{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 "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 24 0 0 0 0 0 0 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 269 "" 0 1 0 0 0 0 1 0 0 0 0 0 0 0 0 0 } {CSTYLE "" 261 270 "" 1 18 0 0 0 0 0 0 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 271 "" 1 12 0 0 0 0 0 0 0 0 0 0 0 0 0 0 }{CSTYLE "" 258 272 "" 0 1 0 0 0 0 0 0 1 0 0 0 0 0 0 0 }{CSTYLE "" 258 273 "" 0 1 0 0 0 0 0 0 1 0 0 0 0 0 0 0 }{CSTYLE "" 258 274 "" 0 1 0 0 0 0 0 0 1 0 0 0 0 0 0 0 } {CSTYLE "" 258 275 "" 0 1 0 0 0 0 0 0 1 0 0 0 0 0 0 0 }{CSTYLE "" 258 276 "" 0 1 0 0 0 0 0 0 1 0 0 0 0 0 0 0 }{CSTYLE "" 258 277 "" 0 1 0 0 0 0 0 0 1 0 0 0 0 0 0 0 }{CSTYLE "" 258 278 "" 0 1 0 0 0 0 0 0 1 0 0 0 0 0 0 0 }{CSTYLE "" 258 279 "" 0 1 0 0 0 0 0 0 1 0 0 0 0 0 0 0 } {CSTYLE "" 258 280 "" 0 1 0 0 0 0 0 0 1 0 0 0 0 0 0 0 }{CSTYLE "" 258 281 "" 0 1 0 0 0 0 0 0 1 0 0 0 0 0 0 0 }{CSTYLE "" 258 282 "" 0 1 0 0 0 0 0 0 1 0 0 0 0 0 0 0 }{CSTYLE "" 258 283 "" 0 1 0 0 0 0 0 0 1 0 0 0 0 0 0 0 }{CSTYLE "" 258 284 "" 0 1 0 0 0 0 0 0 1 0 0 0 0 0 0 0 } {CSTYLE "" 258 285 "" 0 1 0 0 0 0 0 0 1 0 0 0 0 0 0 0 }{CSTYLE "" 258 286 "" 0 1 0 0 0 0 0 0 1 0 0 0 0 0 0 0 }{CSTYLE "" 258 287 "" 0 1 0 0 0 0 0 0 1 0 0 0 0 0 0 0 }{CSTYLE "" 258 288 "" 0 1 0 0 0 0 0 0 1 0 0 0 0 0 0 0 }{CSTYLE "" 258 289 "" 0 1 0 0 0 0 0 0 1 0 0 0 0 0 0 0 } {CSTYLE "" 258 290 "" 0 1 0 0 0 0 0 0 1 0 0 0 0 0 0 0 }{CSTYLE "" 258 291 "" 0 1 0 0 0 0 0 0 1 0 0 0 0 0 0 0 }{CSTYLE "" 258 292 "" 0 1 0 0 0 0 0 0 1 0 0 0 0 0 0 0 }{PSTYLE "Normal" -1 0 1 {CSTYLE "" -1 -1 "Tim es" 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 "Heading 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 "Bullet Item" -1 15 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 3 3 1 0 1 0 2 2 15 2 }{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 "Bullet Item" -1 258 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 3 3 1 0 1 0 2 2 15 2 }{PSTYLE "Text Output" -1 259 1 {CSTYLE "" -1 -1 "Courier" 1 10 0 0 255 1 2 2 2 2 2 1 3 1 3 1 }3 1 0 0 0 0 1 0 1 0 2 2 0 1 }} {SECT 0 {PARA 3 "" 0 "" {TEXT -1 51 "Derivation of 7 stage, order 6 Ru nge-Kutta schemes\n" }{TEXT 271 24 "(stage-order 2 examples)" }}{PARA 0 "" 0 "" {TEXT -1 45 "by Peter Stone, Gabriola Island, B.C., Canada" }}{PARA 0 "" 0 "" {TEXT -1 18 "Version: 5.12.2011" }}{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 procedures for constructing Runge-Kutta schemes et c. " }}{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 sessi on by commands similar to those that follow, where each file path give s the location of the m-file." }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 102 "read \"C:\\\\Maple/procdrs/butcher.m\";\nread \"C:\\\\Maple/pro cdrs/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 26 "#================== =======" }}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 28 "alternative order cond itions" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 5 " Let " }{XPPEDIT 18 0 "[A]" "6#7#%\"AG" }{TEXT -1 133 " be the 7 by 7 lower triangular matrix of linking coefficients from the Butcher tabl eau of an order 6 expilicit Runge-Kutta scheme. " }}{PARA 0 "" 0 "" {TEXT -1 5 "Let " }{XPPEDIT 18 0 "[b] = [b[1], b[2] = 0, b[3], b[4], \+ b[5], b[6], b[7]];" "6#/7#%\"bG7)&F%6#\"\"\"/&F%6#\"\"#\"\"!&F%6#\"\"$ &F%6#\"\"%&F%6#\"\"&&F%6#\"\"'&F%6#\"\"(" }{TEXT -1 2 ". " }}{PARA 0 " " 0 "" {TEXT -1 5 "Let " }{XPPEDIT 18 0 "[C]" "6#7#%\"CG" }{TEXT -1 39 " be diagonal matrix whose entries are " }{XPPEDIT 18 0 "c[1]=0" " 6#/&%\"cG6#\"\"\"\"\"!" }{TEXT -1 3 ", " }{XPPEDIT 18 0 "c[2]" "6#&% \"cG6#\"\"#" }{TEXT -1 3 ", " }{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 3 ", " } {XPPEDIT 18 0 "c[6]" "6#&%\"cG6#\"\"'" }{TEXT -1 3 ", " }{XPPEDIT 18 0 "c[7] = 1;" "6#/&%\"cG6#\"\"(\"\"\"" }{TEXT -1 11 " and let " } {TEXT 269 2 "Id" }{TEXT -1 32 " be the 7 by 7 identity matrix." }} {PARA 0 "" 0 "" {TEXT -1 5 "Let " }{XPPEDIT 18 0 "[c]" "6#7#%\"cG" } {TEXT -1 38 " be the row vector whose entries are " }{XPPEDIT 18 0 "c [1]=0" "6#/&%\"cG6#\"\"\"\"\"!" }{TEXT -1 3 ", " }{XPPEDIT 18 0 "c[2] " "6#&%\"cG6#\"\"#" }{TEXT -1 3 ", " }{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 3 " , " }{XPPEDIT 18 0 "c[6]" "6#&%\"cG6#\"\"'" }{TEXT -1 3 ", " } {XPPEDIT 18 0 "c[7] = 1;" "6#/&%\"cG6#\"\"(\"\"\"" }{TEXT -1 2 ". " }} {PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 197 "A := matrix([seq([seq(a[i,j],j=1..i-1),seq(0,j=i..7)],i=1..7)]): \nB := matrix([[seq(b[i],i=1..7)]]):\nId := linalg[diag](1$7):\nC := l inalg[diag](seq(c[i],i=1..7)):\nc_ := matrix([seq([c[i]],i=1..7)]):" } }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 49 "#====== ==========================================" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 38 "ee: coefficients of Butcher's scheme A" } }{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 409 "ee := \{c[2]=1/2,c[3]=2/3,c[4]=1/3,c[5]=5/6,c[6]=1/6,c[7]=1,\na [2,1]=1/2,a[3,1]=2/9,a[3,2]=4/9,\na[4,1]=7/36,a[4,2]=2/9,a[4,3]=-1/12, \na[5,1]=-35/144,a[5,2]=-55/36,a[5,3]=35/48,a[5,4]=15/8,\na[6,1]=-1/36 0,a[6,2]=-11/36,a[6,3]=-1/8,a[6,4]=1/2,a[6,5]=1/10,\na[7,1]=-41/260,a[ 7,2]=22/13,a[7,3]=43/156,a[7,4]=-118/39,a[7,5]=32/195,a[7,6]=80/39,\nb [1]=13/200,b[2]=0,b[3]=11/40,b[4]=11/40,b[5]=4/25,b[6]=4/25,b[7]=13/20 0\}:" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}{PARA 0 "" 0 "" {TEXT -1 35 "#-------- --------------------------" }}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 25 "The first order condition" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 30 "The first order condition is: " }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "[b]^T*([C]-Id)*[A]*(C-c[3]*Id)*[c] = In t((x-1)*Int((t-c[3])*t,t = 0 .. x),x = 0 .. 1);" "6#/*,)7#%\"bG%\"TG\" \"\",&7#%\"CGF)%#IdG!\"\"F)7#%\"AGF),&F,F)*&&%\"cG6#\"\"$F)F-F)F.F)7#F 4F)-%$IntG6$*&,&%\"xGF)F)F.F)-F96$*&,&%\"tGF)&F46#F6F.F)FBF)/FB;\"\"!F =F)/F=;FGF)" }{TEXT -1 2 ". " }}{PARA 0 "" 0 "" {TEXT -1 36 "In summat ion form this condition is:" }}{PARA 256 "" 0 "" {TEXT -1 1 " " } {XPPEDIT 18 0 "Sum(Sum(b[i]*(c[i]-1)*a[i,j-1],i = j .. 7)*(c[j-1]-c[3] )*c[j-1],j = 2 .. 7) = -1/60+c[3]/24;" "6#/-%$SumG6$*(-F%6$*(&%\"bG6#% \"iG\"\"\",&&%\"cG6#F.F/F/!\"\"F/&%\"aG6$F.,&%\"jGF/F/F4F//F.;F9\"\"(F /,&&F26#,&F9F/F/F4F/&F26#\"\"$F4F/&F26#,&F9F/F/F4F//F9;\"\"#F<,&*&F/F/ \"#gF4F4*&&F26#FCF/\"#CF4F/" }{TEXT -1 2 ". " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 13 "We can omit " }{XPPEDIT 18 0 " i=7" "6#/%\"iG\"\"(" }{TEXT -1 9 " in the " }{TEXT 260 15 "inner summ ation" }{TEXT -1 28 " because of the zero factor " }{XPPEDIT 18 0 "``( c[7]-1)" "6#-%!G6#,&&%\"cG6#\"\"(\"\"\"F+!\"\"" }{TEXT -1 2 ". " }} {PARA 0 "" 0 "" {TEXT -1 13 "We can omit " }{XPPEDIT 18 0 "j=2" "6#/% \"jG\"\"#" }{TEXT -1 9 " in the " }{TEXT 260 15 "outer summation" } {TEXT -1 9 " because " }{XPPEDIT 18 0 "c[1]=0" "6#/&%\"cG6#\"\"\"\"\"! " }{TEXT -1 2 ". " }}{PARA 0 "" 0 "" {TEXT -1 13 "We can omit " } {XPPEDIT 18 0 "j = 3;" "6#/%\"jG\"\"$" }{TEXT -1 30 " because (it tur ns out that) " }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "Sum( b[i]*(c[i]-1)*a[i,2],i = 3 .. 6) = 0;" "6#/-%$SumG6$*(&%\"bG6#%\"iG\" \"\",&&%\"cG6#F+F,F,!\"\"F,&%\"aG6$F+\"\"#F,/F+;\"\"$\"\"'\"\"!" } {TEXT -1 2 ". " }}{PARA 0 "" 0 "" {TEXT -1 13 "We can omit " } {XPPEDIT 18 0 "j = 4;" "6#/%\"jG\"\"%" }{TEXT -1 7 " and " } {XPPEDIT 18 0 "j=7" "6#/%\"jG\"\"(" }{TEXT -1 38 " because of the obv ious zero factors " }{XPPEDIT 18 0 "``(c[j-1]-c[3])" "6#-%!G6#,&&%\"cG 6#,&%\"jG\"\"\"F,!\"\"F,&F(6#\"\"$F-" }{TEXT -1 7 " and " }{XPPEDIT 18 0 "``(c[7]-1)" "6#-%!G6#,&&%\"cG6#\"\"(\"\"\"F+!\"\"" }{TEXT -1 15 " respectively." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 11 "This gives " }}{PARA 256 "" 0 "" {TEXT -1 1 " " } {XPPEDIT 18 0 "Sum(Sum(b[i]*(c[i]-1)*a[i,j-1],i = j .. 6)*(c[j-1]-c[3] )*c[j-1],j = 5 .. 6) = -1/60+c[3]/24;" "6#/-%$SumG6$*(-F%6$*(&%\"bG6#% \"iG\"\"\",&&%\"cG6#F.F/F/!\"\"F/&%\"aG6$F.,&%\"jGF/F/F4F//F.;F9\"\"'F /,&&F26#,&F9F/F/F4F/&F26#\"\"$F4F/&F26#,&F9F/F/F4F//F9;\"\"&F<,&*&F/F/ \"#gF4F4*&&F26#FCF/\"#CF4F/" }{TEXT -1 2 ". " }}{PARA 0 "" 0 "" {TEXT -1 9 "that is, " }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "(b [5]*(c[5]-1)*a[5,4]+b[6]*(c[6]-1)*a[6,4])*(c[4]-c[3])*c[4]+b[6]*(c[6]- 1)*a[6,5]*(c[5]-c[3])*c[5] = -1/60+c[3]/24" "6#/,&*(,&*(&%\"bG6#\"\"& \"\"\",&&%\"cG6#F+F,F,!\"\"F,&%\"aG6$F+\"\"%F,F,*(&F)6#\"\"'F,,&&F/6#F 9F,F,F1F,&F36$F9F5F,F,F,,&&F/6#F5F,&F/6#\"\"$F1F,&F/6#F5F,F,*,&F)6#F9F ,,&&F/6#F9F,F,F1F,&F36$F9F+F,,&&F/6#F+F,&F/6#FDF1F,&F/6#F+F,F,,&*&F,F, \"#gF1F1*&&F/6#FDF,\"#CF1F," }{TEXT -1 2 ". " }}{PARA 0 "" 0 "" {TEXT -1 3 "or " }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "b[6]*(1- c[6])*(a[6,5]*c[5]*(c[5]-c[3])+a[6,4]*c[4]*(c[4]-c[3]))+b[5]*(1-c[5])* a[5,4]*c[4]*(c[4]-c[3]) = 1/60-c[3]/24" "6#/,&*(&%\"bG6#\"\"'\"\"\",&F *F*&%\"cG6#F)!\"\"F*,&*(&%\"aG6$F)\"\"&F*&F-6#F5F*,&&F-6#F5F*&F-6#\"\" $F/F*F**(&F36$F)\"\"%F*&F-6#FAF*,&&F-6#FAF*&F-6#F=F/F*F*F*F**,&F'6#F5F *,&F*F*&F-6#F5F/F*&F36$F5FAF*&F-6#FAF*,&&F-6#FAF*&F-6#F=F/F*F*,&*&F*F* \"#gF/F**&&F-6#F=F*\"#CF/F/" }{TEXT -1 2 ". " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 48 "#------------------------------ -----------------" }}{PARA 0 "" 0 "" {TEXT -1 124 "We can check some o f the above symbolically as follows and also verify that Butcher's fir st scheme satisfies the condition. " }}{PARA 0 "" 0 "" {TEXT -1 0 "" } }{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 53 "Int((x-1)*Int((t-c[3])*t,t= 0..x),x=0..1):\n%=value(%);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/-%$Int G6$*&,&%\"xG\"\"\"F*!\"\"F*-F%6$*&,&%\"tGF*&%\"cG6#\"\"$F+F*F0F*/F0;\" \"!F)F*/F);F7F*,&#F*\"#gF+*&#F*\"#CF*F1F*F*" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 116 "evalm(B &* (C-Id) &* A &* (C-c[3]*Id) &* c_)[1,1]=int((x-1)*int((t-c[3])*t,t=0..x),x=0. .1):\nsubs(\{c[1]=0,c[7]=1\},%);" }}{PARA 12 "" 1 "" {XPPMATH 20 "6#/, (*(,**(&%\"bG6#\"\"$\"\"\",&&%\"cGF*F,F,!\"\"F,&%\"aG6$F+\"\"#F,F,*(&F )6#\"\"%F,,&&F/F7F,F,F0F,&F26$F8F4F,F,*(&F)6#\"\"&F,,&&F/F?F,F,F0F,&F2 6$F@F4F,F,*(&F)6#\"\"'F,,&&F/FGF,F,F0F,&F26$FHF4F,F,F,,&&F/6#F4F,F.F0F ,FNF,F,*(,&*(F>F,FAF,&F26$F@F8F,F,*(FFF,FIF,&F26$FHF8F,F,F,,&F:F,F.F0F ,F:F,F,*,FFF,FIF,&F26$FHF@F,,&FBF,F.F0F,FBF,F,,&#F,\"#gF0*&#F,\"#CF,F. F,F," }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 107 "add(add(b[i]*(c[i]-1)*a[i,j-1],i=j..7)*(c[j-1]-c[3]) *c[j-1],j=2..7)=-1/60+c[3]/24:\nsubs(\{c[1]=0,c[7]=1\},%);" }}{PARA 12 "" 1 "" {XPPMATH 20 "6#/,(*(,**(&%\"bG6#\"\"$\"\"\",&&%\"cGF*F,F,! \"\"F,&%\"aG6$F+\"\"#F,F,*(&F)6#\"\"%F,,&&F/F7F,F,F0F,&F26$F8F4F,F,*(& F)6#\"\"&F,,&&F/F?F,F,F0F,&F26$F@F4F,F,*(&F)6#\"\"'F,,&&F/FGF,F,F0F,&F 26$FHF4F,F,F,,&&F/6#F4F,F.F0F,FNF,F,*(,&*(F>F,FAF,&F26$F@F8F,F,*(FFF,F IF,&F26$FHF8F,F,F,,&F:F,F.F0F,F:F,F,*,FFF,FIF,&F26$FHF@F,,&FBF,F.F0F,F BF,F,,&#F,\"#gF0*&#F,\"#CF,F.F,F," }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 129 "b[6]*(1-c[6])*(a[6,5]*c[5]* (c[5]-c[3])+a[6,4]*c[4]*(c[4]-c[3]))+b[5]*(1-c[5])*a[5,4]*c[4]*(c[4]-c [3])=1/60-1/24*c[3];\nsubs(ee,%);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/ ,&*(&%\"bG6#\"\"'\"\"\",&F*F*&%\"cGF(!\"\"F*,&*(&%\"aG6$F)\"\"&F*&F-6# F4F*,&F5F*&F-6#\"\"$F.F*F**(&F26$F)\"\"%F*&F-6#F>F*,&F?F*F8F.F*F*F*F** ,&F'F6F*,&F*F*F5F.F*&F26$F4F>F*F?F*FAF*F*,&#F*\"#gF**&#F*\"#CF*F8F*F. " }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/#!\"\"\"#!*F$" }}}{PARA 0 "" 0 " " {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}} {PARA 0 "" 0 "" {TEXT -1 35 "#----------------------------------" }} {SECT 1 {PARA 4 "" 0 "" {TEXT -1 23 "The 2nd order condition" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 28 "The 2nd order c ondition is: " }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "[b]^ T*([C]-Id)*[A]*(C-c[3]*Id)*(C-c[4]*Id)*[c] = Int((x-1)*Int((t-c[3])*(t -c[4])*t,t=0..x),x=0..1)" "6#/*.)7#%\"bG%\"TG\"\"\",&7#%\"CGF)%#IdG!\" \"F)7#%\"AGF),&F,F)*&&%\"cG6#\"\"$F)F-F)F.F),&F,F)*&&F46#\"\"%F)F-F)F. F)7#F4F)-%$IntG6$*&,&%\"xGF)F)F.F)-F>6$*(,&%\"tGF)&F46#F6F.F),&FGF)&F4 6#F;F.F)FGF)/FG;\"\"!FBF)/FB;FOF)" }{TEXT -1 2 ". " }}{PARA 0 "" 0 "" {TEXT -1 37 "In summation form this condition is: " }}{PARA 256 "" 0 " " {TEXT -1 1 " " }{XPPEDIT 18 0 "Sum(Sum(b[i]*(c[i]-1)*a[i,j-1],i = j \+ .. 7)*(c[j-1]-c[3])*(c[j-1]-c[4])*c[j-1],j = 2 .. 7) = -1/120+c[3]/60+ c[4]/60-c[3]*c[4]/24;" "6#/-%$SumG6$**-F%6$*(&%\"bG6#%\"iG\"\"\",&&%\" cG6#F.F/F/!\"\"F/&%\"aG6$F.,&%\"jGF/F/F4F//F.;F9\"\"(F/,&&F26#,&F9F/F/ F4F/&F26#\"\"$F4F/,&&F26#,&F9F/F/F4F/&F26#\"\"%F4F/&F26#,&F9F/F/F4F//F 9;\"\"#F<,**&F/F/\"$?\"F4F4*&&F26#FCF/\"#gF4F/*&&F26#FJF/FWF4F/*(&F26# FCF/&F26#FJF/\"#CF4F4" }{TEXT -1 2 ". " }}{PARA 0 "" 0 "" {TEXT -1 0 " " }}{PARA 0 "" 0 "" {TEXT -1 13 "We can omit " }{XPPEDIT 18 0 "i=7" " 6#/%\"iG\"\"(" }{TEXT -1 9 " in the " }{TEXT 260 15 "inner summation " }{TEXT -1 28 " because of the zero factor " }{XPPEDIT 18 0 "``(c[7]- 1)" "6#-%!G6#,&&%\"cG6#\"\"(\"\"\"F+!\"\"" }{TEXT -1 2 ". " }}{PARA 0 "" 0 "" {TEXT -1 13 "We can omit " }{XPPEDIT 18 0 "j=2" "6#/%\"jG\"\" #" }{TEXT -1 9 " in the " }{TEXT 260 15 "outer summation" }{TEXT -1 9 " because " }{XPPEDIT 18 0 "c[1]=0" "6#/&%\"cG6#\"\"\"\"\"!" }{TEXT -1 2 ". " }}{PARA 0 "" 0 "" {TEXT -1 13 "We can omit " }{XPPEDIT 18 0 "j = 3;" "6#/%\"jG\"\"$" }{TEXT -1 30 " because (it turns out that) " }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "Sum(b[i]*(c[i]-1 )*a[i,2],i = 3 .. 6) = 0;" "6#/-%$SumG6$*(&%\"bG6#%\"iG\"\"\",&&%\"cG6 #F+F,F,!\"\"F,&%\"aG6$F+\"\"#F,/F+;\"\"$\"\"'\"\"!" }{TEXT -1 2 ". " } }{PARA 0 "" 0 "" {TEXT -1 13 "We can omit " }{XPPEDIT 18 0 "j = 4;" " 6#/%\"jG\"\"%" }{TEXT -1 7 " and " }{XPPEDIT 18 0 "j = 5;" "6#/%\"jG \"\"&" }{TEXT -1 38 " because of the obvious zero factors " } {XPPEDIT 18 0 "``(c[j-1]-c[3])" "6#-%!G6#,&&%\"cG6#,&%\"jG\"\"\"F,!\" \"F,&F(6#\"\"$F-" }{TEXT -1 7 " and " }{XPPEDIT 18 0 "``(c[j-1]-c[4] );" "6#-%!G6#,&&%\"cG6#,&%\"jG\"\"\"F,!\"\"F,&F(6#\"\"%F-" }{TEXT -1 15 " respectively." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 " " {TEXT -1 11 "This gives " }}{PARA 256 "" 0 "" {TEXT -1 1 " " } {XPPEDIT 18 0 "Sum(Sum(b[i]*(c[i]-1)*a[i,j-1],i = j .. 6)*(c[j-1]-c[3] )*(c[j-1]-c[4])*c[j-1],j = 6 .. 6) = -1/120+c[3]/60+c[4]/60-c[3]*c[4]/ 24;" "6#/-%$SumG6$**-F%6$*(&%\"bG6#%\"iG\"\"\",&&%\"cG6#F.F/F/!\"\"F/& %\"aG6$F.,&%\"jGF/F/F4F//F.;F9\"\"'F/,&&F26#,&F9F/F/F4F/&F26#\"\"$F4F/ ,&&F26#,&F9F/F/F4F/&F26#\"\"%F4F/&F26#,&F9F/F/F4F//F9;FF)FGF.F)*(&F,6#F8F)&F,6#F>F)\"#CF.F." }{TEXT -1 1 "," }}{PARA 0 "" 0 "" {TEXT -1 3 "or " }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "b[6]*(1-c[6])*a[6,5]*(c[5]-c[3])*(c[5]-c[4])* c[5] = 1/120-c[3]/60-c[4]/60+c[3]*c[4]/24;" "6#/*.&%\"bG6#\"\"'\"\"\", &F)F)&%\"cG6#F(!\"\"F)&%\"aG6$F(\"\"&F),&&F,6#F2F)&F,6#\"\"$F.F),&&F,6 #F2F)&F,6#\"\"%F.F)&F,6#F2F),**&F)F)\"$?\"F.F)*&&F,6#F8F)\"#gF.F.*&&F, 6#F>F)FGF.F.*(&F,6#F8F)&F,6#F>F)\"#CF.F)" }{TEXT -1 1 "." }}{PARA 0 " " 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 " " {TEXT -1 48 "#-----------------------------------------------" }} {PARA 0 "" 0 "" {TEXT -1 123 "We can check some of the above symbolica lly as follows and also verify that Butcher's first scheme satisfies t he condition." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 62 "Int((x-1)*Int((t-c[3])*(t-c[4])*t,t=0..x),x=0..1 ):\n%=value(%);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/-%$IntG6$*&,&%\"xG \"\"\"F*!\"\"F*-F%6$*(,&%\"tGF*&%\"cG6#\"\"$F+F*,&F0F*&F26#\"\"%F+F*F0 F*/F0;\"\"!F)F*/F);F;F*,*#F*\"$?\"F+*&#F*\"#gF*F1F*F**&FBF*F6F*F**&#F* \"#CF**&F1F*F6F*F*F+" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 140 "evalm(B &* (C-Id) &* A &* (C-c[3]*Id) &* (C-c[4]*Id) &* c_)[1,1]=int((x-1)*int((t-c[3])*(t-c[4])*t,t=0..x),x=0 ..1):\nsubs(\{c[1]=0,c[7]=1\},%);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/ ,&**,**(&%\"bG6#\"\"$\"\"\",&&%\"cGF*F,F,!\"\"F,&%\"aG6$F+\"\"#F,F,*(& F)6#\"\"%F,,&&F/F7F,F,F0F,&F26$F8F4F,F,*(&F)6#\"\"&F,,&&F/F?F,F,F0F,&F 26$F@F4F,F,*(&F)6#\"\"'F,,&&F/FGF,F,F0F,&F26$FHF4F,F,F,,&&F/6#F4F,F.F0 F,,&FNF,F:F0F,FNF,F,*.FFF,FIF,&F26$FHF@F,,&FBF,F.F0F,,&FBF,F:F0F,FBF,F ,,*#F,\"$?\"F0*&#F,\"#gF,F.F,F,*&FZF,F:F,F,*&#F,\"#CF,*&F.F,F:F,F,F0" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 146 "add(add(b[i]*(c[i]-1)*a[i,j-1],i=j..7)*(c[j-1]-c[3])*(c[j-1]- c[4])*c[j-1],j=2..7)=\n -1/120+c[3]/60+c[4]/60-c[3]*c[4]/24:\nsubs(\{ c[1]=0,c[7]=1\},%);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/,&**,**(&%\"bG 6#\"\"$\"\"\",&&%\"cGF*F,F,!\"\"F,&%\"aG6$F+\"\"#F,F,*(&F)6#\"\"%F,,&& F/F7F,F,F0F,&F26$F8F4F,F,*(&F)6#\"\"&F,,&&F/F?F,F,F0F,&F26$F@F4F,F,*(& F)6#\"\"'F,,&&F/FGF,F,F0F,&F26$FHF4F,F,F,,&&F/6#F4F,F.F0F,,&FNF,F:F0F, FNF,F,*.FFF,FIF,&F26$FHF@F,,&FBF,F.F0F,,&FBF,F:F0F,FBF,F,,*#F,\"$?\"F0 *&#F,\"#gF,F.F,F,*&FZF,F:F,F,*&#F,\"#CF,*&F.F,F:F,F,F0" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 99 "b[6]*(1 -c[6])*a[6,5]*c[5]*(c[5]-c[3])*(c[5]-c[4]) = 1/120-c[3]/60-c[4]/60+c[3 ]*c[4]/24;\nsubs(ee,%);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/*.&%\"bG6# \"\"'\"\"\",&F)F)&%\"cGF'!\"\"F)&%\"aG6$F(\"\"&F)&F,6#F1F),&F2F)&F,6# \"\"$F-F),&F2F)&F,6#\"\"%F-F),*#F)\"$?\"F)*&#F)\"#gF)F5F)F-*&#F)FAF)F9 F)F-*&#F)\"#CF)*&F5F)F9F)F)F)" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/#\" \"\"\"%!3\"F$" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}}{PARA 0 "" 0 "" {TEXT -1 35 "#------------ ----------------------" }}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 23 "The 3rd order condition" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 28 "The 3rd order condition is: " }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "[b]^T*([C]-Id)*(C-c[6]*Id)*[A]*(C-c[3]*Id)*[c ] = Int((x-1)*(x-c[6])*Int((t-c[3])*t,t = 0 .. x),x = 0 .. 1);" "6#/*. )7#%\"bG%\"TG\"\"\",&7#%\"CGF)%#IdG!\"\"F),&F,F)*&&%\"cG6#\"\"'F)F-F)F .F)7#%\"AGF),&F,F)*&&F26#\"\"$F)F-F)F.F)7#F2F)-%$IntG6$*(,&%\"xGF)F)F. F),&FBF)&F26#F4F.F)-F>6$*&,&%\"tGF)&F26#F;F.F)FJF)/FJ;\"\"!FBF)/FB;FOF )" }{TEXT -1 2 ". " }}{PARA 0 "" 0 "" {TEXT -1 37 "In summation form t his condition is: " }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "Sum(Sum(b[i]*(c[i]-1)*(c[i]-c[6])*a[i,j-1],i = j .. 7)*(c[j-1]-c[3])* c[j-1],j = 2 .. 7) = -1/90+c[3]/40+c[6]/60-c[3]*c[6]/24;" "6#/-%$SumG6 $*(-F%6$**&%\"bG6#%\"iG\"\"\",&&%\"cG6#F.F/F/!\"\"F/,&&F26#F.F/&F26#\" \"'F4F/&%\"aG6$F.,&%\"jGF/F/F4F//F.;F?\"\"(F/,&&F26#,&F?F/F/F4F/&F26# \"\"$F4F/&F26#,&F?F/F/F4F//F?;\"\"#FB,**&F/F/\"#!*F4F4*&&F26#FIF/\"#SF 4F/*&&F26#F:F/\"#gF4F/*(&F26#FIF/&F26#F:F/\"#CF4F4" }{TEXT -1 1 " " }} {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 13 "We can om it " }{XPPEDIT 18 0 "i=7" "6#/%\"iG\"\"(" }{TEXT -1 9 " in the " } {TEXT 260 15 "inner summation" }{TEXT -1 28 " because of the zero fact or " }{XPPEDIT 18 0 "``(c[7]-1)" "6#-%!G6#,&&%\"cG6#\"\"(\"\"\"F+!\"\" " }{TEXT -1 2 ". " }}{PARA 0 "" 0 "" {TEXT -1 13 "We can omit " } {XPPEDIT 18 0 "j=2" "6#/%\"jG\"\"#" }{TEXT -1 9 " in the " }{TEXT 260 15 "outer summation" }{TEXT -1 9 " because " }{XPPEDIT 18 0 "c[1]= 0" "6#/&%\"cG6#\"\"\"\"\"!" }{TEXT -1 2 ". " }}{PARA 0 "" 0 "" {TEXT -1 13 "We can omit " }{XPPEDIT 18 0 "j = 3;" "6#/%\"jG\"\"$" }{TEXT -1 30 " because (it turns out that) " }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "Sum(b[i]*(c[i]-1)*(c[i]-c[6])*a[i,2],i = 3 .. 6) = 0;" "6#/-%$SumG6$**&%\"bG6#%\"iG\"\"\",&&%\"cG6#F+F,F,!\"\"F,,&&F/6#F +F,&F/6#\"\"'F1F,&%\"aG6$F+\"\"#F,/F+;\"\"$F7\"\"!" }{TEXT -1 2 ", " } }{PARA 0 "" 0 "" {TEXT -1 13 "We can omit " }{XPPEDIT 18 0 "j = 4;" " 6#/%\"jG\"\"%" }{TEXT -1 37 " because of the obvious zero factor " } {XPPEDIT 18 0 "``(c[j-1]-c[3])" "6#-%!G6#,&&%\"cG6#,&%\"jG\"\"\"F,!\" \"F,&F(6#\"\"$F-" }{TEXT -1 1 "." }}{PARA 0 "" 0 "" {TEXT -1 13 "We ca n omit " }{XPPEDIT 18 0 "j = 6;" "6#/%\"jG\"\"'" }{TEXT -1 76 " beca use in this case the inner sum has a single term with the zero factor \+ " }{XPPEDIT 18 0 "``(c[i]-c[6]);" "6#-%!G6#,&&%\"cG6#%\"iG\"\"\"&F(6# \"\"'!\"\"" }{TEXT -1 1 "." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 12 "This gives: " }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "Sum(Sum(b[i]*(c[i]-1)*(c[i]-c[6])*a[i,j-1],i = j .. 6) *(c[j-1]-c[3])*c[j-1],j = 5 .. 5) = -1/90+c[3]/40+c[6]/60-c[3]*c[6]/24 ;" "6#/-%$SumG6$*(-F%6$**&%\"bG6#%\"iG\"\"\",&&%\"cG6#F.F/F/!\"\"F/,&& F26#F.F/&F26#\"\"'F4F/&%\"aG6$F.,&%\"jGF/F/F4F//F.;F?F:F/,&&F26#,&F?F/ F/F4F/&F26#\"\"$F4F/&F26#,&F?F/F/F4F//F?;\"\"&FN,**&F/F/\"#!*F4F4*&&F2 6#FHF/\"#SF4F/*&&F26#F:F/\"#gF4F/*(&F26#FHF/&F26#F:F/\"#CF4F4" }{TEXT -1 3 " , " }}{PARA 0 "" 0 "" {TEXT -1 9 "that is, " }}{PARA 256 "" 0 " " {TEXT -1 1 " " }{XPPEDIT 18 0 "b[5]*(c[5]-1)*(c[5]-c[6])*a[5,4]*(c[4 ]-c[3])*c[4]=-1/90+c[3]/40+c[6]/60-c[3]*c[6]/24" "6#/*.&%\"bG6#\"\"&\" \"\",&&%\"cG6#F(F)F)!\"\"F),&&F,6#F(F)&F,6#\"\"'F.F)&%\"aG6$F(\"\"%F), &&F,6#F8F)&F,6#\"\"$F.F)&F,6#F8F),**&F)F)\"#!*F.F.*&&F,6#F>F)\"#SF.F)* &&F,6#F4F)\"#gF.F)*(&F,6#F>F)&F,6#F4F)\"#CF.F." }{TEXT -1 2 ", " }} {PARA 0 "" 0 "" {TEXT -1 3 "or " }}{PARA 256 "" 0 "" {TEXT -1 1 " " } {XPPEDIT 18 0 "b[5]*(1-c[5])*(c[5]-c[6])*a[5,4]*(c[4]-c[3])*c[4]=1/90- c[3]/40-c[6]/60+c[3]*c[6]/24" "6#/*.&%\"bG6#\"\"&\"\"\",&F)F)&%\"cG6#F (!\"\"F),&&F,6#F(F)&F,6#\"\"'F.F)&%\"aG6$F(\"\"%F),&&F,6#F8F)&F,6#\"\" $F.F)&F,6#F8F),**&F)F)\"#!*F.F)*&&F,6#F>F)\"#SF.F.*&&F,6#F4F)\"#gF.F.* (&F,6#F>F)&F,6#F4F)\"#CF.F)" }{TEXT -1 1 "." }}{PARA 256 "" 0 "" {TEXT -1 1 " " }}{PARA 0 "" 0 "" {TEXT -1 48 "#----------------------- ------------------------" }}{PARA 0 "" 0 "" {TEXT -1 123 "We can check some of the above symbolically as follows and also verify that Butche r's first scheme satisfies the condition." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 72 "Int((x-1)*(x-c[6])*In t((t-c[3])*t,t=0..x),x=0..1):\n%=simplify(value(%));" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/-%$IntG6$*(,&%\"xG\"\"\"F*!\"\"F*,&F)F*&%\"cG6#\"\" 'F+F*-F%6$*&,&%\"tGF*&F.6#\"\"$F+F*F5F*/F5;\"\"!F)F*/F);F;F*,*#F*\"#!* F+*&#F*\"#SF*F6F*F**&#F*\"#gF*F-F*F**&#F*\"#CF**&F6F*F-F*F*F+" }}} {PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 150 "evalm(B &* (C-Id) &* (C-c[6]*Id) &* A &* (C-c[3]*Id)&* c_)[1,1]= \nsimplify(int((x-1)*(x-c[6])*int((t-c[3])*t,t=0..x),x=0..1)):\nsubs( \{c[1]=0,c[7]=1\},%);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/,&*(,(**&%\" bG6#\"\"$\"\"\",&&%\"cGF*F,F,!\"\"F,,&F.F,&F/6#\"\"'F0F,&%\"aG6$F+\"\" #F,F,**&F)6#\"\"%F,,&&F/F;F,F,F0F,,&F>F,F2F0F,&F66$FF,F.F0F,F>F,F,,*#F,\"#!*F0*&#F,\"#gF,F2F,F, *&#F,\"#SF,F.F,F,*&#F,\"#CF,*&F2F,F.F,F,F0" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 140 "add(add(b[i]*(c[i ]-1)*(c[i]-c[6])*a[i,j-1],i=j..7)*(c[j-1]-c[3])*c[j-1],j=2..7)=-1/90+c [3]/40+c[6]/60-c[3]*c[6]/24:\nsubs(\{c[1]=0,c[7]=1\},%);" }}{PARA 11 " " 1 "" {XPPMATH 20 "6#/,&*(,(**&%\"bG6#\"\"$\"\"\",&&%\"cGF*F,F,!\"\"F ,,&F.F,&F/6#\"\"'F0F,&%\"aG6$F+\"\"#F,F,**&F)6#\"\"%F,,&&F/F;F,F,F0F,, &F>F,F2F0F,&F66$FF,F.F0F, F>F,F,,*#F,\"#!*F0*&#F,\"#gF,F2F,F,*&#F,\"#SF,F.F,F,*&#F,\"#CF,*&F2F,F .F,F,F0" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 102 "b[5]*(1-c[5])*(c[5]- c[6])*a[5,4]*c[4]*(c[4]-c[3])=1/90-1/40*c[3]-1/60*c[6]+1/24*c[3]*c[6]; \nsubs(ee,%);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/*.&%\"bG6#\"\"&\"\" \",&F)F)&%\"cGF'!\"\"F),&F+F)&F,6#\"\"'F-F)&%\"aG6$F(\"\"%F)&F,6#F5F), &F6F)&F,6#\"\"$F-F),*#F)\"#!*F)*&#F)\"#SF)F9F)F-*&#F)\"#gF)F/F)F-*&#F) \"#CF)*&F9F)F/F)F)F)" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/#!\"\"\"$q#F$ " }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 54 "#--- --------------------------------------------------" }}{EXCHG {PARA 0 " > " 0 "" {MPLTEXT 1 0 1 ";" }}}}{PARA 0 "" 0 "" {TEXT -1 35 "#-------- --------------------------" }}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 23 "The 4th order condition" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 " " {TEXT -1 31 "The fourth order condition is: " }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "[b]^T*([C]-Id)*[A]^2*(C-c[3]*Id)*[c] = \+ Int((x-1)*Int(Int((s-c[3])*s,s=0..t),t=0..x),x=0..1)" "6#/*,)7#%\"bG% \"TG\"\"\",&7#%\"CGF)%#IdG!\"\"F)7#%\"AG\"\"#,&F,F)*&&%\"cG6#\"\"$F)F- F)F.F)7#F5F)-%$IntG6$*&,&%\"xGF)F)F.F)-F:6$-F:6$*&,&%\"sGF)&F56#F7F.F) FEF)/FE;\"\"!%\"tG/FK;FJF>F)/F>;FJF)" }{TEXT -1 2 ". " }}{PARA 0 "" 0 "" {TEXT -1 37 "In summation form this condition is: " }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "Sum(Sum(b[j]*(c[j]-1)*Sum(a[j,k]*a [k,i-1],k = i .. j-1),j = i+1 .. 7)*(c[i-1]-c[3])*c[i-1],i = 2 .. 6) = -1/360+c[3]/120;" "6#/-%$SumG6$*(-F%6$*(&%\"bG6#%\"jG\"\"\",&&%\"cG6# F.F/F/!\"\"F/-F%6$*&&%\"aG6$F.%\"kGF/&F96$F;,&%\"iGF/F/F4F//F;;F?,&F.F /F/F4F//F.;,&F?F/F/F/\"\"(F/,&&F26#,&F?F/F/F4F/&F26#\"\"$F4F/&F26#,&F? F/F/F4F//F?;\"\"#\"\"',&*&F/F/\"$g$F4F4*&&F26#FMF/\"$?\"F4F/" }{TEXT -1 1 " " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 13 "We can omit " }{XPPEDIT 18 0 "j = 7 ;" "6#/%\"jG\"\"(" }{TEXT -1 9 " in the " }{TEXT 260 15 "inner summat ion" }{TEXT -1 28 " because of the zero factor " }{XPPEDIT 18 0 "``(c[ 7]-1)" "6#-%!G6#,&&%\"cG6#\"\"(\"\"\"F+!\"\"" }{TEXT -1 2 ". " }} {PARA 0 "" 0 "" {TEXT -1 13 "We can omit " }{XPPEDIT 18 0 "i = 2;" "6 #/%\"iG\"\"#" }{TEXT -1 9 " in the " }{TEXT 260 15 "outer summation" }{TEXT -1 9 " because " }{XPPEDIT 18 0 "c[1]=0" "6#/&%\"cG6#\"\"\"\"\" !" }{TEXT -1 1 "." }}{PARA 0 "" 0 "" {TEXT -1 13 "We can omit " } {XPPEDIT 18 0 "i = 3;" "6#/%\"iG\"\"$" }{TEXT -1 30 " because (it tur ns out that) " }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "Sum( b[j]*(c[j]-1)*Sum(a[j,k]*a[k,2],k = 3 .. j-1),j = 4 .. 6) = 0;" "6#/-% $SumG6$*(&%\"bG6#%\"jG\"\"\",&&%\"cG6#F+F,F,!\"\"F,-F%6$*&&%\"aG6$F+% \"kGF,&F66$F8\"\"#F,/F8;\"\"$,&F+F,F,F1F,/F+;\"\"%\"\"'\"\"!" }{TEXT -1 2 ". " }}{PARA 0 "" 0 "" {TEXT -1 13 "We can omit " }{XPPEDIT 18 0 "i = 4;" "6#/%\"iG\"\"%" }{TEXT -1 37 " because of the obvious zero factor " }{XPPEDIT 18 0 "``(c[i-1]-c[3]);" "6#-%!G6#,&&%\"cG6#,&%\"iG \"\"\"F,!\"\"F,&F(6#\"\"$F-" }{TEXT -1 2 " ." }}{PARA 0 "" 0 "" {TEXT -1 13 "We can omit " }{XPPEDIT 18 0 "i = 6;" "6#/%\"iG\"\"'" }{TEXT -1 56 " because we have seen that the inner sum can end with " } {XPPEDIT 18 0 "j=6" "6#/%\"jG\"\"'" }{TEXT -1 1 "." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 10 "This gives" }}{PARA 256 " " 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "Sum(Sum(b[j]*(c[j]-1)*Sum(a[j,k] *a[k,i-1],k = i .. j-1),j = i+1 .. 6)*(c[i-1]-c[3])*c[i-1],i = 5 .. 5) = -1/360+c[3]/120;" "6#/-%$SumG6$*(-F%6$*(&%\"bG6#%\"jG\"\"\",&&%\"cG 6#F.F/F/!\"\"F/-F%6$*&&%\"aG6$F.%\"kGF/&F96$F;,&%\"iGF/F/F4F//F;;F?,&F .F/F/F4F//F.;,&F?F/F/F/\"\"'F/,&&F26#,&F?F/F/F4F/&F26#\"\"$F4F/&F26#,& F?F/F/F4F//F?;\"\"&FS,&*&F/F/\"$g$F4F4*&&F26#FMF/\"$?\"F4F/" }{TEXT -1 2 " ," }}{PARA 0 "" 0 "" {TEXT -1 9 "that is, " }}{PARA 256 "" 0 " " {TEXT -1 1 " " }{XPPEDIT 18 0 "b[6]*(c[6]-1)*a[6,5]*a[5,4]*(c[4]-c[3 ])*c[4] = -1/360+c[3]/120" "6#/*.&%\"bG6#\"\"'\"\"\",&&%\"cG6#F(F)F)! \"\"F)&%\"aG6$F(\"\"&F)&F06$F2\"\"%F),&&F,6#F5F)&F,6#\"\"$F.F)&F,6#F5F ),&*&F)F)\"$g$F.F.*&&F,6#F;F)\"$?\"F.F)" }{TEXT -1 2 ", " }}{PARA 0 " " 0 "" {TEXT -1 3 "or " }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "b[6]*(1-c[6])*a[6,5]*a[5,4]*(c[4]-c[3])*c[4] = 1/360-c[3]/120" " 6#/*.&%\"bG6#\"\"'\"\"\",&F)F)&%\"cG6#F(!\"\"F)&%\"aG6$F(\"\"&F)&F06$F 2\"\"%F),&&F,6#F5F)&F,6#\"\"$F.F)&F,6#F5F),&*&F)F)\"$g$F.F)*&&F,6#F;F) \"$?\"F.F." }{TEXT -1 1 "." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 48 "#----------------------------------------------- " }}{PARA 0 "" 0 "" {TEXT -1 123 "We can check some of the above symbo lically as follows and also verify that Butcher's first scheme satisfi es the condition." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 65 "Int((x-1)*Int(Int((s-c[3])*s,s=0..t),t=0..x), x=0..1):\n%=value(%);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/-%$IntG6$*&, &%\"xG\"\"\"F*!\"\"F*-F%6$-F%6$*&,&%\"sGF*&%\"cG6#\"\"$F+F*F2F*/F2;\" \"!%\"tG/F:;F9F)F*/F);F9F*,&#F*\"$g$F+*&#F*\"$?\"F*F3F*F*" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 130 "eva lm(B &* (C-Id) &* A^2 &* (C-c[3]*Id) &* c_)[1,1]=int((x-1)*int(int((s- c[3])*s,s=0..t),t=0..x),x=0..1):\nsubs(\{c[1]=0,c[7]=1\},%);" }}{PARA 12 "" 1 "" {XPPMATH 20 "6#/,&*(,(**&%\"bG6#\"\"%\"\"\",&&%\"cGF*F,F,! \"\"F,&%\"aG6$F+\"\"$F,&F26$F4\"\"#F,F,*(&F)6#\"\"&F,,&&F/F:F,F,F0F,,& *&&F26$F;F4F,F5F,F,*&&F26$F;F+F,&F26$F+F7F,F,F,F,*(&F)6#\"\"'F,,&&F/FI F,F,F0F,,(*&&F26$FJF4F,F5F,F,*&&F26$FJF+F,FEF,F,*&&F26$FJF;F,&F26$F;F7 F,F,F,F,F,,&&F/6#F7F,&F/6#F4F0F,FZF,F,*.FHF,FKF,FUF,FCF,,&F.F,FfnF0F,F .F,F,,&#F,\"$g$F0*&#F,\"$?\"F,FfnF,F," }}}{PARA 0 "" 0 "" {TEXT -1 0 " " }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 130 "add(add(b[j]*(c[j]-1)*a dd(a[j,k]*a[k,i-1],k=i..j-1),j=i+1..7)*(c[i-1]-c[3])*c[i-1],i=2..6)=-1 /360+c[3]/120:\nsubs(c[1]=0,c[7]=1,%);" }}{PARA 12 "" 1 "" {XPPMATH 20 "6#/,&*(,(**&%\"bG6#\"\"%\"\"\",&&%\"cGF*F,F,!\"\"F,&%\"aG6$F+\"\"$ F,&F26$F4\"\"#F,F,*(&F)6#\"\"&F,,&&F/F:F,F,F0F,,&*&&F26$F;F4F,F5F,F,*& &F26$F;F+F,&F26$F+F7F,F,F,F,*(&F)6#\"\"'F,,&&F/FIF,F,F0F,,(*&&F26$FJF4 F,F5F,F,*&&F26$FJF+F,FEF,F,*&&F26$FJF;F,&F26$F;F7F,F,F,F,F,,&&F/6#F7F, &F/6#F4F0F,FZF,F,*.FHF,FKF,FUF,FCF,,&F.F,FfnF0F,F.F,F,,&#F,\"$g$F0*&#F ,\"$?\"F,FfnF,F," }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 72 "b[6]*(1-c[6])*a[6,5]*a[5,4]*c[4]*(c[4]-c[3])= 1/360-c[3]/120;\nsubs(ee,%);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/*.&% \"bG6#\"\"'\"\"\",&F)F)&%\"cGF'!\"\"F)&%\"aG6$F(\"\"&F)&F/6$F1\"\"%F)& F,6#F4F),&F5F)&F,6#\"\"$F-F),&#F)\"$g$F)*&#F)\"$?\"F)F8F)F-" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/#!\"\"\"$g$F$" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}}{PARA 0 "" 0 " " {TEXT -1 35 "#----------------------------------" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 59 "The following \"alternati ve\" order conditions are satisfied." }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "b[6]*(1-c[6])*(a[6,5]*c[5]*(c[5]-c[3])+a[6,4]*c[4]* (c[4]-c[3]))+b[5]*(1-c[5])*a[5,4]*c[4]*(c[4]-c[3]) = 1/60-c[3]/24" "6# /,&*(&%\"bG6#\"\"'\"\"\",&F*F*&%\"cG6#F)!\"\"F*,&*(&%\"aG6$F)\"\"&F*&F -6#F5F*,&&F-6#F5F*&F-6#\"\"$F/F*F**(&F36$F)\"\"%F*&F-6#FAF*,&&F-6#FAF* &F-6#F=F/F*F*F*F**,&F'6#F5F*,&F*F*&F-6#F5F/F*&F36$F5FAF*&F-6#FAF*,&&F- 6#FAF*&F-6#F=F/F*F*,&*&F*F*\"#gF/F**&&F-6#F=F*\"#CF/F/" }{TEXT -1 2 ". " }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "b[6]*(1-c[6])*a[ 6,5]*(c[5]-c[3])*(c[5]-c[4])*c[5] = 1/120-c[3]/60-c[4]/60+c[3]*c[4]/24 ;" "6#/*.&%\"bG6#\"\"'\"\"\",&F)F)&%\"cG6#F(!\"\"F)&%\"aG6$F(\"\"&F),& &F,6#F2F)&F,6#\"\"$F.F),&&F,6#F2F)&F,6#\"\"%F.F)&F,6#F2F),**&F)F)\"$? \"F.F)*&&F,6#F8F)\"#gF.F.*&&F,6#F>F)FGF.F.*(&F,6#F8F)&F,6#F>F)\"#CF.F) " }{TEXT -1 1 "." }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "b [5]*(1-c[5])*(c[5]-c[6])*a[5,4]*(c[4]-c[3])*c[4]=1/90-c[3]/40-c[6]/60+ c[3]*c[6]/24" "6#/*.&%\"bG6#\"\"&\"\"\",&F)F)&%\"cG6#F(!\"\"F),&&F,6#F (F)&F,6#\"\"'F.F)&%\"aG6$F(\"\"%F),&&F,6#F8F)&F,6#\"\"$F.F)&F,6#F8F),* *&F)F)\"#!*F.F)*&&F,6#F>F)\"#SF.F.*&&F,6#F4F)\"#gF.F.*(&F,6#F>F)&F,6#F 4F)\"#CF.F)" }{TEXT -1 1 "." }}{PARA 256 "" 0 "" {TEXT -1 1 " " } {XPPEDIT 18 0 "b[6]*(1-c[6])*a[6,5]*a[5,4]*(c[4]-c[3])*c[4] = 1/360-c[ 3]/120" "6#/*.&%\"bG6#\"\"'\"\"\",&F)F)&%\"cG6#F(!\"\"F)&%\"aG6$F(\"\" &F)&F06$F2\"\"%F),&&F,6#F5F)&F,6#\"\"$F.F)&F,6#F5F),&*&F)F)\"$g$F.F)*& &F,6#F;F)\"$?\"F.F." }{TEXT -1 1 "." }}{PARA 0 "" 0 "" {TEXT -1 0 "" } }{PARA 0 "" 0 "" {TEXT -1 65 "The following \"alternative\" simplifyin g conditions are satisfied." }}{PARA 256 "" 0 "" {TEXT -1 1 " " } {XPPEDIT 18 0 "Sum(b[i]*(c[i]-1)*a[i,2],i = 3 .. 7) = 0;" "6#/-%$SumG6 $*(&%\"bG6#%\"iG\"\"\",&&%\"cG6#F+F,F,!\"\"F,&%\"aG6$F+\"\"#F,/F+;\"\" $\"\"(\"\"!" }{TEXT -1 2 ", " }}{PARA 0 "" 0 "" {TEXT -1 9 "that is, \+ " }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "b[3]*(c[3]-1)*a[3 ,2]+b[4]*(c[4]-1)*a[4,2]+b[5]*(c[5]-1)*a[5,2]+b[6]*(c[6]-1)*a[6,2] = 0 " "6#/,**(&%\"bG6#\"\"$\"\"\",&&%\"cG6#F)F*F*!\"\"F*&%\"aG6$F)\"\"#F*F **(&F'6#\"\"%F*,&&F-6#F7F*F*F/F*&F16$F7F3F*F**(&F'6#\"\"&F*,&&F-6#F@F* F*F/F*&F16$F@F3F*F**(&F'6#\"\"'F*,&&F-6#FIF*F*F/F*&F16$FIF3F*F*\"\"!" }{TEXT -1 2 ". " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "Sum(b[j]*(c[j]-1)*Sum(a[j,k]*a[k,2],k = 3 .. j-1),j = 4 .. 7) = 0;" "6#/-%$SumG6$*(&%\"bG6#%\"jG\"\"\",&&%\"c G6#F+F,F,!\"\"F,-F%6$*&&%\"aG6$F+%\"kGF,&F66$F8\"\"#F,/F8;\"\"$,&F+F,F ,F1F,/F+;\"\"%\"\"(\"\"!" }{TEXT -1 2 ", " }}{PARA 0 "" 0 "" {TEXT -1 8 "that is," }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "b[4]*( c[4]-1)*a[4,3]*a[3,2]+b[5]*(c[5]-1)*(a[5,3]*a[3,2]+a[5,4]*a[4,2])+b[6] *(c[6]-1)*(a[6,3]*a[3,2]+a[6,4]*a[4,2]+a[6,5]*a[5,2]) = 0;" "6#/,(**&% \"bG6#\"\"%\"\"\",&&%\"cG6#F)F*F*!\"\"F*&%\"aG6$F)\"\"$F*&F16$F3\"\"#F *F**(&F'6#\"\"&F*,&&F-6#F:F*F*F/F*,&*&&F16$F:F3F*&F16$F3F6F*F**&&F16$F :F)F*&F16$F)F6F*F*F*F**(&F'6#\"\"'F*,&&F-6#FLF*F*F/F*,(*&&F16$FLF3F*&F 16$F3F6F*F**&&F16$FLF)F*&F16$F)F6F*F**&&F16$FLF:F*&F16$F:F6F*F*F*F*\" \"!" }{TEXT -1 2 ". " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "Sum(b[i]*(c[i]-1)*(c[i]-c[6])*a[i, 2],i = 3 .. 7) = 0;" "6#/-%$SumG6$**&%\"bG6#%\"iG\"\"\",&&%\"cG6#F+F,F ,!\"\"F,,&&F/6#F+F,&F/6#\"\"'F1F,&%\"aG6$F+\"\"#F,/F+;\"\"$\"\"(\"\"! " }{TEXT -1 2 ", " }}{PARA 0 "" 0 "" {TEXT -1 9 "that is, " }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "b[3]*(c[3]-1)*(c[3]-c[6])*a [3,2]+b[4]*(c[4]-1)*(c[4]-c[6])*a[4,2]+b[5]*(c[5]-1)*(c[5]-c[6])*a[5,2 ]=0" "6#/,(**&%\"bG6#\"\"$\"\"\",&&%\"cG6#F)F*F*!\"\"F*,&&F-6#F)F*&F-6 #\"\"'F/F*&%\"aG6$F)\"\"#F*F***&F'6#\"\"%F*,&&F-6#F=F*F*F/F*,&&F-6#F=F *&F-6#F5F/F*&F76$F=F9F*F***&F'6#\"\"&F*,&&F-6#FKF*F*F/F*,&&F-6#FKF*&F- 6#F5F/F*&F76$FKF9F*F*\"\"!" }{TEXT -1 2 ". " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT 260 4 "Note" }{TEXT -1 69 ": The upper limit in each (outer) sum can be 6 instead of 7 because " }{XPPEDIT 18 0 "c[7]=1" "6#/&%\"cG6#\"\"(\"\"\"" }{TEXT -1 1 "." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}} {PARA 0 "" 0 "" {TEXT -1 26 "#=========================" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 28 "#=================== ========" }}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 28 "A relation between th e nodes" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 62 "See: On Runge-Kutta Processes of High Order, by J. C. Butcher," }} {PARA 0 "" 0 "" {TEXT -1 87 " Journal of the Australian Mathemat ical Society, Vol. 4, (1964) pages 179 to 194." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 124 "We consider 7 stage orde r 6 Runge-Kutta schemes that, in addition to the order conditions, sat isfy the following conditions." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }} {PARA 15 "" 0 "" {TEXT -1 26 "1. The row-sum conditions:" }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "Sum(a[i,j],j=1..i-1)=c[i]" "6#/ -%$SumG6$&%\"aG6$%\"iG%\"jG/F+;\"\"\",&F*F.F.!\"\"&%\"cG6#F*" }{TEXT -1 5 ", " }{XPPEDIT 18 0 "i=2" "6#/%\"iG\"\"#" }{TEXT -1 9 " . . . \+ 7 " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 15 "" 0 "" {TEXT -1 64 "2. The stage-order conditions to ensure that stages 3 to 7 have " } {TEXT 260 13 "stage-order 2" }{TEXT -1 1 "." }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "Sum(a[i,j]*c[j],j=2..i-1)=1/2" "6#/-%$S umG6$*&&%\"aG6$%\"iG%\"jG\"\"\"&%\"cG6#F,F-/F,;\"\"#,&F+F-F-!\"\"*&F-F -F3F5" }{TEXT -1 1 " " }{XPPEDIT 18 0 "c[i]^2" "6#*$&%\"cG6#%\"iG\"\"# " }{TEXT -1 6 ", " }{XPPEDIT 18 0 "i=3" "6#/%\"iG\"\"$" }{TEXT -1 8 " . . . 7" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 15 "" 0 "" {TEXT -1 40 "3. The (column) simplifying conditions: " }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "Sum(b[i]*a[i,1],i=2..7)=b[1]" "6#/-%$Su mG6$*&&%\"bG6#%\"iG\"\"\"&%\"aG6$F+F,F,/F+;\"\"#\"\"(&F)6#F," }{TEXT -1 6 ", " }}{PARA 257 "" 0 "" {TEXT -1 10 " and" }}{PARA 256 "" 0 "" {TEXT -1 2 " " }{XPPEDIT 18 0 "Sum(b[i]*a[i,j],i = j+1 .. 7) = b[j]*(1-c[j]);" "6#/-%$SumG6$*&&%\"bG6#%\"iG\"\"\"&%\"aG6$F+%\"j GF,/F+;,&F0F,F,F,\"\"(*&&F)6#F0F,,&F,F,&%\"cG6#F0!\"\"F," }{TEXT -1 1 " " }}{PARA 0 "" 0 "" {TEXT -1 12 " for " }{XPPEDIT 18 0 "j = 3 ;" "6#/%\"jG\"\"$" }{TEXT -1 10 ", 4, 5, 6." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 15 "" 0 "" {TEXT -1 42 "4. The additional simplifying \+ conditions: " }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "Sum(b [i]*c[i]*a[i,2],i = 3 .. 7) = 0;" "6#/-%$SumG6$*(&%\"bG6#%\"iG\"\"\"&% \"cG6#F+F,&%\"aG6$F+\"\"#F,/F+;\"\"$\"\"(\"\"!" }{TEXT -1 8 ", \+ " }{XPPEDIT 18 0 "Sum(b[i]*c[i]*Sum(a[i,j]*a[j,2],j = 3 .. i-1),i = 3 \+ .. 7) = 0;" "6#/-%$SumG6$*(&%\"bG6#%\"iG\"\"\"&%\"cG6#F+F,-F%6$*&&%\"a G6$F+%\"jGF,&F46$F6\"\"#F,/F6;\"\"$,&F+F,F,!\"\"F,/F+;F<\"\"(\"\"!" } {TEXT -1 8 ", " }{XPPEDIT 18 0 "Sum(b[i]*c[i]^2*a[i,2],i = 3 .. \+ 7) = 0;" "6#/-%$SumG6$*(&%\"bG6#%\"iG\"\"\"*$&%\"cG6#F+\"\"#F,&%\"aG6$ F+F1F,/F+;\"\"$\"\"(\"\"!" }{TEXT -1 2 ". " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 83 "Now consider the four simple or der conditions given in abbreviated form as follows." }}{PARA 0 "" 0 " " {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 170 "SO6 := Si mpleOrderConditions(6):\n[seq([i,SO6[i]],i=[13,24,28,29])]:\nlinalg[au gment](linalg[delcols](%,2..2),matrix([[` `]$(linalg[rowdim](%))]),li nalg[delcols](%,1..1));" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#K%'matrixG6 #7&7%\"#8%#~~G/*(%\"bG\"\"\"%\"cGF--%!G6#*&)F.\"\"#F-%\"aGF-F-#F-\"#:7 %\"#CF)/*(F,F-F.F--F06#*&F5F-F/F-F-#F-\"#s7%\"#GF)/*(F,F-F3F-F/F-#F-\" #=7%\"#HF)/*(F,F-F.F--F06#*&)F.\"\"$F-F5F-F-#F-F9Q)pprint906\"" }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 21 "In detail these are: " }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "Sum(b[i]*c[i]*Sum(a[i,j]*c[ j]^2,j = 2 .. i-1),i = 3 .. 7) = 1/15" "6#/-%$SumG6$*(&%\"bG6#%\"iG\" \"\"&%\"cG6#F+F,-F%6$*&&%\"aG6$F+%\"jGF,*$&F.6#F6\"\"#F,/F6;F:,&F+F,F, !\"\"F,/F+;\"\"$\"\"(*&F,F,\"#:F>" }{TEXT -1 2 ", " }}{PARA 256 "" 0 " " {TEXT -1 1 " " }{XPPEDIT 18 0 "Sum(b[i]*c[i]*Sum(a[i,j]*Sum(a[j,k]*c [k]^2,k = 2 .. j-1),j = 3 .. i-1),i = 4 .. 7) = 1/72" "6#/-%$SumG6$*(& %\"bG6#%\"iG\"\"\"&%\"cG6#F+F,-F%6$*&&%\"aG6$F+%\"jGF,-F%6$*&&F46$F6% \"kGF,*$&F.6#F<\"\"#F,/F<;F@,&F6F,F,!\"\"F,/F6;\"\"$,&F+F,F,FDF,/F+;\" \"%\"\"(*&F,F,\"#sFD" }{TEXT -1 2 ", " }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "Sum(b[i]*c[i]^2*Sum(a[i,j]*c[j]^2,j = 2 .. i-1),i \+ = 3 .. 7) = 1/18" "6#/-%$SumG6$*(&%\"bG6#%\"iG\"\"\"*$&%\"cG6#F+\"\"#F ,-F%6$*&&%\"aG6$F+%\"jGF,*$&F/6#F8F1F,/F8;F1,&F+F,F,!\"\"F,/F+;\"\"$\" \"(*&F,F,\"#=F?" }{TEXT -1 2 ", " }}{PARA 256 "" 0 "" {TEXT -1 1 " " } {XPPEDIT 18 0 "Sum(b[i]*c[i]*Sum(a[i,j]*c[j]^3,j = 2 .. i-1),i = 3 .. \+ 7) = 1/24" "6#/-%$SumG6$*(&%\"bG6#%\"iG\"\"\"&%\"cG6#F+F,-F%6$*&&%\"aG 6$F+%\"jGF,*$&F.6#F6\"\"$F,/F6;\"\"#,&F+F,F,!\"\"F,/F+;F:\"\"(*&F,F,\" #CF?" }{TEXT -1 1 "." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 " " {TEXT -1 92 "According to Butcher, these four order conditions toget her with the simplifying conditions: " }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "Sum(b[i]*c[i]*a[i,2],i = 3 .. 7) = 0;" "6#/-%$SumG 6$*(&%\"bG6#%\"iG\"\"\"&%\"cG6#F+F,&%\"aG6$F+\"\"#F,/F+;\"\"$\"\"(\"\" !" }{TEXT -1 8 ", " }{XPPEDIT 18 0 "Sum(b[i]*c[i]*Sum(a[i,j]*a[j ,2],j = 3 .. i-1),i = 3 .. 7) = 0;" "6#/-%$SumG6$*(&%\"bG6#%\"iG\"\"\" &%\"cG6#F+F,-F%6$*&&%\"aG6$F+%\"jGF,&F46$F6\"\"#F,/F6;\"\"$,&F+F,F,!\" \"F,/F+;F<\"\"(\"\"!" }{TEXT -1 8 ", " }{XPPEDIT 18 0 "Sum(b[i]* c[i]^2*a[i,2],i = 3 .. 7) = 0;" "6#/-%$SumG6$*(&%\"bG6#%\"iG\"\"\"*$&% \"cG6#F+\"\"#F,&%\"aG6$F+F1F,/F+;\"\"$\"\"(\"\"!" }{TEXT -1 2 ". " }} {PARA 0 "" 0 "" {TEXT -1 4 "are " }{TEXT 260 39 "equivalent to the fol lowing 7 equations" }{TEXT -1 1 "." }}{PARA 0 "" 0 "" {TEXT -1 62 "The se consist of the following 4 alternative order conditions:" }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "b[6]*(1-c[6])*(a[6,5]*c[5]* (c[5]-c[3])+a[6,4]*c[4]*(c[4]-c[3]))+b[5]*(1-c[5])*a[5,4]*c[4]*(c[4]-c [3])=1/60-c[3]/24" "6#/,&*(&%\"bG6#\"\"'\"\"\",&F*F*&%\"cG6#F)!\"\"F*, &*(&%\"aG6$F)\"\"&F*&F-6#F5F*,&&F-6#F5F*&F-6#\"\"$F/F*F**(&F36$F)\"\"% F*&F-6#FAF*,&&F-6#FAF*&F-6#F=F/F*F*F*F**,&F'6#F5F*,&F*F*&F-6#F5F/F*&F3 6$F5FAF*&F-6#FAF*,&&F-6#FAF*&F-6#F=F/F*F*,&*&F*F*\"#gF/F**&&F-6#F=F*\" #CF/F/" }{TEXT -1 2 ", " }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "b[6]*(1-c[6])*a[6,5]*c[5]*(c[5]-c[3])*(c[5]-c[4])=1/120-(c[3]+c[ 4])/60+c[3]*c[4]/24" "6#/*.&%\"bG6#\"\"'\"\"\",&F)F)&%\"cG6#F(!\"\"F)& %\"aG6$F(\"\"&F)&F,6#F2F),&&F,6#F2F)&F,6#\"\"$F.F),&&F,6#F2F)&F,6#\"\" %F.F),(*&F)F)\"$?\"F.F)*&,&&F,6#F:F)&F,6#F@F)F)\"#gF.F.*(&F,6#F:F)&F,6 #F@F)\"#CF.F)" }{TEXT -1 3 ",\n " }{XPPEDIT 18 0 "b[5]*(1-c[5])*(c[6]- c[5])*a[5,4]*c[4]*(c[4]-c[3])=-1/90+c[3]/40+c[6]/60-c[3]*c[6]/24" "6#/ *.&%\"bG6#\"\"&\"\"\",&F)F)&%\"cG6#F(!\"\"F),&&F,6#\"\"'F)&F,6#F(F.F)& %\"aG6$F(\"\"%F)&F,6#F8F),&&F,6#F8F)&F,6#\"\"$F.F),**&F)F)\"#!*F.F.*&& F,6#F@F)\"#SF.F)*&&F,6#F2F)\"#gF.F)*(&F,6#F@F)&F,6#F2F)\"#CF.F." } {TEXT -1 3 ",\n " }{XPPEDIT 18 0 "b[6]*(1-c[6])*a[6,5]*a[5,4]*c[4]*(c[ 4]-c[3])=1/360-c[3]/120" "6#/*.&%\"bG6#\"\"'\"\"\",&F)F)&%\"cG6#F(!\" \"F)&%\"aG6$F(\"\"&F)&F06$F2\"\"%F)&F,6#F5F),&&F,6#F5F)&F,6#\"\"$F.F), &*&F)F)\"$g$F.F)*&&F,6#F=F)\"$?\"F.F." }{TEXT -1 2 ", " }}{PARA 0 "" 0 "" {TEXT -1 65 "together with the following 3 alternative simplifyin g conditions:" }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "Sum( b[i]*(1-c[i])*a[i,2],i=3..6)=0" "6#/-%$SumG6$*(&%\"bG6#%\"iG\"\"\",&F, F,&%\"cG6#F+!\"\"F,&%\"aG6$F+\"\"#F,/F+;\"\"$\"\"'\"\"!" }{TEXT -1 2 " , " }}{PARA 0 "" 0 "" {TEXT -1 8 "that is," }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "b[3]*(1-c[3])*a[3,2]+b[4]*(1-c[4])*a[4,2]+b[5 ]*(1-c[5])*a[5,2]+b[6]*(1-c[6])*a[6,2] = 0" "6#/,**(&%\"bG6#\"\"$\"\" \",&F*F*&%\"cG6#F)!\"\"F*&%\"aG6$F)\"\"#F*F**(&F'6#\"\"%F*,&F*F*&F-6#F 7F/F*&F16$F7F3F*F**(&F'6#\"\"&F*,&F*F*&F-6#F@F/F*&F16$F@F3F*F**(&F'6# \"\"'F*,&F*F*&F-6#FIF/F*&F16$FIF3F*F*\"\"!" }{TEXT -1 2 ", " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "Sum(b[i]*(1-c[i])*Sum(a[i,j]*a[j,2],j = 3 .. i-1),i = 3 .. 6) = \+ 0" "6#/-%$SumG6$*(&%\"bG6#%\"iG\"\"\",&F,F,&%\"cG6#F+!\"\"F,-F%6$*&&% \"aG6$F+%\"jGF,&F66$F8\"\"#F,/F8;\"\"$,&F+F,F,F1F,/F+;F>\"\"'\"\"!" } {TEXT -1 2 ", " }}{PARA 0 "" 0 "" {TEXT -1 8 "that is," }}{PARA 256 " " 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "b[4]*(1-c[4])*a[4,3]*a[3,2]+b[5] *(1-c[5])*(a[5,3]*a[3,2]+a[5,4]*a[4,2])+b[6]*(1-c[6])*(a[6,3]*a[3,2]+a [6,4]*a[4,2]+a[6,5]*a[5,2]) = 0" "6#/,(**&%\"bG6#\"\"%\"\"\",&F*F*&%\" cG6#F)!\"\"F*&%\"aG6$F)\"\"$F*&F16$F3\"\"#F*F**(&F'6#\"\"&F*,&F*F*&F-6 #F:F/F*,&*&&F16$F:F3F*&F16$F3F6F*F**&&F16$F:F)F*&F16$F)F6F*F*F*F**(&F' 6#\"\"'F*,&F*F*&F-6#FLF/F*,(*&&F16$FLF3F*&F16$F3F6F*F**&&F16$FLF)F*&F1 6$F)F6F*F**&&F16$FLF:F*&F16$F:F6F*F*F*F*\"\"!" }{TEXT -1 2 ", " }} {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 256 "" 0 "" {TEXT -1 1 " " } {XPPEDIT 18 0 "Sum(b[i]*(1-c[i])*(c[6]-c[i])*a[i,2],i=3..6)=0" "6#/-%$ SumG6$**&%\"bG6#%\"iG\"\"\",&F,F,&%\"cG6#F+!\"\"F,,&&F/6#\"\"'F,&F/6#F +F1F,&%\"aG6$F+\"\"#F,/F+;\"\"$F5\"\"!" }{TEXT -1 3 "., " }}{PARA 257 "" 0 "" {TEXT -1 10 "that is, " }}{PARA 256 "" 0 "" {TEXT -1 1 " " } {XPPEDIT 18 0 "b[3]*(1-c[3])*(c[6]-c[3])*a[3,2]+b[4]*(1-c[4])*(c[6]-c[ 4])*a[4,2]+b[5]*(1-c[5])*(c[6]-c[5])*a[5,2] = 0" "6#/,(**&%\"bG6#\"\"$ \"\"\",&F*F*&%\"cG6#F)!\"\"F*,&&F-6#\"\"'F*&F-6#F)F/F*&%\"aG6$F)\"\"#F *F***&F'6#\"\"%F*,&F*F*&F-6#F=F/F*,&&F-6#F3F*&F-6#F=F/F*&F76$F=F9F*F** *&F'6#\"\"&F*,&F*F*&F-6#FKF/F*,&&F-6#F3F*&F-6#FKF/F*&F76$FKF9F*F*\"\"! " }{TEXT -1 2 ". " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 5 "Let " }{XPPEDIT 18 0 "theta = b[5]*(c[5]-1)*(c[5]-c[6])*( c[5]-c[4])*(c[5]-c[3])*c[5]" "6#/%&thetaG*.&%\"bG6#\"\"&\"\"\",&&%\"cG 6#F)F*F*!\"\"F*,&&F-6#F)F*&F-6#\"\"'F/F*,&&F-6#F)F*&F-6#\"\"%F/F*,&&F- 6#F)F*&F-6#\"\"$F/F*&F-6#F)F*" }{TEXT -1 2 ". " }}{PARA 0 "" 0 "" {TEXT -1 82 "The product of the left hand sides of the 2nd and 3rd equ ations is the product of " }{XPPEDIT 18 0 "theta" "6#%&thetaG" }{TEXT -1 44 " with the left hand side of the 4th equation" }}{PARA 0 "" 0 " " {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 6 "Hence " }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "(1/360-c[3]/120)*theta = (1/120-c[ 3]/60-c[4]/60+c[3]*c[4]/24)*(-1/90+c[3]/40+c[6]/60-c[3]*c[6]/24);" "6# /*&,&*&\"\"\"F'\"$g$!\"\"F'*&&%\"cG6#\"\"$F'\"$?\"F)F)F'%&thetaGF'*&,* *&F'F'F/F)F'*&&F,6#F.F'\"#gF)F)*&&F,6#\"\"%F'F7F)F)*(&F,6#F.F'&F,6#F;F '\"#CF)F'F',**&F'F'\"#!*F)F)*&&F,6#F.F'\"#SF)F'*&&F,6#\"\"'F'F7F)F'*(& F,6#F.F'&F,6#FLF'FAF)F)F'" }{TEXT -1 1 "." }}{PARA 0 "" 0 "" {TEXT -1 1 " " }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 553 "cdns := [b[6]*(1-c[ 6])*(a[6,5]*c[5]*(c[5]-c[3])+a[6,4]*c[4]*(c[4]-c[3]))+\n b[5]*(1-c[5 ])*a[5,4]*c[4]*(c[4]-c[3])=1/60-c[3]/24,\n b[6]*(1-c[6])*a[6,5]*c[5] *(c[5]-c[3])*(c[5]-c[4])=1/120-(c[3]+c[4])/60+c[3]*c[4]/24,\n b[5]*( 1-c[5])*(c[6]-c[5])*a[5,4]*c[4]*(c[4]-c[3])=-1/90+c[3]/40+c[6]/60-c[3] *c[6]/24,\n b[6]*(1-c[6])*a[6,5]*a[5,4]*c[4]*(c[4]-c[3])=1/360-c[3]/ 120,\n add(b[i]*(1-c[i])*a[i,2],i=3..6)=0,add(b[i]*(1-c[i])*add(a[i, j]*a[j,2],j=3..i-1),i=3..6)=0,\n add(b[i]*(1-c[i])*(c[6]-c [i])*a[i,2],i=3..6)=0]:\nlinalg[transpose]([cdns]);" }}{PARA 11 "" 1 " " {XPPMATH 20 "6#K%'matrixG6#7)7#/,&*(&%\"bG6#\"\"'\"\"\",&F/F/&%\"cGF -!\"\"F/,&*(&%\"aG6$F.\"\"&F/&F26#F9F/,&F:F/&F26#\"\"$F3F/F/*(&F76$F. \"\"%F/&F26#FCF/,&FDF/F=F3F/F/F/F/*,&F,F;F/,&F/F/F:F3F/&F76$F9FCF/FDF/ FFF/F/,&#F/\"#gF/*&#F/\"#CF/F=F/F37#/*.F+F/F0F/F6F/F:F/FF/,&F/F/F=F3F/&F76$F?\"\"#F/F/*(&F,FEF/,&F/F /FDF3F/&F76$FCFhpF/F/*(FHF/FIF/&F76$F9FhpF/F/*(F+F/F0F/&F76$F.FhpF/F/ \"\"!7#/,(**FjpF/F[qF/&F76$FCF?F/FfpF/F/*(FHF/FIF/,&*&&F76$F9F?F/FfpF/ F/*&FJF/F\\qF/F/F/F/*(F+F/F0F/,(*&&F76$F.F?F/FfpF/F/*&FAF/F\\qF/F/*&F6 F/F_qF/F/F/F/Fdq7#/,(**FdpF/FepF/,&F1F/F=F3F/FfpF/F/**FjpF/F[qF/,&F1F/ FDF3F/F\\qF/F/**FHF/FIF/F]oF/F_qF/F/FdqQ)pprint136\"" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 70 "(b[5]*(c[5]-1)*(c[5]-c[6])*( c[5]-c[4])*(c[5]-c[3])*c[5])*lhs(cdns[4]);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#*:&%\"bG6#\"\"&\"\"\",&&%\"cGF&F(F(!\"\"F(,&F*F(&F+6#\" \"'F,F(,&F*F(&F+6#\"\"%F,F(,&F*F(&F+6#\"\"$F,F(F*F(&F%F/F(,&F(F(F.F,F( &%\"aG6$F0F'F(&F<6$F'F4F(F2F(,&F2F(F6F,F(" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 26 "lhs(cdns[2])*lhs(c dns[3]);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#*:&%\"bG6#\"\"'\"\"\",&F(F (&%\"cGF&!\"\"F(&%\"aG6$F'\"\"&F(&F+6#F0F(,&F1F(&F+6#\"\"$F,F(,&F1F(&F +6#\"\"%F,F(&F%F2F(,&F(F(F1F,F(,&F*F(F1F,F(&F.6$F0F:F(F8F(,&F8F(F4F,F( " }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 135 "theta_eq1 := theta=b[5]*(c[5]-1)*(c[5]-c[6])*(c[5]-c [4])*(c[5]-c[3])*c[5];\ntheta_eq2 := rhs(cdns[4])*theta=rhs(cdns[2])*r hs(cdns[3]);\n\n" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%*theta_eq1G/%&th etaG*.&%\"bG6#\"\"&\"\"\",&&%\"cGF*F,F,!\"\"F,,&F.F,&F/6#\"\"'F0F,,&F. F,&F/6#\"\"%F0F,,&F.F,&F/6#\"\"$F0F,F.F," }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%*theta_eq2G/*&,&#\"\"\"\"$g$F)*&#F)\"$?\"F)&%\"cG6#\"\"$F)!\" \"F)%&thetaGF)*&,*#F)F-F)*&#F)\"#gF)F.F)F2*&#F)F9F)&F/6#\"\"%F)F2*&#F) \"#CF)*&F.F)F " 0 "" {MPLTEXT 1 0 73 "Qeqs := [add(b[i],i =1..7)=1,seq(add(b[i]*c[i]^(k-1),i=2..7)=1/k,k=2..6)]:" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 103 "quadeq ns := subs(\{b[2]=0,c[7]=1\},Qeqs):\nnops(%);\nindets(quadeqns) minus \+ \{c[3],c[4],c[5],c[6]\};\nnops(%);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6# \"\"'" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#<(&%\"bG6#\"\"'&F%6#\"\"(&F%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 0 142 "sol := factor(solve(\{op(quadeqns)\},indets(quadeqns ) minus \{c[3],c[4],c[5],c[6]\})):\nfor ii in [1,$3..7] do print(b[ii] =subs(sol,b[ii])) end do;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/&%\"bG6# \"\"\",$*&#F'\"#gF'*,,B**\"#5F'&%\"cG6#\"\"$F'&F16#\"\"'F'&F16#\"\"&F' !\"\"*(F9F'F0F'F7F'F'*&F3F'F7F'F:*(F9F'F4F'F7F'F'*,\"#IF'&F16#\"\"%F'F 0F'F4F'F7F'F'*(F9F'F@F'F7F'F'**F/F'F@F'F4F'F7F'F:**F/F'F@F'F0F'F7F'F: \"\"#F'*&F3F'F4F'F:*&F3F'F0F'F:*&F3F'F@F'F:**F/F'F0F'F4F'F@F'F:*(F9F'F 0F'F4F'F'*(F9F'F4F'F@F'F'*(F9F'F0F'F@F'F'F'F@F:F0F:F4F:F7F:F'F'" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#/&%\"bG6#\"\"$,$*&#\"\"\"\"#gF+*.,2*&F 'F+&%\"cG6#\"\"&F+F+**\"#5F+&F16#\"\"%F+&F16#\"\"'F+F0F+F+*(F3F+F9F+F0 F+!\"\"*(F3F+F6F+F0F+F=\"\"#F=*&F'F+F9F+F+*&F'F+F6F+F+*(F3F+F9F+F6F+F= F+,&F0F+&F1F&F=F=,&F6F=FDF+F=FDF=,&FDF+F+F=F=,&F9F+FDF=F=F+F+" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#/&%\"bG6#\"\"%,$*&#\"\"\"\"#gF+*.,2*& \"\"$F+&%\"cG6#\"\"&F+F+*(F4F+&F26#\"\"'F+F1F+!\"\"*(F4F+&F26#F0F+F1F+ F9**\"#5F+F;F+F6F+F1F+F+\"\"#F9*&F0F+F6F+F+*&F0F+F;F+F+*(F4F+F;F+F6F+F 9F+,&F1F+&F2F&F9F9FDF9,&FDF+F+F9F9,&F6F+FDF9F9,&FDF9F;F+F9F+F9" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#/&%\"bG6#\"\"&,$*&#\"\"\"\"#gF+*.,2\" \"#!\"\"*&\"\"$F+&%\"cG6#\"\"'F+F+*&F2F+&F46#F2F+F+*&F2F+&F46#\"\"%F+F +**\"#5F+F8F+F3F+F;F+F+*(F'F+F8F+F3F+F0*(F'F+F3F+F;F+F0*(F'F+F8F+F;F+F 0F+&F4F&F0,&FCF+F+F0F0,&FCF+F8F0F0,&FCF+F;F0F0,&FCF+F3F0F0F+F+" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#/&%\"bG6#\"\"',$*&#\"\"\"\"#gF+*.,2*& \"\"$F+&%\"cG6#\"\"&F+F+**\"#5F+&F26#\"\"%F+&F26#F0F+F1F+F+*(F4F+F:F+F 1F+!\"\"*(F4F+F7F+F1F+F=*&F0F+F7F+F+*(F4F+F:F+F7F+F=\"\"#F=*&F0F+F:F+F +F+,&F1F+&F2F&F=F=,&FDF+F7F=F=,&FDF+F:F=F=,&FDF+F+F=F=FDF=F+F=" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#/&%\"bG6#\"\"(,$*&#\"\"\"\"#gF+*,,B*( \"#:F+&%\"cG6#\"\"$F+&F26#\"\"&F+F+*(F0F+&F26#\"\"'F+F5F+F+**\"#?F+F1F +F9F+F5F+!\"\"**F=F+&F26#\"\"%F+F1F+F5F+F>*(F0F+F@F+F5F+F+*&\"#7F+F5F+ F>**F=F+F@F+F9F+F5F+F>*,\"#IF+F@F+F1F+F9F+F5F+F+\"#5F+*&FEF+F9F+F>*&FE F+F1F+F>*&FEF+F@F+F>**F=F+F1F+F9F+F@F+F>*(F0F+F9F+F@F+F+*(F0F+F1F+F@F+ F+*(F0F+F1F+F9F+F+F+,&F5F+F+F>F>,&F@F+F+F>F>,&F1F+F+F>F>,&F9F+F+F>F>F+ F+" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 32 "Th is means that we can express " }{XPPEDIT 18 0 "theta" "6#%&thetaG" } {TEXT -1 45 " in terms of the nodes by substituting for " }{XPPEDIT 18 0 "b[5]" "6#&%\"bG6#\"\"&" }{TEXT -1 31 " in the equation that def ines " }{XPPEDIT 18 0 "theta" "6#%&thetaG" }{TEXT -1 1 "." }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 33 "theta_eq3 := subs(sol,theta_eq1);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%*theta_eq3G/%&thetaG,2#\"\"\"\"#I! \"\"*&#F)\"#?F)&%\"cG6#\"\"'F)F)*&F-F)&F06#\"\"$F)F)*&F-F)&F06#\"\"%F) F)*&#F)F2F)*(F4F)F/F)F8F)F)F)*&#F)\"#7F)*&F4F)F/F)F)F+*&#F)F@F)*&F/F)F 8F)F)F+*&#F)F@F)*&F4F)F8F)F)F+" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }} {PARA 0 "" 0 "" {TEXT -1 54 "We can now obtain an equation that relate s the nodes " }{XPPEDIT 18 0 "c[3]" "6#&%\"cG6#\"\"$" }{TEXT -1 7 " \+ and " }{XPPEDIT 18 0 "c[4]" "6#&%\"cG6#\"\"%" }{TEXT -1 9 ", namely: " }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "c[4]=c[3]/(15*c[3 ]^2-10*c[3]+2)" "6#/&%\"cG6#\"\"%*&&F%6#\"\"$\"\"\",(*&\"#:F,*$&F%6#F+ \"\"#F,F,*&\"#5F,&F%6#F+F,!\"\"F3F,F8" }{TEXT -1 2 ". " }}{PARA 256 " " 0 "" {TEXT -1 1 " " }{TEXT 268 12 "____________" }{TEXT -1 1 " " }} {PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 46 "subs(theta_eq3,theta_eq2);\nc[4]=solve(%,c[4]);" }}{PARA 11 "" 1 " " {XPPMATH 20 "6#/*&,&#\"\"\"\"$g$F'*&#F'\"$?\"F'&%\"cG6#\"\"$F'!\"\"F ',2#F'\"#IF0*&#F'\"#?F'&F-6#\"\"'F'F'*&F5F'F,F'F'*&F5F'&F-6#\"\"%F'F'* &#F'F9F'*(F,F'F7F'F " 0 "" {MPLTEXT 1 0 1 ";" }}} }{PARA 0 "" 0 "" {TEXT -1 28 "#===========================" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 44 "#================ ===========================" }}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 48 "St ep by step construction of Butcher's scheme A " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 62 "See: On Runge-Kutta Proce sses of High Order, by J. C. Butcher," }}{PARA 0 "" 0 "" {TEXT -1 88 " Journal of the Australian Mathematical Society, Vol. 4, (1964), pages 179 to 194." }}{PARA 0 "" 0 "" {TEXT -1 87 "------------------- --------------------------------------------------------------------" }}{PARA 0 "" 0 "" {TEXT -1 17 "We specify that " }{XPPEDIT 18 0 "c[2] =1/2" "6#/&%\"cG6#\"\"#*&\"\"\"F)F'!\"\"" }{TEXT -1 3 ", " }{XPPEDIT 18 0 "c[3]=2/3" "6#/&%\"cG6#\"\"$*&\"\"#\"\"\"F'!\"\"" }{TEXT -1 3 ", \+ " }{XPPEDIT 18 0 "c[5]=5/6" "6#/&%\"cG6#\"\"&*&F'\"\"\"\"\"'!\"\"" } {TEXT -1 3 ", " }{XPPEDIT 18 0 "c[6]=1/6" "6#/&%\"cG6#\"\"'*&\"\"\"F) F'!\"\"" }{TEXT -1 3 ", " }{XPPEDIT 18 0 "c[7]=1" "6#/&%\"cG6#\"\"(\" \"\"" }{TEXT -1 7 " and " }{XPPEDIT 18 0 "b[2]=0" "6#/&%\"bG6#\"\"# \"\"!" }{TEXT -1 1 "." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 15 "" 0 "" {TEXT 272 6 "Step 1" }{TEXT -1 1 " " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 21 "We use the relation " }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "c[4]=c[3]/(15*c[3]^2-10*c[3 ]+2)" "6#/&%\"cG6#\"\"%*&&F%6#\"\"$\"\"\",(*&\"#:F,*$&F%6#F+\"\"#F,F,* &\"#5F,&F%6#F+F,!\"\"F3F,F8" }{TEXT -1 1 " " }}{PARA 0 "" 0 "" {TEXT -1 14 "to determine " }{XPPEDIT 18 0 "c[4]" "6#&%\"cG6#\"\"%" }{TEXT -1 36 " and use the quadrature conditions:" }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "Sum(b[i],i = 1 .. 7) = 1;" "6#/-%$SumG6 $&%\"bG6#%\"iG/F*;\"\"\"\"\"(F-" }{TEXT -1 15 ", " } {XPPEDIT 18 0 "Sum(b[i]*c[i]^(k-1),i = 2 .. 7) = 1/k;" "6#/-%$SumG6$*& &%\"bG6#%\"iG\"\"\")&%\"cG6#F+,&%\"kGF,F,!\"\"F,/F+;\"\"#\"\"(*&F,F,F2 F3" }{TEXT -1 7 ", " }{XPPEDIT 18 0 "k = 2;" "6#/%\"kG\"\"#" } {TEXT -1 8 " . . 6, " }}{PARA 0 "" 0 "" {TEXT -1 21 "to find the weigh ts " }{XPPEDIT 18 0 "b[1]" "6#&%\"bG6#\"\"\"" }{TEXT -1 2 ", " } {XPPEDIT 18 0 "b[3]" "6#&%\"bG6#\"\"$" }{TEXT -1 3 ", " }{XPPEDIT 18 0 "b[4];" "6#&%\"bG6#\"\"%" }{TEXT -1 3 ", " }{XPPEDIT 18 0 "b[5];" " 6#&%\"bG6#\"\"&" }{TEXT -1 3 ", " }{XPPEDIT 18 0 "b[6]" "6#&%\"bG6#\" \"'" }{TEXT -1 3 ", " }{XPPEDIT 18 0 "b[7]" "6#&%\"bG6#\"\"(" }{TEXT -1 1 "." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 329 "e0 := \{b[2]=0,c[2]=1/2,c[3]=2/3,c[5]=5/6,c[6]=1/6,c [7]=1\}:\ne1 := `union`(e0,\{subs(e0,c[4]=c[3]/(15*c[3]^2-10*c[3]+2)) \}):\nc[4]=subs(e1,c[4]);\n``;\nquad_cdns := [add(b[i],i=1..7)=1,seq(a dd(b[i]*c[i]^(k-1),i=2..7)=1/k,k=2..6)]:\nquad_eqns := subs(e1,quad_cd ns):\nmatrix(ListTools[Enumerate](quad_eqns));\n``;\nindets(quad_eqns) ;\nnops(%);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/&%\"cG6#\"\"%#\"\"\"\" \"$" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%!G" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#K%'matrixG6#7(7$\"\"\"/,.&%\"bG6#F(F(&F,6#\"\"$F(&F,6# \"\"%F(&F,6#\"\"&F(&F,6#\"\"'F(&F,6#\"\"(F(F(7$\"\"#/,,*&#F>F0F(F.F(F( *&#F(F0F(F1F(F(*&#F6F9F(F4F(F(*&#F(F9F(F7F(F(F:F(#F(F>7$F0/,,*&#F3\"\" *F(F.F(F(*&#F(FOF(F1F(F(*&#\"#D\"#OF(F4F(F(*&#F(FUF(F7F(F(F:F(FD7$F3/, ,*&#\"\")\"#FF(F.F(F(*&#F(FhnF(F1F(F(*&#\"$D\"\"$;#F(F4F(F(*&#F(F^oF(F 7F(F(F:F(#F(F37$F6/,,*&#\"#;\"#\")F(F.F(F(*&#F(FhoF(F1F(F(*&#\"$D'\"%' H\"F(F4F(F(*&#F(F^pF(F7F(F(F:F(#F(F67$F9/,,*&#\"#K\"$V#F(F.F(F(*&#F(Fh pF(F1F(F(*&#\"%DJ\"%wxF(F4F(F(*&#F(F^qF(F7F(F(F:F(FHQ)pprint176\"" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#%!G" }}{PARA 11 "" 1 "" {XPPMATH 20 "6 #<(&%\"bG6#\"\"\"&F%6#\"\"%&F%6#\"\"$&F%6#\"\"(&F%6#\"\"'&F%6#\"\"&" } }{PARA 11 "" 1 "" {XPPMATH 20 "6#\"\"'" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 83 "e2 := solve(\{op(quad_e qns)\},\{b[1],b[3],b[4],b[5],b[6],b[7]\}):\ne3 := `union`(e1,e2):" }}} {PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 2 "e3" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 171 "e3 := \{c[7 ] = 1, b[2] = 0, c[4] = 1/3, c[5] = 5/6, c[6] = 1/6, b[4] = 11/40, b[1 ] = 13/200, b[7] = 13/200, b[5] = 4/25, b[3] = 11/40, b[6] = 4/25, c[2 ] = 1/2, c[3] = 2/3\}:" }{TEXT -1 0 "" }}}{PARA 0 "" 0 "" {TEXT -1 0 " " }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 41 "The weights of the scheme are as follows." }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 31 "seq(b[i] =subs(e3,b[i]),i=1..7);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6)/&%\"bG6#\" \"\"#\"#8\"$+#/&F%6#\"\"#\"\"!/&F%6#\"\"$#\"#6\"#S/&F%6#\"\"%F4/&F%6# \"\"&#F:\"#D/&F%6#\"\"'F?/&F%6#\"\"(F(" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 15 "" 0 "" {TEXT -1 1 " " }{TEXT 273 6 "Step 2" }{TEXT -1 1 " " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 51 "W e can obtain values for the linking coefficients " }{XPPEDIT 18 0 "a[ 5,4]" "6#&%\"aG6$\"\"&\"\"%" }{TEXT -1 4 " , " }{XPPEDIT 18 0 "a[6,5] " "6#&%\"aG6$\"\"'\"\"&" }{TEXT -1 7 " and " }{XPPEDIT 18 0 "a[6,4] " "6#&%\"aG6$\"\"'\"\"%" }{TEXT -1 57 " using the following three alt ernative order conditions:" }}{PARA 256 "" 0 "" {TEXT -1 1 " " } {XPPEDIT 18 0 "b[6]*(1-c[6])*(a[6,5]*c[5]*(c[5]-c[3])+a[6,4]*c[4]*(c[4 ]-c[3]))+b[5]*(1-c[5])*a[5,4]*c[4]*(c[4]-c[3])=1/60-c[3]/24" "6#/,&*(& %\"bG6#\"\"'\"\"\",&F*F*&%\"cG6#F)!\"\"F*,&*(&%\"aG6$F)\"\"&F*&F-6#F5F *,&&F-6#F5F*&F-6#\"\"$F/F*F**(&F36$F)\"\"%F*&F-6#FAF*,&&F-6#FAF*&F-6#F =F/F*F*F*F**,&F'6#F5F*,&F*F*&F-6#F5F/F*&F36$F5FAF*&F-6#FAF*,&&F-6#FAF* &F-6#F=F/F*F*,&*&F*F*\"#gF/F**&&F-6#F=F*\"#CF/F/" }{TEXT -1 2 ", " }} {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 256 "" 0 "" {TEXT -1 1 " " } {XPPEDIT 18 0 "b[6]*(1-c[6])*a[6,5]*c[5]*(c[5]-c[3])*(c[5]-c[4])=1/120 -(c[3]+c[4])/60+c[3]*c[4]/24" "6#/*.&%\"bG6#\"\"'\"\"\",&F)F)&%\"cG6#F (!\"\"F)&%\"aG6$F(\"\"&F)&F,6#F2F),&&F,6#F2F)&F,6#\"\"$F.F),&&F,6#F2F) &F,6#\"\"%F.F),(*&F)F)\"$?\"F.F)*&,&&F,6#F:F)&F,6#F@F)F)\"#gF.F.*(&F,6 #F:F)&F,6#F@F)\"#CF.F)" }{TEXT -1 2 ", " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "b[6]*(1-c[6])*a[ 6,5]*a[5,4]*c[4]*(c[4]-c[3])=1/360-c[3]/120" "6#/*.&%\"bG6#\"\"'\"\"\" ,&F)F)&%\"cG6#F(!\"\"F)&%\"aG6$F(\"\"&F)&F06$F2\"\"%F)&F,6#F5F),&&F,6# F5F)&F,6#\"\"$F.F),&*&F)F)\"$g$F.F)*&&F,6#F=F)\"$?\"F.F." }{TEXT -1 3 ". " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 385 "cdns1 := [b[6]*(1-c[6])*(a[6,5]*c[5]*(c[5]-c[3])+a[6 ,4]*c[4]*(c[4]-c[3]))+\n b[5]*(1-c[5])*a[5,4]*c[4]*(c[4]-c[3])=1/ 60-c[3]/24,\n b[6]*(1-c[6])*a[6,5]*c[5]*(c[5]-c[3])*(c[5]-c[4])=1/12 0-(c[3]+c[4])/60+c[3]*c[4]/24,\n b[6]*(1-c[6])*a[6,5]*a[5,4]*c[4]*(c [4]-c[3])=1/360-c[3]/120]:\neqns1 := simplify(subs(e3,cdns1)):\nmatrix (ListTools[Enumerate](eqns1));\n``;\nindets(eqns1);\nnops(%);" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#K%'matrixG6#7%7$\"\"\"/,(*&#F(\"#aF(&% \"aG6$\"\"'\"\"&F(F(*&#\"\"#\"$N\"F(&F/6$F1\"\"%F(!\"\"*&#F5\"$v'F(&F/ 6$F2F9F(F:#F:\"#!*7$F5/,$*&#F(\"$3\"F(F.F(F(#F(\"%!3\"7$\"\"$/,$*&#F5F 6F(*&F.F(F>F(F(F:#F:\"$g$Q(pprint56\"" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%!G" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#<%&%\"aG6$\"\"'\"\"%&F%6$\" \"&F(&F%6$F'F+" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#\"\"$" }}}{PARA 0 " " 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 140 "e4 : = solve(\{op(eqns1)\},\{a[5,4],a[6,4],a[6,5]\}):\ne5 := `union`(e3,e4) :\na[5,4]=subs(e5,a[5,4]),a[6,4]=subs(e5,a[6,4]),a[6,5]=subs(e5,a[6,5] );" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 2 "e5" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 215 "e 5 := \{c[7] = 1, b[2] = 0, c[4] = 1/3, a[6,4] = 1/2, a[5,4] = 15/8, a[ 6,5] = 1/10, c[5] = 5/6, c[6] = 1/6, b[4] = 11/40, b[1] = 13/200, b[7] = 13/200, b[5] = 4/25, b[3] = 11/40, b[6] = 4/25, c[2] = 1/2, c[3] = \+ 2/3\}:" }{TEXT -1 0 "" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}}{PARA 0 "" 0 "" {TEXT -1 0 " " }}{PARA 15 "" 0 "" {TEXT -1 1 " " }{TEXT 277 6 "Step 3" }{TEXT -1 1 " " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 18 "We \+ can determine " }{XPPEDIT 18 0 "a[3,2]" "6#&%\"aG6$\"\"$\"\"#" } {TEXT -1 32 " from the stage-order condition" }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "a[3,2]*c[2] = 1/2" "6#/*&&%\"aG6$\"\"$ \"\"#\"\"\"&%\"cG6#F)F**&F*F*F)!\"\"" }{TEXT -1 1 " " }{XPPEDIT 18 0 " c[3]^2" "6#*$&%\"cG6#\"\"$\"\"#" }{TEXT -1 1 "." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 113 "a[3,2]*c[2] =1/2*c[3]^2:\nsubs(e5,%);\ne6 := \{a[3,2]=solve(%,a[3,2])\}:\ne7 := `u nion`(e5,e6):\na[3,2]=subs(e7,a[3,2]);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/,$*&#\"\"\"\"\"#F'&%\"aG6$\"\"$F(F'F'#F(\"\"*" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/&%\"aG6$\"\"$\"\"##\"\"%\"\"*" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 2 "e7" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 229 "e7 := \{b[7] = 13/200, b[5] = 4/25, b[4] = 11/40, b[1] = 13/200, c[5] = 5/6, c[6] = 1/6, c[2] = 1 /2, c[3] = 2/3, b[3] = 11/40, b[6] = 4/25, a[6,4] = 1/2, a[6,5] = 1/10 , a[5,4] = 15/8, c[4] = 1/3, c[7] = 1, b[2] = 0, a[3,2] = 4/9\}:" }}} {PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 15 "" 0 "" {TEXT -1 1 " " }{TEXT 278 6 "Step 4" }{TEXT -1 1 " " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 41 "We can now determine the 6 coeffic ients " }{XPPEDIT 18 0 "a[4,2]" "6#&%\"aG6$\"\"%\"\"#" }{TEXT -1 3 ", " }{XPPEDIT 18 0 "a[5, 2];" "6#&%\"aG6$\"\"&\"\"#" }{TEXT -1 3 ", \+ " }{XPPEDIT 18 0 "a[6, 2];" "6#&%\"aG6$\"\"'\"\"#" }{TEXT -1 3 ", " } {XPPEDIT 18 0 "a[4,3];" "6#&%\"aG6$\"\"%\"\"$" }{TEXT -1 3 ", " } {XPPEDIT 18 0 "a[5,3];" "6#&%\"aG6$\"\"&\"\"$" }{TEXT -1 3 ", " } {XPPEDIT 18 0 "a[6,3]" "6#&%\"aG6$\"\"'\"\"$" }{TEXT -1 56 " by using the three alternative simplifying conditions:" }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "Sum(b[i]*(1-c[i])*a[i,2],i=3..6)=0" "6# /-%$SumG6$*(&%\"bG6#%\"iG\"\"\",&F,F,&%\"cG6#F+!\"\"F,&%\"aG6$F+\"\"#F ,/F+;\"\"$\"\"'\"\"!" }{TEXT -1 2 ", " }}{PARA 0 "" 0 "" {TEXT -1 8 "t hat is," }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "b[3]*(1-c[ 3])*a[3,2]+b[4]*(1-c[4])*a[4,2]+b[5]*(1-c[5])*a[5,2]+b[6]*(1-c[6])*a[6 ,2] = 0" "6#/,**(&%\"bG6#\"\"$\"\"\",&F*F*&%\"cG6#F)!\"\"F*&%\"aG6$F) \"\"#F*F**(&F'6#\"\"%F*,&F*F*&F-6#F7F/F*&F16$F7F3F*F**(&F'6#\"\"&F*,&F *F*&F-6#F@F/F*&F16$F@F3F*F**(&F'6#\"\"'F*,&F*F*&F-6#FIF/F*&F16$FIF3F*F *\"\"!" }{TEXT -1 2 ", " }}{PARA 0 "" 0 "" {TEXT -1 3 "and" }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "Sum(b[i]*(1-c[i])*Sum(a[i,j ]*a[j,2],j = 3 .. i-1),i = 3 .. 6) = 0" "6#/-%$SumG6$*(&%\"bG6#%\"iG\" \"\",&F,F,&%\"cG6#F+!\"\"F,-F%6$*&&%\"aG6$F+%\"jGF,&F66$F8\"\"#F,/F8; \"\"$,&F+F,F,F1F,/F+;F>\"\"'\"\"!" }{TEXT -1 2 ", " }}{PARA 0 "" 0 "" {TEXT -1 8 "that is," }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "b[4]*(1-c[4])*a[4,3]*a[3,2]+b[5]*(1-c[5])*(a[5,3]*a[3,2]+a[5,4]*a[4 ,2])+b[6]*(1-c[6])*(a[6,3]*a[3,2]+a[6,4]*a[4,2]+a[6,5]*a[5,2]) = 0" "6 #/,(**&%\"bG6#\"\"%\"\"\",&F*F*&%\"cG6#F)!\"\"F*&%\"aG6$F)\"\"$F*&F16$ F3\"\"#F*F**(&F'6#\"\"&F*,&F*F*&F-6#F:F/F*,&*&&F16$F:F3F*&F16$F3F6F*F* *&&F16$F:F)F*&F16$F)F6F*F*F*F**(&F'6#\"\"'F*,&F*F*&F-6#FLF/F*,(*&&F16$ FLF3F*&F16$F3F6F*F**&&F16$FLF)F*&F16$F)F6F*F**&&F16$FLF:F*&F16$F:F6F*F *F*F*\"\"!" }{TEXT -1 2 ", " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "Sum(b[i]*(c[i]-1)*(c[i]-c[6 ])*a[i,2],i = 3 .. 5) = 0;" "6#/-%$SumG6$**&%\"bG6#%\"iG\"\"\",&&%\"cG 6#F+F,F,!\"\"F,,&&F/6#F+F,&F/6#\"\"'F1F,&%\"aG6$F+\"\"#F,/F+;\"\"$\"\" &\"\"!" }{TEXT -1 2 ", " }}{PARA 0 "" 0 "" {TEXT -1 9 "that is, " }} {PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "b[3]*(c[3]-1)*(c[3]-c [6])*a[3,2]+b[4]*(c[4]-1)*(c[4]-c[6])*a[4,2]+b[5]*(c[5]-1)*(c[5]-c[6]) *a[5,2]=0" "6#/,(**&%\"bG6#\"\"$\"\"\",&&%\"cG6#F)F*F*!\"\"F*,&&F-6#F) F*&F-6#\"\"'F/F*&%\"aG6$F)\"\"#F*F***&F'6#\"\"%F*,&&F-6#F=F*F*F/F*,&&F -6#F=F*&F-6#F5F/F*&F76$F=F9F*F***&F'6#\"\"&F*,&&F-6#FKF*F*F/F*,&&F-6#F KF*&F-6#F5F/F*&F76$FKF9F*F*\"\"!" }{TEXT -1 2 ", " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 48 "together with the three s tage-order conditions: " }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "Sum(a[4,j]*c[j],j = 2 .. 3) = 1/2;" "6#/-%$SumG6$*&&%\"aG6$\"\"% %\"jG\"\"\"&%\"cG6#F,F-/F,;\"\"#\"\"$*&F-F-F3!\"\"" }{TEXT -1 1 " " } {XPPEDIT 18 0 "c[4]^2;" "6#*$&%\"cG6#\"\"%\"\"#" }{TEXT -1 8 " , \+ " }{XPPEDIT 18 0 "Sum(a[5,j]*c[j],j=2..4)=1/2" "6#/-%$SumG6$*&&%\"aG6$ \"\"&%\"jG\"\"\"&%\"cG6#F,F-/F,;\"\"#\"\"%*&F-F-F3!\"\"" }{TEXT -1 1 " " }{XPPEDIT 18 0 "c[5]^2" "6#*$&%\"cG6#\"\"&\"\"#" }{TEXT -1 2 " " } }{PARA 0 "" 0 "" {TEXT -1 4 "and " }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "Sum(a[6,j]*c[j],j=2..5)=1/2" "6#/-%$SumG6$*&&%\"aG6$\" \"'%\"jG\"\"\"&%\"cG6#F,F-/F,;\"\"#\"\"&*&F-F-F3!\"\"" }{TEXT -1 1 " \+ " }{XPPEDIT 18 0 "c[6]^2" "6#*$&%\"cG6#\"\"'\"\"#" }{TEXT -1 1 "." }} {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 291 "cdns2 := [add(b[i]*(1-c[i]) *a[i,2],i=3..6)=0,add(b[i]*(1-c[i])*add(a[i,j]*a[j,2],j=3..i-1),i=3..6 )=0,\n add(b[i]*(1-c[i])*(c[6]-c[i])*a[i,2],i=3..5)=0,seq(add(a[i,j]* c[j],j=2..i-1)=1/2*c[i]^2,i=4..6)]:\neqns2 := subs(e7,cdns2):\nmatrix( ListTools[Enumerate](eqns2));\n``;\nindets(eqns2);\nnops(%);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#K%'matrixG6#7(7$\"\"\"/,*#\"#6\"$q#F(*&#F,\" #gF(&%\"aG6$\"\"%\"\"#F(F(*&#F5\"#vF(&F26$\"\"&F5F(F(*&#F5\"#:F(&F26$ \"\"'F5F(F(\"\"!7$F5/,,*&#F,\"$N\"F(&F26$F4\"\"$F(F(*&#\"\")\"$v'F(&F2 6$F;FKF(F(*&#\"\"(F0F(F1F(F(*&#FNFHF(&F26$FAFKF(F(*&#F(F8F(F9F(F(FB7$F K/,(#F,\"$S&!\"\"*&#F,\"$g$F(F1F(Fjn*&#F4\"$D#F(F9F(FjnFB7$F4/,&*&#F(F 5F(F1F(F(*&#F5FKF(FIF(F(#F(\"#=7$F;/,(*&FeoF(F9F(F(*&FgoF(FPF(F(#F;FNF (#\"#D\"#s7$FA/,(*&FeoF(F?F(F(*&FgoF(FWF(F(#F(F4F(#F(FbpQ(pprint06\"" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%!G" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#<(&%\"aG6$\"\"&\"\"#&F%6$\"\"%F(&F%6$\"\"'F(&F%6$F+\"\"$&F%6$F'F1&F %6$F.F1" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#\"\"'" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 164 "e8 := solve (\{op(eqns2)\},\{a[4,2],a[5,2],a[6,2],a[4,3],a[5,3],a[6,3]\}):\ne9 := \+ `union`(e7,e8):\nseq(a[i,2]=subs(e9,a[i,2]),i=4..6);\nseq(a[i,3]=subs( e9,a[i,3]),i=4..6);\n" }}{PARA 11 "" 1 "" {XPPMATH 20 "6%/&%\"aG6$\"\" %\"\"##F(\"\"*/&F%6$\"\"&F(#!#b\"#O/&F%6$\"\"'F(#!#6F1" }}{PARA 11 "" 1 "" {XPPMATH 20 "6%/&%\"aG6$\"\"%\"\"$#!\"\"\"#7/&F%6$\"\"&F(#\"#N\"# [/&F%6$\"\"'F(#F*\"\")" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 2 "e9" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 324 "e9 := \{a[6,2] = -11/36, b[7] = 13/200, a[5,3] = 35/ 48, a[6,3] = -1/8, b[5] = 4/25, b[4] = 11/40, b[1] = 13/200, a[4,3] = \+ -1/12, c[5] = 5/6, c[6] = 1/6, c[2] = 1/2, c[3] = 2/3, a[4,2] = 2/9, a [5,2] = -55/36, b[3] = 11/40, b[6] = 4/25, a[6,4] = 1/2, a[6,5] = 1/10 , a[5,4] = 15/8, c[4] = 1/3, c[7] = 1, b[2] = 0, a[3,2] = 4/9\}:" }}} {PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 15 "" 0 "" {TEXT -1 1 " " }{TEXT 274 6 "Step 5" }{TEXT -1 1 " " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 39 "The four column simplifying condit ions " }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "Sum(b[i]*a[i ,j],i=j+1..7)=b[j]*(1-c[j])" "6#/-%$SumG6$*&&%\"bG6#%\"iG\"\"\"&%\"aG6 $F+%\"jGF,/F+;,&F0F,F,F,\"\"(*&&F)6#F0F,,&F,F,&%\"cG6#F0!\"\"F," } {TEXT -1 5 ", " }{XPPEDIT 18 0 "j = 2;" "6#/%\"jG\"\"#" }{TEXT -1 10 ", 4, 5, 6 " }}{PARA 0 "" 0 "" {TEXT -1 19 "give the equations " }} {PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "b[3]*a[3,2]+b[4]*a[4, 2]+b[5]*a[5,2]+b[6]*a[6,2]+b[7]*a[7,2] = b[2]*(1-c[2])" "6#/,,*&&%\"bG 6#\"\"$\"\"\"&%\"aG6$F)\"\"#F*F**&&F'6#\"\"%F*&F,6$F2F.F*F**&&F'6#\"\" &F*&F,6$F8F.F*F**&&F'6#\"\"'F*&F,6$F>F.F*F**&&F'6#\"\"(F*&F,6$FDF.F*F* *&&F'6#F.F*,&F*F*&%\"cG6#F.!\"\"F*" }{TEXT -1 2 ", " }}{PARA 0 "" 0 " " {TEXT -1 0 "" }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "b[5 ]*a[5,4]+b[6]*a[6,4]+b[7]*a[7,4] = b[4]*(1-c[4])" "6#/,(*&&%\"bG6#\"\" &\"\"\"&%\"aG6$F)\"\"%F*F**&&F'6#\"\"'F*&F,6$F2F.F*F**&&F'6#\"\"(F*&F, 6$F8F.F*F**&&F'6#F.F*,&F*F*&%\"cG6#F.!\"\"F*" }{TEXT -1 2 ", " }} {PARA 257 "" 0 "" {TEXT -1 0 "" }}{PARA 256 "" 0 "" {TEXT -1 1 " " } {XPPEDIT 18 0 "b[6]*a[6,5]+b[7]*a[7,5] = b[5]*(1-c[5])" "6#/,&*&&%\"bG 6#\"\"'\"\"\"&%\"aG6$F)\"\"&F*F**&&F'6#\"\"(F*&F,6$F2F.F*F**&&F'6#F.F* ,&F*F*&%\"cG6#F.!\"\"F*" }{TEXT -1 2 ", " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "b[7]*a[7,6] = \+ b[6]*(1-c[6])" "6#/*&&%\"bG6#\"\"(\"\"\"&%\"aG6$F(\"\"'F)*&&F&6#F-F),& F)F)&%\"cG6#F-!\"\"F)" }{TEXT -1 2 ", " }}{PARA 0 "" 0 "" {TEXT -1 26 "from which the values of " }{XPPEDIT 18 0 "a[7,2]" "6#&%\"aG6$\"\"( \"\"#" }{TEXT -1 3 ", " }{XPPEDIT 18 0 "a[7,4]" "6#&%\"aG6$\"\"(\"\"% " }{TEXT -1 3 ", " }{XPPEDIT 18 0 "a[7,5]" "6#&%\"aG6$\"\"(\"\"&" } {TEXT -1 7 " and " }{XPPEDIT 18 0 "a[7,6]" "6#&%\"aG6$\"\"(\"\"'" } {TEXT -1 18 " can be obtained." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 158 "cdns3 := [seq(add(b[i]*a[i, j],i=j+1..7)=b[j]*(1-c[j]),j=[2,4,5,6])]:\neqns3 := subs(e9,cdns3):\nm atrix(ListTools[Enumerate](eqns3));\n``;\nindets(eqns3);\nnops(%);" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#K%'matrixG6#7&7$\"\"\"/,&#\"#6\"$+\"! \"\"*&#\"#8\"$+#F(&%\"aG6$\"\"(\"\"#F(F(\"\"!7$F7/,&#\"#>\"#]F(*&F0F(& F46$F6\"\"%F(F(#F,\"#g7$\"\"$/,&#F7\"$D\"F(*&F0F(&F46$F6\"\"&F(F(#F7\" #v7$FB/,$*&F0F(&F46$F6\"\"'F(F(#F7\"#:Q(pprint16\"" }}{PARA 11 "" 1 " " {XPPMATH 20 "6#%!G" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#<&&%\"aG6$\"\" (\"\"'&F%6$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 0 122 "e10 := solve(\{op(eqns3)\},\{a[7,2],a[7, 4],a[7,5],a[7,6]\}):\ne11 := `union`(e9,e10):\nseq(a[7,j]=subs(e11,a[7 ,j]),j=[2,4,5,6]);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6&/&%\"aG6$\"\"(\" \"##\"#A\"#8/&F%6$F'\"\"%#!$=\"\"#R/&F%6$F'\"\"&#\"#K\"$&>/&F%6$F'\"\" '#\"#!)F2" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 " " {MPLTEXT 1 0 1 ";" }}}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 3 "e11" }} {PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 392 "e11 := \{a[6,2] = -11/36, b[7] = 13/200, a[5,3] = 35/48, a[6,3] = -1/8, b[5] = 4/25, b[4] = 11/40, b[1] = 13/200, a[4,3] = -1/12, c[5] \+ = 5/6, c[6] = 1/6, c[2] = 1/2, c[3] = 2/3, a[4,2] = 2/9, a[5,2] = -55/ 36, b[3] = 11/40, b[6] = 4/25, a[6,4] = 1/2, a[6,5] = 1/10, a[5,4] = 1 5/8, a[7,4] = -118/39, a[7,5] = 32/195, a[7,2] = 22/13, a[7,6] = 80/39 , c[4] = 1/3, c[7] = 1, b[2] = 0, a[3,2] = 4/9\}:" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 15 "" 0 "" {TEXT -1 1 " " }{TEXT 275 6 "Step 6" }{TEXT -1 1 " " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 " " 0 "" {TEXT -1 27 "The stage-order condition: " }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "Sum(a[7,j]*c[j],j = 2 .. 6) = 1/2;" "6# /-%$SumG6$*&&%\"aG6$\"\"(%\"jG\"\"\"&%\"cG6#F,F-/F,;\"\"#\"\"'*&F-F-F3 !\"\"" }{TEXT -1 1 " " }{XPPEDIT 18 0 "c[7]^2;" "6#*$&%\"cG6#\"\"(\"\" #" }{TEXT -1 1 " " }}{PARA 0 "" 0 "" {TEXT -1 12 "determines " } {XPPEDIT 18 0 "a[7,3]" "6#&%\"aG6$\"\"(\"\"$" }{TEXT -1 1 "." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 128 "s ubs(e11,add(a[7,j]*c[j],j=2..6)=1/2*c[7]^2);\ne12 := \{a[7,3]=solve(%, a[7,3])\}:\ne13 := `union`(e11,e12):\na[7,3]=subs(e13,a[7,3]);" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#/,&#\"#P\"$<\"\"\"\"*&#\"\"#\"\"$F(&% \"aG6$\"\"(F,F(F(#F(F+" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/&%\"aG6$\" \"(\"\"$#\"#V\"$c\"" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 3 " e13" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 409 "e13 := \{a[6,2] = -11/36, b[7] = 13/200, a[5,3] = 35 /48, a[6,3] = -1/8, b[5] = 4/25, b[4] = 11/40, b[1] = 13/200, a[4,3] = -1/12, c[5] = 5/6, c[6] = 1/6, c[2] = 1/2, c[3] = 2/3, a[4,2] = 2/9, \+ a[5,2] = -55/36, b[3] = 11/40, b[6] = 4/25, a[6,4] = 1/2, a[6,5] = 1/1 0, a[5,4] = 15/8, a[7,4] = -118/39, a[7,5] = 32/195, a[7,2] = 22/13, a [7,6] = 80/39, c[4] = 1/3, c[7] = 1, b[2] = 0, a[7,3] = 43/156, a[3,2] = 4/9\}:" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 " " {MPLTEXT 1 0 1 ";" }}}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 15 "" 0 "" {TEXT -1 1 " " }{TEXT 276 6 "Step 7" }{TEXT -1 1 " " }}{PARA 0 " " 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 24 "The row-sum condit ions: " }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "Sum(a[i,j], j=1..i-1)=c[i]" "6#/-%$SumG6$&%\"aG6$%\"iG%\"jG/F+;\"\"\",&F*F.F.!\"\" &%\"cG6#F*" }{TEXT -1 8 ", for " }{XPPEDIT 18 0 "i=2" "6#/%\"iG\"\"# " }{TEXT -1 8 " . . 7, " }}{PARA 0 "" 0 "" {TEXT -1 21 "can be used to find " }{XPPEDIT 18 0 "a[2,1]" "6#&%\"aG6$\"\"#\"\"\"" }{TEXT -1 3 " , " }{XPPEDIT 18 0 "a[3,1]" "6#&%\"aG6$\"\"$\"\"\"" }{TEXT -1 3 ", \+ " }{XPPEDIT 18 0 "a[4,1]" "6#&%\"aG6$\"\"%\"\"\"" }{TEXT -1 11 ", . . \+ . , " }{XPPEDIT 18 0 "a[7,1];" "6#&%\"aG6$\"\"(\"\"\"" }{TEXT -1 2 ". " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 140 "cdns4 := [seq(add(a[i,j],j=1..i-1)=c[i],i=2..7)]:\ne qns4 := subs(e13,cdns4):\nmatrix(ListTools[Enumerate](eqns4));\n``;\ni ndets(eqns4);\nnops(%);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#K%'matrixG6 #7(7$\"\"\"/&%\"aG6$\"\"#F(#F(F-7$F-/,&&F+6$\"\"$F(F(#\"\"%\"\"*F(#F-F 47$F4/,&&F+6$F6F(F(#\"\"&\"#OF(#F(F47$F6/,&&F+6$F?F(F(#\"$b\"\"$W\"F(# F?\"\"'7$F?/,&&F+6$FKF(F(#\"#h\"$g$F(#F(FK7$FK/,&&F+6$\"\"(F(F(#\"$,$ \"$g#F(F(Q(pprint36\"" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%!G" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#<(&%\"aG6$\"\"&\"\"\"&F%6$\"\"$F(&F%6$\"\"#F (&F%6$\"\"'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 0 109 "e14 := solve(\{op(eqns4)\},\{seq(a[i,1],i=2..7)\}): \ne15 := `union`(e13,e14):\nseq(a[i,1]=subs(e15,a[i,1]),i=2..7);" }} {PARA 11 "" 1 "" {XPPMATH 20 "6(/&%\"aG6$\"\"#\"\"\"#F(F'/&F%6$\"\"$F( #F'\"\"*/&F%6$\"\"%F(#\"\"(\"#O/&F%6$\"\"&F(#!#N\"$W\"/&F%6$\"\"'F(#! \"\"\"$g$/&F%6$F5F(#!#T\"$g#" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}{SECT 1 {PARA 4 "" 0 " " {TEXT -1 3 "e15" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 505 "e15 := \{a[2,1] = 1/2, a[6,2] = -11/36, b[7] = 13/200, a[5,3] = 35/48, a[6,3] = -1/8, b[5] = 4/25, b[4] = 11/40, b [1] = 13/200, a[4,3] = -1/12, c[5] = 5/6, c[6] = 1/6, c[2] = 1/2, c[3] = 2/3, a[4,2] = 2/9, a[5,2] = -55/36, b[3] = 11/40, b[6] = 4/25, a[6, 4] = 1/2, a[6,5] = 1/10, a[5,4] = 15/8, a[3,1] = 2/9, a[4,1] = 7/36, a [6,1] = -1/360, a[7,1] = -41/260, a[5,1] = -35/144, a[7,4] = -118/39, \+ a[7,5] = 32/195, a[7,2] = 22/13, a[7,6] = 80/39, c[4] = 1/3, c[7] = 1, b[2] = 0, a[7,3] = 43/156, a[3,2] = 4/9\}:" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 92 "subs(e1 5,matrix([seq([c[i],seq(a[i,j],j=1..i-1),``$(8-i)],i=2..7),\n[``,seq(b [i],i=1..7)]]));" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#K%'matrixG6#7)7*# \"\"\"\"\"#F(%!GF+F+F+F+F+7*#F*\"\"$#F*\"\"*#\"\"%F0F+F+F+F+F+7*#F)F.# \"\"(\"#OF/#!\"\"\"#7F+F+F+F+7*#\"\"&\"\"'#!#N\"$W\"#!#bF7#\"#N\"#[#\" #:\"\")F+F+F+7*#F)F>#F9\"$g$#!#6F7#F9FIF(#F)\"#5F+F+7*F)#!#T\"$g##\"#A \"#8#\"#V\"$c\"#!$=\"\"#R#\"#K\"$&>#\"#!)FinF+7*F+#FY\"$+#\"\"!#\"#6\" #SFco#F2\"#DFfoF`oQ(pprint46\"" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }} {PARA 0 "" 0 "" {TEXT -1 7 "Check: " }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 149 "RK6_7eqs := [op(RowSumConditions(7,'expanded')),op(O rderConditions(6,7,'expanded'))]:\nsimplify(subs(e15,RK6_7eqs)):\nmap( u->lhs(u)-rhs(u),%);\nnops(%);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#7M\" \"!F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F $F$F$F$F$F$F$F$F$" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#\"#V" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }} }}{PARA 0 "" 0 "" {TEXT -1 44 "#====================================== =====" }}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 46 "Step by step constructio n of a general scheme " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 117 "We construct expressions for the weights and linkin g coefficients of a general order 6 scheme in terms of the nodes " } {XPPEDIT 18 0 "c[2]" "6#&%\"cG6#\"\"#" }{TEXT -1 3 ", " }{XPPEDIT 18 0 "c[3]" "6#&%\"cG6#\"\"$" }{TEXT -1 3 ", " }{XPPEDIT 18 0 "c[5]" "6# &%\"cG6#\"\"&" }{TEXT -1 7 " and " }{XPPEDIT 18 0 "c[6]" "6#&%\"cG6# \"\"'" }{TEXT -1 1 "." }}{PARA 0 "" 0 "" {TEXT -1 17 "We specify that \+ " }{XPPEDIT 18 0 "b[2]=0" "6#/&%\"bG6#\"\"#\"\"!" }{TEXT -1 7 " and \+ " }{XPPEDIT 18 0 "c[7]=1" "6#/&%\"cG6#\"\"(\"\"\"" }{TEXT -1 1 "." }} {PARA 0 "" 0 "" {TEXT -1 34 "We also incorporate the relation " }} {PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "c[4]=c[3]/(15*c[3]^2- 10*c[3]+2)" "6#/&%\"cG6#\"\"%*&&F%6#\"\"$\"\"\",(*&\"#:F,*$&F%6#F+\"\" #F,F,*&\"#5F,&F%6#F+F,!\"\"F3F,F8" }{TEXT -1 1 "." }}{PARA 15 "" 0 "" {TEXT 286 6 "Step 1" }{TEXT -1 1 " " }}{PARA 0 "" 0 "" {TEXT -1 0 "" } }{PARA 0 "" 0 "" {TEXT -1 33 "We use the quadrature conditions:" }} {PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "Sum(b[i],i = 1 .. 7) \+ = 1;" "6#/-%$SumG6$&%\"bG6#%\"iG/F*;\"\"\"\"\"(F-" }{TEXT -1 15 ", \+ " }{XPPEDIT 18 0 "Sum(b[i]*c[i]^(k-1),i = 2 .. 7) = 1/k;" "6 #/-%$SumG6$*&&%\"bG6#%\"iG\"\"\")&%\"cG6#F+,&%\"kGF,F,!\"\"F,/F+;\"\"# \"\"(*&F,F,F2F3" }{TEXT -1 7 ", " }{XPPEDIT 18 0 "k = 2;" "6#/%\" kG\"\"#" }{TEXT -1 8 " . . 6, " }}{PARA 0 "" 0 "" {TEXT -1 21 "to find the weights " }{XPPEDIT 18 0 "b[1]" "6#&%\"bG6#\"\"\"" }{TEXT -1 2 " , " }{XPPEDIT 18 0 "b[3]" "6#&%\"bG6#\"\"$" }{TEXT -1 3 ", " } {XPPEDIT 18 0 "b[4];" "6#&%\"bG6#\"\"%" }{TEXT -1 3 ", " }{XPPEDIT 18 0 "b[5];" "6#&%\"bG6#\"\"&" }{TEXT -1 3 ", " }{XPPEDIT 18 0 "b[6] " "6#&%\"bG6#\"\"'" }{TEXT -1 3 ", " }{XPPEDIT 18 0 "b[7]" "6#&%\"bG6 #\"\"(" }{TEXT -1 25 " in terms of the nodes " }{XPPEDIT 18 0 "c[3] " "6#&%\"cG6#\"\"$" }{TEXT -1 3 ", " }{XPPEDIT 18 0 "c[5]" "6#&%\"cG6 #\"\"&" }{TEXT -1 7 " and " }{XPPEDIT 18 0 "c[6]" "6#&%\"cG6#\"\"'" }{TEXT -1 1 "." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 239 "e1 := \{b[2]=0,c[4]=c[3]/(15*c[3]^2-10*c[3]+2), c[7]=1\}:\nquad_cdns := [add(b[i],i=1..7)=1,seq(add(b[i]*c[i]^(k-1),i= 2..7)=1/k,k=2..6)]:\nquad_eqns := subs(e1,quad_cdns):\nmatrix(ListTool s[Enumerate](quad_eqns));\n``;\nindets(quad_eqns);\nnops(%);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#K%'matrixG6#7(7$\"\"\"/,.&%\"bG6#F(F(&F,6#\" \"$F(&F,6#\"\"%F(&F,6#\"\"&F(&F,6#\"\"'F(&F,6#\"\"(F(F(7$\"\"#/,,*&F.F (&%\"cGF/F(F(*(F1F(FBF(,(*&\"#:F()FBF>F(F(*&\"#5F(FBF(!\"\"F>F(FKF(*&F 4F(&FCF5F(F(*&F7F(&FCF8F(F(F:F(#F(F>7$F0/,,*&F.F(FHF(F(*(F1F(FBF>FE!\" #F(*&F4F()FMF>F(F(*&F7F()FOF>F(F(F:F(#F(F07$F3/,,*&F.F()FBF0F(F(*(F1F( FBF0FE!\"$F(*&F4F()FMF0F(F(*&F7F()FOF0F(F(F:F(#F(F37$F6/,,*&F.F()FBF3F (F(*(F1F(FBF3FE!\"%F(*&F4F()FMF3F(F(*&F7F()FOF3F(F(F:F(#F(F67$F9/,,*&F .F()FBF6F(F(*(F1F(FBF6FE!\"&F(*&F4F()FMF6F(F(*&F7F()FOF6F(F(F:F(#F(F9Q )pprint136\"" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%!G" }}{PARA 11 "" 1 " " {XPPMATH 20 "6#<+&%\"cG6#\"\"$&%\"bG6#\"\"%&F)6#\"\"(&F)6#\"\"\"&F%6 #\"\"'&F%6#\"\"&&F)F6&F)F3&F)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 0 114 "e2 := factor(solve(\{op(quad_eqns)\},\{b[1],b[3],b[4 ],b[5],b[6],b[7]\})):\ne3 := `union`(e1,e2):\ninfolevel[solve] := 0:" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 2 "e3" }}{PARA 0 "" 0 " " {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1345 "e3 := \{ b[6] = 1/60*(1-5*c[3]+5*c[3]^2)*(15*c[5]*c[3]-9*c[3]+4-6*c[5])/c[6]/(- 1+c[6])/(2*c[6]-10*c[6]*c[3]+15*c[6]*c[3]^2-c[3])/(c[3]-c[6])/(c[6]-c[ 5]), b[3] = -1/60*(75*c[5]*c[6]*c[3]^2-60*c[5]*c[6]*c[3]+10*c[5]*c[6]- 45*c[5]*c[3]^2+35*c[5]*c[3]-6*c[5]-45*c[6]*c[3]^2+35*c[6]*c[3]-6*c[6]+ 30*c[3]^2-23*c[3]+4)/(c[3]-c[6])/(-c[5]+c[3])/c[3]^2/(c[3]-1)/(15*c[3] ^2-10*c[3]+1), b[7] = 1/60*(300*c[5]*c[3]^3*c[6]-225*c[3]^3*c[6]+180*c [3]^3-225*c[3]^3*c[5]-455*c[5]*c[6]*c[3]^2+350*c[6]*c[3]^2-285*c[3]^2+ 350*c[5]*c[3]^2+210*c[5]*c[6]*c[3]-165*c[6]*c[3]+136*c[3]-165*c[5]*c[3 ]-30*c[5]*c[6]+24*c[6]-20+24*c[5])/(c[3]-1)/(-1+c[6])/(c[5]-1)/(3*c[3] -1)/(5*c[3]-2), c[7] = 1, b[4] = 1/60*(15*c[3]^2-10*c[3]+2)^5*(3*c[3]- 5*c[5]*c[3]-5*c[6]*c[3]+10*c[5]*c[6]*c[3]-5*c[5]*c[6]-2+3*c[6]+3*c[5]) /(15*c[3]^2-10*c[3]+1)/c[3]^2/(3*c[3]-1)/(5*c[3]-2)/(2*c[6]-10*c[6]*c[ 3]+15*c[6]*c[3]^2-c[3])/(15*c[5]*c[3]^2-10*c[5]*c[3]+2*c[5]-c[3]), c[4 ] = c[3]/(15*c[3]^2-10*c[3]+2), b[2] = 0, b[1] = -1/60*1/c[3]^2*(150*c [5]*c[3]^3*c[6]-75*c[3]^3*c[6]+45*c[3]^3-75*c[3]^3*c[5]+105*c[5]*c[3]^ 2-205*c[5]*c[6]*c[3]^2+105*c[6]*c[3]^2-65*c[3]^2+29*c[3]-45*c[5]*c[3]+ 80*c[5]*c[6]*c[3]-45*c[6]*c[3]+6*c[5]-10*c[5]*c[6]-4+6*c[6])/c[5]/c[6] , b[5] = -1/60*(1-5*c[3]+5*c[3]^2)*(15*c[6]*c[3]-9*c[3]+4-6*c[6])/(c[6 ]-c[5])/c[5]/(c[5]-1)/(15*c[5]*c[3]^2-10*c[5]*c[3]+2*c[5]-c[3])/(-c[5] +c[3])\}:" }}}{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 41 "The weights of the scheme are as follows." }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 60 "for ii to 7 do print(b[ii]=subs(e3, b[ii]));print(``) end do:" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/&%\"bG6# \"\"\",$*&#F'\"#gF'**&%\"cG6#\"\"$!\"#,B**\"$]\"F'&F.6#\"\"&F')F-F0F'& F.6#\"\"'F'F'*(\"#vF'F8F'F9F'!\"\"*&\"#XF'F8F'F'*(F=F'F8F'F5F'F>*(\"$0 \"F'F5F')F-\"\"#F'F'**\"$0#F'F5F'F9F'FDF'F>*(FCF'F9F'FDF'F'*&\"#lF'FDF 'F>*&\"#HF'F-F'F'*(F@F'F5F'F-F'F>**\"#!)F'F5F'F9F'F-F'F'*(F@F'F9F'F-F' F>*&F;F'F5F'F'*(\"#5F'F5F'F9F'F>\"\"%F>*&F;F'F9F'F'F'F5F>F9F>F'F>" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#%!G" }}{PARA 11 "" 1 "" {XPPMATH 20 "6 #/&%\"bG6#\"\"#\"\"!" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%!G" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/&%\"bG6#\"\"$,$*&#\"\"\"\"#gF+*.,:**\"#vF+& %\"cG6#\"\"&F+&F26#\"\"'F+)&F2F&\"\"#F+F+**F,F+F1F+F5F+F9F+!\"\"*(\"#5 F+F1F+F5F+F+*(\"#XF+F1F+F8F+F<*(\"#NF+F1F+F9F+F+*&F7F+F1F+F<*(F@F+F5F+ F8F+F<*(FBF+F5F+F9F+F+*&F7F+F5F+F<*&\"#IF+F8F+F+*&\"#BF+F9F+F<\"\"%F+F +,&F9F+F5FF+F9F+F F+FAF+F2F+F+*(F:F+F>F+FAF+F9F6F9* &F5F+FAF+F+*&F5F+F>F+F+F+,(*&F0F+F1F+F+*&F8F+F2F+F9F+F+F9F2!\"#,&*&F5F +F2F+F+F+F9F9,&*&F:F+F2F+F+F6F9F9,**&F6F+FAF+F+*(F8F+FAF+F2F+F9*(F0F+F AF+F1F+F+F2F9F9,**(F0F+F>F+F1F+F+*(F8F+F>F+F2F+F9*&F6F+F>F+F+F2F9F9F+F +" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%!G" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/&%\"bG6#\"\"&,$*&#\"\"\"\"#gF+*0,(F+F+*&F'F+&%\"cG6#\"\"$F+!\" \"*&F'F+)F0\"\"#F+F+F+,**(\"#:F+&F16#\"\"'F+F0F+F+*&\"\"*F+F0F+F4\"\"% F+*&F=F+F;F+F4F+,&F;F+&F1F&F4F4FCF4,&FCF+F+F4F4,**(F:F+FCF+F6F+F+*(\"# 5F+FCF+F0F+F4*&F7F+FCF+F+F0F4F4,&FCF4F0F+F4F+F4" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%!G" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/&%\"bG6#\"\"',$ *&#\"\"\"\"#gF+*0,(F+F+*&\"\"&F+&%\"cG6#\"\"$F+!\"\"*&F0F+)F1\"\"#F+F+ F+,**(\"#:F+&F26#F0F+F1F+F+*&\"\"*F+F1F+F5\"\"%F+*&F'F+F**\"$b%F+F1F+F9F+)F6\"\"#F+F>*(\"$] $F+F9F+FDF+F+*&\"$&GF+FDF+F>*(FGF+F1F+FDF+F+**\"$5#F+F1F+F9F+F6F+F+*( \"$l\"F+F9F+F6F+F>*&\"$O\"F+F6F+F+*(FNF+F1F+F6F+F>*(\"#IF+F1F+F9F+F>*& \"#CF+F9F+F+\"#?F>*&FUF+F1F+F+F+,&F6F+F+F>F>,&F+F>F9F+F>,&F1F+F+F>F>,& *&F8F+F6F+F+F+F>F>,&*&F4F+F6F+F+FEF>F>F+F+" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%!G" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 15 "" 0 " " {TEXT -1 1 " " }{TEXT 287 6 "Step 2" }{TEXT -1 1 " " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 56 "We can obtain expressi ons for the linking coefficients " }{XPPEDIT 18 0 "a[6,5]" "6#&%\"aG6 $\"\"'\"\"&" }{TEXT -1 3 ", " }{XPPEDIT 18 0 "a[5,4]" "6#&%\"aG6$\"\" &\"\"%" }{TEXT -1 7 " and " }{XPPEDIT 18 0 "a[6,4]" "6#&%\"aG6$\"\"' \"\"%" }{TEXT -1 57 " using the following three alternative order con ditions:" }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "b[6]*(1-c [6])*(a[6,5]*c[5]*(c[5]-c[3])+a[6,4]*c[4]*(c[4]-c[3]))+b[5]*(1-c[5])*a [5,4]*c[4]*(c[4]-c[3])=1/60-c[3]/24" "6#/,&*(&%\"bG6#\"\"'\"\"\",&F*F* &%\"cG6#F)!\"\"F*,&*(&%\"aG6$F)\"\"&F*&F-6#F5F*,&&F-6#F5F*&F-6#\"\"$F/ F*F**(&F36$F)\"\"%F*&F-6#FAF*,&&F-6#FAF*&F-6#F=F/F*F*F*F**,&F'6#F5F*,& F*F*&F-6#F5F/F*&F36$F5FAF*&F-6#FAF*,&&F-6#FAF*&F-6#F=F/F*F*,&*&F*F*\"# gF/F**&&F-6#F=F*\"#CF/F/" }{TEXT -1 2 ", " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "b[6]*(1-c[6])* a[6,5]*c[5]*(c[5]-c[3])*(c[5]-c[4])=1/120-(c[3]+c[4])/60+c[3]*c[4]/24 " "6#/*.&%\"bG6#\"\"'\"\"\",&F)F)&%\"cG6#F(!\"\"F)&%\"aG6$F(\"\"&F)&F, 6#F2F),&&F,6#F2F)&F,6#\"\"$F.F),&&F,6#F2F)&F,6#\"\"%F.F),(*&F)F)\"$?\" F.F)*&,&&F,6#F:F)&F,6#F@F)F)\"#gF.F.*(&F,6#F:F)&F,6#F@F)\"#CF.F)" } {TEXT -1 2 ", " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "b[6]*(1-c[6])*a[6,5]*a[5,4]*c[4]*(c[4]- c[3])=1/360-c[3]/120" "6#/*.&%\"bG6#\"\"'\"\"\",&F)F)&%\"cG6#F(!\"\"F) &%\"aG6$F(\"\"&F)&F06$F2\"\"%F)&F,6#F5F),&&F,6#F5F)&F,6#\"\"$F.F),&*&F )F)\"$g$F.F)*&&F,6#F=F)\"$?\"F.F." }{TEXT -1 3 ". " }}{PARA 0 "" 0 " " {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 380 "cdns1 := \+ [b[6]*(1-c[6])*(a[6,5]*c[5]*(c[5]-c[3])+a[6,4]*c[4]*(c[4]-c[3]))+\n \+ b[5]*(1-c[5])*a[5,4]*c[4]*(c[4]-c[3])=1/60-c[3]/24,\n b[6]*(1-c[6 ])*a[6,5]*c[5]*(c[5]-c[3])*(c[5]-c[4])=1/120-(c[3]+c[4])/60+c[3]*c[4]/ 24,\n b[6]*(1-c[6])*a[6,5]*a[5,4]*c[4]*(c[4]-c[3])=1/360-c[3]/120]: \neqns1 := simplify(subs(e3,cdns1)):\nnops(eqns1);\nindets(eqns1) minu s \{c[3],c[5],c[6]\};\nnops(%);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#\" \"$" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#<%&%\"aG6$\"\"'\"\"%&F%6$\"\"&F (&F%6$F'F+" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#\"\"$" }}}{PARA 0 "" 0 " " {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 78 "e4 := fact or(solve(\{op(eqns1)\},\{a[5,4],a[6,4],a[6,5]\})):\ne5 := `union`(e3,e 4):" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 2 "e5" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 2079 " e5 := \{b[6] = 1/60*(1-5*c[3]+5*c[3]^2)*(15*c[5]*c[3]-9*c[3]+4-6*c[5]) /c[6]/(-1+c[6])/(2*c[6]-10*c[6]*c[3]+15*c[6]*c[3]^2-c[3])/(c[3]-c[6])/ (c[6]-c[5]), a[6,4] = 1/6*c[6]*(15*c[3]^2-10*c[3]+2)^2*(2*c[6]-10*c[6] *c[3]+15*c[6]*c[3]^2-c[3])*(c[3]-c[6])*(90*c[3]^3*c[6]-120*c[6]*c[3]^2 +48*c[6]*c[3]-6*c[6]-225*c[3]^3*c[5]+225*c[5]^2*c[3]^3-240*c[5]^2*c[3] ^2+9*c[3]^2+255*c[5]*c[3]^2-100*c[5]*c[3]-4*c[3]+90*c[5]^2*c[3]+14*c[5 ]-12*c[5]^2)/c[3]^2/(1-5*c[3]+5*c[3]^2)/(15*c[3]^2-10*c[3]+1)/(15*c[5] *c[3]^2-10*c[5]*c[3]+2*c[5]-c[3])/(15*c[5]*c[3]-9*c[3]+4-6*c[5]), b[3] = -1/60*(75*c[5]*c[6]*c[3]^2-60*c[5]*c[6]*c[3]+10*c[5]*c[6]-45*c[5]*c [3]^2+35*c[5]*c[3]-6*c[5]-45*c[6]*c[3]^2+35*c[6]*c[3]-6*c[6]+30*c[3]^2 -23*c[3]+4)/(c[3]-c[6])/(-c[5]+c[3])/c[3]^2/(c[3]-1)/(15*c[3]^2-10*c[3 ]+1), b[7] = 1/60*(300*c[5]*c[3]^3*c[6]-225*c[3]^3*c[6]+180*c[3]^3-225 *c[3]^3*c[5]-455*c[5]*c[6]*c[3]^2+350*c[6]*c[3]^2-285*c[3]^2+350*c[5]* c[3]^2+210*c[5]*c[6]*c[3]-165*c[6]*c[3]+136*c[3]-165*c[5]*c[3]-30*c[5] *c[6]+24*c[6]-20+24*c[5])/(c[3]-1)/(-1+c[6])/(c[5]-1)/(3*c[3]-1)/(5*c[ 3]-2), c[7] = 1, b[4] = 1/60*(15*c[3]^2-10*c[3]+2)^5*(3*c[3]-5*c[5]*c[ 3]-5*c[6]*c[3]+10*c[5]*c[6]*c[3]-5*c[5]*c[6]-2+3*c[6]+3*c[5])/(15*c[3] ^2-10*c[3]+1)/c[3]^2/(3*c[3]-1)/(5*c[3]-2)/(2*c[6]-10*c[6]*c[3]+15*c[6 ]*c[3]^2-c[3])/(15*c[5]*c[3]^2-10*c[5]*c[3]+2*c[5]-c[3]), c[4] = c[3]/ (15*c[3]^2-10*c[3]+2), b[2] = 0, b[1] = -1/60*1/c[3]^2*(150*c[5]*c[3]^ 3*c[6]-75*c[3]^3*c[6]+45*c[3]^3-75*c[3]^3*c[5]+105*c[5]*c[3]^2-205*c[5 ]*c[6]*c[3]^2+105*c[6]*c[3]^2-65*c[3]^2+29*c[3]-45*c[5]*c[3]+80*c[5]*c [6]*c[3]-45*c[6]*c[3]+6*c[5]-10*c[5]*c[6]-4+6*c[6])/c[5]/c[6], a[5,4] \+ = 1/6*(15*c[3]^2-10*c[3]+2)^2*c[5]*(-c[5]+c[3])*(15*c[5]*c[3]^2-10*c[5 ]*c[3]+2*c[5]-c[3])/c[3]^2/(1-5*c[3]+5*c[3]^2)/(15*c[3]^2-10*c[3]+1), \+ b[5] = -1/60*(1-5*c[3]+5*c[3]^2)*(15*c[6]*c[3]-9*c[3]+4-6*c[6])/(c[6]- c[5])/c[5]/(c[5]-1)/(15*c[5]*c[3]^2-10*c[5]*c[3]+2*c[5]-c[3])/(-c[5]+c [3]), a[6,5] = -(3*c[3]-1)*(2*c[6]-10*c[6]*c[3]+15*c[6]*c[3]^2-c[3])*( c[3]-c[6])*(c[6]-c[5])*c[6]/(15*c[5]*c[3]-9*c[3]+4-6*c[5])/(-c[5]+c[3] )/(15*c[5]*c[3]^2-10*c[5]*c[3]+2*c[5]-c[3])/c[5]\}:" }}}{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 18 "The coeff icients " }{XPPEDIT 18 0 "a[6,5]" "6#&%\"aG6$\"\"'\"\"&" }{TEXT -1 3 ", " }{XPPEDIT 18 0 "a[5,4]" "6#&%\"aG6$\"\"&\"\"%" }{TEXT -1 7 " an d " }{XPPEDIT 18 0 "a[6,4]" "6#&%\"aG6$\"\"'\"\"%" }{TEXT -1 17 " ar e as follows." }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 77 "a[5,4]=subs (e5,a[5,4]);``;\na[6,4]=subs(e5,a[6,4]);``;\na[6,5]=subs(e5,a[6,5]);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/&%\"aG6$\"\"&\"\"%,$*&#\"\"\"\"\"'F ,*0,(*&\"#:F,)&%\"cG6#\"\"$\"\"#F,F,*&\"#5F,F3F,!\"\"F7F,F7&F46#F'F,,& F;F:F3F,F,,**(F1F,F;F,F2F,F,*(F9F,F;F,F3F,F:*&F7F,F;F,F,F3F:F,F3!\"#,( F,F,*&F'F,F3F,F:*&F'F,F2F,F,F:,(*&F1F,F2F,F,*&F9F,F3F,F:F,F,F:F,F," }} {PARA 11 "" 1 "" {XPPMATH 20 "6#%!G" }}{PARA 12 "" 1 "" {XPPMATH 20 "6 #/&%\"aG6$\"\"'\"\"%,$*&#\"\"\"F'F,*6&%\"cG6#F'F,,(*&\"#:F,)&F/6#\"\"$ \"\"#F,F,*&\"#5F,F5F,!\"\"F8F,F8,**&F8F,F.F,F,*(F:F,F.F,F5F,F;*(F3F,F. F,F4F,F,F5F;F,,&F5F,F.F;F,,>*(\"#!*F,)F5F7F,F.F,F,*(\"$?\"F,F.F,F4F,F; *(\"#[F,F.F,F5F,F,*&F'F,F.F,F;*(\"$D#F,FDF,&F/6#\"\"&F,F;*(FKF,)FLF8F, FDF,F,*(\"$S#F,FPF,F4F,F;*&\"\"*F,F4F,F,*(\"$b#F,FLF,F4F,F,*(\"$+\"F,F LF,F5F,F;*&F(F,F5F,F;*(FCF,FPF,F5F,F,*&\"#9F,FLF,F,*&\"#7F,FPF,F;F,F5! \"#,(F,F,*&FNF,F5F,F;*&FNF,F4F,F,F;,(*&F3F,F4F,F,*&F:F,F5F,F;F,F,F;,** (F3F,FLF,F4F,F,*(F:F,FLF,F5F,F;*&F8F,FLF,F,F5F;F;,**(F3F,FLF,F5F,F,*&F TF,F5F,F;F(F,*&F'F,FLF,F;F;F,F," }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%!G " }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/&%\"aG6$\"\"'\"\"&,$*4,&*&\"\"$\" \"\"&%\"cG6#F-F.F.F.!\"\"F.,**&\"\"#F.&F06#F'F.F.*(\"#5F.F6F.F/F.F2*( \"#:F.F6F.)F/F5F.F.F/F2F.,&F/F.F6F2F.,&F6F.&F06#F(F2F.F6F.,**(F;F.F?F. F/F.F.*&\"\"*F.F/F.F2\"\"%F.*&F'F.F?F.F2F2,&F?F2F/F.F2,**(F;F.F?F.F " 0 "" {MPLTEXT 1 0 114 "a[3,2]*c[2]=1/2*c[3]^2:\nsubs(e5,%);\ne6 := \{a[3,2]=solve(%,a[3, 2])\}:\ne7 := `union`(e5,e6):\na[3,2]=subs(e7,a[3,2]); " }}{PARA 11 " " 1 "" {XPPMATH 20 "6#/*&&%\"aG6$\"\"$\"\"#\"\"\"&%\"cG6#F)F*,$*&#F*F) F**$)&F,6#F(F)F*F*F*" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/&%\"aG6$\"\"$ \"\"#,$*&#\"\"\"F(F,*&&%\"cG6#F'F(&F/6#F(!\"\"F,F," }}}{PARA 0 "" 0 " " {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}} {SECT 1 {PARA 4 "" 0 "" {TEXT -1 2 "e7" }}{PARA 0 "" 0 "" {TEXT -1 0 " " }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 2105 "e7 := \{b[6] = 1/60*(1 -5*c[3]+5*c[3]^2)*(15*c[5]*c[3]-9*c[3]+4-6*c[5])/c[6]/(-1+c[6])/(2*c[6 ]-10*c[6]*c[3]+15*c[6]*c[3]^2-c[3])/(c[3]-c[6])/(c[6]-c[5]), a[6,4] = \+ 1/6*c[6]*(15*c[3]^2-10*c[3]+2)^2*(2*c[6]-10*c[6]*c[3]+15*c[6]*c[3]^2-c [3])*(c[3]-c[6])*(90*c[3]^3*c[6]-120*c[6]*c[3]^2+48*c[6]*c[3]-6*c[6]-2 25*c[3]^3*c[5]+225*c[5]^2*c[3]^3-240*c[5]^2*c[3]^2+9*c[3]^2+255*c[5]*c [3]^2-100*c[5]*c[3]-4*c[3]+90*c[5]^2*c[3]+14*c[5]-12*c[5]^2)/c[3]^2/(1 -5*c[3]+5*c[3]^2)/(15*c[3]^2-10*c[3]+1)/(15*c[5]*c[3]^2-10*c[5]*c[3]+2 *c[5]-c[3])/(15*c[5]*c[3]-9*c[3]+4-6*c[5]), b[3] = -1/60*(75*c[5]*c[6] *c[3]^2-60*c[5]*c[6]*c[3]+10*c[5]*c[6]-45*c[5]*c[3]^2+35*c[5]*c[3]-6*c [5]-45*c[6]*c[3]^2+35*c[6]*c[3]-6*c[6]+30*c[3]^2-23*c[3]+4)/(c[3]-c[6] )/(-c[5]+c[3])/c[3]^2/(c[3]-1)/(15*c[3]^2-10*c[3]+1), b[7] = 1/60*(300 *c[5]*c[3]^3*c[6]-225*c[3]^3*c[6]+180*c[3]^3-225*c[3]^3*c[5]-455*c[5]* c[6]*c[3]^2+350*c[6]*c[3]^2-285*c[3]^2+350*c[5]*c[3]^2+210*c[5]*c[6]*c [3]-165*c[6]*c[3]+136*c[3]-165*c[5]*c[3]-30*c[5]*c[6]+24*c[6]-20+24*c[ 5])/(c[3]-1)/(-1+c[6])/(c[5]-1)/(3*c[3]-1)/(5*c[3]-2), c[7] = 1, b[4] \+ = 1/60*(15*c[3]^2-10*c[3]+2)^5*(3*c[3]-5*c[5]*c[3]-5*c[6]*c[3]+10*c[5] *c[6]*c[3]-5*c[5]*c[6]-2+3*c[6]+3*c[5])/(15*c[3]^2-10*c[3]+1)/c[3]^2/( 3*c[3]-1)/(5*c[3]-2)/(2*c[6]-10*c[6]*c[3]+15*c[6]*c[3]^2-c[3])/(15*c[5 ]*c[3]^2-10*c[5]*c[3]+2*c[5]-c[3]), c[4] = c[3]/(15*c[3]^2-10*c[3]+2), b[2] = 0, b[1] = -1/60*1/c[3]^2*(150*c[5]*c[3]^3*c[6]-75*c[3]^3*c[6]+ 45*c[3]^3-75*c[3]^3*c[5]+105*c[5]*c[3]^2-205*c[5]*c[6]*c[3]^2+105*c[6] *c[3]^2-65*c[3]^2+29*c[3]-45*c[5]*c[3]+80*c[5]*c[6]*c[3]-45*c[6]*c[3]+ 6*c[5]-10*c[5]*c[6]-4+6*c[6])/c[5]/c[6], a[3,2] = 1/2*c[3]^2/c[2], a[5 ,4] = 1/6*(15*c[3]^2-10*c[3]+2)^2*c[5]*(-c[5]+c[3])*(15*c[5]*c[3]^2-10 *c[5]*c[3]+2*c[5]-c[3])/c[3]^2/(1-5*c[3]+5*c[3]^2)/(15*c[3]^2-10*c[3]+ 1), b[5] = -1/60*(1-5*c[3]+5*c[3]^2)*(15*c[6]*c[3]-9*c[3]+4-6*c[6])/(c [6]-c[5])/c[5]/(c[5]-1)/(15*c[5]*c[3]^2-10*c[5]*c[3]+2*c[5]-c[3])/(-c[ 5]+c[3]), a[6,5] = -(3*c[3]-1)*(2*c[6]-10*c[6]*c[3]+15*c[6]*c[3]^2-c[3 ])*(c[3]-c[6])*(c[6]-c[5])*c[6]/(15*c[5]*c[3]-9*c[3]+4-6*c[5])/(-c[5]+ c[3])/(15*c[5]*c[3]^2-10*c[5]*c[3]+2*c[5]-c[3])/c[5]\}:" }}}{PARA 0 " " 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}} }{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 15 "" 0 "" {TEXT -1 1 " " } {TEXT 292 6 "Step 4" }{TEXT -1 1 " " }}{PARA 0 "" 0 "" {TEXT -1 0 "" } }{PARA 0 "" 0 "" {TEXT -1 41 "We can now determine the 6 coefficients \+ " }{XPPEDIT 18 0 "a[4,2]" "6#&%\"aG6$\"\"%\"\"#" }{TEXT -1 3 ", " } {XPPEDIT 18 0 "a[5, 2];" "6#&%\"aG6$\"\"&\"\"#" }{TEXT -1 3 ", " } {XPPEDIT 18 0 "a[6, 2];" "6#&%\"aG6$\"\"'\"\"#" }{TEXT -1 3 ", " } {XPPEDIT 18 0 "a[4,3];" "6#&%\"aG6$\"\"%\"\"$" }{TEXT -1 3 ", " } {XPPEDIT 18 0 "a[5,3];" "6#&%\"aG6$\"\"&\"\"$" }{TEXT -1 3 ", " } {XPPEDIT 18 0 "a[6,3]" "6#&%\"aG6$\"\"'\"\"$" }{TEXT -1 56 " by using the three alternative simplifying conditions:" }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "Sum(b[i]*(1-c[i])*a[i,2],i=3..6)=0" "6# /-%$SumG6$*(&%\"bG6#%\"iG\"\"\",&F,F,&%\"cG6#F+!\"\"F,&%\"aG6$F+\"\"#F ,/F+;\"\"$\"\"'\"\"!" }{TEXT -1 2 ", " }}{PARA 0 "" 0 "" {TEXT -1 8 "t hat is," }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "b[3]*(1-c[ 3])*a[3,2]+b[4]*(1-c[4])*a[4,2]+b[5]*(1-c[5])*a[5,2]+b[6]*(1-c[6])*a[6 ,2] = 0" "6#/,**(&%\"bG6#\"\"$\"\"\",&F*F*&%\"cG6#F)!\"\"F*&%\"aG6$F) \"\"#F*F**(&F'6#\"\"%F*,&F*F*&F-6#F7F/F*&F16$F7F3F*F**(&F'6#\"\"&F*,&F *F*&F-6#F@F/F*&F16$F@F3F*F**(&F'6#\"\"'F*,&F*F*&F-6#FIF/F*&F16$FIF3F*F *\"\"!" }{TEXT -1 2 ", " }}{PARA 0 "" 0 "" {TEXT -1 3 "and" }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "Sum(b[i]*(1-c[i])*Sum(a[i,j ]*a[j,2],j = 3 .. i-1),i = 3 .. 6) = 0" "6#/-%$SumG6$*(&%\"bG6#%\"iG\" \"\",&F,F,&%\"cG6#F+!\"\"F,-F%6$*&&%\"aG6$F+%\"jGF,&F66$F8\"\"#F,/F8; \"\"$,&F+F,F,F1F,/F+;F>\"\"'\"\"!" }{TEXT -1 2 ", " }}{PARA 0 "" 0 "" {TEXT -1 8 "that is," }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "b[4]*(1-c[4])*a[4,3]*a[3,2]+b[5]*(1-c[5])*(a[5,3]*a[3,2]+a[5,4]*a[4 ,2])+b[6]*(1-c[6])*(a[6,3]*a[3,2]+a[6,4]*a[4,2]+a[6,5]*a[5,2]) = 0" "6 #/,(**&%\"bG6#\"\"%\"\"\",&F*F*&%\"cG6#F)!\"\"F*&%\"aG6$F)\"\"$F*&F16$ F3\"\"#F*F**(&F'6#\"\"&F*,&F*F*&F-6#F:F/F*,&*&&F16$F:F3F*&F16$F3F6F*F* *&&F16$F:F)F*&F16$F)F6F*F*F*F**(&F'6#\"\"'F*,&F*F*&F-6#FLF/F*,(*&&F16$ FLF3F*&F16$F3F6F*F**&&F16$FLF)F*&F16$F)F6F*F**&&F16$FLF:F*&F16$F:F6F*F *F*F*\"\"!" }{TEXT -1 2 ", " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "Sum(b[i]*(c[i]-1)*(c[i]-c[6 ])*a[i,2],i = 3 .. 5) = 0;" "6#/-%$SumG6$**&%\"bG6#%\"iG\"\"\",&&%\"cG 6#F+F,F,!\"\"F,,&&F/6#F+F,&F/6#\"\"'F1F,&%\"aG6$F+\"\"#F,/F+;\"\"$\"\" &\"\"!" }{TEXT -1 2 ", " }}{PARA 0 "" 0 "" {TEXT -1 9 "that is, " }} {PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "b[3]*(c[3]-1)*(c[3]-c [6])*a[3,2]+b[4]*(c[4]-1)*(c[4]-c[6])*a[4,2]+b[5]*(c[5]-1)*(c[5]-c[6]) *a[5,2]=0" "6#/,(**&%\"bG6#\"\"$\"\"\",&&%\"cG6#F)F*F*!\"\"F*,&&F-6#F) F*&F-6#\"\"'F/F*&%\"aG6$F)\"\"#F*F***&F'6#\"\"%F*,&&F-6#F=F*F*F/F*,&&F -6#F=F*&F-6#F5F/F*&F76$F=F9F*F***&F'6#\"\"&F*,&&F-6#FKF*F*F/F*,&&F-6#F KF*&F-6#F5F/F*&F76$FKF9F*F*\"\"!" }{TEXT -1 2 ", " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 48 "together with the three s tage-order conditions: " }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "Sum(a[4,j]*c[j],j = 2 .. 3) = 1/2;" "6#/-%$SumG6$*&&%\"aG6$\"\"% %\"jG\"\"\"&%\"cG6#F,F-/F,;\"\"#\"\"$*&F-F-F3!\"\"" }{TEXT -1 1 " " } {XPPEDIT 18 0 "c[4]^2;" "6#*$&%\"cG6#\"\"%\"\"#" }{TEXT -1 8 " , \+ " }{XPPEDIT 18 0 "Sum(a[5,j]*c[j],j=2..4)=1/2" "6#/-%$SumG6$*&&%\"aG6$ \"\"&%\"jG\"\"\"&%\"cG6#F,F-/F,;\"\"#\"\"%*&F-F-F3!\"\"" }{TEXT -1 1 " " }{XPPEDIT 18 0 "c[5]^2" "6#*$&%\"cG6#\"\"&\"\"#" }{TEXT -1 2 " " } }{PARA 0 "" 0 "" {TEXT -1 4 "and " }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "Sum(a[6,j]*c[j],j=2..5)=1/2" "6#/-%$SumG6$*&&%\"aG6$\" \"'%\"jG\"\"\"&%\"cG6#F,F-/F,;\"\"#\"\"&*&F-F-F3!\"\"" }{TEXT -1 1 " \+ " }{XPPEDIT 18 0 "c[6]^2" "6#*$&%\"cG6#\"\"'\"\"#" }{TEXT -1 1 "." }} {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 18 "We can de termine " }{XPPEDIT 18 0 "a[4,2];" "6#&%\"aG6$\"\"%\"\"#" }{TEXT -1 44 " from the alternative simplifying condition" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "Sum(b [i]*(c[i]-1)*(c[i]-c[6])*a[i,2],i = 3 .. 7) = 0;" "6#/-%$SumG6$**&%\"b G6#%\"iG\"\"\",&&%\"cG6#F+F,F,!\"\"F,,&&F/6#F+F,&F/6#\"\"'F1F,&%\"aG6$ F+\"\"#F,/F+;\"\"$\"\"(\"\"!" }{TEXT -1 2 ", " }}{PARA 0 "" 0 "" {TEXT -1 9 "that is, " }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "b[3]*(c[3]-1)*(c[3]-c[6])*a[3,2]+b[4]*(c[4]-1)*(c[4]-c[6])*a[4,2 ]+b[5]*(c[5]-1)*(c[5]-c[6])*a[5,2]=0" "6#/,(**&%\"bG6#\"\"$\"\"\",&&% \"cG6#F)F*F*!\"\"F*,&&F-6#F)F*&F-6#\"\"'F/F*&%\"aG6$F)\"\"#F*F***&F'6# \"\"%F*,&&F-6#F=F*F*F/F*,&&F-6#F=F*&F-6#F5F/F*&F76$F=F9F*F***&F'6#\"\" &F*,&&F-6#FKF*F*F/F*,&&F-6#FKF*&F-6#F5F/F*&F76$FKF9F*F*\"\"!" }{TEXT -1 2 ". " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 337 "cdns2 := [add(b[i]*(1-c[i])*a[i,2],i=3..6)=0,add(b[i ]*(1-c[i])*add(a[i,j]*a[j,2],j=3..i-1),i=3..6)=0,\n add(b[i]*(1-c[i]) *(c[6]-c[i])*a[i,2],i=3..5)=0,seq(add(a[i,j]*c[j],j=2..i-1)=1/2*c[i]^2 ,i=4..6)]:\neqns2 := simplify(subs(e7,cdns2)):\ne8 := factor(solve(\{o p(eqns2)\},\{a[4,2],a[5,2],a[6,2],a[4,3],a[5,3],a[6,3]\})):\ne9 := `un ion`(e7,e8):\n" }}}{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 3254 "e9 := \{b[6] = 1/60*(1-5*c[3]+5*c[3]^2)*(15*c[5]*c[3]-9*c[3]+4 -6*c[5])/c[6]/(-1+c[6])/(2*c[6]-10*c[6]*c[3]+15*c[6]*c[3]^2-c[3])/(c[3 ]-c[6])/(c[6]-c[5]), a[6,4] = 1/6*c[6]*(15*c[3]^2-10*c[3]+2)^2*(2*c[6] -10*c[6]*c[3]+15*c[6]*c[3]^2-c[3])*(c[3]-c[6])*(90*c[3]^3*c[6]-120*c[6 ]*c[3]^2+48*c[6]*c[3]-6*c[6]-225*c[3]^3*c[5]+225*c[5]^2*c[3]^3-240*c[5 ]^2*c[3]^2+9*c[3]^2+255*c[5]*c[3]^2-100*c[5]*c[3]-4*c[3]+90*c[5]^2*c[3 ]+14*c[5]-12*c[5]^2)/c[3]^2/(1-5*c[3]+5*c[3]^2)/(15*c[3]^2-10*c[3]+1)/ (15*c[5]*c[3]^2-10*c[5]*c[3]+2*c[5]-c[3])/(15*c[5]*c[3]-9*c[3]+4-6*c[5 ]), a[6,3] = 1/6*(c[3]-c[6])*(-2*c[5]*c[3]+1350*c[3]^5*c[6]^2-2535*c[5 ]*c[3]^3*c[6]+1450*c[5]*c[6]*c[3]^2-10*c[6]*c[3]-450*c[3]^5*c[5]-24*c[ 5]^2*c[6]-328*c[5]*c[6]*c[3]-6*c[5]^2*c[3]+2100*c[6]^2*c[3]^3-4*c[3]^2 +89*c[3]^3-300*c[3]^4+270*c[3]^5-12*c[6]^2-2250*c[5]^2*c[3]^4*c[6]+900 *c[3]^5*c[6]*c[5]+975*c[5]*c[3]^4*c[6]+3075*c[5]^2*c[3]^3*c[6]+300*c[5 ]^2*c[6]*c[3]-1470*c[5]^2*c[3]^2*c[6]+156*c[6]^2*c[3]+1815*c[3]^4*c[6] -35*c[5]*c[3]^2-810*c[6]^2*c[3]^2+135*c[5]^2*c[3]^2+300*c[3]^4*c[5]+45 0*c[5]^2*c[3]^4+45*c[3]^3*c[5]-2700*c[6]^2*c[3]^4-1350*c[3]^5*c[6]+28* c[5]*c[6]-480*c[5]^2*c[3]^3+148*c[6]*c[3]^2-820*c[3]^3*c[6])*c[6]/(1-5 *c[3]+5*c[3]^2)/(15*c[3]^2-10*c[3]+1)/(15*c[5]*c[3]-9*c[3]+4-6*c[5])/( -c[5]+c[3])/c[3]^2, b[3] = -1/60*(75*c[5]*c[6]*c[3]^2-60*c[5]*c[6]*c[3 ]+10*c[5]*c[6]-45*c[5]*c[3]^2+35*c[5]*c[3]-6*c[5]-45*c[6]*c[3]^2+35*c[ 6]*c[3]-6*c[6]+30*c[3]^2-23*c[3]+4)/(c[3]-c[6])/(-c[5]+c[3])/c[3]^2/(c [3]-1)/(15*c[3]^2-10*c[3]+1), b[7] = 1/60*(300*c[5]*c[3]^3*c[6]-225*c[ 3]^3*c[6]+180*c[3]^3-225*c[3]^3*c[5]-455*c[5]*c[6]*c[3]^2+350*c[6]*c[3 ]^2-285*c[3]^2+350*c[5]*c[3]^2+210*c[5]*c[6]*c[3]-165*c[6]*c[3]+136*c[ 3]-165*c[5]*c[3]-30*c[5]*c[6]+24*c[6]-20+24*c[5])/(c[3]-1)/(-1+c[6])/( c[5]-1)/(3*c[3]-1)/(5*c[3]-2), c[7] = 1, b[4] = 1/60*(15*c[3]^2-10*c[3 ]+2)^5*(3*c[3]-5*c[5]*c[3]-5*c[6]*c[3]+10*c[5]*c[6]*c[3]-5*c[5]*c[6]-2 +3*c[6]+3*c[5])/(15*c[3]^2-10*c[3]+1)/c[3]^2/(3*c[3]-1)/(5*c[3]-2)/(2* c[6]-10*c[6]*c[3]+15*c[6]*c[3]^2-c[3])/(15*c[5]*c[3]^2-10*c[5]*c[3]+2* c[5]-c[3]), c[4] = c[3]/(15*c[3]^2-10*c[3]+2), b[2] = 0, a[5,2] = 1/2* c[3]*c[5]/c[2]*(-5*c[5]+1+5*c[5]^2)/(1-5*c[3]+5*c[3]^2), a[4,2] = 1/2* c[3]^2*(4-30*c[3]+45*c[3]^2)/c[2]/(15*c[3]^2-10*c[3]+2)^3, a[6,2] = 1/ 2*c[6]*c[3]/c[2]*(5*c[6]^2-5*c[6]+1)/(1-5*c[3]+5*c[3]^2), a[4,3] = -(1 5*c[3]^2-10*c[3]+1)*c[3]/(15*c[3]^2-10*c[3]+2)^3, b[1] = -1/60*1/c[3]^ 2*(150*c[5]*c[3]^3*c[6]-75*c[3]^3*c[6]+45*c[3]^3-75*c[3]^3*c[5]+105*c[ 5]*c[3]^2-205*c[5]*c[6]*c[3]^2+105*c[6]*c[3]^2-65*c[3]^2+29*c[3]-45*c[ 5]*c[3]+80*c[5]*c[6]*c[3]-45*c[6]*c[3]+6*c[5]-10*c[5]*c[6]-4+6*c[6])/c [5]/c[6], a[3,2] = 1/2*c[3]^2/c[2], a[5,4] = 1/6*(15*c[3]^2-10*c[3]+2) ^2*c[5]*(-c[5]+c[3])*(15*c[5]*c[3]^2-10*c[5]*c[3]+2*c[5]-c[3])/c[3]^2/ (1-5*c[3]+5*c[3]^2)/(15*c[3]^2-10*c[3]+1), b[5] = -1/60*(1-5*c[3]+5*c[ 3]^2)*(15*c[6]*c[3]-9*c[3]+4-6*c[6])/(c[6]-c[5])/c[5]/(c[5]-1)/(15*c[5 ]*c[3]^2-10*c[5]*c[3]+2*c[5]-c[3])/(-c[5]+c[3]), a[5,3] = 1/6*c[5]*(15 0*c[3]^3*c[5]-145*c[5]*c[3]^2+40*c[5]*c[3]-4*c[5]-30*c[3]^3+20*c[3]^2- c[3])*(-c[5]+c[3])/c[3]^2/(1-5*c[3]+5*c[3]^2)/(15*c[3]^2-10*c[3]+1), a [6,5] = -(3*c[3]-1)*(2*c[6]-10*c[6]*c[3]+15*c[6]*c[3]^2-c[3])*(c[3]-c[ 6])*(c[6]-c[5])*c[6]/(15*c[5]*c[3]-9*c[3]+4-6*c[5])/(-c[5]+c[3])/(15*c [5]*c[3]^2-10*c[5]*c[3]+2*c[5]-c[3])/c[5]\}:" }{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 0 158 "a[4,2]=subs(e9,a[4,2]);``;\na[5,2]=subs(e9,a[5,2]);``;\na[6,2 ]=subs(e9,a[6,2]);``;\na[4,3]=subs(e9,a[4,3]);``;\na[5,3]=subs(e9,a[5, 3]);``;\na[6,3]=subs(e9,a[6,3]);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/& %\"aG6$\"\"%\"\"#,$*&#\"\"\"F(F,**&%\"cG6#\"\"$F(,(F'F,*&\"#IF,F.F,!\" \"*&\"#XF,)F.F(F,F,F,&F/6#F(F5,(*&\"#:F,F8F,F,*&\"#5F,F.F,F5F(F,!\"$F, F," }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%!G" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/&%\"aG6$\"\"&\"\"#,$*&#\"\"\"F(F,*,&%\"cG6#\"\"$F,&F/6 #F'F,&F/6#F(!\"\",(*&F'F,F2F,F6F,F,*&F'F,)F2F(F,F,F,,(F,F,*&F'F,F.F,F6 *&F'F,)F.F(F,F,F6F,F," }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%!G" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/&%\"aG6$\"\"'\"\"#,$*&#\"\"\"F(F,*,&%\"cG6# F'F,&F/6#\"\"$F,&F/6#F(!\"\",(*&\"\"&F,)F.F(F,F,*&F9F,F.F,F6F,F,F,,(F, F,*&F9F,F1F,F6*&F9F,)F1F(F,F,F6F,F," }}{PARA 11 "" 1 "" {XPPMATH 20 "6 #%!G" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/&%\"aG6$\"\"%\"\"$,$*(,(*&\"# :\"\"\")&%\"cG6#F(\"\"#F.F.*&\"#5F.F0F.!\"\"F.F.F.F0F.,(*&F-F.F/F.F.*& F5F.F0F.F6F3F.!\"$F6" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%!G" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/&%\"aG6$\"\"&\"\"$,$*&#\"\"\"\"\"'F,*.&%\"c G6#F'F,,0*(\"$]\"F,)&F06#F(F(F,F/F,F,*(\"$X\"F,F/F,)F6\"\"#F,!\"\"*(\" #SF,F/F,F6F,F,*&\"\"%F,F/F,F<*&\"#IF,F5F,F<*&\"#?F,F:F,F,F6FF,FAF,F,*&\"\"%F,FDF,F4*&\"#*)F,FAF,F,*&\"$+$F,)F/FR F,F4*&\"$q#F,F=F,F,*&\"#7F,F>F,F4**\"%]AF,FKF,FWF,F2F,F4**\"$+*F,F=F,F 2F,F8F,F,**\"$v*F,F8F,FWF,F2F,F,**\"%vIF,FKF,FAF,F2F,F,**FVF,FKF,F2F,F /F,F,**\"%q9F,FKF,FDF,F2F,F4*(\"$c\"F,F>F,F/F,F,*(\"%:=F,FWF,F2F,F,*( \"#NF,F8F,FDF,F4*(\"$5)F,F>F,FDF,F4*(\"$N\"F,FKF,FDF,F,*(FVF,FWF,F8F,F ,*(FHF,FKF,FWF,F,*(\"#XF,FAF,F8F,F,*(\"%+FF,F>F,FWF,F4*(FF.F*F**&&F'6#\"\"(F*&F,6 $FDF.F*F**&&F'6#F.F*,&F*F*&%\"cG6#F.!\"\"F*" }{TEXT -1 2 ", " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "b[5]*a[5,4]+b[6]*a[6,4]+b[7]*a[7,4] = b[4]*(1-c[4])" "6#/,(*&&% \"bG6#\"\"&\"\"\"&%\"aG6$F)\"\"%F*F**&&F'6#\"\"'F*&F,6$F2F.F*F**&&F'6# \"\"(F*&F,6$F8F.F*F**&&F'6#F.F*,&F*F*&%\"cG6#F.!\"\"F*" }{TEXT -1 2 ", " }}{PARA 257 "" 0 "" {TEXT -1 0 "" }}{PARA 256 "" 0 "" {TEXT -1 1 " \+ " }{XPPEDIT 18 0 "b[6]*a[6,5]+b[7]*a[7,5] = b[5]*(1-c[5])" "6#/,&*&&% \"bG6#\"\"'\"\"\"&%\"aG6$F)\"\"&F*F**&&F'6#\"\"(F*&F,6$F2F.F*F**&&F'6# F.F*,&F*F*&%\"cG6#F.!\"\"F*" }{TEXT -1 2 ", " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "b[7]*a[7,6] = b[6]*(1-c[6])" "6#/*&&%\"bG6#\"\"(\"\"\"&%\"aG6$F(\"\"'F)*&&F&6#F-F ),&F)F)&%\"cG6#F-!\"\"F)" }{TEXT -1 2 ", " }}{PARA 0 "" 0 "" {TEXT -1 12 "from which " }{XPPEDIT 18 0 "a[7,2]" "6#&%\"aG6$\"\"(\"\"#" } {TEXT -1 3 ", " }{XPPEDIT 18 0 "a[7,4]" "6#&%\"aG6$\"\"(\"\"%" } {TEXT -1 3 ", " }{XPPEDIT 18 0 "a[7,5]" "6#&%\"aG6$\"\"(\"\"&" } {TEXT -1 7 " and " }{XPPEDIT 18 0 "a[7,6]" "6#&%\"aG6$\"\"(\"\"'" } {TEXT -1 20 " can be determined." