJ6F07$$\":nmmmmTg_ d[ge\"Fdu$\"+eoI%4\"F07$F\\\\l$\"+6-nZ5F07$$\":MLLLLeky#)*QU;Fdu$\"*c* 4#z*F07$Fa\\l$\"*@/S3*F07$Fh\\n$\"*KA!f$)F07$Ff\\l$\"*v4=U(F07$F`]n$\" *`j>G'F07$F[]l$\"*ToxI&F07$$\":MLLL$e9m%>(*)zF07$$\":MLLL$ek.HW#)))>Fdu$!*u9:>#F07$$\":+++++D1k; /B+#Fdu$!*fH&GBF07$$\":MLLLe9\"4NS/4?Fdu$!*;UOD#F07$$\":nmmm;/wP!Ry:?F du$!*Y0@?#F07$$\":MLL$e*[=\"QQ:>?Fdu$!*Cq5F#F07$$\":++++v$4YsP_A?Fdu$! *(f!\\H#F07$$\":nmm;aQ.oq$*e-#Fdu$!*#\\BdAF07$Fh_l$!**H??AF07$$\":ommm mT&)e6Bi0#Fdu$!*g'3W?F07$F]`l$!*D@Iz\"F07$Fg`l$!*4X+C\"F07$Febl$!)^n(p 'F07$F_cl$!)\"eD*[F07$Ficl$!)%\\S*GF07$Fcdl$!)\\M5=F07$F]el$!)[gb6F07$ Fbel$!(13b'F07$Fgel$!(nb5$F07$F\\fl$!()48@F07$Fafl$!(&3/ e@F07$Fffl$!(C.p#F07$F[gl$!(cU.$F07$F`gl$!(yTw#F07$Fegl$!(0h&GF07$Fjgl $!()R'>$F07$F_hl$!(Rzt$F07$Fdhl$!(Vt/'F07$Fihl$!(k\"e<&F07$$\":,++++DJ&p2_#)RFdu$!)p8y dF07$$\":ommm;a8ANzf*RFdu$!)8>kbF07$F`gn$!)2BYfF07$$\":ommmmTg-5bj.%Fd u$!)<(e2'F07$Fi]m$!)#f=R'F07$F^^m$!)1vUcF07$Fc^m$!)DdBYF07$F]_m$!)V_#H #F07$Fb_m$!(!et)*F07$Fg_m$!(x_!eF07$F\\`m$!(PSI%F07$Fa`m$!(^.m$F07$Ff` m$!(l.R$F07$F[am$!(9Fb$F07$F`am$!(MB=%F0-Feam6&FgamF\\^s$\"#bF\\jnF(-F ^bm6#%AVerner's~\"most~efficient\"~schemeG-%%FONTG6$%*HELVETICAGF\\bm- %&TITLEG6#%Verror~curves~for~12~stage~order~8~Runge-Kutta~methodsG-%+A XESLABELSG6$Q\"x6\"Q!F\\_v-%%VIEWG6$;F(F`am;$!\"&!#<$\"#dFe_v" 1 2 0 1 10 0 2 9 1 4 2 1.000000 45.000000 45.000000 0 1 "Enright-Verner sche me" "Verner's Maple scheme" "a modification of Verner's Maple scheme" "an \"efficient\" scheme" "a scheme of Tsitouras and Papakostas" "a mo dification of the Tsitouras-Papakostas scheme" "Verner's \"most effici ent\" scheme" }}}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 48 "Test \+ 8 of 12 stage, order 8 Runge-Kutta methods " }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "dy/dx=x*(9-x^2)/(1+y^2)" "6#/*&%#dyG\" \"\"%#dxG!\"\"*(%\"xGF&,&\"\"*F&*$F*\"\"#F(F&,&F&F&*$%\"yGF.F&F(" } {TEXT -1 3 ", " }{XPPEDIT 18 0 "y(0)=0" "6#/-%\"yG6#\"\"!F'" }{TEXT -1 1 " " }}{PARA 0 "" 0 "" {TEXT -1 10 "Solution: " }}{PARA 256 "" 0 " " {TEXT -1 1 " " }{XPPEDIT 18 0 "y = rho(x)/2-2/rho(x);" "6#/%\"yG,&*& -%$rhoG6#%\"xG\"\"\"\"\"#!\"\"F+*&F,F+-F(6#F*F-F-" }{TEXT -1 2 ", " }} {PARA 0 "" 0 "" {TEXT -1 7 "where " }{XPPEDIT 18 0 "rho(x) = (54*x^2- 3*x^4+sqrt(64+9*x^8-324*x^6+2916*x^4))^(1/3);" "6#/-%$rhoG6#%\"xG),(*& \"#a\"\"\"*$F'\"\"#F,F,*&\"\"$F,*$F'\"\"%F,!\"\"-%%sqrtG6#,*\"#kF,*&\" \"*F,*$F'\"\")F,F,*&\"$C$F,*$F'\"\"'F,F3*&\"%;HF,*$F'F2F,F,F,*&F,F,F0F 3" }{TEXT -1 2 ". " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 " > " 0 "" {MPLTEXT 1 0 178 "de := diff(y(x),x)=x*(9-x^2)/(1+y(x)^2);\ni c := y(0)=0;\ndsolve(\{de,ic\},y(x));\nw := unapply(rhs(%),x):\nplot(w (x),x=0..4,0..3.7,numpoints=75,font=[HELVETICA,9],labels=[`x`,`y(x)`]) ;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%#deG/-%%diffG6$-%\"yG6#%\"xGF,* (F,\"\"\",&\"\"*F.*$)F,\"\"#F.!\"\"F.,&F.F.*$)F)F3F.F.F4" }}{PARA 11 " " 1 "" {XPPMATH 20 "6#>%#icG/-%\"yG6#\"\"!F)" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/-%\"yG6#%\"xG,&*&\"\"#!\"\",(*&\"\"$\"\"\")F'\"\"%F/F+ *&\"#aF/)F'F*F/F/*$,*\"#kF/*&\"\"*F/)F'\"\")F/F/*&\"$C$F/)F'\"\"'F/F+* &\"%;HF/F0F/F/#F/F*F/#F/F.F/*&F*F/F,#F+F.F+" }}{PARA 13 "" 1 "" {GLPLOT2D 503 318 318 {PLOTDATA 2 "6&-%'CURVESG6$7io7$$\"\"!F)F(7$$\"3 ()=*=*=*Qx#G!#>$\"3_LLtbH2)f$!#?7$$\"3uPy$y$yZbcF-$\"3ZF^'eEW*Q9F-7$$ \"3;_8N^$ye6)F-$\"3C$\"3aT8Yqv-h6F>7$$\"3oKCVKs3o@F>$\"3c?q**e5wz?F>7$$\"3$4\" 3\"3T.Ds#F>$\"3+#H`Y\")*G6KF>7$$\"3jy$y$y\"=lB$F>$\"3L\\!fpl0?S%F>7$$ \"3G(H(H(p](oPF>$\"3XzNO%Rmzr&F>7$$\"3/I(H(Hj=>VF>$\"3n)f#*4g%))4rF>7$ $\"373\"3\"3n&y'[F>$\"3[oK(4I9f[)F>7$$\"3oKCVK;BKaF>$\"3y!44&)e3J')*F> 7$$\"3etH(HPL$HfF>$\"3xCnB_*)['F>$\"3g[#>`Yr+B\" Fho7$$\"3Ul['[')o30(F>$\"3?yA^q$e=N\"Fho7$$\"3)yH(H(pzBf(F>$\"3)GF.C0: VY\"Fho7$$\"3m53\"3TBT3)F>$\"3ipKusL^i:Fho7$$\"3u\\'['[U&)o')F>$\"3_Io w_)\\Zn\"Fho7$$\"3ynvcnz>k\"*F>$\"31*=-!RxBm$\"3 +Ep.yT!)o=Fho7$$\"3\">*=*=tV]-\"Fho$\"3eT(\\)>LNc>Fho7$$\"3!z$y$y&G+\" 3\"Fho$\"3!4zI&*RE\"\\?Fho7$$\"37>*=*y\"*GM6Fho$\"3tVizMbXM@Fho7$$\"3' \\'['['y))*=\"Fho$\"3fa-R]1_?AFho7$$\"3s%f%f9[%4C\"Fho$\"3#y2[?x6qH#Fh o7$$\"3[KCVKm,'H\"Fho$\"3)*e,ywZ!pP#Fho7$$\"3c8N^t2A`8Fho$\"3WUCG,38dC Fho7$$\"3Y'['[Yr,.9Fho$\"3@#)=I[fvCDFho7$$\"3q8N^$f)zc9Fho$\"3%3DX;Fho$\"3iG:8:1'ez#Fho7$$\"3c%f%fuKqx;Fho$\"3 [Z4\"[C.I'GFho7$$\"3=^8N\"ft,t\"Fho$\"3!\\dJru-7#HFho7$$\"3EaS0ao>'y\" Fho$\"3!*)Rs,)f8\")HFho7$$\"3tcnvcA'p$=Fho$\"3`s'\\F\\'[LIFho7$$\"3363 \"3DiC*=Fho$\"3wC-.o]f)3$Fho7$$\"3$)******>MoW>Fho$\"3Mca,!\\@%QJFho7$ $\"3!*['[')ep#**>Fho$\"3Or<<>%)R)=$Fho7$$\"31Yf%faPE0#Fho$\"3!)Qt2wy;N KFho7$$\"3C^8N^)3&3@Fho$\"3Ik#>&=s*=G$Fho7$$\"3E>*=*e&>B;#Fho$\"3HYm&4 R?ZK$Fho7$$\"3_f%f%z([t@#Fho$\"3-!>y6]piO$Fho7$$\"3oq-FIB#>F#Fho$\"3%> b4%)\\*>0MFho7$$\"3vvcnb(p?K#Fho$\"35nW;n-%*QMFho7$$\"3eCVKkVazBFho$\" 3))))QO[-#QDFho$\"3]n/[Ud8hNFho7$$\"3AdnvO#pkf #Fho$\"3[Y'[5$oF(e$Fho7$$\"3'>;i@A4pk#Fho$\"3e\"*)HR3Wug$Fho7$$\"3$ovc n0fTq#Fho$\"3E1Ex&>gui$Fho7$$\"3;Yf%f1Ojv#Fho$\"3)*3G:eW%Hk$Fho7$$\"3[ aS09&4M\"GFho$\"3#G5xO+Znl$Fho7$$\"3OnvcZUliGFho$\"3?xe_?\\\"fm$Fho7$$ \"3163\"3DQ(=HFho$\"3I;`e!RAJn$Fho7$$\"3k*=*=HE\"H(HFho$\"3IP\"\\xLemn $Fho7$$\"3%pvcnh^q-$Fho$\"3e^O4S8lwOFho7$$\"3-dnvO9*43$Fho$\"3y,I^j&GH n$Fho7$$\"3m(H(H<2\"G8$Fho$\"3]\\w?2VllOFho7$$\"3'>*=*=tG))=$Fho$\"3#R \\Gi]bMl$Fho7$$\"3-A;ihz@UKFho$\"3q^o+4DNPOFho7$$\"3!>*=*=h2%)H$Fho$\" 3%>QOHhJ`h$Fho7$$\"3A;i@wFF\\LFho$\"3YxQ[8&f0f$Fho7$$\"3)\\'['[I)[0MFh o$\"3q>yn%)HPdNFho7$$\"3c0aS&)HLfMFho$\"3a59(yGT$>NFho7$$\"3['['[1m/8N Fho$\"3uI!*3pTpuMFho7$$\"3E#*=*=p]\"pNFho$\"3s<,a='3,U$Fho7$$\"37.Fq-Y #3i$Fho$\"3w<#3Q&zxhLFho7$$\"31wcnbdutOFho$\"3?^'pWBqHH$Fho7$$\"32dnv' *p'o+i*\\/FFFho7$$\"\"%F)$\"3CxC=rRoRDFho-%'COLOU RG6&%$RGBG$\"#5!\"\"F(F(-%%FONTG6$%*HELVETICAG\"\"*-%+AXESLABELSG6$%\" xG%%y(x)G-%%VIEWG6$;F(F`cl;F($\"#PFjcl" 1 2 0 1 10 0 2 9 1 4 2 1.000000 45.000000 45.000000 0 0 "Curve 1" }}}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 32 "The following code constructs a " }{TEXT 260 17 "discrete solution" }{TEXT -1 44 " based on each of t he methods and gives the " }{TEXT 260 22 "root mean square error" } {TEXT -1 18 " of each solution." }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 795 "W := (x,y) -> x*(9-x^2)/(1+y^2): hh := 0.01: numsteps := 400: x0 := 0: y0 := 0:\nmatrix([[`slope field: `,W(x,y)],[`initial point : `,``(x0,y0)],[`step width: `,hh],\n[`no. of steps: `,numsteps]]) ;``;\nmthds := [`Enright-Verner scheme`,`Verner's Maple scheme`,`a mod ification of Verner's Maple scheme`,`an \"efficient\" scheme`,`a schem e of Tsitouras and Papakostas`,`a modification of the Tsitouras-Papako stas scheme`,`Verner's \"most efficient\" scheme`]: errs := []:\nDigit s := 30:\nfor ct to 7 do\n Wn_RK8_||ct := RK8_||ct(W(x,y),x,y,x0,y0, hh,numsteps,false);\n sm := 0: numpts := nops(Wn_RK8_||ct):\n for \+ ii to numpts do\n sm := sm+(Wn_RK8_||ct[ii,2]-w(Wn_RK8_||ct[ii,1] ))^2;\n end do:\n errs := [op(errs),sqrt(sm/numpts)];\nend do:\nDi gits := 10:\nlinalg[transpose]([mthds,evalf(errs)]);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#K%'matrixG6#7&7$%0slope~field:~~~G*(%\"xG\"\"\",&\" \"*F+*$)F*\"\"#F+!\"\"F+,&F+F+*$)%\"yGF0F+F+F17$%0initial~point:~G-%!G 6$\"\"!F;7$%/step~width:~~~G$F+!\"#7$%1no.~of~steps:~~~G\"$+%Q)pprint3 56\"" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%!G" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#K%'matrixG6#7)7$%6Enright-Verner~schemeG$\"+`K4W@!#G7$% 6Verner's~Maple~schemeG$\"+$*)[mp&!#H7$%Ha~modification~of~Verner's~Ma ple~schemeG$\"+PducJF07$%6an~\"efficient\"~schemeG$\"+Fe\"*yEF07$%Ea~s cheme~of~Tsitouras~and~PapakostasG$\"++Mo7=F07$%Ra~modification~of~the ~Tsitouras-Papakostas~schemeG$\"+`s\"*f@F07$%AVerner's~\"most~efficien t\"~schemeG$\"+.9\"*39F0Q)pprint366\"" }}}{PARA 0 "" 0 "" {TEXT -1 0 " " }}{PARA 0 "" 0 "" {TEXT -1 30 "The following code constructs " } {TEXT 260 20 "numerical procedures" }{TEXT -1 56 " for solutions based on each of the Runge-Kutta schemes." }}{PARA 0 "" 0 "" {TEXT -1 75 "T he error in the value obtained by each of the methods at the point whe re " }{XPPEDIT 18 0 "x = 3.499;" "6#/%\"xG-%&FloatG6$\"%*\\$!\"$" } {TEXT -1 16 " is also given." }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 726 "W := (x,y) -> x*(9-x^2)/(1+y^2): hh := 0.01: numsteps := 400: x 0 := 0: y0 := 0:\nmatrix([[`slope field: `,W(x,y)],[`initial point: \+ `,``(x0,y0)],[`step width: `,hh],\n[`no. of steps: `,numsteps]]);` `;\nmthds := [`Enright-Verner scheme`,`Verner's Maple scheme`,`a modif ication of Verner's Maple scheme`,`an \"efficient\" scheme`,`a scheme \+ of Tsitouras and Papakostas`,`a modification of the Tsitouras-Papakost as scheme`,`Verner's \"most efficient\" scheme`]: errs := []:\nDigits \+ := 30:\nfor ct to 7 do\n wn_RK8_||ct := RK8_||ct(W(x,y),x,y,x0,y0,hh ,numsteps,true);\nend do:\nxx := 3.499: wxx := evalf(w(xx)):\nfor ct t o 7 do\n errs := [op(errs),abs(wn_RK8_||ct(xx)-wxx)];\nend do:\nDigi ts := 10:\nlinalg[transpose]([mthds,evalf(errs)]);" }}{PARA 11 "" 1 " " {XPPMATH 20 "6#K%'matrixG6#7&7$%0slope~field:~~~G*(%\"xG\"\"\",&\"\" *F+*$)F*\"\"#F+!\"\"F+,&F+F+*$)%\"yGF0F+F+F17$%0initial~point:~G-%!G6$ \"\"!F;7$%/step~width:~~~G$F+!\"#7$%1no.~of~steps:~~~G\"$+%Q)pprint376 \"" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%!G" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#K%'matrixG6#7)7$%6Enright-Verner~schemeG$\"+>&*Qbt!#H7$ %6Verner's~Maple~schemeG$\"+Eke)3#F+7$%Ha~modification~of~Verner's~Map le~schemeG$\"+8VP-7F+7$%6an~\"efficient\"~schemeG$\"*fowg*F+7$%Ea~sche me~of~Tsitouras~and~PapakostasG$\"*KAxD'F+7$%Ra~modification~of~the~Ts itouras-Papakostas~schemeG$\"*Q6vr(F+7$%AVerner's~\"most~efficient\"~s chemeG$\"*@Z3%RF+Q)pprint386\"" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }} {PARA 0 "" 0 "" {TEXT -1 4 "The " }{TEXT 260 22 "root mean square erro r" }{TEXT -1 20 " over the interval " }{XPPEDIT 18 0 "[0,4]" "6#7$\" \"!\"\"%" }{TEXT -1 82 " of each Runge-Kutta method is estimated as f ollows using the special procedure " }{TEXT 0 5 "NCint" }{TEXT -1 98 " to perform numerical integration by the 7 point Newton-Cotes method over 200 equal subintervals." }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 476 "mthds := [`Enright-Verner scheme`,`Verner's Maple scheme`,`a mo dification of Verner's Maple scheme`,`an \"efficient\" scheme`,`a sche me of Tsitouras and Papakostas`,`a modification of the Tsitouras-Papak ostas scheme`,`Verner's \"most efficient\" scheme`]: errs := []:\nDigi ts := 20:\nfor ct to 7 do\n sm := NCint((w(x)-'wn_RK8_||ct'(x))^2,x= 0..4,adaptive=false,numpoints=7,factor=200);\n errs := [op(errs),sqr t(sm/4)];\nend do:\nDigits := 10:\nlinalg[transpose]([mthds,evalf(errs )]);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#K%'matrixG6#7)7$%6Enright-Vern er~schemeG$\"+owT@@!#G7$%6Verner's~Maple~schemeG$\"+)GEtm&!#H7$%Ha~mod ification~of~Verner's~Maple~schemeG$\"+H3*H9$F07$%6an~\"efficient\"~sc hemeG$\"+)zwkk#F07$%Ea~scheme~of~Tsitouras~and~PapakostasG$\"+LiAn " 0 "" {MPLTEXT 1 0 614 "evalf[25](plot([ 'wn_RK8_1'(x)-w(x),'wn_RK8_2'(x)-w(x),'wn_RK8_3'(x)-w(x),'wn_RK8_4'(x) -w(x),\n'wn_RK8_5'(x)-w(x),'wn_RK8_6'(x)-w(x),'wn_RK8_7'(x)-w(x)],x=0. .4,-4e-19..7.5e-19,font=[HELVETICA,9],\ncolor=[COLOR(RGB,.4,0,.9),COLO R(RGB,0,.65,0),brown,red,COLOR(RGB,0,.75,.95),magenta,COLOR(RGB,.95,.5 5,0)],\nlegend=[`Enright-Verner scheme`,`Verner's Maple scheme`,`a mod ification of Verner's Maple scheme`,`an \"efficient\" scheme`,`a schem e of Tsitouras and Papakostas`,`a modification of the Tsitouras-Papako stas scheme`,`Verner's \"most efficient\" scheme`],title=`error curves for 12 stage order 8 Runge-Kutta methods`));" }}{PARA 13 "" 1 "" {GLPLOT2D 1026 651 651 {PLOTDATA 2 "6--%'CURVESG6%7gp7$$\"\"!F)F(7$$\" :mmmmmmmmmh)=()!#E$\"'zNZ!#D7$$\":LLLLLLL$e'40j\"F0$\"(4B2\"F07$$\":mm mmmm;H_(zV=F0$\"'wy)*F07$$\":+++++++vQ&3d?F0$\"'&>!))F07$$\":MLLLL$ek` tS5@F0$\"'!Qu)F07$$\":nmmmmm\"z>$HP;#F0$\"'<>()F07$$\":MLLLLekGI!R!>#F 0$\"',M*)F07$$\":+++++]PfG^q@#F0$\"'u4$*F07$$\":nmmmmT5!pArVAF0$\"'`!H *F07$$\":MLLLLL$3_KPqAF0$\"'6H$*F07$$\":++++++vV=,\"f#F0$\"(l3#>F07$$\":nm mmmmm;zi$)p#F0$\"(jps#F07$$\":nmmmmmm\"HOr0GF0$\"(f*yMF07$$\":nmmmmmmm YkI\"HF0$\"([c<%F07$$\":nmmmmmmTI:/-$F0$\"(*Qi[F07$$\":nmmmmmm;9mx7$F0 $\"(T'*\\&F07$$\":nmmmmmm\"zp6NKF0$\"()pagF07$$\":nmmmmmmm\"yYULF0$\"( !*\\]'F07$$\":++++++vVV3$\\MF0$\"(D,%oF07$$\":MLLLLL$3_!\\hb$F0$\"(R$f qF07$$\":nmmmmT5l?fGe$F0$\"(F02(F07$$\":+++++]P4Op&4OF0$\"(O&[sF07$$\" :MLLLLek`^zij$F0$\"(J%3sF07$$\":nmmmmm\"zp'*)Hm$F0$\"(&HrrF07$$\":++++ +v=U#)*p*o$F0$\"(o!)>(F07$$\":MLLLL$eky*4kr$F0$\"(?(osF07$$\":nmmmm\"H 2L,7VPF0$\"(bsA(F07$$\":+++++++vGI)pPF0$\"(3/>(F07$$\":mmmmmm;H_6N)RF0 $\"(q$=qF07$$\":LLLLLLL$eF>(>%F0$\"((QpmF07$$\":+++++++]Z7Mf%F0$\"(+E( fF07$$\":mmmmmmm;>K'*)\\F0$\"(2XQ&F07$$\":NLLLLLL$eZ*)*R&F0$\"(m$))[F0 7$$\":+++++++]Kd,\"eF0$\"(a\"eWF07$$\":mmmmmmm;fX(emF0$\"(!R8PF07$$\": +++++++]U7Y](F0$\"(=r8$F07$$\":LLLLLLLLV!pu$)F0$\"(e-o#F07$$\":mmmmmmm mhb59*F0$\"(QWO#F07$$\":++++++++8!Q+5!#C$\"(U03#F07$$\":++++++++&*3q3 \"Fax$\"(GD&=F07$$\":++++++++(=\\q6Fax$\"(/Mn\"F07$$\":nmmmmmm\"fBIY7F ax$\"(ku`\"F07$$\":LLLLLLLLO[kL\"Fax$\"(EFax$\"'!=5 *F07$$\":nmmmmmm;kD!)*>Fax$\"'U(p)F07$$\":nmmmmmm\"f`@'3#Fax$\"'MI$)F0 7$$\":+++++++vw%)H;#Fax$\"'C\\!)F07$$\":nmmmmmm;$y*eC#Fax$\"'/#y(F07$$ \":++++++++9b:L#Fax$\"'oTvF07$$\":+++++++]5a`T#Fax$\"'sQtF07$$\":+++++ ++D\"RV'\\#Fax$\"'3qrF07$$\":+++++++]@fke#Fax$\"'Q8qF07$$\":LLLLLLLL&4 NnEFax$\"'w)*oF07$$\":++++++++:?Pv#Fax$\"')G!oF07$$\":nmmmmmm\"zM)>$GF ax$\"'wRnF07$$\":++++++++(faV_QFax$\"'_SzF07$$\":++++++++u]E(QFax$\"'mEtF07$$ \":++++++]PdpG*QFax$\"'kEiF07$$\":+++++++vS)38RFax$\"'['G%F07$$\":++++ ++DcI;[$RFax$\"&_e)F07$$\":++++++]P?Wl&RFax$!''QT&F07$$\":+++++]7G:3u' RFax$!(_T0\"F07$$\":++++++v=5s#yRFax$!('4C=F07$$\":+++++D1k2/P)RFax$!( eqb#F07$$\":+++++]P40O\"*)RFax$!(3:5$F07$$\":++++]7.#Q?&=*RFax$!(+,z$F 07$$\":+++++voa-oX*RFax$!(qp\"QF07$$\":++++]PMF,%G(*RFax$!(-1$RF07$$\" \"%F)$!(c_^&F0-%&COLORG6&%$RGBG$Ffgl!\"\"F($\"\"*F^hl-%'LEGENDG6#%6Enr ight-Verner~schemeG-F$6%7[pF'7$F+$!'S@5F07$$\":+++++++D\"z>^7F0$!'L[?F 07$F2$!'m5HF07$F<$!'kfNF07$F^o$!'HqbF07$Fco$!'9/nF07$Fho$!'NJ$)F07$F]p $!'@n(*F07$Fbp$!(!G16F07$Fgp$!(0LB\"F07$F\\q$!(\\>N\"F07$Faq$!(LtX\"F0 7$Ffq$!(Gga\"F07$F`r$!(P\"o;F07$Fht$!(T9s\"F07$$\":LLLLLL3_!4nwQF0$!(: ys\"F07$F]u$!(YWs\"F07$$\":++++++D19_.4%F0$!(\"=9F07$Fd^l$!'Cd>F07$Fi^l$!'7I>F07$F^_l$ !'C7>F07$Fc_l$!'3+>F07$Fh_l$!';'*=F07$F]`l$!'5+>F07$Fb`l$!'/8>F07$Fg`l $!'AP>F07$F\\al$!'ws>F07$Faal$!'qB?F07$Ffal$!'!=4#F07$F[bl$!'Ct@F07$F` bl$!'+%H#F07$Febl$!'q>CF07$Fjbl$!'+rCF07$F_cl$!'A$\\#F07$$\":++++++v$* [AB%QFax$!'I&[#F07$Fdcl$!'[qCF07$$\":++++++DJKTD'QFax$!')RW#F07$Ficl$! '%[S#F07$F^dl$!'OmAF07$Fcdl$!'K.?F07$Fhdl$!'GC:F07$F]el$!&cH'F07$Fbel$ \"&y3\"F07$Fgel$\"'%GA\"F07$F\\fl$\"'#RD#F07$Fafl$\"'U_IF07$Fffl$\"'!4 +%F07$F[gl$\"'!)HSF07$F`gl$\"'Q*>%F07$Fegl$\"'C=kF0-Fjgl6&F\\hlF($\"#l !\"#F(-Fbhl6#%6Verner's~Maple~schemeG-F$6%7joF'7$F+$!&q\"QF07$F2$!&pc( F07$F7$!&El'F07$F<$!&6j&F07$FA$!&__&F07$FF$!&F^&F07$FP$!&X&fF07$FZ$!&w (fF07$Fin$!&wL(F07$F^o$!&6***F07$Fco$!'Ak9F07$Fho$!'eNAF07$F]p$!'G1IF0 7$Fbp$!'7uPF07$Fgp$!'/$f%F07$F\\q$!'\"zU&F07$Faq$!'MViF07$Ffq$!'P2qF07 $F[r$!'\"[p(F07$F`r$!'+)G)F07$Fds$!'/y()F07$Fht$!')[;*F07$F]u$!'Ve'*F0 7$Fbu$!'Ci)*F07$Fgu$!'s8%*F07$F\\v$!'29()F07$Fav$!''y'zF07$Ffv$!'YksF0 7$$\":NLLLLLL$e9XMiF0$!'#fh'F07$F[w$!'+YgF07$$\":NLLLLLL$3!z;3(F0$!'C[ bF07$F`w$!'U7^F07$Few$!'srVF07$Fjw$!'ieQF07$F_x$!'['R$F07$Fex$!'iCIF07 $Fjx$!'YKFF07$F_y$!'Y5DF07$Fdy$!'%*)G#F07$Fiy$!''*H@F07$F^z$!'%=(>F07$ Fcz$!'Y^=F07$Fhz$!''pt\"F07$F][l$!'CU;F07$Fb[l$!'#eb\"F07$Fg[l$!'I'[\" F07$F\\\\l$!'G?9F07$Fa\\l$!'Eg8F07$Ff\\l$!'c98F07$F[]l$!''3F\"F07$F`]l $!'_J7F07$Fe]l$!'Q)>\"F07$Fj]l$!'#4<\"F07$F_^l$!'KX6F07$Fd^l$!'aE6F07$ Fi^l$!'#46\"F07$F^_l$!'k+6F07$Fc_l$!'e$4\"F07$Fh_l$!'O\"4\"F07$F]`l$!' g$4\"F07$Fb`l$!'/,6F07$Fg`l$!'#\\6\"F07$F\\al$!'ON6F07$Faal$!'![;\"F07 $Ffal$!'?/7F07$F[bl$!'%=D\"F07$F`bl$!'IC8F07$Febl$!'+29F07$F_cl$!'76:F 07$Ficl$!'ki:F07$Fcdl$!'_3;F07$Fhdl$!'GP;F07$F]el$!'Yy;F07$Fbel$!'#3r \"F07$Fgel$!'Eh\"F07$$\":nmmmmm;/@Q?v #F0$\"'Cs5F07$F]p$\"%!f&F07$$\":nmmmmmT&Q8bKGF0$\"%mbF07$$\":nmmmmm;z/ *QfGF0$\"%BXF07$$\":nmmmmm\"HdnA')GF0$!&\\`#F07$Fbp$!'&)Q6F07$$\":nmmm mmTg<-*RHF0$!'jM6F07$$\":nmmmmm;a))Rn'HF0$!'*H;\"F07$$\":nmmmmm\"z%fxN *HF0$!'(4%=F07$Fgp$!'WDCF07$$\":nmmmmmTN,`s/$F0$!'o;CF07$$\":nmmmmm;Hs !4uIF0$!'!Q\\#F07$$\":nmmmmm\"HK%G45$F0$!'**QPF07$F\\q$!'GAPF07$$\":nm mmmm;/cT9=$F0$!''R*QF07$Faq$!'(H&\\F07$$\":nmmmmm;zR#z)G$F0$!'z?`F07$F fq$!''[0'F07$F[r$!'#[)pF07$F`r$!'G?xF07$Fds$!'-e#)F07$Fht$!'o5')F07$$ \":LLLLL3F>WSlz$F0$!'VG))F07$$\":mmmmm;ajf]K#QF0$!'jy))F07$$\":+++++D \"y]2'*\\QF0$!'+F))F07$Fh[m$!'3+))F07$$\":mmmmmTN'f5Q.RF0$!'%)z*)F07$$ \":+++++]iS@\"4IRF0$!':E*)F07$$\":LLLLLe*[o8!o&RF0$!'wt))F07$F]u$!'[_) )F07$F`\\m$!'Q$z)F07$Fbu$!'IX')F07$Fgu$!'S\"z(F07$F\\v$!'U')pF07$Fav$! '];jF07$Ffv$!''4w&F07$F\\^n$!'-f_F07$F[w$!'5;[F07$F`w$!'#>3%F07$Few$!' i#\\$F07$Fjw$!'7$3$F07$F_x$!')Qr#F07$Fex$!'s;CF07$Fjx$!'E$=#F07$F_y$!' 'e+#F07$Fdy$!'%)G=F07$Fiy$!'w,()F07$F]`l$!&qt)F07$Fb`l$!&uz)F07$Fg`l$!&s!*) F07$F\\al$!&;2*F07$Faal$!&qI*F07$Ffal$!&Ii*F07$F[bl$!'k+5F07$F`bl$!'Sg 5F07$Febl$!'qL6F07$Fjbl$!'!))=\"F07$F_cl$!'#[E\"F07$Fdcl$!'G=8F07$Ficl $!'a\"R\"F07$F^dl$!'')*\\\"F07$Fcdl$!'#Rn\"F07$Fhdl$!'3z>F07$F]el$!'cY DF07$Fbel$!'UCIF07$Fgel$!'OkPF07$F\\fl$!'y=XF07$Fafl$!'on]F07$Fffl$!'! z#eF07$F[gl$!']oeF07$F`gl$!'A&)fF07$Fegl$!'EmxF0-Fcgn6&F\\hl$\"*++++\" FggnF(F(-Fbhl6#%6an~\"efficient\"~schemeG-F$6%7fpF'7$Fahn$!%b=F07$Ffhn $!%daF07$F[in$!&v2\"F07$F`in$!&Zx\"F07$Fein$!&5h$F07$F+$!&U'eF07$F]jn$ !&/(yF07$F\\il$!&b())F07$Fd[o$!&?l(F07$F2$!&B,$F07$F7$\"&i`&F07$F<$\"' ]*p\"F07$FZ$\"'[aGF07$F^o$\"'nyNF07$$\":nmmmmmTg#))\\5DF0$\"'*pl$F07$$ \":nmmmmm;a`Ot`#F0$\"'RYOF07$$\":nmmmmm\"zWU\"F07$Fiy$!'w36F07$F^z$!'aE5F07$Fcz$!&'Q'*F0 7$Fhz$!&E/*F07$F][l$!&%\\&)F07$Fb[l$!&#*4)F07$Fg[l$!&qt(F07$F\\\\l$!&Q R(F07$Fa\\l$!&;3(F07$Ff\\l$!&O%oF07$F[]l$!&mh'F07$F`]l$!&7T'F07$Fe]l$! &yB'F07$Fj]l$!&_4'F07$F_^l$!&A'fF07$Fd^l$!&W'eF07$Fi^l$!&Ky&F07$F^_l$! &%HdF07$Fc_l$!&Gp&F07$Fh_l$!&;o&F07$F]`l$!&?p&F07$Fb`l$!&9t&F07$Fg`l$! &A!eF07$F\\al$!&'4fF07$Faal$!&?1'F07$Ffal$!&!oiF07$F[bl$!&%=lF07$F`bl$ !&]\"pF07$Febl$!&!HuF07$Fjbl$!&S(yF07$F_cl$!&sb)F07$Fdcl$!&[4*F07$Ficl $!&M))*F07$F^dl$!'O66F07$Fcdl$!'s:8F07$Fhdl$!'=y;F07$F]el$!'YaBF07$Fbe l$!'AAHF07$Fgel$!'w'z$F07$F\\fl$!'Q!o%F07$Fafl$!')4K&F07$Fffl$!'S,iF07 $F[gl$!'qWiF07$F`gl$!'#)yjF07$Fegl$!'mH%)F0-Fjgl6&F\\hlF($\"#vF`hm$\"# &*F`hm-Fbhl6#%Ea~scheme~of~Tsitouras~and~PapakostasG-F$6%7hpF'7$Fahn$! %cF07$Fein$!&9/%F07$F+$!& uq'F07$F]jn$!&yC*F07$F\\il$!'L(4\"F07$Fjjn$!'!o6\"F07$Fd[o$!'8r5F07$Fi [o$!&SY*F07$F2$!&OK(F07$F7$!%WHF07$F<$\"&=`*F07$FZ$\"'yj>F07$F^o$\"'!* )f#F07$F\\dp$\"'$*fEF07$Fadp$\"'A_EF07$Ffdp$\"'TVEF07$Fco$\"'3$f#F07$F ^ep$\"',.DF07$Fho$\"'H(>#F07$F]p$\"'rk:F07$Fbp$\"&^+)F07$Fgp$!%`)*F07$ F\\q$!'Rw5F07$Faq$!'Tu?F07$Ffq$!'.QIF07$F[r$!'HARF07$F`r$!'F&p%F07$Fds $!'tQ`F07$Fht$!'+[eF07$F]u$!'v!\\'F07$Fbu$!'w?nF07$Fgu$!'&3J'F07$F\\v$ !'N3dF07$Ffv$!''4l%F07$F[w$!']mQF07$F`w$!'iuKF07$Few$!'s-GF07$Fjw$!'su CF07$F_x$!'yy@F07$Fex$!'US>F07$Fjx$!''Hv\"F07$F_y$!'c5;F07$Fdy$!'Wo9F0 7$Fiy$!'Ym8F07$F^z$!'/l7F07$Fcz$!''y=\"F07$Fhz$!'O96F07$F][l$!'k`5F07$ Fb[l$!&7)**F07$Fg[l$!&]`*F07$F\\\\l$!&=6*F07$Fa\\l$!&ms)F07$Ff\\l$!&OV )F07$F[]l$!&O:)F07$F`]l$!&7!zF07$Fe]l$!&yo(F07$Fj]l$!&A^(F07$F_^l$!&#[ tF07$Fd^l$!&uA(F07$Fi^l$!&s7(F07$F^_l$!&/1(F07$Fc_l$!&e,(F07$Fh_l$!&;+ (F07$F]`l$!&],(F07$Fb`l$!&M1(F07$Fg`l$!&A:(F07$F\\al$!&YG(F07$Faal$!&S Z(F07$Ffal$!&+t(F07$F[bl$!&W/)F07$F`bl$!&qa)F07$Febl$!&+@*F07$Fjbl$!&+ z*F07$F_cl$!'Kn5F07$Fdcl$!'QN6F07$Ficl$!'aL7F07$F^dl$!'w$Q\"F07$Fcdl$! 'iF;F07$Fhdl$!'o]?F07$F]el$!''G#GF07$Fbel$!'ihMF07$Fgel$!'ENWF07$F\\fl $!'Q+aF07$Fafl$!'=3hF07$Fffl$!'gbqF07$F[gl$!'![5(F07$F`gl$!'AcsF07$Feg l$!'Oc%*F0-Fcgn6&F\\hlFi`pF(Fi`p-Fbhl6#%Ra~modification~of~the~Tsitour as-Papakostas~schemeG-F$6%7`pF'7$F`in$!%ZRF07$F+$!&3I\"F07$$\":0+++++] 7.Trm*F-$!&tX\"F07$F]jn$!&M^\"F07$$\":nmmmmmTg(*pj:\"F0$!&tS\"F07$F\\i l$!&o1\"F07$Fd[o$\"%#>'F07$F2$\"&]4%F07$F<$\"'BS;F07$F^o$\"'%Rq#F07$Fh o$\"'F(Q#F07$Fbp$\"'m&Q\"F07$Fgp$\"&2V(F07$F\\q$\"%8WF07$Faq$!&Xp'F07$ Ffq$!'Je8F07$F`r$!'PRDF07$Fht$!'M^LF07$Fh[m$!'D8OF07$F]u$!'i)y$F07$$\" :+++++vVtn@-,%F0$!'83RF07$$\":LLLLL$3xJ=$p.%F0$!'N%)QF07$$\":mmmmm\"z> ')>kjSF0$!'#H'QF07$F`\\m$!'6()QF07$$\":mmmmm;z%\\CxVTF0$!'q3RF07$Fbu$! ']V\"F07$Fjw$!'ii7F07$F_x$!')46\"F07$Fex$!&A*)*F07$Fjx$!&c $*)F07$F_y$!&'3#)F07$Fdy$!&W[(F07$Fiy$!&Y'pF07$F^z$!&%[kF07$Fcz$!&Y0'F 07$Fhz$!&'zcF07$F][l$!&/P&F07$Fb[l$!&s3&F07$Fg[l$!&+'[F07$F\\\\l$!&Qk% F07$Fa\\l$!&wW%F07$Ff\\l$!&')H%F07$F[]l$!&c:%F07$F`]l$!&s-%F07$Fe]l$!& y\"RF07$Fj]l$!&#HQF07$F_^l$!&_u$F07$Fd^l$!&Mo$F07$Fi^l$!&Aj$F07$F^_l$! &%*f$F07$Fc_l$!&ed$F07$Fh_l$!&'pNF07$F]`l$!&]d$F07$Fb`l$!&/g$F07$Fg`l$ !&_k$F07$F\\al$!&Or$F07$Faal$!&I\"QF07$Ffal$!&![RF07$F[bl$!&%=TF07$F`b l$!&ST%F07$Febl$!&q([F07$Fjbl$!&]S&F07$F_cl$!&KJ'F07$Fdcl$!&))3(F07$Fi cl$!&aE)F07$F^dl$!'c75F07$Fcdl$!'-?8F07$Fhdl$!'o`=F07$F]el$!'wCGF07$Fb el$!'UBOF07$Fgel$!''Q$[F07$F\\fl$!'G9gF07$Fafl$!'Q*)oF07$Fffl$!']H!)F0 7$F[gl$!'!e3)F07$F`gl$!'it#)F07$Fegl$!(O?4\"F0-Fjgl6&F\\hlFhbq$\"#bF`h mF(-Fbhl6#%AVerner's~\"most~efficient\"~schemeG-%%FONTG6$%*HELVETICAGF `hl-%&TITLEG6#%Verror~curves~for~12~stage~order~8~Runge-Kutta~methodsG -%+AXESLABELSG6$Q\"x6\"Q!Fafs-%%VIEWG6$;F(Fegl;$!\"%!#>$Fgbq!#?" 1 2 0 1 10 0 2 9 1 4 2 1.000000 45.000000 45.000000 0 1 "Enright-Verner sc heme" "Verner's Maple scheme" "a modification of Verner's Maple scheme " "an \"efficient\" scheme" "a scheme of Tsitouras and Papakostas" "a \+ modification of the Tsitouras-Papakostas scheme" "Verner's \"most effi cient\" scheme" }}}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> \+ " 0 "" {MPLTEXT 1 0 554 "evalf[25](plot(['wn_RK8_2'(x)-w(x),'wn_RK8_3' (x)-w(x),'wn_RK8_4'(x)-w(x),\n'wn_RK8_5'(x)-w(x),'wn_RK8_6'(x)-w(x),'w n_RK8_7'(x)-w(x)],x=0..4,-1.8e-19..0.5e-19,font=[HELVETICA,9],\ncolor= [COLOR(RGB,0,.65,0),brown,red,COLOR(RGB,0,.75,.95),magenta,COLOR(RGB,. 95,.55,0)],\nlegend=[`Verner's Maple scheme`,`a modification of Verner 's Maple scheme`,`an \"efficient\" scheme`,`a scheme of Tsitouras and \+ Papakostas`,`a modification of the Tsitouras-Papakostas scheme`,`Verne r's \"most efficient\" scheme`],title=`error curves for 12 stage order 8 Runge-Kutta methods`));" }}{PARA 13 "" 1 "" {GLPLOT2D 991 535 535 {PLOTDATA 2 "6,-%'CURVESG6%7[p7$$\"\"!F)F(7$$\":mmmmmmmmmh)=()!#E$!'S@ 5!#D7$$\":+++++++D\"z>^7F0$!'L[?F07$$\":LLLLLLL$e'40j\"F0$!'m5HF07$$\" :+++++++vQ&3d?F0$!'kfNF07$$\":nmmmmmmm6hO[#F0$!'HqbF07$$\":nmmmmmmT&>, \"f#F0$!'9/nF07$$\":nmmmmmm;zi$)p#F0$!'NJ$)F07$$\":nmmmmmm\"HOr0GF0$!' @n(*F07$$\":nmmmmmmmYkI\"HF0$!(!G16F07$$\":nmmmmmmTI:/-$F0$!(0LB\"F07$ $\":nmmmmmm;9mx7$F0$!(\\>N\"F07$$\":nmmmmmm\"zp6NKF0$!(LtX\"F07$$\":nm mmmmmm\"yYULF0$!(Gga\"F07$$\":MLLLLL$3_!\\hb$F0$!(P\"o;F07$$\":+++++++ vGI)pPF0$!(T9s\"F07$$\":LLLLLL3_!4nwQF0$!(:ys\"F07$$\":mmmmmm;H_6N)RF0 $!(YWs\"F07$$\":++++++D19_.4%F0$!(\"=9(>%F0$!(%y*p \"F07$$\":+++++++]Z7Mf%F0$!(eaf\"F07$$\":mmmmmmm;>K'*)\\F0$!(&R#[\"F07 $$\":+++++++]Kd,\"eF0$!(1gD\"F07$$\":mmmmmmm;fX(emF0$!(!R]5F07$$\":+++ ++++]U7Y](F0$!'_()))F07$$\":LLLLLLLLV!pu$)F0$!'_)f(F07$$\":mmmmmmmmhb5 9*F0$!'_0nF07$$\":++++++++8!Q+5!#C$!'y,fF07$$\":++++++++&*3q3\"F[t$!'U b_F07$$\":++++++++(=\\q6F[t$!'cZZF07$$\":nmmmmmm\"fBIY7F[t$!''=O%F07$$ \":LLLLLLLLO[kL\"F[t$!'%o(RF07$$\":LLLLLLLL&Q\"GT\"F[t$!'m+PF07$$\":++ +++++D2X;]\"F[t$!'%fU$F07$$\":LLLLLLLLvv-e\"F[t$!'w;KF07$$\":+++++++D2 Ylm\"F[t$!'wF[t$!'I#e#F07$$\":nmmmmmm;kD!)*>F[t$!'enCF0 7$$\":nmmmmmm\"f`@'3#F[t$!'YjBF07$$\":+++++++vw%)H;#F[t$!''QG#F07$$\": nmmmmmm;$y*eC#F[t$!''z?#F07$$\":++++++++9b:L#F[t$!'sR@F07$$\":+++++++] 5a`T#F[t$!'3#3#F07$$\":+++++++D\"RV'\\#F[t$!'KM?F07$$\":+++++++]@fke#F [t$!'#**)>F07$$\":LLLLLLLL&4NnEF[t$!'Cd>F07$$\":++++++++:?Pv#F[t$!'7I> F07$$\":nmmmmmm\"zM)>$GF[t$!'C7>F07$$\":++++++++(faF07$$\": LLLLLLLeg`!)*HF[t$!';'*=F07$$\":+++++++DG2A3$F[t$!'5+>F07$$\":LLLLLLLL )G[kJF[t$!'/8>F07$$\":+++++++D\"yh]KF[t$!'AP>F07$$\":nmmmmmmm)fdLLF[t$ !'ws>F07$$\":nmmmmmm;q7%=MF[t$!'qB?F07$$\":LLLLLLLe#pa-NF[t$!'!=4#F07$ $\":++++++++ad)zNF[t$!'Ct@F07$$\":LLLLLLL$GUYoOF[t$!'+%H#F07$$\":nmmmm mmm5:xu$F[t$!'q>CF07$$\":MLLLLL$e*3k**y$F[t$!'+rCF07$$\":+++++++D28A$Q F[t$!'A$\\#F07$$\":++++++v$*[AB%QF[t$!'I&[#F07$$\":++++++]i!>V_QF[t$!' [qCF07$$\":++++++DJKTD'QF[t$!')RW#F07$$\":++++++++u]E(QF[t$!'%[S#F07$$ \":++++++]PdpG*QF[t$!'OmAF07$$\":+++++++vS)38RF[t$!'K.?F07$$\":++++++D cI;[$RF[t$!'GC:F07$$\":++++++]P?Wl&RF[t$!&cH'F07$$\":+++++]7G:3u'RF[t$ \"&y3\"F07$$\":++++++v=5s#yRF[t$\"'%GA\"F07$$\":+++++D1k2/P)RF[t$\"'#R D#F07$$\":+++++]P40O\"*)RF[t$\"'U_IF07$$\":++++]7.#Q?&=*RF[t$\"'!4+%F0 7$$\":+++++voa-oX*RF[t$\"'!)HSF07$$\":++++]PMF,%G(*RF[t$\"'Q*>%F07$$\" \"%F)$\"'C=kF0-%&COLORG6&%$RGBGF($\"#l!\"#F(-%'LEGENDG6#%6Verner's~Map le~schemeG-F$6%7joF'7$F+$!&q\"QF07$F7$!&pc(F07$$\":mmmmmm;H_(zV=F0$!&E l'F07$F<$!&6j&F07$$\":MLLLL$ek`tS5@F0$!&__&F07$$\":nmmmmm\"z>$HP;#F0$! &F^&F07$$\":+++++]PfG^q@#F0$!&X&fF07$$\":MLLLLL$3_KPqAF0$!&w(fF07$$\": ++++++vV=F07$F]v$!'Y^=F07$Fbv$! ''pt\"F07$Fgv$!'CU;F07$F\\w$!'#eb\"F07$Faw$!'I'[\"F07$Ffw$!'G?9F07$F[x $!'Eg8F07$F`x$!'c98F07$Fex$!''3F\"F07$Fjx$!'_J7F07$F_y$!'Q)>\"F07$Fdy$ !'#4<\"F07$Fiy$!'KX6F07$F^z$!'aE6F07$Fcz$!'#46\"F07$Fhz$!'k+6F07$F][l$ !'e$4\"F07$Fb[l$!'O\"4\"F07$Fg[l$!'g$4\"F07$F\\\\l$!'/,6F07$Fa\\l$!'# \\6\"F07$Ff\\l$!'ON6F07$F[]l$!'![;\"F07$F`]l$!'?/7F07$Fe]l$!'%=D\"F07$ Fj]l$!'IC8F07$F_^l$!'+29F07$Fi^l$!'76:F07$F]`l$!'ki:F07$Fg`l$!'_3;F07$ F\\al$!'GP;F07$Faal$!'Yy;F07$Ffal$!'#3r\"F07$F[bl$!'Eh\"F07$$\":nmmmmm;/@Q?v#F0$\"'Cs5F07$FP$\"%!f&F07$$\":nmm mmmT&Q8bKGF0$\"%mbF07$$\":nmmmmm;z/*QfGF0$\"%BXF07$$\":nmmmmm\"HdnA')G F0$!&\\`#F07$FU$!'&)Q6F07$$\":nmmmmmTg<-*RHF0$!'jM6F07$$\":nmmmmm;a))R n'HF0$!'*H;\"F07$$\":nmmmmm\"z%fxN*HF0$!'(4%=F07$FZ$!'WDCF07$$\":nmmmm mTN,`s/$F0$!'o;CF07$$\":nmmmmm;Hs!4uIF0$!'!Q\\#F07$$\":nmmmmm\"HK%G45$ F0$!'**QPF07$Fin$!'GAPF07$$\":nmmmmm;/cT9=$F0$!''R*QF07$F^o$!'(H&\\F07 $$\":nmmmmm;zR#z)G$F0$!'z?`F07$Fco$!''[0'F07$F^il$!'#[)pF07$Fho$!'G?xF 07$Ffil$!'-e#)F07$F]p$!'o5')F07$$\":LLLLL3F>WSlz$F0$!'VG))F07$$\":mmmm m;ajf]K#QF0$!'jy))F07$$\":+++++D\"y]2'*\\QF0$!'+F))F07$Fbp$!'3+))F07$$ \":mmmmmTN'f5Q.RF0$!'%)z*)F07$$\":+++++]iS@\"4IRF0$!':E*)F07$$\":LLLLL e*[o8!o&RF0$!'wt))F07$Fgp$!'[_))F07$F\\q$!'Q$z)F07$Faq$!'IX')F07$Ffq$! 'S\"z(F07$F[r$!'U')pF07$Fjjl$!'];jF07$F`r$!''4w&F07$Fb[m$!'-f_F07$Fer$ !'5;[F07$Fjr$!'#>3%F07$F_s$!'i#\\$F07$Fds$!'7$3$F07$Fis$!')Qr#F07$F_t$ !'s;CF07$Fdt$!'E$=#F07$Fit$!''e+#F07$F^u$!'%)G=F07$Fcu$!'w,()F07$Fg[l$!&qt)F07$F\\\\l$!&uz)F 07$Fa\\l$!&s!*)F07$Ff\\l$!&;2*F07$F[]l$!&qI*F07$F`]l$!&Ii*F07$Fe]l$!'k +5F07$Fj]l$!'Sg5F07$F_^l$!'qL6F07$Fd^l$!'!))=\"F07$Fi^l$!'#[E\"F07$Fc_ l$!'G=8F07$F]`l$!'a\"R\"F07$Fb`l$!'')*\\\"F07$Fg`l$!'#Rn\"F07$F\\al$!' 3z>F07$Faal$!'cYDF07$Ffal$!'UCIF07$F[bl$!'OkPF07$F`bl$!'y=XF07$Febl$!' on]F07$Fjbl$!'!z#eF07$F_cl$!']oeF07$Fdcl$!'A&)fF07$Ficl$!'EmxF0-Fidm6& F`dl$\"*++++\"F]emF(F(-Fedl6#%6an~\"efficient\"~schemeG-F$6%7fpF'7$Fge m$!%b=F07$F\\fm$!%daF07$Fafm$!&v2\"F07$Fffm$!&Zx\"F07$F[gm$!&5h$F07$F+ $!&U'eF07$Fcgm$!&/(yF07$F2$!&b())F07$Fjhm$!&?l(F07$F7$!&B,$F07$Fbel$\" &i`&F07$F<$\"']*p\"F07$Fifl$\"'[aGF07$FA$\"'nyNF07$$\":nmmmmmTg#))\\5D F0$\"'*pl$F07$$\":nmmmmm;a`Ot`#F0$\"'RYOF07$$\":nmmmmm\"zWU\"F07$Fcu$!'w36F07$Fhu$!'aE5F07$F ]v$!&'Q'*F07$Fbv$!&E/*F07$Fgv$!&%\\&)F07$F\\w$!&#*4)F07$Faw$!&qt(F07$F fw$!&QR(F07$F[x$!&;3(F07$F`x$!&O%oF07$Fex$!&mh'F07$Fjx$!&7T'F07$F_y$!& yB'F07$Fdy$!&_4'F07$Fiy$!&A'fF07$F^z$!&W'eF07$Fcz$!&Ky&F07$Fhz$!&%HdF0 7$F][l$!&Gp&F07$Fb[l$!&;o&F07$Fg[l$!&?p&F07$F\\\\l$!&9t&F07$Fa\\l$!&A! eF07$Ff\\l$!&'4fF07$F[]l$!&?1'F07$F`]l$!&!oiF07$Fe]l$!&%=lF07$Fj]l$!&] \"pF07$F_^l$!&!HuF07$Fd^l$!&S(yF07$Fi^l$!&sb)F07$Fc_l$!&[4*F07$F]`l$!& M))*F07$Fb`l$!'O66F07$Fg`l$!'s:8F07$F\\al$!'=y;F07$Faal$!'YaBF07$Ffal$ !'AAHF07$F[bl$!'w'z$F07$F`bl$!'Q!o%F07$Febl$!')4K&F07$Fjbl$!'S,iF07$F_ cl$!'qWiF07$Fdcl$!'#)yjF07$Ficl$!'mH%)F0-F^dl6&F`dlF($\"#vFcdl$\"#&*Fc dl-Fedl6#%Ea~scheme~of~Tsitouras~and~PapakostasG-F$6%7hpF'7$Fgem$!%cF07$F[gm$!&9/%F07$F+$!&uq' F07$Fcgm$!&yC*F07$F2$!'L(4\"F07$F`hm$!'!o6\"F07$Fjhm$!'8r5F07$F_im$!&S Y*F07$F7$!&OK(F07$Fbel$!%WHF07$F<$\"&=`*F07$Fifl$\"'yj>F07$FA$\"'!*)f# F07$Fbao$\"'$*fEF07$Fgao$\"'A_EF07$F\\bo$\"'TVEF07$FF$\"'3$f#F07$Fdbo$ \"',.DF07$FK$\"'H(>#F07$FP$\"'rk:F07$FU$\"&^+)F07$FZ$!%`)*F07$Fin$!'Rw 5F07$F^o$!'Tu?F07$Fco$!'.QIF07$F^il$!'HARF07$Fho$!'F&p%F07$Ffil$!'tQ`F 07$F]p$!'+[eF07$Fgp$!'v!\\'F07$Faq$!'w?nF07$Ffq$!'&3J'F07$F[r$!'N3dF07 $F`r$!''4l%F07$Fer$!']mQF07$Fjr$!'iuKF07$F_s$!'s-GF07$Fds$!'suCF07$Fis $!'yy@F07$F_t$!'US>F07$Fdt$!''Hv\"F07$Fit$!'c5;F07$F^u$!'Wo9F07$Fcu$!' Ym8F07$Fhu$!'/l7F07$F]v$!''y=\"F07$Fbv$!'O96F07$Fgv$!'k`5F07$F\\w$!&7) **F07$Faw$!&]`*F07$Ffw$!&=6*F07$F[x$!&ms)F07$F`x$!&OV)F07$Fex$!&O:)F07 $Fjx$!&7!zF07$F_y$!&yo(F07$Fdy$!&A^(F07$Fiy$!&#[tF07$F^z$!&uA(F07$Fcz$ !&s7(F07$Fhz$!&/1(F07$F][l$!&e,(F07$Fb[l$!&;+(F07$Fg[l$!&],(F07$F\\\\l $!&M1(F07$Fa\\l$!&A:(F07$Ff\\l$!&YG(F07$F[]l$!&SZ(F07$F`]l$!&+t(F07$Fe ]l$!&W/)F07$Fj]l$!&qa)F07$F_^l$!&+@*F07$Fd^l$!&+z*F07$Fi^l$!'Kn5F07$Fc _l$!'QN6F07$F]`l$!'aL7F07$Fb`l$!'w$Q\"F07$Fg`l$!'iF;F07$F\\al$!'o]?F07 $Faal$!''G#GF07$Ffal$!'ihMF07$F[bl$!'ENWF07$F`bl$!'Q+aF07$Febl$!'=3hF0 7$Fjbl$!'gbqF07$F_cl$!'![5(F07$Fdcl$!'AcsF07$Ficl$!'Oc%*F0-Fidm6&F`dlF _^oF(F_^o-Fedl6#%Ra~modification~of~the~Tsitouras-Papakostas~schemeG-F $6%7`pF'7$Fffm$!%ZRF07$F+$!&3I\"F07$$\":0+++++]7.Trm*F-$!&tX\"F07$Fcgm $!&M^\"F07$$\":nmmmmmTg(*pj:\"F0$!&tS\"F07$F2$!&o1\"F07$Fjhm$\"%#>'F07 $F7$\"&]4%F07$F<$\"'BS;F07$FA$\"'%Rq#F07$FK$\"'F(Q#F07$FU$\"'m&Q\"F07$ FZ$\"&2V(F07$Fin$\"%8WF07$F^o$!&Xp'F07$Fco$!'Je8F07$Fho$!'PRDF07$F]p$! 'M^LF07$Fbp$!'D8OF07$Fgp$!'i)y$F07$$\":+++++vVtn@-,%F0$!'83RF07$$\":LL LLL$3xJ=$p.%F0$!'N%)QF07$$\":mmmmm\"z>')>kjSF0$!'#H'QF07$F\\q$!'6()QF0 7$$\":mmmmm;z%\\CxVTF0$!'q3RF07$Faq$!']V\"F07$Fds$!'ii7F07$ Fis$!')46\"F07$F_t$!&A*)*F07$Fdt$!&c$*)F07$Fit$!&'3#)F07$F^u$!&W[(F07$ Fcu$!&Y'pF07$Fhu$!&%[kF07$F]v$!&Y0'F07$Fbv$!&'zcF07$Fgv$!&/P&F07$F\\w$ !&s3&F07$Faw$!&+'[F07$Ffw$!&Qk%F07$F[x$!&wW%F07$F`x$!&')H%F07$Fex$!&c: %F07$Fjx$!&s-%F07$F_y$!&y\"RF07$Fdy$!&#HQF07$Fiy$!&_u$F07$F^z$!&Mo$F07 $Fcz$!&Aj$F07$Fhz$!&%*f$F07$F][l$!&ed$F07$Fb[l$!&'pNF07$Fg[l$!&]d$F07$ F\\\\l$!&/g$F07$Fa\\l$!&_k$F07$Ff\\l$!&Or$F07$F[]l$!&I\"QF07$F`]l$!&![ RF07$Fe]l$!&%=TF07$Fj]l$!&ST%F07$F_^l$!&q([F07$Fd^l$!&]S&F07$Fi^l$!&KJ 'F07$Fc_l$!&))3(F07$F]`l$!&aE)F07$Fb`l$!'c75F07$Fg`l$!'-?8F07$F\\al$!' o`=F07$Faal$!'wCGF07$Ffal$!'UBOF07$F[bl$!''Q$[F07$F`bl$!'G9gF07$Febl$! 'Q*)oF07$Fjbl$!']H!)F07$F_cl$!'!e3)F07$Fdcl$!'it#)F07$Ficl$!(O?4\"F0-F ^dl6&F`dlF^`p$\"#bFcdlF(-Fedl6#%AVerner's~\"most~efficient\"~schemeG-% %FONTG6$%*HELVETICAG\"\"*-%&TITLEG6#%Verror~curves~for~12~stage~order~ 8~Runge-Kutta~methodsG-%+AXESLABELSG6$Q\"x6\"Q!Fhcr-%%VIEWG6$;F(Ficl;$ !#=!#?$\"\"&Fadr" 1 2 0 1 10 0 2 9 1 4 2 1.000000 45.000000 45.000000 0 1 "Verner's Maple scheme" "a modification of Verner's Maple scheme" "an \"efficient\" scheme" "a scheme of Tsitouras and Papakostas" "a mo dification of the Tsitouras-Papakostas scheme" "Verner's \"most effici ent\" scheme" }}}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 48 "Test \+ 9 of 12 stage, order 8 Runge-Kutta methods " }}{PARA 256 "" 0 "" {TEXT -1 3 " " }{XPPEDIT 18 0 "dy/dx=-(1+cos(2*x))*y^3" "6#/*&%#dyG \"\"\"%#dxG!\"\",$*&,&F&F&-%$cosG6#*&\"\"#F&%\"xGF&F&F&*$%\"yG\"\"$F&F (" }{TEXT -1 3 ", " }{XPPEDIT 18 0 "y(0) = sqrt(2);" "6#/-%\"yG6#\"\" !-%%sqrtG6#\"\"#" }{TEXT -1 1 " " }}{PARA 0 "" 0 "" {TEXT -1 10 "Solut ion: " }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "y = 1/sqrt(s in(2*x)+2*x+1/2)" "6#/%\"yG*&\"\"\"F&-%%sqrtG6#,(-%$sinG6#*&\"\"#F&%\" xGF&F&*&F/F&F0F&F&*&F&F&F/!\"\"F&F3" }{TEXT -1 2 ". " }}{PARA 0 "" 0 " " {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 172 "de := dif f(y(x),x)=-(1+cos(2*x))*y(x)^3;\nic := y(0)=sqrt(2);\ndsolve(\{de,ic\} ,y(x));\nm := unapply(rhs(%),x):\nplot(m(x),x=0..3,0..1.42,font=[HELVE TICA,9],labels=[`x`,`y(x)`]);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%#de G/-%%diffG6$-%\"yG6#%\"xGF,,$*&,&\"\"\"F0-%$cosG6#,$*&\"\"#F0F,F0F0F0F 0)F)\"\"$F0!\"\"" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%#icG/-%\"yG6#\" \"!*$\"\"##\"\"\"F+" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/-%\"yG6#%\"xG* &\"\"\"F)*$,(*(\"\"#F)-%$cosGF&F)-%$sinGF&F)F)*&F-F)F'F)F)#F)F-F)#F)F- !\"\"" }}{PARA 13 "" 1 "" {GLPLOT2D 503 318 318 {PLOTDATA 2 "6&-%'CURV ESG6$7Y7$$\"\"!F)$\"3:&4tBc8UT\"!#<7$$\"3$*****\\ilyM;!#>$\"3ozW7@k#*H 8F,7$$\"3')*****\\7t&pKF0$\"3!G<)\\ef9f7F,7$$\"3z****\\(ofV!\\F0$\"3oe \"F,7$$\"3s******\\i9RlF0$\"3kESFh\"zh9\"F,7$$\"33++vVV)RQ*F0$ \"3'f)*)e-w\\p5F,7$$\"3/++vVA)GA\"!#=$\"3V)o6<$fq15F,7$$\"3;+]iSS\"Ga \"FJ$\"3IyW%eHk>[*FJ7$$\"3+++]Peui=FJ$\"3#4`!o2+#G**)FJ7$$\"37+++]$)z% =#FJ$\"3OGH4wwYu&)FJ7$$\"3A++]i3&o]#FJ$\"3=1g%=M2W@)FJ7$$\"3%)***\\(oX *y9$FJ$\"31u2v$Q9&GwFJ7$$\"3z***\\P9CAu$FJ$\"3=XIMTf7+sFJ7$$\"3!)***\\ P*zhdVFJ$\"3P$G(zQ8#4%oFJ7$$\"31++v$>fS*\\FJ$\"3X'3%RcqqPlFJ7$$\"3$)** *\\(=$f%GcFJ$\"3mYY%G?7\"*G'FJ7$$\"3Q+++Dy,\"G'FJ$\"3?u@-35zxgFJ7$$\"3 3++]7[X$ocbFJ7 $$\"3))***\\PpnsM*FJ$\"3!\\;$Q)fJR[&FJ7$$\"3,++]siL-5F,$\"3&3j/q(Qq8aF J7$$\"3-+++!R5'f5F,$\"3q`:6QhHm`FJ7$$\"3)***\\P/QBE6F,$\"3@Igj*yDKK&FJ 7$$\"3!******\\\"o?&=\"F,$\"3i/K.-M\\%H&FJ7$$\"31+]Pa&4*\\7F,$\"3OjcS# )ygr_FJ7$$\"33+]7j=_68F,$\"3'e4m\")R`oD&FJ7$$\"33++vVy!eP\"F,$\"3a@U-1 /NZ_FJ7$$\"34+](=WU[V\"F,$\"3Nrr*HO\"oU_FJ7$$\"3)****\\7B>&)\\\"F,$\"3 'HX%)zwR1C&FJ7$$\"3)***\\P>:mk:F,$\"3<^\"Q\"4\"y-C&FJ7$$\"3'***\\iv&QA i\"F,$\"3:*4?^OZ,C&FJ7$$\"31++vtLU%o\"F,$\"3\"3gSMou)Q_FJ7$$\"3!****** \\Nm'[F,$\"3[h+0^h(R>&FJ 7$$\"3z*****\\@80+#F,$\"3!zBIi>A%o^FJ7$$\"31++]7,Hl?F,$\"3<)30`]&>L^FJ 7$$\"3()**\\P4w)R7#F,$\"3!Qwx>a)*Q4&FJ7$$\"3;++]x%f\")=#F,$\"3q$pQbJ#) G/&FJ7$$\"3!)**\\P/-a[AF,$\"3gJla\"HTu)\\FJ7$$\"3/+](=Yb;J#F,$\"3c:[>; ?IA\\FJ7$$\"3')****\\i@OtBF,$\"3m09))4iC_[FJ7$$\"3')**\\PfL'zV#F,$\"3% Gjf])o8tZFJ7$$\"3>+++!*>=+DF,$\"3[G/4+_V#p%FJ7$$\"3-++DE&4Qc#F,$\"3!** R*=7x[1YFJ7$$\"3=+]P%>5pi#F,$\"3f7E:iH**=XFJ7$$\"39+++bJ*[o#F,$\"3cgVv c$ovV%FJ7$$\"33++Dr\"[8v#F,$\"3Ln\\jDQ5WVFJ7$$\"3++++Ijy5GF,$\"3OZ!)Q% zK7E%FJ7$$\"31+]P/)fT(GF,$\"3)*4_&egIW<%FJ7$$\"31+]i0j\"[$HF,$\"3qns]& )H\\$4%FJ7$$\"\"$F)$\"3ntdq;jW4SFJ-%'COLOURG6&%$RGBG$\"#5!\"\"F(F(-%%F ONTG6$%*HELVETICAG\"\"*-%+AXESLABELSG6$%\"xG%%y(x)G-%%VIEWG6$;F(Fc\\l; F($\"$U\"!\"#" 1 2 0 1 10 0 2 9 1 4 2 1.000000 45.000000 45.000000 0 0 "Curve 1" }}}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 32 "The following code constructs a " }{TEXT 260 17 "discrete solut ion" }{TEXT -1 44 " based on each of the methods and gives the " } {TEXT 260 22 "root mean square error" }{TEXT -1 18 " of each solution. " }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 808 "M := (x,y) -> -(1+cos(2 *x))*y^3: hh := 0.01: numsteps := 300: x0 := 0: y0 := sqrt(2):\nmatrix ([[`slope field: `,M(x,y)],[`initial point: `,``(x0,y0)],[`step widt h: `,hh],\n[`no. of steps: `,numsteps]]);``;\nmthds := [`Enright-V erner scheme`,`Verner's Maple scheme`,`a modification of Verner's Mapl e scheme`,`an \"efficient\" scheme`,`a scheme of Tsitouras and Papakos tas`,`a modification of the Tsitouras-Papakostas scheme`,`Verner's \"m ost efficient\" scheme`]: errs := []:\nDigits := 20:\nfor ct to 7 do\n Mn_RK8_||ct := RK8_||ct(M(x,y),x,y,x0,evalf(y0),hh,numsteps,false); \n sm := 0: numpts := nops(Mn_RK8_||ct):\n for ii to numpts do\n \+ sm := sm+(Mn_RK8_||ct[ii,2]-m(Mn_RK8_||ct[ii,1]))^2;\n end do:\n errs := [op(errs),sqrt(sm/numpts)];\nend do:\nDigits := 10:\nlinalg [transpose]([mthds,evalf(errs)]);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#K %'matrixG6#7&7$%0slope~field:~~~G,$*&,&\"\"\"F,-%$cosG6#,$*&\"\"#F,%\" xGF,F,F,F,)%\"yG\"\"$F,!\"\"7$%0initial~point:~G-%!G6$\"\"!*$F2#F,F27$ %/step~width:~~~G$F,!\"#7$%1no.~of~steps:~~~G\"$+$Q)pprint406\"" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#%!G" }}{PARA 11 "" 1 "" {XPPMATH 20 "6 #K%'matrixG6#7)7$%6Enright-Verner~schemeG$\"+MW'Qw'!#E7$%6Verner's~Map le~schemeG$\"+VEGkFF+7$%Ha~modification~of~Verner's~Maple~schemeG$\"+K N1uEF+7$%6an~\"efficient\"~schemeG$\"+MUbB;F+7$%Ea~scheme~of~Tsitouras ~and~PapakostasG$\"+7K`] " 0 "" {MPLTEXT 1 0 739 "M := (x,y) -> -(1+cos (2*x))*y^3: hh := 0.01: numsteps := 300: x0 := 0: y0 := sqrt(2):\nmatr ix([[`slope field: `,M(x,y)],[`initial point: `,``(x0,y0)],[`step wi dth: `,hh],\n[`no. of steps: `,numsteps]]);``;\nmthds := [`Enright -Verner scheme`,`Verner's Maple scheme`,`a modification of Verner's Ma ple scheme`,`an \"efficient\" scheme`,`a scheme of Tsitouras and Papak ostas`,`a modification of the Tsitouras-Papakostas scheme`,`Verner's \+ \"most efficient\" scheme`]: errs := []:\nDigits := 30:\nfor ct to 7 d o\n mn_RK8_||ct := RK8_||ct(M(x,y),x,y,x0,evalf(y0),hh,numsteps,true );\nend do:\nxx := 2.999: mxx := evalf(m(xx)):\nfor ct to 7 do\n err s := [op(errs),abs(mn_RK8_||ct(xx)-mxx)];\nend do:\nDigits := 10:\nlin alg[transpose]([mthds,evalf(errs)]);" }}{PARA 11 "" 1 "" {XPPMATH 20 " 6#K%'matrixG6#7&7$%0slope~field:~~~G,$*&,&\"\"\"F,-%$cosG6#,$*&\"\"#F, %\"xGF,F,F,F,)%\"yG\"\"$F,!\"\"7$%0initial~point:~G-%!G6$\"\"!*$F2#F,F 27$%/step~width:~~~G$F,!\"#7$%1no.~of~steps:~~~G\"$+$Q)pprint426\"" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#%!G" }}{PARA 11 "" 1 "" {XPPMATH 20 "6 #K%'matrixG6#7)7$%6Enright-Verner~schemeG$\"+kXi;7!#E7$%6Verner's~Mapl e~schemeG$\"+X[-^\\!#F7$%Ha~modification~of~Verner's~Maple~schemeG$\"+ g*3#)y%F07$%6an~\"efficient\"~schemeG$\"+-^d,HF07$%Ea~scheme~of~Tsitou ras~and~PapakostasG$\"+&*f:JJF07$%Ra~modification~of~the~Tsitouras-Pap akostas~schemeG$\"+4%*)*=PF07$%AVerner's~\"most~efficient\"~schemeG$\" +b!*e5hF0Q)pprint436\"" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 4 "The " }{TEXT 260 22 "root mean square error" }{TEXT -1 20 " over the interval " }{XPPEDIT 18 0 "[0, 3];" "6#7$\"\"!\"\"$ " }{TEXT -1 82 " of each Runge-Kutta method is estimated as follows u sing the special procedure " }{TEXT 0 5 "NCint" }{TEXT -1 98 " to pe rform numerical integration by the 7 point Newton-Cotes method over 15 0 equal subintervals." }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 476 "mt hds := [`Enright-Verner scheme`,`Verner's Maple scheme`,`a modificatio n of Verner's Maple scheme`,`an \"efficient\" scheme`,`a scheme of Tsi touras and Papakostas`,`a modification of the Tsitouras-Papakostas sch eme`,`Verner's \"most efficient\" scheme`]: errs := []:\nDigits := 20: \nfor ct to 7 do\n sm := NCint((m(x)-'mn_RK8_||ct'(x))^2,x=0..3,adap tive=false,numpoints=7,factor=150);\n errs := [op(errs),sqrt(sm/3)]; \nend do:\nDigits := 10:\nlinalg[transpose]([mthds,evalf(errs)]);" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#K%'matrixG6#7)7$%6Enright-Verner~schem eG$\"+iE@mm!#E7$%6Verner's~Maple~schemeG$\"+\")fCCFF+7$%Ha~modificatio n~of~Verner's~Maple~schemeG$\"+O+SNEF+7$%6an~\"efficient\"~schemeG$\"+ uC#**f\"F+7$%Ea~scheme~of~Tsitouras~and~PapakostasG$\"+:!*4D " 0 "" {MPLTEXT 1 0 609 "evalf[30](plot(['mn_RK8 _1'(x)-m(x),'mn_RK8_2'(x)-m(x),'mn_RK8_3'(x)-m(x),'mn_RK8_4'(x)-m(x), \n'mn_RK8_5'(x)-m(x),'mn_RK8_6'(x)-m(x),'mn_RK8_7'(x)-m(x)],x=0..3,0.. 3.3e-16,font=[HELVETICA,9],\ncolor=[COLOR(RGB,.4,0,.9),COLOR(RGB,0,.65 ,0),brown,red,COLOR(RGB,0,.75,.95),magenta,COLOR(RGB,.95,.55,0)],\nleg end=[`Enright-Verner scheme`,`Verner's Maple scheme`,`a modification o f Verner's Maple scheme`,`an \"efficient\" scheme`,`a scheme of Tsitou ras and Papakostas`,`a modification of the Tsitouras-Papakostas scheme `,`Verner's \"most efficient\" scheme`],title=`error curves for 12 sta ge order 8 Runge-Kutta methods`));" }}{PARA 13 "" 1 "" {GLPLOT2D 1044 629 629 {PLOTDATA 2 "6--%'CURVESG6%7do7$$\"\"!F)F(7$$\"?++++++++]7.K[V ?!#K$\")#)*\\I\"!#H7$$\"?+++++++++D1k'p3%F-$\"+mK*RV'F07$$\"?++++++++] P4'\\/8'F-$\"-t$*)=zX#F07$$\"?+++++++++]7G$R<)F-$\".@!3W#pK$F07$$\"?++ ++++++Dc,;u@5!#J$\"/4cad]#4#F07$$\"?++++++++](=#**3E7FC$\"/jO\"zef/#F0 7$$\"?++++++++v=U#Q/V\"FC$\"/5wTNg,?F07$$\"?+++++++++]ilyM;FC$\"/_,Bp* R(>F07$$\"?++++++++D\"G)[8R=FC$\"/`.xgR<@F07$$F,FC$\"/EMdGXmGF07$$\"?+ +++++++vVB:$yC#FC$\"/&=t=5r!GF07$$\"?+++++++++vV)z@X#FC$\"/\\$=vv,v#F0 7$$\"?++++++++]P%[w3'GFC$\"/*)\\6uAoFF07$$\"?++++++++++DJdpKFC$\"/2<3z &R-$F07$$\"?+++++++++](ofV!\\FC$\"/+(*pSL1HF07$$\"?++++++++++]i9RlFC$ \"/3vFNV$p#F07$$\"?++++++++](oHl:'zFC$\"/F07$$\"?++++++++v=U\"[GQ\"Fcq$\"0U&[CIVY< Fcq7$$\"?++++++++]iSS\"Ga\"Fcq$\"0\\wOrLQg\"Fcq7$$\"?++++++++D1R*zFq\" Fcq$\"0:[#RzUz9Fcq7$$\"?+++++++++]Peui=Fcq$\"0S.f>f+P\"Fcq7$$\"?++++++ ++++]$)z%=#Fcq$\"0/\"*HW!H)=\"Fcq7$$\"?+++++++++]i3&o]#Fcq$\"0(e6S52X5 Fcq7$$\"?++++++++]i:FPFGFcq$\"/F-f\\r0$*Fcq7$$\"?+++++++++voX*y9$Fcq$ \"/?W\"y88P)Fcq7$$\"?+++++++++vVTAUPFcq$\"/*4G=`*QqFcq7$$\"?+++++++++v $*zhdVFcq$\"/8G>$)>PgFcq7$$\"?+++++++++v$>fS*\\Fcq$\"/#)>Zs\\p_Fcq7$$ \"?+++++++++v=$f%GcFcq$\"/m8@O&4p%Fcq7$$\"?++++++++++Dy,\"G'Fcq$\"/hM5 7!QB%Fcq7$$\"?+++++++++]7Dq-\\NKFcq7$$\"?+++++++++v$pnsM*Fcq$\"/()46J15JFcq7 $$\"?+++++++++]siL-5F0$\"/#pRR.@*HFcq7$$\"?++++++++++!R5'f5F0$\"/E:#G% =9HFcq7$$\"?++++++++]P/QBE6F0$\"/^8*\\xX%GFcq7$$\"?++++++++++:o?&=\"F0 $\"/8!4Wk()z#Fcq7$$\"?++++++++]Pa&4*\\7F0$\"/yr\\!GEw#Fcq7$$\"?+++++++ +]7j=_68F0$\"/u6Oj\\RFFcq7$$\"?+++++++++vVy!eP\"F0$\"/2AcjmCFFcq7$$\"? ++++++++](=WU[V\"F0$\"/sdv'*R&)\\\"F0$\"/-:mk:F0$\"/koQVm8FFcq7$$\"?++++++++]iv&QAi\"F0$\" /W417Y8FFcq7$$\"?+++++++++vtLU%o\"F0$\"/x1[Y[6FFcq7$$\"?++++++++++bjm[ F0$\"/W,KsOUEFcq7$$\"?++++ ++++++:K^+?F0$\"/jAH&eNg#Fcq7$$\"?+++++++++]7,Hl?F0$\"/^Fc_o]DFcq7$$\" ?++++++++]P4w)R7#F0$\"/%[Gv_D\\#Fcq7$$\"?+++++++++]x%f\")=#F0$\"/$>ZE5 %=CFcq7$$\"?++++++++]P/-a[AF0$\"/b9%G?&RBFcq7$$\"?++++++++](=Yb;J#F0$ \"/g,RZ/\\AFcq7$$\"?+++++++++]i@OtBF0$\"/z8kpPa@Fcq7$$\"?++++++++]PfL' zV#F0$\"/--sBr]?Fcq7$$\"?++++++++++!*>=+DF0$\"/M9`MW[>Fcq7$$\"?+++++++ ++DE&4Qc#F0$\"/;*\\(zKV=Fcq7$$\"?++++++++]P%>5pi#F0$\"/%H(QYFSX\"f9Fcq7$$\"?++++++++]P/)fT(G F0$\"/DNH\"Fcq7$$\"\"$F)$\" /@#))zua@\"Fcq-%&COLORG6&%$RGBG$\"\"%!\"\"F($\"\"*F`bl-%'LEGENDG6#%6En right-Verner~schemeG-F$6%7doF'7$F+$\"(vhN%F07$F2$\"+)po_F#F07$F7$\",OE WQC*F07$F<$\".!*f>)GC8F07$FA$\".S3Vxvp)F07$FG$\".Ei)>5/&)F07$FL$\".wgI E%>$)F07$FQ$\".\\\\!)G\")>)F07$FV$\".oZG&>d()F07$Fen$\"/;aTf\"[=\"F07$ Fin$\"/u]vrGg6F07$F^o$\"/mJb)Hn8\"F07$Fco$\"/n\"H0b79\"F07$Fho$\"/$el, (yW7F07$F]p$\"/b\"*zLC\">\"F07$Fbp$\"/N\"HKA75\"F07$Fgp$\"/4my\"[F,\"F 07$F\\q$\".1rg3OD*F07$Faq$\".L(\\6Ln%)F07$Fgq$\".<(4X!3z(F07$F\\r$\"/u T(>wA6(Fcq7$Far$\"/2[7M**HlFcq7$Ffr$\"/knDg[AgFcq7$F[s$\"/+>R9%pd&Fcq7 $F`s$\"/5`=5ZO[Fcq7$Fes$\"/qM.WE`UFcq7$Fjs$\"/8os3;(y$Fcq7$F_t$\"/V3rb #oS$Fcq7$Fdt$\"/J(RDZX'GFcq7$Fit$\"/=\"zb`oX#Fcq7$F^u$\"/Ll'pBW9#Fcq7$ Fcu$\"/BXR\"z*3>Fcq7$Fhu$\"/59\\\"QHs\"Fcq7$F]v$\"/#yRPERf\"Fcq7$Fbv$ \"/=m@^Hy9Fcq7$Fgv$\"/()3X&)o'Q\"Fcq7$F\\w$\"/)Q.PtmJ\"Fcq7$Faw$\"/I5 \")3jl7Fcq7$Ffw$\"/kNCti<7Fcq7$F[x$\"/5&3F=f=\"Fcq7$F`x$\"/C4>>fd6Fcq7 $Fex$\"/zgx$[*Q6Fcq7$Fjx$\"/21RFCC6Fcq7$F_y$\"/^AT$H[6\"Fcq7$Fdy$\"/bg KVz36Fcq7$Fiy$\"/w%=@Pe5\"Fcq7$F^z$\"/!=]'ea/6Fcq7$Fcz$\"/'HAF07$F7$\",9fa`**)F07$F<$\".*GK;9&G\"F07$FA$\". \"[7v!HU)F07$FG$\".X(f?aN#)F07$FL$\".K)>sqc!)F07$FQ$\".\"yUGURzF07$FV$ \".T`x;8[)F07$Fen$\"/<,7V2Z6F07$Fin$\"/b!y*oKB6F07$F^o$\"/kWR/_+6F07$F co$\"/Gg^@*[5\"F07$Fho$\"/oGQ&op%H$Fcq7$Fdt$\"/ox9**GqFFcq7$Fit$\"/vCNh-wB Fcq7$F^u$\"/vY=Fcq7$Fhu$\"/7E]@Fm;Fcq7$F]v$\"/ wmIb]T:Fcq7$Fbv$\"/9(=oy'H9Fcq7$Fgv$\"/P/wb3T8Fcq7$F\\w$\"/ncvMPt7Fcq7 $Faw$\"/>;)*)4SA\"Fcq7$Ffw$\"/x5<_ex6Fcq7$F[x$\"/+7c!>p9\"Fcq7$F`x$\"/ 3K:V_>6Fcq7$Fex$\"/U5CR\\,6Fcq7$Fjx$\"/9F=>F(3\"Fcq7$F_y$\"/3i&4o\"y5F cq7$Fdy$\"/vP^:Ls5Fcq7$Fiy$\"/02p;Zp5Fcq7$F^z$\"/VP9&F07$FG$\". r!G\")HH]F07$FL$\"..#3r.?\\F07$FQ$\".)4s$Hu%[F07$FV$\".%*o4JG<&F07$Fen $\".8!*\\&>$*pF07$Fin$\".!**\\zT[oF07$F^o$\".ujpO$4nF07$$\"?++++++++D1 k\"Gll#FC$\".y__rnf'F07$Fco$\".a_id6t'F07$$\"?++++++++vo/[AlIFC$\".yNF V/[(F07$Fho$\".%3`9lNtF07$F]p$\".i7`v!3qF07$Fbp$\".*HUG`qkF07$Fgp$\".y `WA`%fF07$F\\q$\".P.91)HaF07$Faq$\".E.nMp'\\F07$Fgq$\".F8(>boXF07$F\\r $\"/LtXg/qTFcq7$Far$\"/$fRzQ!GQFcq7$Ffr$\"/(*ePG8INFcq7$F[s$\"/(3&yL%) oKFcq7$F`s$\"/!G@p.Y$GFcq7$Fes$\"/gZdlo#\\#Fcq7$Fjs$\"/\"oj4'[>AFcq7$F _t$\"/]MtTd'*>Fcq7$Fdt$\"/F`1^wy;Fcq7$Fit$\"/kVK%Q)R9Fcq7$F^u$\"/FDD?u c7Fcq7$Fcu$\"/?sM=w=6Fcq7$Fhu$\"/8gMOt45Fcq7$F]v$\".X7to7M*Fcq7$Fbv$\" .Rc3:Om)Fcq7$Fgv$\".2R'\\vE\")Fcq7$F\\w$\".ypeIkr(Fcq7$Faw$\".dVR&H3['Fcq7$F^z$\".6,2^KZ'Fcq7$Fcz$\".3l36>Z'Fcq7$Fhz$\" .&[inUrkFcq7$F][l$\".2fy7nY'Fcq7$Fb[l$\".C>-&3_kFcq7$Fg[l$\".NTQ4KU'Fc q7$F\\\\l$\".$\\6sQxjFcq7$Fa\\l$\".ur;o=I'Fcq7$Ff\\l$\".?%G-J4iFcq7$F[ ]l$\".klY4K3'Fcq7$F`]l$\"..lImX%fFcq7$Fe]l$\".s))>Txw&Fcq7$Fj]l$\".v'f XfzbFcq7$F_^l$\".)oz\"=QO&Fcq7$Fd^l$\".o%[S/Q^Fcq7$Fi^l$\".OYR83*[Fcq7 $F^_l$\".W/^6pk%Fcq7$Fc_l$\".x>+?iR%Fcq7$Fh_l$\".4N.Y/:%Fcq7$F]`l$\".& yzH6IRFcq7$Fb`l$\".p**)y(po$Fcq7$Fg`l$\".!=<@(*zMFcq7$F\\al$\".^Zd6;F$ Fcq7$Faal$\".O9by\\3$Fcq7$Ffal$\".qf'H$))*GFcq-F`^n6&F]bl$\"*++++\"Fd^ nF(F(-Fdbl6#%6an~\"efficient\"~schemeG-F$6%7doF'7$F+$\"(BU0$F07$F2$\"+ $>>Gd\"F07$F7$\",`?Fc@'F07$F<$\"-4'*>MC')F07$FA$\".Pg?:8^&F07$FG$\".c4 o<()Q&F07$FL$\".J!eWyr_F07$FQ$\".$[y[#p>&F07$FV$\".GRY!fhbF07$Fen$\".K >eDK^(F07$Fin$\".E0h#odtF07$F^o$\".kYge$3sF07$Fco$\".6$QV6VsF07$Fho$\" .]Zpb^*yF07$F]p$\".P&*)o`bvF07$Fbp$\".H)fN5!)pF07$Fgp$\".`AczgT'F07$F \\q$\".D'>7BgeF07$Faq$\".!3lf!4O&F07$Fgq$\".e`T$)3$\\F07$F\\r$\"/$4)\\ (42]%Fcq7$Far$\"/m.LoVJTFcq7$Ffr$\"/V]P!*y4QFcq7$F[s$\"/,]$4Zx_$Fcq7$F `s$\"/s*[h-!fIFcq7$Fes$\"/#>?l\\**o#Fcq7$Fjs$\"/?mfJ5&R#Fcq7$F_t$\"/^, %fSX:#Fcq7$Fdt$\"/!z(e%z:\"=Fcq7$Fit$\"/$e7j]Pb\"Fcq7$F^u$\"/id43g$*pFcq7$F^z$\".:X:Na)pFcq7$Fcz$\".i[U*)R)pFcq7$Fhz$\" .T5.nM)pFcq7$F][l$\".cwI!QypFcq7$Fb[l$\".1qN&fipFcq7$Fg[l$\".H@8N9$pFc q7$F\\\\l$\".&z)R()>)oFcq7$Fa\\l$\".\\T'Fcq7$Fe]l$\".gP)H5CiFcq7$Fj]l$\".n q;p5-'Fcq7$F_^l$\".-%*R>#)y&Fcq7$Fd^l$\".RT!4eWbFcq7$Fi^l$\".fpA)yx_Fc q7$F^_l$\".L&\\xe9]Fcq7$Fc_l$\".XWggSu%Fcq7$Fh_l$\".:1@S)yWFcq7$F]`l$ \".<$*zt5C%Fcq7$Fb`l$\".yeI,(yRFcq7$Fg`l$\".v*GoJbPFcq7$F\\al$\".iYTq/ `$Fcq7$Faal$\".Hvlq!HLFcq7$Ffal$\".y*4p>GJFcq-F[bl6&F]blF($\"#vFc`m$\" #&*Fc`m-Fdbl6#%Ea~scheme~of~Tsitouras~and~PapakostasG-F$6%7doF'7$F+$\" (XLW$F07$F2$\"+xG#yy\"F07$F7$\",P&*[*orF07$F<$\".(=e%z0,\"F07$FA$\".IZ IC+a'F07$FG$\".m[EVXR'F07$FL$\".#y!yHdD'F07$FQ$\".x&ygrlhF07$FV$\".OB) =i#f'F07$Fen$\".x!*3-G\"*)F07$Fin$\".q*4TGG()F07$F^o$\"._[c16b)F07$Fco $\".%zm#z()e)F07$Fho$\".AXLM\\O*F07$F]p$\".%p:=zh*)F07$Fbp$\".Moc/;G)F 07$Fgp$\".1v.]Sh(F07$F\\q$\".1G*3hbpF07$Faq$\".!\\Z[pjjF07$Fgq$\".'zbe $Fcq7$Fjs$\"/X8& R#pWGFcq7$F_t$\"/Vn!G***eDFcq7$Fdt$\"/jCajn^@Fcq7$Fit$\"/ylT&\\a%=Fcq7 $F^u$\"/d1yux5;Fcq7$Fcu$\"/-C#=HRV\"Fcq7$Fhu$\"/k@8y=%H\"Fcq7$F]v$\"/9 \\vCG(>\"Fcq7$Fbv$\"/R3!HF/6\"Fcq7$Fgv$\"/:2&[<;/\"Fcq7$F\\w$\".u67e-* )*Fcq7$Faw$\".=wF`o]*Fcq7$Ffw$\".N<[ui9*Fcq7$F[x$\".!34943*)Fcq7$F`x$ \".hKG<`p)Fcq7$Fex$\".?&fdFb&)Fcq7$Fjx$\".3w%Q\"[W)Fcq7$F_y$\".6Pr/TP) Fcq7$Fdy$\".N_Vs(G$)Fcq7$Fiy$\".o\")zflI)Fcq7$F^z$\".M!Q'foH)Fcq7$Fcz$ \".MsMU^H)Fcq7$Fhz$\".^yu@XH)Fcq7$F][l$\".At$*z%)G)Fcq7$Fb[l$\".:5[J(p #)Fcq7$Fg[l$\".(*eJ@FB)Fcq7$F\\\\l$\".7\\\\!*R<)Fcq7$Fa\\l$\".4Ju'>x!) Fcq7$Ff\\l$\".wp#RcezFcq7$F[]l$\".Z%3(Qpz(Fcq7$F`]l$\".AWhP#>wFcq7$Fe] l$\".]!G!)f#R(Fcq7$Fj]l$\".#fQtW^rFcq7$F_^l$\".Aw-$)[(oFcq7$Fd^l$\".(e aW]&e'Fcq7$Fi^l$\".%)[uC'oiFcq7$F^_l$\".fEe6g&fFcq7$Fc_l$\".>NI'pMcFcq 7$Fh_l$\".f%ySo>`Fcq7$F]`l$\".>:+!GP]Fcq7$Fb`l$\".nYR]cs%Fcq7$Fg`l$\". SLMG.Y%Fcq7$F\\al$\".]*=*pK>%Fcq7$Faal$\".-7(*fS&RFcq7$Ffal$\".(G$zuar $Fcq-F`^n6&F]blF^]oF(F^]o-Fdbl6#%Ra~modification~of~the~Tsitouras-Papa kostas~schemeG-F$6%7foF'7$F+$\"(h3P%F07$F2$\"+XC*=R#F07$F7$\"-`\"Hp<. \"F07$F<$\".#*4S!px:F07$FA$\"/?-e&\\))4\"F07$FG$\"//h!z0W2\"F07$FL$\"/ !f+HP50\"F07$FQ$\"/]0c)G[.\"F07$FV$\"/X\\FHn*4\"F07$Fen$\"/y#=kEg[\"F0 7$Fin$\"/ad()=Eb9F07$F^o$\"/D$3N)oD9F07$Fban$\"/6NCcL,9F07$Fco$\"/^UMO 8F9F07$Fjan$\"/c\\\"3BTe\"F07$Fho$\"/l([igMb\"F07$F]p$\"/pMerfz9F07$Fb p$\"/v??eAk8F07$Fgp$\"/lNtR__7F07$F\\q$\"/$4ybLO9\"F07$Faq$\"/],%\\=g/ \"F07$Fgq$\".?lQK0i*F07$F\\r$\"/e,^[E\"y)Fcq7$Far$\"/h#p?M61)Fcq7$Ffr$ \"/[5bm$RV(Fcq7$F[s$\"/,/@bw$)oFcq7$F`s$\"/kM-#e%pfFcq7$Fes$\"/k#>1$\\ \\_Fcq7$Fjs$\"/*Hz1tTn%Fcq7$F_t$\"/]+_xt/UFcq7$Fdt$\"/06c$Ra`$Fcq7$Fit $\"/zp!))eA.$Fcq7$F^u$\"/P$fycmk#Fcq7$Fcu$\"/Z$[2rgN#Fcq7$Fhu$\"/#)G'R fk7#Fcq7$F]v$\"/a29KBn>Fcq7$Fbv$\"/R**Q7_C=Fcq7$Fgv$\"/6,F.Y6F #pOY\"Fcq7$F`x$\"/Q@B(3(G9Fcq7$Fex$\"/T7z()p09Fcq7$Fjx$\"/q!=+\\vQ\"Fc q7$F_y$\"/9^=4$fP\"Fcq7$Fdy$\"/rTXC[o8Fcq7$Fiy$\"/+;-F$[O\"Fcq7$F^z$\" /$yn%)QKO\"Fcq7$Fcz$\"/#)>Hm&HO\"Fcq7$Fhz$\"/pE8Y&GO\"Fcq7$F][l$\"/*z! f='=O\"Fcq7$Fb[l$\"/&\\BK\"ye8Fcq7$Fg[l$\"/#GtF+FN\"Fcq7$F\\\\l$\"/(* \\%R]IM\"Fcq7$Fa\\l$\"/\"\\[_YrK\"Fcq7$Ff\\l$\"/!G`MawI\"Fcq7$F[]l$\"/ %*R(=)4\"G\"Fcq7$F`]l$\"/BVJ1!>D\"Fcq7$Fe]l$\"/VPw?m97Fcq7$Fj]l$\"/JK% >R]<\"Fcq7$F_^l$\"/G.LvfH6Fcq7$Fd^l$\"/UaR.0#3\"Fcq7$Fi^l$\"/u*yX%)*H5 Fcq7$F^_l$\".MKr&>'y*Fcq7$Fc_l$\".DBg[#e#*Fcq7$Fh_l$\".f9%zlS()Fcq7$F] `l$\".lb:XmF)Fcq7$Fb`l$\"..Fh7Yw(Fcq7$Fg`l$\".evIn'GtFcq7$F\\al$\".\"* p " 0 "" {MPLTEXT 1 0 1 ";" }}}}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 48 "Test \+ 10 of 12 stage, order 8 Runge-Kutta methods" }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "dy/dx = -(2*sin(5*x)+3*cos(7*x))*sinh(y );" "6#/*&%#dyG\"\"\"%#dxG!\"\",$*&,&*&\"\"#F&-%$sinG6#*&\"\"&F&%\"xGF &F&F&*&\"\"$F&-%$cosG6#*&\"\"(F&F3F&F&F&F&-%%sinhG6#%\"yGF&F(" }{TEXT -1 5 " , " }{XPPEDIT 18 0 "y(0)=sqrt(5)/2" "6#/-%\"yG6#\"\"!*&-%%sqr tG6#\"\"&\"\"\"\"\"#!\"\"" }{TEXT -1 1 " " }}{PARA 0 "" 0 "" {TEXT -1 10 "Solution: " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 207 "de := diff(y(x),x)=-(2*sin(5*x)+3*cos(7*x))*sin h(y(x));\nic := y(0)=sqrt(5)/2;\ndsolve(\{de,ic\},y(x));\nsimplify(con vert(%,exp));\np := unapply(rhs(%),x):\nplot(p(x),x=0..5,font=[HELVETI CA,9],labels=[`x`,`y(x)`]);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%#deG/ -%%diffG6$-%\"yG6#%\"xGF,,$*&,&*&\"\"#\"\"\"-%$sinG6#,$*&\"\"&F2F,F2F2 F2F2*&\"\"$F2-%$cosG6#,$*&\"\"(F2F,F2F2F2F2F2-%%sinhG6#F)F2!\"\"" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#>%#icG/-%\"yG6#\"\"!,$*&\"\"#!\"\"\"\" &#\"\"\"F,F0" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/-%\"yG6#%\"xG-%#lnG6# -%%tanhG6#,*#\"\"\"\"\"&F0*&#F0\"\"#F0-F)6#,$*&,&-%$expG6#,$*&F4!\"\"F 1F3F0F0F0F0F0,&F:F0F0F?F?F?F0F0*&#\"\"$\"#9F0-%$sinG6#,$*&\"\"(F0F'F0F 0F0F0*&#F0F1F0-%$cosG6#,$*&F1F0F'F0F0F0F?" }}{PARA 12 "" 1 "" {XPPMATH 20 "6#/-%\"yG6#%\"xG-%#lnG6#,$*&,*-%$expG6#,4#\"\"#\"\"&\"\" \"*&#\"$#>\"\"(F5*&)-%$cosGF&\"\"'F5-%$sinGF&F5F5F5*&#\"$S#F9F5*&)F<\" \"%F5F?F5F5!\"\"*&#\"#sF9F5*&)F$\"3`uw, WSt45F,7$$\"3WmmmT&)G\\aF0$\"3cSB%H69F5*!#=7$$\"3m****\\7G$R<)F0$\"3OF 6@7@G;#)F87$$\"3GLLL3x&)*3\"F8$\"3_]\"Hlv:eW(F87$$\"3))**\\i!R(*Rc\"F8 $\"3WS]N\"zi(yjF87$$\"3umm\"H2P\"Q?F8$\"3WN:-NK+KcF87$$\"3YLek.pu/BF8$ \"3:Vt%)yf.P`F87$$\"3!***\\PMnNrDF8$\"3!okU]7kD7&F87$$\"37$eR(\\;m/FF8 $\"3q$>&4'\\tL/&F87$$\"3MmT5ll'z$GF8$\"3;>k>%*z$=)\\F87$$\"37](o/[r7(H F8$\"3dP2*>TRs$\\F87$$\"3MLL$eRwX5$F8$\"3*H\">\\xg!*3\\F87$$\"3:L$3F% \\wQKF8$\"3wA2?_M<'*[F87$$\"3_LLe*[`HP$F8$\"37`)o(RAn)*[F87$$\"3*QLek. Ur]$F8$\"3K()))eR[\"e\"\\F87$$\"3rLLL$eI8k$F8$\"35_F/H[-Z\\F87$$\"3*QL $3xwq4RF8$\"3y)RBYN;$\\]F87$$\"33ML$3x%3yTF8$\"3;0fX,$G1?&F87$$\"3h+]P fyG7ZF8$\"3\"Hyxl/-Si&F87$$\"3emm\"z%4\\Y_F8$\"3;'Q!R(Q;g:'F87$$\"32++ v$flMLe*)>VB$)F8$\"3R-PW?nJGpF87$$\"3wmm Tg()4_))F8$\"3*[/**oz*pDlF87$$\"3Y++DJbw!Q*F8$\"3?xIZl^+ohF87$$\"3=nT& )3\\m_'*F8$\"3P'=[TB`#HgF87$$\"3+N$ekGkX#**F8$\"30<=#[laH$fF87$$\"3nTg _(R^g+\"F,$\"3EgYLpC*H!fF87$$\"31]iSmjk>5F,$\"3K[/nW4A')eF87$$\"3XekGN 8CL5F,$\"3u8jkI7P$)eF87$$\"3%ommTIOo/\"F,$\"3]\"*RMEK8&*eF87$$\"3cLe9; _yq5F,$\"3_[.xr\")Q`fF87$$\"3E+]7GTt%4\"F,$\"3_%yw2JYC1'F87$$\"3(p;/,/ $o=6F,$\"3[(3')fymbA'F87$$\"3YLL3_>jU6F,$\"3aRGM!H;eW'F87$$\"3ym;HdNb' >\"F,$\"33P8#3QfS;(F87$$\"37++]i^Z]7F,$\"3[rQ!Q)F87$$\"35+++v\"=Y I\"F,$\"3&441r_RP_*F87$$\"33++](=h(e8F,$\"3G(\\BrLl\"*4\"F,7$$\"3&**** *\\7!Q4T\"F,$\"39o3wy`oD7F,7$$\"3/++]P[6j9F,$\"3;)>\\NC[mH\"F,7$$\"3'= HKkAg\\Z\"F,$\"3icb1EJ%4I\"F,7$$\"3W$ek`h0o[\"F,$\"3Pbt7ruA+8F,7$$\"3/ voH/5l)\\\"F,$\"3Otx0vuR%H\"F,7$$\"3%o;HKR'\\5:F,$\"3Mz4l5FZ$G\"F,7$$ \"3-]P4rr=M:F,$\"31-%*Q\"yBqC\"F,7$$\"3UL$e*[z(yb\"F,$\"3o\")o!f;2L>\" F,7$$\"3w;/Ev&[ge\"F,$\"3#3#R!QDA@6\"F,7$$\"34+Dc,#>Uh\"F,$\"3O\"HI%*[ n*=5F,7$$\"3V$eky#)*QU;F,$\"3mms=Lk!y?*F87$$\"3wmm;a/cq;F,$\"3Pe)[!HQi L#)F87$$\"3\"pm;a)))G=F,$\"3U@p0?YWZMF87$$\"3KLe9;0?E>F,$\"34:XWhm;,LF87$$ \"3pTg-gl[Q>F,$\"3qsZpoe$*\\KF87$$\"31]i!Rgs2&>F,$\"3VKI#4&[)H@$F87$$ \"3WekyZ'eI'>F,$\"3=g(Q?ez,>$F87$$\"3gmmm\"pW`(>F,$\"3q(HR/(>[\"=$F87$ $\"3_ek.HW#)))>F,$\"3SdFyRyC)=$F87$$\"3?]iSmTI-?F,$\"3]%GeCR0B@$F87$$ \"3*=/wP!Ry:?F,$\"3/I\"o!ej/aKF87$$\"3dLe9TOEH?F,$\"3u\"yA?()GSJ$F87$$ \"3'pT&)e6Bi0#F,$\"3#fXkrX5=\\$F87$$\"3K+]i!f#=$3#F,$\"3?\\niL?f`PF87$ $\"3/++D\"=EX8#F,$\"3)=7zi$ePDXF87$$\"3?+](=xpe=#F,$\"3#GkoxRDBu&F87$$ \"3$pTNrfbE@#F,$\"3I&**Rb6vOf'F87$$\"3mLeRA9WRAF,$\"3N6cI=h\\:wF87$$\" 3S]ilZsAmAF,$\"3D)zb%4IS@))F87$$\"37nm\"H28IH#F,$\"3v'yDR:j=-\"F,7$$\" 3!oTN@#3hF,7 $$\"3WeRseStdCF,$\"3,,$en#[n??F,7$$\"3)oT5l0+5Z#F,$\"3%)[_;#*)R!\\?F,7 $$\"35YOSbIjxCF,$\"3i%>URy8d0#F,7$$\"3IvoHagE%[#F,$\"3K3\\y.l?d?F,7$$ \"3_/,>`!**3\\#F,$\"3+HUb4&*\\`?F,7$$\"3GLL3_?`(\\#F,$\"3'*zGRRomW?F,7 $$\"3.$3-)Q84DDF,$\"3/_^lOB%z&>F,7$$\"3AL3_D1l_DF,$\"3VH*H8v%))4=F,7$$ \"3I3-))o-VmDF,$\"3o]!f;TM@s\"F,7$$\"3S$eRA\"*4-e#F,$\"3lf!)pZ(f(H;F,7 $$\"3[e*)fb&*)Rf#F,$\"3-n>0Ar[N:F,7$$\"3fL$e*)>pxg#F,$\"3T[_^!fj9W\"F, 7$$\"3V+D1R'f:&H/%HG\"F,7$$\"3%omm\"z+vbEF,$\"3Q#=@:,\\a8 \"F,7$$\"3AL3F>0uzEF,$\"3C!3Cbi?=+\"F,7$$\"33+]Pf4t.FF,$\"3(3T(3()HVJ) )F87$$\"3Q$3F>HT'HFF,$\"3y\"y5tq@xr(F87$$\"3om\"zWi^bv#F,$\"3?:fooRnqn F87$$\"3)*\\7.d>Y\"y#F,$\"3!QE&f3vnxfF87$$\"3uLLe*Gst!GF,$\"3eVA%fwpRK &F87$$\"3)om\"H2\"34'GF,$\"3Yo]3R$*[ONF87$$\"3C+voaa+$*HF,$\"3_,\"*[foPON F87$$\"3>](oH/*41IF,$\"33E#)e$)41_NF87$$\"3:++DJE>>IF,$\"3!z;8D.rLe$F8 7$$\"3A+v$4^n)pIF,$\"3q;[s.$y8&QF87$$\"3F+]i!RU07$F,$\"3%Q(yiwjedVF87$ $\"39+vo/#3o<$F,$\"3.u`zNU*o?&F87$$\"3+++v=S2LKF,$\"3rC&z7e?,M'F87$$\" 3;L$3_NJOG$F,$\"3'oVq=V<'GvF87$$\"3Jmmm\"p)=MLF,$\"3**GlY8#))ft)F87$$ \"3GLLeR%p\")Q$F,$\"3.cJG%fz/\"F,7$$\"3C$ekyZ2mY$F,$\"3%zJ4cdg.0\" F,7$$\"3=vo/Bh$)yMF,$\"3i,C<-Qm\\5F,7$$\"3mm\"H#oZ1\"\\$F,$\"3;\"3uAU \"*f/\"F,7$$\"36]Pfe?_:NF,$\"3t?:\"y&yYI5F,7$$\"35L$e*[$z*RNF,$\"3AZcw FIo05F,7$$\"3%o;Hd!fX$f$F,$\"3\\R'zK#G91$*F87$$\"3e++]iC$pk$F,$\"3[AIx (pVQY)F87$$\"3ILe*[t\\sp$F,$\"3C%z!\\!\\E.v(F87$$\"3[m;H2qcZPF,$\"3Co* ep\"3v-sF87$$\"3s***\\7.lQx$F,$\"3_n\"\\:&*zo*pF87$$\"3UL$3_0j,!QF,$\" 3WVvs*pf#\\oF87$$\"3F+v=n?J8QF,$\"3KU^fg!Quz'F87$$\"36nm;z5YEQF,$\"3-) RYLJ)=gnF87$$\"3^Le9\"45'RQF,$\"39oisVpNPnF87$$\"3O+]7.\"fF&QF,$\"3VEh 1S\\sGnF87$$\"3i3_vlYhlQF,$\"33cF9)GTPt'F87$$\"3)oT&QG-ZyQF,$\"3kS=Q3G t^nF87$$\"39Dc,\"zD8*QF,$\"3!zI/$G#3By'F87$$\"3TLek`8=/RF,$\"3IQP7dc+D oF87$$\"3$*\\i!*yC*)HRF,$\"3$4'G3V'=X%pF87$$\"3Ymm;/OgbRF,$\"33DC(y#>% [5(F87$$\"3*G$e*[$zV4SF,$\"3O_/NuwfQvF87$$\"3w**\\ilAFjSF,$\"3D8xFXpB@ !)F87$$\"3#G3_]p'>*3%F,$\"3qve-u54K#)F87$$\"3ym\"zW7@^6%F,$\"31QL&Rp%o 0%)F87$$\"3w3F>RL3GTF,$\"3!)Q$yF^SJZ)F87$$\"3t]i!RbX59%F,$\"3R'pQA'3/D &)F87$$\"3#=z>'ox+aTF,$\"3M&4T=0@'f&)F87$$\"3yLLL$)*pp;%F,$\"3E>(p'zRP v&)F87$$\"3!Q3_+sD-=%F,$\"3Ga;6C(p2d)F87$$\"3#Q$3xc9[$>%F,$\"3MZZFn2MW &)F87$$\"3'Qe*[$>Pn?%F,$\"31`G!HhJc\\)F87$$\"3)QL3-$H**>UF,$\"3[:#3'R6 kC%)F87$$\"3#R$ek.W]YUF,$\"3+Xc4Am3=#)F87$$\"3)RL$3xe,tUF,$\"3s/#zi!3V LzF87$$\"3Cn;HdO=yVF,$\"3E$>X)HduejF87$$\"3a+++D>#[Z%F,$\"3gxAO;Hx\") \\F87$$\"3TM$3_5,-`%F,$\"3%GSzCT^zW%F87$$\"3SnmT&G!e&e%F,$\"3=g#R;G9D: %F87$$\"3/]i:NK'zf%F,$\"3*f`:i,h67%F87$$\"3fLe*[=Y.h%F,$\"33)>WD]()H5% F87$$\"386but%F,$\"3_@+X\\xqwZF87$$\"37+]iSjE!z%F,$\"3<\"z**[W(=J cF87$$\"3y*\\7G))Rb\"[F,$\"3OXJ\"GK]h>'F87$$\"3L+++DM\"3%[F,$\"3!**fZR -9n(oF87$$\"3)3](=np3m[F,$\"3QTt^2I*Ho(F87$$\"3a+]P40O\"*[F,$\"3k:x`tb ,B')F87$$\"3>]7.#Q?&=\\F,$\"3'4(*[g6ply*F87$$\"3s+voa-oX\\F,$\"3?(Q=b( [U56F,7$$\"3O]PMF,%G(\\F,$\"3_;`pzy]b7F,7$$\"\"&F)$\"3.ug')yo>49F,-%'C OLOURG6&%$RGBG$\"#5!\"\"F(F(-%%FONTG6$%*HELVETICAG\"\"*-%+AXESLABELSG6 $%\"xG%%y(x)G-%%VIEWG6$;F(Ficn%(DEFAULTG" 1 2 0 1 10 0 2 9 1 4 2 1.000000 45.000000 45.000000 0 0 "Curve 1" }}}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 93 "The following code constructs a discrete solution based on each of the methods and gives the " } {TEXT 260 22 "root mean square error" }{TEXT -1 18 " of each solution. " }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 825 "P := (x,y) -> -(2*sin(5 *x)+3*cos(7*x))*sinh(y): hh := 0.01: numsteps := 500: x0 := 0: y0 := s qrt(5)/2:\nmatrix([[`slope field: `,P(x,y)],[`initial point: `,``(x0 ,y0)],[`step width: `,hh],\n[`no. of steps: `,numsteps]]);``;\nmth ds := [`Enright-Verner scheme`,`Verner's Maple scheme`,`a modification of Verner's Maple scheme`,`an \"efficient\" scheme`,`a scheme of Tsit ouras and Papakostas`,`a modification of the Tsitouras-Papakostas sche me`,`Verner's \"most efficient\" scheme`]: errs := []:\nDigits := 30: \nfor ct to 7 do\n Pn_RK8_||ct := RK8_||ct(P(x,y),x,y,x0,evalf(y0),h h,numsteps,false);\n sm := 0: numpts := nops(Pn_RK8_||ct):\n for i i to numpts do\n sm := sm+(Pn_RK8_||ct[ii,2]-p(Pn_RK8_||ct[ii,1]) )^2;\n end do:\n errs := [op(errs),sqrt(sm/numpts)];\nend do:\nDig its := 10:\nlinalg[transpose]([mthds,evalf(errs)]);" }}{PARA 11 "" 1 " " {XPPMATH 20 "6#K%'matrixG6#7&7$%0slope~field:~~~G,$*&,&*&\"\"#\"\"\" -%$sinG6#,$*&\"\"&F.%\"xGF.F.F.F.*&\"\"$F.-%$cosG6#,$*&\"\"(F.F5F.F.F. F.F.-%%sinhG6#%\"yGF.!\"\"7$%0initial~point:~G-%!G6$\"\"!,$*&F-FBF4#F. F-F.7$%/step~width:~~~G$F.!\"#7$%1no.~of~steps:~~~G\"$+&Q)pprint456\" " }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%!G" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#K%'matrixG6#7)7$%6Enright-Verner~schemeG$\"+tiGQ!*!#>7$%6Verner' s~Maple~schemeGF)7$%Ha~modification~of~Verner's~Maple~schemeGF)7$%6an~ \"efficient\"~schemeGF)7$%Ea~scheme~of~Tsitouras~and~PapakostasGF)7$%R a~modification~of~the~Tsitouras-Papakostas~schemeGF)7$%AVerner's~\"mos t~efficient\"~schemeGF)Q)pprint466\"" }}}{PARA 0 "" 0 "" {TEXT -1 0 " " }}{PARA 0 "" 0 "" {TEXT -1 30 "The following code constructs " } {TEXT 260 20 "numerical procedures" }{TEXT -1 56 " for solutions based on each of the Runge-Kutta schemes." }}{PARA 0 "" 0 "" {TEXT -1 75 "T he error in the value obtained by each of the methods at the point whe re " }{XPPEDIT 18 0 "x = 4.999;" "6#/%\"xG-%&FloatG6$\"%**\\!\"$" } {TEXT -1 16 " is also given." }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 756 "P := (x,y) -> -(2*sin(5*x)+3*cos(7*x))*sinh(y): hh := 0.01: num steps := 500: x0 := 0: y0 := sqrt(5)/2:\nmatrix([[`slope field: `,P( x,y)],[`initial point: `,``(x0,y0)],[`step width: `,hh],\n[`no. of s teps: `,numsteps]]);``;\nmthds := [`Enright-Verner scheme`,`Verner's Maple scheme`,`a modification of Verner's Maple scheme`,`an \"efficie nt\" scheme`,`a scheme of Tsitouras and Papakostas`,`a modification of the Tsitouras-Papakostas scheme`,`Verner's \"most efficient\" scheme` ]: errs := []:\nDigits := 30:\nfor ct to 7 do\n pn_RK8_||ct := RK8_| |ct(P(x,y),x,y,x0,evalf(y0),hh,numsteps,true);\nend do:\nxx := 4.999: \+ pxx := evalf(p(xx)):\nfor ct to 7 do\n errs := [op(errs),abs(pn_RK8_ ||ct(xx)-pxx)];\nend do:\nDigits := 10:\nlinalg[transpose]([mthds,eval f(errs)]);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#K%'matrixG6#7&7$%0slope~ field:~~~G,$*&,&*&\"\"#\"\"\"-%$sinG6#,$*&\"\"&F.%\"xGF.F.F.F.*&\"\"$F .-%$cosG6#,$*&\"\"(F.F5F.F.F.F.F.-%%sinhG6#%\"yGF.!\"\"7$%0initial~poi nt:~G-%!G6$\"\"!,$*&F-FBF4#F.F-F.7$%/step~width:~~~G$F.!\"#7$%1no.~of~ steps:~~~G\"$+&Q)pprint486\"" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%!G" } }{PARA 11 "" 1 "" {XPPMATH 20 "6#K%'matrixG6#7)7$%6Enright-Verner~sche meG$\"+w2p4R!#D7$%6Verner's~Maple~schemeG$\"+_(HW$=!#F7$%Ha~modificati on~of~Verner's~Maple~schemeG$\"+fy6I')F07$%6an~\"efficient\"~schemeG$ \"+S:b2?!#E7$%Ea~scheme~of~Tsitouras~and~PapakostasG$\"+Gq4xmF07$%Ra~m odification~of~the~Tsitouras-Papakostas~schemeG$\"+&32Bp(F07$%AVerner' s~\"most~efficient\"~schemeG$\"+(z+^6'F9Q)pprint496\"" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 4 "The " }{TEXT 260 22 " root mean square error" }{TEXT -1 20 " over the interval " }{XPPEDIT 18 0 "[0, 5];" "6#7$\"\"!\"\"&" }{TEXT -1 82 " of each Runge-Kutta me thod is estimated as follows using the special procedure " }{TEXT 0 5 "NCint" }{TEXT -1 98 " to perform numerical integration by the 7 po int Newton-Cotes method over 200 equal subintervals." }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 476 "mthds := [`Enright-Verner scheme`,`Verne r's Maple scheme`,`a modification of Verner's Maple scheme`,`an \"effi cient\" scheme`,`a scheme of Tsitouras and Papakostas`,`a modification of the Tsitouras-Papakostas scheme`,`Verner's \"most efficient\" sche me`]: errs := []:\nDigits := 20:\nfor ct to 7 do\n sm := NCint((p(x) -'pn_RK8_||ct'(x))^2,x=0..5,adaptive=false,numpoints=7,factor=200);\n \+ errs := [op(errs),sqrt(sm/5)];\nend do:\nDigits := 10:\nlinalg[trans pose]([mthds,evalf(errs)]);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#K%'matr ixG6#7)7$%6Enright-Verner~schemeG$\"+K]^RT!#D7$%6Verner's~Maple~scheme G$\"+(zHbJ\"F+7$%Ha~modification~of~Verner's~Maple~schemeG$\"+_Xy\"=\" F+7$%6an~\"efficient\"~schemeG$\"+xb2#>\"F+7$%Ea~scheme~of~Tsitouras~a nd~PapakostasG$\"+eh:,#)!#E7$%Ra~modification~of~the~Tsitouras-Papakos tas~schemeG$\"+WxJz5F+7$%AVerner's~\"most~efficient\"~schemeG$\"+Y%z<; \"F+Q)pprint506\"" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 92 "The following error graphs are constructed using the nume rical procedures for the solutions." }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 618 "evalf[20](plot(['pn_RK8_1'(x)-p(x),'pn_RK8_2'(x)-p(x ),'pn_RK8_3'(x)-p(x),'pn_RK8_4'(x)-p(x),\n'pn_RK8_5'(x)-p(x),'pn_RK8_6 '(x)-p(x),'pn_RK8_7'(x)-p(x)],x=0..2.2,-2.5e-17..8.2e-17,font=[HELVETI CA,9],\ncolor=[COLOR(RGB,.4,0,.9),COLOR(RGB,0,.65,0),brown,red,COLOR(R GB,0,.75,.95),magenta,COLOR(RGB,.95,.55,0)],\nlegend=[`Enright-Verner \+ scheme`,`Verner's Maple scheme`,`a modification of Verner's Maple sche me`,`an \"efficient\" scheme`,`a scheme of Tsitouras and Papakostas`,` a modification of the Tsitouras-Papakostas scheme`,`Verner's \"most ef ficient\" scheme`],title=`error curves for 12 stage order 8 Runge-Kutt a methods`));" }}{PARA 13 "" 1 "" {GLPLOT2D 997 652 652 {PLOTDATA 2 "6 --%'CURVESG6%7bs7$$\"\"!F)F(7$$\"5mmmm;zM%))>\"!#@$\"\"*!#>7$$\"5LLLLL epo(R#F-$\"#=F07$$\"5++++]P/`'f$F-$\"$E#!#?7$$\"5mmmmm;RP&z%F-$\"$f#F; 7$$\"5NL$3F>u]h0&F-$\"$$HF;7$$\"5+++v=nv#pJ&F-$\"$&HF;7$$\"5qm;zW#R/xd &F-FD7$$\"5NLL$3x@\"[QeF-$\"$1$F;7$$\"5qmm\"H#o[.gjF-$\"$*HF;7$$\"5+++ +v=&)e\")oF-FQ7$$\"5lmm;z>epCzF-$\"$=$F;7$$\"5LLLL$37.y'*)F-$\"$6$F;7$ $\"5mmmTgP')395F;$\"$A$F;7$$\"5+++]7jpRJ6F;$\"$C$F;7$$\"5MLLek)G0([7F; FD7$$\"5nmmm;9O,m8F;$\"$q#F;7$$\"5nmmm;*Hd$Q=F;$\"$u#F;7$$\"5LLLL3$F;$\"$T#F;7$$\"5 mmmmTv+JiOF;$\"$r#F;7$$\"5++++vLo`FTF;$\"$v#F;7$$\"5LLLLLQ(zgg%F;$\"$) GF;7$$\"5mmmm;*e!eF]F;$\"$;$F;7$$\"5NLLL3_c$[E&F;$\"$G$F;7$$\"5+++++:2 4-bF;$\"$i$F;7$$\"5+++++q*>.u&F;$\"$@%F;7$$\"5+++++D#\\&yfF;$\"$\"\\F; 7$$\"5+++++&G0xV'F;$\"$N'F;7$$\"5qmmTg2Z%>a'F;$\"$o'F;7$$\"5NLL$3-8%=Y mF;$\"$#pF;7$$\"5+++D\"GbB/v'F;$\"$.(F;7$$\"5mmmmTvHmaoF;$\"$<(F;7$$\" 5+++](o[kD5(F;$\"$0(F;7$$\"5LLLLL)*fY]tF;$\"$!pF;7$$\"5NLLL$e4r/c(F;$ \"$U'F;7$$\"5LLLLL$>w/x(F;$\"$%eF;7$$\"5*******\\()*y/f#)F;$\"$#[F;7$$ \"5LLLLLVm^\"p)F;$\"$K%F;7$$\"5*******\\()R.g;*F;$\"$$QF;7$$\"5******* \\i*p#yh*F;Fas7$$\"5LLLL3_d#*35F0$\"$Y$F;7$$\"5LLL$32z,q9F0$\"$5(F07$$\" 5LLek.Ok\\r9F0$\"$6(F07$$\"5mmmTg14)HZ\"F0Facl7$$\"5++v=:F0$\"$,(F07$$\"5nm;/E!H2`_\"F0$F_vF07$$\"5M$3_ +Af(*z_\"F0$\"$w'F07$$\"5++D19%*yoI:F0$\"$1'F07$$\"5n;H23'>yL`\"F0$\"$ -'F07$$\"5MLL3-)\\og`\"F0$\"$'fF07$$\"5+]P4'**ze(Q:F0$\"$p&F07$$\"5nmT 5!>5\\9a\"F0$\"$0&F07$$\"5M$e9TQSRTa\"F0$\"$-&F07$$\"5++]7y0(Hoa\"F0$ \"$'\\F07$$\"5n;a8s2+_\\:F0$\"$T%F07$$\"5MLe9m4.@_:F0$\"$5%F07$$\"5+]i :g61!\\b\"F0$\"$1%F07$$\"5nmm;a84fd:F0$\"$(RF07$$\"5MLekyMLZj:F0$\"$B$ F07$$\"5++]7.cdNp:F0$\"$&GF07$$\"5nmTgFx\"Q_d\"F0$\"$_#F07$$\"5MLL3_)f ?6e\"F0$\"$.#F07$$\"5nm;/,Ta)Gf\"F0$\"$g\"F07$$\"5++++]$G]Yg\"F0$\"$J \"F07$$\"5mmmTN$))*yE;F0$\"%/5F;7$$\"5LLL$3K[H*[;F0$\"$z)F;7$$\"5mmm;H W<2s;F0$\"$^)F;7$$\"5+++]P0S@&p\"F0$\"$5)F;7$$\"5LLLL$eel/u\"F0$\"$`(F ;7$$\"5+++](ozRyy\"F0$\"$W'F;7$$\"5nmmmm#zmM$=F0$\"$!eF;7$$\"5nmmm\"f) p7!)=F0$\"$W&F;7$$\"5LLL$3#43SE>F0$\"$H&F;7$$\"5+++++Z;#*o>F0$\"$D&F;7 $$\"5LLLLeD`l_@F0$\"$6'F;7$$\"5+++D17$*4w@F0$\"$I(F;7$$\"#A!\"\"$\"$^*F;-%&COL ORG6&%$RGBG$\"\"%FbcmF($F/Fbcm-%'LEGENDG6#%6Enright-Verner~schemeG-F$6 %7i^lF'7$F=$!$$=F;7$FY$!$u#F;7$F[o$!$j$F;7$$\"5&****\\P%[ZMa&*F-$!$m$F ;7$F`o$!$u$F;7$$\"5L$3F%Hy@vG5F;$!$o$F;7$$\"5++vV)*=dTV5F;Fjdm7$$\"5m; zWnf#z!e5F;$!$.%F;7$$\"5LL$ek.!Gus5F;$!$$QF;7$$\"5mm\"zW<))p?5\"F;$!$! QF;7$Feo$!$t$F;7$$\"5n;/^\"Q]gg9\"F;$!$2%F;7$$\"5ML3_]WSsg6F;$!$i$F;7$ $\"5+]7`>&e(Qv6F;$!$d$F;7$$\"5nm;a)e7^+>\"F;Fbem7$$\"5M$3_vlm9Z?\"F;$! $:%F;7$$\"5++DcE2#y$>7F;$!$$RF;7$$\"5n;Hd&zuTSB\"F;Fdfm7$Fjo$!$'RF;7$$ \"5+]PfLH)oLE\"F;$!$\"RF;7$$\"5nmTg-qB.y7F;$!$v$F;7$$\"5M$e9;2\"fp#H\" F;$!$#RF;7$$\"5++]iS^%ftI\"F;$!$=%F;7$$\"5n;aj4#*H-A8F;$!$*QF;7$$\"5ML ekyKloO8F;$!$-%F;7$$\"5+]ilZt+N^8F;$!$!RF;7$F]pFagm7$$\"5nmmmmca=-;F;$ !$l$F;7$Fbp$!$9$F;7$$\"5+++]7emSt?F;$!$(GF;7$Fgp$!$n#F;7$$\"5+++]7Gi$H O#F;$!$l#F;7$$\"5mmmm;RkTqBnDF;Fc^n7$$\"5mmm;Hsq&3 e#F;F\\]n7$$\"5LL$3_]7xWf#F;Fc^n7$$\"5+++D\"y<(43EF;$!$b#F;7$$\"5mm;Hd Isr@EF;$!$M#F;7$$\"5LLLLL$GP`j#F;$!$P#F;7$$\"5++]P4Ot&*[EF;Fc^n7$$\"5m mmT&))QxDm#F;$!$]#F;7$$\"5LL$e9;W(>wEF;$!$T#F;7$$\"5+++]P%\\<)*o#F;$!$ F#F;7$$\"5mm;a8ZvV.FF;Fe`n7$$\"5LLLe*)*fdqr#F;$!$a#F;7$$\"5++]il_wnIFF ;$!$N#F;7$F\\q$!$W#F;7$$\"5LLLe9\"[+%eFF;$!$K#F;7$$\"5+++](oD.Dx#F;$!$ G#F;7$$\"5mmmTgKgg'y#F;$!$\\#F;7$$\"5LLLLL3)32!GF;F\\cn7$$\"5mmm;zfV\" *GGF;Fj`n7$$\"5++++D6*>r&GF;Ffcn7$$\"5mmmm;95`8HF;$!$Q#F;7$$\"5LLLL3<@ %*pHF;Fibn7$$\"5mmm;aow9)*HF;$!$E#F;7$$\"5+++++?KNEIF;$!$H#F;7$$\"5LLL $e9xeX0$F;Fh^n7$$\"5mmmm\"HKkF3$F;F\\cn7$$\"5LLLek)4no4$F;$!$U#F;7$$\" 5+++]Pu)p46$F;$!$[#F;7$$\"5mmmT5]E2DJF;F[^n7$$\"5LLLL$eUv\"RJF;$!$C#F; 7$$\"5+++Dc,#yK:$F;Ffen7$$\"5mmm;Hx4QnJF;Fc^n7$$\"5LLL3-`P[\"=$F;F^gn7 $FaqF]_n7$$\"5LLLL3-$[*GMF;F]_n7$FfqFban7$$\"5+++D\"yu'Q\"p$F;$!$c#F;7 $$\"5LLL$3-Uj/s$F;$!$i#F;7$$\"5++]iSc<+NPF;F\\]n7$$\"5mmmTg#4S&\\PF;$! $y#F;7$$\"5LL$3-)G%ySw$F;Fg\\n7$$\"5+++++lnhyPF;Fdbn7$$\"5mm;z>,^:$z$F ;Fban7$$\"5LLLeRPMp2QF;$!$s#F;7$$\"5++]Pfto:qoF;$!$b%F;7$$\"5NLe*[Vm]c)oF;$!$-&F;7$$\"5q;/^\")3X9,pF;$!$ s%F;7$$\"5++]7G`$Qm\"pF;$!$r%F;7$$\"5N$eRZx>K@$pF;$!$(\\F;7$$\"5qmTN@U giZpF;$!$)[F;7$$\"5+](ozm))>J'pF;$!$j%F;7$$\"5NLLe9JPhypF;Fgbo7$$\"5q; z>hvv5%*pF;$!$&[F;7$$\"5++D\"y+U,'4qF;$!$w%F;7$$\"5N$3FWXE&4DqF;$!$5&F ;7$$\"5qm;/,4\"*eSqF;$!$3&F;7$$\"5+]ilZ`H3cqF;$!$'\\F;7$$\"5NL3F%zzw:2 (F;$!$\"\\F;7$$\"5q;a)3Ckqq3(F;$!$\"[F;7$Fgu$!$2&F;7$$\"5I$e9T8Le!=rF; $!$x%F;7$$\"5lm\"H2e<_N8(F;$!$6&F;7$$\"5+]PMF?g/\\rF;F]eo7$$\"5IL$eRZ' )RX;(F;$!$*\\F;7$$\"5l;Hd?4P.!=(F;$!$&\\F;7$$\"5++v=n`v_&>(F;$!$$\\F;7 $$\"5I$3-Q\")R@5@(F;Fago7$$\"5lmmTgU_^EsF;$!$X%F;7$$\"5++](o/i!\\)G(F; F]bo7$F\\v$!$$[F;7$Fav$!$p%F;7$Ffv$!$a%F;7$$\"5lmm;/Y?w9!)F;$!$G%F;7$F [w$!$)RF;7$$\"5lmm;/rAGv%)F;$!$(QF;7$F`wF\\gm7$$\"5l;/,WaVM1()F;$!$r$F ;7$$\"5++voal?<@()F;$!$\"QF;7$$\"5I$ek`mx**ft)F;Fbio7$$\"5lm;/w([F3v)F ;$!$e$F;7$$\"5+](=n))>bcw)F;$!$b$F;7$$\"5ILeR(*4H[!y)F;F]jm7$$\"5l;H23 @1J&z)F;$!$&RF;7$$\"5+++v=K$Q,\"))F;$!$p$F;7$$\"5lmT5SaPzR))F;Fg[n7$$ \"5IL$e9m<\\%p))F;$!$f$F;7$$\"5l;a8s()oF%)))F;$!$y$F;7$$\"5++D\"G))f/ \"**))F;$!$k$F;7$$\"5I$e*[$*4B$R\"*)F;$!$z$F;7$$\"5lmm;/@+wG*)F;$!$#QF ;7$$\"5+]P%[@t(eV*)F;Fjjo7$$\"5IL3_DVaTe*)F;$!$^$F;7$$\"5l;z>OaJCt*)F; Fhim7$$\"5++](oa'32))*)F;F_\\p7$$\"5I$3_vld)*G+*F;$!$%QF;7$$\"5lm\"H#o (GEx,*F;F^]p7$$\"5+]i!*y)*RbK!*F;$!$g$F;7$$\"5ILLe*)42$*F;F_em7$$\"5++D1RgxJ@$*F;F]jo7$$\"5+++D\"yCPa L*F;$!$q$F;7$$\"5++]ilAinj$*F;F_\\p7$$\"5++++](>:>R*F;$!$[$F;7$$\"5++v =#\\oMgS*F;F[^p7$$\"5++]PMsT:?%*F;F_em7$$\"5++DcwfOFM%*F;$!$n$F;7$$\"5 +++v=ZJR[%*F;Fjdm7$$\"5++v$4Yj7DY*F;F[^p7$$\"5++]7.A@jw%*F;$!$]$F;7$$ \"5++DJX4;v!\\*F;F_\\p7$$\"5+++](o4r[]*F;Fejo7$$\"5+++DcY!\\8c*F;Fi_o7 $FjwFi_o7$F]x$!$L$F;7$$\"5LLLeRrF07$Fidl$!$.#F07$F^el$! $9#F07$Fcel$FgcnF07$FhelF]jp7$F]fl$F]bnF07$Fcfl$FbcnF07$F[gl$FjbnF07$F `gl$Ff`nF07$Fegl$!$L#F07$F_hl$F_gnF07$Fbil$!$3#F07$F\\jl$!$!>F07$Ffjl$ !$'=F07$F`[m$!$l\"F07$Fj[m$!$g\"F07$F_\\m$!$S\"F07$Fd\\m$!$E\"F07$Fi\\ m$!$:\"F07$F^]m$!#$*F07$Fc]m$!#uF07$Fh]m$!#dF07$$\"5LL$3FM3?dh\"F0$!#T F07$F]^mFf[o7$$\"5++]7G$ofyj\"F0$Fd[qF;7$Fb^m$!#**F;7$$\"5++++v81]g;F0 $!#DF;7$Fg^m$FjcmF;7$$\"5+++Dc4t&yn\"F0F(7$$\"5LLLL$[(Gk$o\"F0F(7$$\"5 mmmT5S%G%*o\"F0$!\"'F;7$F\\_m$!#7F;7$$\"5mmmTg&zRyr\"F0$!#yF;7$Fa_mFga n7$$\"5mmmTN\"p_Tw\"F0F[^p7$Ff_mFc`o7$$\"5nm;H#eaY#*z\"F0$!$#\\F;7$$\" 5MLL3x%H`1\"=F0$!$O&F;7$$\"5+]7y+#)\\]8=F0$!$G&F;7$$\"5nm\"zW#pmN;=F0$ !$L&F;7$$\"5+D\"GjG^#y<=F0Fiep7$$\"5M$3x\"[c$3#>=F0$!$H&F;7$$\"5nTg-5+ Uj?=F0$!$V&F;7$$\"5++](=P/g?#=F0$!$^&F;7$$\"5MeRsL()e[B=F0$!$N&F;7$$\" 5n;Hd&4t6\\#=F0Fbbq7$$\"5+v=UduvLE=F0F[aq7$$\"5ML3F>=MwF=F0Fa`q7$$\"5+ ](oHa5:1$=F0$!$Z&F;7$F[`m$!$S&F;7$$\"5nmm;H*)ozc=F0$!$<&F;7$F``mFbeo7$ $\"5+++Dc(*QE.>F0$!$N%F;7$Fe`m$!$4%F;7$$\"5mmmT5G7mZ>F0Fcim7$Fj`mFifm7 $$\"5mm;H#oN8](>F0Fifm7$$\"5LLLekm]5\")>F0Fifm7$$\"5++](okx'>()>F0$!$x $F;7$$\"5mmm;H'[)G$*>F0F\\gm7$$\"5LL$e9h>!Q**>F0Fd\\p7$$\"5+++v$f!>Z0? F0Fhio7$$\"5mm;/w:Oc6?F0F\\gm7$F_amF]jo7$$\"5++](oM#Q5B?F0F\\gm7$$\"5m mmTN@BbG?F0Fhim7$$\"5LL$eR#>3+M?F0$!$w$F;7$$\"5+++]7<$\\%R?F0F`eq7$$\" 5nm;/,:y*[/#F0F]jo7$$\"5MLLe*GJY.0#F0Fcim7$$\"5++]7y5[zb?F0Fgjm7$FdamF ^im7$Fiam$!$R%F;7$F\\bm$!$%\\F;7$$\"5++Dc^zs\\5@F0Fhdo7$$\"5++]i:ptF8@ F0$!$)\\F;7$$\"5+]il(RTnY6#F0Fgfo7$$\"5++vozeu0;@F0$!$8&F;7$$\"5+](=\"F;-Ffcm6&FhcmF($\"#l!\"#F(-F]dm6#%6Verner's~Maple~schemeG -F$6%7[`lF'7$$\"5KLL$e*[Vb)\\\"!#A$\"\"\"F07$$\"5lmmm\"zp3r*HFe^rF(7$$ \"5)*****\\(o/jc\\%Fe^rFf^r7$$\"5ILLL$eRF-F(7 $$\"5mmm;a)3wz4#F-F[`r7$$\"5+++vVB:$yC#F-F(7$F2Ff^r7$$\"5mmm\"HKRUva#F -F[`r7$$\"5+++]7GyR(p#F-F[`r7$$\"5LLL3-jKDZGF-Ff^r7$$\"5mmmm\"zp3r*HF- $\"#8F;7$$\"5LLL$3xc>oH$F-$\"\"(F;7$F7$F\\`rF;7$$\"5mmm;H28C'*QF-Fj^q7 $$\"5LLLL3x@&f>%F-$FbcmF;7$$\"5mmm\"z>h2eM%F-$\"\"$F;7$$\"5+++](o/jc\\ %F-$!\"$F;7$$\"5LLL3x\"[=bk%F-Fbbr7$F=$!#;F;7$FY$!#SF;7$F[o$F`hpF;7$F` oF^dr7$Feo$!#zF;7$Fjo$!$0\"F;7$F]p$FihpF;7$Fe[n$!#(*F;7$Fbp$!#oF;7$F] \\n$!#gF;7$Fgp$!#cF;7$F\\q$!#[F;7$Faq$!#XF;7$F[hn$Fa]qF;7$Ffq$!#UF;7$F _hn$!#YF;7$Fdhn$!#^F;7$FihnFeer7$F\\in$FcgpF;7$Fain$!#`F;7$FdinF]fr7$F gin$!#PF;7$Fjin$!#eF;7$F_jn$!#fF;7$FdjnFher7$FgjnF\\er7$F\\[o$!#qF;7$F a[o$!#rF;7$Fd[o$Fi\\qF;7$Fi[oF_er7$F\\\\o$!#VF;7$F_\\o$!#aF;7$Fb\\oF`g r7$Fg\\o$F^gpF;7$F\\]oF\\er7$Fa]oFeer7$Ff]oFber7$F[^oF[hr7$F`^o$F\\]qF ;7$Fe^o$!#hF;7$F[r$FggpF;7$$\"5LLLe*)fDka*[rWF;Fcir7$$\"5+++v=7SW'[%F;F\\er7$$\"5m;Hd&zZ)R,XF ;$FjgpF;7$$\"5LLeRsVHN;XF;$!#*)F;7$$\"5+](=#\\4uIJXF;$!#vF;7$$\"5mm;/E v=EYXF;F[\\s7$$\"5L$ekG5M;7c%F;Fhgr7$$\"5++voz13UH'F;F^_s7$$\"5+++++qi\"HK'F;$!$;\"F;7$$\"5+++](o*[EP jF;Fidr7$$\"5++++vBNh^jF;F^_s7$$\"5+++]i]@'fO'F;Fidr7$$\"5++++]x2J!Q'F ;$!#!*F;7$$\"5+++]P/%fYR'F;$!$9\"F;7$$\"5++++DJ!3!4kF;$!$2\"F;7$$\"5++ +]7emNBkF;$!$G\"F;7$F^t$!$4\"F;7$Fct$!$5\"F;7$FhtF]cs7$$\"5q;aQ.eS@fmF ;F]cs7$$\"5++v$fe)RCsmF;Fd^s7$$\"5N$e*[o8RF&o'F;F]ds7$$\"5qm;/^TQI)p'F ;$!$H\"F;7$$\"5+]PfLpPL6nF;$!#%)F;7$$\"5NLe9;(pjVs'F;Fh]q7$$\"5q;zp)\\ i$RPnF;$!$=\"F;7$F]u$F`\\qF;7$$\"5qmTNY3M[wnF;Fj`s7$$\"5NL$e9TEVD!oF;$ !$8\"F;7$$\"5++Dcw>JgGoF;Fj`s7$Fbu$!$F\"F;7$F\\aoFi^s7$Faao$!$R\"F;7$F fao$FfhpF;7$F[bo$!$1\"F;7$F`bo$!$K\"F;7$Febo$!$B\"F;7$Fjbo$!#)*F;7$F_c oFhgs7$Fbco$!$?\"F;7$Fgco$!$6\"F;7$F\\do$!$X\"F;7$Fado$!$U\"F;7$Ffdo$! $J\"F;7$F[eoF\\fs7$F`eo$Fc\\qF;7$FguFhhs7$FheoFbhs7$F]foFehs7$FbfoF\\f s7$Fefo$!$M\"F;7$Fjfo$!$I\"F;7$F_goFgcs7$FdgoF\\es7$Fggo$!#\")F;7$F\\h oFjcs7$F\\vFj`s7$Fav$!$<\"F;7$FfvFcfs7$FhhoFj_s7$F[w$!#')F;7$F`ioFaes7 $F`wFadr7$FfioFjir7$F[jo$!#))F;7$F`joFe_s7$FcjoFgfr7$Fhjo$FifpF;7$F][p Fd^s7$F`[pF_as7$Fe[pFadr7$Fj[p$F]hpF;7$F]\\pF[hr7$Fg\\pFjir7$Fa]p$Fchp F;7$Ff]pFcir7$Fi]pFhhr7$F^^pFhbs7$Fa^pF^hr7$Fd^pFh]q7$Fi^pF[\\t7$F\\_p Fg[t7$Fa_p$!#&)F;7$Fd_pF\\^s7$Fg_p$!#!)F;7$Fj_pFe]s7$FewF``s7$F^`pFd\\ t7$Fa`pF`gr7$Ff`pF^_s7$Fi`p$!#\"*F;7$F\\apFddr7$F_apF[\\t7$FdapFd\\t7$ FgapF`\\s7$F\\bpFb_q7$F_bpFe_s7$FbbpF[\\t7$FgbpF`]t7$FjbpFh\\t7$F]cpFa dr7$FbcpF[\\s7$FecpF[\\s7$FhcpFi^s7$FjwF[\\t7$F]xFadr7$FbxF`]s7$FgxFji r7$F]epFadr7$F\\yF^dr7$Fcep$!#'*F;7$FayF_hs7$Ffy$!$]\"F;7$F`z$!$%=F;7$ Fez$!$1#F;7$Fjz$!#BF07$Fd[l$!#EF07$Ff\\l$!#KF07$F^]l$F\\drF07$Fh]l$Fdg rF07$Fb^l$F]erF07$F[_l$FbdrF07$Fi`l$Fj^sF07$Febl$FcgsF07$FiclFhhp7$Fid l$!$D\"F07$F^el$FfisF07$Fcel$!$Y\"F07$$\"5++vV)R-zA\\\"F0$FfhsF07$Fhel F^at7$$\"5mm\"z>^'zC&\\\"F0Fcat7$F]flFcat7$Fcfl$F[_tF07$F[gl$!$a\"F07$ $\"5mmTg_)yaT]\"F0$!$`\"F07$$\"5++]P4f#Rc]\"F0Fabt7$$\"5LLe9mHP72:F0$! $_\"F07$F`glF\\bt7$$\"5++vozqE45:F0F^ip7$$\"5LL$ek89x:^\"F0F^ip7$$\"5m m\"HK>hhI^\"F0$!$b\"F07$FeglFect7$F_hl$!$^\"F07$FbilF\\\\q7$F\\jlF_\\q 7$Ffjl$!$C\"F07$F`[m$FccsF07$Fj[m$Ff`sF07$F_\\m$Ff]sF07$Fd\\m$Fa\\sF07 $Fi\\mFbgp7$F^]m$!#ZF07$Fc]m$!#IF07$Fh]m$FicrF07$F^]q$F[^rF07$F]^m$\"# \")F;7$Fd]q$\"$p\"F;7$Fb^m$\"$Z#F;7$F[^q$\"$D$F;7$Fg^m$\"$c$F;7$$\"5LL $3Fp_k\\n\"F0$\"$^$F;7$Fb^qF_x7$$\"5LL3-)3q.$z;F0$\"$e$F;7$$\"5mm;z>#4 ]2o\"F0$\"$s$F;7$$\"5++Dc^$['>#o\"F0$\"$o$F;7$Fe^q$\"$h$F;7$Fh^q$\"$l$ F;7$F\\_mFas7$$\"5mm\"HKHXq3q\"F0Fegt7$$\"5LL$e*[+p_1'yWd2%=F0F]_q7$$\"5nmT5!*yb@U=F0$F;F;7$$\"5n;/,%*4PnV= F0$!#5F;7$$\"5nmm\"z4%=8X=F0$!#=F;7$$\"5nm;a8lV'4&=F0Fhcr7$FfcqFhcr7$$ \"5nmmTgP>Yo=F0Fhcr7$F``m$!#9F;7$$\"5++D\"GQ@6f)=F0$!#8F;7$$\"5ML$eRF0Ffjt7$F\\dqFbbr7$$\"5mm;aQ`B$[\">F0F]cr7$Fe`m$\"\"'F;7$Fd dq$\"#7F;7$Fj`m$\"#;F;7$FhdqF`jt7$F[eq$\"#9F;7$F^eq$\"#F;7$FdamF[_u7$FiamF( 7$F\\bm$!#:F;7$Fahq$!\"(F;7$FaiqF^]u7$FfiqFj^q7$Fabm$\"\"&F;7$F\\jqFbb r7$FajqFcau7$Ffjq$F/F;7$FijqFibr7$F^[rFf`u7$Fa[rF[_u7$Fi[rF[`u7$Fa\\rF hau7$Ff\\r$\"#JF;7$Ffbm$\"#\\F;7$Fj\\r$\"##*F;7$F[cm$\"$h\"F;7$F`]rFdi t7$F`cm$\"$S%F;-%'COLOURG6&Fhcm$\")#)eqkF\\^u$\"))eqk\"F\\^uFccu-F]dm6 #%Ha~modification~of~Verner's~Maple~schemeG-F$6%7d]lF'Fb^rFh^rF[_rF^_r Fe_r7$F+Fd_r7$$\"5+++D19*)p[8F-F(F^`r7$$\"5mmmT&Qy4%[;F-Fd_r7$Fb`rFd_r 7$Fe`rFd_r7$Fh`rFf^r7$F[arFd_r7$F2F(7$F7F[au7$F=F]gr7$FYFhgr7$F[oFbjs7 $F`oFe]r7$FeoFies7$Fjo$F_atF;7$F]p$!$k\"F;7$Fe[n$!$Z\"F;7$FbpFgdr7$F] \\nFbjs7$FgpFjas7$F\\qFbcs7$FaqF[hs7$$\"5LLLLeR;<5KF;F``s7$$\"5mmmmT]n vCKF;Ff^t7$$\"5++++Dh=MRKF;F\\^s7$$\"5LLLL3sp#RD$F;Fd\\t7$$\"5mmmm\"H3 7&oKF;Fjcs7$$\"5++++v$>(4$G$F;Fcfs7$$\"5LLLLe/Bo(H$F;Fe`s7$$\"5mmmmT:u E7LF;Fh]q7$$\"5++++veygqLF;Ff^t7$F[hnFhbs7$$\"5++++v)=Hca$F;Fhbs7$FfqF [\\s7$F_hnFi^s7$FdhnFf^t7$FihnF^_s7$F\\inFbhs7$FainFh]q7$FdinFe]s7$Fgi nF^dr7$FjinFj_s7$F_jnFe`s7$FdjnF[\\s7$Fgjn$!$7\"F;7$F\\[oF]cs7$Fa[oF_i s7$Fd[oFbjs7$Fi[oFe`s7$F\\\\oFgjs7$F_\\oFidr7$Fb\\oFd^s7$Fg\\oF`gs7$F \\]oF]ds7$Fa]oFhbs7$Ff]oF[hs7$F[^oF]cs7$F`^oFh]q7$Fe^oFe`s7$F[rFbhs7$F `rFj`s7$F^_oFhgs7$Fer$F_dtF;7$Fjr$!$Q\"F;7$F_s$FjctF;7$FdsF^eu7$FisF^e u7$Fj]sF^ju7$F_^s$F_ipF;7$Fb^s$!$q\"F;7$Fg^s$!$h\"F;7$F\\_s$!$i\"F;7$F `_s$!$[\"F;7$Fc_sF[[v7$Fh_sFeju7$F]`sF[[v7$F^`o$!$o\"F;7$Fc`s$!$r\"F;7 $Fh`s$Fa[qF;7$F]as$!$t\"F;7$Fbas$!$T\"F;7$FeasFhju7$FhasFddm7$F]bs$Fg[ qF;7$F`bsFhju7$FcbsF`eu7$Ffbs$!$d\"F;7$F[cs$!$\"=F;7$F`cs$!$u\"F;7$Fec s$!$&>F;7$F^t$!$w\"F;7$$\"5NL$3-j*\\#)*['F;$!$!=F;7$FctFd]v7$$\"5+](oH aju\\b'F;Fceu7$$\"5NL3_DjX+olF;F]_t7$$\"5q;H23\"\\M5e'F;$!$#>F;7$$\"5+ +]i!*=W1%f'F;$!$n\"F;7$$\"5N$3xJnM%42mF;$!$v\"F;7$$\"5qm\"HdXFC,i'F;Fa ]v7$$\"5+]7GQ-U:LmF;$!$f\"F;7$FhtFi]v7$Fads$!$z\"F;7$FddsFj^v7$FgdsFd] v7$FjdsFa]v7$F_es$!$\\\"F;7$FdesF`eu7$FgesFddm7$F]u$!$\">F;7$F^fs$!$(= F;7$Fafs$!$y\"F;7$FffsFi`v7$FbuFe^v7$F\\aoFh\\v7$Faao$!$/#F;7$FfaoF\\ \\v7$F[boFg[v7$F`bo$FeipF;7$Febo$!$)=F;7$Fjbo$!$j\"F;7$F_coFiav7$Fbco$ !$&=F;7$FgcoFd]v7$F\\do$!$5#F;7$Fado$F^[qF;7$Ffdo$!$'>F;7$F[eoFf`v7$F` eoF[]v7$FguFgbv7$Fheo$FbipF;7$F]fo$!$6#F;7$FbfoFe^v7$Fefo$!$+#F;7$Fjfo Fibv7$F_goFa]v7$FdgoFibv7$FggoF^[v7$F\\hoFd]v7$F\\vFj[v7$FavFf]q7$FfvF ]_t7$FhhoF\\\\v7$F[w$FfctF;7$F`ioFj^t7$F`w$F]\\qF;7$FfioF[ju7$F[joF^[v 7$F`jo$F]btF;7$FcjoF\\fs7$FhjoFhgs7$F][p$Fj[qF;7$F`[pF\\bv7$Fe[p$F\\ip F;7$Fj[pFeis7$F]\\pF\\es7$Fb\\pFceu7$Fg\\p$!$L\"F;7$F\\]pF^[v7$Fa]pF^j u7$Ff]p$Fj`tF;7$Fi]pFgdr7$F^^pF^eu7$Fa^pFhis7$Fd^pFhdv7$Fi^pFj^t7$F\\_ pFhis7$Fa_pFddv7$Fd_pF_ev7$Fg_pFdev7$Fj_pFddv7$Few$FjbtF;7$F^`p$!$O\"F ;7$Fa`pF]ds7$Ff`p$!$W\"F;7$Fi`pFhhs7$F\\apFcju7$F_apF^eu7$FdapFgfv7$Fg apF\\fs7$FecpF]gs7$FjwF^eu7$F]xFdev7$FbxFdev7$FgxF_\\v7$F]ep$!$V\"F;7$ F\\yF^eu7$FcepFh\\v7$FayF_cv7$Ffy$FhipF;7$F`zFidn7$Fez$!$k#F;7$Fjz$!#H F07$Fd[l$!#NF07$Ff\\l$FierF07$Fi\\l$!#]F07$F^]lF[]q7$Fc]l$FigrF07$Fh]l $Fi\\tF07$Fb^l$F\\\\sF07$F[_l$Fi]qF07$Fi`l$F^dsF07$Febl$F[asF07$Fidl$F \\juF07$FcelF^ip7$FaatF^ip7$Fhel$Fi\\vF07$FfatFect7$F]flFect7$FcflFi[q 7$F[gl$FaeuF07$F_btFejv7$Fdbt$F]bvF07$FgbtFhjv7$F`glFejv7$F]ct$!$m\"F0 7$F`ct$F[_vF07$FcctF\\[w7$FeglFf[q7$F_hl$F\\[vF07$FbilFabt7$FfjlF\\jv7 $Fj[m$Ff]rF07$F_\\mFcdt7$Fd\\m$FibsF07$Fi\\m$F^jsF07$F^]m$!#iF07$Fc]m$ !#WF07$Fh]mFdhv7$F^]q$F\\auF07$F]^m$!#OF;7$Fd]q$\"#dF;7$Fb^m$\"$R\"F;7 $F[^q$\"$=#F;7$Fg^m$\"$^#F;7$Fe^qFdit7$F\\_m$\"$i#F;7$F_ht$\"$n#F;7$Fb htFg]w7$Fght$\"$b#F;7$F`_q$\"$V#F;7$F_it$\"$:#F;7$Fa_mFget7$Ff_mFi_u7$ Fj_qF^au7$F_`qF`fr7$$\"5n\"HK*QQ\"z?\"=F0$Fb\\wF;7$Fd`qFahr7$$\"5M3-ji D3$\\\"=F0Fdhr7$Fi`qFifr7$F^aqF[dr7$Faaq$!#\\F;7$FfaqF_er7$F[bqF[hr7$F [cqFhhr7$F[`mFg[s7$FijtF^hr7$F\\[uFg[s7$Fa[uF]js7$Fd[uF[\\s7$Fh[uFb_q7 $F]\\uFd\\t7$Fb\\uFd\\t7$FfcqFaes7$Ff\\uFd\\t7$F``mF][t7$F\\]uFgjs7$Fa ]uFaes7$Fi]uF^dr7$F\\dqF`\\s7$Fb^uF`\\s7$Fe`mF`]s7$Fj`mFhgr7$F_amFh\\t 7$F`fqF^hr7$FcfqF^hr7$FffqF`\\s7$F[gqF`\\s7$F^gqFjir7$FagqFd\\t7$FdgqF h\\t7$FdamF^dr7$FiamFd^s7$F\\bmFies7$FabmFh]q7$FfbmFb[t7$Fj\\r$FehvF;7 $F[cm$\"#KF;7$F`]r$\"$<\"F;7$F`cm$\"$$GF;-F_cu6&Fhcm$\"*++++\"F\\^uF(F (-F]dm6#%6an~\"efficient\"~schemeG-F$6%7balF'7$F+F[`r7$F2$FdauF07$Fbar $Ff^uF07$Fhar$Fa]lF;7$F]br$\"#pF;7$F7Fjit7$$\"5LLLeRseQYPF-$\"##)F;7$F dbrFjit7$$\"5+++v=Un4YSF-$\"#vF;7$Fgbr$\"#%)F;7$F[cr$\"#()F;7$F`crFdet 7$Fecr$\"#&)F;7$F=Ffcw7$FGF\\ew7$FO$\"#(*F;7$$\"5qmT&Q.jp)ofF-$\"#uF;7 $$\"5++](oH/e#*4'F-Fdet7$$\"5NLe*)fbkkHiF-$\"#$)F;7$FT$\"#xF;7$$\"5++v $f3GB/\\'F-Fgit7$$\"5NL$e*[$p63i'F-$\"#))F;7$$\"5qm\"z>h5+7v'F-F\\dw7$ FYF^fw7$$\"5ILL3Fp@9.uF-Fifw7$FfnF\\ew7$$\"5++v=UKU3b!)F-$\"#*)F;7$$\" 5IL$3_]ksa=)F-$\"#yF;7$$\"5lm\"H#od5'eJ)F-F\\ew7$$\"5+++DJq%\\iW)F-$\" #!*F;7$$\"5lm;Hd&HEqq)F-F^fw7$F[o$\"#qF;7$F`oF[hw7$FeoF\\dw7$Fjo$\"#bF ;7$F]p$\"#OF;7$Fe[n$\"#_F;7$FbpFfcw7$F]\\n$\"#nF;7$Fgp$\"#iF;7$F\\q$Fj ]rF;7$FaqFfcw7$F[fuFhiw7$F^fu$\"#rF;7$FafuFedw7$FdfuF\\dw7$Fgfu$\"#eF; 7$Fjfu$\"#aF;7$F]guFjit7$F`gu$\"#oF;7$FcguFbjw7$F[hnFafw7$FgguFafw7$Ff q$\"#zF;7$F_hnFdcw7$Fdhn$\"#sF;7$FihnFbdw7$F\\inFj\\w7$FainFihw7$FdinF \\dw7$FginFifw7$FjinF^[x7$F_jnFhiw7$Fdjn$\"#!)F;7$FgjnFgjw7$F\\[o$\"#c F;7$Fa[oF^iw7$Fd[oFjjw7$Fi[oFhiw7$F\\\\oF\\ew7$F_\\oFfew7$Fb\\oFbjw7$F g\\oFjit7$F\\]oF[jw7$Fa]oF^fw7$Ff]oFbdw7$F[^o$\"#gF;7$F`^oFfew7$Fe^oFb jw7$F[rFjit7$$\"5mm;H#opat=%F;Fjit7$Feir$\"#hF;7$$\"5+]iSmDq7iUF;Fhiw7 $$\"5mm\"HK9\\\"3xUF;F[hw7$$\"5L$3_+s&f.#H%F;F^jw7$$\"5++](oHU!*pI%F;F dcw7$$\"5m;zpt))[%>K%F;F\\ew7$$\"5LL3_]a$**oL%F;F[jw7$$\"5+]PMF?Q&=N%F ;Fh[x7$FhirF\\^x7$F]jrFedw7$F`jrFd[x7$FcjrFc]x7$FfjrFb\\x7$FijrFbjw7$F \\[sFbjw7$F_[sFbjw7$Fb[sFh[x7$Fe[sF^[x7$Fi[sFdiw7$F^\\s$\"#mF;7$Fc\\sF diw7$Ff\\sFh[x7$Fi\\sFedw7$F\\]sFfew7$F`rF_`x7$F^_o$\"#tF;7$FerFb\\x7$ FjrFdcw7$F_sFdcw7$FdsFebu7$Fis$\"$0\"F;7$Fj]s$\"$-\"F;7$F_^s$\"#**F;7$ Fb^sF\\ew7$Fg^s$\"#%*F;7$F\\_s$\"#&*F;7$F`_s$\"$4\"F;7$Fc_sFaew7$Fh_sF chw7$$\"5+++]Pu83lhF;$\"#)*F;7$F]`sFdax7$$\"5+++]7G'yP>'F;Faax7$F^`oFg it7$Fc`s$\"#$*F;7$Fh`sFfew7$F]asFebu7$Fbas$\"$C\"F;7$Feas$\"$1\"F;7$Fh asF\\ew7$F]bs$\"$/\"F;7$F`bsF^bx7$FcbsFecx7$Ffbs$\"$8\"F;7$F[cs$\"#\"* F;7$F`csFebx7$FecsF[hw7$F^tFaew7$FbuF]cx7$F\\vFebx7$FavFebx7$FfvFhax7$ $\"5l;/w7fSu&y(F;Fhax7$$\"5++v=#\\#>,,yF;$\"$3\"F;7$$\"5I$e9;2zzi\"yF; $\"$6\"F;7$$\"5lm;/^cwaJyF;Fadx7$$\"5+](o/B_:o%yF;Fhax7$$\"5ILe*)4)Q$3 iyF;Fhax7$$\"5l;HK*QD^t(yF;Feex7$$\"5+++vo>\">E*yF;$\"$;\"F;7$$\"5I$3x \"[&)p)y!zF;Fifw7$$\"5lmTgF^[:BzF;F_bw7$$\"5+]7.2)F;Fecx7$F[w$\"$.\"F;7$F `ioFdax7$F`wFdax7$Ffio$\"$+\"F;7$F[joFfgw7$F`joF^fw7$FcjoFhix7$Fhjo$\" $7\"F;7$F][pFbdw7$F`[pFbjw7$Fe[pFgit7$Fj[pFebx7$F]\\pFdjx7$Fg\\pFgit7$ Fa]pFafw7$Ff]pFdjx7$Fi]p$\"$2\"F;7$F^^pFdet7$Fa^pFaew7$Fd^pFh[x7$Fi^pF dcw7$F\\_pFhax7$Fa_pF\\ew7$Fd_p$\"#')F;7$Fg_pFfgw7$Fj_pFdet7$FewF^[x7$ F^`pF\\dw7$Fa`pFj[y7$Ff`pFg`x7$Fi`pFfew7$F\\ap$\"#fF;7$F_apFfcw7$FdapF d[x7$FgapFifw7$FecpFh[x7$Fjw$\"#jF;7$$\"5lmm;aeAa`)*F;Ffcw7$F]xFihw7$F bxFbjw7$FgxFihw7$F]epFfew7$F\\yFb\\x7$$\"5ML$ekVM'p_6F0Fafw7$$\"5+++v= TU(z:\"F0Ffew7$$\"5n;HKR:PHf6F0Fb\\x7$$\"5MLe*)f*=81;\"F0F\\dw7$$\"5+] (o/QmK>;\"F0Fdcw7$$\"5nm;/,Q@Dj6F0F^fw7$$\"5++v=U'3\"*e;\"F0F[hw7$Fcep Fafw7$$\"5nm\"zWK)*o6<\"F0F^fw7$$\"5++]ilJz!Q<\"F0Fdcw7$$\"5n;z>'eSF^< \"F0Fihw7$$\"5ML3x1!)oWw6F0Fd[x7$$\"5+]PMFajwx6F0F^fw7$$\"5nmm\"z%Ge3z 6F0Fhiw7$$\"5M$e*[o-`S!=\"F0Fb\\x7$$\"5++D1*oxC<=\"F0Fafw7$$\"5n;aj4^U /$=\"F0Fafw7$$\"5ML$3-`sjV=\"F0Fbdw7$$\"5+]7y]*>$o&=\"F0Fbdw7$$\"5nmTN rtE+(=\"F0F\\ew7$$\"5M$3F>z9A$)=\"F0Fb\\x7$FayFhiw7$$\"5n;Hdqe<\\#>\"F 0Fafw7$$\"5MLekG&*=M&>\"F0Fc\\y7$$\"5n\"H#odjpw'>\"F0Fd[x7$$\"5+](=n=. #>)>\"F0F\\ew7$$\"5M3_v:+rh*>\"F0F_`x7$$\"5nm;zWo@/,7F0F^[x7$$\"5M$ekG ]I#*Q?\"F0Fg`x7$$\"5++v$4;WUn?\"F0Fbdw7$$\"5n;/,>yDf47F0Fafw7$$\"5MLL3 x9FW77F0Fg`x7$$\"5n\"z>hIynQ@\"F0F^jw7$$\"5+]i:N^GH:7F0Fbjw7$$\"5M3F>k >zr;7F0Fh[x7$$\"5nm\"HKz)H9=7F0Fdet7$$\"5+DcEAc!o&>7F0Fh[x7$$\"5M$3-8X 7$*4A\"F0Fc]x7$$\"5nT&Q.G>=CA\"F0F_`x7$$\"5++]P4hK%QA\"F0Fd[x7$$\"5n;z Wn(R$pE7F0Fbjw7$$\"5ML3_DMNaH7F0Fd]y7$$\"5n\"HdXDgo4B\"F0F_`x7$$\"5+]P f$3n$RK7F0Fbbu7$$\"5M3-j7R(=QB\"F0Fbdw7$FfyFjit7$F[zFf\\x7$F`z$\"#YF;7 $Fez$\"#IF;7$FjzFf^r7$Fd[lFbet7$Ff\\l$F[_qF07$F^]l$F[\\uF07$Fh]l$F0F07 $F[_lFf_t7$FeblF\\`t7$Ficl$Ff_wF07$Fidl$!#_F07$F^elF^`t7$FcelFfgp7$F[g lFb`t7$Fegl$F]^sF07$$\"5n;/,W%QOs^\"F0$FbesF07$Fjgl$Fe\\tF07$$\"5+]7.K ))phA:F0Fbhy7$F_hlFbhy7$FchlF`hy7$FhhlF\\\\w7$F]ilF\\\\w7$FbilF\\\\w7$ Ffjl$F\\hrF07$Fj[m$FagrF07$F_\\m$FafrF07$Fd\\m$Fh\\wF07$Fi\\mF]et7$F^] m$Fj\\uF07$Fc]mFd_r7$Fh]m$\"#6F07$F^]q$\"#BF07$F]^m$\"$9$F;7$Fd]q$\"$ \"RF;7$Fb^m$\"$i%F;7$F[^qFb`m7$Fg^m$\"$x&F;7$Fe^q$\"$(eF;7$F\\_m$FeilF ;7$$\"5m\"HKks6Gmp\"F0$\"$)fF;7$$\"5L$ek`\"HA/)p\"F0$\"$&fF;7$$\"5+voH /TjX*p\"F0$\"$2'F;7$F_ht$\"$>'F;7$$\"5+]P4rw')p.O*fuf8!o1s\"F0$\"$ .'F;7$$\"5LLe9;Vi\\Bt\"F0$\"$$eF;7$$\"5mmTgFQ\"4[t\"F0$\"$#eF;7$$\"5+](oa? OPwt\"F0$\"$u&F;7$Fa_m$\"$k&F;7$Ff_q$\"$*\\F;7$Ff_m$\"$B%F;7$F_`q$\"$d $F;7$F[`m$\"$8$F;7$FfcqFbbw7$F``m$\"$m#F;7$F\\]u$\"$j#F;7$Fa]uFhaz7$Ff ]uFeaz7$Fi]u$\"$h#F;7$F^^uFhaz7$F\\dq$\"$o#F;7$$\"5m;H#=:,ch!>F0$\"$l# F;7$$\"5LLeRZD\"[!4>F0Fdit7$$\"5m\"H#=X#=%\\5>F0Fhaz7$$\"5+](oH%R-%>\" >F0$\"$p#F;7$$\"5L3_vS'H'Q8>F0$\"$s#F;7$Fb^uFhaz7$$\"5++voH\"e;1#>F0Fj ]w7$Fe`mFg]w7$Fj`m$\"$k#F;7$F_amF?7$FdamF^dz7$$\"5ML3x\"Qa_q1#F0Fhaz7$ $\"5++](o*y<'G2#F0$\"$d#F;7$$\"5nm\"z>T,r'y?F0$\"$g#F;7$FiamF?7$$\"5++ v=U%[*G!4#F0F]bz7$$\"5nm;Hd>()4'4#F0Ffbz7$$\"5MLeRsaz!>5#F0Feaz7$F\\bm F^dz7$Fabm$\"$3$F;7$Ffbm$\"$y$F;7$Fj\\r$\"$M%F;7$F[cm$\"$:&F;7$$\"5++v o/%[u?=#F0$\"$'eF;7$F`]r$\"$='F;7$$\"5++Dc,G[-%>#F0$\"$$oF;7$F`cm$\"$# yF;-Ffcm6&FhcmF($FcdwF[^r$F\\bxF[^r-F]dm6#%Ea~scheme~of~Tsitouras~and~ PapakostasG-F$6%7^^lF'F^cw7$F2Fbcw7$F_arFbcwFacw7$FearFbcw7$FharFd[x7$ F]brFh[x7$F7Fhiw7$FjcwFedw7$FdbrF^jw7$F`dwFfew7$FgbrF^fw7$F[crFc\\y7$F `crFb\\x7$FecrFedw7$F=F^[x7$FGFb\\x7$FOFadx7$FdewF^jw7$FiewFbjw7$F\\fw Ffew7$FTFhiw7$FdfwFhdw7$FgfwF[hw7$F\\gwFh[x7$FYFh[x7$F`gwFfew7$FfnFfcw 7$FdgwFh[x7$FigwF\\^x7$F^hwF^[x7$FahwFh[x7$FfhwF^jw7$F[o$\"#^F;7$F`oF^ iw7$FeoFj\\w7$FjoFgfy7$F]p$\"#5F;7$Fe[n$\"#EF;7$Fbp$\"#WF;7$F]\\n$\"#U F;7$Fgp$F\\\\lF;7$F\\q$F][lF;7$Faq$\"#ZF;7$F[hnFjjw7$FfqFf\\x7$F_hnFdi w7$Fdhn$\"#[F;7$FihnF`jz7$F\\in$\"#LF;7$FainFdfy7$FdinFgjw7$FginFd]y7$ FjinF\\[[l7$F_jnF_[[l7$FdjnFf\\x7$Fgjn$\"#MF;7$F\\[oF\\bw7$Fa[oF_bu7$F d[o$\"#HF;7$Fi[oFd[[l7$F\\\\oF\\^x7$F_\\o$\"#]F;7$Fb\\oFdfy7$Fg\\o$Fb[ lF;7$F\\]o$\"#PF;7$Fa]oFgjw7$Ff]oFc][l7$F[^o$\"#NF;7$F`^oFbbu7$Fe^oFdf y7$F[rFb[[l7$FeirF^^[l7$FhirFi\\[l7$F]jrFgjw7$F`jrFdiw7$FcjrFi\\[l7$Ff jr$F\\]lF;7$FijrF\\[[l7$F\\[sF\\[[l7$F_[s$\"#XF;7$Fb[sF]_[l7$Fe[s$Fd\\ lF;7$Fi[s$\"#DF;7$F^\\sFb[[l7$Fc\\sFc_[l7$Ff\\sF\\[[l7$Fi\\sFj\\w7$F\\ ]sFf[[l7$F`rFb[[l7$F^_oF\\[[l7$FerFc][l7$FjrF\\[[l7$F_sFa_[l7$FdsF^iw7 $FisF^jw7$Fj]sF[jw7$F_^sFgjw7$Fb^sF\\[[l7$Fg^sFi^[l7$F\\_sFjjw7$F`_sF^ [x7$Fc_sFf\\x7$Fh_sF\\\\[l7$FcbxFj\\w7$F]`sFj\\w7$FibxFc]x7$F^`oFjjw7$ Fc`sF`jz7$Fh`sF\\bw7$F]asFc][l7$FbasF\\dw7$FeasFjit7$FhasFd[[l7$F]bsF[ jw7$F`bsF_`x7$FcbsFd]y7$FfbsFihw7$F[csF\\\\[l7$F`csF^iw7$FecsF^^[l7$F^ tFjjw7$FbuFbbu7$F\\vFi^[l7$FavFi^[l7$FfvF`jz7$F[exFc][l7$F^exFjit7$Fce xF^[x7$FhexF\\\\[l7$F[fxF`jz7$F^fxF`jz7$FafxF^[x7$FdfxFg`x7$FifxFdfy7$ F\\gxFbdw7$F_gxFdcw7$FdgxFf\\x7$FggxFdiw7$FjgxFf[[l7$F]hxFhiw7$FhhoFi^ [l7$FahxFd[x7$FfhxF\\ew7$F[ixF_[[l7$F^ixF^[x7$FfixFfcw7$F^jxF[jw7$Fajx F_`x7$F[wFjit7$$\"5lm;H#=\\1JJ)F;F]]y7$$\"5ILLe*[3lrO)F;F_`x7$$\"5++]( oznB7U)F;Fjit7$F`ioF\\^x7$$\"5++D\"yvc6B])F;Fd]y7$$\"5IL$e9T'3MH&)F;Fh iw7$$\"5+]7GQ7b&Ga)F;Fc][l7$$\"5lmT5lg,Pc&)F;F`jz7$$\"5I$3F>*3[))p&)F; F`jz7$$\"5+++v=d%*R$e)F;Fi^[l7$$\"5lm;/E]!euj)F;Fgjw7$F`wFjit7$FfioF^j w7$F[joFjjw7$F`joF\\\\[l7$FcjoFbdw7$FhjoFafw7$F][pFg][l7$F`[pFaiw7$Fe[ pF[jw7$Fj[pF^jw7$F]\\pFfcw7$Fg\\pFd]y7$Fa]pF\\[[l7$Ff]pFihw7$Fi]pFfew7 $F^^pF\\\\[l7$Fa^pFjit7$Fd^pFb[[l7$Fi^pFd[[l7$F\\_pF[jw7$Fa_pFdiw7$Fd_ pFjjw7$Fg_pFj\\w7$Fj_pFbbu7$FewFaiw7$F^`pF`jz7$Fa`pFdcw7$Ff`pF_[[l7$Fi `pFd[[l7$F\\ap$\"#GF;7$F_ap$Fg[lF;7$FdapF\\\\[l7$FgapFj\\w7$FecpF_[[l7 $FjwF`\\[l7$F]xFa_[l7$FbxFd[[l7$FgxF_[[l7$F]epF]_[l7$F\\yFbbu7$F_^yFdf y7$Fb^yFd[[l7$Fh^yF`jz7$F^_yF`jz7$Fa_yFdfy7$FcepF]_[l7$Fe_yF`jz7$Fh_yF d[[l7$F[`yFi][l7$F^`yFdfy7$Fa`yF`jz7$Fd`yF^^[l7$Fg`yFf[[l7$Fj`yF\\[[l7 $F]ayF\\[[l7$F`ayF_[[l7$FcayF_[[l7$FfayF`jz7$FiayFf[[l7$FayFi\\[l7$Ffy Fdg[l7$F[zFe_u7$F`zFhau7$FezFj[u7$Fjz$FccrF07$Fd[l$F\\^uF07$Ff\\lFe\\w 7$F^]l$Fe^rF07$Fh]lFghv7$F[_lFa\\w7$Febl$FcerF07$Fidl$!#nF07$FcelF`hy7 $F[glFe\\q7$Fegl$FjdrF07$F^hy$F\\hsF07$FjglFfiv7$$\"5nTg-NP=F@:F0$Fa`s F07$FdhyFa[\\l7$$\"5Mek.HR@'R_\"F0Ffiv7$F_hlFfiv7$FchlF\\[\\l7$FhhlFbh p7$F]ilFbhp7$FbilFbhp7$Ffjl$FhjsF07$Fj[mFigp7$F_\\mF^`t7$Fd\\mFdgy7$$ \"5n\"zWU8OE3d\"F0$F^frF07$$\"5M$ek`m'pHs:F0F\\`t7$$\"5+vV['>dnPd\"F0F `]q7$Fi\\mFc\\\\l7$$\"5+]P%)*yQz\"y:F0$!#RF07$F^]m$!#CF07$Fc]m$Fd]uF07 $Fh]mF`cw7$F^]q$Fg`uF07$F]^m$\"$\"HF;7$Fd]q$\"$%QF;7$Fb^mFbz7$F[^qFe`z 7$Fg^m$\"$,'F;7$Fe^qFd\\z7$F\\_m$FhblF;7$F`[z$\"$A'F;7$Fe[zF_\\z7$Fj[z $\"$J'F;7$F_htFcv7$Fb\\zF`t7$Fbht$\"$F'F;7$F_]zF`_\\l7$FghtF`t7$Fi]z$ \"$G'F;7$F^^zFe]z7$Fa^zFe^\\l7$F`_qFg]z7$F_it$\"$+'F;7$Fa_m$\"$q&F;7$F f_q$FijlF;7$Ff_m$\"$8%F;7$F_`q$\"$S$F;7$F[`mFD7$FfcqF]bz7$F``m$\"$W#F; 7$F\\dq$\"$Y#F;7$Fe`mFa^w7$Fj`m$\"$\\#F;7$F_amF[a\\l7$FdamFjet7$FcdzF^ a\\l7$Ffdz$\"$S#F;7$F[ezFa^w7$Fiam$\"$U#F;7$FaezF[a\\l7$Fdez$\"$[#F;7$ Fgez$\"$]#F;7$F\\bmFjet7$FabmFD7$Ffbm$\"$m$F;7$Fj\\rF[az7$F[cm$Fb`lF;7 $F`]rFhbmFcgz-F_cu6&FhcmFfbwF(Ffbw-F]dm6#%Ra~modification~of~the~Tsito uras-Papakostas~schemeG-F$6%7_alF'Fb^rFh^rF[_rF^_rFa_rFe_r7$Fi_r$F^crF 0F]`r7$F]duFf^r7$F_`rFf^r7$F`duF(Fa`r7$Fe`rFf^r7$Fh`rFgc\\l7$F[arF[`r7 $F2F[`r7$F_arFgc\\l7$FbarFgc\\l7$FearF[`r7$FharF_bu7$F]brF]jt7$F7Ff`u7 $FjcwFaiw7$FdbrF__u7$F`dwF[_u7$FgbrFc_[l7$F[crF^][l7$F`cr$F\\jyF;7$Fec r$\"#FF;7$F=$FiiyF;7$FGF[`u7$FOF^][l7$Fdew$Fg^rF;7$FiewFe^u7$F\\fwFhau 7$FTF]cr7$FdfwF[e\\l7$FgfwF`jt7$F\\gw$\"\")F;7$FYF[f\\l7$F`gwF`e\\l7$F fnF_br7$FdgwFhau7$FigwFibr7$F^hwFe^u7$FahwF`e\\l7$FfhwFcau7$F[oFc]u7$F `oF(7$FeoFe^u7$Fjo$F0F;7$F]pFg\\w7$Fe[n$Fe^rF;7$FbpFj^q7$F]\\nF]_q7$Fg p$F-F;7$Fe\\nF\\g\\l7$Fj\\nF\\g\\l7$$\"5LL$3F>\\O5V#F;Ffjt7$F_]n$Fd_tF ;7$$\"5mm;zW(fw#eCF;F]^q7$Fd]n$!#JF;7$Fi]nF`^q7$F^^nF]cr7$Fa^nF[au7$Ff ^nF[^u7$F[_nF]_q7$F`_n$F_]\\lF;7$Fe_n$!#&Q$F;Ffjt7$$\"5mmmmT!3y(*R$F;F]cr7$$\"5+ +++D\">jVT$F;Ffjz7$F[hnFbcr7$FgguFbcr7$FfqFibr7$F_hnF^[u7$FdhnF[^u7$Fi hnFfjt7$F\\inFgg\\l7$FainFj[u7$FdinFbbr7$FginF[f\\l7$FjinF]_q7$F_jnF^] u7$FdjnF(7$FgjnF\\g\\l7$F\\[oF`j\\l7$Fa[oF]^q7$Fd[oF\\i\\l7$Fi[oF^]u7$ F\\\\oFcau7$F_\\oFj^q7$Fb\\oFj[u7$Fg\\oFfh\\l7$F\\]oFif\\l7$Fa]oFbbr7$ Ff]oFj^q7$F[^oF\\g\\l7$F`^oF^au7$$\"5++]i:D=#R3%F;F[f\\l7$Fe^oFj[u7$$ \"5LL$3_v\\)*H6%F;F[au7$F[rF_\\u7$FeirF\\g\\l7$FhirF`j\\l7$F]jrFibr7$F `jrF^au7$Fcjr$Fg_tF;7$FfjrFj^q7$FijrF[au7$F\\[sFhcr7$F_[sF[au7$Fb[sF[a u7$Fe[sFif\\l7$Fi[s$FhhvF;7$F^\\sF\\g\\l7$Fc\\sFg\\w7$Ff\\sFfh\\l7$Fi \\sFfjt7$F\\]sFi\\u7$F`rFgg\\l7$F^_oF`g\\l7$FerF_\\u7$FjrFbj\\l7$F_sFi ^]l7$FdsF`j\\l7$FisFfh\\l7$Fj]sF`g\\l7$F_^sF`j\\l7$Fb^sFdh\\l7$Fg^sFih \\l7$F\\_sFjaw7$F`_sF[au7$Fc_sFbj\\l7$Fh_sFg\\w7$FcbxFbj\\l7$F]`sF\\i \\l7$FibxF`j\\l7$F^`oF\\h\\l7$Fc`s$!#MF;7$Fh`sFifr7$F]asFi^]l7$FbasFbc r7$FeasF`g\\l7$FhasFahr7$F]bsF`j\\l7$F`bsFf[u7$FcbsFgg\\l7$FfbsFhcr7$F [cs$!#QF;7$F`csF\\h\\l7$Fecs$FggyF;7$F^t$Fj_tF;7$FbuF[dr7$F\\vF[dr7$Fa vF[fr7$FfvF]_w7$F[exFher7$F^exF\\h\\l7$FcexFbj\\l7$Fhex$F[etF;7$F[fxF] _w7$F^fxF]_w7$FafxF\\i\\l7$FdfxF\\g\\l7$FifxFe_w7$F\\gxFf[u7$F_gxFif\\ l7$FdgxFdh\\l7$FggxFahr7$FjgxF^b]l7$F]hxF\\i\\l7$FhhoF[fr7$FahxF[au7$F fhxFc]u7$F[ixFca]l7$F^ixF]^q7$FfixF`j\\l7$F^jxF\\h\\l7$FajxFa^]l7$F[wF \\i\\l7$F`ioFa^]l7$F`wFf[u7$FfioF_\\u7$F[joFbj\\l7$F`joFc`]l7$FcjoF^au 7$FhjoF^[u7$F][pF[fr7$F`[pFher7$Fe[pF_\\u7$Fj[pFhcr7$F]\\pF[j\\l7$Fg\\ pFhcr7$Fa]pFc`]l7$Ff]pF[^u7$Fi]pF^[u7$F^^pFih\\l7$Fa^pFi\\u7$Fd^pF]gr7 $Fi^pFc`]l7$F\\_pF[au7$Fa_pF`j\\l7$Fd_pF\\g\\l7$Fg_pF_\\u7$Fj_pFa^]l7$ FewF_a]l7$F^`pFgg\\l7$Fa`pF]cr7$Fi`pFjaw7$F_apFc`]l7$FdapF`j\\l7$FgapF i\\u7$FecpFjaw7$FjwF]gr7$F]xFjaw7$FbxFih\\l7$FgxFi^]l7$F]ep$!#LF;7$F\\ yFih\\l7$F_^yFie]l7$Fb^yF]gr7$Fe^yF\\h\\l7$Fh^yFbj\\l7$F[_yFi^]l7$F^_y Fjaw7$Fa_yFc`]l7$FcepFi^]l7$$\"5+]i!R!4&\\)p6F0Fih\\l7$Fe_yFjaw7$$\"5M $3_]uX)[s6F0F\\h\\l7$Fh_yF]gr7$F[`yFahr7$F^`yFc`]l7$Fa`yFih\\l7$Fd`yF` fr7$Fg`yFie]l7$Fj`yFg\\w7$F]ayF]gr7$F`ayF_a]l7$FcayFdh\\l7$FfayFih\\l7 $FiayFi^]l7$FayF^b]l7$FfyFber7$F[zFhhr7$F`zF`]s7$FezF][t7$Fjz$F\\j\\lF 07$Fd[lF`et7$Ff\\l$F^^qF07$Fi\\l$F]h\\lF07$F^]l$F^grF07$Fc]lFdgy7$Fh]l Fcj[l7$F[_lF]gp7$FeblFh\\q7$FidlFbiv7$Fcel$Fa]tF07$F[glF\\[\\l7$Fegl$F [`sF07$$\"5M3_+ZL7*e^\"F0$Fe^sF07$F^hyF^i]l7$$\"5+Dc,TN:e=:F0Fbi]l7$Fj gl$F`asF07$FdhyFhi]l7$F_hlFhi]l7$FchlFcdt7$FhhlFa[\\l7$F]ilFa[\\l7$Fbi lFa[\\l7$FfjlFj[w7$Fj[mF\\hp7$F_\\mF_fp7$Fd\\mFfgy7$Fa\\\\lFa\\w7$Fe\\ \\l$FbhrF07$Fh\\\\lFa\\w7$Fi\\mFa\\w7$F\\]\\lF`]q7$F^]mFf_t7$$\"5+DJ?$ Q?\"f#e\"F0Ff_t7$$\"5n;HK94=1%e\"F0Ff_t7$$\"5M3FWX9C`&e\"F0F`h]l7$$\"5 ++Dcw>I+(e\"F0F`h]l7$$\"5M$3-)QIU%**e\"F0F^gyFc]\\l7$$\"5ML3_Diyw)f\"F 0$F_[uF0Fe]\\lFf]\\l7$F]^m$\"$'GF;7$Fd]qFegt7$Fb^m$\"$N%F;7$F[^q$\"$+& F;7$Fg^m$\"$?&F;7$Fe^q$\"$<&F;7$F\\_m$\"$9&F;7$F`[zFaam7$Fe[z$\"$7&F;7 $Fj[zFfam7$F_htF\\am7$Fb\\z$\"$>&F;7$Fbht$\"$8&F;7$F_]z$\"$4&F;7$FghtF ^]^l7$Fi]zF]^^l7$F^^zFh`z7$Fa^z$\"$.&F;7$F`_q$\"$(\\F;7$F_it$\"$%[F;7$ Fa_mFdjy7$Ff_mFhgt7$F[`mF^\\^l7$FfcqFdit7$F``m$\"$L#F;7$F\\dqF97$Fe`m$ \"$8#F;7$Fj`m$Fa]mF;7$FceqFg_^l7$F_am$\"$)>F;7$F[gqFd^w7$Fdam$\"$F#F;7 $FiamF`_^l7$F\\bmFa^w7$FabmFeaz7$Ffbm$\"$%GF;7$F[cm$\"$E$F;7$F`cmFajy- Ffcm6&FhcmFigz$F_iwF[^rF(-F]dm6#%AVerner's~\"most~efficient\"~schemeG- %%FONTG6$%*HELVETICAGF/-%&TITLEG6#%Verror~curves~for~12~stage~order~8~ Runge-Kutta~methodsG-%+AXESLABELSG6$Q\"x6\"Q!F\\b^l-%%VIEWG6$;F(F`cm;$ F^^qF`\\u$F]dwF`\\u" 1 2 0 1 10 0 2 9 1 4 2 1.000000 45.000000 45.000000 0 1 "Enright-Verner scheme" "Verner's Maple scheme" "a modif ication of Verner's Maple scheme" "an \"efficient\" scheme" "a scheme \+ of Tsitouras and Papakostas" "a modification of the Tsitouras-Papakost as scheme" "Verner's \"most efficient\" scheme" }}}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 620 "evalf[20](p lot(['pn_RK8_1'(x)-p(x),'pn_RK8_2'(x)-p(x),'pn_RK8_3'(x)-p(x),'pn_RK8_ 4'(x)-p(x),\n'pn_RK8_5'(x)-p(x),'pn_RK8_6'(x)-p(x),'pn_RK8_7'(x)-p(x)] ,x=2.2..2.8,-1.5e-15..4.2e-15,font=[HELVETICA,9],\ncolor=[COLOR(RGB,.4 ,0,.9),COLOR(RGB,0,.65,0),brown,red,COLOR(RGB,0,.75,.95),magenta,COLOR (RGB,.95,.55,0)],\nlegend=[`Enright-Verner scheme`,`Verner's Maple sch eme`,`a modification of Verner's Maple scheme`,`an \"efficient\" schem e`,`a scheme of Tsitouras and Papakostas`,`a modification of the Tsito uras-Papakostas scheme`,`Verner's \"most efficient\" scheme`],title=`e rror curves for 12 stage order 8 Runge-Kutta methods`));" }}{PARA 13 " " 1 "" {GLPLOT2D 1073 689 689 {PLOTDATA 2 "6--%'CURVESG6%7^u7$$\"#A!\" \"$\"$^*!#?7$$\"5++++]#HyI@#!#>$\"%\"4\"F-7$$\"5+++]([kdWA#F1$\"%E7F-7 $$\"5++++v;\\DPAF1$\"%99F-7$$\"5++++Dq0]BF1$\"$f#F17$$\"5++ ++]U80jBF1$\"$`\"F17$$\"5++++]!ytbP#F1$!#uF17$$\"5+++](QNXpQ#F1$!$s%F1 7$$\"5+++v=/jq$R#F1$!$B*F17$$\"5++++]asY+CF1$!%'R\"F17$$\"5++++DmY>1CF 1$!%b9F17$$\"5+++++y?#>T#F1$!%Q:F17$$\"5++v$fLm(e8CF1$!%k:F17$$\"5++]( =([KD:CF1$!%y:F17$$\"5+]P%)RTg3;CF1$!%$e\"F17$$\"5++D\"yS$)=pT#F1$!%h: F17$$\"5+]7yvE;vWe=CF1$!%j8F17$$\"5+](=<@@<%>CF 1$!%X5F17$$\"5++voz/+D?CF1$!$A'F17$$\"5+]ilZ(z#3@CF1$!$B'F17$$\"5++]i: !f:>U#F1$!$N'F17$$\"5++Dc^v6eBCF1$!$T'F17$$\"5+++](3wY_U#F1$!$U'F17$$ \"5+]7y]BR)fU#F1$!$>'F17$$\"5++D19'3@nU#F1$!$!eF17$$\"5+]PMx[#euU#F1$! $j%F17$$\"5++]iS6a>GCF1$!$$=F17$$\"5++v=nO(p'HCF1$\"%%\\\"F17$$\"5+++v $>1W6V#F1$\"%(G#F17$$\"5++DJ?(Q=EV#F1$\"%9BF17$$\"5++](oCr#4MCF1$\"%IB F17$$\"5+]i:5v)H[V#F1$\"%UBF17$$\"5++vVtPqcNCF1$\"%mBF17$$\"5+](=n.?/j V#F1$\"%PCF17$$\"5+++++j8/PCF1$\"%'e#F17$$\"5++vV[J*e'QCF1$\"%4RF17$$ \"5++](o**\\w-W#F1$\"%F\")F17$$\"5++DJXoS*=W#F1$\"%%=)F17$$\"5+++v$pj6 NW#F1$\"%_#)F17$$\"5+](oz6U?VW#F1$\"%!G)F17$$\"5++v=U0#H^W#F1$\"%?$)F1 7$$\"5+]iSm*)z$fW#F1$\"%9%)F17$$\"5++]i!RxYnW#F1$\"%%e)F17$$\"5+]P%[\" ebbZCF1$\"%i!*F17$$\"5++D1RUVO[CF1$\"&4,\"F17$$\"5+v=<^M(o([CF1$\"&C5 \"F17$$\"5+]7GjEJ<\\CF1$\"&^B\"F17$$\"5+D1Rv=vd\\CF1$\"&PU\"F17$$\"5++ +](3\">)*\\CF1$\"&6p\"F17$$\"5+++DJED1`CF1$\"&Ks\"F17$$\"5++++vTJ9cCF1 $\"&Dv\"F17$$\"5++](o%\\ModCF1$\"&@%=F17$$\"5+++v=dPAfCF1$\"&5C#F17$$ \"5+](=Z#F1$\"&9p$F17$$\"5++vVB>WbtCF1$\"&!*p$F17$$ \"5++++vo:;vCF1$\"&jq$F17$$\"5++]il)H8\"yCF1$\"&Ey$F17$$\"5+++DcG]1\"[ #F1$\"&y6%F17$$\"5++Dc^$*3a#[#F1$\"&#=TF17$$\"5++](o%en,%[#F1$\"&%>TF1 7$$\"5++v=UBE\\&[#F1$\"&e6%F17$$\"5+++]P)[op[#F1$\"&*4TF17$$\"5++v$4\" 3/c)[#F1$\"&T2%F17$$\"5++]P%yK_,\\#F1$\"&0\"RF17$$\"5++D\"yvCW<\\#F1$ \"&[!RF17$$\"5+++DJnhL$\\#F1$\"&r*QF17$$\"5++vo/(3G\\\\#F1$\"&k)QF17$$ \"5++]7y1+_'\\#F1$\"&q'QF17$$\"5++Dc^E>6)\\#F1$\"&9x$F17$$\"5++++DYQq* \\#F1$\"&aN$F17$$\"5+D\"y]bB<,]#F1$\"&1>$F17$$\"5+]i:&[iI0]#F1$\"&&)=$ F17$$\"5+vVB:9S%4]#F1$\"&]=$F17$$\"5++DJX.uN,DF1$\"&L=$F17$$\"5+](oa?= %=-DF1$\"&r<$F17$$\"5++]ilg4,.DF1$\"&8<$F17$$\"5++v$fy^kY]#F1$\"&e:$F1 7$$\"5+++D1v!=j]#F1$\"&'HJF17$$\"5+]iSm`[92DF1$\"&b4$F17$$\"5++DcEK;(z ]#F1$\"&t,$F17$$\"5+](=n3T)z3DF1$\"&A&GF17$$\"5++](o%*=D'4DF1$\"&(>DF1 7$$\"5+DJ&p(y&Q+^#F1$\"&$zAF17$$\"5+]7.2o>X5DF1$\"&^F#F17$$\"5+v$4rtNl 3^#F1$\"&F17 $$\"5++DJqA(4%>DF1$\"&\"om1`#F1$\"%'4*F 17$$\"5++D1*[\"3AKDF1$\"%F!*F17$$\"5++]P%[VvP`#F1$\"%^*)F17$$\"5+]7.#[ u_X`#F1$\"%.*)F17$$\"5++voza+LNDF1$\"%k))F17$$\"5+]PMxkt5ODF1$\"%'y)F1 7$$\"5++++vuY)o`#F1$\"%v')F17$$\"5+++](G(*3La#F1$\"%bfF17$$\"5+++++rKt \\DF1$\"%P\\F17$$\"5+++]P$>=gb#F1$\"%#e%F17$$\"5++++v:JIiDF1$\"%OVF17$ $\"5++]7`3OMlDF1$\"%[UF17$$\"5+++DJ,TQoDF1$\"%1UF17$$\"5++DJqZV!*pDF1$ \"%xVF17$$\"5++]P4%fC9d#F1$\"%iVF17$$\"5++vV[S[%Hd#F1$\"%>VF17$$\"5+++ ](o3lWd#F1$\"%nUF17$$\"5++++D#))oze#F1F_fm7$$\"5+++]iid.%f#F1$\"%6VF17 $$\"5+++++VE5+EF1$\"%XVF17$$\"5++++vK.e1EF1$\"%VTF17$$\"5++++]A!eIh#F1 $\"%!4%F17$$\"5+++](=_(zCEF1$\"%HQF17$$\"5++++]&*=jPEF1$\"%LNF17$$\"5+ ++](3/3(\\EF1$\"%#G$F17$$\"5+++]P#4JBm#F1$\"%1IF17$$\"5++++]KCnuEF1$\" %oFF17$$\"5+++](=n#f(o#F1$\"&Y`#F-7$$\"5+++++)RO+q#F1$\"&;L#F-7$$\"5++ ++D0>w7FF1$\"&!R@F-7$$\"5+++]()Q?QDFF1$\"&m'>F-7$$\"5+++++J'ypt#F1$\"& q#=F-7$$\"5++++DM'p-v#F1$\"&xn\"F-7$$\"5+++++ms:iFF1$\"&Cc\"F-7$$\"5++ +](3'>$[x#F1$\"&HX\"F-7$$\"5+++]7hK'py#F1$\"&*f8F-7$$\"#GF*$\"&(p7F--% &COLORG6&%$RGBG$\"\"%F*$\"\"!Fc^n$\"\"*F*-%'LEGENDG6#%6Enright-Verner~ schemeG-F$6%7fu7$F($!$>\"F-7$F/$\"#`F-7$F5$\"$]#F-7$F:$\"$:&F-7$F?$\"$ ()*F-7$FD$\"%&G\"F-7$FI$\"%Z;F-7$FN$\"%)*>F-7$FS$\"$Z#F17$FX$\"$w#F17$ Fgn$\"$1$F17$F\\o$\"$=$F17$Fao$\"$%HF17$Ffo$\"$a#F17$$\"5++++]hDJpBF1$ \"$5#F17$F[p$\"$f\"F17$$\"5++v=&F17$Fh\\l$!%AcF17$Fb]l$!%jkF17$Fg]l$!%tlF17$F\\^l$!%lmF17$Ff^ l$!%RvF17$Fd`l$!%N&)F17$Fi`l$!%h&)F17$F^al$!%2')F17$Fcal$!%,()F17$Fhal $!%v\"*F17$F]bl$!%J)*F17$Fbbl$!&![5F17$Fgbl$!&,0\"F17$F\\cl$!&-0\"F17$ Facl$!&@0\"F17$Ffcl$!&U0\"F17$$\"5++DJqLujwCF1$!&01\"F17$F[dl$!&r2\"F1 7$$\"5++v$4O;*ezCF1$!&p:\"F17$F`dl$!&P?\"F17$Fedl$!&V?\"F17$Fjdl$!&F? \"F17$F_el$!&W?\"F17$Fdel$!&^?\"F17$Fiel$!&Q@\"F17$F^fl$!&wD\"F17$Fcfl $!&^D\"F17$Fhfl$!&KD\"F17$$\"5+](ozr7KT\\#F1$!&BD\"F17$F]gl$!&;D\"F17$ $\"5+]iS\"p/Cd\\#F1$!&*[7F17$Fbgl$!&]C\"F17$Fggl$!&VB\"F17$F\\hl$!&&)= \"F17$Fjil$!&V;\"F17$Fdjl$!&F:\"F17$F^[m$!&%G6F17$Fh[m$!&(=5F17$F]\\m$ !%7'*F17$Fb\\mF`bo7$Fg\\m$!%-'*F17$F\\]m$!%'e*F17$Fa]m$!%f&*F17$Ff]m$! %M&*F17$F[^m$!%'[*F17$F`^m$!%@%*F17$Fe^m$!%j#*F17$Fj^m$!%3()F17$F__m$! %%*zF17$Fd_m$!%LpF17$Fi_m$!%4pF17$F^`m$!%#)oF17$Fc`m$!%RoF17$Fh`m$!%1o F17$Fbam$!%DnF17$F\\bm$!%NlF17$$\"5+]7y+lULGDF1$!%riF17$Fabm$!%HdF17$$ \"5+]P4'\\)))))HDF1$!%8YF17$Ffbm$!%rVF17$F`cm$!%*H%F17$Fddm$!%zTF17$$ \"5++Dc,7xoPDF1$!%nSF17$$\"5++]7G\\2\\QDF1$!%HQF17$$\"5++voa'y$HRDF1$! %9LF17$$\"5+++D\"Q#o4SDF1$!%bCF17$$\"5++]PM)*GqTDF1$!%HCF17$Fidm$!%6CF 17$$\"5++]iSZ]\"\\a#F1$!%$R#F17$$\"5+++v$>7@la#F1$!%[BF17$$\"5++](ok>F \"[DF1$!%$>#F17$F^em$!%\"[\"F17$Fcem$!%y6F17$Fhem$!$P&F17$$\"5++D19iL# Qc#F1$!$M&F17$F]fm$!$A&F17$$\"5+]ilsJP5mDF1$!$=&F17$$\"5++v=#\\&Q'oc#F 1$!$0&F17$$\"5+](=<\"yRinDF1$!$)[F17$Fbfm$!$Z%F17$$\"5+]7y]CU9pDF1$!$v $F17$Fgfm$!$=#F17$$\"5+]P%)*3Zk1d#F1$!$#>F17$F\\gm$!$&=F17$Fagm$!$y\"F 17$Ffgm$!$\"=F17$$\"5++voHhI:wDF1Ffv7$$\"5++](=d.Tyd#F1$!$j\"F17$$\"5+ +D195!H&zDF1$!#pF17$$\"5+++Dc%)p@\"e#F1$!\"$F17$$\"5++]iSLHf%e#F1$\"\" 'F17$F[hm$\"#:F17$F^hm$\"#'*F17$Fchm$\"$s\"F17$$\"5+++](y[TLg#F1$\"$l \"F17$Fhhm$\"$n\"F17$$\"5+++v=b(*>3EF1$\"$o\"F17$$\"5+++]ix\">)4EF1$\" $8#F17$$\"5+++D1+'Q9h#F1$\"$>#F17$F]im$\"$?#F17$Fbim$\"$S#F17$Fgim$\"$ Y#F17$F\\jm$\"$^#F17$Fajm$\"$@#F17$Ffjm$\"$+#F17$F[[n$\"%g;F-7$F`[n$\" %u6F-7$Fe[n$\"$G)F-7$Fj[n$\"$7&F-7$F_\\n$\"$q#F-7$Fd\\n$!$8\"F-7$Fi\\n $!$o#F-7$F^]n$!$)QF-7$Fc]n$!$u%F-7$Fh]n$!$*eF--F]^n6&F_^nFb^n$\"#l!\"# Fb^n-Fg^n6#%6Verner's~Maple~schemeG-F$6%7\\t7$F($\"$S%F-7$F/$\"$%fF-7$ F5$\"$a(F-7$F:$\"$q*F-7$F?$\"%#H\"F-7$FD$\"%::F-7$FI$\"%%z\"F-7$FN$\"% c?F-7$FS$\"$R#F17$FX$\"$j#F17$Fgn$\"$$HF17$F\\o$\"$3$F17$FaoFafp7$Ffo$ \"$&GF17$Fhan$Fe_nF17$F[p$\"$2#F17$F`bnFjan7$Febn$\"$#=F17$Fjbn$\"$V\" F17$F_cn$\"#nF17$Fdcn$\"#ZF17$Ficn$\"#WF17$F^dn$Fb_nF17$Fbdn$\"#^F17$F gdn$\"#VF17$F`p$\"#FF17$Fep$!$H#F17$Fjp$!$y'F17$Fdq$!%W8F17$Fbs$!%#f\" F17$F`u$!%*H#F17$Fdv$!%zCF17$F^w$!%cLF17$Fcw$!%uLF17$Fhw$!%.MF17$Fbx$! %TMF17$F\\y$!%/NF17$Fay$!%sPF17$Ffy$!%:YF17$F[z$!%aYF17$F`z$!%zYF17$Fj z$!%CZF17$Fd[l$!%'z%F17$F^\\l$!%-]F17$Fb]l$!%$)fF17$Fg]l$!%#3'F17$F\\^ l$!%khF17$Ff^l$!%8oF17$Fd`l$!%cvF17$Fhal$!%`!)F17$Ffcl$!%/\"*F17$F[dl$ !%$H*F17$F`dl$!&!Q5F17$Fedl$!&&Q5F17$Fjdl$!&q.\"F17$F_el$!&)Q5F17$Fdel $!&(R5F17$Fiel$!&/0\"F17$F^fl$!&F5\"F17$Fcfl$!&05\"F17$Fhfl$!&))4\"F17 $F^`o$!&\")4\"F17$F]gl$!&w4\"F17$Ff`o$!&`4\"F17$Fbgl$!&@4\"F17$$\"5+]P %[m'fJ(\\#F1$!&34\"F17$Fggl$!&k3\"F17$$\"5+]7GQ')y!*)\\#F1$!&!z5F17$F \\hl$!&K1\"F17$Fjil$!&([5F17$FdjlF_^q7$Fh[m$!%u$*F17$Ff]m$!%h))F17$F[^ m$!%;))F17$F`^m$!%e()F17$Fe^m$!%A')F17$Fj^m$!%d\")F17$F__mFj\\q7$Fd_m$ !%amF17$Fi_m$!%KmF17$F^`m$!%0mF17$Fc`m$!%klF17$Fh`m$!%KlF17$Fbam$!%ckF 17$F\\bm$!%%G'F17$Fdeo$!%YgF17$Fabm$!%dbF17$F\\fo$!%VXF17$Ffbm$!%AVF17 $F`cm$!%^UF17$Fddm$!%NTF17$Fjfo$!%FSF17$F_go$!%*z$F17$FdgoF_hn7$FigoFg ip7$F^ho$!%`CF17$Fidm$!%NCF17$Ffho$!%;CF17$F[io$!%rBF17$F`ioFign7$F^em $!%)[\"F17$Fcem$!%\"=\"F17$Fhem$!$\"\\F17$F^jo$!$*[F17$F]fm$!$y%F17$Ff jo$!$t%F17$F[[p$!$g%F17$F`[p$!$S%F17$Fbfm$!$%RF17$Fh[p$!$7$F17$Fgfm$!$ O\"F17$F`\\p$!$2\"F17$F\\gm$!$,\"F17$Fagm$!#&*F17$Ffgm$!#**F17$F^]p$!$ -\"F17$Fa]p$!#!)F17$Ff]p$\"#DF17$F[^p$\"#(*F17$F`^p$\"$.\"F17$F[hm$\"$ 6\"F17$F^hm$\"$\">F17$FchmFU7$F^_pFf`n7$FhhmFf`n7$Ff_p$\"$X#F17$F[`p$ \"$$GF17$F``pFefp7$F]im$\"$'GF17$Fbim$\"$#HF17$Fgim$\"$)GF17$F\\jmF_[r 7$Fajm$\"$m#F17$FfjmFhfp7$F[[n$\"%HAF-7$F`[n$\"%3>F-7$Fe[n$\"%7;F-7$Fj [n$\"%N8F-7$F_\\n$\"%H6F-7$Fd\\n$\"$D)F-7$Fi\\n$\"$n'F-7$F^]n$\"$J&F-7 $Fc]n$\"$C%F-7$Fh]n$\"$)HF--%'COLOURG6&F_^n$\")#)eqk!\")$\"))eqk\"F\\^ rF]^r-Fg^n6#%Ha~modification~of~Verner's~Maple~schemeG-F$6%7cs7$F($Fij qF-7$F/$\"$?%F-7$F5$\"$j&F-7$F:$\"$f(F-7$F?$\"%d5F-7$FD$\"%d7F-7$FI$\" %7:F-7$FN$\"%[2\"F17$Fhfl$!&.2\"F 17$Fbgl$!&U1\"F17$F\\hl$!&V0\"F17$Fjil$!&j/\"F17$Fdjl$!&o.\"F17$Fh[mFj bo7$Ff]m$!%P\"*F17$F[^m$!%#4*F17$F`^m$!%L!*F17$Fe^m$!%/*)F17$Fj^m$!%)[ )F17$F__m$!%hzF17$Fd_m$!%vrF17$Fi_m$!%^rF17$F^`m$!%BrF17$Fc`m$!%yqF17$ Fh`m$!%WqF17$Fbam$!%kpF17$F\\bm$!%$z'F17$FdeoF]cq7$Fabm$!%0hF17$F\\fo$ !%o^F17$Ffbm$!%i\\F17$F`cm$!%\")[F17$Fddm$!%_ZF17$Fjfo$!%SYF17$F_go$!% 4WF17$Fdgo$!%@RF17$Figo$!%DJF17$F^ho$!%$4$F17$Fidm$!%oIF17$$\"5++D195? 6WDF1$!%^IF17$Ffho$!%VIF17$$\"5++v=n%3=da#F1$!%@IF17$F[io$!%!*HF17$F`i o$!%:GF17$F^em$!%p?F17$Fcem$!%PF-7$FN$\"%J@F-7$FS$\"$P#F17$FX$\"$d#F17$FgnF \\[r7$F\\o$\"$.$F17$Fao$\"$<$F17$FfoFin7$Fhan$\"$z#F17$F[p$\"$W#F17$F` bnFf`n7$Febn$\"$A#F17$Fjbn$\"$%=F17$F_cn$\"$8\"F17$Fdcn$\"#&*F17$Ficn$ \"#$*F17$F^dn$\"$-\"F17$Fbdn$\"$+\"F17$FgdnF_js7$F`p$\"#yF17$Fep$!$m\" F17$Fjp$!$#fF17$$\"5+++]Pg4L.CF1$!$4'F17$F_q$!$C'F17$$\"5+++]7s$e!4CF1 $!$Z)F17$Fdq$!%*=\"F17$Fbs$!%/9F17$F`u$!%n>F17$Fdv$!%0@F17$F^w$!%&o#F1 7$Fcw$!%)p#F17$Fhw$!%AFF17$Fbx$!%`FF17$F\\y$!%*z#F17$F`z$!%EMF17$Fb]l$ !%8SF17$Fg]l$!%wSF17$F\\^l$!%JTF17$Ff^l$!%NWF17$Fd`l$!%rZF17$Fhal$!%l] F17$Ffcl$!%%p&F17$Ff]o$!%PdF17$F[dl$!%LeF17$F^^o$!%yjF17$F`dl$!%1nF17$ Fedl$!%6nF17$Fjdl$!%'p'F17$F_el$!%:nF17$Fdel$!%HnF17$Fiel$!%doF17$F^fl $!%vuF17$Fcfl$!%fuF17$Fhfl$!%\\uF17$F^`o$!%YuF17$F]gl$!%XuF17$Ff`o$!%G uF17$Fbgl$!%1uF17$F]`q$!%8uF17$Fggl$!%7uF17$Fe`q$!%JuF17$F\\hl$!%iuF17 $F`il$!%![(F17$Fjil$!%]uF17$$\"5+]7yDRx$Q]#F1$!%ZuF17$F_jl$!%NuF17$$\" 5+]P4Y'H\"\\0DF1$!%dF17$Fd_m$! %[^F17$Fi_m$!%H^F17$F^`m$!%3^F17$Fc`m$!%w]F17$Fh`m$!%a]F17$Fbam$!%$*\\ F17$F\\bm$!%m[F17$Fdeo$!%'p%F17$Fabm$!%dVF17$F\\fo$!%gOF17$Ffbm$!%4NF1 7$F`cm$!%^MF17$Fddm$!%eLF17$Fjfo$!%uKF17$F_go$!%+JF17$Fdgo$!%LFF17$Fig o$!%Y@F17$F^ho$!%B@F17$Fidm$!%3@F17$Ff^s$!%&4#F17$Ffho$!%#4#F17$F^_s$! %v?F17$F[io$!%`?F17$F`io$!%B>F17$F^em$!%o8F17$Fcem$!%G6F17$Fhem$FjesF1 7$F^jo$!$l&F17$F]fm$!$a&F17$Ffjo$!$\\&F17$F[[p$!$N&F17$F`[p$!$;&F17$Fb fm$!$v%F17$Fh[p$!$2%F17$Fgfm$!$j#F17$F`\\p$!$R#F17$F\\gm$!$K#F17$Fagm$ !$D#F17$Ffgm$!$F#F17$F[hm$!#LF17$F^hm$\"#PF17$FchmF\\js7$F^_p$\"#!*F17 $FhhmF_js7$Ff_p$\"##*F17$F[`p$\"$C\"F17$F``p$\"$E\"F17$F]im$\"$H\"F17$ Fbim$\"$T\"F17$Fgim$\"$Z\"F17$F\\jm$\"$j\"F17$FajmFho7$Ffjm$\"$_\"F17$ F[[n$\"%*R\"F-7$F`[n$\"%k7F-7$Fe[nF^gs7$Fj[n$\"$#*)F-7$F_\\n$\"$j(F-7$ Fd\\n$\"$e&F-7$Fi\\n$\"$W%F-7$F^]n$\"$W$F-7$Fc]n$Fg[rF-7$Fh]n$FaarF--F ]^n6&F_^nFb^n$\"#vFhcp$F]jsFhcp-Fg^n6#%Ea~scheme~of~Tsitouras~and~Papa kostasG-F$6%7_tFgfs7$F/$\"$O*F-7$F5$\"%(3\"F-7$F:$\"%*G\"F-7$F?$\"%Z:F -7$FD$\"%RF-7$FN$\"%!>#F-7$FS$\"$U#F17$FX$\"$g#F17$FgnFhj q7$F\\oF_[r7$FaoFi`n7$$\"5++++D+cI`BF1$\"$#GF17$$\"5+++++\"=alN#F1Fefp 7$$\"5+++]Pr%y\"eBF1Fhjq7$$\"5++++vhF!)fBF1Fejq7$$\"5+++]7_qUhBF1$\"$G #F17$FfoF^hs7$Fhan$\"$'=F17$F[p$\"$F\"F17$F`bnF[eu7$FebnFh]u7$Fjbn$\"# TF17$F_cn$!#bF17$Fdcn$!##)F17$Ficn$!#')F17$F^dn$F[esF17$FbdnFfeu7$Fgdn $!##*F17$F`p$!$6\"F17$Fep$!$[%F17$Fjp$!%75F17$Fb[t$!%P5F17$F_q$!%i5F17 $Fj[t$!%d8F17$Fdq$!%?=F17$Fbs$!%7@F17$F`u$!%#*GF17$Fdv$!%zIF17$F^w$!%7 RF17$Fhw$!%pRF17$F\\y$!%qSF17$F`z$!%\"*\\F17$Fb]l$!%*y&F17$Fg]l$!%weF1 7$F\\^l$!%[fF17$Ff^l$!%#G'F17$Fd`l$!%imF17$FhalF^\\s7$Ffcl$!%uvF17$F[d l$!%4xF17$F`dl$!%Z&)F17$Fedl$!%_&)F17$Fjdl$!%P&)F17$F_el$!%b&)F17$Fdel $!%p&)F17$Fiel$!%+()F17$F^fl$!%R$*F17$Fcfl$!%?$*F17$Fhfl$!%1$*F17$F^`o $!%,$*F17$F]gl$!%)H*F17$Ff`o$!%z#*F17$Fbgl$!%c#*F17$F]`q$!%k#*F17$Fggl $!%m#*F17$Fe`q$!%&H*F17$F\\hl$!%\\$*F17$F`il$!%t$*F17$Fjil$!%O$*F17$Fd ct$!%H$*F17$F_jl$!%8$*F17$F\\dt$!%\"H*F17$Fdjl$!%d#*F17$Fh[m$!%.()F17$ Ff]m$!%(Q)F17$F[^m$!%Y$)F17$F`^m$!%%H)F17$Fe^m$!%'=)F17$Fj^m$!%_yF17$F __m$!%>uF17$Fd_m$!%gnF17$Fi_m$!%PnF17$F^`m$!%5nF17$Fc`m$!%omF17$Fh`m$! %PmF17$Fbam$!%jlF17$F\\bm$!%9kF17$Fdeo$!%9iF17$Fabm$!%3eF17$F\\fo$!%h \\F17$Ffbm$!%tZF17$F`cm$!%%p%F17$Fddm$!%vXF17$Fjfo$!%sWF17$F_go$!%eUF1 7$Fdgo$!%)z$F17$Figo$!%OIF17$F^ho$!%/IF17$Fidm$!%\")HF17$Ff^s$!%kHF17$ Ffho$!%dHF17$F^_s$!%NHF17$F[io$!%1HF17$F`io$!%QFF17$F^em$!%&*>F17$Fcem $!%j;F17$Fhem$!$n)F17$F^jo$!$g)F17$F]fm$!$X)F17$Ffjo$!$Q)F17$F[[p$!$?) F17$F`[p$!$$zF17$Fbfm$!$L(F17$Fh[p$!$E'F17$Fgfm$!$/%F17$F`\\p$!$m$F17$ F\\gm$!$e$F17$Fagm$!$\\$F17$Ffgm$!$^$F17$F[hm$!#QF17$F^hmF\\cn7$FchmF_ jq7$F^_pFfis7$Fhhm$\"$&=F17$$\"5++](oR/!R2EF1$\"$!>F17$Ff_p$\"$(=F17$$ \"5++]iSm%4!4EF1Fjfp7$F[`pF^hs7$F``p$\"$V#F17$F]imF_is7$Fbim$\"$k#F17$ Fgim$\"$r#F17$F\\jmF\\[r7$Fajm$\"$p#F17$FfjmFco7$F[[n$\"%pBF-7$F`[n$\" %T@F-7$Fe[n$\"%v=F-7$Fj[n$\"%?;F-7$F_\\n$\"%K9F-7$Fd\\n$\"%j6F-7$Fi\\n $\"%05F-7$F^]n$\"$j)F-7$Fc]n$\"$](F-7$Fh]n$\"$='F--Fh]r6&F_^nF_fsFb^nF _fs-Fg^n6#%Ra~modification~of~the~Tsitouras-Papakostas~schemeG-F$6%7[t 7$F($\"$\"RF-7$F/$\"$Y%F-7$F5$\"$'\\F-7$F:$\"$%eF-7$F?$\"$,(F-7$FD$\"$ 3)F-7$FIFj_n7$FN$\"%q6F-7$FSF^_u7$FX$\"$\"=F17$FgnFe`p7$F\\o$\"$`#F17$ Fao$\"$(HF17$FfoF^fp7$Fhan$\"$h#F17$F[pFaap7$F`bn$\"$B#F17$Febn$\"$#>F 17$FjbnFg^u7$F_cn$\"#hF17$Fdcn$\"#QF17$Ficn$\"#NF17$F^dnFjgp7$FbdnF`eu 7$Fgdn$\"#LF17$F`pFe^p7$Fep$!$4$F17$Fjp$!$\"*)F17$Fb[t$!$8*F17$F_q$!$P *F17$Fj[t$!%e7F17$Fdq$!%m$F17$F^ w$!%.UF17$Fhw$!%lUF17$F\\y$!%wVF17$Fay$!%MYF17$Ffy$!%KaF17$F[z$!%xaF17 $F`z$!%3bF17$FjzF\\dq7$Fd[l$!%CcF17$F^\\l$!%ndF17$Fb]l$!%JkF17$Fg]l$!% FlF17$F\\^lFjbq7$Ff^l$!%?pF17$Fd`l$!%\"H(F17$Ffcl$!%s!)F17$Fdel$!%z*)F 17$Fiel$!%B\"*F17$F^fl$!%<)*F17$Fcfl$!%(z*F17$Fhfl$!%$y*F17$F]gl$!%u(* F17$Fbgl$!%M(*F17$F]`q$!%[(*F17$Fggl$!%l(*F17$Fe`q$!%E)*F17$F\\hl$!%V* *F17$Fahl$!&R+\"F17$Ffhl$!&C+\"F17$F[il$!&A+\"F17$F`il$!&++\"F17$Feil$ !%&)**F17$Fjil$!%h**F17$Fdct$!%`**F17$F_jl$!%O**F17$F\\dt$!%9**F17$Fdj l$!%z)*F17$Fh[m$!%l$*F17$Ff]m$!%\\!*F17$F[^m$!%/!*F17$F`^m$!%[*)F17$Fe ^m$!%P))F17$Fj^m$!%w%)F17$F__m$!%#*zF17$Fd_m$!%SsF17$Fi_m$!%:sF17$F^`m $!%(=(F17$Fc`m$!%UrF17$Fh`m$!%2rF17$Fbam$!%HqF17$F\\bm$!%joF17$Fdeo$!% LmF17$Fabm$!%^hF17$F\\fo$!%E^F17$Ffbm$!%)*[F17$F`cm$!%<[F17$Fddm$!%)o% F17$Fjfo$!%pXF17$F_go$!%7VF17$Fdgo$!%[PF17$Figo$!%\"z#F17$F^ho$!%hFF17 $Fidm$!%SFF17$Ff^s$!%DFF17$Ffho$!%=FF17$F^_s$!%(p#F17$F[io$!%mEF17$F`i o$!%oCF17$F^em$!%W:F17$Fcem$!%j6F17$FhemF]\\p7$F^joF]\\p7$F]fm$!$4#F17 $Ffjo$!$0#F17$F[[p$F^esF17$F`[p$!$g\"F17$FbfmFghq7$Fh[p$Fi]nF17$FgfmFh jq7$F`\\p$\"$C$F17$F\\gm$\"$G$F17$Fagm$\"$H$F17$Ffgm$\"$?$F17$F^]p$\"$ 9$F17$Fa]p$\"$T$F17$Ff]p$\"$!\\F17$F[^p$\"$'eF17$F`^p$\"$\"eF17$F[hm$ \"$#eF17$F^hm$F]]rF17$Fchm$\"$3(F17$Fhhm$\"$x'F17$F]imFdax7$Fbim$\"$F' F17$Fgim$\"$q&F17$F\\jm$\"$D&F17$Fajm$\"$q%F17$Ffjm$FddsF17$F[[n$\"%`R F-7$F`[n$\"%MOF-7$Fe[n$\"%+LF-7$Fj[n$\"%,IF-7$F_\\n$\"%$z#F-7$Fd\\n$\" %nDF-7$Fi\\n$\"%)Q#F-7$F^]n$\"%;AF-7$Fc]n$\"%v?F-7$Fh]n$\"%P>F--F]^n6& F_^nF^au$FadsFhcpFb^n-Fg^n6#%AVerner's~\"most~efficient\"~schemeG-%%FO NTG6$%*HELVETICAGFe^n-%&TITLEG6#%Verror~curves~for~12~stage~order~8~Ru nge-Kutta~methodsG-%+AXESLABELSG6$Q\"x6\"Q!Feex-%%VIEWG6$;F(Fh]n;$!#:! #;$\"#UF^fx" 1 2 0 1 10 0 2 9 1 4 2 1.000000 45.000000 45.000000 0 1 " Enright-Verner scheme" "Verner's Maple scheme" "a modification of Vern er's Maple scheme" "an \"efficient\" scheme" "a scheme of Tsitouras an d Papakostas" "a modification of the Tsitouras-Papakostas scheme" "Ver ner's \"most efficient\" scheme" }}}}{PARA 0 "" 0 "" {TEXT -1 0 "" }} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 618 "evalf[20](plot(['pn_RK8_1'( x)-p(x),'pn_RK8_2'(x)-p(x),'pn_RK8_3'(x)-p(x),'pn_RK8_4'(x)-p(x),\n'pn _RK8_5'(x)-p(x),'pn_RK8_6'(x)-p(x),'pn_RK8_7'(x)-p(x)],x=2.8..5,-0.3e- 16..3.9e-16,font=[HELVETICA,9],\ncolor=[COLOR(RGB,.4,0,.9),COLOR(RGB,0 ,.65,0),brown,red,COLOR(RGB,0,.75,.95),magenta,COLOR(RGB,.95,.55,0)], \nlegend=[`Enright-Verner scheme`,`Verner's Maple scheme`,`a modificat ion of Verner's Maple scheme`,`an \"efficient\" scheme`,`a scheme of T sitouras and Papakostas`,`a modification of the Tsitouras-Papakostas s cheme`,`Verner's \"most efficient\" scheme`],title=`error curves for 1 2 stage order 8 Runge-Kutta methods`));" }}{PARA 13 "" 1 "" {GLPLOT2D 995 833 833 {PLOTDATA 2 "6--%'CURVESG6%7cq7$$\"#G!\"\"$\"&(p7!#?7$$\"5 nmmm;RP&z%G!#>$\"&8.\"F-7$$\"5LLL$37.y'*)GF1$\"%q!*F-7$$\"5+++DJ'pRJ\" HF1$\"%*f)F-7$$\"5nmmmTh8gOHF1$\"%w#)F-7$$\"5nmm;a`*4%[HF1$\"%m\")F-7$ $\"5nmmmmX&=-'HF1$\"%%4)F-7$$\"5nmm;zPr-sHF1$\"%T!)F-7$$\"5nmmm\"*Hd$Q )HF1$\"%J!)F-7$$\"5+++D\"emSt+$F1$\"%#3)F-7$$\"5LLL$3JF1$\"%)*)*F-7$$\"5nmm;a25BmJF1$\"&%\\6F-7$$\"5+++]P$o`F@$F1$\"&;O \"F-7$$\"5LLLL$Q(zggKF1$\"&Dk\"F-7$$\"5+++]P;Io\"G$F1$\"&hy\"F-7$$\"5n mmm\"*e!eFI$F1$\"&P$>F-7$$\"5MLL$3_c$[ELF1$\"&y6#F-7$$\"5++++]r!4-N$F1 $\"&6J#F-7$$\"5+++++(*>.uLF1$\"&d\\#F-7$$\"5++++]A\\&yR$F1$\"&hm#F-7$$ \"5++++]DF\"3U$F1$\"%3GF17$$\"5++++]G0xVMF1$\"%()GF17$$\"5ML3-j*\\#)*[ MF1$\"%(*GF17$$\"5nm;/wqW>aMF1$\"%**GF17$$\"5++D1*=W1%fMF1$\"%.HF17$$ \"5MLL3-8%=YY$F1$\"%,HF17$$\"5nmT5:%QI)pMF1$\"%#*GF17$$\"5++]7GbB/vMF1 $\"%*)GF17$$\"5M3-Q1[`MwMF1$\"%))GF17$$\"5n;aj%3M[wZ$F1Fbu7$$\"5+D1*GO L^*yMF1$\"%\")GF17$$\"5MLe9TEVD![$F1$\"%pGF17$$\"5+]il(>JgG[$F1$\"%qGF 17$$\"5nmm;a(Hma[$F1$\"%lGF17$$\"5+++vo[kD5NF1$\"%pFF17$$\"5LLLL$)*fY] `$F1$\"%eEF17$$\"5LLLLL>w/xNF1$\"&FW#F-7$$\"5+++]()*y/fi$F1$\"&V>#F-7$ $\"5LLLLLk;:pOF1$\"&a*>F-7$$\"5+++]()R.g;PF1$\"&a\"=F-7$$\"5+++]i*p#yh PF1$\"&5p\"F-7$$\"5mmmT&eAa`y$F1$\"&[k\"F-7$$\"5LLLL3_d#*3QF1$\"&Th\"F -7$$\"5LL$eRQF1$\"&Zg\"F-7$$\"5LLLeRrx\"F-7$$\"5nmmmT2QCNSF1$\"&'*)=F -7$$\"5+++++F`N#3%F1$\"&7,#F-7$$\"5++++vdZWGTF1$\"&C6#F-7$$\"5++]7GJKf RTF1$\"&W8#F-7$$\"5+++D\"[qT2:%F1$\"&3:#F-7$$\"5++]PMy,*=;%F1$\"&(f@F- 7$$\"5+++](=lQI<%F1$\"&s;#F-7$$\"5++](o48%F1$\"&(e@F-7$$\"5++]i:*3u,@%F1$\"&T9#F-7$$\"5++++DoDbAUF1$\"&47#F-7 $$\"5mmm;H'z(zWUF1$\"&>1#F-7$$\"5LLLLLCI/nUF1$\"&V)>F-7$$\"5++++]#3YXJ %F1$\"&ix\"F-7$$\"5nmm;a84fdVF1$\"&jd\"F-7$$\"5++++]$G]YS%F1$\"&QP\"F- 7$$\"5LLL$3K[H*[WF1$\"&F@\"F-7$$\"5+++]P0S@&\\%F1$\"&)z5F-7$$\"5LLLL$e el/a%F1$\"%x)*F-7$$\"5mmmTN\"p_Tc%F1$\"%W&*F-7$$\"5+++](ozRye%F1$\"%9$ *F-7$$\"5nm;H#eaY#*f%F1$\"%_#*F-7$$\"5MLL3x%H`1h%F1$\"%,#*F-7$$\"5++]( =P/g?i%F1$\"%%>*F-7$$\"5nmmmm#zmMj%F1$\"%8#*F-7$$\"5nmm\"z4%=8XYF1$\"% `#*F-7$$\"5nmm;H*)ozcYF1$\"%B$*F-7$$\"5nmmTgP>YoYF1$\"%@%*F-7$$\"5nmmm \"f)p7!o%F1$\"%n&*F-7$$\"5+++Dc(*QE.ZF1$\"%V**F-7$$\"5LLL$3#43SEZF1$\" &a/\"F-7$$\"5+++++Z;#*oZF1$\"&u=\"F-7$$\"5mmm;H'[)G$z%F1$\"&$*H\"F-7$$ \"5LLLLeD`l<[F1$\"&)R9F-7$$\"5+++]7<$\\%R[F1$\"&Of\"F-7$$\"5nmmmm3LCh[ F1$\"&+y\"F-7$$\"5MLL3F\\-[%)[F1$\"&,-#F-7$$\"5+++]()*=_\\F1$\"%CIF17$$\"5++]P4o*[T'\\F1$\"%aK F17$$\"5+++D17$*4w\\F1$\"%#\\$F17$$\"5++]7.c'\\!))\\F1$\"%GPF17$$\"\"& \"\"!$\"%wQF1-%&COLORG6&%$RGBG$\"\"%F*$Fc[mFc[m$\"\"*F*-%'LEGENDG6#%6E nright-Verner~schemeG-F$6%7e[l7$F($!$*eF-7$$\"5nmm;zM%))>\"GF1$!$-'F-7 $$\"5MLLLepo(R#GF1$!$F'F-7$$\"5+++]P/`'f$GF1$!$(fF-7$F/$!$o&F-7$$\"5+v oHHBwD\\GF1$!$t&F-7$$\"5M$3F>u]h0&GF1$!$v&F-7$$\"5n\"HdX:Rl=&GF1$!$^&F -7$$\"5++v=nv#pJ&GF1$!$m&F-7$$\"5M3x\")zfJZaGF1$!$r&F-7$$\"5n;zW#R/xd& GF1$!$s&F-7$$\"5+D\"y]!G43dGF1$!$_&F-7$$\"5ML$3x@\"[QeGF1F\\`m7$$\"5nT &Q.jp)ofGF1$!$h&F-7$$\"5+](oH/e#*4'GF1$!$f&F-7$$\"5Me*)fbkkHiGF1$!$g&F -7$$\"5nm\"H#o[.gjGF1$!$U&F-7$$\"5M$e*[$p63i'GF1$!$Y&F-7$$\"5+++v=&)e \")oGF1$!$W&F-7$$\"5mm;z>epCzGF1$!$#\\F-7$F5$!$m%F-7$F:$!$S%F-7$F?$!$E %F-7$FI$!$A%F-7$FSF^cm7$$\"5++D1*Q'>r*)HF1$!$G%F-7$$\"5ML$eky>)e&*HF1$ !$J%F-7$$\"5n\"H2ejDdq*HF1Ficm7$$\"5+]i:&[JE&)*HF1$!$I%F-7$$\"5M3_]Mt` ****HF1$!$P%F-7$$\"5nmT&Q=Vk9+$F1$!$R%F-7$$\"5M$3_D)[DS/IF1$!$O%F-7$FX $!$K%F-7$$\"5Le*)fIC(4)3IF1$!$U%F-7$$\"5m;z%*z#yy-,$F1$!$M%F-7$$\"5+vo HHTyu6IF1Fhem7$$\"5LLeky**o@8IF1$!$[%F-7$$\"5+]PMx;]:;IF1$!$X%F-7$$\"5 mm;/wLJ4>IF1$!$Q%F-7$$\"5+D1RD#>i0-$F1$!$T%F-7$$\"5L$eRZ2DJ?-$F1Fefm7$ $\"5mT&)3C4.]BIF1$!$^%F-7$$\"5++vVtn$p\\-$F1Fhem7$$\"5m;a8s%[2z-$F1$!$ c%F-7$Fgn$!$p%F-7$Fao$!$<&F-7$Ffo$!$2'F-7$$\"5MLL$3-$[*G9$F1$!$V'F-7$F [p$!$o'F-7$$\"5MLL$eaM#\\*=$F1$!$$oF-7$F`p$!$%pF-7$$\"5LL$e*)fDz'z88LF1$!$s*F-7$$\" 5+++D173i9LF1$!$e*F-7$$\"5MLe*[.]'eUHMF1F]an7$$\"5+++DJk n&3V$F1Fgan7$$\"5+++++F;HKMF1F]an7$$\"5++++vx5.QMF1Fban7$F]s$!$w\"F17$ $\"5M3_DG@N2XMF1Facn7$$\"5n;/^19lPYMF1F]an7$$\"5+Dcw%o]zwW$F1$!$r\"F17 $Fbs$!$u\"F17$$\"5nTgFT#\\&G]MF1$!$x\"F17$$\"5+]7`>&[)e^MF1$!$y\"F17$$ \"5Meky(zZ\"*GX$F1Fban7$FgsFidn7$$\"5+voHaju\\bMF1$!$z\"F17$$\"5M$3_Dj X+oX$F1F_dn7$$\"5n\"H23\"\\M5eMF1Fben7$F\\tFban7$$\"5M3xJnM%42Y$F1Facn 7$$\"5n;HdXFC,iMF1Fddn7$$\"5+D\"GQ-U:LY$F1Fben7$FatFban7$$\"5nT&Q.eS@f Y$F1$!$!=F17$$\"5+]Pfe)RCsY$F1F_dn7$$\"5Me*[o8RF&oMF1Fddn7$FftF_dn7$$ \"5+v$fLpPL6Z$F1Fddn7$$\"5M$e9;(pjVsMF1F]an7$$\"5n\"zp)\\i$RPZ$F1Fddn7 $F[uF_dn7$F]vF]an7$FgvF\\dn7$F\\w$!$d\"F17$Faw$!$[\"F17$$\"5LLLLe4r/cN F1$!%t8F-7$Ffw$!%B8F-7$F[x$!%.7F-7$F`x$!%26F-7$Fex$!%05F-7$FjxF_\\n7$F _y$!$(*)F-7$Fdy$!$m)F-7$$\"5Lek.HG'y-\"QF1$!$f)F-7$$\"5L$eR(\\/:j6QF1$ !$n)F-7$$\"5L3FWq!Q%)H\"QF1$!$^)F-7$$\"5LLe9\"pDPV\"QF1$!$Z)F-7$$\"5Le *[=J8!p:QF1$!$w)F-7$$\"5L$3_D$4I/-=FgSF1F`go7$$\"5nm\" H#=PSuhSF1$!%_5F-7$$\"5M$3Fp@F;K1%F1$!%v5F-7$$\"5++]i:2&)okSF1$!%g5F-7 $$\"5n;HK9U2;mSF1$!%(3\"F-7$$\"5ML3-8xHjnSF1$!%j5F-7$$\"5+](=<@@0\"pSF 1$!%[5F-7$$\"5nmmT5ZudqSF1$!%W5F-7$$\"5M$e9\"4#o\\?2%F1Fbho7$$\"5++D\" yq\">_tSF1$!%Z5F-7$$\"5n;/^1_T*\\2%F1$!%)3\"F-7$$\"5ML$3_qQmk2%F1$!%,6 F-7$$\"5+]i!R?iQz2%F1$!%u5F-7$$\"5nmTg-d3TzSF1$!%z5F-7$$\"5M$3-8?4$)33 %F1$!%w5F-7$Fe]l$!%\"3\"F-7$Fj]l$!%_6F-7$Fd^l$!%t6F-7$F^_l$!%%>\"F-7$$ \"5+]P%[;FLh<%F1$!%47F-7$$\"5++v=U\"*yAzTF1$!%=7F-7$$\"5+]7`>6DK#=%F1$ !%<7F-7$Fc_l$!%D7F-7$$\"5++Dc^qjg\">%F1$!%?7F-7$Fh_l$!%:7F-7$$\"5++v$4 '\\[)R?%F1$!%@7F-7$F]`lFb^p7$$\"5++DJqGLO;UF1$!%;7F-7$Fb`lF]^p7$Fg`l$! %$=\"F-7$F\\al$!%C6F-7$FaalFe[n7$Ffal$!$1)F-7$F[bl$!$)pF-7$F`bl$!$W'F- 7$Febl$!$.'F-7$Fjbl$!$\\&F-7$Fdcl$!$7&F-7$F^dl$!$&\\F-7$Fhdl$!$u%F-7$F bel$!$y%F-7$F\\fl$!$x%F-7$$\"5ML$eRr%F1$!$4&F-7$$\"5mm;aQ` B$[r%F1F]cp7$$\"5++voH\"e;1s%F1$!$A&F-7$Fffl$!$T&F-7$F[gl$!$H'F-7$F`gl $!$(pF-7$FeglF_[n7$$\"5+](oa+X[F1$!$Z(F-7$$\"5mmTg_u&z.#[F1F\\[n7$$ \"5L$eR(**)pT<#[F1$!$[(F-7$$\"5++](oM#Q5B[F1F\\ep7$$\"5LLe9Ts!Ge#[F1$! $c(F-7$$\"5mmmTN@BbG[F1$!$g(F-7$$\"5LL$eR#>3+M[F1$!$a(F-7$Fjgl$!$^(F-7 $$\"5MLLe*GJY.&[F1$!$3(F-7$F_hl$!$!oF-7$Fdhl$!$w&F-7$Fihl$!$,%F-7$F^il $!$z#F-7$Fcil$!#5F17$Fhil$\"\"\"F17$F]jl$F^\\mF17$$\"5++v$4hzt\"e\\F1$ \"#6F17$Fbjl$\"#;F17$$\"5+]Pf3al8n\\F1Fbhp7$$\"5++D\"y+9C,(\\F1Fbhp7$$ \"5+v=U2Lzhr\\F1Fbhp7$$\"5+]7.2E<6t\\F1Fbhp7$$\"5+D1k1>bgu\\F1Fbhp7$Fg jl$\"#:F17$$\"5++vo/%[u?)\\F1$\"#5F17$F\\[m$\"\")F17$$\"5+vVt-\\Ma*)\\ F1$F*F17$$\"5+]PM-Us.\"*\\F1$!\")F17$$\"5+DJ&>].JD*\\F1$!\"(F17$$\"5++ Dc,G[-%*\\F1$!\"'F17$$\"5+v=<,@'=b*\\F1Fjjp7$$\"5+]7y+9C,(*\\F1Fejp7$$ \"5+D1R+2i])*\\F1$!#9F17$Fa[m$!#RF1-Fg[m6&Fi[mF\\\\m$\"#l!\"#F\\\\m-F` \\m6#%6Verner's~Maple~schemeG-F$6%7\\bm7$F($\"$)HF-7$Fj\\m$\"$[#F-7$F_ ]m$\"$'=F-7$Fd]m$\"$!=F-7$F/$\"$t\"F-7$F_`m$\"$g\"F-7$F[bm$\"$V\"F-7$$ \"5LL3Fp@9.uGF1$\"$^\"F-7$F`bm$\"$r\"F-7$$\"5+v=UKU3b!)GF1$\"$j\"F-7$$ \"5L$3_]ksa=)GF1$\"$d\"F-7$$\"5m\"H#od5'eJ)GF1$\"$_\"F-7$$\"5++DJq%\\i W)GF1$\"$e\"F-7$$\"5m;Hd&HEqq)GF1$\"$p\"F-7$F5$\"$x\"F-7$F:Fe^q7$F?$\" $o\"F-7$FI$\"$n\"F-7$FSFe`q7$$\"5+Dc,T)y/`)HF1$\"$f\"F-7$$\"5M$ek.p%Qx ')HF1$\"$`\"F-7$$\"5nTNrR0HC))HF1Fj^q7$FbcmFj^q7$$\"5n;/w(33]E*HF1Fj^q 7$Fgcm$\"$h\"F-7$F\\dm$\"$i\"F-7$F_dm$\"$k\"F-7$FddmF^aq7$FidmF__q7$F^ emF]bq7$FX$\"$m\"F-7$FfemF__q7$F[fmFh`q7$F`fmFj]q7$Fcfm$\"$a\"F-7$Fhfm Fi_q7$F]gmFe`q7$FbgmFh`q7$FggmFj]q7$FjgmF^aq7$F_hmFe`q7$Fbhm$\"$c\"F-7 $Fgn$\"$Z\"F-7$$\"5++vV)pg2A.$F1F`bq7$$\"5mm;/E7'pN.$F1F^`q7$$\"5LLek` <;$\\.$F1Fj^q7$$\"5+++D\"Gi$HOIF1$\"$b\"F-7$$\"5mmT&)3GclPIF1Fj^q7$$\" 5LL$ekLj w1$F1Fcaq7$$\"5+++vV\\<)*oIF1$\"$[\"F-7$$\"5ML$e*)*fdqrIF1$\"$]\"F-7$F aoF_cq7$$\"5+++]7\"*>r&3$F1F_gq7$$\"5MLL$3<@%*p4$F1$\"$S\"F-7$$\"5++++ +A`j-JF1Fgfq7$$\"5nmm;HKkF3JF1Fb^q7$$\"5ML$ek)4no4JF1Fj]q7$$\"5+++vV() p46JF1Fdgq7$$\"5nm;/,ls]7JF1F_cq7$$\"5MLLLeUv\"R6$F1F]^q7$$\"5nmm\"Hx4 Qn6$F1Fgfq7$Ffo$\"$\\\"F-7$$\"5nmm;/vcZAJF1$\"$X\"F-7$$\"5MLL$3sp#RDJF 1Fjiq7$$\"5nmm;H37&o7$F1F`bq7$$\"5+++]P>(4$GJF1Fi_q7$$\"5MLL$e/Bo(HJF1 Fcaq7$$\"5nmm;aTnAJJF1F__q7$$\"5+++]i__oKJF1F^aq7$$\"5MLL$3PwVT8$F1F^a q7$$\"5nmm;zuAgNJF1Fcbq7$$\"5+++](eygq8$F1Feiq7$$\"5MLL$epH>&QJF1$\"$U \"F-7$$\"5nmm;/3y(*RJF1Fdgq7$$\"5+++]7>jVTJF1Fcaq7$F`imFi_q7$$\"5+++]( )=HcaJF1Fh`q7$F[p$\"$#=F-7$F`p$\"$M#F-7$F`jm$\"$_#F-7$Fejm$\"$q#F-7$Fj jm$\"$w#F-7$Fep$\"$\"GF-7$$\"5+](=#*R;D>E$F1$\"$$HF-7$$\"5mmT5:aBCjKF1 F`]r7$$\"5L$e*)4VafXE$F1$\"$z#F-7$$\"5++](oWtweE$F1F`]r7$$\"5m;/wiCR>n KF1$\"$\"HF-7$$\"5LLeky96^oKF1F[]q7$$\"5+]7`%\\IG)pKF1$\"$(GF-7$$\"5mm mT5&\\X6F$F1$\"$&HF-7$$\"5L$3-j_oiCF$F1Fc_r7$$\"5++v=Uv)zPF$F1$\"$o#F- 7$$\"5m;H2elq4vKF1$\"$)GF-7$$\"5LL$eRdD9kF$F1Ff]r7$$\"5mm\"Hdgj[!zKF1$ \"$*GF-7$FjpFh_r7$$\"5nm;/,x<&pG$F1$\"$,$F-7$$\"5MLLekP0A#H$F1$\"$5$F- 7$$\"5++]7G)H*[(H$F1$\"$h#F-7$F_q$\"$Q#F-7$Fg\\nF``r7$F[^nFc^r7$F`^n$ \"$f#F-7$Fe^n$\"$X#F-7$Fj^nFj\\r7$F__n$\"$U#F-7$Fb_n$\"$R#F-7$FdqF]br7 $Fj_n$\"$N#F-7$Fiq$\"$w\"F-7$F^r$\"$>\"F-7$Fcr$\"#dF-7$$\"5+++](ykC2S$ F1$\"\"'F17$$\"5++++DtVf.MF1Fadr7$$\"5+++v$fBH]S$F1$\"\"$F17$$\"5+++]i )4kkS$F1$F[\\mF17$$\"5+++DJh*)*yS$F1F[hp7$Fh`n$\"\"(F17$$\"5++++vuK2:M F1F^er7$FhrF\\\\m7$F`an$Fc\\qF17$FeanFhgp7$FjanF^er7$F_bnFjer7$FbbnFaj p7$Febn$\"\"#F17$FhbnFidr7$F[cnF`fr7$$\"5+++vo*[EPV$F1Fcer7$$\"5+++]P_ 8;NMF1Fidr7$$\"5+++D1:ifOMF1F\\\\m7$F^cnFhgp7$$\"5+++vVSfYRMF1$!\"$F17 $$\"5+++]7.3!4W$F1$Fb[mF17$$\"5+++D\"emNBW$F1Ffgr7$F]sF`fr7$FatFidr7$F gvFfgr7$$\"5ML$e9JPhy\\$F1Fcer7$F\\w$\"#8F17$$\"5Le9T8Le!=^$F1Fiip7$$ \"5m;H2e<_N8NF1F[hp7$$\"5+vVt--Y!\\^$F1$\"#7F17$$\"5LLeRZ')RX;NF1$\"#< F17$$\"5m\"Hd?4P.!=NF1Fiip7$$\"5+](=n`v_&>NF1$\"#9F17$$\"5L3-Q\")R@5@N F1Fbhp7$$\"5mm;/EC:lANF1Fiir7$$\"5+DJqq34?CNF1Fdip7$$\"5L$ek`JH]d_$F1F iir7$$\"5mTg-gx'*HFNF1Fair7$$\"5++vo/i!\\)GNF1$\"#=F17$$\"5Le*[$\\Y%)R INF1Fbhp7$$\"5m;/,%4$y%>`$F1F\\ir7$$\"5+v=nQ:s\\LNF1Fair7$FawFiir7$$\" 5LL3FpJ\"fj`$F1F][s7$$\"5LL$3_Nmrw`$F1Fair7$$\"5LLe9T&>%)*QNF1$\"#>F17 $$\"5LLL3FFnHSNF1Fair7$$\"5LL3-8f#4;a$F1Fiir7$$\"5LL$e*)4z@Ha$F1Fb\\s7 $$\"5LLe*[GKMUa$F1F][s7$$\"5LLL$3Z&oaXNF1$\"$P\"F-7$$\"5LL3xc'Qfoa$F1F e`q7$$\"5LL$3F%=><[NF1$\"$&>F-7$$\"5LLekG]W[\\NF1$\"$\"=F-7$$\"5LLLe9# )pz]NF1$\"$$=F-7$$\"5LL3_+9&4@b$F1F^`q7$$\"5LL$eke/AMb$F1$\"$\">F-7$$ \"5LLeRsxXtaNF1$\"$q\"F-7$FdhnFe^s7$$\"5LL$3-LF-7$$\"5LLL 3-PsHhNF1$\"$4#F-7$$\"5LL3-))o(4Ec$F1$\"$$>F-7$$\"5LL$eR2IARc$F1$\"$%= F-7$$\"5LLe*)fK[BlNF1$\"$#>F-7$$\"5LLL$eWOZlc$F1$\"$/#F-7$$\"5LLLe*=\\ (zrNF1F]_s7$FfwFe`q7$$\"5mmT5llZ:$e$F1$\"$!>F-7$$\"5++](o>\">E*e$F1$\" $+#F-7$$\"5m;/w7&[:Bf$F1$\"$'>F-7$$\"5LLekGe!p`f$F1Fhcq7$$\"5mT&)e'[%e *of$F1Faas7$$\"5+]7`WJEU)f$F1$\"$(=F-7$$\"5LeRZ-=%\\**f$F1F_cs7$$\"5mm mTg/iZ,OF1Fg`s7$$\"5+v$f$=\"*H+.OF1Fj[r7$$\"5L$3-jxxHXg$F1Fj^q7$$\"5m \"zWUVccgg$F1$\"$z\"F-7$$\"5++v=#4N$e2OF1Fd]q7$$\"5L3-8]P,64OF1Fa]q7$$ \"5m;H23Cpj5OF1Fcbq7$$\"5+Dc,m5P;7OF1$\"$)>F-7$$\"5LL$eRs\\!p8OF1F_bs7 $$\"5+]P%)RqSu;OF1F\\as7$$\"5mm\"HdNk(z>OF1F_cq7$$\"5+v=n8IWK@OF1Fidq7 $$\"5L$e9;n@^Gi$F1F^es7$$\"5m\"Hd&H.!yVi$F1Fcaq7$F[xF_gq7$F`x$\"$I\"F- 7$$\"5mT5SWNWjqOF1$\"$?\"F-7$$\"5+](oal?<@n$F1Ficr7$$\"5Lek`mx**ftOF1$ \"$K\"F-7$$\"5mmTgx[F3vOF1$\"$W\"F-7$$\"5+v=n))>bcwOF1Fdgq7$$\"5L$eR(* 4H[!yOF1Fi_q7$$\"5m\"H23@1J&zOF1$\"$A\"F-7$$\"5++](=K$Q,\"o$F1$\"$.\"F -7$$\"5LLe9m<\\%po$F1Fc]s7$$\"5mmmT5-g(Gp$F1Fdgq7$$\"5+vV[@t(eVp$F1$\" $6\"F-7$$\"5L$3_DVaTep$F1Fais7$$\"5m\"z>OaJCtp$F1$\"$R\"F-7$$\"5++voa' 32))p$F1F_gq7$$\"5L3_vld)*G+PF1Fgfq7$$\"5m;H#o(GEx,PF1$\"#%)F-7$$\"5+D 1*y)*RbKq$F1$\"$L\"F-7$$\"5LL$e*)4:>RPF1$ \"$=\"F-7$FjxFifs7$$\"5m\"HK9+#fDjPF1$\"$;\"F-7$$\"5L$ek./9HZw$F1$\"$H \"F-7$$\"5+voHzgB?mPF1$\"#\"*F-7$$\"5mm\"H#=\"evww$F1Fifs7$$\"5+]P4'>- A1x$F1$\"$0\"F-7$$\"5LL$eRFYoNx$F1$\"$+\"F-7$$\"5m;H#=N!\\^wPF1Fc\\t7$ $\"5++voHW8YzPF1Fiis7$$\"5L$3_v]y2Cy$F1Fijs7$F_yFc\\t7$$\"5+]7Gjm1I)y$ F1$\"$9\"F-7$$\"5LLe9T2rC\"z$F1$\"$M\"F-7$$\"5+D\"y+yK?Fz$F1Fj_t7$$\"5 m;/,>[N>%z$F1Fiis7$$\"5L3F%z&onm&z$F1$\"$E\"F-7$$\"5++](o*))*Rrz$F1$\" $C\"F-7$$\"5m\"H2e$4Kh)z$F1Fais7$$\"5L$eRZ(Hk3+QF1$\"$/\"F-7$$\"5+v=n8 ]'f:!QF1Fc]s7$$\"5mmTg_qG..QF1F[\\t7$$\"5Lek`\"441X!QF1F[\\t7$$\"5+](o /8Jzf!QF1$\"$7\"F-7$$\"5mT5SpJDX2QF1$\"$4\"F-7$FdyF_]t7$F\\jnFc[t7$Faj n$\"$8\"F-7$Ffjn$\"$G\"F-7$F[[oF[\\t7$F`[o$\"$,\"F-7$Fe[oFj_t7$Fj[oFe_ t7$Fiy$\"$B\"F-7$Fb\\o$\"#$*F-7$Fg\\o$\"$Q\"F-7$F\\]oFijs7$Fa]oFc\\t7$ $\"5LeRZxUJ^EQF1$\"$O\"F-7$Ff]o$\"$:\"F-7$$\"5L3-))=&*)=#HQF1Ffgs7$F^z Fc\\t7$FczFjbt7$FhzFicr7$Fb^o$\"$T\"F-7$Fg^oFahs7$F\\_o$\"$5\"F-7$F__o Fict7$Fd_oFc[t7$Fi_oF[\\t7$F\\`oF^[t7$F][lFijs7$Fd`o$\"$1\"F-7$Fg`oFfg s7$F\\aoFicr7$FaaoF\\dt7$FfaoFjbt7$Fb[lFa^t7$Fg[lFicr7$F\\\\lFgfq7$$\" 5n\"z>'oj)H/!RF1Fe`t7$$\"5n;HdXCc%>!RF1F`bt7$$\"5nTg_A&QhM!RF1Fc\\t7$$ \"5nm\"z%*f9x\\!RF1Fe_t7$$\"5n\"HKkn!H\\1RF1Ficr7$$\"5n;aQ`n'3!3RF1F]c t7$$\"5nT&Q.$GW_4RF1Fifs7$$\"5nm;H2*=S5\"RF1Fifs7$$\"5n\"zWU)\\fb7RF1F `bt7$$\"5n;z>h5<29RF1Fd]t7$$\"5nT5:Qrue:RF1Ffct7$$\"5nmT5:KK5'RF1Ffct7$$\"5nm;/,Q@DjRF1Feiq7$$\"5M$e9;AhrX'RF1$ \"#%*F-7$$\"5++v=U'3\"*e'RF1F[ft7$$\"5n;/wig0@nRF1Ffdt7$$\"5MLLL$[.I&o RF1Fd]t7$$\"5++]ilJz!Q(RF1Fe_t7$$\"5nmm\"z%Ge3zRF1Ffhs7$$\"5M$e*[o-`S! )RF1$\"#)*F-7$$\"5++D1*oxC<)RF1$\"#**F-7$$\"5n;aj4^U/$)RF1$\"##*F-7$$ \"5ML$3-`sjV)RF1F[ft7$$\"5+]7y]*>$o&)RF1$\"$<\"F-7$$\"5nmTNrtE+()RF1F] ct7$$\"5M$3F>z9A$))RF1Fc]s7$F[]lFi\\t7$$\"5Mek`T!pm5*RF1F^it7$$\"5n;Hd qe<\\#*RF1Fbat7$$\"5+v$4'*p#o\"R*RF1F`bt7$$\"5MLekG&*=M&*RF1Ficr7$$\"5 n\"H#odjpw'*RF1Fcet7$$\"5+](=n=.#>)*RF1Fjbt7$$\"5M3_v:+rh**RF1Fg^u7$Fg eoF^it7$$\"5M$ekG]I#*Q+%F1F\\dt7$$\"5++v$4;WUn+%F1F_cq7$$\"5n;/,>yDf4S F1Fcdt7$F\\foFfct7$FafoF]hq7$F`]lFcaq7$$\"5M$ek.C/;n.%F1Fe`t7$$\"5++D1 Rx#)=QSF1Fjbt7$$\"5n;/wP70mRSF1F[\\t7$$\"5ML$ektuK6/%F1Fijs7$$\"5+]i:N #)\\gUSF1Fc]s7$$\"5nmT&Qt@xS/%F1Fhcq7$$\"5M$3_DBX\\b/%F1F]^q7$FifoFeiq 7$$\"5MLekGdh'*\\SF1Fiis7$$\"5nm;/EF1\"H0%F1Fdfs7$$\"5M$eRZA'GQaSF1Fag s7$$\"5++vVB(4be0%F1Fiis7$$\"5n;a8AKtKdSF1Feiq7$F^goF^jt7$F`hoF_et7$Fd ioFibq7$FiioFcaq7$F\\joFe`q7$FajoF]ct7$FfjoFi\\t7$F[[pF[dq7$F`[pFjiq7$ Fe[pFdgq7$Fe]lF_gq7$Fj]lFj[r7$$\"5+D19%pJQ)HTF1F_et7$$\"5+]7G8w=BJTF1F ^aq7$$\"5+v=UKNaiKTF1Fb^q7$$\"5++Dc^%**=S8%F1F\\dt7$$\"5+DJqq`DTNTF1F \\dt7$$\"5+]P%)*G61o8%F1Fidq7$$\"5+vV)*3s'*>QTF1Fgfq7$F_^lFd_q7$$\"5++ vo/ou;XTF1F_gq7$Fd^lF_gq7$$\"5+D1R+k_8_TF1F_gq7$$\"5+]7`>B)GN:%F1F]ct7 $$\"5+v=nQ#QA\\:%F1Fidq7$$\"5++D\"y:%fJcTF1Fjiq7$$\"5+]P4'*fI5fTF1Fcdt 7$Fi^lF^jt7$$\"5+Dc^`PPGjTF1$\"$N\"F-7$$\"5+]ils'HxY;%F1Fj[r7$$\"5+voz \"f&32mTF1Fj_t7$$\"5++v$4^Tku;%F1Fj`t7$$\"5+](=#\\L:DqTF1Fd]t7$F^_lF]h q7$Fh_lFfdt7$Fb`lF_^u7$$\"5m;/E+&*G%RA%F1Fj]u7$$\"5LL3_v@KLDUF1$\"#(*F -7$$\"5+]7y][NsEUF1$\"#&)F-7$$\"5mm;/EvQ6GUF1$\"#'*F-7$$\"5L$3-8??/&HU F1Fa\\u7$$\"5++DcwGX*3B%F1Ffhs7$$\"5m;H#=b&[GKUF1Fghu7$$\"5LLL3F#=vOB% F1F\\iu7$$\"5mmTgxNeXOUF1$\"#))F-7$$\"5++]7G*[O#RUF1$\"#!*F-7$$\"5LLek yUr,UUF1$\"#()F-7$Fg`l$\"#$)F-7$$\"5LL$3-L5f.D%F1$\"#*)F-7$$\"5+++DJ5/ #fD%F1$\"#&*F-7$$\"5LL3x\"Q1,(eUF1Fj]u7$$\"5mm;HK<<[hUF1Faiu7$$\"5L$3_ vS/sGE%F1Fbat7$$\"5++D\"G3PiUE%F1Fi[v7$$\"5m;H2e(p_cE%F1Fa\\u7$F\\alFd [v7$$\"5mmT5!\\\\F&oUF1Fict7$$\"5++](oa'>,qUF1Faiu7$$\"5LLek.Ok\\rUF1F act7$$\"5mmmTg14)HF%F1Faiu7$$\"5++v=^'zC&H%F1Fbat7$$\"5+++voNCt'H%F1Fbat7$$\"5LL3_D1 p@)H%F1Fcet7$$\"5mm;H#oP,(*H%F1Fi\\t7$$\"5++D1RZe=,VF1Ffdt7$$\"5LLL$ez JqEI%F1Fjbt7$$\"5mmTg_)yaTI%F1Fc[t7$$\"5++]P4f#RcI%F1F_]t7$$\"5LLe9mHP 72VF1Febt7$$\"5mmm\"H-?3'3VF1Fags7$$\"5LL$ek89x:J%F1F]ct7$FaalFfct7$$ \"5MLL3-)\\ogL%F1F[\\t7$FfalF\\dt7$$\"5+]iS;C@`gVF1Fj[r7$$\"5MLekyMLZj VF1Fd_q7$$\"5+Dcw4SR%\\O%F1F]^q7$$\"5n;a)3aa9kO%F1Fj[r7$$\"5M3_+s]^)yO %F1Ffgs7$$\"5++]7.cdNpVF1Feiq7$$\"5n\"zWU8OE3P%F1F_gq7$$\"5M$ek`m'pHsV F1F_cq7$$\"5+vV['>dnPP%F1F[dq7$$\"5nmTgFx\"Q_P%F1F]^q7$$\"5MeRse#y3nP% F1F_et7$$\"5+]P%)*yQz\"yVF1F`gu7$$\"5nTN'4K**\\'zVF1F]^q7$$\"5MLL3_)f? 6Q%F1F[dq7$$\"5n;HK94=1%Q%F1F[dq7$$\"5++Dcw>I+(Q%F1F[dq7$$\"5n\"H#o2DO Z)Q%F1Fjiq7$$\"5M$3-)QIU%**Q%F1F]hq7$$\"5+v=#*pN[T\"R%F1F_et7$$\"5nm;/ ,Ta)GR%F1Fiis7$$\"5ML3_Diyw)R%F1F]hq7$F[blFcdt7$$\"5mmmTN$))*yEWF1F]ct 7$F`blFe_t7$$\"5mmm;HW<2sWF1Fe]u7$FeblFbju7$$\"5m\"HKks6Gm\\%F1Fd[v7$$ \"5L$ek`\"HA/)\\%F1Fi]t7$$\"5+voH/TjX*\\%F1F_^u7$$\"5mm\"HKHXq3]%F1$\" #\")F-7$$\"5Le9;#[c%G-XF1F\\iu7$$\"5+]P4rw')p.XF1F_^u7$$\"5mTg-g)y7^]% F1Fd[v7$$\"5LL$e*[+p_1XF1Fd[v7$$\"5+D1*yB,Tz]%F1$\"#yF-7$$\"5m;H#oU7b$ 4XF1Fict7$$\"5L3_v:O#p2^%F1Fict7$$\"5++vo/[L=7XF1$\"#wF-7$$\"5m\"z>O*f uf8XF1Fict7$$\"5L$3_D=d6]^%F1F_^u7$$\"5+vV[r$oDk^%F1Fdjs7$$\"5mmmTg&zR y^%F1$\"#xF-7$$\"5Le*[$\\2RD>XF1Feiv7$$\"5+]7GQ>!o1_%F1$\"#vF-7$$\"5mT N@FJ@3AXF1Fd[v7$$\"5LLe9;Vi\\BXF1F\\iu7$$\"5m;/,%pYCj_%F1F\\iu7$$\"5++ ](=2p_\"HXF1Fdjs7$$\"5mmTgFQ\"4[`%F1$\"#!)F-7$FjblF[gv7$$\"5mmTN@itQYX F1F\\iu7$$\"5++]PfQ\"4Bb%F1F_[v7$$\"5L3-))o#e*y`XF1F[gv7$$\"5m;aQyE+Fb XF1Fdjs7$$\"5+D1*y3Z]nb%F1$\"##)F-7$$\"5LLeR(\\\"4BeXF1$\"#sF-7$$\"5mT 5!p!f8rfXF1Fe\\w7$$\"5+]iS;.=>hXF1$\"#')F-7$$\"5Le9\"fsCsEc%F1F\\iu7$F _clF^[w7$$\"5++vVtnW2qXF1F[gv7$$\"5LL$e9TC'*fd%F1F\\iu7$$\"5mTN'4#)owu d%F1F[gv7$$\"5+](o/B8d*yXF1F]jv7$$\"5LeR(*RwvV!e%F1F\\iu7$$\"5mm\"z%\\ ?!=>e%F1$\"#tF-7$$\"5L$e*[o3*y[e%F1Feiv7$Fdcl$\"#uF-7$$\"5Me*[$\\ScE*e %F1F`\\w7$$\"5n;z>6%[\"p!f%F1F_[v7$$\"5+vo/tFt6#f%F1F^[w7$$\"5MLe*[8'yWd2k%F1F`\\w7$$\"5nmT5!*yb@UYF1Fd[v7$$\"5n;/ ,%*4PnVYF1F_^u7$F]elFa\\u7$$\"5nm;a8lV'4l%F1Fict7$FbelF_^u7$$\"5n;H2L? ]DeYF1Fict7$$\"5nm\"zp8:8(fYF1Fj]u7$$\"5n;a)3CGr6m%F1F]]w7$$\"5nm;zW8% HEm%F1Fghu7$$\"5n;zp[Wv3kYF1F[gv7$$\"5nmTg_vcalYF1Faiu7$$\"5n;/^c1Q+nY F1Faiu7$FgelF\\[v7$$\"5nm\"H#o*>y8n%F1Faiu7$$\"5nm;/whWHuYF1Ff^t7$$\"5 nmT&QQs5sn%F1Fact7$F\\flFact7$$\"5+DJX*G/t:o%F1Fgju7$$\"5M$eRs)*4>Io%F 1Fj]u7$$\"5nTg-&o:lWo%F1Fact7$$\"5++D\"GQ@6fo%F1Fbju7$$\"5Me*)f!3Fdto% F1Fj]u7$$\"5n;aQyFL!))o%F1Fghu7$$\"5+v=&)QjbZF1Fghu7$$\"5+v$4r8mipv%F1Fa\\u7$$\"5LL$3_v V\"HeZF1Faiu7$$\"5mmTgFUlgjZF1Fict7$F[glF_^u7$$\"5mm;H#oN8]x%F1Fi]t7$$ \"5LLLekm]5\"y%F1Fd[v7$$\"5++](okx'>(y%F1F_^u7$F`glFict7$$\"5+++v$f!