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }} {PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 179 "[seq(add(b[i]*a[i,j],i=j+1..7)=b[j]*(1-c[j]),j=[2,4,5,6])]:\neqns 3 := simplify(subs(e9,%)):\ne10 := factor(solve(\{op(eqns3)\},\{a[7,2] ,a[7,4],a[7,5],a[7,6]\})):\ne11 := `union`(e9,e10):" }}}{PARA 0 "" 0 " " {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}} {SECT 1 {PARA 4 "" 0 "" {TEXT -1 3 "e11" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 5825 "e11 := \{b[6] = 1/60* (1-5*c[3]+5*c[3]^2)*(15*c[5]*c[3]-9*c[3]+4-6*c[5])/c[6]/(-1+c[6])/(2*c [6]-10*c[6]*c[3]+15*c[6]*c[3]^2-c[3])/(c[3]-c[6])/(c[6]-c[5]), a[7,2] \+ = 1/2*c[3]*(14*c[6]-20*c[5]*c[6]-30*c[6]*c[3]+45*c[5]*c[6]*c[3]+21*c[3 ]-10+14*c[5]-30*c[5]*c[3])/c[2]/(300*c[5]*c[3]^3*c[6]-225*c[3]^3*c[6]+ 180*c[3]^3-225*c[3]^3*c[5]-455*c[5]*c[6]*c[3]^2+350*c[6]*c[3]^2-285*c[ 3]^2+350*c[5]*c[3]^2+210*c[5]*c[6]*c[3]-165*c[6]*c[3]+136*c[3]-165*c[5 ]*c[3]-30*c[5]*c[6]+24*c[6]-20+24*c[5]), a[6,4] = 1/6*c[6]*(15*c[3]^2- 10*c[3]+2)^2*(2*c[6]-10*c[6]*c[3]+15*c[6]*c[3]^2-c[3])*(c[3]-c[6])*(90 *c[3]^3*c[6]-120*c[6]*c[3]^2+48*c[6]*c[3]-6*c[6]-225*c[3]^3*c[5]+225*c [5]^2*c[3]^3-240*c[5]^2*c[3]^2+9*c[3]^2+255*c[5]*c[3]^2-100*c[5]*c[3]- 4*c[3]+90*c[5]^2*c[3]+14*c[5]-12*c[5]^2)/c[3]^2/(1-5*c[3]+5*c[3]^2)/(1 5*c[3]^2-10*c[3]+1)/(15*c[5]*c[3]^2-10*c[5]*c[3]+2*c[5]-c[3])/(15*c[5] *c[3]-9*c[3]+4-6*c[5]), a[7,6] = -(15*c[5]*c[3]-9*c[3]+4-6*c[5])*(1-5* c[3]+5*c[3]^2)*(c[3]-1)*(-1+c[6])*(c[5]-1)*(3*c[3]-1)*(5*c[3]-2)/c[6]/ (2*c[6]-10*c[6]*c[3]+15*c[6]*c[3]^2-c[3])/(c[3]-c[6])/(c[6]-c[5])/(300 *c[5]*c[3]^3*c[6]-225*c[3]^3*c[6]+180*c[3]^3-225*c[3]^3*c[5]-455*c[5]* c[6]*c[3]^2+350*c[6]*c[3]^2-285*c[3]^2+350*c[5]*c[3]^2+210*c[5]*c[6]*c [3]-165*c[6]*c[3]+136*c[3]-165*c[5]*c[3]-30*c[5]*c[6]+24*c[6]-20+24*c[ 5]), a[7,4] = 1/6*(c[3]-1)*(3*c[3]-1)*(5*c[3]-2)*(15*c[3]^2-10*c[3]+2) ^2*(-48-1418*c[5]*c[3]+6750*c[3]^5*c[6]^2+37020*c[5]*c[3]^3*c[6]-14810 *c[5]*c[6]*c[3]^2-1256*c[6]*c[3]-10800*c[3]^5*c[5]-24750*c[6]^2*c[3]^4 *c[5]^2-16875*c[5]^2*c[3]^5*c[6]+168*c[5]^2*c[6]+3044*c[5]*c[6]*c[3]-2 4600*c[6]^2*c[3]^3*c[5]+9720*c[6]^2*c[3]^2*c[5]+852*c[5]^2*c[3]-1980*c [6]^2*c[3]*c[5]+10440*c[6]^2*c[3]^3+558*c[3]-2678*c[3]^2+6576*c[3]^3-8 190*c[3]^4+4050*c[3]^5-72*c[5]^2+108*c[6]-72*c[6]^2+120*c[5]+31950*c[5 ]^2*c[3]^4*c[6]+24300*c[3]^5*c[6]*c[5]-47250*c[5]*c[3]^4*c[6]-16875*c[ 3]^5*c[6]^2*c[5]+13500*c[3]^5*c[6]^2*c[5]^2-7200*c[6]^2*c[5]^2*c[3]^2- 24600*c[5]^2*c[3]^3*c[6]-1980*c[5]^2*c[6]*c[3]+9720*c[5]^2*c[3]^2*c[6] +31950*c[6]^2*c[3]^4*c[5]+18600*c[6]^2*c[3]^3*c[5]^2+1440*c[6]^2*c[5]^ 2*c[3]+852*c[6]^2*c[3]+18810*c[3]^4*c[6]+168*c[6]^2*c[5]+6910*c[5]*c[3 ]^2-4170*c[6]^2*c[3]^2-4170*c[5]^2*c[3]^2+21600*c[3]^4*c[5]-120*c[6]^2 *c[5]^2-13275*c[5]^2*c[3]^4-17175*c[3]^3*c[5]+6750*c[3]^5*c[5]^2-13275 *c[6]^2*c[3]^4-9450*c[3]^5*c[6]-260*c[5]*c[6]+10440*c[5]^2*c[3]^3+6052 *c[6]*c[3]^2-14955*c[3]^3*c[6])/c[3]^2/(15*c[3]^2-10*c[3]+1)/(2*c[6]-1 0*c[6]*c[3]+15*c[6]*c[3]^2-c[3])/(15*c[5]*c[3]^2-10*c[5]*c[3]+2*c[5]-c [3])/(300*c[5]*c[3]^3*c[6]-225*c[3]^3*c[6]+180*c[3]^3-225*c[3]^3*c[5]- 455*c[5]*c[6]*c[3]^2+350*c[6]*c[3]^2-285*c[3]^2+350*c[5]*c[3]^2+210*c[ 5]*c[6]*c[3]-165*c[6]*c[3]+136*c[3]-165*c[5]*c[3]-30*c[5]*c[6]+24*c[6] -20+24*c[5]), a[6,3] = 1/6*(c[3]-c[6])*(-2*c[5]*c[3]+1350*c[3]^5*c[6]^ 2-2535*c[5]*c[3]^3*c[6]+1450*c[5]*c[6]*c[3]^2-10*c[6]*c[3]-450*c[3]^5* c[5]-24*c[5]^2*c[6]-328*c[5]*c[6]*c[3]-6*c[5]^2*c[3]+2100*c[6]^2*c[3]^ 3-4*c[3]^2+89*c[3]^3-300*c[3]^4+270*c[3]^5-12*c[6]^2-2250*c[5]^2*c[3]^ 4*c[6]+900*c[3]^5*c[6]*c[5]+975*c[5]*c[3]^4*c[6]+3075*c[5]^2*c[3]^3*c[ 6]+300*c[5]^2*c[6]*c[3]-1470*c[5]^2*c[3]^2*c[6]+156*c[6]^2*c[3]+1815*c [3]^4*c[6]-35*c[5]*c[3]^2-810*c[6]^2*c[3]^2+135*c[5]^2*c[3]^2+300*c[3] ^4*c[5]+450*c[5]^2*c[3]^4+45*c[3]^3*c[5]-2700*c[6]^2*c[3]^4-1350*c[3]^ 5*c[6]+28*c[5]*c[6]-480*c[5]^2*c[3]^3+148*c[6]*c[3]^2-820*c[3]^3*c[6]) *c[6]/(1-5*c[3]+5*c[3]^2)/(15*c[3]^2-10*c[3]+1)/(15*c[5]*c[3]-9*c[3]+4 -6*c[5])/(-c[5]+c[3])/c[3]^2, b[3] = -1/60*(75*c[5]*c[6]*c[3]^2-60*c[5 ]*c[6]*c[3]+10*c[5]*c[6]-45*c[5]*c[3]^2+35*c[5]*c[3]-6*c[5]-45*c[6]*c[ 3]^2+35*c[6]*c[3]-6*c[6]+30*c[3]^2-23*c[3]+4)/(c[3]-c[6])/(-c[5]+c[3]) /c[3]^2/(c[3]-1)/(15*c[3]^2-10*c[3]+1), b[7] = 1/60*(300*c[5]*c[3]^3*c [6]-225*c[3]^3*c[6]+180*c[3]^3-225*c[3]^3*c[5]-455*c[5]*c[6]*c[3]^2+35 0*c[6]*c[3]^2-285*c[3]^2+350*c[5]*c[3]^2+210*c[5]*c[6]*c[3]-165*c[6]*c [3]+136*c[3]-165*c[5]*c[3]-30*c[5]*c[6]+24*c[6]-20+24*c[5])/(c[3]-1)/( -1+c[6])/(c[5]-1)/(3*c[3]-1)/(5*c[3]-2), c[7] = 1, b[4] = 1/60*(15*c[3 ]^2-10*c[3]+2)^5*(3*c[3]-5*c[5]*c[3]-5*c[6]*c[3]+10*c[5]*c[6]*c[3]-5*c [5]*c[6]-2+3*c[6]+3*c[5])/(15*c[3]^2-10*c[3]+1)/c[3]^2/(3*c[3]-1)/(5*c [3]-2)/(2*c[6]-10*c[6]*c[3]+15*c[6]*c[3]^2-c[3])/(15*c[5]*c[3]^2-10*c[ 5]*c[3]+2*c[5]-c[3]), c[4] = c[3]/(15*c[3]^2-10*c[3]+2), b[2] = 0, a[5 ,2] = 1/2*c[3]*c[5]/c[2]*(-5*c[5]+1+5*c[5]^2)/(1-5*c[3]+5*c[3]^2), a[4 ,2] = 1/2*c[3]^2*(4-30*c[3]+45*c[3]^2)/c[2]/(15*c[3]^2-10*c[3]+2)^3, a [6,2] = 1/2*c[6]*c[3]/c[2]*(5*c[6]^2-5*c[6]+1)/(1-5*c[3]+5*c[3]^2), a[ 4,3] = -(15*c[3]^2-10*c[3]+1)*c[3]/(15*c[3]^2-10*c[3]+2)^3, b[1] = -1/ 60*1/c[3]^2*(150*c[5]*c[3]^3*c[6]-75*c[3]^3*c[6]+45*c[3]^3-75*c[3]^3*c [5]+105*c[5]*c[3]^2-205*c[5]*c[6]*c[3]^2+105*c[6]*c[3]^2-65*c[3]^2+29* c[3]-45*c[5]*c[3]+80*c[5]*c[6]*c[3]-45*c[6]*c[3]+6*c[5]-10*c[5]*c[6]-4 +6*c[6])/c[5]/c[6], a[3,2] = 1/2*c[3]^2/c[2], a[5,4] = 1/6*(15*c[3]^2- 10*c[3]+2)^2*c[5]*(-c[5]+c[3])*(15*c[5]*c[3]^2-10*c[5]*c[3]+2*c[5]-c[3 ])/c[3]^2/(1-5*c[3]+5*c[3]^2)/(15*c[3]^2-10*c[3]+1), b[5] = -1/60*(1-5 *c[3]+5*c[3]^2)*(15*c[6]*c[3]-9*c[3]+4-6*c[6])/(c[6]-c[5])/c[5]/(c[5]- 1)/(15*c[5]*c[3]^2-10*c[5]*c[3]+2*c[5]-c[3])/(-c[5]+c[3]), a[5,3] = 1/ 6*c[5]*(150*c[3]^3*c[5]-145*c[5]*c[3]^2+40*c[5]*c[3]-4*c[5]-30*c[3]^3+ 20*c[3]^2-c[3])*(-c[5]+c[3])/c[3]^2/(1-5*c[3]+5*c[3]^2)/(15*c[3]^2-10* c[3]+1), a[7,5] = (c[3]-1)*(3*c[3]-1)*(5*c[3]-2)*(1-5*c[3]+5*c[3]^2)*( c[5]-1)*(c[5]-3*c[5]*c[3]+15*c[6]^2*c[3]-21*c[6]*c[3]-6*c[6]^2+9*c[3]- 4+9*c[6])/c[5]/(15*c[5]*c[3]^2-10*c[5]*c[3]+2*c[5]-c[3])/(-c[5]+c[3])/ (c[6]-c[5])/(300*c[5]*c[3]^3*c[6]-225*c[3]^3*c[6]+180*c[3]^3-225*c[3]^ 3*c[5]-455*c[5]*c[6]*c[3]^2+350*c[6]*c[3]^2-285*c[3]^2+350*c[5]*c[3]^2 +210*c[5]*c[6]*c[3]-165*c[6]*c[3]+136*c[3]-165*c[5]*c[3]-30*c[5]*c[6]+ 24*c[6]-20+24*c[5]), a[6,5] = -(3*c[3]-1)*(2*c[6]-10*c[6]*c[3]+15*c[6] *c[3]^2-c[3])*(c[3]-c[6])*(c[6]-c[5])*c[6]/(15*c[5]*c[3]-9*c[3]+4-6*c[ 5])/(-c[5]+c[3])/(15*c[5]*c[3]^2-10*c[5]*c[3]+2*c[5]-c[3])/c[5]\}:" }} }{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 18 "The coefficients " }{XPPEDIT 18 0 "a[7,2]" "6#&%\"aG6$\"\"(\"\"# " }{TEXT -1 3 ", " }{XPPEDIT 18 0 "a[7,4]" "6#&%\"aG6$\"\"(\"\"%" } {TEXT -1 3 ", " }{XPPEDIT 18 0 "a[7,5]" "6#&%\"aG6$\"\"(\"\"&" } {TEXT -1 7 " and " }{XPPEDIT 18 0 "a[7,6]" "6#&%\"aG6$\"\"(\"\"'" } {TEXT -1 17 " are as follows." }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 108 "a[7,2]=subs(e11,a[7,2]);``;\na[7,4]=subs(e11,a[7,4]);``;\na[7,5 ]=subs(e11,a[7,5]);``;\na[7,6]=subs(e11,a[7,6]);" }}{PARA 12 "" 1 "" {XPPMATH 20 "6#/&%\"aG6$\"\"(\"\"#,$*&#\"\"\"F(F,**&%\"cG6#\"\"$F,,2*& \"#9F,&F/6#\"\"'F,F,*(\"#?F,&F/6#\"\"&F,F5F,!\"\"*(\"#IF,F5F,F.F,F=** \"#XF,F:F,F5F,F.F,F,*&\"#@F,F.F,F,\"#5F=*&F4F,F:F,F,*(F?F,F:F,F.F,F=F, &F/6#F(F=,B**\"$+$F,F:F,)F.F1F,F5F,F,*(\"$D#F,FLF,F5F,F=*&\"$!=F,FLF,F ,*(FNF,FLF,F:F,F=**\"$b%F,F:F,F5F,)F.F(F,F=*(\"$]$F,F5F,FTF,F,*&\"$&GF ,FTF,F=*(FVF,F:F,FTF,F,**\"$5#F,F:F,F5F,F.F,F,*(\"$l\"F,F5F,F.F,F=*&\" $O\"F,F.F,F,*(FgnF,F:F,F.F,F=*(F?F,F:F,F5F,F=*&\"#CF,F5F,F,F9F=*&F]oF, F:F,F,F=F,F," }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%!G" }}{PARA 12 "" 1 " " {XPPMATH 20 "6#/&%\"aG6$\"\"(\"\"%,$*&#\"\"\"\"\"'F,*6,&&%\"cG6#\"\" $F,F,!\"\"F,,&*&F3F,F0F,F,F,F4F,,&*&\"\"&F,F0F,F,\"\"#F4F,,(*&\"#:F,)F 0F:F,F,*&\"#5F,F0F,F4F:F,F:,hq\"#[F4*(\"%=9F,&F16#F9F,F0F,F4*(\"%]nF,) F0F9F,)&F16#F-F:F,F,**\"&?q$F,FEF,)F0F3F,FKF,F,**\"&5[\"F,FEF,FKF,F>F, F4*(\"%c7F,FKF,F0F,F4*(\"&+3\"F,FIF,FEF,F4**\"&]Z#F,FJF,)F0F(F,)FEF:F, F4**\"&vo\"F,FYF,FIF,FKF,F4*(\"$o\"F,FYF,FKF,F,**\"%WIF,FEF,FKF,F0F,F, **\"&+Y#F,FJF,FOF,FEF,F4**\"%?(*F,FJF,F>F,FEF,F,*(\"$_)F,FYF,F0F,F,** \"%!)>F,FJF,F0F,FEF,F4*(\"&S/\"F,FJF,FOF,F,*&\"$e&F,F0F,F,*&\"%yEF,F>F ,F4*&\"%wlF,FOF,F,*&\"%!>)F,FXF,F4*&\"%]SF,FIF,F,*&\"#sF,FYF,F4*&\"$3 \"F,FKF,F,*&F_pF,FJF,F4*&\"$?\"F,FEF,F,**\"&]>$F,FYF,FXF,FKF,F,**\"&+V #F,FIF,FKF,FEF,F,**\"&]s%F,FEF,FXF,FKF,F4**FenF,FIF,FJF,FEF,F4**\"&+N \"F,FIF,FJF,FYF,F,**\"%+sF,FJF,FYF,F>F,F4**F[oF,FYF,FOF,FKF,F4**FaoF,F YF,FKF,F0F,F4**F]oF,FYF,F>F,FKF,F,**FfpF,FJF,FXF,FEF,F,**\"&+'=F,FJF,F OF,FYF,F,**\"%S9F,FJF,FYF,F0F,F,*(F_oF,FJF,F0F,F,*(\"&5)=F,FXF,FKF,F,* (FgnF,FJF,FEF,F,*(\"%5pF,FEF,F>F,F,*(\"%qTF,FJF,F>F,F4*(F_rF,FYF,F>F,F 4*(\"&+;#F,FXF,FEF,F,*(FdpF,FJF,FYF,F4*(\"&vK\"F,FYF,FXF,F4*(\"&vr\"F, FOF,FEF,F4*(FHF,FIF,FYF,F,*(FerF,FJF,FXF,F4*(\"%]%*F,FIF,FKF,F4*(\"$g# F,FEF,FKF,F4*(FcoF,FYF,FOF,F,*(\"%_gF,FKF,F>F,F,*(\"&b\\\"F,FOF,FKF,F4 F,F0!\"#,(*&F=F,F>F,F,*&F@F,F0F,F4F,F,F4,**&F:F,FKF,F,*(F@F,FKF,F0F,F4 *(F=F,FKF,F>F,F,F0F4F4,**(F=F,FEF,F>F,F,*(F@F,FEF,F0F,F4*&F:F,FEF,F,F0 F4F4,B**\"$+$F,FEF,FOF,FKF,F,*(\"$D#F,FOF,FKF,F4*&\"$!=F,FOF,F,*(FctF, FOF,FEF,F4**\"$b%F,FEF,FKF,F>F,F4*(\"$]$F,FKF,F>F,F,*&\"$&GF,F>F,F4*(F jtF,FEF,F>F,F,**\"$5#F,FEF,FKF,F0F,F,*(\"$l\"F,FKF,F0F,F4*&\"$O\"F,F0F ,F,*(FauF,FEF,F0F,F4*(\"#IF,FEF,FKF,F4*&\"#CF,FKF,F,\"#?F4*&FhuF,FEF,F ,F4F,F," }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%!G" }}{PARA 12 "" 1 "" {XPPMATH 20 "6#/&%\"aG6$\"\"(\"\"&*8,&&%\"cG6#\"\"$\"\"\"F/!\"\"F/,&*& F.F/F+F/F/F/F0F/,&*&F(F/F+F/F/\"\"#F0F/,(F/F/*&F(F/F+F/F0*&F(F/)F+F5F/ F/F/,&&F,6#F(F/F/F0F/,2F;F/*(F.F/F;F/F+F/F0*(\"#:F/)&F,6#\"\"'F5F/F+F/ F/*(\"#@F/FBF/F+F/F0*&FDF/FAF/F0*&\"\"*F/F+F/F/\"\"%F0*&FIF/FBF/F/F/F; F0,**(F@F/F;F/F9F/F/*(\"#5F/F;F/F+F/F0*&F5F/F;F/F/F+F0F0,&F;F0F+F/F0,& FBF/F;F0F0,B**\"$+$F/F;F/)F+F.F/FBF/F/*(\"$D#F/FVF/FBF/F0*&\"$!=F/FVF/ F/*(FXF/FVF/F;F/F0**\"$b%F/F;F/FBF/F9F/F0*(\"$]$F/FBF/F9F/F/*&\"$&GF/F 9F/F0*(FinF/F;F/F9F/F/**\"$5#F/F;F/FBF/F+F/F/*(\"$l\"F/FBF/F+F/F0*&\"$ O\"F/F+F/F/*(F`oF/F;F/F+F/F0*(\"#IF/F;F/FBF/F0*&\"#CF/FBF/F/\"#?F0*&Fg oF/F;F/F/F0" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%!G" }}{PARA 12 "" 1 " " {XPPMATH 20 "6#/&%\"aG6$\"\"(\"\"',$*:,**(\"#:\"\"\"&%\"cG6#\"\"&F.& F06#\"\"$F.F.*&\"\"*F.F3F.!\"\"\"\"%F.*&F(F.F/F.F8F.,(F.F.*&F2F.F3F.F8 *&F2F.)F3\"\"#F.F.F.,&F3F.F.F8F.,&F.F8&F06#F(F.F.,&F/F.F.F8F.,&*&F5F.F 3F.F.F.F8F.,&*&F2F.F3F.F.F?F8F.FBF8,**&F?F.FBF.F.*(\"#5F.FBF.F3F.F8*(F -F.FBF.F>F.F.F3F8F8,&F3F.FBF8F8,&FBF.F/F8F8,B**\"$+$F.F/F.)F3F5F.FBF.F .*(\"$D#F.FSF.FBF.F8*&\"$!=F.FSF.F.*(FUF.FSF.F/F.F8**\"$b%F.F/F.FBF.F> F.F8*(\"$]$F.FBF.F>F.F.*&\"$&GF.F>F.F8*(FfnF.F/F.F>F.F.**\"$5#F.F/F.FB F.F3F.F.*(\"$l\"F.FBF.F3F.F8*&\"$O\"F.F3F.F.*(F]oF.F/F.F3F.F8*(\"#IF.F /F.FBF.F8*&\"#CF.FBF.F.\"#?F8*&FdoF.F/F.F.F8F8" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 15 "" 0 "" {TEXT -1 1 " " }{TEXT 289 6 "Step 6" }{TEXT -1 1 " " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 27 "The stage-order condition: " }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "Sum(a[7,j]*c[j],j = 2 .. 6) = 1/2;" "6#/-%$Su mG6$*&&%\"aG6$\"\"(%\"jG\"\"\"&%\"cG6#F,F-/F,;\"\"#\"\"'*&F-F-F3!\"\" " }{TEXT -1 1 " " }{XPPEDIT 18 0 "c[7]^2;" "6#*$&%\"cG6#\"\"(\"\"#" } {TEXT -1 1 " " }}{PARA 0 "" 0 "" {TEXT -1 12 "determines " }{XPPEDIT 18 0 "a[7,3]" "6#&%\"aG6$\"\"(\"\"$" }{TEXT -1 1 "." }}{PARA 0 "" 0 " " {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 121 "simplify( subs(e11,add(a[7,j]*c[j],j=2..6)=1/2*c[7]^2)):\ne12 := \{a[7,3]=factor (solve(%,a[7,3]))\}:\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 3 "e13" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 7146 "e13 := \{b[6] = 1/60*(1-5* c[3]+5*c[3]^2)*(15*c[5]*c[3]-9*c[3]+4-6*c[5])/c[6]/(-1+c[6])/(2*c[6]-1 0*c[6]*c[3]+15*c[6]*c[3]^2-c[3])/(c[3]-c[6])/(c[6]-c[5]), a[7,2] = 1/2 *c[3]*(14*c[6]-20*c[5]*c[6]-30*c[6]*c[3]+45*c[5]*c[6]*c[3]+21*c[3]-10+ 14*c[5]-30*c[5]*c[3])/c[2]/(300*c[5]*c[3]^3*c[6]-225*c[3]^3*c[6]+180*c [3]^3-225*c[3]^3*c[5]-455*c[5]*c[6]*c[3]^2+350*c[6]*c[3]^2-285*c[3]^2+ 350*c[5]*c[3]^2+210*c[5]*c[6]*c[3]-165*c[6]*c[3]+136*c[3]-165*c[5]*c[3 ]-30*c[5]*c[6]+24*c[6]-20+24*c[5]), a[6,4] = 1/6*c[6]*(15*c[3]^2-10*c[ 3]+2)^2*(2*c[6]-10*c[6]*c[3]+15*c[6]*c[3]^2-c[3])*(c[3]-c[6])*(90*c[3] ^3*c[6]-120*c[6]*c[3]^2+48*c[6]*c[3]-6*c[6]-225*c[3]^3*c[5]+225*c[5]^2 *c[3]^3-240*c[5]^2*c[3]^2+9*c[3]^2+255*c[5]*c[3]^2-100*c[5]*c[3]-4*c[3 ]+90*c[5]^2*c[3]+14*c[5]-12*c[5]^2)/c[3]^2/(1-5*c[3]+5*c[3]^2)/(15*c[3 ]^2-10*c[3]+1)/(15*c[5]*c[3]^2-10*c[5]*c[3]+2*c[5]-c[3])/(15*c[5]*c[3] -9*c[3]+4-6*c[5]), a[7,6] = -(15*c[5]*c[3]-9*c[3]+4-6*c[5])*(1-5*c[3]+ 5*c[3]^2)*(c[3]-1)*(-1+c[6])*(c[5]-1)*(3*c[3]-1)*(5*c[3]-2)/c[6]/(2*c[ 6]-10*c[6]*c[3]+15*c[6]*c[3]^2-c[3])/(c[3]-c[6])/(c[6]-c[5])/(300*c[5] *c[3]^3*c[6]-225*c[3]^3*c[6]+180*c[3]^3-225*c[3]^3*c[5]-455*c[5]*c[6]* c[3]^2+350*c[6]*c[3]^2-285*c[3]^2+350*c[5]*c[3]^2+210*c[5]*c[6]*c[3]-1 65*c[6]*c[3]+136*c[3]-165*c[5]*c[3]-30*c[5]*c[6]+24*c[6]-20+24*c[5]), \+ a[7,4] = 1/6*(c[3]-1)*(3*c[3]-1)*(5*c[3]-2)*(15*c[3]^2-10*c[3]+2)^2*(- 48-1418*c[5]*c[3]+6750*c[3]^5*c[6]^2+37020*c[5]*c[3]^3*c[6]-14810*c[5] *c[6]*c[3]^2-1256*c[6]*c[3]-10800*c[3]^5*c[5]-24750*c[6]^2*c[3]^4*c[5] ^2-16875*c[5]^2*c[3]^5*c[6]+168*c[5]^2*c[6]+3044*c[5]*c[6]*c[3]-24600* c[6]^2*c[3]^3*c[5]+9720*c[6]^2*c[3]^2*c[5]+852*c[5]^2*c[3]-1980*c[6]^2 *c[3]*c[5]+10440*c[6]^2*c[3]^3+558*c[3]-2678*c[3]^2+6576*c[3]^3-8190*c [3]^4+4050*c[3]^5-72*c[5]^2+108*c[6]-72*c[6]^2+120*c[5]+31950*c[5]^2*c [3]^4*c[6]+24300*c[3]^5*c[6]*c[5]-47250*c[5]*c[3]^4*c[6]-16875*c[3]^5* c[6]^2*c[5]+13500*c[3]^5*c[6]^2*c[5]^2-7200*c[6]^2*c[5]^2*c[3]^2-24600 *c[5]^2*c[3]^3*c[6]-1980*c[5]^2*c[6]*c[3]+9720*c[5]^2*c[3]^2*c[6]+3195 0*c[6]^2*c[3]^4*c[5]+18600*c[6]^2*c[3]^3*c[5]^2+1440*c[6]^2*c[5]^2*c[3 ]+852*c[6]^2*c[3]+18810*c[3]^4*c[6]+168*c[6]^2*c[5]+6910*c[5]*c[3]^2-4 170*c[6]^2*c[3]^2-4170*c[5]^2*c[3]^2+21600*c[3]^4*c[5]-120*c[6]^2*c[5] ^2-13275*c[5]^2*c[3]^4-17175*c[3]^3*c[5]+6750*c[3]^5*c[5]^2-13275*c[6] ^2*c[3]^4-9450*c[3]^5*c[6]-260*c[5]*c[6]+10440*c[5]^2*c[3]^3+6052*c[6] *c[3]^2-14955*c[3]^3*c[6])/c[3]^2/(15*c[3]^2-10*c[3]+1)/(2*c[6]-10*c[6 ]*c[3]+15*c[6]*c[3]^2-c[3])/(15*c[5]*c[3]^2-10*c[5]*c[3]+2*c[5]-c[3])/ (300*c[5]*c[3]^3*c[6]-225*c[3]^3*c[6]+180*c[3]^3-225*c[3]^3*c[5]-455*c [5]*c[6]*c[3]^2+350*c[6]*c[3]^2-285*c[3]^2+350*c[5]*c[3]^2+210*c[5]*c[ 6]*c[3]-165*c[6]*c[3]+136*c[3]-165*c[5]*c[3]-30*c[5]*c[6]+24*c[6]-20+2 4*c[5]), a[6,3] = 1/6*(c[3]-c[6])*(-2*c[5]*c[3]+1350*c[3]^5*c[6]^2-253 5*c[5]*c[3]^3*c[6]+1450*c[5]*c[6]*c[3]^2-10*c[6]*c[3]-450*c[3]^5*c[5]- 24*c[5]^2*c[6]-328*c[5]*c[6]*c[3]-6*c[5]^2*c[3]+2100*c[6]^2*c[3]^3-4*c [3]^2+89*c[3]^3-300*c[3]^4+270*c[3]^5-12*c[6]^2-2250*c[5]^2*c[3]^4*c[6 ]+900*c[3]^5*c[6]*c[5]+975*c[5]*c[3]^4*c[6]+3075*c[5]^2*c[3]^3*c[6]+30 0*c[5]^2*c[6]*c[3]-1470*c[5]^2*c[3]^2*c[6]+156*c[6]^2*c[3]+1815*c[3]^4 *c[6]-35*c[5]*c[3]^2-810*c[6]^2*c[3]^2+135*c[5]^2*c[3]^2+300*c[3]^4*c[ 5]+450*c[5]^2*c[3]^4+45*c[3]^3*c[5]-2700*c[6]^2*c[3]^4-1350*c[3]^5*c[6 ]+28*c[5]*c[6]-480*c[5]^2*c[3]^3+148*c[6]*c[3]^2-820*c[3]^3*c[6])*c[6] /(1-5*c[3]+5*c[3]^2)/(15*c[3]^2-10*c[3]+1)/(15*c[5]*c[3]-9*c[3]+4-6*c[ 5])/(-c[5]+c[3])/c[3]^2, b[3] = -1/60*(75*c[5]*c[6]*c[3]^2-60*c[5]*c[6 ]*c[3]+10*c[5]*c[6]-45*c[5]*c[3]^2+35*c[5]*c[3]-6*c[5]-45*c[6]*c[3]^2+ 35*c[6]*c[3]-6*c[6]+30*c[3]^2-23*c[3]+4)/(c[3]-c[6])/(-c[5]+c[3])/c[3] ^2/(c[3]-1)/(15*c[3]^2-10*c[3]+1), b[7] = 1/60*(300*c[5]*c[3]^3*c[6]-2 25*c[3]^3*c[6]+180*c[3]^3-225*c[3]^3*c[5]-455*c[5]*c[6]*c[3]^2+350*c[6 ]*c[3]^2-285*c[3]^2+350*c[5]*c[3]^2+210*c[5]*c[6]*c[3]-165*c[6]*c[3]+1 36*c[3]-165*c[5]*c[3]-30*c[5]*c[6]+24*c[6]-20+24*c[5])/(c[3]-1)/(-1+c[ 6])/(c[5]-1)/(3*c[3]-1)/(5*c[3]-2), c[7] = 1, b[4] = 1/60*(15*c[3]^2-1 0*c[3]+2)^5*(3*c[3]-5*c[5]*c[3]-5*c[6]*c[3]+10*c[5]*c[6]*c[3]-5*c[5]*c [6]-2+3*c[6]+3*c[5])/(15*c[3]^2-10*c[3]+1)/c[3]^2/(3*c[3]-1)/(5*c[3]-2 )/(2*c[6]-10*c[6]*c[3]+15*c[6]*c[3]^2-c[3])/(15*c[5]*c[3]^2-10*c[5]*c[ 3]+2*c[5]-c[3]), c[4] = c[3]/(15*c[3]^2-10*c[3]+2), b[2] = 0, a[5,2] = 1/2*c[3]*c[5]/c[2]*(-5*c[5]+1+5*c[5]^2)/(1-5*c[3]+5*c[3]^2), a[4,2] = 1/2*c[3]^2*(4-30*c[3]+45*c[3]^2)/c[2]/(15*c[3]^2-10*c[3]+2)^3, a[6,2] = 1/2*c[6]*c[3]/c[2]*(5*c[6]^2-5*c[6]+1)/(1-5*c[3]+5*c[3]^2), a[4,3] \+ = -(15*c[3]^2-10*c[3]+1)*c[3]/(15*c[3]^2-10*c[3]+2)^3, b[1] = -1/60*1/ c[3]^2*(150*c[5]*c[3]^3*c[6]-75*c[3]^3*c[6]+45*c[3]^3-75*c[3]^3*c[5]+1 05*c[5]*c[3]^2-205*c[5]*c[6]*c[3]^2+105*c[6]*c[3]^2-65*c[3]^2+29*c[3]- 45*c[5]*c[3]+80*c[5]*c[6]*c[3]-45*c[6]*c[3]+6*c[5]-10*c[5]*c[6]-4+6*c[ 6])/c[5]/c[6], a[3,2] = 1/2*c[3]^2/c[2], a[5,4] = 1/6*(15*c[3]^2-10*c[ 3]+2)^2*c[5]*(-c[5]+c[3])*(15*c[5]*c[3]^2-10*c[5]*c[3]+2*c[5]-c[3])/c[ 3]^2/(1-5*c[3]+5*c[3]^2)/(15*c[3]^2-10*c[3]+1), b[5] = -1/60*(1-5*c[3] +5*c[3]^2)*(15*c[6]*c[3]-9*c[3]+4-6*c[6])/(c[6]-c[5])/c[5]/(c[5]-1)/(1 5*c[5]*c[3]^2-10*c[5]*c[3]+2*c[5]-c[3])/(-c[5]+c[3]), a[5,3] = 1/6*c[5 ]*(150*c[3]^3*c[5]-145*c[5]*c[3]^2+40*c[5]*c[3]-4*c[5]-30*c[3]^3+20*c[ 3]^2-c[3])*(-c[5]+c[3])/c[3]^2/(1-5*c[3]+5*c[3]^2)/(15*c[3]^2-10*c[3]+ 1), a[7,5] = (c[3]-1)*(3*c[3]-1)*(5*c[3]-2)*(1-5*c[3]+5*c[3]^2)*(c[5]- 1)*(c[5]-3*c[5]*c[3]+15*c[6]^2*c[3]-21*c[6]*c[3]-6*c[6]^2+9*c[3]-4+9*c [6])/c[5]/(15*c[5]*c[3]^2-10*c[5]*c[3]+2*c[5]-c[3])/(-c[5]+c[3])/(c[6] -c[5])/(300*c[5]*c[3]^3*c[6]-225*c[3]^3*c[6]+180*c[3]^3-225*c[3]^3*c[5 ]-455*c[5]*c[6]*c[3]^2+350*c[6]*c[3]^2-285*c[3]^2+350*c[5]*c[3]^2+210* c[5]*c[6]*c[3]-165*c[6]*c[3]+136*c[3]-165*c[5]*c[3]-30*c[5]*c[6]+24*c[ 6]-20+24*c[5]), a[6,5] = -(3*c[3]-1)*(2*c[6]-10*c[6]*c[3]+15*c[6]*c[3] ^2-c[3])*(c[3]-c[6])*(c[6]-c[5])*c[6]/(15*c[5]*c[3]-9*c[3]+4-6*c[5])/( -c[5]+c[3])/(15*c[5]*c[3]^2-10*c[5]*c[3]+2*c[5]-c[3])/c[5], a[7,3] = 1 /6*(c[3]-1)*(48+1408*c[5]*c[3]-900*c[3]^5*c[6]^2-23250*c[5]*c[3]^3*c[6 ]+12622*c[5]*c[6]*c[3]^2+1234*c[6]*c[3]+3225*c[3]^5*c[5]-450*c[3]^6*c[ 5]+7650*c[6]^2*c[3]^4*c[5]^2-900*c[5]^2*c[3]^5*c[6]-168*c[5]^2*c[6]-29 82*c[5]*c[6]*c[3]+14565*c[6]^2*c[3]^3*c[5]-8145*c[6]^2*c[3]^2*c[5]-840 *c[5]^2*c[3]+1932*c[6]^2*c[3]*c[5]+900*c[3]^6*c[6]*c[5]-7095*c[6]^2*c[ 3]^3-552*c[3]+2360*c[3]^2-4452*c[3]^3-630*c[3]^6+3094*c[3]^4+360*c[3]^ 5+72*c[5]^2-108*c[6]+72*c[6]^2-120*c[5]-8850*c[5]^2*c[3]^4*c[6]-1500*c [3]^5*c[6]*c[5]+16230*c[5]*c[3]^4*c[6]-900*c[3]^5*c[6]^2*c[5]+6120*c[6 ]^2*c[5]^2*c[3]^2+14565*c[5]^2*c[3]^3*c[6]+1932*c[5]^2*c[6]*c[3]-8145* c[5]^2*c[3]^2*c[6]-8850*c[6]^2*c[3]^4*c[5]-11400*c[6]^2*c[3]^3*c[5]^2- 1410*c[6]^2*c[5]^2*c[3]-840*c[6]^2*c[3]-6135*c[3]^4*c[6]-168*c[6]^2*c[ 5]-6222*c[5]*c[3]^2+900*c[3]^6*c[6]+3654*c[6]^2*c[3]^2+3654*c[5]^2*c[3 ]^2-11145*c[3]^4*c[5]+120*c[6]^2*c[5]^2+5595*c[5]^2*c[3]^4+12590*c[3]^ 3*c[5]-900*c[3]^5*c[5]^2+5595*c[6]^2*c[3]^4-915*c[3]^5*c[6]+260*c[5]*c [6]-7095*c[5]^2*c[3]^3-5202*c[6]*c[3]^2+9512*c[3]^3*c[6])/(c[3]-c[6])/ (-c[5]+c[3])/(15*c[3]^2-10*c[3]+1)/c[3]^2/(300*c[5]*c[3]^3*c[6]-225*c[ 3]^3*c[6]+180*c[3]^3-225*c[3]^3*c[5]-455*c[5]*c[6]*c[3]^2+350*c[6]*c[3 ]^2-285*c[3]^2+350*c[5]*c[3]^2+210*c[5]*c[6]*c[3]-165*c[6]*c[3]+136*c[ 3]-165*c[5]*c[3]-30*c[5]*c[6]+24*c[6]-20+24*c[5])\}:" }}}{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 24 "a[7,3]=subs(e13,a[7,3]);" }}{PARA 12 "" 1 "" {XPPMATH 20 "6#/&%\"a G6$\"\"(\"\"$,$*&#\"\"\"\"\"'F,*0,&&%\"cG6#F(F,F,!\"\"F,,^r\"#[F,*(\"% 39F,&F16#\"\"&F,F0F,F,*(\"$+*F,)F0F:F,)&F16#F-\"\"#F,F3**\"&]K#F,F8F,) F0F(F,F?F,F3**\"&AE\"F,F8F,F?F,)F0FAF,F,*(\"%M7F,F?F,F0F,F,*(\"%DKF,F= F,F8F,F,*(\"$]%F,)F0F-F,F8F,F3**\"%]wF,F>F,)F0\"\"%F,)F8FAF,F,**FF ,FDF,F8F,F,**\"%X\")F,F>F,FGF,F8F,F3*(\"$S)F,FSF,F0F,F3**\"%K>F,F>F,F0 F,F8F,F,**FF,FDF,F3*&\"$_&F,F0F,F3*&\"%gB F,FGF,F,*&\"%_WF,FDF,F3*&\"$I'F,FNF,F3*&\"%%4$F,FQF,F,*&\"$g$F,F=F,F,* &\"#sF,FSF,F,*&\"$3\"F,F?F,F3*&F[pF,F>F,F,*&\"$?\"F,F8F,F3**\"%]))F,FS F,FQF,F?F,F3**\"%+:F,F=F,F?F,F8F,F3**\"&Ii\"F,F8F,FQF,F?F,F,**FF,F8F,F3**\"%?hF,F>F,FSF,FGF,F,**FZF,FSF,FDF,F?F,F,**FjnF,FSF,F?F,F0 F,F,**FfnF,FSF,FGF,F?F,F3**FbpF,F>F,FQF,F8F,F3**\"&+9\"F,F>F,FDF,FSF,F 3**\"%59F,F>F,FSF,F0F,F3*(FhnF,F>F,F0F,F3*(\"%NhF,FQF,F?F,F3*(FVF,F>F, F8F,F3*(\"%AiF,F8F,FGF,F3*(FF,FGF,F,*(FjqF,FSF ,FGF,F,*(\"&X6\"F,FQF,F8F,F3*(F`pF,F>F,FSF,F,*(\"%&f&F,FSF,FQF,F,*(\"& !f7F,FDF,F8F,F,*(FF,FQF,F,*(\"$:*F,F=F,F?F,F3*( \"$g#F,F8F,F?F,F,*(F]oF,FSF,FDF,F3*(\"%-_F,F?F,FGF,F3*(\"%7&*F,FDF,F?F ,F,F,,&F0F,F?F3F3,&F8F3F0F,F3,(*&\"#:F,FGF,F,*&\"#5F,F0F,F3F,F,F3F0!\" #,B**\"$+$F,F8F,FDF,F?F,F,*(\"$D#F,FDF,F?F,F3*&\"$!=F,FDF,F,*(FjsF,FDF ,F8F,F3**\"$b%F,F8F,F?F,FGF,F3*(\"$]$F,F?F,FGF,F,*&\"$&GF,FGF,F3*(FatF ,F8F,FGF,F,**\"$5#F,F8F,F?F,F0F,F,*(\"$l\"F,F?F,F0F,F3*&\"$O\"F,F0F,F, *(FhtF,F8F,F0F,F3*(\"#IF,F8F,F?F,F3*&\"#CF,F?F,F,\"#?F3*&F_uF,F8F,F,F3 F,F," }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 15 "" 0 "" {TEXT -1 1 " " }{TEXT 290 6 "Step 7" }{TEXT -1 1 " " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 24 "The row-sum conditions: " }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "Sum(a[i,j],j=1..i-1)=c[i]" "6#/-%$SumG6$&%\"aG6$%\"iG%\"jG/F+;\"\"\",&F*F.F.!\"\"&%\"cG6#F*" } {TEXT -1 8 ", for " }{XPPEDIT 18 0 "i=2" "6#/%\"iG\"\"#" }{TEXT -1 8 " . . 7, " }}{PARA 0 "" 0 "" {TEXT -1 21 "can be used to find " } {XPPEDIT 18 0 "a[2,1]" "6#&%\"aG6$\"\"#\"\"\"" }{TEXT -1 3 ", " } {XPPEDIT 18 0 "a[3,1]" "6#&%\"aG6$\"\"$\"\"\"" }{TEXT -1 3 ", " } {XPPEDIT 18 0 "a[4,1]" "6#&%\"aG6$\"\"%\"\"\"" }{TEXT -1 11 ", . . . , " }{XPPEDIT 18 0 "a[7,1];" "6#&%\"aG6$\"\"(\"\"\"" }{TEXT -1 2 ". " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 137 "simplify(subs(e13,[seq(add(a[i,j],j=1..i-1)=c[i],i=2..7)])):\ne 14 := factor(solve(\{op(%)\},\{seq(a[i,1],i=2..7)\})):\ne15 := `union` (e13,e14):" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 " " {MPLTEXT 1 0 1 ";" }}}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 3 "e15" }} {PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 11161 "e15 := \{b[6] = 1/60*(1-5*c[3]+5*c[3]^2)*(15*c[5]*c[3]-9*c[3]+4 -6*c[5])/c[6]/(-1+c[6])/(2*c[6]-10*c[6]*c[3]+15*c[6]*c[3]^2-c[3])/(c[3 ]-c[6])/(c[6]-c[5]), a[7,2] = 1/2*c[3]*(14*c[6]-20*c[5]*c[6]-30*c[6]*c [3]+45*c[5]*c[6]*c[3]+21*c[3]-10+14*c[5]-30*c[5]*c[3])/c[2]/(300*c[5]* c[3]^3*c[6]-225*c[3]^3*c[6]+180*c[3]^3-225*c[3]^3*c[5]-455*c[5]*c[6]*c [3]^2+350*c[6]*c[3]^2-285*c[3]^2+350*c[5]*c[3]^2+210*c[5]*c[6]*c[3]-16 5*c[6]*c[3]+136*c[3]-165*c[5]*c[3]-30*c[5]*c[6]+24*c[6]-20+24*c[5]), a [6,4] = 1/6*c[6]*(15*c[3]^2-10*c[3]+2)^2*(2*c[6]-10*c[6]*c[3]+15*c[6]* c[3]^2-c[3])*(c[3]-c[6])*(90*c[3]^3*c[6]-120*c[6]*c[3]^2+48*c[6]*c[3]- 6*c[6]-225*c[3]^3*c[5]+225*c[5]^2*c[3]^3-240*c[5]^2*c[3]^2+9*c[3]^2+25 5*c[5]*c[3]^2-100*c[5]*c[3]-4*c[3]+90*c[5]^2*c[3]+14*c[5]-12*c[5]^2)/c [3]^2/(1-5*c[3]+5*c[3]^2)/(15*c[3]^2-10*c[3]+1)/(15*c[5]*c[3]^2-10*c[5 ]*c[3]+2*c[5]-c[3])/(15*c[5]*c[3]-9*c[3]+4-6*c[5]), a[7,6] = -(15*c[5] *c[3]-9*c[3]+4-6*c[5])*(1-5*c[3]+5*c[3]^2)*(c[3]-1)*(-1+c[6])*(c[5]-1) *(3*c[3]-1)*(5*c[3]-2)/c[6]/(2*c[6]-10*c[6]*c[3]+15*c[6]*c[3]^2-c[3])/ (c[3]-c[6])/(c[6]-c[5])/(300*c[5]*c[3]^3*c[6]-225*c[3]^3*c[6]+180*c[3] ^3-225*c[3]^3*c[5]-455*c[5]*c[6]*c[3]^2+350*c[6]*c[3]^2-285*c[3]^2+350 *c[5]*c[3]^2+210*c[5]*c[6]*c[3]-165*c[6]*c[3]+136*c[3]-165*c[5]*c[3]-3 0*c[5]*c[6]+24*c[6]-20+24*c[5]), a[7,4] = 1/6*(c[3]-1)*(3*c[3]-1)*(5*c [3]-2)*(15*c[3]^2-10*c[3]+2)^2*(-48-1418*c[5]*c[3]+6750*c[3]^5*c[6]^2+ 37020*c[5]*c[3]^3*c[6]-14810*c[5]*c[6]*c[3]^2-1256*c[6]*c[3]-10800*c[3 ]^5*c[5]-24750*c[6]^2*c[3]^4*c[5]^2-16875*c[5]^2*c[3]^5*c[6]+168*c[5]^ 2*c[6]+3044*c[5]*c[6]*c[3]-24600*c[6]^2*c[3]^3*c[5]+9720*c[6]^2*c[3]^2 *c[5]+852*c[5]^2*c[3]-1980*c[6]^2*c[3]*c[5]+10440*c[6]^2*c[3]^3+558*c[ 3]-2678*c[3]^2+6576*c[3]^3-8190*c[3]^4+4050*c[3]^5-72*c[5]^2+108*c[6]- 72*c[6]^2+120*c[5]+31950*c[5]^2*c[3]^4*c[6]+24300*c[3]^5*c[6]*c[5]-472 50*c[5]*c[3]^4*c[6]-16875*c[3]^5*c[6]^2*c[5]+13500*c[3]^5*c[6]^2*c[5]^ 2-7200*c[6]^2*c[5]^2*c[3]^2-24600*c[5]^2*c[3]^3*c[6]-1980*c[5]^2*c[6]* c[3]+9720*c[5]^2*c[3]^2*c[6]+31950*c[6]^2*c[3]^4*c[5]+18600*c[6]^2*c[3 ]^3*c[5]^2+1440*c[6]^2*c[5]^2*c[3]+852*c[6]^2*c[3]+18810*c[3]^4*c[6]+1 68*c[6]^2*c[5]+6910*c[5]*c[3]^2-4170*c[6]^2*c[3]^2-4170*c[5]^2*c[3]^2+ 21600*c[3]^4*c[5]-120*c[6]^2*c[5]^2-13275*c[5]^2*c[3]^4-17175*c[3]^3*c [5]+6750*c[3]^5*c[5]^2-13275*c[6]^2*c[3]^4-9450*c[3]^5*c[6]-260*c[5]*c [6]+10440*c[5]^2*c[3]^3+6052*c[6]*c[3]^2-14955*c[3]^3*c[6])/c[3]^2/(15 *c[3]^2-10*c[3]+1)/(2*c[6]-10*c[6]*c[3]+15*c[6]*c[3]^2-c[3])/(15*c[5]* c[3]^2-10*c[5]*c[3]+2*c[5]-c[3])/(300*c[5]*c[3]^3*c[6]-225*c[3]^3*c[6] +180*c[3]^3-225*c[3]^3*c[5]-455*c[5]*c[6]*c[3]^2+350*c[6]*c[3]^2-285*c [3]^2+350*c[5]*c[3]^2+210*c[5]*c[6]*c[3]-165*c[6]*c[3]+136*c[3]-165*c[ 5]*c[3]-30*c[5]*c[6]+24*c[6]-20+24*c[5]), a[6,3] = 1/6*(c[3]-c[6])*(-2 *c[5]*c[3]+1350*c[3]^5*c[6]^2-2535*c[5]*c[3]^3*c[6]+1450*c[5]*c[6]*c[3 ]^2-10*c[6]*c[3]-450*c[3]^5*c[5]-24*c[5]^2*c[6]-328*c[5]*c[6]*c[3]-6*c [5]^2*c[3]+2100*c[6]^2*c[3]^3-4*c[3]^2+89*c[3]^3-300*c[3]^4+270*c[3]^5 -12*c[6]^2-2250*c[5]^2*c[3]^4*c[6]+900*c[3]^5*c[6]*c[5]+975*c[5]*c[3]^ 4*c[6]+3075*c[5]^2*c[3]^3*c[6]+300*c[5]^2*c[6]*c[3]-1470*c[5]^2*c[3]^2 *c[6]+156*c[6]^2*c[3]+1815*c[3]^4*c[6]-35*c[5]*c[3]^2-810*c[6]^2*c[3]^ 2+135*c[5]^2*c[3]^2+300*c[3]^4*c[5]+450*c[5]^2*c[3]^4+45*c[3]^3*c[5]-2 700*c[6]^2*c[3]^4-1350*c[3]^5*c[6]+28*c[5]*c[6]-480*c[5]^2*c[3]^3+148* c[6]*c[3]^2-820*c[3]^3*c[6])*c[6]/(1-5*c[3]+5*c[3]^2)/(15*c[3]^2-10*c[ 3]+1)/(15*c[5]*c[3]-9*c[3]+4-6*c[5])/(-c[5]+c[3])/c[3]^2, b[3] = -1/60 *(75*c[5]*c[6]*c[3]^2-60*c[5]*c[6]*c[3]+10*c[5]*c[6]-45*c[5]*c[3]^2+35 *c[5]*c[3]-6*c[5]-45*c[6]*c[3]^2+35*c[6]*c[3]-6*c[6]+30*c[3]^2-23*c[3] +4)/(c[3]-c[6])/(-c[5]+c[3])/c[3]^2/(c[3]-1)/(15*c[3]^2-10*c[3]+1), b[ 7] = 1/60*(300*c[5]*c[3]^3*c[6]-225*c[3]^3*c[6]+180*c[3]^3-225*c[3]^3* c[5]-455*c[5]*c[6]*c[3]^2+350*c[6]*c[3]^2-285*c[3]^2+350*c[5]*c[3]^2+2 10*c[5]*c[6]*c[3]-165*c[6]*c[3]+136*c[3]-165*c[5]*c[3]-30*c[5]*c[6]+24 *c[6]-20+24*c[5])/(c[3]-1)/(-1+c[6])/(c[5]-1)/(3*c[3]-1)/(5*c[3]-2), c [7] = 1, b[4] = 1/60*(15*c[3]^2-10*c[3]+2)^5*(3*c[3]-5*c[5]*c[3]-5*c[6 ]*c[3]+10*c[5]*c[6]*c[3]-5*c[5]*c[6]-2+3*c[6]+3*c[5])/(15*c[3]^2-10*c[ 3]+1)/c[3]^2/(3*c[3]-1)/(5*c[3]-2)/(2*c[6]-10*c[6]*c[3]+15*c[6]*c[3]^2 -c[3])/(15*c[5]*c[3]^2-10*c[5]*c[3]+2*c[5]-c[3]), c[4] = c[3]/(15*c[3] ^2-10*c[3]+2), b[2] = 0, a[5,2] = 1/2*c[3]*c[5]/c[2]*(-5*c[5]+1+5*c[5] ^2)/(1-5*c[3]+5*c[3]^2), a[4,2] = 1/2*c[3]^2*(4-30*c[3]+45*c[3]^2)/c[2 ]/(15*c[3]^2-10*c[3]+2)^3, a[6,2] = 1/2*c[6]*c[3]/c[2]*(5*c[6]^2-5*c[6 ]+1)/(1-5*c[3]+5*c[3]^2), a[4,3] = -(15*c[3]^2-10*c[3]+1)*c[3]/(15*c[3 ]^2-10*c[3]+2)^3, b[1] = -1/60*1/c[3]^2*(150*c[5]*c[3]^3*c[6]-75*c[3]^ 3*c[6]+45*c[3]^3-75*c[3]^3*c[5]+105*c[5]*c[3]^2-205*c[5]*c[6]*c[3]^2+1 05*c[6]*c[3]^2-65*c[3]^2+29*c[3]-45*c[5]*c[3]+80*c[5]*c[6]*c[3]-45*c[6 ]*c[3]+6*c[5]-10*c[5]*c[6]-4+6*c[6])/c[5]/c[6], a[3,2] = 1/2*c[3]^2/c[ 2], a[5,1] = -1/6*c[5]*(225*c[2]*c[3]^5*c[5]-300*c[5]*c[2]*c[3]^4-45*c [2]*c[3]^4-225*c[5]^2*c[2]*c[3]^4+15*c[5]^2*c[3]^3+3*c[3]^3-15*c[3]^3* c[5]+300*c[5]^2*c[2]*c[3]^3+190*c[5]*c[2]*c[3]^3+40*c[3]^3*c[2]-50*c[5 ]*c[2]*c[3]^2-11*c[3]^2*c[2]-175*c[5]^2*c[2]*c[3]^2+9*c[2]*c[5]*c[3]+4 0*c[5]^2*c[2]*c[3]-4*c[2]*c[5]^2)/c[3]^2/c[2]/(1-5*c[3]+5*c[3]^2), a[3 ,1] = -1/2*c[3]*(c[3]-2*c[2])/c[2], a[5,4] = 1/6*(15*c[3]^2-10*c[3]+2) ^2*c[5]*(-c[5]+c[3])*(15*c[5]*c[3]^2-10*c[5]*c[3]+2*c[5]-c[3])/c[3]^2/ (1-5*c[3]+5*c[3]^2)/(15*c[3]^2-10*c[3]+1), a[2,1] = c[2], b[5] = -1/60 *(1-5*c[3]+5*c[3]^2)*(15*c[6]*c[3]-9*c[3]+4-6*c[6])/(c[6]-c[5])/c[5]/( c[5]-1)/(15*c[5]*c[3]^2-10*c[5]*c[3]+2*c[5]-c[3])/(-c[5]+c[3]), a[4,1] = 1/2*c[3]*(-4*c[3]+30*c[3]^2-45*c[3]^3+350*c[3]^2*c[2]-100*c[3]*c[2] +10*c[2]+450*c[2]*c[3]^4-600*c[3]^3*c[2])/c[2]/(15*c[3]^2-10*c[3]+2)^3 , a[5,3] = 1/6*c[5]*(150*c[3]^3*c[5]-145*c[5]*c[3]^2+40*c[5]*c[3]-4*c[ 5]-30*c[3]^3+20*c[3]^2-c[3])*(-c[5]+c[3])/c[3]^2/(1-5*c[3]+5*c[3]^2)/( 15*c[3]^2-10*c[3]+1), a[7,5] = (c[3]-1)*(3*c[3]-1)*(5*c[3]-2)*(1-5*c[3 ]+5*c[3]^2)*(c[5]-1)*(c[5]-3*c[5]*c[3]+15*c[6]^2*c[3]-21*c[6]*c[3]-6*c [6]^2+9*c[3]-4+9*c[6])/c[5]/(15*c[5]*c[3]^2-10*c[5]*c[3]+2*c[5]-c[3])/ (-c[5]+c[3])/(c[6]-c[5])/(300*c[5]*c[3]^3*c[6]-225*c[3]^3*c[6]+180*c[3 ]^3-225*c[3]^3*c[5]-455*c[5]*c[6]*c[3]^2+350*c[6]*c[3]^2-285*c[3]^2+35 0*c[5]*c[3]^2+210*c[5]*c[6]*c[3]-165*c[6]*c[3]+136*c[3]-165*c[5]*c[3]- 30*c[5]*c[6]+24*c[6]-20+24*c[5]), a[6,5] = -(3*c[3]-1)*(2*c[6]-10*c[6] *c[3]+15*c[6]*c[3]^2-c[3])*(c[3]-c[6])*(c[6]-c[5])*c[6]/(15*c[5]*c[3]- 9*c[3]+4-6*c[5])/(-c[5]+c[3])/(15*c[5]*c[3]^2-10*c[5]*c[3]+2*c[5]-c[3] )/c[5], a[7,3] = 1/6*(c[3]-1)*(48+1408*c[5]*c[3]-900*c[3]^5*c[6]^2-232 50*c[5]*c[3]^3*c[6]+12622*c[5]*c[6]*c[3]^2+1234*c[6]*c[3]+3225*c[3]^5* c[5]-450*c[3]^6*c[5]+7650*c[6]^2*c[3]^4*c[5]^2-900*c[5]^2*c[3]^5*c[6]- 168*c[5]^2*c[6]-2982*c[5]*c[6]*c[3]+14565*c[6]^2*c[3]^3*c[5]-8145*c[6] ^2*c[3]^2*c[5]-840*c[5]^2*c[3]+1932*c[6]^2*c[3]*c[5]+900*c[3]^6*c[6]*c [5]-7095*c[6]^2*c[3]^3-552*c[3]+2360*c[3]^2-4452*c[3]^3-630*c[3]^6+309 4*c[3]^4+360*c[3]^5+72*c[5]^2-108*c[6]+72*c[6]^2-120*c[5]-8850*c[5]^2* c[3]^4*c[6]-1500*c[3]^5*c[6]*c[5]+16230*c[5]*c[3]^4*c[6]-900*c[3]^5*c[ 6]^2*c[5]+6120*c[6]^2*c[5]^2*c[3]^2+14565*c[5]^2*c[3]^3*c[6]+1932*c[5] ^2*c[6]*c[3]-8145*c[5]^2*c[3]^2*c[6]-8850*c[6]^2*c[3]^4*c[5]-11400*c[6 ]^2*c[3]^3*c[5]^2-1410*c[6]^2*c[5]^2*c[3]-840*c[6]^2*c[3]-6135*c[3]^4* c[6]-168*c[6]^2*c[5]-6222*c[5]*c[3]^2+900*c[3]^6*c[6]+3654*c[6]^2*c[3] ^2+3654*c[5]^2*c[3]^2-11145*c[3]^4*c[5]+120*c[6]^2*c[5]^2+5595*c[5]^2* c[3]^4+12590*c[3]^3*c[5]-900*c[3]^5*c[5]^2+5595*c[6]^2*c[3]^4-915*c[3] ^5*c[6]+260*c[5]*c[6]-7095*c[5]^2*c[3]^3-5202*c[6]*c[3]^2+9512*c[3]^3* c[6])/(c[3]-c[6])/(-c[5]+c[3])/(15*c[3]^2-10*c[3]+1)/c[3]^2/(300*c[5]* c[3]^3*c[6]-225*c[3]^3*c[6]+180*c[3]^3-225*c[3]^3*c[5]-455*c[5]*c[6]*c [3]^2+350*c[6]*c[3]^2-285*c[3]^2+350*c[5]*c[3]^2+210*c[5]*c[6]*c[3]-16 5*c[6]*c[3]+136*c[3]-165*c[5]*c[3]-30*c[5]*c[6]+24*c[6]-20+24*c[5]), a [6,1] = -1/6*c[6]*(-5100*c[6]*c[2]*c[5]*c[3]^4+12*c[6]^3*c[2]-60*c[5]* c[3]^3*c[6]+225*c[6]^2*c[3]^4*c[5]^2+204*c[3]^2*c[6]^2*c[2]-675*c[5]^2 *c[2]*c[3]^5+60*c[6]^2*c[3]^3*c[5]-18*c[2]*c[6]^2*c[3]+495*c[2]*c[3]^5 *c[5]-660*c[5]*c[2]*c[3]^4+870*c[5]^2*c[2]*c[3]^4+24*c[6]^2*c[2]*c[5]^ 2-28*c[6]^2*c[2]*c[5]-405*c[5]^2*c[2]*c[3]^3+307*c[5]*c[2]*c[3]^3-50*c [5]*c[2]*c[3]^2+66*c[5]^2*c[2]*c[3]^2+120*c[2]*c[3]^4*c[6]-225*c[5]^2* c[3]^4*c[6]+135*c[5]*c[3]^4*c[6]+90*c[5]^2*c[3]^3*c[6]-135*c[6]^2*c[3] ^4*c[5]-90*c[6]^2*c[3]^3*c[5]^2-27*c[3]^4*c[5]+45*c[5]^2*c[3]^4+12*c[3 ]^3*c[5]-18*c[5]^2*c[3]^3-1890*c[6]*c[2]*c[5]^2*c[3]^3+3375*c[6]*c[2]* c[5]^2*c[3]^6-300*c[3]*c[6]^2*c[2]*c[5]^2-1870*c[3]^2*c[6]^2*c[2]*c[5] +435*c[3]^2*c[6]*c[2]*c[5]^2+2140*c[3]^3*c[6]*c[2]*c[5]+54*c[2]*c[5]*c [6]*c[3]+4650*c[6]*c[2]*c[5]^2*c[3]^4+1350*c[6]^2*c[2]*c[3]^6+2190*c[6 ]^2*c[2]*c[3]^4-2700*c[6]^2*c[2]*c[3]^5-90*c[2]*c[3]^5*c[6]-930*c[3]^3 *c[6]^2*c[2]+6*c[2]*c[6]*c[3]^2-48*c[2]*c[3]^3*c[6]-3375*c[6]*c[2]*c[5 ]*c[3]^6-485*c[3]^2*c[6]*c[2]*c[5]+352*c[3]*c[6]^2*c[2]*c[5]+3375*c[6] ^2*c[2]*c[5]*c[3]^5-5850*c[6]*c[2]*c[5]^2*c[3]^5-4425*c[6]^2*c[2]*c[5] ^2*c[3]^3+1650*c[6]^2*c[2]*c[5]^2*c[3]^2+4875*c[6]^2*c[2]*c[5]*c[3]^3+ 5850*c[6]^2*c[2]*c[5]^2*c[3]^4+6300*c[6]*c[2]*c[5]*c[3]^5-3375*c[6]^2* c[2]*c[5]^2*c[3]^5-6300*c[6]^2*c[2]*c[5]*c[3]^4-54*c[2]*c[5]^2*c[6]*c[ 3]-2100*c[6]^3*c[2]*c[3]^3+2700*c[6]^3*c[2]*c[3]^4-1350*c[6]^3*c[2]*c[ 3]^5+810*c[3]^2*c[6]^3*c[2]-156*c[3]*c[6]^3*c[2])/c[5]/c[3]^2/(15*c[5] *c[3]-9*c[3]+4-6*c[5])/c[2]/(1-5*c[3]+5*c[3]^2), a[7,1] = -1/6*(94320* c[6]*c[2]*c[5]*c[3]^4-30*c[5]*c[3]^3*c[6]+135*c[6]^2*c[3]^4*c[5]^2+570 6*c[3]^2*c[6]^2*c[2]-108*c[2]*c[6]-21150*c[5]^2*c[2]*c[3]^5-12870*c[2] *c[3]^5+42*c[6]^2*c[3]^3*c[5]-1008*c[5]^2*c[2]*c[3]+72*c[2]*c[5]^2+405 0*c[2]*c[3]^6-9450*c[2]*c[6]*c[3]^6+3666*c[3]^2*c[2]-10494*c[3]^3*c[2] +16260*c[2]*c[3]^4-1008*c[2]*c[6]^2*c[3]+72*c[6]^2*c[2]+34020*c[2]*c[3 ]^5*c[5]-42540*c[5]*c[2]*c[3]^4+26280*c[5]^2*c[2]*c[3]^4+260*c[2]*c[5] *c[6]+120*c[6]^2*c[2]*c[5]^2-168*c[6]^2*c[2]*c[5]-168*c[6]*c[2]*c[5]^2 -16650*c[5]^2*c[2]*c[3]^3+27144*c[5]*c[2]*c[3]^3-9372*c[5]*c[2]*c[3]^2 +5706*c[5]^2*c[2]*c[3]^2+1668*c[2]*c[5]*c[3]-37530*c[2]*c[3]^4*c[6]+67 50*c[5]^2*c[2]*c[3]^6-10800*c[5]*c[2]*c[3]^6+48*c[2]-90*c[5]^2*c[3]^4* c[6]+63*c[5]*c[3]^4*c[6]+42*c[5]^2*c[3]^3*c[6]-90*c[6]^2*c[3]^4*c[5]-6 0*c[6]^2*c[3]^3*c[5]^2-660*c[3]*c[2]-120*c[5]*c[2]+39960*c[6]*c[2]*c[5 ]^2*c[3]^3-16875*c[6]*c[2]*c[5]^2*c[3]^6-1710*c[3]*c[6]^2*c[2]*c[5]^2- 13545*c[3]^2*c[6]^2*c[2]*c[5]-13545*c[3]^2*c[6]*c[2]*c[5]^2-59782*c[3] ^3*c[6]*c[2]*c[5]-3626*c[2]*c[5]*c[6]*c[3]-63840*c[6]*c[2]*c[5]^2*c[3] ^4+6750*c[6]^2*c[2]*c[3]^6+26280*c[6]^2*c[2]*c[3]^4-21150*c[6]^2*c[2]* c[3]^5+29880*c[2]*c[3]^5*c[6]-16650*c[3]^3*c[6]^2*c[2]-8352*c[2]*c[6]* c[3]^2+24066*c[2]*c[3]^3*c[6]+1494*c[2]*c[6]*c[3]+24300*c[6]*c[2]*c[5] *c[3]^6+20512*c[3]^2*c[6]*c[2]*c[5]+2364*c[3]*c[6]^2*c[2]*c[5]+52200*c [6]^2*c[2]*c[5]*c[3]^5+13500*c[6]^2*c[2]*c[5]^2*c[3]^6-16875*c[6]^2*c[ 2]*c[5]*c[3]^6+52200*c[6]*c[2]*c[5]^2*c[3]^5-30030*c[6]^2*c[2]*c[5]^2* c[3]^3+9990*c[6]^2*c[2]*c[5]^2*c[3]^2+39960*c[6]^2*c[2]*c[5]*c[3]^3+48 930*c[6]^2*c[2]*c[5]^2*c[3]^4-76050*c[6]*c[2]*c[5]*c[3]^5-40950*c[6]^2 *c[2]*c[5]^2*c[3]^5-63840*c[6]^2*c[2]*c[5]*c[3]^4+2364*c[2]*c[5]^2*c[6 ]*c[3])/c[6]/c[5]/c[3]^2/c[2]/(300*c[5]*c[3]^3*c[6]-225*c[3]^3*c[6]+18 0*c[3]^3-225*c[3]^3*c[5]-455*c[5]*c[6]*c[3]^2+350*c[6]*c[3]^2-285*c[3] ^2+350*c[5]*c[3]^2+210*c[5]*c[6]*c[3]-165*c[6]*c[3]+136*c[3]-165*c[5]* c[3]-30*c[5]*c[6]+24*c[6]-20+24*c[5])\}:" }}}{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 9 "Examples:" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 71 "for ii in [2,3,5] do print(a[ii,1]= subs(e15,a[ii,1]));print(``) end do:" }}{PARA 11 "" 1 "" {XPPMATH 20 " 6#/&%\"aG6$\"\"#\"\"\"&%\"cG6#F'" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%! G" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/&%\"aG6$\"\"$\"\"\",$*&#F(\"\"#F (*(&%\"cG6#F'F(,&F.F(*&F,F(&F/6#F,F(!\"\"F(F3F5F(F5" }}{PARA 11 "" 1 " " {XPPMATH 20 "6#%!G" }}{PARA 12 "" 1 "" {XPPMATH 20 "6#/&%\"aG6$\"\"& \"\"\",$*&#F(\"\"'F(*,&%\"cG6#F'F(,B**\"$D#F(&F/6#\"\"#F()&F/6#\"\"$F' F(F.F(F(**\"$+$F(F.F(F4F()F8\"\"%F(!\"\"*(\"#XF(F4F(F=F(F?**F3F()F.F6F (F4F(F=F(F?*(\"#:F(FCF()F8F:F(F(*&F:F(FFF(F(*(FEF(FFF(F.F(F?**FF(F.F(F4F(FFF(F(*(\"#SF(FFF(F4F(F(**\"#]F(F.F(F4F()F8 F6F(F?*(\"#6F(FPF(F4F(F?**\"$v\"F(FCF(F4F(FPF(F?**\"\"*F(F4F(F.F(F8F(F (**FMF(FCF(F4F(F8F(F(*(F>F(F4F(FCF(F?F(F8!\"#F4F?,(F(F(*&F'F(F8F(F?*&F 'F(FPF(F(F?F(F?" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%!G" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 85 "RK6_7eq s := [op(RowSumConditions(7,'expanded')),op(OrderConditions(6,7,'expan ded'))]:" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 7 "Check: " }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 140 "ind := []:\nf or ct to nops(RK6_7eqs) do\n eq := simplify(subs(e15,RK6_7eqs[ct])); \n ind := [op(ind),lhs(eq)-rhs(eq)];\nend do:\nind;\nnops(%);" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#7M\"\"!F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F $F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#\"#V" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 " " 0 "" {TEXT -1 10 "Examples: " }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 20 "c[4]=subs(e17,c[4]);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/&%\"cG6 #\"\"%*&&F%6#\"\"$\"\"\",(*&\"#:F,)F)\"\"#F,F,*&\"#5F,F)F,!\"\"F1F,F4 " }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 24 "a[5,4]=subs(e17,a[5,4]);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/&%\"aG6$\"\"&\"\"%,$*&#\"\"\"\"\"'F,*0,(*&\"#:F,)&%\"c G6#\"\"$\"\"#F,F,*&\"#5F,F3F,!\"\"F7F,F7&F46#F'F,,**&F7F,F;F,F,*(F9F,F ;F,F3F,F:*(F1F,F;F,F2F,F,F3F:F,,&F;F,F3F:F,,(F,F,*&F'F,F3F,F:*&F'F,F2F ,F,F:,(*&F1F,F2F,F,*&F9F,F3F,F:F,F,F:F3!\"#F,F:" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 24 "a[6,2]=subs( e17,a[6,2]);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/&%\"aG6$\"\"'\"\"#,$* &#\"\"\"F(F,*,&%\"cG6#\"\"$F,&F/6#F'F,,(*&\"\"&F,)F2F(F,F,*&F6F,F2F,! \"\"F,F,F,,(F,F,*&F6F,F.F,F9*&F6F,)F.F(F,F,F9&F/6#F(F9F,F," }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 20 "b[ 6]=subs(e17,b[6]);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/&%\"bG6#\"\"',$ *&#\"\"\"\"#gF+*0,(F+F+*&\"\"&F+&%\"cG6#\"\"$F+!\"\"*&F0F+)F1\"\"#F+F+ F+,**&\"\"*F+F1F+F5*(\"#:F+&F26#F0F+F1F+F+\"\"%F+*&F'F+F>F+F5F+&F2F&F5 ,&F+F5FBF+F5,**&F8F+FBF+F+*(F=F+FBF+F7F+F+*(\"#5F+F1F+FBF+F5F1F5F5,&FB F5F1F+F5,&FBF5F>F+F5F+F5" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 20 "printed coefficients" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 357 "for ii from 2 to 7 do\n if ii<>4 and ii<>7 then print(c[ii]*` is a parameter`)\n else print(c[ii]=su bs(e15,c[ii]));\n end if;print(``); \n for jj to ii-1 do\n pr int(a[ii,jj]=subs(e15,a[ii,jj]));\n end do:\n print(``);print(`*** *********************************`);\nend do:print(``);\nfor ii to 7 d o\n print(b[ii]=subs(e15,b[ii]));\nend do:print(``);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#*&&%\"cG6#\"\"#\"\"\"%0~is~a~parameterGF(" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#%!G" }}{PARA 11 "" 1 "" {XPPMATH 20 "6 #/&%\"aG6$\"\"#\"\"\"&%\"cG6#F'" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%!G " }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%E******************************** ****G" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#*&&%\"cG6#\"\"$\"\"\"%0~is~a~ parameterGF(" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%!G" }}{PARA 11 "" 1 " " {XPPMATH 20 "6#/&%\"aG6$\"\"$\"\"\",$*&#F(\"\"#F(*(&%\"cG6#F'F(,&F.F (*&F,F(&F/6#F,F(!\"\"F(F3F5F(F5" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/&% \"aG6$\"\"$\"\"#,$*&#\"\"\"F(F,*&&%\"cG6#F'F(&F/6#F(!\"\"F,F," }} {PARA 11 "" 1 "" {XPPMATH 20 "6#%!G" }}{PARA 11 "" 1 "" {XPPMATH 20 "6 #%E************************************G" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/&%\"cG6#\"\"%*&&F%6#\"\"$\"\"\",(*&\"#:F,)F)\"\"#F,F,*&\"#5F,F) F,!\"\"F1F,F4" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%!G" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/&%\"aG6$\"\"%\"\"\",$*&#F(\"\"#F(**&%\"cG6#\"\"$F(, 2*&F'F(F.F(!\"\"*&\"#IF()F.F,F(F(*&\"#XF()F.F1F(F4*(\"$]$F(F7F(&F/6#F, F(F(*(\"$+\"F(F.F(F=F(F4*&\"#5F(F=F(F(*(\"$]%F(F=F()F.F'F(F(*(\"$+'F(F :F(F=F(F4F(F=F4,(*&\"#:F(F7F(F(*&FBF(F.F(F4F,F(!\"$F(F(" }}{PARA 11 " " 1 "" {XPPMATH 20 "6#/&%\"aG6$\"\"%\"\"#,$*&#\"\"\"F(F,**&%\"cG6#\"\" $F(,(F'F,*&\"#IF,F.F,!\"\"*&\"#XF,)F.F(F,F,F,&F/6#F(F5,(*&\"#:F,F8F,F, *&\"#5F,F.F,F5F(F,!\"$F,F," }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/&%\"aG6 $\"\"%\"\"$,$*(,(*&\"#:\"\"\")&%\"cG6#F(\"\"#F.F.*&\"#5F.F0F.!\"\"F.F. F.F0F.,(*&F-F.F/F.F.*&F5F.F0F.F6F3F.!\"$F6" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%!G" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%E************** **********************G" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#*&&%\"cG6# \"\"&\"\"\"%0~is~a~parameterGF(" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%!G " }}{PARA 12 "" 1 "" {XPPMATH 20 "6#/&%\"aG6$\"\"&\"\"\",$*&#F(\"\"'F( *,&%\"cG6#F'F(,B**\"$D#F(&F/6#\"\"#F()&F/6#\"\"$F'F(F.F(F(**\"$+$F(F.F (F4F()F8\"\"%F(!\"\"*(\"#XF(F4F(F=F(F?**F3F()F.F6F(F4F(F=F(F?*(\"#:F(F CF()F8F:F(F(*&F:F(FFF(F(*(FEF(FFF(F.F(F?**FF(F .F(F4F(FFF(F(*(\"#SF(FFF(F4F(F(**\"#]F(F.F(F4F()F8F6F(F?*(\"#6F(FPF(F4 F(F?**\"$v\"F(FCF(F4F(FPF(F?**\"\"*F(F4F(F.F(F8F(F(**FMF(FCF(F4F(F8F(F (*(F>F(F4F(FCF(F?F(F8!\"#F4F?,(F(F(*&F'F(F8F(F?*&F'F(FPF(F(F?F(F?" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#/&%\"aG6$\"\"&\"\"#,$*&#\"\"\"F(F,*,&% \"cG6#\"\"$F,&F/6#F'F,&F/6#F(!\"\",(*&F'F,F2F,F6F,F,*&F'F,)F2F(F,F,F,, (F,F,*&F'F,F.F,F6*&F'F,)F.F(F,F,F6F,F," }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/&%\"aG6$\"\"&\"\"$,$*&#\"\"\"\"\"'F,*.&%\"cG6#F'F,,0*(\"$X\"F,F /F,)&F06#F(\"\"#F,!\"\"*(\"$]\"F,)F6F(F,F/F,F,*(\"#SF,F/F,F6F,F,*&\"\" %F,F/F,F9*&\"#?F,F5F,F,*&\"#IF,FF(F3F(F(*(\"#7F(F@F(F8F(F(** \"#gF(F8F(F@F(F-F(F;**\"$N\"F(F8F(F3F(F-F(F(**FFF()F-F?F(F@F(F8F(F(** \"#!*F(F>F(F@F(F-F(F(**\"$D#F(F>F(F3F(F-F(F;**FNF(FJF(F3F(F>F(F(**FHF( FJF(F3F(F8F(F;**FLF(FJF(F@F(F>F(F;*,\"%]YF(F-F(&F.6#F?F(F>F(F3F(F(*,\" #aF(FTF(F8F(F-F(F4F(F(*,\"%S@F(F@F(F-F(FTF(F8F(F(*,\"$N%F()F4F?F(F-F(F TF(F>F(F(*,\"%q=F(FfnF(FJF(FTF(F8F(F;*,\"$+$F(F4F(FJF(FTF(F>F(F;*,\"%v LF(F-F(FTF(F>F()F4F'F(F(*,\"%!*=F(F-F(FTF(F>F(F@F(F;*,\"%+^F(F-F(FTF(F 8F(F3F(F;*,F\\oF(F-F(FTF(F8F(F]oF(F;*,\"%+jF(F-F(FTF(F8F()F4F:F(F(*,\" %]eF(FJF(FTF(F>F(F3F(F(*,\"%v[F(FJF(FTF(F8F(F@F(F(*,\"%];F(FJF(FTF(F>F (FfnF(F(*,\"%DWF(FJF(FTF(F>F(F@F(F;*,FgoF(F-F(FTF(F>F(FeoF(F;*,F\\oF(F JF(FTF(F8F(FeoF(F(*,\"$_$F(F4F(FJF(FTF(F8F(F(*,\"$&[F(FfnF(F-F(FTF(F8F (F;*,FWF(FTF(F>F(F-F(F4F(F;*,FdoF(FJF(FTF(F8F(F3F(F;*,F\\oF(FJF(FTF(F> F(FeoF(F;**\"$&\\F(FTF(FeoF(F8F(F(**\"$g'F(F8F(FTF(F3F(F;**\"$q)F(F>F( FTF(F3F(F(**\"$0%F(F>F(FTF(F@F(F;**\"$2$F(F8F(FTF(F@F(F(**\"#]F(F8F(FT F(FfnF(F;**\"#mF(F>F(FTF(FfnF(F(*(FDF()F-F6F(FTF(F(**\"$/#F(FfnF(FJF(F TF(F(**\"$v'F(F>F(FTF(FeoF(F;**F=F(FTF(FJF(F4F(F;**\"#CF(FJF(FTF(F>F(F (**\"#GF(FJF(FTF(F8F(F;**\"$?\"F(FTF(F3F(F-F(F(**\"%]8F(FJF(FTF(F]oF(F (**\"%!>#F(FJF(FTF(F3F(F(**\"%+FF(FJF(FTF(FeoF(F;**FLF(FTF(FeoF(F-F(F; **\"$I*F(F@F(FJF(FTF(F;**F'F(FTF(F-F(FfnF(F(**\"#[F(FTF(F@F(F-F(F;**\" %+@F(FfqF(FTF(F@F(F;**FgrF(FfqF(FTF(F3F(F(**FcrF(FfqF(FTF(FeoF(F;**\"$ 5)F(FfnF(FfqF(FTF(F(**\"$c\"F(F4F(FfqF(FTF(F;F(F8F;F4!\"#,**(\"#:F(F8F (F4F(F(*&\"\"*F(F4F(F;F7F(*&F'F(F8F(F;F;FTF;,(F(F(*&F:F(F4F(F;*&F:F(Ff nF(F(F;F(F;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/&%\"aG6$\"\"'\"\"#,$*& #\"\"\"F(F,*,&%\"cG6#F'F,&F/6#\"\"$F,&F/6#F(!\"\",(*&\"\"&F,)F.F(F,F,F ,F,*&F9F,F.F,F6F,,(F,F,*&F9F,F1F,F6*&F9F,)F1F(F,F,F6F,F," }}{PARA 12 " " 1 "" {XPPMATH 20 "6#/&%\"aG6$\"\"'\"\"$,$*&#\"\"\"F'F,*2&%\"cG6#F'F, ,&&F/6#F(F,F.!\"\"F,,bo*(\"$N\"F,)&F/6#\"\"&\"\"#F,)F2F*(\"#!*F,)F2F5F,F:F,F,*(\"$D#F,FDF,&F36#\"\"&F,F9*(FFF,)FGF6F, FDF,F,*(\"$?\"F,F:F,F1F,F9*&\"\"*F,F1F,F,*(\"$S#F,FKF,F1F,F9*(\"$b#F,F GF,F1F,F,*(\"#[F,F:F,F2F,F,*(\"$+\"F,FGF,F2F,F9*&F(F,F2F,F9*(FCF,FKF,F 2F,F,*&F'F,F:F,F9*&\"#7F,FKF,F9*&\"#9F,FGF,F,F,,**(F0F,FGF,F1F,F,*(F8F ,FGF,F2F,F9*&F6F,FGF,F,F2F9F9,(*&F0F,F1F,F,*&F8F,F2F,F9F,F,F9F2!\"#,(F ,F,*&FIF,F2F,F9*&FIF,F1F,F,F9,**(F0F,FGF,F2F,F,*&FOF,F2F,F9F(F,*&F'F,F GF,F9F9F,F," }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/&%\"aG6$\"\"'\"\"&,$*4 ,&*&\"\"$\"\"\"&%\"cG6#F-F.F.F.!\"\"F.,**&\"\"#F.&F06#F'F.F.*(\"#5F.F6 F.F/F.F2*(\"#:F.F6F.)F/F5F.F.F/F2F.,&F/F.F6F2F.,&F6F.&F06#F(F2F.F6F.,* *(F;F.F?F.F/F.F.*&\"\"*F.F/F.F2\"\"%F.*&F'F.F?F.F2F2,&F?F2F/F.F2,**(F; F.F?F.FF(F(**\"#UF()F>F4F(F:F(F7F(F(**FFF()F7F4F(F:F(F>F(F(**\" #!*F(FIF(FCF(F>F(F@**\"$N\"F(FGF(FCF(FIF(F(**FKF(FGF(FCF(F7F(F@**\"#gF (FGF(F:F(FIF(F@**\"%%\\\"F(F1F(F>F(F;F(F(*,\"&SQ'F(F>F(F1F(FIF(FCF(F@* ,\"%EOF(F1F(F7F(F>F(F;F(F@*,\"&#yfF(F:F(F>F(F1F(F7F(F@*,\"&XN\"F()F;F4 F(F>F(F1F(FIF(F@*,FZF(FenF(FGF(F1F(F7F(F@*,\"%5F(F1F(FIF()F;F,F(F@*,\"&g*RF(F>F(F1F(FIF(F:F(F(*,\"&?V*F(F> F(F1F(F7F(FCF(F(*,\"&+V#F(F>F(F1F(F7F(F[oF(F(*,\"&]g(F(F>F(F1F(F7F()F; F9F(F@*,\"&I*[F(FGF(F1F(FIF(FCF(F(*,F]oF(FGF(F1F(F7F(F:F(F(*,\"%!***F( FGF(F1F(FIF(FenF(F(*,\"&I+$F(FGF(F1F(FIF(F:F(F@*,\"&+A&F(F>F(F1F(FIF(F doF(F(*,F]pF(FGF(F1F(F7F(FdoF(F(*,\"%kBF(F;F(FGF(F1F(F7F(F(*,\"&70#F(F enF(F>F(F1F(F7F(F(*,F`pF(F1F(FIF(F>F(F;F(F(*,FTF(FGF(F1F(F7F(FCF(F@*, \"&]4%F(FGF(F1F(FIF(FdoF(F@*,FjnF(FGF(F1F(F7F(F[oF(F@*,\"&+N\"F(FGF(F1 F(FIF(F[oF(F(*(\"$?\"F(F7F(F1F(F@**\"&?S$F(F1F(FdoF(F7F(F(**\"&SD%F(F7 F(F1F(FCF(F@**\"&!GEF(FIF(F1F(FCF(F(**\"&]m\"F(FIF(F1F(F:F(F@**\"&Wr#F (F7F(F1F(F:F(F(**\"%s$*F(F7F(F1F(FenF(F@**\"%1dF(FIF(F1F(FenF(F(*(\"&g i\"F(F1F(FCF(F(*(\"&%\\5F(F:F(F1F(F@*(\"#sF(F1F(FIF(F(*(\"%mOF(FenF(F1 F(F(**\"%o;F(F1F(F7F(F;F(F(**\"%35F(FIF(F1F(F;F(F@*(\"$g'F(F;F(F1F(F@* *\"%]%*F(F1F(F>F(F[oF(F@**\"%]nF(FIF(F1F(F[oF(F(**\"&+3\"F(F7F(F1F(F[o F(F@*(\"%]SF(F1F(F[oF(F(*(F_rF(FGF(F1F(F(**FiqF(FenF(FGF(F1F(F(**\"&]6 #F(FIF(F1F(FdoF(F@**FerF(F1F(FGF(F;F(F@**F[qF(FGF(F1F(FIF(F(**\"$o\"F( FGF(F1F(F7F(F@**\"$g#F(F1F(F7F(F>F(F(**\"&Iv$F(F1F(FCF(F>F(F@*(\"$3\"F (F1F(F>F(F@*(\"&qG\"F(F1F(FdoF(F@**F[sF(FGF(F1F(F[oF(F(**FaqF(FGF(F1F( FCF(F(**FcsF(FGF(F1F(FdoF(F@**\"&!))HF(F1F(FdoF(F>F(F(**FcqF(F:F(FGF(F 1F(F@**\"%_$)F(F1F(F>F(FenF(F@**\"&mS#F(F1F(F:F(F>F(F(**FgsF(F>F(F1F(F IF(F@F(F>F@F7F@F;!\"#F1F@,B**\"$+$F(F7F(F:F(F>F(F(*(\"$D#F(F:F(F>F(F@* &\"$!=F(F:F(F(*(F`uF(F:F(F7F(F@**\"$b%F(F7F(F>F(FenF(F@*(\"$]$F(F>F(Fe nF(F(*&\"$&GF(FenF(F@*(FguF(F7F(FenF(F(**\"$5#F(F7F(F>F(F;F(F(*(\"$l\" F(F>F(F;F(F@*&\"$O\"F(F;F(F(*(F^vF(F7F(F;F(F@*(F6F(F7F(F>F(F@*&\"#CF(F >F(F(\"#?F@*&FdvF(F7F(F(F@F(F@" }}{PARA 12 "" 1 "" {XPPMATH 20 "6#/&% \"aG6$\"\"(\"\"#,$*&#\"\"\"F(F,**&%\"cG6#\"\"$F,,2*&\"#9F,&F/6#\"\"'F, F,*(\"#?F,&F/6#\"\"&F,F5F,!\"\"*(\"#IF,F5F,F.F,F=**\"#XF,F:F,F5F,F.F,F ,*&\"#@F,F.F,F,\"#5F=*&F4F,F:F,F,*(F?F,F:F,F.F,F=F,&F/6#F(F=,B**\"$+$F ,F:F,)F.F1F,F5F,F,*(\"$D#F,FLF,F5F,F=*&\"$!=F,FLF,F,*(FNF,FLF,F:F,F=** \"$b%F,F:F,F5F,)F.F(F,F=*(\"$]$F,F5F,FTF,F,*&\"$&GF,FTF,F=*(FVF,F:F,FT F,F,**\"$5#F,F:F,F5F,F.F,F,*(\"$l\"F,F5F,F.F,F=*&\"$O\"F,F.F,F,*(FgnF, F:F,F.F,F=*(F?F,F:F,F5F,F=*&\"#CF,F5F,F,F9F=*&F]oF,F:F,F,F=F,F," }} {PARA 12 "" 1 "" {XPPMATH 20 "6#/&%\"aG6$\"\"(\"\"$,$*&#\"\"\"\"\"'F,* 0,&&%\"cG6#F(F,F,!\"\"F,,^r\"#[F,*(\"%aOF,)&F16#\"\"&\"\"#F,)F0FF,FXF,F0F ,F9F,F,**FiqF,F8F,FKF,FPF,F,**F[rF,F8F,F=F,FPF,F3**F]rF,F8F,FPF,F0F,F, **\"%?hF,FXF,F8F,F=F,F,**\"%]))F,F8F,FGF,FPF,F3**\"%]wF,FXF,FGF,F8F,F, **FdrF,FXF,FGF,F9F,F3**FCF,F8F,FDF,FPF,F3**FCF,FVF,FPF,F9F,F,**\"&+9\" F,FXF,FKF,F8F,F3**\"%59F,FXF,F8F,F0F,F3F,,B**\"$+$F,F9F,FKF,FPF,F,*(\" $D#F,FKF,FPF,F3*&\"$!=F,FKF,F,*(FbsF,FKF,F9F,F3**\"$b%F,F9F,FPF,F=F,F3 *(\"$]$F,FPF,F=F,F,*&\"$&GF,F=F,F3*(FisF,F9F,F=F,F,**\"$5#F,F9F,FPF,F0 F,F,*(\"$l\"F,FPF,F0F,F3*&\"$O\"F,F0F,F,*(F`tF,F9F,F0F,F3*(\"#IF,F9F,F PF,F3*&\"#CF,FPF,F,\"#?F3*&FgtF,F9F,F,F3,&F0F,FPF3F3,&F9F3F0F,F3,(*&\" #:F,F=F,F,*&\"#5F,F0F,F3F,F,F3F0!\"#F,F," }}{PARA 12 "" 1 "" {XPPMATH 20 "6#/&%\"aG6$\"\"(\"\"%,$*&#\"\"\"\"\"'F,*6,&*&\"\"$F,&%\"cG6#F1F,F, F,!\"\"F,,&F2F,F,F5F,,&*&\"\"&F,F2F,F,\"\"#F5F,,hq\"#[F5*(\"%qTF,)&F36 #F9F:F,)F2F:F,F5*&\"$e&F,F2F,F,*&\"%yEF,FBF,F5*(\"%]nF,)F2F9F,F?F,F,*( \"&+;#F,)F2F(F,F@F,F,*(\"&S/\"F,F?F,)F2F1F,F,*&\"$?\"F,F@F,F,*&\"$3\"F ,&F36#F-F,F,*&\"%wlF,FOF,F,*(FQF,)FTF:F,F?F,F5*(\"&5)=F,FLF,FTF,F,*(\" &+3\"F,FIF,F@F,F5*(\"%]%*F,FIF,FTF,F5**\"&+V#F,FIF,FTF,F@F,F,**\"&vo\" F,FIF,FYF,F@F,F5*(\"&vK\"F,F?F,FLF,F5*(FHF,FIF,FYF,F,*(\"$_)F,F?F,F2F, F,*(\"$o\"F,F?F,FTF,F,*&\"%]SF,FIF,F,*&\"%!>)F,FLF,F5*&\"#sF,F?F,F5*&F joF,FYF,F5*(\"&b\\\"F,FOF,FTF,F5*(\"&vr\"F,FOF,F@F,F5*(\"%_gF,FTF,FBF, F,*(\"%5pF,F@F,FBF,F,*(\"%c7F,FTF,F2F,F5*(\"%=9F,F@F,F2F,F5*(\"$g#F,F@ F,FTF,F5*(F_oF,FYF,FLF,F5*(FNF,FYF,FOF,F,*(F>F,FYF,FBF,F5*(FboF,FYF,F2 F,F,*(FdoF,FYF,F@F,F,**\"&?q$F,F@F,FOF,FTF,F,**\"&5[\"F,F@F,FTF,FBF,F5 **\"%WIF,F@F,FTF,F2F,F,**\"&]s%F,F@F,FLF,FTF,F5**\"&+Y#F,FYF,FOF,F@F,F 5**\"%?(*F,FYF,FBF,F@F,F,**\"%!)>F,FYF,F2F,F@F,F5**FhqF,F?F,FOF,FTF,F5 **FjqF,F?F,FBF,FTF,F,**F\\rF,F?F,FTF,F2F,F5**\"&+N\"F,FIF,FYF,F?F,F,** \"%+sF,FYF,F?F,FBF,F5**\"&]>$F,F?F,FLF,FTF,F,**\"&]Z#F,FYF,FLF,F?F,F5* *FerF,FYF,FLF,F@F,F,**F]oF,F?F,FIF,FTF,F5**\"&+'=F,FYF,FOF,F?F,F,**\"% S9F,FYF,F?F,F2F,F,F,,(*&\"#:F,FBF,F,*&\"#5F,F2F,F5F:F,F:,B**\"$+$F,F@F ,FOF,FTF,F,*(\"$D#F,FOF,FTF,F5*&\"$!=F,FOF,F,*(FgsF,FOF,F@F,F5**\"$b%F ,F@F,FTF,FBF,F5*(\"$]$F,FTF,FBF,F,*&\"$&GF,FBF,F5*(F^tF,F@F,FBF,F,**\" $5#F,F@F,FTF,F2F,F,*(\"$l\"F,FTF,F2F,F5*&\"$O\"F,F2F,F,*(FetF,F@F,F2F, F5*(\"#IF,F@F,FTF,F5*&\"#CF,FTF,F,\"#?F5*&F\\uF,F@F,F,F5,**&F:F,FTF,F, *(FbsF,FTF,F2F,F5*(F`sF,FTF,FBF,F,F2F5F5,(*&F`sF,FBF,F,*&FbsF,F2F,F5F, F,F5,**(F`sF,F@F,FBF,F,*(FbsF,F@F,F2F,F5*&F:F,F@F,F,F2F5F5F2!\"#F,F," }}{PARA 12 "" 1 "" {XPPMATH 20 "6#/&%\"aG6$\"\"(\"\"&*8,&&%\"cG6#\"\"$ \"\"\"F/!\"\"F/,&*&F.F/F+F/F/F/F0F/,&*&F(F/F+F/F/\"\"#F0F/,(F/F/*&F(F/ F+F/F0*&F(F/)F+F5F/F/F/,&&F,6#F(F/F/F0F/,2F;F/*(F.F/F;F/F+F/F0*(\"#:F/ )&F,6#\"\"'F5F/F+F/F/*(\"#@F/FBF/F+F/F0*&FDF/FAF/F0*&\"\"*F/F+F/F/\"\" %F0*&FIF/FBF/F/F/F;F0,**(F@F/F;F/F9F/F/*(\"#5F/F;F/F+F/F0*&F5F/F;F/F/F +F0F0,&F;F0F+F/F0,&FBF/F;F0F0,B**\"$+$F/F;F/)F+F.F/FBF/F/*(\"$D#F/FVF/ FBF/F0*&\"$!=F/FVF/F/*(FXF/FVF/F;F/F0**\"$b%F/F;F/FBF/F9F/F0*(\"$]$F/F BF/F9F/F/*&\"$&GF/F9F/F0*(FinF/F;F/F9F/F/**\"$5#F/F;F/FBF/F+F/F/*(\"$l \"F/FBF/F+F/F0*&\"$O\"F/F+F/F/*(F`oF/F;F/F+F/F0*(\"#IF/F;F/FBF/F0*&\"# CF/FBF/F/\"#?F0*&FgoF/F;F/F/F0" }}{PARA 12 "" 1 "" {XPPMATH 20 "6#/&% \"aG6$\"\"(\"\"',$*:,**(\"#:\"\"\"&%\"cG6#\"\"&F.&F06#\"\"$F.F.*&\"\"* F.F3F.!\"\"\"\"%F.*&F(F.F/F.F8F.,(F.F.*&F2F.F3F.F8*&F2F.)F3\"\"#F.F.F. ,&F3F.F.F8F.,&F.F8&F06#F(F.F.,&F/F.F.F8F.,&*&F5F.F3F.F.F.F8F.,&*&F2F.F 3F.F.F?F8F.FBF8,**&F?F.FBF.F.*(\"#5F.FBF.F3F.F8*(F-F.FBF.F>F.F.F3F8F8, &F3F.FBF8F8,&FBF.F/F8F8,B**\"$+$F.F/F.)F3F5F.FBF.F.*(\"$D#F.FSF.FBF.F8 *&\"$!=F.FSF.F.*(FUF.FSF.F/F.F8**\"$b%F.F/F.FBF.F>F.F8*(\"$]$F.FBF.F>F .F.*&\"$&GF.F>F.F8*(FfnF.F/F.F>F.F.**\"$5#F.F/F.FBF.F3F.F.*(\"$l\"F.FB F.F3F.F8*&\"$O\"F.F3F.F.*(F]oF.F/F.F3F.F8*(\"#IF.F/F.FBF.F8*&\"#CF.FBF .F.\"#?F8*&FdoF.F/F.F.F8F8" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%!G" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#%E************************************ G" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%!G" }}{PARA 12 "" 1 "" {XPPMATH 20 "6#/&%\"bG6#\"\"\",$*&#F'\"#gF'**&%\"cG6#\"\"$!\"#,B**\"$]\"F'&F.6# \"\"&F')F-F0F'&F.6#\"\"'F'F'*(\"#vF'F8F'F9F'!\"\"*&\"#XF'F8F'F'*(F=F'F 8F'F5F'F>*(\"$0\"F'F5F')F-\"\"#F'F'**\"$0#F'F5F'F9F'FDF'F>*(FCF'F9F'FD F'F'*&\"#lF'FDF'F>*&\"#HF'F-F'F'*(F@F'F5F'F-F'F>**\"#!)F'F5F'F9F'F-F'F '*(F@F'F9F'F-F'F>*&F;F'F5F'F'*(\"#5F'F5F'F9F'F>\"\"%F>*&F;F'F9F'F'F'F5 F>F9F>F'F>" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/&%\"bG6#\"\"#\"\"!" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#/&%\"bG6#\"\"$,$*&#\"\"\"\"#gF+*.,:** \"#vF+&%\"cG6#\"\"&F+&F26#\"\"'F+)&F2F&\"\"#F+F+**F,F+F1F+F5F+F9F+!\" \"*(\"#5F+F1F+F5F+F+*(\"#XF+F1F+F8F+F<*(\"#NF+F1F+F9F+F+*&F7F+F1F+F<*( F@F+F5F+F8F+F<*(FBF+F5F+F9F+F+*&F7F+F5F+F<*&\"#IF+F8F+F+*&\"#BF+F9F+F< \"\"%F+F+,&F9F+F+FF+F9F +FF+F2F+F+*(F:F+FBF+F>F+F9*&F5F+FBF+F+F6F9*&F5F+F>F+F+F+F2!\"#,* *(F0F+FBF+F1F+F+*(F8F+FBF+F2F+F9*&F6F+FBF+F+F2F9F9,(*&F0F+F1F+F+*&F8F+ F2F+F9F+F+F9,&*&F5F+F2F+F+F+F9F9,&*&F:F+F2F+F+F6F9F9,**&F6F+F>F+F+*(F8 F+F>F+F2F+F9*(F0F+F>F+F1F+F+F2F9F9F+F+" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/&%\"bG6#\"\"&,$*&#\"\"\"\"#gF+*0,(F+F+*&F'F+&%\"cG6#\"\"$F+!\" \"*&F'F+)F0\"\"#F+F+F+,**&\"\"*F+F0F+F4*(\"#:F+&F16#\"\"'F+F0F+F+\"\"% F+*&F?F+F=F+F4F+,&F=F+&F1F&F4F4FCF4,&FCF+F+F4F4,**(F**\"$b%F+F1F+F9F+)F6\"\"#F+F>*(\"$]$F+F9F+FDF+F +*&\"$&GF+FDF+F>*(FGF+F1F+FDF+F+**\"$5#F+F1F+F9F+F6F+F+*(\"$l\"F+F9F+F 6F+F>*&\"$O\"F+F6F+F+*(FNF+F1F+F6F+F>*(\"#IF+F1F+F9F+F>*&\"#CF+F9F+F+ \"#?F>*&FUF+F1F+F+F+,&F+F>F9F+F>,&F1F+F+F>F>,&F6F+F+F>F>,&*&F8F+F6F+F+ F+F>F>,&*&F4F+F6F+F+FEF>F>F+F+" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%!G " }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 " " {TEXT 260 4 "Note" }{TEXT -1 2 ": " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 54 "(1) The stability function is determine d entirely by " }{XPPEDIT 18 0 "c[3]" "6#&%\"cG6#\"\"$" }{TEXT -1 1 " ." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 58 "map(simplify,subs(e15,StabilityFunction(6,7,'expanded ')));" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#,2\"\"\"F$%\"zGF$*&\"\"#!\"\" F%F'F$*&\"\"'F(F%\"\"$F$*&\"#CF(F%\"\"%F$*&\"$?\"F(F%\"\"&F$*&\"$?(F(F %F*F$*&#F$F3F$**,&*&F+F$&%\"cG6#F+F$F$F$F(F$,(*&\"#:F$)F9F'F$F$*&\"#5F $F9F$F(F'F$F(F9F$F%\"\"(F$F(" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }} {PARA 0 "" 0 "" {TEXT 260 43 "A strategy for constructing order 6 sche mes" }{TEXT -1 21 " is to first choose " }{XPPEDIT 18 0 "c[3]" "6#&% \"cG6#\"\"$" }{TEXT -1 56 " to give a desirable stability region and \+ then choose " }{XPPEDIT 18 0 "c[2]" "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 39 " to minimize the principal error norm." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 75 "(2) The first principal error term is given in abre viated form as follows." }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 26 "P rincipalErrorTerms(6)[1];" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#,&*&%\"bG \"\"\"-%!G6#*&%\"aGF&-F(6#*&F+F&-F(6#*&F+F&-F(6#*&F+F&-F(6#*&F+F&%\"cG F&F&F&F&F&F&F&#F&\"%S]!\"\"" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 70 "For the general order 6 scheme this first term \+ error depends only on " }{XPPEDIT 18 0 "c[3]" "6#&%\"cG6#\"\"$" } {TEXT -1 1 "." }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 67 "factor(simp lify(subs(e15,PrincipalErrorTerms(6,7,'expanded')[1])));" }}{PARA 11 " " 1 "" {XPPMATH 20 "6#,$*&#\"\"\"\"%S]F&*(,&*&\"\"*F&&%\"cG6#\"\"$F&F& \"\"#!\"\"F&,&*&\"\"%F&F,F&F&F&F1F&,(*&\"#:F&)F,F0F&F&*&\"#5F&F,F&F1F0 F&F1F&F1" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 45 "The first principal error term is zero when " }{XPPEDIT 18 0 "c[3 ]=2/9" "6#/&%\"cG6#\"\"$*&\"\"#\"\"\"\"\"*!\"\"" }{TEXT -1 76 " which means that the scheme has order 7 for linear differential equations. " }}{PARA 0 "" 0 "" {TEXT -1 38 "In this case the stability function i s" }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "Sum(z^i/i!,i=0.. 7)=1+1/2" "6#/-%$SumG6$*&)%\"zG%\"iG\"\"\"-%*factorialG6#F*!\"\"/F*;\" \"!\"\"(,&F+F+*&F+F+\"\"#F/F+" }{TEXT -1 1 " " }{XPPEDIT 18 0 "z^2+1/6 " "6#,&*$%\"zG\"\"#\"\"\"*&F'F'\"\"'!\"\"F'" }{TEXT -1 1 " " } {XPPEDIT 18 0 "z^3+1/24" "6#,&*$%\"zG\"\"$\"\"\"*&F'F'\"#C!\"\"F'" } {TEXT -1 1 " " }{XPPEDIT 18 0 "z^4+1/120" "6#,&*$%\"zG\"\"%\"\"\"*&F'F '\"$?\"!\"\"F'" }{TEXT -1 1 " " }{XPPEDIT 18 0 "z^5+1/720" "6#,&*$%\"z G\"\"&\"\"\"*&F'F'\"$?(!\"\"F'" }{TEXT -1 1 " " }{XPPEDIT 18 0 "z^6+1/ 5040" "6#,&*$%\"zG\"\"'\"\"\"*&F'F'\"%S]!\"\"F'" }{TEXT -1 1 " " } {XPPEDIT 18 0 "z^7" "6#*$%\"zG\"\"(" }{TEXT -1 1 "." }}{PARA 0 "" 0 " " {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 73 "subs(c[3]= 2/9,map(simplify,subs(e15,StabilityFunction(6,7,'expanded'))));" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#,2\"\"\"F$%\"zGF$*&#F$\"\"#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%F-F$F$F$*&#F$\"%S]F$*$)F%\"\"(F$F$F $" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 56 "#-- -----------------------------------------------------" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}}{PARA 0 "" 0 "" {TEXT 270 33 "_____ ____________________________" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 44 "#===========================================" } }{SECT 1 {PARA 4 "" 0 "" {TEXT -1 48 "Step by step construction of But cher's scheme B " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 62 "See: On Runge-Kutta Processes of High Order, by J. C. But cher," }}{PARA 0 "" 0 "" {TEXT -1 88 " Journal of the Australian Mathematical Society, Vol. 4, (1964), pages 179 to 194." }}{PARA 0 " " 0 "" {TEXT -1 87 "-------------------------------------------------- -------------------------------------" }}{PARA 0 "" 0 "" {TEXT -1 67 " The first complete solution suggested by Butcher involves setting " } {XPPEDIT 18 0 "c[3]=2/3" "6#/&%\"cG6#\"\"$*&\"\"#\"\"\"F'!\"\"" } {TEXT -1 3 ". " }}{PARA 0 "" 0 "" {TEXT -1 19 "Using the relation " } }{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "c[3]/(15*c[3]^2-10*c [3]+2)" "6#*&&%\"cG6#\"\"$\"\"\",(*&\"#:F(*$&F%6#F'\"\"#F(F(*&\"#5F(&F %6#F'F(!\"\"F/F(F4" }{TEXT -1 2 " " }}{PARA 0 "" 0 "" {TEXT -1 12 "th is gives " }{XPPEDIT 18 0 "c[4]=1/3" "6#/&%\"cG6#\"\"%*&\"\"\"F)\"\"$ !\"\"" }{TEXT -1 1 "." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 47 "c[4]=subs(c[3]=2/3,c[3]/(15*c[3]^2-10*c[3 ]+2));" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/&%\"cG6#\"\"%#\"\"\"\"\"$" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 30 "Then B utcher suggests taking " }{XPPEDIT 18 0 "c[5]=c[6]" "6#/&%\"cG6#\"\"& &F%6#\"\"'" }{TEXT -1 37 " in the alternative order condition " }} {PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "b[5]*(1-c[5])*(c[5]-c [6])*a[5,4]*(c[4]-c[3])*c[4] = 1/90-c[3]/40-c[6]/60+c[3]*c[6]/24" "6#/ *.&%\"bG6#\"\"&\"\"\",&F)F)&%\"cG6#F(!\"\"F),&&F,6#F(F)&F,6#\"\"'F.F)& %\"aG6$F(\"\"%F),&&F,6#F8F)&F,6#\"\"$F.F)&F,6#F8F),**&F)F)\"#!*F.F)*&& F,6#F>F)\"#SF.F.*&&F,6#F4F)\"#gF.F.*(&F,6#F>F)&F,6#F4F)\"#CF.F)" } {TEXT -1 2 ". " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 11 "which gives" }}{PARA 256 "" 0 "" {TEXT -1 1 " " } {XPPEDIT 18 0 "1/90-c[3]/40-c[6]/60+c[3]*c[6]/24=0" "6#/,**&\"\"\"F&\" #!*!\"\"F&*&&%\"cG6#\"\"$F&\"#SF(F(*&&F+6#\"\"'F&\"#gF(F(*(&F+6#F-F&&F +6#F2F&\"#CF(F&\"\"!" }{TEXT -1 1 "." }}{PARA 0 "" 0 "" {TEXT -1 9 "so that " }{XPPEDIT 18 0 "c[5]=c[6]" "6#/&%\"cG6#\"\"&&F%6#\"\"'" } {XPPEDIT 18 0 "``=1/2" "6#/%!G*&\"\"\"F&\"\"#!\"\"" }{TEXT -1 2 ". " } }{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 71 "subs(c[3]=2/3,1/90-c[3]/40-c[6]/60+c[3]*c[6]/24=0);\nc[6]=solve( %,c[6]);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/,&#\"\"\"\"$!=!\"\"*&#F& \"#!*F&&%\"cG6#\"\"'F&F&\"\"!" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/&%\" cG6#\"\"'#\"\"\"\"\"#" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 31 " Butcher also suggests taking " }{XPPEDIT 18 0 "c[2 ] = 1/3;" "6#/&%\"cG6#\"\"#*&\"\"\"F)\"\"$!\"\"" }{TEXT -1 25 ". In \+ addition we have " }{XPPEDIT 18 0 "c[7]=1" "6#/&%\"cG6#\"\"(\"\"\"" } {TEXT -1 7 " and " }{XPPEDIT 18 0 "b[2]=0" "6#/&%\"bG6#\"\"#\"\"!" } {TEXT -1 1 "." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 15 "" 0 "" {TEXT 279 6 "Step 1" }{TEXT -1 1 " " }}{PARA 0 "" 0 "" {TEXT -1 33 "We use the quadrature conditions:" }}{PARA 256 "" 0 "" {TEXT -1 1 " " } {XPPEDIT 18 0 "Sum(b[i],i = 1 .. 7) = 1;" "6#/-%$SumG6$&%\"bG6#%\"iG/F *;\"\"\"\"\"(F-" }{TEXT -1 15 ", " }{XPPEDIT 18 0 "Sum(b[ i]*c[i]^(k-1),i = 2 .. 7) = 1/k;" "6#/-%$SumG6$*&&%\"bG6#%\"iG\"\"\")& %\"cG6#F+,&%\"kGF,F,!\"\"F,/F+;\"\"#\"\"(*&F,F,F2F3" }{TEXT -1 7 ", \+ " }{XPPEDIT 18 0 "k = 2;" "6#/%\"kG\"\"#" }{TEXT -1 8 " . . 6, " }} {PARA 0 "" 0 "" {TEXT -1 21 "to find the weights " }{XPPEDIT 18 0 "b[ 1]" "6#&%\"bG6#\"\"\"" }{TEXT -1 2 ", " }{XPPEDIT 18 0 "b[3]" "6#&%\"b G6#\"\"$" }{TEXT -1 3 ", " }{XPPEDIT 18 0 "b[4];" "6#&%\"bG6#\"\"%" } {TEXT -1 3 ", " }{XPPEDIT 18 0 "b[5];" "6#&%\"bG6#\"\"&" }{TEXT -1 3 ", " }{XPPEDIT 18 0 "b[6]" "6#&%\"bG6#\"\"'" }{TEXT -1 3 ", " } {XPPEDIT 18 0 "b[7]" "6#&%\"bG6#\"\"(" }{TEXT -1 1 "." }}{PARA 0 "" 0 "" {TEXT -1 32 "In the current situation where " }{XPPEDIT 18 0 "c[5] =c[6]" "6#/&%\"cG6#\"\"&&F%6#\"\"'" }{TEXT -1 117 " the expessions f or the weights given in the general solution end up have zero in the d enominator for the weights " }{XPPEDIT 18 0 "b[5]" "6#&%\"bG6#\"\"&" }{TEXT -1 7 " and " }{XPPEDIT 18 0 "b[6]" "6#&%\"bG6#\"\"'" }{TEXT -1 1 "." }}{PARA 0 "" 0 "" {TEXT -1 62 "The problem disappears if we i ncorporate the extra condition " }{XPPEDIT 18 0 "b[5]=b[6]" "6#/&%\"b G6#\"\"&&F%6#\"\"'" }{TEXT -1 40 " along with the quadrature conditio ns. " }}{PARA 0 "" 0 "" {TEXT -1 66 "This ensures that we obtain Butch er's values for the two weights " }{XPPEDIT 18 0 "b[5]" "6#&%\"bG6#\" \"&" }{TEXT -1 7 " and " }{XPPEDIT 18 0 "b[6]" "6#&%\"bG6#\"\"'" } {TEXT -1 31 " along with the other weights." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 224 "e1 := \{b[2]=0,c[ 2]=1/3,c[3]=2/3,c[4]=1/3,c[5]=1/2,c[6]=1/2,c[7]=1\}:\nquad_cdns := [ad d(b[i],i=1..7)=1,seq(add(b[i]*c[i]^(k-1),i=2..7)=1/k,k=2..6)]:\nquad_e qns := subs(e1,quad_cdns):\nnops(quad_eqns);\nindets(quad_eqns);\nnops (%);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#\"\"'" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#<(&%\"bG6#\"\"&&F%6#\"\"'&F%6#\"\"(&F%6#\"\"\"&F%6#\"\" $&F%6#\"\"%" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#\"\"'" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 252 "e1 := \{ b[2]=0,c[2]=1/3,c[3]=2/3,c[4]=1/3,c[5]=1/2,c[6]=1/2,c[7]=1\}:\nquad_cd ns := [add(b[i],i=1..7)=1,seq(add(b[i]*c[i]^(k-1),i=2..7)=1/k,k=2..6)] :\nquad_eqns := subs(e1,quad_cdns):\nmatrix(ListTools[Enumerate](quad_ eqns));\n``;\nindets(quad_eqns);\nnops(%);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#K%'matrixG6#7(7$\"\"\"/,.&%\"bG6#F(F(&F,6#\"\"$F(&F,6# \"\"%F(&F,6#\"\"&F(&F,6#\"\"'F(&F,6#\"\"(F(F(7$\"\"#/,,*&#F>F0F(F.F(F( *&#F(F0F(F1F(F(*&#F(F>F(F4F(F(*&FFF(F7F(F(F:F(FF7$F0/,,*&#F3\"\"*F(F.F (F(*&#F(FMF(F1F(F(*&#F(F3F(F4F(F(*&FQF(F7F(F(F:F(FD7$F3/,,*&#\"\")\"#F F(F.F(F(*&#F(FYF(F1F(F(*&#F(FXF(F4F(F(*&FgnF(F7F(F(F:F(FQ7$F6/,,*&#\"# ;\"#\")F(F.F(F(*&#F(F_oF(F1F(F(*&#F(F^oF(F4F(F(*&FcoF(F7F(F(F:F(#F(F67 $F9/,,*&#\"#K\"$V#F(F.F(F(*&#F(F\\pF(F1F(F(*&#F(F[pF(F4F(F(*&F`pF(F7F( F(F:F(#F(F9Q(pprint76\"" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%!G" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#<(&%\"bG6#\"\"%&F%6#\"\"(&F%6#\"\"\"&F %6#\"\"&&F%6#\"\"'&F%6#\"\"$" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#\"\"' " }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 93 "e2 := solve(\{op(quad_eqns),b[5]=b[6]\},\{b[1],b[3],b [4],b[5],b[6],b[7]\});\ne3 := `union`(e1,e2);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%#e2G<(/&%\"bG6#\"\"\"#\"#6\"$?\"/&F(6#\"\"$#\"#F\"#S/ &F(6#\"\"(F+/&F(6#\"\"%F2/&F(6#\"\"'#!\"%\"#:/&F(6#\"\"&FA" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%#e3G " 0 "" {MPLTEXT 1 0 1 ";" }}}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 2 "e3" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 173 "e 3 := \{b[1] = 11/120, b[3] = 27/40, b[7] = 11/120, c[3] = 2/3, b[4] = \+ 27/40, b[6] = -4/15, b[5] = -4/15, c[6] = 1/2, c[5] = 1/2, c[4] = 1/3, c[7] = 1, b[2] = 0, c[2] = 1/3\}:" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" } }{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 41 "The weights of the scheme are as follows." }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 31 "seq(b[i] =subs(e3,b[i]),i=1..7);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6)/&%\"bG6#\" \"\"#\"#6\"$?\"/&F%6#\"\"#\"\"!/&F%6#\"\"$#\"#F\"#S/&F%6#\"\"%F4/&F%6# \"\"&#!\"%\"#:/&F%6#\"\"'F?/&F%6#\"\"(F(" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 15 "" 0 "" {TEXT -1 1 " " }{TEXT 280 6 "Step 2" }{TEXT -1 1 " " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 51 "We can obtain values for the linking coefficients " }{XPPEDIT 18 0 "a[5,4]" "6#&%\"aG6$\"\"&\"\"%" }{TEXT -1 4 " , " }{XPPEDIT 18 0 "a [6,5]" "6#&%\"aG6$\"\"'\"\"&" }{TEXT -1 7 " and " }{XPPEDIT 18 0 "a[ 6,4]" "6#&%\"aG6$\"\"'\"\"%" }{TEXT -1 57 " using the following three alternative order conditions:" }}{PARA 256 "" 0 "" {TEXT -1 1 " " } {XPPEDIT 18 0 "b[6]*(1-c[6])*(a[6,5]*c[5]*(c[5]-c[3])+a[6,4]*c[4]*(c[4 ]-c[3]))+b[5]*(1-c[5])*a[5,4]*c[4]*(c[4]-c[3])=1/60-c[3]/24" "6#/,&*(& %\"bG6#\"\"'\"\"\",&F*F*&%\"cG6#F)!\"\"F*,&*(&%\"aG6$F)\"\"&F*&F-6#F5F *,&&F-6#F5F*&F-6#\"\"$F/F*F**(&F36$F)\"\"%F*&F-6#FAF*,&&F-6#FAF*&F-6#F =F/F*F*F*F**,&F'6#F5F*,&F*F*&F-6#F5F/F*&F36$F5FAF*&F-6#FAF*,&&F-6#FAF* &F-6#F=F/F*F*,&*&F*F*\"#gF/F**&&F-6#F=F*\"#CF/F/" }{TEXT -1 2 ", " }} {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 256 "" 0 "" {TEXT -1 1 " " } {XPPEDIT 18 0 "b[6]*(1-c[6])*a[6,5]*c[5]*(c[5]-c[3])*(c[5]-c[4])=1/120 -(c[3]+c[4])/60+c[3]*c[4]/24" "6#/*.&%\"bG6#\"\"'\"\"\",&F)F)&%\"cG6#F (!\"\"F)&%\"aG6$F(\"\"&F)&F,6#F2F),&&F,6#F2F)&F,6#\"\"$F.F),&&F,6#F2F) &F,6#\"\"%F.F),(*&F)F)\"$?\"F.F)*&,&&F,6#F:F)&F,6#F@F)F)\"#gF.F.*(&F,6 #F:F)&F,6#F@F)\"#CF.F)" }{TEXT -1 2 ", " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "b[6]*(1-c[6])*a[ 6,5]*a[5,4]*c[4]*(c[4]-c[3])=1/360-c[3]/120" "6#/*.&%\"bG6#\"\"'\"\"\" ,&F)F)&%\"cG6#F(!\"\"F)&%\"aG6$F(\"\"&F)&F06$F2\"\"%F)&F,6#F5F),&&F,6# F5F)&F,6#\"\"$F.F),&*&F)F)\"$g$F.F)*&&F,6#F=F)\"$?\"F.F." }{TEXT -1 3 ". " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 385 "cdns1 := [b[6]*(1-c[6])*(a[6,5]*c[5]*(c[5]-c[3])+a[6 ,4]*c[4]*(c[4]-c[3]))+\n b[5]*(1-c[5])*a[5,4]*c[4]*(c[4]-c[3])=1/ 60-c[3]/24,\n b[6]*(1-c[6])*a[6,5]*c[5]*(c[5]-c[3])*(c[5]-c[4])=1/12 0-(c[3]+c[4])/60+c[3]*c[4]/24,\n b[6]*(1-c[6])*a[6,5]*a[5,4]*c[4]*(c [4]-c[3])=1/360-c[3]/120]:\neqns1 := simplify(subs(e3,cdns1)):\nmatrix (ListTools[Enumerate](eqns1));\n``;\nindets(eqns1);\nnops(%);" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#K%'matrixG6#7%7$\"\"\"/,(*&#F(\"#!*F(& %\"aG6$\"\"'\"\"&F(F(*&#\"\"#\"$N\"F(&F/6$F1\"\"%F(F(*&F4F(&F/6$F2F9F( F(#!\"\"F-7$F5/,$*&#F(\"$S&F(F.F(F(#F(\"%!3\"7$\"\"$/,$*&F4F(*&F.F(F;F (F(F(#F>\"$g$Q(pprint86\"" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%!G" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#<%&%\"aG6$\"\"'\"\"%&F%6$\"\"&F(&F%6$F 'F+" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#\"\"$" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 140 "e4 := solve (\{op(eqns1)\},\{a[5,4],a[6,4],a[6,5]\}):\ne5 := `union`(e3,e4):\na[5, 4]=subs(e5,a[5,4]),a[6,4]=subs(e5,a[6,4]),a[6,5]=subs(e5,a[6,5]);" }} {PARA 11 "" 1 "" {XPPMATH 20 "6%/&%\"aG6$\"\"&\"\"%#!\"$\"\")/&F%6$\" \"'F(#F*F(/&F%6$F/F'#\"\"\"\"\"#" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}{SECT 1 {PARA 4 "" 0 " " {TEXT -1 2 "e5" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> \+ " 0 "" {MPLTEXT 1 0 217 "e5 := \{b[1] = 11/120, b[3] = 27/40, b[7] = 1 1/120, c[3] = 2/3, b[4] = 27/40, b[6] = -4/15, b[5] = -4/15, a[6,4] = \+ -3/4, c[6] = 1/2, a[5,4] = -3/8, c[5] = 1/2, c[4] = 1/3, c[7] = 1, b[2 ] = 0, c[2] = 1/3, a[6,5] = 1/2\}:" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" } }{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 15 "" 0 "" {TEXT -1 1 " " }{TEXT 284 6 "Step 3" }{TEXT -1 1 " " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 18 "We can determine " }{XPPEDIT 18 0 "a[3,2]" "6#&%\"aG6$\" \"$\"\"#" }{TEXT -1 32 " from the stage-order condition" }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "a[3,2]*c[2] = 1/2" "6#/*&&%\"aG 6$\"\"$\"\"#\"\"\"&%\"cG6#F)F**&F*F*F)!\"\"" }{TEXT -1 1 " " } {XPPEDIT 18 0 "c[3]^2" "6#*$&%\"cG6#\"\"$\"\"#" }{TEXT -1 1 "." }} {PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 113 "a[3,2]*c[2]=1/2*c[3]^2:\nsubs(e5,%);\ne6 := \{a[3,2]=solve(%,a[3, 2])\}:\ne7 := `union`(e5,e6):\na[3,2]=subs(e7,a[3,2]);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/,$*&#\"\"\"\"\"$F'&%\"aG6$F(\"\"#F'F'#F,\"\"*" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#/&%\"aG6$\"\"$\"\"##F(F'" }}}{PARA 0 " " 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}} {SECT 1 {PARA 4 "" 0 "" {TEXT -1 2 "e7" }}{PARA 0 "" 0 "" {TEXT -1 0 " " }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 231 "e7 := \{b[2] = 0, c[7] \+ = 1, c[3] = 2/3, c[4] = 1/3, b[3] = 27/40, b[7] = 11/120, b[4] = 27/40 , b[6] = -4/15, a[6,4] = -3/4, b[5] = -4/15, c[6] = 1/2, a[5,4] = -3/8 , c[5] = 1/2, a[6,5] = 1/2, c[2] = 1/3, b[1] = 11/120, a[3,2] = 2/3\}: " }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 15 "" 0 " " {TEXT -1 1 " " }{TEXT 285 6 "Step 4" }{TEXT -1 1 " " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 41 "We can now determine t he 6 coefficients " }{XPPEDIT 18 0 "a[4,2]" "6#&%\"aG6$\"\"%\"\"#" } {TEXT -1 3 ", " }{XPPEDIT 18 0 "a[5, 2];" "6#&%\"aG6$\"\"&\"\"#" } {TEXT -1 3 ", " }{XPPEDIT 18 0 "a[6, 2];" "6#&%\"aG6$\"\"'\"\"#" } {TEXT -1 3 ", " }{XPPEDIT 18 0 "a[4,3];" "6#&%\"aG6$\"\"%\"\"$" } {TEXT -1 3 ", " }{XPPEDIT 18 0 "a[5,3];" "6#&%\"aG6$\"\"&\"\"$" } {TEXT -1 3 ", " }{XPPEDIT 18 0 "a[6,3]" "6#&%\"aG6$\"\"'\"\"$" } {TEXT -1 56 " by using the three alternative simplifying conditions: " }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "Sum(b[i]*(1-c[i]) *a[i,2],i=3..6)=0" "6#/-%$SumG6$*(&%\"bG6#%\"iG\"\"\",&F,F,&%\"cG6#F+! \"\"F,&%\"aG6$F+\"\"#F,/F+;\"\"$\"\"'\"\"!" }{TEXT -1 2 ", " }}{PARA 0 "" 0 "" {TEXT -1 8 "that is," }}{PARA 256 "" 0 "" {TEXT -1 1 " " } {XPPEDIT 18 0 "b[3]*(1-c[3])*a[3,2]+b[4]*(1-c[4])*a[4,2]+b[5]*(1-c[5]) *a[5,2]+b[6]*(1-c[6])*a[6,2] = 0" "6#/,**(&%\"bG6#\"\"$\"\"\",&F*F*&% \"cG6#F)!\"\"F*&%\"aG6$F)\"\"#F*F**(&F'6#\"\"%F*,&F*F*&F-6#F7F/F*&F16$ F7F3F*F**(&F'6#\"\"&F*,&F*F*&F-6#F@F/F*&F16$F@F3F*F**(&F'6#\"\"'F*,&F* F*&F-6#FIF/F*&F16$FIF3F*F*\"\"!" }{TEXT -1 2 ", " }}{PARA 0 "" 0 "" {TEXT -1 3 "and" }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "Su m(b[i]*(1-c[i])*Sum(a[i,j]*a[j,2],j = 3 .. i-1),i = 3 .. 6) = 0" "6#/- %$SumG6$*(&%\"bG6#%\"iG\"\"\",&F,F,&%\"cG6#F+!\"\"F,-F%6$*&&%\"aG6$F+% \"jGF,&F66$F8\"\"#F,/F8;\"\"$,&F+F,F,F1F,/F+;F>\"\"'\"\"!" }{TEXT -1 2 ", " }}{PARA 0 "" 0 "" {TEXT -1 8 "that is," }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "b[4]*(1-c[4])*a[4,3]*a[3,2]+b[5]*(1-c[5 ])*(a[5,3]*a[3,2]+a[5,4]*a[4,2])+b[6]*(1-c[6])*(a[6,3]*a[3,2]+a[6,4]*a [4,2]+a[6,5]*a[5,2]) = 0" "6#/,(**&%\"bG6#\"\"%\"\"\",&F*F*&%\"cG6#F)! \"\"F*&%\"aG6$F)\"\"$F*&F16$F3\"\"#F*F**(&F'6#\"\"&F*,&F*F*&F-6#F:F/F* ,&*&&F16$F:F3F*&F16$F3F6F*F**&&F16$F:F)F*&F16$F)F6F*F*F*F**(&F'6#\"\"' F*,&F*F*&F-6#FLF/F*,(*&&F16$FLF3F*&F16$F3F6F*F**&&F16$FLF)F*&F16$F)F6F *F**&&F16$FLF:F*&F16$F:F6F*F*F*F*\"\"!" }{TEXT -1 2 ", " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 " Sum(b[i]*(c[i]-1)*(c[i]-c[6])*a[i,2],i = 3 .. 5) = 0;" "6#/-%$SumG6$** &%\"bG6#%\"iG\"\"\",&&%\"cG6#F+F,F,!\"\"F,,&&F/6#F+F,&F/6#\"\"'F1F,&% \"aG6$F+\"\"#F,/F+;\"\"$\"\"&\"\"!" }{TEXT -1 2 ", " }}{PARA 0 "" 0 " " {TEXT -1 9 "that is, " }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "b[3]*(c[3]-1)*(c[3]-c[6])*a[3,2]+b[4]*(c[4]-1)*(c[4]-c[6])*a[4,2 ]+b[5]*(c[5]-1)*(c[5]-c[6])*a[5,2]=0" "6#/,(**&%\"bG6#\"\"$\"\"\",&&% \"cG6#F)F*F*!\"\"F*,&&F-6#F)F*&F-6#\"\"'F/F*&%\"aG6$F)\"\"#F*F***&F'6# \"\"%F*,&&F-6#F=F*F*F/F*,&&F-6#F=F*&F-6#F5F/F*&F76$F=F9F*F***&F'6#\"\" &F*,&&F-6#FKF*F*F/F*,&&F-6#FKF*&F-6#F5F/F*&F76$FKF9F*F*\"\"!" }{TEXT -1 2 ", " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 48 "together with the three stage-order conditions: " }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "Sum(a[4,j]*c[j],j = 2 .. 3) = 1/2; " "6#/-%$SumG6$*&&%\"aG6$\"\"%%\"jG\"\"\"&%\"cG6#F,F-/F,;\"\"#\"\"$*&F -F-F3!\"\"" }{TEXT -1 1 " " }{XPPEDIT 18 0 "c[4]^2;" "6#*$&%\"cG6#\"\" %\"\"#" }{TEXT -1 8 " , " }{XPPEDIT 18 0 "Sum(a[5,j]*c[j],j=2..4) =1/2" "6#/-%$SumG6$*&&%\"aG6$\"\"&%\"jG\"\"\"&%\"cG6#F,F-/F,;\"\"#\"\" %*&F-F-F3!\"\"" }{TEXT -1 1 " " }{XPPEDIT 18 0 "c[5]^2" "6#*$&%\"cG6# \"\"&\"\"#" }{TEXT -1 2 " " }}{PARA 0 "" 0 "" {TEXT -1 4 "and " }} {PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "Sum(a[6,j]*c[j],j=2.. 5)=1/2" "6#/-%$SumG6$*&&%\"aG6$\"\"'%\"jG\"\"\"&%\"cG6#F,F-/F,;\"\"#\" \"&*&F-F-F3!\"\"" }{TEXT -1 1 " " }{XPPEDIT 18 0 "c[6]^2" "6#*$&%\"cG6 #\"\"'\"\"#" }{TEXT -1 1 "." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 291 "c dns2 := [add(b[i]*(1-c[i])*a[i,2],i=3..6)=0,add(b[i]*(1-c[i])*add(a[i, j]*a[j,2],j=3..i-1),i=3..6)=0,\n add(b[i]*(1-c[i])*(c[6]-c[i])*a[i,2] ,i=3..5)=0,seq(add(a[i,j]*c[j],j=2..i-1)=1/2*c[i]^2,i=4..6)]:\neqns2 : = subs(e7,cdns2):\nmatrix(ListTools[Enumerate](eqns2));\n``;\nindets(e qns2);\nnops(%);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#K%'matrixG6#7(7$\" \"\"/,*#\"\"$\"#?F(*&#\"\"*F-F(&%\"aG6$\"\"%\"\"#F(F(*&#F5\"#:F(&F26$ \"\"&F5F(!\"\"*&#F5F8F(&F26$\"\"'F5F(F<\"\"!7$F5/,,*&#F,\"#5F(&F26$F4F ,F(F(*&#F4\"#XF(&F26$F;F,F(F<*&F+F(F1F(F(*&#F4FMF(&F26$FAF,F(F<*&#F(F8 F(F9F(F " 0 "" {MPLTEXT 1 0 164 "e8 := solve(\{op(eqns2)\}, \{a[4,2],a[5,2],a[6,2],a[4,3],a[5,3],a[6,3]\}):\ne9 := `union`(e7,e8): \nseq(a[i,2]=subs(e9,a[i,2]),i=4..6);\nseq(a[i,3]=subs(e9,a[i,3]),i=4. .6);\n" }}{PARA 11 "" 1 "" {XPPMATH 20 "6%/&%\"aG6$\"\"%\"\"##\"\"\"\" \"$/&F%6$\"\"&F(#\"\"*\"\")/&F%6$\"\"'F(F0" }}{PARA 11 "" 1 "" {XPPMATH 20 "6%/&%\"aG6$\"\"%\"\"$#!\"\"\"#7/&F%6$\"\"&F(#!\"$\"#;/&F% 6$\"\"'F(#F1\"\")" }}}{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 0 320 "e9 := \{b[1] = 11/120, a[5,2] = 9/8, b[3] = 27/40, b[7] = 11/ 120, a[4,3] = -1/12, a[4,2] = 1/3, c[3] = 2/3, b[4] = 27/40, a[3,2] = \+ 2/3, b[6] = -4/15, b[5] = -4/15, a[6,4] = -3/4, c[6] = 1/2, a[5,4] = - 3/8, c[5] = 1/2, c[4] = 1/3, c[7] = 1, b[2] = 0, a[6,3] = -3/8, c[2] = 1/3, a[6,5] = 1/2, a[5,3] = -3/16, a[6,2] = 9/8\}:" }}}{PARA 0 "" 0 " " {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}} {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 15 "" 0 "" {TEXT -1 1 " " } {TEXT 281 6 "Step 5" }{TEXT -1 1 " " }}{PARA 0 "" 0 "" {TEXT -1 0 "" } }{PARA 0 "" 0 "" {TEXT -1 39 "The four column simplifying conditions \+ " }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "Sum(b[i]*a[i,j],i =j+1..7)=b[j]*(1-c[j])" "6#/-%$SumG6$*&&%\"bG6#%\"iG\"\"\"&%\"aG6$F+% \"jGF,/F+;,&F0F,F,F,\"\"(*&&F)6#F0F,,&F,F,&%\"cG6#F0!\"\"F," }{TEXT -1 5 ", " }{XPPEDIT 18 0 "j = 2;" "6#/%\"jG\"\"#" }{TEXT -1 10 ", 4 , 5, 6 " }}{PARA 0 "" 0 "" {TEXT -1 19 "give the equations " }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "b[3]*a[3,2]+b[4]*a[4,2]+b[5 ]*a[5,2]+b[6]*a[6,2]+b[7]*a[7,2] = b[2]*(1-c[2])" "6#/,,*&&%\"bG6#\"\" $\"\"\"&%\"aG6$F)\"\"#F*F**&&F'6#\"\"%F*&F,6$F2F.F*F**&&F'6#\"\"&F*&F, 6$F8F.F*F**&&F'6#\"\"'F*&F,6$F>F.F*F**&&F'6#\"\"(F*&F,6$FDF.F*F**&&F'6 #F.F*,&F*F*&%\"cG6#F.!\"\"F*" }{TEXT -1 2 ", " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "b[5]* a[5,4]+b[6]*a[6,4]+b[7]*a[7,4] = b[4]*(1-c[4])" "6#/,(*&&%\"bG6#\"\"& \"\"\"&%\"aG6$F)\"\"%F*F**&&F'6#\"\"'F*&F,6$F2F.F*F**&&F'6#\"\"(F*&F,6 $F8F.F*F**&&F'6#F.F*,&F*F*&%\"cG6#F.!\"\"F*" }{TEXT -1 2 ", " }}{PARA 257 "" 0 "" {TEXT -1 0 "" }}{PARA 256 "" 0 "" {TEXT -1 1 " " } {XPPEDIT 18 0 "b[6]*a[6,5]+b[7]*a[7,5] = b[5]*(1-c[5])" "6#/,&*&&%\"bG 6#\"\"'\"\"\"&%\"aG6$F)\"\"&F*F**&&F'6#\"\"(F*&F,6$F2F.F*F**&&F'6#F.F* ,&F*F*&%\"cG6#F.!\"\"F*" }{TEXT -1 2 ", " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "b[7]*a[7,6] = \+ b[6]*(1-c[6])" "6#/*&&%\"bG6#\"\"(\"\"\"&%\"aG6$F(\"\"'F)*&&F&6#F-F),& F)F)&%\"cG6#F-!\"\"F)" }{TEXT -1 2 ", " }}{PARA 0 "" 0 "" {TEXT -1 26 "from which the values of " }{XPPEDIT 18 0 "a[7,2]" "6#&%\"aG6$\"\"( \"\"#" }{TEXT -1 3 ", " }{XPPEDIT 18 0 "a[7,4]" "6#&%\"aG6$\"\"(\"\"% " }{TEXT -1 3 ", " }{XPPEDIT 18 0 "a[7,5]" "6#&%\"aG6$\"\"(\"\"&" } {TEXT -1 7 " and " }{XPPEDIT 18 0 "a[7,6]" "6#&%\"aG6$\"\"(\"\"'" } {TEXT -1 18 " can be obtained." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 158 "cdns3 := [seq(add(b[i]*a[i, j],i=j+1..7)=b[j]*(1-c[j]),j=[2,4,5,6])]:\neqns3 := subs(e9,cdns3):\nm atrix(ListTools[Enumerate](eqns3));\n``;\nindets(eqns3);\nnops(%);" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#K%'matrixG6#7&7$\"\"\"/,&#\"\"$\"#SF(* &#\"#6\"$?\"F(&%\"aG6$\"\"(\"\"#F(F(\"\"!7$F6/,&#F,\"#5F(*&F/F(&F36$F5 \"\"%F(F(#\"\"*\"#?7$F,/,&#F6\"#:!\"\"*&F/F(&F36$F5\"\"&F(F(#!\"#FH7$F @/,$*&F/F(&F36$F5\"\"'F(F(FNQ)pprint106\"" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%!G" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#<&&%\"aG6$\"\"( \"\"'&F%6$F'\"\"&&F%6$F'\"\"%&F%6$F'\"\"#" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#\"\"%" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 122 "e10 := solve(\{op(eqns3)\},\{a[7,2],a[7, 4],a[7,5],a[7,6]\}):\ne11 := `union`(e9,e10):\nseq(a[7,j]=subs(e11,a[7 ,j]),j=[2,4,5,6]);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6&/&%\"aG6$\"\"(\" \"##!\"*\"#6/&F%6$F'\"\"%#\"#=F+/&F%6$F'\"\"&\"\"!/&F%6$F'\"\"'#!#;F+ " }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 3 "e11" }} {PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 382 "e11 := \{b[1] = 11/120, a[5,2] = 9/8, b[3] = 27/40, b[7] = 11/120 , a[4,3] = -1/12, a[4,2] = 1/3, c[3] = 2/3, b[4] = 27/40, a[3,2] = 2/3 , b[6] = -4/15, b[5] = -4/15, a[6,4] = -3/4, c[6] = 1/2, a[5,4] = -3/8 , c[5] = 1/2, a[7,5] = 0, c[4] = 1/3, a[7,6] = -16/11, c[7] = 1, b[2] \+ = 0, a[7,4] = 18/11, a[7,2] = -9/11, a[6,3] = -3/8, c[2] = 1/3, a[6,5] = 1/2, a[5,3] = -3/16, a[6,2] = 9/8\}:" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}}{PARA 0 "" 0 " " {TEXT -1 0 "" }}{PARA 15 "" 0 "" {TEXT -1 1 " " }{TEXT 282 6 "Step 6 " }{TEXT -1 1 " " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 27 "The stage-order condition: " }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "Sum(a[7,j]*c[j],j = 2 .. 6) = 1/2;" "6#/-%$Su mG6$*&&%\"aG6$\"\"(%\"jG\"\"\"&%\"cG6#F,F-/F,;\"\"#\"\"'*&F-F-F3!\"\" " }{TEXT -1 1 " " }{XPPEDIT 18 0 "c[7]^2;" "6#*$&%\"cG6#\"\"(\"\"#" } {TEXT -1 1 " " }}{PARA 0 "" 0 "" {TEXT -1 12 "determines " }{XPPEDIT 18 0 "a[7,3]" "6#&%\"aG6$\"\"(\"\"$" }{TEXT -1 1 "." }}{PARA 0 "" 0 " " {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 128 "subs(e11, add(a[7,j]*c[j],j=2..6)=1/2*c[7]^2);\ne12 := \{a[7,3]=solve(%,a[7,3]) \}:\ne13 := `union`(e11,e12):\na[7,3]=subs(e13,a[7,3]);" }}{PARA 11 " " 1 "" {XPPMATH 20 "6#/,&#\"\"&\"#6!\"\"*&#\"\"#\"\"$\"\"\"&%\"aG6$\" \"(F,F-F-#F-F+" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/&%\"aG6$\"\"(\"\"$# \"#j\"#W" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 3 "e13" }} {PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 398 "e13 := \{b[1] = 11/120, a[5,2] = 9/8, b[3] = 27/40, b[7] = 11/120 , a[4,3] = -1/12, a[4,2] = 1/3, c[3] = 2/3, b[4] = 27/40, a[3,2] = 2/3 , b[6] = -4/15, b[5] = -4/15, a[6,4] = -3/4, c[6] = 1/2, a[5,4] = -3/8 , c[5] = 1/2, a[7,5] = 0, c[4] = 1/3, a[7,6] = -16/11, c[7] = 1, b[2] \+ = 0, a[7,4] = 18/11, a[7,2] = -9/11, a[6,3] = -3/8, c[2] = 1/3, a[6,5] = 1/2, a[5,3] = -3/16, a[6,2] = 9/8, a[7,3] = 63/44\}:" }}}{PARA 0 " " 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}} }{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 15 "" 0 "" {TEXT -1 1 " " } {TEXT 283 6 "Step 7" }{TEXT -1 1 " " }}{PARA 0 "" 0 "" {TEXT -1 0 "" } }{PARA 0 "" 0 "" {TEXT -1 24 "The row-sum conditions: " }}{PARA 256 " " 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "Sum(a[i,j],j=1..i-1)=c[i]" "6#/- %$SumG6$&%\"aG6$%\"iG%\"jG/F+;\"\"\",&F*F.F.!\"\"&%\"cG6#F*" }{TEXT -1 8 ", for " }{XPPEDIT 18 0 "i=2" "6#/%\"iG\"\"#" }{TEXT -1 8 " . . 7, " }}{PARA 0 "" 0 "" {TEXT -1 21 "can be used to find " }{XPPEDIT 18 0 "a[2,1]" "6#&%\"aG6$\"\"#\"\"\"" }{TEXT -1 3 ", " }{XPPEDIT 18 0 "a[3,1]" "6#&%\"aG6$\"\"$\"\"\"" }{TEXT -1 3 ", " }{XPPEDIT 18 0 "a [4,1]" "6#&%\"aG6$\"\"%\"\"\"" }{TEXT -1 11 ", . . . , " }{XPPEDIT 18 0 "a[7,1];" "6#&%\"aG6$\"\"(\"\"\"" }{TEXT -1 2 ". " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 140 "cdns4 \+ := [seq(add(a[i,j],j=1..i-1)=c[i],i=2..7)]:\neqns4 := subs(e13,cdns4): \nmatrix(ListTools[Enumerate](eqns4));\n``;\nindets(eqns4);\nnops(%); " }}{PARA 11 "" 1 "" {XPPMATH 20 "6#K%'matrixG6#7(7$\"\"\"/&%\"aG6$\" \"#F(#F(\"\"$7$F-/,&&F+6$F/F(F(#F-F/F(F57$F//,&&F+6$\"\"%F(F(#F(F;F(F. 7$F;/,&&F+6$\"\"&F(F(#\"\"*\"#;F(#F(F-7$FB/,&&F+6$\"\"'F(F(FFF(FF7$FL/ ,&&F+6$\"\"(F(F(#\"#N\"#WF(F(Q)pprint116\"" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%!G" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#<(&%\"aG6$\"\"& \"\"\"&F%6$\"\"$F(&F%6$\"\"#F(&F%6$\"\"'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 0 109 "e14 := solve(\{op(eqns4 )\},\{seq(a[i,1],i=2..7)\}):\ne15 := `union`(e13,e14):\nseq(a[i,1]=sub s(e15,a[i,1]),i=2..7);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6(/&%\"aG6$\" \"#\"\"\"#F(\"\"$/&F%6$F*F(\"\"!/&F%6$\"\"%F(#F(\"#7/&F%6$\"\"&F(#!\" \"\"#;/&F%6$\"\"'F(F./&F%6$\"\"(F(#\"\"*\"#W" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 3 "e15" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 482 "e15 := \{b[1] = 11/120, a[5 ,2] = 9/8, b[3] = 27/40, b[7] = 11/120, a[4,3] = -1/12, a[4,2] = 1/3, \+ c[3] = 2/3, b[4] = 27/40, a[2,1] = 1/3, a[3,2] = 2/3, b[6] = -4/15, a[ 3,1] = 0, a[6,1] = 0, b[5] = -4/15, a[6,4] = -3/4, c[6] = 1/2, a[4,1] \+ = 1/12, a[5,4] = -3/8, c[5] = 1/2, a[7,5] = 0, c[4] = 1/3, a[5,1] = -1 /16, a[7,1] = 9/44, a[7,6] = -16/11, c[7] = 1, b[2] = 0, a[7,4] = 18/1 1, a[7,2] = -9/11, a[6,3] = -3/8, c[2] = 1/3, a[6,5] = 1/2, a[5,3] = - 3/16, a[6,2] = 9/8, a[7,3] = 63/44\}:" }}}{PARA 0 "" 0 "" {TEXT -1 0 " " }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 92 "subs(e15,mat rix([seq([c[i],seq(a[i,j],j=1..i-1),``$(8-i)],i=2..7),\n[``,seq(b[i],i =1..7)]]));" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#K%'matrixG6#7)7*#\"\"\" \"\"$F(%!GF+F+F+F+F+7*#\"\"#F*\"\"!F-F+F+F+F+F+7*F(#F)\"#7F(#!\"\"F2F+ F+F+F+7*#F)F.#F4\"#;#\"\"*\"\")#!\"$F8#F=F;F+F+F+7*F6F/F9F>#F=\"\"%F6F +F+7*F)#F:\"#W#!\"*\"#6#\"#jFD#\"#=FGF/#!#;FGF+7*F+#FG\"$?\"F/#\"#F\"# SFQ#!\"%\"#:FTFOQ)pprint126\"" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }} {PARA 0 "" 0 "" {TEXT -1 7 "Check: " }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 149 "RK6_7eqs := [op(RowSumConditions(7,'expanded')),op(O rderConditions(6,7,'expanded'))]:\nsimplify(subs(e15,RK6_7eqs)):\nmap( u->lhs(u)-rhs(u),%);\nnops(%);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#7M\" \"!F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F $F$F$F$F$F$F$F$F$" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#\"#V" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }} }}{PARA 0 "" 0 "" {TEXT -1 44 "#====================================== =====" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 20 " #===================" }}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 18 "Butcher's scheme A" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 62 "See: On Runge-Kutta Processes of High Order, by J. C. Butcher," }} {PARA 0 "" 0 "" {TEXT -1 88 " Journal of the Australian Mathemat ical Society, Vol. 4, (1964), pages 179 to 194." }}{EXCHG {PARA 0 "> \+ " 0 "" {MPLTEXT 1 0 1 ";" }}}{PARA 0 "" 0 "" {TEXT -1 39 "#----------- ---------------------------" }}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 19 "ch ecking the scheme" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 26 "coe fficients of the scheme" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 438 "ee := \{c[2]=1/2,\nc[3]=2/3,\nc[4] =1/3,\nc[5]=5/6,\nc[6]=1/6,\nc[7]=1,\n\na[2,1]=1/2,\na[3,1]=2/9,\na[3, 2]=4/9,\na[4,1]=7/36,\na[4,2]=2/9,\na[4,3]=-1/12,\na[5,1]=-35/144,\na[ 5,2]=-55/36,\na[5,3]=35/48,\na[5,4]=15/8,\na[6,1]=-1/360,\na[6,2]=-11/ 36,\na[6,3]=-1/8,\na[6,4]=1/2,\na[6,5]=1/10,\na[7,1]=-41/260,\na[7,2]= 22/13,\na[7,3]=43/156,\na[7,4]=-118/39,\na[7,5]=32/195,\na[7,6]=80/39, \n\nb[1]=13/200,\nb[2]=0,\nb[3]=11/40,\nb[4]=11/40,\nb[5]=4/25,\nb[6]= 4/25,\nb[7]=13/200\}:" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}}{PARA 0 "" 0 "" {TEXT -1 0 " " }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 91 "subs(ee,matrix([seq([c[i ],seq(a[i,j],j=1..i-1),``$(8-i)],i=2..7),\n[``,seq(b[i],i=1..7)]]));" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#K%'matrixG6#7)7*#\"\"\"\"\"#F(%!GF+F +F+F+F+7*#F*\"\"$#F*\"\"*#\"\"%F0F+F+F+F+F+7*#F)F.#\"\"(\"#OF/#!\"\"\" #7F+F+F+F+7*#\"\"&\"\"'#!#N\"$W\"#!#bF7#\"#N\"#[#\"#:\"\")F+F+F+7*#F)F >#F9\"$g$#!#6F7#F9FIF(#F)\"#5F+F+7*F)#!#T\"$g##\"#A\"#8#\"#V\"$c\"#!$= \"\"#R#\"#K\"$&>#\"#!)FinF+7*F+#FY\"$+#\"\"!#\"#6\"#SFco#F2\"#DFfoF`oQ (pprint06\"" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 7 "Check: " }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 148 "RK6_7eqs : = [op(RowSumConditions(7,'expanded')),op(OrderConditions(6,7,'expanded '))]:\nsimplify(subs(ee,RK6_7eqs)):\nmap(u->lhs(u)-rhs(u),%);\nnops(%) ;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#7M\"\"!F$F$F$F$F$F$F$F$F$F$F$F$F$ F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#\"#V" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }} {PARA 0 "" 0 "" {TEXT -1 86 "Next we set-up stage-order condtions to c heck for stage-orders from 2 to 4 inclusive. " }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 85 "for ct from 2 to 4 do\n so||ct||_7 := StageOrd erConditions(ct,7,'expanded');\nend do:" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 58 "Stages 3 to 7 have the following r espective stage-orders. " }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 173 "[seq([seq(expand(subs(ee,so||i||_7[j])),i=2..4)],j=1..5)]:\nmap(proc( L) local i; for i to nops(L) do if not evalb(L[i]) then break end if e nd do; i end proc,%):\nsimplify(%); " }}{PARA 11 "" 1 "" {XPPMATH 20 " 6#7'\"\"#F$F$F$F$" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 42 "Thus stages 5, 6 and 7 have stage-order 3." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 32 "#------------------- ------------" }}{PARA 0 "" 0 "" {TEXT -1 27 "The simplifying condition s:" }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "Sum(b[i]*a[i,j] ,i = j+1 .. 7) = b[j]*(1-c[j]);" "6#/-%$SumG6$*&&%\"bG6#%\"iG\"\"\"&% \"aG6$F+%\"jGF,/F+;,&F0F,F,F,\"\"(*&&F)6#F0F,,&F,F,&%\"cG6#F0!\"\"F," }{TEXT -1 3 ", " }{XPPEDIT 18 0 "j=1" "6#/%\"jG\"\"\"" }{TEXT -1 7 " \+ . . 7 " }}{PARA 0 "" 0 "" {TEXT -1 8 "(where " }{XPPEDIT 18 0 "c[1]=0 " "6#/&%\"cG6#\"\"\"\"\"!" }{TEXT -1 17 " ) are satisfied." }}{PARA 256 "" 0 "" {TEXT -1 1 " " }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 162 "[Sum(b[i]*a[i,1],i=2..7)=b[1],seq(Sum(b[i]*a[i,j],i=j+1..7)=b[j]* (1-c[j]),j=2..6)];\neval(subs(Sum=add,%)):\nsubs(ee,%):\nmap(u->`if`(l hs(u)=rhs(u),0,1),%);\nnops(%);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#7(/ -%$SumG6$*&&%\"bG6#%\"iG\"\"\"&%\"aG6$F,F-F-/F,;\"\"#\"\"(&F*6#F-/-F&6 $*&F)F-&F/6$F,F3F-/F,;\"\"$F4*&&F*6#F3F-,&F-F-&%\"cGFB!\"\"F-/-F&6$*&F )F-&F/6$F,F?F-/F,;\"\"%F4*&&F*6#F?F-,&F-F-&FEFRFFF-/-F&6$*&F)F-&F/6$F, FOF-/F,;\"\"&F4*&&F*6#FOF-,&F-F-&FEFjnFFF-/-F&6$*&F)F-&F/6$F,FgnF-/F,; \"\"'F4*&&F*6#FgnF-,&F-F-&FEFhoFFF-/-F&6$*&F)F-&F/6$F,FeoF-/F,;F4F4*&& F*6#FeoF-,&F-F-&FEFepFFF-" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#7(\"\"!F$ F$F$F$F$" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#\"\"'" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 27 "The simplifying condition : " }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "Sum(b[i]*a[i,2] ,i = 3 .. 7) = 0;" "6#/-%$SumG6$*&&%\"bG6#%\"iG\"\"\"&%\"aG6$F+\"\"#F, /F+;\"\"$\"\"(\"\"!" }{TEXT -1 1 " " }}{PARA 0 "" 0 "" {TEXT -1 13 "is satisfied." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 " " {MPLTEXT 1 0 59 "Sum(b[i]*a[i,2],i=3..7);\neval(subs(Sum=add,%));\ns ubs(ee,%);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#-%$SumG6$*&&%\"bG6#%\"iG \"\"\"&%\"aG6$F*\"\"#F+/F*;\"\"$\"\"(" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#,,*&&%\"bG6#\"\"$\"\"\"&%\"aG6$F(\"\"#F)F)*&&F&6#\"\"%F)&F+6$F1F-F) F)*&&F&6#\"\"&F)&F+6$F7F-F)F)*&&F&6#\"\"'F)&F+6$F=F-F)F)*&&F&6#\"\"(F) &F+6$FCF-F)F)" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#\"\"!" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 27 "The simplifying cond ition: " }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "Sum(b[i]*c [i]*a[i,2],i = 3 .. 7) = 0;" "6#/-%$SumG6$*(&%\"bG6#%\"iG\"\"\"&%\"cG6 #F+F,&%\"aG6$F+\"\"#F,/F+;\"\"$\"\"(\"\"!" }{TEXT -1 1 " " }}{PARA 0 " " 0 "" {TEXT -1 13 "is satisfied." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 64 "Sum(b[i]*c[i]*a[i,2],i=3..7) ;\neval(subs(Sum=add,%));\nsubs(ee,%);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#-%$SumG6$*(&%\"bG6#%\"iG\"\"\"&%\"cGF)F+&%\"aG6$F*\"\"#F+/F*;\" \"$\"\"(" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#,,*(&%\"bG6#\"\"$\"\"\"&% \"cGF'F)&%\"aG6$F(\"\"#F)F)*(&F&6#\"\"%F)&F+F2F)&F-6$F3F/F)F)*(&F&6#\" \"&F)&F+F9F)&F-6$F:F/F)F)*(&F&6#\"\"'F)&F+F@F)&F-6$FAF/F)F)*(&F&6#\"\" (F)&F+FGF)&F-6$FHF/F)F)" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#\"\"!" }}} {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 27 "The simpl ifying condition: " }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "Sum(b[i]*c[i]^2*a[i,2],i = 3 .. 7) = 0;" "6#/-%$SumG6$*(&%\"bG6#%\"iG \"\"\"*$&%\"cG6#F+\"\"#F,&%\"aG6$F+F1F,/F+;\"\"$\"\"(\"\"!" }{TEXT -1 1 " " }}{PARA 0 "" 0 "" {TEXT -1 13 "is satisfied." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 66 "Sum(b[i]*c[i ]^2*a[i,2],i=3..7);\neval(subs(Sum=add,%));\nsubs(ee,%);" }}{PARA 11 " " 1 "" {XPPMATH 20 "6#-%$SumG6$*(&%\"bG6#%\"iG\"\"\")&%\"cGF)\"\"#F+&% \"aG6$F*F/F+/F*;\"\"$\"\"(" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#,,*(&%\" bG6#\"\"$\"\"\")&%\"cGF'\"\"#F)&%\"aG6$F(F-F)F)*(&F&6#\"\"%F))&F,F3F-F )&F/6$F4F-F)F)*(&F&6#\"\"&F))&F,F;F-F)&F/6$F " 0 "" {MPLTEXT 1 0 85 "Sum(b[i]*c[i]*Sum(a[i,j]*a[j,2],j=3..i-1),i=3..7);\ne val(subs(Sum=add,%));\nsubs(ee,%);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6# -%$SumG6$*(&%\"bG6#%\"iG\"\"\"&%\"cGF)F+-F$6$*&&%\"aG6$F*%\"jGF+&F26$F 4\"\"#F+/F4;\"\"$,&F*F+F+!\"\"F+/F*;F:\"\"(" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#,***&%\"bG6#\"\"%\"\"\"&%\"cGF'F)&%\"aG6$F(\"\"$F)&F-6$ F/\"\"#F)F)*(&F&6#\"\"&F)&F+F5F),&*&&F-6$F6F/F)F0F)F)*&&F-6$F6F(F)&F-6 $F(F2F)F)F)F)*(&F&6#\"\"'F)&F+FCF),(*&&F-6$FDF/F)F0F)F)*&&F-6$FDF(F)F? F)F)*&&F-6$FDF6F)&F-6$F6F2F)F)F)F)*(&F&6#\"\"(F)&F+FTF),**&&F-6$FUF/F) F0F)F)*&&F-6$FUF(F)F?F)F)*&&F-6$FUF6F)FPF)F)*&&F-6$FUFDF)&F-6$FDF2F)F) F)F)" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#\"\"!" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 114 "We can calculate the pri ncipal error norm of the order 6 scheme, that is, the 2-norm of the pr incipal error terms." }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 158 "err terms6_7 := PrincipalErrorTerms(6,7,'expanded'):\nsm := 0:\nfor ct to \+ nops(errterms6_7) do\n sm := sm+(evalf(subs(ee,errterms6_7[ct])))^2; \nend do:\nsqrt(sm);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#$\"+sq,W\\!#7 " }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}}{PARA 0 "" 0 "" {TEXT -1 39 "#----------------- ---------------------" }}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 32 "short co nstruction of the scheme" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 " " 0 "" {TEXT 260 4 "Note" }{TEXT -1 123 ": This scheme was constructed in a step by step manner in a previous section making use of \"altern ative\" order conditions. " }}{PARA 0 "" 0 "" {TEXT -1 77 "In this sub section we construct the scheme using \"standard\" order conditions." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 192 "We set up a system of equations using a selection of \"simple\" order condit ions and incorporate the row-sum conditions together with the stage-or der equations to ensure that stages 3 to 7 have " }{TEXT 260 13 "stage -order 2" }{TEXT -1 1 "." }}{PARA 0 "" 0 "" {TEXT -1 48 "We also incor porate the simplifying conditions: " }}{PARA 256 "" 0 "" {TEXT -1 2 " \+ " }{XPPEDIT 18 0 "Sum(b[i]*a[i,j],i = j+1 .. 7) = b[j]*(1-c[j]);" "6# /-%$SumG6$*&&%\"bG6#%\"iG\"\"\"&%\"aG6$F+%\"jGF,/F+;,&F0F,F,F,\"\"(*&& F)6#F0F,,&F,F,&%\"cG6#F0!\"\"F," }{TEXT -1 1 " " }}{PARA 0 "" 0 "" {TEXT -1 5 "for " }{XPPEDIT 18 0 "j = 3;" "6#/%\"jG\"\"$" }{TEXT -1 62 ", 4, 5, 6, together with the further simplifying conditions: " }} {PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "Sum(b[i]*c[i]*a[i,2], i = 3 .. 7) = 0;" "6#/-%$SumG6$*(&%\"bG6#%\"iG\"\"\"&%\"cG6#F+F,&%\"aG 6$F+\"\"#F,/F+;\"\"$\"\"(\"\"!" }{TEXT -1 8 ", " }{XPPEDIT 18 0 "Sum(b[i]*c[i]*Sum(a[i,j]*a[j,2],j = 3 .. i-1),i = 3 .. 7) = 0;" "6#/- %$SumG6$*(&%\"bG6#%\"iG\"\"\"&%\"cG6#F+F,-F%6$*&&%\"aG6$F+%\"jGF,&F46$ F6\"\"#F,/F6;\"\"$,&F+F,F,!\"\"F,/F+;F<\"\"(\"\"!" }{TEXT -1 8 ", \+ " }{XPPEDIT 18 0 "Sum(b[i]*c[i]^2*a[i,2],i = 3 .. 7) = 0;" "6#/-%$Su mG6$*(&%\"bG6#%\"iG\"\"\"*$&%\"cG6#F+\"\"#F,&%\"aG6$F+F1F,/F+;\"\"$\" \"(\"\"!" }{TEXT -1 2 ". " }}{PARA 0 "" 0 "" {TEXT -1 1 " " }}{PARA 0 "" 0 "" {TEXT -1 76 "The simple order conditions used are given (in ab reviated form) as follows. " }}{PARA 0 "" 0 "" {TEXT 260 4 "Note" } {TEXT -1 86 ": We use one additional order condition #28 beyond those \+ used for the previous scheme." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 184 "SO6 := SimpleOrderCondition s(6):\n[seq([i,SO6[i]],i=[1,2,4,8,13,16,24,28,29,32])]:\nlinalg[augmen t](linalg[delcols](%,2..2),matrix([[` `]$(linalg[rowdim](%))]),linalg [delcols](%,1..1));" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#K%'matrixG6#7,7 %\"\"\"%#~~G/*&%\"bGF(%\"eGF(F(7%\"\"#F)/*&F,F(%\"cGF(#F(F/7%\"\"%F)/* &F,F()F2F/F(#F(\"\"$7%\"\")F)/*&F,F()F2F:F(#F(F57%\"#8F)/*(F,F(F2F(-%! G6#*&F8F(%\"aGF(F(#F(\"#:7%\"#;F)/*&F,F()F2F5F(#F(\"\"&7%\"#CF)/*(F,F( F2F(-FF6#*&FIF(FEF(F(#F(\"#s7%\"#GF)/*(F,F(F8F(FEF(#F(\"#=7%\"#HF)/*(F ,F(F2F(-FF6#*&F?F(FIF(F(#F(FT7%\"#KF)/*&F,F()F2FRF(#F(\"\"'Q(pprint06 \"" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 462 "SO6_7 := SimpleOrderConditions(6,7,'expanded'):\nord _cdns := [seq(SO6_7[i],i=[1,2,4,8,13,16,24,28,29,32])]:\nSO_eqs := [op (RowSumConditions(7,'expanded')),op(StageOrderConditions(2,7,'expanded '))]:\nsimp_eqs := [seq(add(b[i]*a[i,j],i=j+1..7)=b[j]*(1-c[j]),j=[3,4 ,5,6]),\n add(b[i]*c[i]*a[i,2],i=3..7)=0,add(b[i]*c[i]*ad d(a[i,j]*a[j,2],j=3..i-1),i=3..7)=0,\n add(b[i]*c[i]^2*a [i,2],i=3..7)=0]:\ncdns := [op(ord_cdns),op(simp_eqs),op(SO_eqs)]:" }} }{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 21 "We speci fy the nodes " }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "c[2] = 1/2;" "6#/&%\"cG6#\"\"#*&\"\"\"F)F'!\"\"" }{TEXT -1 3 ", " } {XPPEDIT 18 0 "c[3]=2/3" "6#/&%\"cG6#\"\"$*&\"\"#\"\"\"F'!\"\"" } {TEXT -1 3 ", " }{XPPEDIT 18 0 "c[5] = 5/6;" "6#/&%\"cG6#\"\"&*&F'\" \"\"\"\"'!\"\"" }{TEXT -1 3 ", " }{XPPEDIT 18 0 "c[6] = 1/6;" "6#/&% \"cG6#\"\"'*&\"\"\"F)F'!\"\"" }{TEXT -1 3 ", " }{XPPEDIT 18 0 "c[7]=1 " "6#/&%\"cG6#\"\"(\"\"\"" }{TEXT -1 1 "," }}{PARA 0 "" 0 "" {TEXT -1 16 " and the weight " }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "b[2]=0" "6#/&%\"bG6#\"\"#\"\"!" }{TEXT -1 2 ". " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 116 "e1 := \{c[2 ]=1/2,c[3]=2/3,c[5]=5/6,c[6]=1/6,c[7]=1,b[2]=0\}:\neqns := subs(e1,cdn s):\nnops(eqns);\nindets(eqns);\nnops(%);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#\"#G" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#<>&%\"aG6$\"\"' \"\"\"&F%6$F'\"\"#&F%6$\"\"&F+&F%6$\"\"%F+&F%6$F1\"\"$&F%6$F.F1&%\"bG6 #F4&F86#F1&F86#F'&F86#\"\"(&F%6$F1F(&F%6$F+F(&%\"cGF;&F%6$F4F+&F%6$F.F 4&F%6$F@F+&F%6$F@F4&F%6$F@F1&F%6$F'F4&F%6$F'F1&F%6$F'F.&F%6$F@F(&F%6$F @F.&F%6$F@F'&F%6$F.F(&F86#F.&F%6$F4F(&F86#F(" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#\"#G" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 " " {TEXT -1 51 "There are 28 equations and 28 unknown coefficients." }} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 22 "infolevel[solve] := 4:" }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 69 "e2 := solve(\{op(eqns)\}):\n infolevel[solve] := 0:\ne3 := `union`(e1,e2):" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 2 "e3" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 504 "e3 := \{a[5,3] = 35/48, b[6 ] = 4/25, a[2,1] = 1/2, a[7,1] = -41/260, a[4,3] = -1/12, a[6,5] = 1/1 0, a[5,1] = -35/144, b[3] = 11/40, a[7,2] = 22/13, b[5] = 4/25, a[6,2] = -11/36, c[2] = 1/2, c[5] = 5/6, c[3] = 2/3, c[6] = 1/6, c[4] = 1/3, a[3,1] = 2/9, a[3,2] = 4/9, a[7,6] = 80/39, a[4,2] = 2/9, a[5,2] = -5 5/36, b[4] = 11/40, c[7] = 1, b[2] = 0, a[6,3] = -1/8, a[6,4] = 1/2, b [1] = 13/200, a[6,1] = -1/360, a[7,5] = 32/195, a[7,3] = 43/156, a[4,1 ] = 7/36, a[5,4] = 15/8, b[7] = 13/200, a[7,4] = -118/39\}:" }{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 91 "subs(e3,matrix([seq([c[i],seq(a[i,j],j=1..i -1),``$(8-i)],i=2..7),\n[``,seq(b[i],i=1..7)]]));" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#K%'matrixG6#7)7*#\"\"\"\"\"#F(%!GF+F+F+F+F+7*#F*\"\"$#F *\"\"*#\"\"%F0F+F+F+F+F+7*#F)F.#\"\"(\"#OF/#!\"\"\"#7F+F+F+F+7*#\"\"& \"\"'#!#N\"$W\"#!#bF7#\"#N\"#[#\"#:\"\")F+F+F+7*#F)F>#F9\"$g$#!#6F7#F9 FIF(#F)\"#5F+F+7*F)#!#T\"$g##\"#A\"#8#\"#V\"$c\"#!$=\"\"#R#\"#K\"$&># \"#!)FinF+7*F+#FY\"$+#\"\"!#\"#6\"#SFco#F2\"#DFfoF`oQ(pprint16\"" }}} {PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 85 "RK6_7eqs := [op(RowSumConditions(7,'expanded')),op(OrderConditions (6,7,'expanded'))]:" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 " " {TEXT -1 7 "Check: " }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 62 "sim plify(subs(e3,RK6_7eqs)):\nmap(u->lhs(u)-rhs(u),%);\nnops(%);" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#7M\"\"!F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F $F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#\"#V" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 " " 0 "" {TEXT -1 65 "#------------------------------------------------- ---------------" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}} {PARA 0 "" 0 "" {TEXT -1 39 "#--------------------------------------" }}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 25 "absolute stability region" }} {PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 26 "coefficients of the sche me" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 438 "ee := \{c[2]=1/2,\nc[3]=2/3,\nc[4]=1/3,\nc[5]=5/6,\n c[6]=1/6,\nc[7]=1,\n\na[2,1]=1/2,\na[3,1]=2/9,\na[3,2]=4/9,\na[4,1]=7/ 36,\na[4,2]=2/9,\na[4,3]=-1/12,\na[5,1]=-35/144,\na[5,2]=-55/36,\na[5, 3]=35/48,\na[5,4]=15/8,\na[6,1]=-1/360,\na[6,2]=-11/36,\na[6,3]=-1/8, \na[6,4]=1/2,\na[6,5]=1/10,\na[7,1]=-41/260,\na[7,2]=22/13,\na[7,3]=43 /156,\na[7,4]=-118/39,\na[7,5]=32/195,\na[7,6]=80/39,\n\nb[1]=13/200, \nb[2]=0,\nb[3]=11/40,\nb[4]=11/40,\nb[5]=4/25,\nb[6]=4/25,\nb[7]=13/2 00\}:" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 " " {TEXT -1 68 "The stability function R for the order 6 scheme is give n as follows." }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 75 "subs(ee,Sta bilityFunction(6,7,'expanded')):\nR := unapply(%,z):\n'R(z)'=R(z);" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#/-%\"RG6#%\"zG,2\"\"\"F)F'F)*&#F)\"\"# F)*$)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)*$)F'F1F)F)F)*&#F)\"%g@F)*$)F '\"\"(F)F)!\"\"" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{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#/-%\"RG6#%\"zG\"\"\"" }{TEXT -1 2 ". " }} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 26 "z0 := newton(R(z)=1,z=-3);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%#z0G$!+z*3h&G!\"*" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 302 "z0 := \+ newton(R(z)=1,z=-3):\np1 := plot([R(z),1],z=-3.29..0.49,color=[red,blu e]):\np2 := plot([[[z0,1]]$3],style=point,symbol=[circle,cross,diamond ],color=black):\np3 := plot([[z0,0],[z0,1]],linestyle=3,color=COLOR(RG B,0,.5,0)):\nplots[display]([p1,p2,p3],view=[-3.29..0.49,-0.07..1.47], font=[HELVETICA,9]);" }}{PARA 13 "" 1 "" {GLPLOT2D 360 253 253 {PLOTDATA 2 "6+-%'CURVESG6$7X7$$!3/++++++!H$!#<$\"3)p'RR\"o@$\\DF*7$$! 3O+]iyL!)[KF*$\"3a[97!)4&3M#F*7$$!3A++Ddng2KF*$\"3->p*)[ikZ@F*7$$!3=]( oszh<<$F*$\"3m#Q$**o6C\"*>F*7$$!3:+vGPo\"f8$F*$\"3WBiP([:^%=F*7$$!3;]7 )3V0c4$F*$\"3ka#G!=RP#p\"F*7$$!3?+]ZCSHbIF*$\"3uNyxl<=^:F*7$$!3\")**** *yS:Z,$F*$\"36NX5$*e**>9F*7$$!3')**\\K\"zOT(HF*$\"3'of-S,*4*H\"F*7$$!3 8]7$y.^P$HF*$\"3e#3!HzHM)=\"F*7$$!3%**\\PVGlL*GF*$\"3mWX8Mt^'3\"F*7$$! 3F+v)yvz%=GF*$\"3%[0W`D2R>*!#=7$$!3E+vyK,%4u#F*$\"3b(f#)RY2)HxF^o7$$!3 3+veT&[2m#F*$\"3=d9uPC'\\Y'F^o7$$!39+v$e793e#F*$\"3>E/cVrXBaF^o7$$!3++ +0aTw:cLF^o7$$!39++DUlxiAF*$\"3bgqIuO!Q\"HF^o7$$!35+ +&G=&)Q=#F*$\"3o%G1DxR,f#F^o7$$!3(**\\(eqUC7@F*$\"3=Yo#>6[5P#F^o7$$!34 ++l'Hcq-#F*$\"3%oA#R)>)y*=#F^o7$$!3#*****f34*[&>F*$\"31g)RNG,T4#F^o7$$ !3'**\\([1a%4(=F*$\"3`o!fj+#yR?F^o7$$!33++58%Rmz\"F*$\"3&fS!H*=>e.#F^o 7$$!3&**\\([hR6:FF^o7$$!3?+Dr%RzfC\"F*$\"3,yG+b/;1HF^o7$$!3'***\\2\\lin6F*$\"3*pk]gZ c)HJF^o7$$!3%*****p#R!o'3\"F*$\"337)*[a&\\ZQ$F^o7$$!3D+]xq.\\25F*$\"3w 9.XCP2eOF^o7$$!3Y**\\(o_*p3$*F^o$\"3Y(QWX>&)f%RF^o7$$!3+***\\#=/'zX)F^ o$\"3#G:tH0WTH%F^o7$$!3%>++5\\LNp(F^o$\"3*R:n#))G:MYF^o7$$!39++]#eWt(o F^o$\"33')=@QngF]F^o7$$!33-](=7cx8'F^o$\"3=6='>!\\C8aF^o7$$!3T'***\\$e !>H`F^o$\"3%3m/@Cs*oeF^o7$$!3M,]([U$RoXF^o$\"3W<[E_x%GL'F^o7$$!3C**\\P !=TJx$F^o$\"3!R()eSf8q&oF^o7$$!3#Q++DvOc*HF^o$\"3/>Qb,nT6uF^o7$$!3s.]( =nh;=#F^o$\"3IAVQ*[=*R!)F^o7$$!3o*****f#pq(R\"F^o$\"3Gb4>#4wbp)F^o7$$! 3I%)**\\#p***ff!#>$\"32!Rxk<89U*F^o7$$\"3c;+D\"\\%o!*>F\\z$\"3;2\"3vI1 ,-\"F*7$$\"3YP++Iv`'H*F\\z$\"3EbFgOPU(4\"F*7$$\"3-.+vdp)pw\"F^o$\"3ko1 j(\\rK>\"F*7$$\"3*******zv2f^#F^o$\"3+vh:'fpgG\"F*7$$\"3P+]7NNT9LF^o$ \"3)3*\\ZcT(HR\"F*7$$\"3Y,](3X&oySF^o$\"3HpF_H#3O]\"F*7$$\"3!********* ******[F^o$\"3_MD@Z;JK;F*-%'COLOURG6&%$RGBG$\"*++++\"!\")$\"\"!Fj\\lFi \\l-F$6$7S7$F($\"\"\"Fj\\l7$F3F_]l7$F=F_]l7$FGF_]l7$FQF_]l7$FenF_]l7$F jnF_]l7$F`oF_]l7$FeoF_]l7$FjoF_]l7$F_pF_]l7$FdpF_]l7$FipF_]l7$F^qF_]l7 $FcqF_]l7$FhqF_]l7$F]rF_]l7$FbrF_]l7$FgrF_]l7$F\\sF_]l7$FasF_]l7$FfsF_ ]l7$F[tF_]l7$F`tF_]l7$FetF_]l7$FjtF_]l7$F_uF_]l7$FduF_]l7$FiuF_]l7$F^v F_]l7$FcvF_]l7$FhvF_]l7$F]wF_]l7$FbwF_]l7$FgwF_]l7$F\\xF_]l7$FaxF_]l7$ FfxF_]l7$F[yF_]l7$F`yF_]l7$FeyF_]l7$FjyF_]l7$F`zF_]l7$FezF_]l7$FjzF_]l 7$F_[lF_]l7$Fd[lF_]l7$Fi[lF_]l7$F^\\lF_]l-Fc\\l6&Fe\\lFi\\lFi\\lFf\\l- F$6&7#7$$!3/+++z*3h&GF*F_]l-%'SYMBOLG6#%'CIRCLEG-Fc\\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-%&COLORG6&Fe\\lFi\\l$\"\"& !\"\"Fi\\l-%*LINESTYLEG6#\"\"$-%+AXESLABELSG6%Q\"z6\"Q!F_cl-%%FONTG6#% (DEFAULTG-Fbcl6$%*HELVETICAG\"\"*-%%VIEWG6$;$!$H$!\"#$\"#\\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 follo wing picture shows the stability region. " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1640 "R := z -> 1+z+1/2*z ^2+1/6*z^3+1/24*z^4+1/120*z^5+1/720*z^6-1/2160*z^7:\npts := []: z0 := \+ 0: tt := 0: \nwhile tt<=161/20 do\n zz := newton(`R`(z)=exp(tt*Pi*I) ,z=z0):\n z0 := zz:\n if (7/20<=tt and tt<=27/20) or (133/20<=tt a nd tt<=153/20) then\n hh := 1/40\n else \n hh := 1/20\n \+ end if;\n tt := tt+hh;\n pts := [op(pts),[Re(zz),Im(zz)]]:\nend do :\np1 := plot(pts,color=COLOR(RGB,.48,.1,0)):\np2 := plots[polygonplot ]([seq([pts[i-1],pts[i],[-1.45,0]],i=2..nops(pts))],\n style= patchnogrid,color=COLOR(RGB,.95,.2,0)):\npts := []: z0 := 0.8+3.3*I: t t := 0: \nwhile tt<=41/20 do\n zz := newton(R(z)=exp(tt*Pi*I),z=z0): \n z0 := zz:\n if (9/20<=tt and tt<=6/5) then\n hh := 1/60\n \+ else \n hh := 1/20\n end if;\n tt := tt+hh;\n pts := [op( pts),[Re(zz),Im(zz)]]:\nend do:\np3 := plot(pts,color=COLOR(RGB,.48,.1 ,0)):\np4 := plots[polygonplot]([seq([pts[i-1],pts[i],[0.75,3.03]],i=2 ..nops(pts))],\n style=patchnogrid,color=COLOR(RGB,.95,.2,0)) :\npts := []: z0 := 0.8-3.3*I: tt := 0: \nwhile tt<=41/20 do\n zz := newton(R(z)=exp(tt*Pi*I),z=z0):\n z0 := zz:\n if (17/20<=tt and t t<=8/5) then\n hh := 1/60\n else \n hh := 1/20\n end if; \n tt := tt+hh;\n pts := [op(pts),[Re(zz),Im(zz)]]:\nend do:\np5 : = plot(pts,color=COLOR(RGB,.48,.1,0)):\np6 := plots[polygonplot]([seq( [pts[i-1],pts[i],[0.75,-3.03]],i=2..nops(pts))],\n style=patc hnogrid,color=COLOR(RGB,.95,.2,0)):\np7 := plot([[[-3.49,0],[1.29,0]], [[0,-3.59],[0,3.59]]],color=black,linestyle=3):\nplots[display]([p||(1 ..7)],view=[-3.49..1.29,-3.59..3.59],font=[HELVETICA,9],\n \+ labels=[`Re(z)`,`Im(z)`],axes=boxed,scaling=constrained);" }}{PARA 13 "" 1 "" {GLPLOT2D 381 565 565 {PLOTDATA 2 "6/-%'CURVESG6$7fw7$$\"\" !F)F(7$$!33+++8]B]B!#F$\"31+++6jzq:!#=7$$!3s*****zW3m%f!#D$\"3&)*****p u!fTJF07$$!3/+++QAH%\\\"!#B$\"3:+++E$eBr%F07$$!3)******H227X\"!#A$\"37 +++*GvHG'F07$$!3U******RBpL$)F@$\"3E+++?l4`yF07$$!3:+++]S.=M!#@$\"3\\* *****)o\"4A%*F07$$!3'******>Hgr5\"!#?$\"33+++JV#*)4\"!#<7$$!3'*******H ?Xg=FQ$\"3'******fQ.s<\"FT7$$!3;+++iRW4IFQ$\"3/+++v=Wb7FT7$$!3#****** \\FPzq%FQ$\"3(******RsmOL\"FT7$$!3=+++(=N=:(FQ$\"3'******z^C>T\"FT7$$! 33+++4.')e5!#>$\"35+++WXG!\\\"FT7$$!3%*******4!*3L:Ffo$\"3)******R+R)o :FT7$$!33+++'pzw<#Ffo$\"3!******Ra(pZ;FT7$$!3-+++!RqR/$Ffo$\"34+++u0(p s\"FT7$$!33+++ln%*)>%Ffo$\"3*******pQIn!=FT7$$!3:+++ZW,IdFfo$\"3!***** *RMHp)=FT7$$!37+++aPlZxFfo$\"33+++@BCn>FT7$$!31+++/VuP5F0$\"3')******4 $3o/#FT7$$!37+++q**[s8F0$\"3&*******3!\\R7#FT7$$!3%******f%GZ!y\"F0$\" 35+++FO@'>#FT7$$!31+++JR6[AF0$\"3;+++\"3\"=hAFT7$$!3G+++@z`]FF0$\"3/++ +-:gcF0$\"38+++ &z!*3_#FT7$$!3`++++0$z.'F0$\"3/+++s;0TDFT7$$!3R+++n$G2W'F0$\"35+++,s\" )eDFT7$$!3f******fzlHoF0$\"3))*****f.\"[uDFT7$$!3O+++o?/1sF0$\"3%)**** *fEt#)e#FT7$$!3W+++$yh5d(F0$\"3y*****R3x.g#FT7$$!3z*****>gdd#zF0$\"3$* *****\\cS4h#FT7$$!3W+++l+/r#)F0$\"3/+++\"3&3?EFT7$$!3R+++xOq2')F0$\"30 +++v8\"zi#FT7$$!3z******y6WO*)F0$\"3?+++6S]MEFT7$$!3?+++.p&yD*F0$\"3<+ ++\"oM*REFT7$$!35+++G(yCd*F0$\"36+++(pkUk#FT7$$!3Y+++X&o2))*F0$\"3))** ***z*oaZEFT7$$!3!*******HJJ=5FT$\"3)******f6F)\\EFT7$$!3))******zA*z/ \"FT$\"3#)*****\\LX6l#FT7$$!3!******>jXr2\"FT$\"3')*****zfO:l#FT7$$!3+ +++W3!e5\"FT$\"3#******ztJ5l#FT7$$!3$******\\Q#)R8\"FT$\"37+++jzl\\EFT 7$$!3!******4(>rh6FT$\"3'*******Q$Ruk#FT7$$!3!******R#*3!*=\"FT$\"3-++ +wrRWEFT7$$!3#******H%H\"FT$\"3#** *****y&RVi#FT7$$!3,+++&))fXM\"FT$\"32+++Nsw4EFT7$$!3#******p)Hc$R\"FT$ \"37+++gIF#f#FT7$$!3*******zZr7W\"FT$\"3#)*****>O6>d#FT7$$!3++++*>_x[ \"FT$\"3=+++`(=([DFT7$$!34+++%QrI`\"FT$\"3/+++.hrADFT7$$!3++++r#*Hx:FT $\"3/+++o-\"R\\#FT7$$!33+++&3:0i\"FT$\"3)******f\"fHiCFT7$$!3-+++EE\"G m\"FT$\"35+++$*y&yU#FT7$$!3'******pj1Vq\"FT$\"3!******Hru0R#FT7$$!3'** ****\\&*R^u\"FT$\"3')*****p-C/N#FT7$$!3!******\\6\"\\&y\"FT$\"30+++@7R 2BFT7$$!3)******fO\"eD=FT$\"31+++ME[hAFT7$$!3#******\\%)yc'=FT$\"3%)** ***z\"\\u7AFT7$$!3$******py&41>FT$\"35+++T8Hh@FT7$$!3-+++;EFT$\"38+ ++\\LL2@FT7$$!31+++7?C*)>FT$\"31+++N0@^?FT7$$!3,+++>+dK?FT$\"3#******H '\\S$*>FT7$$!3;+++TGGx?FT$\"3$*******H8_M>FT7$$!3++++#[/L7#FT$\"33+++C :Av=FT7$$!3)*******y!Q.<#FT$\"3*******>*\\7;=FT7$$!3#******H)z!z@#FT$ \"3++++P9rd8FT7$$!3!******f$\\P%f#FT$\"3'*******y#>kE\"FT7$$!3& *******\\D2EEFT$\"33+++RNn77FT7$$!3A+++PD\\bEFT$\"3++++V#3&e6FT7$$!3'* ******\\nk#o#FT$\"36+++8k\"Q5\"FT7$$!3#********\\cvq#FT$\"33+++'y/&[5F T7$$!3(*******42DIFFT$\"3Z******[%3\\#**F07$$!3!)*****4Dl2v#FT$\"3U+++ 6E+d$*F07$$!3++++(>V\"pFFT$\"3a*****fM[1y)F07$$!3/+++eeV&y#FT$\"3!**** **pC5_>)F07$$!3'******zW/(*z#FT$\"3z*******y\"4+wF07$$!3))******zA-7GF T$\"3'******4@OZ*pF07$$!39+++\\tZAGFT$\"3/+++P\"Q'yjF07$$!3#)******H]< JGFT$\"3W+++;(o8v&F07$$!3&******R[S#QGFT$\"3R+++,sh7^F07$$!32+++#4?Q%G FT$\"3%*******37DiWF07$$!3;+++U43[GFT$\"3w*****pb&R+QF07$$!3))*****\\< 27&GFT$\"3?+++vY_FJF07$$!3)******H'>R`GFT$\"3/+++()ebWCF07$$!37+++WW#[ &GFT$\"3/+++L:#Hv\"F07$$!39+++IAnbGFT$\"3-+++25fa5F07$$!3-+++C>1cGFT$ \"3$******H^b,_$Ffo7$F][m$!3$******H^b,_$Ffo7$Fhjl$!3-+++25fa5F07$Fcjl $!3/+++L:#Hv\"F07$F^jl$!3/+++()ebWCF07$Fiil$!3?+++vY_FJF07$Fdil$!3w*** **pb&R+QF07$F_il$!3%*******37DiWF07$Fjhl$!3R+++,sh7^F07$Fehl$!3W+++;(o 8v&F07$F`hl$!3/+++P\"Q'yjF07$F[hl$!3'******4@OZ*pF07$Ffgl$!3z*******y \"4+wF07$Fagl$!3!******pC5_>)F07$F\\gl$!3a*****fM[1y)F07$Fgfl$!3U+++6E +d$*F07$Fbfl$!3Z******[%3\\#**F07$F]fl$!33+++'y/&[5FT7$Fhel$!36+++8k\" Q5\"FT7$Fcel$!3++++V#3&e6FT7$F^el$!33+++RNn77FT7$Fidl$!3'*******y#>kE \"FT7$Fddl$!3++++W$p)>8FT7$F_dl$!3!******4inJP\"FT7$Fjcl$!34+++`.[E9FT 7$Fecl$!33+++,m**z9FT7$F`cl$!30+++k[#R`\"FT7$F[cl$!3#******4'G[)e\"FT7 $Ffbl$!3!******ziwQk\"FT7$Fabl$!34+++3`E+*\\7;=FT7$Fbal$!33+++C:Av=FT7$F]al$!3$*******H8_M>FT7$Fh`l $!3#******H'\\S$*>FT7$Fc`l$!31+++N0@^?FT7$F^`l$!38+++\\LL2@FT7$Fi_l$!3 5+++T8Hh@FT7$Fd_l$!3%)*****z\"\\u7AFT7$F__l$!31+++ME[hAFT7$Fj^l$!30+++ @7R2BFT7$Fe^l$!3')*****p-C/N#FT7$F`^l$!3!******Hru0R#FT7$F[^l$!35+++$* y&yU#FT7$Ff]l$!3)******f\"fHiCFT7$Fa]l$!3/+++o-\"R\\#FT7$F\\]l$!3/+++. hrADFT7$Fg\\l$!3=+++`(=([DFT7$Fb\\l$!3#)*****>O6>d#FT7$F]\\l$!37+++gIF #f#FT7$Fh[l$!32+++Nsw4EFT7$Fc[l$!3#*******y&RVi#FT7$F^[l$!3.+++)e:fj#F T7$Fiz$!3-+++wrRWEFT7$F_z$!37+++jzl\\EFT7$Fjy$!3#******ztJ5l#FT7$Fey$! 3')*****zfO:l#FT7$F`y$!3#)*****\\LX6l#FT7$F[y$!3)******f6F)\\EFT7$Ffx$ !3))*****z*oaZEFT7$Fax$!36+++(pkUk#FT7$F\\x$!3<+++\"oM*REFT7$Fgw$!3?++ +6S]MEFT7$Fbw$!30+++v8\"zi#FT7$F]w$!3/+++\"3&3?EFT7$Fhv$!3$******\\cS4 h#FT7$Fcv$!3y*****R3x.g#FT7$F^v$!3%)*****fEt#)e#FT7$Fiu$!3))*****f.\"[ uDFT7$Fdu$!35+++,s\")eDFT7$F_u$!3/+++s;0TDFT7$Fjt$!38+++&z!*3_#FT7$Fet $!35+++\"y`z\\#FT7$F`t$!3!)*****z')R#FT7$F]r$!3&*******3!\\R7#FT7$Fhq$!3')***** *4$3o/#FT7$Fcq$!33+++@BCn>FT7$F^q$!3!******RMHp)=FT7$Fip$!3*******pQIn !=FT7$Fdp$!34+++u0(ps\"FT7$F_p$!3!******Ra(pZ;FT7$Fjo$!3)******R+R)o:F T7$Fdo$!35+++WXG!\\\"FT7$F_o$!3'******z^C>T\"FT7$Fjn$!3(******RsmOL\"F T7$Fen$!3/+++v=Wb7FT7$FV$!3'******fQ.s<\"FT7$FO$!33+++JV#*)4\"FT7$$!3X +++\"*f&zI'FK$!36+++kJf?5FT7$$!33+++QZ)pu\"FK$!3N+++IGyP')F07$$!3-+++Y /;fOF@$!3[+++O58oqF07$$!3y*****HT4E1&F:$!3;+++o0q(\\&F07$$!3!******>Qe K^$!#C$!3>+++p>)p#RF07$$!3m*****z))*>%*f!#E$!3'*******HU>cBF07$F($!3!) *****Hj\")R&yFfo7$F($\"3!)*****Hj\")R&yFfo-%&COLORG6&%$RGBG$\"#[!\"#$ \"\"\"!\"\"F(-%)POLYGONSG6gw7%F'7$$!+8]B]BFfo$\"+6jzq:!#57$$!$X\"Fd_nF (7%F\\`n7$$!+[%3m%fFT$\"+Z2fTJFa`nFb`n7%Ff`n7$$!+QAH%\\\"!#:$\"+E$eBr% Fa`nFb`n7%F\\an7$$!+tq?^9!#9$\"+*GvHG'Fa`nFb`n7%Fcan7$$!+SBpL$)Ffan$\" +?l4`yFa`nFb`n7%Fjan7$$!+]S.=M!#8$\"+*o\"4A%*Fa`nFb`n7%F`bn7$$!+#Hgr5 \"!#7$\"+JV#*)4\"!\"*Fb`n7%Fgbn7$$!+I?Xg=Fjbn$\"+'Q.s<\"F]cnFb`n7%F_cn 7$$!+iRW4IFjbn$\"+v=Wb7F]cnFb`n7%Fecn7$$!+vs$zq%Fjbn$\"+CnmL8F]cnFb`n7 %F[dn7$$!+(=N=:(Fjbn$\"+=X#>T\"F]cnFb`n7%Fadn7$$!+4.')e5!#6$\"+WXG!\\ \"F]cnFb`n7%Fgdn7$$!+5!*3L:Fjdn$\"+/!R)o:F]cnFb`n7%F^en7$$!+'pzw<#Fjdn $\"+WvpZ;F]cnFb`n7%Fden7$$!+!RqR/$Fjdn$\"+u0(ps\"F]cnFb`n7%Fjen7$$!+ln %*)>%Fjdn$\"+(QIn!=F]cnFb`n7%F`fn7$$!+ZW,IdFjdn$\"+W$Hp)=F]cnFb`n7%Fff n7$$!+aPlZxFjdn$\"+@BCn>F]cnFb`n7%F\\gn7$$!+/VuP5Fa`n$\"+5$3o/#F]cnFb` n7%Fbgn7$$!+q**[s8Fa`n$\"+4!\\R7#F]cnFb`n7%Fhgn7$$!+YGZ!y\"Fa`n$\"+FO@ '>#F]cnFb`n7%F^hn7$$!+JR6[AFa`n$\"+\"3\"=hAF]cnFb`n7%Fdhn7$$!+@z`]FFa` n$\"+-:gcFa`n$\"+&z!*3_#F]cnFb`n7%F^[o7$$!++0$z.'Fa`n$\"+s ;0TDF]cnFb`n7%Fd[o7$$!+n$G2W'Fa`n$\"+,s\")eDF]cnFb`n7%Fj[o7$$!+gzlHoFa `n$\"+O5[uDF]cnFb`n7%F`\\o7$$!+o?/1sFa`n$\"+mKF)e#F]cnFb`n7%Ff\\o7$$!+ $yh5d(Fa`n$\"+%3x.g#F]cnFb`n7%F\\]o7$$!+-wvDzFa`n$\"+l0%4h#F]cnFb`n7%F b]o7$$!+l+/r#)Fa`n$\"+\"3&3?EF]cnFb`n7%Fh]o7$$!+xOq2')Fa`n$\"+v8\"zi#F ]cnFb`n7%F^^o7$$!+z6WO*)Fa`n$\"+6S]MEF]cnFb`n7%Fd^o7$$!+.p&yD*Fa`n$\"+ \"oM*REF]cnFb`n7%Fj^o7$$!+G(yCd*Fa`n$\"+(pkUk#F]cnFb`n7%F`_o7$$!+X&o2) )*Fa`n$\"+)*oaZEF]cnFb`n7%Ff_o7$$!+IJJ=5F]cn$\"+;r#)\\EF]cnFb`n7%F\\`o 7$$!+!G#*z/\"F]cn$\"+N`9^EF]cnFb`n7%Fb`o7$$!+Kc9x5F]cn$\"+)fO:l#F]cnFb `n7%Fh`o7$$!+W3!e5\"F]cn$\"+Q<.^EF]cnFb`n7%F^ao7$$!+&Q#)R8\"F]cn$\"+jz l\\EF]cnFb`n7%Fdao7$$!+r>rh6F]cn$\"+R$Ruk#F]cnFb`n7%Fjao7$$!+C*3!*=\"F ]cn$\"+wrRWEF]cnFb`n7%F`bo7$$!+t@PU7F]cn$\"+)e:fj#F]cnFb`n7%Ffbo7$$!+5 ->%H\"F]cn$\"+z&RVi#F]cnFb`n7%F\\co7$$!+&))fXM\"F]cn$\"+Nsw4EF]cnFb`n7 %Fbco7$$!+()Hc$R\"F]cn$\"+gIF#f#F]cnFb`n7%Fhco7$$!+y9FT9F]cn$\"+i8\">d #F]cnFb`n7%F^do7$$!+*>_x[\"F]cn$\"+`(=([DF]cnFb`n7%Fddo7$$!+%QrI`\"F]c n$\"+.hrADF]cnFb`n7%Fjdo7$$!+r#*Hx:F]cn$\"+o-\"R\\#F]cnFb`n7%F`eo7$$!+ &3:0i\"F]cn$\"+;fHiCF]cnFb`n7%Ffeo7$$!+EE\"Gm\"F]cn$\"+$*y&yU#F]cnFb`n 7%F\\fo7$$!+PmI/F]cn$\"+T8Hh@F]cnFb`n7%F`ho7$$!+;EF]cn$\"+\\LL2@F]cnFb`n7%Ffho7$$ !+7?C*)>F]cn$\"+N0@^?F]cnFb`n7%F\\io7$$!+>+dK?F]cn$\"+j\\S$*>F]cnFb`n7 %Fbio7$$!+TGGx?F]cn$\"+I8_M>F]cnFb`n7%Fhio7$$!+#[/L7#F]cn$\"+C:Av=F]cn Fb`n7%F^jo7$$!+z!Q.<#F]cn$\"+#*\\7;=F]cnFb`n7%Fdjo7$$!+$)z!z@#F]cn$\"+ P9rd8F]cnFb`n7%Fj]p7$$!+O\\P%f#F]cn$ \"+z#>kE\"F]cnFb`n7%F`^p7$$!+]D2EEF]cn$\"+RNn77F]cnFb`n7%Ff^p7$$!+PD\\ bEF]cn$\"+V#3&e6F]cnFb`n7%F\\_p7$$!+]nk#o#F]cn$\"+8k\"Q5\"F]cnFb`n7%Fb _p7$$!++lb2FF]cn$\"+'y/&[5F]cnFb`n7%Fh_p7$$!+52DIFF]cn$\"+\\%3\\#**Fa` nFb`n7%F^`p7$$!+^_w]FF]cn$\"+6E+d$*Fa`nFb`n7%Fd`p7$$!+(>V\"pFF]cn$\"+Y $[1y)Fa`nFb`n7%Fj`p7$$!+eeV&y#F]cn$\"+Z-@&>)Fa`nFb`n7%F`ap7$$!+[Wq*z#F ]cn$\"+!z\"4+wFa`nFb`n7%Ffap7$$!+!GA?\"GF]cn$\"+6it%*pFa`nFb`n7%F\\bp7 $$!+\\tZAGF]cn$\"+P\"Q'yjFa`nFb`n7%Fbbp7$$!