>Z 0[F1Ffhs7$FeglFe_t7$FgepFc]s7$FjglF^`q7$Fdfp$\"$F#F-7$F_hl$\"$t#F-7$$ \"5++](o*y<'G([F1$\"$N$F-7$Fdhl$\"$2%F-7$$\"5nm;Hd>()4'*[F1$\"$\"\\F-7 $Fihl$\"$&eF-7$$\"5++]i:ptF8\\F1$\"$^'F-7$F^il$\"$'oF-7$$\"5++](=xs(RC \\F1$F\\_wF17$Fcil$Fa\\wF17$Fhil$Fj]tF17$F]jl$Fg^tF17$Fbjl$FfbtF17$Fgj l$FabtF17$$\"5+](oa!)*o3z\\F1$FdetF17$Fgip$Fd\\tF17$$\"5+voH/x#oN)\\F1 F\\^x7$$\"5+]i!R+2i])\\F1Fd]x7$$\"5+Dc^.jeb')\\F1$FbisF17$F\\[mFd]x7$F _jpFb]x7$Fcjp$FjctF17$Fhjp$Fb\\uF17$F][q$FbiuF17$Fb[qF^_x7$Fe[qF^_x7$F h[qF`]x7$Fa[m$\"#kF1-%'COLOURG6&Fi[m$\")#)eqkFfjp$\"))eqk\"FfjpFj_x-F` \\m6#%Ha~modification~of~Verner's~Maple~schemeG-F$6%7^[l7$F($!$N'F-7$$ \"5MLe*[Vb)\\,GF1Fc`x7$$\"5nm;zp3r*H!GF1Fc`x7$$\"5++vo/jc\\/GF1$!$7'F- 7$$\"5MLLeRGF 1$!$G'F-7$$\"5nm;a)3wz4#GF1$!$a'F-7$$\"5++vVB:$yC#GF1Fjbx7$F_]mF_cx7$F d]m$!$B'F-7$F/$!$'fF-7$F\\^mF\\]m7$Fa^mF]im7$Ff^m$!$$eF-7$F[_m$!$)fF-7 $F`_mF\\`p7$Fe_m$!$/'F-7$Fj_mF^dx7$F_`m$!$&eF-7$Faam$!$x&F-7$F[bm$!$!e F-7$F`bm$!$L&F-7$F5$!$5&F-7$F:$!$$[F-7$F?$!$i%F-7$FI$!$Y%F-7$FSFfdm7$F bcmFhbm7$FgcmFhem7$FidmFjfm7$FXF_gm7$Ffem$!$Z%F-7$F[fmFfdm7$F`fm$!$W%F -7$FcfmF\\hm7$$\"5m\"z%*z#efo9IF1Fjfm7$FhfmFefm7$$\"5L3FpEvSi\"F-7$F[^n$!%(>\"F-7$Fe^n$!%g7F-7$Fdq$!%v7F-7$Fj_n$!%e8F-7$Fiq$!%s9F- 7$F^r$!%K;F-7$Fcr$!%\"y\"F-7$F_drFben7$FddrFhfn7$Fgdr$!$$=F17$F\\er$!$ #=F17$F`erFidn7$Fh`nFhfn7$Ffer$!$&=F17$Fhr$!$!>F17$F`an$!$$>F17$FeanF_ ]y7$Fjan$!$(=F17$F_bnFb]y7$FbbnFb]y7$Febn$!$\">F17$Fhbn$!$*=F17$F[cnF[ ^y7$F^cn$!$#>F17$F]sFb]y7$Fdcn$!$%>F17$FgcnF_]y7$FjcnF^^y7$FbsFb^y7$Fb dn$!$&>F17$Fgdn$!$'>F17$F\\enFb]y7$FgsF__y7$F`en$!$(>F17$FeenFb^y7$Fhe nFd_y7$F\\tFb]y7$F\\fnF\\_y7$F_fnF__y7$Fbfn$!$*>F17$FatF\\_y7$F[uFf^y7 $FgvFb]y7$F\\w$!$%=F17$FawFddn7$Ffw$!%)f\"F-7$F[x$!%C9F-7$F`x$!%\"H\"F -7$FexF]jx7$Fjx$!%*4\"F-7$F_y$!%$3\"F-7$Fdy$!%b5F-7$F\\jn$!%]5F-7$Fajn $!%d5F-7$Ffjn$!%T5F-7$F[[oF\\_n7$F`[o$!%m5F-7$Fe[o$!%L5F-7$Fj[oFhgo7$F iy$!%U5F-7$Fb\\o$!%r5F-7$Fg\\o$!%E5F-7$F\\]o$!%I5F-7$Fa]oFfay7$FadtF[g o7$Ff]oFd_n7$Fidt$!%<5F-7$F^zFhgo7$FczFfio7$Fhz$!%O5F-7$Fb^o$!%95F-7$F g^oFfby7$F\\_o$!%X5F-7$F__o$!%i5F-7$Fd_o$!%M5F-7$Fi_o$!%C5F-7$F\\`oFai o7$F][l$!%A5F-7$Fd`oFiay7$Fg`o$!%75F-7$F\\aoF\\_n7$FaaoFicy7$Ffao$!%V5 F-7$Fb[lFhgo7$Fg[lF_by7$F\\\\l$!%?5F-7$Feft$!%S5F-7$Fhft$!%a5F-7$F[gtF bho7$F^gtFfay7$Fagt$!%^5F-7$FdgtF[fy7$FggtF[gy7$Fjgt$!%`5F-7$F`htFd_n7 $FfhtFffy7$FihtFhgo7$F\\itFffy7$FaitF]ho7$Fa\\lF_by7$FeitF]ho7$Ff\\l$! %)4\"F-7$F[]lF]\\p7$F`]l$!%07F-7$Fe]l$!%08F-7$$\"5+++]PU+S0TF1$!%S8F-7 $Fj]l$!%w8F-7$Fd^l$!%#Q\"F-7$F^_l$!%\"R\"F-7$Ff\\p$!%29F-7$F[]p$!%89F- 7$F`]p$!%49F-7$Fc_l$!%<9F-7$Fh]p$!%39F-7$Fh_l$!%-9F-7$F`^p$!%09F-7$F]` l$!%/9F-7$Ff^p$!%)R\"F-7$Fb`l$!%(R\"F-7$Fg`l$!%q8F-7$F\\al$!%>8F-7$Faa l$!%j6F-7$FfalFicy7$F[bl$!$s)F-7$F`bl$!$r(F-7$FeblFf_p7$Fciv$!$s'F-7$F jbl$!$U'F-7$$\"5mT&QG*Hg%>a%F1$!$P'F-7$$\"5+]PM-ukUVXF1$!$['F-7$$\"5Le *[=\"=p!\\a%F1$!$K'F-7$Fb[wFhax7$$\"5L$ek./D[$\\XF1$!$M'F-7$Fe[wF[dp7$ Fh[w$!$I'F-7$F[\\w$!$D'F-7$F^\\w$!$E'F-7$Fc\\wFc`x7$Fh\\wFc`x7$F[]w$!$ ?'F-7$F`]w$!$>'F-7$F_clFfcx7$$\"5+v=#\\a8Lcc%F1F[_z7$$\"5L$3FW&zN6nXF1 F[dp7$$\"5m\"HKRO-%foXF1F]^z7$Fd]wFh^z7$$\"5L3F%H=\"\\brXF1$!$<'F-7$$ \"5m;zW#fNNId%F1Fjbx7$$\"5+DJ&>+!e^uXF1$!$9'F-7$Fg]wF^ax7$F]^wFh^z7$Fc ^w$!$A'F-7$Fh^w$!$;'F-7$Fdcl$!$='F-7$F^dl$!$8'F-7$Fhdl$!$,'F-7$Fbel$!$ :'F-7$F\\flFa]m7$FaflF_cx7$Fffl$!$1(F-7$F[gl$!$)zF-7$Fegl$!$`*F-7$F_hl $!%R5F-7$Fihl$!%76F-7$$\"5+]7`pMs54\\F1Fhjo7$$\"5++Dc^zs\\5\\F1$!%&3\" F-7$$\"5+]PfLCt)=\"\\F1$!%$4\"F-7$Fb\\x$!%+6F-7$$\"5++vozeu0;\\F1$!%<6 F-7$F^ilF__p7$$\"5+]7yD$fF-#\\F1$!%&4\"F-7$$\"5++D\"y!Qwh@\\F1F`ay7$$ \"5+]P%)*Go2I#\\F1F_in7$Fj\\x$!$7\"F17$$\"5+]i!RDx(yD\\F1$!$8\"F17$$\" 5++v$ft\"y\"F17$Fbjl$!$D\"F17$Fgjl$!$Q\"F17$Fgip$!$]\"F17$F\\[m$!$f \"F17$F_jp$!$q\"F17$FcjpFben7$FhjpFhfn7$F][q$!$\"=F17$Fb[qFb`y7$Fe[q$! $'=F17$Fh[qF\\_y7$Fa[m$!$E#F1-Ff_x6&Fi[m$\"*++++\"FfjpF\\\\mF\\\\m-F` \\m6#%6an~\"efficient\"~schemeG-F$6%7_`m7$F(F^`q7$Fj\\mFags7$F_]mFeaw7 $$\"5nm\"HKRUva#GF1Fd[v7$$\"5++]7GyR(p#GF1Fgju7$$\"5ML3-jKDZGGF1Fgju7$ $\"5nmm\"zp3r*HGF1Fict7$$\"5ML$3xc>oH$GF1Fdjs7$Fd]mF^[w7$$\"5MLeRseQYP GF1$Fb\\qF-7$$\"5nm;H28C'*QGF1F\\hv7$$\"5++v=Un4YSGF1$\"#hF-7$$\"5MLL3 x@&f>%GF1$\"#jF-7$$\"5++](o/jc\\%GF1F]jv7$F/Feaw7$F_`m$\"#qF-7$F[bmF\\ dr7$F`bmFd[v7$F5Ff^t7$F:Fbat7$F?Fc\\t7$FIFi\\t7$FSFfct7$FbcmFifs7$Fgcm Fi\\t7$F\\dmFicr7$F_dmFc[t7$FddmFg^u7$FidmFfdt7$F^emFicr7$FXFj`t7$Ffem Ffdt7$F[fmFe`t7$F`fmFi\\t7$FcfmFjbt7$FhfmFg^u7$F]gmF^jt7$FbgmFe`t7$Fgg mFifs7$FjgmFi\\t7$F_hmF^jt7$FbhmFfdt7$FgnF^[t7$F^dqFahs7$FadqFd]t7$Fdd qFahs7$FgdqFe_t7$F\\eqFahs7$F_eqFd]t7$FbeqFc[t7$FeeqF_]t7$FheqFi\\t7$F \\oF_]t7$F\\fqF_]t7$F_fqF_]t7$FbfqFicr7$FefqFa^t7$FjfqF`bt7$F]gqF^[t7$ FbgqFebt7$FaoFjbt7$FhgqF[ft7$F[hqFghu7$F`hqFj]v7$FchqF[ft7$FihqFfhs7$F _iqFghu7$FbiqFj]u7$FfoFf^t7$FhiqFaiu7$F]jqFi[v7$F`jqFais7$FcjqF^[t7$Ff jqFf^t7$FijqFfhs7$F\\[rFa^t7$F_[rFa^t7$Fb[rFcet7$Fe[rFa\\u7$Fh[rF]]w7$ F]\\rFi]t7$F`\\rFa\\u7$F`imFj]u7$$\"5nmm;au)G([JF1Fj]u7$Fd\\rFj]u7$$\" 5nmm;/T*zu:$F1Faiu7$$\"5MLL$3K'pRgJF1Fict7$$\"5nmm;Hua&=;$F1Fj]v7$$\"5 +++]P&)RJjJF1Fa\\u7$$\"5MLL$ek\\sZ;$F1F]]w7$F[pFa^t7$FhimFaiu7$F`pFbju 7$F`jmF\\[v7$FejmF\\iu7$Fjjm$\"#pF-7$Fep$\"#[F-7$F^^r$\"#NF-7$Ff^r$\"# KF-7$Fi^r$\"#_F-7$F^_rF\\dr7$Fa_r$\"#SF-7$Ff_r$\"#YF-7$F[`r$\"#PF-7$F^ `r$FchpF-7$Fc`r$\"#MF-7$Fh`r$\"#EF-7$F[ar$\"#IF-7$FjpF`c[l7$FaarF`c[l7 $Ffar$\"#JF-7$F[br$!#BF-7$F_q$!#aF-7$Fh[n$!#6F-7$F]\\n$!#8F-7$Fb\\n$Fc \\qF-7$Fg\\n$!#IF-7$F\\]n$!#TF-7$Fa]nFfd[l7$Ff]n$!#VF-7$F[^n$!#CF-7$F` ^n$!#[F-7$Fe^n$!#iF-7$Fj^n$!#uF-7$F__n$!#pF-7$Fb_n$!#sF-7$Fdq$!#`F-7$F j_n$!#&)F-7$Fiq$!$W\"F-7$F^r$!$-#F-7$Fcr$!$e#F-7$Fddr$!#DF17$Fh`n$Fic[ lF17$Ffer$!#FF17$Fhr$Fgd[lF17$F`an$!#KF17$Fean$!#HF17$Fjan$!#EF17$F_bn $!#JF17$Febn$!#GF17$F[cnFfg[l7$Fefr$!#AF17$FhfrFfg[l7$F[grF^h[l7$F^cnF gh[l7$F_grF[h[l7$Fdgr$Fae[lF17$FhgrFbi[l7$F]sFgh[l7$FatFfg[l7$FgvFah[l 7$$\"5+D\"G))>o:q[$F1Fig[l7$$\"5M$e*[Vm]c)[$F1Fbi[l7$$\"5nT5:)3X9,\\$F 1$!#@F17$$\"5++D\"G`$Qm\"\\$F1Fbi[l7$$\"5MeRZx>K@$\\$F1$F-F17$$\"5n;a8 A/Ew%\\$F1Fah[l7$$\"5+vozm))>J'\\$F1F`j[l7$F^hrFag[l7$$\"5nmT5!4\"*eS] $F1F[i[l7$F\\wF`j[l7$FdhrFdg[l7$FghrFag[l7$FjhrF[i[l7$F_ir$!#;F17$Fdir Fdg[l7$FgirFhj[l7$F\\jr$!#=F17$F_jrFhj[l7$Fbjr$F1F17$FejrF`j[l7$Fhjr$! #)4%F1F^b\\l7$$\"5++]7`j)Q'*4%F1F\\a\\l7$$\"5+](=U#e\"z55 %F1Fcb\\l7$$\"5++DJ&HX>D5%F1$!$M\"F-7$$\"5+]iSmZ(fR5%F1F[`\\l7$FdhyF_a \\l7$$\"5+]Pf3P.%o5%F1F\\a]l7$$\"5++vozJ1G3TF1Fic\\l7$$\"5+]7y]E4s4TF1 Feg\\l7$$\"5++](=7Ah66%F1F[`\\l7$$\"5+](oHf^,E6%F1Ffc\\l7$$\"5++D1k5=/ 9TF1F\\d\\l7$$\"5+]i:N0@[:TF1F__\\l7$$\"5+++D1+C#p6%F1Fhf[l7$$\"5+]PMx %pi$=TF1F\\a\\l7$$\"5++vV[*)H!)>TF1Fbc]l7$$\"5+]7`>%GV77%F1F_a\\l7$$\" 5++]i!*yNoATF1Fbe\\l7$$\"5+](=&e[F1 Fe\\]l7$F_hlFcg\\l7$FdhlFd[^l7$FihlF[_^l7$F^czF^`^l7$Fb\\xFdb^l7$F[dzF jc^l7$F^ilFac^l7$Fj\\xF_[q7$FcilF_[q7$FhilFjjp7$F]jl$!\"*F17$Fbjl$!#7F 17$FgjlFhj[l7$FgipFig[l7$F\\[m$!#MF17$F_jp$!#XF17$Fcjp$Fcf[lF17$FhjpF \\j^l7$F][q$F`a^lF17$Fb[q$F\\d[lF17$Fe[q$Fj`^lF17$Fh[q$Fge[lF17$Fa[m$F e[^lF1-Fg[m6&Fi[mF\\\\m$F^jvFc\\q$Fj[vFc\\q-F`\\m6#%Ea~scheme~of~Tsito uras~and~PapakostasG-F$6%7`fl7$F($\"$='F-7$Fj\\m$\"$b&F-7$F_]m$\"$z%F- 7$Fd]m$\"$f%F-7$F/$\"$S%F-7$F_`m$\"$<%F-7$F[bm$\"$\"RF-7$F`^q$\"$'RF-7 $F`bm$\"$7%F-7$Fh^q$\"$.%F-7$F]_q$\"$(RF-7$Fb_qFe\\_l7$Fg_qFh\\_l7$F\\ `q$\"$1%F-7$F5$\"$8%F-7$F:$\"$+%F-7$F?$\"$$RF-7$FIFh\\_l7$FSFa]_l7$Fbc m$\"$%RF-7$Fgcm$\"$#RF-7$F\\dmFd^_l7$F_dm$\"$&RF-7$FddmFe\\_l7$Fidm$\" $!RF-7$F^emFd^_l7$FX$\"$*RF-7$FfemF___l7$F[fmF\\^_l7$F`fmF_^_l7$Fcfm$ \"$)QF-7$FhfmF_^_l7$F]gmF^]_l7$Fbgm$\"$-%F-7$FggmFh\\_l7$FjgmF[__l7$F_ hm$\"$0%F-7$FbhmF_^_l7$Fgn$\"$&QF-7$F^dq$\"$,%F-7$Fadq$\"$3%F-7$FddqF^ `_l7$FgdqF[__l7$F\\eqF^]_l7$F_eq$\"$5%F-7$FbeqF^`_l7$Feeq$\"$)RF-7$$\" 5++](oWllW/$F1F^`_l7$FheqF^]_l7$$\"5mm;HdvO\"*\\IF1Fj`_l7$F\\oF^]_l7$F _fqF]a_l7$FaoF[]_l7$FhgqF[]_l7$F[hqFi]_l7$F`hq$\"$B%F-7$Fchq$\"$L%F-7$ Fihq$\"$M%F-7$F_iq$\"$I%F-7$Fbiq$\"$N%F-7$Ffo$\"$R%F-7$F`im$\"$n%F-7$F [p$\"$2&F-7$F`p$\"$w&F-7$Fejm$\"$C'F-7$Fep$\"$U'F-7$Ff_r$\"$k'F-7$Fjp$ \"$u'F-7$Faar$\"$(oF-7$Ffar$\"$*pF-7$F[br$\"$e'F-7$F_q$\"$R'F-7$Fg\\n$ \"$y'F-7$F[^n$\"$(pF-7$F`^n$\"$\"oF-7$Fe^n$\"$s'F-7$Fj^nFgd_l7$F__n$\" $t'F-7$Fb_nFff_l7$Fdq$\"$'pF-7$$\"5nm\"z%\\`#\\%HLF1$\"$h'F-7$$\"5++]7 yT\\TKLF1$\"$b'F-7$$\"5n;zW#fy(*QL$F1$\"$v'F-7$$\"5ML3x1I1QNLF1Fdg_l7$ $\"5+]P4@uM'oL$F1$\"$\"pF-7$Fj_nFah_l7$$\"5ML$3F\\pxUM$F1$\"$&oF-7$Fiq Fce_l7$F^r$\"$_'F-7$Fcr$\"$V'F-7$Fddr$\"#mF17$Fh`n$F`a[lF17$FferFai_l7 $FhrFc_x7$F`an$\"#iF17$Fean$Fb\\qF17$Fjan$\"#oF17$F_bn$F[[[lF17$Febn$ \"#nF17$F[cnF]j_l7$Fefr$Ff^wF17$FhfrFdi_l7$F[grFai_l7$F^cnFbj_l7$F_grF `j_l7$Fdgr$Ff\\wF17$FhgrF\\[`l7$F]sF]j_l7$FdcnF]j_l7$FgcnF\\[`l7$FjcnF fj_l7$Fbs$\"#rF17$FbdnF]j_l7$FgdnFbj_l7$F\\en$Fb[[lF17$FgsFbj_l7$F`enF ai_l7$FeenFc[`l7$FhenFai_l7$F\\tFc[`l7$F\\fnFdi_l7$F_fnF]j_l7$FbfnFai_ l7$FatFh[`l7$F[uFh[`l7$FgvFh[`l7$Fhi[lFai_l7$F[j[lF\\[`l7$F^j[lF\\]x7$ Fcj[lF\\[`l7$Ffj[l$FhhvF17$Fjj[lFdi_l7$F][\\lF\\]x7$F^hrFh[`l7$Fa[\\lF \\[`l7$F\\wFfj_l7$FdhrFc[`l7$FghrFdi_l7$FjhrF\\[`l7$F_ir$FfivF17$FdirF h[`l7$FgirFfj_l7$F\\jr$F^jvF17$F_jrFfj_l7$FbjrF\\]x7$FejrF\\[`l7$FhjrF i\\`l7$F[[sFi\\`l7$F`[sFfj_l7$Fc[sFdi_l7$Ff[sF\\]x7$FawFc[`l7$Fj[sF\\] x7$F]\\sFfj_l7$F`\\sFg]`l7$Fe\\sFfj_l7$Fh\\sFh[`l7$F[]sFg]`l7$F^]sFfj_ l7$Fa]s$\"$!pF-7$Ff]s$\"$>(F-7$Fi]s$\"$W(F-7$F^^s$\"$G(F-7$Fc^s$\"$F(F -7$Fh^s$\"$7(F-7$F[_s$\"$L(F-7$F`_s$\"$5(F-7$Fdhn$\"$@(F-7$Ff_s$\"$I(F -7$F[`s$\"$S(F-7$F``s$\"$B(F-7$Fe`sFh_`l7$Fj`sF\\_`l7$F_asFd``l7$Fdas$ \"$6(F-7$Ffw$\"$$oF-7$F[xFc[_l7$F`x$\"$j&F-7$Fex$\"$5&F-7$Fjx$\"$v%F-7 $F]]t$\"$r%F-7$Fb]t$\"$$[F-7$Fg]t$\"$W%F-7$F\\^t$\"$s%F-7$F_^t$\"$b%F- 7$Fd^t$\"$[%F-7$Fi^tF\\c`l7$F\\_t$\"$%[F-7$F__t$\"$x%F-7$F_y$\"$]%F-7$ Fc_tF\\c`l7$Fh_t$\"$t%F-7$F]`tF]d`l7$F``tFfc`l7$Fc`t$\"$k%F-7$Fh`t$\"$ h%F-7$F]atF_c`l7$F`atF_\\_l7$FeatF]d`l7$Fhat$\"$m%F-7$F[bt$\"$l%F-7$F^ bt$\"$Y%F-7$Fcbt$\"$V%F-7$Fdy$\"$\\%F-7$F\\jn$\"$a%F-7$Fajn$\"$X%F-7$F fjn$\"$g%F-7$F[[o$\"$i%F-7$F`[oFib_l7$Fe[oF^e`l7$Fj[oF]f`l7$Fiy$\"$`%F -7$Fb\\oFfb_l7$Fg\\oFhc_l7$F\\]o$\"$j%F-7$Fa]o$\"$P%F-7$FadtF[e`l7$Ff] oFfb`l7$FidtFib`l7$F^zFag`l7$Fcz$\"$T%F-7$Fhz$\"$Z%F-7$Fb^o$\"$p%F-7$F g^oFge`l7$F\\_o$\"$Q%F-7$F__o$\"$@%F-7$Fd_oFge`l7$Fi_oF\\\\_l7$F\\`oFb c_l7$F][lFed`l7$Fd`oF\\c_l7$Fg`oFib`l7$F\\aoF[h`l7$FaaoFhc_l7$Ffao$\"$ U%F-7$Fb[lFbc_l7$Fg[l$\"$^%F-7$F\\\\l$\"$\"[F-7$FeftF^g`l7$FhftFge`l7$ F[gtF]f`l7$F^gt$\"$_%F-7$Fagt$\"$d%F-7$FdgtFhc_l7$FggtF`f`l7$FjgtF`f`l 7$FfhtF[e`l7$Fa\\l$\"$'[F-7$$\"5n\"zp)*f.#oCRF1F\\\\x7$$\"5n;H#onz(>ER F1Fae`l7$$\"5nTgx`dNrFRF1$\"$$\\F-7$$\"5nm\"H2$=$H#HRF1$\"$!\\F-7$$\"5 n\"H#o2z]uIRF1F[e`l7$$\"5n;aj%)R3EKRF1$\"$q%F-7$$\"5nT&)eh+mxLRF1F_\\a l7$FeitF^h`l7$$\"5n;zW#H)QKQRF1Fib`l7$$\"5nmTNY/aNTRF1$\"$w%F-7$$\"5n \"H2L_;rG%RF1Fib`l7$$\"5n;/E+EpQWRF1Fge`l7$$\"5nTN@x'o-f%RF1Fhc_l7$Ff \\lF^e`l7$FiitF[]al7$F\\jt$\"$&[F-7$FajtFhc_l7$FdjtF\\\\_l7$FjjtFgj`l7 $F`[u$\"$*[F-7$Fc[u$\"$'\\F-7$Ff[uF`b`l7$Fi[u$\"$([F-7$F\\\\u$\"$8&F-7 $F_\\uF\\\\_l7$Fd\\uF`b`l7$Fg\\u$\"$![F-7$Fj\\uFb^al7$F]]uFcb`l7$F`]uF ]b`l7$Fb^uFgi`l7$F[]l$\"$&\\F-7$F\\fo$\"$7&F-7$F`]l$\"$`&F-7$Fcau$\"$F &F-7$Ffau$\"$:&F-7$Fiau$\"$L&F-7$F\\bu$\"$N&F-7$F_bu$\"$S&F-7$Fbbu$\"$ g&F-7$Febu$\"$[&F-7$Fifo$\"$a&F-7$Fibu$\"$X&F-7$F\\cu$\"$P&F-7$F_cuFj` al7$Fbcu$\"$\\&F-7$Fecu$\"$f&F-7$F^go$\"$Q&F-7$F`hoFcaal7$Fdio$\"$$eF- 7$Fiio$\"$q&F-7$F\\jo$\"$'eF-7$Fajo$\"$Z&F-7$FfjoFiaal7$F`[p$\"$m&F-7$ Fe]lFjbal7$Fj]l$\"$*eF-7$Fddu$\"$!fF-7$Fgdu$\"$4'F-7$Fjdu$\"$,'F-7$F]e uFhcal7$F`euF[dal7$Fceu$\"$2'F-7$Ffeu$\"$*fF-7$F_^lFfdal7$$\"5+DcEZ!z' )49%F1$\"$7'F-7$$\"5+]iSm\\.QUTF1$\"$8'F-7$$\"5+voa&)3RxVTF1$\"$)eF-7$ Fjeu$\"$0'F-7$$\"5+D\"GQs-hl9%F1$\"$5'F-7$$\"5+](oHkeaz9%F1Fdeal7$$\"5 +v$4@c9[$\\TF1Fidal7$Fd^lF^dal7$F^fuF^dal7$FafuFhcal7$Fdfu$\"$<'F-7$Fg fuFfdal7$Fjfu$\"$+'F-7$Fi^l$\"$$fF-7$F^guFagal7$Fcgu$\"$3'F-7$FfguFaga l7$FiguF[dal7$F\\hu$\"$)fF-7$F^_lF^dal7$Fh_lF_\\x7$Fb`l$\"$_&F-7$F\\al $\"$4&F-7$FaalFgi`l7$FfalFge`l7$F[bl$\"$9%F-7$F`bl$\"$r$F-7$Febl$\"$C$ F-7$Fjbl$\"$(HF-7$F_cl$\"$'GF-7$FdclF_[x7$F^_wFf]r7$Fa_wFf]r7$Fd_w$\"$ y#F-7$Fg_w$\"$!GF-7$$\"5n\"zWn\\,p\\f%F1Ff]r7$Fj_w$\"$%GF-7$$\"5M3FW?- 2#yf%F1$\"$#GF-7$FiclFc^r7$F^`wFf]r7$Fa`wFjjal7$Fd`wF_jal7$Fg`wF``r7$F j`w$\"$$GF-7$F^dl$\"$s#F-7$FcdlFc^r7$FhdlFfial7$F]el$\"$&GF-7$FbelFeja l7$F\\cwFj[bl7$F_cw$\"$!HF-7$Fbcw$\"$x#F-7$FecwFe`r7$FhcwF_[x7$F[dwFe` r7$F^dwFe`r7$FgelFc^r7$FbdwFc_r7$Fedw$\"$#HF-7$FhdwF[^r7$F\\flF[^r7$F \\ewFjjal7$F_ewF[]bl7$FbewF[^r7$FeewFjjal7$FhewF[^r7$F[fwF[_r7$F^fwFb[ bl7$FaapF[]bl7$$\"5n\"zW<()\\TJp%F1F[]bl7$$\"5+]7`pbve%p%F1Ffial7$$\"5 M3xJn7O.'p%F1F]ar7$Fbfw$\"$'HF-7$$\"5+D1*GmsD*)p%F1Fa^bl7$$\"5M$3x1Oyr .q%F1Fejal7$$\"5nTNYeSy\"=q%F1F[]q7$FaflF[]q7$FiapFfial7$F^bp$\"$+$F-7 $FcbpF[^r7$FhbpFfial7$F[cpFcial7$F`cp$\"$.$F-7$Fccp$\"$*HF-7$FfflF[]bl 7$F]hw$\"$6$F-7$F[glF`ial7$F`gl$\"$\\$F-7$FeglFd^_l7$Fgep$\"$E%F-7$Fjg lF^g`l7$Fdfp$\"$@&F-7$F_hl$\"$o&F-7$Fb[x$\"$H'F-7$FdhlFjf_l7$Fj[x$\"$r (F-7$Fihl$\"$])F-7$F^il$\"$C*F-7$Fcil$FbctF17$FhilF\\^x7$F]jl$Ff_tF17$ F]hp$Fj\\tF17$Fbjl$FjfsF17$$\"5+vV)*3hFkl\\F1Fbbbl7$Fehp$FbhsF17$$\"5+ DJ?3Z.jo\\F1Fbbbl7$Fhhp$F`]tF17$F^ipF`bbl7$Fgjl$FjcrF17$Fh]xF\\cbl7$Fg ipFj]x7$F^^xFj]x7$Fa^xFf^x7$Fd^x$F[ctF17$F\\[mFf^x7$F_jp$FhhuF17$Fcjp$ FcjuF17$Fhjp$Fe[vF17$F][qF`]x7$Fb[qF`]x7$Fe[qF`]x7$Fh[q$F`[vF17$Fa[m$F ca[lF1-Ff_x6&Fi[mF`hzF\\\\mF`hz-F`\\m6#%Ra~modification~of~the~Tsitour as-Papakostas~schemeG-F$6%7jv7$F($\"%P>F-7$F/$\"%p:F-7$F5$\"%$Q\"F-7$F :$\"%-8F-7$F?$\"%]7F-7$FD$\"%K7F-7$FI$\"%B7F-7$$\"5nmm\"H<%G7mHF1$\"%? 7F-7$FN$\"%67F-7$$\"5nmmT&QVJz(HF1$\"%57F-7$FS$\"%<7F-7$FX$\"%E7F-7$Fg n$\"%O7F-7$F\\o$\"%'G\"F-7$Fao$\"%P8F-7$Ffo$\"%'[\"F-7$F[p$\"%JGF-7$Fiq$\"%FKF-7$Fcr$\"%QNF-7$Fhr$\"$ k$F17$F]s$\"$v$F17$FdcnFfibl7$Fgcn$\"$z$F17$Fjcn$\"$\"QF17$Fbs$\"$y$F1 7$FbdnFfibl7$Fgdn$\"$u$F17$F\\en$\"$x$F17$FgsFfibl7$F`enFdjbl7$FeenFji bl7$FhenFdjbl7$F\\tF`jbl7$F\\fnFgjbl7$F_fnFfibl7$Fbfn$\"$t$F17$FatFgjb l7$FftFgjbl7$F[uFgjbl7$F`uFgjbl7$FeuF`jbl7$Fhu$\"$w$F17$F]vFdjbl7$$\"5 nT5S>>tb\"[$F1Fa[cl7$FbvFgjbl7$$\"5Me9\"fZIjT[$F1Fa[cl7$FgvFfibl7$F[j[ lFi[cl7$Fcj[lFfibl7$Ffj[lFjibl7$Fjj[l$\"$s$F17$F][\\lFgjbl7$F^hrFh\\cl 7$$\"5+]7y+U,'4]$F1Fh\\cl7$Fa[\\lFa[cl7$$\"5+DcwM&H3c]$F1$\"$p$F17$$\" 5M$3F%zzw:2NF1Fdjbl7$$\"5nT&)3Ckqq3NF1Fh\\cl7$F\\w$F^ialF17$F_jr$\"$m$ F17$Faw$\"$g$F17$Ffw$\"%yLF-7$F[x$\"%hIF-7$F`x$\"%)y#F-7$Fex$\"%MDF-7$ Fjx$\"%hBF-7$F\\^t$\"%WBF-7$Fd^t$\"%2BF-7$$\"5+D1*GJoT]x$F1$\"%ABF-7$F i^t$\"%3BF-7$$\"5L3_v!R7))zx$F1Ff_cl7$F\\_t$\"%JBF-7$F__t$\"%=BF-7$F_y $\"%'G#F-7$Fc_tFj`cl7$Fh_t$\"%)H#F-7$F``tF^acl7$Fh`t$\"%xAF-7$F]at$\"% iAF-7$F`at$\"%^AF-7$Feat$\"%$G#F-7$Fhat$\"%uAF-7$F[bt$\"%rAF-7$F^btFha cl7$Fcbt$\"%XAF-7$Fdy$\"%]AF-7$F[[o$\"%dAF-7$Fiy$\"%VAF-7$Fa]o$\"%BAF- 7$F^z$\"%>AF-7$Fcz$\"%=AF-7$FhzFaccl7$Fb^o$\"%WAF-7$Fg^o$\"%DAF-7$F\\_ o$\"%9AF-7$F__o$\"%'>#F-7$Fd_oF^dcl7$Fi_o$\"%OAF-7$F\\`o$\"%7AF-7$F][l $\"%QAF-7$Fg`oFhbcl7$Faao$\"%ZAF-7$$\"5+voHan\")=rQF1$\"%TAF-7$FfaoFac cl7$$\"5Le*)f0(o1T(QF1$\"%YAF-7$Fb[l$\"%BF-7$Ff\\l$\"%UBF-7$Fj\\u$\"%>CF-7$F[]l$\"%qCF-7$F`]l$\"%_EF -7$Fe]l$\"%'z#F-7$Fj]l$\"%7HF-7$Fddu$\"%;HF-7$Fgdu$\"%PHF-7$Fjdu$\"%KH F-7$F]eu$\"%@HF-7$Fceu$\"%VHF-7$F_^l$\"%ZHF-7$Fjeu$\"%_HF-7$Fd^l$\"%jH F-7$Fgfu$\"%lHF-7$Fi^l$\"%bHF-7$Fcgu$\"%sHF-7$Figu$\"%aHF-7$F\\huFbicl 7$F^_l$\"%vHF-7$Ff\\p$\"%fHF-7$F[]p$\"%`HF-7$F`]p$\"%cHF-7$Fc_l$\"%XHF -7$Fh]p$\"%[HF-7$Fh_l$\"%UHF-7$F]`l$\"%6HF-7$Fb`l$\"%tGF-7$F\\al$\"%'p #F-7$Faal$\"%fCF-7$FfalFadcl7$F[bl$\"%`>F-7$F`bl$\"%E " 0 "" {MPLTEXT 1 0 1 ";" }} }}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 49 "Test 11 of 12 stage, order 8 Ru nge-Kutta methods " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 143 "This example is similar to one that appears in an articl e by F. G. Lether: Mathematics of Computation, Vol. 20, no. 95, (July 1966) page 382. " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 256 "" 0 " " {TEXT -1 1 " " }{XPPEDIT 18 0 "dy/dx=exp(-x)/(x-1)^2" "6#/*&%#dyG\" \"\"%#dxG!\"\"*&-%$expG6#,$%\"xGF(F&*$,&F.F&F&F(\"\"#F(" }{TEXT -1 2 " " }{XPPEDIT 18 0 "cos(1/(x-1))-y" "6#,&-%$cosG6#*&\"\"\"F(,&%\"xGF(F (!\"\"F+F(%\"yGF+" }{TEXT -1 5 ", " }{XPPEDIT 18 0 "y(0)=sin*1" "6# /-%\"yG6#\"\"!*&%$sinG\"\"\"F*F*" }{TEXT -1 1 " " }}{PARA 0 "" 0 "" {TEXT -1 10 "Solution: " }}{PARA 256 "" 0 "" {TEXT -1 2 " " } {XPPEDIT 18 0 "y = -exp(-x)*sin(1/(x-1))" "6#/%\"yG,$*&-%$expG6#,$%\"x G!\"\"\"\"\"-%$sinG6#*&F-F-,&F+F-F-F,F,F-F," }{TEXT -1 1 " " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 185 "d e := diff(y(x),x)=exp(-x)/(x-1)^2*cos(1/(x-1))-y(x);\nic := y(0)=sin(1 );\ndsolve(\{de,ic\},y(x));\nq := unapply(rhs(%),x):\nplot(q(x),x=0..1 -1/(6*Pi),font=[HELVETICA,9],labels=[`x`,`y(x)`]);" }}{PARA 11 "" 1 " " {XPPMATH 20 "6#>%#deG/-%%diffG6$-%\"yG6#%\"xGF,,&*(-%$expG6#,$F,!\" \"\"\"\",&F,F4F4F3!\"#-%$cosG6#*&F4F4F5F3F4F4F)F3" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%#icG/-%\"yG6#\"\"!-%$sinG6#\"\"\"" }}{PARA 11 "" 1 " " {XPPMATH 20 "6#/-%\"yG6#%\"xG,$*&-%$expG6#,$F'!\"\"\"\"\"-%$sinG6#*& F/F/,&F'F/F/F.F.F/F." }}{PARA 13 "" 1 "" {GLPLOT2D 603 396 396 {PLOTDATA 2 "6&-%'CURVESG6$7[r7$$\"\"!F)$\"30l*y![)4ZT)!#=7$$\"3#>=\"* )>z2k?!#>$\"3m%p=Qu6DN)F,7$$\"3kMQU\"3@+'QF0$\"3m=O<,5$yO#)F,7$$\"38UQ!)p4'G\"zF0$\"3/Z^$zTlU<)F,7$$\" 3BY$*R0>JO**F0$\"36ty1)z*36\")F,7$$\"3wbXC%*4B\"=\"F,$\"3A;o(=P!Q^!)F, 7$$\"3M!)fxC(zaP\"F,$\"3E8^!ydw!))zF,7$$\"3kgswR?Pw:F,$\"3T8>lD8j?zF,7 $$\"3Qnlb!4?mx\"F,$\"3%p4\\s^P4&yF,7$$\"3OsvSC)*f#)>F,$\"3/$H(=wa6wxF, 7$$\"3)Rk+h)o-k@F,$\"3LR8d$>`qq(F,7$$\"3Q^Vo'yq#oBF,$\"3YB)Qc;#3DwF,7$ $\"3?0sMKLNtDF,$\"3,;%fG`C(F,7$$ \"3S+dSsVlWLF,$\"3&36sy[X09(F,7$$\"3EOur83&\\b$F,$\"37)QgTzpp+(F,7$$\" 3c#**Qwz)4TPF,$\"3M]xfz#>l(oF,7$$\"3wx#p)QELXRF,$\"3UR-VbS%zr'F,7$$\"3 \"yy;Gb6)RTF,$\"3%40quzD%\\lF,7$$\"3p2KM(*)HFM%F,$\"3W'4!o9@F_jF,7$$\" 3`G+(=Gs!HXF,$\"3S.Rv)o&))[hF,7$$\"3&\\%Rcec>>`F,7$$\"39 pTj@J(oJ&F,$\"3-j_%RM!Rk\\F,7$$\"3QD(p)Qdl>bF,$\"34_#)R=svWXF,7$$\"3#) Qm@o*Q!=dF,$\"3Iba)Q0\")Q2%F,7$$\"3#oP&GV\\)*4fF,$\"3;HPYk\"[Jb$F,7$$ \"3qUjqA#3J7'F,$\"3iI9Us9n*)GF,7$$\"3-koIo*3YJ'F,$\"3G0-$>\\f\"3AF,7$$ \"3fQ6D*)o2>lF,$\"3CG]Vmkt$Q\"F,7$$\"35Y]`:_N/nF,$\"33Dv;1s&pZ&F07$$\" 3.\"Q#ekL\"p!pF,$!3OST&zF\\gd%F07$$\"3%yB5rz/v4(F,$!3cn!**p^J2Z\"F,7$$ \"33-p_gxs'H(F,$!3A9e\"Q#e(4b#F,7$$\"3%324>i/:\\(F,$!3S!=W0GJd`$F,7$$ \"39FEN$GhMf(F,$!3U'\\`Wd^A(RF,7$$\"3c%='zWzT&p(F,$!3f\\'**z(4Q8VF,7$$ \"3,%e*>Oh^WxF,$!3G(>*pycjJWF,7$$\"3K#)HgFVh$z(F,$!3)Q%ReLd!G^%F,7$$\" 3h#o/LUj\"=yF,$!3p>V&oBmw`%F,7$$\"3y\"Q1!>DrUyF,$!3MH+5_!>5b%F,7$$\"3% 433Zhhs'yF,$!3KtI_'\\k?b%F,7$$\"3?\"y4/r5=*yF,$!3)*fMuc7)*RXF,7$$\"3@o 'GDq??%zF,$!3)R70$\\E%3Z%F,7$$\"3=bvk%pIA*zF,$!3(yFHocWfL%F,7$$\"3IVkw 'oSC/)F,$!3&[P'\\m/)z7%F,7$$\"3II`))y1l#4)F,$!3E`XP\"[()*RQF,7$$\"3?gE &\\8RA>)F,$!3'*pxl2UD4IF,7$$\"37!**>5fF=H)F,$!3Fu3OFE=:=F,7$$\"3$)fW:: LeP$)F,$!374'4ii(p\\6F,7$$\"3kI*)GR!RLQ)F,$!3q\\t;%4rO@%F07$$\"3Z,MUjZ 4H%)F,$\"3c]$R4W&G]NF07$$\"3=ryb([][Z)F,$\"37Ccp$Rpv:\"F,7$$\"3%QzY4r \"HF&)F,$\"3CcG)))y\"yp?F,7$$\"3g()F,$\"3y_cgRCTxTF,7$$\"3Wx*okV>:t)F,$\"37VUul#=X<%F,7$$\"3q% oYTYr\\v)F,$\"3Ri.A'QV25%F,7$$\"3'>RC=\\B%y()F,$\"3x'=W7Fx&HRF,7$$\"3] 1)zraF`#))F,$\"3QFG%plB4F$F,7$$\"39A_`-;Bs))F,$\"3700_RQTz@F,7$$\"31^1 [bjB(*))F,$\"3OuNH?R:M9F,7$$\"34\"3E%36CA*)F,$\"3E<)GsFg<(fF07$$\"376: PheCZ*)F,$!3ocHHl.]EIF07$$\"39TpJ91Ds*)F,$!38U'y=D^^A\"F,7$$\"3=rBEn`D (**)F,$!3j&4Nt%[@=@F,7$$\"3@,y??,EA!*F,$!3!e@9jLE#=HF,7$$\"3EJK:t[EZ!* F,$!3I(>(y]J<_NF,7$$\"3Gh')4E'pA2*F,$!3U`$=&457URF,7$$\"3'3]u3\"GDy!*F ,$!3U'ykJ>F2*RF,7$$\"3cT.l&*fB%3*F,$!3AH^N$RP+-%F,7$$\"3E#=E/=>-4*F,$! 3&Rn/8H]\"HSF,7$$\"3'=--_O-i4*F,$!39=u$p*4B>*F,$!39yV6L@mxvF07$$\"3oYr;\"p**Q?*F,$!3_9 %**Gult/#!#?7$$\"3(p#)=21me@*F,$\"3=-0hZ^rztF07$$\"3E20FIC$yA*F,$\"3ML bSaq^*[\"F,7$$\"3b(=A)*z)zR#*F,$\"3Qccz;X$>?#F,7$$\"3%*oQPp^w^#*F,$\"3 \\RTgHcmRGF,7$$\"3C\\b#*Q:tj#*F,$\"3M$o6)eV:lLF,7$$\"3oE3CP5fw#*F,$\"3 Ma)HpV]]I!H!G$*F,$\"3a%4t 07BP*GF,7$$\"3J=s\")G&))3M*F,$\"3f(3)H;[%o+#F,7$$\"3w&\\Kr-[PN*F,$\"33 n1kl[]%*F,$!3m (=[SoWqQ#F,7$$\"3%>saO,CmX*F,$!3GDS5eP#en\"F,7$$\"3AhB\"Gw`IY*F,$!3U$3 !Gg0_(o)F07$$\"3]++(>^$[p%*F,$!3V'=8$[D+C:!#C-%'COLOURG6&%$RGBG$\"#5! \"\"F(F(-%%FONTG6$%*HELVETICAG\"\"*-%+AXESLABELSG6$%\"xG%%y(x)G-%%VIEW G6$;F($\"+BN[p%*!#5%(DEFAULTG" 1 2 0 1 10 0 2 9 1 4 2 1.000000 45.000000 45.000000 0 0 "Curve 1" }}}}{PARA 0 "" 0 "" {TEXT -1 0 "" }} {PARA 0 "" 0 "" {TEXT -1 93 "The following code constructs a discrete \+ solution based on each of the methods and gives the " }{TEXT 260 22 "r oot mean square error" }{TEXT -1 18 " of each solution." }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 848 "Q := (x,y) -> exp(-x)/(x-1)^2*cos( 1/(x-1))-y: hh := 1/500-1/(3000*Pi): numsteps := 500: x0 := 0: y0 := s in(1):\nmatrix([[`slope field: `,Q(x,y)],[`initial point: `,``(x0,y0 )],[`step width: `,hh],\n[`no. of steps: `,numsteps]]);``;\nmthds \+ := [`Enright-Verner scheme`,`Verner's Maple scheme`,`a modification of Verner's Maple scheme`,`an \"efficient\" scheme`,`a scheme of Tsitour as and Papakostas`,`a modification of the Tsitouras-Papakostas scheme` ,`Verner's \"most efficient\" scheme`]: errs := []:\nDigits := 30:\nfo r ct to 7 do\n Qn_RK8_||ct := RK8_||ct(Q(x,y),x,y,x0,evalf[33](y0),e valf[33](hh),numsteps,false);\n sm := 0: numpts := nops(Qn_RK8_||ct) :\n for ii to numpts do\n sm := sm+(Qn_RK8_||ct[ii,2]-q(Qn_RK8_ ||ct[ii,1]))^2;\n end do:\n errs := [op(errs),sqrt(sm/numpts)];\ne nd do:\nDigits := 10:\nlinalg[transpose]([mthds,evalf(errs)]);" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#K%'matrixG6#7&7$%0slope~field:~~~G,&*( -%$expG6#,$%\"xG!\"\"\"\"\",&F/F1F1F0!\"#-%$cosG6#*&F1F1F2F0F1F1%\"yGF 07$%0initial~point:~G-%!G6$\"\"!-%$sinG6#F17$%/step~width:~~~G,&#F1\"$ +&F1*&F1F1*&\"%+IF1%#PiGF1F0F07$%1no.~of~steps:~~~GFFQ)pprint516\"" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#%!G" }}{PARA 11 "" 1 "" {XPPMATH 20 "6 #K%'matrixG6#7)7$%6Enright-Verner~schemeG$\"+KT#fj*!#B7$%6Verner's~Map le~schemeG$\"+Z&=[B&F+7$%Ha~modification~of~Verner's~Maple~schemeG$\"+ BV=fhF+7$%6an~\"efficient\"~schemeG$\"+`Sue$)F+7$%Ea~scheme~of~Tsitour as~and~PapakostasG$\"+A7/byF+7$%Ra~modification~of~the~Tsitouras-Papak ostas~schemeG$\"+I1,4yF+7$%AVerner's~\"most~efficient\"~schemeG$\"+\\% *=&>)F+Q)pprint526\"" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 30 "The following code constructs " }{TEXT 260 20 "numeric al procedures" }{TEXT -1 56 " for solutions based on each of the Runge -Kutta schemes." }}{PARA 0 "" 0 "" {TEXT -1 75 "The error in the value obtained by each of the methods at the point where " }{XPPEDIT 18 0 "x = 0;" "6#/%\"xG\"\"!" }{TEXT -1 21 ".9469 is also given." }} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 772 "Q := (x,y) -> exp(-x)/(x-1) ^2*cos(1/(x-1))-y: hh := 1/500-1/(3000*Pi): numsteps := 500: x0 := 0: \+ y0 := sin(1):\nmatrix([[`slope field: `,Q(x,y)],[`initial point: `,` `(x0,y0)],[`step width: `,hh],\n[`no. of steps: `,numsteps]]);``; \nmthds := [`Enright-Verner scheme`,`Verner's Maple scheme`,`a modific ation of Verner's Maple scheme`,`an \"efficient\" scheme`,`a scheme of Tsitouras and Papakostas`,`a modification of the Tsitouras-Papakostas scheme`,`Verner's \"most efficient\" scheme`]: errs := []:\nDigits := 30:\nfor ct to 7 do\n qn_RK8_||ct := RK8_||ct(Q(x,y),x,y,x0,evalf(y 0),evalf(hh),numsteps,true);\nend do:\nxx := 0.9469: qxx := evalf(q(xx )):\nfor ct to 7 do\n errs := [op(errs),abs(qn_RK8_||ct(xx)-qxx)];\n end do:\nDigits := 10:\nlinalg[transpose]([mthds,evalf(errs)]);" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#K%'matrixG6#7&7$%0slope~field:~~~G,&*( -%$expG6#,$%\"xG!\"\"\"\"\",&F/F1F1F0!\"#-%$cosG6#*&F1F1F2F0F1F1%\"yGF 07$%0initial~point:~G-%!G6$\"\"!-%$sinG6#F17$%/step~width:~~~G,&#F1\"$ +&F1*&F1F1*&\"%+IF1%#PiGF1F0F07$%1no.~of~steps:~~~GFFQ)pprint536\"" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#%!G" }}{PARA 11 "" 1 "" {XPPMATH 20 "6 #K%'matrixG6#7)7$%6Enright-Verner~schemeG$\"+wZ)\\p\"!#@7$%6Verner's~M aple~schemeG$\"+&RHk/)!#A7$%Ha~modification~of~Verner's~Maple~schemeG$ \"+;qd\\5F+7$%6an~\"efficient\"~schemeG$\"+RZl\"[\"F+7$%Ea~scheme~of~T sitouras~and~PapakostasG$\"+s*H1R\"F+7$%Ra~modification~of~the~Tsitour as-Papakostas~schemeG$\"+/7?#Q\"F+7$%AVerner's~\"most~efficient\"~sche meG$\"+Tqq_9F+Q)pprint546\"" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 4 "The " }{TEXT 260 22 "root mean square error" } {TEXT -1 19 " over the interval " }{XPPEDIT 18 0 " [0, 1-1/(6*Pi)] " " 6#7$\"\"!,&\"\"\"F&*&F&F&*&\"\"'F&%#PiGF&!\"\"F+" }{TEXT -1 82 " of e ach Runge-Kutta method is estimated as follows using the special proce dure " }{TEXT 0 5 "NCint" }{TEXT -1 98 " to perform numerical integr ation by the 7 point Newton-Cotes method over 200 equal subintervals. " }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 496 "mthds := [`Enright-Vern er scheme`,`Verner's Maple scheme`,`a modification of Verner's Maple s cheme`,`an \"efficient\" scheme`,`a scheme of Tsitouras and Papakostas `,`a modification of the Tsitouras-Papakostas scheme`,`Verner's \"most efficient\" scheme`]: errs := []:\nDigits := 20:\nfor ct to 7 do\n \+ sm := NCint((q(x)-'qn_RK8_||ct'(x))^2,x=0..1-1/(6*Pi),adaptive=false,n umpoints=7,factor=200);\n errs := [op(errs),sqrt(sm/(1-1/(6*Pi)))]; \nend do:\nDigits := 10:\nlinalg[transpose]([mthds,evalf(errs)]);" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#K%'matrixG6#7)7$%6Enright-Verner~schem eG$\"+S:!4X%!#B7$%6Verner's~Maple~schemeG$\"+ai4XNF+7$%Ha~modification ~of~Verner's~Maple~schemeG$\"+x!HMU$F+7$%6an~\"efficient\"~schemeG$\"+ QJG9RF+7$%Ea~scheme~of~Tsitouras~and~PapakostasG$\"+<))QRPF+7$%Ra~modi fication~of~the~Tsitouras-Papakostas~schemeG$\"+%)H)Rs$F+7$%AVerner's~ \"most~efficient\"~schemeG$\"+\\CPDQF+Q)pprint556\"" }}}{PARA 0 "" 0 " " {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 93 "The following error gra phs are constructed using the numerical procedures for the solutions. \+ " }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 618 "evalf[30](plot(['qn_RK8 _1'(x)-q(x),'qn_RK8_2'(x)-q(x),'qn_RK8_3'(x)-q(x),'qn_RK8_4'(x)-q(x), \n'qn_RK8_5'(x)-q(x),'qn_RK8_6'(x)-q(x),'qn_RK8_7'(x)-q(x)],x=0..0.5,- 4.3e-25..4.3e-25,font=[HELVETICA,9],\ncolor=[COLOR(RGB,.4,0,.9),COLOR( RGB,0,.65,0),brown,red,COLOR(RGB,0,.75,.95),magenta,COLOR(RGB,.95,.55, 0)],\nlegend=[`Enright-Verner scheme`,`Verner's Maple scheme`,`a modif ication of Verner's Maple scheme`,`an \"efficient\" scheme`,`a scheme \+ of Tsitouras and Papakostas`,`a modification of the Tsitouras-Papakost as scheme`,`Verner's \"most efficient\" scheme`],title=`error curves f or 12 stage order 8 Runge-Kutta methods`));" }}{PARA 13 "" 1 "" {GLPLOT2D 783 536 536 {PLOTDATA 2 "6--%'CURVESG6%7gn7$$\"\"!F)F(7$$\"? MLLLLLLLLL3x&)*3\"!#J$\"#g!#I7$$\"?nmmmmmmmm\"H2P\"Q?F-$\"$B\"F07$$\"? MLLLLLLLL$eRwX5$F-$\"$5#F07$$\"?NLLLLLLLL$3x%3yTF-$\"$*HF07$$\"?ommmmm mmm\"z%4\\Y_F-$\"$&QF07$$\"?NLLLLLLLLeR-/PiF-$\"$%[F07$$\"?-++++++++Dc mpisF-$\"$&fF07$$\"?OLLLLLLLLe*)>VB$)F-$\"$8(F07$$\"?.++++++++DJbw!Q*F -$\"$W)F07$$\"?nmmmmmmmm;/j$o/\"F0$\"$)**F07$$\"?MLLLLLLLL3_>jU6F0$\"% R6F07$$\"?+++++++++]i^Z]7F0$\"%@8F07$$\"?+++++++++](=h(e8F0$\"%*[\"F07 $$\"?+++++++++]P[6j9F0$\"%0F07$$\"? nmmmmmmmm;a/cq;F0$\"%c@F07$$\"?nmmmmmmmmm;t,mF0$\"%WHF07$$\"?,+++++++]i!f #=$3#F0$\"%!G$F07$$\"?,+++++++](=xpe=#F0$\"%!e$F07$$\"?nmmmmmmmm\"H28I H#F0$\"%dRF07$$\"?nmmmmmmm;zpSS\"R#F0$\"%!G%F07$$\"?MLLLLLLLL3_?`(\\#F 0$\"%GYF07$$\"?MLLLLLLL$e*)>pxg#F0$\"%4]F07$$\"?,+++++++]Pf4t.FF0$\"%L `F07$$\"?MLLLLLLLLe*Gst!GF0$\"%&p&F07$$\"?,+++++++++DRW9HF0$\"%mfF07$$ \"?,++++++++DJE>>IF0$\"%$='F07$$\"?,+++++++]i!RU07$F0$\"%wiF07$$\"?,++ ++++++v=S2LKF0$\"%6iF07$$\"?ommmmmmmmm\"p)=MLF0$\"%#)eF07$$\"?,+++++++ +](=]@W$F0$\"%)G&F07$$\"?MLLLLLLL$e*[$z*RNF0$\"%`UF07$$\"?,++++++++]iC $pk$F0$\"%\"e#F07$$\"?ommmmmmm;H2qcZPF0$\"$>#F07$$\"?,+++++++]7.\"fF&Q F0$!%RMF07$$\"?ommmmmmmm;/OgbRF0$!%#=)F07$$\"?,+++++++]ilAFjSF0$!&o`\" F07$$\"?NLLLLLLLLL$)*pp;%F0$!&#)e#F07$$\"?NLLLLLLLL3xe,tUF0$!&Ox$F07$$ \"?ommmmmmm;HdO=yVF0$!&Bp&F07$$\"?,+++++++++D>#[Z%F0$!&>$yF07$$\"?ommm mmmmmT&G!e&e%F0$!'&o7\"F07$$\"?-+++++++]P%37^j%F0$!'*3F\"F07$$\"?NLLLL LLLLL$)Qk%o%F0$!'k1:F07$$\"?ommmmmmm\"z>6but%F0$!'L(y\"F07$$\"?-++++++ +]iSjE!z%F0$!'pe?F07$$\"?-+++++++D\"G))Rb\"[F0$!'0MAF07$$\"?-+++++++++ DM\"3%[F0$!'rgBF07$$\"?-+++++++v=np3m[F0$!'0qDF07$$\"?-+++++++]P40O\"* [F0$!'`#y#F07$$\"?-++++++]7.#Q?&=\\F0$!'bUHF07$$\"?,+++++++voa-oX\\F0$ !'auKF07$$\"?+++++++Dc,\">g#f\\F0$!'.:LF07$$\"?+++++++]PMF,%G(\\F0$!'/ aMF07$$\"?+++++++v=nj+U')\\F0$!'8XOF07$$\"\"&!\"\"$!'dZQF0-%&COLORG6&% $RGBG$\"\"%Ff]lF($\"\"*Ff]l-%'LEGENDG6#%6Enright-Verner~schemeG-F$6%7g nF'7$F+$!#fF07$F2$!$A\"F07$F7$!$)>F07$F<$!$!HF07$FA$!$p$F07$FF$!$n%F07 $FK$!$o&F07$FP$!$*oF07$FU$!$2)F07$FZ$!$e*F07$Fin$!%$4\"F07$F^o$!%u7F07 $Fco$!%N9F07$Fho$!%W;F07$F]p$!%O=F07$Fbp$!%%3#F07$Fgp$!%0BF07$F\\q$!%% f#F07$Faq$!%^GF07$Ffq$!%zJF07$F[r$!%mMF07$F`r$!%JQF07$Fer$!%STF07$Fjr$ !%qWF07$F_s$!%E[F07$Fds$!%H^F07$Fis$!%^aF07$F^t$!%$o&F07$Fct$!%\\eF07$ Fht$!%zeF07$F]u$!%BdF07$Fbu$!%s_F07$Fgu$!%YXF07$F\\v$!%KLF07$Fav$!%E9F 07$Ffv$\"%G7F07$F[w$\"%\"G&F07$F`w$\"&50\"F07$Few$\"&l$=F07$Fjw$\"&7)H F07$F_x$\"&pE%F07$Fdx$\"&:M'F07$Fix$\"&'\\')F07$F^y$\"'+N7F07$Fcy$\"'5 !R\"F07$Fhy$\"'DV;F07$F]z$\"'kW>F07$Fbz$\"'gOAF07$Fgz$\"''RU#F07$F\\[l $\"'\"*fDF07$Fa[l$\"'>&y#F07$Ff[l$\"'17IF07$F[\\l$\"'(Q=$F07$F`\\l$\"' IRNF07$Fe\\l$\"'#Qe$F07$Fj\\l$\"'oJPF07$F_]l$\"'JORF07$Fd]l$\"'7`TF0-F j]l6&F\\^lF($\"#l!\"#F(-Fb^l6#%6Verner's~Maple~schemeG-F$6%7gnF'7$F+$! #MF07$F2$!#qF07$F7$!$6\"F07$F<$!$l\"F07$FA$!$4#F07$FF$!$l#F07$FK$!$@$F 07$FP$!$\"RF07$FU$!$c%F07$FZ$!$U&F07$Fin$!$>'F07$F^o$!$A(F07$Fco$!$8)F 07$Fho$!$J*F07$F]p$!%T5F07$Fbp$!%#=\"F07$Fgp$!%38F07$F\\q$!%t9F07$Faq$ !%?;F07$Ffq$!%3=F07$F[r$!%t>F07$F`r$!%%=#F07$Fer$!%jBF07$Fjr$!%bDF07$F _s$!%kFF07$Fds$!%XHF07$Fis$!%RJF07$F^t$!%'G$F07$Fct$!%,MF07$Fht$!%TMF0 7$F]u$!%*Q$F07$Fbu$!%\"=$F07$Fgu$!%AGF07$F\\v$!%0AF07$Fav$!%@7F07$Ffv$ \"$m\"F07$F[w$\"%(H#F07$F`w$\"%l]F07$Few$\"%M#*F07$Fjw$\"&J`\"F07$F_x$ \"&&>AF07$Fdx$\"&)GLF07$Fix$\"&Yc%F07$F^y$\"&\"[lF07$Fcy$\"&1Q(F07$Fhy $\"&$Q()F07$F]z$\"'eN5F07$Fbz$\"']#>\"F07$Fgz$\"'$HH\"F07$F\\[l$\"'(fO \"F07$Fa[l$\"':([\"F07$Ff[l$\"'!)3;F07$F[\\l$\"'>,F07$Fj\\l$\"'Z&*>F07$F_]l$\"'T0@F07$Fd]l$\"'#>A#F0-%'COL OURG6&F\\^l$\")#)eqk!\")$\"))eqk\"FdemFeem-Fb^l6#%Ha~modification~of~V erner's~Maple~schemeG-F$6%7enF'7$F+$\"#5F07$F2$\"#?F07$F7$\"#RF07$F<$ \"#`F07$FA$\"#qF07$FF$\"#')F07$FK$\"$5\"F07$FP$\"$G\"F07$FU$\"$b\"F07$ FZ$\"$#=F07$Fin$\"$3#F07$F^o$\"$S#F07$Fco$\"$r#F07$Fho$\"$5$F07$F]p$\" $W$F07$Fbp$\"$*QF07$Fgp$\"$I%F07$F\\q$\"$\"[F07$Faq$\"$B&F07$Ffq$\"$x& F07$F[r$\"$E'F07$F`r$\"$$oF07$Fer$\"$H(F07$Fjr$\"$v(F07$F_s$\"$?)F07$F ds$\"$\\)F07$Fis$\"$r)F07$F^t$\"$n)F07$Fct$\"$J)F07$Fht$\"$i(F07$F]u$ \"$<'F07$Fbu$\"$z$F07$Fgu$\"#vF07$F\\v$!$#QF07$Fav$!%[5F07$Ffv$!%D>F07 $F[w$!%@KF07$F`w$!%M[F07$Few$!%@sF07$Fjw$!&K1\"F07$F_x$!&6W\"F07$Fdx$! &c/#F07$Fix$!&Dr#F07$F^y$!&\\x$F07$Fcy$!&o@%F07$Fhy$!&5%\\F07$F]z$!&)* z&F07$Fbz$!&]i'F07$Fgz$!&@;(F07$F\\[l$!&qa(F07$Fa[l$!&2=)F07$Ff[l$!&'H ))F07$F[\\l$!&QJ*F07$F`\\l$!'AK5F07$Fj\\l$!'`'3\"F07$Fd]l$!'t07F0-F`em 6&F\\^l$\"*++++\"FdemF(F(-Fb^l6#%6an~\"efficient\"~schemeG-F$6%7enF'7$ F+$\"\"#F07$F2$F^^lF07$F7$\"#7F07$F<$\"#8F07$FA$\"#>F07$FF$\"#AF07$FK$ \"#JF07$FP$\"#LF07$FU$\"#VF07$FZ$\"#]F07$Fin$\"#dF07$F^o$\"#kF07$Fco$ \"#tF07$Fho$\"#$)F07$F]p$\"#\"*F07$Fbp$\"$,\"F07$Fgp$\"$6\"F07$F\\qF47 $Faq$\"$J\"F07$Ffq$\"$T\"F07$F[r$\"$^\"F07$F`r$\"$f\"F07$Fer$\"$l\"F07 $Fjr$\"$o\"F07$F_s$\"$p\"F07$Fds$\"$i\"F07$Fis$\"$\\\"F07$F^t$\"$C\"F0 7$Fct$\"#!)F07$Fht$\"#DF07$F]u$!#rF07$Fbu$!$3#F07$Fgu$!$q$F07$F\\v$!$- 'F07$Fav$!$F*F07$Ffv$!%S8F07$F[w$!%S>F07$F`w$!%oEF07$Few$!%NPF07$Fjw$! %R_F07$F_x$!%()oF07$Fdx$!%3&*F07$Fix$!&#Q7F07$F^y$!&Pp\"F07$Fcy$!&>)=F 07$Fhy$!&=>#F07$F]z$!&!eDF07$Fbz$!&q!HF07$Fgz$!&x8$F07$F\\[l$!&4I$F07$ Fa[l$!&yc$F07$Ff[l$!&g%QF07$F[\\l$!&00%F07$F`\\l$!&$zWF07$Fj\\l$!ZF 07$Fd]l$!&Z@&F0-Fj]l6&F\\^lF($F_\\nFjil$\"#&*Fjil-Fb^l6#%Ea~scheme~of~ Tsitouras~and~PapakostasG-F$6%7enF'7$F+$\"\"\"F07$F2F`an7$F7$F`^lF07$F [F07$F_x$!%!H'F07$Fdx$!%E')F07$Fix$!&#=6F07$F^y$!&I_\"F07$Fcy$! &**o\"F07$Fhy$!&]'>F07$F]z$!&**G#F07$Fbz$!&!*f#F07$Fgz$!&R!GF07$F\\[l$ !&'[HF07$Fa[l$!&Y=$F07$Ff[l$!&%F07$Fd]l$!&Sk%F0-F`em6&F\\^lFg`nF(Fg`n-Fb^l6#%Ra~modification ~of~the~Tsitouras-Papakostas~schemeG-F$6%7enF'7$F+Fhan7$F2$\"#GF07$F7$ \"#_F07$F<$\"#rF07$FAF[`o7$FF$\"$:\"F07$FK$\"$X\"F07$FP$\"$q\"F07$FU$ \"$/#F07$FZ$\"$T#F07$Fin$\"$v#F07$F^o$\"$<$F07$Fco$\"$e$F07$Fho$\"$5%F 07$F]p$\"$c%F07$Fbp$\"$;&F07$Fgp$\"$q&F07$F\\q$\"$Q'F07$Faq$\"$(pF07$F fq$\"$s(F07$F[r$\"$R)F07$F`r$\"$>*F07$Fer$\"$')*F07$Fjr$\"%b5F07$F_s$ \"%F6F07$Fds$\"%z6F07$Fis$\"%I7F07$F^t$\"%^7F07$Fct$\"%S7F07$Fht$\"%\" >\"F07$F]u$\"%m5F07$Fbu$\"$S)F07$Fgu$\"$O&F07$F\\v$\"#nF07$Fav$!$J'F07 $Ffv$!%l:F07$F[w$!%gHF07$F`w$!%8ZF07$Few$!%AtF07$Fjw$!&t5\"F07$F_x$!&X _\"F07$Fdx$!&T>#F07$Fix$!&[$HF07$F^y$!&t6%F07$Fcy$!&+h%F07$Fhy$!&vT&F0 7$F]z$!&gP'F07$Fbz$!&$)H(F07$Fgz$!&y*yF07$F\\[l$!&\"G$)F07$Fa[l$!&r.*F 07$Ff[l$!&?w*F07$F[\\l$!'RI5F07$F`\\l$!';V6F07$Fj\\l$!''R?\"F07$Fd]l$! 'OP8F0-Fj]l6&F\\^lFg[o$\"#bFjilF(-Fb^l6#%AVerner's~\"most~efficient\"~ schemeG-%%FONTG6$%*HELVETICAGF`^l-%&TITLEG6#%Verror~curves~for~12~stag e~order~8~Runge-Kutta~methodsG-%+AXESLABELSG6$Q\"x6\"Q!Ffap-%%VIEWG6$; F(Fd]l;$!#V!#E$FhbnF_bp" 1 2 0 1 10 0 2 9 1 4 2 1.000000 45.000000 45.000000 0 1 "Enright-Verner scheme" "Verner's Maple scheme" "a modif ication of Verner's Maple scheme" "an \"efficient\" scheme" "a scheme \+ of Tsitouras and Papakostas" "a modification of the Tsitouras-Papakost as scheme" "Verner's \"most efficient\" scheme" }}}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 622 "evalf[25](p lot(['qn_RK8_1'(x)-q(x),'qn_RK8_2'(x)-q(x),'qn_RK8_3'(x)-q(x),'qn_RK8_ 4'(x)-q(x),\n'qn_RK8_5'(x)-q(x),'qn_RK8_6'(x)-q(x),'qn_RK8_7'(x)-q(x)] ,x=0.5..0.76,-11.9e-22..7.9e-22,font=[HELVETICA,9],\ncolor=[COLOR(RGB, .4,0,.9),COLOR(RGB,0,.65,0),brown,red,COLOR(RGB,0,.75,.95),magenta,COL OR(RGB,.95,.55,0)],\nlegend=[`Enright-Verner scheme`,`Verner's Maple s cheme`,`a modification of Verner's Maple scheme`,`an \"efficient\" sch eme`,`a scheme of Tsitouras and Papakostas`,`a modification of the Tsi touras-Papakostas scheme`,`Verner's \"most efficient\" scheme`],title= `error curves for 12 stage order 8 Runge-Kutta methods`));" }}{PARA 13 "" 1 "" {GLPLOT2D 854 449 449 {PLOTDATA 2 "6--%'CURVESG6%7^o7$$\"\" &!\"\"$!\"%!#D7$$\":LLLLLLL3gsm0&F-$!\"'F-7$$\":nmmmmm\"zFJ)f5&F-$!\"& F-7$$\":LLLLLL$eszVh^F-F17$$\":LLLLLL$33/E<_F-$!\")F-7$$\":nmmmmm\"HHv \"GF&F-$!\"*F-7$$\":LLLLLLeC4EVK&F-$!#5F-7$$\":++++++DhAgwP&F-FC7$$\": LLLLLLeMY=GV&F-$!#8F-7$$\":++++++Dw!)*z([&F-$!#:F-7$$\":nmmmmmm\"y[NWb F-$!#>F-7$$\":LLLLLL$3:'oTf&F-$!#@F-7$$\":+++++++X3Z-l&F-Fin7$$\":++++ +++v\"eb1dF-$F-F-7$$\":+++++++br>3w&F-$!#GF-7$$\":LLLLLLeM`'45eF-$!#MF -7$$\":nmmmmmmhV\"poeF-$!#QF-7$$\":nmmmmmmY+H$=fF-$!#WF-7$$\":++++++Dr Hpg(fF-$!#^F-7$$\":nmmmmmm'R#zr-'F-$!#eF-7$$\":++++++Dr%\\D$3'F-$!#mF- 7$$\":++++++v8G_m8'F-$!#wF-7$$\":nmmmmm;zzmB>'F-$!#()F-7$$\":nmmmmm\"H ;,`ViF-$!#$*F-7$$\":LLLLLL$3nmr)H'F-$!$2\"F-7$$\":LLLLLLeM)*RgN'F-$!$B \"F-7$$\":++++++v))4SfS'F-$!$R\"F-7$$\":LLLLLL$e!fL)fkF-$!$\\\"F-7$$\" :+++++++5%3^:lF-$!$n\"F-7$$\":++++++]#o,)*plF-$!$w\"F-7$$\":++++++DJ/# oAmF-$!%I=!#E7$$\":++++++](*[)>\"o'F-$!%5>Fgt7$$\":nmmmmmm'>\"yPt'F-$! %#)=Fgt7$$\":+++++++v4=**y'F-$!%`A(F-$\"%O[,OuF-$\"%$>(F-7$$\":lmmmm;H#emZjuF-$\"%M!*F-7 $$\":++++++Dr\\Q4\\(F-$\"&&45F-7$$\":+++++]i!R23/vF-$\"&t7\"F-7$$\":++ +++++5)HAF--%&COLORG6&%$RGBG$\"\"%F*$\"\"!F_`l $\"\"*F*-%'LEGENDG6#%6Enright-Verner~schemeG-F$6%7ao7$F($F]`lF-7$F/Fj` l7$F4$F)F-7$F9$\"\"'F-7$FF-7$F\\o$\"#DF-7$F_o$\"#HF-7$Fco $\"#MF-7$Fho$\"#NF-7$F]p$\"#UF-7$Fbp$\"#[F-7$Fgp$\"#bF-7$F\\q$\"#kF-7$ Faq$\"#uF-7$Ffq$\"#&)F-7$F[r$\"#%*F-7$F`r$\"$2\"F-7$Fer$\"$?\"F-7$Fjr$ \"$M\"F-7$F_s$\"$\\\"F-7$Fds$\"$s\"F-7$Fis$\"$'=F-7$F^t$\"$-#F-7$Fct$ \"%v@Fgt7$Fit$\"%'G#Fgt7$F^u$\"%gBFgt7$Fcu$\"%:BFgt7$Fhu$\"%E@Fgt7$F]v $\"%)p\"Fgt7$Fbv$\"%-6Fgt7$Fgv$\"#hFgt7$F\\w$!$l\"F-7$Faw$!$H%F-7$Ffw$ !$A)F-7$F[x$!%'R\"F-7$F`x$!%5AF-7$Fex$!%dIF-7$F_y$!%JYF-7$Fdy$!%JcF-7$ Fiy$!%rjF-7$F^z$!%$3)F-7$Fcz$!%#3*F-7$Fhz$!&*=5F-7$F][l$!&e2\"F-7$Fb[l $!&@9\"F-7$Fg[l$!&cF\"F-7$F\\\\l$!&bF\"F-7$Fa\\l$!&(z7F-7$Ff\\l$!&hJ\" F-7$F[]l$!&ZU\"F-7$F`]l$!&SU\"F-7$Fe]l$!&PU\"F-7$Fj]lFa[m7$F_^l$!&]U\" F-7$Fd^l$!&-V\"F-7$Fi^l$!&[W\"F-7$$\":++++D\"G)e^%GvvF-$!&fc\"F-7$$\": ++++]7G$*H:)yvF-$!&te\"F-7$$\":++++vVtF3YBe(F-$!&pe\"F-7$F^_lFj\\m7$$ \":++++]P4JVQHf(F-$!&%o;F-7$Fc_l$!&dw\"F--Fi_l6&F[`lF^`l$\"#lFe_lF^`l- Fc`l6#%6Verner's~Maple~schemeG-F$6%7ao7$F($\"\"$F-7$F/$\"\"\"F-7$F4F`^ m7$F9F`^m7$F'F-7$F`x$!%45F-7$Fex$F[vF-7$F_y$!%$>#F-7$Fd y$!%$o#F-7$Fiy$!%ZIF-7$F^z$!%$*QF-7$Fcz$!%*Q%F-7$Fhz$!%S\\F-7$F][l$!%G _F-7$Fb[l$!%bbF-7$Fg[l$!%AiF-7$F\\\\lF_fm7$Fa\\l$!%XiF-7$Ff\\l$!%KkF-7 $F[]l$!%spF-7$F`]l$!%opF-7$Fe]l$!%mpF-7$Fj]lF_gm7$F_^l$!%tpF-7$Fd^l$!% ,qF-7$Fi^l$!%wqF-7$F^\\m$!%&o(F-7$Fc\\m$!%!z(F-7$Fh\\m$!%)y(F-7$F^_lFb hm7$F^]m$!%0#)F-7$Fc_l$!%#p)F--%'COLOURG6&F[`l$\")#)eqkF?$\"))eqk\"F?F `im-Fc`l6#%Ha~modification~of~Verner's~Maple~schemeG-F$6%7^o7$F($F*F-7 $F/$!\"$F-7$F4$Fe_lF-7$F9F^jm7$FFgt7$Fbv$\"$S &Fgt7$Fgv$\"%,6Fgt7$F\\w$\"$&>F-7$FawF^w7$Ffw$\"$2&F-7$F[x$\"$j(F-7$F` x$\"%D6F-7$Fex$\"%#[\"F-7$Fjx$\"%==F-7$F_y$\"%R@F-7$Fdy$\"%cDF-7$Fiy$ \"%hGF-7$F^z$\"%`NF-7$Fcz$\"%\\RF-7$Fhz$\"%(Q%F-7$F][l$\"%(f%F-7$Fb[l$ \"%l[F-7$Fg[l$\"%&Q&F-7$F\\\\l$\"%$Q&F-7$Fa\\l$\"%%R&F-7$Ff\\l$\"%CbF- 7$F[]l$\"%`fF-7$F`]lFjan7$Fe]l$\"%]fF-7$Fj]l$\"%_fF-7$F_^l$\"%bfF-7$Fd ^l$\"%pfF-7$Fi^l$\"%>gF-7$F^_l$\"%nlF-7$Fc_l$\"%NsF--F\\im6&F[`l$\"*++ ++\"F?F^`lF^`l-Fc`l6#%6an~\"efficient\"~schemeG-F$6%7^o7$F(F^`l7$F/F^j m7$F4Fiim7$F9Fiim7$F7$F[rFH7$F`rFC7$FerF^[n7$Fjr$!#9F-7$F_sFd[n7$FdsFP7$ Fis$!#Fgt7$F]v$\"$&QFgt7$Fbv$\"$,'Fgt7$Fgv$\"$P*Fgt7$F\\w$\"$U\"F-7$Faw$\" $<#F-7$Ffw$\"$>$F-7$F[x$\"$g%F-7$F`x$\"$d'F-7$Fex$\"$Y)F-7$Fjx$\"%E5F- 7$F_y$\"%#>\"F-7$Fdy$\"%69F-7$Fiy$\"%r:F-7$F^z$\"%G>F-7$Fcz$\"%H@F-7$F hz$\"%_BF-7$F][l$\"%aCF-7$Fb[l$\"%#f#F-7$Fg[l$\"%aGF-7$F\\\\l$\"%`GF-7 $Fa\\l$\"%cGF-7$Ff\\l$\"%=HF-7$F[]l$\"%NJF-7$F`]l$\"%OJF-7$Fe]l$\"%MJF -7$Fj]lFfco7$F_^l$\"%PJF-7$Fd^l$\"%UJF-7$Fi^l$\"%lJF-7$F^_l$\"%RMF-7$F c_l$\"%lPF--F\\im6&F[`lFdcnF^`lFdcn-Fc`l6#%Ra~modification~of~the~Tsit ouras-Papakostas~schemeG-F$6%7^oFhimFjimF]jmF_jmF`jm7$FAF+FbjmFcjm7$FN F67$FSF17$FXF>7$FgnF>7$F\\oFgjm7$F_oF>7$FcoF>7$FhoF]^o7$F]pF]^o7$FbpFf en7$FgpF[fn7$F\\qFZ7$FaqFin7$Ffq$!#CF-7$F[r$!#HF-7$F`rFefo7$FerFjo7$Fj rFa\\n7$F_s$!#XF-7$FdsF[go7$Fis$!#_F-7$F^t$!#`F-7$Fct$!$>&Fgt7$Fit$!$G &Fgt7$F^u$!$t%Fgt7$Fcu$!$\"QFgt7$Fhu$!$$>Fgt7$F]v$\"$;\"Fgt7$Fbv$\"$\" [Fgt7$Fgv$\"%q5Fgt7$F\\w$\"$'>F-7$Faw$\"$M$F-7$Ffw$\"$H&F-7$F[x$\"$-)F -7$F`x$\"%)=\"F-7$FexFeao7$Fjx$\"%J>F-7$F_y$\"%wAF-7$Fdy$\"%BFF-7$Fiy$ \"%^IF-7$F^z$\"%(z$F-7$Fcz$\"%BUF-7$Fhz$\"%&p%F-7$F][l$\"%B\\F-7$Fb[l$ \"%6_F-7$Fg[l$\"%tdF-7$F\\\\l$\"%rdF-7$Fa\\l$\"%$y&F-7$Ff\\l$\"%CfF-7$ F[]l$\"%(Q'F-7$F`]l$\"%'Q'F-7$Fe]l$\"%%Q'F-7$Fj]l$\"%&Q'F-7$F_^lFfy7$F d^l$\"%0kF-7$Fi^l$\"%fkF-7$F^_l$\"%_qF-7$Fc_l$\"%vxF--Fi_l6&F[`lF\\]o$ FjclFe_lF^`l-Fc`l6#%AVerner's~\"most~efficient\"~schemeG-%%FONTG6$%*HE LVETICAGFa`l-%&TITLEG6#%Verror~curves~for~12~stage~order~8~Runge-Kutta ~methodsG-%+AXESLABELSG6$Q\"x6\"Q!F^_p-%%VIEWG6$;F(Fc_l;$!$>\"!#B$\"#z Fg_p" 1 2 0 1 10 0 2 9 1 4 2 1.000000 45.000000 45.000000 0 1 "Enright -Verner scheme" "Verner's Maple scheme" "a modification of Verner's Ma ple scheme" "an \"efficient\" scheme" "a scheme of Tsitouras and Papak ostas" "a modification of the Tsitouras-Papakostas scheme" "Verner's \+ \"most efficient\" scheme" }}}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 624 "evalf[25](plot(['qn_RK8_1'(x)-q(x) ,'qn_RK8_2'(x)-q(x),'qn_RK8_3'(x)-q(x),'qn_RK8_4'(x)-q(x),\n'qn_RK8_5' (x)-q(x),'qn_RK8_6'(x)-q(x),'qn_RK8_7'(x)-q(x)],x=0.76..0.86,-1.05e-18 ..1.05e-18,font=[HELVETICA,9],\ncolor=[COLOR(RGB,.4,0,.9),COLOR(RGB,0, .65,0),brown,red,COLOR(RGB,0,.75,.95),magenta,COLOR(RGB,.95,.55,0)],\n legend=[`Enright-Verner scheme`,`Verner's Maple scheme`,`a modificatio n of Verner's Maple scheme`,`an \"efficient\" scheme`,`a scheme of Tsi touras and Papakostas`,`a modification of the Tsitouras-Papakostas sch eme`,`Verner's \"most efficient\" scheme`],title=`error curves for 12 \+ stage order 8 Runge-Kutta methods`));" }}{PARA 13 "" 1 "" {GLPLOT2D 981 600 600 {PLOTDATA 2 "6--%'CURVESG6%7in7$$\"#w!\"#$\"&u!>!#D7$$\":n mmmmmmT:(z@wF-$\"&y5#F-7$$\":LLLLLLe9ui2k(F-$\"&cK#F-7$$\":nmmmmm;z_\" 4iwF-$\"&Ac#F-7$$\":nmmmmm;aphNo(F-$\"&[#GF-7$$\":LLLLLLe*=)H\\q(F-$\" &)RJF-7$$\":nmmmmm\"z/3uCxF-$\"&3Z$F-7$$\":++++++DJ$RDXxF-$\"&T\"RF-7$ $\":nmmmmm\"zR'okw(F-$\"&_O%F-7$$\":++++++D1J:wy(F-$\"&Ks%F-7$$\":LLLL LLL3En$4yF-$\"&B4&F-7$$\":nmmmmm;/RE&GyF-$\"&%paF-7$$\":+++++++D.&4]yF -$\"&r%eF-7$$\":+++++++vB_<(yF-$\"&LA'F-7$$\":+++++++v'Hi#*yF-$\"&Zf'F -7$$\":nmmmmm\"z*ev:\"zF-$\"&&>pF-7$$\":LLLLLLL347T$zF-$\"&LI(F-7$$\": LLLLLLLLY.K&zF-$\"&9[(F-7$$\":++++++D\"o7TvzF-$\"&Ea(F-7$$\":LLLLLLL$Q *o]*zF-$\"&=R(F-7$$\":++++++D\"=lj;!)F-$\"&j.(F-7$$\":++++++vV&RY29)F-$!&$)>%F-7$$\":nmmmmm;zXu9;)F-$!'JJ5F-7$$\":+++++++]y)) G=)F-$!'\\I>F-7$$\":++++++]i_QQ?)F-$!'\\TFF-7$$\":++++++D\"y%3TA)F-$!' %zt$F-7$$\":++++++]P![hY#)F-$!'$>&\\F-7$$\":LLLLLLL$Qx$oE)F-$!'ABkF-7$ $\":+++++++v.I%)G)F-$!'A<#)F-7$$\":nmmmmm\"zpe*zI)F-$!(x_.\"F-7$$\":++ +++++D\\'QH$)F-$!(@hJ\"F-7$$\":LLLLLLe9S8&\\$)F-$!)llg;!#E7$$\":++++++ D1#=bq$)F-$!)SWIAF\\w7$$\":LLLLLLL3s?6R)F-$!)@')oFF\\w7$$\":++++++DJXa ET)F-$!*$4<%G$!#F7$$\":mmmmmmmm*RRL%)F-$!)WUhQF\\w7$$\":mmmmmm;a<.YX)F -$!)$fz\\%F\\w7$$\":LLLLLLe9tOcZ)F-$!((=)=&F-7$$\":+++++++]Qk\\\\)F-$! (u\")F-7$$\":+++++++]oi\"o&)F-$!(?w])F-7$$ \":++++++v=5s#y&)F-$!(Sdl)F-7$$\":+++]iS\"p)4^*y&)F-$!(&*Rq)F-7$$\":++ ++D\"y]&4I'z&)F-$!((4P()F-7$$\":+++](=UK#44.e)F-$!(/lt)F-7$$\":++++]iS \"*3))4e)F-$!(7ft)F-7$$\":++++vVtF3YBe)F-$!(BZt)F-7$$\":+++++D1k2/Pe)F -$!(RNt)F-7$$\":++++](=nj+U'e)F-$!(+7t)F-7$$\":+++++]P40O\"*e)F-$!(A#H ()F-7$$\":++++]7.#Q?&=f)F-$!(&4G()F-7$$\":+++++voa-oXf)F-$!(&4?()F-7$$ \":++++]PMF,%G(f)F-$!(M-k)F-7$$\"#')F*$!('pX&)F--%&COLORG6&%$RGBG$\"\" %!\"\"$\"\"!F^_l$\"\"*F\\_l-%'LEGENDG6#%6Enright-Verner~schemeG-F$6%7g n7$F($!&dw\"F-7$F/$!&7'>F-7$F4$!&`<#F-7$F9$!&&4CF-7$F>$!&Fn#F-7$FC$!&F *HF-7$FH$!&JL$F-7$FM$!&^z$F-7$FR$!&XF%F-7$FW$!&pm%F-7$Ffn$!&;3&F-7$F[o $!&#>bF-7$F`o$!&[(fF-7$Feo$!&BX'F-7$Fjo$!&j&pF-7$F_p$!&*RuF-7$Fdp$!&$G \")F-7$Fip$!&te)F-7$F^q$!&1:*F-7$Fcq$!&EW*F-7$Fhq$!&Ji*F-7$F]r$!&rl*F- 7$Fbr$!&\\\\*F-7$Fgr$!&@3*F-7$F\\s$!&7K)F-7$Fas$!&v*oF-7$Ffs$!&a2&F-7$ F[t$!&\"=;F-7$F`t$\"&Jn$F-7$Fet$\"&\"y()F-7$Fjt$\"'VD:F-7$F_u$\"']PBF- 7$Fdu$\"'RZLF-7$Fiu$\"'p8YF-7$F^v$\"'3ahF-7$Fcv$\"'N\\#)F-7$Fhv$\")!)f (3\"F\\w7$F^w$\")m#4`\"F\\w7$Fcw$\")ybj>F\\w7$Fhw$\"*Q#e&R#F\\x7$F^x$ \")q$p*GF\\w7$Fcx$\")5OtMF\\w7$Fhx$\"(v58%F-7$F]y$\"(H.([F-7$Fby$\"(jm s&F-7$$\":lmmmm;zpBp?_)F-$\"(Q`)F-$\"(]'ylF-7$$\": qmmm;zW<*Q@L&)F-$\"(e9e'F-7$$\":+++++v=n=_W`)F-$\"('y*e'F-7$$\":NLLL$3 Fp\"[!pN&)F-$\"(X*4mF-7$Fgy$\"(.Pl'F-7$$\":NLLLL$eRA5\\Z&)F-$\"(#4GvF- 7$F\\z$\"(DY\"yF-7$Faz$\"(*41&)F-7$Ffz$\"(Jy1*F-7$F^]l$\"(%3s%*F-7$Fb^ l$\")6rO5F--Fg^l6&Fi^lF]_l$\"#lF*F]_l-Fb_l6#%6Verner's~Maple~schemeG-F $6%7co7$F($!%#p)F-7$F/$!%%o*F-7$F4$!&v2\"F-7$F9$!&t>\"F-7$F>$!&GL\"F-7 $FC$!&()\\\"F-7$FH$!&hn\"F-7$FM$!&'=>F-7$FR$!&E<#F-7$FW$!&MQ#F-7$Ffn$! &(3EF-7$F[o$!&'\\GF-7$F`o$!&W5$F-7$Feo$!&kP$F-7$Fjo$!&/n$F-7$F_p$!&2'R F-7$Fdp$!&XR%F-7$Fip$!&Tq%F-7$F^q$!&_7&F-7$Fcq$!&7R&F-7$Fhq$!&Ui&F-7$F ]r$!&7\"eF-7$Fbr$!&<$fF-7$Fgr$!&V'fF-7$F\\s$!&#peF-7$Fas$!&p`&F-7$Ffs$ !&\\.&F-7$F[t$!&A&RF-7$F`t$!&#)>#F-7$Fet$!%\"f$F-7$Fjt$\"&%e?F-7$F_u$ \"&A=&F-7$Fdu$\"&q;*F-7$Fiu$\"'_H9F-7$F^v$\"']l?F-7$Fcv$\"'XbHF-7$Fhv$ \"(E14%F\\w7$F^w$\"(:r.'F\\w7$Fcw$\"(B@)zF\\w7$Fhw$\")Z'=)**F\\x7$F^x$ \")&*eN7F\\w7$Fcx$\")(=b^\"F\\w7$Fhx$\"(!>W=F-7$F]y$\"(%zDAF-7$Fby$\"( &>'o#F-7$F`hl$\"(m[1$F-7$Fehl$\"(I_;$F-7$Fjhl$\"(4X;$F-7$F_il$\"('okJF -7$Fdil$\"(%RmJF-7$Fiil$\"(@8<$F-7$F^jl$\"(vJ=$F-7$Fgy$\"(8*3KF-7$Ffjl $\"(-&HPF-7$F\\z$\"(M1\"RF-7$Faz$\"(qnN%F-7$Ffz$\"(22v%F-7$F[[l$\"(4!> \\F-7$F`[l$\"(\"4Y]F-7$Fe[l$\"(\\d/&F-7$Fj[l$\"(0a/&F-7$F_\\l$\"(BZ/&F -7$Fd\\l$\"(OS/&F-7$Fi\\l$\"(KF/&F-7$F^]l$\"([E/&F-7$$\":++++DJqX/%\\! f)F-$\"(=_/&F-7$Fc]l$\"(UO0&F-7$$\":++++v$f$=.5Kf)F-$\"(&*e2&F-7$Fh]l$ \"(y#G^F-7$F]^l$\"(?TZ&F-7$Fb^l$\"(Hyx&F--%'COLOURG6&Fi^l$\")#)eqk!\") $\"))eqk\"FajmFbjm-Fb_l6#%Ha~modification~of~Verner's~Maple~schemeG-F$ 6%7jo7$F($\"%NsF-7$F/$\"%_zF-7$F4$\"%E()F-7$F9$\"%f&*F-7$F>$\"&s/\"F-7 $FC$\"&^:\"F-7$FH$\"&pE\"F-7$FM$\"&TT\"F-7$FR$\"&/c\"F-7$FW$\"&?n\"F-7 $Ffn$\"&Jy\"F-7$F[o$\"&:*=F-7$F`o$\"&L*>F-7$Feo$\"&f3#F-7$Fjo$\"&[;#F- 7$F_p$\"&w@#F-7$Fdp$\"&cB#F-7$Fip$\"&7>#F-7$F^q$\"&C-#F-7$Fcq$\"&A!=F- 7$Fhq$\"&aZ\"F-7$F]r$\"&k,\"F-7$Fbr$\"%KRF-7$Fgr$!%4VF-7$F\\s$!&!>:F-7 $Fas$!&f7$F-7$Ffs$!&%p\\F-7$F[t$!&a5)F-7$F`t$!'_j7F-7$Fet$!'!)f;F-7$Fj t$!'1R@F-7$F_u$!'O9FF-7$Fdu$!'.-MF-7$Fiu$!'pFUF-7$F^v$!'K)>&F-7$Fcv$!' c\\kF-7$Fhv$!(tQ'zF\\w7$F^w$!)E]V5F\\w7$Fcw$!)8.s7F\\w7$Fhw$!*i)H%[\"F \\x7$F^x$!)8#er\"F\\w7$Fcx$!))\\E'>F\\w7$Fhx$!(&e=AF-7$F]y$!(\\CZ#F-7$ Fby$!(\"f:FF-7$Fgy$!(,9\"HF-7$$\":NLLL$e*)4Q$p&R&)F-$!(x(\\HF-7$$\":++ +++DJ&**)4Aa)F-$!(X7-$F-7$$\":qmmm;aj4Y][a)F-$!(\\/-$F-7$Ffjl$!(b'>IF- 7$$\":++++]7GQeJ,b)F-$!(]*=IF-7$$\":qmmmmTg_9sFb)F-$!(!*)=IF-7$$\":NLL L$3Fp1FTb&)F-$!('4@IF-7$F\\z$!(%*e-$F-7$$\":+++++vVBi!eg&)F-$!(N#=IF-7 $$\":+++++]il(z5j&)F-$!(tu,$F-7$$\":+++++D\"yI`jl&)F-$!(4n,$F-7$Faz$!( ef,$F-7$$\":+++++v=#R+pq&)F-$!(i^,$F-7$$\":+++++]PMR$\"%,dF-7$FC$\"%KiF-7$FH$\"%rnF-7$FM$\"%luF- 7$FR$\"%G\")F-7$FW$\"%+')F-7$Ffn$\"%R!*F-7$F[o$\"%E%*F-7$F`o$\"%M(*F-7 $Feo$\"%M**F-7$Fjo$\"%\")**F-7$F_p$\"%G)*F-7$Fdp$\"%:\"*F-7$Fip$\"%\\ \")F-7$F^q$\"%ufF-7$Fcq$\"%&p$F-7$Fhq$\"$M'F-7$F]r$!%!R$F-7$Fbr$!%*e)F -7$Fgr$!&:_\"F-7$F\\s$!&*oBF-7$Fas$!&De$F-7$Ffs$!&4&\\F-7$F[t$!&OB(F-7 $F`t$!'N\\5F-7$Fet$!'\\G8F-7$Fjt$!'Fi;F-7$F_u$!'ae?F-7$Fdu$!'CFDF-7$Fi u$!'<$3$F-7$F^v$!';IPF-7$Fcv$!'G]XF-7$Fhv$!(&HJbF\\w7$F^w$!(aN6(F\\w7$ Fcw$!(J![&)F\\w7$Fhw$!)m/U)*F\\x7$F^x$!)5^@6F\\w7$Fcx$!)5ci7F\\w7$Fhx$ !(y6S\"F-7$F]y$!(=w_\"F-7$Fby$!(Y)H;F-7$$\":+++++DJqk#f>&)F-$!([%R;F-7 $F`hl$!(`Dn\"F-7$$\":ILLLe9mWQ)oA&)F-$!(dQo\"F-7$$\":++++]7`>`2L_)F-$! (`Po\"F-7$$\":lmmmT5S%zm#R_)F-$!(\\Oo\"F-7$$\":ILLLL3Fp#eaC&)F-$!(UNo \"F-7$$\":lmmm;/,>7%yD&)F-$!(MLo\"F-7$Fehl$!(DJo\"F-7$F_il$!(MBo\"F-7$ Fgy$!(e>o\"F-7$Ffjl$!(3Zm\"F-7$F\\z$!(nOj\"F-7$Faz$!(=q`\"F-7$Ffz$!('e #Q\"F-7$F[[l$!(l&48F-7$F`[l$!(jGD\"F-7$Fe[l$!(xFD\"F-7$Fj[l$!(%p_7F-7$ F_\\l$!(@DD\"F-7$Fd\\l$!(`BD\"F-7$Fi\\l$!(!*>D\"F-7$F^]l$!(,6D\"F-7$Fg hm$!(R$\\7F-7$Fc]l$!(\"HW7F-7$F_im$!(q3B\"F-7$Fh]l$!(wz>\"F-7$$\":++++ Dc,\">g#ff)F-$!(CJ7\"F-7$F]^l$!'1M'*F-7$$\":+++]7y]&4I'zf)F-$!',$H)F-7 $$\":++++v=nj+U')f)F-$!'.huF-7$$\":+++]Pf$=.5K*f)F-$!'_guF-7$Fb^l$!'** fuF--Fg^l6&Fi^lF]_l$\"#vF*$\"#&*F*-Fb_l6#%Ea~scheme~of~Tsitouras~and~P apakostasG-F$6%7ep7$F($\"%lPF-7$F/$\"%5TF-7$F4$\"%zWF-7$F9$\"%r[F-7$F> $\"%\"H&F-7$FC$\"%vdF-7$FH$\"%liF-7$FM$\"%\"*oF-7$FR$\"%&[(F-7$FW$\"%. zF-7$Ffn$\"%&G)F-7$F[o$\"%7')F-7$F`o$\"%f))F-7$Feo$\"%(**)F-7$Fjo$\"%# )*)F-7$F_p$\"%r()F-7$Fdp$\"%$)zF-7$Fip$\"%upF-7$F^q$\"%fZF-7$Fcq$\"%wC F-7$Fhq$!$m&F-7$F]r$!%QXF-7$Fbr$!%X'*F-7$Fgr$!&Gh\"F-7$F\\s$!&!RCF-7$F as$!&\"=OF-7$Ffs$!&^%\\F-7$F[t$!&L:(F-7$F`t$!'DI5F-7$Fet$!'@*H\"F-7$Fj t$!'M?;F-7$F_u$!'1,?F-7$Fdu$!'v]CF-7$Fiu$!'H$)HF-7$F^v$!'A-OF-7$Fcv$!' *\\Q%F-7$Fhv$!(H)>`F\\w7$F^w$!(j^#oF\\w7$Fcw$!(hg=)F\\w7$Fhw$!)6Z3%*F \\x7$F^x$!)y,q5F\\w7$Fcx$!)5%>?\"F\\w7$Fhx$!(X/L\"F-7$F]y$!(:fW\"F-7$F by$!(Jg`\"F-7$$\":lmmm;Hd?Na$=&)F-$!(L&Q:F-7$Fbdo$!($fV:F-7$$\":ILLL$3 _+U4$3_)F-$!(`Ib\"F-7$F`hl$!(+(p:F-7$Fjdo$!(z&y:F-7$F_eo$!(#[y:F-7$Fde o$!(&Qy:F-7$Fieo$!(%Gy:F-7$F^fo$!(*3y:F-7$Fehl$!($*yd\"F-7$F_il$!(Krd \"F-7$Fgy$!(?ed\"F-7$Ffjl$!(Sya\"F-7$F\\z$!(DO^\"F-7$$\":++++]PM_%pJf& )F-$!(fXZ\"F-7$Fien$!(12T\"F-7$$\":++++]7`%*HW=c)F-$!(G0T\"F-7$F^fn$!( ].T\"F-7$Fcfn$!(#**49F-7$Faz$!(3'49F-7$F[gn$!(o(39F-7$F`gn$!(BRS\"F-7$ Fjgn$!(xhP\"F-7$Ffz$!(9-D\"F-7$F[[l$!(G_<\"F-7$F`[l$!(yq6\"F-7$Fe[l$!( -q6\"F-7$Fj[l$!(Fp6\"F-7$F_\\l$!(sn6\"F-7$Fd\\l$!(Bm6\"F-7$Fi\\l$!('H; 6F-7$F^]l$!(Ha6\"F-7$Fghm$!(_O6\"F-7$Fc]l$!(P&36F-7$F_im$!(T\\4\"F-7$F h]l$!(k;1\"F-7$$\":+++]i:&GAqC&f)F-$!(H7.\"F-7$F\\jo$!'ag)*F-7$$\":+++ ](oz\"f,0mf)F-$!'P+#*F-7$F]^l$!'X\\#)F-7$Fdjo$!'%z*oF-7$Fijo$!'*)fgF-7 $F^[p$!'ZfgF-7$Fb^l$!'/fgF--F]jm6&Fi^lFb[oF]_lFb[o-Fb_l6#%Ra~modificat ion~of~the~Tsitouras-Papakostas~schemeG-F$6%7\\p7$F($\"%vxF-7$F/$\"%`& )F-7$F4$\"%%R*F-7$F9$\"&+.\"F-7$F>$\"&'H6F-7$FC$\"&vC\"F-7$FH$\"&*p8F- 7$FM$\"&<`\"F-7$FR$\"&Kp\"F-7$FW$\"&s\"=F-7$Ffn$\"&:%>F-7$F[o$\"&Q1#F- 7$F`o$\"&.=#F-7$Feo$\"&$)G#F-7$Fjo$\"&OQ#F-7$F_p$\"&AX#F-7$Fdp$\"&J\\# F-7$Fip$\"&TY#F-7$F^q$\"&]J#F-7$Fcq$\"&e5#F-7$Fhq$\"&py\"F-7$F]r$\"&;L \"F-7$Fbr$\"%lqF-7$Fgr$!%o7F-7$F\\s$!&TB\"F-7$Fas$!&$zGF-7$Ffs$!&Jx%F- 7$F[t$!&l+)F-7$F`t$!'to7F-7$Fet$!'xz;F-7$Fjt$!'*y<#F-7$F_u$!'&px#F-7$F du$!'J%\\$F-7$Fiu$!'OdVF-7$F^v$!'kt`F-7$Fcv$!'B(o'F-7$Fhv$!([*z#)F\\w7 $F^w$!)rQ)3\"F\\w7$Fcw$!)<#*H8F\\w7$Fhw$!*/H_b\"F\\x7$F^x$!)\\$>!=F\\w 7$Fcx$!)TGm?F\\w7$Fhx$!(%HUBF-7$F]y$!(E)=EF-7$Fby$!(#Q))GF-7$Fgy$!(%*H 6$F-7$Fecn$!(@.;$F-7$Fjcn$!(e.D$F-7$F_dn$!(,&\\KF-7$Ffjl$!(['[KF-7$Fgd n$!(2zC$F-7$F\\en$!(W![KF-7$Faen$!(;=D$F-7$F\\z$!(\\RE$F-7$Fggp$!()[sK F-7$Fien$!(K%yKF-7$F_hp$!(=!yKF-7$F^fn$!(/wF$F-7$Fcfn$!(vnF$F-7$Faz$!( mfF$F-7$F[gn$!(4_F$F-7$F`gn$!(*ftKF-7$Fegn$!(%fqKF-7$Fjgn$!(/EE$F-7$F_ hn$!(%)HC$F-7$Ffz$!(x&)>$F-7$F[[l$!(bl:$F-7$F`[l$!(OG7$F-7$Fe[l$!(BE7$ F-7$Fj[l$!(8C7$F-7$F_\\l$!(')>7$F-7$Fd\\l$!(k:7$F-7$Fi\\l$!(327$F-7$F^ ]l$!(]&>JF-7$Fghm$!(%3=JF-7$Fc]l$!(1U6$F-7$F_im$!(]Q5$F-7$Fh]l$!(@y2$F -7$F]^l$!(JQ)GF-7$Fb^l$!(e*)p#F--Fg^l6&Fi^lFi[p$\"#bF*F]_l-Fb_l6#%AVer ner's~\"most~efficient\"~schemeG-%%FONTG6$%*HELVETICAGF`_l-%&TITLEG6#% Verror~curves~for~12~stage~order~8~Runge-Kutta~methodsG-%+AXESLABELSG6 $Q\"x6\"Q!Fc^r-%%VIEWG6$;F(Fb^l;$!$0\"!#?$\"$0\"F\\_r" 1 2 0 1 10 0 2 9 1 4 2 1.000000 45.000000 45.000000 0 1 "Enright-Verner scheme" "Vern er's Maple scheme" "a modification of Verner's Maple scheme" "an \"eff icient\" scheme" "a scheme of Tsitouras and Papakostas" "a modificatio n of the Tsitouras-Papakostas scheme" "Verner's \"most efficient\" sch eme" }}}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 617 "evalf[20](plot(['qn_RK8_1'(x)-q(x),'qn_RK8_2'(x)-q(x ),'qn_RK8_3'(x)-q(x),'qn_RK8_4'(x)-q(x),\n'qn_RK8_5'(x)-q(x),'qn_RK8_6 '(x)-q(x),'qn_RK8_7'(x)-q(x)],x=0.86..0.9,-7e-17..5e-17,font=[HELVETIC A,9],\ncolor=[COLOR(RGB,.4,0,.9),COLOR(RGB,0,.65,0),brown,red,COLOR(RG B,0,.75,.95),magenta,COLOR(RGB,.95,.55,0)],\nlegend=[`Enright-Verner s cheme`,`Verner's Maple scheme`,`a modification of Verner's Maple schem e`,`an \"efficient\" scheme`,`a scheme of Tsitouras and Papakostas`,`a modification of the Tsitouras-Papakostas scheme`,`Verner's \"most eff icient\" scheme`],title=`error curves for 12 stage order 8 Runge-Kutta methods`));" }}{PARA 13 "" 1 "" {GLPLOT2D 979 630 630 {PLOTDATA 2 "6- -%'CURVESG6%7iw7$$\"#')!\"#$!#%)!#?7$$\"5nmmmmh)=(3')F-$!#&)F-7$$\"5LL LLe'40jh)F-$!#!)F-7$$\"5nmmm;6m$[i)F-$!#zF-7$$\"5nmmm;yYUL')F-$!#tF-7$ $\"5LLLLeF>(>k)F-$!#hF-7$$\"5nmmm\">K'*)\\')F-$!#gF-7$$\"5++++Dt:5e')F -$!#LF-7$$\"5NLLLe9XMi')F-$!#JF-7$$\"5nmmm\"fX(em')F-$!#KF-7$$\"5+++++ B@qo')F-$!#HF-7$$\"5NLLL3!z;3n)F-$!#DF-7$$\"5qmmm;d9$Hn)F-$!\"(F-7$$\" 5++++DCh/v')F-$\"#9F-7$$\"5lmm;H9lRz')F-Fgo7$$\"5LLLLL/pu$o)F-Fgo7$$\" 5nmmm;c0T\"p)F-$\"#UF-7$$\"5+++++8!Q+q)F-$\"#\")F-7$$\"5+++++a%pVq)F-F gp7$$\"5+++++&*3q3()F-$\"#'*F-7$$\"5+++++ozy5()F-$\"$L\"F-7$$\"5+++++T ](Gr)F-$\"$w\"F-7$$\"5+++++9@'\\r)F-$\"$v\"F-7$$\"5+++++(=\\qr)F-$\"$x \"F-7$$\"5NLL$e9rR3s)F-Fiq7$$\"5nmmm\"fBIYs)F-$\"$z\"F-7$$\"5+++]7Ov8H ()F-$\"$G#F-7$$\"5LLLLLO[kL()F-$\"$2$F-7$$\"5NLLL$36jut)F-Fes7$$\"5LLL LL&Q\"GT()F-Fes7$$\"5lm;HdvMw)F-$\"$$\\F-7$$\"5ILL3F\\y\\k()F-$\"$.&F-7$$\"5lm;/EGidl( )F-$\"$B&F-7$$\"5++++D2Ylm()F-$\"$f&F-7$$\"5+++D1z9on()F-$\"$:'F-7$$\" 5+++](3N3(o()F-$\"$4(F-7$$\"5+++voA_tp()F-$\"$9(F-7$$\"5++++]%4i2x)F-$ \"$7(F-7$$\"5+++]7Qe\"Gx)F-$\"$8(F-7$$\"5++++v\"ep[x)F-Fay7$$\"5ILL3xZ C,x()F-Fay7$$\"5lmm;z8`:z()F-F\\y7$$\"5IL$3-ouE-y)F-$\"$;(F-7$$\"5+++D \")z\")H\"y)F-$\"$=(F-7$$\"5lm;H#GhpBy)F-$\"$C(F-7$$\"5LLLL$e/TMy)F-$ \"$R(F-7$$\"5NLL3xl)3ay)F-$\"$+)F-7$$\"5NLL$3dowty)F-$\"$k*F-7$$\"5N$3 _+KmAwy)F-$\"$)**F-7$$\"5NL3FpS'oyy)F-$\"%15F-7$$\"5N$e*[==Y6)y)F-Fg\\ l7$$\"5NL$3xcfg$)y)F-$\"%05F-7$$\"5NLe9m]D&))y)F-Fg\\l7$$\"5NLLek0XM*y )F-Fg\\l7$$\"5NL$e9cTG.z)F-Fg\\l7$$\"5LLLLeDBJ\"z)F-Fg\\l7$$\"5+++]([W dbz)F-Fg\\l7$$\"5nmmm;kD!)*z)F-$\"%65F-7$$\"5lm;ajN\\!4!))F-$\"%<5F-7$ $\"5lmmT52t+-))F-$\"%K5F-7$$\"5lm;Hdy'4J!))F-$\"%e5F-7$$\"5lmm;/]?@/)) F-$\"%26F-7$$\"5lm;/^@WJ0))F-$\"%$>\"F-7$$\"5lmm\"zHz;k!))F-$\"%P8F-7$ $\"5l;aj%eQ#p1))F-$\"%r8F-7$$\"5lmTNryz'p!))F-Ff`l7$$\"5l;H2erNC2))F-F f`l7$$\"5lm;zWk\">v!))F-Ff`l7$$\"5lm\"H#=].23))F-Ff`l7$$\"5nmmm\"f`@'3 ))F-Ff`l7$$\"5NLLLL1+Y7))F-Ff`l7$$\"5++++vw%)H;))F-$\"%s8F-7$$\"5NL$3_ +*[L<))F-F]bl7$$\"5qmmTN.8P=))F-$\"%v8F-7$$\"5++]il;xS>))F-$\"%\"Q\"F- 7$$\"5NLL$e*HTW?))F-$\"%#R\"F-7$$\"5+++Dccp^A))F-$\"%c9F-7$$\"5nmmm;$y *eC))F-$\"%S;F-7$$\"5+++++9b:L))F-$\"%6=F-7$$\"5+++]7)[]_$))F-$\"%8=F- 7$$\"5++++DiaMP))F-$\"%<=F-7$$\"5+++DJ\\HRQ))F-$\"%D=F-7$$\"5+++]PO/WR ))F-$\"%R=F-7$$\"5+++vVBz[S))F-$\"%n=F-7$$\"5++++]5a`T))F-$\"%<>F-7$$ \"5+++vo0CcV))F-$\"%K@F-7$$\"5+++](3S*eX))F-$\"%FBF-7$$\"5+++D1'R;w%)) F-Fffl7$$\"5++++D\"RV'\\))F-$\"%EBF-7$$\"5+++D\"Q-%*=&))F-$\"%CBF-7$$ \"5+++]PcY9a))F-$\"%DBF-7$$\"5++]ils*p_&))F-Fffl7$$\"5+++v$*)G&Rc))F-$ \"%JBF-7$$\"5++D\"yq%z&p&))F-$\"%PBF-7$$\"5++](=_g?v&))F-$\"%SBF-7$$\" 5++v$fLE$3e))F-$\"%]BF-7$$\"5++++]@fke))F-$\"%iBF-7$$\"5lmmmT30pi))F-$ \"%>FF-7$$\"5LLLLL&4Nn'))F-$\"%(*GF-7$$\"5lmmm;bN0r))F-$\"%'*GF-7$$\"5 +++++:?Pv))F-$\"%0HF-7$$\"5NL$e9mI]j())F-$\"%8HF-7$$\"5qmm\"H#)fGt())F -$\"%IHF-7$$\"5++]P%)*)oIy))F-$\"%fHF-7$$\"5NLL$e9=&Gz))F-$\"%7IF-7$$ \"5+++vok$)))F-$\"%#\\$F-7$$\"5NLL$eC Zwu)))F-$\"%*[$F-7$$\"5+++++(fa<*))F-$\"%\"\\$F-7$$\"5lm;zWU4w#*))F-F` ]m7$$\"5ILLe*yGnP*))F-Ff\\m7$$\"5++]PMLOx%*))F-$\"%*\\$F-7$$\"5lmm;zy* zd*))F-$\"%6NF-7$$\"5+++vopEz(*))F-$\"%vNF-7$$\"5LLLLeg`!)**))F-$\"%YP F-7$$\"5lmm;a\"Gd3!*)F-$\"%HRF-7$$\"5++++]-#4>!*)F-$\"%QSF-7$$\"5ILL$e M7hH!*)F-$\"%MSF-7$$\"5lmmmTWI,/*)F-$\"%PSF-7$$\"5ILLLL')o61*)F-$\"%NS F-7$$\"5++++DG2A3*)F-Fc`m7$$\"5ILL3F=wF5*)F-$\"%KSF-7$$\"5lmm;H3XL7*)F -$\"&_.%!#@7$$\"5IL$3-L&HO8*)F-$\"&%QSFbam7$$\"5+++DJ)R\"R9*)F-$\"&'\\ SFbam7$$\"5lm;HKV)>a\"*)F-$\"&D1%Fbam7$$\"5LLLLL)G[k\"*)F-$\"&n3%Fbam7 $$\"5+++Dch;g=*)F-$\"&?@%Fbam7$$\"5lmm;zM]v?*)F-$\"&,U%Fbam7$$\"5ILL3- 3%3H#*)F-$\"&kT%Fbam7$$\"5++++D\"yh]#*)F-$\"&!=WFbam7$$\"5NLL$e**o4#H* )F-$\"&jT%Fbam7$$\"5nmmmm)fdL$*)F-$\"&-U%Fbam7$$\"5lmm;a;&ya$*)F-$\"&c V%Fbam7$$\"5lmmmTM%*fP*)F-$\"'T_W!#A7$$\"5lmm;H_.sR*)F-$\"';`WF_em7$$ \"5nmmm;q7%=%*)F-$\"&AX%Fbam7$$\"5+++]P\")z/Y*)F-$\"&7X%Fbam7$$\"5LLLL e#pa-&*)F-$\"&sW%Fbam7$$\"5lm;/Ev5A^*)F-$\"&[W%Fbam7$$\"5+++v$zX(=_*)F -$\"&1W%Fbam7$$\"5IL$e91%Q:`*)F-$\"&$HWFbam7$$\"5lmm;HB-7a*)F-$\"&zS%F bam7$$\"5ILLek))H0c*)F-$\"&))G%Fbam7$$\"5+++++ad)z&*)F-$\"&u\"RFbam7$$ \"5lmT5!*\\EEe*)F-$\"&9(QFbam7$$\"5IL$3-eaR&e*)F-$\"&&pQFbam7$$\"5++DJ qTk\")e*)F-$\"&#pQFbam7$$\"5lmmTgPL4f*)F-$\"&,(QFbam7$$\"5++]iSHrkf*)F -$\"&.(QFbam7$$\"5ILL$37#4?g*)F-$\"&*oQFbam7$$\"5+++D\"[]38'*)F-$\"&:( QFbam7$$\"5lmmmT)3;C'*)F-$\"&s'QFbam7$$\"5+++]ib7jk*)F-$\"&p'QFbam7$$ \"5LLLL$GUYo'*)F-$\"%oQF-7$$\"5+++D\")eq$y'*)F-$\"%nQF-7$$\"5lmm;z%pF) o*)F-$\"%jQF-7$$\"5ILL3xI$=)p*)F-$\"%bQF-7$$\"5++++vm*33(*)F-$\"%RQF-7 $$\"5NLL$3(Q-zs*)F-Fj^m7$$\"5nmmmm5:xu*)F-$\"%QMF-7$$\"5++DJ&Hi*Hv*)F- $\"%!G$F-7$$\"5NL$eR_tFe(*)F-$\"%pIF-7$$\"5qmTg_ZeNw*)F-$\"%\"z#F-7$$ \"5+++D\")fR)o(*)F-$\"%ECF-7$$\"5q;Hd&f,[r(*)F-$\"%uAF-7$$\"5NLe*)4s?T x*)F-$\"%wAF-7$$\"5+](=U#Ghnx*)F-$\"%xAF-7$$\"5qm;aQ%=Sz(*)F-$\"%vAF-7 $$\"5++v=n'Ho%y*)F-F\\_n7$$\"5NLL$e*3k**y*)F-F\\_n7$$\"5qmmT5e)36)*)F- Fg^n7$$\"5++++D28A$)*)F-$\"%sAF-7$$\"5+++]i!>V_)*)F-Fd`n7$$\"5+++++u]E ()*)F-$\"%nAF-7$$\"5+++vo:gF))*)F-$\"%bAF-7$$\"5+++]PdpG*)*)F-$\"%LAF- 7$$\"5+++D1**yH!**)F-$\"%\"=#F-7$$\"5++++vS)38**)F-$\"%%3#F-7$$\"5++]i lN_R#**)F-$\"%')=F-7$$\"5+++DcI;[$**)F-$\"%::F-7$$\"5++Dc,G[-%**)F-$\" %D7F-7$$\"5++](oa-oX**)F-$\"$W)F-7$$\"5+]7`>C'R[**)F-$\"$1'F-7$$\"5++v =#HA6^**)F-$\"$O$F-7$$\"5+]P%[;#GQ&**)F-$\"#HF-7$$\"5+++]P?Wl&**)F-$!$ E$F-7$$\"5+]i:5>g#f**)F-$!$F(F-7$$\"5++D\"Gyh(>'**)F-$!$f)F-7$$\"5+](o al@pk**)F-$!$i)F-7$$\"5++]7G:3u'**)F-$!$h)F-7$$\"5++vVt7SG(**)F-Fgen7$ $\"5+++v=5s#y**)F-$!$j)F-7$$\"5++]P40O\"*)**)F-$!$g)F-7$$\"\"*!\"\"F\\ fn-%&COLORG6&%$RGBG$\"\"%F^gn$\"\"!FfgnF\\gn-%'LEGENDG6#%6Enright-Vern er~schemeG-F$6%7gu7$F($\"$1\"F-7$F/$\"$/\"F-7$F4$\"$3\"F-7$F9$\"$4\"F- 7$F>$\"$7\"F-7$FC$\"$:\"F-7$FH$\"$9\"F-7$FM$\"$8\"F-7$FWFain7$FeoF_hn7 $F]p$\"$2\"F-7$F`pF_q7$Fep$\"#*)F-7$F]q$\"#!)F-7$Far$\"#bF-7$FfrFcjn7$ Fir$\"#`F-7$F^s$\"#IF-7$Fcs$\"\"\"F-7$FhsF][o7$F[tFegn7$Fct$F*F-7$F]u$ !\"*F-7$$\"5++]7`OL$ou)F-$!#=F-7$FbuFO7$$\"5lm;/,F-7$Fjx$!$&>F-7$F_ y$!$'>F-7$FdyFa]o7$FiyFa]o7$F_zF^]o7$Fa[l$!$9#F-7$$\"5NL$3-e&\\U%y)F-$ !$G#F-7$Ff[l$!$\\#F-7$$\"5NL$eRdx#R'y)F-$!$%GF-7$F[\\l$!$P$F-7$F]]l$!$ e$F-7$Fe]l$!$d$F-7$Fh]lFj^o7$F[^lFj^o7$F^^lF]_o7$Fa^l$!$i$F-7$Ff^l$!$o $F-7$F[_l$!$w$F-7$F`_l$!$%RF-7$Fe_l$!$D%F-7$Fj_l$!$u%F-7$F_`l$!$c&F-7$ F_al$!$x&F-7$FealFh`o7$Fhal$!$v&F-7$F[bl$!$w&F-7$Fcbl$!$!eF-7$F]cl$!$& fF-7$$\"5qm;/EV0[@))F-$!$7'F-7$Fbcl$!$Q'F-7$$\"5NL$ek)pLbB))F-$!$&oF-7 $Fgcl$!$f(F-7$F\\dl$!$l)F-7$FadlFhbo7$Ffdl$!$r)F-7$F[el$!$w)F-7$F`el$! $*))F-7$Feel$!$5*F-7$Fjel$!$[*F-7$F_fl$!%06F-7$Fdfl$!%R7F-7$FiflF^do7$ F\\glF^do7$Fagl$!%S7F-7$FfglFcdo7$F[hl$!%U7F-7$F^hl$!%Y7F-7$Fhhl$!%d7F -7$Fbil$!%u7F-7$Fgil$!%q:F-7$F\\jl$!%6'F-7$Ffdn$ !%UjF-7$F[en$!%'R'F-7$F`en$!%9kF-7$Feen$!%$\"#nF-7$FC$\"#tF-7$FHFfgp7$FM$\"#zF-7$FWFjgp7$Feo$\"#&)F-7$ F]p$F)F-7$F`pF^hpF\\jn7$F]qFahp7$Far$\"#))F-7$FirFahp7$FcsF`jn7$F[tFjg p7$Fgu$\"#iF-7$FavF[ip7$F[x$\"#]F-7$Fiy$\"#KF-7$F_z$\"#LF-7$Fa[l$\"#DF -7$Fj]o$\"#?F-7$Ff[lFgo7$Fb^o$\"\"$F-7$F[\\l$!#6F-7$Fe]l$!#;F-7$F[^l$! #F-7$Fe_l$!#VF-7$Feal$!##*F-7$Fhal$!#\"*F-7$F[b lFe[q7$Fcbl$!#$*F-7$F]cl$!$+\"F-7$Fhao$!$3\"F-7$Fbcl$!$<\"F-7$F`bo$!$O \"F-7$Fgcl$!$j\"F-7$F\\dl$!$,#F-7$Fadl$!$+#F-7$Ffdl$!$.#F-7$F[el$!$0#F -7$F`el$!$7#F-7$Feel$!$@#F-7$Fjel$!$P#F-7$F_fl$!$.$F-7$Fdfl$!$c$F-7$Fi fl$!$b$F-7$F\\glFc^q7$FfglF]_o7$Fbil$!$u$F-7$$\"5IL$ek[\\^\"f))F-$!$!Q F-7$$\"5lmm\"H#oqlf))F-$!$(QF-7$$\"5++]PfTE;g))F-$!$*RF-7$$\"5ILL$e\\@ o1'))F-$!$8%F-7$$\"5+++voh$z;'))F-$!$\\%F-7$Fgil$!$4&F-7$$\"5++]7y\"3' >j))F-$!$\\&F-7$$\"5ILLe9b;qj))F-$!$q&F-7$$\"5lm;/^Gs?k))F-$!$s&F-7$$ \"5+++](=!Grk))F-Ffaq7$$\"5lmmTg[Rsl))F-Ffaq7$F\\jlFfaq7$Fajl$!$r&F-7$ FfjlF\\ao7$F[[m$!$\"eF-7$F`[m$!$*eF-7$Fe[m$!$1'F-7$Fj[m$!$J'F-7$F_\\m$ !$R(F-7$Fd\\mFgen7$Fi\\m$!$k)F-7$F^]mFgen7$Ff]m$!$o)F-7$F^^m$!$!))F-7$ $\"5IL$eRUK'y'*))F-$!$&*)F-7$Fc^m$!$@*F-7$$\"5lm;a8:!*z)*))F-$!$o*F-7$ Fh^m$!%Z5F-7$F]_m$!%u6F-7$Fb_m$!%]7F-7$Fg_m$!%a7F-7$F\\`mFaeq7$Fa`mFae q7$Ff`mFaeq7$Fi`m$!%_7F-7$F^am$!&UD\"Fbam7$Fdam$!&\"f7Fbam7$Fiam$!&VE \"Fbam7$F^bm$!&DG\"Fbam7$Fcbm$!&TJ\"Fbam7$Fhbm$!&NY\"Fbam7$F]cm$!&Qv\" Fbam7$Fbcm$!&iv\"Fbam7$Fgcm$!&Lv\"Fbam7$Fe[p$!&Ov\"Fbam7$F\\dmFegq7$Fb \\p$!&(e$!#7F-7$FCFb[o7$FHFb[o7$FM$\"##F-7$Fhs$\"$=#F-7$F[tFcer7$Fct$\"$?#F-7$F]u$\"$ F#F-7$Fg[o$\"$O#F-7$Fbu$\"$`#F-7$F]\\o$\"$$GF-7$Fgu$\"$9$F-7$F\\vFjt7$ FavFjt7$FdvFffr7$Fgv$\"$<$F-7$F\\w$\"$?$F-7$FawF_u7$$\"5++DcwQq.l()F-$ \"$H$F-7$Ffw$\"$M$F-7$$\"5IL3_v#))F-$\"$'zF-7$Fbcl$\"$3)F- 7$F`bo$\"$R)F-7$Fgcl$\"$'*)F-7$F\\dl$\"$z*F-7$Fadl$\"$!)*F-7$Ffdl$\"$ \")*F-7$F[el$\"$')*F-7$F`el$\"$\"**F-7$Feel$\"%.5F-7$Fjel$\"%D5F-7$F_f l$\"%A6F-7$Fdfl$\"%87F-7$FiflF_`s7$F\\gl$\"%77F-7$FfglFc`s7$Fbil$\"%E7 F-7$Fgil$\"%u8F-7$F\\jl$\"%]9F-7$Fajl$\"%^9F-7$Ffjl$\"%a9F-7$F[[mFdcl7 $F`[m$\"%i9F-7$Fe[m$\"%r9F-7$Fj[m$\"%\"\\\"F-7$F_\\m$\"%t:F-7$Fd\\m$\" %p;F-7$Fi\\m$\"%n;F-7$F^]m$\"%o;F-7$Fc]mFfbs7$Ff]m$\"%m;F-7$Fi]mFcbs7$ F^^m$\"%t;F-7$F^dq$\"%z;F-7$Fc^m$\"%\"p\"F-7$Ffdq$\"%3F-7$Fadn$!%(>#F -7$Ffdn$!%lCF-7$F[en$!%oFF-7$F`en$!%oGF-7$Feen$!%rGF-7$Fjen$!%pGF-7$F_ fnF__t7$FbfnF__t7$FgfnF\\_t7$F\\gnFb_t-Ffar6&Fbgn$\"*++++\"FjarFegnFeg n-Fhgn6#%6an~\"efficient\"~schemeG-F$6%7du7$F($!\"'F-7$F/Fbo7$F4Fb[o7$ F9$F^gnF-7$F>$FdgnF-7$FC$\"#8F-7$FHF\\at7$FMFcdn7$FWFjjn7$FeoFcjn7$F]p Fcjn7$F`p$\"#oF-F\\jn7$FjpFehp7$F]q$\"#&*F-7$Fbq$\"$6\"F-7$Fgq$\"$K\"F -7$F\\rF^bt7$FarFdq7$FfrFdq7$Fir$\"$M\"F-7$F^s$\"$a\"F-7$Fcs$\"$!>F-7$ FhsFjbt7$F[tFjbt7$Fct$\"$\">F-7$F]u$\"$'>F-7$Fg[o$\"$.#F-7$Fbu$\"$;#F- 7$F]\\o$\"$R#F-7$Fgu$\"$j#F-7$F\\vF^dt7$FavF^dt7$F[x$\"$#HF-7$F`xFffr7 $Fex$\"$`$F-7$Fjx$\"$c$F-7$F_yF`hr7$Fdy$\"$b$F-7$FiyF`hr7$F_zFjdt7$Fa[ l$\"$j$F-7$Fj]o$\"$r$F-7$Ff[l$\"$&QF-7$Fb^o$\"$4%F-7$F[\\lFdir7$F]]l$ \"$k%F-7$Fe]l$\"$m%F-7$Fh]l$\"$l%F-7$F[^lFfft7$F^^lFcft7$Fa^lFcft7$Ff^ l$\"$n%F-7$F[_l$\"$t%F-7$F`_lFiu7$Fe_l$\"$)\\F-7$Fj_l$\"$I&F-7$F_`l$\" $#eF-7$F_al$\"$&fF-7$FealF\\ht7$Fhal$\"$'fF-7$F[blF\\ht7$FcblF`ht7$F]c lFd[s7$FhaoFicn7$Fbcl$\"$?'F-7$F`bo$\"$U'F-7$Fgcl$\"$\"oF-7$F\\dl$\"$S (F-7$Fadl$\"$T(F-7$Ffdl$\"$U(F-7$F[el$\"$X(F-7$F`el$\"$[(F-7$Feel$\"$c (F-7$FjelFg\\s7$F_fl$\"$M)F-7$FdflFd^s7$FiflFd^s7$F\\gl$\"$&*)F-7$Ffgl Fhjt7$Fbil$\"$-*F-7$Fgil$\"$%**F-7$F\\jl$\"%U5F-7$FajlFb[u7$Ffjl$\"%W5 F-7$F[[m$\"%X5F-7$F`[m$\"%[5F-7$Fe[m$\"%_5F-7$Fj[m$\"%j5F-7$F_\\mFg_l7 $Fd\\m$\"%e6F-7$Fi\\m$\"%c6F-7$F^]mFf\\u7$Fh^m$\"%u6F-7$F]_m$\"%)=\"F- 7$Fb_m$\"%'>\"F-7$Fg_m$\"%#>\"F-7$F\\`m$\"%&>\"F-7$Fa`mFi]u7$Ff`mFi]u7 $F^am$\"&E>\"Fbam7$Fcbm$\"&J=\"Fbam7$$\"5lm;z%\\(\\_<*)F-$\"&b<\"Fbam7 $Fhbm$\"&v:\"Fbam7$$\"5IL$3x\"[$y'>*)F-$\"&(=6Fbam7$F]cm$\"&84\"Fbam7$ Fbcm$\"&$)3\"Fbam7$Fgcm$\"&14\"Fbam7$F\\dm$\"&**3\"Fbam7$Fadm$\"&33\"F bam7$F[em$\"&=^*F_em7$Ffem$\"%SuFbam7$F[fm$\"%YuFbam7$F`fm$\"%1uFbam7$ Fefm$\"%otFbam7$Fjfm$\"%+tFbam7$F_gm$\"%PrFbam7$Fdgm$\"%SoFbam7$Figm$ \"%-`Fbam7$F^hm$\"$:)Fbam7$Fchm$\"$r#Fbam7$Fhhm$FafrFbam7$F]im$\"$^#Fb am7$Fbim$\"$h#Fbam7$Fgim$\"$m#Fbam7$F\\jmFabu7$Fajm$\"$%GFbam7$Ffjm$\" $X#Fbam7$F[[nFcbu7$F`[n$\"#FF-7$Fe[nFdcu7$Fj[n$\"#BF-7$F_\\nFccr7$Fd\\ n$\"\"'F-7$Fi\\n$!#nF-7$F\\]n$!$7$F-7$Fa]n$!$R%F-7$Ff]n$!$3'F-7$F[^n$! $L)F-7$F`^n$!%G6F-7$Fe^nFjeq7$Fj^nFaeq7$F__n$!%[7F-7$Fd_n$!%^7F-7$Fi_n $!%\\7F-7$F\\`nFieu7$F_`nFaeq7$Fb`nFfeu7$Fg`nFaeq7$Fj`nFfeu7$F_an$!%f7 F-7$Fdan$!%r7F-7$Fian$!%.8F-7$F^bn$!%i8F-7$Fcbn$!%([\"F-7$Fhbn$!%EF-7$Fbcn$!%o@F-7$F\\dn$!%2DF-7$Ffdn$!%_HF-7$F[en$!%AKF-7$F`e n$!%6LF-7$Feen$!%9LF-7$Fjen$!%7LF-7$F_fnFehu7$FbfnFehu7$FgfnFbhu7$F\\g nFhhu-F`gn6&FbgnFegn$\"#vF*$FiatF*-Fhgn6#%Ea~scheme~of~Tsitouras~and~P apakostasG-F$6%7cu7$F($!\"%F-7$F/$!\"&F-7$F4Fh`t7$F9F][o7$F>$\"\"&F-7$ FCFgo7$FHFgo7$FMFjjn7$FW$\"#JF-Faat7$F]p$\"#cF-FcatF\\jnFfat7$F]q$\"#% *F-Fjat7$Fgq$\"$J\"F-7$F\\rFa[v7$FarF^bt7$FfrF^bt7$FirF^bt7$F^s$\"$`\" F-7$Fcs$\"$)=F-7$Fhs$\"$(=F-7$F[tF^\\v7$FctF[\\v7$F]u$\"$%>F-7$Fg[o$\" $+#F-7$Fbu$\"$7#F-7$F]\\o$\"$N#F-7$Fgu$\"$e#F-7$F\\v$\"$f#F-7$FavFb]v7 $F[x$\"$(GF-7$F`xF`t7$Fex$\"$Y$F-7$Fjx$\"$[$F-7$F_y$\"$Z$F-7$FdyF`^v7$ FiyF`^v7$F_z$\"$\\$F-7$Fa[lF^et7$Fj]oFcet7$Ff[l$\"$x$F-7$Fb^o$\"$+%F-7 $F[\\lFihr7$F]]l$\"$a%F-7$Fe]l$\"$b%F-7$Fh]lFa_v7$F[^lFa_v7$F^^lFd_v7$ Fa^lFd_v7$Ff^l$\"$c%F-7$F[_l$\"$i%F-7$F`_l$\"$q%F-7$Fe_l$\"$'[F-7$Fj_l $\"$<&F-7$F_`l$\"$o&F-7$F_al$\"$!eF-7$FealF]av7$FhalF]av7$F[blF]av7$Fc bl$\"$\"eF-7$F]cl$\"$%eF-7$FhaoF^[s7$Fbcl$\"$/'F-7$F`bo$\"$D'F-7$Fgcl$ \"$j'F-7$F\\dl$\"$?(F-7$Fadl$\"$@(F-7$FfdlFfbv7$F[elF^[l7$F`elFajs7$Fe el$\"$M(F-7$Fjel$\"$Z(F-7$F_fl$\"$4)F-7$Fdfl$\"$p)F-7$Fifl$\"$o)F-7$F \\glFhcv7$Ffgl$\"$n)F-7$Fbil$\"$u)F-7$Fgil$\"$h*F-7$F\\jl$\"%25F-7$Faj lFedv7$Ffjl$\"%45F-7$F[[mFidv7$F`[m$\"%75F-7$Fe[m$\"%;5F-7$Fj[m$\"%E5F -7$F_\\m$\"%m5F-7$Fd\\m$\"%96F-7$Fi\\m$\"%76F-7$F^]mFiev7$Fh^m$\"%D6F- 7$Ff`m$\"%T6F-7$F^am$\"&\"R6Fbam7$Fcbm$\"&&G6Fbam7$$\"5++D1kJm)p\"*)F- $\"&t7\"Fbam7$Fd^u$\"&+7\"Fbam7$$\"5IL3_D=L1=*)F-$\"&(46Fbam7$Fhbm$\"& .5\"Fbam7$F\\_u$\"&)e5Fbam7$F]cm$\"&%H5Fbam7$Fbcm$\"&k-\"Fbam7$Fgcm$\" &(G5Fbam7$F\\dm$\"&!G5Fbam7$Fadm$\"&(=5Fbam7$F[em$\"&Z'))F_em7$Ffem$\" %hnFbam7$F[fm$\"%mnFbam7$F`fm$\"%FnFbam7$Fefm$\"%*o'Fbam7$Fjfm$\"%?mFb am7$F_gm$\"%dkFbam7$Fdgm$\"%fhFbam7$$\"5++](ofg'3b*)F-$\"%'f&Fbam7$Fig m$\"%>YFbam7$$\"5lm;HKr$>q&*)F-$\"%8F-7$Fdan$!%I8F-7$FianFjfu7$F^bn$! %@9F-7$Fcbn$!%W:F-7$Fhbn$!%!y\"F-7$F]cn$!%p>F-7$Fbcn$!%$!#:F-7$FCFc`t7$FHF]ju7$FMFgo7$FWFg o7$Feo$\"#WF-7$F]p$\"#XF-7$F`p$\"#hF-7$FepFahp7$FjpFahpF][v7$Fbq$\"$; \"F-7$Fgq$\"$U\"F-7$F\\rFdew7$Far$\"$V\"F-7$FfrFhew7$FirFbdr7$F^s$\"$s \"F-7$FcsFcerFberFeer7$FctF`erFierF\\frF_fr7$F]\\o$F^cuF-7$Fgu$\"$;$F- 7$F\\vFdfw7$FavFdfw7$FdvFdfw7$Fgv$\"$>$F-7$F\\w$\"$A$F-7$Faw$\"$E$F-7$ Ffw$\"$P$F-7$F[x$\"$d$F-7$F`x$\"$(QF-7$Fex$\"$S%F-7$Fjx$\"$V%F-7$F_y$ \"$T%F-7$Fdy$\"$U%F-7$FiyFbhw7$F_zF_hw7$Fa[lFa_v7$Fj]oFcft7$Ff[lFd`v7$ Fb^o$\"$>&F-7$F[\\l$\"$v&F-7$F`\\l$\"$$fF-7$Fe\\l$\"$)fF-7$Fj\\l$\"$(f F-7$F]]lFiiw7$Fb]lFiiw7$Fe]lFfiw7$Fh]lFiiw7$F[^lFiiw7$F^^lFfiw7$Fa^l$ \"$*fF-7$Ff^l$\"$-'F-7$F[_l$\"$5'F-7$F`_l$\"$B'F-7$Fe_l$\"$Z'F-7$Fj_l$ \"$#pF-7$F_`l$\"$n(F-7$F_al$\"$&yF-7$FealFg[x7$Fhal$\"$'yF-7$F[blF[\\x 7$FcblFf]s7$F]cl$\"$%zF-7$Fa]sFh[l7$Fhao$\"$/)F-7$Fi]s$\"$9)F-7$Fbcl$ \"$E)F-7$F`bo$\"$f)F-7$Fgcl$\"$<*F-7$F\\dl$\"%/5F-7$FadlF_]l7$FfdlFg\\ l7$F[elFc^l7$F`elF`ev7$Feel$\"%H5F-7$FjelF_\\u7$F_fl$\"%`6F-7$Fdfl$\"% \\7F-7$FiflFa^x7$F\\gl$\"%[7F-7$Ffgl$\"%Z7F-7$Fbil$\"%i7F-7$Fgil$\"%?9 F-7$F\\jl$\"%+:F-7$FajlFa_x7$Ffjl$\"%/:F-7$F[[m$\"%1:F-7$F`[m$\"%8:F-7 $$\"5NLek.Wx\"y())F-$\"%;:F-7$Fe[m$\"%B:F-7$$\"5qmT5lNgzy))F-$\"%L:F-7 $Fj[m$\"%X:F-7$F_\\m$\"%L;F-7$Fd\\m$\"%OF-7$Fg_m$\"%) *=F-7$F\\`m$\"%,>F-7$Fa`m$\"%+>F-7$Ff`mFfcx7$F^am$\"&')*=Fbam7$Fcbm$\" &X!>Fbam7$Fgcm$\"&*>>Fbam7$F\\dm$\"&*=>Fbam7$Fadm$\"&J\">Fbam7$F[em$\" 'hD=F_em7$Ffem$\"&Fn\"Fbam7$F[fm$\"&Hn\"Fbam7$F`fm$\"&*o;Fbam7$Fefm$\" &bm\"Fbam7$Fjfm$\"&%f;Fbam7$F_gm$\"&Yk\"Fbam7$Fdgm$\"&sh\"Fbam7$Figm$ \"&PZ\"Fbam7$F^hm$\"&*[5Fbam7$Fchm$\"%r**Fbam7$Fhhm$\"%`**Fbam7$F]im$ \"%^**Fbam7$Fbim$\"%h**Fbam7$Fgim$\"%l**Fbam7$F\\jm$\"%_**Fbam7$Fajm$ \"%\")**Fbam7$Ffjm$\"%U**Fbam7$F[[n$\"%X**Fbam7$F`[n$\"$'**F-7$Fe[nFbh x7$Fj[n$\"$#**F-7$F_\\nF`_s7$Fd\\n$\"$t*F-7$Fi\\nFd^s7$F\\]n$\"$P'F-7$ Fa]n$\"$/&F-7$Ff]nF`gw7$F[^n$\"#!*F-7$F`^n$!$?#F-7$Fe^n$!$\\$F-7$Fj^n$ !$[$F-7$F__n$!$Y$F-7$Fd_nF[jx7$Fi_n$!$Z$F-7$F\\`nFejx7$F_`nF^jx7$Fb`n$ !$]$F-7$Fg`nF[jx7$Fj`n$!$^$F-7$F_an$!$g$F-7$FdanF[_q7$FianF^[t7$F^bn$! $z%F-7$Fcbn$!$@'F-7$Fhbn$!$\"*)F-7$F]cnF[do7$Fbcn$!%'Q\"F-7$Fgcn$!%j:F -7$F\\dn$!%lF-7$Ffdn$!%gAF-7$F[en$!%gDF-7$F`en$!%gEF-7$ Feen$!%jEF-7$Fjen$!%hEF-7$F_fnFe]y7$FbfnFe]y7$FgfnFb]y7$F\\gnFh]y-F`gn 6&FbgnFbiu$FdjnF*Fegn-Fhgn6#%AVerner's~\"most~efficient\"~schemeG-%%FO NTG6$%*HELVETICAGF]gn-%&TITLEG6#%Verror~curves~for~12~stage~order~8~Ru nge-Kutta~methodsG-%+AXESLABELSG6$Q\"x6\"Q!F`_y-%%VIEWG6$;F(F\\gn;$Fco Fijp$FcjuFijp" 1 2 0 1 10 0 2 9 1 4 2 1.000000 45.000000 45.000000 0 1 "Enright-Verner scheme" "Verner's Maple scheme" "a modification of V erner's Maple scheme" "an \"efficient\" scheme" "a scheme of Tsitouras and Papakostas" "a modification of the Tsitouras-Papakostas scheme" " Verner's \"most efficient\" scheme" }}}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 622 "evalf[20](plot(['qn_RK8_1 '(x)-q(x),'qn_RK8_2'(x)-q(x),'qn_RK8_3'(x)-q(x),'qn_RK8_4'(x)-q(x),\n' qn_RK8_5'(x)-q(x),'qn_RK8_6'(x)-q(x),'qn_RK8_7'(x)-q(x)],x=0.9..0.925, -1.7e-15..7.8e-15,font=[HELVETICA,9],\ncolor=[COLOR(RGB,.4,0,.9),COLOR (RGB,0,.65,0),brown,red,COLOR(RGB,0,.75,.95),magenta,COLOR(RGB,.95,.55 ,0)],\nlegend=[`Enright-Verner scheme`,`Verner's Maple scheme`,`a modi fication of Verner's Maple scheme`,`an \"efficient\" scheme`,`a scheme of Tsitouras and Papakostas`,`a modification of the Tsitouras-Papakos tas scheme`,`Verner's \"most efficient\" scheme`],title=`error curves \+ for 12 stage order 8 Runge-Kutta methods`));" }}{PARA 13 "" 1 "" {GLPLOT2D 979 617 617 {PLOTDATA 2 "6--%'CURVESG6%7`t7$$\"\"*!\"\"$!$h) !#?7$$\"5nmm;a)G\\a+*F-$!$_)F-7$$\"5LL$ek`o!>5!*F-$!%$>\"F-7$$\"5nmm\" z>)G_:!*F-$!%@iF-7$$\"5nmmT&QU!*3-*F-$!%IiF-7$$\"5LL$eRZXKi-*F-$!%kiF- 7$$\"5nm;z>,_=J!*F-$!%&4)F-7$$\"5++](=F-7$$\"5l\"HKkn1i;0*F-$!&e7#F-7$$\"5LLL$3_\"=M_!*F-$!&`T#F -7$$\"5nmmTg(fJr0*F-$!&_o#F-7$$\"5+++]7eP_i!*F-$!&po#F-7$$\"5++++v34Bl !*F-$!&Nr#F-7$$\"5+++]Pf!Qz1*F-$!&!zGF-7$$\"5+++]i+paq!*F-$!&]d$F-7$$ \"5+++](=ubJ2*F-$!&UO%F-7$$\"5NLe9m>[_v!*F-$!&IO%F-7$$\"5nm;zW(*Q*y2*F -$!&=O%F-7$$\"5++D\"y+'4r!3*F-$!&DO%F-7$$\"5LLL$3F-GN3*F-$!&QQ%F-7$$\" 5NL$e*[L7s%3*F-$!&$>WF-7$$\"5NLL3FWW\"f3*F-$!&g\\%F-7$$\"5NL$3_]l2r3*F -$!&Dl%F-7$$\"5LLLL$e'3I)3*F-$!&M&\\F-7$$\"5+]7Gj`))o*3*F-$!&8j&F-7$$ \"5lm\"HK9%o2\"4*F-$!&5]'F-7$$\"5I$3xJ#H[Y#4*F-$!&,]'F-7$$\"5++]7. FW0\"*F-$!&At'F-7$$\"5+++D14js1\"*F-$!&O,(F-7$$\"5++D\"G))*)4!3\"*F-$! &3c(F-7$$\"5++]Pf)[$H4\"*F-$!&Md)F-7$$\"5!3_]49JG'4\"*F-$!&/&*)F-7$$\" 5lTg_AMJ'*4\"*F-$!&'o*)F-7$$\"5]i:5/dzH5\"*F-$!&%o*)F-7$$\"5I$3xc)zFj5 \"*F-$!&t'*)F-7$$\"5+D\"G)[DCI6\"*F-$!&q'*)F-7$$\"5lm\"z>62s>6*F-$!&f' *)F-7$$\"5+]7GQi8J8\"*F-$!&`'*)F-7$$\"5LLLek`1l9\"*F-$!&T'*)F-7$$\"5LL $e*[.-d>\"*F-$!&n'*)F-7$$\"5++vo/.MAA\"*F-$!&d,*F-7$$\"5nmmTg-m([7*F-$ !&DH*F-7$$\"5lmTgFJDjF\"*F-$!'*z/\"F-7$$\"5nm;z%*f%)QI\"*F-$!'cQ6F-7$$ \"5NLL$eR](yK\"*F-$!'LQ6F-7$$\"5++](oza'=N\"*F-$!'0Q6F-7$$\"5NLeRA\"ex x8*F-$!')z8\"F-7$$\"5nmm\"zWho.9*F-$!';S6F-7$$\"5++v=#*4qqT\"*F-$!'=W6 F-7$$\"5NL$ek`SXI9*F-$!'V`6F-7$$\"5qm\"H23!QQW\"*F-$!'*=<\"F-7$$\"5+++ +D'>Ad9*F-$!';17F-7$$\"5+]iSmvoPY\"*F-$!'cJ7F-7$$\"5++D\"y]bJq9*F-$!'p k7F-7$$\"5+](=#\\MioZ\"*F-$!'[&H\"F-7$$\"5++]i!R\"4M[\"*F-Fbal7$$\"5++ vVts-l\\\"*F-$!'@&H\"F-7$$\"5+++DcJ'f4:*F-$!'6&H\"F-7$$\"5++voavL\\`\" *F-$!'![H\"F-7$$\"5++]7`>r-c\"*F-$!'`%H\"F-7$$\"5++vVB5/%)e\"*F-$!'m%H \"F-7$$\"5+++v$4q`;;*F-$!')[H\"F-7$$\"5ILe*[VjZ3%p\"*F-$!'2@7F-7$$\"5+++]P4v5s\"*F-$!'&3A\"F-7$ $\"5NLe9TQKbu\"*F-$!'_?7F-7$$\"5nm;zWn*)*p<*F-$!'4>7F-7$$\"5+](=n8)eLy \"*F-$!'I;7F-7$$\"5NLekG&zs'z\"*F-$!'&z?\"F-7$$\"5+v$4YADT.=*F-$!'O*> \"F-7$$\"5q;Hd?4(45=*F-$!'Z'=\"F-7$$\"5Nek`;m\"y;=*F-$!'1m6F-7$$\"5+++ ]7BmM#=*F-$!'VN6F-7$$\"5lm\"zWn[i[=*F-$!&iA)F-7$$\"5LL$ek.Nyt=*F-$!&(> oF-7$$\"5lm;/w_\"3+>*F-$!'6@o!#@7$$\"5++]i:bzj#>*F-$!'U;oFjhl7$$\"5lm \"H#on!4_>*F-$!'n&z'Fjhl7$$\"5LLL$3-=!y(>*F-$!'DCmFjhl7$$\"5l\"zWU$4JX )>*F-$!'-$\\'Fjhl7$$\"5+]ilZQg7*>*F-$!'D#G'Fjhl7$$\"5I3x1hn*)z*>*F-$!' ddfFjhl7$$\"5lm\"zWn*=Z+#*F-$!'&3Y&Fjhl7$$\"5I$3-8]v<=?*F-$!'\"yi$Fjhl 7$$\"5++]7G8O;.#*F-$\"'>BA!#A7$$\"5N3Fp,C<\"Q?*F-$\"(F0G$Fh[m7$$\"5q;/ EvM)fW?*F-$\"(\")yf'Fh[m7$$\"5+D\"G)[Xz50#*F-$\"($y%f'Fh[m7$$\"5NLeRAc gv0#*F-$\"(DWf'Fh[m7$$\"5+]7`pxA02#*F-$\"'D'f'Fjhl7$$\"5nmmm;*\\[$3#*F -$\"'Y#f'Fjhl7$$\"5lm;/^Y'**4@*F-$\"'C#f'Fjhl7$$\"5nmmT&Qz]O@*F-$\"'E: mFjhl7$$\"5N$3x16Rl\\@*F-$\"'.#o'Fjhl7$$\"5++v$f$))*zi@*F-$\"'%)poFjhl 7$$\"5I3xc)pGPp@*F-$\"'EgqFjhl7$$\"5l;z>h&e%f<#*F-$\"'&*ftFjhl7$$\"5+D \"GQU)=D=#*F-$\"'^EyFjhl7$$\"5LL$ekG=4*=#*F-$\"'NK&)Fjhl7$$\"5+]P4@hr6 ?#*F-$\"'H$3\"F-7$$\"5lm\"Hd&R^K@#*F-$\"'1N:F-7$$\"5I$ek.z6LDA*F-$\"'V $Q#F-7$$\"5++++D'4TPA*F-$\"'(4A$F-7$$\"5lm;/Eb+^E#*F-$\"'B?KF-7$$\"5LL L3F9!z#H#*F-$\"'>>KF-7$$\"5++](=UgbKUB*F-$ \"'4RKF-7$$\"5NeRZ_MB*[B*F-$\"'eaKF-7$$\"5+]7G)[Z_bB*F-$\"'^!G$F-7$$\" 5qT&)3C:E@O#*F-$\"'t@LF-7$$\"5NLe*)fbF(oB*F-$\"'!eQ$F-7$$\"5q;/^JOI>Q# *F-$\"'\")GOF-7$$\"5++]7.vCU#*F-$\"'u[tF-7$$\"5+vVt_!fcBC*F-$\"'T&Q(F-7$$\"5]i:g ;]X^U#*F-$\"'?&Q(F-7$$\"5+](o/)4DnU#*F-$\"'D&Q(F-7$$\"5+DJ?3H%))HC*F-$ \"'6&Q(F-7$$\"5++v$f$[VIV#*F-$\"'y%Q(F-7$$\"5+]iS\"p=ORC*F-$\"'<%Q(F-7 $$\"5++](oa-oXC*F-$\"'*RQ(F-7$$\"5++vVt7SGZ#*F-$\"'$>Q(F-7$$\"$D*!\"$$ \"'B\"Q(F--%&COLORG6&%$RGBG$\"\"%F*$\"\"!F]jmF(-%'LEGENDG6#%6Enright-V erner~schemeG-F$6%7ft7$F($!%8kF-7$F/$!%#R'F-7$F4$!%ZjF-7$F9$!%)*fF-7$F >$!%3gF-7$FC$!%dfF-7$FH$!%baF-7$Ffn$!%KVF-7$F`o$!%LVF-7$Fco$!%CVF-7$Fh o$!%rTF-7$F]p$!%cRF-7$Fbp$!%@NF-7$F\\q$!%MFF-7$Ffq$!%'Q\"F-7$$\"5++]iS 1nta!*F-$!$8(F-7$F[r$!$<(F-7$F`r$!$'pF-7$Fer$!$M&F-7$Fjr$\"$s#F-7$F_s$ \"%oIF-7$Fds$\"%peF-7$Fis$\"%qeF-7$F^t$\"%reF-7$Fct$\"%$)eF-7$Fht$\"%T gF-7$F]u$\"%piF-7$Fbu$\"%PnF-7$Fgu$\"%EwF-7$F\\v$\"%L#*F-7$Fav$\"&6E\" F-7$Ffv$\"&'o;F-7$F[w$\"&$o;F-7$F`w$\"&\"o;F-7$Few$\"&ym\"F-7$FjwF[an7 $F_x$\"&#y;F-7$Fdx$\"&vu\"F-7$$\"5+]iSTC4![5*F-$\"&))y\"F-7$Fix$\"&x%= F-7$$\"5+](ozT^%31\"*F-$\"&/$>F-7$F^y$\"&^/#F-7$Fcy$\"&OT#F-7$Fhy$\"&) pIF-7$$\"5SgF;++4Y4\"*F-$\"&[=$F-7$F]z$\"&%3LF-7$$\"5?\"GQLF-7$Fbz$\"&#>LF-7$Fgz$\"&\">LF-7$F\\[l$\"&(>LF-7$Fa[lF[dn7$Ff[l$ \"&%>LF-7$F[\\l$\"&%=LF-7$F`\\l$\"&!=LF-7$$\"5NL3xcG/6<\"*F-$\"&wJ$F-7 $Fe\\l$\"&LK$F-7$$\"5l;H#oK!o*37*F-$\"&tL$F-7$Fj\\l$\"&hP$F-7$$\"5q\"z >OHq')G7*F-$\"&.T$F-7$$\"5N$3_DG+]N7*F-$\"&>Y$F-7$$\"5+vV[r-L@C\"*F-$ \"&#RNF-7$F_]l$\"&$[OF-7$Fd]l$\"&m![F-7$Fi]l$\"&go&F-7$F^^l$\"&Uo&F-7$ Fc^l$\"&Lo&F-7$Fh^l$\"&lo&F-7$F]_l$\"&8s&F-7$$\"5N$3_+A\"y.T\"*F-$\"&% [dF-7$Fb_l$\"&**y&F-7$$\"5q;HKk2iPU\"*F-$\"&C&eF-7$Fg_l$\"&U%fF-7$F\\` l$\"&cF'F-7$Fa`l$\"&L%pF-7$F[al$\"&#z\")F-7$Feal$\"&]'))F-7$Fhal$\"&\\ '))F-7$F]bl$\"&J'))F-7$Fbbl$\"&2'))F-7$FgblFdjn7$$\"5+]7G)[wLu:*F-$\"& z'))F-7$F\\cl$\"&q)))F-7$$\"5+Dc,\"HtV&f\"*F-$\"&)4*)F-7$$\"5+]PfebqCg \"*F-$\"&e%*)F-7$$\"5+v=7 Fh[m7$F_\\m$\")%R63#Fh[m7$Fd\\m$\")vt!3#Fh[m7$ Fi\\m$\")'413#Fh[m7$F^]m$\"(313#Fjhl7$Fc]m$\"(W+3#Fjhl7$Fh]m$\"($\\z?F jhl7$F]^m$\"(X#y?Fjhl7$Fg^m$\"(#[t?Fjhl7$F[`m$\"(c81#Fjhl7$$\"5lTgx.sJ ^>#*F-$\"(r*e?Fjhl7$F``m$\"'2e?F-7$$\"5Ie9TQ]6s?#*F-$\"'Ig?F-7$Fe`m$\" '_n?F-7$Fj`m$\"'+8@F-7$F_am$\"'ex@F-7$Fdam$\"'8x@F-7$Fiam$\"'Dw@F-7$F^ bm$\"'\"\\<#F-7$Fcbm$\"'zm@F-7$F]cm$\"'!H:#F-7$Fgcm$\"'qC@F-7$F\\dm$\" 'Ir?F-7$Fadm$\"'*)z>F-7$F[em$\"'3W=F-7$F_fm$\"'Sa;F-7$Fdfm$\"'4F;F-7$F ifm$\"'#Ri\"F-7$F^gm$\"'!Qi\"F-7$Fcgm$\"'%Ri\"F-7$Fhgm$\"'*Ri\"F-7$F]h m$\"'$Qi\"F-7$Fbhm$\"'gB;F-7$Fghm$\"'yB;F-7$F\\im$\"'BB;F-7$Faim$\"'\" >i\"F--Fgim6&FiimF\\jm$\"#l!\"#F\\jm-F_jm6#%6Verner's~Maple~schemeG-F$ 6%7cr7$F($!%S^F-7$F/$!%A^F-7$F4$!%u^F-7$F9$!%\">'F-7$F>$!%,iF-7$FC$!%r hF-7$FH$!%aiF-7$Ffn$!%_qF-7$F`o$!%`qF-7$Fho$!%9qF-7$Ffq$!%MsF-7$F[r$!% 5uF-7$F`r$!%)R(F-7$Fjr$!%YrF-7$Fds$!%ZmF-7$F^t$!%QmF-7$Fht$!%llF-7$F\\ v$!%rbF-7$F`w$!%1QF-7$Fjw$!%+QF-7$F_x$!%[PF-7$Fdx$!%WMF-7$Fix$!%CIF-7$ F^y$!%GAF-7$Fcy$!$+)F-7$Fhy$\"%[;F-7$Ff[l$\"%gDF-7$F`\\l$\"%aDF-7$Fe\\ l$\"%)e#F-7$Fj\\l$\"%nGF-7$F_]l$\"%\\UF-7$Fd]l$\"&*45F-7$Fi]l$\"&KX\"F -7$F^^l$\"&CX\"F-7$Fc^l$\"&DX\"F-7$Fh^l$\"&^X\"F-7$F]_l$\"&iZ\"F-7$Fb_ l$\"&&=:F-7$Fg_l$\"&Lh\"F-7$F\\`l$\"&.#=F-7$Fa`l$\"&\\C#F-7$F[al$\"&L/ $F-7$Feal$\"&2\\$F-7$Fhal$\"&8\\$F-7$F]bl$\"&,\\$F-7$Fbbl$\"&\"*[$F-7$ Fgbl$\"&)*[$F-7$Fhjn$\"&d\\$F-7$F\\cl$\"&'4NF-7$F`[o$\"&o_$F-7$Fe[o$\" &Pb$F-7$Fj[o$\"&$)f$F-7$Facl$\"&?n$F-7$F[dl$\"&rR%F-7$Fedl$\"&@q'F-7$F jdl$\"&6q'F-7$F_el$\"&#)p'F-7$Fdel$\"&rp'F-7$Fiel$\"&\"3nF-7$F^fl$\"&V t'F-7$Fcfl$\"&N\"oF-7$Fhfl$\"&Q*oF-7$F]gl$\"&C,(F-7$Fbgl$\"&z>(F-7$Fgg l$\"&DZ(F-7$F\\hl$\"'A>5F-7$Fahl$\"'&*Q6F-7$Ffhl$\"(H$Q6Fjhl7$F\\il$\" (R#Q6Fjhl7$Fail$\"(^&Q6Fjhl7$Ffil$\"(^U9\"Fjhl7$F[jl$\"(6#\\6Fjhl7$F`j l$\"(*pd6Fjhl7$Fejl$\"(26<\"Fjhl7$Fjjl$\"(AC>\"Fjhl7$F_[m$\"(kVF\"Fjhl 7$Fd[m$\")d%[X\"Fh[m7$Fj[m$\")#e@g\"Fh[m7$F_\\m$\")'HWw\"Fh[m7$Fd\\m$ \")\"[Sw\"Fh[m7$Fi\\m$\")3%Rw\"Fh[m7$F^]m$\"(!)Rw\"Fjhl7$Fc]m$\"(dMw\" Fjhl7$Fh]m$\"(GIw\"Fjhl7$F]^m$\"(eEw\"Fjhl7$Fb^m$\"(\\Kw\"Fjhl7$Fg^m$ \"(.bw\"Fjhl7$F\\_m$\"(/&oF-7$Fj`m$\"'*)GAF-7$F_a m$\"'@#\\#F-7$Fdam$\"'n\"\\#F-7$Fiam$\"'t!\\#F-7$F^bm$\"'***[#F-7$Fcbm $\"'b)[#F-7$F]cm$\"'[)[#F-7$Fgcm$\"'u$\\#F-7$$\"5+DJq&f*G`P#*F-$\"'B,D F-7$F\\dm$\"'$[^#F-7$$\"5N3xJnwJ&)Q#*F-$\"'tQDF-7$Fadm$\"'HyDF-7$Ffdm$ \"'.QEF-7$F[em$\"'3HFF-7$Feem$\"'WkGF-7$F_fm$\"',iIF-7$Fdfm$\"'pBJF-7$ Fifm$\"'iIJF-7$F^gm$\"'[IJF-7$Fcgm$\"'gIJF-7$Fhgm$\"'fIJF-7$F]hm$\"'RI JF-7$Fbhm$\"'1IJF-7$Fghm$\"':IJF-7$F\\im$\"'?HJF-7$Faim$\"',GJF--%'COL OURG6&Fiim$\")#)eqk!\")$\"))eqk\"FhaqFiaq-F_jm6#%Ha~modification~of~Ve rner's~Maple~schemeG-F$6%7hr7$F($!%pGF-7$F/$!%cGF-7$F4$!%!3$F-7$F9$!%> lF-7$F>$!%HlF-7$FC$!%OlF-7$FH$!%cwF-7$Ffn$!&k=\"F-7$F`oFgcq7$Fco$!&%*= \"F-7$Fho$!&W?\"F-7$F]p$!&[B\"F-7$Fbp$!&RI\"F-7$F\\q$!&TX\"F-7$Ffq$!&( fF-7$F`r$!&x#>F-7$Fer$!&3%>F-7$Fjr$!&(H?F-7$F_s$!&UU#F-7 $Fds$!&X)GF-7$Fis$!&O)GF-7$F^t$!&F)GF-7$Fht$!&B*GF-7$F\\v$!&N=$F-7$Fav $!&Ya$F-7$Ffv$!&r,%F-7$F[w$!&l,%F-7$F`w$!&f,%F-7$Few$!&\\,%F-7$Fjw$!&V ,%F-7$F_x$!&!=SF-7$Fdx$!&l0%F-7$Fix$!&z6%F-7$F^y$!&xC%F-7$Fcy$!&c]%F-7 $Fhy$!&=*\\F-7$Ff[l$!&>=&F-7$F`\\l$!&6=&F-7$Fe\\l$!&5=&F-7$Fj\\l$!&t>& F-7$F_]l$!&vH&F-7$Fd]l$!&6t&F-7$Fi]l$!&I1'F-7$F^^l$!&?1'F-7$Fc^l$!&.1' F-7$F]_lFdjq7$Fa`l$!&)*4'F-7$F]bl$!&Y0'F-7$Fgbl$!&80'F-7$F\\cl$!&J/'F- 7$Facl$!&,(fF-7$$\"5l;HKkncGi\"*F-$!&D#fF-7$Ffcl$!&0&eF-7$$\"5+](oa5g \\N;*F-$!&(RdF-7$F[dl$!&$zbF-7$F`dl$!&p+&F-7$Fedl$!&![TF-7$Fjdl$!&g9%F -7$F_elF`]r7$Fdel$!&Y9%F-7$Fiel$!&68%F-7$F^fl$!&D5%F-7$Fcfl$!&p,%F-7$F hfl$!&&HRF-7$F]gl$!&*)z$F-7$Fbgl$!&Mf$F-7$Fggl$!&mG$F-7$F\\hl$!%C>F-7$ Fahl$\"&s=\"F-7$Ffhl$\"'l$=\"Fjhl7$F\\il$\"'#f=\"Fjhl7$Fail$\"'c*>\"Fj hl7$Ffil$\"'1F8Fjhl7$F[jl$\"'QF9Fjhl7$F`jl$\"'v!f\"Fjhl7$Fejl$\"'1W=Fj hl7$Fjjl$\"'\"Fh[m7$Fd\\m$\")yl(>\"Fh[m7$Fi\\m$\")se(>\"Fh[m7$F^ ]m$\"(+x>\"Fjhl7$Fc]m$\"(^s>\"Fjhl7$Fh]m$\"(Rq>\"Fjhl7$F]^m$\"(G#)>\"F jhl7$Fb^m$\"(MC?\"Fjhl7$Fg^m$\"(rX@\"Fjhl7$F\\_m$\"(,sA\"Fjhl7$Fa_m$\" (QtC\"Fjhl7$Ff_m$\"(?#z7Fjhl7$F[`m$\"(/!G8Fjhl7$F``m$\"'n*[\"F-7$Fe`m$ \"'u7=F-7$Fj`m$\"'8HCF-7$F_am$\"'aWIF-7$Fdam$\"'&Q/$F-7$Fiam$\"'#G/$F- 7$F^bm$\"':VIF-7$Fcbm$\"'d`IF-7$Fhbm$\"'^iIF-7$F]cm$\"'.yIF-7$Fbcm$\"' 5.JF-7$Fgcm$\"'lUJF-7$F\\dm$\"'!fH$F-7$Fadm$\"'4MOF-7$Ffdm$\"'o3RF-7$F [em$\"'=$H%F-7$F`em$\"')y`%F-7$Feem$\"'.D[F-7$Fjem$\"'.h^F-7$F_fm$\"'* Gb&F-7$Fdfm$\"'PsdF-7$Fifm$\"'@(z&F-7$F^gm$\"'-(z&F-7$Fcgm$\"'5(z&F-7$ Fhgm$\"',(z&F-7$F]hm$\"'t'z&F-7$Fbhm$\"'B'z&F-7$Fghm$\"':'z&F-7$F\\im$ \"']%z&F-7$Faim$\"'a$z&F--Fdaq6&Fiim$\"*++++\"FhaqF\\jmF\\jm-F_jm6#%6a n~\"efficient\"~schemeG-F$6%7er7$F($!%7LF-7$F/$!%)H$F-7$F4$!%([$F-7$F9 $!%JkF-7$F>$!%TkF-7$FC$!%SkF-7$FH$!%]tF-7$Ffn$!&j3\"F-7$F`o$!&k3\"F-7$ Fco$!&))3\"F-7$Fho$!&**4\"F-7$F]p$!&N7\"F-7$Fbp$!&\"y6F-7$F\\q$!&&)H\" F-7$Ffq$!&pa\"F-7$F[r$!&So\"F-7$F`r$!&To\"F-7$Fer$!&Mp\"F-7$Fjr$!&/w\" F-7$F_s$!&v1#F-7$Fds$!&=V#F-7$Fis$!&5V#F-7$F^t$!&-V#F-7$Fht$!&kV#F-7$F \\v$!&'[EF-7$Fav$!&%>HF-7$Ffv$!&zF$F-7$F[w$!&uF$F-7$F`w$!&pF$F-7$Few$! &hF$F-7$Fjw$!&bF$F-7$F_xF__s7$Fdx$!&@I$F-7$Fix$!&EM$F-7$F^y$!&-V$F-7$F cy$!&tg$F-7$Fhy$!&f%RF-7$Ff[l$!&$zSF-7$F`\\l$!&)ySF-7$Fe\\l$!&!ySF-7$F _]l$!&y8%F-7$Fi]l$!&Fa%F-7$Fc^l$!&2a%F-7$F]_l$!&s`%F-7$Fa`l$!&ST%F-7$F ]bl$!&$HTF-7$Fgbl$!&o7%F-7$F\\cl$!&j6%F-7$Facl$!&d-%F-7$Fj[r$!&%oRF-7$ Ffcl$!&K)QF-7$Fb\\r$!&Uv$F-7$F[dl$!&&pNF-7$F`dl$!&C#HF-7$Fedl$!&$o>F-7 $Fjdl$!&q'>F-7$F_el$!&v'>F-7$Fdel$!&m'>F-7$Fiel$!&P&>F-7$F^fl$!&d#>F-7 $Fcfl$!&<%=F-7$Fhfl$!&gv\"F-7$F]gl$!&#G;F-7$Fbgl$!&tU\"F-7$Fggl$!&!G6F -7$F\\hl$\"&9)=F-7$Fahl$\"&,A$F-7$Ffhl$\"'0;KFjhl7$F\\il$\"'pHKFjhl7$Ffil$\"'UOF-7$F[em$\"'%o'RF-7$F`em$\"'3uTF-7$Feem$\"'y$!%IkF-7$FC$!%HkF-7$FH$!%?tF-7$Ffn$!&s2\"F-7$F`o$ !&t2\"F-7$Fco$!&(z5F-7$Fho$!&04\"F-7$F]p$!&N6\"F-7$Fbp$!&n;\"F-7$F\\q$ !&YG\"F-7$Ffq$!&z_\"F-7$F[r$!&Bm\"F-7$F`r$!&Cm\"F-7$Fer$!&9n\"F-7$Fjr$ !<\"F-7$F_s$!&g.#F-7$Fds$!&>R#F-7$Fis$!&5R#F-7$F^t$!&.R#F-7$Fht$!&iR #F-7$F\\v$!&;g#F-7$Fav$!&W'GF-7$Ffv$!&I@$F-7$F[w$!&E@$F-7$F`w$!&@@$F-7 $Few$!&8@$F-7$Fjw$!&2@$F-7$F_x$!&D@$F-7$Fdx$!&fB$F-7$Fix$!&YF$F-7$F^y$ !&'eLF-7$Fcy$!&'GNF-7$Fhy$!&X&QF-7$Ff[l$!&H)RF-7$F`\\l$!&C)RF-7$Fe\\l$ !&;)RF-7$F_]l$!&l.%F-7$Fi]l$!&.T%F-7$Fc^l$!&$3WF-7$F]_l$!&XS%F-7$Fa`l$ !&vE%F-7$F]bl$!&D'RF-7$Fgbl$!&+'RF-7$F\\cl$!&$\\RF-7$Facl$!&s&QF-7$Fj[ r$!&#*z$F-7$Ffcl$!&Gr$F-7$Fb\\r$!&Be$F-7$F[dl$!&cR$F-7$F`dl$!&Cu#F-7$F edl$!&2y\"F-7$Fjdl$!&$z8Fjhl7$Fg^m$\"(w!H8Fjhl7$ F\\_m$\"(p&R8Fjhl7$Fa_m$\"(1kN\"Fjhl7$Ff_m$\"($H$Q\"Fjhl7$F[`m$\"(dYU \"Fjhl7$F``m$\"'$Gc\"F-7$Fe`m$\"';T=F-7$Fj`m$\"',wBF-7$F_am$\"'o7HF-7$ Fdam$\"'-7HF-7$Fiam$\"',6HF-7$F^bm$\"'66HF-7$Fcbm$\"'#*=HF-7$Fhbm$\"'x DHF-7$F]cm$\"''z$HF-7$Fbcm$\"'%y&HF-7$Fgcm$\"'Y*)HF-7$F\\dm$\"'Z8JF-7$ Fadm$\"'x!R$F-7$Ffdm$\"'q&F-7 $F^gm$\"'+'>&F-7$Fcgm$\"'4'>&F-7$Fhgm$\"','>&F-7$F]hm$\"'v&>&F-7$Fbhm$ \"'H&>&F-7$Fghm$\"'C&>&F-7$F\\im$\"'v$>&F-7$Faim$\"'v#>&F--Fdaq6&FiimF eirF\\jmFeir-F_jm6#%Ra~modification~of~the~Tsitouras-Papakostas~scheme G-F$6%7jr7$F($!%hEF-7$F/$!%[EF-7$F4$!%uGF-7$F9$!%:jF-7$F>$!%CjF-7$FC$! %KjF-7$FH$!%muF-7$Ffn$!&)p6F-7$F`o$!&*p6F-7$Fco$!&H<\"F-7$Fho$!&$)=\"F -7$F]p$!&$>7F-7$Fbp$!&'*G\"F-7$Fgp$!&?N\"F-7$F\\q$!&=W\"F-7$Faq$!&3d\" F-7$Ffq$!&4v\"F-7$F[r$!&-#>F-7$F`r$!&2#>F-7$Fer$!&V$>F-7$Fjr$!&b-#F-7$ F_s$!&yU#F-7$Fds$!&e*GF-7$Fis$!&[*GF-7$F^t$!&R*GF-7$Fht$!&T!HF-7$F\\v$ !&X?$F-7$Fav$!&ad$F-7$Ffv$!&&fSF-7$F[w$!&!fSF-7$F`w$!&%eSF-7$Few$!&u0% F-7$Fjw$!&o0%F-7$F_x$!&21%F-7$Fdx$!&85%F-7$Fix$!&d;%F-7$F^y$!&9I%F-7$F cy$!&-d%F-7$Fhy$!&e2&F-7$Ff[l$!&MF&F-7$F`\\l$!&EF&F-7$Fe\\l$!&FF&F-7$F j\\l$!&0H&F-7$F_]l$!&!*R&F-7$Fd]l$!&!oeF-7$Fi]l$!&oA'F-7$F^^l$!&dA'F-7 $Fc^l$!&SA'F-7$F]_l$!&pA'F-7$Fa`l$!&II'F-7$F]bl$!&&=jF-7$Fgbl$!&^J'F-7 $F\\cl$!&xI'F-7$Facl$!&:C'F-7$Fj[r$!&y>'F-7$Ffcl$!&<8'F-7$Fb\\r$!&\"Hg F-7$F[dl$!&/)eF-7$F`dl$!&iM&F-7$FedlFjas7$Fjdl$!&'QXF-7$F_el$!&&QXF-7$ Fdel$!&q`%F-7$Fiel$!&Q_%F-7$F^fl$!&i\\%F-7$Fcfl$!&JT%F-7$Fhfl$!&#GVF-7 $F]gl$!&:?%F-7$Fbgl$!&=+%F-7$Fggl$!&Rq$F-7$F\\hl$!%ipF-7$Fahl$\"%bkF-7 $Ffhl$\"&5U'Fjhl7$F\\il$\"&^W'Fjhl7$Fail$\"&De'Fjhl7$Ffil$\"&%[yFjhl7$ F[jl$\"&B%))Fjhl7$F`jl$\"')f/\"Fjhl7$Fejl$\"'b'H\"Fjhl7$Fjjl$\"'?$o\"F jhl7$F_[m$Fg_qFjhl7$Fd[m$\"(zQ='Fh[m7$Fj[m$\"(@5j)Fh[m7$F_\\m$\")\"z&H 6Fh[m7$Fd\\m$\"))Q#H6Fh[m7$Fi\\m$\")EJdF-7$F\\im$\"'cHdF-7$Faim$\"'kGdF--Fgim 6&FiimFj`t$\"#bFcjoF\\jm-F_jm6#%AVerner's~\"most~efficient\"~schemeG-% %FONTG6$%*HELVETICAGF)-%&TITLEG6#%Verror~curves~for~12~stage~order~8~R unge-Kutta~methodsG-%+AXESLABELSG6$Q\"x6\"Q!F_aw-%%VIEWG6$;F(Faim;$!#< !#;$\"#yFhaw" 1 2 0 1 10 0 2 9 1 4 2 1.000000 45.000000 45.000000 0 1 "Enright-Verner scheme" "Verner's Maple scheme" "a modification of Ver ner's Maple scheme" "an \"efficient\" scheme" "a scheme of Tsitouras a nd Papakostas" "a modification of the Tsitouras-Papakostas scheme" "Ve rner's \"most efficient\" scheme" }}}}{PARA 0 "" 0 "" {TEXT -1 0 "" }} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 613 "evalf[20](plot(['qn_RK8_1'( x)-q(x),'qn_RK8_2'(x)-q(x),'qn_RK8_3'(x)-q(x),'qn_RK8_4'(x)-q(x),\n'qn _RK8_5'(x)-q(x),'qn_RK8_6'(x)-q(x),'qn_RK8_7'(x)-q(x)],x=0.925..0.9469 483523,font=[HELVETICA,9],\ncolor=[COLOR(RGB,.4,0,.9),COLOR(RGB,0,.65, 0),brown,red,COLOR(RGB,0,.75,.95),magenta,COLOR(RGB,.95,.55,0)],\nlege nd=[`Enright-Verner scheme`,`Verner's Maple scheme`,`a modification of Verner's Maple scheme`,`an \"efficient\" scheme`,`a scheme of Tsitour as and Papakostas`,`a modification of the Tsitouras-Papakostas scheme` ,`Verner's \"most efficient\" scheme`],title=`error curves for 12 stag e order 8 Runge-Kutta methods`));" }}{PARA 13 "" 1 "" {GLPLOT2D 986 617 617 {PLOTDATA 2 "6--%'CURVESG6%7gr7$$\"$D*!\"$$\"'B\"Q(!#?7$$\"5#H T(y=;Tya#*F-$\"']%\\(F-7$$\"5x:8C5]n%*e#*F-$\"'\\h\"*F-7$$\"5/&Gg&Hn!G OE*F-$\"(Q5K\"F-7$$\"5H&H!zH:/Mo#*F-$\"(*\\?8F-7$$\"5->FFLm..t#*F-$\"( .&G8F-7$$\"55GIvH^&ytF*F-$\"(%Rx9F-7$$\"5%*R\")=(\\%3)=G*F-$\"(.`)>F-7 $$\"5g)H\\IN7PlG*F-$\"(7W)>F-7$$\"5>L9ZMp%y6H*F-$\"(Cy)>F-7$$\"5e\"*f& 3Sm_fH*F-$\"(jh3#F-7$$\"5zVBrExx:+$*F-$\"(yhX#F-7$$\"5v\\**y\")Q<*[I*F -$\"(@]X#F-7$$\"5DrK'p%Q^k4$*F-$\"(%paCF-7$$\"5D]9w\")>fA9$*F-$\"(&=UC F-7$$\"5g$yFG\"3dQ=$*F-$\"(AV8#F-7$$\"5e+'*Hx)4KLK*F-$\"(hK8#F-7$$\"5L K,#o/MAvK*F-$\"(#pI@F-7$$\"5S'p3&3k%f*H$*F-$\"(FQ6#F-7$$\"5Wgs>q(e'RK$ *F-$\"(Jr+#F-7$$\"50*ozz8AvML*F-$\"(>t'=F-7$$\"5l<@w0bQbM$*F-$\"(Gzf\" F-7$$\"5DYXat)[KcL*F-$\"(?**4\"F-7$$\"5$[(pKTA6rO$*F-$\"'k>@F-7$$\"5:0 **3h')z2R$*F-$!&iN\"F-7$$\"5WNG&33&[WT$*F-$!&uN\"F-7$$\"5JEm+nzC&fM*F- $!&Jq\"F-7$$\"5+s8ze)4/$[$*F-$!&)GZF-7$$\"5rnUL'f$*F-$!)&43N&F^u7$$\"5?C$\\b/&H0i$*F-$!).r\\`F^u7$$\"5ge_;>uCZk$* F-$!)4$HN&F^u7$$\"55y0mrw'ylO*F-$!)JP&Q&F^u7$$\"5c(*e:Cz[oo$*F-$!)Phhb F^u7$$\"5?a&obuDA)p$*F-$!).?OeF^u7$$\"5!3@\")pcjf4P*F-$!)6U%R'F^u7$$\" 5SnQR)Q,(4s$*F-$!)\\wruF^u7$$\"5/Cl!)4#RMKP*F-$!)5[g%*F^u7$$\"5!>jz&3: %4WP*F-$!)G]98F-7$$\"5vRFN2QWev$*F-$!)W;&[\"F-7$$\"5gZe71h%fnP*F-$!)1* \\[\"F-7$$\"5]b*)*[S[MzP*F-$!)f\"[[\"F-7$$\"5qwLnbXNB!Q*F-$!)a[%[\"F-7 $$\"5)yzZkqgKDQ*F-$!)hS%[\"F-7$$\"5lqg()Gnqv%Q*F-$!)8q'[\"F-7$$\"5WVVI ^F:)pQ*F-$!)*=F]\"F-7$$\"5&yM`A%pk@)Q*F-$!)m.J:F-7$$\"5I_B?L69X*Q*F-$! )Ba#f\"F-7$$\"5qc8:C`jo!R*F-$!)Pa<R*F-$!)k\"y&>F-7$$ \"5buSHQX4.$R*F-$!)Y$4M#F-7$$\"5+)y([h&fSTR*F-$!)%eTq#F-7$$\"5S,:o%eC] _R*F-$!)(fQq#F-7$$\"5$[@vyg*)fjR*F-$!)#fNq#F-7$$\"508PT-t%H()R*F-$!)a# Hq#F-7$$\"5D6A&p*\\!*4,%*F-$!)qY-FF-7$$\"5S>B/()QiC.%*F-$!)Fe.FF-7$$\" 5gFC8xFMR0%*F-$!)]l7FF-7$$\"5lZ@\"z+:nlS*F-$!)OjFFF-7$$\"5qn=pQs3u2%*F -$!)\"ezv#F-7$$\"5q(er%p%f9*3%*F-$!)<$[\"GF-7$$\"5v28D+<$)35%*F-$!)!*o 9HF-7$$\"5D*=$QA)p#>6%*F-$!)%ox1$F-7$$\"5vq]^WzqH7%*F-$!)tO,LF-7$$\"5] 65e0q#\\GT*F-$!)xu,MF-7$$\"5D_pkmg9S8%*F-$!)0c,MF-7$$\"5+$*GrF^O&RT*F- $!)EP,MF-7$$\"5xL)y()=%e]9%*F-$!)X=,MF-7$$\"5+q8j1NY\"oT*F-$!))*R+MF-7 $$\"5>1R[CGM7>%*F-$!)&G'*R$F-7$$\"5!z#>ee&p!Q@%*F-$!)b#))R$F-7$$\"5e\\ *zEH'zjB%*F-$!)6t&R$F-7$$\"5:'zZ$yP&>[U*F-$!)Te)Q$F-7$$\"5vUc,k76+E%*F -$!)3RoLF-7$$\"50m&\\o+!>fE%*F-$!)kB[LF-7$$\"5N*[$o\\(o#=F%*F-$!))4mJ$ F-7$$\"5l7u^#\\ZtxU*F-$!)$**zE$F-7$$\"5%fL^`BEk$G%*F-$!):o%>$F-7$$\"5! 3=[%Rc-kI%*F-$!),lICF-7$$\"5nD]aV]i\"HV*F-$!)#H6k\"F-7$$\"5]iV92$yV_V* F-$!)yuS;F-7$$\"5H*pV2dJrvV*F-$!)_IS;F-7$$\"50Ye!)>v&z)R%*F-$!)\\PQ;F- 7$$\"5x#*z')oMy=U%*F-$!)NR?;F-7$$\"5X.U81g$[KW*F-$!)m\"*)e\"F-7$$\"5:9 /SV&))3VW*F-$!)Y(o^\"F-7$$\"5![im13Tp`W*F-$!)SLi8F-7$$\"5]NG$zh$*HkW*F -$!)oz[5F-7$$\"5D/Em$zTXwW*F-$!(Fu<$F-7$$\"5+tBRp**3')[%*F-$\")\\Ja6F- 7$$\"5SdsDdS'o%\\%*F-$\"):=YBF-7$$\"5vT@7X\"Qw+X*F-$\")(H9)RF-7$$\"5!R ea!*=D!Q]%*F-$\")pU3]F-7$$\"55Eq)HB7%o]%*F-$\")'3=j&F-7$$\"5Io%>pF*z)4 X*F-$\")\"Q;j&F-7$$\"5Y5>&3K'=H^%*F-$\")tYJcF-7$$\"50l*pbf9mMX*F-$\")O CIcF-7$$\"5n>!)GqG/kb%*F-$\")O1HcF-7$$\"50S4jgn'ezX*F-$\")wIHcF-7$$\"5 WgQ(4l!pFg%*F-$\")zM[cF-7$$\"5qwQv6>;$3Y*F-$\")>ijcF-7$$\"5&H*Q`sJjQh% *F-$\")uB)o&F-7$$\"5?4RJLW5%>Y*F-$\")F)os&F-7$$\"5]DR4%pv&\\i%*F-$\"*, !>'y&F^u7$$\"50eRl:#=0OY*F-$\"*`6r+'F^u7$$\"5c!*R@P2Yrk%*F-$\"*\"*RCZ' F^u7$$\"5&H\\5zi$o!fY*F-$\"*HCX\\(F^u7$$\"5I&*pg=l!*4n%*F-$\"*A&><&*F^ u7$$\"5]Y_&R'z^pn%*F-$\"+DbR86F^u7$$\"5l(\\.$4%H\"Ho%*F-$\"+=;;L8F^u7$ $\"5?Bw(>8N*eo%*F-$\"+!eM-Z\"F^u7$$\"5!)[\"=Faam7$$\"+BN[p%*!#5$\"3y=?J$GFH-#!#H-%&COL ORG6&%$RGBG$\"\"%!\"\"$\"\"!Ffbm$\"\"*Fdbm-%'LEGENDG6#%6Enright-Verner ~schemeG-F$6%7[r7$F($\"'\">i\"F-7$F/$\"'1j:F-7$F4$\"&Tk*F-7$F9$!&G%>F- 7$F>$!&%\\>F-7$FC$!&Xe#F-7$FH$!'%\\?\"F-7$FM$!'RfTF-7$FR$!'&y:%F-7$FW$ !'93UF-7$Ffn$!'=PbF-7$$\"5qnTyj?_0)H*F-$!'%*G&)F-7$F[o$!(l?8\"F-7$$\"5 vY6D/eZ_-$*F-$!(,=8\"F-7$F`o$!(^:8\"F-7$Feo$!(Ca8\"F-7$$\"5vgBO9Hb$>J* F-$!(O&e6F-7$Fjo$!(YBE\"F-7$$\"5!ph%H(R\"eI;$*F-$!('f%e\"F-7$F_p$!($>W AF-7$$\"55#pj]M!*e3K*F-$!(IOC#F-7$Fdp$!(*3VAF-7$Fip$!(9NC#F-7$F^q$!(z] D#F-7$Fcq$!($4HBF-7$Fhq$!(cYU#F-7$F]r$!(rog#F-7$Fbr$!(B1%HF-7$Fgr$!(L6 `$F-7$F\\s$!(d(zOF-7$Fas$!(&*)yOF-7$Ffs$!(3qn$F-7$F[t$!(*yxOF-7$F`t$!( R=q$F-7$Fet$!(x[u$F-7$Fjt$!)goSQF^u7$F`u$!)*)eQSF^u7$Feu$!):@CWF^u7$Fc w$!)t!z'\\F^u7$F]x$!)*oO'\\F^u7$Fgx$!)wk@\\F^u7$F[z$!)Q\\UZF^u7$F_[l$! (cq![F-7$Fi[l$!(vH![F-7$F^\\l$!(tny%F-7$Fc\\l$!(M_q%F-7$Fh\\l$!(G\"yXF -7$F]]l$!(_VK%F-7$Fb]l$!(\">YQF-7$Fg]l$!(^z)HF-7$F\\^l$!(slp\"F-7$Fa^l $!'%>>&F-7$Ff^l$!']\">&F-7$F[_l$!')3>&F-7$F`_l$!'6*=&F-7$Fe_l$!'.n^F-7 $Fj_l$!'pr\\F-7$F_`l$!'M6QF-7$$\"5g(GAD*)G!)fS*F-$!'GIIF-7$Fd`l$!'^s=F -7$$\"5q2?IB6S:2%*F-$!&e$=F-7$Fi`l$\"'XVAF-7$F^al$\"(J-0\"F-7$Fcal$\"( `5j#F-7$F]bl$\"(9I$**F-7$Facl$\")?\")47F-7$Ffcl$\")T`47F-7$F[dl$\")3L4 7F-7$F`dl$\")lE57F-7$Fedl$\")cD?7F-7$$\"5&G(Q^N](GUU*F-$\"),xF7F-7$Fjd l$\")3aR7F-7$$\"5X><=@D.TD%*F-$\")\")fd7F-7$F_el$\")=z%G\"F-7$Fiel$\") w\"QQ\"F-7$Fcfl$\")&)R)e\"F-7$$\"5Ne(*RPfA]H%*F-$\")[(4(>F-7$Fhfl$\")b 0xEF-7$$\"5+#Rs/\\D47V*F-$\")$*[=KF-7$$\"5D.m\\T`#yDZBV*F-$\")up'p$F-7$F]gl$\")x['p$F-7$Fbgl$\")ki&p$F-7$Fggl$\" )rx%p$F-7$F\\hl$\")jO%p$F-7$Fahl$\")jn*p$F-7$$\"55)4,vt4=FW*F-$\")f2/P F-7$Ffhl$\")v=6PF-7$$\"5!)3twuA'yPW*F-$\")=OAPF-7$F[il$\")o]RPF-7$F`il $\")%oL!QF-7$Feil$\")([!QRF-7$Fjil$\")T.jUF-7$F_jl$\")d]P\\F-7$Fdjl$\" )j'R\\&F-7$Fijl$\")PrmiF-7$F^[m$\")/4cnF-7$Fc[m$\")>KaqF-7$Fh[m$\")#3T 0(F-7$F]\\m$\")V*Q0(F-7$Fb\\m$\")4O_qF-7$Fg\\m$\")a!30(F-7$Fa]m$\")0hV qF-7$Fe^m$\"*=S@.(F^u7$F__m$\"*zLN.(F^u7$$\"5vTAc#=s5`Y*F-$\"*Ba\"\\qF ^u7$Fd_m$\"*a-43(F^u7$$\"55W(eK2&H]m%*F-$\"*L,!QrF^u7$Fi_m$\"*([\\LsF^ u7$Fc`m$\"*e2$=wF^u7$Fham$\"2y=!eo:Bo%)F]bm-F_bm6&FabmFebm$\"#l!\"#Feb m-Fjbm6#%6Verner's~Maple~schemeG-F$6%7cr7$F($\"',GJF-7$F/$\"'\"=6$F-7$ F4$\"'DyIF-7$F9$\"'-BKF-7$F>$\"'8@KF-7$FC$\"'(G>$F-7$FH$\"'7ZGF-7$FM$ \"'[x=F-7$FR$\"'Rw=F-7$FW$\"'aY=F-7$Ffn$\"'Sy5F-7$F[o$!'38CF-7$F`o$!'4 7CF-7$Feo$!'hUCF-7$Fjo$!'zZMF-7$F]gm$!'V@hF-7$F_p$!(;k<\"F-7$Fegm$!(F-7$Fgr$!(^Sl#F-7$F\\s$!(1c#GF -7$Fas$!(Z\\#GF-7$Ffs$!($QCGF-7$F[t$!(/F$GF-7$F`t$!(vx*GF-7$Fet$!(!z#* HF-7$Fjt$!)r`&=$F^u7$F`u$!)(zrb$F^u7$Feu$!)&)[VUF^u7$Fju$!)H%o\"[F^u7$ F_v$!)#pt<&F^u7$Fdv$!)Y1x^F^u7$Fiv$!)qxw^F^u7$F^w$!)C=w^F^u7$Fcw$!)tcv ^F^u7$Fhw$!)CFu^F^u7$F]x$!)\"fG<&F^u7$Fgx$!)***Q>&F^u7$F[z$!)cP^gF^u7$ $\"5&z2$>f.>#QP*F-$!(*R`kF-7$F`z$!((R6qF-7$$\"5!e=mzl#p*\\P*F-$!(uwZ(F -7$Fez$!(7sZ(F-7$Fjz$!(TjZ(F-7$F_[l$!(faZ(F-7$Fd[l$!(3PZ(F-7$Fi[l$!(%G ruF-7$Fc\\l$!(\"3VuF-7$Fg]l$!(^rG(F-7$F[_l$!(QbH(F-7$Fe_l$!(B4H(F-7$Fj _l$!('zysF-7$F_`l$!(#)H@(F-7$Fd`l$!(OD5(F-7$Fi`l$!(rU'oF-7$$\"5qF<3aLx K3%*F-$!($>imF-7$F^al$!(nfP'F-7$$\"5qZ9'[eX,&4%*F-$!(G[(fF-7$Fcal$!(w \"=aF-7$Fhal$!(R)\\PF-7$F]bl$!'#f-)F-7$$\"55T![]Z)R6Ol`78%*F-$\"'NTgF-7$Fgbl$\"':TgF-7$F\\cl$\"'#3/'F -7$Facl$\"'_SgF-7$Ffcl$\"'ARgF-7$F[dl$\"'IVgF-7$F`dl$\"']NhF-7$Fedl$\" 'K!)pF-7$Fjdl$\"'YT')F-7$F_el$\"('oi7F-7$Fiel$\"(!Ga@F-7$Fcfl$\"(gX.%F -7$Fhfl$\")&\\NV\"F-7$F]gl$\")+8Nv#F-7$Fjil$ \")KI*=$F-7$F_jl$\")=A#3%F-7$Fdjl$\")k98[F-7$Fijl$\")&3J#eF-7$F^[m$\") L_gkF-7$Fc[m$\")#R$[oF-7$Fh[m$\")<8[oF-7$F]\\m$\")T#z%oF-7$Fb\\m$\")cV YoF-7$Fg\\m$\")7%\\%oF-7$F\\]m$\")dQVoF-7$Fa]m$\")\\MWoF-7$F[^m$\")fs] oF-7$Fe^m$\"*P?2(oF^u7$$\"5!=%R([&p/0j%*F-$\"*f66*oF^u7$Fj^m$\"*-YK#pF ^u7$$\"5IuRVw%*)fTY*F-$\"*^GE(pF^u7$F__m$\"*cho/(F^u7$Fd_m$\"*Z0RM(F^u 7$Fi_m$\"*lGo(zF^u7$F^`m$\"*M)o0&)F^u7$Fc`m$\"*gYZC*F^u7$Fh`m$\"*bVWr* F^u7$F]am$\",:*)fk-\"Faam7$Fcam$\",#)3924\"Faam7$Fham$\"3y=7Iu-bl6F]bm -%'COLOURG6&Fabm$\")#)eqk!\")$\"))eqk\"FdbpFebp-Fjbm6#%Ha~modification ~of~Verner's~Maple~schemeG-F$6%7cr7$F($\"'a$z&F-7$F/$\"'2`eF-7$F4$\"'g AoF-7$F9$\"'c!G*F-7$F>$\"'^w#*F-7$FC$\"'*3J*F-7$FH$\"(DK+\"F-7$FM$\"(, )e7F-7$FR$\"(C#e7F-7$FW$\"(w)e7F-7$Ffn$\"(3=G\"F-7$F[o$\"(p.M\"F-7$F`o $\"(J(R8F-7$Feo$\"(U\"Q8F-7$Fjo$\"(RZG\"F-7$F_p$\"'W=sF-7$Fdp$\"'\"[@( F-7$Fip$\"'9&>(F-7$F^q$\"'BNqF-7$Fcq$\"'X?gF-7$Fhq$\"'f&p%F-7$F]r$\"'W \\@F-7$Fbr$!'%pa#F-7$Fgr$!(204\"F-7$F\\s$!(@BI\"F-7$Fas$!(D?I\"F-7$Ffs $!(5QI\"F-7$F[t$!(&\\D8F-7$F`t$!(6KY\"F-7$Fet$!(Y-l\"F-7$Fjt$!)&oX,#F^ u7$F`u$!)Gl$p#F^u7$Feu$!)z#>\"RF^u7$Fju$!)Eo5\\F^u7$F_v$!)z#H`&F^u7$Fd v$!)DgKbF^u7$Fiv$!)VHKbF^u7$F^w$!)%e;`&F^u7$Fcw$!)L+JbF^u7$Fhw$!);wHbF ^u7$F]x$!)z,JbF^u7$Fbx$!)A[]bF^u7$Fgx$!)s$Rm&F^u7$F\\y$!)AGYeF^u7$Fay$ !)bYCiF^u7$Ffy$!)Y7npF^u7$F[z$!)dEf$)F^u7$Fbco$!(/fZ*F-7$F`z$!)GZ(4\"F -7$$\"5&)3HF$3<.ZP*F-$!)@:!>\"F-7$Fjco$!)]l>7F-7$$\"5!GYfEBo!Hv$*F-$!) ph>7F-7$Fez$!)6e>7F-7$Fjz$!)&Q%>7F-7$F_[l$!)\\H>7F-7$Fd[l$!)&>!>7F-7$F i[l$!)>')=7F-7$F^\\l$!)!=*>7F-7$Fc\\l$!)4RG7F-7$Fh\\l$!)\\'RC\"F-7$F]] l$!)pcy7F-7$$\"5]aonG#))o+R*F-$!)o=38F-7$Fb]l$!)f>]8F-7$$\"5!*eei>CQI \"R*F-$!)K$*39F-7$Fg]l$!)d)**[\"F-7$F\\^l$!)Ea:F-7$Ff^ l$!)syI>F-7$F[_l$!)DdI>F-7$F`_l$!)!=,$>F-7$Fe_l$!)krH>F-7$F_`l$!)-F -7$Fcal$!);Cp>F-7$Facl$!)`7!)>F-7$F[dl$!)T>z>F-7$F`dl$!)&)Hy>F-7$Fedl$ !)jgs>F-7$Fjdl$!)Dyg>F-7$F_el$!)*31$>F-7$Fdel$!)^?->F-7$Fiel$!)1>f=F-7 $F^fl$!)%*4&z\"F-7$Fcfl$!)8.,f$F-7$Fijl$\")\")R?]F-7$F^[m$ \")=$*=fF-7$Fc[m$\")hvkkF-7$Fh[m$\").ckkF-7$F]\\m$\")VOkkF-7$Fb\\m$\") %fHY'F-7$Fg\\m$\")wdhkF-7$F\\]m$\")$45Y'F-7$Fa]m$\")_pskF-7$Ff]m$\")$> F['F-7$F[^m$\")o<*\\'F-7$F`^m$\")qSDlF-7$Fe^m$\"*W!=mlF^u7$Fj^m$\"*#=p ?nF^u7$F__m$\"*7i?0(F^u7$Fd_m$\"*@]:z(F^u7$Fi_m$\"*8=bF*F^u7$F^`m$\"+D N@Z5F^u7$Fc`m$\"+%4)y57F^u7$Fh`m$\"+\"H5KJ\"F^u7$F]am$\",#>f.K9Faam7$F cam$\",'R`bp:Faam7$Fham$\"3y=G,zHLG$\"'!p.)F-7$FC$\"'me!)F-7$FH$\"'ck&)F- 7$FM$\"(e2/\"F-7$FR$\"(w-/\"F-7$FW$\"(+-/\"F-7$Ffn$\"(NJ/\"F-7$F[o$\"( $**>5F-7$F`o$\"(/&>5F-7$Feo$\"(Hw,\"F-7$Fjo$\"'()\\&*F-7$F_p$\"''3S$F- 7$Fdp$\"'7*R$F-7$Fip$\"'[\"Q$F-7$F^q$\"'uFKF-7$Fcq$\"'<^AF-7$Fhq$\"&^x *F-7$F]r$!'Zo9F-7$Fbr$!'AxfF-7$Fgr$!(@(*R\"F-7$F\\s$!(QGg\"F-7$Fas$!(r Cg\"F-7$Ffs$!(\"y.;F-7$F[t$!(EFi\"F-7$F`t$!(&[XF-7$Fjt $!)bTUAF^u7$F`u$!))4z&GF^u7$Feu$!)h-mRF^u7$Fju$!)lhw[F^u7$F_v$!)vfWaF^ u7$Fdv$!)tFWaF^u7$Fiv$!)U(RW&F^u7$F^w$!)'[LW&F^u7$Fcw$!)NqUaF^u7$Fhw$! )]XTaF^u7$F]x$!)T%))F-7$F`z$! )iy95F-7$Fa\\q$!)ip&4\"F-7$Fjco$!)BZ@6F-7$Fi\\q$!)rV@6F-7$Fez$!)US@6F- 7$Fjz$!)KF@6F-7$F_[l$!)69@6F-7$Fd[l$!)n)37\"F-7$Fi[l$!)Er?6F-7$F^\\l$! )MU@6F-7$Fc\\l$!)VuF6F-7$Fh\\l$!)FlR6F-7$F]]l$!)^Zm6F-7$Fi^q$!)qh*=\"F -7$Fb]l$!)')fA7F-7$Fa_q$!)[\"*o7F-7$Fg]l$!)!yIL\"F-7$F\\^l$!)C_7:F-7$F a^l$!)'RYo\"F-7$Ff^l$!)TX%o\"F-7$F[_l$!)nE%o\"F-7$F`_l$!)%pQo\"F-7$Fe_ l$!)B\\$o\"F-7$F_`l$!):*Qo\"F-7$Fcal$!)G_x;F-7$Facl$!)L^l:F-7$F[dl$!)f wk:F-7$F`dl$!)#oQc\"F-7$Fedl$!)Vkd:F-7$Fjdl$!)=!\\a\"F-7$F_el$!)z)G^\" F-7$Fdel$!)u/$[\"F-7$Fiel$!)?9Q9F-7$F^fl$!)rhr8F-7$Fcfl$!)b[u7F-7$Fhfl $!(6TK$F-7$F]gl$\"(Jb&fF-7$Fbgl$\"(QT&fF-7$Fggl$\"(tJ&fF-7$F\\hl$\"(^K 'fF-7$Fahl$\"(qn4'F-7$Ffhl$\"(>,M'F-7$F[il$\"(cc!pF-7$F`il$\"(nB8)F-7$ Feil$\")XMk5F-7$Fjil$\")kha;F-7$F_jl$\")y&>&GF-7$Fdjl$\")c#f#QF-7$Fijl $\")+Fm^F-7$F^[m$\")<$)4gF-7$Fc[m$\")hQAlF-7$Fh[m$\")&)=AlF-7$F]\\m$\" )2*>_'F-7$Fb\\m$\")Kd?lF-7$Fg\\m$\")I<>lF-7$F\\]m$\")!4%=lF-7$Fa]m$\") Q\"z_'F-7$Ff]m$\")-MOlF-7$F[^m$\")OH]lF-7$F`^m$\")UoslF-7$Fe^m$\"*y'o2 mF^u7$Fj^m$\"*:;8u'F^u7$F__m$\"*i(3IqF^u7$Fd_m$\"*mH(ywF^u7$Fi_m$\"*^= x)*)F^u7$F^`m$\"+%R\"p/5F^u7$Fc`m$\"+t2!)\\6F^u7$Fh`m$\"+y&)zS7F^u7$F] am$\",y*QZY8Faam7$Fcam$\",q!)*))o9Faam7$Fham$\"3y=+vnxN5;F]bm-F_bm6&Fa bmFebm$\"#vFcjn$\"#&*Fcjn-Fjbm6#%Ea~scheme~of~Tsitouras~and~Papakostas G-F$6%7er7$F($\"'v#>&F-7$F/$\"'jN_F-7$F4$\"'\"R)fF-7$F9$\"'mIzF-7$F>$ \"'4FzF-7$FC$\"'wZzF-7$FH$\"')\\V)F-7$FM$\"(<;-\"F-7$FR$\"(W6-\"F-7$FW $\"(05-\"F-7$Ffn$\"(EA-\"F-7$F[o$\"'5@**F-7$F`o$\"'N;**F-7$Feo$\"'O(*) *F-7$Fjo$\"'Jj#*F-7$F_p$\"'ysIF-7$Fdp$\"'?rIF-7$Fip$\"'v`IF-7$F^q$\"'g +HF-7$Fcq$\"'uF>F-7$Fhq$\"&1f'F-7$F]r$!'@xF-7$Fjt$!)M%3E#F^u7$F`u$!)AhqGF^u7$Feu$!)'*yoRF^u 7$Fju$!)TTr[F^u7$F_v$!)&)[MaF^u7$Fdv$!)*oTV&F^u7$Fiv$!)k'QV&F^u7$F^w$! )>CLaF^u7$Fcw$!)!)fKaF^u7$Fhw$!)&\\8V&F^u7$F]x$!)A-KaF^u7$Fbx$!)0LZaF^ u7$Fgx$!)R;SbF^u7$F\\y$!)az\"p&F^u7$Fay$!)%R%4gF^u7$Ffy$!)'*\\QmF^u7$F [z$!)(Hj#yF^u7$Fbco$!(\"e$y)F-7$F`z$!)SC25F-7$Fa\\q$!)46(3\"F-7$Fjco$! )ob76F-7$Fi\\q$!)=_76F-7$Fez$!)\"*[76F-7$Fjz$!)#fB6\"F-7$F_[l$!)#GA6\" F-7$Fd[l$!)c(>6\"F-7$Fi[l$!)-!=6\"F-7$F^\\l$!)6[76F-7$Fc\\l$!)[h=6F-7$ Fh\\l$!)T?I6F-7$F]]l$!)'[j:\"F-7$Fi^q$!)h#*y6F-7$Fb]l$!)5767F-7$Fa_q$! )YNc7F-7$Fg]l$!)d/>8F-7$F\\^l$!)7Z%\\\"F-7$Fa^l$!)<\"Gm\"F-7$Ff^l$!)'G Em\"F-7$F[_l$!)PWi;F-7$F`_l$!):0i;F-7$Fe_l$!)mnh;F-7$F_`l$!)'**=m\"F-7 $Fcal$!)P#=l\"F-7$F]bl$!);Rk:F-7$Facl$!)cSH:F-7$Ffcl$!)A0H:F-7$F[dl$!) UnG:F-7$F`dl$!)nxF:F-7$Fedl$!)+^@:F-7$Fjdl$!)Xp3:F-7$F_el$!)!QlZ\"F-7$ Fdel$!)')eY9F-7$Fiel$!)Ha,9F-7$F^fl$!)?%[L\"F-7$Fcfl$!)e\\P7F-7$Fhfl$! (0I%HF-7$F]gl$\"(o4M'F-7$Fbgl$\"(&[RjF-7$Fggl$\"(E%QjF-7$F\\hl$\"(=$[j F-7$Fahl$\"(C2['F-7$Ffhl$\"(vAs'F-7$F[il$\"(JQG(F-7$F`il$\"(6A])F-7$Fe il$\")In*4\"F-7$Fjil$\")x:'o\"F-7$F_jl$\")x+wGF-7$Fdjl$\")'zR%QF-7$Fij l$\")&fh<&F-7$F^[m$\")3i9gF-7$Fc[m$\")r3ClF-7$Fh[m$\")&*)Q_'F-7$F]\\m$ \")$ \"'$4E*F-7$FC$\"'@(H*F-7$FH$\"(rY+\"F-7$FM$\"(Z!p7F-7$FR$\"(n%o7F-7$FW $\"(u#p7F-7$Ffn$\"(WjH\"F-7$F[o$\"((>t8F-7$F`o$\"(WDP\"F-7$Feo$\"(z5P \"F-7$Fjo$\"(.=K\"F-7$F_p$\"'H:zF-7$Fdp$\"'K6zF-7$Fip$\"'$=*yF-7$F^q$ \"'\"ft(F-7$Fcq$\"'%ou'F-7$Fhq$\"'MbaF-7$F]r$\"'*H(HF-7$Fbr$!';1;F-7$F gr$!'Fc(*F-7$F\\s$!('>#=\"F-7$Fas$!(H>=\"F-7$Ffs$!(pP=\"F-7$F[t$!(z`? \"F-7$F`t$!(=?M\"F-7$Fet$!(es_\"F-7$Fjt$!)Uq()=F^u7$F`u$!)/%*eDF^u7$Fe u$!);:iPF^u7$Fju$!)],[ZF^u7$F_v$!)?1i`F^u7$Fdv$!)nuh`F^u7$Fiv$!)%[9O&F ^u7$F^w$!)B$3O&F^u7$Fcw$!)s>g`F^u7$Fhw$!)++f`F^u7$F]x$!)`Mg`F^u7$Fbx$! )o2!Q&F^u7$Fgx$!)%zU\\&F^u7$F\\y$!)AGxcF^u7$Fay$!)[7cgF^u7$Ffy$!)g*))z 'F^u7$F[z$!)HK*=)F^u7$Fbco$!('e.$*F-7$F`z$!)Q$)z5F-7$$\"5Sqi#fHHcXP*F- $!)\")RC6F-7$Fa\\q$!)NAs6F-7$$\"5IZ&>1([+&[P*F-$!)mk,7F-7$Fjco$!)$H;? \"F-7$Fi\\q$!)=f,7F-7$Fez$!)lb,7F-7$Fjz$!)hT,7F-7$F_[l$!)YF,7F-7$Fd[l$ !)O+,7F-7$Fi[l$!)a&3?\"F-7$F^\\l$!)(f>?\"F-7$Fc\\l$!)*p1@\"F-7$Fh\\l$! )0hE7F-7$F]]l$!):%>E\"F-7$Fi^q$!)I9#H\"F-7$Fb]l$!)-%\\L\"F-7$Fa_q$!)Kt %R\"F-7$Fg]l$!)e=x9F-7$F\\^l$!)?W1F-7$Ff^l$!)r1D>F-7$ F[_l$!)J&[#>F-7$F`_l$!),SC>F-7$Fe_l$!)i+C>F-7$F_`l$!)0)o#>F-7$Fcal$!)0 ^t>F-7$Facl$!)z6:?F-7$F[dl$!)I<9?F-7$F`dl$!)$3L,#F-7$Fedl$!)G%z+#F-7$F jdl$!)_v'*>F-7$F_el$!)l3o>F-7$Fdel$!)'H5%>F-7$Fiel$!)v)***=F-7$F^fl$!) [uQ=F-7$Fcfl$!)yt[F-7$Fbgl$\"'$3(>F-7 $Fggl$\"',v>F-7$F\\hl$\"'<)4#F-7$Fahl$\"'bONF-7$Ffhl$\"'0DhF-7$F[il$\" (G7@\"F-7$F`il$\"(%\\0DF-7$Feil$\"(Ww9&F-7$Fjil$\")5JM6F-7$F_jl$\")zB) Q#F-7$Fdjl$\")%*z1MF-7$Fijl$\")AB2[F-7$F^[m$\")C0)o&F-7$Fc[m$\")G3BiF- 7$Fh[m$\")V*GA'F-7$F]\\m$\")cqAiF-7$Fb\\m$\")KN@iF-7$Fg\\m$\")Y-?iF-7$ F\\]m$\")]_>iF-7$Fa]m$\")dMJiF-7$Ff]m$\")zTTiF-7$F[^m$\")`#zD'F-7$F`^m $\")!)>%G'F-7$Fe^m$\"*c*)\\K'F^u7$Fj^m$\"**)H$zkF^u7$F__m$\"*O0)4oF^u7 $Fd_m$\"*(>GYvF^u7$Fi_m$\"*SbC-*F^u7$F^`m$\"+y9>@5F^u7$Fc`m$\"+#47P=\" F^u7$Fh`m$\"+r?W&G\"F^u7$F]am$\",VNSMS\"Faam7$Fcam$\",ILv*R:Faam7$Fham $\"3y=<#Q_'e(p\"F]bm-F_bm6&FabmF]as$\"#bFcjnFebm-Fjbm6#%AVerner's~\"mo st~efficient\"~schemeG-%%FONTG6$%*HELVETICAGFhbm-%&TITLEG6#%Verror~cur ves~for~12~stage~order~8~Runge-Kutta~methodsG-%+AXESLABELSG6$Q\"x6\"Q! Fj`v-%%VIEWG6$;F(Fham%(DEFAULTG" 1 2 0 1 10 0 2 9 1 4 2 1.000000 45.000000 45.000000 0 1 "Enright-Verner scheme" "Verner's Maple scheme " "a modification of Verner's Maple scheme" "an \"efficient\" scheme" "a scheme of Tsitouras and Papakostas" "a modification of the Tsitoura s-Papakostas scheme" "Verner's \"most efficient\" scheme" }}}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }} }}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 50 "Test 12 of 12 stage, order 8 Ru nge-Kutta methods " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 " " {TEXT -1 0 "" }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "dy/ dx=exp(-x)/(x-1)^2" "6#/*&%#dyG\"\"\"%#dxG!\"\"*&-%$expG6#,$%\"xGF(F&* $,&F.F&F&F(\"\"#F(" }{TEXT -1 2 " " }{XPPEDIT 18 0 "5*y*sin^7*7*x;" " 6#*,\"\"&\"\"\"%\"yGF%%$sinG\"\"(F(F%%\"xGF%" }{TEXT -1 5 ", " } {XPPEDIT 18 0 "y(0) = 1;" "6#/-%\"yG6#\"\"!\"\"\"" }{TEXT -1 1 " " }} {PARA 0 "" 0 "" {TEXT -1 10 "Solution: " }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "y = exp(16/49+5/3136*cos*49*x-cos*35*x/64+5/64*c os*21*x-25/64*cos*7*x);" "6#/%\"yG-%$expG6#,,*&\"#;\"\"\"\"#\\!\"\"F+* ,\"\"&F+\"%OJF-%$cosGF+F,F+%\"xGF+F+**F1F+\"#NF+F2F+\"#kF-F-*,F/F+F5F- F1F+\"#@F+F2F+F+*,\"#DF+F5F-F1F+\"\"(F+F2F+F-" }{TEXT -1 2 " " }} {PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 213 "de := diff(y(x),x)=5*y(x)*sin(7*x)^7;\nic := y(0)=1;\ndsolve(\{de ,ic\},y(x)):\ny(x)=combine((numer(rhs(%))/convert(denom(rhs(%)),exp))) ;\nr := unapply(rhs(%),x):\nplot(r(x),x=0..5,font=[HELVETICA,9],labels =[`x`,`y(x)`]);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%#deG/-%%diffG6$-% \"yG6#%\"xGF,,$*(\"\"&\"\"\"F)F0)-%$sinG6#,$*&\"\"(F0F,F0F0F7F0F0" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#>%#icG/-%\"yG6#\"\"!\"\"\"" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/-%\"yG6#%\"xG-%$expG6#,,*&#\"#D\"#k\"\"\"-% $cosG6#,$*&\"\"(F0F'F0F0F0!\"\"*&#F0F/F0-F26#,$*&\"#NF0F'F0F0F0F7*&#\" \"&\"%OJF0-F26#,$*&\"#\\F0F'F0F0F0F0*&#FAF/F0-F26#,$*&\"#@F0F'F0F0F0F0 #\"#;FGF0" }}{PARA 13 "" 1 "" {GLPLOT2D 806 286 286 {PLOTDATA 2 "6&-%' CURVESG6$7_^l7$$\"\"!F)$\"\"\"F)7$$\"3ALL$3FWYs#!#>$\"3RX36^,++5!#<7$$ \"3WmmmT&)G\\aF/$\"3/h:lL\\.+5F27$$\"3MKL3x1h6oF/$\"3N>!>$zG>+5F27$$\" 3m****\\7G$R<)F/$\"3^$*H^L`v+5F27$$\"3±z%\\DO&*F/$\"3J?*f5+AB+\"F27 $$\"3GLLL3x&)*3\"!#=$\"3x]lWM_&f+\"F27$$\"3em\"z%\\v#pK\"FJ$\"3$\\ClDT `A-\"F27$$\"3))**\\i!R(*Rc\"FJ$\"3%z>L#y^ah5F27$$\"3_;z>6B`#o\"FJ$\"3c ]f\"H$G$Q4\"F27$$\"3!**HO6F27$$\"34]PM_@g>>FJ$\"33 RW!o::'*=\"F27$$\"3umm\"H2P\"Q?FJ$\"3?q+*)GVj`7F27$$\"3C]7G))>Wr@FJ$\" 3L$)HfnG\"pL\"F27$$\"3YLek.pu/BFJ$\"3Gv`o46\")G9F27$$\"33D\"G8O*RrBFJ$ \"3+PI&HSekZ\"F27$$\"3o;/,>=0QCFJ$\"3mjV0*4cU_\"F27$$\"3c3FpwUq/DFJ$\" 3#pc5I%eTr:F27$$\"3!***\\PMnNrDFJ$\"3U+AysC:<;F27$$\"37$eR(\\;m/FFJ$\" 3)f:eA`w9q\"F27$$\"3MmT5ll'z$GFJ$\"35TSzD!3Dx\"F27$$\"37](o/[r7(HFJ$\" 3e:[6>4vF=F27$$\"3MLL$eRwX5$FJ$\"3ICf`%)[>n=F27$$\"3_LLe*[`HP$FJ$\"3!e Z8'p.63>F27$$\"3rLLL$eI8k$FJ$\"3%[F&)3\"\\`>>F27$$\"3_L$3-8>bx$FJ$\"3I prJEM*3#>F27$$\"3*QL$3xwq4RFJ$\"3rrv?(f28#>F27$$\"3EM$eRA'*Q/%FJ$\"3X^ 'fP/+9#>F27$$\"33ML$3x%3yTFJ$\"3cOdy3HT@>F27$$\"3h+]PfyG7ZFJ$\"3VH&=@l 89#>F27$$\"3emm\"z%4\\Y_FJ$\"3ml4uC:e?>F27$$\"3C+]P4'4.P&FJ$\"3`*yZBqt )=>F27$$\"3'QLL3FGT\\&FJ$\"3t^RtM))*[\">F27$$\"3Um;HKp%zh&FJ$\"3!*H'3: A\\o!>F27$$\"32++v$flV;F27$$\"3I++vVVX$\\'FJ$\"3s/S7/D%eb\"F27$$\"3pLL$eRh;i'FJ$\"3)z%[YI ??k9F27$$\"31nm\"zWo)\\nFJ$\"31\")Q9\"48QP\"F27$$\"3W++++b2yoFJ$\"3k#R e\"fbk*G\"F27$$\"3%QL$3_DG1qFJ$\"3jY.gupS:7F27$$\"3Anm;/'*[MrFJ$\"3y@x $3A]K:\"F27$$\"3]***\\il'pisFJ$\"3_zleRG$Q5\"F27$$\"3+MLe*[!)y_(FJ$\"3 :2l<5ljQ5F27$$\"3Qnm\"HKkIz(FJ$\"3z[=lFuU55F27$$\"35MLeRilDzFJ$\"3tR>* >I/Y+\"F27$$\"3!3+]i:[#e!)FJ$\"3ZybuD9w,5F27$$\"3[nm\"H2S3>)FJ$\"36f+O Z/c+5F27$$\"3>MLe*)>VB$)FJ$\"3q[OmV#R,+\"F27$$\"3wmmTg()4_))FJ$\"3!z[$ G++++5F27$$\"3Y++DJbw!Q*FJ$\"3(f!Q*>W.++\"F27$$\"3E$3_+A:n^*FJ$\"3$o[9 !4H.+5F27$$\"3=nT&)3\\m_'*FJ$\"3qlL!>[$=+5F27$$\"33^il(f9')y*FJ$\"39t! )=KMs+5F27$$\"3+N$ekGkX#**FJ$\"3)[%4Vj`B-5F27$$\"31]iSmjk>5F2$\"3?UoC! *)fG,\"F27$$\"3%ommTIOo/\"F2$\"3%pJ)[*fok/\"F27$$\"3?]i:g2\")e5F2$\"3' ob!*>J(\\t5F27$$\"3cLe9;_yq5F2$\"3!f)3IJ$>-6\"F27$$\"3!pTN@nfF3\"F2$\" 3O\\W6vrpd6F27$$\"3E+]7GTt%4\"F2$\"3&p\"fsjqH;7F27$$\"3i$e9Te3n5\"F2$ \"3e&f$*[!\\V&G\"F27$$\"3(p;/,/$o=6F2$\"3K1G\">I$Qj8F27$$\"3L]P4'\\d18 \"F2$\"3mAQZ9DAZ9F27$$\"3YLL3_>jU6F2$\"3A//J![AI`\"F27$$\"3)\\PMF:s$\\ 6F2$\"3%**=P`q)\\!e\"F27$$\"3u;aQ`B6c6F2$\"3vn0oImOE;F27$$\"3[ek.aD&G; \"F2$\"3q#GT!4W()p;F27$$\"3C+voaFfp6F2$\"3Hrm)p6m.r\"F27$$\"3]$e*)f:tI =\"F2$\"3C=5u0pN!y\"F27$$\"3ym;HdNb'>\"F2$\"3oD;8U'oS$=F27$$\"3NLe*)fV ^B7F2$\"3Q\\+g\"\\,f*=F27$$\"37++]i^Z]7F2$\"3!y&=TMDx;>F27$$\"37+]i:4, k7F2$\"34bF!Hpw)>>F27$$\"37++vomax7F2$\"3s-.5q=,@>F27$$\"35+](=U#3\"H \"F2$\"3EfS@r$Q8#>F27$$\"35+++v\"=YI\"F2$\"3=\"f$>R_S@>F27$$\"34++D\"o *oJ8F2$\"36A3*=r89#>F27$$\"33++](=h(e8F2$\"3#z)y\\8PT@>F27$$\"3!****** **f\\[Q\"F2$\"34cw3'H49#>F27$$\"3&*****\\7!Q4T\"F2$\"3n&=!o'on6#>F27$$ \"3(****\\(=A)RU\"F2$\"3bRtq7'G/#>F27$$\"3*******\\UEqV\"F2$\"3yBj&H)z K=>F27$$\"3-++DJ12]9F2$\"32T7QjDP8>F27$$\"3/++]P[6j9F2$\"3+dVO')[G.>F2 7$$\"3W$ek`h0o[\"F2$\"3I$zM;9\\?'=F27$$\"3%o;HKR'\\5:F2$\"3-heuX3wwd\"F2$\"3yrf8;r3y8F 27$$\"3#f*[o=f+z:F2$\"3lUh\"F2$\"3=x^]S+%)R6F27$ $\"3V$eky#)*QU;F2$\"3k!o/)p1f`5F27$$\"3wmm;a/cq;F2$\"3nOMw%)=y95F27$$ \"3im\"z>c#\\#o\"F2$\"3VZh#ea.v+\"F27$$\"3sm;zpYU%p\"F2$\"3He\"pN-SM+ \"F27$$\"3#o;/wxcjq\"F2$\"3/ne\\$[\"R,5F27$$\"3\"pm;a)))G=F2$\"3:l7r\" p]]+\"F27$$\"3%\\ilAZ9R\">F2$\"3^hCaoQo55F27$$\"3KLe9;0?E>F2$\"3:4Zh8k [?5F27$$\"31]i!Rgs2&>F2$\"3wO#Q8N_'f5F27$$\"3gmmm\"pW`(>F2$\"3,$4k)\\, ZO6F27$$\"3_ek.HW#)))>F2$\"3Q;r;:R*z>\"F27$$\"3?]iSmTI-?F2$\"3_fyny')> t7F27$$\"3EY64NS/4?F2$\"3+2!G,Td`J\"F27$$\"3*=/wP!Ry:?F2$\"3/M$G\"4m+g 8F27$$\"3^P4YsP_A?F2$\"3$R*>L\"f'e19F27$$\"3dLe9TOEH?F2$\"3%\\u#Gw;Va9 F27$$\"3kH2$)4N+O?F2$\"3\\;?B!3#z-:F27$$\"3EDc^yLuU?F2$\"37f,s,Z'3b\"F 27$$\"3)3_+sC$[\\?F2$\"3C:ye37$yf\"F27$$\"3'pT&)e6Bi0#F2$\"3VLP%*y0!Hk \"F27$$\"3k3_D`Gqp?F2$\"3;-/UsbdC+v$fQa)3@F2$\"3_4a5;!3X(=F27$$\"3/++D\"=EX8#F2$\"3XR)Rsy(o4 >F27$$\"3?]i!*y?OZ@F2$\"3#=X-%G8_;>F27$$\"3M+Dcwz>g@F2$\"3\"HQ**f-s'>> F27$$\"3/](=U(Q.t@F2$\"3jDRpm*34#>F27$$\"3?+](=xpe=#F2$\"3.\"p)[NLI@>F 27$$\"3mLeRA9WRAF2$\"3HST.9PT@>F27$$\"37nm\"H28IH#F2$\"3`7Ny#=%Q@>F27$ $\"3=Ug_Z>J0BF2$\"3F27$$\"3!oTN@#3hF27$$\" 3(=zWnp4*HBF2$\"3aFU&R3A$>>F27$$\"3$p;a8d3AM#F2$\"3/F6dAB#f\">F27$$\"3 1gc(z*f*=F27$$\"3um;zpSS\"R#F2$\"3->e1#==?%=F27$$\"3 uC\"yv1qYS#F2$\"3[KdPD&>Az\"F27$$\"3;$ek`1OzT#F2$\"3m9DqP!=ls\"F27$$\" 3fT5:j??JCF2$\"3!\\()pJuklk\"F27$$\"3-+v$41oWW#F2$\"3y%HsO_ejb\"F27$$ \"3CH2$)f55^CF2$\"3(>FHM)z84:F27$$\"3WeRseStdCF2$\"3/?hWfY`h9F27$$\"3m (=9\"F27$$\"3.$3-)Q84DDF2$\"3St1.wX6#4\"F27$$ \"37e9;#)4()QDF2$\"3A!)Gqgm(f0\"F27$$\"3AL3_D1l_DF2$\"3/w0[YW]J5F27$$ \"3I3-))o-VmDF2$\"3hGW>'Q(=;5F27$$\"3S$eRA\"*4-e#F2$\"38vW*f>\\u+\"F27 $$\"3%4F>Rt*4(e#F2$\"34I&o$G5\"[+\"F27$$\"3[e*)fb&*)Rf#F2$\"3*Rd232\"* H+\"F27$$\"3/Y'ysPz3g#F2$\"3EN#z,7\"y,5F27$$\"3fL$e*)>pxg#F2$\"3cLdJH' 45+\"F27$$\"3%omm\"z+vbEF2$\"3VA\\(Rr,++\"F27$$\"33+]Pf4t.FF2$\"3EB[5+ +++5F27$$\"3Q$3F>HT'HFF2$\"3PG;$Qk,++\"F27$$\"3om\"zWi^bv#F2$\"3tr+Z;M 5+5F27$$\"3M3_v!z1&oFF2$\"33DW)H[F/+\"F27$$\"3)*\\7.d>Y\"y#F2$\"3?pw0F @P,5F27$$\"3#3Fp,aRzy#F2$\"3V/:%[T%G-5F27$$\"3k\"H2L7ST$GF2$\"3>]6Z^*QK.\"F27$$\"3)om\"H2\"34'GF2$\"3c89;GxR $4\"F27$$\"3=](=<1#HuGF2$\"3H$eWMp\"*>9\"F27$$\"3YLe9;gn()GF2$\"3LgFbL %HW?\"F27$$\"3w;Hdq*f5!HF2$\"337;\"\\lV-G\"F27$$\"30+++DRW9HF2$\"3/%p0 (oZBn8F27$$\"3-D19>2*4#HF2$\"3.aKDn)*p79F27$$\"3X]7G8v`FHF2$\"39c[>!pr #f9F27$$\"3Uv=U2V3MHF2$\"3FI^JzcD1:F27$$\"3S+Dc,6jSHF2$\"3)fI*>\"*G\"H b\"F27$$\"3O]P%)*oCP&HF2$\"3kq@\"z(y\"F27$$\"3C+voaa+$*HF2$\"3](oH/*41IF2$\"3/1YYU*QP(=F27$$\"3:++DJE>>IF2$\"3w5uUgAb'* =F27$$\"3Fv=<^8'=.$F2$\"3=[IQQI\\4>F27$$\"3S]P4r+`WIF2$\"3bz\"Rx1kj\"> F27$$\"33Dc,\"z)>dIF2$\"3\")Hman(y&>>F27$$\"3A+v$4^n)pIF2$\"3P!f*z`l'3 #>F27$$\"3-]7y]\\?&4$F2$\"3W+7R*\\%R@>F27$$\"3F+]i!RU07$F2$\"3Lg5nwOT@ >F27$$\"39+vo/#3o<$F2$\"3oIo*Q]69#>F27$$\"3+++v=S2LKF2$\"39dX:D?6=>F27 $$\"3Om\"zpo_$eKF2$\"3#zYTO,TK!>F27$$\"3;L$3_NJOG$F2$\"3=J`$Q^]x&=F27$ $\"3y;HK*oqiH$F2$\"3h#pBbd4n\"=F27$$\"3'**\\PM-5*3LF2$\"3uG&fHPn7w\"F2 7$$\"39$3_vN\\:K$F2$\"3Y(o1g2R=p\"F27$$\"3Jmmm\"p)=MLF2$\"3896'o\"f'3h \"F27$$\"3&)\\i:&GO4M$F2$\"3=*)4prhIk:F27$$\"3%H$ekyQoZLF2$\"3IceZ)o3k ^\"F27$$\"3/;a8s9VaLF2$\"3G6b_$H\"*zY\"F27$$\"3e**\\il!z6O$F2$\"3q?/^g #o)>9F27$$\"35$e9\"fm#zO$F2$\"3#o/$41/\"GP\"F27$$\"3kmTg_UnuLF2$\"3\\t !fI&=^F8F27$$\"3=]P4Y=U\"Q$F2$\"3]X:oIac%G\"F27$$\"3GLLeR%p\")Q$F2$\"3 F&[%Q!RZWC\"F27$$\"3!**\\ilik;S$F2$\"3_Z%zlm()R<\"F27$$\"3)pmTN\")f^T$ F2$\"331)eDsVv6\"F27$$\"3hL3_+]lGMF2$\"3CD\\786*[2\"F27$$\"3B++](=]@W$ F2$\"3.\">ZcV5Y/\"F27$$\"3C$ekyZ2mY$F2$\"3bMcTbf795F27$$\"3mm\"H#oZ1\" \\$F2$\"38K^#3O0J+\"F27$$\"35+5F27$$\"35L$e*[$z*RNF2$\" 3:J>88\">++\"F27$$\"3%o;Hd!fX$f$F2$\"3+/++++++5F27$$\"3e++]iC$pk$F2$\" 3>3i#3WY++\"F27$$\"3)*e*)f!y6&fOF2$\"3/sYmoa@+5F27$$\"3%p\"zp)4\"4sOF2 $\"3/b\">zh_2+\"F27$$\"3!\\(oz;/n%o$F2$\"3FA+oZ!R@+\"F27$$\"3ILe*[t\\s p$F2$\"3]TEK8J>05F27$$\"36]P4r$3Cs$F2$\"3cyY&>9K:-\"F27$$\"3[m;H2qcZPF 2$\"3yA^kd)oL1\"F27$$\"3KL3F>grgPF2$\"3kx1i!p\"\\+6F27$$\"3s***\\7.lQx $F2$\"3WMY`\\)G.:\"F27$$\"3em\"HK/9qy$F2$\"3j!GcKQeN@\"F27$$\"3UL$3_0j ,!QF2$\"3%*G.x'3G&*G\"F27$$\"3i;z>hvt1QF2$\"3uM!o>\"3kJ8F27$$\"3F+v=n? J8QF2$\"3yKp#[4RfP\"F27$$\"3\"R3xJd'))>QF2$\"3/CVy4j'=U\"F27$$\"36nm;z 5YEQF2$\"3%GUA'*pv(o9F27$$\"3J]i:&eNI$QF2$\"3eJ%\\YU_f^\"F27$$\"3^Le9 \"45'RQF2$\"3][!)RoHki:F27$$\"3s;a8(f%=YQF2$\"3Zk-ND-43;F27$$\"3O+]7. \"fF&QF2$\"3]5UW.;d^;F27$$\"3i3_vlYhlQF2$\"3S)o\\Z'>`GF27$$\"3Ymm;/O gbRF2$\"3e5rkQQq>>F27$$\"3*G$e*[$zV4SF2$\"3^Ar&3889#>F27$$\"3w**\\ilAF jSF2$\"3Q')**40OT@>F27$$\"3#G3_]p'>*3%F2$\"3!o$)[rAz8#>F27$$\"3ym\"zW7 @^6%F2$\"3371C6\\d?>F27$$\"3@Qf$=B-;7%F2$\"39!e*f[J()>>F27$$\"3w3F>RL3 GTF2$\"3A(\\vy/K(=>F27$$\"3Iz%\\lWkX8%F2$\"38-([etep\">F27$$\"3t]i!RbX 59%F2$\"38M:lS7J9>F27$$\"3#=z>'ox+aTF2$\"3&>*GfbB=0>F27$$\"3yLLL$)*pp; %F2$\"3C=Lu(zB&))=F27$$\"3!Q3_+sD-=%F2$\"3ug(ptnA/'=F27$$\"3#Q$3xc9[$> %F2$\"3:,22Ku<==F27$$\"3'Qe*[$>Pn?%F2$\"3i6,,)=G,w\"F27$$\"3)QL3-$H**> UF2$\"33wUWwLu'o\"F27$$\"3X3xc)z?mA%F2$\"3/e\"zYY>^k\"F27$$\"3\"R3Fpm[ KB%F2$\"3x%eAK2J4g\"F27$$\"3OfkGNl()RUF2$\"3/aZW5%e[b\"F27$$\"3#R$ek.W ]YUF2$\"3_pHqN-l2:F27$$\"3]3_+sA8`UF2$\"3?puckU3g9F27$$\"3'Rek.9g(fUF2 $\"30z2N6y#HT\"F27$$\"3SfRs3!)QmUF2$\"3E%zqA:-pO\"F27$$\"3)RL$3xe,tUF2 $\"3Hzu(eD`EK\"F27$$\"3kv$4'\\=;'G%F2$\"32(*p'*eu>U7F27$$\"3G#[Z %F2$\"3S)\\h/++++\"F27$$\"3TM$3_5,-`%F2$\"3y/++\"F27$$\"3SnmT&G!e &e%F2$\"3$eA?a3RF+\"F27$$\"3fLe*[=Y.h%F2$\"3M:+/vy285F27$$\"3m+]P%37^j %F2$\"356M^f6but%F2$\"3%\\.QNQyVc\"F27$$\"3ID19>zl]ZF2$\"3S[]>-K'z\"F27$$\"37+]i SjE!z%F2$\"3&)4L&Q*oyW=F27$$\"3y*\\7G))Rb\"[F2$\"3)fCVDB`!)*=F27$$\"3L +++DM\"3%[F2$\"3$Gxb\\1an\">F27$$\"3i]P4'>]M&[F2$\"3p-qe54u>>F27$$\"3) 3](=np3m[F2$\"3zS9J5F#4#>F27$$\"3G]7GQPsy[F2$\"3j:7^7UI@>F27$$\"3a+]P4 0O\"*[F2$\"3L:$**HR(R@>F27$$\"3s+voa-oX\\F2$\"3%e\"e*Rr89#>F27$$\"\"&F )$\"3n\\kX'z.7#>F2-%'COLOURG6&%$RGBG$\"#5!\"\"F(F(-%%FONTG6$%*HELVETIC AG\"\"*-%+AXESLABELSG6$%\"xG%%y(x)G-%%VIEWG6$;F(F_[q%(DEFAULTG" 1 2 0 1 10 0 2 9 1 4 2 1.000000 45.000000 45.000000 0 0 "Curve 1" }}}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 32 "The following c ode constructs a " }{TEXT 260 17 "discrete solution" }{TEXT -1 44 " ba sed on each of the methods and gives the " }{TEXT 260 22 "root mean sq uare error" }{TEXT -1 18 " of each solution." }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 792 "R := (x,y) -> 5*y*sin(7*x)^7: hh := 0.01: numstep s := 500: x0 := 0: y0 := 1:\nmatrix([[`slope field: `,R(x,y)],[`init ial point: `,``(x0,y0)],[`step width: `,hh],\n[`no. of steps: `,nu msteps]]);``;\nmthds := [`Enright-Verner scheme`,`Verner's Maple schem e`,`a modification of Verner's Maple scheme`,`an \"efficient\" scheme` ,`a scheme of Tsitouras and Papakostas`,`a modification of the Tsitour as-Papakostas scheme`,`Verner's \"most efficient\" scheme`]: errs := [ ]:\nDigits := 30:\nfor ct to 7 do\n Rn_RK8_||ct := RK8_||ct(R(x,y),x ,y,x0,y0,hh,numsteps,false);\n sm := 0: numpts := nops(Rn_RK8_||ct): \n for ii to numpts do\n sm := sm+(Rn_RK8_||ct[ii,2]-r(Rn_RK8_| |ct[ii,1]))^2;\n end do:\n errs := [op(errs),sqrt(sm/numpts)];\nen d do:\nDigits := 10:\nlinalg[transpose]([mthds,evalf(errs)]);" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#K%'matrixG6#7&7$%0slope~field:~~~G,$*( \"\"&\"\"\"%\"yGF,)-%$sinG6#,$*&\"\"(F,%\"xGF,F,F4F,F,7$%0initial~poin t:~G-%!G6$\"\"!F,7$%/step~width:~~~G$F,!\"#7$%1no.~of~steps:~~~G\"$+&Q )pprint566\"" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%!G" }}{PARA 11 "" 1 " " {XPPMATH 20 "6#K%'matrixG6#7)7$%6Enright-Verner~schemeG$\"+FE\"fh#!# B7$%6Verner's~Maple~schemeG$\"+8b>bD!#C7$%Ha~modification~of~Verner's~ Maple~schemeG$\"+&G\"4&3#F07$%6an~\"efficient\"~schemeG$\"+Ys&=X$F07$% Ea~scheme~of~Tsitouras~and~PapakostasG$\"+.Rg\\EF07$%Ra~modification~o f~the~Tsitouras-Papakostas~schemeG$\"+jNpN=F07$%AVerner's~\"most~effic ient\"~schemeG$\"+=fk3AF0Q)pprint576\"" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 30 "The following code constructs " } {TEXT 260 20 "numerical procedures" }{TEXT -1 56 " for solutions based on each of the Runge-Kutta schemes." }}{PARA 0 "" 0 "" {TEXT -1 75 "T he error in the value obtained by each of the methods at the point whe re " }{XPPEDIT 18 0 "x = 4.999;" "6#/%\"xG-%&FloatG6$\"%**\\!\"$" } {TEXT -1 16 " is also given." }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 723 "R := (x,y) -> 5*y*sin(7*x)^7: hh := 0.01: numsteps := 500: x0 : = 0: y0 := 1:\nmatrix([[`slope field: `,R(x,y)],[`initial point: `,` `(x0,y0)],[`step width: `,hh],\n[`no. of steps: `,numsteps]]);``; \nmthds := [`Enright-Verner scheme`,`Verner's Maple scheme`,`a modific ation of Verner's Maple scheme`,`an \"efficient\" scheme`,`a scheme of Tsitouras and Papakostas`,`a modification of the Tsitouras-Papakostas scheme`,`Verner's \"most efficient\" scheme`]: errs := []:\nDigits := 25:\nfor ct to 7 do\n rn_RK8_||ct := RK8_||ct(R(x,y),x,y,x0,y0,hh,n umsteps,true);\nend do:\nxx := 4.999: rxx := evalf(r(xx)):\nfor ct to \+ 7 do\n errs := [op(errs),abs(rn_RK8_||ct(xx)-rxx)];\nend do:\nDigits := 10:\nlinalg[transpose]([mthds,evalf(errs)]);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#K%'matrixG6#7&7$%0slope~field:~~~G,$*(\"\"&\"\"\"%\"yGF ,)-%$sinG6#,$*&\"\"(F,%\"xGF,F,F4F,F,7$%0initial~point:~G-%!G6$\"\"!F, 7$%/step~width:~~~G$F,!\"#7$%1no.~of~steps:~~~G\"$+&Q)pprint586\"" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#%!G" }}{PARA 11 "" 1 "" {XPPMATH 20 "6 #K%'matrixG6#7)7$%6Enright-Verner~schemeG$\"+j#GB@&!#B7$%6Verner's~Map le~schemeG$\"+`\"R/f#!#C7$%Ha~modification~of~Verner's~Maple~schemeG$ \"+\"H\"38XF07$%6an~\"efficient\"~schemeG$\"+mE1MvF07$%Ea~scheme~of~Ts itouras~and~PapakostasG$\"+cD4GbF07$%Ra~modification~of~the~Tsitouras- Papakostas~schemeG$\"+Y(y$=RF07$%AVerner's~\"most~efficient\"~schemeG$ \"+si\"=y%F0Q)pprint596\"" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 4 "The " }{TEXT 260 22 "root mean square error" } {TEXT -1 20 " over the interval " }{XPPEDIT 18 0 "[0, 5];" "6#7$\"\"! \"\"&" }{TEXT -1 82 " of each Runge-Kutta method is estimated as foll ows using the special procedure " }{TEXT 0 5 "NCint" }{TEXT -1 98 " \+ to perform numerical integration by the 7 point Newton-Cotes method ov er 200 equal subintervals." }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 476 "mthds := [`Enright-Verner scheme`,`Verner's Maple scheme`,`a modi fication of Verner's Maple scheme`,`an \"efficient\" scheme`,`a scheme of Tsitouras and Papakostas`,`a modification of the Tsitouras-Papakos tas scheme`,`Verner's \"most efficient\" scheme`]: errs := []:\nDigits := 20:\nfor ct to 7 do\n sm := NCint((r(x)-'rn_RK8_||ct'(x))^2,x=0. .5,adaptive=false,numpoints=7,factor=200);\n errs := [op(errs),sqrt( sm/5)];\nend do:\nDigits := 10:\nlinalg[transpose]([mthds,evalf(errs)] );" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#K%'matrixG6#7)7$%6Enright-Verner ~schemeG$\"+33_5E!#B7$%6Verner's~Maple~schemeG$\"+r>ZdD!#C7$%Ha~modifi cation~of~Verner's~Maple~schemeG$\"+E.z!3#F07$%6an~\"efficient\"~schem eG$\"+3RIWMF07$%Ea~scheme~of~Tsitouras~and~PapakostasG$\"+thKVEF07$%Ra ~modification~of~the~Tsitouras-Papakostas~schemeG$\"+b'Q:$=F07$%AVerne r's~\"most~efficient\"~schemeG$\"+sz//AF0Q)pprint606\"" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 92 "The following error \+ graphs are constructed using the numerical procedures for the solution s." }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 598 "evalf[20](plot(['rn_R K8_1'(x)-r(x),'rn_RK8_2'(x)-r(x),'rn_RK8_3'(x)-r(x),'rn_RK8_4'(x)-r(x) ,\n'rn_RK8_5'(x)-r(x),'rn_RK8_6'(x)-r(x),'rn_RK8_7'(x)-r(x)],x=0..5,fo nt=[HELVETICA,9],\ncolor=[COLOR(RGB,.4,0,.9),COLOR(RGB,0,.65,0),brown, red,COLOR(RGB,0,.75,.95),magenta,COLOR(RGB,.95,.55,0)],\nlegend=[`Enri ght-Verner scheme`,`Verner's Maple scheme`,`a modification of Verner's Maple scheme`,`an \"efficient\" scheme`,`a scheme of Tsitouras and Pa pakostas`,`a modification of the Tsitouras-Papakostas scheme`,`Verner' s \"most efficient\" scheme`],title=`error curves for 12 stage order 8 Runge-Kutta methods`));" }}{PARA 13 "" 1 "" {GLPLOT2D 1036 658 658 {PLOTDATA 2 "6--%'CURVESG6%7[al7$$\"\"!F)F(7$$\"5NLLL$3FWYs#!#@$\"%A5! #>7$$\"5qmmmmT&)G\\aF-$\"%$o\"F07$$\"50+++]7G$R<)F-$\"%i8F07$$\"5MLLLL 3x&)*3\"!#?$\"%y7F07$$\"5+++]i!R(*Rc\"F>$\"&i9\"F07$$\"5nmmm\"H2P\"Q?F >$\"&yQ%F07$$\"5MLe*)fI&*y/@F>$\"&\"*=%F07$$\"5++]7G))>Wr@F>$\"&\\K%F0 7$$\"5M$eRAr@oZ?#F>$\"&ZO%F07$$\"5nmTN'fW%4QAF>$\"&\"QWF07$$\"5+](o/[n ?9F#F>$\"&;b%F07$$\"5MLLek.pu/BF>$\"&1k&F07$$\"5nm;/,>=0QCF>$\"&^_)F07 $$\"5+++]PMnNrDF>$\"'1n7F07$$\"5nm\"Hd?>4!QEF>$\"'w1;F07$$\"5ML$eR(\\; m/FF>$\"'*G%=F07$$\"5++v=U2TJrFF>$\"')o)=F07$$\"5nmmT5ll'z$GF>$\"'Lo>F 07$$\"5MLekyA!>Y!HF>$\"'(*[>F07$$\"5++](o/[r7(HF>$\"'\"G(>F07$$\"5nmT5 :QR#z.$F>$\"'[D>F07$$\"5MLLL$eRwX5$F>$\"'J.>F07$$\"5MLL$3F%\\wQKF>$\"' 3K>F07$$\"5MLLLe*[`HP$F>$\"'%H(>F07$$\"5MLL3-jx/SMF>$\"'N'*>F07$$\"5ML L$ek.Ur]$F>$\"'F.?F07$$\"5MLLe*)4jBuNF>$\"'+0?F07$$\"5MLLLL$eI8k$F>$\" '0(*>F07$$\"5MLL$3-8>bx$F>$\"'-%)>F07$$\"5MLLL3xwq4RF>$\"'(o(>F07$$\"5 MLL$eRA'*Q/%F>$\"'(Q)>F07$$\"5NLLL$3x%3yTF>$\"'r%*>F07$$\"5-+](oHaN;J% F>$\"'Z2?F07$$\"5ommT5:j=XWF>$\"'!p+#F07$$\"5NL$eRs3P(yXF>$\"'E+?F07$$ \"5-++]PfyG7ZF>$\"'u\")>F07$$\"5om;/^J'Qe%[F>$\"'+u>F07$$\"5NLLek.%*Qz \\F>$\"'>s>F07$$\"5++]7yv,%H6&F>$\"'[$*>F07$$\"5ommm\"z%4\\Y_F>$\"'&o, #F07$$\"5NLLeky-S3`F>$\"'$>/#F07$$\"5+++]P4'4.P&F>$\"'eT?F07$$\"5qmmT5 S*=AV&F>$\"'%f0#F07$$\"5NLLL$3FGT\\&F>$\"'v_?F07$$\"5qmm;HKp%zh&F>$\"' o@?F07$$\"5++++v$fl$\"'$R)>F07$$\"5qmmmm;HS*)fF>$\"'GM?F07$$\"5NL LLeR-/PiF>$\"'_4@F07$$\"5qmmmm\"HZ_O'F>$\"'d**=F07$$\"5++++vVVX$\\'F>$ \"'$HW\"F07$$\"5NLLL$eRh;i'F>$\"'ge5F07$$\"5qmmm\"zWo)\\nF>$\"&,+*F07$ $\"5NLLL3_DG1qF>$\"&T'pF07$$\"5-+++DcmpisF>$\"&`'\\F07$$\"5qmmm\"Hd)G& R(F>$\"&l'RF07$$\"5NLLLe*[!)y_(F>$\"&HY$F07$$\"5++++D1CZgwF>$\"&(pKF07 $$\"5qmmm\"HKkIz(F>$\"&CG$F07$$\"5SLLLeRilDzF>$\"&*\\MF07$$\"50+++Dc\" [#e!)F>$\"&%[NF07$$\"5qmmm\"H2S3>)F>$\"&sf$F07$$\"5OLLLe*)>VB$)F>$\"&E c$F07$$\"50++++v`w(e)F>$\"&kV$F07$$\"5qmmmTg()4_))F>$\"&*GLF07$$\"5NLL L$e9Kk6*F>$\"&zT$F07$$\"5.+++DJbw!Q*F>$\"&J^$F07$$\"5qmmT&)3\\m_'*F>$ \"&=^$F07$$\"5NLL$ekGkX#**F>$\"&`\\$F07$$\"5nmTg_(R^g+\"F0$\"&V]$F07$$ \"5++]iSmjk>5F0$\"&hg$F07$$\"5MLekGN8CL5F0$\"&s'QF07$$\"5nmmm;/j$o/\"F 0$\"&%*G%F07$$\"5MLLe9;_yq5F0$\"&Me'F07$$\"5+++]7GTt%4\"F0$\"&PR)F07$$ \"5ML$e9Te3n5\"F0$\"&4u)F07$$\"5nmmT5SIo=6F0$\"&k&*)F07$$\"5++]P4'\\d1 8\"F0$\"'[.6F07$$\"5MLLL3_>jU6F0$\"'6[9F07$$\"5nm;aQ`B6c6F0$\"'#=*=F07 $$\"5+++voaFfp6F0$\"'i.CF07$$\"5nmT&Q`&HLw6F0$\"'\\/DF07$$\"5ML$e*)f:t I=\"F0$\"'hoDF07$$\"5++D1kcL\")*=\"F0$\"':cDF07$$\"5nmm;HdNb'>\"F0$\"' %*yDF07$$\"5+](=\"F0$\"'(4`#F07$$\"5ML3F%zv$H.7F0$\"';SDF07$$\" 5n;H#o#eQm17F0$\"'=^DF07$$\"5++]PfeR.57F0$\"'yLDF07$$\"5nm\"zW#fTx;7F0 $\"'2_DF07$$\"5MLLe*)fV^B7F0$\"'jwDF07$$\"5nm;z>hZ*pB\"F0$\"'P@EF07$$ \"5++++]i^Z]7F0$\"'LYEF07$$\"5+++]i:4,k7F0$\"'%)QEF07$$\"5++++vomax7F0 $\"'pEEF07$$\"5+++](=U#3\"H\"F0$\"'HBEF07$$\"5+++++v\"=YI\"F0$\"'TJEF0 7$$\"5+++]7GR:=8F0$\"'pUEF07$$\"5++++D\"o*oJ8F0$\"'6`EF07$$\"5+++]PMaA X8F0$\"'A^EF07$$\"5++++](=h(e8F0$\"'rTEF07$$\"5++++]7!Q4T\"F0$\"'yVEF0 7$$\"5++++]P[6j9F0$\"'w]EF07$$\"5M$e9m>`P!p9F0$\"'+LEF07$$\"5nm\"HKkAg \\Z\"F0$\"'B5EF07$$\"5+]P%)*3#H)3[\"F0$\"'&\\h#F07$$\"5ML$ek`h0o[\"F0$ \"'^(f#F07$$\"5++voH/5l)\\\"F0$\"'kaEF07$$\"5nmm\"HKR'\\5:F0$\"'WmFF07 $$\"5MLe9;#yTB_\"F0$\"'SkEF07$$\"5++]P4rr=M:F0$\"'u3CF07$$\"5nmTg-gD.Y :F0$\"'Q]?F07$$\"5MLL$e*[z(yb\"F0$\"'^!o\"F07$$\"5nm;/Ev&[ge\"F0$\"'SR 7F07$$\"5+++Dc,#>Uh\"F0$\"&G]*F07$$\"5nmTNr9XIG;F0$\"&3J)F07$$\"5ML$ek y#)*QU;F0$\"&tC(F07$$\"5n;/,W%[K%\\;F0$\"&e+(F07$$\"5++Dc,T^Zc;F0$\"&; %oF07$$\"5M$e9\"f(zF0$ \"&&yoF07$$\"5,+DcEsW\"R\">F0$\"&([pF07$$\"5MLLe9;0?E>F0$\"&P7(F07$$\" 5++]i!Rgs2&>F0$\"&X`)F07$$\"5nmmmm\"pW`(>F0$\"'_v5F07$$\"5++]iSmTI-?F0 $\"'n&G\"F07$$\"5MLLe9TOEH?F0$\"'_h:F07$$\"5,+Dc^yLuU?F0$\"'wl?F07$$\" 5om;a)e6Bi0#F0$\"'GVDF07$$\"5ML3_D`Gqp?F0$\"'oQIF07$$\"5,++]i!f#=$3#F0 $\"'SvJF07$$\"5,+]7G)[=g4#F0$\"'!y=$F07$$\"5,++v$fQa)3@F0$\"'!p<$F07$$ \"5,+]Pf$G!p@@F0$\"'O?KF07$$\"5,+++D\"=EX8#F0$\"':nKF07$$\"5,+D\"y+8W4 9#F0$\"'`'G$F07$$\"5,+]i!*y?OZ@F0$\"'N\"H$F07$$\"5,+vVtF+y`@F0$\"'q!H$ F07$$\"5,++Dcwz>g@F0$\"'()zKF07$$\"5,+](=U(Q.t@F0$\"'4qKF07$$\"5,++](= xpe=#F0$\"'EmKF07$$\"5ML3_]%oi#*>#F0$\"'BuKF07$$\"5om;a8(fbE@#F0$\"'!* )G$F07$$\"5,+Dcw4&[gA#F0$\"':'H$F07$$\"5MLLeRA9WRAF0$\"'%oH$F07$$\"5++ ]ilZsAmAF0$\"'JxKF07$$\"5nmmm\"H28IH#F0$\"',kKF07$$\"5nm;a8A3h0\"zBF0$\"'#\\B$F0 7$$\"5nTNY$Q\"*z@Q#F0$\"':WKF07$$\"5n;zp[KYD&Q#F0$\"'\"4B$F07$$\"5n\"H KR6NH$)Q#F0$\"'#>B$F07$$\"5nmm;zpSS\"R#F0$\"'f#H$F07$$\"5+D\"GQ#p0s%R# F0$\"'7tKF07$$\"5M$e*[ooq.)R#F0$\"'!zE$F07$$\"5nT5:8oNN,CF0$\"'!)\\LF0 7$$\"5++D\"yv1qYS#F0$\"'-CLF07$$\"5MeRZ-nl)zS#F0$\"'Q2LF07$$\"5n;a8ZmI I6CF0$\"'KWLF07$$\"5+voz\"fc>YT#F0$\"'=7LF07$$\"5ML$ek`1OzT#F0$\"'WvKF 07$$\"5nmT5:j??JCF0$\"'B9HF07$$\"5+++v$41oWW#F0$\"'>0DF07$$\"5MLeRseSt dCF0$\"'V4@F07$$\"5nm;/^c++rCF0$\"'w&z\"F07$$\"5++voHagE%[#F0$\"'Xh;F0 7$$\"5MLLL3_?`(\\#F0$\"'fE:F07$$\"5ML$3-)Q84DDF0$\"'G'>\"F07$$\"5MLL3_ D1l_DF0$\"'_G5F07$$\"5M$3_+sWS&fDF0$\"'#[,\"F07$$\"5ML3-))o-VmDF0$\"'T 15F07$$\"5M$e*)f04?Ld#F0$\"'\"o+\"F07$$\"5ML$eRA\"*4-e#F0$\"'h:5F07$$ \"5MLe*)fb&*)Rf#F0$\"'\"R-\"F07$$\"5MLL$e*)>pxg#F0$\"'UI5F07$$\"5ommm; z+vbEF0$\"'795F07$$\"5,++]Pf4t.FF0$\"'t55F07$$\"5omm\"zWi^bv#F0$\"'hB5 F07$$\"5MLLLe*Gst!GF0$\"'?H5F07$$\"5omm;H2\"34'GF0$\"'\\S8F07$$\"5,+++ +DRW9HF0$\"'g4=F07$$\"5,+D19>2*4#HF0$\"'SQ>F07$$\"5,+]7G8v`FHF0$\"'W;? F07$$\"5,+v=U2V3MHF0$\"':zAF07$$\"5,++Dc,6jSHF0$\"'rrEF07$$\"5,+]P%)*o CP&HF0$\"'vkJF07$$\"5,++]7y#=o'HF0$\"'+kNF07$$\"5,++voaa+$*HF0$\"'j%y$ F07$$\"5,+++DJE>>IF0$\"'N^QF07$$\"5,]P4@\"*p_DIF0$\"'SyQF07$$\"5,+v=<^ 8'=.$F0$\"'f7RF07$$\"5,]7G86d>QIF0$\"'vARF07$$\"5,+]P4r+`WIF0$\"'2NRF0 7$$\"5,+Dc,\"z)>dIF0$\"'#R$RF07$$\"5,++v$4^n)pIF0$\"'H9RF07$$\"5,+v$f3 BOD3$F0$\"'p6RF07$$\"5,+]7y]\\?&4$F0$\"'C$F0$\"'$4\"RF07$$\"5,+](=<6T\\?$F0$\"'pBRF07$$\"5,+ vVBlv+>KF0$\"':aRF07$$\"5,+++v=S2LKF0$\"'?')RF07$$\"5omm\"zpo_$eKF0$\" 'bPRF07$$\"5MLL$3_NJOG$F0$\"'BoQF07$$\"5om;HK*oqiH$F0$\"'#y)QF07$$\"5, ++vVB+\"*3LF0$\"'40RF07$$\"5ML$3_vN\\:K$F0$\"'@6PF07$$\"5ommmm\"p)=MLF 0$\"'pMLF07$$\"5MLLekyQoZLF0$\"'1nGF07$$\"5,++]il!z6O$F0$\"'@zBF07$$\" 5ommTg_UnuLF0$\"')>=#F07$$\"5MLLLeR%p\")Q$F0$\"'!*>?F07$$\"5omm;a8)f^T $F0$\"'E[;F07$$\"5,+++](=]@W$F0$\"'*pS\"F07$$\"5om\"H#oK)yVX$F0$\"'yk8 F07$$\"5ML$ekyZ2mY$F0$\"'KX8F07$$\"5,+vo/Bh$)yMF0$\"'RZ8F07$$\"5omm\"H #oZ1\"\\$F0$\"'Rj8F07$$\"5MLe9T8MH.NF0$\"'nn8F07$$\"5,+]Pfe?_:NF0$\"'! zO\"F07$$\"5omTgx.2vFNF0$\"'hk8F07$$\"5MLL$e*[$z*RNF0$\"'ha8F07$$\"5om T5:)[[Lb$F0$\"'i[8F07$$\"5,+]PMFwrmNF0$\"'cV8F07$$\"5MLek`mn3!e$F0$\"' bS8F07$$\"5omm\"Hd!fX$f$F0$\"'?OF0$\"'ad8F07$$\"5om\"H2LKjNj$F0$\"'(*f8F07$$\"5,+++]iC$p k$F0$\"'Bg8F07$$\"5om;zp)4\"4sOF0$\"''zN\"F07$$\"5MLLe*[t\\sp$F0$\"'(G O\"F07$$\"5,]7`%RRRNq$F0$\"'Qn8F07$$\"5om\"z%*H0H)4PF0$\"'2x8F07$$\"5M $3FW?r=hr$F0$\"'X%Q\"F07$$\"5,+]P4r$3Cs$F0$\"'(yS\"F07$$\"5ML3F>*o()\\ t$F0$\"'mf9F07$$\"5omm;H2qcZPF0$\"'\"[a\"F07$$\"5,+++DJ]'Qx$F0$\"'h(*= F07$$\"5MLL$3_0j,!QF0$\"''4:#F07$$\"5,++v=n?J8QF0$\"'o)G#F07$$\"5ommm; z5YEQF0$\"'))\\DF07$$\"5,+]i:&eNI$QF0$\"'RsGF07$$\"5MLLe9\"45'RQF0$\"' B$>$F07$$\"5om;a8(f%=YQF0$\"'W&R$F07$$\"5,++]7.\"fF&QF0$\"'%f!QF07$$\" 5ML3_vlYhlQF0$\"'()*=%F07$$\"5om;aQG-ZyQF0$\"'%yQ%F07$$\"5M$3_+(4!)*[) QF0$\"'Z6WF07$$\"5,+Dc,\"zD8*QF0$\"'C\"R%F07$$\"5M3xJn\"oRX*QF0$\"'n9W F07$$\"5o;H2LsNv(*QF0$\"'dIWF07$$\"5,D\"G))HYn4!RF0$\"'*GT%F07$$\"5MLL ek`8=/RF0$\"'=IWF07$$\"5omTgF;p.*3%F0$\"'%\\b%F07$$ \"5,+Dcw4*e@5%F0$\"'ZtXF07$$\"5omm\"zW7@^6%F0$\"'y&f%F07$$\"5-]Pf$=B-; 7%F0$\"'$*>YF07$$\"5NL3F>RL3GTF0$\"'*)>YF07$$\"5o;z%\\lWkX8%F0$\"'PIYF 07$$\"5-+]i!RbX59%F0$\"'qAYF07$$\"5om\"z>'ox+aTF0$\"'?!e%F07$$\"5NLLLL $)*pp;%F0$\"':XF07$$\"5NLL3xc9[$>%F0$\"'m HXF07$$\"5N$3_]4R&z'>%F0$\"'l+XF07$$\"5NL3-8D$4,?%F0$\"'StXF07$$\"5N$e *)4$fKU.UF0$\"'0NXF07$$\"5NL$e*[$>Pn?%F0$\"'`&\\%F07$$\"5NLe*[=1lL@%F0 $\"']LWF07$$\"5NLL$3-$H**>UF0$\"'CGUF07$$\"5NL$3Fpm[KB%F0$\"'%4z$F07$$ \"5NLLek.W]YUF0$\"'%fJ$F07$$\"5NL$ek.9g(fUF0$\"'AVGF07$$\"5NLLL3xe,tUF 0$\"';8EF07$$\"5om;a8AyI*H%F0$\"'Z;AF07$$\"5-++v=n(*fDVF0$\"'qw=F07$$ \"5omTNrRduQVF0$\"'%Rx\"F07$$\"5NL$eRAr\"*=N%F0$\"'U2#[Z%F0$\"'Cw;F07$$ \"5ommT5::^-XF0$\"'*[o\"F07$$\"5MLL$3_5,-`%F0$\"'/'p\"F07$$\"5,++DJ&p! *yb%F0$\"'![p\"F07$$\"5ommmT&G!e&e%F0$\"'q'p\"F07$$\"5-+]i:NK'zf%F0$\" 'Z-6but%F0$\"'d*e$F07$$\"5N$e *)fb^cSu%F0$\"'o=SF07$$\"5-+D19>zl]ZF0$\"'!oU%F07$$\"5o;a8sA$fsv%F0$\" '(oa%F07$$\"5NL$3-jsgQw%F0$\"'N0[F07$$\"5-]7G))H@YqZF0$\"'+A\\F07$$\"5 omTNYLN1xZF0$\"'h.]F07$$\"5N$3FWq$\\m$y%F0$\"'y:]F07$$\"5-++]iSjE!z%F0 $\"'m1]F07$$\"5-++D\"G))Rb\"[F0$\"'YG^F07$$\"5-++++DM\"3%[F0$\"'O@_F07 $$\"5-+]P4'>]M&[F0$\"'(z@&F07$$\"5-++v=np3m[F0$\"')z?&F07$$\"5-+]7GQPs y[F0$\"'>-_F07$$\"5-++]P40O\"*[F0$\"'+7_F07$$\"5,++voa-oX\\F0$\"'xD_F0 7$$\"\"&F)$\"'uB_F0-%&COLORG6&%$RGBG$\"\"%!\"\"F($\"\"*Fciq-%'LEGENDG6 #%6Enright-Verner~schemeG-F$6%7_\\mF'7$$\"5ommmTN@Ki8F-$\"$>#F07$F+$\" #')F07$$\"5-+++D1k'p3%F-$!$'*)F07$F2$!%*\\\"F07$$\"5SLL3_v!p)*y&F-$!%X :F07$$\"50++]P4'\\/8'F-$!%Q>F07$$\"5qmm\"HK9I5Z'F-$!%P>F07$$\"5SLLL3x1 h6oF-$!%`>F07$$\"5qmm;zW'o^7\\6F>$\"$1'F07$$\"5++]i!*GER37F>$\" %#Q\"F07$$\"5ML3F>*3gwE\"F>$\"%\"R\"F07$$\"5nmm\"z%\\v#pK\"F>$\"%i;F07 $$\"5++Dcw4]>'Q\"F>$\"%z:F07$$\"5ML$3_+ZiaW\"F>$\"%b8F07$$\"5+]7`>+i4v 9F>$\"%+8F07$$\"5nmT&Q.$*HZ]\"F>$\"$*fF07$$\"5M$3x\"[gOOM:F>$\"$-'F07$ FB$\"$$eF07$$\"5nm;z>6B`#o\"F>$!$m)F07$$\"5MLL3xJs1,=F>$!%_eF07$$\"5++ ]PM_@g>>F>$!&18\"F07$FG$!&dv\"F07$F_o$!&&y=F07$Fio$!&Y+#F07$$\"5M$e9;K 'Ho/EF>$!&eo#F07$F^p$!&5s#F07$$\"5+]P%)*3UN8n#F>$!&iz#F07$Fcp$!&2n$F07 $Fhp$!&Cy$F07$F]q$!&q\\%F07$Fbq$!&U$[F07$Fgq$!&@!\\F07$F\\r$!&D$\\F07$ Far$!&G\"\\F07$F[s$!&GF&F07$F_t$!&6\\&F07$Fdt$!&&=`F07$Fit$!&u)\\F07$$ \"5ML$3-Q\"[DVRF>$!&w)\\F07$$\"5MLL3_]>!o(RF>$!&4)\\F07$$\"5ML$eRs3\\. ,%F>$!&W!\\F07$F^uFder7$$\"5MLLeR(\\!*46%F>$!&I!\\F07$Fcu$!&-\"\\F07$F ]v$!&t>&F07$Fgv$!&9H&F07$Faw$!&`2&F07$F[x$!&$))[F07$Fex$!&`'\\F07$F_y$ !&P<&F07$$\"5+++Dc,w.cbF>$!&)H_F07$Fdy$!&$3`F07$$\"5NLL3-ji&)zcF>$!&xG &F07$Fiy$!&\"[_F07$F^z$!&*\\[F07$Fcz$!&uu%F07$$\"5+++]ilP9,jF>$!&AN%F0 7$Fhz$!&8B%F07$$\"5+++voa!*H(R'F>$!&wa$F07$$\"5NLL$3x\"3NHkF>$!&YJ$F07 $$\"5qmm\"H2e-9Y'F>$!&fD$F07$F][l$!&Tf#F07$$\"5NLL3x1h]DlF>$!&v+#F07$$ \"5qmm;zpybdlF>$!&8(>F07$$\"5+++D\"Gj4'*e'F>$!&;e\"F07$Fb[l$!%<\"*F07$ $\"5+++](=#\\w&o'F>$!%**yF07$Fg[l$!%I[F07$$\"5++++++b2yoF>$!%!*yF07$F \\\\l$!&#Q:F07$$\"5qmmm;/'*[MrF>$!&\"38F07$Fa\\l$!%;wF07$Ff\\l$\"$&pF0 7$F[]l$\"%hZF07$F`]l$\"%.jF07$Fe]l$\"%WyF07$$\"50+++D\"Gg$fyF>$\"%X#)F 07$Fj]l$\"%\\')F07$$\"5qmmm\"z>_>*zF>$\"%#f)F07$F_^l$\"%Y&)F07$Fd^l$\" %sxF07$Fi^l$\"%WmF07$Fc_l$\"%!)oF07$F]`l$\"%5tF07$$\"5qmT5lv.u[%*F>$\" %\"y'F07$$\"5NL$3_+A:n^*F>$\"%/iF07$$\"5++DJXk+p%e*F>$\"%GhF07$Fb`l$\" %GeF07$$\"5NL3_D`(R1s*F>$\"%rdF07$$\"5++]il(f9')y*F>$\"%!)eF07$$\"5N$3 xc)>?gA)*F>$\"%!4'F07$$\"5qm\"Hd?W*ec)*F>$\"%'4'F07$$\"5+]7yDkod!*)*F> $\"%ljF07$Fg`l$\"%tnF07$F\\al$\"%SxF07$Faal$\"%J#*F07$$\"5n;aj%3&QWE5F 0$\"%#[*F07$Ffal$\"%p'*F07$$\"5+]ils>)Q+/\"F0$\"%-$*F07$F[bl$\"%v$*F07 $$\"5++]i:g2\")e5F0$\"%m$)F07$F`bl$\"%ScF07$$\"5nm;a8s'fF3\"F0$\"%4?F0 7$Febl$!%KMF07$Fjbl$!%+\"*F07$F_cl$!&k:\"F07$Fdcl$!%FnF07$Ficl$!%K\\F0 7$$\"5++vVt_@P\\6F0$!%=oF07$F^dl$!%_%)F07$$\"5MLek.aD&G;\"F0$!&'Q;F07$ Fcdl$!&'pAF07$Fhdl$!&Ug#F07$F]el$!&DA$F07$Fbel$!&5U$F07$Fgel$!&N\\$F07 $Fegl$!&5g$F07$F_hl$!&I/%F07$Fihl$!&$pPF07$Fcil$!&DR$F07$F]jl$!&_g$F07 $Fgjl$!&.!QF07$$\"5++++++'\\[Q\"F0$!&`n$F07$F\\[m$!&JQ$F07$$\"5+++++Dk -P9F0$!&8\\$F07$Fa[m$!&Y\"QF07$Fe\\m$!&Rb$F07$F_]m$!&+Z$F07$Fd]m$!&FO$ F07$Fi]m$!&,\"GF07$$\"5n\"H#oK=&[r`\"F0$!&Vt#F07$$\"5M$e*)fb')4,a\"F0$ !&Mz\"F07$$\"5+voHz772V:F0$!&2x\"F07$F^^m$!&kt\"F07$$\"5Me9\"fs!R**[:F 0$!&)z7F07$$\"5+](=#\\a_&>b\"F0$!%l^F07$$\"5nTg_s,m\"\\b\"F0$!%!3&F07$ Fc^m$!%XSF07$$\"5n;aQ`01#\\c\"F0$\"%\"*QF07$$\"5++v$4@Ej>d\"F0$\"%ndF0 7$$\"5M$e*[o=f+z:F0$\"%ySF07$Fh^m$\"%+9F07$$\"5+]Pf$=B\"4$f\"F0$!%?RF0 7$$\"5MLe9T))Q8+;F0$!%$)eF07$$\"5+v=#*p;_l.;F0$!