+I]R`GF]cn$\"+()ebWCFa`nFb`n7%Ffdp7$$!+WW#[&GF]cn $\"+L:#Hv\"Fa`nFb`n7%F\\ep7$$!+IAnbGF]cn$\"+25fa5Fa`nFb`n7%Fbep7$$!+C> 1cGF]cn$\"+8b:?NFjdnFb`n7%Fhep7$Fiep$!+8b:?NFjdnFb`n7%F^fp7$Fcep$!+25f a5Fa`nFb`n7%Fbfp7$F]ep$!+L:#Hv\"Fa`nFb`n7%Fffp7$Fgdp$!+()ebWCFa`nFb`n7 %Fjfp7$Fadp$!+vY_FJFa`nFb`n7%F^gp7$F[dp$!+dbR+QFa`nFb`n7%Fbgp7$Fecp$!+ 47DiWFa`nFb`n7%Ffgp7$F_cp$!+,sh7^Fa`nFb`n7%Fjgp7$Fibp$!+;(o8v&Fa`nFb`n 7%F^hp7$Fcbp$!+P\"Q'yjFa`nFb`n7%Fbhp7$F]bp$!+6it%*pFa`nFb`n7%Ffhp7$Fga p$!+!z\"4+wFa`nFb`n7%Fjhp7$Faap$!+Z-@&>)Fa`nFb`n7%F^ip7$F[ap$!+Y$[1y)F a`nFb`n7%Fbip7$Fe`p$!+6E+d$*Fa`nFb`n7%Ffip7$F_`p$!+\\%3\\#**Fa`nFb`n7% Fjip7$Fi_p$!+'y/&[5F]cnFb`n7%F^jp7$Fc_p$!+8k\"Q5\"F]cnFb`n7%Fbjp7$F]_p $!+V#3&e6F]cnFb`n7%Ffjp7$Fg^p$!+RNn77F]cnFb`n7%Fjjp7$Fa^p$!+z#>kE\"F]c nFb`n7%F^[q7$F[^p$!+W$p)>8F]cnFb`n7%Fb[q7$Fe]p$!+@w;t8F]cnFb`n7%Ff[q7$ F_]p$!+`.[E9F]cnFb`n7%Fj[q7$Fi\\p$!+,m**z9F]cnFb`n7%F^\\q7$Fc\\p$!+k[# R`\"F]cnFb`n7%Fb\\q7$F]\\p$!+hG[)e\"F]cnFb`n7%Ff\\q7$Fg[p$!+Gm(Qk\"F]c nFb`n7%Fj\\q7$Fa[p$!+3`E+ F]cnFb`n7%F^^q7$Fcio$!+j\\S$*>F]cnFb`n7%Fb^q7$F]io$!+N0@^?F]cnFb`n7%Ff ^q7$Fgho$!+\\LL2@F]cnFb`n7%Fj^q7$Faho$!+T8Hh@F]cnFb`n7%F^_q7$F[ho$!+= \\u7AF]cnFb`n7%Fb_q7$Fego$!+ME[hAF]cnFb`n7%Ff_q7$F_go$!+@7R2BF]cnFb`n7 %Fj_q7$Fifo$!+FSU]BF]cnFb`n7%F^`q7$Fcfo$!+8Zd!R#F]cnFb`n7%Fb`q7$F]fo$! +$*y&yU#F]cnFb`n7%Ff`q7$Fgeo$!+;fHiCF]cnFb`n7%Fj`q7$Faeo$!+o-\"R\\#F]c nFb`n7%F^aq7$F[eo$!+.hrADF]cnFb`n7%Fbaq7$Fedo$!+`(=([DF]cnFb`n7%Ffaq7$ F_do$!+i8\">d#F]cnFb`n7%Fjaq7$Fico$!+gIF#f#F]cnFb`n7%F^bq7$Fcco$!+Nsw4 EF]cnFb`n7%Fbbq7$F]co$!+z&RVi#F]cnFb`n7%Ffbq7$Fgbo$!+)e:fj#F]cnFb`n7%F jbq7$Fabo$!+wrRWEF]cnFb`n7%F^cq7$Feao$!+jzl\\EF]cnFb`n7%Fbcq7$F_ao$!+Q <.^EF]cnFb`n7%Ffcq7$Fi`o$!+)fO:l#F]cnFb`n7%Fjcq7$Fc`o$!+N`9^EF]cnFb`n7 %F^dq7$F]`o$!+;r#)\\EF]cnFb`n7%Fbdq7$Fg_o$!+)*oaZEF]cnFb`n7%Ffdq7$Fa_o $!+(pkUk#F]cnFb`n7%Fjdq7$F[_o$!+\"oM*REF]cnFb`n7%F^eq7$Fe^o$!+6S]MEF]c nFb`n7%Fbeq7$F_^o$!+v8\"zi#F]cnFb`n7%Ffeq7$Fi]o$!+\"3&3?EF]cnFb`n7%Fje q7$Fc]o$!+l0%4h#F]cnFb`n7%F^fq7$F]]o$!+%3x.g#F]cnFb`n7%Fbfq7$Fg\\o$!+m KF)e#F]cnFb`n7%Fffq7$Fa\\o$!+O5[uDF]cnFb`n7%Fjfq7$F[\\o$!+,s\")eDF]cnF b`n7%F^gq7$Fe[o$!+s;0TDF]cnFb`n7%Fbgq7$F_[o$!+&z!*3_#F]cnFb`n7%Ffgq7$F ijn$!+\"y`z\\#F]cnFb`n7%Fjgq7$Fcjn$!+o)R#F]cnFb`n7%Ffiq7$Fign$!+4!\\R7#F]cnFb`n7%Fjiq7$Fcgn$!+5$3o/#F]cnF b`n7%F^jq7$F]gn$!+@BCn>F]cnFb`n7%Fbjq7$Fgfn$!+W$Hp)=F]cnFb`n7%Ffjq7$Fa fn$!+(QIn!=F]cnFb`n7%Fjjq7$F[fn$!+u0(ps\"F]cnFb`n7%F^[r7$Feen$!+WvpZ;F ]cnFb`n7%Fb[r7$F_en$!+/!R)o:F]cnFb`n7%Ff[r7$Fhdn$!+WXG!\\\"F]cnFb`n7%F j[r7$Fbdn$!+=X#>T\"F]cnFb`n7%F^\\r7$F\\dn$!+CnmL8F]cnFb`n7%Fb\\r7$Ffcn $!+v=Wb7F]cnFb`n7%Ff\\r7$F`cn$!+'Q.s<\"F]cnFb`n7%Fj\\r7$Fhbn$!+JV#*)4 \"F]cnFb`n7%F^]r7$$!+\"*f&zI'Fcbn$!+kJf?5F]cnFb`n7%Fb]r7$$!+QZ)pu\"Fcb n$!+IGyP')Fa`nFb`n7%Fh]r7$$!+Y/;fOFfan$!+O58oqFa`nFb`n7%F^^r7$$!+8%4E1 &F_an$!+o0q(\\&Fa`nFb`n7%Fd^r7$$!+#QeK^$!#;$!+p>)p#RFa`nFb`n7%Fj^r7$$! +)))*>%*fF0$!+IU>cBFa`nFb`n7%Fa_r7$F($!+L;)R&yFjdnFb`n7%Fg_r7$F($\"+L; )R&yFjdnFb`n-F__n6&Fa_n$\"#&*Fd_n$\"\"#Fg_nF(-%&STYLEG6#%,PATCHNOGRIDG -F$6$7do7$$\"3/+++=qZtwF0$\"3'******H\\J!pKFT7$$\"33+++hY8EuF0$\"3/+++ w$owE$FT7$$\"3s*****R!>HurF0$\"3;+++s7QjKFT7$$\"30+++zE%)>pF0$\"35+++a y'fD$FT7$$\"35+++6!f\\m'F0$\"3#)*****z\\'=XKFT7$$\"3++++GJB7kF0$\"3$)* *****QMuIKFT7$$\"3E+++&3-\\;'F0$\"3;+++/^F7KFT7$$\"38+++x(>s#fF0$\"3!* *****fuB$*=$FT7$$\"3&)******>[50dF0$\"35+++]XIhJFT7$$\"3W++++=L2bF0$\" 31++++_YFJFT7$$\"3q*****>92!\\aF0$\"3$*******[Kt9JFT7$$\"3[+++WNc&R&F0 $\"3&)*****f6575$FT7$$\"3o*****p8YyM&F0$\"31+++2R&o3$FT7$$\"3:+++/\"yo I&F0$\"3?+++)QA;2$FT7$$\"3m*****\\71RF&F0$\"3%)******o`ZbIFT7$$\"3s*** **4J?IFT7$$\"3Y+++p )=+C&F0$\"36+++\"Qy7+$FT7$$\"3#)*****>jr!e_F0$\"3#)******QTK\")HFT7$$ \"3]*****RR!z&H&F0$\"3))*****>fk0'HFT7$$\"3;+++8*pnN&F0$\"37+++\">@#RH FT7$$\"3x*****zr6YW&F0$\"3\"******42kw\"HFT7$$\"3[*****p\")3Ac&F0$\"3A +++MTW'*GFT7$$\"3]*****zzw1r&F0$\"3))*****z-#GwGFT7$$\"3M+++=/G))eF0$ \"3++++['yz&GFT7$$\"3g*****>kb+4'F0$\"3&)*****Hk]A%GFT7$$\"3D+++x'Q'3j F0$\"39+++XScHGFT7$$\"3F+++SG'f`'F0$\"33+++t31?GFT7$$\"3[++++w&[w'F0$ \"3')******3Wg8GFT7$$\"3?+++mt()*)pF0$\"3*********p())4GFT7$$\"3_+++iQ P2sF0$\"3()*****RyP&3GFT7$$\"3]+++`'o^T(F0$\"3()*****zP%=4GFT7$$\"3d** ***\\zM@h(F0$\"39+++&y$\\6GFT7$$\"3!******4FPyz(F0$\"3@+++[+=:GFT7$$\" 3E+++u*pA(zF0$\"35+++cV+?GFT7$$\"3*)******G(yc8)F0$\"3')*****pfpd#GFT7 $$\"3y*****p\"yW)G)F0$\"3,+++aOJKGFT7$$\"3X*****4XH5V)F0$\"3=+++vf #*F0$\"31+++np75HFT7$$\"3.+++&4m.L*F0$\"32+++`Gy>HFT7$$\"3[+++GTw%R*F0 $\"39+++V[]HHFT7$$\"3o*****\\9=KX*F0$\"3*******\\?p#RHFT7$$\"31+++e;)f ]*F0$\"3'******4([0\\HFT7$$\"3Q+++T7H`&*F0$\"3++++RI%)eHFT7$$\"3e***** *)*zO&f*F0$\"37+++XohoHFT7$$\"35+++O*=Cj*F0$\"3%********3h$yHFT7$$\"3m ******)>QYm*F0$\"3!)*****4,i!))HFT7$$\"3%)*******>3Ap*F0$\"3A+++(32x*H FT7$$\"37+++X1I:(*F0$\"3!)*****\\)[G2IFT7$$\"3')*****f)yGf(*F0$\"3=+++ WP^NIFT7$$\"3Q+++1#3$o(*F0$\"3')*****p`EI R*F0$\"3#******fFz#zJFT7$$\"3-+++xsK]#*F0$\"3'******4Y'o(>$FT7$$\"3q** ***\\e*e*3*F0$\"3z*****f1)>SoKFT7$$\"3w**** *z8Q;f(F0$\"3%******H$3*)oKFTF^_n-Fi_n6eo7%7$$\"+=qZtwFa`n$\"+$\\J!pKF ]cn7$$\"+hY8EuFa`n$\"+w$owE$F]cn7$$\"#vFd_n$\"$.$Fd_n7%F[hs7$$\"+/>Hur Fa`n$\"+s7QjKF]cnF`hs7%Ffhs7$$\"+zE%)>pFa`n$\"+ay'fD$F]cnF`hs7%F\\is7$ $\"+6!f\\m'Fa`n$\"+)\\'=XKF]cnF`hs7%Fbis7$$\"+GJB7kFa`n$\"+RMuIKF]cnF` hs7%Fhis7$$\"+&3-\\;'Fa`n$\"+/^F7KF]cnF`hs7%F^js7$$\"+x(>s#fFa`n$\"+YP K*=$F]cnF`hs7%Fdjs7$$\"+?[50dFa`n$\"+]XIhJF]cnF`hs7%Fjjs7$$\"++=L2bFa` n$\"++_YFJF]cnF`hs7%F`[t7$$\"+Ur+\\aFa`n$\"+\\Kt9JF]cnF`hs7%Ff[t7$$\"+ WNc&R&Fa`n$\"+;,@,JF]cnF`hs7%F\\\\t7$$\"+Ph%yM&Fa`n$\"+2R&o3$F]cnF`hs7 %Fb\\t7$$\"+/\"yoI&Fa`n$\"+)QA;2$F]cnF`hs7%Fh\\t7$$\"+Dh!RF&Fa`n$\"+p` ZbIF]cnF`hs7%F^]t7$$\"+rhX]_Fa`n$\"+^)y$QIF]cnF`hs7%Fd]t7$$\"+z))RQ_Fa `n$\"+u>J?IF]cnF`hs7%Fj]t7$$\"+p)=+C&Fa`n$\"+\"Qy7+$F]cnF`hs7%F`^t7$$ \"+K;2e_Fa`n$\"+RTK\")HF]cnF`hs7%Ff^t7$$\"+%R!z&H&Fa`n$\"+#fk0'HF]cnF` hs7%F\\_t7$$\"+8*pnN&Fa`n$\"+\">@#RHF]cnF`hs7%Fb_t7$$\"+=vf#*Fa`n$\"+np75HF]cnF`hs7%Fhht7$$\"+ &4m.L*Fa`n$\"+`Gy>HF]cnF`hs7%F^it7$$\"+GTw%R*Fa`n$\"+V[]HHF]cnF`hs7%Fd it7$$\"+X\"=KX*Fa`n$\"+0#p#RHF]cnF`hs7%Fjit7$$\"+e;)f]*Fa`n$\"+r[0\\HF ]cnF`hs7%F`jt7$$\"+T7H`&*Fa`n$\"+RI%)eHF]cnF`hs7%Ffjt7$$\"+**zO&f*Fa`n $\"+XohoHF]cnF`hs7%F\\[u7$$\"+O*=Cj*Fa`n$\"+!4h$yHF]cnF`hs7%Fb[u7$$\"+ *>QYm*Fa`n$\"+6?1))HF]cnF`hs7%Fh[u7$$\"++#3Ap*Fa`n$\"+(32x*HF]cnF`hs7% F^\\u7$$\"+X1I:(*Fa`n$\"+&)[G2IF]cnF`hs7%Fd\\u7$$\"+')yGf(*Fa`n$\"+WP^ NIF]cnF`hs7%Fj\\u7$$\"+1#3$o(*Fa`n$\"+xT!G1$F]cnF`hs7%F`]u7$$\"+UUsX(* Fa`n$\"+f)f*)3$F]cnF`hs7%Ff]u7$$\"+8)eWp*Fa`n$\"+Du\"Q6$F]cnF`hs7%F\\^ u7$$\"+6))3<'*Fa`n$\"+koBPJF]cnF`hs7%Fb^u7$$\"+#y;f^*Fa`n$\"+\\Z4fJF]c nF`hs7%Fh^u7$$\"+Kl-$R*Fa`n$\"+w#z#zJF]cnF`hs7%F^_u7$$\"+xsK]#*Fa`n$\" +hko(>$F]cnF`hs7%Fd_u7$$\"+&e*e*3*Fa`n$\"+mq@9KF]cnF`hs7%Fj_u7$$\"+kbZ 7*)Fa`n$\"+uRxGKF]cnF`hs7%F``u7$$\"+'=l0s)Fa`n$\"+0)f7C$F]cnF`hs7%Ff`u 7$$\"+X.Q:&)Fa`n$\"+(fu:D$F]cnF`hs7%F\\au7$$\"+5\"3%)H)Fa`n$\"+(R8'fKF ]cnF`hs7%Fbau7$$\"+(oC62)Fa`n$\"+SoKF]cnF`hs7%F^bu7$$\"+Q\"Q;f(Fa`n$\"+L3*)oKF]cnF`hsF^`rFd`r-F$6$7 do7$F\\ar$!3'******H\\J!pKFT7$$\"37+++1\"4Y\"zF0$!3%******46VwE$FT7$$ \"3=+++n+%z9)F0$!3%*******fHljKFT7$$\"3#)*****p#3&>P)F0$!31+++YM>dKFT7 $$\"3!)*****z(z:&e)F0$!35+++3RQ[KFT7$$\"3_+++oW3'y)F0$!3)******H2MtB$F T7$$\"3'******\\@MK(*)F0$!3#)*****4&p9CKFT7$$\"35+++d&o]9*F0$!3,+++&Q@ *3KFT7$$\"3/++++H)**H*F0$!31+++4Wv\">$FT7$$\"3')*****>1#GO%*F0$!3)**** **HdVF<$FT7$$\"37+++bG:_&*F0$!31+++)Q*)>:$FT7$$\"3=+++4AjX'*F0$!3\")** ***p2)fHJFT7$$\"3^*****zzoXr*F0$!3')*****\\%[o0JFT7$$\"3))*****p[vlv*F 0$!3#)*****R-y.3$FT7$$\"39+++!)*o*o(*F0$!3!)*****RaCQ0$FT7$$\"3%)***** *4;p[(*F0$!3<+++4v>EIFT7$Fjas$!3A+++(32x*HFT7$F[as$!37+++XohoHFT7$Ff`s $!3++++RI%)eHFT7$Fa`s$!3'******4([0\\HFT7$F\\`s$!3*******\\?p#RHFT7$Fg _s$!39+++V[]HHFT7$Fb_s$!32+++`Gy>HFT7$F]_s$!31+++np75HFT7$Fh^s$!30+++u Sc+HFT7$Fc^s$!3#)*****z![7\"*GFT7$F^^s$!3))*****HHW=)GFT7$Fi]s$!3=+++^ IwsGFT7$Fd]s$!37+++Y\"GR'GFT7$F_]s$!3))*****fh%RbGFT7$Fj\\s$!3!)****** zsAZGFT7$Fe\\s$!3=+++@#RHFT7$F[gr$ !3))*****>fk0'HFT7$Fffr$!3#)******QTK\")HFT7$Fafr$!36+++\"Qy7+$FT7$F\\ fr$!3=+++u>J?IFT7$Fger$!30+++^)y$QIFT7$Fber$!3%)******o`ZbIFT7$F]er$!3 ?+++)QA;2$FT7$Fhdr$!31+++2R&o3$FT7$Fcdr$!3&)*****f6575$FT7$F^dr$!3$*** ****[Kt9JFT7$Ficr$!31++++_YFJFT7$$\"3#)*****4HM)pbF0$!3#)*****H@Y%RJFT 7$$\"3u*****H%[#fj&F0$!3)******p@92:$FT7$Fdcr$!35+++]XIhJFT7$$\"3=+++H #\\px&F0$!3;+++/*\\7<$FT7$$\"33+++6R00gF0$!3)******\\P0v>$FT7$$\"33+++ %>5lC'F0$!30+++lW!*=KFT7$$\"3_+++M?0'\\'F0$!3++++?a)fB$FT7$$\"3%****** R9>)\\nF0$!3))*******Qr\"\\KFT7$$\"3\\+++P;#[+(F0$!3'******RM+)eKFT7$$ \"39+++(GF'esF0$!37+++xp9lKFT7$$\"3W+++j'p\"4vF0$!3?+++!eQ%oKFT7$$\"3] *****p!\\iaxF0$!3/+++P)Fa`n$!+YM>dKF]cnFcdv7%F]ev7$$\"+yz:&e)Fa`n$!+3 RQ[KF]cnFcdv7%Fcev7$$\"+oW3'y)Fa`n$!+tSLPKF]cnFcdv7%Fiev7$$\"+:UBt*)Fa `n$!+^p9CKF]cnFcdv7%F_fv7$$\"+d&o]9*Fa`n$!+&Q@*3KF]cnFcdv7%Fefv7$$\"++ H)**H*Fa`n$!+4Wv\">$F]cnFcdv7%F[gv7$$\"+i?GO%*Fa`n$!+tNusJF]cnFcdv7%Fa gv7$$\"+bG:_&*Fa`n$!+)Q*)>:$F]cnFcdv7%Fggv7$$\"+4AjX'*Fa`n$!+x!)fHJF]c nFcdv7%F]hv7$$\"+)zoXr*Fa`n$!+X[o0JF]cnFcdv7%Fchv7$$\"+([vlv*Fa`n$!+C! y.3$F]cnFcdv7%Fihv7$$\"+!)*o*o(*Fa`n$!+WX#Q0$F]cnFcdv7%F_iv7$$\"+5;p[( *Fa`n$!+4v>EIF]cnFcdv7%Feiv7$F_\\u$!+(32x*HF]cnFcdv7%F[jv7$F][u$!+Xoho HF]cnFcdv7%F_jv7$Fgjt$!+RI%)eHF]cnFcdv7%Fcjv7$Fajt$!+r[0\\HF]cnFcdv7%F gjv7$F[jt$!+0#p#RHF]cnFcdv7%F[[w7$Feit$!+V[]HHF]cnFcdv7%F_[w7$F_it$!+` Gy>HF]cnFcdv7%Fc[w7$Fiht$!+np75HF]cnFcdv7%Fg[w7$Fcht$!+uSc+HF]cnFcdv7% F[\\w7$F]ht$!+3[7\"*GF]cnFcdv7%F_\\w7$Fggt$!+$HW=)GF]cnFcdv7%Fc\\w7$Fa gt$!+^IwsGF]cnFcdv7%Fg\\w7$F[gt$!+Y\"GR'GF]cnFcdv7%F[]w7$Feft$!+;YRbGF ]cnFcdv7%F_]w7$F_ft$!+!GFs%GF]cnFcdv7%Fc]w7$Fiet$!+@#RHF]cnFcdv7%F[bw7$F]_ t$!+#fk0'HF]cnFcdv7%F_bw7$Fg^t$!+RTK\")HF]cnFcdv7%Fcbw7$Fa^t$!+\"Qy7+$ F]cnFcdv7%Fgbw7$F[^t$!+u>J?IF]cnFcdv7%F[cw7$Fe]t$!+^)y$QIF]cnFcdv7%F_c w7$F_]t$!+p`ZbIF]cnFcdv7%Fccw7$Fi\\t$!+)QA;2$F]cnFcdv7%Fgcw7$Fc\\t$!+2 R&o3$F]cnFcdv7%F[dw7$F]\\t$!+;,@,JF]cnFcdv7%F_dw7$Fg[t$!+\\Kt9JF]cnFcd v7%Fcdw7$Fa[t$!++_YFJF]cnFcdv7%Fgdw7$$\"+\"HM)pbFa`n$!+8iWRJF]cnFcdv7% F[ew7$$\"+V[#fj&Fa`n$!+$F]cnF cdv7%Fafw7$$\"+%>5lC'Fa`n$!+lW!*=KF]cnFcdv7%Fgfw7$$\"+M?0'\\'Fa`n$!+?a )fB$F]cnFcdv7%F]gw7$$\"+W\">)\\nFa`n$!+!Rr\"\\KF]cnFcdv7%Fcgw7$$\"+P;# [+(Fa`n$!+W.!)eKF]cnFcdv7%Figw7$$\"+(GF'esFa`n$!+xp9lKF]cnFcdv7%F_hw7$ $\"+j'p\"4vFa`n$!+!eQ%oKF]cnFcdv7%Fehw7$$\"+2\\iaxFa`n$!+ " 0 "" {MPLTEXT 1 0 321 "R := z -> 1+z+1/2*z^2+1/6*z^3+1/24*z^4+1/120*z^5+1/720*z^6-1/2160 *z^7:\nDigits := 25:\npts := []: z0 := 0:\nfor ct from 0 to 80 do\n \+ zz := newton(R(z)=exp(ct*Pi/100*I),z=z0):\n z0 := zz:\n pts := [op (pts),[surd(Re(zz),11),Im(zz)]]:\nend do:\nplot(pts,color=COLOR(RGB,.9 5,.1,0),thickness=2,font=[HELVETICA,9]);\nDigits := 10:" }}{PARA 13 " " 1 "" {GLPLOT2D 220 305 305 {PLOTDATA 2 "6(-%'CURVESG6#7]p7$$\"\"!F)F (7$$!:\\*F-$\":qaqYv$)=k&zxC%*F-7$$!:o &oGgO#=pY!)H8\"!#D$\":%oW5'=A(=eqjc7F=7$$!:6$z?*RL!fo#QCL\"F=$\":J#GZD #Ro8J'zq:F=7$$!:F5DgK@#)y1G6_\"F=$\":YR^=(z)fr`b\\)=F=7$$!:TkFbw`H'RTF ,#F=7$$!:%f5fU)3z![1Qu=F=$\":M)z'G=cUirtK^#F=7$ $!:Sz'R1(\\(pL\\_T?F=$\":0(Hzr=%H#oCVFGF=7$$!:u(*Q$oA3ps=^.AF=$\":')>I (>]>LZ2fTJF=7$$!:R$y6Kn[w0D'4O#F=$\":z!3A?s>?T#[dX$F=7$$!:\"fN'Rc[GTPo V^#F=$\":x=91)G^<'\\/*pPF=7$$!:eu?%H,+\"\\eETm#F=$\":Mz(3?JfIy)eS3%F=7 $$!:S'HVV:Sf%Rj0\"GF=$\":6kN1jKeBr%F=7$$!:-P$R!QKr?Q@%zO$F=$\":1[[$pu.x!GiZl&F=7$$! :%=]^!e(*y*)G76]$F=$\":\"3#Q`*=]3dj()ofF=7$$!:t4OFPxax9&4KOF=$\":&=bx& yn!)*)GvHG'F=7$$!:Aa)y\")G*GZw+5w$F=$\":(z4&34CjGTbqf'F=7$$!:,f4*)y'y/ \"zDz)QF=$\":WK8aPYVJa76\"pF=7$$!:Yfo`6%fg@a&H,%F=$\":kJ;@%oNkw>9DsF=7 $$!:Bph],+N.,lh8%F=$\":`WiUI5ym^Q\"RvF=7$$!:Fox8\"[Y\")>9idUF=$\":sUvM *4:K?l4`yF=7$$!:Z^_/2Gnf;%QxVF=$\":4\"\\CCp'GB(*4q;)F=7$$!:3)QW&**o*>/ k]&\\%F=$\":Ia$fQSd,*es3[)F=7$$!:P9IWvz2FzN?h%F=$\":CKMmJ\"G7bzn%z)F=7 $$!:NN^_%4vkg_,FZF=$\":'G5)3;sbip>%3\"*F=7$$!:R<>)pv$HFq$[S[F=$\":^`S3 v5M*)o\"4A%*F=7$$!:M&G3B#G%*phwC&\\F=$\":nb\"*3]Q`&4$)oN(*F=7$$!:o!=bc U/r)pEI1&F=$\":8C6?]>qAZ?\\+\"!#C7$$!:%)GXiqY[-Sk@<&F=$\":zCyU2'p#3sji .\"Ffu7$$!:yU:t.roUY=*z_F=$\":\\W'\\%y9lDN)fn5Ffu7$$!:\"3VCus]ZNkJ'Q&F =$\":h(z$zZ(4@JV#*)4\"Ffu7$$!:I]SNH?rW:&Q\"\\&F=$\":j3NGI.-6&=CI6Ffu7$ $!:0,%\\F3J%eB^^f&F=$\":ZoWz]lJCQ^:;\"Ffu7$$!:U$oY=&fY9`Twp&F=$\":!)f> 'pD_>=P&G>\"Ffu7$$!:9%f)=#)[m*yN)))z&F=$\"::u;hG@j>-]TA\"Ffu7$$!:EMqj \\&e5Ng!*)*eF=$\":=H\")\\4LOa(=Wb7Ffu7$$!:E&=zh?5A#4Rx*fF=$\":)4x\"HxA ]XKJnG\"Ffu7$$!:.%[zqzZzw[T&4'F=$\":DKk9n6zE#4-=8Ffu7$$!:pXurcVsx#y'>> 'F=$\":0oh*>PEApPJ\\8Ffu7$$!:7d4&)Q](Q_]V(G'F=$\":AG\\-C3B:a81Q\"Ffu7$ $!:SC^qeWhllc=Q'F=$\":p;!3SnkhT\"Ffu7$$!:&p&*=(3BAh/w_Z'F=$\":=PTM S*Qs'e^KW\"Ffu7$$!:!pzM:8zu$>Sxc'F=$\":?*oK29pdV-gu9Ffu7$$!:Y_kCtCOW') *HfmF=$\":$R!enU.g:cwf]\"Ffu7$$!:xnm_Rcu!*y4+v'F=$\":8AD&3A&Q\"QrQP:Ff u7$$!:$eV9x+R'3xG*RoF=$\":]J\"p+jO)R+R)o:Ffu7$$!:(>cl(H[7Bq>\"HpF=$\": %RD2n3\"RW[R.g\"Ffu7$$!:J:%=\\]_Vg%\\w,(F=$\":u0`ysa#G)*f*=j\"Ffu7$$!: P!fMRICiL')e0rF=$\":&H#HAhD()Qx:Nm\"Ffu7$$!:11[$**)*R195,$>(F=$\":nE^$ pVt\\UHX(F=$\":2Hy\"p!*= \\c!R2z\"Ffu7$$!:_/bi&otj#pf!RvF=$\"::G8.S,TV7SF#=Ffu7$$!:Tdl&ecf*=,G] i(F=$\":e0\"4[&=Ffu7$$!:(funN\"4&e(*)**3r(F=$\":e#ekR8z\"QMHp)= Ffu7$$!:=>]FRDhcw2nz(F=$\":dl1XF[;6rt!>>Ffu7$$!:rYd(3c,RkZX#)yF=$\":yw vrvi!zx;?^>Ffu7$$!:fm&*Q%Q3U>M5ozF=$\":H2v%zUT=:eD$)>Ffu7$$!:bry_d&=YD Ec`!)F=$\":7!)*\\!*eHo_$e^,#Ffu7$$!:yJB`ShN3Iu'Q\")F=$\":9)*H:qv@-J3o/ #Ffu7$$!:$)HJ5p\\jZA-KA)F=$\":f=Rg0ZA>=!3y?Ffu7$$!:*=+**>gS5fo#oI)F=$ \":tt2&pbdA'RD)3@Ffu7$$!:)4#35bP%*Qk]\"*Q)F=$\":cam7\"Qj-Dx()Q@Ffu7$$! :G%\\6x)pmQO<(p%)F=$\":R!z#pBS'>yJ1o@Ffu7$$!:(z9)>Dl&f'*f/[&)F=$\":n(f db)4*>FO@'>#Ffu7$$!:kNg)o5e<*[uOi)F=$\":`GiP^kXH$==BAFfu7$$!:h()zSz.I` P/ip)F=$\":@w\\a!pDCJX&)[AFfu7$$!:*o\\*R@98G%eLl()F=$\":vHQ%*[_L6^gJF# Ffu7$$!:0QG))e'ygdf)3$))F=$\":sZC(['f))4AsgH#Ffu7$$!:L_)*=L8vD " 0 "" {MPLTEXT 1 0 1 ";" }}}}{PARA 0 "" 0 "" {TEXT -1 38 " #-------------------------------------" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}}{PARA 0 "" 0 "" {TEXT -1 26 "#================= ========" }}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 26 "a scheme with simple \+ nodes" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 85 " For early schemes constructed by hand it was convenient to work with \+ \"simple\" nodes. " }}{PARA 0 "" 0 "" {TEXT -1 123 "The two Butcher sc hemes have nodes which are rational numbers with numerator and denomin ator each a digit between 1 and 6. " }}{PARA 0 "" 0 "" {TEXT -1 81 "Th e scheme constructed in this section has nodes which are simple in thi s sense. " }}{PARA 0 "" 0 "" {TEXT -1 73 "Like the previous scheme of \+ Butcher, this scheme has no negative weights." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}{PARA 0 "" 0 "" {TEXT -1 39 "#--------------------------------------" }} {SECT 1 {PARA 4 "" 0 "" {TEXT -1 19 "checking the scheme" }}{PARA 0 " " 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}} {SECT 1 {PARA 4 "" 0 "" {TEXT -1 26 "coefficients of the scheme" }} {PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 450 "ee := \{c[2]=1/6,\nc[3]=1/5,\nc[4]=1/3,\nc[5]=2/3,\nc[6]=3/4,\nc[ 7]=1,\n\na[2,1]=1/6,\na[3,1]=2/25,\na[3,2]=3/25,\na[4,1]=2/27,\na[4,2] =-1/9,\na[4,3]=10/27,\na[5,1]=10/27,\na[5,2]=-2/9,\na[5,3]=-35/54,\na[ 5,4]=7/6,\na[6,1]=-9/256,\na[6,2]=9/64,\na[6,3]=165/448,\na[6,4]=0,\na [6,5]=495/1792,\na[7,1]=4/19,\na[7,2]=-3/19,\na[7,3]=-305/1463,\na[7,4 ]=81/95,\na[7,5]=-90/133,\na[7,6]=1024/1045,\n\nb[1]=3/40,\nb[2]=0,\nb [3]=625/3696,\nb[4]=27/100,\nb[5]=27/280,\nb[6]=256/825,\nb[7]=19/240 \}:" }}}{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 91 "subs(ee,matrix([seq([c[i],seq(a[i,j],j=1..i -1),``$(8-i)],i=2..7),\n[``,seq(b[i],i=1..7)]]));" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#K%'matrixG6#7)7*#\"\"\"\"\"'F(%!GF+F+F+F+F+7*#F)\"\"&# \"\"#\"#D#\"\"$F1F+F+F+F+F+7*#F)F3#F0\"#F#!\"\"\"\"*#\"#5F7F+F+F+F+7*# F0F3F;#!\"#F:#!#N\"#a#\"\"(F*F+F+F+7*#F3\"\"%#!\"*\"$c##F:\"#k#\"$l\" \"$[%\"\"!#\"$&\\\"%#z\"F+F+7*F)#FH\"#>#!\"$FW#!$0$\"%j9#\"#\")\"#&*#! #!*\"$L\"#\"%C5\"%X5F+7*F+#F3\"#SFQ#\"$D'\"%'p$#F7\"$+\"#F7\"$!G#FK\"$ D)#FW\"$S#Q)pprint176\"" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 " " 0 "" {TEXT -1 7 "Check: " }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 148 "RK6_7eqs := [op(RowSumConditions(7,'expanded')),op(OrderCondition s(6,7,'expanded'))]:\nsimplify(subs(ee,RK6_7eqs)):\nmap(u->lhs(u)-rhs( u),%);\nnops(%);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#7M\"\"!F$F$F$F$F$F $F$F$F$F$F$F$F$F$F$F$F$F$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#\"#V" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 86 "Next we set-up stage-order cond tions to check for stage-orders from 2 to 4 inclusive. " }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 85 "for ct from 2 to 4 do\n so||ct||_ 7 := StageOrderConditions(ct,7,'expanded');\nend do:" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 58 "Stages 3 to 7 have the following respective stage-orders. " }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 173 "[seq([seq(expand(subs(ee,so||i||_7[j])),i=2..4)],j=1 ..5)]:\nmap(proc(L) local i; for i to nops(L) do if not evalb(L[i]) th en break end if end do; i end proc,%):\nsimplify(%); " }}{PARA 11 "" 1 "" {XPPMATH 20 "6#7'\"\"#F$F$F$F$" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 42 "Thus stages 5, 6 and 7 have stage-order 3." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 32 "#- ------------------------------" }}{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 .. 7) = b[j]*(1-c[j]);" "6#/-%$SumG6$*&&%\" bG6#%\"iG\"\"\"&%\"aG6$F+%\"jGF,/F+;,&F0F,F,F,\"\"(*&&F)6#F0F,,&F,F,&% \"cG6#F0!\"\"F," }{TEXT -1 3 ", " }{XPPEDIT 18 0 "j=1" "6#/%\"jG\"\" \"" }{TEXT -1 7 " . . 7 " }}{PARA 0 "" 0 "" {TEXT -1 8 "(where " } {XPPEDIT 18 0 "c[1]=0" "6#/&%\"cG6#\"\"\"\"\"!" }{TEXT -1 17 " ) are s atisfied." }}{PARA 256 "" 0 "" {TEXT -1 1 " " }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 162 "[Sum(b[i]*a[i,1],i=2..7)=b[1],seq(Sum(b[i]*a[i,j] ,i=j+1..7)=b[j]*(1-c[j]),j=2..6)];\neval(subs(Sum=add,%)):\nsubs(ee,%) :\nmap(u->`if`(lhs(u)=rhs(u),0,1),%);\nnops(%);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#7(/-%$SumG6$*&&%\"bG6#%\"iG\"\"\"&%\"aG6$F,F-F-/F,;\"\" #\"\"(&F*6#F-/-F&6$*&F)F-&F/6$F,F3F-/F,;\"\"$F4*&&F*6#F3F-,&F-F-&%\"cG FB!\"\"F-/-F&6$*&F)F-&F/6$F,F?F-/F,;\"\"%F4*&&F*6#F?F-,&F-F-&FEFRFFF-/ -F&6$*&F)F-&F/6$F,FOF-/F,;\"\"&F4*&&F*6#FOF-,&F-F-&FEFjnFFF-/-F&6$*&F) F-&F/6$F,FgnF-/F,;\"\"'F4*&&F*6#FgnF-,&F-F-&FEFhoFFF-/-F&6$*&F)F-&F/6$ F,FeoF-/F,;F4F4*&&F*6#FeoF-,&F-F-&FEFepFFF-" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#7(\"\"!F$F$F$F$F$" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#\" \"'" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 27 "T he simplifying condition: " }}{PARA 256 "" 0 "" {TEXT -1 1 " " } {XPPEDIT 18 0 "Sum(b[i]*a[i,2],i = 3 .. 7) = 0;" "6#/-%$SumG6$*&&%\"bG 6#%\"iG\"\"\"&%\"aG6$F+\"\"#F,/F+;\"\"$\"\"(\"\"!" }{TEXT -1 1 " " }} {PARA 0 "" 0 "" {TEXT -1 13 "is satisfied." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 59 "Sum(b[i]*a[i,2],i= 3..7);\neval(subs(Sum=add,%));\nsubs(ee,%);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#-%$SumG6$*&&%\"bG6#%\"iG\"\"\"&%\"aG6$F*\"\"#F+/F*;\"\" $\"\"(" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#,,*&&%\"bG6#\"\"$\"\"\"&%\"a G6$F(\"\"#F)F)*&&F&6#\"\"%F)&F+6$F1F-F)F)*&&F&6#\"\"&F)&F+6$F7F-F)F)*& &F&6#\"\"'F)&F+6$F=F-F)F)*&&F&6#\"\"(F)&F+6$FCF-F)F)" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#\"\"!" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 " " 0 "" {TEXT -1 27 "The simplifying condition: " }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "Sum(b[i]*c[i]*a[i,2],i = 3 .. 7) = 0;" "6#/-%$SumG6$*(&%\"bG6#%\"iG\"\"\"&%\"cG6#F+F,&%\"aG6$F+\"\"#F,/F+;\" \"$\"\"(\"\"!" }{TEXT -1 1 " " }}{PARA 0 "" 0 "" {TEXT -1 13 "is satis fied." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 64 "Sum(b[i]*c[i]*a[i,2],i=3..7);\neval(subs(Sum=add,%)); \nsubs(ee,%);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#-%$SumG6$*(&%\"bG6#% \"iG\"\"\"&%\"cGF)F+&%\"aG6$F*\"\"#F+/F*;\"\"$\"\"(" }}{PARA 11 "" 1 " " {XPPMATH 20 "6#,,*(&%\"bG6#\"\"$\"\"\"&%\"cGF'F)&%\"aG6$F(\"\"#F)F)* (&F&6#\"\"%F)&F+F2F)&F-6$F3F/F)F)*(&F&6#\"\"&F)&F+F9F)&F-6$F:F/F)F)*(& F&6#\"\"'F)&F+F@F)&F-6$FAF/F)F)*(&F&6#\"\"(F)&F+FGF)&F-6$FHF/F)F)" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#\"\"!" }}}{PARA 0 "" 0 "" {TEXT -1 0 " " }}{PARA 0 "" 0 "" {TEXT -1 27 "The simplifying condition: " }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "Sum(b[i]*c[i]^2*a[i,2],i = \+ 3 .. 7) = 0;" "6#/-%$SumG6$*(&%\"bG6#%\"iG\"\"\"*$&%\"cG6#F+\"\"#F,&% \"aG6$F+F1F,/F+;\"\"$\"\"(\"\"!" }{TEXT -1 1 " " }}{PARA 0 "" 0 "" {TEXT -1 13 "is satisfied." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 66 "Sum(b[i]*c[i]^2*a[i,2],i=3..7);\nev al(subs(Sum=add,%));\nsubs(ee,%);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#- %$SumG6$*(&%\"bG6#%\"iG\"\"\")&%\"cGF)\"\"#F+&%\"aG6$F*F/F+/F*;\"\"$\" \"(" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#,,*(&%\"bG6#\"\"$\"\"\")&%\"cGF '\"\"#F)&%\"aG6$F(F-F)F)*(&F&6#\"\"%F))&F,F3F-F)&F/6$F4F-F)F)*(&F&6#\" \"&F))&F,F;F-F)&F/6$F " 0 "" {MPLTEXT 1 0 85 "Su m(b[i]*c[i]*Sum(a[i,j]*a[j,2],j=3..i-1),i=3..7);\neval(subs(Sum=add,%) );\nsubs(ee,%);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#-%$SumG6$*(&%\"bG6# %\"iG\"\"\"&%\"cGF)F+-F$6$*&&%\"aG6$F*%\"jGF+&F26$F4\"\"#F+/F4;\"\"$,& F*F+F+!\"\"F+/F*;F:\"\"(" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#,***&%\"bG 6#\"\"%\"\"\"&%\"cGF'F)&%\"aG6$F(\"\"$F)&F-6$F/\"\"#F)F)*(&F&6#\"\"&F) &F+F5F),&*&&F-6$F6F/F)F0F)F)*&&F-6$F6F(F)&F-6$F(F2F)F)F)F)*(&F&6#\"\"' F)&F+FCF),(*&&F-6$FDF/F)F0F)F)*&&F-6$FDF(F)F?F)F)*&&F-6$FDF6F)&F-6$F6F 2F)F)F)F)*(&F&6#\"\"(F)&F+FTF),**&&F-6$FUF/F)F0F)F)*&&F-6$FUF(F)F?F)F) *&&F-6$FUF6F)FPF)F)*&&F-6$FUFDF)&F-6$FDF2F)F)F)F)" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#\"\"!" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 114 "We can calculate the principal error norm of the orde r 6 scheme, that is, the 2-norm of the principal error terms." }} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 158 "errterms6_7 := PrincipalErr orTerms(6,7,'expanded'):\nsm := 0:\nfor ct to nops(errterms6_7) do\n \+ sm := sm+(evalf(subs(ee,errterms6_7[ct])))^2;\nend do:\nsqrt(sm);" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#$\"+'3V\\[#!#8" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}}{PARA 0 "" 0 "" {TEXT -1 39 "#--------------------------------------" }} {SECT 1 {PARA 4 "" 0 "" {TEXT -1 32 "short construction of the scheme " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT 260 4 "Note " }{TEXT -1 123 ": This scheme was constructed in a step by step manne r in a previous section making use of \"alternative\" order conditions . " }}{PARA 0 "" 0 "" {TEXT -1 77 "In this subsection we construct the scheme using \"standard\" order conditions." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 192 "We set up a system of equation s using a selection of \"simple\" order conditions and incorporate the row-sum conditions together with the stage-order equations to ensure \+ that stages 3 to 7 have " }{TEXT 260 13 "stage-order 2" }{TEXT -1 1 ". " }}{PARA 0 "" 0 "" {TEXT -1 48 "We also incorporate the simplifying c onditions: " }}{PARA 256 "" 0 "" {TEXT -1 2 " " }{XPPEDIT 18 0 "Sum(b [i]*a[i,j],i = j+1 .. 7) = b[j]*(1-c[j]);" "6#/-%$SumG6$*&&%\"bG6#%\"i G\"\"\"&%\"aG6$F+%\"jGF,/F+;,&F0F,F,F,\"\"(*&&F)6#F0F,,&F,F,&%\"cG6#F0 !\"\"F," }{TEXT -1 1 " " }}{PARA 0 "" 0 "" {TEXT -1 5 "for " } {XPPEDIT 18 0 "j = 3;" "6#/%\"jG\"\"$" }{TEXT -1 62 ", 4, 5, 6, toget her with the further simplifying conditions: " }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "Sum(b[i]*c[i]*a[i,2],i = 3 .. 7) = 0;" "6#/-%$SumG6$*(&%\"bG6#%\"iG\"\"\"&%\"cG6#F+F,&%\"aG6$F+\"\"#F,/F+;\" \"$\"\"(\"\"!" }{TEXT -1 8 ", " }{XPPEDIT 18 0 "Sum(b[i]*c[i]*Su m(a[i,j]*a[j,2],j = 3 .. i-1),i = 3 .. 7) = 0;" "6#/-%$SumG6$*(&%\"bG6 #%\"iG\"\"\"&%\"cG6#F+F,-F%6$*&&%\"aG6$F+%\"jGF,&F46$F6\"\"#F,/F6;\"\" $,&F+F,F,!\"\"F,/F+;F<\"\"(\"\"!" }{TEXT -1 8 ", " }{XPPEDIT 18 0 "Sum(b[i]*c[i]^2*a[i,2],i = 3 .. 7) = 0;" "6#/-%$SumG6$*(&%\"bG6#%\" iG\"\"\"*$&%\"cG6#F+\"\"#F,&%\"aG6$F+F1F,/F+;\"\"$\"\"(\"\"!" }{TEXT -1 2 ". " }}{PARA 0 "" 0 "" {TEXT -1 1 " " }}{PARA 0 "" 0 "" {TEXT -1 76 "The simple order conditions used are given (in abreviated form) as follows. " }}{PARA 0 "" 0 "" {TEXT 260 4 "Note" }{TEXT -1 86 ": We us e one additional order condition #28 beyond those used for the previou s scheme." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 184 "SO6 := SimpleOrderConditions(6):\n[seq([i,SO6[i]],i= [1,2,4,8,13,16,24,28,29,32])]:\nlinalg[augment](linalg[delcols](%,2..2 ),matrix([[` `]$(linalg[rowdim](%))]),linalg[delcols](%,1..1));" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#K%'matrixG6#7,7%\"\"\"%#~~G/*&%\"bGF(% \"eGF(F(7%\"\"#F)/*&F,F(%\"cGF(#F(F/7%\"\"%F)/*&F,F()F2F/F(#F(\"\"$7% \"\")F)/*&F,F()F2F:F(#F(F57%\"#8F)/*(F,F(F2F(-%!G6#*&F8F(%\"aGF(F(#F( \"#:7%\"#;F)/*&F,F()F2F5F(#F(\"\"&7%\"#CF)/*(F,F(F2F(-FF6#*&FIF(FEF(F( #F(\"#s7%\"#GF)/*(F,F(F8F(FEF(#F(\"#=7%\"#HF)/*(F,F(F2F(-FF6#*&F?F(FIF (F(#F(FT7%\"#KF)/*&F,F()F2FRF(#F(\"\"'Q(pprint06\"" }}}{PARA 0 "" 0 " " {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 462 "SO6_7 := \+ SimpleOrderConditions(6,7,'expanded'):\nord_cdns := [seq(SO6_7[i],i=[1 ,2,4,8,13,16,24,28,29,32])]:\nSO_eqs := [op(RowSumConditions(7,'expand ed')),op(StageOrderConditions(2,7,'expanded'))]:\nsimp_eqs := [seq(add (b[i]*a[i,j],i=j+1..7)=b[j]*(1-c[j]),j=[3,4,5,6]),\n add( b[i]*c[i]*a[i,2],i=3..7)=0,add(b[i]*c[i]*add(a[i,j]*a[j,2],j=3..i-1),i =3..7)=0,\n add(b[i]*c[i]^2*a[i,2],i=3..7)=0]:\ncdns := \+ [op(ord_cdns),op(simp_eqs),op(SO_eqs)]:" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 21 "We specify the nodes " }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "c[2] = 1/6;" "6#/&%\"cG6#\" \"#*&\"\"\"F)\"\"'!\"\"" }{TEXT -1 3 ", " }{XPPEDIT 18 0 "c[3] = 1/5; " "6#/&%\"cG6#\"\"$*&\"\"\"F)\"\"&!\"\"" }{TEXT -1 3 ", " }{XPPEDIT 18 0 "c[5] = 2/3;" "6#/&%\"cG6#\"\"&*&\"\"#\"\"\"\"\"$!\"\"" }{TEXT -1 3 ", " }{XPPEDIT 18 0 "c[6] = 3/4;" "6#/&%\"cG6#\"\"'*&\"\"$\"\"\" \"\"%!\"\"" }{TEXT -1 3 ", " }{XPPEDIT 18 0 "c[7]=1" "6#/&%\"cG6#\"\" (\"\"\"" }{TEXT -1 1 "," }}{PARA 0 "" 0 "" {TEXT -1 16 " and the weigh t " }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "b[2]=0" "6#/&% \"bG6#\"\"#\"\"!" }{TEXT -1 2 ". " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 116 "e1 := \{c[2]=1/6,c[3]=1/5,c [5]=2/3,c[6]=3/4,c[7]=1,b[2]=0\}:\neqns := subs(e1,cdns):\nnops(eqns); \nindets(eqns);\nnops(%);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#\"#G" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#<>&%\"aG6$\"\"%\"\"#&F%6$F'\"\"$&F%6$F (\"\"\"&F%6$F+F.&F%6$\"\"&F(&F%6$F3F+&F%6$F+F(&F%6$F'F.&%\"cG6#F'&F%6$ F3F.&%\"bG6#F3&F@6#\"\"'&F@6#\"\"(&F@6#F.&F@6#F+&F@F<&F%6$FGF3&F%6$FGF D&F%6$FGF+&F%6$FGF'&F%6$FGF.&F%6$FGF(&F%6$FDF'&F%6$FDF3&F%6$FDF(&F%6$F DF+&F%6$F3F'&F%6$FDF." }}{PARA 11 "" 1 "" {XPPMATH 20 "6#\"#G" }}} {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 51 "There are 28 equations and 28 unknown coefficients." }}{EXCHG {PARA 0 "> " 0 " " {MPLTEXT 1 0 22 "infolevel[solve] := 4:" }}}{EXCHG {PARA 0 "> " 0 " " {MPLTEXT 1 0 69 "e2 := solve(\{op(eqns)\}):\ninfolevel[solve] := 0: \ne3 := `union`(e1,e2):" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 2 "e3" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 516 "e3 := \{c[7] = 1, c[4] = 1/3, a[3,2] = 3/25, b[2] = \+ 0, a[4,3] = 10/27, a[4,1] = 2/27, a[7,3] = -305/1463, a[6,3] = 165/448 , a[6,4] = 0, a[5,3] = -35/54, a[4,2] = -1/9, b[3] = 625/3696, a[6,5] \+ = 495/1792, a[3,1] = 2/25, a[7,4] = 81/95, b[5] = 27/280, a[7,2] = -3/ 19, b[1] = 3/40, b[7] = 19/240, c[3] = 1/5, c[5] = 2/3, c[2] = 1/6, a[ 5,2] = -2/9, a[6,1] = -9/256, a[7,1] = 4/19, a[2,1] = 1/6, b[6] = 256/ 825, a[7,6] = 1024/1045, c[6] = 3/4, a[5,1] = 10/27, a[6,2] = 9/64, a[ 5,4] = 7/6, a[7,5] = -90/133, b[4] = 27/100\}:" }{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 91 "subs(e3,matrix([seq([c[i],seq(a[i,j],j=1..i-1),``$(8- i)],i=2..7),\n[``,seq(b[i],i=1..7)]]));" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#K%'matrixG6#7)7*#\"\"\"\"\"'F(%!GF+F+F+F+F+7*#F)\"\"&#\"\"#\"#D# \"\"$F1F+F+F+F+F+7*#F)F3#F0\"#F#!\"\"\"\"*#\"#5F7F+F+F+F+7*#F0F3F;#!\" #F:#!#N\"#a#\"\"(F*F+F+F+7*#F3\"\"%#!\"*\"$c##F:\"#k#\"$l\"\"$[%\"\"!# \"$&\\\"%#z\"F+F+7*F)#FH\"#>#!\"$FW#!$0$\"%j9#\"#\")\"#&*#!#!*\"$L\"# \"%C5\"%X5F+7*F+#F3\"#SFQ#\"$D'\"%'p$#F7\"$+\"#F7\"$!G#FK\"$D)#FW\"$S# Q)pprint166\"" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 85 "RK6_7eqs := [op(RowSumConditions(7,'expanded')), op(OrderConditions(6,7,'expanded'))]:" }}}{PARA 0 "" 0 "" {TEXT -1 0 " " }}{PARA 0 "" 0 "" {TEXT -1 7 "Check: " }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 62 "simplify(subs(e3,RK6_7eqs)):\nmap(u->lhs(u)-rhs(u),%) ;\nnops(%);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#7M\"\"!F$F$F$F$F$F$F$F$ F$F$F$F$F$F$F$F$F$F$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#\"#V" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 65 "#----------------------------------- -----------------------------" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}}{PARA 0 "" 0 "" {TEXT -1 39 "#---------------------------- ----------" }}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 25 "absolute stability \+ region" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 26 "coefficient s of the scheme" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> \+ " 0 "" {MPLTEXT 1 0 450 "ee := \{c[2]=1/6,\nc[3]=1/5,\nc[4]=1/3,\nc[5] =2/3,\nc[6]=3/4,\nc[7]=1,\n\na[2,1]=1/6,\na[3,1]=2/25,\na[3,2]=3/25,\n a[4,1]=2/27,\na[4,2]=-1/9,\na[4,3]=10/27,\na[5,1]=10/27,\na[5,2]=-2/9, \na[5,3]=-35/54,\na[5,4]=7/6,\na[6,1]=-9/256,\na[6,2]=9/64,\na[6,3]=16 5/448,\na[6,4]=0,\na[6,5]=495/1792,\na[7,1]=4/19,\na[7,2]=-3/19,\na[7, 3]=-305/1463,\na[7,4]=81/95,\na[7,5]=-90/133,\na[7,6]=1024/1045,\n\nb[ 1]=3/40,\nb[2]=0,\nb[3]=625/3696,\nb[4]=27/100,\nb[5]=27/280,\nb[6]=25 6/825,\nb[7]=19/240\}:" }}}{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 77 "The stability function R for the 7 st age, order 6 scheme is given as follows." }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 75 "subs(ee,StabilityFunction(6,7,'expanded')):\nR := una pply(%,z):\n'R(z)'=R(z);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/-%\"RG6#% \"zG,2\"\"\"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)\"%+aF)*$)F'\"\"(F)F)F)" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 124 "We can find the point where the b oundary of the stability region intersects the negative real axis by s olving 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 27 "z0 := newton(R(z)=-1,z=-4); " }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%#z0G$!+TuxkS!\"*" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 306 "z0 := \+ newton(R(z)=-1,z=-4):\np1 := plot([R(z),-1],z=-4.59..0.49,color=[red,b lue]):\np2 := plot([[[z0,-1]]$3],style=point,symbol=[circle,cross,diam ond],color=black):\np3 := plot([[z0,0],[z0,-1]],linestyle=3,color=COLO R(RGB,0,.5,0)):\nplots[display]([p1,p2,p3],view=[-4.59..0.49,-1.47..1. 47],font=[HELVETICA,9]);" }}{PARA 13 "" 1 "" {GLPLOT2D 360 253 253 {PLOTDATA 2 "6+-%'CURVESG6$7U7$$!3')*************e%!#<$!3xR/y#z[rh#F*7 $$!3mmmTG_jMXF*$!35fG?dEozBF*7$$!3[LL$oXq#zWF*$!3_&>()3;#F*7$$!39+D 6ly4JWF*$!3UF(fA1?Z)>F*7$$!3#om\"Rt_#HQ%F*$!3$*yI*3XU4#=F*7$$!3SLL=Q]d uUF*$!3%zY1N[AS\\\"F*7$$!3yKLG(e1b;%F*$!3j)QPME4l@\"F*7$$!3/n;p>l&p0%F *$!3m0v$pSGj%)*!#=7$$!3KL$e;r;j&RF*$!3)*3n'GOOp.)FP7$$!3++]s>+6_QF*$!3 Ygj/(*\\CgkFP7$$!3IL$e)4$RVu$F*$!3yeFjk^o,^FP7$$!39+]-AU\"pj$F*$!3eip8 _o?\")RFP7$$!3Ymm'\\J9k_$F*$!3['oKuMQM.$FP7$$!3GLLoOf3HMF*$!3AhL(Q^cJM #FP7$$!3t*****[B<&>LF*$!3x2-')=G+,7$$!3>L$eQ+'>2IF*$!3sCvD`)!#?7$$!3 y**\\#zv5Ho#F*$\"3W,@,$Qu,1$Fhp7$$!3]mmE$z\\Ie#F*$\"37$p(Qj\">0x%Fhp7$ $!3#)**\\#z['[tCF*$\"3[W2#45sgY'Fhp7$$!3')**\\x6j:pBF*$\"3Y'3=%GM\\qzF hp7$$!3Emm\"zr)HgAF*$\"3E'[SccGf[*Fhp7$$!37m;4lMLg@F*$\"3))>,)[Y1u3\"F P7$$!3ALL3Nu]_?F*$\"3N>Fw7a/T7FP7$$!3SL$eQ\\10%>F*$\"3vu[3VI849FP7$$!3 /+]FX$4I%=F*$\"3![zQI'pjl:FP7$$!3ELLy`(4xt\"F*$\"39$H4*y;K[FP7$$!39++l'G+D_\"F*$\"3#RT&3qY0y@FP7$$!3*)** \\7J*G&>9F*$\"3;T-!)H-H;CFP7$$!3z***\\pz'>08F*$\"37V.X3TA5FFP7$$!3/nmE *GkC?\"F*$\"3@PNfk[0/IFP7$$!3u****\\4ax#4\"F*$\"3CFC@ZMk_LFP7$$!3BMLe9 )4Q$**FP$\"3cA^Bj)RJq$FP7$$!3'o****4=mr%))FP$\"3X3@[%)GGGTFP7$$!3Oqm\" f?>Z#yFP$\"3S#GagQMFd%FP7$$!3E'**\\A^nfv'FP$\"3ahXD0__)3&FP7$$!3%zmm;Q n5r&FP$\"3M9p6Y*)**[cFP7$$!3J'**\\7y`rh%FP$\"3^b\"*fSp,-jFP7$$!3!)GLLp \\ejNFP$\"3=J$fS^:A+(FP7$$!39LL$))oeh[#FP$\"3!RF#eA\\z)z(FP7$$!3yim\"> /awT\"FP$\"3SU'Qo?[#y')FP7$$!3Mf****>C4eVFhp$\"3'\\uBwx]Nd*FP7$$\"3wML L)*p&\\*oFhp$\"3?fZ'z<#Qr5F*7$$\"31mmma=)fp\"FP$\"3M@y?u'G[=\"F*7$$\"3 70+v?g5pFFP$\"3)o\"R.V%[!>8F*7$$\"3g1+DvFA'z$FP$\"3QEfyHBth9F*7$$\"3!* **************[FP$\"3k@G5VgJK;F*-%'COLOURG6&%$RGBG$\"*++++\"!\")$\"\"! F\\\\lF[\\l-F$6$7S7$F($!\"\"F\\\\l7$F3Fa\\l7$F=Fa\\l7$FBFa\\l7$FGFa\\l 7$FLFa\\l7$FRFa\\l7$FWFa\\l7$FfnFa\\l7$F[oFa\\l7$F`oFa\\l7$FeoFa\\l7$F joFa\\l7$F_pFa\\l7$FdpFa\\l7$FjpFa\\l7$F_qFa\\l7$FdqFa\\l7$FjqFa\\l7$F _rFa\\l7$FdrFa\\l7$FirFa\\l7$F^sFa\\l7$FcsFa\\l7$FhsFa\\l7$F]tFa\\l7$F btFa\\l7$FgtFa\\l7$F\\uFa\\l7$FauFa\\l7$FfuFa\\l7$F[vFa\\l7$F`vFa\\l7$ FevFa\\l7$FjvFa\\l7$F_wFa\\l7$FdwFa\\l7$FiwFa\\l7$F^xFa\\l7$FcxFa\\l7$ FhxFa\\l7$F]yFa\\l7$FbyFa\\l7$FgyFa\\l7$F\\zFa\\l7$FazFa\\l7$FfzFa\\l7 $F[[lFa\\l7$F`[lFa\\l-Fe[l6&Fg[lF[\\lF[\\lFh[l-F$6&7#7$$!3Q+++TuxkSF*F a\\l-%'SYMBOLG6#%'CIRCLEG-Fe[l6&Fg[lF\\\\lF\\\\lF\\\\l-%&STYLEG6#%&POI NTG-F$6&Fg_l-F\\`l6#%&CROSSGF_`lFa`l-F$6&Fg_l-F\\`l6#%(DIAMONDGF_`lFa` l-F$6%7$7$Fi_lF[\\lFh_l-%&COLORG6&Fg[lF[\\l$\"\"&Fb\\lF[\\l-%*LINESTYL EG6#\"\"$-%%FONTG6$%*HELVETICAG\"\"*-%+AXESLABELSG6%Q\"z6\"Q!Febl-F]bl 6#%(DEFAULTG-%%VIEWG6$;$!$f%!\"#$\"#\\F`cl;$!$Z\"F`cl$\"$Z\"F`cl" 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 1501 "R := z -> 1+z+1/2*z^2+1/6*z^3+1/24*z^4+1/ 120*z^5+1/720*z^6+1/5400*z^7:\npts := []:\nz0 := 0:\nfor ct from 0 to \+ 200 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,.23,0,.48)):\np2 := plots[polygonplot]([seq([pts[i-1],pts[i],[-2, 0]],i=2..nops(pts))],\n style=patchnogrid,color=COLOR(RGB,.45 ,0,.95)):\npts := []: z0 := 1.4+4.3*I: tt := 0: \nwhile tt<=51/25 do\n zz := newton(R(z)=exp(tt*Pi*I),z=z0):\n z0 := zz:\n if (11/25<= tt and tt<=43/25) then\n hh := 1/50\n else \n hh := 1/25\n end if;\n tt := tt+hh;\n pts := [op(pts),[Re(zz),Im(zz)]]:\nend do:\np3 := plot(pts,color=COLOR(RGB,.23,0,.48)):\np4 := plots[polygon plot]([seq([pts[i-1],pts[i],[1.32,4.2]],i=2..nops(pts))],\n s tyle=patchnogrid,color=COLOR(RGB,.45,0,.95)):\npts := []: z0 := 1.4-4. 3*I: tt := 0: \nwhile tt<=51/25 do\n zz := newton(R(z)=exp(tt*Pi*I), z=z0):\n z0 := zz:\n if (8/25<=tt and tt<=8/5) then\n hh := 1 /50\n else \n hh := 1/25\n end if;\n tt := tt+hh;\n pts : = [op(pts),[Re(zz),Im(zz)]]:\nend do:\np5 := plot(pts,color=COLOR(RGB, .23,0,.48)):\np6 := plots[polygonplot]([seq([pts[i-1],pts[i],[1.32,-4. 2]],i=2..nops(pts))],\n style=patchnogrid,color=COLOR(RGB,.45 ,0,.95)):\np7 := plot([[[-4.59,0],[1.89,0]],[[0,-4.79],[0,4.79]]],colo r=black,linestyle=3):\nplots[display]([p||(1..7)],view=[-4.59..1.89,-4 .79..4.79],font=[HELVETICA,9],\n labels=[`Re(z)`,`Im(z)`] ,axes=boxed,scaling=constrained);" }}{PARA 13 "" 1 "" {GLPLOT2D 381 565 565 {PLOTDATA 2 "6/-%'CURVESG6$7ew7$$\"\"!F)F(7$F($\"3#******fK'zq :!#=7$$\"3)******RS$[E5!#E$\"39+++4EfTJF-7$$\"3*)*****RG&z/C!#D$\"34++ +F\")Q7ZF-7$$\"3++++Hy4$4#!#C$\"3;+++%)z<$G'F-7$$\"36+++l0%y,\"!#B$\"3 7+++z8%R&yF-7$$\"3()*****pKPX@$FC$\"3@+++lBhC%*F-7$$\"3I+++$obk`'FC$\" 3%*******)4-&*4\"!#<7$$\"3y*****z)f;iZFC$\"3'******RTzkD\"FP7$$!3)**** **HVFnk#!#A$\"3#*******f9K89FP7$$!3#******Hn*)p`\"!#@$\"3)******Hh0*p: FP7$$!3q*****4&pxd_Fin$\"3)******H.Wgs\"FP7$$!3(*******[-4J9!#?$\"33++ +g%y9)=FP7$$!39+++J(onO$Fdo$\"37+++U+*e.#FP7$$!3#******\\6_t5(Fdo$\"3% )******>>#*)=#FP7$$!3'******fbccP\"!#>$\"3%)*****4#f>SBFP7$$!3;+++*H3n Z#Fdp$\"37+++6cG*[#FP7$$!3u*****4M24>%Fdp$\"3*******HR)fNEFP7$$!3=+++B Gy8nFdp$\"3;+++7\"Q\"yFFP7$$!3%*******eDaA5F-$\"31+++cZ?:HFP7$$!3#**** **zT;F[\"F-$\"3@+++6vFWIFP7$$!39+++'Ruq/#F-$\"3A+++\"zoC;$FP7$$!33+++n $fXp#F-$\"3%******>r^vE$FP7$$!3G+++9V(eR$F-$\"3A+++d[xeLFP7$$!3!****** *HM:CTF-$\"3#******>,(yOMFP7$$!3w*****pb?/'[F-$\"3>+++Ee)H]$FP7$$!3_++ +p;Y$f&F-$\"39+++D`#*eNFP7$$!3E+++e$HvJ'F-$\"37+++f2,1OFP7$$!30+++9P9I qF-$\"3y*****>01ak$FP7$$!3#)******HrfIxF-$\"3#*********)R!yOFP7$$!3M++ +.72>%)F-$\"3@+++uLk/PFP7$$!3s*****>\\^h4*F-$\"32+++%p!zDPFP7$$!3i**** *\\*Qdi(*F-$\"31+++kS$>u$FP7$$!3%******\\&*4>/\"FP$\"3%)*****f.JMv$FP7 $$!3!******H&fk16FP$\"3++++)Gl0w$FP7$$!33+++P$\\PFP7$$!34+++-a%)=9FP$\"3$)* ****p!4BPPFP7$$!3%******>?@%z9FP$\"3$)*****\\-o:s$FP7$$!3-+++_K`R:FP$ \"3;+++%e'R-PFP7$$!3'******pPm#*f\"FP$\"3%)*****f8_(zOFP7$$!3%******> \"4se;FP$\"35+++%HiOl$FP7$$!33+++Xq,=FP$\"3/+++s(3GZ$FP7$$!3 8+++)Qa\"=?FP$\"3y*****zc[rU$FP7$$!3%******zpm/3#FP$\"3))*****pok(yLFP 7$$!3)******\\w$3W@FP$\"3y*****\\45!GLFP7$$!3&******fJ_\"4AFP$\"3'**** **HeK`F$FP7$$!3))*****H`,dF#FP$\"32+++\\CD@KFP7$$!3:+++u#3OM#FP$\"3!** ****4f9j;$FP7$$!36+++^%*e7CFP$\"3=+++&4B56$FP7$$!3?+++3(HA[#FP$\"3'*** ***Rkxd0$FP7$$!31+++hq._DFP$\"3'*******eG$3+$FP7$$!3&)******Rr]@EFP$\" 39+++46HYHFP7$$!37+++3k@( )y)GFP$\"3.+++a+TIFFP7$$!3)******R\"yD]HFP$\"3%)******=+7wEFP7$$!31+++ VCk5IFP$\"34+++u9J@EFP7$$!3=+++14)*oIFP$\"37+++];\"ec#FP7$$!3y*****\\# [CDJFP$\"3%********fo%4DFP7$$!3?+++[$H%zJFP$\"35+++#>^@X#FP7$$!3>+++yq aJKFP$\"3!)*****f#yu$R#FP7$$!35+++/Qi\"G$FP$\"3')*****\\mkTL#FP7$$!3.+ ++lcpHLFP$\"3?+++SQKtAFP7$$!3'******HM2eP$FP$\"3\"******4$>;6AFP7$$!3( ******z*4,?MFP$\"39+++a)Gw9#FP7$$!3y*****>nlBY$FP$\"33+++Vso#3#FP7$$!3 !******f\"p$H]$FP$\"3!)*****>P7j,#FP7$$!3(******pU'zTNFP$\"33+++)f#\\[ >FP7$$!3++++$[@!zNFP$\"35+++*GI#z=FP7$$!3'******\\%Rp9OFP$\"35+++aJa3= FP7$$!31+++H')*)[OFP$\"3!******fylkt\"FP7$$!3))*****4*3s\"o$FP$\"3#*** ****)=^Im\"FP7$$!3,+++CLC8PFP$\"3!******\\#>P)e\"FP7$$!3=+++I;aVPFP$\" 30+++)[?D^\"FP7$$!3++++6*zEx$FP$\"3.+++i%3cV\"FP7$$!3=+++Eeq+QFP$\"3%* *****>9kxN\"FP7$$!39+++eikFQFP$\"31+++'[G\"z7FP7$$!3')*****\\\"FP7$$!3/+++6tDyQFP$\"3$*******\\f2?6FP7$$!3++++z*e=!RF P$\"3%******p-]*R5FP7$$!3@+++))4CCRFP$\"3%)*****z*p,'f*F-7$$!37+++B#=` %RFP$\"3'******4q!*e\"F-7$$!3U+++R&GP1%FP$\"3l******zOpVzFdp7$$!3Q+++TuxkSFPF(7 $Fgjl$!3l******zOpVzFdp7$Fbjl$!3%******z>q!*e\"F-7$F]jl$!3%******HR@WQ #F-7$Fhil$!3u*****Hf:2=$F-7$Fcil$!3)******fC.#yRF-7$F^il$!3u*****Rjtqx %F-7$Fihl$!3Y+++bBVxbF-7$Fdhl$!3]******QaFzjF-7$F_hl$!3e******)3hC=(F- 7$Fjgl$!3_+++A7o')zF-7$Fegl$!3'******4 \"FP7$F\\fl$!31+++'[G\"z7FP7$Fgel$!3%******>9kxN\"FP7$Fbel$!3.+++i%3cV \"FP7$F]el$!30+++)[?D^\"FP7$Fhdl$!3!******\\#>P)e\"FP7$Fcdl$!3#******* )=^Im\"FP7$F^dl$!3!******fylkt\"FP7$Ficl$!35+++aJa3=FP7$Fdcl$!35+++*GI #z=FP7$F_cl$!33+++)f#\\[>FP7$Fjbl$!3!)*****>P7j,#FP7$Febl$!33+++Vso#3# FP7$F`bl$!39+++a)Gw9#FP7$F[bl$!3\"******4$>;6AFP7$Ffal$!3?+++SQKtAFP7$ Faal$!3')*****\\mkTL#FP7$F\\al$!3!)*****f#yu$R#FP7$Fg`l$!35+++#>^@X#FP 7$Fb`l$!3%********fo%4DFP7$F]`l$!37+++];\"ec#FP7$Fh_l$!34+++u9J@EFP7$F c_l$!3%)******=+7wEFP7$F^_l$!3.+++a+TIFFP7$Fi^l$!3%)******fEP%y#FP7$Fd ^l$!3\"******HW5#QGFP7$F_^l$!3!******fBC@*GFP7$Fj]l$!39+++46HYHFP7$Fe] l$!3'*******eG$3+$FP7$F`]l$!3'******Rkxd0$FP7$F[]l$!3=+++&4B56$FP7$Ff \\l$!3!******4f9j;$FP7$Fa\\l$!32+++\\CD@KFP7$F\\\\l$!3'******HeK`F$FP7 $Fg[l$!3y*****\\45!GLFP7$Fb[l$!3))*****pok(yLFP7$F][l$!3y*****zc[rU$FP 7$Fhz$!3/+++s(3GZ$FP7$Fcz$!32+++Kr[:NFP7$F^z$!39+++'f1]b$FP7$Fiy$!3!** ****\\@_7f$FP7$Fdy$!3%******RH_Ti$FP7$F_y$!35+++%HiOl$FP7$Fjx$!3%)**** *f8_(zOFP7$Fex$!3;+++%e'R-PFP7$F`x$!3$)*****\\-o:s$FP7$F[x$!3$)*****p! 4BPPFP7$Ffw$!3'******4>P$\\PFP7$Faw$!37+++fA#yv$FP7$F\\w$!3!)*****zz+E w$FP7$Fgv$!3#)*****4CiNw$FP7$Fbv$!3++++)Gl0w$FP7$F]v$!3%)*****f.JMv$FP 7$Fhu$!31+++kS$>u$FP7$Fcu$!32+++%p!zDPFP7$F^u$!3@+++uLk/PFP7$Fit$!3#** *******)R!yOFP7$Fdt$!3y*****>01ak$FP7$F_t$!37+++f2,1OFP7$Fjs$!39+++D`# *eNFP7$Fes$!3>+++Ee)H]$FP7$F`s$!3#******>,(yOMFP7$F[s$!3A+++d[xeLFP7$F fr$!3%******>r^vE$FP7$Far$!3A+++\"zoC;$FP7$F\\r$!3@+++6vFWIFP7$Fgq$!31 +++cZ?:HFP7$Fbq$!3;+++7\"Q\"yFFP7$F]q$!3*******HR)fNEFP7$Fhp$!37+++6cG *[#FP7$Fbp$!3%)*****4#f>SBFP7$F]p$!3%)******>>#*)=#FP7$Fho$!37+++U+*e. #FP7$Fbo$!33+++g%y9)=FP7$F]o$!3)******H.Wgs\"FP7$Fgn$!3)******Hh0*p:FP 7$FW$!3#*******f9K89FP7$FR$!3'******RTzkD\"FP7$$\"3)******>obk`'FC$!3% *******)4-&*4\"FP7$$\"35+++Lt`9KFC$!3@+++lBhC%*F-7$FA$!37+++z8%R&yF-7$ $\"3/+++Gy4$4#F=$!3;+++%)z<$G'F-7$$\"3/+++&G&z/CF7$!34+++F\")Q7ZF-7$$ \"32+++1M[E5F1$!39+++4EfTJF-7$F($!3#******fK'zq:F-F'-%&COLORG6&%$RGBG$ \"#B!\"#F($\"#[Fg^n-%)POLYGONSG6fw7%F'7$F($\"+Ejzq:!#57$$Fg^nF)F(7%F^_ n7$$\"+/M[E5F-$\"+4EfTJFa_nFb_n7%Fe_n7$$\"+%G&z/CFP$\"+F\")Q7ZFa_nFb_n 7%F[`n7$$\"+Hy4$4#!#;$\"+%)z<$G'Fa_nFb_n7%Fa`n7$$\"+l0%y,\"!#:$\"+z8%R &yFa_nFb_n7%Fh`n7$$\"+Ft`9KF[an$\"+lBhC%*Fa_nFb_n7%F_an7$$\"+$obk`'F[a n$\"+*4-&*4\"!\"*Fb_n7%Fean7$$\"+))f;iZF[an$\"+9%zkD\"FjanFb_n7%F\\bn7 $$!+LusYE!#9$\"+g9K89FjanFb_n7%Fbbn7$$!+t'*)p`\"!#8$\"+8c!*p:FjanFb_n7 %Fibn7$$!+^pxd_F\\cn$\"+LS/E #*)=#FjanFb_n7%Fcdn7$$!+cllv8!#6$\"+@f>SBFjanFb_n7%Fidn7$$!+*H3nZ#F\\e n$\"+6cG*[#FjanFb_n7%F`en7$$!+Tt!4>%F\\en$\"+$R)fNEFjanFb_n7%Ffen7$$!+ BGy8nF\\en$\"+7\"Q\"yFFjanFb_n7%F\\fn7$$!+fDaA5Fa_n$\"+cZ?:HFjanFb_n7% Fbfn7$$!+=kr#[\"Fa_n$\"+6vFWIFjanFb_n7%Fhfn7$$!+'Ruq/#Fa_n$\"+\"zoC;$F janFb_n7%F^gn7$$!+n$fXp#Fa_n$\"+7%)Fa_n$\"+uLk/PFjanFb_n7 %Fdjn7$$!+#\\^h4*Fa_n$\"+%p!zDPFjanFb_n7%Fjjn7$$!+&*Qdi(*Fa_n$\"+kS$>u $FjanFb_n7%F`[o7$$!+b*4>/\"Fjan$\"+O5V`PFjanFb_n7%Ff[o7$$!+`fk16Fjan$ \"+)Gl0w$FjanFb_n7%F\\\\o7$$!+P$\\PFjanFb_n7%Fd]o7$$!+-a%)=9Fjan$\"+24BPPFjan Fb_n7%Fj]o7$$!+-7Uz9Fjan$\"+D!o:s$FjanFb_n7%F`^o7$$!+_K`R:Fjan$\"+%e'R -PFjanFb_n7%Ff^o7$$!+xjE*f\"Fjan$\"+O@vzOFjanFb_n7%F\\_o7$$!+74se;Fjan $\"+%HiOl$FjanFb_n7%Fb_o7$$!+Xq,=Fjan$\"+s(3GZ$Fja nFb_n7%F`ao7$$!+)Qa\"=?Fjan$\"+o&[rU$FjanFb_n7%Ffao7$$!+)pm/3#Fjan$\"+ (ok(yLFjanFb_n7%F\\bo7$$!+lP3W@Fjan$\"+&45!GLFjanFb_n7%Fbbo7$$!+;B:4AF jan$\"+$eK`F$FjanFb_n7%Fhbo7$$!+L:qvAFjan$\"+\\CD@KFjanFb_n7%F^co7$$!+ u#3OM#Fjan$\"+\"f9j;$FjanFb_n7%Fdco7$$!+^%*e7CFjan$\"+&4B56$FjanFb_n7% Fjco7$$!+3(HA[#Fjan$\"+WwxbIFjanFb_n7%F`do7$$!+hq._DFjan$\"+fG$3+$Fjan Fb_n7%Ffdo7$$!+Sr]@EFjan$\"+46HYHFjanFb_n7%F\\eo7$$!+3k+7wEFjanFb_n7%Fjfo7$$!+VCk5IFjan$\"+u9J@EFjanFb_n7%F`go7$$!+ 14)*oIFjan$\"+];\"ec#FjanFb_n7%Ffgo7$$!+D[CDJFjan$\"++'o%4DFjanFb_n7%F \\ho7$$!+[$H%zJFjan$\"+#>^@X#FjanFb_n7%Fbho7$$!+yqaJKFjan$\"+Eyu$R#Fja nFb_n7%Fhho7$$!+/Qi\"G$Fjan$\"+lY;MBFjanFb_n7%F^io7$$!+lcpHLFjan$\"+SQ KtAFjanFb_n7%Fdio7$$!+Vt!eP$Fjan$\"+J>;6AFjanFb_n7%Fjio7$$!+)*4,?MFjan $\"+a)Gw9#FjanFb_n7%F`jo7$$!+scOiMFjan$\"+Vso#3#FjanFb_n7%Ffjo7$$!+;p$ H]$Fjan$\"+sBJ;?FjanFb_n7%F\\[p7$$!+FkzTNFjan$\"+)f#\\[>FjanFb_n7%Fb[p 7$$!+$[@!zNFjan$\"+*GI#z=FjanFb_n7%Fh[p7$$!+XRp9OFjan$\"+aJa3=FjanFb_n 7%F^\\p7$$!+H')*)[OFjan$\"+'ylkt\"FjanFb_n7%Fd\\p7$$!+\"*3s\"o$Fjan$\" +*=^Im\"FjanFb_n7%Fj\\p7$$!+CLC8PFjan$\"+D>P)e\"FjanFb_n7%F`]p7$$!+I;a VPFjan$\"+)[?D^\"FjanFb_n7%Ff]p7$$!+6*zEx$Fjan$\"+i%3cV\"FjanFb_n7%F\\ ^p7$$!+Eeq+QFjan$\"+UTwd8FjanFb_n7%Fb^p7$$!+eikFQFjan$\"+'[G\"z7FjanFb _n7%Fh^p7$$!+vT]`QFjan$\"+a&\\)*>\"FjanFb_n7%F^_p7$$!+6tDyQFjan$\"+]f2 ?6FjanFb_n7%Fd_p7$$!+z*e=!RFjan$\"+F+&*R5FjanFb_n7%Fj_p7$$!+))4CCRFjan $\"+)*p,'f*Fa_nFb_n7%F``p7$$!+B#=`%RFjan$\"+rhV\"z)Fa_nFb_n7%Ff`p7$$!+ 8U*\\'RFjan$\"+A7o')zFa_nFb_n7%F\\ap7$$!+Cp;$)RFjan$\"+*3hC=(Fa_nFb_n7 %Fbap7$$!+ETt**RFjan$\"+RaFzjFa_nFb_n7%Fhap7$$!+%*yf9SFjan$\"+bBVxbFa_ nFb_n7%F^bp7$$!+M!ow-%Fjan$\"+MO2xZFa_nFb_n7%Fdbp7$$!+eW')QSFjan$\"+YK ?yRFa_nFb_n7%Fjbp7$$!+O'=\"[SFjan$\"+$f:2=$Fa_nFb_n7%F`cp7$$!+cVPbSFja n$\"+$R@WQ#Fa_nFb_n7%Ffcp7$$!+)*zegSFjan$\"+)>q!*e\"Fa_nFb_n7%F\\dp7$$ !+R&GP1%Fjan$\"+!o$pVzF\\enFb_n7%Fbdp7$$!+TuxkSFjanF(Fb_n7%Fhdp7$Fcdp$ !+!o$pVzF\\enFb_n7%F\\ep7$F]dp$!+)>q!*e\"Fa_nFb_n7%F`ep7$Fgcp$!+$R@WQ# Fa_nFb_n7%Fdep7$Facp$!+$f:2=$Fa_nFb_n7%Fhep7$F[cp$!+YK?yRFa_nFb_n7%F\\ fp7$Febp$!+MO2xZFa_nFb_n7%F`fp7$F_bp$!+bBVxbFa_nFb_n7%Fdfp7$Fiap$!+RaF zjFa_nFb_n7%Fhfp7$Fcap$!+*3hC=(Fa_nFb_n7%F\\gp7$F]ap$!+A7o')zFa_nFb_n7 %F`gp7$Fg`p$!+rhV\"z)Fa_nFb_n7%Fdgp7$Fa`p$!+)*p,'f*Fa_nFb_n7%Fhgp7$F[` p$!+F+&*R5FjanFb_n7%F\\hp7$Fe_p$!+]f2?6FjanFb_n7%F`hp7$F__p$!+a&\\)*> \"FjanFb_n7%Fdhp7$Fi^p$!+'[G\"z7FjanFb_n7%Fhhp7$Fc^p$!+UTwd8FjanFb_n7% F\\ip7$F]^p$!+i%3cV\"FjanFb_n7%F`ip7$Fg]p$!+)[?D^\"FjanFb_n7%Fdip7$Fa] p$!+D>P)e\"FjanFb_n7%Fhip7$F[]p$!+*=^Im\"FjanFb_n7%F\\jp7$Fe\\p$!+'ylk t\"FjanFb_n7%F`jp7$F_\\p$!+aJa3=FjanFb_n7%Fdjp7$Fi[p$!+*GI#z=FjanFb_n7 %Fhjp7$Fc[p$!+)f#\\[>FjanFb_n7%F\\[q7$F][p$!+sBJ;?FjanFb_n7%F`[q7$Fgjo $!+Vso#3#FjanFb_n7%Fd[q7$Fajo$!+a)Gw9#FjanFb_n7%Fh[q7$F[jo$!+J>;6AFjan Fb_n7%F\\\\q7$Feio$!+SQKtAFjanFb_n7%F`\\q7$F_io$!+lY;MBFjanFb_n7%Fd\\q 7$Fiho$!+Eyu$R#FjanFb_n7%Fh\\q7$Fcho$!+#>^@X#FjanFb_n7%F\\]q7$F]ho$!++ 'o%4DFjanFb_n7%F`]q7$Fggo$!+];\"ec#FjanFb_n7%Fd]q7$Fago$!+u9J@EFjanFb_ n7%Fh]q7$F[go$!+>+7wEFjanFb_n7%F\\^q7$Fefo$!+a+TIFFjanFb_n7%F`^q7$F_fo $!+gEP%y#FjanFb_n7%Fd^q7$Fieo$!+V/@QGFjanFb_n7%Fh^q7$Fceo$!+OU7#*GFjan Fb_n7%F\\_q7$F]eo$!+46HYHFjanFb_n7%F`_q7$Fgdo$!+fG$3+$FjanFb_n7%Fd_q7$ Fado$!+WwxbIFjanFb_n7%Fh_q7$F[do$!+&4B56$FjanFb_n7%F\\`q7$Feco$!+\"f9j ;$FjanFb_n7%F``q7$F_co$!+\\CD@KFjanFb_n7%Fd`q7$Fibo$!+$eK`F$FjanFb_n7% Fh`q7$Fcbo$!+&45!GLFjanFb_n7%F\\aq7$F]bo$!+(ok(yLFjanFb_n7%F`aq7$Fgao$ !+o&[rU$FjanFb_n7%Fdaq7$Faao$!+s(3GZ$FjanFb_n7%Fhaq7$F[ao$!+Kr[:NFjanF b_n7%F\\bq7$Fe`o$!+'f1]b$FjanFb_n7%F`bq7$F_`o$!+:AD\"f$FjanFb_n7%Fdbq7 $Fi_o$!+%H_Ti$FjanFb_n7%Fhbq7$Fc_o$!+%HiOl$FjanFb_n7%F\\cq7$F]_o$!+O@v zOFjanFb_n7%F`cq7$Fg^o$!+%e'R-PFjanFb_n7%Fdcq7$Fa^o$!+D!o:s$FjanFb_n7% Fhcq7$F[^o$!+24BPPFjanFb_n7%F\\dq7$Fe]o$!+\">P$\\PFjanFb_n7%F`dq7$F_]o $!+fA#yv$FjanFb_n7%Fddq7$Fi\\o$!+)z+Ew$FjanFb_n7%Fhdq7$Fc\\o$!+TAcjPFj anFb_n7%F\\eq7$F]\\o$!+)Gl0w$FjanFb_n7%F`eq7$Fg[o$!+O5V`PFjanFb_n7%Fde q7$Fa[o$!+kS$>u$FjanFb_n7%Fheq7$F[[o$!+%p!zDPFjanFb_n7%F\\fq7$Fejn$!+u Lk/PFjanFb_n7%F`fq7$F_jn$!++*R!yOFjanFb_n7%Fdfq7$Fiin$!+_gSXOFjanFb_n7 %Fhfq7$Fcin$!+f2,1OFjanFb_n7%F\\gq7$F]in$!+D`#*eNFjanFb_n7%F`gq7$Fghn$ !+Ee)H]$FjanFb_n7%Fdgq7$Fahn$!+7qyOMFjanFb_n7%Fhgq7$F[hn$!+d[xeLFjanFb _n7%F\\hq7$Fegn$!+7SBFjanFb_n7%F\\jq7$Fddn$!+?>#*)=#FjanFb_n7%F`jq7$F^dn$! +U+*e.#FjanFb_n7%Fdjq7$Fgcn$!+g%y9)=FjanFb_n7%Fhjq7$Facn$!+LS/E#\\8@8FP$\"33+++ix7iVFP7$$\"3$******4n9[I\"FP$\"3e*****\\ bP4O%FP7$$\"3++++C*4$)G\"FP$\"3=+++Dh+eVFP7$$\"31+++&zI+G\"FP$\"3s**** *>+eeN%FP7$$\"31+++O)eZ7FP$\"3w*****p3 ECM%FP7$$\"33+++r*y\"R7FP$\"3v*******p$zPVFP7$$\"3\"******H%oGJ7FP$\"3 i*****4DDEL%FP7$$\"3-+++*QaNA\"FP$\"36+++Uw!pK%FP7$$\"3\"******H(e,;7F P$\"3P+++y!G1K%FP7$$\"35+++D!4(37FP$\"3'******pKuPJ%FP7$$\"3*******pZv ;?\"FP$\"3N+++r]L1VFP7$$\"3!******>Dh\\>\"FP$\"3S+++n.I)H%FP7$$\"3'*** ***HF<')=\"FP$\"3)*******eAm*G%FP7$$\"3#******f*)*p#=\"FP$\"3Y+++obT!G %FP7$$\"3#******4Urs<\"FP$\"3i*****\\\"*e0F%FP7$$\"3-+++T0Ss6FP$\"3o** ****=g4gUFP7$$\"3!*******fE;o6FP$\"3E+++Jt.\\UFP7$$\"3#*******f*RY;\"F P$\"3?+++_?SPUFP7$$\"33+++j5#>;\"FP$\"3-+++c0ADUFP7$$\"31+++&3+,;\"FP$ \"3J+++;s`7UFP7$$\"3%******f\"[Ff6FP$\"3S+++LNT*>%FP7$$\"3++++QPaf6FP$ \"3I+++d8$f=%FP7$$\"3#********e,5;\"FP$\"3)******4!e>sTFP7$$\"3-+++PLt j6FP$\"3-+++`sLeTFP7$$\"3)******Hh1y;\"FP$\"3:+++s;^WTFP7$$\"3)****** \\i.^5%FP7$$\"3'******px'z(>\"FP$\"3s*****\\ $*GL4%FP7$$\"3%******z=N%37FP$\"3o******4Rb#3%FP7$$\"3/+++I/2?7FP$\"3K +++wG$H2%FP7$$\"3)******>o;DB\"FP$\"33+++.6ekSFP7$$\"35+++DXdX7FP$\"3y *****\\Fqv0%FP7$$\"3-+++oQ/f7FP$\"3k******p*G>0%FP7$$\"31+++f[ts7FP$\" 3c*****pCZw/%FP7$$\"35+++xcZ'G\"FP$\"3t*****H#QoWSFP7$$\"3%******f%p6+ 8FP$\"3>+++%=uH/%FP7$$\"3/+++_K`88FP$\"3>+++\"QQC/%FP7$$\"32+++.DiE8FP $\"3!******fq()H/%FP7$$\"3$******H&RIR8FP$\"37+++f*HX/%FP7$$\"3/+++Xb^ ^8FP$\"3A+++)Gtp/%FP7$$\"3)******>16KO\"FP$\"3!)*****f!)G-0%FP7$$\"35+ ++ztNu8FP$\"3'******p?7U0%FP7$$\"31+++#*>$\\Q\"FP$\"3))*****zmW)eSFP7$ $\"3'******z(3#\\R\"FP$\"3A+++8K0kSFP7$$\"3&******Hp;VS\"FP$\"3)****** *Q2xpSFP7$$\"35+++0t689FP$\"3;+++9e$f2%FP7$$\"3-+++$fC8U\"FP$\"3))**** **\\B\\#3%FP7$$\"34+++%\\V*G9FP$\"3))*****f?*Q*3%FP7$$\"35+++Y7)fV\"FP $\"3U+++l(zl4%FP7$$\"3*)******)yYCW\"FP$\"3E+++9:-/TFP7$$\"3)********H ]$[9FP$\"3!)*****>kv;6%FP7$$\"37+++IGq`9FP$\"3'******4m1&>TFP7$$\"3%** *****=g^e9FP$\"3&)*****f0#[FTFP7$$\"3%******>)=!GY\"FP$\"3w*****z(>dNT FP7$$\"3%******fkslY\"FP$\"3********\\*[P9%FP7$$\"3%******fhS)p9FP$\"3 s*****zh()>:%FP7$$\"3'******44=EZ\"FP$\"3Q+++BXEgTFP7$$\"3))*****zG<\\ Z\"FP$\"3E+++'*yboTFP7$$\"31+++$H]nZ\"FP$\"3#******pZZo<%FP7$$\"3/+++F !H\"y9FP$\"3q*****fN9^=%FP7$$\"3-+++A_1z9FP$\"3U+++J3M$>%FP7$$\"3!**** **fSq&z9FP$\"3x*****zF5:?%FP7$$\"3++++(*elz9FP$\"3T+++Oqg4UFP7$$\"31++ +-GLz9FP$\"3P+++5jhjBI%FP7$$\"31+++q()oP9FP$\"3i*****fS $p8VFPFa^n-F[_n6ap7%7$$\"+EXmU9Fjan$\"+j#\\\"3VFjan7$$\"+PAXK9Fjan$\"+ 6l)*=VFjan7$$\"$K\"Fg^n$\"#U!\"\"7%F[js7$$\"+qSB@9Fjan$\"+8NyGVFjanF`j s7%Fgjs7$$\"+$*\\349Fjan$\"+u%ouL%FjanF`js7%F][t7$$\"+[83'R\"Fjan$\"+0 (p\\M%FjanF`js7%Fc[t7$$\"+6Z7Fjan$\"+(3ECM% FjanF`js7%F[`t7$$\"+r*y\"R7Fjan$\"++PzPVFjanF`js7%Fa`t7$$\"+VoGJ7Fjan$ \"+^_iKVFjanF`js7%Fg`t7$$\"+*QaNA\"Fjan$\"+Uw!pK%FjanF`js7%F]at7$$\"+t e,;7Fjan$\"+y!G1K%FjanF`js7%Fcat7$$\"+D!4(37Fjan$\"+FVx8VFjanF`js7%Fia t7$$\"+xan,7Fjan$\"+r]L1VFjanF`js7%F_bt7$$\"+_7'\\>\"Fjan$\"+n.I)H%Fja nF`js7%Febt7$$\"+tsh)=\"Fjan$\"+fAm*G%FjanF`js7%F[ct7$$\"+'*)*p#=\"Fja n$\"+obT!G%FjanF`js7%Fact7$$\"+@9Fx6Fjan$\"+:*e0F%FjanF`js7%Fgct7$$\"+ T0Ss6Fjan$\"+>g4gUFjanF`js7%F]dt7$$\"+gE;o6Fjan$\"+Jt.\\UFjanF`js7%Fcd t7$$\"+g*RY;\"Fjan$\"+_?SPUFjanF`js7%Fidt7$$\"+j5#>;\"Fjan$\"+c0ADUFja nF`js7%F_et7$$\"+&3+,;\"Fjan$\"+;s`7UFjanF`js7%Feet7$$\"+;[Ff6Fjan$\"+ LNT*>%FjanF`js7%F[ft7$$\"+QPaf6Fjan$\"+d8$f=%FjanF`js7%Faft7$$\"+!f,5; \"Fjan$\"+,e>sTFjanF`js7%Fgft7$$\"+PLtj6Fjan$\"+`sLeTFjanF`js7%F]gt7$$ \"+8m!y;\"Fjan$\"+s;^WTFjanF`js7%Fcgt7$$\"+vJEt6Fjan$\"+r#)*38%FjanF`j s7%Figt7$$\"+8/6!=\"Fjan$\"+iQp\"Fjan$\"+N*GL4%FjanF`js7%F[it7$$\"+)=N%37 Fjan$\"+5Rb#3%FjanF`js7%Fait7$$\"+I/2?7Fjan$\"+wG$H2%FjanF`js7%Fgit7$$ \"+#o;DB\"Fjan$\"+.6ekSFjanF`js7%F]jt7$$\"+DXdX7Fjan$\"+v-ddSFjanF`js7 %Fcjt7$$\"+oQ/f7Fjan$\"+q*G>0%FjanF`js7%Fijt7$$\"+f[ts7Fjan$\"+ZskZSFj anF`js7%F_[u7$$\"+xcZ'G\"Fjan$\"+BQoWSFjanF`js7%Fe[u7$$\"+Yp6+8Fjan$\" +%=uH/%FjanF`js7%F[\\u7$$\"+_K`88Fjan$\"+\"QQC/%FjanF`js7%Fa\\u7$$\"+. DiE8Fjan$\"+1x)H/%FjanF`js7%Fg\\u7$$\"+`RIR8Fjan$\"+f*HX/%FjanF`js7%F] ]u7$$\"+Xb^^8Fjan$\"+)Gtp/%FjanF`js7%Fc]u7$$\"+i5@j8Fjan$\"+1)G-0%Fjan F`js7%Fi]u7$$\"+ztNu8Fjan$\"+2A@aSFjanF`js7%F_^u7$$\"+#*>$\\Q\"Fjan$\" +oY%)eSFjanF`js7%Fe^u7$$\"+y3#\\R\"Fjan$\"+8K0kSFjanF`js7%F[_u7$$\"+$p ;VS\"Fjan$\"+R2xpSFjanF`js7%Fa_u7$$\"+0t689Fjan$\"+9e$f2%FjanF`js7%Fg_ u7$$\"+$fC8U\"Fjan$\"+]B\\#3%FjanF`js7%F]`u7$$\"+%\\V*G9Fjan$\"+1#*Q*3 %FjanF`js7%Fc`u7$$\"+Y7)fV\"Fjan$\"+l(zl4%FjanF`js7%Fi`u7$$\"+*yYCW\"F jan$\"+9:-/TFjanF`js7%F_au7$$\"++.N[9Fjan$\"+Ucn6TFjanF`js7%Feau7$$\"+ IGq`9Fjan$\"+hm]>TFjanF`js7%F[bu7$$\"+>g^e9Fjan$\"+c?[FTFjanF`js7%Fabu 7$$\"+#)=!GY\"Fjan$\"+y>dNTFjanF`js7%Fgbu7$$\"+YEdm9Fjan$\"+]*[P9%Fjan F`js7%F]cu7$$\"+;1%)p9Fjan$\"+=w)>:%FjanF`js7%Fccu7$$\"+\"4=EZ\"Fjan$ \"+BXEgTFjanF`js7%Ficu7$$\"+)G<\\Z\"Fjan$\"+'*yboTFjanF`js7%F_du7$$\"+ $H]nZ\"Fjan$\"+xu%o<%FjanF`js7%Fedu7$$\"+F!H\"y9Fjan$\"+cV6&=%FjanF`js 7%F[eu7$$\"+A_1z9Fjan$\"+J3M$>%FjanF`js7%Faeu7$$\"+1/dz9Fjan$\"+y-^,UF janF`js7%Fgeu7$$\"+(*elz9Fjan$\"+Oqg4UFjanF`js7%F]fu7$$\"+-GLz9Fjan$\" +5jhjBI%FjanF`js7%Fghu7$$\"+q()oP9Fjan$\"+1Mp 8VFjanF`jsFb^rFh^r-F$6$7`p7$F`_r$!3z*****HE\\\"3VFP7$$\"33+++xlz^9FP$! 3))*****RVWjH%FP7$$\"35+++iRxf9FP$!31+++(RYOG%FP7$$\"3)******Hb@lY\"FP $!3Q+++T@8qUFP7$$\"33+++^H'>Z\"FP$!3E+++2B)eD%FP7$$\"3++++X*>gZ\"FP$!3 ++++yB)4C%FP7$$\"3'******\\,7'y9FP$!38+++wS_DUFP7$Fgfs$!3T+++Oqg4UFP7$ F]fs$!3U+++J3M$>%FP7$Fhes$!3q*****fN9^=%FP7$Fces$!3#******pZZo<%FP7$F^ es$!3E+++'*yboTFP7$Fids$!3Q+++BXEgTFP7$Fdds$!3s*****zh()>:%FP7$F_ds$!3 ********\\*[P9%FP7$Fjcs$!3w*****z(>dNTFP7$Fecs$!3&)*****f0#[FTFP7$F`cs $!3'******4m1&>TFP7$F[cs$!3!)*****>kv;6%FP7$Ffbs$!3E+++9:-/TFP7$Fabs$! 3U+++l(zl4%FP7$F\\bs$!3))*****f?*Q*3%FP7$Fgas$!3))******\\B\\#3%FP7$Fb as$!3;+++9e$f2%FP7$F]as$!3)*******Q2xpSFP7$Fh`s$!3A+++8K0kSFP7$Fc`s$!3 ))*****zmW)eSFP7$F^`s$!3'******p?7U0%FP7$Fi_s$!3!)*****f!)G-0%FP7$Fd_s $!3A+++)Gtp/%FP7$F__s$!37+++f*HX/%FP7$Fj^s$!3!******fq()H/%FP7$Fe^s$!3 >+++\"QQC/%FP7$F`^s$!3>+++%=uH/%FP7$F[^s$!3t*****H#QoWSFP7$Ff]s$!3c*** **pCZw/%FP7$Fa]s$!3k******p*G>0%FP7$F\\]s$!3y*****\\Fqv0%FP7$Fg\\s$!33 +++.6ekSFP7$Fb\\s$!3K+++wG$H2%FP7$F]\\s$!3o******4Rb#3%FP7$Fh[s$!3s*** **\\$*GL4%FP7$Fc[s$!3n*****>i.^5%FP7$F^[s$!3K+++iQpsTF P7$Feir$!3I+++d8$f=%FP7$F`ir$!3S+++LNT*>%FP7$F[ir$!3J+++;s`7UFP7$Ffhr$ !3-+++c0ADUFP7$Fahr$!3?+++_?SPUFP7$F\\hr$!3E+++Jt.\\UFP7$Fggr$!3o***** *=g4gUFP7$Fbgr$!3i*****\\\"*e0F%FP7$F]gr$!3Y+++obT!G%FP7$Fhfr$!3)***** **eAm*G%FP7$Fcfr$!3S+++n.I)H%FP7$F^fr$!3N+++r]L1VFP7$Fier$!3'******pKu PJ%FP7$Fder$!3P+++y!G1K%FP7$F_er$!36+++Uw!pK%FP7$Fjdr$!3i*****4DDEL%FP 7$Fedr$!3v*******p$zPVFP7$F`dr$!3w*****p3ECM%FP7$F[dr$!3W+++9c`YVFP7$F fcr$!3/+++`a8]VFP7$Facr$!3m*****zjQKN%FP7$F\\cr$!3s*****>+eeN%FP7$Fgbr $!3=+++Dh+eVFP7$$\"3)*******[rd'H\"FP$!3/+++3`pfVFP7$Fbbr$!3e*****\\bP 4O%FP7$$\"35+++?e+88FP$!3-+++rXuhVFP7$$\"3-+++%=(=H8FP$!3C+++W#)4iVFP7 $$\"3$******H#e+X8FP$!3m*****\\:W3O%FP7$$\"3++++@uNg8FP$!3++++Df1eVFP7 $$\"31+++2j9v8FP$!3U+++;N%QN%FP7$$\"3++++/NG*Q\"FP$!3I+++mQD[VFP7$$\"3 6+++W^o-9FP$!35+++`:PTVFP7$$\"3\"******f;r_T\"FP$!3l*****HZpKL%FP7$$\" 3&******fAkpU\"FP$!3>+++T&>SK%FP7$F_is$!3i*****fS$p8VFP7$Fjhs$!36+++)> jBI%FPFa^n-F[_n6ap7%7$Fgis$!+j#\\\"3VFjan7$$\"+xlz^9Fjan$!+MWM'H%Fjan7 $Fajs$!#UFejs7%Fg[w7$$\"+iRxf9Fjan$!+(RYOG%FjanF\\\\w7%F`\\w7$$\"+`:_m 9Fjan$!+T@8qUFjanF\\\\w7%Ff\\w7$$\"+^H'>Z\"Fjan$!+2B)eD%FjanF\\\\w7%F \\]w7$$\"+X*>gZ\"Fjan$!+yB)4C%FjanF\\\\w7%Fb]w7$$\"+:?hy9Fjan$!+wS_DUF janF\\\\w7%Fh]w7$F^fu$!+Oqg4UFjanF\\\\w7%F^^w7$Fbeu$!+J3M$>%FjanF\\\\w 7%Fb^w7$F\\eu$!+cV6&=%FjanF\\\\w7%Ff^w7$Ffdu$!+xu%o<%FjanF\\\\w7%Fj^w7 $F`du$!+'*yboTFjanF\\\\w7%F^_w7$Fjcu$!+BXEgTFjanF\\\\w7%Fb_w7$Fdcu$!+= w)>:%FjanF\\\\w7%Ff_w7$F^cu$!+]*[P9%FjanF\\\\w7%Fj_w7$Fhbu$!+y>dNTFjan F\\\\w7%F^`w7$Fbbu$!+c?[FTFjanF\\\\w7%Fb`w7$F\\bu$!+hm]>TFjanF\\\\w7%F f`w7$Ffau$!+Ucn6TFjanF\\\\w7%Fj`w7$F`au$!+9:-/TFjanF\\\\w7%F^aw7$Fj`u$ !+l(zl4%FjanF\\\\w7%Fbaw7$Fd`u$!+1#*Q*3%FjanF\\\\w7%Ffaw7$F^`u$!+]B\\# 3%FjanF\\\\w7%Fjaw7$Fh_u$!+9e$f2%FjanF\\\\w7%F^bw7$Fb_u$!+R2xpSFjanF\\ \\w7%Fbbw7$F\\_u$!+8K0kSFjanF\\\\w7%Ffbw7$Ff^u$!+oY%)eSFjanF\\\\w7%Fjb w7$F`^u$!+2A@aSFjanF\\\\w7%F^cw7$Fj]u$!+1)G-0%FjanF\\\\w7%Fbcw7$Fd]u$! +)Gtp/%FjanF\\\\w7%Ffcw7$F^]u$!+f*HX/%FjanF\\\\w7%Fjcw7$Fh\\u$!+1x)H/% FjanF\\\\w7%F^dw7$Fb\\u$!+\"QQC/%FjanF\\\\w7%Fbdw7$F\\\\u$!+%=uH/%Fjan F\\\\w7%Ffdw7$Ff[u$!+BQoWSFjanF\\\\w7%Fjdw7$F`[u$!+ZskZSFjanF\\\\w7%F^ ew7$Fjjt$!+q*G>0%FjanF\\\\w7%Fbew7$Fdjt$!+v-ddSFjanF\\\\w7%Ffew7$F^jt$ !+.6ekSFjanF\\\\w7%Fjew7$Fhit$!+wG$H2%FjanF\\\\w7%F^fw7$Fbit$!+5Rb#3%F janF\\\\w7%Fbfw7$F\\it$!+N*GL4%FjanF\\\\w7%Fffw7$Ffht$!+AO50TFjanF\\\\ w7%Fjfw7$F`ht$!+iQpsTFjanF\\\\w7%F^hw7$Fbft$!+d8$f=%FjanF\\\\w7%Fbhw7$F\\ft$!+LNT*>%Fjan F\\\\w7%Ffhw7$Ffet$!+;s`7UFjanF\\\\w7%Fjhw7$F`et$!+c0ADUFjanF\\\\w7%F^ iw7$Fjdt$!+_?SPUFjanF\\\\w7%Fbiw7$Fddt$!+Jt.\\UFjanF\\\\w7%Ffiw7$F^dt$ !+>g4gUFjanF\\\\w7%Fjiw7$Fhct$!+:*e0F%FjanF\\\\w7%F^jw7$Fbct$!+obT!G%F janF\\\\w7%Fbjw7$F\\ct$!+fAm*G%FjanF\\\\w7%Ffjw7$Ffbt$!+n.I)H%FjanF\\ \\w7%Fjjw7$F`bt$!+r]L1VFjanF\\\\w7%F^[x7$Fjat$!+FVx8VFjanF\\\\w7%Fb[x7 $Fdat$!+y!G1K%FjanF\\\\w7%Ff[x7$F^at$!+Uw!pK%FjanF\\\\w7%Fj[x7$Fh`t$!+ ^_iKVFjanF\\\\w7%F^\\x7$Fb`t$!++PzPVFjanF\\\\w7%Fb\\x7$F\\`t$!+(3ECM%F janF\\\\w7%Ff\\x7$Ff_t$!+9c`YVFjanF\\\\w7%Fj\\x7$F`_t$!+`a8]VFjanF\\\\ w7%F^]x7$Fj^t$!+Q'QKN%FjanF\\\\w7%Fb]x7$Fd^t$!+-!eeN%FjanF\\\\w7%Ff]x7 $F^^t$!+Dh+eVFjanF\\\\w7%Fj]x7$$\"+\\rd'H\"Fjan$!+3`pfVFjanF\\\\w7%F^^ x7$Fh]t$!+bv$4O%FjanF\\\\w7%Fd^x7$$\"+?e+88Fjan$!+rXuhVFjanF\\\\w7%Fh^ x7$$\"+%=(=H8Fjan$!+W#)4iVFjanF\\\\w7%F^_x7$$\"+Be+X8Fjan$!+bT%3O%Fjan F\\\\w7%Fd_x7$$\"+@uNg8Fjan$!+Df1eVFjanF\\\\w7%Fj_x7$$\"+2j9v8Fjan$!+; N%QN%FjanF\\\\w7%F``x7$$\"+/NG*Q\"Fjan$!+mQD[VFjanF\\\\w7%Ff`x7$$\"+W^ o-9Fjan$!+`:PTVFjanF\\\\w7%F\\ax7$$\"+m6F:9Fjan$!+t%pKL%FjanF\\\\w7%Fb ax7$$\"+EU'pU\"Fjan$!+T&>SK%FjanF\\\\w7%Fhax7$F^iu$!+1Mp8VFjanF\\\\w7% F^bx7$Fhhu$!+)>jBI%FjanF\\\\wFb^rFh^r-F$6%7$7$$!3')*************e%FPF( 7$$\"3!**************)=FPF(-%'COLOURG6&Fd^nF)F)F)-%*LINESTYLEG6#\"\"$- F$6%7$7$F($!3/++++++!z%FP7$F($\"3/++++++!z%FPF^cxFacx-%%FONTG6$%*HELVE TICAG\"\"*-%+AXESLABELSG6%%&Re(z)G%&Im(z)G-F_dx6#%(DEFAULTG-%*AXESSTYL EG6#%$BOXG-%(SCALINGG6#%,CONSTRAINEDG-%%VIEWG6$;$!$f%Fg^n$\"$*=Fg^n;$! $z%Fg^n$\"$z%Fg^n" 1 2 0 1 10 0 2 9 1 2 1 1.000000 45.000000 45.000000 0 0 "Curve 1" "Curve 2" "Curve 3" "Curve 4" "Curve 5" "Curve 6" "Curve 7" "Curve 8" }}}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 " " 0 "" {TEXT -1 4 "The " }{TEXT 260 30 "interval of absolute stability " }{TEXT -1 89 " (or stability interval) is the intersection of the s tability region with the real line." }}{PARA 0 "" 0 "" {TEXT -1 59 "Fo r this scheme the stability interval is (approximately) " }{XPPEDIT 18 0 "[-4.0648, 0];" "6#7$,$-%&FloatG6$\"&[1%!\"%!\"\"\"\"!" }{TEXT -1 2 ". " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 260 "We can distort the boundary curve horizontally by taking the 11th root of the real part of points along the curve. In this way we see t hat the largest interval on the nonnegative imaginary axis that contai ns the origin and lies inside the stability region is " }{XPPEDIT 18 0 "[0, 1.76];" "6#7$\"\"!-%&FloatG6$\"$w\"!\"#" }{TEXT -1 18 " approx imately. " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 " " {MPLTEXT 1 0 322 "R := z -> 1+z+1/2*z^2+1/6*z^3+1/24*z^4+1/120*z^5+1 /720*z^6+1/5400*z^7:\nDigits := 25:\npts := []: z0 := 0:\nfor ct from \+ 0 to 45 do\n zz := newton(R(z)=exp(ct*Pi/100*I),z=z0):\n z0 := zz: \n pts := [op(pts),[surd(Re(zz),11),Im(zz)]]:\nend do:\nplot(pts,col or=COLOR(RGB,.45,0,.95),thickness=2,font=[HELVETICA,9]);\nDigits := 10 :" }}{PARA 13 "" 1 "" {GLPLOT2D 261 276 276 {PLOTDATA 2 "6(-%'CURVESG6 #7P7$$\"\"!F)F(7$$\":JMK>8$)>GAsG!#E$\":[&*\\TKv*e`EfTJF-7$$\":$*))) y$yDJtVHTv%F-$\":JL'[@\\W<2`=$G'F-7$$\":^3$yB&pJ\\\\QFQ'F-$\":Q&fYx-6o gzxC%*F-7$$\":DqEjw#Q/mh\"['yF-$\":KH)*[(e#p81PmD\"!#D7$$\":TZ`^\\&p$= lSbC*F-$\":0\\4,-$QZEjzq:F?7$$\":u'QPtBz1T:%\\0\"F?$\":OtGQqS*)4fb\\)= F?7$$\":K:O]W)4LGt:z6F?$\":BbH!)H^QS&[6*>#F?7$$\":K34b*3#*4kW@)H\"F?$ \":i')*oHsA)Q6uK^#F?7$$\":E@s4=&eAX!HGT\"F?$\":9vSkpxitOLu#GF?7$$\":<< +LPP8B2FN_\"F?$\":t9u0Td5\"4EfTJF?7$$\":]&=5G`l:\\cqI;F?$\":C)z=ifj@c.nMb>J\")4*>_'eQ$>F?$\":76()\\AmWm\"*>2eqCF?$\":P#GoRDse$)z<$G'F?7$$\":!eff6iht)ewh#>a3]'Hz RvF?7$$\":j10f>6KF(zh_GF?$\":]&*>_64s'y8%R&yF?7$$\":w3gys#3)Q%oZ@HF?$ \":sjuL\"*eG7T'3o\")F?7$$\":!fbRS!4KOQ`v)HF?$\":g/YY'4SD*=FA[)F?7$$\": b@Zd`pdfAk10$F?$\":SyA3*\\u%ylijz)F?7$$\":'Hy)*3QTuAkd5JF?$\":=@M(y^B0 _:\\5\"*F?7$$\":;vF!ybB:AK*p;$F?$\":e,0W#>DOlBhC%*F?7$$\":y*eB0U6IUM`> KF?$\":AR,r'oPD:LsQ(*F?7$$\":aw_*He;\\q2qnKF?$\":eU]%*p#>dIAG05!#C7$$ \":484qZuX,aO3J$F?$\":Y&H$Hg*Ho)o!pO5Ffu7$$\":D1b&)3)>'p0W![LF?$\":-j@ GU5R6U(4o5Ffu7$$\":P\"*y@uur$p41yLF?$\":6gDh@;#)))4-&*4\"Ffu7$$\":okj \\3H-vdH!*R$F?$\":ttLx*)pWOM/48\"Ffu7$$\":WBMVxI%>Gp/3MF?$\":,bC-Y\"Ffu7$$\":w0l!e8JKeI (eO$F?$\":U$3(pK4+2&=4D7Ffu7$$\":F.C2+&\\!\\\\#>#G$F?$\":8Ffu7$$!:^R!4=(R\\)e@/@MF?$\":;q.B+)[(**Q21N\"Ffu7 $$!:F\\!*QjSW9!='\\l$F?$\":t')4Q#ePd$*)o>Q\"Ffu7$$!:#pQStH)[6%e.OQF?$ \":j>$*)HEKSg9K89Ffu-%%FONTG6$%*HELVETICAG\"\"*-%+AXESLABELSG6$Q!6\"F` z-%&COLORG6&%$RGBG$\"#X!\"#F($\"#&*Fhz-%*THICKNESSG6#\"\"#-%%VIEWG6$%( DEFAULTGFb[l" 1 2 0 1 10 2 2 6 1 4 2 1.000000 45.000000 45.000000 0 0 "Curve 1" }}}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 123 "The relevant intersection point of the boundary curve with the imaginary axis can be determined more accurately as follows." }} {PARA 0 "" 0 "" {TEXT -1 86 "First we look for points on the boundary \+ curve either side of the intersection point. " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 111 "Digits := 1 5:\nz0 := 1.3*I:\nfor ct from 40 to 43 do\n newton(R(z)=exp(ct*Pi/10 0*I),z=z0);\nend do;\nDigits := 10:" }}{PARA 11 "" 1 "" {XPPMATH 20 "6 #^$$\"0h8!))f;iZ!#?$\"0'R:9%zkD\"!#9" }}{PARA 11 "" 1 "" {XPPMATH 20 " 6#^$$\"0v]k*eZ#>#!#?$\"0be/thyG\"!#9" }}{PARA 11 "" 1 "" {XPPMATH 20 " 6#^$$!0FOQU`Dw\"!#?$\"0S[N,Q#>8!#9" }}{PARA 11 "" 1 "" {XPPMATH 20 "6# ^$$!0z:k-x<^(!#?$\"0)[(**Q21N\"!#9" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" } }{PARA 0 "" 0 "" {TEXT -1 107 "Then we apply the bisection method to c alculate the parameter value associated with the intersection point." }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 176 "real_part := proc(u)\n \+ Re(newton(R(z)=exp(u*Pi*I),z=1.3*I))\nend proc:\nDigits := 15:\nu0 := \+ bisect('real_part'(u),u=0.40..0.43);\nnewton(R(z)=exp(u0*Pi*I),z=1.3*I );\nDigits := 10:" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%#u0G$\"0P(f@gCg T!#:" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#^#$\"0.$3paw18!#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, 1.3068];" "6#7$\"\"!-%&FloatG6$\"&oI\"!\"%" }{TEXT -1 18 " (approximately)." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 " " 0 "" {TEXT -1 37 "#------------------------------------" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}}{PARA 0 "" 0 "" {TEXT -1 38 " #-------------------------------------" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}}{PARA 0 "" 0 "" {TEXT -1 26 "#================= ========" }}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 38 "a scheme with a large stability region" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}} {PARA 0 "" 0 "" {TEXT -1 39 "#--------------------------------------" }}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 19 "checking the scheme" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }} }{SECT 1 {PARA 4 "" 0 "" {TEXT -1 26 "coefficients of the scheme" }} {PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 668 "ee := \{c[2]=1/40,\nc[3]=2/11,\nc[4]=11/41,\nc[5]=8/13,\nc[6]=10/ 13,\nc[7]=1,\n\na[2,1]=1/40,\na[3,1]=-58/121,\na[3,2]=80/121,\na[4,1]= 2695/275684,\na[4,2]=4840/68921,\na[4,3]=51909/275684,\na[5,1]=129980/ 72501,\na[5,2]=-3520/2197,\na[5,3]=-343156/257049,\na[5,4]=4975760/282 7539,\na[6,1]=-2066341/1498354,\na[6,2]=83600/68107,\na[6,3]=1551704/8 85391,\na[6,4]=-2379381536/1801770685,\na[6,5]=5607/11470,\na[7,1]=157 3235/820072,\na[7,2]=-11440/9319,\na[7,3]=-52267281/13568464,\na[7,4]= 36832433025/8776615562,\na[7,5]=-296595/344803,\na[7,6]=76620375/92891 792,\n\nb[1]=1667/21120,\nb[2]=0,\nb[3]=161051/7076160,\nb[4]=70672282 61/19071409500,\nb[5]=314171/1776000,\nb[6]=885391/3229632,\nb[7]=9319 /121500\}:" }}}{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 0 109 "subs(ee,matrix([seq([c[i],seq(a[i,j],j=1 ..i-1),``$(8-i)],i=2..7),\n[``,seq(b[i],i=1..7)]]));\n``;\nevalf[8](%% );" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#K%'matrixG6#7)7*#\"\"\"\"#SF(%!G F+F+F+F+F+7*#\"\"#\"#6#!#e\"$@\"#\"#!)F2F+F+F+F+F+7*#F/\"#T#\"%&p#\"'% ov##\"%S[\"&@*o#\"&4>&F:F+F+F+F+7*#\"\")\"#8#\"'!)*H\"\"&,D(#!%?N\"%(> ##!'cJM\"'\\qD#\"(gd(\\\"(Rv#GF+F+F+7*#\"#5FC#!(Tj1#\"(a$)\\\"#\"&+O) \"&2\"o#\"(/$*#!)\"GnA&\")k%oN\"#\",DIVKo$\"+ibhw()#!'&f'H\"'. [M#\")v.iw\")#z\"*G*F+7*F+#\"%n;\"&?6#\"\"!#\"'^5;\"(gh2(#\"+h#Gs1(\", +&492>#\"'rTJ\"(+gx\"#Fen\"(K'HK#Fbo\"'+:7Q)pprint206\"" }}{PARA 11 " " 1 "" {XPPMATH 20 "6#%!G" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#K%'matrix G6#7)7*$\")+++D!\"*F(%!GF+F+F+F+F+7*$\")====!\")$!)%)Q$z%F/$\")-d6mF/F +F+F+F+F+7*$\")o#Ho#F/$\")_ov(*!#5$\")I`AqF*$\")n\"H)=F/F+F+F+F+7*$\") i%Q:'F/$\")H!Gz\"!\"($!)[=-;FC$!)G)\\L\"FC$\")%\\(fFC$!)&*fF7FC$!)[6_QFC$\")cl'>%FC$!)$o=g)F/$\")sM[#)F/F+7*F+ $\")C*H*yF*$FZFZ$\")h'fF#F*$\")lm0PF/$\")4)*o " 0 "" {MPLTEXT 1 0 148 "RK6_ 7eqs := [op(RowSumConditions(7,'expanded')),op(OrderConditions(6,7,'ex panded'))]:\nsimplify(subs(ee,RK6_7eqs)):\nmap(u->lhs(u)-rhs(u),%);\nn ops(%);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#7M\"\"!F$F$F$F$F$F$F$F$F$F$ F$F$F$F$F$F$F$F$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#\"#V" }}}{PARA 0 "" 0 "" {TEXT -1 0 " " }}{PARA 0 "" 0 "" {TEXT -1 86 "Next we set-up stage-order condtions \+ to check for stage-orders from 2 to 4 inclusive. " }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 85 "for ct from 2 to 4 do\n so||ct||_7 := Stage OrderConditions(ct,7,'expanded');\nend do:" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 58 "Stages 3 to 7 have the followin g respective stage-orders. " }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 173 "[seq([seq(expand(subs(ee,so||i||_7[j])),i=2..4)],j=1..5)]:\nmap(p roc(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$" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 42 "Thus stages 5, 6 and 7 have stage-order 3." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 32 "#-------------- -----------------" }}{PARA 0 "" 0 "" {TEXT -1 27 "The simplifying cond itions:" }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "Sum(b[i]*a [i,j],i = j+1 .. 7) = b[j]*(1-c[j]);" "6#/-%$SumG6$*&&%\"bG6#%\"iG\"\" \"&%\"aG6$F+%\"jGF,/F+;,&F0F,F,F,\"\"(*&&F)6#F0F,,&F,F,&%\"cG6#F0!\"\" F," }{TEXT -1 3 ", " }{XPPEDIT 18 0 "j=1" "6#/%\"jG\"\"\"" }{TEXT -1 7 " . . 7 " }}{PARA 0 "" 0 "" {TEXT -1 8 "(where " }{XPPEDIT 18 0 "c[ 1]=0" "6#/&%\"cG6#\"\"\"\"\"!" }{TEXT -1 17 " ) are satisfied." }} {PARA 256 "" 0 "" {TEXT -1 1 " " }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 162 "[Sum(b[i]*a[i,1],i=2..7)=b[1],seq(Sum(b[i]*a[i,j],i=j+1..7)=b [j]*(1-c[j]),j=2..6)];\neval(subs(Sum=add,%)):\nsubs(ee,%):\nmap(u->`i f`(lhs(u)=rhs(u),0,1),%);\nnops(%);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6 #7(/-%$SumG6$*&&%\"bG6#%\"iG\"\"\"&%\"aG6$F,F-F-/F,;\"\"#\"\"(&F*6#F-/ -F&6$*&F)F-&F/6$F,F3F-/F,;\"\"$F4*&&F*6#F3F-,&F-F-&%\"cGFB!\"\"F-/-F&6 $*&F)F-&F/6$F,F?F-/F,;\"\"%F4*&&F*6#F?F-,&F-F-&FEFRFFF-/-F&6$*&F)F-&F/ 6$F,FOF-/F,;\"\"&F4*&&F*6#FOF-,&F-F-&FEFjnFFF-/-F&6$*&F)F-&F/6$F,FgnF- /F,;\"\"'F4*&&F*6#FgnF-,&F-F-&FEFhoFFF-/-F&6$*&F)F-&F/6$F,FeoF-/F,;F4F 4*&&F*6#FeoF-,&F-F-&FEFepFFF-" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#7(\" \"!F$F$F$F$F$" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#\"\"'" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 27 "The simplifying cond ition: " }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "Sum(b[i]*a [i,2],i = 3 .. 7) = 0;" "6#/-%$SumG6$*&&%\"bG6#%\"iG\"\"\"&%\"aG6$F+\" \"#F,/F+;\"\"$\"\"(\"\"!" }{TEXT -1 1 " " }}{PARA 0 "" 0 "" {TEXT -1 13 "is satisfied." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 59 "Sum(b[i]*a[i,2],i=3..7);\neval(subs(Sum=add,% ));\nsubs(ee,%);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#-%$SumG6$*&&%\"bG6 #%\"iG\"\"\"&%\"aG6$F*\"\"#F+/F*;\"\"$\"\"(" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#,,*&&%\"bG6#\"\"$\"\"\"&%\"aG6$F(\"\"#F)F)*&&F&6#\"\"%F )&F+6$F1F-F)F)*&&F&6#\"\"&F)&F+6$F7F-F)F)*&&F&6#\"\"'F)&F+6$F=F-F)F)*& &F&6#\"\"(F)&F+6$FCF-F)F)" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#\"\"!" }} }{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 27 "The simp lifying condition: " }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "Sum(b[i]*c[i]*a[i,2],i = 3 .. 7) = 0;" "6#/-%$SumG6$*(&%\"bG6#%\"iG \"\"\"&%\"cG6#F+F,&%\"aG6$F+\"\"#F,/F+;\"\"$\"\"(\"\"!" }{TEXT -1 1 " \+ " }}{PARA 0 "" 0 "" {TEXT -1 13 "is satisfied." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 64 "Sum(b[i]*c[i ]*a[i,2],i=3..7);\neval(subs(Sum=add,%));\nsubs(ee,%);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#-%$SumG6$*(&%\"bG6#%\"iG\"\"\"&%\"cGF)F+&%\"aG6$F* \"\"#F+/F*;\"\"$\"\"(" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#,,*(&%\"bG6# \"\"$\"\"\"&%\"cGF'F)&%\"aG6$F(\"\"#F)F)*(&F&6#\"\"%F)&F+F2F)&F-6$F3F/ F)F)*(&F&6#\"\"&F)&F+F9F)&F-6$F:F/F)F)*(&F&6#\"\"'F)&F+F@F)&F-6$FAF/F) F)*(&F&6#\"\"(F)&F+FGF)&F-6$FHF/F)F)" }}{PARA 11 "" 1 "" {XPPMATH 20 " 6#\"\"!" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 27 "The simplifying condition: " }}{PARA 256 "" 0 "" {TEXT -1 1 " " } {XPPEDIT 18 0 "Sum(b[i]*c[i]^2*a[i,2],i = 3 .. 7) = 0;" "6#/-%$SumG6$* (&%\"bG6#%\"iG\"\"\"*$&%\"cG6#F+\"\"#F,&%\"aG6$F+F1F,/F+;\"\"$\"\"(\" \"!" }{TEXT -1 1 " " }}{PARA 0 "" 0 "" {TEXT -1 13 "is satisfied." }} {PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 66 "Sum(b[i]*c[i]^2*a[i,2],i=3..7);\neval(subs(Sum=add,%));\nsubs(ee,% );" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#-%$SumG6$*(&%\"bG6#%\"iG\"\"\")& %\"cGF)\"\"#F+&%\"aG6$F*F/F+/F*;\"\"$\"\"(" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#,,*(&%\"bG6#\"\"$\"\"\")&%\"cGF'\"\"#F)&%\"aG6$F(F-F)F) *(&F&6#\"\"%F))&F,F3F-F)&F/6$F4F-F)F)*(&F&6#\"\"&F))&F,F;F-F)&F/6$F " 0 "" {MPLTEXT 1 0 85 "Sum(b[i]*c[i]*Sum(a[i,j]*a[j,2],j=3 ..i-1),i=3..7);\neval(subs(Sum=add,%));\nsubs(ee,%);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#-%$SumG6$*(&%\"bG6#%\"iG\"\"\"&%\"cGF)F+-F$6$*&&%\"a G6$F*%\"jGF+&F26$F4\"\"#F+/F4;\"\"$,&F*F+F+!\"\"F+/F*;F:\"\"(" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#,***&%\"bG6#\"\"%\"\"\"&%\"cGF'F)&%\"a G6$F(\"\"$F)&F-6$F/\"\"#F)F)*(&F&6#\"\"&F)&F+F5F),&*&&F-6$F6F/F)F0F)F) *&&F-6$F6F(F)&F-6$F(F2F)F)F)F)*(&F&6#\"\"'F)&F+FCF),(*&&F-6$FDF/F)F0F) F)*&&F-6$FDF(F)F?F)F)*&&F-6$FDF6F)&F-6$F6F2F)F)F)F)*(&F&6#\"\"(F)&F+FT F),**&&F-6$FUF/F)F0F)F)*&&F-6$FUF(F)F?F)F)*&&F-6$FUF6F)FPF)F)*&&F-6$FU FDF)&F-6$FDF2F)F)F)F)" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#\"\"!" }}} {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 114 "We can c alculate the principal error norm of the order 6 scheme, that is, the \+ 2-norm of the principal error terms." }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 158 "errterms6_7 := PrincipalErrorTerms(6,7,'expanded'): \nsm := 0:\nfor ct to nops(errterms6_7) do\n sm := sm+(evalf(subs(ee ,errterms6_7[ct])))^2;\nend do:\nsqrt(sm);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#$\"+cO-]H!#8" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}}{PARA 0 "" 0 "" {TEXT -1 39 "#--------------------------------------" }}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 32 "short construction of the scheme" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT 260 4 "Note" }{TEXT -1 123 ": Th is scheme was constructed in a step by step manner in a previous secti on making use of \"alternative\" order conditions. " }}{PARA 0 "" 0 " " {TEXT -1 77 "In this subsection we construct the scheme using \"stan dard\" order conditions." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 " " 0 "" {TEXT -1 192 "We set up a system of equations using a selection of \"simple\" order conditions and incorporate the row-sum conditions together with the stage-order equations to ensure that stages 3 to 7 \+ have " }{TEXT 260 13 "stage-order 2" }{TEXT -1 1 "." }}{PARA 0 "" 0 " " {TEXT -1 48 "We also incorporate the simplifying conditions: " }} {PARA 256 "" 0 "" {TEXT -1 2 " " }{XPPEDIT 18 0 "Sum(b[i]*a[i,j],i = \+ j+1 .. 7) = b[j]*(1-c[j]);" "6#/-%$SumG6$*&&%\"bG6#%\"iG\"\"\"&%\"aG6$ F+%\"jGF,/F+;,&F0F,F,F,\"\"(*&&F)6#F0F,,&F,F,&%\"cG6#F0!\"\"F," } {TEXT -1 1 " " }}{PARA 0 "" 0 "" {TEXT -1 5 "for " }{XPPEDIT 18 0 "j \+ = 3;" "6#/%\"jG\"\"$" }{TEXT -1 62 ", 4, 5, 6, together with the furt her simplifying conditions: " }}{PARA 256 "" 0 "" {TEXT -1 1 " " } {XPPEDIT 18 0 "Sum(b[i]*c[i]*a[i,2],i = 3 .. 7) = 0;" "6#/-%$SumG6$*(& %\"bG6#%\"iG\"\"\"&%\"cG6#F+F,&%\"aG6$F+\"\"#F,/F+;\"\"$\"\"(\"\"!" } {TEXT -1 8 ", " }{XPPEDIT 18 0 "Sum(b[i]*c[i]*Sum(a[i,j]*a[j,2], j = 3 .. i-1),i = 3 .. 7) = 0;" "6#/-%$SumG6$*(&%\"bG6#%\"iG\"\"\"&%\" cG6#F+F,-F%6$*&&%\"aG6$F+%\"jGF,&F46$F6\"\"#F,/F6;\"\"$,&F+F,F,!\"\"F, /F+;F<\"\"(\"\"!" }{TEXT -1 8 ", " }{XPPEDIT 18 0 "Sum(b[i]*c[i] ^2*a[i,2],i = 3 .. 7) = 0;" "6#/-%$SumG6$*(&%\"bG6#%\"iG\"\"\"*$&%\"cG 6#F+\"\"#F,&%\"aG6$F+F1F,/F+;\"\"$\"\"(\"\"!" }{TEXT -1 2 ". " }} {PARA 0 "" 0 "" {TEXT -1 1 " " }}{PARA 0 "" 0 "" {TEXT -1 76 "The simp le order conditions used are given (in abreviated form) as follows. " }}{PARA 0 "" 0 "" {TEXT 260 4 "Note" }{TEXT -1 86 ": We use one additi onal order condition #28 beyond those used for the previous scheme." } }{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 184 "SO6 := SimpleOrderConditions(6):\n[seq([i,SO6[i]],i=[1,2,4,8,13 ,16,24,28,29,32])]:\nlinalg[augment](linalg[delcols](%,2..2),matrix([[ ` `]$(linalg[rowdim](%))]),linalg[delcols](%,1..1));" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#K%'matrixG6#7,7%\"\"\"%#~~G/*&%\"bGF(%\"eGF(F(7%\" \"#F)/*&F,F(%\"cGF(#F(F/7%\"\"%F)/*&F,F()F2F/F(#F(\"\"$7%\"\")F)/*&F,F ()F2F:F(#F(F57%\"#8F)/*(F,F(F2F(-%!G6#*&F8F(%\"aGF(F(#F(\"#:7%\"#;F)/* &F,F()F2F5F(#F(\"\"&7%\"#CF)/*(F,F(F2F(-FF6#*&FIF(FEF(F(#F(\"#s7%\"#GF )/*(F,F(F8F(FEF(#F(\"#=7%\"#HF)/*(F,F(F2F(-FF6#*&F?F(FIF(F(#F(FT7%\"#K F)/*&F,F()F2FRF(#F(\"\"'Q(pprint06\"" }}}{PARA 0 "" 0 "" {TEXT -1 0 " " }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 462 "SO6_7 := SimpleOrderCon ditions(6,7,'expanded'):\nord_cdns := [seq(SO6_7[i],i=[1,2,4,8,13,16,2 4,28,29,32])]:\nSO_eqs := [op(RowSumConditions(7,'expanded')),op(Stage OrderConditions(2,7,'expanded'))]:\nsimp_eqs := [seq(add(b[i]*a[i,j],i =j+1..7)=b[j]*(1-c[j]),j=[3,4,5,6]),\n add(b[i]*c[i]*a[i, 2],i=3..7)=0,add(b[i]*c[i]*add(a[i,j]*a[j,2],j=3..i-1),i=3..7)=0,\n \+ add(b[i]*c[i]^2*a[i,2],i=3..7)=0]:\ncdns := [op(ord_cdns), op(simp_eqs),op(SO_eqs)]:" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 21 "We specify the nodes " }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "c[2] = 1/40;" "6#/&%\"cG6#\"\"#*&\"\"\"F)\"#S !\"\"" }{TEXT -1 3 ", " }{XPPEDIT 18 0 "c[3] = 2/11;" "6#/&%\"cG6#\" \"$*&\"\"#\"\"\"\"#6!\"\"" }{TEXT -1 3 ", " }{XPPEDIT 18 0 "c[5] = 8/ 13;" "6#/&%\"cG6#\"\"&*&\"\")\"\"\"\"#8!\"\"" }{TEXT -1 3 ", " } {XPPEDIT 18 0 "c[6] = 10/13;" "6#/&%\"cG6#\"\"'*&\"#5\"\"\"\"#8!\"\"" }{TEXT -1 3 ", " }{XPPEDIT 18 0 "c[7]=1" "6#/&%\"cG6#\"\"(\"\"\"" } {TEXT -1 1 "," }}{PARA 0 "" 0 "" {TEXT -1 16 " and the weight " }} {PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "b[2]=0" "6#/&%\"bG6# \"\"#\"\"!" }{TEXT -1 2 ". " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 121 "e1 := \{c[2]=1/40,c[3]=2/11,c[5]=8 /13,c[6]=10/13,c[7]=1,b[2]=0\}:\neqns := subs(e1,cdns):\nnops(eqns);\n indets(eqns);\nnops(%);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#\"#G" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#<>&%\"aG6$\"\"%\"\"#&F%6$F'\"\"$&F%6$F (\"\"\"&F%6$F+F.&F%6$\"\"&F(&F%6$F3F+&F%6$F+F(&F%6$F'F.&%\"cG6#F'&F%6$ F3F.&%\"bG6#F3&F@6#\"\"'&F@6#\"\"(&F@6#F.&F@6#F+&F@F<&F%6$FGF3&F%6$FGF D&F%6$FGF+&F%6$FGF'&F%6$FGF.&F%6$FGF(&F%6$FDF'&F%6$FDF3&F%6$FDF(&F%6$F DF+&F%6$F3F'&F%6$FDF." }}{PARA 11 "" 1 "" {XPPMATH 20 "6#\"#G" }}} {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 51 "There are 28 equations and 28 unknown coefficients." }}{EXCHG {PARA 0 "> " 0 " " {MPLTEXT 1 0 22 "infolevel[solve] := 4:" }}}{EXCHG {PARA 0 "> " 0 " " {MPLTEXT 1 0 69 "e2 := solve(\{op(eqns)\}):\ninfolevel[solve] := 0: \ne3 := `union`(e1,e2):" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 2 "e3" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 734 "e3 := \{c[7] = 1, a[5,1] = 129980/72501, c[4] = 11/4 1, a[3,1] = -58/121, b[2] = 0, b[6] = 885391/3229632, a[5,3] = -343156 /257049, a[5,2] = -3520/2197, b[5] = 314171/1776000, a[5,4] = 4975760/ 2827539, a[4,3] = 51909/275684, a[2,1] = 1/40, a[7,2] = -11440/9319, a [6,4] = -2379381536/1801770685, a[4,2] = 4840/68921, a[7,1] = 1573235/ 820072, b[1] = 1667/21120, b[3] = 161051/7076160, b[7] = 9319/121500, \+ a[7,6] = 76620375/92891792, a[6,1] = -2066341/1498354, a[7,5] = -29659 5/344803, a[7,3] = -52267281/13568464, a[6,5] = 5607/11470, a[4,1] = 2 695/275684, a[6,3] = 1551704/885391, a[7,4] = 36832433025/8776615562, \+ b[4] = 7067228261/19071409500, c[2] = 1/40, c[3] = 2/11, c[5] = 8/13, \+ c[6] = 10/13, a[6,2] = 83600/68107, a[3,2] = 80/121\}:" }{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 91 "subs(e3,matrix([seq([c[i],seq(a[i,j],j=1..i -1),``$(8-i)],i=2..7),\n[``,seq(b[i],i=1..7)]]));" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#K%'matrixG6#7)7*#\"\"\"\"#SF(%!GF+F+F+F+F+7*#\"\"#\"#6# !#e\"$@\"#\"#!)F2F+F+F+F+F+7*#F/\"#T#\"%&p#\"'%ov##\"%S[\"&@*o#\"&4>&F :F+F+F+F+7*#\"\")\"#8#\"'!)*H\"\"&,D(#!%?N\"%(>##!'cJM\"'\\qD#\"(gd(\\ \"(Rv#GF+F+F+7*#\"#5FC#!(Tj1#\"(a$)\\\"#\"&+O)\"&2\"o#\"(/$*#! )\"GnA&\")k%oN\"#\",DIVKo$\"+ibhw()#!'&f'H\"'.[M#\")v.iw\")#z\"*G*F+7* F+#\"%n;\"&?6#\"\"!#\"'^5;\"(gh2(#\"+h#Gs1(\",+&492>#\"'rTJ\"(+gx\"#Fe n\"(K'HK#Fbo\"'+:7Q)pprint186\"" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 85 "RK6_7eqs := [op(RowSumCondit ions(7,'expanded')),op(OrderConditions(6,7,'expanded'))]:" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 7 "Check: " }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 62 "simplify(subs(e3,RK6_7eqs)):\nmap(u ->lhs(u)-rhs(u),%);\nnops(%);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#7M\" \"!F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$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#\"#V" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 65 "#---------------- ------------------------------------------------" }}{EXCHG {PARA 0 "> \+ " 0 "" {MPLTEXT 1 0 1 ";" }}}}{PARA 0 "" 0 "" {TEXT -1 39 "#---------- ----------------------------" }}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 25 "a bsolute stability region" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 26 "coefficients of the scheme" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 668 "ee := \{c[2]=1/40,\nc[3]=2/ 11,\nc[4]=11/41,\nc[5]=8/13,\nc[6]=10/13,\nc[7]=1,\n\na[2,1]=1/40,\na[ 3,1]=-58/121,\na[3,2]=80/121,\na[4,1]=2695/275684,\na[4,2]=4840/68921, \na[4,3]=51909/275684,\na[5,1]=129980/72501,\na[5,2]=-3520/2197,\na[5, 3]=-343156/257049,\na[5,4]=4975760/2827539,\na[6,1]=-2066341/1498354, \na[6,2]=83600/68107,\na[6,3]=1551704/885391,\na[6,4]=-2379381536/1801 770685,\na[6,5]=5607/11470,\na[7,1]=1573235/820072,\na[7,2]=-11440/931 9,\na[7,3]=-52267281/13568464,\na[7,4]=36832433025/8776615562,\na[7,5] =-296595/344803,\na[7,6]=76620375/92891792,\n\nb[1]=1667/21120,\nb[2]= 0,\nb[3]=161051/7076160,\nb[4]=7067228261/19071409500,\nb[5]=314171/17 76000,\nb[6]=885391/3229632,\nb[7]=9319/121500\}:" }}}{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 77 "The stability f unction R for the 7 stage, order 6 scheme is given as follows." }} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 75 "subs(ee,StabilityFunction(6, 7,'expanded')):\nR := unapply(%,z):\n'R(z)'=R(z);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/-%\"RG6#%\"zG,2\"\"\"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)\"%/fF)*$)F'\"\"(F)F)F)" }}} {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 124 "We can f ind the point where the boundary of the stability region intersects th e 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 29 "z0 \+ := newton(R(z)=-1,z=-4.2);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%#z0G$! +a-CMU!\"*" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 " " {MPLTEXT 1 0 308 "z0 := newton(R(z)=-1,z=-4.2):\np1 := plot([R(z),-1 ],z=-4.69..0.49,color=[red,blue]):\np2 := plot([[[z0,-1]]$3],style=poi nt,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=[-4.69..0.49,-1.47..1.47],font=[HELVETICA,9]);" }}{PARA 13 "" 1 "" {GLPLOT2D 360 253 253 {PLOTDATA 2 "6+-%'CURVESG6$7U7$$!3Q++++++!p%! #<$!3*44psCy'3BF*7$$!3i)!#>7$$!3_++NGI@uJF*$!3S$R.'HUIY^Fcp7$$!3) o;z[WQg2$F*$!3c:2WCiHJDFcp7$$!3QLL[p$*HfHF*$\"39O=ceNJL^!#@7$$!3kLL$*f gSgGF*$\"37T1HJ#\\D*=Fcp7$$!3A+D6J'p`u#F*$\"3([NPQB<)QPFcp7$$!3lLLV**G aVEF*$\"3kJK5#)[o$=&Fcp7$$!39+D6OG#=`#F*$\"3SD7j%Q\\lj'Fcp7$$!3J+vL;*Q aU#F*$\"3e3nrg4@\\zFcp7$$!3NL$ekXQWJ#F*$\"3]YJMY&z!)H*Fcp7$$!3\\Lep$Q0 D@#F*$\"3o4_(e#=yb5FP7$$!3&pmTSzcD5#F*$\"3/4#yBj)o(>\"FP7$$!3'o;z)46N) )>F*$\"3C&*\\?54Vb8FP7$$!3C+v3EZ$*))=F*$\"30#HvMTBT]\"FP7$$!3#pm\"*zIi :y\"F*$\"3'pF)3!zQ$z;FP7$$!3?++q$4O1n\"F*$\"3sgkq4$f\"z=FP7$$!3W+]-Mk6 i:F*$\"3#Hc%4hD.'4#FP7$$!3O+DJ$3=rX\"F*$\"3wC\"oUz&yGBFP7$$!3-+]d;``S8 F*$\"3z)>lj*p1Hls'f$FP7$$!3?'***\\)ozx6*FP $\"3_AIW![O\"=SFP7$$!3eQ$eW!e?v!)FP$\"3+8jsO\"['fWFP7$$!3w)*\\i\"p:a)p FP$\"3E%\\,%*>6J(\\FP7$$!3`RL$3mY*>fFP$\"3G5e%*y2BKbFP7$$!3e-]7G$*\\/[ FP$\"3)oX>B[^]='FP7$$!3Wgmms4>IPFP$\"39>f)f,@l)oFP7$$!3$zm;M^b:j#FP$\" 3%pP4wHBio(FP7$$!3NK$e/J-#R)QD1\"F*7$$\"3.LLLJ1\"Hj \"FP$\"31_bVJ$zt<\"F*7$$\"3q+]())Gfrs#FP$\"3dq()o_q_88F*7$$\"3'3+Dr([ \\uPFP$\"3iV%)QC(f&e9F*7$$\"3!***************[FP$\"3M&)f)e$fJK;F*-%'CO LOURG6&%$RGBG$\"*++++\"!\")$\"\"!F\\\\lF[\\l-F$6$7S7$F($!\"\"F\\\\l7$F 3Fa\\l7$F=Fa\\l7$FBFa\\l7$FGFa\\l7$FLFa\\l7$FRFa\\l7$FWFa\\l7$FfnFa\\l 7$F[oFa\\l7$F`oFa\\l7$FeoFa\\l7$FjoFa\\l7$F_pFa\\l7$FepFa\\l7$FjpFa\\l 7$F_qFa\\l7$FeqFa\\l7$FjqFa\\l7$F_rFa\\l7$FdrFa\\l7$FirFa\\l7$F^sFa\\l 7$FcsFa\\l7$FhsFa\\l7$F]tFa\\l7$FbtFa\\l7$FgtFa\\l7$F\\uFa\\l7$FauFa\\ l7$FfuFa\\l7$F[vFa\\l7$F`vFa\\l7$FevFa\\l7$FjvFa\\l7$F_wFa\\l7$FdwFa\\ l7$FiwFa\\l7$F^xFa\\l7$FcxFa\\l7$FhxFa\\l7$F]yFa\\l7$FbyFa\\l7$FgyFa\\ l7$F\\zFa\\l7$FazFa\\l7$FfzFa\\l7$F[[lFa\\l7$F`[lFa\\l-Fe[l6&Fg[lF[\\l F[\\lFh[l-F$6&7#7$$!3P+++a-CMUF*Fa\\l-%'SYMBOLG6#%'CIRCLEG-Fe[l6&Fg[lF \\\\lF\\\\lF\\\\l-%&STYLEG6#%&POINTG-F$6&Fg_l-F\\`l6#%&CROSSGF_`lFa`l- F$6&Fg_l-F\\`l6#%(DIAMONDGF_`lFa`l-F$6%7$7$Fi_lF[\\lFh_l-%&COLORG6&Fg[ lF[\\l$\"\"&Fb\\lF[\\l-%*LINESTYLEG6#\"\"$-%%FONTG6$%*HELVETICAG\"\"*- %+AXESLABELSG6%Q\"z6\"Q!Febl-F]bl6#%(DEFAULTG-%%VIEWG6$;$!$p%!\"#$\"# \\F`cl;$!$Z\"F`cl$\"$Z\"F`cl" 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 1648 "R := z -> 1+z+1/2*z^2+1/6*z^3+1/24*z^4+1/120*z^5+1/720*z^6+1/590 4*z^7:\npts := []: z0 := 0: tt := 0: \nwhile tt<=201/20 do\n zz := n ewton(`R`(z)=exp(tt*Pi*I),z=z0):\n z0 := zz:\n if (17/20<=tt and t t<=29/20) or (171/20<=tt and tt<=183/20) then\n hh := 1/40\n el se \n hh := 1/20\n end if;\n tt := tt+hh;\n pts := [op(pts) ,[Re(zz),Im(zz)]]:\nend do:\np1 := plot(pts,color=COLOR(RGB,0,.23,.48) ):\np2 := plots[polygonplot]([seq([pts[i-1],pts[i],[-2.1,0]],i=2..nops (pts))],\n style=patchnogrid,color=COLOR(RGB,0,.45,.95)):\npt s := []: z0 := 1.4+4.3*I: tt := 0: \nwhile tt<=51/25 do\n zz := newt on(R(z)=exp(tt*Pi*I),z=z0):\n z0 := zz:\n if (11/25<=tt and tt<=43 /25) then\n hh := 1/50\n else \n hh := 1/25\n end if;\n \+ tt := tt+hh;\n pts := [op(pts),[Re(zz),Im(zz)]]:\nend do:\np3 := p lot(pts,color=COLOR(RGB,0,.23,.48)):\np4 := plots[polygonplot]([seq([p ts[i-1],pts[i],[1.23,4.19]],i=2..nops(pts))],\n style=patchno grid,color=COLOR(RGB,0,.45,.95)):\npts := []: z0 := 1.4-4.3*I: tt := 0 : \nwhile tt<=51/25 do\n zz := newton(R(z)=exp(tt*Pi*I),z=z0):\n z 0 := zz:\n if (8/25<=tt and tt<=8/5) then\n hh := 1/50\n else \n hh := 1/25\n end if;\n tt := tt+hh;\n pts := [op(pts),[ Re(zz),Im(zz)]]:\nend do:\np5 := plot(pts,color=COLOR(RGB,0,.23,.48)): \np6 := plots[polygonplot]([seq([pts[i-1],pts[i],[1.23,-4.19]],i=2..no ps(pts))],\n style=patchnogrid,color=COLOR(RGB,0,.45,.95)):\n p7 := plot([[[-4.79,0],[1.89,0]],[[0,-4.79],[0,4.79]]],color=black,lin estyle=3):\nplots[display]([p||(1..7)],view=[-4.79..1.89,-4.79..4.79], 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^y7$$\"\"!F)F(7$F($\"3#******fK'zq:!#=7$$! 33+++pQd*[%!#F$\"3v*****Rc#fTJF-7$$!3#*******y6!))H\"!#D$\"3*)*******R (Q7ZF-7$$!3)******HH)3*\\\"!#C$\"3c******QI<$G'F-7$$!33+++FphU5!#B$\"3 c+++u2#R&yF-7$$!3R+++FSiI_FC$\"3J+++05bC%*F-7$$!3')*****fvt53#!#A$\"3- +++h\")[*4\"!#<7$$!31+++2y)z%pFN$\"3%******HDbkD\"FQ7$$!38+++k74>?!#@$ \"3%*******=MH89FQ7$$!34+++KQ\"pB&FZ$\"3#******4C/*p:FQ7$$!3-+++rTFM7! #?$\"33+++%)*\\hs\"FQ7$$!3?+++:5')yEF_o$\"3%******H')e=)=FQ7$$!3-+++R) >(4aF_o$\"3y*****4=To.#FQ7$$!34+++$3p]-\"!#>$\"3%******Hy44>#FQ7$$!3') *****p7Y!)4&F_p$\"31+++64VXEFQ7$$!3!)*****>1[(3kF_p$\"33+++$*GW>FFQ7 $$!3]+++*GAp(zF_p$\"35+++7Qd#z#FQ7$$!3S+++pc%3$)*F_p$\"37+++UA`kGFQ7$$ !3'******R\\F$*>\"F-$\"3.+++qo$\\$HFQ7$$!3%******\\k6xW\"F-$\"38+++wVK .IFQ7$$!3-+++f\">\"G%owQ?F-$\"36++ +bz*>8$FQ7$$!3))*****H/miP#F-$\"37+++\"4R8>$FQ7$$!3t******Hg.OFF-$\"37 +++z$)*oC$FQ7$$!3=+++8T%H6$F-$\"3;+++i%G&)H$FQ7$$!3*******\\.`>]$F-$\" 3#******fBMiM$FQ7$$!37+++oaf)*QF-$\"3!)******z59!R$FQ7$$!3'*******R@@* H%F-$\"3y******pMXIMFQ7$$!3:+++*R!*4q%F-$\"3%)*****>s;uY$FQ7$$!3h***** RP!)=5&F-$\"3'******R4)G,NFQ7$$!3!)*****z9f/]&F-$\"33+++F!=B`$FQ7$$!3[ ******4()y&*eF-$\"3#******HpR2c$FQ7$$!3[+++C/H(G'F-$\"3=+++'Gkne$FQ7$$ !3q******)fVYn'F-$\"32+++#*)z0h$FQ7$$!39+++5yqdqF-$\"3%)*****>N_Bj$FQ7 $$!3t*****4WlkV(F-$\"3!******f3FAl$FQ7$$!3]+++s,)4\"yF-$\"3;+++(HJ.n$F Q7$$!3o*******\\n8=)F-$\"3!******f=wno$FQ7$$!3_******)4uxa)F-$\"3)**** **\\/f;q$FQ7$$!3K+++nNO5*)F-$\"3')*****47l]r$FQ7$$!3!******>CvZi*F-$\" 3++++@ttPPFQ7$$!35+++?bfK5FQ$\"31+++%y!HbPFQ7$$!3#********)p^,6FQ$\"3) ******4M9\"oPFQ7$$!31+++%4c$p6FQ$\"31+++cR^wPFQ7$$!3++++TW@O7FQ$\"3'** ****\\yJ2y$FQ7$$!36+++2G=-8FQ$\"3y*****p6g4y$FQ7$$!3#******4oVtO\"FQ$ \"3y*****Hs^tx$FQ7$$!34+++xTxJ9FQ$\"3++++6x-qPFQ7$$!3#******\\7\\b\\\" FQ$\"3)******>o$3fPFQ7$$!3#******fIW(e:FQ$\"3)******pz%fWPFQ7$$!3-+++* zR9i\"FQ$\"31+++J-iEPFQ7$$!3!******4[BPo\"FQ$\"3&*******pu?0PFQ7$$!3++ ++FYpXFQ$\"3#*******QY*fe$FQ7$$! 3#*******3t-$*>FQ$\"31+++!fn![NFQ7$$!37+++m?ab?FQ$\"35+++bZ22NFQ7$$!3/ +++C-s=@FQ$\"3=+++r5=jMFQ7$$!33+++\\dw#=#FQ$\"3%******H3:mT$FQ7$$!3'** ****f\\dyC#FQ$\"3=+++p&ywO$FQ7$$!36+++k)=TJ#FQ$\"3!******>mYnJ$FQ7$$!3 #******Hf%e\"Q#FQ$\"33+++DHDkKFQ7$$!3++++/P<]CFQ$\"3-+++9ol5KFQ7$$!3.+ ++<$z'>DFQ$\"3%)*****pe(RcJFQ7$$!39+++;Ay*e#FQ$\"3&)*****\\*e%=5$FQ7$$ !35+++Ii3gEFQ$\"3%)*****f@ns/$FQ7$$!3<+++/[;IFFQ$\"31+++so!G*HFQ7$$!3A +++*=0'*z#FQ$\"30+++st\\QHFQ7$$!3%)*****4VS!oGFQ$\"3#)*****p9\"G%)GFQ7 $$!3.+++&4m^$HFQ$\"31+++a\"R+$GFQ7$$!3/+++iRu+IFQ$\"3'******HyF(fkIFQ$\"3<+++s+&3s#FQ7$$!3&******4#=gEJFQ$\"3!)*****fPqbm #FQ7$$!3=+++Jgn'=$FQ$\"3?+++3Li4EFQ7$$!3\"******zsrZC$FQ$\"3;+++>%oGb# FQ7$$!3(******>Tm3I$FQ$\"3%)******[?=&\\#FQ7$$!39+++#=d\\N$FQ$\"3,+++u uXOCFQ7$$!3'*******)=cqS$FQ$\"3!******f;/mP#FQ7$$!3?+++qt=dMFQ$\"3'*** ***RqYbJ#FQ7$$!3-+++uTQ0NFQ$\"39+++\"oBKD#FQ7$$!3;+++C!)o^NFQ$\"3))*** **f!oe*=#FQ7$$!3;+++%GZhf$FQ$\"3!)******=.gC@FQ7$$!3z*****R];)QOFQ$\"3 =+++&yS#e?FQ7$$!3))*****Rvb(zOFQ$\"31+++ns\\!*>FQ7$$!3:+++n)H!>PFQ$\"3 ,+++l=P@>FQ7$$!3++++8uqcPFQ$\"3-+++O0)3&=FQ7$$!3)******pCfGz$FQ$\"3!** ****\\2a!z7M>gF-7$$!3#**** **H$[0\">%FQ$\"32+++\"RTT@&F-7$$!3y*****H;aJ?%FQ$\"3&)*****z\"H')4WF-7 $$!3C+++#fPL@%FQ$\"3$******\\4Clg$F-7$$!38+++C/b@UFQ$\"3++++))y0/GF-7$ $!3.+++L'[xA%FQ$\"3*******H'>L-?F-7$$!3U+++T\"**=B%FQ$\"30+++[k;,7F-7$ $!3m*****4!*zRB%FQ$\"3\"******zF%\\.SF_p7$Fh^m$!3\"******zF%\\.SF_p7$F c^m$!30+++[k;,7F-7$F^^m$!3*******H'>L-?F-7$Fi]m$!3++++))y0/GF-7$Fd]m$! 3$******\\4Clg$F-7$F_]m$!3&)*****z\"H')4WF-7$Fj\\m$!32+++\"RTT@&F-7$Fe \\m$!3g******>7M>gF-7$F`\\m$!3X+++qqLDoF-7$F[\\m$!3?+++]W(=j(F-7$Ff[m$ !3]+++Y$\\&Q%)F-7$Fa[m$!3%)*****RV#zW#*F-7$F\\[m$!3&******47')\\+\"FQ7 $Fgjl$!3))******=VG&3\"FQ7$Fbjl$!3/+++OsEl6FQ7$F]jl$!3,+++Uf\"[C\"FQ7$ Fhil$!3-+++%R.QK\"FQ7$Fcil$!33+++W%*4-9FQ7$F^il$!3-+++%4w&z9FQ7$Fihl$! 3!*******4C6c:FQ7$Fdhl$!3#******\\h)fJ;FQ7$F_hl$!3++++u!Rfq\"FQ7$Fjgl$ !3!******\\2a!zFQ7$F[gl$!31+ ++ns\\!*>FQ7$Fffl$!3=+++&yS#e?FQ7$Fafl$!3!)******=.gC@FQ7$F\\fl$!3))** ***f!oe*=#FQ7$Fgel$!39+++\"oBKD#FQ7$Fbel$!3'******RqYbJ#FQ7$F]el$!3!** ****f;/mP#FQ7$Fhdl$!3,+++uuXOCFQ7$Fcdl$!3%)******[?=&\\#FQ7$F^dl$!3;++ +>%oGb#FQ7$Ficl$!3?+++3Li4EFQ7$Fdcl$!3!)*****fPqbm#FQ7$F_cl$!3<+++s+&3 s#FQ7$Fjbl$!3'******HymYnJ$FQ7$Fc_l$!3=+++p&ywO$FQ7$ F^_l$!3%******H3:mT$FQ7$Fi^l$!3=+++r5=jMFQ7$Fd^l$!35+++bZ22NFQ7$F_^l$! 31+++!fn![NFQ7$Fj]l$!3#*******QY*fe$FQ7$Fe]l$!3*******p%3u?OFQ7$F`]l$! 3#)******RpA_OFQ7$F[]l$!3/+++qoR!o$FQ7$Ff\\l$!3&*******pu?0PFQ7$Fa\\l$ !31+++J-iEPFQ7$F\\\\l$!3)******pz%fWPFQ7$Fg[l$!3)******>o$3fPFQ7$Fb[l$ !3++++6x-qPFQ7$F][l$!3y*****Hs^tx$FQ7$Fhz$!3y*****p6g4y$FQ7$Fcz$!3'*** ***\\yJ2y$FQ7$F^z$!31+++cR^wPFQ7$Fiy$!3)******4M9\"oPFQ7$Fdy$!31+++%y! HbPFQ7$F_y$!3++++@ttPPFQ7$Fjx$!3')*****47l]r$FQ7$F`x$!3!******f=wno$FQ 7$F[x$!3;+++(HJ.n$FQ7$Ffw$!3!******f3FAl$FQ7$Faw$!3%)*****>N_Bj$FQ7$F \\w$!32+++#*)z0h$FQ7$Fgv$!3=+++'Gkne$FQ7$Fbv$!3#******HpR2c$FQ7$F]v$!3 3+++F!=B`$FQ7$Fhu$!3'******R4)G,NFQ7$Fcu$!3%)*****>s;uY$FQ7$F^u$!3y*** ***pMXIMFQ7$Fit$!3!)******z59!R$FQ7$Fdt$!3#******fBMiM$FQ7$F_t$!3;+++i %G&)H$FQ7$Fjs$!37+++z$)*oC$FQ7$Fes$!37+++\"4R8>$FQ7$F`s$!36+++bz*>8$FQ 7$F[s$!3)*******y4=pIFQ7$Ffr$!38+++wVK.IFQ7$Far$!3.+++qo$\\$HFQ7$F\\r$ !37+++UA`kGFQ7$Fgq$!35+++7Qd#z#FQ7$Fbq$!33+++$*GW>FFQ7$F]q$!31+++64VXE FQ7$$!3++++^v\\9SF_p$!3/+++iwuqDFQ7$$!36+++EPa5CF_p$!3*)*****44&*)>CFQ 7$$!31+++UV&4Q\"F_p$!3#)*****>_XvE#FQ7$$!3\")*****fp8E](F_o$!3#******* G4+9@FQ7$$!3?+++308RQF_o$!33+++,GXf>FQ7$$!35+++3-\"f$=F_o$!3-+++>K3/=F Q7$$!3W+++2#)4H\")FZ$!3++++EG3[;FQ7$$!3#)*****\\\"*\\TH$FZ$!3!******>% \\j\"\\\"FQ7$$!3&******4i:I?\"FZ$!3*******Hp&*[L\"FQ7$$!3%******Hg=m(Q FN$!3++++XJ)z<\"FQ7$$!31+++f'p&p5FN$!3\"******f5x4-\"FQ7$$!3/+++3^E:CF C$!3A+++&ye#R')F-7$$!3#)*****\\c?F:%F=$!3G+++S\\boqF-7$$!3?+++$pJBy%F7 $!3')*****\\T#y(\\&F-7$$!35+++Mg+KG!#E$!3D+++$Q!*p#RF-7$$!3)******HUS! *G%!#G$!3%******zZ%>cBF-7$F($!3))*****Rj\")R&yF_p7$F($\"3))*****Rj\")R &yF_p-%&COLORG6&%$RGBGF($\"#B!\"#$\"#[Fgfn-%)POLYGONSG6_y7%F'7$F($\"+E jzq:!#57$$FZ!\"\"F(7%F^gn7$$!+pQd*[%F_p$\"+kDfTJFagnFbgn7%Ffgn7$$!+z6! ))H\"FQ$\"++uQ7ZFagnFbgn7%F\\hn7$$!+$H)3*\\\"!#;$\"+RI<$G'FagnFbgn7%Fb hn7$$!+FphU5!#:$\"+u2#R&yFagnFbgn7%Fihn7$$!+FSiI_F\\in$\"+05bC%*FagnFb gn7%F`in7$$!+cP2\"3#!#9$\"+h\")[*4\"!\"*Fbgn7%Ffin7$$!+2y)z%pFiin$\"+` _Xc7F\\jnFbgn7%F^jn7$$!+k74>?!#8$\"+>MH89F\\jnFbgn7%Fdjn7$$!+KQ\"pB&Fg jn$\"+TU!*p:F\\jnFbgn7%F[[o7$$!+rTFM7!#7$\"+%)*\\hs\"F\\jnFbgn7%Fa[o7$ $!+:5')yEFd[o$\"+j)e=)=F\\jnFbgn7%Fh[o7$$!+R)>(4aFd[o$\"+\"=To.#F\\jnF bgn7%F^\\o7$$!+$3p]-\"!#6$\"+$y44>#F\\jnFbgn7%Fd\\o7$$!+Fr\"f$=Fg\\o$ \"+UR)QM#F\\jnFbgn7%F[]o7$$!+y%)4GJFg\\o$\"+*=Pb\\#F\\jnFbgn7%Fa]o7$$! +*>Y!)4&Fg\\o$\"+64VXEF\\jnFbgn7%Fg]o7$$!+i![(3kFg\\o$\"+$*GW>FF\\jnFb gn7%F]^o7$$!+*GAp(zFg\\o$\"+7Qd#z#F\\jnFbgn7%Fc^o7$$!+pc%3$)*Fg\\o$\"+ UA`kGF\\jnFbgn7%Fi^o7$$!+%\\F$*>\"Fagn$\"+qo$\\$HF\\jnFbgn7%F__o7$$!+X ;rZ9Fagn$\"+wVK.IF\\jnFbgn7%Fe_o7$$!+f\">\"G8$F\\jnFbgn7%Fa`o7$$!+VgEwBFagn$\"+\"4R8>$F\\ jnFbgn7%Fg`o7$$!+Ig.OFFagn$\"+z$)*oC$F\\jnFbgn7%F]ao7$$!+8T%H6$Fagn$\" +i%G&)H$F\\jnFbgn7%Fcao7$$!+NI&>]$Fagn$\"+OUBYLF\\jnFbgn7%Fiao7$$!+oaf )*QFagn$\"+!3T,R$F\\jnFbgn7%F_bo7$$!+S@@*H%Fagn$\"+qMXIMF\\jnFbgn7%Feb o7$$!+*R!*4q%Fagn$\"+AnTnMF\\jnFbgn7%F[co7$$!+u.)=5&Fagn$\"+%4)G,NF\\j nFbgn7%Faco7$$!+[\"f/]&Fagn$\"+F!=B`$F\\jnFbgn7%Fgco7$$!+5()y&*eFagn$ \"+$pR2c$F\\jnFbgn7%F]do7$$!+C/H(G'Fagn$\"+'Gkne$F\\jnFbgn7%Fcdo7$$!+* fVYn'Fagn$\"+#*)z0h$F\\jnFbgn7%Fido7$$!+5yqdqFagn$\"+_BNKOF\\jnFbgn7%F _eo7$$!+TaYOuFagn$\"+'3FAl$F\\jnFbgn7%Feeo7$$!+s,)4\"yFagn$\"+(HJ.n$F \\jnFbgn7%F[fo7$$!++vO\"=)Fagn$\"+'=wno$F\\jnFbgn7%Fafo7$$!+*4uxa)Fagn $\"+X!f;q$F\\jnFbgn7%Fgfo7$$!+nNO5*)Fagn$\"+@^1:PF\\jnFbgn7%F]go7$$!+U _xC'*Fagn$\"+@ttPPF\\jnFbgn7%Fcgo7$$!+?bfK5F\\jn$\"+%y!HbPF\\jnFbgn7%F igo7$$!+!*p^,6F\\jn$\"+TV6oPF\\jnFbgn7%F_ho7$$!+%4c$p6F\\jn$\"+cR^wPF \\jnFbgn7%Feho7$$!+TW@O7F\\jn$\"+&yJ2y$F\\jnFbgn7%F[io7$$!+2G=-8F\\jn$ \"+<,'4y$F\\jnFbgn7%Faio7$$!+\"oVtO\"F\\jn$\"+BF\\jn$\"+RY*fe$ F\\jnFbgn7%F]]p7$$!+4t-$*>F\\jn$\"+!fn![NF\\jnFbgn7%Fc]p7$$!+m?ab?F\\j n$\"+bZ22NF\\jnFbgn7%Fi]p7$$!+C-s=@F\\jn$\"+r5=jMF\\jnFbgn7%F_^p7$$!+ \\dw#=#F\\jn$\"+$3:mT$F\\jnFbgn7%Fe^p7$$!+'\\dyC#F\\jn$\"+p&ywO$F\\jnF bgn7%F[_p7$$!+k)=TJ#F\\jn$\"+imu;LF\\jnFbgn7%Fa_p7$$!+$f%e\"Q#F\\jn$\" +DHDkKF\\jnFbgn7%Fg_p7$$!+/P<]CF\\jn$\"+9ol5KF\\jnFbgn7%F]`p7$$!+<$z'> DF\\jn$\"+(e(RcJF\\jnFbgn7%Fc`p7$$!+;Ay*e#F\\jn$\"+&*e%=5$F\\jnFbgn7%F i`p7$$!+Ii3gEF\\jn$\"+;sEZIF\\jnFbgn7%F_ap7$$!+/[;IFF\\jn$\"+so!G*HF\\ jnFbgn7%Feap7$$!+*=0'*z#F\\jn$\"+st\\QHF\\jnFbgn7%F[bp7$$!+J//oGF\\jn$ \"+Z6G%)GF\\jnFbgn7%Fabp7$$!+&4m^$HF\\jn$\"+a\"R+$GF\\jnFbgn7%Fgbp7$$! +iRu+IF\\jn$\"+$y%oGb#F\\jnFbgn7%Fedp7$$!+7k'3I$F\\j n$\"+\\?=&\\#F\\jnFbgn7%F[ep7$$!+#=d\\N$F\\jn$\"+uuXOCF\\jnFbgn7%Faep7 $$!+*=cqS$F\\jn$\"+mTgwBF\\jnFbgn7%Fgep7$$!+qt=dMF\\jn$\"+/na:BF\\jnFb gn7%F]fp7$$!+uTQ0NF\\jn$\"+\"oBKD#F\\jnFbgn7%Fcfp7$$!+C!)o^NF\\jn$\"+1 oe*=#F\\jnFbgn7%Fifp7$$!+%GZhf$F\\jn$\"+>.gC@F\\jnFbgn7%F_gp7$$!+/l\") QOF\\jn$\"+&yS#e?F\\jnFbgn7%Fegp7$$!+advzOF\\jn$\"+ns\\!*>F\\jnFbgn7%F [hp7$$!+n)H!>PF\\jn$\"+l=P@>F\\jnFbgn7%Fahp7$$!+8uqcPF\\jn$\"+O0)3&=F \\jnFbgn7%Fghp7$$!+Z#fGz$F\\jn$\"+vS0zVG&3\"F\\jnFbgn7% Fc\\q7$$!+HEO#3%F\\jn$\"+@h)\\+\"F\\jnFbgn7%Fi\\q7$$!+(4fV5%F\\jn$\"+M CzW#*FagnFbgn7%F_]q7$$!+ts\"\\7%F\\jn$\"+Y$\\&Q%)FagnFbgn7%Fe]q7$$!+%= _R9%F\\jn$\"+]W(=j(FagnFbgn7%F[^q7$$!+ykPhTF\\jn$\"+qqLDoFagnFbgn7%Fa^ q7$$!+fV5xTF\\jn$\"+?7M>gFagnFbgn7%Fg^q7$$!+L[0\">%F\\jn$\"+\"RTT@&Fag nFbgn7%F]_q7$$!+jT:.UF\\jn$\"+=H')4WFagnFbgn7%Fc_q7$$!+#fPL@%F\\jn$\"+ &4Clg$FagnFbgn7%Fi_q7$$!+C/b@UF\\jn$\"+))y0/GFagnFbgn7%F_`q7$$!+L'[xA% F\\jn$\"+j>L-?FagnFbgn7%Fe`q7$$!+T\"**=B%F\\jn$\"+[k;,7FagnFbgn7%F[aq7 $$!+,*zRB%F\\jn$\"+yU\\.SFg\\oFbgn7%Faaq7$Fbaq$!+yU\\.SFg\\oFbgn7%Fgaq 7$F\\aq$!+[k;,7FagnFbgn7%F[bq7$Ff`q$!+j>L-?FagnFbgn7%F_bq7$F``q$!+))y0 /GFagnFbgn7%Fcbq7$Fj_q$!+&4Clg$FagnFbgn7%Fgbq7$Fd_q$!+=H')4WFagnFbgn7% F[cq7$F^_q$!+\"RTT@&FagnFbgn7%F_cq7$Fh^q$!+?7M>gFagnFbgn7%Fccq7$Fb^q$! +qqLDoFagnFbgn7%Fgcq7$F\\^q$!+]W(=j(FagnFbgn7%F[dq7$Ff]q$!+Y$\\&Q%)Fag nFbgn7%F_dq7$F`]q$!+MCzW#*FagnFbgn7%Fcdq7$Fj\\q$!+@h)\\+\"F\\jnFbgn7%F gdq7$Fd\\q$!+>VG&3\"F\\jnFbgn7%F[eq7$F^\\q$!+OsEl6F\\jnFbgn7%F_eq7$Fh[ q$!+Uf\"[C\"F\\jnFbgn7%Fceq7$Fb[q$!+%R.QK\"F\\jnFbgn7%Fgeq7$F\\[q$!+W% *4-9F\\jnFbgn7%F[fq7$Ffjp$!+%4w&z9F\\jnFbgn7%F_fq7$F`jp$!+5C6c:F\\jnFb gn7%Fcfq7$Fjip$!+:')fJ;F\\jnFbgn7%Fgfq7$Fdip$!+u!Rfq\"F\\jnFbgn7%F[gq7 $F^ip$!+vS0zF\\jnFbgn7%Fggq7$F\\hp$!+ns\\!*>F\\jnFbgn7%F[hq7$Ffgp$!+&yS#e?F\\jnF bgn7%F_hq7$F`gp$!+>.gC@F\\jnFbgn7%Fchq7$Fjfp$!+1oe*=#F\\jnFbgn7%Fghq7$ Fdfp$!+\"oBKD#F\\jnFbgn7%F[iq7$F^fp$!+/na:BF\\jnFbgn7%F_iq7$Fhep$!+mTg wBF\\jnFbgn7%Fciq7$Fbep$!+uuXOCF\\jnFbgn7%Fgiq7$F\\ep$!+\\?=&\\#F\\jnF bgn7%F[jq7$Ffdp$!+>%oGb#F\\jnFbgn7%F_jq7$F`dp$!+3Li4EF\\jnFbgn7%Fcjq7$ Fjcp$!+w.dlEF\\jnFbgn7%Fgjq7$Fdcp$!+s+&3s#F\\jnFbgn7%F[[r7$F^cp$!+$y$F\\jnFbgn7%F_gr7$Fb`o$!+bz*>8 $F\\jnFbgn7%Fcgr7$F\\`o$!+z4=pIF\\jnFbgn7%Fggr7$Ff_o$!+wVK.IF\\jnFbgn7 %F[hr7$F`_o$!+qo$\\$HF\\jnFbgn7%F_hr7$Fj^o$!+UA`kGF\\jnFbgn7%Fchr7$Fd^ o$!+7Qd#z#F\\jnFbgn7%Fghr7$F^^o$!+$*GW>FF\\jnFbgn7%F[ir7$Fh]o$!+64VXEF \\jnFbgn7%F_ir7$$!+^v\\9SFg\\o$!+iwuqDF\\jnFbgn7%Fcir7$$!+EPa5CFg\\o$! +\"4&*)>CF\\jnFbgn7%Fiir7$$!+UV&4Q\"Fg\\o$!+AbanAF\\jnFbgn7%F_jr7$$!+' p8E](Fd[o$!+H4+9@F\\jnFbgn7%Fejr7$$!+308RQFd[o$!+,GXf>F\\jnFbgn7%F[[s7 $$!+3-\"f$=Fd[o$!+>K3/=F\\jnFbgn7%Fa[s7$$!+2#)4H\")Fgjn$!+EG3[;F\\jnFb gn7%Fg[s7$$!+:*\\TH$Fgjn$!+U\\j\"\\\"F\\jnFbgn7%F]\\s7$$!+@c,.7Fgjn$!+ $p&*[L\"F\\jnFbgn7%Fc\\s7$$!+.'=m(QFiin$!+XJ)z<\"F\\jnFbgn7%Fi\\s7$$!+ f'p&p5Fiin$!+1r(4-\"F\\jnFbgn7%F_]s7$$!+3^E:CF\\in$!+&ye#R')FagnFbgn7% Fe]s7$$!+l0s_TFehn$!+S\\boqFagnFbgn7%F[^s7$$!+$pJBy%FQ$!+:Cy(\\&FagnFb gn7%Fa^s7$$!+Mg+KGF-$!+$Q!*p#RFagnFbgn7%Fg^s7$$!+B//*G%F_o$!+yW>cBFagn Fbgn7%F]_s7$F($!+M;)R&yFg\\oFbgn7%Fc_s7$F($\"+M;)R&yFg\\oFbgn-Fbfn6&Fd fnF($\"#XFgfn$\"#&*Fgfn-%&STYLEG6#%,PATCHNOGRIDG-F$6$7`p7$$\"33+++QN7FQ$\"3m*****pNtzN%FQ7$$\"33+++R!)R,7FQ$\"3# ******>A%)[N%FQ7$$\"3/+++!GsC>\"FQ$\"3Q+++$H:EN%FQ7$$\"35+++\"GTN=\"FQ $\"3))******4Y%)\\VFQ7$$\"3)*******>biu6FQ$\"39+++!QdlM%FQ7$$\"3/+++1s ul6FQ$\"3R+++w#QFM%FQ7$$\"3%******\\_Ip:\"FQ$\"3N+++]9PQVFQ7$$\"3#**** **z$>?[6FQ$\"3A+++c0WLVFQ7$$\"31+++70fR6FQ$\"3&)*****R\")GzK%FQ7$$\"3% ******pJG68\"FQ$\"31+++\"4>=K%FQ7$$\"31+++u3&G7\"FQ$\"32+++_S4:VFQ7$$ \"32+++uwz96FQ$\"3B+++ljt2VFQ7$$\"34+++yF,26FQ$\"3e*****R+H(*H%FQ7$$\" 3#******H\\X&*4\"FQ$\"3/+++wc0\"H%FQ7$$\"3/+++M6X#4\"FQ$\"3S+++89q\"G% FQ7$$\"3'*******e=z&3\"FQ$\"3m*****fQ`;F%FQ7$$\"3!******4dP'z5FQ$\"3K+ ++2?!4E%FQ7$$\"3#******f#o1u5FQ$\"3%******fTU%\\UFQ7$$\"3/+++&on\"p5FQ $\"3%)*****HWwsB%FQ7$$\"3#******4QQ]1\"FQ$\"3;+++o^TCUFQ7$$\"31+++!e(y h5FQ$\"3%******\\>#)3@%FQ7$$\"3/+++ZR`f5FQ$\"3q******4xr'>%FQ7$$\"3'** *****eXSe5FQ$\"3m******4L)>=%FQ7$$\"3++++;;`e5FQ$\"39+++>uwmTFQ7$$\"3' ******R!o/g5FQ$\"3U+++g2>^TFQ7$$\"3'******fvsI1\"FQ$\"3&******4%4TNTFQ 7$$\"35+++>9rn5FQ$\"3$******fnC'>TFQ7$$\"3)*******e*HS2\"FQ$\"3/+++fd1 /TFQ7$$\"3)******\\,Y?3\"FQ$\"3o*****p+(**)3%FQ7$$\"3%******zT:<4\"FQ$ \"3%)*****4O&puSFQ7$$\"32+++-h#H5\"FQ$\"3')*****pBJ91%FQ7$$\"3&******* H6]:6FQ$\"3%******RdX%\\SFQ7$$\"31+++,9@H6FQ$\"3f*****\\]H*QSFQ7$$\"3( ******4b&zV6FQ$\"3)******zE6+.%FQ7$$\"3)******pU#)*e6FQ$\"3I+++TEvASFQ 7$$\"3/+++N8^u6FQ$\"3C+++VU:\"FQ$\"36+++:D]7FQ$\"3<+++x)R 8,%FQ7$$\"3'******>e%3k7FQ$\"3u*****\\\")>Q,%FQ7$$\"3%******4F-tF\"FQ$ \"3p*****f6Qs,%FQ7$$\"31+++*Qq)*G\"FQ$\"3k*****f&z[@SFQ7$$\"3%******H8 o!Gvy8F Q$\"3-+++,'o@4%FQ7$$\"3'******p-cTQ\"FQ$\"3c******3'*y+TFQ7$$\"35+++EC )*)Q\"FQ$\"3))*****p:H&4TFQ7$$\"35+++(RZKR\"FQ$\"32+++hzN=TFQ7$$\"3/++ +Lh'pR\"FQ$\"3K+++(=\\s7%FQ7$$\"30+++fN:+9FQ$\"3\"*******o\"yh8%FQ7$$ \"3,+++tU#GS\"FQ$\"3)******H:A^9%FQ7$$\"30+++@D*\\S\"FQ$\"3I+++v+1aTFQ 7$$\"3\"******\\=smS\"FQ$\"3e*****>PsH;%FQ7$$\"3.+++pn(yS\"FQ$\"3')*** ***33%=<%FQ7$$\"32+++3%>'39FQ$\"35+++F$[1=%FQ7$$\"3$******>(G\"*39FQ$ \"31+++=*y$*=%FQ7$$\"3\"******>ep(39FQ$\"3!)*****4]%FQ7$$\"3'***** **=;?39FQ$\"3m*****R#)\\l?%FQ7$$\"31+++J%QeS\"FQ$\"3o*****fFUKA%FQ7$$ \"3,+++8R\">S\"FQ$\"3k*****fab$RUFQ7$$\"3#*******yW^'R\"FQ$\"3C+++'=(z aUFQ7$$\"3)******R*Hs*Q\"FQ$\"3%******\\?\"[pUFQ7$$\"3#******>V>;Q\"FQ $\"3)******4cELG%FQ7$$\"3%******p_\"Gs8FQ$\"3W+++\")eD'H%FQ7$$\"3!**** **fV&yh8FQ$\"3\")******eV>3VFQFafn-F[gn6ap7%7$$\"+QN%Fgfn7%Fc[u7$$\" +fU.W8F\\jn$\"+/V3CVF\\jnFh[u7%F^\\u7$$\"+e<(4L\"F\\jn$\"+bmALVF\\jnFh [u7%Fd\\u7$$\"+X`,<8F\\jn$\"+JJ7TVF\\jnFh[u7%Fj\\u7$$\"+XJC-8F\\jn$\"+ Q$*pZVF\\jnFh[u7%F`]u7$$\"+*=OnG\"F\\jn$\"+;(yGN%F\\jnFh[u7%Ff]u7$$\"+ P'z0F\"F\\jn$\"+17F\\jn $\"+dL(zN%F\\jnFh[u7%F^_u7$$\"+R!)R,7F\\jn$\"+AU)[N%F\\jnFh[u7%Fd_u7$$ \"+!GsC>\"F\\jn$\"+$H:EN%F\\jnFh[u7%Fj_u7$$\"+\"GTN=\"F\\jn$\"+5Y%)\\V F\\jnFh[u7%F``u7$$\"+?biu6F\\jn$\"+!QdlM%F\\jnFh[u7%Ff`u7$$\"+1sul6F\\ jn$\"+w#QFM%F\\jnFh[u7%F\\au7$$\"+D0$p:\"F\\jn$\"+]9PQVF\\jnFh[u7%Fbau 7$$\"+Q>?[6F\\jn$\"+c0WLVF\\jnFh[u7%Fhau7$$\"+70fR6F\\jn$\"+9)GzK%F\\j nFh[u7%F^bu7$$\"+<$G68\"F\\jn$\"+\"4>=K%F\\jnFh[u7%Fdbu7$$\"+u3&G7\"F \\jn$\"+_S4:VF\\jnFh[u7%Fjbu7$$\"+uwz96F\\jn$\"+ljt2VF\\jnFh[u7%F`cu7$ $\"+yF,26F\\jn$\"+/!H(*H%F\\jnFh[u7%Ffcu7$$\"+$\\X&*4\"F\\jn$\"+wc0\"H %F\\jnFh[u7%F\\du7$$\"+M6X#4\"F\\jn$\"+89q\"G%F\\jnFh[u7%Fbdu7$$\"+f=z &3\"F\\jn$\"+'Q`;F%F\\jnFh[u7%Fhdu7$$\"+rvjz5F\\jn$\"+2?!4E%F\\jnFh[u7 %F^eu7$$\"+Eo1u5F\\jn$\"+;CW\\UF\\jnFh[u7%Fdeu7$$\"+&on\"p5F\\jn$\"+Vk FPUF\\jnFh[u7%Fjeu7$$\"+\"QQ]1\"F\\jn$\"+o^TCUF\\jnFh[u7%F`fu7$$\"+!e( yh5F\\jn$\"+&>#)3@%F\\jnFh[u7%Fffu7$$\"+ZR`f5F\\jn$\"+5xr'>%F\\jnFh[u7 %F\\gu7$$\"+fXSe5F\\jn$\"+5L)>=%F\\jnFh[u7%Fbgu7$$\"+;;`e5F\\jn$\"+>uw mTF\\jnFh[u7%Fhgu7$$\"+/o/g5F\\jn$\"+g2>^TF\\jnFh[u7%F^hu7$$\"+cF2j5F \\jn$\"+T4TNTF\\jnFh[u7%Fdhu7$$\"+>9rn5F\\jn$\"+wYi>TF\\jnFh[u7%Fjhu7$ $\"+f*HS2\"F\\jn$\"+fd1/TF\\jnFh[u7%F`iu7$$\"+:g/#3\"F\\jn$\"+2q**)3%F \\jnFh[u7%Ffiu7$$\"+=ar\"4\"F\\jn$\"+h`puSF\\jnFh[u7%F\\ju7$$\"+-h#H5 \"F\\jn$\"+P7VhSF\\jnFh[u7%Fbju7$$\"+I6]:6F\\jn$\"+ubW\\SF\\jnFh[u7%Fh ju7$$\"+,9@H6F\\jn$\"+0&H*QSF\\jnFh[u7%F^[v7$$\"+^bzV6F\\jn$\"+o7,ISF \\jnFh[u7%Fd[v7$$\"+FC)*e6F\\jn$\"+TEvASF\\jnFh[u7%Fj[v7$$\"+N8^u6F\\j n$\"+VU:\"F\\jn$\"+Q,%F\\jnFh[u7%Fd^ v7$$\"+rAIx7F\\jn$\"+;\"Qs,%F\\jnFh[u7%Fj^v7$$\"+*Qq)*G\"F\\jn$\"+cz[@ SF\\jnFh[u7%F`_v7$$\"+L\"o'39F\\j n$\"+F$[1=%F\\jnFh[u7%Fhfv7$$\"+sG\"*39F\\jn$\"+=*y$*=%F\\jnFh[u7%F^gv 7$$\"+#ep(39F\\jn$\"+,v,)>%F\\jnFh[u7%Fdgv7$$\"+>;?39F\\jn$\"+C)\\l?%F \\jnFh[u7%Fjgv7$$\"+J%QeS\"F\\jn$\"+wACBUF\\jnFh[u7%F`hv7$$\"+8R\">S\" F\\jn$\"+YbNRUF\\jnFh[u7%Ffhv7$$\"+zW^'R\"F\\jn$\"+'=(zaUF\\jnFh[u7%F \\iv7$$\"+%*Hs*Q\"F\\jn$\"+07[pUF\\jnFh[u7%Fbiv7$$\"+K%>;Q\"F\\jn$\"+h lK$G%F\\jnFh[u7%Fhiv7$$\"+F:Gs8F\\jn$\"+\")eD'H%F\\jnFh[u7%F^jv7$$\"+O ayh8F\\jn$\"+fV>3VF\\jnFh[uFj_sF``s-F$6$7`p7$Fh`s$!3()*****pd`BI%FQ7$$ \"3++++%)**4x8FQ$!3u******[/\"**G%FQ7$$\"3%*******>.$eQ\"FQ$!3o*****Ri 8lF%FQ7$$\"3!******Ro(G$R\"FQ$!3++++j!RAE%FQ7$$\"3))*****fI$R*R\"FQ$!3 #)*****f#e;ZUFQ7$$\"3.+++Ze1/9FQ$!3?+++esPJUFQ7$$\"3%*******[2A29FQ$!3 P+++rB'\\@%FQ7$F_ht$!3!)*****4]%FQ7$Fegt$!35+++F$[1=%FQ7$F`gt$!3') ******33%=<%FQ7$F[gt$!3e*****>PsH;%FQ7$Ffft$!3I+++v+1aTFQ7$Faft$!3)*** ***H:A^9%FQ7$F\\ft$!3\"*******o\"yh8%FQ7$Fget$!3K+++(=\\s7%FQ7$Fbet$!3 2+++hzN=TFQ7$F]et$!3))*****p:H&4TFQ7$Fhdt$!3c******3'*y+TFQ7$Fcdt$!3-+ ++,'o@4%FQ7$F^dt$!3)******H?)p$3%FQ7$Fict$!3#******4k8a2%FQ7$Fdct$!3E+ ++DPNnSFQ7$F_ct$!3))*****pLh&fSFQ7$Fjbt$!3#)*****z'R3_SFQ7$Febt$!3U+++ HV(\\/%FQ7$F`bt$!3,+++55HQSFQ7$F[bt$!3y*****R=*4KSFQ7$Ffat$!3E+++P8ZES FQ7$Faat$!3k*****f&z[@SFQ7$F\\at$!3p*****f6Qs,%FQ7$Fg`t$!3u*****\\\")> Q,%FQ7$Fb`t$!3<+++x)R8,%FQ7$F]`t$!3*)*****f68*4SFQ7$Fh_t$!3L+++x.m4SFQ 7$Fc_t$!38+++^\\q5SFQ7$F^_t$!36+++TFQ7$Fg[ t$!3&******4%4TNTFQ7$Fb[t$!3U+++g2>^TFQ7$F][t$!39+++>uwmTFQ7$Fhjs$!3m* *****4L)>=%FQ7$Fcjs$!3q******4xr'>%FQ7$F^js$!3%******\\>#)3@%FQ7$Fiis$ !3;+++o^TCUFQ7$Fdis$!3%)*****HWwsB%FQ7$F_is$!3%******fTU%\\UFQ7$Fjhs$! 3K+++2?!4E%FQ7$Fehs$!3m*****fQ`;F%FQ7$F`hs$!3S+++89q\"G%FQ7$F[hs$!3/++ +wc0\"H%FQ7$Ffgs$!3e*****R+H(*H%FQ7$Fags$!3B+++ljt2VFQ7$F\\gs$!32+++_S 4:VFQ7$Fgfs$!31+++\"4>=K%FQ7$Fbfs$!3&)*****R\")GzK%FQ7$F]fs$!3A+++c0WL VFQ7$Fhes$!3N+++]9PQVFQ7$Fces$!3R+++w#QFM%FQ7$F^es$!39+++!QdlM%FQ7$Fid s$!3))******4Y%)\\VFQ7$Fdds$!3Q+++$H:EN%FQ7$F_ds$!3#******>A%)[N%FQ7$$ \"3!*******)e*H57FQ$!3++++gcmcVFQ7$Fjcs$!3m*****pNtzN%FQ7$$\"3'******* Q1'zA\"FQ$!3M+++m0#)eVFQ7$$\"3'*******4tKX7FQ$!3q*****4?%=fVFQ7$$\"3/+ ++eeGi7FQ$!37+++iO&yN%FQ7$$\"3%******4fL(y7FQ$!31+++t)>\\N%FQ7$$\"3)** ****4IwXH\"FQ$!3)******H_o/N%FQ7$$\"32+++Nis48FQ$!3.+++#*4eWVFQ7$$\"3& ******HY+TK\"FQ$!3H+++N_LPVFQ7$$\"3*******fa>wL\"FQ$!3W+++Em!)GVFQ7$$ \"3/+++>k?]8FQ$!3F+++!yo!>VFQ7$Fgjt$!3\")******eV>3VFQ7$Fbjt$!3W+++\") eD'H%FQFafn-F[gn6ap7%7$F_[u$!+xNN-VF\\jn7$$\"+%)**4x8F\\jn$!+\\/\"**G% F\\jn7$Fi[u$!$>%Fgfn7%F^]x7$$\"+?.$eQ\"F\\jn$!+CO^wUF\\jnFc]x7%Fg]x7$$ \"+%o(G$R\"F\\jn$!+j!RAE%F\\jnFc]x7%F]^x7$$\"+1LR*R\"F\\jn$!+Ee;ZUF\\j nFc]x7%Fc^x7$$\"+Ze1/9F\\jn$!+esPJUF\\jnFc]x7%Fi^x7$$\"+\\2A29F\\jn$!+ rB'\\@%F\\jnFc]x7%F__x7$Fegv$!+,v,)>%F\\jnFc]x7%Fe_x7$Fifv$!+F$[1=%F\\ jnFc]x7%Fi_x7$Fcfv$!+43%=<%F\\jnFc]x7%F]`x7$F]fv$!+sB(H;%F\\jnFc]x7%Fa `x7$Fgev$!+v+1aTF\\jnFc]x7%Fe`x7$Faev$!+`@7XTF\\jnFc]x7%Fi`x7$F[ev$!+p \"yh8%F\\jnFc]x7%F]ax7$Fedv$!+(=\\s7%F\\jnFc]x7%Faax7$F_dv$!+hzN=TF\\j nFc]x7%Feax7$Ficv$!+d\"H&4TF\\jnFc]x7%Fiax7$Fccv$!+4'*y+TF\\jnFc]x7%F] bx7$F]cv$!+,'o@4%F\\jnFc]x7%Fabx7$Fgbv$!+.#)p$3%F\\jnFc]x7%Febx7$Fabv$ !+TOTvSF\\jnFc]x7%Fibx7$F[bv$!+DPNnSF\\jnFc]x7%F]cx7$Feav$!+P8cfSF\\jn Fc]x7%Facx7$F_av$!+oR3_SF\\jnFc]x7%Fecx7$Fi`v$!+HV(\\/%F\\jnFc]x7%Ficx 7$Fc`v$!+55HQSF\\jnFc]x7%F]dx7$F]`v$!+%=*4KSF\\jnFc]x7%Fadx7$Fg_v$!+P8 ZESF\\jnFc]x7%Fedx7$Fa_v$!+cz[@SF\\jnFc]x7%Fidx7$F[_v$!+;\"Qs,%F\\jnFc ]x7%F]ex7$Fe^v$!+:)>Q,%F\\jnFc]x7%Faex7$F_^v$!+x)R8,%F\\jnFc]x7%Feex7$ Fi]v$!+;J\"*4SF\\jnFc]x7%Fiex7$Fc]v$!+x.m4SF\\jnFc]x7%F]fx7$F]]v$!+^\\ q5SF\\jnFc]x7%Fafx7$Fg\\v$!+TF\\jnFc]x7%F]ix7$Fehu $!+T4TNTF\\jnFc]x7%Faix7$F_hu$!+g2>^TF\\jnFc]x7%Feix7$Figu$!+>uwmTF\\j nFc]x7%Fiix7$Fcgu$!+5L)>=%F\\jnFc]x7%F]jx7$F]gu$!+5xr'>%F\\jnFc]x7%Faj x7$Fgfu$!+&>#)3@%F\\jnFc]x7%Fejx7$Fafu$!+o^TCUF\\jnFc]x7%Fijx7$F[fu$!+ VkFPUF\\jnFc]x7%F][y7$Feeu$!+;CW\\UF\\jnFc]x7%Fa[y7$F_eu$!+2?!4E%F\\jn Fc]x7%Fe[y7$Fidu$!+'Q`;F%F\\jnFc]x7%Fi[y7$Fcdu$!+89q\"G%F\\jnFc]x7%F] \\y7$F]du$!+wc0\"H%F\\jnFc]x7%Fa\\y7$Fgcu$!+/!H(*H%F\\jnFc]x7%Fe\\y7$F acu$!+ljt2VF\\jnFc]x7%Fi\\y7$F[cu$!+_S4:VF\\jnFc]x7%F]]y7$Febu$!+\"4>= K%F\\jnFc]x7%Fa]y7$F_bu$!+9)GzK%F\\jnFc]x7%Fe]y7$Fiau$!+c0WLVF\\jnFc]x 7%Fi]y7$Fcau$!+]9PQVF\\jnFc]x7%F]^y7$F]au$!+w#QFM%F\\jnFc]x7%Fa^y7$Fg` u$!+!QdlM%F\\jnFc]x7%Fe^y7$Fa`u$!+5Y%)\\VF\\jnFc]x7%Fi^y7$F[`u$!+$H:EN %F\\jnFc]x7%F]_y7$Fe_u$!+AU)[N%F\\jnFc]x7%Fa_y7$$\"+*e*H57F\\jn$!+gcmc VF\\jnFc]x7%Fe_y7$F__u$!+dL(zN%F\\jnFc]x7%F[`y7$$\"+R1'zA\"F\\jn$!+m0# )eVF\\jnFc]x7%F_`y7$$\"+5tKX7F\\jn$!+,U=fVF\\jnFc]x7%Fe`y7$$\"+eeGi7F \\jn$!+iO&yN%F\\jnFc]x7%F[ay7$$\"+\"fL(y7F\\jn$!+t)>\\N%F\\jnFc]x7%Faa y7$$\"+,jd%H\"F\\jn$!+B&o/N%F\\jnFc]x7%Fgay7$$\"+Nis48F\\jn$!+#*4eWVF \\jnFc]x7%F]by7$$\"+j/5C8F\\jn$!+N_LPVF\\jnFc]x7%Fcby7$$\"+Y&>wL\"F\\j n$!+Em!)GVF\\jnFc]x7%Fiby7$$\"+>k?]8F\\jn$!+!yo!>VF\\jnFc]x7%F_cy7$Fej v$!+fV>3VF\\jnFc]x7%Fecy7$F_jv$!+\")eD'H%F\\jnFc]xFj_sF``s-F$6%7$7$$!3 /++++++!z%FQF(7$$\"3!**************)=FQF(-%'COLOURG6&FdfnF)F)F)-%*LINE STYLEG6#\"\"$-F$6%7$7$F(F`dy7$F($\"3/++++++!z%FQFedyFhdy-%%FONTG6$%*HE LVETICAG\"\"*-%+AXESLABELSG6%%&Re(z)G%&Im(z)G-Fdey6#%(DEFAULTG-%*AXESS TYLEG6#%$BOXG-%(SCALINGG6#%,CONSTRAINEDG-%%VIEWG6$;$!$z%Fgfn$\"$*=Fgfn ;F\\gy$\"$z%Fgfn" 1 2 0 1 10 0 2 9 1 2 1 1.000000 45.000000 45.000000 0 0 "Curve 1" "Curve 2" "Curve 3" "Curve 4" "Curve 5" "Curve 6" "Curve 7" "Curve 8" }}}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 4 "The " }{TEXT 260 30 "interval of absolute stability" } {TEXT -1 89 " (or stability interval) is the intersection of the stab ility region with the real line." }}{PARA 0 "" 0 "" {TEXT -1 59 "For t his scheme the stability interval is (approximately) " }{XPPEDIT 18 0 "[-4.2342, 0];" "6#7$,$-%&FloatG6$\"&UB%!\"%!\"\"\"\"!" }{TEXT -1 2 ". " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 148 "W e can distort the boundary curve horizontally by taking the 11th root \+ of the real part of points along the curve. In this way we see that th ere is " }{TEXT 260 19 "no largest interval" }{TEXT -1 99 " on the non negative imaginary axis that contains the origin and lies inside the s tability region. " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 " > " 0 "" {MPLTEXT 1 0 319 "R := z -> 1+z+1/2*z^2+1/6*z^3+1/24*z^4+1/12 0*z^5+1/720*z^6+1/5904*z^7:\nDigits := 25:\npts := []: z0 := 0:\nfor c t from 0 to 45 do\n zz := newton(R(z)=exp(ct*Pi/100*I),z=z0):\n z0 := zz:\n pts := [op(pts),[surd(Re(zz),11),Im(zz)]]:\nend do:\nplot( pts,color=COLOR(RGB,0,.2,1),thickness=2,font=[HELVETICA,9]);\nDigits : = 10:" }}{PARA 13 "" 1 "" {GLPLOT2D 254 303 303 {PLOTDATA 2 "6(-%'CURV ESG6#7P7$$\"\"!F)F(7$$!:L(z_d^cgI9v@E!#E$\":$))z(=bq*e`EfTJF-7$$!::_j$=R$eF-$\":g]&e3Xrdgz xC%*F-7$$!:9zMT#\\Pf\")4#p>(F-$\":0>'=4[;Hhqjc7!#D7$$!:ifFCiQCXszHZ)F- $\":$=F='[S0hK'zq:F?7$$!:([/EN*eM(zOP&o*F-$\":)o%Q#\\\"\\w'*eb\\)=F?7$ $!:uT02L*4)yiu[3\"F?$\":8$Gvl*4,-&[6*>#F?7$$!:\"F?$\":T$ o93LP=/TF8DF?7$$!:.#)R$37:!G,i\"F?$\":!=FWd$Q%))e3\"*pPF?7$$!:j\"o&\\3T66>Y5s\"F?$\": G0$G_.L_w)pS3%F?7$$!:@B:F?)4U.Bd?=F?$\":wvRe9.PjuG#)R%F?7$$!:>N\"z)*=Z g0n))=>F?$\":1de#[y]D+uQ7ZF?7$$!:^I?)QVu\"y)*Qh,#F?$\":Yvy7(emiYdaE]F? 7$$!:)p&)z`@\"4?\"RX7@F?$\":QmNx\\`?Vm.2M&F?7$$!:@!yk?t%[#ef$z?#F?$\": !G^-uoA$e*4'[l&F?7$$!:FwPFe;bh(Gn-BF?$\"::Y[\"HI`IRv,pfF?7$$!:J/-0e)=G 0&QnR#F?$\":X$[si\"zs$RI<$G'F?7$$!:%45+8?G(e/&>!\\#F?$\":6G`&=S#Gng')e!yE+x(RvF?7$$!:#fkeS5 *Gdzh)eGF?$\":%*)=`N\"ous\"Gk0/$F?$\":6vW()zK7`;%>#[)F?7$$!:$)fK&G'Q**=\")Q3 8$F?$\":S@os\\ok+g@jz)F?7$$!:GmJF?F*GJ'\\2A$F?$\":3:$[#Gx&>$3T/6*F?7$$ !:ZW\"4z>6%zt3.J$F?$\":*['G,OHnZ+^XU*F?7$$!:$Q4'>OC&3Id_*R$F?$\":ptOay oRD]\\'Q(*F?7$$!:yQITs(\\v?\"3%)[$F?$\":PKn?\\ZZMWt_+\"!#C7$$!:54'4i'e mOghpd$F?$\":$p6(*e))\\*\\L!oO5Ffu7$$!:HT]J%\\!eyF!>lOF?$\":'[\">T<9() fM&3o5Ffu7$$!:J2uPG+fer>fg\")[*4\"Ffu7$$!:m_[ekWt;t#oS QF?$\":0M33+t?HU))38\"Ffu7$$!:N'*QId-Y/I[z#RF?$\":(G$eZ4;oQt&Gi6Ffu7$$ !:DmGgNJz#))G*[,%F?$\":W+m/&Qc'zkzO>\"Ffu7$$!:OB-#z%4o5%\\^,TF?$\":\"Q JR4U3Hq'p]A\"Ffu7$$!:R3T#oZJ,)37y=%F?