%'z&F07$$\"5n;zp)\\awr g\"F0$!%\"f&F07$$\"5MeRZFtyp5;F0$!%tIF07$F]_m$!%KIF07$Fb_m$\"%dKF07$Fg _m$\"&)>6F07$Fa`m$\"&fK\"F07$F[am$\"&@c\"F07$$\"5nm\"z>6[VNn\"F0$\"&(e :F07$$\"5nm;H23l_w;F0$\"&rb\"F07$$\"5nmTg-N&4&z;F0$\"&7g\"F07$$\"5nmm \"z>c#\\#o\"F0$\"&\"G;F07$$\"5nm;a)ehe%)o\"F0$\"&+j\"F07$F`am$\"&@l\"F 07$$\"5nm;zpB2R+V\"F07$Fjam$\"&=Q\"F07$$ \"5nmmT&)3_3a(*)z9F07$$\"5,]7GjBlBc=F0$ \"&;R\"F07$$\"5ML3-jikF0$\"&pw\"F 07$$\"5nmTg-gl[Q>F0$\"&!oeHY%>F0$\"&Fu\"F07$Fbdm$\"&8n \"F07$$\"5MLekyZ'eI'>F0$\"&Ge\"F07$Fgdm$\"&-R\"F07$$\"5MLek.HW#)))>F0$ \"%\"\\)F07$F\\em$\"#%)F07$$\"5M$e9\"4NS/4?F0$!$<%F07$$\"5nmTgx.Ry:?F0 $!%58F07$$\"5+]P4YsP_A?F0$\"%g8F07$Faem$\"%'Q$F07$$\"5o;H2$)4N+O?F0$\" %ccF07$Ffem$\"%SpF07$$\"5M$3_+sC$[\\?F0$\"%=WF07$F[fm$\"%XFF07$$\"5,]7 .d%)H'H1#F0$!%)\\&F07$F`fm$!&m=\"F07$$\"5o;/,%>sUk2#F0$!&.U\"F07$Fefm$ !&a\">F07$F_gm$!&l,#F07$Figm$!&OP#F07$$\"5,]iSmb^tP@F0$!&))Q#F07$F^hm$ !&(>DF07$$\"5,](=#\\/J:W@F0$!&=_#F07$Fchm$!&P_#F07$$\"5,]7.K`5d]@F0$!& C`#F07$Fhhm$!&N`#F07$$\"5,]P%[@+*)p:#F0$!&,`#F07$F]im$!&gT#F07$Fbim$!& DB#F07$Fgim$!&>0#F07$$\"5M3_D.+!=#*=#F0$!&`*>F07$$\"5o;/,>Gic#>#F0$!& \"H>F07$$\"5,DcwMcW\"f>#F0$!&*G>F07$F\\jm$!&%3>F07$$\"5,]7.#39ff?#F0$! &e)=F07$Fajm$!&+#>F07$Ffjm$!&L,#F07$F[[n$!&<>#F07$$\"5+]P4ryy8YAF0$!&s B#F07$$\"5nmTg-NV$GD#F0$!&=I#F07$$\"5M$e9T8zI&fAF0$!&PI#F07$F`[n$!&_I# F07$$\"5MLekGg,izAF0$!&[<#F07$Fe[n$!&<-#F07$$\"5nmTg_Z>J0BF0$!&-#>F07$ Fj[n$!&!y=F07$$\"5nm\"zWnp4*HBF0$!&$3?F07$F_\\n$!&g;#F07$$\"5nT5l+!H$G XBF0$!&\\;#F07$$\"5n;a)e'3!e$[BF0$!&r=#F07$$\"5n\"z>6tsK9N#F0$!&;H#F07 $Fd\\n$!&!*G#F07$$\"5n;H#oK)olgBF0$!&RK#F07$Fi\\n$!&FJ#F07$F]^n$!&%f@F 07$Fa_n$!&2(>F07$F[`n$!&T&>F07$Fe`n$!&o)>F07$Fj`n$!&Z)>F07$F_an$!&z3#F 07$Fdan$!&y1#F07$Fian$!&k.#F07$$\"5+]7yDk!pXU#F0$!&l$>F07$F^bn$!&GM\"F 07$$\"5+Dcwfi&=XV#F0$!&WK\"F07$$\"5M$3FW?1NyV#F0$!&x>\"F07$$\"5nT&)3\\ h::TCF0$!%PCF07$Fcbn$!%'R#F07$$\"5Me9TQgXyZCF0$!%A7F07$$\"5n;H2$)f55^C F0$\"%c$*F07$$\"5+vVtFfvTaCF0$\"%7#*F07$Fhbn$\"%^(*F07$$\"5n\"Hdq\"e00 hCF0$\"&tj\"F07$$\"5+](=Rt*4(e#F0$\"&HU#F07$Fjen$\"&lV #F07$$\"5M$ekysPz3g#F0$\"&YS#F07$F_fn$\"&mR#F07$$\"5om;/,>Ww>EF0$\"&#F07$$\"5ML$e 9\"f[vVEF0$\"&s:#F07$$\"5,+D19pCv\\EF0$\"&#\\@F07$Fdfn$\"&q9#F07$$\"5M LL3F>0uzEF0$\"&,F#F07$Fifn$\"&wO#F07$$\"5ML$3F>HT'HFF0$\"&DH#F07$F^gn$ \"&Z;#F07$$\"5,](=Y\"y#F0$\"&a?#F07$$\"5om\"H2L7ST$GF0$\"&gd#F07$$\"5omT5lv@$3% GF0$\"&m`#F07$$\"5ML$ekG:Cv%GF0$\"&4c#F07$$\"5,+D\"y+8;U&GF0$\"&r]#F07 $Fhgn$\"&NR#F07$$\"5MLLe9;gn()GF0$\"&Qz\"F07$F]hn$\"%)p*F07$Fghn$\"&FT \"F07$Fain$\"&p&=F07$Ffin$\"&RR\"F07$F[jn$\"%faF07$$\"5,+DcEs]OtHF0$!% a=F07$$\"5,+]iSm=\"*zHF0$!%Y`F07$$\"5,+voag'ek)HF0$!%EdF07$F`jn$!%$3'F 07$$\"5,+D\"G)[Ab**HF0$!%CcF07$$\"5,+](oH/*41IF0$!%&R&F07$$\"5,]i!R+Ws $4IF0$!%+bF07$$\"5,+v$4r$ek7IF0$!%9cF07$$\"5,](ozTB>f,$F0$!%UcF07$Fejn $!%VjF07$F_[o$!%>#*F07$Fi[o$!&!Q5F07$$\"5,](oa5Vk30$F0$!&G,\"F07$F^\\o $!&n+\"F07$$\"5,]il(4:LN1$F0$!%A()F07$Fc\\o$!%ZqF07$Fh\\o$!%B^F07$F]]o $!%mSF07$$\"5,](=U2JR:5$F0$!%GQF07$Fb]o$!%()QF07$$\"5,]iSmI!3U6$F0$!%P VF07$Fg]o$!%^`F07$F\\^o$!%NlF07$Fa^o$!%dwF07$$\"5,]iSTC&3d:$F0$!%V!)F0 7$Ff^o$!%KzF07$$\"5,D\"y]XOei;$F0$!%AzF07$$\"5,](oHz(\\xpJF0$!%_tF07$$ \"5,v$f38f\"HtJF0$!%>sF07$F[_o$!%(=(F07$F`_o$!%**[F07$Fe_o$!%*)RF07$$ \"5,]ilZQV(>@$F0$!%5PF07$Fj_o$!%GRF07$$\"5,](=#*>zSgA$F0$!%`UF07$F_`o$ !%([&F07$$\"5ML$ekGN8dC$F0$!%>qF07$Fd`o$!%T\")F07$$\"5,+]P4@?*4F$F0$!% ;sF07$Fi`o$!%+eF07$$\"5o;zpt)=\"z'G$F0$!%ddF07$$\"5,+DcEA5&**G$F0$!%C` F07$$\"5M$3F%zb36$H$F0$!%w_F07$F^ao$!%`_F07$$\"5ML3-Qc.f-LF0$!%thF07$F cao$!%LkF07$Fhao$!%c`F07$F]bo$\"%<;F07$$\"5MLe9T)[ivL$F0$\"%JCF07$$\"5 ,+]i:&GO4M$F0$\"&'y7F07$$\"5omT5!>35VM$F0$\"&)f7F07$Fbbo$\"&hL\"F07$$ \"5,+D1Rvw0^LF0$\"&VL#F07$$\"5om;a8s9VaLF0$\"&pH#F07$$\"5ML3-))o_!yN$F 0$\"&yJ#F07$Fgbo$\"&Q!GF07$F\\co$\"&O\\#F07$Faco$\"&kv\"F07$$\"5,+D\"G jBV:R$F0$\"&AP\"F07$$\"5om;H2Lq\"\\R$F0$\"&AN\"F07$$\"5ML3x\")H3H)R$F0 $\"&tL\"F07$$\"5,++DcEYm,MF0$\"&nL\"F07$$\"5ML$3_+A7%3MF0$\"&\")Q\"F07 $Ffco$\"&6p\"F07$$\"5MLL3_+]lGMF0$\"&#*H#F07$F[do$\"&?(GF07$Fedo$\"&o4 $F07$F_eo$\"&!>KF07$Fieo$\"&g5$F07$Fcfo$\"&^$HF07$Fhfo$\"&E$HF07$F]go$ \"&d'HF07$Fbgo$\"&13$F07$Fggo$\"&,8$F07$$\"5M$3_D`ZS,g$F0$\"&K:$F07$F \\ho$\"&G:$F07$$\"5o;H#=Xh4Nh$F0$\"&M9$F07$Faho$\"&K5$F07$Ffho$\"&U/$F 07$F[io$\"&D)HF07$Feio$\"&06$F07$Fc[p$\"&VQ$F07$$\"5MLL3F>grgPF0$\"&n@ $F07$Fh[p$\"&k-$F07$$\"5omm\"HK/9qy$F0$\"&\"\\EF07$F]\\p$\"&!4>F07$Fb \\p$\"&(3@F07$Fg\\p$\"&gl#F07$$\"5MLe9;K$[(HQF0$\"&5%HF07$F\\]p$\"&r0$ F07$$\"5omT5:QGKOQF0$\"&45$F07$Fa]p$\"&c/$F07$$\"5,+D19Wt*G%QF0$\"&&\\ IF07$Ff]p$\"&O3$F07$$\"5ML3-8]=Z\\QF0$\"&Us#F07$F[^p$\"&!*\\#F07$F`^p$ \"&Hr\"F07$Fe^p$\"&8-\"F07$$\"5,voH/>To\")QF0$\"%R\")F07$Fj^p$\"%(>)F0 7$$\"5o\"H2e.!>6))QF0$\"%p#)F07$F__p$\"%5&)F07$Fd_p$\"%e&)F07$Fi_p$\"% m')F07$F^`p$\"%N$*F07$Fc`p$\"%u$*F07$$\"5oT&Q.VC&R2RF0$\"%y$*F07$$\"5, ]P4'\\841\"RF0$\"%')))F07$$\"5Me*[=c-BQ\"RF0$\"%7*)F07$Fh`p$\"%p))F07$ $\"5M$e9\"f(pkM#RF0$\"%%=(F07$F]ap$\"%IbF07$$\"5M3-Qcpj5LRF0$\"%d`F07$ $\"5o;a8Ag-KORF0$\"%^`F07$$\"5,D1*y3:M&RRF0$\"%Z[F07$Fbap$\"%dXF07$$\" 5,]i:&G#e<\\RF0$\"%D[F07$Fgap$\"%e^F07$F\\bp$\"%9vF07$Fabp$\"&V-\"F07$ $\"5M$eRs31]#*)RF0$\"&a1\"F07$Ffbp$\"&16\"F07$$\"5,](oaNk3F+%F0$\"&a6 \"F07$F[cp$\"&l2\"F07$Fecp$\"%G#)F07$F_dp$\"%JsF07$Fidp$\"%l(*F07$Fcep $\"&t7\"F07$F]fp$\"&P.\"F07$Fgfp$\"%FwF07$$\"5oTNYe2hGWTF0$\"%BwF07$$ \"5N$3-j7mEv9%F0$\"%YvF07$$\"5-D19%\\@n2:%F0$\"%]mF07$F\\gp$\"%VmF07$$ \"5-]il(f())[gTF0$\"%knF07$FagpF_bs7$$\"5NL3Fp^yftTF0$\"%LyF07$Ffgp$\" %$)*)F07$$\"5NLe9T)e`o=%F0$\"%%*))F07$F[hp$\"%g))F07$F`hp$\"%n()F07$Fe hp$\"%#[(F07$Fjhp$\"%=uF07$F_ip$\"%StF07$$\"5N$3Fpw7^+@%F0$\"%$['F07$F dip$\"%kF07$Fiip$\"%!)*)F07$$\"5N$3-)QkoI BUF0$\"%*)))F07$$\"5NL3xc)z?mA%F0$\"%e*)F07$$\"5N$eRZFtM*HUF0$\"&-j\"F 07$F^jp$\"&al\"F07$$\"5N$3x16gilB%F0F]\\y7$$\"5NLekGNl()RUF0$\"&_l#F07 $$\"5N$e9m%p/>VUF0$\"&Fu#F07$Fcjp$\"&$>FF07$$\"5N$3_DyL=)\\UF0$\"&&GNF 07$$\"5NL3_+sA8`UF0$\"&Ej$F07$$\"5N$e*[=1iWcUF0$\"&4e$F07$Fhjp$\"&W%QF 07$$\"5NLeRs3!)QmUF0$\"&_z$F07$F][q$\"&\"3MF07$$\"5-+v$4'\\=;'G%F0$\"& el#F07$Fb[q$\"&QE#F07$$\"5NLe9m%z`CJ%F0$\"&5q#F07$Fg[q$\"&&\\JF07$Fa\\ q$\"&.%QF07$F[]q$\"&&yRF07$$\"5,]PM-\\NA%Q%F0$\"&(3SF07$F`]q$\"&s*RF07 $$\"5,v$4@mQ$G$R%F0$\"&]*RF07$$\"5o;zp[KLI'R%F0$\"&D*RF07$$\"5MekGNyKK *R%F0$\"&n'RF07$Fe]q$\"&P%RF07$Fj]q$\"&<(QF07$F_^q$\"&zz$F07$$\"5,v=<, PF_HWF0$\"&Iw$F07$$\"5o;/w(GoUDV%F0$\"&>u$F07$$\"5Me*[V(GEcNWF0$\"&=u$ F07$Fd^q$\"&at$F07$$\"5M$e9TjYAYW%F0$\"&Nr$F07$Fi^q$\"&-s$F07$F^_q$\"& *ePF07$Fc_q$\"&r\"QF07$F]`q$\"&@\"QF07$Fg`q$\"&[\"QF07$Faaq$\"&T6%F07$ F[bq$\"&X?%F07$F`bq$\"&n0%F07$Febq$\"&7[$F07$$\"5o\"zpB^eWzo%F0$\"&BF07$$\"5-+v=9#F07$F[gq$\"&z&>F07$Fegq$\"&k A#F07$F_hq$\"&Ii#F07$$\"5-+]7.#Q?&=\\F0$\"&W]#F07$Fdhq$\"&OA#F07$$\"5+ +]PMF,%G(\\F0$\"&rL#F07$Fihq$\"&@j#F0-F^iq6&F`iqF($\"#l!\"#F(-Fgiq6#%6 Verner's~Maple~schemeG-F$6%7_]mF'7$$F^\\r!#A$\"\"'F07$F^jq$\"$(GF07$$ \"5-++]7.K[V?F-$\"$P$F07$F+$\"$E$F07$Ffjq$!$n#F07$F2$!$Y(F07$F^[r$!$(y F07$Fc[r$!%f6F07$Fh[rFd_z7$F]\\r$!%&=\"F07$Fb\\r$!%!Q\"F07$F7$!%?8F07$ Fj\\r$!$U*F07$F<$\"#FF07$Fg]r$\"%$R\"F07$Fa^r$\"%_>F07$F[_r$\"%.AF07$F B$\"%cBF07$$\"5MLe9\"4&[EB;F>$\"%4EF07$Fb`r$\"%\\EF07$$\"5M$e9T8/m@r\" F>$\"%*z#F07$$\"5++vV[r(*zT$\"%CGF07$$\"5n;/wi,NVr$\"%8GF07$Fg`r $\"%'3#F07$F\\ar$!$h$F07$FG$!%xVF07$FL$!%!)zF07$FQ$!%j#)F07$FV$!%[\"*F 07$Fen$!%,$*F07$Fjn$!%f$*F07$F_o$!%xrF07$Fdo$!%YSF07$Fio$!%)y#F07$F^p$ !%9ZF07$Fcp$!%=*)F07$Fhp$!%h#*F07$F]q$!&aG\"F07$$\"5+]7`%Rz#HrGF>$!&9I \"F07$Fbq$!&uU\"F07$$\"5n;/wi^_%z$HF>$!&!Q9F07$Fgq$!&WW\"F07$F\\r$!&eQ \"F07$Far$!&`G\"F07$F[s$!&`S\"F07$F_t$!&]g\"F07$Fit$!&HC\"F07$Fcu$!&J8 \"F07$F]v$!&>K\"F07$Fgv$!&\"e9F07$Faw$!&(=8F07$F[x$!&%*3\"F07$F_y$!&4A \"F07$Fiy$!&dO\"F07$$\"5NLL$3_D%eleF>$!&tG\"F07$F^z$!&v<\"F07$$\"5+++] 7y:A8hF>$!&GH\"F07$Fcz$!&bY\"F07$$\"5qmmTg-?4piF>$!&5X\"F07$Fhhr$!&\\ \\\"F07$$\"5NLLekGb>LjF>$!&tZ\"F07$Fhz$!&MX\"F07$Feir$!&;6\"F07$F][l$! %ZtF07$Fbjr$!%sSF07$Fgjr$!%uRF07$F\\[s$!%7:F07$Fb[l$\"%]IF07$Fd[s$\"%] PF07$Fg[l$\"%.iF07$$\"5NLL$eR(>(R\"oF>$\"%OVF07$F\\\\s$\"%uPF07$$\"5qm m;/E!z@%pF>$!$c&F07$F\\\\l$!%tXF07$$\"5qmmT5:VLQqF>$!%4XF07$$\"5+++]7y gQqqF>$!%$[%F07$$\"5NLLe9TyV-rF>$!%y`F07$Fd\\s$!%7`F07$$\"5NLL$3-8$f)> (F>$!%(e$F07$Fa\\l$!%WKF07$Ff\\l$\"%\">\"F07$F[]l$\"%aNF07$F`]l$\"%&\\ %F07$Fe]l$\"%#p&F07$Fh]s$\"%pgF07$Fj]l$\"%\"f'F07$F`^sF[_s7$F_^l$\"%Gn F07$Fd^l$\"%;jF07$Fi^l$\"%qaF07$$\"5qmm;H#o)fb%)F>$\"%b\\F07$F^_l$\"%+ YF07$$\"5qmmTN@([Ql)F>$\"%UXF07$$\"5SLL$3x1K*>()F>$\"%JZF07$$\"5qm;a)3 utHv)F>$\"%KZF07$$\"50++D19a,'y)F>$\"%E[F07$$\"5SL$eRs3d!>))F>$\"%C^F0 7$Fc_l$\"%D^F07$$\"5+++]7`aE%)*)F>$\"%xcF07$Fh_l$\"%]iF07$$\"5+++v=#\\ :D=*F>$\"%XiF07$$\"5qmm;aQ))f[#*F>$\"%PiF07$$\"5NLLe*[=#o9$*F>$\"%yfF0 7$F]`l$\"%AfF07$Fi_s$\"%r]F07$Fb`l$\"%%p%F07$Ff`s$\"%OXF07$F[as$\"%&e% F07$F`asF_\\v7$Feas$\"%yYF07$Fjas$\"%d[F07$Fg`l$\"%M^F07$Faal$\"%HtF07 $F[bl$\"%X%)F07$Fhcs$\"%J))F07$F`bl$\"%G$*F07$F`ds$\"%_&)F07$Febl$\"%p fF07$Fjbl$\"%*4#F07$F_cl$!$\"zF07$Fdcl$\"%l@F07$Ficl$\"%mcF07$Fcdl$\"% W:F07$Fgel$!%xNF07$Fafl$!%*f#F07$F[gl$!%S;F07$F`gl$!%`;F07$Fegl$!%D=F0 7$Fjgl$!%`IF07$F_hl$!%&)[F07$Fihl$!%zKF07$Fcil$!#mF07$F]jl$!%M6F07$Fgj l$!%>HF07$$\"5+++D1RA-i8F0$!%8KF07$$\"5+++]i!H$Gl8F0Faj[l7$$\"5+++v=UV ao8F0$!%(=$F07$$\"5++++v$R0=P\"F0$!%FJF07$$\"5+++DJXk1v8F0F^[\\l7$$\"5 +++](o\\F$y8F0$!%FIF07$$\"5+++vV[&)e\"Q\"F0$!%nDF07$Fbhs$!%mDF07$$\"5+ +++D1Q*yR\"F0$!%q:F07$F\\[m$\"$^\"F07$$\"5+++D1k!*>99F0Ff\\\\l7$$\"5++ +]i:,Y<9F0$\"$n\"F07$$\"5+++v=n6s?9F0$\"$F&F07$$\"5++++v=A)RU\"F0Fc]\\ l7$$\"5+++](=K/0V\"F0$\"$%HF07$Fjhs$\"$k#F07$$\"5++++DJ12]9F0$!%u:F07$ Fa[m$!%bBF07$Ff[m$!%`BF07$F[\\m$!%dBF07$$\"5Mek`mt:#zZ\"F0$!%tAF07$F` \\m$!%'f\"F07$$\"5nT5:8oU%Q[\"F0$!%!f\"F07$Fe\\m$!%m:F07$$\"5+Dcwfipw* [\"F0$!$k*F07$$\"5n;H2$)4$GF\\\"F0$!$*zF07$$\"5M3-Q1d'*o&\\\"F0$!$(zF0 7$Fj\\m$!$p)F07$$\"5M$3-j()ptX]\"F0$!%$4\"F07$F_]m$!%#)GF07$$\"5MeRAYS xX8:F0$!%gGF07$$\"5+]7`p(3>k^\"F0$!%jGF07$$\"5nT&QG\\V!Q>:F0$!%5RF07$F d]m$!%O[F07$$\"5n;/wiwWEG:F0$!%,ZF07$Fi]m$!%'[%F07$F^js$!%kUF07$Fcjs$! $f#F07$Fhjs$!$c#F07$F^^m$!$&>F07$F`[t$\"%1BF07$Fe[t$\"%-oF07$Fj[t$\"%: nF07$Fc^m$\"%nsF07$Fb\\t$\"&-F\"F07$Fg\\t$\"&NW\"F07$F\\]t$\"&rH\"F07$ Fh^m$\"&#z5F07$Fd]t$\"%_cF07$Fi]t$\"%\\?F07$Fc^t$\"%$)>F07$F]_m$\"%W7F07$F`am$\"&1E\"F07$Fhat$\"&_E\"F07$F]bt$\"&PE\"F07$Fgbt$\"&YB\" F07$Feam$\"&aA\"F07$Fdct$\"&l7\"F07$Fjam$\"&\"y5F07$$\"5nm;z%\\:>\"[7F 07$Fdbm$\"&B@\"F07$Feft$\"&H6\"F07$Fibm$\"&\"[5F07$F]dm$\"&2Q\"F07$Fgd m$\"&Sh\"F07$F\\jt$\"&8W\"F07$F\\em$\"%\"y)F07$Fdjt$\"%0#)F07$Fijt$\"% PpF07$F^[u$\"%c!)F07$Faem$\"%<**F07$Ffem$\"&5a\"F07$F[fm$\"&sh\"F07$F` fm$\"&D0\"F07$Fefm$\"%\\tF07$$\"5,+DJXR0g*3#F0$\"%\"R(F07$Fjfm$\"%nuF0 7$$\"5,+v$4rVOC5#F0$\"%7()F07$F_gm$\"%P!*F07$$\"5,]i:NgL17@F0$\"%z&*F0 7$$\"5,+DcwMBF:@F0$\"%3'*F07$$\"5,](oz\"48[=@F0$\"%`&*F07$Fdgm$\"%)>*F 07$$\"5,]7y+e#**[7#F0$\"%;#*F07$$\"5,+v=UK#3\"G@F0$\"%n!*F07$$\"5,]Pf$ o?<88#F0$\"%fzF07$Figm$\"%pzF07$F\\^u$\"%#)yF07$F^hm$\"%;oF07$Fd^u$\"% AoF07$Fchm$\"%5oF07$F\\_u$\"%&['F07$Fhhm$\"%*['F07$Fd_u$\"%7lF07$F]im$ \"%+rF07$Fbim$\"%Q$)F07$Fgim$\"%B(*F07$Fg`u$\"&A3\"F07$F\\jm$\"&*46F07 $$\"5oTgFm74h-AF0$\"&39\"F07$Fdau$\"&29\"F07$$\"5Meky(*otI4AF0$\"&)R6F 07$Fajm$\"&+9\"F07$Ffjm$\"&q3\"F07$F[[n$\"%?&*F07$Fbbu$\"%I\"*F07$Fgbu $\"%Q%)F07$F\\cu$\"%i#)F07$F`[n$\"%`\")F07$Fdcu$\"%6))F07$Fe[n$\"%e**F 07$F\\du$\"&U4\"F07$Fj[n$\"&%o6F07$$\"5nTgxySbo?BF0$\"&7>\"F07$$\"5n;/ ,Wf-wBBF0$\"&5>\"F07$$\"5n\"zW#4y\\$oK#F0$\"&%*=\"F07$Fddu$\"&_:\"F07$ $\"5nTNrR:W)HL#F0$\"&=:\"F07$$\"5n;z%\\S8fgL#F0$\"&1:\"F07$$\"5n\"H#=q _Q8RBF0$\"&D6\"F07$F_\\n$\"&(f5F07$Fd\\n$\"%2&*F07$Fi\\n$\"%5))F07$F]^ nFebs7$Fa_n$\"&!=5F07$Fe`n$\"%A#*F07$Fian$\"%.nF07$$\"5n\"z>6[c_7U#F0$ \"%R]F07$F^hu$\"%\")\\F07$$\"5M3FWqjb)yU#F0$\"%N]F07$F^bn$\"%veF07$F[i u$\"%0iF07$Fcbn$\"&40\"F07$Fhiu$\"&45\"F07$F]ju$\"&)\\#F07$Fd[v$\"&%p@F07$F]c n$\"&\"=AF07$Fbcn$\"&ts\"F07$Fgcn$\"&,9\"F07$Fadn$\"&ti\"F07$F_fn$\"&$ \\=F07$F]_v$\"&kq\"F07$Fdfn$\"&-k\"F07$Fd`v$\"&0r\"F07$Fifn$\"&4!=F07$ $\"5M$3-8Da3-r#F0$\"&I\"=F07$$\"5omT5lDho;FF0$\"&F\"=F07$$\"5,]i!*y3P; BFF0$\"&$*z\"F07$F\\av$\"&Vx\"F07$$\"5,+DJ?ekfUFF0$\"&or\"F07$F^gn$\"& Bn\"F07$Fdav$\"&Sk\"F07$Fiav$\"&Wk\"F07$F^bv$\"&Bk\"F07$Fcbv$\"&Ek\"F0 7$Fhbv$\"&yk\"F07$F]cv$\"&Cn\"F07$Fbcv$\">\"F07$Fcgn$\"&1$=F07$Fidv$ \"&E0#F07$Fhgn$\"&GA#F07$$\"5,]P%)*e4aU'GF0$\"&[C#F07$$\"5ML3_]%3+w'GF 0$\"&WE#F07$$\"5o;z>6tg%4(GF0$\"&.D#F07$$\"5,+](=<1#HuGF0$\"&rF#F07$$ \"5M$3_D.0Qw(GF0$\"&0H#F07$$\"5om\"HK*QS)4)GF0$\"&W8#F07$$\"5,]i!Rv-IV )GF0$\"&P;#F07$F`fv$\"&s;#F07$$\"5,+v$fL*zO%*GF0$\"&r)=F07$$\"5om;Hdq* f5!HF0$\"&=c\"F07$$\"5,](oz\"ffS/HF0$\"&ue\"F07$$\"5MLekyZ>v2HF0$\"&8g \"F07$$\"5o;HKROz46HF0$\"&?Z\"F07$F]hn$\"&r\\\"F07$Fbhn$\"&Br\"F07$Fgh n$\"&7z\"F07$F\\in$\"&!)=#F07$Fain$\"&N\\#F07$$\"5,]7Gj)\\/R%HF0$\"&.` #F07$$\"5,+DJq&*yF07$Figv$\"&gy\"F07$$ \"5,]ilZj_=$)HF0$\"&Zy\"F07$F^hv$\"&$)z\"F07$$\"5,](= F07$Fh\\o$\"&29#F07$F]]o$\"&1C#F07$$\"5,v=F07$Ff^o$\"&T&>F07$Fa^w$\"&/)>F07 $F[_o$\"&%))>F07$F`_o$\"&!f@F07$Fe_o$\"&RD#F07$$\"5,Dcw4DxX3KF0$\"&_E# F07$Fd_w$\"&sJ#F07$$\"5,voa&=&4\\:KF0$\"&oJ#F07$Fj_o$\"&(=BF07$$\"5,D \"G8'yT_AKF0$\"&cK#F07$F\\`w$\"&WK#F07$$\"5,v$4r`Sd&HKF0$\"&%*G#F07$F_ `o$\"&0F#F07$Fi`o$\"&z3#F07$F]bo$\"&^h\"F07$Fdcw$\"&?j\"F07$Ficw$\"&Q: #F07$F^dw$\"&97#F07$Fbbo$\"&h9#F07$F[ew$\"&)RFF07$Fgbo$\"&.7$F07$$\"5M L$e9\"fm#zO$F0$\"&X,$F07$F\\co$\"&c)GF07$$\"5,+]P4Y=U\"Q$F0$\"&:P#F07$ Faco$\"&BB#F07$Fcfw$\"&8y\"F07$F]gw$\"&1a\"F07$$\"5om\"H2LUQ]S$F0$\"&1 _\"F07$Fbgw$\"&P_\"F07$$\"5,+voz;gy6MF0$\"&Eg\"F07$Ffco$\"&]e\"F07$Fjg w$\"&O'=F07$F[do$\"&&z@F07$Fedo$\"&HJ#F07$F_eo$\"&sX#F07$Fieo$\"&rR#F0 7$Fcfo$\"&zC#F07$Fhfo$\"&;B#F07$F]go$\"&UC#F07$Fbgo$\"&ZK#F07$Fggo$\"& $pBF07$$\"5,vVt_!>)z'f$F0$\"&*pBF07$Fjiw$\"&()R#F07$$\"5o\"zpB,w#[.OF0 F^g_l7$F\\ho$\"&*)R#F07$Fbjw$\"&[S#F07$Faho$\"&_Q#F07$Ffho$\"&dM#F07$F [ioF]ew7$$\"5MeRZ-#Hx+l$F0$\"&iD#F07$$\"5o;z%\\:7AKl$F0$\"&jD#F07$$\"5 ,v=U2^pOcOF0Fhh_l7$$\"5MLe*)f!y6&fOF0$\"&GC#F07$$\"5,]P%['R9!em$F0$\"& UB#F07$F`io$\"&'RAF07$$\"5,+voz;/n%o$F0$\"&$zAF07$Feio$\"&kN#F07$Fijo$ \"&_h#F07$Fc[p$\"&mv#F07$Ff[x$\"&'))GF07$Fh[p$\"&T%HF07$F^\\x$\"&9%GF0 7$F]\\p$\"&6F#F07$Fb\\p$\"&CL#F07$Fg\\p$\"&_t#F07$F\\]p$\"&'zJF07$Fa]p $\"&*)Q$F07$F\\^x$\"&a\\$F07$Ff]p$\"&Aa$F07$Fd^x$\"&O[$F07$F[^p$\"&(fM F07$Fc`p$\"&5:$F07$Fgap$\"&Q%HF07$Fabp$\"&xI$F07$F[cp$\"&\"4MF07$Fecp$ \"&fB$F07$F_dp$\"&i4$F07$Fidp$\"&kE$F07$FcepF]]l7$Fgfp$\"&dF$F07$Fagp$ \"&]6$F07$Fbhx$\"&+:$F07$Ffgp$\"&<@$F07$Fjhx$\"&'yJF07$F[hp$\"&r9$F07$ F_ip$\"&1!HF07$Fiip$\"&w\\#F07$F^jp$\"&Im#F07$Fcjp$\"&e:$F07$Fe]y$\"&A u$F07$Fhjp$\"&)3RF07$Fb^y$\"&@&QF07$F][q$\"&7e$F07$$\"5o;ajMj))ezUF0$ \"&&eJF07$Fj^y$\"&`!HF07$$\"5N$eRse$[t#H%F0$\"&pT#F07$Fb[q$\"&]H#F07$$ \"5N3FpE:Vf-VF0$\"&3@#F07$$\"5-]P%)R33)eI%F0$\"&\\=#F07$$\"5o\"z%*H:In \"4VF0$\"&WB#F07$Fb_y$\"&!4BF07$$\"5o;zW#4yE!>VF0$\"&6P#F07$Fg[q$\"&z] #F07$Fa\\q$\"&,)GF07$F[]q$\"&o*HF07$$\"5M3_v:.O?\"Q%F0$\"&6/$F07$F``y$ \"&$QIF07$$\"5o\"HK*)[\\VsQ%F0$\"&k.$F07$F`]q$\"&60$F07$F]ay$\"&y/$F07 $Fe]q$\"&d-$F07$Fj]q$\"&w(HF07$F_^q$\"&%>HF07$Fd^q$\"&/'GF07$Fi^q$\"&Z #GF07$$\"5,](o/)\\AqcWF0$\"&^#GF07$F^_q$\"&G%GF07$$\"5o;H#oK.#yoWF0$\" &T&GF07$Fc_q$\"&:)GF07$$\"5ML$3_+sm')[%F0$\"&$RHF07$Fh_q$\"&`*HF07$$\" 5ML3-j7RV4XF0$\"&[*HF07$$\"5,+]i:5jN;XF0$\"&Y*HF07$$\"5om\"H#o2(yK_%F0 $\"&*pHF07$F]`q$\"&s#HF07$$\"5om;/E+f/WXF0$\"&(zGF07$Fb`q$\"&/%GF07$$ \"5omT&QG48[c%F0$\"&c#GF07$$\"5ML$ek.\\Nv XF0$\"&,%GF07$$\"5,+D1*y)ylyXF0$\"&D&GF07$$\"5M$ek`m3>@e%F0$\"&n)GF07$ Fg`q$\"&())GF07$F[bq$\"&IJ$F07$Febq$\"&z[$F07$Fjbq$\"&GE$F07$F_cq$\"&. =$F07$Fdcq$\"&8p$F07$Ficq$\"&'zUF07$$\"5-DJ&pP\"evSZF0$\"&*GWF07$F^dq$ \"&=\\%F07$$\"5oTg-N9%F07$F bhy$\"&'pTF07$Fbeq$\"&$4RF07$Fhhy$\"&Q%RF07$Fgeq$\"&G(RF07$F`iy$\"&Z$R F07$F\\fq$\"&>'RF07$Fhiy$\"&+*RF07$Fafq$\"&/6%F07$F`jy$\"&@:%F07$Fejy$ \"&SG%F07$Fjjy$\"&qI%F07$Fffq$\"&(>VF07$Fb[z$\"&W@%F07$F[gq$\"&d0%F07$ Fegq$\"&@@%F07$F_hq$\"&ja%F07$$\"5-]i!RvZ]\")*[F0$\"&\"[XF07$$\"5-+DJq X/%\\!\\F0$\"&>c%F07$$\"5-](=nQTI<\"\\F0$\"&E_%F07$F`\\z$\"&e]%F07$$\" 5-+v$f$=.5K\\F0$\"&TN%F07$Fdhq$\"&iF%F07$$\"5+DJ&p(Q_2\\\\F0$\"&0E%F07 $$\"5+]i:&GAqC&\\F0$\"&XB%F07$$\"5+v$fLp?le&\\F0$\"&YB%F07$$\"5++Dc,\" >g#f\\F0$\"&$QUF07$$\"5+Dcw4v^li\\F0$\"&6C%F07$$\"5+](oz\"f,0m\\F0$\"& $!$X(F07$F< $!$O&F07$Fa^r$\"%s$!$!GF07$$\"5MLL3Fp 1nrJF>$!$:#F07$$\"5ML$e*)f!y@0KF>$\"$I*F07$Ffr$\"$M*F07$$\"5ML$3F%z?Js KF>$\"$G*F07$$\"5MLLe9;#feI$F>$\"$u%F07$$\"5ML$ekGN1%RLF>$\"$v%F07$F[s $\"$5%F07$Fes$!%$\\#F07$F_t$!%8LF07$Fit$!%[;F07$Fcu$\"$i\"F07$$\"5omT5 !p:g[C%F>$\"$?&F07$Fhu$\"$z%F07$$\"5o;/E+OK-XVF>Fa_bl7$$\"5NLek.H4TyVF >$\"$Y%F07$$\"5-]7.2A')z6WF>$\"#KF07$F]v$\"#JF07$Fbv$!$f(F07$Fgv$!%w?F 07$F\\w$!%JBF07$Faw$!%l?F07$Ffw$!$'[F07$F[x$\"$\\&F07$$\"5++]7G8cWx_F> $\"$;'F07$F`x$\"%T8F07$$\"5qm;/,W\\NR`F>$\"%S8F07$FexF]_r7$Fjx$\"%m;F0 7$F_y$\"%'f\"F07$Fdy$\"%g5F07$Fiy$\"$Z(F07$$\"5qmm\"zW#\\n.eF>$\"$h(F0 7$F[iz$\"$\\(F07$$\"5qm;Hd?*Ql*eF>$\"$*pF07$$\"5+++v$fe$\\FfF>$\"$,(F0 7$$\"5NL$3-8D[%efF>$\"$(oF07$F^z$\"$*GF07$Fciz$!%MLF07$Fcz$!%ftF07$F[j z$!%vtF07$Fhhr$!&.-\"F07$Fcjz$!&$35F07$Fhz$!%\\**F07$Feir$!%@')F07$F][ l$!%&G&F07$Fbjr$!%rAF07$Fgjr$!%%>#F07$F\\[s$\"$7(F07$Fb[l$\"%nhF07$Fd[ s$\"%\\uF07$Fg[l$\"&o=\"F07$$\"5+++v$4@?>y'F>$\"&(*=\"F07$Fc\\[l$\"&'[ 7F07$$\"5qmm\"zptBg%oF>$\"&!H7F07$F\\\\s$\"&A=\"F07$F[][l$\"%$G)F07$F \\\\l$\"%-MF07$Fc][l$\"%cLF07$Fh][l$\"%5KF07$F]^[l$\"$#fF07$Fd\\s$\"$& eF07$Fe^[l$\"$u'F07$Fa\\l$\"$8(F07$F[]l$\"%?cF07$Fe]l$\"%dqF07$Fj]l$\" %;zF07$F_^l$\"%%R)F07$Fd^l$\"%M()F07$Fi^l$\"%7')F07$Fi`[l$\"%v#)F07$F^ _l$\"%rxF07$Faa[l$\"%CvF07$Ffa[l$\"%^tF07$F`b[l$\"%ctF07$Fc_l$\"%,uF07 $F]c[l$\"%BxF07$Fh_l$\"%S%)F07$Fjc[l$\"%U()F07$F]`l$\"%W))F07$Fb`l$\"% _zF07$Fg`l$\"%@qF07$Faal$\"%8))F07$F[bl$\"&u4\"F07$Fhcs$\"&!)>\"F07$F` bl$\"&eQ\"F07$$\"5++D19WCxw5F0$\"&ZT\"F07$F`ds$\"&3[\"F07$$\"5+]7G8'G` d3\"F0$\"&\")\\\"F07$$\"5ML3-8+pu)3\"F0$\"&.\\\"F07$$\"5n;/w790u\"4\"F 0$\"&iT\"F07$Febl$\"&YV\"F07$Fjbl$\"&D@\"F07$F_cl$\"%%)*)F07$$\"5+]i:5 amn@6F0$\"%XxF07$$\"5MLe*)4o-nC6F0$\"%iyF07$$\"5n;aj4#)QmF6F0$\"%uzF07 $Fdcl$\"%V&)F07$$\"5nmT&)3CZkO6F0$\"%-))F07$Ficl$\"&O1\"F07$Fdes$\"&L7 \"F07$F^dl$\"&H>\"F07$F\\fs$\"&O5\"F07$Fcdl$\"%J**F07$Fhdl$\"%L%*F07$F ]el$\"%6')F07$Fbel$\"%z'*F07$Fgel$\"&L+\"F07$F[gl$\"&D]\"F07$Fegl$\"&= g\"F07$Fjgl$\"&\"H:F07$F_hl$\"&aC\"F07$Fgjl$\"&tT\"F07$F\\[m$\"&B]\"F0 7$Fa[m$\"&sg\"F07$F[\\m$\"&2d\"F07$Fe\\m$\"&]b\"F07$Fj\\m$\"&hY\"F07$F _]m$\"%C)*F07$Fd]m$\"%K`F07$Fi]m$\"%*)GF07$F^^m$\"%p]F07$Fc^m$\"&=B\"F 07$Fb\\t$\"&4#>F07$Fg\\t$\"&8O#F07$F\\]tFb]v7$Fh^m$\"&?<#F07$Fd]t$\"& \\p\"F07$Fi]tFhj\\l7$Fc^t$\"&*e6F07$F]_m$\"%C%*F07$Fg_m$\"&:J\"F07$F[a m$\"&**\\\"F07$F`am$\"&2g\"F07$Feam$\"&*o;F07$Fjam$\"&?h\"F07$F_bm$\"& .a\"F07$Fddt$\"&t`\"F07$Fidt$\"&!o:F07$Fcet$\"&2h\"F07$Fdbm$\"&Gn\"F07 $F`ft$\"&&y;F07$Feft$\"&Tk\"F07$F_gt$\"&vd\"F07$Fibm$\"&.`\"F07$F]dm$ \"&9v\"F07$Fgdm$\"&'QBF07$F\\em$\"&[8#F07$Faem$\"&J*>F07$Ffem$\"&uL#F0 7$F[fm$\"&b\\#F07$Ff\\u$\"&HS#F07$F`fm$\"&CJ#F07$F^]u$\"&xI#F07$Fefm$ \"&gH#F07$F_gm$\"&W*GF07$Figm$\"&a,$F07$Fchm$\"&#fGF07$F]im$\"&%3FF07$ Fbim$\"&qu#F07$Fgim$\"&Q$GF07$F\\jm$\"&(yHF07$Fajm$\"&x3$F07$$\"5M$3_] M0_$>AF0$\"&!*4$F07$Ffjm$\"&n5$F07$$\"5o;H23m\\uKAF0$\"&@3$F07$F[[n$\" &#\\IF07$Fgbu$\"&Y%HF07$F`[n$\"&:(GF07$$\"5M3-Q\"eZv&pAF0$\"&7%GF07$$ \"5n;a8(RqBHF#F0$\"&g#GF07$$\"5+D1*G@$>FwAF0$\"&d#GF07$Fdcu$\"&B#GF07$ $\"5nT5SW)QoHG#F0$\"&?#GF07$$\"5+]i:g;mJ'G#F0$\"&C#GF07$$\"5Me9\"fZ%[m *G#F0$\"&A&GF07$Fe[n$\"&c'GF07$Fj[n$\"&!fIF07$F_\\n$\"&l?$F07$Fi\\n$\" &x4$F07$Fa_n$\"&>'HF07$Fian$\"&/=#F07$Fcbn$\"&!o=F07$Fhiu$\"&m!>F07$F] ju$\"&$yDF07$Fbju$\"&z`#F07$Fhbn$\"&]c#F07$F_[v$\"&B>$F07$F]cn$\"&XZ$F 07$Fbcn$\"&%4JF07$Fgcn$\"&0Y#F07$$\"5M$3-jP(=U/DF0$\"&h-#F07$Fb\\v$\"& m%=F07$$\"5M3_DG1mv9DF0$\"&V#=F07$$\"5M$eRAr^,#=DF0$\"&o\"=F07$$\"5MeR A'zUY;_#F0$\"&()*=F07$F\\dn$\"&)z=F07$Fj\\v$\"&52#F07$Fadn$\"&vB#F07$F een$\"&JO#F07$F_fn$\"&#[CF07$Fifn$\"&;T#F07$F`^^l$\"&*[CF07$F\\av$\"&4 Z#F07$F]_^l$\"&\"[CF07$F^gn$\"&\"3CF07$Fiav$\"&5N#F07$F]cv$\"&BH#F07$$ \"5M$3Fp,aRzy#F0$\"&XH#F07$Fbcv$\"&hI#F07$$\"5M3-j([\"fl(z#F0$\"&8J#F0 7$$\"5,]7`W1Z*3!GF0$\"&EO#F07$$\"5o\"HK9!)\\LT!GF0$\"&cO#F07$Fcgn$\"&J P#F07$Fidv$\"&Mp#F07$Fhgn$\"&a0$F07$Fbb^l$\"&&pKF07$F`fv$\"&oS$F07$F^d ^l$\"&43$F07$F]hn$\"&D+$F07$Fain$\"&bf$F07$F[jn$\"&iw$F07$F`jn$\"&>.%F 07$Fejn$\"&vf%F07$F_[o$\"&P\\%F07$Fi[o$\"&@N%F07$$\"5,v=U2^spZIF0$\"&` M%F07$Fhjv$\"&CD%F07$$\"5,Dc^.6;.aIF0$\"&RD%F07$F^\\o$\"&OD%F07$$\"5,v $4'*4(fOgIF0$\"&hB%F07$F`[w$\"&pB%F07$$\"5,DJq&4L+n1$F0$\"&\"QUF07$Fc \\o$\"&EG%F07$F]]o$\"&_[%F07$Fg]o$\"&/j%F07$Fa^o$\"&5^%F07$F[_o$\"&iM% F07$Fe_o$\"&!*\\%F07$F_`o$\"&@t%F07$Fd`o$\"&pk%F07$Fi`o$\"&F]%F07$F^ao $\"&9L%F07$Fcao$\"&G$QF07$Fhao$\"&)fIF07$F]boFgj`l7$Fdcw$\"&[(GF07$Fic w$\"&&zKF07$F^dw$\"&*HKF07$Fbbo$\"&EC$F07$F[ew$\"&.)QF07$Fgbo$\"&iZ%F0 7$$\"5om\"zpB'GbkLF0$\"&>S%F07$Feb_l$\"&6O%F07$$\"5,+v$feX+8P$F0$\"&-^ %F07$F\\co$\"&bV%F07$F]c_l$\"&+/%F07$Faco$\"&\"\\QF07$Fcfw$\"&`L$F07$F ]gw$\"&F!HF07$Fbgw$\"&F\"GF07$Ffco$\"&]q#F07$Fjgw$\"&*)z#F07$F[do$\"&N -$F07$F`do$\"&v0$F07$Fedo$\"&53$F07$Fjdo$\"&G8$F07$F_eo$\"&v@$F07$Fdeo $\"&&[KF07$Fieo$\"&.E$F07$$\"5M$e*[=\"QO;_$F0$\"&0D$F07$F^fo$\"&#[KF07 $$\"5,](=nj-lQ`$F0$\"&/A$F07$Fcfo$\"&8=$F07$Fhfo$\"&H9$F07$F]go$\"&'>J F07$Fbgo$\"&(QJF07$Fggo$\"&X<$F07$F\\ho$\"&i@$F07$FahoF_^`l7$Ffho$\"&5 E$F07$F[io$\"&GB$F07$F^i_l$\"&:;$F07$F`io$\"&F5$F07$F[j_l$\"&\")3$F07$ Feio$\"&%=JF07$Fijo$\"&\"=MF07$Fc[p$\"&zo$F07$Ff[x$\"&7(RF07$Fh[p$\"&N @%F07$$\"5ML$eRs`R/y$F0$\"&ZD%F07$F^\\x$\"&!pVF07$$\"5,+](=#\\&)e$z$F0 Fh[el7$F]\\p$\"&([SF07$Fg\\p$\"&PL%F07$F[^p$\"&\\2&F07$F`^p$\"&c5&F07$ Fe^p$\"&t:&F07$F__p$\"&*3ei HSF0$\"&E7'F07$Fecp$\"&u5'F07$Fjcp$\"&B(fF07$F_dp$\"&n*eF07$$\"5M$3x\" )HP`(pSF0$\"&[(eF07$Fddp$\"&'peF07$$\"5,]7`p(e:F3%F0$\"&&))eF07$Fidp$ \"&a\"fF07$F^ep$\"&%\\gF07$Fcep$\"&6:'F07$$\"5N3_v:y;O=TF0$\"&T;'F07$F hep$\"&$HiF07$$\"5o\"HK9byU[7%F0$\"&zA'F07$F]fp$\"&5B'F07$Fbfp$\"&\\D' F07$Fgfp$\"&4B'F07$F\\gp$\"&(ehF07$Fagp$\"&G2'F07$Ffgp$\"&/)fF07$F[hp$ \"&9u&F07$Fehp$\"&!p`F07$F_ip$\"&fE&F07$Fdip$\"&!eZF07$Fiip$\"&EM%F07$ Fc[y$\"&UB%F07$F^jp$\"&z?%F07$F`\\y$\"&s:%F07$Fc\\y$\"&ug%F07$Fh\\y$\" &bg%F07$Fcjp$\"&va%F07$Fe]y$\"&*>_F07$Fhjp$\"&/c&F07$Fb^y$\"bF07$F] [q$\"&![aF07$F]a`l$\"&i0&F07$Fj^y$\"&>y%F07$Fea`l$\"&_=%F07$Fb[q$\"&!H RF07$Fbb`l$\"&xp$F07$Fb_y$\"&'=OF07$$\"5-voHz(GSdJ%F0$\"&8e$F07$F_c`l$ \"&&3OF07$$\"5Ne*)f0uKJAVF0$\"&.p$F07$Fg[q$\"&#fOF07$F\\\\q$\"&xy$F07$ Fa\\q$\"&y)QF07$$\"5o;/E])pk%eVF0$\"&;(QF07$Ff\\q$\"&i*QF07$$\"5oTNr*y SF07$Fe]q$ \"&&[SF07$Fj]q$\"&W0%F07$F_^q$\"&#QSF07$Fd^q$\"&l*RF07$Fi^q$\"&'HRF07$ F^_q$\"&:\"RF07$Fc_q$\"&c\"RF07$F_g`l$\"&#\\RF07$Fh_q$\"&'=SF07$F\\h`l $\"&&\\SF07$F]`q$\"&/0%F07$Fb`q$\"&\"pRF07$Fg`q$\"&o)QF07$Faaq$\"&W7%F 07$F[bq$\"&'F07$FQ$\"%(y$F07$F_o$\"%uHF07$Fdo$\"%$y&F0 7$Fio$\"%=))F07$Fjar$\"%X**F07$F^p$\"&w+\"F07$Fbbr$\"&1-\"F07$Fcp$\"&Z +\"F07$Fhp$\"&)G5F07$F]q$\"&o.\"F07$Fgq$\"&y?\"F07$Far$\"&0o\"F07$F\\ \\bl$\"&uo\"F07$Fa\\bl$\"&\")p\"F07$Ff\\bl$\"&!zo\"F07$F]v$\"&=o\"F07$F bv$\"&9g\"F07$Fgv$\"&n[\"F07$$\"5NL3FWXK1zZF>$\"&][\"F07$F\\w$\"&aZ\"F 07$$\"5N$3FWXKE#z[F>$\"&!y9F07$$\"5-+D\"yv,9E\"\\F>$\"&'3:F07$$\"5o;z> h5<+Y\\F>$\"&&3:F07$Faw$\"&_^\"F07$Ffw$\"&mn\"F07$F[x$\"&3x\"F07$Ffabl $\"&hx\"F07$F`x$\"&#Q=F07$F^bbl$\"&x$=F07$Fex$\"&$Q=F07$Fjx$\"&1'=F07$ F_y$\"&?&=F07$$\"5qm;z>OH3DbF>$\"&=%=F07$Fbgr$\"&,%=F07$$\"5NL$3FpE#*p e&F>$\"&G$=F07$Fdy$\"&n!=F07$Fjgr$\"&8!=F07$Fiy$\"&\")y\"F07$F`cbl$\"& 7\"=F07$F[iz$\"&')z\"F07$Fhcbl$\"&k!=F07$F]dbl$\"&h!=F07$Fbdbl$\"&iz\" F07$F^z$\"&Hv\"F07$Fciz$\"&&[8F07$Fcz$\"%2&)F07$F[jz$\"%!H)F07$Fhhr$\" %xWF07$Fcjz$\"%BWF07$Fhz$\"%[VF07$F`ir$\"%kRF07$Feir$\"%zQF07$Fjir$\"% _QF07$F][l$\"%XdF07$Fbjr$\"%UwF07$Fgjr$\"%bvF07$F\\[s$\"%`%*F07$Fb[l$ \"&eK\"F07$Fd[s$\"&nP\"F07$Fg[l$\"&5j\"F07$Fc\\[l$\"&$*\\\"F07$F\\\\s$ \"&6T\"F07$$\"5NLL3-js75pF>$\"%q%*F07$F[][lFff[l7$$\"5+++D1*yIU(pF>$\" %n))F07$F\\\\l$\"%tLF07$Fd\\sF]cz7$Fa\\l$!%c5F07$F[]l$\"%BBF07$Fe]l$\" %VNF07$Fj]l$\"%8VF07$F_^l$\"%%o%F07$$\"5SLLLe9TaC\")F>$\"%^[F07$Fd^l$ \"%H[F07$$\"50+++DJg8d#)F>$\"%(y%F07$Fi^l$\"%EXF07$Fi`[l$\"%STF07$F^_l $\"%fOF07$Faa[l$\"%[MF07$Ffa[l$\"%ZLF07$F`b[l$\"%qLF07$Fc_l$\"%lMF07$F ]c[l$\"%KQF07$Fh_l$\"%=XF07$Fjc[l$\"%cZF07$F]`l$\"%uZF07$Fb`l$\"%DF07$$\"5+++vo/=# \\K\"F0$\"&>`#F07$F]jl$\"&Y]#F07$Fbjl$\"&VW#F07$Fgjl$\"&/N#F07$F\\[m$ \"&&zCF07$Fa[m$\"&0d#F07$F[\\m$\"&Ob#F07$Fe\\m$\"&pb#F07$Fj\\m$\"&kZ#F 07$F_]m$\"&<'>F07$Fd]m$\"&GV\"F07$Fi]m$\"&Q0\"F07$F^js$\"&4/\"F07$Fcjs $\"&&)4\"F07$Fhjs$\"&X3\"F07$F^^m$\"&L2\"F07$F`[t$\"&*=7F07$Fe[t$\"&g^ \"F07$Fj[t$\"&b\\\"F07$Fc^m$\"&4`\"F07$Fb\\t$\"&/+#F07$Fg\\t$\"&JA#F07 $F\\]t$\"&73#F07$Fh^m$\"&y(=F07$$\"5M3x\"[N!*p&*e\"F0$\"&,]\"F07$Fd]t$ \"&%)G\"F07$$\"5n\"zpB,c7mf\"F0$\"&wD\"F07$Fi]t$\"%&3(F07$Fc^t$\"%SnF0 7$F]_m$\"%XQF07$Fg_m$\"%RfF07$F[am$\"%WvF07$F`am$\"%&[)F07$Feam$\"%)*) )F07$Fjam$\"%'4)F07$F_bm$\"%fuF07$Fidt$\"%2zF07$Fdbm$\"%R))F07$$\"5o;H KknnZG=F0$\"%S))F07$F`ft$\"%@))F07$$\"5M$3-QckcB%=F0$\"%$f)F07$Feft$\" %5%)F07$F_gt$\"%oxF07$FibmF]^]l7$Fccm$\"%)*zF07$F]dm$\"&U+\"F07$Fbdm$ \"&F07$F[fm$\"&xH#F07$F`fm$\"&RT#F07$Fefm$\"&Ba#F07$Fjfm$\"&r!GF07$F_gm $\"&(fJF07$F[_]l$\"&wF$F07$F`_]l$\"&zG$F07$Fe_]l$\"&**H$F07$Fdgm$\"&gI $F07$F]`]l$\"&HJ$F07$Fb`]l$\"&MI$F07$Fg`]l$\"&')>$F07$Figm$\"&G?$F07$F chm$\"&\"\\IF07$F]im$\"&q#HF07$Fbim$\"&%yHF07$Fgim$\"&;2$F07$F\\jm$\"& ;@$F07$Fajm$\"&NI$F07$Fa]dl$\"&pI$F07$Ffjm$\"&!3LF07$Fi]dl$\"&F07$F^bn$ \"&$)p\"F07$Ffhu$\"&dn\"F07$F[iu$\"&Rm\"F07$F`iu$\"&x!=F07$Fcbn$\"&7y \"F07$Fhbn$\"&0C#F07$F]cn$\"&)fFF07$$\"5M$ek.a0LwZ#F0$\"&3l#F07$Fbcn$ \"&gG#F07$$\"5Me*[VPb#e([#F0$\"&<@#F07$$\"5n;/,>`!**3\\#F0$\"&mk\"F07$ $\"5+v=nj_b@%\\#F0$\"&>i\"F07$Fgcn$\"&Zc\"F07$Fecdl$\"&r1\"F07$Fb\\v$ \"%F#)F07$Fbddl$\"%J!)F07$F\\dn$\"%#>)F07$Fj\\v$\"%K'*F07$Fadn$\"&x4\" F07$Feen$\"&OA\"F07$F_fn$\"&;H\"F07$F]_v$\"&iC\"F07$Fdfn$\"&!p6F07$$\" 5,+](=#*HXxm#F0$\"&\"[6F07$Fd`v$\"&\\;\"F07$$\"5om;HKRdt\"p#F0$\"&a?\" F07$Fifn$\"&YC\"F07$$\"5oT5S%4vppq#F0$\"&bC\"F07$F[^^l$\"&jF\"F07$$\"5 ,DJ?3MtW8FF0Fafjl7$F`^^l$\"&nF\"F07$Fe^^l$\"&5H\"F07$F\\av$\"&lG\"F07$ F]_^l$\"&nD\"F07$F^gn$\"&e@\"F07$Fiav$\"&b;\"F07$F]cv$\"&*H6F07$Fbcv$ \"&&e6F07$Fcgn$\"&aB\"F07$Fidv$\"&;`\"F07$Fhgn$\"&#y=F07$Fbb^l$\"&Y4#F 07$F`fv$\"&rC#F07$F^d^l$\"&g*>F07$F]hn$\"&/\">F07$Fbhn$\"&+(>F07$Fghn$ \"&m/#F07$F\\in$\"&PK#F07$Fain$\"&Pn#F07$Ffin$\"&H)HF07$F[jn$\"&_7$F07 $F`jn$\"&jg$F07$Fejn$\"&f2%F07$F_[o$\"&z$RF07$Fi[o$\"&#*z$F07$F^[el$\" &Kz$F07$Fhjv$\"&Mr$F07$Ff[el$\"&Zr$F07$F^\\o$\"&]r$F07$F^\\el$\"&Ar$F0 7$F`[w$\"&Hr$F07$Ff\\el$\"&Wr$F07$Fc\\o$\"&'oPF07$F]]o$\"&T(RF07$Fg]o$ \"&@3%F07$Fa^o$\"&*[RF07$F[_o$\"&<\"QF07$Fe_o$\"&i)RF07$F_`o$\"&F>%F07 $Fd`o$\"&\\7%F07$Fi`o$\"&92%F07$F^ao$\"&%HRF07$Fcao$\"&\\V$F07$$\"5om \"z%\\!pH_J$F0$\"&))3$F07$Fhao$\"&Gf#F07$$\"5,+v$4Y-pyK$F0$\"&>^#F07$F ]bo$\"&bJ#F07$Fdcw$\"&nH#F07$Ficw$\"&C`#F07$F^dw$\"&T\\#F07$Fbbo$\"&u \\#F07$F[ew$\"&%[HF07$Fgbo$\"&>M$F07$Feb_l$\"&6C$F07$F\\co$\"&S=$F07$F ]c_l$\"&Wo#F07$Faco$\"&\"=DF07$Fcfw$\"&E&>F07$F]gw$\"&RY\"F07$Fbgw$\"& ?R\"F07$Ffco$\"&zC\"F07$Fjgw$\"&GI\"F07$F[do$\"&E[\"F07$Fedo$\"&8b\"F0 7$F_eoF]ggl7$Fdeo$\"&3q\"F07$Fieo$\"&'*p\"F07$F\\delF_icl7$F^foFb`jl7$ Fddel$\"&@k\"F07$Fcfo$\"&Ag\"F07$Fggo$\"&0i\"F07$F[io$\"&Al\"F07$F^i_l Fhd_l7$F`io$\"&*R:F07$$\"5M$eRZxv!QyOF0$\"&)R:F07$F[j_l$\"&'R:F07$$\"5 Me9;KY_\"yo$F0$\"&La\"F07$$\"5o;aj%e2g4p$F0$\"&ud\"F07$$\"5,v$4r`!\\5% p$F0$\"&)y:F07$FeioF\\ht7$Fijo$\"&](=F07$Fc[p$\"&^3#F07$Ff[xF[g_l7$Fh[ p$\"&()f#F07$Fjgel$\"&1n#F07$F^\\x$\"&Nu#F07$Fbhel$\"&]l#F07$F]\\p$\"& 5Z#F07$$\"5,+DJ?3.X.QF0$\"&3^#F07$$\"5om;z>hvt1QF0$\"&\"\\DF07$$\"5ML3 F>9[-5QF0$\"&mT#F07$Fb\\p$\"&mX#F07$$\"5ML$3xJd'))>QF0$\"&&)e#F07$Fg\\ p$\"&Lo#F07$F\\]p$\"&E-$F07$Fa]p$\"&(*G$F07$Ff]p$\"&WZ$F07$F[^p$\"&*zO F07$F`^p$\"&l#QF07$Fe^p$\"&H)RF07$F__p$\"&yT%F07$Fc`p$\"&>s%F07$Fcax$ \"&$R[F07$Fh`p$\"&E'[F07$F`bx$\"&R\"[F07$F]ap$\"&#*o%F07$Fbap$\"&za%F0 7$Fgap$\"&e[%F07$Fabp$\"&(yYF07$F[cp$\"&$e[F07$F\\\\fl$\"&%p[F07$F`cp$ \"&P&[F07$Fd\\fl$\"&\"=[F07$Fecp$\"&.![F07$Fjcp$\"&vm%F07$F_dp$\"&Xg%F 07$Fb]fl$\"&Sf%F07$Fddp$\"&=f%F07$Fj]fl$\"&Hi%F07$Fidp$\"&Jl%F07$F^ep$ \"&!*y%F07$Fcep$\"&G)[F07$Fh^fl$\"&Q*[F07$Fhep$\"&,&\\F07$F`_fl$\"&\" \\\\F07$F]fp$\"&8&\\F07$Fbfp$\"&)p\\F07$Fgfp$\"&([\\F07$F\\gp$\"&k*[F0 7$Fagp$\"&y%[F07$Fbhx$\"&o%[F07$Ffgp$\"&=$[F07$Fjhx$\"&)yZF07$F[hp$\"& ]j%F07$Fehp$\"&[F%F07$F_ip$\"&'*=%F07$Fdip$\"&%pOF07$Fiip$\"&q>$F07$Fc [y$\"&H6$F07$F^jp$\"&u(HF07$F`\\y$\"&*RHF07$Fc\\y$\"&!GKF07$Fh\\y$\"&] A$F07$Fcjp$\"&P=$F07$Fe]y$\"&vm$F07$Fhjp$\"&g(QF07$Fb^y$\"&B$QF07$F][q $\"&yk$F07$F]a`l$\"&-A$F07$Fj^y$\"&%eHF07$$\"5oT&)3uU$[%*G%F0$\"&ye#F0 7$Fea`l$\"&9J#F07$$\"5-D1R+H8-'H%F0$\"&gF#F07$Fb[q$\"&q0#F07$Fbb`l$\"& z#=F07$Fb_y$\"&`q\"F07$Ffdfl$\"&vo\"F07$F_c`l$\"&Oq\"F07$F^efl$\"&rv\" F07$Fg[q$\"&Bu\"F07$Fa\\q$\"&)[>F07$F[]q$\"&r,#F07$F``y$\"&$e?F07$F`]q $\"&U4#F07$F]ay$\"&C4#F07$Fe]q$\"&'3@F07$$\"5M$3_]f6$Q3WF0$\"&s5#F07$F j]q$\"&@5#F07$$\"5,]iST**GY?WF0$\"&j2#F07$F_^q$\"&d2#F07$Fd^q$\"&!H?F0 7$Fi^q$\"&x'>F07$F^_q$\"&o&>F07$Fc_qF[h\\m7$F_g`l$\"&g+#F07$Fh_q$\"&D2 #F07$Fgg`l$\"&k3#F07$F\\h`l$\"&q4#F07$Fah`l$\"&95#F07$F]`q$\"&T3#F07$F b`q$\"&>+#F07$Fg`q$\"&@&>F07$Faaq$\"&@?#F07$F[bq$\"&\">CF07$F`bq$\"&t% GF07$Febq$\"&K@$F07$Fbey$\"&`C$F07$Fgey$\"&@5$F07$F\\fy$\"&/:$F07$Fjbq $\"&3=$F07$Fify$\"&a.$F07$F_cq$\"&v+$F07$Fdcq$\"&&GLF07$Ficq$\"&e$QF07 $Fcdq$\"&rN%F07$F]eq$\"&k_%F07$Fgeq$\"&Mr%F07$Fafq$\"&hB&F07$Fejy$\"& \\_&F07$Fffq$\"&Hj&F07$Fb[z$\"&ya&F07$F[gq$\"&4I&F07$$\"5-]PMxaF07$Fhcal$\"&))R&F07$Fbdal$\"&mP&F07$Fh\\z$\"&TP&F07$F _eal$\"&nT&F07$Fihq$\"&>f&F0-F^iq6&F`iqF($\"#vFc]z$\"#&*Fc]z-Fgiq6#%Ea ~scheme~of~Tsitouras~and~PapakostasG-F$6%7[]mF'7$F^jq$\"$!QF07$F+$\"$d 'F07$F^bgl$\"$M(F07$Ffjq$\"$)fF07$FfbglFec]m7$Fibgl$\"$v&F07$F^cgl$\"$ r#F07$F2F\\d]m7$F]\\r$!$?#F07$F7$!$-*F07$Ffgal$!$;*F07$Fj\\r$!$P*F07$F ^hal$!$)eF07$F<$!$O$F07$FB$\"%KIF07$FG$\"%VYF07$FQ$\"%R'F07$FitF_e]l7$Fcu$\"%**)*F07$F]v$\"%(Q*F07$Fgv$\"%]uF07$Ff jgl$\"%MuF07$F\\wFh]]l7$F^[hl$\"%wtF07$Fc[hl$\"%!p(F07$Fh[hlF[[^m7$Faw $\"%cxF07$Ffw$\"%^$*F07$F[x$\"&_-\"F07$Ffabl$\"&+.\"F07$F`x$\"&d3\"F07 $F^bbl$\"&a3\"F07$Fex$\"&f3\"F07$Fjx$\"&,5\"F07$F_y$\"&s3\"F07$Fbgr$\" &F2\"F07$Fdy$\"&V.\"F07$Fjgr$\"&6.\"F07$Fiy$\"&#>5F07$$\"5NL$e9\"f-ssd F>$\"&$=5F07$F`cbl$\"&20\"F07$$\"5++]P%)*eHY$eF>$\"&s/\"F07$F[iz$\"&O/ \"F07$Fhcbl$\"&:1\"F07$F]dblF^cjl7$Fbdbl$\"&61\"F07$F^z$\"&f-\"F07$$\" 5NLLeRZAJ^gF>$\"%'f*F07$Fciz$\"%?lF07$$\"5NL$e*[Vi$\"%lkF07$$\"5q mmT&)348vhF>$\"%,hF07$$\"5++](=Ud&31iF>$\"%Q$\"%PQ F07$F\\fhl$\"%OQF07$$\"5qmmmT5SG!H)F>$\"%^PF07$Fi^l$\"%&e$F07$Fi`[l$\" %1KF07$F^_l$\"%LFF07$Faa[l$\"%FDF07$Ffa[l$\"%LCF07$F`b[l$\"%cCF07$Fc_l $\"%cDF07$F]c[l$\"%FHF07$Fh_l$\"%5OF07$Fec[l$\"%YOF07$Fjc[l$\"%UQF07$F _d[lFee^m7$F]`l$\"%`QF07$Fb`l$\"%xGF07$Fg`l$\"%nAF07$Faal$\"%RUF07$F[b l$\"%')fF07$F`bl$\"%s()F07$Febl$\"%_**F07$Fjbl$\"%p!)F07$F_cl$\"%$)\\F 07$F^`cl$\"%lOF07$Fc`cl$\"%@PF07$Fh`cl$\"%%y$F07$Fdcl$\"%:VF07$F`acl$ \"%lWF07$Ficl$\"%\"o'F07$Fcdl$\"%D'*F07$Fgel$\"&!o6F07$F[gl$\"&(*e\"F0 7$Fegl$\"&wj\"F07$Fjgl$\"&X`\"F07$F_hl$\"&0E\"F07$F^[il$\"&7E\"F07$Fc[ il$\"&$f7F07$Fh[ilFg\\v7$Fdhl$\"&DA\"F07$F^\\il$\"&UA\"F07$Fc\\il$\"&E E\"F07$Fh\\il$\"&GE\"F07$Fihl$\"&\"o7F07$F^il$\"&fX\"F07$Fcil$\"&ea\"F 07$Ff]il$\"&Cb\"F07$F[^il$\"&Dg\"F07$F`^ilFe^_m7$Fhil$\"&]g\"F07$$\"5+ +]iSmy`@8F0$\"&mh\"F07$Fh^ilF^__m7$$\"5++](oHu0$G8F0$\"&Gh\"F07$F]jl$ \"&#)e\"F07$Fbjl$\"&s_\"F07$Fgjl$\"&OV\"F07$F\\[m$\"&Mc\"F07$Fa[m$\"&j i\"F07$Ff[m$\"&Gi\"F07$F[\\m$\"&lh\"F07$F__\\l$\"&ih\"F07$F`\\m$\"&.l \"F07$Fg_\\l$\"&Xk\"F07$Fe\\m$\"&\"Q;F07$F_`\\l$\"&Mk\"F07$Fd`\\l$\"&) R;F07$Fi`\\l$\"&2j\"F07$Fj\\m$\"&()e\"F07$Faa\\l$\"&TZ\"F07$F_]m$\"&n5 \"F07$Fia\\l$\"&y4\"F07$F^b\\l$\"&53\"F07$Fcb\\l$\"%*>)F07$Fd]m$\"%#)f F07$F[c\\l$\"%q_F07$Fi]mFfcw7$F^js$\"%+CF07$Fcjs$\"%BJF07$Fhjs$\"%$3$F 07$F^^m$\"%wIF07$F`[t$\"%oZF07$Fe[t$\"%u!)F07$Fj[t$\"%ozF07$Fc^m$\"%' \\)F07$$\"5+v$4YsF*Rh:F0$\"&ZU\"F07$Fb\\t$\"&2S\"F07$$\"5Me9;#Q$>Wo:F0 $\"&LX\"F07$Fg\\t$\"&,s\"F07$F\\]t$\"&4i\"F07$Fh^m$\"&/Z\"F07$Fbcil$\" &c7\"F07$Fd]t$\"%>$*F07$Fjcil$\"%t!*F07$Fi]tFbhhl7$Fc^t$\"%tNF07$F]_m$ \"$f)F07$Fg_m$\"%:OF07$F[am$\"%3cF07$F`am$\"%(f'F07$Feam$\"%NqF07$Fjam $\"%fiF07$F_bm$\"%NcF07$Fidt$\"%(4'F07$Fdbm$\"%>qF07$Feft$\"%ulF07$Fib m$\"%FbF07$Fccm$\"%dgF07$F]dm$\"%'*zF07$Fbdm$\"%d**F07$Fgdm$\"&QG\"F07 $F\\em$\"&=8\"F07$Faem$\"%R()F07$Ffem$\"&s@\"F07$F[fm$\"&#\\9F07$F`fm$ \"&6]\"F07$Fefm$\"&,e\"F07$Fjfm$\"&/!=F07$F_gm$\"&&3@F07$F[_]l$\"&8@#F 07$F`_]l$\"&#=AF07$Fe_]l$\"&ZA#F07$Fdgm$\"&%>AF07$F]`]l$\"&SA#F07$Fb`] lF^bjl7$Fg`]l$\"&/5#F07$Figm$\"&J5#F07$Fchm$\"&a%>F07$F]im$\"&.$=F07$F bim$\"&i)=F07$Fgim$\"&G)>F07$F\\jm$\"&V7#F07$Fajm$\"&_@#F07$Fa]dl$\"&z @#F07$Ffjm$\"&&=AF07$Fi]dl$\"&8=#F07$F[[n$\"&W9#F07$Fgbu$\"&&R?F07$F`[ n$\"&j(>F07$Fg^dl$\"&d&>F07$F\\_dl$\"&`%>F07$Fa_dlFg_`m7$Fdcu$\"&A&>F0 7$Fi_dl$\"&j&>F07$F^`dlF^h\\m7$Fc`dl$\"&V*>F07$Fe[n$\"&.,#F07$Fj[n$\"& f>#F07$F_\\n$\"&()H#F07$Fd\\n$\"&dD#F07$Fi\\n$\"&Y@#F07$F]^n$\"&=A#F07 $Fa_n$\"&R?#F07$F[`n$\"&M:#F07$Fe`n$\"&n'>F07$F_an$\"&7a\"F07$Fian$\"& uX\"F07$Fei]l$\"&t-\"F07$F^hu$\"&_,\"F07$F]j]l$\"%C(*F07$F^bn$\"%dtF07 $Ffhu$\"%fsF07$F[iu$\"%#G(F07$F`iu$\"%g*)F07$Fcbn$\"%H))F07$Fhiu$\"%z \"*F07$F]ju$\"&1W\"F07$Fbju$\"&!=9F07$Fhbn$\"&vW\"F07$Fjju$\"&\\*>F07$ F_[v$\"&C'>F07$Fd[v$\"&R&>F07$F]cn$\"&S:#F07$Fdajl$\"&,2#F07$Fbcn$\"& \\x\"F07$F\\bjl$\"&=r\"F07$Fabjl$\"&S=\"F07$Ffbjl$\"&j;\"F07$Fgcn$\"&r 6\"F07$$\"5M3xJ#H'p(4]#F0$\"%`lF07$Fecdl$\"%hkF07$$\"5MekGg%ymy]#F0F`` s7$Fb\\v$\"%GUF07$F]ddl$\"%wTF07$Fbddl$\"%YTF07$Fgddl$\"%PXF07$F\\dn$ \"%$\\%F07$Fj\\v$\"%wiF07$Fadn$\"%[zF07$Feen$\"%D%*F07$F_fn$\"&Q,\"F07 $F]_v$\"%:(*F07$Fdfn$\"%c*)F07$$\"5ML3F>*oZ7F07$Fhgn$\"&/ \\\"F07$Fbb^l$\"&Rn\"F07$F`fv$\"&.z\"F07$F^d^l$\"&pY\"F07$F]hn$\"&=H\" F07$$\"5,]7.2AtrF07$F[jn$\" &=1#F07$F`jn$\"&(HCF07$Fejn$\"&7#GF07$F_[o$\"&^m#F07$Fi[o$\"&E_#F07$F^ [el$\"&j^#F07$Fhjv$\"&(RCF07$Ff[el$\"&1W#F07$F^\\o$\"&3W#F07$F^\\el$\" &HW#F07$F`[w$\"&MW#F07$Ff\\el$\"&[W#F07$Fc\\o$\"&K]#F07$F]]o$\"&Jr#F07 $Fg]o$\"&&=GF07$Fa^o$\"&Uo#F07$F[_o$\"&%\\DF07$Fe_o$\"&Ss#F07$F_`o$\"& R\"HF07$Fd`w$\"&8*GF07$Fd`o$\"&#QGF07$F\\aw$\"&>#GF07$Fi`o$\"&H#GF07$F iaw$\"&yv#F07$F^ao$\"&*>FF07$Ffbw$\"&GX#F07$Fcao$\"&%oAF07$$\"5M$e9mp& )p?J$F0$\"&7'>F07$Fh^[m$\"&7%>F07$$\"5,]PM-C&*Q=LF0$\"&c$=F07$Fhao$\"& sX\"F07$F`_[m$\"&>S\"F07$F]bo$\"&#G7F07$Fdcw$\"&jA\"F07$Ficw$\"&e\\\"F 07$F^dw$\"&LZ\"F07$FbboFbe^l7$Ffdw$\"&21#F07$F[ew$\"&x-#F07$F`ew$\"&r/ #F07$Fgbo$\"&f_#F07$Fb`el$\"&T[#F07$Feb_l$\"&9Y#F07$Fj`el$\"&U`#F07$F \\co$\"&A\\#F07$F]c_l$\"&j1#F07$Faco$\"&V#>F07$F^fw$\"&,U\"F07$Fcfw$\" &()R\"F07$FhfwFe^am7$F]gw$\"%'Q*F07$Fbgw$\"%C))F07$Ffco$\"%#e(F07$Fjgw $\"%(\\)F07$F[do$\"&42\"F07$Fedo$\"&A<\"F07$F_eo$\"&1J\"F07$$\"5MeRZ_H >7%\\$F0$\"&)48F07$$\"5,]7.#34zr\\$F0$\"&,J\"F07$$\"5oT&)e6_iB+NF0$\"& GL\"F07$Fdeo$\"&CL\"F07$$\"5o;/E+OxS4NF0$\"&IL\"F07$Fieo$\"&FL\"F07$F^ fo$\"&+J\"F07$Fcfo$\"&yB\"F07$Fggo$\"&'e7F07$F[io$\"&xG\"F07$F^i_l$\"& !>7F07$F`io$\"&;<\"F07$Fdd[m$\"&3<\"F07$F[j_l$\"&&o6F07$F\\e[m$\"&<<\" F07$Fae[m$\"&M?\"F07$Ffe[m$\"&W?\"F07$Feio$\"&%37F07$Fijo$\"&>[\"F07$F c[p$\"&Yk\"F07$Ff[x$\"&f(=F07$Fh[p$\"&y1#F07$Fjgel$\"&Y6#F07$F^\\x$\"& 7<#F07$Fbhel$\"&)[?F07$F]\\p$\"&6\"=F07$Fbg[m$\"&.%=F07$Fgg[m$\"&h'=F0 7$F\\h[m$\"&hm\"F07$Fb\\p$\"&Op\"F07$Fdh[m$\"&Kt\"F07$Fg\\p$\"&3z\"F07 $F\\]p$\"&..#F07$Fa]p$\"&0B#F07$Ff]p$\"&)oBF07$F[^p$\"&)=DF07$F`^p$\"& 3g#F07$Fe^p$\"&-q#F07$F__p$\"&$oIF07$Fc`p$\"&\"GLF07$F]ap$\"&NC$F07$Fg ap$\"&)QIF07$Fabp$\"&KC$F07$F[cp$\"&AU$F07$Fecp$\"&=O$F07$F_dp$\"&u;$F 07$Fidp$\"&#=KF07$Fcep$\"&BW$F07$Fh^fl$\"&AX$F07$Fhep$\"&K]$F07$F`_fl$ \"&D]$F07$F]fp$\"&S]$F07$$\"5-v$4rG*QKJTF0$\"&v^$F07$Fbfp$\"&g^$F07$$ \"5NekyA+]!y8%F0$\"&C^$F07$Fgfp$\"&y[$F07$F\\gp$\"&aV$F07$Fagp$\"&sR$F 07$Fbhx$\"&%4MF07$Ffgp$\"&fT$F07$Fjhx$\"&!yLF07$F[hp$\"&@E$F07$Fehp$\" &t#HF07$F_ip$\"&a'GF07$F^jx$\"&(*Q#F07$Fdip$\"&_O#F07$Ffjx$\"&(HBF07$F iip$\"&%4>F07$Fc[y$\"&q&=F07$F^jp$\"&Ov\"F07$F`\\y$\"&Tt\"F07$Fc\\y$\" &Q1#F07$Fh\\y$\"&93#F07$Fcjp$\"&!f?F07$F`]y$\"&(pDF07$Fe]y$\"&lj#F07$F j]y$\"&9g#F07$Fhjp$\"&+%HF07$$\"5N$3FWX2uIE%F0$\"&1*HF07$Fb^y$\"&F%HF0 7$$\"5N$ek.H%>qpUF0$\"&V!HF07$F][q$\"&O&GF07$F]a`l$\"&G[#F07$Fj^y$\"&s D#F07$Fdc\\m$\"&5\">F07$Fea`l$\"&Il\"F07$F\\d\\m$\"&oi\"F07$Fb[q$\"&FU \"F07$Fbb`l$\"&Z@\"F07$Fb_y$\"&n6\"F07$F_c`l$\"&F8\"F07$Fg[q$\"&/>\"F0 7$Fa\\q$\"&oY\"F07$F[]q$\"&Rb\"F07$F``y$\"&mf\"F07$F`]q$\"&Rj\"F07$F]a y$\"&Dj\"F07$Fe]q$\"&+l\"F07$Fhf\\m$\"&\"\\;F07$Fj]q$\"&\\k\"F07$$\"5M 3x\"[N&HW;F07$Fd^q$\"&Rd\"F07$Fi^q$\"&O^\"F07$F^_q$\"&M]\"F07$Fc_ q$\"&\\^\"F07$F_g`l$\"&Ob\"F07$Fh_q$\"&)>;F07$Fgg`l$\"&Mj\"F07$F\\h`l$ \"&Qk\"F07$Fah`l$\"&wk\"F07$F]`q$\"&'H;F07$Fb`qF_^_m7$Fg`q$\"&%*[\"F07 $Faaq$\"&es\"F07$F[bq$\"&!3>F07$F`bq$\"&)[AF07$Febq$\"&%GDF07$Fjbq$\"& zU#F07$F_cq$\"&G5#F07$Fdcq$\"&`G#F07$Ficq$\"&=l#F07$Fcdq$\"&-%F07$FdaalFdfcm7$Fiaal$\"&x.%F07$F`\\z$\"&K.%F07$Fabal$\"&8&RF07$ FdhqFeb\\m7$F^cal$\"&x!QF07$Fhcal$\"&wy$F07$F]dal$\"&fw$F07$FbdalFhgcm 7$Fgdal$\"&Ww$F07$Fh\\z$\"&Yw$F07$F_eal$\"&!3QF07$Fihq$\"&2)RF0-Fgeal6 &F`iqF_aglF(F_agl-Fgiq6#%Ra~modification~of~the~Tsitouras-Papakostas~s chemeG-F$6%7ijlF'7$F^jq$\"$+%F07$F+Fecbl7$F^bgl$\"$:*F07$Ffjq$\"$v)F07 $Fibgl$\"$g)F07$F2$\"$F'F07$F]\\r$\"$\"=F07$F7$!$p&F07$Ffgal$!$;'F07$F j\\r$!$Z(F07$F^hal$!$,'F07$F<$!$D%F07$Fa^r$\"%G=F07$FB$\"%BPF07$Fb`r$ \"%\"4&F07$Fg`r$\"%kqF07$F\\ar$\"%YkF07$FG$\"%,PF07$FQ$!$#GF07$F_o$!%* z#F07$Fdo$!$=&F07$Fio$\"%%)=F07$Fcp$\"$a(F07$F]q$!%I5F07$Fgez$!%S5F07$ Fbq$!%'R\"F07$F_fz$!%19F07$Fgq$!%e8F07$F\\r$!#UF07$Far$\"%#)>F07$F\\\\ bl$\"%!*>F07$Fa\\bl$\"%e?F07$Ff\\bl$\"%$H$F07$Ffr$\"%.LF07$Fc]bl$\"%aK F07$F[s$\"%AKF07$Fes$\"$e*F07$F_t$\"$\">F07$Fit$\"%R;F07$Fcu$\"%@MF07$ $\"5-](=nQYs9@%F>$\"%*z$F07$F\\_blFh`dm7$$\"5N$e*[$*\\yCyUF>$\"%3QF07$ Fhu$\"%$z$F07$Fd_blFaadm7$Fg_bl$\"%kPF07$F\\`bl$\"%$Q$F07$F]v$\"%#Q$F0 7$Fbv$\"%AEF07$Fgv$\"%28F07$$\"5o;a)3Cbvcu%F>$\"%18F07$Ffjgl$\"%q7F07$ $\"5-]ilZQ4X7[F>$\"%F5F07$F\\wF^cdm7$Fc[hl$\"%37F07$Faw$\"%k7F07$Ffw$ \"%tGF07$F[x$\"%#*RF07$Ffabl$\"%oSF07$F`x$\"%!)[F07$F^bbl$\"%y[F07$Fex $\"%%*[F07$$\"5NL$eRZFk7S&F>$\"%N_F07$Fjx$\"%K_F07$$\"5++](oagtJY&F>$ \"%I_F07$F_y$\"%+^F07$Fdy$\"%vUF07$Fiy$\"%ZPF07$F`cbl$\"%!)QF07$F[iz$ \"%dQF07$Fhcbl$\"%$H%F07$F]dbl$\"%#\\%F07$Fbdbl$\"%oWF07$F^z$\"%*[%F07 $Fciz$\"%eHF07$Fcz$!$i'F07$F[jz$!$n(F07$FhhrFj][l7$Fcjz$!%JWF07$Fhz$!% 0WF07$F`ir$!%!H&F07$Feir$!%baF07$Fjir$!%g`F07$F][l$!%UPF07$Fbjr$!%s?F0 7$Fgjr$!%8?F07$F\\[s$\"#\"*F07$Fb[l$\"%=UF07$Fd[s$\"%[`F07$Fg[l$\"%J$* F07$Fdgbl$\"%X%*F07$Fc\\[l$\"&K/\"F07$F\\hbl$\"&p-\"F07$F\\\\s$\"%&*)* F07$F[][l$\"%WpF07$F\\\\l$\"%h@F07$Fd\\s$!%c6F07$Fa\\l$!%(y\"F07$F[]l$ \"%q>F07$Fe]l$\"%COF07$Fj]l$\"%ZYF07$F_^l$\"%)=&F07$$\"5qmmmTNhR\"4)F> $\"%H`F07$Fdehl$\"%9bF07$$\"50+++v$4#pd\")F>Fi\\em7$Fd^l$\"%`bF07$F\\f hl$\"%$f&F07$Fi^l$\"%UaF07$F^_l$\"%>YF07$Fc_lFgidm7$Fh_l$\"%>_F07$F]`l $\"%[cF07$Fb`l$\"%f[F07$Fg`l$\"%$*RF07$Faal$\"%mbF07$F[bl$\"%nzF07$Fhc sF`ax7$F`bl$\"&><\"F07$F^^cl$\"&r>\"F07$F`ds$\"&IE\"F07$Ff^cl$\"&xF\"F 07$F[_cl$\"&;E\"F07$F`_cl$\"&J;\"F07$Febl$\"&#y6F07$FjblFbfil7$F_cl$\" %\\[F07$F^`cl$\"%7MF07$Fc`cl$\"%kMF07$Fh`cl$\"%RNF07$Fdcl$\"%_WF07$$\" 5M$e9\"456lL6F0Fighl7$F`acl$\"%;YF07$$\"5+]Pf3Q$Q'R6F0$\"%UkF07$Ficl$ \"%)Q(F07$Fdes$\"%^')F07$F^dl$\"%n)*F07$$\"5+]P4r`C[f6F0F\\]]l7$F\\fs$ \"%I**F07$$\"5n;z>OaEAm6F0$\"&T+\"F07$Fcdl$\"%2!*F07$Fhdl$\"%J&)F07$F] el$\"%GrF07$Fbel$\"%QtF07$Fgel$\"%![(F07$F[gl$\"&^7\"F07$Fegl$\"&zB\"F 07$$\"5n;a8AgW)oA\"F0$\"&1C\"F07$$\"5++voagXDI7F0$\"&l?\"F07$$\"5M$eRs 3mCOB\"F0$\"&&37F07$Fjgl$\"&q?\"F07$$\"5MLe*[='\\tV7F0$\"&R4\"F07$F_hl $\"%v(*F07$Fgjl$\"&'Q6F07$F\\[mF]]_m7$Fa[m$\"&2J\"F07$F_]m$\"&Q1\"F07$ Fc^m$\"%@pF07$Fbe_m$\"&AE\"F07$Fb\\t$\"&6C\"F07$Fje_m$\"&RJ\"F07$Fg\\t $\"&km\"F07$F\\]t$\"&wi\"F07$Fh^m$\"&td\"F07$Fd]t$\"&,<\"F07$Fi]t$\"%Q oF07$Fc^t$\"%AlF07$F]_m$\"%ZQF07$Fg_m$\"%\\hF07$F[amF_e]l7$F`am$\"%N&* F07$Feam$\"&&H5F07$Fjam$\"%](*F07$F_bm$\"%O!*F07$$\"5+]iSmbs&Hx\"F0$\" %,*)F07$Fddt$\"%v*)F07$$\"5+Dc,;krO$y\"F0$\"%&)*)F07$$\"5n;a)eO8Poy\"F 0$\"%!**)F07$$\"5M3_v:.rI!z\"F0$\"%h#*F07$FidtFa[fm7$Fcet$\"%w'*F07$Fd bm$\"&..\"F07$F`fil$\"&C.\"F07$F`ft$\"&x.\"F07$Fhfil$\"&L-\"F07$Feft$ \"&!35F07$F_gt$\"%\"[*F07$Fibm$\"%c!*F07$$\"5M3_]f^y7!)=F0$\"%w()F07$$ \"5o;/^ci$*>$)=F0$\"%z()F07$$\"5,Dc^`t3F')=F0F`\\]l7$F^cmF`\\am7$$\"5o Tg_Z&*QT#*=F0Ffi`m7$$\"5,]7`W1a[&*=F0$\"%h()F07$$\"5Mek`T9F07$ Fgdm$\"&$eF07$Figm$\"&')3#F07$F chm$\"&U'>F07$F]im$\"&I#=F07$Fbim$\"&C&=F07$Fgim$\"&3$>F07$F\\jmFfe`m7 $Fajm$\"&'z@F07$Fa]dl$\"&C>#F07$Ffjm$\"&;?#F07$Fi]dl$\"&2=#F07$F[[n$\" &(\\@F07$Fgbu$\"&([?F07$F`[n$\"&e(>F07$Fg^dl$\"&W%>F07$F\\_dl$\"&'G>F0 7$Fa_dl$\"&$G>F07$Fdcu$\"&G#>F07$Fi_dl$\"&<#>F07$F^`dl$\"&?#>F07$Fc`dl $\"&.&>F07$Fe[n$\"&J'>F07$Fj[n$\"&\\;#F07$F_\\n$\"&'GBF07$Fd\\n$\"&*oA F07$Fi\\n$\"&t=#F07$F^]n$\"&G;#F07$Fc]n$\"&_:#F07$Fh]n$\"&)[@F07$F]^n$ \"&t:#F07$Fg^n$\"&@<#F07$Fa_n$\"&A@#F07$Fe`n$\"&15#F07$Fian$\"&ms\"F07 $F^hu$\"&TL\"F07$F^bn$\"&)45F07$Ffhu$\"%j**F07$F[iu$\"%c)*F07$F`iu$\"& N0\"F07$Fcbn$\"&\"Q5F07$Fhiu$\"&?1\"F07$F]ju$\"&x]\"F07$Fbju$\"&T[\"F0 7$Fhbn$\"&@^\"F07$Fjju$\"&f2#F07$F_[v$\"&@/#F07$Fd[v$\"&4/#F07$F]cn$\" &KN#F07$Fdajl$\"&`F#F07$Fbcn$\"&+7#F07$Fabjl$\"&>k\"F07$Fgcn$\"&qc\"F0 7$Fecdl$\"&?8\"F07$Fb\\v$\"%0!*F07$Fbddl$\"%+))F07$F\\dn$\"%S*)F07$Fj \\vFd[]l7$Fadn$\"&*47F07$Feen$\"&OQ\"F07$F_fn$\"&g[\"F07$Fifn$\"&rW\"F 07$F`^^l$\"&W[\"F07$F\\av$\"&*3:F07$F]_^lFh^am7$F^gn$\"&OX\"F07$Fiav$ \"&BS\"F07$F]cv$\"&xM\"F07$F`gdl$\"&'[8F07$Fbcv$\"&_N\"F07$Fhgdl$\"&*e 8F07$F]hdl$\"&8S\"F07$Fbhdl$\"&IS\"F07$Fcgn$\"&(39F07$Fidv$\"&pr\"F07$ Fhgn$\"&(H@F07$Fbb^l$\"&=K#F07$F`fv$\"&YR#F07$F^d^l$\"&O!>F07$F]hn$\"& **p\"F07$Fh]am$\"&(>F07$F]bo$\"&%GF07$Fjiam$\"&t#>F07$F_jam$\"&\"G>F07$Fdjam$\"&`'>F07$Fdeo$\"&Z '>F07$F\\[bm$\"&H(>F07$Fieo$\"&$y>F07$F\\del$\"&\"p>F07$F^fo$\"&q'>F07 $Fddel$\"&-%>F07$Fcfo$\"&?!>F07$Fhfo$\"&Q'=F07$F]go$\"&(R=F07$$\"5o;/, %p>-Md$F0$\"&q$=F07$Fbgo$\"&_&=F07$$\"5oTNYL^!HMe$F0Fe\\hm7$$\"5,]7G8O 8x'e$F0$\"&e&=F07$$\"5Me*)4$4i8,f$F0$\"'J*)=F>7$Fggo$\"&%*)=F07$F\\ho$ \"&.$>F07$Faho$\"&?)>F07$$\"5,]P4r`(yoi$F0$\"&A)>F07$Ffho$\"&)y>F07$$ \"5M$ek.H*yCSOF0$\"&[&>F07$F[io$\"&S&>F07$F^i_l$\"&**)=F07$F`io$\"&l$= F07$F[j_l$\"&.#=F07$Feio$\"&4%=F07$Fijo$\"&.6#F07$Fc[pFjg_l7$Ff[x$\"&C p#F07$Fh[p$\"&)*)GF07$$\"5om\"zWUG_rx$F0$\"&^#HF07$Fjgel$\"&)yGF07$$\" 5,+vVB!zEPy$F0$\"&%=HF07$F^\\x$\"&J&HF07$Fbhel$\"&vu#F07$F]\\p$\"&`V#F 07$Fbg[m$\"&XZ#F07$Fgg[m$\"&(3DF07$F\\h[m$\"&/G#F07$Fb\\p$\"&\"=BF07$F dh[m$\"&3S#F07$Fg\\p$\"&\\[#F07$F\\]p$\"&S!GF07$Fa]p$\"&-0$F07$Ff]p$\" &.A$F07$F[^p$\"&=N$F07$$\"5o;/,W%)o=fQF0$\"&@P$F07$F`^p$\"&EN$F07$$\"5 oTgFTc&G)oQF0$\"&rL$F07$$\"5,]7.2ZC/sQF0$\"&VB$F07$$\"5MekysPjDvQF0Fc[ al7$Fe^p$\"&NG$F07$F__p$\"&$oMF07$Fc`p$\"&nt$F07$Fcax$\"&\\*QF07$Fh`p$ \"&g\"RF07$F`jel$\"&>#RF07$F`bx$\"&\"HRF07$Fhjel$\"&O$RF07$F]ap$\"&t%Q F07$Fbap$\"&Ss$F07$Fgap$\"&/l$F07$Fabp$\"&Iz$F07$F[cp$\"&U*RF07$F\\\\f l$\"&B,%F07$F`cp$\"&L,%F07$Fd\\fl$\"&r)RF07$Fecp$\"&I(RF07$Fjcp$\"&=%Q F07$F_dp$\"&dw$F07$Fb]flFgby7$Fddp$\"&ht$F07$Fj]fl$\"&Av$F07$Fidp$\"&' yPF07$F^ep$\"&e\"RF07$Fcep$\"&n-%F07$Fh^fl$\"&?/%F07$Fhep$\"&a6%F07$F` _fl$\"&Y6%F07$F]fp$\"&\">TF07$F^dbm$\"&(\\TF07$Fbfp$\"&z9%F07$Ffdbm$\" &T9%F07$Fgfp$\"&p6%F07$F\\gp$\"&C.%F07$Fagp$\"&`%RF07$$\"5N$3-8v\"RGqT F0$\"&e%RF07$Fbhx$\"&?$RF07$$\"5N$eRsey6p<%F0$\"&u\"RF07$Ffgp$\"&h'RF0 7$Fjhx$\"&]#RF07$F[hp$\"&(**QF07$Fehp$\"&Sp$F07$F_ip$\"&Ci$F07$Fdip$\" &>@$F07$Fiip$\"&[v#F07$Fc[y$\"&2o#F07$F^jp$\"&!yCF07$F`\\y$\"&iW#F07$F c\\y$\"&:n#F07$Fh\\y$\"&\"oEF07$Fcjp$\"&aj#F07$F`]yFij\\m7$Fe]y$\"&]:$ F07$Fj]y$\"&@6$F07$Fhjp$\"&w\\$F07$F`ibm$\"&cb$F07$Fb^y$\"&/]$F07$Fhib m$\"&Uc$F07$F][q$\"&8a$F07$F]a`l$\"&9D$F07$Fj^y$\"&70$F07$Fdc\\m$\"&![ FF07$Fea`l$\"&G^#F07$F\\d\\m$\"&`Z#F07$Fb[q$\"&bF#F07$Fbb`l$\"&k0#F07$ Fb_y$\"&>$>F07$Ffdfl$\"&=\">F07$F_c`l$\"&r#>F07$F^efl$\"&+)>F07$Fg[q$ \"&L'>F07$Fa\\q$\"&n?#F07$F[]qFegdl7$F`]q$\"&NT#F07$Fe]q$\"&NW#F07$Fj] q$\"&6X#F07$F_^q$\"&fV#F07$Fd^q$\"&aR#F07$Fi^q$\"&#HBF07$F^_q$\"&+J#F0 7$Fc_q$\"&AJ#F07$F_g`l$\"&NM#F07$Fh_q$\"&8T#F07$Fgg`l$\"&#HCF07$F\\h`l $\"&GW#F07$Fah`l$\"&hX#F07$F]`q$\"&wW#F07$Fb`q$\"&ZP#F07$Fg`q$\"&`H#F0 7$Faaq$\"&o\\#F07$F[bq$\"&,w#F07$F`bq$\"&1C$F07$Febq$\"&3[$F07$Fjbq$\" &^J$F07$F_cq$\"&c%HF07$Fdcq$\"&VA$F07$Ficq$\"&6r$F07$F]eq$\"&.8%F07$Fa fq$\"&mR%F07$F[gq$\"&_g%F07$F_hq$\"&)=[F07$Fdaal$\"&0*[F07$F`\\z$\"&S# \\F07$Fabal$\"&v'[F07$Fdhq$\"&^z%F07$Fhcal$\"&Cp%F07$Fh\\z$\"&Qk%F07$F ^`gl$\"&=m%F07$F_eal$\"&An%F07$Ff`gl$\"&bu%F07$Fihq$\"&4&[F0-F^iq6&F`i qFcb]m$\"#bFc]zF(-Fgiq6#%AVerner's~\"most~efficient\"~schemeG-%%FONTG6 $%*HELVETICAGFeiq-%&TITLEG6#%Verror~curves~for~12~stage~order~8~Runge- Kutta~methodsG-%+AXESLABELSG6$Q\"x6\"Q!Fjjim-%%VIEWG6$;F(Fihq%(DEFAULT G" 1 2 0 1 10 0 2 9 1 4 2 1.000000 45.000000 45.000000 0 1 "Enright-Ve rner scheme" "Verner's Maple scheme" "a modification of Verner's Maple scheme" "an \"efficient\" scheme" "a scheme of Tsitouras and Papakost as" "a modification of the Tsitouras-Papakostas scheme" "Verner's \"mo st efficient\" scheme" }}}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 536 "evalf[20](plot(['rn_RK8_2'(x)-r(x) ,'rn_RK8_3'(x)-r(x),'rn_RK8_4'(x)-r(x),\n'rn_RK8_5'(x)-r(x),'rn_RK8_6' (x)-r(x),'rn_RK8_7'(x)-r(x)],x=0..5,font=[HELVETICA,9],\ncolor=[COLOR( RGB,0,.65,0),brown,red,COLOR(RGB,0,.75,.95),magenta,COLOR(RGB,.95,.55, 0)],\nlegend=[`Verner's Maple scheme`,`a modification of Verner's Mapl e scheme`,`an \"efficient\" scheme`,`a scheme of Tsitouras and Papakos tas`,`a modification of the Tsitouras-Papakostas scheme`,`Verner's \"m ost efficient\" scheme`],title=`error curves for 12 stage order 8 Rung e-Kutta methods`));" }}{PARA 13 "" 1 "" {GLPLOT2D 1044 510 510 {PLOTDATA 2 "6,-%'CURVESG6%7_\\m7$$\"\"!F)F(7$$\"5ommmTN@Ki8!#@$\"$>#! #>7$$\"5NLLL$3FWYs#F-$\"#')F07$$\"5-+++D1k'p3%F-$!$'*)F07$$\"5qmmmmT&) G\\aF-$!%*\\\"F07$$\"5SLL3_v!p)*y&F-$!%X:F07$$\"50++]P4'\\/8'F-$!%Q>F0 7$$\"5qmm\"HK9I5Z'F-$!%P>F07$$\"5SLLL3x1h6oF-$!%`>F07$$\"5qmm;zW'o^7\\6F`o$\"$1'F07$$\"5++]i!*GER37F `o$\"%#Q\"F07$$\"5ML3F>*3gwE\"F`o$\"%\"R\"F07$$\"5nmm\"z%\\v#pK\"F`o$ \"%i;F07$$\"5++Dcw4]>'Q\"F`o$\"%z:F07$$\"5ML$3_+ZiaW\"F`o$\"%b8F07$$\" 5+]7`>+i4v9F`o$\"%+8F07$$\"5nmT&Q.$*HZ]\"F`o$\"$*fF07$$\"5M$3x\"[gOOM: F`o$\"$-'F07$$\"5+++]i!R(*Rc\"F`o$\"$$eF07$$\"5nm;z>6B`#o\"F`o$!$m)F07 $$\"5MLL3xJs1,=F`o$!%_eF07$$\"5++]PM_@g>>F`o$!&18\"F07$$\"5nmmm\"H2P\" Q?F`o$!&dv\"F07$$\"5MLLek.pu/BF`o$!&&y=F07$$\"5+++]PMnNrDF`o$!&Y+#F07$ $\"5M$e9;K'Ho/EF`o$!&eo#F07$$\"5nm\"Hd?>4!QEF`o$!&5s#F07$$\"5+]P%)*3UN 8n#F`o$!&iz#F07$$\"5ML$eR(\\;m/FF`o$!&2n$F07$$\"5++v=U2TJrFF`o$!&Cy$F0 7$$\"5nmmT5ll'z$GF`o$!&q\\%F07$$\"5MLekyA!>Y!HF`o$!&U$[F07$$\"5++](o/[ r7(HF`o$!&@!\\F07$$\"5nmT5:QR#z.$F`o$!&D$\\F07$$\"5MLLL$eRwX5$F`o$!&G \"\\F07$$\"5MLLLe*[`HP$F`o$!&GF&F07$$\"5MLLLL$eI8k$F`o$!&6\\&F07$$\"5M LL$3-8>bx$F`o$!&&=`F07$$\"5MLLL3xwq4RF`o$!&u)\\F07$$\"5ML$3-Q\"[DVRF`o $!&w)\\F07$$\"5MLL3_]>!o(RF`o$!&4)\\F07$$\"5ML$eRs3\\.,%F`o$!&W!\\F07$ $\"5MLL$eRA'*Q/%F`oFfy7$$\"5MLLeR(\\!*46%F`o$!&I!\\F07$$\"5NLLL$3x%3yT F`o$!&-\"\\F07$$\"5ommT5:j=XWF`o$!&t>&F07$$\"5-++]PfyG7ZF`o$!&9H&F07$$ \"5NLLek.%*Qz\\F`o$!&`2&F07$$\"5ommm\"z%4\\Y_F`o$!&$))[F07$$\"5+++]P4' 4.P&F`o$!&`'\\F07$$\"5NLLL$3FGT\\&F`o$!&P<&F07$$\"5+++Dc,w.cbF`o$!&)H_ F07$$\"5qmm;HKp%zh&F`o$!&$3`F07$$\"5NLL3-ji&)zcF`o$!&xG&F07$$\"5++++v$ flF07$$\"5+++D\"Gj4'*e'F`o$!&;e\"F07$$\"5NLLL$eRh;i'F`o $!%<\"*F07$$\"5+++](=#\\w&o'F`o$!%**yF07$$\"5qmmm\"zWo)\\nF`o$!%I[F07$ $\"5++++++b2yoF`o$!%!*yF07$$\"5NLLL3_DG1qF`o$!&#Q:F07$$\"5qmmm;/'*[MrF `o$!&\"38F07$$\"5-+++DcmpisF`o$!%;wF07$$\"5qmmm\"Hd)G&R(F`o$\"$&pF07$$ \"5NLLLe*[!)y_(F`o$\"%hZF07$$\"5++++D1CZgwF`o$\"%.jF07$$\"5qmmm\"HKkIz (F`o$\"%WyF07$$\"50+++D\"Gg$fyF`o$\"%X#)F07$$\"5SLLLeRilDzF`o$\"%\\')F 07$$\"5qmmm\"z>_>*zF`o$\"%#f)F07$$\"50+++Dc\"[#e!)F`o$\"%Y&)F07$$\"5qm mm\"H2S3>)F`o$\"%sxF07$$\"5OLLLe*)>VB$)F`o$\"%WmF07$$\"5qmmmTg()4_))F` o$\"%!)oF07$$\"5.+++DJbw!Q*F`o$\"%5tF07$$\"5qmT5lv.u[%*F`o$\"%\"y'F07$ $\"5NL$3_+A:n^*F`o$\"%/iF07$$\"5++DJXk+p%e*F`o$\"%GhF07$$\"5qmmT&)3\\m _'*F`o$\"%GeF07$$\"5NL3_D`(R1s*F`o$\"%rdF07$$\"5++]il(f9')y*F`o$\"%!)e F07$$\"5N$3xc)>?gA)*F`o$\"%!4'F07$$\"5qm\"Hd?W*ec)*F`o$\"%'4'F07$$\"5+ ]7yDkod!*)*F`o$\"%ljF07$$\"5NLL$ekGkX#**F`o$\"%tnF07$$\"5nmTg_(R^g+\"F 0$\"%SxF07$$\"5++]iSmjk>5F0$\"%J#*F07$$\"5n;aj%3&QWE5F0$\"%#[*F07$$\"5 MLekGN8CL5F0$\"%p'*F07$$\"5+]ils>)Q+/\"F0$\"%-$*F07$$\"5nmmm;/j$o/\"F0 $\"%v$*F07$$\"5++]i:g2\")e5F0$\"%m$)F07$$\"5MLLe9;_yq5F0$\"%ScF07$$\"5 nm;a8s'fF3\"F0$\"%4?F07$$\"5+++]7GTt%4\"F0$!%KMF07$$\"5ML$e9Te3n5\"F0$ !%+\"*F07$$\"5nmmT5SIo=6F0$!&k:\"F07$$\"5++]P4'\\d18\"F0$!%FnF07$$\"5M LLL3_>jU6F0$!%K\\F07$$\"5++vVt_@P\\6F0$!%=oF07$$\"5nm;aQ`B6c6F0$!%_%)F 07$$\"5MLek.aD&G;\"F0$!&'Q;F07$$\"5+++voaFfp6F0$!&'pAF07$$\"5nmT&Q`&HL w6F0$!&Ug#F07$$\"5ML$e*)f:tI=\"F0$!&DA$F07$$\"5++D1kcL\")*=\"F0$!&5U$F 07$$\"5nmm;HdNb'>\"F0$!&N\\$F07$$\"5MLLe*)fV^B7F0$!&5g$F07$$\"5++++]i^ Z]7F0$!&I/%F07$$\"5++++vomax7F0$!&$pPF07$$\"5+++++v\"=YI\"F0$!&DR$F07$ $\"5++++D\"o*oJ8F0$!&_g$F07$$\"5++++](=h(e8F0$!&.!QF07$$\"5++++++'\\[Q \"F0$!&`n$F07$$\"5++++]7!Q4T\"F0$!&JQ$F07$$\"5+++++Dk-P9F0$!&8\\$F07$$ \"5++++]P[6j9F0$!&Y\"QF07$$\"5ML$ek`h0o[\"F0$!&Rb$F07$$\"5nmm\"HKR'\\5 :F0$!&+Z$F07$$\"5MLe9;#yTB_\"F0$!&FO$F07$$\"5++]P4rr=M:F0$!&,\"GF07$$ \"5n\"H#oK=&[r`\"F0$!&Vt#F07$$\"5M$e*)fb')4,a\"F0$!&Mz\"F07$$\"5+voHz7 72V:F0$!&2x\"F07$$\"5nmTg-gD.Y:F0$!&kt\"F07$$\"5Me9\"fs!R**[:F0$!&)z7F 07$$\"5+](=#\\a_&>b\"F0$!%l^F07$$\"5nTg_s,m\"\\b\"F0$!%!3&F07$$\"5MLL$ e*[z(yb\"F0$!%XSF07$$\"5n;aQ`01#\\c\"F0$\"%\"*QF07$$\"5++v$4@Ej>d\"F0$ \"%ndF07$$\"5M$e*[o=f+z:F0$\"%ySF07$$\"5nm;/Ev&[ge\"F0$\"%+9F07$$\"5+] Pf$=B\"4$f\"F0$!%?RF07$$\"5MLe9T))Q8+;F0$!%$)eF07$$\"5+v=#*p;_l.;F0$!% 'z&F07$$\"5n;zp)\\awrg\"F0$!%\"f&F07$$\"5MeRZFtyp5;F0$!%tIF07$$\"5+++D c,#>Uh\"F0$!%KIF07$$\"5nmTNr9XIG;F0$\"%dKF07$$\"5ML$eky#)*QU;F0$\"&)>6 F07$$\"5++Dc,T^Zc;F0$\"&fK\"F07$$\"5nmmm;a/cq;F0$\"&@c\"F07$$\"5nm\"z> 6[VNn\"F0$\"&(e:F07$$\"5nm;H23l_w;F0$\"&rb\"F07$$\"5nmTg-N&4&z;F0$\"&7 g\"F07$$\"5nmm\"z>c#\\#o\"F0$\"&\"G;F07$$\"5nm;a)ehe%)o\"F0$\"&+j\"F07 $$\"5nmm;zpYU%p\"F0$\"&@l\"F07$$\"5nm;zpB2R+V\"F07$$\"5nmm;/,J:U(*)z9F07$$\"5,]7GjB lBc=F0$\"&;R\"F07$$\"5ML3-jikF0$\"&Le\"F07$$\"5MLLe9 ;0?E>F0$\"&sv\"F07$$\"5+]Pf3QNMK>F0$\"&pw\"F07$$\"5nmTg-gl[Q>F0$\"&!o< F07$$\"5M$e9m>eHY%>F0$\"&Fu\"F07$$\"5++]i!Rgs2&>F0$\"&8n\"F07$$\"5MLek yZ'eI'>F0$\"&Ge\"F07$$\"5nmmmm\"pW`(>F0$\"&-R\"F07$$\"5MLek.HW#)))>F0$ \"%\"\\)F07$$\"5++]iSmTI-?F0$\"#%)F07$$\"5M$e9\"4NS/4?F0$!$<%F07$$\"5n mTgx.Ry:?F0$!%58F07$$\"5+]P4YsP_A?F0$\"%g8F07$$\"5MLLe9TOEH?F0$\"%'Q$F 07$$\"5o;H2$)4N+O?F0$\"%ccF07$$\"5,+Dc^yLuU?F0$\"%SpF07$$\"5M$3_+sC$[ \\?F0$\"%=WF07$$\"5om;a)e6Bi0#F0$\"%XFF07$$\"5,]7.d%)H'H1#F0$!%)\\&F07 $$\"5ML3_D`Gqp?F0$!&m=\"F07$$\"5o;/,%>sUk2#F0$!&.U\"F07$$\"5,++]i!f#=$ 3#F0$!&a\">F07$$\"5,++v$fQa)3@F0$!&l,#F07$$\"5,+++D\"=EX8#F0$!&OP#F07$ $\"5,]iSmb^tP@F0$!&))Q#F07$$\"5,+D\"y+8W49#F0$!&(>DF07$$\"5,](=#\\/J:W @F0$!&=_#F07$$\"5,+]i!*y?OZ@F0$!&P_#F07$$\"5,]7.K`5d]@F0$!&C`#F07$$\"5 ,+vVtF+y`@F0$!&N`#F07$$\"5,]P%[@+*)p:#F0$!&,`#F07$$\"5,++Dcwz>g@F0$!&g T#F07$$\"5,+](=U(Q.t@F0$!&DB#F07$$\"5,++](=xpe=#F0$!&>0#F07$$\"5M3_D.+ !=#*=#F0$!&`*>F07$$\"5o;/,>Gic#>#F0$!&\"H>F07$$\"5,DcwMcW\"f>#F0$!&*G> F07$$\"5ML3_]%oi#*>#F0$!&%3>F07$$\"5,]7.#39ff?#F0$!&e)=F07$$\"5om;a8(f bE@#F0$!&+#>F07$$\"5,+Dcw4&[gA#F0$!&L,#F07$$\"5MLLeRA9WRAF0$!&<>#F07$$ \"5+]P4ryy8YAF0$!&sB#F07$$\"5nmTg-NV$GD#F0$!&=I#F07$$\"5M$e9T8zI&fAF0$ !&PI#F07$$\"5++]ilZsAmAF0$!&_I#F07$$\"5MLekGg,izAF0$!&[<#F07$$\"5nmmm \"H28IH#F0$!&<-#F07$$\"5nmTg_Z>J0BF0$!&-#>F07$$\"5nm;a8A3h6tsK9N#F0$!&;H#F07 $$\"5nmTN'fW2XN#F0$!&!*G#F07$$\"5n;H#oK)olgBF0$!&RK#F07$$\"5nm;Hd?j!oO #F0$!&FJ#F07$$\"5nm\"H#=&>0\"zBF0$!&%f@F07$$\"5nmm;zpSS\"R#F0$!&2(>F07 $$\"5M$e*[ooq.)R#F0$!&T&>F07$$\"5++D\"yv1qYS#F0$!&o)>F07$$\"5MeRZ-nl)z S#F0$!&Z)>F07$$\"5n;a8ZmII6CF0$!&z3#F07$$\"5+voz\"fc>YT#F0$!&y1#F07$$ \"5ML$ek`1OzT#F0$!&k.#F07$$\"5+]7yDk!pXU#F0$!&l$>F07$$\"5nmT5:j??JCF0$ !&GM\"F07$$\"5+Dcwfi&=XV#F0$!&WK\"F07$$\"5M$3FW?1NyV#F0$!&x>\"F07$$\"5 nT&)3\\h::TCF0$!%PCF07$$\"5+++v$41oWW#F0$!%'R#F07$$\"5Me9TQgXyZCF0$!%A 7F07$$\"5n;H2$)f55^CF0$\"%c$*F07$$\"5+vVtFfvTaCF0$\"%7#*F07$$\"5MLeRse StdCF0$\"%^(*F07$$\"5n\"Hdq\"e00hCF0$\"&tj\"F07$$\"5+](=Rt*4(e#F0$\"&HU#F07$$\"5MLe*)fb&*)Rf#F0$\"&lV# F07$$\"5M$ekysPz3g#F0$\"&YS#F07$$\"5MLL$e*)>pxg#F0$\"&mR#F07$$\"5om;/, >Ww>EF0$\"& #F07$$\"5ML$e9\"f[vVEF0$\"&s:#F07$$\"5,+D19pCv\\EF0$\"&#\\@F07$$\"5omm m;z+vbEF0$\"&q9#F07$$\"5MLL3F>0uzEF0$\"&,F#F07$$\"5,++]Pf4t.FF0$\"&wO# F07$$\"5ML$3F>HT'HFF0$\"&DH#F07$$\"5omm\"zWi^bv#F0$\"&Z;#F07$$\"5,](=< w?H?w#F0$\"&;9#F07$$\"5ML3_v!z1&oFF0$\"&_9#F07$$\"5,v=UK#eXY\"y#F0$\"&a?#F07$$\"5om\"H2L7ST$GF0$\"&gd#F07$$\"5omT5lv@$3%GF0$\" &m`#F07$$\"5ML$ekG:Cv%GF0$\"&4c#F07$$\"5,+D\"y+8;U&GF0$\"&r]#F07$$\"5o mm;H2\"34'GF0$\"&NR#F07$$\"5MLLe9;gn()GF0$\"&Qz\"F07$$\"5,++++DRW9HF0$ \"%)p*F07$$\"5,+]7G8v`FHF0$\"&FT\"F07$$\"5,++Dc,6jSHF0$\"&p&=F07$$\"5, +]P%)*oCP&HF0$\"&RR\"F07$$\"5,++]7y#=o'HF0$\"%faF07$$\"5,+DcEs]OtHF0$! %a=F07$$\"5,+]iSm=\"*zHF0$!%Y`F07$$\"5,+voag'ek)HF0$!%EdF07$$\"5,++voa a+$*HF0$!%$3'F07$$\"5,+D\"G)[Ab**HF0$!%CcF07$$\"5,+](oH/*41IF0$!%&R&F0 7$$\"5,]i!R+Ws$4IF0$!%+bF07$$\"5,+v$4r$ek7IF0$!%9cF07$$\"5,](ozTB>f,$F 0$!%UcF07$$\"5,+++DJE>>IF0$!%VjF07$$\"5,+v=<^8'=.$F0$!%>#*F07$$\"5,+]P 4r+`WIF0$!&!Q5F07$$\"5,](oa5Vk30$F0$!&G,\"F07$$\"5,+Dc,\"z)>dIF0$!&n+ \"F07$$\"5,]il(4:LN1$F0$!%A()F07$$\"5,++v$4^n)pIF0$!%ZqF07$$\"5,+v$f3B OD3$F0$!%B^F07$$\"5,+]7y]\\?&4$F0$!%mSF07$$\"5,](=U2JR:5$F0$!%GQF07$$ \"5,+DJqqO(y5$F0$!%()QF07$$\"5,]iSmI!3U6$F0$!%PVF07$$\"5,++]i!RU07$F0$ !%^`F07$$\"5,+D19W)3Y8$F0$!%NlF07$$\"5,+]il(Hv'[JF0$!%dwF07$$\"5,]iSTC &3d:$F0$!%V!)F07$$\"5,+v=<^sF07$$\"5,++vo/#3o<$F0 $!%(=(F07$$\"5,+DJ?eY(3>$F0$!%**[F07$$\"5,+](=<6T\\?$F0$!%*)RF07$$\"5, ]ilZQV(>@$F0$!%5PF07$$\"5,+vVBlv+>KF0$!%GRF07$$\"5,](=#*>zSgA$F0$!%`UF 07$$\"5,+++v=S2LKF0$!%([&F07$$\"5ML$ekGN8dC$F0$!%>qF07$$\"5omm\"zpo_$e KF0$!%T\")F07$$\"5,+]P4@?*4F$F0$!%;sF07$$\"5MLL$3_NJOG$F0$!%+eF07$$\"5 o;zpt)=\"z'G$F0$!%ddF07$$\"5,+DcEA5&**G$F0$!%C`F07$$\"5M$3F%zb36$H$F0$ !%w_F07$$\"5om;HK*oqiH$F0$!%`_F07$$\"5ML3-Qc.f-LF0$!%thF07$$\"5,++vVB+ \"*3LF0$!%LkF07$$\"5ML$3_vN\\:K$F0$!%c`F07$$\"5ommmm\"p)=MLF0$\"%<;F07 $$\"5MLe9T)[ivL$F0$\"%JCF07$$\"5,+]i:&GO4M$F0$\"&'y7F07$$\"5omT5!>35VM $F0$\"&)f7F07$$\"5MLLekyQoZLF0$\"&hL\"F07$$\"5,+D1Rvw0^LF0$\"&VL#F07$$ \"5om;a8s9VaLF0$\"&pH#F07$$\"5ML3-))o_!yN$F0$\"&yJ#F07$$\"5,++]il!z6O$ F0$\"&Q!GF07$$\"5ommTg_UnuLF0$\"&O\\#F07$$\"5MLLLeR%p\")Q$F0$\"&kv\"F0 7$$\"5,+D\"GjBV:R$F0$\"&AP\"F07$$\"5om;H2Lq\"\\R$F0$\"&AN\"F07$$\"5ML3 x\")H3H)R$F0$\"&tL\"F07$$\"5,++DcEYm,MF0$\"&nL\"F07$$\"5ML$3_+A7%3MF0$ \"&\")Q\"F07$$\"5omm;a8)f^T$F0$\"&6p\"F07$$\"5MLL3_+]lGMF0$\"&#*H#F07$ $\"5,+++](=]@W$F0$\"&?(GF07$$\"5ML$ekyZ2mY$F0$\"&o4$F07$$\"5omm\"H#oZ1 \"\\$F0$\"&!>KF07$$\"5,+]Pfe?_:NF0$\"&g5$F07$$\"5MLL$e*[$z*RNF0$\"&^$H F07$$\"5omT5:)[[Lb$F0$\"&E$HF07$$\"5,+]PMFwrmNF0$\"&d'HF07$$\"5MLek`mn 3!e$F0$\"&13$F07$$\"5omm\"Hd!fX$f$F0$\"&,8$F07$$\"5M$3_D`ZS,g$F0$\"&K: $F07$$\"5,+v=#\\/Dog$F0$\"&G:$F07$$\"5o;H#=Xh4Nh$F0$\"&M9$F07$$\"5ML$e 9T=%>?OF0$\"&K5$F07$$\"5om\"H2LKjNj$F0$\"&U/$F07$$\"5,+++]iC$pk$F0$\"& D)HF07$$\"5MLLe*[t\\sp$F0$\"&06$F07$$\"5omm;H2qcZPF0$\"&VQ$F07$$\"5MLL 3F>grgPF0$\"&n@$F07$$\"5,+++DJ]'Qx$F0$\"&k-$F07$$\"5omm\"HK/9qy$F0$\"& \"\\EF07$$\"5MLL$3_0j,!QF0$\"&!4>F07$$\"5,++v=n?J8QF0$\"&(3@F07$$\"5om mm;z5YEQF0$\"&gl#F07$$\"5MLe9;K$[(HQF0$\"&5%HF07$$\"5,+]i:&eNI$QF0$\"& r0$F07$$\"5omT5:QGKOQF0$\"&45$F07$$\"5MLLe9\"45'RQF0$\"&c/$F07$$\"5,+D 19Wt*G%QF0$\"&&\\IF07$$\"5om;a8(f%=YQF0$\"&O3$F07$$\"5ML3-8]=Z\\QF0$\" &Us#F07$$\"5,++]7.\"fF&QF0$\"&!*\\#F07$$\"5ML3_vlYhlQF0$\"&Hr\"F07$$\" 5om;aQG-ZyQF0$\"&8-\"F07$$\"5,voH/>To\")QF0$\"%R\")F07$$\"5M$3_+(4!)*[ )QF0$\"%(>)F07$$\"5o\"H2e.!>6))QF0$\"%p#)F07$$\"5,+Dc,\"zD8*QF0$\"%5&) F07$$\"5M3xJn\"oRX*QF0$\"%e&)F07$$\"5o;H2LsNv(*QF0$\"%m')F07$$\"5,D\"G ))HYn4!RF0$\"%N$*F07$$\"5MLLek`8=/RF0$\"%u$*F07$$\"5oT&Q.VC&R2RF0$\"%y $*F07$$\"5,]P4'\\841\"RF0$\"%')))F07$$\"5Me*[=c-BQ\"RF0$\"%7*)F07$$\"5 omTgF;p.*3%F0$\"%l (*F07$$\"5omm\"zW7@^6%F0$\"&t7\"F07$$\"5NL3F>RL3GTF0$\"&P.\"F07$$\"5-+ ]i!RbX59%F0$\"%FwF07$$\"5oTNYe2hGWTF0$\"%BwF07$$\"5N$3-j7mEv9%F0$\"%Yv F07$$\"5-D19%\\@n2:%F0$\"%]mF07$$\"5om\"z>'ox+aTF0$\"%VmF07$$\"5-]il(f ())[gTF0$\"%knF07$$\"5NLLLL$)*pp;%F0F]jl7$$\"5NL3Fp^yftTF0$\"%LyF07$$ \"5NL$3_+sD-=%F0$\"%$)*)F07$$\"5NLe9T)e`o=%F0$\"%%*))F07$$\"5NLL3xc9[$ >%F0$\"%g))F07$$\"5N$3_]4R&z'>%F0$\"%n()F07$$\"5NL3-8D$4,?%F0$\"%#[(F0 7$$\"5N$e*)4$fKU.UF0$\"%=uF07$$\"5NL$e*[$>Pn?%F0$\"%StF07$$\"5N$3Fpw7^ +@%F0$\"%$['F07$$\"5NLe*[=1lL@%F0$\"%kF07 $$\"5NLL$3-$H**>UF0$\"%!)*)F07$$\"5N$3-)QkoIBUF0$\"%*)))F07$$\"5NL3xc) z?mA%F0$\"%e*)F07$$\"5N$eRZFtM*HUF0$\"&-j\"F07$$\"5NL$3Fpm[KB%F0$\"&al \"F07$$\"5N$3x16gilB%F0F][t7$$\"5NLekGNl()RUF0$\"&_l#F07$$\"5N$e9m%p/> VUF0$\"&Fu#F07$$\"5NLLek.W]YUF0$\"&$>FF07$$\"5N$3_DyL=)\\UF0$\"&&GNF07 $$\"5NL3_+sA8`UF0$\"&Ej$F07$$\"5N$e*[=1iWcUF0$\"&4e$F07$$\"5NL$ek.9g(f UF0$\"&W%QF07$$\"5NLeRs3!)QmUF0$\"&_z$F07$$\"5NLLL3xe,tUF0$\"&\"3MF07$ $\"5-+v$4'\\=;'G%F0$\"&el#F07$$\"5om;a8AyI*H%F0$\"&QE#F07$$\"5NLe9m%z` CJ%F0$\"&5q#F07$$\"5-++v=n(*fDVF0$\"&&\\JF07$$\"5NL$eRAr\"*=N%F0$\"&.% QF07$$\"5omm;HdO=yVF0$\"&&yRF07$$\"5,]PM-\\NA%Q%F0$\"&(3SF07$$\"5ML3_v SME!R%F0$\"&s*RF07$$\"5,v$4@mQ$G$R%F0$\"&]*RF07$$\"5o;zp[KLI'R%F0$\"&D *RF07$$\"5MekGNyKK*R%F0$\"&n'RF07$$\"5,+](=UAVBS%F0$\"&P%RF07$$\"5om\" H#o2IU9WF0$\"&<(QF07$$\"5MLLe9\"z-lU%F0$\"&zz$F07$$\"5,v=<,PF_HWF0$\"& Iw$F07$$\"5o;/w(GoUDV%F0$\"&>u$F07$$\"5Me*[V(GEcNWF0$\"&=u$F07$$\"5,+v $4Yd#eQWF0$\"&at$F07$$\"5M$e9TjYAYW%F0$\"&Nr$F07$$\"5om;H2eBm]WF0$\"&- s$F07$$\"5MLek`T@uiWF0$\"&*ePF07$$\"5,++++D>#[Z%F0$\"&r\"QF07$$\"5MLL$ 3_5,-`%F0$\"&@\"QF07$$\"5ommmT&G!e&e%F0$\"&[\"QF07$$\"5NLLe*[=Y.h%F0$ \"&T6%F07$$\"5-++]P%37^j%F0$\"&X?%F07$$\"5ommT&Q)z()fYF0$\"&n0%F07$$\" 5NLLLL$)Qk%o%F0$\"&7[$F07$$\"5o\"zpB^eWzo%F0$\"&6but%F0$\"&LP%F07$$\"5-+D19>zl]ZF0$\"&+f$F07$$\"5 NL$3-jsgQw%F0$\"&@)GF07$$\"5o\"zW#4G9;nZF0Fejt7$$\"5-]7G))H@YqZF0$\"&) [BF07$$\"5N3xJnJGwtZF0$\"&&pBF07$$\"5omTNYLN1xZF0$\"&*zBF07$$\"5-D1RDN UO!y%F0$\"&/A#F07$$\"5N$3FWq$\\m$y%F0$\"&dB#F07$$\"5oTNY$)Qc'py%F0$\"& 6D#F07$$\"5-++]iSjE!z%F0$\"&#>BF07$$\"5-+v=9#F07$$\"5-++++DM\"3%[F0$\"&z&>F07$$ \"5-++v=np3m[F0$\"&kA#F07$$\"5-++]P40O\"*[F0$\"&Ii#F07$$\"5-+]7.#Q?&= \\F0$\"&W]#F07$$\"5,++voa-oX\\F0$\"&OA#F07$$\"5++]PMF,%G(\\F0$\"&rL#F0 7$$\"\"&F)$\"&@j#F0-%&COLORG6&%$RGBGF($\"#l!\"#F(-%'LEGENDG6#%6Verner' s~Maple~schemeG-F$6%7_]mF'7$$FQ!#A$\"\"'F07$F+$\"$(GF07$$\"5-++]7.K[V? F-$\"$P$F07$F2$\"$E$F07$F7$!$n#F07$F<$!$Y(F07$FA$!$(yF07$FF$!%f6F07$FK Facu7$FP$!%&=\"F07$FU$!%!Q\"F07$FZ$!%?8F07$Fin$!$U*F07$F^o$\"#FF07$Fio $\"%$R\"F07$Fcp$\"%_>F07$F]q$\"%.AF07$Far$\"%cBF07$$\"5MLe9\"4&[EB;F`o $\"%4EF07$Ffr$\"%\\EF07$$\"5M$e9T8/m@r\"F`o$\"%*z#F07$$\"5++vV[r(*zTWr@F`o$!%j#)F07$$\"5M$ eRAr@oZ?#F`o$!%[\"*F07$$\"5nmTN'fW%4QAF`o$!%,$*F07$$\"5+](o/[n?9F#F`o$ !%f$*F07$Fjs$!%xrF07$$\"5nm;/,>=0QCF`o$!%YSF07$F_t$!%)y#F07$Fit$!%9ZF0 7$Fcu$!%=*)F07$Fhu$!%h#*F07$F]v$!&aG\"F07$$\"5+]7`%Rz#HrGF`o$!&9I\"F07 $Fbv$!&uU\"F07$$\"5n;/wi^_%z$HF`o$!&!Q9F07$Fgv$!&WW\"F07$F\\w$!&eQ\"F0 7$Faw$!&`G\"F07$Ffw$!&`S\"F07$F[x$!&]g\"F07$Fex$!&HC\"F07$Faz$!&J8\"F0 7$Ffz$!&>K\"F07$F[[l$!&\"e9F07$F`[l$!&(=8F07$Fe[l$!&%*3\"F07$F_\\l$!&4 A\"F07$Fc]l$!&dO\"F07$$\"5NLL$3_D%eleF`o$!&tG\"F07$Fh]l$!&v<\"F07$$\"5 +++]7y:A8hF`o$!&GH\"F07$F]^l$!&bY\"F07$$\"5qmmTg-?4piF`o$!&5X\"F07$Fb^ l$!&\\\\\"F07$$\"5NLLekGb>LjF`o$!&tZ\"F07$Fg^l$!&MX\"F07$Fa_l$!&;6\"F0 7$F[`l$!%ZtF07$F``l$!%sSF07$Fe`l$!%uRF07$Fj`l$!%7:F07$F_al$\"%]IF07$Fd al$\"%]PF07$Fial$\"%.iF07$$\"5NLL$eR(>(R\"oF`o$\"%OVF07$F^bl$\"%uPF07$ $\"5qmm;/E!z@%pF`o$!$c&F07$Fcbl$!%tXF07$$\"5qmmT5:VLQqF`o$!%4XF07$$\"5 +++]7ygQqqF`o$!%$[%F07$$\"5NLLe9TyV-rF`o$!%y`F07$Fhbl$!%7`F07$$\"5NLL$ 3-8$f)>(F`o$!%(e$F07$F]cl$!%WKF07$Fbcl$\"%\">\"F07$Fgcl$\"%aNF07$F\\dl $\"%&\\%F07$Fadl$\"%#p&F07$Ffdl$\"%pgF07$F[el$\"%\"f'F07$F`elFafl7$Fee l$\"%GnF07$Fjel$\"%;jF07$F_fl$\"%qaF07$$\"5qmm;H#o)fb%)F`o$\"%b\\F07$$ \"50++++v`w(e)F`o$\"%+YF07$$\"5qmmTN@([Ql)F`o$\"%UXF07$$\"5SLL$3x1K*>( )F`o$\"%JZF07$$\"5qm;a)3utHv)F`o$\"%KZF07$$\"50++D19a,'y)F`o$\"%E[F07$ $\"5SL$eRs3d!>))F`o$\"%C^F07$Fdfl$\"%D^F07$$\"5+++]7`aE%)*)F`o$\"%xcF0 7$$\"5NLLL$e9Kk6*F`o$\"%]iF07$$\"5+++v=#\\:D=*F`o$\"%XiF07$$\"5qmm;aQ) )f[#*F`o$\"%PiF07$$\"5NLLe*[=#o9$*F`o$\"%yfF07$Fifl$\"%AfF07$Fcgl$\"%r ]F07$F]hl$\"%%p%F07$Fbhl$\"%OXF07$Fghl$\"%&e%F07$F\\ilFe^p7$Fail$\"%yY F07$Ffil$\"%d[F07$F[jl$\"%M^F07$Fejl$\"%HtF07$Fi[m$\"%X%)F07$F^\\m$\"% J))F07$Fc\\m$\"%G$*F07$Fh\\m$\"%_&)F07$F]]m$\"%pfF07$Fb]m$\"%*4#F07$Fg ]m$!$\"zF07$F\\^m$\"%l@F07$Fa^m$\"%mcF07$Fe_m$\"%W:F07$Fi`m$!%xNF07$$ \"5ML3F%zv$H.7F0$!%*f#F07$$\"5++]PfeR.57F0$!%S;F07$$\"5nm\"zW#fTx;7F0$ !%`;F07$F^am$!%D=F07$$\"5nm;z>hZ*pB\"F0$!%`IF07$Fcam$!%&)[F07$Fham$!%z KF07$F]bm$!#mF07$Fbbm$!%M6F07$Fgbm$!%>HF07$$\"5+++D1RA-i8F0$!%8KF07$$ \"5+++]i!H$Gl8F0Ff_w7$$\"5+++v=UVao8F0$!%(=$F07$$\"5++++v$R0=P\"F0$!%F JF07$$\"5+++DJXk1v8F0Fc`w7$$\"5+++](o\\F$y8F0$!%FIF07$$\"5+++vV[&)e\"Q \"F0$!%nDF07$F\\cm$!%mDF07$$\"5++++D1Q*yR\"F0$!%q:F07$Facm$\"$^\"F07$$ \"5+++D1k!*>99F0F[bw7$$\"5+++]i:,Y<9F0$\"$n\"F07$$\"5+++v=n6s?9F0$\"$F &F07$$\"5++++v=A)RU\"F0Fhbw7$$\"5+++](=K/0V\"F0$\"$%HF07$Ffcm$\"$k#F07 $$\"5++++DJ12]9F0$!%u:F07$F[dm$!%bBF07$$\"5M$e9m>`P!p9F0$!%`BF07$$\"5n m\"HKkAg\\Z\"F0$!%dBF07$$\"5Mek`mt:#zZ\"F0$!%tAF07$$\"5+]P%)*3#H)3[\"F 0$!%'f\"F07$$\"5nT5:8oU%Q[\"F0$!%!f\"F07$F`dm$!%m:F07$$\"5+Dcwfipw*[\" F0$!$k*F07$$\"5n;H2$)4$GF\\\"F0$!$*zF07$$\"5M3-Q1d'*o&\\\"F0$!$(zF07$$ \"5++voH/5l)\\\"F0$!$p)F07$$\"5M$3-j()ptX]\"F0$!%$4\"F07$Fedm$!%#)GF07 $$\"5MeRAYSxX8:F0$!%gGF07$$\"5+]7`p(3>k^\"F0$!%jGF07$$\"5nT&QG\\V!Q>:F 0$!%5RF07$Fjdm$!%O[F07$$\"5n;/wiwWEG:F0$!%,ZF07$F_em$!%'[%F07$Fdem$!%k UF07$Fiem$!$f#F07$F^fm$!$c#F07$Fcfm$!$&>F07$Fhfm$\"%1BF07$F]gm$\"%-oF0 7$Fbgm$\"%:nF07$Fggm$\"%nsF07$F\\hm$\"&-F\"F07$Fahm$\"&NW\"F07$Ffhm$\" &rH\"F07$F[im$\"&#z5F07$F`im$\"%_cF07$Feim$\"%\\?F07$F_jm$\"%$)>F07$Fi jm$\"%W7F07$F[^n$\"&1E\"F07$F`^n$\"&_E\"F07$Fe^n$\"&PE\"F07$F__ n$\"&YB\"F07$Fi_n$\"&aA\"F07$F^`n$\"&l7\"F07$Fc`n$\"&\"y5F07$$\"5nm;z% \\:>\"[7F07$F[cn$\"&B@\"F07$Fecn$\"&H6\"F07$Fidn$\"&\"[5F07$Fcen$\"&2Q\" F07$Fagn$\"&Sh\"F07$Ffgn$\"&8W\"F07$F[hn$\"%\"y)F07$F`hn$\"%0#)F07$Feh n$\"%PpF07$Fjhn$\"%c!)F07$F_in$\"%<**F07$Fiin$\"&5a\"F07$Fcjn$\"&sh\"F 07$F][o$\"&D0\"F07$Fg[o$\"%\\tF07$$\"5,+DJXR0g*3#F0$\"%\"R(F07$$\"5,+] 7G)[=g4#F0$\"%nuF07$$\"5,+v$4rVOC5#F0$\"%7()F07$F\\\\o$\"%P!*F07$$\"5, ]i:NgL17@F0$\"%z&*F07$$\"5,+DcwMBF:@F0$\"%3'*F07$$\"5,](oz\"48[=@F0$\" %`&*F07$$\"5,+]Pf$G!p@@F0$\"%)>*F07$$\"5,]7y+e#**[7#F0$\"%;#*F07$$\"5, +v=UK#3\"G@F0$\"%n!*F07$$\"5,]Pf$o?<88#F0$\"%fzF07$Fa\\o$\"%pzF07$Ff\\ o$\"%#)yF07$F[]o$\"%;oF07$F`]o$\"%AoF07$Fe]o$\"%5oF07$Fj]o$\"%&['F07$F _^o$\"%*['F07$Fd^o$\"%7lF07$Fi^o$\"%+rF07$F^_o$\"%Q$)F07$Fc_o$\"%B(*F0 7$F]`o$\"&A3\"F07$Fg`o$\"&*46F07$$\"5oTgFm74h-AF0$\"&39\"F07$F\\ao$\"& 29\"F07$$\"5Meky(*otI4AF0$\"&)R6F07$Faao$\"&+9\"F07$Ffao$\"&q3\"F07$F[ bo$\"%?&*F07$F`bo$\"%I\"*F07$Febo$\"%Q%)F07$Fjbo$\"%i#)F07$F_co$\"%`\" )F07$Fdco$\"%6))F07$Fico$\"%e**F07$F^do$\"&U4\"F07$Fcdo$\"&%o6F07$$\"5 nTgxySbo?BF0$\"&7>\"F07$$\"5n;/,Wf-wBBF0$\"&5>\"F07$$\"5n\"zW#4y\\$oK# F0$\"&%*=\"F07$Fhdo$\"&_:\"F07$$\"5nTNrR:W)HL#F0$\"&=:\"F07$$\"5n;z%\\ S8fgL#F0$\"&1:\"F07$$\"5n\"H#=q_Q8RBF0$\"&D6\"F07$F]eo$\"&(f5F07$Fafo$ \"%2&*F07$F[go$\"%5))F07$F`goFgjl7$Fego$\"&!