$\":dN[uV(G[`_Xc7Ffu7$$!:Lvz-9@v9 @\"ytUF?$\":%[f&[+w;\\zNyG\"Ffu7$$!:d\"o!e,Xwp]=%fVF?$\":Q%\\nEHf4M1@> 8Ffu7$$!:N%p05<2LN&>ZW%F?$\":s#p&[2RD.0z0N\"Ffu7$$!:L1Z=y%flo#z'HXF?$ \":=PQ\"Ffu7$$!:ob$=_)3E=8#H9YF?$\":?F8\\oQ(p=MH89Ffu-%%FONTG 6$%*HELVETICAG\"\"*-%+AXESLABELSG6$Q!6\"F`z-%&COLORG6&%$RGBGF($\"\"#! \"\"$\"\"\"F)-%*THICKNESSG6#Fgz-%%VIEWG6$%(DEFAULTGFa[l" 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 "#--------------------- ---------------" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}} {PARA 0 "" 0 "" {TEXT -1 38 "#-------------------------------------" } }{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}}{PARA 0 "" 0 "" {TEXT -1 42 "#=========================================" }}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 25 "Butcher's scheme B with " }{XPPEDIT 18 0 "c[5]=c[6]" "6#/&%\"cG6#\"\"&&F%6#\"\"'" }{XPPEDIT 18 0 "``=1/2" "6# /%!G*&\"\"\"F&\"\"#!\"\"" }{TEXT -1 7 " and " }{XPPEDIT 18 0 "b[5]=b [6]" "6#/&%\"bG6#\"\"&&F%6#\"\"'" }{TEXT -1 1 " " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 62 "See: On Runge-Kutta Proce sses of High Order, by J. C. Butcher," }}{PARA 0 "" 0 "" {TEXT -1 87 " Journal of the Australian Mathematical Society, Vol. 4, (1964) \+ pages 179 to 194." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}{PARA 0 "" 0 "" {TEXT -1 42 "#---------- -------------------------------" }}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 19 "checking the scheme" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 26 "coefficients of the scheme" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 415 "ee := \{c[2]=1/3,\nc[3]=2/3 ,\nc[4]=1/3,\nc[5]=1/2,\nc[6]=1/2,\nc[7]=1,\n\na[2,1]=1/3,\na[3,1]=0, \na[3,2]=2/3,\na[4,1]=1/12,\na[4,2]=1/3,\na[4,3]=-1/12,\na[5,1]=-1/16, \na[5,2]=9/8,\na[5,3]=-3/16,\na[5,4]=-3/8,\na[6,1]=0,\na[6,2]=9/8,\na[ 6,3]=-3/8,\na[6,4]=-3/4,\na[6,5]=1/2,\na[7,1]=9/44,\na[7,2]=-9/11,\na[ 7,3]=63/44,\na[7,4]=18/11,\na[7,5]=0,\na[7,6]=-16/11,\n\nb[1]=11/120, \nb[2]=0,\nb[3]=27/40,\nb[4]=27/40,\nb[5]=-4/15,\nb[6]=-4/15,\nb[7]=11 /120\}:" }}}{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 91 "subs(ee,matrix([seq([c[i],seq(a[i,j],j=1..i -1),``$(8-i)],i=2..7),\n[``,seq(b[i],i=1..7)]]));" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#K%'matrixG6#7)7*#\"\"\"\"\"$F(%!GF+F+F+F+F+7*#\"\"#F*\" \"!F-F+F+F+F+F+7*F(#F)\"#7F(#!\"\"F2F+F+F+F+7*#F)F.#F4\"#;#\"\"*\"\")# !\"$F8#F=F;F+F+F+7*F6F/F9F>#F=\"\"%F6F+F+7*F)#F:\"#W#!\"*\"#6#\"#jFD# \"#=FGF/#!#;FGF+7*F+#FG\"$?\"F/#\"#F\"#SFQ#!\"%\"#:FTFOQ)pprint186\"" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 7 "Check: \+ " }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 148 "RK6_7eqs := [op(RowSumC onditions(7,'expanded')),op(OrderConditions(6,7,'expanded'))]:\nsimpli fy(subs(ee,RK6_7eqs)):\nmap(u->lhs(u)-rhs(u),%);\nnops(%);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#7M\"\"!F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$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#\"#V" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 " " {TEXT -1 86 "Next we set-up stage-order condtions to check for stage -orders from 2 to 4 inclusive. " }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 85 "for ct from 2 to 4 do\n so||ct||_7 := StageOrderConditions(c t,7,'expanded');\nend do:" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 58 "Stages 3 to 7 have the following respective stage -orders. " }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 173 "[seq([seq(expa nd(subs(ee,so||i||_7[j])),i=2..4)],j=1..5)]:\nmap(proc(L) local i; for i to nops(L) do if not evalb(L[i]) then break end if end do; i end pr oc,%):\nsimplify(%); " }}{PARA 11 "" 1 "" {XPPMATH 20 "6#7'\"\"#F$F$F$ F$" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 42 "Th us stages 5, 6 and 7 have stage-order 3." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 32 "#-------------------------------" }}{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 .. \+ 7) = b[j]*(1-c[j]);" "6#/-%$SumG6$*&&%\"bG6#%\"iG\"\"\"&%\"aG6$F+%\"jG F,/F+;,&F0F,F,F,\"\"(*&&F)6#F0F,,&F,F,&%\"cG6#F0!\"\"F," }{TEXT -1 3 " , " }{XPPEDIT 18 0 "j=1" "6#/%\"jG\"\"\"" }{TEXT -1 7 " . . 7 " }} {PARA 0 "" 0 "" {TEXT -1 8 "(where " }{XPPEDIT 18 0 "c[1]=0" "6#/&%\" cG6#\"\"\"\"\"!" }{TEXT -1 17 " ) are satisfied." }}{PARA 256 "" 0 "" {TEXT -1 1 " " }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 162 "[Sum(b[i]* a[i,1],i=2..7)=b[1],seq(Sum(b[i]*a[i,j],i=j+1..7)=b[j]*(1-c[j]),j=2..6 )];\neval(subs(Sum=add,%)):\nsubs(ee,%):\nmap(u->`if`(lhs(u)=rhs(u),0, 1),%);\nnops(%);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#7(/-%$SumG6$*&&%\" bG6#%\"iG\"\"\"&%\"aG6$F,F-F-/F,;\"\"#\"\"(&F*6#F-/-F&6$*&F)F-&F/6$F,F 3F-/F,;\"\"$F4*&&F*6#F3F-,&F-F-&%\"cGFB!\"\"F-/-F&6$*&F)F-&F/6$F,F?F-/ F,;\"\"%F4*&&F*6#F?F-,&F-F-&FEFRFFF-/-F&6$*&F)F-&F/6$F,FOF-/F,;\"\"&F4 *&&F*6#FOF-,&F-F-&FEFjnFFF-/-F&6$*&F)F-&F/6$F,FgnF-/F,;\"\"'F4*&&F*6#F gnF-,&F-F-&FEFhoFFF-/-F&6$*&F)F-&F/6$F,FeoF-/F,;F4F4*&&F*6#FeoF-,&F-F- &FEFepFFF-" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#7(\"\"!F$F$F$F$F$" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#\"\"'" }}}{PARA 0 "" 0 "" {TEXT -1 0 " " }}{PARA 0 "" 0 "" {TEXT -1 27 "The simplifying condition: " }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "Sum(b[i]*a[i,2],i = 3 .. 7) = 0;" "6#/-%$SumG6$*&&%\"bG6#%\"iG\"\"\"&%\"aG6$F+\"\"#F,/F+;\"\"$\" \"(\"\"!" }{TEXT -1 1 " " }}{PARA 0 "" 0 "" {TEXT -1 13 "is satisfied. " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 59 "Sum(b[i]*a[i,2],i=3..7);\neval(subs(Sum=add,%));\nsubs(ee,%); " }}{PARA 11 "" 1 "" {XPPMATH 20 "6#-%$SumG6$*&&%\"bG6#%\"iG\"\"\"&%\" aG6$F*\"\"#F+/F*;\"\"$\"\"(" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#,,*&&% \"bG6#\"\"$\"\"\"&%\"aG6$F(\"\"#F)F)*&&F&6#\"\"%F)&F+6$F1F-F)F)*&&F&6# \"\"&F)&F+6$F7F-F)F)*&&F&6#\"\"'F)&F+6$F=F-F)F)*&&F&6#\"\"(F)&F+6$FCF- F)F)" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#\"\"!" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 27 "The simplifying condition : " }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "Sum(b[i]*c[i]*a [i,2],i = 3 .. 7) = 0;" "6#/-%$SumG6$*(&%\"bG6#%\"iG\"\"\"&%\"cG6#F+F, &%\"aG6$F+\"\"#F,/F+;\"\"$\"\"(\"\"!" }{TEXT -1 1 " " }}{PARA 0 "" 0 " " {TEXT -1 13 "is satisfied." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 64 "Sum(b[i]*c[i]*a[i,2],i=3..7) ;\neval(subs(Sum=add,%));\nsubs(ee,%);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#-%$SumG6$*(&%\"bG6#%\"iG\"\"\"&%\"cGF)F+&%\"aG6$F*\"\"#F+/F*;\" \"$\"\"(" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#,,*(&%\"bG6#\"\"$\"\"\"&% \"cGF'F)&%\"aG6$F(\"\"#F)F)*(&F&6#\"\"%F)&F+F2F)&F-6$F3F/F)F)*(&F&6#\" \"&F)&F+F9F)&F-6$F:F/F)F)*(&F&6#\"\"'F)&F+F@F)&F-6$FAF/F)F)*(&F&6#\"\" (F)&F+FGF)&F-6$FHF/F)F)" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#\"\"!" }}} {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 27 "The simpl ifying condition: " }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "Sum(b[i]*c[i]^2*a[i,2],i = 3 .. 7) = 0;" "6#/-%$SumG6$*(&%\"bG6#%\"iG \"\"\"*$&%\"cG6#F+\"\"#F,&%\"aG6$F+F1F,/F+;\"\"$\"\"(\"\"!" }{TEXT -1 1 " " }}{PARA 0 "" 0 "" {TEXT -1 13 "is satisfied." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 66 "Sum(b[i]*c[i ]^2*a[i,2],i=3..7);\neval(subs(Sum=add,%));\nsubs(ee,%);" }}{PARA 11 " " 1 "" {XPPMATH 20 "6#-%$SumG6$*(&%\"bG6#%\"iG\"\"\")&%\"cGF)\"\"#F+&% \"aG6$F*F/F+/F*;\"\"$\"\"(" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#,,*(&%\" bG6#\"\"$\"\"\")&%\"cGF'\"\"#F)&%\"aG6$F(F-F)F)*(&F&6#\"\"%F))&F,F3F-F )&F/6$F4F-F)F)*(&F&6#\"\"&F))&F,F;F-F)&F/6$F " 0 "" {MPLTEXT 1 0 85 "Sum(b[i]*c[i]*Sum(a[i,j]*a[j,2],j=3..i-1),i=3..7);\ne val(subs(Sum=add,%));\nsubs(ee,%);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6# -%$SumG6$*(&%\"bG6#%\"iG\"\"\"&%\"cGF)F+-F$6$*&&%\"aG6$F*%\"jGF+&F26$F 4\"\"#F+/F4;\"\"$,&F*F+F+!\"\"F+/F*;F:\"\"(" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#,***&%\"bG6#\"\"%\"\"\"&%\"cGF'F)&%\"aG6$F(\"\"$F)&F-6$ F/\"\"#F)F)*(&F&6#\"\"&F)&F+F5F),&*&&F-6$F6F/F)F0F)F)*&&F-6$F6F(F)&F-6 $F(F2F)F)F)F)*(&F&6#\"\"'F)&F+FCF),(*&&F-6$FDF/F)F0F)F)*&&F-6$FDF(F)F? F)F)*&&F-6$FDF6F)&F-6$F6F2F)F)F)F)*(&F&6#\"\"(F)&F+FTF),**&&F-6$FUF/F) F0F)F)*&&F-6$FUF(F)F?F)F)*&&F-6$FUF6F)FPF)F)*&&F-6$FUFDF)&F-6$FDF2F)F) F)F)" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#\"\"!" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 114 "We can calculate the pri ncipal error norm of the order 6 scheme, that is, the 2-norm of the pr incipal error terms." }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 158 "err terms6_7 := PrincipalErrorTerms(6,7,'expanded'):\nsm := 0:\nfor ct to \+ nops(errterms6_7) do\n sm := sm+(evalf(subs(ee,errterms6_7[ct])))^2; \nend do:\nsqrt(sm);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#$\"+]\"!#7 " }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}}{PARA 0 "" 0 "" {TEXT -1 73 "#----------------- -------------------------------------------------------" }}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 32 "short construction of the scheme" }} {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT 260 4 "Note" } {TEXT -1 123 ": This scheme was constructed in a step by step manner i n a previous section making use of \"alternative\" order conditions. \+ " }}{PARA 0 "" 0 "" {TEXT -1 77 "In this subsection we construct the s cheme using \"standard\" order conditions." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 192 "We set up a system of equation s using a selection of \"simple\" order conditions and incorporate the row-sum conditions together with the stage-order equations to ensure \+ that stages 3 to 7 have " }{TEXT 260 13 "stage-order 2" }{TEXT -1 1 ". " }}{PARA 0 "" 0 "" {TEXT -1 48 "We also incorporate the simplifying c onditions: " }}{PARA 256 "" 0 "" {TEXT -1 2 " " }{XPPEDIT 18 0 "Sum(b [i]*a[i,j],i = j+1 .. 7) = b[j]*(1-c[j]);" "6#/-%$SumG6$*&&%\"bG6#%\"i G\"\"\"&%\"aG6$F+%\"jGF,/F+;,&F0F,F,F,\"\"(*&&F)6#F0F,,&F,F,&%\"cG6#F0 !\"\"F," }{TEXT -1 1 " " }}{PARA 0 "" 0 "" {TEXT -1 5 "for " } {XPPEDIT 18 0 "j = 3;" "6#/%\"jG\"\"$" }{TEXT -1 62 ", 4, 5, 6, toget her with the further simplifying conditions: " }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "Sum(b[i]*c[i]*a[i,2],i = 3 .. 7) = 0;" "6#/-%$SumG6$*(&%\"bG6#%\"iG\"\"\"&%\"cG6#F+F,&%\"aG6$F+\"\"#F,/F+;\" \"$\"\"(\"\"!" }{TEXT -1 8 ", " }{XPPEDIT 18 0 "Sum(b[i]*c[i]*Su m(a[i,j]*a[j,2],j = 3 .. i-1),i = 3 .. 7) = 0;" "6#/-%$SumG6$*(&%\"bG6 #%\"iG\"\"\"&%\"cG6#F+F,-F%6$*&&%\"aG6$F+%\"jGF,&F46$F6\"\"#F,/F6;\"\" $,&F+F,F,!\"\"F,/F+;F<\"\"(\"\"!" }{TEXT -1 8 ", " }{XPPEDIT 18 0 "Sum(b[i]*c[i]^2*a[i,2],i = 3 .. 7) = 0;" "6#/-%$SumG6$*(&%\"bG6#%\" iG\"\"\"*$&%\"cG6#F+\"\"#F,&%\"aG6$F+F1F,/F+;\"\"$\"\"(\"\"!" }{TEXT -1 2 ". " }}{PARA 0 "" 0 "" {TEXT -1 1 " " }}{PARA 0 "" 0 "" {TEXT -1 76 "The simple order conditions used are given (in abreviated form) as follows. " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 " " {MPLTEXT 1 0 184 "SO6 := SimpleOrderConditions(6):\n[seq([i,SO6[i]], i=[1,2,4,8,13,16,24,28,29,32])]:\nlinalg[augment](linalg[delcols](%,2. .2),matrix([[` `]$(linalg[rowdim](%))]),linalg[delcols](%,1..1));" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#K%'matrixG6#7,7%\"\"\"%#~~G/*&%\"bGF(% \"eGF(F(7%\"\"#F)/*&F,F(%\"cGF(#F(F/7%\"\"%F)/*&F,F()F2F/F(#F(\"\"$7% \"\")F)/*&F,F()F2F:F(#F(F57%\"#8F)/*(F,F(F2F(-%!G6#*&F8F(%\"aGF(F(#F( \"#:7%\"#;F)/*&F,F()F2F5F(#F(\"\"&7%\"#CF)/*(F,F(F2F(-FF6#*&FIF(FEF(F( #F(\"#s7%\"#GF)/*(F,F(F8F(FEF(#F(\"#=7%\"#HF)/*(F,F(F2F(-FF6#*&F?F(FIF (F(#F(FT7%\"#KF)/*&F,F()F2FRF(#F(\"\"'Q(pprint96\"" }}}{PARA 0 "" 0 " " {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 462 "SO6_7 := \+ SimpleOrderConditions(6,7,'expanded'):\nord_cdns := [seq(SO6_7[i],i=[1 ,2,4,8,13,16,24,28,29,32])]:\nSO_eqs := [op(RowSumConditions(7,'expand ed')),op(StageOrderConditions(2,7,'expanded'))]:\nsimp_eqs := [seq(add (b[i]*a[i,j],i=j+1..7)=b[j]*(1-c[j]),j=[3,4,5,6]),\n add( b[i]*c[i]*a[i,2],i=3..7)=0,add(b[i]*c[i]*add(a[i,j]*a[j,2],j=3..i-1),i =3..7)=0,\n add(b[i]*c[i]^2*a[i,2],i=3..7)=0]:\ncdns := \+ [op(ord_cdns),op(simp_eqs),op(SO_eqs)]:" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 21 "We specify the nodes " }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "c[2]=1/3" "6#/&%\"cG6#\"\"# *&\"\"\"F)\"\"$!\"\"" }{TEXT -1 3 ", " }{XPPEDIT 18 0 "c[3]=2/3" "6#/ &%\"cG6#\"\"$*&\"\"#\"\"\"F'!\"\"" }{TEXT -1 3 ", " }{XPPEDIT 18 0 "c [7]=1" "6#/&%\"cG6#\"\"(\"\"\"" }{TEXT -1 1 "," }}{PARA 0 "" 0 "" {TEXT -1 16 " and the weight " }}{PARA 256 "" 0 "" {TEXT -1 1 " " } {XPPEDIT 18 0 "b[2]=0" "6#/&%\"bG6#\"\"#\"\"!" }{TEXT -1 2 ". " }} {PARA 0 "" 0 "" {TEXT -1 26 "We include the equations " }{XPPEDIT 18 0 "c[5]=c[6]" "6#/&%\"cG6#\"\"&&F%6#\"\"'" }{TEXT -1 7 " and " } {XPPEDIT 18 0 "b[5]=b[6]" "6#/&%\"bG6#\"\"&&F%6#\"\"'" }{TEXT -1 38 " \+ in the system of equations to solve." }}{PARA 0 "" 0 "" {TEXT -1 0 " " }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 124 "e1 := \{c[2]=1/3,c[3]=2 /3,c[7]=1,b[2]=0\}:\neqns := subs(e1,[op(cdns),c[5]=c[6],b[5]=b[6]]): \nnops(eqns);\nindets(eqns);\nnops(%);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#\"#I" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#<@&%\"aG6$\"\"%\"\"#&F%6 $F'\"\"$&%\"cG6#\"\"&&F-6#\"\"'&F%6$F(\"\"\"&F%6$F+F5&F%6$F/F(&F%6$F/F +&F%6$F+F(&F%6$F'F5&F-6#F'&F%6$F/F5&%\"bGF.&FEF1&FE6#\"\"(&FE6#F5&FE6# F+&FEFA&F%6$FIF/&F%6$FIF2&F%6$FIF+&F%6$FIF'&F%6$FIF5&F%6$FIF(&F%6$F2F' &F%6$F2F/&F%6$F2F(&F%6$F2F+&F%6$F/F'&F%6$F2F5" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#\"#I" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 " " {TEXT -1 51 "There are 30 equations and 30 unknown coefficients." }} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 22 "infolevel[solve] := 4:" }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 69 "e2 := solve(\{op(eqns)\}):\n infolevel[solve] := 0:\ne3 := `union`(e1,e2):" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 2 "e3" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 481 "e3 := \{a[4,3] = -1/12, c[7 ] = 1, a[7,6] = -16/11, b[3] = 27/40, a[3,1] = 0, c[3] = 2/3, c[4] = 1 /3, a[7,3] = 63/44, a[6,4] = -3/4, a[5,1] = -1/16, a[4,2] = 1/3, a[5,4 ] = -3/8, a[7,1] = 9/44, a[6,3] = -3/8, b[1] = 11/120, b[7] = 11/120, \+ a[5,3] = -3/16, a[6,2] = 9/8, a[7,2] = -9/11, a[4,1] = 1/12, a[7,4] = \+ 18/11, a[5,2] = 9/8, c[5] = 1/2, c[2] = 1/3, c[6] = 1/2, b[2] = 0, b[4 ] = 27/40, b[6] = -4/15, b[5] = -4/15, a[6,5] = 1/2, a[7,5] = 0, a[3,2 ] = 2/3, a[2,1] = 1/3, a[6,1] = 0\}:" }{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 91 "subs(e3,matrix([seq([c[i],seq(a[i,j],j=1..i-1),``$(8-i)],i=2..7), \n[``,seq(b[i],i=1..7)]]));" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#K%'matr ixG6#7)7*#\"\"\"\"\"$F(%!GF+F+F+F+F+7*#\"\"#F*\"\"!F-F+F+F+F+F+7*F(#F) \"#7F(#!\"\"F2F+F+F+F+7*#F)F.#F4\"#;#\"\"*\"\")#!\"$F8#F=F;F+F+F+7*F6F /F9F>#F=\"\"%F6F+F+7*F)#F:\"#W#!\"*\"#6#\"#jFD#\"#=FGF/#!#;FGF+7*F+#FG \"$?\"F/#\"#F\"#SFQ#!\"%\"#:FTFOQ)pprint106\"" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 85 "RK6_7eqs := \+ [op(RowSumConditions(7,'expanded')),op(OrderConditions(6,7,'expanded') )]:" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 7 "Ch eck: " }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 62 "simplify(subs(e3,RK 6_7eqs)):\nmap(u->lhs(u)-rhs(u),%);\nnops(%);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#7M\"\"!F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$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#\"#V" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 65 "#---------------------------------------------------------------- " }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}}{PARA 0 "" 0 "" {TEXT -1 45 "#--------------------------------------------" }}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 25 "absolute stability region" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}} {SECT 1 {PARA 4 "" 0 "" {TEXT -1 26 "coefficients of the scheme" }} {PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 415 "ee := \{c[2]=1/3,\nc[3]=2/3,\nc[4]=1/3,\nc[5]=1/2,\nc[6]=1/2,\nc[ 7]=1,\n\na[2,1]=1/3,\na[3,1]=0,\na[3,2]=2/3,\na[4,1]=1/12,\na[4,2]=1/3 ,\na[4,3]=-1/12,\na[5,1]=-1/16,\na[5,2]=9/8,\na[5,3]=-3/16,\na[5,4]=-3 /8,\na[6,1]=0,\na[6,2]=9/8,\na[6,3]=-3/8,\na[6,4]=-3/4,\na[6,5]=1/2,\n a[7,1]=9/44,\na[7,2]=-9/11,\na[7,3]=63/44,\na[7,4]=18/11,\na[7,5]=0,\n a[7,6]=-16/11,\n\nb[1]=11/120,\nb[2]=0,\nb[3]=27/40,\nb[4]=27/40,\nb[5 ]=-4/15,\nb[6]=-4/15,\nb[7]=11/120\}:" }}}{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 68 "The stability function R \+ for the order 6 scheme is given as follows." }}{EXCHG {PARA 0 "> " 0 " " {MPLTEXT 1 0 75 "subs(ee,StabilityFunction(6,7,'expanded')):\nR := u napply(%,z):\n'R(z)'=R(z);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/-%\"RG6 #%\"zG,2\"\"\"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)\"%g@F)*$)F'\"\"(F)F)!\"\"" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT 260 4 "Note" }{TEXT -1 74 ": Thi s stability function is the same as that of the other Butcher scheme. " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 124 "We c an find the point where the boundary of the stability region intersect s 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 := newt on(R(z)=1,z=-3);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%#z0G$!+z*3h&G!\" *" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 302 "z0 := newton(R(z)=1,z=-3):\np1 := plot([R(z),1],z=-3 .29..0.49,color=[red,blue]):\np2 := plot([[[z0,1]]$3],style=point,symb ol=[circle,cross,diamond],color=black):\np3 := plot([[z0,0],[z0,1]],li nestyle=3,color=COLOR(RGB,0,.5,0)):\nplots[display]([p1,p2,p3],view=[- 3.29..0.49,-0.07..1.47],font=[HELVETICA,9]);" }}{PARA 13 "" 1 "" {GLPLOT2D 360 253 253 {PLOTDATA 2 "6+-%'CURVESG6$7X7$$!3/++++++!H$!#<$ \"3)p'RR\"o@$\\DF*7$$!3O+]iyL!)[KF*$\"3a[97!)4&3M#F*7$$!3A++Ddng2KF*$ \"3->p*)[ikZ@F*7$$!3=](oszh<<$F*$\"3m#Q$**o6C\"*>F*7$$!3:+vGPo\"f8$F*$ \"3WBiP([:^%=F*7$$!3;]7)3V0c4$F*$\"3ka#G!=RP#p\"F*7$$!3?+]ZCSHbIF*$\"3 uNyxl<=^:F*7$$!3\")*****yS:Z,$F*$\"36NX5$*e**>9F*7$$!3')**\\K\"zOT(HF* $\"3'of-S,*4*H\"F*7$$!38]7$y.^P$HF*$\"3e#3!HzHM)=\"F*7$$!3%**\\PVGlL*G F*$\"3mWX8Mt^'3\"F*7$$!3F+v)yvz%=GF*$\"3%[0W`D2R>*!#=7$$!3E+vyK,%4u#F* $\"3b(f#)RY2)HxF^o7$$!33+veT&[2m#F*$\"3=d9uPC'\\Y'F^o7$$!39+v$e793e#F* $\"3>E/cVrXBaF^o7$$!3+++0aTw:cLF^o7$$!39++DUlxiAF*$ \"3bgqIuO!Q\"HF^o7$$!35++&G=&)Q=#F*$\"3o%G1DxR,f#F^o7$$!3(**\\(eqUC7@F *$\"3=Yo#>6[5P#F^o7$$!34++l'Hcq-#F*$\"3%oA#R)>)y*=#F^o7$$!3#*****f34*[ &>F*$\"31g)RNG,T4#F^o7$$!3'**\\([1a%4(=F*$\"3`o!fj+#yR?F^o7$$!33++58%R mz\"F*$\"3&fS!H*=>e.#F^o7$$!3&**\\([hR6:FF^o7$$!3?+Dr%RzfC\"F*$\"3,yG+b/;1HF^o7$$!3 '***\\2\\lin6F*$\"3*pk]gZc)HJF^o7$$!3%*****p#R!o'3\"F*$\"337)*[a&\\ZQ$ F^o7$$!3D+]xq.\\25F*$\"3w9.XCP2eOF^o7$$!3Y**\\(o_*p3$*F^o$\"3Y(QWX>&)f %RF^o7$$!3+***\\#=/'zX)F^o$\"3#G:tH0WTH%F^o7$$!3%>++5\\LNp(F^o$\"3*R:n #))G:MYF^o7$$!39++]#eWt(oF^o$\"33')=@QngF]F^o7$$!33-](=7cx8'F^o$\"3=6= '>!\\C8aF^o7$$!3T'***\\$e!>H`F^o$\"3%3m/@Cs*oeF^o7$$!3M,]([U$RoXF^o$\" 3W<[E_x%GL'F^o7$$!3C**\\P!=TJx$F^o$\"3!R()eSf8q&oF^o7$$!3#Q++DvOc*HF^o $\"3/>Qb,nT6uF^o7$$!3s.](=nh;=#F^o$\"3IAVQ*[=*R!)F^o7$$!3o*****f#pq(R \"F^o$\"3Gb4>#4wbp)F^o7$$!3I%)**\\#p***ff!#>$\"32!Rxk<89U*F^o7$$\"3c;+ D\"\\%o!*>F\\z$\"3;2\"3vI1,-\"F*7$$\"3YP++Iv`'H*F\\z$\"3EbFgOPU(4\"F*7 $$\"3-.+vdp)pw\"F^o$\"3ko1j(\\rK>\"F*7$$\"3*******zv2f^#F^o$\"3+vh:'fp gG\"F*7$$\"3P+]7NNT9LF^o$\"3)3*\\ZcT(HR\"F*7$$\"3Y,](3X&oySF^o$\"3HpF_ H#3O]\"F*7$$\"3!***************[F^o$\"3_MD@Z;JK;F*-%'COLOURG6&%$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^qF_]l7$FcqF_]l7$FhqF_]l7$F]rF_]l7$FbrF_]l7$FgrF_]l 7$F\\sF_]l7$FasF_]l7$FfsF_]l7$F[tF_]l7$F`tF_]l7$FetF_]l7$FjtF_]l7$F_uF _]l7$FduF_]l7$FiuF_]l7$F^vF_]l7$FcvF_]l7$FhvF_]l7$F]wF_]l7$FbwF_]l7$Fg wF_]l7$F\\xF_]l7$FaxF_]l7$FfxF_]l7$F[yF_]l7$F`yF_]l7$FeyF_]l7$FjyF_]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$$!3/+++z*3h&GF*F_]l-%'SYMBOLG6#%'CIR CLEG-Fc\\l6&Fe\\lFj\\lFj\\lFj\\l-%&STYLEG6#%&POINTG-F$6&Fe`l-Fj`l6#%&C ROSSGF]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#\"\"$-%+AXESLABELSG 6%Q\"z6\"Q!F_cl-%%FONTG6#%(DEFAULTG-Fbcl6$%*HELVETICAG\"\"*-%%VIEWG6$; $!$H$!\"#$\"#\\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 1640 "R := z -> 1+z+1/2*z^2+1/6*z^3+1/24*z^4+1/120*z^5+1/720*z^6-1/2 160*z^7:\npts := []: z0 := 0: tt := 0: \nwhile tt<=161/20 do\n zz := newton(`R`(z)=exp(tt*Pi*I),z=z0):\n z0 := zz:\n if (7/20<=tt and \+ tt<=27/20) or (133/20<=tt and tt<=153/20) then\n hh := 1/40\n e lse \n hh := 1/20\n end if;\n tt := tt+hh;\n pts := [op(pts ),[Re(zz),Im(zz)]]:\nend do:\np1 := plot(pts,color=COLOR(RGB,0,.38,.1) ):\np2 := plots[polygonplot]([seq([pts[i-1],pts[i],[-1.45,0]],i=2..nop s(pts))],\n style=patchnogrid,color=COLOR(RGB,0,.75,.2)):\npt s := []: z0 := 0.8+3.3*I: tt := 0: \nwhile tt<=41/20 do\n zz := newt on(R(z)=exp(tt*Pi*I),z=z0):\n z0 := zz:\n if (9/20<=tt and tt<=6/5 ) then\n hh := 1/60\n else \n hh := 1/20\n end if;\n t t := tt+hh;\n pts := [op(pts),[Re(zz),Im(zz)]]:\nend do:\np3 := plot (pts,color=COLOR(RGB,0,.38,.1)):\np4 := plots[polygonplot]([seq([pts[i -1],pts[i],[0.75,3.03]],i=2..nops(pts))],\n style=patchnogrid ,color=COLOR(RGB,0,.75,.2)):\npts := []: z0 := 0.8-3.3*I: tt := 0: \nw hile tt<=41/20 do\n zz := newton(R(z)=exp(tt*Pi*I),z=z0):\n z0 := \+ zz:\n if (17/20<=tt and tt<=8/5) then\n hh := 1/60\n else \n \+ hh := 1/20\n end if;\n tt := tt+hh;\n pts := [op(pts),[Re(z z),Im(zz)]]:\nend do:\np5 := plot(pts,color=COLOR(RGB,0,.38,.1)):\np6 \+ := plots[polygonplot]([seq([pts[i-1],pts[i],[0.75,-3.03]],i=2..nops(pt s))],\n style=patchnogrid,color=COLOR(RGB,0,.75,.2)):\np7 := \+ plot([[[-3.49,0],[1.29,0]],[[0,-3.59],[0,3.59]]],color=black,linestyle =3):\nplots[display]([p||(1..7)],view=[-3.49..1.29,-3.59..3.59],font=[ HELVETICA,9],\n labels=[`Re(z)`,`Im(z)`],axes=boxed,scali ng=constrained);" }}{PARA 13 "" 1 "" {GLPLOT2D 381 565 565 {PLOTDATA 2 "6/-%'CURVESG6$7fw7$$\"\"!F)F(7$$!33+++8]B]B!#F$\"31+++6jzq:!#=7$$!3 s*****zW3m%f!#D$\"3&)*****pu!fTJF07$$!3/+++QAH%\\\"!#B$\"3:+++E$eBr%F0 7$$!3)******H227X\"!#A$\"37+++*GvHG'F07$$!3U******RBpL$)F@$\"3E+++?l4` yF07$$!3:+++]S.=M!#@$\"3\\******)o\"4A%*F07$$!3'******>Hgr5\"!#?$\"33+ ++JV#*)4\"!#<7$$!3'*******H?Xg=FQ$\"3'******fQ.s<\"FT7$$!3;+++iRW4IFQ$ \"3/+++v=Wb7FT7$$!3#******\\FPzq%FQ$\"3(******RsmOL\"FT7$$!3=+++(=N=:( FQ$\"3'******z^C>T\"FT7$$!33+++4.')e5!#>$\"35+++WXG!\\\"FT7$$!3%****** *4!*3L:Ffo$\"3)******R+R)o:FT7$$!33+++'pzw<#Ffo$\"3!******Ra(pZ;FT7$$! 3-+++!RqR/$Ffo$\"34+++u0(ps\"FT7$$!33+++ln%*)>%Ffo$\"3*******pQIn!=FT7 $$!3:+++ZW,IdFfo$\"3!******RMHp)=FT7$$!37+++aPlZxFfo$\"33+++@BCn>FT7$$ !31+++/VuP5F0$\"3')******4$3o/#FT7$$!37+++q**[s8F0$\"3&*******3!\\R7#F T7$$!3%******f%GZ!y\"F0$\"35+++FO@'>#FT7$$!31+++JR6[AF0$\"3;+++\"3\"=h AFT7$$!3G+++@z`]FF0$\"3/+++-:gcF0$\"38+++&z!*3_#FT7$$!3`++++0$z.'F0$\"3/+++s;0TDFT7$$ !3R+++n$G2W'F0$\"35+++,s\")eDFT7$$!3f******fzlHoF0$\"3))*****f.\"[uDFT 7$$!3O+++o?/1sF0$\"3%)*****fEt#)e#FT7$$!3W+++$yh5d(F0$\"3y*****R3x.g#F T7$$!3z*****>gdd#zF0$\"3$******\\cS4h#FT7$$!3W+++l+/r#)F0$\"3/+++\"3&3 ?EFT7$$!3R+++xOq2')F0$\"30+++v8\"zi#FT7$$!3z******y6WO*)F0$\"3?+++6S]M EFT7$$!3?+++.p&yD*F0$\"3<+++\"oM*REFT7$$!35+++G(yCd*F0$\"36+++(pkUk#FT 7$$!3Y+++X&o2))*F0$\"3))*****z*oaZEFT7$$!3!*******HJJ=5FT$\"3)******f6 F)\\EFT7$$!3))******zA*z/\"FT$\"3#)*****\\LX6l#FT7$$!3!******>jXr2\"FT $\"3')*****zfO:l#FT7$$!3++++W3!e5\"FT$\"3#******ztJ5l#FT7$$!3$******\\ Q#)R8\"FT$\"37+++jzl\\EFT7$$!3!******4(>rh6FT$\"3'*******Q$Ruk#FT7$$!3 !******R#*3!*=\"FT$\"3-+++wrRWEFT7$$!3#******H%H\"FT$\"3#*******y&RVi#FT7$$!3,+++&))fXM\"FT$\"32+++Nsw4E FT7$$!3#******p)Hc$R\"FT$\"37+++gIF#f#FT7$$!3*******zZr7W\"FT$\"3#)*** **>O6>d#FT7$$!3++++*>_x[\"FT$\"3=+++`(=([DFT7$$!34+++%QrI`\"FT$\"3/+++ .hrADFT7$$!3++++r#*Hx:FT$\"3/+++o-\"R\\#FT7$$!33+++&3:0i\"FT$\"3)***** *f\"fHiCFT7$$!3-+++EE\"Gm\"FT$\"35+++$*y&yU#FT7$$!3'******pj1Vq\"FT$\" 3!******Hru0R#FT7$$!3'******\\&*R^u\"FT$\"3')*****p-C/N#FT7$$!3!****** \\6\"\\&y\"FT$\"30+++@7R2BFT7$$!3)******fO\"eD=FT$\"31+++ME[hAFT7$$!3# ******\\%)yc'=FT$\"3%)*****z\"\\u7AFT7$$!3$******py&41>FT$\"35+++T8Hh@ FT7$$!3-+++;EFT$\"38+++\\LL2@FT7$$!31+++7?C*)>FT$\"31+++N0@^?FT7$$! 3,+++>+dK?FT$\"3#******H'\\S$*>FT7$$!3;+++TGGx?FT$\"3$*******H8_M>FT7$ $!3++++#[/L7#FT$\"33+++C:Av=FT7$$!3)*******y!Q.<#FT$\"3*******>*\\7;=F T7$$!3#******H)z!z@#FT$\"3++++P9rd8FT7$$!3!******f$\\P%f#FT$\"3 '*******y#>kE\"FT7$$!3&*******\\D2EEFT$\"33+++RNn77FT7$$!3A+++PD\\bEFT $\"3++++V#3&e6FT7$$!3'*******\\nk#o#FT$\"36+++8k\"Q5\"FT7$$!3#******** \\cvq#FT$\"33+++'y/&[5FT7$$!3(*******42DIFFT$\"3Z******[%3\\#**F07$$!3 !)*****4Dl2v#FT$\"3U+++6E+d$*F07$$!3++++(>V\"pFFT$\"3a*****fM[1y)F07$$ !3/+++eeV&y#FT$\"3!******pC5_>)F07$$!3'******zW/(*z#FT$\"3z*******y\"4 +wF07$$!3))******zA-7GFT$\"3'******4@OZ*pF07$$!39+++\\tZAGFT$\"3/+++P \"Q'yjF07$$!3#)******H]R`GFT$\"3/ +++()ebWCF07$$!37+++WW#[&GFT$\"3/+++L:#Hv\"F07$$!39+++IAnbGFT$\"3-+++2 5fa5F07$$!3-+++C>1cGFT$\"3$******H^b,_$Ffo7$F][m$!3$******H^b,_$Ffo7$F hjl$!3-+++25fa5F07$Fcjl$!3/+++L:#Hv\"F07$F^jl$!3/+++()ebWCF07$Fiil$!3? +++vY_FJF07$Fdil$!3w*****pb&R+QF07$F_il$!3%*******37DiWF07$Fjhl$!3R+++ ,sh7^F07$Fehl$!3W+++;(o8v&F07$F`hl$!3/+++P\"Q'yjF07$F[hl$!3'******4@OZ *pF07$Ffgl$!3z*******y\"4+wF07$Fagl$!3!******pC5_>)F07$F\\gl$!3a*****f M[1y)F07$Fgfl$!3U+++6E+d$*F07$Fbfl$!3Z******[%3\\#**F07$F]fl$!33+++'y/ &[5FT7$Fhel$!36+++8k\"Q5\"FT7$Fcel$!3++++V#3&e6FT7$F^el$!33+++RNn77FT7 $Fidl$!3'*******y#>kE\"FT7$Fddl$!3++++W$p)>8FT7$F_dl$!3!******4inJP\"F T7$Fjcl$!34+++`.[E9FT7$Fecl$!33+++,m**z9FT7$F`cl$!30+++k[#R`\"FT7$F[cl $!3#******4'G[)e\"FT7$Ffbl$!3!******ziwQk\"FT7$Fabl$!34+++3`E+*\\7;=FT7$Fbal$!33+++C:Av=FT7$F]al$!3 $*******H8_M>FT7$Fh`l$!3#******H'\\S$*>FT7$Fc`l$!31+++N0@^?FT7$F^`l$!3 8+++\\LL2@FT7$Fi_l$!35+++T8Hh@FT7$Fd_l$!3%)*****z\"\\u7AFT7$F__l$!31++ +ME[hAFT7$Fj^l$!30+++@7R2BFT7$Fe^l$!3')*****p-C/N#FT7$F`^l$!3!******Hr u0R#FT7$F[^l$!35+++$*y&yU#FT7$Ff]l$!3)******f\"fHiCFT7$Fa]l$!3/+++o-\" R\\#FT7$F\\]l$!3/+++.hrADFT7$Fg\\l$!3=+++`(=([DFT7$Fb\\l$!3#)*****>O6> d#FT7$F]\\l$!37+++gIF#f#FT7$Fh[l$!32+++Nsw4EFT7$Fc[l$!3#*******y&RVi#F T7$F^[l$!3.+++)e:fj#FT7$Fiz$!3-+++wrRWEFT7$F_z$!37+++jzl\\EFT7$Fjy$!3# ******ztJ5l#FT7$Fey$!3')*****zfO:l#FT7$F`y$!3#)*****\\LX6l#FT7$F[y$!3) ******f6F)\\EFT7$Ffx$!3))*****z*oaZEFT7$Fax$!36+++(pkUk#FT7$F\\x$!3<++ +\"oM*REFT7$Fgw$!3?+++6S]MEFT7$Fbw$!30+++v8\"zi#FT7$F]w$!3/+++\"3&3?EF T7$Fhv$!3$******\\cS4h#FT7$Fcv$!3y*****R3x.g#FT7$F^v$!3%)*****fEt#)e#F T7$Fiu$!3))*****f.\"[uDFT7$Fdu$!35+++,s\")eDFT7$F_u$!3/+++s;0TDFT7$Fjt $!38+++&z!*3_#FT7$Fet$!35+++\"y`z\\#FT7$F`t$!3!)*****z')R#FT7$F]r$!3&*******3!\\ R7#FT7$Fhq$!3')******4$3o/#FT7$Fcq$!33+++@BCn>FT7$F^q$!3!******RMHp)=F T7$Fip$!3*******pQIn!=FT7$Fdp$!34+++u0(ps\"FT7$F_p$!3!******Ra(pZ;FT7$ Fjo$!3)******R+R)o:FT7$Fdo$!35+++WXG!\\\"FT7$F_o$!3'******z^C>T\"FT7$F jn$!3(******RsmOL\"FT7$Fen$!3/+++v=Wb7FT7$FV$!3'******fQ.s<\"FT7$FO$!3 3+++JV#*)4\"FT7$$!3X+++\"*f&zI'FK$!36+++kJf?5FT7$$!33+++QZ)pu\"FK$!3N+ ++IGyP')F07$$!3-+++Y/;fOF@$!3[+++O58oqF07$$!3y*****HT4E1&F:$!3;+++o0q( \\&F07$$!3!******>QeK^$!#C$!3>+++p>)p#RF07$$!3m*****z))*>%*f!#E$!3'*** ****HU>cBF07$F($!3!)*****Hj\")R&yFfo7$F($\"3!)*****Hj\")R&yFfo-%&COLOR G6&%$RGBGF($\"#Q!\"#$\"\"\"!\"\"-%)POLYGONSG6gw7%F'7$$!+8]B]BFfo$\"+6j zq:!#57$$!$X\"Fd_nF(7%F\\`n7$$!+[%3m%fFT$\"+Z2fTJFa`nFb`n7%Ff`n7$$!+QA H%\\\"!#:$\"+E$eBr%Fa`nFb`n7%F\\an7$$!+tq?^9!#9$\"+*GvHG'Fa`nFb`n7%Fca n7$$!+SBpL$)Ffan$\"+?l4`yFa`nFb`n7%Fjan7$$!+]S.=M!#8$\"+*o\"4A%*Fa`nFb `n7%F`bn7$$!+#Hgr5\"!#7$\"+JV#*)4\"!\"*Fb`n7%Fgbn7$$!+I?Xg=Fjbn$\"+'Q. s<\"F]cnFb`n7%F_cn7$$!+iRW4IFjbn$\"+v=Wb7F]cnFb`n7%Fecn7$$!+vs$zq%Fjbn $\"+CnmL8F]cnFb`n7%F[dn7$$!+(=N=:(Fjbn$\"+=X#>T\"F]cnFb`n7%Fadn7$$!+4. ')e5!#6$\"+WXG!\\\"F]cnFb`n7%Fgdn7$$!+5!*3L:Fjdn$\"+/!R)o:F]cnFb`n7%F^ en7$$!+'pzw<#Fjdn$\"+WvpZ;F]cnFb`n7%Fden7$$!+!RqR/$Fjdn$\"+u0(ps\"F]cn Fb`n7%Fjen7$$!+ln%*)>%Fjdn$\"+(QIn!=F]cnFb`n7%F`fn7$$!+ZW,IdFjdn$\"+W$ Hp)=F]cnFb`n7%Fffn7$$!+aPlZxFjdn$\"+@BCn>F]cnFb`n7%F\\gn7$$!+/VuP5Fa`n $\"+5$3o/#F]cnFb`n7%Fbgn7$$!+q**[s8Fa`n$\"+4!\\R7#F]cnFb`n7%Fhgn7$$!+Y GZ!y\"Fa`n$\"+FO@'>#F]cnFb`n7%F^hn7$$!+JR6[AFa`n$\"+\"3\"=hAF]cnFb`n7% Fdhn7$$!+@z`]FFa`n$\"+-:gcFa`n$\"+&z!*3_#F]cnFb`n7%F^[o7$$ !++0$z.'Fa`n$\"+s;0TDF]cnFb`n7%Fd[o7$$!+n$G2W'Fa`n$\"+,s\")eDF]cnFb`n7 %Fj[o7$$!+gzlHoFa`n$\"+O5[uDF]cnFb`n7%F`\\o7$$!+o?/1sFa`n$\"+mKF)e#F]c nFb`n7%Ff\\o7$$!+$yh5d(Fa`n$\"+%3x.g#F]cnFb`n7%F\\]o7$$!+-wvDzFa`n$\"+ l0%4h#F]cnFb`n7%Fb]o7$$!+l+/r#)Fa`n$\"+\"3&3?EF]cnFb`n7%Fh]o7$$!+xOq2' )Fa`n$\"+v8\"zi#F]cnFb`n7%F^^o7$$!+z6WO*)Fa`n$\"+6S]MEF]cnFb`n7%Fd^o7$ $!+.p&yD*Fa`n$\"+\"oM*REF]cnFb`n7%Fj^o7$$!+G(yCd*Fa`n$\"+(pkUk#F]cnFb` n7%F`_o7$$!+X&o2))*Fa`n$\"+)*oaZEF]cnFb`n7%Ff_o7$$!+IJJ=5F]cn$\"+;r#) \\EF]cnFb`n7%F\\`o7$$!+!G#*z/\"F]cn$\"+N`9^EF]cnFb`n7%Fb`o7$$!+Kc9x5F] cn$\"+)fO:l#F]cnFb`n7%Fh`o7$$!+W3!e5\"F]cn$\"+Q<.^EF]cnFb`n7%F^ao7$$!+ &Q#)R8\"F]cn$\"+jzl\\EF]cnFb`n7%Fdao7$$!+r>rh6F]cn$\"+R$Ruk#F]cnFb`n7% Fjao7$$!+C*3!*=\"F]cn$\"+wrRWEF]cnFb`n7%F`bo7$$!+t@PU7F]cn$\"+)e:fj#F] cnFb`n7%Ffbo7$$!+5->%H\"F]cn$\"+z&RVi#F]cnFb`n7%F\\co7$$!+&))fXM\"F]cn $\"+Nsw4EF]cnFb`n7%Fbco7$$!+()Hc$R\"F]cn$\"+gIF#f#F]cnFb`n7%Fhco7$$!+y 9FT9F]cn$\"+i8\">d#F]cnFb`n7%F^do7$$!+*>_x[\"F]cn$\"+`(=([DF]cnFb`n7%F ddo7$$!+%QrI`\"F]cn$\"+.hrADF]cnFb`n7%Fjdo7$$!+r#*Hx:F]cn$\"+o-\"R\\#F ]cnFb`n7%F`eo7$$!+&3:0i\"F]cn$\"+;fHiCF]cnFb`n7%Ffeo7$$!+EE\"Gm\"F]cn$ \"+$*y&yU#F]cnFb`n7%F\\fo7$$!+PmI/F]cn$\"+T8Hh@F]cnFb`n7%F`ho7$$!+;EF]cn$\"+\\LL2@ F]cnFb`n7%Ffho7$$!+7?C*)>F]cn$\"+N0@^?F]cnFb`n7%F\\io7$$!+>+dK?F]cn$\" +j\\S$*>F]cnFb`n7%Fbio7$$!+TGGx?F]cn$\"+I8_M>F]cnFb`n7%Fhio7$$!+#[/L7# F]cn$\"+C:Av=F]cnFb`n7%F^jo7$$!+z!Q.<#F]cn$\"+#*\\7;=F]cnFb`n7%Fdjo7$$ !+$)z!z@#F]cn$\"+P9rd8F]cnFb`n7%Fj]p 7$$!+O\\P%f#F]cn$\"+z#>kE\"F]cnFb`n7%F`^p7$$!+]D2EEF]cn$\"+RNn77F]cnFb `n7%Ff^p7$$!+PD\\bEF]cn$\"+V#3&e6F]cnFb`n7%F\\_p7$$!+]nk#o#F]cn$\"+8k \"Q5\"F]cnFb`n7%Fb_p7$$!++lb2FF]cn$\"+'y/&[5F]cnFb`n7%Fh_p7$$!+52DIFF] cn$\"+\\%3\\#**Fa`nFb`n7%F^`p7$$!+^_w]FF]cn$\"+6E+d$*Fa`nFb`n7%Fd`p7$$ !+(>V\"pFF]cn$\"+Y$[1y)Fa`nFb`n7%Fj`p7$$!+eeV&y#F]cn$\"+Z-@&>)Fa`nFb`n 7%F`ap7$$!+[Wq*z#F]cn$\"+!z\"4+wFa`nFb`n7%Ffap7$$!+!GA?\"GF]cn$\"+6it% *pFa`nFb`n7%F\\bp7$$!+\\tZAGF]cn$\"+P\"Q'yjFa`nFb`n7%Fbbp7$$!+I]R`GF]cn$\"+()ebWCFa`nFb`n7%F fdp7$$!+WW#[&GF]cn$\"+L:#Hv\"Fa`nFb`n7%F\\ep7$$!+IAnbGF]cn$\"+25fa5Fa` nFb`n7%Fbep7$$!+C>1cGF]cn$\"+8b:?NFjdnFb`n7%Fhep7$Fiep$!+8b:?NFjdnFb`n 7%F^fp7$Fcep$!+25fa5Fa`nFb`n7%Fbfp7$F]ep$!+L:#Hv\"Fa`nFb`n7%Fffp7$Fgdp $!+()ebWCFa`nFb`n7%Fjfp7$Fadp$!+vY_FJFa`nFb`n7%F^gp7$F[dp$!+dbR+QFa`nF b`n7%Fbgp7$Fecp$!+47DiWFa`nFb`n7%Ffgp7$F_cp$!+,sh7^Fa`nFb`n7%Fjgp7$Fib p$!+;(o8v&Fa`nFb`n7%F^hp7$Fcbp$!+P\"Q'yjFa`nFb`n7%Fbhp7$F]bp$!+6it%*pF a`nFb`n7%Ffhp7$Fgap$!+!z\"4+wFa`nFb`n7%Fjhp7$Faap$!+Z-@&>)Fa`nFb`n7%F^ ip7$F[ap$!+Y$[1y)Fa`nFb`n7%Fbip7$Fe`p$!+6E+d$*Fa`nFb`n7%Ffip7$F_`p$!+ \\%3\\#**Fa`nFb`n7%Fjip7$Fi_p$!+'y/&[5F]cnFb`n7%F^jp7$Fc_p$!+8k\"Q5\"F ]cnFb`n7%Fbjp7$F]_p$!+V#3&e6F]cnFb`n7%Ffjp7$Fg^p$!+RNn77F]cnFb`n7%Fjjp 7$Fa^p$!+z#>kE\"F]cnFb`n7%F^[q7$F[^p$!+W$p)>8F]cnFb`n7%Fb[q7$Fe]p$!+@w ;t8F]cnFb`n7%Ff[q7$F_]p$!+`.[E9F]cnFb`n7%Fj[q7$Fi\\p$!+,m**z9F]cnFb`n7 %F^\\q7$Fc\\p$!+k[#R`\"F]cnFb`n7%Fb\\q7$F]\\p$!+hG[)e\"F]cnFb`n7%Ff\\q 7$Fg[p$!+Gm(Qk\"F]cnFb`n7%Fj\\q7$Fa[p$!+3`E+F]cnFb`n7%F^^q7$Fcio$!+j\\S$*>F]cnFb`n7%Fb^q7$F]io$ !+N0@^?F]cnFb`n7%Ff^q7$Fgho$!+\\LL2@F]cnFb`n7%Fj^q7$Faho$!+T8Hh@F]cnFb `n7%F^_q7$F[ho$!+=\\u7AF]cnFb`n7%Fb_q7$Fego$!+ME[hAF]cnFb`n7%Ff_q7$F_g o$!+@7R2BF]cnFb`n7%Fj_q7$Fifo$!+FSU]BF]cnFb`n7%F^`q7$Fcfo$!+8Zd!R#F]cn Fb`n7%Fb`q7$F]fo$!+$*y&yU#F]cnFb`n7%Ff`q7$Fgeo$!+;fHiCF]cnFb`n7%Fj`q7$ Faeo$!+o-\"R\\#F]cnFb`n7%F^aq7$F[eo$!+.hrADF]cnFb`n7%Fbaq7$Fedo$!+`(=( [DF]cnFb`n7%Ffaq7$F_do$!+i8\">d#F]cnFb`n7%Fjaq7$Fico$!+gIF#f#F]cnFb`n7 %F^bq7$Fcco$!+Nsw4EF]cnFb`n7%Fbbq7$F]co$!+z&RVi#F]cnFb`n7%Ffbq7$Fgbo$! +)e:fj#F]cnFb`n7%Fjbq7$Fabo$!+wrRWEF]cnFb`n7%F^cq7$Feao$!+jzl\\EF]cnFb `n7%Fbcq7$F_ao$!+Q<.^EF]cnFb`n7%Ffcq7$Fi`o$!+)fO:l#F]cnFb`n7%Fjcq7$Fc` o$!+N`9^EF]cnFb`n7%F^dq7$F]`o$!+;r#)\\EF]cnFb`n7%Fbdq7$Fg_o$!+)*oaZEF] cnFb`n7%Ffdq7$Fa_o$!+(pkUk#F]cnFb`n7%Fjdq7$F[_o$!+\"oM*REF]cnFb`n7%F^e q7$Fe^o$!+6S]MEF]cnFb`n7%Fbeq7$F_^o$!+v8\"zi#F]cnFb`n7%Ffeq7$Fi]o$!+\" 3&3?EF]cnFb`n7%Fjeq7$Fc]o$!+l0%4h#F]cnFb`n7%F^fq7$F]]o$!+%3x.g#F]cnFb` n7%Fbfq7$Fg\\o$!+mKF)e#F]cnFb`n7%Fffq7$Fa\\o$!+O5[uDF]cnFb`n7%Fjfq7$F[ \\o$!+,s\")eDF]cnFb`n7%F^gq7$Fe[o$!+s;0TDF]cnFb`n7%Fbgq7$F_[o$!+&z!*3_ #F]cnFb`n7%Ffgq7$Fijn$!+\"y`z\\#F]cnFb`n7%Fjgq7$Fcjn$!+o)R#F]cnFb`n7%Ffiq7$Fign$!+4!\\R7#F]cnFb`n7%Fjiq7$ Fcgn$!+5$3o/#F]cnFb`n7%F^jq7$F]gn$!+@BCn>F]cnFb`n7%Fbjq7$Fgfn$!+W$Hp)= F]cnFb`n7%Ffjq7$Fafn$!+(QIn!=F]cnFb`n7%Fjjq7$F[fn$!+u0(ps\"F]cnFb`n7%F ^[r7$Feen$!+WvpZ;F]cnFb`n7%Fb[r7$F_en$!+/!R)o:F]cnFb`n7%Ff[r7$Fhdn$!+W XG!\\\"F]cnFb`n7%Fj[r7$Fbdn$!+=X#>T\"F]cnFb`n7%F^\\r7$F\\dn$!+CnmL8F]c nFb`n7%Fb\\r7$Ffcn$!+v=Wb7F]cnFb`n7%Ff\\r7$F`cn$!+'Q.s<\"F]cnFb`n7%Fj \\r7$Fhbn$!+JV#*)4\"F]cnFb`n7%F^]r7$$!+\"*f&zI'Fcbn$!+kJf?5F]cnFb`n7%F b]r7$$!+QZ)pu\"Fcbn$!+IGyP')Fa`nFb`n7%Fh]r7$$!+Y/;fOFfan$!+O58oqFa`nFb `n7%F^^r7$$!+8%4E1&F_an$!+o0q(\\&Fa`nFb`n7%Fd^r7$$!+#QeK^$!#;$!+p>)p#R Fa`nFb`n7%Fj^r7$$!+)))*>%*fF0$!+IU>cBFa`nFb`n7%Fa_r7$F($!+L;)R&yFjdnFb `n7%Fg_r7$F($\"+L;)R&yFjdnFb`n-F__n6&Fa_nF($\"#vFd_n$\"\"#Fg_n-%&STYLE G6#%,PATCHNOGRIDG-F$6$7do7$$\"3/+++=qZtwF0$\"3'******H\\J!pKFT7$$\"33+ ++hY8EuF0$\"3/+++w$owE$FT7$$\"3s*****R!>HurF0$\"3;+++s7QjKFT7$$\"30+++ zE%)>pF0$\"35+++ay'fD$FT7$$\"35+++6!f\\m'F0$\"3#)*****z\\'=XKFT7$$\"3+ +++GJB7kF0$\"3$)******QMuIKFT7$$\"3E+++&3-\\;'F0$\"3;+++/^F7KFT7$$\"38 +++x(>s#fF0$\"3!******fuB$*=$FT7$$\"3&)******>[50dF0$\"35+++]XIhJFT7$$ \"3W++++=L2bF0$\"31++++_YFJFT7$$\"3q*****>92!\\aF0$\"3$*******[Kt9JFT7 $$\"3[+++WNc&R&F0$\"3&)*****f6575$FT7$$\"3o*****p8YyM&F0$\"31+++2R&o3$ FT7$$\"3:+++/\"yoI&F0$\"3?+++)QA;2$FT7$$\"3m*****\\71RF&F0$\"3%)****** o`ZbIFT7$$\"3s*****4J?IFT7$$\"3Y+++p)=+C&F0$\"36+++\"Qy7+$FT7$$\"3#)*****>jr!e_F0$\"3#)** ****QTK\")HFT7$$\"3]*****RR!z&H&F0$\"3))*****>fk0'HFT7$$\"3;+++8*pnN&F 0$\"37+++\">@#RHFT7$$\"3x*****zr6YW&F0$\"3\"******42kw\"HFT7$$\"3[**** *p\")3Ac&F0$\"3A+++MTW'*GFT7$$\"3]*****zzw1r&F0$\"3))*****z-#GwGFT7$$ \"3M+++=/G))eF0$\"3++++['yz&GFT7$$\"3g*****>kb+4'F0$\"3&)*****Hk]A%GFT 7$$\"3D+++x'Q'3jF0$\"39+++XScHGFT7$$\"3F+++SG'f`'F0$\"33+++t31?GFT7$$ \"3[++++w&[w'F0$\"3')******3Wg8GFT7$$\"3?+++mt()*)pF0$\"3*********p()) 4GFT7$$\"3_+++iQP2sF0$\"3()*****RyP&3GFT7$$\"3]+++`'o^T(F0$\"3()*****z P%=4GFT7$$\"3d*****\\zM@h(F0$\"39+++&y$\\6GFT7$$\"3!******4FPyz(F0$\"3 @+++[+=:GFT7$$\"3E+++u*pA(zF0$\"35+++cV+?GFT7$$\"3*)******G(yc8)F0$\"3 ')*****pfpd#GFT7$$\"3y*****p\"yW)G)F0$\"3,+++aOJKGFT7$$\"3X*****4XH5V) F0$\"3=+++vf#*F0$\"31+++np75HFT7$$\"3.+++&4m.L*F0$\"32+++`Gy>HFT7$ $\"3[+++GTw%R*F0$\"39+++V[]HHFT7$$\"3o*****\\9=KX*F0$\"3*******\\?p#RH FT7$$\"31+++e;)f]*F0$\"3'******4([0\\HFT7$$\"3Q+++T7H`&*F0$\"3++++RI%) eHFT7$$\"3e******)*zO&f*F0$\"37+++XohoHFT7$$\"35+++O*=Cj*F0$\"3%****** **3h$yHFT7$$\"3m******)>QYm*F0$\"3!)*****4,i!))HFT7$$\"3%)*******>3Ap* F0$\"3A+++(32x*HFT7$$\"37+++X1I:(*F0$\"3!)*****\\)[G2IFT7$$\"3')*****f )yGf(*F0$\"3=+++WP^NIFT7$$\"3Q+++1#3$o(*F0$\"3')*****p`EIR*F0$\"3#******fFz#zJFT7$$\"3-+++xsK]#*F0$\"3'******4Y 'o(>$FT7$$\"3q*****\\e*e*3*F0$\"3z*****f1) >SoKFT7$$\"3w*****z8Q;f(F0$\"3%******H$3*)oKFTF^_n-Fi_n6eo7%7$$\"+=qZt wFa`n$\"+$\\J!pKF]cn7$$\"+hY8EuFa`n$\"+w$owE$F]cn7$F``r$\"$.$Fd_n7%F[h s7$$\"+/>HurFa`n$\"+s7QjKF]cnF`hs7%Fdhs7$$\"+zE%)>pFa`n$\"+ay'fD$F]cnF `hs7%Fjhs7$$\"+6!f\\m'Fa`n$\"+)\\'=XKF]cnF`hs7%F`is7$$\"+GJB7kFa`n$\"+ RMuIKF]cnF`hs7%Ffis7$$\"+&3-\\;'Fa`n$\"+/^F7KF]cnF`hs7%F\\js7$$\"+x(>s #fFa`n$\"+YPK*=$F]cnF`hs7%Fbjs7$$\"+?[50dFa`n$\"+]XIhJF]cnF`hs7%Fhjs7$ $\"++=L2bFa`n$\"++_YFJF]cnF`hs7%F^[t7$$\"+Ur+\\aFa`n$\"+\\Kt9JF]cnF`hs 7%Fd[t7$$\"+WNc&R&Fa`n$\"+;,@,JF]cnF`hs7%Fj[t7$$\"+Ph%yM&Fa`n$\"+2R&o3 $F]cnF`hs7%F`\\t7$$\"+/\"yoI&Fa`n$\"+)QA;2$F]cnF`hs7%Ff\\t7$$\"+Dh!RF& Fa`n$\"+p`ZbIF]cnF`hs7%F\\]t7$$\"+rhX]_Fa`n$\"+^)y$QIF]cnF`hs7%Fb]t7$$ \"+z))RQ_Fa`n$\"+u>J?IF]cnF`hs7%Fh]t7$$\"+p)=+C&Fa`n$\"+\"Qy7+$F]cnF`h s7%F^^t7$$\"+K;2e_Fa`n$\"+RTK\")HF]cnF`hs7%Fd^t7$$\"+%R!z&H&Fa`n$\"+#f k0'HF]cnF`hs7%Fj^t7$$\"+8*pnN&Fa`n$\"+\">@#RHF]cnF`hs7%F`_t7$$\"+=vf#*Fa`n$\"+np75HF]cnF`hs7%F fht7$$\"+&4m.L*Fa`n$\"+`Gy>HF]cnF`hs7%F\\it7$$\"+GTw%R*Fa`n$\"+V[]HHF] cnF`hs7%Fbit7$$\"+X\"=KX*Fa`n$\"+0#p#RHF]cnF`hs7%Fhit7$$\"+e;)f]*Fa`n$ \"+r[0\\HF]cnF`hs7%F^jt7$$\"+T7H`&*Fa`n$\"+RI%)eHF]cnF`hs7%Fdjt7$$\"+* *zO&f*Fa`n$\"+XohoHF]cnF`hs7%Fjjt7$$\"+O*=Cj*Fa`n$\"+!4h$yHF]cnF`hs7%F `[u7$$\"+*>QYm*Fa`n$\"+6?1))HF]cnF`hs7%Ff[u7$$\"++#3Ap*Fa`n$\"+(32x*HF ]cnF`hs7%F\\\\u7$$\"+X1I:(*Fa`n$\"+&)[G2IF]cnF`hs7%Fb\\u7$$\"+')yGf(*F a`n$\"+WP^NIF]cnF`hs7%Fh\\u7$$\"+1#3$o(*Fa`n$\"+xT!G1$F]cnF`hs7%F^]u7$ $\"+UUsX(*Fa`n$\"+f)f*)3$F]cnF`hs7%Fd]u7$$\"+8)eWp*Fa`n$\"+Du\"Q6$F]cn F`hs7%Fj]u7$$\"+6))3<'*Fa`n$\"+koBPJF]cnF`hs7%F`^u7$$\"+#y;f^*Fa`n$\"+ \\Z4fJF]cnF`hs7%Ff^u7$$\"+Kl-$R*Fa`n$\"+w#z#zJF]cnF`hs7%F\\_u7$$\"+xsK ]#*Fa`n$\"+hko(>$F]cnF`hs7%Fb_u7$$\"+&e*e*3*Fa`n$\"+mq@9KF]cnF`hs7%Fh_ u7$$\"+kbZ7*)Fa`n$\"+uRxGKF]cnF`hs7%F^`u7$$\"+'=l0s)Fa`n$\"+0)f7C$F]cn F`hs7%Fd`u7$$\"+X.Q:&)Fa`n$\"+(fu:D$F]cnF`hs7%Fj`u7$$\"+5\"3%)H)Fa`n$ \"+(R8'fKF]cnF`hs7%F`au7$$\"+(oC62)Fa`n$\"+SoKF]cnF`hs7%F\\bu7$$\"+Q\"Q;f(Fa`n$\"+L3*)oKF]cnF`hsF^` rFd`r-F$6$7do7$F\\ar$!3'******H\\J!pKFT7$$\"37+++1\"4Y\"zF0$!3%******4 6VwE$FT7$$\"3=+++n+%z9)F0$!3%*******fHljKFT7$$\"3#)*****p#3&>P)F0$!31+ ++YM>dKFT7$$\"3!)*****z(z:&e)F0$!35+++3RQ[KFT7$$\"3_+++oW3'y)F0$!3)*** ***H2MtB$FT7$$\"3'******\\@MK(*)F0$!3#)*****4&p9CKFT7$$\"35+++d&o]9*F0 $!3,+++&Q@*3KFT7$$\"3/++++H)**H*F0$!31+++4Wv\">$FT7$$\"3')*****>1#GO%* F0$!3)******HdVF<$FT7$$\"37+++bG:_&*F0$!31+++)Q*)>:$FT7$$\"3=+++4AjX'* F0$!3\")*****p2)fHJFT7$$\"3^*****zzoXr*F0$!3')*****\\%[o0JFT7$$\"3))** ***p[vlv*F0$!3#)*****R-y.3$FT7$$\"39+++!)*o*o(*F0$!3!)*****RaCQ0$FT7$$ \"3%)******4;p[(*F0$!3<+++4v>EIFT7$Fjas$!3A+++(32x*HFT7$F[as$!37+++Xoh oHFT7$Ff`s$!3++++RI%)eHFT7$Fa`s$!3'******4([0\\HFT7$F\\`s$!3*******\\? p#RHFT7$Fg_s$!39+++V[]HHFT7$Fb_s$!32+++`Gy>HFT7$F]_s$!31+++np75HFT7$Fh ^s$!30+++uSc+HFT7$Fc^s$!3#)*****z![7\"*GFT7$F^^s$!3))*****HHW=)GFT7$Fi ]s$!3=+++^IwsGFT7$Fd]s$!37+++Y\"GR'GFT7$F_]s$!3))*****fh%RbGFT7$Fj\\s$ !3!)******zsAZGFT7$Fe\\s$!3=+++@#R HFT7$F[gr$!3))*****>fk0'HFT7$Fffr$!3#)******QTK\")HFT7$Fafr$!36+++\"Qy 7+$FT7$F\\fr$!3=+++u>J?IFT7$Fger$!30+++^)y$QIFT7$Fber$!3%)******o`ZbIF T7$F]er$!3?+++)QA;2$FT7$Fhdr$!31+++2R&o3$FT7$Fcdr$!3&)*****f6575$FT7$F ^dr$!3$*******[Kt9JFT7$Ficr$!31++++_YFJFT7$$\"3#)*****4HM)pbF0$!3#)*** **H@Y%RJFT7$$\"3u*****H%[#fj&F0$!3)******p@92:$FT7$Fdcr$!35+++]XIhJFT7 $$\"3=+++H#\\px&F0$!3;+++/*\\7<$FT7$$\"33+++6R00gF0$!3)******\\P0v>$FT 7$$\"33+++%>5lC'F0$!30+++lW!*=KFT7$$\"3_+++M?0'\\'F0$!3++++?a)fB$FT7$$ \"3%******R9>)\\nF0$!3))*******Qr\"\\KFT7$$\"3\\+++P;#[+(F0$!3'******R M+)eKFT7$$\"39+++(GF'esF0$!37+++xp9lKFT7$$\"3W+++j'p\"4vF0$!3?+++!eQ%o KFT7$$\"3]*****p!\\iaxF0$!3/+++P)Fa`n$!+YM>dKF]cnFadv7%F[ev7$$\"+yz: &e)Fa`n$!+3RQ[KF]cnFadv7%Faev7$$\"+oW3'y)Fa`n$!+tSLPKF]cnFadv7%Fgev7$$ \"+:UBt*)Fa`n$!+^p9CKF]cnFadv7%F]fv7$$\"+d&o]9*Fa`n$!+&Q@*3KF]cnFadv7% Fcfv7$$\"++H)**H*Fa`n$!+4Wv\">$F]cnFadv7%Fifv7$$\"+i?GO%*Fa`n$!+tNusJF ]cnFadv7%F_gv7$$\"+bG:_&*Fa`n$!+)Q*)>:$F]cnFadv7%Fegv7$$\"+4AjX'*Fa`n$ !+x!)fHJF]cnFadv7%F[hv7$$\"+)zoXr*Fa`n$!+X[o0JF]cnFadv7%Fahv7$$\"+([vl v*Fa`n$!+C!y.3$F]cnFadv7%Fghv7$$\"+!)*o*o(*Fa`n$!+WX#Q0$F]cnFadv7%F]iv 7$$\"+5;p[(*Fa`n$!+4v>EIF]cnFadv7%Fciv7$F]\\u$!+(32x*HF]cnFadv7%Fiiv7$ F[[u$!+XohoHF]cnFadv7%F]jv7$Fejt$!+RI%)eHF]cnFadv7%Fajv7$F_jt$!+r[0\\H F]cnFadv7%Fejv7$Fiit$!+0#p#RHF]cnFadv7%Fijv7$Fcit$!+V[]HHF]cnFadv7%F][ w7$F]it$!+`Gy>HF]cnFadv7%Fa[w7$Fght$!+np75HF]cnFadv7%Fe[w7$Faht$!+uSc+ HF]cnFadv7%Fi[w7$F[ht$!+3[7\"*GF]cnFadv7%F]\\w7$Fegt$!+$HW=)GF]cnFadv7 %Fa\\w7$F_gt$!+^IwsGF]cnFadv7%Fe\\w7$Fift$!+Y\"GR'GF]cnFadv7%Fi\\w7$Fc ft$!+;YRbGF]cnFadv7%F]]w7$F]ft$!+!GFs%GF]cnFadv7%Fa]w7$Fget$!+@#RHF]cnFadv 7%Fiaw7$F[_t$!+#fk0'HF]cnFadv7%F]bw7$Fe^t$!+RTK\")HF]cnFadv7%Fabw7$F_^ t$!+\"Qy7+$F]cnFadv7%Febw7$Fi]t$!+u>J?IF]cnFadv7%Fibw7$Fc]t$!+^)y$QIF] cnFadv7%F]cw7$F]]t$!+p`ZbIF]cnFadv7%Facw7$Fg\\t$!+)QA;2$F]cnFadv7%Fecw 7$Fa\\t$!+2R&o3$F]cnFadv7%Ficw7$F[\\t$!+;,@,JF]cnFadv7%F]dw7$Fe[t$!+\\ Kt9JF]cnFadv7%Fadw7$F_[t$!++_YFJF]cnFadv7%Fedw7$$\"+\"HM)pbFa`n$!+8iWR JF]cnFadv7%Fidw7$$\"+V[#fj&Fa`n$!+$F]cnFadv7%F_fw7$$\"+%>5lC'Fa`n$!+lW!*=KF]cnFadv7%Fefw7$$\"+M?0' \\'Fa`n$!+?a)fB$F]cnFadv7%F[gw7$$\"+W\">)\\nFa`n$!+!Rr\"\\KF]cnFadv7%F agw7$$\"+P;#[+(Fa`n$!+W.!)eKF]cnFadv7%Fggw7$$\"+(GF'esFa`n$!+xp9lKF]cn Fadv7%F]hw7$$\"+j'p\"4vFa`n$!+!eQ%oKF]cnFadv7%Fchw7$$\"+2\\iaxFa`n$!+< m')oKF]cnFadvF^`rFd`r-F$6%7$7$$!3A++++++!\\$FTF(7$$\"3/++++++!H\"FTF(- %'COLOURG6&Fa_nF)F)F)-%*LINESTYLEG6#\"\"$-F$6%7$7$F($!3')************* e$FT7$F($\"3')*************e$FTFgiwFjiw-%%FONTG6$%*HELVETICAG\"\"*-%+A XESLABELSG6%%&Re(z)G%&Im(z)G-Fhjw6#%(DEFAULTG-%*AXESSTYLEG6#%$BOXG-%(S CALINGG6#%,CONSTRAINEDG-%%VIEWG6$;$!$\\$Fd_n$\"$H\"Fd_n;$!$f$Fd_n$\"$f $Fd_n" 1 2 0 1 10 0 2 9 1 2 1 1.000000 45.000000 45.000000 0 0 "Curve \+ 1" "Curve 2" "Curve 3" "Curve 4" "Curve 5" "Curve 6" "Curve 7" "Curve \+ 8" }}}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 4 "The " }{TEXT 260 30 "interval of absolute stability" }{TEXT -1 89 " (or \+ stability interval) is the intersection of the stability region with t he real line." }}{PARA 0 "" 0 "" {TEXT -1 59 "For this scheme the stab ility interval is (approximately) " }{XPPEDIT 18 0 "[-2.8561, 0];" "6 #7$,$-%&FloatG6$\"&h&G!\"%!\"\"\"\"!" }{TEXT -1 2 ". " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 148 "We can distort the bo undary curve horizontally by taking the 11th root of the real part of \+ points along the curve. In this way we see that there is " }{TEXT 260 19 "no largest interval" }{TEXT -1 99 " on the nonnegative imaginary a xis that contains the origin and lies inside the stability region. " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 322 "R := z -> 1+z+1/2*z^2+1/6*z^3+1/24*z^4+1/120*z^5+1/720*z^6-1/21 60*z^7:\nDigits := 25:\npts := []: z0 := 0:\nfor ct from 0 to 80 do\n \+ zz := newton(R(z)=exp(ct*Pi/100*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,.15),thickness=2,font=[HELVETICA,9]);\nDigits := 10:" }}{PARA 13 "" 1 "" {GLPLOT2D 220 305 305 {PLOTDATA 2 "6(-%'CURVESG6#7]p7$$\"\" !F)F(7$$!:\\*F-$\":qaqYv$)=k&zxC%*F-7$$ !:o&oGgO#=pY!)H8\"!#D$\":%oW5'=A(=eqjc7F=7$$!:6$z?*RL!fo#QCL\"F=$\":J# GZD#Ro8J'zq:F=7$$!:F5DgK@#)y1G6_\"F=$\":YR^=(z)fr`b\\)=F=7$$!:TkFbw`H' RTF,#F=7$$!:%f5fU)3z![1Qu=F=$\":M)z'G=cUirtK^#F =7$$!:Sz'R1(\\(pL\\_T?F=$\":0(Hzr=%H#oCVFGF=7$$!:u(*Q$oA3ps=^.AF=$\":' )>I(>]>LZ2fTJF=7$$!:R$y6Kn[w0D'4O#F=$\":z!3A?s>?T#[dX$F=7$$!:\"fN'Rc[G TPoV^#F=$\":x=91)G^<'\\/*pPF=7$$!:eu?%H,+\"\\eETm#F=$\":Mz(3?JfIy)eS3% F=7$$!:S'HVV:Sf%Rj0\"GF=$\":6kN1jKeBr%F=7$$!:-P$R!QKr?Q@%zO$F=$\":1[[$pu.x!GiZl&F=7 $$!:%=]^!e(*y*)G76]$F=$\":\"3#Q`*=]3dj()ofF=7$$!:t4OFPxax9&4KOF=$\":&= bx&yn!)*)GvHG'F=7$$!:Aa)y\")G*GZw+5w$F=$\":(z4&34CjGTbqf'F=7$$!:,f4*)y 'y/\"zDz)QF=$\":WK8aPYVJa76\"pF=7$$!:Yfo`6%fg@a&H,%F=$\":kJ;@%oNkw>9Ds F=7$$!:Bph],+N.,lh8%F=$\":`WiUI5ym^Q\"RvF=7$$!:Fox8\"[Y\")>9idUF=$\":s UvM*4:K?l4`yF=7$$!:Z^_/2Gnf;%QxVF=$\":4\"\\CCp'GB(*4q;)F=7$$!:3)QW&**o *>/k]&\\%F=$\":Ia$fQSd,*es3[)F=7$$!:P9IWvz2FzN?h%F=$\":CKMmJ\"G7bzn%z) F=7$$!:NN^_%4vkg_,FZF=$\":'G5)3;sbip>%3\"*F=7$$!:R<>)pv$HFq$[S[F=$\":^ `S3v5M*)o\"4A%*F=7$$!:M&G3B#G%*phwC&\\F=$\":nb\"*3]Q`&4$)oN(*F=7$$!:o! =bcU/r)pEI1&F=$\":8C6?]>qAZ?\\+\"!#C7$$!:%)GXiqY[-Sk@<&F=$\":zCyU2'p#3 sji.\"Ffu7$$!:yU:t.roUY=*z_F=$\":\\W'\\%y9lDN)fn5Ffu7$$!:\"3VCus]ZNkJ' Q&F=$\":h(z$zZ(4@JV#*)4\"Ffu7$$!:I]SNH?rW:&Q\"\\&F=$\":j3NGI.-6&=CI6Ff u7$$!:0,%\\F3J%eB^^f&F=$\":ZoWz]lJCQ^:;\"Ffu7$$!:U$oY=&fY9`Twp&F=$\":! )f>'pD_>=P&G>\"Ffu7$$!:9%f)=#)[m*yN)))z&F=$\"::u;hG@j>-]TA\"Ffu7$$!:EM qj\\&e5Ng!*)*eF=$\":=H\")\\4LOa(=Wb7Ffu7$$!:E&=zh?5A#4Rx*fF=$\":)4x\"H xA]XKJnG\"Ffu7$$!:.%[zqzZzw[T&4'F=$\":DKk9n6zE#4-=8Ffu7$$!:pXurcVsx#y' >>'F=$\":0oh*>PEApPJ\\8Ffu7$$!:7d4&)Q](Q_]V(G'F=$\":AG\\-C3B:a81Q\"Ffu 7$$!:SC^qeWhllc=Q'F=$\":p;!3SnkhT\"Ffu7$$!:&p&*=(3BAh/w_Z'F=$\":=P TMS*Qs'e^KW\"Ffu7$$!:!pzM:8zu$>Sxc'F=$\":?*oK29pdV-gu9Ffu7$$!:Y_kCtCOW ')*HfmF=$\":$R!enU.g:cwf]\"Ffu7$$!:xnm_Rcu!*y4+v'F=$\":8AD&3A&Q\"QrQP: Ffu7$$!:$eV9x+R'3xG*RoF=$\":]J\"p+jO)R+R)o:Ffu7$$!:(>cl(H[7Bq>\"HpF=$ \":%RD2n3\"RW[R.g\"Ffu7$$!:J:%=\\]_Vg%\\w,(F=$\":u0`ysa#G)*f*=j\"Ffu7$ $!:P!fMRICiL')e0rF=$\":&H#HAhD()Qx:Nm\"Ffu7$$!:11[$**)*R195,$>(F=$\":n E^$pVt\\UHX(F=$\":2Hy\"p! *=\\c!R2z\"Ffu7$$!:_/bi&otj#pf!RvF=$\"::G8.S,TV7SF#=Ffu7$$!:Tdl&ecf*=, G]i(F=$\":e0\"4[&=Ffu7$$!:(funN\"4&e(*)**3r(F=$\":e#ekR8z\"QMHp )=Ffu7$$!:=>]FRDhcw2nz(F=$\":dl1XF[;6rt!>>Ffu7$$!:rYd(3c,RkZX#)yF=$\": ywvrvi!zx;?^>Ffu7$$!:fm&*Q%Q3U>M5ozF=$\":H2v%zUT=:eD$)>Ffu7$$!:bry_d&= YDEc`!)F=$\":7!)*\\!*eHo_$e^,#Ffu7$$!:yJB`ShN3Iu'Q\")F=$\":9)*H:qv@-J3 o/#Ffu7$$!:$)HJ5p\\jZA-KA)F=$\":f=Rg0ZA>=!3y?Ffu7$$!:*=+**>gS5fo#oI)F= $\":tt2&pbdA'RD)3@Ffu7$$!:)4#35bP%*Qk]\"*Q)F=$\":cam7\"Qj-Dx()Q@Ffu7$$ !:G%\\6x)pmQO<(p%)F=$\":R!z#pBS'>yJ1o@Ffu7$$!:(z9)>Dl&f'*f/[&)F=$\":n( fdb)4*>FO@'>#Ffu7$$!:kNg)o5e<*[uOi)F=$\":`GiP^kXH$==BAFfu7$$!:h()zSz.I `P/ip)F=$\":@w\\a!pDCJX&)[AFfu7$$!:*o\\*R@98G%eLl()F=$\":vHQ%*[_L6^gJF #Ffu7$$!:0QG))e'ygdf)3$))F=$\":sZC(['f))4AsgH#Ffu7$$!:L_)*=L8vD " 0 "" {MPLTEXT 1 0 1 ";" }}}}{PARA 0 "" 0 "" {TEXT -1 42 " #-----------------------------------------" }}{EXCHG {PARA 0 "> " 0 " " {MPLTEXT 1 0 1 ";" }}}}{PARA 0 "" 0 "" {TEXT -1 42 "#=============== ==========================" }}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 13 "sch eme with " }{XPPEDIT 18 0 "c[5]=c[6]" "6#/&%\"cG6#\"\"&&F%6#\"\"'" } {XPPEDIT 18 0 "``=3/4" "6#/%!G*&\"\"$\"\"\"\"\"%!\"\"" }{TEXT -1 7 " \+ and " }{XPPEDIT 18 0 "b[5]=b[6]" "6#/&%\"bG6#\"\"&&F%6#\"\"'" }{TEXT -1 1 " " }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}{PARA 0 "" 0 "" {TEXT -1 39 "#--------------------------------------" }}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 19 "checking the scheme" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 26 "coefficients of the scheme" }}{PARA 0 " " 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 516 "ee : = \{c[2]=3/17,\nc[3]=2/9,\nc[4]=3/7,\nc[5]=3/4,\nc[6]=3/4,\nc[7]=1,\n \na[2,1]=3/17,\na[3,1]=20/243,\na[3,2]=34/243,\na[4,1]=3/28,\na[4,2]=- 153/343,\na[4,3]=1053/1372,\na[5,1]=51/176,\na[5,2]=153/704,\na[5,3]=- 1539/2288,\na[5,4]=8379/9152,\na[6,1]=-219/1408,\na[6,2]=153/704,\na[6 ,3]=11745/18304,\na[6,4]=-931/4576,\na[6,5]=1/4,\na[7,1]=229/4644,\na[ 7,2]=-17/43,\na[7,3]=40887/42484,\na[7,4]=-9604/45279,\na[7,5]=0,\na[7 ,6]=39424/66177,\n\nb[1]=79/1080,\nb[2]=0,\nb[3]=19683/69160,\nb[4]=16 807/84240,\nb[5]=1408/7695,\nb[6]=1408/7695,\nb[7]=43/560\}:" }}} {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 0 109 "subs(ee,matrix([seq([c[i],seq(a[i,j],j=1..i-1),``$(8 -i)],i=2..7),\n[``,seq(b[i],i=1..7)]]));\n``;\nevalf[8](%%);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#K%'matrixG6#7)7*#\"\"$\"##\"%39FF#\"&X<\"\"&/$=#!$J*\"%wX#\"\"\"FBF+F+7*FZ#\"$ H#\"%WY#!#<\"#V#\"&()3%\"&%[U#!%/'*\"&z_%\"\"!#\"&C%R\"&xh'F+7*F+#\"#z \"%!3\"Fbo#\"&$o>\"&g\"p#\"&2o\"\"&SU)#FR\"%&p(F`p#F[o\"$g&Q)pprint146 \"" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%!G" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#K%'matrixG6#7)7*$\")fqk$!)xRb:F*FB$\")-j;kF*$!)!GX.#F*$\")+++DF*F+F+7*$\"\"\" \"\"!$\")R4J\\F1$!)%)[`RF*$\")Q4C'*F*$!);2@@F*$FTFT$\")oNdfF*F+7*F+$\" )[\"[J(F1Fgn$\")$4g%GF*$\")I8&*>F*$\")'f(H=F*Fao$\")9dywF1Q)pprint156 \"" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 7 "Che ck: " }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 148 "RK6_7eqs := [op(Row SumConditions(7,'expanded')),op(OrderConditions(6,7,'expanded'))]:\nsi mplify(subs(ee,RK6_7eqs)):\nmap(u->lhs(u)-rhs(u),%);\nnops(%);" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#7M\"\"!F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F $F$F$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#\"#V" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 " " 0 "" {TEXT -1 86 "Next we set-up stage-order condtions to check for \+ stage-orders from 2 to 4 inclusive. " }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 85 "for ct from 2 to 4 do\n so||ct||_7 := StageOrderCon ditions(ct,7,'expanded');\nend do:" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" } }{PARA 0 "" 0 "" {TEXT -1 58 "Stages 3 to 7 have the following respect ive stage-orders. " }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 173 "[seq( [seq(expand(subs(ee,so||i||_7[j])),i=2..4)],j=1..5)]:\nmap(proc(L) loc al 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$" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 42 "Thus stages 5, 6 and 7 have stage-order 3." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 32 "#------------------------ -------" }}{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 .. 7) = b[j]*(1-c[j]);" "6#/-%$SumG6$*&&%\"bG6#%\"iG\"\"\"&%\"aG6$ F+%\"jGF,/F+;,&F0F,F,F,\"\"(*&&F)6#F0F,,&F,F,&%\"cG6#F0!\"\"F," } {TEXT -1 3 ", " }{XPPEDIT 18 0 "j=1" "6#/%\"jG\"\"\"" }{TEXT -1 7 " . . 7 " }}{PARA 0 "" 0 "" {TEXT -1 8 "(where " }{XPPEDIT 18 0 "c[1]=0 " "6#/&%\"cG6#\"\"\"\"\"!" }{TEXT -1 17 " ) are satisfied." }}{PARA 256 "" 0 "" {TEXT -1 1 " " }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 162 "[Sum(b[i]*a[i,1],i=2..7)=b[1],seq(Sum(b[i]*a[i,j],i=j+1..7)=b[j]* (1-c[j]),j=2..6)];\neval(subs(Sum=add,%)):\nsubs(ee,%):\nmap(u->`if`(l hs(u)=rhs(u),0,1),%);\nnops(%);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#7(/ -%$SumG6$*&&%\"bG6#%\"iG\"\"\"&%\"aG6$F,F-F-/F,;\"\"#\"\"(&F*6#F-/-F&6 $*&F)F-&F/6$F,F3F-/F,;\"\"$F4*&&F*6#F3F-,&F-F-&%\"cGFB!\"\"F-/-F&6$*&F )F-&F/6$F,F?F-/F,;\"\"%F4*&&F*6#F?F-,&F-F-&FEFRFFF-/-F&6$*&F)F-&F/6$F, FOF-/F,;\"\"&F4*&&F*6#FOF-,&F-F-&FEFjnFFF-/-F&6$*&F)F-&F/6$F,FgnF-/F,; \"\"'F4*&&F*6#FgnF-,&F-F-&FEFhoFFF-/-F&6$*&F)F-&F/6$F,FeoF-/F,;F4F4*&& F*6#FeoF-,&F-F-&FEFepFFF-" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#7(\"\"!F$ F$F$F$F$" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#\"\"'" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 27 "The simplifying condition : " }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "Sum(b[i]*a[i,2] ,i = 3 .. 7) = 0;" "6#/-%$SumG6$*&&%\"bG6#%\"iG\"\"\"&%\"aG6$F+\"\"#F, /F+;\"\"$\"\"(\"\"!" }{TEXT -1 1 " " }}{PARA 0 "" 0 "" {TEXT -1 13 "is satisfied." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 " " {MPLTEXT 1 0 59 "Sum(b[i]*a[i,2],i=3..7);\neval(subs(Sum=add,%));\ns ubs(ee,%);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#-%$SumG6$*&&%\"bG6#%\"iG \"\"\"&%\"aG6$F*\"\"#F+/F*;\"\"$\"\"(" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#,,*&&%\"bG6#\"\"$\"\"\"&%\"aG6$F(\"\"#F)F)*&&F&6#\"\"%F)&F+6$F1F-F) F)*&&F&6#\"\"&F)&F+6$F7F-F)F)*&&F&6#\"\"'F)&F+6$F=F-F)F)*&&F&6#\"\"(F) &F+6$FCF-F)F)" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#\"\"!" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 27 "The simplifying cond ition: " }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "Sum(b[i]*c [i]*a[i,2],i = 3 .. 7) = 0;" "6#/-%$SumG6$*(&%\"bG6#%\"iG\"\"\"&%\"cG6 #F+F,&%\"aG6$F+\"\"#F,/F+;\"\"$\"\"(\"\"!" }{TEXT -1 1 " " }}{PARA 0 " " 0 "" {TEXT -1 13 "is satisfied." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 64 "Sum(b[i]*c[i]*a[i,2],i=3..7) ;\neval(subs(Sum=add,%));\nsubs(ee,%);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#-%$SumG6$*(&%\"bG6#%\"iG\"\"\"&%\"cGF)F+&%\"aG6$F*\"\"#F+/F*;\" \"$\"\"(" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#,,*(&%\"bG6#\"\"$\"\"\"&% \"cGF'F)&%\"aG6$F(\"\"#F)F)*(&F&6#\"\"%F)&F+F2F)&F-6$F3F/F)F)*(&F&6#\" \"&F)&F+F9F)&F-6$F:F/F)F)*(&F&6#\"\"'F)&F+F@F)&F-6$FAF/F)F)*(&F&6#\"\" (F)&F+FGF)&F-6$FHF/F)F)" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#\"\"!" }}} {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 27 "The simpl ifying condition: " }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "Sum(b[i]*c[i]^2*a[i,2],i = 3 .. 7) = 0;" "6#/-%$SumG6$*(&%\"bG6#%\"iG \"\"\"*$&%\"cG6#F+\"\"#F,&%\"aG6$F+F1F,/F+;\"\"$\"\"(\"\"!" }{TEXT -1 1 " " }}{PARA 0 "" 0 "" {TEXT -1 13 "is satisfied." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 66 "Sum(b[i]*c[i ]^2*a[i,2],i=3..7);\neval(subs(Sum=add,%));\nsubs(ee,%);" }}{PARA 11 " " 1 "" {XPPMATH 20 "6#-%$SumG6$*(&%\"bG6#%\"iG\"\"\")&%\"cGF)\"\"#F+&% \"aG6$F*F/F+/F*;\"\"$\"\"(" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#,,*(&%\" bG6#\"\"$\"\"\")&%\"cGF'\"\"#F)&%\"aG6$F(F-F)F)*(&F&6#\"\"%F))&F,F3F-F )&F/6$F4F-F)F)*(&F&6#\"\"&F))&F,F;F-F)&F/6$F " 0 "" {MPLTEXT 1 0 85 "Sum(b[i]*c[i]*Sum(a[i,j]*a[j,2],j=3..i-1),i=3..7);\ne val(subs(Sum=add,%));\nsubs(ee,%);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6# -%$SumG6$*(&%\"bG6#%\"iG\"\"\"&%\"cGF)F+-F$6$*&&%\"aG6$F*%\"jGF+&F26$F 4\"\"#F+/F4;\"\"$,&F*F+F+!\"\"F+/F*;F:\"\"(" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#,***&%\"bG6#\"\"%\"\"\"&%\"cGF'F)&%\"aG6$F(\"\"$F)&F-6$ F/\"\"#F)F)*(&F&6#\"\"&F)&F+F5F),&*&&F-6$F6F/F)F0F)F)*&&F-6$F6F(F)&F-6 $F(F2F)F)F)F)*(&F&6#\"\"'F)&F+FCF),(*&&F-6$FDF/F)F0F)F)*&&F-6$FDF(F)F? F)F)*&&F-6$FDF6F)&F-6$F6F2F)F)F)F)*(&F&6#\"\"(F)&F+FTF),**&&F-6$FUF/F) F0F)F)*&&F-6$FUF(F)F?F)F)*&&F-6$FUF6F)FPF)F)*&&F-6$FUFDF)&F-6$FDF2F)F) F)F)" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#\"\"!" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 114 "We can calculate the pri ncipal error norm of the order 6 scheme, that is, the 2-norm of the pr incipal error terms." }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 158 "err terms6_7 := PrincipalErrorTerms(6,7,'expanded'):\nsm := 0:\nfor ct to \+ nops(errterms6_7) do\n sm := sm+(evalf(subs(ee,errterms6_7[ct])))^2; \nend do:\nsqrt(sm);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#$\"+(f6f;#!#8 " }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}}{PARA 0 "" 0 "" {TEXT -1 39 "#----------------- ---------------------" }}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 26 "construc tion of the scheme" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 192 "We set up a system of equations using a selection of \"s imple\" order conditions and incorporate the row-sum conditions togeth er with the stage-order equations to ensure that stages 3 to 7 have " }{TEXT 260 13 "stage-order 2" }{TEXT -1 1 "." }}{PARA 0 "" 0 "" {TEXT -1 48 "We also incorporate the simplifying conditions: " }}{PARA 256 " " 0 "" {TEXT -1 2 " " }{XPPEDIT 18 0 "Sum(b[i]*a[i,j],i = j+1 .. 7) = b[j]*(1-c[j]);" "6#/-%$SumG6$*&&%\"bG6#%\"iG\"\"\"&%\"aG6$F+%\"jGF,/F +;,&F0F,F,F,\"\"(*&&F)6#F0F,,&F,F,&%\"cG6#F0!\"\"F," }{TEXT -1 1 " " } }{PARA 0 "" 0 "" {TEXT -1 5 "for " }{XPPEDIT 18 0 "j = 3;" "6#/%\"jG \"\"$" }{TEXT -1 62 ", 4, 5, 6, together with the further simplifying conditions: " }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "Sum( b[i]*c[i]*a[i,2],i = 3 .. 7) = 0;" "6#/-%$SumG6$*(&%\"bG6#%\"iG\"\"\"& %\"cG6#F+F,&%\"aG6$F+\"\"#F,/F+;\"\"$\"\"(\"\"!" }{TEXT -1 8 ", \+ " }{XPPEDIT 18 0 "Sum(b[i]*c[i]*Sum(a[i,j]*a[j,2],j = 3 .. i-1),i = 3 \+ .. 7) = 0;" "6#/-%$SumG6$*(&%\"bG6#%\"iG\"\"\"&%\"cG6#F+F,-F%6$*&&%\"a G6$F+%\"jGF,&F46$F6\"\"#F,/F6;\"\"$,&F+F,F,!\"\"F,/F+;F<\"\"(\"\"!" } {TEXT -1 8 ", " }{XPPEDIT 18 0 "Sum(b[i]*c[i]^2*a[i,2],i = 3 .. \+ 7) = 0;" "6#/-%$SumG6$*(&%\"bG6#%\"iG\"\"\"*$&%\"cG6#F+\"\"#F,&%\"aG6$ F+F1F,/F+;\"\"$\"\"(\"\"!" }{TEXT -1 2 ". " }}{PARA 0 "" 0 "" {TEXT -1 1 " " }}{PARA 0 "" 0 "" {TEXT -1 76 "The simple order conditions us ed are given (in abreviated form) as follows. " }}{PARA 0 "" 0 "" {TEXT 260 4 "Note" }{TEXT -1 86 ": We use one additional order conditi on #28 beyond those used for the previous scheme." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 184 "SO6 := Simp leOrderConditions(6):\n[seq([i,SO6[i]],i=[1,2,4,8,13,16,24,28,29,32])] :\nlinalg[augment](linalg[delcols](%,2..2),matrix([[` `]$(linalg[rowd im](%))]),linalg[delcols](%,1..1));" }}{PARA 11 "" 1 "" {XPPMATH 20 "6 #K%'matrixG6#7,7%\"\"\"%#~~G/*&%\"bGF(%\"eGF(F(7%\"\"#F)/*&F,F(%\"cGF( #F(F/7%\"\"%F)/*&F,F()F2F/F(#F(\"\"$7%\"\")F)/*&F,F()F2F:F(#F(F57%\"#8 F)/*(F,F(F2F(-%!G6#*&F8F(%\"aGF(F(#F(\"#:7%\"#;F)/*&F,F()F2F5F(#F(\"\" &7%\"#CF)/*(F,F(F2F(-FF6#*&FIF(FEF(F(#F(\"#s7%\"#GF)/*(F,F(F8F(FEF(#F( \"#=7%\"#HF)/*(F,F(F2F(-FF6#*&F?F(FIF(F(#F(FT7%\"#KF)/*&F,F()F2FRF(#F( \"\"'Q(pprint06\"" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 " > " 0 "" {MPLTEXT 1 0 462 "SO6_7 := SimpleOrderConditions(6,7,'expande d'):\nord_cdns := [seq(SO6_7[i],i=[1,2,4,8,13,16,24,28,29,32])]:\nSO_e qs := [op(RowSumConditions(7,'expanded')),op(StageOrderConditions(2,7, 'expanded'))]:\nsimp_eqs := [seq(add(b[i]*a[i,j],i=j+1..7)=b[j]*(1-c[j ]),j=[3,4,5,6]),\n add(b[i]*c[i]*a[i,2],i=3..7)=0,add(b[i ]*c[i]*add(a[i,j]*a[j,2],j=3..i-1),i=3..7)=0,\n add(b[i] *c[i]^2*a[i,2],i=3..7)=0]:\ncdns := [op(ord_cdns),op(simp_eqs),op(SO_e qs)]:" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 21 "We specify the nodes " }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "c[2] = 3/17;" "6#/&%\"cG6#\"\"#*&\"\"$\"\"\"\"# " 0 "" {MPLTEXT 1 0 125 "e1 := \{c[2]=3/17,c[3]=2/9,c[7]=1,b[2]=0\}:\neqns := subs(e1, [op(cdns),c[5]=c[6],b[5]=b[6]]):\nnops(eqns);\nindets(eqns);\nnops(%); " }}{PARA 11 "" 1 "" {XPPMATH 20 "6#\"#I" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#<@&%\"aG6$\"\"%\"\"#&F%6$F'\"\"$&%\"cG6#\"\"&&F-6#\"\"'&F%6$F(\" \"\"&F%6$F+F5&F%6$F/F(&F%6$F/F+&F%6$F+F(&F%6$F'F5&F-6#F'&F%6$F/F5&%\"b GF.&FEF1&FE6#\"\"(&FE6#F5&FE6#F+&FEFA&F%6$FIF/&F%6$FIF2&F%6$FIF+&F%6$F IF'&F%6$FIF5&F%6$FIF(&F%6$F2F'&F%6$F2F/&F%6$F2F(&F%6$F2F+&F%6$F/F'&F%6 $F2F5" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#\"#I" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 51 "There are 30 equations an d 30 unknown coefficients." }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 22 "infolevel[solve] := 4:" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 69 "e2 := solve(\{op(eqns)\}):\ninfolevel[solve] := 0:\ne3 := `union`( e1,e2):" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 2 "e3" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 582 "e 3 := \{c[7] = 1, a[5,3] = -1539/2288, a[7,2] = -17/43, a[3,1] = 20/243 , a[6,3] = 11745/18304, a[4,1] = 3/28, a[7,1] = 229/4644, b[2] = 0, a[ 7,5] = 0, a[2,1] = 3/17, a[4,2] = -153/343, a[4,3] = 1053/1372, a[6,2] = 153/704, a[3,2] = 34/243, a[5,2] = 153/704, b[7] = 43/560, b[4] = 1 6807/84240, a[6,1] = -219/1408, b[1] = 79/1080, a[6,5] = 1/4, a[5,1] = 51/176, a[7,6] = 39424/66177, a[7,4] = -9604/45279, b[3] = 19683/6916 0, b[5] = 1408/7695, a[5,4] = 8379/9152, a[7,3] = 40887/42484, c[3] = \+ 2/9, c[4] = 3/7, c[5] = 3/4, c[6] = 3/4, b[6] = 1408/7695, c[2] = 3/17 , a[6,4] = -931/4576\}:" }{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 0 109 "subs(e3,mat rix([seq([c[i],seq(a[i,j],j=1..i-1),``$(8-i)],i=2..7),\n[``,seq(b[i],i =1..7)]]));\n``;\nevalf[8](%%);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#K%' matrixG6#7)7*#\"\"$\"##\"%39FF#\"&X <\"\"&/$=#!$J*\"%wX#\"\"\"FBF+F+7*FZ#\"$H#\"%WY#!#<\"#V#\"&()3%\"&%[U# !%/'*\"&z_%\"\"!#\"&C%R\"&xh'F+7*F+#\"#z\"%!3\"Fbo#\"&$o>\"&g\"p#\"&2o \"\"&SU)#FR\"%&p(F`p#F[o\"$g&Q)pprint116\"" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%!G" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#K%'matrixG6#7)7* $\")fqk$!)xRb:F*FB$\")-j ;kF*$!)!GX.#F*$\")+++DF*F+F+7*$\"\"\"\"\"!$\")R4J\\F1$!)%)[`RF*$\")Q4C '*F*$!);2@@F*$FTFT$\")oNdfF*F+7*F+$\")[\"[J(F1Fgn$\")$4g%GF*$\")I8&*>F *$\")'f(H=F*Fao$\")9dywF1Q)pprint126\"" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 85 "RK6_7eqs := [op(RowSumC onditions(7,'expanded')),op(OrderConditions(6,7,'expanded'))]:" }}} {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 7 "Check: " } }{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 62 "simplify(subs(e3,RK6_7eqs)) :\nmap(u->lhs(u)-rhs(u),%);\nnops(%);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#7M\"\"!F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$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#\"#V" }}} {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 65 "#-------- --------------------------------------------------------" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}}{PARA 0 "" 0 "" {TEXT -1 39 " #--------------------------------------" }}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 26 "absolute stability region " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}{SECT 1 {PARA 4 " " 0 "" {TEXT -1 26 "coefficients of the scheme" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 516 "ee := \{c[2 ]=3/17,\nc[3]=2/9,\nc[4]=3/7,\nc[5]=3/4,\nc[6]=3/4,\nc[7]=1,\n\na[2,1] =3/17,\na[3,1]=20/243,\na[3,2]=34/243,\na[4,1]=3/28,\na[4,2]=-153/343, \na[4,3]=1053/1372,\na[5,1]=51/176,\na[5,2]=153/704,\na[5,3]=-1539/228 8,\na[5,4]=8379/9152,\na[6,1]=-219/1408,\na[6,2]=153/704,\na[6,3]=1174 5/18304,\na[6,4]=-931/4576,\na[6,5]=1/4,\na[7,1]=229/4644,\na[7,2]=-17 /43,\na[7,3]=40887/42484,\na[7,4]=-9604/45279,\na[7,5]=0,\na[7,6]=3942 4/66177,\n\nb[1]=79/1080,\nb[2]=0,\nb[3]=19683/69160,\nb[4]=16807/8424 0,\nb[5]=1408/7695,\nb[6]=1408/7695,\nb[7]=43/560\}:" }}}{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 77 "The stabi lity function R for the 7 stage, order 6 scheme is given as follows." }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 75 "subs(ee,StabilityFunction( 6,7,'expanded')):\nR := unapply(%,z):\n'R(z)'=R(z);" }}{PARA 11 "" 1 " " {XPPMATH 20 "6#/-%\"RG6#%\"zG,2\"\"\"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)" } }}{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 27 "z0 \+ := newton(R(z)=-1,z=-4);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%#z0G$!+J (HT&R!\"*" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 " " {MPLTEXT 1 0 306 "z0 := newton(R(z)=-1,z=-4):\np1 := plot([R(z),-1], z=-4.49..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],v iew=[-4.49..0.49,-1.47..1.47],font=[HELVETICA,9]);" }}{PARA 13 "" 1 " " {GLPLOT2D 360 253 253 {PLOTDATA 2 "6+-%'CURVESG6$7U7$$!3A++++++!\\%! #<$!3iK'p5:D&)o#F*7$$!3;+]7'3DdV%F*$!3'yBEJ]7![CF*7$$!35++Ds,X\"Q%F*$! 33#HMw%y!=%F*$!3[%)Q[\"\\_ya\"F*7$$!35+]#oviQ2%F*$!3? Vq%yrMWE\"F*7$$!3A+ve,&\\u'RF*$!3=u.%e0+,T)!#=7$$!3;+v.8ajmPF*$!33s5ua7+&y'FU7$$!3C+v$Q<')4m$F*$!3'oly[$*y) z`FU7$$!3G+v3`dnbNF*$!3rXMsSCn;UFU7$$!37++0T5NZMF*$!31(G3#p8$)GKFU7$$! 3?+]sv&Q>N$F*$!3lAf+n4Z1DFU7$$!3E++:Qn_WKF*$!3YB],/wpJ=FU7$$!3#****\\s &QnOJF*$!3?]#*f2#*ew7FU7$$!3&****\\=iPF.$F*$!3udd/!fT_O)!#>7$$!3Q+v$Gc `$QHF*$!3[J')\\MFVE]Fhp7$$!3+++l(y@h#GF*$!3OVJc7l=4dQp\"F*$\"3%>lkG9%)R#=FU7$$!3)*****p]Q@(e\"F*$\"3pL`rFNZO?FU7$$! 3I+]FRT)G[\"F*$\"3[txl0H&[E#FU7$$!3')*\\P*y(R>Q\"F*$\"3xZAkgb03DFU7$$! 3++]Kx#e)p7F*$\"3%fe2:*pC2GFU7$$!3=++5j![\"p6F*$\"3>kW?L%\\b5$FU7$$!3s ***\\KT=;1\"F*$\"3UL#R=.)feMFU7$$!3'***\\P%o0=k*FU$\"39v_l:!)y7QFU7$$! 3](***\\tEbw&)FU$\"3D$***QX#)[TUFU7$$!39-]P2EBuvFU$\"3sh<4*e0()o%FU7$$ !3>)*\\(GL>l_'FU$\"3#p9(z$>Cm?&FU7$$!3z++]-\")=-bFU$\"3C)eGR&QBodFU7$$ !3$*)*\\PM#3)HWFU$\"3O%[Bd&[>@kFU7$$!3;(****f'*ypR$FU$\"3^sjV#z`)>rFU7 $$!3L)**\\U'=wSBFU$\"3D'>n%p_,8zFU7$$!3?$*\\Pt2H$H\"FU$\"3+V,>Fw%oy)FU 7$$!33R****pit2LFhp$\"3+r)*)36PYn*FU7$$\"3Z^+]FkzBxFhp$\"3QUWS=\"*H!3 \"F*7$$\"35*****z2`!f\"F*7$$\"3l+]i_F06GFU$\"35LwZUIf C8F*7$$\"3\\.]Pt1&z\"QFU$\"3]WLRM=\"\\Y\"F*7$$\"3!***************[FU$ \"3Ko[\"G8;Bj\"F*-%'COLOURG6&%$RGBG$\"*++++\"!\")$\"\"!F\\\\lF[\\l-F$6 $7S7$F($!\"\"F\\\\l7$F3Fa\\l7$F=Fa\\l7$FBFa\\l7$FGFa\\l7$FLFa\\l7$FQFa \\l7$FWFa\\l7$FfnFa\\l7$F[oFa\\l7$F`oFa\\l7$FeoFa\\l7$FjoFa\\l7$F_pFa \\l7$FdpFa\\l7$FjpFa\\l7$F_qFa\\l7$FdqFa\\l7$FjqFa\\l7$F_rFa\\l7$FdrFa \\l7$FirFa\\l7$F^sFa\\l7$FcsFa\\l7$FhsFa\\l7$F]tFa\\l7$FbtFa\\l7$FgtFa \\l7$F\\uFa\\l7$FauFa\\l7$FfuFa\\l7$F[vFa\\l7$F`vFa\\l7$FevFa\\l7$FjvF a\\l7$F_wFa\\l7$FdwFa\\l7$FiwFa\\l7$F^xFa\\l7$FcxFa\\l7$FhxFa\\l7$F]yF a\\l7$FbyFa\\l7$FgyFa\\l7$F\\zFa\\l7$FazFa\\l7$FfzFa\\l7$F[[lFa\\l7$F` [lFa\\l-Fe[l6&Fg[lF[\\lF[\\lFh[l-F$6&7#7$$!35+++J(HT&RF*Fa\\l-%'SYMBOL G6#%'CIRCLEG-Fe[l6&Fg[lF\\\\lF\\\\lF\\\\l-%&STYLEG6#%&POINTG-F$6&Fg_l- F\\`l6#%&CROSSGF_`lFa`l-F$6&Fg_l-F\\`l6#%(DIAMONDGF_`lFa`l-F$6%7$7$Fi_ lF[\\lFh_l-%&COLORG6&Fg[lF[\\l$\"\"&Fb\\lF[\\l-%*LINESTYLEG6#\"\"$-%%F ONTG6$%*HELVETICAG\"\"*-%+AXESLABELSG6%Q\"z6\"Q!Febl-F]bl6#%(DEFAULTG- %%VIEWG6$;$!$\\%!\"#$\"#\\F`cl;$!$Z\"F`cl$\"$Z\"F`cl" 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 1281 "R := z -> 1+z+1/2*z^2+1/6*z^3+1/24*z^4+1/120*z^5+1/ 720*z^6+1/5040*z^7:\npts := []:\nz0 := 0:\nfor ct from 0 to 200 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,.55,. 28,0)):\np2 := plots[polygonplot]([seq([pts[i-1],pts[i],[-2,0]],i=2..n ops(pts))],\n style=patchnogrid,color=COLOR(RGB,.9,.55,0)):\n pts := []: z0 := 1.5+4.3*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),I m(zz)]]:\nend do:\np3 := plot(pts,color=COLOR(RGB,.55,.28,0)):\np4 := \+ plots[polygonplot]([seq([pts[i-1],pts[i],[1.39,4.2]],i=2..nops(pts))], \n style=patchnogrid,color=COLOR(RGB,.9,.55,0)):\npts := []: \+ z0 := 1.5-4.3*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)]]:\ne nd do:\np5 := plot(pts,color=COLOR(RGB,.55,.28,0)):\np6 := plots[polyg onplot]([seq([pts[i-1],pts[i],[1.39,-4.2]],i=2..nops(pts))],\n \+ style=patchnogrid,color=COLOR(RGB,.9,.55,0)):\np7 := plot([[[-4.49,0 ],[1.79,0]],[[0,-4.69],[0,4.69]]],color=black,linestyle=3):\nplots[dis play]([p||(1..7)],view=[-4.49..1.79,-4.69..4.69],font=[HELVETICA,9],\n labels=[`Re(z)`,`Im(z)`],axes=boxed,scaling=constrained) ;" }}{PARA 13 "" 1 "" {GLPLOT2D 381 565 565 {PLOTDATA 2 "6/-%'CURVESG6 $7ew7$$\"\"!F)F(7$F($\"3++++Fjzq:!#=7$$\"3!)*****\\QO5E#!#E$\"3'****** pk#fTJF-7$$\"3!******4`NP]&!#D$\"3))*****\\t)Q7ZF-7$$\"3Y+++5i'))4&!#C $\"3]+++@@=$G'F-7$$\"3*)*****498?u#!#B$\"3#*******='eR&yF-7$$\"3(***** *HR0#G5!#A$\"3B+++.PmC%*F-7$$\"36+++IKeUHFI$\"3!******fv8&*4\"!#<7$$\" 3M*****>c%*4p'FI$\"3/+++&e*\\c7FQ7$$\"30+++YW#Q?\"!#@$\"3-+++)oWLT\"FQ 7$$\"3#******pUgyb\"FZ$\"3)******\\#e!*p:FQ7$$\"3_+++tN#Gd'FI$\"3++++d C&fs\"FQ7$$!39+++*>;_$RFZ$\"3++++KH:\")=FQ7$$!3-+++0t1!o\"!#?$\"3*)*** **zJ!3N?FQ7$$!3:+++okGTXFio$\"3=+++b+C(=#FQ7$$!3%******\\9yf+\"!#>$\"3 -+++'>-rL#FQ7$$!3!******zLXZ'>Fdp$\"3A++++*)4%[#FQ7$$!3/+++zx3(\\$Fdp$ \"31+++q?aFEFQ7$$!3#)******=O1ydFdp$\"3%******H3Ykw#FQ7$$!3Q+++sKCf*)F dp$\"3))*****z]@$**GFQ7$$!38+++=g-78F-$\"3%)*****\\!)HT-$FQ7$$!3****** *fg'=A=F-$\"3%******zRK'QJFQ7$$!3.+++!Qf+T#F-$\"38+++JI4TKFQ7$$!3%)*** **fF\\B0$F-$\"34+++#oX3L$FQ7$$!3z*****Rbb[T%F-$\"3')*****p'\\`uMFQ7$$!35+++-+u$FQ7$$!3'*******4\\tg6FQ$\"36+++R^TR PFQ7$$!3.+++5fR@7FQ$\"37+++Cc)\\t$FQ7$$!3#*******z-V\"G\"FQ$\"3?+++%>C os$FQ7$$!33+++E&))3M\"FQ$\"34+++a[,:PFQ7$$!3/+++-t#)*R\"FQ$\"3#******f U>'*p$FQ7$$!3%******pf7$e9FQ$\"3/+++P9o!o$FQ7$$!33+++ENU;:FQ$\"3%)**** **p*H#eOFQ7$$!3-+++@nDu:FQ$\"3'******H&RGKOFQ7$$!36+++*HJ>j\"FQ$\"3,++ +8v&Gg$FQ7$$!3!******R:%f*o\"FQ$\"3/+++Wr'*pNFQ7$$!3-+++>]UZFQ$\"3(******\\'e%[S$FQ7$$!3'***** *p[Y_)>FQ$\"3-+++?c)eN$FQ7$$!3?+++lBrZ?FQ$\"3!)*****z.QWI$FQ7$$!36+++J s&=6#FQ$\"3%)*****R)=*4D$FQ7$$!3'******H4Xx<#FQ$\"3;+++FF8'>$FQ7$$!3%* *****R!*e_C#FQ$\"30+++&f'[SJFQ7$$!3(******4'f39BFQ$\"3'******z`PY3$FQ7 $$!37+++s1v$Q#FQ$\"36+++T^/HIFQ7$$!33+++!f&o`CFQ$\"37+++_@*R(HFQ7$$!3' ******fW8L_#FQ$\"3(******>Bx&>HFQ7$$!3/+++TG6#f#FQ$\"3=+++RbulGFQ7$$!3 3+++vYlfEFQ$\"3++++6sL7GFQ7$$!3#******H`6cs#FQ$\"3;+++E_8fFFQ7$$!3/+++ `&\\(*y#FQ$\"3%)*****\\h-fq#FQ7$$!3=+++5C\">&GFQ$\"3-+++@oS_EFQ7$$!3&) *****z#R+7HFQ$\"3#)*****4EL%)f#FQ7$$!3;+++[E(*pHFQ$\"33+++;8zVDFQ7$$!3 >+++;&*zDIFQ$\"39+++/dJ)[#FQ7$$!3/+++4()[zIFQ$\"32+++P`'=V#FQ7$$!30+++ Z61JJFQ$\"3!******p1@VP#FQ7$$!39+++G,b!=$FQ$\"37+++RJe:BFQ7$$!3;+++X&) *zA$FQ$\"35+++9)obD#FQ7$$!35+++esXtKFQ$\"3/+++9/@%>#FQ7$$!3*)******[U) pJ$FQ$\"3++++GRXJ@FQ7$$!33+++'GW'eLFQ$\"3=+++>#es1#FQ7$$!33+++;)3&)R$F Q$\"3#)*****\\)[f,?FQ7$$!3********zelOMFQ$\"36+++[([W$>FQ7$$!3)******R ypJZ$FQ$\"3!*******[*=e'=FQ7$$!3))*****zGS\"3NFQ$\"3#******Ra?dz\"FQ7$ $!3&)*****R2h;a$FQ$\"35+++Ok=C(4YMOFQ$\"30+++Jbg ,:FQ7$$!3))*****ps)3jOFQ$\"34+++KN2D9FQ7$$!31+++S6n!p$FQ$\"3!******z8& eZ8FQ7$$!3!******H1Tsr$FQ$\"37++++GHp7FQ7$$!3?+++\\U!Gu$FQ$\"3-+++4vN! >\"FQ7$$!31+++3wLnPFQ$\"3/+++a6%46\"FQ7$$!3#******\\1!z!z$FQ$\"33+++P& )>J5FQ7$$!3*******f(f38QFQ$\"3k*****f>,F^*F-7$$!3#)*****4cIT$QFQ$\"3k* ****f)f.RFQ$\"3$*** ****HQKAbF-7$$!3++++jhu;RFQ$\"3!)*****4Q@!HZF-7$$!3#******\\o?Rt$RFQ$\"3))******3=%y9$F-7$$!32+++R2lWRF Q$\"3-+++UlafBF-7$$!3/+++'*f!*\\RFQ$\"3++++iFQs:F-7$$!3))*****R,F^*F-7$F[gl$!33+++P&)>J5FQ7$Fffl$!3/+++a6%46\"FQ7$Faf l$!3-+++4vN!>\"FQ7$F\\fl$!37++++GHp7FQ7$Fgel$!3!******z8&eZ8FQ7$Fbel$! 34+++KN2D9FQ7$F]el$!30+++Jbg,:FQ7$Fhdl$!35+++dN/x:FQ7$Fcdl$!3'******fO p7l\"FQ7$F^dl$!35+++Ok=CFQ7$Fjbl$!3#)*****\\)[f,?FQ7$Febl$!3=+++>#es 1#FQ7$F`bl$!3++++GRXJ@FQ7$F[bl$!3/+++9/@%>#FQ7$Ffal$!35+++9)obD#FQ7$Fa al$!37+++RJe:BFQ7$F\\al$!3!******p1@VP#FQ7$Fg`l$!32+++P`'=V#FQ7$Fb`l$! 39+++/dJ)[#FQ7$F]`l$!33+++;8zVDFQ7$Fh_l$!3#)*****4EL%)f#FQ7$Fc_l$!3-++ +@oS_EFQ7$F^_l$!3%)*****\\h-fq#FQ7$Fi^l$!3;+++E_8fFFQ7$Fd^l$!3++++6sL7 GFQ7$F_^l$!3=+++RbulGFQ7$Fj]l$!3(******>Bx&>HFQ7$Fe]l$!37+++_@*R(HFQ7$ F`]l$!36+++T^/HIFQ7$F[]l$!3'******z`PY3$FQ7$Ff\\l$!30+++&f'[SJFQ7$Fa\\ l$!3;+++FF8'>$FQ7$F\\\\l$!3%)*****R)=*4D$FQ7$Fg[l$!3!)*****z.QWI$FQ7$F b[l$!3-+++?c)eN$FQ7$F][l$!3(******\\'e%[S$FQ7$Fhz$!3')*****z[[4X$FQ7$F cz$!31+++Od$R\\$FQ7$F^z$!3\")*****ziTO`$FQ7$Fiy$!3/+++Wr'*pNFQ7$Fdy$!3 ,+++8v&Gg$FQ7$F_y$!3'******H&RGKOFQ7$Fjx$!3%)******p*H#eOFQ7$Fex$!3/++ +P9o!o$FQ7$F`x$!3#******fU>'*p$FQ7$F[x$!34+++a[,:PFQ7$Ffw$!3?+++%>Cos$ FQ7$Faw$!37+++Cc)\\t$FQ7$F\\w$!36+++R^TRPFQ7$Fgv$!3()*****4>-+u$FQ7$Fb v$!34+++AWgOPFQ7$F]v$!3<+++R5/HPFQ7$Fhu$!34+++$H%3-rL #FQ7$F]p$!3=+++b+C(=#FQ7$Fgo$!3*)*****zJ!3N?FQ7$Fbo$!3++++KH:\")=FQ7$F ]o$!3++++dC&fs\"FQ7$Fhn$!3)******\\#e!*p:FQ7$$\"3'******pWCQ?\"FZ$!3-+ ++)oWLT\"FQ7$FS$!3/+++&e*\\c7FQ7$FM$!3!******fv8&*4\"FQ7$FG$!3B+++.PmC %*F-7$FA$!3#*******='eR&yF-7$F;$!3]+++@@=$G'F-7$$\"34+++Ibt.bF7$!3))** ***\\t)Q7ZF-7$$\"3#******HQO5E#F1$!3'******pk#fTJF-7$F($!3++++Fjzq:F-F '-%&COLORG6&%$RGBG$\"#b!\"#$\"#GFc^nF(-%)POLYGONSG6fw7%F'7$F($\"+Fjzq: !#57$$Fc^nF)F(7%Fj^n7$$\"+&QO5E#F-$\"+ZEfTJF]_nF^_n7%Fa_n7$$\"+Jbt.bFQ $\"+N()Q7ZF]_nF^_n7%Fg_n7$$\"+5i'))4&!#;$\"+@@=$G'F]_nF^_n7%F]`n7$$\"+ TJ,UF!#:$\"+>'eR&yF]_nF^_n7%Fd`n7$$\"+$R0#G5!#9$\"+.PmC%*F]_nF^_n7%F[a n7$$\"+IKeUHF^an$\"+cP^*4\"!\"*F^_n7%Fban7$$\"+iX*4p'F^an$\"+&e*\\c7Fg anF^_n7%Fian7$$\"+YW#Q?\"!#8$\"+)oWLT\"FganF^_n7%F_bn7$$\"+F/'yb\"Fbbn $\"+De!*p:FganF^_n7%Ffbn7$$\"+tN#Gd'F^an$\"+dC&fs\"FganF^_n7%F\\cn7$$! +*>;_$RFbbn$\"+KH:\")=FganF^_n7%Fbcn7$$!+0t1!o\"!#7$\"+=.3N?FganF^_n7% Fhcn7$$!+okGTXF[dn$\"+b+C(=#FganF^_n7%F_dn7$$!+X\"yf+\"!#6$\"+'>-rL#Fg anF^_n7%Fedn7$$!+Q`uk>Fhdn$\"++*)4%[#FganF^_n7%F\\en7$$!+zx3(\\$Fhdn$ \"+q?aFEFganF^_n7%Fben7$$!+>O1ydFhdn$\"+$3Ykw#FganF^_n7%Fhen7$$!+sKCf* )Fhdn$\"+3:K**GFganF^_n7%F^fn7$$!+=g-78F]_n$\"+0)HT-$FganF^_n7%Fdfn7$$ !+1m=A=F]_n$\"+)RK'QJFganF^_n7%Fjfn7$$!+!Qf+T#F]_n$\"+JI4TKFganF^_n7%F `gn7$$!+w#\\B0$F]_n$\"+#oX3L$FganF^_n7%Ffgn7$$!+uTSEPF]_n$\"+s=I3MFgan F^_n7%F\\hn7$$!+_b&[T%F]_n$\"+n\\`uMFganF^_n7%Fbhn7$$!+-+u$FganF^_n7%F^\\o7$$!+5\\tg6Fgan$\"+R^TRPFganF^_n7%Fd\\o7$$!+5fR @7Fgan$\"+Cc)\\t$FganF^_n7%Fj\\o7$$!+!GI9G\"Fgan$\"+%>Cos$FganF^_n7%F` ]o7$$!+E&))3M\"Fgan$\"+a[,:PFganF^_n7%Ff]o7$$!+-t#)*R\"Fgan$\"+E%>'*p$ FganF^_n7%F\\^o7$$!+(f7$e9Fgan$\"+P9o!o$FganF^_n7%Fb^o7$$!+ENU;:Fgan$ \"+q*H#eOFganF^_n7%Fh^o7$$!+@nDu:Fgan$\"+`RGKOFganF^_n7%F^_o7$$!+*HJ>j \"Fgan$\"+8v&Gg$FganF^_n7%Fd_o7$$!+aTf*o\"Fgan$\"+Wr'*pNFganF^_n7%Fj_o 7$$!+>]UZFgan$\"+le%[ S$FganF^_n7%Fbao7$$!+([Y_)>Fgan$\"+?c)eN$FganF^_n7%Fhao7$$!+lBrZ?Fgan$ \"+Q!QWI$FganF^_n7%F^bo7$$!+Js&=6#Fgan$\"+%)=*4D$FganF^_n7%Fdbo7$$!+$4 Xx<#Fgan$\"+FF8'>$FganF^_n7%Fjbo7$$!+/*e_C#Fgan$\"+&f'[SJFganF^_n7%F`c o7$$!+hf39BFgan$\"+Qvj%3$FganF^_n7%Ffco7$$!+s1v$Q#Fgan$\"+T^/HIFganF^_ n7%F\\do7$$!+!f&o`CFgan$\"+_@*R(HFganF^_n7%Fbdo7$$!+YMJBDFgan$\"+Ksd>H FganF^_n7%Fhdo7$$!+TG6#f#Fgan$\"+RbulGFganF^_n7%F^eo7$$!+vYlfEFgan$\"+ 6sL7GFganF^_n7%Fdeo7$$!+L:hDFFgan$\"+E_8fFFganF^_n7%Fjeo7$$!+`&\\(*y#F gan$\"+:E!fq#FganF^_n7%F`fo7$$!+5C\">&GFgan$\"+@oS_EFganF^_n7%Fffo7$$! +GR+7HFgan$\"+hKV)f#FganF^_n7%F\\go7$$!+[E(*pHFgan$\"+;8zVDFganF^_n7%F bgo7$$!+;&*zDIFgan$\"+/dJ)[#FganF^_n7%Fhgo7$$!+4()[zIFgan$\"+P`'=V#Fga nF^_n7%F^ho7$$!+Z61JJFgan$\"+n5KuBFganF^_n7%Fdho7$$!+G,b!=$Fgan$\"+RJe :BFganF^_n7%Fjho7$$!+X&)*zA$Fgan$\"+9)obD#FganF^_n7%F`io7$$!+esXtKFgan $\"+9/@%>#FganF^_n7%Ffio7$$!+\\U)pJ$Fgan$\"+GRXJ@FganF^_n7%F\\jo7$$!+' GW'eLFgan$\"+>#es1#FganF^_n7%Fbjo7$$!+;)3&)R$Fgan$\"+&)[f,?FganF^_n7%F hjo7$$!+!)elOMFgan$\"+[([W$>FganF^_n7%F^[p7$$!+%ypJZ$Fgan$\"+\\*=e'=Fg anF^_n7%Fd[p7$$!+)GS\"3NFgan$\"+W0s&z\"FganF^_n7%Fj[p7$$!+u5mTNFgan$\" +Ok=C\"Fga nF^_n7%Fj^p7$$!+3wLnPFgan$\"+a6%46\"FganF^_n7%F`_p7$$!+l+z!z$Fgan$\"+P &)>J5FganF^_n7%Ff_p7$$!+wf38QFgan$\"+'>,F^*F]_nF^_n7%F\\`p7$$!+h08MQFg an$\"+wTw7()F]_nF^_n7%Fb`p7$$!+([;Q&QFgan$\"++h98zF]_nF^_n7%Fh`p7$$!+V 1.sQFgan$\"+SVd9rF]_nF^_n7%F^ap7$$!+1.m))QFgan$\"+\"fSvJ'F]_nF^_n7%Fda p7$$!+\">)f.RFgan$\"+IQKAbF]_nF^_n7%Fjap7$$!+jhu;RFgan$\"+\"Q@!HZF]_nF ^_n7%F`bp7$$!+&o,F ^*F]_nF^_n7%Fdgp7$Fg_p$!+P&)>J5FganF^_n7%Fhgp7$Fa_p$!+a6%46\"FganF^_n7 %F\\hp7$F[_p$!+4vN!>\"FganF^_n7%F`hp7$Fe^p$!++GHp7FganF^_n7%Fdhp7$F_^p $!+Q^eZ8FganF^_n7%Fhhp7$Fi]p$!+KN2D9FganF^_n7%F\\ip7$Fc]p$!+Jbg,:FganF ^_n7%F`ip7$F]]p$!+dN/x:FganF^_n7%Fdip7$Fg\\p$!+m$p7l\"FganF^_n7%Fhip7$ Fa\\p$!+Ok=CFganF^_n7%Fhjp7$Fijo$!+&)[f,?FganF^_ n7%F\\[q7$Fcjo$!+>#es1#FganF^_n7%F`[q7$F]jo$!+GRXJ@FganF^_n7%Fd[q7$Fgi o$!+9/@%>#FganF^_n7%Fh[q7$Faio$!+9)obD#FganF^_n7%F\\\\q7$F[io$!+RJe:BF ganF^_n7%F`\\q7$Feho$!+n5KuBFganF^_n7%Fd\\q7$F_ho$!+P`'=V#FganF^_n7%Fh \\q7$Figo$!+/dJ)[#FganF^_n7%F\\]q7$Fcgo$!+;8zVDFganF^_n7%F`]q7$F]go$!+ hKV)f#FganF^_n7%Fd]q7$Fgfo$!+@oS_EFganF^_n7%Fh]q7$Fafo$!+:E!fq#FganF^_ n7%F\\^q7$F[fo$!+E_8fFFganF^_n7%F`^q7$Feeo$!+6sL7GFganF^_n7%Fd^q7$F_eo $!+RbulGFganF^_n7%Fh^q7$Fido$!+Ksd>HFganF^_n7%F\\_q7$Fcdo$!+_@*R(HFgan F^_n7%F`_q7$F]do$!+T^/HIFganF^_n7%Fd_q7$Fgco$!+Qvj%3$FganF^_n7%Fh_q7$F aco$!+&f'[SJFganF^_n7%F\\`q7$F[co$!+FF8'>$FganF^_n7%F``q7$Febo$!+%)=*4 D$FganF^_n7%Fd`q7$F_bo$!+Q!QWI$FganF^_n7%Fh`q7$Fiao$!+?c)eN$FganF^_n7% F\\aq7$Fcao$!+le%[S$FganF^_n7%F`aq7$F]ao$!+)[[4X$FganF^_n7%Fdaq7$Fg`o$ !+Od$R\\$FganF^_n7%Fhaq7$Fa`o$!+G;kLNFganF^_n7%F\\bq7$F[`o$!+Wr'*pNFga nF^_n7%F`bq7$Fe_o$!+8v&Gg$FganF^_n7%Fdbq7$F__o$!+`RGKOFganF^_n7%Fhbq7$ Fi^o$!+q*H#eOFganF^_n7%F\\cq7$Fc^o$!+P9o!o$FganF^_n7%F`cq7$F]^o$!+E%>' *p$FganF^_n7%Fdcq7$Fg]o$!+a[,:PFganF^_n7%Fhcq7$Fa]o$!+%>Cos$FganF^_n7% F\\dq7$F[]o$!+Cc)\\t$FganF^_n7%F`dq7$Fe\\o$!+R^TRPFganF^_n7%Fddq7$F_\\ o$!+\">-+u$FganF^_n7%Fhdq7$Fi[o$!+AWgOPFganF^_n7%F\\eq7$Fc[o$!+R5/HPFg anF^_n7%F`eq7$F][o$!+$H%3-rL#FganF^_n7%Fhiq7$F`dn$!+b+C(=#FganF^_n7 %F\\jq7$Ficn$!+=.3N?FganF^_n7%F`jq7$Fccn$!+KH:\")=FganF^_n7%Fdjq7$F]cn $!+dC&fs\"FganF^_n7%Fhjq7$Fgbn$!+De!*p:FganF^_n7%F\\[r7$$\"+ZW#Q?\"Fbb n$!+)oWLT\"FganF^_n7%F`[r7$Fjan$!+&e*\\c7FganF^_n7%Ff[r7$Fcan$!+cP^*4 \"FganF^_n7%Fj[r7$F\\an$!+.PmC%*F]_nF^_n7%F^\\r7$Fe`n$!+>'eR&yF]_nF^_n 7%Fb\\r7$F^`n$!+@@=$G'F]_nF^_n7%Ff\\r7$$\"+Ibt.bFQ$!+N()Q7ZF]_nF^_n7%F j\\r7$$\"+$QO5E#F-$!+ZEfTJF]_nF^_n7%F`]r7$F($!+Fjzq:F]_nF^_n7%Ff]rF'F^ _n-F^^n6&F`^n$\"\"*!\"\"Fa^nF(-%&STYLEG6#%,PATCHNOGRIDG-F$6$7in7$$\"31 +++>\"fX]\"FQ$\"32+++h[;1VFQ7$$\"3)******pE!f'\\\"FQ$\"3c*****\\)\\#[J %FQ7$$\"3++++i*Rz[\"FQ$\"3,+++B1!GK%FQ7$$\"3%******\\9^'y9FQ$\"3q***** *z/0IVFQ7$$\"31+++ruwo9FQ$\"3%******4kLlL%FQ7$$\"3++++;OLe9FQ$\"3))*** **RH4AM%FQ7$$\"31+++XcRZ9FQ$\"3=+++tk.ZVFQ7$$\"3++++:9+O9FQ$\"3C+++6Q( 4N%FQ7$$\"3(*******\\5?C9FQ$\"3))*****>JzRN%FQ7$$\"3,+++'\\Z?T\"FQ$\"3 3+++O,,cVFQ7$$\"3'*******orf*R\"FQ$\"3Q+++0C-dVFQ7$$\"33+++I2\"pQ\"FQ$ \"3))*****\\/rpN%FQ7$$\"33+++8S0u8FQ$\"3-+++o'4eN%FQ7$$\"3)******f8*4h 8FQ$\"3')*****f_!\\`VFQ7$$\"3'*******\\e7[8FQ$\"3s*****pdk*\\VFQ7$$\"3 4+++vJAN8FQ$\"39+++3<=XVFQ7$$\"3%******>Q\"\\A8FQ$\"3;+++>74RVFQ7$$\"3 &******RRW+J\"FQ$\"3D+++uEkJVFQ7$$\"3-+++sE,)H\"FQ$\"3V+++GtyAVFQ7$$\" 31+++Pma'G\"FQ$\"3-+++*R![7VFQ7$$\"31+++81#eF\"FQ$\"3&******\\e%o+VFQ7 $$\"3-+++(=PgE\"FQ$\"3M+++-aP(G%FQ7$$\"3/+++%fJuD\"FQ$\"3A+++#4\\DF%FQ 7$$\"3-+++A`F]7FQ$\"3)******R2MiD%FQ7$$\"3#******\\Mx[C\"FQ$\"3$)***** *3p]QUFQ7$$\"33+++1-eT7FQ$\"3\")*****pV8&>UFQ7$$\"3#******>sY2C\"FQ$\" 31+++#R%\\*>%FQ7$$\"34+++*)=tU7FQ$\"3!)******G=\")yTFQ7$$\"35+++#\\NyC \"FQ$\"3g*****>rnz:%FQ7$$\"3-+++zkBc7FQ$\"3*******RM+w8%FQ7$$\"3)***** *>9AzE\"FQ$\"3l*****p-V%=TFQ7$$\"3++++\"**QEG\"FQ$\"3')*****fjR75%FQ7$ $\"3'*******4]*)*H\"FQ$\"3/+++:[j'3%FQ7$$\"3#*******zO->8FQ$\"3Q+++ux2 vSFQ7$$\"3++++,yGR8FQ$\"3Q+++.nxmSFQ7$$\"3-+++9B)*f8FQ$\"3M+++$f8<1%FQ 7$$\"3++++GV]!Q\"FQ$\"39+++)***pfSFQ7$$\"3%******R9#Q+9FQ$\"3S+++LVWgS FQ7$$\"3*******4)HF>9FQ$\"3G+++c&4O1%FQ7$$\"33+++aT%pV\"FQ$\"3Q+++KK&) oSFQ7$$\"3!******pE\\KX\"FQ$\"3I+++q4&e2%FQ7$$\"3%*******\\e5o9FQ$\"3E +++fwI%3%FQ7$$\"3!******z(pZ\"[\"FQ$\"3o*****p.hR4%FQ7$$\"33+++MsN$\\ \"FQ$\"3w*****>+\"e/TFQ7$$\"3*)*****4WiP]\"FQ$\"3!)*****\\(o'f6%FQ7$$ \"3,+++xBs7:FQ$\"3%)*****z*Q%z7%FQ7$$\"3$******4pv-_\"FQ$\"3&******>kf .9%FQ7$$\"35+++;kYE:FQ$\"3')*****Ro!3`TFQ7$$\"3\"******>gT8`\"FQ$\"3E+ ++S'*)f;%FQ7$$\"31+++S(\\\\`\"FQ$\"3))*****pi#)*yTFQ7$$\"3)******\\tRt `\"FQ$\"3C+++*3n>>%FQ7$$\"3-+++[-cQ:FQ$\"3D+++\")*f[?%FQ7$$\"3&******z If'Q:FQ$\"3m*****RB'eAio H%FQFf^rF]^n-Fg^n6jn7%7$$\"+>\"fX]\"Fgan$\"+h[;1VFgan7$$\"+n-f'\\\"Fga n$\"+&)\\#[J%Fgan7$$\"$R\"Fc^n$\"#UF^^r7%Fjas7$$\"+i*Rz[\"Fgan$\"+B1!G K%FganF_bs7%Febs7$$\"+X6ly9Fgan$\"+![]+L%FganF_bs7%F[cs7$$\"+ruwo9Fgan $\"+TO`OVFganF_bs7%Facs7$$\"+;OLe9Fgan$\"+%H4AM%FganF_bs7%Fgcs7$$\"+Xc RZ9Fgan$\"+tk.ZVFganF_bs7%F]ds7$$\"+:9+O9Fgan$\"+6Q(4N%FganF_bs7%Fcds7 $$\"+]5?C9Fgan$\"+7$zRN%FganF_bs7%Fids7$$\"+'\\Z?T\"Fgan$\"+O,,cVFganF _bs7%F_es7$$\"+prf*R\"Fgan$\"+0C-dVFganF_bs7%Fees7$$\"+I2\"pQ\"Fgan$\" +X5(pN%FganF_bs7%F[fs7$$\"+8S0u8Fgan$\"+o'4eN%FganF_bs7%Fafs7$$\"+O\"* 4h8Fgan$\"+E0\\`VFganF_bs7%Fgfs7$$\"+]e7[8Fgan$\"+xX'*\\VFganF_bs7%F]g s7$$\"+vJAN8Fgan$\"+3<=XVFganF_bs7%Fcgs7$$\"+#Q\"\\A8Fgan$\"+>74RVFgan F_bs7%Figs7$$\"+%RW+J\"Fgan$\"+uEkJVFganF_bs7%F_hs7$$\"+sE,)H\"Fgan$\" +GtyAVFganF_bs7%Fehs7$$\"+Pma'G\"Fgan$\"+*R![7VFganF_bs7%F[is7$$\"+81# eF\"Fgan$\"+&e%o+VFganF_bs7%Fais7$$\"+(=PgE\"Fgan$\"+-aP(G%FganF_bs7%F gis7$$\"+%fJuD\"Fgan$\"+#4\\DF%FganF_bs7%F]js7$$\"+A`F]7Fgan$\"+uSBcUF ganF_bs7%Fcjs7$$\"+Xt([C\"Fgan$\"+4p]QUFganF_bs7%Fijs7$$\"+1-eT7Fgan$ \"+PM^>UFganF_bs7%F_[t7$$\"+AnuS7Fgan$\"+#R%\\*>%FganF_bs7%Fe[t7$$\"+* )=tU7Fgan$\"+H=\")yTFganF_bs7%F[\\t7$$\"+#\\NyC\"Fgan$\"+7x'z:%FganF_b s7%Fa\\t7$$\"+zkBc7Fgan$\"+W.gPTFganF_bs7%Fg\\t7$$\"+U@#zE\"Fgan$\"+FI W=TFganF_bs7%F]]t7$$\"+\"**QEG\"Fgan$\"+O'R75%FganF_bs7%Fc]t7$$\"+5]*) *H\"Fgan$\"+:[j'3%FganF_bs7%Fi]t7$$\"+!oB!>8Fgan$\"+ux2vSFganF_bs7%F_^ t7$$\"+,yGR8Fgan$\"+.nxmSFganF_bs7%Fe^t7$$\"+9B)*f8Fgan$\"+$f8<1%FganF _bs7%F[_t7$$\"+GV]!Q\"Fgan$\"+)***pfSFganF_bs7%Fa_t7$$\"+W@Q+9Fgan$\"+ LVWgSFganF_bs7%Fg_t7$$\"+\")HF>9Fgan$\"+c&4O1%FganF_bs7%F]`t7$$\"+aT%p V\"Fgan$\"+KK&)oSFganF_bs7%Fc`t7$$\"+n#\\KX\"Fgan$\"+q4&e2%FganF_bs7%F i`t7$$\"+]e5o9Fgan$\"+fwI%3%FganF_bs7%F_at7$$\"+ypZ\"[\"Fgan$\"+P5'R4% FganF_bs7%Feat7$$\"+MsN$\\\"Fgan$\"+-5e/TFganF_bs7%F[bt7$$\"+TCw.:Fgan $\"+vo'f6%FganF_bs7%Fabt7$$\"+xBs7:Fgan$\"+)*Q%z7%FganF_bs7%Fgbt7$$\"+ \"pv-_\"Fgan$\"+U'f.9%FganF_bs7%F]ct7$$\"+;kYE:Fgan$\"+%o!3`TFganF_bs7 %Fcct7$$\"+-;MJ:Fgan$\"+S'*)f;%FganF_bs7%Fict7$$\"+S(\\\\`\"Fgan$\"+FE )*yTFganF_bs7%F_dt7$$\"+N(Rt`\"Fgan$\"+*3n>>%FganF_bs7%Fedt7$$\"+[-cQ: Fgan$\"+\")*f[?%FganF_bs7%F[et7$$\"+3$f'Q:Fgan$\"+MieAioH%FQ7$Fi`s$! 3!******\\.gpG%FQ7$Fd`s$!3C+++>%FQ7$F\\^s$!3))*****pi# )*yTFQ7$Fg]s$!3E+++S'*)f;%FQ7$Fb]s$!3')*****Ro!3`TFQ7$F]]s$!3&******>k f.9%FQ7$Fh\\s$!3%)*****z*Q%z7%FQ7$Fc\\s$!3!)*****\\(o'f6%FQ7$F^\\s$!3w *****>+\"e/TFQ7$Fi[s$!3o*****p.hR4%FQ7$Fd[s$!3E+++fwI%3%FQ7$F_[s$!3I++ +q4&e2%FQ7$Fjjr$!3Q+++KK&)oSFQ7$Fejr$!3G+++c&4O1%FQ7$F`jr$!3S+++LVWgSF Q7$F[jr$!39+++)***pfSFQ7$Ffir$!3M+++$f8<1%FQ7$Fair$!3Q+++.nxmSFQ7$F\\i r$!3Q+++ux2vSFQ7$Fghr$!3/+++:[j'3%FQ7$Fbhr$!3')*****fjR75%FQ7$F]hr$!3l *****p-V%=TFQ7$Fhgr$!3*******RM+w8%FQ7$Fcgr$!3g*****>rnz:%FQ7$F^gr$!3! )******G=\")yTFQ7$Fifr$!31+++#R%\\*>%FQ7$Fdfr$!3\")*****pV8&>UFQ7$F_fr $!3$)******3p]QUFQ7$Fjer$!3)******R2MiD%FQ7$Feer$!3A+++#4\\DF%FQ7$F`er $!3M+++-aP(G%FQ7$F[er$!3&******\\e%o+VFQ7$Ffdr$!3-+++*R![7VFQ7$Fadr$!3 V+++GtyAVFQ7$F\\dr$!3D+++uEkJVFQ7$Fgcr$!3;+++>74RVFQ7$Fbcr$!39+++3<=XV FQ7$F]cr$!3s*****pdk*\\VFQ7$Fhbr$!3')*****f_!\\`VFQ7$Fcbr$!3-+++o'4eN% FQ7$F^br$!3))*****\\/rpN%FQ7$Fiar$!3Q+++0C-dVFQ7$Fdar$!33+++O,,cVFQ7$F _ar$!3))*****>JzRN%FQ7$Fj`r$!3C+++6Q(4N%FQ7$Fe`r$!3=+++tk.ZVFQ7$F``r$! 3))*****RH4AM%FQ7$F[`r$!3%******4kLlL%FQ7$Ff_r$!3q******z/0IVFQ7$Fa_r$ !3,+++B1!GK%FQ7$F\\_r$!3c*****\\)\\#[J%FQFdhtF]^n-Fg^n6jn7%7$Ffas$!+h[ ;1VFgan7$F\\ht$!+AA'oH%Fgan7$F`bs$!#UF^^r7%F^du7$Ffgt$!+N+'pG%FganFadu 7%Fedu7$F`gt$!+>%FganFadu7%Fefu7$F`dt$!+FE)*yTFganFadu7%Fifu7$Fjct$!+S'*)f ;%FganFadu7%F]gu7$Fdct$!+%o!3`TFganFadu7%Fagu7$F^ct$!+U'f.9%FganFadu7% Fegu7$Fhbt$!+)*Q%z7%FganFadu7%Figu7$Fbbt$!+vo'f6%FganFadu7%F]hu7$F\\bt $!+-5e/TFganFadu7%Fahu7$Ffat$!+P5'R4%FganFadu7%Fehu7$F`at$!+fwI%3%Fgan Fadu7%Fihu7$Fj`t$!+q4&e2%FganFadu7%F]iu7$Fd`t$!+KK&)oSFganFadu7%Faiu7$ F^`t$!+c&4O1%FganFadu7%Feiu7$Fh_t$!+LVWgSFganFadu7%Fiiu7$Fb_t$!+)***pf SFganFadu7%F]ju7$F\\_t$!+$f8<1%FganFadu7%Faju7$Ff^t$!+.nxmSFganFadu7%F eju7$F`^t$!+ux2vSFganFadu7%Fiju7$Fj]t$!+:[j'3%FganFadu7%F][v7$Fd]t$!+O 'R75%FganFadu7%Fa[v7$F^]t$!+FIW=TFganFadu7%Fe[v7$Fh\\t$!+W.gPTFganFadu 7%Fi[v7$Fb\\t$!+7x'z:%FganFadu7%F]\\v7$F\\\\t$!+H=\")yTFganFadu7%Fa\\v 7$Ff[t$!+#R%\\*>%FganFadu7%Fe\\v7$F`[t$!+PM^>UFganFadu7%Fi\\v7$Fjjs$!+ 4p]QUFganFadu7%F]]v7$Fdjs$!+uSBcUFganFadu7%Fa]v7$F^js$!+#4\\DF%FganFad u7%Fe]v7$Fhis$!+-aP(G%FganFadu7%Fi]v7$Fbis$!+&e%o+VFganFadu7%F]^v7$F\\ is$!+*R![7VFganFadu7%Fa^v7$Ffhs$!+GtyAVFganFadu7%Fe^v7$F`hs$!+uEkJVFga nFadu7%Fi^v7$Fjgs$!+>74RVFganFadu7%F]_v7$Fdgs$!+3<=XVFganFadu7%Fa_v7$F ^gs$!+xX'*\\VFganFadu7%Fe_v7$Fhfs$!+E0\\`VFganFadu7%Fi_v7$Fbfs$!+o'4eN %FganFadu7%F]`v7$F\\fs$!+X5(pN%FganFadu7%Fa`v7$Ffes$!+0C-dVFganFadu7%F e`v7$F`es$!+O,,cVFganFadu7%Fi`v7$Fjds$!+7$zRN%FganFadu7%F]av7$Fdds$!+6 Q(4N%FganFadu7%Faav7$F^ds$!+tk.ZVFganFadu7%Feav7$Fhcs$!+%H4AM%FganFadu 7%Fiav7$Fbcs$!+TO`OVFganFadu7%F]bv7$F\\cs$!+![]+L%FganFadu7%Fabv7$Ffbs $!+B1!GK%FganFadu7%Febv7$F[bs$!+&)\\#[J%FganFadu7%FibvF[duFaduFj]rF_^r -F$6%7$7$$!3A++++++!\\%FQF(7$$\"3/++++++!z\"FQF(-%'COLOURG6&F`^nF)F)F) -%*LINESTYLEG6#\"\"$-F$6%7$7$F($!3Q++++++!p%FQ7$F($\"3Q++++++!p%FQFfcv Ficv-%%FONTG6$%*HELVETICAGF]^r-%+AXESLABELSG6%%&Re(z)G%&Im(z)G-Fgdv6#% (DEFAULTG-%*AXESSTYLEG6#%$BOXG-%(SCALINGG6#%,CONSTRAINEDG-%%VIEWG6$;$! $\\%Fc^n$\"$z\"Fc^n;$!$p%Fc^n$\"$p%Fc^n" 1 2 0 1 10 0 2 9 1 2 1 1.000000 45.000000 45.000000 0 0 "Curve 1" "Curve 2" "Curve 3" "Curve \+ 4" "Curve 5" "Curve 6" "Curve 7" "Curve 8" }}}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 4 "The " }{TEXT 260 30 "interval of absolute stability" }{TEXT -1 89 " (or stability interval) is the in tersection of the stability region with the real line." }}{PARA 0 "" 0 "" {TEXT -1 59 "For this scheme the stability interval is (approxima tely) " }{XPPEDIT 18 0 "[-3.9541, 0];" "6#7$,$-%&FloatG6$\"&T&R!\"%! \"\"\"\"!" }{TEXT -1 2 ". " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 260 "We can distort the boundary curve horizontally b y taking the 11th root of the real part of points along the curve. In \+ this way we see that the largest interval on the nonnegative imaginary axis that contains the origin and lies inside the stability region is " }{XPPEDIT 18 0 "[0, 1.76];" "6#7$\"\"!-%&FloatG6$\"$w\"!\"#" } {TEXT -1 18 " approximately. " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 322 "R := z -> 1+z+1/2*z^2+1/6*z ^3+1/24*z^4+1/120*z^5+1/720*z^6+1/5040*z^7:\nDigits := 25:\npts := []: z0 := 0:\nfor ct from 0 to 60 do\n zz := newton(R(z)=exp(ct*Pi/100* I),z=z0):\n z0 := zz:\n pts := [op(pts),[surd(Re(zz),11),Im(zz)]]: \nend do:\nplot(pts,color=COLOR(RGB,.85,.45,0),thickness=2,font=[HELVE TICA,9]);\nDigits := 10:" }}{PARA 13 "" 1 "" {GLPLOT2D 261 276 276 {PLOTDATA 2 "6(-%'CURVESG6#7in7$$\"\"!F)F(7$$\":(GMFj2VC9eNyI!#E$\":yZ vsJz*e`EfTJF-7$$\":=SG%)Gykc@*o&4&F-$\":w13[GbzrI&=$G'F-7$$\":!)3))4>t ?9qG@%oF-$\":!y*ezx3o2'zxC%*F-7$$\":A+QI!=3it^IK%)F-$\":d8Yy#)>M91PmD \"!#D7$$\":P)>V*eS67jI[\"**F-$\":CoYlW5#yEjzq:F?7$$\":obXc>7S_R:;8\"F? $\":'o/^.pz3#fb\\)=F?7$$\":-C)pR\"f*)Q1o_E\"F?$\":sb85*4%\\s&[6*>#F?7$ $\":BxXKDBKxyXNR\"F?$\":qcv5pP(*>7uK^#F?7$$\":\")ya<9'Q3^cB<:F?$\":/E/ 5]Z7dQLu#GF?7$$\":`#fswJ!y/!>#pj\"F?$\":Sx3Px>1rk#fTJF?7$$\":)3\"orXVL N#=0`vbMF?7$$\":7=K)y)pe\"3h(f'=F?$\":tc!G!=d_5:6*p PF?7$$\":V-d8&)zz2'[(f(>F?$\":'HrDDI8b\"QqS3%F?7$$\":3Z1;+9g!y]F$3#F?$ \":1=2]nS+GeH#)R%F?7$$\":.*GNchB@wF1)=#F?$\":#\\y2`DfKN()Q7ZF?7$$\":=S .17J!GX;\\!H#F?$\":Qo#Q+x[`4yaE]F?7$$\":92+c%))*>_O*o!R#F?$\":\"[Y*))e 3G'eR&yF?7 $$\":#e*[ypTY`WbZ?$F?$\":(=*p@)z#*4u$3\"o\")F?7$$\":*o)*)z_7DE&)=iG$F? $\":$4%3^;1X4#[D#[)F?7$$\":YFq!Q%=snbNfO$F?$\":Lh([eqW06qR'z)F?7$$\":H FMZ`e6An&*QW$F?$\":hpFExG*fyP`5\"*F?7$$\":(or$oWGI)y93?NF?$\":qr]p'=,E .PmC%*F?7$$\":p,g_r>zQ2nWf$F?$\":)Hu'R!=omo]yQ(*F?7$$\":!\\/.rl\"Ffu7$$\":*or%4d75p1\"H=TF?$\":crMI=1y(y.6D7Ffu7$$\":u,0qbVG)R y[tTF?$\":'[y&HZym_e*\\c7Ffu7$$\":.BK_a()=f%4\"fA%F?$\":!38Ffu7$$\":\"4M([@ZU(3%e9K%F ?$\":\"y;5zPltw4j]8Ffu7$$\":24d-,$4jtL&RO%F?$\":$ej\\b=!Q$yE*>Q\"Ffu7$ $\":w[$3ox#))\\L!Q-WF?$\":6qt.Htd\")oWLT\"Ffu7$$\":B'*Qv:g,*\\+>OWF?$ \":U[\\Okk`Di&oW9Ffu7$$\":JEMO$yF-*orYY%F?$\":PG)=^MV)e&R,w9Ffu7$$\":2 o-:pO$pp?)o[%F?$\":[&fP>#e.P,Gt]\"Ffu7$$\":(3;:2S))p#GI:]%F?$\":RJ%H6' )4omfiQ:Ffu7$$\":Y/@x)ezm#oyn]%F?$\":Czo'QWBTDe!*p:Ffu7$$\":^K#=r^T]'4 u)*\\%F?$\":X3cJ\">XGOa;,;Ffu7$$\":.mk\"3u_kVmRwWF?$\":Y\\#HbrabyCSK;F fu7$$\":Di#*Hae;=/:&GWF?$\":[!p,&*4/!GY9Om\"Ffu7$$\":DgMc=qsT-\\+M%F?$ \":Rs&yR5'e3t)z%p\"Ffu7$$\":lMP\"Qh'yWjKn;%F?$\":))4keas\"RdC&fs\"Ffu7 $$\":ye43b)4t(3B6j$F?$\":-_9>CrKJlsqv\"Ffu7$$!:![sV6%4$>0k\"34%F?$\":c RZc'\\X')ph:)y\"Ffu7$$!:y]OyGQ:9QF)oWF?$\":rq*>>=Ffu7$$!:[XNpEO \"33:R6ZF?$\":lFWI%*[M0#)*>]=Ffu7$$!:l!\\o*4H$e=[)G!\\F?$\":I/Ka`&4tJH :\")=Ffu-%*THICKNESSG6#\"\"#-%%FONTG6$%*HELVETICAG\"\"*-%&COLORG6&%$RG BG$\"#&)!\"#$\"#XFb_lF(-%+AXESLABELSG6$Q!6\"Fh_l-%%VIEWG6$%(DEFAULTGF] `l" 1 2 0 1 10 2 2 6 1 4 2 1.000000 45.000000 45.000000 0 0 "Curve 1" }}}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 123 "The \+ relevant intersection point of the boundary curve with the imaginary a xis can be determined more accurately as follows." }}{PARA 0 "" 0 "" {TEXT -1 86 "First we look for points on the boundary curve either sid e of the intersection point. " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 112 "Digits := 15:\nz0 := 1.76*I :\nfor ct from 55 to 58 do\n newton(R(z)=exp(ct*Pi/100*I),z=z0);\nen d do;\nDigits := 10:" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#^$$\"02+GdBGd' !#>$\"0t\"RdC&fs\"!#9" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#^$$\"0o0E_SpW \"!#>$\"0rKJlsqv\"!#9" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#^$$!05]am.#p` !#>$\"0bk)ph:)y\"!#9" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#^$$!0-k9J![>9! #=$\"0j>rq*>>=!#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 0 178 "real_part := proc(u)\n Re(newton(R(z)=ex p(u*Pi*I),z=1.76*I))\nend proc:\nDigits := 15:\nu0 := bisect('real_par t'(u),u=0.55..0.58);\nnewton(R(z)=exp(u0*Pi*I),z=1.76*I);\nDigits := 1 0:" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%#u0G$\"0')fUmIOi&!#:" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#^#$\"0T`XK@Ww\"!#9" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 4 "The " }{TEXT 260 112 "larg est interval on the nonnegative imaginary axis that contains the origi n and lies inside the stability region" }{TEXT -1 5 " is " }{XPPEDIT 18 0 "[0, 1.7644];" "6#7$\"\"!-%&FloatG6$\"&Ww\"!\"%" }{TEXT -1 18 " \+ (approximately)." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 37 "#------------------------------------" }}{EXCHG {PARA 0 " > " 0 "" {MPLTEXT 1 0 1 ";" }}}}{PARA 0 "" 0 "" {TEXT -1 38 "#-------- -----------------------------" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}}{PARA 0 "" 0 "" {TEXT -1 42 "#============================ =============" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 23 "#======================" }}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 23 "Abreviated calculations" }}{PARA 257 "" 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 0 548 "SO6_7 := SimpleOr derConditions(6,7,'expanded'):\ncdns := [seq(SO6_7[i],i=[1,2,4,8,13,16 ,24,28,29,32])]:\nSO_eqs := [op(RowSumConditions(7,'expanded')),op(Sta geOrderConditions(2,7,'expanded'))]:\nsimp_eqs := [add(b[i]*a[i,1],i=2 ..7)=b[1],seq(add(b[i]*a[i,j],i=j+1..7)=b[j]*(1-c[j]),j=[3,4,5,6])]:\n simp_eqs2 := [add(b[i]*c[i]*a[i,2],i=3..7)=0,add(b[i]*c[i]*add(a[i,j]* a[j,2],j=3..i-1),i=3..7)=0,\n add(b[i]*c[i]^2*a[i,2],i=3.. 7)=0]:\ncdns := [op(cdns),op(simp_eqs),op(simp_eqs2),op(SO_eqs)]:\nerr terms6_7 := PrincipalErrorTerms(6,7,'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 3 " \+ " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1315 "calc_RKcoeffs := proc()\n local c4,c5,c6,eqns,sol,sm,ct,Rz ,stb6,nrm;\n global cdns,e1,e2,e3;\n\n c4 := simplify(c_3/(15*c_3^ 2-10*c_3+2)); \n if nargs>0 and args[1]='equal_nodes' then\n \+ c6 := 1/3*(9*c_3-4)/(5*c_3-2);\n c5 := c6;\n e1 := \{c[2] =c_2,c[3]=c_3,c[4]=c4,c[5]=c5,c[6]=c6,c[7]=1,b[2]=0\};\n eqns := subs(e1,cdns);\n sol := solve(\{op(eqns),b[5]=b[6]\});\n else \n c5 := c_5;\n c6 := c_6;\n e1 := \{c[2]=c_2,c[3]=c_3, c[4]=c4,c[5]=c5,c[6]=c6,c[7]=1,b[2]=0\};\n eqns := subs(e1,cdns); \n sol := solve(\{op(eqns)\});\n end if;\n if sol<>NULL then \+ e2 := sol\n else error \"try different nodes\" end if;\n e3 := `un ion`(e1,e2);\n Digits := 14;\n sm := 0;\n for ct to nops(errterm s6_7) do\n sm := sm+evalf(subs(e3,errterms6_7[ct]))^2;\n end do ;\n Rz := subs(e3,StabilityFunction(6,7,'expanded'));\n stb6 := ma x(fsolve(Rz=1,z=-8..-1e-7),fsolve(Rz=-1,z=-8..-1e-7));\n stb6 := eva lf[8](stb6);\n print(`nodes:`,c[2]=c_2,c[3]=c_3,c[4]=c4,c[5]=c5,c[6] =c6); \n print(`weights:`,seq(b[i]=evalf[6](subs(e3,b[i])),i=[1,$3 ..7]));\n nrm := max(seq(seq(subs(e3,abs(a[i,j])),j=1..i-1),i=2..7)) ;\n print(infinity*`-norm of linking coeffs`=evalf[10](nrm));\n pr int(`2-norm of principal error` = evalf[10](sqrt(sm)));\n print(`sta bility interval` = [stb6,0]);\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 68 "#---------------------- ---------------------------------------------" }}{PARA 0 "" 0 "" {TEXT -1 18 "Butcher's scheme A" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 64 "c_2 := 1/2: c_3 := 2/3: c_5 := 5/6: c_6 := 1/6:\ncalc_RKcoeffs ();" }}{PARA 11 "" 1 "" {XPPMATH 20 "6(%'nodes:G/&%\"cG6#\"\"##\"\"\"F (/&F&6#\"\"$#F(F./&F&6#\"\"%#F*F./&F&6#\"\"&#F8\"\"'/&F&6#F:#F*F:" }} {PARA 11 "" 1 "" {XPPMATH 20 "6)%)weights:G/&%\"bG6#\"\"\"$\"'++l!\"(/ &F&6#\"\"$$\"'+]F!\"'/&F&6#\"\"%F0/&F&6#\"\"&$\"'++;F2/&F&6#\"\"'F;/&F &6#\"\"(F)" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/*&%)infinityG\"\"\"%8-n orm~of~linking~coeffsGF&$\"+E5kDI!\"*" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/%:2-norm~of~principal~errorG$\"+sq,W\\!#7" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/%3stability~intervalG7$$!)!4h&G!\"(\"\"!" }}}{PARA 0 " " 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 27 "A scheme with simp le nodes " }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 64 "c_2 := 1/6: c_3 := 1/5: c_5 := 2/3: c_6 := 3/4:\ncalc_RKcoeffs();" }}{PARA 11 "" 1 " " {XPPMATH 20 "6(%'nodes:G/&%\"cG6#\"\"##\"\"\"\"\"'/&F&6#\"\"$#F*\"\" &/&F&6#\"\"%#F*F//&F&6#F1#F(F//&F&6#F+#F/F5" }}{PARA 11 "" 1 "" {XPPMATH 20 "6)%)weights:G/&%\"bG6#\"\"\"$\"'++v!\"(/&F&6#\"\"$$\"'-\" p\"!\"'/&F&6#\"\"%$\"'++FF2/&F&6#\"\"&$\"''Gk*F+/&F&6#\"\"'$\"'..JF2/& F&6#\"\"($\"'n;zF+" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/*&%)infinityG\" \"\"%8-norm~of~linking~coeffsGF&$\"+nmmm6!\"*" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/%:2-norm~of~principal~errorG$\"+'3V\\[#!#8" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/%3stability~intervalG7$$!)uxkS!\"(\"\"!" }} }{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 18 "Butcher' s scheme B" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 53 "c_2 := 1/3: c_ 3 := 2/3:\ncalc_RKcoeffs('equal_nodes');" }}{PARA 11 "" 1 "" {XPPMATH 20 "6(%'nodes:G/&%\"cG6#\"\"##\"\"\"\"\"$/&F&6#F+#F(F+/&F&6#\"\"%F)/&F &6#\"\"&#F*F(/&F&6#\"\"'F8" }}{PARA 11 "" 1 "" {XPPMATH 20 "6)%)weight s:G/&%\"bG6#\"\"\"$\"'nm\"*!\"(/&F&6#\"\"$$\"'+]n!\"'/&F&6#\"\"%F0/&F& 6#\"\"&$!'nmEF2/&F&6#\"\"'F;/&F&6#\"\"(F)" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/*&%)infinityG\"\"\"%8-norm~of~linking~coeffsGF&$\"+OOO O;!\"*" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/%:2-norm~of~principal~error G$\"+]\"!#7" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/%3stability~interv alG7$$!)!4h&G!\"(\"\"!" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 21 "Another scheme with " }{XPPEDIT 18 0 "c[5]=c[6]" "6 #/&%\"cG6#\"\"&&F%6#\"\"'" }{TEXT -1 1 "." }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 54 "c_2 := 3/17: c_3 := 2/9:\ncalc_RKcoeffs('equal_nodes' );" }}{PARA 11 "" 1 "" {XPPMATH 20 "6(%'nodes:G/&%\"cG6#\"\"##\"\"$\"# F2/&F&6#\"\"&$\"'wH=F2/&F&6# \"\"'F=/&F&6#\"\"($\"'dywF+" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/*&%)in finityG\"\"\"%8-norm~of~linking~coeffsGF&$\"+wP4C'*!#5" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/%:2-norm~of~principal~errorG$\"+(f6f;#!#8" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#/%3stability~intervalG7$$!)(HT&R!\"(\" \"!" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 41 "A scheme with small principal error norm." }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 66 "c_2 := 2/11: c_3 := 2/9: c_5 := 7/10: c_6 := 7/9:\nca lc_RKcoeffs();" }}{PARA 11 "" 1 "" {XPPMATH 20 "6(%'nodes:G/&%\"cG6#\" \"##F(\"#6/&F&6#\"\"$#F(\"\"*/&F&6#\"\"%#F.\"\"(/&F&6#\"\"&#F6\"#5/&F& 6#\"\"'#F6F0" }}{PARA 11 "" 1 "" {XPPMATH 20 "6)%)weights:G/&%\"bG6#\" \"\"$\"'Cts!\"(/&F&6#\"\"$$\"'PyG!\"'/&F&6#\"\"%$\"'8!*=F2/&F&6#\"\"&$ \"'_E:F2/&F&6#\"\"'$\"'jJAF2/&F&6#\"\"($\"'KguF+" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/*&%)infinityG\"\"\"%8-norm~of~linking~coeffsGF&$\"+9r# \\n(!#5" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/%:2-norm~of~principal~erro rG$\"+h(Rm5#!#8" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/%3stability~interv alG7$$!)(HT&R!\"(\"\"!" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 36 "minimizing the principal error norm " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 50 "eG: coefficients of a general scheme in t erms of " }{XPPEDIT 18 0 "c[2]" "6#&%\"cG6#\"\"#" }{TEXT -1 3 ", " } {XPPEDIT 18 0 "c[3]" "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 1 " " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 11160 "eG := \{b[6] = 1/60*(1-5* c[3]+5*c[3]^2)*(15*c[5]*c[3]-9*c[3]+4-6*c[5])/c[6]/(-1+c[6])/(2*c[6]-1 0*c[6]*c[3]+15*c[6]*c[3]^2-c[3])/(c[3]-c[6])/(c[6]-c[5]), a[7,2] = 1/2 *c[3]*(14*c[6]-20*c[5]*c[6]-30*c[6]*c[3]+45*c[5]*c[6]*c[3]+21*c[3]-10+ 14*c[5]-30*c[5]*c[3])/c[2]/(300*c[5]*c[3]^3*c[6]-225*c[3]^3*c[6]+180*c [3]^3-225*c[3]^3*c[5]-455*c[5]*c[6]*c[3]^2+350*c[6]*c[3]^2-285*c[3]^2+ 350*c[5]*c[3]^2+210*c[5]*c[6]*c[3]-165*c[6]*c[3]+136*c[3]-165*c[5]*c[3 ]-30*c[5]*c[6]+24*c[6]-20+24*c[5]), a[6,4] = 1/6*c[6]*(15*c[3]^2-10*c[ 3]+2)^2*(2*c[6]-10*c[6]*c[3]+15*c[6]*c[3]^2-c[3])*(c[3]-c[6])*(90*c[3] ^3*c[6]-120*c[6]*c[3]^2+48*c[6]*c[3]-6*c[6]-225*c[3]^3*c[5]+225*c[5]^2 *c[3]^3-240*c[5]^2*c[3]^2+9*c[3]^2+255*c[5]*c[3]^2-100*c[5]*c[3]-4*c[3 ]+90*c[5]^2*c[3]+14*c[5]-12*c[5]^2)/c[3]^2/(1-5*c[3]+5*c[3]^2)/(15*c[3 ]^2-10*c[3]+1)/(15*c[5]*c[3]^2-10*c[5]*c[3]+2*c[5]-c[3])/(15*c[5]*c[3] -9*c[3]+4-6*c[5]), a[7,6] = -(15*c[5]*c[3]-9*c[3]+4-6*c[5])*(1-5*c[3]+ 5*c[3]^2)*(c[3]-1)*(-1+c[6])*(c[5]-1)*(3*c[3]-1)*(5*c[3]-2)/c[6]/(2*c[ 6]-10*c[6]*c[3]+15*c[6]*c[3]^2-c[3])/(c[3]-c[6])/(c[6]-c[5])/(300*c[5] *c[3]^3*c[6]-225*c[3]^3*c[6]+180*c[3]^3-225*c[3]^3*c[5]-455*c[5]*c[6]* c[3]^2+350*c[6]*c[3]^2-285*c[3]^2+350*c[5]*c[3]^2+210*c[5]*c[6]*c[3]-1 65*c[6]*c[3]+136*c[3]-165*c[5]*c[3]-30*c[5]*c[6]+24*c[6]-20+24*c[5]), \+ a[7,4] = 1/6*(c[3]-1)*(3*c[3]-1)*(5*c[3]-2)*(15*c[3]^2-10*c[3]+2)^2*(- 48-1418*c[5]*c[3]+6750*c[3]^5*c[6]^2+37020*c[5]*c[3]^3*c[6]-14810*c[5] *c[6]*c[3]^2-1256*c[6]*c[3]-10800*c[3]^5*c[5]-24750*c[6]^2*c[3]^4*c[5] ^2-16875*c[5]^2*c[3]^5*c[6]+168*c[5]^2*c[6]+3044*c[5]*c[6]*c[3]-24600* c[6]^2*c[3]^3*c[5]+9720*c[6]^2*c[3]^2*c[5]+852*c[5]^2*c[3]-1980*c[6]^2 *c[3]*c[5]+10440*c[6]^2*c[3]^3+558*c[3]-2678*c[3]^2+6576*c[3]^3-8190*c [3]^4+4050*c[3]^5-72*c[5]^2+108*c[6]-72*c[6]^2+120*c[5]+31950*c[5]^2*c [3]^4*c[6]+24300*c[3]^5*c[6]*c[5]-47250*c[5]*c[3]^4*c[6]-16875*c[3]^5* c[6]^2*c[5]+13500*c[3]^5*c[6]^2*c[5]^2-7200*c[6]^2*c[5]^2*c[3]^2-24600 *c[5]^2*c[3]^3*c[6]-1980*c[5]^2*c[6]*c[3]+9720*c[5]^2*c[3]^2*c[6]+3195 0*c[6]^2*c[3]^4*c[5]+18600*c[6]^2*c[3]^3*c[5]^2+1440*c[6]^2*c[5]^2*c[3 ]+852*c[6]^2*c[3]+18810*c[3]^4*c[6]+168*c[6]^2*c[5]+6910*c[5]*c[3]^2-4 170*c[6]^2*c[3]^2-4170*c[5]^2*c[3]^2+21600*c[3]^4*c[5]-120*c[6]^2*c[5] ^2-13275*c[5]^2*c[3]^4-17175*c[3]^3*c[5]+6750*c[3]^5*c[5]^2-13275*c[6] ^2*c[3]^4-9450*c[3]^5*c[6]-260*c[5]*c[6]+10440*c[5]^2*c[3]^3+6052*c[6] *c[3]^2-14955*c[3]^3*c[6])/c[3]^2/(15*c[3]^2-10*c[3]+1)/(2*c[6]-10*c[6 ]*c[3]+15*c[6]*c[3]^2-c[3])/(15*c[5]*c[3]^2-10*c[5]*c[3]+2*c[5]-c[3])/ (300*c[5]*c[3]^3*c[6]-225*c[3]^3*c[6]+180*c[3]^3-225*c[3]^3*c[5]-455*c [5]*c[6]*c[3]^2+350*c[6]*c[3]^2-285*c[3]^2+350*c[5]*c[3]^2+210*c[5]*c[ 6]*c[3]-165*c[6]*c[3]+136*c[3]-165*c[5]*c[3]-30*c[5]*c[6]+24*c[6]-20+2 4*c[5]), a[6,3] = 1/6*(c[3]-c[6])*(-2*c[5]*c[3]+1350*c[3]^5*c[6]^2-253 5*c[5]*c[3]^3*c[6]+1450*c[5]*c[6]*c[3]^2-10*c[6]*c[3]-450*c[3]^5*c[5]- 24*c[5]^2*c[6]-328*c[5]*c[6]*c[3]-6*c[5]^2*c[3]+2100*c[6]^2*c[3]^3-4*c [3]^2+89*c[3]^3-300*c[3]^4+270*c[3]^5-12*c[6]^2-2250*c[5]^2*c[3]^4*c[6 ]+900*c[3]^5*c[6]*c[5]+975*c[5]*c[3]^4*c[6]+3075*c[5]^2*c[3]^3*c[6]+30 0*c[5]^2*c[6]*c[3]-1470*c[5]^2*c[3]^2*c[6]+156*c[6]^2*c[3]+1815*c[3]^4 *c[6]-35*c[5]*c[3]^2-810*c[6]^2*c[3]^2+135*c[5]^2*c[3]^2+300*c[3]^4*c[ 5]+450*c[5]^2*c[3]^4+45*c[3]^3*c[5]-2700*c[6]^2*c[3]^4-1350*c[3]^5*c[6 ]+28*c[5]*c[6]-480*c[5]^2*c[3]^3+148*c[6]*c[3]^2-820*c[3]^3*c[6])*c[6] /(1-5*c[3]+5*c[3]^2)/(15*c[3]^2-10*c[3]+1)/(15*c[5]*c[3]-9*c[3]+4-6*c[ 5])/(-c[5]+c[3])/c[3]^2, b[3] = -1/60*(75*c[5]*c[6]*c[3]^2-60*c[5]*c[6 ]*c[3]+10*c[5]*c[6]-45*c[5]*c[3]^2+35*c[5]*c[3]-6*c[5]-45*c[6]*c[3]^2+ 35*c[6]*c[3]-6*c[6]+30*c[3]^2-23*c[3]+4)/(c[3]-c[6])/(-c[5]+c[3])/c[3] ^2/(c[3]-1)/(15*c[3]^2-10*c[3]+1), b[7] = 1/60*(300*c[5]*c[3]^3*c[6]-2 25*c[3]^3*c[6]+180*c[3]^3-225*c[3]^3*c[5]-455*c[5]*c[6]*c[3]^2+350*c[6 ]*c[3]^2-285*c[3]^2+350*c[5]*c[3]^2+210*c[5]*c[6]*c[3]-165*c[6]*c[3]+1 36*c[3]-165*c[5]*c[3]-30*c[5]*c[6]+24*c[6]-20+24*c[5])/(c[3]-1)/(-1+c[ 6])/(c[5]-1)/(3*c[3]-1)/(5*c[3]-2), c[7] = 1, b[4] = 1/60*(15*c[3]^2-1 0*c[3]+2)^5*(3*c[3]-5*c[5]*c[3]-5*c[6]*c[3]+10*c[5]*c[6]*c[3]-5*c[5]*c [6]-2+3*c[6]+3*c[5])/(15*c[3]^2-10*c[3]+1)/c[3]^2/(3*c[3]-1)/(5*c[3]-2 )/(2*c[6]-10*c[6]*c[3]+15*c[6]*c[3]^2-c[3])/(15*c[5]*c[3]^2-10*c[5]*c[ 3]+2*c[5]-c[3]), c[4] = c[3]/(15*c[3]^2-10*c[3]+2), b[2] = 0, a[5,2] = 1/2*c[3]*c[5]/c[2]*(-5*c[5]+1+5*c[5]^2)/(1-5*c[3]+5*c[3]^2), a[4,2] = 1/2*c[3]^2*(4-30*c[3]+45*c[3]^2)/c[2]/(15*c[3]^2-10*c[3]+2)^3, a[6,2] = 1/2*c[6]*c[3]/c[2]*(5*c[6]^2-5*c[6]+1)/(1-5*c[3]+5*c[3]^2), a[4,3] \+ = -(15*c[3]^2-10*c[3]+1)*c[3]/(15*c[3]^2-10*c[3]+2)^3, b[1] = -1/60*1/ c[3]^2*(150*c[5]*c[3]^3*c[6]-75*c[3]^3*c[6]+45*c[3]^3-75*c[3]^3*c[5]+1 05*c[5]*c[3]^2-205*c[5]*c[6]*c[3]^2+105*c[6]*c[3]^2-65*c[3]^2+29*c[3]- 45*c[5]*c[3]+80*c[5]*c[6]*c[3]-45*c[6]*c[3]+6*c[5]-10*c[5]*c[6]-4+6*c[ 6])/c[5]/c[6], a[3,2] = 1/2*c[3]^2/c[2], a[5,1] = -1/6*c[5]*(225*c[2]* c[3]^5*c[5]-300*c[5]*c[2]*c[3]^4-45*c[2]*c[3]^4-225*c[5]^2*c[2]*c[3]^4 +15*c[5]^2*c[3]^3+3*c[3]^3-15*c[3]^3*c[5]+300*c[5]^2*c[2]*c[3]^3+190*c [5]*c[2]*c[3]^3+40*c[3]^3*c[2]-50*c[5]*c[2]*c[3]^2-11*c[3]^2*c[2]-175* c[5]^2*c[2]*c[3]^2+9*c[2]*c[5]*c[3]+40*c[5]^2*c[2]*c[3]-4*c[2]*c[5]^2) /c[3]^2/c[2]/(1-5*c[3]+5*c[3]^2), a[3,1] = -1/2*c[3]*(c[3]-2*c[2])/c[2 ], a[5,4] = 1/6*(15*c[3]^2-10*c[3]+2)^2*c[5]*(-c[5]+c[3])*(15*c[5]*c[3 ]^2-10*c[5]*c[3]+2*c[5]-c[3])/c[3]^2/(1-5*c[3]+5*c[3]^2)/(15*c[3]^2-10 *c[3]+1), a[2,1] = c[2], b[5] = -1/60*(1-5*c[3]+5*c[3]^2)*(15*c[6]*c[3 ]-9*c[3]+4-6*c[6])/(c[6]-c[5])/c[5]/(c[5]-1)/(15*c[5]*c[3]^2-10*c[5]*c [3]+2*c[5]-c[3])/(-c[5]+c[3]), a[4,1] = 1/2*c[3]*(-4*c[3]+30*c[3]^2-45 *c[3]^3+350*c[3]^2*c[2]-100*c[3]*c[2]+10*c[2]+450*c[2]*c[3]^4-600*c[3] ^3*c[2])/c[2]/(15*c[3]^2-10*c[3]+2)^3, a[5,3] = 1/6*c[5]*(150*c[3]^3*c [5]-145*c[5]*c[3]^2+40*c[5]*c[3]-4*c[5]-30*c[3]^3+20*c[3]^2-c[3])*(-c[ 5]+c[3])/c[3]^2/(1-5*c[3]+5*c[3]^2)/(15*c[3]^2-10*c[3]+1), a[7,5] = (c [3]-1)*(3*c[3]-1)*(5*c[3]-2)*(1-5*c[3]+5*c[3]^2)*(c[5]-1)*(c[5]-3*c[5] *c[3]+15*c[6]^2*c[3]-21*c[6]*c[3]-6*c[6]^2+9*c[3]-4+9*c[6])/c[5]/(15*c [5]*c[3]^2-10*c[5]*c[3]+2*c[5]-c[3])/(-c[5]+c[3])/(c[6]-c[5])/(300*c[5 ]*c[3]^3*c[6]-225*c[3]^3*c[6]+180*c[3]^3-225*c[3]^3*c[5]-455*c[5]*c[6] *c[3]^2+350*c[6]*c[3]^2-285*c[3]^2+350*c[5]*c[3]^2+210*c[5]*c[6]*c[3]- 165*c[6]*c[3]+136*c[3]-165*c[5]*c[3]-30*c[5]*c[6]+24*c[6]-20+24*c[5]), a[6,5] = -(3*c[3]-1)*(2*c[6]-10*c[6]*c[3]+15*c[6]*c[3]^2-c[3])*(c[3]- c[6])*(c[6]-c[5])*c[6]/(15*c[5]*c[3]-9*c[3]+4-6*c[5])/(-c[5]+c[3])/(15 *c[5]*c[3]^2-10*c[5]*c[3]+2*c[5]-c[3])/c[5], a[7,3] = 1/6*(c[3]-1)*(48 +1408*c[5]*c[3]-900*c[3]^5*c[6]^2-23250*c[5]*c[3]^3*c[6]+12622*c[5]*c[ 6]*c[3]^2+1234*c[6]*c[3]+3225*c[3]^5*c[5]-450*c[3]^6*c[5]+7650*c[6]^2* c[3]^4*c[5]^2-900*c[5]^2*c[3]^5*c[6]-168*c[5]^2*c[6]-2982*c[5]*c[6]*c[ 3]+14565*c[6]^2*c[3]^3*c[5]-8145*c[6]^2*c[3]^2*c[5]-840*c[5]^2*c[3]+19 32*c[6]^2*c[3]*c[5]+900*c[3]^6*c[6]*c[5]-7095*c[6]^2*c[3]^3-552*c[3]+2 360*c[3]^2-4452*c[3]^3-630*c[3]^6+3094*c[3]^4+360*c[3]^5+72*c[5]^2-108 *c[6]+72*c[6]^2-120*c[5]-8850*c[5]^2*c[3]^4*c[6]-1500*c[3]^5*c[6]*c[5] +16230*c[5]*c[3]^4*c[6]-900*c[3]^5*c[6]^2*c[5]+6120*c[6]^2*c[5]^2*c[3] ^2+14565*c[5]^2*c[3]^3*c[6]+1932*c[5]^2*c[6]*c[3]-8145*c[5]^2*c[3]^2*c [6]-8850*c[6]^2*c[3]^4*c[5]-11400*c[6]^2*c[3]^3*c[5]^2-1410*c[6]^2*c[5 ]^2*c[3]-840*c[6]^2*c[3]-6135*c[3]^4*c[6]-168*c[6]^2*c[5]-6222*c[5]*c[ 3]^2+900*c[3]^6*c[6]+3654*c[6]^2*c[3]^2+3654*c[5]^2*c[3]^2-11145*c[3]^ 4*c[5]+120*c[6]^2*c[5]^2+5595*c[5]^2*c[3]^4+12590*c[3]^3*c[5]-900*c[3] ^5*c[5]^2+5595*c[6]^2*c[3]^4-915*c[3]^5*c[6]+260*c[5]*c[6]-7095*c[5]^2 *c[3]^3-5202*c[6]*c[3]^2+9512*c[3]^3*c[6])/(c[3]-c[6])/(-c[5]+c[3])/(1 5*c[3]^2-10*c[3]+1)/c[3]^2/(300*c[5]*c[3]^3*c[6]-225*c[3]^3*c[6]+180*c [3]^3-225*c[3]^3*c[5]-455*c[5]*c[6]*c[3]^2+350*c[6]*c[3]^2-285*c[3]^2+ 350*c[5]*c[3]^2+210*c[5]*c[6]*c[3]-165*c[6]*c[3]+136*c[3]-165*c[5]*c[3 ]-30*c[5]*c[6]+24*c[6]-20+24*c[5]), a[6,1] = -1/6*c[6]*(-5100*c[6]*c[2 ]*c[5]*c[3]^4+12*c[6]^3*c[2]-60*c[5]*c[3]^3*c[6]+225*c[6]^2*c[3]^4*c[5 ]^2+204*c[3]^2*c[6]^2*c[2]-675*c[5]^2*c[2]*c[3]^5+60*c[6]^2*c[3]^3*c[5 ]-18*c[2]*c[6]^2*c[3]+495*c[2]*c[3]^5*c[5]-660*c[5]*c[2]*c[3]^4+870*c[ 5]^2*c[2]*c[3]^4+24*c[6]^2*c[2]*c[5]^2-28*c[6]^2*c[2]*c[5]-405*c[5]^2* c[2]*c[3]^3+307*c[5]*c[2]*c[3]^3-50*c[5]*c[2]*c[3]^2+66*c[5]^2*c[2]*c[ 3]^2+120*c[2]*c[3]^4*c[6]-225*c[5]^2*c[3]^4*c[6]+135*c[5]*c[3]^4*c[6]+ 90*c[5]^2*c[3]^3*c[6]-135*c[6]^2*c[3]^4*c[5]-90*c[6]^2*c[3]^3*c[5]^2-2 7*c[3]^4*c[5]+45*c[5]^2*c[3]^4+12*c[3]^3*c[5]-18*c[5]^2*c[3]^3-1890*c[ 6]*c[2]*c[5]^2*c[3]^3+3375*c[6]*c[2]*c[5]^2*c[3]^6-300*c[3]*c[6]^2*c[2 ]*c[5]^2-1870*c[3]^2*c[6]^2*c[2]*c[5]+435*c[3]^2*c[6]*c[2]*c[5]^2+2140 *c[3]^3*c[6]*c[2]*c[5]+54*c[2]*c[5]*c[6]*c[3]+4650*c[6]*c[2]*c[5]^2*c[ 3]^4+1350*c[6]^2*c[2]*c[3]^6+2190*c[6]^2*c[2]*c[3]^4-2700*c[6]^2*c[2]* c[3]^5-90*c[2]*c[3]^5*c[6]-930*c[3]^3*c[6]^2*c[2]+6*c[2]*c[6]*c[3]^2-4 8*c[2]*c[3]^3*c[6]-3375*c[6]*c[2]*c[5]*c[3]^6-485*c[3]^2*c[6]*c[2]*c[5 ]+352*c[3]*c[6]^2*c[2]*c[5]+3375*c[6]^2*c[2]*c[5]*c[3]^5-5850*c[6]*c[2 ]*c[5]^2*c[3]^5-4425*c[6]^2*c[2]*c[5]^2*c[3]^3+1650*c[6]^2*c[2]*c[5]^2 *c[3]^2+4875*c[6]^2*c[2]*c[5]*c[3]^3+5850*c[6]^2*c[2]*c[5]^2*c[3]^4+63 00*c[6]*c[2]*c[5]*c[3]^5-3375*c[6]^2*c[2]*c[5]^2*c[3]^5-6300*c[6]^2*c[ 2]*c[5]*c[3]^4-54*c[2]*c[5]^2*c[6]*c[3]-2100*c[6]^3*c[2]*c[3]^3+2700*c [6]^3*c[2]*c[3]^4-1350*c[6]^3*c[2]*c[3]^5+810*c[3]^2*c[6]^3*c[2]-156*c [3]*c[6]^3*c[2])/c[5]/c[3]^2/(15*c[5]*c[3]-9*c[3]+4-6*c[5])/c[2]/(1-5* c[3]+5*c[3]^2), a[7,1] = -1/6*(94320*c[6]*c[2]*c[5]*c[3]^4-30*c[5]*c[3 ]^3*c[6]+135*c[6]^2*c[3]^4*c[5]^2+5706*c[3]^2*c[6]^2*c[2]-108*c[2]*c[6 ]-21150*c[5]^2*c[2]*c[3]^5-12870*c[2]*c[3]^5+42*c[6]^2*c[3]^3*c[5]-100 8*c[5]^2*c[2]*c[3]+72*c[2]*c[5]^2+4050*c[2]*c[3]^6-9450*c[2]*c[6]*c[3] ^6+3666*c[3]^2*c[2]-10494*c[3]^3*c[2]+16260*c[2]*c[3]^4-1008*c[2]*c[6] ^2*c[3]+72*c[6]^2*c[2]+34020*c[2]*c[3]^5*c[5]-42540*c[5]*c[2]*c[3]^4+2 6280*c[5]^2*c[2]*c[3]^4+260*c[2]*c[5]*c[6]+120*c[6]^2*c[2]*c[5]^2-168* c[6]^2*c[2]*c[5]-168*c[6]*c[2]*c[5]^2-16650*c[5]^2*c[2]*c[3]^3+27144*c [5]*c[2]*c[3]^3-9372*c[5]*c[2]*c[3]^2+5706*c[5]^2*c[2]*c[3]^2+1668*c[2 ]*c[5]*c[3]-37530*c[2]*c[3]^4*c[6]+6750*c[5]^2*c[2]*c[3]^6-10800*c[5]* c[2]*c[3]^6+48*c[2]-90*c[5]^2*c[3]^4*c[6]+63*c[5]*c[3]^4*c[6]+42*c[5]^ 2*c[3]^3*c[6]-90*c[6]^2*c[3]^4*c[5]-60*c[6]^2*c[3]^3*c[5]^2-660*c[3]*c [2]-120*c[5]*c[2]+39960*c[6]*c[2]*c[5]^2*c[3]^3-16875*c[6]*c[2]*c[5]^2 *c[3]^6-1710*c[3]*c[6]^2*c[2]*c[5]^2-13545*c[3]^2*c[6]^2*c[2]*c[5]-135 45*c[3]^2*c[6]*c[2]*c[5]^2-59782*c[3]^3*c[6]*c[2]*c[5]-3626*c[2]*c[5]* c[6]*c[3]-63840*c[6]*c[2]*c[5]^2*c[3]^4+6750*c[6]^2*c[2]*c[3]^6+26280* c[6]^2*c[2]*c[3]^4-21150*c[6]^2*c[2]*c[3]^5+29880*c[2]*c[3]^5*c[6]-166 50*c[3]^3*c[6]^2*c[2]-8352*c[2]*c[6]*c[3]^2+24066*c[2]*c[3]^3*c[6]+149 4*c[2]*c[6]*c[3]+24300*c[6]*c[2]*c[5]*c[3]^6+20512*c[3]^2*c[6]*c[2]*c[ 5]+2364*c[3]*c[6]^2*c[2]*c[5]+52200*c[6]^2*c[2]*c[5]*c[3]^5+13500*c[6] ^2*c[2]*c[5]^2*c[3]^6-16875*c[6]^2*c[2]*c[5]*c[3]^6+52200*c[6]*c[2]*c[ 5]^2*c[3]^5-30030*c[6]^2*c[2]*c[5]^2*c[3]^3+9990*c[6]^2*c[2]*c[5]^2*c[ 3]^2+39960*c[6]^2*c[2]*c[5]*c[3]^3+48930*c[6]^2*c[2]*c[5]^2*c[3]^4-760 50*c[6]*c[2]*c[5]*c[3]^5-40950*c[6]^2*c[2]*c[5]^2*c[3]^5-63840*c[6]^2* c[2]*c[5]*c[3]^4+2364*c[2]*c[5]^2*c[6]*c[3])/c[6]/c[5]/c[3]^2/c[2]/(30 0*c[5]*c[3]^3*c[6]-225*c[3]^3*c[6]+180*c[3]^3-225*c[3]^3*c[5]-455*c[5] *c[6]*c[3]^2+350*c[6]*c[3]^2-285*c[3]^2+350*c[5]*c[3]^2+210*c[5]*c[6]* c[3]-165*c[6]*c[3]+136*c[3]-165*c[5]*c[3]-30*c[5]*c[6]+24*c[6]-20+24*c [5])\}:" }}}{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 20 "indets(map(rhs,eG));" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#<&&%\"cG6#\"\"$&F%6#\"\"&&F%6#\"\"'&F%6#\"\"#" }}} {PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 60 "errterms6_7 := PrincipalErrorTerms(6,7,'expanded'):\nnops(%);" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#\"#[" }}}{PARA 0 "" 0 "" {TEXT -1 0 " " }}{PARA 0 "" 0 "" {TEXT -1 24 "The following procedure " }{TEXT 0 13 "prin_err_norm" }{TEXT -1 112 " calculates the principal error norm of stage-order 2, 7 stage order 6 Runge-Kutta scheme given the four n odes " }{XPPEDIT 18 0 "c[2]" "6#&%\"cG6#\"\"#" }{TEXT -1 3 ", " } {XPPEDIT 18 0 "c[3]" "6#&%\"cG6#\"\"$" }{TEXT -1 3 ", " }{XPPEDIT 18 0 "c[5]" "6#&%\"cG6#\"\"&" }{TEXT -1 7 " and " }{XPPEDIT 18 0 "c[6] " "6#&%\"cG6#\"\"'" }{TEXT -1 1 "." }}{PARA 0 "" 0 "" {TEXT -1 63 "It \+ makes use of formulas for all the coefficients in terms of " } {XPPEDIT 18 0 "c[2]" "6#&%\"cG6#\"\"#" }{TEXT -1 3 ", " }{XPPEDIT 18 0 "c[3]" "6#&%\"cG6#\"\"$" }{TEXT -1 3 ", " }{XPPEDIT 18 0 "c[5]" "6# &%\"cG6#\"\"&" }{TEXT -1 7 " and " }{XPPEDIT 18 0 "c[6]" "6#&%\"cG6# \"\"'" }{TEXT -1 1 "." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 270 "prin_err_norm := proc(c2,c3,c5,c6)\n l ocal sm,ct;\n global eA,eB;\n eA := \{c[2]=c2,c[3]=c3,c[5]=c5,c[6] =c6\};\n eB := `union`(evalf(simplify(subs(eA,eG))),eA);\n sm := 0 :\n for ct to 48 do\n sm := sm+subs(eB,errterms6_7[ct])^2;\n \+ end do:\n sqrt(sm);\nend proc:" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }} {PARA 0 "" 0 "" {TEXT -1 70 "A near minimum value for the principal er ror norm is given by taking " }{XPPEDIT 18 0 "c[2]=2/11" "6#/&%\"cG6# \"\"#*&F'\"\"\"\"#6!\"\"" }{TEXT -1 3 ", " }{XPPEDIT 18 0 "c[3]=2/9" "6#/&%\"cG6#\"\"$*&\"\"#\"\"\"\"\"*!\"\"" }{TEXT -1 3 ", " }{XPPEDIT 18 0 "c[5]=7/10" "6#/&%\"cG6#\"\"&*&\"\"(\"\"\"\"#5!\"\"" }{TEXT -1 7 " and " }{XPPEDIT 18 0 "c[6]=7/9" "6#/&%\"cG6#\"\"'*&\"\"(\"\"\"\"\" *!\"\"" }{TEXT -1 1 "." }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 95 "Di gits := 10:\nc_2 := 2/11: c_3 := 2/9: c_5 := 7/10: c_6 := 7/9:\nprin_e rr_norm(c_2,c_3,c_5,c_6);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#$\"+#yRm5 #!#8" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 81 " The following graph shows that the principal error norm has a minimum \+ for 0.75 < " }{XPPEDIT 18 0 "c[6]" "6#&%\"cG6#\"\"'" }{TEXT -1 13 " < \+ 0.8 with " }{XPPEDIT 18 0 "c[2]" "6#&%\"cG6#\"\"#" }{TEXT -1 3 ", " }{XPPEDIT 18 0 "c[3]" "6#&%\"cG6#\"\"$" }{TEXT -1 7 " and " } {XPPEDIT 18 0 "c[5]" "6#&%\"cG6#\"\"&" }{TEXT -1 18 " remaining fixed ." }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 165 "Digits := 20:\nc_2 := \+ 2/11: c_3 := 2/9: c_5 := 7/10: c_6 := 7/9:\nplot('prin_err_norm'(c_2,c _3,c_5,c[6]),c[6]=0.75..0.8,0.2105e-3..0.21245e-3,color=COLOR(RGB,.5,0 ,1));" }}{PARA 13 "" 1 "" {GLPLOT2D 359 354 354 {PLOTDATA 2 "6&-%'CURV ESG6#7S7$$\"#v!\"#$\"5>68x(>xXX7#!#B7$$\"5LLLL3x&)*3^(!#?$\"51N6yK)=BI 7#F-7$$\"5nmm\"H2P\"Q?vF1$\"5ZONq_C@F-7$$\"5nmm\"z%4\\Y_vF1$\"5o n=_8$p>y6#F-7$$\"5LLLeR-/PivF1$\"5[asx*Rd@n6#F-7$$\"5+++DcmpisvF1$\"5( [)=N\"HSUc6#F-7$$\"5LLLe*)>VB$e(F1$\"5$H*[uQ9%)e9@F-7$$\"5+++DJbw!Qf(F 1$\"5#e\"p#oB!3g8@F-7$$\"5nmmmTIOo/wF1$\"5**e%ylUr]E6#F-7$$\"5LLL$3_>j Uh(F1$\"5!e!Rk\\x#p=6#F-7$$\"5++++D;v/DwF1$\"5wnA$o]!=06@F-7$$\"5++++v =h(ej(F1$\"5,PjuiiuH5@F-7$$\"5++++v$[6jk(F1$\"59SL%e?qL'4@F-7$$\"5LLLe *[z(ybwF1$\"5\"Q_!R&fz%34@F-7$$\"5nmmmTXg0nwF1$\"5^w54ib*)\\3@F-7$$\"5 nmmmmJ!y5hSWh2@F-7$$\" 5nmmm;pW`(p(F1$\"5)GRa1Cgzs5#F-7$$\"5+++D1f#=$3xF1$\"5!*G#*Q)zMwp5#F-7 $$\"5+++v=xpe=xF1$\"5sS%pHy0]n5#F-7$$\"5nmm;H28IHxF1$\"55/!psH0zl5#F-7 $$\"5nmm\"zpSS\"RxF1$\"5vI2\"=0p![1@F-7$$\"5LLL$3_?`(\\xF1$\"5CN88q&oP k5#F-7$$\"5LLLe*)>pxgxF1$\"5a\"\\+P`Zik5#F-7$$\"5+++v$f4t.x(F1$\"5g'RV \\ByTl5#F-7$$\"5LLL$e*Gst!y(F1$\"5<3d3'\\*yo1@F-7$$\"5++++]#RW9z(F1$\" 5d1p*=C*[!p5#F-7$$\"5+++]7j#>>!yF1$\"5L(\\IL&\\A=2@F-7$$\"5+++D1RU07yF 1$\"5()3:ROl>^2@F-7$$\"5+++](=S2L#yF1$\"5Az-5ry)[z5#F-7$$\"5nmmm;p)=M$ yF1$\"5+d\\dHM^S3@F-7$$\"5++++v=]@WyF1$\"5#fkEY7#*e*3@F-7$$\"5LLLe*[$z *R&yF1$\"5O$H'3Bh-_4@F-7$$\"5++++DYKpkyF1$\"5v#pq\\c\"))>5@F-7$$\"5nmm \"H2qcZ(yF1$\"59$>%=hU#**36#F-7$$\"5+++DJ5fF&)yF1$\"5$\\X>#42dp6@F-7$$ \"5nmmmTg.c&*yF1$\"5+m!4J*\\!QD6#F-7$$\"5+++DcEsK1zF1$\"5`K.45Uu[8@F-7 $$\"5MLLLL)*pp;zF1$\"5fO.kyIsY9@F-7$$\"5MLL$3xe,t#zF1$\"5#yrPs!*oNb6#F -7$$\"5nmm\"HdO=y$zF1$\"5WvXL4)zhm6#F-7$$\"5++++]#>#[ZzF1$\"5$43iE#[]v <@F-7$$\"5nmm;aG!e&ezF1$\"5uCF?v@F-7$$\"5MLLLL)Qk%ozF1$\"5*3j^$))> >K?@F-7$$\"5+++D1Mm-zzF1$\"5B(*f]dvVr@@F-7$$\"5+++v$40O\"*)zF1$\"5.d\" 4O4>5J7#F-7$$\"\")!\"\"$\"5r3#*yAa!zY7#F--%+AXESLABELSG6$Q%c[6]6\"Q!F^ [l-%&COLORG6&%$RGBG$\"\"&Fgz$\"\"!Fg[l$\"\"\"Fg[l-%%VIEWG6$;F(Fez;$\"% 0@!\"($\"&X7#!\")" 1 2 0 1 10 0 2 6 1 4 2 1.000000 45.000000 45.000000 0 0 "Curve 1" }}}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 " " 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 33 "We can use the spe cial procedure " }{TEXT 0 7 "findmin" }{TEXT -1 55 " to minimize the p rincipal error norm with respect to " }{XPPEDIT 18 0 "c[6]" "6#&%\"cG 6#\"\"'" }{TEXT -1 8 " with " }{XPPEDIT 18 0 "c[2]" "6#&%\"cG6#\"\"# " }{TEXT -1 3 ", " }{XPPEDIT 18 0 "c[3]" "6#&%\"cG6#\"\"$" }{TEXT -1 7 " and " }{XPPEDIT 18 0 "c[5]" "6#&%\"cG6#\"\"&" }{TEXT -1 18 " re maining fixed." }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 122 "Digits := 14:\nc_2 := 2/11: c_3 := 2/9: c_5 := 7/10: c_6 := 7/9:\nfindmin('prin _err_norm'(c_2,c_3,c_5,c[6]),c[6]=0.75..0.8);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#7$$\"/(o>I29v(!#9$\"/Vy))oV1@!#<" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 82 "The following graph shows that the principal error norm has a minimum for 0.664 < " }{XPPEDIT 18 0 "c[5];" "6#&%\"cG6#\"\"&" }{TEXT -1 14 " < 0.73 with " } {XPPEDIT 18 0 "c[2]" "6#&%\"cG6#\"\"#" }{TEXT -1 3 ", " }{XPPEDIT 18 0 "c[3]" "6#&%\"cG6#\"\"$" }{TEXT -1 7 " and " }{XPPEDIT 18 0 "c[6]; " "6#&%\"cG6#\"\"'" }{TEXT -1 18 " remaining fixed." }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 180 "Digits := 20:\nc_2 := 2/11: c_3 := 2/9: \+ c_5 := 7/10: c_6 := .77514073019687:\nplot('prin_err_norm'(c_2,c_3,c[5 ],c_6),c[5]=0.664..0.73,0.2105e-3..0.21233e-3,color=COLOR(RGB,0,.65,0) );" }}{PARA 13 "" 1 "" {GLPLOT2D 359 354 354 {PLOTDATA 2 "6&-%'CURVESG 6#7S7$$\"$k'!\"$$\"5-([.(HTFLB@!#B7$$\"5++++v@hQam!#?$\"5viN?w!=\"*=7# F-7$$\"5+++DO4M!pm'F1$\"5i]&yC`>)o?@F-7$$\"5+++]U3/)4o'F1$\"5X+Tq*)eCR >@F-7$$\"5+++](*=2:&p'F1$\"5L4e3@W$\\\"=@F-7$$\"5+++D^!o`#4nF1$\"5A'[i .DAtp6#F-7$$\"5+++D;J*GBs'F1$\"59lH.[Pt$f6#F-7$$\"5+++D'efnet'F1$\"5A) of:a6?\\6#F-7$$\"5+++DE-$p)\\nF1$\"5g)zc**[XFR6#F-7$$\"5+++D,0h#Qw'F1$ \"52^<0aB\")*H6#F-7$$\"5++++:#R#=ynF1$\"5m-@x')H\\57@F-7$$\"5+++]noF1$\"577ckWgo*)4@F-7$$\"5++++be68LoF1$\"5=rq*fsVv#4@F-7$$\"5+++DE*)) Rc%oF1$\"5@(pyB#\\@w3@F-7$$\"5++++&*zR^goF1$\"5ewVQ%Q1:#3@F-7$$\"5++++ !eG9J(oF1$\"5*ef$R$HU0y5#F-7$$\"5+++D'pVrx)oF1$\"5!=&3(=\"f5R2@F-7$$\" 5++++I*\\X2!pF1$\"5'=T^2T2!32@F-7$$\"5+++D'>5!)\\\"pF1$\"5Ml0&R))=*z1@ F-7$$\"5+++v))4[`GpF1$\"5/DHQ+0/f1@F-7$$\"5+++]iDxnUpF1$\"5^z[&=PhLk5# F-7$$\"5+++D@P`mbpF1$\"5$o'p%3McWj5#F-7$$\"5+++](3Bu'ppF1$\"5k,\\im#[2 j5#F-7$$\"5+++DEMbA%)pF1$\"5)Q@'GjxPL1@F-7$$\"5+++vj'[#*o*pF1$\"5Ab>1V P/T1@F-7$$\"5+++]AUJd5qF1$\"55at@TX%\\l5#F-7$$\"5++++5)f1Z-(F1$\"5d7Im M`Vv1@F-7$$\"5+++]KFM`QqF1$\"5rOE`3)3:q5#F-7$$\"5+++Dc&f6>0(F1$\"5kfb3 w(3Ct5#F-7$$\"5+++]ZqdwmqF1$\"5**f)GZv^Kx5#F-7$$\"5++++I2H6!3(F1$\"5Zu XTlG\"e\"3@F-7$$\"5++++vCQO%4(F1$\"5CL#*flvPn3@F-7$$\"5+++D1usF2rF1$\" 5A6g8a&f&>4@F-7$$\"5++++0&3&R@rF1$\"5P42\">fSD)4@F-7$$\"5+++D'\\%)yY8( F1$\"50upzw\\XZ5@F-7$$\"5+++Dh,Uc[rF1$\"5&[Yh[Ok676#F-7$$\"5++++vv'R@; (F1$\"5?Gu%[Z4!*>6#F-7$$\"5+++D1R>NwrF1$\"5[\"*HA`Qi'G6#F-7$$\"5++++!y .S+>(F1$\"5S.GQZl\"pP6#F-7$$\"5+++]x&4QS?(F1$\"55TWX;RCv9@F-7$$\"5+++D wU-#z@(F1$\"5_q$o8wJ(y:@F-7$$\"5++++5%\\w1B(F1$\"5WrC,v^1z;@F-7$$\"5++ +]n(f'HXsF1$\"53IOg$p?-!=@F-7$$\"5++++g#*HPesF1$\"5!)fP2$e^T\">@F-7$$ \"5+++D'p::BF(F1$\"5([+i\"4!39/7#F-7$$\"5+++vB(efcG(F1$\"5eq8@-#*yo@@F -7$$\"#t!\"#$\"5;0#=\\(>v6B@F--%+AXESLABELSG6$Q%c[5]6\"Q!F^[l-%&COLORG 6&%$RGBG$\"\"!Fe[l$\"#lFgzFd[l-%%VIEWG6$;F(Fez;$\"%0@!\"($\"&L7#!\")" 1 2 0 1 10 0 2 6 1 4 2 1.000000 45.000000 45.000000 0 0 "Curve 1" }}}} {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 33 "We can us e the special procedure " }{TEXT 0 7 "findmin" }{TEXT -1 55 " to minim ize the principal error norm with respect to " }{XPPEDIT 18 0 "c[5]; " "6#&%\"cG6#\"\"&" }{TEXT -1 8 " with " }{XPPEDIT 18 0 "c[2]" "6#&% \"cG6#\"\"#" }{TEXT -1 3 ", " }{XPPEDIT 18 0 "c[3]" "6#&%\"cG6#\"\"$ " }{TEXT -1 7 " and " }{XPPEDIT 18 0 "c[6];" "6#&%\"cG6#\"\"'" } {TEXT -1 18 " remaining fixed." }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 136 "Digits := 14:\nc_2 := 2/11: c_3 := 2/9: c_5 := 7/10: c_6 := . 77514073019687:\nfindmin('prin_err_norm'(c_2,c_3,c[5],c_6),c[5]=0.664. .0.73);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#7$$\"/(\\KUc6(p!#9$\"/Q0SrI 1@!#<" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 10 "We leave " }{XPPEDIT 18 0 "c[3] = 2/9;" "6#/&%\"cG6#\"\"$*&\"\"#\"\" \"\"\"*!\"\"" }{TEXT -1 63 " because this value makes the first princ ipal error term zero." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 81 "The following graph shows that the principal error nor m has a minimum for 0.16 < " }{XPPEDIT 18 0 "c[2];" "6#&%\"cG6#\"\"#" }{TEXT -1 14 " < 0.2 with " }{XPPEDIT 18 0 "c[3];" "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 18 " remaining fixed." }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 176 "Digit s := 20:\nc_2 := 2/11: c_3 := 2/9: c_5 := .69711564232497: c_6 := .775 14073019687:\nplot('prin_err_norm'(c[2],c_3,c_5,c_6),c[2]=0.16..0.2,0. 2105e-3..0.21111e-3,color=brown);" }}{PARA 13 "" 1 "" {GLPLOT2D 359 354 354 {PLOTDATA 2 "6&-%'CURVESG6#7S7$$\"#;!\"#$\"5P.B5()y*G56#!#B7$$ \"5nmmmmh)=(3;!#?$\"50XfrI7Ei5@F-7$$\"5LLLLe'40jh\"F1$\"5PP?mP_PG5@F-7 $$\"5nmmm;6m$[i\"F1$\"5rO7ZYN!>*4@F-7$$\"5nmmm;yYUL;F1$\"5y:<9T0%p&4@F -7$$\"5LLLLeF>(>k\"F1$\"5x1QHdz)Q#4@F-7$$\"5nmmm\">K'*)\\;F1$\"50BCe7* )z%*3@F-7$$\"5++++Dt:5e;F1$\"5m#HRn*eDm3@F-7$$\"5nmmm\"fX(em;F1$\"5Y3U pqZUQ3@F-7$$\"5++++DCh/v;F1$\"5*H(Hw+:R73@F-7$$\"5LLLLL/pu$o\"F1$\"5o% eQ+B$R(y5#F-7$$\"5nmmm;c0T\"p\"F1$\"5N%)**Q0.(ow5#F-7$$\"5+++++8!Q+q\" F1$\"5>+&*p>:dR32@F-7$$\"5nmmm\"fBIYs\"F1$\"5yIX2')*4Tp5#F-7$$\"5LLL LLO[kLh\"px$z!*y1@F-7$$\"5LLLLL&Q\"GT$)=F1$\"5ihu\"*zLg42@F-7$$\"5++ +++(fa<*=F1$\"5*=o0Q<$[F2@F-7$$\"5LLLLeg`!)**=F1$\"5SdHZ\"f(*eu5#F-7$$ \"5++++DG2A3>F1$\"5]P$G(eUzm2@F-7$$\"5LLLLL)G[k\">F1$\"5@?Ce#z_))y5#F- 7$$\"5++++D\"yh]#>F1$\"5BHL@\\-n83@F-7$$\"5nmmmm)fdL$>F1$\"5%G42J3R#R3 @F-7$$\"5nmmm;q7%=%>F1$\"5;w9J/v2n3@F-7$$\"5LLLLe#pa-&>F1$\"5$y_ql*HP' *3@F-7$$\"5+++++ad)z&>F1$\"5PlAxxMxC4@F-7$$\"5LLLL$GUYo'>F1$\"50Khbhr1 f4@F-7$$\"5nmmmm5:xu>F1$\"5s(3Djs;8*4@F-7$$\"5++++D28A$)>F1$\"5a'3>PvS t-6#F-7$$\"5++++vS)38*>F1$\"5?*\\L+0/M16#F-7$$\"\"#!\"\"$\"5r(p\"G]Z)Q 56#F--%+AXESLABELSG6$Q%c[2]6\"Q!F^[l-%'COLOURG6&%$RGBG$\")#)eqk!\")$\" ))eqk\"Ff[lFg[l-%%VIEWG6$;F(Fez;$\"%0@!\"($\"&66#Ff[l" 1 2 0 1 10 0 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 33 "We can use the special \+ procedure " }{TEXT 0 7 "findmin" }{TEXT -1 55 " to minimize the princi pal error norm with respect to " }{XPPEDIT 18 0 "c[2];" "6#&%\"cG6#\" \"#" }{TEXT -1 8 " with " }{XPPEDIT 18 0 "c[3];" "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 18 " rem aining fixed." }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 146 "Digits := \+ 14:\nc_2 := 2/11: c_3 := 2/9: c_5 := .69711564232497: c_6 := .77514073 019687:\nfindmin('prin_err_norm'(c[2],c_3,c_5,c_6),c[2]=0.163..0.2);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#7$$\"/\\+pG!)*z\"!#9$\"/'))*[oE1@!#< " }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 34 "By c ycling around the parameters " }{XPPEDIT 18 0 "c[2]" "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 61 " we can obtain a small reduction in the principal error norm." }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 487 "Digits := 14:\nc_2 := .17998028690 049: c_3 := 2/9: c_5 := .69711564232497: c_6 := .77514073019687:\nfor \+ ct to 7 do\n c_6 := op(1,findmin('prin_err_norm'(c_2,c_3,c_5,c[6]),c [6]=\{c_6-.3e-1,c_6,c_6+.3e-1\}));\n c_5 := op(1,findmin('prin_err_n orm'(c_2,c_3,c[5],c_6),c[5]=\{c_5-.3e-1,c_5,c_5+.3e-1\}));\n mn := f indmin('prin_err_norm'(c[2],c_3,c_5,c_6),c[2]=\{c_2-.3e-1,c_2,c_2+.3e- 1\});\n c_2 := op(1,mn);\n print(c[2]=c_2,c[5]=c_5,c[6]=c_6);\n \+ print(`principal error norm`=op(2,mn));\nend do:" }}{PARA 11 "" 1 "" {XPPMATH 20 "6%/&%\"cG6#\"\"#$\"/X$)y1%pz\"!#9/&F%6#\"\"&$\"/$[]2(4mpF */&F%6#\"\"'$\"/Tfsm5[xF*" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/%5princi pal~error~normG$\"/d:S(ei5#!#<" }}{PARA 11 "" 1 "" {XPPMATH 20 "6%/&% \"cG6#\"\"#$\"/2l))H\\'z\"!#9/&F%6#\"\"&$\"/[u8#=`'pF*/&F%6#\"\"'$\"/@ ]2(yvu(F*" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/%5principal~error~normG$ \"/cMT&ei5#!#<" }}{PARA 11 "" 1 "" {XPPMATH 20 "6%/&%\"cG6#\"\"#$\"/pI MJU'z\"!#9/&F%6#\"\"&$\"/o6Rm>lpF*/&F%6#\"\"'$\"/QF\"Q'\\ZxF*" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#/%5principal~error~normG$\"/n]O&ei5#!# <" }}{PARA 11 "" 1 "" {XPPMATH 20 "6%/&%\"cG6#\"\"#$\"/r)fB7kz\"!#9/&F %6#\"\"&$\"/8Gvw " 0 "" {MPLTEXT 1 0 120 "[c[2] = .17964102291106, c[5] = .6965174183 2593, c[6] = .77474811680069]:\nconvert(%,rational,4);\nconvert(%%,rat ional,5);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#7%/&%\"cG6#\"\"##\"\"(\"# R/&F&6#\"\"&#\"#B\"#L/&F&6#\"\"'#\"#C\"#J" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#7%/&%\"cG6#\"\"##\"#I\"$n\"/&F&6#\"\"&#\"$,\"\"$X\"/&F& 6#\"\"'#\"#')\"$6\"" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 98 "Digits := 10:\nc_2 := 7/39: c_3 := 2/9: c _5 := 23/33: c_6 := 24/31:\nprin_err_norm(c_2,c_3,c_5,c_6);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#$\"+I6F1@!#8" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 91 "This does not give much reduction in the principal error norm given by the initial values " }{XPPEDIT 18 0 "c[2]=2/11" "6#/&%\"cG6#\"\"#*&F'\"\"\"\"#6!\"\"" }{TEXT -1 3 ", " }{XPPEDIT 18 0 "c[3]=2/9" "6#/&%\"cG6#\"\"$*&\"\"#\"\"\"\"\"*!\"\"" } {TEXT -1 3 ", " }{XPPEDIT 18 0 "c[5]=7/10" "6#/&%\"cG6#\"\"&*&\"\"(\" \"\"\"#5!\"\"" }{TEXT -1 7 " and " }{XPPEDIT 18 0 "c[6]=7/9" "6#/&% \"cG6#\"\"'*&\"\"(\"\"\"\"\"*!\"\"" }{TEXT -1 1 "." }}{EXCHG {PARA 0 " > " 0 "" {MPLTEXT 1 0 95 "Digits := 10:\nc_2 := 2/11: c_3 := 2/9: c_5 \+ := 7/10: c_6 := 7/9:\nprin_err_norm(c_2,c_3,c_5,c_6);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#$\"+#yRm5#!#8" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }} {PARA 0 "" 0 "" {TEXT -1 77 "If we minimize the principal error norm w ith respect to all four parameters " }{XPPEDIT 18 0 "c[2]" "6#&%\"cG6 #\"\"#" }{TEXT -1 3 ", " }{XPPEDIT 18 0 "c[3]" "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 38 ". we ob tain a further small reduction." }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 588 "Digits := 14:\nc_2 := .17964102291106: c_3 := 2/9: c_5 := .69 651741832593: c_6 := .77474811680069:\nfor ct to 20 do\n c_6 := op(1 ,findmin('prin_err_norm'(c_2,c_3,c_5,c[6]),c[6]=\{c_6-.3e-1,c_6,c_6+.3 e-1\}));\n c_5 := op(1,findmin('prin_err_norm'(c_2,c_3,c[5],c_6),c[5 ]=\{c_5-.3e-1,c_5,c_5+.3e-1\}));\n c_3 := op(1,findmin('prin_err_nor m'(c_2,c[3],c_5,c_6),c[3]=\{c_3-.3e-1,c_3,c_3+.3e-1\}));\n mn := fin dmin('prin_err_norm'(c[2],c_3,c_5,c_6),c[2]=\{c_2-.3e-1,c_2,c_2+.3e-1 \});\n c_2 := op(1,mn);\n print(c[2]=c_2,c[3]=c_3,c[5]=c_5,c[6]=c_ 6);\n print(`principal error norm`=op(2,mn));\nend do:" }}{PARA 11 " " 1 "" {XPPMATH 20 "6&/&%\"cG6#\"\"#$\"/>%Qo,&4=!#9/&F%6#\"\"$$\"/!*=> x(*HAF*/&F%6#\"\"&$\"/U.sT?\"RqF*/&F%6#\"\"'$\"/?3e))*[v (F*" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/%5principal~error~normG$\"/\\q 5@>0@!#<" }}{PARA 11 "" 1 "" {XPPMATH 20 "6&/&%\"cG6#\"\"#$\"/?g[&Rh%= !#9/&F%6#\"\"$$\"/'\\A$\\!eB#F*/&F%6#\"\"&$\"/T.[17RqF*/&F%6#\"\"'$\"/ x!H&*)*[v(F*" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/%5principal~error~nor mG$\"/[q5@>0@!#<" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 120 "[c[2] = .18461395486020, c[3] = .22358049322 496, c[5] = .70391206480341, c[6] = .77548989529077]:\nconvert(%,ratio nal,4);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#7&/&%\"cG6#\"\"##\"#7\"#l/& F&6#\"\"$#\"#<\"#w/&F&6#\"\"&#\"#>\"#F/&F&6#\"\"'#\"#J\"#S" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 101 "D igits := 10:\nc_2 := 12/65: c_3 := 17/76: c_5 := 19/27: c_6 := 31/40: \nprin_err_norm(c_2,c_3,c_5,c_6);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#$ \"+\"*H@0@!#8" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 32 "#-------------------------------" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 " " 0 "" {TEXT -1 65 "A scheme with near minimal principal error norm wi th respect to " }{XPPEDIT 18 0 "c[2]" "6#&%\"cG6#\"\"#" }{TEXT -1 3 " , " }{XPPEDIT 18 0 "c[5]" "6#&%\"cG6#\"\"&" }{TEXT -1 7 " and " } {XPPEDIT 18 0 "c[6]" "6#&%\"cG6#\"\"'" }{TEXT -1 1 "." }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 69 "c_2 := 7/39: c_3 := 2/9: c_5 := 23/33: c_ 6 := 24/31:\ncalc_RKcoeffs();" }}{PARA 11 "" 1 "" {XPPMATH 20 "6(%'nod es:G/&%\"cG6#\"\"##\"\"(\"#R/&F&6#\"\"$#F(\"\"*/&F&6#\"\"%#F/F*/&F&6# \"\"&#\"#B\"#L/&F&6#\"\"'#\"#C\"#J" }}{PARA 11 "" 1 "" {XPPMATH 20 "6) %)weights:G/&%\"bG6#\"\"\"$\"'2ws!\"(/&F&6#\"\"$$\"'HwG!\"'/&F&6#\"\"% $\"'-'*=F2/&F&6#\"\"&$\"'4b8F2/&F&6#\"\"'$\"'z'R#F2/&F&6#\"\"($\"'9#[( F+" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/*&%)infinityG\"\"\"%8-norm~of~l inking~coeffsGF&$\"+9r#\\n(!#5" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/%:2 -norm~of~principal~errorG$\"+;6F1@!#8" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/%3stability~intervalG7$$!)(HT&R!\"(\"\"!" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 65 "A scheme with near minima l principal error norm with respect to " }{XPPEDIT 18 0 "c[2]" "6#&% \"cG6#\"\"#" }{TEXT -1 3 ", " }{XPPEDIT 18 0 "c[3];" "6#&%\"cG6#\"\"$ " }{TEXT -1 3 ", " }{XPPEDIT 18 0 "c[5]" "6#&%\"cG6#\"\"&" }{TEXT -1 7 " and " }{XPPEDIT 18 0 "c[6]" "6#&%\"cG6#\"\"'" }{TEXT -1 1 "." }} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 72 "c_2 := 12/65: c_3 := 17/76: \+ c_5 := 19/27: c_6 := 31/40:\ncalc_RKcoeffs();" }}{PARA 11 "" 1 "" {XPPMATH 20 "6(%'nodes:G/&%\"cG6#\"\"##\"#7\"#l/&F&6#\"\"$#\"#<\"#w/&F &6#\"\"%#\"%#H\"\"%nH/&F&6#\"\"&#\"#>\"#F/&F&6#\"\"'#\"#J\"#S" }} {PARA 11 "" 1 "" {XPPMATH 20 "6)%)weights:G/&%\"bG6#\"\"\"$\"'`(G(!\"( /&F&6#\"\"$$\"'=FH!\"'/&F&6#\"\"%$\"'v$)=F2/&F&6#\"\"&$\"'8.9F2/&F&6# \"\"'$\"'Z4BF2/&F&6#\"\"($\"'>xuF+" }}{PARA 11 "" 1 "" {XPPMATH 20 "6# /*&%)infinityG\"\"\"%8-norm~of~linking~coeffsGF&$\"+,'3e-)!#5" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#/%:2-norm~of~principal~errorG$\"+\")H@ 0@!#8" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/%3stability~intervalG7$$!)(* 4]R!\"(\"\"!" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 66 "A scheme with a larger stability region than the previous schemes." }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 69 "c_2 := 1/40: c_ 3 := 2/11: c_5 := 8/13: c_6 := 10/13:\ncalc_RKcoeffs();" }}{PARA 11 " " 1 "" {XPPMATH 20 "6(%'nodes:G/&%\"cG6#\"\"##\"\"\"\"#S/&F&6#\"\"$#F( \"#6/&F&6#\"\"%#F1\"#T/&F&6#\"\"&#\"\")\"#8/&F&6#\"\"'#\"#5F>" }} {PARA 11 "" 1 "" {XPPMATH 20 "6)%)weights:G/&%\"bG6#\"\"\"$\"'*H*y!\"( /&F&6#\"\"$$\"'(fF#F+/&F&6#\"\"%$\"'n0P!\"'/&F&6#\"\"&$\"')*o%!\"*" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#/%:2-norm~of~principal~errorG$\"+cO-]H !#8" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/%3stability~intervalG7$$!).CMU !\"(\"\"!" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 55 "#------------------------------------------------------" }} {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 34 "#=================================" }} {SECT 1 {PARA 4 "" 0 "" {TEXT -1 36 "Test-bed procedures for the examp les" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 7 "RK6step" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 2662 "rk6step := proc(x_rk6step::realco ns)\n local c2,c3,c4,c5,c6,c7,a21,a31,a32,a41,a42,a43,a51,a52,a53,a5 4,\n a61,a62,a63,a64,a65,a71,a72,a73,a74,a75,a76,\n f1,f2,f3,f4,f5 ,f6,f7,b1,b2,b3,b4,b5,b6,b7,\n xk,yk,t,jF,jM,jS,n,h,data,fn,xx,ys,sa veDigits;\n options `Copyright 2004 by Peter Stone`;\n \n data : = SOLN_;\n\n saveDigits := Digits;\n Digits := max(trunc(evalhf(Di gits)),Digits+5);\n\n # procedure to evaluate the slope field\n fn := proc(X_,Y_)\n local val; \n val := traperror(evalf(FXY_) );\n if val=lasterror or not type(val,numeric) then\n err or \"evaluation of slope field failed at %1\",evalf([X_,Y_],saveDigits );\n end if;\n val;\n end proc;\n\n xx := evalf(x_rk6 step);\n n := nops(data);\n\n if (data[1,1]data [n,1] or xxdata[1,1])) then\n error \"independent variable is outsi de the interpolation interval: %1\",evalf(data[1,1])..evalf(data[n,1]) ;\n end if;\n\n c2 := c2_; c3 := c3_; c4 := c4_; c5 := c5_; c6 := \+ c6_; c7 := c7_; \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_; a64 := a64_; a 65 := a65_;\n a71 := a71_; a72 := a72_; a73 := a73_; a74 := a74_; a7 5 := a75_; a76 := a76_;\n b1 := b1_; b2 := b2_; b3 := b3_; b4 := b4_ ; b5 := b5_; b6 := b6_; b7 := b7_;\n # Perform a binary search for t he interval containing x.\n n := nops(data);\n jF := 0;\n jS := \+ n+1;\n\n if data[1,1]1 do\n \+ jM := trunc((jF+jS)/2);\n if xx>=data[jM,1] then jF := jM els e 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 := trunc((j F+jS)/2);\n if xx<=data[jM,1] then jF := jM else jS := jM end i f;\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 := data[j F,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 + a74*f4 + a75*f5 + a76*f6;\n f7 := fn(xk + c7*h,yk + t*h);\n \n ys \+ := yk + (b1*f1 + b2*f2 + b3*f3 + b4*f4 + b5*f5 + b6*f6 + b7*f7)*h;\n\n evalf[saveDigits](ys);\nend proc: # of rk7_6step" }}}{PARA 0 "" 0 " " {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}} {SECT 1 {PARA 4 "" 0 "" {TEXT -1 26 "RK6_1 Butcher's scheme A " }} {PARA 0 "" 0 "" {TEXT -1 1 " " }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 2728 "RK6_1 := proc(fxy,x,y,xx,yy,h,stps,bb)\n local c2,c3,c4,c5,c 6,c7,a21,a31,a32,a41,a42,a43,\n a51,a52,a53,a54,a61,a62,a63,a64,a65, a71,a72,a73,a74,a75,a76,\n b1,b2,b3,b4,b5,b6,b7,f1,f2,f3,f4,f5,f6,f7 ,t,k,fn,xk,yk,soln,\n eqns,A,saveDigits;\n \n saveDigits := Digi ts;\n Digits := max(trunc(evalhf(Digits)),Digits+5);\n\n fn := una pply(fxy,x,y);\n \n A := matrix([[1/2,1/2,0,0,0,0,0,0],\n \+ [2/3,2/9,4/9,0,0,0,0,0],\n [1/3,7/36,2/9,-1/12,0,0,0,0],\n \+ [5/6,-35/144,-55/36,35/48,15/8,0,0,0],\n [1/6,-1/360 ,-11/36,-1/8,1/2,1/10,0,0],\n [1,-41/260,22/13,43/156,-118/39 ,32/195,80/39,0],\n [0,13/200,0,11/40,11/40,4/25,4/25,13/200] ]);\n \n c2 := evalf(A[1,1]);\n c3 := evalf(A[2,1]);\n c4 := eva lf(A[3,1]);\n c5 := evalf(A[4,1]);\n c6 := evalf(A[5,1]);\n c7 : = evalf(A[6,1]);\n a21 := c2;\n a31 := evalf(A[2,2]);\n a32 := e valf(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 := eva lf(A[5,2]);\n a62 := evalf(A[5,3]);\n a63 := evalf(A[5,4]);\n a6 4 := 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 b1 : = evalf(A[7,2]);\n b2 := evalf(A[7,3]);\n b3 := evalf(A[7,4]);\n \+ b4 := evalf(A[7,5]);\n b5 := evalf(A[7,6]);\n b6 := evalf(A[7,7]) ;\n b7 := evalf(A[7,8]);\n\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 + a 74*f4 + a75*f5 + a76*f6;\n f7 := fn(xk + c7*h,yk + t*h);\n \+ \n yk := yk + (b1*f1 + b2*f2 + b3*f3 + b4*f4 + b5*f5 + b6*f6 + b7 *f7)*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,a31_=a31,a32_=a32 ,a41_=a41,\n a42_=a42,a43_=a43,a51_=a51,a52_=a52,a53_=a53,a54_ =a54,\n a61_=a61,a62_=a62,a63_=a63,a64_=a64,a65_=a65,\n \+ a71_=a71,a72_=a72,a73_=a73,a74_=a74,a75_=a75,a76_=a76,\n b1_ =b1,b2_=b2,b3_=b3,b4_=b4,b5_=b5,b6_=b6,b7_=b7\};\n return subs(eq ns,eval(rk6step)); \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 32 "RK6_2 scheme with simple nodes " }}{PARA 0 "" 0 "" {TEXT -1 1 " " }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 2740 "RK6_2 := \+ proc(fxy,x,y,xx,yy,h,stps,bb)\n local c2,c3,c4,c5,c6,c7,a21,a31,a32, a41,a42,a43,\n a51,a52,a53,a54,a61,a62,a63,a64,a65,a71,a72,a73,a74,a 75,a76,\n b1,b2,b3,b4,b5,b6,b7,f1,f2,f3,f4,f5,f6,f7,t,k,fn,xk,yk,sol n,\n 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([[1/6,1/6,0,0,0,0,0,0],\n [1/5,2/25,3/25, 0,0,0,0,0],\n [1/3,2/27,-1/9,10/27,0,0,0,0],\n [2/3, 10/27,-2/9,-35/54,7/6,0,0,0],\n [3/4,-9/256,9/64,165/448,0,49 5/1792,0,0],\n [1,4/19,-3/19,-305/1463,81/95,-90/133,1024/104 5,0],\n [0,3/40,0,625/3696,27/100,27/280,256/825,19/240]]);\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 := eva lf(A[6,1]);\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 := evalf(A[6,4]);\n a74 := evalf(A[6, 5]);\n a75 := evalf(A[6,6]);\n a76 := evalf(A[6,7]);\n b1 := eva lf(A[7,2]);\n b2 := evalf(A[7,3]);\n b3 := evalf(A[7,4]);\n b4 : = evalf(A[7,5]);\n b5 := evalf(A[7,6]);\n b6 := evalf(A[7,7]);\n \+ b7 := evalf(A[7,8]);\n\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*f 4 + a75*f5 + a76*f6;\n f7 := fn(xk + c7*h,yk + t*h);\n \n \+ yk := yk + (b1*f1 + b2*f2 + b3*f3 + b4*f4 + b5*f5 + b6*f6 + b7*f7) *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_=c 2,c3_=c3,\n c4_=c4,c5_=c5,c6_=c6,c7_=c7,a31_=a31,a32_=a32,a41_ =a41,\n a42_=a42,a43_=a43,a51_=a51,a52_=a52,a53_=a53,a54_=a54, \n a61_=a61,a62_=a62,a63_=a63,a64_=a64,a65_=a65,\n a71 _=a71,a72_=a72,a73_=a73,a74_=a74,a75_=a75,a76_=a76,\n b1_=b1,b 2_=b2,b3_=b3,b4_=b4,b5_=b5,b6_=b6,b7_=b7\};\n return subs(eqns,ev al(rk6step)); \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 55 "RK6_3 scheme with a relatively large stability region " }}{PARA 0 "" 0 "" {TEXT -1 1 " " }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 2944 "RK6_3 := proc(fxy,x,y,xx,yy,h,stps,bb)\n local c2,c3,c4,c5 ,c6,c7,a21,a31,a32,a41,a42,a43,\n a51,a52,a53,a54,a61,a62,a63,a64,a6 5,a71,a72,a73,a74,a75,a76,\n b1,b2,b3,b4,b5,b6,b7,f1,f2,f3,f4,f5,f6, f7,t,k,fn,xk,yk,soln,\n eqns,A,saveDigits;\n \n saveDigits := Di gits;\n Digits := max(trunc(evalhf(Digits)),Digits+5);\n\n fn := u napply(fxy,x,y);\n \n A := matrix([[1/40,1/40,0,0,0,0,0,0],\n \+ [2/11,-58/121,80/121,0,0,0,0,0],\n [11/41,2695/275684,4840/68921, 51909/275684,0,0,0,0],\n [8/13,129980/72501,-3520/2197,-343156/257 049,4975760/2827539,0,0,0],\n [10/13,-2066341/1498354,83600/68107, 1551704/885391,-2379381536/1801770685,5607/11470,0,0],\n [1,157323 5/820072,-11440/9319,-52267281/13568464,36832433025/8776615562,\n \+ -296595/344803,76620375/92891792,0],\n [0,1667/21120,0,1 61051/7076160,7067228261/19071409500,314171/1776000,885391/3229632,931 9/121500]]);\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 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 := ev alf(A[4,3]);\n a53 := evalf(A[4,4]);\n a54 := evalf(A[4,5]);\n a 61 := 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 := eval f(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 b1 := evalf(A[7,2]);\n b2 := evalf(A[7,3]);\n b3 := evalf(A[7 ,4]);\n b4 := evalf(A[7,5]);\n b5 := evalf(A[7,6]);\n b6 := eval f(A[7,7]);\n b7 := evalf(A[7,8]);\n\n xk := evalf(xx);\n yk := e valf(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 + a 73*f3 + a74*f4 + a75*f5 + a76*f6;\n f7 := fn(xk + c7*h,yk + t*h); \n \n yk := yk + (b1*f1 + b2*f2 + b3*f3 + b4*f4 + b5*f5 + b 6*f6 + b7*f7)*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,a31_=a31, a32_=a32,a41_=a41,\n a42_=a42,a43_=a43,a51_=a51,a52_=a52,a53_= a53,a54_=a54,\n a61_=a61,a62_=a62,a63_=a63,a64_=a64,a65_=a65, \n a71_=a71,a72_=a72,a73_=a73,a74_=a74,a75_=a75,a76_=a76,\n \+ b1_=b1,b2_=b2,b3_=b3,b4_=b4,b5_=b5,b6_=b6,b7_=b7\};\n retur n subs(eqns,eval(rk6step)); \n else \n return evalf[saveDigit s]([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 32 "RK6_4 Butcher's scheme B with " }{XPPEDIT 18 0 "c[ 5]=c[6]" "6#/&%\"cG6#\"\"&&F%6#\"\"'" }{XPPEDIT 18 0 "``=1/2" "6#/%!G* &\"\"\"F&\"\"#!\"\"" }{TEXT -1 7 " and " }{XPPEDIT 18 0 "b[5]=b[6]" "6#/&%\"bG6#\"\"&&F%6#\"\"'" }{TEXT -1 1 " " }}{PARA 0 "" 0 "" {TEXT -1 1 " " }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 2696 "RK6_4 := proc(f xy,x,y,xx,yy,h,stps,bb)\n local c2,c3,c4,c5,c6,c7,a21,a31,a32,a41,a4 2,a43,\n a51,a52,a53,a54,a61,a62,a63,a64,a65,a71,a72,a73,a74,a75,a76 ,\n b1,b2,b3,b4,b5,b6,b7,f1,f2,f3,f4,f5,f6,f7,t,k,fn,xk,yk,soln,\n \+ eqns,A,saveDigits;\n \n saveDigits := Digits;\n Digits := max(t runc(evalhf(Digits)),Digits+5);\n\n fn := unapply(fxy,x,y);\n\n A \+ := matrix([[1/3,1/3,0,0,0,0,0,0],\n [2/3,0,2/3,0,0,0,0,0],\n \+ [1/3,1/12,1/3,-1/12,0,0,0,0],\n [1/2,-1/16,9/8,-3/16 ,-3/8,0,0,0],\n [1/2,0,9/8,-3/8,-3/4,1/2,0,0],\n [1, 9/44,-9/11,63/44,18/11,0,-16/11,0],\n [0,11/120,0,27/40,27/40 ,-4/15,-4/15,11/120]]);\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 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 := ev alf(A[5,4]);\n a64 := evalf(A[5,5]);\n a65 := evalf(A[5,6]);\n a 71 := 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 := eval f(A[6,7]);\n b1 := evalf(A[7,2]);\n b2 := evalf(A[7,3]);\n b3 := evalf(A[7,4]);\n b4 := evalf(A[7,5]);\n b5 := evalf(A[7,6]);\n \+ b6 := evalf(A[7,7]);\n b7 := evalf(A[7,8]);\n\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 \n yk := yk + (b1*f1 + b2*f2 + b3*f3 + b4*f4 \+ + b5*f5 + b6*f6 + b7*f7)*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_=c 7,a31_=a31,a32_=a32,a41_=a41,\n a42_=a42,a43_=a43,a51_=a51,a52 _=a52,a53_=a53,a54_=a54,\n a61_=a61,a62_=a62,a63_=a63,a64_=a64 ,a65_=a65,\n a71_=a71,a72_=a72,a73_=a73,a74_=a74,a75_=a75,a76_ =a76,\n b1_=b1,b2_=b2,b3_=b3,b4_=b4,b5_=b5,b6_=b6,b7_=b7\};\n \+ return subs(eqns,eval(rk6step));\n else\n return evalf[sav eDigits]([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 20 "RK6_5 scheme with " }{XPPEDIT 18 0 "c[5 ]=c[6]" "6#/&%\"cG6#\"\"&&F%6#\"\"'" }{XPPEDIT 18 0 "``=3/4" "6#/%!G*& \"\"$\"\"\"\"\"%!\"\"" }{TEXT -1 7 " and " }{XPPEDIT 18 0 "b[5]=b[6] " "6#/&%\"bG6#\"\"&&F%6#\"\"'" }{TEXT -1 1 " " }}{PARA 0 "" 0 "" {TEXT -1 1 " " }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 2797 "RK6_5 := \+ proc(fxy,x,y,xx,yy,h,stps,bb)\n local c2,c3,c4,c5,c6,c7,a21,a31,a32, a41,a42,a43,\n a51,a52,a53,a54,a61,a62,a63,a64,a65,a71,a72,a73,a74,a 75,a76,\n b1,b2,b3,b4,b5,b6,b7,f1,f2,f3,f4,f5,f6,f7,t,k,fn,xk,yk,sol n,\n 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([[3/17,3/17,0,0,0,0,0,0],\n [2/9,20/243,34/2 43,0,0,0,0,0],\n [3/7,3/28,-153/343,1053/1372,0,0,0,0],\n \+ [3/4,51/176,153/704,-1539/2288,8379/9152,0,0,0],\n [3/4 ,-219/1408,153/704,11745/18304,-931/4576,1/4,0,0],\n [1,229/4 644,-17/43,40887/42484,-9604/45279,0,39424/66177,0],\n [0,79/ 1080,0,19683/69160,16807/84240,1408/7695,1408/7695,43/560]]);\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 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 := evalf(A[6,4]);\n a74 := evalf(A[6,5]);\n \+ a75 := evalf(A[6,6]);\n a76 := evalf(A[6,7]);\n b1 := evalf(A[7,2 ]);\n b2 := evalf(A[7,3]);\n b3 := evalf(A[7,4]);\n b4 := evalf( A[7,5]);\n b5 := evalf(A[7,6]);\n b6 := evalf(A[7,7]);\n b7 := e valf(A[7,8]);\n\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*f 5 + a76*f6;\n f7 := fn(xk + c7*h,yk + t*h);\n \n yk := yk + (b1*f1 + b2*f2 + b3*f3 + b4*f4 + b5*f5 + b6*f6 + b7*f7)*h;\n \+ xk := xk + h:\n soln := soln,[xk,yk];\n end do;\n if bb=tru e then\n eqns := \{SOLN_=[soln],FXY_=fxy,X_=x,Y_=y,c2_=c2,c3_=c3, \n c4_=c4,c5_=c5,c6_=c6,c7_=c7,a31_=a31,a32_=a32,a41_=a41,\n \+ a42_=a42,a43_=a43,a51_=a51,a52_=a52,a53_=a53,a54_=a54,\n \+ a61_=a61,a62_=a62,a63_=a63,a64_=a64,a65_=a65,\n a71_=a71,a72 _=a72,a73_=a73,a74_=a74,a75_=a75,a76_=a76,\n b1_=b1,b2_=b2,b3_ =b3,b4_=b4,b5_=b5,b6_=b6,b7_=b7\};\n return subs(eqns,eval(rk6ste p));\n else\n return evalf[saveDigits]([soln]);\n end if;\nen d proc: " }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}}{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 "T hese tests do not make use of the embedded order 5 method for error co rrection." }}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 46 "Test 1 of 7 stage, o rder 6 Runge-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 "Solution: " }}{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)*s in(4*x)*x+96/289*exp(-x)*sin(4*x)-180/289)" "6#/%\"yG-%$expG6#,,*,\"#7 \"\"\"\"# " 0 "" {MPLTEXT 1 0 229 "de := diff(y(x),x)=12*x*cos(4*x)*e xp(-x)*y(x);\nic := y(0)=1;\ndsolve(\{de,ic\},y(x)):\ny(x)=simplify(nu mer(rhs(%))/convert(denom(rhs(%)),exp));\nf := unapply(rhs(%),x):\nplo t(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-%$cosG6#,$*&\"\"%F0F'F0F0F0-F)6#,$F'!\"\"F0F0F;*&#\"$!=\"$*GF0*&F 8F0F2F0F0F0*&#\"#[F/F0*(F8F0-%$sinGF4F0F'F0F0F0*&#\"#'*F?F0*&F8F0FEF0F 0F0#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$3FWYs#F/$\"3!H*fm:2P/5F27$$\"3%)***\\iSmp3%F/$\"3Qn()\\Dat45F 27$$\"3WmmmT&)G\\aF/$\"34$Q7t`Dr,\"F27$$\"3m****\\7G$R<)F/$\"3S2-*\\9j w.\"F27$$\"3GLLL3x&)*3\"!#=$\"3U([#>C\\El5F27$$\"3))**\\i!R(*Rc\"FJ$\" 3>&=^@[0u7\"F27$$\"3umm\"H2P\"Q?FJ$\"3k\\#o#G?)=?\"F27$$\"3!***\\PMnNr DFJ$\"3s_j<)f!R*G\"F27$$\"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$$\"3h+]Pfy G7ZFJ$\"3e=U+Y19h8F27$$\"3emm\"z%4\\Y_FJ$\"3Yii#4W6uD\"F27$$\"3'QLL3FG T\\&FJ$\"3c!QStI8]>\"F27$$\"32++v$flWv*FJ7$$\"3I++vVVX $\\'FJ$\"3w/21T*\\F&*)FJ7$$\"31nm\"zWo)\\nFJ$\"3E>3;k'H:;)FJ7$$\"3%QL$ 3_DG1qFJ$\"31le1yn9(R(FJ7$$\"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$$\"3wmm Tg()4_))FJ$\"3,.mLb#*p3MFJ7$$\"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!Q4 T\"F2$\"3Ig`=6c[gEFJ7$$\"3/++]P[6j9F2$\"31r>cjB_'>$FJ7$$\"3%o;HKR'\\5: F2$\"3XwZW,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$eky#)*Q U;F2$\"3y9nBQhKrjFJ7$$\"3wmm;a/cq;F2$\"3A`yB*3;b/(FJ7$$\"3\"pm;a)))G=< F2$\"3w18fBz!3C)FJ7$$\"3%ommmJF 2$\"3)3HOInxF>\"F27$$\"3KLe9;0?E>F2$\"3`!yI!pI]77F27$$\"3pTg-gl[Q>F2$ \"3>kb3F2$\"3OF\\_#G6DA\"F27$$\"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+))GPMfE5F27$$\"3?+](=xpe=#F2$\"3ES3-I\\16#*FJ7$$\"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$41oWW#F2$\"3 QVRl9U0BZFJ7$$\"3GLL3_?`(\\#F2$\"3/\\cKWs=$>%FJ7$$\"3AL3_D1l_DF2$\"3o6 E$fFc$yPFJ7$$\"3fL$e*)>pxg#F2$\"3ym)*p(*f`&[$FJ7$$\"3%omm\"z+vbEF2$\"3 jG&[,$f<=LFJ7$$\"33+]Pf4t.FF2$\"3%R>3YHT'HFF2$\"3/C% )f*f*e+KFJ7$$\"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$$\"3+++v=S2LKF2$\"3K% \\5FaXpw&FJ7$$\"3Jmmm\"p)=MLF2$\"3))zmB`6`OlFJ7$$\"3GLLeR%p\")Q$F2$\"3 #o,C;(=8foFJ7$$\"3B++](=]@W$F2$\"3#G%=QV$\\;4(FJ7$$\"3C$ekyZ2mY$F2$\"3 u,muc\"4C(FJ7$$\"3hTgx.2vFNF2$ \"3/^M\"Q[;lC(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`&3dArFJ7$$\"3e++]iC$pk$F2$\"3ma\\oRiHQqFJ7$$\"3ILe*[t\\sp$F2$ \"3'e9/wG(3MoFJ7$$\"3[m;H2qcZPF2$\"3CYQ8S*3be'FJ7$$\"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&*))oXb%FJ7$$\"3h+v=n(*fDVF2$\"3kIpK$)H$3a%FJ7$$\"3Cn;HdO=yVF2 $\"3u&G6!oNOhXFJ7$$\"3MMe9\"z-lU%F2$\"3kC\">#=Lu2YFJ7$$\"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\"*[F 2$\"3!>+$=fU-gcFJ7$$\"\"&F)$\"3h(Q0fOqh\"eFJ-%'COLOURG6&%$RGBG$\"#5!\" \"F(F(-%%FONTG6$%*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 "Curve 1" }}}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 " " 0 "" {TEXT -1 32 "The following code constructs a " }{TEXT 260 17 "d iscrete solution" }{TEXT -1 44 " based on each of the methods and give s the " }{TEXT 260 22 "root mean square error" }{TEXT -1 18 " of each \+ solution." }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 761 "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 := [` Butcher's scheme A`,`scheme with simple nodes`,`scheme with a relative ly large stability region`,`Butcher's scheme B with `*(c[5]=1/2,c[6]=1 /2,b[5]=b[6]),`scheme with `*(c[5]=3/4,c[6]=3/4,b[5]=b[6])]:\nerrs := \+ []:\nDigits := 25:\nfor ct to 5 do\n Fn_RK6_||ct := RK6_||ct(F(x,y), x,y,x0,y0,hh,numsteps,false);\n sm := 0: numpts := nops(Fn_RK6_||ct) :\n for ii to numpts do\n sm := sm+(Fn_RK6_||ct[ii,2]-f(Fn_RK6_ ||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,$*, \"#7\"\"\"%\"xGF,-%$cosG6#,$*&\"\"%F,F-F,F,F,-%$expG6#,$F-!\"\"F,%\"yG F,F,7$%0initial~point:~G-%!G6$\"\"!F,7$%/step~width:~~~G$F,!\"#7$%1no. ~of~steps:~~~G\"$+&Q(pprint06\"" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%!G " }}{PARA 11 "" 1 "" {XPPMATH 20 "6#K%'matrixG6#7'7$%3Butcher's~scheme ~AG$\"+q17UF!#@7$%9scheme~with~simple~nodesG$\"+'zAm4$!#A7$%Pscheme~wi th~a~relatively~large~stability~regionG$\"+pt%3f#F07$*&%9Butcher's~sch eme~B~with~G\"\"\"6%/&%\"cG6#\"\"&#F8\"\"#/&F<6#\"\"'F?/&%\"bGF=&FGFCF 8$\"+9%HA0$F07$*&%-scheme~with~GF86%/F;#\"\"$\"\"%/FBFPFEF8$\"+N/!f*GF 0Q(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 0 692 "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 := [`Butche r's scheme A`,`scheme with simple nodes`,`scheme with a relatively lar ge stability region`,`Butcher's scheme B with `*(c[5]=1/2,c[6]=1/2,b[5 ]=b[6]),`scheme with `*(c[5]=3/4,c[6]=3/4,b[5]=b[6])]:\nerrs := []:\nD igits := 25:\nfor ct to 5 do\n fn_RK6_||ct := RK6_||ct(F(x,y),x,y,x0 ,y0,hh,numsteps,true);\nend do:\nxx := 4.999: fxx := evalf(f(xx)):\nfo r ct to 5 do\n errs := [op(errs),abs(fn_RK6_||ct(xx)-fxx)];\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$%0in itial~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$%3Butcher's~scheme~AG$\"+\"4 qN&H!#@7$%9scheme~with~simple~nodesG$\"+E:KcL!#A7$%Pscheme~with~a~rela tively~large~stability~regionG$\"+=9&4t#F07$*&%9Butcher's~scheme~B~wit h~G\"\"\"6%/&%\"cG6#\"\"&#F8\"\"#/&F<6#\"\"'F?/&%\"bGF=&FGFCF8$\"+ff$> $GF07$*&%-scheme~with~GF86%/F;#\"\"$\"\"%/FBFPFEF8$\"+_GUNJF0Q(pprint3 6\"" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 4 "Th e " }{TEXT 260 22 "root mean square error" }{TEXT -1 20 " over the int erval " }{XPPEDIT 18 0 "[0, 5];" "6#7$\"\"!\"\"&" }{TEXT -1 82 " of \+ each Runge-Kutta method is estimated as follows using the special proc edure " }{TEXT 0 5 "NCint" }{TEXT -1 98 " to perform numerical integ ration by the 7 point Newton-Cotes method over 200 equal subintervals. " }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 436 "mthds := [`Butcher's sc heme A`,`scheme with simple nodes`,`scheme with a relatively large sta bility region`,`Butcher's scheme B with `*(c[5]=1/2,c[6]=1/2,b[5]=b[6] ),`scheme with `*(c[5]=3/4,c[6]=3/4,b[5]=b[6])]:\nerrs := []:\nDigits \+ := 20:\nfor ct to 5 do\n sm := NCint((f(x)-'fn_RK6_||ct'(x))^2,x=0.. 5,adaptive=false,numpoints=7,factor=200);\n errs := [op(errs),sqrt(s m/5)];\nend do:\nDigits := 10:\nlinalg[transpose]([mthds,evalf(errs)]) ;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#K%'matrixG6#7'7$%3Butcher's~schem e~AG$\"+>s_UF!#@7$%9scheme~with~simple~nodesG$\"+R\\G(4$!#A7$%Pscheme~ with~a~relatively~large~stability~regionG$\"+ICT\"f#F07$*&%9Butcher's~ scheme~B~with~G\"\"\"6%/&%\"cG6#\"\"&#F8\"\"#/&F<6#\"\"'F?/&%\"bGF=&FG FCF8$\"+:%=S/$F07$*&%-scheme~with~GF86%/F;#\"\"$\"\"%/FBFPFEF8$\"+14b' *GF0Q(pprint46\"" }}}{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 0 516 "evalf[20](plot([f(x)-'fn_RK6_1'(x),f(x)-'fn_RK6_2'(x ),f(x)-'fn_RK6_3'(x),f(x)-'fn_RK6_4'(x),\nf(x)-'fn_RK6_5'(x)],x=0..5,f ont=[HELVETICA,9],\ncolor=[COLOR(RGB,.95,.2,0),COLOR(RGB,.45,0,.95),CO LOR(RGB,0,.25,1),COLOR(RGB,0,.75,.2),COLOR(RGB,.95,.45,0)],\nlegend=[` Butcher's scheme A`,`scheme with simple nodes`,`scheme with a relative ly large stability region`,`Butcher's scheme B with c[5]=c[6]=1/2 and \+ b[5]=b[6]`,`scheme with c[5]=c[6]=3/4 and b[5]=b[6]`],title=`error cur ves for 7 stage order 6 Runge-Kutta methods`));" }}{PARA 13 "" 1 "" {GLPLOT2D 970 553 553 {PLOTDATA 2 "6+-%'CURVESG6%7`s7$$\"\"!F)F(7$$\"5 qmmmmT&)G\\a!#@$!%uW!#>7$$\"5MLLLL3x&)*3\"!#?$\"'F5:F07$$\"5+++]i!R(*R c\"F4$\"';#)[F07$$\"5nmmm\"H2P\"Q?F4$\"(jj-\"F07$$\"5MLLek.pu/BF4$\"(+ OK\"F07$$\"5+++]PMnNrDF4$\"(!Rq9F07$$\"5ML$eR(\\;m/FF4$\"(uB]\"F07$$\" 5nmmT5ll'z$GF4$\"(`B\\\"F07$$\"5++](o/[r7(HF4$\"(]rW\"F07$$\"5MLLL$eRw X5$F4$\"(%=&G\"F07$$\"5MLLLe*[`HP$F4$\"(Do.\"F07$$\"5MLLLL$eI8k$F4$\"' &oi'F07$$\"5MLL$3-8>bx$F4$\"'Y[bF07$$\"5MLLL3xwq4RF4$\"'cL[F07$$\"5MLL $eRA'*Q/%F4$\"'b\\_F07$$\"5NLLL$3x%3yTF4$\"'&\\n'F07$$\"5-++]PfyG7ZF4$ \"(lr?$F07$$\"5ommm\"z%4\\Y_F4$\"(o)RwF07$$\"5++++v$flVB$)F4$!*$[#*>>F47$ $\"50++++v`w(e)F4$!*u*zo@F47$$\"5qmmmTg()4_))F4$!*V@'eBF47$$\"5+++]7`a E%)*)F4$!*e._T#F47$$\"5NLLL$e9Kk6*F4$!*b*z+DF47$$\"5qmm;aQ))f[#*F4$!*8 nd[#F47$$\"5.+++DJbw!Q*F4$!*'=^#[#F47$$\"5qmT5lv.u[%*F4$!*_42\\#F47$$ \"5NL$3_+A:n^*F4$!*Kp/^#F47$$\"5q;/EDUEq]&*F4$!*!Q#=[#F47$$\"5++DJXk+p %e*F4$!*M?]Z#F47$$\"5N$ek`m[x'='*F4$!*!\\W&\\#F47$$\"5qmmT&)3\\m_'*F4$ !*ZJ!oCF47$$\"5++]il(f9')y*F4$!*4\\![CF47$$\"5NLL$ekGkX#**F4$!*'=sICF4 7$$\"5++]iSmjk>5F0$!*AgjO#F47$$\"5nmmm;/j$o/\"F0$!*P$R)G#F47$$\"5+++]7 GTt%4\"F0$!*Zc2?#F47$$\"5MLLL3_>jU6F0$!*U%H\\@F47$$\"5nm;aQ`B6c6F0$!*6 ![S@F47$$\"5+++voaFfp6F0$!*^**z8#F47$$\"5ML$e*)f:tI=\"F0$!*;Iz8#F47$$ \"5nmm;HdNb'>\"F0$!*dKB9#F47$$\"5MLLe*)fV^B7F0$!*%Gmg@F47$$\"5++++]i^Z ]7F0$!*!y^\">#F47$$\"5+++++v\"=YI\"F0$!*#f$HJ#F47$$\"5++++](=h(e8F0$!* 81g[#F47$$\"5++++]7!Q4T\"F0$!*$\\GeEF47$$\"5++++]P[6j9F0$!*PQ%HGF47$$ \"5nmm\"HKR'\\5:F0$!*%3hpGF47$$\"5MLL$e*[z(yb\"F0$!*H*p9HF47$$\"5nm;/E v&[ge\"F0$!*pCf(GF47$$\"5+++Dc,#>Uh\"F0$!*$zIEGF47$$\"5nmTNr9XIG;F0$!* K'zOGF47$$\"5ML$eky#)*QU;F0$!**Qs5GF47$$\"5+vVB:c6\"fk\"F0$!*#H![%GF47 $$\"5n;/,W%[K%\\;F0$!*J?Z#GF47$$\"5Mekys7Q&Hl\"F0$!*]95$GF47$$\"5++Dc, T^Zc;F0$!*s`J'GF47$$\"5M$e9\"f(zF0$!)'QfT& F07$$\"5MLLe9;0?E>F0$!)$*oobF07$$\"5nmTg-gl[Q>F0$!)?SBcF07$$\"5++]i!Rg s2&>F0$!)!o-m&F07$$\"5MLekyZ'eI'>F0$!)#y#pcF07$$\"5nmmmm\"pW`(>F0$!)!o 4m&F07$$\"5MLLe9TOEH?F0$!)e-2aF07$$\"5,++]i!f#=$3#F0$!)s\\=\\F07$$\"5, +++D\"=EX8#F0$!)/3%Q%F07$$\"5,++](=xpe=#F0$!*Jo\"zQF47$$\"5MLLeRA9WRAF 0$!*Y0'QMF47$$\"5nmmm\"H28IH#F0$!*4>\"zIF47$$\"5nmmTNr&3AM#F0$!*E10!GF 47$$\"5nmm;zpSS\"R#F0$!*rU5b#F47$$\"5MLLL3_?`(\\#F0$!*DpV4#F47$$\"5MLL 3_D1l_DF0$!*R8B#>F47$$\"5MLL$e*)>pxg#F0$!*!Gf#z\"F47$$\"5ommm;z+vbEF0$ !*4uyr\"F47$$\"5,++]Pf4t.FF0$!*CK`n\"F47$$\"5ML$3F>HT'HFF0$!*bx\\m\"F4 7$$\"5omm\"zWi^bv#F0$!*2@Jm\"F47$$\"5,+]7.d>Y\"y#F0$!*R*op;F47$$\"5MLL Le*Gst!GF0$!*FB[o\"F47$$\"5omm;H2\"34'GF0$!*Iu3u\"F47$$\"5,++++DRW9HF0 $!*dV9$=F47$$\"5,+++DJE>>IF0$!*aV')4#F47$$\"5,++]i!RU07$F0$!*xyRX#F47$ $\"5,+++v=S2LKF0$!*J3f\"HF47$$\"5ommmm\"p)=MLF0$!*3mIJ$F47$$\"5MLLLeR% p\")Q$F0$!*P?OF0$!*wiWi$F47$$\"5,+++]iC$pk$F0$!*n. 2e$F47$$\"5MLLe*[t\\sp$F0$!*YN\\Z$F47$$\"5omm;H2qcZPF0$!*;OtM$F47$$\"5 ,++]7.\"fF&QF0$!*kJ)\\IF47$$\"5ommm;/OgbRF0$!*\"p,rFF47$$\"5,++]ilAFjS F0$!*zml`#F47$$\"5omm\"zW7@^6%F0$!*KbUF0$!*zk/M#F47$$\"5NLLL3xe,tUF0$!*\"[S9BF47$$\"5-++ v=n(*fDVF0$!*5.uI#F47$$\"5omm;HdO=yVF0$!*+HyJ#F47$$\"5MLLe9\"z-lU%F0$! *Et8M#F47$$\"5,++++D>#[Z%F0$!*YenP#F47$$\"5ommmT&G!e&e%F0$!*v7I\\#F47$ $\"5NLLLL$)Qk%o%F0$!*k,>i#F47$$\"5-++]iSjE!z%F0$!*sf9w#F47$$\"5-++]P40 O\"*[F0$!*I)ouGF47$$\"\"&F)$!*:jS&HF4-%&COLORG6&%$RGBG$\"#&*!\"#$\"\"# !\"\"F(-%'LEGENDG6#%3Butcher's~scheme~AG-F$6%7_tF'7$F+$!'q>RF07$F8$!'l*=%F07$$\"5nm;z>6B`#o\"F4$! 'MkUF07$$\"5MLL3xJs1,=F4$!'<^UF07$$\"5++]PM_@g>>F4$!'iF 47$Far$\"(z=F47$ F`s$\")Lr*4#F47$$\"5++++D1CZgwF4$\")lPzAF47$Fes$\")X)Re#F47$Fjs$\")ms% )GF47$F_t$\").!z:$F47$$\"50++v$fL:&*Q)F4$\")iR\\JF47$$\"5qmm;H#o)fb%)F 4$\")>+bJF47$$\"50+](oaNS')[)F4$\")I:vJF47$$\"5SLLekG?o@&)F4$\")]&z@$F 47$$\"5qm;H#=qBZb)F4$\")hmvJF47$Fdt$\")7L&=$F47$$\"5SL$3x\"[q!3i)F4$\" )*pHA$F47$$\"5qmmTN@([Ql)F4$\")6k!=$F47$$\"50+]7`%R!*oo)F4$\"):d\"=$F4 7$$\"5SLL$3x1K*>()F4$\")I#Q@$F47$$\"5qm;a)3utHv)F4$\")5!=<$F47$$\"50++ D19a,'y)F4$\")AhlJF47$$\"5SL$eRs3d!>))F4$\")G\\#>$F47$Fit$\")!z5:$F47$ Fcu$\")IyuIF47$F]v$\")#\\&4HF47$F_y$\")G'\\P#F47$Fdy$\")1oPAF47$Fiy$\" )qlg@F47$F^z$\")(*f[@F47$Fcz$\")T7U@F47$Fhz$\")\"z+9#F47$F][l$\")#fN9# F47$$\"5++]PfeR.57F0$\")\"y'[@F47$Fb[l$\")/ri@F47$$\"5nm;z>hZ*pB\"F0$ \")<-#=#F47$Fg[l$\")qL+AF47$F\\\\l$\"))z9N#F47$Fa\\l$\")Bk(f#F47$Ff\\l $\")[z9HF47$F[]l$\")jk%H$F47$F`]l$\")J8zNF47$Fe]l$\")g2KQF47$$\"5n;aQ` 01#\\c\"F0$\")ooNQF47$$\"5++v$4@Ej>d\"F0$\")?A6QF47$$\"5M$e*[o=f+z:F0$ \")*[8&QF47$Fj]l$\")E\"4)QF47$$\"5+]Pf$=B\"4$f\"F0$\")GOZQF47$$\"5MLe9 T))Q8+;F0$\")H^0QF47$$\"5+v=#*p;_l.;F0$\")%Rs&QF47$$\"5n;zp)\\awrg\"F0 $\")!3]*QF47$$\"5MeRZFtyp5;F0$\")Gj6QF47$F_^l$\")?oiQF47$Fi^l$\")\"**o \"QF47$Fg`l$\")_e'y$F47$F\\al$\")9#\\%RF47$Faal$\"),))oUF47$F[bl$\"(A. \"\\F07$Febl$\"(!>ZbF07$Fjbl$\"(R%ydF07$F_cl$\"(+j\"fF07$Fdcl$\"(c;'fF 07$Ficl$\"(`p)fF07$F^dl$\"(*)**)fF07$Fcdl$\"(Ve(fF07$Fhdl$\"(e\"*p&F07 $F]el$\"((G3_F07$Fbel$\"(_zp%F07$Fgel$\")@*>C%F47$F\\fl$\")*RR'QF47$Fa fl$\")_>INF47$Fffl$\")#HvB$F47$F[gl$\")SFYHF47$F`gl$\")>k(Q#F47$Fegl$ \")U]x@F47$Fjgl$\")h0A?F47$F_hl$\")z,J>F47$Fdhl$\")SSw=F47$Fihl$\")H2h =F47$F^il$\")27c=F47$Fcil$\")C-g=F47$Fhil$\")e'[(=F47$F]jl$\")7GM>F47$ Fbjl$\")pYO?F47$Fgjl$\")E%*\\BF47$F\\[m$\")fxsFF47$Fa[m$\")15%F47$F_]m$\")om.UF47$$\"5omTgx .2vFNF0$\")v+1UF47$Fd]m$\")tL/UF47$$\"5omT5:)[[Lb$F0$\")yw*>%F47$Fi]m$ \")3W\">%F47$$\"5MLek`mn3!e$F0$\")2JyTF47$F^^m$\")p,jTF47$Fc^m$\")]J@T F47$Fh^m$\")`VpSF47$F]_m$\")*pX%RF47$Fb_m$\")A,'z$F47$Fg_m$\")1%[X$F47 $F\\`m$\")&f'RJF47$Fa`m$\")5!o(GF47$Ff`m$\")h\"=y#F47$F[am$\")/x3FF47$ F`am$\")5qcEF47$Feam$\")(4pi#F47$Fjam$\")hU=EF47$F_bm$\")!f&HEF47$Fdbm $\")!)pbEF47$Fibm$\")#yap#F47$F^cm$\")J!y#GF47$Fccm$\")myvHF47$Fhcm$\" )2_OJF47$F]dm$\")q^mKF47$Fbdm$\")&eoN$F4-Fgdm6&Fidm$\"#XF\\emF(Fjdm-Fa em6#%9scheme~with~simple~nodesG-F$6%7atF'7$F+$!'J6=F07$F2$!'vZFF07$$\" 5++]i!*GER37F4$!'I4FF07$F^fm$!';:EF07$$\"5ML$3_+ZiaW\"F4$!'6fCF07$F8$! '!3B#F07$F[gm$!'8=7F07$F=$!&ys#F07$FG$\"'P*3$F07$Fen$\"'1Q$)F07$F_o$\" (FHG\"F07$Fcp$\"(r:Z\"F07$$\"5ommT5:j=XWF4$\"(TCN\"F07$Fhp$\"(@@-\"F07 $$\"5NLLek.%*Qz\\F4$\"'%[_'F07$F]q$\"'&z-\"F07$Fjhm$!'H%>%F07$Fbq$!'@n zF07$Fajm$!'6^\"*F07$F[[n$!(Nf-\"F07$$\"5NLLeRZAJ^gF4$!(=90\"F07$F`[n$ !(ly2\"F07$$\"5qmmT&)348vhF4$!).(*e5F47$Fgq$!)v!f/\"F47$F\\r$!(J?*zF47 $Far$!(_my$F47$F^\\n$!(!G(=\"F47$Ffr$\"(=jG$F47$Ff\\n$\"(=l\"eF47$F[s$ \"(FOS)F47$F^]n$\")F47$Fiy$\")y&p*=F47$F^z$\")\"3n)=F 47$Fcz$\")qc!)=F47$Fhz$\")\"Hx(=F47$F][l$\")1Xz=F47$Fb[l$\")'*\\!*=F47 $Fg[l$\")F-:>F47$F\\\\l$\")))oG?F47$Fa\\l$\")RrAAF47$F[]l$\")Yv:GF47$F `]l$\")>ZhIF47$Fe]l$\")W,dKF47$$\"5+v$4YsF*Rh:F0$\")i?3KF47$Fcfn$\")'R ?D$F47$$\"5Me9;#Q$>Wo:F0$\")-mgKF47$Fhfn$\")$H&>KF47$F]gn$\")!ynC$F47$ Fj]l$\")(*\\jKF47$Fegn$\")X->KF47$Fjgn$\")M=lJF47$F_hn$\")%*=3KF47$Fdh n$\")wjPKF47$Fihn$\")iE\\JF47$F_^l$\")MR\">$F47$Fi^l$\")g(43$F47$Fg`l$ \")ej#)HF47$$\"5nmm\"z>c#\\#o\"F0$\")Fq()HF47$$\"5nmm;zpYU%p\"F0$\")Ma +IF47$$\"5nmmTgxnN1F47$Fegl$\")mxO#F47$F^cm$\")MH)H#F47$Fccm$\")yT>CF47$ Fhcm$\")'f6b#F47$F]dm$\")GidEF47$Fbdm$\")sQJFF4-Fgdm6&FidmF($\"#DF\\em $\"\"\"F)-Faem6#%Pscheme~with~a~relatively~large~stability~regionG-F$6 %7hrF'7$F+$!'jRgF07$F2$!(Ql3\"F07$Fbgo$!(nI:\"F07$F^fm$!(X/=\"F07$Fjgo $!(a2>\"F07$F8$!(#p\"=\"F07$F[gm$!()4P5F07$F=$!'$F07$ Fen$\"&X^#F07$F_o$!'B`GF07$Fcp$!'UZ%*F07$$\"5omT5!p:g[C%F4$!(tk.\"F07$ $\"5-+](oHaN;J%F4$!(9!G6F07$$\"5NLek.H4TyVF4$!(:p8\"F07$Fdio$!(-&)=\"F 07$$\"5NL$eRs3P(yXF4$!(!)R@\"F07$Fhp$!(hS:\"F07$F\\jo$!']G%)F07$F]q$!& CC(F07$Fjhm$\"(V/2\"F07$Fbq$\"(b>L#F07$F[[n$\"(0O\"QF07$Fgq$\")qDS`F47 $F\\r$\")PcomF47$Far$\"):/5uF47$F^\\n$\")\\sovF47$Ffr$\")I^uxF47$Ff\\n $\")^4pvF47$F[s$\")#z&psF47$F`s$\")/@IiF47$Fes$\")o43[F47$$\"5SLLLeRil DzF4$\")3H2SF47$Fjs$\"),_hLF47$$\"5qmmm\"H2S3>)F4$\")GlXDF47$F_t$\")H9 (y\"F47$Fdt$\"(n$etF47$Fit$!'5LfF47$F^u$!(NcA$F47$Fcu$!(]%=dF47$Fhu$!( mob'F47$F]v$!(1e?(F47$Fex$!(>4J(F47$F_y$!(F%GpF47$Fdy$!(!e;zF47$Fiy$!( k\"4$*F47$Fcz$!(#[:)*F47$F][l$!)0&Q+\"F47$Fb[l$!)kX/5F47$Fg[l$!(B0')*F 47$Fa\\l$!)gCC6F47$$\"5++++++'\\[Q\"F0$!)U&)G7F47$Ff\\l$!)#fdK\"F47$$ \"5++++v=A)RU\"F0$!)\\-m8F47$$\"5+++++Dk-P9F0$!)t[%R\"F47$$\"5++++DJ12 ]9F0$!)Gre8F47$F[]l$!)q5Z8F47$F`]l$!(1_6*F47$Fe]l$!(leV#F47$Fj]l$\"(P( \\DF47$F_^l$\"(&GMjF47$$\"5M$3-Q\"e=E@;F0$\"(`U%pF47$Fd^l$\"(Sn9(F47$$ \"5+]i!*GrrMN;F0$\"(1KN(F47$Fi^l$\"(_v:(F47$Fc_l$\"(pkz'F47$F]`l$\"(?A i'F47$Fb`l$\"(#*eb&F47$Fg`l$\"(-n(RF47$F^gp$!'6^]F47$F\\al$!(I**y(F47$ F[hp$!)Qrl=F47$Faal$!)ls0GF47$Ffal$!(\\B<%F07$F[bl$!(<_X&F07$F`bl$!($4 UkF07$Febl$!(#H5sF07$Fjbl$!(0\"yxF07$F_cl$!($)R/)F07$$\"5+]Pf3QNMK>F0$ !(To4)F07$Fdcl$!(Y(=\")F07$$\"5M$e9m>eHY%>F0$!(5!R\")F07$Ficl$!(ct8)F0 7$F^dl$!('G9\")F07$Fcdl$!(`i0)F07$Fhdl$!(Y'=tF07$F]el$!(*[shF07$Fbel$! (3O,&F07$Fgel$!)my&)RF47$F\\fl$!)[\"o<$F47$Fafl$!)6q*o#F47$F[gl$!)$z') =#F47$$\"5+++v$41oWW#F0$!)ITf>F47$F`gl$!)XTdp\"\\$F47$Fh^m$!)mx9MF47$Fb_m$!)RK'>$F47$Fg_m$!)!**H#HF47$F\\`m$!) (z,m#F47$Fa`m$!)?JLCF47$F[am$!)i])G#F47$Feam$!)*zCA#F47$F_bm$!)t%3B#F4 7$Fibm$!)r&4H#F47$F^cm$!)_e-CF47$Fccm$!)W#G_#F47$Fhcm$!)uw^EF47$F]dm$! )cFdFF47$Fbdm$!)6TKGF4-Fgdm6&FidmF($\"#vF\\emF]em-Faem6#%TButcher's~sc heme~B~with~c[5]=c[6]=1/2~and~b[5]=b[6]G-F$6%7gsF'7$F+$!'l->F07$F2$!'= 4SF07$F^fm$!'$4m%F07$F8$!'82]F07$Fffm$!')=6&F07$F[gm$!'GL^F07$F`gm$!'- M^F07$F=$!'`4^F07$$\"5++]7G))>Wr@F4$!'(Q1&F07$FB$!'x'*[F07$$\"5nm;/,>= 0QCF4$!'Xa[F07$FG$!'/:[F07$FL$!'RbZF07$FQ$!'%o![F07$FV$!''4!\\F07$Fen$ !'%39&F07$F_o$!'?+mF07$Fcp$!'&=i)F07$Fdio$!'y)[*F07$Fhp$!'V/**F07$$\"5 om;/^J'Qe%[F4$!'>I)*F07$F\\jo$!'bL'*F07$$\"5++]7yv,%H6&F4$!'\"y<*F07$F ]q$!'$Qn)F07$Fjhm$!'[MpF07$Fbq$!')Ho%F07$F[[n$!'hR;F07$Fgq$\"(z\\7#F47 $F\\r$\"(#*Hd'F47$Far$\")Kgv5F47$Ffr$\")UqH;F47$F[s$\")`aF>F47$F`s$\") -_sBF47$Fes$\")\\1]EF47$Fjs$\")cS/GF47$F_t$\"))Rs'HF47$Fi^n$\")c`ZHF47 $Fdt$\")x'R'HF47$F``n$\")3`'*HF47$Fe`n$\")];dHF47$Fj`n$\")\")3eHF47$F_ an$\")r_))HF47$Fdan$\")0\\\\HF47$Fian$\")VnFF47$F_y$\")L5SB F47$Fdy$\")_%Q?#F47$Fiy$\")X#)=@F47$F^z$\")kE0@F47$Fcz$\")>a'4#F47$Fhz $\")P@$4#F47$F][l$\")O\"e4#F47$Fadn$\")O%**4#F47$Fb[l$\")*\\M6#F47$Fid n$\")#)>K@F47$Fg[l$\")w()\\@F47$F\\\\l$\")>&HH#F47$Fa\\l$\")'R7^#F47$F f\\l$\").VtFF47$F[]l$\")>&e4$F47$F`]l$\")\")ykLF47$Fe]l$\")UrjOF47$F_^ l$\")gnoQF47$Fg`l$\")gKsRF47$Faal$\")P'=F%F47$Febl$\"(J'yYF07$F_cl$\"( Sg#[F07$Fcdl$\"(*y,\\F07$$\"5MLek.HW#)))>F0$\"(+q!\\F07$$\"5++]iSmTI-? F0$\"((R2\\F07$$\"5nmTgx.Ry:?F0$\"([!*)[F07$Fhdl$\"(;N([F07$$\"5om;a)e 6Bi0#F0$\"(c@![F07$F]el$\"(TCr%F07$Fbel$\"(mMX%F07$Fgel$\")M&48%F47$F \\fl$\")QPuPF47$Fafl$\")TK/MF47$Fffl$\"))\\/3$F47$F[gl$\")9J#y#F47$F^h r$\")^R'\\#F47$F`gl$\")CahAF47$Fjgl$\")\"H$H>F47$F_hl$\")?GV=F47$Fdhl$ \")L]*y\"F47$Fihl$\")wIt'oF47$Fgjl$\")3%Q@#F47$F\\[m$\")(z(3EF47$F a[m$\")F)z5$F47$Ff[m$\")%=m^$F47$F[\\m$\")^#po$F47$F`\\m$\")Fm4QF47$Fe \\m$\")t*y%QF47$Fj\\m$\")miuQF47$Ff`o$\")Kk$)QF47$F_]m$\")Ru*)QF47$F^a o$\")'yH*QF47$Fd]m$\")s_$*QF47$Ffao$\")Im!*QF47$Fi]m$\")C]%)QF47$F^bo$ \")$3a(QF47$F^^m$\")a*G'QF47$Fc^m$\")>JHQF47$Fh^m$\")ob%y$F47$F]_m$\") *)fvOF47$Fb_m$\")[/UNF47$Fg_m$\")Q!pA$F47$F\\`m$\")&=@$HF47$Fa`m$\")N) po#F47$Ff`m$\"),w)f#F47$F[am$\")a/JDF47$F`am$\")!)*G[#F47$Feam$\")qMbC F47$Fjam$\")>fZCF47$F_bm$\")F4eCF47$Fdbm$\")Pd#[#F47$Fibm$\")jz>DF47$F ^cm$\")*QNk#F47$Fccm$\")!p;y#F47$Fhcm$\")S>JHF47$F]dm$\")u(=0$F47$Fbdm $\")g#f8$F4-Fgdm6&FidmFjdmFcfoF(-Faem6#%Hscheme~with~c[5]=c[6]=3/4~and ~b[5]=b[6]G-%%FONTG6$%*HELVETICAG\"\"*-%+AXESLABELSG6$Q\"x6\"Q!F^[u-%& TITLEG6#%Uerror~curves~for~7~stage~order~6~Runge-Kutta~methodsG-%%VIEW G6$;F(Fbdm%(DEFAULTG" 1 2 0 1 10 0 2 9 1 4 2 1.000000 45.000000 45.000000 0 1 "Butcher's scheme A" "scheme with simple nodes" "scheme \+ with a relatively large stability region" "Butcher's scheme B with c[5 ]=c[6]=1/2 and b[5]=b[6]" "scheme with c[5]=c[6]=3/4 and b[5]=b[6]" }} }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 455 "evalf[20](plot([f(x)-'fn_RK6_2'(x),f(x)-'fn_RK6_3'(x),f(x)-'fn_ RK6_4'(x),f(x)-'fn_RK6_5'(x)],\nx=0..5,font=[HELVETICA,9],color=[COLOR (RGB,.45,0,.95),COLOR(RGB,0,.25,1),COLOR(RGB,0,.75,.2),COLOR(RGB,.95,. 45,0)],\nlegend=[`scheme with simple nodes`,`scheme with a relatively \+ large stability region`,`Butcher's scheme B with c[5]=c[6]=1/2 and b[5 ]=b[6]`,`scheme with c[5]=c[6]=3/4 and b[5]=b[6]`],title=`error curves for 7 stage order 6 Runge-Kutta methods`));" }}{PARA 13 "" 1 "" {GLPLOT2D 919 523 523 {PLOTDATA 2 "6*-%'CURVESG6%7_t7$$\"\"!F)F(7$$\"5 qmmmmT&)G\\a!#@$!'q>7$$\"5MLLLL3x&)*3\"!#?$!'P4MF07$$\"5nmm\"z%\\v #pK\"F4$!'`>RF07$$\"5+++]i!R(*Rc\"F4$!'l*=%F07$$\"5nm;z>6B`#o\"F4$!'Mk UF07$$\"5MLL3xJs1,=F4$!'<^UF07$$\"5++]PM_@g>>F4$!'iF 47$$\"5qmmm\"zWo)\\nF4$\"(z=F47$$\"5NLLLe*[!)y_(F4$\")Lr*4# F47$$\"5++++D1CZgwF4$\")lPzAF47$$\"5qmmm\"HKkIz(F4$\")X)Re#F47$$\"50++ +Dc\"[#e!)F4$\")ms%)GF47$$\"5OLLLe*)>VB$)F4$\").!z:$F47$$\"50++v$fL:&* Q)F4$\")iR\\JF47$$\"5qmm;H#o)fb%)F4$\")>+bJF47$$\"50+](oaNS')[)F4$\")I :vJF47$$\"5SLLekG?o@&)F4$\")]&z@$F47$$\"5qm;H#=qBZb)F4$\")hmvJF47$$\"5 0++++v`w(e)F4$\")7L&=$F47$$\"5SL$3x\"[q!3i)F4$\")*pHA$F47$$\"5qmmTN@([ Ql)F4$\")6k!=$F47$$\"50+]7`%R!*oo)F4$\"):d\"=$F47$$\"5SLL$3x1K*>()F4$ \")I#Q@$F47$$\"5qm;a)3utHv)F4$\")5!=<$F47$$\"50++D19a,'y)F4$\")AhlJF47 $$\"5SL$eRs3d!>))F4$\")G\\#>$F47$$\"5qmmmTg()4_))F4$\")!z5:$F47$$\"5NL LL$e9Kk6*F4$\")IyuIF47$$\"5.+++DJbw!Q*F4$\")#\\&4HF47$$\"5nmmm;/j$o/\" F0$\")G'\\P#F47$$\"5+++]7GTt%4\"F0$\")1oPAF47$$\"5MLLL3_>jU6F0$\")qlg@ F47$$\"5nm;aQ`B6c6F0$\")(*f[@F47$$\"5+++voaFfp6F0$\")T7U@F47$$\"5ML$e* )f:tI=\"F0$\")\"z+9#F47$$\"5nmm;HdNb'>\"F0$\")#fN9#F47$$\"5++]PfeR.57F 0$\")\"y'[@F47$$\"5MLLe*)fV^B7F0$\")/ri@F47$$\"5nm;z>hZ*pB\"F0$\")<-#= #F47$$\"5++++]i^Z]7F0$\")qL+AF47$$\"5+++++v\"=YI\"F0$\"))z9N#F47$$\"5+ +++](=h(e8F0$\")Bk(f#F47$$\"5++++]7!Q4T\"F0$\")[z9HF47$$\"5++++]P[6j9F 0$\")jk%H$F47$$\"5nmm\"HKR'\\5:F0$\")J8zNF47$$\"5MLL$e*[z(yb\"F0$\")g2 KQF47$$\"5n;aQ`01#\\c\"F0$\")ooNQF47$$\"5++v$4@Ej>d\"F0$\")?A6QF47$$\" 5M$e*[o=f+z:F0$\")*[8&QF47$$\"5nm;/Ev&[ge\"F0$\")E\"4)QF47$$\"5+]Pf$=B \"4$f\"F0$\")GOZQF47$$\"5MLe9T))Q8+;F0$\")H^0QF47$$\"5+v=#*p;_l.;F0$\" )%Rs&QF47$$\"5n;zp)\\awrg\"F0$\")!3]*QF47$$\"5MeRZFtyp5;F0$\")Gj6QF47$ $\"5+++Dc,#>Uh\"F0$\")?oiQF47$$\"5ML$eky#)*QU;F0$\")\"**o\"QF47$$\"5nm mm;a/cq;F0$\")_e'y$F47$$\"5nmmmT&)))G=ZbF0 7$$\"5om;aQG%G;!>F0$\"(R%ydF07$$\"5MLLe9;0?E>F0$\"(+j\"fF07$$\"5nmTg-g l[Q>F0$\"(c;'fF07$$\"5++]i!Rgs2&>F0$\"(`p)fF07$$\"5MLekyZ'eI'>F0$\"(*) **)fF07$$\"5nmmmm\"pW`(>F0$\"(Ve(fF07$$\"5MLLe9TOEH?F0$\"(e\"*p&F07$$ \"5,++]i!f#=$3#F0$\"((G3_F07$$\"5,+++D\"=EX8#F0$\"(_zp%F07$$\"5,++](=x pe=#F0$\")@*>C%F47$$\"5MLLeRA9WRAF0$\")*RR'QF47$$\"5nmmm\"H28IH#F0$\") _>INF47$$\"5nmmTNr&3AM#F0$\")#HvB$F47$$\"5nmm;zpSS\"R#F0$\")SFYHF47$$ \"5MLLL3_?`(\\#F0$\")>k(Q#F47$$\"5MLL3_D1l_DF0$\")U]x@F47$$\"5MLL$e*)> pxg#F0$\")h0A?F47$$\"5ommm;z+vbEF0$\")z,J>F47$$\"5,++]Pf4t.FF0$\")SSw= F47$$\"5ML$3F>HT'HFF0$\")H2h=F47$$\"5omm\"zWi^bv#F0$\")27c=F47$$\"5,+] 7.d>Y\"y#F0$\")C-g=F47$$\"5MLLLe*Gst!GF0$\")e'[(=F47$$\"5omm;H2\"34'GF 0$\")7GM>F47$$\"5,++++DRW9HF0$\")pYO?F47$$\"5,+++DJE>>IF0$\")E%*\\BF47 $$\"5,++]i!RU07$F0$\")fxsFF47$$\"5,+++v=S2LKF0$\")15%F47$$\"5,+]Pfe?_:NF0$\")om.UF47$$\"5omTg x.2vFNF0$\")v+1UF47$$\"5MLL$e*[$z*RNF0$\")tL/UF47$$\"5omT5:)[[Lb$F0$\" )yw*>%F47$$\"5,+]PMFwrmNF0$\")3W\">%F47$$\"5MLek`mn3!e$F0$\")2JyTF47$$ \"5omm\"Hd!fX$f$F0$\")p,jTF47$$\"5ML$e9T=%>?OF0$\")]J@TF47$$\"5,+++]iC $pk$F0$\")`VpSF47$$\"5MLLe*[t\\sp$F0$\")*pX%RF47$$\"5omm;H2qcZPF0$\")A ,'z$F47$$\"5,++]7.\"fF&QF0$\")1%[X$F47$$\"5ommm;/OgbRF0$\")&f'RJF47$$ \"5,++]ilAFjSF0$\")5!o(GF47$$\"5omm\"zW7@^6%F0$\")h\"=y#F47$$\"5NLLLL$ )*pp;%F0$\")/x3FF47$$\"5NLL$3-$H**>UF0$\")5qcEF47$$\"5NLLL3xe,tUF0$\") (4pi#F47$$\"5-++v=n(*fDVF0$\")hU=EF47$$\"5omm;HdO=yVF0$\")!f&HEF47$$\" 5MLLe9\"z-lU%F0$\")!)pbEF47$$\"5,++++D>#[Z%F0$\")#yap#F47$$\"5ommmT&G! e&e%F0$\")J!y#GF47$$\"5NLLLL$)Qk%o%F0$\")myvHF47$$\"5-++]iSjE!z%F0$\") 2_OJF47$$\"5-++]P40O\"*[F0$\")q^mKF47$$\"\"&F)$\")&eoN$F4-%&COLORG6&%$ RGBG$\"#X!\"#F($\"#&*Fgim-%'LEGENDG6#%9scheme~with~simple~nodesG-F$6%7 atF'7$F+$!'J6=F07$F2$!'vZFF07$$\"5++]i!*GER37F4$!'I4FF07$F8$!';:EF07$$ \"5ML$3_+ZiaW\"F4$!'6fCF07$F=$!'!3B#F07$FG$!'8=7F07$FQ$!&ys#F07$FV$\"' P*3$F07$Fen$\"'1Q$)F07$Fjn$\"(FHG\"F07$F_o$\"(r:Z\"F07$$\"5ommT5:j=XWF 4$\"(TCN\"F07$Fdo$\"(@@-\"F07$$\"5NLLek.%*Qz\\F4$\"'%[_'F07$Fio$\"'&z- \"F07$F^p$!'H%>%F07$Fcp$!'@nzF07$Fgq$!'6^\"*F07$Far$!(Nf-\"F07$$\"5NLL eRZAJ^gF4$!(=90\"F07$Ffr$!(ly2\"F07$$\"5qmmT&)348vhF4$!).(*e5F47$F[s$! )v!f/\"F47$F`s$!(J?*zF47$Fes$!(_my$F47$Fjs$!(!G(=\"F47$F_t$\"(=jG$F47$ Fdt$\"(=l\"eF47$Fit$\"(FOS)F47$F^u$\") F47$Ff\\l$\")y&p*=F47$F[]l$\")\"3n)=F47$F`]l$\")qc!)=F47$Fe]l$\")\"Hx( =F47$Fj]l$\")1Xz=F47$Fd^l$\")'*\\!*=F47$F^_l$\")F-:>F47$Fc_l$\")))oG?F 47$Fh_l$\")RrAAF47$Fb`l$\")Yv:GF47$Fg`l$\")>ZhIF47$F\\al$\")W,dKF47$$ \"5+v$4YsF*Rh:F0$\")i?3KF47$Faal$\")'R?D$F47$$\"5Me9;#Q$>Wo:F0$\")-mgK F47$Ffal$\")$H&>KF47$F[bl$\")!ynC$F47$F`bl$\")(*\\jKF47$Febl$\")X->KF4 7$Fjbl$\")M=lJF47$F_cl$\")%*=3KF47$Fdcl$\")wjPKF47$Ficl$\")iE\\JF47$F^ dl$\")MR\">$F47$Fcdl$\")g(43$F47$Fhdl$\")ej#)HF47$$\"5nmm\"z>c#\\#o\"F 0$\")Fq()HF47$$\"5nmm;zpYU%p\"F0$\")Ma+IF47$$\"5nmmTgxnN1F47$F\\[m$\")mxO#F47$Figm$\")MH)H#F47$F^hm$\")yT>CF47$Fchm $\")'f6b#F47$Fhhm$\")GidEF47$F]im$\")sQJFF4-Fbim6&FdimF($\"#DFgim$\"\" \"F)-F[jm6#%Pscheme~with~a~relatively~large~stability~regionG-F$6%7hrF '7$F+$!'jRgF07$F2$!(Ql3\"F07$Fhjm$!(nI:\"F07$F8$!(X/=\"F07$F`[n$!(a2> \"F07$F=$!(#p\"=\"F07$FG$!()4P5F07$FQ$!'$F07$Fen$\"&X ^#F07$Fjn$!'B`GF07$F_o$!'UZ%*F07$$\"5omT5!p:g[C%F4$!(tk.\"F07$$\"5-+]( oHaN;J%F4$!(9!G6F07$$\"5NLek.H4TyVF4$!(:p8\"F07$Fj\\n$!(-&)=\"F07$$\"5 NL$eRs3P(yXF4$!(!)R@\"F07$Fdo$!(hS:\"F07$Fb]n$!']G%)F07$Fio$!&CC(F07$F ^p$\"(V/2\"F07$Fcp$\"(b>L#F07$Far$\"(0O\"QF07$F[s$\")qDS`F47$F`s$\")Pc omF47$Fes$\"):/5uF47$Fjs$\")\\sovF47$F_t$\")I^uxF47$Fdt$\")^4pvF47$Fit $\")#z&psF47$Fcu$\")/@IiF47$F]v$\")o43[F47$$\"5SLLLeRilDzF4$\")3H2SF47 $Fbv$\"),_hLF47$$\"5qmmm\"H2S3>)F4$\")GlXDF47$Fgv$\")H9(y\"F47$Fex$\"( n$etF47$F][l$!'5LfF47$$\"5+++]7`aE%)*)F4$!(NcA$F47$Fb[l$!(]%=dF47$$\"5 qmm;aQ))f[#*F4$!(mob'F47$Fg[l$!(1e?(F47$Fddn$!(>4J(F47$F\\\\l$!(F%GpF4 7$Fa\\l$!(!e;zF47$Ff\\l$!(k\"4$*F47$F`]l$!(#[:)*F47$Fj]l$!)0&Q+\"F47$F d^l$!)kX/5F47$F^_l$!(B0')*F47$Fh_l$!)gCC6F47$$\"5++++++'\\[Q\"F0$!)U&) G7F47$F]`l$!)#fdK\"F47$$\"5++++v=A)RU\"F0$!)\\-m8F47$$\"5+++++Dk-P9F0$ !)t[%R\"F47$$\"5++++DJ12]9F0$!)Gre8F47$Fb`l$!)q5Z8F47$Fg`l$!(1_6*F47$F \\al$!(leV#F47$F`bl$\"(P(\\DF47$F^dl$\"(&GMjF47$$\"5M$3-Q\"e=E@;F0$\"( `U%pF47$$\"5nmTNr9XIG;F0$\"(Sn9(F47$$\"5+]i!*GrrMN;F0$\"(1KN(F47$Fcdl$ \"(_v:(F47$$\"5n;/,W%[K%\\;F0$\"(pkz'F47$$\"5++Dc,T^Zc;F0$\"(?Ai'F47$$ \"5M$e9\"f(zF0$!(To4)F07$F[gl$!(Y(=\")F07$$\"5M$e9m>eHY%>F0$!(5!R\")F07$F` gl$!(ct8)F07$Fegl$!('G9\")F07$Fjgl$!(`i0)F07$F_hl$!(Y'=tF07$Fdhl$!(*[s hF07$Fihl$!(3O,&F07$F^il$!)my&)RF47$Fcil$!)[\"o<$F47$Fhil$!)6q*o#F47$F bjl$!)$z')=#F47$$\"5+++v$41oWW#F0$!)ITf>F47$Fgjl$!)XTdp\"\\$F47$Fccm$!)mx9MF47$F]dm$!)RK'>$F47$Fbdm$!)!**H#HF47$ Fgdm$!)(z,m#F47$F\\em$!)?JLCF47$Ffem$!)i])G#F47$F`fm$!)*zCA#F47$Fjfm$! )t%3B#F47$Fdgm$!)r&4H#F47$Figm$!)_e-CF47$F^hm$!)W#G_#F47$Fchm$!)uw^EF4 7$Fhhm$!)cFdFF47$F]im$!)6TKGF4-Fbim6&FdimF($\"#vFgim$\"\"#!\"\"-F[jm6# %TButcher's~scheme~B~with~c[5]=c[6]=1/2~and~b[5]=b[6]G-F$6%7gsF'7$F+$! 'l->F07$F2$!'=4SF07$F8$!'$4m%F07$F=$!'82]F07$FB$!')=6&F07$FG$!'GL^F07$ FL$!'-M^F07$FQ$!'`4^F07$$\"5++]7G))>Wr@F4$!'(Q1&F07$$\"5MLLek.pu/BF4$! 'x'*[F07$$\"5nm;/,>=0QCF4$!'Xa[F07$FV$!'/:[F07$$\"5ML$eR(\\;m/FF4$!'Rb ZF07$$\"5nmmT5ll'z$GF4$!'%o![F07$$\"5++](o/[r7(HF4$!''4!\\F07$Fen$!'%3 9&F07$Fjn$!'?+mF07$F_o$!'&=i)F07$Fj\\n$!'y)[*F07$Fdo$!'V/**F07$$\"5om; /^J'Qe%[F4$!'>I)*F07$Fb]n$!'bL'*F07$$\"5++]7yv,%H6&F4$!'\"y<*F07$Fio$! '$Qn)F07$F^p$!'[MpF07$Fcp$!')Ho%F07$Far$!'hR;F07$F[s$\"(z\\7#F47$F`s$ \"(#*Hd'F47$Fes$\")Kgv5F47$F_t$\")UqH;F47$Fit$\")`aF>F47$Fcu$\")-_sBF4 7$F]v$\")\\1]EF47$Fbv$\")cS/GF47$Fgv$\"))Rs'HF47$Faw$\")c`ZHF47$Fex$\" )x'R'HF47$Fjx$\")3`'*HF47$F_y$\")];dHF47$Fdy$\")\")3eHF47$Fiy$\")r_))H F47$F^z$\")0\\\\HF47$Fcz$\")VnFF47$F\\\\l$\")L5SBF47$Fa\\ l$\")_%Q?#F47$Ff\\l$\")X#)=@F47$F[]l$\")kE0@F47$F`]l$\")>a'4#F47$Fe]l$ \")P@$4#F47$Fj]l$\")O\"e4#F47$F_^l$\")O%**4#F47$Fd^l$\")*\\M6#F47$Fi^l $\")#)>K@F47$F^_l$\")w()\\@F47$Fc_l$\")>&HH#F47$Fh_l$\")'R7^#F47$F]`l$ \").VtFF47$Fb`l$\")>&e4$F47$Fg`l$\")\")ykLF47$F\\al$\")UrjOF47$F^dl$\" )gnoQF47$Fhdl$\")gKsRF47$Fbel$\")P'=F%F47$F\\fl$\"(J'yYF07$Fffl$\"(Sg# [F07$Fjgl$\"(*y,\\F07$$\"5MLek.HW#)))>F0$\"(+q!\\F07$$\"5++]iSmTI-?F0$ \"((R2\\F07$$\"5nmTgx.Ry:?F0$\"([!*)[F07$F_hl$\"(;N([F07$$\"5om;a)e6Bi 0#F0$\"(c@![F07$Fdhl$\"(TCr%F07$Fihl$\"(mMX%F07$F^il$\")M&48%F47$Fcil$ \")QPuPF47$Fhil$\")TK/MF47$F]jl$\"))\\/3$F47$Fbjl$\")9J#y#F47$Ff\\q$\" )^R'\\#F47$Fgjl$\")CahAF47$Fa[m$\")\"H$H>F47$Ff[m$\")?GV=F47$F[\\m$\") L]*y\"F47$F`\\m$\")wIt'oF47$F^^m$\")3%Q@#F47$Fc^m$\")(z(3EF47$Fh ^m$\")F)z5$F47$F]_m$\")%=m^$F47$Fb_m$\")^#po$F47$Fg_m$\")Fm4QF47$F\\`m $\")t*y%QF47$Fa`m$\")miuQF47$Ff`m$\")Kk$)QF47$F[am$\")Ru*)QF47$F`am$\" )'yH*QF47$Feam$\")s_$*QF47$Fjam$\")Im!*QF47$F_bm$\")C]%)QF47$Fdbm$\")$ 3a(QF47$Fibm$\")a*G'QF47$F^cm$\")>JHQF47$Fccm$\")ob%y$F47$Fhcm$\")*)fv OF47$F]dm$\")[/UNF47$Fbdm$\")Q!pA$F47$Fgdm$\")&=@$HF47$F\\em$\")N)po#F 47$Faem$\"),w)f#F47$Ffem$\")a/JDF47$F[fm$\")!)*G[#F47$F`fm$\")qMbCF47$ Fefm$\")>fZCF47$Fjfm$\")F4eCF47$F_gm$\")Pd#[#F47$Fdgm$\")jz>DF47$Figm$ \")*QNk#F47$F^hm$\")!p;y#F47$Fchm$\")S>JHF47$Fhhm$\")u(=0$F47$F]im$\") g#f8$F4-Fbim6&FdimFhimFeimF(-F[jm6#%Hscheme~with~c[5]=c[6]=3/4~and~b[5 ]=b[6]G-%%FONTG6$%*HELVETICAG\"\"*-%+AXESLABELSG6$Q\"x6\"Q!Fa`s-%&TITL EG6#%Uerror~curves~for~7~stage~order~6~Runge-Kutta~methodsG-%%VIEWG6$; F(F]im%(DEFAULTG" 1 2 0 1 10 0 2 9 1 4 2 1.000000 45.000000 45.000000 0 1 "scheme with simple nodes" "scheme with a relatively large stabili ty region" "Butcher's scheme B with c[5]=c[6]=1/2 and b[5]=b[6]" "sche me with c[5]=c[6]=3/4 and b[5]=b[6]" }}}}{PARA 0 "" 0 "" {TEXT -1 0 " " }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}}{SECT 1 {PARA 4 " " 0 "" {TEXT -1 46 "Test 2 of 7 stage, order 6 Runge-Kutta methods" }} {PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "dy/dx=x/y" "6#/*&%#dy G\"\"\"%#dxG!\"\"*&%\"xGF&%\"yGF(" }{TEXT -1 10 ", " } {XPPEDIT 18 0 "y(0)=1" "6#/-%\"yG6#\"\"!\"\"\"" }{TEXT -1 1 " " }} {PARA 0 "" 0 "" {TEXT -1 10 "Solution: " }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "y=sqrt(1+x^2)" "6#/%\"yG-%%sqrtG6#,&\"\"\"F)*$% \"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 methods and gives the " }{TEXT 260 22 "root mean square error" }{TEXT -1 18 " of \+ each solution." }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 764 "G := (x,y ) -> x/y: hh := 0.05: numsteps := 200: x0 := 0: y0 := 1:\nmatrix([[`sl ope field: `,G(x,y)],[`initial point: `,``(x0,y0)],[`step width: ` ,hh],\n[`no. of steps: `,numsteps]]);``;\nmthds := [`Butcher's schem e A`,`scheme with simple nodes`,`scheme with a relatively large stabil ity region`,`Butcher's scheme B with `*(c[5]=1/2,c[6]=1/2,b[5]=b[6]),` scheme with `*(c[5]=3/4,c[6]=3/4,b[5]=b[6])]:\nerrs := []:\nDigits := \+ 20:\ng := x -> sqrt(1+x^2):\nfor ct to 5 do\n Gn_RK6_||ct := RK6_||c t(G(x,y),x,y,x0,y0,hh,numsteps,false);\n sm := 0: numpts := nops(Gn_ RK6_||ct):\n for ii to numpts do\n sm := sm+(Gn_RK6_||ct[ii,2]- g(Gn_RK6_||ct[ii,1]))^2;\n end do:\n errs := [op(errs),sqrt(sm/num pts)];\nend do:\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~widt h:~~~G$\"\"&!\"#7$%1no.~of~steps:~~~G\"$+#Q(pprint56\"" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%!G" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#K%'matrixG6 #7'7$%3Butcher's~scheme~AG$\"+dy'\\e\"!#@7$%9scheme~with~simple~nodesG $\"+-=)y9\"!#A7$%Pscheme~with~a~relatively~large~stability~regionG$\"+ >G;1 " 0 "" {MPLTEXT 1 0 694 "G := (x, y) -> x/y: hh := 0.05: numsteps := 200: x0 := 0: y0 := 1:\nmatrix([[`s lope field: `,G(x,y)],[`initial point: `,``(x0,y0)],[`step width: \+ `,hh],\n[`no. of steps: `,numsteps]]);``;\nmthds := [`Butcher's sche me A`,`scheme with simple nodes`,`scheme with a relatively large stabi lity region`,`Butcher's scheme B with `*(c[5]=1/2,c[6]=1/2,b[5]=b[6]), `scheme with `*(c[5]=3/4,c[6]=3/4,b[5]=b[6])]:\nerrs := []:\nDigits := 30:\nfor ct to 5 do\n gn_RK6_||ct := RK6_||ct(G(x,y),x,y,x0,y0,hh,n umsteps,true);\nend do:\ng := x -> sqrt(1+x^2):\nxx := 9.99: gxx := ev alf(g(xx)):\nfor ct to 5 do\n errs := [op(errs),abs(gn_RK6_||ct(xx)- gxx)];\nend do:\nDigits := 10:\nlinalg[transpose]([mthds,evalf(errs)]) ;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#K%'matrixG6#7&7$%0slope~field:~~~ G*&%\"xG\"\"\"%\"yG!\"\"7$%0initial~point:~G-%!G6$\"\"!F+7$%/step~widt h:~~~G$\"\"&!\"#7$%1no.~of~steps:~~~G\"$+#Q(pprint76\"" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%!G" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#K%'matrixG6 #7'7$%3Butcher's~scheme~AG$\"+:plX`!#A7$%9scheme~with~simple~nodesG$\" +&z.T\"G!#B7$%Pscheme~with~a~relatively~large~stability~regionG$\"+ayy 4YF07$*&%9Butcher's~scheme~B~with~G\"\"\"6%/&%\"cG6#\"\"&#F8\"\"#/&F<6 #\"\"'F?/&%\"bGF=&FGFCF8$\"+zo+ZBF07$*&%-scheme~with~GF86%/F;#\"\"$\" \"%/FBFPFEF8$\"+^i]PK!#CQ(pprint86\"" }}}{PARA 0 "" 0 "" {TEXT -1 0 " " }}{PARA 0 "" 0 "" {TEXT -1 4 "The " }{TEXT 260 22 "root mean square \+ error" }{TEXT -1 20 " over the interval " }{XPPEDIT 18 0 "[0, 10]" "6 #7$\"\"!\"#5" }{TEXT -1 82 " of each Runge-Kutta method is estimated \+ as 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 0 461 "mthds := [`Butcher's scheme A`,`scheme with simple n odes`,`scheme with a relatively large stability region`,`Butcher's sch eme B with `*(c[5]=1/2,c[6]=1/2,b[5]=b[6]),`scheme with `*(c[5]=3/4,c[ 6]=3/4,b[5]=b[6])]:\nerrs := []:\nDigits := 20:\ng := x -> sqrt(1+x^2) :\nfor ct to 5 do\n sm := NCint((g(x)-'gn_RK6_||ct'(x))^2,x=0..10,ad aptive=false,numpoints=7,factor=100);\n errs := [op(errs),sqrt(sm/10 )];\nend do:\nDigits := 10:\nlinalg[transpose]([mthds,evalf(errs)]);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#K%'matrixG6#7'7$%3Butcher's~scheme~A G$\"+%3Y/e\"!#@7$%9scheme~with~simple~nodesG$\"+5VOW6!#A7$%Pscheme~wit h~a~relatively~large~stability~regionG$\"+%y+5q\"F07$*&%9Butcher's~sch eme~B~with~G\"\"\"6%/&%\"cG6#\"\"&#F8\"\"#/&F<6#\"\"'F?/&%\"bGF=&FGFCF 8$\"+Afz07F07$*&%-scheme~with~GF86%/F;#\"\"$\"\"%/FBFPFEF8$\"+<09YH!#B 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 0 517 "evalf[20](plot([g(x)-'gn_RK6_1'(x),g(x)-'gn_RK6_2'(x ),g(x)-'gn_RK6_3'(x),g(x)-'gn_RK6_4'(x),\ng(x)-'gn_RK6_5'(x)],x=0..10, font=[HELVETICA,9],\ncolor=[COLOR(RGB,.95,.2,0),COLOR(RGB,.45,0,.95),C OLOR(RGB,0,.25,1),COLOR(RGB,0,.75,.2),COLOR(RGB,.95,.45,0)],\nlegend=[ `Butcher's scheme A`,`scheme with simple nodes`,`scheme with a relativ ely large stability region`,`Butcher's scheme B with c[5]=c[6]=1/2 and b[5]=b[6]`,`scheme with c[5]=c[6]=3/4 and b[5]=b[6]`],title=`error cu rves for 7 stage order 6 Runge-Kutta methods`));" }}{PARA 13 "" 1 "" {GLPLOT2D 935 526 526 {PLOTDATA 2 "6+-%'CURVESG6%7]o7$$\"\"!F)F(7$$\"5 lmmmmT&)G\\a!#@$!'iqZ!#>7$$\"5LLLLL3x&)*3\"!#?$!(?6s\"F07$$\"5++++]ily M;F4$!(%RbOF07$$\"5mmmmm;arz@F4$!(v)fhF07$$\"5LLL$e*)4bQl#F4$!(3H4*F07 $$\"5++++D\"y%*z7$F4$!),EH7F07$$\"5mmm;ajW8-OF4$!)5lf:F07$$\"5LLLL$e9u i2%F4$!)&[b)=F07$$\"5mmm;H2Q\\4YF4$!)^\\*=#F07$$\"5++++voMrU^F4$!)?#[Y #F07$$\"5NLL$3-8Lfn&F4$!)hI0FF07$$\"5mmmmm\"z_\"4iF4$!)%*)y!HF07$$\"5l mmmmm6m#G(F4$!)nX*>$F07$$\"5lmmmmT&phN)F4$!)t(*fLF07$$\"5ILL$3-js.*))F 4$!)'p7S$F07$$\"5++++v=ddC%*F4$!)+CAMF07$$\"5ILL3-jsn\"p*F4$!)D@MMF07$ $\"5lmm;H2)y(e**F4$!)w'oU$F07$$\"5++]i:N!)eA5F0$!)%oET$F07$$\"5LLLLe*= )H\\5F0$!)Nl=MF07$$\"5++++v=JN[6F0$!)<$RL$F07$$\"5mmmm\"z/3uC\"F0$!)@@ HKF07$$\"5++++DJ$RDX\"F0$!)0%3*HF07$$\"5mmmm\"zR'ok;F0$!)\\0SFF07$$\"5 ++++D1J:w=F0$!)()Q8DF07$$\"5LLLLL3En$4#F0$!)R03BF07$$\"5mmmm;/RE&G#F0$ !)'Q)[@F07$$\"5+++++D.&4]#F0$!)TM\"*>F07$$\"5+++++vB_Y\"F07$$\"5*******\\7o7Tv$F0$!)GT \"Q\"F07$$\"5LLLLL$Q*o]RF0$!)?$pJ\"F07$$\"5*******\\7=lj;%F0$!)zd_7F07 $$\"5*******\\PaR\"F07$$\"5LLLL$e9Ege%F0$!)rTV6F07$$\"5LLL LeR\"3Gy%F0$!)hQ)4\"F07$$\"5mmmm;/T1&*\\F0$!)5b`5F07$$\"5mmmm\"zRQb@&F 0$!)&G1,\"F07$$\"5*******\\(=>Y2aF0$!(q'f(*F07$$\"5mmmm;zXu9cF0$!(a1T* F07$$\"5**********\\y))GeF0$!(&*\\2*F07$$\"5********\\i_QQgF0$!()po()F 07$$\"5*******\\7y%3TiF0$!(a6\\)F07$$\"5********\\P![hY'F0$!(LE?)F07$$ \"5KLLLL$Qx$omF0$!(J%fzF07$$\"5*********\\P+V)oF0$!(F]r(F07$$\"5mmmm\" zpe*zqF0$!([g](F07$$\"5*********\\#\\'QH(F0$!(T+H(F07$$\"5KLLLe9S8&\\( F0$!(]x4(F07$$\"5*******\\i?=bq(F0$!(9s!pF07$$\"5KLLLL3s?6zF0$!(!\\InF 07$$\"5*******\\7`Wl7)F0$!(T[b'F07$$\"5lmmmmm*RRL)F0$!(lSR'F07$$\"5lmm m;a<.Y&)F0$!(\\vB'F07$$\"5KLLLe9tOc()F0$!(q'*3'F07$$\"5**********\\Qk \\*)F0$!((zffF07$$\"5KLLL$3dg6<*F0$!(fv\"eF07$$\"5lmmmmmxGp$*F0$!(Qfp& F07$$\"5)******\\7oK0e*F0$!(:kNF07$$\"5+++](=#**3E7F4$\"'#[d$F07$$\"5LLLekGg?%H\"F4$\"'y^ OF07$$\"5mmmmTN@Ki8F4$\"'8pRF07$$\"5+++v=U#Q/V\"F4$\"'1l\\F07$$\"5LLL$ e*[Vb)\\\"F4$\"'`yvF07$$\"5mmm\"HdXqmc\"F4$\"'(\\l(F07$F8$\"'LZwF07$$ \"5LLL3FpE!Hq\"F4$\"'F4$\"'=*z)F07$$\"5mmmTN'4n`(>F4$\"(zR6 \"F07$$\"5+++]7.K[V?F4$\"(&>k7F07$$\"5LLLe*)4$*f6@F4$\"((\\i7F07$F=$\" (O6E\"F07$$\"5++DJ&p(G)*QAF4$\"(>AE\"F07$$\"5LL$eRsL]#)H#F4$\"(m?F\"F0 7$$\"5mmTg_(z+RsFF4$\"(m7z\"F07$$\"5LL3x\")zulJGF4$\"(B6\"=F07$$\"5mmmT5S\\#4*G F4$\"(5M(=F07$$\"5++D1R+C>]HF4$\"(p/.#F07$$\"5LL$3x1')f%4IF4$\"(kEI#F0 7$$\"5mmTN'4KF(oIF4$\"(q))H#F07$FG$\"(e]H#F07$$\"5LLek`TAE(=$F4$\"(J;H #F07$$\"5mm;H#=qHlC$F4$\"(W1H#F07$$\"5++v$4@;(z0LF4$\"(r()H#F07$$\"5LL LeRAY1lLF4$\"(uEL#F07$$\"5++](oHa*f$[$F4$\"(5'[EF07$FL$\"(Tpt#F07$$\"5 LL$e9TQp1s$F4$\"(Sys#F07$$\"5+++vo/V?RQF4$\"(%yUFF07$$\"5mm;/ED#Rx&RF4 $\"(`(>HF07$FQ$\"('3)3$F07$FV$\"(D)HLF07$Fen$\"(f'pMF07$$\"5NL$e9TQ=gF &F4$\"(]HX$F07$$\"5qmm\"z%*HB$4aF4$\"(5WY$F07$$\"5NLe9;dd(fZ&F4$\"(SJ^ $F07$$\"5++]P%[@GEa&F4$\"(=#RNF07$$\"5qmTg_s1G4cF4$\"(,#HNF07$Fjn$\"(% >>NF07$$\"5+++v$4'HaUfF4$\"(fW]$F07$F_o$\"(g]\\$F07$Fdo$\"(\\dH$F07$Fi o$\"(Ia*HF07$Fcp$\"(YGn#F07$Fgq$\"(hLO#F07$F\\r$\"(=V:#F07$Far$\"(G(z> F07$$\"5LLLLe*ot*\\8F0$\"(/<#=F07$Ffr$\"(3))p\"F07$F[s$\"(<7]\"F07$F`s $\"(w:N\"F07$Fes$\"(!\\G7F07$Fjs$\"(R&Q6F07$F_t$\"(k<0\"F07$Fdt$\"'rs( *F07$Fit$\"'SX\"*F07$F^u$\"'VR')F07$Fcu$\"'K/\")F07$Fhu$\"'k)p(F07$F]v $\"'atsF07$Fbv$\"'hLpF07$Fgv$\"'_%f'F07$F\\w$\"'N+jF07$Faw$\"'b>gF07$F fw$\"'T#y&F07$F[x$\"'JYbF07$F`x$\"'K?`F07$Fex$\"'#y8&F07$Fjx$\"'3a\\F0 7$F_y$\"'OxZF07$Fdy$\"'6;YF07$Fiy$\"'**pWF07$F^z$\"'6=VF07$Fcz$\"'3!>% F07$Fhz$\"'UhSF07$F][l$\"'S^RF07$Fb[l$\"'pPQF07$Fg[l$\"'YOPF07$F\\\\l$ \"';OOF07$Fa\\l$\"'7VNF07$Ff\\l$\"'l]MF07$F[]l$\"',mLF07$F`]l$\"'i$G$F 07$Fe]l$\"'y0KF07$Fj]l$\"'SPJF07$F_^l$\"'_iIF07$Fd^l$\"'])*HF07$Fi^l$ \"'6LHF07$F^_l$\"'7tGF07$Fc_l$\"&8\"GFg_l-Fi_l6&F[`l$\"#XF^`lF(F\\`l-F c`l6#%9scheme~with~simple~nodesG-F$6%7fqF'7$F+$\"'Y]5F07$F2$\"'EvVF07$ F_bl$\"'#o)[F07$F8$\"'rK&*F07$Ffcl$\"'\"G`*F07$F[dl$\"'&pf*F07$F`dl$\" '(y\"**F07$Fedl$\"(u25\"F07$Fjdl$\"(X**R\"F07$F_el$\"(g=f\"F07$Fdel$\" (A(*e\"F07$F=$\"(C!)e\"F07$F\\fl$\"()[*e\"F07$Fafl$\"(5Bg\"F07$Fffl$\" (3&[;F07$F[gl$\"(Ifx\"F07$F`gl$\"(Zk2#F07$Fegl$\"(E'yAF07$Fjgl$\"($QvA F07$FB$\"(bAF#F07$Fbhl$\"(60F#F07$Fghl$\"(P`F#F07$F\\il$\"(,;I#F07$Fai l$\"(CPQ#F07$Ffil$\"(L1f#F07$F[jl$\"(w\"\\HF07$F`jl$\"(=V%HF07$FG$\"(P %RHF07$Fhjl$\"(x]$HF07$F][m$\"(**R$HF07$Fb[m$\"(u^%HF07$Fg[m$\"(>5*HF0 7$F\\\\m$\"(9oT$F07$FL$\"(Pj`$F07$Fd\\m$\"()pCNF07$Fi\\m$\"(jja$F07$F^ ]m$\"(-Hz$F07$FQ$\"(Sy-%F07$FV$\"(qvQ%F07$Fen$\"(a@i%F07$$\"5qm\"HKk#f O4_F4$\"(?,h%F07$F\\^m$\"(_2g%F07$$\"5++vozT3nU`F4$\"(%=,YF07$Fa^m$\"( >)GYF07$Ff^m$\"(r$>ZF07$F[_m$\"()=qZF07$F`_m$\"()ocZF07$Fjn$\"(SKu%F07 $$\"5++D1*ye&eUdF4$\"(A0t%F07$$\"5qm;HdX!Q#4eF4$\"(26s%F07$$\"5NL3_D.0 *e(eF4$\"(\")>s%F07$Fh_m$\"([%[ZF07$$\"5qm\"z>'=a>4gF4$\"(,;\"[F07$$\" 5NL$3-j(y%e2'F4$\"(Fuz%F07$$\"5++vV)RL+D9'F4$\"(NKy%F07$F_o$\"(v\"pZF0 7$Fdo$\"(Wkh%F07$Fio$\"(oBJ%F07$Fcp$\"(tm&RF07$Fgq$\"(t()f$F07$F\\r$\" (\\fL$F07$Far$\"(Tx5$F07$Fbam$\"($H'*GF07$Ffr$\"(.Fs#F07$F[s$\"(B2V#F0 7$F`s$\"(`-?#F07$Fes$\"(ve+#F07$Fjs$\"(%\\h=F07$F_t$\"(>7s\"F07$Fdt$\" (P**f\"F07$Fit$\"(xv\\\"F07$F^u$\"(5\\T\"F07$Fcu$\"($QF8F07$Fhu$\"((*4 E\"F07$F]v$\"(79>\"F07$Fbv$\"(^d8\"F07$Fgv$\"(?-3\"F07$F\\w$\"(U?.\"F0 7$Faw$\"'\\g)*F07$Ffw$\"'2s%*F07$F[x$\"'O&3*F07$F`x$\"'>:()F07$Fex$\"' C;%)F07$Fjx$\"'E:\")F07$F_y$\"'zDyF07$Fdy$\"'lhvF07$Fiy$\"'IAtF07$F^z$ \"']tqF07$Fcz$\"'xjoF07$Fhz$\"',`mF07$F][l$\"'zskF07$Fb[l$\"'_'G'F07$F g[l$\"'q?hF07$F\\\\l$\"'RcfF07$Fa\\l$\"'*R!eF07$Ff\\l$\"'__cF07$F[]l$ \"'(Q^&F07$F`]l$\"'\"*y`F07$Fe]l$\"'Q^_F07$Fj]l$\"'RR^F07$F_^l$\"'s;]F 07$Fd^l$\"'&=\"\\F07$Fi^l$\"'t/[F07$F^_l$\"'Y1ZF07$Fc_l$\"&_g%Fg_l-Fi_ l6&F[`lF($\"#DF^`l$\"\"\"F)-Fc`l6#%Pscheme~with~a~relatively~large~sta bility~regionG-F$6%7hqF'7$F+$\"(*F07$F2$\"'nMWF07$F`al$\"'*GV%F07$F eal$\"'xZWF07$Fjal$\"'G_XF07$F_bl$\"';y\\F07$Fdbl$\"'z,jF07$Fibl$\"'QV (*F07$F^cl$\"'%Q%)*F07$F8$\"'1M)*F07$Ffcl$\"'_M)*F07$F[dl$\"'_-**F07$F `dl$\"(&*Q-\"F07$Fedl$\"(!>P6F07$Fjdl$\"(icW\"F07$F_el$\"(SDk\"F07$Fde l$\"(M.k\"F07$F=$\"()eQ;F07$Fafl$\"(!H`;F07$F[gl$\"(\\$H=F07$F`gl$\"(% 4J@F07$Fegl$\"(-IL#F07$Fjgl$\"($oHBF07$FB$\"(ykK#F07$Fbhl$\"(jYK#F07$F ghl$\"(%QHBF07$F\\il$\"(C_N#F07$Fail$\"(=bV#F07$Ffil$\"(uhj#F07$F[jl$ \"(\"3\")HF07$F`jl$\"(qh(HF07$FG$\"(M7(HF07$Fhjl$\"()zmHF07$F][m$\"($[ lHF07$Fb[m$\"(+d(HF07$Fg[m$\"(c\"=IF07$F\\\\m$\"(,zS$F07$FL$\"(z`^$F07 $Fd\\m$\"(FO]$F07$Fi\\m$\"(f4_$F07$F^]m$\"(%RHPF07$FQ$\"(fY#RF07$FV$\" (m!yTF07$Fen$\"(e(*G%F07$Fgan$\"(\\$yUF07$F\\^m$\"(%4oUF07$F_bn$\"(!*= E%F07$Fa^m$\"(=iE%F07$Ff^m$\"(@JH%F07$F[_m$\"($4/VF07$F`_m$\"(4>H%F07$ Fjn$\"(%pzUF07$Fh_m$\"(b$HUF07$F_o$\"(VR<%F07$$\"5lmmm;zp!fu'F4$\"(B)) *RF07$Fdo$\"()QzPF07$$\"5lmmm;a`T>yF4$\"()[NNF07$Fio$\"(`=G$F07$F^p$\" (f+.$F07$Fcp$\"(ndy#F07$F]q$\"(nDb#F07$Fgq$\"([?L#F07$F\\r$\"(tG0#F07$ Far$\"(\\5$=F07$Fbam$\"($pO;F07$Ffr$\"(:u\\\"F07$F[s$\"('Q!H\"F07$F`s$ \"(Df9\"F07$Fes$\"(xM.\"F07$Fjs$\"'IW&*F07$F_t$\"'1&z)F07$Fdt$\"'^j\") F07$Fit$\"'$[j(F07$F^u$\"'%)4sF07$Fcu$\"'thnF07$Fhu$\"'WAkF07$F]v$\"'? ngF07$Fbv$\"'O$y&F07$Fgv$\"'M+bF07$F\\w$\"'([D&F07$Faw$\"'e?]F07$Ffw$ \"'vA[F07$F[x$\"'!ei%F07$F`x$\"'IPWF07$Fex$\"'1&G%F07$Fjx$\"'\"=8%F07$ F_y$\"'U%)RF07$Fdy$\"'$*\\QF07$Fiy$\"'1GPF07$F^z$\"'Q,OF07$Fcz$\"'g%\\ $F07$Fhz$\"'H(Q$F07$F][l$\"'`&H$F07$Fb[l$\"'p+KF07$Fg[l$\"'F;JF07$F\\ \\l$\"'hKIF07$Fa\\l$\"'-bHF07$Ff\\l$\"'!z(GF07$F[]l$\"'J2GF07$F`]l$\"' fQFF07$Fe]l$\"'ntEF07$Fj]l$\"'l;EF07$F_^l$\"'>aDF07$Fd^l$\"'!3]#F07$Fi ^l$\"'EYCF07$F^_l$\"'A'R#F07$Fc_l$\"&ZM#Fg_l-Fi_l6&F[`lF($\"#vF^`lF_`l -Fc`l6#%TButcher's~scheme~B~with~c[5]=c[6]=1/2~and~b[5]=b[6]G-F$6%7jqF '7$F+$\"&[6%F07$F2$\"'sB:F07$F`al$\"'-B:F07$Feal$\"'?F:F07$Fjal$\"'*zb \"F07$F_bl$\"'k&o\"F07$Fdbl$\"'l(3#F07$Fibl$\"'nXJF07$F^cl$\"'dwJF07$F 8$\"'RtJF07$Ffcl$\"'8tJF07$F[dl$\"'X#>$F07$F`dl$\"'f!H$F07$Fedl$\"']DO F07$Fjdl$\"'pZXF07$F_el$\"'.S^F07$Fdel$\"'7L^F07$F=$\"'\\F^F07$Fafl$\" 'Jo^F07$F[gl$\"'XycF07$F`gl$\"''Qc'F07$Fegl$\"'ifrF07$Fjgl$\"'U\\rF07$ FB$\"'`RrF07$Fbhl$\"'YLrF07$Fghl$\"'lXrF07$F\\il$\"'Sh!z&4UF4$\"(O$\\6F07$$\"5+++DcwR)GM%F 4$\"(S'\\6F07$$\"5LL$3F>*))=wWF4$\"(c\\>\"F07$FV$\"(R8@\"F07$$\"5LLeR( \\EYhn%F4$\"(*G37F07$$\"5++]ilA()zUZF4$\"(Za?\"F07$$\"5mmT&Q.=^%4[F4$ \"(^N?\"F07$$\"5LLL3-QO5w[F4$\"(dX?\"F07$$\"5***\\7.d4cF%\\F4$\"(IE@\" F07$$\"5lm;aQ`&3%4]F4$\"(K/B\"F07$$\"5IL3x1651w]F4$\"(SrA\"F07$Fen$\"( KQA\"F07$Fjn$\"(c!*>\"F07$F_o$\"((F07$Fcp$\"'#>O'F07$F]q $\"'iwbF07$Fgq$\"'ZW[F07$F\\r$\"']$)RF07$Far$\"'5KLF07$Fbam$\"'f!y#F07 $Ffr$\"',BCF07$F[s$\"'d[>F07$F`s$\"' " 0 "" {MPLTEXT 1 0 476 "evalf[20](plot([g (x)-'gn_RK6_2'(x),g(x)-'gn_RK6_3'(x),g(x)-'gn_RK6_4'(x),g(x)-'gn_RK6_5 '(x)],x=0..10,\ncolor=[COLOR(RGB,.95,.2,0),COLOR(RGB,.45,0,.95),COLOR( RGB,0,.25,1),COLOR(RGB,0,.75,.2),COLOR(RGB,.95,.45,0)],\nlegend=[`sche me with simple nodes`,`scheme with a relatively large stability region `,`Butcher's scheme B with c[5]=c[6]=1/2 and b[5]=b[6]`,`scheme with c [5]=c[6]=3/4 and b[5]=b[6]`],font=[HELVETICA,9],title=`error curves fo r 7 stage order 6 Runge-Kutta methods`));" }}{PARA 13 "" 1 "" {GLPLOT2D 969 539 539 {PLOTDATA 2 "6*-%'CURVESG6%7dq7$$\"\"!F)F(7$$\"5 lmmmmT&)G\\a!#@$\"&n%*)!#>7$$\"5LLLLL3x&)*3\"!#?$\"'ulNF07$$\"5mmmT5:Q (z:\"F4$\"'>kNF07$$\"5+++](=#**3E7F4$\"'#[d$F07$$\"5LLLekGg?%H\"F4$\"' y^OF07$$\"5mmmmTN@Ki8F4$\"'8pRF07$$\"5+++v=U#Q/V\"F4$\"'1l\\F07$$\"5LL L$e*[Vb)\\\"F4$\"'`yvF07$$\"5mmm\"HdXqmc\"F4$\"'(\\l(F07$$\"5++++]ilyM ;F4$\"'LZwF07$$\"5LLL3FpE!Hq\"F4$\"'F4$\"'=*z)F07$$\"5mmmTN '4n`(>F4$\"(zR6\"F07$$\"5+++]7.K[V?F4$\"(&>k7F07$$\"5LLLe*)4$*f6@F4$\" ((\\i7F07$$\"5mmmmm;arz@F4$\"(O6E\"F07$$\"5++DJ&p(G)*QAF4$\"(>AE\"F07$ $\"5LL$eRsL]#)H#F4$\"(m?F\"F07$$\"5mmTg_(z+RsFF4$\"(m7z\"F07$$ \"5LL3x\")zulJGF4$\"(B6\"=F07$$\"5mmmT5S\\#4*GF4$\"(5M(=F07$$\"5++D1R+ C>]HF4$\"(p/.#F07$$\"5LL$3x1')f%4IF4$\"(kEI#F07$$\"5mmTN'4KF(oIF4$\"(q ))H#F07$$\"5++++D\"y%*z7$F4$\"(e]H#F07$$\"5LLek`TAE(=$F4$\"(J;H#F07$$ \"5mm;H#=qHlC$F4$\"(W1H#F07$$\"5++v$4@;(z0LF4$\"(r()H#F07$$\"5LLLeRAY1 lLF4$\"(uEL#F07$$\"5++](oHa*f$[$F4$\"(5'[EF07$$\"5mmm;ajW8-OF4$\"(Tpt# F07$$\"5LL$e9TQp1s$F4$\"(Sys#F07$$\"5+++vo/V?RQF4$\"(%yUFF07$$\"5mm;/E D#Rx&RF4$\"(`(>HF07$$\"5LLLL$e9ui2%F4$\"('3)3$F07$$\"5mmm;H2Q\\4YF4$\" (D)HLF07$$\"5++++voMrU^F4$\"(f'pMF07$$\"5NL$e9TQ=gF&F4$\"(]HX$F07$$\"5 qmm\"z%*HB$4aF4$\"(5WY$F07$$\"5NLe9;dd(fZ&F4$\"(SJ^$F07$$\"5++]P%[@GEa &F4$\"(=#RNF07$$\"5qmTg_s1G4cF4$\"(,#HNF07$$\"5NLL$3-8Lfn&F4$\"(%>>NF0 7$$\"5+++v$4'HaUfF4$\"(fW]$F07$$\"5mmmmm\"z_\"4iF4$\"(g]\\$F07$$\"5lmm mmm6m#G(F4$\"(\\dH$F07$$\"5lmmmmT&phN)F4$\"(Ia*HF07$$\"5++++v=ddC%*F4$ \"(YGn#F07$$\"5LLLLe*=)H\\5F0$\"(hLO#F07$$\"5++++v=JN[6F0$\"(=V:#F07$$ \"5mmmm\"z/3uC\"F0$\"(G(z>F07$$\"5LLLLe*ot*\\8F0$\"(/<#=F07$$\"5++++DJ $RDX\"F0$\"(3))p\"F07$$\"5mmmm\"zR'ok;F0$\"(<7]\"F07$$\"5++++D1J:w=F0$ \"(w:N\"F07$$\"5LLLLL3En$4#F0$\"(!\\G7F07$$\"5mmmm;/RE&G#F0$\"(R&Q6F07 $$\"5+++++D.&4]#F0$\"(k<0\"F07$$\"5+++++vB_gF07$$\"5LLLLeR\"3Gy%F0$ \"'T#y&F07$$\"5mmmm;/T1&*\\F0$\"'JYbF07$$\"5mmmm\"zRQb@&F0$\"'K?`F07$$ \"5*******\\(=>Y2aF0$\"'#y8&F07$$\"5mmmm;zXu9cF0$\"'3a\\F07$$\"5****** ****\\y))GeF0$\"'OxZF07$$\"5********\\i_QQgF0$\"'6;YF07$$\"5*******\\7 y%3TiF0$\"'**pWF07$$\"5********\\P![hY'F0$\"'6=VF07$$\"5KLLLL$Qx$omF0$ \"'3!>%F07$$\"5*********\\P+V)oF0$\"'UhSF07$$\"5mmmm\"zpe*zqF0$\"'S^RF 07$$\"5*********\\#\\'QH(F0$\"'pPQF07$$\"5KLLLe9S8&\\(F0$\"'YOPF07$$\" 5*******\\i?=bq(F0$\"';OOF07$$\"5KLLLL3s?6zF0$\"'7VNF07$$\"5*******\\7 `Wl7)F0$\"'l]MF07$$\"5lmmmmm*RRL)F0$\"',mLF07$$\"5lmmm;a<.Y&)F0$\"'i$G $F07$$\"5KLLLe9tOc()F0$\"'y0KF07$$\"5**********\\Qk\\*)F0$\"'SPJF07$$ \"5KLLL$3dg6<*F0$\"'_iIF07$$\"5lmmmmmxGp$*F0$\"'])*HF07$$\"5)******\\7 oK0e*F0$\"'6LHF07$$\"5)******\\(=5s#y*F0$\"'7tGF07$$\"#5F)$\"&8\"G!#=- %&COLORG6&%$RGBG$\"#&*!\"#$\"\"#!\"\"F(-%'LEGENDG6#%9scheme~with~simpl e~nodesG-F$6%7fqF'7$F+$\"'Y]5F07$F2$\"'EvVF07$FG$\"'#o)[F07$Fen$\"'rK& *F07$Fjn$\"'\"G`*F07$F_o$\"'&pf*F07$Fdo$\"'(y\"**F07$Fio$\"(u25\"F07$F ^p$\"(X**R\"F07$Fcp$\"(g=f\"F07$Fhp$\"(A(*e\"F07$F]q$\"(C!)e\"F07$Fbq$ \"()[*e\"F07$Fgq$\"(5Bg\"F07$F\\r$\"(3&[;F07$Far$\"(Ifx\"F07$Ffr$\"(Zk 2#F07$F[s$\"(E'yAF07$F`s$\"($QvAF07$Fes$\"(bAF#F07$Fjs$\"(60F#F07$F_t$ \"(P`F#F07$Fdt$\"(,;I#F07$Fit$\"(CPQ#F07$F^u$\"(L1f#F07$Fcu$\"(w\"\\HF 07$Fhu$\"(=V%HF07$F]v$\"(P%RHF07$Fbv$\"(x]$HF07$Fgv$\"(**R$HF07$F\\w$ \"(u^%HF07$Faw$\"(>5*HF07$Ffw$\"(9oT$F07$F[x$\"(Pj`$F07$F`x$\"()pCNF07 $Fex$\"(jja$F07$Fjx$\"(-Hz$F07$F_y$\"(Sy-%F07$Fdy$\"(qvQ%F07$Fiy$\"(a@ i%F07$$\"5qm\"HKk#fO4_F4$\"(?,h%F07$F^z$\"(_2g%F07$$\"5++vozT3nU`F4$\" (%=,YF07$Fcz$\"(>)GYF07$Fhz$\"(r$>ZF07$F][l$\"()=qZF07$Fb[l$\"()ocZF07 $Fg[l$\"(SKu%F07$$\"5++D1*ye&eUdF4$\"(A0t%F07$$\"5qm;HdX!Q#4eF4$\"(26s %F07$$\"5NL3_D.0*e(eF4$\"(\")>s%F07$F\\\\l$\"([%[ZF07$$\"5qm\"z>'=a>4g F4$\"(,;\"[F07$$\"5NL$3-j(y%e2'F4$\"(Fuz%F07$$\"5++vV)RL+D9'F4$\"(NKy% F07$Fa\\l$\"(v\"pZF07$Ff\\l$\"(Wkh%F07$F[]l$\"(oBJ%F07$F`]l$\"(tm&RF07 $Fe]l$\"(t()f$F07$Fj]l$\"(\\fL$F07$F_^l$\"(Tx5$F07$Fd^l$\"($H'*GF07$Fi ^l$\"(.Fs#F07$F^_l$\"(B2V#F07$Fc_l$\"(`-?#F07$Fh_l$\"(ve+#F07$F]`l$\"( %\\h=F07$Fb`l$\"(>7s\"F07$Fg`l$\"(P**f\"F07$F\\al$\"(xv\\\"F07$Faal$\" (5\\T\"F07$Ffal$\"($QF8F07$F[bl$\"((*4E\"F07$F`bl$\"(79>\"F07$Febl$\"( ^d8\"F07$Fjbl$\"(?-3\"F07$F_cl$\"(U?.\"F07$Fdcl$\"'\\g)*F07$Ficl$\"'2s %*F07$F^dl$\"'O&3*F07$Fcdl$\"'>:()F07$Fhdl$\"'C;%)F07$F]el$\"'E:\")F07 $Fbel$\"'zDyF07$Fgel$\"'lhvF07$F\\fl$\"'IAtF07$Fafl$\"']tqF07$Fffl$\"' xjoF07$F[gl$\"',`mF07$F`gl$\"'zskF07$Fegl$\"'_'G'F07$Fjgl$\"'q?hF07$F_ hl$\"'RcfF07$Fdhl$\"'*R!eF07$Fihl$\"'__cF07$F^il$\"'(Q^&F07$Fcil$\"'\" *y`F07$Fhil$\"'Q^_F07$F]jl$\"'RR^F07$Fbjl$\"'s;]F07$Fgjl$\"'&=\"\\F07$ F\\[m$\"'t/[F07$Fa[m$\"'Y1ZF07$Ff[m$\"&_g%Fj[m-F\\\\m6&F^\\m$\"#XFa\\m F(F_\\m-Ff\\m6#%Pscheme~with~a~relatively~large~stability~regionG-F$6% 7hqF'7$F+$\"(*F07$F2$\"'nMWF07$F8$\"'*GV%F07$F=$\"'xZWF07$FB$\"'G_X F07$FG$\"';y\\F07$FL$\"'z,jF07$FQ$\"'QV(*F07$FV$\"'%Q%)*F07$Fen$\"'1M) *F07$Fjn$\"'_M)*F07$F_o$\"'_-**F07$Fdo$\"(&*Q-\"F07$Fio$\"(!>P6F07$F^p $\"(icW\"F07$Fcp$\"(SDk\"F07$Fhp$\"(M.k\"F07$F]q$\"()eQ;F07$Fgq$\"(!H` ;F07$Far$\"(\\$H=F07$Ffr$\"(%4J@F07$F[s$\"(-IL#F07$F`s$\"($oHBF07$Fes$ \"(ykK#F07$Fjs$\"(jYK#F07$F_t$\"(%QHBF07$Fdt$\"(C_N#F07$Fit$\"(=bV#F07 $F^u$\"(uhj#F07$Fcu$\"(\"3\")HF07$Fhu$\"(qh(HF07$F]v$\"(M7(HF07$Fbv$\" ()zmHF07$Fgv$\"($[lHF07$F\\w$\"(+d(HF07$Faw$\"(c\"=IF07$Ffw$\"(,zS$F07 $F[x$\"(z`^$F07$F`x$\"(FO]$F07$Fex$\"(f4_$F07$Fjx$\"(%RHPF07$F_y$\"(fY #RF07$Fdy$\"(m!yTF07$Fiy$\"(e(*G%F07$Fedm$\"(\\$yUF07$F^z$\"(%4oUF07$F ]em$\"(!*=E%F07$Fcz$\"(=iE%F07$Fhz$\"(@JH%F07$F][l$\"($4/VF07$Fb[l$\"( 4>H%F07$Fg[l$\"(%pzUF07$F\\\\l$\"(b$HUF07$Fa\\l$\"(VR<%F07$$\"5lmmm;zp !fu'F4$\"(B))*RF07$Ff\\l$\"()QzPF07$$\"5lmmm;a`T>yF4$\"()[NNF07$F[]l$ \"(`=G$F07$$\"5ILL$3-js.*))F4$\"(f+.$F07$F`]l$\"(ndy#F07$$\"5lmm;H2)y( e**F4$\"(nDb#F07$Fe]l$\"([?L#F07$Fj]l$\"(tG0#F07$F_^l$\"(\\5$=F07$Fd^l $\"($pO;F07$Fi^l$\"(:u\\\"F07$F^_l$\"('Q!H\"F07$Fc_l$\"(Df9\"F07$Fh_l$ \"(xM.\"F07$F]`l$\"'IW&*F07$Fb`l$\"'1&z)F07$Fg`l$\"'^j\")F07$F\\al$\"' $[j(F07$Faal$\"'%)4sF07$Ffal$\"'thnF07$F[bl$\"'WAkF07$F`bl$\"'?ngF07$F ebl$\"'O$y&F07$Fjbl$\"'M+bF07$F_cl$\"'([D&F07$Fdcl$\"'e?]F07$Ficl$\"'v A[F07$F^dl$\"'!ei%F07$Fcdl$\"'IPWF07$Fhdl$\"'1&G%F07$F]el$\"'\"=8%F07$ Fbel$\"'U%)RF07$Fgel$\"'$*\\QF07$F\\fl$\"'1GPF07$Fafl$\"'Q,OF07$Fffl$ \"'g%\\$F07$F[gl$\"'H(Q$F07$F`gl$\"'`&H$F07$Fegl$\"'p+KF07$Fjgl$\"'F;J F07$F_hl$\"'hKIF07$Fdhl$\"'-bHF07$Fihl$\"'!z(GF07$F^il$\"'J2GF07$Fcil$ \"'fQFF07$Fhil$\"'ntEF07$F]jl$\"'l;EF07$Fbjl$\"'>aDF07$Fgjl$\"'!3]#F07 $F\\[m$\"'EYCF07$Fa[m$\"'A'R#F07$Ff[m$\"&ZM#Fj[m-F\\\\m6&F^\\mF($\"#DF a\\m$\"\"\"F)-Ff\\m6#%TButcher's~scheme~B~with~c[5]=c[6]=1/2~and~b[5]= b[6]G-F$6%7jqF'7$F+$\"&[6%F07$F2$\"'sB:F07$F8$\"'-B:F07$F=$\"'?F:F07$F B$\"'*zb\"F07$FG$\"'k&o\"F07$FL$\"'l(3#F07$FQ$\"'nXJF07$FV$\"'dwJF07$F en$\"'RtJF07$Fjn$\"'8tJF07$F_o$\"'X#>$F07$Fdo$\"'f!H$F07$Fio$\"']DOF07 $F^p$\"'pZXF07$Fcp$\"'.S^F07$Fhp$\"'7L^F07$F]q$\"'\\F^F07$Fgq$\"'Jo^F0 7$Far$\"'XycF07$Ffr$\"''Qc'F07$F[s$\"'ifrF07$F`s$\"'U\\rF07$Fes$\"'`Rr F07$Fjs$\"'YLrF07$F_t$\"'lXrF07$Fdt$\"'Sh!z&4UF4$\"(O$\\6F07$$\"5+++DcwR)GM%F4$\"(S'\\6F07$$\"5LL$3F>* ))=wWF4$\"(c\\>\"F07$Fdy$\"(R8@\"F07$$\"5LLeR(\\EYhn%F4$\"(*G37F07$$\" 5++]ilA()zUZF4$\"(Za?\"F07$$\"5mmT&Q.=^%4[F4$\"(^N?\"F07$$\"5LLL3-QO5w [F4$\"(dX?\"F07$$\"5***\\7.d4cF%\\F4$\"(IE@\"F07$$\"5lm;aQ`&3%4]F4$\"( K/B\"F07$$\"5IL3x1651w]F4$\"(SrA\"F07$Fiy$\"(KQA\"F07$Fg[l$\"(c!*>\"F0 7$Fa\\l$\"((F07$F`]l$\"'#>O'F07$F\\^o$\"'iwbF07$Fe ]l$\"'ZW[F07$Fj]l$\"']$)RF07$F_^l$\"'5KLF07$Fd^l$\"'f!y#F07$Fi^l$\"',B CF07$F^_l$\"'d[>F07$Fc_l$\"' " 0 "" {MPLTEXT 1 0 1 ";" }}}}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 46 "Test 3 of 7 stage, order 6 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 m ethods and gives the " }{TEXT 260 22 "root mean square error" }{TEXT -1 18 " of each solution." }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 764 "H := (x,y) -> -x*y: hh := 0.1: numsteps := 100: x0 := 0: y0 := 1: \nmatrix([[`slope field: `,H(x,y)],[`initial point: `,``(x0,y0)],[`s tep width: `,hh],\n[`no. of steps: `,numsteps]]);``;\nmthds := [`B utcher's scheme A`,`scheme with simple nodes`,`scheme with a relativel y large stability region`,`Butcher's scheme B with `*(c[5]=1/2,c[6]=1/ 2,b[5]=b[6]),`scheme with `*(c[5]=3/4,c[6]=3/4,b[5]=b[6])]:\nerrs := [ ]:\nDigits := 20:\nh := x -> exp(-x^2/2):\nfor ct to 5 do\n Hn_RK6_| |ct := RK6_||ct(H(x,y),x,y,x0,y0,hh,numsteps,false);\n sm := 0: nump ts := nops(Hn_RK6_||ct):\n for ii to numpts do\n sm := sm+(Hn_R K6_||ct[ii,2]-h(Hn_RK6_||ct[ii,1]))^2;\n end do:\n errs := [op(err s),sqrt(sm/numpts)];\nend do:\nDigits := 10:\nlinalg[transpose]([mthds ,evalf(errs)]);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#K%'matrixG6#7&7$%0s lope~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$%3Butcher's~scheme~AG$\"+!Qb2!G!#=7$%9scheme~wit h~simple~nodesG$\"+]3$R5\"!#>7$%Pscheme~with~a~relatively~large~stabil ity~regionG$\"+XPF$y)!#?7$*&%9Butcher's~scheme~B~with~G\"\"\"6%/&%\"cG 6#\"\"&#F9\"\"#/&F=6#\"\"'F@/&%\"bGF>&FHFDF9$\"+,#)piAF+7$*&%-scheme~w ith~GF96%/F<#\"\"$\"\"%/FCFQFFF9$\"+,d-/&*F5Q)pprint116\"" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 30 "The following cod e 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 each of the methods \+ at the point where " }{XPPEDIT 18 0 "9.99;" "6#-%&FloatG6$\"$***!\"# " }{TEXT -1 16 " is also given." }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 694 "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 := [`Butcher's scheme A`,`scheme with simple nodes`,`scheme with a relat ively large stability region`,`Butcher's scheme B with `*(c[5]=1/2,c[6 ]=1/2,b[5]=b[6]),`scheme with `*(c[5]=3/4,c[6]=3/4,b[5]=b[6])]:\nerrs \+ := []:\nDigits := 20:\nfor ct to 5 do\n hn_RK6_||ct := RK6_||ct(H(x, y),x,y,x0,y0,hh,numsteps,true);\nend do:\nh := x -> exp(-x^2/2):\nxx : = 9.99: hxx := evalf(h(xx)):\nfor ct to 5 do\n errs := [op(errs),abs (hn_RK6_||ct(xx)-hxx)];\nend do:\nDigits := 10:\nlinalg[transpose]([mt hds,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)pprint126 \"" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%!G" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#K%'matrixG6#7'7$%3Butcher's~scheme~AG$\"+C%fr+%!#L7$%9s cheme~with~simple~nodesG$\"+)[)f;^!#N7$%Pscheme~with~a~relatively~larg e~stability~regionG$\"+*>uDm%F07$*&%9Butcher's~scheme~B~with~G\"\"\"6% /&%\"cG6#\"\"&#F8\"\"#/&F<6#\"\"'F?/&%\"bGF=&FGFCF8$\"+$3h;'RF+7$*&%-s cheme~with~GF86%/F;#\"\"$\"\"%/FBFPFEF8$\"+g]Im7!#MQ)pprint136\"" }}} {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 4 "The " } {TEXT 260 22 "root mean square error" }{TEXT -1 110 " over the interva l [0, 0.5] of each Runge-Kutta method is estimated as follows using \+ the special procedure " }{TEXT 0 5 "NCint" }{TEXT -1 97 " to perform numerical integration by the 7 point Newton-Cotes method over 50 equa l subintervals." }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 460 "mthds := [`Butcher's scheme A`,`scheme with simple nodes`,`scheme with a relat ively large stability region`,`Butcher's scheme B with `*(c[5]=1/2,c[6 ]=1/2,b[5]=b[6]),`scheme with `*(c[5]=3/4,c[6]=3/4,b[5]=b[6])]:\nerrs \+ := []:\nDigits := 20:\nh := x -> exp(-x^2/2):\nfor ct to 5 do\n sm : = NCint((h(x)-'hn_RK6_||ct'(x))^2,x=0..10,adaptive=false,numpoints=7,f actor=50);\n errs := [op(errs),sqrt(sm/10)];\nend do:\nDigits := 10: \nlinalg[transpose]([mthds,evalf(errs)]);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#K%'matrixG6#7'7$%3Butcher's~scheme~AG$\"+f%GJh#!#=7$%9s cheme~with~simple~nodesG$\"+C#z7$%Pscheme~with~a~relatively~lar ge~stability~regionG$\"+4_VR$)!#?7$*&%9Butcher's~scheme~B~with~G\"\"\" 6%/&%\"cG6#\"\"&#F9\"\"#/&F=6#\"\"'F@/&%\"bGF>&FHFDF9$\"+')p51@F+7$*&% -scheme~with~GF96%/F<#\"\"$\"\"%/FCFQFFF9$\"+IbbY*)F5Q)pprint146\"" }} }{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 92 "The foll owing error graphs are constructed using the numerical procedures for \+ the solutions." }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 530 "evalf[20] (plot(['hn_RK6_1'(x)-h(x),'hn_RK6_2'(x)-h(x),'hn_RK6_3'(x)-h(x),'hn_RK 6_4'(x)-h(x),\n'hn_RK6_5'(x)-h(x)],x=0..6,numpoints=100,font=[HELVETIC A,9],\ncolor=[COLOR(RGB,.95,.2,0),COLOR(RGB,.45,0,.95),COLOR(RGB,0,.25 ,1),COLOR(RGB,0,.75,.2),COLOR(RGB,.95,.45,0)],\nlegend=[`Butcher's sch eme A`,`scheme with simple nodes`,`scheme with a relatively large stab ility region`,`Butcher's scheme B with c[5]=c[6]=1/2 and b[5]=b[6]`,`s cheme with c[5]=c[6]=3/4 and b[5]=b[6]`],title=`error curves for 7 sta ge order 6 Runge-Kutta methods`));" }}{PARA 13 "" 1 "" {GLPLOT2D 886 574 574 {PLOTDATA 2 "6+-%'CURVESG6%7eal7$$\"\"!F)F(7$$\"5.......*4M'!# @$!)3*QB#!#?7$$\"5111111_#e=\"F0$!*EC?_)F07$$\"5%RRRRR**)H1=F0$!+SX&=G \"F07$$\"5&[[[[[o&)3V#F0$!+EfJdHF07$$\"5qpppp>-H'e#F0$!+Re,4IF07$$\"5a aaaaaZpTFF0$!+[')o4LF07$$\"5RRRRR*G*4(*GF0$!+G)))[U%F07$$\"5CCCCCCQ]_I F0$!+Z)f&4iF07$$\"5======OmSLF0$!+6&eY:'F07$$\"5777777M#)GOF0$!+WW6[iF 07$$\"5)yyyyy='>FRF0$!+qD;L&)F07$$\"5kjjjjj*obA%F0$!,Pn%eA5F07$$\"5SRR RRRx&)zVF0$!,4!\\L;5F07$$\"5;:::::l9MXF0$!,P?MZ,\"F07$$\"5#44444HN%)o% F0$!,Imps.\"F07$$\"5nmmmmmSsU[F0$!,yZ*3W6F07$$\"5YXXXXX***yX&F0$!,*pgP g9F07$$\"5+(pppp%f4;cF0$!,t8HKY\"F07$$\"5][[[[[>HudF0$!,21Q?_\"F07$$\" 5+++++]z[KfF0$!,zQbTv\"F07$$\"5_^^^^^Ro!4'F0$!,pggJ&>F07$$\"5++++++AOp jF0$!,`:K+#>F07$$\"5\\[[[[[//[mF0$!,s=X2\">F07$$\"5++++++L!\\!oF0$!,M) Q?%)>F07$$\"5]^^^^^hwhpF0$!,PKg[E#F07$$\"5+.....!H'=rF0$!,(\\UbtBF07$$ \"5baaaaa=\\vsF0$!,w;mpM#F07$$\"5++++++))*HV(F0$!,u&>E@BF07$$\"5XXXXXX d]!f(F0$!,eZa[I#F07$$\"5!44444p7![xF0$!,&p'Q*HBF07$$\"5POOOOO'>b!zF0$! ,6ELr[#F07$$\"5YXXXXX\"oE^)F0$!,Gn$z9EF07$$\"5][[[[)R00l)F0$!,r*HO*f#F 07$$\"5]^^^^^EM)y)F0$!,!ok=AEF07$$\"5baaaa/*zh#*)F0$!,M)GHVFF07$$\"5ed ddddr,k!*F0$!,/z&4qGF07$$\"55444443#=R*F0$!,L**[[y#F07$$\"5hgggggWi>(* F0$!,J-dpr#F07$$\"5CCCCC%)4]F5!#>$!,NDb\"Fgw$!,N&e\"pi\"F07$$ \"5[[[[)4v!>/7Fgw$!,W.5&e9F07$$\"5FFFFF2N.77Fgw$!,$GcvW9F07$$\"5#===== 91>C\"Fgw$!,dCm,R\"F07$$\"5OOOOOw(y)=?_[F07$$\"5\"44444$)zId\"Fgw$\",K?MH8&F07$$\"5[[[[[L\\:)e\"Fgw$ \",=/%f5kF07$$\"511111O+B.;Fgw$\",10n%Q!*F07$$\"5kjjjjQ^I=;Fgw$\",q0r; #))F07$$\"5@@@@@T-QL;Fgw$\",ut/1h)F07$$\"5GFFFF(Ha*[;Fgw$\",[Wh\"H%)F0 7$$\"5MLLLL`$GXm\"Fgw$\",hH\\$\\%)F07$$\"5SRRRR4C5!o\"Fgw$\",$4\\yO#*F 07$$\"5YXXXXlkn&p\"Fgw$\"-7CJEG7F07$$\"5#====$4b[*p\"Fgw$\"-OIXBp8F07$ $\"5=====`XH.LQ\"F07$$\"5aaaa/(f.rq\"Fgw$\"-geKOu8F07$$\"5 \"44444k74r\"Fgw$\"-#=wXaO\"F07$$\"5kjjjjG2`=F07$$\"5#====o`c>$=Fgw$\"-Q@m2a=F07$$\"5kjjjjtZK[=F gw$\"-x.WP/=F07$$\"5baaa/#*)3l&=Fgw$\"-_dOn)y\"F07$$\"5YXXXX5Ipk=Fgw$ \"-.U`A*y\"F07$$\"5OOOO')Gr(G(=Fgw$\"-7]ms@=F07$$\"5FFFFFZ71\")=Fgw$\" -3$>%f7>F07$$\"511111O)od*=Fgw$\"-9*>d,P#F07$$\"5&[[[[[Uw/\">Fgw$\"-0m /knDF07$$\"5kjjjj8S=D>Fgw$\"-;PcH'\\#F07$$\"5VUUUU-;*)R>Fgw$\"-3@_,GCF 07$$\"5eddddZLuZ>Fgw$\"-mLfr&R#F07$$\"5ussss#4&fb>Fgw$\"-tI)p8P#F07$$ \"5*yyyyy$oWj>Fgw$\"-$\\6UOO#F07$$\"5/....$e)Hr>Fgw$\"-[=^%zQ#F07$$\"5 MLLLLt?+()>Fgw$\"-tgd!*\\EF07$$\"5kjjjjjbq-?Fgw$\"-U!pt2K$F07$$\"51111 1c\\;J?Fgw$\"-&*eH)e8$F07$$\"5\\[[[[[Vif?Fgw$\"-KA)zI*HF07$$\"5_^^^^\" )zt!4#Fgw$\"-vs?RiMF07$$\"5baaaa9;&=7#Fgw$\"-qf#)39RF07$$\"5_^^^^m$*[O @Fgw$\"-K9Q=&z$F07$$\"5\\[[[[=r7^@Fgw$\".g))p%4!p$F-7$$\"5)ppppW*fWe@F gw$\".R*4Di`OF-7$$\"5YXXXXq[wl@Fgw$\".%[W\"z#Fgw$\".% [F)[0[%F-7$$\"5SRRRRfR+6AFgw$\"./,l\"4`ZF-7$$\"5)yyyyyi/jA#Fgw$\".@yM$ e%f%F-7$$\"5POOOO'H0;C#Fgw$\".T*GE^VWF-7$$\"5%RRRRk*[3\\AFgw$\".hnG9sP %F-7$$\"5_^^^^'\\klD#Fgw$\".u,'*>MK%F-7$$\"55444f'4WSE#Fgw$\".Lyw!4%H% F-7$$\"5nmmmm'pB:F#Fgw$\".A^sA(4VF-7$$\"5CCCCu'H.!zAFgw$\".Tp_!R.WF-7$ $\"5#====o*G['G#Fgw$\".:0Jyfi%F-7$$\"5SRRR*o\\iRH#Fgw$\".M\\t$y_]F-7$$ \"5(ppppp4U9I#Fgw$\".'**f&F-7$$\"5SRRRR*)QwKBFgw$\".+D3L'3_F-7$$\" 5#=====o&3kBFgw$\".<;;ky\"\\F-7$$\"5A@@@@')*o\"zBFgw$\".%fI!RL-&F-7$$ \"5hgggg!H_UR#Fgw$\".BC@h#HdF-7$$\"5YXXXq;J-)R#Fgw$\".?z)o\"Q3'F-7$$\" 5IIII!G%Rz,CFgw$\".K%4UQ$G'F-7$$\"5::::!*oZc0CFgw$\".7.%)*oEiF-7$$\"5+ ++++&fN$4CFgw$\".^PQ>/<'F-7$$\"5qppp>Zs(oT#Fgw$\".5-\" yO#)fF-7$$\"5hggg5o:RoDFgw$\"..H#pL]fF-7$$\"5wvvvv0)>ad#Fgw$\".*>GWu$) fF-7$$\"5\"4444M/[Ce#Fgw$\".3G!GlBhF-7$$\"511111\"Gw%*e#Fgw$\".jj.G.V' F-7$$\"5A@@@r=X]'f#Fgw$\".Hdl!\\*)pF-7$$\"5POOOOcF`.EFgw$\".w5]5#HtF-7 $$\"5IIIIIqLvNEFgw$\".=Zk+ut'F-7$$\"5CCCCC%)R(zm#Fgw$\".z3h(*yJ'F-7$$ \"5RRRR*oHw:n#Fgw$\".(f#HewJ'F-7$$\"5aaaaa4'y^n#Fgw$\".1j8S!RjF-7$$\"5 qppp>A4yyEFgw$\".0$pj0)Q'F-7$$\"5&[[[[[B$Q#o#Fgw$\".\"zxg.skF-7$$\"5;: :::gye*o#Fgw$\".W\"\\X1#y'F-7$$\"5YXXXX&[#z'p#Fgw$\".N1wJEO(F-7$$\"5/. ..yTOf)p#Fgw$\".H$)*>QkvF-7$$\"5hggg5)z%R+FFgw$\".;EJ0Dt(F-7$$\"5====V af>-FFgw$\".C-`u\\p(F-7$$\"5wvvvv5r*Rq#Fgw$\"./k1,wl(F-7$$\"5\"4444MU* f2FFgw$\".m@.CLe(F-7$$\"511111OvLcFFgw$\".(e&ybJn'F-7$$\"5YXXX&H:>Sw#Fgw$\".e<; \"e&e'F-7$$\"5A@@@@'y+CQzFFgw$\".V!=k4 EmF-7$$\"5tssss_S1(y#Fgw$\".i%o%[G'oF-7$$\"5#4444fN;Wz#Fgw$\".#QB;lNtF -7$$\"554444f'onF-7$$\"5%RRR*oZ>HmGFgw$\".R&*pk5o'F-7$$\"55444fTQ PqGFgw$\".^9[0;m'F-7$$\"5DCCC\\NdXuGFgw$\".7\\=Sqm'F-7$$\"5SRRRRHw`yGF gw$\".I5&pP0nF-7$$\"5+++++0_'[*GFgw$\".=h)R-FuF-7$$\"5hgggg!y#>6HFgw$ \".t_>tmq(F-7$$\"577777U(p%RHFgw$\".(p*>:v4(F-7$$\"5kjjjj.nunHFgw$\". \")yh?uk'F-7$$\"5nmmm;H&e<(HFgw$\".^.[-4j'F-7$$\"5qppppa.xvHFgw$\".\\# )e&yRmF-7$$\"5tsssA!=#yzHFgw$\".,fa0>o'F-7$$\"5wvvvv0Sz$)HFgw$\".FLW1q w'F-7$$\"5#====ol<=*HFgw$\".2!Q&Gn6(F-7$$\"5)yyyyyIT)**HFgw$\".OC^p)=y F-7$$\"5+++++5'))e,$Fgw$\".4O'4rruF-7$$\"5777777f$>.$Fgw$\".4aFi$=rF-7 $$\"5\"4444fGhl/$Fgw$\"/\"fhN6k\"o!#A7$$\"5qppppfm=hIFgw$\"/m14GQmlFc[ o7$$\"5SRRR9.I%[1$Fgw$\"/))H$))f>_'Fc[o7$$\"54444fY$*\\oIFgw$\"/SKg]r* ['Fc[o7$$\"5yyyy.!pb@2$Fgw$\"/-4Q1KtkFc[o7$$\"5[[[[[L?\"e2$Fgw$\"/sBI% [tZ'Fc[o7$$\"5)yyyy.sCJ3$Fgw$\"/R6KwUqlFc[o7$$\"5FFFFF2uV!4$Fgw$\"/\"G DAI+$oFc[o7$$\"5kjjjQsoV%4$Fgw$\"/4_j&QE2(Fc[o7$$\"5++++]PjV)4$Fgw$\"/ *e=a_\"4uFc[o7$$\"5===o0qgV+JFgw$\"/%4/3n1c(Fc[o7$$\"5OOOOh-eV-JFgw$\" /k791#R^(Fc[o7$$\"5aaa/MW6$Fgw$\"/Y8xK%)QsFc[o7$$\"5=====GJVAJF gw$\"/[;q?bgqFc[o7$$\"5kjjjj))4VQJFgw$\"/6-V!evr'Fc[o7$$\"544444\\)GW: $Fgw$\"//e.ER3kFc[o7$$\"5gggg5t0LhJFgw$\"/^#)*Ht.I'Fc[o7$$\"577777(HK# oJFgw$\"/s^t*)*QA'Fc[o7$$\"5)yyyG\"fJorJFgw$\"/@'>sJN?'Fc[o7$$\"5kjjj8 @S8vJFgw$\"/(HRY%[*>'Fc[o7$$\"5SRRR9$)[eyJFgw$\"/u7tE)f@'Fc[o7$$\"5::: ::Xd.#=$Fgw$\"/'3E.I\"eiFc[o7$$\"5=====$>Re>$Fgw$\"/bH@M%Q#oFc[o7$$\"5 @@@@@TEk4KFgw$\"/+-(4=@'pFc[o7$$\"5MLLLLtO3TKFgw$\"/S'[uGNH'Fc[o7$$\"5 YXXXX0Z_sKFgw$\"/Ot@B1LeFc[o7$$\"5)yyyyy%*4xG$Fgw$\"/_s@Hu&*fFc[o7$$\" 5IIIII!>&*GI$Fgw$\"/#H+5Hni'Fc[o7$$\"5sssssK/3=LFgw$\"/XKIB%=I'Fc[o7$$ \"5:::::vcELLFgw$\"/KCtn0#*fFc[o7$$\"5UUUUU(*4W[LFgw$\"/z#QNjhjLFgw$\"/hXY$esY&Fc[o7$$\"5_^^^E],TnLFgw$\"/Y+MBCEaFc[o7$$ \"5MLLL$3)R?rLFgw$\"/+k![jsR&Fc[o7$$\"5;:::S6y*\\P$Fgw$\"/$)4>7\"RQ&Fc [o7$$\"5(pppp>k\"zyLFgw$\"/I*pO^1R&Fc[o7$$\"5gggg5.$zjQ$Fgw$\"/*fJY'f( [&Fc[o7$$\"5CCCCCkp'RR$Fgw$\"/hEbPmZdFc[o7$$\"5CCCCCaf?CMFgw$\"/J(fUY4 k&Fc[o7$$\"5CCCCCW\\WaMFgw$\"/^syww*4&Fc[o7$$\"5UUUU#\\\\2\\Fc[o7$$\"5yyyyG'fKiZ$Fgw$ \"/0o_ck')[Fc[o7$$\"5(ppppp9&\\$[$Fgw$\"/T@iy![#\\Fc[o7$$\"5;:::l(pd2 \\$Fgw$\"/NhR*HR2&Fc[o7$$\"5MLLLL[--)\\$Fgw$\"/MmDm*4R&Fc[o7$$\"5)yyy. g)e$)*\\$Fgw$\"/_Z@:@0bFc[o7$$\"5VUUUnB:l,NFgw$\"/E:g!QX[&Fc[o7$$\"5)p ppW8;nM]$Fgw$\"/%pIZq(\\aFc[o7$$\"5_^^^,*z#G0NFgw$\"/8>Ma?:aFc[o7$$\"5 hgggNuS\"*3NFgw$\"/2.7)ymM&Fc[o7$$\"5qpppp\\`a7NFgw$\"/R0I,&*y_Fc[o7$$ \"5=====)Q\\Ra$Fgw$\"/v`tPNGZFc[o7$$\"5nmmmmEMNvNFgw$\"/(=5f%3jVFc[o7$ $\"5aaaaa*e=.f$Fgw$\"/$[QNB**\\%Fc[o7$$\"5UUUUU_PG0OFgw$\"/8)*>(3Nz%Fc [o7$$\"5IIIII:*[-i$Fgw$\"/F.GtBTXFc[o7$$\"5=====yS@NOFgw$\"/u\"Q')4=I% Fc[o7$$\"5UUUUU2U'4l$Fgw$\"/p3h9%)oSFc[o7$$\"5mmmmmOVrmOFgw$\"/'*)z\"4 f$)QFc[o7$$\"5ssss(*o=lqOFgw$\"/*)GxnM`QFc[o7$$\"5yyyyG,%*euOFgw$\"/X. )epP$QFc[o7$$\"5%[[[)fLp_yOFgw$\"/k=wF*z#QFc[o7$$\"5!4444fYkCo$Fgw$\"/ geNj!*RQFc[o7$$\"5-...`I&R.p$Fgw$\"/&>k!4&p$RFc[o7$$\"5:::::&f9#)p$Fgw $\"/Ks&\\Jl<%Fc[o7$$\"5aaaaa9*Hns$Fgw$\"0yF$Q&4b&Q!#B7$$\"5%RRRRRBX_v$ Fgw$\"0H'3GlNwMFh`p7$$\"5YXXXXDR7jPFgw$\"0#Gv#z%R#R$Fh`p7$$\"5(ppppph- 5x$Fgw$\"0Op\"opUILFh`p7$$\"5[[[[[38))yPFgw$\"0<(*Q&=)eI$Fh`p7$$\"5+++ ++++w'y$Fgw$\"0kTIl!*HM$Fh`p7$$\"5_^^^^\"pQYz$Fgw$\"0_\"QV%o\"yMFh`p7$ $\"5.....$QNU1'y1MFh`p7$$\"5IIIII?/Y[QFgw$\"0A<5e[\"QIFh`p 7$$\"5baaaaugkyQFgw$\"0Jxg*HJ5GFh`p7$$\"5IIIIIl@q$*QFgw$\"0Nt(R7_DHFh` p7$$\"511111c#e(3RFgw$\"0UjS:b6)HFh`p7$$\"5#====oM9Q#RFgw$\"0w94Zw/\"G Fh`p7$$\"5eddddP/()QRFgw$\"0VO9Bd&\\EFh`p7$$\"5tssss#e'faRFgw$\"04P[\"R*RFgw$ \"0_]jwsBW#Fh`p7$$\"5======]x,SFgw$\"0[\"R8>)za#Fh`p7$$\"5wvvvvDPuISFg w$\"0&ew>qAoAFh`p7$$\"5MLLLLLCrfSFgw$\"0!H\\\\AiF?Fh`p7$$\"5hggg58%Hr1 %Fgw$\"0*px*fG4)>Fh`p7$$\"5)yyyyGRYX2%Fgw$\"0!4=GCS[>Fh`p7$$\"5;:::lsL '>3%Fgw$\"0>.41p'Q>Fh`p7$$\"5VUUUU_.Q*3%Fgw$\"09(y(fjY'>Fh`p7$$\"5qppp >Ktz'4%Fgw$\"0hfpN(4X?Fh`p7$$\"5)pppp>J9U5%Fgw$\"0+-40zo1#Fh`p7$$\"5DC CCu\"HJ;6%Fgw$\"0JwHPa[+#Fh`p7$$\"5_^^^^r#[!>TFgw$\"0u80^'eW>Fh`p7$$\" 5SRRRRz^!=:%Fgw$\"0\\HB3E7q\"Fh`p7$$\"5GFFFF(3iX=%Fgw$\"0t93I&=w:Fh`p7 $$\"5@@@@@\"[zQC%Fgw$\"0$=P\"*[n69Fh`p7$$\"5sssssn_/fUFgw$\"0Ycn9A(G8F h`p7$$\"5CCCCCa5@uUFgw$\"0%=TC,Pp7Fh`p7$$\"5++++]ZRz\"G%Fgw$\"0/B_1%ye 7Fh`p7$$\"5wvvvvSoP*G%Fgw$\"0tF^;s6F\"Fh`p7$$\"5kjjjQ(GoJH%Fgw$\"1L%=0 nH'*G\"!#C7$$\"5_^^^,M(fpH%Fgw$\"1&e_r+#z=8F\\]q7$$\"5SRRRk!=^2I%Fgw$ \"1P]&yR8rM\"F\\]q7$$\"5GFFFFFEa/VFgw$\"1O#*zpeJD8F\\]q7$$\"5wvvvvb!*Q NVFgw$\"1M#3;=&4g6F\\]q7$$\"5DCCCC%[NiO%Fgw$\"1')zP_%[W-\"F\\]q7$$\"5S RRRR>CT!Q%Fgw$\"0sS^!GdJ**F\\]q7$$\"5baaaaa$*e%R%Fgw$\"1M+`nJ5@5F\\]q7 $$\"577777Ayn,WFgw$\"1)H!RMU^_5F\\]q7$$\"5qpppp*Gm(3WFgw$\"1O%e_)f:?5F \\]q7$$\"5GFFFFdZ&eT%Fgw$\"02K#\\>V())*F\\]q7$$\"5&[[[[[AVHU%Fgw$\"0@_ onLDe*F\\]q7$$\"5UUUUUKv;`WFgw$\"0C]N\\EgR)F\\]q7$$\"5+++++S=R$[%Fgw$ \"0QKT0v(fMmUu'F\\]q7$$\"577777sDUfXF gw$\"0?9qlZNH'F\\]q7$$\"5)yyyyy#[YvXFgw$\"0$4c$z!fofF\\]q7$$\"5kjjjj$3 2:f%Fgw$\"0B[J@(3ufF\\]q7$$\"5SRRRRR$\\vg%Fgw$\"0ow>XgM/'F\\]q7$$\"5#4 4444`iej%Fgw$\"0Y+QNqFI&F\\]q7$$\"5VUUUUAd4#F_eq7$$\"5&[[[[[wDD(\\Fgw$\"1:&f%3pB>=F_eq7$$\"577777ZE&o )\\Fgw$\"1A%>=w%3tQ*)og4l!#E7 $$\"5\"44444V`TL&Fgw$\"1X6LBjp!Q&Fghq7$$\"5A@@@@T**H%R&Fgw$\"1!f6!fCwC VFghq7$$\"5#=====]lQX&Fgw$\"13`PvHi4LFghq7$$\"5GFFFFZ%yM^&Fgw$\"1Y:$) \\0K$y#Fghq7$$\"5>====y)zUd&Fgw$\"1)RYYOL>-#Fghq7$$\"5POOOOw$G(QcFgw$ \"1*ewW!e18;Fghq7$$\"5>====Q8#yp&Fgw$\"2G:8)4z#>J\"!#F7$$\"5SRRRRzO:cd Fgw$\"1dp\">-+0i*F[[r7$$\"577777#**4&=eFgw$\"1J0$R1Btv(F[[r7$$\"5hgggg g0t!)eFgw$\"1jIMZ(4xb&F[[r7$$\"5wvvvv:sdOfFgw$\"1/K$yq;C[%F[[r7$$\"\"' F)$\"1j3U_:!=a$F[[r-%&COLORG6&%$RGBG$\"#&*!\"#$\"\"#!\"\"F(-%'LEGENDG6 #%3Butcher's~scheme~AG-F$6%7_`lF'7$F+$\"(F/!=F07$F2$\")^v0pF07$F7$\")! \\Lv*F07$F<$\"*w,y6#F07$FA$\"*3(*z9#F07$FF$\"*2/GL#F07$FK$\"*ZWH.$F07$ FP$\"*dS\"oTF07$FU$\"*2R58%F07$FZ$\"*ivo<%F07$Fin$\"*TKn[&F07$F^o$\"* \\bnY'F07$Fco$\"*-WkU'F07$Fho$\"*pp$3kF07$F]p$\"*<=\"3lF07$Fbp$\"*'R#R -(F07$Fgp$\"*%yL\\&)F07$F\\q$\"*<=2`)F07$Faq$\"*te=r)F07$Ffq$\"*l#z5&* F07$F[r$\"+6y\"*=5F07$F`r$\"+BdX,5F07$Fer$\"*R*Hq)*F07$F_s$\"+gsv:5F07 $Fis$\"+&4i*35F07$F^t$\"*$HYq**F07$Fct$\"*:Ym#)*F07$Fht$\"*ZD(o&*F07$F ]u$\"*A4m\"*)F07$$\"5ljjjjjnId!)F0$\"*]&[J!)F07$$\"5!44444*Q44#)F0$\"* $>)G$zF07$$\"5?=====5)3O)F0$\"*ay2$yF07$Fbu$\"*=cpp(F07$Fgu$\"*$zSPuF0 7$F\\v$\"*!*)REnF07$Fav$\"*)*>pz%F07$Ffv$\"*h%>LFF07$$\"5NLLLL$)*=zA*F 0$\"*$[H#p#F07$F[w$\"*qH4k#F07$$\"5&[[[[[jAdb*F0$\"*uz/[#F07$F`w$\"*Lp $RD'F07$$\"5q pppp\\jh85Fgw$!*>atL'F07$Few$!*B<'\\iF07$F[x$!+[e*3P\"F07$$\"5)yyyG\"H 'zc4\"Fgw$!+_cA<;F07$$\"5[[[[)4v`#*4\"Fgw$!+nswG>F07$$\"54444%G(y#G5\" Fgw$!+jf1(*>F07$$\"5qpppp%*>S16Fgw$!+Tq>*)>F07$$\"5\"4444%Q-b86Fgw$!+! )yZt>F07$F`x$!+%[3y&>F07$$\"5aaaaap\\*\\8\"Fgw$!+]AiF>F07$Fex$!+DhO3>F 07$$\"5wvvvD8U8d6Fgw$!+\")Hi:>F07$Fjx$!+mRta>F07$$\"5LLLL$es>G<\"Fgw$! +%=V_0#F07$F_y$!+`*zfE#F07$Fiy$!+Z.\"RO$F07$Fcz$!+)e/Gy$F07$$\"5aaaaaC )ppA\"Fgw$!+OPx9PF07$Fhz$!+#GwAl$F07$$\"544444fC%oD\"Fgw$!+f;,GOF07$F] [l$!+qr-pPF07$Fb[l$!+#p2&)[%F07$Fg[l$!+**Hq=hF07$F\\\\l$!+kd+&*fF07$Fa \\l$!+I)=S(eF07$Ff\\l$!+#>AjE(F07$F`]l$!+B1b`')F07$Fj]l$!+i#opy)F07$F_ ^l$!+2&>=p)F07$Fd^l$!+E(zsf)F07$Fi^l$!+2*GOT)F07$F^_l$!+/5*4F)F07$Fc_l $!+!=%>+#*F07$Fh_l$!,&zZRj6F07$F]`l$!,f)=.R6F07$Fb`l$!,q8ok6\"F07$$\"5 aaaaa>F9_:Fgw$!,2n'e26F07$Fg`l$!,wQeE5\"F07$$\"5yyyyy$z+hc\"Fgw$!,,wJ^ 5\"F07$F\\al$!,(f%317\"F07$Faal$!,GL&HX7F07$Ffal$!,9mn^^\"F07$F[bl$!,) o@#)y9F07$F`bl$!,H,]KW\"F07$Febl$!,HpI,T\"F07$Fjbl$!,$4))z&R\"F07$F_cl $!,v()H?X\"F07$Fdcl$!,o(=)[r\"F07$Ficl$!,'o\"Q)R=F07$F^dl$!,RU#*)[=F07 $Fcdl$!,!fC#p$=F07$Fhdl$!,0g.]#=F07$F]el$!,]#GK,=F07$Fbel$!,lm%)yx\"F0 7$Fgel$!,c\\wEt\"F07$F\\fl$!,[Pnop\"F07$Fafl$!,PNc]\"=F07$Fffl$!,>BHe7 #F07$F[gl$!,'*e5N1#F07$F`gl$!,6)**)e+#F07$Fegl$!,o$f[$)>F07$Fjgl$!,jqa A(>F07$F_hl$!,o,uG)>F07$Fdhl$!,9t@L.#F07$Fihl$!,'3;dCBF07$F^il$!,l@O$R CF07$Fcil$!,CzJ:P#F07$Fhil$!,%\\z01BF07$F]jl$!,&HZwtAF07$Fbjl$!,!*[7jC #F07$$\"5KIIIIl4_f>Fgw$!,W()oeB#F07$Fgjl$!,ECH)GAF07$$\"5YXXXX5FPn>Fgw $!,D(GZEAF07$F\\[m$!,J7a/B#F07$$\"5>====G.:z>Fgw$!,(H1TmAF07$Fa[m$!,*o fugBF07$$\"5\\[[[[=Q&[*>Fgw$!,#yQ&)\\DF07$Ff[m$!,y^V]t#F07$F[\\m$!,YV_ Ee#F07$F`\\m$!,LfcVX#F07$Fe\\m$!,jB?=k#F07$Fj\\m$!,Ha&pCGF07$F_]m$!,aV _&QFF07$Fd]m$!-'>q)RfEF-7$Fi]m$!-nG,PFEF-7$F^^m$!-pTpr1EF-7$$\"5qpppW3 VUp@Fgw$!-UevF.EF-7$Fc^m$!-,giT1EF-7$$\"5====V%=Vn<#Fgw$!-TUQ4=EF-7$Fh ^m$!-zG5tSEF-7$F]_m$!-:4/4KHF-7$Fb_m$!-;u'\\#HIF-7$Fg_m$!-\"yX%>GHF-7$ F\\`m$!-&HO#4JGF-7$Ff`m$!-*)pe/[FF-7$F`am$!->.gv#F-7$Fccm$!-R[?KpFF-7$$ \"5#4444%Q1r'Q#Fgw$!-(HaV#QGF-7$Fhcm$!-*4zb3*HF-7$F]dm$!-gmw37JF-7$Fbd m$!-\"[uDr<$F-7$Fgdm$!-KX)e%[JF-7$F\\em$!-()pi+?JF-7$Faem$!-E%RTP1$F-7 $Ffem$!-RGLM3IF-7$F[fm$!-d+.U)*GF-7$F`fm$!-l\\p:*z#F-7$Fjfm$!-qQDegFF- 7$Fdgm$!-e?*GF-7$F\\jm $!-3y+N%y#F-7$Ffjm$!-:NI\\FF-7$F^]n$!-*)3[McDF-7$Fc]n$!-AY5*oa#F -7$Fh]n$!-TNw2VDF-7$F]^n$!-m2$>la#F-7$Fb^n$!-h#o0#fDF-7$Fg^n$!-wAaaAEF -7$F\\_n$!-.M-?fFF-7$Fa_n$!-,>n))3GF-7$Ff_n$!-'=U\\)\\GF-7$F[`n$!-8%G< g$GF-7$F``n$!-#o7VA#GF-7$Fe`n$!-c))z'[z#F-7$Fj`n$!-$H`Axw#F-7$F_an$!-^ im69FF-7$Fdan$!-e>AVhEF-7$Fian$!-@s\"4Fb#F-7$F^bn$!-GFi\"QX#F-7$Fcbn$! -P:*)p8CF-7$Fhbn$!-+s*4qQ#F-7$$\"55444%GgTbx#Fgw$!-9@-o\"Q#F-7$F]cn$!- u3U#F-7$Fgcn$!-$ eavj^#F-7$F\\dn$!-]%3)oMEF-7$Fadn$!-l)Q+4e#F-7$Ffdn$!-I]]2GDF-7$F[en$! -/c(R`U#F-7$F`en$!-ZH]aFBF-7$Fjen$!-VP%\\KB#F-7$F^gn$!-\"HJ)4!>#F-7$Fc gn$!-@]y13BF-7$Fhgn$!-S3XnPBF-7$F]hn$!-`f:\\_@F-7$Fbhn$!-3$\\O.+#F-7$F ghn$!-t:/x()>F-7$F\\in$!-hT`6!)>F-7$Fain$!-*pKs*y>F-7$Ffin$!-k)otj)>F- 7$F[jn$!-wNOjP?F-7$F`jn$!-)>NPC;#F-7$Fejn$!-jD*H[1#F-7$Fjjn$!-cgh4n>F- 7$F_[o$!.nh94E)=Fc[o7$Fe[o$!.V<9\"\\2=Fc[o7$F_\\o$!.$fPkBygFc[o7$Fg^o$!.83!*3p*=Fc[o 7$F\\_o$!.3+1t^)=Fc[o7$Fa_o$!.%HNB]t=Fc[o7$Ff_o$!.>Afku#=Fc[o7$F[`o$!. QK8[Cy\"Fc[o7$F``o$!.@x(\\f&p\"Fc[o7$Fe`o$!.#3%\\g\\h\"Fc[o7$F_ao$!.Ez q\\$e:Fc[o7$Fcbo$!.(e(y$zO:Fc[o7$$\"5mmmm;pu$*)=$Fgw$!.\"=AZaa:Fc[o7$F hbo$!.k_TSgg\"Fc[o7$$\"5111c=CYc(>$Fgw$!.Oq\\rgi\"Fc[o7$$\"5%RRR*=b+H* >$Fgw$!.@#*4(e\\;Fc[o7$$\"5#===$>'[:5?$Fgw$!.G&R*p\\l\"Fc[o7$$\"5qppp> <4u-KFgw$!.*>p?&ek\"Fc[o7$$\"5YXXX?z<>1KFgw$!./d@_xi\"Fc[o7$F]co$!.+tC K)4;Fc[o7$Fbco$!.]tlj[X\"Fc[o7$Fgco$!.%[.2HK8Fc[o7$$\"5nmmmmEt6!G$Fgw$ !.:W\\o;K\"Fc[o7$F\\do$!.&H'QV0L\"Fc[o7$$\"544444pDI&H$Fgw$!.taf>9P\"F c[o7$Fado$!.N7r9rS\"Fc[o7$Ffdo$!.,)Ht7Q8Fc[o7$F[eo$!.M#R?Fs7Fc[o7$Fcgo $!.^Ws6'\\6Fc[o7$Fhgo$!.zg')3G5\"Fc[o7$F]ho$!-09i=_**Fc[o7$Fgho$!-lu*) *Q`*Fc[o7$Faio$!-7#)HzZ$*Fc[o7$Ffio$!-#eESHX*Fc[o7$F[jo$!-`GOl,)*Fc[o7 $Fejo$!-R$*R/)*)*Fc[o7$F_[p$!-yg\"=Hx*Fc[o7$Fd[p$!-\"o(oC\\'*Fc[o7$Fi[ p$!-h)3:q_*Fc[o7$F^\\p$!-,#)>rH&)Fc[o7$Fc\\p$!-y96B`xFc[o7$$\"5gggg53g $Ge$Fgw$!-rrS8#p(Fc[o7$Fh\\p$!-6gf`cxFc[o7$$\"5^^^^E!))fSf$Fgw$!-k&=f$ fyFc[o7$$\"5[[[[)4<,yf$Fgw$!-]-CPE!)Fc[o7$$\"5XXXXqhCa,OFgw$!-U-*Rd6)F c[o7$F]]p$!-lXC12!)Fc[o7$Fb]p$!-0'H`ce(Fc[o7$Fg]p$!-P$)e/&=(Fc[o7$F_`p $!-=VUu7lFc[o7$Fd`p$!.*Qu7&p(fFh`p7$Fj`p$!.X>.q\"y`Fh`p7$F_ap$!.-;AkIB &Fh`p7$Fdap$!.]Px(o3^Fh`p7$Fiap$!.IB$H\">-&Fh`p7$F^bp$!.l8[FB+&Fh`p7$F cbp$!.3>p+!*4&Fh`p7$Fhbp$!.l/(fG>_Fh`p7$F]cp$!.#f@R3l]Fh`p7$Fbcp$!.#GJ G8:\\Fh`p7$Fgcp$!.%\\n=ZzVFh`p7$F\\dp$!.I^Ru\\(RFh`p7$Fadp$!.>\"\\!Ry+ %Fh`p7$Ffdp$!.vKCxT,%Fh`p7$F[ep$!.qU2HVy$Fh`p7$F`ep$!.<\\$*fpc$Fh`p7$F eep$!.&oF&*zaLFh`p7$Fjep$!.o8!\\ZsJFh`p7$Fdfp$!.j=I!>$3$Fh`p7$F^gp$!.p 5RU9@$Fh`p7$Fcgp$!.;6YC(eGFh`p7$Fhgp$!.wA1)\\YDFh`p7$Fbhp$!.)=[\"o3U#F h`p7$F\\ip$!./NWI7Q#Fh`p7$Faip$!.lI5)RPCFh`p7$Ffip$!.>:L&zXCFh`p7$F[jp $!.&4(o*RsBFh`p7$F`jp$!.YOaz5I#Fh`p7$Fjjp$!.CF.q=\"=Fh`p7$F_[q$!.'GwF_eq7$F[fq$!/oNAe)fW\"F_eq7$F_gq$!/^AX^**R7F_eq7$Fdgq$!.aoN\"49'*F _eq7$Figq$!.k$3P3Fghq7$F]jq$!/%yIF&H#R\"Fghq7$Fbjq$!/p+%z#*36\"Fghq7$Fgjq$! /h6'e\"z0*)F[[r7$F][r$!/`b(o(f\\lF[[r7$Fb[r$!/r#z8)Rh_F[[r7$Fg[r$!/2%) GU&yq$F[[r7$F\\\\r$!/[,t'[(GIF[[r7$Fa\\r$!/VL2&**4R#F[[r-Ff\\r6&Fh\\r$ \"#XF[]rF(Fi\\r-F`]r6#%9scheme~with~simple~nodesG-F$6%7]^lF'7$$\"5_^^^ ^^^\\qJF-$\"&kQ\"F07$F+$\"()zaNF07$$\"5!======@'*4*F-$\")QP3kF07$F2$\" *=<F0$\"*LR)pLF07$$\"5/.....!>0/#F0$\"*#R[R SF07$$\"5SRRRRRBf=@F0$\"*qJH.%F07$$\"5777777!RZF#F0$\"*5h%>SF07$F<$\"* 8jG,%F07$FA$\"*B:o1%F07$FF$\"*H\\=S%F07$FK$\"*HSpn&F07$FP$\"*\"=**[xF0 7$FU$\"*5X*zwF07$FZ$\"*n40w(F07$Fin$\"+3eb85F07$F^o$\"+#e'e\">\"F07$Fc o$\"+;P:%=\"F07$Fho$\"+(Q*y!=\"F07$F]p$\"+2u2*>\"F07$Fbp$\"+p;8%H\"F07 $$\"5_^^^^^5i>\\F0$\"+]#3zS\"F07$$\"5OOOOOO!=l*\\F0$\"+sUy,;F07$$\"5?@ @@@@]Tt]F0$\"+1@72;F07$$\"5011111?J]^F0$\"+4x\"3g\"F07$$\"5vvvvvvf5/`F 0$\"+qR3)e\"F07$Fgp$\"+eF07$F`r$\"+N>Pu=F07$Fer$\"++Lu]=F07$Fjr$\"+ ([$=o=F07$F_s$\"+qFpl>F07$Fds$\"+kVs&*>F07$Fis$\"+)o:L(>F07$Fct$\"+!** *HE>F07$F]u$\"+9UXX=F07$Fccr$\"+)f%yf_m'e\"F07$Fav$\"+66nq8F07$Ffv$\"+&og79\"F07$F\\er$\"+(4_U 7\"F07$F[w$\"+(Gef5\"F07$Fder$\"+TOpu5F07$F`w$\"**HW^(*F07$F\\fr$\"*P9 *=rF07$Fafr$\")v!p\\#F07$Fffr$\"(91q%F07$Few$\"(1'eWF07$F[x$!*uQLM*F07 $Fex$!+S\"QGh\"F07$F_y$!+A^&Q1#F07$Fcz$!+Q\\]xQF07$F[[s$!+.r\"y!QF07$F hz$!+q&)pWPF07$$\"5YXXXX+VP\\7Fgw$!+_g\"[s$F07$Fc[s$!+hK*ps$F07$$\"5ss sss<1Jk7Fgw$!+o%\\Qx$F07$F][l$!+[9,/RF07$Fb[l$!+^-hcZF07$Fg[l$!+s\\;gm F07$F\\\\l$!+vP_DlF07$Fa\\l$!+yd)RR'F07$Ff\\l$!+Y!od,)F07$F`]l$!+\"G>k*F07$Fd^l$!+N40P&*F07$Fi^l$!+z?\\L $*F07$F^_l$!+)3jl<*F07$Fc_l$!,kT&)>-\"F07$Fh_l$!,3\"e(3H\"F07$F]`l$!,, \"[%QE\"F07$Fb`l$!,r\"4\")Q7F07$Fh^s$!,1^R*G7F07$Fg`l$!,5_&RB7F07$$\"5 ssssAv7hi:Fgw$!,!f>LB7F07$F`_s$!,@02fA\"F07$$\"5%[[[[BJ!fp:Fgw$!,nr\\> B\"F07$F\\al$!,mu)[U7F07$Faal$!,.)H=w8F07$Ffal$!,&=laj;F07$F[bl$!,@tTO i\"F07$F`bl$!,)4Be%e\"F07$Febl$!,41F\"[:F07$Fjbl$!,nK@:`\"F07$F_cl$!,y b9%)e\"F07$Fdcl$!,enVv&=F07$Ficl$!,od3^)>F07$F^dl$!,mgKP*>F07$Fcdl$!,y #\\#3)>F07$Fhdl$!,sNsz'>F07$F]el$!,sMOC%>F07$Fbel$!,Glgr\">F07$Fgel$!, vYf$o=F07$F\\fl$!,K-$=H=F07$$\"5OOOO')e?*Rw\"Fgw$!,Z=:5#=F07$$\"5YXXXX ]HOrF07$$ \"5tsssADcZ$z\"Fgw$!,^\\%)o3#F07$$\"5#====o^Y3!=Fgw$!,\")Gc&)H#F07$$\" 5\"4444%3u@3=Fgw$!,@h&=oAF07$Fffl$!,^\"f4QAF07$F[gl$!,6BuC<#F07$F`gl$! ,\\A\\:6#F07$Fegl$!,eMJt3#F07$Fjgl$!,NM`Q2#F07$$\"5\"444f'p]yo=Fgw$!,7 5`S2#F07$F_hl$!,_Ap63#F07$$\"5#===o!)=pp(=Fgw$!,GLAv4#F07$Fdhl$!,4RIg7 #F07$Fihl$!,7)>T$R#F07$F^il$!,()QzR\\#F07$Fcil$!,.8`YU#F07$Fhil$!,#Goi dBF07$Fbjl$!,kxgbH#F07$F\\[m$!,L$\\LtAF07$Fbfs$!,5XE9I#F07$Fa[m$!,[7U5 Q#F07$Fjfs$!,\"4]]VDF07$Ff[m$!,fQ*H,FF07$F[\\m$!,f#)p2b#F07$F`\\m$!,dn ,=U#F07$Fe\\m$!,>0&oaDF07$Fj\\m$!,(\\WC!o#F07$Fd]m$!-)p[KD_#F-7$Fh^m$! -zXW@![#F-7$F]_m$!-C[72(o#F-7$Fb_m$!-?\\byWFF-7$Fg_m$!-D3>@`EF-7$F\\`m $!-N3>,lDF-7$Ff`m$!-7zu&y[#F-7$F`am$!-$H)G5UCF-7$Fjam$!-*)33q!\\#F-7$F dbm$!-iUx.DFF-7$Fibm$!-*4KGW`#F-7$F^cm$!-!3*HKsBF-7$F^\\t$!-#4Nz-O#F-7 $Fc\\t$!-<['f>N#F-7$Fh\\t$!-NM$)\\[BF-7$Fccm$!-;1eI^BF-7$F`]t$!-]Ar/$Q #F-7$Fhcm$!-ra)))fY#F-7$F]dm$!-L#Q2Z`#F-7$Fbdm$!-)yL3)oDF-7$Fgdm$!-dF- jXDF-7$F\\em$!-d)RDE_#F-7$Faem$!-C*)H8xCF-7$Ffem$!-\\UgLKCF-7$F`fm$!-i \">!)4E#F-7$F^hm$!-j;#R!3AF-7$Fchm$!-t0W1nAF-7$Fhhm$!-)f5.NN#F-7$F]im$ !-x+e$)3BF-7$Fbim$!-\\7X)[E#F-7$Fgim$!-(y=p\"z@F-7$F\\jm$!-&RA-t4#F-7$ Ffjm$!-5C%z+.#F-7$F`[n$!-yF-7$$\"5MLLLeCR$*yDFgw$!-t-Z3\")>F-7$F e[n$!-C5tO\")>F-7$$\"5[[[[Bi@'fe#Fgw$!-6e.;()>F-7$Fj[n$!->w+y**>F-7$F_ \\n$!-2ov1_?F-7$Fd\\n$!-[/LMw?F-7$Fi\\n$!-i!Q\"R3>F-7$F^]n$!-DD%*Qj$p#Fgw$!-D(HI5v\"F-7$F\\_n$!-!*=Eir>\"oI\"F- 7$Fhgn$!-iT$G;9\"F-7$Fbhn$!,v/N!e'*F-7$Fjjn$!,M7\"oj!)F-7$Fc]o$!-p()QK _lFc[o7$Fe`o$!-k7c))[_Fc[o7$F]co$!-'z!fFEQFc[o7$Fgco$!-uD:s7IFc[o7$F[e o$!-:orBF=Fc[o7$Fcgo$!,My\")Hg)Fc[o7$$\"5CCCC*z$ou(R$Fgw$!,'>bdamFc[o7 $$\"5CCCCu6n_,MFgw$!,ZvXGH&Fc[o7$$\"5CCCC\\&e1`S$Fgw$!,i4#>D_Fc[o7$$\" 5CCCCCfk34MFgw$!,,sG$e^Fc[o7$$\"5CCCCu1ik;MFgw$!,I8!*o-&Fc[o7$Fhgo$!,4 y**y*[Fc[o7$$\"5CCCCC\\aKRMFgw$!,NLr1j%Fc[o7$F]ho$!,!G#R:B%Fc[o7$Faio$ !+%>wP.*Fc[o7$Fi[p$\",VK8nr&Fc[o7$$\"5%RRRR*otCGNFgw$\",9898T&Fc[o7$F^ \\p$\",z%y.f^Fc[o7$$\"5UUUUU29lfNFgw$\",y>s)o^Fc[o7$Fc\\p$\",P!z\"Fc[o7$Fe_p$\"-vt6$yG\"Fc[o7$Fj_p$\"-\\ADBr9Fc[o7$F_`p$\"-R!Qc* zFh`p7$$\"5=====$fnL$QFgw$\".@sj!pm=Fh`p7$Fgcp$\".,%))* 3sw\"Fh`p7$$\"5UUUUUZKbjQFgw$\".h*RRW*p\"Fh`p7$F\\dp$\".*[\\U$Hs\"Fh`p 7$Fadp$\".?0%*)pk>Fh`p7$Ffdp$\".HWArn5#Fh`p7$F[ep$\".r;]zh)>Fh`p7$F`ep $\".x(\\G;t=Fh`p7$Feep$\".q#>g6rFh`p7$F^gp$\".)R#Q 4+4#Fh`p7$Fcgp$\".\\kEc1'=Fh`p7$Fhgp$\".$o9Z'Gn\"Fh`p7$F\\ip$\".rhx-os \"Fh`p7$F`jp$\".YyNVOz\"Fh`p7$$\"5YXXXXDnUNTFgw$\".$=@<$on\"Fh`p7$Fejp $\".7'fW:s:Fh`p7$$\"5MLLLLLO=oTFgw$\"..KSC&*\\\"Fh`p7$Fjjp$\".Ao_-j^\" Fh`p7$$\"5_^^^^hn(>>%Fgw$\".=5l%Fgw$\".