=5F07$F_ho$\"%A#*F07$Fcio$ \"%.nF07$$\"5n\"z>6[c_7U#F0$\"%R]F07$Fhio$\"%\")\\F07$$\"5M3FWqjb)yU#F 0$\"%N]F07$F]jo$\"%veF07$Fgjo$\"%0iF07$Fa[p$\"&40\"F07$Ff[p$\"&45\"F07 $F[\\p$\"&)\\#F07$Fd]p$\"&%p@F07$Fi]p$\"&\"=AF07$F^^p$\"&ts\"F07$Fc^p$\"&,9 \"F07$Fg_p$\"&ti\"F07$Feap$\"&$\\=F07$F_bp$\"&kq\"F07$Fccp$\"&-k\"F07$ Fhcp$\"&0r\"F07$F]dp$\"&4!=F07$$\"5M$3-8Da3-r#F0$\"&I\"=F07$$\"5omT5lD ho;FF0$\"&F\"=F07$$\"5,]i!*y3P;BFF0$\"&$*z\"F07$Fbdp$\"&Vx\"F07$$\"5,+ DJ?ekfUFF0$\"&or\"F07$Fgdp$\"&Bn\"F07$F\\ep$\"&Sk\"F07$Faep$\"&Wk\"F07 $Ffep$\"&Bk\"F07$F[fp$\"&Ek\"F07$F`fp$\"&yk\"F07$Fefp$\"&Cn\"F07$Fjfp$ \">\"F07$F_gp$\"&1$=F07$Fchp$\"&E0#F07$Fgip$\"&GA#F07$$\"5,]P%)*e4aU 'GF0$\"&[C#F07$$\"5ML3_]%3+w'GF0$\"&WE#F07$$\"5o;z>6tg%4(GF0$\"&.D#F07 $$\"5,+](=<1#HuGF0$\"&rF#F07$$\"5M$3_D.0Qw(GF0$\"&0H#F07$$\"5om\"HK*QS )4)GF0$\"&W8#F07$$\"5,]i!Rv-IV)GF0$\"&P;#F07$F\\jp$\"&s;#F07$$\"5,+v$f L*zO%*GF0$\"&r)=F07$$\"5om;Hdq*f5!HF0$\"&=c\"F07$$\"5,](oz\"ffS/HF0$\" &ue\"F07$$\"5MLekyZ>v2HF0$\"&8g\"F07$$\"5o;HKROz46HF0$\"&?Z\"F07$Fajp$ \"&r\\\"F07$$\"5,+D19>2*4#HF0$\"&Br\"F07$Ffjp$\"&7z\"F07$$\"5,+v=U2V3M HF0$\"&!)=#F07$F[[q$\"&N\\#F07$$\"5,]7Gj)\\/R%HF0$\"&.`#F07$$\"5,+DJq& *yF07$F_\\q$\"&gy\"F07$$\"5,]ilZj_=$)H F0$\"&Zy\"F07$Fd\\q$\"&$)z\"F07$$\"5,](=F07$Fj`q$\"&29 #F07$F_aq$\"&1C#F07$$\"5,v=F07$Fgcq$\"&T&>F07$Fadq$\"&/)>F07$F[eq$\"&%))>F07$F`eq$ \"&!f@F07$Feeq$\"&RD#F07$$\"5,Dcw4DxX3KF0$\"&_E#F07$Fjeq$\"&sJ#F07$$\" 5,voa&=&4\\:KF0$\"&oJ#F07$F_fq$\"&(=BF07$$\"5,D\"G8'yT_AKF0$\"&cK#F07$ Fdfq$\"&WK#F07$$\"5,v$4r`Sd&HKF0$\"&%*G#F07$Fifq$\"&0F#F07$F]hq$\"&z3# F07$Fejq$\"&^h\"F07$Fjjq$\"&?j\"F07$F_[r$\"&Q:#F07$Fd[r$\"&97#F07$Fi[r $\"&h9#F07$Fc\\r$\"&)RFF07$F]]r$\"&.7$F07$$\"5ML$e9\"fm#zO$F0$\"&X,$F0 7$Fb]r$\"&c)GF07$$\"5,+]P4Y=U\"Q$F0$\"&:P#F07$Fg]r$\"&BB#F07$Fa^r$\"&8 y\"F07$F[_r$\"&1a\"F07$$\"5om\"H2LUQ]S$F0$\"&1_\"F07$F`_r$\"&P_\"F07$$ \"5,+voz;gy6MF0$\"&Eg\"F07$Fe_r$\"&]e\"F07$Fj_r$\"&O'=F07$F_`r$\"&&z@F 07$Fd`r$\"&HJ#F07$Fi`r$\"&sX#F07$F^ar$\"&rR#F07$Fcar$\"&zC#F07$Fhar$\" &;B#F07$F]br$\"&UC#F07$Fbbr$\"&ZK#F07$Fgbr$\"&$pBF07$$\"5,vVt_!>)z'f$F 0$\"&*pBF07$F\\cr$\"&()R#F07$$\"5o\"zpB,w#[.OF0Fc][l7$Facr$\"&*)R#F07$ Ffcr$\"&[S#F07$F[dr$\"&_Q#F07$F`dr$\"&dM#F07$FedrFe\\r7$$\"5MeRZ-#Hx+l $F0$\"&iD#F07$$\"5o;z%\\:7AKl$F0$\"&jD#F07$$\"5,v=U2^pOcOF0F]_[l7$$\"5 MLe*)f!y6&fOF0$\"&GC#F07$$\"5,]P%['R9!em$F0$\"&UB#F07$$\"5om;zp)4\"4sO F0$\"&'RAF07$$\"5,+voz;/n%o$F0$\"&$zAF07$Fjdr$\"&kN#F07$$\"5,+]P4r$3Cs $F0$\"&_h#F07$F_er$\"&mv#F07$Fder$\"&'))GF07$Fier$\"&T%HF07$F^fr$\"&9% GF07$Fcfr$\"&6F#F07$Fhfr$\"&CL#F07$F]gr$\"&_t#F07$Fggr$\"&'zJF07$Fahr$ \"&*)Q$F07$Ffhr$\"&a\\$F07$F[ir$\"&Aa$F07$F`ir$\"&O[$F07$Feir$\"&(fMF0 7$Fg\\s$\"&5:$F07$Fc`s$\"&Q%HF07$F]as$\"&xI$F07$Fabs$\"&\"4MF07$Ffbs$ \"&fB$F07$F[cs$\"&i4$F07$F`cs$\"&kE$F07$Fecs$\"&HY$F07$F_ds$\"&dF$F07$ F]fs$\"&]6$F07$F`fs$\"&+:$F07$Fefs$\"&<@$F07$Fjfs$\"&'yJF07$F_gs$\"&r9 $F07$Fchs$\"&1!HF07$Fgis$\"&w\\#F07$F[[t$\"&Im#F07$F]\\t$\"&e:$F07$Fg \\t$\"&Au$F07$Fa]t$\"&)3RF07$Ff]t$\"&@&QF07$F[^t$\"&7e$F07$$\"5o;ajMj) )ezUF0$\"&&eJF07$F`^t$\"&`!HF07$$\"5N$eRse$[t#H%F0$\"&pT#F07$Fe^t$\"&] H#F07$$\"5N3FpE:Vf-VF0$\"&3@#F07$$\"5-]P%)R33)eI%F0$\"&\\=#F07$$\"5o\" z%*H:In\"4VF0$\"&WB#F07$Fj^t$\"&!4BF07$$\"5o;zW#4yE!>VF0$\"&6P#F07$F__ t$\"&z]#F07$Fd_t$\"&,)GF07$Fi_t$\"&o*HF07$$\"5M3_v:.O?\"Q%F0$\"&6/$F07 $F^`t$\"&$QIF07$$\"5o\"HK*)[\\VsQ%F0$\"&k.$F07$Fc`t$\"&60$F07$F]at$\"& y/$F07$Fgat$\"&d-$F07$F\\bt$\"&w(HF07$Fabt$\"&%>HF07$Fect$\"&/'GF07$F_ dt$\"&Z#GF07$$\"5,](o/)\\AqcWF0$\"&^#GF07$Fddt$\"&G%GF07$$\"5o;H#oK.#y oWF0$\"&T&GF07$Fidt$\"&:)GF07$$\"5ML$3_+sm')[%F0$\"&$RHF07$$\"5ommT5:: ^-XF0$\"&`*HF07$$\"5ML3-j7RV4XF0$\"&[*HF07$$\"5,+]i:5jN;XF0$\"&Y*HF07$ $\"5om\"H#o2(yK_%F0$\"&*pHF07$F^et$\"&s#HF07$$\"5om;/E+f/WXF0$\"&(zGF0 7$$\"5,++DJ&p!*yb%F0$\"&/%GF07$$\"5omT&QG48[c%F0$\"&c#GF07$$\"5ML$ek. \\NvXF0$\"&,%GF07$$\"5,+D1*y)ylyXF0$\"&D& GF07$$\"5M$ek`m3>@e%F0$\"&n)GF07$Fcet$\"&())GF07$F]ft$\"&IJ$F07$Fgft$ \"&z[$F07$F[ht$\"&GE$F07$F_it$\"&.=$F07$Fdit$\"&8p$F07$Fiit$\"&'zUF07$ $\"5-DJ&pP\"evSZF0$\"&*GWF07$$\"5N$e*)fb^cSu%F0$\"&=\\%F07$$\"5oTg-N9%F07$Fhjt$\"&'p TF07$F[[u$\"&$4RF07$F`[u$\"&Q%RF07$Fe[u$\"&G(RF07$Fj[u$\"&Z$RF07$F_\\u $\"&>'RF07$Fd\\u$\"&+*RF07$Fi\\u$\"&/6%F07$F^]u$\"&@:%F07$Fc]u$\"&SG%F 07$Fh]u$\"&qI%F07$F]^u$\"&(>VF07$Fb^u$\"&W@%F07$Fg^u$\"&d0%F07$F\\_u$ \"&@@%F07$Fa_u$\"&ja%F07$$\"5-]i!RvZ]\")*[F0$\"&\"[XF07$$\"5-+DJqX/%\\ !\\F0$\"&>c%F07$$\"5-](=nQTI<\"\\F0$\"&E_%F07$Ff_u$\"&e]%F07$$\"5-+v$f $=.5K\\F0$\"&TN%F07$F[`u$\"&iF%F07$$\"5+DJ&p(Q_2\\\\F0$\"&0E%F07$$\"5+ ]i:&GAqC&\\F0$\"&XB%F07$$\"5+v$fLp?le&\\F0$\"&YB%F07$$\"5++Dc,\">g#f\\ F0$\"&$QUF07$$\"5+Dcw4v^li\\F0$\"&6C%F07$$\"5+](oz\"f,0m\\F0$\"&#F07$Fj`l$\"$7(F07$F_al$\"%nhF07$Fdal$ \"%\\uF07$Fial$\"&o=\"F07$$\"5+++v$4@?>y'F`o$\"&(*=\"F07$F\\av$\"&'[7F 07$$\"5qmm\"zptBg%oF`o$\"&!H7F07$F^bl$\"&A=\"F07$Fdav$\"%$G)F07$Fcbl$ \"%-MF07$F\\bv$\"%cLF07$Fabv$\"%5KF07$Ffbv$\"$#fF07$Fhbl$\"$&eF07$F^cv $\"$u'F07$F]cl$\"$8(F07$Fgcl$\"%?cF07$Fadl$\"%dqF07$F[el$\"%;zF07$Feel $\"%%R)F07$Fjel$\"%M()F07$F_fl$\"%7')F07$Fbev$\"%v#)F07$Fgev$\"%rxF07$ F\\fv$\"%CvF07$Fafv$\"%^tF07$F[gv$\"%ctF07$Fdfl$\"%,uF07$Fhgv$\"%BxF07 $F]hv$\"%S%)F07$Fghv$\"%U()F07$Fifl$\"%W))F07$F]hl$\"%_zF07$F[jl$\"%@q F07$Fejl$\"%8))F07$Fi[m$\"&u4\"F07$F^\\m$\"&!)>\"F07$Fc\\m$\"&eQ\"F07$ $\"5++D19WCxw5F0$\"&ZT\"F07$Fh\\m$\"&3[\"F07$$\"5+]7G8'G`d3\"F0$\"&\") \\\"F07$$\"5ML3-8+pu)3\"F0$\"&.\\\"F07$$\"5n;/w790u\"4\"F0$\"&iT\"F07$ F]]m$\"&YV\"F07$Fb]m$\"&D@\"F07$Fg]m$\"%%)*)F07$$\"5+]i:5amn@6F0$\"%Xx F07$$\"5MLe*)4o-nC6F0$\"%iyF07$$\"5n;aj4#)QmF6F0$\"%uzF07$F\\^m$\"%V&) F07$$\"5nmT&)3CZkO6F0$\"%-))F07$Fa^m$\"&O1\"F07$Ff^m$\"&L7\"F07$F[_m$ \"&H>\"F07$F`_m$\"&O5\"F07$Fe_m$\"%J**F07$Fj_m$\"%L%*F07$F_`m$\"%6')F0 7$Fd`m$\"%z'*F07$Fi`m$\"&L+\"F07$Fc]w$\"&D]\"F07$F^am$\"&=g\"F07$F`^w$ \"&\"H:F07$Fcam$\"&aC\"F07$Fgbm$\"&tT\"F07$Facm$\"&B]\"F07$F[dm$\"&sg \"F07$Fcdw$\"&2d\"F07$F`dm$\"&]b\"F07$Fifw$\"&hY\"F07$Fedm$\"%C)*F07$F jdm$\"%K`F07$F_em$\"%*)GF07$Fcfm$\"%p]F07$Fggm$\"&=B\"F07$F\\hm$\"&4#> F07$Fahm$\"&8O#F07$FfhmF^`p7$F[im$\"&?<#F07$F`im$\"&\\p\"F07$FeimFe`x7 $F_jm$\"&*e6F07$Fijm$\"%C%*F07$Fc[n$\"&:J\"F07$F]\\n$\"&**\\\"F07$F[^n $\"&2g\"F07$Fi_n$\"&*o;F07$Fc`n$\"&?h\"F07$F]an$\"&.a\"F07$Fban$\"&t` \"F07$Fgan$\"&!o:F07$Fabn$\"&2h\"F07$F[cn$\"&Gn\"F07$F`cn$\"&&y;F07$Fe cn$\"&Tk\"F07$F_dn$\"&vd\"F07$Fidn$\"&.`\"F07$Fcen$\"&9v\"F07$Fagn$\"& 'QBF07$F[hn$\"&[8#F07$F_in$\"&J*>F07$Fiin$\"&uL#F07$Fcjn$\"&b\\#F07$Fh jn$\"&HS#F07$F][o$\"&CJ#F07$Fb[o$\"&xI#F07$Fg[o$\"&gH#F07$F\\\\o$\"&W* GF07$Fa\\o$\"&a,$F07$Fe]o$\"&#fGF07$Fi^o$\"&%3FF07$F^_o$\"&qu#F07$Fc_o $\"&Q$GF07$Fg`o$\"&(yHF07$Faao$\"&x3$F07$$\"5M$3_]M0_$>AF0$\"&!*4$F07$ Ffao$\"&n5$F07$$\"5o;H23m\\uKAF0$\"&@3$F07$F[bo$\"&#\\IF07$Febo$\"&Y%H F07$F_co$\"&:(GF07$$\"5M3-Q\"eZv&pAF0$\"&7%GF07$$\"5n;a8(RqBHF#F0$\"&g #GF07$$\"5+D1*G@$>FwAF0$\"&d#GF07$Fdco$\"&B#GF07$$\"5nT5SW)QoHG#F0$\"& ?#GF07$$\"5+]i:g;mJ'G#F0$\"&C#GF07$$\"5Me9\"fZ%[m*G#F0$\"&A&GF07$Fico$ \"&c'GF07$Fcdo$\"&!fIF07$F]eo$\"&l?$F07$F[go$\"&x4$F07$Fego$\"&>'HF07$ Fcio$\"&/=#F07$Fa[p$\"&!o=F07$Ff[p$\"&m!>F07$F[\\p$\"&$yDF07$F`\\p$\"& z`#F07$Fe\\p$\"&]c#F07$F_]p$\"&B>$F07$Fi]p$\"&XZ$F07$F^^p$\"&%4JF07$Fc ^p$\"&0Y#F07$$\"5M$3-jP(=U/DF0$\"&h-#F07$Fh^p$\"&m%=F07$$\"5M3_DG1mv9D F0$\"&V#=F07$$\"5M$eRAr^,#=DF0$\"&o\"=F07$$\"5MeRA'zUY;_#F0$\"&()*=F07 $F]_p$\"&)z=F07$Fb_p$\"&52#F07$Fg_p$\"&vB#F07$Fa`p$\"&JO#F07$Feap$\"&# [CF07$F]dp$\"&;T#F07$Fady$\"&*[CF07$Fbdp$\"&4Z#F07$F^ey$\"&\"[CF07$Fgd p$\"&\"3CF07$Faep$\"&5N#F07$Fefp$\"&BH#F07$$\"5M$3Fp,aRzy#F0$\"&XH#F07 $Fjfp$\"&hI#F07$$\"5M3-j([\"fl(z#F0$\"&8J#F07$$\"5,]7`W1Z*3!GF0$\"&EO# F07$$\"5o\"HK9!)\\LT!GF0$\"&cO#F07$F_gp$\"&JP#F07$Fchp$\"&Mp#F07$Fgip$ \"&a0$F07$Fchy$\"&&pKF07$F\\jp$\"&oS$F07$F_jy$\"&43$F07$Fajp$\"&D+$F07 $F[[q$\"&bf$F07$Fe[q$\"&iw$F07$Fi\\q$\"&>.%F07$Fg^q$\"&vf%F07$F\\_q$\" &P\\%F07$Fa_q$\"&@N%F07$$\"5,v=U2^spZIF0$\"&`M%F07$Ff_q$\"&CD%F07$$\"5 ,Dc^.6;.aIF0$\"&RD%F07$F[`q$\"&OD%F07$$\"5,v$4'*4(fOgIF0$\"&hB%F07$F`` q$\"&pB%F07$$\"5,DJq&4L+n1$F0$\"&\"QUF07$Fe`q$\"&EG%F07$F_aq$\"&_[%F07 $Fcbq$\"&/j%F07$F]cq$\"&5^%F07$F[eq$\"&iM%F07$Feeq$\"&!*\\%F07$Fifq$\" &@t%F07$Fcgq$\"&pk%F07$F]hq$\"&F]%F07$Faiq$\"&9L%F07$F[jq$\"&G$QF07$F` jq$\"&)fIF07$FejqFfa\\l7$Fjjq$\"&[(GF07$F_[r$\"&&zKF07$Fd[r$\"&*HKF07$ Fi[r$\"&EC$F07$Fc\\r$\"&.)QF07$F]]r$\"&iZ%F07$$\"5om\"zpB'GbkLF0$\"&>S %F07$Fjhz$\"&6O%F07$$\"5,+v$feX+8P$F0$\"&-^%F07$Fb]r$\"&bV%F07$Fbiz$\" &+/%F07$Fg]r$\"&\"\\QF07$Fa^r$\"&`L$F07$F[_r$\"&F!HF07$F`_r$\"&F\"GF07 $Fe_r$\"&]q#F07$Fj_r$\"&*)z#F07$F_`r$\"&N-$F07$$\"5om\"H#oK)yVX$F0$\"& v0$F07$Fd`r$\"&53$F07$$\"5,+vo/Bh$)yMF0$\"&G8$F07$Fi`r$\"&v@$F07$$\"5M Le9T8MH.NF0$\"&&[KF07$F^ar$\"&.E$F07$$\"5M$e*[=\"QO;_$F0$\"&0D$F07$$\" 5omTgx.2vFNF0$\"&#[KF07$$\"5,](=nj-lQ`$F0$\"&/A$F07$Fcar$\"&8=$F07$Fha r$\"&H9$F07$F]br$\"&'>JF07$Fbbr$\"&(QJF07$Fgbr$\"&X<$F07$Facr$\"&i@$F0 7$F[drFhd[l7$F`dr$\"&5E$F07$Fedr$\"&GB$F07$Fc_[l$\"&:;$F07$F]`[l$\"&F5 $F07$Fb`[l$\"&\")3$F07$Fjdr$\"&%=JF07$Fj`[l$\"&\"=MF07$F_er$\"&zo$F07$ Fder$\"&7(RF07$Fier$\"&N@%F07$$\"5ML$eRs`R/y$F0$\"&ZD%F07$F^fr$\"&!pVF 07$$\"5,+](=#\\&)e$z$F0F[d`l7$Fcfr$\"&([SF07$F]gr$\"&PL%F07$Feir$\"&\\ 2&F07$Fjir$\"&c5&F07$F_jr$\"&t:&F07$Fc[s$\"&*3eiHSF0$\"&E7'F07$Ffbs$\"&u5'F07$$\" 5ML3F%Ho8)\\SF0$\"&B(fF07$F[cs$\"&n*eF07$$\"5M$3x\")HP`(pSF0$\"&[(eF07 $$\"5omT&Q.[Mi2%F0$\"&'peF07$$\"5,]7`p(e:F3%F0$\"&&))eF07$F`cs$\"&a\"f F07$$\"5,+Dcw4*e@5%F0$\"&%\\gF07$Fecs$\"&6:'F07$$\"5N3_v:y;O=TF0$\"&T; 'F07$$\"5-]Pf$=B-;7%F0$\"&$HiF07$$\"5o\"HK9byU[7%F0$\"&zA'F07$Fjcs$\"& 5B'F07$$\"5o;z%\\lWkX8%F0$\"&\\D'F07$F_ds$\"&4B'F07$Fces$\"&(ehF07$F]f s$\"&G2'F07$Fefs$\"&/)fF07$F_gs$\"&9u&F07$Figs$\"&!p`F07$Fchs$\"&fE&F0 7$F]is$\"&!eZF07$Fgis$\"&EM%F07$Fajs$\"&UB%F07$F[[t$\"&z?%F07$F`[t$\"& s:%F07$Fc[t$\"&ug%F07$Fh[t$\"&bg%F07$F]\\t$\"&va%F07$Fg\\t$\"&*>_F07$F a]t$\"&/c&F07$Ff]t$\"bF07$F[^t$\"&![aF07$Fhg[l$\"&i0&F07$F`^t$\"&>y %F07$F`h[l$\"&_=%F07$Fe^t$\"&!HRF07$F]i[l$\"&xp$F07$Fj^t$\"&'=OF07$$\" 5-voHz(GSdJ%F0$\"&8e$F07$Fji[l$\"&&3OF07$$\"5Ne*)f0uKJAVF0$\"&.p$F07$F __t$\"&#fOF07$$\"5omTNrRduQVF0$\"&xy$F07$Fd_t$\"&y)QF07$$\"5o;/E])pk%e VF0$\"&;(QF07$$\"5-+Dcw%oP]O%F0$\"&i*QF07$$\"5oTNr*ySF07$Fgat$\"&&[SF07$F\\bt$\"& W0%F07$Fabt$\"&#QSF07$Fect$\"&l*RF07$F_dt$\"&'HRF07$Fddt$\"&:\"RF07$Fi dt$\"&c\"RF07$Fj]\\l$\"&#\\RF07$F_^\\l$\"&'=SF07$Fi^\\l$\"&&\\SF07$F^e t$\"&/0%F07$F[`\\l$\"&\"pRF07$Fcet$\"&o)QF07$Fhet$\"&W7%F07$F]ft$\"&'F07$Fegu$\"%(y$F07$Fjs$\"%uHF07$F\\iu$\"%$y&F07$F_t$ \"%=))F07$Fdt$\"%X**F07$Fit$\"&w+\"F07$F^u$\"&1-\"F07$Fcu$\"&Z+\"F07$F hu$\"&)G5F07$F]v$\"&o.\"F07$Fgv$\"&y?\"F07$Faw$\"&0o\"F07$F_c]l$\"&uo \"F07$Fdc]l$\"&\")p\"F07$Fic]l$\"&!zo\"F07$Ffz$\"&=o\"F07$F]h]l$\"& 9g\"F07$F[[l$\"&n[\"F07$$\"5NL3FWXK1zZF`o$\"&][\"F07$Feh]l$\"&aZ\"F07$ $\"5N$3FWXKE#z[F`o$\"&!y9F07$$\"5-+D\"yv,9E\"\\F`o$\"&'3:F07$$\"5o;z>h 5<+Y\\F`o$\"&&3:F07$F`[l$\"&_^\"F07$F]i]l$\"&mn\"F07$Fe[l$\"&3x\"F07$F ei]l$\"&hx\"F07$Fji]l$\"&#Q=F07$F_j]l$\"&x$=F07$Fj[l$\"&$Q=F07$Fej]l$ \"&1'=F07$F_\\l$\"&?&=F07$$\"5qm;z>OH3DbF`o$\"&=%=F07$Fd\\l$\"&,%=F07$ $\"5NL$3FpE#*pe&F`o$\"&G$=F07$Fi\\l$\"&n!=F07$F^]l$\"&8!=F07$Fc]l$\"& \")y\"F07$Fc[^l$\"&7\"=F07$Fd]v$\"&')z\"F07$F[\\^l$\"&k!=F07$F`\\^l$\" &h!=F07$Fe\\^l$\"&iz\"F07$Fh]l$\"&Hv\"F07$F\\^v$\"&&[8F07$F]^l$\"%2&)F 07$Fd^v$\"%!H)F07$Fb^l$\"%xWF07$F\\_v$\"%BWF07$Fg^l$\"%[VF07$F\\_l$\"% kRF07$Fa_l$\"%zQF07$Ff_l$\"%_QF07$F[`l$\"%XdF07$F``l$\"%UwF07$Fe`l$\"% bvF07$Fj`l$\"%`%*F07$F_al$\"&eK\"F07$Fdal$\"&nP\"F07$Fial$\"&5j\"F07$F \\av$\"&$*\\\"F07$F^bl$\"&6T\"F07$$\"5NLL3-js75pF`o$\"%q%*F07$FdavFc[w 7$$\"5+++D1*yIU(pF`o$\"%n))F07$Fcbl$\"%tLF07$FhblFjfu7$F]cl$!%c5F07$Fg cl$\"%BBF07$Fadl$\"%VNF07$F[el$\"%8VF07$Feel$\"%%o%F07$$\"5SLLLe9TaC\" )F`o$\"%^[F07$Fjel$\"%H[F07$$\"50+++DJg8d#)F`o$\"%(y%F07$F_fl$\"%EXF07 $Fbev$\"%STF07$Fgev$\"%fOF07$F\\fv$\"%[MF07$Fafv$\"%ZLF07$F[gv$\"%qLF0 7$Fdfl$\"%lMF07$Fhgv$\"%KQF07$F]hv$\"%=XF07$Fghv$\"%cZF07$Fifl$\"%uZF0 7$F]hl$\"%DF07$$\"5+++vo/=#\\K\"F0$\"&> `#F07$Fbbm$\"&Y]#F07$$\"5+++]PMaAX8F0$\"&VW#F07$Fgbm$\"&/N#F07$Facm$\" &&zCF07$F[dm$\"&0d#F07$Fcdw$\"&Ob#F07$F`dm$\"&pb#F07$Fifw$\"&kZ#F07$Fe dm$\"&<'>F07$Fjdm$\"&GV\"F07$F_em$\"&Q0\"F07$Fdem$\"&4/\"F07$Fiem$\"&& )4\"F07$F^fm$\"&X3\"F07$Fcfm$\"&L2\"F07$Fhfm$\"&*=7F07$F]gm$\"&g^\"F07 $Fbgm$\"&b\\\"F07$Fggm$\"&4`\"F07$F\\hm$\"&/+#F07$Fahm$\"&JA#F07$Ffhm$ \"&73#F07$F[im$\"&y(=F07$$\"5M3x\"[N!*p&*e\"F0$\"&,]\"F07$F`im$\"&%)G \"F07$$\"5n\"zpB,c7mf\"F0$\"&wD\"F07$Feim$\"%&3(F07$F_jm$\"%SnF07$Fijm $\"%XQF07$Fc[n$\"%RfF07$F]\\n$\"%WvF07$F[^n$\"%&[)F07$Fi_n$\"%)*))F07$ Fc`n$\"%'4)F07$F]an$\"%fuF07$Fgan$\"%2zF07$F[cn$\"%R))F07$$\"5o;HKknnZ G=F0$\"%S))F07$F`cn$\"%@))F07$$\"5M$3-QckcB%=F0$\"%$f)F07$Fecn$\"%5%)F 07$F_dn$\"%oxF07$FidnFjcx7$F^en$\"%)*zF07$Fcen$\"&U+\"F07$Fgfn$\"&F07$Fc jn$\"&xH#F07$F][o$\"&RT#F07$Fg[o$\"&Ba#F07$F]dx$\"&r!GF07$F\\\\o$\"&(f JF07$Fjdx$\"&wF$F07$F_ex$\"&zG$F07$Fdex$\"&**H$F07$Fiex$\"&gI$F07$F^fx $\"&HJ$F07$Fcfx$\"&MI$F07$Fhfx$\"&')>$F07$Fa\\o$\"&G?$F07$Fe]o$\"&\"\\ IF07$Fi^o$\"&q#HF07$F^_o$\"&%yHF07$Fc_o$\"&;2$F07$Fg`o$\"&;@$F07$Faao$ \"&NI$F07$Fde_l$\"&pI$F07$Ffao$\"&!3LF07$F\\f_l$\"&F07$F]jo$\"&$)p\"F07$F bjo$\"&dn\"F07$Fgjo$\"&Rm\"F07$F\\[p$\"&x!=F07$Fa[p$\"&7y\"F07$Fe\\p$ \"&0C#F07$Fi]p$\"&)fFF07$$\"5M$ek.a0LwZ#F0$\"&3l#F07$F^^p$\"&gG#F07$$ \"5Me*[VPb#e([#F0$\"&<@#F07$$\"5n;/,>`!**3\\#F0$\"&mk\"F07$$\"5+v=nj_b @%\\#F0$\"&>i\"F07$Fc^p$\"&Zc\"F07$Fh[`l$\"&r1\"F07$Fh^p$\"%F#)F07$Fe \\`l$\"%J!)F07$F]_p$\"%#>)F07$Fb_p$\"%K'*F07$Fg_p$\"&x4\"F07$Fa`p$\"&O A\"F07$Feap$\"&;H\"F07$F_bp$\"&iC\"F07$Fccp$\"&!p6F07$$\"5,+](=#*HXxm# F0$\"&\"[6F07$Fhcp$\"&\\;\"F07$$\"5om;HKRdt\"p#F0$\"&a?\"F07$F]dp$\"&Y C\"F07$$\"5oT5S%4vppq#F0$\"&bC\"F07$F\\dy$\"&jF\"F07$$\"5,DJ?3MtW8FF0F d`fl7$Fady$\"&nF\"F07$Ffdy$\"&5H\"F07$Fbdp$\"&lG\"F07$F^ey$\"&nD\"F07$ Fgdp$\"&e@\"F07$Faep$\"&b;\"F07$Fefp$\"&*H6F07$Fjfp$\"&&e6F07$F_gp$\"& aB\"F07$Fchp$\"&;`\"F07$Fgip$\"&#y=F07$Fchy$\"&Y4#F07$F\\jp$\"&rC#F07$ F_jy$\"&g*>F07$Fajp$\"&/\">F07$Ff[z$\"&+(>F07$Ffjp$\"&m/#F07$F^\\z$\"& PK#F07$F[[q$\"&Pn#F07$F`[q$\"&H)HF07$Fe[q$\"&_7$F07$Fi\\q$\"&jg$F07$Fg ^q$\"&f2%F07$F\\_q$\"&z$RF07$Fa_q$\"&#*z$F07$Fac`l$\"&Kz$F07$Ff_q$\"&M r$F07$Fic`l$\"&Zr$F07$F[`q$\"&]r$F07$Fad`l$\"&Ar$F07$F``q$\"&Hr$F07$Fi d`l$\"&Wr$F07$Fe`q$\"&'oPF07$F_aq$\"&T(RF07$Fcbq$\"&@3%F07$F]cq$\"&*[R F07$F[eq$\"&<\"QF07$Feeq$\"&i)RF07$Fifq$\"&F>%F07$Fcgq$\"&\\7%F07$F]hq $\"&92%F07$Faiq$\"&%HRF07$F[jq$\"&\\V$F07$$\"5om\"z%\\!pH_J$F0$\"&))3$ F07$F`jq$\"&Gf#F07$$\"5,+v$4Y-pyK$F0$\"&>^#F07$Fejq$\"&bJ#F07$Fjjq$\"& nH#F07$F_[r$\"&C`#F07$Fd[r$\"&T\\#F07$Fi[r$\"&u\\#F07$Fc\\r$\"&%[HF07$ F]]r$\"&>M$F07$Fjhz$\"&6C$F07$Fb]r$\"&S=$F07$Fbiz$\"&Wo#F07$Fg]r$\"&\" =DF07$Fa^r$\"&E&>F07$F[_r$\"&RY\"F07$F`_r$\"&?R\"F07$Fe_r$\"&zC\"F07$F j_r$\"&GI\"F07$F_`r$\"&E[\"F07$Fd`r$\"&8b\"F07$Fi`rFh`cl7$F]\\al$\"&3q \"F07$F^ar$\"&'*p\"F07$Fe\\alFba_l7$Fj\\alFejel7$F_]al$\"&@k\"F07$Fcar $\"&Ag\"F07$Fgbr$\"&0i\"F07$Fedr$\"&Al\"F07$Fc_[lF][[l7$F]`[l$\"&*R:F0 7$$\"5M$eRZxv!QyOF0$\"&)R:F07$Fb`[l$\"&'R:F07$$\"5Me9;KY_\"yo$F0$\"&La \"F07$$\"5o;aj%e2g4p$F0$\"&ud\"F07$$\"5,v$4r`!\\5%p$F0$\"&)y:F07$FjdrF `en7$Fj`[l$\"&](=F07$F_er$\"&^3#F07$FderF`][l7$Fier$\"&()f#F07$Fe`al$ \"&1n#F07$F^fr$\"&Nu#F07$F]aal$\"&]l#F07$Fcfr$\"&5Z#F07$$\"5,+DJ?3.X.Q F0$\"&3^#F07$$\"5om;z>hvt1QF0$\"&\"\\DF07$$\"5ML3F>9[-5QF0$\"&mT#F07$F hfr$\"&mX#F07$$\"5ML$3xJd'))>QF0$\"&&)e#F07$F]gr$\"&Lo#F07$Fggr$\"&E-$ F07$Fahr$\"&(*G$F07$F[ir$\"&WZ$F07$Feir$\"&*zOF07$Fjir$\"&l#QF07$F_jr$ \"&H)RF07$Fc[s$\"&yT%F07$Fg\\s$\"&>s%F07$Fa]s$\"&$R[F07$F[^s$\"&E'[F07 $F`^s$\"&R\"[F07$Fe^s$\"&#*o%F07$Fi_s$\"&za%F07$Fc`s$\"&e[%F07$F]as$\" &(yYF07$Fabs$\"&$e[F07$Fgdal$\"&%p[F07$F\\eal$\"&P&[F07$Faeal$\"&\"=[F 07$Ffbs$\"&.![F07$Fieal$\"&vm%F07$F[cs$\"&Xg%F07$Fafal$\"&Sf%F07$Fffal $\"&=f%F07$F[gal$\"&Hi%F07$F`cs$\"&Jl%F07$Fcgal$\"&!*y%F07$Fecs$\"&G)[ F07$F[hal$\"&Q*[F07$F`hal$\"&,&\\F07$Fehal$\"&\"\\\\F07$Fjcs$\"&8&\\F0 7$F]ial$\"&)p\\F07$F_ds$\"&([\\F07$Fces$\"&k*[F07$F]fs$\"&y%[F07$F`fs$ \"&o%[F07$Fefs$\"&=$[F07$Fjfs$\"&)yZF07$F_gs$\"&]j%F07$Figs$\"&[F%F07$ Fchs$\"&'*=%F07$F]is$\"&%pOF07$Fgis$\"&q>$F07$Fajs$\"&H6$F07$F[[t$\"&u (HF07$F`[t$\"&*RHF07$Fc[t$\"&!GKF07$Fh[t$\"&]A$F07$F]\\t$\"&P=$F07$Fg \\t$\"&vm$F07$Fa]t$\"&g(QF07$Ff]t$\"&B$QF07$F[^t$\"&yk$F07$Fhg[l$\"&-A $F07$F`^t$\"&%eHF07$$\"5oT&)3uU$[%*G%F0$\"&ye#F07$F`h[l$\"&9J#F07$$\"5 -D1R+H8-'H%F0$\"&gF#F07$Fe^t$\"&q0#F07$F]i[l$\"&z#=F07$Fj^t$\"&`q\"F07 $F]^bl$\"&vo\"F07$Fji[l$\"&Oq\"F07$Fe^bl$\"&rv\"F07$F__t$\"&Bu\"F07$Fd _t$\"&)[>F07$Fi_t$\"&r,#F07$F^`t$\"&$e?F07$Fc`t$\"&U4#F07$F]at$\"&C4#F 07$Fgat$\"&'3@F07$$\"5M$3_]f6$Q3WF0$\"&s5#F07$F\\bt$\"&@5#F07$$\"5,]iS T**GY?WF0$\"&j2#F07$Fabt$\"&d2#F07$Fect$\"&!H?F07$F_dt$\"&x'>F07$Fddt$ \"&o&>F07$FidtF^bhl7$Fj]\\l$\"&g+#F07$F_^\\l$\"&D2#F07$Fd^\\l$\"&k3#F0 7$Fi^\\l$\"&q4#F07$F^_\\l$\"&95#F07$F^et$\"&T3#F07$F[`\\l$\"&>+#F07$Fc et$\"&@&>F07$Fhet$\"&@?#F07$F]ft$\"&\">CF07$Fbft$\"&t%GF07$Fgft$\"&K@$ F07$F\\gt$\"&`C$F07$Fagt$\"&@5$F07$Ffgt$\"&/:$F07$F[ht$\"&3=$F07$Feht$ \"&a.$F07$F_it$\"&v+$F07$Fdit$\"&&GLF07$Fiit$\"&e$QF07$F^jt$\"&rN%F07$ Fcjt$\"&k_%F07$Fe[u$\"&Mr%F07$Fi\\u$\"&hB&F07$Fc]u$\"&\\_&F07$F]^u$\"& Hj&F07$Fb^u$\"&ya&F07$Fg^u$\"&4I&F07$$\"5-]PMx]M&[F0$\"& /E&F07$$\"5-+D1k\"eo(f[F0$\"&mG&F07$F\\_u$\"&dH&F07$$\"5-+]7GQPsy[F0$ \"&fS&F07$Fa_u$\"&Ad&F07$Fggbl$\"&Bd&F07$Fbh\\l$\"&0e&F07$F_hbl$\"&8j& F07$Fgh\\l$\"&9j&F07$F\\i\\l$\"&%[cF07$Ff_u$\"&Tk&F07$Fdi\\l$\"&Pc&F07 $F[`u$\"&%)[&F07$Faj\\l$\"&%>aF07$F[[]l$\"&))R&F07$Fe[]l$\"&mP&F07$F`` u$\"&TP&F07$Fb\\]l$\"&nT&F07$Fe`u$\"&>f&F0-Fj`u6&F\\auF($\"#vF_au$\"#& *F_au-Faau6#%Ea~scheme~of~Tsitouras~and~PapakostasG-F$6%7[]mF'7$F+$\"$ !QF07$F2$\"$d'F07$Fi[cl$\"$M(F07$F7$\"$)fF07$Fa\\clF\\^il7$Fd\\cl$\"$v &F07$Fi\\cl$\"$r#F07$F'F07$FexF`[y7$Faz$\"%**)*F07$Ffz$\"%(Q*F07$F[ [l$\"%]uF07$Fadcl$\"%MuF07$Feh]lFecx7$Fidcl$\"%wtF07$F^ecl$\"%!p(F07$F ceclFbeil7$F`[l$\"%cxF07$F]i]l$\"%^$*F07$Fe[l$\"&_-\"F07$Fei]l$\"&+.\" F07$Fji]l$\"&d3\"F07$F_j]l$\"&a3\"F07$Fj[l$\"&f3\"F07$Fej]l$\"&,5\"F07 $F_\\l$\"&s3\"F07$Fd\\l$\"&F2\"F07$Fi\\l$\"&V.\"F07$F^]l$\"&6.\"F07$Fc ]l$\"&#>5F07$$\"5NL$e9\"f-ssdF`o$\"&$=5F07$Fc[^l$\"&20\"F07$$\"5++]P%) *eHY$eF`o$\"&s/\"F07$Fd]v$\"&O/\"F07$F[\\^l$\"&:1\"F07$F`\\^lFa]fl7$Fe \\^l$\"&61\"F07$Fh]l$\"&f-\"F07$$\"5NLLeRZAJ^gF`o$\"%'f*F07$F\\^v$\"%? lF07$$\"5NL$e*[Vi)F07$F jdm$\"%#)fF07$Fhhw$\"%q_F07$F_emF\\[r7$Fdem$\"%+CF07$Fiem$\"%BJF07$F^f m$\"%$3$F07$Fcfm$\"%wIF07$Fhfm$\"%oZF07$F]gm$\"%u!)F07$Fbgm$\"%ozF07$F ggm$\"%'\\)F07$$\"5+v$4YsF*Rh:F0$\"&ZU\"F07$F\\hm$\"&2S\"F07$$\"5Me9;# Q$>Wo:F0$\"&LX\"F07$Fahm$\"&,s\"F07$Ffhm$\"&4i\"F07$F[im$\"&/Z\"F07$Fe ]el$\"&c7\"F07$F`im$\"%>$*F07$F]^el$\"%t!*F07$FeimF]bdl7$F_jm$\"%tNF07 $Fijm$\"$f)F07$Fc[n$\"%:OF07$F]\\n$\"%3cF07$F[^n$\"%(f'F07$Fi_n$\"%NqF 07$Fc`n$\"%fiF07$F]an$\"%NcF07$Fgan$\"%(4'F07$F[cn$\"%>qF07$Fecn$\"%ul F07$Fidn$\"%FbF07$F^en$\"%dgF07$Fcen$\"%'*zF07$Fgfn$\"%d**F07$Fagn$\"& QG\"F07$F[hn$\"&=8\"F07$F_in$\"%R()F07$Fiin$\"&s@\"F07$Fcjn$\"&#\\9F07 $F][o$\"&6]\"F07$Fg[o$\"&,e\"F07$F]dx$\"&/!=F07$F\\\\o$\"&&3@F07$Fjdx$ \"&8@#F07$F_ex$\"&#=AF07$Fdex$\"&ZA#F07$Fiex$\"&%>AF07$F^fx$\"&SA#F07$ FcfxFa\\fl7$Fhfx$\"&/5#F07$Fa\\o$\"&J5#F07$Fe]o$\"&a%>F07$Fi^o$\"&.$=F 07$F^_o$\"&i)=F07$Fc_o$\"&G)>F07$Fg`o$\"&V7#F07$Faao$\"&_@#F07$Fde_l$ \"&z@#F07$Ffao$\"&&=AF07$F\\f_l$\"&8=#F07$F[bo$\"&W9#F07$Febo$\"&&R?F0 7$F_co$\"&j(>F07$Fjf_l$\"&d&>F07$F_g_l$\"&`%>F07$Fdg_lF^j[m7$Fdco$\"&A &>F07$F\\h_l$\"&j&>F07$Fah_lFabhl7$Ffh_l$\"&V*>F07$Fico$\"&.,#F07$Fcdo $\"&f>#F07$F]eo$\"&()H#F07$Fafo$\"&dD#F07$F[go$\"&Y@#F07$F`go$\"&=A#F0 7$Fego$\"&R?#F07$Fjgo$\"&M:#F07$F_ho$\"&n'>F07$Fiho$\"&7a\"F07$Fcio$\" &uX\"F07$Ff_y$\"&t-\"F07$Fhio$\"&_,\"F07$F^`y$\"%C(*F07$F]jo$\"%dtF07$ Fbjo$\"%fsF07$Fgjo$\"%#G(F07$F\\[p$\"%g*)F07$Fa[p$\"%H))F07$Ff[p$\"%z \"*F07$F[\\p$\"&1W\"F07$F`\\p$\"&!=9F07$Fe\\p$\"&vW\"F07$Fj\\p$\"&\\*> F07$F_]p$\"&C'>F07$Fd]p$\"&R&>F07$Fi]p$\"&S:#F07$Fg[fl$\"&,2#F07$F^^p$ \"&\\x\"F07$F_\\fl$\"&=r\"F07$Fd\\fl$\"&S=\"F07$Fi\\fl$\"&j;\"F07$Fc^p $\"&r6\"F07$$\"5M3xJ#H'p(4]#F0$\"%`lF07$Fh[`l$\"%hkF07$$\"5MekGg%ymy]# F0Fjgl7$Fh^p$\"%GUF07$F`\\`l$\"%wTF07$Fe\\`l$\"%YTF07$Fj\\`l$\"%PXF07$ F]_p$\"%$\\%F07$Fb_p$\"%wiF07$Fg_p$\"%[zF07$Fa`p$\"%D%*F07$Feap$\"&Q, \"F07$F_bp$\"%:(*F07$Fccp$\"%c*)F07$$\"5ML3F>*oZ7F07$Fgip$\"&/ \\\"F07$Fchy$\"&Rn\"F07$F\\jp$\"&.z\"F07$F_jy$\"&pY\"F07$Fajp$\"&=H\"F 07$$\"5,]7.2AtrF07$Fe[q$\"& =1#F07$Fi\\q$\"&(HCF07$Fg^q$\"&7#GF07$F\\_q$\"&^m#F07$Fa_q$\"&E_#F07$F ac`l$\"&j^#F07$Ff_q$\"&(RCF07$Fic`l$\"&1W#F07$F[`q$\"&3W#F07$Fad`l$\"& HW#F07$F``q$\"&MW#F07$Fid`l$\"&[W#F07$Fe`q$\"&K]#F07$F_aq$\"&Jr#F07$Fc bq$\"&&=GF07$F]cq$\"&Uo#F07$F[eq$\"&%\\DF07$Feeq$\"&Ss#F07$Fifq$\"&R\" HF07$F^gq$\"&8*GF07$Fcgq$\"&#QGF07$Fhgq$\"&>#GF07$F]hq$\"&H#GF07$Fghq$ \"&yv#F07$Faiq$\"&*>FF07$Ffiq$\"&GX#F07$F[jq$\"&%oAF07$$\"5M$e9mp&)p?J $F0$\"&7'>F07$F[ifl$\"&7%>F07$$\"5,]PM-C&*Q=LF0$\"&c$=F07$F`jq$\"&sX\" F07$Fcifl$\"&>S\"F07$Fejq$\"&#G7F07$Fjjq$\"&jA\"F07$F_[r$\"&e\\\"F07$F d[r$\"&LZ\"F07$Fi[rFc[z7$F^\\r$\"&21#F07$Fc\\r$\"&x-#F07$Fh\\r$\"&r/#F 07$F]]r$\"&f_#F07$Feh`l$\"&T[#F07$Fjhz$\"&9Y#F07$F]i`l$\"&U`#F07$Fb]r$ \"&A\\#F07$Fbiz$\"&j1#F07$Fg]r$\"&V#>F07$F\\^r$\"&,U\"F07$Fa^r$\"&()R \"F07$Ff^rF\\i\\m7$F[_r$\"%'Q*F07$F`_r$\"%C))F07$Fe_r$\"%#e(F07$Fj_r$ \"%(\\)F07$F_`r$\"&42\"F07$Fd`r$\"&A<\"F07$Fi`r$\"&1J\"F07$$\"5MeRZ_H> 7%\\$F0$\"&)48F07$$\"5,]7.#34zr\\$F0$\"&,J\"F07$$\"5oT&)e6_iB+NF0$\"&G L\"F07$F]\\al$\"&CL\"F07$$\"5o;/E+OxS4NF0$\"&IL\"F07$F^ar$\"&FL\"F07$F j\\al$\"&+J\"F07$Fcar$\"&yB\"F07$Fgbr$\"&'e7F07$Fedr$\"&xG\"F07$Fc_[l$ \"&!>7F07$F]`[l$\"&;<\"F07$Fg^gl$\"&3<\"F07$Fb`[l$\"&&o6F07$F__gl$\"&< <\"F07$Fd_gl$\"&M?\"F07$Fi_gl$\"&W?\"F07$Fjdr$\"&%37F07$Fj`[l$\"&>[\"F 07$F_er$\"&Yk\"F07$Fder$\"&f(=F07$Fier$\"&y1#F07$Fe`al$\"&Y6#F07$F^fr$ \"&7<#F07$F]aal$\"&)[?F07$Fcfr$\"&6\"=F07$Feagl$\"&.%=F07$Fjagl$\"&h'= F07$F_bgl$\"&hm\"F07$Fhfr$\"&Op\"F07$Fgbgl$\"&Kt\"F07$F]gr$\"&3z\"F07$ Fggr$\"&..#F07$Fahr$\"&0B#F07$F[ir$\"&)oBF07$Feir$\"&)=DF07$Fjir$\"&3g #F07$F_jr$\"&-q#F07$Fc[s$\"&$oIF07$Fg\\s$\"&\"GLF07$Fe^s$\"&NC$F07$Fc` s$\"&)QIF07$F]as$\"&KC$F07$Fabs$\"&AU$F07$Ffbs$\"&=O$F07$F[cs$\"&u;$F0 7$F`cs$\"&#=KF07$Fecs$\"&BW$F07$F[hal$\"&AX$F07$F`hal$\"&K]$F07$Fehal$ \"&D]$F07$Fjcs$\"&S]$F07$$\"5-v$4rG*QKJTF0$\"&v^$F07$F]ial$\"&g^$F07$$ \"5NekyA+]!y8%F0$\"&C^$F07$F_ds$\"&y[$F07$Fces$\"&aV$F07$F]fs$\"&sR$F0 7$F`fs$\"&%4MF07$Fefs$\"&fT$F07$Fjfs$\"&!yLF07$F_gs$\"&@E$F07$Figs$\"& t#HF07$Fchs$\"&a'GF07$Fhhs$\"&(*Q#F07$F]is$\"&_O#F07$Fbis$\"&(HBF07$Fg is$\"&%4>F07$Fajs$\"&q&=F07$F[[t$\"&Ov\"F07$F`[t$\"&Tt\"F07$Fc[t$\"&Q1 #F07$Fh[t$\"&93#F07$F]\\t$\"&!f?F07$Fb\\t$\"&(pDF07$Fg\\t$\"&lj#F07$F \\]t$\"&9g#F07$Fa]t$\"&+%HF07$$\"5N$3FWX2uIE%F0$\"&1*HF07$Ff]t$\"&F%HF 07$$\"5N$ek.H%>qpUF0$\"&V!HF07$F[^t$\"&O&GF07$Fhg[l$\"&G[#F07$F`^t$\"& sD#F07$Fg]hl$\"&5\">F07$F`h[l$\"&Il\"F07$F_^hl$\"&oi\"F07$Fe^t$\"&FU\" F07$F]i[l$\"&Z@\"F07$Fj^t$\"&n6\"F07$Fji[l$\"&F8\"F07$F__t$\"&/>\"F07$ Fd_t$\"&oY\"F07$Fi_t$\"&Rb\"F07$F^`t$\"&mf\"F07$Fc`t$\"&Rj\"F07$F]at$ \"&Dj\"F07$Fgat$\"&+l\"F07$F[ahl$\"&\"\\;F07$F\\bt$\"&\\k\"F07$$\"5M3x \"[N&HW;F07$Fect$\"&Rd\"F07$F_dt$\"&O^\"F07$Fddt$\"&M]\"F07$Fidt$ \"&\\^\"F07$Fj]\\l$\"&Ob\"F07$F_^\\l$\"&)>;F07$Fd^\\l$\"&Mj\"F07$Fi^\\ l$\"&Qk\"F07$F^_\\l$\"&wk\"F07$F^et$\"&'H;F07$F[`\\lFfhjl7$Fcet$\"&%*[ \"F07$Fhet$\"&es\"F07$F]ft$\"&!3>F07$Fbft$\"&)[AF07$Fgft$\"&%GDF07$F[h t$\"&zU#F07$F_it$\"&G5#F07$Fdit$\"&`G#F07$Fiit$\"&=l#F07$F^jt$\"&-%F07$Fgh\\lF[a_m7$F\\i\\l$\"&x.%F07$Ff_u$\"&K.%F07$Fdi \\l$\"&8&RF07$F[`uFh\\hl7$Faj\\l$\"&x!QF07$F[[]l$\"&wy$F07$F`[]l$\"&fw $F07$Fe[]lF_b_m7$Fj[]l$\"&Ww$F07$F``u$\"&Yw$F07$Fb\\]l$\"&!3QF07$Fe`u$ \"&2)RF0-Fj\\]l6&F\\auFjjblF(Fjjbl-Faau6#%Ra~modification~of~the~Tsito uras-Papakostas~schemeG-F$6%7ijlF'7$F+$\"$+%F07$F2Fh[^l7$Fi[cl$\"$:*F0 7$F7$\"$v)F07$Fd\\cl$\"$g)F07$F<$\"$F'F07$FP$\"$\"=F07$FZ$!$p&F07$Fi^] l$!$;'F07$Fin$!$Z(F07$Fa_]l$!$,'F07$F^o$!$D%F07$Fcp$\"%G=F07$Far$\"%BP F07$Ffr$\"%\"4&F07$F[s$\"%kqF07$F`s$\"%YkF07$Fes$\"%,PF07$Fegu$!$#GF07 $Fjs$!%*z#F07$F\\iu$!$=&F07$F_t$\"%%)=F07$Fcu$\"$a(F07$F]v$!%I5F07$F`j u$!%S5F07$Fbv$!%'R\"F07$Fhju$!%19F07$Fgv$!%e8F07$F\\w$!#UF07$Faw$\"%#) >F07$F_c]l$\"%!*>F07$Fdc]l$\"%e?F07$Fic]l$\"%$H$F07$F^d]l$\"%.LF07$Fhd ]l$\"%aKF07$Ffw$\"%AKF07$Fee]l$\"$e*F07$F[x$\"$\">F07$Fex$\"%R;F07$Faz $\"%@MF07$$\"5-](=nQYs9@%F`o$\"%*z$F07$Fcf]lF_[`m7$$\"5N$e*[$*\\yCyUF` o$\"%3QF07$Fhf]l$\"%$z$F07$F]g]lFh[`m7$F`g]l$\"%kPF07$Feg]l$\"%$Q$F07$ Ffz$\"%#Q$F07$F]h]l$\"%AEF07$F[[l$\"%28F07$$\"5o;a)3Cbvcu%F`o$\"%18F07 $Fadcl$\"%q7F07$$\"5-]ilZQ4X7[F`o$\"%F5F07$Feh]lFe]`m7$F^ecl$\"%37F07$ F`[l$\"%k7F07$F]i]l$\"%tGF07$Fe[l$\"%#*RF07$Fei]l$\"%oSF07$Fji]l$\"%!) [F07$F_j]l$\"%y[F07$Fj[l$\"%%*[F07$$\"5NL$eRZFk7S&F`o$\"%N_F07$Fej]l$ \"%K_F07$$\"5++](oagtJY&F`o$\"%I_F07$F_\\l$\"%+^F07$Fi\\l$\"%vUF07$Fc] l$\"%ZPF07$Fc[^l$\"%!)QF07$Fd]v$\"%dQF07$F[\\^l$\"%$H%F07$F`\\^l$\"%# \\%F07$Fe\\^l$\"%oWF07$Fh]l$\"%*[%F07$F\\^v$\"%eHF07$F]^l$!$i'F07$Fd^v $!$n(F07$Fb^lFcbv7$F\\_v$!%JWF07$Fg^l$!%0WF07$F\\_l$!%!H&F07$Fa_l$!%ba F07$Ff_l$!%g`F07$F[`l$!%UPF07$F``l$!%s?F07$Fe`l$!%8?F07$Fj`l$\"#\"*F07 $F_al$\"%=UF07$Fdal$\"%[`F07$Fial$\"%J$*F07$Fg_^l$\"%X%*F07$F\\av$\"&K /\"F07$F_`^l$\"&p-\"F07$F^bl$\"%&*)*F07$Fdav$\"%WpF07$Fcbl$\"%h@F07$Fh bl$!%c6F07$F]cl$!%(y\"F07$Fgcl$\"%q>F07$Fadl$\"%COF07$F[el$\"%ZYF07$Fe el$\"%)=&F07$$\"5qmmmTNhR\"4)F`o$\"%H`F07$F__dl$\"%9bF07$$\"50+++v$4#p d\")F`oF`g`m7$Fjel$\"%`bF07$Fg_dl$\"%$f&F07$F_fl$\"%UaF07$Fgev$\"%>YF0 7$FdflF^d`m7$F]hv$\"%>_F07$Fifl$\"%[cF07$F]hl$\"%f[F07$F[jl$\"%$*RF07$ Fejl$\"%mbF07$Fi[m$\"%nzF07$F^\\mF^]s7$Fc\\m$\"&><\"F07$Faf^l$\"&r>\"F 07$Fh\\m$\"&IE\"F07$Fif^l$\"&xF\"F07$F^g^l$\"&;E\"F07$Fcg^l$\"&J;\"F07 $F]]m$\"&#y6F07$Fb]mFe`el7$Fg]m$\"%\\[F07$Fah^l$\"%7MF07$Ffh^l$\"%kMF0 7$F[i^l$\"%RNF07$F\\^m$\"%_WF07$$\"5M$e9\"456lL6F0Fdadl7$Fci^l$\"%;YF0 7$$\"5+]Pf3Q$Q'R6F0$\"%UkF07$Fa^m$\"%)Q(F07$Ff^m$\"%^')F07$F[_m$\"%n)* F07$$\"5+]P4r`C[f6F0Fibx7$F`_m$\"%I**F07$$\"5n;z>OaEAm6F0$\"&T+\"F07$F e_m$\"%2!*F07$Fj_m$\"%J&)F07$F_`m$\"%GrF07$Fd`m$\"%QtF07$Fi`m$\"%![(F0 7$Fc]w$\"&^7\"F07$F^am$\"&zB\"F07$$\"5n;a8AgW)oA\"F0$\"&1C\"F07$$\"5++ voagXDI7F0$\"&l?\"F07$$\"5M$eRs3mCOB\"F0$\"&&37F07$F`^w$\"&q?\"F07$$\" 5MLe*[='\\tV7F0$\"&R4\"F07$Fcam$\"%v(*F07$Fgbm$\"&'Q6F07$FacmFdgjl7$F[ dm$\"&2J\"F07$Fedm$\"&Q1\"F07$Fggm$\"%@pF07$Fi_[m$\"&AE\"F07$F\\hm$\"& 6C\"F07$Fa`[m$\"&RJ\"F07$Fahm$\"&km\"F07$Ffhm$\"&wi\"F07$F[im$\"&td\"F 07$F`im$\"&,<\"F07$Feim$\"%QoF07$F_jm$\"%AlF07$Fijm$\"%ZQF07$Fc[n$\"% \\hF07$F]\\nF`[y7$F[^n$\"%N&*F07$Fi_n$\"&&H5F07$Fc`n$\"%](*F07$F]an$\" %O!*F07$$\"5+]iSmbs&Hx\"F0$\"%,*)F07$Fban$\"%v*)F07$$\"5+Dc,;krO$y\"F0 $\"%&)*)F07$$\"5n;a)eO8Poy\"F0$\"%!**)F07$$\"5M3_v:.rI!z\"F0$\"%h#*F07 $FganFheam7$Fabn$\"%w'*F07$F[cn$\"&..\"F07$Fc`el$\"&C.\"F07$F`cn$\"&x. \"F07$F[ael$\"&L-\"F07$Fecn$\"&!35F07$F_dn$\"%\"[*F07$Fidn$\"%c!*F07$$ \"5M3_]f^y7!)=F0$\"%w()F07$$\"5o;/^ci$*>$)=F0$\"%z()F07$$\"5,Dc^`t3F') =F0F]bx7$$\"5ML3_]%QU$*)=F0Fgf\\m7$$\"5oTg_Z&*QT#*=F0F]d\\m7$$\"5,]7`W 1a[&*=F0$\"%h()F07$$\"5Mek`TF0$\"%*))*F07$Fcen$\"&m4\"F07$Fgfn$\"&\">9F07$Fagn$\"&$eF07$Fa\\o$\"&')3#F07$Fe]o$\"&U'>F07 $Fi^o$\"&I#=F07$F^_o$\"&C&=F07$Fc_o$\"&3$>F07$Fg`oF]`\\m7$Faao$\"&'z@F 07$Fde_l$\"&C>#F07$Ffao$\"&;?#F07$F\\f_l$\"&2=#F07$F[bo$\"&(\\@F07$Feb o$\"&([?F07$F_co$\"&e(>F07$Fjf_l$\"&W%>F07$F_g_l$\"&'G>F07$Fdg_l$\"&$G >F07$Fdco$\"&G#>F07$F\\h_l$\"&<#>F07$Fah_l$\"&?#>F07$Ffh_l$\"&.&>F07$F ico$\"&J'>F07$Fcdo$\"&\\;#F07$F]eo$\"&'GBF07$Fafo$\"&*oAF07$F[go$\"&t= #F07$$\"5nTg_AR5))pBF0$\"&G;#F07$$\"5n;/w(yvbHP#F0$\"&_:#F07$$\"5n\"z% *HlZIgP#F0$\"&)[@F07$F`go$\"&t:#F07$$\"5n;zp[KYD&Q#F0$\"&@<#F07$Fego$ \"&A@#F07$F_ho$\"&15#F07$Fcio$\"&ms\"F07$Fhio$\"&TL\"F07$F]jo$\"&)45F0 7$Fbjo$\"%j**F07$Fgjo$\"%c)*F07$F\\[p$\"&N0\"F07$Fa[p$\"&\"Q5F07$Ff[p$ \"&?1\"F07$F[\\p$\"&x]\"F07$F`\\p$\"&T[\"F07$Fe\\p$\"&@^\"F07$Fj\\p$\" &f2#F07$F_]p$\"&@/#F07$Fd]p$\"&4/#F07$Fi]p$\"&KN#F07$Fg[fl$\"&`F#F07$F ^^p$\"&+7#F07$Fd\\fl$\"&>k\"F07$Fc^p$\"&qc\"F07$Fh[`l$\"&?8\"F07$Fh^p$ \"%0!*F07$Fe\\`l$\"%+))F07$F]_p$\"%S*)F07$Fb_pFaax7$Fg_p$\"&*47F07$Fa` p$\"&OQ\"F07$Feap$\"&g[\"F07$F]dp$\"&rW\"F07$Fady$\"&W[\"F07$Fbdp$\"&* 3:F07$F^eyF_i\\m7$Fgdp$\"&OX\"F07$Faep$\"&BS\"F07$Fefp$\"&xM\"F07$Fc_` l$\"&'[8F07$Fjfp$\"&_N\"F07$F[``l$\"&*e8F07$F```l$\"&8S\"F07$Fe``l$\"& IS\"F07$F_gp$\"&(39F07$Fchp$\"&pr\"F07$Fgip$\"&(H@F07$Fchy$\"&=K#F07$F \\jp$\"&YR#F07$F_jy$\"&O!>F07$Fajp$\"&**p\"F07$F_h\\m$\"&(>F07$Fejq$\"&%GF07$Fad]m$\"&t#>F07$F fd]m$\"&\"G>F07$F[e]m$\"&`'>F07$F]\\al$\"&Z'>F07$Fce]m$\"&H(>F07$F^ar$ \"&$y>F07$Fe\\al$\"&\"p>F07$Fj\\al$\"&q'>F07$F_]al$\"&-%>F07$Fcar$\"&? !>F07$Fhar$\"&Q'=F07$F]br$\"&(R=F07$$\"5o;/,%p>-Md$F0$\"&q$=F07$Fbbr$ \"&_&=F07$$\"5oTNYL^!HMe$F0Fhgcm7$$\"5,]7G8O8x'e$F0$\"&e&=F07$$\"5Me*) 4$4i8,f$F0$\"'J*)=F`o7$Fgbr$\"&%*)=F07$Facr$\"&.$>F07$F[dr$\"&?)>F07$$ \"5,]P4r`(yoi$F0$\"&A)>F07$F`dr$\"&)y>F07$$\"5M$ek.H*yCSOF0$\"&[&>F07$ Fedr$\"&S&>F07$Fc_[l$\"&**)=F07$F]`[l$\"&l$=F07$Fb`[l$\"&.#=F07$Fjdr$ \"&4%=F07$Fj`[l$\"&.6#F07$F_erF_^[l7$Fder$\"&Cp#F07$Fier$\"&)*)GF07$$ \"5om\"zWUG_rx$F0$\"&^#HF07$Fe`al$\"&)yGF07$$\"5,+vVB!zEPy$F0$\"&%=HF0 7$F^fr$\"&J&HF07$F]aal$\"&vu#F07$Fcfr$\"&`V#F07$Feagl$\"&XZ#F07$Fjagl$ \"&(3DF07$F_bgl$\"&/G#F07$Fhfr$\"&\"=BF07$Fgbgl$\"&3S#F07$F]gr$\"&\\[# F07$Fggr$\"&S!GF07$Fahr$\"&-0$F07$F[ir$\"&.A$F07$Feir$\"&=N$F07$$\"5o; /,W%)o=fQF0$\"&@P$F07$Fjir$\"&EN$F07$$\"5oTgFTc&G)oQF0$\"&rL$F07$$\"5, ]7.2ZC/sQF0$\"&VB$F07$$\"5MekysPjDvQF0Fbb\\l7$F_jr$\"&NG$F07$Fc[s$\"&$ oMF07$Fg\\s$\"&nt$F07$Fa]s$\"&\\*QF07$F[^s$\"&g\"RF07$F[cal$\"&>#RF07$ F`^s$\"&\"HRF07$Fccal$\"&O$RF07$Fe^s$\"&t%QF07$Fi_s$\"&Ss$F07$Fc`s$\"& /l$F07$F]as$\"&Iz$F07$Fabs$\"&U*RF07$Fgdal$\"&B,%F07$F\\eal$\"&L,%F07$ Faeal$\"&r)RF07$Ffbs$\"&I(RF07$Fieal$\"&=%QF07$F[cs$\"&dw$F07$FafalF]c t7$Fffal$\"&ht$F07$F[gal$\"&Av$F07$F`cs$\"&'yPF07$Fcgal$\"&e\"RF07$Fec s$\"&n-%F07$F[hal$\"&?/%F07$F`hal$\"&a6%F07$Fehal$\"&Y6%F07$Fjcs$\"&\" >TF07$Fe^^m$\"&(\\TF07$F]ial$\"&z9%F07$F]_^m$\"&T9%F07$F_ds$\"&p6%F07$ Fces$\"&C.%F07$F]fs$\"&`%RF07$$\"5N$3-8v\"RGqTF0$\"&e%RF07$F`fs$\"&?$R F07$$\"5N$eRsey6p<%F0$\"&u\"RF07$Fefs$\"&h'RF07$Fjfs$\"&]#RF07$F_gs$\" &(**QF07$Figs$\"&Sp$F07$Fchs$\"&Ci$F07$F]is$\"&>@$F07$Fgis$\"&[v#F07$F ajs$\"&2o#F07$F[[t$\"&!yCF07$F`[t$\"&iW#F07$Fc[t$\"&:n#F07$Fh[t$\"&\"o EF07$F]\\t$\"&aj#F07$Fb\\tF\\ehl7$Fg\\t$\"&]:$F07$F\\]t$\"&@6$F07$Fa]t $\"&w\\$F07$Fgc^m$\"&cb$F07$Ff]t$\"&/]$F07$F_d^m$\"&Uc$F07$F[^t$\"&8a$ F07$Fhg[l$\"&9D$F07$F`^t$\"&70$F07$Fg]hl$\"&![FF07$F`h[l$\"&G^#F07$F_^ hl$\"&`Z#F07$Fe^t$\"&bF#F07$F]i[l$\"&k0#F07$Fj^t$\"&>$>F07$F]^bl$\"&= \">F07$Fji[l$\"&r#>F07$Fe^bl$\"&+)>F07$F__t$\"&L'>F07$Fd_t$\"&n?#F07$F i_tFh_`l7$Fc`t$\"&NT#F07$Fgat$\"&NW#F07$F\\bt$\"&6X#F07$Fabt$\"&fV#F07 $Fect$\"&aR#F07$F_dt$\"&#HBF07$Fddt$\"&+J#F07$Fidt$\"&AJ#F07$Fj]\\l$\" &NM#F07$F_^\\l$\"&8T#F07$Fd^\\l$\"&#HCF07$Fi^\\l$\"&GW#F07$F^_\\l$\"&h X#F07$F^et$\"&wW#F07$F[`\\l$\"&ZP#F07$Fcet$\"&`H#F07$Fhet$\"&o\\#F07$F ]ft$\"&,w#F07$Fbft$\"&1C$F07$Fgft$\"&3[$F07$F[ht$\"&^J$F07$F_it$\"&c%H F07$Fdit$\"&VA$F07$Fiit$\"&6r$F07$Fcjt$\"&.8%F07$Fi\\u$\"&mR%F07$Fg^u$ \"&_g%F07$Fa_u$\"&)=[F07$Fgh\\l$\"&0*[F07$Ff_u$\"&S#\\F07$Fdi\\l$\"&v' [F07$F[`u$\"&^z%F07$F[[]l$\"&Cp%F07$F``u$\"&Qk%F07$Fiibl$\"&=m%F07$Fb \\]l$\"&An%F07$Fajbl$\"&bu%F07$Fe`u$\"&4&[F0-Fj`u6&F\\auFj\\il$\"#bF_a uF(-Faau6#%AVerner's~\"most~efficient\"~schemeG-%%FONTG6$%*HELVETICAG \"\"*-%&TITLEG6#%Verror~curves~for~12~stage~order~8~Runge-Kutta~method sG-%+AXESLABELSG6$Q\"x6\"Q!F^fem-%%VIEWG6$;F(Fe`u%(DEFAULTG" 1 2 0 1 10 0 2 9 1 4 2 1.000000 45.000000 45.000000 0 1 "Verner's Maple scheme " "a modification of Verner's Maple scheme" "an \"efficient\" scheme" "a scheme of Tsitouras and Papakostas" "a modification of the Tsitoura s-Papakostas scheme" "Verner's \"most efficient\" scheme" }}}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }} }}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 49 "Test 13 of 12 stage, order 8 Ru nge-Kutta methods " }}{PARA 256 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 " " {TEXT -1 72 "See: \"Mathematica in Action\" by Stan Wagon, Springer -Verlag, page 302. " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 256 "" 0 "" {TEXT -1 3 " " }{XPPEDIT 18 0 "dy/dx = cos*x+2*y;" "6#/*&%#dyG\" \"\"%#dxG!\"\",&*&%$cosGF&%\"xGF&F&*&\"\"#F&%\"yGF&F&" }{TEXT -1 8 ", \+ " }{XPPEDIT 18 0 "y(0) = -2/5;" "6#/-%\"yG6#\"\"!,$*&\"\"#\"\"\" \"\"&!\"\"F-" }{TEXT -1 1 " " }}{PARA 0 "" 0 "" {TEXT -1 10 "Solution: " }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "y = 1/5;" "6#/% \"yG*&\"\"\"F&\"\"&!\"\"" }{TEXT -1 1 " " }{XPPEDIT 18 0 "sin*x-2/5" " 6#,&*&%$sinG\"\"\"%\"xGF&F&*&\"\"#F&\"\"&!\"\"F+" }{TEXT -1 1 " " } {XPPEDIT 18 0 "cos*x" "6#*&%$cosG\"\"\"%\"xGF%" }{TEXT -1 3 " . " }} {PARA 0 "" 0 "" {TEXT 260 4 "Note" }{TEXT -1 2 ": " }}{PARA 0 "" 0 "" {TEXT -1 53 "The general solution of the differential equation " } {XPPEDIT 18 0 "dy/dx = cos*x+2*y;" "6#/*&%#dyG\"\"\"%#dxG!\"\",&*&%$co sGF&%\"xGF&F&*&\"\"#F&%\"yGF&F&" }{TEXT -1 64 " contains an exponenti al term, but with the initial condition " }{XPPEDIT 18 0 "y(0) = -2/5 " "6#/-%\"yG6#\"\"!,$*&\"\"#\"\"\"\"\"&!\"\"F-" }{TEXT -1 23 " this t erm disappears." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> \+ " 0 "" {MPLTEXT 1 0 50 "de := diff(y(x),x)=cos(x)+2*y(x);\ndsolve(de,y (x));" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%#deG/-%%diffG6$-%\"yG6#%\"x GF,,&-%$cosGF+\"\"\"*&\"\"#F0F)F0F0" }}{PARA 11 "" 1 "" {XPPMATH 20 "6 #/-%\"yG6#%\"xG,(*&#\"\"#\"\"&\"\"\"-%$cosGF&F-!\"\"*&#F-F,F--%$sinGF& F-F-*&-%$expG6#,$*&F+F-F'F-F-F-%$_C1GF-F-" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 105 "Any slight deviation of a nume rical solution from the correct solution tends to become rapidly magni fied." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 154 "de := diff(y(x),x)=cos(x)+2*y(x);\nic := y(0)=-2/5; \ndsolve(\{de,ic\},y(x));\ne := unapply(rhs(%),x):\nplot(e(x),x=0..8,f ont=[HELVETICA,9],labels=[`x`,`y(x)`]);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%#deG/-%%diffG6$-%\"yG6#%\"xGF,,&-%$cosGF+\"\"\"*&\"\"#F0F)F0F0 " }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%#icG/-%\"yG6#\"\"!#!\"#\"\"&" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#/-%\"yG6#%\"xG,&*&#\"\"#\"\"&\"\"\"-%$ cosGF&F-!\"\"*&#F-F,F--%$sinGF&F-F-" }}{PARA 13 "" 1 "" {GLPLOT2D 703 312 312 {PLOTDATA 2 "6&-%'CURVESG6$7gn7$$\"\"!F)$!3A+++++++S!#=7$$\"3E LLLLBxV5E$F,$!35;%fC]([[JF,7$$\"3MLL LLAKn\\F,$!3C&4%=OwYjDF,7$$\"3=LLLLc$\\o'F,$!31c1[)*fT**=F,7$$\"3)emmm ^&Q%R)F,$!39J7$$\"3))** ***\\YJ?;\"!#<$\"3m!=?Y3*>`CFK7$$\"3?LLL=\"\\g**FK7 $$\"3\")*****\\[A4]\"FO$\"3Xgu?U;&er\"F,7$$\"3wmmm'3Q\\n\"FO$\"3S\\4.g (y\\S#F,7$$\"3OLLLB6@G=FO$\"3e*[f2BGC&HF,7$$\"3&)******f-w+?FO$\"375@E VOJ&[$F,7$$\"3%*********y,u@FO$\"3VG2]n#=i\"RF,7$$\"3)*******RP)4M#FO$ \"3ym!)\\t%R1A%F,7$$\"3Umm;HUz;CFO$\"3:@(\\YT,0K%F,7$$\"3ILLL=Zg#\\#FO $\"3++xVHVa&R%F,7$$\"3;++]A2v#e#FO$\"3+<'Hh4))=X%F,7$$\"3cmmmEn*Gn#FO$ \"3a5#zx'*y?Z%F,7$$\"3qmmm;AE\\FFO$\"35^%H>#ywgWF,7$$\"3Tmmm1xiDGFO$\" 3(3\\(>4bXBWF,7$$\"3LLL$e#*eW\"HFO$\"3![MOl&\\jZVF,7$$\"3!)*****\\9!H. IFO$\"37X%)HL)HvB%F,7$$\"3Immm1:bgJFO$\"37d#H1rl8'RF,7$$\"3<+++X@4LLFO $\"3G,Fnxt@YNF,7$$\"31+++N;R(\\$FO$\"3+2Ml_]z_IF,7$$\"3wmmm;4#)oOFO$\" 3?6K>$)*R0X#F,7$$\"3jmmm6lCEQFO$\"3xp`%>:UP$=F,7$$\"3ELLL$G^g*RFO$\"38 $\\Qkcw!=6F,7$$\"3oKLL=2VsTFO$\"39U#4i*[S4MFK7$$\"3f*****\\`pfK%FO$!3E &Q)=hjJ]MFK7$$\"3!HLLLm&z\"\\%FO$!3u$z\"\\\">,j2\"F,7$$\"3s******z-6jY FO$!3_=%f%oq`+=F,7$$\"3<******4#32$[FO$!3Gvm#oI!>eCF,7$$\"3O*****\\#y' G*\\FO$!3Ak5yX#4\"HIF,7$$\"3G******H%=H<&FO$!3EIq1&[C$pNF,7$$\"35mmm1> qM`FO$!3%z'[2h*Gn&RF,7$$\"3%)*******HSu]&FO$!3%*oc=HW4cUF,7$$\"3'fmm\" HOq&e&FO$!3oqc`'[F/N%F,7$$\"3'HLL$ep'Rm&FO$!3$e%**GFr7=WF,7$$\"3D***\\ P?[nq&FO$!3TlAsE+sVWF,7$$\"3Umm;\\%H&\\dFO$!3y[ey96=hWF,7$$\"3eLLe%p5B z&FO$!3=)zg%Q%y/Z%F,7$$\"3')******R>4NeFO$!3waa0%)\\frWF,7$$\"3HLL$ed* f:fFO$!3]_J$4k<:X%F,7$$\"3#emm;@2h*fFO$!3V5vHeMg-WF,7$$\"37LLL))3E!3'F O$!3=l`a'y%*4K%F,7$$\"3]*****\\c9W;'FO$!3>=$e-d.)3UF,7$$\"3Lmmmmd'*GjF O$!3Gy*y<4!G/RF,7$$\"3j*****\\iN7]'FO$!3;B6I^7jsMF,7$$\"3aLLLt>:nmFO$! 37+2hu:afHF,7$$\"35LLL.a#o$oFO$!3;\"e/Z#4*3N#F,7$$\"3ammm^Q40qFO$!3!4` 1I$pa!o\"F,7$$\"3y******z]rfrFO$!3pfL'*)RTA-\"F,7$$\"3gmmmc%GpL(FO$!3? j;%3XMsQ#FK7$$\"3/LLL8-V&\\(FO$\"3qi(R>/(R\"p%FK7$$\"3=+++XhUkwFO$\"3Z X^U-))=F,7$$\"\")F)$\"3s<7[GmrgDF,- %'COLOURG6&%$RGBG$\"#5!\"\"F(F(-%%FONTG6$%*HELVETICAG\"\"*-%+AXESLABEL SG6$%\"xG%%y(x)G-%%VIEWG6$;F(Fg]l%(DEFAULTG" 1 2 0 1 10 0 2 9 1 4 2 1.000000 45.000000 45.000000 0 0 "Curve 1" }}}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 32 "The following code constructs a " }{TEXT 260 17 "discrete solution" }{TEXT -1 44 " based on each of t he methods and gives the " }{TEXT 260 22 "root mean square error" } {TEXT -1 18 " of each solution." }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 798 "E := (x,y) -> cos(x)+2*y: hh := 0.02: numsteps := 400: x0 := \+ 0: y0 := -2/5:\nmatrix([[`slope field: `,E(x,y)],[`initial point: `, ``(x0,y0)],[`step width: `,hh],\n[`no. of steps: `,numsteps]]);``; \nmthds := [`Enright-Verner scheme`,`Verner's Maple scheme`,`a modific ation of Verner's Maple scheme`,`an \"efficient\" scheme`,`a scheme of Tsitouras and Papakostas`,`a modification of the Tsitouras-Papakostas scheme`,`Verner's \"most efficient\" scheme`]: errs := []:\nDigits := 20:\nfor ct to 7 do\n En_RK8_||ct := RK8_||ct(E(x,y),x,y,x0,evalf(y 0),hh,numsteps,false);\n sm := 0: numpts := nops(En_RK8_||ct):\n f or ii to numpts do\n sm := sm+(En_RK8_||ct[ii,2]-e(En_RK8_||ct[ii ,1]))^2;\n end do:\n errs := [op(errs),sqrt(sm/numpts)];\nend do: \nDigits := 10:\nlinalg[transpose]([mthds,evalf(errs)]);" }}{PARA 11 " " 1 "" {XPPMATH 20 "6#K%'matrixG6#7&7$%0slope~field:~~~G,&-%$cosG6#%\" xG\"\"\"*&\"\"#F.%\"yGF.F.7$%0initial~point:~G-%!G6$\"\"!#!\"#\"\"&7$% /step~width:~~~G$F0F97$%1no.~of~steps:~~~G\"$+%Q)pprint616\"" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%!G" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#K%'ma trixG6#7)7$%6Enright-Verner~schemeG$\"+'y'z-f!#B7$%6Verner's~Maple~sch emeG$\"+UA2XrF+7$%Ha~modification~of~Verner's~Maple~schemeG$\"+,a(*GcF +7$%6an~\"efficient\"~schemeG$\"+]o>MkF+7$%Ea~scheme~of~Tsitouras~and~ PapakostasG$\"+7'f7@'F+7$%Ra~modification~of~the~Tsitouras-Papakostas~ schemeG$\"+G>QR%)F+7$%AVerner's~\"most~efficient\"~schemeG$\"+4GE'[\"F +Q)pprint626\"" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 30 "The following code constructs " }{TEXT 260 20 "numerical \+ procedures" }{TEXT -1 56 " for solutions based on each of the Runge-Ku tta schemes." }}{PARA 0 "" 0 "" {TEXT -1 75 "The error in the value ob tained by each of the methods at the point where " }{XPPEDIT 18 0 "x \+ = 7.999;" "6#/%\"xG-%&FloatG6$\"%**z!\"$" }{TEXT -1 16 " is also give n." }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 729 "E := (x,y) -> cos(x)+ 2*y: hh := 0.02: numsteps := 400: x0 := 0: y0 := -2/5:\nmatrix([[`slop e field: `,E(x,y)],[`initial point: `,``(x0,y0)],[`step width: `,h h],\n[`no. of steps: `,numsteps]]);``;\nmthds := [`Enright-Verner sc heme`,`Verner's Maple scheme`,`a modification of Verner's Maple scheme `,`an \"efficient\" scheme`,`a scheme of Tsitouras and Papakostas`,`a \+ modification of the Tsitouras-Papakostas scheme`,`Verner's \"most effi cient\" scheme`]: errs := []:\nDigits := 25:\nfor ct to 7 do\n en_RK 8_||ct := RK8_||ct(E(x,y),x,y,x0,evalf(y0),hh,numsteps,true);\nend do: \nxx := 7.999: exx := evalf(e(xx)):\nfor ct to 7 do\n errs := [op(er rs),abs(en_RK8_||ct(xx)-exx)];\nend do:\nDigits := 10:\nlinalg[transpo se]([mthds,evalf(errs)]);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#K%'matrix G6#7&7$%0slope~field:~~~G,&-%$cosG6#%\"xG\"\"\"*&\"\"#F.%\"yGF.F.7$%0i nitial~point:~G-%!G6$\"\"!#!\"#\"\"&7$%/step~width:~~~G$F0F97$%1no.~of ~steps:~~~G\"$+%Q)pprint636\"" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%!G" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#K%'matrixG6#7)7$%6Enright-Verner~sch emeG$\"+kQ(4F$!#A7$%6Verner's~Maple~schemeG$\"+9zNfRF+7$%Ha~modificati on~of~Verner's~Maple~schemeG$\"+s$4#>JF+7$%6an~\"efficient\"~schemeG$ \"+\"\\Pac$F+7$%Ea~scheme~of~Tsitouras~and~PapakostasG$\"+^8\">W$F+7$% Ra~modification~of~the~Tsitouras-Papakostas~schemeG$\"+YggwYF+7$%AVern er's~\"most~efficient\"~schemeG$\"+#e%eN#)!#BQ)pprint646\"" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 4 "The " }{TEXT 260 22 "root mean square error" }{TEXT -1 20 " over the interval " } {XPPEDIT 18 0 "[0, 8];" "6#7$\"\"!\"\")" }{TEXT -1 82 " of each Runge -Kutta method is estimated as follows using the special procedure " } {TEXT 0 5 "NCint" }{TEXT -1 98 " to perform numerical integration by \+ the 7 point Newton-Cotes method over 200 equal subintervals." }} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 476 "mthds := [`Enright-Verner s cheme`,`Verner's Maple scheme`,`a modification of Verner's Maple schem e`,`an \"efficient\" scheme`,`a scheme of Tsitouras and Papakostas`,`a modification of the Tsitouras-Papakostas scheme`,`Verner's \"most eff icient\" scheme`]: errs := []:\nDigits := 20:\nfor ct to 7 do\n sm : = NCint((e(x)-'en_RK8_||ct'(x))^2,x=0..8,adaptive=false,numpoints=7,fa ctor=200);\n errs := [op(errs),sqrt(sm/8)];\nend do:\nDigits := 10: \nlinalg[transpose]([mthds,evalf(errs)]);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#K%'matrixG6#7)7$%6Enright-Verner~schemeG$\"+Xf*Qz&!#B7$ %6Verner's~Maple~schemeG$\"+rZB8qF+7$%Ha~modification~of~Verner's~Mapl e~schemeG$\"+eX2DbF+7$%6an~\"efficient\"~schemeG$\"+e5[:jF+7$%Ea~schem e~of~Tsitouras~and~PapakostasG$\"+%[ym4'F+7$%Ra~modification~of~the~Ts itouras-Papakostas~schemeG$\"+M0q$G)F+7$%AVerner's~\"most~efficient\"~ schemeG$\"+3Txe9F+Q)pprint656\"" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }} {PARA 0 "" 0 "" {TEXT -1 92 "The following error graphs are constructe d using the numerical procedures for the solutions." }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 610 "evalf[20](plot([e(x)-'en_RK8_1'(x),e(x)-'e n_RK8_2'(x),e(x)-'en_RK8_3'(x),e(x)-'en_RK8_4'(x),\ne(x)-'en_RK8_5'(x) ,e(x)-'en_RK8_6'(x),e(x)-'en_RK8_7'(x)],x=0..2,0..2.63e-18,font=[HELVE TICA,9],\ncolor=[COLOR(RGB,.4,0,.9),COLOR(RGB,0,.65,0),brown,red,COLOR (RGB,0,.75,.95),magenta,COLOR(RGB,.95,.55,0)],\nlegend=[`Enright-Verne r scheme`,`Verner's Maple scheme`,`a modification of Verner's Maple sc heme`,`an \"efficient\" scheme`,`a scheme of Tsitouras and Papakostas` ,`a modification of the Tsitouras-Papakostas scheme`,`Verner's \"most \+ efficient\" scheme`],title=`error curves for 12 stage order 8 Runge-Ku tta methods`));" }}{PARA 13 "" 1 "" {GLPLOT2D 979 587 587 {PLOTDATA 2 "6--%'CURVESG6%7]o7$$\"\"!F)F(7$$\"5MLLLLL3VfV!#@F(7$$\"5nmmmm\"H[D:)F -$\"\"\"!#?7$$\"5LLLLLe0$=C\"F3$\"\"#F37$$\"5LLLLL3RBr;F3F77$$\"5nmmm; zjf)4#F3$\"\"$F37$$\"5++++]Piq'H#F3F?7$$\"5MLLL$e4;[\\#F3F?7$$\"5nmmm; zt%**p#F3$\"\"%F37$$\"5++++]i'y]!HF3$\"\"&F37$$\"5nmmm;HdA+&F3 $\"#5F37$$\"5+++++]Z/NaF3$\"#6F37$$\"5+++++]$fC&eF3$\"#7F37$$\"5MLLL$e z6:B'F3$\"#9F37$$\"5nmmmm;=C#o'F3$\"#:F37$$\"5nmmmmm#pS1(F3$\"#$\"$D$F- 7$$\"5++++v$Q#\\\"3\"Fis$\"$^$F-7$$\"5LLLL$e\"*[H7\"Fis$\"%,Q!#A7$$\"5 +++++qvxl6Fis$\"$B%F-7$$\"5++++]_qn27Fis$\"$a%F-7$$\"5++vV['z5-@\"Fis$ \"$e%F-7$$\"5++](o/aWF@\"Fis$\"$d%F-7$$\"5++DJX%Gy_@\"Fis$\"$f%F-7$$\" 5+++vVG?\"y@\"Fis$\"$o%F-7$$\"5+](oH/!*y!>7Fis$\"$i%F-7$$\"5++v=UsdM?7 FisFcv7$$\"5+]iSTWEh@7Fis$\"$k%F-7$$\"5++]iS;&zGA\"FisFiu7$$\"5++D1RgK TD7FisF`w7$$\"5+++]P/q%zA\"Fis$\"$u%F-7$$\"5+++DJ!)>3Q7Fis$\"$x%F-7$$ \"5++++Dcp@[7Fis$\"$z%F-7$$\"5+++](=GB2F\"Fis$\"$3&F-7$$\"5++++]2'HKH \"Fis$\"$M&F-7$$\"5nmmmmwanL8Fis$\"#eF37$$\"5+++++v+'oP\"Fis$\"#iF37$$ \"5LLLLeR<*fT\"Fis$\"#oF37$$\"5+++++&)Hxe9Fis$\"#tF37$$\"5nmmm\"H!o-* \\\"Fis$\"#zF37$$\"5++++DTO5T:Fis$\"#')F37$$\"5nmmmmT9C#e\"Fis$\"#%*F3 7$$\"5++++D1*3`i\"Fis$\"$,\"F37$$\"5LLLLL$*zym;Fis$\"$4\"F37$$\"5LLLL$ 3N1#4\"F37$$\"5nmmm\"HYt7v\"Fis$\"$G\"F37$$\"5+++++q(G**y\"Fi s$\"$R\"F37$$\"5nmmm;9@BM=Fis$\"$^\"F37$$\"5LLLLL`v&Q(=Fis$\"$j\"F37$$ \"5++++DOl5;>Fis$\"$x\"F37$$\"5++++v.Uac>Fis$\"$\">F37$$F8F)$\"$3#F3-% &COLORG6&%$RGBG$FK!\"\"F($FdoFj^l-%'LEGENDG6#%6Enright-Verner~schemeG- F$6%7boF'F*F.7$$\"5NLLLe9w)*=#*F-F17$$\"5++++v$pU&G5F3F17$$\"5LLLe*)fY '=3\"F3F77$$\"5mmm;/Em=N6F3F17$$\"5LL$e9\"4w%=;\"F3F77$$\"5+++v=#f3&)= \"F3F17$$\"5mm;/Ev&p^@\"F3F77$F5F?7$$\"5LLLL3xf]&H\"F3F?7$$\"5LLLL$eR \"=\\8F3F77$$\"5LLLLe9o&GS\"F3F?7$$\"5LLLLLLA`c9F3F?7$$\"5LLLL3_w?5:F3 F77$$\"5LLLL$32$)Qc\"F3F77$$\"5LLLLe*[evh\"F3F7F97$$\"5+++v=F3F?F<7$FEFJFL7$ FUFZ7$FgnF^o7$F\\oFco7$FaoFho7$FfoFbp7$F[p$\"#8F37$F`pFgp7$Fep$\"#;F37 $FjpFfq7$F_qF[r7$FdqF`r7$FiqFer7$F^r$\"#DF37$Fcr$\"#FF37$Fhr$\"$3$F-7$ F]s$\"$J$F-7$Fbs$\"$d$F-7$Fgs$\"$\"RF-7$F]t$\"$A%F-7$Fbt$\"%zXFft7$Fht Fjx7$F]u$\"$Y&F-7$Fcx$\"$z&F-7$F]y$\"$V'F-7$Fby$\"#pF37$Fgy$\"#vF37$F \\z$\"#\")F37$Faz$\"#))F37$Ffz$\"#&*F37$F[[l$\"$/\"F37$F`[l$\"$8\"F37$ Fe[l$\"$A\"F37$Fj[l$\"$K\"F37$F_\\l$\"$V\"F37$Fd\\l$\"$b\"F37$Fi\\l$\" $n\"F37$F^]l$\"$#=F37$Fc]l$\"$(>F37$Fh]l$\"$9#F37$F]^l$\"$J#F37$Fb^l$ \"$_#F3-Ff^l6&Fh^lF($\"#l!\"#F(-F]_l6#%6Verner's~Maple~schemeG-F$6%7[w F'F*7$$\"5++]i!RvlzZ%F-F17$$\"5nmm\"zWn+lf%F-F(7$$\"5ML$3_]fN]r%F-F(7$ $\"5+++]i:0dL[F-F17$$\"5ILL3xc.kq]F-F17$$\"5lmmm\"z>5xI&F-F17$$\"5IL$e *[=^CEaF-F17$$\"5+++D1R+yWbF-F(7$$\"5lm;ajf\\JjcF-F17$$\"5ILL$3-))\\=y &F-F(7$$\"5++]7y+[Q+fF-F(7$$\"5lmmTN@(>*=gF-F(7$$\"5IL$3F>kau8'F-F17$$ \"5++++]i&*)fD'F-F17$$\"5qmm;zW#H,t'F-F17$$\"5NLLL3F*oU?(F-F17$$\"5qmm \"H#o(Q8W(F-F17$$\"5+++]P4'3%ywF-F17$$\"5qm;z%*HN%pz(F-F77$$\"5NLL3_]% ya\"zF-F17$$\"5++]P4rL,M!)F-F17$F/F(Ff_lF47$FcalF7F9F_6,$F3FJ7$$\"5++++](Q6W-$F3FJ7$$\"5nmmm;z0nPI F3FO7$$\"5MLLL$3xH40$F3FO7$$\"5++++]i*)=kIF3FO7$$\"5MLLL$eM224$F3FOFQ7 $$\"5MLLL$e\\i-<$F3FO7$$\"5++++]i#*HBKF3FO7$$\"5MLLL$ek<)\\KF3FO7$$\"5 nmmm;HgLwKF3FO7$$\"5MLLL$3A&f*G$F3FO7$$\"5++++]7W&GI$F3FZ7$$\"5nmmm;/O 6;LF3FO7$FUFJ7$$\"5+++]PMh!eN$F3FO7$$\"5nmmm\"HZRAQ$F3FZ7$$\"5+++v=UhX &R$F3FO7$$\"5MLL$e9\"Gn3MF3FO7$$\"5nmm\"H2[*)=U$F3FO7$$\"5+++++]h5NMF3 FO7$$\"5MLL3F>GK[MF3FO7$$\"5nmm;a)[R:Y$F3FO7$$\"5+++D\"y:cZZ$F3FO7$$\" 5MLLL3FG(z[$F3FZ7$$\"5nmmTN'\\*=,NF3FZ7$$\"5+++]ilhS9NF3FZ7$$\"5MLLe*[ $GiFNF3FZ7$FXFO7$$\"5MLLLLeGdYOF3FZFfn7$$\"5ML3-Q\"=,fw$F3FZ7$$\"5nm;/ E]h\\zPF3FZ7$$\"5++D19>64$z$F3Fin7$$\"5MLL3-)3'o1QF3Fin7$$\"5nmT5!p0\" G?QF3FZ7$$\"5++]7yDg(Q$QF3FZ7$$\"5MLe9m%*4ZZQF3Fin7$$\"5nmm;ajf1hQF3FZ 7$$\"5+++D1ReW:RF3FZ7$$\"5MLLLe9d#)pRF3FZ7$$\"5nmTNY$o?M)RF3FZ7$$\"5++ ]PM_c,(*RF3Fin7$$\"5MLeRA@1h5SF3Fin7$$\"5nmmT5!f0U-%F3FZ7$$\"5++vV)*e0 !y.%F3Fin7$$\"5ML$eky_&R^SF3Fin7$$\"5nm\"zWn\\!*\\1%F3Fin7$$\"5+++]ila eySF3Fin7$$\"5ML3_]M/=#4%F3Fin7$$\"5nm;aQ.ax0TF3Fin7$$\"5++DcEs.P>TF3F in7$$\"5MLLe9T`'H8%F3FZ7$$\"5nmTg-5.cYTF3Fin7$$\"5++]i!*y_:gTF3F^o7$$ \"5MLekyZ-vtTF3Fin7$F\\oFin7$FaoF^o7$$\"5++++v=5PyYF3F^o7$$\"5nmmm;HU@ 'y%F3F^o7$$\"5MLLLeRu0%*[F3FcoFeo7$$\"5++++Dc@(*G]F3Fco7$$\"5++++]iO/c ]F3F^o7$$\"5+++]i:%z&p]F3Fco7$$\"5++++vo^6$3&F3Fho7$$\"5+++](=#4l'4&F3 Fho7$$\"5+++++vm=5^F3Fho7$$\"5+++]7GCsB^F3Fco7$$\"5++++D\"=es8&F3Fco7$ $\"5+++]PMRz]^F3F^o7$$\"5++++](oHV;&F3Fco7$$\"5++++v$>,9>&F3Fco7$$\"5+ +++++FZ=_F3Fco7$$\"5++++]7dhs_F3Fco7$$\"5+++++D(enK&F3Fco7$$\"5+++]7yW HS`F3Fho7$$\"5++++DJ-$QN&F3Fco7$$\"5+++]P%)fOn`F3Fho7$$\"5++++]PF37$F^rF[r7$$\"5++]i:gicX$)F3F`r7$$\"5+++D\"y:-%e$)F3F `r7$$\"5++](oa0Q7P)F3F[r7$$\"5+++]7`R2%Q)F3F`r7$$\"5++]7y])4pR)F3F[r7$ $\"5+++vV[du4%)F3F[r7$$\"5++]P4Y;eA%)F3F[r7$$\"5++++vVvTN%)F3F[r7$$\"5 ++]iSTMD[%)F3F`r7$$\"5+++D1R$*3h%)F3F`r7$$\"5++](=nBDRZ)F3F[r7$$\"5+++ ]PM6w'[)F3F`r7$$\"5+++voHHV7&)F3F`r7$$\"5+++++DZ5Q&)F3F`r7$$\"5+++DJ?l xj&)F3F`r7$$\"5+++]i:$[%*e)F3F`r7$$\"5++]7G8UG-')F3$\"#AF37$$\"5+++v$4 6?^h)F3Fh`n7$$\"5++]Pf3g&zi)F3F`r7$$\"5++++D1>zS')F3Fh`n7$$\"5++]i!R!y i`')F3F`r7$$\"5+++Dc,PYm')F3Fh`n7$$\"5++](=#*f*Hz')F3Fh`n7$$\"5+++](o \\N@p)F3F`r7$$\"5++]7`%Rr\\q)F3Fer7$$\"5+++v=#H2yr)F3Fh`n7$$\"5++]P%)* =V1t)F3F`r7$FcrF`r7$$\"5qmm;a)QA1&))F3$\"$>#F-7$$\"5NLLLe*olx&*)F3$\"$ N#F-7$$\"5+++]i!**3\\1*F3$\"$T#F-7$Fhr$\"$X#F-7$F]s$\"$i#F-7$Fbs$\"$\" GF-7$Fgs$\"$4$F-7$F]t$\"$L$F-7$Fbt$\"%6OFft7$Fht$\"$-%F-7$F]u$\"$K%F-7 $Fbu$\"$O%F-7$Fgu$\"$N%F-7$F\\vFaen7$Fav$\"$X%F-7$Ffv$\"$R%F-7$F[wFhen 7$F^w$\"$T%F-7$Fcw$\"$M%F-7$FfwF_fn7$Fiw$\"$^%F-7$F^xF_u7$Fcx$\"$b%F-7 $Fhx$\"$$[F-7$F]yFjx7$Fby$\"#bF37$Fgy$\"#fF37$F\\z$\"#kF37$FazFgfl7$Ff zFjfl7$F[[l$\"##)F37$F`[l$\"#*)F37$Fe[l$\"#'*F37$Fj[lFfgl7$F_\\lFigl7$ Fd\\lF\\hl7$Fi\\lF_hl7$F^]l$\"$W\"F37$Fc]l$\"$c\"F37$Fh]l$\"$o\"F37$F] ^lF[il7$Fb^l$\"$)>F3-%'COLOURG6&Fh^l$\")#)eqk!\")$\"))eqk\"FjinF[jn-F] _l6#%Ha~modification~of~Verner's~Maple~schemeG-F$6%7fqF'F*F.Fc_lFf_l7$ $\"5mm;H#on._0\"F3F7Fi_l7$$\"5++](oHkD&36F3F1F\\`lF_`lFb`lFe`lFh`l7$$ \"5LLL$3xEo'o7F3F77$Fj`lF77$$\"5LLL$ekoVBK\"F3F7F\\al7$F`alF7Fd^mFhalF 97$F_blF7FablFdbl7$FhblF?7$F[clF7F]clF`clF<7$$\"5MLL$e96gL7#F3F?7$$\"5 ++++vVQ7[@F3FJ7$$\"5nmm;/wv)G<#F3FJ7$$\"5MLLLL38l(>#F3F?7$$\"5+++]iS]T AAF3FJ7$$\"5nmmm\"HxyrC#F3F?7$$\"5MLL$3_]U>F#F3F?7$FBFJ7$$\"5nmm;zp*p9 K#F3FJ7$$\"5MLLL3-PBYBF3F?7$$\"5+++]PMu*4P#F3FJ7$$\"5nmmmmm6w&R#F3F?7$ $\"5MLL$e*)*[_?CF3FJ7$$\"5++++DJ')GXCF3FJ7$$\"5nmm;ajB0qCF3FJFDFGFLFQF TFWFfnF[oF`o7$FjhmFco7$F]imFco7$F`imFho7$FfoF]p7$FcimFho7$FfimFcoF[jm7 $FbjmF]p7$FhjmFho7$F^[nFho7$Fa[nFho7$Fd[nFho7$Fg[nFho7$Fj[nFho7$F`\\nF ho7$Ff\\nF]p7$Fi\\nF]pFjoF_pFdp7$FjpF^dl7$F_qFfq7$FdqFb]n7$FiqF`r7$F^r Fh`n7$Fcr$\"#CF37$Fhr$\"$t#F-7$F]s$\"$%HF-7$Fbs$\"$;$F-7$Fgs$\"$Z$F-7$ F]t$\"$v$F-7$Fbt$\"%rSFft7$Fht$\"$`%F-7$F]u$\"$([F-7$Fcx$\"$:&F-7$F]y$ \"$u&F-7$FbyFiy7$Fgy$\"#nF37$F\\zFcz7$Faz$\"#yF37$Ffz$\"#&)F37$F[[l$\" #$*F37$F`[lFg[l7$Fe[lF\\\\l7$Fj[l$\"$=\"F37$F_\\lFf\\l7$Fd\\lF[]l7$Fi \\l$\"$]\"F37$F^]lFe]l7$Fc]lFj]l7$Fh]l$\"$#>F37$F]^l$\"$2#F37$Fb^l$\"$ E#F3-Ffin6&Fh^l$\"*++++\"FjinF(F(-F]_l6#%6an~\"efficient\"~schemeG-F$6 %7goF'F*F.F4F9FF3FJF`cl7$$\"5 +++DJqgF37$F^]l$\"$8#F37$Fc]lFdil7$Fh]l$\"$^ #F37$F]^l$\"$r#F37$Fb^l$FesF3-Ffin6&Fh^lFjcoF(Fjco-F]_l6#%Ra~modificat ion~of~the~Tsitouras-Papakostas~schemeG-F$6%7^[mF'F*7$FejlF(FgjlFjjlF] [m7$$\"5lm;z>Oa5_\\F-F(7$Fa[mF(7$$\"5++]PMx_<*=&F-F(7$Fd[mF(Fi[mF_\\mF b\\m7$Ff\\m$Fj^lF37$Fi\\mF(F[]m7$$\"5qm;H2$[CXP'F-F(7$$\"5NLLek.%fI\\' F-F17$$\"5++](=UK%f6mF-F17$F_]mF(7$$\"5+++v$f3*>npF-F(7$Fb]mF(7$$\"5++ ]ilZQ!GK(F-F(7$Fe]mF(7$$\"5NL$3-))ot)fvF-F1Fg]m7$F[^mF1F]^m7$Fa^mF(Fc^ m7$Fd_lF(7$Fg_lF(7$Fj_lF(7$F]`lF(7$F``lF(7$Fc`lF(7$$\"5LLeRs$3R=?\"F3F (7$Ff`lF(7$$\"5++vozm+]G7F3F(7$F5F17$FjjnF17$Fj`lF17$$\"5LLL3xJ[#*38F3 F17$F^[oF17$$\"5LLLe9TDwN8F3F(7$F]alF(7$$\"5LLL3_]-gi8F3F17$$\"5LLL$3_ 5>gP\"F3F17$$\"5LLLe*)fzV*Q\"F3F17$F`alF17$$\"5LLL3FpcF;9F3F(7$$\"5LLL $eR_%pH9F3F(7$$\"5LLLekyL6V9F3F17$FcalF17$$\"5LLL3-)3^*p9F3F17$$\"5LLL $3F%*pL[\"F3F17$$\"5LLLeR(z)y'\\\"F3F(7$FfalF(7$$\"5LLL3x1liB:F3F17$$ \"5LLL$e9OXq`\"F3F17$$\"5LLLe9;UY]:F3F17$FialF(7$F\\blF(7$F:F(7$F_blF( 7$FbblF(7$FhjoF17$FeblF17$F\\[pF17$FhblF17$$\"5++v$4@\"*\\[%=F3F(7$F[c lF(7$$\"5LLek`m+cr=F3F17$F^clF17$$\"5nmTN'4Aq#)*=F3F(7$$\"5ML$3x\")HD; \">F3F(7$$\"5++D1Rv.)\\#>F3F(7$F`[pF17$$\"5ML3x\")H0p^>F3F17$$\"5++]7. 2c/l>F3F(7$$\"5nm\"zWUo+%y>F3F(7$FaclF(7$Fc[pF(7$F=F(7$$\"5+++DJX#y46# F3F17$Fe[oF17$$\"5nmmTgx>uN@F3F17$Fh[oF17$F[\\oF17$F^\\oF17$$\"5nmm\"z W*eDF3F(7$$\"5MLLL3F.urDF3F77 $$\"5nmm;HK5c%e#F3F17$$\"5++++]P$F3F17$F_` mF17$Fe`mF17$FUF17$FabmF17$FXF17$$\"5+++vVth0aNF3F77$$\"5MLL$3F%GFnNF3 F77$$\"5nmm\"z>^*[!e$F3F77$$\"5++++D\"=1Pf$F3F17$$\"5MLL3_]G#pg$F3F77$ $\"5nmm;z>&R,i$F3F17$$\"5+++D1*=cLj$F3F77$FjcmF77$$\"5nmmmTN&R%*p$F3F7 7$FgnF77$FhemF77$F\\oF77$$\"5MLLLek39$G%F3F77$$\"5++++]7l$*yVF3F77$$\" 5nmm\"zWU&)GS%F3F77$$\"5MLL$ekLMoU%F3F77$$\"5nm;zW#z3)QWF3F77$$\"5+++v V[Ky]WF3F77$$\"5ML$3FWqdFY%F3F77$$\"5nmmmTg@tuWF3F?7$$\"5++]iS;mq'[%F3 F77$$\"5MLLeRs5o)\\%F3F77$$\"5nm;aQGbl5XF3F77$$\"5+++]P%)*HE_%F3F77$$ \"5ML$ek.W/Y`%F3F77$$\"5nmmTN'*)yla%F3F77$$\"5++]PM_LbeXF3F77$FaoF?7$$ \"5nm;aj4#3Se%F3F77$$\"5+++v$4h)[(f%F3F77$$\"5ML$eRA,p4h%F3F77$$\"5nmm ;a8%\\Wi%F3F77$$\"5++]P%[\")Hzj%F3F77$$\"5MLLe9;-T^YF3F?7$$\"5nm;zW<1* [m%F3F77$FjhmF77$$\"5MLL$eRi#HKZF3F77$F]imF77$$\"5++](o/j%p*z%F3F77$$ \"5MLL3xJ]<8[F3F?7$$\"5nm;H2LalE[F3F77$$\"5+++]PMe8S[F3F77$$\"5ML$3xcB ;O&[F3F77$$\"5nmm\"zpj'4n[F3F77$$\"5++]7GQqd!)[F3F77$F`imF?7$$\"5nmm;z W!zz%\\F3F?7$FfoF?7$F[pF?7$$\"5+++++]?vVcF3F?7$F`pF?7$$\"5lm\"HK*QZIke F3F?7$$\"5IL$eky7]h(eF3F?7$$\"5++voz;b*z)eF3F?7$$\"5lmm\"Hd!4%)**eF3FJ 7$$\"5++]Pf$oJN#fF3F?7$$\"5ILL$e9YAs%fF3F77$$\"5++D1R]y1ffF3FJ7$$\"5lm ;HKRK\"4(fF3F?7$$\"5IL3_DG'eF)fF3FJ7$$\"5+++v=hS \\k+'F3FJ7$$\"5IL$3_]z%H=gF3FJ7$$\"5++vV)R=S,.'F3F?7$$\"5lmmm\"Hd&)>/' F3F?7$$\"5ILLekGrO*3'F3F?7$$\"5+++]P%o[n8'F3F?7$$\"5IL$eRAYR/;'F3F?7$$ \"5lmmT5S-8%='F3F?7$$\"5ILek.Hc(f>'F3F?7$$\"5++](oz,@y?'F3F77$$\"5lmT5 !pSm'>iF3F?7$FepFJ7$$\"5++++D1o(oX'F3FJ7$FjpFJ7$$\"5lmmm;z')pxnF3FJ7$$ \"5lmmmmTb:toF3FJ7$$\"5lmmm\"H(R)3#pF3FJ7$$\"5lmmm;/ChopF3FJ7$$\"5lmm; z>mZ#*pF3FJ7$$\"5lmmmTN3M;qF3FJ7$$\"5lmm\"HK%HFGqF3F?7$$\"5lmm;/^]?SqF 3FO7$$\"5lmmT&)er8_qF3FJ7$F_qFO7$$\"5NLe9mW\"\\z2(F3FJ7$$\"5++]ilA!H=4 (F3FJ7$$\"5qmT5l+*3d5(F3FJ7$$\"5NLLeky()e>rF3FJ7$$\"5qm;ajM&[t9(F3FJ7$ $\"5+++]i!H3^<(F3FJ7$$\"5qmmTg-yiIsF3FJ7$$\"5NLLLe9t9'G(F3FJ7$$\"5++D \"yD>F+I(F3FO7$$\"5qm;Hdqq!RJ(F3FJ7$$\"5NL3xc[pyFtF3FJ7$$\"5+++DcEomTt F3FO7$$\"5qm\"HdXqYbN(F3FJ7$$\"5NL$3_DeE%ptF3FJ7$$\"5++voagkI$Q(F3FJ7$ $\"5qmm;aQj=(R(F3FO7$$\"5NLek`;i16uF3FJ7$$\"5++]7`%4Y\\U(F3FJ7$$\"5qmT g_sf#)QuF3FO7$$\"5NLL3_]eq_uF3FJ7$$\"5++Dc^GdemuF3FO7$$\"5qm;/^1cY![(F 3FO7$$\"5NL3_]%[XV\\(F3FJ7$FdqFJ7$$\"5qmm;a8P^1wF3FJ7$$\"5NLLLek?![q(F 3FJ7$$\"5qmTNY3\")3g;u(F3FO7$$ \"5qmmT5Si%Rv(F3FO7$$\"5++vV)RGKiw(F3FO7$$\"5NL$ekyK=&yxF3FJ7$$\"5qm\" zW

/*)F3Fdir7$$\"5qm;HKk)z4$*)F3$\"#j F-7$F_cnFiir7$$\"5qmTN@-'e6(*)F3$\"#dF-7$$\"5++]P%[^^X)*)F3Fiir7$$\"5N LeRZFW%z**)F3Fajr7$$\"5qmmT5StL6!*F3F]ir7$$\"5NL$ek`;B\"Q!*F3$F\\jlF-7 $Fdcn$F_zF-7$$\"5qm;a)e\"[p\"4*F3F]ir7$$\"5NLLe9T1[=\"*F3F^[s7$$\"5qmT gx`N(=8*F3F^[s7$$\"5++]iSmkEX\"*F3$\"#cF-7$$\"5NLek.z$f'e\"*F3Fhjr7$Fh rF^\\s7$$\"5lmm;a)GV/F*F3F\\\\s7$$\"5lmmmT&GM)o$*F3Fhjr7$$\"5lm;ajM?V$ R*F3Fiir7$$\"5lmmT&QyH!=%*F3F^\\s7$$\"5lm;H2LviU%*F3Fajr7$$\"5lmm;H#GD sY*F3$FjyF-7$$\"5lmT5!p:C&z%*F3$\"#rF-7$$\"5lm;/^JI#=\\*F3$FhflF-7$$\" 5lm\"z>h!>7/&*F3$FdzF-7$$\"5lmm\"H2y?k^*F3F]ir7$$\"5lmT&Q`l>(G&*F3Fg^s 7$$\"5lm;z%*H&=5a*F3$\"#uF-7$$\"5lm\"HdXS#)zB**F3F_cs7$$\"5+++v=k;;N5\"Fis$ \"$j*Fft7$$\"5++vo/$o6[5\"Fis$\"$H*Fft7$$\"5m;zWn*>2h5\"Fis$\"%]))!#B7 $$\"5LL$3-jr-u5\"Fis$\"%W%*F]js7$$\"5+](oHHB)p36Fis$\"%y&)F]js7$$\"5mm \"Hd&\\P**46Fis$\"$7*Fft7$$\"5L$e*[=m#*G66Fis$\"$J)Fft7$$\"5+++D\"Gy%e 76Fis$\"$P*Fft7$$\"5LL3x1;e<:6Fis$\"$a*Fft7$$\"5mm;HK\\ow<6Fis$\"$f*Ff t7$$\"5++D\"yD)yN?6Fis$\"$N*Fft7$Fbt$\"$I*Fft7$$\"5mmTgF6tGC6Fis$\"$e* Fft7$$\"5++](=nqDc7\"Fis$\"%:5Fft7$$\"5LLe9;-T'p7\"Fis$\"$@*Fft7$$\"5m mmTg(\\-$G6Fis$\"$%)*Fft7$$\"5++vo/$*3kH6Fis$FdglF-7$$\"5LL$e*[)Gz48\" Fis$\"#**F-7$$\"5mm\"HKRo>\"Fis$\"$<\"F-7$$\"5++D1R0TF$ >\"Fis$\"$5\"F-7$$\"5++](=UY$e%>\"FisFfat7$$\"5++vo/BG*e>\"FisF^at7$$ \"5+++](==-s>\"FisFfat7$$\"5++DJqS:^)>\"Fis$FdhoF-7$$\"5++]7`**3#)*>\" Fis$FgglF-7$$\"5++v$f$e-8,7FisF`ct7$$\"5+++v=<'RC?\"FisFgdt7$$\"5++Dc, w*[P?\"Fis$\"$;\"F-7$$\"5++]P%[Le]?\"Fis$FgboF-7$$\"5++v=n$pnj?\"Fis$F jglF-7$F]u$\"$:\"F-7$$\"5+](=#\\CR%*37FisFgdt7$FbuFjet7$$\"5+]ilZowZ67 Fis$\"$7\"F-7$FguF`ft7$$\"5+]P4Y79,97FisF`ft7$F\\vF`ft7$$\"5+]7`Wc^a;7 FisF`ct7$Fav$F]hlF-7$F[w$\"$?\"F-7$FcwF[bt7$FfwF^ft7$FiwFdgt7$F^xFjet7 $FcxFift7$Fhx$\"$D\"F-7$F]y$FjhoF-7$$\"5MLLL3UDX88Fis$\"$Q\"F-7$FbyFgp 7$$\"5ML$ek`*\\-N8FisF\\q7$$\"5+++D19XPO8FisFgp7$$\"5nm;/wKSsP8FisFgp7 $$\"5MLL$e9bt!R8FisF\\q7$$\"5++]i:qIUS8FisF\\q7$$\"5nmmT&))es#>7N\"FisF\\q7$$\"5+++vVQ(oDN\"FisF\\q7$$\"5nm;a8d#=R N\"FisF\\q7$$\"5MLLL$exn_N\"FisF\\q7$$\"5nmm\"HK\"o'zN\"FisF\\q7$$\"5+ ++]i]emg8FisF\\q7$$\"5nm;HKp`,i8FisF\\q7$$\"5MLL3-))[Oj8FisF\\q7$$\"5+ +](=nS9ZO\"FisF^dl7$$\"5nmmmTDR1m8FisF^dl7$$\"5ML$e9TW8uO\"FisF\\q7$$ \"5+++D\"G'Hwo8FisF\\q7$$\"5nm;/^\"[7,P\"FisF^dl7$$\"5MLL$3-+i9P\"FisF \\q7$$\"5++]i!*=:\"GP\"FisF^dl7$$\"5nmmTgP5;u8FisF^dl7$$\"5ML$3-jb5bP \"FisF^dl7$FgyF\\q7$$\"5mmm;H2fU'R\"FisF^dl7$F\\zFaq7$$\"5m;/E]`'GtT\" FisFaq7$$\"5++v=Unbm=9FisFaq7$$\"5L$e9T8[-+U\"FisFaq7$$\"5mm;/E&RR8U\" FisFfq7$$\"5+](oz\"4jnA9FisFaq7$$\"5LLe*)4BK,C9FisF^dl7$$\"5m;H#=q8]`U \"FisFfq7$$\"5+++v$40(oE9FisFaq7$$\"5L$3xc['R-G9FisFaq7$$\"5mmTgxy3OH9 FisFfq7$$\"5+]7`p#z(pI9FisFfq7$$\"5LL$e9mqM?V\"FisFfq7$$\"5m;aQ`?;PL9F isFfq7$$\"5++DJXM&3ZV\"FisFaq7$$\"5L$eRs$[a/O9FisFfq7$$\"5mmm;HiBQP9Fi sFfq7$$\"5LLLektw2[9FisFfq7$FazFfq7$$\"5M$e*)4o\"4.g9FisFb]n7$$\"5nm\" z>'[))Gh9FisFb]n7$$\"5+](oH/yYDY\"FisFfq7$$\"5ML$eRAr/QY\"FisFb]n7$$\" 5n;z%\\Ski]Y\"FisFb]n7$$\"5++v$fed?jY\"FisF[r7$$\"5M$3Fpw]yvY\"FisFb]n 7$$\"5nmm\"z%Rk$)o9FisFb]n7$$\"5++](=n;oQZ\"FisFb]n7$$\"5MLL$eR*)**)y9 FisFb]n7$$\"5+++vV[L'*)[\"FisFb]n7$FfzFb]n7$$\"5MLLL3A_1?:FisF[r7$F[[l Fh`n7$$\"5MLL$e9as;c\"FisFer7$F`[lFe_o7$$\"5M$e*[$**H(e$e\"FisFe_o7$$ \"5++DJ?eJ$\\e\"FisFe_o7$$\"5n;a8Z;!zie\"FisFer7$$\"5ML$eRZ([i(e\"FisF er7$$\"5+]7y+L2(*)e\"FisFe_o7$$\"5nmTgF\"f;.f\"FisFe_o7$$\"5M$3FW&\\Cm \"f\"FisFe_o7$$\"5+++D\"yI3If\"FisFer7$$\"5n;H23mTN%f\"FisFe_o7$$\"5ML e*[V-+df\"FisFe_o7$$\"5+](= " 0 "" {MPLTEXT 1 0 598 "evalf[20](p lot([e(x)-'en_RK8_1'(x),e(x)-'en_RK8_2'(x),e(x)-'en_RK8_3'(x),e(x)-'en _RK8_4'(x),\ne(x)-'en_RK8_5'(x),e(x)-'en_RK8_6'(x),e(x)-'en_RK8_7'(x)] ,x=2..8,font=[HELVETICA,9],\ncolor=[COLOR(RGB,.4,0,.9),COLOR(RGB,0,.65 ,0),brown,red,COLOR(RGB,0,.75,.95),magenta,COLOR(RGB,.95,.55,0)],\nleg end=[`Enright-Verner scheme`,`Verner's Maple scheme`,`a modification o f Verner's Maple scheme`,`an \"efficient\" scheme`,`a scheme of Tsitou ras and Papakostas`,`a modification of the Tsitouras-Papakostas scheme `,`Verner's \"most efficient\" scheme`],title=`error curves for 12 sta ge order 8 Runge-Kutta methods`));" }}{PARA 13 "" 1 "" {GLPLOT2D 900 643 643 {PLOTDATA 2 "6--%'CURVESG6%7Z7$$\"\"#\"\"!$\"$3#!#?7$$\"5+++++ DHyI@!#>$\"$p#F-7$$\"5++++v[kdWAF1$\"$N$F-7$$\"5++++]n\"\\DP#F1$\"$J%F -7$$\"5++++]s,P,DF1$\"$b&F-7$$\"5++++v8*y&HEF1$\"$:(F-7$$\"5++++vG[W[F F1$\"$1*F-7$$\"5++++v)fB:(GF1$\"%c6F-7$$\"5++++vQ=\"))*HF1$\"%*[\"F-7$ $\"5++++vj=pDJF1$\"%<>F-7$$\"5+++++lN?cKF1$\"%([#F-7$$\"5++++]U$e6P$F1 $\"%HJF-7$$\"5+++++&>q0]$F1$\"%^SF-7$$\"5+++++DM^IOF1$\"%__F-7$$\"5+++ ++0ytbPF1$\"%XnF-7$$\"5++++vQNXpQF1$\"%o%)F-7$$\"5+++++XDn/SF1$\"&(46F -7$$\"5+++++!y?#>TF1$\"'K&R\"!#@7$$\"5++++v3wY_UF1$\"($R@=!#A7$$\"5+++ ++IOTqVF1$\"'*fI#F^q7$$\"5++++v3\">)*\\%F1$\"&r)HF-7$$\"5++++DEP/BYF1$ \"&@#QF-7$$\"5++++](o:;v%F1$\"&H%\\F-7$$\"5++++v$)[op[F1$\"&'fiF-7$$\" 5++++]i%Qq*\\F1$\"&c2)F-7$$\"5++++vQIKH^F1$\"';_5F-7$$\"5++++D^rZW_F1$ \"'nC8F-7$$\"5++++]Zn%)o`F1$\"'y)p\"F-7$$\"5+++++5FL(\\&F1$\"'c'>#F-7$ $\"5++++]d6.BcF1$\"'SCGF-7$$\"5++++vo3lWdF1$\"'>-OF-7$$\"5++++]A))ozeF 1$\"'7>ZF-7$$\"5+++++Ik-,gF1$\"'G:gF-7$$\"5+++++D-eIhF1$\"'X%z(F-7$$\" 5++++v=_(zC'F1$\"'@d)*F-7$$\"5+++++b*=jP'F1$\"('=u7F-7$$\"5++++v3/3(\\ 'F1$\"(zAi\"F-7$$\"5++++vB4JBmF1$\"(#=)3#F-7$$\"5+++++DVsYnF1$\"(&ysEF -7$$\"5++++v=n#f(oF1$\"(\"*3Y$F-7$$\"5+++++!)RO+qF1$\"(u)QWF-7$$\"5+++ +]_!>w7(F1$\"($RDdF-7$$\"5++++v)Q?QD(F1$\")WBptF^q7$$\"5+++++5jyptF1$ \"*L`GH*Fdq7$$\"5++++DE8COuF1$\"*q!Qh5F^q7$$\"5++++]Ujp-vF1$\"*,`A@\"F ^q7$$\"5++++D,X8ivF1$\"*Mw_O\"F^q7$$\"5+++++gEd@wF1$\")hhP:F-7$$\"5+++ ]PMh%\\o(F1$\")ORX$[xF1$\")#[7)>F-7$$\"5+++++5h(*3yF1 $\")hyOAF-7$$\"5++++D6EjpyF1$\")FGDDF-7$$\"5+++vVeWA-zF1$\")LP&p#F-7$$ \"5+++]i0j\"[$zF1$\")-#p(GF-7$$\"5+++D\"G:3u'zF1$\")bpqIF-7$$\"\")F*$ \")C_xKF--%&COLORG6&%$RGBG$\"\"%!\"\"$F*F*$\"\"*Fd]l-%'LEGENDG6#%6Enri ght-Verner~schemeG-F$6%7Z7$F($\"$_#F-7$F/$\"$D$F-7$F5$\"$1%F-7$F:$\"$A &F-7$F?$\"$r'F-7$FD$\"$m)F-7$FI$\"%'4\"F-7$FN$\"%*R\"F-7$FS$\"%-=F-7$F X$\"%@BF-7$Fgn$\"%6IF-7$F\\o$\"%(y$F-7$Fao$\"%/\\F-7$Ffo$\"%djF-7$F[p$ \"%l\")F-7$F`p$\"&]-\"F-7$Fep$\"&KM\"F-7$Fjp$\"'**)o\"F^q7$F`q$\"(MZ?# Fdq7$Ffq$\"'K\"z#F^q7$F[r$\"&eh$F-7$F`r$\"&li%F-7$Fer$\"&K)fF-7$Fjr$\" &qd(F-7$F_s$\"&^x*F-7$Fds$\"'ft7F-7$Fis$\"'X.;F-7$F^t$\"'Ic?F-7$Fct$\" '$)eEF-7$Fht$\"'!)=MF-7$F]u$\"'GgVF-7$Fbu$\"'F7dF-7$Fgu$\"'@\"G(F-7$F \\v$\"'\"[V*F-7$Fav$\"(pJ>\"F-7$Ffv$\"(UBa\"F-7$F[w$\"(#pj>F-7$F`w$\"( Xw_#F-7$Few$\"(z_B$F-7$Fjw$\"(V#*=%F-7$F_x$\"(YIP&F-7$Fdx$\"(:.$pF-7$F ix$\")m5?*)F^q7$F^y$\"+pb&[7\"Fdq7$Fcy$\"*r]ZG\"F^q7$Fhy$\"*WutY\"F^q7 $F]z$\"*\">g_;F^q7$Fbz$\")4@h=F-7$Fgz$\")dr7@F-7$F\\[l$\")k?)R#F-7$Fa[ l$\")H_2FF-7$Ff[l$\")VtcIF-7$F[\\l$\")3iiKF-7$F`\\l$\")[P#[$F-7$Fe\\l$ \")0$pr$F-7$Fj\\l$\")YGnRF--F_]l6&Fa]lFe]l$\"#l!\"#Fe]l-Fi]l6#%6Verner 's~Maple~schemeG-F$6%7Z7$F($\"$)>F-7$F/$\"$c#F-7$F5$\"$?$F-7$F:$\"$6%F -7$F?$\"$H&F-7$FD$\"$#oF-7$FI$\"$k)F-7$FN$\"%-6F-7$FS$\"%?9F-7$FX$\"%G =F-7$Fgn$\"%sBF-7$F\\o$\"%$)HF-7$Fao$\"%jQF-7$Ffo$\"%3]F-7$F[p$\"%KkF- 7$F`p$\"%v!)F-7$Fep$\"&#e5F-7$Fjp$\"'fI8F^q7$F`q$\"()*ot\"Fdq7$Ffq$\"' -*>#F^q7$F[r$\"&'[GF-7$F`r$\"&[k$F-7$Fer$\"&Or%F-7$Fjr$\"&#pfF-7$F_s$ \"&4q(F-7$Fds$\"'N.5F-7$Fis$\"'@j7F-7$F^t$\"''*>;F-7$Fct$\"'k%4#F-7$Fh t$\"'O$p#F-7$F]u$\"'1NMF-7$Fbu$\"'<+XF-7$Fgu$\"'>OdF-7$F\\v$\"'\"GV(F- 7$Fav$\"'()*R*F-7$Ffv$\"(n]@\"F-7$F[w$\"(5qa\"F-7$F`w$\"('H\"*>F-7$Few $\"(v([DF-7$Fjw$\"(9.I$F-7$F_x$\"(AHB%F-7$Fdx$\"(](faF-7$Fix$\")9KFqF^ q7$F^y$\"*c*oh))Fdq7$Fcy$\"*XN@,\"F^q7$Fhy$\"*o2g:\"F^q7$F]z$\"*9J>I\" F^q7$Fbz$\")[Fm9F-7$Fgz$\")?Tk;F-7$F\\[l$\")NK*)=F-7$Fa[l$\")_+L@F-7$F f[l$\")j63CF-7$F[\\l$\")^JqDF-7$F`\\l$\"))QMu#F-7$Fe\\l$\")MAGHF-7$Fj \\l$\")SXDJF--%'COLOURG6&Fa]l$\")#)eqk!\")$\"))eqk\"F_dmF`dm-Fi]l6#%Ha ~modification~of~Verner's~Maple~schemeG-F$6%7Z7$F($\"$E#F-7$F/$\"$#HF- 7$F5$\"$l$F-7$F:$\"$p%F-7$F?$\"$/'F-7$FD$\"$z(F-7$FI$\"$()*F-7$FN$\"%g 7F-7$FS$\"%B;F-7$FX$\"%!4#F-7$Fgn$\"%6FF-7$F\\o$\"%5MF-7$Fao$\"%;WF-7$ Ffo$\"%DdF-7$F[p$\"%`tF-7$F`p$\"%I#*F-7$Fep$\"&'47F-7$Fjp$\"'+@:F^q7$F `q$\"(Ma)>Fdq7$Ffq$\"'m8DF^q7$F[r$\"&hD$F-7$F`r$\"&j;%F-7$Fer$\"&!)Q&F -7$Fjr$\"&K#oF-7$F_s$\"&E!))F-7$Fds$\"'*o9\"F-7$Fis$\"'$RW\"F-7$F^t$\" 's^=F-7$Fct$\"'I%R#F-7$Fht$\"'nyIF-7$F]u$\"'ZERF-7$Fbu$\"''R9&F-7$Fgu$ \"'!ob'F-7$F\\v$\"'8'\\)F-7$Fav$\"(fW2\"F-7$Ffv$\"($*))Q\"F-7$F[w$\"(A $oD!>F-7$F\\[l$ \")lgf@F-7$Fa[l$\"))[\"QCF-7$Ff[l$\")oh_FF-7$F[\\l$\")&>!QHF-7$F`\\l$ \")*4f8$F-7$Fe\\l$\")%HrM$F-7$Fj\\l$\")bdsNF--F[dm6&Fa]l$\"*++++\"F_dm Fe]lFe]l-Fi]l6#%6an~\"efficient\"~schemeG-F$6%7Z7$F($\"$=#F-7$F/$\"$#G F-7$F5$\"$_$F-7$F:$\"$`%F-7$F?$\"$$eF-7$FD$\"$_(F-7$FI$\"$`*F-7$FN$\"% ;7F-7$FS$\"%n:F-7$FX$\"%=?F-7$Fgn$\"%Fdq7$Ffq$\"'cECF^q7$F[r$\"&L9$F-7$F`r$\"&>-%F-7$Fer$\"&8 ?&F-7$Fjr$\"&oe'F-7$F_s$\"&w\\)F-7$Fds$\"':26F-7$Fis$\"'!RR\"F-7$F^t$ \"'c(y\"F-7$Fct$\"'N6BF-7$Fht$\"'+sHF-7$F]u$\"'W!z$F-7$Fbu$\"'ul\\F-7$ Fgu$\"'kHjF-7$F\\v$\"'y,#)F-7$Fav$\"(Ms.\"F-7$Ffv$\"(u2M\"F-7$F[w$\"(e qq\"F-7$F`w$\"(3t>#F-7$Few$\"(hC\"GF-7$Fjw$\"(`F^q7$F`q$\"($G/EFdq7$Ffq$\"'8(H$F^q7$F[r$\"&5F%F-7$F`r$\" &[Y&F-7$Fer$\"&t1(F-7$Fjr$\"&)\\*)F-7$F_s$\"'ha6F-7$Fds$\"'K/:F-7$Fis$ \"'$R*=F-7$F^t$\"'#)GCF-7$Fct$\"']SJF-7$Fht$\"'9QSF-7$F]u$\"'<]^F-7$Fb u$\"'2ZnF-7$Fgu$\"'B+')F-7$F\\v$\"(&R96F-7$Fav$\"(:$49F-7$Ffv$\"(T<#=F -7$F[w$\"(>%>BF-7$F`w$\"(Mb)HF-7$Few$\"(d8#QF-7$Fjw$\"(N\"[\\F-7$F_x$ \"('QYjF-7$Fdx$\"(fd=)F-7$Fix$\"*k+O0\"F^q7$F^y$\"+9giG8Fdq7$Fcy$\"*d' [<:F^q7$Fhy$\"*4$>L&>F^q7$Fbz$\")TP)>#F-7$Fgz$\")(Ra\\ #F-7$F\\[l$\")ykKGF-7$Fa[l$\")y*z>$F-7$Ff[l$\")(p/h$F-7$F[\\l$\")Kl`QF -7$F`\\l$\")i@8TF-7$Fe\\l$\")BE!R%F-7$Fj\\l$\")'ofo%F--F[dm6&Fa]lFb_nF e]lFb_n-Fi]l6#%Ra~modification~of~the~Tsitouras-Papakostas~schemeG-F$6 %7Z7$F($\"#_F-7$F/$\"#oF-7$F5$\"#%)F-7$F:$\"$4\"F-7$F?