*)fr(o 3\"F\\]q7 $F``q$\"/9ta34A6F\\]q7$$\"5kjjjjy`0QWFgw$\"/WbGww\\5F\\]q7$Fe`q$\".Qb) **oY)*F\\]q7$$\"5@@@@@'oz#oWFgw$\".5A)*=hN*F\\]q7$Fj`q$\"._j&*4dD*F\\] q7$F_aq$\".?Z>S(\\$)F\\]q7$Fdaq$\".Wo;)R6yF\\]q7$Fiaq$\".egS,+[(F\\]q7 $F^bq$\".]\"Gpe\\wF\\]q7$Fcbq$\".&\\\"o!fSyF\\]q7$Fhbq$\".XNM#G!)oF\\] q7$F]cq$\".G*ogr+hF\\]q7$Fbcq$\".fzd/2!fF\\]q7$Fgcq$\".r//'o/iF\\]q7$F \\dq$\".V)\\:txfF\\]q7$Fadq$\".[xITV`&F\\]q7$Ffdq$\".>3_')yg%F\\]q7$F[ eq$\"/*y31r[!RF_eq7$Faeq$\"/OM%Qn._$F_eq7$F[fq$\"/kl;zmSEF_eq7$F_gq$\" //lFt$fN#F_eq7$Fdgq$\"/GfeIP<>F_eq7$Figq$\"/gcclWY:F_eq7$F^hq$\"/(p>0I $G8F_eq7$Fchq$\"0+S**Hp*35Fghq7$Fihq$\"/Fc\"*)y+W)Fghq7$F^iq$\"//0rJs& )oFghq7$Fciq$\"/]>#=v1F&Fghq7$Fhiq$\"/;wf3#G[%Fghq7$F]jq$\"/n$)*3nZF$F ghq7$Fbjq$\"/>tvX4DEFghq7$Fgjq$\"0)>tVs9c@F[[r7$F][r$\"0_h[9(f!e\"F[[r 7$Fb[r$\"0,=$f95#G\"F[[r7$Fg[r$\"/#pi.'z]#*F[[r7$F\\\\r$\"/#GD*pQ]uF[[ r7$Fa\\r$\"/tXm<'>\"fF[[r-Ff\\r6&Fh\\rF($\"#DF[]r$\"\"\"F)-F`]r6#%Psch eme~with~a~relatively~large~stability~regionG-F$6%7bblF'7$F+$!(]d`$F07 $F2$!*e]PN\"F07$F7$!*_.@l#F07$F<$!*f/f^(F07$FP$!+W^X5>F07$FZ$!+87iT>F0 7$F^o$!+%zLnl$F07$Fbp$!+r\"R+M%F07$Fgp$!+`W@dgF07$F\\q$!+HSQ1hF07$Faq$ !+&o>^_'F07$Ffq$!+RJ;'4)F07$F[r$!+[-(\\X*F07$F`r$!+nZK'H*F07$Fer$!+`#) HV$*F07$Fjr$!,i)fq/5F07$F_s$!,!eXPb7F07$Fds$!,!*HuyN\"F07$Fis$!,ts&pU8 F07$F]u$!,UBQZd\"F07$F^cr$!,9/y,'=F07$Fccr$!,b?mt$=F07$Fhcr$!,d)R\"\\ \"=F07$Fbu$!,*Q!*\\(z\"F07$Fgu$!,oo>O!=F07$F\\v$!,8#)G&z=F07$Fav$!,Ih*RBF07$F`w$!,j>&eUBF07$$\"5NLLLLLw/*y* F0$!,&)[?qR#F07$F\\fr$!,$=R\"3]#F07$$\"5!)yyyyyR*y#**F0$!,Hqj/o#F07$Fa fr$!,K?tH(HF07$Fffr$!,oC3l%HF07$Few$!,%ex80HF07$$\"5+++++:@lV5Fgw$!,PH H\"fGF07$$\"5wvvvvXK!)f5Fgw$!,i\\Q&HGF07$$\"5_^^^^wV&f2\"Fgw$!,m%y/tGF 07$F[x$!,qXj;:$F07$Ffgr$!,nI_kV$F07$F`hr$!,Qq_.X$F07$Fehr$!,X;&3BMF07$ F`x$!,*R#peR$F07$F]ir$!,oCo>M$F07$Fex$!,*HU8#H$F07$Fjx$!,5.Y5E$F07$F_y $!,zb=*=LF07$Fdy$!,:v)*oU$F07$Fiy$!,nKc9j$F07$F^z$!,a=MVv$F07$Fcz$!,7H I*=PF07$Fhz$!,K`'3'e$F07$F][l$!,^\\*p%\\$F07$Fb[l$!,:b5la$F07$Fg[l$!,' f'*QCPF07$F\\\\l$!,&eI3\\OF07$Fa\\l$!,k\"HXuNF07$$\"5UUUUUK&RFO\"Fgw$! ,4zF\"RMF07$Ff\\l$!,z3wAG$F07$Fd^l$!,`$Q#z5$F07$F^_l$!,;L^b'HF07$Fc_l$ !,')**eAg#F07$Fh_l$!,g.&zd>F07$F]`l$!,S]!=;>F07$Fb`l$!,(4;Aq=F07$Fg`l$ !,>hboz\"F07$F\\al$!,UH_xh\"F07$Faal$!,&*=c?1\"F07$Ffal$\")2yaxF07$F[b l$\")s\\&e(F07$F`bl$\")^w]&)F07$Febl$\"*+[^[#F07$Fjbl$\"+BPuO8F07$F_cl $\"+u&fo.'F07$Fdcl$\",w$3_`@F07$$\"5kjjjQ()4e(p\"Fgw$\",m*G1#[#F07$Fic l$\",lhtK&GF07$$\"5++++DJ+R,5*H_'F07$F`gl$\",5\"F07$Fa[m$\"-'Hc-=H\"F07$Ff[m$\"-&4PQev\"F07$F[\\m$ \"-\"44S\"e;F07$F`\\m$\"-Y$H6()e\"F07$Fe\\m$\"-orala>F07$Fj\\m$\"-5Gg! yI#F07$F_]m$\"-m/i(yB#F07$Fd]m$\".F)=%Gx<#F-7$Fi]m$\".zm#yRf@F-7$F^^m$ \".`V$*H#f@F-7$Fc^m$\".xcCw<>#F-7$Fh^m$\".dt;e+G#F-7$F]_m$\".ww8sw\"GF -7$Fb_m$\".\"4'*o#z.$F-7$Fg_m$\".t\">TjOHF-7$F\\`m$\".d\"\\a]SGF-7$Fa` m$\".iQ&G7*z#F-7$Ff`m$\".2A&yBnFF-7$F[am$\".r#Q\"ySv#F-7$F`am$\".\"o(z sax#F-7$Feam$\".$RQJ*p&GF-7$Fjam$\".%)RIA!QIF-7$F_bm$\".,V$=!pP$F-7$Fd bm$\".j:**4!4QF-7$Fibm$\"._U27Ha$F-7$F^cm$\".7-v9TN$F-7$Fccm$\".lEQdwX $F-7$Fhcm$\".oeD/#RSF-7$F]dm$\".L)fncEVF-7$Fbdm$\".$)\\i)R!\\%F-7$Fgdm $\".!)>>#))\\WF-7$F\\em$\".jnro'4WF-7$Faem$\".8hQ\\,L%F-7$Ffem$\".xv$p )=D%F-7$F[fm$\".=!*=Qw4%F-7$F`fm$\".MLA@v'RF-7$Fefm$\".xwJ(*\\%RF-7$Fj fm$\".[rY,*HRF-7$F_gm$\".mc_Q\\#RF-7$Fdgm$\".*Qr#*eLRF-7$Figm$\".5b8;0 ,%F-7$F^hm$\".U?k$o5UF-7$Fchm$\".x#3!yZg%F-7$Fhhm$\".!)QOSU7&F-7$F]im$ \".Z&*3()p-&F-7$Fbim$\".z%\\lHJ\\F-7$Fgim$\".\\>:O]u%F-7$F\\jm$\".%[Mp 90XF-7$Ffjm$\".@u>\")fX%F-7$$\"5/...G\\u([c#Fgw$\". Y(*)Q@UWF-7$F[[n$\".[d^$yQWF-7$$\"5====$po0>d#Fgw$\".+l3&y[WF-7$F`[n$ \".:maWgZ%F-7$Fe[n$\"./a`i?g%F-7$Fj[n$\".ac$o>n[F-7$F_\\n$\".Td12JM&F- 7$Fd\\n$\".KG3]dj&F-7$Fi\\n$\".*\\Uaz!=&F-7$F^]n$\".FD9<)o[F-7$Fh]n$\" .Ws1ri*[F-7$Fb^n$\".74fN'=]F-7$Fg^n$\".d,&3N\"H&F-7$F\\_n$\".'z$pvRz&F -7$Fa_n$\".e#zkvnfF-7$Ff_n$\".oa\"y%H6'F-7$F[`n$\".\\!zxF$3'F-7$F``n$ \".o66KP0'F-7$Fe`n$\".`CD7]*fF-7$Fj`n$\".J>B&yOfF-7$F_an$\".:KA1=#eF-7 $Fdan$\".*p4J&)3dF-7$Fian$\".wU$)*=xaF-7$F^bn$\".d(Hf+y_F-7$$\"5edddKO $y,w#Fgw$\".(oNTtS_F-7$Fcbn$\".`Nc'\\7_F-7$$\"5MLLLep*fyw#Fgw$\".:&ewV '>&F-7$Fhbn$\".1M[!e'>&F-7$F]cn$\".0SuOqE&F-7$Fbcn$\".W+<54[&F-7$Fgcn$ \".y>R:!)*eF-7$F\\dn$\".Z^u5mQ'F-7$Fadn$\".1'GoAciF-7$Ffdn$\".glay\"Gh F-7$F[en$\".7*)=(QzeF-7$F`en$\".rWc\\ck&F-7$Feen$\".#HcGKJbF-7$Fjen$\" .Q#y'yBW&F-7$F_fn$\".'3k:99aF-7$Fdfn$\".p'*y0@S&F-7$$\"5omm;a)y9C(GFgw $\".>SELQS&F-7$Fifn$\".f/^!z6aF-7$$\"5#===VCo'\\wGFgw$\".Vts,pU&F-7$F^ gn$\".]ix,-X&F-7$Fcgn$\".j3xw/5'F-7$Fhgn$\".imU_MO'F-7$F]hn$\"._V$Qige F-7$Fbhn$\".wx2[f\\&F-7$Fghn$\".!3Z8*e[&F-7$F\\in$\".mk@g#)\\&F-7$Fain $\".:sf))*RbF-7$Ffin$\".cF-7$F[jn$\".FC4Hy$fF-7$F`jn$\".m_'*Gwc' F-7$Fejn$\".x?a!)pF'F-7$Fjjn$\".Msoi,)fF-7$F_[o$\"/7&4r&*os&Fc[o7$Fe[o $\"/Q&4X$Q>bFc[o7$Fj[o$\"/96A$oN[&Fc[o7$F_\\o$\"/eGnyfeaFc[o7$Fd\\o$\" /qD,WvZaFc[o7$Fi\\o$\"/eq$=C^X&Fc[o7$F^]o$\"/a$e`#)GdFc[o7$ Fe`o$\"/&=TPshY&Fc[o7$Fj`o$\"/1.2P]v`Fc[o7$F_ao$\"/h`es98`Fc[o7$Fdao$ \"/%>,5L!)H&Fc[o7$Fiao$\"/>!y!=g(H&Fc[o7$F^bo$\"/`S/Lm:`Fc[o7$Fcbo$\"/ ]UO#>oN&Fc[o7$Fhbo$\"/b>A3=zeFc[o7$F]co$\"/$zZWlc,'Fc[o7$Fbco$\"/$[Vi# 4QaFc[o7$Fgco$\"/v.w#>q/&Fc[o7$F\\do$\"/LD(ztr?&Fc[o7$Fado$\"/vJR%GaP_Fc[o7$F`eo$\"/%H$G+Z&)\\Fc[ o7$Feeo$\"/9Y821\"y%Fc[o7$Fjeo$\"/m)\\SYju%Fc[o7$F_fo$\"/SiyVhAZFc[o7$ Fdfo$\"/(z.**[Jr%Fc[o7$Fifo$\"/t10i.AZFc[o7$F^go$\"/(\\C&>3;[Fc[o7$Fcg o$\"/OI<6Cf]Fc[o7$Fa\\y$\"/b]R\\=f_Fc[o7$Ff\\y$\"/Kgc1%>Q&Fc[o7$F[]y$ \"/U5+#[JJ&Fc[o7$F`]y$\"/fej,;X_Fc[o7$Fe]y$\"/.Q`sc6^Fc[o7$Fhgo$\"/KtM [8\")\\Fc[o7$F]^y$\"/]X%oA2t%Fc[o7$F]ho$\"/K\"QamQ]%Fc[o7$Fbho$\"/?f&) o\\8WFc[o7$Fgho$\"/jjSi4GVFc[o7$F\\ io$\"/]?r!)f@VFc[o7$$\"5)yyyG;(Q')zMFgw$\"/e=BYCJVFc[o7$Faio$\"/QABB*))z%Fc[o7$F`jo$\"/@qC%o]!\\ Fc[o7$Fejo$\"/xa*Qoq)[Fc[o7$Fjjo$\"/Ud#G)3c[Fc[o7$F_[p$\"/j#)e')GD[Fc[ o7$Fd[p$\"/>iHrAkZFc[o7$Fi[p$\"/e()Gm(Qq%Fc[o7$F^\\p$\"/O5`%pL@%Fc[o7$ Fc\\p$\"/?jemf#*QFc[o7$Fh\\p$\"/rz8)[p-%Fc[o7$F]]p$\"/+[J)pfI%Fc[o7$Fb ]p$\"/RHroNzSFc[o7$Fg]p$\"/ok^*)HkQFc[o7$F\\^p$\"/2Dc1DbOFc[o7$Fa^p$\" /tJK?@!\\$Fc[o7$Ff^p$\"/%>*e[%QY$Fc[o7$F[_p$\"/%z=JztW$Fc[o7$F`_p$\"/r .+.tVMFc[o7$Fe_p$\"/BD`h^cMFc[o7$Fj_p$\"/E70%H,b$Fc[o7$F_`p$\"/]aK#Hix $Fc[o7$Fd`p$\"0F_5)Ra)[$Fh`p7$Fj`p$\"0\"z5v:\"e9$Fh`p7$F_ap$\"0,y^()f. 2$Fh`p7$Fdap$\"0G/.K&Q:IFh`p7$$\"5sssssi>%\\x$Fgw$\"07]KV&z**HFh`p7$Fi ap$\"0FN?N2`*HFh`p7$$\"5CCCCCa1#Gy$Fgw$\"05a'es-0IFh`p7$F^bp$\"0N@z[mF .$Fh`p7$Fcbp$\"0>dy')z<;$Fh`p7$Fhbp$\"0ll@InYH$Fh`p7$F]cp$\"0=m_`Ft>$F h`p7$Fbcp$\"0@J3^uE5$Fh`p7$Fgcp$\"0kqr(o/nFFh`p7$F\\dp$\"0GvCU=Dc#Fh`p 7$Fadp$\"0N1!z+([n#Fh`p7$Ffdp$\"0^A`Tb5t#Fh`p7$F[ep$\"0_Do6'puDFh`p7$F `ep$\"0&=boJHFCFh`p7$Feep$\"0MA\"y05(G#Fh`p7$Fjep$\"0$*\\7Yj6=#Fh`p7$F _fp$\"0]giB)yd@Fh`p7$Fdfp$\"0%>))*[(*=<#Fh`p7$Fifp$\"03By*>cXAFh`p7$F^ gp$\"0O\"oFh`p7$F[j p$\"0.$4E/Wb=Fh`p7$F`jp$\"0gaK:k'*z\"Fh`p7$Fejp$\"0mYPM5Xd\"Fh`p7$Fjjp $\"0tkVUs3Y\"Fh`p7$F_[q$\"0/\"pjlE78Fh`p7$Fd[q$\"0R`&y#o_B\"Fh`p7$Fi[q $\"0O>BZ,1=\"Fh`p7$F^\\q$\"0c!\\DrPr6Fh`p7$Fc\\q$\"0;w`xjR=\"Fh`p7$Fh \\q$\"1$p6R:->?\"F\\]q7$F^]q$\"1@A]Sj+I7F\\]q7$Fc]q$\"1,oP?>Id7F\\]q7$ Fh]q$\"1a-DCw&pB\"F\\]q7$F]^q$\"1+brEZv#3\"F\\]q7$Fb^q$\"0[1m`hNc*F\\] q7$Fg^q$\"0)z%GOiuF*F\\]q7$F\\_q$\"05MgMl^b*F\\]q7$Fa_q$\"0K]%z#e)f)*F \\]q7$Ff_q$\"0rF_eq7$F[fq$ \"1'eO[J?(HFghq7$Fbjq$\"1')3H5'QPb\"Fghq7$Fg jq$\"2uOd(>vDl7F[[r7$F][r$\"1$)Qx?,C!G*F[[r7$Fb[r$\"1Uq'f#*\\H\\(F[[r7 $Fg[r$\"17a\\Rr))p`F[[r7$F\\\\r$\"18V1$*f>NVF[[r7$Fa\\r$\"1ibh!f.'HMF[ [r-Ff\\r6&Fh\\rF($\"#vF[]rF\\]r-F`]r6#%TButcher's~scheme~B~with~c[5]=c [6]=1/2~and~b[5]=b[6]G-F$6%7ealF'7$F+$\"'ZYzF07$F2$\")9VZIF07$F7$\")_& [?$F07$F<$\")U'HO%F07$FP$\")$pS\"HF07$FZ$\")(y\"*f#F07$F^o$!)h:1LF07$F bp$!)al]tF07$F^jv$!*l#>f;F07$Fhjv$!*s&[z;F07$F][w$!*5'fm;F07$Fgp$!*%=H i;F07$F\\q$!*V^(=F07$F`w$!+Lu:a?F0 7$F\\fr$!+._d0BF07$Fafr$!+$\\dY-$F07$Fffr$!+#o#p/IF07$Few$!+:PdiHF07$F [x$!+c()f+OF07$Fagr$!+d2(4&QF07$Ffgr$!+bp6pTF07$F[hr$!+vo#\\*[TF07$F]ir$!+v6l$3%F07$Fex$!+Qs sGSF07$Feir$!+\")fF:SF07$Fjx$!+?IXJSF07$F]jr$!+`w\\/TF07$F_y$!+*R\\#zU F07$Fiy$!+*[`(\\_F07$Fcz$!+JM&Qf&F07$F[[s$!+De4$\\&F07$Fhz$!+J1H(R&F07 $Fc[s$!+cj0M`F07$F][l$!+a`G5aF07$Fb[l$!+$>H1(fF07$Fg[l$!+z=P$H(F07$F\\ \\l$!+-!=f9(F07$Fa\\l$!+-?.,qF07$Ff\\l$!+$\\Kq\"zF07$F^_l$!+'>++_)F07$ Fc_l$!+)pl36*F07$Fh_l$!,ln&*R3\"F07$F]`l$!,x;o71\"F07$Fb`l$!,XQ**)R5F0 7$Fh^s$!,n'H(4.\"F07$Fg`l$!,@0F\\-\"F07$F`_s$!,.MuU-\"F07$F\\al$!,r#)3 J.\"F07$Faal$!,hQw*=6F07$Ffal$!,C;6JL3:F07$Fhdl$!,vH Y&)\\\"F07$F]el$!,!**45z9F07$Fbel$!,i#)\\)f9F07$Fgel$!,@v$eA9F07$F\\fl $!,Ws3?R\"F07$Fafl$!,p()3oY\"F07$Fffl$!,R'38#o\"F07$F[gl$!,P<)zK;F07$F `gl$!,7Pioe\"F07$Fegl$!,AaW%o:F07$Fjgl$!,G$=)zb\"F07$Fh\\x$!,\"fK&zb\" F07$F_hl$!,7QjJc\"F07$F`]x$!,^REad\"F07$Fdhl$!,IKXqf\"F07$Fihl$!,[=AQ! =F07$F^il$!,n_WS)=F07$Fcil$!,#fF07$Ff[m$!,()H:T3#F07$F[\\m$!,(R9)z'>F07$F`\\m$!,Yb!\\p=F07$Fe\\ m$!,a50g+#F07$Fj\\m$!,Bv'=V@F07$F_]m$!,7r1y2#F07$Fd]m$!-H/DhF-7$F^^m$!-^lYFx>F-7$Fjhs$!-QlEdu>F-7$Fc^m$!-daK!p(>F-7$Fbis $!-%3M)z&)>F-7$Fh^m$!-5f'fJ+#F-7$F]_m$!-!3i'QHAF-7$Fb_m$!-^9o-2BF-7$Fg _m$!-O]K1IAF-7$F\\`m$!-t*3$4c@F-7$Ff`m$!-<&)R&G4#F-7$F`am$!-et!*ej?F-7 $Feam$!-$**z*[\"3#F-7$Fjam$!-dV,'=9#F-7$F_bm$!-4NRdqAF-7$Fdbm$!-$)y\\^ TCF-7$Fibm$!--`qxqAF-7$F^cm$!-QzElJ@F-7$F^\\t$!-\\x]AC@F-7$Fc\\t$!-!G@ U;7#F-7$Fh\\t$!-1[sYD@F-7$Fccm$!-*eEnw8#F-7$F`]t$!-Sd%fv>#F-7$Fhcm$!-, )>w$GBF-7$F]dm$!-Hfj=KCF-7$Fbdm$!-K$*4$*)[#F-7$Fgdm$!-w3OZmCF-7$F\\em$ !-EBT=WCF-7$Faem$!-?Jn5+CF-7$Ffem$!-bb&3nN#F-7$F[fm$!-K\\phqAF-7$F`fm$ !-9UK@$>#F-7$Fjfm$!-RX*yP;#F-7$Fdgm$!-Q+6M[@F-7$Figm$!-'))y4\"e@F-7$F^ hm$!-0-m76AF-7$Fchm$!-()>jyLBF-7$Fhhm$!-OcV(R]#F-7$F]im$!-L`7XcCF-7$Fb im$!-6#F-7$Fc]n$!-0N,,1@F-7$Fh]n$!-a&o%[0@F-7$F]^n$!-naA)>6#F -7$Fb^n$!-=D2[F@F-7$Fg^n$!-v(>@d>#F-7$F\\_n$!-xCl]OBF-7$Fa_n$!-,qo;(Q# F-7$Ff_n$!-w-0MHCF-7$F[`n$!-9j$\\vT#F-7$F``n$!-l[w!eS#F-7$Fe`n$!-LV=Z# Q#F-7$Fj`n$!-PM=LfBF-7$F_an$!-%*ffj8BF-7$Fdan$!-o3vsoAF-7$Fian$!-'*))y 7w@F-7$F^bn$!-Y.$4F4#F-7$Fcbn$!-&)Q!\\,1#F-7$Fhbn$!-aD\"*3T?F-7$F`ft$! -F_RrR?F-7$F]cn$!-1R99Y?F-7$Fhft$!-S:!GF1#F-7$Fbcn$!-`CcS#4#F-7$Fgcn$! -@R5b*>#F-7$F\\dn$!-t4B?JBF-7$Fadn$!-fR.h$G#F-7$Ffdn$!-B&)*poB#F-7$F[e n$!-v;5)f9#F-7$F`en$!-fBcmf?F-7$Fjen$!-D/5Iy>F-7$F^gn$!-\\;%z:&>F-7$Fc gn$!-kzuf,@F-7$Fhgn$!-$HqV]:#F-7$F]hn$!-?l&4W)>F-7$Fbhn$!-Rkeq[=F-7$Fg hn$!-H#4f'R=F-7$F\\in$!-'evvi$=F-7$Fain$!-&3HU/%=F-7$Ffin$!-X0(fX&=F-7 $F[jn$!-3psZD>F-7$F`jn$!-Yc;(H3#F-7$Fejn$!-r2D')*)>F-7$Fjjn$!-#eo&p&*= F-7$F_[o$!.hC+_X\"=Fc[o7$Fe[o$!.Q'Q04W\"GyJ2bYVG>Fc[o7$Fg^o$!.aSV6l\">Fc [o7$F\\_o$!.\\eJaY!>Fc[o7$Fa_o$!.Lo)H'G*=Fc[o7$Ff_o$!.g,c\\j%=Fc[o7$F[ `o$!.JLSp3!=Fc[o7$F``o$!.UT\\#>8'zMOd\"Fc[o7$Ffdo$!.f67\"Fc[o7$Ffio$!.%ycdN[6Fc [o7$F[jo$!.72#4t87Fc[o7$F`jo$!.#[,uEQ7Fc[o7$Fejo$!.wBjCNB\"Fc[o7$Fjjo$ !.UA20dA\"Fc[o7$F_[p$!.5q1Jz@\"Fc[o7$Fd[p$!.Og3]0\"Fc[o7$Fb]p$!-TVh%[***Fc[o7$Fg]p$!->GmOn%*Fc[o7$F\\ ^p$!-&e8q$\\*)Fc[o7$Fa^p$!-ibR9=&)Fc[o7$Ff^p$!-\"\\:[/W)Fc[o7$F[_p$!-` ouf$Q)Fc[o7$F`_p$!-B]rXa$)Fc[o7$Fe_p$!-%4)G$=O)Fc[o7$Fj_p$!-&)3nhL&)Fc [o7$F_`p$!-W@GmD!*Fc[o7$Fd`p$!.Np%yqI$)Fh`p7$Fj`p$!.NTE;(Fh`p7$Fiap$!.Aa,ic3(Fh`p7$F^bp$!.d'f#Gi8( Fh`p7$Fcbp$!.\\IkO9S(Fh`p7$Fhbp$!.[o7)o'p(Fh`p7$F]cp$!.t-8#HpuFh`p7$Fb cp$!.9q1o\"[sFh`p7$Fgcp$!.,B[75Y'Fh`p7$F\\dp$!.7[vZW$fFh`p7$Fadp$!.*[I *e<9'Fh`p7$Ffdp$!.'G\"pJ>E'Fh`p7$F[ep$!.'\\:!)R.fFh`p7$F`ep$!.3`.R[c&F h`p7$Feep$!._Fz)yQ_Fh`p7$Fjep$!.\"**\\2Wy\\Fh`p7$F_fp$!.Y4kI%3\\Fh`p7$ Fdfp$!.3zDC-#\\Fh`p7$Fifp$!.:A%))**p]Fh`p7$F^gp$!.S)Rqx'H&Fh`p7$Fcgp$! .x%R-7:ZFh`p7$Fhgp$!.jx]v&3UFh`p7$F]hp$!.=wBdb5%Fh`p7$Fbhp$!.E)\\NCHSF h`p7$Fghp$!.h*opR)*RFh`p7$F\\ip$!..$oj8VSFh`p7$$\"51111JU))3$4%Fgw$!.d RMbv5%Fh`p7$Faip$!.k[_z)4UFh`p7$$\"5MLLL3Ae]+TFgw$!.]I()RpK%Fh`p7$Ffip $!.Q3#*4;E%Fh`p7$F[jp$!.F%pPsLTFh`p7$F`jp$!.=g'pX4SFh`p7$Fejp$!.m\">0 \\0NFh`p7$Fjjp$!.e!3&=LA$Fh`p7$F_[q$!.rRbT?*GFh`p7$Fd[q$!.(*\\#>**=FFh `p7$Fi[q$!.cVfL*)e#Fh`p7$F^\\q$!.Y=fE:c#Fh`p7$Fc\\q$!.Zr(*>Ie#Fh`p7$F^ ]q$!/rs#4\"G%o#F\\]q7$Fh]q$!/XF\\]q7$Fe`q$!/uC]_'oq\"F\\]q7$Fj`q$!/3Npele:F\\]q7$$\"544444f*)Q) \\%Fgw$!/;2#4kEk\"F\\]q7$$\"5=====ygQ8XFgw$!/CdObzo:F\\]q7$$\"5GFFFF(> $QGXFgw$!/!=HVqfY\"F\\]q7$F_aq$!/V(oy%*)p8F\\]q7$Fdaq$!/;N@Z#pF\"F\\]q 7$Fiaq$!/Wh%4%>27F\\]q7$F^bq$!/[K1bX17F\\]q7$Fcbq$!/z5:FrF7F\\]q7$Fhbq $!/u6yp=x5F\\]q7$F]cq$!.fZBRM\\*F\\]q7$Fbcq$!.#Hzf)G/*F\\]q7$Fgcq$!.F' pN=U$*F\\]q7$F\\dq$!.M'yFQ***)F\\]q7$Fadq$!.eZ9e:L)F\\]q7$Ffdq$!.#oYs1 UnF\\]q7$F[eq$!/M**R#Gco&F_eq7$Faeq$!/D%G7@M*\\F_eq7$F[fq$!/7**3$>uq$F _eq7$F_gq$!/+%HR!*)oKF_eq7$Fdgq$!/`Yl82-EF_eq7$Figq$!/!yW?te5#F_eq7$F^ hq$!/&\\?3!f&y\"F_eq7$Fchq$!0DeoN6jM\"Fghq7$Fihq$!0f;k*HRA6Fghq7$F^iq$ !/L&R$)*>=!*Fghq7$Fciq$!/(e'y)*3XpFghq7$Fhiq$!/]!Hwh_)eFghq7$F]jq$!/,z VpYiUFghq7$Fbjq$!/UQzw(fV$Fghq7$Fgjq$!0!R4)Q\"=2GF[[r7$F][r$!0&f5.E(R1 #F[[r7$Fb[r$!0$p![!=/z;F[[r7$Fg[r$!0GC#ep!*)>\"F[[r7$F\\\\r$!/\\kD%*e$ y*F[[r7$Fa\\r$!/Zg%*RC)z(F[[r-Ff\\r6&Fh\\rFi\\rF^cvF(-F`]r6#%Hscheme~w ith~c[5]=c[6]=3/4~and~b[5]=b[6]G-%%FONTG6$%*HELVETICAG\"\"*-%+AXESLABE LSG6$Q\"x6\"Q!F[bcl-%&TITLEG6#%Uerror~curves~for~7~stage~order~6~Runge -Kutta~methodsG-%%VIEWG6$;F(Fa\\r%(DEFAULTG" 1 2 0 1 10 0 2 9 1 4 2 1.000000 45.000000 45.000000 0 1 "Butcher's scheme A" "scheme with sim ple nodes" "scheme with a relatively large stability region" "Butcher' s scheme B with c[5]=c[6]=1/2 and b[5]=b[6]" "scheme with c[5]=c[6]=3/ 4 and b[5]=b[6]" }}}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 376 "evalf[20](plot(['hn_RK6_2'(x)-h(x),'hn_RK6_3 '(x)-h(x),'hn_RK6_5'(x)-h(x)],x=0..6,numpoints=100,font=[HELVETICA,9], \ncolor=[COLOR(RGB,.45,0,.95),COLOR(RGB,0,.25,1),COLOR(RGB,.95,.45,0)] ,\nlegend=[`scheme with simple nodes`,`scheme with a relatively large \+ stability region`,`scheme with c[5]=c[6]=3/4 and b[5]=b[6]`],title=`er ror curves for 7 stage order 6 Runge-Kutta methods`));" }}{PARA 13 "" 1 "" {GLPLOT2D 1022 417 417 {PLOTDATA 2 "6)-%'CURVESG6%7_`l7$$\"\"!F)F (7$$\"5.......*4M'!#@$\"(F/!=!#?7$$\"5111111_#e=\"F0$\")^v0pF07$$\"5%R RRRR**)H1=F0$\")!\\Lv*F07$$\"5&[[[[[o&)3V#F0$\"*w,y6#F07$$\"5qpppp>-H' e#F0$\"*3(*z9#F07$$\"5aaaaaaZpTFF0$\"*2/GL#F07$$\"5RRRRR*G*4(*GF0$\"*Z WH.$F07$$\"5CCCCCCQ]_IF0$\"*dS\"oTF07$$\"5======OmSLF0$\"*2R58%F07$$\" 5777777M#)GOF0$\"*ivo<%F07$$\"5)yyyyy='>FRF0$\"*TKn[&F07$$\"5kjjjjj*ob A%F0$\"*\\bnY'F07$$\"5SRRRRRx&)zVF0$\"*-WkU'F07$$\"5;:::::l9MXF0$\"*pp $3kF07$$\"5#44444HN%)o%F0$\"*<=\"3lF07$$\"5nmmmmmSsU[F0$\"*'R#R-(F07$$ \"5YXXXXX***yX&F0$\"*%yL\\&)F07$$\"5+(pppp%f4;cF0$\"*<=2`)F07$$\"5][[[ [[>HudF0$\"*te=r)F07$$\"5+++++]z[KfF0$\"*l#z5&*F07$$\"5_^^^^^Ro!4'F0$ \"+6y\"*=5F07$$\"5++++++AOpjF0$\"+BdX,5F07$$\"5\\[[[[[//[mF0$\"*R*Hq)* F07$$\"5]^^^^^hwhpF0$\"+gsv:5F07$$\"5baaaaa=\\vsF0$\"+&4i*35F07$$\"5++ ++++))*HV(F0$\"*$HYq**F07$$\"5XXXXXXd]!f(F0$\"*:Ym#)*F07$$\"5!44444p7! [xF0$\"*ZD(o&*F07$$\"5POOOOO'>b!zF0$\"*A4m\"*)F07$$\"5ljjjjjnId!)F0$\" *]&[J!)F07$$\"5!44444*Q44#)F0$\"*$>)G$zF07$$\"5?=====5)3O)F0$\"*ay2$yF 07$$\"5YXXXXX\"oE^)F0$\"*=cpp(F07$$\"5][[[[)R00l)F0$\"*$zSPuF07$$\"5]^ ^^^^EM)y)F0$\"*!*)REnF07$$\"5baaaa/*zh#*)F0$\"*)*>pz%F07$$\"5edddddr,k !*F0$\"*h%>LFF07$$\"5NLLLL$)*=zA*F0$\"*$[H#p#F07$$\"55444443#=R*F0$\"* qH4k#F07$$\"5&[[[[[jAdb*F0$\"*uz/[#F07$$\"5hgggggWi>(*F0$\"*Lp$RD'F07$$\"5qpppp\\jh 85!#>$!*>atL'F07$$\"5CCCCC%)4]F5F`y$!*B<'\\iF07$$\"5FFFFF2b5#4\"F`y$!+ [e*3P\"F07$$\"5)yyyG\"H'zc4\"F`y$!+_cA<;F07$$\"5[[[[)4v`#*4\"F`y$!+nsw G>F07$$\"54444%G(y#G5\"F`y$!+jf1(*>F07$$\"5qpppp%*>S16F`y$!+Tq>*)>F07$ $\"5\"4444%Q-b86F`y$!+!)yZt>F07$$\"577777#[)p?6F`y$!+%[3y&>F07$$\"5aaa aap\\*\\8\"F`y$!+]AiF>F07$$\"5(pppppX\"H\\6F`y$!+DhO3>F07$$\"5wvvvD8U8 d6F`y$!+\")Hi:>F07$$\"5aaaaapp(\\;\"F`y$!+mRta>F07$$\"5LLLL$es>G<\"F`y $!+%=V_0#F07$$\"577777#[i1=\"F`y$!+`*zfE#F07$$\"5qpppp%*zM'>\"F`y$!+Z. \"RO$F07$$\"5FFFFF2N.77F`y$!+)e/Gy$F07$$\"5aaaaaC)ppA\"F`y$!+OPx9PF07$ $\"5#=====91>C\"F`y$!+#GwAl$F07$$\"544444fC%oD\"F`y$!+f;,GOF07$$\"5OOO OOw(yAjE(F07$$\"54444fm23*R\"F`y$!+B1b` ')F07$$\"5IIIIIl\"*z19F`y$!+i#opy)F07$$\"5_^^^,kv^99F`y$!+2&>=p)F07$$ \"5tssssifBA9F`y$!+E(zsf)F07$$\"5;::::gFnP9F`y$!+2*GOT)F07$$\"5eddddd& 4JX\"F`y$!+/5*4F)F07$$\"5wvvvvX&y^[\"F`y$!+!=%>+#*F07$$\"5%RRRRR`Zs^\" F`y$!,&zZRj6F07$$\"5=====3c?J:F`y$!,f)=.R6F07$$\"5UUUUU#oj^a\"F`y$!,q8 ok6\"F07$$\"5aaaaa>F9_:F`y$!,2n'e26F07$$\"5mmmmmc<7f:F`y$!,wQeE5\"F07$ $\"5yyyyy$z+hc\"F`y$!,,wJ^5\"F07$$\"5\"44444$)zId\"F`y$!,(f%317\"F07$$ \"5[[[[[L\\:)e\"F`y$!,GL&HX7F07$$\"511111O+B.;F`y$!,9mn^^\"F07$$\"5kjj jjQ^I=;F`y$!,)o@#)y9F07$$\"5@@@@@T-QL;F`y$!,H,]KW\"F07$$\"5GFFFF(Ha*[; F`y$!,HpI,T\"F07$$\"5MLLLL`$GXm\"F`y$!,$4))z&R\"F07$$\"5SRRRR4C5!o\"F` y$!,v()H?X\"F07$$\"5YXXXXlkn&p\"F`y$!,o(=)[r\"F07$$\"5#====$4b[*p\"F`y $!,'o\"Q)R=F07$$\"5=====`XH.BHe7#F07$$\"5#====o`c>$=F`y$!,'*e5N1#F07$$\"5kjjjjtZK[ =F`y$!,6)**)e+#F07$$\"5baaa/#*)3l&=F`y$!,o$f[$)>F07$$\"5YXXXX5Ipk=F`y$ !,jqaA(>F07$$\"5OOOO')Gr(G(=F`y$!,o,uG)>F07$$\"5FFFFFZ71\")=F`y$!,9t@L .#F07$$\"511111O)od*=F`y$!,'3;dCBF07$$\"5&[[[[[Uw/\">F`y$!,l@O$RCF07$$ \"5kjjjj8S=D>F`y$!,CzJ:P#F07$$\"5VUUUU-;*)R>F`y$!,%\\z01BF07$$\"5edddd ZLuZ>F`y$!,&HZwtAF07$$\"5ussss#4&fb>F`y$!,!*[7jC#F07$$\"5KIIIIl4_f>F`y $!,W()oeB#F07$$\"5*yyyyy$oWj>F`y$!,ECH)GAF07$$\"5YXXXX5FPn>F`y$!,D(GZE AF07$$\"5/....$e)Hr>F`y$!,J7a/B#F07$$\"5>====G.:z>F`y$!,(H1TmAF07$$\"5 MLLLLt?+()>F`y$!,*ofugBF07$$\"5\\[[[[=Q&[*>F`y$!,#yQ&)\\DF07$$\"5kjjjj jbq-?F`y$!,y^V]t#F07$$\"511111c\\;J?F`y$!,YV_Ee#F07$$\"5\\[[[[[Vif?F`y $!,LfcVX#F07$$\"5_^^^^\")zt!4#F`y$!,jB?=k#F07$$\"5baaaa9;&=7#F`y$!,Ha& pCGF07$$\"5_^^^^m$*[O@F`y$!,aV_&QFF07$$\"5\\[[[[=r7^@F`y$!-'>q)RfEF-7$ $\"5)ppppW*fWe@F`y$!-nG,PFEF-7$$\"5YXXXXq[wl@F`y$!-pTpr1EF-7$$\"5qpppW 3VUp@F`y$!-UevF.EF-7$$\"5%RRRRku$3t@F`y$!-,giT1EF-7$$\"5====V%=Vn<#F`y $!-TUQ4=EF-7$$\"5VUUUUAES!=#F`y$!-zG5tSEF-7$$\"5#44444H.d>#F`y$!-:4/4K HF-7$$\"5SRRRRfR+6AF`y$!-;u'\\#HIF-7$$\"5)yyyyyi/jA#F`y$!-\"yX%>GHF-7$ $\"5POOOO'H0;C#F`y$!-&HO#4JGF-7$$\"5_^^^^'\\klD#F`y$!-*)pe/[FF-7$$\"5n mmmm'pB:F#F`y$!->.gv#F-7$$\"5A@@@@')*o\"zBF`y$!-R[?KpFF- 7$$\"5#4444%Q1r'Q#F`y$!-(HaV#QGF-7$$\"5hgggg!H_UR#F`y$!-*4zb3*HF-7$$\" 5YXXXq;J-)R#F`y$!-gmw37JF-7$$\"5IIII!G%Rz,CF`y$!-\"[uDr<$F-7$$\"5::::! *oZc0CF`y$!-KX)e%[JF-7$$\"5+++++&fN$4CF`y$!-()pi+?JF-7$$\"5qppp>Zs(oT# F`y$!-E%RTP1$F-7$$\"5SRRRR**)=WU#F`y$!-RGLM3IF-7$$\"5nmmmmcP%)RCF`y$!- d+.U)*GF-7$$\"5%RRRRRho_X#F`y$!-l\\p:*z#F-7$$\"5eddddU5)HY#F`y$!-qQDeg FF-7$$\"5A@@@@rMpqCF`y$!-e?*GF-7$$\"5:::::boIZDF`y$!-3y+N%y#F -7$$\"5YXXXXILOhDF`y$!-:ad#F`y$!-%y/(zkEF-7$$\"5 11111\"Gw%*e#F`y$!-q@%)4fFF-7$$\"5POOOOcF`.EF`y$!-cgV3\"*HF-7$$\"5IIII IqLvNEF`y$!-c>NI\\FF-7$$\"5CCCCC%)R(zm#F`y$!-*)3[McDF-7$$\"5RRRR*oHw:n #F`y$!-AY5*oa#F-7$$\"5aaaaa4'y^n#F`y$!-TNw2VDF-7$$\"5qppp>A4yyEF`y$!-m 2$>la#F-7$$\"5&[[[[[B$Q#o#F`y$!-h#o0#fDF-7$$\"5;::::gye*o#F`y$!-wAaaAE F-7$$\"5YXXXX&[#z'p#F`y$!-.M-?fFF-7$$\"5/...yTOf)p#F`y$!-,>n))3GF-7$$ \"5hggg5)z%R+FF`y$!-'=U\\)\\GF-7$$\"5====Vaf>-FF`y$!-8%GAVhEF-7$$\"5=====`U(4u#F`y$!-@s\"4Fb#F-7$$\"5qpppp>vLcFF`y$!-GFi\" QX#F-7$$\"5YXXX&H:>Sw#F`y$!-P:*)p8CF-7$$\"5A@@@@'y+CQzFF`y$!-u3U#F-7$$\"5#4444 fN;Wz#F`y$!-$eavj^#F-7$$\"554444f'o#F-7$$\"5+++++0_'[*GF`y$!-@]y13BF-7$$\"5h gggg!y#>6HF`y$!-S3XnPBF-7$$\"577777U(p%RHF`y$!-`f:\\_@F-7$$\"5kjjjj.nu nHF`y$!-3$\\O.+#F-7$$\"5nmmm;H&e<(HF`y$!-t:/x()>F-7$$\"5qppppa.xvHF`y$ !-hT`6!)>F-7$$\"5tsssA!=#yzHF`y$!-*pKs*y>F-7$$\"5wvvvv0Sz$)HF`y$!-k)ot j)>F-7$$\"5#====ol<=*HF`y$!-wNOjP?F-7$$\"5)yyyyyIT)**HF`y$!-)>NPC;#F-7 $$\"5+++++5'))e,$F`y$!-jD*H[1#F-7$$\"5777777f$>.$F`y$!-cgh4n>F-7$$\"5 \"4444fGhl/$F`y$!.nh94E)=!#A7$$\"5qppppfm=hIF`y$!.V<9\"\\2=Fc`o7$$\"54 444fY$*\\oIF`y$!.$fPkBygFc`o7$$\"5OOOOh-eV-JF`y$!.83!*3p*=Fc`o7$$\"5aaa/ M W6$F`y$!.>Afku#=Fc`o7$$\"5=====GJVAJF`y$!.QK8[Cy\"Fc`o7$$\"5kjjjj))4VQ JF`y$!.@x(\\f&p\"Fc`o7$$\"544444\\)GW:$F`y$!.#3%\\g\\h\"Fc`o7$$\"57777 7(HK#oJF`y$!.Ezq\\$e:Fc`o7$$\"5:::::Xd.#=$F`y$!.(e(y$zO:Fc`o7$$\"5mmmm ;pu$*)=$F`y$!.\"=AZaa:Fc`o7$$\"5=====$>Re>$F`y$!.k_TSgg\"Fc`o7$$\"5111 c=CYc(>$F`y$!.Oq\\rgi\"Fc`o7$$\"5%RRR*=b+H*>$F`y$!.@#*4(e\\;Fc`o7$$\"5 #===$>'[:5?$F`y$!.G&R*p\\l\"Fc`o7$$\"5qppp><4u-KF`y$!.*>p?&ek\"Fc`o7$$ \"5YXXX?z<>1KF`y$!./d@_xi\"Fc`o7$$\"5@@@@@TEk4KF`y$!.+tCK)4;Fc`o7$$\"5 MLLLLtO3TKF`y$!.]tlj[X\"Fc`o7$$\"5YXXXX0Z_sKF`y$!.%[.2HK8Fc`o7$$\"5nmm mmEt6!G$F`y$!.:W\\o;K\"Fc`o7$$\"5)yyyyy%*4xG$F`y$!.&H'QV0L\"Fc`o7$$\"5 44444pDI&H$F`y$!.taf>9P\"Fc`o7$$\"5IIIII!>&*GI$F`y$!.N7r9rS\"Fc`o7$$\" 5sssssK/3=LF`y$!.,)Ht7Q8Fc`o7$$\"5:::::vcELLF`y$!.M#R?Fs7Fc`o7$$\"5CCC CCkp'RR$F`y$!.^Ws6'\\6Fc`o7$$\"5CCCCCaf?CMF`y$!.zg')3G5\"Fc`o7$$\"5CCC CCW\\WaMF`y$!-09i=_**Fc`o7$$\"5gggggX+(*oMF`y$!-lu*)*Q`*Fc`o7$$\"5(ppp pp9&\\$[$F`y$!-7#)HzZ$*Fc`o7$$\"5;:::l(pd2\\$F`y$!-#eESHX*Fc`o7$$\"5ML LLL[--)\\$F`y$!-`GOl,)*Fc`o7$$\"5VUUUnB:l,NF`y$!-R$*R/)*)*Fc`o7$$\"5_^ ^^,*z#G0NF`y$!-yg\"=Hx*Fc`o7$$\"5hgggNuS\"*3NF`y$!-\"o(oC\\'*Fc`o7$$\" 5qpppp\\`a7NF`y$!-h)3:q_*Fc`o7$$\"5=====)Q\\Ra$F`y$!-,#)>rH&)Fc`o7$$\" 5nmmmmEMNvNF`y$!-y96B`xFc`o7$$\"5gggg53g$Ge$F`y$!-rrS8#p(Fc`o7$$\"5aaa aa*e=.f$F`y$!-6gf`cxFc`o7$$\"5^^^^E!))fSf$F`y$!-k&=f$fyFc`o7$$\"5[[[[) 4<,yf$F`y$!-]-CPE!)Fc`o7$$\"5XXXXqhCa,OF`y$!-U-*Rd6)Fc`o7$$\"5UUUUU_PG 0OF`y$!-lXC12!)Fc`o7$$\"5IIIII:*[-i$F`y$!-0'H`ce(Fc`o7$$\"5=====yS@NOF `y$!-P$)e/&=(Fc`o7$$\"5:::::&f9#)p$F`y$!-=VUu7lFc`o7$$\"5aaaaa9*Hns$F` y$!.*Qu7&p(f!#B7$$\"5%RRRRRBX_v$F`y$!.X>.q\"y`Ffbp7$$\"5YXXXXDR7jPF`y$ !.-;AkIB&Ffbp7$$\"5(ppppph-5x$F`y$!.]Px(o3^Ffbp7$$\"5[[[[[38))yPF`y$!. IB$H\">-&Ffbp7$$\"5+++++++w'y$F`y$!.l8[FB+&Ffbp7$$\"5_^^^^\"pQYz$F`y$! .3>p+!*4&Ffbp7$$\"5.....$Q_Ffbp7$$\"5aaaaaugR5QF`y$!.# f@R3l]Ffbp7$$\"511111mZF=QF`y$!.#GJG8:\\Ffbp7$$\"5IIIII?/Y[QF`y$!.%\\n =ZzVFfbp7$$\"5baaaaugkyQF`y$!.I^Ru\\(RFfbp7$$\"5IIIIIl@q$*QF`y$!.>\"\\ !Ry+%Ffbp7$$\"511111c#e(3RF`y$!.vKCxT,%Ffbp7$$\"5#====oM9Q#RF`y$!.qU2H Vy$Ffbp7$$\"5eddddP/()QRF`y$!.<\\$*fpc$Ffbp7$$\"5tssss#e'faRF`y$!.&oF& *zaLFfbp7$$\"5)yyyyysA.(RF`y$!.o8!\\ZsJFfbp7$$\"5.....t)[g)RF`y$!.j=I! >$3$Ffbp7$$\"5======]x,SF`y$!.p5RU9@$Ffbp7$$\"5wvvvvDPuISF`y$!.;6YC(eG Ffbp7$$\"5MLLLLLCrfSF`y$!.wA1)\\YDFfbp7$$\"5)yyyyGRYX2%F`y$!.)=[\"o3U# Ffbp7$$\"5VUUUU_.Q*3%F`y$!./NWI7Q#Ffbp7$$\"5qppp>Ktz'4%F`y$!.lI5)RPCFf bp7$$\"5)pppp>J9U5%F`y$!.>:L&zXCFfbp7$$\"5DCCCu\"HJ;6%F`y$!.&4(o*RsBFf bp7$$\"5_^^^^r#[!>TF`y$!.YOaz5I#Ffbp7$$\"5GFFFF(3iX=%F`y$!.CF.q=\"=Ffb p7$$\"5@@@@@\"[zQC%F`y$!.'GwFe_q7$$\"5&[[[[[wDD(\\F`y$!/oNAe)fW\"Fe_q7$$\"5%RRRRRHL)H]F`y$!/^AX^* *R7Fe_q7$$\"5DCCCCW8p!4&F`y$!.aoN\"49'*Fe_q7$$\"5MLLLL`G3]^F`y$!.k$3P3 Fiaq7$$\"5>====y)zUd&F`y$!/%yIF&H#R\"Fiaq7$$\"5POOOOw$G(QcF`y$ !/p+%z#*36\"Fiaq7$$\"5>====Q8#yp&F`y$!/h6'e\"z0*)!#F7$$\"5SRRRRzO:cdF` y$!/`b(o(f\\lF]dq7$$\"577777#**4&=eF`y$!/r#z8)Rh_F]dq7$$\"5hggggg0t!)e F`y$!/2%)GU&yq$F]dq7$$\"5wvvvv:sdOfF`y$!/[,t'[(GIF]dq7$$\"\"'F)$!/VL2& **4R#F]dq-%&COLORG6&%$RGBG$\"#X!\"#F($\"#&*F]fq-%'LEGENDG6#%9scheme~wi th~simple~nodesG-F$6%7]^lF'7$$\"5_^^^^^^\\qJF-$\"&kQ\"F07$F+$\"()zaNF0 7$$\"5!======@'*4*F-$\")QP3kF07$F2$\"*=<F0$ \"*LR)pLF07$$\"5/.....!>0/#F0$\"*#R[RSF07$$\"5SRRRRRBf=@F0$\"*qJH.%F07 $$\"5777777!RZF#F0$\"*5h%>SF07$F<$\"*8jG,%F07$FA$\"*B:o1%F07$FF$\"*H\\ =S%F07$FK$\"*HSpn&F07$FP$\"*\"=**[xF07$FU$\"*5X*zwF07$FZ$\"*n40w(F07$F in$\"+3eb85F07$F^o$\"+#e'e\">\"F07$Fco$\"+;P:%=\"F07$Fho$\"+(Q*y!=\"F0 7$F]p$\"+2u2*>\"F07$Fbp$\"+p;8%H\"F07$$\"5_^^^^^5i>\\F0$\"+]#3zS\"F07$ $\"5OOOOOO!=l*\\F0$\"+sUy,;F07$$\"5?@@@@@]Tt]F0$\"+1@72;F07$$\"5011111 ?J]^F0$\"+4x\"3g\"F07$$\"5vvvvvvf5/`F0$\"+qR3)e\"F07$Fgp$\"+eF07$F`r $\"+N>Pu=F07$Fer$\"++Lu]=F07$$\"5++++++L!\\!oF0$\"+([$=o=F07$Fjr$\"+qF pl>F07$$\"5+.....!H'=rF0$\"+kVs&*>F07$F_s$\"+)o:L(>F07$Fis$\"+!***HE>F 07$Fct$\"+9UXX=F07$F]u$\"+)f%yf_m'e\"F07$Ffv$\"+66nq8F07$F[w$\"+&og79\"F07$F`w$\"+(4_U7\"F07$F ew$\"+(Gef5\"F07$Fjw$\"+TOpu5F07$F_x$\"**HW^(*F07$Fdx$\"*P9*=rF07$Fix$ \")v!p\\#F07$F^y$\"(91q%F07$Fdy$\"(1'eWF07$Fiy$!*uQLM*F07$Fa\\l$!+S\"Q Gh\"F07$Fe]l$!+A^&Q1#F07$F_^l$!+Q\\]xQF07$Fd^l$!+.r\"y!QF07$Fi^l$!+q&) pWPF07$$\"5YXXXX+VP\\7F`y$!+_g\"[s$F07$F^_l$!+hK*ps$F07$$\"5sssss<1Jk7 F`y$!+o%\\Qx$F07$Fc_l$!+[9,/RF07$Fh_l$!+^-hcZF07$F]`l$!+s\\;gmF07$Fb`l $!+vP_DlF07$Fg`l$!+yd)RR'F07$F\\al$!+Y!od,)F07$Faal$!+\"G>k*F07$F`bl$!+N40P&*F07$Febl$!+z?\\L$*F07$Fjbl$ !+)3jl<*F07$F_cl$!,kT&)>-\"F07$Fdcl$!,3\"e(3H\"F07$Ficl$!,,\"[%QE\"F07 $F^dl$!,r\"4\")Q7F07$Fcdl$!,1^R*G7F07$Fhdl$!,5_&RB7F07$$\"5ssssAv7hi:F `y$!,!f>LB7F07$F]el$!,@02fA\"F07$$\"5%[[[[BJ!fp:F`y$!,nr\\>B\"F07$Fbel $!,mu)[U7F07$Fgel$!,.)H=w8F07$F\\fl$!,&=laj;F07$Fafl$!,@tTOi\"F07$Fffl $!,)4Be%e\"F07$F[gl$!,41F\"[:F07$F`gl$!,nK@:`\"F07$Fegl$!,yb9%)e\"F07$ Fjgl$!,enVv&=F07$F_hl$!,od3^)>F07$Fdhl$!,mgKP*>F07$Fihl$!,y#\\#3)>F07$ F^il$!,sNsz'>F07$Fcil$!,sMOC%>F07$Fhil$!,Glgr\">F07$F]jl$!,vYf$o=F07$F bjl$!,K-$=H=F07$$\"5OOOO')e?*Rw\"F`y$!,Z=:5#=F07$$\"5YXXXX]HOrF07$$\"5tsssADcZ$ z\"F`y$!,^\\%)o3#F07$$\"5#====o^Y3!=F`y$!,\")Gc&)H#F07$$\"5\"4444%3u@3 =F`y$!,@h&=oAF07$F\\[m$!,^\"f4QAF07$Fa[m$!,6BuC<#F07$Ff[m$!,\\A\\:6#F0 7$F[\\m$!,eMJt3#F07$F`\\m$!,NM`Q2#F07$$\"5\"444f'p]yo=F`y$!,75`S2#F07$ Fe\\m$!,_Ap63#F07$$\"5#===o!)=pp(=F`y$!,GLAv4#F07$Fj\\m$!,4RIg7#F07$F_ ]m$!,7)>T$R#F07$Fd]m$!,()QzR\\#F07$Fi]m$!,.8`YU#F07$F^^m$!,#GoidBF07$F h^m$!,kxgbH#F07$F\\`m$!,L$\\LtAF07$Fa`m$!,5XE9I#F07$Ff`m$!,[7U5Q#F07$F [am$!,\"4]]VDF07$F`am$!,fQ*H,FF07$Feam$!,f#)p2b#F07$Fjam$!,dn,=U#F07$F _bm$!,>0&oaDF07$Fdbm$!,(\\WC!o#F07$F^cm$!-)p[KD_#F-7$F\\em$!-zXW@![#F- 7$Faem$!-C[72(o#F-7$Ffem$!-?\\byWFF-7$F[fm$!-D3>@`EF-7$F`fm$!-N3>,lDF- 7$Fefm$!-7zu&y[#F-7$Fjfm$!-$H)G5UCF-7$Fdgm$!-*)33q!\\#F-7$F^hm$!-iUx.D FF-7$Fchm$!-*4KGW`#F-7$Fhhm$!-!3*HKsBF-7$F]im$!-#4Nz-O#F-7$Fbim$!-<['f >N#F-7$Fgim$!-NM$)\\[BF-7$F\\jm$!-;1eI^BF-7$Fajm$!-]Ar/$Q#F-7$Ffjm$!-r a)))fY#F-7$F[[n$!-L#Q2Z`#F-7$F`[n$!-)yL3)oDF-7$Fe[n$!-dF-jXDF-7$Fj[n$! -d)RDE_#F-7$F_\\n$!-C*)H8xCF-7$Fd\\n$!-\\UgLKCF-7$F^]n$!-i\">!)4E#F-7$ F\\_n$!-j;#R!3AF-7$Fa_n$!-t0W1nAF-7$Ff_n$!-)f5.NN#F-7$F[`n$!-x+e$)3BF- 7$F``n$!-\\7X)[E#F-7$Fe`n$!-(y=p\"z@F-7$Fj`n$!-&RA-t4#F-7$F_an$!-5C%z+ .#F-7$Fdan$!-yF-7$$\"5MLLLeCR$*yDF`y$!-t-Z3\")>F-7$$\"5\"4444M/[ Ce#F`y$!-C5tO\")>F-7$$\"5[[[[Bi@'fe#F`y$!-6e.;()>F-7$Fian$!->w+y**>F-7 $$\"5A@@@r=X]'f#F`y$!-2ov1_?F-7$F^bn$!-[/LMw?F-7$Fcbn$!-i!Q\"R3>F-7$Fh bn$!-DD%*Qj$p#F`y$!-D(HI5v\"F-7$Ffdn $!-!*=Eir>\"oI\"F-7$Fh\\o$!-iT$G;9\"F-7$Fb]o$!,v/N!e'*F-7$Fj_o$!,M7\"oj!)F -7$Fcbo$!-p()QK_lFc`o7$Feeo$!-k7c))[_Fc`o7$Fgho$!-'z!fFEQFc`o7$Faio$!- uD:s7IFc`o7$F_[p$!-:orBF=Fc`o7$Fd[p$!,My\")Hg)Fc`o7$$\"5CCCC*z$ou(R$F` y$!,'>bdamFc`o7$$\"5CCCCu6n_,MF`y$!,ZvXGH&Fc`o7$$\"5CCCC\\&e1`S$F`y$!, i4#>D_Fc`o7$$\"5CCCCCfk34MF`y$!,,sG$e^Fc`o7$$\"5CCCCu1ik;MF`y$!,I8!*o- &Fc`o7$Fi[p$!,4y**y*[Fc`o7$$\"5CCCCC\\aKRMF`y$!,NLr1j%Fc`o7$F^\\p$!,!G #R:B%Fc`o7$Fh\\p$!+%>wP.*Fc`o7$Ff^p$\",VK8nr&Fc`o7$$\"5%RRRR*otCGNF`y$ \",9898T&Fc`o7$F[_p$\",z%y.f^Fc`o7$$\"5UUUUU29lfNF`y$\",y>s)o^Fc`o7$F` _p$\",P!z\"Fc`o7$$ \"5!4444fYkCo$F`y$\"-vt6$yG\"Fc`o7$$\"5-...`I&R.p$F`y$\"-\\ADBr9Fc`o7$ F]bp$\"-R!Qc*zFfbp7$$\"5=====$fnL$QF`y$\".@sj!pm=Ffbp 7$Feep$\".,%))*3sw\"Ffbp7$$\"5UUUUUZKbjQF`y$\".h*RRW*p\"Ffbp7$Fjep$\". *[\\U$Hs\"Ffbp7$F_fp$\".?0%*)pk>Ffbp7$Fdfp$\".HWArn5#Ffbp7$Fifp$\".r;] zh)>Ffbp7$F^gp$\".x(\\G;t=Ffbp7$Fcgp$\".q#>g6r\"R*RF`y$\".4)*oW/$>Ffbp7$Fbhp$\".)R#Q4+4#Ffbp7$Fghp$\".\\kEc1'=Ff bp7$F\\ip$\".$o9Z'Gn\"Ffbp7$Ffip$\".rhx-os\"Ffbp7$Fjjp$\".YyNVOz\"Ffbp 7$$\"5YXXXXDnUNTF`y$\".$=@<$on\"Ffbp7$$\"5SRRRRz^!=:%F`y$\".7'fW:s:Ffb p7$$\"5MLLLLLO=oTF`y$\"..KSC&*\\\"Ffbp7$F_[q$\".Ao_-j^\"Ffbp7$$\"5_^^^ ^hn(>>%F`y$\".=5l%F`y$\".*)fr(o3CT!Q%F`y$\"/:4%Q3Z6\"F]\\q7$Fi\\q$ \"/I:AuU!=\"F]\\q7$$\"5qpppp*Gm(3WF`y$\"/Qeh6d%>\"F]\\q7$F^]q$\"/9ta34 A6F]\\q7$$\"5kjjjjy`0QWF`y$\"/WbGww\\5F]\\q7$$\"5UUUUUKv;`WF`y$\".Qb)* *oY)*F]\\q7$$\"5@@@@@'oz#oWF`y$\".5A)*=hN*F]\\q7$Fc]q$\"._j&*4dD*F]\\q 7$Fh]q$\".?Z>S(\\$)F]\\q7$$\"577777sDUfXF`y$\".Wo;)R6yF]\\q7$$\"5)yyyy y#[YvXF`y$\".egS,+[(F]\\q7$$\"5kjjjj$32:f%F`y$\".]\"Gpe\\wF]\\q7$F]^q$ \".&\\\"o!fSyF]\\q7$$\"5#44444`iej%F`y$\".XNM#G!)oF]\\q7$Fb^q$\".G*ogr +hF]\\q7$$\"5;::::&\\40o%F`y$\".fzd/2!fF]\\q7$$\"5)yyyyyEVop%F`y$\".r/ /'o/iF]\\q7$$\"5hggggSq<8ZF`y$\".V)\\:txfF]\\q7$Fg^q$\".[xITV`&F]\\q7$ F\\_q$\".>3_')yg%F]\\q7$Fa_q$\"/*y31r[!RFe_q7$Fg_q$\"/OM%Qn._$Fe_q7$F \\`q$\"/kl;zmSEFe_q7$Fa`q$\"//lFt$fN#Fe_q7$Ff`q$\"/GfeIP<>Fe_q7$F[aq$ \"/gcclWY:Fe_q7$F`aq$\"/(p>0I$G8Fe_q7$Feaq$\"0+S**Hp*35Fiaq7$F[bq$\"/F c\"*)y+W)Fiaq7$F`bq$\"//0rJs&)oFiaq7$Febq$\"/]>#=v1F&Fiaq7$Fjbq$\"/;wf 3#G[%Fiaq7$F_cq$\"/n$)*3nZF$Fiaq7$Fdcq$\"/>tvX4DEFiaq7$Ficq$\"0)>tVs9c @F]dq7$F_dq$\"0_h[9(f!e\"F]dq7$Fddq$\"0,=$f95#G\"F]dq7$Fidq$\"/#pi.'z] #*F]dq7$F^eq$\"/#GD*pQ]uF]dq7$Fceq$\"/tXm<'>\"fF]dq-Fheq6&FjeqF($\"#DF ]fq$\"\"\"F)-Fafq6#%Pscheme~with~a~relatively~large~stability~regionG- F$6%7ealF'7$F+$\"'ZYzF07$F2$\")9VZIF07$F7$\")_&[?$F07$F<$\")U'HO%F07$F P$\")$pS\"HF07$FZ$\")(y\"*f#F07$F^o$!)h:1LF07$Fbp$!)al]tF07$F_]r$!*l#> f;F07$Fi]r$!*s&[z;F07$F^^r$!*5'fm;F07$Fgp$!*%=Hi;F07$F\\q$!*V^(=F07$F_x$!+Lu:a?F07$Fdx$!+._d0BF07$Fix$!+$\\d Y-$F07$F^y$!+#o#p/IF07$Fdy$!+:PdiHF07$Fiy$!+c()f+OF07$F^z$!+d2(4&QF07$ Fcz$!+bp6pTF07$Fhz$!+vo#\\*[TF07$F\\\\l$!+v6l$3%F07$Fa\\l$!+QssGSF07$Ff\\l$!+\")fF:SF07$F[] l$!+?IXJSF07$F`]l$!+`w\\/TF07$Fe]l$!+*R\\#zUF07$Fj]l$!+*[`(\\_F07$F_^l $!+JM&Qf&F07$Fd^l$!+De4$\\&F07$Fi^l$!+J1H(R&F07$F^_l$!+cj0M`F07$Fc_l$! +a`G5aF07$Fh_l$!+$>H1(fF07$F]`l$!+z=P$H(F07$Fb`l$!+-!=f9(F07$Fg`l$!+-? .,qF07$F\\al$!+$\\Kq\"zF07$Fjbl$!+'>++_)F07$F_cl$!+)pl36*F07$Fdcl$!,ln &*R3\"F07$Ficl$!,x;o71\"F07$F^dl$!,XQ**)R5F07$Fcdl$!,n'H(4.\"F07$Fhdl$ !,@0F\\-\"F07$F]el$!,.MuU-\"F07$Fbel$!,r#)3J.\"F07$Fgel$!,hQw*=6F07$F \\fl$!,C;6JL3:F07$F^il$!,vHY&)\\\"F07$Fcil$!,!**45z9 F07$Fhil$!,i#)\\)f9F07$F]jl$!,@v$eA9F07$Fbjl$!,Ws3?R\"F07$Fgjl$!,p()3o Y\"F07$F\\[m$!,R'38#o\"F07$Fa[m$!,P<)zK;F07$Ff[m$!,7Pioe\"F07$F[\\m$!, AaW%o:F07$F`\\m$!,G$=)zb\"F07$F]`s$!,\"fK&zb\"F07$Fe\\m$!,7QjJc\"F07$F e`s$!,^REad\"F07$Fj\\m$!,IKXqf\"F07$F_]m$!,[=AQ!=F07$Fd]m$!,n_WS)=F07$ Fi]m$!,#f< F07$Fa`m$!,(*fBNu\"F07$Ff`m$!,FDm3\"=F07$F[am$!,JPg%[>F07$F`am$!,()H:T 3#F07$Feam$!,(R9)z'>F07$Fjam$!,Yb!\\p=F07$F_bm$!,a50g+#F07$Fdbm$!,Bv'= V@F07$Fibm$!,7r1y2#F07$F^cm$!-H/DhF-7$Fhcm$!-^lYFx >F-7$F]dm$!-QlEdu>F-7$Fbdm$!-daK!p(>F-7$Fgdm$!-%3M)z&)>F-7$F\\em$!-5f' fJ+#F-7$Faem$!-!3i'QHAF-7$Ffem$!-^9o-2BF-7$F[fm$!-O]K1IAF-7$F`fm$!-t*3 $4c@F-7$Fefm$!-<&)R&G4#F-7$Fjfm$!-et!*ej?F-7$F_gm$!-$**z*[\"3#F-7$Fdgm $!-dV,'=9#F-7$Figm$!-4NRdqAF-7$F^hm$!-$)y\\^TCF-7$Fchm$!--`qxqAF-7$Fhh m$!-QzElJ@F-7$F]im$!-\\x]AC@F-7$Fbim$!-!G@U;7#F-7$Fgim$!-1[sYD@F-7$F\\ jm$!-*eEnw8#F-7$Fajm$!-Sd%fv>#F-7$Ffjm$!-,)>w$GBF-7$F[[n$!-Hfj=KCF-7$F `[n$!-K$*4$*)[#F-7$Fe[n$!-w3OZmCF-7$Fj[n$!-EBT=WCF-7$F_\\n$!-?Jn5+CF-7 $Fd\\n$!-bb&3nN#F-7$Fi\\n$!-K\\phqAF-7$F^]n$!-9UK@$>#F-7$Fc]n$!-RX*yP; #F-7$Fh]n$!-Q+6M[@F-7$Fb^n$!-'))y4\"e@F-7$F\\_n$!-0-m76AF-7$Fa_n$!-()> jyLBF-7$Ff_n$!-OcV(R]#F-7$F[`n$!-L`7XcCF-7$F``n$!-6#F-7$F]cn$!-0 N,,1@F-7$Fbcn$!-a&o%[0@F-7$Fgcn$!-naA)>6#F-7$F\\dn$!-=D2[F@F-7$Fadn$!- v(>@d>#F-7$Ffdn$!-xCl]OBF-7$F[en$!-,qo;(Q#F-7$F`en$!-w-0MHCF-7$Feen$!- 9j$\\vT#F-7$Fjen$!-l[w!eS#F-7$F_fn$!-LV=Z#Q#F-7$Fdfn$!-PM=LfBF-7$Fifn$ !-%*ffj8BF-7$F^gn$!-o3vsoAF-7$Fcgn$!-'*))y7w@F-7$Fhgn$!-Y.$4F4#F-7$F]h n$!-&)Q!\\,1#F-7$Fbhn$!-aD\"*3T?F-7$Fghn$!-F_RrR?F-7$F\\in$!-1R99Y?F-7 $Fain$!-S:!GF1#F-7$Ffin$!-`CcS#4#F-7$F[jn$!-@R5b*>#F-7$F`jn$!-t4B?JBF- 7$Fejn$!-fR.h$G#F-7$Fjjn$!-B&)*poB#F-7$F_[o$!-v;5)f9#F-7$Fd[o$!-fBcmf? F-7$Fi[o$!-D/5Iy>F-7$F^\\o$!-\\;%z:&>F-7$Fc\\o$!-kzuf,@F-7$Fh\\o$!-$Hq V]:#F-7$F]]o$!-?l&4W)>F-7$Fb]o$!-Rkeq[=F-7$Fg]o$!-H#4f'R=F-7$F\\^o$!-' evvi$=F-7$Fa^o$!-&3HU/%=F-7$Ff^o$!-X0(fX&=F-7$F[_o$!-3psZD>F-7$F`_o$!- Yc;(H3#F-7$Fe_o$!-r2D')*)>F-7$Fj_o$!-#eo&p&*=F-7$F_`o$!.hC+_X\"=Fc`o7$ Fe`o$!.Q'Q04W\"GyJ2bYVG>Fc`o7$Fgco$!.aSV6l\">Fc`o7$F\\do$!.\\eJaY!>Fc`o7$F ado$!.Lo)H'G*=Fc`o7$Ffdo$!.g,c\\j%=Fc`o7$F[eo$!.JLSp3!=Fc`o7$F`eo$!.UT \\#>8'zMOd\"Fc`o7$Fjjo $!.fjhjLF`y$!.%4UAZ%H\"Fc`o7$$\"5MLLL$3)R?rLF`y$!.&RK_=u7Fc `o7$$\"5(pppp>k\"zyLF`y$!.D/[$zm7Fc`o7$$\"5gggg5.$zjQ$F`y$!.Yb&yX\"G\" Fc`o7$Fd[p$!.bHS0DL\"Fc`o7$Fi[p$!.zyX$4,8Fc`o7$F^\\p$!.c@2M]<\"Fc`o7$$ \"5UUUU#\\\\267\"Fc`o7$F]]p$!.%ycdN[6Fc`o7$Fb]p$! .72#4t87Fc`o7$$\"5)yyy.g)e$)*\\$F`y$!.#[,uEQ7Fc`o7$Fg]p$!.wBjCNB\"Fc`o 7$$\"5)pppW8;nM]$F`y$!.UA20dA\"Fc`o7$F\\^p$!.5q1Jz@\"Fc`o7$Fa^p$!.Og3]0\"Fc`o7$Fcap$!-TVh%[***Fc`o7$Fhap $!->GmOn%*Fc`o7$F`dt$!-&e8q$\\*)Fc`o7$Fedt$!-ibR9=&)Fc`o7$$\"5ssss(*o= lqOF`y$!-\"\\:[/W)Fc`o7$Fjdt$!-`ouf$Q)Fc`o7$$\"5%[[[)fLp_yOF`y$!-B]rXa $)Fc`o7$F_et$!-%4)G$=O)Fc`o7$Fdet$!-&)3nhL&)Fc`o7$F]bp$!-W@GmD!*Fc`o7$ Fbbp$!.Np%yqI$)Ffbp7$Fhbp$!.NTE;(Ffbp7$Fgcp$!.Aa,ic3(Ffbp7$F\\dp$!.d'f#Gi8(Ffbp7$Fadp$!.\\IkO9S(Ff bp7$Ffdp$!.[o7)o'p(Ffbp7$F[ep$!.t-8#HpuFfbp7$F`ep$!.9q1o\"[sFfbp7$Feep $!.,B[75Y'Ffbp7$Fjep$!.7[vZW$fFfbp7$F_fp$!.*[I*e<9'Ffbp7$Fdfp$!.'G\"pJ >E'Ffbp7$Fifp$!.'\\:!)R.fFfbp7$F^gp$!.3`.R[c&Ffbp7$Fcgp$!._Fz)yQ_Ffbp7 $Fhgp$!.\"**\\2Wy\\Ffbp7$F\\jt$!.Y4kI%3\\Ffbp7$F]hp$!.3zDC-#\\Ffbp7$Fd jt$!.:A%))**p]Ffbp7$Fbhp$!.S)Rqx'H&Ffbp7$Fghp$!.x%R-7:ZFfbp7$F\\ip$!.j x]v&3UFfbp7$$\"5hggg58%Hr1%F`y$!.=wBdb5%Ffbp7$Faip$!.E)\\NCHSFfbp7$$\" 5;:::lsL'>3%F`y$!.h*opR)*RFfbp7$Ffip$!..$oj8VSFfbp7$$\"51111JU))3$4%F` y$!.dRMbv5%Ffbp7$F[jp$!.k[_z)4UFfbp7$$\"5MLLL3Ae]+TF`y$!.]I()RpK%Ffbp7 $F`jp$!.Q3#*4;E%Ffbp7$Fejp$!.F%pPsLTFfbp7$Fjjp$!.=g'pX4SFfbp7$F]\\u$!. m\">0\\0NFfbp7$F_[q$!.e!3&=LA$Ffbp7$Fd[q$!.rRbT?*GFfbp7$Ff^u$!.(*\\#>* *=FFfbp7$F[_u$!.cVfL*)e#Ffbp7$$\"5++++]ZRz\"G%F`y$!.Y=fE:c#Ffbp7$F`_u$ !.Zr(*>Ie#Ffbp7$$\"5_^^^,M(fpH%F`y$!/rs#4\"G%o#F]\\q7$Fi[q$!/XF] \\q7$Fcau$!/uC]_'oq\"F]\\q7$Fc]q$!/3Npele:F]\\q7$$\"544444f*)Q)\\%F`y$ !/;2#4kEk\"F]\\q7$$\"5=====ygQ8XF`y$!/CdObzo:F]\\q7$$\"5GFFFF(>$QGXF`y $!/!=HVqfY\"F]\\q7$Fh]q$!/V(oy%*)p8F]\\q7$Fcbu$!/;N@Z#pF\"F]\\q7$Fhbu$ !/Wh%4%>27F]\\q7$F]cu$!/[K1bX17F]\\q7$F]^q$!/z5:FrF7F]\\q7$Fecu$!/u6yp =x5F]\\q7$Fb^q$!.fZBRM\\*F]\\q7$F]du$!.#Hzf)G/*F]\\q7$Fbdu$!.F'pN=U$*F ]\\q7$Fgdu$!.M'yFQ***)F]\\q7$Fg^q$!.eZ9e:L)F]\\q7$F\\_q$!.#oYs1UnF]\\q 7$Fa_q$!/M**R#Gco&Fe_q7$Fg_q$!/D%G7@M*\\Fe_q7$F\\`q$!/7**3$>uq$Fe_q7$F a`q$!/+%HR!*)oKFe_q7$Ff`q$!/`Yl82-EFe_q7$F[aq$!/!yW?te5#Fe_q7$F`aq$!/& \\?3!f&y\"Fe_q7$Feaq$!0DeoN6jM\"Fiaq7$F[bq$!0f;k*HRA6Fiaq7$F`bq$!/L&R$ )*>=!*Fiaq7$Febq$!/(e'y)*3XpFiaq7$Fjbq$!/]!Hwh_)eFiaq7$F_cq$!/,zVpYiUF iaq7$Fdcq$!/UQzw(fV$Fiaq7$Ficq$!0!R4)Q\"=2GF]dq7$F_dq$!0&f5.E(R1#F]dq7 $Fddq$!0$p![!=/z;F]dq7$Fidq$!0GC#ep!*)>\"F]dq7$F^eq$!/\\kD%*e$y*F]dq7$ Fceq$!/Zg%*RC)z(F]dq-Fheq6&FjeqF^fqF[fqF(-Fafq6#%Hscheme~with~c[5]=c[6 ]=3/4~and~b[5]=b[6]G-%%FONTG6$%*HELVETICAG\"\"*-%+AXESLABELSG6$Q\"x6\" Q!F[bz-%&TITLEG6#%Uerror~curves~for~7~stage~order~6~Runge-Kutta~method sG-%%VIEWG6$;F(Fceq%(DEFAULTG" 1 2 0 1 10 0 2 9 1 4 2 1.000000 45.000000 45.000000 0 1 "scheme with simple nodes" "scheme with a rela tively large stability region" "scheme with c[5]=c[6]=3/4 and b[5]=b[6 ]" }}}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 536 "evalf[20](plot(['hn_RK6_1'(x)/h(x)-1,'hn_RK6_2'(x)/h (x)-1,'hn_RK6_3'(x)/h(x)-1,'hn_RK6_4'(x)/h(x)-1,\n'hn_RK6_5'(x)/h(x)-1 ],x=0..10,font=[HELVETICA,9],\ncolor=[COLOR(RGB,.95,.2,0),COLOR(RGB,.4 5,0,.95),COLOR(RGB,0,.25,1),COLOR(RGB,0,.75,.2),COLOR(RGB,.95,.45,0)], \nlegend=[`Butcher's scheme A`,`scheme with simple nodes`,`scheme with a relatively large stability region`,`Butcher's scheme B with c[5]=c[ 6]=1/2 and b[5]=b[6]`,`scheme with c[5]=c[6]=3/4 and b[5]=b[6]`],title =`relative error curves for 7 stage order 6 Runge-Kutta methods`));" } }{PARA 13 "" 1 "" {GLPLOT2D 850 579 579 {PLOTDATA 2 "6+-%'CURVESG6%7Y7 $$\"\"!F)F(7$$\"5mmmmm;arz@!#?$!+*o\"*y.$F-7$$\"5LLLL$e9ui2%F-$!,5li!= 6F-7$$\"5mmmmm\"z_\"4iF-$!,E5B7N#F-7$$\"5lmmmmT&phN)F-$!,u!z*=v$F-7$$ \"5LLLLe*=)H\\5!#>$!,W3DHh%F-7$$\"5mmmm\"z/3uC\"FB$!,)p!zz*HF-7$$\"5++ ++DJ$RDX\"FB$\"+D#GY%fFB7$$\"5mmmm\"zR'ok;FB$\",G.6%yLFB7$$\"5++++D1J: w=FB$\"-r:(zZ2\"FB7$$\"5LLLLL3En$4#FB$\"-q(zy5C$FB7$$\"5mmmm;/RE&G#FB$ \"-_v1PLiFB7$$\"5+++++D.&4]#FB$\".%zOfMt:FB7$$\"5+++++vB__#FB7$$\"5** *****\\7o7Tv$FB$\"/[rp#=(4SFB7$$\"5LLLLL$Q*o]RFB$\"/yq1_(\\?'FB7$$\"5* ******\\7=lj;%FB$\"/h))*zp&*\\*FB7$$\"5*******\\PaRKC9FB 7$$\"5LLLL$e9Ege%FB$\"0<]#y.g\"=#FB7$$\"5LLLLeR\"3Gy%FB$\"0R6+T>x3$FB7 $$\"5mmmm;/T1&*\\FB$\"0,*y'p*\\;ZFB7$$\"5mmmm\"zRQb@&FB$\"0H3#3hM4pFB7 $$\"5*******\\(=>Y2aFB$\"0&fA*G4k]*FB7$$\"5mmmm;zXu9cFB$\"1)ozc#Hs$H\" FB7$$\"5**********\\y))GeFB$\"1&Q%G4c\"Gu\"FB7$$\"5********\\i_QQgFB$ \"1rm\\jk$eK#FB7$$\"5*******\\7y%3TiFB$\"1_\"e?Fuj2$FB7$$\"5********\\ P![hY'FB$\"1fs3&G!efSFB7$$\"5KLLLL$Qx$omFB$\"1sdg$*G>\"H&FB7$$\"5***** ****\\P+V)oFB$\"1c@%e6;!HqFB7$$\"5mmmm\"zpe*zqFB$\"1\\Jko%4%3*)FB7$$\" 5*********\\#\\'QH(FB$\"2+)\\hN%Qc>\"FB7$$\"5KLLLe9S8&\\(FB$\"2m%z]Jq[ B:FB7$$\"5*******\\i?=bq(FB$\"2*f(otu![$)>FB7$$\"5KLLLL3s?6zFB$\"2ttF! H`EyCFB7$$\"5*******\\7`Wl7)FB$\"20u9WDH33$FB7$$\"5lmmmmm*RRL)FB$\"2Y+ )y/rf6QFB7$$\"5lmmm;a<.Y&)FB$\"2Zlnku:`p%FB7$$\"5KLLLe9tOc()FB$\"2%o$p p;AJw&FB7$$\"5**********\\Qk\\*)FB$\"2)zQY&pR>.(FB7$$\"5KLLL$3dg6<*FB$ \"2Lm'*zKz0h)FB7$$\"5++++voTAq#*FB$\"2)R2d(f6fZ*FB7$$\"5lmmmmmxGp$*FB$ \"3E)*y#yh`>/\"FB7$$\"5ILL$eRA5\\Z*FB$\"3v>`j[/o]6FB7$$\"5)******\\7oK 0e*FB$\"3Q(pwKH>LF\"FB7$$\"5+++++]oi\"o*FB$\"3rF)Gu#o()*R\"FB7$$\"5)** ****\\(=5s#y*FB$\"3TJ'o`e$)z`\"FB7$$\"5+++D1k2/P)*FB$\"3@*ehU=$)*\\;FB 7$$\"5+++]P40O\"*)*FB$\"3n$HVt8Mis\"FB7$$\"5+++voa-oX**FB$\"3v+1O)opo! =FB7$$\"#5F)$\"3W0'GFE%Qw>FB-%&COLORG6&%$RGBG$\"#&*!\"#$\"\"#!\"\"F(-% 'LEGENDG6#%3Butcher's~scheme~AG-F$6%7[oF'7$F+$\"))pk<#FB7$F1$\")[dqqFB 7$F6$\"*qtlA\"FB7$F;$\"*[Z26\"FB7$F@$!+l%*)[2\"F-7$FF$!+#)fS9zF-7$FK$! ,p2;iP#F-7$FP$!,,Ay&zbF-7$FU$!-&)=Npg6F-7$FZ$!-gYW1MCF-7$Fin$!-x[bx%z$ F-7$F^o$!-ud_$Q8(F-7$Fco$!.=rLx#F-7$Fbu$!0wvdkGRo$F-7$Fgu$!0DxszJ.*\\F-7$F\\v$!0G.nsA`_'F -7$Fav$!08tc_!RK\"*F-7$Ffv$!1g0&*f2a57F-7$F[w$!1ldox&eWm\"F-7$F`w$!1:h 3)oWF;#F-7$Few$!1\"F-7$F]z$!2:N]o[Y+M\"F-7$Fbz$!2\"Rve#>f%4:F-7$Fgz$ !2I#4cw=(**o\"F-7$F\\[l$!2;([#G;l/*=F-7$Fa[l$!2kX>\"H?Xt?F-7$Ff[l$!2(3 !>:K@*o@F-7$$\"5+]i!RvZ]\")*)*FB$!2>kn-ON(pAF-7$$\"5++DJqX/%\\!**FB$!2 kvH.\"='*4BF-7$$\"5+](=nQTI<\"**FB$!2gyzByh*4BF-7$$\"5++]7.#Q?&=**FB$! 2!REd27'*4BF-7$$\"5++v$f$=.5K**FB$!2g1-#*=V*4BF-7$F[\\l$!2l\"*yx$H*)4B F-7$$\"5++Dc,\">g#f**FB$!2gZT>9X2J#F-7$$\"5++]PMF,%G(**FB$!2qaWQ'p\")> BF-7$$\"5++v=nj+U')**FB$!2w([Q/z#)pBF-7$F`\\l$!2(H/oX.!3d#F--Fe\\l6&Fg \\l$\"#XFj\\lF(Fh\\l-F_]l6#%9scheme~with~simple~nodesG-F$6%7WF'7$F+$\" ))yW7%FB7$F1$\"*f[GI\"FB7$F6$\"*,XcH#FB7$F;$\"*E[WY#FB7$F@$!)_r86F-7$F F$!+#\\.w6)F-7$FK$!,!zfNOEF-7$FP$!,$4u.AhF-7$FU$!-L.;u;7F-7$FZ$!-W#[p< M#F-7$Fin$!-/*3;tP$F-7$F^o$!-W$HUbP&F-7$Fco$!-)p0gS)oF-7$Fho$!-!QKcT!z F-7$F]p$!-+%>8!>wF-7$Fbp$!-mgc$*>ZF-7$Fgp$\",?wcLt#FB7$F\\q$\"-&)=m2v< FB7$Faq$\"->Zb]&R%FB7$Ffq$\"-)H55u%))FB7$F[r$\".!=:0l!e\"FB7$F`r$\".r) 3gupFFB7$Fer$\".=GvlAC%FB7$Fjr$\".kl3w<'pFB7$F_s$\"/Ef_M!e1\"FB7$Fds$ \"/:;(RrD^\"FB7$Fis$\"/i'o!4>0@FB7$F^t$\"/y'>g'\\!)GFB7$Fct$\"/j6\"y1E )QFB7$Fht$\"/q\"QQw@;&FB7$F]u$\"/Vs7%=V$oFB7$Fbu$\"/Gvi#o!*)))FB7$Fgu$ \"0C$*HT(fx6FB7$F\\v$\"0p]8\"*R!z9FB7$Fav$\"0'p%G?rx&>FB7$Ffv$\"0Gm.v# fdCFB7$F[w$\"0tDiLJ\\7$FB7$F`w$\"0RWf'*F B7$F^y$\"1EgBM>d^6FB7$Fcy$\"1e\"[dW\"yZ7FB7$Fhy$\"1IT;7GD]8FB7$F]z$\"1 9P)H-2%o9FB7$Fbz$\"1mG01Wj)f\"FB7$Fgz$\"1]H=\"flss\"FB7$F\\[l$\"1YRST' *)Q'=FB7$Ff[l$\"1!zknv!\\W?FB7$F`\\l$\"1d%\\j/?HD#FB-Fe\\l6&Fg\\lF($\" #DFj\\l$\"\"\"F)-F_]l6#%Pscheme~with~a~relatively~large~stability~regi onG-F$6%7YF'7$F+$!*Vu6r(F-7$F1$!+)3O\")*RF-7$F6$!,A8(>Q6F-7$F;$!,@P)=u DF-7$F@$!,.erU$\\F-7$FF$!,t30rv(F-7$FK$!,L>B^_)F-7$FP$\"*_3DV&FB7$FU$ \",'*oJc*RFB7$FZ$\"-Fk><`=FB7$Fin$\"-Zsj3%3%FB7$F^o$\".4AC2/<\"FB7$Fco $\".O+G0FM#FB7$Fho$\".Vmy**eS%FB7$F]p$\".1j)ef%)yFB7$Fbp$\"/$o*4J(\\N \"FB7$Fgp$\"/;-0(4sC#FB7$F\\q$\"/p,zdSGOFB7$Faq$\"/.=Y6o%o&FB7$Ffq$\"/ f333$Rz)FB7$F[r$\"0Iy/Cu)H8FB7$F`r$\"0N%G#[cI0#FB7$Fer$\"0KZO:i@#HFB7$ Fjr$\"0\\5WPp2\\%FB7$F_s$\"0s%GsA?5mFB7$Fds$\"0A0S!yvF\"*FB7$Fis$\"11r s)yQhC\"FB7$F^t$\"1n(\\n%*>Mo\"FB7$Fct$\"1pkh()3;_AFB7$Fht$\"1n,!\\:Ib )HFB7$F]u$\"1\"QAJl8x%RFB7$Fbu$\"1W[U/b&HJq6FB7$Ffv$\"2c$*RN4AI\\\"FB7$F[w$\"2V 7Q2\"*)=Y>FB7$F`w$\"2nc\"f\"*\\.MCFB7$Few$\"2TACsh^&GIFB7$Fjw$\"2mg:&> 1,]PFB7$F_x$\"2d![5%)o(Hi%FB7$Fdx$\"2JNd&=(e$ycFB7$Fix$\"2&3Q-VT-LpFB7 $F^y$\"2#ffmqO\"[\\)FB7$Fcy$\"2Ohx*=/<^$*FB7$Fhy$\"3w;7fW'>&G5FB7$F]z$ \"3d(3G)4T;O6FB7$Fbz$\"3_z7#)>ogd7FB7$Fgz$\"3xb#>))=_HQ\"FB7$F\\[l$\"3 Fj%=S]g(>:FB7$Fa[l$\"3%*Q*e!4KrI;FB7$Ff[l$\"3?Wa,hyF1FB-Fe\\l6&Fg\\lF($\"#vFj\\lF[]l-F_]l6#%T Butcher's~scheme~B~with~c[5]=c[6]=1/2~and~b[5]=b[6]G-F$6%7YF'7$F+$\"(c :]%FB7$F1$!)GX9OF-7$F6$!*CBw*[F-7$F;$!+_qvO=F-7$F@$!+'QX#R]F-7$FF$!,)) R)\\o6F-7$FK$!,ObR![CF-7$FP$!,-O>W\"[F-7$FU$!,W@d(Q\"*F-7$FZ$!-NL&>w%= F-7$Fin$!-3a$Hw*GF-7$F^o$!-!plF#>dF-7$Fco$!-'3H:,J*F-7$Fho$!.PJW%3#\\ \"F-7$F]p$!.r**4\"=eBF-7$Fbp$!.;Z>h/o$F-7$Fgp$!.rJ%3!>n&F-7$F\\q$!.\"G Ga-c')F-7$Faq$!/`g#z4EI\"F-7$Ffq$!/7!QbFL&>F-7$F[r$!/$)e;gi%*GF-7$F`r$ !/)3\">Ys.WF-7$Fer$!/&HZu[dC'F-7$Fjr$!/>1`YqF'*F-7$F_s$!0]/V'>pK9F-7$F ds$!0$4&Hkmg*>F-7$Fis$!0wk1/Zfv#F-7$F^t$!04Q*4%>Ax$F-7$Fct$!0-2OtR17&F -7$Fht$!0H&pJPm&*oF-7$F]u$!0C)o()zUd#*F-7$Fbu$!1\"z-vkO'H7F-7$Fgu$!17z E*zbWm\"F-7$F\\v$!13T(zx.5:#F-7$Fav$!1Abvl%yy&HF-7$Ffv$!1W&R_JP([QF-7$ F[w$!1S38z8hV^F-7$F`w$!1piYtD.`lF-7$Few$!1V:!ow`[I)F-7$Fjw$!2I#4_Y^@Z5 F-7$F_x$!2K=ibw8VJ\"F-7$Fdx$!2C**[D]&yU;F-7$Fix$!2LZZjT&oT?F-7$F^y$!2t WgM]*)=a#F-7$Fcy$!2G(3p=>=AGF-7$Fhy$!2))ec@Ng/8$F-7$F]z$!28vz/Fla[$F-7 $Fbz$!2k=EA_W-s\\T&F-7$F[\\l$!2@Vj9yE6r&F-7$F`\\l$!2A'\\\" p4(z'H'F--Fe\\l6&Fg\\lFh\\lFajlF(-F_]l6#%Hscheme~with~c[5]=c[6]=3/4~an d~b[5]=b[6]G-%%FONTG6$%*HELVETICAG\"\"*-%+AXESLABELSG6$Q\"x6\"Q!F][o-% &TITLEG6#%hnrelative~error~curves~for~7~stage~order~6~Runge-Kutta~meth odsG-%%VIEWG6$;F(F`\\l%(DEFAULTG" 1 2 0 1 10 0 2 9 1 4 2 1.000000 45.000000 45.000000 0 1 "Butcher's scheme A" "scheme with simple nodes " "scheme with a relatively large stability region" "Butcher's scheme \+ B with c[5]=c[6]=1/2 and b[5]=b[6]" "scheme with c[5]=c[6]=3/4 and b[5 ]=b[6]" }}}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 378 "evalf[20](plot(['hn_RK6_2'(x)/h(x)-1,'hn_RK6_3'(x)/h (x)-1,'hn_RK6_5'(x)/h(x)-1],x=0..10,font=[HELVETICA,9],\ncolor=[COLOR( RGB,.45,0,.95),COLOR(RGB,0,.25,1),COLOR(RGB,.95,.45,0)],\nlegend=[`sch eme with simple nodes`,`scheme with a relatively large stability regio n`,`scheme with c[5]=c[6]=3/4 and b[5]=b[6]`],title=`relative error cu rves for 7 stage order 6 Runge-Kutta methods`));" }}{PARA 13 "" 1 "" {GLPLOT2D 949 486 486 {PLOTDATA 2 "6)-%'CURVESG6%7[o7$$\"\"!F)F(7$$\"5 mmmmm;arz@!#?$\"))pk<#!#>7$$\"5LLLL$e9ui2%F-$\")[dqqF07$$\"5mmmmm\"z_ \"4iF-$\"*qtlA\"F07$$\"5lmmmmT&phN)F-$\"*[Z26\"F07$$\"5LLLLe*=)H\\5F0$ !+l%*)[2\"F-7$$\"5mmmm\"z/3uC\"F0$!+#)fS9zF-7$$\"5++++DJ$RDX\"F0$!,p2; iP#F-7$$\"5mmmm\"zR'ok;F0$!,,Ay&zbF-7$$\"5++++D1J:w=F0$!-&)=Npg6F-7$$ \"5LLLLL3En$4#F0$!-gYW1MCF-7$$\"5mmmm;/RE&G#F0$!-x[bx%z$F-7$$\"5+++++D .&4]#F0$!-ud_$Q8(F-7$$\"5+++++vB_Y2aF0$!/:D9CTJnF-7$$\"5mmmm ;zXu9cF0$!/\"\\.)e!=\"*)F-7$$\"5**********\\y))GeF0$!0'pT4e-#=\"F-7$$ \"5********\\i_QQgF0$!0uI(=Jyp:F-7$$\"5*******\\7y%3TiF0$!0&Hf-e4'3#F- 7$$\"5********\\P![hY'F0$!08C>=rLx#F-7$$\"5KLLLL$Qx$omF0$!0wvdkGRo$F-7 $$\"5*********\\P+V)oF0$!0DxszJ.*\\F-7$$\"5mmmm\"zpe*zqF0$!0G.nsA`_'F- 7$$\"5*********\\#\\'QH(F0$!08tc_!RK\"*F-7$$\"5KLLLe9S8&\\(F0$!1g0&*f2 a57F-7$$\"5*******\\i?=bq(F0$!1ldox&eWm\"F-7$$\"5KLLLL3s?6zF0$!1:h3)oW F;#F-7$$\"5*******\\7`Wl7)F0$!1\" F-7$$\"5ILL$eRA5\\Z*F0$!2:N]o[Y+M\"F-7$$\"5)******\\7oK0e*F0$!2\"Rve#> f%4:F-7$$\"5+++++]oi\"o*F0$!2I#4cw=(**o\"F-7$$\"5)******\\(=5s#y*F0$!2 ;([#G;l/*=F-7$$\"5+++D1k2/P)*F0$!2kX>\"H?Xt?F-7$$\"5+++]P40O\"*)*F0$!2 (3!>:K@*o@F-7$$\"5+]i!RvZ]\")*)*F0$!2>kn-ON(pAF-7$$\"5++DJqX/%\\!**F0$ !2kvH.\"='*4BF-7$$\"5+](=nQTI<\"**F0$!2gyzByh*4BF-7$$\"5++]7.#Q?&=**F0 $!2!REd27'*4BF-7$$\"5++v$f$=.5K**F0$!2g1-#*=V*4BF-7$$\"5+++voa-oX**F0$ !2l\"*yx$H*)4BF-7$$\"5++Dc,\">g#f**F0$!2gZT>9X2J#F-7$$\"5++]PMF,%G(**F 0$!2qaWQ'p\")>BF-7$$\"5++v=nj+U')**F0$!2w([Q/z#)pBF-7$$\"#5F)$!2(H/oX. !3d#F--%&COLORG6&%$RGBG$\"#X!\"#F($\"#&*Fb_l-%'LEGENDG6#%9scheme~with~ simple~nodesG-F$6%7WF'7$F+$\"))yW7%F07$F2$\"*f[GI\"F07$F7$\"*,XcH#F07$ F<$\"*E[WY#F07$FA$!)_r86F-7$FF$!+#\\.w6)F-7$FK$!,!zfNOEF-7$FP$!,$4u.Ah F-7$FU$!-L.;u;7F-7$FZ$!-W#[p8!>wF-7$Fbp$!-mgc$*>ZF-7$Fgp $\",?wcLt#F07$F\\q$\"-&)=m2vZb]&R%F07$Ffq$\"-)H55u%))F07$ F[r$\".!=:0l!e\"F07$F`r$\".r)3gupFF07$Fer$\".=GvlAC%F07$Fjr$\".kl3w<'p F07$F_s$\"/Ef_M!e1\"F07$Fds$\"/:;(RrD^\"F07$Fis$\"/i'o!4>0@F07$F^t$\"/ y'>g'\\!)GF07$Fct$\"/j6\"y1E)QF07$Fht$\"/q\"QQw@;&F07$F]u$\"/Vs7%=V$oF 07$Fbu$\"/Gvi#o!*)))F07$Fgu$\"0C$*HT(fx6F07$F\\v$\"0p]8\"*R!z9F07$Fav$ \"0'p%G?rx&>F07$Ffv$\"0Gm.v#fdCF07$F[w$\"0tDiLJ\\7$F07$F`w$\"0RWf'*F07$F^y$\"1EgBM>d^6F07$Fcy$\"1e\"[dW\"yZ7F0 7$Fhy$\"1IT;7GD]8F07$F]z$\"19P)H-2%o9F07$Fbz$\"1mG01Wj)f\"F07$Fgz$\"1] H=\"flss\"F07$F\\[l$\"1YRST'*)Q'=F07$Ff[l$\"1!zknv!\\W?F07$Fh^l$\"1d% \\j/?HD#F0-F]_l6&F__lF($\"#DFb_l$\"\"\"F)-Ff_l6#%Pscheme~with~a~relati vely~large~stability~regionG-F$6%7YF'7$F+$\"(c:]%F07$F2$!)GX9OF-7$F7$! *CBw*[F-7$F<$!+_qvO=F-7$FA$!+'QX#R]F-7$FF$!,))R)\\o6F-7$FK$!,ObR![CF-7 $FP$!,-O>W\"[F-7$FU$!,W@d(Q\"*F-7$FZ$!-NL&>w%=F-7$Fin$!-3a$Hw*GF-7$F^o $!-!plF#>dF-7$Fco$!-'3H:,J*F-7$Fho$!.PJW%3#\\\"F-7$F]p$!.r**4\"=eBF-7$ Fbp$!.;Z>h/o$F-7$Fgp$!.rJ%3!>n&F-7$F\\q$!.\"GGa-c')F-7$Faq$!/`g#z4EI\" F-7$Ffq$!/7!QbFL&>F-7$F[r$!/$)e;gi%*GF-7$F`r$!/)3\">Ys.WF-7$Fer$!/&HZu [dC'F-7$Fjr$!/>1`YqF'*F-7$F_s$!0]/V'>pK9F-7$Fds$!0$4&Hkmg*>F-7$Fis$!0w k1/Zfv#F-7$F^t$!04Q*4%>Ax$F-7$Fct$!0-2OtR17&F-7$Fht$!0H&pJPm&*oF-7$F]u $!0C)o()zUd#*F-7$Fbu$!1\"z-vkO'H7F-7$Fgu$!17zE*zbWm\"F-7$F\\v$!13T(zx. 5:#F-7$Fav$!1Abvl%yy&HF-7$Ffv$!1W&R_JP([QF-7$F[w$!1S38z8hV^F-7$F`w$!1p iYtD.`lF-7$Few$!1V:!ow`[I)F-7$Fjw$!2I#4_Y^@Z5F-7$F_x$!2K=ibw8VJ\"F-7$F dx$!2C**[D]&yU;F-7$Fix$!2LZZjT&oT?F-7$F^y$!2tWgM]*)=a#F-7$Fcy$!2G(3p=> =AGF-7$Fhy$!2))ec@Ng/8$F-7$F]z$!28vz/Fla[$F-7$Fbz$!2k=EA_W -s\\T&F-7$Fd]l$!2@Vj9yE6r&F-7$Fh^l$!2A'\\\"p4(z'H'F--F]_l6&F__lFc_lF`_ lF(-Ff_l6#%Hscheme~with~c[5]=c[6]=3/4~and~b[5]=b[6]G-%%FONTG6$%*HELVET ICAG\"\"*-%+AXESLABELSG6$Q\"x6\"Q!Fdem-%&TITLEG6#%hnrelative~error~cur ves~for~7~stage~order~6~Runge-Kutta~methodsG-%%VIEWG6$;F(Fh^l%(DEFAULT G" 1 2 0 1 10 0 2 9 1 4 2 1.000000 45.000000 45.000000 0 1 "scheme wit h simple nodes" "scheme with a relatively large stability region" "sch eme with c[5]=c[6]=3/4 and b[5]=b[6]" }}}}{PARA 0 "" 0 "" {TEXT -1 0 " " }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}}{SECT 1 {PARA 4 " " 0 "" {TEXT -1 46 "Test 4 of 7 stage, order 6 Runge-Kutta methods" }} {PARA 0 "" 0 "" {TEXT -1 81 "F. G. Lether: 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 "de := 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):\nplot(k(x),x=-1..1,font=[HEL VETICA,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#>%#icG/-%\"yG6#!\"\"#\"\"\"\"\") " }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/-%\"yG6#%\"xG)\"\"#,&\"#8\"\"\"*& \"#;F,)F'F)F,!\"\"" }}{PARA 13 "" 1 "" {GLPLOT2D 577 286 286 {PLOTDATA 2 "6&-%'CURVESG6$7eo7$$!\"\"\"\"!$\"3+++++++]7!#=7$$!3ommm;p 0k&*F-$\"3!Hg[[\"[$)=KF-7$$!3wKL$3$3(F-7$$!3mmmmT%p \"e()F-$\"3!=E-TWD`l\"!#<7$$!3:mmm\"4m(G$)F-$\"3M\"fONp()[t$F=7$$!3\"Q LL3i.9!zF-$\"3A!e'4A%*Rg!)F=7$$!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$$!3 hmmm\">s%HaF-$\"3dBW_P%Gb6$Fgn7$$!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$$!3c****\\P'psm\"F-$\"38i9x*[!p=gFap7$$!3 s*****\\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$$!3KMLL3s$QM%Fft$\"3<%pUH&HNA!)Fap 7$$!3'ym;aQdDG$Fft$\"3%eWuwq(o%4)Fap7$$!3T,+]ivF@AFft$\"3[vW[$G&HZ\")F ap7$$!3=o;/^wj!p\"Fft$\"3]3j^OK2m\")Fap7$$!3'\\L$eRx**f6Fft$\"355#oX4% yz\")Fap7$$!3S<+D\"GyNH'!#?$\"3#QE)R9AS)=)Fap7$$!3]^omm;zr)*!#@$\"3;#) *eHY6>>)Fap7$$\"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@nl+<)Fap7$$\"3@CL$3-Dg5#F ft$\"3kJX?)*G!=:)Fap7$$\"3e?Le*['R3KFft$\"3E'yGoI5!*4)Fap7$$\"3%pJL$ez w5VFft$\"3U_-I6(**[-)Fap7$$\"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$$\"3I*******pvxl\"F-$\"3S%z:5s)zRgFap7$$\"3g)***\\7JFn=F-$\"3 1))p(30[[c&Fap7$$\"3#z****\\_qn2#F-$\"3ae5F\"zuv2&Fap7$$\"3=)**\\P/q%z AF-$\"3ZUhzOe!Rg%Fap7$$\"3U)***\\i&p@[#F-$\"3r&f%4uLbOTFap7$$\"3L)**\\ (=GB2FF-$\"3WV]5@%**Rj$Fap7$$\"3B)****\\2'HKHF-$\"3ul]=$GLo:$Fap7$$\"3 uJL$3UDX8$F-$\"3sKZjodBbFFap7$$\"3ElmmmZvOLF-$\"3!>\\-t_7IQ#Fap7$$\"3i ******\\2goPF-$\"3Q>G9F7l&p\"Fap7$$\"3UKL$eR<*fTF-$\"3?\"Fap 7$$\"3m******\\)Hxe%F-$\"3V-?C_;$p$zFgn7$$\"3ckm;H!o-*\\F-$\"31MiF2c]v ^Fgn7$$\"3y)***\\7k.6aF-$\"3#pB[/J``=$Fgn7$$\"3#emmmT9C#eF-$\"3&*=.D]9 +3>Fgn7$$\"33****\\i!*3`iF-$\"3%HX+j$our5Fgn7$$\"3%QLLL$*zym'F-$\"3!o4 *yfd(\\\"fFM7$$\"3wKLL3N1#4(F-$\"3!\\\\K5**)='4$FM7$$\"3Nmm;HYt7vF-$\" 3%o[)olFVm:FM7$$\"3Y*******p(G**yF-$\"3)3H-pcT.4)F=7$$\"3]mmmT6KU$)F-$ \"35omE\\#[Ck$F=7$$\"3fKLLLbdQ()F-$\"3TxwT%Qu%>ei< " 0 "" {MPLTEXT 1 0 771 "K := (x,y) -> -32*x*y(x)*ln(2): hh := 0.01: numsteps := 200: \+ x0 := -1: y0 := 1/8:\nmatrix([[`slope field: `,K(x,y)],[`initial poi nt: `,``(x0,y0)],[`step width: `,hh],\n[`no. of steps: `,numsteps] ]);``;\nmthds := [`Butcher's scheme A`,`scheme with simple nodes`,`sch eme with a relatively large stability region`,`Butcher's scheme B with `*(c[5]=1/2,c[6]=1/2,b[5]=b[6]),`scheme with `*(c[5]=3/4,c[6]=3/4,b[5 ]=b[6])]:\nerrs := []:\nDigits := 20:\nfor ct to 5 do\n Kn_RK6_||ct \+ := RK6_||ct(evalf(K(x,y)),x,y,x0,evalf(y0),hh,numsteps,false);\n sm \+ := 0: numpts := nops(Kn_RK6_||ct):\n for ii to numpts do\n sm : = sm+(Kn_RK6_||ct[ii,2]-k(Kn_RK6_||ct[ii,1]))^2;\n end do:\n errs \+ := [op(errs),sqrt(sm/numpts)];\nend do:\nDigits := 10:\nlinalg[transpo se]([mthds,evalf(errs)]);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#K%'matrix G6#7&7$%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\"$+#Q)pprint156\"" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%!G" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#K%'matrixG6#7'7$%3Butcher 's~scheme~AG$\"+%*z)fV'!#87$%9scheme~with~simple~nodesG$\"+%)p&>a'!#:7 $%Pscheme~with~a~relatively~large~stability~regionG$\"+/j-2>!#97$*&%9B utcher's~scheme~B~with~G\"\"\"6%/&%\"cG6#\"\"&#F9\"\"#/&F=6#\"\"'F@/&% \"bGF>&FHFDF9$\"+\\+74hF+7$*&%-scheme~with~GF96%/F<#\"\"$\"\"%/FCFQFFF 9$\"+&4))z2&!#;Q)pprint166\"" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }} {PARA 0 "" 0 "" {TEXT -1 30 "The following code constructs " }{TEXT 260 20 "numerical procedures" }{TEXT -1 56 " for solutions based on ea ch of the Runge-Kutta schemes." }}{PARA 0 "" 0 "" {TEXT -1 75 "The err or in the value obtained by each of the methods at the point where " }{XPPEDIT 18 0 "x = 0;" "6#/%\"xG\"\"!" }{TEXT -1 20 ".995 is also gi ven." }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 702 "K := (x,y) -> -32*x *y(x)*ln(2): hh := 0.01: numsteps := 200: x0 := -1: y0 := 1/8:\nmatrix ([[`slope field: `,K(x,y)],[`initial point: `,``(x0,y0)],[`step widt h: `,hh],\n[`no. of steps: `,numsteps]]);``;\nmthds := [`Butcher's scheme A`,`scheme with simple nodes`,`scheme with a relatively large \+ stability region`,`Butcher's scheme B with `*(c[5]=1/2,c[6]=1/2,b[5]=b [6]),`scheme with `*(c[5]=3/4,c[6]=3/4,b[5]=b[6])]:\nerrs := []:\nDigi ts := 20:\nfor ct to 5 do\n kn_RK6_||ct := RK6_||ct(evalf(K(x,y)),x, y,x0,evalf(y0),hh,numsteps,true);\nend do:\nxx := 0.995: kxx := evalf( k(xx)):\nfor ct to 5 do\n errs := [op(errs),abs(kn_RK6_||ct(xx)-kxx) ];\nend do:\nDigits := 10:\nlinalg[transpose]([mthds,evalf(errs)]);" } }{PARA 11 "" 1 "" {XPPMATH 20 "6#K%'matrixG6#7&7$%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\"$+#Q) pprint176\"" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%!G" }}{PARA 11 "" 1 " " {XPPMATH 20 "6#K%'matrixG6#7'7$%3Butcher's~scheme~AG$\"+R`IA()!#=7$% 9scheme~with~simple~nodesG$\"+0/PxL!#>7$%Pscheme~with~a~relatively~lar ge~stability~regionG$\"+@t726F07$*&%9Butcher's~scheme~B~with~G\"\"\"6% /&%\"cG6#\"\"&#F8\"\"#/&F<6#\"\"'F?/&%\"bGF=&FGFCF8$\"+A?:m\")F+7$*&%- scheme~with~GF86%/F;#\"\"$\"\"%/FBFPFEF8$\"+GuDRYF0Q)pprint186\"" }}} {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 "[-1,1]" "6#7$,$\"\"\"!\"\"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 100 equal subintervals. " }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 437 "mthds := [`Butcher's sc heme A`,`scheme with simple nodes`,`scheme with a relatively large sta bility region`,`Butcher's scheme B with `*(c[5]=1/2,c[6]=1/2,b[5]=b[6] ),`scheme with `*(c[5]=3/4,c[6]=3/4,b[5]=b[6])]:\nerrs := []:\nDigits \+ := 20:\nfor ct to 5 do\n sm := NCint((k(x)-'kn_RK6_||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]([mthds,evalf(errs)] );" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#K%'matrixG6#7'7$%3Butcher's~sche me~AG$\"+b!e?X'!#87$%9scheme~with~simple~nodesG$\"+\"R(Gel!#:7$%Pschem e~with~a~relatively~large~stability~regionG$\"+\")yy6>!#97$*&%9Butcher 's~scheme~B~with~G\"\"\"6%/&%\"cG6#\"\"&#F9\"\"#/&F=6#\"\"'F@/&%\"bGF> &FHFDF9$\"+:RPChF+7$*&%-scheme~with~GF96%/F<#\"\"$\"\"%/FCFQFFF9$\"+#f -14&!#;Q)pprint196\"" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 92 "The following error graphs are constructed using the n umerical procedures for the solutions." }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 517 "evalf[20](plot([k(x)-'kn_RK6_1'(x),k(x)-'kn_RK6_2'(x ),k(x)-'kn_RK6_3'(x),k(x)-'kn_RK6_4'(x),\nk(x)-'kn_RK6_5'(x)],x=-1..1, font=[HELVETICA,9],\ncolor=[COLOR(RGB,.95,.2,0),COLOR(RGB,.45,0,.95),C OLOR(RGB,0,.25,1),COLOR(RGB,0,.75,.2),COLOR(RGB,.95,.45,0)],\nlegend=[ `Butcher's scheme A`,`scheme with simple nodes`,`scheme with a relativ ely large stability region`,`Butcher's scheme B with c[5]=c[6]=1/2 and b[5]=b[6]`,`scheme with c[5]=c[6]=3/4 and b[5]=b[6]`],title=`error cu rves for 7 stage order 6 Runge-Kutta methods`));" }}{PARA 13 "" 1 "" {GLPLOT2D 916 651 651 {PLOTDATA 2 "6+-%'CURVESG6%7eo7$$!\"\"\"\"!$F*F* 7$$!5nmmmm;p0k&*!#?$\".s4LyLf\"F/7$$!5LLLL$37$$!5nmmmm\"4m(G$)F/$\".(3g_6d]F<7$$!5LL LL$3i.9!zF/$\"/-\">9q)G7F<7$$!5mmmm;/R=0vF/$\".h\"QRElD!#=7$$!5++++]P8 #\\4(F/$\".c.Z;1@&FL7$$!5mmmm;/siqmF/$\"/HU,(\\F-\"FL7$$!5++++](y$pZiF /$\".U.wWv!>!#<7$$!5LLLLL$yaE\"eF/$\".`?]sTX$Ffn7$$!5mmmmm\">s%HaF/$\" .y_n+Xg&Ffn7$$!5+++++]$*4)*\\F/$\".m\">eeu#*Ffn7$$!5+++++]_&\\c%F/$\"/ zYaeSr9Ffn7$$!5+++++]1aZTF/$\".Gb\"G,0A!#;7$$!5mmmm;/#)[oPF/$\".Qj^Rv2 $F`p7$$!5LLLLL$=exJ$F/$\".*)y[;rQ%F`p7$$!5LLLLLeW%o7$F/$\".Nm9&))G]F`p 7$$!5LLLLLL2$f$HF/$\".&)=-')zr&F`p7$$!5mmmmT&o_Qr#F/$\".38p+>d'F`p7$$! 5********\\PYx\"\\#F/$\".^!)z[5Z(F`p7$$!5mmmmTNz>&H#F/$\".B-RJHH)F`p7$ $!5LLLLLL7i)4#F/$\".**QNBm7*F`p7$$!5mmmmTNa%H)=F/$\"/f.uN&Q+\"F`p7$$!5 ********\\P'psm\"F/$\"/qg75#G4\"F`p7$$!5*********\\F&*=Y\"F/$\"/iiB\"[ N<\"F`p7$$!5********\\74_c7F/$\"/\")=1I][7F`p7$$!5mmmmT5VBU5F/$\"/,GDB g=8F`p7$$!4LLLLL3x%z#)F/$\"/m'RfD&y8F`p7$$!5ILLL$e9d;J'!#@$\"/\\.\"eDJ U\"F`p7$$!4LLLL$3s$QM%F/$\"/^'G_/mX\"F`p7$$!5immmT&QdDG$Fjt$\"/dO=rtp9 F`p7$$!5&*******\\ivF@AFjt$\"/D,6\")Gz9F`p7$$!5imm;/^wj!p\"Fjt$\"/[0)4 (p#[\"F`p7$$!5GLLLeRx**f6Fjt$\"/W+Sk=&[\"F`p7$$!5S*****\\7GyNH'!#A$\"/ J&p(4v'[\"F`p7$$!2mmmmm\"zr)*F/$\"/*RB4)Q([\"F`p7$$\"5]LLL3x\"yY_%Fiv$ \"/KF`k1([\"F`p7$$\"5ILLL3_Nl.5Fjt$\"/I*ocVd[\"F`p7$$\"5DLL$ekGR[b\"Fj t$\"/ojM#*F`p7$$\"5++++vV+ZzAF/$ \".b9<>%f$)F`p7$$\"4++++Dcp@[#F<$\".KA\\-3^(F`p7$$\"5++++v=GB2FF/$\".a (H>>)f'F`p7$$\"4++++]2'HKHF<$\".MH5b;t&F`p7$$\"5NLLL$3UDX8$F/$\".9&[@F -]F`p7$$\"4nmmmmwanL$F<$\".8f!Q;EVF`p7$$\"4+++++v+'oPF<$\".E&*f=u2$F`p 7$$\"4LLLLeR<*fTF<$\".!HTh;!=#F`p7$$\"4+++++&)Hxe%F<$\"/A%z9DwV\"Ffn7$ $\"4nmmm\"H!o-*\\F<$\".l,vp9N*Ffn7$$\"4++++DTO5T&F<$\".>RMNos&Ffn7$$\" 4nmmmmT9C#eF<$\".+t0!y.MFfn7$$\"4++++D1*3`iF<$\".j'fgd()=Ffn7$$\"4LLLL L$*zym'F<$\"/y8:f??5FL7$$\"4LLLL$3N1#4(F<$\".&ex7\">9&FL7$$\"4nmmm\"HY t7vF<$\".,6\"f;YCFL7$$\"4+++++q(G**yF<$\"//8_<.d6F<7$$\"4nmmm;9@BM)F<$ \".If&*)3/XF<7$$\"4LLLLL`v&Q()F<$\".BAVg7m\"F<7$$\"4++++DOl5;*F<$\".%f #pkAO%F/7$$\"4++++v.Uac*F<$\"-nuble@F/7$$\"\"\"F*$!.(\\kTES5F/-%&COLOR G6&%$RGBG$\"#&*!\"#$\"\"#F)F+-%'LEGENDG6#%3Butcher's~scheme~AG-F$6%7eo F'7$F-$\",C\\4\"yDF<7$F>$\",=We?e'F<7$ FC$\"-C%)faD:F<7$FH$\",^)*Qf2$FL7$FN$\",9HC1.'FL7$FS$\"--p@*)\\6FL7$FX $\",/o'4#4#Ffn7$Fhn$\",]7>^q$Ffn7$F]o$\",mA%pAfFfn7$Fbo$\",b&))zb'*Ffn 7$Fgo$\"-&pC(p=:Ffn7$F\\p$\",,5l?E#F`p7$Fbp$\",ye!RWJF`p7$Fgp$\",B\"QM qWF`p7$F\\q$\",E5[-7&F`p7$Faq$\",xLG#=eF`p7$Ffq$\",&R#GNo'F`p7$F[r$\", ['o&[f(F`p7$F`r$\",VfB(G%)F`p7$Fer$\",,:9\\F*F`p7$Fjr$\"-78%*3?5F`p7$F _s$\"-)*HQX56F`p7$Fds$\"-PjpY#>\"F`p7$Fis$\"-8*fJ'o7F`p7$F^t$\"-%4'Q() R8F`p7$Fct$\"-r&z#y+9F`p7$Fht$\"-0?<7Y9F`p7$F^u$\"-8B\"e,[\"F`p7$Fcu$ \"-@Y*4N\\\"F`p7$Fhu$\"-j^3A.:F`p7$F]v$\"-a'*zo1:F`p7$Fbv$\"-e$G=#4:F` p7$Fgv$\"-!4\\43^\"F`p7$F]w$\"-!H6d9^\"F`p7$Fbw$\"-@F.86:F`p7$Fgw$\"-G +Xy4:F`p7$F\\x$\"-ab]U2:F`p7$Fax$\"-u:E0/:F`p7$Ffx$\"-oGxI%\\\"F`p7$F[ y$\"-I5#G1[\"F`p7$F`y$\"-G1PwZ9F`p7$Fey$\"-.\">1TS\"F`p7$Fjy$\"-)[6BgM \"F`p7$F_z$\"-+&=1\"y7F`p7$Fdz$\"-Tv/[*>\"F`p7$Fiz$\"-!)yHJ96F`p7$F^[l $\"-/HUrE5F`p7$Fc[l$\",8!4qo$*F`p7$Fh[l$\",%ofl&\\)F`p7$F]\\l$\",#G(4Y j(F`p7$Fb\\l$\",>!4p4nF`p7$Fg\\l$\",&R1?JeF`p7$F\\]l$\",vGOC4&F`p7$Fa] l$\",@%)>!3WF`p7$Ff]l$\",(3NOWJF`p7$F[^l$\",uf+wB#F`p7$F`^l$\"-.uAH([ \"Ffn7$Fe^l$\",=$z'fy*Ffn7$Fj^l$\",O'442hFfn7$F__l$\",Q5a#=PFfn7$Fd_l$ \",HWg<8#Ffn7$Fi_l$\"-1IU617FL7$F^`l$\",eS5#=lFL7$Fc`l$\",qmdgT$FL7$Fh `l$\"-X@76%F<7$Fgal$\"-(=:*)=\" =F/7$F\\bl$\",w0J'pyF/7$Fabl$\",0#3t+IF/-Ffbl6&Fhbl$\"#XF[clF+Fibl-F_c l6#%9scheme~with~simple~nodesG-F$6%7eoF'7$F-$\",p0USi&F/7$F3$\"-%f%=+C @F/7$F8$\",Rh'G$\"-LXg!fn\"F<7$FC$\"-`s]2$)RF<7$FH$\",N(\\R\"[5 Ffn7$F]o$\"-PoC]*o\"Ffn7$Fbo$\"-q\\+axFFfn7$Fgo$\"-[:Nx*Q%Ffn7$F\\p$\" ,%Q\"z1c'F`p7$Fbp$\",d,)>S\"*F`p7$Fgp$\"-d-2Q,8F`p7$F\\q$\"-RA+A\"\\\" F`p7$Faq$\"-z%[x]p\"F`p7$Ffq$\"-#[5Tx%>F`p7$F[r$\"-7+N\"Q@#F`p7$F`r$\" -N0h8dCF`p7$Fer$\"-G1C+/FF`p7$Fjr$\"-f_f3uHF`p7$F_s$\"-%[;>wB$F`p7$Fds $\"-Ya=xwMF`p7$Fis$\"-VK([))p$F`p7$F^t$\"-63xb1RF`p7$Fct$\"-'f'*GT3%F` p7$Fht$\"-)H&=I;UF`p7$F^u$\"-to@_:VF`p7$Fcu$\"-?-PWaVF`p7$Fhu$\"-BY6v# Q%F`p7$F]v$\"-F`p7$Fg\\ l$\"-\"HW[*)p\"F`p7$F\\]l$\"-p?l;$[\"F`p7$Fa]l$\"-c:A>$G\"F`p7$Ff]l$\" ,J(R!*Q\"*F`p7$F[^l$\",1F%o'['F`p7$F`^l$\"-\"\\wM;H%Ffn7$Fe^l$\"-nN1![ !GFfn7$Fj^l$\"->lK%3t\"Ffn7$F__l$\"-,&y1(Q5Ffn7$Fd_l$\",RGmo$eFfn7$Fi_ l$\"-kjlS7KFL7$F^`l$\"-$oyBYm\"FL7$Fc`l$\",*z?/N#)FL7$Fh`l$\"-W3C&f4%F <7$F]al$\"-qR5#[s\"F<7$Fbal$\",:'=UssF<7$Fgal$\"-)\\#y!Qd#F/7$F\\bl$\" ,7\"QLMvF/7$Fabl$\"+T^TWRF/-Ffbl6&FhblF+$\"#DF[clFabl-F_cl6#%Pscheme~w ith~a~relatively~large~stability~regionG-F$6%7eoF'7$F-$\".\"oXkkM:F/7$ F3$\".a*HS37fF/7$F8$\".$=WI0@=F<7$F>$\".b$)yIn%[F<7$FC$\"/;^]=lv6F<7$F H$\".f8B-5X#FL7$FN$\".-w6=>(\\FL7$FS$\".m4$)e$[(*FL7$FX$\".y6Bmk\"=Ffn 7$Fhn$\".%>y>U'G$Ffn7$F]o$\".e81g\"H`Ffn7$Fbo$\".=<)3_8))Ffn7$Fgo$\"/A #G5;D\"F`p7$Fct$\"/?/J/\\38F`p7$Fht$\"/vusf#3N\"F`p7$F^u$\"/_>)R0EQ \"F`p7$Fcu$\"/LW'Rr]R\"F`p7$Fhu$\"/*zijPTS\"F`p7$F]v$\"/>3'pttS\"F`p7$ Fbv$\"/bMKmt49F`p7$Fgv$\"/$eN!=A69F`p7$F]w$\"/`zol#=T\"F`p7$Fbw$\"/fat 7_69F`p7$Fgw$\"/t)yXl-T\"F`p7$F\\x$\"/+nu:139F`p7$Fax$\"/%)[8U\"\\S\"F `p7$Ffx$\"/ezUf\"eR\"F`p7$F[y$\"/YCSP/$Q\"F`p7$F`y$\"/Kq(=gBN\"F`p7$Fe y$\"/!)p0!)f68F`p7$Fjy$\"/))=r6Od7F`p7$F_z$\"/i<;^$R>\"F`p7$Fdz$\"/gnh z\\?6F`p7$Fiz$\"/y!>/P4/\"F`p7$F^[l$\".X@6j3f*F`p7$Fc[l$\".\"zc:7^()F` p7$Fh[l$\".y/t@[$zF`p7$F]\\l$\".I*=wRHrF`p7$Fb\\l$\".y^-#HjiF`p7$Fg\\l $\".YD'p)3W&F`p7$F\\]l$\".k^pz'[ZF`p7$Fa]l$\".p@)G/2TF`p7$Ff]l$\".]VXd >#HF`p7$F[^l$\".)Gb[Yq?F`p7$F`^l$\"/Ql'zbdO\"Ffn7$Fe^l$\".D^yK!))))Ffn 7$Fj^l$\".`2()3oW&Ffn7$F__l$\".$zRD**RKFfn7$Fd_l$\".L,(Gi)z\"Ffn7$Fi_l $\".*G`,4M(*FL7$F^`l$\".L^A**\\\"\\FL7$Fc`l$\".M.-JNM#FL7$Fh`l$\"/ZBU^ K66F<7$F]al$\".!Q/)R8M%F<7$Fbal$\".fTH3)4;F<7$Fgal$\".([KQKyUF/7$F\\bl $\"-l/&>x`#F/7$Fabl$!-hGpD0)*F/-Ffbl6&FhblF+$\"#vF[clF\\cl-F_cl6#%TBut cher's~scheme~B~with~c[5]=c[6]=1/2~and~b[5]=b[6]G-F$6%7eoF'7$F-$\"+9fG b^F/7$F3$\",%*)4%*y$\",Zj(fX6F<7$FC$\",=Ra/S#F<7 $FH$\"+>\\?ZWFL7$FN$\"+^^rDzFL7$FS$\",zUVJQ\"FL7$FX$\"+Y\">AJ#Ffn7$Fhn $\"+JC1jPFfn7$F]o$\"+uU$Hl&Ffn7$Fbo$\"+\")*eug)Ffn7$Fgo$\",(z<^'H\"Ffn 7$F\\p$\"+$QX(o=F`p7$Fbp$\"+LL/PDF`p7$Fgp$\"+1HYXNF`p7$F\\q$\"+$yR)QSF `p7$Faq$\"+$p&RoXF`p7$Ffq$\"+;e\\D_F`p7$F[r$\"+aLf;fF`p7$F`r$\"+%\\BJb 'F`p7$Fer$\"+h7>+sF`p7$Fjr$\"+U=A5zF`p7$F_s$\"+$[lRg)F`p7$Fds$\"+<$oSB *F`p7$Fis$\"+tC*)>)*F`p7$F^t$\",&Hv%o.\"F`p7$Fct$\",fyyP3\"F`p7$Fht$\" ,1sH(=6F`p7$F^u$\",^'y(\\9\"F`p7$Fcu$\",v;w_:\"F`p7$Fhu$\",_CkF;\"F`p7 $F]v$\",f`Ja;\"F`p7$Fbv$\",n9&Qn6F`p7$Fgv$\",Vj1'o6F`p7$F]w$\",\"QN5p6 F`p7$Fbw$\",\"ef%)o6F`p7$Fgw$\",q,+y;\"F`p7$F\\x$\",Y.vf;\"F`p7$Fax$\" ,d^lL;\"F`p7$Ffx$\",^XJe:\"F`p7$F[y$\",?=f_9\"F`p7$F`y$\",33w)>6F`p7$F ey$\",H-\"='3\"F`p7$Fjy$\",fNy8/\"F`p7$F_z$\"+CJO!*)*F`p7$Fdz$\"+\"*>% \\G*F`p7$Fiz$\"+#pW*H')F`p7$F^[l$\"+ZuPdzF`p7$Fc[l$\"+Wo'*osF`p7$Fh[l$ \"+$fF9g'F`p7$F]\\l$\"+\\ceWfF`p7$Fb\\l$\"+i.%[C&F`p7$Fg\\l$\"+F28wXF` p7$F\\]l$\"+u'oo,%F`p7$Fa]l$\"+S?#**\\$F`p7$Ff]l$\"+#oB^a#F`p7$F[^l$\" +n**\\k=F`p7$F`^l$\",iUv5I\"Ffn7$Fe^l$\"+3DTZ\"*Ffn7$Fj^l$\"+\\GZCjFfn 7$F__l$\"+T%HIM%Ffn7$Fd_l$\"+*HuH*GFfn7$Fi_l$\",uE$>g>FL7$F^`l$\",*\\ \"QJL\"FL7$Fc`l$\"+F1j***)FL7$Fh`l$\",Oqa%4hF<7$F]al$\",;i#f$o$F<7$Fba l$\",WwucL#F<7$Fgal$\"-P+!fGO\"F/7$F\\bl$\",V&)*\\UzF/7$Fabl$\",YVK-Z% F/-Ffbl6&FhblFiblF_amF+-F_cl6#%Hscheme~with~c[5]=c[6]=3/4~and~b[5]=b[6 ]G-%%FONTG6$%*HELVETICAG\"\"*-%+AXESLABELSG6$Q\"x6\"Q!Fa\\p-%&TITLEG6# %Uerror~curves~for~7~stage~order~6~Runge-Kutta~methodsG-%%VIEWG6$;F(Fa bl%(DEFAULTG" 1 2 0 1 10 0 2 9 1 4 2 1.000000 45.000000 45.000000 0 1 "Butcher's scheme A" "scheme with simple nodes" "scheme with a relativ ely large stability region" "Butcher's scheme B with c[5]=c[6]=1/2 and b[5]=b[6]" "scheme with c[5]=c[6]=3/4 and b[5]=b[6]" }}}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 363 "evalf[ 20](plot([k(x)-'kn_RK6_2'(x),k(x)-'kn_RK6_3'(x),k(x)-'kn_RK6_4'(x)],x= -1..1,font=[HELVETICA,9],\ncolor=[COLOR(RGB,.45,0,.95),COLOR(RGB,0,.25 ,1),COLOR(RGB,.95,.45,0)],\nlegend=[`scheme with simple nodes`,`scheme with a relatively large stability region`,`scheme with c[5]=c[6]=3/4 \+ and b[5]=b[6]`],title=`error curves for 7 stage order 6 Runge-Kutta me thods`));" }}{PARA 13 "" 1 "" {GLPLOT2D 851 586 586 {PLOTDATA 2 "6)-%' CURVESG6%7eo7$$!\"\"\"\"!$F*F*7$$!5nmmmm;p0k&*!#?$\",C\\4\"yD!#>7$$!5nmmmm\"4m (G$)F/$\",=We?e'F<7$$!5LLLL$3i.9!zF/$\"-C%)faD:F<7$$!5mmmm;/R=0vF/$\", ^)*Qf2$!#=7$$!5++++]P8#\\4(F/$\",9HC1.'FL7$$!5mmmm;/siqmF/$\"--p@*)\\6 FL7$$!5++++](y$pZiF/$\",/o'4#4#!#<7$$!5LLLLL$yaE\"eF/$\",]7>^q$Ffn7$$! 5mmmmm\">s%HaF/$\",mA%pAfFfn7$$!5+++++]$*4)*\\F/$\",b&))zb'*Ffn7$$!5++ +++]_&\\c%F/$\"-&pC(p=:Ffn7$$!5+++++]1aZTF/$\",,5l?E#!#;7$$!5mmmm;/#)[ oPF/$\",ye!RWJF`p7$$!5LLLLL$=exJ$F/$\",B\"QMqWF`p7$$!5LLLLLeW%o7$F/$\" ,E5[-7&F`p7$$!5LLLLLL2$f$HF/$\",xLG#=eF`p7$$!5mmmmT&o_Qr#F/$\",&R#GNo' F`p7$$!5********\\PYx\"\\#F/$\",['o&[f(F`p7$$!5mmmmTNz>&H#F/$\",VfB(G% )F`p7$$!5LLLLLL7i)4#F/$\",,:9\\F*F`p7$$!5mmmmTNa%H)=F/$\"-78%*3?5F`p7$ $!5********\\P'psm\"F/$\"-)*HQX56F`p7$$!5*********\\F&*=Y\"F/$\"-PjpY# >\"F`p7$$!5********\\74_c7F/$\"-8*fJ'o7F`p7$$!5mmmmT5VBU5F/$\"-%4'Q()R 8F`p7$$!4LLLLL3x%z#)F/$\"-r&z#y+9F`p7$$!5ILLL$e9d;J'!#@$\"-0?<7Y9F`p7$ $!4LLLL$3s$QM%F/$\"-8B\"e,[\"F`p7$$!5immmT&QdDG$Fjt$\"-@Y*4N\\\"F`p7$$ !5&*******\\ivF@AFjt$\"-j^3A.:F`p7$$!5imm;/^wj!p\"Fjt$\"-a'*zo1:F`p7$$ !5GLLLeRx**f6Fjt$\"-e$G=#4:F`p7$$!5S*****\\7GyNH'!#A$\"-!4\\43^\"F`p7$ $!2mmmmm\"zr)*F/$\"-!H6d9^\"F`p7$$\"5]LLL3x\"yY_%Fiv$\"-@F.86:F`p7$$\" 5ILLL3_Nl.5Fjt$\"-G+Xy4:F`p7$$\"5DLL$ekGR[b\"Fjt$\"-ab]U2:F`p7$$\"5?LL L$3-Dg5#Fjt$\"-u:E0/:F`p7$$\"55LLLe*['R3KFjt$\"-oGxI%\\\"F`p7$$\"3LLLL ezw5VF<$\"-I5#G1[\"F`p7$$\"5]mmmmmJ+IiFjt$\"-G1PwZ9F`p7$$\"3++++v$Q#\\ \")F<$\"-.\">1TS\"F`p7$$\"5lmmm\"z\\1A-\"F/$\"-)[6BgM\"F`p7$$\"4LLLL$e \"*[H7F<$\"-+&=1\"y7F`p7$$\"5lmmm;HCjV9F/$\"-Tv/[*>\"F`p7$$\"4+++++qvx l\"F<$\"-!)yHJ96F`p7$$\"5++++]7JFn=F/$\"-/HUrE5F`p7$$\"4++++]_qn2#F<$ \",8!4qo$*F`p7$$\"5++++vV+ZzAF/$\",%ofl&\\)F`p7$$\"4++++Dcp@[#F<$\",#G (4Yj(F`p7$$\"5++++v=GB2FF/$\",>!4p4nF`p7$$\"4++++]2'HKHF<$\",&R1?JeF`p 7$$\"5NLLL$3UDX8$F/$\",vGOC4&F`p7$$\"4nmmmmwanL$F<$\",@%)>!3WF`p7$$\"4 +++++v+'oPF<$\",(3NOWJF`p7$$\"4LLLLeR<*fTF<$\",uf+wB#F`p7$$\"4+++++&)H xe%F<$\"-.uAH([\"Ffn7$$\"4nmmm\"H!o-*\\F<$\",=$z'fy*Ffn7$$\"4++++DTO5T &F<$\",O'442hFfn7$$\"4nmmmmT9C#eF<$\",Q5a#=PFfn7$$\"4++++D1*3`iF<$\",H Wg<8#Ffn7$$\"4LLLLL$*zym'F<$\"-1IU617FL7$$\"4LLLL$3N1#4(F<$\",eS5#=lFL 7$$\"4nmmm\"HYt7vF<$\",qmdgT$FL7$$\"4+++++q(G**yF<$\"-X@76%F<7$$\"4++++DOl5; *F<$\"-(=:*)=\"=F/7$$\"4++++v.Uac*F<$\",w0J'pyF/7$$\"\"\"F*$\",0#3t+IF /-%&COLORG6&%$RGBG$\"#X!\"#F+$\"#&*F[cl-%'LEGENDG6#%9scheme~with~simpl e~nodesG-F$6%7eoF'7$F-$\",p0USi&F/7$F3$\"-%f%=+C@F/7$F8$\",Rh'G$\"-LXg!fn\"F<7$FC$\"-`s]2$)RF<7$FH$\",N(\\R\"[5Ffn7$F]o$\"-PoC]*o\"Ff n7$Fbo$\"-q\\+axFFfn7$Fgo$\"-[:Nx*Q%Ffn7$F\\p$\",%Q\"z1c'F`p7$Fbp$\",d ,)>S\"*F`p7$Fgp$\"-d-2Q,8F`p7$F\\q$\"-RA+A\"\\\"F`p7$Faq$\"-z%[x]p\"F` p7$Ffq$\"-#[5Tx%>F`p7$F[r$\"-7+N\"Q@#F`p7$F`r$\"-N0h8dCF`p7$Fer$\"-G1C +/FF`p7$Fjr$\"-f_f3uHF`p7$F_s$\"-%[;>wB$F`p7$Fds$\"-Ya=xwMF`p7$Fis$\"- VK([))p$F`p7$F^t$\"-63xb1RF`p7$Fct$\"-'f'*GT3%F`p7$Fht$\"-)H&=I;UF`p7$ F^u$\"-to@_:VF`p7$Fcu$\"-?-PWaVF`p7$Fhu$\"-BY6v#Q%F`p7$F]v$\"-F`p7$Fg\\l$\"-\"HW[*)p\"F`p7$F \\]l$\"-p?l;$[\"F`p7$Fa]l$\"-c:A>$G\"F`p7$Ff]l$\",J(R!*Q\"*F`p7$F[^l$ \",1F%o'['F`p7$F`^l$\"-\"\\wM;H%Ffn7$Fe^l$\"-nN1![!GFfn7$Fj^l$\"->lK%3 t\"Ffn7$F__l$\"-,&y1(Q5Ffn7$Fd_l$\",RGmo$eFfn7$Fi_l$\"-kjlS7KFL7$F^`l$ \"-$oyBYm\"FL7$Fc`l$\",*z?/N#)FL7$Fh`l$\"-W3C&f4%F<7$F]al$\"-qR5#[s\"F <7$Fbal$\",:'=UssF<7$Fgal$\"-)\\#y!Qd#F/7$F\\bl$\",7\"QLMvF/7$Fabl$\"+ T^TWRF/-Ffbl6&FhblF+$\"#DF[clFabl-F_cl6#%Pscheme~with~a~relatively~lar ge~stability~regionG-F$6%7eoF'7$F-$\".\"oXkkM:F/7$F3$\".a*HS37fF/7$F8$ \".$=WI0@=F<7$F>$\".b$)yIn%[F<7$FC$\"/;^]=lv6F<7$FH$\".f8B-5X#FL7$FN$ \".-w6=>(\\FL7$FS$\".m4$)e$[(*FL7$FX$\".y6Bmk\"=Ffn7$Fhn$\".%>y>U'G$Ff n7$F]o$\".e81g\"H`Ffn7$Fbo$\".=<)3_8))Ffn7$Fgo$\"/A#G5;D\"F`p7$Fct$ \"/?/J/\\38F`p7$Fht$\"/vusf#3N\"F`p7$F^u$\"/_>)R0EQ\"F`p7$Fcu$\"/LW'Rr ]R\"F`p7$Fhu$\"/*zijPTS\"F`p7$F]v$\"/>3'pttS\"F`p7$Fbv$\"/bMKmt49F`p7$ Fgv$\"/$eN!=A69F`p7$F]w$\"/`zol#=T\"F`p7$Fbw$\"/fat7_69F`p7$Fgw$\"/t)y Xl-T\"F`p7$F\\x$\"/+nu:139F`p7$Fax$\"/%)[8U\"\\S\"F`p7$Ffx$\"/ezUf\"eR \"F`p7$F[y$\"/YCSP/$Q\"F`p7$F`y$\"/Kq(=gBN\"F`p7$Fey$\"/!)p0!)f68F`p7$ Fjy$\"/))=r6Od7F`p7$F_z$\"/i<;^$R>\"F`p7$Fdz$\"/gnhz\\?6F`p7$Fiz$\"/y! >/P4/\"F`p7$F^[l$\".X@6j3f*F`p7$Fc[l$\".\"zc:7^()F`p7$Fh[l$\".y/t@[$zF `p7$F]\\l$\".I*=wRHrF`p7$Fb\\l$\".y^-#HjiF`p7$Fg\\l$\".YD'p)3W&F`p7$F \\]l$\".k^pz'[ZF`p7$Fa]l$\".p@)G/2TF`p7$Ff]l$\".]VXd>#HF`p7$F[^l$\".)G b[Yq?F`p7$F`^l$\"/Ql'zbdO\"Ffn7$Fe^l$\".D^yK!))))Ffn7$Fj^l$\".`2()3oW& Ffn7$F__l$\".$zRD**RKFfn7$Fd_l$\".L,(Gi)z\"Ffn7$Fi_l$\".*G`,4M(*FL7$F^ `l$\".L^A**\\\"\\FL7$Fc`l$\".M.-JNM#FL7$Fh`l$\"/ZBU^K66F<7$F]al$\".!Q/ )R8M%F<7$Fbal$\".fTH3)4;F<7$Fgal$\".([KQKyUF/7$F\\bl$\"-l/&>x`#F/7$Fab l$!-hGpD0)*F/-Ffbl6&FhblF\\clFiblF+-F_cl6#%Hscheme~with~c[5]=c[6]=3/4~ and~b[5]=b[6]G-%%FONTG6$%*HELVETICAG\"\"*-%+AXESLABELSG6$Q\"x6\"Q!F]`n -%&TITLEG6#%Uerror~curves~for~7~stage~order~6~Runge-Kutta~methodsG-%%V IEWG6$;F(Fabl%(DEFAULTG" 1 2 0 1 10 0 2 9 1 4 2 1.000000 45.000000 45.000000 0 1 "scheme with simple nodes" "scheme with a relatively lar ge stability region" "scheme with c[5]=c[6]=3/4 and b[5]=b[6]" }}}} {PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 46 "Test 5 of 7 stage, orde r 6 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 "d e := 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=[HELV ETICA,9],labels=[`x`,`y(x)`]);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%#d eG/-%%diffG6$-%\"yG6#%\"xGF,,$*(\"#;\"\"\",&*&F/F0F,F0F0F0F0!\"\"F)F3F 0" }}{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$$\"3 MLL$eRwX5$F6$\"3!Qb+fY'3W8F27$$\"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\"F 27$$\"3!))**\\7`l2Q*F6$\"3#HUv\"fmC$o\"F27$$\"3smm;/j$o/\"!#=$\"3:'H!f >cuAjU6Fco$\"3K$o8QC!za=F27$$\"3)*****\\P[6j9 Fco$\"39iuo+OIZ=F27$$\"3KL$e*[z(yb\"Fco$\"3Q:]fA\\>F27$$\"3))**\\iSj0 x=Fco$\"3-5Hbh&QF%>F27$$\"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)4US+%ypO?F27$$\"3GL L3_?`(\\#Fco$\"3#4Cj+a0O0#F27$$\"3#HLe*)>pxg#Fco$\"3ab\\mG7Vq?F27$$\"3 u**\\Pf4t.FFco$\"3Cx7m@=^%3#F27$$\"32LLe*Gst!GFco$\"3Q>IFco$\"3&ocGC'[]F@ F27$$\"3h**\\i!RU07$Fco$\"3HCH$Q\")f.9#F27$$\"3b***\\(=S2LKFco$\"3C`wr Wc9a@F27$$\"3Kmmm\"p)=MLFco$\"3;=S,IA7m@F27$$\"3!*****\\(=]@W$Fco$\"3w 4%eC\"p]y@F27$$\"35L$e*[$z*RNFco$\"3UyOr,.R*=#F27$$\"3#*****\\iC$pk$Fc o$\"3wIdFs1%4?#F27$$\"39m;H2qcZPFco$\"3Qbx\"QY%\\6AF27$$\"3q**\\7.\"fF &QFco$\"3f+!e(oz@AAF27$$\"3Ymm;/OgbRFco$\"36qG(yA8CB#F27$$\"3y**\\ilAF jSFco$\"3v.zLgjzUAF27$$\"3YLLL$)*pp;%Fco$\"3IImU*yHDD#F27$$\"3?LL3xe,t UFco$\"3I%R!fhiAiAF27$$\"3em;HdO=yVFco$\"3?ogo1xfrAF27$$\"3))*****\\#> #[Z%Fco$\"3EO%fx0/+G#F27$$\"3immT&G!e&e%Fco$\"3)zsS%e\"3%*G#F27$$\"3;L LL$)Qk%o%Fco$\"35e8d5)>wH#F27$$\"37+]iSjE!z%Fco$\"3e%4h.zwhI#F27$$\"35 +]P40O\"*[Fco$\"3Nwd2,K=9BF27$$\"3++++++++]Fco$\"3m'>())[`fABF2-%'COLO URG6&%$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 code constr ucts a " }{TEXT 260 17 "discrete solution" }{TEXT -1 44 " based on eac h 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 0 754 "S := (x,y) -> 16/((16*x+1)*y): hh := 0.005: numsteps := 100: x0 := 0: y0 := 1:\nmatrix([[`slope field: `,S(x,y)],[`initi al point: `,``(x0,y0)],[`step width: `,hh],\n[`no. of steps: `,num steps]]);``;\nmthds := [`Butcher's scheme A`,`scheme with simple nodes `,`scheme with a relatively large stability region`,`Butcher's scheme \+ B with `*(c[5]=1/2,c[6]=1/2,b[5]=b[6]),`scheme with `*(c[5]=3/4,c[6]=3 /4,b[5]=b[6])]:\nerrs := []:\nDigits := 20:\nfor ct to 5 do\n Sn_RK6 _||ct := RK6_||ct(S(x,y),x,y,x0,y0,hh,numsteps,false);\n sm := 0: nu mpts := nops(Sn_RK6_||ct):\n for ii to numpts do\n sm := sm+(Sn _RK6_||ct[ii,2]-s(Sn_RK6_||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,$*(\"#;\"\"\",&*&F+F,%\"xGF,F,F,F,!\"\"%\"yGF0F,7$%0 initial~point:~G-%!G6$\"\"!F,7$%/step~width:~~~G$\"\"&!\"$7$%1no.~of~s teps:~~~G\"$+\"Q)pprint206\"" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%!G" } }{PARA 11 "" 1 "" {XPPMATH 20 "6#K%'matrixG6#7'7$%3Butcher's~scheme~AG $\"+)>5h5\"!#?7$%9scheme~with~simple~nodesG$\"+$>(Q&f\"!#>7$%Pscheme~w ith~a~relatively~large~stability~regionG$\"+N!*yw,rBF07$*&%-scheme~with~GF86%/F;#\"\"$\"\"%/FBFPFEF8$\"+v2#Q])F +Q)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-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 \+ = 0;" "6#/%\"xG\"\"!" }{TEXT -1 21 ".4995 is also given." }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 686 "S := (x,y) -> 16/((16*x+1)*y): hh \+ := 0.005: numsteps := 100: x0 := 0: y0 := 1:\nmatrix([[`slope field: \+ `,S(x,y)],[`initial point: `,``(x0,y0)],[`step width: `,hh],\n[`no. of steps: `,numsteps]]);``;\nmthds := [`Butcher's scheme A`,`scheme with simple nodes`,`scheme with a relatively large stability region`, `Butcher's scheme B with `*(c[5]=1/2,c[6]=1/2,b[5]=b[6]),`scheme with \+ `*(c[5]=3/4,c[6]=3/4,b[5]=b[6])]:\nerrs := []:\nDigits := 25:\nfor ct \+ to 5 do\n sn_RK6_||ct := RK6_||ct(S(x,y),x,y,x0,y0,hh,numsteps,true) ;\nend do:\nxx := 0.4995: sxx := evalf(s(xx)):\nfor ct to 5 do\n err s := [op(errs),abs(sn_RK6_||ct(xx)-sxx)];\nend do:\nDigits := 10:\nlin alg[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)pprint226\"" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%!G" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#K%'matrixG6#7'7$ %3Butcher's~scheme~AG$\"+s5pv))!#@7$%9scheme~with~simple~nodesG$\"+%)p ==8!#>7$%Pscheme~with~a~relatively~large~stability~regionG$\"+n&))yY\" F07$*&%9Butcher's~scheme~B~with~G\"\"\"6%/&%\"cG6#\"\"&#F8\"\"#/&F<6# \"\"'F?/&%\"bGF=&FGFCF8$\"+v#49'>F07$*&%-scheme~with~GF86%/F;#\"\"$\" \"%/FBFPFEF8$\"+#)o#3.(!#?Q)pprint236\"" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 4 "The " }{TEXT 260 22 "root mean squa re error" }{TEXT -1 110 " over the interval [0, 0.5] of each Runge-K utta method is estimated as follows 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 0 439 "mthds := [`Butcher's scheme A`,`sc heme with simple nodes`,`scheme with a relatively large stability regi on`,`Butcher's scheme B with `*(c[5]=1/2,c[6]=1/2,b[5]=b[6]),`scheme w ith `*(c[5]=3/4,c[6]=3/4,b[5]=b[6])]:\nerrs := []:\nDigits := 20:\nfor ct to 5 do\n sm := NCint((s(x)-'sn_RK6_||ct'(x))^2,x=0..0.5,adaptiv e=false,numpoints=7,factor=50);\n errs := [op(errs),sqrt(sm/0.5)];\n end do:\nDigits := 10:\nlinalg[transpose]([mthds,evalf(errs)]);" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#K%'matrixG6#7'7$%3Butcher's~scheme~AG$ \"+'3zk5\"!#?7$%9scheme~with~simple~nodesG$\"+sW%[f\"!#>7$%Pscheme~wit h~a~relatively~large~stability~regionG$\"+D[=wqBF07$*&%-scheme~with~GF86%/F;#\"\"$\"\"%/FBFPFEF8$\"+g[#4])F+Q) pprint246\"" }}}{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 0 518 "evalf[20](plot(['sn_RK6_1'(x)-s(x),'sn_RK6_2'(x)-s(x),'sn_RK6_3 '(x)-s(x),'sn_RK6_4'(x)-s(x),\n'sn_RK6_5'(x)-s(x)],x=0..0.5,font=[HELV ETICA,9],\ncolor=[COLOR(RGB,.95,.2,0),COLOR(RGB,.45,0,.95),COLOR(RGB,0 ,.25,1),COLOR(RGB,0,.75,.2),COLOR(RGB,.95,.45,0)],\nlegend=[`Butcher's scheme A`,`scheme with simple nodes`,`scheme with a relatively large \+ stability region`,`Butcher's scheme B with c[5]=c[6]=1/2 and b[5]=b[6] `,`scheme with c[5]=c[6]=3/4 and b[5]=b[6]`],title=`error curves for 7 stage order 6 Runge-Kutta methods`));" }}{PARA 13 "" 1 "" {GLPLOT2D 1029 527 527 {PLOTDATA 2 "6+-%'CURVESG6%7`o7$$\"\"!F)F(7$$\"5SLLL3x1h6 o!#B$\"$Y#!#>7$$\"5ommmTN@Ki8!#A$\"&/-$F07$$\"5NLL3FpE!Hq\"F4$\"'P39F0 7$$\"5-++]7.K[V?F4$\"'2I\\F07$$\"5omm\"zptjSQ#F4$\"(SdT\"F07$$\"5NLLL$ 3FWYs#F4$\"(oj^$F07$$\"5-++vo/[AlIF4$\"(zm\"yF07$$\"5omm;aQ`!eS$F4$\") h&=f\"F07$$\"5NLLeRseQYPF4$\")ww>IF07$$\"5-+++D1k'p3%F4$\")5G-aF07$$\" 5pmmT5SpaFWF4$\")=y*>*F07$$\"5OLL$eRZF\"oZF4$\"*\"eA-:F07$$\"50++D\"y+ 3(3^F4$\"**=2^?F07$$\"5qmmmmT&)G\\aF4$\"*rAA/#F07$$F,F4$\"*e,%3?F07$$ \"50+++]7G$R<)F4$\"*HP1)>F07$$\"5qmmm\"z%\\DO&*F4$\"*VVZ+#F07$$\"5MLLL L3x&)*3\"!#@$\"*A'\\O?F07$$\"5+++]i!R(*Rc\"F^q$\"*Gd'f=F07$$\"5nmmm\"H 2P\"Q?F^q$\"*:S'=VB$)F^q$\"*'z,`7F07$$\"5.+++DJbw!Q*F^q$\"*'\\YA7F07$$\"5 nmmm;/j$o/\"!#?$\"*+\"3&>\"F07$$\"5MLLL3_>jU6Fft$\"*SgN<\"F07$$\"5++++ ]i^Z]7Fft$\"*N[>:\"F07$$\"5++++](=h(e8Fft$\"*l1C8\"F07$$\"5++++]P[6j9F ft$\"*t!Q:6F07$$\"5MLL$e*[z(yb\"Fft$\"*lN75\"F07$$\"5nmmm;a/cq;Fft$\"* )**z&3\"F07$$\"5nmmmm;t,mFft$\"*7p,0\"F07$$\"5,++]i!f#=$3#Fft$\"*m\"GR5F07$ $\"5,++](=xpe=#Fft$\"*@*fH5F07$$\"5nmmm\"H28IH#Fft$\"*^K,-\"F07$$\"5nm m;zpSS\"R#Fft$\"*!G&>,\"F07$$\"5MLLL3_?`(\\#Fft$\"*qHO+\"F07$$\"5MLL$e *)>pxg#Fft$\")sza**F07$$\"5,++]Pf4t.FFft$\")if())*F07$$\"5MLLLe*Gst!GF ft$\"):f=)*F07$$\"5,++++DRW9HFft$\")5)3v*F07$$\"5,+++DJE>>IFft$\"))pyo *F07$$\"5,++]i!RU07$Fft$\")NqH'*F07$$\"5,+++v=S2LKFft$\")C4o&*F07$$\"5 ommmm\"p)=MLFft$\")G?:&*F07$$\"5,+++](=]@W$Fft$\")77h%*F07$$\"5MLL$e*[ $z*RNFft$\")()49%*F07$$\"5,+++]iC$pk$Fft$\")Jqk$*F07$$\"5omm;H2qcZPFft $\")(=+K*F07$$\"5,++]7.\"fF&QFft$\"),0v#*F07$$\"5ommm;/OgbRFft$\")fpK# *F07$$\"5,++]ilAFjSFft$\")k&**=*F07$$\"5NLLLL$)*pp;%Fft$\")([-:*F07$$ \"5NLLL3xe,tUFft$\")6.6\"*F07$$\"5omm;HdO=yVFft$\")hWt!*F07$$\"5,++++D >#[Z%Fft$\")V**R!*F07$$\"5ommmT&G!e&e%Fft$\")I'G+*F07$$\"5NLLLL$)Qk%o% Fft$\")&)oq*)F07$$\"5-++]iSjE!z%Fft$\")XSP*)F07$$\"5-++]P40O\"*[Fft$\" )\\[1*)F07$$\"\"&!\"\"$\")!GU())F0-%&COLORG6&%$RGBG$\"#&*!\"#$\"\"#Fe` lF(-%'LEGENDG6#%3Butcher's~scheme~AG-F$6%7^pF'7$F+$\"%KCF07$F2$\"'QkGF 07$F8$\"(\")3J\"F07$F=$\"(_)4XF07$FB$\")FMu7F07$FG$\")\\4=JF07$FL$\")e fNoF07$FQ$\"*q;UP\"F07$FV$\"*>Lfd#F07$Fen$\"*\"GgdXF07$Fjn$\"*B([#o(F0 7$F_o$\"+@VrU7F07$Fdo$\"+$\\Smo\"F07$Fio$\"+pOOz;F07$$\"50++]P4'\\/8'F 4$\"+!**=_m\"F07$F^p$\"+W*e@l\"F07$$\"5qmm;zWF07$$\"5NLL3 x\"[No()*F4$\"+zB\\*3#F07$$\"5+++Dc,;u@5F^q$\"+JE'*p@F07$$\"5nm;z%\\l* zb5F^q$\"+4;Ai@F07$F\\q$\"+P,ga@F07$$\"5++]i!*GER37F^q$\"+(R'fH@F07$$ \"5nmm\"z%\\v#pK\"F^q$\"+G[R=@F07$$\"5ML$3_+ZiaW\"F^q$\"+7rv!>#F07$Fbq $\"+#)4&GH#F07$$\"5nm;z>6B`#o\"F^q$\"+Z8ZpAF07$$\"5MLL3xJs1,=F^q$\"+qg V]AF07$$\"5nm\"Hd?pM.'=F^q$\"+qJ&zC#F07$$\"5++]PM_@g>>F^q$\"+MXbeAF07$ $\"5ML3-j7'p)y>F^q$\"+h;6&H#F07$Fgq$\"+BsE6BF07$F\\r$\"+Nz=xAF07$Far$ \"+Cy>IAF07$Ffr$\"+)3X@8#F07$F[s$\"+59zX?F07$F`s$\"+cmrx>F07$Fes$\"+UP H<>F07$Fjs$\"+jguj=F07$F_t$\"+(**fw\"=F07$Fdt$\"+\")*[jx\"F07$Fjt$\"+y \"eSu\"F07$F_u$\"+?ef6G'[O\"F07$F[^l$\"+k^'*e8F07$F`^l$\"+b.9`8F07$Fe^l$\"+W!evM\"F07$F j^l$\"+a'*eU8F07$F__l$\"+(zuqL\"F07$Fd_l$\"+[hHK8F07$Fi_l$\"+XFNF8F07$ F^`l$\"+50wA8F07$Fc`l$\"+m'pzJ\"F0-Fi`l6&F[al$\"#XF^alF(F\\al-Fbal6#%9 scheme~with~simple~nodesG-F$6%7^pF'7$F+$\"%!p#F07$F2$\"'_rJF07$F8$\"(8 CX\"F07$F=$\"(E-+&F07$FB$\")\\\"RT\"F07$FG$\")Z9iMF07$FL$\")1d&f(F07$F Q$\"*i(=G:F07$FV$\"*bNo'GF07$Fen$\"*(pXw]F07$Fjn$\"*t\"Hk&)F07$F_o$\"+ 8-b'Q\"F07$Fdo$\"+$p')H)=F07$Fio$\"+bF'[(=F07$Fcdl$\"+b62f=F07$F^p$\"+ fT[W=F07$F[el$\"+65FN=F07$Fbp$\"+gHdY=F07$Fcel$\"+bLQ<>F07$Fgp$\"+-e2H @F07$F[fl$\"+fE`IBF07$F`fl$\"+*=Z,U#F07$Fefl$\"+pO^6CF07$F\\q$\"+gM,.C F07$F]gl$\"+W(>^P#F07$Fbgl$\"+Cs]iBF07$Fggl$\"+F&eDW#F07$Fbq$\"+riwbDF 07$F_hl$\"+0VqHDF07$Fdhl$\"+y\"\\%3DF07$Fihl$\"+))**f0DF07$F^il$\"+/6B F07$Fjt$\"+g$R@%>F07$F_ u$\"+I\\)f!>F07$Fdu$\"+o:Xt=F07$Fiu$\"+\"=P^%=F07$F^v$\"+)3H;#=F07$Fcv $\"+k`,'z\"F07$Fhv$\"+'=agx\"F07$F]w$\"+SSda(o\"F07$Ffx$\"+hUlt;F07$F[y $\"+#4x)f;F07$F`y$\"+!\\*QY;F07$Fey$\"+l%p_j\"F07$Fjy$\"+zA&Qi\"F07$F_ z$\"+`+l7;F07$Fdz$\"+qdA-;F07$Fiz$\"+1Lg#f\"F07$F^[l$\"+a:T#e\"F07$Fc[ l$\"+WEmt:F07$Fh[l$\"+]nrk:F07$F]\\l$\"+$zQpb\"F07$Fb\\l$\"+2$o([:F07$ Fg\\l$\"+`tPT:F07$F\\]l$\"+S%RR`\"F07$Fa]l$\"+cT$p_\"F07$Ff]l$\"+E_')> :F07$F[^l$\"+$z(H8:F07$F`^l$\"+y9\"o]\"F07$Fe^l$\"+-_f+:F07$Fj^l$\"+oD 1&\\\"F07$F__l$\"+19#*)[\"F07$Fd_l$\"+i+g$[\"F07$Fi_l$\"+__4y9F07$F^`l $\"+$\\\")HZ\"F07$Fc`l$\"+plkn9F0-Fi`l6&F[alF($\"#DF^al$\"\"\"F)-Fbal6 #%Pscheme~with~a~relatively~large~stability~regionG-F$6%7bpF'7$F+$\"%* )RF07$F2$\"'J3YF07$F8$\"(K')3#F07$F=$\"(nc6(F07$FB$\")<&4*>F07$FG$\")+ QB[F07$FL$\"*\\qo/\"F07$FQ$\"*P)\\$3#F07$FV$\"*7#*f'QF07$Fen$\"*gk/x'F 07$Fjn$\"+BMbH6F07$F_o$\"+QOG3=F07$Fdo$\"+60&oV#F07$Fio$\"+**pLECF07$F cdl$\"+0f!fS#F07$F^p$\"+Xb9(Q#F07$$\"50++v$4@\">_rF4$\"+3(H)zBF07$F[el $\"+d8.wBF07$$\"5SLLekyANLyF4$\"+\\B&*yBF07$Fbp$\"+#\\XPR#F07$Fcel$\"+ UfX$\\#F07$Fgp$\"+6E:$y#F07$F[fl$\"+2IpaIF07$F`fl$\"+*eE[<$F07$Fefl$\" +y3]jJF07$F\\q$\"+d/N_JF07$F]gl$\"+(*3)e6$F07$Fbgl$\"+A,N,JF07$Fggl$\" +%e#\\'4qI=F^q$\"+)pUEJ$F07$Fihl$\"++N:8LF07$$\"5M$3_+AUo**)=F^q$\"+)\\ V(=LF07$F^il$\"+zcdJLF07$Fcil$\"+/i)4R$F07$Fgq$\"+y1nqJP#F07$F]w$\"+)RrWM#F07$Fbw$ \"++,\"4K#F07$Fgw$\"+9O\"oH#F07$F\\x$\"+\"f)QvAF07$Fax$\"+*pWWD#F07$Ff x$\"+AFNOAF07$F[y$\"+,U%z@#F07$F`y$\"+2C#**>#F07$Fey$\"+QS1&=#F07$Fjy$ \"+I&3)p@F07$F_z$\"+/-%[:#F07$Fdz$\"+L8\"49#F07$Fiz$\"+HQ0G@F07$F^[l$ \"+9cV9@F07$Fc[l$\"+<`u-@F07$Fh[l$\"+*z\"z!4#F07$F]\\l$\"+n))R!3#F07$F b\\l$\"+N9[p?F07$Fg\\l$\"+/cgf?F07$F\\]l$\"+Aqm\\?F07$Fa]l$\"+8lIS?F07 $Ff]l$\"+a4'3.#F07$F[^l$\"+)\\&3A?F07$F`^l$\"+I%=M,#F07$Fe^l$\"+)=7^+# F07$Fj^l$\"+L%>x*>F07$F__l$\"+cN^*)>F07$Fd_l$\"+EJS#)>F07$Fi_l$\"+[v/v >F07$F^`l$\"+#\\9#o>F07$Fc`l$\"+:f3h>F0-Fi`l6&F[alF($\"#vF^alF_al-Fbal 6#%TButcher's~scheme~B~with~c[5]=c[6]=1/2~and~b[5]=b[6]G-F$6%7^pF'7$F+ $\"%i7F07$F2$\"'#**[\"F07$F8$\"'RFoF07$F=$\"(m>N#F07$FB$\"(l]l'F07$FG$ \").pI;F07$FL$\")/3!e$F07$FQ$\")R93sF07$FV$\"*#*HKN\"F07$Fen$\"*$>0)R# F07$Fjn$\"*#fy[SF07$F_o$\"*R(4glF07$Fdo$\"*f'o8*)F07$Fio$\"*ZH_())F07$ Fcdl$\"*kv/!))F07$F^p$\"*\"*R9t)F07$F[el$\"*\"y*zo)F07$Fbp$\"**o[U()F0 7$Fcel$\"*4\\;3*F07$Fgp$\"+y*>'45F07$F[fl$\"+y>E16F07$F`fl$\"+*4-$\\6F 07$Fefl$\"+K@?X6F07$F\\q$\"+)[l69\"F07$F]gl$\"+%pCz7\"F07$Fbgl$\"+6#o? 7\"F07$Fggl$\"+9$*>h6F07$Fbq$\"+@Aa;7F07$F_hl$\"+d&QT?\"F07$Fdhl$\"+?# zS>\"F07$Fihl$\"+JL(G>\"F07$F^il$\"++!3))>\"F07$Fcil$\"+(e^*=7F07$Fgq$ \"+F]'zA\"F07$F\\r$\"+jC167F07$Far$\"+S`$p=\"F07$Ffr$\"+$)F07$Fgw$\"*%z3L#)F07$F\\x$\"*s#H c\")F07$Fax$\"*Y@73)F07$Ffx$\"*(=P;!)F07$F[y$\"*c(Q]zF07$F`y$\"*%))y&) yF07$Fey$\"*4HD$yF07$Fjy$\"*aXyx(F07$F_z$\"*A\">CxF07$Fdz$\"*wiUn(F07$ Fiz$\"*fu\"GwF07$F^[l$\"*]f$zvF07$Fc[l$\"*4bu`(F07$Fh[l$\"*72Y\\(F07$F ]\\l$\"*3`tX(F07$Fb\\l$\"*&*=#=uF07$Fg\\l$\"*W=GQ(F07$F\\]l$\"**H>ZtF0 7$Fa]l$\"*kROJ(F07$Ff]l$\"*S\"yzsF07$F[^l$\"*=D$[sF07$F`^l$\"*Zds@(F07 $Fe^l$\"*;$[(=(F07$Fj^l$\"*I$)4;(F07$F__l$\"*uo:8(F07$Fd_l$\"**331rF07 $Fi_l$\"*G9(zqF07$F^`l$\"*p?_0(F07$Fc`l$\"*yn'HqF0-Fi`l6&F[alF\\alFibm F(-Fbal6#%Hscheme~with~c[5]=c[6]=3/4~and~b[5]=b[6]G-%%FONTG6$%*HELVETI CAG\"\"*-%+AXESLABELSG6$Q\"x6\"Q!Fbdp-%&TITLEG6#%Uerror~curves~for~7~s tage~order~6~Runge-Kutta~methodsG-%%VIEWG6$;F(Fc`l%(DEFAULTG" 1 2 0 1 10 0 2 9 1 4 2 1.000000 45.000000 45.000000 0 1 "Butcher's scheme A" " scheme with simple nodes" "scheme with a relatively large stability re gion" "Butcher's scheme B with c[5]=c[6]=1/2 and b[5]=b[6]" "scheme wi th c[5]=c[6]=3/4 and b[5]=b[6]" }}}}{PARA 0 "" 0 "" {TEXT -1 0 "" }} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}}{SECT 1 {PARA 4 "" 0 " " {TEXT -1 46 "Test 6 of 7 stage, order 6 Runge-Kutta methods" }} {PARA 256 "" 0 "" {TEXT -1 2 " " }{XPPEDIT 18 0 "dy/dx = (1+2*(x+1)*s in(3*x))*exp(-y);" "6#/*&%#dyG\"\"\"%#dxG!\"\"*&,&F&F&*(\"\"#F&,&%\"xG F&F&F&F&-%$sinG6#*&\"\"$F&F.F&F&F&F&-%$expG6#,$%\"yGF(F&" }{TEXT -1 6 ", " }{XPPEDIT 18 0 "y(0) = 0;" "6#/-%\"yG6#\"\"!F'" }{TEXT -1 1 " " }}{PARA 0 "" 0 "" {TEXT -1 10 "Solution: " }}{PARA 256 "" 0 "" {TEXT -1 2 " " }{XPPEDIT 18 0 "y=ln(x+2/9*sin(3*x)-2/3*x*cos(3*x)-2/3 *cos(3*x)+5/3)" "6#/%\"yG-%#lnG6#,,%\"xG\"\"\"*(\"\"#F*\"\"*!\"\"-%$si nG6#*&\"\"$F*F)F*F*F***F,F*F3F.F)F*-%$cosG6#*&F3F*F)F*F*F.*(F,F*F3F.-F 66#*&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,i c\},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$$\"3umm\"H2P\"Q?F-$\"3pUCE&Gm M$HF-7$$\"3MLL$eRwX5$F-$\"3l!G\"yWq,6\\F-7$$\"33ML$3x%3yTF-$\"3dz%)zau hMpF-7$$\"3emm\"z%4\\Y_F-$\"3,G5kQO>C))F-7$$\"3`LLeR-/PiF-$\"36YrjIBvP 5!#<7$$\"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%ommTIOo/\"FI$\"3i$3slO!o09FI7$$\"3E+]7GTt%4\"FI$\"3 qC'f&H6%RT\"FI7$$\"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%ommmJFI$\"3yKBC%\\'\\`dF-7$$\"3gmmm\"pW`(>FI$\"3$ )>nU%[*3T`F-7$$\"3_ek.HW#)))>FI$\"3kKE$Q%*GSE&F-7$$\"3?]iSmTI-?FI$\"3N xPe(47T?&F-7$$\"3*=/wP!Ry:?FI$\"3MId=-(\\?;&F-7$$\"3dLe9TOEH?FI$\"3+V! )\\%**R%Q^F-7$$\"3EDc^yLuU?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$$\"3-+v$41oWW# FI$\"3if$>8HM6:\"FI7$$\"3GLL3_?`(\\#FI$\"31,>EjcGm7FI7$$\"3AL3_D1l_DFI $\"3+C:%e=o-Q\"FI7$$\"3fL$e*)>pxg#FI$\"3%R*>K#Rqn[\"FI7$$\"33+]Pf4t.FF I$\"3A89rRoB^;FI7$$\"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$3_NJOG$FI$\"3:>!e))R4?FI7$$\"3Jmmm\"p)=MLFI$\"3Or POD8'y)>FI7$$\"3GLLeR%p\")Q$FI$\"3S(z'\\,LGb>FI7$$\"3B++](=]@W$FI$\"3K :aBe2s7>FI7$$\"35L$e*[$z*RNFI$\"3!>**fc?u*4=FI7$$\"3e++]iC$pk$FI$\"3'e \\^`F#Qg;FI7$$\"3[m;H2qcZPFI$\"351&p,mkr[\"FI7$$\"3O+]7.\"fF&QFI$\"3o; 2PnR[#G\"FI7$$\"3Ymm;/OgbRFI$\"3/lp%G(QG#3\"FI7$$\"3*G$e*[$zV4SFI$\"3P 9Az61+7**F-7$$\"3w**\\ilAFjSFI$\"3!p>ERLl'*>*F-7$$\"3#G3_]p'>*3%FI$\"3 O9*z2=!e_*)F-7$$\"3ym\"zW7@^6%FI$\"3w4XE;/Rz()F-7$$\"3w3F>RL3GTFI$\"3J eP:9JjA()F-7$$\"3t]i!RbX59%FI$\"3mH1#H$\\k'o)F-7$$\"3#=z>'ox+aTFI$\"3o Lr_-o*=n)F-7$$\"3yLLL$)*pp;%FI$\"3A7j1wipy')F-7$$\"3!Q3_+sD-=%FI$\"32p cM,k23()F-7$$\"3#Q$3xc9[$>%FI$\"3Gri,**=4g()F-7$$\"3'Qe*[$>Pn?%FI$\"3s e,X+?^M))F-7$$\"3)QL3-$H**>UFI$\"3Z**e,OD#4$*)F-7$$\"3#R$ek.W]YUFI$\"3 i#fiyx0s=*F-7$$\"3)RL$3xe,tUFI$\"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$\"3_eaMDR_K " 0 "" {MPLTEXT 1 0 766 "U := (x,y) -> \+ (1+2*(x+1)*sin(3*x))*exp(-y): hh := 0.01: numsteps := 500: x0 := 0: y0 := 0:\nmatrix([[`slope field: `,U(x,y)],[`initial point: `,``(x0,y0 )],[`step width: `,hh],\n[`no. of steps: `,numsteps]]);``;\nmthds \+ := [`Butcher's scheme A`,`scheme with simple nodes`,`scheme with a rel atively large stability region`,`Butcher's scheme B with `*(c[5]=1/2,c [6]=1/2,b[5]=b[6]),`scheme with `*(c[5]=3/4,c[6]=3/4,b[5]=b[6])]:\nerr s := []:\nDigits := 25:\nfor ct to 5 do\n Un_RK6_||ct := RK6_||ct(U( x,y),x,y,x0,y0,hh,numsteps,false);\n sm := 0: numpts := nops(Un_RK6_ ||ct):\n for ii to numpts do\n sm := sm+(Un_RK6_||ct[ii,2]-u(Un _RK6_||ct[ii,1]))^2;\n end do:\n errs := [op(errs),sqrt(sm/numpts) ];\nend do:\nDigits := 10:\nlinalg[transpose]([mthds,evalf(errs)]);" } }{PARA 11 "" 1 "" {XPPMATH 20 "6#K%'matrixG6#7&7$%0slope~field:~~~G*&, &\"\"\"F+*(\"\"#F+,&%\"xGF+F+F+F+-%$sinG6#,$*&\"\"$F+F/F+F+F+F+F+-%$ex pG6#,$%\"yG!\"\"F+7$%0initial~point:~G-%!G6$\"\"!FA7$%/step~width:~~~G $F+!\"#7$%1no.~of~steps:~~~G\"$+&Q)pprint256\"" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%!G" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#K%'matrixG6#7'7$ %3Butcher's~scheme~AG$\"+Goqm=!#A7$%9scheme~with~simple~nodesG$\"+_\"y /a(!#C7$%Pscheme~with~a~relatively~large~stability~regionG$\"+x.l)R\"! #B7$*&%9Butcher's~scheme~B~with~G\"\"\"6%/&%\"cG6#\"\"&#F9\"\"#/&F=6# \"\"'F@/&%\"bGF>&FHFDF9$\"+QgG09F57$*&%-scheme~with~GF96%/F<#\"\"$\"\" %/FCFQFFF9$\"+Z@sXrF0Q)pprint266\"" }}}{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 ea ch of the Runge-Kutta schemes." }}{PARA 0 "" 0 "" {TEXT -1 75 "The err or 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 0 697 " U := (x,y) -> (1+2*(x+1)*sin(3*x))*exp(-y): hh := 0.01: numsteps := 50 0: x0 := 0: y0 := 0:\nmatrix([[`slope field: `,U(x,y)],[`initial poi nt: `,``(x0,y0)],[`step width: `,hh],\n[`no. of steps: `,numsteps] ]);``;\nmthds := [`Butcher's scheme A`,`scheme with simple nodes`,`sch eme with a relatively large stability region`,`Butcher's scheme B with `*(c[5]=1/2,c[6]=1/2,b[5]=b[6]),`scheme with `*(c[5]=3/4,c[6]=3/4,b[5 ]=b[6])]:\nerrs := []:\nDigits := 25:\nfor ct to 5 do\n un_RK6_||ct \+ := RK6_||ct(U(x,y),x,y,x0,y0,hh,numsteps,true);\nend do:\nxx := 4.999: uxx := evalf(u(xx)):\nfor ct to 5 do\n errs := [op(errs),abs(un_RK6 _||ct(xx)-uxx)];\nend do:\nDigits := 10:\nlinalg[transpose]([mthds,eva lf(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$%/s tep~width:~~~G$F+!\"#7$%1no.~of~steps:~~~G\"$+&Q)pprint276\"" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%!G" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#K%'ma trixG6#7'7$%3Butcher's~scheme~AG$\"+^;$oS#!#A7$%9scheme~with~simple~no desG$\"+-W)4G#!#C7$%Pscheme~with~a~relatively~large~stability~regionG$ \"+eA_DsF07$*&%9Butcher's~scheme~B~with~G\"\"\"6%/&%\"cG6#\"\"&#F8\"\" #/&F<6#\"\"'F?/&%\"bGF=&FGFCF8$\"+^SL5#)F07$*&%-scheme~with~GF86%/F;# \"\"$\"\"%/FBFPFEF8$\"+keTFGF0Q)pprint286\"" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 4 "The " }{TEXT 260 22 "root mean s quare error" }{TEXT -1 20 " over the interval " }{XPPEDIT 18 0 "[0, 5 ];" "6#7$\"\"!\"\"&" }{TEXT -1 82 " of each Runge-Kutta method is est imated 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 0 436 "mthds := [`Butcher's scheme A`,`scheme with simple n odes`,`scheme with a relatively large stability region`,`Butcher's sch eme B with `*(c[5]=1/2,c[6]=1/2,b[5]=b[6]),`scheme with `*(c[5]=3/4,c[ 6]=3/4,b[5]=b[6])]:\nerrs := []:\nDigits := 20:\nfor ct to 5 do\n sm := NCint((u(x)-'un_RK6_||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$%3Butcher's~scheme~AG$\"+#\\zm&=!#A7$%9 scheme~with~simple~nodesG$\"+oWNAv!#C7$%Pscheme~with~a~relatively~larg e~stability~regionG$\"+3j\\%R\"!#B7$*&%9Butcher's~scheme~B~with~G\"\" \"6%/&%\"cG6#\"\"&#F9\"\"#/&F=6#\"\"'F@/&%\"bGF>&FHFDF9$\"+-Rf-9F57$*& %-scheme~with~GF96%/F<#\"\"$\"\"%/FCFQFFF9$\"+=,EOrF0Q)pprint296\"" }} }{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 92 "The foll owing error graphs are constructed using the numerical procedures for \+ the solutions." }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 516 "evalf[20] (plot([u(x)-'un_RK6_1'(x),u(x)-'un_RK6_2'(x),u(x)-'un_RK6_3'(x),u(x)-' un_RK6_4'(x),\nu(x)-'un_RK6_5'(x)],x=0..5,font=[HELVETICA,9],\ncolor=[ COLOR(RGB,.95,.2,0),COLOR(RGB,.45,0,.95),COLOR(RGB,0,.25,1),COLOR(RGB, 0,.75,.2),COLOR(RGB,.95,.45,0)],\nlegend=[`Butcher's scheme A`,`scheme with simple nodes`,`scheme with a relatively large stability region`, `Butcher's scheme B with c[5]=c[6]=1/2 and b[5]=b[6]`,`scheme with c[5 ]=c[6]=3/4 and b[5]=b[6]`],title=`error curves for 7 stage order 6 Run ge-Kutta methods`));" }}{PARA 13 "" 1 "" {GLPLOT2D 934 601 601 {PLOTDATA 2 "6+-%'CURVESG6%7at7$$\"\"!F)F(7$$\"5MLLLL3x&)*3\"!#?$!(4*) )[F-7$$\"5+++]i!R(*Rc\"F-$!(SvW'F-7$$\"5nmmm\"H2P\"Q?F-$!(()4L(F-7$$\" 5++]7G))>Wr@F-$!(;+P(F-7$$\"5MLLek.pu/BF-$!(ve[(F-7$$\"5nm;/,>=0QCF-$! (G5U(F-7$$\"5+++]PMnNrDF-$!(E)RtF-7$$\"5nmmT5ll'z$GF-$!(JM9(F-7$$\"5ML LL$eRwX5$F-$!(i]&oF-7$$\"5MLLLL$eI8k$F-$!(ldA'F-7$$\"5NLLL$3x%3yTF-$!( m@u&F-7$$\"5-++]PfyG7ZF-$!(89^&F-7$$\"5ommm\"z%4\\Y_F-$!(0vR&F-7$$\"5N LLLeR-/PiF-$!'_(Q&!#>7$$\"5-+++DcmpisF-$!'AQ_F[p7$$\"5OLLLe*)>VB$)F-$! '*=*[F[p7$$\"5.+++DJbw!Q*F-$!'7lWF[p7$$\"5nmmm;/j$o/\"F[p$!'UITF[p7$$ \"5MLLL3_>jU6F[p$!'>LSF[p7$$\"5++++]i^Z]7F[p$!'.+UF[p7$$\"5++++](=h(e8 F[p$!'#ec%F[p7$$\"5++++]P[6j9F[p$!'_h[F[p7$$\"5nmm\"HKR'\\5:F[p$!'bf[F [p7$$\"5MLL$e*[z(yb\"F[p$!'r6[F[p7$$\"5+++Dc,#>Uh\"F[p$!'`mXF[p7$$\"5n mmm;a/cq;F[p$!(E;F%F-7$$\"5nmmmT&)))G=(*)zF[p$\"'R& f*F-7$$\"5nmmmm\"pW`(>F[p$\"(%)ze&F-7$$\"5++]iSmTI-?F[p$\"(io(zF-7$$\" 5MLLe9TOEH?F[p$\"(eS3*F-7$$\"5o;H2$)4N+O?F[p$\"(aq;*F-7$$\"5,+Dc^yLuU? F[p$\"(*)o:*F-7$$\"5M$3_+sC$[\\?F[p$\"(1Q**)F-7$$\"5om;a)e6Bi0#F[p$\"( I1*))F-7$$\"5ML3_D`Gqp?F[p$\"(<))z(F-7$$\"5,++]i!f#=$3#F[p$\"(+?h'F-7$ $\"5,++v$fQa)3@F[p$\"(\\,,$F-7$$\"5,+++D\"=EX8#F[p$!(Q%y@F-7$$\"5,+]i! *y?OZ@F[p$!(39u%F-7$$\"5,++Dcwz>g@F[p$!(S]O*F-7$$\"5,+](=U(Q.t@F[p$!)V $*y6F-7$$\"5,++](=xpe=#F[p$!)1s?9F-7$$\"5ML3_]%oi#*>#F[p$!)WX#y\"F-7$$ \"5om;a8(fbE@#F[p$!)Cq!3#F-7$$\"5,+Dcw4&[gA#F[p$!),(*fAF-7$$\"5MLLeRA9 WRAF[p$!)HJ0BF[p$!)k;eHF-7$$\"5nm;a8A3h#F[p7$$\"5MLL$e*)>pxg# F[p$!(%H))>F[p7$$\"5,++]Pf4t.FF[p$!(BZ'=F[p7$$\"5MLLLe*Gst!GF[p$!(Qtq \"F[p7$$\"5,++++DRW9HF[p$!(l'[:F[p7$$\"5,+++DJE>>IF[p$!(g_U\"F[p7$$\"5 ,++v$4^n)pIF[p$!(XOQ\"F[p7$$\"5,++]i!RU07$F[p$!(VoN\"F[p7$$\"5,++vo/#3 o<$F[p$!(.^M\"F[p7$$\"5,+++v=S2LKF[p$!(cAN\"F[p7$$\"5MLL$3_NJOG$F[p$!( &=v8F[p7$$\"5ommmm\"p)=MLF[p$!(6JT\"F[p7$$\"5,+++](=]@W$F[p$!(Ot`\"F[p 7$$\"5MLL$e*[$z*RNF[p$!(Men\"F[p7$$\"5,+++]iC$pk$F[p$!(MC#=F[p7$$\"5om m;H2qcZPF[p$!(:o$>F[p7$$\"5,++]7.\"fF&QF[p$!(2%p@F[p7$$\"5om;aQG-ZyQF[ p$!(t-E#F[p7$$\"5MLLek`8=/RF[p$!(gBM#F[p7$$\"5oT&Q.VC&R2RF[p$!(b[N#F[p 7$$\"5,]P4'\\841\"RF[p$!(!4_BF[p7$$\"5Me*[=c-BQ\"RF[p$!([oO#F[p7$$\"5o mTgF;p.F-7$$ \"5,+D\"yv^'*G-%F[p$!),>y*3%F[p$!(R6E%F-7$$\"5,+Dcw4*e@5%F[p$!(cRL3GTF[p$\"(e2V#F-7$$\"5-+]i!Rb X59%F[p$\"(YV@%F-7$$\"5oTNYe2hGWTF[p$\"(e#=UF-7$$\"5N$3-j7mEv9%F[p$\"( TIG%F-7$$\"5-D19%\\@n2:%F[p$\"(G\"=YF-7$$\"5om\"z>'ox+aTF[p$\"(#4>YF-7 $$\"5-]il(f())[gTF[p$\"(#>WXF-7$$\"5NLLLL$)*pp;%F[p$\"(\"[.XF-7$$\"5NL $3_+sD-=%F[p$\"(m:(GF-7$$\"5NLL3xc9[$>%F[p$\"(Y(f7F-7$$\"5NL$e*[$>Pn?% F[p$!'.S**F-7$$\"5NLL$3-$H**>UF[p$!(#4ujF-7$$\"5NL$3Fpm[KB%F[p$!(T9p*F -7$$\"5NLLek.W]YUF[p$!)2UU8F-7$$\"5NL3_+sA8`UF[p$!)#[Z%F[p$!(%>9XF[p7$$\"5MLL$3 _5,-`%F[p$!(?x,%F[p7$$\"5ommmT&G!e&e%F[p$!(=Kk$F[p7$$\"5-++]P%37^j%F[p $!(yMU$F[p7$$\"5NLLLL$)Qk%o%F[p$!(6%fKF[p7$$\"5-++]iSjE!z%F[p$!((fkHF[ p7$$\"5-++]P40O\"*[F[p$!(Zbn#F[p7$$\"\"&F)$!(\\XS#F[p-%&COLORG6&%$RGBG $\"#&*!\"#$\"\"#!\"\"F(-%'LEGENDG6#%3Butcher's~scheme~AG-F$6%7dvF'7$F+ $\"'eaPF-7$F1$\"'P<`F-7$F6$\"'4)G'F-7$F;$\"'J[jF-7$F@$\"'F-7$Fgo$\"&Aq \"F[p7$$\"5qmmm\"zWo)\\nF-$\"&z_\"F[p7$F]p$\"&bT\"F[p7$Fbp$\"&JH\"F[p7 $Fgp$\"&HB\"F[p7$F\\q$\"&u>\"F[p7$Faq$\"&`<\"F[p7$Ffq$\"&\"\\6F[p7$F[r $\"&t6\"F[p7$F`r$\"&$46F[p7$Fer$\"&(z6F[p7$Fjr$\"&.O\"F[p7$$\"5nm;/Ev& [ge\"F[p$\"&\\d\"F[p7$F_s$\"&K#>F[p7$$\"5ML$eky#)*QU;F[p$\"'#*[CF-7$Fd s$\"')))>$F-7$F^t$\"'5IrF-7$Fht$\"'h.()F-7$Fbu$\"(hh+\"F-7$$\"5,+D1k1n TN=F[p$\"(&pY5F-7$Fgu$\"(Ur2\"F-7$$\"5ML3-jikF[p$\"'lo\"*F-7$Fav$\"'M[rF-7$$\"5++]i!Rgs2 &>F[p$\"'Q(o#F-7$Ffv$!'KS5F-7$$\"5MLek.HW#)))>F[p$!'L$y$F-7$F[w$!'U/oF -7$$\"5nmTgx.Ry:?F[p$!'l7')F-7$F`w$!(\\V4\"F-7$Fjw$!(GKD\"F-7$Fdx$!(\" *RK\"F-7$$\"5MekyA]Iff?F[p$!(W$[8F-7$$\"5,]7.d%)H'H1#F[p$!(`bN\"F-7$$ \"5oTgF\"*=HLm?F[p$!(sWN\"F-7$Fix$!(%z]8F-7$$\"5,Dcwf(ysI2#F[p$!(b\"[8 F-7$$\"5o;/,%>sUk2#F[p$!(6[M\"F-7$$\"5M3_DGcE\")z?F[p$!(JyI\"F-7$F^y$! (q-I\"F-7$$\"5,+]7G)[=g4#F[p$!(#f27F-7$Fcy$!(F'>5F-7$$\"5,+]Pf$G!p@@F[ p$!'\"eT(F-7$Fhy$!'us_F-7$F]z$!'Z3FF-7$Fbz$\"'4S=F-7$Fgz$\"'fUUF-7$F\\ [l$\"'b\"f'F-7$Fa[l$\"'*R\"**F-7$Ff[l$\"(v>D\"F-7$F[\\l$\"(`4S\"F-7$F` \\l$\"(_xf\"F-7$$\"5nmTg-NV$GD#F[p$\"(z=q\"F-7$Fe\\l$\"(j'QFw AF[p$\"(h!fWw>EF[p$!%\"*pF[p7$$\"5,++D1R'f0uzEF[p $!&*45F[p7$F[al$!&@/\"F[p7$$\"5omm\"zWi^bv#F[p$!&$R5F[p7$F`al$!%*y*F[p 7$Feal$!%-#)F[p7$Fjal$!%/rF[p7$Fdbl$!%;mF[p7$F^cl$!%3nF[p7$Fhcl$!%AuF[ p7$F]dl$!%$3*F[p7$Fbdl$!&x6\"F[p7$$\"5omm\"Hd!fX$f$F[p$!&z=\"F[p7$Fgdl $!&%=6F[p7$$\"5om;zp)4\"4sOF[p$!%W$*F[p7$$\"5MLLe*[t\\sp$F[p$!%CrF[p7$ $\"5,+]P4r$3Cs$F[p$!%Z?F[p7$F\\el$\"%5OF[p7$$\"5MLL$3_0j,!QF[p$\"&I0$F [p7$Fael$\"&49(F[p7$Ffel$\"&fh*F[p7$F[fl$\"'rm7F[p7$Figl$\"'Sw:F[p7$Fc hl$\"'qqy9zvRF[p$\"'hl=F[p7$Fhhl$\"'$Q'=F[p7$$\"5M$eRs31]#*)RF[ p$\"'\"Q&=F[p7$$\"5omTN@_$zf*RF[p$\"'Z[=F[p7$$\"5,](oaNk3F+%F[p$\"'n\" y\"F[p7$F]il$\"(/tr\"F-7$Fbil$\"(fZa\"F-7$Fgil$\"(')*f8F-7$F\\jl$\"'E& 4*F-7$Fajl$\"'#*=eF-7$$\"5M$3x\")HP`(pSF[p$\"'P$3$F-7$Ffjl$\"'%=W#F-7$ $\"5,]7`p(e:F3%F[p$!&<.*F-7$F[[m$!'\\()GF-7$$\"5o;a)3C!yn&4%F[p$!'$4b% F-7$F`[m$!'_M!)F-7$$\"5M$eRAr,S'3TF[p$!'i)G*F-7$Fe[m$!(c::\"F-7$Fj[m$! (^.`\"F-7$F_\\m$!(w8(>F-7$Fc]m$!(:+9#F-7$F]^m$!(r1C#F-7$$\"5N$3-8v\"RG qTF[p$!('GkAF-7$$\"5NL3Fp^yftTF[p$!(mFE#F-7$$\"5N$eRsey6p<%F[p$!($QeAF -7$Fb^m$!(BI@#F-7$$\"5N$3xJUlRN=%F[p$!(]0@#F-7$$\"5NLe9T)e`o=%F[p$!(q- ?#F-7$$\"5N$e9\"fAv;!>%F[p$!(gf3#F-7$Fg^m$!(/F3#F-7$$\"5NL3-8D$4,?%F[p $!(9r)=F-7$F\\_m$!(hV'=F-7$$\"5NLe*[=1lL@%F[p$!(k&>;F-7$Fa_m$!(iiI\"F- 7$Ff_m$!'$GR*F-7$F[`m$!'Ne_F-7$F``m$!'T)Q\"F-7$Fe`m$\"',$4#F-7$Fj`m$\" '%>*GF-7$F_am$\"'!*[nF-7$Fdam$\"'55'*F-7$Fiam$\"(%>j5F-7$F^bm$\"(%G19F -7$Fcbm$\"([\\f\"F-7$Fhbm$\"'b&)>F[p7$F]cm$\"'o&>#F[p7$Fbcm$\"'a6CF[p7 $Fgcm$\"'OqDF[p7$$\"5o;/E])pk%eVF[p$\"''>c#F[p7$F\\dm$\"']#e#F[p7$$\"5 oTNr*y6but%F[p$!&#=HF[p7$$\" 5NL$3-jsgQw%F[p$!&&zHF[p7$F\\im$!&!oHF[p7$$\"5-++++DM\"3%[F[p$!&E#GF[p 7$Faim$!&kj#F[p7$Ffim$!&_F#F[p-F[jm6&F]jm$\"#XF`jmF(F^jm-Fejm6#%9schem e~with~simple~nodesG-F$6%7isF'7$F+$\"'M(H'F-7$F1$\"'Vp))F-7$F6$\"(0'p5 F-7$$\"5MLe*)fI&*y/@F-$\"(Px4\"F-7$F;$\"($>)3\"F-7$$\"5nmTN'fW%4QAF-$ \"(p\"36F-7$F@$\"(Cu7\"F-7$FE$\"(2)G6F-7$FJ$\"(-\"G6F-7$$\"5ML$eR(\\;m /FF-$\"(]!Q6F-7$FO$\"(+`7\"F-7$$\"5++](o/[r7(HF-$\"(h46\"F-7$FT$\"(\"z +6F-7$FY$\"(F5,\"F-7$Fhn$\"'84\"*F-7$Fbo$\"'87uF-7$Fgo$\"&NO'F[p7$F]p$ \"&oj&F[p7$Fbp$\"&q6&F[p7$Fgp$\"&^v%F[p7$F\\q$\"&n_%F[p7$Faq$\"&_V%F[p 7$Ffq$\"&5W%F[p7$F[r$\"&(QXF[p7$F`r$\"&km%F[p7$Fer$\"&4v%F[p7$Fjr$\"&z #\\F[p7$F^`n$\"&V3&F[p7$F_s$\"&.K&F[p7$Ff`n$\"'%Qn&F-7$Fds$\"'C!='F-7$ Fis$\"'#3H(F-7$F^t$\"'_!y)F-7$Fct$\"'\"RE*F-7$Fht$\"'Kw&*F-7$F]u$\"'qC )*F-7$Fbu$\"'r\"*)*F-7$$\"5o;HKknnZG=F[p$\"'#p#**F-7$Fgan$\"'`n)*F-7$$ \"5M$3-QckcB%=F[p$\"'i;'*F-7$Fgu$\"'-8%*F-7$F_bn$\"'_6()F-7$F\\v$\"'@5 zF-7$Fgbn$\"'V+VF-7$Fav$\"&gR&F-7$F_cn$!'(>j'F-7$Ffv$!(+$G7F-7$F[w$!(& 4]?F-7$F`w$!(!H/EF-7$Fjw$!(*Q*z#F-7$Fdx$!(?A(GF-7$Fben$!(U$*)GF-7$Fix$ !(kt&GF-7$F_fn$!(Aq%GF-7$Fdfn$!(5(RGF-7$Fifn$!(84w#F-7$F^y$!(a]u#F-7$F cy$!(LlD#F-7$Fhy$!($)\\X\"F-7$F]z$!(5I/\"F-7$Fbz$!'Q$>$F-7$Fgz$\"&5(pF -7$F\\[l$\"'?QXF-7$Fa[l$\"(QD+\"F-7$Ff[l$\"(+qW\"F-7$F[\\l$\"(=vr\"F-7 $F`\\l$\"(<=4#F-7$Fe\\l$\"(O%QCF-7$Fj\\l$\"(z6k#F-7$F_]l$\"(-#QEF-7$Fd ]l$\"(jbh#F-7$F^^l$\"(C^d#F-7$Fb_l$\"(PD]#F-7$Fg_l$\"(p*GBF-7$F\\`l$\" 'z.@F[p7$Fa`l$\"'JT8F[p7$F[]o$\"'i%4\"F[p7$Ff`l$\"&SV*F[p7$Fb^o$\"&g_) F[p7$F[al$\"&A(yF[p7$F`al$\"&%4pF[p7$Feal$\"&qD'F[p7$Fjal$\"&f$eF[p7$F dbl$\"&8h&F[p7$F^cl$\"&bc&F[p7$Fhcl$\"&dp&F[p7$F]dl$\"&o)fF[p7$Fbdl$\" &OP'F[p7$F\\ao$\"&Ao'F[p7$Fgdl$\"&q7(F[p7$Fiao$\"&o\"yF[p7$F\\el$\"&&[ !*F[p7$Ffbo$\"']a6F[p7$Fael$\"'[=:F[p7$F[fl$\"'Ha>F[p7$Fchl$\"'V0AF[p7 $Fhhl$\"'%4/#F[p7$F]il$\"(mrd\"F-7$Fbil$\"(JQ@\"F-7$Fgil$\"'yC')F-7$F \\jl$\"&C#**F-7$Fajl$!'(fO%F-7$Ffjl$!'+9(*F-7$F[[m$!(*G$y\"F-7$F`[m$!( <*\\DF-7$Fe[m$!(,31$F-7$Fj[m$!()z.OF-7$F_\\m$!(=`@%F-7$Fc]m$!(AyV%F-7$ F]^m$!(lRb%F-7$Fb^m$!(zNX%F-7$Fg^m$!()*pA%F-7$Fj[p$!(\"=2RF-7$F\\_m$!( Tu'QF-7$Fb\\p$!(6jZ$F-7$Fa_m$!(uA)HF-7$Ff_m$!(::S#F-7$F[`m$!(ezu\"F-7$ F``m$!(x/9\"F-7$Fe`m$!'M/fF-7$Fj`m$!'hyXF-7$F_am$\"'wg:F-7$Fdam$\"'G!> 'F-7$Fiam$\"'.0zF-7$F^bm$\"(zLN\"F-7$Fcbm$\"(3Un\"F-7$Fhbm$\"'w_BF[p7$ F]cm$\"'w\\FF[p7$Fgcm$\"'g#e$F[p7$Fadm$\"'[*z$F[p7$Fifm$\"'Y.OF[p7$Fcg m$\"'IRHF[p7$Fhgm$\"'Fa@F[p7$F]hm$\"'nX;F[p7$Fbhm$\"'!QL\"F[p7$Fghm$\" '-P6F[p7$F_dp$\"'3/5F[p7$F\\im$\"&K9*F[p7$Faim$\"&Y,)F[p7$Ffim$\"&@A(F [p-F[jm6&F]jmF($\"#DF`jm$\"\"\"F)-Fejm6#%Pscheme~with~a~relatively~lar ge~stability~regionG-F$6%7iuF'7$$\"5qmmmmT&)G\\a!#@$\"(XV[#F_br7$F+$\" '#*\\`F-7$F1$\"'!Q3(F-7$F6$\"'HbyF-7$F;$\"'p=yF-7$F@$\"'$\\x(F-7$FE$\" 'z1wF-7$FJ$\"'Z/uF-7$FO$\"'yboF-7$FT$\"'z'='F-7$FY$\"'o5]F-7$Fhn$\"'L] TF-7$$\"5ommT5:j=XWF-$\"':uQF-7$F]o$\"')>r$F-7$$\"5om;/^J'Qe%[F-$\"'n_ OF-7$$\"5NLLek.%*Qz\\F-$\"',8OF-7$$\"5++]7yv,%H6&F-$\"'Nd\"F[p$\" &%pKF[p7$F^`n$\"&3K$F[p7$$\"5MLe9T))Q8+;F[p$\"&SS$F[p7$F_s$\"&&3NF[p7$ Ff`n$\"'8%*QF-7$Fds$\"'e\\XF-7$Fis$\"'nOhF-7$F^t$\"'/!o)F-7$Fbu$\"((z= 7F-7$F\\v$\"(!f:9F-7$$\"5ML3_]%QU$*)=F[p$\"(gqU\"F-7$Fgbn$\"(YcU\"F-7$ $\"5,+DcEsW\"R\">F[p$\"(;F-7$Fbz$\"(O(RB F-7$F\\[l$\"(GAc#F-7$Ff[l$\"(Co&GF-7$F`\\l$\"(Aw'HF-7$Fe\\l$\"(`!3HF-7 $Fj\\l$\"(sco#F-7$F_]l$\"(WSb#F-7$Fd]l$\"(gwR#F-7$F^^l$\"(IB;#F-7$Fb_l $\"(.2*>F-7$Fg_l$\"(eql\"F-7$F\\`l$\"'Ve7F[p7$$\"5ML$ek`1OzT#F[p$\"&C) )*F[p7$Fc\\o$\"&*ztF[p7$$\"5nm;/^c++rCF[p$\"&'fcF[p7$Fa`l$\"&R'[F[p7$$ \"5M$3-jP(=U/DF[p$\"&Ho%F[p7$$\"5ML3FW&p68^#F[p$\"&,b%F[p7$$\"5M$eRAr^ ,#=DF[p$\"&@[%F[p7$$\"5ML$3-)Q84DDF[p$\"&HT%F[p7$$\"5MLe9;#)4()QDF[p$ \"&pO%F[p7$F[]o$\"&rT%F[p7$$\"5ML$eRA\"*4-e#F[p$\"&8r%F[p7$Ff`l$\"&-& \\F[p7$Fb^o$\"&qn&F[p7$F[al$\"&!RjF[p7$F__o$\"&kx'F[p7$F`al$\"&*RqF[p7 $$\"5,++vV)>ST$GF[p$\"&N9(F[p7$$\"5omm;H2\"34'GF[p$\"&!>sF[p7$$\"5MLLe 9;gn()GF[p$\"&$*>(F[p7$Feal$\"&*>sF[p7$Fjal$\"&b<(F[p7$Fdbl$\"&'=rF[p7 $F^cl$\"&\"yqF[p7$Fhcl$\"&N0(F[p7$$\"5MLLLeR%p\")Q$F[p$\"&v-(F[p7$F]dl $\"&e#pF[p7$$\"5omm\"H#oZ1\"\\$F[p$\"&Tx'F[p7$Fbdl$\"&:_'F[p7$F\\ao$\" &0='F[p7$Fgdl$\"&9u&F[p7$$\"5MLe*)f!y6&fOF[p$\"&Wc&F[p7$Fdao$\"&!eaF[p 7$$\"5,+voz;/n%o$F[p$\"&WT&F[p7$Fiao$\"&\\P&F[p7$$\"5om\"z%*H0H)4PF[p$ \"&LH&F[p7$F^bo$\"&5L&F[p7$$\"5ML3F>*o()\\t$F[p$\"&&=aF[p7$F\\el$\"&+c &F[p7$$\"5,+++DJ]'Qx$F[p$\"&F='F[p7$Ffbo$\"&<\\(F[p7$$\"5ommm;z5YEQF[p $\"&g-*F[p7$Fael$\"'s'=\"F[p7$Ffel$\"'+*[\"F[p7$F[fl$\"'u&)=F[p7$F_gl$ \"'^k?F[p7$Figl$\"'')HBF[p7$F^hl$\"'X+DF[p7$Fchl$\"'9cEF[p7$Fhhl$\"'Dv HF[p7$F]il$\"(<+8$F-7$$\"5,D1kc!e-G,%F[p$\"(%HTJF-7$$\"5o;zpBEs;;SF[p$ \"(5i:$F-7$$\"5M3_v!>(=`>SF[p$\"(;4:$F-7$Fbil$\"(rs:$F-7$$\"5M$3F>*3ei HSF[p$\"(8F:$F-7$Fgil$\"(u%oJF-7$F\\jl$\"(XH6$F-7$Fajl$\"(/w3$F-7$Ffjl $\"((fiIF-7$F[[m$\"(Z\\,$F-7$$\"5,vo/t[sV#4%F[p$\"(V6+$F-7$Fjgo$\"(py+ $F-7$$\"5MeRs3c$=*)4%F[p$\"('4**HF-7$F`[m$\"(Q`)HF-7$Fbho$\"(!)*))HF-7 $Fe[m$\"(XT)HF-7$F_\\m$\"(ot,$F-7$F]^m$\"(DU3$F-7$Fg^m$\"(%*[A$F-7$Fa_ m$\"(JuX$F-7$F[`m$\"(u%\\OF-7$F_am$\"(#[=SF-7$Fcbm$\"(a$*G%F-7$F]cm$\" 'w(Q%F[p7$Fgcm$\"'RuUF[p7$Fadm$\"'t6RF[p7$F[em$\"'m&f$F[p7$F_fm$\"'9TL F[p7$Fdfm$\"'5tIF[p7$Fifm$\"'***y#F[p7$$\"5,+v$4Yd#eQWF[p$\"'^bCF[p7$F ^gm$\"'Ca?F[p7$$\"5MLek`T@uiWF[p$\"'B,=F[p7$Fcgm$\"'2k:F[p7$$\"5ML$3_+ sm')[%F[p$\"'+$G\"F[p7$Fjap$\"&O(**F[p7$$\"5,+]i:5jN;XF[p$\"&xK)F[p7$F hgm$\"&X/'F[p7$Fbbp$\"&><%F[p7$F]hm$\"&n%HF[p7$$\"5NLekGg6B`#o\"F-$\"'\"\\v\"F-7$$\"5MLL3xJs1,=F-$\"'%3$=F-7$$\"5++]PM_@g>>F -$\"'kS=F-7$F6$\"'\\P=F-7$F;$\"'1==F-7$F@$\"'!\\y\"F-7$FE$\"'?JPF-7$Fc]n$!&QV&F-7$Fgo$!%'['F[p7$F[^n$!%XqF[p7$F]p$!%)G(F[p7$Fbp$!% .tF[p7$Fgp$!%zqF[p7$F\\q$!%SpF[p7$Faq$!%XpF[p7$Ffq$!%lqF[p7$F[r$!%&=(F [p7$$\"5++++]7!Q4T\"F[p$!%eqF[p7$F`r$!%(p'F[p7$Fer$!%rdF[p7$Fjr$!%=VF[ p7$F_s$!$d$F[p7$Fds$\"&MT(F-7$Fis$\"'-O;F-7$F^t$\"'81IF-7$Fbu$\"'Jn^F- 7$F\\v$\"',wrF-7$Fav$\"'7&y)F-7$Ffv$\"'*))*)*F-7$F`w$\"(#\\y5F-7$F^y$ \"(,x<\"F-7$Fhy$\"(&)QG\"F-7$F\\[l$\"(\"fV8F-7$F`\\l$\"(?&o7F-7$Fj\\l$ \"(?\"G5F-7$Fd]l$\"'J`*)F-7$Fb_l$\"'9]sF-7$Fg_l$\"'o8fF-7$F\\`l$\"&%eU F[p7$Fc\\o$\"&%*z\"F[p7$Fa`l$\"$\"QF[p7$F[]o$!&f=\"F[p7$Ff`l$!&ru\"F[p 7$$\"5,+vV)*3ow8EF[p$!&]\"=F[p7$Fc]o$!&p(=F[p7$$\"5MLek.H?wDEF[p$!&J(= F[p7$Fh]o$!&'Q>F[p7$F]^o$!&s(>F[p7$Fb^o$!&$4?F[p7$$\"5,+](=#*HXxm#F[p$ !&B/#F[p7$Fg^o$!&`4#F[p7$$\"5om;HKRdt\"p#F[p$!&A6#F[p7$F[al$!&f6#F[p7$ F__o$!&d6#F[p7$F`al$!&(p?F[p7$Feal$!&W$>F[p7$Fjal$!&P#=F[p7$Fdbl$!&tw \"F[p7$F^cl$!&#oF[p7$Fffs$!&X*>F[p7$Fbd l$!&O.#F[p7$$\"5,+]PMFwrmNF[p$!&^0#F[p7$F\\ao$!&k.#F[p7$$\"5ML$e9T=%>? OF[p$!&x(>F[p7$Fgdl$!&\"H>F[p7$Fdao$!&5w\"F[p7$Fiao$!&#*f\"F[p7$F^bo$! &-C\"F[p7$F\\el$!%=))F[p7$Ffbo$\"%%\\(F[p7$Fael$\"&s6$F[p7$Ffel$\"&Ne% F[p7$F[fl$\"&d^'F[p7$Figl$\"&&=))F[p7$Fchl$\"'en5F[p7$Fhhl$\"'U68F[p7$ F]il$\"(dd^\"F-7$Fgil$\"(.&z;F-7$Fajl$\"($RE=F-7$Fe[m$\"(hT*>F-7$F]^m$ \"(0o5#F-7$Fg^m$\"(bY<#F-7$Fa_m$\"(qjC#F-7$F[`m$\"(v`E#F-7$F_am$\"(-&y AF-7$Fcbm$\"(u9A#F-7$F]cm$\"'v-@F[p7$Fgcm$\"'J+>F[p7$Fadm$\"'Lw;F[p7$F [em$\"'W9:F[p7$F_fm$\"'e*R\"F[p7$Fdfm$\"''QG\"F[p7$Fifm$\"'Jl6F[p7$Fbb t$\"'4G5F[p7$F^gm$\"&pk)F[p7$Fjbt$\"&8g(F[p7$Fcgm$\"&@g'F[p7$Fjap$\"&i /%F[p7$Fhgm$\"&f$>F[p7$Fbbp$\"%9nF[p7$F]hm$!%FfF[p7$Fbhm$!&(f?F[p7$Fgh m$!&k#GF[p7$F`cp$!&j#HF[p7$Fecp$!&@3$F[p7$Fjcp$!&k6$F[p7$F_dp$!&2:$F[p 7$$\"5-+D19>zl]ZF[p$!&=B$F[p7$Fddp$!&+B$F[p7$$\"5omTNYLN1xZF[p$!&mA$F[ p7$F\\im$!&!fKF[p7$F\\ep$!&N>$F[p7$Faim$!&;3$F[p7$Ffim$!&r#GF[p-F[jm6& F]jmF^jmFhepF(-Fejm6#%Hscheme~with~c[5]=c[6]=3/4~and~b[5]=b[6]G-%%FONT G6$%*HELVETICAG\"\"*-%+AXESLABELSG6$Q\"x6\"Q!F[cv-%&TITLEG6#%Uerror~cu rves~for~7~stage~order~6~Runge-Kutta~methodsG-%%VIEWG6$;F(Ffim%(DEFAUL TG" 1 2 0 1 10 0 2 9 1 4 2 1.000000 45.000000 45.000000 0 1 "Butcher's scheme A" "scheme with simple nodes" "scheme with a relatively large \+ stability region" "Butcher's scheme B with c[5]=c[6]=1/2 and b[5]=b[6] " "scheme with c[5]=c[6]=3/4 and b[5]=b[6]" }}}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 455 "evalf[20](plot([u (x)-'un_RK6_2'(x),u(x)-'un_RK6_3'(x),u(x)-'un_RK6_4'(x),u(x)-'un_RK6_5 '(x)],x=0..5,\ncolor=[COLOR(RGB,.45,0,.95),COLOR(RGB,0,.25,1),COLOR(RG B,0,.75,.2),COLOR(RGB,.95,.45,0)],\nlegend=[`scheme with simple nodes` ,`scheme with a relatively large stability region`,`Butcher's scheme B with c[5]=c[6]=1/2 and b[5]=b[6]`,`scheme with c[5]=c[6]=3/4 and b[5] =b[6]`],font=[HELVETICA,9],title=`error curves for 7 stage order 6 Run ge-Kutta methods`));" }}{PARA 13 "" 1 "" {GLPLOT2D 991 549 549 {PLOTDATA 2 "6*-%'CURVESG6%7dv7$$\"\"!F)F(7$$\"5MLLLL3x&)*3\"!#?$\"'ea PF-7$$\"5+++]i!R(*Rc\"F-$\"'P<`F-7$$\"5nmmm\"H2P\"Q?F-$\"'4)G'F-7$$\"5 ++]7G))>Wr@F-$\"'J[jF-7$$\"5MLLek.pu/BF-$\"'=0QCF-$\" '1;kF-7$$\"5+++]PMnNrDF-$\"'QNjF-7$$\"5nmmT5ll'z$GF-$\"';'3'F-7$$\"5ML LL$eRwX5$F-$\"'XwcF-7$$\"5MLLLL$eI8k$F-$\"'2EZF-7$$\"5NLLL$3x%3yTF-$\" 'F-7$$\"5NLLLeR-/PiF-$\"&Aq\"!#>7$$\"5qmmm\"zWo)\\nF -$\"&z_\"F`p7$$\"5-+++DcmpisF-$\"&bT\"F`p7$$\"5OLLLe*)>VB$)F-$\"&JH\"F `p7$$\"5.+++DJbw!Q*F-$\"&HB\"F`p7$$\"5nmmm;/j$o/\"F`p$\"&u>\"F`p7$$\"5 MLLL3_>jU6F`p$\"&`<\"F`p7$$\"5++++]i^Z]7F`p$\"&\"\\6F`p7$$\"5++++](=h( e8F`p$\"&t6\"F`p7$$\"5++++]P[6j9F`p$\"&$46F`p7$$\"5nmm\"HKR'\\5:F`p$\" &(z6F`p7$$\"5MLL$e*[z(yb\"F`p$\"&.O\"F`p7$$\"5nm;/Ev&[ge\"F`p$\"&\\d\" F`p7$$\"5+++Dc,#>Uh\"F`p$\"&K#>F`p7$$\"5ML$eky#)*QU;F`p$\"'#*[CF-7$$\" 5nmmm;a/cq;F`p$\"')))>$F-7$$\"5nmmmm;t,mF`p$\"'lo\"*F-7$$\"5MLLe9; 0?E>F`p$\"'M[rF-7$$\"5++]i!Rgs2&>F`p$\"'Q(o#F-7$$\"5nmmmm\"pW`(>F`p$!' KS5F-7$$\"5MLek.HW#)))>F`p$!'L$y$F-7$$\"5++]iSmTI-?F`p$!'U/oF-7$$\"5nm Tgx.Ry:?F`p$!'l7')F-7$$\"5MLLe9TOEH?F`p$!(\\V4\"F-7$$\"5,+Dc^yLuU?F`p$ !(GKD\"F-7$$\"5om;a)e6Bi0#F`p$!(\"*RK\"F-7$$\"5MekyA]Iff?F`p$!(W$[8F-7 $$\"5,]7.d%)H'H1#F`p$!(`bN\"F-7$$\"5oTgF\"*=HLm?F`p$!(sWN\"F-7$$\"5ML3 _D`Gqp?F`p$!(%z]8F-7$$\"5,Dcwf(ysI2#F`p$!(b\"[8F-7$$\"5o;/,%>sUk2#F`p$ !(6[M\"F-7$$\"5M3_DGcE\")z?F`p$!(JyI\"F-7$$\"5,++]i!f#=$3#F`p$!(q-I\"F -7$$\"5,+]7G)[=g4#F`p$!(#f27F-7$$\"5,++v$fQa)3@F`p$!(F'>5F-7$$\"5,+]Pf $G!p@@F`p$!'\"eT(F-7$$\"5,+++D\"=EX8#F`p$!'us_F-7$$\"5,+]i!*y?OZ@F`p$! 'Z3FF-7$$\"5,++Dcwz>g@F`p$\"'4S=F-7$$\"5,+](=U(Q.t@F`p$\"'fUUF-7$$\"5, ++](=xpe=#F`p$\"'b\"f'F-7$$\"5ML3_]%oi#*>#F`p$\"'*R\"**F-7$$\"5om;a8(f bE@#F`p$\"(v>D\"F-7$$\"5,+Dcw4&[gA#F`p$\"(`4S\"F-7$$\"5MLLeRA9WRAF`p$ \"(_xf\"F-7$$\"5nmTg-NV$GD#F`p$\"(z=q\"F-7$$\"5++]ilZsAmAF`p$\"(j'QFwAF`p$\"(h!fpxg#F`p$!%H_F`p7$$\"5om;/,>Ww>EF`p$!%\"*pF`p7$$\"5,++D1R'f0uzEF`p$!&*45F`p7$$\"5,++]Pf4t.FF`p$!&@/\"F`p7$$\"5omm\"zWi^bv#F`p$ !&$R5F`p7$$\"5MLLLe*Gst!GF`p$!%*y*F`p7$$\"5,++++DRW9HF`p$!%-#)F`p7$$\" 5,+++DJE>>IF`p$!%/rF`p7$$\"5,++]i!RU07$F`p$!%;mF`p7$$\"5,+++v=S2LKF`p$ !%3nF`p7$$\"5ommmm\"p)=MLF`p$!%AuF`p7$$\"5,+++](=]@W$F`p$!%$3*F`p7$$\" 5MLL$e*[$z*RNF`p$!&x6\"F`p7$$\"5omm\"Hd!fX$f$F`p$!&z=\"F`p7$$\"5,+++]i C$pk$F`p$!&%=6F`p7$$\"5om;zp)4\"4sOF`p$!%W$*F`p7$$\"5MLLe*[t\\sp$F`p$! %CrF`p7$$\"5,+]P4r$3Cs$F`p$!%Z?F`p7$$\"5omm;H2qcZPF`p$\"%5OF`p7$$\"5ML L$3_0j,!QF`p$\"&I0$F`p7$$\"5,++]7.\"fF&QF`p$\"&49(F`p7$$\"5om;aQG-ZyQF `p$\"&fh*F`p7$$\"5MLLek`8=/RF`p$\"'rm7F`p7$$\"5,+]i!*yC*)HRF`p$\"'Sw:F `p7$$\"5ommm;/OgbRF`p$\"'qqy9zvRF`p$\"'hl=F`p7$$\"5,+]7`p2_#)RF `p$\"'$Q'=F`p7$$\"5M$eRs31]#*)RF`p$\"'\"Q&=F`p7$$\"5omTN@_$zf*RF`p$\"' Z[=F`p7$$\"5,](oaNk3F+%F`p$\"'n\"y\"F`p7$$\"5MLLe*[$zV4SF`p$\"(/tr\"F- 7$$\"5,+D\"yv^'*G-%F`p$\"(fZa\"F-7$$\"5om;/E+^NOSF`p$\"(')*f8F-7$$\"5M L3F%Ho8)\\SF`p$\"'E&4*F-7$$\"5,++]ilAFjSF`p$\"'#*=eF-7$$\"5M$3x\")HP`( pSF`p$\"'P$3$F-7$$\"5omT&Q.[Mi2%F`p$\"'%=W#F-7$$\"5,]7`p(e:F3%F`p$!&<. *F-7$$\"5ML$3_]p'>*3%F`p$!'\\()GF-7$$\"5o;a)3C!yn&4%F`p$!'$4b%F-7$$\"5 ,+Dcw4*e@5%F`p$!'_M!)F-7$$\"5M$eRAr,S'3TF`p$!'i)G*F-7$$\"5omm\"zW7@^6% F`p$!(c::\"F-7$$\"5NL3F>RL3GTF`p$!(^.`\"F-7$$\"5-+]i!RbX59%F`p$!(w8(>F -7$$\"5om\"z>'ox+aTF`p$!(:+9#F-7$$\"5NLLLL$)*pp;%F`p$!(r1C#F-7$$\"5N$3 -8v\"RGqTF`p$!('GkAF-7$$\"5NL3Fp^yftTF`p$!(mFE#F-7$$\"5N$eRsey6p<%F`p$ !($QeAF-7$$\"5NL$3_+sD-=%F`p$!(BI@#F-7$$\"5N$3xJUlRN=%F`p$!(]0@#F-7$$ \"5NLe9T)e`o=%F`p$!(q-?#F-7$$\"5N$e9\"fAv;!>%F`p$!(gf3#F-7$$\"5NLL3xc9 [$>%F`p$!(/F3#F-7$$\"5NL3-8D$4,?%F`p$!(9r)=F-7$$\"5NL$e*[$>Pn?%F`p$!(h V'=F-7$$\"5NLe*[=1lL@%F`p$!(k&>;F-7$$\"5NLL$3-$H**>UF`p$!(iiI\"F-7$$\" 5NL$3Fpm[KB%F`p$!'$GR*F-7$$\"5NLLek.W]YUF`p$!'Ne_F-7$$\"5NL3_+sA8`UF`p $!'T)Q\"F-7$$\"5NL$ek.9g(fUF`p$\"',$4#F-7$$\"5NLeRs3!)QmUF`p$\"'%>*GF- 7$$\"5NLLL3xe,tUF`p$\"'!*[nF-7$$\"5o;ajMj))ezUF`p$\"'55'*F-7$$\"5-+v$4 '\\=;'G%F`p$\"(%>j5F-7$$\"5N$eRse$[t#H%F`p$\"(%G19F-7$$\"5om;a8AyI*H%F `p$\"([\\f\"F-7$$\"5NLe9m%z`CJ%F`p$\"'b&)>F`p7$$\"5-++v=n(*fDVF`p$\"'o &>#F`p7$$\"5omTNrRduQVF`p$\"'a6CF`p7$$\"5NL$eRAr\"*=N%F`p$\"'OqDF`p7$$ \"5o;/E])pk%eVF`p$\"''>c#F`p7$$\"5-+Dcw%oP]O%F`p$\"']#e#F`p7$$\"5oTNr* y#[Z%F`p$\"';58F`p7$$\"5ommT5::^-XF`p$\"&q$))F`p7$$\"5MLL$3_5,-`%F`p $\"&%\\_F`p7$$\"5,++DJ&p!*yb%F`p$\"&^2$F`p7$$\"5ommmT&G!e&e%F`p$\"%F'* F`p7$$\"5-++]P%37^j%F`p$!&(*Q\"F`p7$$\"5NLLLL$)Qk%o%F`p$!&T^#F`p7$$\"5 om\"z%\\!pYyp%F`p$!&3l#F`p7$$\"5-+]il(\\\\5r%F`p$!&r$GF`p7$$\"5NL3x\"[ I_Us%F`p$!&4)GF`p7$$\"5omm\"z>6but%F`p$!&#=HF`p7$$\"5NL$3-jsgQw%F`p$!& &zHF`p7$$\"5-++]iSjE!z%F`p$!&!oHF`p7$$\"5-++++DM\"3%[F`p$!&E#GF`p7$$\" 5-++]P40O\"*[F`p$!&kj#F`p7$$\"\"&F)$!&_F#F`p-%&COLORG6&%$RGBG$\"#X!\"# F($\"#&*F_en-%'LEGENDG6#%9scheme~with~simple~nodesG-F$6%7isF'7$F+$\"'M (H'F-7$F1$\"'Vp))F-7$F6$\"(0'p5F-7$$\"5MLe*)fI&*y/@F-$\"(Px4\"F-7$F;$ \"($>)3\"F-7$$\"5nmTN'fW%4QAF-$\"(p\"36F-7$F@$\"(Cu7\"F-7$FE$\"(2)G6F- 7$FJ$\"(-\"G6F-7$$\"5ML$eR(\\;m/FF-$\"(]!Q6F-7$FO$\"(+`7\"F-7$$\"5++]( o/[r7(HF-$\"(h46\"F-7$FT$\"(\"z+6F-7$FY$\"(F5,\"F-7$Fhn$\"'84\"*F-7$Fb o$\"'87uF-7$F\\p$\"&NO'F`p7$Fgp$\"&oj&F`p7$F\\q$\"&q6&F`p7$Faq$\"&^v%F `p7$Ffq$\"&n_%F`p7$F[r$\"&_V%F`p7$F`r$\"&5W%F`p7$Fer$\"&(QXF`p7$Fjr$\" &km%F`p7$F_s$\"&4v%F`p7$Fds$\"&z#\\F`p7$Fis$\"&V3&F`p7$F^t$\"&.K&F`p7$ Fct$\"'%Qn&F-7$Fht$\"'C!='F-7$$\"5nmmmT&)))G=(*)zj'F-7$F_x$!(+$G7F-7$Fix$ !(&4]?F-7$Fcy$!(!H/EF-7$Fhy$!(*Q*z#F-7$F]z$!(?A(GF-7$Fgz$!(U$*)GF-7$Fa [l$!(kt&GF-7$Ff[l$!(Aq%GF-7$F[\\l$!(5(RGF-7$F`\\l$!(84w#F-7$Fe\\l$!(a] u#F-7$F_]l$!(LlD#F-7$Fi]l$!($)\\X\"F-7$F^^l$!(5I/\"F-7$Fc^l$!'Q$>$F-7$ Fh^l$\"&5(pF-7$F]_l$\"'?QXF-7$Fb_l$\"(QD+\"F-7$Fg_l$\"(+qW\"F-7$F\\`l$ \"(=vr\"F-7$Fa`l$\"(<=4#F-7$F[al$\"(O%QCF-7$Fibl$\"(z6k#F-7$$\"5nmTg_Z >J0BF`p$\"(-#QEF-7$$\"5nm;a8A3hF`p7$F^]m$\"'V0AF`p7$Fb^m$\" '%4/#F`p7$Ff_m$\"(mrd\"F-7$F[`m$\"(JQ@\"F-7$F``m$\"'yC')F-7$Fe`m$\"&C# **F-7$Fj`m$!'(fO%F-7$Fdam$!'+9(*F-7$F^bm$!(*G$y\"F-7$Fhbm$!(<*\\DF-7$F bcm$!(,31$F-7$Fgcm$!()z.OF-7$F\\dm$!(=`@%F-7$Fadm$!(AyV%F-7$Ffdm$!(lRb %F-7$Fjem$!(zNX%F-7$F^gm$!()*pA%F-7$Fcgm$!(\"=2RF-7$Fhgm$!(Tu'QF-7$F]h m$!(6jZ$F-7$Fbhm$!(uA)HF-7$Fghm$!(::S#F-7$F\\im$!(ezu\"F-7$Faim$!(x/9 \"F-7$Ffim$!'M/fF-7$F[jm$!'hyXF-7$F`jm$\"'wg:F-7$Fejm$\"'G!