$\"$R\"F-7$FD$\" $!=F-7$FI$\"$G#F-7$FN$\"$\"HF-7$FS$\"$v$F-7$FX$\"$$[F-7$Fgn$\"$E'F-7$F \\o$\"$)yF-7$Fao$\"%?5F-7$Ffo$\"%A8F-7$F[p$\"%)p\"F-7$F`p$\"%K@F-7$Fep $\"%%z#F-7$Fjp$\"&M^$F^q7$F`q$\"'/'e%Fdq7$Ffq$\"&j!eF^q7$F[r$\"%@vF-7$ F`r$\"%B'*F-7$Fer$\"&XC\"F-7$Fjr$\"&hd\"F-7$F_s$\"&L.#F-7$Fds$\"&#\\EF -7$Fis$\"&_L$F-7$F^t$\"&sF%F-7$Fct$\"&/`&F-7$Fht$\"&76(F-7$F]u$\"&&p!* F-7$Fbu$\"'<)=\"F-7$Fgu$\"'_9:F-7$F\\v$\"'Zi>F-7$Fav$\"'#=[#F-7$Ffv$\" '73KF-7$F[w$\"'a%3%F-7$F`w$\"'ed_F-7$Few$\"'[HnF-7$Fjw$\"'v8()F-7$F_x$ \"(5w6\"F-7$Fdx$\"(F:W\"F-7$Fix$\")&4a&=F^q7$F^y$\"*\"QtRBFdq7$Fcy$\") 3KsEF^q7$Fhy$\")P=_IF^q7$F]z$\") " 0 "" {MPLTEXT 1 0 1 ";" }}}}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 49 "Test \+ 14 of 12 stage, order 8 Runge-Kutta methods " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 256 "" 0 "" {TEXT -1 3 " " }{XPPEDIT 18 0 "dy/dx = 1 0*x*cos*x-10*y;" "6#/*&%#dyG\"\"\"%#dxG!\"\",&**\"#5F&%\"xGF&%$cosGF&F ,F&F&*&F+F&%\"yGF&F(" }{TEXT -1 8 ", " }{XPPEDIT 18 0 "y(0) = sq rt(5);" "6#/-%\"yG6#\"\"!-%%sqrtG6#\"\"&" }{TEXT -1 1 " " }}{PARA 0 " " 0 "" {TEXT -1 10 "Solution: " }}{PARA 256 "" 0 "" {TEXT -1 2 " " } {XPPEDIT 18 0 "y=100/101" "6#/%\"yG*&\"$+\"\"\"\"\"$,\"!\"\"" }{TEXT -1 1 " " }{XPPEDIT 18 0 "x*cos*x-990/10201" "6#,&*(%\"xG\"\"\"%$cosGF& F%F&F&*&\"$!**F&\"&,-\"!\"\"F+" }{TEXT -1 1 " " }{XPPEDIT 18 0 "cos*x+ 10/101" "6#,&*&%$cosG\"\"\"%\"xGF&F&*&\"#5F&\"$,\"!\"\"F&" }{TEXT -1 1 " " }{XPPEDIT 18 0 "x*sin*x-200/10201" "6#,&*(%\"xG\"\"\"%$sinGF&F%F &F&*&\"$+#F&\"&,-\"!\"\"F+" }{TEXT -1 1 " " }{XPPEDIT 18 0 "sin*x+(990 /10201+sqrt(5))*exp(-10*x)" "6#,&*&%$sinG\"\"\"%\"xGF&F&*&,&*&\"$!**F& \"&,-\"!\"\"F&-%%sqrtG6#\"\"&F&F&-%$expG6#,$*&\"#5F&F'F&F-F&F&" } {TEXT -1 1 "." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 163 "de := diff(y(x),x)=10*x*cos(x)-10*y(x);\nic := \+ y(0)=sqrt(5);\ndsolve(\{de,ic\},y(x));\nb := unapply(rhs(%),x):\nplot( b(x),x=0..5,font=[HELVETICA,9],labels=[`x`,`y(x)`]);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%#deG/-%%diffG6$-%\"yG6#%\"xGF,,&*(\"#5\"\"\"F,F0-% $cosGF+F0F0*&F/F0F)F0!\"\"" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%#icG/- %\"yG6#\"\"!*$\"\"&#\"\"\"\"\"#" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/-% \"yG6#%\"xG,,*&#\"$+\"\"$,\"\"\"\"*&F'F--%$cosGF&F-F-F-*&#\"$!**\"&,- \"F-F/F-!\"\"*&#\"#5F,F-*&-%$sinGF&F-F'F-F-F-*&#\"$+#F4F-F:F-F5*&-%$ex pG6#,$*&F8F-F'F-F5F-,&#F3F4F-*$\"\"&#F-\"\"#F-F-F-" }}{PARA 13 "" 1 " " {GLPLOT2D 703 312 312 {PLOTDATA 2 "6&-%'CURVESG6$7hn7$$\"\"!F)$\"3\" )*y*\\xz1OA!#<7$$\"3ALL$3FWYs#!#>$\"3*pc)\\jF;12h\"H\"48F,7$$\"3m****\\7G$R<)F0$\"3<_u(oLbK,\"F,7$$\"3GLLL3x &)*3\"!#=$\"3(**[ro!GyVzF@7$$\"3))**\\i!R(*Rc\"F@$\"3A'ysO]2xW&F@7$$\" 3umm\"H2P\"Q?F@$\"3/$)oqvSKmSF@7$$\"3YLek.pu/BF@$\"3$Qjx*Gs<7OF@7$$\"3 !***\\PMnNrDF@$\"3M:4%*3rt@LF@7$$\"3MmT5ll'z$GF@$\"3/?Np5C\\bJF@7$$\"3 MLL$eRwX5$F@$\"3)GTJ!oG0$3$F@7$$\"3rLLL$eI8k$F@$\"3FyHM$p'GKJF@7$$\"33 ML$3x%3yTF@$\"3n$**Q]`\"yRLF@7$$\"3emm\"z%4\\Y_F@$\"3%)*)G8T#p2%RF@7$$ \"3`LLeR-/PiF@$\"3PZ.%R2Cm^%F@7$$\"3]***\\il'pisF@$\"31e'*fKlL9]F@7$$ \"3>MLe*)>VB$)F@$\"3%3)yy-pAk`F@7$$\"3Y++DJbw!Q*F@$\"3Kg$RQBm7^&F@7$$ \"3%ommTIOo/\"F,$\"3xrTB'zz$GaF@7$$\"3YLL3_>jU6F,$\"3!p\\3Dp!RX^F@7$$ \"37++]i^Z]7F,$\"3Kvxv\"*=%>e%F@7$$\"33++](=h(e8F,$\"3C^UQ-(*\\]PF@7$$ \"3/++]P[6j9F,$\"3icMTpV\"zp#F@7$$\"3UL$e*[z(yb\"F,$\"3C-jVJM+L:F@7$$ \"3wmm;a/cq;F,$!33XIKjA5+5F07$$\"3%ommmJF,$!3!\\F1VDv$)p&F@7$$\"3K+] i!f#=$3#F,$!3V$*[LN4F4!)F@7$$\"3?+](=xpe=#F,$!3$[6ral`?.\"F,7$$\"37nm \"H28IH#F,$!3s6ToLL*4G\"F,7$$\"3um;zpSS\"R#F,$!3/xdj'GyI^\"F,7$$\"3GLL 3_?`(\\#F,$!3MGc,i$*4jpxg#F,$!3XXbRF+x>IF,$!3r\"fP(*)y]OGF,7$$\"3F+]i!RU07$F, $!3[Tb]!*H%e)HF,7$$\"3+++v=S2LKF,$!3Tyi#Q\\'[=JF,7$$\"3Jmmm\"p)=MLF,$! 3N`)=cl>V?$F,7$$\"3B++](=]@W$F,$!3Y_BiN[odKF,7$$\"3mm\"H#oZ1\"\\$F,$!3 )o4&)z%=-oKF,7$$\"35L$e*[$z*RNF,$!3%)Q61)ek$pKF,7$$\"3%o;Hd!fX$f$F,$!3 #*y!45Ut-E$F,7$$\"3e++]iC$pk$F,$!3LO[nw')*)RKF,7$$\"3ILe*[t\\sp$F,$!3D 1>x`HA5KF,7$$\"3[m;H2qcZPF,$!3/[q%\\V.-<$F,7$$\"3O+]7.\"fF&QF,$!3KL?tX &>E0$F,7$$\"3Ymm;/OgbRF,$!3KQEMNc$G*GF,7$$\"3w**\\ilAFjSF,$!3/QR)44g!y EF,7$$\"3yLLL$)*pp;%F,$!30,GW_`#fU#F,7$$\"3)RL$3xe,tUF,$!3*G#*H@1([B@F ,7$$\"3Cn;HdO=yVF,$!35Q!)*4x]5y\"F,7$$\"3a+++D>#[Z%F,$!3(y*pyl_QJ9F,7$ $\"3SnmT&G!e&e%F,$!3]X/0\"RC%G**F@7$$\"3#RLLL)Qk%o%F,$!3u!*)Q\"4WH+dF@ 7$$\"37+]iSjE!z%F,$!3+r[gfMO'=*F07$$\"3a+]P40O\"*[F,$\"3+2*eSHde(QF@7$ $\"\"&F)$\"3&Q8`\">jC3#*F@-%'COLOURG6&%$RGBG$\"#5!\"\"F(F(-%%FONTG6$%* HELVETICAG\"\"*-%+AXESLABELSG6$%\"xG%%y(x)G-%%VIEWG6$;F(F\\^l%(DEFAULT G" 1 2 0 1 10 0 2 9 1 4 2 1.000000 45.000000 45.000000 0 0 "Curve 1" } }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 32 "The fo llowing code constructs a " }{TEXT 260 17 "discrete solution" }{TEXT -1 44 " based on each of the methods and gives the " }{TEXT 260 22 "ro ot mean square error" }{TEXT -1 18 " of each solution." }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 807 "B := (x,y) -> 10*x*cos(x)-10*y: hh := 0.01: numsteps := 500: x0 := 0: y0 := sqrt(5):\nmatrix([[`slope fi eld: `,B(x,y)],[`initial point: `,``(x0,y0)],[`step width: `,hh], \n[`no. of steps: `,numsteps]]);``;\nmthds := [`Enright-Verner schem e`,`Verner's Maple scheme`,`a modification of Verner's Maple scheme`,` an \"efficient\" scheme`,`a scheme of Tsitouras and Papakostas`,`a mod ification of the Tsitouras-Papakostas scheme`,`Verner's \"most efficie nt\" scheme`]: errs := []:\nDigits := 20:\nfor ct to 7 do\n Bn_RK8_| |ct := RK8_||ct(B(x,y),x,y,x0,evalf(y0),hh,numsteps,false);\n sm := \+ 0: numpts := nops(Bn_RK8_||ct):\n for ii to numpts do\n sm := s m+(Bn_RK8_||ct[ii,2]-b(Bn_RK8_||ct[ii,1]))^2;\n end do:\n errs := \+ [op(errs),sqrt(sm/numpts)];\nend do:\nDigits := 10:\nlinalg[transpose] ([mthds,evalf(errs)]);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#K%'matrixG6# 7&7$%0slope~field:~~~G,&*(\"#5\"\"\"%\"xGF,-%$cosG6#F-F,F,*&F+F,%\"yGF ,!\"\"7$%0initial~point:~G-%!G6$\"\"!*$\"\"&#F,\"\"#7$%/step~width:~~~ G$F,!\"#7$%1no.~of~steps:~~~G\"$+&Q)pprint666\"" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%!G" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#K%'matrixG6#7)7$ %6Enright-Verner~schemeG$\"+\\cCDq!#?7$%6Verner's~Maple~schemeG$\"+bcC DqF+7$%Ha~modification~of~Verner's~Maple~schemeGF.7$%6an~\"efficient\" ~schemeGF)7$%Ea~scheme~of~Tsitouras~and~PapakostasGF.7$%Ra~modificatio n~of~the~Tsitouras-Papakostas~schemeGF)7$%AVerner's~\"most~efficient\" ~schemeGF)Q)pprint676\"" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 " " 0 "" {TEXT -1 30 "The following code constructs " }{TEXT 260 20 "num erical procedures" }{TEXT -1 56 " for solutions based on each of the R unge-Kutta schemes." }}{PARA 0 "" 0 "" {TEXT -1 75 "The error in the v alue obtained by each of the methods at the point where " }{XPPEDIT 18 0 "x = 4.999;" "6#/%\"xG-%&FloatG6$\"%**\\!\"$" }{TEXT -1 16 " is \+ also given." }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 738 "B := (x,y) - > 10*x*cos(x)-10*y: hh := 0.01: numsteps := 500: x0 := 0: y0 := sqrt(5 ):\nmatrix([[`slope field: `,B(x,y)],[`initial point: `,``(x0,y0)],[ `step width: `,hh],\n[`no. of steps: `,numsteps]]);``;\nmthds := [ `Enright-Verner scheme`,`Verner's Maple scheme`,`a modification of Ver ner's Maple scheme`,`an \"efficient\" scheme`,`a scheme of Tsitouras a nd Papakostas`,`a modification of the Tsitouras-Papakostas scheme`,`Ve rner's \"most efficient\" scheme`]: errs := []:\nDigits := 30:\nfor ct to 7 do\n bn_RK8_||ct := RK8_||ct(B(x,y),x,y,x0,evalf(y0),hh,numste ps,true);\nend do:\nxx := 4.999: bxx := evalf(b(xx)):\nfor ct to 7 do \n errs := [op(errs),abs(bn_RK8_||ct(xx)-bxx)];\nend do:\nDigits := \+ 10:\nlinalg[transpose]([mthds,evalf(errs)]);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#K%'matrixG6#7&7$%0slope~field:~~~G,&*(\"#5\"\"\"%\"xGF, -%$cosG6#F-F,F,*&F+F,%\"yGF,!\"\"7$%0initial~point:~G-%!G6$\"\"!*$\"\" &#F,\"\"#7$%/step~width:~~~G$F,!\"#7$%1no.~of~steps:~~~G\"$+&Q(pprint8 6\"" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%!G" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#K%'matrixG6#7)7$%6Enright-Verner~schemeG$\"+#)zr6;!#E7$ %6Verner's~Maple~schemeG$\"+yXQ`=F+7$%Ha~modification~of~Verner's~Mapl e~schemeG$\"+Ok0(\\\"F+7$%6an~\"efficient\"~schemeG$\"+RH+K:F+7$%Ea~sc heme~of~Tsitouras~and~PapakostasG$\"+;wK]:F+7$%Ra~modification~of~the~ Tsitouras-Papakostas~schemeG$\"+HcVp " 0 "" {MPLTEXT 1 0 476 "mthds := [`Enright-Verner scheme`,`Verner's Maple scheme`,`a \+ modification of Verner's Maple scheme`,`an \"efficient\" scheme`,`a sc heme of Tsitouras and Papakostas`,`a modification of the Tsitouras-Pap akostas scheme`,`Verner's \"most efficient\" scheme`]: errs := []:\nDi gits := 20:\nfor ct to 7 do\n sm := NCint((b(x)-'bn_RK8_||ct'(x))^2, x=0..5,adaptive=false,numpoints=7,factor=200);\n errs := [op(errs),s qrt(sm/5)];\nend do:\nDigits := 10:\nlinalg[transpose]([mthds,evalf(er rs)]);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#K%'matrixG6#7)7$%6Enright-Ve rner~schemeG$\"+_mIg9!#D7$%6Verner's~Maple~schemeG$\"+9!Q1Q\"F+7$%Ha~m odification~of~Verner's~Maple~schemeG$\"+IF%HA\"F+7$%6an~\"efficient\" ~schemeG$\"++1>K6F+7$%Ea~scheme~of~Tsitouras~and~PapakostasG$\"+Pt+i7F +7$%Ra~modification~of~the~Tsitouras-Papakostas~schemeG$\"+.))RV7F+7$% AVerner's~\"most~efficient\"~schemeG$\"+X&QrO\"!#EQ)pprint706\"" }}} {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 92 "The follo wing error graphs are constructed using the numerical procedures for t he solutions." }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 631 "evalf[25]( plot(['bn_RK8_1'(x)-b(x),'bn_RK8_2'(x)-b(x),'bn_RK8_3'(x)-b(x),'bn_RK8 _4'(x)-b(x),\n'bn_RK8_5'(x)-b(x),'bn_RK8_6'(x)-b(x),'bn_RK8_7'(x)-b(x) ],x=0..0.65,-9e-17..7.9e-16,numpoints=100,\ncolor=[COLOR(RGB,.4,0,.9), COLOR(RGB,0,.65,0),brown,red,COLOR(RGB,0,.75,.95),magenta,COLOR(RGB,.9 5,.55,0)],\nlegend=[`Enright-Verner scheme`,`Verner's Maple scheme`,`a modification of Verner's Maple scheme`,`an \"efficient\" scheme`,`a s cheme of Tsitouras and Papakostas`,`a modification of the Tsitouras-Pa pakostas scheme`,`Verner's \"most efficient\" scheme`],font=[HELVETICA ,9],title=`error curves for 12 stage order 8 Runge-Kutta methods`));" }}{PARA 13 "" 1 "" {GLPLOT2D 1041 574 574 {PLOTDATA 2 "6--%'CURVESG6%7 i[l7$$\"\"!F)F(7$$\":////////a^tr\"!#F$\"#D!#C7$$\":333333333.ZV$F-$\" &pH\"F07$$\":45555555&)yLH%F-$\"&nk*F07$$\":57777777ia?:&F-$\"'Cp\\F07 $$\":5888888j]#R\"e&F-$\"(h/-\"F07$$\"::9999999RI2,'F-$\"(*e')>F07$$\" ::::::::lFo+W'F-$\"(=Mp$F07$$\":<;;;;;;;;1%poF-$\"(1pf'F07$$\":baaaaaa zNNlh(F-$\")$[$o;F07$$\":&HHHHHHHaXmj$)F-$\")DlnQF07$$\":577777iC:Hst) F-$\")pFFdF07$$\":IJJJJJJ1v$z5\"*F-$\")3bU$)F07$$\":]]]]]]+)[$eV[*F-$ \"*gXp>\"F07$$\":qppppppp%H#z&)*F-$\"*&=\\$p\"F07$$\":#HHHHHagC0Z/5!#E $\"*17\\#>F07$$\":)))))))))))Q^a([J-\"F_p$\"*.?8#>F07$$\":%[[[[[BU%)p# =/\"F_p$\"*pMx\">F07$$\":!333333L9_]g5F_p$\"*/cT\">F07$$\":ssssssZTnhy 4\"F_p$\"*x=q!>F07$$\":kkkkkkkR8=_8\"F_p$\"*93***=F07$$\":[[[[[[)f`5$* 47F_p$\"*knd)=F07$$\":KKKKKKKK(Rk%G\"F_p$\"*H^<(=F07$$\":\\\\\\\\\\\\ \\u&*)o_9F_p$\"*EW>%=F07$$\":mmmmmmm;%Rt?;F_p$\"*%[(Q$=F07$$\":DDDDDDv PVcZq\"F_p$\"*Zi)p=F07$$\":%QQQQQQ)e#*y()y\"F_p$\"*f_k)>F07$$\":UUUUUU #*zT,G(=F_p$\"*Um\"yAF07$$\":,,,,,,,,\"R#o&>F_p$\"*6oA#HF07$$\":#GGGGG G.s(oz(>F_p$\"*O')G<$F07$$\":kkkkkkkRj8\"**>F_p$\"*I)RrMF07$$\":YYYYYY '*e\\e--#F_p$\"*O:!yMF07$$\":GGGGGGGyN.9/#F_p$\"*\"*o1Z$F07$$\":#>>>>> >p\"3$p$3#F_p$\"*bAgX$F07$$\":cbbbbbbb!G)f7#F_p$\"**zVTMF07$$\":$GGGGG GG`Ac5AF_p$\"*`aCT$F07$$\":55555555qT^H#F_p$\"*wRPQ$F07$$\":kkkkkkkkf+ VY#F_p$\"*E_&GLF07$$\":>>>>>>>>\\fMj#F_p$\"*e>sH$F07$$\":YXXXXX&H[pj]Nk4$F_p$\"*stZo%F 07$$\":xwwwwww,BC&QJF_p$\"*a(4lYF07$$\":/.....`lo,FA$F_p$\"*o%*fi%F07$ $\":IHHHHHHH9zoI$F_p$\"*i]se%F07$$\":!)zzzzzzzplHY$F_p$\"*Czv^%F07$$\" :JIIIIIIID_!>OF_p$\"*/4aY%F07$$\":#33333333)Q^x$F_p$\"*3y?_%F07$$\":KJ JJJJJJOD7$RF_p$\"*/5=1&F07$$\":\\[[[[[[)4+jrRF_p$\"*'*zQR&F07$$\":mlll lllllM?,%F_p$\"*>l)*p&F07$$\":$GGGGGGG.$RC0%F_p$\"*a\")on&F07$$\":++++ ++++&R%G4%F_p$\"*e!*Rl&F07$$\":MMMMMMMMC`O<%F_p$\"*P&[3cF07$$\":pooooo oo`iWD%F_p$\"*!>NjbF07$$\":QPPPPPPP7\"3;WF_p$\"*YUYZ&F07$$\":11111111r *pxXF_p$\"*O;dR&F07$$\":\"3333333e?*>\"\\F_p$\"*_!etdF07$$\":cbbbbbbbS %GY_F_p$\"*rKvH'F07$$\":&RRRRRRRRX*GT&F_p$\"*h')Q>'F07$$\":IKKKKKKKn/& zbF_p$\"*Q:)*4'F07$$\":]^^^^^^,u4Gm&F_p$\"*q@'pgF07$$\":qqqqqqqq![6YdF _p$\"*;tX2'F07$$\":!**)*)*)*)*)*)R()>%HeF_p$\"*vP%ehF07$$\":44444444% \\s7fF_p$\"**>w0kF07$$\":vuuuuuuu%4C)f'F_p$\"*O#y6mF07$$\":NLLLLLLLQU, !pF_p$\"*vOd\"oF07$$\":#>>>>>>>>Q/-sF_p$\"*8\\,D(F07$$\":+++++++]dy>P( F_p$\"*=;\"GrF07$$\":53333333L8>a(F_p$\"*Yk<,(F07$$\"::777777(32)oi(F_ p$\"*-yH'pF07$$\":?;;;;;;m3[=r(F_p$\"*)G!\\$pF07$$\":?=====obxJVv(F_p$ \"*\")Gl$pF07$$\":???????XY:oz(F_p$\"*?`d&pF07$$\":?AAAAAsM:*HRyF_p$\" *#3p+qF07$$\":DCCCCCCC%Gy\")yF_p$\"*>#\\#3(F07$$\":]XXXXXXX5*4nzF_p$\" *4uSU(F07$$\":qmmmmmmmO:C0)F_p$\"+dD<5w!#D7$$\":!zyyyyyyG;tP\")F_p$\"+ ;;_XvFfel7$$\":54444444*y/B#)F_p$\"+_5U\"[(Ffel7$$\":]^^^^^^^T!o$R)F_p $\"+WT/btFfel7$$\":&RRRRRRRRHJk&)F_p$\"+mAZNsFfel7$$\":qmmmmmmm'4`Y')F _p$\"+z:R)=(Ffel7$$\":SRRRRRRR**[(G()F_p$\"+mw(G;(Ffel7$$\":vvvvvvvv+e )p()F_p$\"+(pCf=(Ffel7$$\":][[[[ [[[.w?&))F_p$\"+I(Q1B(Ffel7$$\":&[[[[[[[[]=$*))F_p$\"+3*G+J(Ffel7$$\": gddddddd2.a(*)F_p$\"+20xIwFfel7$$\":IIIIIIII5@w0*F_p$\"+H\"eBu(Ffel7$$ \":0.......8R)R\"*F_p$\"+rH'*ywFfel7$$\":wvvvvvvv:d?A*F_p$\"+4z3;wFfel 7$$\":STTTTTT\"RXq?&*F_p$\"+!e=UR(Ffel7$$\":32222222#>N>)*F_p$\"+n/:0t Ffel7$$\":vvvvvvDj2K\"3**F_p$\"+U2LSuFfel7$$\":XWWWWWW>B7p***F_p$\"+*p g#4yFfel7$$\":JJJJJJc(Q#p&35Ffel$\"+-JshxFfel7$$\":=======VDZu,\"Ffel$ \"+#\\>Jp(Ffel7$$\":#>>>>>>W&G._.\"Ffel$\"+)=&ydvFfel7$$\":mlllllllJfH 0\"Ffel$\"+([aqU(Ffel7$$\":yxxxxxx-;![g5Ffel$\"+gK!oP(Ffel7$$\":!**)*) *)*)*)*)R+,!o5Ffel$\"+]^+OtFfel7$$\":-------x%=_v5Ffel$\"+_$))eJ(Ffel7 $$\":99999999pUI3\"Ffel$\"+)3EuFfel7$$\":YXXXXXX&*H9J=\"Ffel$\"+r^\\\"H(Ffel 7$$\":feeeeeL-Be3>\"Ffel$\"+([P_Q(Ffel7$$\":srrrrr@4;-')>\"Ffel$\"+#eR Ri(Ffel7$$\":+++++v$fV\"Q0?\"Ffel$\"+hY)fo(Ffel7$$\":GGGGGyli7uC?\"Ffe l$\"+B\">6n(Ffel7$$\":ccccc\"y$*3,T/7Ffel$\"+MBGcwFfel7$$\":&[[[[[)4;4 Yj?\"Ffel$\"+VUZTwFfel7$$\":UTTTT\"R&p0=-@\"Ffel$\"+IR%>h(Ffel7$$\":)z zzzzzHA+497Ffel$\"+Wx_#e(Ffel7$$\":CCCCCCuO)ydH7Ffel$\"+3b+muFfel7$$\" :^]]]]]]]ul]C\"Ffel$\"+qBq^tFfel7$$\":CCCCCCuOQe?E\"Ffel$\"+gBNNsFfel7 $$\":)zzzzzzHA50z7Ffel$\"+meovrFfel7$$\":srrrrr@4mVgH\"Ffel$\"+eD$))Q( Ffel7$$\":YXXXXXX&*HOIJ\"Ffel$\"+>ftRuFfel7$$\":/.......K)RX8Ffel$\"+x %eK?(Ffel7$$\":hgggggg5MgxP\"Ffel$\"+lI6GqFfel7$$\":UTTTTT\"*y_VYR\"Ff el$\"+?&yg<(Ffel7$$\":AAAAAAAZrE:T\"Ffel$\"+keefsFfel7$$\":......`:!*4 %G9Ffel$\"+_b0QrFfel7$$\":%QQQQQQQ)3$HX9Ffel$\"+1@$*=qFfel7$$\":000000 bU7(zg9Ffel$\"+&[%)f\"pFfel7$$\":EEEEEEE,;,jZ\"Ffel$\"+9E%f%oFfel7$$\" :[ZZZZZ(*f>0=\\\"Ffel$\"+$*fD>pFfel7$$\":poooooo=B4t]\"Ffel$\"+fnemqFf el7$$\":)yyyyyyGEcvS:Ffel$\"+p!yU$oFfel7$$\":22222222--Ud\"Ffel$\"+!** z'QmFfel7$$\":))))))))))))QEss:f\"Ffel$\"+(Qx'*o'Ffel7$$\":qqqqqqqXUV* 3;Ffel$\"+&[%*z!oFfel7$$\":______-l79ji\"Ffel$\"+=%e2p'Ffel7$$\":MMMMM MM%G[oV;Ffel$\"+^JwvlFfel7$$\":IIIIIIIb21)e;Ffel$\"+PS8!['Ffel7$$\":EE EEEEEEKFRn\"Ffel$\"+R8I0kFfel7$$\":CCCCCCuh%z[\"o\"Ffel$\"+@`7$R'Ffel7 $$\":AAAAAAA(p&[!*o\"Ffel$\"+suW?kFfel7$$\":??????qK>4mp\"Ffel$\"+D*)[ ?lFfel7$$\":=======o\")pTq\"Ffel$\"+C(G]d'Ffel7$$\":mllllll!RD$ot\"Ffe l$\"+vRyjjFfel7$$\":88888888E&\\pA'Ffel7$$\":WWWWWWW%\\!RK!=Ffel$\"+=*fNI'Ffel7$ $\":5555555N%46?=Ffel$\"+f#*4)>'Ffel7$$\":wvvvvvvv$G)p$=Ffel$\"+^SW%4' Ffel7$$\":=======Vp)[`=Ffel$\"+jxx&*fFfel7$$\":hgggggg5b%**p=Ffel$\"+C (=$4fFfel7$$\":#======VzuCy=Ffel$\"+@j$H)eFfel7$$\":/......yS+l)=Ffel$ \"+M(*)e)eFfel7$$\":DCCCCCuhL`Z*=Ffel$\"+hL6ZfFfel7$$\":YXXXXXXXE1I!>F fel$\"+nV>?gFfel7$$\":tssssssZzY\\$>Ffel$\"+.r'4$eFfel7$$\":+++++++]K( )o'>Ffel$\"+IXMbcFfel7$$\":_^^^^^wKX`d(>Ffel$\"+S.O@cFfel7$$\":......` :e>Y)>Ffel$\"+gC47cFfel7$$\":aaaaaaH)4d[$*>Ffel$\"+zINbcFfel7$$\":1111 111\"Q=N-?Ffel$\"+42zNdFfel7$$\":444444fY4%3??Ffel$\"+\"zy\\j&Ffel7$$ \":77777777N;y.#Ffel$\"+Z(zf`&Ffel7$$\":**)*)*)*)*)*)Rs&\\P0#Ffel$\"+8 kS\\aFfel7$$\":'eeeeeee$z#op?Ffel$\"+RzNs`Ffel7$$\":IHHHHHzTS\\w2#Ffel $\"++4MY`Ffel7$$\":tssssssZ,;c3#Ffel$\"+XZZT`Ffel7$$\":;;;;;;m`i#e$4#F fel$\"+`8%yP&Ffel7$$\":gfffffffB\\:5#Ffel$\"+VAo^aFfel7$$\":]\\\\\\\\ \\\\\\Ytb8#Ffel$\"+\\XLp_Ffel7$$\":SRRRRRRRp(fp@Ffel$\"+'=>?5&Ffel7$$ \":IHHHHHagW0t<#Ffel$\"+yG+x]Ffel7$$\":?>>>>>p\")>8]=#Ffel$\"+FwIp]Ffe l7$$\":544444%G]4s#>#Ffel$\"+,Yw%4&Ffel7$$\":+**)*)*)*)*)R-(G/?#Ffel$ \"+y_Br^Ffel7$$\":!)yyyyyGm?We@#Ffel$\"+)QH@4&Ffel7$$\":feeeeee3rf7B#F fel$\"+Y'RU,&Ffel7$$\":CCCCCCCCes')H#Ffel$\"+21\"Q([Ffel7$$\":#>>>>>>p 6xQIBFfel$\"+5H(\\u%Ffel7$$\":gffffff4%G5iBFfel$\"+ezY*f%Ffel7$$\":cbb bbbbIq!RqBFfel$\"+4^*pc%Ffel7$$\":_^^^^^^^cy'yBFfel$\"+,!yKa%Ffel7$$\" :[ZZZZZZsUmpQ#Ffel$\"+ke1QXFfel7$$\":WVVVVVV$*Ga_R#Ffel$\"+v'=&pXFfel7 $$\":VUUUUUn[DEtR#Ffel$\"+l$\\ke%Ffel7$$\":UTTTTT\"R?#)R*R#Ffel$\"+WuB 3YFfel7$$\":TSSSSS:f=q9S#Ffel$\"+.@!)3YFfel7$$\":SRRRRRR9:UNS#Ffel$\"+ 4FE*f%Ffel7$$\":QPPPPP([#3'o2CFfel$\"+;JC!e%Ffel7$$\":ONNNNNNN,I=T#Ffe l$\"+x@IhXFfel7$$\":KJJJJJJc(y6?CFfel$\"+]]lBXFfel7$$\":FFFFFFFxt0%GCF fel$\"+\\0K'[%Ffel7$$\":%QQQQQQQQ*GK\\#Ffel$\"+D'feH%Ffel7$$\":qpppppp >#G4hDFfel$\"+%4;;4%Ffel7$$\":xwwwwwww!QXEEFfel$\"+%y!=\")RFfel7$$\":f eeeeeee&[H$p#Ffel$\"+g=C)z$Ffel7$$\":9999999kU#efFFfel$\"+7Y*ph$Ffel7$ $\":%RRRRRRR>Q\\?GFfel$\"+2:jCNFfel7$$\":'fffffff\\^I!*GFfel$\"+K'p>L$ Ffel7$$\":cbbbbbbbBXF&HFfel$\"+r@,&>$Ffel7$$\":baaaaaa/R>$>IFfel$\"+!* G1)3$Ffel7$$\":kjjjjjj8NRI3$Ffel$\"+9;_9HFfel7$$\":!**)*)*)*)*)*)*y@z` JFfel$\",n0'z&y#F_p7$$\":SRRRRRR*G*e]@$Ffel$\",D))y+q#F_p7$$\":KJJJJJJ \"Qsf%G$Ffel$\",1T:\\`#F_p7$$\":zyyyyyyy&Q(zM$Ffel$\",HTKNV#F_p7$$\": \\[[[[[[)>zH_MX?#F_p7$$ \":wvvvvvvv4N_a$Ffel$\",d\"Rl4@F_p7$$\":[ZZZZZZZ\"y.6OFfel$\",K)>SE?F_ p7$$\":'fffffffWvzwOFfel$\",h3CF_p7$$\":jiiiiiiiN:Bu$Ffel$\",HkJ6#= F_p7$$\":srrrrrr@Yd_!QFfel$\",=rb/v\"F_p7$$\":cbbbbbbba*HtQFfel$\",_() =vj\"F_p7$$\":(ppppppp%e[\"QRFfel$\",T^]rc\"F_p7$$\":[ZZZZZZZ9*R1SFfel $\",rO`]\\\"F_p7$$\":onnnnnnnL#=oSFfel$\",:8ljS\"F_p7$$\":**)*)*)*)*)* )*)HVYOTFfel$\",(=IER8F_p7$$\":edddddd2em=?%Ffel$\",QOY$y7F_p7$$\":222 22222k4rE%Ffel$\",v)==)>\"F_p7$$\":(ppppppppiDNVFfel$\",*ROiQ6F_p7$$\" :yxxxxxxF!=-)R%Ffel$\",)3_3%3\"F_p7$$\":UTTTTTTTH-BY%Ffel$\",ixZ&>5F_p 7$$\":YXXXXXXXfvK`%Ffel$\"+!*))*)Q'*F_p7$$\":ZYYYYYYY5Ovf%Ffel$\"+cX(z 8*F_p7$$\":zyyyyyyG^aKm%Ffel$\"+O'**=e)F_p7$$\":jiiiiii7W)3IZFfel$\"+1 $>M7)F_p7$$\":?>>>>>>p#=_\"z%Ffel$\"+#[Ecn(F_p7$$\":nmmmmmmm#y+d[Ffel$ \"+&\\p5B(F_p7$$\":GFFFFFFFM&*>#\\Ffel$\"+\"42D&Ffel$\"+J<$*))\\F_p7$$\":00000000co'=`Ffel$\"+$Qd)fYF_p7$$\":?> >>>>>>\"H!pQ&Ffel$\"+]sv[VF_p7$$\":NMMMMMM%og)*[aFfel$\",DNvs2%F-7$$\" :IHHHHHHHi:\\^&Ffel$\",o5Q@z$F-7$$\":XWWWWWWWUc#zbFfel$\",pV&\\^NF-7$$ \":rqqqqqq?`-!\\cFfel$\",(>!f=G$F-7$$\":nmmmmmmmI_$4dFfel$\",<1======o@m'ydFfel$\",'*)36MGF-7$$\":9888888j$\\#Q%eFfel$\",6`U%3E F-7$$\":POOOOOOOHa$3fFfel$\",!ymY#Q#F-7$$\":YXXXXXXX)\\$H(fFfel$\",Nvl $HAF-7$$\":kjjjjjj8?.)QgFfel$\",7\"))RA?F-7$$\":GFFFFFFxSA'3hFfel$\",q 7*)z!=F-7$$\":JIIIIIIIyRE<'Ffel$\",s$=7\"p\"F-7$$\":xwwwwwwE:LeB'Ffel$ \",!H7\\4:F-7$$\":)zzzzzzzCeQ.jFfel$\",Kn5$>8F-7$$\":dcccccc1W\"zqjFfe l$\",'))z**G7F-7$$\":QPPPPPPP)>HJkFfel$\",a(42m5F-7$$\"#l!\"#$\"+G]f=* )F--%&COLORG6&%$RGBG$\"\"%!\"\"F($\"\"*F[`p-%'LEGENDG6#%6Enright-Verne r~schemeG-F$6%7i[lF'7$F+$\"#BF07$F2$\"&g@\"F07$F7$\"&r/*F07$F<$\"'ghYF 07$FA$\"'4u&*F07$FF$\"(!3k=F07$FK$\"()4mMF07$FP$\"(!o\">'F07$FU$\")x@m :F07$FZ$\")BqJOF07$Fin$\")@[y`F07$F^o$\")SONyF07$Fco$\"*W.V7\"F07$Fho$ \"*R$*3f\"F07$F]p$\"*iq$3=F07$Fcp$\"*2'*\\!=F07$Fhp$\"*!yi,=F07$F]q$\" *\"eE)z\"F07$Fbq$\"*mg:z\"F07$Fgq$\"*a!)[y\"F07$F\\r$\"*:'fr>Gs\"F07$F`s$\"*X!fc0E$F07$Fhu$\"*KgnC$F07$F]v$\"*&)eIB$F07$ Fbv$\"*MIe?$F07$Fgv$\"*?a)yJF07$F\\w$\"*\"f+FJF07$Faw$\"*KXv4$F07$Ffw$ \"*A=C7$F07$F[x$\"*D%GDKF07$F`x$\"*Y%f&\\$F07$Fex$\"*'4o*4%F07$Fjx$\"* @-ML%F07$F_y$\"*6w#QWF07$Fdy$\"*'e%*GWF07$Fiy$\"*AN'>WF07$F^z$\"*ks5S% F07$Fcz$\"*-)e#Q%F07$Fhz$\"*:`eM%F07$F][l$\"*Hb%4VF07$Fb[l$\"*(4+WUF07 $Fg[l$\"*#4(\\>%F07$F\\\\l$\"*(=8[UF07$Fa\\l$\"*]*4bZF07$Ff\\l$\"*X.r1 &F07$F[]l$\"*$Rja`F07$F`]l$\"*OUIL&F07$Fe]l$\"*'y`6`F07$Fj]l$\"*$))yo_ F07$F_^l$\"*+*QE_F07$Fd^l$\"*t^I9&F07$Fi^l$\"*A&*)o]F07$F^_l$\"*q3PU&F 07$Fc_l$\"*RAg\"fF07$Fh_l$\"*ca'=eF07$F]`l$\"*#>FIdF07$Fb`l$\"*D))=q&F 07$Fg`l$\"*5-lq&F07$F\\al$\"*HH_y&F07$Faal$\"*;@v,'F07$Ffal$\"*9?6@'F0 7$F[bl$\"*HgDS'F07$F`bl$\"*9%o5oF07$Febl$\"*\"z/'p'F07$Fjbl$\"*QVne'F0 7$F_cl$\"*>-4a'F07$Fdcl$\"*(>]9lF07$Ficl$\"*-3g^'F07$F^dl$\"*6SS`'F07$ Fcdl$\"*IAid'F07$Fhdl$\"*zKIl'F07$F]el$\"*=xQ(pF07$Fbel$\"+2&=([rFfel7 $Fhel$\"+wx)z3(Ffel7$F]fl$\"+USxFqFfel7$Fbfl$\"+%zf!4pFfel7$Fgfl$\"+i8 t'z'Ffel7$F\\gl$\"+(z\"\\_nFfel7$Fagl$\"+!*e\\GnFfel7$Ffgl$\"+8_BJnFfe l7$F[hl$\"+![%4]nFfel7$F`hl$\"+B^1#z'Ffel7$Fehl$\"+?ugmoFfel7$Fjhl$\"+ u*ey;(Ffel7$F_il$\"+b\"*ossFfel7$Fdil$\"+\"pRJ@(Ffel7$Fiil$\"+q(yS:(Ff el7$F^jl$\"+')[mXpFfel7$Fcjl$\"+iw*='oFfel7$Fhjl$\"++))z))pFfel7$F][m$ \"+\"f'HNtFfel7$Fb[m$\"+*)[k!H(Ffel7$Fg[m$\"+-]?EsFfel7$F\\\\m$\"+UV3* 4(Ffel7$Fa\\m$\"+bTGwpFfel7$Ff\\m$\"+]_2HpFfel7$F[]m$\"+zjt!*oFfel7$F` ]m$\"+6z!=(oFfel7$Fe]m$\"+v%RD*oFfel7$Fj]m$\"+BSV*)pFfel7$F_^m$\"+C0$ \\A(Ffel7$Fd^m$\"+QmJ;tFfel7$Fi^m$\"+XVq/tFfel7$F^_m$\"+kG)4H(Ffel7$Fc _m$\"+drGxsFfel7$Fh_m$\"+xG(*\\sFfel7$F]`m$\"+E6wAsFfel7$Fb`m$\"+]k@vp Ffel7$Fg`m$\"+BCe[oFfel7$F\\am$\"+MqaOpFfel7$Faam$\"+umlgrFfel7$Ffam$ \"+]*=*=sFfel7$F[bm$\"+pn&\\?(Ffel7$F`bm$\"+*e@5>(Ffel7$Febm$\"+jL6xrF fel7$Fjbm$\"+jvP\\rFfel7$F_cm$\"+a*[<7(Ffel7$Fdcm$\"+svI7qFfel7$Ficm$ \"+G&\\\\!pFfel7$F^dm$\"+rvl&z'Ffel7$Fcdm$\"+dhaRnFfel7$Fhdm$\"+WC`RpF fel7$F]em$\"+)y$G()pFfel7$Fbem$\"+zo=lnFfel7$Fgem$\"+(\\=1g'Ffel7$F\\f m$\"+g,PRnFfel7$Fafm$\"+?hp&Ffel7 $F^cn$\"+tx![6&Ffel7$Fccn$\"+W%G%Ffel7$Fhhn$\"+],7iUFfel7$F]in$\"+rE2dUFfel7$Fbin$\"+G/E'G%Ffe l7$Fgin$\"+%eD?I%Ffel7$F\\jn$\"+5xKAVFfel7$Fajn$\"+(\\9GK%Ffel7$Ffjn$ \"+Yq'QJ%Ffel7$F[[o$\"+mw-'H%Ffel7$F`[o$\"+g?EyUFfel7$Fe[o$\"+H5&HC%Ff el7$Fj[o$\"+7K$z?%Ffel7$F_\\o$\"+S(o(GSFfel7$Fd\\o$\"+iT\"o$QFfel7$Fi \\o$\"+-:HKPFfel7$F^]o$\"+;v?gNFfel7$Fc]o$\"+1;')*Q$Ffel7$Fh]o$\"+!e:A I$Ffel7$F]^o$\"+#)*H77$Ffel7$Fb^o$\"+#exA*HFfel7$Fg^o$\"+J'44*GFfel7$F \\_o$\"+Yt>GFFfel7$Fa_o$\",d<6ng#F_p7$Ff_o$\",l+z^_#F_p7$F[`o$\",;3%Rq BF_p7$F``o$\",zM^XF#F_p7$Fe`o$\",cy$\\!=#F_p7$Fj`o$\",pK)**e?F_p7$F_ao $\",xef!p>F_p7$Fdao$\",#Q\"z(*)=F_p7$Fiao$\",hI]Lx\"F_p7$F^bo$\",Hldnp \"F_p7$Fcbo$\",QY&>H;F_p7$Fhbo$\",AJrR_\"F_p7$F]co$\",,*p'oX\"F_p7$Fbc o$\",T6,!)Q\"F_p7$Fgco$\",bM,cI\"F_p7$F\\do$\",F-7$Fh\\p$\",N<%GD=F-7$F]]p$\",7SOwh\"F-7$Fb]p$\",qbT:S \"F-7$Fg]p$\",s1A#38F-7$F\\^p$\",!zVEB6F-7$Fa^p$\"+Kq4*G*F-7$Ff^p$\"+' Qxri)F-7$F[_p$\"+a-(y$pF-7$F`_p$\"+GSYW^F--Ff_p6&Fh_pF(F`_pF(-F_`p6#%6 Verner's~Maple~schemeG-F$6%7i[lF'7$F+$\"#@F07$F2$\"&T3\"F07$F7$\"&E1)F 07$F<$\"'g_TF07$FA$\"'(o_)F07$FF$\"(P)f;F07$FK$\"(yc3$F07$FP$\"(c4^&F0 7$FU$\")_^$R\"F07$FZ$\")J1IKF07$Fin$\")xy#y%F07$F^o$\")QGmpF07$Fco$\") k:%***F07$Fho$\"*E=RT\"F07$F]p$\"*0\"42;F07$Fcp$\"*5#4/;F07$Fhp$\"*u)4 ,;F07$F]q$\"*'46)f\"F07$Fbq$\"*7_@f\"F07$Fgq$\"*_:ie\"F07$F\\r$\"*v4Wd \"F07$Far$\"*l2Fc\"F07$Ffr$\"*BBy`\"F07$F[s$\"*:*4J:F07$F`s$\"*9l6c\"F 07$Fes$\"*'z`e;F07$Fjs$\"*y=@!>F07$F_t$\"*9X)RCF07$Fdt$\"*%>0\\EF07$Fi t$\"*dF#)*GF07$F^u$\"*h\\P!HF07$Fcu$\"*;;w*GF07$Fhu$\"*5)Q&)GF07$F]v$ \"*l6K(GF07$Fbv$\"*u8!\\GF07$Fgv$\"*=S]#GF07$F\\w$\"*;m*yFF07$Faw$\"*K ?Gv#F07$Ffw$\"*Zy\\x#F07$F[x$\"*^zk'GF07$F`x$\"*-yn5$F07$Fex$\"*JKNk$F 07$Fjx$\"*tZ6&QF07$F_y$\"*d(HWRF07$Fdy$\"*x0g$RF07$Fiy$\"*SJx#RF07$F^z $\"*%[B6RF07$Fcz$\"*a2[*QF07$Fhz$\"*Dh@'QF07$F][l$\"*c9)HQF07$Fb[l$\"* H[;x$F07$Fg[l$\"*F-\"GPF07$F\\\\l$\"*>^ax$F07$Fa\\l$\"*@!3EUF07$Ff\\l$ \"*RzK]%F07$F[]l$\"*j%oeZF07$F`]l$\"*6'\\RZF07$Fe]l$\"*)\\Q?ZF07$Fj]l$ \"*v$R#o%F07$F_^l$\"*(GrWYF07$Fd^l$\"*'=lqXF07$Fi^l$\"*yjZ]%F07$F^_l$ \"*!oG?[F07$Fc_l$\"*U?wD&F07$Fh_l$\"*y*3r^F07$F]`l$\"*`dD4&F07$Fb`l$\" *Wet1&F07$Fg`l$\"*$y]r]F07$F\\al$\"*bF07$F[bl$\"*([B!p&F07$F`bl$\"*N-G0'F07$Febl$\"*+B4&fF07$Fjbl$\"*( ))y`eF07$F_cl$\"*:kI\"eF07$Fdcl$\"*%[j*y&F07$Ficl$\"*S)*4z&F07$F^dl$\" *%R02eF07$Fcdl$\"**QdWeF07$Fhdl$\"*BjG\"fF07$F]el$\"*`wz>'F07$Fbel$\"+ (\\#G`jFfel7$Fhel$\"+m#4$*H'Ffel7$F]fl$\"+-bzXiFfel7$Fbfl$\"+k5HShFfel 7$Fgfl$\"+9.ZSgFfel7$F\\gl$\"+F,<,gFfel7$Fagl$\"+M\"y)zfFfel7$Ffgl$\"+ vrL#)fFfel7$F[hl$\"+dj7**fFfel7$F`hl$\"+TdXOgFfel7$Fehl$\"+/fs-hFfel7$ Fjhl$\"+&*HUqjFfel7$F_il$\"+;S`jkFfel7$Fdil$\"+\"**41T'Ffel7$Fiil$\"+d ,7ejFfel7$F^jl$\"+x))*G<'Ffel7$Fcjl$\"+!He&)4'Ffel7$Fhjl$\"+?/R6iFfel7 $F][m$\"+KGF>lFfel7$Fb[m$\"+e6ezkFfel7$Fg[m$\"+F+JAkFfel7$F\\\\m$\"+\\ 8L4jFfel7$Fa\\m$\"+5q>+iFfel7$Ff\\m$\"+A([#ehFfel7$F[]m$\"+WN>ChFfel7$ F`]m$\"+FQS2hFfel7$Fe]m$\"+cf(e7'Ffel7$Fj]m$\"+bU.7iFfel7$F_^m$\"+9BJ@ kFfel7$Fd^m$\"+[H]-lFfel7$Fi^m$\"+5;=#\\'Ffel7$F^_m$\"+3k)*zkFfel7$Fc_ m$\"+8T\"yY'Ffel7$Fh_m$\"+x!QNW'Ffel7$F]`m$\"+lJN>kFfel7$Fb`m$\"+ScM*> 'Ffel7$Fg`m$\"+:A!p3'Ffel7$F\\am$\"+Wx7lhFfel7$Faam$\"+ufHkjFfel7$Ffam $\"+c(fgT'Ffel7$F[bm$\"+#R]OS'Ffel7$F`bm$\"+F]E\"R'Ffel7$Febm$\"+>O!*y jFfel7$Fjbm$\"+$[_UN'Ffel7$F_cm$\"+n'eFfel7$F\\f m$\"+c79!*fFfel7$Fafm$\"+$)pufgFfel7$Fffm$\"+yFIefFfel7$F[gm$\"+%zn)ee Ffel7$F`gm$\"+Od$Hx&Ffel7$Fegm$\"+rdY9dFfel7$Fjgm$\"+DGfvdFfel7$F_hm$ \"+n)G%)*eFfel7$Fdhm$\"+PD_/dFfel7$Fihm$\"+RQDTbFfel7$F^im$\"+BCu$e&Ff el7$Fcim$\"+y`M#o&Ffel7$Fhim$\"+)Q$\\%e&Ffel7$F]jm$\"+Z<^)[&Ffel7$Fbjm $\"+nSp3aFfel7$Fgjm$\"+mxAY`Ffel7$F\\[n$\"+Q\"[gL&Ffel7$Fa[n$\"+Tv!)e` Ffel7$Ff[n$\"+#G*>UaFfel7$F[\\n$\"+'*pj([&Ffel7$F`\\n$\"+X$G8J&Ffel7$F e\\n$\"+'\\4<:&Ffel7$Fj\\n$\"+7B\\G^Ffel7$F_]n$\"+'38G8&Ffel7$Fd]n$\"+ e[w#>&Ffel7$Fi]n$\"+&fz2E&Ffel7$F^^n$\"+)*\\ws^Ffel7$Fc^n$\"+%=di3&Ffe l7$Fh^n$\"+JF\"R+&Ffel7$F]_n$\"+Q.vJ\\Ffel7$Fb_n$\"+'y;(4\\Ffel7$Fg_n$ \"+S#Q@\"\\Ffel7$F\\`n$\"+n^7j\\Ffel7$Fa`n$\"+&[#)R-&Ffel7$Ff`n$\"+O#o g'[Ffel7$F[an$\"+#f-&>ZFfel7$F`an$\"+>(G6p%Ffel7$Fean$\"+^VN$o%Ffel7$F jan$\"+A%\\$>ZFfel7$F_bn$\"+z'4jy%Ffel7$Fdbn$\"+$y&=-ZFfel7$Fibn$\"+^Y d>YFfel7$F^cn$\"+YdWFfel7$Fbdn$\"+%=Ku[%Ffel7$Fgdn$\"+\"Gv)[XFfel7$F\\en$\"+MYs' R%Ffel7$Faen$\"+un5dUFfel7$Ffen$\"+mU@OUFfel7$F[fn$\"+pruHUFfel7$F`fn$ \"+)\\%)3D%Ffel7$Fefn$\"+u'*[9VFfel7$Fjfn$\"+q'*[[UFfel7$F_gn$\"+CV]$= %Ffel7$Fdgn$\"+rZ)f1%Ffel7$Fign$\"+:WXeRFfel7$F^hn$\"+:Q1PQFfel7$Fchn$ \"+=N'*4QFfel7$Fhhn$\"+7*\\,z$Ffel7$F]in$\"+TEt&y$Ffel7$Fbin$\"+3T\"= \"QFfel7$Fgin$\"+&)e(e#QFfel7$F\\jn$\"+Wr(R%QFfel7$Fajn$\"+uUUWQFfel7$ Ffjn$\"+0qYOQFfel7$F[[o$\"+W=g?QFfel7$F`[o$\"+$H-[!QFfel7$Fe[o$\"+2!*R tPFfel7$Fj[o$\"+&\\cAu$Ffel7$F_\\o$\"+#y]Je$Ffel7$Fd\\o$\"+A)zDT$Ffel7 $Fi\\o$\"+qi+?LFfel7$F^]o$\"+!f%=nJFfel7$Fc]o$\"+d#=e,$Ffel7$Fh]o$\"+h hGQHFfel7$F]^o$\"+ivXxFFfel7$Fb^o$\"+Mg'Hm#Ffel7$Fg^o$\"+$z\\Kd#Ffel7$ F\\_o$\"+$))G&GCFfel7$Fa_o$\",57h-#F_p7$Fe`o$\",'**o#H%>F_p7$Fj`o$\",fUNZ$=F_p7$F_ ao$\",20Q^v\"F_p7$Fdao$\",i$o7&o\"F_p7$Fiao$\",T>v8e\"F_p7$F^bo$\",fqr O^\"F_p7$Fcbo$\",y<:TX\"F_p7$Fhbo$\",s?Z-O\"F_p7$F]co$\",hDE5I\"F_p7$F bco$\",rW*HS7F_p7$Fgco$\",N8&pm6F_p7$F\\do$\",x#z<56F_p7$Fado$\",Q#4ue 5F_p7$Ffdo$\"+:L@B**F_p7$F[eo$\"+z[z?%*F_p7$F`eo$\"+=-8h*)F_p7$Feeo$\" +_7&fU)F_p7$Fjeo$\"+I$4g&zF_p7$F_fo$\"+w)>R`(F_p7$Fdfo$\"+EGBtqF_p7$Fi fo$\"+c6n%o'F_p7$F^go$\"+7`)4J'F_p7$Fcgo$\"+voNRfF_p7$Fhgo$\"+(o\">EF- 7$Fd[p$\",d)z7F-7$F\\^p$\",!\\\\`@6F-7$Fa^p$\"+K$3Wb*F-7$Ff^p$\"+')y9))))F -7$F[_p$\"+a)yQX(F-7$F`_p$\"+G$pj#fF--%'COLOURG6&Fh_p$\")#)eqk!\")$\") )eqk\"FjevF[fv-F_`p6#%Ha~modification~of~Verner's~Maple~schemeG-F$6%7i [lF'7$F+$\"#>F07$F2$\"%'***F07$F7$\"&RV(F07$F<$\"'cGQF07$FA$\"'KhyF07$ FF$\"(X-`\"F07$FK$\"((pWGF07$FP$\"(\\/3&F07$FU$\")Qg%G\"F07$FZ$\")Q\\x HF07$Fin$\")kr3WF07$F^o$\")sI@kF07$Fco$\")=77#*F07$Fho$\"*2aKI\"F07$F] p$\"*L'H\"[\"F07$Fcp$\"*7K&y9F07$Fhp$\"*1tdZ\"F07$F]q$\"*:>IZ\"F07$Fbq $\"*uEvY\"F07$Fgq$\"*$[0i9F07$F\\r$\"*:t6X\"F07$Far$\"*.(QS9F07$Ffr$\" *t]uT\"F07$F[s$\"*2c7T\"F07$F`s$\"*>t*Q9F07$Fes$\"*xI(G:F07$Fjs$\"*#)[ Kv\"F07$F_t$\"*\"o')[AF07$Fdt$\"*Y&oTCF07$Fit$\"*5R8n#F07$F^u$\"*4Gkn# F07$Fcu$\"*![xqEF07$Fhu$\"*-/&fEF07$F]v$\"*!3G[EF07$Fbv$\"*Cxfi#F07$Fg v$\"*Z!)Qg#F07$F\\w$\"*o89c#F07$Faw$\"*eN&F07$Fcdl$\"*.ClQ&F07$Fhdl$ \"*jX%\\aF07$F]el$\"*`W@r&F07$Fbel$\"+2+BbeFfel7$Fhel$\"+'*y[0eFfel7$F ]fl$\"+K#phv&Ffel7$Fbfl$\"+/c$*ecFfel7$Fgfl$\"+#**Rpc&Ffel7$F\\gl$\"+4 )>2`&Ffel7$Fagl$\"+FL46bFfel7$Ffgl$\"+2_N8bFfel7$F[hl$\"+d0#)GbFfel7$F `hl$\"+M1@jbFfel7$Fehl$\"+\\XECcFfel7$Fjhl$\"+s0*3(eFfel7$F_il$\"+U,mc fFfel7$Fdil$\"+[k)y!fFfel7$Fiil$\"+&*G^feFfel7$F^jl$\"+_m\"))o&Ffel7$F cjl$\"+b?G?cFfel7$Fhjl$\"+kn@CdFfel7$F][m$\"+5+$y+'Ffel7$Fb[m$\"+7hCrf Ffel7$Fg[m$\"+1!o%=fFfel7$F\\\\m$\"+6FN9eFfel7$Fa\\m$\"+'))zPr&Ffel7$F f\\m$\"+?;7vcFfel7$F[]m$\"+-_tVcFfel7$F`]m$\"+bVDGcFfel7$Fe]m$\"+$>c_k &Ffel7$Fj]m$\"+QhgCdFfel7$F_^m$\"+o*ft\"fFfel7$Fd^m$\"+F-9#*fFfel7$Fi^ m$\"+-#GE)fFfel7$F^_m$\"+o,RrfFfel7$Fc_m$\"+VKbFfel7$Fhdm$\"+yxO$o&Ffel7$F]em$\"+zJOAdFfel7$Fbem$\"+L PZSbFfel7$Fgem$\"+p-t0aFfel7$F\\fm$\"+D&*H>bFfel7$Fafm$\"+S0J$e&Ffel7$ Fffm$\"+(>U)*[&Ffel7$F[gm$\"+H\\A)R&Ffel7$F`gm$\"+)QY!>`Ffel7$Fegm$\"+ E5:l_Ffel7$Fjgm$\"+R![8K&Ffel7$F_hm$\"+t%QVV&Ffel7$Fdhm$\"+\"o)ob_Ffel 7$Fihm$\"+hVC0^Ffel7$F^im$\"+3@DW^Ffel7$Fcim$\"+q7*[B&Ffel7$Fhim$\"+&f WZ9&Ffel7$F]jm$\"+@3Kc]Ffel7$Fbjm$\"+'z&y#)\\Ffel7$Fgjm$\"+6z@D\\Ffel7 $F\\[n$\"+:_!e\"\\Ffel7$Fa[n$\"+:qpO\\Ffel7$Ff[n$\"+S\"oL,&Ffel7$F[\\n $\"+N!>^0&Ffel7$F`\\n$\"+Fkq#*[Ffel7$Fe\\n$\"+rRlXZFfel7$Fj\\n$\"+V%RU s%Ffel7$F_]n$\"+&y_\"GZFfel7$Fd]n$\"+K&GKy%Ffel7$Fi]n$\"+#y:d%[Ffel7$F ^^n$\"+U`kkZFfel7$Fc^n$\"+bF'\\o%Ffel7$Fh^n$\"+`P64YFfel7$F]_n$\"+]&HE a%Ffel7$Fb_n$\"+$G/B_%Ffel7$Fg_n$\"+?GYCXFfel7$F\\`n$\"+<(o7d%Ffel7$Fa `n$\"+N`9FYFfel7$Ff`n$\"+uWq\"[%Ffel7$F[an$\"+3SqYVFfel7$F`an$\"+Nea?V Ffel7$Fean$\"+D!>LJ%Ffel7$Fjan$\"+*[8jM%Ffel7$F_bn$\"+%=fxS%Ffel7$Fdbn $\"+p')GIVFfel7$Fibn$\"+.7@aUFfel7$F^cn$\"+b.o(=%Ffel7$Fccn$\"+*pY%GTF fel7$Fhcn$\"+_]S3TFfel7$F]dn$\"+&[\\X5%Ffel7$Fbdn$\"+?)QA8%Ffel7$Fgdn$ \"+6ce)=%Ffel7$F\\en$\"+xe[[SFfel7$Faen$\"+_m!*>RFfel7$Ffen$\"+)*pj+RF fel7$F[fn$\"+A-h%*QFfel7$F`fn$\"+KE#R\"RFfel7$Fefn$\"+yY@sRFfel7$Fjfn$ \"+b0X6RFfel7$F_gn$\"+!e?;&QFfel7$Fdgn$\"+V)4Hu$Ffel7$Fign$\"+..'Qk$Ff el7$F^hn$\"+,#4@`$Ffel7$Fchn$\"+8e92NFfel7$Fhhn$\"+SK'))[$Ffel7$F]in$ \"+AYp%[$Ffel7$Fbin$\"+,8[3NFfel7$Fgin$\"+-+M@NFfel7$F\\jn$\"+24!z`$Ff el7$Fajn$\"+H/GQNFfel7$Ffjn$\"+Bo&4`$Ffel7$F[[o$\"+c]N;NFfel7$F`[o$\"+ uO\"=]$Ffel7$Fe[o$\"+K6\"HZ$Ffel7$Fj[o$\"+\"f[UW$Ffel7$F_\\o$\"+EIZ(H$ Ffel7$Fd\\o$\"+U.@SJFfel7$Fi\\o$\"+gIPaIFfel7$F^]o$\"+l7S8HFfel7$Fc]o$ \"+Vr%Qx#Ffel7$Fh]o$\"+eW\"=q#Ffel7$F]^o$\"+u>i`DFfel7$Fb^o$\"+&=8zW#F fel7$Fg^o$\"+\\/lkBFfel7$F\\_o$\"+D\\]JAFfel7$Fa_o$\",PK^=8#F_p7$Ff_o$ \",lE8[1#F_p7$F[`o$\",O[v\"Q>F_p7$F``o$\",4J>&f=F_p7$Fe`o$\",YdWAy\"F_ p7$Fj`o$\",H72Ho\"F_p7$F_ao$\",(zn/4;F_p7$Fdao$\",7KdQa\"F_p7$Fiao$\", Tg1([9F_p7$F^bo$\",f6ldQ\"F_p7$Fcbo$\",eec,L\"F_p7$Fhbo$\",APAUC\"F_p7 $F]co$\",,vJ!*=\"F_p7$Fbco$\",TVtB8\"F_p7$Fgco$\",l\"f8l5F_p7$F\\do$\" ,nz6C,\"F_p7$Fado$\"+G*3Ek*F_p7$Ffdo$\"+0HLP!*F_p7$F[eo$\"+p0nn&)F_p7$ F`eo$\"+)eD&Q\")F_p7$Feeo$\"+-OA]wF_p7$Fjeo$\"+5(G/@(F_p7$F_fo$\"+1$\\ l\"oF_p7$Fdfo$\"+'R,oR'F_p7$Fifo$\"+'[f8.'F_p7$F^go$\"+K!fto&F_p7$Fcgo $\"+b8UW`F_p7$Fhgo$\"+F-7$F^\\p$ \",6tYhs\"F-7$Fc\\p$\",![DHO:F-7$Fh\\p$\",NXc]V\"F-7$F]]p$\",78,DE\"F- 7$Fb]p$\",qBuF3\"F-7$Fg]p$\",s0))*45F-7$F\\^p$\"+!pD\"e&)F-7$Fa^p$\"+K &4r$pF-7$Ff^p$\"+'oLbV'F-7$F[_p$\"+aF6F07$F7$\"&qK)F07$F<$\"''yG%F07$FA$\"'p.))F07$FF$\"(VNr\"F07$FK$ \"($=&=$F07$FP$\"(t!)o&F07$FU$\")(Q!Q9F07$FZ$\")slKLF07$Fin$\").DM\\F0 7$F^o$\")+C'=(F07$Fco$\"*'y(3.\"F07$Fho$\"*#oHe9F07$F]p$\"*]vul\"F07$F cp$\"*_#Qa;F07$Fhp$\"*J&H^;F07$F]q$\"*'Q@[;F07$Fbq$\"*?o?k\"F07$Fgq$\" *\\Xfj\"F07$F\\r$\"*fpPi\"F07$Far$\"*l+$)R F07$F][l$\"*iI)\\RF07$Fb[l$\"*3V)*)QF07$Fg[l$\"*xX\\%QF07$F\\\\l$\"*2G Q*QF07$Fa\\l$\"*I-'eVF07$Ff\\l$\"*yNWk%F07$F[]l$\"*dqx!\\F07$F`]l$\"*! 4)z)[F07$Fe]l$\"*.r#o[F07$Fj]l$\"*c*3H[F07$F_^l$\"*R'F07$Fbel$\"+Zh1_lFfel7$Fhel$\"+wTS'\\'Ffel7$F]fl$\"+_g @TkFfel7$Fbfl$\"+%z5CL'Ffel7$Fgfl$\"+e2ZHiFfel7$F\\gl$\"+Hz%*)='Ffel7$ Fagl$\"+XD+nhFfel7$Ffgl$\"+@maphFfel7$F[hl$\"+t(oo='Ffel7$F`hl$\"+J0PD iFfel7$Fehl$\"+Z-r$H'Ffel7$Fjhl$\"+/>rplFfel7$F_il$\"+0UolmFfel7$Fdil$ \"+a\\56mFfel7$Fiil$\"+xM(pb'Ffel7$F^jl$\"+.8'fO'Ffel7$Fcjl$\"+m!G$*G' Ffel7$Fhjl$\"+'fzcS'Ffel7$F][m$\"+<:3BnFfel7$Fb[m$\"+%yT@o'Ffel7$Fg[m$ \"+z-3BmFfel7$F\\\\m$\"+0)pl]'Ffel7$Fa\\m$\"+0d-%R'Ffel7$Ff\\m$\"+>%p2 N'Ffel7$F[]m$\"+/hl:jFfel7$F`]m$\"+*)4N)H'Ffel7$Fe]m$\"+C!3uJ'Ffel7$Fj ]m$\"+NuC1kFfel7$F_^m$\"+C1*>i'Ffel7$Fd^m$\"+S,o0nFfel7$Fi^m$\"+$[N]p' Ffel7$F^_m$\"+K#fCo'Ffel7$Fc_m$\"+-m!*pmFfel7$Fh_m$\"+S?([k'Ffel7$F]`m $\"+\\9$*>mFfel7$Fb`m$\"+J/0$R'Ffel7$Fg`m$\"+La6xiFfel7$F\\am$\"+s&oxN 'Ffel7$Faam$\"+2U2jlFfel7$Ffam$\"+>eU;mFfel7$F[bm$\"+E*GOg'Ffel7$F`bm$ \"+!yc3f'Ffel7$Febm$\"+R$4\"ylFfel7$Fjbm$\"+x$)o_lFfel7$F_cm$\"+gcOFlF fel7$Fdcm$\"+M$eqU'Ffel7$Ficm$\"+g>mGjFfel7$F^dm$\"+N%4&GiFfel7$Fcdm$ \"+Pv:xhFfel7$Fhdm$\"+U^ZgjFfel7$F]em$\"+6w9/kFfel7$Fbem$\"+Bxe+iFfel7 $Fgem$\"+xF$)\\gFfel7$F\\fm$\"+UN0xhFfel7$Fafm$\"+7wu[iFfel7$Fffm$\"+= %RT9'Ffel7$F[gm$\"+PMgTgFfel7$F`gm$\"+dQ*H&fFfel7$Fegm$\"+ukq#*eFfel7$ Fjgm$\"+VnqbfFfel7$F_hm$\"+zXE#3'Ffel7$Fdhm$\"+<\\J#)eFfel7$Fihm$\"+NR 'Rr&Ffel7$F^im$\"+g0uddFfel7$Fcim$\"+R]IfeFfel7$Fhim$\"+WdSedFfel7$F]j m$\"+D_VfcFfel7$Fbjm$\"+1L8xbFfel7$Fgjm$\"+>Ws7bFfel7$F\\[n$\"+qQA-bFf el7$Fa[n$\"+E/nDbFfel7$Ff[n$\"+!H!f6cFfel7$F[\\n$\"+jGQecFfel7$F`\\n$ \"+s%)ewaFfel7$Fe\\n$\"+;e+7`Ffel7$Fj\\n$\"+DV1)G&Ffel7$F_]n$\"+qO]#H& Ffel7$Fd]n$\"+7&3af%Ffel7$Fbdn$\"+B!Hli%Ffel7$Fgdn$\"+ngx*o%Ffel7$F\\en$ \"+(e7H`%Ffel7$Faen$\"+k(o*)Q%Ffel7$Ffen$\"+;JUnVFfel7$F[fn$\"+-mtgVFf el7$F`fn$\"+btZ#Q%Ffel7$Fefn$\"+())RzW%Ffel7$Fjfn$\"+d%)*)zVFfel7$F_gn $\"+sI!HJ%Ffel7$Fdgn$\"+*4y:>%Ffel7$Fign$\"+a(*p!3%Ffel7$F^hn$\"+[(fb& RFfel7$Fchn$\"+_(>w#RFfel7$Fhhn$\"+.V=2RFfel7$F]in$\"+C5g-RFfel7$Fbin$ \"+W2THRFfel7$Fgin$\"+QU(Q%RFfel7$F\\jn$\"+SV\\iRFfel7$Fajn$\"+wA%H'RF fel7$Ffjn$\"+'pRZ&RFfel7$F[[o$\"+NaQQRFfel7$F`[o$\"+2))4ARFfel7$Fe[o$ \"+.us*)QFfel7$Fj[o$\"+9[idQFfel7$F_\\o$\"+!>9Np$Ffel7$Fd\\o$\"+7CeU$Ffel7$F^]o$\"+(R?VE$Ffel7$Fc]o$\"+70?3JFfel7$Fh]o$ \"+IC1GIFfel7$F]^o$\"+7EBiGFfel7$Fb^o$\"++y4WFFfel7$Fg^o$\"+:8S^EFfel7 $F\\_o$\"+[NC-DFfel7$Fa_o$\",()fC5R#F_p7$Ff_o$\",0rRlJ#F_p7$F[`o$\",md LY<#F_p7$F``o$\",z$>$p3#F_p7$Fe`o$\",w)>(4+#F_p7$Fj`o$\",>(Q_*)=F_p7$F _ao$\",(\\]H2=F_p7$Fdao$\",7P3\\t\"F_p7$Fiao$\",\"*Gm!G;F_p7$F^bo$\",4 U%4e:F_p7$Fcbo$\",oUxk\\\"F_p7$Fhbo$\",A7d)*R\"F_p7$F]co$\",h4='Q8F_p7 $Fbco$\",h!z!eF\"F_p7$Fgco$\",&em3+7F_p7$F\\do$\",x#QjT6F_p7$Fado$\",3 '=R)3\"F_p7$Ffdo$\",&QJ5?5F_p7$F[eo$\"+**G=\"o*F_p7$F`eo$\"+)3ld?*F_p7 $Feeo$\"+#pZ`l)F_p7$Fjeo$\"+ge,p\")F_p7$F_fo$\"+OEbKxF_p7$Fdfo$\"+wV\" *esF_p7$Fifo$\"+;9NcoF_p7$F^go$\"+_\\DrkF_p7$Fcgo$\"+lv+)3'F_p7$Fhgo$ \"+dxkQdF_p7$F]ho$\"+MF7i`F_p7$Fbho$\"+#)4*H0&F_p7$Fgho$\"+&4rDr%F_p7$ F\\io$\"+5at=WF_p7$Faio$\"+^r]UTF_p7$Ffio$\"+L_i]QF_p7$F[jo$\"+S\"*)ze $F_p7$F`jo$\",DDN.N$F-7$Fejo$\",oIRd4$F-7$Fjjo$\",p-^m*GF-7$F_[p$\",(H FgeEF-7$Fd[p$\",<6JPW#F-7$Fi[p$\",'>vMsAF-7$F^\\p$\",6!f8r?F-7$Fc\\p$ \",!)*4zm=F-7$Fh\\p$\",Nz#GX0<56F-7$Fa^p$\"+KV-o$*F-7$Ff^p$\"+')pJ5() F-7$F[_p$\"+aD#*4sF-7$F`_p$\"+Ge=9cF--Ff_p6&Fh_pF($\"#vFb_p$\"#&*Fb_p- F_`p6#%Ea~scheme~of~Tsitouras~and~PapakostasG-F$6%7g[lF'Facs7$F2$\"&[4 \"F07$F7$\"&19)F07$F<$\"'u\">%F07$FA$\"'E1')F07$FF$\"((4v;F07$FK$\"(&o 8JF07$FP$\"(K.c&F07$FU$\")wr09F07$FZ$\")?pdKF07$Fin$\")M@B[F07$F^o$\") )fW-(F07$Fco$\"*sgw+\"F07$Fho$\"*.SaU\"F07$F]p$\"*>D,i\"F07$Fcp$\"*#>5 <;F07$Fhp$\"*G%39;F07$F]q$\"*Es5h\"F07$Fbq$\"*4l]g\"F07$Fgq$\"*N!3*f\" F07$F\\r$\"*$)yre\"F07$Far$\"*&=Qv:F07$Ffr$\"*D(H]:F07$F[s$\"*QONa\"F0 7$F`s$\"*F07$F_t$\"*4'ofCF07$F dt$\"*)G`qEF07$Fit$\"*'fk@HF07$F^u$\"*y3s#HF07$Fcu$\"*yD5#HF07$Fhu$\"* $*)p3HF07$F]v$\"*5Ck*GF07$Fbv$\"*qI?(GF07$Fgv$\"*]jy%GF07$F\\w$\"*->9! GF07$Faw$\"*az]x#F07$Ffw$\"*[Tuz#F07$F[x$\"*t:(*)GF07$F`x$\"*Wn>8$F07$ Fex$\"*GUHn$F07$Fjx$\"*7g@)QF07$F_y$\"*-Bg(RF07$Fdy$\"*`kw'RF07$Fiy$\" *hB$fRF07$F^z$\"*N%pURF07$Fcz$\"*$\\8ERF07$Fhz$\"*0EK*QF07$F][l$\"*B>1 'QF07$Fb[l$\"*f')>!QF07$Fg[l$\"*&G5ePF07$F\\\\l$\"*7te!QF07$Fa\\l$\"*w )3gUF07$Ff\\l$\"*FC%RXF07$F[]l$\"*opnz%F07$F`]l$\"*fFux%F07$Fe]l$\"*^j \"eZF07$Fj]l$\"*Ao)>ZF07$F_^l$\"*y&)=o%F07$Fd^l$\"*fKsg%F07$Fi^l$\"*'Q #3a%F07$F^_l$\"*Pr)e[F07$Fc_l$\"*8h%*H&F07$Fh_l$\"*JUA@&F07$F]`l$\"*y! 4L^F07$Fb`l$\"****p2^F07$Fg`l$\"*w%*=6&F07$F\\al$\"*Q)[#=&F07$Faal$\"* ([b!R&F07$Ffal$\"*-%ejbF07$F[bl$\"*!f=NdF07$F`bl$\"*6=.5'F07$Febl$\"*7 Rw*fF07$Fjbl$\"*xW(**eF07$F_cl$\"*5/(eeF07$Fdcl$\"*A&4NeF07$Ficl$\"*Ep k$eF07$F^dl$\"*EXE&eF07$Fcdl$\"*AV/*eF07$Fhdl$\"*%>BffF07$F]el$\"*D#QY iF07$Fbel$\"+(**\\FS'Ffel7$Fhel$\"+;lN[jFfel7$F]fl$\"+#4EWH'Ffel7$Fbfl $\"+a.5)='Ffel7$Fgfl$\"+sX](3'Ffel7$F\\gl$\"+j2!z/'Ffel7$Fagl$\"+XDWEg Ffel7$Ffgl$\"+Fc\"*GgFfel7$F[hl$\"+]D#e/'Ffel7$F`hl$\"+)R:M3'Ffel7$Feh l$\"+[-:]hFfel7$Fjhl$\"+w&*o>kFfel7$F_il$\"+YbR8lFfel7$Fdil$\"+]K1gkFf el7$Fiil$\"+&\\orS'Ffel7$F^jl$\"+T$>0A'Ffel7$Fcjl$\"+A5dXhFfel7$Fhjl$ \"+V`:fiFfel7$F][m$\"+I\"e!plFfel7$Fb[m$\"+?\\/HlFfel7$Fg[m$\"+$fO8Z'F fel7$F\\\\m$\"++b\\djFfel7$Fa\\m$\"+T'GvC'Ffel7$Ff\\m$\"+9-E0iFfel7$F[ ]m$\"+2D%4<'Ffel7$F`]m$\"+e<,ahFfel7$Fe]m$\"+J9@'Ffel7$Faam$ \"+,Xv6kFfel7$Ffam$\"+()Q\"QY'Ffel7$F[bm$\"+i@J^kFfel7$F`bm$\"+9Y$)QkF fel7$Febm$\"++7QEkFfel7$Fjbm$\"+*eY:S'Ffel7$F_cm$\"+bz!oP'Ffel7$Fdcm$ \"+sU\")yiFfel7$Ficm$\"+)3(o#='Ffel7$F^dm$\"+=x$[3'Ffel7$Fcdm$\"+\"))4 Y.'Ffel7$Fhdm$\"+S#eL@'Ffel7$F]em$\"+Dg%eD'Ffel7$Fbem$\"+J&**p0'Ffel7$ Fgem$\"+L?n4fFfel7$F\\fm$\"+z0iLgFfel7$Fafm$\"+hRS.hFfel7$Fffm$\"+4*G7 +'Ffel7$F[gm$\"+\"Qx5!fFfel7$F`gm$\"+\\>_9eFfel7$Fegm$\"+]`ebdFfel7$Fj gm$\"+\"Reo\"eFfel7$F_hm$\"+H]4SfFfel7$Fdhm$\"+n)=[u&Ffel7$Fihm$\"+aON !e&Ffel7$F^im$\"+&e^Ffel7$Fh^n$\"+bKiO]Ffel7$F]_n$\"+=q&R'\\Ffel7$Fb_n$\"+*Q7 <%\\Ffel7$Fg_n$\"+:J)R%\\Ffel7$F\\`n$\"+eH#\\*\\Ffel7$Fa`n$\"+))etb]Ff el7$Ff`n$\"+4N#o*[Ffel7$F[an$\"+0oI\\ZFfel7$F`an$\"+2sp?ZFfel7$Fean$\" +-5s7ZFfel7$Fjan$\"+$pp&[ZFfel7$F_bn$\"+%=.a\"[Ffel7$Fdbn$\"+UzwIZFfel 7$Fibn$\"+JXlZYFfel7$F^cn$\"+p\"p\\d%Ffel7$Fccn$\"+V%R-^%Ffel7$Fhcn$\" +SyI)[%Ffel7$F]dn$\"+'p1S[%Ffel7$Fbdn$\"+`j09XFfel7$Fgdn$\"+>.IvXFfel7 $F\\en$\"+CdEAWFfel7$Faen$\"+#G%z\"G%Ffel7$Ffen$\"+y(32E%Ffel7$F[fn$\" +/*QSD%Ffel7$F`fn$\"+b[%\\F%Ffel7$Fefn$\"+W%f#QVFfel7$Fjfn$\"+Fe*=F%Ff el7$F_gn$\"+ICb1UFfel7$Fdgn$\"+BW=(3%Ffel7$Fign$\"+T;%*yRFfel7$F^hn$\" +-e!p&QFfel7$Fchn$\"+eziHQFfel7$Fhhn$\"+yLh4QFfel7$F]in$\"+by$\\!QFfel 7$Fbin$\"+I[jIQFfel7$F\\jn$\"+h1_iQFfel7$Ffjn$\"+W#)*[&QFfel7$F[[o$\"+ ko&*QQFfel7$F`[o$\"+393BQFfel7$Fe[o$\"+`s_\"z$Ffel7$Fj[o$\"+7^BgPFfel7 $F_\\o$\"+_Ze*f$Ffel7$Fd\\o$\"+Y\"[vU$Ffel7$Fi\\o$\"+)zeIL$Ffel7$F^]o$ \"+%pu(yJFfel7$Fc]o$\"+0%=h-$Ffel7$Fh]o$\"+6]mYHFfel7$F]^o$\"+!3#o%y#F fel7$Fb^o$\"+!4q)oEFfel7$Fg^o$\"+&f\\rd#Ffel7$F\\_o$\"+n\"f=V#Ffel7$Fa _o$\",2)4[ABF_p7$Ff_o$\",v#[V[AF_p7$F[`o$\",1^0.6#F_p7$F``o$\",fZlQ-#F _p7$Fe`o$\",1Wy'Q>F_p7$Fj`o$\",*=h^I=F_p7$F_ao$\",()>*=\\^ot:F_p7$F^bo$\",*QbD/:F_p7$Fcbo$\",eO\\EW\"F_ p7$Fhbo$\",sXr$\\8F_p7$F]co$\",^2\\$)G\"F_p7$Fbco$\",,**QcA\"F_p7$Fgco $\",&*\\@G:\"F_p7$F\\do$\",nGxW4\"F_p7$Fado$\",[d:5/\"F_p7$Ffdo$\"+b7J c(*F_p7$F[eo$\"+\\^hN#*F_p7$F`eo$\"+Q7^g()F_p7$Feeo$\"+#=UBB)F_p7$Fjeo $\"+5\">Vu(F_p7$F_fo$\"+E]a3tF_p7$Fdfo$\"+;q>boF_p7$Fifo$\"+mL!yW'F_p7 $F^go$\"+sRTsgF_p7$Fcgo$\"+0#4sp&F_p7$Fhgo$\"+(\\-1M&F_p7$F]ho$\"+W&)* f(\\F_p7$Fbho$\"+srRsYF_p7$Fgho$\"+&*)[cK%F_p7$F\\io$\"+ItqXSF_p7$Faio $\"+6E')pPF_p7$Ffio$\"+8)*RpMF_p7$F[jo$\"++)4BA$F_p7$F`jo$\",DR1K)HF-7 $Fejo$\",o-G#=FF-7$Fjjo$\",pw.%QDF-7$F_[p$\",(ppP&H#F-7$Fd[p$\",F-7$F^\\p$\",64e!3\"F-7$Fb]p$\"+qC2$*)*F-7$Fg]p$\"+su#f?* F-7$F\\^p$\"+!\\ucV(F-7$Fa^p$\"+KoNrbF-7$Ff^p$\"+')4]W^F-7$F[_p$\"+a)4 c]$F-7$F`_p$\"+G%zrx\"F--Ffev6&Fh_pFihyF(Fihy-F_`p6#%Ra~modification~o f~the~Tsitouras-Papakostas~schemeG-F$6%7i[lF'7$F+$!\"$F07$F2$!%O6F07$F 7$!%)\\)F07$F<$!&SS%F07$FA$!&12*F07$FF$!'/r?&F07$F^o$!(Ilf(F07$Fco$!)pn#4 \"F07$Fho$!)k))\\:F07$F]p$!)yPjIF07$Ffw$!)*fK/$F07$F[x$!)gMUJF07$F`x$!)h%[S$F 07$Fex$!)HG&*RF07$Fjx$!)<_CUF07$F_y$!).`FVF07$Fdy$!)GV=VF07$Fiy$!)YN4V F07$F^z$!)]D\"H%F07$Fcz$!)=BtUF07$Fhz$!)NTPUF07$F][l$!)E#>?%F07$Fb[l$! )$e!QTF07$Fg[l$!)X')*3%F07$F\\\\l$!)0?STF07$Fa\\l$!)>MLYF07$Ff\\l$!):v Q\\F07$F[]l$!)-%4A&F07$F`]l$!)y))*>&F07$Fe]l$!)-#*y^F07$Fj]l$!)'Qs8&F0 7$F_^l$!)r*e4&F07$Fd^l$!)Ri9]F07$Fi^l$!)S5U\\F07$F^_l$!)[N&G&F07$Fc_l$ !)3KodF07$Fh_l$!))pLn&F07$F]`l$!)T*pe&F07$Fb`l$!)&f*ebF07$Fg`l$!)6yibF 07$F\\al$!)ViQcF07$Faal$!).pkeF07$Ffal$!))Qf0'F07$F[bl$!))Q0C'F07$F`bl $!)54TmF07$Febl$!)ZIHlF07$Fjbl$!)RiAkF07$F_cl$!)zrxjF07$Fdcl$!)2a^jF07 $Ficl$!)2p_jF07$F^dl$!)#4*pjF07$Fcdl$!)!*o5kF07$Fhdl$!)%o`['F07$F]el$! )[@*z'F07$Fbel$!*j67(pFfel7$Fhel$!*u))>\"pFfel7$F]fl$!*=qK&oFfel7$Fbfl $!*'o\\PnFfel7$Fgfl$!*%*Qyi'Ffel7$F\\gl$!*zzWe'Ffel7$Fagl$!*4f1c'Ffel7 $Ffgl$!*$G/jlFfel7$F[hl$!*.E6e'Ffel7$F`hl$!*-#z@mFfel7$Fehl$!*#**Q%p'F fel7$Fjhl$!*!zX*)pFfel7$F_il$!**p$G4(Ffel7$Fdil$!*=gZ.(Ffel7$Fiil$!*zf r(pFfel7$F^jl$!*6SQx'Ffel7$Fcjl$!*yu4p'Ffel7$Fhjl$!*J&f9oFfel7$F][m$!* gEZ:(Ffel7$Fb[m$!*IP86(Ffel7$Fg[m$!*G#[[qFfel7$F\\\\m$!*(e[CpFfel7$Fa \\m$!*VZY!oFfel7$Ff\\m$!*H%\\enFfel7$F[]m$!*D')3s'Ffel7$F`]m$!*O#3-nFf el7$Fe]m$!*1F>s'Ffel7$Fj]m$!*VPk\"oFfel7$F_^m$!*SUw/(Ffel7$Fd^m$!*iUw8 (Ffel7$Fi^m$!*ZKj7(Ffel7$F^_m$!*/YH6(Ffel7$Fc_m$!*y%e*4(Ffel7$Fh_m$!*[ PH2(Ffel7$F]`m$!*=!RYqFfel7$Fb`m$!*;l[!oFfel7$Fg`m$!*(oV!o'Ffel7$F\\am $!*!RPmnFfel7$Faam$!*ZSn)pFfel7$Ffam$!*]OU/(Ffel7$F[bm$!*;71.(Ffel7$F` bm$!*@9q,(Ffel7$Febm$!*`UM+(Ffel7$Fjbm$!*%yPwpFfel7$F_cm$!*!yT\\pFfel7 $Fdcm$!*GCE%oFfel7$Ficm$!*Y^yt'Ffel7$F^dm$!*Zl5j'Ffel7$Fcdm$!*&)Ged'Ff el7$Fhdm$!*]3?x'Ffel7$F]em$!*$=6?oFfel7$Fbem$!*c;Lg'Ffel7$Fgem$!*H*eFfel7$Ff[n$!*uq^)f Ffel7$F[\\n$!*l!fOgFfel7$F`\\n$!*rVE%eFfel7$Fe\\n$!*GRpm&Ffel7$Fj\\n$! *w$QTcFfel7$F_]n$!*0Mkk&Ffel7$Fd]n$!*9fPr&Ffel7$Fi]n$!*3s1z&Ffel7$F^^n $!*?#z$p&Ffel7$Fc^n$!*!)p&)f&Ffel7$Fh^n$!*D8z]&Ffel7$F]_n$!*a@%GaFfel7 $Fb_n$!*!R@/aFfel7$Fg_n$!*RzsS&Ffel7$F\\`n$!*e*)[Y&Ffel7$Fa`n$!*1xS`&F fel7$Ff`n$!*\"*G,O&Ffel7$F[an$!*'>k)>&Ffel7$F`an$!*iNu;&Ffel7$Fean$!*t V#f^Ffel7$Fjan$!*FB.?&Ffel7$F_bn$!*8jnF&Ffel7$Fdbn$!*/>S=&Ffel7$Fibn$! *\">%H4&Ffel7$F^cn$!*A(G8]Ffel7$Fccn$!*b/C%\\Ffel7$Fhcn$!*@#e=\\Ffel7$ F]dn$!*&Hc9\\Ffel7$Fbdn$!*t/%\\\\Ffel7$Fgdn$!*()R*>]Ffel7$F\\en$!*qK?& [Ffel7$Faen$!*q\"*zp%Ffel7$Ffen$!*)H5vYFfel7$F[fn$!*G/&oYFfel7$F`fn$!* j%G$p%Ffel7$Fefn$!*S)omZFfel7$Fjfn$!*)4x$p%Ffel7$F_gn$!*]u>i%Ffel7$Fdg n$!*\\%>(\\%Ffel7$Fign$!*.T!zVFfel7$F^hn$!*ZxZC%Ffel7$Fchn$!*&=*[@%Ffe l7$Fhhn$!*6$H$>%Ffel7$F]in$!*DK$*=%Ffel7$Fbin$!*Bf0A%Ffel7$Fgin$!*24rB %Ffel7$F\\jn$!*Cd$eUFfel7$Fajn$!*JZ#fUFfel7$Ffjn$!*WJ/D%Ffel7$F[[o$!*S aGB%Ffel7$F`[o$!*/]`@%Ffel7$Fe[o$!*De0=%Ffel7$Fj[o$!*Rbg9%Ffel7$F_\\o$ !*jQI(RFfel7$Fd\\o$!*-)H(y$Ffel7$Fi\\o$!*#4o\"p$Ffel7$F^]o$!*f[c_$Ffel 7$Fc]o$!*%4!4O$Ffel7$Fh]o$!*ajEG$Ffel7$F]^o$!*xsi5$Ffel7$Fb^o$!*[]N)HF fel7$Fg^o$!*U\"G#*GFfel7$F\\_o$!*@:8t#Ffel7$Fa_o$!+.--=EF_p7$Ff_o$!+:& 4ra#F_p7$F[`o$!+u=X$R#F_p7$F``o$!+\"pObI#F_p7$Fe`o$!+uHMAAF_p7$Fj`o$!+ 6pp*4#F_p7$F_ao$!+$)*p#>?F_p7$Fdao$!+)\\;9&>F_p7$Fiao$!+R3YK=F_p7$F^bo $!+@xvl7F_p7$F[eo$!+@\"HV<\"F_p7$F`eo$!+sS1K6F_p7$Feeo$!+Qs]n5F_p7 $Fjeo$!+gE&e-\"F_p7$F_fo$!*%4**p)*F_p7$Fdfo$!*kNmI*F_p7$Fifo$!*%)G#*)* )F_p7$F^go$!*3I=e)F_p7$Fcgo$!*D=z=)F_p7$Fhgo$!*$QZPzF_p7$F]ho$!*c'eAvF _p7$Fbho$!*Q)38sF_p7$Fgho$!*l)GmpF_p7$F\\io$!*5t#4mF_p7$Faio$!*4Z#ojF_ p7$Ffio$!*xcG='F_p7$F[jo$!*IJ5%eF_p7$F`jo$!+v)y'[cF-7$Fejo$!+KdX6bF-7$ Fjjo$!+J)*z&>&F-7$F_[p$!+.h:K]F-7$Fd[p$!+$yVS&\\F-7$Fi[p$!+/FbZYF-7$F^ \\p$!+*o\"ePXF-7$Fc\\p$!+?+rlWF-7$Fh\\p$!+l&G!*>%F-7$F]]p$!+)=LK7%F-7$ Fb]p$!+I#oN0%F-7$Fg]p$!+G " 0 "" {MPLTEXT 1 0 615 "evalf[25](plot(['bn_RK8_1'(x)-b(x),'bn_R K8_2'(x)-b(x),'bn_RK8_3'(x)-b(x),'bn_RK8_4'(x)-b(x),\n'bn_RK8_5'(x)-b( x),'bn_RK8_6'(x)-b(x),'bn_RK8_7'(x)-b(x)],x=0.65..5,-3e-17..5e-17,font =[HELVETICA,9],\ncolor=[COLOR(RGB,.4,0,.9),COLOR(RGB,0,.65,0),brown,re d,COLOR(RGB,0,.75,.95),magenta,COLOR(RGB,.95,.55,0)],\nlegend=[`Enrigh t-Verner scheme`,`Verner's Maple scheme`,`a modification of Verner's M aple scheme`,`an \"efficient\" scheme`,`a scheme of Tsitouras and Papa kostas`,`a modification of the Tsitouras-Papakostas scheme`,`Verner's \+ \"most efficient\" scheme`],title=`error curves for 12 stage order 8 R unge-Kutta methods`));" }}{PARA 13 "" 1 "" {GLPLOT2D 836 519 519 {PLOTDATA 2 "6--%'CURVESG6%7ay7$$\"#l!\"#$\"+G]f=*)!#F7$$\":+++++]i:0W qt'!#D$\"+P+F%*\\F-7$$\":++++++DJ5)3upF1$\"+z8=C%F-7$$\":+++++vV)HxngyF1$! ,mwi(ew!#G7$$\":+++++]PMDzJF)F1$!-Tx([6.\"FH7$$\":+++++D1*e.3P()F1$!-_ ]jN@8FH7$$\":++++++vVY\")4?*F1$!-jjc!\\d\"FH7$$\":+++++]Pf`pWV*F1$!-WI 1&))e\"FH7$$\":+++++++vgdzm*F1$!-sxL(**f\"FH7$$\":++++DJXkhVrp*F1$!-N- (H`p\"FH7$$\":++++]i!RD'Hjs*F1$!-uL8!)*o\"FH7$$\":++++v$fLMc^b(*F1$!-r Jp9U;FH7$$\":+++++D\"GV;q%y*F1$!-UO_*pj\"FH7$$\":++++](=po]Sx\"F`q7$$\":+++++voCrWk5 \"F]q$!.4'**\\?*)=F`q7$$\":++++]i:g$p@<6F]q$!.-n+Mh*=F`q7$$\":+++++]i& f\"*)z7\"F]q$!.7k;y=\">F`q7$$\":++++]P4JQh(Q6F]q$!.i!)eTj%>F`q7$$\":++ +++DcmgL&\\6F]q$!.T2D$>\\:\"F]q$!.e1l_B)>F`q7$$\":+++]iS;Vx7w:\"F]q$!.(3AzIZ>F`q7$ $\":++++]7.-$eIg6F]q$!.v[d+!)3#F`q7$$\":+++]P%)*3')))*H;\"F]q$!.-z-CD. #F`q7$$\":++++Dcw>%>pl6F]q$!.Ff-B)z>F`q7$$\":+++]7Gjy*\\Qo6F]q$!.COeJ! p>F`q7$$\":++++++]P0y5<\"F]q$!.&[5\"F]q$!.Hm:hL2#F`q7$$\":+++++D1z_I\\@\"F] q$!.\"4#GUb/#F`q7$$\":+++++vo\\bQsB\"F]q$!/tg_h'p-#!#I7$$\":++++Dc^$eq -S7F]q$!/'p`]qV<#F[w7$$\":++++]PMGed?F[w7$$\":+++++++&oDR[7F]q$!/g_Y.1W?F[w7$$\":++++] il_dpRD\"F]q$!/Y$fO.z4#F[w7$$\":+++++DJ?eY&f7F]q$!/*\\'yEbD@F[w7$$\":+ +++D19a3NBE\"F]q$!/t%Hjm?F[w7$$\":++++v$4r0h1z7F]q$!/Q94Xi&4#F[w7$$\":+++++v$44Y&=G\"F]q$! /)z%GU/j@F[w7$$\":+++++vVe5!R$H\"F]q$!/,y,>*e8#F[w7$$\":+++++v$f-c#\\I \"F]q$!/_iCW(*3@F[w7$$\":+++++vV$*4hkJ\"F]q$!/*)*y-ca3#F[w7$$\":+++++v $4'f'*zK\"F]q$!/je,*eD3#F[w7$$\":+++++vVG4K&R8F]q$!/*zAifk;#F[w7$$\":+ ++++v$f*en5N\"F]q$!/!\\Gw&R7AF[w7$$\":+++++vVj3.EO\"F]q$!/0v!H7:=#F[w7 $$\":+++++v$4$eQTP\"F]q$!/&GKf92:#F[w7$$\":++++]Pf`3PcQ\"F]q$!/MMN3!=7 #F[w7$$\":++++++DweNrR\"F]q$!/(GCqi25#F[w7$$\":+++]iS\">8-,+9F]q$!/:$z 76hC#F[w7$$\":++++D\"yvQ[)GS\"F]q$!/n&G(fY#=#F[w7$$\":+++](=UKk%fdS\"F ]q$!/w8=z4A@F[w7$$\":++++]i!*)*3M'39F]q$!/\"3D#F\\al7 $$\":+++++vo9p=Cg\"F]q$!0()e%fJ>H@F\\al7$$\":+++++]7$)z*3W;F]q$!0&[cJL Fi?F\\al7$$\":++++DJXuu?F\\al7$$\":++++]7ylpXvm\"F] q$!0?C(=Nt*)>F\\al7$$\":++++v$4rXOFz;F]q$!0$yGftJG?F\\al7$$\":+++++vV[ f,5p\"F]q$!0)QFSQ)o2#F\\al7$$\":++++]P4J\\dWr\"F]q$!1#zEIC7W)>!#K7$$\" :++++++v8R8zt\"F]q$!1/\\,1Q8;>Fjcl7$$\":++++++]A\"y,&y\"F]q$!1d'pHFdZ) =Fjcl7$$\":++++++DJBA@$=F]q$!1faA2*pG'=Fjcl7$$\":+++++++?:>[&=F]q$!1y7 ]g&))4y\"Fjcl7$$\":++++++v32;v(=F]q$!1%e-:Mf@r\"Fjcl7$$\":++++]PMd=`.) =F]q$!1+\")yz+@3=Fjcl7$$\":+++++v$f+.>$)=F]q$!1KB]!=Nwv\"Fjcl7$$\":+++ +]7`aTFg)=F]q$!1d*y#fR15F] q$!1y^D6g)Gx\"Fjcl7$$\":+++++](=\\9c6>F]q$!1fm^v5\"*HF]q$!2<.b/E6uo\"!#L7$$\":+++++v$f:QN0?F]q$!2NQ=\\G9Q[\"Ffhl7$$ \":++++++D^f(Q.@F]q$!2pXB`f!\\u7Ffhl7$$\":+++++++b1Nk=#F]q$!3J+YPE!Q5. \"!#M7$$\":+++++vVj,RIG#F]q$!29&45.dmFxFfil7$$\":++++++]<))\\&oBF]q$!2 ;$=5/o9t\\Ffil7$$\":+++++vVQ&)oBY#F]q$!3$G-%\\daj&)=!#N7$$\":+++++DJ:q 1d8>dUW!#O7$$ \":+++++v=2a@0t#F]q$\"42y/(44-nT!)Fa[m7$$\":+++++]7`)G&G#GF]q$\"5p\\Yc L/6q?6Fa[m7$$\":+++++v$4.#f(=HF]q$\"6(*>$z&>(*[peU\"!#P7$$\":+++++DcY$ fC-IF]q$\"6to**=:e#ym%z\"Fa\\m7$$\":+++++]P>*QT#4$F]q$\"7a],zj=#=dK6#! #Q7$$\":++++++]Z@mb=$F]q$\"7S#oSK'**z)=#[BF\\]m7$$\":+++++](=*e(pwKF]q $\"7oK^^miw0A3EF\\]m7$$\":+++++vV)z=([O$F]q$\"8-fmT;-a4[O\"H!#R7$$\":+ ++++]7jRuFY$F]q$\"8h5fLHLy`gtC$F\\^m7$$\":++++](=nPow%[$F]q$\"8AID(z48;'p/j/LF\\^m7$$\":++++++]<;W2b$F]q$\"9i\"=7Q2yOt*[MN!# S7$$\":+++++v$4(yDUd$F]q$\"9qt+^kYgt<,dMFf_m7$$\":+++++]PCT2xf$F]q$\"9 EO%QY?O&zR29MFf_m7$$\":++++]P4^A[%4OF]q$\"9dvQ9zz^r7hhNFf_m7$$\":+++++ D\"yP!*=@OF]q$\"9MkuV9fG>^1oOFf_m7$$\":++++]7`/&)HHj$F]q$\"9.]4i$eSbUC Ti$Ff_m7$$\":++++++DJmqYk$F]q$\"9j*GEk^dG;%e!e$Ff_m7$$\":+++]PfLWe4`l$ F]q$\"9^6lS'*p?XFtwNFf_m7$$\":++++v=Ud][fm$F]q$\"9o`&)4/SCagxtNFf_m7$$ \":+++]7y]qU(ewOF]q$\"9+T[l#=CYY#RtNFf_m7$$\":++++]Pf$[jA(o$F]q$\"9iZN i[2(Q@o(yNFf_m7$$\":++]PM_'oyg))*o$F]q$\"9b&Ha[dfE)z*)3QFf_m7$$\":+++v =n8!4ea#p$F]q$\"9)*p%)>vq^ER$eu$Ff_m7$$\":++]7.#3MRb?&p$F]q$\"9WR@a4yv pmd[OFf_m7$$\":+++](ozmp_'yp$F]q$\"9)=MK3ojV+,cf$Ff_m7$$\":++](=<^**** \\_+PF]q$\"9Uy/&3PF]q$\" 9ra)H%p'pe#\\eLOFf_m7$$\":+++]i:&G7J9>PF]q$\"94$3E**>N31K&3PFf_m7$$\": +++++v$fL?yHPF]q$\"9zZ/QNf3@QKXQFf_m7$$\":++]iS;o8#)*oKPF]q$\"9O_Qq4R6 Dta.QFf_m7$$\":+++D\"yD94wfNPF]q$\"9Ek!37Rj^\\,mp$Ff_m7$$\":++](=#*p\" pR0&QPF]q$\"9K:Zs\"HA^6F%yOFf_m7$$\":+++]iS\"p%=89u$F]q$\"91cYX(R?X/ms 'QFf_m7$$\":++]7.#eYs4KWPF]q$\"9\\MpJf.IZRjcPFf_m7$$\":+++vVBS-wGsu$F] q$\"9gaZ\")>()Ra2`pOFf_m7$$\":++]P%[Y,[l8]PF]q$\"9xq84%prL4M:$RFf_m7$$ \":++++D1*yNV/`PF]q$\"9kC=.'p$H9\"o)=QFf_m7$$\":+++](=n)o[vYw$F]q$\":, %R+](G;w'>(4x$!#T7$$\":++++]P%)zj1jx$F]q$\":hM3>'zDx#Q*[GPFhhm7$$\":++ ]i!*yedU9#zPF]q$\":qL!>\\+#pm(3J-QFhhm7$$\":+++DJ?LN@A@y$F]q$\":SpzeDZ a8BlV*QFhhm7$$\":++](=]*=)*)QFhhm7$$\":++]i:5azswC !QF]q$\":pGRzN%f.^R#\\!RFhhm7$$\":+++Dc^Gd^%Q0QF]q$\":X9)*Qp;&>Co[%z$F hhm7$$\":++](oHH].BH3QF]q$\":B$3Q(Gl6c2ynv$Fhhm7$$\":+++]PMx74+7\"QF]q $\":x8`M$zF.\"f1n'RFhhm7$$\":++]7yv^!zy59QF]q$\":XD,7$pbA%)H:`QFhhm7$$ \":+++v=QF]q$\":$*H zyQ!Q&\\*HH-SFhhm7$$\":++++++vBCJG#QF]q$\":&H?H()**3nPf[8RFhhm7$$\":++ ++v=neD>Z%QF]q$\":c)eVz3&p,=O3'QFhhm7$$\":++++]Pf$psgmQF]q$\":M[m_88^i Coa\"QFhhm7$$\":++]P%)*e/rKMpQF]q$\":%z:B&e2F:q%pzxsG!)QF ]q$\":Q`8)[QqlKzSrSFhhm7$$\":+++Dc^y%zK-$)QF]q$\":kF3uu&)p*eC`hRFhhm7$ $\":++]i!R];\"Gfd)QF]q$\":8Pj'H8Ac9(Qt&QFhhm7$$\":++++Dc^GG&\\))QF]q$ \":g,R/fEq(o*f:%QFhhm7$$\":+++]il(f*GR%**QF]q$\":Wt&>-sgs\\d,eRFhhm7$$ \":+++++vVjH$Q5RF]q$\":%4%\\GWn&4o?F*3%Fhhm7$$\":+++++](=xDELRF]q$\":f 7y!)*z`azf&[)RFhhm7$$\":+++++DJ!e=9cRF]q$\":V'[!)[-5%4S')f)QFhhm7$$\": +++](=n8o<+fRF]q$\":JE/MyE9lhC`#RFhhm7$$\":++++v=U#y;'='RF]q$\":-ku&3l z$p\"G%\\0%Fhhm7$$\":+++]ilZ$)e@Z'RF]q$\":J+AL >m\")[Bg;TFhhm7$$\":++++D1k'=8ItRF]q$\":c_#)pV')\\a\"ob+SFhhm7$$\":+++ ]7`p(G7;wRF]q$\":!=,pQ4kGq6x#*QFhhm7$$\":++++++v)Q6-zRF]q$\":5%fs%R'*> ^&onMRFhhm7$$\":+++](o/)*[5)=)RF]q$\":54NF;&e/SI&31%Fhhm7$$\":++++v$f3 f4u%)RF]q$\":`IxW\"p=>kU$o%RFhhm7$$\":+++]iS\">p3g()RF]q$\":'z'HEs2!yh TXoQFhhm7$$\":++++](oHz2Y!*RF]q$\":gAsPpQQ:M]:7%Fhhm7$$\":+++]PM-%*o?L *RF]q$\":q5x`LF![^iO0SFhhm7$$\":++++D\"y]*f!='*RF]q$\":&[EpfJ/2i/!>+%F] q$\":6QK15*R9`[pkSFhhm7$$\":++++DJ&p)4pU-%F]q$\":KVJUv1N&H!=A(RFhhm7$$ \":++++](=nxxjYSF]q$\":*3*G(=UC%\\Od#*)QFhhm7$$\":++]7`%RaiQV\\SF]q$\" :r&e2'p%H\\JyK0SFhhm7$$\":+++Dc,;u%*HA0%F]q$\":=h8Lk,>GVbP0%Fhhm7$$\": ++]Pf3)GKg-bSF]q$\":h9`y'\\\"3L2tF%RFhhm7$$\":+++]i:gr6Ay0%F]q$\":>\"y ;wQFhhm7$$\":++]ilAK??=11%F]q$\":;s_CM>8__;&=TFhhm7$$\":+++vo H/pG9M1%F]q$\":J!)>bDi[dvy\\+%Fhhm7$$\":++](=njxr.@mSF]q$\":'=sidWSpPp ')**QFhhm7$$\":++++vV[mX1!pSF]q$\":\"HLd7;+#Q'*p)RRFhhm7$$\":++]7y]?:a -=2%F]q$\":r0O\"QmYQoiSoSFhhm7$$\":+++D\"yDRE')fuSF]q$\":Qf7>zx](y7ecR Fhhm7$$\":++]P%[YE6ZRxSF]q$\":#)o^cRElF4!)G(QFhhm7$$\":+++](=n8'z!>!3% F]q$\":Vb%)eNf2f2BB8%Fhhm7$$\":++]i!*y35)o)H3%F]q$\":m\"fitNN@9uQ=SFhh m7$$\":+++v$f3)e'Hy&3%F]q$\":Ma***or(pKx])HQz^RFhhm7$$\": +++](oHHq*>*\\TF]q$\":QX0KKOw]mPc3%Fhhm7$$\":++](=nj-7s%G:%F]q$\":0!)4 6/**>FxzU*RFhhm7$$\":+++DcwfPXud:%F]q$\":e\\#f.G-h?Cw\")QFhhm7$$\":++] iS;$\\&p,(eTF]q$\":2\"zHjK4o'Rkn(QFhhm7$$\":++++DcEs$*G;;%F]q$\":GTo$4 0*o&4?yOSFhhm7$$\":+++]i:gT!zLtTF]q$\":.6gmDUS(fc_hRFhhm7$$\":+++++v$4 ro/&=%F]q$\":BS2Ch!G&Ru(*y)QFhhm7$$\":+++]7`%*y*RK'>%F]q$\":l#[`An<\\A i9OQFhhm7$$\":++++DJ&pC6g2UF]q$\":X-H6j'4@*Rq^!QFhhm7$$\":+++]P4'\\^#y )=UF]q$\":m_$4o!*33J$fc%QFhhm7$$\":++++](oHy`:IUF]q$\":UB.9?LWUkna.%Fh hm7$$\":++++vV)*=j4FD%F]q$\":%Q)HP7T!*f)\\!=\"RFhhm7$$\":+++++++b)QEvU F]q$\":EEKqtcj6@K3z$Fhhm7$$\":+++++]iI\"Q_nVF]q$\":&e+;jBi8voG.OFhhm7$ $\":+++++vV=y>!fWF]q$\":[<\\[Rjc*=&G[[$Fhhm7$$\":++++++]Z2&4VXF]q$\":v f[a'4.)*Q39TLFhhm7$$\":+++++]7$[[XRYF]q$\":\\-(yV#)3'o)*fH0$Fhhm7$$\": +++++++&y,kDZF]q$\":.5pgb_m&3[(3r#Fhhm7$$\":+++++vVjrJv\"[F]q$\":))[;0 #*oi(>jj]BFhhm7$$\":+++++DcJk$[0\\F]q$\":mj\"*Gh,pk#p()>?Fhhm7$$\"\"& \"\"!$\":2$fA\"f)Q+S%*em;Fhhm-%&COLORG6&%$RGBG$\"\"%!\"\"$FjcoFjco$\" \"*Fcdo-%'LEGENDG6#%6Enright-Verner~schemeG-F$6%7jx7$F($\"+GSYW^F-7$F5 $!+@ToB;F-7$F?$!+6'*o.vF-7$FD$!-mEr$=4\"FH7$FJ$!-T2<FH7$FY$!-WSD'[%>FH7$Fhn$!-s(Hq\"\\>FH7$F]o$!-N_SAi?FH 7$Fbo$!-u.wfa?FH7$Fgo$!-rJ)Qm*>FH7$F\\p$!-U1>c*)>FH7$Fap$!-X8\">([?FH7 $Ffp$!-\"fb^P;#FH7$F[q$!.i#f#oU:#F`q7$Fbq$!.>p#>WV@F`q7$Fgq$!.4'*yORE# F`q7$F\\r$!.-n4F3F#F`q7$Far$!.7kO,$)G#F`q7$Ffr$!.i!e!H\"GBF`q7$F[s$!.T 2='H6CF`q7$F`s$!.&ft\")[LCF`q7$Fes$!.e1oX#pBF`q7$Fjs$!.(3-GHFBF`q7$F_t $!.v[u5W\\#F`q7$Fdt$!.-z=O\"GCF`q7$Fit$!.Ff?u^O#F`q7$F^u$!.COj6@N#F`q7 $Fcu$!.&[5[B)[#F`q7$Fhu$!.^dY89[#F`q7$F]v$!.Hm(Q(RZ#F`q7$Fbv$!.\"4Kz1R CF`q7$Fgv$!/tg77M:CF[w7$F]w$!/'p`oH-f#F[w7$Fbw$!/\\+z:+>DF[w7$Fgw$!/ca fY5^CF[w7$F\\x$!/g_'G#)[V#F[w7$Fax$!/Y$f23%)\\#F[w7$Ffx$!/*\\'y*))3`#F [w7$F[y$!/t%H+xta#F[w7$F`y$!/e`J'RzZ#F[w7$Fey$!/F9*yC,W#F[w7$Fjy$!/T,4 O\"of#F[w7$F_z$!/s!H*4UDDF[w7$Fdz$!/KdYjzfCF[w7$Fiz$!/Q94\"pS\\#F[w7$F ^[l$!/)z%y4(Qd#F[w7$Fc[l$!/,y\"y04a#F[w7$Fh[l$!/_iMJD3DF[w7$F]\\l$!/*) *yHp'zCF[w7$Fb\\l$!/je\"p$evCF[w7$Fg\\l$!/*zA=.Wd#F[w7$F\\]l$!/!\\G%3: GEF[w7$Fa]l$!/0v?J)3f#F[w7$Ff]l$!/&GKaTPb#F[w7$F[^l$!/MMb\"f)=DF[w7$F` ^l$!/(GCKNL\\#F[w7$Fe^l$!/:$zS&GlEF[w7$Fj^l$!/n&Gli(*e#F[w7$F__l$!/w8= $G\"=DF[w7$Fd_l$!/\"3DG(f7DF[w7$Fi_l$!/7ig[q^DF[w7$F^`l$!/gjj2=iEF[w7$ Fc`l$!/_GP_z\"e#F[w7$Fh`l$!0,yn0k\\]#F\\al7$F^al$!0SjYEgCb#F\\al7$Fcal $!0HSz0W:f#F\\al7$Fhal$!0()e%*Hff^#F\\al7$F]bl$!0&[c\")zrMCF\\al7$Fbbl $!0+`eX!y$Q#F\\al7$Fgbl$!0?C(Gm(zM#F\\al7$F\\cl$!0$yGRur#R#F\\al7$Facl $!0)QFqy5\\CF\\al7$Ffcl$!1#zEI_@*QBFjcl7$F\\dl$!1/\\,YKEdAFjcl7$Fadl$! 1d'pH_*RFjcl7$Fhgl$!1y^DJ!>z2#Fjcl7$F]hl$!1fm^l&4o-#Fjcl7 $Fbhl$!2<.b/Hui(>Ffhl7$Fhhl$!2NQ=\\+#Ffjl7$F][m$\"47r(>d8pnOdF a[m7$Fc[m$\"52y/(44iyM+\"Fa[m7$Fh[m$\"5p\\YcL/$[!y8Fa[m7$F]\\m$\"6(*>$ z&>(*e0%Qm<#Fa\\m7$Fh\\m$\"7a],zj=#o48;'p\"o3&RF\\^m7$Fb_m$\"9i \"=7Q2yOB?PA%Ff_m7$Fh_m$\"9qt+^kYgt$)zHTFf_m7$F]`m$\"9EO%QY?O&z,>xSFf_ m7$Fb`m$\"9dvQ9zz^rpR_UFf_m7$Fg`m$\"9MkuV9fG>8\\yVFf_m7$F\\am$\"9.]4i$ eSb7y`K%Ff_m7$Faam$\"9j*GEk^dG;kFF%Ff_m7$Ffam$\"9^6lS'*p?X]_nUFf_m7$F[ bm$\"9o`&)4/SCa$fLE%Ff_m7$F`bm$\"9+T[l#=CY'pEiUFf_m7$Febm$\"9iZNi[2(Qr U!oUFf_m7$Fjbm$\"9b&Ha[dfEG9>a%Ff_m7$F_cm$\"9)*p%)>vq^E?jmWFf_m7$Fdcm$ \"9WR@a4yvp4m]VFf_m7$Ficm$\"9)=MK3ojVqnuG%Ff_m7$F^dm$\"9UyN3c\\1U%Ff_m7$Fgem$\"9zZ/QNf3@xv#e%Ff_ m7$F\\fm$\"9O_Qq4R6D4#G`%Ff_m7$Fafm$\"9Ek!37Rj^\\r`S%Ff_m7$Fffm$\"9K:Z s\"HA^JLOQ%Ff_m7$F[gm$\"91cYX(R?Xk'43YFf_m7$F`gm$\"9\\MpJf.IZ()Ra*zCP%Ff_m7$Fjgm$\"9xq84%prLR3So%Ff_m7$F_hm$\"9kC=. 'p$H9(y(\\XFf_m7$Fdhm$\":,%R+](G;w;y?\\%Fhhm7$Fjhm$\":hM3>'zDx#yX3W%Fh hm7$F_im$\":qL!>\\+#pmnM&GXFhhm7$Fdim$\":SpzeDZa88dxj%Fhhm7$Fiim$\":`# e&=,i69L\"z0XFhhm7$F^jm$\":Ry/cVl/'4WpLWFhhm7$Fcjm$\":\\(y9u4q'RbInd%F hhm7$Fhjm$\":&oI;B`Q>],JJYFhhm7$F][n$\":pGRzN%f.^w-\\YFhhm7$Fb[n$\":X9 )*Qp;&>Cvas %Fhhm7$Fa\\n$\":[D,7$pbA%G^ne%Fhhm7$Ff\\n$\":-MN.iR8T3xUZ%Fhhm7$F[]n$ \":'*HzyQ!Q&\\pzOw%Fhhm7$F`]n$\":#H?H()**3nPB#zl%Fhhm7$Fe]n$\":c)eVz3& p,G$)Rf%Fhhm7$Fj]n$\":M[m_88^ikZ(QXFhhm7$F_^n$\":%z:B&e2F:Ip+m%Fhhm7$F d^n$\":1(zJQ'y&HyP'ou%Fhhm7$Fi^n$\":A)HGv/z.!G(R>YFhhm7$F^_n$\":FaTuNs ;RR`9`%Fhhm7$Fc_n$\":Q`8)[QqlKo'=%[Fhhm7$Fh_n$\":kF3uu&)p*e*)>6ZFhhm7$ F]`n$\":8Pj'H8Ac9,H(e%Fhhm7$Fb`n$\":g,R/fEq(o*[%oXFhhm7$Fg`n$\":Wt&>-s gs\\u/1ZFhhm7$F\\an$\":%4%\\GWn&4o_5h[Fhhm7$Faan$\":f7y!)*z`az*yct%Fhh m7$Ffan$\":V'[!)[-5%4g7ph%Fhhm7$F[bn$\":JE/MyE9lhuMm%Fhhm7$F`bn$\":-ku &3lz$p@#)p\"[Fhhm7$Febn$\":J+AL>m\")[\\b*)[Fhhm7$Fdcn$\":c_#)pV')\\aJ?^:\"HtYFhhm7$Fcdn$\":54N F;&e/S>nA[Fhhm7$Fhdn$\":`IxW\"p=>kPE(o%Fhhm7$F]en$\":'z'HEs2!yh!oTf%Fh hm7$Fben$\":gAsPpQQ:/zS*[Fhhm7$Fgen$\":q5x`LF![^z6cZFhhm7$F\\fn$\":&[E pfJ/2iJ8GYFhhm7$Fafn$\":7L*o*zo$z;lx!o%Fhhm7$Fffn$\":6QK15*R9`!)*e#[Fh hm7$F[gn$\":KVJUv1N&H=z9ZFhhm7$F`gn$\":*3*G(=UC%\\cL]h%Fhhm7$Fegn$\":r &e2'p%H\\J+V_ZFhhm7$Fjgn$\":=h8Lk,>G8^&4[Fhhm7$F_hn$\":h9`y'\\\"3Lnxyn %Fhhm7$Fdhn$\":>\"y8_#zp&)[Fhhm7$F^i n$\":J!)>bDi[dv55v%Fhhm7$Fcin$\":'=sidWSpP9KEYFhhm7$Fhin$\":\"HLd7;+#Q E\"ftYFhhm7$F]jn$\":r0O\"QmYQo0dD[Fhhm7$Fbjn$\":Qf7>zx](yW$Hp%Fhhm7$Fg jn$\":#)o^cRElF\\[Of%Fhhm7$F\\[o$\":Vb%)eNf2fx'o+\\Fhhm7$Fa[o$\":m\"fi tNN@9fclZFhhm7$Ff[o$\":Ma***or(pKx])HA`#o%Fhhm7$Fd]o$\":QX0KKOw]'eZS[F hhm7$Fi]o$\":0!)46/**>F2y@t%Fhhm7$F^^o$\":e\\#f.G-h?g())f%Fhhm7$Fc^o$ \":2\"zHjK4o'RIGf%Fhhm7$Fh^o$\":GTo$40*o&4U#=y%Fhhm7$F]_o$\":.6gmDUS(f $z>p%Fhhm7$Fb_o$\":BS2Ch!G&Rk&3/YFhhm7$F`ao$\":EEKqtcj6hXG[%Fhhm7$Feao $\":&e+;jBi8vd_aUFhhm7$Fjao$\":[<\\[Rjc*=6P2TFhhm7$F_bo$\":vf[a'4.)*Q? _IRFhhm7$Fdbo$\":\\-(yV#)3'oF*[#e$Fhhm7$Fibo$\":.5pgb_m&[lNsJFhhm7$F^c o$\":))[;0#*oi(H>JSFFhhm7$Fcco$\":mj\"*Gh,pk2UCM#Fhhm7$Fhco$\":2$fA\"f )Q+g(>^\">Fhhm-F^do6&F`doFddoF(Fddo-Fhdo6#%6Verner's~Maple~schemeG-F$6 %7\\y7$F($\"+G$pj#fF-7$F/$\"+Piw`EF-7$F5$!)@F2#)F-7$F:$!+]R7MMF-7$F?$! +6YYp^F-7$FD$!,m'=iN\")FH7$FJ$!-Tp(z)Gq\"F`q7$Fgq$!.4'RwN/=F`q7$F\\r$!.-n4x- \"=F`q7$Far$!.7k!QiC=F`q7$Ffr$!.i!)y1o&=F`q7$F[s$!.T2^/O#>F`q7$F`s$!.& f8cUT>F`q7$Fes$!.e1gt,*=F`q7$Fjs$!.(3_nsc=F`q7$F_t$!.v[5`.*>F`q7$Fdt$! .-z'4ZP>F`q7$Fit$!.FfLLs)=F`q7$F^u$!.COako(=F`q7$Fcu$!.&[q'Gd)>F`q7$Fh u$!.^d.q0)>F`q7$F]v$!.HmY+\\(>F`q7$Fbv$!.\"4AX_Z>F`q7$Fgv$!/tg-_-H>F[w 7$F]w$!/'p`]5*o?F[w7$Fbw$!/\\+H!=?,#F[w7$Fgw$!/ca**oyd>F[w7$F\\x$!/g_O \\'[%>F[w7$Fax$!/Y$fard*>F[w7$Ffx$!/*\\')pW=-#F[w7$F[y$!/t%H%o3N?F[w7$ F`y$!/e`\")Qhz>F[w7$Fey$!/F9*o@%\\>F[w7$Fjy$!/T,4fxu?F[w7$F_z$!/s!HONx ,#F[w7$Fdz$!/KdY[Il>F[w7$Fiz$!/Q9zev#*>F[w7$F^[l$!/)z%y8jc?F[w7$Fc[l$! /,yJ$o/.#F[w7$Fh[l$!/_ik^a/?F[w7$F]\\l$!/*)*yLm=)>F[w7$Fb\\l$!/je@Qxy> F[w7$Fg\\l$!/*zAG/!e?F[w7$F\\]l$!/!\\Gy(=,@F[w7$Fa]l$!/0vSWar?F[w7$Ff] l$!/&GK,&*>/#F[w7$F[^l$!/MM0fC9?F[w7$F`^l$!/(GC^yR*>F[w7$Fe^l$!/:$z\"R jJ@F[w7$Fj^l$!/n&GcK72#F[w7$F__l$!/w8G9%R,#F[w7$Fd_l$!/\"3DA[&4?F[w7$F i_l$!/7iqT$4/#F[w7$F^`l$!/gjj-WH@F[w7$Fc`l$!/_GZpTl?F[w7$Fh`l$!0,ynxF\\al7$Fbbl$!0+`e*[o4>F\\al7$Fgbl$!0?C(=P9\")=F \\al7$F\\cl$!0$yG\\o=<>F\\al7$Facl$!0)QF+8fi>F\\al7$Ffcl$!1#zEIIxX(=Fj cl7$F\\dl$!1/\\,mgU4=Fjcl7$Fadl$!1d'pHZ*=yG7;Fjcl7$Feel$!1+\")yR?N-\"Ffhl7$Fbil$!2J+YPEK))e*Ffil7$Fhil$!29&45.Zr 7rFfil7$F]jl$!2;$=5/)*>#\\%Ffil7$Fbjl$!3$G-%\\dM=S:Ffjl7$Fhjl$\"3ajz6H T8([\"Ffjl7$F][m$\"47r(>d8pCvWFa[m7$Fc[m$\"42y/(44U`1zFa[m7$Fh[m$\"5p \\YcL/w^!4\"Fa[m7$F]\\m$\"6(*>$z&>(*o&>z8Fa\\m7$Fc\\m$\"6to**=:eKs&H48;'pD4`JF\\^m7$Fb_m$\"9i\"=7Q2yOB?8P$Ff_m7$Fh_m$\"9qt+^kYgt&ymH$Ff_m 7$F]`m$\"9EO%QY?O&zq+bKFf_m7$Fb`m$\"9dvQ9zz^r^7&R$Ff_m7$Fg`m$\"9MkuV9f G>&Rg\\$Ff_m7$F\\am$\"9.]4i$eSb_#z`MFf_m7$Faam$\"9j*GEk^dG;R>T$Ff_m7$F fam$\"9^6lS'*p?XF\"zS$Ff_m7$F[bm$\"9o`&)4/SCaCu/MFf_m7$F`bm$\"9+T[l#=C Y'e-/MFf_m7$Febm$\"9iZNi[2(Q@&z3MFf_m7$Fjbm$\"9b&Ha[dfE)>nFOFf_m7$F_cm $\"9)*p%)>vq^E:cnNFf_m7$Fdcm$\"9WR@a4yvpP$\\Z$Ff_m7$Ficm$\"9)=MK3ojV+q WU$Ff_m7$F^dm$\"9UyN3Y37`$ Ff_m7$Fgem$\"9zZ/QNf3@#H4m$Ff_m7$F\\fm$\"9O_Qq4R6D22@OFf_m7$Fafm$\"9Ek !37Rj^*pD>NFf_m7$Fffm$\"9K:Zs\"HA^,;>]$Ff_m7$F[gm$\"91cYX(R?Xum8o$Ff_m 7$F`gm$\"9\\MpJf.IZH0wNFf_m7$Fegm$\"9gaZ\")>()Raw8$\\$Ff_m7$Fjgm$\"9xq 84%prLRv@u$Ff_m7$F_hm$\"9kC=.'p$H9e$\\j$Ff_m7$Fdhm$\":,%R+](G;wY$**)e$ Fhhm7$Fjhm$\":hM3>'zDx#)\\@[NFhhm7$F_im$\":qL!>\\+#pmFT$=OFhhm7$Fdim$ \":SpzeDZa8$fq0PFhhm7$Fiim$\":`#e&=,i69Bh-g$Fhhm7$F^jm$\":Ry/cVl/'4TmU NFhhm7$Fcjm$\":\\(y9u4q'RD,rl$Fhhm7$Fhjm$\":&oI;B`Q>]C\"3q$Fhhm7$F][n$ \":pGRzN%f.^2.:PFhhm7$Fb[n$\":X9)*Qp;&>C^'*4OFhhm7$Fg[n$\":B$3Q(Gl6c(e 1uNFhhm7$F\\\\n$\":x8`M$zF.\"pVMx$Fhhm7$Fa\\n$\":XD,7$pbA%oAam$Fhhm7$F f\\n$\":/MN.iR8TyUbd$Fhhm7$F[]n$\":$*HzyQ!Q&\\php!QFhhm7$F`]n$\":&H?H( )**3nPnXAPFhhm7$Fe]n$\":c)eVz3&p,3r;n$Fhhm7$Fj]n$\":M[m_88^iCLyi$Fhhm7 $F_^n$\":%z:B&e2F:5!)[s$Fhhm7$Fd^n$\":1(zJQ'y&Hy3M%z$Fhhm7$Fi^n$\":A)H Gv/z.!=_Cp$Fhhm7$F^_n$\":FaTuNs;R*3;AOFhhm7$Fc_n$\":Q`8)[QqlKRWqQFhhm7 $Fh_n$\":kF3uu&)p*e=*fw$Fhhm7$F]`n$\":8Pj'H8Ac9G%pm$Fhhm7$Fb`n$\":g,R/ fEq(oe!>l$Fhhm7$Fg`n$\":Wt&>-sgs\\u6iPFhhm7$F\\an$\":%4%\\GWn&4oZJ')QF hhm7$Faan$\":f7y!)*z`az'fjy$Fhhm7$Ffan$\":V'[!)[-5%4S8+AL>m\")[405RFhhm7$Fdcn $\":c_#)pV')\\a6G)*z$Fhhm7$Ficn$\":!=,pQ4kGqFX(p$Fhhm7$F^dn$\":5%fs%R' *>^&4;PPFhhm7$Fcdn$\":54NF;&e/S*Hn&QFhhm7$Fhdn$\":`IxW\"p=>kGW[PFhhm7$ F]en$\":'z'HEs2!yh&**Rn$Fhhm7$Fben$\":gAsPpQQ:9,S\"RFhhm7$Fgen$\":q5x` LF![^ym.QFhhm7$F\\fn$\":&[EpfJ/2iKJ,PFhhm7$Fafn$\":7L*o*zo$z;jYVPFhhm7 $Fffn$\":6QK15*R9`#Q'fQFhhm7$F[gn$\":KVJUv1N&H/5rPFhhm7$F`gn$\":*3*G(= UC%\\cF;p$Fhhm7$Fegn$\":r&e2'p%H\\J#>;!QFhhm7$Fjgn$\":=h8Lk,>G$**QZQFh hm7$F_hn$\":h9`y'\\\"3L(z0UPFhhm7$Fdhn$\":>\"y8_ss%3RFhhm7$F^in$\":J!)>bDi[dND2!QFhhm7$Fcin$\":'=sidWSpPj(4 q$Fhhm7$Fhin$\":\"HLd7;+#QcU)QPFhhm7$F]jn$\":r0O\"QmYQo)R0'QFhhm7$Fbjn $\":Qf7>zx](y'GWv$Fhhm7$Fgjn$\":#)o^cRElFz-]n$Fhhm7$F\\[o$\":Vb%)eNf2f P03#RFhhm7$Fa[o$\":m\"fitNN@9$\\t$Fhhm7$Fj\\o$\":lb`@Q'p1N].iOFhhm7$F_]o$\":$>r(pKx])H*osu$Fh hm7$Fd]o$\":QX0KKOw]'o#Q(QFhhm7$Fi]o$\":0!)46/**>F pwf)R=F`q7$$\":+++]7GQAH7^-\"F]q$!./-Gl-$=F`q7$$\":++++D19%47tO5F]q$!. ^JLPX#=F`q7$$\":++]7y+3(QfjR5F]q$!.qeO9F\">F`q7$$\":+++DJ&>+o1aU5F]q$! .[t[@]\">F`q7$$\":++]P%)*eH(RXa/\"F]q$!.7LL\\5'=F`q7$$\":+++]P%)*eE,N[ 5F]q$!.#pQ&es%=F`q7$$\":+++vVtx^efT0\"F]q$!.Hta1F`q7$$\":++++]ilP/p *f5F]q$!.LvrQY*>F`q7$$\":++++v=<\"yo?$3\"F]q$!.6s3z#o>F`q7$Fgq$!.4'\\- tP>F`q7$F\\r$!.-n<5J%>F`q7$Far$!.7kMNv&>F`q7$Ffr$!.i!y>/\"*>F`q7$F[s$! .T2T@:1#F`q7$F`s$!.&f$yN.3#F`q7$Fes$!.e1_:a-#F`q7$Fjs$!.(3A=^*)>F`q7$F _t$!.v[[S>8#F`q7$Fdt$!.-zY'Hv?F`q7$Fit$!.Ff!4[@?F`q7$F^u$!.CO.D-,#F`q7 $Fcu$!.&[Iu>E@F`q7$Fhu$!.^dA:*>@F`q7$F]v$!.HmY;J6#F`q7$Fbv$!.\"4i8X#3# F`q7$Fgv$!/tg#)pNh?F[w7$F]w$!/'p`h5-@#F[w7$Fbw$!/\\+fDV\\@F[w7$Fgw$!/c a*y%\\\"4#F[w7$F\\x$!/g_1!pv2#F[w7$Fax$!/Y$fs_98#F[w7$Ffx$!/*\\'Qz))e@ F[w7$F[y$!/t%Hh=G<#F[w7$F`y$!/e`^,f8@F[w7$Fey$!/F9HsG\"3#F[w7$Fjy$!/T, H=d9AF[w7$F_z$!/s!H\"zo`@F[w7$Fdz$!/Kdmqr(4#F[w7$Fiz$!/Q9z#F[w7$Fc[l$!/,y\"f$4m@F[w7$Fh[l$!/_i9p'y8#F[w7$F]\\l$!/*) *y59J6#F[w7$Fb\\l$!/je^v?4@F[w7$Fg\\l$!/*zAg.G>#F[w7$F\\]l$!/!\\Gj_!QA F[w7$Fa]l$!/0vgc#f?#F[w7$Ff]l$!/&GK==R<#F[w7$F[^l$!/MMlN%Q9#F[w7$F`^l$ !/(GCdH<7#F[w7$Fe^l$!/:$zRswE#F[w7$Fj^l$!/n&GG;M?#F[w7$F__l$!/w83fYU@F [w7$Fd_l$!/\"3DFew8#F[w7$Fi_l$!/7i5Mlq@F[w7$F^`l$!/gjBGBkAF[w7$Fc`l$!/ _GF!)3&>#F[w7$Fh`l$!0,ynz/!H@F\\al7$F^al$!0SjY:Hu;#F\\al7$Fcal$!0HSz$ \\e)>#F\\al7$Fhal$!0()e%px#G8#F\\al7$F]bl$!0&[cT^Ji?F\\al7$Fbbl$!0+`eY _(=?F\\al7$Fgbl$!0?C([$))z)>F\\al7$F\\cl$!0$yGzVDD?F\\al7$Facl$!0)QF]v Ks?F\\al7$Ffcl$!1#zEI>@#y>Fjcl7$F\\dl$!1/\\,ETB3>Fjcl7$Fadl$!1d'pH,NB( =Fjcl7$Ffdl$!1faAn0YX=Fjcl7$F[el$!1y7]5sBip\"Fjcl 7$Feel$!1+\")yR:y&y\"Fjcl7$Fjel$!1KB]IR$et\"Fjcl7$F_fl$!1d*y#4Q%))o\"F jcl7$Fdfl$!1k=l%)*G4p\"Fjcl7$Fifl$!1`?$4eoCu\"Fjcl7$F^gl$!1O4@zu%Qp\"F jcl7$Fcgl$!1(QXtL'*el\"Fjcl7$Fhgl$!1y^D^Ma[d8>_J_ Fa[m7$Fc[m$\"42y/(44_%)3*)Fa[m7$Fh[m$\"5p\\YcL/$o(37Fa[m7$F]\\m$\"6(*> $z&>(*[!)R^\"Fa\\m7$Fc\\m$\"6to**=:eUbs)=Fa\\m7$Fh\\m$\"7a],zj=#e@i?#F \\]m7$F^]m$\"7S#oSK'**zP)yV#F\\]m7$Fc]m$\"7oK^^miwNj&p#F\\]m7$Fh]m$\"8 -fmT;-apf***HF\\^m7$F^^m$\"8h5fLHLy8y9L$F\\^m7$Fc^m$\"8AID(z48;'p=K$Q$F\\^m7$Fb_m$\"9 i\"=7Q2yOLlbh$Ff_m7$Fh_m$\"9qt+^kYgt8;MNFf_m7$F]`m$\"9EO%QY?O&z&=\")[$ Ff_m7$Fb`m$\"9dvQ9zz^rFDPOFf_m7$Fg`m$\"9MkuV9fG>.SWPFf_m7$F\\am$\"9.]4 i$eSb7![)p$Ff_m7$Faam$\"9j*GEk^dG;-Il$Ff_m7$Ffam$\"9^6lS'*p?Xl.