>'F-7$Fjjm$ \"'.0zF-7$F_[n$\"(zLN\"F-7$Fd[n$\"(3Un\"F-7$Fi[n$\"'w_BF`p7$F^\\n$\"'w \\FF`p7$Fh\\n$\"'g#e$F`p7$Ff^n$\"'[*z$F`p7$F`_n$\"'Y.OF`p7$Fj_n$\"'IRH F`p7$Fd`n$\"'Fa@F`p7$F^an$\"'nX;F`p7$Fcan$\"'!QL\"F`p7$Fhan$\"'-P6F`p7 $F\\cn$\"'3/5F`p7$Ffcn$\"&K9*F`p7$F`dn$\"&Y,)F`p7$Fedn$\"&@A(F`p-Fjdn6 &F\\enF($\"#DF_en$\"\"\"F)-Fcen6#%Pscheme~with~a~relatively~large~stab ility~regionG-F$6%7iuF'7$$\"5qmmmmT&)G\\a!#@$\"(XV[#Ffbp7$F+$\"'#*\\`F -7$F1$\"'!Q3(F-7$F6$\"'HbyF-7$F;$\"'p=yF-7$F@$\"'$\\x(F-7$FE$\"'z1wF-7 $FJ$\"'Z/uF-7$FO$\"'yboF-7$FT$\"'z'='F-7$FY$\"'o5]F-7$Fhn$\"'L]TF-7$$ \"5ommT5:j=XWF-$\"':uQF-7$F]o$\"')>r$F-7$$\"5om;/^J'Qe%[F-$\"'n_OF-7$$ \"5NLLek.%*Qz\\F-$\"',8OF-7$$\"5++]7yv,%H6&F-$\"'Nd\"F`p$\"&%pKF` p7$Fis$\"&3K$F`p7$$\"5MLe9T))Q8+;F`p$\"&SS$F`p7$F^t$\"&&3NF`p7$Fct$\"' 8%*QF-7$Fht$\"'e\\XF-7$F_\\o$\"'nOhF-7$F]u$\"'/!o)F-7$Fgu$\"((z=7F-7$F [w$\"(!f:9F-7$$\"5ML3_]%QU$*)=F`p$\"(gqU\"F-7$F`w$\"(YcU\"F-7$$\"5,+Dc EsW\"R\">F`p$\"(;F-7$Fc^l$\"(O(RBF-7$F]_ l$\"(GAc#F-7$Fg_l$\"(Co&GF-7$Fa`l$\"(Aw'HF-7$F[al$\"(`!3HF-7$Fibl$\"(s co#F-7$F[do$\"(WSb#F-7$F`do$\"(gwR#F-7$Fedo$\"(IB;#F-7$F^cl$\"(.2*>F-7 $F]eo$\"(eql\"F-7$Fccl$\"'Ve7F`p7$$\"5ML$ek`1OzT#F`p$\"&C))*F`p7$Fhcl$ \"&*ztF`p7$$\"5nm;/^c++rCF`p$\"&'fcF`p7$F]dl$\"&R'[F`p7$$\"5M$3-jP(=U/ DF`p$\"&Ho%F`p7$$\"5ML3FW&p68^#F`p$\"&,b%F`p7$$\"5M$eRAr^,#=DF`p$\"&@[ %F`p7$$\"5ML$3-)Q84DDF`p$\"&HT%F`p7$$\"5MLe9;#)4()QDF`p$\"&pO%F`p7$Fbd l$\"&rT%F`p7$$\"5ML$eRA\"*4-e#F`p$\"&8r%F`p7$Fgdl$\"&-&\\F`p7$F[fl$\"& qn&F`p7$Fefl$\"&!RjF`p7$Fjfl$\"&kx'F`p7$F_gl$\"&*RqF`p7$$\"5,++vV)>ST$ GF`p$\"&N9(F`p7$$\"5omm;H2\"34'GF`p$\"&!>sF`p7$$\"5MLLe9;gn()GF`p$\"&$ *>(F`p7$Fdgl$\"&*>sF`p7$Figl$\"&b<(F`p7$F^hl$\"&'=rF`p7$Fchl$\"&\"yqF` p7$Fhhl$\"&N0(F`p7$$\"5MLLLeR%p\")Q$F`p$\"&v-(F`p7$F]il$\"&e#pF`p7$$\" 5omm\"H#oZ1\"\\$F`p$\"&Tx'F`p7$Fbil$\"&:_'F`p7$Fgil$\"&0='F`p7$F\\jl$ \"&9u&F`p7$$\"5MLe*)f!y6&fOF`p$\"&Wc&F`p7$Fajl$\"&!eaF`p7$$\"5,+voz;/n %o$F`p$\"&WT&F`p7$Ffjl$\"&\\P&F`p7$$\"5om\"z%*H0H)4PF`p$\"&LH&F`p7$F[[ m$\"&5L&F`p7$$\"5ML3F>*o()\\t$F`p$\"&&=aF`p7$F`[m$\"&+c&F`p7$$\"5,+++D J]'Qx$F`p$\"&F='F`p7$Fe[m$\"&<\\(F`p7$$\"5ommm;z5YEQF`p$\"&g-*F`p7$Fj[ m$\"'s'=\"F`p7$F_\\m$\"'+*[\"F`p7$Fd\\m$\"'u&)=F`p7$$\"5omTgF;p.(=`>SF`p$\"(;4:$F-7$F[`m$\"(rs:$F- 7$$\"5M$3F>*3eiHSF`p$\"(8F:$F-7$F``m$\"(u%oJF-7$Fe`m$\"(XH6$F-7$Fj`m$ \"(/w3$F-7$Fdam$\"((fiIF-7$F^bm$\"(Z\\,$F-7$$\"5,vo/t[sV#4%F`p$\"(V6+$ F-7$Fcbm$\"(py+$F-7$$\"5MeRs3c$=*)4%F`p$\"('4**HF-7$Fhbm$\"(Q`)HF-7$F] cm$\"(!)*))HF-7$Fbcm$\"(XT)HF-7$F\\dm$\"(ot,$F-7$Ffdm$\"(DU3$F-7$F^gm$ \"(%*[A$F-7$Fbhm$\"(JuX$F-7$F\\im$\"(u%\\OF-7$F`jm$\"(#[=SF-7$Fd[n$\"( a$*G%F-7$F^\\n$\"'w(Q%F`p7$Fh\\n$\"'RuUF`p7$Ff^n$\"'t6RF`p7$$\"5ML3_vS ME!R%F`p$\"'m&f$F`p7$F[_n$\"'9TLF`p7$$\"5om\"H#o2IU9WF`p$\"'5tIF`p7$F` _n$\"'***y#F`p7$$\"5,+v$4Yd#eQWF`p$\"'^bCF`p7$Fe_n$\"'Ca?F`p7$$\"5MLek `T@uiWF`p$\"'B,=F`p7$Fj_n$\"'2k:F`p7$$\"5ML$3_+sm')[%F`p$\"'+$G\"F`p7$ F_`n$\"&O(**F`p7$$\"5,+]i:5jN;XF`p$\"&xK)F`p7$Fd`n$\"&X/'F`p7$Fi`n$\"& ><%F`p7$F^an$\"&n%HF`p7$$\"5NLekGg6B`#o\"F-$\"' \"\\v\"F-7$$\"5MLL3xJs1,=F-$\"'%3$=F-7$$\"5++]PM_@g>>F-$\"'kS=F-7$F6$ \"'\\P=F-7$F;$\"'1==F-7$F@$\"'!\\y\"F-7$FE$\"'?JPF-7$Fgo$!& QV&F-7$F\\p$!%'['F`p7$Fbp$!%XqF`p7$Fgp$!%)G(F`p7$F\\q$!%.tF`p7$Faq$!%z qF`p7$Ffq$!%SpF`p7$F[r$!%XpF`p7$F`r$!%lqF`p7$Fer$!%&=(F`p7$$\"5++++]7! Q4T\"F`p$!%eqF`p7$Fjr$!%(p'F`p7$F_s$!%rdF`p7$Fds$!%=VF`p7$F^t$!$d$F`p7 $Fht$\"&MT(F-7$F_\\o$\"'-O;F-7$F]u$\"'81IF-7$Fgu$\"'Jn^F-7$F[w$\"',wrF -7$Few$\"'7&y)F-7$F_x$\"'*))*)*F-7$Fcy$\"(#\\y5F-7$Fe\\l$\"(,x<\"F-7$F i]l$\"(&)QG\"F-7$F]_l$\"(\"fV8F-7$Fa`l$\"(?&o7F-7$Fibl$\"(?\"G5F-7$F`d o$\"'J`*)F-7$F^cl$\"'9]sF-7$F]eo$\"'o8fF-7$Fccl$\"&%eUF`p7$Fhcl$\"&%*z \"F`p7$F]dl$\"$\"QF`p7$Fbdl$!&f=\"F`p7$Fgdl$!&ru\"F`p7$$\"5,+vV)*3ow8E F`p$!&]\"=F`p7$F\\el$!&p(=F`p7$$\"5MLek.H?wDEF`p$!&J(=F`p7$Fael$!&'Q>F `p7$Ffel$!&s(>F`p7$F[fl$!&$4?F`p7$$\"5,+](=#*HXxm#F`p$!&B/#F`p7$F`fl$! &`4#F`p7$$\"5om;HKRdt\"p#F`p$!&A6#F`p7$Fefl$!&f6#F`p7$Fjfl$!&d6#F`p7$F _gl$!&(p?F`p7$Fdgl$!&W$>F`p7$Figl$!&P#=F`p7$F^hl$!&tw\"F`p7$Fchl$!&#o< F`p7$Fhhl$!&m#=F`p7$F]il$!&$R>F`p7$F]gq$!&X*>F`p7$Fbil$!&O.#F`p7$$\"5, +]PMFwrmNF`p$!&^0#F`p7$Fgil$!&k.#F`p7$$\"5ML$e9T=%>?OF`p$!&x(>F`p7$F\\ jl$!&\"H>F`p7$Fajl$!&5w\"F`p7$Ffjl$!&#*f\"F`p7$F[[m$!&-C\"F`p7$F`[m$!% =))F`p7$Fe[m$\"%%\\(F`p7$Fj[m$\"&s6$F`p7$F_\\m$\"&Ne%F`p7$Fd\\m$\"&d^' F`p7$Fi\\m$\"&&=))F`p7$F^]m$\"'en5F`p7$Fb^m$\"'U68F`p7$Ff_m$\"(dd^\"F- 7$F``m$\"(.&z;F-7$Fj`m$\"($RE=F-7$Fbcm$\"(hT*>F-7$Ffdm$\"(0o5#F-7$F^gm $\"(bY<#F-7$Fbhm$\"(qjC#F-7$F\\im$\"(v`E#F-7$F`jm$\"(-&yAF-7$Fd[n$\"(u 9A#F-7$F^\\n$\"'v-@F`p7$Fh\\n$\"'J+>F`p7$Ff^n$\"'Lw;F`p7$Fabr$\"'W9:F` p7$F[_n$\"'e*R\"F`p7$Fibr$\"''QG\"F`p7$F`_n$\"'Jl6F`p7$Facr$\"'4G5F`p7 $Fe_n$\"&pk)F`p7$Ficr$\"&8g(F`p7$Fj_n$\"&@g'F`p7$F_`n$\"&i/%F`p7$Fd`n$ \"&f$>F`p7$Fi`n$\"%9nF`p7$F^an$!%FfF`p7$Fcan$!&(f?F`p7$Fhan$!&k#GF`p7$ F]bn$!&j#HF`p7$Fbbn$!&@3$F`p7$Fgbn$!&k6$F`p7$F\\cn$!&2:$F`p7$$\"5-+D19 >zl]ZF`p$!&=B$F`p7$Facn$!&+B$F`p7$$\"5omTNYLN1xZF`p$!&mA$F`p7$Ffcn$!&! fKF`p7$F[dn$!&N>$F`p7$F`dn$!&;3$F`p7$Fedn$!&r#GF`p-Fjdn6&F\\enF`enF]en F(-Fcen6#%Hscheme~with~c[5]=c[6]=3/4~and~b[5]=b[6]G-%%FONTG6$%*HELVETI CAG\"\"*-%+AXESLABELSG6$Q\"x6\"Q!F]dt-%&TITLEG6#%Uerror~curves~for~7~s tage~order~6~Runge-Kutta~methodsG-%%VIEWG6$;F(Fedn%(DEFAULTG" 1 2 0 1 10 0 2 9 1 4 2 1.000000 45.000000 45.000000 0 1 "scheme with simple no des" "scheme with a relatively large stability region" "Butcher's sche me B with c[5]=c[6]=1/2 and b[5]=b[6]" "scheme with c[5]=c[6]=3/4 and \+ b[5]=b[6]" }}}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 " " {MPLTEXT 1 0 1 ";" }}}}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 46 "Test 7 o f 7 stage, order 6 Runge-Kutta methods" }}{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 "Sol ution: " }}{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,F 1F*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 := dif f(y(x),x)=-(1+4*cos(3*x))*(y(x)-1/3);\nic := y(0)=1;\nsimplify(dsolve( \{de,ic\},y(x)));\nv := unapply(rhs(%),x):\nplot(v(x),x=0..5,0..1.1,fo nt=[HELVETICA,9],labels=[`x`,`y(x)`]);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%#deG/-%%diffG6$-%\"yG6#%\"xGF,,$*&,&\"\"\"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#,&F'F9*&#F3F,F+F4F+F9F+F+" }}{PARA 13 "" 1 "" {GLPLOT2D 503 318 318 {PLOTDATA 2 "6&-%'CURVESG6$7ap7$$\"\" !F)$\"\"\"F)7$$\"3gmmTN@Ki8!#>$\"3W+7cSy5h&*!#=7$$\"3ALL$3FWYs#F/$\"3K tP[t*Q;:*F27$$\"3%)***\\iSmp3%F/$\"3g.\"H>f!3q()F27$$\"3WmmmT&)G\\aF/$ \"36p*p.:G\\T)F27$$\"3m****\\7G$R<)F/$\"3a?glh]$zx(F27$$\"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$$\"33++](=h(e8Fgt$\"3l&QV-<82M)F27 $$\"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$$\"3UL$e*[z(yb\"Fg t$\"3m)))[\\1qQl)F27$$\"34+Dc,#>Uh\"Fgt$\"3(fTb\\\\y3J)F27$$\"3wmm;a/c q;Fgt$\"3-!y\"yF27$$\"3\"pm;a)))G=BtF27$$\"3%om mmJFgt$\"3%RlX>.MR=&F27$$\"3gmmm \"pW`(>Fgt$\"3+6YS9:C2[F27$$\"3dLe9TOEH?Fgt$\"3!eWte3T%oWF27$$\"3K+]i! f#=$3#Fgt$\"3:XZ<;2j,UF27$$\"3?+](=xpe=#Fgt$\"3E#Q(H44MbQF27$$\"37nm\" H28IH#Fgt$\"3MH4)f2==l$F27$$\"3um;zpSS\"R#Fgt$\"3wpxg#Fgt$\"37l*=e[EHY$F27$$ \"33+]Pf4t.FFgt$\"35!4Ne]qiX$F27$$\"3uLLe*Gst!GFgt$\"3U+pq))z7kMF27$$ \"30+++DRW9HFgt$\"37'z:1TS%*[$F27$$\"3:++DJE>>IFgt$\"3N!o4Joz]`$F27$$ \"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*RNFgt$\"3UAxt;S)>+% F27$$\"3%o;Hd!fX$f$Fgt$\"3+h91z&\\y+%F27$$\"3e++]iC$pk$Fgt$\"3eIRs#H!Q \"*RF27$$\"3ILe*[t\\sp$Fgt$\"3m\"Rx)H&*[cRF27$$\"3[m;H2qcZPFgt$\"3w))) [$RF!f!RF27$$\"3O+]7.\"fF&QFgt$\"3+Efp,iIqPF27$$\"3Ymm;/OgbRFgt$\"3W-T ml[`MOF27$$\"3w**\\ilAFjSFgt$\"3&zNMj#[Z%Fgt$\"3ADU\\K%G5O$F27$$\"3SnmT&G!e&e%Fgt$\"3 5gRzc#\\LF27$$\"\"&F)$\"3Ii# 4)y!3AN$F2-%'COLOURG6&%$RGBG$\"#5!\"\"F(F(-%%FONTG6$%*HELVETICAG\"\"*- %+AXESLABELSG6$%\"xG%%y(x)G-%%VIEWG6$;F(Fiel;F($\"#6Fcfl" 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 c onstructs a " }{TEXT 260 17 "discrete solution" }{TEXT -1 44 " based o n 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 0 761 "V := (x,y) -> -(1+4*cos(3*x))*(y-1/3): hh := 0.02: n umsteps := 250: x0 := 0: y0 := 1:\nmatrix([[`slope field: `,V(x,y)], [`initial point: `,``(x0,y0)],[`step width: `,hh],\n[`no. of steps: \+ `,numsteps]]);``;\nmthds := [`Butcher's scheme A`,`scheme with simpl e nodes`,`scheme with a relatively large stability region`,`Butcher's \+ scheme B with `*(c[5]=1/2,c[6]=1/2,b[5]=b[6]),`scheme with `*(c[5]=3/4 ,c[6]=3/4,b[5]=b[6])]:\nerrs := []:\nDigits := 30:\nfor ct to 5 do\n \+ Vn_RK6_||ct := RK6_||ct(V(x,y),x,y,x0,y0,hh,numsteps,false);\n sm : = 0: numpts := nops(Vn_RK6_||ct):\n for ii to numpts do\n sm := sm+(Vn_RK6_||ct[ii,2]-v(Vn_RK6_||ct[ii,1]))^2;\n end do:\n errs : = [op(errs),sqrt(sm/numpts)];\nend do:\nDigits := 10:\nlinalg[transpos e]([mthds,evalf(errs)]);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#K%'matrixG 6#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)pprint306\"" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#%!G" }}{PARA 11 "" 1 "" {XPPMATH 20 "6 #K%'matrixG6#7'7$%3Butcher's~scheme~AG$\"+cOJ3@!#>7$%9scheme~with~simp le~nodesG$\"+5b(zz'!#@7$%Pscheme~with~a~relatively~large~stability~reg ionG$\"+5NdRhF07$*&%9Butcher's~scheme~B~with~G\"\"\"6%/&%\"cG6#\"\"&#F 8\"\"#/&F<6#\"\"'F?/&%\"bGF=&FGFCF8$\"+cG$=3\"F+7$*&%-scheme~with~GF86 %/F;#\"\"$\"\"%/FBFPFEF8$\"+y.iB%)F0Q)pprint316\"" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 30 "The following code constr ucts " }{TEXT 260 20 "numerical procedures" }{TEXT -1 56 " for solutio ns 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 p oint where " }{XPPEDIT 18 0 "x = 4.999;" "6#/%\"xG-%&FloatG6$\"%**\\! \"$" }{TEXT -1 16 " is also given." }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 692 "V := (x,y) -> -(1+4*cos(3*x))*(y-1/3): hh := 0.02: n umsteps := 250: x0 := 0: y0 := 1:\nmatrix([[`slope field: `,V(x,y)], [`initial point: `,``(x0,y0)],[`step width: `,hh],\n[`no. of steps: \+ `,numsteps]]);``;\nmthds := [`Butcher's scheme A`,`scheme with simpl e nodes`,`scheme with a relatively large stability region`,`Butcher's \+ scheme B with `*(c[5]=1/2,c[6]=1/2,b[5]=b[6]),`scheme with `*(c[5]=3/4 ,c[6]=3/4,b[5]=b[6])]:\nerrs := []:\nDigits := 30:\nfor ct to 5 do\n \+ vn_RK6_||ct := RK6_||ct(V(x,y),x,y,x0,y0,hh,numsteps,true);\nend do: \nxx := 4.999: vxx := evalf(v(xx)):\nfor ct to 5 do\n errs := [op(er rs),abs(vn_RK6_||ct(xx)-vxx)];\nend do:\nDigits := 10:\nlinalg[transpo se]([mthds,evalf(errs)]);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#K%'matrix G6#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)pprint326\"" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%!G" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#K%'matrixG6#7'7$%3Butcher's~scheme~AG$\"+V,\"*e\"*!#@7$%9scheme~wit h~simple~nodesG$\"+_Phv[!#A7$%Pscheme~with~a~relatively~large~stabilit y~regionG$\"+G\"*4]YF07$*&%9Butcher's~scheme~B~with~G\"\"\"6%/&%\"cG6# \"\"&#F8\"\"#/&F<6#\"\"'F?/&%\"bGF=&FGFCF8$\"+qu(f:%F+7$*&%-scheme~wit h~GF86%/F;#\"\"$\"\"%/FBFPFEF8$\"+An(*))[F0Q)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 0 436 "mthds := [`Butcher's scheme A`,`scheme with simple nodes`,`scheme with a relatively large stabili ty region`,`Butcher's scheme B with `*(c[5]=1/2,c[6]=1/2,b[5]=b[6]),`s cheme with `*(c[5]=3/4,c[6]=3/4,b[5]=b[6])]:\nerrs := []:\nDigits := 2 0:\nfor ct to 5 do\n sm := NCint((v(x)-'vn_RK6_||ct'(x))^2,x=0..5,ad aptive=false,numpoints=7,factor=100);\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$%3Butcher's~scheme~AG $\"+-lf1@!#>7$%9scheme~with~simple~nodesG$\"+2WWkn!#@7$%Pscheme~with~a ~relatively~large~stability~regionG$\"+_N!)4hF07$*&%9Butcher's~scheme~ B~with~G\"\"\"6%/&%\"cG6#\"\"&#F8\"\"#/&F<6#\"\"'F?/&%\"bGF=&FGFCF8$\" +s%e73\"F+7$*&%-scheme~with~GF86%/F;#\"\"$\"\"%/FBFPFEF8$\"+(z)>(R)F0Q )pprint346\"" }}}{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 0 516 "evalf[20](plot(['vn_RK6_1'(x)-v(x),'vn_RK6_2'(x)-v(x ),'vn_RK6_3'(x)-v(x),'vn_RK6_4'(x)-v(x),\n'vn_RK6_5'(x)-v(x)],x=0..5,f ont=[HELVETICA,9],\ncolor=[COLOR(RGB,.95,.2,0),COLOR(RGB,.45,0,.95),CO LOR(RGB,0,.25,1),COLOR(RGB,0,.75,.2),COLOR(RGB,.95,.45,0)],\nlegend=[` Butcher's scheme A`,`scheme with simple nodes`,`scheme with a relative ly large stability region`,`Butcher's scheme B with c[5]=c[6]=1/2 and \+ b[5]=b[6]`,`scheme with c[5]=c[6]=3/4 and b[5]=b[6]`],title=`error cur ves for 7 stage order 6 Runge-Kutta methods`));" }}{PARA 13 "" 1 "" {GLPLOT2D 1049 534 534 {PLOTDATA 2 "6+-%'CURVESG6%7cr7$$\"\"!F)F(7$$\" 5qmm;aQ`!eS$!#A$\"&mr#!#?7$$\"5SLLL3x1h6oF-$\"(A&oMF07$$\"5,++Dc,;u@5! #@$\")()[5fF07$$\"5ommmTN@Ki8F9$\"*C4`T%F07$$\"5NLL3FpE!Hq\"F9$\"+dm0* 4#F07$$\"5-++]7.K[V?F9$\"+zV;QkF07$$\"5omm\"zptjSQ#F9$\"+uTqHjF07$$\"5 NLLL$3FWYs#F9$\"+dd$yA'F07$$\"5omm;aQ`!eS$F9$\"+z/44lF07$$\"5-+++D1k'p 3%F9$\",hijo:\"F07$$\"5pmmT5SpaFWF9$\",O3nu8\"F07$$\"5OLL$eRZF\"oZF9$ \",1\"f.>6F07$$\"50++D\"y+3(3^F9$\",P`*)z5\"F07$$\"5qmmmmT&)G\\aF9$\", $)*[?N6F07$$F3F9$\",F_>3]\"F07$$\"50+++]7G$R<)F9$\",=vD/%=F07$$\"5qmmT NYL^9&)F9$\",03[,\"=F07$$\"5SLL$3-)Q4b))F9$\",z>&Q\"y\"F07$$\"50++D19W n&>*F9$\",MXj;w\"F07$$\"5qmmm\"z%\\DO&*F9$\",l*\\g#y\"F07$$\"5NLL3x\"[ No()*F9$\",a22j$>F07$$\"5+++Dc,;u@5F0$\",\\gP\"R?F07$$\"5nm;z%\\l*zb5F 0$\",uOmg+#F07$$\"5MLLLL3x&)*3\"F0$\",%*oQZ(>F07$$\"5nmm\"z%\\v#pK\"F0 $\",AYhF2#F07$$\"5+++]i!R(*Rc\"F0$\",-k!RW@F07$$\"5nm;z>6B`#o\"F0$\",I b'e(=#F07$$\"5MLL3xJs1,=F0$\",v#RKZAF07$$\"5nm\"Hd?pM.'=F0$\",%G!e$*=# F07$$\"5++]PM_@g>>F0$\",b*[@P@F07$$\"5ML3-j7'p)y>F0$\",;n2I9#F07$$\"5n mmm\"H2P\"Q?F0$\",)>%4G;#F07$$\"5+++]PMnNrDF0$\",Hlju$>F07$$\"5MLLL$eR wX5$F0$\",wl3%>VB$)F0$\",mJx-X\"F07$$\"5qmmmTg()4_))F0$\",#)RDji\"F07$$\"5.+++DJbw !Q*F0$\",*4pa^=F07$$\"5NLL$ekGkX#**F0$\",9o!oN@F07$$\"5nmmm;/j$o/\"!#> $\",v;g(oCF07$$\"5+++]7GTt%4\"Fc[l$\",mu1^\"GF07$$\"5MLLL3_>jU6Fc[l$\" ,<3#f$>$F07$$\"5nmm;HdNb'>\"Fc[l$\",o`gln$F07$$\"5++++]i^Z]7Fc[l$\",jc %R%=%F07$$\"5+++++v\"=YI\"Fc[l$\",Cw`]p%F07$$\"5++++](=h(e8Fc[l$\",>%[ z`^F07$$\"5++++]7!Q4T\"Fc[l$\",Q'*=,\\&F07$$\"5++++]P[6j9Fc[l$\",.i&Qx cF07$$\"5nm\"HKkAg\\Z\"Fc[l$\",N259p&F07$$\"5ML$ek`h0o[\"Fc[l$\",X(f+) p&F07$$\"5++voH/5l)\\\"Fc[l$\",JP,Ep&F07$$\"5nmm\"HKR'\\5:Fc[l$\",%>H& yn&F07$$\"5++]P4rr=M:Fc[l$\",Cm)y:cF07$$\"5MLL$e*[z(yb\"Fc[l$\",)>Fy3b F07$$\"5nm;/Ev&[ge\"Fc[l$\",&z$H'H`F07$$\"5+++Dc,#>Uh\"Fc[l$\",XL9(3^F 07$$\"5ML$eky#)*QU;Fc[l$\",!>wiF[F07$$\"5nmmm;a/cq;Fc[l$\",0\\Dm`%F07$ $\"5nmmmT&)))G=Fc[l$\",#obp:@F07$$\"5nmmmm\"pW`(>Fc[l$\",^&H(*o=F07$$\"5,++]i!f# =$3#Fc[l$\"-2kOcf:F97$$\"5,++](=xpe=#Fc[l$\"-\"*4Sh27F97$$\"5nmmm\"H28 IH#Fc[l$\",(f=Jv&)F97$$\"5nmm;zpSS\"R#Fc[l$\",,%F97$$\"5MLL$e*)>pxg#Fc[l$\",(*)3H))QF97$$\"5ommm;z+vbEFc[l$ \",X6&\\UPF97$$\"5,++]Pf4t.FFc[l$\",#=E\\)p$F97$$\"5omm\"zWi^bv#Fc[l$ \",>\"HydPF97$$\"5MLLLe*Gst!GFc[l$\",2cuz#RF97$$\"5omm;H2\"34'GFc[l$\" ,*e&RjA%F97$$\"5,++++DRW9HFc[l$\",08fjm%F97$$\"5,+++DJE>>IFc[l$\",/C=7 )fF97$$\"5,++]i!RU07$Fc[l$\",HL6Z$zF97$$\"5,+++v=S2LKFc[l$\"-6E#\\l4\" F97$$\"5ommmm\"p)=MLFc[l$\"-$pvoXU\"F97$$\"5,+++](=]@W$Fc[l$\"-TU#R5w \"F97$$\"5omm\"H#oZ1\"\\$Fc[l$\"-9(y%Gu=F97$$\"5MLL$e*[$z*RNFc[l$\"-In eOY>F97$$\"5omT5:)[[Lb$Fc[l$\"-B]?lc>F97$$\"5,+]PMFwrmNFc[l$\"-PQ$HN'> F97$$\"5MLek`mn3!e$Fc[l$\"-xPM-m>F97$$\"5omm\"Hd!fX$f$Fc[l$\"-IE^Bk>F9 7$$\"5,+v=#\\/Dog$Fc[l$\"-`4P4e>F97$$\"5ML$e9T=%>?OFc[l$\"-qPXMZ>F97$$ \"5om\"H2LKjNj$Fc[l$\"-K+lAL>F97$$\"5,+++]iC$pk$Fc[l$\"-H3>+9>F97$$\"5 MLLe*[t\\sp$Fc[l$\"-c.\\w2=F97$$\"5omm;H2qcZPFc[l$\"->CDw`;F97$$\"5,++ ]7.\"fF&QFc[l$\"-D$*[6_7F97$$\"5ommm;/OgbRFc[l$\",*Ht#et)F97$$\"5,++]i lAFjSFc[l$\",#)>5K*eF97$$\"5NLLLL$)*pp;%Fc[l$\",`#3N^SF97$$\"5NLLL3xe, tUFc[l$\"-)4S2Xu#F-7$$\"5omm;HdO=yVFc[l$\"-QMkYd=F-7$$\"5,++++D>#[Z%Fc [l$\"-<#)=X78F-7$$\"5ommmT&G!e&e%Fc[l$\",sFmm`*F-7$$\"5NLLLL$)Qk%o%Fc[ l$\",D3I;%zF-7$$\"5-++]iSjE!z%Fc[l$\",Z$e+\"R(F-7$$\"5-++]P40O\"*[Fc[l $\",(o&Giw(F-7$$\"\"&F)$\",vYbc<*F--%&COLORG6&%$RGBG$\"#&*!\"#$\"\"#! \"\"F(-%'LEGENDG6#%3Butcher's~scheme~AG-F$6%7bvF'7$F+$!%c8F07$F2$!'()H #F07$FB$!*HtQ/\"F07$FG$!*L*H*>$F07$FL$!*z*R XJF07$FQ$!*5_X4$F07$$\"5-++vo/[AlIF9$!*#HGsIF07$FV$!*@@5@$F07$$\"5NLLe RseQYPF9$!*`:t%RF07$Fen$!*3%*[Z&F07$Fjn$!*)=4$Q&F07$F_o$!*?GaH&F07$Fdo $!*VcyB&F07$Fio$!*7FaL&F07$F^p$!*66/x'F07$Fbp$!*e+x#zF07$F\\q$!*5*HswF 07$Ffq$!*$43;wF07$F[r$!*L(Q#4)F07$F`r$!*:ZMS)F07$Fer$!*$R5n#)F07$Fjr$! *Gsl8)F07$$\"5+]ilZQ9\\>6F0$!*\"3CW!)F07$$\"5nm\"z>'o^7\\6F0$!*e\"[E!) F07$$\"5M$3-j()*)e(y6F0$!*GlBA)F07$$\"5++]i!*GER37F0$!*qDyh)F07$$\"5ML 3F>*3gwE\"F0$!*`(3!Q)F07$F_s$!*R(ey\")F07$$\"5++Dcw4]>'Q\"F0$!*F74N)F0 7$$\"5ML$3_+ZiaW\"F0$!*u)4'R)F07$$\"5nmT&Q.$*HZ]\"F0$!*kg^<)F07$Fds$!* \\))*z!)F07$Fis$!*9wa.)F07$F^t$!*h-i+)F07$Fht$!*!ex2wF07$Fbu$!*O*G1vF0 7$Fgu$!*uX^W'F07$F\\v$!*G/8h&F07$$\"5MLLLL$eI8k$F0$!*JA!)*\\F07$Fav$!* =nV^%F07$Ffv$!*1/Q=%F07$F[w$!*%e1sRF07$F`w$!*+.>!RF07$Few$!*L3j&QF07$F jw$!+8V$H$QF97$F_x$!+!yM3$QF97$Fdx$!*iT`&QF07$Fix$!*^%)[!RF07$F^y$!*ic 2(RF07$Fcy$!*!)[]2%F07$Fhy$!*B30P%F07$F]z$!*7\"Q2[F07$Fbz$!*#\\@e`F07$ Fgz$!*8XZ/'F07$F\\[l$!*A_j&oF07$Fa[l$!*1xGm(F07$Fg[l$!*xUuW)F07$F\\\\l $!*;v-3*F07$Fa\\l$!+lI0.5F07$Ff\\l$!+c#3%)4\"F07$F[]l$!+,'*437F07$F`]l $!+qV0D8F07$Fe]l$!+U7$)>9F07$Fj]l$!+>h4z9F07$F_^l$!+u(yH[\"F07$Fd^l$!+ :g!f[\"F07$Fi^l$!+X!zV[\"F07$F^_l$!+K![.[\"F07$Fc_l$!+C$>;Y\"F07$Fh_l$ !+lk:F9F07$F]`l$!+G,lp8F07$Fb`l$!+I9^.8F07$Fg`l$!+&4%R;7F07$F\\al$!+`/ 5Q6F07$Faal$!+`V)4,\"F07$Ffal$!*K6mH*F07$$\"5Mek`;'G([p*F0 7$$\"5+]iSmbs&Hx\"Fc[l$!*f+'4\"*F07$$\"5nTgF;DsUw(*)z6s# *F07$$\"5o;HKknnZG=Fc[l$!*(f*\\.*F07$$\"5,+D1k1nTN=Fc[l$!**f+E*)F07$$ \"5o\"HKRhn'))Q=Fc[l$!*nDMA*F07$$\"5M$3-QckcB%=Fc[l$!*\"okr$*F07$$\"5, v=n8:m#e%=Fc[l$!*q^]C*F07$$\"5om;aj%e'H\\=Fc[l$!*?)3B\"*F07$$\"5Me9T8a lw_=Fc[l$!*4IB.*F07$$\"5,]7GjBlBc=Fc[l$!*Bu#z!*F07$$\"5oT5:8$\\1(f=Fc[ l$!*_t*p&*F07$$\"5ML3-jik>*F07$F`bl$ !*$f'zJ*F07$$\"5ML3_]%QU$*)=Fc[l$!*16w\\*F07$$\"5om;aQG%G;!>Fc[l$!+(3J N+\"F07$$\"5M$3_D.Xrx!>Fc[l$!*\\34x*F07$$\"5,+DcEsW\"R\">Fc[l$!*([=\"f *F07$$\"5M3xcB$)f)p\">Fc[l$!*8$)fr*F07$$\"5o;Hd?%\\d+#>Fc[l$!+_!G8.\"F 07$$\"5,D\"yv^+HJ#>Fc[l$!+Q9D<5F07$Febl$!+p!RL+\"F07$$\"5nT&)e6F?FH>Fc [l$!*8Y)**)*F07$$\"5+]Pf3QNMK>Fc[l$!*(f)Rz*F07$$\"5Me*)f0\\]TN>Fc[l$!* (>_#z*F07$$\"5nmTg-gl[Q>Fc[l$!+!o_'45F07$$\"5+v$4'*42e:%>Fc[l$!+6W^S5F 07$$\"5M$e9m>eHY%>Fc[l$!+ye$f-\"F07$$\"5n\"z>OH4,x%>Fc[l$!+\"fJ;,\"F07 $$\"5++]i!Rgs2&>Fc[l$!*Shc)**F07$$\"5M3-j([6WQ&>Fc[l$!*4k'3**F07$$\"5n ;aj%ei:p&>Fc[l$!+&o&y+5F07$$\"5+D1k\"o8()*f>Fc[l$!+BiTd5F07$$\"5MLekyZ 'eI'>Fc[l$!+ljwU5F07$$\"5+]ilsp;?p>Fc[l$!+BUY85F07$Fjbl$!*?_<***F07$$ \"5MLek.HW#)))>Fc[l$!+/v@;5F07$$\"5++]iSmTI-?Fc[l$!+`L@T5F07$$\"5M$e9 \"4NS/4?Fc[l$!+\\I)z+\"F07$$\"5nmTgx.Ry:?Fc[l$!*wQp!**F07$$\"5Me*[=\"Q Q:>?Fc[l$!+Xbo>5F07$$\"5+]P4YsP_A?Fc[l$!+bWOC5F07$$\"5nT&Q.oq$*e-#Fc[l $!+#*Gc25F07$$\"5MLLe9TOEH?Fc[l$!*PfJ\"**F07$$\"5om;a)e6Bi0#Fc[l$!+t4H \"\\*F97$F_cl$!+LB;o#*F97$$\"5,+++D\"=EX8#Fc[l$!+!epR$zF97$Fdcl$!+>&*o mnF97$$\"5MLLeRA9WRAFc[l$!+^BLEbF97$Ficl$!+jy<`WF97$$\"5nmmTNr&3AM#Fc[ l$!+_#zSq$F97$F^dl$!+RI=7JF97$Fcdl$!+:VE^EF97$Fhdl$!+R<7 A6!3#F97$Fbel$!+B$)\\I>F97$Fgel$!+v\\#z&=F97$F\\fl$!+e'[Z$=F97$Fafl$!+ \"oxF'=F97$Fffl$!+j'3a%>F97$F[gl$!+@U$>4#F97$F`gl$!+&)pN4BF97$Fegl$!+w =%y&HF97$Fjgl$!+@r^)*QF97$$\"5,++vo/#3o<$Fc[l$!+:wokXF97$F_hl$!+*z\" yF97$$\"5MLL$3_0j,!QFc[l$!+V8]9pF97$Fj\\m$!+0uU!*fF97$$\"5MLLek`8=/RFc [l$!+lpn'>&F97$F_]m$!+6P\")fWF97$$\"5MLLe*[$zV4SFc[l$!+dv-eQF97$Fd]m$! +Q(p&fLF97$Fi]m$!+dz?/CF97$F^^m$!,-w/le\"F-7$Fc^m$!,in#3A5F-7$Fh^m$!+$ o%>XqF-7$F]_m$!+GR(H3&F-7$Fb_m$!+v(zYB%F-7$Fg_m$!+`\"H$RRF-7$F\\`m$!+8 JgNTF-7$Fa`m$!+Dub%)[F--Ff`m6&Fh`m$\"#XF[amF(Fi`m-F`am6#%9scheme~with~ simple~nodesG-F$6%7isF'7$F+$!%)=\"F07$F2$!'#e]\"F07$F7$!(Nva#F07$F=$!) a]*)=F07$FB$!)&[\">*)F07$FG$!*+b)=FF07$FL$!*Z\\In#F07$FQ$!*(=\")HEF07$ F_cm$!*&\\O5EF07$FV$!*g$)Qs#F07$Fgcm$!*;SqK$F07$Fen$!*&RhsXF07$Fjn$!*! *Rf\\%F07$F_o$!*vVEU%F07$Fdo$!*PJMP%F07$Fio$!*$[EZWF07$F^p$!*IT#fbF07$ Fbp$!*S$R0kF07$F\\q$!*d+))>'F07$Ffq$!*iz$QhF07$F[r$!*E$eukF07$F`r$!*\" H3!p'F07$Fer$!*dF:e'F07$Fjr$!*x3tZ'F07$Fds$!*daFF'F07$Fbu$!*h8Lx&F07$F \\v$!*Q*pCXF07$Fav$!*:DS\"RF07$Ffv$!*85Yq$F07$F[w$!*?![gNF07$F`w$!*1OS ]$F07$Few$!*B$emMF07$Fjw$!+8\\qXMF97$F_x$!+SR(3W$F97$Fdx$!*NU#fMF07$Fi x$!*(yo)\\$F07$F^y$!*,E#[NF07$Fcy$!*lNij$F07$Fhy$!*^uq)QF07$F]z$!*1F(p UF07$Fbz$!*Q8hF07$Fa[l$!**Q%)QoF07$ Fg[l$!*skH#)F07$Ffal$!*Pwqd(F07$F [bl$!*qt8)zF07$F`bl$!*.#Hv%)F07$Fahn$!*X=)e()F07$Ffhn$!*pKBU*F07$F[in$ !*p)Gu\"*F07$F`in$!*(>1g@Fc[l$!+9\\tqqF97$Fdcl$!+**z&3P'F97$Fcbo$!+@(\\l8&F97$ Ficl$!+$)pQ:TF97$F[co$!+_8#3U$F97$F^dl$!+HrS$)GF97$Fcdl$!+:u]uCF97$Fhd l$!+R@dv@F97$F]el$!+p[+j>F97$Fbel$!+$>>x#=F97$Fgel$!+&*eogHT'HFFc[l$!+&)zJXY\"y#Fc[l$!+>\"faz\"F97$Fffl$!+VIHT=F97$F[gl$!+\"y2&y>F97$F`gl$!+b(HO =#F97$Fegl$!+E2j*z#F97$$\"5,++v$4^n)pIFc[l$!+#*G^9KF97$Fjgl$!+\")4;'p$ F97$Fgeo$!+0m@GVF97$F_hl$!+*>Q$H]F97$F_fo$!+$e>.q&F97$Fdhl$!+(F97$Fihl$!+f\"3y\"zF97$F^il$!+1BUY%)F97$Fcil$!++W_+))F97 $Fhil$!+n]YZ))F97$F]jl$!+VN`$)))F97$Fbjl$!+$=Vj*))F97$Fgjl$!+]s9))))F9 7$Fa[m$!+gd2,))F97$F[\\m$!+^[wP')F97$F`\\m$!+uQ!R6)F97$Fe\\m$!+J@vmtF9 7$F]io$!+j#*HykF97$Fj\\m$!+0r..cF97$Feio$!+:0:!)[F97$F_]m$!+\"4$[?UF97 $F]jo$!+x&[ep$F97$Fd]m$!+)3S.D$F97$Fi]m$!+(G(zEBF97$F^^m$!,-8(y8:F-7$F c^m$!+i'3;j*F-7$Fh^m$!+$oG,k'F-7$F]_m$!+G(p!G[F-7$Fb_m$!+v]ISSF-7$Fg_m $!+`K(>w$F-7$F\\`m$!+8(Qo%RF-7$Fa`m$!+D]eeYF--Ff`m6&Fh`mF($\"#DF[am$\" \"\"F)-F`am6#%Pscheme~with~a~relatively~large~stability~regionG-F$6%7i rF'7$F+$\"&s[\"F07$F2$\"(*[,>F07$F7$\")zfWKF07$F=$\"*K-qU#F07$FB$\"+CA Gb6F07$FG$\"+ViGZNF07$FL$\"+^t_([$F07$FQ$\"+PgUJMF07$FV$\"+3*)[*e$F07$ Fen$\"+TuY9kF07$Fjn$\"+z*>pI'F07$F_o$\"+%fjZ?'F07$Fdo$\"+8=+WhF07$Fio$ \"+$4byH'F07$F^p$\"+ie&QN)F07$Fbp$\",pg&*e-\"F07$Fgp$\",;8=!45F07$F\\q $\"+*4V)H**F07$Faq$\"+$z!y>)*F07$Ffq$\"+*)[]N**F07$F[r$\",v;D(y5F07$F` r$\",rYAc8\"F07$Fer$\",eJ/s6\"F07$Fjr$\",]4Z(*4\"F07$F_s$\",#[$=/:\"F0 7$Fds$\",g#p&==\"F07$Fis$\",\"HW,*>\"F07$F^t$\",+w)\\?7F07$Fct$\",qU:! *=\"F07$Fht$\",;2o.;\"F07$F]u$\",E9q!e6F07$Fbu$\",EC7?;\"F07$Fgu$\",rJ ]0,\"F07$F\\v$\"+1WY\\')F07$Fcjm$\"+Vrr0uF07$Fav$\"+0@4%['F07$Ffv$\"+B G\"[(eF07$F[w$\"+9:M4T7F07$Fg[l$\",!\\%>& G9F07$F\\\\l$\",1Kw4k\"F07$Fa\\l$\",RUag!>F07$Ff\\l$\",STUz=#F07$F[]l$ \",x.t#oCF07$F`]l$\",%4H*Rr#F07$Fe]l$\",p\"4t\"*GF07$Fj]l$\",yEe:*HF07 $F_^l$\",=D;\"**HF07$Fd^l$\",v)\\$R+$F07$Fi^l$\",yXaC+$F07$F^_l$\",6)* Hc*HF07$Fc_l$\",T**Qi'HF07$Fh_l$\",!)\\/]\"HF07$F]`l$\",ERNn#GF07$Fb`l $\",F\\AOr#F07$Fg`l$\",<<))zc#F07$F\\al$\",cBa>T#F07$$\"5nmm;zpYU%p\"F c[l$\",+DlTE#F07$Faal$\",]0@35#F07$$\"5nmm;/,J:UF07$Ffa l$\",'e77rF97$Fbel $\",n@R&pF97$F`gl$\",bHc&R@F97$Fegl$\",W4 >^w#F97$Fjgl$\",Hog&)p$F97$F_hl$\",\"o.^^^F97$Fdhl$\",./^$QnF97$Fihl$ \",\"))y^g$)F97$F^il$\",Ctw1!*)F97$Fcil$\",g<^aC*F97$Fhil$\",.I$R%H*F9 7$F]jl$\",FIO&G$*F97$Fbjl$\",(4$4EM*F97$Fgjl$\",?$eFM$*F97$F\\[m$\",.* )*)yI*F97$Fa[m$\",+rY/E*F97$Ff[m$\",KKkN>*F97$F[\\m$\",RwEg5*F97$F`\\m $\",c&ou4')F97$Fe\\m$\",4,$eyyF97$Fj\\m$\",b^Nk!fF97$F_]m$\",HAUR)RF97 $Fd]m$\",-j?Xe#F97$Fi]m$\",`RgDz\"F97$F^^m$\"-)*pDl_7F-7$Fc^m$\",Q-!)f g)F-7$Fh^m$\",&F07$Fjn$!*Oim5&F07$F_o$!*h(\\F07$Fio$!*V%f#3&F07$F^p$!*qFQj'F07$Fbp$!*:]4.)F07$Fgp$!*q<))*y F07$F\\q$!*,\\Hx(F07$Faq$!*o1Qo(F07$Ffq$!*W<5w(F07$F[r$!*FDrQ)F07$F`r$ !*Ltr!))F07$Fer$!*<>Vm)F07$Fjr$!*v(fG&)F07$F_s$!*p+4)))F07$Fds$!*`oT8* F07$Fis$!*W'y,$*F07$F^t$!*,v#\\&*F07$Fct$!*HiHI*F07$Fht$!*3k63*F07$F]u $!*0?'3\"*F07$Fbu$!*qM1?*F07$Fgu$!*G8kI)F07$F\\v$!*!HxluF07$Fav$!*I(># *fF07$Ffv$!*xChZ&F07$F[w$!*uKG7&F07$F`w$!*bJL,&F07$Few$!*+L&Q\\F07$$\" 5NLL$3_D%eleF0$!*]\"G5\\F07$Fjw$!+$)3:#*[F97$$\"5+++]7y:A8hF0$!+!*HY') [F97$F_x$!+!zx6)[F97$Fdx$!*7t!4\\F07$Fix$!*fi1(\\F07$F^y$!*h7_0&F07$Fc y$!*X\"4*=&F07$Fhy$!*:bnc&F07$F]z$!*<^-7'F07$Fbz$!*\"HjF07$$\"5++++v =A)RU\"Fc[l$!+6**yE>F07$$\"5+++++Dk-P9Fc[l$!+4B7S>F07$$\"5++++DJ12]9Fc [l$!+`![o%>F07$Fj]l$!+4j;[>F07$F_^l$!+7=*>&>F07$Fd^l$!+!z%yZ>F07$Fi^l$ !+&HfF%>F07$F^_l$!+*=>d$>F07$Fc_l$!+%oK?\">F07$Fh_l$!+OyOx=F07$Fb`l$!+ @ezg28F07$F`bl$!+s%y3?\"F07$Fjbl$!+inFm5F07$F^ao$!*;U9')*F07$F_cl$!+L aU.*)F97$F[bo$!+!=lWe(F97$Fdcl$!+f()*Ga'F97$Fcbo$!+rMOlaF97$Ficl$!+$4< 5\\%F97$F[co$!+_yG=QF97$F^dl$!+f,%3C$F97$Fcdl$!+NO,\"y#F97$Fhdl$!+4Q%> V#F97$F]el$!+fcIy@F97$Fbel$!+.YR9?F97$Fgel$!+DyKM>F97$F\\fl$!+)=zl!>F9 7$Fafl$!+@NCM>F97$Fffl$!+V]->?F97$F[gl$!+@n*)p@F97$F`gl$!+&\\%)QR#F97$ Fegl$!+;fZgIF97$Fjgl$!+^G5WSF97$F_hl$!+z?HYbF97$Fdhl$!+FPn?rF97$Fihl$! +HO=q')F97$F^il$!+Ew9'=*F97$Fcil$!+goQ*[*F97$Fhil$!+x)=(Q&*F97$F]jl$!+ Lx3h&*F97$Fbjl$!+8L`k&*F97$Fgjl$!++]\\b&*F97$Fa[m$!+?kYn%*F97$F[\\m$!+ Jsd3$*F97$F`\\m$!+arE=))F97$Fe\\m$!+hhYG\")F97$Fj\\m$!+&4H]N'F97$F_]m$ !+^(\\Wp%F97$Fd]m$!+yHI_LF97$Fi]m$!+F+M6BF97$F^^m$!,-?\\P_\"F-7$Fc^m$! ,i&fT15F-7$Fh^m$!+$o_z1(F-7$F]_m$!+Gg*Q8&F-7$Fb_m$!+vtYlUF-7$Fg_m$!+`% Ht&RF-7$F\\`m$!+80]]TF-7$Fa`m$!+DWh(*[F--Ff`m6&Fh`mFi`mFa\\pF(-F`am6#% Hscheme~with~c[5]=c[6]=3/4~and~b[5]=b[6]G-%%FONTG6$%*HELVETICAG\"\"*-% +AXESLABELSG6$Q\"x6\"Q!Ffit-%&TITLEG6#%Uerror~curves~for~7~stage~order ~6~Runge-Kutta~methodsG-%%VIEWG6$;F(Fa`m%(DEFAULTG" 1 2 0 1 10 0 2 9 1 4 2 1.000000 45.000000 45.000000 0 1 "Butcher's scheme A" "scheme wi th simple nodes" "scheme with a relatively large stability region" "Bu tcher's scheme B with c[5]=c[6]=1/2 and b[5]=b[6]" "scheme with c[5]=c [6]=3/4 and b[5]=b[6]" }}}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 362 "evalf[20](plot(['vn_RK6_2'(x)-v(x) ,'vn_RK6_3'(x)-v(x),'vn_RK6_5'(x)-v(x)],x=0..5,font=[HELVETICA,9],\nco lor=[COLOR(RGB,.45,0,.95),COLOR(RGB,0,.25,1),COLOR(RGB,.95,.45,0)],\nl egend=[`scheme with simple nodes`,`scheme with a relatively large stab ility region`,`scheme with c[5]=c[6]=3/4 and b[5]=b[6]`],title=`error \+ curves for 7 stage order 6 Runge-Kutta methods`));" }}{PARA 13 "" 1 " " {GLPLOT2D 1032 384 384 {PLOTDATA 2 "6)-%'CURVESG6%7bv7$$\"\"!F)F(7$$ \"5qmm;aQ`!eS$!#A$!%c8!#?7$$\"5SLLL3x1h6oF-$!'()H#F07$$\"5NLL3FpE!Hq\"F9$!*HtQ/\"F07 $$\"5-++]7.K[V?F9$!*L*H*>$F07$$\"5omm\"zptjSQ#F9$!*z*RXJF07$$\"5NLLL$3 FWYs#F9$!*5_X4$F07$$\"5-++vo/[AlIF9$!*#HGsIF07$$\"5omm;aQ`!eS$F9$!*@@5 @$F07$$\"5NLLeRseQYPF9$!*`:t%RF07$$\"5-+++D1k'p3%F9$!*3%*[Z&F07$$\"5pm mT5SpaFWF9$!*)=4$Q&F07$$\"5OLL$eRZF\"oZF9$!*?GaH&F07$$\"50++D\"y+3(3^F 9$!*VcyB&F07$$\"5qmmmmT&)G\\aF9$!*7FaL&F07$$F3F9$!*66/x'F07$$\"50+++]7 G$R<)F9$!*e+x#zF07$$\"5SLL$3-)Q4b))F9$!*5*HswF07$$\"5qmmm\"z%\\DO&*F9$ !*$43;wF07$$\"5NLL3x\"[No()*F9$!*L(Q#4)F07$$\"5+++Dc,;u@5F0$!*:ZMS)F07 $$\"5nm;z%\\l*zb5F0$!*$R5n#)F07$$\"5MLLLL3x&)*3\"F0$!*Gsl8)F07$$\"5+]i lZQ9\\>6F0$!*\"3CW!)F07$$\"5nm\"z>'o^7\\6F0$!*e\"[E!)F07$$\"5M$3-j()*) e(y6F0$!*GlBA)F07$$\"5++]i!*GER37F0$!*qDyh)F07$$\"5ML3F>*3gwE\"F0$!*`( 3!Q)F07$$\"5nmm\"z%\\v#pK\"F0$!*R(ey\")F07$$\"5++Dcw4]>'Q\"F0$!*F74N)F 07$$\"5ML$3_+ZiaW\"F0$!*u)4'R)F07$$\"5nmT&Q.$*HZ]\"F0$!*kg^<)F07$$\"5+ ++]i!R(*Rc\"F0$!*\\))*z!)F07$$\"5nm;z>6B`#o\"F0$!*9wa.)F07$$\"5MLL3xJs 1,=F0$!*h-i+)F07$$\"5++]PM_@g>>F0$!*!ex2wF07$$\"5nmmm\"H2P\"Q?F0$!*O*G 1vF07$$\"5+++]PMnNrDF0$!*uX^W'F07$$\"5MLLL$eRwX5$F0$!*G/8h&F07$$\"5MLL LL$eI8k$F0$!*JA!)*\\F07$$\"5NLLL$3x%3yTF0$!*=nV^%F07$$\"5-++]PfyG7ZF0$ !*1/Q=%F07$$\"5ommm\"z%4\\Y_F0$!*%e1sRF07$$\"5NLLL$3FGT\\&F0$!*+.>!RF0 7$$\"5++++v$flVB$)F0$!*7\"Q2[F07$$\"5qmmmTg ()4_))F0$!*#\\@e`F07$$\"5.+++DJbw!Q*F0$!*8XZ/'F07$$\"5NLL$ekGkX#**F0$! *A_j&oF07$$\"5nmmm;/j$o/\"!#>$!*1xGm(F07$$\"5+++]7GTt%4\"Ff]l$!*xUuW)F 07$$\"5MLLL3_>jU6Ff]l$!*;v-3*F07$$\"5nmm;HdNb'>\"Ff]l$!+lI0.5F07$$\"5+ +++]i^Z]7Ff]l$!+c#3%)4\"F07$$\"5+++++v\"=YI\"Ff]l$!+,'*437F07$$\"5++++ ](=h(e8Ff]l$!+qV0D8F07$$\"5++++]7!Q4T\"Ff]l$!+U7$)>9F07$$\"5++++]P[6j9 Ff]l$!+>h4z9F07$$\"5nm\"HKkAg\\Z\"Ff]l$!+u(yH[\"F07$$\"5ML$ek`h0o[\"Ff ]l$!+:g!f[\"F07$$\"5++voH/5l)\\\"Ff]l$!+X!zV[\"F07$$\"5nmm\"HKR'\\5:Ff ]l$!+K![.[\"F07$$\"5++]P4rr=M:Ff]l$!+C$>;Y\"F07$$\"5MLL$e*[z(yb\"Ff]l$ !+lk:F9F07$$\"5nm;/Ev&[ge\"Ff]l$!+G,lp8F07$$\"5+++Dc,#>Uh\"Ff]l$!+I9^. 8F07$$\"5ML$eky#)*QU;Ff]l$!+&4%R;7F07$$\"5nmmm;a/cq;Ff]l$!+`/5Q6F07$$ \"5nmmmT&)))G=*F07$$\"5+]iSmbs&Hx\"Ff]l$!*f+'4\"*F07$$\"5nTgF;DsU w(*)z6s#*F07$$\"5o;HKknnZG=Ff]l$!*(f*\\.*F 07$$\"5,+D1k1nTN=Ff]l$!**f+E*)F07$$\"5o\"HKRhn'))Q=Ff]l$!*nDMA*F07$$\" 5M$3-QckcB%=Ff]l$!*\"okr$*F07$$\"5,v=n8:m#e%=Ff]l$!*q^]C*F07$$\"5om;aj %e'H\\=Ff]l$!*?)3B\"*F07$$\"5Me9T8alw_=Ff]l$!*4IB.*F07$$\"5,]7GjBlBc=F f]l$!*Bu#z!*F07$$\"5oT5:8$\\1(f=Ff]l$!*_t*p&*F07$$\"5ML3-jik>*F07$$\"5,++]iSj0x=Ff]l$!*$f'zJ*F07$$\"5ML3_] %QU$*)=Ff]l$!*16w\\*F07$$\"5om;aQG%G;!>Ff]l$!+(3JN+\"F07$$\"5M$3_D.Xrx !>Ff]l$!*\\34x*F07$$\"5,+DcEsW\"R\">Ff]l$!*([=\"f*F07$$\"5M3xcB$)f)p\" >Ff]l$!*8$)fr*F07$$\"5o;Hd?%\\d+#>Ff]l$!+_!G8.\"F07$$\"5,D\"yv^+HJ#>Ff ]l$!+Q9D<5F07$$\"5MLLe9;0?E>Ff]l$!+p!RL+\"F07$$\"5nT&)e6F?FH>Ff]l$!*8Y )**)*F07$$\"5+]Pf3QNMK>Ff]l$!*(f)Rz*F07$$\"5Me*)f0\\]TN>Ff]l$!*(>_#z*F 07$$\"5nmTg-gl[Q>Ff]l$!+!o_'45F07$$\"5+v$4'*42e:%>Ff]l$!+6W^S5F07$$\"5 M$e9m>eHY%>Ff]l$!+ye$f-\"F07$$\"5n\"z>OH4,x%>Ff]l$!+\"fJ;,\"F07$$\"5++ ]i!Rgs2&>Ff]l$!*Shc)**F07$$\"5M3-j([6WQ&>Ff]l$!*4k'3**F07$$\"5n;aj%ei: p&>Ff]l$!+&o&y+5F07$$\"5+D1k\"o8()*f>Ff]l$!+BiTd5F07$$\"5MLekyZ'eI'>Ff ]l$!+ljwU5F07$$\"5+]ilsp;?p>Ff]l$!+BUY85F07$$\"5nmmmm\"pW`(>Ff]l$!*?_< ***F07$$\"5MLek.HW#)))>Ff]l$!+/v@;5F07$$\"5++]iSmTI-?Ff]l$!+`L@T5F07$$ \"5M$e9\"4NS/4?Ff]l$!+\\I)z+\"F07$$\"5nmTgx.Ry:?Ff]l$!*wQp!**F07$$\"5M e*[=\"QQ:>?Ff]l$!+Xbo>5F07$$\"5+]P4YsP_A?Ff]l$!+bWOC5F07$$\"5nT&Q.oq$* e-#Ff]l$!+#*Gc25F07$$\"5MLLe9TOEH?Ff]l$!*PfJ\"**F07$$\"5om;a)e6Bi0#Ff] l$!+t4H\"\\*F97$$\"5,++]i!f#=$3#Ff]l$!+LB;o#*F97$$\"5,+++D\"=EX8#Ff]l$ !+!epR$zF97$$\"5,++](=xpe=#Ff]l$!+>&*omnF97$$\"5MLLeRA9WRAFf]l$!+^BLEb F97$$\"5nmmm\"H28IH#Ff]l$!+jy<`WF97$$\"5nmmTNr&3AM#Ff]l$!+_#zSq$F97$$ \"5nmm;zpSS\"R#Ff]l$!+RI=7JF97$$\"5+++v$41oWW#Ff]l$!+:VE^EF97$$\"5MLLL 3_?`(\\#Ff]l$!+R<7A6!3#F97$$\"5MLL$e*)>pxg #Ff]l$!+B$)\\I>F97$$\"5ommm;z+vbEFf]l$!+v\\#z&=F97$$\"5,++]Pf4t.FFf]l$ !+e'[Z$=F97$$\"5omm\"zWi^bv#Ff]l$!+\"oxF'=F97$$\"5MLLLe*Gst!GFf]l$!+j' 3a%>F97$$\"5omm;H2\"34'GFf]l$!+@U$>4#F97$$\"5,++++DRW9HFf]l$!+&)pN4BF9 7$$\"5,+++DJE>>IFf]l$!+w=%y&HF97$$\"5,++]i!RU07$Ff]l$!+@r^)*QF97$$\"5, ++vo/#3o<$Ff]l$!+:wokXF97$$\"5,+++v=S2LKFf]l$!+*z?OFf]l$!+5tLD#*F9 7$$\"5,+++]iC$pk$Ff]l$!+T-\"yF97$$\"5MLL$3_0j,!QFf]l$!+V8]9pF97$$\"5,++ ]7.\"fF&QFf]l$!+0uU!*fF97$$\"5MLLek`8=/RFf]l$!+lpn'>&F97$$\"5ommm;/Ogb RFf]l$!+6P\")fWF97$$\"5MLLe*[$zV4SFf]l$!+dv-eQF97$$\"5,++]ilAFjSFf]l$! +Q(p&fLF97$$\"5NLLLL$)*pp;%Ff]l$!+dz?/CF97$$\"5NLLL3xe,tUFf]l$!,-w/le \"F-7$$\"5omm;HdO=yVFf]l$!,in#3A5F-7$$\"5,++++D>#[Z%Ff]l$!+$o%>XqF-7$$ \"5ommmT&G!e&e%Ff]l$!+GR(H3&F-7$$\"5NLLLL$)Qk%o%Ff]l$!+v(zYB%F-7$$\"5- ++]iSjE!z%Ff]l$!+`\"H$RRF-7$$\"5-++]P40O\"*[Ff]l$!+8JgNTF-7$$\"\"&F)$! +Dub%)[F--%&COLORG6&%$RGBG$\"#X!\"#F($\"#&*Ffdn-%'LEGENDG6#%9scheme~wi th~simple~nodesG-F$6%7isF'7$F+$!%)=\"F07$F2$!'#e]\"F07$F7$!(Nva#F07$F= $!)a]*)=F07$FB$!)&[\">*)F07$FG$!*+b)=FF07$FL$!*Z\\In#F07$FQ$!*(=\")HEF 07$FV$!*&\\O5EF07$Fen$!*g$)Qs#F07$Fjn$!*;SqK$F07$F_o$!*&RhsXF07$Fdo$!* !*Rf\\%F07$Fio$!*vVEU%F07$F^p$!*PJMP%F07$Fcp$!*$[EZWF07$Fhp$!*IT#fbF07 $F\\q$!*S$R0kF07$Faq$!*d+))>'F07$Ffq$!*iz$QhF07$F[r$!*E$eukF07$F`r$!* \"H3!p'F07$Fer$!*dF:e'F07$Fjr$!*x3tZ'F07$F\\v$!*daFF'F07$F`w$!*h8Lx&F0 7$Fjw$!*Q*pCXF07$Fdx$!*:DS\"RF07$Fix$!*85Yq$F07$F^y$!*?![gNF07$Fcy$!*1 OS]$F07$Fhy$!*B$emMF07$F]z$!+8\\qXMF97$Fbz$!+SR(3W$F97$Fgz$!*NU#fMF07$ F\\[l$!*(yo)\\$F07$Fa[l$!*,E#[NF07$Ff[l$!*lNij$F07$F[\\l$!*^uq)QF07$F` \\l$!*1F(pUF07$Fe\\l$!*Q8hF07$Fd]l $!**Q%)QoF07$Fj]l$!*skH#)F07$Ficl$!*P wqd(F07$Fafl$!*qt8)zF07$Fgjl$!*.#Hv%)F07$F\\[m$!*X=)e()F07$Fa[m$!*pKBU *F07$Ff[m$!*p)Gu\"*F07$F[\\m$!*(>1g@Ff]l$!+9\\tqqF97$Faem$!+**z&3P'F97$Ffe m$!+@(\\l8&F97$F[fm$!+$)pQ:TF97$F`fm$!+_8#3U$F97$Fefm$!+HrS$)GF97$Fjfm $!+:u]uCF97$F_gm$!+R@dv@F97$Fdgm$!+p[+j>F97$Figm$!+$>>x#=F97$F^hm$!+&* eogHT'HFFf]l$!+&)zJXY\"y#Ff]l$!+>\"faz\"F97$F]im$!+VIHT=F97$Fbim$!+\"y2&y>F 97$Fgim$!+b(HO=#F97$F\\jm$!+E2j*z#F97$$\"5,++v$4^n)pIFf]l$!+#*G^9KF97$ Fajm$!+\")4;'p$F97$Ffjm$!+0m@GVF97$F[[n$!+*>Q$H]F97$F`[n$!+$e>.q&F97$F e[n$!+(F97$F_\\n$!+f\"3y\"zF97$Fd\\n$!+1BUY%)F97 $Fi\\n$!++W_+))F97$F^]n$!+n]YZ))F97$Fc]n$!+VN`$)))F97$Fh]n$!+$=Vj*))F9 7$F]^n$!+]s9))))F97$Fb^n$!+gd2,))F97$Fg^n$!+^[wP')F97$F\\_n$!+uQ!R6)F9 7$Fa_n$!+J@vmtF97$Ff_n$!+j#*HykF97$F[`n$!+0r..cF97$F``n$!+:0:!)[F97$Fe `n$!+\"4$[?UF97$Fj`n$!+x&[ep$F97$F_an$!+)3S.D$F97$Fdan$!+(G(zEBF97$Fia n$!,-8(y8:F-7$F^bn$!+i'3;j*F-7$Fcbn$!+$oG,k'F-7$Fhbn$!+G(p!G[F-7$F]cn$ !+v]ISSF-7$Fbcn$!+`K(>w$F-7$Fgcn$!+8(Qo%RF-7$F\\dn$!+D]eeYF--Fadn6&Fcd nF($\"#DFfdn$\"\"\"F)-Fjdn6#%Pscheme~with~a~relatively~large~stability ~regionG-F$6%7jrF'7$F+$!%97F07$F2$!'od:F07$F7$!('emEF07$F=$!)&*=,?F07$ FB$!)`Vd&*F07$FG$!*2WJ%HF07$FL$!*&4c$*GF07$FQ$!*&3(o%GF07$FV$!*.jx#GF0 7$Fen$!*?<`'HF07$Fjn$!*CE&)o$F07$F_o$!*#pu$>&F07$Fdo$!*Oim5&F07$Fio$!* h(\\F07$Fcp$!*V%f#3&F07$Fhp$!*qFQj'F07$F\\q$!*:]4.)F0 7$$\"5qmmTNYL^9&)F9$!*q<))*yF07$Faq$!*,\\Hx(F07$$\"50++D19Wn&>*F9$!*o1 Qo(F07$Ffq$!*W<5w(F07$F[r$!*FDrQ)F07$F`r$!*Ltr!))F07$Fer$!*<>Vm)F07$Fj r$!*v(fG&)F07$Fht$!*p+4)))F07$F\\v$!*`oT8*F07$Fav$!*W'y,$*F07$Ffv$!*,v #\\&*F07$$\"5nm\"Hd?pM.'=F0$!*HiHI*F07$F[w$!*3k63*F07$$\"5ML3-j7'p)y>F 0$!*0?'3\"*F07$F`w$!*qM1?*F07$Few$!*G8kI)F07$Fjw$!*!HxluF07$Fdx$!*I(># *fF07$Fix$!*xChZ&F07$F^y$!*uKG7&F07$Fcy$!*bJL,&F07$Fhy$!*+L&Q\\F07$$\" 5NLL$3_D%eleF0$!*]\"G5\\F07$F]z$!+$)3:#*[F97$$\"5+++]7y:A8hF0$!+!*HY') [F97$Fbz$!+!zx6)[F97$Fgz$!*7t!4\\F07$F\\[l$!*fi1(\\F07$Fa[l$!*h7_0&F07 $Ff[l$!*X\"4*=&F07$F[\\l$!*:bnc&F07$F`\\l$!*<^-7'F07$Fe\\l$!*\"HjF07$$ \"5++++v=A)RU\"Ff]l$!+6**yE>F07$$\"5+++++Dk-P9Ff]l$!+4B7S>F07$$\"5++++ DJ12]9Ff]l$!+`![o%>F07$F]`l$!+4j;[>F07$Fb`l$!+7=*>&>F07$Fg`l$!+!z%yZ>F 07$F\\al$!+&HfF%>F07$Faal$!+*=>d$>F07$Ffal$!+%oK?\">F07$F[bl$!+OyOx=F0 7$Febl$!+@ezg28F07$Fgjl$!+s%y3?\"F07$Feam$!+inFm5F07$F]dm$!*;U9')*F07$ Fgdm$!+LaU.*)F97$F\\em$!+!=lWe(F97$Faem$!+f()*Ga'F97$Ffem$!+rMOlaF97$F [fm$!+$4<5\\%F97$F`fm$!+_yG=QF97$Fefm$!+f,%3C$F97$Fjfm$!+NO,\"y#F97$F_ gm$!+4Q%>V#F97$Fdgm$!+fcIy@F97$Figm$!+.YR9?F97$F^hm$!+DyKM>F97$Fchm$!+ )=zl!>F97$Fhhm$!+@NCM>F97$F]im$!+V]->?F97$Fbim$!+@n*)p@F97$Fgim$!+&\\% )QR#F97$F\\jm$!+;fZgIF97$Fajm$!+^G5WSF97$F[[n$!+z?HYbF97$Fe[n$!+FPn?rF 97$F_\\n$!+HO=q')F97$Fd\\n$!+Ew9'=*F97$Fi\\n$!+goQ*[*F97$F^]n$!+x)=(Q& *F97$Fc]n$!+Lx3h&*F97$Fh]n$!+8L`k&*F97$F]^n$!++]\\b&*F97$Fb^n$!+?kYn%* F97$Fg^n$!+Jsd3$*F97$F\\_n$!+arE=))F97$Fa_n$!+hhYG\")F97$F[`n$!+&4H]N' F97$Fe`n$!+^(\\Wp%F97$F_an$!+yHI_LF97$Fdan$!+F+M6BF97$Fian$!,-?\\P_\"F -7$F^bn$!,i&fT15F-7$Fcbn$!+$o_z1(F-7$Fhbn$!+Gg*Q8&F-7$F]cn$!+vtYlUF-7$ Fbcn$!+`%Ht&RF-7$Fgcn$!+80]]TF-7$F\\dn$!+DWh(*[F--Fadn6&FcdnFgdnFddnF( -Fjdn6#%Hscheme~with~c[5]=c[6]=3/4~and~b[5]=b[6]G-%%FONTG6$%*HELVETICA G\"\"*-%+AXESLABELSG6$Q\"x6\"Q!Fcjq-%&TITLEG6#%Uerror~curves~for~7~sta ge~order~6~Runge-Kutta~methodsG-%%VIEWG6$;F(F\\dn%(DEFAULTG" 1 2 0 1 10 0 2 9 1 4 2 1.000000 45.000000 45.000000 0 1 "scheme with simple no des" "scheme with a relatively large stability region" "scheme with c[ 5]=c[6]=3/4 and b[5]=b[6]" }}}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 46 "Test 8 of 7 stage, order 6 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#/-%$rhoG 6#%\"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,F0F3" }{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);\nic := y(0)=0;\ndsolve(\{de,ic\},y(x));\nw := unapply(rh s(%),x):\nplot(w(x),x=0..4,0..3.7,numpoints=75,font=[HELVETICA,9],labe ls=[`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$$\"3u\\'['[U&)o')F>$\"3 _Iow_)\\Zn\"Fho7$$\"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$$\"3s%f%f9[%4C\"Fho$\"3#y2[?x6q H#Fho7$$\"3[KCVKm,'H\"Fho$\"3)*e,ywZ!pP#Fho7$$\"3c8N^t2A`8Fho$\"3WUCG, 38dCFho7$$\"3Y'['[Yr,.9Fho$\"3@#)=I[fvCDFho7$$\"3q8N^$f)zc9Fho$\"3%3DX ;Fho$\"3iG:8:1'ez#Fho7$$\"3c%f%fuKqx;Fh o$\"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$$\"3$)******>MoW>Fho$\"3Mca,!\\@%Q JFho7$$\"3!*['[')ep#**>Fho$\"3Or<<>%)R)=$Fho7$$\"31Yf%faPE0#Fho$\"3!)Q t2wy;NKFho7$$\"3C^8N^)3&3@Fho$\"3Ik#>&=s*=G$Fho7$$\"3E>*=*e&>B;#Fho$\" 3HYm&4R?ZK$Fho7$$\"3_f%f%z([t@#Fho$\"3-!>y6]piO$Fho7$$\"3oq-FIB#>F#Fho $\"3%>b4%)\\*>0MFho7$$\"3vvcnb(p?K#Fho$\"35nW;n-%*QMFho7$$\"3eCVKkVazB Fho$\"3))))QO[-#QDFho$\"3]n/[Ud8hNFho7$$\"3Adn vO#pkf#Fho$\"3[Y'[5$oF(e$Fho7$$\"3'>;i@A4pk#Fho$\"3e\"*)HR3Wug$Fho7$$ \"3$ovcn0fTq#Fho$\"3E1Ex&>gui$Fho7$$\"3;Yf%f1Ojv#Fho$\"3)*3G:eW%Hk$Fho 7$$\"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$$\"3-dnvO9*43$Fho$\"3y ,I^j&GHn$Fho7$$\"3m(H(H<2\"G8$Fho$\"3]\\w?2VllOFho7$$\"3'>*=*=tG))=$Fh o$\"3#R\\Gi]bMl$Fho7$$\"3-A;ihz@UKFho$\"3q^o+4DNPOFho7$$\"3!>*=*=h2%)H $Fho$\"3%>QOHhJ`h$Fho7$$\"3A;i@wFF\\LFho$\"3YxQ[8&f0f$Fho7$$\"3)\\'['[ I)[0MFho$\"3q>yn%)HPdNFho7$$\"3c0aS&)HLfMFho$\"3a59(yGT$>NFho7$$\"3['[ '[1m/8NFho$\"3uI!*3pTpuMFho7$$\"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)$\"3CxC=rRoRDFho -%'COLOURG6&%$RGBG$\"#5!\"\"F(F(-%%FONTG6$%*HELVETICAG\"\"*-%+AXESLABE LSG6$%\"xG%%y(x)G-%%VIEWG6$;F(F`cl;F($\"#PFjcl" 1 2 0 1 10 0 2 9 1 4 2 1.000000 45.000000 45.000000 0 0 "Curve 1" }}}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 32 "The following code constr ucts a " }{TEXT 260 17 "discrete solution" }{TEXT -1 44 " based on eac h of the methods and gives the " }{TEXT 260 22 "root mean square error " }{TEXT -1 18 " of each solution." }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 755 "W := (x,y) -> x*(9-x^2)/(1+y^2): hh := 0.01: numstep s := 400: x0 := 0: y0 := 0:\nmatrix([[`slope field: `,W(x,y)],[`init ial point: `,``(x0,y0)],[`step width: `,hh],\n[`no. of steps: `,nu msteps]]);``;\nmthds := [`Butcher's scheme A`,`scheme with simple node s`,`scheme with a relatively large stability region`,`Butcher's scheme B with `*(c[5]=1/2,c[6]=1/2,b[5]=b[6]),`scheme with `*(c[5]=3/4,c[6]= 3/4,b[5]=b[6])]:\nerrs := []:\nDigits := 30:\nfor ct to 5 do\n Wn_RK 6_||ct := RK6_||ct(W(x,y),x,y,x0,y0,hh,numsteps,false);\n sm := 0: n umpts := nops(Wn_RK6_||ct):\n for ii to numpts do\n sm := sm+(W n_RK6_||ct[ii,2]-w(Wn_RK6_||ct[ii,1]))^2;\n end do:\n errs := [op( errs),sqrt(sm/numpts)];\nend do:\nDigits := 10:\nlinalg[transpose]([mt hds,evalf(errs)]);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#K%'matrixG6#7&7$ %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)pprint356\"" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%!G" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#K%'matrixG6#7'7$%3Butcher 's~scheme~AG$\"+SY.di!#B7$%9scheme~with~simple~nodesG$\"+&3]^y)!#C7$%P scheme~with~a~relatively~large~stability~regionG$\"+=u>Y\"*F07$*&%9But cher's~scheme~B~with~G\"\"\"6%/&%\"cG6#\"\"&#F8\"\"#/&F<6#\"\"'F?/&%\" bGF=&FGFCF8$\"+2T%y3)F07$*&%-scheme~with~GF86%/F;#\"\"$\"\"%/FBFPFEF8$ \"+q')Rr()F0Q)pprint366\"" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 30 "The following code constructs " }{TEXT 260 20 "nu merical 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 = 3.499;" "6#/%\"xG-%&FloatG6$\"%*\\$!\"$" }{TEXT -1 16 " is \+ also given." }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 686 "W := (x,y) - > x*(9-x^2)/(1+y^2): hh := 0.01: numsteps := 400: x0 := 0: y0 := 0:\nm atrix([[`slope field: `,W(x,y)],[`initial point: `,``(x0,y0)],[`step width: `,hh],\n[`no. of steps: `,numsteps]]);``;\nmthds := [`Butc her's scheme A`,`scheme with simple nodes`,`scheme with a relatively l arge stability region`,`Butcher's scheme B with `*(c[5]=1/2,c[6]=1/2,b [5]=b[6]),`scheme with `*(c[5]=3/4,c[6]=3/4,b[5]=b[6])]:\nerrs := []: \nDigits := 30:\nfor ct to 5 do\n wn_RK6_||ct := RK6_||ct(W(x,y),x,y ,x0,y0,hh,numsteps,true);\nend do:\nxx := 3.499: wxx := evalf(w(xx)): \nfor ct to 5 do\n errs := [op(errs),abs(wn_RK6_||ct(xx)-wxx)];\nend do:\nDigits := 10:\nlinalg[transpose]([mthds,evalf(errs)]);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#K%'matrixG6#7&7$%0slope~field:~~~G*(%\"xG\" \"\",&\"\"*F+*$)F*\"\"#F+!\"\"F+,&F+F+*$)%\"yGF0F+F+F17$%0initial~poin t:~G-%!G6$\"\"!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$%3Butcher's~scheme~AG$\"+=15dF!#B7$%9 scheme~with~simple~nodesG$\"+`xdiH!#C7$%Pscheme~with~a~relatively~larg e~stability~regionG$\"+Am6NL!#D7$*&%9Butcher's~scheme~B~with~G\"\"\"6% /&%\"cG6#\"\"&#F9\"\"#/&F=6#\"\"'F@/&%\"bGF>&FHFDF9$\"+AsmT=F07$*&%-sc heme~with~GF96%/F<#\"\"$\"\"%/FCFQFFF9$\"+VMgEOF0Q)pprint386\"" }}} {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 4 "The " } {TEXT 260 22 "root mean square error" }{TEXT -1 20 " over the interval " }{XPPEDIT 18 0 "[0,4]" "6#7$\"\"!\"\"%" }{TEXT -1 82 " of each Ru nge-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 0 436 "mthds := [`Butcher's scheme A`,`scheme with simple nodes`,`scheme with a relatively large stabili ty region`,`Butcher's scheme B with `*(c[5]=1/2,c[6]=1/2,b[5]=b[6]),`s cheme with `*(c[5]=3/4,c[6]=3/4,b[5]=b[6])]:\nerrs := []:\nDigits := 2 0:\nfor ct to 5 do\n sm := NCint((w(x)-'wn_RK6_||ct'(x))^2,x=0..4,ad aptive=false,numpoints=7,factor=200);\n errs := [op(errs),sqrt(sm/4) ];\nend do:\nDigits := 10:\nlinalg[transpose]([mthds,evalf(errs)]);" } }{PARA 11 "" 1 "" {XPPMATH 20 "6#K%'matrixG6#7'7$%3Butcher's~scheme~AG $\"+Wy`,i!#B7$%9scheme~with~simple~nodesG$\"+M[dG&)!#C7$%Pscheme~with~ a~relatively~large~stability~regionG$\"+))fP*o)F07$*&%9Butcher's~schem e~B~with~G\"\"\"6%/&%\"cG6#\"\"&#F8\"\"#/&F<6#\"\"'F?/&%\"bGF=&FGFCF8$ \"+q#)z " 0 "" {MPLTEXT 1 0 516 "evalf[30](plot(['wn_RK6_1'(x)-w(x),'wn_RK6_2'(x)-w(x ),'wn_RK6_3'(x)-w(x),'wn_RK6_4'(x)-w(x),\n'wn_RK6_5'(x)-w(x)],x=0..4,f ont=[HELVETICA,9],\ncolor=[COLOR(RGB,.95,.2,0),COLOR(RGB,.45,0,.95),CO LOR(RGB,0,.25,1),COLOR(RGB,0,.75,.2),COLOR(RGB,.95,.45,0)],\nlegend=[` Butcher's scheme A`,`scheme with simple nodes`,`scheme with a relative ly large stability region`,`Butcher's scheme B with c[5]=c[6]=1/2 and \+ b[5]=b[6]`,`scheme with c[5]=c[6]=3/4 and b[5]=b[6]`],title=`error cur ves for 7 stage order 6 Runge-Kutta methods`));" }}{PARA 13 "" 1 "" {GLPLOT2D 919 527 527 {PLOTDATA 2 "6+-%'CURVESG6%7]p7$$\"\"!F)F(7$$\"? LLLLLLLLLLL3VfV!#J$\"0(\\g#Q8uF\"!#I7$$\"?mmmmmmmmmmm;')=()F-$\"0bz\"z eJ-sF07$$\"?MLLLLLLLLeR?ah5F0$\"1i%=$Q=Bf8F07$$\"?+++++++++]7z>^7F0$\" 1a%)fn?k,DF07$$\"?mmmmmmmmmT&y`3W\"F0$\"1g)QRr!RNXF07$$\"?LLLLLLLLLLe' 40j\"F0$\"1x9z#G/%4\")F07$$\"?mmmmmmmmm\"H_(zV=F0$\"2$zPe*pc&=9F07$$\" ?+++++++++](Q&3d?F0$\"2o6Wv#o#RS#F07$$\"?MLLLLLLLL3_KPqAF0$\"2[c]%>cYB RF07$$\"?nmmmmmmmmm;6m$[#F0$\"2I;E!>g&)yhF07$$\"?nmmmmmmmm;a>,\"f#F0$ \"2;/%Qhpi.xF07$$\"?nmmmmmmmmm\"zi$)p#F0$\"2@Z$eo#\\Jd*F07$$\"?nmmmmmm mm;HOr0GF0$\"3aD$*G?3f76F07$$\"?nmmmmmmmmmmW18HF0$\"3?%>Ld)y6]7F07$$\" ?nmmmmmmmmmThwFJF0$\"3.SXBX$H?]\"F07$$\"?nmmmmmmmmm;yYULF0$\"3#[V3YzWk p\"F07$$\"?MLLLLLLLL3_!\\hb$F0$\"3aAV=pf:9=F07$$\"?+++++++++](GI)pPF0$ \"3\"[mSz*3P_=F07$$\"?mmmmmmmmm\"H_6N)RF0$\"3#\\(H^0$o9#=F07$$\"?LLLLL LLLLLeF>(>%F0$\"3;cs(3WOtt\"F07$$\"?++++++++++vCT$f%F0$\"3CeDOL&4.a\"F 07$$\"?mmmmmmmmmm\">K'*)\\F0$\"3([;1Pl([s8F07$$\"?NLLLLLLLLLeZ*)*R&F0$ \"3d1pI0\")Ge7F07$$\"?++++++++++Dt:5eF0$\"3alM(G%y2&=\"F07$$\"?mmmmmmm mmm\"fX(emF0$\"3I+]6Qo*)y5F07$$\"?++++++++++DCh/vF0$\"23x2)Qf.?**F07$$ \"?LLLLLLLLLLL/pu$)F0$\"21B0H%zWC*)F07$$\"?mmmmmmmmmm;c0T\"*F0$\"2q)) \\[ipb8)F07$$\"?++++++++++I,Q+5!#H$\"2#3bww8,MtF07$$\"?++++++++++]*3q3 \"Fet$\"2#\\E>t)QNi'F07$$\"?++++++++++q=\\q6Fet$\"2CoB'=@!o/'F07$$\"?n mmmmmmmm;fBIY7Fet$\"2]%z9U_])e&F07$$\"?LLLLLLLLLLj$[kL\"Fet$\"2A#))QE* fK7&F07$$\"?LLLLLLLLLL`Q\"GT\"Fet$\"2g\"fvLsR%y%F07$$\"?+++++++++]s]k, :Fet$\"2UMf%4jEUWF07$$\"?LLLLLLLLLL`dF!e\"Fet$\"2[U/+8_%zTF07$$\"?++++ +++++]sgam;Fet$\"2-G4![!eq#RF07$$\"?+++++++++]_aFet$\"2/ iLIvI0P$F07$$\"?nmmmmmmmmmTc-)*>Fet$\"2!Ho=%oBEA$F07$$\"?nmmmmmmmm;f`@ '3#Fet$\"2M-U#3q'z3$F07$$\"?+++++++++]nZ)H;#Fet$\"2%=M91Ro%)HF07$$\"?n mmmmmmmmmJy*eC#Fet$\"2cA!>!*>8')GF07$$\"?++++++++++S^bJBFet$\"2!GE!Gkr sz#F07$$\"?++++++++++0TN:CFet$\"2)*eh21t>s#F07$$\"?+++++++++]7RV'\\#Fe t$\"2!)zJn0?$fEF07$$\"?++++++++++:#fke#Fet$\"2u#G\"p*Q$3g#F07$$\"?LLLL LLLLLL`4NnEFet$\"2m&z]fb$yb#F07$$\"?++++++++++],s`FFet$\"2oKr8z\"o@DF0 7$$\"?nmmmmmmmm;zM)>$GFet$\"2IIL8(4m(\\#F07$$\"?++++++++++qfal![#F07$$\"?LLLLLLLLLL$)G[kJFet$\"21#)ol7$*z\\#F07$$ \"?+++++++++]7yh]KFet$\"2;`b^1W:`#F07$$\"?nmmmmmmmmm')fdLLFet$\"2ybPPX hq)*GF07$$\"?LLLLL LLLLLGUYoOFet$\"2-2*p#[9U6$F07$$\"?nmmmmmmmmm1^rZPFet$\"25#f***[XWP$F0 7$$\"?MLLLLLLLLe*3k**y$Fet$\"2sw`OpH$>NF07$$\"?+++++++++]sI@KQFet$\"2i SPN$**p7OF07$$\"?++++++++]P*[AB%QFet$\"2)p1:26/8OF07$$\"?+++++++++D1>V _QFet$\"2#o>=S!*G)f$F07$$\"?++++++++]7B8aiQFet$\"2KAoD&[FjNF07$$\"?+++ +++++++S2lsQFet$\"2O`O2`%=,NF07$$\"?+++++++++vt&pG*QFet$\"2o#[*)H\"pnD $F07$$\"?+++++++++]2%)38RFet$\"2)=H.w2$3v#F07$$\"?++++++++]i0j\"[$RFet $\"2)=dtXCIy@%fj(fr\"F07$$\"?++++++++](=5s#yRFet$!2q_QxH^D9%F07$$ \"?+++++++]iSwSq$)RFet$!2kB&o33i@hF07$$\"?++++++++v$40O\"*)RFet$!2#=I) \\6+,0)F07$$\"?+++++++DJ?Q?&=*RFet$!2#Gv\"f-!4s(*F07$$\"?+++++++](oa-o X*RFet$!2q2U\"z6&p&)*F07$$\"?+++++++vVt7SG(*RFet$!3I)*)>Ku\"eT5F07$$\" \"%F)$!3[2NjCsF![\"F0-%&COLORG6&%$RGBG$\"#&*!\"#$\"\"#!\"\"F(-%'LEGEND G6#%3Butcher's~scheme~AG-F$6%7`qF'7$$\"?mmmmmmmmmm;arz@F-$\"/bQzm!**=# F07$F+$\"0bh]'Rp2:F07$$\"?lmmmmmmmmmT&)G\\aF-$\"0#)H6-/vc9CF07$FP$\"2I>^,Y!y')HF07$$\"?nm mmmmmm;z>$HP;#F0$\"2xCy(3C%))=$F07$FU$\"2m@F0hlPI$F07$$\"?nmmmmmmm\"H# =_pBBF0$\"2YvD(o2/JLF07$$\"?++++++++]P%=B'=F07$Fho$\"2@$f5)fw)[8F07$$\" ?nmmmmmmmm;/`T?IF0$\"1v,fhlE6F07$Faq$!1MMe\")R!z6*F07$Ffq$!1nr1,Xt,TF07$$\"?mmmmmmmmmm; EI&R%F0$\"0u!H\")3)G=$F07$F[r$\"1!=Z@M.k_%F07$$\"?LLLLLLLLLLLB_\"z%F0$ \"1G&*Q-*f$[\")F07$F`r$\"2zcI\"*oFC5\"F07$$\"?++++++++++vMw%>&F0$\"2x` ;nFOrM\"F07$Fer$\"2,y6hP1R^\"F07$$\"?+++++++++++/Y-bF0$\"2)>13eDTd:F07 $$\"?qmmmmmmmmmTg-0cF0$\"2+q[Gox!)e\"F07$$\"?NLLLLLLLLL$o\"f2dF0$\"2-( GDx%f#3;F07$Fjr$\"2M\\b;>5&>;F07$$\"?NLLLLLLLLLe3B;fF0$\"2#>D/d9#>i\"F 07$$\"?qmmmmmmmmm\"R/B-'F0$\"2?&>7Ak3=;F07$$\"?++++++++++DzPGhF0$\"2_> )3I!=\"4;F07$$\"?NLLLLLLLLLe9XMiF0$\"2eMD]o$)ff\"F07$$\"?++++++++++D&) fYkF0$\"2w$*R,/L/c\"F07$F_s$\"2I@uSrUn^\"F07$$\"?NLLLLLLLLL3!z;3(F0$\" 23xDMfj\">9F07$Fds$\"2G!Q?*p!p>8F07$Fis$\"2E;ra%3SK6F07$F^t$\"1I1xOO&o )**F07$Fct$\"1-[pq2Yx()F07$Fit$\"1-B(zOJ%3yF07$F^u$\"1M!e=*Rg[qF07$Fcu $\"1!G(GPvhtkF07$Fhu$\"1sx*\\)eI+fF07$F]v$\"1+q(y\")Q'*[&F07$Fbv$\"17Y 5'el73&F07$Fgv$\"1)QK><\\2x%F07$F\\w$\"1_'>!H%HaZ%F07$Faw$\"1eo^hN;JUF 07$Ffw$\"1Q\"Hc.D(3SF07$F[x$\"1kknsJF 07$Fiy$\"13/gHz2(3$F07$F^z$\"1S8SCf(f,$F07$Fcz$\"1%fH4S)p\\HF07$Fhz$\" 11)oL(*))4!HF07$F][l$\"1)*)ffnE*fGF07$Fb[l$\"1ILbWRNKGF07$Fg[l$\"1Yl]e (QG\"GF07$F\\\\l$\"1]n(H#)*f/GF07$Fa\\l$\"1_-BEcj1GF07$Ff\\l$\"1'f$3&* y#*>GF07$F[]l$\"1mYOR]![%GF07$F`]l$\"1=tHz%f-)GF07$Fe]l$\"1_zKw'fU#HF0 7$Fj]l$\"19`s^MQiHF07$F_^l$\"199!3&yaoHF07$Fd^l$\"1_>L:nX!)GF07$$\"?++ +++++++]n'*33PFet$\"19='[FhLu#F07$Fi^l$\"1gf(=\\S#zCF07$F^_l$\"1#p3$)) \\u;=F07$Fc_l$\"0i*4^;oDtF07$Fg`l$!1%R]hi5SP\"F07$Faal$!1iAP#punm&F07$ Ffal$!1#[EN^x?W*F07$F[bl$!2!3(z!RL>G:F07$F`bl$!2!*p9&H)HF&>F07$Febl$!2 5uc7gP/`#F07$Fjbl$!2Mn/uIYp'HF07$F_cl$!2_N5rK)\\kLF07$Fdcl$!2-uS<^\")4 r$F07$Ficl$!2!oQRNuTRPF07$F^dl$!2IN[=6gJ&QF07$Fcdl$!2)4Z6%R!HbYF0-Fhdl 6&Fjdl$\"#XF]elF(F[el-Fbel6#%9scheme~with~simple~nodesG-F$6%7^qF'7$Fie l$\"/.x\\\"*35>F07$F+$\"0D$*oG_?X\"F07$Fafl$\"0R*z;*H$QIF07$Fffl$\"0ff DP#eKdF07$F[gl$\"1DCwh3Q/5F07$F2$\"12Ia(HY9n\"F07$F7$\"1Z<:jFLHOF07$F< $\"1\"R!\\qL%H!pF07$FA$\"2%*=Gw3P*HWMA,$F07$$\"?+++++++](=nLoq8#F0$\"2+h4) )G&o1IF07$Fehl$\"2$>xqkI=/IF07$$\"?MLLLLLL$ekGI!R!>#F0$\"2\"G(f\"\\P]F IF07$$\"?++++++++v$fG^q@#F0$\"2hN%oa>,]IF07$$\"?nmmmmmm;/,pArVAF0$\"2v \"4%o$*fM/$F07$FU$\"2mf7^G[O.$F07$Fbil$\"2?(=%y(RwEHF07$FZ$\"2#47/(e:T l#F07$Fin$\"2-%3EL!*Ht@F07$F^o$\"2E'y,M%[[W\"F07$Fco$\"1=/!47#oFtF07$F ho$\"0a$[LMnwPF07$F[[m$!1&351*fAPmF07$F]p$!2@CHc>r^L\"F07$Fc[m$!2$f*)y !fuS%>F07$Fbp$!26c9:&fmkCF07$Fgp$!2Fs\\-3=&zJF07$F\\q$!2^!H,$ywSV$F07$ Faq$!2e>Vv=(pqKF07$Ffq$!2\"oSkdfwbFF07$Fg\\m$!2cV;$y$)[SAF07$F[r$!2X%p \\>(ops\"F07$F_]m$!2)[SN@SFel!yDF07 $Fjr$\"0k=w)z2KAF07$Ff_m$\"1!zV+d(f05F07$F``m$\"1)y\"R'GQy^\"F07$Fe`m$ \"1m=,F)4$Q=F07$F_s$\"1+`\\D@'e-#F07$F]am$\"13w&\\I07;#F07$Fds$\"13daj >VE@F07$Fis$\"1mW%4HXJ'=F07$F^t$\"1q-Jt(Qcj\"F07$Fct$\"1U'p%49*[U\"F07 $Fit$\"1#yi!H7rf7F07$F^u$\"1C4IF?,K6F07$Fcu$\"1g%poast.\"F07$Fhu$\"0Kt ;P!RR%*F07$F]v$\"0q46`7fx)F07$Fbv$\"07Qyp./7)F07$Fgv$\"0Q/6Y&4CwF07$F \\w$\"0A!*=PqK:(F07$Faw$\"0or&)z\"oknF07$Ffw$\"0y2k'GH6kF07$F[x$\"0%3^ f6wEhF07$F`x$\"0?aWX8n&eF07$Fex$\"0%3'phX;h&F07$Fjx$\"0%okM,>CaF07$F_y $\"0wmF#GjX_F07$Fdy$\"0+sO:][3&F07$Fiy$\"03()zLv%[\\F07$F^z$\"0!)He.]Y $[F07$Fcz$\"0%oA:DUFZF07$Fhz$\"01Z)pv%pk%F07$F][l$\"03Tl\\yfd%F07$Fb[l $\"0]&)HP)pBXF07$Fg[l$\"0Ye%p%\\*yWF07$F\\\\l$\"0g(Qo_3WWF07$Fa\\l$\"0 i/H:Q%3WF07$Ff\\l$\"0w&f1biqVF07$F[]l$\"0m?&=r='H%F07$F`]l$\"0)e\"GpqH <%F07$Fe]l$\"0#***pd\">1RF07$Fj]l$\"0Mxi%Q'*zKF07$F_^l$\"0W'Hdp&e,#F07 $Fd^l$!0=P(3QJR6F07$F^hm$!0'e#*o8!p*RF07$Fi^l$!0!p/B&3^h)F07$F^_l$!131 _Kmn\\=F07$Fc_l$!1oyEp_B&R$F07$Fg`l$!1%pXez(\\TiF07$Faal$!2A7Gt6fi=\"F 07$Ffal$!2-$*yApsZn\"F07$F[bl$!2!42\"pThXU#F07$F`bl$!2!>N?fy4nHF07$Feb l$!2!*[g(o1Q.PF07$Fjbl$!2%4p,#f\"fdUF07$F_cl$!2_/\\Bk7Uw%F07$Fdcl$!2_ \"H+LwM._F07$Ficl$!25d*)e;PFC&F07$F^dl$!2];!G1vo*Q&F07$Fcdl$!2)*p'\\3] O/kF0-Fhdl6&FjdlF($\"#DF]el$\"\"\"F)-Fbel6#%Pscheme~with~a~relatively~ large~stability~regionG-F$6%7bsF'7$$\"?LLLLLLLLLL3x&)*3\"F-$\".OX_;vd) F07$Fiel$\"/U.vm'QI%F07$$\"?++++++++++DJdpKF-$\"0o\"oRiq)>\"F07$F+$\"0 qe^WeQn#F07$Fffl$\"0]a\"pI<+&*F07$F2$\"1cdJ'>jUi#F07$F7$\"1-QIK\\M\\bF 07$F<$\"21lL^0[#R5F07$$\"?LLLLLLLL$e*[e-Y8F0$\"2@f4::E?O\"F07$FA$\"227 $)>)okO%e:#F07$FF$\"2d#p\"HS0ng# F07$$\"?++++++++]i!f`rt\"F0$\"2WM@!fsHpIF07$FK$\"2nH6\"o)H%>NF07$$\"?L LLLLLLL$3_XT/&>F0$\"2jh]vuIx#RF07$FP$\"2//j)RnVhUF07$Fehl$\"2JHK**H:f[ %F07$FU$\"2svpm'z\"fc%F07$Fbil$\"2E$G48[SmWF07$FZ$\"2;Y*GNSL`TF07$$\"? nmmmmmmmmTNlLPDF0$\"2BAWai)pnQF07$Fin$\"2B7/F:r4e$F07$$\"?nmmmmmmm;aj' \\yh#F0$\"2f()yFF0$\"2WEj7y#GwCF07$Fco$\"2mu4A%[;t=F07$$\"?nmmmmmmm;aQ8b KGF0$\"2*31()\\w$f'=F07$$\"?nmmmmmmmm\"z/*QfGF0$\"2NQWuVB*Q=F07$$\"?nm mmmmmm;HdnA')GF0$\"2VSR'Rjbw:F07$Fho$\"2+gz)[N$)p5F07$$\"?nmmmmmmm;/w@ !*RHF0$\"2,G*R/&*Hk5F07$$\"?nmmmmmmmmT&))Rn'HF0$\"2FKx'4*z],\"F07$$\"? nmmmmmmm;z%fxN*HF0$\"1#Qv(zfD-dF07$F[[m$\"1[[32KzDGF07$$\"?nmmmmmmm;a8 IDZIF0$\"1;l_9UPuFF07$$\"?nmmmmmmmm\"Hs!4uIF0$\"11<&*oq&*)*=F07$$\"?nm mmmmmm;HK%G45$F0$!1VC9CN^IWF07$F]p$!1'4gY`&f6WF07$$\"?nmmmmmmmmTg:W\"= $F0$!11Q=\"Hwf!fF07$Fc[m$!2M[9Xd'oh5F07$$\"?nmmmmmmmm\"zR#z)G$F0$!2*3$ **p(GRy7F07$Fbp$!2[>02*GP]:F07$$\"?++++++++]PM%3$\\MF0$!22PtC))e6*=F07 $Fgp$!2p/(>3s*)z?F07$$\"?nmmmmmm;/^1#fGe$F0$!2r#p;4W#[4#F07$$\"?++++++ ++v$4Op&4OF0$!2@QC$\\PeY@F07$$\"?MLLLLLL$ek`^zij$F0$!2O)*['yLrM@F07$$ \"?nmmmmmmm;zp'*)Hm$F0$!2$=@0!QcD7#F07$$\"?MLLLLLLLeky*4kr$F0$!2Eo:;$3 ]l?F07$F\\q$!2UUrC=e@.#F07$$\"?LLLLLLLL$3_!4nwQF0$!2^g!RP#*eD=F07$Faq$ !2\\&)Gehm4_\"F07$$\"?++++++++]iS@N!4%F0$!2UD9.ubg8\"F07$Ffq$!1:!zLTj! zoF07$$\"?+++++++++](oZiH%F0$!1@@,'4A/r$F07$Fg\\m$!0Nbpy&etuF07$$\"?LL LLLLLLL$eadV\\%F0$\"1\\w()GGYB>F07$F[r$\"1c#f,%RiXUF07$F_]m$\"1d`WO'= \"[xF07$F`r$\"13@>wPuk(*F07$$\"?NLLLLLLLLLLy>#4&F0$\"21vfl6a_.\"F07$Fg ]m$\"2Q[GEzR\"f5F07$$\"?NLLLLLLLL$eHYgC&F0$\"2VU@()Rfk0\"F07$$\"?qmmmm mmmmm;\"HtH&F0$\"2QG0c?nE0\"F07$$\"?+++++++++]P>h[`F0$\"2-8D7!*G9/\"F0 7$Fer$\"2aTw:qj/-\"F07$Fd^m$\"1q()Q-$)R<\"*F07$Fjr$\"1%)ebm@75xF07$F`` m$\"1y74u/a_YF07$F_s$\"1I`(Hz2G%>F07$F]am$!0_jWL_zC#F07$Fds$!1U>zz_]C> F07$$\"?lmmmmmmmm;H9lRzF0$!1k8gcKAGGF07$Fis$!1u)e*R?/8MF07$$\"?+++++++ +++DI(yv)F0$!1!*o.Zdw^PF07$F^t$!1+G=TG.XRF07$$\"?NLLLLLLLLLe%GCd*F0$!1 ')oJ@y-3SF07$Fct$!1Q6,S(>U.%F07$Fit$!1o!y%H2^iQF07$F^u$!1')=jK_%el$F07 $Fcu$!1I(4;f5rV$F07$Fhu$!1)\\Q&**)zU>$F07$F]v$!1!Q!\\mW<0IF07$Fbv$!13u [za\"\\!GF07$Fgv$!1KU[-&>mk#F07$F\\w$!1yB'e(>R\"\\#F07$Faw$!1_lC`TAhBF 07$Ffw$!1_7mJ\\'4C#F07$F[x$!1O(=j$)[I9#F07$F`x$!1?[R'[+$\\?F07$Fex$!1w Rp&f(oj>F07$Fjx$!1ORi3V(y*=F07$F_y$!19/uGZ+N=F07$Fdy$!1]vTT'f#yI&*HDA!p\"F07$Fcz$!1;X*)3$))Hl\"F07$Fhz$!1%Gk ,(otD;F07$F][l$!1AQLP$[Ig\"F07$Fb[l$!1S#o%fbI)e\"F07$Fg[l$!1M@J]Zwy:F0 7$F\\\\l$!1!)z8Q-jw:F07$Fa\\l$!1[J-%RSDe\"F07$Ff\\l$!1%H5jTUvf\"F07$F[ ]l$!1CYxaXWE;F07$F`]l$!1Knx=W2q;F07$Fe]l$!1y0$*fB)*QWI#F07$Fi^l$!1+ez4jQxFF07$F^_l$! 1yv.8P!Q>$F07$Fc_l$!13$z`7*z\\OF07$F]`l$!1=()3N/-xQF07$Fg`l$!1%*4#\\! \\QjSF07$$\"?++++++++](o:gF)QFet$!1w>_L>j>TF07$F\\al$!1K5F&Rv?8%F07$$ \"?++++++++]i!**yH!RFet$!1yZT=$R(ySF07$Faal$!1#*H!pDF&GRF07$Ffal$!1KwS c&pm9$F07$F[bl$!1+!H#R2ne5F07$F`bl$\"1qD2Q3:,5F07$Febl$\"1Squ/)*=;WF07 $Fjbl$\"1c>=qvX?tF07$F_cl$\"2QY;W],z.\"F07$Fdcl$\"2Qr]EK5mI\"F07$Ficl$ \"25`*ykGz=8F07$F^dl$\"2gS!e=Ix79F07$Fcdl$\"2sC%*\\yyd9#F0-Fhdl6&FjdlF ($\"#vF]elF^el-Fbel6#%TButcher's~scheme~B~with~c[5]=c[6]=1/2~and~b[5]= b[6]G-F$6%7aqF'7$Fe_o$\".DXs_(=NF07$Fiel$\"/!od<3_l\"F07$F]`o$\"/RigIX 4YF07$F+$\"0c*)*yd$4/\"F07$Fffl$\"0u)[9_'[z$F07$F2$\"1/)y-4([o5F07$F7$ \"1TH3ER5!H#F07$F<$\"1p%[/O)4ZVF07$Faao$\"1?1Ex`e\"F07$Fibo$\"2\"zUR%z[\\!=F07$FP$\"2tu@&epy7?F07$Fehl $\"2!oGQi)>M>#F07$FU$\"2F6v)RJVMBF07$Fbil$\"2<#*[WPiLU#F07$FZ$\"2U$\\> 8I#zW#F07$Fin$\"2zYQ62F[R#F07$F^o$\"2;P)GJ#H*\\AF07$Fco$\"2nM*y,zQp?F0 7$Fho$\"2$>7j;0vu=F07$F[[m$\"24i'\\E_Gq;F07$F]p$\"2!*)[7`FVr9F07$Fc[m$ \"2w$pz+!G?H\"F07$Fbp$\"2j$o1`\")4V6F07$F_[p$\"2'RxEG$HB.\"F07$Fgp$\"1 J&oVn**fj*F07$F\\\\p$\"1![(o6cBs%*F07$Ff\\p$\"1E3)R-/\"y$*F07$$\"?++++ +++](=U#)*p*o$F0$\"1MXCZI$GY*F07$F[]p$\"1#>n(ebY$f*F07$$\"?nmmmmmm;H2L ,7VPF0$\"1+M9)4P.a*F07$F\\q$\"1@F6YO'f`*F07$Faq$\"2WN&[*3Z!*4\"F07$Ffq $\"2q'osX.oy8F07$Fg\\m$\"2h8mDu$>*e\"F07$F[r$\"2MMf\">_9wF07$F`r$\"2([Q3=d3D?F07$Fg]m$\"2n&=ERuS+@F07$Fer$\"2Fp!3wakK @F07$F_^m$\"2GU)ROg*o7#F07$Fd^m$\"25<<(4#F07$ Fjr$\"2%ygR2^av?F07$F``m$\"2)>lGn/Cb>F07$F_s$\"2gJ,,n!=<=F07$Fds$\"2eg \"Hluoa:F07$Fis$\"2c'4v1TqI8F07$F^t$\"25z(G8Sut6F07$Fct$\"2#)H(\\a:IK5 F07$Fit$\"1KG-Dh>)=*F07$F^u$\"1/bV'>@vH)F07$Fcu$\"1!Gk/F[Bi(F07$Fhu$\" 1Uf/OYk[pF07$F]v$\"1qlZn,vlkF07$Fbv$\"1-/Qx@I&)fF07$Fgv$\"1)=tt6p)>cF0 7$F\\w$\"1i,pYcBs_F07$Faw$\"1G^*[Z!p%)\\F07$Ffw$\"1GDU%zIGs%F07$F[x$\" 1a%)\\^&*y6XF07$F`x$\"1q-lI.Y6VF07$Fex$\"1%HD@>!eHTF07$Fjx$\"1aYMK,X!* RF07$F_y$\"1;9%yd2\"eQF07$Fdy$\"15E(\\r*4RPF07$Fiy$\"1Qw)*3DdQOF07$F^z $\"1+%pw&\\?bNF07$Fcz$\"1u7R6HqxMF07$Fhz$\"11za\"Q45U$F07$F][l$\"1y!fr esOP$F07$Fb[l$\"15Q9s,\\ULF07$Fg[l$\"1wl1h)e8K$F07$F\\\\l$\"1!)f)3DAVJ $F07$Fa\\l$\"1#=VyH(4@LF07$Ff\\l$\"11F4yh&HM$F07$F[]l$\"1O'=8Y\"e$Q$F0 7$F`]l$\"1)Rl9w4CW$F07$Fe]l$\"1A!Q^RD^_$F07$Fj]l$\"1/aE+2?JOF07$F_^l$ \"1/&f+cD&\\PF07$Fd^l$\"1#o$y\\$yZ!RF07$F^hm$\"1%49sU,%oRF07$Fi^l$\"15 /%[/IU,%F07$F^_l$\"1#ojZ71J)RF07$Fc_l$\"1_[(*f(GC(QF07$F]`l$\"1#*z)=Tc 3u$F07$Fg`l$\"11#G-)R8INF07$F\\al$\"1ol3S81.KF07$Faal$\"1y%p'oMR.FF07$ Ffal$\"1G>BqQbZ>F07$F[bl$\"0!z$yOrff(F07$F`bl$!0IFY]r)[5F07$Febl$!1!=i 2X[LF\"F07$F_cl$!1-BH=\"GP$HF07$Fcdl$!1y;-?LQHaF0-Fhdl6&FjdlF[elFa[nF( -Fbel6#%Hscheme~with~c[5]=c[6]=3/4~and~b[5]=b[6]G-%%FONTG6$%*HELVETICA G\"\"*-%+AXESLABELSG6$Q\"x6\"Q!Fabr-%&TITLEG6#%Uerror~curves~for~7~sta ge~order~6~Runge-Kutta~methodsG-%%VIEWG6$;F(Fcdl%(DEFAULTG" 1 2 0 1 10 0 2 9 1 4 2 1.000000 45.000000 45.000000 0 1 "Butcher's scheme A" " scheme with simple nodes" "scheme with a relatively large stability re gion" "Butcher's scheme B with c[5]=c[6]=1/2 and b[5]=b[6]" "scheme wi th c[5]=c[6]=3/4 and b[5]=b[6]" }}}}{PARA 0 "" 0 "" {TEXT -1 0 "" }} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 455 "evalf[30](plot(['wn_RK6_2'( x)-w(x),'wn_RK6_3'(x)-w(x),'wn_RK6_4'(x)-w(x),'wn_RK6_5'(x)-w(x)],x=0. .4,\ncolor=[COLOR(RGB,.45,0,.95),COLOR(RGB,0,.25,1),COLOR(RGB,0,.75,.2 ),COLOR(RGB,.95,.45,0)],\nlegend=[`scheme with simple nodes`,`scheme w ith a relatively large stability region`,`Butcher's scheme B with c[5] =c[6]=1/2 and b[5]=b[6]`,`scheme with c[5]=c[6]=3/4 and b[5]=b[6]`],fo nt=[HELVETICA,9],title=`error curves for 7 stage order 6 Runge-Kutta m ethods`));" }}{PARA 13 "" 1 "" {GLPLOT2D 935 447 447 {PLOTDATA 2 "6*-% 'CURVESG6%7`q7$$\"\"!F)F(7$$\"?mmmmmmmmmm;arz@!#J$\"/bQzm!**=#!#I7$$\" ?LLLLLLLLLLL3VfVF-$\"0bh]'Rp2:F07$$\"?lmmmmmmmmmT&)G\\aF-$\"0#)H6-^7F0$\"1M/\\rW`NoF07$$\"?mmmmmmmmmT &y`3W\"F0$\"2\"R$f?!GEd6F07$$\"?LLLLLLLLLLe'40j\"F0$\"2t([IHW?g/vc9CF07$$\"?+++++++++](Q&3d?F0$\"2I>^,Y !y')HF07$$\"?nmmmmmmm;z>$HP;#F0$\"2xCy(3C%))=$F07$$\"?MLLLLLLLL3_KPqAF 0$\"2m@F0hlPI$F07$$\"?nmmmmmmm\"H#=_pBBF0$\"2YvD(o2/JLF07$$\"?++++++++ ]P%=,\"f#F0$\"2_T&yy '*=vGF07$$\"?nmmmmmmmmm\"zi$)p#F0$\"2Iz*3![uiP#F07$$\"?nmmmmmmmm;HOr0G F0$\"2AP25W>B'=F07$$\"?nmmmmmmmmmmW18HF0$\"2@$f5)fw)[8F07$$\"?nmmmmmmm m;/`T?IF0$\"1v,fhlE6F07$$\"?mmmmmmmmm\"H_6N)RF0$!1MMe\")R!z6*F07$$\"?LLLLLLLLLLeF>( >%F0$!1nr1,Xt,TF07$$\"?mmmmmmmmmm;EI&R%F0$\"0u!H\")3)G=$F07$$\"?++++++ ++++vCT$f%F0$\"1!=Z@M.k_%F07$$\"?LLLLLLLLLLLB_\"z%F0$\"1G&*Q-*f$[\")F0 7$$\"?mmmmmmmmmm\">K'*)\\F0$\"2zcI\"*oFC5\"F07$$\"?++++++++++vMw%>&F0$ \"2x`;nFOrM\"F07$$\"?NLLLLLLLLLeZ*)*R&F0$\"2,y6hP1R^\"F07$$\"?++++++++ +++/Y-bF0$\"2)>13eDTd:F07$$\"?qmmmmmmmmmTg-0cF0$\"2+q[Gox!)e\"F07$$\"? NLLLLLLLLL$o\"f2dF0$\"2-(GDx%f#3;F07$$\"?++++++++++Dt:5eF0$\"2M\\b;>5& >;F07$$\"?NLLLLLLLLLe3B;fF0$\"2#>D/d9#>i\"F07$$\"?qmmmmmmmmm\"R/B-'F0$ \"2?&>7Ak3=;F07$$\"?++++++++++DzPGhF0$\"2_>)3I!=\"4;F07$$\"?NLLLLLLLLL e9XMiF0$\"2eMD]o$)ff\"F07$$\"?++++++++++D&)fYkF0$\"2w$*R,/L/c\"F07$$\" ?mmmmmmmmmm\"fX(emF0$\"2I@uSrUn^\"F07$$\"?NLLLLLLLLL3!z;3(F0$\"23xDMfj \">9F07$$\"?++++++++++DCh/vF0$\"2G!Q?*p!p>8F07$$\"?LLLLLLLLLLL/pu$)F0$ \"2E;ra%3SK6F07$$\"?mmmmmmmmmm;c0T\"*F0$\"1I1xOO&o)**F07$$\"?+++++++++ +I,Q+5!#H$\"1-[pq2Yx()F07$$\"?++++++++++]*3q3\"Fc[l$\"1-B(zOJ%3yF07$$ \"?++++++++++q=\\q6Fc[l$\"1M!e=*Rg[qF07$$\"?nmmmmmmmm;fBIY7Fc[l$\"1!G( GPvhtkF07$$\"?LLLLLLLLLLj$[kL\"Fc[l$\"1sx*\\)eI+fF07$$\"?LLLLLLLLLL`Q \"GT\"Fc[l$\"1+q(y\")Q'*[&F07$$\"?+++++++++]s]k,:Fc[l$\"17Y5'el73&F07$ $\"?LLLLLLLLLL`dF!e\"Fc[l$\"1)QK><\\2x%F07$$\"?+++++++++]sgam;Fc[l$\"1 _'>!H%HaZ%F07$$\"?+++++++++]Fc[l$\"1kFc[l$\"1+PSj:EfOF07$$\"?nmmmmmmmm;f`@'3#Fc[l$\"1k'pV2knsJF07$$\"?+++++++ +++0TN:CFc[l$\"13/gHz2(3$F07$$\"?+++++++++]7RV'\\#Fc[l$\"1S8SCf(f,$F07 $$\"?++++++++++:#fke#Fc[l$\"1%fH4S)p\\HF07$$\"?LLLLLLLLLL`4NnEFc[l$\"1 1)oL(*))4!HF07$$\"?++++++++++],s`FFc[l$\"1)*)ffnE*fGF07$$\"?nmmmmmmmm; zM)>$GFc[l$\"1ILbWRNKGF07$$\"?++++++++++qfaGF07$$\"?+++++++++]7yh ]KFc[l$\"1mYOR]![%GF07$$\"?nmmmmmmmmm')fdLLFc[l$\"1=tHz%f-)GF07$$\"?nm mmmmmmmm,FT=MFc[l$\"1_zKw'fU#HF07$$\"?LLLLLLLLL$e#pa-NFc[l$\"19`s^MQiH F07$$\"?++++++++++Sv&)zNFc[l$\"199!3&yaoHF07$$\"?LLLLLLLLLLGUYoOFc[l$ \"1_>L:nX!)GF07$$\"?+++++++++]n'*33PFc[l$\"19='[FhLu#F07$$\"?nmmmmmmmm m1^rZPFc[l$\"1gf(=\\S#zCF07$$\"?MLLLLLLLLe*3k**y$Fc[l$\"1#p3$))\\u;=F0 7$$\"?+++++++++]sI@KQFc[l$\"0i*4^;oDtF07$$\"?++++++++++S2lsQFc[l$!1%R] hi5SP\"F07$$\"?+++++++++]2%)38RFc[l$!1iAP#punm&F07$$\"?++++++++]i0j\"[ $RFc[l$!1#[EN^x?W*F07$$\"?+++++++++v.UacRFc[l$!2!3(z!RL>G:F07$$\"?++++ ++++D\"G:3u'RFc[l$!2!*p9&H)HF&>F07$$\"?++++++++](=5s#yRFc[l$!25uc7gP/` #F07$$\"?+++++++]iSwSq$)RFc[l$!2Mn/uIYp'HF07$$\"?++++++++v$40O\"*)RFc[ l$!2_N5rK)\\kLF07$$\"?+++++++DJ?Q?&=*RFc[l$!2-uS<^\")4r$F07$$\"?++++++ +](oa-oX*RFc[l$!2!oQRNuTRPF07$$\"?+++++++vVt7SG(*RFc[l$!2IN[=6gJ&QF07$ $\"\"%F)$!2)4Z6%R!HbYF0-%&COLORG6&%$RGBG$\"#X!\"#F($\"#&*F\\[m-%'LEGEN DG6#%9scheme~with~simple~nodesG-F$6%7^qF'7$F+$\"/.x\\\"*35>F07$F2$\"0D $*oG_?X\"F07$F7$\"0R*z;*H$QIF07$F<$\"0ffDP#eKdF07$FA$\"1DCwh3Q/5F07$FF $\"12Ia(HY9n\"F07$FK$\"1Z<:jFLHOF07$FP$\"1\"R!\\qL%H!pF07$FU$\"2%*=Gw3 P* HWMA,$F07$$\"?+++++++](=nLoq8#F0$\"2+h4))G&o1IF07$Fco$\"2$>xqkI=/IF07$ $\"?MLLLLLL$ekGI!R!>#F0$\"2\"G(f\"\\P]FIF07$$\"?++++++++v$fG^q@#F0$\"2 hN%oa>,]IF07$$\"?nmmmmmm;/,pArVAF0$\"2v\"4%o$*fM/$F07$Fho$\"2mf7^G[O.$ F07$Fbp$\"2?(=%y(RwEHF07$F\\q$\"2#47/(e:Tl#F07$Faq$\"2-%3EL!*Ht@F07$Ff q$\"2E'y,M%[[W\"F07$F[r$\"1=/!47#oFtF07$F`r$\"0a$[LMnwPF07$Fer$!1&351* fAPmF07$Fjr$!2@CHc>r^L\"F07$F_s$!2$f*)y!fuS%>F07$Fds$!26c9:&fmkCF07$Fi s$!2Fs\\-3=&zJF07$F^t$!2^!H,$ywSV$F07$Fct$!2e>Vv=(pqKF07$Fht$!2\"oSkdf wbFF07$F]u$!2cV;$y$)[SAF07$Fbu$!2X%p\\>(ops\"F07$Fgu$!2)[SN@SFel!yDF07$Fjw$\"0k=w)z2KAF07$Fdx$\"1!zV+d( f05F07$F^y$\"1)y\"R'GQy^\"F07$Fcy$\"1m=,F)4$Q=F07$Fhy$\"1+`\\D@'e-#F07 $F]z$\"13w&\\I07;#F07$Fbz$\"13daj>VE@F07$Fgz$\"1mW%4HXJ'=F07$F\\[l$\"1 q-Jt(Qcj\"F07$Fa[l$\"1U'p%49*[U\"F07$Fg[l$\"1#yi!H7rf7F07$F\\\\l$\"1C4 IF?,K6F07$Fa\\l$\"1g%poast.\"F07$Ff\\l$\"0Kt;P!RR%*F07$F[]l$\"0q46`7fx )F07$F`]l$\"07Qyp./7)F07$Fe]l$\"0Q/6Y&4CwF07$Fj]l$\"0A!*=PqK:(F07$F_^l $\"0or&)z\"oknF07$Fd^l$\"0y2k'GH6kF07$Fi^l$\"0%3^f6wEhF07$F^_l$\"0?aWX 8n&eF07$Fc_l$\"0%3'phX;h&F07$Fh_l$\"0%okM,>CaF07$F]`l$\"0wmF#GjX_F07$F b`l$\"0+sO:][3&F07$Fg`l$\"03()zLv%[\\F07$F\\al$\"0!)He.]Y$[F07$Faal$\" 0%oA:DUFZF07$Ffal$\"01Z)pv%pk%F07$F[bl$\"03Tl\\yfd%F07$F`bl$\"0]&)HP)p BXF07$Febl$\"0Ye%p%\\*yWF07$Fjbl$\"0g(Qo_3WWF07$F_cl$\"0i/H:Q%3WF07$Fd cl$\"0w&f1biqVF07$Ficl$\"0m?&=r='H%F07$F^dl$\"0)e\"GpqH<%F07$Fcdl$\"0# ***pd\">1RF07$Fhdl$\"0Mxi%Q'*zKF07$F]el$\"0W'Hdp&e,#F07$Fbel$!0=P(3QJR 6F07$Fgel$!0'e#*o8!p*RF07$F\\fl$!0!p/B&3^h)F07$Fafl$!131_Kmn\\=F07$Fff l$!1oyEp_B&R$F07$F[gl$!1%pXez(\\TiF07$F`gl$!2A7Gt6fi=\"F07$Fegl$!2-$*y ApsZn\"F07$Fjgl$!2!42\"pThXU#F07$F_hl$!2!>N?fy4nHF07$Fdhl$!2!*[g(o1Q.P F07$Fihl$!2%4p,#f\"fdUF07$F^il$!2_/\\Bk7Uw%F07$Fcil$!2_\"H+LwM._F07$Fh il$!25d*)e;PFC&F07$F]jl$!2];!G1vo*Q&F07$Fbjl$!2)*p'\\3]O/kF0-Fgjl6&Fij lF($\"#DF\\[m$\"\"\"F)-F`[m6#%Pscheme~with~a~relatively~large~stabilit y~regionG-F$6%7bsF'7$$\"?LLLLLLLLLL3x&)*3\"F-$\".OX_;vd)F07$F+$\"/U.vm 'QI%F07$$\"?++++++++++DJdpKF-$\"0o\"oRiq)>\"F07$F2$\"0qe^WeQn#F07$F<$ \"0]a\"pI<+&*F07$FF$\"1cdJ'>jUi#F07$FK$\"1-QIK\\M\\bF07$FP$\"21lL^0[#R 5F07$$\"?LLLLLLLL$e*[e-Y8F0$\"2@f4::E?O\"F07$FU$\"227$)>)okO%e:#F07$FZ$\"2d#p\"HS0ng#F07$$\"?++++++++] i!f`rt\"F0$\"2WM@!fsHpIF07$Fin$\"2nH6\"o)H%>NF07$$\"?LLLLLLLL$3_XT/&>F 0$\"2jh]vuIx#RF07$F^o$\"2//j)RnVhUF07$Fco$\"2JHK**H:f[%F07$Fho$\"2svpm 'z\"fc%F07$Fbp$\"2E$G48[SmWF07$F\\q$\"2;Y*GNSL`TF07$$\"?nmmmmmmmmTNlLP DF0$\"2BAWai)pnQF07$Faq$\"2B7/F:r4e$F07$$\"?nmmmmmmm;aj'\\yh#F0$\"2f()yF F0$\"2WEj7y#GwCF07$F[r$\"2mu4A%[;t=F07$$\"?nmmmmmmm;aQ8bKGF0$\"2*31() \\w$f'=F07$$\"?nmmmmmmmm\"z/*QfGF0$\"2NQWuVB*Q=F07$$\"?nmmmmmmm;HdnA') GF0$\"2VSR'Rjbw:F07$F`r$\"2+gz)[N$)p5F07$$\"?nmmmmmmm;/w@!*RHF0$\"2,G* R/&*Hk5F07$$\"?nmmmmmmmmT&))Rn'HF0$\"2FKx'4*z],\"F07$$\"?nmmmmmmm;z%fx N*HF0$\"1#Qv(zfD-dF07$Fer$\"1[[32KzDGF07$$\"?nmmmmmmm;a8IDZIF0$\"1;l_9 UPuFF07$$\"?nmmmmmmmm\"Hs!4uIF0$\"11<&*oq&*)*=F07$$\"?nmmmmmmm;HK%G45$ F0$!1VC9CN^IWF07$Fjr$!1'4gY`&f6WF07$$\"?nmmmmmmmmTg:W\"=$F0$!11Q=\"Hwf !fF07$F_s$!2M[9Xd'oh5F07$$\"?nmmmmmmmm\"zR#z)G$F0$!2*3$**p(GRy7F07$Fds $!2[>02*GP]:F07$$\"?++++++++]PM%3$\\MF0$!22PtC))e6*=F07$Fis$!2p/(>3s*) z?F07$$\"?nmmmmmm;/^1#fGe$F0$!2r#p;4W#[4#F07$$\"?++++++++v$4Op&4OF0$!2 @QC$\\PeY@F07$$\"?MLLLLLL$ek`^zij$F0$!2O)*['yLrM@F07$$\"?nmmmmmmm;zp'* )Hm$F0$!2$=@0!QcD7#F07$$\"?MLLLLLLLeky*4kr$F0$!2Eo:;$3]l?F07$F^t$!2UUr C=e@.#F07$$\"?LLLLLLLL$3_!4nwQF0$!2^g!RP#*eD=F07$Fct$!2\\&)Gehm4_\"F07 $$\"?++++++++]iS@N!4%F0$!2UD9.ubg8\"F07$Fht$!1:!zLTj!zoF07$$\"?+++++++ ++](oZiH%F0$!1@@,'4A/r$F07$F]u$!0Nbpy&etuF07$$\"?LLLLLLLLL$eadV\\%F0$ \"1\\w()GGYB>F07$Fbu$\"1c#f,%RiXUF07$Fgu$\"1d`WO'=\"[xF07$F\\v$\"13@>w Puk(*F07$$\"?NLLLLLLLLLLy>#4&F0$\"21vfl6a_.\"F07$Fav$\"2Q[GEzR\"f5F07$ $\"?NLLLLLLLL$eHYgC&F0$\"2VU@()Rfk0\"F07$$\"?qmmmmmmmmm;\"HtH&F0$\"2QG 0c?nE0\"F07$$\"?+++++++++]P>h[`F0$\"2-8D7!*G9/\"F07$Ffv$\"2aTw:qj/-\"F 07$F`w$\"1q()Q-$)R<\"*F07$Fjw$\"1%)ebm@75xF07$F^y$\"1y74u/a_YF07$Fhy$ \"1I`(Hz2G%>F07$F]z$!0_jWL_zC#F07$Fbz$!1U>zz_]C>F07$$\"?lmmmmmmmm;H9lR zF0$!1k8gcKAGGF07$Fgz$!1u)e*R?/8MF07$$\"?++++++++++DI(yv)F0$!1!*o.Zdw^ PF07$F\\[l$!1+G=TG.XRF07$$\"?NLLLLLLLLLe%GCd*F0$!1')oJ@y-3SF07$Fa[l$!1 Q6,S(>U.%F07$Fg[l$!1o!y%H2^iQF07$F\\\\l$!1')=jK_%el$F07$Fa\\l$!1I(4;f5 rV$F07$Ff\\l$!1)\\Q&**)zU>$F07$F[]l$!1!Q!\\mW<0IF07$F`]l$!13u[za\"\\!G F07$Fe]l$!1KU[-&>mk#F07$Fj]l$!1yB'e(>R\"\\#F07$F_^l$!1_lC`TAhBF07$Fd^l $!1_7mJ\\'4C#F07$Fi^l$!1O(=j$)[I9#F07$F^_l$!1?[R'[+$\\?F07$Fc_l$!1wRp& f(oj>F07$Fh_l$!1ORi3V(y*=F07$F]`l$!19/uGZ+N=F07$Fb`l$!1]vTT'f#yI&*HDA!p\"F07$Faal$!1;X*)3$))Hl\"F07$Ffal$ !1%Gk,(otD;F07$F[bl$!1AQLP$[Ig\"F07$F`bl$!1S#o%fbI)e\"F07$Febl$!1M@J]Z wy:F07$Fjbl$!1!)z8Q-jw:F07$F_cl$!1[J-%RSDe\"F07$Fdcl$!1%H5jTUvf\"F07$F icl$!1CYxaXWE;F07$F^dl$!1Knx=W2q;F07$Fcdl$!1y0$*fB)*QWI#F07$F\\fl$!1+ez4jQxFF07$Fafl $!1yv.8P!Q>$F07$Fffl$!13$z`7*z\\OF07$$\"?+++++++++D1>V_QFc[l$!1=()3N/- xQF07$F[gl$!1%*4#\\!\\QjSF07$$\"?++++++++](o:gF)QFc[l$!1w>_L>j>TF07$$ \"?+++++++++vt&pG*QFc[l$!1K5F&Rv?8%F07$$\"?++++++++]i!**yH!RFc[l$!1yZT =$R(ySF07$F`gl$!1#*H!pDF&GRF07$Fegl$!1KwSc&pm9$F07$Fjgl$!1+!H#R2ne5F07 $F_hl$\"1qD2Q3:,5F07$Fdhl$\"1Squ/)*=;WF07$Fihl$\"1c>=qvX?tF07$F^il$\"2 QY;W],z.\"F07$Fcil$\"2Qr]EK5mI\"F07$Fhil$\"25`*ykGz=8F07$F]jl$\"2gS!e= Ix79F07$Fbjl$\"2sC%*\\yyd9#F0-Fgjl6&FijlF($\"#vF\\[m$\"\"#!\"\"-F`[m6# %TButcher's~scheme~B~with~c[5]=c[6]=1/2~and~b[5]=b[6]G-F$6%7aqF'7$Fb_n $\".DXs_(=NF07$F+$\"/!od<3_l\"F07$Fj_n$\"/RigIX4YF07$F2$\"0c*)*yd$4/\" F07$F<$\"0u)[9_'[z$F07$FF$\"1/)y-4([o5F07$FK$\"1TH3ER5!H#F07$FP$\"1p%[ /O)4ZVF07$F^an$\"1?1Ex`e\"F07$F fbn$\"2\"zUR%z[\\!=F07$F^o$\"2tu@&epy7?F07$Fco$\"2!oGQi)>M>#F07$Fho$\" 2F6v)RJVMBF07$Fbp$\"2<#*[WPiLU#F07$F\\q$\"2U$\\>8I#zW#F07$Faq$\"2zYQ62 F[R#F07$Ffq$\"2;P)GJ#H*\\AF07$F[r$\"2nM*y,zQp?F07$F`r$\"2$>7j;0vu=F07$ Fer$\"24i'\\E_Gq;F07$Fjr$\"2!*)[7`FVr9F07$F_s$\"2w$pz+!G?H\"F07$Fds$\" 2j$o1`\")4V6F07$F\\[o$\"2'RxEG$HB.\"F07$Fis$\"1J&oVn**fj*F07$Fi[o$\"1! [(o6cBs%*F07$Fc\\o$\"1E3)R-/\"y$*F07$$\"?+++++++](=U#)*p*o$F0$\"1MXCZI $GY*F07$Fh\\o$\"1#>n(ebY$f*F07$$\"?nmmmmmm;H2L,7VPF0$\"1+M9)4P.a*F07$F ^t$\"1@F6YO'f`*F07$Fct$\"2WN&[*3Z!*4\"F07$Fht$\"2q'osX.oy8F07$F]u$\"2h 8mDu$>*e\"F07$Fbu$\"2MMf\">_9wF07$F\\v$\"2([Q3=d 3D?F07$Fav$\"2n&=ERuS+@F07$Ffv$\"2Fp!3wakK@F07$F[w$\"2GU)ROg*o7#F07$F` w$\"25<<(4#F07$Fjw$\"2%ygR2^av?F07$F^y$\"2)>lG n/Cb>F07$Fhy$\"2gJ,,n!=<=F07$Fbz$\"2eg\"Hluoa:F07$Fgz$\"2c'4v1TqI8F07$ F\\[l$\"25z(G8Sut6F07$Fa[l$\"2#)H(\\a:IK5F07$Fg[l$\"1KG-Dh>)=*F07$F\\ \\l$\"1/bV'>@vH)F07$Fa\\l$\"1!Gk/F[Bi(F07$Ff\\l$\"1Uf/OYk[pF07$F[]l$\" 1qlZn,vlkF07$F`]l$\"1-/Qx@I&)fF07$Fe]l$\"1)=tt6p)>cF07$Fj]l$\"1i,pYcBs _F07$F_^l$\"1G^*[Z!p%)\\F07$Fd^l$\"1GDU%zIGs%F07$Fi^l$\"1a%)\\^&*y6XF0 7$F^_l$\"1q-lI.Y6VF07$Fc_l$\"1%HD@>!eHTF07$Fh_l$\"1aYMK,X!*RF07$F]`l$ \"1;9%yd2\"eQF07$Fb`l$\"15E(\\r*4RPF07$Fg`l$\"1Qw)*3DdQOF07$F\\al$\"1+ %pw&\\?bNF07$Faal$\"1u7R6HqxMF07$Ffal$\"11za\"Q45U$F07$F[bl$\"1y!fresO P$F07$F`bl$\"15Q9s,\\ULF07$Febl$\"1wl1h)e8K$F07$Fjbl$\"1!)f)3DAVJ$F07$ F_cl$\"1#=VyH(4@LF07$Fdcl$\"11F4yh&HM$F07$Ficl$\"1O'=8Y\"e$Q$F07$F^dl$ \"1)Rl9w4CW$F07$Fcdl$\"1A!Q^RD^_$F07$Fhdl$\"1/aE+2?JOF07$F]el$\"1/&f+c D&\\PF07$Fbel$\"1#o$y\\$yZ!RF07$Fgel$\"1%49sU,%oRF07$F\\fl$\"15/%[/IU, %F07$Fafl$\"1#ojZ71J)RF07$Fffl$\"1_[(*f(GC(QF07$Fcjo$\"1#*z)=Tc3u$F07$ F[gl$\"11#G-)R8INF07$F`[p$\"1ol3S81.KF07$F`gl$\"1y%p'oMR.FF07$Fegl$\"1 G>BqQbZ>F07$Fjgl$\"0!z$yOrff(F07$F_hl$!0IFY]r)[5F07$Fdhl$!1!=i2X[LF\"F 07$F^il$!1-BH=\"GP$HF07$Fbjl$!1y;-?LQHaF0-Fgjl6&FijlF][mFjjlF(-F`[m6#% Hscheme~with~c[5]=c[6]=3/4~and~b[5]=b[6]G-%%FONTG6$%*HELVETICAG\"\"*-% +AXESLABELSG6$Q\"x6\"Q!Febq-%&TITLEG6#%Uerror~curves~for~7~stage~order ~6~Runge-Kutta~methodsG-%%VIEWG6$;F(Fbjl%(DEFAULTG" 1 2 0 1 10 0 2 9 1 4 2 1.000000 45.000000 45.000000 0 1 "scheme with simple nodes" "sch eme with a relatively large stability region" "Butcher's scheme B with c[5]=c[6]=1/2 and b[5]=b[6]" "scheme with c[5]=c[6]=3/4 and b[5]=b[6] " }}}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 46 "Test 9 of \+ 7 stage, order 6 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 "Solution: " }} {PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "y = 1/sqrt(sin(2*x)+2 *x+1/2)" "6#/%\"yG*&\"\"\"F&-%%sqrtG6#,(-%$sinG6#*&\"\"#F&%\"xGF&F&*&F /F&F0F&F&*&F&F&F/!\"\"F&F3" }{TEXT -1 2 ". " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 172 "de := diff(y(x),x )=-(1+cos(2*x))*y(x)^3;\nic := y(0)=sqrt(2);\ndsolve(\{de,ic\},y(x)); \nm := unapply(rhs(%),x):\nplot(m(x),x=0..3,0..1.42,font=[HELVETICA,9] ,labels=[`x`,`y(x)`]);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%#deG/-%%di ffG6$-%\"yG6#%\"xGF,,$*&,&\"\"\"F0-%$cosG6#,$*&\"\"#F0F,F0F0F0F0)F)\" \"$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&-%'CURVESG6$7Y 7$$\"\"!F)$\"3:&4tBc8UT\"!#<7$$\"3$*****\\ilyM;!#>$\"3ozW7@k#*H8F,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$\"3 IyW%eHk>[*FJ7$$\"3+++]Peui=FJ$\"3#4`!o2+#G**)FJ7$$\"37+++]$)z%=#FJ$\"3 OGH4wwYu&)FJ7$$\"3A++]i3&o]#FJ$\"3=1g%=M2W@)FJ7$$\"3%)***\\(oX*y9$FJ$ \"31u2v$Q9&GwFJ7$$\"3z***\\P9CAu$FJ$\"3=XIMTf7+sFJ7$$\"3!)***\\P*zhdVF J$\"3P$G(zQ8#4%oFJ7$$\"31++v$>fS*\\FJ$\"3X'3%RcqqPlFJ7$$\"3$)***\\(=$f %GcFJ$\"3mYY%G?7\"*G'FJ7$$\"3Q+++Dy,\"G'FJ$\"3?u@-35zxgFJ7$$\"33++]7[X$ocbFJ7$$\"3)) ***\\PpnsM*FJ$\"3!\\;$Q)fJR[&FJ7$$\"3,++]siL-5F,$\"3&3j/q(Qq8aFJ7$$\"3 -+++!R5'f5F,$\"3q`:6QhHm`FJ7$$\"3)***\\P/QBE6F,$\"3@Igj*yDKK&FJ7$$\"3! ******\\\"o?&=\"F,$\"3i/K.-M\\%H&FJ7$$\"31+]Pa&4*\\7F,$\"3OjcS#)ygr_FJ 7$$\"33+]7j=_68F,$\"3'e4m\")R`oD&FJ7$$\"33++vVy!eP\"F,$\"3a@U-1/NZ_FJ7 $$\"34+](=WU[V\"F,$\"3Nrr*HO\"oU_FJ7$$\"3)****\\7B>&)\\\"F,$\"3'HX%)zw R1C&FJ7$$\"3)***\\P>:mk:F,$\"3<^\"Q\"4\"y-C&FJ7$$\"3'***\\iv&QAi\"F,$ \"3:*4?^OZ,C&FJ7$$\"31++vtLU%o\"F,$\"3\"3gSMou)Q_FJ7$$\"3!******\\Nm'[ F,$\"3[h+0^h(R>&FJ7$$\"3z *****\\@80+#F,$\"3!zBIi>A%o^FJ7$$\"31++]7,Hl?F,$\"3<)30`]&>L^FJ7$$\"3( )**\\P4w)R7#F,$\"3!Qwx>a)*Q4&FJ7$$\"3;++]x%f\")=#F,$\"3q$pQbJ#)G/&FJ7$ $\"3!)**\\P/-a[AF,$\"3gJla\"HTu)\\FJ7$$\"3/+](=Yb;J#F,$\"3c:[>;?IA\\FJ 7$$\"3')****\\i@OtBF,$\"3m09))4iC_[FJ7$$\"3')**\\PfL'zV#F,$\"3%Gjf])o8 tZFJ7$$\"3>+++!*>=+DF,$\"3[G/4+_V#p%FJ7$$\"3-++DE&4Qc#F,$\"3!**R*=7x[1 YFJ7$$\"3=+]P%>5pi#F,$\"3f7E:iH**=XFJ7$$\"39+++bJ*[o#F,$\"3cgVvc$ovV%F J7$$\"33++Dr\"[8v#F,$\"3Ln\\jDQ5WVFJ7$$\"3++++Ijy5GF,$\"3OZ!)Q%zK7E%FJ 7$$\"31+]P/)fT(GF,$\"3)*4_&egIW<%FJ7$$\"31+]i0j\"[$HF,$\"3qns]&)H\\$4% FJ7$$\"\"$F)$\"3ntdq;jW4SFJ-%'COLOURG6&%$RGBG$\"#5!\"\"F(F(-%%FONTG6$% *HELVETICAG\"\"*-%+AXESLABELSG6$%\"xG%%y(x)G-%%VIEWG6$;F(Fc\\l;F($\"$U \"!\"#" 1 2 0 1 10 0 2 9 1 4 2 1.000000 45.000000 45.000000 0 0 "Curve 1" }}}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 32 "T he following code constructs a " }{TEXT 260 17 "discrete solution" } {TEXT -1 44 " based on each of the methods and gives the " }{TEXT 260 22 "root mean square error" }{TEXT -1 18 " of each solution." }} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 768 "M := (x,y) -> -(1+cos(2*x)) *y^3: hh := 0.01: numsteps := 300: x0 := 0: y0 := sqrt(2):\nmatrix([[` slope field: `,M(x,y)],[`initial point: `,``(x0,y0)],[`step width: \+ `,hh],\n[`no. of steps: `,numsteps]]);``;\nmthds := [`Butcher's sch eme A`,`scheme with simple nodes`,`scheme with a relatively large stab ility region`,`Butcher's scheme B with `*(c[5]=1/2,c[6]=1/2,b[5]=b[6]) ,`scheme with `*(c[5]=3/4,c[6]=3/4,b[5]=b[6])]:\nerrs := []:\nDigits : = 20:\nfor ct to 5 do\n Mn_RK6_||ct := RK6_||ct(M(x,y),x,y,x0,evalf( y0),hh,numsteps,false);\n sm := 0: numpts := nops(Mn_RK6_||ct):\n \+ for ii to numpts do\n sm := sm+(Mn_RK6_||ct[ii,2]-m(Mn_RK6_||ct[i i,1]))^2;\n end do:\n errs := [op(errs),sqrt(sm/numpts)];\nend do: \nDigits := 10:\nlinalg[transpose]([mthds,evalf(errs)]);" }}{PARA 11 " " 1 "" {XPPMATH 20 "6#K%'matrixG6#7&7$%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$%3Butcher's~scheme~AG$\"+r+KG8!# >7$%9scheme~with~simple~nodesG$\"+cV]>M!#@7$%Pscheme~with~a~relatively ~large~stability~regionG$\"+DA%y'QF07$*&%9Butcher's~scheme~B~with~G\" \"\"6%/&%\"cG6#\"\"&#F8\"\"#/&F<6#\"\"'F?/&%\"bGF=&FGFCF8$\"+@'y(ob!#? 7$*&%-scheme~with~GF86%/F;#\"\"$\"\"%/FBFQFEF8$\"+t;DP8F0Q)pprint416\" " }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 30 "The \+ following code constructs " }{TEXT 260 20 "numerical procedures" } {TEXT -1 56 " for solutions based on each of the Runge-Kutta schemes. " }}{PARA 0 "" 0 "" {TEXT -1 75 "The error in the value obtained by ea ch of the methods at the point where " }{XPPEDIT 18 0 "x = 2.999;" "6 #/%\"xG-%&FloatG6$\"%**H!\"$" }{TEXT -1 16 " is also given." }} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 699 "M := (x,y) -> -(1+cos(2*x)) *y^3: hh := 0.01: numsteps := 300: x0 := 0: y0 := sqrt(2):\nmatrix([[` slope field: `,M(x,y)],[`initial point: `,``(x0,y0)],[`step width: \+ `,hh],\n[`no. of steps: `,numsteps]]);``;\nmthds := [`Butcher's sch eme A`,`scheme with simple nodes`,`scheme with a relatively large stab ility region`,`Butcher's scheme B with `*(c[5]=1/2,c[6]=1/2,b[5]=b[6]) ,`scheme with `*(c[5]=3/4,c[6]=3/4,b[5]=b[6])]:\nerrs := []:\nDigits : = 25:\nfor ct to 5 do\n mn_RK6_||ct := RK6_||ct(M(x,y),x,y,x0,evalf( y0),hh,numsteps,true);\nend do:\nxx := 2.999: mxx := evalf(m(xx)):\nfo r ct to 5 do\n errs := [op(errs),abs(mn_RK6_||ct(xx)-mxx)];\nend do: \nDigits := 10:\nlinalg[transpose]([mthds,evalf(errs)]);" }}{PARA 11 " " 1 "" {XPPMATH 20 "6#K%'matrixG6#7&7$%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)pprint426\"" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%!G" }}{PARA 11 " " 1 "" {XPPMATH 20 "6#K%'matrixG6#7'7$%3Butcher's~scheme~AG$\"+L)y@a#! #?7$%9scheme~with~simple~nodesG$\"+9%RUc'!#A7$%Pscheme~with~a~relative ly~large~stability~regionG$\"+%Qv9T(F07$*&%9Butcher's~scheme~B~with~G \"\"\"6%/&%\"cG6#\"\"&#F8\"\"#/&F<6#\"\"'F?/&%\"bGF=&FGFCF8$\"+kDPp5F+ 7$*&%-scheme~with~GF86%/F;#\"\"$\"\"%/FBFPFEF8$\"+&o))*yDF0Q)pprint436 \"" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 4 "The " }{TEXT 260 22 "root mean square error" }{TEXT -1 20 " over the inte rval " }{XPPEDIT 18 0 "[0, 3];" "6#7$\"\"!\"\"$" }{TEXT -1 82 " of e ach Runge-Kutta method is estimated as follows using the special proce dure " }{TEXT 0 5 "NCint" }{TEXT -1 98 " to perform numerical integr ation by the 7 point Newton-Cotes method over 150 equal subintervals. " }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 436 "mthds := [`Butcher's sc heme A`,`scheme with simple nodes`,`scheme with a relatively large sta bility region`,`Butcher's scheme B with `*(c[5]=1/2,c[6]=1/2,b[5]=b[6] ),`scheme with `*(c[5]=3/4,c[6]=3/4,b[5]=b[6])]:\nerrs := []:\nDigits \+ := 20:\nfor ct to 5 do\n sm := NCint((m(x)-'mn_RK6_||ct'(x))^2,x=0.. 3,adaptive=false,numpoints=7,factor=150);\n errs := [op(errs),sqrt(s m/3)];\nend do:\nDigits := 10:\nlinalg[transpose]([mthds,evalf(errs)]) ;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#K%'matrixG6#7'7$%3Butcher's~schem e~AG$\"+\\7$%9scheme~with~simple~nodesG$\"+q')RuL!#@7$%Pscheme~ with~a~relatively~large~stability~regionG$\"+'Hfn\"QF07$*&%9Butcher's~ scheme~B~with~G\"\"\"6%/&%\"cG6#\"\"&#F8\"\"#/&F<6#\"\"'F?/&%\"bGF=&FG FCF8$\"+?$p^\\&!#?7$*&%-scheme~with~GF86%/F;#\"\"$\"\"%/FBFQFEF8$\"+EI s>8F0Q)pprint446\"" }}}{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 0 516 "evalf[25](plot(['mn_RK6_1'(x)-m(x),'mn_RK6_2'(x)-m(x ),'mn_RK6_3'(x)-m(x),'mn_RK6_4'(x)-m(x),\n'mn_RK6_5'(x)-m(x)],x=0..3,f ont=[HELVETICA,9],\ncolor=[COLOR(RGB,.95,.2,0),COLOR(RGB,.45,0,.95),CO LOR(RGB,0,.25,1),COLOR(RGB,0,.75,.2),COLOR(RGB,.95,.45,0)],\nlegend=[` Butcher's scheme A`,`scheme with simple nodes`,`scheme with a relative ly large stability region`,`Butcher's scheme B with c[5]=c[6]=1/2 and \+ b[5]=b[6]`,`scheme with c[5]=c[6]=3/4 and b[5]=b[6]`],title=`error cur ves for 7 stage order 6 Runge-Kutta methods`));" }}{PARA 13 "" 1 "" {GLPLOT2D 904 583 583 {PLOTDATA 2 "6+-%'CURVESG6%7\\p7$$\"\"!F)F(7$$\" :++++++DJ?$[V?!#F$!+C#zv<&!#C7$$\":++++++]iSmp3%F-$!--Ie8NlF07$$\":+++ +++v$4'\\/8'F-$!/))*\\KH25\"F07$$\":+++++++D\"G$R<)F-$!/\\)p(olE\")F07 $$\":+++++]i:gT<-\"!#E$!0Z(*39L:G$F07$$\":++++++v=#**3E7FC$!0&\\e8Mf3K F07$$\":+++++](=U#Q/V\"FC$!0Q**)=a\\VJF07$$\":+++++++Dc'yM;FC$!0KR4\"G g\\JF07$$\":+++++]7G)[8R=FC$!0W]I!o5cNF07$$F,FC$!08<:hDVz%F07$$\":++++ +]PM_JyC#FC$!0(zSoS8&p%F07$$\":++++++]P%)z@X#FC$!0#)GIFwNg%F07$$\":+++ +++vV[w3'GFC$!0I@\"3]5/[F07$$\":+++++++]7t&pKFC$!0q^la&y8`F07$$\":++++ +]7`W@RZ$FC$!0IW%**\\(e@&F07$$\":++++++Dcwp#yOFC$!0OL,!pue^F07$$\":+++ ++]Pf3=E)QFC$!0w.y@m))H&F07$$F3FC$!0E%))e:4)e&F07$$\":+++++]ils98H%FC$ !0r85PKm[&F07$$\":++++++vo/jc\\%FC$!0Ux+?P7R&F07$$\":+++++](=n8,+ZFC$! 0b9Xz#eF`F07$$\":+++++++vofV!\\FC$!0JxwLR!3aF07$$\":++++++](oHv@dFC$!0 1?$[Oyw_F07$$\":++++++++DY\"RlFC$!0(R=c/_w^F07$$\":++++++voHl:'zFC$!0R I;sVN!\\F07$$\":++++++]PM%)RQ*FC$!0GTIP&zUXF07$$\":+++++]i!R.k!3\"!#D$ !0r#\\J?W+UF07$$\":++++++]PC#)GA\"Fjs$!0::'yzh0RF07$$\":+++++](=U\"[GQ \"Fjs$!1&pUx+YXe$Fjs7$$\":++++++D1/9Ga\"Fjs$!1DxP9ZQ3LFjs7$$\":+++++]i !R*zFq\"Fjs$!1;k2**4yjIFjs7$$\":+++++++v$eui=Fjs$!1X!>GLZ:%GFjs7$$\":+ +++++++N)z%=#Fjs$!1R**fz#yBZ#Fjs7$$\":+++++++D'3&o]#Fjs$!1ia')=hay@Fjs 7$$\":++++++Dcrst#GFjs$!1YdeABbT>Fjs7$$\":++++++](oX*y9$Fjs$!1SIS#pOvu \"Fjs7$$\":++++++]P9CAu$Fjs$!1@#3Su8-Z\"Fjs7$$\":++++++]P*zhdVFjs$!1q6 6D0Fh7Fjs7$$\":++++++]P>fS*\\Fjs$!1?k\"Qw,55\"Fjs7$$\":++++++](=$f%GcF js$!0&[tuLi,)*Fjs7$$\":+++++++]#y,\"G'Fjs$!0Van!fjgnFjs7$$ \":++++++]PpnsM*Fjs$!0$pIY#\\&)\\'Fjs7$$\":+++++++DFOB+\"F0$!0(f$o@m?D 'Fjs7$$\":++++++++R5'f5F0$!034h]]#*3'Fjs7$$\":++++++vV!QBE6F0$!0$f@$>/ Q%fFjs7$$\":+++++++]\"o?&=\"F0$!02z=)f2[eFjs7$$\":++++++vVb4*\\7F0$!0) Q)\\jnDx&Fjs7$$\":++++++DJ'=_68F0$!0f=&>JBCdFjs7$$\":++++++]P%y!eP\"F0 $!0SSOPXKp&Fjs7$$\":++++++v=WU[V\"F0$!0DL@Rh!ycFjs7$$\":++++++]7B>&)\\ \"F0$!04F%z/VrcFjs7$$\":++++++v$>:mk:F0$!0%4\\rjDqcFjs7$$\":++++++DcdQ Ai\"F0$!0x)=.?$)pcFjs7$$\":++++++]PPBWo\"F0$!0'pb))>qlcFjs7$$\":++++++ +]Nm'[F0$!0\"GQh*y7_&Fjs7$$\":++ +++++]@80+#F0$!06e#4w=SaFjs7$$\":+++++++D6!Hl?F0$!0%)[kt3(H`Fjs7$$\":+ +++++v$4w)R7#F0$!0-b&y9C3_Fjs7$$\":+++++++vZf\")=#F0$!0!\\w'4AL0&Fjs7$ $\":++++++vV?S&[AF0$!0u(=jM[))[Fjs7$$\":++++++v=Yb;J#F0$!0uu.QP%*p%Fjs 7$$\":+++++++D;iLP#F0$!0SRN\"4j,XFjs7$$\":++++++v$fL'zV#F0$!0'48S[-&G% Fjs7$$\":++++++++*>=+DF0$!0)[&>OM82%Fjs7$$\":++++++]i_4Qc#F0$!01O-M%p^ QFjs7$$\":++++++vV>5pi#F0$!0,N3Eijj$Fjs7$$\":+++++++]:$*[o#F0$!0T3Uo>L W$Fjs7$$\":++++++]7<[8v#F0$!0WuwI*HIKFjs7$$\":++++++++L'y5GF0$!0dpbYL* [IFjs7$$\":++++++vV!)fT(GF0$!0L$yL4QmGFjs7$$\":++++++DcI;[$HF0$!0<0A( \\'Gq#Fjs7$$\"\"$F)$!0#[$eRw(RDFjs-%&COLORG6&%$RGBG$\"#&*!\"#$\"\"#!\" \"F(-%'LEGENDG6#%3Butcher's~scheme~AG-F$6%7\\pF'7$F+$\"*4w>i\"F07$F2$ \",YL9`$>F07$F7$\"-wdM'Q3$F07$F<$\".T_@(eb@F07$FA$\".,J+TQH)F07$FG$\". `M!e_4\")F07$FL$\".n04Ks%zF07$FQ$\".,+7LE)zF07$FV$\".&zt0pl!*F07$Fen$ \"/gV`Bo=7F07$Fin$\"/nU`AZ$>\"F07$F^o$\"/qqCTPq6F07$Fco$\"//oXT`E7F07$ Fho$\"/Ik$ynkN\"F07$F]p$\"/'3]#RhJ8F07$Fbp$\"/BX9)\\!=8F07$Fgp$\"/VW.- Yc8F07$F\\q$\"/OlVD)4V\"F07$F`q$\"/'ff/0]S\"F07$Feq$\"/\\l9-p!Q\"F07$F jq$\"/[$fhW^O\"F07$F_r$\"/N=K2j(Q\"F07$Fdr$\"/'>'*[#4b8F07$Fir$\"/^QbT 3J8F07$F^s$\"/81p\"QNE\"F07$Fcs$\"/$3$ey_r6F07$Fhs$\"/4)z*f(R3\"F07$F^ t$\"/C))H,T35F07$Fct$\"/6D$zIwD*Fjs7$Fht$\"/;0cW!fa)Fjs7$F]u$\"/V%ob6 \\\"zFjs7$Fbu$\"/&)GH8@Fjs7$Fix$\"/6:5N(*f>Fjs7$F^y$\"/-D0B^Q=Fjs7$Fcy$\"/\"f )R,oXy=Bd\"Fj s7$Fgz$\"/*3)HOwM:Fjs7$F\\[l$\"/WYOk/5:Fjs7$Fa[l$\"/[%y[]0\\\"Fjs7$Ff[ l$\"/z)R'42y9Fjs7$F[\\l$\"/EOs/2q9Fjs7$F`\\l$\"/=R=0:m9Fjs7$Fe\\l$\"/v m9)QWY\"Fjs7$Fj\\l$\"/7M&pNTY\"Fjs7$F_]l$\"/B^^c-k9Fjs7$Fd]l$\"/BW/#eH Y\"Fjs7$Fi]l$\"/!H.LZ'f9Fjs7$F^^l$\"/gG)f7JX\"Fjs7$Fc^l$\"/S*p2WFW\"Fj s7$Fh^l$\"/=4QrlD9Fjs7$F]_l$\"/(*[Ekr/9Fjs7$Fb_l$\"/my)Q)=w8Fjs7$Fg_l$ \"/u_1T#[M\"Fjs7$F\\`l$\"/T3XL#[I\"Fjs7$Fa`l$\"/N*QOiAE\"Fjs7$Ff`l$\"/ \\*\\x^M@\"Fjs7$F[al$\"/2Re!zB;\"Fjs7$F`al$\"/>(o)=X16Fjs7$Feal$\"/;z< rF^5Fjs7$Fjal$\".\"*=#ykX**Fjs7$F_bl$\".#*fsQ'*Q*Fjs7$Fdbl$\".P2pv6*)) Fjs7$Fibl$\".9Iy=6M)Fjs7$F^cl$\".4dh%zsyFjs7$Fccl$\".%\\RSS,uFjs7$Fhcl $\".1wDo\"zpFjs7$F]dl$\".QsyM!elFjs-Fbdl6&Fddl$\"#XFgdlF(Fedl-F\\el6#% 9scheme~with~simple~nodesG-F$6%7\\pF'7$F+$\"*5[&f=F07$F2$\",q5NP@#F07$ F7$\"-mmw>=NF07$F<$\"..j6A*F07$FL $\".I@\">Z6!*F07$FQ$\".lC+0F0*F07$FV$\"/qqV(*GG5F07$Fen$\"/&Hhn*f\"Q\" F07$Fin$\"/$Q[f>IN\"F07$F^o$\"/I$*e?%oK\"F07$Fco$\"/j$pQ&f!R\"F07$Fho$ \"/!oj)zUP:F07$F]p$\"/bY8NE4:F07$Fbp$\"/x[x>\"R\\\"F07$Fgp$\"/5:dOOP:F 07$F\\q$\"/XT-FW@;F07$F`q$\"/Na\"y2?f\"F07$Feq$\"/*Q;-fWc\"F07$Fjq$\"/ [LMG%oa\"F07$F_r$\"/R!G#)p@d\"F07$Fdr$\"/p%o&e+N:F07$Fir$\"/&GVb7u]\"F 07$F^s$\"/_s9wGI9F07$Fcs$\"/cLf_uD8F07$Fhs$\"/T!ogjjA\"F07$F^t$\"/Bq+ \"*\\S6F07$Fct$\"0hNTIAo/\"Fjs7$Fht$\"/t/2z9h'*Fjs7$F]u$\"/)Qe(4#e%*)F js7$Fbu$\"/3[b6K'H)Fjs7$Fgu$\"/\\DnRi;sFjs7$F\\v$\"/\"RH8.sN'Fjs7$Fav$ \"/WyisdkcFjs7$Ffv$\"/ep7.u(4&Fjs7$F[w$\"/4J)R#z(G%Fjs7$F`w$\"/YeR[)yn $Fjs7$Few$\"/UKdj@5KFjs7$Fjw$\"/Yg&38x&GFjs7$F_x$\"/(=jun\"zDFjs7$Fdx$ \"/!o@\"*4gQ#Fjs7$Fix$\"/d(Qs!*G@#Fjs7$F^y$\"/&pt;\\d2#Fjs7$Fcy$\"/fk! eM4(>Fjs7$Fhy$\"/S'=(\\_%*=Fjs7$F]z$\"/+NAomA=Fjs7$Fbz$\"/G\\$z,_x\"Fj s7$Fgz$\"/VX=7!Gt\"Fjs7$F\\[l$\"/Uoy^*[q\"Fjs7$Fa[l$\"/RoFT)Go\"Fjs7$F f[l$\"/lc$o%zo;Fjs7$F[\\l$\"/]./@wf;Fjs7$F`\\l$\"/\"3'*[O`l\"Fjs7$Fe\\ l$\"/kS#*RS`;Fjs7$Fj\\l$\"/fP#zhIl\"Fjs7$F_]l$\"/\"z\"yv$Hl\"Fjs7$Fd]l $\"/xJiCt^;Fjs7$Fi]l$\"/p.IX*zk\"Fjs7$F^^l$\"/oS,phS;Fjs7$Fc^l$\"/0sP4 \"*G;Fjs7$Fh^l$\"/Rn&)*>'4;Fjs7$F]_l$\"/4y'=yfe\"Fjs7$Fb_l$\"/$**)*HqP b\"Fjs7$Fg_l$\"/h()y/O=:Fjs7$F\\`l$\"/Xsu+?t9Fjs7$Fa`l$\"/\"*49#\\^U\" Fjs7$Ff`l$\"/#*p*eU+P\"Fjs7$F[al$\"/H*Rg\"Q78Fjs7$F`al$\"/*yljR#\\7Fjs 7$Feal$\"/&R`;Zp=\"Fjs7$Fjal$\"/$)y6\"=H7\"Fjs7$F_bl$\"/`NfO9g5Fjs7$Fd bl$\"/+^kh'Q+\"Fjs7$Fibl$\".\\xQMwT*Fjs7$F^cl$\".k@v#))))))Fjs7$Fccl$ \".)RVkmc$)Fjs7$Fhcl$\".mHnY*zyFjs7$F]dl$\".7>yqWS(Fjs-Fbdl6&FddlF($\" #DFgdl$\"\"\"F)-F\\el6#%Pscheme~with~a~relatively~large~stability~regi onG-F$6%7\\pF'7$F+$\"+E%esT#F07$F2$\"-a27GaHF07$F7$\".d$oR*=#[F07$F<$ \"/Y!zXxBX$F07$FA$\"0FvXH$>d8F07$FG$\"0a6)4'HqK\"F07$FL$\"0r:?#zI+8F07 $FQ$\"0C>ajmYI\"F07$FV$\"0_DSD?\"y9F07$Fen$\"0C'47'*R!*>F07$Fin$\"0l=0 \"GA\\>F07$F^o$\"0,+\\.s8\">F07$Fco$\"0g'[r?r**>F07$Fho$\"0pO(3*\\C@#F 07$F]p$\"0]Qn$R\"=<#F07$Fbp$\"0n&zV_+\\@F07$Fgp$\"0)*H,`7,@#F07$F\\q$ \"0??m`6=L#F07$F`q$\"0L:(3!y%*G#F07$Feq$\"0H#3f*y(\\AF07$Fjq$\"0y5C?iR A#F07$F_r$\"0Jad&=ifAF07$Fdr$\"037N6Kj?#F07$Fir$\"0Ow.5sm;#F07$F^s$\"0 Q5![]#e0#F07$Fcs$\"0gtfX()f!>F07$Fhs$\"0Rcj\\EMw\"F07$F^t$\"0\"f]`%)eS ;F07$Fct$\"1c)QE,#>1:Fjs7$Fht$\"1K-4'e\\0R\"Fjs7$F]u$\"1UQT@u.)G\"Fjs7 $Fbu$\"1oUNrXr%>\"Fjs7$Fgu$\"16a>?1pR5Fjs7$F\\v$\"0^V-*3Ei\"*Fjs7$Fav$ \"0A]$f?&f;)Fjs7$Ffv$\"0,G8&)z,N(Fjs7$F[w$\"0RE_g\")R='Fjs7$F`w$\"0Agd xS_I&Fjs7$Few$\"0FtJ**z6j%Fjs7$Fjw$\"0o@`]NH7%Fjs7$F_x$\"0Jvu,L7s$Fjs7 $Fdx$\"0t#\\V&QEW$Fjs7$Fix$\"0.(=_C#H>$Fjs7$F^y$\"0;)ol&y]*HFjs7$Fcy$ \"0#HxlL'Q%GFjs7$Fhy$\"0d8J9BOt#Fjs7$F]z$\"0#*3!4a%*HEFjs7$Fbz$\"0i:G& *f9c#Fjs7$Fgz$\"0@]d>!G+DFjs7$F\\[l$\"0O/!GO,gCFjs7$Fa[l$\"0$z]sADGCFj s7$Ff[l$\"0=;%*4@zS#Fjs7$F[\\l$\"0FRmi')[R#Fjs7$F`\\l$\"0'[&G%)*\\)Q#F js7$Fe\\l$\"0n'oe2r&Q#Fjs7$Fj\\l$\"08]r\"p@&Q#Fjs7$F_]l$\"0>fxIQ]Q#Fjs 7$Fd]l$\"0C#Fjs7$Fg_l$\"0l@3Tl3>#Fjs7$F\\`l$\"0\"\\)[;(pD@Fjs7$Fa`l$\"0iX uEcj0#Fjs7$Ff`l$\"0IVGIKo(>Fjs7$F[al$\"0e^gWBO*=Fjs7$F`al$\"0zqch1D!=F js7$Feal$\"0i5PS;Er\"Fjs7$Fjal$\"0\")zX$QA?;Fjs7$F_bl$\"0'R()4OkH:Fjs7 $Fdbl$\"0I>'p'R%[9Fjs7$Fibl$\"0FC)4@$)e8Fjs7$F^cl$\"0Cp6:SDG\"Fjs7$Fcc l$\"0\"*)*)3!\\d?\"Fjs7$Fhcl$\"0B'e=c'p8\"Fjs7$F]dl$\"0ioV'>Oo5Fjs-Fbd l6&FddlF($\"#vFgdlFhdl-F\\el6#%TButcher's~scheme~B~with~c[5]=c[6]=1/2~ and~b[5]=b[6]G-F$6%7\\pF'7$F+$\")B^coF07$F2$\"+>$3k+)F07$F7$\"-t0n#pC \"F07$F<$\"-ed'4l])F07$FA$\".>f+-%)>$F07$FG$\".o$Q-LFJF07$FL$\".x(z8=l IF07$FQ$\".du2yF3$F07$FV$\".S@!\\*=^$F07$Fen$\".8>fiir%F07$Fin$\".qiZ2 (=YF07$F^o$\".J;El'HXF07$Fco$\".(4i#z!eZF07$Fho$\".Tz\"R(QE&F07$F]p$\" .6&Q*4x;&F07$Fbp$\".2e:3s6&F07$Fgp$\".&>J)fCF&F07$F\\q$\".YnMV\\c&F07$ F`q$\".i5VFRY&F07$Feq$\".Q6^1'p`F07$Fjq$\".Sy$)33J&F07$F_r$\".l&ou4.aF 07$Fdr$\".LxK$H!G&F07$Fir$\"._6nOC>&F07$F^s$\".)H'Hxx$\\F07$Fcs$\".Eo- !>#e%F07$Fhs$\".L.@rGC%F07$F^t$\".!f,v5]RF07$Fct$\"/w\"\\9jzi$Fjs7$Fht $\"/&H)zDc]LFjs7$F]u$\"/yT:HT/JFjs7$Fbu$\"/Pb:yyzGFjs7$Fgu$\"/qIbXj1DF js7$F\\v$\"/a#)=`@4AFjs7$Fav$\"/m\"3Rz!p>Fjs7$Ffv$\"/UIw&>Cx\"Fjs7$F[w $\"/k8TVC\"\\\"Fjs7$F`w$\"/O]dlNz7Fjs7$Few$\"/)3Lj=o6\"Fjs7$Fjw$\".f1) =jU**Fjs7$F_x$\".J4a&)R(*)Fjs7$Fdx$\".UP'>?-$)Fjs7$Fix$\".3*e81+xFjs7$ F^y$\".K(pV*HA(Fjs7$Fcy$\"./p8q$eoFjs7$Fhy$\".#GbAb#f'Fjs7$F]z$\".0MFh DM'Fjs7$Fbz$\".7iDGu<'Fjs7$Fgz$\".]dU6*HgFjs7$F\\[l$\".UNj$Gs!eFjs7$F[\\l$\".Y#>I!ex&Fjs7$F`\\l$\".f 131/w&Fjs7$Fe\\l$\".jrM#o`dFjs7$Fj\\l$\".,Yr!\\_dFjs7$F_]l$\".7p-d?v&F js7$Fd]l$\".#z1B'yu&Fjs7$Fi]l$\".#H-]&[t&Fjs7$F^^l$\".NE5%=4dFjs7$Fc^l $\".R-c_%ocFjs7$Fh^l$\".(>EvK,cFjs7$F]_l$\".GnWg!>bFjs7$Fb_l$\".X0`\") pS&Fjs7$Fg_l$\".m%[iv$G&Fjs7$F\\`l$\".5c\"[fE^Fjs7$Fa`l$\".!zS/Pf\\Fjs 7$Ff`l$\".S>c(enZFjs7$F[al$\".jQr:pc%Fjs7$F`al$\".'y_+ " 0 "" {MPLTEXT 1 0 362 "evalf[25](plot(['mn_RK6_2'(x)-m(x),'mn_R K6_3'(x)-m(x),'mn_RK6_5'(x)-m(x)],x=0..3,font=[HELVETICA,9],\ncolor=[C OLOR(RGB,.45,0,.95),COLOR(RGB,0,.25,1),COLOR(RGB,.95,.45,0)],\nlegend= [`scheme with simple nodes`,`scheme with a relatively large stability \+ region`,`scheme with c[5]=c[6]=3/4 and b[5]=b[6]`],title=`error curves for 7 stage order 6 Runge-Kutta methods`));" }}{PARA 13 "" 1 "" {GLPLOT2D 953 502 502 {PLOTDATA 2 "6)-%'CURVESG6%7\\p7$$\"\"!F)F(7$$\" :++++++DJ?$[V?!#F$\"*4w>i\"!#C7$$\":++++++]iSmp3%F-$\",YL9`$>F07$$\":+ +++++v$4'\\/8'F-$\"-wdM'Q3$F07$$\":+++++++D\"G$R<)F-$\".T_@(eb@F07$$\" :+++++]i:gT<-\"!#E$\".,J+TQH)F07$$\":++++++v=#**3E7FC$\".`M!e_4\")F07$ $\":+++++](=U#Q/V\"FC$\".n04Ks%zF07$$\":+++++++Dc'yM;FC$\".,+7LE)zF07$ $\":+++++]7G)[8R=FC$\".&zt0pl!*F07$$F,FC$\"/gV`Bo=7F07$$\":+++++]PM_Jy C#FC$\"/nU`AZ$>\"F07$$\":++++++]P%)z@X#FC$\"/qqCTPq6F07$$\":++++++vV[w 3'GFC$\"//oXT`E7F07$$\":+++++++]7t&pKFC$\"/Ik$ynkN\"F07$$\":+++++]7`W@ RZ$FC$\"/'3]#RhJ8F07$$\":++++++Dcwp#yOFC$\"/BX9)\\!=8F07$$\":+++++]Pf3 =E)QFC$\"/VW.-Yc8F07$$F3FC$\"/OlVD)4V\"F07$$\":+++++]ils98H%FC$\"/'ff/ 0]S\"F07$$\":++++++vo/jc\\%FC$\"/\\l9-p!Q\"F07$$\":+++++](=n8,+ZFC$\"/ [$fhW^O\"F07$$\":+++++++vofV!\\FC$\"/N=K2j(Q\"F07$$\":++++++](oHv@dFC$ \"/'>'*[#4b8F07$$\":++++++++DY\"RlFC$\"/^QbT3J8F07$$\":++++++voHl:'zFC $\"/81p\"QNE\"F07$$\":++++++]PM%)RQ*FC$\"/$3$ey_r6F07$$\":+++++]i!R.k! 3\"!#D$\"/4)z*f(R3\"F07$$\":++++++]PC#)GA\"Fjs$\"/C))H,T35F07$$\":++++ +](=U\"[GQ\"Fjs$\"/6D$zIwD*Fjs7$$\":++++++D1/9Ga\"Fjs$\"/;0cW!fa)Fjs7$ $\":+++++]i!R*zFq\"Fjs$\"/V%ob6\\\"zFjs7$$\":+++++++v$eui=Fjs$\"/&)GfS*\\Fjs$\"/b/+(QJ%GFjs7$$\":++ ++++](=$f%GcFjs$\"/Qz=N,JDFjs7$$\":+++++++]#y,\"G'Fjs$\"/?)f3]VG#Fjs7$ $\":+++++++Dr\"zboFjs$\"/v7<>H8@Fjs7$$\":+++++++](4&G](Fjs$\"/6:5N(*f> Fjs7$$\":+++++++]7nD:)Fjs$\"/-D0B^Q=Fjs7$$\":+++++++]-*oy()Fjs$\"/\"f) R,oXy=Bd\"Fjs7$$\":++++++vV!QBE6F 0$\"/*3)HOwM:Fjs7$$\":+++++++]\"o?&=\"F0$\"/WYOk/5:Fjs7$$\":++++++vVb4 *\\7F0$\"/[%y[]0\\\"Fjs7$$\":++++++DJ'=_68F0$\"/z)R'42y9Fjs7$$\":+++++ +]P%y!eP\"F0$\"/EOs/2q9Fjs7$$\":++++++v=WU[V\"F0$\"/=R=0:m9Fjs7$$\":++ ++++]7B>&)\\\"F0$\"/vm9)QWY\"Fjs7$$\":++++++v$>:mk:F0$\"/7M&pNTY\"Fjs7 $$\":++++++DcdQAi\"F0$\"/B^^c-k9Fjs7$$\":++++++]PPBWo\"F0$\"/BW/#eHY\" Fjs7$$\":+++++++]Nm'[F0$\"/=4 QrlD9Fjs7$$\":+++++++]@80+#F0$\"/(*[Ekr/9Fjs7$$\":+++++++D6!Hl?F0$\"/m y)Q)=w8Fjs7$$\":++++++v$4w)R7#F0$\"/u_1T#[M\"Fjs7$$\":+++++++vZf\")=#F 0$\"/T3XL#[I\"Fjs7$$\":++++++vV?S&[AF0$\"/N*QOiAE\"Fjs7$$\":++++++v=Yb ;J#F0$\"/\\*\\x^M@\"Fjs7$$\":+++++++D;iLP#F0$\"/2Re!zB;\"Fjs7$$\":++++ ++v$fL'zV#F0$\"/>(o)=X16Fjs7$$\":++++++++*>=+DF0$\"/;z5pi#F0$\".#*fsQ'*Q*Fjs7$$ \":+++++++]:$*[o#F0$\".P2pv6*))Fjs7$$\":++++++]7<[8v#F0$\".9Iy=6M)Fjs7 $$\":++++++++L'y5GF0$\".4dh%zsyFjs7$$\":++++++vV!)fT(GF0$\".%\\RSS,uFj s7$$\":++++++DcI;[$HF0$\".1wDo\"zpFjs7$$\"\"$F)$\".QsyM!elFjs-%&COLORG 6&%$RGBG$\"#X!\"#F($\"#&*Fgdl-%'LEGENDG6#%9scheme~with~simple~nodesG-F $6%7\\pF'7$F+$\"*5[&f=F07$F2$\",q5NP@#F07$F7$\"-mmw>=NF07$F<$\"..j6A*F07$FL$\".I@\">Z6!*F07$FQ$\".lC+0F 0*F07$FV$\"/qqV(*GG5F07$Fen$\"/&Hhn*f\"Q\"F07$Fin$\"/$Q[f>IN\"F07$F^o$ \"/I$*e?%oK\"F07$Fco$\"/j$pQ&f!R\"F07$Fho$\"/!oj)zUP:F07$F]p$\"/bY8NE4 :F07$Fbp$\"/x[x>\"R\\\"F07$Fgp$\"/5:dOOP:F07$F\\q$\"/XT-FW@;F07$F`q$\" /Na\"y2?f\"F07$Feq$\"/*Q;-fWc\"F07$Fjq$\"/[LMG%oa\"F07$F_r$\"/R!G#)p@d \"F07$Fdr$\"/p%o&e+N:F07$Fir$\"/&GVb7u]\"F07$F^s$\"/_s9wGI9F07$Fcs$\"/ cLf_uD8F07$Fhs$\"/T!ogjjA\"F07$F^t$\"/Bq+\"*\\S6F07$Fct$\"0hNTIAo/\"Fj s7$Fht$\"/t/2z9h'*Fjs7$F]u$\"/)Qe(4#e%*)Fjs7$Fbu$\"/3[b6K'H)Fjs7$Fgu$ \"/\\DnRi;sFjs7$F\\v$\"/\"RH8.sN'Fjs7$Fav$\"/WyisdkcFjs7$Ffv$\"/ep7.u( 4&Fjs7$F[w$\"/4J)R#z(G%Fjs7$F`w$\"/YeR[)yn$Fjs7$Few$\"/UKdj@5KFjs7$Fjw $\"/Yg&38x&GFjs7$F_x$\"/(=jun\"zDFjs7$Fdx$\"/!o@\"*4gQ#Fjs7$Fix$\"/d(Q s!*G@#Fjs7$F^y$\"/&pt;\\d2#Fjs7$Fcy$\"/fk!eM4(>Fjs7$Fhy$\"/S'=(\\_%*=F js7$F]z$\"/+NAomA=Fjs7$Fbz$\"/G\\$z,_x\"Fjs7$Fgz$\"/VX=7!Gt\"Fjs7$F\\[ l$\"/Uoy^*[q\"Fjs7$Fa[l$\"/RoFT)Go\"Fjs7$Ff[l$\"/lc$o%zo;Fjs7$F[\\l$\" /]./@wf;Fjs7$F`\\l$\"/\"3'*[O`l\"Fjs7$Fe\\l$\"/kS#*RS`;Fjs7$Fj\\l$\"/f P#zhIl\"Fjs7$F_]l$\"/\"z\"yv$Hl\"Fjs7$Fd]l$\"/xJiCt^;Fjs7$Fi]l$\"/p.IX *zk\"Fjs7$F^^l$\"/oS,phS;Fjs7$Fc^l$\"/0sP4\"*G;Fjs7$Fh^l$\"/Rn&)*>'4;F js7$F]_l$\"/4y'=yfe\"Fjs7$Fb_l$\"/$**)*HqPb\"Fjs7$Fg_l$\"/h()y/O=:Fjs7 $F\\`l$\"/Xsu+?t9Fjs7$Fa`l$\"/\"*49#\\^U\"Fjs7$Ff`l$\"/#*p*eU+P\"Fjs7$ F[al$\"/H*Rg\"Q78Fjs7$F`al$\"/*yljR#\\7Fjs7$Feal$\"/&R`;Zp=\"Fjs7$Fjal $\"/$)y6\"=H7\"Fjs7$F_bl$\"/`NfO9g5Fjs7$Fdbl$\"/+^kh'Q+\"Fjs7$Fibl$\". \\xQMwT*Fjs7$F^cl$\".k@v#))))))Fjs7$Fccl$\".)RVkmc$)Fjs7$Fhcl$\".mHnY* zyFjs7$F]dl$\".7>yqWS(Fjs-Fbdl6&FddlF($\"#DFgdl$\"\"\"F)-F[el6#%Pschem e~with~a~relatively~large~stability~regionG-F$6%7\\pF'7$F+$\")B^coF07$ F2$\"+>$3k+)F07$F7$\"-t0n#pC\"F07$F<$\"-ed'4l])F07$FA$\".>f+-%)>$F07$F G$\".o$Q-LFJF07$FL$\".x(z8=lIF07$FQ$\".du2yF3$F07$FV$\".S@!\\*=^$F07$F en$\".8>fiir%F07$Fin$\".qiZ2(=YF07$F^o$\".J;El'HXF07$Fco$\".(4i#z!eZF0 7$Fho$\".Tz\"R(QE&F07$F]p$\".6&Q*4x;&F07$Fbp$\".2e:3s6&F07$Fgp$\".&>J) fCF&F07$F\\q$\".YnMV\\c&F07$F`q$\".i5VFRY&F07$Feq$\".Q6^1'p`F07$Fjq$\" .Sy$)33J&F07$F_r$\".l&ou4.aF07$Fdr$\".LxK$H!G&F07$Fir$\"._6nOC>&F07$F^ s$\".)H'Hxx$\\F07$Fcs$\".Eo-!>#e%F07$Fhs$\".L.@rGC%F07$F^t$\".!f,v5]RF 07$Fct$\"/w\"\\9jzi$Fjs7$Fht$\"/&H)zDc]LFjs7$F]u$\"/yT:HT/JFjs7$Fbu$\" /Pb:yyzGFjs7$Fgu$\"/qIbXj1DFjs7$F\\v$\"/a#)=`@4AFjs7$Fav$\"/m\"3Rz!p>F js7$Ffv$\"/UIw&>Cx\"Fjs7$F[w$\"/k8TVC\"\\\"Fjs7$F`w$\"/O]dlNz7Fjs7$Few $\"/)3Lj=o6\"Fjs7$Fjw$\".f1)=jU**Fjs7$F_x$\".J4a&)R(*)Fjs7$Fdx$\".UP'> ?-$)Fjs7$Fix$\".3*e81+xFjs7$F^y$\".K(pV*HA(Fjs7$Fcy$\"./p8q$eoFjs7$Fhy $\".#GbAb#f'Fjs7$F]z$\".0MFhDM'Fjs7$Fbz$\".7iDGu<'Fjs7$Fgz$\".]dU6*HgF js7$F\\[l$\".UNj$Gs!eFjs7$F[\\ l$\".Y#>I!ex&Fjs7$F`\\l$\".f131/w&Fjs7$Fe\\l$\".jrM#o`dFjs7$Fj\\l$\"., Yr!\\_dFjs7$F_]l$\".7p-d?v&Fjs7$Fd]l$\".#z1B'yu&Fjs7$Fi]l$\".#H-]&[t&F js7$F^^l$\".NE5%=4dFjs7$Fc^l$\".R-c_%ocFjs7$Fh^l$\".(>EvK,cFjs7$F]_l$ \".GnWg!>bFjs7$Fb_l$\".X0`\")pS&Fjs7$Fg_l$\".m%[iv$G&Fjs7$F\\`l$\".5c \"[fE^Fjs7$Fa`l$\".!zS/Pf\\Fjs7$Ff`l$\".S>c(enZFjs7$F[al$\".jQr:pc%Fjs 7$F`al$\".'y_+ " 0 "" {MPLTEXT 1 0 1 ";" }}}}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 47 "Test 10 of 7 stage, order 6 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#\"\"!*&-%%sqrtG6#\"\"&\"\"\"\"\"#!\"\"" }{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))*sinh(y(x));\nic := y(0)=sqrt(5)/2;\nd solve(\{de,ic\},y(x));\nsimplify(convert(%,exp));\np := unapply(rhs(%) ,x):\nplot(p(x),x=0..5,font=[HELVETICA,9],labels=[`x`,`y(x)`]);" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#>%#deG/-%%diffG6$-%\"yG6#%\"xGF,,$*&,& *&\"\"#\"\"\"-%$sinG6#,$*&\"\"&F2F,F2F2F2F2*&\"\"$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#,**&#\"\"\"\"\"&F1-%$co sG6#,$*&F2F1F'F1F1F1!\"\"*&#\"\"$\"#9F1-%$sinG6#,$*&\"\"(F1F'F1F1F1F1# F1F2F1*&#F1\"\"#F1-F)6#,$*&,&-%$expG6#,$*&FFF8F2FEF1F1F1F1F1,&FLF1F1F8 F8F8F1F1" }}{PARA 12 "" 1 "" {XPPMATH 20 "6#/-%\"yG6#%\"xG-%#lnG6#,$*& ,*-%$expG6#,4#\"\"#\"\"&!\"\"*&#\"\"$\"\"(\"\"\"-%$sinGF&F:F:*&#\"$#>F 9F:*&)-%$cosGF&\"\"'F:F;F:F:F5*&#\"$S#F9F:*&)FB\"\"%F:F;F:F:F:*&#\"#sF 9F:*&)FBF3F:F;F:F:F5*&#\"#KF4F:*$)FBF4F:F:F:*&\"\")F:)FBF8F:F5*&F3F:FB F:F:*&F3F5F4#F:F3F:F:-F/6#,2#F3F4F5*&F7F:F;F:F:*&#F?F9F:F@F:F5*&FFF:FH F:F:*&#FMF9F:FNF:F5*&FQF:FSF:F:*&FVF:FWF:F5*&F3F:FBF:F:F5-F/6#,$*&F3F5 F4FZF:F:F:F:F:,*F.F:FenF5FboF5F:F5F5F5" }}{PARA 13 "" 1 "" {GLPLOT2D 552 388 388 {PLOTDATA 2 "6&-%'CURVESG6$7av7$$\"\"!F)$\"3!\\*)\\())R.=6 !#<7$$\"3ALL$3FWYs#!#>$\"33uw,WSt45F,7$$\"3WmmmT&)G\\aF0$\"3yUB%H69F5* !#=7$$\"3m****\\7G$R<)F0$\"3[G6@7@G;#)F87$$\"3GLLL3x&)*3\"F8$\"3u_\"Hl v:eW(F87$$\"3))**\\i!R(*Rc\"F8$\"3aT]N\"zi(yjF87$$\"3umm\"H2P\"Q?F8$\" 3@V:-NK+KcF87$$\"3YLek.pu/BF8$\"3:Vt%)yf.P`F87$$\"3!***\\PMnNrDF8$\"3 \"zkU]7kD7&F87$$\"37$eR(\\;m/FF8$\"3#f>&4'\\tL/&F87$$\"3MmT5ll'z$GF8$ \"3OEk>%*z$=)\\F87$$\"37](o/[r7(HF8$\"3oQ2*>TRs$\\F87$$\"3MLL$eRwX5$F8 $\"359>\\xg!*3\\F87$$\"3:L$3F%\\wQKF8$\"3wA2?_M<'*[F87$$\"3_LLe*[`HP$F 8$\"3-d)o(RAn)*[F87$$\"3*QLek.Ur]$F8$\"3K()))eR[\"e\"\\F87$$\"3rLLL$eI 8k$F8$\"3kdF/H[-Z\\F87$$\"3*QL$3xwq4RF8$\"3!**RBYN;$\\]F87$$\"33ML$3x% 3yTF8$\"3r5fX,$G1?&F87$$\"3h+]PfyG7ZF8$\"38&yxl/-Si&F87$$\"3emm\"z%4\\ Y_F8$\"3[*Q!R(Q;g:'F87$$\"32++v$flMLe*)>V B$)F8$\"3R-PW?nJGpF87$$\"3wmmTg()4_))F8$\"3XS!**oz*pDlF87$$\"3Y++DJbw! Q*F8$\"3wsIZl^+ohF87$$\"3=nT&)3\\m_'*F8$\"3:%=[TB`#HgF87$$\"3+N$ekGkX# **F8$\"3;==#[laH$fF87$$\"3nTg_(R^g+\"F,$\"3PhYLpC*H!fF87$$\"31]iSmjk>5 F,$\"3?Z/nW4A')eF87$$\"3XekGN8CL5F,$\"3S5jkI7P$)eF87$$\"3%ommTIOo/\"F, $\"3i#*RMEK8&*eF87$$\"3cLe9;_yq5F,$\"3_[.xr\")Q`fF87$$\"3E+]7GTt%4\"F, $\"3U$yw2JYC1'F87$$\"3(p;/,/$o=6F,$\"3q*3')fymbA'F87$$\"3YLL3_>jU6F,$ \"3?OGM!H;eW'F87$$\"3ym;HdNb'>\"F,$\"33P8#3QfS;(F87$$\"37++]i^Z]7F,$\" 3[rQ!Q)F87$$\"35+++v\"=YI\"F,$\"3&441r_RP_*F87$$\"33++](=h(e8F,$ \"3G(\\BrLl\"*4\"F,7$$\"3&*****\\7!Q4T\"F,$\"39o3wy`oD7F,7$$\"3/++]P[6 j9F,$\"3%z>\\NC[mH\"F,7$$\"3'=HKkAg\\Z\"F,$\"3=cb1EJ%4I\"F,7$$\"3W$ek` h0o[\"F,$\"3:bt7ruA+8F,7$$\"3/voH/5l)\\\"F,$\"39tx0vuR%H\"F,7$$\"3%o;H KR'\\5:F,$\"37z4l5FZ$G\"F,7$$\"3-]P4rr=M:F,$\"3i,%*Q\"yBqC\"F,7$$\"3UL $e*[z(yb\"F,$\"3W\")o!f;2L>\"F,7$$\"3w;/Ev&[ge\"F,$\"3r>R!QDA@6\"F,7$$ \"34+Dc,#>Uh\"F,$\"3#4HI%*[n*=5F,7$$\"3V$eky#)*QU;F,$\"3mms=Lk!y?*F87$ $\"3wmm;a/cq;F,$\"3Pe)[!HQiL#)F87$$\"3\"pm;a)))G=F,$\"35Bp0?YWZMF87$$\"3KL e9;0?E>F,$\"34:XWhm;,LF87$$\"3pTg-gl[Q>F,$\"3-rZpoe$*\\KF87$$\"31]i!Rg s2&>F,$\"35MI#4&[)H@$F87$$\"3WekyZ'eI'>F,$\"3=g(Q?ez,>$F87$$\"3gmmm\"p W`(>F,$\"3Q*HR/(>[\"=$F87$$\"3_ek.HW#)))>F,$\"3SdFyRyC)=$F87$$\"3?]iSm TI-?F,$\"3]%GeCR0B@$F87$$\"3*=/wP!Ry:?F,$\"3OG\"o!ej/aKF87$$\"3dLe9TOE H?F,$\"32!yA?()GSJ$F87$$\"3'pT&)e6Bi0#F,$\"3EaW;d/\"=\\$F87$$\"3K+]i!f #=$3#F,$\"35[niL?f`PF87$$\"3/++D\"=EX8#F,$\"3)=7zi$ePDXF87$$\"3?+](=xp e=#F,$\"3rT'oxRDBu&F87$$\"3$pTNrfbE@#F,$\"3S'**Rb6vOf'F87$$\"3mLeRA9WR AF,$\"3Y7cI=h\\:wF87$$\"3S]ilZsAmAF,$\"3.'zb%4IS@))F87$$\"37nm\"H28IH# F,$\"3I'yDR:j=-\"F,7$$\"3!oTN@#3hF,7$$\"3WeRseStdCF,$\"3!>Ien#[n??F,7$$\"3)oT5l0+5Z#F, $\"3=]_;#*)R!\\?F,7$$\"35YOSbIjxCF,$\"3'f>URy8d0#F,7$$\"3IvoHagE%[#F,$ \"3+2\\y.l?d?F,7$$\"3_/,>`!**3\\#F,$\"3WHUb4&*\\`?F,7$$\"3GLL3_?`(\\#F ,$\"3&3)GRRomW?F,7$$\"3.$3-)Q84DDF,$\"3%H:bmLUz&>F,7$$\"3AL3_D1l_DF,$ \"3aG*H8v%))4=F,7$$\"3I3-))o-VmDF,$\"3Y]!f;TM@s\"F,7$$\"3S$eRA\"*4-e#F ,$\"34g!)pZ(f(H;F,7$$\"3[e*)fb&*)Rf#F,$\"3\"z'>0Ar[N:F,7$$\"3fL$e*)>px g#F,$\"3')[_^!fj9W\"F,7$$\"3V+D1R'f:&H/%HG\"F,7$$\"3%omm\" z+vbEF,$\"3;#=@:,\\a8\"F,7$$\"3AL3F>0uzEF,$\"3C!3Cbi?=+\"F,7$$\"33+]Pf 4t.FF,$\"3l3u3()HVJ))F87$$\"3Q$3F>HT'HFF,$\"3A'y5tq@xr(F87$$\"3om\"zWi ^bv#F,$\"3w5fooRnqnF87$$\"3)*\\7.d>Y\"y#F,$\"3Op_f3vnxfF87$$\"3uLLe*Gs t!GF,$\"3eVA%fwpRK&F87$$\"3)om\"H2\"34'GF,$\"3!H2v@w6,N%F87$$\"30+++DR W9HF,$\"3UKzOA[p(y$F87$$\"3S+Dc,6jSHF,$\"3l.`HJu\"\\j$F87$$\"3K+]7y#=o 'HF,$\"3w)4GNSsEb$F87$$\"3G]iSm=\"*zHF,$\"354>3R$*[ONF87$$\"3C+voaa+$* HF,$\"3>.\"*[foPONF87$$\"3>](oH/*41IF,$\"3UC#)e$)41_NF87$$\"3:++DJE>>I F,$\"37lJ^K5P$e$F87$$\"3A+v$4^n)pIF,$\"3q;[s.$y8&QF87$$\"3F+]i!RU07$F, $\"3'\\(yiwjedVF87$$\"39+vo/#3o<$F,$\"3\"=P&zNU*o?&F87$$\"3+++v=S2LKF, $\"3rC&z7e?,M'F87$$\"3;L$3_NJOG$F,$\"3'oVq=V<'GvF87$$\"3Jmmm\"p)=MLF,$ \"35IlY8#))ft)F87$$\"3GLLeR%p\")Q$F,$\"3\"\\w,G=fT\")*F87$$\"3B++](=]@ W$F,$\"3p.a#\\O(RU5F,7$$\"3u\"H#oK)yVX$F,$\"37R>JG%fz/\"F,7$$\"3C$ekyZ 2mY$F,$\"3i=$4cdg.0\"F,7$$\"3=vo/Bh$)yMF,$\"3i,C<-Qm\\5F,7$$\"3mm\"H#o Z1\"\\$F,$\"3%43uAU\"*f/\"F,7$$\"36]Pfe?_:NF,$\"3]?:\"y&yYI5F,7$$\"35L $e*[$z*RNF,$\"3WZcwFIo05F,7$$\"3%o;Hd!fX$f$F,$\"3\\R'zK#G91$*F87$$\"3e ++]iC$pk$F,$\"3!e-txpVQY)F87$$\"3ILe*[t\\sp$F,$\"3o)z!\\!\\E.v(F87$$\" 3[m;H2qcZPF,$\"3-m*ep\"3v-sF87$$\"3s***\\7.lQx$F,$\"3Il\"\\:&*zo*pF87$ $\"3UL$3_0j,!QF,$\"35]vs*pf#\\oF87$$\"3F+v=n?J8QF,$\"3+R^fg!Quz'F87$$ \"36nm;z5YEQF,$\"3-)RYLJ)=gnF87$$\"3^Le9\"45'RQF,$\"3qjisVpNPnF87$$\"3 O+]7.\"fF&QF,$\"3VEh1S\\sGnF87$$\"3i3_vlYhlQF,$\"3(\\vU\")GTPt'F87$$\" 3)oT&QG-ZyQF,$\"3aR=Q3Gt^nF87$$\"39Dc,\"zD8*QF,$\"3o0VIG#3By'F87$$\"3T Lek`8=/RF,$\"33OP7dc+DoF87$$\"3$*\\i!*yC*)HRF,$\"3\\cG3V'=X%pF87$$\"3Y mm;/OgbRF,$\"33DC(y#>%[5(F87$$\"3*G$e*[$zV4SF,$\"39]/NuwfQvF87$$\"3w** \\ilAFjSF,$\"3D8xFXpB@!)F87$$\"3#G3_]p'>*3%F,$\"3\"o(e-u54K#)F87$$\"3y m\"zW7@^6%F,$\"3%eL`Rp%o0%)F87$$\"3w3F>RL3GTF,$\"3-T$yF^SJZ)F87$$\"3t] i!RbX59%F,$\"3i)pQA'3/D&)F87$$\"3#=z>'ox+aTF,$\"37$4T=0@'f&)F87$$\"3yL LL$)*pp;%F,$\"3![sp'zRPv&)F87$$\"3!Q3_+sD-=%F,$\"3se;6C(p2d)F87$$\"3#Q $3xc9[$>%F,$\"3X[ZFn2MW&)F87$$\"3'Qe*[$>Pn?%F,$\"3'>&G!HhJc\\)F87$$\"3 )QL3-$H**>UF,$\"3[:#3'R6kC%)F87$$\"3#R$ek.W]YUF,$\"3+Xc4Am3=#)F87$$\"3 )RL$3xe,tUF,$\"3%p?zi!3VLzF87$$\"3Cn;HdO=yVF,$\"3;#>X)HduejF87$$\"3a++ +D>#[Z%F,$\"3gxAO;Hx\")\\F87$$\"3TM$3_5,-`%F,$\"3'RSzCT^zW%F87$$\"3Snm T&G!e&e%F,$\"3%=ER;G9D:%F87$$\"3/]i:NK'zf%F,$\"3*f`:i,h67%F87$$\"3fLe* [=Y.h%F,$\"3v*>WD]()H5%F87$$\"386but%F,$\"3')>+X\\xqwZF87$$\"37+]i SjE!z%F,$\"3<\"z**[W(=JcF87$$\"3y*\\7G))Rb\"[F,$\"3[YJ\"GK]h>'F87$$\"3 L+++DM\"3%[F,$\"3!**fZR-9n(oF87$$\"3)3](=np3m[F,$\"3gVt^2I*Ho(F87$$\"3 a+]P40O\"*[F,$\"3`9x`tb,B')F87$$\"3>]7.#Q?&=\\F,$\"3Ik*[g6ply*F87$$\"3 s+voa-oX\\F,$\"3)pQ=b([U56F,7$$\"3O]PMF,%G(\\F,$\"3_;`pzy]b7F,7$$\"\"& F)$\"3ftg')yo>49F,-%'COLOURG6&%$RGBG$\"#5!\"\"F(F(-%%FONTG6$%*HELVETIC AG\"\"*-%+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 c ode constructs a discrete solution based on each of the methods and gi ves the " }{TEXT 260 22 "root mean square error" }{TEXT -1 18 " of eac h solution." }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 785 "P := (x,y) - > -(2*sin(5*x)+3*cos(7*x))*sinh(y): hh := 0.01: numsteps := 500: x0 := 0: y0 := sqrt(5)/2:\nmatrix([[`slope field: `,P(x,y)],[`initial poi nt: `,``(x0,y0)],[`step width: `,hh],\n[`no. of steps: `,numsteps] ]);``;\nmthds := [`Butcher's scheme A`,`scheme with simple nodes`,`sch eme with a relatively large stability region`,`Butcher's scheme B with `*(c[5]=1/2,c[6]=1/2,b[5]=b[6]),`scheme with `*(c[5]=3/4,c[6]=3/4,b[5 ]=b[6])]:\nerrs := []:\nDigits := 30:\nfor ct to 5 do\n Pn_RK6_||ct \+ := RK6_||ct(P(x,y),x,y,x0,evalf(y0),hh,numsteps,false);\n sm := 0: n umpts := nops(Pn_RK6_||ct):\n for ii to numpts do\n sm := sm+(P n_RK6_||ct[ii,2]-p(Pn_RK6_||ct[ii,1]))^2;\n end do:\n errs := [op( errs),sqrt(sm/numpts)];\nend do:\nDigits := 10:\nlinalg[transpose]([mt hds,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$%0ini tial~point:~G-%!G6$\"\"!,$*&F-FBF4#F.F-F.7$%/step~width:~~~G$F.!\"#7$% 1no.~of~steps:~~~G\"$+&Q)pprint456\"" }}{PARA 11 "" 1 "" {XPPMATH 20 " 6#%!G" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#K%'matrixG6#7'7$%3Butcher's~s cheme~AG$\"+k7$%9scheme~with~simple~nodesG$\"+kr!=*oF+7$%Pschem e~with~a~relatively~large~stability~regionG$\"+bKT%*oF+7$*&%9Butcher's ~scheme~B~with~G\"\"\"6%/&%\"cG6#\"\"&#F7\"\"#/&F;6#\"\"'F>/&%\"bGF<&F FFBF7$\"+m!\\;(oF+7$*&%-scheme~with~GF76%/F:#\"\"$\"\"%/FAFOFDF7F.Q)pp rint466\"" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 30 "The following code constructs " }{TEXT 260 20 "numerical proced ures" }{TEXT -1 56 " for solutions based on each of the Runge-Kutta sc hemes." }}{PARA 0 "" 0 "" {TEXT -1 75 "The error in the value obtained by each of the methods at the point where " }{XPPEDIT 18 0 "x = 4.99 9;" "6#/%\"xG-%&FloatG6$\"%**\\!\"$" }{TEXT -1 16 " is also given." } }{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 716 "P := (x,y) -> -(2*sin(5*x) +3*cos(7*x))*sinh(y): hh := 0.01: numsteps := 500: x0 := 0: y0 := sqrt (5)/2:\nmatrix([[`slope field: `,P(x,y)],[`initial point: `,``(x0,y0 )],[`step width: `,hh],\n[`no. of steps: `,numsteps]]);``;\nmthds \+ := [`Butcher's scheme A`,`scheme with simple nodes`,`scheme with a rel atively large stability region`,`Butcher's scheme B with `*(c[5]=1/2,c [6]=1/2,b[5]=b[6]),`scheme with `*(c[5]=3/4,c[6]=3/4,b[5]=b[6])]:\nerr s := []:\nDigits := 30:\nfor ct to 5 do\n pn_RK6_||ct := RK6_||ct(P( x,y),x,y,x0,evalf(y0),hh,numsteps,true);\nend do:\nxx := 4.999: pxx := evalf(p(xx)):\nfor ct to 5 do\n errs := [op(errs),abs(pn_RK6_||ct(x x)-pxx)];\nend do:\nDigits := 10:\nlinalg[transpose]([mthds,evalf(errs )]);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#K%'matrixG6#7&7$%0slope~field: ~~~G,$*&,&*&\"\"#\"\"\"-%$sinG6#,$*&\"\"&F.%\"xGF.F.F.F.*&\"\"$F.-%$co sG6#,$*&\"\"(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\"$+&Q)pprint476\"" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%!G" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#K%'matrixG6#7'7$%3Butcher's~scheme~AG$ \"+(zKWw&!#?7$%9scheme~with~simple~nodesG$\"+z(*3k7F+7$%Pscheme~with~a ~relatively~large~stability~regionG$\"+rLFS:F+7$*&%9Butcher's~scheme~B ~with~G\"\"\"6%/&%\"cG6#\"\"&#F7\"\"#/&F;6#\"\"'F>/&%\"bGF<&FFFBF7$\"+ A[Q*>#F+7$*&%-scheme~with~GF76%/F:#\"\"$\"\"%/FAFOFDF7$\"+%)*4Y3*!#@Q) pprint486\"" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 4 "The " }{TEXT 260 22 "root mean square error" }{TEXT -1 20 " over the interval " }{XPPEDIT 18 0 "[0, 5];" "6#7$\"\"!\"\"&" }{TEXT -1 82 " of each Runge-Kutta method is estimated as follows using the spe cial procedure " }{TEXT 0 5 "NCint" }{TEXT -1 98 " to perform numeri cal integration by the 7 point Newton-Cotes method over 200 equal subi ntervals." }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 436 "mthds := [`But cher's scheme A`,`scheme with simple nodes`,`scheme with a relatively \+ large stability region`,`Butcher's scheme B with `*(c[5]=1/2,c[6]=1/2, b[5]=b[6]),`scheme with `*(c[5]=3/4,c[6]=3/4,b[5]=b[6])]:\nerrs := []: \nDigits := 20:\nfor ct to 5 do\n sm := NCint((p(x)-'pn_RK6_||ct'(x) )^2,x=0..5,adaptive=false,numpoints=7,factor=200);\n errs := [op(err s),sqrt(sm/5)];\nend do:\nDigits := 10:\nlinalg[transpose]([mthds,eval f(errs)]);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#K%'matrixG6#7'7$%3Butche r's~scheme~AG$\"+=zRnz!#?7$%9scheme~with~simple~nodesG$\"+d6jw#*!#@7$% Pscheme~with~a~relatively~large~stability~regionG$\"+kk?W\")F07$*&%9Bu tcher's~scheme~B~with~G\"\"\"6%/&%\"cG6#\"\"&#F8\"\"#/&F<6#\"\"'F?/&% \"bGF=&FGFCF8$\"+vvjF'*F07$*&%-scheme~with~GF86%/F;#\"\"$\"\"%/FBFPFEF 8$\"+WNe$z)F0Q)pprint496\"" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 92 "The following error graphs are constructed usin g the numerical procedures for the solutions." }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 518 "evalf[30](plot(['pn_RK6_1'(x)-p(x),'pn_RK6_2'(x )-p(x),'pn_RK6_3'(x)-p(x),'pn_RK6_4'(x)-p(x),\n'pn_RK6_5'(x)-p(x)],x=0 ..2.2,font=[HELVETICA,9],\ncolor=[COLOR(RGB,.95,.2,0),COLOR(RGB,.45,0, .95),COLOR(RGB,0,.25,1),COLOR(RGB,0,.75,.2),COLOR(RGB,.95,.45,0)],\nle gend=[`Butcher's scheme A`,`scheme with simple nodes`,`scheme with a r elatively large stability region`,`Butcher's scheme B with c[5]=c[6]=1 /2 and b[5]=b[6]`,`scheme with c[5]=c[6]=3/4 and b[5]=b[6]`],title=`er ror curves for 7 stage order 6 Runge-Kutta methods`));" }}{PARA 13 "" 1 "" {GLPLOT2D 868 572 572 {PLOTDATA 2 "6+-%'CURVESG6%7`u7$$\"\"!F)F(7 $$\"?mmmmmmmmmm;RP&z%!#J$!4T%zT'fg9Na\"!#I7$$\"?+++++++++v=&)e\")oF-$! 4:NOwdjH>V#F07$$\"?LLLLLLLLL$37.y'*)F-$!4!4ddH.B,#4$F07$$\"?lmmmmmm;aj MR2h#*F-$!4Jva)HY_r=JF07$$\"?&**************\\P%[ZMa&*F-$!4N#>.@eLW$3$ F07$$\"?gmmmmmmT&Q`:!)4q*F-$!4>Y%*exO'[\"3$F07$$\"?ILLLLLL$eRAc:w%)*F- $!4[.V4)oO(>7$F07$$\"?&************\\iS\"p4D%***F-$!4mHCLX0H%oKF07$$\" ?mmmmmmmmTgP')395F0$!4AKM$zhs>eKF07$$\"?+++++++vV)*=dTV5F0$!4xvM(\\$op '=KF07$$\"?LLLLLLL$ek.!Gus5F0$!4#3:mg5#H(*>$F07$$\"?++++++](oa5M1u3\"F 0$!4.O\"zW!)[AMKF07$$\"?mmmmmmm\"zW<))p?5\"F0$!4`sm>;d\"4ILF07$$\"?LLL LLL$e*[VAMt;6F0$!4/&e.].v45LF07$$\"?++++++++]7jpRJ6F0$!4[Qw%>=zO!H$F07 $$\"?nmmmmmm;a)e7^+>\"F0$!4y<3yT%HnvKF07$$\"?MLLLLLLLek)G0([7F0$!4`lS+ %*3DjE$F07$$\"?+++++++]iS^%ftI\"F0$!4x:T^da@rE$F07$$\"?nmmmmmmmm;9O,m8 F0$!4]$pStzSv(>$F07$$\"?nmmmmmmmmmca=-;F0$!4grMI%*>!HGHF07$$\"?nmmmmmm mm;*Hd$Q=F0$!4(4/'QCviGg#F07$$\"?LLLLLLLLL3F07$$\" ?+++++++++DhoPEDF0$!4&)G1Hbbf.x\"F07$$\"?mmmmmmmmmT0xHWFF0$!43(GcnwZEJ ;F07$$\"?LLLLLLLLL3<@%*pHF0$!4S\")\\7\"oy7Q:F07$$\"?+++++++++vGle&>$F0 $!4(z*yD(p>R%[\"F07$$\"?LLLLLLLLL3-$[*GMF0$!4c$>d.#Q'fy9F07$$\"?mmmmmm mmmTv+JiOF0$!4\"=x*eHy?Q]\"F07$$\"?+++++++++vLo`FTF0$!4*y#ov#GM?L;F07$ $\"?LLLLLLLLLLQ(zgg%F0$!42+:,V4@+%=F07$$\"?mmmmmmmmm;*e!eF]F0$!4XeE>W \"f1g?F07$$\"?++++++++++:24-bF0$!4h$*33jymKK#F07$$\"?++++++++++q*>.u&F 0$!4`tur(HYqHCF07$$\"?++++++++++D#\\&yfF0$!4EW%e%p\\)o4DF07$$\"?++++++ ++++bs73iF0$!4aIS')f,Q-a#F07$$\"?++++++++++&G0xV'F0$!4nrf(\\_E/cDF07$$ \"?mmmmmmmmmTvHmaoF0$!48(>H#)e=XODF07$$\"?LLLLLLLLLL)*fY]tF0$!4D)pw%*G 2e8DF07$$\"?LLLLLLLLLL$>w/x(F0$!4kN%=$=pwt[#F07$$\"?***************** \\()*y/f#)F0$!4j+&yJN/!fQ#F07$$\"?LLLLLLLLLLVm^\"p)F0$!4NB^fJ5DvB#F07$ $\"?*****************\\()R.g;*F0$!43nCUg'Rx\"3#F07$$\"?*************** **\\i*p#yh*F0$!4'G9$zf$R^v>F07$$\"?LLLLLLLLL3_d#*35!#H$!4%p_nNW*RK#>F0 7$$\"?LLLLLLLL$32zF07$$\"?nmmmmmmmm\"H59*)4\"F `y$!4%4n#[`)z:j?F07$$\"?nmmmmmmm;aZ%=u9\"F`y$!4'f6A-=A<@BF07$$\"?+++++ +++]7A;k*=\"F`y$!4Go'R!>]L`k#F07$$\"?nmmmmmmmmT2QCN7F`y$!4yQ`%\\<)>M*H F07$$\"?++++++++++F`N#G\"F`y$!4^:MUQnshH$F07$$\"?++++++++]PU+S08F`y$!4 b!)os!eJ@$\\$F07$$\"?+++++++++vdZWG8F`y$!3Lq@:n#)QJPF`y7$$\"?+++++++]7 GJKfR8F`y$!3#=%\\yy9&G%QF`y7$$\"?++++++++D\"[qT2N\"F`y$!38![(RYG**GRF` y7$$\"?+++++++D\"y:%fJc8F`y$!3(*\\u@$pX-+%F`y7$$\"?+++++++]PMy,*=O\"F` y$!3bT$o\\`!4sRF`y7$$\"?++++++]ils'HxYO\"F`y$!3mY!))3-,!3SF`y7$$\"?+++ ++++v$4^TkuO\"F`y$!3=eaDn?\\ASF`y7$$\"?++++++D\"y+V(z&)o8F`y$!3\\Yy)\\ =\"z&)RF`y7$$\"?++++++](=#\\L:Dq8F`y$!3EC$G\"\\tH'*QF`y7$$\"?++++++v$f $o#4X;P\"F`y$!3$42.KoZP\"RF`y7$$\"?++++++++](=lQIP\"F`y$!31]:Val5JRF`y 7$$\"?+++++++v=U\"*yAz8F`y$!3Lh.B^==TQF`y7$$\"?+++++++](o48]gOzOF`y7$$\"?+++++++D c^qjg\"R\"F`y$!3y$\\P8O]eT$F`y7$$\"?++++++]i!*G!*4q%R\"F`y$!3g]h3=?(HW $F`y7$$\"?++++++++D15cz(R\"F`y$!3TMbo%ybKP$F`y7$$\"?+++++++]i:*3u,T\"F `y$!3^AU'Q\"F`y7$$\"?LLLLLL LL3F#=vOV\"F`y$!2ND\\(QKI[TF`y7$$\"?mmmmmmmm;H'z(zW9F`y$\"2l%*>j.G<3'F `y7$$\"?++++++++DJ5/#fX\"F`y$\"3#eNogk?Xe\"F`y7$$\"?LLLLLLLLLLCI/n9F`y $\"3!=M!on*GDR#F`y7$$\"?mmmmmmmT5!\\\\F&o9F`y$\"3a=rh$f(H9DF`y7$$\"?++ +++++](oa'>,q9F`y$\"3;\\*\\.$ecAGF`y7$$\"?LLLLLLLek.Ok\\r9F`y$\"3(R!)p KlRS#GF`y7$$\"?mmmmmmmmTg14)HZ\"F`y$\"3rX3%Q2:`#GF`y7$$\"?LLLLLLL$eRx% )\\fZ\"F`y$\"3lE6d))G0KGF`y7$$\"?++++++++]())y=*y9F`y$\"3aq,$H^)*p(GF` y7$$\"?mmmmmmm;/,Ix)=[\"F`y$\"3L\"z9\\f&41HF`y7$$\"?LLLLLLLLe9rm&[[\"F `y$\"3_p*>E#R8.HF`y7$$\"?+++++++]7G7c#y[\"F`y$\"3,(RTwncQ%GF`y7$$\"?mm mmmmmmmT`Xz!\\\"F`y$\"3FbDTlYX/DF`y7$$\"?LLLLLLLL$ezJqE]\"F`y$\"3rRr\" 4^R\\d\"F`y7$$\"?+++++++++]#3YX^\"F`y$\"2C$[GWk`48F`y7$$\"?nmmmmm;/,W% QOs^\"F`y$!1\"*ej/KYJ\\F`y7$$\"?MLLLLLL3-Q'oE*>:F`y$!3&)))*yv;P7l\"F`y 7$$\"?nmmmmmTg-NP=F@:F`y$!3!pA)4'\\cUu\"F`y7$$\"?++++++]7.K))phA:F`y$! 3yAFL()eASyL`\"F`y$!3>*pn\"4dmLRF`y7$$\"?MLLLLLLL3-)\\og`\"F`y$!3$R[CtFA0 )RF`y7$$\"?++++++]P4'**ze(Q:F`y$!3bb3c2cO\\[F`y7$$\"?nmmmmmmT5!>5\\9a \"F`y$!3V)z7c=*y=jF`y7$$\"?++++++v$4rGD%zU:F`y$!3q%3zv_ItH'F`y7$$\"?ML LLLL$e9TQSRTa\"F`y$!3`N<6V>)*ziF`y7$$\"?nmmmmm\"z>6[b%[X:F`y$!3c&Ht)Q9 C*G'F`y7$$\"?+++++++]7y0(Hoa\"F`y$!3Br#e+62?S'F`y7$$\"?nmmmmm;a8s2+_\\ :F`y$!3NM@etp&\\(zF`y7$$\"?MLLLLLLe9m4.@_:F`y$!3*Hy%z:m3b')F`y7$$\"?nm mmmmT5:jgab`:F`y$!3[\\b`BUbA')F`y7$$\"?++++++]i:g61!\\b\"F`y$!3l3*33rS Dg)F`y7$$\"?MLLLLLe9;didCc:F`y$!3Y%\\UtxV1k)F`y7$$\"?nmmmmmmm;a84fd:F` y$!3;3&>I-Tw'))F`y7$$\"?MLLLLLekG&)=:1f:F`y$!3!4sNVIQ!)p*F`y7$$\"?++++ ++]iS;C@`g:F`y$!4mSD?6-Rv3\"F`y7$$\"?nmmmmmTg_ZHF+i:F`y$!4-F584$\\T#3 \"F`y7$$\"?MLLLLLLekyMLZj:F`y$!4(o!)3O)yqt2\"F`y7$$\"?++++++Dcw4SR%\\c \"F`y$!4(4HRe&3-O2\"F`y7$$\"?nmmmmm;a)3aa9kc\"F`y$!4bx,s@.*ew5F`y7$$\" ?MLLLLL3_+s]^)yc\"F`y$!4Pzj/Sz_K5\"F`y7$$\"?+++++++]7.cdNp:F`y$!4Q%R/ \"HfRZ>\"F`y7$$\"?nmmmmmmTgFx\"Q_d\"F`y$!4$)QOv^\"\\V`7F`y7$$\"?MLLLLL LL3_)f?6e\"F`y$!4mp%z1/I&=T\"F`y7$$\"?nmmmmmm;/,Ta)Gf\"F`y$!46keRVtjl \\\"F`y7$$\"?+++++++++]$G]Yg\"F`y$!4ITdcvLRU`\"F`y7$$\"?mmmmmmT&)3\\3S .1;F`y$!4(\\f'**=XNq_\"F`y7$$\"?LLLLLL$3x\"[LxT2;F`y$!4V\">:nw-AD:F`y7 $$\"?++++++DcEZe9!)3;F`y$!44;6\"G)Gsy`\"F`y7$$\"?mmmmmmmTNY$=&=5;F`y$! 41k!eNJC8t:F`y7$$\"?++++++]7`WLE&Hh\"F`y$!422pg.K=Sb\"F`y7$$\"?LLLLLLL $3FM3?dh\"F`y$!4t)4N%QQ**e`\"F`y7$$\"?++++++vozT3Q5<;F`y$!48Uc&o2.xH:F `y7$$\"?mmmmmm;a)3M`([=;F`y$!4*[iqw7D\"3`\"F`y7$$\"?LLLLLLeR(*Re7()>;F `y$!5Iz&>iq=s&[:F07$$\"?