[OFf_m7 $F[bm$\"9o`&)4/SCa6*Rk$Ff_m7$F`bm$\"9+T[l#=CYYsDk$Ff_m7$Febm$\"9iZNi[2 (Qrvq^EGC;QFf_m7$Fd cm$\"9WR@a4yvpq:N3chdx$Ff_m7$Fgem$\"9zZ/QNf3@&QN\" RFf_m7$F\\fm$\"9O_Qq4R6D,\"3(QFf_m7$Fafm$\"9Ek!37Rj^*=(>w$Ff_m7$Fffm$ \"9K:Zs\"HA^,(HVPFf_m7$F[gm$\"91cYX(R?Xu$fMRFf_m7$F`gm$\"9\\MpJf.IZc.A QFf_m7$Fegm$\"9gaZ\")>()Ra#)QLPFf_m7$Fjgm$\"9xq84%prLH1*)*RFf_m7$F_hm$ \"9kC=.'p$H9'4V)QFf_m7$Fdhm$\":,%R+](G;wcmX$QFhhm7$Fjhm$\":hM3>'zDx#)> N!z$Fhhm7$F_im$\":qL!>\\+#pmPm\\'QFhhm7$Fdim$\":SpzeDZa8.Mz&RFhhm7$Fii m$\":`#e&=,i69V6`%QFhhm7$F^jm$\":Ry/cVl/'4Ls$y$Fhhm7$Fcjm$\":\\(y9u4q' RXn`!RFhhm7$Fhjm$\":&oI;B`Q>]\\g^RFhhm7$F][n$\":pGRzN%f.^wcmRFhhm7$Fb[ n$\":X9)*Qp;&>CmQaQFhhm7$Fg[n$\":B$3Q(Gl6cF\\f\"QFhhm7$F\\\\n$\":x8`M$ zF.\"Hl#GSFhhm7$Fa\\n$\":[D,7$pbA%[\\H\"RFhhm7$Ff\\n$\":/MN.iR8Teyp\"Q Fhhm7$F[]n$\":$*HzyQ!Q&\\H-sgs\\s[5SFhhm7$F\\an$\":%4%\\GWn&4oQ#>9%Fhhm7$Faan$\":f7y!)*z `az(oS.%Fhhm7$Ffan$\":V'[!)[-5%4!4$>$RFhhm7$F[bn$\":JE/MyE9lJ'RrRFhhm7 $F`bn$\":-ku&3lz$pT@=5%Fhhm7$Febn$\":J+AL>m\")[\"4J;%Fhhm7$Fdcn$\":c_#)pV')\\ah_d/ %Fhhm7$Ficn$\":!=,pQ4kGqauORFhhm7$F^dn$\":5%fs%R'*>^vw(yRFhhm7$Fcdn$\" :54NF;&e/S(ec5%Fhhm7$Fhdn$\":`IxW\"p=>k>Q!*RFhhm7$F]en$\":'z'HEs2!yhj3 6RFhhm7$Fben$\":gAsPpQQ:%o$f;%Fhhm7$Fgen$\":q5x`LF![^;][SFhhm7$F\\fn$ \":&[EpfJ/2iAbRRFhhm7$Fafn$\":7L*o*zo$z;b;%)RFhhm7$Fffn$\":6QK15*R9`MR 2TFhhm7$F[gn$\":KVJUv1N&Hg$=,%Fhhm7$F`gn$\":*3*G(=UC%\\'='f#RFhhm7$Feg n$\":r&e2'p%H\\J,bUSFhhm7$Fjgn$\":=h8Lk,>G$p$44%Fhhm7$F_hn$\":h9`y'\\ \"3LPO*yRFhhm7$Fdhn$\":>\"y8__%=bTFhhm 7$F^in$\":J!)>bDi[d&ejSSFhhm7$Fcin$\":'=sidWSpPNeMRFhhm7$Fhin$\":\"HLd 7;+#Q1\"fuRFhhm7$F]jn$\":r0O\"QmYQo-`.TFhhm7$Fbjn$\":Qf7>zx](y(R2*RFhh m7$Fgjn$\":#)o^cRElF4xi!RFhhm7$F\\[o$\":Vb%)eNf2fZzo;%Fhhm7$Fa[o$\":m \"fitNN@95*>0%Fhhm7$Ff[o$\":Ma***or(pKx])H]JyRFhhm7$Fd]o$\":QX0KKOw]m +?6%Fhhm7$Fi]o$\":0!)46/**>Fdm*>SFhhm7$F^^o$\":e\\#f.G-h?Ms1RFhhm7$Fc^ o$\":2\"zHjK4o'4R9!RFhhm7$Fh^o$\":GTo$40*o&4RghSFhhm7$F]_o$\":.6gmDUS( f3w%)RFhhm7$Fb_o$\":BS2Ch!G&RM!f4RFhhm7$F`ao$\":EEKqtcj6@j=!QFhhm7$Fea o$\":&e+;jBi8voB.OFhhm7$Fjao$\":[<\\[Rjc*=U/tMFhhm7$F_bo$\":vf[a'4.)*Q U&yJ$Fhhm7$Fdbo$\":\\-(yV#)3'ohhr,$Fhhm7$Fibo$\":.5pgb_m&oR7lEFhhm7$F^ co$\":))[;0#*oi(4:7%H#Fhhm7$Fcco$\":mj\"*Gh,pk>9;&>Fhhm7$Fhco$\":2$fA \"f)Q+53*>e\"Fhhm-Fait6&F`do$\"*++++\"FeitFddoFddo-Fhdo6#%6an~\"effici ent\"~schemeG-F$6%7jx7$F($\"+Ge=9cF-7$F5$!*@%o)o&F-7$F?$!+6OIYeF-7$FD$ !,mmS[\"*)FH7$FJ$!-T2eWG6FH7$FO$!-_![HwR\"FH7$FT$!-jtZ#pj\"FH7$FY$!-W? 5yV;FH7$Fhn$!-sP,'*[;FH7$F]o$!-N-))3XpvLf\"=F`q7$Fgq$!.4'fjw>>F`q7$F\\r$!.-nWac#>F`q7$Far$!.7 kw&\\S>F`q7$Ffr$!.i!eZFu>F`q7$F[s$!.T2B)zW?F`q7$F`s$!.&f$pOO1#F`q7$Fes $!.e1Id\"4?F`q7$Fjs$!.(3UDct>F`q7$F_t$!.v[Xr_6#F`q7$Fdt$!.-zJq!f?F`q7$ Fit$!.Ff)en0?F`q7$F^u$!.COQmX*>F`q7$Fcu$!.&[gO**4@F`q7$Fhu$!.^d\"z:/@F `q7$F]v$!.HmV)y(4#F`q7$Fbv$!.\"4-S/o?F`q7$Fgv$!/tgswtZ?F[w7$F]w$!/'p`? Ef>#F[w7$Fbw$!/\\+R4aN@F[w7$Fgw$!/caps(z2#F[w7$F\\x$!/g_cy'H%o?F[w7$Fjy$!/T,H0<,AF[w7$F_z$!/s!H.b19#F[w7$Fdz$!/Kd'RA ]3#F[w7$Fiz$!/Q9Hz*R6#F[w7$F^[l$!/)z%)3.;=#F[w7$Fc[l$!/,yT%RN:#F[w7$Fh [l$!/_iW'Rd7#F[w7$F]\\l$!/*)*yo\"Q,@F[w7$Fb\\l$!/jer)ex4#F[w7$Fg\\l$!/ *zA#\\G\"=#F[w7$F\\]l$!/!\\GkfmA#F[w7$Fa]l$!/0v+7%\\>#F[w7$Ff]l$!/&GKd JL;#F[w7$F[^l$!/MM0bjL@F[w7$F`^l$!/(GCbb=6#F[w7$Fe^l$!/:$zRhtD#F[w7$Fj ^l$!/n&GX(R$>#F[w7$F__l$!/w8eLsK@F[w7$Fd_l$!/\"3D:rz7#F[w7$Fi_l$!/7i?M ,h@F[w7$F^`l$!/gjVfTaAF[w7$Fc`l$!/_Gm?@F\\al7$F ^al$!0SjY)[1g@F\\al7$Fcal$!0HSzfwA>#F\\al7$Fhal$!0()e%p_jF@F\\al7$F]bl $!0&[crHAe?F\\al7$Fbbl$!0+`eaz\\,#F\\al7$Fgbl$!0?C(G3^%)>F\\al7$F\\cl$ !0$yGRi/A?F\\al7$Facl$!0)QF]GWp?F\\al7$Ffcl$!1#zEI?ef(>Fjcl7$F\\dl$!1/ \\,')Gb1>Fjcl7$Fadl$!1d'pH\"*R>(=Fjcl7$Ffdl$!1faA2[WY=Fjcl7$F[el$!1y7] !*=ujd8z&G,&Fa[m7$Fc[m$\"42y/(44-[i')Fa[m7$Fh[m$\"5p\\ YcL/6<$=\"Fa[m7$F]\\m$\"6(*>$z&>(*)fx([\"Fa\\m7$Fc\\m$\"6to**=:eK!=f=F a\\m7$Fh\\m$\"7a],zj=#QZu<#F\\]m7$F^]m$\"7S#oSK'**z&[&4CF\\]m7$Fc]m$\" 7oK^^miw%3um#F\\]m7$Fh]m$\"8-fmT;-a*eYrHF\\^m7$F^^m$\"8h5fLHLyV]HI$F\\ ^m7$Fc^m$\"8AID(z48;'pL5cLF\\^m7$Fb_m$\"9i\"=7Q2yOtY$Ff_m7$Fb`m$\"9dvQ9zz^rgP5OFf_m7$Fg`m$\"9MkuV9f G>H<vq^EpX!z$Ff_m7$Fdcm$\"9WR@a4yvp$R?p$Ff_m7$Ficm$\"9)=MK3ojVSa$ QOFf_m7$F^dm$\"9UyN3'o&3v $Ff_m7$Fgem$\"9zZ/QNf3@s8))QFf_m7$F\\fm$\"9O_Qq4R6D\"fd%QFf_m7$Fafm$\" 9Ek!37Rj^>Bwt$Ff_m7$Fffm$\"9K:Zs\"HA^\"H3>PFf_m7$F[gm$\"91cYX(R?XM2%4R Ff_m7$F`gm$\"9\\MpJf.IZ%pvz$Ff_m7$Fegm$\"9gaZ\")>()Ra<[4PFf_m7$Fjgm$\" 9xq84%prL*))etRFf_m7$F_hm$\"9kC=.'p$H9xrfQFf_m7$Fdhm$\":,%R+](G;w;d0\" QFhhm7$Fjhm$\":hM3>'zDx#[zow$Fhhm7$F_im$\":qL!>\\+#pm<76%QFhhm7$Fdim$ \":SpzeDZa8V.P$RFhhm7$Fiim$\":`#e&=,i69Bp<#QFhhm7$F^jm$\":Ry/cVl/'4Sbg PFhhm7$Fcjm$\":\\(y9u4q'R&*G<)QFhhm7$Fhjm$\":&oI;B`Q>],$y#RFhhm7$F][n$ \":pGRzN%f.^F$G%RFhhm7$Fb[n$\":X9)*Qp;&>C8KJQFhhm7$Fg[n$\":B$3Q(Gl6c(z 6$z$Fhhm7$F\\\\n$\":x8`M$zF.\"pQW+%Fhhm7$Fa\\n$\":XD,7$pbA%o/)*)QFhhm7 $Ff\\n$\":/MN.iR8T)RR%z$Fhhm7$F[]n$\":$*HzyQ!Q&\\pL'RSFhhm7$F`]n$\":&H ?H()**3nPy%*\\RFhhm7$Fe]n$\":c)eVz3&p,=0`*QFhhm7$Fj]n$\":M[m_88^iuR![Q Fhhm7$F_^n$\":%z:B&e2F:?W2&RFhhm7$Fd^n$\":1(zJQ'y&HynGCSFhhm7$Fi^n$\": A)HGv/z.!3Ai\"RFhhm7$F^_n$\":FaTuNs;R4?;%QFhhm7$Fc_n$\":Q`8)[QqlK\"3Y5 %Fhhm7$Fh_n$\":kF3uu&)p*em$Q*RFhhm7$F]`n$\":8Pj'H8Ac9\"*y))QFhhm7$Fb`n $\":g,R/fEq(o1ssQFhhm7$Fg`n$\":Wt&>-sgs\\Z2*)RFhhm7$F\\an$\":%4%\\GWn& 4oYD?TFhhm7$Faan$\":f7y!)*z`azN^8SFhhm7$Ffan$\":V'[!)[-5%4!*GC\"RFhhm7 $F[bn$\":JE/MyE9l@_<&RFhhm7$F`bn$\":-ku&3lz$pTh<3%Fhhm7$Febn$\":J+AL>m\")[W.VTF hhm7$Fdcn$\":c_#)pV')\\a\"RCESFhhm7$Ficn$\":!=,pQ4kGq#ex\"RFhhm7$F^dn$ \":5%fs%R'*>^DV'fRFhhm7$Fcdn$\":54NF;&e/SG9'3%Fhhm7$Fhdn$\":`IxW\"p=>k KTrRFhhm7$F]en$\":'z'HEs2!yhv[#*QFhhm7$Fben$\":gAsPpQQ:u?k9%Fhhm7$Fgen $\":q5x`LF![^c`HSFhhm7$F\\fn$\":&[EpfJ/2iD4@RFhhm7$Fafn$\":7L*o*zo$z;h blRFhhm7$Fffn$\":6QK15*R9`XV)3%Fhhm7$F[gn$\":KVJUv1N&HH(Q*RFhhm7$F`gn$ \":*3*G(=UC%\\1C*3RFhhm7$Fegn$\":r&e2'p%H\\JH8DSFhhm7$Fjgn$\":=h8Lk,>G VvM2%Fhhm7$F_hn$\":h9`y'\\\"3L(>&>'RFhhm7$Fdhn$\":>\"y8_iSx8%Fhhm7$F^in$\":J!)>bDi[d&GnBSFhhm7$Fcin$\":'=s idWSpP;1=RFhhm7$Fhin$\":\"HLd7;+#Qcfz&RFhhm7$F]jn$\":r0O\"QmYQoDf'3%Fh hm7$Fbjn$\":Qf7>zx](yrEuRFhhm7$Fgjn$\":#)o^cRElF>Y,*QFhhm7$F\\[o$\":Vb %)eNf2fPv*\\TFhhm7$Fa[o$\":m\"fitNN@9IbNSFhhm7$Ff[o$\":Ma***or(pKx ])Hk*Q'RFhhm7$Fd]o$\":QX0KKOw]Ept4%Fhhm7$Fi]o$\":0!)46/**>F(Gp0SFhhm7$ F^^o$\":e\\#f.G-h?%\\G*QFhhm7$Fc^o$\":2\"zHjK4o'*oh()QFhhm7$Fh^o$\":GT o$40*o&4J[ZSFhhm7$F]_o$\":.6gmDUS(fU?rRFhhm7$Fb_o$\":BS2Ch!G&R/#e'*QFh hm7$F`ao$\":EEKqtcj6r))=z$Fhhm7$Feao$\":&e+;jBi8v,c'f$Fhhm7$Fjao$\":[< \\[Rjc*=BrpMFhhm7$F_bo$\":vf[a'4.)*Q#**yJ$Fhhm7$Fdbo$\":\\-(yV#)3'o.l5 -$Fhhm7$Fibo$\":.5pgb_m&oSMsEFhhm7$F^co$\":))[;0#*oi(4(y[I#Fhhm7$Fcco$ \":mj\"*Gh,pk7Lh'>Fhhm7$Fhco$\":2$fA\"f)Q+g0`,;Fhhm-F^do6&F`doFddo$\"# vF*$\"#&*F*-Fhdo6#%Ea~scheme~of~Tsitouras~and~PapakostasG-F$6%7cy7$F($ \"+G%zrx\"F-7$F5$!+@bcKUF-7$F?$!+6MX/(*F-7$FD$!-mY73%G\"FH7$FJ$!-TdjzH :FH7$FO$!-_qq\"e#=FH7$FT$!-jVvZ&4#FH7$FY$!-W]5f#4#FH7$Fhn$!-snGT*3#FH7 $F]o$!-Ns4(y?#FH7$Fbo$!-u$*3'*)>#FH7$Fgo$!-r\"o5p8#FH7$F\\p$!-Uc5bG@FH 7$Fap$!-XB]M*=#FH7$Ffp$!-\"fF5!4BFH7$F[q$!.i#*pbeH#F`q7$Fbq$!.>p4!\\\" G#F`q7$Fb]u$!./-#G\"yE#F`q7$Fg]u$!.^Ji+!fAF`q7$F\\^u$!.qe/+qO#F`q7$Fa^ u$!.[tWU%pBF`q7$Ff^u$!.7LhaEI#F`q7$F[_u$!.#poDD&G#F`q7$F`_u$!.Hth>9N#F `q7$Fe_u$!.LvAD[Y#F`q7$Fj_u$!.6sp$RHCF`q7$Fgq$!.4'4z6*Q#F`q7$F\\r$!.-n R0XR#F`q7$Far$!.7kB;5T#F`q7$Ffr$!.i!3:%4X#F`q7$F[s$!.T2o$>ODF`q7$F`s$! .&f$z,!fDF`q7$Fes$!.e1@U9\\#F`q7$Fjs$!.(37`;ZCF`q7$F_t$!.v[op8i#F`q7$F dt$!.-z(=s^DF`q7$Fit$!.FfcUb[#F`q7$F^u$!.CO#zXrCF`q7$Fcu$!.&[]1B8EF`q7 $Fhu$!.^dGiWg#F`q7$F]v$!.HmR!4&f#F`q7$Fbv$!.\"47S[bDF`q7$Fgv$!/tgK4oFD F[w7$F]w$!/'p`)>J4FF[w7$Fbw$!/\\+R#4[j#F[w7$Fgw$!/ca>2yjDF[w7$F\\x$!/g _Y&)[YDF[w7$Fax$!/Y$f\")[=h#F[w7$Ffx$!/*\\')z:[k#F[w7$F[y$!/t%H78;m#F[ w7$F`y$!/e`\"Hd!*e#F[w7$Fey$!/F9\\HN\\DF[w7$Fjy$!/T,Hy#=r#F[w7$F_z$!/s !H/tsj#F[w7$Fdz$!/Kd1`roDF[w7$Fiz$!/Q9pU!Rg#F[w7$F^[l$!/)z%[%eko#F[w7$ Fc[l$!/,y\"Rs1l#F[w7$Fh[l$!/_i%>d_h#F[w7$F]\\l$!/*)*yk+Te#F[w7$Fb\\l$! /jeT`NyDF[w7$Fg\\l$!/*zAc6#zEF[w7$F\\]l$!/!\\GS]Lt#F[w7$Fa]l$!/0v!HYKp #F[w7$Ff]l$!/&GKc:Ll#F[w7$F[^l$!/MMXRw:EF[w7$F`^l$!/(GCO1ze#F[w7$Fe^l$ !/:$zT!4lFF[w7$Fj^l$!/n&GfRno#F[w7$F__l$!/w8[2T7EF[w7$Fd_l$!/\"3D)yG1E F[w7$Fi_l$!/7i?Q%fk#F[w7$F^`l$!/gj.C8fFF[w7$Fc`l$!/_G(zhJn#F[w7$Fh`l$! 0,ynp.5f#F\\al7$F^al$!0SjY)[[LEF\\al7$Fcal$!0HSz(y&om#F\\al7$Fhal$!0() e%\\4\\$e#F\\al7$F]bl$!0&[c\"y^W\\#F\\al7$Fbbl$!0+`eDQ3W#F\\al7$Fgbl$! 0?C(=xk-CF\\al7$F\\cl$!0$yG>5JYCF\\al7$Facl$!0)QFqGv,DF\\al7$Ffcl$!1#z EI\\;D#Fjcl7$Ffdl$!1fa A23*Q@#Fjcl7$F[el$!1y7]SH#=6#Fjcl7$F`el$!1%e-:\"4CD?Fjcl7$Feel$!1+\")y z)[k8#Fjcl7$Fjel$!1KB]!>$pw?Fjcl7$F_fl$!1d*y#*=g/-#Fjcl7$Fdfl$!1k=lk\\ _A?Fjcl7$Fifl$!1`?$4`.M3#Fjcl7$F^gl$!1O4@>yED?Fjcl7$Fcgl$!1(QXtI+)z>Fj cl7$Fhgl$!1y^D6KS*3#Fjcl7$F]hl$!1fm^ll1O?Fjcl7$Fbhl$!2<.b/&HN$)>Ffhl7$ Fhhl$!2NQ=\\:jCs\"Ffhl7$F]il$!2pXB`h[zW\"Ffhl7$Fbil$!3J+YPEX7T6Ffil7$F hil$!29&45.2Rp!)Ffil7$F]jl$!2;$=5/e\"yj%Ffil7$Fbjl$!2$G-%\\dW?'pFfjl7$ Fhjl$\"3ajz6H\"ewG$Ffjl7$F][m$\"47r(>d8pN`rFa[m7$Fc[m$\"52y/(44-(Go6Fa [m7$Fh[m$\"5p\\YcL/(oTb\"Fa[m7$F]\\m$\"6(*>$z&>(*['GB>Fa\\m7$Fc\\m$\"6 to**=:e_,&zBFa\\m7$Fh\\m$\"7a],zj=#Gpcw#F\\]m7$F^]m$\"7S#oSK'**z+RUIF \\]m7$Fc]m$\"7oK^^miwYu^LF\\]m7$Fh]m$\"8-fmT;-afz'=PF\\^m7$F^^m$\"8h5f LHLyVNt6%F\\^m7$Fc^m$\"8AID(z48;'p/AuTF\\^m7$Fb_m$\"9i\"=7Q2yO`NxX%Ff_m7$Fh_m$\"9qt+ ^kYgt)o^N%Ff_m7$F]`m$\"9EO%QY?O&z5:'H%Ff_m7$Fb`m$\"9dvQ9zz^r5:yWFf_m7$ Fg`m$\"9MkuV9fG>\")e3YFf_m7$F\\am$\"9.]4i$eSb#o(4b%Ff_m7$Faam$\"9j*GEk ^dG1UR\\%Ff_m7$Ffam$\"9^6lS'*p?Xdw'[%Ff_m7$F[bm$\"9o`&)4/SCaWs![%Ff_m7 $F`bm$\"9+T[l#=CYc3zZ%Ff_m7$Febm$\"9iZNi[2(Q\"RG#[%Ff_m7$Fjbm$\"9b&Ha[ dfE=V%oZFf_m7$F_cm$\"9)*p%)>vq^E'p#*o%Ff_m7$Fdcm$\"9WR@a4yvp5^nXFf_m7$ Ficm$\"9)=MK3ojVqN4]%Ff_m7$F^dm$\"9UyN3wLoj%Ff_m7$Fgem$\"9zZ/QNf3@id/[Ff_m7$F\\fm$\"9O_Qq4R6D+(> v%Ff_m7$Fafm$\"9Ek!37Rj^fX$=YFf_m7$Fffm$\"9K:Zs\"HA^\"46&f%Ff_m7$F[gm$ \"91cYX(R?Xu!=H[Ff_m7$F`gm$\"9\\MpJf.IZ)H5p%Ff_m7$Fegm$\"9gaZ\")>()RaD 9#e%Ff_m7$Fjgm$\"9xq84%prL\\4q!\\Ff_m7$F_hm$\"9kC=.'p$H9$*QmZFf_m7$Fdh m$\":,%R+](G;wO\"H/ZFhhm7$Fjhm$\":hM3>'zDx#3#)*[YFhhm7$F_im$\":qL!>\\+ #pmFJ*RZFhhm7$Fdim$\":SpzeDZa88sM&[Fhhm7$Fiim$\":`#e&=,i69.j`r%Fhhm7$F ^jm$\":Ry/cVl/'4-mRYFhhm7$Fcjm$\":\\(y9u4q'RvXzy%Fhhm7$Fhjm$\":&oI;B`Q >]x$Q%[Fhhm7$F][n$\":pGRzN%f.^w*='[Fhhm7$Fb[n$\":X9)*Qp;&>CzQCZFhhm7$F g[n$\":B$3Q(Gl6c2@qn%Fhhm7$F\\\\n$\":x8`M$zF.\"HGk$\\Fhhm7$Fa\\n$\":[D ,7$pbA%39^z%Fhhm7$Ff\\n$\":-MN.iR8TQSun%Fhhm7$F[]n$\":'*HzyQ!Q&\\*)f%y \\Fhhm7$F`]n$\":#H?H()**3nPF%y'[Fhhm7$Fe]n$\":c)eVz3&p,QBxz%Fhhm7$Fj]n $\":M[m_88^iaTnt%Fhhm7$F_^n$\":%z:B&e2F:q:C'[Fhhm7$Fd^n$\":1(zJQ'y&Hy. I_\\Fhhm7$Fi^n$\":A)HGv/z.!y8$>[Fhhm7$F^_n$\":FaTuNs;RR(RFZFhhm7$Fc_n$ \":Q`8)[QqlK(*o\\]Fhhm7$Fh_n$\":kF3uu&)p*eJT8\\Fhhm7$F]`n$\":8Pj'H8Ac9 !pTy%Fhhm7$Fb`n$\":g,R/fEq(o-3kZFhhm7$Fg`n$\":Wt&>-sgs\\(y_!\\Fhhm7$F \\an$\":%4%\\GWn&4oKck]Fhhm7$Faan$\":f7y!)*z`azn^I\\Fhhm7$Ffan$\":V'[! )[-5%4SXN![Fhhm7$F[bn$\":JE/MyE9l,x7&[Fhhm7$F`bn$\":-ku&3lz$pJ:+,&Fhhm 7$Febn$\":J+ AL>m\")[*\\P3&Fhhm7$Fdcn$\":c_#)pV')\\a\"4WS\\Fhhm7$Ficn$\":!=,pQ4kGqX I2[Fhhm7$F^dn$\":5%fs%R'*>^lQ\"e[Fhhm7$Fcdn$\":54NF;&e/Sc[7]Fhhm7$Fhdn $\":`IxW\"p=>k[ur[Fhhm7$F]en$\":'z'HEs2!yhf![x%Fhhm7$Fben$\":gAsPpQQ:k \\\\3&Fhhm7$Fgen$\":q5x`LF![^ygT\\Fhhm7$F\\fn$\":&[EpfJ/2i/g3[Fhhm7$Fa fn$\":7L*o*zo$z;Kbi[Fhhm7$Fffn$\":6QK15*R9`oP7]Fhhm7$F[gn$\":KVJUv1N&H 'yN*[Fhhm7$F`gn$\":*3*G(=UC%\\1Omy%Fhhm7$Fegn$\":r&e2'p%H\\J!)3G\\Fhhm 7$Fjgn$\":=h8Lk,>G8'p')\\Fhhm7$F_hn$\":h9`y'\\\"3Lnn,&[Fhhm7$Fdhn$\":> \"y8_UiQ1&Fhhm7$F^in$\":J!)>bDi[dbjU# \\Fhhm7$Fcin$\":'=sidWSpPV*\\z%Fhhm7$Fhin$\":\"HLd7;+#QccK%[Fhhm7$F]jn $\":r0O\"QmYQoEx**\\Fhhm7$Fbjn$\":Qf7>zx](yfMi[Fhhm7$Fgjn$\":#)o^cRElF >J$fZFhhm7$F\\[o$\":Vb%)eNf2fn&zv]Fhhm7$Fa[o$\":m\"fitNN@9n%e$\\Fhhm7$ Ff[o$\":Ma***o-,&Fhhm7$Fe\\o$\":G8^GwCb[#3KH[Fhhm7$Fj\\o$\":lb`@Q'p1Nd\"3t% Fhhm7$F_]o$\":$>r(pKx])HJKR[Fhhm7$Fd]o$\":QX0KKOw]O:3+&Fhhm7$Fi]o$\":0 !)46/**>FPD)))[Fhhm7$F^^o$\":e\\#f.G-h?F4^ZFhhm7$Fc^o$\":2\"zHjK4o'HtU u%Fhhm7$Fh^o$\":GTo$40*o&4zFQ\\Fhhm7$F]_o$\":.6gmDUS(f*yO%[Fhhm7$Fb_o$ \":BS2Ch!G&RaX6v%Fhhm7$F`ao$\":EEKqtcj6,#p4YFhhm7$Feao$\":&e+;jBi8v[!y N%Fhhm7$Fjao$\":[<\\[Rjc*=R+)=%Fhhm7$F_bo$\":vf[a'4.)*Q]R))RFhhm7$Fdbo $\":\\-(yV#)3'o%ea6OFhhm7$Fibo$\":.5pgb_m&3EWvJFhhm7$F^co$\":))[;0#*oi (*zWar#Fhhm7$Fcco$\":mj\"*Gh,pkBY*)G#Fhhm7$Fhco$\":2$fA\"f)Q+q\\#[#=Fh hm-Fait6&F`doFgfwFddoFgfw-Fhdo6#%Ra~modification~of~the~Tsitouras-Papa kostas~schemeG-F$6%7i[l7$F($!+sF,jMF-7$F5$!+@!*pVGF-7$F?$!+6=e0EF-7$$ \":++++](o/opUawF1$!+&H[i^#F-7$FD$!,m19%[CFH7$$\":++++]iS;\\Gp1)F1$!,D .z'*R#FH7$FJ$!,T(3DFH7$$\":+++]P4@;* o;J$)F1$!,9OdzW#FH7$$\":+++D1*Gr52;g$)F1$!,;\"3_!Q#FH7$$\":++++vo/)HX: *Q)F1$!,:EQ*H9*F1$!,ku+EV#FH7$FT$!,j$**[XDFH7$FY$!,W5i7[#FH7$Fhn$!,s(*=(HCFH 7$F]o$!,N-`@b#FH7$Fbo$!,u`!oODFH7$Fgo$!,r\"H3lCFH7$F\\p$!,U'*zBX#FH7$F ap$!,XL,t]#FH7$Ffp$!,\"flGFEFH7$F[q$!-i#pKtf#F`q7$Fbq$!->p#Qzc#F`q7$Fb ]u$!-/-u/TDF`q7$Fg]u$!-^J#Q6_#F`q7$F\\^u$!-qe9tNEF`q7$Fa^u$!-[t\"[]j#F `q7$Ff^u$!-7Ly!3c#F`q7$F[_u$!-#py\\4a#F`q7$F`_u$!-Htg&pg#F`q7$Fe_u$!-L v#=cs#F`q7$Fj_u$!-6s5VuEF`q7$Fgq$!-4'p>=i#F`q7$F\\r$!--n#oYi#F`q7$Far$ !-7kkJSEF`q7$Ffr$!-i![?>o#F`q7$F[s$!-T2!oEx#F`q7$F`s$!-&f$p'fz#F`q7$Fe s$!-e1>=AFF`q7$Fjs$!-(3KhWn#F`q7$F_t$!-v[>/jGF`q7$Fdt$!--zM(py#F`q7$Fi t$!-Ff2z9FF`q7$F^u$!-COpU+FF`q7$Fcu$!-&[S`M&GF`q7$Fhu$!-^d@^VGF`q7$F]v $!-Hm=@LGF`q7$Fbv$!-\"4AU1z#F`q7$Fgv$!.tg_$QiFF[w7$F]w$!.'p`lGhHF[w7$F bw$!.\\+4b)zGF[w7$Fgw$!.ca4?B!GF[w7$F\\x$!.g_;t[y#F[w7$Fax$!.Y$f#*)f&G F[w7$Ffx$!.*\\')49%*GF[w7$F[y$!.t%H\"GF[w7$Fi z$!.Q94+C&GF[w7$F^[l$!.)z%QIA%HF[w7$Fc[l$!.,yhB[!HF[w7$Fh[l$!._ia:z'GF [w7$F]\\l$!.*)*yw'f$GF[w7$Fb\\l$!.je^DF$GF[w7$Fg\\l$!.*zA?.ZHF[w7$F\\] l$!.!\\GO_3IF[w7$Fa]l$!.0v!p!o'HF[w7$Ff]l$!.&GK;GDHF[w7$F[^l$!.MMDhk)G F[w7$F`^l$!.(GC.yeGF[w7$Fe^l$!.:$z+`cIF[w7$Fj^l$!.n&G;#*pHF[w7$F__l$!. w8Q#*y)GF[w7$Fd_l$!.\"3DF:$)GF[w7$Fi_l$!.7i?0v#HF[w7$F^`l$!.gj8`b0$F[w 7$Fc`l$!._G(4(f'HF[w7$Fh`l$!/,yn\\l!)GF\\al7$F^al$!/SjY!*oUHF\\al7$Fca l$!/HSzhD'*HF\\al7$Fhal$!/()e%p/g\"HF\\al7$F]bl$!/&[c@L#HGF\\al7$Fbbl$ !/+`eT/sFF\\al7$Fgbl$!/?C(30It#F\\al7$F\\cl$!/$yGH0&)y#F\\al7$Facl$!/) QF]nf&GF\\al7$Ffcl$!0#zEIRcJFFjcl7$F\\dl$!0/\\,'yOTEFjcl7$Fadl$!0d'pH2 U/EFjcl7$Ffdl$!0faAZ6Ae#Fjcl7$F[el$!0y7]q#4sCFjcl7$F`el$!0%e-:)G1Q#Fjc l7$Feel$!0+\")yH/`^#Fjcl7$Fjel$!0KB]?`\\W#Fjcl7$F_fl$!0d*y#>>*yBFjcl7$ Fdfl$!0k=l%y(RQ#Fjcl7$Fifl$!0`?$4#)zdCFjcl7$F^gl$!0O4@4K#*Q#Fjcl7$Fcgl $!0(QXtdQOBFjcl7$Fhgl$!0y^D^y*pCFjcl7$F]hl$!0fm^NX?T#Fjcl7$Fbhl$!1<.b/ \\taBFfhl7$Fhhl$!1NQ=\\KZ'3#Ffhl7$F]il$!1pXB`Co:=Ffhl7$Fbil$!2J+YPEY> \\\"Ffil7$Fhil$!29&45.nNb6Ffil7$F]jl$!1;$=5/)4oyFfil7$Fbjl$!2$G-%\\d%o 6QFfjl7$Fhjl$\"1ajz6H6^TFfjl7$F][m$\"37r(>d8pnm%Fa[m7$Fc[m$\"32y/(44sA V*Fa[m7$Fh[m$\"4p\\YcL/[$z&>(*3-)y\"Fa\\m7$Fc\\m $\"5to**=:e#*Q!G#Fa\\m7$Fh\\m$\"6a],zj=#ea7FF\\]m7$F^]m$\"6S#oSK'**zOs .$F\\]m7$Fc]m$\"6oK^^miw;UR$F\\]m7$Fh]m$\"7-fmT;-ap*)4QF\\^m7$F^^m$\"7 h5fLHLy$\\fE%F\\^m7$Fc^m$\"7AID(z48;'p5\\N%F\\^m7$Fb_m$\"8i\"=7Q2yOB5gYFf_m7$Fh_m$\"8qt +^kYgtE9c%Ff_m7$F]`m$\"8EO%QY?O&zQ(4XFf_m7$Fb`m$\"8dvQ9zz^r@zq%Ff_m7$F g`m$\"8MkuV9fG>b([[Ff_m7$F\\am$\"8.]4i$eSbsP#z%Ff_m7$Faam$\"8j*GEk^dGO ^OZFf_m7$Ffam$\"8^6lS'*p?XYJt%Ff_m7$F[bm$\"8o`&)4/SCaQ5t%Ff_m7$F`bm$\" 8+T[l#=CYEZKZFf_m7$Febm$\"8iZNi[2(Q^uTZFf_m7$Fjbm$\"8b&Ha[dfE[tZ]Ff_m7 $F_cm$\"8)*p%)>vq^E7O'\\Ff_m7$Fdcm$\"8WR@a4yvpL[$[Ff_m7$Ficm$\"8)=MK3o jV?WmZFf_m7$F^dm$\"8Uy[Ff_m7$Fbem$\"84$3E**>N3w# 4#\\Ff_m7$Fgem$\"8zZ/QNf3@FK5&Ff_m7$F\\fm$\"8O_Qq4R6D@o/&Ff_m7$Fafm$\" 8Ek!37Rj^H40\\Ff_m7$Fffm$\"8K:Zs\"HA^\"[O)[Ff_m7$F[gm$\"81cYX(R?XW0L^F f_m7$F`gm$\"8\\MpJf.IZLi)\\Ff_m7$Fegm$\"8gaZ\")>()RaT<([Ff_m7$Fjgm$\"8 xq84%prL\\1?_Ff_m7$F_hm$\"8kC=.'p$H9u/2&Ff_m7$Fdhm$\"9,%R+](G;w'\\&3]F hhm7$Fjhm$\"9iM3>'zDx#[7a\\Fhhm7$F_im$\"9sL!>\\+#pmdDb]Fhhm7$Fdim$\"9T pzeDZa8jtv^Fhhm7$Fiim$\"9[#e&=,i69t`G]Fhhm7$F^jm$\"9Qy/cVl/'4x+&\\Fhhm 7$Fcjm$\"9[(y9u4q'RD**4^Fhhm7$Fhjm$\"9$oI;B`Q>]lX<&Fhhm7$F][n$\"9pGRzN %f.^\"4$>&Fhhm7$Fb[n$\"9U9)*Qp;&>Cbj/&Fhhm7$Fg[n$\"9=$3Q(Gl6cPE`Fhhm7$F`]n$\"9&H?H()**3nP% y2_Fhhm7$Fe]n$\"9c)eVz3&p,y+T^Fhhm7$Fj]n$\"9M[m_88^ikO%3&Fhhm7$F_^n$\" 9%z:B&e2F:?WB_Fhhm7$Fd^n$\"91(zJQ'y&HyW#>`Fhhm7$Fi^n$\"9A)HGv/z.!=Yw^F hhm7$F^_n$\"9FaTuNs;R\\Tz]Fhhm7$Fc_n$\"9Q`8)[QqlKV\"GaFhhm7$Fh_n$\"9kF 3uu&)p*ea;G&Fhhm7$F]`n$\"98Pj'H8Ac9jH9&Fhhm7$Fb`n$\"9g,R/fEq(o+Y7&Fhhm 7$Fg`n$\"9Wt&>-sgs\\P?G&Fhhm7$F\\an$\"9%4%\\GWn&4o\\qX&Fhhm7$Faan$\"9f 7y!)*z`az45K&Fhhm7$Ffan$\"9V'[!)[-5%4Id#>&Fhhm7$F[bn$\"9JE/MyE9lJC[_Fh hm7$F`bn$\"9-ku&3lz$pJj>aFhhm7$Febn$\"9J+AL>m\")[^P]&Fhhm7$Fdcn$\"9c_#)pV')\\a^g[` Fhhm7$Ficn$\"9!=,pQ4kGqu[?&Fhhm7$F^dn$\"95%fs%R'*>^b,k_Fhhm7$Fcdn$\"95 4NF;&e/S,4V&Fhhm7$Fhdn$\"9`IxW\"p=>kj%y_Fhhm7$F]en$\"9'z'HEs2!yh3_<&Fh hm7$Fben$\"9gAsPpQQ:9z8bFhhm7$Fgen$\"9q5x`LF![^i$e`Fhhm7$F\\fn$\":Z[Ep fJ/2i_X@&!#U7$Fafn$\":AJ$*o*zo$z;\">x_F\\g_l7$Fffn$\":8\"QK15*R9`@%RaF \\g_l7$F[gn$\":CL9Ban]`H/!>`F\\g_l7$F`gn$\":$*3*G(=UC%\\'4=@&F\\g_l7$F egn$\":9de2'p%H\\JO,P&F\\g_l7$Fjgn$\":%=h8Lk,>GjFLaF\\g_l7$F_hn$\":9YJ &y'\\\"3L&F\\g_l7$Fihn$\":e@FXU$> 8_i\"=_&F\\g_l7$F^in$\":8.)>bDi[dbfp`F\\g_l7$Fcin$\":k=sidWSpPj!H_F\\g _l7$Fhin$\":5HLd7;+#Qwx&G&F\\g_l7$F]jn$\":4dg8QmYQocjX&F\\g_l7$Fbjn$\" :%Qf7>zx](y>kI&F\\g_l7$Fgjn$\":<)o^cRElF4[&>&F\\g_l7$F\\[o$\":LaX)eNf2 f(RQa&F\\g_l7$Fa[o$\":c;fitNN@9')4R&F\\g_l7$Ff[o$\":WV&***o&F\\g_l7$F_]o$\":H> r(pKx])H:>J&F\\g_l7$Fd]o$\":(QX0KKOw]O-%\\&F\\g_l7$Fi]o$\":a+)46/**>F2 xq`F\\g_l7$F^^o$\":\"e\\#f.G-h?-(>_F\\g_l7$Fc^o$\":p5zHjK4o'R!f@&F\\g_ l7$Fh^o$\":\"GTo$40*o&44(HaF\\g_l7$F]_o$\":I5,mcA/ufa-L&F\\g_l7$Fb_o$ \":L-uS71G&Ru+L_F\\g_l7$Fg_o$\":XE[`An<\\Aaa;&F\\g_l7$F\\`o$\":^C!H6j' 4@**3m7&F\\g_l7$Fa`o$\":fEN4o!*33JZY=&F\\g_l7$Ff`o$\":;MKS,KVCW;3W&F\\ g_l7$F[ao$\":TQ)HP7T!*fQyx_F\\g_l7$F`ao$\":kiA.PnN;6=$=^F\\g_l7$$\":++ +D\"GjB3q9yUF]q$\":k-6h'z\\s/6`^]F\\g_l7$$\":+++]ils48I5G%F]q$\":n>p\" =1STiiDH%F]q$\":d(*oI#4!3Xud^A&F\\g _l7$$\":+++voHaYuXaH%F]q$\":%GP!GYoap\\Q)y]F\\g_l7$$\":++++]i!Rn)G$)H% F]q$\":9J#o]*zm_et.&F\\g_l7$Feao$\":de+;jBi8v4H)[F\\g_l7$Fjao$\":#[<\\[Rjc*=s@u%F\\ g_l7$F_bo$\":S(f[a'4.)*QMLc%F\\g_l7$Fdbo$\":)[-(yV#)3'o`l%>%F\\g_l7$Fi bo$\":G+\"pgb_m&3'fXPF\\g_l7$F^co$\":\"))[;0#*oi(RQcF$F\\g_l7$Fcco$\": kO;*Gh,pkP)f%GF\\g_l7$Fhco$\":lIfA\"f)Q+!40%R#F\\g_l-F^do6&F`doF]az$\" #bF*Fddo-Fhdo6#%AVerner's~\"most~efficient\"~schemeG-%%FONTG6$%*HELVET ICAGFfdo-%&TITLEG6#%Verror~curves~for~12~stage~order~8~Runge-Kutta~met hodsG-%+AXESLABELSG6$Q\"x6\"Q!F_e`l-%%VIEWG6$;F(Fhco;$!\"$!#<$FicoFhe` l" 1 2 0 1 10 0 2 9 1 4 2 1.000000 45.000000 45.000000 0 1 "Enright-Ve rner scheme" "Verner's Maple scheme" "a modification of Verner's Maple scheme" "an \"efficient\" scheme" "a scheme of Tsitouras and Papakost as" "a modification of the Tsitouras-Papakostas scheme" "Verner's \"mo st efficient\" scheme" }}}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 49 "Test 15 of 12 stage, order 8 Runge-Kutta methods " }} {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 198 "See: Com paring Numerical Methods for Ordinary Differential Equations, Hull, En right, Fellen and Sedgwick,\n Siam Journal on Numerical Analysi s, Vol. 9, No. 4 (Dec. 1972), page 617, Example A5." }}{PARA 0 "" 0 " " {TEXT -1 0 "" }}{PARA 256 "" 0 "" {TEXT -1 3 " " }{XPPEDIT 18 0 "d y/dx = (y-x)/(y+x);" "6#/*&%#dyG\"\"\"%#dxG!\"\"*&,&%\"yGF&%\"xGF(F&,& F+F&F,F&F(" }{TEXT -1 8 ", " }{XPPEDIT 18 0 "y(1) = 1;" "6#/-%\" yG6#\"\"\"F'" }{TEXT -1 1 " " }}{PARA 0 "" 0 "" {TEXT -1 10 "Solution: " }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "2*ln((x^2+y^2)/( x^2))+4*arctan(y/x)+4*ln*x-2*ln*2-Pi = 0;" "6#/,,*&\"\"#\"\"\"-%#lnG6# *&,&*$%\"xGF&F'*$%\"yGF&F'F'*$F.F&!\"\"F'F'*&\"\"%F'-%'arctanG6#*&F0F' F.F2F'F'*(F4F'F)F'F.F'F'*(F&F'F)F'F&F'F2%#PiGF2\"\"!" }{TEXT -1 2 ". \+ " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 73 "de := diff(y(x),x)=(y(x)-x)/(y(x)+x);\nic := y(1)=1;\ndsolve( \{de,ic\},y(x));" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%#deG/-%%diffG6$- %\"yG6#%\"xGF,*&,&F)\"\"\"F,!\"\"F/,&F)F/F,F/F0" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%#icG/-%\"yG6#\"\"\"F)" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/-%\"yG6#%\"xG-%'RootOfG6#,,*&\"\"#\"\"\"-%#lnG6#*&,&*$)F'F-F.F. *$)%#_ZGF-F.F.F.F'!\"#F.!\"\"*&\"\"%F.-%'arctanG6#*&F8F.F'F:F.F:*&F " 0 "" {MPLTEXT 1 0 1 ";" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 41 "The solution can be given more simply as " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 " ln(x^2+y^2)+2*arctan(y/x)=ln*2+Pi/2" "6#/,&-%#lnG6#,&*$%\"xG\"\"#\"\" \"*$%\"yGF+F,F,*&F+F,-%'arctanG6#*&F.F,F*!\"\"F,F,,&*&F&F,F+F,F,*&%#Pi GF,F+F4F," }{TEXT -1 2 ". " }}{PARA 0 "" 0 "" {TEXT -1 18 "The solutio n (for " }{TEXT 267 1 "x" }{TEXT -1 47 " increasing) is the section of the polar curve " }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 " r=sqrt(2)*exp(Pi/4-theta)" "6#/%\"rG*&-%%sqrtG6#\"\"#\"\"\"-%$expG6#,& *&%#PiGF*\"\"%!\"\"F*%&thetaGF2F*" }{TEXT -1 5 ", " }{XPPEDIT 18 0 "-Pi/4<=theta" "6#1,$*&%#PiG\"\"\"\"\"%!\"\"F)%&thetaG" }{XPPEDIT 18 0 "``<=Pi/4" "6#1%!G*&%#PiG\"\"\"\"\"%!\"\"" }{TEXT -1 2 ". " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 7 "Check: " }} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 60 "ln((x^2+y^2))+2*arctan(y/x)= ln(2)+Pi/2;\nimplicitdiff(%,y,x);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/ ,&-%#lnG6#,&*$)%\"xG\"\"#\"\"\"F-*$)%\"yGF,F-F-F-*&F,F--%'arctanG6#*&F 0F-F+!\"\"F-F-,&-F&6#F,F-*&F,F6%#PiGF-F-" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#,$*&,&%\"xG\"\"\"%\"yG!\"\"F',&F(F'F&F'F)F)" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 471 "p1 := plot( [sqrt(2)*exp(Pi/4-t),t,t=-Pi/4..Pi/4],coords=polar,thickness=2,color=r ed):\np2 := plot([sqrt(2)*exp(Pi/4-t),t,t=Pi/4..2*Pi],coords=polar,col or=black,linestyle=2):\np3 := plot([sqrt(2)*exp(Pi/4-t),t,t=-Pi/3..-Pi /4],coords=polar,color=black,linestyle=2):\np4 := plot([[[1,1],[uu,-uu ]]$4],style=point,symbol=[circle$2,diamond,cross],\n \+ symbolsize=[12,10$3],color=[black,green$3]):\nplots[display]([p1,p2,p3 ,p4],font=[HELVETICA,9],labels=[`x`,`y(x)`]);" }}{PARA 13 "" 1 "" {GLPLOT2D 567 520 520 {PLOTDATA 2 "6,-%'CURVESG6%7S7$$\"3F_`'4Qx/\"[!# <$!3\"o*>&>Qx/\"[F*7$$\"3.\"H4?nl\\![F*$!3+OcK;[q'[%F*7$$\"3[WM^6Oe\"z %F*$!34+VrCq:9UF*7$$\"3m#o2Wj>ww%F*$!3B=.>Xj!y\"RF*7$$\"3bf)4;S8Yt%F*$ !3:ymd`!*HIOF*7$$\"3U*G$3T;c$p%F*$!3i\"z4>R][N$F*7$$\"3)>>'Q`Nt[YF*$!3 pIdcz?N/uDVF*$!3Mk<%4i8_'>F*7$$\"3neCR^2fVUF*$!3U`V:L2,dAqSF*$!3]0%Rz+R[P\"F*7 $$\"3CpGt*G2)))RF*$!3Vje1<#>j@\"F*7$$\"31&*>Xy@@*)QF*$!3KrYY*z8q.\"F*7 $$\"3m&*QGM(*y-QF*$!3Z+5yDwaF*)!#=7$$\"3x^E')y`D+PF*$!3%[->)G'=aL(F\\q 7$$\"3v7s?m@*zg$F*$!3c,0?p.L,gF\\q7$$\"3'*yXFYgX0NF*$!3H?*=wI4nh%F\\q7 $$\"3W4K\"RWEoS$F*$!3%3mwkU\\IP$F\\q7$$\"3sJbIp3<.LF*$!3OMj)e)4)4:#F\\ q7$$\"3$p#*\\.zXv?$F*$!3gM#f*yWs%4\"F\\q7$$\"368R(=GrT5$F*$!3oLM#4e`nS #!#?7$$\"3O349%H!z'*HF*$\"3?)\\7Z#>+:5F\\q7$$\"3_&4uxb.N!HF*$\"3?;dqg# **3'=F\\q7$$\"3Q!f))4wMJ!GF*$\"3Rn1jA(f_r#F\\q7$$\"3`xr$)p%\\+q#F*$\"3 7>6@EK]NNF\\q7$$\"3;#Rcs]s**f#F*$\"3]^E27l!)yUF\\q7$$\"3Y 0BF*$\"3I%H$\\j$zL='F\\q7$$\"3**>nP`dw1AF*$\"30SJ7l:PHnF\\q7$$\"30Xcw+ 8\"*=@F*$\"3CV>J\"e5-=(F\\q7$$\"3Xb+T*Q]zR\"\\ti(F\\q7$$ \"3\"GM^D8wq$>F*$\"3\\$HR%\\7H1!)F\\q7$$\"3OsIgb1\\Z=F*$\"3#Q*3z#yPy=*F\\q7$$\"3tj&H(oDW3:F*$\"3 !)e5j3rW%R*F\\q7$$\"3Hm+#y!yfG9F*$\"3U,&*3jpEm&*F\\q7$$\"3YmX)QzoqN\"F *$\"3]k9op*4mp*F\\q7$$\"3s5rrvvGx7F*$\"3P5fR'>0`\")*F\\q7$$\"3XZ9deE%z ?\"F*$\"3Gc$*zYS?&*)*F\\q7$$\"30pa^\"yv$))F\\q$\"3]-mK&*3^l**F\\q7$$\"3eM?/#4dlu(F\\q$\"3uEm +DD_n)*F\\q7$$\"3%*yjGB$o\\&oF\\q$\"3AaVWo?rO(*F\\q7$$\"3[>li88H;gF\\q $\"3w@=8&*eNF\\q$\"3yNhC*4%R#y)F\\q7$$ \"3#*HGP\"4-3'GF\\q$\"3/%=\\C*>ai%)F\\q7$$\"3E\"4(yFv\\CAF\\q$\"3&z[> \"H4QC\")F\\q7$$\"3&3B;*phCW;F\\q$\"3u\"3vLbP3x(F\\q7$$\"3SY8\"*=2Ba6F \\q$\"3unuMyI\"GV(F\\q7$$\"3'Qw\"f#\\*)H3(!#>$\"3)pj-cf?z3(F\\q7$$\"3f (3R5[t$3HFc_l$\"3f7r>seQEnF\\q7$$!37`UU,+,&Q)F`s$\"3yN-Jd*yHO'F\\q7$$! 3$)3\"RF\\q$\"3#4c;J=K@(HF\\q7$$!3iuBr]$3f1#F\\ q$\"3Gm]`0-<5CF\\q7$$!3G4mSr>(e2#F\\q$\"30UK<,&yQ#>F\\q7$$!3#o3.QX91/# F\\q$\"3P$*fjX!f&H:F\\q7$$!3#y(eD!G(*f&>F\\q$\"3UEc#G.d\"=6F\\q7$$!3=q le`Y)o&=F\\q$\"37JSjs:(p;)Fc_l7$$!30utN:SE>B.UUNGWgF`s7$$!3zWz&))>R47\"F\\q$!3Unq>(fHx'=F c_l7$$!3_y,&)zs9)3t![Fc_l$!3_$o;W4rzI%Fc_l7$$!3E\\<<**)*)Rz$Fc_l$!3 e#*))[vi%\\I%Fc_l7$$!3>P#z)H$>z\"HFc_l$!3)y!G_rWw*>%Fc_l7$$!3At>+u$>i< #Fc_l$!3'oQ0i'\\&Q-%Fc_l7$$!3Q^mSZRTp9Fc_l$!3_uFr$)*ycw$Fc_l7$$!3-[9Nr!)oebDFc_l7$$ \"3Wcg/C\"='=WF`s$!37)o9o#3(GD#Fc_l7$$\"33dhy:jnDiF`s$!3?g29(Qlz%>Fc_l 7$$\"3-K8`c=f([(F`s$!35q&z[NHem\"Fc_l7$$\"3ckwjCv\"HN)F`s$!3EJf'zD?4R \"Fc_l7$$\"3*>iL/L$Q;))F`s$!3I#Q\\]N!=[6Fc_l7$$\"3[Kf]CL_\")*)F`s$!36H bzi()GSn;F`s7$$\"3/Xu8A%33 ['F`s$!3;Is^'z2(yxFehl7$$\"3)Q4tG5rBz&F`s$\"3c0'H(*oS@v%!#H-Fjz6&F\\[l Fa[lFa[lFa[l-%*LINESTYLEGFd[l-F$6%7S7$$\"3W'4ORO![>WF*$!3Ar[\"*HXwawF* 7$$\"3VdsTJRpPWF*$!3M'>\"=)=2ge(F*7$$\"3'G4%*HMRJX%F*$!38,ZT))oTEvF*7$ $\"3_L#[)Gh1qWF*$!3SWWx3PmfuF*7$$\"3_-)f6JNm[%F*$!3#o7ACd\\FR(F*7$$\"3 akH:B>m-XF*$!3:r-1\\\\VEtF*7$$\"3C*o#*G!46\" Q)[RgXF*$!35r.'eb*[sqF*7$$\"3I*G!GQ%[pF*7$$\"3u*H#3E-h*f%F*$!3wdnf&y\"pUX'F*7$$\"3CLFg!*\\T'o%F*$!3WZZJr$p:R'F*7$$\"39!\\a\\())R&p% F*$!3JUA:p?9KjF*7$$\"3*yZODQ#Q/ZF*$!3eR\\wA`UqiF*7$$\"3#Gs'H4VG7ZF*$!3 @Fo#z)3,9iF*7$$\"3zqGL?zV?ZF*$!3[?z6=2W`hF*7$$\"3,4vY0\\]GZF*$!3')*GM% 4S$34'F*7$$\"3<`G9ym>NZF*$!3T9'Qi<\"fOgF*7$$\"3J@Tq\"p\"3UZF*$!3!y4%3& *fFyfF*7$$\"3w'3pk%[#)[ZF*$!3<#*G&H-C$=fF*7$$\"31ni$>nh]v%F*$!3CB5%zdh *feF*7$$\"3G2i'*30wgZF*$!3$H7(43_w.eF*7$$\"3)Q\"Q9p_qmZF*$!3a]Y_qGoTdF *7$$\"3[]LaAnqrZF*$!3kKO%\\U\"='o&F*7$$\"3!zPO+/&pwZF*$!3R2\")Hxq@FcF* 7$$\"3oO**41X!4y%F*$!3Ayw?\\-0ubF*7$$\"3G3]Qj><&y%F*$!3t<&zB%>@;bF*7$$ \"3?1\"fYJr))y%F*$!3Crd]bY1iaF*7$$\"3eA/XIST#z%F*$!3Z>]2+%\\dS&F*7$$\" 392#[Idgbz%F*$!3Y'QI)[?(4N&F*7$$\"3gZd(yDA&)z%F*$!3#*=D?KK#RH&F*7$$\"3 ZJ;\"pbc5![F*$!34Ww)4ym#R_F*7$$\"3d)\\>CyGL![F*$!3AMPAGam$=&F*7$$\"3B_ c%y]m_![F*$!3C%yzJ*p\")G^F*7$$\"3'*o#foxsn![F*$!3?qs#R*Qny]F*7$$\"37Ot >[%z\"3[F*$!3@8Kvdp]@]F*7$$\"3$**\\xVK^\"4[F*$!3[*Q1)f,lq\\F*7$$\"3s=V +(*H*)4[F*$!3ES2X[7r;\\F*7$$\"3@WmjN6K5[F*$!3i,7R(Rg`'[F*F'F_]mFa]m-F$ 6&7$7$$\"\"\"Fa[lFj\\n7$$\"3>+++#Qx/\"[F*$!3>+++#Qx/\"[F*F_]m-%'SYMBOL G6$%'CIRCLEG\"#7-%&STYLEG6#%&POINTG-F$6&Fh\\n-Fjz6&F\\[lF`[lF][lF`[l-F b]n6$Fd]n\"#5Ff]n-F$6&Fh\\nF\\^n-Fb]n6$%(DIAMONDGF`^nFf]n-F$6&Fh\\nF\\ ^n-Fb]n6$%&CROSSGF`^nFf]n-%+AXESLABELSG6%%\"xG%%y(x)G-%%FONTG6#%(DEFAU LTG-Fa_n6$%*HELVETICAG\"\"*-%%VIEWG6$Fc_nFc_n" 1 2 0 1 10 0 2 9 1 4 2 1.000000 45.000000 45.000000 0 0 "Curve 1" "Curve 2" "Curve 3" "Curve \+ 4" "Curve 5" "Curve 6" "Curve 7" }}}}{PARA 0 "" 0 "" {TEXT -1 0 "" }} {PARA 0 "" 0 "" {TEXT -1 29 "The following procedure uses " }{TEXT 0 6 "fsolve" }{TEXT -1 23 " to solve the equation " }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 " ln(x^2+y^2)+2*arctan(y/x)=ln*2+Pi/2" " 6#/,&-%#lnG6#,&*$%\"xG\"\"#\"\"\"*$%\"yGF+F,F,*&F+F,-%'arctanG6#*&F.F, F*!\"\"F,F,,&*&F&F,F+F,F,*&%#PiGF,F+F4F," }{TEXT -1 1 " " }}{PARA 0 " " 0 "" {TEXT -1 4 "for " }{TEXT 265 1 "y" }{TEXT -1 25 " numerically i n terms of " }{TEXT 266 1 "x" }{TEXT -1 1 "." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 209 "phi := proc(x) lo cal y;\n fsolve(ln(x^2+y^2)+2*arctan(y/x)=ln(2)+Pi/2,y=-x..7/ 2-x);\n end proc:\nuu := evalf(exp(Pi/2)):\nplot('phi'(x),x=1..u u,numpoints=100,font=[HELVETICA,9],labels=[`x`,`y(x)`]);" }}{PARA 13 " " 1 "" {GLPLOT2D 515 404 404 {PLOTDATA 2 "6&-%'CURVESG6$7jq7$$\"\"\"\" \"!$\"+++++5!\"*7$$\"+M.FS5F-$\"+!Hsf***!#57$$\"+N$4`2\"F-$\"+]6*f)**F 37$$\"+eVr96F-$\"+Zyon**F37$$\"+#e!Qa6F-$\"+z@\"=%**F37$$\"+!GeQ>\"F-$ \"+^qu3**F37$$\"+f\"f/B\"F-$\"+$)4mr)*F37$$\"+\\sNo7F-$\"+m$=o#)*F37$$ \"+4:b28F-$\"+-Eit(*F37$$\"+s+iY8F-$\"+@hw8(*F37$$\"+\"*o!oQ\"F-$\"+ig 9X'*F37$$\"+jM?A9F-$\"+N0$)y&*F37$$\"+>;0i9F-$\"+tvj(\\*F37$$\"+'Rj?] \"F-$\"+2v=4%*F37$$\"+d@iS:F-$\"+c3U<$*F37$$\"+1sjv:F-$\"+R4cG#*F37$$ \"+'[tsh\"F-$\"+gf2;\"*F37$$\"+x[a_;F-$\"+=V+:!*F37$$\"+*)Qd$p\"F-$\"+ F-$\"+rA&\\1)F37$$\"+-Hdj>F-$\"+oiz%*yF37$$\"+`3.**>F-$\"+4o( 3u(F37$$\"+6kKP?F-$\"+KCbovF37$$\"+q%*)o2#F-$\"+LT&QQ(F37$$\"+0Uf:@F-$ \"+Okc'>(F37$$\"+EI/`@F-$\"+7w54qF37$$\"+cNi%>#F-$\"+Snr$z'F37$$\"+8b) >B#F-$\"+;Bi$f'F37$$\"+wt(=F#F-$\"+v62tjF37$$\"+L`-3BF-$\"+'*\\+nhF37$ $\"+)fWvM#F-$\"+7i*[$fF37$$\"+Y\"HZQ#F-$\"+Yr#*4dF37$$\"+9xfBCF-$\"+\\ X)yY&F37$$\"+V))fhCF-$\"+8POC_F37$$\"+.DQ,DF-$\"+iJ/i\\F37$$\"+(*))pRD F-$\"+qG<-ZF37$$\"+VH))yDF-$\"+%H()*GWF37$$\"+EDuLBF37$$\"+#GbZ)GF-$\"+E=zC?F37$$\"+O0_DHF-$\"+Z[&em\"F37$$\"+SOni HF-$\"+<&>.L\"F37$$\"+(H8L+$F-$\"+L)p\"R&*!#67$$\"+[%y$QIF-$\"+H`(4@'F ]\\l7$$\"+iNJyIF-$\"+j(QwK#F]\\l7$$\"+6*))o6$F-$!+[(['>:F]\\l7$$\"+f!R a:$F-$!+RiBhaF]\\l7$$\"+)QZQ>$F-$!+d^2([*F]\\l7$$\"+oduIKF-$!+O[*\\M\" F37$$\"+%*QjqKF-$!+%Q!>%y\"F37$$\"+r,l3LF-$!+B!>N@#F37$$\"+70m[LF-$!+? 8/xEF37$$\"+0&z[Q$F-$!+_aG2JF37$$\"+t#3\\U$F-$!+e$R\\f$F37$$\"+d)[KY$F -$!+\")RRuSF37$$\"+>h\\,NF-$!+)*o;lXF37$$\"+LbWTNF-$!+6&)e\"4&F37$$\"+ ,/CyNF-$!+4mR*e&F37$$\"+zL#fh$F-$!+w:g7hF37$$\"+l)Hvl$F-$!+Zam1nF37$$ \"+I6?&p$F-$!+q5-gsF37$$\"+aqsLPF-$!+g\\%=%yF37$$\"+Fp!Hx$F-$!+)QD3X)F 37$$\"+a3#*3QF-$!+'*3rE!*F37$$\"+M0JZQF-$!+?;We'*F37$$\"+*)zS&)QF-$!+6 MXI5F-7$$\"+C/;ERF-$!+B0!=5\"F-7$$\"+$oA@'RF-$!+qNzm6F-7$$\"+kch.SF-$! +'>,VC\"F-7$$\"+!))f5/%F-$!+-[r;8F-7$$\"+v*3\"ySF-$!+Z4$3R\"F-7$$\"+b' [z6%F-$!+zwYt9F-7$$\"+$\\\\z:%F-$!+'fX(f:F-7$$\"+MVM%>%F-$!+DqOT;F-7$$ \"+QS*HB%F-$!+A:fJF- 7$$\"+G\"ypM%F-$!+AUgA?F-7$$\"+M=h(Q%F-$!+:96P@F-7$$\"+`'4eU%F-$!+)[X5 D#F-7$$\"+C&QOY%F-$!+d+zqBF-7$$\"+([(\\,XF-$!+HVk)\\#F-7$$\"+L76SXF-$! +'GZ)QEF-7$$\"+d6/\"e%F-$!+$pm1!GF-7$$\"+!*)p&=YF-$!+?m5kHF-7$$\"+obhb YF-$!+9%4S9$F-7$$\"+(y;_p%F-$!+/&R[O$F-7$$\"+*zJZt%F-$!+`i>JOF-7$$\"+M `Y_ZF-$!+El5vPF-7$$\"+o))>qZF-$!+iFfWRF-7$$\"+'\\o-y%F-$!+G<\")eSF-7$$ \"+D\"Q.z%F-$!+!p\\]>%F-7$$\"+SHP&z%F-$!+M@kwUF-7$$\"+axS+[F-$!+ " 0 " " {MPLTEXT 1 0 860 "C := (x,y) -> (y-x)/(y+x): hh := 0.01: numsteps := 375: x0 := 1: y0 := 1:\nmatrix([[`slope field: `,C(x,y)],[`initial \+ point: `,``(x0,y0)],[`step width: `,hh],\n[`no. of steps: `,numste ps]]);``;\nmthds := [`Enright-Verner scheme`,`Verner's Maple scheme`,` a modification of Verner's Maple scheme`,`an \"efficient\" scheme`,`a \+ scheme of Tsitouras and Papakostas`,`a modification of the Tsitouras-P apakostas scheme`,`Verner's \"most efficient\" scheme`]: \nerrs := []: vals := []:\nDigits := 25:\nfor ct to 7 do\n Cn_RK8_||ct := RK8_|| ct(C(x,y),x,y,x0,y0,hh,numsteps,false);\n sm := 0: numpts := nops(Cn _RK8_||ct):\n for ii to numpts do\n if ct=1 then vals := [op(va ls),phi(Cn_RK8_||ct[ii,1])] end if;\n sm := sm+(Cn_RK8_||ct[ii,2] -vals[ii])^2;\n end do:\n errs := [op(errs),sqrt(sm/numpts)];\nend do:\nDigits := 10:\nlinalg[transpose]([mthds,evalf(errs)]);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#K%'matrixG6#7&7$%0slope~field:~~~G*&,&%\"yG \"\"\"%\"xG!\"\"F,,&F+F,F-F,F.7$%0initial~point:~G-%!G6$F,F,7$%/step~w idth:~~~G$F,!\"#7$%1no.~of~steps:~~~G\"$v$Q(pprint06\"" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%!G" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#K%'matrixG6 #7)7$%6Enright-Verner~schemeG$\"+eO4JI!#D7$%6Verner's~Maple~schemeG$\" +eI62H!#E7$%Ha~modification~of~Verner's~Maple~schemeG$\"+bWhsCF07$%6an ~\"efficient\"~schemeG$\"+=vvtTF07$%Ea~scheme~of~Tsitouras~and~Papakos tasG$\"+R-5wpF07$%Ra~modification~of~the~Tsitouras-Papakostas~schemeG$ \"+&[&)*e^F07$%AVerner's~\"most~efficient\"~schemeG$\"+UZ*)G=F0Q(pprin t16\"" }}}{PARA 0 "" 0 "" {TEXT -1 2 " " }}{PARA 0 "" 0 "" {TEXT -1 30 "The following code constructs " }{TEXT 260 20 "numerical procedure s" }{TEXT -1 56 " for solutions based on each of the Runge-Kutta schem es." }}{PARA 0 "" 0 "" {TEXT -1 75 "The error in the value obtained by each of the methods at the point where " }{XPPEDIT 18 0 "x = 4.749; " "6#/%\"xG-%&FloatG6$\"%\\Z!\"$" }{TEXT -1 16 " is also given." }} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 722 "C := (x,y) -> (y-x)/(y+x): \+ hh := 0.01: numsteps := 375: x0 := 1: y0 := 1:\nmatrix([[`slope field: `,C(x,y)],[`initial point: `,``(x0,y0)],[`step width: `,hh],\n[`n o. of steps: `,numsteps]]);``;\nmthds := [`Enright-Verner scheme`,`V erner's Maple scheme`,`a modification of Verner's Maple scheme`,`an \" efficient\" scheme`,`a scheme of Tsitouras and Papakostas`,`a modifica tion of the Tsitouras-Papakostas scheme`,`Verner's \"most efficient\" \+ scheme`]: errs := []:\nDigits := 25:\nfor ct to 7 do\n cn_RK8_||ct : = RK8_||ct(C(x,y),x,y,x0,y0,hh,numsteps,true);\nend do:\nxx := 4.749: \+ cxx := evalf(phi(xx)):\nfor ct to 7 do\n errs := [op(errs),abs(cn_RK 8_||ct(xx)-cxx)];\nend do:\nDigits := 10:\nlinalg[transpose]([mthds,ev alf(errs)]);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#K%'matrixG6#7&7$%0slop e~field:~~~G*&,&%\"yG\"\"\"%\"xG!\"\"F,,&F+F,F-F,F.7$%0initial~point:~ G-%!G6$F,F,7$%/step~width:~~~G$F,!\"#7$%1no.~of~steps:~~~G\"$v$Q(pprin t26\"" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%!G" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#K%'matrixG6#7)7$%6Enright-Verner~schemeG$\"+G&4,F$!#C7$ %6Verner's~Maple~schemeG$\"*V/`\"GF+7$%Ha~modification~of~Verner's~Map le~schemeG$\"*:56%GF+7$%6an~\"efficient\"~schemeG$\"*hhcC%F+7$%Ea~sche me~of~Tsitouras~and~PapakostasG$\"*e%oxqF+7$%Ra~modification~of~the~Ts itouras-Papakostas~schemeG$\"**z`>bF+7$%AVerner's~\"most~efficient\"~s chemeG$\"*H**HM#F+Q(pprint36\"" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }} {PARA 0 "" 0 "" {TEXT -1 4 "The " }{TEXT 260 22 "root mean square erro r" }{TEXT -1 20 " over the interval " }{XPPEDIT 18 0 "[1, 4.75];" "6# 7$\"\"\"-%&FloatG6$\"$v%!\"#" }{TEXT -1 82 " of each Runge-Kutta meth od is estimated as follows using the special procedure " }{TEXT 0 5 " NCint" }{TEXT -1 98 " to perform numerical integration by the 7 point Newton-Cotes method over 200 equal subintervals." }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 483 "mthds := [`Enright-Verner scheme`,`Verner's \+ Maple scheme`,`a modification of Verner's Maple scheme`,`an \"efficien t\" scheme`,`a scheme of Tsitouras and Papakostas`,`a modification of \+ the Tsitouras-Papakostas scheme`,`Verner's \"most efficient\" scheme`] : errs := []:\nDigits := 20:\nfor ct to 7 do\n sm := NCint(('phi'(x) -'cn_RK8_||ct'(x))^2,x=1..4.75,adaptive=false,numpoints=7,factor=200); \n errs := [op(errs),sqrt(sm/5)];\nend do:\nDigits := 10:\nlinalg[tr anspose]([mthds,evalf(errs)]);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#K%'m atrixG6#7)7$%6Enright-Verner~schemeG$\"+'pi-A\"!#D7$%6Verner's~Maple~s chemeG$\"+-Z?x5!#E7$%Ha~modification~of~Verner's~Maple~schemeG$\"+$*QF e5F07$%6an~\"efficient\"~schemeG$\"+:(pSh\"F07$%Ea~scheme~of~Tsitouras ~and~PapakostasG$\"+$f/Zo#F07$%Ra~modification~of~the~Tsitouras-Papako stas~schemeG$\"+96;!3#F07$%AVerner's~\"most~efficient\"~schemeG$\"+5lo B()!#FQ(pprint46\"" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 " " {TEXT -1 92 "The following error graphs are constructed using the nu merical procedures for the solutions." }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 647 "evalf[30](plot(['cn_RK8_1'(x)-'phi'(x),'cn_RK8_2'(x) -'phi'(x),'cn_RK8_3'(x)-'phi'(x),'cn_RK8_4'(x)-'phi'(x),\n'cn_RK8_5'(x )-'phi'(x),'cn_RK8_6'(x)-'phi'(x),'cn_RK8_7'(x)-'phi'(x)],x=1..3.75,-2 .9e-24..5.8e-24,\ncolor=[COLOR(RGB,.4,0,.9),COLOR(RGB,0,.65,0),brown,r ed,COLOR(RGB,0,.75,.95),magenta,COLOR(RGB,.95,.55,0)],\nlegend=[`Enrig ht-Verner scheme`,`Verner's Maple scheme`,`a modification of Verner's \+ Maple scheme`,`an \"efficient\" scheme`,`a scheme of Tsitouras and Pap akostas`,`a modification of the Tsitouras-Papakostas scheme`,`Verner's \"most efficient\" scheme`],font=[HELVETICA,9],title=`error curves fo r 12 stage order 8 Runge-Kutta methods`));" }}{PARA 13 "" 1 "" {GLPLOT2D 745 499 499 {PLOTDATA 2 "6--%'CURVESG6%7S7$$\"\"\"\"\"!$F*F* 7$$\"?LLLLLLLL$eR(G$o &=F/$!(mt:\"F27$$\"?nmmmmmmm;z\\#3)=>F/$!(k1?\"F27$$\"?nmmmmmmmm;C&48( >F/$!('\\P7F27$$\"?+++++++]PM()4QK?F/$!(h,G\"F27$$\"?nmmmmmmmmT!eRk3#F /$!($G=8F27$$\"?+++++++]P%[U]d9#F/$!(H3O\"F27$$\"?+++++++]7`u$GA?#F/$! (*)>S\"F27$$\"?nmmmmmmmT5!>d6E#F/$!(VdW\"F27$$\"?nmmmmmm;aQQAF:BF/$!(O p[\"F27$$\"?LLLLLLLLekGEktBF/$!(*>K:F27$$\"?LLLLLLL$3F%fIFMCF/$!(80e\" F27$$\"?+++++++]ilF?0([#F/$!(\"yB;F27$$\"?LLLLLLLL3FfZ0WDF/$!('Gr;F27$ $\"?+++++++++veT%Hg#F/$!(iyFF/$!(x4)=F27$$\"?nmmm mmmmmT!y.Q$GF/$!(uL$>F27$$\"?++++++++]7.E=$*GF/$!((*3*>F27$$\"?LLLLLLL $3F>k))p%HF/$!(-]/#F27$$\"?++++++++]PaG\"e+$F/$!)-'R5#!#J7$$\"?nmmmmmm ;/,a=;hIF/$!)Lke@Fgv7$$\"?+++++++](=n]JF/$!)2:>AFgv7$$\"?nmmmmmmm;H #)>evJF/$!)b.xAFgv7$$\"?+++++++]P4Y(*zMKF/$!(-iL#F27$$\"?LLLLLLLLL$3\\ L=H$F/$!(K**Q#F27$$\"?LLLLLLLLeRK(e,N$F/$!(,EW#F27$$\"?nmmmmmm;/^65+3M F/$!(![)\\#F27$$\"?+++++++++ve?:hMF/$!(aU`#F27$$\"?nmmmmmmm\"zp:p?_$F/ $!(xhd#F27$$\"?LLLLLLLLL$e8ald$F/$!(sGh#F27$$\"?+++++++]PM([YYj$F/$!(6 tj#F27$$\"?+++++++]i:!G[-p$F/$!(*yaEF27$$\"$v$!\"#$!(9kp#F2-%&COLORG6& %$RGBG$\"\"%!\"\"F+$\"\"*F`[l-%'LEGENDG6#%6Enright-Verner~schemeG-F$6% 7SF'7$F-$\"'dQGF27$F4$\"'8bYF27$F9$\"'D=jF27$F>$\"'X)e(F27$FC$\"'%=d)F 27$FH$\"'UC%*F27$FM$\"(`b,\"F27$FR$\"(g93\"F27$FW$\"(nQ9\"F27$Ffn$\"(? L?\"F27$F[o$\"(UID\"F27$F`o$\"(XuI\"F27$Feo$\"(.7O\"F27$Fjo$\"(/IT\"F2 7$F_p$\"(4&f9F27$Fdp$\"(Ce^\"F27$Fip$\"(iYc\"F27$F^q$\"(b=i\"F27$Fcq$ \"(qMn\"F27$Fhq$\"(l@t\"F27$F]r$\"(=+z\"F27$Fbr$\"(g@&=F27$Fgr$\"(D3\" >F27$F\\s$\"(.p(>F27$Fas$\"(j\"[?F27$Ffs$\"(!e7@F27$F[t$\"(#Q&=#F27$F` t$\"(YQE#F27$Fet$\"(HFgv7$Fiv$\")<'>.$Fgv7$F^w$ \")))fbJFgv7$Fcw$\"):F&G$Fgv7$Fhw$\"(D8V$F27$F]x$\"(&*Qe$F27$Fbx$\"($\"'kvXF27$FC$\"'jo^F27$FH$\"'v$o&F27$FM$\"'IEhF27$FR$\"'NElF2 7$FW$\"'O1pF27$Ffn$\"'\\psF27$F[o$\"'0uvF27$F`o$\"'B3zF27$Feo$\"'TR#)F 27$Fjo$\"'`f&)F27$F_p$\"'@Z))F27$Fdp$\"'j'>*F27$Fip$\"'q+&*F27$F^q$\"' tc)*F27$Fcq$\"(Ay,\"F27$Fhq$\"(5X0\"F27$F]r$\"(&y!4\"F27$Fbr$\"(x(H6F2 7$Fgr$\"(hl;\"F27$F\\s$\"(t\"37F27$Fas$\"(!4`7F27$Ffs$\"(EPH\"F27$F[t$ \"(h)R8F27$F`t$\"(^'*Q\"F27$Fet$\"(p3W\"F27$Fjt$\"(EA\\\"F27$F_u$\"(+G b\"F27$Fdu$\"(f0h\"F27$Fiu$\"(?bn\"F27$F^v$\"(Ytt\"F27$Fcv$\")`c4=Fgv7 $Fiv$\")*RC)=Fgv7$F^w$\")hOi>Fgv7$Fcw$\")r`Y?Fgv7$Fhw$\"(]:9#F27$F]x$ \"(u4C#F27$Fbx$\"(45N#F27$Fgx$\"(XsY#F27$F\\y$\"(?!)e#F27$Fay$\"(#\\PF F27$Ffy$\"(gN)GF27$F[z$\"(e11$F27$F`z$\"(-OD$F27$Fez$\"(+W[$F2-%'COLOU RG6&F][l$\")#)eqk!\")$\"))eqk\"Fi^mFj^m-Fd[l6#%Ha~modification~of~Vern er's~Maple~schemeG-F$6%7^oF'7$$\"?mmmmmmmm\"zp3r*H5F/$!&8]\"F27$F-$!&u x#F27$$\"?+++++++vV[c)>g3\"F/$!&$fNF27$F4$!&f`%F27$$\"?+++++++D1R?YUT6 F/$!&WQ&F27$F9$!&p6'F27$$\"?LLLLLLLLL3#=t-?\"F/$!&Ov'F27$F>$!&rG(F27$F C$!&T:)F27$FH$!&v&))F27$FM$!&oU*F27$FR$!&J!**F27$FW$!'MJ5F27$Ffn$!':n5 F27$F[o$!'\"[4\"F27$F`o$!'dA6F27$Feo$!'mZ6F27$Fjo$!'Up6F27$F_p$!'n)=\" F27$Fdp$!'g47F27$Fip$!'HD7F27$F^q$!'oW7F27$Fcq$!'Si7F27$Fhq$!'jz7F27$F ]r$!'T%H\"F27$Fbr$!'O58F27$Fgr$!'#oK\"F27$F\\s$!'_T8F27$Fas$!'Gd8F27$F fs$!'Vr8F27$F[t$!'N$Q\"F27$F`t$!'b&R\"F27$Fet$!'`09F27$Fjt$!'.>9F27$F_ u$!'!)G9F27$Fdu$!'aL9F27$Fiu$!'*yV\"F27$F^v$!'(GW\"F27$Fcv$!(/:W\"Fgv7 $Fiv$!(RHV\"Fgv7$F^w$!(:)G9Fgv7$Fcw$!(cqT\"Fgv7$Fhw$!'5)R\"F27$F]x$!'R r8F27$Fbx$!'MS8F27$$\"?+++++++DJ&>()z!zLF/$!'eJ8F27$Fgx$!'\"yJ\"F27$$ \"?MLLLLLL3-8NldMMF/$!'0*H\"F27$F\\y$!'Hy7F27$$\"?MLLLLLL$ekyg5;\\$F/$ !'6i7F27$Fay$!'6[7F27$$\"?+++++++]iSY;J\\NF/$!'XS7F27$Ffy$!'QP7F27$$\" ?mmmmmmTN@xzc\"Qe$F/$!'IM7F27$$\"?++++++]P4rBs2\"f$F/$!'2K7F27$$\"?LLL LLLeR(\\wwQ$)f$F/$!'=P7F27$$\"?mmmmmmmT&)e6.g0OF/$!'#zB\"F27$$\"?+++++ +vVt_b='Gh$F/$!'4Q7F27$$\"?LLLLLL$e9m%*RB,i$F/$!'%)R7F27$$\"?mmmmmm\"z %\\SV\\QFOF/$!'dY7F27$F[z$!'O]7F27$$\"?+++++++++v$QZCm$F/$!'$*y7F27$F` z$!'PK8F27$$\"?+++++++D\"y+9C,s$F/$!'YB9F27$Fez$!'Bk:F2-Fe^m6&F][l$\"* ++++\"Fi^mF+F+-Fd[l6#%6an~\"efficient\"~schemeG-F$6%7SF'7$F-$\"%jhF27$ F4$\"&D.\"F27$F9$\"&RW\"F27$F>$\"&`z\"F27$FC$\"&Z5#F27$FH$\"&!>CF27$FM $\"&7s#F27$FR$\"&'GIF27$FW$\"&*eLF27$Ffn$\"&fq$F27$F[o$\"&$>SF27$F`o$ \"&tQ%F27$Feo$\"&ax%F27$Fjo$\"&_<&F27$F_p$\"&&QbF27$Fdp$\"&Y+'F27$Fip$ \"&hV'F27$F^q$\"&S$pF27$Fcq$\"&PQ(F27$Fhq$\"&(GzF27$F]r$\"&I\\)F27$Fbr $\"&C5*F27$Fgr$\"&fm*F27$F\\s$\"'sM5F27$Fas$\"'!)36F27$Ffs$\"'Iw6F27$F [t$\"'wd7F27$F`t$\"'$pM\"F27$Fet$\"'/U9F27$Fjt$\"'3M:F27$F_u$\"'7];F27 $Fdu$\"'mm$\"&EO#F27$FC$\"&?%GF27$FH$\"&FL$F27$F M$\"&V!QF27$FR$\"&gF%F27$FW$\"&[x%F27$Ffn$\"&\")G&F27$F[o$\"&Du&F27$F` o$\"&gE'F27$Feo$\"&y!oF27$Fjo$\"&dN(F27$F_p$\"&o%yF27$Fdp$\"&vY)F27$Fi p$\"&K.*F27$F^q$\"&*z'*F27$Fcq$\"'*e-\"F27$Fhq$\"';&4\"F27$F]r$\"',m6F 27$Fbr$\"')>C\"F27$Fgr$\"''>J\"F27$F\\s$\"']&R\"F27$Fas$\"'z&[\"F27$Ff s$\"'ln:F27$F[t$\"'Kl;F27$F`t$\"'krF27$ F_u$\"'2H@F27$Fdu$\"'PkAF27$Fiu$\"';=CF27$F^v$\"'+kDF27$Fcv$\"($!'wW5F27$FC$!'\"o>\"F27$ FH$!'FK8F27$FM$!')4X\"F27$FR$!'ge:F27$FW$!'^h;F27$Ffn$!'YfF27$Feo$!'g:?F27$Fjo$!'*y4#F27$F_p$!'iq@F27$Fdp$!'pd AF27$Fip$!'8KBF27$F^q$!'*zT#F27$Fcq$!'\\%\\#F27$Fhq$!'>!e#F27$F]r$!'Dj EF27$Fbr$!'E^FF27$Fgr$!'\\LGF27$F\\s$!'7CHF27$Fas$!'J?IF27$Ffs$!'.1JF2 7$F[t$!'0+KF27$F`t$!'I*H$F27$Fet$!'#yR$F27$Fjt$!'m'\\$F27$F_u$!'t1OF27 $Fdu$!'\\0PF27$Fiu$!'*>\"QF27$F^v$!'\\5RF27$Fcv$!(;V,%Fgv7$Fiv$!(Zb5%F gv7$F^w$!(-S?%Fgv7$Fcw$!(!>!H%Fgv7$Fhw$!'4nVF27$F]x$!'!)>WF27$Fbx$!''4 X%F27$$\"?mmmmm;z%\\S[(Qx`LF/$!'PiWF27$$\"?++++++Dc^GYO$F/$!'(zY%F27$$ \"?LLLLLL3-Q1(e\\=P$F/$!'WF27$Fay$!'g=VF27$Ffy$!'7xTF2 7$Fa[n$!'LWSF27$F[z$!')f(QF27$Fh\\n$!'!Hm$F27$F`z$!'G-MF27$F`]n$!'7#4$ F27$Fez$!'p;FF2-F[[l6&F][lFegn$\"#bFgzF+-Fd[l6#%AVerner's~\"most~effic ient\"~schemeG-%+AXESLABELSG6$Q\"x6\"Q!Fg^p-%%FONTG6$%*HELVETICAGFb[l- %&TITLEG6#%Verror~curves~for~12~stage~order~8~Runge-Kutta~methodsG-%%V IEWG6$;F(Fez;$F/!#D$\"#eFg_p" 1 2 0 1 10 0 2 9 1 4 2 1.000000 45.000000 45.000000 0 1 "Enright-Verner scheme" "Verner's Maple scheme " "a modification of Verner's Maple scheme" "an \"efficient\" scheme" "a scheme of Tsitouras and Papakostas" "a modification of the Tsitoura s-Papakostas scheme" "Verner's \"most efficient\" scheme" }}}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 645 "eva lf[30](plot(['cn_RK8_1'(x)-'phi'(x),'cn_RK8_2'(x)-'phi'(x),'cn_RK8_3'( x)-'phi'(x),'cn_RK8_4'(x)-'phi'(x),\n'cn_RK8_5'(x)-'phi'(x),'cn_RK8_6' (x)-'phi'(x),'cn_RK8_7'(x)-'phi'(x)],x=3.75..4.6,-7e-20..5e-20,\ncolor =[COLOR(RGB,.4,0,.9),COLOR(RGB,0,.65,0),brown,red,COLOR(RGB,0,.75,.95) ,magenta,COLOR(RGB,.95,.55,0)],\nlegend=[`Enright-Verner scheme`,`Vern er's Maple scheme`,`a modification of Verner's Maple scheme`,`an \"eff icient\" scheme`,`a scheme of Tsitouras and Papakostas`,`a modificatio n of the Tsitouras-Papakostas scheme`,`Verner's \"most efficient\" sch eme`],font=[HELVETICA,9],title=`error curves for 12 stage order 8 Rung e-Kutta methods`));" }}{PARA 13 "" 1 "" {GLPLOT2D 823 617 617 {PLOTDATA 2 "6--%'CURVESG6%7co7$$\"$v$!\"#$!(9kp#!#I7$$\"?nmmmmmmm;/\" eF&oP!#H$!(KYs#F-7$$\"?LLLLLLL$eR-L[Yy$F1$!(Pbu#F-7$$\"?nmmmmmmm\"H()z xF!QF1$!(:!yFF-7$$\"?nmmmmmmmT5Tu-@QF1$!(\\S#GF-7$$\"?LLLLLLL$e9hM!>RQ F1$!(W>)GF-7$$\"?nmmmmmm;H2%oHg&QF1$!(P,&HF-7$$\"?+++++++]i:VeYtQF1$!' x_IF17$$\"?nmmmmmm;H#QM)\\\"*QF1$!'s&>$F17$$\"?+++++++]7.9IZ4RF1$!'(*f LF17$$\"?LLLLLLLL$3<F17$$\"?LLLLLLLLeRAA\")RT F1$!(X)QDF17$$\"?LLLLLLL$ek=pQl:%F1$!(W]*HF17$$\"?nmmmmmmmT&)[/euTF1$! (s,%RF17$$\"?nmmmmmm;H#Qw?L>%F1$!(i$\\_F17$$\"?+++++++]P4jUj4UF1$!((Rt nF17$$\"?nmmmmmmm\"H#*G`sA%F1$!(#*eL)F17$$\"?+++++++++DnaXXUF1$!)lmD6F 17$$\"?++++++++DJZFEjUF1$!)rkU:F17$$\"?+++++++]iS1A\\!G%F1$!)mKK@F17$$ \"?++++++++v=$eA'*H%F1$!)`CcGF17$$\"?LLLLLLLLLex?\"oJ%F1$!)10-OF17$$\" ?++++++++](=`l^L%F1$!)(>**3&F17$$\"?nmmmmmm;HK*['z^VF1$!)wS'G(F17$$\"? ++++++++]i=&y*pVF1$!*EGg0\"F17$$\"?LLLLLLL$eR7R'3(Q%F1$!*ufEJ\"F17$$\" ?+++++++]7`Z!p\\S%F1$!*!Q\"R$>F17$$\"?LLLLLLLL$3Fh_CU%F1$!*q2^%F1$!+OxWfCF17$$\"?LLLLLLLL3_['[&HXF1$!+dy'e#RF17$ $\"?+++++++]PMa!pz`%F1$!+2%yA^%F17$$\"?mmmmmmmmm;g%*QYXF1$!+GQ2?eF17$$ \"?LLLLLLLek.ptObXF1$!+uv;1yF17$$\"?+++++++]i!zFXVc%F1$!,%o/nh5F17$$\" ?+++++++D\"y+yT'oXF1$!,GA@,<\"F17$$\"?+++++++++D#GQHd%F1$!,my9/Y\"F17$ $\"?++++++voHz2C,uXF1$!,hIX\\Y\"F17$$\"?++++++]PfLLl3vXF1$!,9ZK.Z\"F17 $$\"?++++++D1*y)e1;wXF1$!,p/)*)z9F17$$\"?+++++++v=U%yMsd%F1$!,_V4_]\"F 17$$\"?++++++]7y]NIQzXF1$!,VWPWy\"F17$$\"?+++++++]Pf'GJ:e%F1$!,FRUZ.#F 17$$\"?++++++DJ&pd()RQe%F1$!,:Tp&[?F17$$\"?++++++]7`%\\Y[he%F1$!,f&[@r ?F17$$\"?+++++]7.K`fFI(e%F1$!,xtsC6#F17$$\"?++++++v$4@T0d%)e%F1$!,@X7L C#F17$$\"?+++++]P%)*3([8h*e%F1$!,Tvm'GEF17$$\"?+++++++voHVcw!f%F1$!,%H mrzGF17$$\"?++++++DcEZKU2$f%F1$!,GZ[)**GF17$$\"?++++++]P%[;#GQ&f%F1$!, HiUT#HF17$$\"?+++++]7GjB;r`'f%F1$!,2$\\t`HF17$$\"?++++++v=U#3T\"p(f%F1 $!,J)3dWIF17$$\"?+++++D1k\"=\"e&o#)f%F1$!,Bh/$[JF17$$\"?+++++]P4@T0d%) )f%F1$!,S)))oGLF17$$\"?+++++voagq_GU*f%F1$!,rq$=MOF17$$\"#Y!\"\"$!,N* \\)y8%F1-%&COLORG6&%$RGBG$\"\"%Feal$\"\"!F_bl$\"\"*Feal-%'LEGENDG6#%6E nright-Verner~schemeG-F$6%7co7$F($\"(x![cF-7$F/$\"(S4y&F-7$F5$\"(-x#fF -7$F:$\"(vw5'F-7$F?$\"(KcI'F-7$FD$\"(GE\\'F-7$FI$\"(M&*o'F-7$FN$\"'DTp F17$FS$\"'-GsF17$FX$\"'&=^(F17$Fgn$\"'U;yF17$F\\o$\"'D$=)F17$Fao$\"'TS ')F17$Ffo$\"'1q\"*F17$F[p$\"'w>'*F17$F`p$\"(I.-\"F17$Fep$\"(535\"F17$F jp$\"(HG>\"F17$F_q$\"(X1G\"F17$Fdq$\"(:[Q\"F17$Fiq$\"(-z`\"F17$F^r$\"( $fCF17$Fhr$\"(Cx6#F17$F]s$\"(%>XCF17$Fbs$\"(&)o'GF17$F gs$\"(&[FLF17$F\\t$\"(HSy$F17$Fat$\"(cOd%F17$Fft$\"(f:j&F17$F[u$\"(w#Q qF17$F`u$\"(=[p)F17$Feu$\")K@L5F17$Fju$\")ThT8F17$F_v$\"):^sSF17$Fcal$\"+\")Rr)\\%F1-Fial6&F[b lF^bl$\"#lF*F^bl-Fcbl6#%6Verner's~Maple~schemeG-F$6%7cp7$F($\"(+W[$F-7 $F/$\"(\"HbNF-7$F5$\"(!)pi$F-7$F:$\"(.7r$F-7$F?$\"(d%*z$F-7$FD$\"(GQ)Q F-7$FI$\"(0q'RF-7$FN$\"'FiSF17$FS$\"'XjTF17$FX$\"'/iUF17$Fgn$\"'wjVF17 $F\\o$\"'^jWF17$Fao$\"'*fd%F17$Ffo$\"'*))o%F17$F[p$\"'4\"z%F17$F`p$\"' V&)[F17$Fep$\"'k\"*\\F17$Fjp$\"'mo]F17$F_q$\"'oa^F17$Fdq$\"'e0_F17$Fiq $\"'1>_F17$F^r$\"'N&[%F1$!)iI%=*F17$Fgx$ !*2&)z=\"F17$$\"?mmmmmmm\"z%\\ua]-XF1$!*3oYa\"F17$F\\y$!*6!\\A?F17$Fay $!*9vuE$F17$$\"?+++++]ilZ)Q#\\2IXF1$!*qOTd$F17$$\"?mmmmmm\"zp[#*>,1`%F 1$!*XC(yNF17$$\"?LLLLL$3-j7YZF6`%F1$!*WDLe$F17$$\"?++++++]il(*\\PlJXF1 $!*vRze$F17$$\"?LLLLLL3FWq+jqKXF1$!*07sf$F17$$\"?mmmmmmm\"HK9&)eP`%F1$ !*X F17$Ff^l$!+nm-5BF17$F[_l$!+%G[ra#F17$F`_l$!+Xg%\\c#F17$Fe_l$!+$[hbe#F1 7$Fj_l$!+>V/4EF17$F_`l$!+TP(Qo#F17$Fd`l$!+yTGtFF17$Fi`l$!+@)oT$HF17$F^ al$!+*HF-7$F?$!'K:AF-7$FD$!'1XCF- 7$FI$!'ZIFF-7$FN$!&:<$F17$FS$!&Dt$F17$FX$!&UI%F17$Fgn$!&e&\\F17$F\\o$! &y!fF17$Fao$!&U=(F17$Ffo$!&xz)F17$F[p$!',75F17$F`p$!'\"R@\"F17$Fep$!'9 4:F17$Fjp$!'ez=F17$F_q$!'3AAF17$Fdq$!'(4n#F17$Fiq$!'\">P$F17$F^r$!'IvU F17$Fcr$!'Ae`F17$Fhr$!'\"4C'F17$F]s$!'/0!)F17$Fbs$!(Nd.\"F17$Fgs$!(r)) H\"F17$F\\t$!(]Xc\"F17$Fat$!(@^/#F17$Fft$!(>qq#F17$F[u$!(3=h$F17$F`u$! (6^o%F17$Feu$!(/zx&F17$Fju$!(,&*)yF17$F_v$!)\"y14\"F17$Fdv$!)\\BF:F17$ Fiv$!);Wi=F17$F^w$!)'pvl#F17$Fcw$!)D*y(QF17$Fhw$!)[`xdF17$F]x$!)v&=^(F 17$Fbx$!*cFr6\"F17$Fgx$!*G\\,w\"F17$F\\y$!*'pRbGF17$Fay$!*UA$RWF17$Ffy $!*U8G2&F17$F[z$!*80e\\'F17$F`z$!*$)=?i)F17$Fez$!+?\"o/;\"F17$Fjz$!+F4 %fE\"F17$F_[l$!+Bzo!e\"F17$Fd[l$!+8g`&e\"F17$Fi[l$!+T!)*4f\"F17$F^_n$! +jzh%f\"F17$F^\\l$!+_$>)*f\"F17$Ff_n$!+?\"G#3;F17$Fc\\l$!+.IzA;F17$Fh \\l$!+c*>w!>F17$F]]l$!+/.j#=#F17$Fb]l$!+L#F17$Fg]l$!+at#*=AF17$F\\ ^l$!+'Q9hD#F17$Fa^l$!+-3^#Q#F17$Ff^l$!+h_p(y#F17$F[_l$!+DWgkIF17$F`_l$ !+o_,'3$F17$Fe_l$!+.yR5JF17$Fj_l$!+]&*pOJF17$F_`l$!+(zZ&>KF17$Fd`l$!+* >u$>LF17$Fi`l$!++#=3]$F17$F^al$!+z@%=#QF17$Fcal$!+\\B#RP%F1-F^cn6&F[bl $\"*++++\"FbcnF^blF^bl-Fcbl6#%6an~\"efficient\"~schemeG-F$6%7co7$F($\" 't)z&F-7$F/$\"'=\\eF-7$F5$\"'lTeF-7$F:$\"'7'y&F-7$F?$\"'2ocF-7$FD$\"'V bbF-7$FI$\"'\"yO&F-7$FN$\"&u*\\F17$FS$\"&iZ%F17$FX$\"&%GRF17$Fgn$\"&$z KF17$F\\o$\"&k@#F17$Fao$\"%3tF17$Ffo$!&\\A\"F17$F[p$!&/#GF17$F`p$!&mQ& F17$Fep$!&7C*F17$Fjp$!'F 17$Fjz$!+$=#of@F17$F_[l$!+>,=\"p#F17$Fd[l$!+EoZ*p#F17$Fi[l$!++<04FF17$ F^\\l$!+;]:(F-7$F5$ \"'e*>(F-7$F:$\"'#R@(F-7$F?$\"':vrF-7$FD$\"'[MrF-7$FI$\"'rEqF-7$FN$\"& aw'F17$FS$\"&3P'F17$FX$\"&h%fF17$Fgn$\"&'HaF17$F\\o$\"&y`%F17$Fao$\"&i E$F17$Ffo$\"&2c\"F17$F[p$\"%z;F17$F`p$!&07#F17$Fep$!&[f&F17$Fjp$!'\"[, \"F17$F_q$!'#oV\"F17$Fdq$!'e0?F17$Fiq$!'Y@HF17$F^r$!'eITF17$Fcr$!'91cF 17$Fhr$!'K7oF17$F]s$!'O&G*F17$Fbs$!(CQE\"F17$Fgs$!($QX;F17$F\\t$!(y>.# F17$Fat$!(8Fu#F17$Fft$!(mgt$F17$F[u$!(\">7^F17$F`u$!(Zfw'F17$Feu$!(v\\ X)F17$Fju$!)15w6F17$F_v$!)8Ja;F17$Fdv$!)*=ON#F17$Fiv$!)Vf%*GF17$F^w$!) *Hh=%F17$Fcw$!)`K*='F17$Fhw$!)7?N$*F17$F]x$!*aGYA\"F17$Fbx$!*x=j$=F17$ Fgx$!*It/#HF17$F\\y$!*5Z*yZF17$Fay$!*/tT\\(F17$Ffy$!*gQud)F17$F[z$!+oh D*4\"F17$F`z$!+(f\"4i9F17$Fez$!+o?ir>F17$Fjz$!+Decg@F17$F_[l$!+NWT*o#F 17$Fd[l$!+cGr(p#F17$Fi[l$!+oEL2FF17$F^\\l$!+n!pOs#F17$Fc\\l$!+qTUmFF17 $Fh\\l$!+qr!*fKF17$F]]l$!+&3Mlr$F17$Fb]l$!+H7uTPF17$Fg]l$!+pw\"3y$F17$ F\\^l$!+Hk,]QF17$Fa^l$!+'oF_2%F17$Ff^l$!+Qe*=w%F17$F[_l$!+Zsc=_F17$F`_ l$!+q!R]D&F17$Fe_l$!+NK(yH&F17$Fj_l$!+$ofrM&F17$F_`l$!+/)=%*\\&F17$Fd` l$!+8^*on&F17$Fi`l$!+u<0\"*fF17$F^al$!+:8!G`'F17$Fcal$!+&Qr?W(F1-F^cn6 &F[blFhaoF^blFhao-Fcbl6#%Ra~modification~of~the~Tsitouras-Papakostas~s chemeG-F$6%7cp7$F($!'p;FF-7$F/$!'c_DF-7$F5$!'2vAF-7$F:$!'#G#>F-7$F?$!' 8H:F-7$FD$!'D97F-7$FI$!&!p&)F-7$FN$!%[MF17$FS$\"%>AF17$FX$\"%csF17$Fgn $\"&/B\"F17$F\\o$\"&\"))=F17$Fao$\"&\\k#F17$Ffo$\"&WW$F17$F[p$\"&N,%F1 7$F`p$\"&-s%F17$Fep$\"&!\\bF17$Fjp$\"&#)H'F17$F_q$\"&<$oF17$Fdq$\"&!Gs F17$Fiq$\"&nV(F17$F^r$\"&t7(F17$Fcr$\"&!phF17$Fhr$\"&c2&F17$F]s$\"&L)= F17$Fbs$!&Ui$F17$Fgs$!'b,6F17$F\\t$!':M>F17$Fat$!'j$o$F17$Fft$!'q6kF17 $F[u$!(!=d5F17$F`u$!(y0f\"F17$Feu$!(lv;#F17$Fju$!(]YP$F17$F_v$!(dGB&F1 7$Fdv$!(T))4)F17$Fiv$!)[.P5F17$F^w$!)[A3;F17$Fcw$!)VLLDF17$Fhw$!)]qXSF 17$F]x$!)'GIZ&F17$Fbx$!)x-)e)F17$Fghm$!*7iK5\"F17$Fgx$!*`<)G9F17$F_im$ !*0Q(f=F17$F\\y$!*1XoV#F17$Fay$!*8\"*Q%RF17$Fjim$!*u.!3VF17$F_jm$!*[LN J%F17$Fdjm$!*=z!>VF17$Fijm$!*&4kCVF17$F^[n$!*<=eL%F17$Fc[n$!*]srM%F17$ Fh[n$!*2?\"yVF17$Ffy$!*hm-b%F17$F`\\n$!*9u<#\\F17$Fe\\n$!*cPc!eF17$Fj \\n$!*SJ5#eF17$F_]n$!*a=l$eF17$Fd]n$!*vj$oeF17$F[z$!*.Jg#fF17$F`z$!*(4 v:!)F17$Fez$!+T&f%)4\"F17$Fjz$!+\")RT37F17$F_[l$!+VlV@:F17$Fd[l$!+2c7E :F17$Fi[l$!+VMbJ:F17$F^_n$!+&>1``\"F17$F^\\l$!+[Y&3a\"F17$Ff_n$!+ev&* \\:F17$Fc\\l$!+)=Adc\"F17$Fh\\l$!+fM%)f=F17$F]]l$!+st0L@F17$Fb]l$!+\"Q =v9#F17$Fg]l$!+v9,q@F17$F\\^l$!+!RL3@#F17$Fa^l$!+n/%eM#F17$Ff^l$!+h@Ng FF17$F[_l$!+4b*f.$F17$F`_l$!+()=@dIF17$Fe_l$!+u05#3$F17$Fj_l$!+o%\\66$ F17$F_`l$!+\\^l-KF17$Fd`l$!+MQB5LF17$Fi`l$!+\"f*G,NF17$F^al$!+F0;JQF17 $Fcal$!+IG\\%Q%F1-Fial6&F[blFi_p$\"#bF*F^bl-Fcbl6#%AVerner's~\"most~ef ficient\"~schemeG-%+AXESLABELSG6$Q\"x6\"Q!F^_r-%%FONTG6$%*HELVETICAGFa bl-%&TITLEG6#%Verror~curves~for~12~stage~order~8~Runge-Kutta~methodsG- %%VIEWG6$;F(Fcal;$!\"(!#?$\"\"&F_`r" 1 2 0 1 10 0 2 9 1 4 2 1.000000 45.000000 45.000000 0 1 "Enright-Verner scheme" "Verner's Maple scheme " "a modification of Verner's Maple scheme" "an \"efficient\" scheme" "a scheme of Tsitouras and Papakostas" "a modification of the Tsitoura s-Papakostas scheme" "Verner's \"most efficient\" scheme" }}}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 646 "eva lf[30](plot(['cn_RK8_1'(x)-'phi'(x),'cn_RK8_2'(x)-'phi'(x),'cn_RK8_3'( x)-'phi'(x),'cn_RK8_4'(x)-'phi'(x),\n'cn_RK8_5'(x)-'phi'(x),'cn_RK8_6' (x)-'phi'(x),'cn_RK8_7'(x)-'phi'(x)],x=4.6..4.75,-14e-16..5e-16,\ncolo r=[COLOR(RGB,.4,0,.9),COLOR(RGB,0,.65,0),brown,red,COLOR(RGB,0,.75,.95 ),magenta,COLOR(RGB,.95,.55,0)],\nlegend=[`Enright-Verner scheme`,`Ver ner's Maple scheme`,`a modification of Verner's Maple scheme`,`an \"ef ficient\" scheme`,`a scheme of Tsitouras and Papakostas`,`a modificati on of the Tsitouras-Papakostas scheme`,`Verner's \"most efficient\" sc heme`],font=[HELVETICA,9],title=`error curves for 12 stage order 8 Run ge-Kutta methods`));" }}{PARA 13 "" 1 "" {GLPLOT2D 831 591 591 {PLOTDATA 2 "6--%'CURVESG6%7^q7$$\"#Y!\"\"$!,N*\\)y8%!#H7$$\"?++++++++ ]7t&pKg%F-$!,o4#e!=%F-7$$\"?+++++++](=7T9hg%F-$!,k%eTPUF-7$$\"?+++++++ +v=HPJ4YF-$!,x<,n=&F-7$$\"?++++++++DJaU`7YF-$!,'3'>56'F-7$$\"?+++++++] P%GZRdh%F-$!,K,ED>'F-7$$\"?+++++++](=276(=YF-$!,b^!G4qF-7$$\"?+++++++] (o**3)y@YF-$!,0)Hl%F-$!-B$R)[tMF-7$$\"?+++++++](=-p6jl%F-$!-SZ#G%\\ NF-7$$\"?+++++++++vS.EfYF-$!-TMW3wXF-7$$\"?+++++++](=xZ&\\iYF-$!-%4L:$ 4dF-7$$\"?+++++++]i:$4wbm%F-$!-p'\\s8!eF-7$$\"?++++++++v=#R!zoYF-$!-E$ *)[*ypF-7$$\"?+++++++]P4A@urYF-$!-o#)\\\\%o*F-7$$\"?++++++++Dchf#\\n%F -$!-))RF/N)*F-7$$\"?+++++++](of2L#yYF-$!.(yE`#)46F-7$$\"?+++++++]7yG>6 \"o%F-$!.&*=4)311ZF-$!.C(G0#*ejF-7$$\"?++++++++](Q(zS4ZF-$!.kjS=5$)*F-7 $$\"?+++++++](=-,FCr%F-$!/%H[=>(p7F-7$$\"?+++++++]P4tFe:ZF-$!/:k\"y')4 I\"F-7$$\"?++++++]i:&o5ajr%F-$!/DFYk`@8F-7$$\"?+++++++v$41WDrr%F-$!/Ld ,v*oO\"F-7$$\"?++++++D\"G)[26^ZF-$!/1=y_(=>#F-7$$\"?++++++ +vo/QJG?ZF-$!/m&=.#3uFF-7$$\"?++++++]7y+`14@ZF-$!/dnt!G!*y#F-7$$\"?+++ ++++](oz;)*=s%F-$!/Z$G5!=/GF-7$$\"?+++++++vVtLOXBZF-$!/^y0)QT$GF-7$$\" ?+++++++++]*44]s%F-$!/9:^MGZF-$!/T]Q;E%\\$F-7$$\"?++++++]iSmM#z*GZF-$!/P (H=.1H%F-7$$\"?+++++++Dc,$*zwHZF-$!/s?D#\\8%fF-7$$\"?+++++DJq&4yem)HZF -$!/96`!Q+C'F-7$$\"?+++++]i:Ng#=l*HZF-$!/%)GO)*)[c'F-7$$\"?+++++v$4Y(R xP1IZF-$!/c#GAp$*o'F-7$$\"?++++++D19>sB;IZF-$!/s@.#GUp'F-7$$\"?+++++]( oHzmWu%F-$!01`#H#z1(=F-7$$\"?++++++ +++v-WAXZF-$!0,:?R*H*)=F-7$$\"?++++++]7G)e].cu%F-$!0\")35B_H!>F-7$$\"? +++++++Dc,4E)fu%F-$!0S!)**fCA#>F-7$$\"?++++++]P%[@rhju%F-$!0$HQG$*f]>F -7$$\"?+++++++]7G:3uYZF-$!0wPZEF-7$$\"?++++++]P4Y6cbZZF-$!0C3kK(Gu @F-7$$\"?+++++++D1k2/P[ZF-$!0[D[8n>g#F-7$$\"?+++++](oa&o1Td[ZF-$!0Co[j vdx#F-7$$\"?++++++vo/t0yx[ZF-$!0@xw[su)HF-7$$\"?+++++]i!RvZ]\")*[ZF-$! 00<5CzUC$F-7$$\"?++++++]7.#Q?&=\\ZF-$!0A&))HjcaNF-7$$\"?+++++]PM_'G!*) Q\\ZF-$!0?f\"R#)*z#RF-7$$\"?++++++Dc,\">g#f\\ZF-$!0!R)H\\\"pvVF-7$$\"? +++++v=$\"+/F6?kF-7$FC$\"+-1_3lF-7$FH$\"+rG9HtF-7$FM$\"+-Uj G#*F-7$FR$\"+)4j\"R$*F-7$FW$\",c')=%45F-7$Ffn$\",U)R:$F-7$F^q$\",O*4eEKF-7$Fcq$\",% \\M\"yIQ@F-7$F^[l$\".utF;L;#F-7$Fc[l$\".a jDWV@#F-7$Fh[l$\".kZT'QgAF-7$F]\\l$\".a.'))[IBF-7$Fb\\l$\".q[0>sV#F-7$ Fg\\l$\".&4M*4!*f#F-7$F\\]l$\".M&4rf*>$F-7$Fa]l$\".D;,/1^%F-7$Ff]l$\". &o\")33cZF-7$F[^l$\".ZRXTc-&F-7$F`^l$\".Ibd0%H^F-7$Fe^l$\".2(R98L^F-7$ Fj^l$\".\"=8RgS^F-7$F__l$\".o\"*=/\"[^F-7$Fd_l$\".%>V))=j^F-7$Fi_l$\". 8[*oQy^F-7$F^`l$\".$33+KN_F-7$Fc`l$\".@R!=u'H&F-7$Fh`l$\"._n]oFM&F-7$F ]al$\".!>l'o0U&F-7$Fbal$\".k^#)Hx[&F-7$Fgal$\".z4S!3\"f&F-7$F\\bl$\".: OZ0Lv&F-7$Fabl$\".Fk?r(4gF-7$Ffbl$\".TPO,G*oF-7$F[cl$\".#4&HfW&))F-7$F `cl$\"/H**G4#*48F-7$Fecl$\"/^,z8W+:F-7$Fjcl$\"/Zz`Qo?:F-7$F_dl$\"/RcZ? #>a\"F-7$Fddl$\"/-'>82Mb\"F-7$Fidl$\"/Z#o@[%o:F-7$F^el$\"/J]o&o\"z:F-7 $Fcel$\"/qo=60%f\"F-7$Fhel$\"/jjW&*z:;F-7$F]fl$\"/wD^Ap[;F-7$Fbfl$\"/+ Oi4Q$z\"F-7$Fgfl$\"/9n*)e%p;#F-7$F\\gl$\"/%)Qey%*GBF-7$Fagl$\"/.zCy)G` #F-7$Ffgl$\"/R*3[X!*y#F-7$F[hl$\"/f26h.5JF-7$F`hl$\"/Vb;wH6NF-7$Fehl$ \"/'y/Qp;,%F-7$Fjhl$\"/eBz`+1VF-7$F_il$\"/^nt$ySj%F-7$Fdil$\"/\"[X\"f_ **\\F-7$Fiil$\"/z?i,N1aF--F_jl6&FajlFdjl$\"#lF[jlFdjl-Fijl6#%6Verner's ~Maple~schemeG-F$6%7aq7$F($!+#p$)Qp$F-7$F/$!+Kf(>t$F-7$F4$!++*4$zPF-7$ F9$!+/'pni%F-7$F>$!+sOh1bF-7$FC$!+;=rwbF-7$FH$!+h$y.H'F-7$FM$!+XY5x#)F -7$FR$!+Pg)HP)F-7$FW$!+Mr,;!*F-7$Ffn$!,&fqfr7F-7$F[o$!,Q^L]G\"F-7$F`o$ !,@G:F-7$Fjo$!,4o&z??F-7$F_p$!,$>I-n?F-7$Fdp$!,X v?l@$F-7$Fip$!,7PJMD$F-7$F^q$!,0,u)>LF-7$Fcq$!,Xjx)zUF-7$Fhq$!,`Lb#*R& F-7$F]r$!,m$R#H[&F-7$Fbr$!,>1S)plF-7$Fgr$!,GeAKC*F-7$F\\s$!,)[?^%Q*F-7 $Fas$!-Ia:Y`5F-7$Ffs$!-'3')e?k\"F-7$F[t$!--TaAn;F-7$F`t$!-o\"e\"*pw\"F -7$Fet$!-$*4HXUIF-7$Fjt$!-SaG)34$F-7$F_u$!-u0dUIKF-7$Fdu$!-HT/E7fF-7$F iu$!-X`&[.-'F-7$$\"?+++++++vo/V>t/ZF-$!-J4:*[2'F-7$F^v$!-)G3;'zhF-7$$ \"?++++++]7.#)G:+2ZF-$!-]9:\\QjF-7$$\"?+++++++v=!4g (F-7$$\"?++++++v=#*p**o+4ZF-$!-ntT\"eQ)F-7$Fcv$!-))GJgH&*F-7$Fhv$!.TlH tcB\"F-7$$\"?+++++++]il\"*[+9ZF-$!.9)Qc)zC\"F-7$F]w$!.k8\"*=^E\"F-7$Fb w$!.f7*Rr$G\"F-7$Fgw$!.#odeVD8F-7$F\\x$!.XX=TQO\"F-7$Fax$!.V4c[>U\"F-7 $Ffx$!._()***y3:F-7$F[y$!.!oa*)pO;F-7$F`y$!.Top%)p6#F-7$Fey$!.8`Z*RyEF -7$Fjy$!.>8+IGp#F-7$F_z$!.Sf1fuq#F-7$Fdz$!.v\"G0LOFF-7$Fiz$!.usZ)=qFF- 7$F^[l$!.%p8q#)*z#F-7$Fc[l$!.%=G(3='GF-7$Fh[l$!.dO'yH>HF-7$F]\\l$!.m.* [:3IF-7$Fb\\l$!.MD-^W9$F-7$Fg\\l$!.n+bL2N$F-7$F\\]l$!.ywm$f)4%F-7$Fa]l $!.^8jApwz.iF-7$F`^l$!.G\"Qek muq2(F-7$Fabl$!.(GQ$[**Q(F-7$Ffbl$!.!*)pBDE%)F-7$F[cl$!/KkuA7a5F-7$F`c l$!/Q5b@Bd9F-7$Fecl$!//J7%pOi\"F-7$Fjcl$!/J*e>tbk\"F-7$F_dl$!/<-#[&[o; F-7$Fddl$!/yU4g\"3o\"F-7$Fidl$!/Wt*QPsp\"F-7$F^el$!/dlS$z#4F-7 $Fgfl$!/VGHQ^3BF-7$Fagl$!/Z#>Kg)>EF-7$F[hl$!/&*fdkXeIF-7$F`hl$!/'f0v.k L$F-7$Fehl$!/(*eb$f&fOF-7$F_il$!/%p+7ZC.%F-7$Fiil$!/j][oIfWF--%'COLOUR G6&Fajl$\")#)eqk!\")$\"))eqk\"F[boF\\bo-Fijl6#%Ha~modification~of~Vern er's~Maple~schemeG-F$6%7\\q7$F($!+\\B#RP%F-7$F/$!+o)>!>WF-7$F4$!+BJ+tW F-7$F9$!+7dC$!+KEwCkF-7$FC$!++hi/lF-7$FH$!+MyI#G(F-7$FM$!+lOjG &*F-7$FR$!+oS5Q'*F-7$FW$!,u7#\\K5F-7$Ffn$!,D&HpY9F-7$F[o$!,7&y$>Y\"F-7 $F`o$!,_B+;_\"F-7$Feo$!,bV?%\\AF-7$Fjo$!,X'GOwAF-7$F_p$!,w\\=\\K#F-7$F dp$!,O#p$ef$F-7$Fip$!,=&[4POF-7$F^q$!,Nr5wq$F-7$Fcq$!,Z)p(=u%F-7$Fhq$! ,1)*)z1gF-7$F]r$!,2NRt4'F-7$Fbr$!,`;QyC(F-7$Fgr$!-eo;*o-\"F-7$F\\s$!-] EFTU5F-7$Fas$!-fW\"[>;\"F-7$Ffs$!-C8!f'H=F-7$F[t$!-'4a/w&=F-7$F`t$!-_y _hf>F-7$Fet$!-'*Qi=>MF-7$Fjt$!-.!ReNZ$F-7$F_u$!-cM0EF-7$F`y$!.nq7Pg^#F-7$Fey$!.>$=9frKF-7$Fjy$!.ex!z@* G$F-7$F_z$!.,Li'32LF-7$Fdz$!.EI#zGULF-7$Fiz$!.lXF@?Q$F-7$F^[l$!.y$3R@9 MF-7$Fc[l$!.v`#Hd![$F-7$Fh[l$!.Hn6>Oa$F-7$F]\\l$!.:1]')Rk$F-7$Fb\\l$!. 'G6RR.QF-7$Fg\\l$!.5c]SS0%F-7$F\\]l$!.4(=pNC]F-7$Fa]l$!.B`%ei(>(F-7$Ff ]l$!.%f_fy0wF-7$F[^l$!.!f01s`!)F-7$F`^l$!.d6s.fA)F-7$Fe^l$!._%*zy=B)F- 7$Fj^l$!.GXAiQC)F-7$F__l$!.MdB!*eD)F-7$Fd_l$!.1E7\"3!G)F-7$Fi_l$!.\"\\ /PX/$)F-7$F^`l$!.iC6hcR)F-7$Fc`l$!.:gs:9\\)F-7$Fh`l$!.\\\"*GRrb)F-7$F] al$!.$f4tkh')F-7$Fbal$!.&)y(yo_()F-7$Fgal$!.\\_R1r*))F-7$F\\bl$!.[&G)H ?8*F-7$Fabl$!._(e<\"e^*F-7$Ffbl$!/$[qxQ#)3\"F-7$F[cl$!/KyC!)>'R\"F-7$F `cl$!/%>e%R5`?F-7$Fecl$!/(o&Q)yEM#F-7$Fjcl$!/+Sg7GuBF-7$F_dl$!/%\\*3]< 2CF-7$Fddl$!/[@d)fUU#F-7$Fidl$!/ka(GdaW#F-7$F^el$!/dn/Q>gCF-7$Fcel$!/! Gu962[#F-7$Fhel$!/UjfK;6DF-7$F]fl$!/aC#))p\"eDF-7$Fbfl$!/2]K8XqFF-7$Fg fl$!/Tj[Q`>LF-7$F\\gl$!/F-?n@aNF-7$Fagl$!/A-!\\ck%QF-7$Ffgl$!/=Nypz3UF -7$F[hl$!/XfM08cYF-7$F`hl$!/g&y+og?&F-7$Fehl$!/[`(=c$zeF-7$Fjhl$!/3fd? rpiF-7$F_il$!/:ZrtM+nF-7$Fdil$!/6`8f&\\<(F-7$Fiil$!/C?2g_(p(F--Fgao6&F ajl$\"*++++\"F[boFdjlFdjl-Fijl6#%6an~\"efficient\"~schemeG-F$6%7\\q7$F ($!+sLbpuF-7$F/$!+1gfYvF-7$F4$!+gkAVwF-7$F9$!+[#*z'G*F-7$F>$!,mjSt4\"F -7$FC$!,!f%496\"F-7$FH$!,#>-B\\7F-7$FM$!,\\ivoi\"F-7$FR$!,*4eyX;F-7$FW $!,r!3]pF-7$Ffs$!-H*))\\\"yIF-7$F[t$!-V'zK`7$F-7$F`t$!-XT3h2LF-7$Fet $!-')\\LgEdF-7$Fjt$!-pJAwncY[cF-7$Fc[ l$!.y9(3([w&F-7$Fh[l$!.>$Qv9ueF-7$F]\\l$!.D\"))*[h/'F-7$Fb\\l$!.u`X$G; jF-7$Fg\\l$!..O![sOnF-7$F\\]l$!.DcuBIM)F-7$Fa]l$!/M:zx%**=\"F-7$Ff]l$! /&\\mB!**4Gg%F-7$F gfl$!/F*>h%*e_&F-7$F\\gl$!/a'o%*p#>fF-7$Fagl$!/%f2L&))3kF-7$Ffgl$!/?3$ 36F-7$Fdil$!0gKIRs)*>\"F-7$Fiil$!0bmbI wzG\"F--F_jl6&FajlFdjl$\"#vF[jl$\"#&*F[jl-Fijl6#%Ea~scheme~of~Tsitoura s~and~PapakostasG-F$6%7^q7$F($!+&Qr?W(F-7$F/$!+\"HN)=vF-7$F4$!+%\\ygh( F-7$F9$!+=BIa#*F-7$F>$!,rB!>\"4\"F-7$FC$!,]Nv_5\"F-7$FH$!,DHUJC\"F-7$F M$!,BRUOh\"F-7$FR$!,A?ZCj\"F-7$FW$!,E1blv\"F-7$Ffn$!,FM\\'RCF-7$F[o$!, \"p_[lCF-7$F`o$!,:_oYd#F-7$Feo$!,cSD=x$F-7$Fjo$!,`g*4W!>F-7$Ffs$!-JOVcQHF-7$F[t$!-2n-m$)HF -7$F`t$!-nO@NhJF-7$Fet$!-C%o'H%R&F-7$Fjt$!-H-f>![&F-7$F_u$!-=G`'os&F-7 $Fdu$!.Su-(4U5F-7$Fiu$!.jFM^61\"F-7$F^v$!.\"3-m?*3\"F-7$Fcv$!./o]MSn\" F-7$Fhv$!.ILpbl<#F-7$F]w$!.l\"\\\\OGAF-7$Fbw$!.3^Z21E#F-7$Fgw$!.*\\a>^ KBF-7$F\\x$!.:mxa()R#F-7$Fax$!.c16H$*\\#F-7$Ffx$!.C?)\\Z]EF-7$F[y$!.O \"[6yuGF-7$F`y$!.OwNG8t$F-7$Fey$!.0mW8Uv%F-7$Fjy$!.aX#z#)zZF-7$F_z$!.U d%\\z0[F-7$Fdz$!.s*=70d[F-7$Fiz$!.cFLtp\"\\F-7$F^[l$!.h$4`io\\F-7$Fc[l $!.qE\")ya2&F-7$Fh[l$!.n56bX<&F-7$F]\\l$!.yCF^$G`F-7$Fb\\l$!.5_@Fgc&F- 7$Fg\\l$!.E)y()\\HfF-7$F\\]l$!.i[*y.xsF-7$Fa]l$!/t%R^4W,\"F-7$Ff]l$!/g b$4Fp1\"F-7$F[^l$!/&)yn3@C6F-7$F`^l$!/;/]o=Y6F-7$Fe^l$!//x4%>q9\"F-7$F j^l$!/c\"[;*o[6F-7$F__l$!/xVK^O]6F-7$Fd_l$!/G%)fet`6F-7$Fi_l$!/,-6>8d6 F-7$F^`l$!/3$[HX)p6F-7$Fc`l$!/$Q>(3N$=\"F-7$Fh`l$!/fUv`1$>\"F-7$F]al$! /co_BA47F-7$Fbal$!/#*3gwLB7F-7$Fgal$!/c**[,OX7F-7$F\\bl$!/y*Rh$H!G\"F- 7$Fabl$!/#[A&prN8F-7$Ffbl$!/[)ol;V_\"F-7$F[cl$!/y)[=1p#>F-7$F`cl$!/BK% Q@>\\94$F-7$Fjcl$!/Q'*oP:LJF-7$F_dl$!/k5M^twJF-7$Fddl $!/&pb#)[**>$F-7$Fidl$!/JkZy3IKF-7$F^el$!/z@<yF%R$F-7$Fbfl$!/rSv*=go$F-7$Fgfl$!/n Zy:+$R%F-7$F\\gl$!/oCT4=%o%F-7$Fagl$!/(zWC(zS]F-7$Ffgl$!/ZEf#zcZ&F-7$F [hl$!/6bw#QP+'F-7$F`hl$!/7%fHIAk'F-7$Fehl$!/c)o7R5T(F-7$Fjhl$!/WB3*=8& yF-7$F_il$!/rd*4!)HL)F-7$Fdil$!/yAcdQf))F-7$Fiil$!/&ptiBTV*F--Fgao6&Fa jlFfdpFdjlFfdp-Fijl6#%Ra~modification~of~the~Tsitouras-Papakostas~sche meG-F$6%7iq7$F($!+IG\\%Q%F-7$F/$!+RUrHWF-7$F4$!+xH<([%F-7$F9$!+Kp.!\\& F-7$F>$!+g#*e.lF-7$FC$!+\"30we'F-7$FH$!+t**=PuF-7$FM$!+&\\n\"=(*F-7$FR $!+QWOJ)*F-7$FW$!,Ff>+1\"F-7$Ffn$!,,F$p#[\"F-7$F[o$!,1h*Q)\\\"F-7$F`o$ !,ngeic\"F-7$Feo$!,lDE'4BF-7$Fjo$!,7)[MPBF-7$F_p$!,q$RU#R#F-7$Fdp$!,uE 6bo$F-7$Fip$!,.Y2ys$F-7$F^q$!,(Gbv0QF-7$Fcq$!,6W(\\#*[F-7$Fhq$!,\\]S*= hF-7$F]r$!,H\"G2:iF-7$Fbr$!,Dyv_W(F-7$Fgr$!-)eW6T.\"F-7$F\\s$!-Sh3,]5F -7$Fas$!-V&Ry,=\"F-7$Ffs$!-O*)\\C4=F-7$F[t$!-)yOCq$=F-7$F`t$!-*)Gz)*\\ >F-7$Fet$!-Ds*p6H$F-7$Fjt$!-FxpeVLF-7$$\"?+++++++D19YUI&p%F-$!-n:L7!Q$ F-7$F_u$!-#RnQ%*\\$F-7$$\"?+++++++v$41e]xp%F-$!-hs>GsOF-7$$\"?+++++++] ilS*3&)p%F-$!-&RCau/%F-7$$\"?+++++++DJq+tE*p%F-$!-!R0g&>[F-7$Fdu$!-lzN l_iF-7$$\"?++++++++v=$3X;q%F-$!-$>!*3\"4jF-7$Fiu$!-@7/*pO'F-7$F`dn$!-3 i\\GDkF-7$F^v$!-'RF+2a'F-7$Fhdn$!-ilLx9nF-7$F]en$!-E'yO57(F-7$Fben$!- \"e_i$*[(F-7$Fgen$!-Pm\"oT.)F-7$F\\fn$!-z,w_E))F-7$Fcv$!-<(*GGf**F-7$$ \"?++++++vozT``y4ZF-$!.O.\"e>W6F-7$$\"?++++++]P4'Hti,r%F-$!.X\"f&*[_7F -7$$\"?++++++D1R]7,a5ZF-$!.>-0T`D\"F-7$$\"?+++++++vo/#\\<4r%F-$!.jy&)3 #e7F-7$$\"?++++++]7G8^An6ZF-$!.n=5%*RE\"F-7$Fhv$!.[YM[)p7F-7$Fgfn$!.rf 1RDG\"F-7$F]w$!.4]3A2I\"F-7$Fbw$!.yU033K\"F-7$Fgw$!.&y\\#*>l8F-7$F\\x$ !.2oDw_S\"F-7$Fax$!.,i+**[Y\"F-7$Ffx$!.G%>1P_:F-7$F[y$!.771e'y;F-7$F`y $!.pz^@f8#F-7$Fey$!.F\"y*3$\\EF-7$Fjy$!.PGx#ejEF-7$F_z$!.1b2`!yEF-7$Fd z$!.3AuRmq#F-7$Fiz$!.ma8I3u#F-7$F^[l$!.Qo!HyrFF-7$Fc[l$!.L%[7aOGF-7$Fh [l$!.Yd3qc*GF-7$F]\\l$!.t-Q%[&)HF-7$Fb\\l$!.)*[!yU?JF-7$Fg\\l$!.Pm+R*> LF-7$F\\]l$!.)f#Q]I,%F-7$Fa]l$!.xikKhM&F-7$Ff]l$!.3X]>fd&F-7$F[^l$!.2z []C#eF-7$F`^l$!.%GJ_W;fF-7$Fe^l$!.#=!yU2#fF-7$Fj^l$!.&>1=OHfF-7$F__l$! .*f+H,QfF-7$Fd_l$!.O>47a&fF-7$Fi_l$!.^\"3?%H(fF-7$F^`l$!.g%))*y&QgF-7$ Fc`l$!.\\r^s*3hF-7$Fh`l$!.*)RCk>;'F-7$F]al$!.@!e>raiF-7$Fbal$!.(3[lcOj F-7$Fgal$!.yCs[DY'F-7$F\\bl$!.!)*)e1ql'F-7$Fabl$!.-\")4tQ&pF-7$Ffbl$!. !p;K9\"*yF-7$F[cl$!.!z6t)ym*F-7$F`cl$!/&\\BMXNF\"F-7$Fecl$!/ijd62#R\"F -7$Fjcl$!/^%*H5&3T\"F-7$F_dl$!/xB9^cI9F-7$Fddl$!/]AK^VT9F-7$Fidl$!/(HT J\\lX\"F-7$Fcel$!/=$>DmT[\"F-7$F]fl$!/Di#[\"3V:F-7$Fbfl$!/4\\(=9Fo\"F- 7$Fgfl$!/!=.]P.(>F-7$Fagl$!/dC\"[;Y>#F-7$F[hl$!/yw&[-2[#F-7$F`hl$!/39N !\\ek#F-7$Fehl$!/^u)e,J#GF-7$F_il$!/A>]:b3IF-7$Fiil$!/@/f/:'>$F--F_jl6 &FajlFbgq$\"#bF[jlFdjl-Fijl6#%AVerner's~\"most~efficient\"~schemeG-%+A XESLABELSG6$Q\"x6\"Q!F^at-%%FONTG6$%*HELVETICAGFgjl-%&TITLEG6#%Verror~ curves~for~12~stage~order~8~Runge-Kutta~methodsG-%%VIEWG6$;F(Fiil;$!#9 !#;$\"\"&F_bt" 1 2 0 1 10 0 2 9 1 4 2 1.000000 45.000000 45.000000 0 1 "Enright-Verner scheme" "Verner's Maple scheme" "a modification of V erner's Maple scheme" "an \"efficient\" scheme" "a scheme of Tsitouras and Papakostas" "a modification of the Tsitouras-Papakostas scheme" " Verner's \"most efficient\" scheme" }}}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}}{PARA 0 "" 0 "" {TEXT -1 29 "#============ ================" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}} {MARK "4 0 0" 0 }{VIEWOPTS 1 1 0 1 1 1803 1 1 1 1 }{PAGENUMBERS 0 1 2 33 1 1 }