+++++++D1R$)\\D@;F`y$!5$=xwd/,dFa\"F07$$\"?LLL LLL$eRsLVASi\"F`y$!5$R.x8+RDL_\"F07$$\"?mmmmmmmmTN$))*yE;F`y$!54Tv^h%= ab]\"F07$$\"?+++++++]7G$ofyj\"F`y$!5*)o'*ozJ0k^9F07$$\"?LLLLLLLL$3K[H* [;F`y$!5[`6`2E&f;Q\"F07$$\"?mmmmmmmm;HW<2s;F`y$!5pitjI:IG>7F07$$\"?+++ +++++]P0S@&p\"F`y$!5U0)obKHN_2\"F07$$\"?mmmmmmmmTg&zRyr\"F`y$!4ylIX&*G SY`*F07$$\"?LLLLLLLLL$eel/u\"F`y$!4!*f&R&)[XT\"R)F07$$\"?mmmmmmmmTN\"p _Tw\"F`y$!4$em*=51)>ysF07$$\"?++++++++](ozRyy\"F`y$!4Fcz!3:1qthF07$$\" ?MLLLLLLL3x%H`1\"=F`y$!4>m?#fvwz&)\\F07$$\"?nmmmmmmmmm#zmM$=F`y$!4$fZw d[\"e72%F07$$\"?nmmmmmmm;H*)ozc=F`y$!4F,*4m#>!\\bKF07$$\"?nmmmmmmmm\"f )p7!)=F`y$!47rV*3e-V(Q#F07$$\"?LLLLLLLL$3#43SE>F`y$!47?Bl*o^)y\\\"F07$ $\"?mmmmmmmmT5G7mZ>F`y$!4(z*pPy=0\\B\"F07$$\"?++++++++++Z;#*o>F`y$!4o& p1>p\\V$4\"F07$$\"?mmmmmmm;H#oN8](>F`y$!4lF`y$!4IhYd8Y%4!3\"F07$$\"?+++++++](okx'>()>F`y$!4t9zvaYKR3\"F07 $$\"?mmmmmmmm;H'[)G$*>F`y$!4I6Cwa$yq76F07$$\"?++++++++v$f!>Z0?F`y$!4G! ='3fbV.=\"F07$$\"?LLLLLLLLLeD`l_@F`y$!4\"GCN$[[.()>(F07$$\"?++++++++D17$*4w@F`y$!4Tjo&p#e4@s)F07 $$\"#A!\"\"$!4\\]j'ppy1m&*F0-%&COLORG6&%$RGBG$\"#&*!\"#$\"\"#Fd^nF(-%' LEGENDG6#%3Butcher's~scheme~AG-F$6%7[vF'7$$\"?lmmmmmmmm\"zp3r*H!#K$!.U Bf$ehhF`y7$$\"?ILLLLLLLL$eR\"F-$!2dmve^**pi #F`y7$$\"?+++++++++v=_E)z\"F-$!2&*H-Qr5%GIF`y7$$\"?LLLLLLLLLLepo(R#F-$ !2NPLiI,Pq%F`y7$$\"?+++++++++]P/`'f$F-$!3#y\\\"\\!y&*F07$FP$!3_gJ6^P^ b(*F07$FU$!3C(GL8!*\\\"4(*F07$FZ$!33-r0vsd!f*F07$Fin$!3:\")fj(>b`]*F07 $F^o$!3,>GIo1/I&*F07$Fco$!3=n,`6F!fk*F07$Fho$!31PJt;w)ze*F07$F]p$!3K_^ ZvmzI&*F07$Fgp$!3:]0\"fjauJ*F07$Faq$!3N#o,\\4jC4*F07$F[r$!3#QHqJ1:ZE)F 07$F`r$!3N**4nX$*)>ZoF07$$\"?LLLLLLLLLe aM#\\*QF0$!3.OAM^^GzoF07$Fct$!3#H=lf\")H-+(F07$$\"?mmmmmmmm;/'G3oO%F0$ !3BG#pvNa\"4sF07$Fht$!3d8[=_?>#\\(F07$F]u$!3?;e_sR#p;)F07$Fbu$!3V?h*Gd JZ<*F07$F\\v$!43=19\"fr!=.\"F07$Ffv$!4w.n7J*GtZ6F07$$\"?NLLLLLLL$3-8%= YmF0$!4\"e^X5KOl'=\"F07$F[w$!45$filnb?77F07$$\"?+++++++]7G`$Qm\"pF0$!4 ])H#)4/Pa<7F07$$\"?NLLLLLLLe9JPhypF0$!4_&HcCqXi>7F07$$\"?qmmmmmm;/,4\" *eSqF0$!4+8%*ymJ57A\"F07$$\"?++++++++](o[kD5(F0$!4VS+M+=>*>7F07$$\"?lm mmmmmmTgU_^EsF0$!4cMNqAK5]@\"F07$F`w$!4md\\P>Pja?\"F07$$\"?NLLLLLLLL$e 4r/c(F0$!4e*Qp]:#=s<\"F07$Few$!4i<&Rmwu9R6F07$Fjw$!4n8Sn\\JVr.\"F07$F_ x$!35w?63^Eo&*F07$Fdx$!30B3k$e?r#*)F07$$\"?+++++++++](>:>R*F0$!3)[5mJ? lvr)F07$Fix$!3v'[@Q[^zd)F07$$\"?lmmmmmmm;aeAa`)*F0$!3UAm,i)>/\\)F07$F^ y$!3#G)4w9K>'[)F07$$\"?LLLLLLLLeRr8F`y$!4%48_W/@er7F07 $$\"?++++++]7`>%GV7K\"F`y$!42`cp&\\,Pr7F07$$\"?+++++++]i!*yNoA8F`y$!3T ,_)RL;zF\"F`y7$$\"?+++++++D\"G$oTcD8F`y$!3b!fhP_E-H\"F`y7$Fg[l$!3HsVNM Ah(G\"F`y7$F\\\\l$!3_Oe.jOkc7F`y7$Fa\\l$!3#[RzYA:eB\"F`y7$$\"?++++++]7 `>B)GNN\"F`y$!3;A[]figZ7F`y7$Ff\\l$!3O6Y&oP[vD\"F`y7$$\"?++++++]P4'*fI 5f8F`y$!3[(3]HKRjC\"F`y7$F[]l$!35+V@4#GR7F`y7$Fi^l$!3D*>%3jWzW7F`y7$Fc_l$!3*yC&HXU2y7F`y7$Fg`l$ !3v^$)Re=AW8F`y7$F\\al$!3Vj%3G4X;^\"F`y7$Faal$!3%=>8[tT%y;F`y7$Ffal$!3 %=L6c<#)Q(=F`y7$F[bl$!3fwH#*\\\"e?4#F`y7$F`bl$!3h&p5d=&=qC!HDF`y7$Fcdl$!3r2!)oycL@FF`y7$Fgel$!37&\\`XY3vz#F`y7$F\\fl$!3M' yNH(*=rs#F`y7$Fafl$!3dLjCSH$ec#F`y7$F[gl$!3@YOQ'\\9\"oBF`y7$F_hl$!3?xm D=T%HL#F`y7$Fcil$!3s'p%*fk\\)y?F`y7$F]jl$!3*\\Q:&*z0k/#F`y7$Fbjl$!3.( \\3qBP5%>F`y7$Fgjl$!3\"H(*Hq$eEut\"F`y7$F`\\m$!3:x9GvfBe:F`y7$Fe\\m$!3AkT\"F`y7$F]_m$!37[vsRoY)>\"F`y7$Fa`m$!3(=o*F`y7$F[am$!2XV&)p0ws<)F`y7$F`am$!2KW/:V\" [tqF`y7$Feam$!2J%G![jj[b'F`y7$Fjam$!2e;4RA0C_'F`y7$F_bm$!2KzS\"p=s5lF` y7$Fdbm$!2-<8X9J1c'F`y7$Fibm$!2S&y\\8AqQqF07$Faem$!3))\\2j4/b;qF07$Ffem$!3?'*>)))f\")\\&yF07$F[fm$! 3b:ocp5J)4*F07$$\"?+++++++++v81]g;F`y$!4WmsBl%)G83\"F07$F`fm$!4MR(3@L[ am6F07$$\"?LLLLLLLLL$[(Gk$o\"F`y$!4()ye3'=*4,C\"F07$Fefm$!4h'=y')RQo+8 F07$Fjfm$!4Zxr]%[<.&R\"F07$F_gm$!4P+w(=y\"3IZ\"F07$Fdgm$!4mH?QYtz\\U\" F07$Figm$!4K%)Hqk'H#4O\"F07$Fchm$!4'RhtgRJsE7F07$F]im$!4ut-))[yho6\"F0 7$$\"?++++++++Dc(*QE.>F`y$!4olIntH]$y5F07$Fbim$!4Fv$R?4/aa5F07$$\"?+++ ++++]il=5.P>F`y$!4:Zpl!)Hu([5F07$Fgim$!4(G44GH]YX5F07$$\"?LLLLLLL$3_vV \"He>F`y$!4@PE#fr;NW5F07$F\\jm$!4YYr0`$p1X5F07$F`[n$!4lwv]9rL60\"F07$F j[n$!4AElX'>\"f`1\"F07$F_\\n$!4aM3;.d*Hz5F07$Fd\\n$!4tkli9bz()4\"F07$F i\\n$!4%e$=W*G]=N6F07$F^]n$!49VvT'40Ay6F07$$\"?++++++]7`pMs54@F`y$!4rQ ,(\\'\\()[<\"F07$$\"?+++++++Dc^zs\\5@F`y$!4VnPGp%*\\\"o6F07$$\"?++++++ ]PfLCt)=6#F`y$!4p=7o_%[_u6F07$$\"?+++++++]i:ptF8@F`y$!4#on-\\M3)4=\"F0 7$$\"?+++++++vozeu0;@F`y$!4&=)H!)RJML>\"F07$$\"?++++++++vV[v$)=@F`y$!4 V'4#3s#*=`>\"F07$$\"?+++++++](=xs(RC@F`y$!471sL7hhc?\"F07$Fc]n$!41\"4! \\5wcH?\"F07$$\"?++++++++Dcl#y59#F`y$!4lc+_FdzmA\"F07$Fh]n$!4r$)f(QUn; ^7F07$$\"?+++++++]P4o*[T;#F`y$!4\\g[j\"HhPz7F07$F]^n$!4Yi[*)[`-II\"F07 $$\"?++++++](oa!)*o3z@F`y$!4e7o5SB,IH\"F07$$\"?+++++++vo/%[u?=#F`y$!4& [(or.Gu=G\"F07$$\"?++++++voH/x#oN=#F`y$!4eg*\\&4Y#R#H\"F07$$\"?++++++] i!R+2i]=#F`y$!4d]m8AcBDI\"F07$$\"?++++++Dc^.jeb'=#F`y$!4H0qb<`w+J\"F07 $$\"?+++++++]7.c'\\!)=#F`y$!4.>/!oK&G#38F07$$\"?+++++++Dc,G[-%>#F`y$!4 JX\"*QdB[1H\"F07$Fb^n$!4A&f=V]t8U7F0-Fh^n6&Fj^n$\"#XF]_nF(F[_n-Fa_n6#% 9scheme~with~simple~nodesG-F$6%7]tF'7$Fh`n$!28&p*p$R`&>#F`y7$Fban$!2?? 947\\y!RF`y7$Fgan$!3_].V.9h-_F07$F+$!3O`$4=5e/C'F07$$\"?NLLLLLLL$3x@\" [QeF-$!36s^e$z1U'oF07$F2$!3-1l?T71JsF07$$\"?ILLLLLLL3Fp@9.uF-$!3i4B,]: C_tF07$$\"?lmmmmmmm;z>epCzF-$!3L%4*e;Y['R(F07$$\"?++++++++DJq%\\iW)F-$ !3vEt%[FVYP(F07$F7$!38cq)oc^-T(F07$F]p$!3,HPy)*3ZFqF07$Faq$!3hPw9E'H)G lF07$F[r$!3BG+c554\"p&F07$F`r$!3k0-*[ijO;&F07$Fjr$!3/DnCoI'[\"[F07$Fds $!39UZYzHN:Z#**F07$Fbin$!3-'F07$Fdy$!3OfB/%>mAI 'F07$Fiy$!3CH6]\\6&Qn'F07$F^z$!3khzl)\\j&zsF07$Fcz$!3WB!>Jvc^4)F07$Fhz $!3f3-3wr(GK*F07$F][l$!4\\noZRmj%p5F07$Fg[l$!3yxXAS+0f6F`y7$$\"?++++++ +Dc^%**=SL\"F`y$!3t&*\\#)y?]h6F`y7$F\\\\l$!3eb!eT_uR;\"F`y7$$\"?++++++ ]iSm\\.QU8F`y$!3c:R@H5Io6F`y7$$\"?+++++++vo/ou;X8F`y$!3sZUon!y%z6F`y7$ $\"?++++++](oHkeazM\"F`y$!3L9v8\"[%y'=\"F`y7$Fa\\l$!3zvu))[B'4=\"F`y7$ F[]l$!37>*e,v#y;7F`y7$Fi^l$!3ie!**4\"z,s7F`y7$Fc_l$!3tS`oxyHh8F`y7$Fg` l$!3SA5Ffh$f]\"F`y7$F\\al$!3(*RZmN1!G#=F`y7$Faal$!3FF`y7$F`bl$!3sLlH!zsp/$F`y7$Feb l$!3D*>4e=2mM$F`y7$Fcdl$!3'QT6*G2([g$F`y7$Fgel$!354[O\\=6#o$F`y7$F\\fl $!3#*fD+h^=cNF`y7$Fafl$!3&pXXC8,/3,#F`y7$F`\\m$!3_E D0xDyL\"e%Hac4>F`y7$Fjam$!2Heh1L/^)=F`y7$F_bm$!2&G&oL(p*e#=F`y7$Fdbm $!29_9do'Gy;F`y7$Fibm$!2X(o&)o*\\7S\"F`y7$F^cm$!2W'=g(\\7UQ\"F`y7$Fccm $!2*o+a<)oxO\"F`y7$Fgdm$!3\"oqOiX%4C9F07$Faem$!39uLD+s4@9F07$Ffem$!3$* 3dUK#4z(>F07$F[fm$!3rO2+>ux+IF07$F`fm$!3CWx\\(\\![.aF07$Fefm$!3*Rql%e- ouoF07$Fjfm$!3DICP01c0zF07$F_gm$!3[f)y$yY;1')F07$Fdgm$!3kkYo=F`y$!3!>#p/MKniiF 07$F]im$!3O6P1N?Q*='F07$F[_p$!3c,mQ-oBRp&F07$$\"?++++++](oz@'ycG@F`y$!3zoAd?>!=l&F07$Fc]n$!3pcU5zj>;8&F07$F[ep$!38!)y`W-Gx\\F07$F`ep$!3q%4mH!*)G6[F07$Fje p$!383NUFC_')[F07$Fdfp$!3qqUh*4:#\\[F07$Fifp$!3*)[fxVv$4b%F07$Fb^n$!3y 2.Uh*)oDSF0-Fh^n6&Fj^nF($\"#DF]_n$\"\"\"F)-Fa_n6#%Pscheme~with~a~relat ively~large~stability~regionG-F$6%7fqF'7$Fh`n$!2,4x'3v'f8#F`y7$Fban$!2 dq:(z'fk1$F`y7$$\"?mmmmmmmmm\"zp3r*HF-$!3H:G6*[J[<$F07$Fgan$!3'\\/?)\\ NauIF07$$\"?LLLLLLLLL3x@&f>%F-$!31!yP(4y&ya#F07$F+$!3n@#*GP2N'H#F07$F2 $\"2Y,3*)f-&\\BF07$F7$\"3ce!Gx*4[EJF07$F]p$\"3(*4DxBE2w]F07$Faq$\"3&y \"y(*fMtChF07$$\"?nmmmmmmmmTN&*4%[\"F0$\"3M'p3'4+uUlF07$Ffq$\"3f9[=NT] 8qF07$$\"?nmmmmmmmm\"zPr-s\"F0$\"3-h:m$=>I5(F07$F[r$\"3_G>=+7,srF07$F` r$\"3vuc)\\*fjRyF07$Fjr$\"3v'z[ZFwag)F07$Fds$\"3a]7^/26](*F07$F^t$\"4q $HC?L!R:3\"F07$Fct$\"4U^J$4ON$f>\"F07$Fht$\"4)43]!=$Gn*G\"F07$F]u$\"4# yH\\CfLCZ8F07$Fbu$\"4q\"p5_.E%pP\"F07$F\\v$\"4&=T$p%z8B39F07$Ffv$\"4dJ a#4I)HvT\"F07$F[w$\"4ZLwbIMOYU\"F07$F`w$\"44,;K1g0[U\"F07$Few$\"4>%4rW _(G[U\"F07$Fjw$\"4<5]vNjg$H9F07$F_x$\"4Y(obcqrnB9F07$Fdx$\"4$)>R&3P9Bz 8F07$Fix$\"4;HO*)H$Q#=I\"F07$F^y$\"4fjN'phUJ47F07$Fdy$\"4dQ\\V&fmbF6F0 7$$\"?++++++++D\"o%fcv5F`y$\"4dV!exB\"4A5\"F07$Fiy$\"4:qsTRYN%z5F07$$ \"?nmmmmmmm\"H_FmJ7\"F`y$\"4TB.***\\YMp5F07$F^z$\"4tu^!GJU7t;7F07$Fi]o$\"4\">TTm+83l7F07$F][l$\"4W))QK69(4D8F07$Fb[l$\"4Y_ o!)*oc'*G9F07$Fg[l$\"3C&\\2E&)[Jg\"F`y7$Fa\\l$\"3)zPKKEF`y7$Fi^l$ \"3M^NlT&*QPAF`y7$Fg`l$\"3osQ`+q&3k#F`y7$Faal$\"3Gb;)R'=')yIF`y7$F[bl$ \"3(Q`Ka@t,P$F`y7$Febl$\"3q:%Q$H0g:OF`y7$Fcdl$\"3u[hITD\"pt$F`y7$Fgel$ \"3#*ooKUp/PQF`y7$$\"?LLLLLLL$3_X\\jP\\\"F`y$\"3w-Y(*QW#3$QF`y7$$\"?++ ++++++voNCt'\\\"F`y$\"3a'Q_2&GtEQF`y7$$\"?mmmmmmm;H#oP,(*\\\"F`y$\"3d? `A`3**eQF`y7$F\\fl$\"39D!=6nq'eQF`y7$$\"?mmmmmmmm\"H-?3'3:F`y$\"3gef@0 sHZQF`y7$Fafl$\"3``&3kx!oSQF`y7$F]jl$\"37iWHbyz#p$F`y7$Fi]m$\"3P#[toEw iL$F`y7$Fa`m$\"3A!RYt([AiHF`y7$F[am$\"3dwn;ZM'Q\\#F`y7$F`am$\"3QXd8YUx a?F`y7$Feam$\"3xJTe#RkTe\"F`y7$Fccm$\"3'49*\\3*zj5\"F`y7$Faem$\"3e&*=7 p3qDjF07$Ffem$\"3)f9qcy[ku\"F07$F[fm$!3KjECA&)f#f#F07$Fi\\p$!3QTcJGA#p b'F07$F`fm$!3[]I<&p)[/!)F07$$\"?LLLLLLL$3Fp_k\\n\"F`y$!33&H1#pNn)*yF07 $$\"?++++++++Dc4t&yn\"F`y$!3Y.FVk9%4+)F07$$\"?LLLLLLL3-)3q.$z;F`y$!3!G #3LNJbO%)F07$$\"?mmmmmmm;z>#4]2o\"F`y$!3LQ_y4>\"H$))F07$$\"?+++++++Dc^ $['>#o\"F`y$!3@0NE.`zo()F07$Fa]p$!3&*3nEsul0()F07$$\"?mmmmmmmmT5S%G%*o \"F`y$!3%QNN%yU;M))F07$Fefm$!31C,a\\'o*f()F07$Fjfm$!3yA#R\\]2#GtF07$F_ gm$!3ZzlnObh\"3%F07$Fdgm$!3$)p@?2QSY;F07$Figm$\"2'e@X>&R>M#F07$$\"?nmm mmmm;H#eaY#*z\"F`y$\"2bRc9\\M;#**F07$F^hm$\"3(f#R([;F07$Fchm$\"3z7y_?=gt;F07$Fhhm$\"3o9\\HRG0 E;F07$F]im$\"3)38]w$*pLf\"F07$Fbim$\"3\\nM-)QLp%=F07$Fgim$\"3?*\\5m(Qn B?F07$F\\jm$\"39fq0h4YN?F07$F`[n$\"3]a]:#[AAn\"F07$Fj[n$\"38T#\\X=4d+ \"F07$F_\\n$!1^y5`b(yr(F07$Fd\\n$!3\"QuXO)zMe6F07$Fi\\n$!3S;/))GO5Q?F0 7$F^]n$!3lYmmjd$H$HF07$Fc]n$!34L\"e*p8drWF07$Fh]n$!3L$[7'*4$)Q2(F07$Fc dp$!3`O#G\"\\-]?$*F07$F]^n$!46`\\OCNpw,(Q;F07$Fdfp $!4vZRX%*Q<I*QQF`y7$F^`n$!0>[\\W\"[hZF`y7$Fc`n$!1V( Q`_\\,)yF`y7$Fh`n$!2Amq'y(GWi\"F`y7$F]an$!2\"[Vh\\4Ct=F`y7$Fban$!2c.[( QbN5HF`y7$Fgan$!3D&>3\")HKb#RF07$F+$!3xGW,d#>9\"[F07$F2$!3)33DM<+5)eF0 7$F7$!3>Z,vSZfekF07$$\"?+++++++vVtF&QW6*F-$!3u/\"Qm$\\X+lF07$F<$!3MfRE aC%*ekF07$$\"?ILLLLLLek`T$4xS*F-$!3bIOgWP^=kF07$FA$!3DQHAaX2$Q'F07$FF$ !3[sZ'*o[0njF07$FK$!3]Vr5(=b#4kF07$FP$!3%*y!)y*Qy^f'F07$FU$!3IZc-/)eyc 'F07$FZ$!3*RLcYP\\z['F07$Fin$!3Eo1?!oi8W'F07$F^o$!3*[[B!e`>)['F07$Fco$ !3Wb9LU.LKmF07$Fho$!3%H4MfX4Df'F07$F]p$!36i%z'Hp@`lF07$Fgp$!38`4lkb*G^ 'F07$Faq$!3\\+D&=*yqjkF07$$\"?nmmmmmm;HKaVx!Q\"F0$!3L#4R!=4njkF07$$\"? nmmmmmmm\"zW4NbR\"F0$!3e`\"GHn)QJlF07$$\"?nmmmmmm;ajMeH59F0$!3ey(R6iq< b'F07$$\"?nmmmmmmm;zul0D9F0$!3oQWazE&o^'F07$$\"?nmmmmmm;z%\\J<)R9F0$!3 f)zt9POE['F07$$\"?nmmmmmmmT5b!yXX\"F0$!3Ob9oQM'4X'F07$$\"?nmmmmmm;/E&z Q$p9F0$!3KhLdDz#*GkF07$Fegr$!3c?aLNR8OkF07$$\"?nmmmmmmm;/'\\UJa\"F0$!3 oQ6:h6BKkF07$Ffq$!34q8%zp1#okF07$F]hr$!3OKJIM([HP'F07$F[r$!3N*HjJP(H!G 'F07$F`r$!3b![1A\"*)*)**fF07$Fjr$!3i:N?CDj'o&F07$Fds$!3@LalCk@u`F07$Fi s$!3JRZ.*[2)[_F07$F^t$!3F$QE:@at<&F07$$\"?++++++++++lnhyPF0$!3gL#G4b]> :&F07$Fdfn$!3(ofjYBk97&F07$$\"?++++++++]P*zwI&RF0$!3,_!)H->PN^F07$$\"? mmmmmmmmm;W,B6SF0$!3*pW*R\\&)[?^F07$$\"?LLLLLLLL$e*)[$QpSF0$!3f;PR.=w_ ^F07$Fct$!3`*f0%z.V^^F07$F\\gn$!3Dz)47I:YE&F07$Fht$!3'p#)y_%RwQaF07$F] u$!3l/J@8]R>fF07$Fbu$!3hilhH%p`g'F07$F\\v$!3.ips)RGQE(F07$Ffv$!3jy`i6' [hx(F07$F`hn$!3.u=Y%4fR$zF07$F[w$!3vA&*=jNzP!)F07$F]in$!3+)[X-,Z=2)F07 $Fgin$!3O#*y*4WCQ3)F07$F\\jn$!3so?iC#RX2)F07$F`w$!3c!G$=q26X!)F07$Fdjn $!3*GdvyHhs%zF07$Few$!3h[4lMlc&z(F07$Fjw$!3[EX@9:g&G(F07$F_x$!3U)Q&ejD ?knF07$$\"?lmmmmmmm;/@+wG*)F0$!3'*>he6[Y2lF07$Fdx$!3qnd(pwH5I'F07$Fe[o $!3>sc^O`]ghF07$Fix$!3`*RJBM[n3'F07$$\"?ILLLLLLLeRFYoN(*F0$!3r^f>sX([1 'F07$F]\\o$!3'o'z&4;vw0'F07$$\"?ILLLLLL$e9T2rC\"**F0$!3;(4y])GT!3'F07$ $\"?++++++++vo*))*Rr**F0$!32zLf%)>_ngF07$$\"?mmmmmmmTg_qG..5F`y$!3\"[K )>r9M%4'F07$F^y$!3G/EDovq.hF07$Fe\\o$!3%GYovLiH?'F07$Fdy$!3\\_,s#))zqI 'F07$Fiy$!3NXG1?K%)3mF07$F^z$!3\\:fe!Q@v+(F07$Fcz$!3&RU7h/r$4uF07$Fhz$ !3LVX!yjlR.)F07$Fi]o$!3hAKRY_en$)F07$F][l$!3>NSl$3Yci)F07$Fa^o$!3zsVAL j6l()F07$Fb[l$!3#G@[k[db())F07$Fi^o$!3c'*)pug$p0*)F07$F^_o$!3Z1LKLsA'* ))F07$Fc_o$!3rhg1+xE$y)F07$Fh_o$!31!*)oc-Ljx)F07$F]`o$!3=>\"*[\"ynm')) F07$Fb`o$!3a(*)*\\'*G*\\$*)F07$Fg`o$!3=9Q)=**fM\"*)F07$F\\ao$!3+=LFpWd n()F07$Faao$!3Yw&[(=Bap()F07$Ffao$!2p#\\ro!)p9))F`y7$$\"?++++++](=#*)F`y7$Fg[l$!2YZz[xUi)))F`y7$Fa\\l$!2)*=\\F'F`y7$Febl$!2Js$3FR$pu&F`y7$Fgel$!2Acw@STW\"[F`y7$Fafl$!20F! GIIYFSF`y7$F]jl$!2yDkq#)z,J$F`y7$Fi]m$!2.U9SdvP(HF`y7$F[am$!2bxsu:;[n$ F`y7$Feam$!2A%4dH@qX[F`y7$Fccm$!2wD91OGMg&F`y7$Faem$!3md.bU+;WkF07$Ffe m$!3L4zIm+hxtF07$F[fm$!3<)3&fWCN_%)F07$$\"?mmmmmmmm\"z%[]ra;F`y$!3!\\Z zT\"*yC&))F07$Fi\\p$!36PCE%p%GS(*F07$$\"?mmmmmmm;aQ'R$Rj;F`y$!33]k!e\\ P>g*F07$$\"?LLLLLLLL3-zhGm;F`y$!3`eHhD+E2&*F07$$\"?+++++++]ilh*y\"p;F` y$!3LUk67JjJ**F07$F`fm$!4,q(>e7msF5F07$Fa]p$!4dVunYe:22\"F07$Fefm$!4(> ul&R*Gz.6F07$Fjfm$!4dKKeh$3;a6F07$F_gm$!4=_L$3yhz/7F07$Fdgm$!4G4S!ya-S q6F07$Figm$!4T'>YLqSwK6F07$Fchm$!4*4C`f!fm'p5F07$F]im$!4$*3$e?E)o$>5F0 7$F[_p$!3.!fNj^4v'**F07$Fbim$!3h(HO_rv4%)*F07$$\"?++++++]7.dh$e!H>F`y$ !3zE6l(e,#e)*F07$$\"?mmmmmmm\"HKR\"frJ>F`y$!3%yA#))G^\\\"))*F07$$\"?LL LLLL$3F%HmMPM>F`y$!3/4G*3S%zZ)*F07$Fc_p$!36B^)G-cZ#)*F07$$\"?LLLLLLL3- QBhMU>F`y$!3vb,glGqi)*F07$Fgim$!3G:*[UC$HF)*F07$F[`p$!3_%fX4)[-[)*F07$ F\\jm$!34Wy5$**[a))*F07$F`[n$!4eW3FIM/2+\"F07$Fj[n$!45F07$Fd \\n$!4kkxFL-u%f5F07$Fi\\n$!4(zieqMvm\"4\"F07$F^]n$!4&R=Zi'**p=7\"F07$F eap$!4o[=9RNIe6\"F07$Fjap$!4oR7ju<-b5\"F07$F_bp$!477ZF[5N:6\"F07$Fdbp$ !4Pb-1%G3k<6F07$$\"?++++++]il(RTnY6#F`y$!4.:PAe>$oB6F07$Fibp$!41GL_Q'y -H6F07$$\"?++++++](=336\"F07$F_ar$!4nH0))ROXs6\"F07$$\"?+ +++++]P%)*Go2I7#F`y$!4.m&\\yrMwB6F07$Fccp$!4'*[j6)e^DI6F07$Fgar$!4gORO V1jh8\"F07$F\\br$!4\\9JLSbQ'R6F07$Fabr$!4TI)o/c`'f8\"F07$Fc]n$!4a7z&\\ t:*\\6\"F07$F[dp$!4?+M$)**)\\f?6F07$Fh]n$!4qs&3aaK1C6F07$$\"?++++++v$4 @rT#p`@F`y$!4)f,\\n&)=DK6F07$$\"?++++++](=<,@'=b@F`y$!4A#e8qGc.S6F07$$ \"?++++++D\"G8J+!oc@F`y$!4^7;E<]%QX6F07$$\"?+++++++v$4hzt\"e@F`y$!4YEr I5ny@9\"F07$$\"?++++++voa5*en'f@F`y$!4N(3yc;!=d6\"F07$$\"?++++++]i:5#Q h6;#F`y$!4(\\=F^X.`56F07$$\"?++++++Dcw4v^li@F`y$!4@#z+5jp)*=6F07$Fcdp$ !43ptlcmAu7\"F07$$\"?++++++vV)*3hFkl@F`y$!4hjL,(yb'\\8\"F07$$\"?++++++ ]Pf3al8n@F`y$!4lq[J6)\\GQ6F07$$\"?++++++DJ?3Z.jo@F`y$!4sey3L_<\"G6F07$ $\"?+++++++D\"y+9C,<#F`y$!4(RSB)GbN**3\"F07$$\"?++++++]7.2E<6t@F`y$!41 RV*Gh " 0 "" {MPLTEXT 1 0 520 "evalf[30](plot(['pn_RK6_1'(x)-p(x),'pn_RK6_2'(x)-p(x),'pn_RK6_3 '(x)-p(x),'pn_RK6_4'(x)-p(x),\n'pn_RK6_5'(x)-p(x)],x=2.2..2.8,font=[HE LVETICA,9],\ncolor=[COLOR(RGB,.95,.2,0),COLOR(RGB,.45,0,.95),COLOR(RGB ,0,.25,1),COLOR(RGB,0,.75,.2),COLOR(RGB,.95,.45,0)],\nlegend=[`Butcher 's scheme A`,`scheme with simple nodes`,`scheme with a relatively larg e stability region`,`Butcher's scheme B with c[5]=c[6]=1/2 and b[5]=b[ 6]`,`scheme with c[5]=c[6]=3/4 and b[5]=b[6]`],title=`error curves for 7 stage order 6 Runge-Kutta methods`));" }}{PARA 13 "" 1 "" {GLPLOT2D 917 638 638 {PLOTDATA 2 "6+-%'CURVESG6%7gq7$$\"#A!\"\"$!4\\] j'ppy1m&*!#I7$$\"?+++++++++]#HyI@#!#H$!41@)opH#p?g*F-7$$\"?++++++++]([ kdWA#F1$!4Yqd\"=k*)o1\"*F-7$$\"?+++++++++v;\\DPAF1$!437)o*3\"e<()yF-7$ $\"?+++++++++D$pMTA`()R$F-7$$\"?++++++++]P\"*y&HE#F1$\"3ZJ 6\"3))fVK\"F-7$$\"?++++++++](G[W[F#F1$\"4fK^,U&GD**[F-7$$\"?++++++++]( )fB:(G#F1$\"5XDl9hFw'Q:\"F-7$$\"?++++++++](Q=\"))*H#F1$\"4r$[$[m1%>nCF 17$$\"?++++++++]P'=pDJ#F1$\"4P*3z?jj@xMF17$$\"?+++++++++]c.iDBF1$\"4D2 4(=Q+`oXF17$$\"?+++++++++DMe6PBF1$\"47w+N.b5/q&F17$$\"?+++++++++]>q0]B F1$\"4JJ0ymm:S?(F17$$\"?+++++++++]U80jBF1$\"4(*H:kviT$)=)F17$$\"?+++++ ++++]!ytbP#F1$\"4(3Tz0HQ`0\"*F17$$\"?++++++++v=n&f7Q#F1$\"4P;GXK#zm)e* F17$$\"?++++++++](QNXpQ#F1$\"5qwv2)*=>/.5F17$$\"?++++++++v=/jq$R#F1$\" 5'4-Y)\\$*Ht'3\"F17$$\"?+++++++++]asY+CF1$\"5DjZ+_,?4T7F17$$\"?+++++++ +++y?#>T#F1$\"5$ol&RG4I+i:F17$$\"?++++++++vV>We=CF1$\"5men&H%e\">'fGCF1$\"55U X^s=-=nAF17$$\"?++++++++v$>1W6V#F1$\"5tU$**)3`JqMFF17$$\"?+++++++DJ?(Q =EV#F1$\"5n+mk.;W_cFF17$$\"?+++++++](oCr#4MCF1$\"5[Yan=x8NzFF17$$\"?++ +++++vVtPqcNCF1$\"5e\")G3lHWJ6GF17$$\"?++++++++++j8/PCF1$\"5WU$\\K]+?J )GF17$$\"?+++++++vV[J*e'QCF1$\"5&\\#f%o*>sV5JF17$$\"?+++++++](o**\\w-W #F1$\"58pV**HSU\"yf$F17$$\"?+++++++DJXoS*=W#F1$\"5`]&>Q$yhzCOF17$$\"?+ +++++++v$pj6NW#F1$\"5T!>M[31p;l$F17$$\"?+++++++v=U0#H^W#F1$\"5b/$y)[%e 9Yo$F17$$\"?+++++++]i!RxYnW#F1$\"5mDv,B`NnaPF17$$\"?+++++++D1RUVO[CF1$ \"5b-i>^hJRhRF17$$\"?++++++++](3\">)*\\CF1$\"5#)ef\\<-'p)\\XF17$$\"?++ ++++++DJED1`CF1$\"5HG.A1'oJ7h%F17$$\"?+++++++++vTJ9cCF1$\"5&\\)G@\\s:! [o%F17$$\"?++++++++v=dPAfCF1$\"5*GA@TP*pfO^F17$$\"?++++++++]isVIiCF1$ \"543OYv=6&**\\&F17$$\"?++++++++voqHtoCF1$\"5Y(eS5F5P$*z&F17$$\"?+++++ ++++vo:;vCF1$\"5&G*QbZ/Mk&='F17$$\"?+++++++]il)H8\"yCF1$\"5S!ROg%y7:ii F17$$\"?++++++++DcG]1\"[#F1$\"5+1*=rmHbeW'F17$$\"?+++++++Dc^$*3a#[#F1$ \"5EN31#[SyqW'F17$$\"?+++++++](o%en,%[#F1$\"56'==HJp*fYkF17$$\"?++++++ +v=UBE\\&[#F1$\"5!Qz]n:**QSW'F17$$\"?++++++++]P)[op[#F1$\"5p$G4,@qHfV' F17$$\"?++++++++DJnhL$\\#F1$\"5q_Sw(=F#3wiF17$$\"?+++++++++DYQq*\\#F1$ \"5p\\G$Ggd#)>#eF17$$\"?+++++++]ilg4,.DF1$\"5V[LoBR-P#o&F17$$\"?++++++ ++D1v!=j]#F1$\"5G/]/=q351cF17$$\"?+++++++](o%*=D'4DF1$\"5'QS/s2*e*)** \\F17$$\"?++++++++](QIKH^#F1$\"5Fx'*yxLJRiZF17$$\"?+++++++++]4+p=DF1$ \"5fLm1`#F1$\"5m 5dk$=v0bk#F17$$\"?+++++++D1*[\"3AKDF1$\"5PA_G@kWWCEF17$$\"?+++++++]P%[ VvP`#F1$\"5'*GaeU1'pAg#F17$$\"?+++++++voza+LNDF1$\"5Y<9NtC@/sDF17$$\"? +++++++++vuY)o`#F1$\"51P!)3&\\0OH]#F17$$\"?+++++++]7G\\2\\QDF1$\"5f$[a =z'ou!H#F17$$\"?++++++++D\"Q#o4SDF1$\"5E)fD)H&\\Z2u\"F17$$\"?+++++++]P M)*GqTDF1$\"58?98knBfCn\"F17$$\"?+++++++ +v$>7@la#F1$\"5il#4?i\\vhk\"F17$$\"?+++++++](ok>F\"[DF1$\"5cSW>l5oEB:F 17$$\"?++++++++++rKt\\DF1$\"5ZUJLISI(p;\"F17$$\"?++++++++]P$>=gb#F1$\" 5+qEq!QN%H95F17$$\"?+++++++++v:JIiDF1$\"4Ep'p9N'\\3@'F17$$\"?++++++++D J,TQoDF1$\"4()[>_/2>mN&F17$$\"?++++++++](o3lWd#F1$\"43Fn'QLaqmPF17$$\" ?+++++++voHhI:wDF1$\"4MQV;*QxA#p$F17$$\"?+++++++](=d.Tyd#F1$\"4TSGM)GB X5NF17$$\"?+++++++D195!H&zDF1$\"4kc8#)e'y$>+$F17$$\"?++++++++Dc%)p@\"e #F1$\"4e.],SWs%)\\_#F17$$\"?++++++++++VE5+EF1$\"4D%pO.sb. ))>F17$$\"?+++++++++]A!eIh#F1$\"4xMRST#)=Gl\"F17$$\"?++++++++](=_(zCEF 1$\"4Ad(zOayK;7F17$$\"?+++++++++]&*=jPEF1$\"3>F:E\\U'*GdF17$$\"?++++++ ++](3/3(\\EF1$!3n(*R(Q+Bi=&F17$$\"?++++++++]P#4JBm#F1$!4MP'G%)*\\BLI\" F17$$\"?+++++++++]KCnuEF1$!4in:Z([(*G$)=F17$$\"?++++++++](=n#f(o#F1$!5 W5$\\y=s()fV#F-7$$\"?++++++++v$\\`9Qp#F1$!5UFymW3P_MGF-7$$\"?+++++++++ +)RO+q#F1$!5fwWTr>H'oB$F-7$$\"?++++++++D\"[xw7FF1$!5'RH]G?%yI3MF-7$$\"?+++++ +++]()Q?QDFF1$!50\"\\)3([tCg\\$F-7$$\"?++++++++++J'ypt#F1$!5.&*prv(z+> a$F-7$$\"?+++++++++DM'p-v#F1$!5;pu%*[!)\\?@OF-7$$\"?++++++++++ms:iFF1$ !5@X/\\)pk\"G&\\$F-7$$\"?++++++++](3'>$[x#F1$!5TU2D^*)*yJL$F-7$$\"?+++ +++++]7hK'py#F1$!5$=JjFgiz1<$F-7$$\"#GF*$!5^n^dc4(>&))HF--%&COLORG6&%$ RGBG$\"#&*!\"#$\"\"#F*$\"\"!Fe]m-%'LEGENDG6#%3Butcher's~scheme~AG-F$6% 7fs7$F($!4A&f=V]t8U7F-7$F/$!4CaC_@`_gA\"F-7$F5$!4\"\\jSqDelq6F-7$F:$!4 t7\"RYD>*z1\"F-7$F?$!32U<*p9FR#pF-7$FD$!3nsQO!4W7;%F-7$FI$!2:9B;#p8x&F-7$FN$\"3A= 1c\")y@5fF-7$$\"?++++++++](=x;NH#F1$\"3)ynS)*ypYB\"F17$FS$\"3]5'pkQ6B1 #F17$FX$\"3q1USS9X=MF17$Fgn$\"3FPPwu&y7>&F17$$\"?++++++++v$fA%\\GBF1$ \"3n[:wl,X,eF17$$\"?++++++++]P&4o8L#F1$\"3cZeFZH\">3(F17$$\"?+++++++]P 4I]!GL#F1$\"3q9feSTXgrF17$$\"?++++++++D\"['>CMBF1$\"3HAT0#flUC(F17$$\" ?+++++++]7`**)ycL#F1$\"3HQ?aPnQdtF17$F\\o$\"3&[`LZXZle(F17$$\"?+++++++ +](oU'eVBF1$\"3yg=#e6I-\"**F17$Fao$\"4TjSgDk#y%H\"F17$$\"?+++++++++D+c I`BF1$\"4%)pm#zYj)yK\"F17$$\"?++++++++++\"=alN#F1$\"4'[C'ft!o\"oP\"F17 $$\"?+++++++++vhF!)fBF1$\"4V\"zW!*yOoe;F17$Ffo$\"45V388278u\"F17$$\"?+ ++++++++]hDJpBF1$\"4!Qw8X&f7&Q?F17$F[p$\"4*4m(4vAY\")G#F17$F`p$\"4+xm. :%4HrFF17$Fep$\"4gLH1H'RmAHF17$Fjp$\"4S!3txdB$eW$F17$F_q$\"40\"*45;Q(o xRF17$$\"?+++++++++DmY>1CF1$\"4`C6.bA>%\\TF17$Fdq$\"4MEl(4]B*ec%F17$Fi q$\"4gq>74LB!)z%F17$F^r$\"49%zc6(QUu/&F17$$\"?++++++]7y]BR)fU#F1$\"47, LpL$*)*p1&F17$$\"?+++++++D19'3@nU#F1$\"4N>6/wytQ3&F17$$\"?++++++]PMx[# euU#F1$\"4CKCUqt&y&4&F17$Fcr$\"4%o4&\\B(*)**)4&F17$$\"?++++++DcEs#*RcG CF1$\"4#=97X\"\\?c4&F17$$\"?++++++]i!RSdK*GCF1$\"44V2e]F17$Fbs $\"45f)>$eZ0u8&F17$F\\t$\"4)y,#\\jsr1=&F17$F`u$\"4e[!*H/]dn$\\F17$Fdv$ \"4y2mHBD)Q@WF17$$\"?+++++++]Pf=A_^CF1$\"4/\"ej`TbJQWF17$Fiv$\"4Qq=#=V IRiWF17$$\"?+++++++]7.MGgaCF1$\"4$R&*4E!*=a#[%F17$F^w$\"4\"f\"*R?w\"GJ [%F17$Fcw$\"4q\"))*e.\"QzDTF17$Fhw$\"4$\\\"G$=@TqRQF17$F]x$\"43'e\")=` ^2VOF17$Fbx$\"4`o$f(ym9LH$F17$Fgx$\"41?*[+dq0I F17$Ffy$\"4\\WwNu.3b+$F17$F`z$\"4a>fQ*RaH2IF17$$\"?++++++](=U#[Ww([#F1 $\"4QI(=XLsR9IF17$$\"?+++++++v$4\"3/c)[#F1$\"4:')\\?j\"y\">.$F17$$\"?+ +++++]il(zOc$*[#F1$\"42hy]#p:apIF17$$\"?+++++++]P%yK_,\\#F1$\"4I_/QXES \\7$F17$$\"?+++++++D\"yvCW<\\#F1$\"4%*oEvbZT+7$F17$Fez$\"4u#Q]+h2Q9JF1 7$$\"?+++++++vo/(3G\\\\#F1$\"4ZGFxspo/6$F17$$\"?+++++++]7y1+_'\\#F1$\" 45QItU.NH7$F17$$\"?+++++++Dc^E>6)\\#F1$\"4%zwUNrUU0KF17$Fjz$\"42J!\\#4 /ap]$F17$$\"?++++++D\"y]bB<,]#F1$\"4qU7%H^5T4OF17$$\"?++++++]i:&[iI0]# F1$\"43'e^61%*G1OF17$$\"?++++++vVB:9S%4]#F1$\"4#=E0FL)*4.OF17$$\"?++++ +++DJX.uN,DF1$\"4a1#3MiJ%)*f$F17$$\"?++++++](oa?=%=-DF1$\"4YT-V&=K9$f$ F17$F_[l$\"4!=()\\XZ8I'e$F17$$\"?+++++++v$fy^kY]#F1$\"4:Ni/r.F[d$F17$F d[l$\"4l\"Qat\\!*3&e$F17$Fi[l$\"4?s(\\-;cy4TF17$F^\\l$\"4S#*f^@;Q&[UF1 7$Fc\\l$\"4z:+xVm]lX%F17$Fh\\l$\"4(y:#e\\,`J%[F17$$\"?++++++]i:5D]ADDF 1$\"4$>=KmHxoI[F17$F]]l$\"4*pFtdXHoV[F17$Fb]l$\"4r*[.e\\Z6j[F17$Fg]l$\"4_&=oGA9\\s]F17$F \\^l$\"4q:D\"oKdlQ`F17$Ff^l$\"49/h)yG0e`_F17$F`_l$\"4%\\_(\\Zx;`>&F17$ Fe_l$\"4'ec4`#=?`C&F17$Fj_l$\"4&)339]*))>caF17$F_`l$\"4*\\&eAxwqbS&F17 $Fd`l$\"4L`y`W0;YN&F17$F^al$\"4!*ec_g,PUI&F17$Fhal$\"4[pq\")33f&e_F17$ $\"?+++++++DJ?fTKZDF1$\"4F-f(=(zT2C&F17$F]bl$\"4v=TR.1$[H_F17$$\"?++++ +++vVtL-$*[DF1$\"4LZF17$Facl$\"4Mj&**GB(=LY%F17$Ffcl$\"4U6'z![: [R4%F17$Fjdl$\"4QN*zN/1tpNF17$Fdel$\"4(pE%zMo&RBLF17$$\"?++++++++]iid. %f#F1$\"4=#yRAhC%Rz^?bD>F17$Fcfl$\"4@#*p9Z&HM=:F17$Fhfl$\"4 Ve3=\"GfIe6F17$F]gl$\"3&H$z=t=Zy\")F17$Fbgl$\"3'pKUDr:m>'F17$Fggl$\"3I Zk1D]!=([F17$F\\hl$\"4[:j9'ywZmPF-7$Fahl$\"4a\"yzADrP)=$F-7$Ffhl$\"4:` ]qcTwAb(f*)F-7$Fc_m$\"3sPi&\\laR@*F-7$Fh_m$\"3 B6Rzj41y**F-7$F]`m$\"4Sa@e=p(>B7F-7$Fb`m$\"4xR)\\\"Q-=BQ\"F-7$Fg`m$\"4 OEJstI?QT\"F-7$FN$\"4FUcpGlLw\\\"F-7$F_am$\"3n$GjJ8:p5#F17$FS$\"3.Mk3S 00(*GF17$FX$\"30HPR'y7.C%F17$Fgn$\"3(zJ**\\&og4gF17$F]bm$\"38:fIWl#Qh' F17$Fbbm$\"3;SVlid:fyF17$Fgbm$\"3(zUf$*Hxi%zF17$F\\cm$\"3A>Ey$3-'Q!)F1 7$Facm$\"3;xQ-9\\Cf\")F17$F\\o$\"3W')[92'pM)\\,Ug/9F17$Ffdm$\"4O`+I7mm]X\"F17$F [em$\"4y2B0]zYBt\"F17$Ffo$\"4xitYFA\"z:=F17$Fcem$\"4U7!*HsPP&4@F17$F[p $\"4rg[rWx\"\\dBF17$F`p$\"4-(*yq(oM)G#GF17$Fep$\"4hD7S@*4jtHF17$Fjp$\" 41!3cRP_+iMF17$F_q$\"4r9fz]roF#RF17$Fgfm$\"4cW:aU6Y#)3%F17$Fdq$\"4l8j$ =76p\"Q%F17$Fiq$\"4E7L;o1v)RXF17$F^r$\"4=DB:4Wz>k%F17$Fegm$\"46&Gv.#3y _l%F17$Fjgm$\"4\\z?(Qm%))3m%F17$$\"?++++++DJqXn'*3FCF1$\"4udMwdCv*eYF1 7$F_hm$\"4M9:'y+Q__YF17$$\"?++++++vV)*3Io#yU#F1$\"4T'y'f@od,k%F17$Fcr$ \"4:Nd!fh`Yn\\\"eWWF17$$\"?++++++](=n.?/jV#F1$\"4/SMqLn(RVWF17 $F\\t$\"4@l@a&HokBWF17$Fat$\"4z.n\"eIb?OUF17$Fft$\"4lapN!Rg3_PF17$F[u$ \"4iTcn5&y?!y$F17$F`u$\"4Md-'pSF'p!QF17$$\"?++++++](oz6U?VW#F1$\"4w>FY +.9\"=QF17$Feu$\"4dy!o`z0OCQF17$$\"?++++++voHa(fLbW#F1$\"4RnLyMviQ#QF1 7$$\"?++++++]iSm*)z$fW#F1$\"4c:?1c?<%>QF17$$\"?++++++Dc^y\"QUjW#F1$\"4 1EDy'z#p$4QF17$Fju$\"4DSIb\"HOa\"z$F17$$\"?++++++]P%[\"ebbZCF1$\"4+$\\ W#fm]0s$F17$F_v$\"4&=rNi$3kQd$F17$$\"?++++++]7GjEJ<\\CF1$\"4(>xHrc6_,L F17$Fdv$\"4CKH17!*z*HGF17$F_[n$\"4<,p)3())4B$GF17$Fiv$\"49CBp:x!\\ZGF1 7$Fg[n$\"46b27!HSMdGF17$F^w$\"4a;.3M9aq$GF17$$\"?+++++++](o%\\ModCF1$ \"4:]zPcnJ9q#F17$Fcw$\"4(f&*QgPe:JAF17$$\"?++++++](=b$R,#F17$$\"?+++++++vo/6R**fCF1$\"4G/?Iet0yt\"F17$$\"?++++++]il(z)*y .Y#F1$\"4.(4`4o0nMQ\"F17$$\"?++ ++++]7`WXl`pCF1$\"4OD9rhSB%p5F17$$\"?+++++++DJ??,MqCF1$\"3*41_by!yf!)F 17$$\"?++++++]P4'\\pV6Z#F1$\"3(o)QP6gIp!)F17$$\"?+++++++](=(ps%>Z#F1$ \"3f;J%f0Y\"y!)F17$$\"?+++++++vVB>WbtCF1$\"3(o\\R\"fEP*3)F17$Fbx$\"3y \"=[40!*o/)F17$Fgx$\"3Yp;\"poc`!pF17$F\\y$\"35=_>r`a%[$F17$Ffy$\"3#on3 `6^X[$F17$F`z$\"3$*z\")*RN(**[NF17$F\\^n$\"3=8bWWt\"o&RF17$Ff^n$\"3$H; $*)>2s7aF17$F[_n$\"3SRc02%fUS&F17$Fez$\"3_pOiD-)oR&F17$$\"?++++++](ozr 7KT\\#F1$\"32$)>LjrM,aF17$Fc_n$\"3S2c-(yN%GaF17$$\"?++++++]iS\"p/Cd\\# F1$\"3[\"*pS^gI7bF17$Fh_n$\"3k=87M\"\\2s&F17$F]`n$\"3>XL(R7]$*3(F17$Fj z$\"4&o)z&>')fR!=\"F17$Fe`n$\"4__W)>#Gr)R8F17$Fj`n$\"4e4+N$yCrQ8F17$F_ an$\"4Q0V2#3&GvL\"F17$Fdan$\"4CI\"HD(=?jL\"F17$Fian$\"43mjK+8^QL\"F17$ F_[l$\"4:6$=dtFZJ8F17$$\"?++++++]7yDRx$Q]#F1$\"4Sg'p;nn')H8F17$Fabn$\" 4Ex!HNHd,J8F17$$\"?++++++Dc,;2z20DF1$\"4x$>o[@Z#RL\"F17$$\"?++++++]P4Y 'H\"\\0DF1$\"4=wO^a<\"fR8F17$$\"?++++++v=X5DF1$\"4M5<&=%[I>\\#F17$$\"?++++++v$4rtNl3^ #F1$\"4)e_>k![.')[#F17$$\"?+++++++v=nY(y7^#F1$\"4lC%zcsqB&[#F17$$\"?++ ++++]PMFDb57DF1$\"4*y&puwG1%yCF17$F^\\l$\"4*=*R$G$G0;Z#F17$$\"?+++++++ ]7GIDF1$\"4--4\"p!*HMOKF17$$\"?+++++++]i!fVH,_#F1$\"4Zv`Of L+\"oOF17$$\"?+++++++v$4\"\\\"\\3_#F1$\"4AB<Yg$F17$Fbdn$ \"43j6VvV@Fj$F17$Fb]l$\"4j$>1tNgz+PF17$Fg]l$\"4Ndh=2k#G$3%F17$F\\^l$\" 4!y#o,9&z^iXF17$Fa^l$\"46#zuoV@DEXF17$Ff^l$\"4rzBS&\\6G!\\%F17$$\"?+++ +++]7.#[u_X`#F1$\"4Ysv*y>2-uWF17$F[_l$\"4VeyT3_z;Y%F17$$\"?++++++]PMxk t5ODF1$\"4*3-W$)fM'yX%F17$F`_l$\"4E\\)Q)Q\"GgqWF17$Fe_l$\"4=e.]'RzG8YF 17$Fj_l$\"4YhB03^Wk1&F17$F_`l$\"4GA.,$RTV>]F17$Fd`l$\"4**RtbGF\\F17$Fhal$\"4pbPk/,&p'*[F17$F]gn$\"46&G`B(*>1(*[F1 7$F]bl$\"4yKAc5>F(=\\F17$Fegn$\"4X<^+mNW\\(\\F17$Fbbl$\"4%QhZh+'*\\%3& F17$Fgbl$\"4CYWU(ejak8b%F1 7$Ffcl$\"4#)*=!Q_zG([UF17$Fjdl$\"4poa&o2_IwPF17$Fdel$\"4$Hd.f-Y4FNF17$ F_in$\"4gjg&>mO'[7$F17$Fiel$\"4$z1hU:?\"Hn#F17$Fgin$\"4et_&)p#yXHDF17$ F^fl$\"4uV\\]Pc*[Y@F17$$\"?++++++++v=sx#*=EF1$\"4#3gLJ(zeF%>F17$Fcfl$ \"4tt_kwF'*)H(\\bF-7$Ffh l$\"4jepbAMNEW%F-7$Fdjl$\"4mB2G(=i;]QF-7$Fijl$\"421[q^g-aP$F-7$F^[m$\" 43Ulh4LdH+$F-7$Fc[m$\"4\"*=o8YP_/g#F-7$Fh[m$\"4\"=F[H6\"eeO#F-7$F]\\m$ \"4qUR#yqI0c@F-7$Fb\\m$\"4k6F*y9a$)y>F-7$Fg\\m$\"4tdH5ii\"F-7$Fb`m$!5v;0dvlux5>F-7$FN$!55/*>'*=)GH8t,#z*pJ#F17$FS$!4do+y#F17$FX$!4vFm#)e^eO6$F17$$\"?++++++++vVrZ4>BF1$!4%>1Ap)=;eN $F17$Fgn$!4H&pTH!y:2h$F17$Fbbm$!4c&ee*3*o+uQF17$F\\o$!4+rF\"HNb#*\\SF1 7$Fao$!4N?'3(3Mm^N%F17$Fadm$!4(Q#o]\")[\"*fY%F17$Ffdm$!4\\F4R\\uScc%F1 7$$\"?++++++++v=EjOdBF1$!4.H!*QTVufd%F17$$\"?++++++++]Pr%y\"eBF1$!4bLB [G`w1uT%F17$$\"?++++++++v$p!\\hgBF1$!4Ux q(*Q058V%F17$$\"?+++++++]7`z4-hBF1$!4OlE+;6b_W%F17$$\"?++++++++]7_qUhB F1$!4\"H)zf\"G:CfWF17$$\"?++++++++DJ(>RAO#F1$!4B&[f-Q'Gt[%F17$Ffo$!4m' z5Q`s\\:XF17$$\"?+++++++++DZmhkBF1$!4.Zm]\"fR_oXF17$$\"?++++++++++_>=m BF1$!4K&pS7lh55YF17$$\"?++++++++]P/Y'pO#F1$!47$QSP7W&oh%F17$$\"?++++++ +++vcsunBF1$!4')eMC524Hg%F17$$\"?++++++++]74*H&oBF1$!4%p'*R**)ppKb%F17 $Fcem$!4Jv]mR))p^W%F17$$\"?++++++++vo()QqpBF1$!4f!eoi:yGfVF17$$\"?++++ ++++](Q@&4qBF1$!4?gSG?&o2zUF17$$\"?++++++++D1Sl[qBF1$!4oy)[Lfz'>H%F17$ $\"?+++++++++Dmy(3P#F1$!4uJl-1U#*[I%F17$$\"?++++++++]i=0mrBF1$!4)\\A*= D%*Q3L%F17$$\"?++++++++++rJWsBF1$!4(3KR@E)))oN%F17$$\"?+++++++++vv%3SP #F1$!4,S%p&)GRR3WF17$F[p$!4SwcB.6$\\^WF17$$\"?+++++++]P%Qn;%yBF1$!4*fJ Mbk4&oN%F17$F`p$!48!*3u1ifJ'RF17$$\"?+++++++v$fQ,\"o#Q#F1$!4b\")f#)eo1 e+%F17$$\"?+++++++]7`gC5%Q#F1$!4`pXGvf0t/%F17$$\"?++++++](=nQ=8[Q#F1$! 4T\"*Qb1xhb1%F17$$\"?+++++++DJ?2R_&Q#F1$!4U[)*))o?!3zSF17$$\"?++++++]i !R0jMiQ#F1$!4wq))Rom;J3%F17$Fep$!4KZAJ=vchT$F17$Fjp$!4eN8\\CtJxX$F17$$\"?+ +++++]il(HU^XR#F1$!4'>1!=/h2hZ$F17$$\"?+++++++DcwTlR&R#F1$!4,(=N%*eFd) [$F17$$\"?++++++Dc,;,\">eR#F1$!4iyG<>)QO!\\$F17$$\"?++++++](oa0mTiR#F1 $!4%>kD#*puG([$F17$$\"?++++++v=#\\*>Um'R#F1$!4q)>$F17$$\"?++++++]P4rN@i*R#F1$!4A J>XG)f!z'GF17$F_q$!40eg$>**3DcEF17$$\"?++++++++vV2\"**=S#F1$!4LGl&>$z9 Ko#F17$$\"?++++++++]Pg4L.CF1$!4'>`)4Q:A)4FF17$$\"?+++++++]P%o)o//CF1$! 4q1Rx;gp?s#F17$$\"?++++++++DJ8Gw/CF1$!4'G^KD0(>>t#F17$$\"?+++++++]7yR( yaS#F1$!4)*y:1_uAkt#F17$Fgfm$!4J)4ffF#e-t#F17$$\"?++++++++]7s$e!4CF1$! 4Sg`[ubx=L#F17$Fdq$!4ACfW/X-)*)=F17$$\"?+++++++v$fLm(e8CF1$!42)fkRfy\" *4>F17$$\"?+++++++](=([KD:CF1$!4EE5\"Gv+i@>F17$$\"?+++++++D\"yS$)=pT#F 1$!4xaA#>:>(f)=F17$Fiq$!4Ws,sx'pRx;F17$$\"?++++++](=<@@<%>CF1$!402ElCX @@U\"F17$$\"?+++++++voz/+D?CF1$!4G#Ri2)*)z?9\"F17$$\"?++++++]ilZ(z#3@C F1$!4\"4z\"p1@Jz9\"F17$$\"?+++++++]i:!f:>U#F1$!40F(*4K`dP:\"F17$$\"?++ +++++Dc^v6eBCF1$!4J*y@CO,%[;\"F17$F^r$!4ro3+bv%Rp6F17$Fcr$!4)=a@9@fGM5 F17$Fhr$!3@UE@'3(z`dF17$F]s$!3Iy$=Uur\"*z&F17$Fbs$!3g-9jFQvNeF17$F_ho$ !3%HZa;'**))QeF17$Fgs$!3l![A5J?V\"eF17$Fgho$!37@\"fL7^`t&F17$F\\t$!3W5 cze\"G#ebF17$F`u$!3X/6#HXf<*>F17$Fdv$\"2,:07I,:b#F17$Fiv$\"2+HgUSGU'GF 17$F^w$\"2\\d;*fW&fY$F17$Fcw$\"3^`4!Q\"f!p;\"F17$Fhw$\"3\"ft>w+A7z\"F1 7$F]x$\"3Y.kro4&HF#F17$Fbx$\"3!Gi\\&))po[HF17$$\"?+++++++DJqLujwCF1$\" 3,,!oG!*QX)HF17$Fgx$\"3qXbx*f,v3$F17$$\"?++++++]7G8J7&)yCF1$\"3oKvw0$ \\[=$F17$$\"?+++++++v$4O;*ezCF1$\"3(>E7EeBeK$F17$$\"?++++++]Pf3'4F.[#F 1$\"3BnvaJX&pU$F17$F\\y$\"3!)>Y:W8iFMF17$Ffy$\"3z=/MS'3xU$F17$F`z$\"3) *f8S-A)RR$F17$Fg]n$\"3av8+[&R\"[LF17$F\\^n$\"3AER=-9/aKF17$Fa^n$\"3T]L HJP#G2$F17$Ff^n$\"3i%[S$42t?GF17$F[_n$\"3RI?4Y9I;GF17$Fez$\"3\"))4)Q8% [/\"GF17$Fc_n$\"3Om/j[SB&z#F17$Fh_n$\"3Y-vQ&R)RCFF17$F]`n$\"3<2=RZ9,HC F17$Fjz$\"3#=\\r;jF@W\"F17$F^\\l$!3Qn$*>L>j77F17$Fh\\l$!3LT8BKWz3QF17$ Fecn$!3tJavUixYMs)oRF17$Fb]l$! 3ixE1x#oL?%F17$Fg]l$!3<+$f:cOkZ&F17$F\\^l$!3'ReUQ&o:;rF17$Fa^l$!3)[TZe Xr'fqF17$Ff^l$!3N&oZdgzo+(F17$Fa_q$!3p#Q;I^B5*pF17$F[_l$!3N@mqa6t)*pF1 7$Fi_q$!3nY\\w#p=)eqF17$F`_l$!3bjd1``jBsF17$Fe_l$!3w'*Q#f$4&oL)F17$Fj_ l$!4Qx!>bkGkw`v\"F17$$\"?+++++++D1*o6!4_DF1$!4@@@(R(3ml u\"F17$$\"?++++++++v=Kd(Gb#F1$!4M_ZX/t\\yt\"F17$$\"?+++++++]7yipWaDF1$ !4s9UQfN+Gs\"F17$Fgbl$!4a0DF`,Rbs\"F17$$\"?++++++++Dca1;fDF1$!4op%3.10 n9@F17$F\\cl$!4#*>oW\"=tv\\CF17$$\"?+++++++D19iL#Qc#F1$!4[m`,;Kh_U#F17 $$\"?+++++++]7`3OMlDF1$!4'=8K$QtF*4CF17$$\"?++++++]ilsJP5mDF1$!4(pRo!3 wxWT#F17$$\"?+++++++v=#\\&Q'oc#F1$!4nM#zGnt6QCF17$$\"?++++++](=<\"yRin DF1$!4)z\"eGF17$$\"?+++++++DJqZV!*pDF1$!4ugKt>Z!>pJF17$$\"?++++++D\"y+$4 WGqDF1$!4-Qh4JY]+A$F17$$\"?++++++]P%)*3Zk1d#F1$!4G,Lf5eU8@$F17$$\"?+++ +++v$4'\\KX/rDF1$!4o(yzg2ek-KF17$$\"?+++++++]P4%fC9d#F1$!4yG;bq2hR>$F1 7$$\"?+++++++vV[S[%Hd#F1$!4#G\\o=@a]fJF17$Ffcl$!4RJF;)yk0GJF17$$\"?+++ +++]Pf3u!4`d#F1$!4LZ(eG)Q\\k6$F17$F[dl$!4kNJ[qpVl6$F17$$\"?++++++]7y][ q*pd#F1$!40>L.V:/29$F17$F`dl$!4pZLb;h(45KF17$Fedl$!45')>[]ft$QOF17$Fjd l$!4$oNdC7B#Q%QF17$$\"?+++++++vV)*e\\!He#F1$!4&y;.aG5R'z$F17$F_el$!4[F lGK+0Dv$F17$$\"?++++++](=<1#pV&e#F1$!4)Hi4/<&=nt$F17$$\"?+++++++D\"Gy! 4G'e#F1$!4Csjg?m;Dt$F17$$\"?++++++]i!R]*[7(e#F1$!45bfx5VQ;v$F17$Fdel$! 4d*4j\"4lrQ\"QF17$F_in$!4)*H$3IcyrMUF17$Fiel$!4*ote!oI+6k%F17$$\"?++++ ++++](y[TLg#F1$!4#*>r%GVz)*GXF17$Fgin$!4moDE7w\\BW%F17$$\"?+++++++v$f$ )=&)pg#F1$!4_&)*>.mDrSWF17$$\"?+++++++](oR/!R2EF1$!4Wa3)y#)4&RW%F17$$ \"?+++++++D\"y&**[z2EF1$!4<4ZHn,(e`WF17$$\"?++++++++v=b(*>3EF1$!4%fMWc tv\\rWF17$$\"?+++++++]iSm%4!4EF1$!4X#))pCQ!p) 4EF1$!4vsJCDWe\"zYF17$$\"?+++++++vVBLSA5EF1$!4))yaV-.UUr%F17$$\"?+++++ ++]P%))))G1h#F1$!4Hn$G^xx!)*p%F17$$\"?+++++++DJXWP.6EF1$!4uv$*QF_=ao%F 17$$\"?++++++++D1+'Q9h#F1$!4aX2rYF17$$\"?+++++++]7G6$[Ah#F1$!4S1F MTdGDk%F17$F^fl$!4z;t@yL;Uh%F17$F]dq$!45\\y'*)QC\\FXF17$Fcfl$!4C,GW\" \\r-hWF17$Fhfl$!4j[w'y_SQ\\TF17$F]gl$!4k-?J)*f+)>PF17$Fbgl$!4Yd0!)oomd C$F17$Fggl$!4V%)Qg`ZwT\"GF17$$\"?++++++++v=_D8\"o#F1$!5nH=E*[`Wn^#F-7$ F\\hl$!5%GDwY7dtAQ#F-7$Fahl$!5?5]gXu4&z8#F-7$Ffhl$!5fnUXBZ5U&)=F-7$F[i l$!5@7o!G5`Lj%=F-7$F`il$!5\"R^tR-PK;!=F-7$Fjil$!5NM_iV%)o%)e;F-7$Fdjl$ !5,)>G%oiQh'e\"F-7$Fijl$!5S!4nH/&pjL8F-7$F^[m$!5[eq=%)HzHC6F-7$Fc[m$!4 3^:x`2)RM()F-7$Fh[m$!4@SFAlXmi^(F-7$F]\\m$!4#QsIL$eqm]'F-7$Fb\\m$!4d%* H%p4I$))p&F-7$Fg\\m$!4F8rw`lzT&[F--F\\]m6&F^]mFd]m$\"#vFa]mFb]m-Fg]m6# %TButcher's~scheme~B~with~c[5]=c[6]=1/2~and~b[5]=b[6]G-F$6%7gp7$F($!3. *RUhC+%R)*F-7$F/$!3t(4w&*>!=h$*F-7$F5$!3oV&>#[>b^&)F-7$F:$!3M$egZgiMK( F-7$F?$!3TpoDiy3$z$F-7$FD$!3(R,=pt;&[8F-7$FI$\"31y)G)4nXL>F-7$Fc_m$\"3 M4c)4*3#\\4#F-7$Fh_m$\"3QHOE'p'f1FF-7$F]`m$\"3(y0n*)>+Qi%F-7$Fb`m$\"32 MP^#=W4&fF-7$Fg`m$\"3;(eeuEs14'F-7$FN$\"3k(zQu$fPhmF-7$F_am$\"3U1p5Pa% )R6F17$FS$\"3BA#fZd@ss\"F17$FX$\"3\"GB_3qKPn#F17$Fgn$\"3uC^4p$p(yQF17$ F\\o$\"3N%*z.W)e;Y&F17$Fao$\"37mMK)f$e%*))F17$Ffo$\"4^_PB,Nrh<\"F17$F[ p$\"4-!oqlb1DG:F17$F`p$\"4m&)>iNH;M%=F17$Fep$\"4m-)[W7oiW>F17$Fjp$\"4r mveh&Q=.BF17$F_q$\"4QBCo%*)e4/FF17$Fgfm$\"4a)*pQ8W>S#GF17$Fdq$\"4#>vUl f0/5KF17$Fiq$\"4T0*3IEH+RMF17$F^r$\"4CA'QixaNdPF17$Fhr$\"4Me)*fq2^45%F 17$F\\t$\"4U\"et\"Ru+GC%F17$Fdv$\"4,2D$o2RVmZF17$Fhw$\"4[(H;0Jk![.&F17 $F]x$\"4SQ)RlwRGA^F17$Fbx$\"4?rN3\")**)y-_F17$F]bs$\"4CASJR)Rz5_F17$Fg x$\"4(ziecr-&GA&F17$Fjbs$\"4N#o]m794^_F17$F\\y$\"4Ktabw!4xm_F17$Ffy$\" 4zgj^-/$Qn_F17$F`z$\"4QZ(*G0)ohk_F17$Fg]n$\"4!3q(*Qcqcl_F17$F\\^n$\"4X 0o#\\Vyip_F17$Fa^n$\"4Vuq;.&\\kz_F17$Ff^n$\"4afY9!H3V%H&F17$F[_n$\"4,X EP:,GhG&F17$Fez$\"44*zvd2IBw_F17$Fc_n$\"481;,\\Ne[E&F17$Fh_n$\"4ed#[wt x$GD&F17$F]`n$\"4C9')p4vHSC&F17$Fjz$\"4m-Id6=_6D&F17$Fd[l$\"4H^QXUK:r< &F17$F^\\l$\"4o**f8Xn(z\"4&F17$Fh\\l$\"4lM01*f&fk%[F17$F\\^l$\"4s<\"Qq np;PYF17$F`_l$\"4p::NU\"f7xWF17$Fd`l$\"4&fB\\drQ(**>%F17$Fbbl$\"4+\"zk &4E;x)QF17$Fgbl$\"4fL(z2n&Rip$F17$F\\cl$\"4'fdk7jb\\JLF17$Facl$\"4ZhV< ')pDD7$F17$Ffcl$\"4,uoPkPyI#GF17$Fjdl$\"4DS&pa2!4\"[CF17$Fdel$\"4oMLF[ ,)4\"G#F17$F_in$\"49I?o8EFF+#F17$Fiel$\"4SR#QWQ(\\vq\"F17$Fcdt$\"45,&4 e\\V8m;F17$Fgin$\"4B;+th;Qfh\"F17$Fjet$\"4MK)H\")fH_g:F17$Fdft$\"4q3Jx sy&QG9F17$Fift$\"4c.'Q$*oxi-9F17$F^gt$\"4*>3PvYJL)R\"F17$Fcgt$\"4nu`YF )=0%R\"F17$Fhgt$\"4>N#z?BQy*Q\"F17$F]ht$\"4v>&\\M9OG\"Q\"F17$F^fl$\"4, 1G?nG0GP\"F17$Fcfl$\"48)>-7'o@\"46F17$Fhfl$\"3)p\")H.9g)H()F17$F]gl$\" 3NRE^qGzzkF17$Fbgl$\"3-W0g&z?o4&F17$Fggl$\"3uW!)4;RwITF17$F\\hl$\"4WK( HF-7$Fijl$\"46C* fi'zGqj\"F-7$F^[m$\"4qx0)*3A4kM\"F-7$Fc[m$\"3W\\i:9:p!*)*F-7$Fh[m$\"3- Y^oS$Rr;)F-7$F]\\m$\"3?VeStF#zp'F-7$Fb\\m$\"3tYD@fH2WaF-7$Fg\\m$\"3(H' Q%4BR,&QF--F\\]m6&F^]mF_]mF_^oFd]m-Fg]m6#%Hscheme~with~c[5]=c[6]=3/4~a nd~b[5]=b[6]G-%%FONTG6$%*HELVETICAG\"\"*-%+AXESLABELSG6$Q\"x6\"Q!F^_v- %&TITLEG6#%Uerror~curves~for~7~stage~order~6~Runge-Kutta~methodsG-%%VI EWG6$;F(Fg\\m%(DEFAULTG" 1 2 0 1 10 0 2 9 1 4 2 1.000000 45.000000 45.000000 0 1 "Butcher's scheme A" "scheme with simple nodes" "scheme \+ with a relatively large stability region" "Butcher's scheme B with c[5 ]=c[6]=1/2 and b[5]=b[6]" "scheme with c[5]=c[6]=3/4 and b[5]=b[6]" }} }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 518 "evalf[30](plot(['pn_RK6_1'(x)-p(x),'pn_RK6_2'(x)-p(x),'pn_RK6_3 '(x)-p(x),'pn_RK6_4'(x)-p(x),\n'pn_RK6_5'(x)-p(x)],x=2.8..5,font=[HELV ETICA,9],\ncolor=[COLOR(RGB,.95,.2,0),COLOR(RGB,.45,0,.95),COLOR(RGB,0 ,.25,1),COLOR(RGB,0,.75,.2),COLOR(RGB,.95,.45,0)],\nlegend=[`Butcher's scheme A`,`scheme with simple nodes`,`scheme with a relatively large \+ stability region`,`Butcher's scheme B with c[5]=c[6]=1/2 and b[5]=b[6] `,`scheme with c[5]=c[6]=3/4 and b[5]=b[6]`],title=`error curves for 7 stage order 6 Runge-Kutta methods`));" }}{PARA 13 "" 1 "" {GLPLOT2D 844 430 430 {PLOTDATA 2 "6+-%'CURVESG6%7ds7$$\"#G!\"\"$!5^n^dc4(>&))H! #I7$$\"?MLLLLLLLLepo(R#G!#H$!5C4V^ImU)pk#F-7$$\"?nmmmmmmmm;RP&z%GF1$!5 `%>hVqtT`M#F-7$$\"?++++++++v=&)e\")oGF1$!5O\\R7!e?6z6#F-7$$\"?LLLLLLLL $37.y'*)GF1$!5Hh(yGL>3.$>F-7$$\"?++++++++DJ'pRJ\"HF1$!5?))fFU:*45y\"F- 7$$\"?nmmmmmmmmTh8gOHF1$!5$HlpJV*e$)z;F-7$$\"?nmmmmmmm;a`*4%[HF1$!5@t* )*RN#R>U;F-7$$\"?nmmmmmmmmmX&=-'HF1$!5wYj=*HR7?h\"F-7$$\"?nmmmmmmm;zPr -sHF1$!5sj)p1sH\\of\"F-7$$\"?nmmmmmmmm\"*Hd$Q)HF1$!5Z\"y?K%fN.!f\"F-7$ $\"?+++++++D1*Q'>r*)HF1$!5c)yc*z^lu*e\"F-7$$\"?MLLLLLL$eky>)e&*HF1$!5Z \"[+A3=r7f\"F-7$$\"?nmmmmmmT&Q=Vk9+$F1$!5Q[teOA)Hdf\"F-7$$\"?++++++++D \"emSt+$F1$!5OY)*>\\z/e+;F-7$$\"?mmmmmmm;/wLJ4>IF1$!5vN:Sv**=B>;F-7$$ \"?LLLLLLLL$3N>F-7$$\"?++++++++](Gle&>JF1$!5.!>\\'QUpX(4#F-7$$\"?MLLL LLLL$3-$[*G9$F1$!5C(**>+V.C3H#F-7$$\"?nmmmmmmm;a25BmJF1$!5]^=O^[\\T8DF -7$$\"?MLLLLLLL$eaM#\\*=$F1$!5UBg&HTm8dx#F-7$$\"?++++++++]P$o`F@$F1$!5 \"y>(Q!Qorb1$F-7$$\"?mmmmmmmmTgG3oOKF1$!5'>?=,5\\rHR$F-7$$\"?LLLLLLLLL $Q(zggKF1$!5ekL1QeAPePF-7$$\"?++++++++]P;Io\"G$F1$!5!o[^_He(Q#4%F-7$$ \"?nmmmmmmmm\"*e!eFI$F1$!5F3+)34zh>V%F-7$$\"?MLLLLLLL$3_c$[ELF1$!5m`R+ b>\"GI![F-7$$\"?+++++++++]r!4-N$F1$!504lDzn@w8^F-7$$\"?++++++++++(*>.u LF1$!52mzM<\"4xAQ&F-7$$\"?+++++++++]A\\&yR$F1$!5Zh-#>ad&F-7$$\"?+++ ++++++]DF\"3U$F1$!4,ww!=NiXfcF17$$\"?+++++++++]G0xVMF1$!4HkV(z#oy8t&F1 7$$\"?MLLLLLL3-j*\\#)*[MF1$!4fR=-L@06u&F17$$\"?nmmmmmm;/wqW>aMF1$!4vmv x+z#*fu&F17$$\"?+++++++D1*=W1%fMF1$!40cpF%[s8\\dF17$$\"?MLLLLLLL3-8%=Y Y$F1$!4!*R.\"o%osw/xNF1$!5Ncqv?y$Go<&F-7$$\"?mmmmmmmm Tg/iZ,OF1$!5du*\\/p?R%G\\F-7$$\"?++++++++]()*y/fi$F1$!52kmrbmdCrYF-7$$ \"?mmmmmmmmT5F#Gvk$F1$!5lcIytZp9_WF-7$$\"?LLLLLLLLLLk;:pOF1$!5rC,**zR] W[UF-7$$\"?mmmmmmmmT5-g(Gp$F1$!5b\"Qb\")eHjq/%F-7$$\"?++++++++]()R.g;P F1$!5\\\"\\A&HB-FrQF-7$$\"?++++++++]i*p#yhPF1$!5&fC!\\!3Z/Mh$F-7$$\"?m mmmmmmmT&eAa`y$F1$!5J8%))HG&HS>NF-7$$\"?LLLLLLLLL3_d#*3QF1$!5D!z`fTs%G `MF-7$$\"?LLLLLLL$eRQF1$!54Wft\")Q35KMF-7$$\"?LLLLLLLLeRr&ed6Ot/MF-7$$\" ?++++++++D\"o%fcvQF1$!5X\\)R)[lWq7MF-7$$\"?MLLLLLL$ek[-Ss)QF1$!5@ion@( *\\rEMF-7$$\"?nmmmmmmmm\"H59*)*QF1$!5K\"\\#*H!>I^YMF-7$$\"?nmmmmmmm\"H _FmJ#RF1$!5X,QTb2Cf0NF-7$$\"?nmmmmmmm;aZ%=u%RF1$!5/M\"pg4-ise$F-7$$\"? ++++++++]7A;k*)RF1$!5h!eW#R\\cGuPF-7$$\"?nmmmmmmmmT2QCNSF1$!5FuTZIh?o> SF-7$$\"?++++++++++F`N#3%F1$!5nG@pZ7:otUF-7$$\"?++++++++]PU+S0TF1$!55F ?uhXO%zP%F-7$$\"?+++++++++vdZWGTF1$!52.P0)e2QsX%F-7$$\"?+++++++]7GJKfR TF1$!5e\"o#>eBz:%[%F-7$$\"?++++++++D\"[qT2:%F1$!5q&fX*yE#4^%F-7$$\"?+++++++]PMy,*=;%F1$!5F5uGH&\\c `^%F-7$$\"?+++++++v$4^Tku;%F1$!5'o_P&HN0,=XF-7$$\"?++++++++](=lQI<%F1$ !5GZ*yzQ&*R&=XF-7$$\"?+++++++v=U\"*yAzTF1$!5l(z)=V3-s;XF-7$$\"?+++++++ ](o48%F1$!5)>6T$y.(fd]%F-7 $$\"?++++++++D15cz(>%F1$!5z!)*Rkzu0X\\%F-7$$\"?+++++++]i:*3u,@%F1$!5N6 \"R31'***4Z%F-7$$\"?+++++++++DoDbAUF1$!5CUEn\"zl&*GV%F-7$$\"?mmmmmmmm; H'z(zWUF1$!5e\"Q&oJF-7$$\"?nmmmmmm;H#eaY#*f%F1$!56\"fWd2g-0'>F-7$$\"?M LLLLLLL3x%H`1h%F1$!517Y78Woo]>F-7$$\"?+++++++](=P/g?i%F1$!5cvJv=I(*G[> F-7$$\"?nmmmmmmmmm#zmMj%F1$!5d,!QlU>xF&>F-7$$\"?nmmmmmmm\"z4%=8XYF1$!5 UTLg%oJFZ'>F-7$$\"?nmmmmmmm;H*)ozcYF1$!5q+)3OOOfY)>F-7$$\"?nmmmmmmmTgP >YoYF1$!5&GE4168vR,#F-7$$\"?nmmmmmmmm\"f)p7!o%F1$!5>8:c!o&Q&f0#F-7$$\" ?++++++++Dc(*QE.ZF1$!5&[k6tQe!fe@F-7$$\"?LLLLLLLL$3#43SEZF1$!5to2y'fGD ?I#F-7$$\"?mmmmmmmmT5G7mZZF1$!5GqJNHJ\"RvZ#F-7$$\"?++++++++++Z;#*oZF1$ !534;))RT?g)p#F-7$$\"?mmmmmmmm;H'[)G$z%F1$!5JQTxsU5^-IF-7$$\"?LLLLLLLL LeD`l<[F1$!53eFA:dy4^LF-7$$\"?++++++++]7<$\\%R[F1$!5qF[eAz#*\\&o$F-7$$ \"?nmmmmmmmmm3LCh[F1$!5<%487>^\"*p+%F-7$$\"?MLLLLLLL3F\\-[%)[F1$!5!H*p u6!zDgM%F-7$$\"?++++++++]()*=8Z_%F-7$$\"?++ ++++]PfLCt)=\"\\F1$!53u=v7Lz`iXF-7$$\"?+++++++]i:ptF8\\F1$!5&)=_@3-Mq+ YF-7$$\"?++++++]il(RTnY\"\\F1$!5.OBKT=)*RQYF-7$$\"?+++++++vozeu0;\\F1$ !5W>QTdt1\">n%F-7$$\"?++++++](=,7)*o%F-7$$\"?+++++ +++vV[v$)=\\F1$!5#R=,2%>&)[kYF-7$$\"?++++++]7yD$fF-#\\F1$!5&*e]v;vpPwX F-7$$\"?+++++++D\"y!Qwh@\\F1$!5M.tvZ8mR:YF-7$$\"?++++++]P%)*Go2I#\\F1$ !5.M\"p^@L+[l%F-7$$\"?+++++++](=xs(RC\\F1$!4wbT#)RgtRp%F17$$\"?++++++] i!RDx(yD\\F1$!4t_t]b(>yHZF17$$\"?+++++++v$ft\"y,ZF17$$\"?++++++++]7C') >_\\F1$!4R#>T%[Q6,![F17$$\"?+++++++]P4o*[T'\\F1$!48['>\">LN(e\\F17$$\" ?++++++++D17$*4w\\F1$!498.N<+!zo^F17$$\"?+++++++]7.c'\\!))\\F1$!4sU*fG ?a`TaF17$$\"\"&\"\"!$!4'zMts(oQCy&F1-%&COLORG6&%$RGBG$\"#&*!\"#$\"\"#F *$FjemFjem-%'LEGENDG6#%3Butcher's~scheme~AG-F$6%7bp7$F($\"3!f(*pnfDC7& F-7$F5$\"3Oz\"p>cJ+%HF-7$F?$\"3rm.KN1&Gs\"F-7$FI$\"3/xDL#HEU8\"F-7$Fgn $\"3J1*G'3hEB5F-7$Fep$\"3jC$=\"=$[xK\"F-7$F_q$\"3.F?KVP/Z**)>X,\"F-7$$\"?+++++++++vfM%fQ$F1$\"3S')* 3*=*fr%)*F-7$Feu$\"3e1B@7dU#Q*F-7$$\"?++++++++++CQL4MF1$\"29T!zI=NA()F 17$Fju$\"26k(y^*HZ;)F17$$\"?++++++++++F;HKMF1$\"2'>)oyZZ:!zF17$F_v$\"2 5A=XLDqu(F17$Fiv$\"2mAydxsZr(F17$Fcw$\"2R=w7/sY\"yF17$F]x$\"2xEtp#pEM! )F17$Fgx$\"2\"G)f&3*e#\\$)F17$F\\y$\"2vU5zXt=c*F17$Fay$\"3A8oC5o$e,\"F 17$Ffy$\"4%\\\\g,4$fv/\"F-7$F[z$\"4:%oLGkL*pF-7$Fh\\l$\"3Em>s4GKQmF-7$F\\^l$\"3%zvmsR:ed'F-7$F`_l $\"3h9W%R7fJx'F-7$Fj_l$\"3%[))4b-c-?(F-7$F_`l$\"3$>3Pi)GR(o(F-7$Fd`l$ \"3n0Ez\\+A$>)F-7$Fi`l$\"3[]'R/iH;X)F-7$Fcal$\"3h)pr5b#fX#)F-7$Facl$\" 3#z))oS<\"**GwF-7$F_el$\"3_ikmOyDXtF-7$Fiel$\"3KP4xc\"Q?b(F-7$F^fl$\"3 x:Hx4\"Ruw(F-7$Fhfl$\"3V_bu%3?-Q(F-7$Fbgl$\"30J!H\"=M0\"f'F-7$F\\hl$\" 3l]\"[M1[G$eF-7$Fahl$\"3'[h6w-R1=&F-7$Ffhl$\"3wiQSJ6lGYF-7$F`il$\"3g% \\II:H;>%F-7$Fdjl$\"3p!z)>gK*[&RF-7$Fh[m$\"3XZoLe`)*QTF-7$Fb\\m$\"3W!Q sp(4FI[F-7$F\\]m$\"3IF+V7!Rp6'F-7$Fa]m$\"3X\"**o34l:@(F-7$Ff]m$\"3$=/6 s^7Ne)F-7$F[^m$\"4#[J/()4hL^5F-7$F`^m$\"4$R)o,g*))HC8F-7$Fe^m$\"4yA\"* =9rM$>yIt_2+IO#F-7$Fbam$\"44?\\:,ih^'GF-7$Fjcm$\"3NPNYHD $Gj$F17$F_dm$\"3:]>p&\\%e*R%F17$Fddm$\"3W]v1$)GEV`F17$$\"?+++++++v$4hz t\"e\\F1$\"336tw1JmLdF17$Fidm$\"3,fSe)3;zaIUt(F17$F^em$\"3)QhOG1h[2)F17$$\"?++++++](oa!)*o3z\\F1$\"3p Xwa#)R\\g()F17$$\"?+++++++vo/%[u?)\\F1$\"3a`(4bxQ?[*F17$$\"?++++++]i!R +2i])\\F1$\"3(e*)*>gJBz'*F17$Fcem$\"4'pK\"Rb()eO,\"F17$$\"?++++++]PM-U s.\"*\\F1$\"4xq)ov8i-N6F17$$\"?+++++++Dc,G[-%*\\F1$\"4!ySCy:L_d6F17$$ \"?++++++v=<,@'=b*\\F1$\"44wX(Rm#\\2<\"F17$$\"?++++++]7y+9C,(*\\F1$\"4 MJDH0Bq<>\"F17$$\"?++++++D1R+2i])*\\F1$\"46s,t-qSwB\"F17$Fhem$\"4z95.! fJ(zM\"F1-F^fm6&F`fm$\"#XFcfmFffmFafm-Fhfm6#%9scheme~with~simple~nodes G-F$6%7_p7$F($\"4Q#*F-7$Fep$\"31'zbt4SJ!)* F-7$F_q$\"4l/xvav(H#3\"F-7$Fiq$\"4BZ^)*>%\\dH7F-7$Fcr$\"4j1e#*)\\^#*H9 F-7$F]s$\"4.H&\\_\"pWNr\"F-7$Fgs$\"4q0jCf1Zf7#F-7$Fat$\"4CNl]A'of>DF-7 $Ffim$\"4A?obJ4/qh#F-7$Fft$\"4GQr$exyD,FF-7$F^jm$\"4Vk?Yu;r4s#F-7$Fcjm $\"4N$obYZ1!)fFF-7$Fhjm$\"4@r6*[-AUvFF-7$F[u$\"4sr1w]ALCw#F-7$F`[n$\"4 SuX^'[Cz!y#F-7$F`u$\"4X'R4#fl!QzFF-7$Fh[n$\"4+sz+q'f(yv#F-7$Feu$\"4go1 L*eUd3FF-7$F`\\n$\"3&=8\\=i\"[CEF17$Fju$\"3@gO?*QV+]#F17$Fcw$\"3eL*Rr%Gq 4DF17$F]x$\"3*fAg,]VK`#F17$Fgx$\"3\\$*\\?%3Qwc#F17$$\"?MLLLLLL$e9JPhy \\$F1$\"3w60*o#fV:EF17$F\\y$\"3nd\\k`O02FF17$$\"?mmmmmmm;/EC:lANF1$\"3 iS]qDUAPFF17$Fay$\"3.:o4wwabFF17$$\"?LLLLLLLL$3Z&oaXNF1$\"4\"R&Q*[))\\ boFF-7$Ffy$\"4/5?Ly'GRpFF-7$$\"?LLLLLLLL$eWOZlc$F1$\"4$oFD$R`!QeFF-7$F [z$\"4r(G(=lU'fOFF-7$F`z$\"4CG6NVwG.m#F-7$Fez$\"4B#\\eoP%*eUDF-7$F_[l$ \"4Z=qkQaS&RBF-7$Fi[l$\"4()4^]su6*\\@F-7$F^\\l$\"4D*z\"yZF+y,#F-7$Fh\\ l$\"4@bjZ$o\\&R$>F-7$F\\^l$\"4#Hq)*4ptZ8>F-7$F`_l$\"4fttCBV#G[>F-7$Fj_ l$\"4RO[WS1q`.#F-7$F_`l$\"4S.VN1xPt8#F-7$Fd`l$\"4t)z?(e(*\\jD#F-7$Fi`l $\"4!>?zV=H'GM#F-7$Fcal$\"4#*GSTTuh$fBF-7$Facl$\"4bWfJQYhxH#F-7$F_el$ \"4_RZaA]oZC#F-7$Fiel$\"4qD(fd1Vf=AF-7$F^fl$\"4:E%fJB'[V;#F-7$Fhfl$\"4 UqI5;szh-#F-7$Fbgl$\"4(ejn#4WWO$=F-7$F\\hl$\"4tm3Cg:d$f;F-7$Fahl$\"4*[ W\"H7#QF3:F-7$Ffhl$\"4lM'e?:wL!R\"F-7$F`il$\"4aQY&)Qsz?I\"F-7$Fdjl$\"4 [zm[,WxzE\"F-7$Fh[m$\"4$*[Ufj^auJ\"F-7$Fb\\m$\"4*[)3wX#f*3Z\"F-7$F\\]m $\"4N'4g(zxm.t\"F-7$Ff]m$\"4oKck#pxK%>#F-7$F`^m$\"4EEA%R3g,**GF-7$Fe^m $\"49E#Rm`%G8X$F-7$Fj^m$\"4[y\\qp\"otlUF-7$Fbam$\"4Pf;Qjyqw%[F-7$Fjcm$ \"3(pM#3//9(o&F17$F_dm$\"3F0a06%>Wb'F17$Fddm$\"3\"4c$H&eA#3wF17$Fgen$ \"3Jvro,NMv!)F17$Fidm$\"3BZPtsF?[*)F17$F_fn$\"4310WONtu,\"F17$F^em$\"4 -^/XpU[51\"F17$F\\gn$\"4(G)e>D!fH17F17$Fcem$\"4Sv+&\\V+\\!G\"F17$Fign$ \"4!3gq+G,k,9F17$F^hn$\"4h&*z%\\l,NH9F17$Fhhn$\"4.$H&e3<[\"o9F17$Fhem$ \"4ZJ()eC?;4i\"F1-F^fm6&F`fmFffm$\"#DFcfm$\"\"\"Fjem-Fhfm6#%Pscheme~wi th~a~relatively~large~stability~regionG-F$6%7bp7$F($!4F8rw`lzT&[F-7$F5 $!4qAH.N\\.Oc$F-7$F?$!4p,XteZqq#HF-7$FI$!4z:hF#otR7 iN#F-7$FS$!4Q(pr#*zG[bAF-7$FX$!4To#F-7$F_q$!4:j_dLI ]mF#F-7$Fiq$!43$*)pBvd2XDF-7$Fcr$!4U)Ho/`C0,IF-7$F]s$!4]DljV^xWq$F-7$F gs$!4xYx8+6Mo`%F-7$Fat$!4675Hvy=c;&F-7$F[u$!4.(4?PjE>:dF-7$Feu$!49N_R( y;L7jF-7$Fju$!3$4ID!)p4(zlF17$F_v$!3#\\>y%[#f#znF17$Fiv$!3@[*[dZ!QVoF1 7$Fcw$!3UV>))fBL()oF17$F]x$!3Na/>!e***4pF17$Fgx$!3(GPb1r>3\"pF17$Fi_o$ !3Gcey2!eF)oF17$F\\y$!3NR(y(*p;1$oF17$Fa`o$!3W;V-1@QUnF17$Fay$!3Eb]fI3 tJmF17$Ffy$!4G)y$>6#[m,kF-7$F[z$!4]v(p:$3v[8'F-7$Fez$!4:'Q^i<96taF-7$F _[l$!4Iufx&)yhY%\\F-7$Fi[l$!4[]Y>@'e$*)\\%F-7$F^\\l$!45z@F;;6!4UF-7$Fh \\l$!4Z>au\\p&H?SF-7$F\\^l$!4tXH*[J`FNRF-7$F`_l$!4#GZ#)=s.#3%RF-7$Fj_l $!4y)eo!*\\0BbSF-7$F_`l$!4&*pf&f664]UF-7$Fd`l$!4\">%=v(=CT_XF-7$Fi`l$! 4sS?NBA^<\"\\F-7$F^al$!4^0\"=[jMdMQAF-7$F[il$!4YJn0s5->=# F-7$F`il$!4=P(HRw_ef@F-7$Fjil$!4$*GYCR)\\x'=#F-7$Fdjl$!4g\"3YW;K^QAF-7 $F^[m$!4r%>A)o%4PMBF-7$Fh[m$!4N\"*4![BK**4DF-7$Fb\\m$!4[-XG=`8J#HF-7$F \\]m$!440)yr*Hk,]$F-7$Fa]m$!4\"[$e/^_84(RF-7$Ff]m$!4$*3mlF)oD:YF-7$F[^ m$!4WVgFJm$zzbF-7$F`^m$!4h!fh**4gz))pF-7$$\"?+++++++](o*y<'G([F1$!4izP #QPbmzyF-7$Fe^m$!4&*3ug#*HV6$*)F-7$$\"?nmmmmmm;Hd>()4'*[F1$!5bq;3;3R1< 5F-7$Fj^m$!5SVU\"zt@\\k;\"F-7$F^`m$!5uZQ5*RX@dE\"F-7$Fbam$!5$G'H%zU'e/ R8F-7$Ffbm$!4T:PJ%[R2G9F17$Fjcm$!41*>f6=-FW:F17$F_dm$!4b%z(e49nrr\"F17 $Fddm$!45GsC!4[u%)=F17$Fidm$!4m0)zy`q!*[?F17$F^em$!4:p,`7MVU=#F17$Fgfn $!4yNM$)y&f+!>#F17$F\\gn$!4M[m,'\\QU$>#F17$Fagn$!4xqi=sv)\\NAF17$Fcem$ !4$3rY$yF&e]AF17$F^hn$!4ZObU#zeaDAF17$Fhem$!4xY<')yjGM5#F1-F^fm6&F`fmF ffm$\"#vFcfmFdfm-Fhfm6#%TButcher's~scheme~B~with~c[5]=c[6]=1/2~and~b[5 ]=b[6]G-F$6%7eo7$F($\"3(H'Q%4BR,&QF-7$F5$\"3=93mN5899F-7$F:$\"2LBlJ'R1 IqF-7$F?$\"2CR#))o9L+?F-7$FD$!16Y@QD!>s(F-7$FI$!2(fSN`xA-?F-7$Fgn$!2_q Z#*fdpA\"F-7$Fep$\"2C:Nv?!R@JF-7$F_q$\"2@F=D;>E?)F-7$Fiq$\"3ulsXDO%)H: F-7$Fcr$\"3!yQgj^1\"[BF-7$F]s$\"3>h;;Y'*=$e$F-7$Fgs$\"3@q_j+AF*[&F-7$F at$\"3scJB*Q3wg(F-7$F[u$\"4&HQ#feLIY,\"F-7$F`u$\"4\"z48%>2yy5\"F-7$Feu $\"4]uenja?y<\"F-7$Fju$\"3@]VURxF;7F17$F_v$\"3Q#3YipMNB\"F17$Fcw$\"3], *z,%z:K7F17$Fgx$\"3@jq'3\"G2=7F17$Fay$\"3JpQ)3mDH9\"F17$F[z$\"4#3_#*\\ X<0\\5F-7$Fez$\"3[:d()[e79$*F-7$F_[l$\"3/'Q<(o$=bN)F-7$Fi[l$\"35D)[S(> 4duF-7$F^\\l$\"3m30Sl\"R'fnF-7$Fh\\l$\"3BS0F1o.VjF-7$F\\^l$\"3'HX0>S$= kiF-7$F`_l$\"3Rt&ftTg8['F-7$Fj_l$\"3;HglA)R\"HpF-7$F_`l$\"3/tg7zJo*R(F -7$Fd`l$\"3OW-m*)ztyyF-7$Fi`l$\"3;Ri_3vr,$)F-7$Fcal$\"3$)*HntyvCh)F-7$ Facl$\"3Ov$3ArrTu)F-7$F_el$\"3&*H.yg8BA')F-7$Fiel$\"3kP%ocTT#*>)F-7$F^ fl$\"3yODSR4U[uF-7$Fhfl$\"3wJ>u0LDDmF-7$Fbgl$\"3D.I'o-3Ow&F-7$F\\hl$\" 3VE'H\"pqmi]F-7$Fahl$\"3pCg2x3z)R%F-7$Ffhl$\"3:)[%z)R()ot$F-7$F`il$\"3 4[.#QS[_>$F-7$Fdjl$\"3FAnY[k`7GF-7$Fh[m$\"3HrhZz/e;\"F-7$Fe^m$\"4;(3iyQtE9:F-7$Fj^m$\"4g.0xd#=LY ?F-7$Fbam$\"4jrPp*3LAMCF-7$Fjcm$\"3:Oi,\")f8'*HF17$F_dm$\"3\\@K;c$4fa$ F17$Fddm$\"3Uyucd\"3h?%F17$Fgen$\"3(*HvzpcN&[%F17$Fidm$\"3\"4Sz./&[O]F 17$F_fn$\"3%pjTq2oB\"eF17$F^em$\"3I5HYEO$R1'F17$F\\gn$\"34'yxB=C-)pF17 $Fcem$\"30SiM=ZnIuF17$Fign$\"3Z&**oK3q6@)F17$F^hn$\"3dtw*pr(pt$)F17$Fc hn$\"3:v0il6en%)F17$Fhhn$\"3\"*GG#[\\&)4h)F17$F]in$\"3w52.Z/u7*)F17$Fh em$\"3ZpPcS&ypi*F1-F^fm6&F`fmFafmFfinFffm-Fhfm6#%Hscheme~with~c[5]=c[6 ]=3/4~and~b[5]=b[6]G-%%FONTG6$%*HELVETICAG\"\"*-%+AXESLABELSG6$Q\"x6\" Q!Fdjq-%&TITLEG6#%Uerror~curves~for~7~stage~order~6~Runge-Kutta~method sG-%%VIEWG6$;F(Fhem%(DEFAULTG" 1 2 0 1 10 0 2 9 1 4 2 1.000000 45.000000 45.000000 0 1 "Butcher's scheme A" "scheme with simple nodes " "scheme with a relatively large stability region" "Butcher's scheme \+ B with c[5]=c[6]=1/2 and b[5]=b[6]" "scheme with c[5]=c[6]=3/4 and b[5 ]=b[6]" }}}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 47 "Test 11 of 7 stage, order 6 Runge-Kutta methods" }}{PARA 0 "" 0 "" {TEXT -1 0 " " }}{PARA 0 "" 0 "" {TEXT -1 143 "This example is similar to one that \+ appears in an article by F. G. Lether: Mathematics of Computation, Vo l. 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#\"\"!*&%$sinG\"\"\"F*F*" }{TEXT -1 1 " \+ " }}{PARA 0 "" 0 "" {TEXT -1 10 "Solution: " }}{PARA 256 "" 0 "" {TEXT -1 2 " " }{XPPEDIT 18 0 "y = -exp(-x)*sin(1/(x-1))" "6#/%\"yG,$ *&-%$expG6#,$%\"xG!\"\"\"\"\"-%$sinG6#*&F-F-,&F+F-F-F,F,F-F," }{TEXT -1 1 " " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 185 "de := diff(y(x),x)=exp(-x)/(x-1)^2*cos(1/(x-1))-y(x) ;\nic := y(0)=sin(1);\ndsolve(\{de,ic\},y(x));\nq := unapply(rhs(%),x) :\nplot(q(x),x=0..1-1/(6*Pi),font=[HELVETICA,9],labels=[`x`,`y(x)`]); " }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%#deG/-%%diffG6$-%\"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$\"3/Z^$z TlU<)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,$\"3 T8>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$$\"3EOur83&\\b$F,$\"37)QgTz pp+(F,7$$\"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$$\"3&\\%Rcec>> `F,7$$\"39pTj@J(oJ&F,$\"3-j_%RM!Rk\\F,7$$\"3QD(p)Qdl>bF,$\"34_#)R=svWX F,7$$\"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$$\"35Y]`:_N/nF,$\"33Dv;1s&p Z&F07$$\"3.\"Q#ekL\"p!pF,$!3OST&zF\\gd%F07$$\"3%yB5rz/v4(F,$!3cn!**p^J 2Z\"F,7$$\"33-p_gxs'H(F,$!3A9e\"Q#e(4b#F,7$$\"3%324>i/:\\(F,$!3S!=W0GJ d`$F,7$$\"39FEN$GhMf(F,$!3U'\\`Wd^A(RF,7$$\"3c%='zWzT&p(F,$!3f\\'**z(4 Q8VF,7$$\"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$$ \"3Z,MUjZ4H%)F,$\"3c]$R4W&G]NF07$$\"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$$\"3'>RC=\\B%y()F,$\"3x'=W7Fx&HR F,7$$\"3]1)zraF`#))F,$\"3QFG%plB4F$F,7$$\"39A_`-;Bs))F,$\"3700_RQTz@F, 7$$\"31^1[bjB(*))F,$\"3OuNH?R:M9F,7$$\"34\"3E%36CA*)F,$\"3E<)GsFg<(fF0 7$$\"376:PheCZ*)F,$!3ocHHl.]EIF07$$\"39TpJ91Ds*)F,$!38U'y=D^^A\"F,7$$ \"3=rBEn`D(**)F,$!3j&4Nt%[@=@F,7$$\"3@,y??,EA!*F,$!3!e@9jLE#=HF,7$$\"3 EJK: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$y A*F,$\"3MLbSaq^*[\"F,7$$\"3b(=A)*z)zR#*F,$\"3Qccz;X$>?#F,7$$\"3%*oQPp^ w^#*F,$\"3\\RTgHcmRGF,7$$\"3C\\b#*Q:tj#*F,$\"3M$o6)eV:lLF,7$$\"3oE3CP5 fw#*F,$\"3Ma)HpV]]I!H!G$* F,$\"3a%4t07BP*GF,7$$\"3J=s\")G&))3M*F,$\"3f(3)H;[%o+#F,7$$\"3w&\\Kr-[ PN*F,$\"33n1kl[ ]%*F,$!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&%$RG BG$\"#5!\"\"F(F(-%%FONTG6$%*HELVETICAG\"\"*-%+AXESLABELSG6$%\"xG%%y(x) G-%%VIEWG6$;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 "root mean square error" }{TEXT -1 18 " of each solution. " }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 808 "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 := [`Butcher's scheme A`,`scheme with simple nodes`,`scheme with a relatively large stability region`,`Butcher's scheme B with `* (c[5]=1/2,c[6]=1/2,b[5]=b[6]),`scheme with `*(c[5]=3/4,c[6]=3/4,b[5]=b [6])]:\nerrs := []:\nDigits := 30:\nfor ct to 5 do\n Qn_RK6_||ct := \+ RK6_||ct(Q(x,y),x,y,x0,evalf[33](y0),evalf[33](hh),numsteps,false);\n \+ sm := 0: numpts := nops(Qn_RK6_||ct):\n for ii to numpts do\n \+ sm := sm+(Qn_RK6_||ct[ii,2]-q(Qn_RK6_||ct[ii,1]))^2;\n end do:\n \+ errs := [op(errs),sqrt(sm/numpts)];\nend do:\nDigits := 10:\nlinalg[tr anspose]([mthds,evalf(errs)]);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#K%'m atrixG6#7&7$%0slope~field:~~~G,&*(-%$expG6#,$%\"xG!\"\"\"\"\",&F/F1F1F 0!\"#-%$cosG6#*&F1F1F2F0F1F1%\"yGF07$%0initial~point:~G-%!G6$\"\"!-%$s inG6#F17$%/step~width:~~~G,&#F1\"$+&F1*&F1F1*&\"%+IF1%#PiGF1F0F07$%1no .~of~steps:~~~GFFQ)pprint506\"" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%!G " }}{PARA 11 "" 1 "" {XPPMATH 20 "6#K%'matrixG6#7'7$%3Butcher's~scheme ~AG$\"+31]([*!#>7$%9scheme~with~simple~nodesG$\"+\">!pg9!#=7$%Pscheme~ with~a~relatively~large~stability~regionG$\"+=XRh9F07$*&%9Butcher's~sc heme~B~with~G\"\"\"6%/&%\"cG6#\"\"&#F8\"\"#/&F<6#\"\"'F?/&%\"bGF=&FGFC F8$\"+co*G^#F07$*&%-scheme~with~GF86%/F;#\"\"$\"\"%/FBFPFEF8$\"+[;if7F 0Q)pprint516\"" }}}{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 \+ = 0;" "6#/%\"xG\"\"!" }{TEXT -1 21 ".9469 is also given." }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 732 "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 \+ := [`Butcher's scheme A`,`scheme with simple nodes`,`scheme with a rel atively large stability region`,`Butcher's scheme B with `*(c[5]=1/2,c [6]=1/2,b[5]=b[6]),`scheme with `*(c[5]=3/4,c[6]=3/4,b[5]=b[6])]:\nerr s := []:\nDigits := 30:\nfor ct to 5 do\n qn_RK6_||ct := RK6_||ct(Q( x,y),x,y,x0,evalf(y0),evalf(hh),numsteps,true);\nend do:\nxx := 0.9469 : qxx := evalf(q(xx)):\nfor ct to 5 do\n errs := [op(errs),abs(qn_RK 6_||ct(xx)-qxx)];\nend do:\nDigits := 10:\nlinalg[transpose]([mthds,ev alf(errs)]);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#K%'matrixG6#7&7$%0slop e~field:~~~G,&*(-%$expG6#,$%\"xG!\"\"\"\"\",&F/F1F1F0!\"#-%$cosG6#*&F1 F1F2F0F1F1%\"yGF07$%0initial~point:~G-%!G6$\"\"!-%$sinG6#F17$%/step~wi dth:~~~G,&#F1\"$+&F1*&F1F1*&\"%+IF1%#PiGF1F0F07$%1no.~of~steps:~~~GFFQ )pprint526\"" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%!G" }}{PARA 11 "" 1 " " {XPPMATH 20 "6#K%'matrixG6#7'7$%3Butcher's~scheme~AG$\"+#pl-m\"!#<7$ %9scheme~with~simple~nodesG$\"+[2>bDF+7$%Pscheme~with~a~relatively~lar ge~stability~regionG$\"+h^&yb#F+7$*&%9Butcher's~scheme~B~with~G\"\"\"6 %/&%\"cG6#\"\"&#F7\"\"#/&F;6#\"\"'F>/&%\"bGF<&FFFBF7$\"+BWb(R%F+7$*&%- scheme~with~GF76%/F:#\"\"$\"\"%/FAFOFDF7$\"+ViE.AF+Q)pprint536\"" }}} {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 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 0 456 "mthds := [`Butcher's scheme A`,`scheme with simple n odes`,`scheme with a relatively large stability region`,`Butcher's sch eme B with `*(c[5]=1/2,c[6]=1/2,b[5]=b[6]),`scheme with `*(c[5]=3/4,c[ 6]=3/4,b[5]=b[6])]:\nerrs := []:\nDigits := 20:\nfor ct to 5 do\n sm := NCint((q(x)-'qn_RK6_||ct'(x))^2,x=0..1-1/(6*Pi),adaptive=false,num points=7,factor=200);\n errs := [op(errs),sqrt(sm/(1-1/(6*Pi)))];\ne nd do:\nDigits := 10:\nlinalg[transpose]([mthds,evalf(errs)]);" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#K%'matrixG6#7'7$%3Butcher's~scheme~AG$ \"+Z7Z@a!#>7$%9scheme~with~simple~nodesG$\"+__$eN)F+7$%Pscheme~with~a~ relatively~large~stability~regionG$\"+G.(fM)F+7$*&%9Butcher's~scheme~B ~with~G\"\"\"6%/&%\"cG6#\"\"&#F7\"\"#/&F;6#\"\"'F>/&%\"bGF<&FFFBF7$\"+ nyvN9!#=7$*&%-scheme~with~GF76%/F:#\"\"$\"\"%/FAFPFDF7$\"+TYm2sF+Q)ppr int546\"" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 92 "The following error graphs are constructed using the numerical pro cedures for the solutions." }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 534 "evalf[20](plot(['qn_RK6_1'(x)-q(x),'qn_RK6_2'(x)-q(x),'qn_RK6_3'( x)-q(x),'qn_RK6_4'(x)-q(x),\n'qn_RK6_5'(x)-q(x)],x=0..0.7,-2.3e-18..1e -17,font=[HELVETICA,9],\ncolor=[COLOR(RGB,.95,.2,0),COLOR(RGB,.45,0,.9 5),COLOR(RGB,0,.25,1),COLOR(RGB,0,.75,.2),COLOR(RGB,.95,.45,0)],\nlege nd=[`Butcher's scheme A`,`scheme with simple nodes`,`scheme with a rel atively large stability region`,`Butcher's scheme B with c[5]=c[6]=1/2 and b[5]=b[6]`,`scheme with c[5]=c[6]=3/4 and b[5]=b[6]`],title=`erro r curves for 7 stage order 6 Runge-Kutta methods`));" }}{PARA 13 "" 1 "" {GLPLOT2D 685 540 540 {PLOTDATA 2 "6+-%'CURVESG6%7fn7$$\"\"!F)F(7$$ \"5mmmmm\"z+e_\"!#@$!\"$!#?7$$\"5LLLL3->R`GF-$!\"\"F07$$\"5mmmm;apSYVF -F.7$$\"5lmmm;z'=$\\eF-F.7$$\"5KLLL3Ft3XtF-F.7$$\"5lmmmTNj&=t)F-$!\"%F 07$$\"5+++](=`xn,\"F0$!\"'F07$$\"5mmm;ay/Gl6F0$!\")F07$$\"5+++]PurI88F 0FL7$$\"5LLLL$e#3dl9F0$!#5F07$$\"5mmmm\"Ht%o*f\"F0FT7$$\"5++++]F_m]F0$!#9F07$$\"5++++]s2O[?F0$!#=F07$$\"5mmm;aG\"H5 =#F0$!#>F07$$\"5LLLL$ej%yQBF0$!#BF07$$\"5LLLLLVUUsCF0$!#DF07$$\"5***** *\\(o()yyi#F0$!#GF07$$\"5LLLLLoD[lFF0$!#LF07$$\"5******\\(oibk\"HF0$!# PF07$$\"5******\\i!o<-1$F0$!#UF07$$\"5LLLL3-$=-@$F0$!#[F07$$\"5LLL$3xp lzM$F0$!#aF07$$\"5mmmm\"H([a'\\$F0$!#jF07$$\"5mmm;ayo(3l$F0$!#pF07$$\" 5******\\7VLA&y$F0$!#vF07$$\"5mmmmT07KIRF0$!#$)F07$$\"5*********\\\\@- 3%F0$!#\"*F07$$\"5*******\\PopoA%F0$!#(*F07$$\"5******\\(oMf(oVF0$!$+ \"F07$$\"5*******\\ii.j_%F0$!#)*F07$$\"5KLLLLoT'ym%F0$!##*F07$$\"5**** ****\\i-,>[F0$!#tF07$$\"5mmm;a)3rf&\\F0Fcq7$$\"5********\\Zaq0^F0$\"#> F07$$\"5KLL$3-\"QfY_F0$\"$>\"F07$$\"5******\\PWF'QR&F0$\"$k#F07$$\"5KL LL$e/Xy`&F0$\"$0&F07$$\"5******\\(=<\"e)o&F0$\"$#))F07$$\"5lmmmmwzvLeF 0$\"%i9F07$$\"5lmmm\"zAAA)fF0$\"%cAF07$$\"5KLL$3-7d%HhF0$\"%nMF07$$\"5 *********\\p]ZE'F0$\"%*3&F07$$\"5lmm;HZ:GUjF0$\"%]jF07$$\"5KLLLe*R7)>k F0$\"%kzF07$$\"5+++]7=p:*['F0$\"%\\$*F07$$\"5lmmmmO9]elF0$\"&c8\"F07$$ \"5ILL3xcrVKmF0$\"'^u8F-7$$\"5)*****\\(o(GP1nF0$\"'&Ql\"F-7$$\"5+++v$f $evTnF0$\"'T%z\"F-7$$\"5+++++&zQrx'F0$\"'28>F-7$$\"5+++D1a<_7oF0$\"(3f 1#!#A7$$\"5)*****\\78Z!z%oF0$\"'-SAF-7$$\"5++]P%[`Gf)oF0$\"'xGCF-7$$\" 5+++DccB&R#pF0$\"'DCEF-7$$\"5++]7Gyh(>'pF0$\"')[#GF-7$$\"\"(F5$\"'9FIF --%&COLORG6&%$RGBG$\"#&*!\"#$\"\"#F5F(-%'LEGENDG6#%3Butcher's~scheme~A G-F$6%7fnF'F*F1F67$F:$F^]lF0FF-7$F[z$\"'PnBF-7$F`z$\"'qjDF-7$Fez$\"'&)GFF -7$Fjz$\"(')4%HF^[l7$F`[l$\"'f\"=$F-7$Fe[l$\"'-TMF-7$Fj[l$\"'13PF-7$F_ \\l$\"'ozRF-7$Fd\\l$\"'U]UF--Fi\\l6&F[]l$\"#XF^]lF(F\\]l-Fb]l6#%9schem e~with~simple~nodesG-F$6%7fnF'F*F1F6Fh]lFF-7$F[z$\"' ZcBF-7$F`z$\"'\\^DF-7$Fez$\"'[:FF-7$Fjz$\"(\"3EHF^[l7$F`[l$\"'$[;$F-7$ Fe[l$\"'6AMF-7$Fj[l$\"'v'o$F-7$F_\\l$\"'nbRF-7$Fd\\l$\"'RBUF--Fi\\l6&F []lF($\"#DF^]l$\"\"\"F)-Fb]l6#%Pscheme~with~a~relatively~large~stabili ty~regionG-F$6%7fnF'F*7$F2Fi]l7$F7FB7$F:FB7$F=F[^l7$F@F^^l7$FEFa^l7$FJ Ffn7$FOFh^l7$FRFhgl7$FWFhgl7$FZ$F0F07$FinFjo7$F^o$!#HF07$Fco$!#KF07$Fh oFchl7$F]pFcq7$FbpFhq7$FgpFjhl7$F\\qFbr7$FaqFcil7$Ffq$!#\")F07$F[r$!#! *F07$F`r$!$.\"F07$Fer$!$8\"F07$Fjr$!$B\"F07$F_s$!$L\"F07$Fds$!$T\"F07$ Fis$!$X\"F07$F^tF_bm7$Fct$!$A\"F07$FhtFfs7$F]uF_p7$Fbu$\"#rF07$Feu$\"$ X#F07$Fju$\"$@&F07$F_v$\"$0*F07$Fdv$\"%I:F07$Fiv$\"%*\\#F07$F^w$\"%rRF 07$Fcw$\"%qfF07$Fhw$\"%***)F07$F]x$\"&WI\"F07$Fbx$\"&sh\"F07$Fgx$\"&r, #F07$F\\y$\"&(eBF07$Fay$\"&O&GF07$Ffy$\"'aSMF-7$F[z$\"'RBTF-7$F`z$\"'[ mWF-7$Fez$\"'*\\v%F-7$Fjz$\"(cd7&F^[l7$F`[l$\"'NYbF-7$Fe[l$\"'@+gF-7$F j[l$\"'7okF-7$F_\\l$\"'TWpF-7$Fd\\l$\"''*>uF--Fi\\l6&F[]lF($\"#vF^]lF_ ]l-Fb]l6#%TButcher's~scheme~B~with~c[5]=c[6]=1/2~and~b[5]=b[6]G-F$6%7f nF'F*F1F6Fh]lFF07$Fcw$\"%!)HF07$Fhw$\"%&\\%F07$F]x$\"%= lF07$Fbx$\"%$3)F07$Fgx$\"&&35F07$F\\y$\"&&z6F07$Fay$\"&rU\"F07$Ffy$\"' x? " 0 "" {MPLTEXT 1 0 538 "evalf[20](plot(['qn_RK6_1'(x)-q(x),'qn_RK6_2'(x)-q(x),'qn_RK6_3 '(x)-q(x),'qn_RK6_4'(x)-q(x),\n'qn_RK6_5'(x)-q(x)],x=0.4..0.5,-1.5e-18 ..1.3e-18,font=[HELVETICA,9],\ncolor=[COLOR(RGB,.95,.2,0),COLOR(RGB,.4 5,0,.95),COLOR(RGB,0,.25,1),COLOR(RGB,0,.75,.2),COLOR(RGB,.95,.45,0)], \nlegend=[`Butcher's scheme A`,`scheme with simple nodes`,`scheme with a relatively large stability region`,`Butcher's scheme B with c[5]=c[ 6]=1/2 and b[5]=b[6]`,`scheme with c[5]=c[6]=3/4 and b[5]=b[6]`],title =`error curves for 7 stage order 6 Runge-Kutta methods`));" }}{PARA 13 "" 1 "" {GLPLOT2D 829 481 481 {PLOTDATA 2 "6+-%'CURVESG6%7idl7$$\" \"%!\"\"$!#')!#?7$$\"5nmmm;arz@SF-$!#))F-7$$\"5LLL$e9ui2/%F-$!#*)F-7$$ \"5nmmm\"z_\"4iSF-$!#!*F-7$$\"5nmm;zp!fu1%F-F;7$$\"5nmmmm6m#G2%F-F;7$$ \"5nmT5Sav\\tSF-$!#\"*F-7$$\"5nm;a8(\\oT2%F-F;7$$\"5nm\"zp)R%R[2%F-FF7 $$\"5nmmTg#Q5b2%F-FF7$$\"5nm;H2oA&o2%F-FF7$$\"5nmm;a`T>ySF-FF7$$\"5nmm \"zW#z(33%F-FF7$$\"5nmmmT&phN3%F-FF7$$\"5+vVtF\\%HU3%F-FF7$$\"5M$3-QJ? (*[3%F-F;7$$\"5n\"zp)*p&\\c&3%F-FF7$$\"5++v$f3rKi3%F-F;7$$\"5M3_+sk/!p 3%F-F;7$$\"5n;H2e=#ov3%F-FF7$$\"5+D19WsfB)3%F-FF7$$\"5ML$3-js.*)3%F-FF 7$$\"5nTgF;![r&*3%F-FF7$$\"5+]PM-M#R-4%F-FF7$$\"5Me9T)y)p!44%F-$!##*F- 7$$\"5nm\"zW1u5%F-Fhp7$$ \"5mmmmTlD))4TF-$!#$*F-7$$\"5LLeRse;]5TF-Fas7$$\"5++]7._276TF-Fas7$$\" 5mmT&Q`%)R<6%F-Fas7$$\"5LLLekQ*eB6%F-Fhp7$$\"5mm;/EDrf8TF-Fhp7$$\"5+++ ](=JN[6%F-Fhp7$$\"5nm\"H#=0WX:TF-Fas7$$\"5ML$e*[)\\tg6%F-Fas7$$\"5++vo z\"f#p;TF-Fhp7$$\"5nmmT5&o6t6%F-Fas7$$\"5++](=<()\\&=TF-Fas7$$\"5MLLLL e!)y>TF-Fas7$$\"5++D1k^rS?TF-Fas7$$\"5nm;z%\\CE57%F-Fhp7$$\"5ML3_DQ`k@ TF-Fas7$$\"5+++DcJWEATF-Fas7$$\"5nm\"zp[_$)G7%F-Fas7$$\"5ML$3x\"=E]BTF -FF7$$\"5++vV[6<7CTF-Fhp7$$\"5nmm;z/3uCTF-Fhp7$$\"5+++]7LRDXTF-Fas7$$ \"5nmm;zR'ok;%F-$!#%*F-7$$\"5+++]i5`h(=%F-$!#&*F-7$$\"5LLLL$3En$4UF-$! #'*F-7$$\"5+]7Gj$)f'*4UF-$!#(*F-7$$\"5mm\"HKkqk0@%F-Fdx7$$\"5L$3xJ#HM; 6UF-F_x7$$\"5++]7._@w6UF-F_x7$$\"5m;H2$[(3O7UF-F_x7$$\"5LL3-j(ffH@%F-F dx7$$\"5+](oH/KeN@%F-F_x7$$\"5mmm\"HK/dT@%F-Fdx7$$\"5L$ekGgwbZ@%F-Fjw7 $$\"5++D\"G))[a`@%F-F_x7$$\"5m;/wi6K&f@%F-Fdx7$$\"5LL$3FW$>b;UF-F_x7$$ \"5mmTg-!Q\\x@%F-F_x7$$\"5+++]iDo%*=UF-F_x7$$\"5MLeRArU9?UF-F_x7$$\"5n m;H#orT8A%F-F_x7$$\"5M$eRA'R/%>A%F-Fdx7$$\"5++v=Ui\"RDA%F-F_x7$$\"5n;a 8A&)y8BUF-Fdx7$$\"5MLL3-3mtBUF-Fdx7$$\"5++](=#*\\JhA%F-Fdx7$$\"5nmmmT! RE&GUF-Fdx7$$\"5MLL$e9r5$RUF-Fdx7$$\"5++++]K]4]UF-Fdx7$$\"5+++]7$=-GD% F-Fdx7$$\"5++++vL$4bD%F-Fdx7$$\"5+++D14H'oD%F-$!#)*F-7$$\"5+++]P%[;#eU F-$!#**F-7$$\"5++]7.sK*)eUF-Fd]l7$$\"5+++vof+dfUF-Fdx7$$\"5++]PMZoCgUF -Fd]l7$$\"5+++++NO#4E%F-Fd]l7$$\"5++++DOzLmUF-Fd]l7$$\"5++++]PAvrUF-Fd ]l7$$\"5+++D\"yW/CF%F-Fdx7$$\"5+++]7em0tUF-Fd]l7$$\"5+++vVo)3PF%F-Fd]l 7$$\"5++++vy5OuUF-Fdx7$$\"5+++]P*\\lcF%F-Fd]l7$$\"5+++++?*ppF%F-Fi]l7$ $\"5+++DJI@ixUF-Fdx7$$\"5+++]iSVFyUF-Fdx7$$\"5+++v$4bE*yUF-Fd]l7$$\"5+ +++Dh(y&zUF-Fd]l7$$\"5+++](==$)3G%F-Fd]l7$$\"5++++]-w=#G%F-Fd]l7$$\"5+ ++++&G0uG%F-Fi]l7$$\"5++++]nHi#H%F-$!$+\"F-7$$\"5MLLeky#*4-VF-Fgal7$$ \"5nmm;z*ev:J%F-Fgal7$$\"5LLLL$347TL%F-Fgal7$$\"5LLLLLjM?`VF-Fgal7$$\" 5m\"H2LsX(*QN%F-Fi]l7$$\"5+]7G8^9faVF-$!$,\"F-7$$\"5L3_D.XaGbVF-F[cl7$ $\"5mm\"HK*Q%zfN%F-F[cl7$$\"5+DJ?$GVtmN%F-F[cl7$$\"5L$3xJnUntN%F-Fgal7 $$\"5mT5:j?91eVF-Fgal7$$\"5++]7`9aveVF-F[cl7$$\"5Le*)4V3%\\%fVF-Fgal7$ $\"5m;H2L-M9gVF-F[cl7$$\"5+vo/B'RP3O%F-Fgal7$$\"5LL3-8!RJ:O%F-Fgal7$$ \"5m\"z%*HSQDAO%F-Fgal7$$\"5+](oHzP>HO%F-Fgal7$$\"5L3F%H=P8OO%F-Fi]l7$ $\"5mmm\"HdO2VO%F-F[cl7$$\"5++D\"G8M$3nVF-Fgal7$$\"5LL$3FpJf)pVF-Fi]l7 $$\"5m\"H#o#3J`0P%F-Fgal7$$\"5+]ils/tCrVF-F[cl7$$\"5L3-ji)HT>P%F-F[cl7 $$\"5mmTg_#HNEP%F-Fgal7$$\"5L$3_D.GBSP%F-Fgal7$$\"5+++]7o7TvVF-Fgal7$$ \"5mT5!>.dDgP%F-F[cl7$$\"5L$3-8D()RmP%F-Fgal7$$\"5+DJqquTDxVF-F[cl7$$ \"5mmT5!pZoyP%F-Fi]l7$$\"5L3_]4zF[yVF-F[cl7$$\"5+]i!*G\"3(4zVF-F[cl7$$ \"5m\"H2$[$Q6(zVF-Fgal7$$\"5LL$3xcoD.Q%F-Fgal7$$\"5+v$4ry)*R4Q%F-Fgal7 $$\"5m;/^1!Ha:Q%F-Fgal7$$\"5Le9\"fAfo@Q%F-Fgal7$$\"5++DJX%*Gy#Q%F-Fi]l 7$$\"5L$e9T))\\6SQ%F-Fgal7$$\"5mmm\"HK5S_Q%F-F[cl7$$\"5++]7y?X:!R%F-F[ cl7$$\"5LLLLLQ*o]R%F-F[cl7$$\"5++]7GL3Y+WF-F[cl7$$\"5mmm\"H#GF&eS%F-F[ cl7$$\"5+Dcw4:n_1WF-Fgal7$$\"5L$e9m>q+sS%F-Fgal7$$\"5mTNY$))ouyS%F-F[c l7$$\"5++DJqv'[&3WF-$!$-\"F-7$$\"5m;/,W\\m*)4WF-F[cl7$$\"5LL$3xJiW7T%F -Fgal7$$\"5m\"HdX+h=>T%F-F[cl7$$\"5+]iS\"pf#f7WF-F[cl7$$\"5L3_Dy$emKT% F-Fgal7$$\"5mmT5lq0%RT%F-F[cl7$$\"5+DJ&>vb9YT%F-F[cl7$$\"5L$3-)QW&)G:W F-F[cl7$$\"5mT5lDJD'fT%F-Fa[m7$$\"5+++]7=lj;WF-Fgal7$$\"5+++v=x3x@WF-F gal7$$\"5++++DO_!pU%F-Fgal7$$\"5++Dc,E))=GWF-F[cl7$$\"5++]7y:CZHWF-Fa[ m7$$\"5+]iSm5U6IWF-Fgal7$$\"5++voa0gvIWF-F[cl7$$\"5+](oH/!yRJWF-F[cl7$ $\"5+++DJ&fR?V%F-F[cl7$$\"5+]7`>!R\"oKWF-Fgal7$$\"5++D\"y]=BLV%F-F[cl7 $$\"5+]P4'*z\\'RV%F-F[cl7$$\"5++]P%[x1YV%F-Fgal7$$\"5+]ilsp&[_V%F-Fa[m 7$$\"5++v$4YO!*eV%F-F[cl7$$\"5+](=#\\f@`OWF-Fi]l7$$\"5+++]PaRTWF-Fgal7$$ \"5m\"Hd&zt9'=W%F-F[cl7$$\"5LL$3F%>6`UWF-F[cl7$$\"5+v$fe]w+KW%F-Fa[m7$ $\"5m;/,p5/(QW%F-Fa[m7$$\"5Le9;Kc+aWWF-F[cl7$$\"5++DJ&>q4_W%F-F[cl7$$ \"5mTNYeZ$zeW%F-F[cl7$$\"5L$e9;K**[lW%F-F[cl7$$\"5+Dcw%)Q'=sW%F-Fgal7$ $\"5mmm\"zWG))yW%F-Fgal7$$\"5++]7`\\aC`WF-Fgal7$$\"5LLLLe9EgeWF-Fgal7$ $\"5LLL3F9@?-\\%F-Fgal 7$$\"5+voa&=^$)3\\%F-Fgal7$$\"5nm\"zW<\"oa\"\\%F-F[cl7$$\"5Me9Tj6,@#\\ %F-F[cl7$$\"5+]PM_6M(G\\%F-Fgal7$$\"5nTgFT6n`$\\%F-F[cl7$$\"5ML$3-8,+U \\%F-Fgal7$$\"5+D19>6L'[\\%F-Fi]l7$$\"5n;H236m_&\\%F-Fgal7$$\"5M3_+(4 \"**='\\%F-F[cl7$$\"5++v$f3@`o\\%F-F[cl7$$\"5n\"zp[2^;v\\%F-Fi]l7$$\"5 M$3-Q1\")z\")\\%F-Fgal7$$\"5+vVt_5J%))\\%F-F[cl7$$\"5nmmmT5k]*\\%F-F[c l7$$\"5nTNYe#R&>+XF-Fgal7$$\"5n;/EvuV)3]%F-Fgal7$$\"5n\"Hd?pNt:]%F-F[c l7$$\"5nmT&)3RBE-XF-Fgal7$$\"5n;zWU..k.XF-Fgal7$$\"5nm;/wn#=]]%F-Fgal7 $$\"5n;aj4KiR1XF-Fgal7$$\"5nm\"HKk>ux]%F-Fgal7$$\"5nTg-gyJY3XF-Fi]l7$$ \"5n;H#o2;_\"4XF-Fgal7$$\"5n\"z>OH9T)4XF-Fgal7$$\"5nmmT5D,`5XF-F[cl7$$ \"5nTN@F2\">7^%F-Fi]l7$$\"5n;/,W*33>^%F-Fgal7$$\"5n\"H23;2(f7XF-F[cl7$ $\"5nmTgx`gG8XF-Fgal7$$\"5n;z>6=Sm9XF-Fgal7$$\"5nm;zW#)>/;XF-Fgal7$$\" 5nm\"z>6\"zz=XF-Fgal7$$\"5nmm;zRQb@XF-Fgal7$$\"5M$3F%z+O:AXF-Fgal7$$\" 5++vozhLvAXF-Fi]l7$$\"5n;z%*zAJNBXF-Fi]l7$$\"5ML$3-Q)G&R_%F-Fgal7$$\"5 +](o/[k_X_%F-Fgal7$$\"5nm\"H2eS_^_%F-F[cl7$$\"5M$e*)4o;_d_%F-Fgal7$$\" 5+++D\"y#>NEXF-Fi]l7$$\"5ML3x\")\\9bFXF-Fi]l7$$\"5nm;H#=(4vGXF-Fi]l7$$ \"5M$3_DGt]$HXF-Fi]l7$$\"5++D\"GQ\\]*HXF-Fi]l7$$\"5n;H2$[D]0`%F-Fgal7$ $\"5MLLL$e,]6`%F-Fgal7$$\"5+]Pf$ox\\<`%F-F[cl7$$\"5nmT&Qy`\\B`%F-Fd]l7 $$\"5M$e9T))H\\H`%F-Fgal7$$\"5++]P%)f!\\N`%F-Fgal7$$\"5n;aj%3#)[T`%F-F i]l7$$\"5MLe*[=e[Z`%F-Fgal7$$\"5+]i:&GM[``%F-Fgal7$$\"5nmmT&Q5[f`%F-Fd ]l7$$\"5ML$eky9Z$QXF-Fd]l7$$\"5+++](=>Y2a%F-Fd]l7$$\"5ML3-8DsLVXF-Fi]l 7$$\"5nm;aQe#Gfa%F-Fgal7$$\"5M$3-8]xBsa%F-Fgal7$$\"5++D1k\"H>&[XF-Fgal 7$$\"5M3FW&*\\q;\\XF-Fd]l7$$\"5n;H#o#3[\")\\XF-Fi]l7$$\"5+DJ?emDY]XF-F i]l7$$\"5MLLe*[K56b%F-Fi]l7$$\"5nTN'4K3e3&Gd%F-Fi]l7$$\"5nmT5!p,?Nd%F- Fd]l7$$\"5M$eRAY@*=uXF-Fi]l7$$\"5++]PM7%e[d%F-Fi]l7$$\"5n;/^15w_vXF-Fd x7$$\"5MLeky2o>wXF-Fi]l7$$\"5+]7y]0g'od%F-Fd]l7$$\"5nmm\"HK?Nvd%F-Fd]l 7$$\"5ML$e9T*>@!e%F-Fdx7$$\"5+++++&y))Ge%F-F_x7$$\"5+]iSTkMa$e%F-Fd]l7 $$\"5++D\"GQ9)>%e%F-F_x7$$\"5+](=UK#G&[e%F-Fdx7$$\"5++]il-v]&e%F-Fd]l7 $$\"5+]7.2#=ihe%F-Fd]l7$$\"5++vV[ho\"oe%F-Fdx7$$\"5+]P%)*3arue%F-F_x7$ $\"5+++DJ?i7)e%F-Fdx7$$\"5++](oz$\\u!f%F-Fdx7$$\"5+++]ibOO$f%F-Fdx7$$ \"5++]7GtB)ff%F-Fdx7$$\"5+++v$44,')f%F-Fdx7$$\"5+]i:NqdD*f%F-Fdx7$$\"5 ++Dcw\\/\"**f%F-Fdx7$$\"5+](oz\"H^c+YF-Fdx7$$\"5++]Pf3)>7g%F-F_x7$$\"5 ++v=Un\"HDg%F-F_x7$$\"5++++DE&QQg%F-F_x7$$\"5+++v=-N(Rh%F-F_x7$$\"5+++ ]7y%3Ti%F-F_x7$$\"5++]7`f]tHYF-Fjw7$$\"5+++v$4kh`j%F-Few7$$\"5+D\"G8O' \\1OYF-Few7$$\"5+]i!*G'Gonj%F-Few7$$\"5+vV['*3;ZPYF-Fjw7$$\"5++D1kJ\\< QYF-Fjw7$$\"5+D1kJa#y)QYF-Fjw7$$\"5+](=#*pd\"eRYF-Fjw7$$\"5+vozm**[GSY F-Few7$$\"5++]PMA#))4k%F-Fhp7$$\"5+DJ&>]a\"pTYF-Few7$$\"5+]7`pn[RUYF-F jw7$$\"5+v$4r.>)4VYF-Fjw7$$\"5++vo/8:!Qk%F-Fjw7$$\"5+]P%)Re\"3_k%F-Fjw 7$$\"5++++v.[hYYF-Fjw7$$\"5mm;HdqE9\\YF-Few7$$\"5LLLeRP0n^YF-Fas7$$\"5 +]i:5/DI_YF-Few7$$\"5mm\"H23ZMHl%F-Fas7$$\"5L$3-8vVmNl%F-Few7$$\"5++]( =US)>aYF-Fjw7$$\"5m;zW#4PI[l%F-Few7$$\"5LL3-jPBYbYF-Fas7$$\"5+]PfL/V4c YF-Few7$$\"5mmm;/riscYF-Fas7$$\"5++DJX/-*zl%F-Fhp7$$\"5LL$eky8a#fYF-FF 7$$\"5+]7.d/h))fYF-Fhp7$$\"5mmTgFr!=0m%F-Fhp7$$\"5L$3x\")z.]6m%F-Fas7$ $\"5+++vo/?yhYF-Fhp7$$\"5m;HKRrRTiYF-Fas7$$\"5LLe*)4Qf/jYF-Fas7$$\"5+] (o/[!znjYF-Fhp7$$\"5mm;/^r)4Vm%F-Fhp7$$\"5L$e9;#Q=%\\m%F-FF7$$\"5++v=# \\!QdlYF-Fas7$$\"5m;/wird?mYF-Fhp7$$\"5LLLLLQx$om%F-Fas7$$\"5+++]78eBs YF-Fhp7$$\"5mmmm\"z)QjxYF-FF7$$\"5+++DJDHL!o%F-FF7$$\"5LLL$3F'>.$o%F-F F7$$\"5++]iS\"[\"Q%o%F-FF7$$\"5mmmT5+5t&o%F-FF7$$\"5++DJXfdS'o%F-FF7$$ \"5LL$3-)=03(o%F-F;7$$\"5mmT5:y_v(o%F-Fhp7$$\"5++++]P+V)o%F-Fhp7$$\"5n m;Hd&\\@Lp%F-FF7$$\"5MLLek`H@)p%F-F17$$\"5++](=Ok:kF\\F-Fhho7$$\"5 LLe*[=rg#G\\F-$!#[F-7$$\"5+]PfLf)z)G\\F-Fhho7$$\"5mm;H#o+*\\H\\F-F`io7 $$\"5++voz,ttI\\F-F`io7$$\"5LLL3x'fv>$\\F-F`io7$$\"5+]7yDWZfK\\F-Fcho7 $$\"5mm\"zW<*Q@L\\F-Fhho7$$\"5L$3xJ#RI$Q$\\F-Fhho7$$\"5++](=n=_W$\\F-F hho7$$\"5m;Hd?M82N\\F-$!#ZF-7$$\"5LL3Fp\"[!pN\\F-F][p7$$\"5+](oz\"H'4j $\\F-F][p7$$\"5mmmmmw(Gp$\\F-F`io7$$\"5LLLeRA5\\Z\\F-$!#VF-7$$\"5+++]7 oK0e\\F-$!#SF-7$$\"5+++++&oi\"o\\F-$!#PF-7$$\"5+++](=5s#y\\F-$!#NF-7$$ \"5+++v$40O\"*)\\F-$!#HF-7$$\"\"&F*$!#BF--%&COLORG6&%$RGBG$\"#&*!\"#$ \"\"#F*$\"\"!F`^p-%'LEGENDG6#%3Butcher's~scheme~AG-F$6%7_\\l7$F(Fefo7$ F/$!#wF-7$$\"5+DJ&p(G)*QASF-Fj^p7$$\"5M$eRsL]#)H-%F-Fj^p7$$\"5nTg_(z+R sFSF-Fh_p7$$\"5nmT5S\\#4*GSF-Fh_p7$$\"5+D1R+C>]HSF-Fh_p7$$\"5M$3x1')f% 4ISF-Fh_p7$$\"5nTN'4KF(oISF-Fh_p7$$\"5+++D\"y%*z7.%F-Fj^p7$$\"5Lek`TAE (=.%F-Fj^p7$$\"5m;H#=qHlC.%F-Fj^p7$$\"5+v$4@;(z0LSF-Fj^p7$$\"5LLeRAY1l LSF-Fh_p7$$\"5+](oHa*f$[.%F-Fh_p7$$\"5mm;ajW8-OSF-Fh_p7$$\"5+D\"GQ#>Sh OSF-Fh_p7$$\"5L$e9TQp1s.%F-Fj^p7$$\"5mT5SWo$*zPSF-Fh_p7$$\"5++vo/V?RQS F-Fj^p7$$\"5LeR(\\wr%)*QSF-Fh_p7$$\"5m;/ED#Rx&RSF-Fj^p7$$\"5+voa&o1q,/ %F-F`fo7$F4Fh_p7$$\"5+++voMrU^SF-Fh_p7$F9Fh_p7$FenF`fo7$FirF`fo7$F]wF` fo7$F`wF`fo7$FcwF`fo7$FhwF[fo7$$\"5mmm\"HdG\"\\)>%F-F[fo7$F]xF[fo7$Fbx Ffeo7$FgxFfeo7$FjxF[fo7$F]yF[fo7$F`yF[fo7$FcyFfeo7$FfyF[fo7$FiyFfeo7$F \\zF`fo7$F_zF[fo7$FbzFfeo7$FezF[fo7$FhzF[fo7$F[[lF[fo7$Fa[lF[fo7$F]\\l F[fo7$$\"5+]7.#3LNVA%F-F`fo7$$\"5nm\"z>O0M\\A%F-Ffeo7$$\"5M$3F>kxKbA%F -F[fo7$F`\\lF[fo7$$\"5n;H#=?AInA%F-F[fo7$$\"5ML3x\"[%*GtA%F-Ffeo7$$\"5 +](=h]%[0B%F-Ffeo7$$\"5ML$3x1ZA7B%F-F[fo7$$\"5+](=U2\\'*=B% F-Ffeo7$$\"5nm\"H23^qDB%F-Ffeo7$$\"5M$eRs3`WKB%F-F[fo7$$\"5+++v$4b=RB% F-Ffeo7$$\"5n;/E+rDfMUF-Ffeo7$$\"5ML3x1\"fm_B%F-Ffeo7$$\"5+]7G861%fB%F -Ffeo7$$\"5nm;z>JYhOUF-F[fo7$$\"5M$3-j7l)GPUF-F`fo7$$\"5++D\"G8nizB%F- Ffeo7$$\"5n;HKR\"pO'QUF-F[fo7$Ff\\lF[fo7$$\"5nmm\"z>(GqWUF-F[fo7$Fi\\l F[fo7$F[_lF[fo7$F^_lF`fo7$Fa_lF[fo7$Fd_lF[fo7$Fg_lF`fo7$$\"5+++D1*G8]F %F-F[fo7$Fj_lF[fo7$$\"5+++vo4xJwUF-Ffeo7$F]`lF[fo7$F``lF`fo7$Fc`lF`fo7 $Ff`lF[fo7$Fi`lF[fo7$$\"5+++Dcr4B!G%F-F[fo7$F\\alF`fo7$$\"5+++v=#RN:G% F-F[fo7$F_alF`fo7$FbalF[fo7$FealFfeo7$$\"5n;HKR1v!QH%F-F[fo7$$\"5MLekG X?*\\H%F-F`fo7$$\"5n\"H2LZJ%e&H%F-F`fo7$$\"5+](ozTewhH%F-F[fo7$$\"5M3- ji`)onH%F-F`fo7$$\"5nm;H2B6O(H%F-F`fo7$$\"5+DJ&>DR`zH%F-F`fo7$$\"5M$e9 m>mX&)H%F-F`fo7$$\"5nTgFTJz8*H%F-F[fo7$$\"5++v$f3?I(*H%F-F[fo7$$\"5n;/ EvRZ\"4I%F-F[fo7$FjalF[fo7$$\"5++](=UVPoI%F-F[fo7$F]blF[fo7$F`blF`fo7$ FcblFh_p7$F]glFj^p7$FgilFj^p7$F]jlFj^p7$F`jlFj^p7$FcjlFj^p7$FfjlFefo7$ FijlFefo7$F\\[mFj^p7$F_[mFj^p7$Fd[mFefo7$Fg[m$!#uF-7$Fc\\mFb_q7$F_]mFb _q7$F_`m$!#tF-7$FecmFjfo7$F[dm$!#qF-7$$\"5++Dc^8S$4[%F-F[`q7$F^dmF[`q7 $$\"5LeRA'H^]U[%F-F[`q7$$\"5+]i:&G\"Q\"\\[%F-Fjfo7$$\"5mT&)3u7rd&[%F-$ !#pF-7$$\"5LL3-j7/C'[%F-Fj`q7$$\"5m;a)3C,nv[%F-Fj`q7$FadmFj`q7$FgdmFj` q7$F]emFj`q7$F`emF[`q7$FcemFj`q7$FfemF[`q7$Fiem$!#oF-7$F_fmFj`q7$FefmF [`q7$FhfmFiaq7$F[gmFj`q7$F^gmF[`q7$FagmF[`q7$F]hmFj`q7$FchmFiaq7$FihmF iaq7$FeimFiaq7$FhimF_go7$F[jmF_go7$F^jmFiaq7$FajmF_go7$$\"5nT5S%f.vR^% F-Fiaq7$FdjmF_go7$$\"5n\"z%*z-+``^%F-Fiaq7$FgjmFiaq7$$\"5nT&)ehk4t;XF- $!#mF-7$$\"5n;aQyY*>u^%F-F_go7$$\"5n\"H#=&*G*3\"=XF-F_go7$FjjmF_go7$$ \"5n;HdXve%)ehc%F-$!#hF-7$F`dnFigq7$$\"5n;a)3uB(\\nXF-Figq7$$ \"5ML3-8Nk;oXF-Figq7$$\"5+]i:&GjN)oXF-Figq7$$\"5nm;HdI[]pXF-Fdgo7$$\"5 M$3F%HGS%>t?h%F- Ffjq7$$\"5+]7`>Imq7YF-F^[r7$$\"5+D19>m+M8YF-Ffjq7$FjinFfjq7$$\"5++v=YF-Ffjq7$$\"5+vVB:EWn>YF-Ffjq7$$\"5+]P%[@'yI ?YF-$!#aF-7$$\"5+DJX9)HT4i%F-F^ho7$$\"5++D19MZd@YF-F^^r7$$\"5+v=n8q\"3 Ai%F-F^^r7$$\"5+]7G81;%Gi%F-F^ho7$$\"5+D1*G@/vMi%F-Ffjq7$F]jnF^^r7$F`j nF^ho7$Fcjn$!#_F-7$FijnFb_r7$F_[oFb_r7$Fe[o$!#^F-7$F[\\oF`io7$F^\\oFch o7$Fa\\oFg_r7$Fd\\oFg_r7$Fg\\oFg_r7$Fj\\oFg_r7$F]]oFg_r7$FeaoF][p7$Fcc o$!#WF-7$F_doF`\\p7$FbdoFj\\p7$Fjdo$!#IF-7$Fdeo$!#AF-7$F^fo$!#:F-7$Fcf o$!#5F-7$Fhfo$!\"$F-7$F]go$F)F-7$Fbgo$\"#8F-7$Fggo$\"#@F-7$F\\ho$\"#HF -7$Ff[p$\"#QF-7$F^\\p$\"#^F-7$Fc\\p$\"#dF-7$Fh\\p$\"#gF-7$F]]p$\"#qF-7 $Fb]p$\"#!)F--Fg]p6&Fi]p$\"#XF\\^pF_^pFj]p-Fb^p6#%9scheme~with~simple~ nodesG-F$6%7[x7$F($!#sF-7$F/Fg_q7$F4Fb_q7$F9Fb_q7$F>Fb_q7$FAFb_q7$FOFb _q7$FUFb_q7$$\"5nm;/,Rg`zSF-Fb_q7$FXFb_q7$$\"5nmTN@n)[:3%F-Fefo7$$\"5n m;z%*4)>A3%F-Fefo7$$\"5nm\"H#o_2*G3%F-Fefo7$FenFefo7$FhnFefo7$F[oFb_q7 $F^oFb_q7$FaoFb_q7$FgoFb_q7$F]pFb_q7$F`pFb_q7$FcpFefo7$FfpFefo7$F[qFef o7$F^qFefo7$FaqFefo7$FdqFefo7$FgqFefo7$FjqFefo7$F]rFefo7$F`rFefo7$FcrF b_q7$FfrFb_q7$FirFb_q7$F]wFb_q7$F`wFefo7$FcwFefo7$FhwFefo7$F]xFj^p7$Fc \\lFj^p7$Fi\\lFefo7$F[_lFefo7$F^_lFb_q7$Fa_lFefo7$Fd_lFefo7$Fg_lFb_q7$ FjjpFefo7$Fj_lFefo7$F^[qFj^p7$F]`lFefo7$F``lFb_q7$Fc`lFb_q7$Ff`lFefo7$ Fi`lFefo7$Ff[qFefo7$F\\alFb_q7$Fj[qFefo7$F_alFb_q7$FbalFefo7$FealFj^p7 $F`\\qFefo7$Fc\\qFb_q7$Ff\\qFb_q7$Fi\\qFefo7$F\\]qFb_q7$F_]qFb_q7$Fb]q Fb_q7$Fe]qFb_q7$Fh]qFb_q7$F[^qFefo7$$\"5Me*)fIqCK+VF-Fb_q7$F^^qFb_q7$$ \"5+v=#*>4q],VF-Fb_q7$FjalFefo7$Fb^qFefo7$F]blFefo7$F`blFb_q7$FcblFg_q 7$F]glF]dr7$F]jlFjfo7$F_]mF[`q7$F_`mFj`q7$FecmF_go7$F[dmFdcq7$FagmFjdq 7$F]hmFbdq7$FchmFcfq7$FihmFcfq7$FeimFcfq7$FgjmFcfq7$F][nFcfq7$Fg]nFigq 7$Fe_n$!#fF-7$FedqF[iq7$FhdqF[iq7$F]eqF[iq7$Fh_nF[iq7$FaeqFgjr7$FdeqFg jr7$FgeqFgjr7$F[`nF[iq7$F[fqF[iq7$F^`nF[iq7$F_fqFgjr7$Fa`nFgjr7$Fd`nFi go7$Fg`nFigo7$Fj`nFgjr7$F]anFgjr7$FebnFigo7$F]dnF^[r7$F[gqF^[r7$F^gqF^ [r7$FagqF^[r7$FdgqF^[r7$FggqFcjq7$F`dnFcjq7$F]hqFcjq7$F`hqFcjq7$FchqFc jq7$FfhqF^[r7$FihqFfjq7$F^iqFfjq7$FaiqF^[r7$FcdnFcjq7$FfdnF^[r7$FidnFc jq7$F\\enF^[r7$F_enF^[r7$FbenFfjq7$FeenF^[r7$FhenFcjq7$F[fnFcjq7$F^fnF fjq7$FafnF^ho7$FignF^ho7$F_hnF^ho7$FbhnF^ho7$FehnF^ho7$FainFg_r7$FginF cho7$$\"5+v$4YA'>Z/YF-Fg_r7$FijqFcho7$$\"5+D\"GQU$)Qdg%F-Fb_r7$F\\[rFg _r7$Fa[rFcho7$Fd[rFcho7$Fg[rFg_r7$Fj[rFg_r7$F]\\rFcho7$F`\\rFcho7$Fc\\ rFcho7$Ff\\rFg_r7$Fi\\rFcho7$F\\]rFb_r7$F_]rFcho7$FjinFcho7$Fc]rFcho7$ Ff]rFcho7$Fi]rFcho7$F\\^rFhho7$Fa^rF`io7$Fd^rFhho7$Fg^rF`io7$Fj^rF`io7 $F]_rFhho7$F]jnFhho7$F`jnF`io7$FcjnF][p7$FijnF][p7$F_[oF][p7$Fe[o$!#YF -7$F[\\oF[\\p7$Fa\\o$!#XF-7$Fg\\oF\\`s7$$\"5+DcEsN[]WYF-F\\`s7$Fj\\oF` `s7$$\"5+v=U2\"[6fk%F-Fb`r7$F]]oF\\`s7$F[_oF[\\p7$Feao$!#TF-7$$\"5mmm \"HdxO&pYF-F]as7$FhaoF]as7$$\"5LLL3_][$\\n%F-$!#RF-7$F[boFe\\p7$Fabo$! #QF-7$FccoFfas7$FfcoFe\\p7$Fico$!#KF-7$F\\do$!#LF-7$F_do$!#MF-7$FbdoF_ ]p7$Fjdo$!#CF-7$Fdeo$!#;F-7$F^foF`ar7$Fcfo$!\"%F-7$Fhfo$\"\"$F-7$F]go$ \"#5F-7$Fbgo$\"#>F-7$Fggo$\"#FF-7$F\\ho$\"#OF-7$Ff[p$\"#WF-7$F^\\pFgbr 7$Fc\\p$\"#kF-7$Fh\\p$\"#nF-7$F]]p$\"#wF-7$Fb]p$\"#()F--Fg]p6&Fi]pF_^p $\"#DF\\^p$\"\"\"F`^p-Fb^p6#%Pscheme~with~a~relatively~large~stability ~regionG-F$6%7bq7$F($!$P\"F-7$F/$!$Q\"F-7$F4$!$R\"F-7$F9$!$S\"F-7$FA$! $T\"F-7$Fen$!$U\"F-7$F[oFjfs7$FaoFgfs7$FgoFjfs7$F]pFjfs7$F[qF]gs7$Faq$ !$V\"F-7$FgqF]gs7$FcrFjfs7$FfrFjfs7$FirFjfs7$F\\sF]gs7$F_sFegs7$F]tF]g s7$FctF]gs7$F_uFegs7$FeuFegs7$FavFegs7$F]wF]gs7$F`wFegs7$FcwFegs7$Fhw$ !$W\"F-7$F]x$!$X\"F-7$Fc\\lFihs7$Fi\\lFfhs7$F[_lFfhs7$F^_lFfhs7$Fa_lFi hs7$Fd_lFihs7$Fg_lFegs7$Fj_lFfhs7$F]`lFihs7$Fi`lFfhs7$F_alFegs7$FbalFf hs7$FealFihs7$$\"5Me9m%pB:KH%F-Ffhs7$F`\\qFihs7$$\"5+vV)Rex*R%H%F-Fihs 7$Fc\\qFegs7$Fi\\qFfhs7$F_]qFegs7$Fe]qFfhs7$F[^qFihs7$F^^qFfhs7$FjalFf hs7$Fb^qFfhs7$F]blFfhs7$F`blFegs7$FcblF]gs7$F]glFgfs7$F]jlFdfs7$F_]mF^ fs7$F_`m$!$O\"F-7$Fecm$!$L\"F-7$F[dm$!$J\"F-7$F^`qFd[t7$F^dm$!$I\"F-7$ F]aq$!$H\"F-7$Fadm$!$G\"F-7$F]emF[\\t7$FiemF^\\t7$FefmF[\\t7$FagmFh[t7 $F]hmF^\\t7$Fchm$!$F\"F-7$FihmFf\\t7$FeimFf\\t7$Fajm$!$E\"F-7$FgjmF[]t 7$FjjmF[]t7$F][nF[]t7$Fe_n$!$@\"F-7$F]dn$!$<\"F-7$Fafn$!$6\"F-7$Fgin$! $1\"F-7$F]jn$!$.\"F-7$F]]oFd]l7$FeaoFF7$FccoF+7$F_doF[fo7$FbdoFjfo7$Fj doFdgo7$FdeoFcho7$F^foFjas7$Fcfo$!#FF-7$FhfoF]ar7$F]goF`cs7$FbgoFfcs7$ Fggo$\"#CF-7$F\\ho$\"#RF-7$Ff[p$\"#aF-7$F^\\p$\"#vF-7$Fc\\pF_es7$Fh\\p $\"#$*F-7$F]]p$\"$2\"F-7$Fb]p$\"$E\"F--Fg]p6&Fi]pF_^p$Fh_tF\\^pF]^p-Fb ^p6#%TButcher's~scheme~B~with~c[5]=c[6]=1/2~and~b[5]=b[6]G-F$6%7h`l7$F (F[`q7$F/Fjfo7$Fd`pFjfo7$FfapFjfo7$FiapFjfo7$F\\bpFjfo7$F_bpFjfo7$Fbbp F]dr7$$\"5m\"H#o#3KVU.%F-Fjfo7$FebpFjfo7$$\"5L3_D.q'Ga.%F-Fjfo7$FhbpFj fo7$F^cpFjfo7$FdcpFjfo7$FgcpF]dr7$FjcpFjfo7$F]dpF]dr7$F4F]dr7$FadpF]dr 7$F9F]dr7$F>F]dr7$FAF]dr7$FIF]dr7$FOF]dr7$$\"5nmT&Q`K\"=wSF-Fg_q7$FRF] dr7$$\"5nm\"H23@Bv2%F-F]dr7$FUFg_q7$FXFg_q7$FenFg_q7$FhnFg_q7$F[oF]dr7 $F^oFg_q7$FaoF]dr7$FdoF]dr7$FgoFg_q7$FjoFg_q7$F]pFg_q7$FcpFg_q7$F[qFg_ q7$$\"5+voag&\\UA4%F-Fg_q7$F^qFg_q7$$\"5n\"H#oK.!yN4%F-Fg_q7$FaqFb_q7$ FgqFg_q7$FcrF]dr7$$\"5+v$fL>ab-5%F-Fg_q7$$\"5L$3F%z&HB45%F-Fg_q7$$\"5m \"z%\\l\\5f,TF-Fg_q7$FfrFg_q7$$\"5L3-jPdl#H5%F-Fg_q7$$\"5m;zpB6Vf.TF-F g_q7$$\"5+Dcw4l?E/TF-Fg_q7$FirF]dr7$F\\sFg_q7$F_sFb_q7$FdsFb_q7$FgsFb_ q7$FjsFg_q7$F]tF]dr7$$\"5++DJ&>.yH6%F-Fg_q7$F`tF]dr7$$\"5LL3xc=i@9TF-F g_q7$FctFg_q7$FeuFg_q7$F]wFg_q7$$\"5MLL$e*ot*\\8%F-Fg_q7$F`wFg_q7$$\"5 MLL$ekGhe:%F-Fg_q7$FcwFg_q7$$\"5++++]2`vrTF-Fb_q7$$\"5MLL$3_(>/xTF-Fef o7$$\"5nm;a8UOOyTF-Fb_q7$$\"5+++D14`ozTF-Fg_q7$$\"5nmTg_UhM!=%F-Fb_q7$ $\"5ML$e*)f(p+\"=%F-Fb_q7$$\"5++DJX4ym\"=%F-Fefo7$$\"5nmmm\"HkGB=%F-Fb _q7$$\"5MLL3xw>(\\=%F-Fb_q7$FhwFb_q7$$\"5LL$3x\")H`I>%F-Fb_q7$F[epFb_q 7$$\"5LL3_]z-@,UF-Fb_q7$$\"5++]7Gt#HR?%F-Fb_q7$$\"5mTg_s@!4Y?%F-Fefo7$ $\"5L$3Fp,x)G0UF-Fefo7$$\"5+D\"G8'=&of?%F-Fefo7$$\"5mm\"HdqE[m?%F-Fb_q 7$$\"5L3-8]:!Gt?%F-Fefo7$$\"5+]7`%Rw2!3UF-Fb_q7$$\"5m\"HK*Q7vo3UF-Fb_q 7$F]xFefo7$F]yFefo7$FiyFefo7$F\\zFg_q7$F_zFb_q7$FbzFefo7$FezFb_q7$FhzF b_q7$F[[lFb_q7$Fa[lFb_q7$F]\\lFb_q7$F_fpFb_q7$FbfpFefo7$FefpFb_q7$F`\\ lFb_q7$FifpFb_q7$F\\gpFefo7$F_gpFb_q7$Fc\\lFefo7$FcgpFefo7$FfgpFj^p7$F igpFefo7$F\\hpFb_q7$F_hpFefo7$FbhpFefo7$FehpFb_q7$FhhpFefo7$F[ipFefo7$ F^ipFefo7$FaipFefo7$FdipFb_q7$FgipFg_q7$FjipFefo7$F]jpFb_q7$Ff\\lFb_q7 $$\"5nmT&)e^(e1C%F-Fb_q7$$\"5++](=&)GSC%F-Fefo7$FajpFefo7$$\"5M$3FW?*o PXUF-Fb_q7$$\"5++v$4@\"40YUF-Fefo7$$\"5n;zW(R$Q%F-F]dr7$F dilF]dr7$$\"5+Dc^.,ei%Q%F-Fg_q7$FgilFg_q7$$\"5L3xJU0W&eQ%F-F]dr7$$\"5+ ](=6)4I3(Q%F-F]dr7$$\"5LL3_+7tp(Q%F-F]dr7$$\"5 +v=#*>9;J)Q%F-F]dr7$$\"5m;HKR;f#*)Q%F-F]dr7$$\"5LeRse=-a*Q%F-F]dr7$Fji lFg_q7$$\"5mTg_(H#)o2R%F-Fg_q7$$\"5L$3Fp^7$Q\"R%F-F]dr7$$\"5+D\"GjtU(* >R%F-F]dr7$$\"5mm\"Hd&H*[,4Z%F-Fiaq7$$\"5L $ek`mPJ@Z%F-Fiaq7$$\"5LL$e9TEhLZ%F-Fiaq7$$\"5LLek.R5#eZ%F-Fiaq7$F[dmFi aq7$$\"5m;zpt8ugzWF-Fiaq7$F^`qFiaq7$$\"5m\"z%\\S8tf\"[%F-Fiaq7$$\"5L$3 F%H81E#[%F-F_go7$$\"5+v$f$=8R#H[%F-Fiaq7$F^dmF_go7$F]aqF_go7$FadmF_go7 $FgdmF_go7$F]emF_go7$F`emFiaq7$FcemF_go7$FfemFiaq7$FiemFdcq7$F_fmF_go7 $FefmFiaq7$FhfmFjdq7$F[gmF_go7$F^gmFiaq7$FagmFiaq7$FeimF_go7$F][nFdcq7 $Fe_nFcfq7$F]dnFigqF]jq7$FginFcjq7$F]jnFfjq7$F`jnF^^r7$FcjnF^ho7$FijnF ^ho7$F_[oF^ho7$Fb[oF^ho7$Fe[oF^ho7$Fh[oFb_r7$F[\\oFcho7$Fa\\oFb_r7$Fg \\oF^ho7$Fd`sF^ho7$Fj\\oFb_r7$Fh`sFg_r7$F]]oF^ho7$F`]oFb_r7$Fc]oFcho7$ Fi]oFg_r7$F_^oFb_r7$Fb^oFg_r7$Fe^oFcho7$Fh^oFg_r7$F[_oFcho7$F^_oFhho7$ Fa_oF`io7$Fg_oF`io7$F]`oF`io7$F``oFhho7$Fc`oFhho7$Ff`oF`io7$Fi`oF`io7$ F\\aoF][p7$F_aoFhho7$FbaoF`io7$FeaoFhho7$$\"5mm\"H#o(\\7vm%F-F`io7$$\" 5++]7.ds=oYF-Fhho7$$\"5LL3-Q;?')oYF-Fhho7$F`asF`io7$$\"5LL$3FWH')3n%F- F`io7$FhaoF`io7$FdasF][p7$F[boF``s7$FaboF\\`s7$FccoF][p7$FfcoF``s7$Fic oF]as7$F\\do$!#UF-7$F_doF[\\p7$FbdoFfas7$FjdoFebs7$Fdeo$!#GF-7$F^foFj` r7$Fcfo$!#=F-7$Fhfo$!#6F-7$F]go$!\"'F-7$Fbgo$F^^pF-7$Fggo$\"\"*F-7$F\\ ho$\"#;F-7$Ff[p$\"#BF-7$F^\\p$\"#MF-7$$\"5++]PMA1eg\\F-Fabr7$$\"5+++Dc wz5j\\F-$\"#TF-7$$\"5+](=<^\")RP'\\F-F[cv7$$\"5++v=n`;Pk\\F-F[cv7$$\"5 +]ilA#\\.]'\\F-F[cv7$$\"5++]7yI`jl\\F-$\"#SF-7$$\"5+]PfLprEm\\F-F[cv7$ $\"5++D1*y+**o'\\F-Fa_t7$$\"5+]7`WY3`n\\F-Fa_t7$Fc\\pFicv7$$\"5+++vV$R $FiarFhfv" 1 2 0 1 10 0 2 9 1 4 2 1.000000 45.000000 45.000000 0 1 "Butcher's sch eme A" "scheme with simple nodes" "scheme with a relatively large stab ility region" "Butcher's scheme B with c[5]=c[6]=1/2 and b[5]=b[6]" "s cheme with c[5]=c[6]=3/4 and b[5]=b[6]" }}}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 539 "evalf[20](plot(['qn_RK6_1'(x)-q(x),'qn_RK6_2 '(x)-q(x),'qn_RK6_3'(x)-q(x),'qn_RK6_4'(x)-q(x),\n'qn_RK6_5'(x)-q(x)], x=0.7..0.87,-1.2e-13..1.4e-13,font=[HELVETICA,9],\ncolor=[COLOR(RGB,.9 5,.2,0),COLOR(RGB,.45,0,.95),COLOR(RGB,0,.25,1),COLOR(RGB,0,.75,.2),CO LOR(RGB,.95,.45,0)],\nlegend=[`Butcher's scheme A`,`scheme with simple nodes`,`scheme with a relatively large stability region`,`Butcher's s cheme B with c[5]=c[6]=1/2 and b[5]=b[6]`,`scheme with c[5]=c[6]=3/4 a nd b[5]=b[6]`],title=`error curves for 7 stage order 6 Runge-Kutta met hods`));" }}{PARA 13 "" 1 "" {GLPLOT2D 823 527 527 {PLOTDATA 2 "6+-%'C URVESG6%7in7$$\"\"(!\"\"$\"'9FI!#@7$$\"5LLLL3i^0Pq!#?$\"&aA$F17$$\"5nm m\"z/m'HpqF1$\"&jT$F17$$\"5LLL$eufbb5(F1$\"&=f$F17$$\"5LLL$3A)[0UrF1$ \"&4u$F17$$\"5nmm\"HAp!QyrF1$\"&3&QF17$$\"5LLLe9o$f?@(F1$\"&e)QF17$$\" 5+++DJ'oJpC(F1$\"&\")*QF17$$\"5LLLek(o'*HG(F1$\"&p#QF17$$\"5+++D1Gg%*= tF1$\"&mj$F17$$\"5nmmmTVV#fN(F1$\"&))G$F17$$\"5LLL$3P'[\\)Q(F1$\"&#RFF 17$$\"5++++Db:;DuF1$\"&h#>F17$$\"5++++v.)y>Y(F1$\"%)z(F17$$\"5++++vW!f u\\(F1$!%>FF17$$\"5LLLek-&y'HvF1$!&7y\"F17$$\"5nmmmTa0*zc(F1$!&#*>%F17 $$\"5nmmmm()eW+wF1$!&]H(F17$$\"5+++D\"e:*>QwF1$!''G8\"F17$$\"5nmmm;&>< ;n(F1$!'BZ;F17$$\"5+++D\"33#G3xF1$!'w&H#F17$$\"5+++vVAd>VxF1$!',7GF17$ $\"5nmm;zWWizxF1$!'XkOF17$$\"5nmm\"HPQxI\"yF1$!']+ZF17$$\"5LLL$3x*3;\\ yF1$!'#y0'F17$$\"5LLLekF:k')yF1$!'w3xF17$$\"5+++v=E&o#>zF1$!''=p*F17$$ \"5LLL$e%yl]azF1$!(UV?\"F17$$\"5++++]M4\"4*zF1$!(s3U\"F17$$\"5+++]i%\\ Dl-)F1$!(Gmm\"F17$$\"5+++D\"GT%)41)F1$!(]7(>F17$$\"5+++]Pm^C*4)F1$!(jR M#F17$$\"5nmmm;bTiL\")F1$!(t4t#F17$$\"5++++vj5Lq\")F1$!(8*GJF17$$\"5LL LekyHf.#)F1$!(()[\\$F17$$\"5++++DPq&*R#)F1$!(Buw$F17$$\"5nmm\"zCysTF)F 1$!(Yi%QF17$$\"5+++D1&4Q*4$)F1$!(2$fPF17$$\"5nmmmTD_!\\M)F1$!(%=2LF17$ $\"5+++DJqD^\"Q)F1$!)bL+AF-7$$\"5LLLLL%zpnT)F1$!(z_I\"F-7$$\"5LLL$3#)R DGX)F1$\")X=dLF-7$$\"5nmm\"zMW#e)[)F1$\"(!3@))F17$$\"5NL$e*)*[4,0&)F1$ \")so]7F17$$\"5++++]a%R9_)F1$\")'e9a\"F17$$\"5NLL3xv$o-a)F1$\")_*p-#F1 7$$\"5nmm;/(H(4f&)F1$\")i'**f#F17$$\"5+++vo3\"Qfd)F1$\")-HFIF17$$\"5LL LLL?*yFf)F1$\")5U^OF17$$\"5lm;H2QZt5')F1$\")dcnWF17$$\"5+++D\"eb!pG')F 1$\")]x0aF17$$\"5++vV)z0()Hj)F1$\")50ocF17$$\"5++]i:gNGP')F1$\")y!>X'F 17$$\"5++D\"GB1!eT')F1$\")eJ\\kF17$$\"5++++]kl(ek)F1$\")(4sX'F17$$\"5+ +]P%)o&pWl)F1$\")pUwtF17$$\"5+++v=tD1j')F1$\")*3$GwF17$$\"5++D1*)HpHs' )F1$\")sgm#)F17$$\"5++]Pf'GJ:o)F1$\")Mg$*))F17$$\"5++voHVcw!p)F1$\")7m T%*F17$$\"#()!\"#$\"*$zO@5F1-%&COLORG6&%$RGBG$\"#&*Fc^l$\"\"#F*$\"\"!F __l-%'LEGENDG6#%3Butcher's~scheme~AG-F$6%7in7$F($\"'U]UF-7$F/$\"&=^%F1 7$F5$\"&zv%F17$F:$\"&h(\\F17$F?$\"&(\\^F17$FD$\"&)e_F17$FI$\"&\\F&F17$ FN$\"&NB&F17$FS$\"&\\0&F17$FX$\"&#*o%F17$Fgn$\"&x2%F17$F\\o$\"&H:$F17$ Fao$\"&>#=F17$Ffo$!$6#F17$F[p$!&;p\"F17$F`p$!&Q2%F17$Fep$!&8'yF17$Fjp$ !'0o7F17$F_q$!'h#*=F17$Fdq$!'u&o#F17$Fiq$!'4#o$F17$F^r$!'#HZ%F17$Fcr$! 'IwdF17$Fhr$!'(oN(F17$F]s$!''HU*F17$Fbs$!(mI>\"F17$Fgs$!(#p$\\\"F17$F \\t$!(V%\\=F17$Fat$!(Oj<#F17$Fft$!(/na#F17$F[u$!(%y/IF17$F`u$!(4Nc$F17 $Feu$!(@79%F17$Fju$!(!RJZF17$F_v$!(?yE&F17$Fdv$!(.cl&F17$Fiv$!(#G^dF17 $F^w$!(cJe&F17$Fcw$!(:o%[F17$Fhw$!)Wr(4$F-7$F]x$\"(#GP8F-7$Fbx$\")4GVb F-7$Fgx$\")QG)R\"F17$F\\y$\")LKm>F17$Fay$\")&RQT#F17$Ffy$\")pcgJF17$F[ z$\")>%3/%F17$F`z$\")b#op%F17$Fez$\"),JacF17$Fjz$\")9C0pF17$F_[l$\")[# >M)F17$Fd[l$\")5:V()F17$Fi[l$\")wQU**F17$F^\\l$\")5RQ**F17$Fc\\l$\")%> /&**F17$Fh\\l$\"*k:b8\"F17$F]]l$\"*R-S<\"F17$Fb]l$\"*d?9F\"F17$Fg]l$\" *Y4rO\"F17$F\\^l$\"*&Ri]9F17$Fa^l$\"*&oDo:F1-Fg^l6&Fi^l$\"#XFc^lF^_lFj ^l-Fa_l6#%9scheme~with~simple~nodesG-F$6%7in7$F($\"'RBUF-7$F/$\"&9[%F1 7$F5$\"&Ps%F17$F:$\"&x$\\F17$F?$\"&j5&F17$FD$\"&+@&F17$FI$\"&DA&F17$FN $\"&^<&F17$FS$\"&#*)\\F17$FX$\"&_h%F17$Fgn$\"&U*RF17$F\\o$\"&'eIF17$Fa o$\"&`r\"F17$Ffo$!%;9F17$F[p$!&I#=F17$F`p$!&\">UF17$Fep$!&h-)F17$Fjp$! 's'G\"F17$F_q$!'\"Q\">F17$Fdq$!'&)4FF17$Fiq$!']4PF17$F^r$!'x-XF17$Fcr$ !'*)4eF17$Fhr$!'m%R(F17$F]s$!'&eY*F17$Fbs$!(Gz>\"F17$Fgs$!(\">*\\\"F17 $F\\t$!(Pc&=F17$Fat$!(@J=#F17$Fft$!(.Tb#F17$F[u$!(oG,$F17$F`u$!(7Bd$F1 7$Feu$!(11:%F17$Fju$!(<6u%F17$F_v$!(^uF&F17$Fdv$!(3Wm&F17$Fiv$!(O(edF1 7$F^w$!('*ze&F17$Fcw$!(Wq%[F17$Fhw$!)\\-!4$F-7$F]x$\"(.v`\"F-7$Fbx$\") Q)>e&F-7$Fgx$\").*[S\"F17$F\\y$\")iru>F17$Fay$\")3mBCF17$Ffy$\")PpsJF1 7$F[z$\")hmbSF17$F`z$\")ll8ZF17$Fez$\")=.ucF17$Fjz$\")JxGpF17$F_[l$\") %p)p$)F17$Fd[l$\")!*Rs()F17$Fi[l$\")#3`(**F17$F^\\l$\")&)Hr**F17$Fc\\l $\")1P$)**F17$Fh\\l$\"*!3ER6F17$F]]l$\"*vmy<\"F17$Fb]l$\"*Z3cF\"F17$Fg ]l$\"*$ogr8F17$F\\^l$\"*K3aX\"F17$Fa^l$\"*.UMd\"F1-Fg^l6&Fi^lF^_l$\"#D Fc^l$\"\"\"F__l-Fa_l6#%Pscheme~with~a~relatively~large~stability~regio nG-F$6%7in7$F($\"''*>uF-7$F/$\"&+)yF17$F5$\"&ZJ)F17$F:$\"&@q)F17$F?$\" &M,*F17$FD$\"&Z@*F17$FI$\"&4D*F17$FN$\"&H>*F17$FS$\"&+!*)F17$FX$\"&hG) F17$Fgn$\"&#\\sF17$F\\o$\"&En&F17$Fao$\"&kR$F17$Ffo$\"%zBF17$F[p$!&&HE F17$F`p$!&=s'F17$Fep$!'UB8F17$Fjp$!'q_@F17$F_q$!'?GKF17$Fdq$!'q%f%F17$ Fiq$!'67jF17$F^r$!'&en(F17$Fcr$!'%R#**F17$Fhr$!($4l7F17$F]s$!(r;i\"F17 $Fbs$!($fa?F17$Fgs$!(KPd#F17$F\\t$!(^#)=$F17$Fat$!(-Jv$F17$Fft$!(VKR%F 17$F[u$!(e_=&F17$F`u$!(y<:'F17$Feu$!([=:(F17$Fju$!(2Y<)F17$F_v$!(9h5*F 17$Fdv$!(@Iy*F17$Fiv$!(9^&**F17$F^w$!(_an*F17$Fcw$!(!4>%)F17$Fhw$!)D7? aF-7$F]x$\"(j\\J\"F-7$Fbx$\")$R`V*F-7$Fgx$\")p2'R#F17$F\\y$\"))>TP$F17 $Fay$\")C([9%F17$Ffy$\")D2JaF17$F[z$\")&*fZpF17$F`z$\")')*y2)F17$Fez$ \")A)ys*F17$Fjz$\"*V'R)=\"F17$F_[l$\"*MqgV\"F17$Fd[l$\"*We_]\"F17$Fi[l $\"*zN?r\"F17$F^\\l$\"*oZ8r\"F17$Fc\\l$\"**QU8F17$F ]]l$\"*Y&3A?F17$Fb]l$\"*Rh,>#F17$Fg]l$\"*4]_N#F17$F\\^l$\"*K#R*\\#F17$ Fa^l$\"*#\\U-FF1-Fg^l6&Fi^lF^_l$\"#vFc^lF\\_l-Fa_l6#%TButcher's~scheme ~B~with~c[5]=c[6]=1/2~and~b[5]=b[6]G-F$6%7in7$F($\"'$Gr$F-7$F/$\"&K%RF 17$F5$\"&4;%F17$F:$\"&]N%F17$F?$\"&6^%F17$FD$\"&@h%F17$FI$\"&.j%F17$FN $\"&:g%F17$FS$\"&bX%F17$FX$\"&'[TF17$Fgn$\"&-j$F17$F\\o$\"&>%GF17$Fao$ \"&Nq\"F17$Ffo$\"%P7F17$F[p$!&1J\"F17$F`p$!&wN$F17$Fep$!&`h'F17$Fjp$!' Rw5F17$F_q$!'U9;F17$Fdq$!'/)H#F17$Fiq$!'BdJF17$F^r$!'YRQF17$Fcr$!'8k\\ F17$Fhr$!'TGjF17$F]s$!'H7\")F17$Fbs$!(2y-\"F17$Fgs$!(9vG\"F17$F\\t$!(E \\f\"F17$Fat$!(![x=F17$Fft$!()o(>#F17$F[u$!(YQf#F17$F`u$!([s2$F17$Feu$ !(ktd$F17$Fju$!(<()3%F17$F_v$!(fUb%F17$Fdv$!(+A*[F17$Fiv$!(Fw(\\F17$F^ w$!(2m$[F17$Fcw$!(;k?%F17$Fhw$!)Wp.FF-7$F]x$\"'a$p(F-7$Fbx$\")(ydt%F-7 $Fgx$\")]!3?\"F17$F\\y$\")'=/p\"F17$Fay$\")!3i2#F17$Ffy$\")x+?FF17$F[z $\")e/zMF17$F`z$\"))QZ/%F17$Fez$\")P]q[F17$Fjz$\")qX\\fF17$F_[l$\")zv) =(F17$Fd[l$\")0*[`(F17$Fi[l$\")/[p&)F17$F^\\l$\")f.m&)F17$Fc\\l$\")pTw &)F17$Fh\\l$\")oU)y*F17$F]]l$\"*I]?,\"F17$Fb]l$\"*?9h4\"F17$Fg]l$\"*L( oy6F17$F\\^l$\"*jl2D\"F17$Fa^l$\"*6\"H_8F1-Fg^l6&Fi^lFj^lF`[mF^_l-Fa_l 6#%Hscheme~with~c[5]=c[6]=3/4~and~b[5]=b[6]G-%%FONTG6$%*HELVETICAG\"\" *-%+AXESLABELSG6$Q\"x6\"Q!Fa`o-%&TITLEG6#%Uerror~curves~for~7~stage~or der~6~Runge-Kutta~methodsG-%%VIEWG6$;F(Fa^l;$!#7!#9$\"#9F^ao" 1 2 0 1 10 0 2 9 1 4 2 1.000000 45.000000 45.000000 0 1 "Butcher's scheme A" " scheme with simple nodes" "scheme with a relatively large stability re gion" "Butcher's scheme B with c[5]=c[6]=1/2 and b[5]=b[6]" "scheme wi th c[5]=c[6]=3/4 and b[5]=b[6]" }}}}{PARA 0 "" 0 "" {TEXT -1 0 "" }} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 523 "evalf[20](plot(['qn_RK6_1'( x)-q(x),'qn_RK6_2'(x)-q(x),'qn_RK6_3'(x)-q(x),'qn_RK6_4'(x)-q(x),\n'qn _RK6_5'(x)-q(x)],x=0.87..0.925,font=[HELVETICA,9],\ncolor=[COLOR(RGB,. 95,.2,0),COLOR(RGB,.45,0,.95),COLOR(RGB,0,.25,1),COLOR(RGB,0,.75,.2),C OLOR(RGB,.95,.45,0)],\nlegend=[`Butcher's scheme A`,`scheme with simpl e nodes`,`scheme with a relatively large stability region`,`Butcher's \+ scheme B with c[5]=c[6]=1/2 and b[5]=b[6]`,`scheme with c[5]=c[6]=3/4 \+ and b[5]=b[6]`],title=`error curves for 7 stage order 6 Runge-Kutta me thods`));" }}{PARA 13 "" 1 "" {GLPLOT2D 747 615 615 {PLOTDATA 2 "6+-%' CURVESG6%7fs7$$\"#()!\"#$\"*$zO@5!#?7$$\"5nmm;zM%))>r)F-$\"*o!=a6F-7$$ \"5LL$3-y]>Cs)F-$\"*?]]:\"F-7$$\"5nmmTNS.:M()F-$\"*M#4y7F-7$$\"5nmm\"z C$*efu)F-$\"*Ek!*H\"F-7$$\"5LL$3F/S6xv)F-$\"*@ODQ\"F-7$$\"5nm;ajUugo() F-$\"*7E(\\9F-7$$\"5++](=Km*))z()F-$\"*:n@X\"F-7$$\"5nm;a)=vd:z)F-$\"* ^f`Y\"F-7$$\"5++]P%3U)=.))F-$\"*c9)[9F-7$$\"5LLL$eM*>::))F-$\"*![4#R\" F-7$$\"5nmm\"HZ^*oD))F-$\"*AA2?\"F-7$$\"5+++](yE_v$))F-$\"*aEh=\"F-7$$ \"5+++]iIPY\\))F-$\")H!)4%)F-7$$\"5+++]7KE%4'))F-$\")0c3jF-7$$\"5nm;aQ ulOr))F-$\");O#f#F-7$$\"5LLL$e*\\;w$)))F-$!)_TTgF-7$$\"5LLLL$[!>E%*))F -$!)phwkF-7$$\"5++](ou>wk!*)F-$!*!>(f#=F-7$$\"5LLLL3;zG<*)F-$!+A:@;B!# @7$$\"5++](o\\3]\"H*)F-$!+kd(*)[$Fgq7$$\"5++]i!\\nX/%*)F-$!,\"fL!)Rc!# A7$$\"5lmTNYc&Qj%*)F-$!+tgDRcFgq7$$\"5LLL3-Q9B_*)F-$!++(*[EeFgq7$$\"5N ekyA9Ve`*)F-$!+[f@@gFgq7$$\"5N$e*[V!>P\\&*)F-$!+#3rkO'Fgq7$$\"5N3F>km+ Hc*)F-$!+APQ\\pFgq7$$\"5NLe*[G%Hkd*)F-$!+AO.%*yFgq7$$\"5N$3-j_p[.'*)F- $!+WTm`$)Fgq7$$\"5LL$3xwWaI'*)F-$!+!=^;N)Fgq7$$\"5nmm\"Hd_GZ(*)F-$!*]# *zj*F-7$$\"5nm;a)=hao)*)F-$!+z/Zm6F-7$$\"5++voHgN\\*)*)F-$!+qi0#=\"F-7 $$\"5NLL$3(3D8#**)F-$!+2=3Q7F-7$$\"5qm\"z>rXrZ**)F-$!+!*[t)R\"F-7$$\"5 ++]7`0/T(**)F-$!+`46V:F-7$$\"5NL3Fpy16.!*F-$!+aolU:F-7$$\"5nmmT&=&4\") 3!*F-$!+zX\"zc\"F-7$$\"5N$e9To=$G5!*F-$!+Iy:)f\"F-7$$\"5++D\"G=Ub<,*F- $!+ef]a;F-7$$\"5q;/^\"olFK,*F-$!+BaX` F-7$$\"5qmTgxhVkF-7$$\"5++++vJ))e?!*F-$!+r'*oa>F-7$$\"5+ ++vV*=6@.*F-$!+!4'Hg@F-7$$\"5++](oHmfK/*F-$!+L\")QlBF-7$$\"5+](=UFGaj/ *F-$!+D&=:Q#F-7$$\"5++Dc^-*[%\\!*F-$!+xSe^CF-7$$\"5+vVBS7i*40*F-$!+f\" =U`#F-7$$\"5+]i!*GANa_!*F-$!+S!*HuEF-7$$\"5+D\"yv@$34a!*F-$!+C,!zq#F-7 $$\"5+++D1U\"Qc0*F-$!+#f\"[2FF-7$$\"5lm;H2\\%*>h!*F-$!+!oynq#F-7$$\"5L LLL3c2wm!*F-$!+s+QHFF-7$$\"5+++]i?ljy!*F-$!+#pC])GF-7$$\"5nm;aQGxR*3*F -$!+S)GW\"GF-7$$\"5NL3-j\\,G&4*F-$!+611bFF-7$$\"5+++](3di65*F-$!+fplXF F-7$$\"5lT&)e'eHYD5*F-$!+XbmLFF-7$$\"5I$3xc3-IR5*F-$!+**[()3FF-7$$\"5+ Dcw%eu8`5*F-$!+Wd\"3m#F-7$$\"5lmT&Q3Z(p1\"*F-$!+]^ssDF-7$$\"5I3F%He>\" 33\"*F-$!+Ju#)=CF-7$$\"5+]7.#3#\\Y4\"*F-$!+.S!3;#F-7$$\"5l\"z>6ek[36*F -$!+Av6<@F-7$$\"5LL$3-3PKA6*F-$!+rO#o6#F-7$$\"5+++vV`^7:\"*F-$!+Zk-;@F -7$$\"5lm;H2Oz,=\"*F-$!+3Yy6@F-7$$\"5++D1RFVY>\"*F-$!+_^L.@F-7$$\"5ILL $3(=2\"47*F-$!+H/&G3#F-7$$\"5lmTg-5rNA\"*F-$!+&fEy.#F-7$$\"5++]PM,N!Q7 *F-$!+FTlY>F-7$$\"5I$eRA^sJm7*F-$!+AZ$G[\"F-7$$\"5lmT5!*[*f%H\"*F-$!*F B\\;(F-7$$\"5+](ozE<)GK\"*F-$!*CyE;(F-7$$\"5LLL$ekR;^8*F-$!*%\\!)[rF-7 $$\"5lT&Q`0%ofO\"*F-$!*Z^$3rF-7$$\"5+]P%[YGx!Q\"*F-$!*Bqm)pF-7$$\"5Ie* [V(GxbR\"*F-$!*P6Mn'F-7$$\"5lmT&QGRLH![\"*F-$\"+&)H8R k\"*F-$\"+t(4>E$F-7$$\"5mmm\"zkuJ+ <*F-$\"+'Qve`&F-7$$\"5N$e*[VgQ#H<*F-$\"+<%Qo`&F-7$$\"5++D1Ruf\"e<*F-$ \"+Od9obF-7$$\"5Ie*[o8.is<*F-$\"+d!=,j&F-7$$\"5l;ajM)33(y\"*F-$\"+`Tnq dF-7$$\"5+v=UKXT:!=*F-$\"+fkxfgF-7$$\"5LL$3-B?+;=*F-$\"+fL46mF-7$$\"5+ ]iSmaxD%=*F-$\"+!4\\Q&*)F-7$$\"5lmTg-2`\"p=*F-$\",/9kl3\"F-7$$\"5I$3-) QfGd*=*F-$\"-y\\7H'3\"Fgq7$$\"5++++v6/B#>*F-$\"-!G>$\\'3\"Fgq7$$\"5l;H d?RLv$>*F-$\"-*yU6z3\"Fgq7$$\"5ILe9mmiF&>*F-$\"-m?KY#4\"Fgq7$$\"5l\"HK *QIx.'>*F-$\"-Zv,2(4\"Fgq7$$\"5+](=*z'>*F-$\"-?)H@U5\"Fgq7$$\"5I3_] %ylgv>*F-$\"-k6@)\\6\"Fgq7$$\"5lm;Hd@@K)>*F-$\"-2J*F-$\"-f3j.&=\"Fgq7$$\"5++vV[wzO,#*F-$\"-?45-)G\"Fgq7$$\"5l;/,% R!4*G?*F-$\".`-I()=Z\"Fbr7$$\"5LLLeRJQT/#*F-$\".)3*H$fkC0`#F-7$$\"5+ ]7.d@&zQB*F-$\",GIy]`#F-7$$\"5+vo/)RauXB*F-$\",70y=a#F-7$$\"5++D1Rm&p_ B*F-$\",u&Gu^DF-7$$\"5+]P4@6'fmB*F-$\",&>Z3&e#F-7$$\"5++]7.c'\\!Q#*F-$ \",$\\O!ok#F-7$$\"5+]PM-Us.T#*F-$\",U=Jn'HF-7$$\"5++Dc,G[-W#*F-$\",=Xq I@$F-7$$\"5+]7y+9C,Z#*F-$\",>'\\97KF-7$$\"$D*!\"$$\",!zC57KF--%&COLORG 6&%$RGBG$\"#&*F*$\"\"#!\"\"$\"\"!Fbgm-%'LEGENDG6#%3Butcher's~scheme~AG -F$6%7fs7$F($\"*&oDo:F-7$F/$\"*d.-x\"F-7$F4$\"*)G]rF-7 $F>$\"*Mq&*)>F-7$FC$\"*jn_6#F-7$FH$\"*;%*[@#F-7$FM$\"*\\n$=AF-7$FR$\"* OnUB#F-7$FW$\"*kLw?#F-7$Ffn$\"*=Ol6#F-7$F[o$\"*#)fm\"=F-7$F`o$\"*QESz \"F-7$Feo$\"*,yuD\"F-7$Fjo$\")4d@$*F-7$F_p$\")+'oc$F-7$Fdp$!)O4r(*F-7$ Fip$!*'3;W5F-7$F^q$!*b<:'GF-7$Fcq$!+:N`;OFgq7$Fiq$!+Yp7BaFgq7$F^r$!,!Q t%Gt)Fbr7$Fdr$!+=t'>t)Fgq7$Fir$!+q0s>!*Fgq7$F^s$!+p30>$*Fgq7$Fcs$!+V- \")\\)*Fgq7$Fhs$!,lx(fu5Fgq7$F]t$!,epO)>7Fgq7$Fbt$!,et00H\"Fgq7$Fgt$!, C`%>!H\"Fgq7$F\\u$!+i$fx[\"F-7$Fau$!+?l1*z\"F-7$Ffu$!+P#zH#=F-7$F[v$!+ Or%*3>F-7$F`v$!+[HZb@F-7$Fev$!+l`-xBF-7$Fjv$!+ZEKwBF-7$F_w$!+'3J]T#F-7 $Fdw$!+,^QhCF-7$Fiw$!+(z\\xa#F-7$F^x$!+1pS*p#F-7$Fcx$!+e&)3`HF-7$Fhx$! +5-d3IF-7$F]y$!+#Q%F-7$Fd\\l$!+pSZCWF-7$F i\\l$!+]GK6VF-7$F^]l$!+=$*y;UF-7$Fc]l$!+E\"GA?%F-7$Fh]l$!+@>i$=%F-7$F] ^l$!+W]9XTF-7$Fb^l$!+S;aqSF-7$Fg^l$!+M:yLRF-7$F\\_l$!+_g$[p$F-7$Fa_l$! +@K>%H$F-7$Ff_l$!+&\\kjA$F-7$F[`l$!+om\"fA$F-7$F``l$!+\">)pCKF-7$Fe`l$ !+9A;=KF-7$Fj`l$!+(>A^?$F-7$F_al$!+-S\\tJF-7$Fdal$!+%\\eR5$F-7$Fial$!+ A17jHF-7$F^bl$!+DdVYAF-7$Fcbl$!+\\!3?1\"F-7$Fhbl$!+JSnh5F-7$F]cl$!+m*[ &f5F-7$Fbcl$!+F-7$F`el$\"+[b?QCF-7$Feel$\"+d$Q[s#F -7$Fjel$\"+17uCFF-7$F_fl$\"+4TkCFF-7$Fdfl$\"+,+XCFF-7$Fifl$\"+tuDCFF-7 $F^gl$\"+rG#Rs#F-7$Fcgl$\"+`z-CFF-7$Fhgl$\"+m&Q^t#F-7$F]hl$\"+i>\\IGF- 7$Fbhl$\"+u-Wp]F-7$Fghl$\"+Sb)=d)F-7$F\\il$\"+a![Ld)F-7$Fail$\"+axY@') F-7$Ffil$\"+!penr)F-7$F[jl$\"+#\\;H$*)F-7$F`jl$\"+:Rdx$*F-7$Fejl$\",dX iD-\"F-7$Fjjl$\",k\"\\)HQ\"F-7$F_[m$\",kzOrn\"F-7$Fd[m$\"-\\jarw;Fgq7$ Fi[m$\"-.lN-x;Fgq7$F^\\m$\"-JA7?z;Fgq7$Fc\\m$\"-95N>'o\"Fgq7$Fh\\m$\"- ks9F$p\"Fgq7$F]]m$\"-5)))eUq\"Fgq7$Fb]m$\"-chKz?Fgq7$Ff^m$\".$H$Q#GpAFbr7$F[_m$ \".CA:Z\">FFbr7$F`_m$\"-x[#3%=FFgq7$Fe_m$\"-D$>syr#Fgq7$Fj_m$\"-,nLE=F Fgq7$F_`m$\"-KX'e.s#Fgq7$Fd`m$\"-K11fEFFgq7$Fi`m$\"-aj*G;u#Fgq7$F^am$ \"-:z^oZGFgq7$Fcam$\",q4tm@$F-7$Fham$\",!3f\"Hh$F-7$F]bm$\",b)o9')QF-7 $Fbbm$\",\"oGe&)QF-7$Fgbm$\",@pH])QF-7$F\\cm$\",StKX)QF-7$Facm$\",FY>, *QF-7$Ffcm$\",b2IY*QF-7$F[dm$\",7k2;!RF-7$F`dm$\",e_D?\"RF-7$Fedm$\",s _Qr#RF-7$Fjdm$\",gW7#yRF-7$F_em$\",zHPF2%F-7$Fdem$\",>oADc%F-7$Fiem$\" ,0+5&R\\F-7$F^fm$\",&Rp3Q\\F-7$Fcfm$\",W:7!Q\\F--Fifm6&F[gm$\"#XF*Fagm F\\gm-Fdgm6#%9scheme~with~simple~nodesG-F$6%7fs7$F($\"*.UMd\"F-7$F/$\" *)H7wF-7$F>$\"*Ouj*>F-7$FC$\"*ozE7#F-7 $FH$\"*7fHA#F-7$FM$\"*b7#F-7$F[o$\"*JIc#=F-7$F`o$\"*FJH!=F-7$Feo$\"*`&)eE\"F-7$Fjo$\")(H#*R *F-7$F_p$\")!fvj$F-7$Fdp$!)#)=C(*F-7$Fip$!*S&oR5F-7$F^q$!*Ho1'GF-7$Fcq $!+EAo#\\w5Fgq7$F]t$!, :x0@A\"Fgq7$Fbt$!,E%3&HH\"Fgq7$Fgt$!,Q8REH\"Fgq7$F\\u$!+d;$3\\\"F-7$Fa u$!+!=WI!=F-7$Ffu$!+K]/F=F-7$F[v$!+*)[J8>F-7$F`v$!+*)4ng@F-7$Fev$!+t,& HQ#F-7$Fjv$!+inC#Q#F-7$F_w$!+$466U#F-7$Fdw$!+VyknCF-7$Fiw$!+]4NaDF-7$F ^x$!+l+g1FF-7$Fcx$!+\">q7'HF-7$Fhx$!+l)op,$F-7$F]y$!+NT@;IF-7$Fby$!+!3 !GKLF-7$Fgy$!+>)Rvk$F-7$F\\z$!+r**GsOF-7$Faz$!+0W\")zPF-7$Ffz$!+Z;j1RF -7$F[[l$!+eph@TF-7$F`[l$!+1R=tTF-7$Fe[l$!+$*)QD<%F-7$Fj[l$!+1WWrTF-7$F _\\l$!+%*z\"f?%F-7$Fd\\l$!+vuEVWF-7$Fi\\l$!+\"HkAL%F-7$F^]l$!+]s[RUF-7 $Fc]l$!+!*Q\"\\A%F-7$Fh]l$!+YXK1UF-7$F]^l$!+50\"z;%F-7$Fb^l$!+AsZ$4%F- 7$Fg^l$!+f.6dRF-7$F\\_l$!+\\])*=PF-7$Fa_l$!+Q$G*>LF-7$Ff_l$!+h9Q_KF-7$ F[`l$!+G+$>D$F-7$F``l$!+CJq]KF-7$Fe`l$!+8u9WKF-7$Fj`l$!+ZR3JKF-7$F_al$ !+d$=%*>$F-7$Fdal$!+(=P)HJF-7$Fial$!+\\[(*))HF-7$F^bl$!+^gosAF-7$Fcbl$ !+ucp*3\"F-7$Fhbl$!+'p`$*3\"F-7$F]cl$!+')\\@(3\"F-7$Fbcl$!+f%o43\"F-7$ Fgcl$!+V$y@1\"F-7$F\\dl$!+z5\"Q,\"F-7$Fadl$!*dlU.*F-7$Ffdl$!*?+mJ#F-7$ F[el$\"++3jd>F-7$F`el$\"+UT<7CF-7$Feel$\"+>/y)p#F-7$Fjel$\"+)>%o)p#F-7 $F_fl$\"+I!)e)p#F-7$Fdfl$\"+zdR)p#F-7$Fifl$\"+;^?)p#F-7$F^gl$\"+WW(yp# F-7$Fcgl$\"+c])zp#F-7$Fhgl$\"+\\/94FF-7$F]hl$\"+t&[Z!GF-7$Fbhl$\"+Fgq7$Ff^m$\".#4Nb&3F#Fbr7$F[_m$\".%z4\"o>s#Fbr7$ F`_m$\"-mX%G7s#Fgq7$Fe_m$\"-H&\\#p?FFgq7$Fj_m$\"-$H\\'3@FFgq7$F_`m$\"- Kr;>BFFgq7$Fd`m$\"-0v0XHFFgq7$Fi`m$\"-$**)>bWFFgq7$F^am$\"-1Zi/^GFgq7$ Fcam$\",A(\\c@KF-7$Fham$\",*=@Z>OF-7$F]bm$\",HFiQ*QF-7$Fbbm$\",l8(H$*Q F-7$Fgbm$\",<*Gu#*QF-7$F\\cm$\",#=mC#*QF-7$Facm$\",ncly*QF-7$Ffcm$\",u :,C!RF-7$F[dm$\",nk<%4RF-7$F`dm$\",4T%*)>RF-7$Fedm$\",xE%4NRF-7$Fjdm$ \",w7rk)RF-7$F_em$\",y_#e\"3%F-7$Fdem$\",&G(*ouXF-7$Fiem$\",jiNW&\\F-7 $F^fm$\",0U3I&\\F-7$Fcfm$\",nwRH&\\F--Fifm6&F[gmFagm$\"#DF*$\"\"\"Fbgm -Fdgm6#%Pscheme~with~a~relatively~large~stability~regionG-F$6%7fs7$F($ \"*#\\U-FF-7$F/$\"*#)p60$F-7$F4$\"*t@M0$F-7$F9$\"*/CdP$F-7$F>$\"*'eSIM F-7$FC$\"*tY![OF-7$FH$\"*Uc7#QF-7$FM$\"*Dft#QF-7$FR$\"*(fucQF-7$FW$\"* l#Q6QF-7$Ffn$\"*3kjl$F-7$F[o$\"*.NE9$F-7$F`o$\"*1@P5$F-7$Feo$\"*8ME=#F -7$Fjo$\"*y#RB;F-7$F_p$\")'QcM'F-7$Fdp$!*u*3f;F-7$Fip$!*%3auAFgq7$Fgt$!,(e3\")=AFgq7$F \\u$!+nE?fDF-7$Fau$!+q<\\&4$F-7$Ffu$!+21rOJF-7$F[v$!+X8)[G$F-7$F`v$!+. hv4PF-7$Fev$!+%4y:4%F-7$Fjv$!+.0P!4%F-7$F_w$!+')*>r:%F-7$Fdw$!+&)*\\qB %F-7$Fiw$!+DC(fQ%F-7$F^x$!+*o%[ZYF-7$Fcx$!+>J$\\3&F-7$Fhx$!+P*31=&F-7$ F]y$!+'*HJz^F-7$Fby$!+mX=AdF-7$Fgy$!+?+njiF-7$F\\z$!+?R=1jF-7$Faz$!+,t &3\\'F-7$Ffz$!+Eik3nF-7$F[[l$!+LK#y2(F-7$F`[l$!+N7PmrF-7$Fe[l$!+*ej_;( F-7$Fj[l$!+/hQjrF-7$F_\\l$!+e5gAsF-7$Fd\\l$!+(*=()HwF-7$Fi\\l$!+HT&)Qu F-7$F^]l$!+300zsF-7$Fc]l$!+9S.asF-7$Fh]l$!+^/6AsF-7$F]^l$!+jK7crF-7$Fb ^l$!+$yG#GqF-7$Fg^l$!+*efQz'F-7$F\\_l$!+Zv]%Q'F-7$Fa_l$!+9wN)p&F-7$Ff_ l$!+NV?#e&F-7$F[`l$!+F&H9e&F-7$F``l$!++RKzbF-7$Fe`l$!+p62obF-7$Fj`l$!+ k@kXbF-7$F_al$!+:ME\"\\&F-7$Fdal$!+S$\\*=:%F-7$Feel$\"+lJ[WYF-7$Fjel$\"+)e!fq9F-7$Fail$\",8\"3))y9F-7$Ffi l$\",pO$H&\\\"F-7$F[jl$\",A&z^K:F-7$F`jl$\",Mj#34;F-7$Fejl$\",$y&*3b.Dfp'F-7 $Ffcm$\",Hj4Pq'F-7$F[dm$\",=C_dr'F-7$F`dm$\",zgLPt'F-7$Fedm$\",E$3#)fn F-7$Fjdm$\",_5&*z%oF-7$F_em$\",8!*>7,(F-7$Fdem$\",\\;!QdyF-7$Fiem$\",M 0e*3&)F-7$F^fm$\",<'o]1&)F-7$Fcfm$\",Q_'Q1&)F--Fifm6&F[gmFagm$\"#vF*F^ gm-Fdgm6#%TButcher's~scheme~B~with~c[5]=c[6]=1/2~and~b[5]=b[6]G-F$6%7f s7$F($\"*6\"H_8F-7$F/$\"*-Am_\"F-7$F4$\"*1Yx_\"F-7$F9$\"*Ir()o\"F-7$F> $\"*^ugr\"F-7$FC$\"*:&oC=F-7$FH$\"*'\\!4\">F-7$FM$\"*iCR\">F-7$FR$\"*s ,!G>F-7$FW$\"*GY^!>F-7$Ffn$\"*'p!p#=F-7$F[o$\"*a*yo:F-7$F`o$\"*=&H\\:F -7$Feo$\"*D2r3\"F-7$Fjo$\")0(z1)F-7$F_p$\")Dy3JF-7$Fdp$!):E(Q)F-7$Fip$ !),?l*)F-7$F^q$!*>YJY#F-7$Fcq$!+mQ-9JFgq7$Fiq$!+YJcrYFgq7$F^r$!,)*Ri^_ (Fbr7$Fdr$!+FdSCvFgq7$Fir$!+f+]sxFgq7$F^s$!+.LeI!)Fgq7$Fcs$!+vU@)[)Fgq 7$Fhs$!+)eA4E*Fgq7$F]t$!,g-D80\"Fgq7$Fbt$!,j`gA6\"Fgq7$Fgt$!,kK#*>6\"F gq7$F\\u$!+*yWBG\"F-7$Fau$!+:By]:F-7$Ffu$!+B>Sr:F-7$F[v$!+7>`X;F-7$F`v $!+^Q6e=F-7$Fev$!+(pi\"\\?F-7$Fjv$!+nqb[?F-7$F_w$!+hq$>3#F-7$Fdw$!+!** 4>7#F-7$Fiw$!+pcQ'>#F-7$F^x$!+'HlrK#F-7$Fcx$!+zP#fa#F-7$Fhx$!+vvw$f#F- 7$F]y$!+$e=Jf#F-7$Fby$!+'*)3W'GF-7$Fgy$!+r(Q\\8$F-7$F\\z$!+$4nh:$F-7$F az$!+()RL[KF-7$Ffz$!+M*zpN$F-7$F[[l$!+vi2TNF-7$F`[l$!+/0A&e$F-7$Fe[l$! +njm%e$F-7$Fj[l$!+eSs$e$F-7$F_\\l$!+!QoJh$F-7$Fd\\l$!+0h\"[\"QF-7$Fi\\ l$!+1\\G:2OF-7$F]^l$! +vR)Rd$F-7$Fb^l$!+Dfm4NF-7$Fg^l$!+,gu\"R$F-7$F\\_l$!+&=$p&=$F-7$Fa_l$! +F=;SGF-7$Ff_l$!++,m\"y#F-7$F[`l$!+**RF\"y#F-7$F``l$!+&eB-y#F-7$Fe`l$! +y**euFF-7$Fj`l$!+t&\\Lw#F-7$F_al$!+>U3OFF-7$Fdal$!+@e8wEF-7$Fial$!+1i qaDF-7$F^bl$!+@sqO>F-7$Fcbl$!*c>D:*F-7$Fhbl$!*#4k\\\"*F-7$F]cl$!*7C88* F-7$Fbcl$!*Vcw2*F-7$Fgcl$!*L$3;*)F-7$F\\dl$!*HZ**\\)F-7$Fadl$!*a#y\\vF -7$Ffdl$!*E,Kw\"F-7$F[el$\"+9V06t\"*\\BF-7$Fifl$\"+b7v\\ BF-7$F^gl$\"+AEY\\BF-7$Fcgl$\"++Ib\\BF-7$Fhgl$\"+C38fBF-7$F]hl$\"+;)Q8 W#F-7$Fbhl$\"+$Q2>P%F-7$Fghl$\"+')\\>#R(F-7$F\\il$\"+w]X$R(F-7$Fail$\" +M-%\\V(F-7$Ffil$\"+$z'4\"F-7$F_[m$\",V0NiW\"F-7$Fd[m$\"-YB<(eW\"F gq7$Fi[m$\"-Dcs8Y9Fgq7$F^\\m$\"-qiY,[9Fgq7$Fc\\m$\"-q\"3VSX\"Fgq7$Fh\\ m$\"-%zXX,Y\"Fgq7$F]]m$\"-79&='p9Fgq7$Fb]m$\"-v@T(Q[\"Fgq7$Fg]m$\"-'eD mZ]\"Fgq7$F\\^m$\"-:#)*)ow:Fgq7$Fa^m$\"-;x[98Fbr7 $F[_m$\".\\#4PmWBFbr7$F`_m$\"-(zcESM#Fgq7$Fe_m$\"-#3JkNM#Fgq7$Fj_m$\"- U5:!RM#Fgq7$F_`m$\"-q-zqXBFgq7$Fd`m$\"->F23^BFgq7$Fi`m$\"-FKf/kBFgq7$F ^am$\"-r+1[bCFgq7$Fcam$\",EI*etFF-7$Fham$\",_b&=:JF-7$F]bm$\",ogI2N$F- 7$Fbbm$\",sHW-N$F-7$Fgbm$\",\\Ln(\\LF-7$F\\cm$\",4\")Q$\\LF-7$Facm$\", 1:bTN$F-7$Ffcm$\",&*pV!eLF-7$F[dm$\",,))eSO$F-7$F`dm$\",/qRIP$F-7$Fedm $\",eqngQ$F-7$Fjdm$\",3w$4IMF-7$F_em$\",,Iq:^$F-7$Fdem$\",=m&pLRF-7$Fi em$\",aXp&eUF-7$F^fm$\",'yDMdUF-7$Fcfm$\",[0ysD%F--Fifm6&F[gmF\\gmFjao Fagm-Fdgm6#%Hscheme~with~c[5]=c[6]=3/4~and~b[5]=b[6]G-%%FONTG6$%*HELVE TICAG\"\"*-%+AXESLABELSG6$Q\"x6\"Q!F`bt-%&TITLEG6#%Uerror~curves~for~7 ~stage~order~6~Runge-Kutta~methodsG-%%VIEWG6$;F(Fcfm%(DEFAULTG" 1 2 0 1 10 0 2 9 1 4 2 1.000000 45.000000 45.000000 0 1 "Butcher's scheme A " "scheme with simple nodes" "scheme with a relatively large stability region" "Butcher's scheme B with c[5]=c[6]=1/2 and b[5]=b[6]" "scheme with c[5]=c[6]=3/4 and b[5]=b[6]" }}}}{PARA 0 "" 0 "" {TEXT -1 0 "" } }{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 531 "evalf[20](plot(['qn_RK6_1' (x)-q(x),'qn_RK6_2'(x)-q(x),'qn_RK6_3'(x)-q(x),'qn_RK6_4'(x)-q(x),\n'q n_RK6_5'(x)-q(x)],x=0.925..0.9469483523,font=[HELVETICA,9],\ncolor=[CO LOR(RGB,.95,.2,0),COLOR(RGB,.45,0,.95),COLOR(RGB,0,.25,1),COLOR(RGB,0, .75,.2),COLOR(RGB,.95,.45,0)],\nlegend=[`Butcher's scheme A`,`scheme w ith simple nodes`,`scheme with a relatively large stability region`,`B utcher's scheme B with c[5]=c[6]=1/2 and b[5]=b[6]`,`scheme with c[5]= c[6]=3/4 and b[5]=b[6]`],title=`error curves for 7 stage order 6 Runge -Kutta methods`));" }}{PARA 13 "" 1 "" {GLPLOT2D 868 671 671 {PLOTDATA 2 "6+-%'CURVESG6%7fr7$$\"$D*!\"$$\",!zC57K!#?7$$\"5#HT(y=;Ty a#*F-$\",jsWMB$F-7$$\"5x:8C5]n%*e#*F-$\",\"QjRgLF-7$$\"5/&Gg&Hn!GOE*F- $\",tn&oNNF-7$$\"5H&H!zH:/Mo#*F-$\",TY9Q`$F-7$$\"5->FFLm..t#*F-$\",OTy 2_$F-7$$\"55GIvH^&ytF*F-$\",\\f7kQ$F-7$$\"5%*R\")=(\\%3)=G*F-$\",R&pU@ IF-7$$\"5g)H\\IN7PlG*F-$\",Crr#>IF-7$$\"5>L9ZMp%y6H*F-$\",kGK/)HF-7$$ \"5e\"*f&3Sm_fH*F-$\",TnUX[#F-7$$\"5qnTyj?_0)H*F-$\",*>94t;F-7$$\"5zVB rExx:+$*F-$\",@\\96-\"F-7$$\"5vY6D/eZ_-$*F-$\",L`\\3-\"F-7$$\"5v\\**y \")Q<*[I*F-$\",f4?(>5F-7$$\"5DrK'p%Q^k4$*F-$\"+w&=sc*F-7$$\"5vgBO9Hb$> J*F-$\"+aO\\3wF-7$$\"5D]9w\")>fA9$*F-$\"+\"*\\gZq(e'RK$*F-$!,F:5z9%F-7$$\"50* ozz8AvML*F-$!,aBJ\\![F-7$$\"5l<@w0bQbM$*F-$!,oi\\X\"eF-7$$\"5DYXat)[Kc L*F-$!,'*plTK(F-7$$\"5$[(pKTA6rO$*F-$!,#ol$y_*F-7$$\"5:0**3h')z2R$*F-$ !-,Ba'G+\"F-7$$\"5WNG&33&[WT$*F-$!-WIcm-5F-7$$\"5JEm+nzC&fM*F-$!-OdK/2 5F-7$$\"5+s8ze)4/$[$*F-$!-&fb4o-\"F-7$$\"5rFgv7$$\"5 5#=0*yeFgv7$$\"5lB3^P#4/tN*F-$!.KvlzF!>Fgv7$$\"5q1@s af(o%e$*F-$!.8$[*eD!>Fgv7$$\"5z*QL>nUL'f$*F-$!.V+J[B!>Fgv7$$\"5?C$\\b/ &H0i$*F-$!.SW(3D->Fgv7$$\"5ge_;>uCZk$*F-$!.h(zB90>Fgv7$$\"55y0mrw'ylO* F-$!.T:$f1=>Fgv7$$\"5c(*e:Cz[oo$*F-$!.\\'=;vf>Fgv7$$\"5?a&obuDA)p$*F-$ !.qDD:o+#Fgv7$$\"5!3@\")pcjf4P*F-$!.)ycF&G3#Fgv7$$\"5SnQR)Q,(4s$*F-$!. 0g1)4,AFgv7$$\"5/Cl!)4#RMKP*F-$!.^q9V!zBFgv7$$\"5&z2$>f.>#QP*F-$!-B)4R :]#F-7$$\"5!>jz&3:%4WP*F-$!-'y#)4'\\EF-7$$\"5&)3HF$3<.ZP*F-$!-rg:_MFF- 7$$\"5!e=mzl#p*\\P*F-$!-QN,mgFF-7$$\"5!GYfEBo!Hv$*F-$!-QS!z0w#F-7$$\"5 vRFN2QWev$*F-$!-6[z\\gFF-7$$\"5gZe71h%fnP*F-$!-q$ot,w#F-7$$\"5]b*)*[S[ MzP*F-$!-)=P^)fFF-7$$\"5qwLnbXNB!Q*F-$!-*[V/$fFF-7$$\"5)yzZkqgKDQ*F-$! -C/[afFF-7$$\"5WVVI^F:)pQ*F-$!-(H1Iex#F-7$$\"57h.5:&H@>R*F-$!-p@>n\"*G F-7$$\"5$[@vyg*)fjR*F-$!-<\"R(\\wHF-7$$\"5D6A&p*\\!*4,%*F-$!-i#=FS(HF- 7$$\"5S>B/()QiC.%*F-$!-_2%>s'HF-7$$\"5gFC8xFMR0%*F-$!-!4p#)*RHF-7$$\"5 lZ@\"z+:nlS*F-$!-5@-5.HF-7$$\"5qn=pQs3u2%*F-$!-'*pbWNGF-7$$\"5q(er%p%f 9*3%*F-$!-j](Gpr#F-7$$\"5v28D+<$)35%*F-$!-%QHOu^#F-7$$\"5D*=$QA)p#>6%* F-$!-NVb\"F-7$$\"5!>)R6Ol`78%*F-$!-'H+7>b \"F-7$$\"5D_pkmg9S8%*F-$!-nb\"p=b\"F-7$$\"5+$*GrF^O&RT*F-$!-.ZMy^:F-7$ $\"5xL)y()=%e]9%*F-$!-6]vp^:F-7$$\"5+q8j1NY\"oT*F-$!-&=1n7b\"F-7$$\"5> 1R[CGM7>%*F-$!-NS)y#\\:F-7$$\"5!z#>ee&p!Q@%*F-$!-`%\\df`\"F-7$$\"5e\\* zEH'zjB%*F-$!-1Edms9F-7$$\"5:'zZ$yP&>[U*F-$!-LEz\"pQ\"F-7$$\"5vUc,k76+ E%*F-$!-CYo2H7F-7$$\"5N*[$o\\(o#=F%*F-$!,[YS3`*F-7$$\"5%fL^`BEk$G%*F-$ !,!eMs4\\F-7$$\"5!3=[%Rc-kI%*F-$\"-7r/y:8F-7$$\"5nD]aV]i\"HV*F-$\"-$R_ (3lEF-7$$\"5]iV92$yV_V*F-$\"-\"G&o_kEF-7$$\"5H*pV2dJrvV*F-$\"-1Fo!em#F -7$$\"5lsZFXXasQ%*F-$\"--r$)RqEF-7$$\"50Ye!)>v&z)R%*F-$\"-Xeo9$o#F-7$$ \"5S>pL%\\qL5W*F-$\"-q#[(H8FF-7$$\"5x#*z')oMy=U%*F-$\"--Rh?xFF-7$$\"5X .U81g$[KW*F-$\"-?F6m))GF-7$$\"5:9/SV&))3VW*F-$\"-zZ-g$3$F-7$$\"5![im13 Tp`W*F-$\"-\"oO_\"4MF-7$$\"5]NG$zh$*HkW*F-$\"-bd^+KRF-7$$\"5D/Em$zTXwW *F-$\"-O5Z*Q*[F-7$$\"5+tBRp**3')[%*F-$\"-!eQ\\wU'F-7$$\"5SdsDdS'o%\\%* F-$\"-y'p-v[(F-7$$\"5vT@7X\"Qw+X*F-$\"-Mb$[\\z)F-7$$\"5!Rea!*=D!Q]%*F- $\"-AKv(eb*F-7$$\"55Eq)HB7%o]%*F-$\".p?90++\"F-7$$\"5Io%>pF*z)4X*F-$\" -S`vu****F-7$$\"5Y5>&3K'=H^%*F-$\"-]'pV%****F-7$$\"50l*pbf9mMX*F-$\"-a (3(G(***F-7$$\"5n>!)GqG/kb%*F-$\"-]9$[f***F-7$$\"50S4jgn'ezX*F-$\".lGv _/+\"F-7$$\"5WgQ(4l!pFg%*F-$\".n_X1p+\"F-7$$\"5&H*Q`sJjQh%*F-$\".3rx?c ,\"F-7$$\"5]DR4%pv&\\i%*F-$\"/n7*\\-<.\"Fgv7$$\"50eRl:#=0OY*F-$\"/Ix^9 kf5Fgv7$$\"5c!*R@P2Yrk%*F-$\"/>&=[Se5\"Fgv7$$\"5&H\\5zi$o!fY*F-$\"/#QN r\"*f=\"Fgv7$$\"5I&*pg=l!*4n%*F-$\"/%f6,=BJ\"Fgv7$$\"5]Y_&R'z^pn%*F-$ \"/%)yevu)R\"Fgv7$$\"5l(\\.$4%H\"Ho%*F-$\"/x!*\\Tb/:Fgv7$$\"5!)[AJj\"!#A7$$\"+BN[p%*!#5$\"5>\"=#p5j(f#)y\"!#F-%&COLORG6& %$RGBG$\"#&*!\"#$\"\"#!\"\"$\"\"!Fdbm-%'LEGENDG6#%3Butcher's~scheme~AG -F$6%7fr7$F($\",W:7!Q\\F-7$F/$\",N?)[q\\F-7$F4$\",g5oH;&F-7$F9$\",fw$e FaF-7$F>$\",un.ZU&F-7$FC$\",28aVS&F-7$FH$\",1tVX>&F-7$FM$\",L#R+DYF-7$ FR$\",%>kp@YF-7$FW$\",Ygq`\"F-7$F_p$\",S;c\\V\"F-7$Fd p$\",*exBL6F-7$Fip$\"+0%z7I#F-7$F^q$!,xW7dx\"F-7$Fcq$!,va46%\\F-7$Fhq$ !,+[/*R\\F-7$F]r$!,&3Z)*R\\F-7$Fbr$!,rT3c,&F-7$Fgr$!,DApTL&F-7$F\\s$!, !ySgIkF-7$Fas$!,/[:F-7$Fjt$!-w\\e)ya\"F-7$F_u$!-q!=3Yb\"F-7$Fdu$! -'G'y(\\e\"F-7$Fiu$!-HtT?!p\"F-7$F^v$!-H'>?;z\"F-7$Fcv$!.H3a*Q\\>Fgv7$ Fiv$!.*4I4b(=#Fgv7$F^w$!.fKt\\x`#Fgv7$Fcw$!.T#pS2(y#Fgv7$Fhw$!.P*4)*QJ HFgv7$F]x$!.K36>7$HFgv7$Fbx$!.$=V%[5$HFgv7$Fgx$!.ipX32$HFgv7$F\\y$!.Fx #RQIHFgv7$Fay$!.kNtJ-$HFgv7$Ffy$!.l;XlY$HFgv7$F[z$!.**Q^([aHFgv7$F`z$! .!4HKT=IFgv7$Fez$!..6Er04$Fgv7$Fjz$!.0e'*Hr?$Fgv7$F_[l$!.h8h]$)Q$Fgv7$ Fd[l$!.)=J2+hOFgv7$Fi[l$!-U'za'[QF-7$F^\\l$!-fmXXvSF-7$Fc\\l$!-Of#*\\0 UF-7$Fh\\l$!-eh)GbC%F-7$F]]l$!-M[TSXUF-7$Fb]l$!-?R%z_C%F-7$Fg]l$!-4j2y WUF-7$F\\^l$!-rx]GWUF-7$Fa^l$!-A_KWVUF-7$Ff^l$!-$Rh.QC%F-7$F[_l$!-mM`` oUF-7$F`_l$!-$Q?3BW%F-7$Fe_l$!-zj(ppc%F-7$Fj_l$!-p-u;jXF-7$F_`l$!-!>AW Eb%F-7$Fd`l$!-aoj\\5XF-7$Fi`l$!-[jfR`WF-7$F^al$!-mAdi[VF-7$Fcal$!-*QOX ];%F-7$Fhal$!-.ys'f&QF-7$F]bl$!-p'3>+R$F-7$Fbbl$!-!p$pNtEF-7$Fgbl$!-]K C-VCF-7$F\\cl$!-'QV='fBF-7$Facl$!-#oG`&fBF-7$Ffcl$!-0T\")[fBF-7$F[dl$! -*y#yNfBF-7$F`dl$!-@DsAfBF-7$Fedl$!-#QUr&eBF-7$Fjdl$!-(3I=bN#F-7$F_el$ !-jzS,NBF-7$Fdel$!-i]__PAF-7$Fiel$!-+n?S0@F-7$F^fl$!-u?#R@'=F-7$Fcfl$! -d#p\"pO9F-7$Fhfl$!,N-j>C(F-7$F]gl$\"-A%)fIi?F-7$Fbgl$\"-7I8vVTF-7$Fgg l$\"-`'\\yG9%F-7$F\\hl$\"-R(zM[9%F-7$Fahl$\"-Vpy)=:%F-7$Ffhl$\"-0DI[rT F-7$F[il$\"-nH#Ry@%F-7$F`il$\"-5*)\\6;VF-7$Feil$\"-05l_([%F-7$Fjil$\"- %[khty%F-7$F_jl$\"-,3\\8)G&F-7$Fdjl$\"-_t8Y#4'F-7$Fijl$\"-:\"[zAd(F-7$ F^[m$\"-9D)G?$**F-7$Fc[m$\".x79si:\"F-7$Fh[m$\".CkZSuN\"F-7$F]\\m$\".R T]@XZ\"F-7$Fb\\m$\".*p:^'Ga\"F-7$Fg\\m$\".7LB=Ga\"F-7$F\\]m$\".GDNrFa \"F-7$Fa]m$\".ycgQCa\"F-7$Ff]m$\".2uUJAa\"F-7$F[^m$\".46yYNa\"F-7$F`^m $\".(>P9X`:F-7$Fe^m$\".[->Doc\"F-7$Fj^m$\"/#[I3.:f\"Fgv7$F__m$\"/qU7.P M;Fgv7$Fd_m$\"/HRsAC0V=t\"G=Fgv7$F^`m$\"/([.=w=-#Fgv7 $Fc`m$\"/D4lkPa@Fgv7$Fh`m$\"/BE'ydlJ#Fgv7$F]am$\"0ajs'ef8DFaam7$Fcam$ \"5>h>2@k=J^FFham-Fjam6&F\\bm$\"#XF_bmFcbmF]bm-Ffbm6#%9scheme~with~sim ple~nodesG-F$6%7fr7$F($\",nwRH&\\F-7$F/$\",gp(z&)\\F-7$F4$\",$\",zSm#\\aF-7$FC$\",-rK\"HaF-7$FH$\",5-*>A_F-7$FM$ \",h!>#4m%F-7$FR$\",r-'fdYF-7$FW$\",hI\"p(f%F-7$Ffn$\",8$[mLQF-7$F[o$ \",(GJ=%e#F-7$F`o$\",X5X0e\"F-7$Feo$\",U3N,e\"F-7$Fjo$\",c;\"Ry:F-7$F_ p$\",\"*pY7[\"F-7$Fdp$\",4`L$z6F-7$Fip$\"+^XSiFF-7$F^q$!,!QGSG?l\\F-7$Fgr$ !,sIaWG&F-7$F\\s$!,$p$[FQ'F-7$Fas$!,9-Z_R(F-7$Ffs$!,*RE)4&*)F-7$F[t$!- %=[)pF6F-7$F`t$!-o#*>@n9F-7$Fet$!-a>&oVa\"F-7$Fjt$!-ZN41W:F-7$F_u$!-\" 4,73b\"F-7$Fdu$!-M/$)G\"e\"F-7$Fiu$!-2UZ'oo\"F-7$F^v$!-#o%3h)y\"F-7$Fc v$!.9'p'))o%>Fgv7$Fiv$!.2\")>8e=#Fgv7$F^w$!.[(p@8PDFgv7$Fcw$!.Dz*\\D(y #Fgv7$Fhw$!.3o$R.KHFgv7$F]x$!.)>+K'=$HFgv7$Fbx$!.V^\\#pJHFgv7$Fgx$!.7- W_8$HFgv7$F\\y$!.FW\"z-JHFgv7$Fay$!..)e#y3$HFgv7$Ffy$!.Q=2Q`$HFgv7$F[z $!.=x?r_&HFgv7$F`z$!.3gVv&>IFgv7$Fez$!..GT&=#4$Fgv7$Fjz$!.?Q&3^4KFgv7$ F_[l$!.qKR))>R$Fgv7$Fd[l$!.]/7Rmm$Fgv7$Fi[l$!-c@4tbQF-7$F^\\l$!-%R*QK% 3%F-7$Fc\\l$!-e59U:UF-7$Fh\\l$!-Vs*ydD%F-7$F]]l$!-5eRlbUF-7$Fb]l$!-(y% *GbD%F-7$Fg]l$!-;o!H]D%F-7$F\\^l$!-W(>KXD%F-7$Fa^l$!-@p!*o`UF-7$Ff^l$! -b\")31aUF-7$F[_l$!-#e5c#zUF-7$F`_l$!-M,\"f'fWF-7$Fe_l$!-$RAkJf%F-7$Fj _l$!-'e*\\N*e%F-7$F_`l$!-xZ[()yXF-7$Fd`l$!-Z\"y&)p`%F-7$Fi`l$!->Z@H![% F-7$F^al$!-5Z[LwVF-7$Fcal$!-MpJG%>%F-7$Fhal$!-o(fQz)QF-7$F]bl$!-`ZgKEM F-7$Fbbl$!-@&)>j;FF-7$Fgbl$!-oK9f)[#F-7$F\\cl$!-'\\9AgS#F-7$Facl$!-v;d &fS#F-7$Ffcl$!-&)*G*)eS#F-7$F[dl$!-U9kv0CF-7$F`dl$!-\\[Ki0CF-7$Fedl$!- 7Jk&\\S#F-7$Fjdl$!-63#))=S#F-7$F_el$!-7&=d8Q#F-7$Fdel$!-N3F'QG#F-7$Fie l$!-_0X\"=:#F-7$F^fl$!-7!\\$z3>F-7$Fcfl$!-$\\;QR[\"F-7$Fhfl$!,&=@,FxF- 7$F]gl$\"-rv.P2?F-7$Fbgl$\"-gt8<$3%F-7$Fggl$\"-pzHJ#3%F-7$F\\hl$\"-LVF H%3%F-7$Fahl$\"-acYP\"4%F-7$Ffhl$\"-Kq-.6TF-7$F[il$\"-1Xy]dTF-7$F`il$ \"-wx:,cUF-7$Feil$\"-!))Q$yFWF-7$Fjil$\"-SIg>GZF-7$F_jl$\"-r$zd)H_F-7$ Fdjl$\"-5h_]NgF-7$Fijl$\"-TKHd+Z\"F-7$Fb\\m$\".euM]%Q:F-7$Fg \\m$\".@#*f.%Q:F-7$F\\]m$\".%e_oNQ:F-7$Fa]m$\".Xu2D!Q:F-7$Ff]m$\".p'R% >y`\"F-7$F[^m$\".'4=O9R:F-7$F`^m$\".\"y5)*4\\:F-7$Fe^m$\".`t.WDc\"F-7$ Fj^m$\"/CGJaip+dbFFham-Fjam6&F\\bmF cbm$\"#DF_bm$\"\"\"Fdbm-Ffbm6#%Pscheme~with~a~relatively~large~stabili ty~regionG-F$6%7fr7$F($\",Q_'Q1&)F-7$F/$\",(\\hpi&)F-7$F4$\",qPnw*))F- 7$F9$\",`Dq,O*F-7$F>$\",1U5_N*F-7$FC$\",wi70K*F-7$FH$\",ZtkM'*)F-7$FM$ \",*\\sq%*zF-7$FR$\",#[)***))zF-7$FW$\",p`Kg)yF-7$Ffn$\",A')\\W!p#F-7$F_ p$\",BG]N_#F-7$Fdp$\",<3zY+#F-7$Fip$\"+ey0AXF-7$F^q$!,PK%p%*HF-7$Fcq$! ,![GqK%)F-7$Fhq$!,6^Y1V)F-7$F]r$!,<\\/3V)F-7$Fbr$!,3*=1h&)F-7$Fgr$!,.j )[4\"*F-7$F\\s$!-7m$f'*4\"F-7$Fas$!-nl7lt7F-7$Ffs$!-Q3&=5a\"F-7$F[t$!- J$=(zS>F-7$F`t$!-:^2QCDF-7$Fet$!-Q7%4ql#F-7$Fjt$!->1+[cEF-7$F_u$!-%Q-p !oEF-7$Fdu$!-i#o,/s#F-7$Fiu$!-\\ikr,HF-7$F^v$!-*)fiYwIF-7$Fcv$!..qSB$[ LFgv7$Fiv$!.[].>(ePFgv7$F^w$!.9rJ,AO%Fgv7$Fcw$!..o1o=z%Fgv7$Fhw$!.!4`s dS]Fgv7$F]x$!.\\Ht$GS]Fgv7$Fbx$!.uoE!**R]Fgv7$Fgx$!.@4m0%R]Fgv7$F\\y$! .4&3x%)Q]Fgv7$Fay$!.Fo$))eQ]Fgv7$Ffy$!.n>ySi/&Fgv7$F[z$!.l$onW!3&Fgv7$ F`z$!.x7H#z!>&Fgv7$Fez$!.WK(\\Q:`Fgv7$Fjz$!.-JH(p;bFgv7$F_[l$!.Y:#eyHe Fgv7$Fd[l$!./`L(*4I'Fgv7$Fi[l$!-W3TSDmF-7$F^\\l$!-r5scF(4xyF-7$F_` l$!-E#ec!fyF-7$Fd`l$!--W#)*oy(F-7$Fi`l$!--]^?*o(F-7$F^al$!-Ds5.5vF-7$F cal$!-B&\\,i>(F-7$Fhal$!-JAW-omF-7$F]bl$!-FgN.seF-7$Fbbl$!-(4M0\"[YF-7 $Fgbl$!-YSC![D%F-7$F\\cl$!-t,FR7TF-7$Facl$!-2i\"zA6%F-7$Ffcl$!-kCc;7TF -7$F[dl$!-n7&Q>6%F-7$F`dl$!---4r6TF-7$Fedl$!-fr-d5TF-7$Fjdl$!-5yRI0TF- 7$F_el$!-qqP-qSF-7$Fdel$!-%*RMT-RF-7$Fiel$!-4.NMvOF-7$F^fl$!-SENPdKF-7 $Fcfl$!-NT-bEDF-7$Fhfl$!-t@N\"HI\"F-7$F]gl$\"-bJ.6\"[$F-7$Fbgl$\"-,&fL Q0(F-7$Fggl$\"->)p\\B0(F-7$F\\hl$\"-;Y%Rd0(F-7$Fahl$\"-Mns*y1(F-7$Ffhl $\"-%GK`;5(F-7$F[il$\"-#>5)[\"=(F-7$F`il$\"-gb?r]tF-7$Feil$\"-)=)*Rek( F-7$Fjil$\"-b:I/i\")F-7$F_jl$\"-*pGQT-*F-7$Fdjl$\".eqhv3/\"F-7$Fijl$\" .QJ$\\h&H\"F-7$F^[m$\".T!*4E=q\"F-7$Fc[m$\".hA.RD)>F-7$Fh[m$\".J@\">%) GBF-7$F]\\m$\".&o5[RIDF-7$Fb\\m$\".R(G#[![EF-7$Fg\\m$\".>Ownzk#F-7$F\\ ]m$\".^7I()yk#F-7$Fa]m$\".RAA;tk#F-7$Ff]m$\".;Vkhpk#F-7$F[^m$\".#pdFB \\EF-7$F`^m$\".'ya*>jm#F-7$Fe^m$\".d\"eZR*o#F-7$Fj^m$\"/[!Rd\")>t#Fgv7 $F__m$\"/ln?d(f!GFgv7$Fd_m$\"/&3[([MGHFgv7$Fi_m$\"/Yx%=v19$Fgv7$F^`m$ \"/EC\\jQvMFgv7$Fc`m$\"/M'z0?Wq$Fgv7$Fh`m$\"/;M2;$[)RFgv7$F]am$\"0A?^% [gDVFaam7$Fcam$\"5>!\\PJPH^ot%Fham-Fjam6&F\\bmFcbm$\"#vF_bmF`bm-Ffbm6# %TButcher's~scheme~B~with~c[5]=c[6]=1/2~and~b[5]=b[6]G-F$6%7fr7$F($\", [0ysD%F-7$F/$\",+Mh_G%F-7$F4$\",HI#)4X%F-7$F9$\",aD.'yYF-7$F>$\",7_?hn %F-7$FC$\",Iwd&eYF-7$FH$\",E&RMxWF-7$FM$\",--j_)RF-7$FR$\",!yCT#)RF-7$ FW$\",LJK2$RF-7$Ffn$\",)pq$*pKF-7$F[o$\",uvBz=#F-7$F`o$\",PL*G=8F-7$Fe o$\",7XYzJ\"F-7$Fjo$\",IuckJ\"F-7$F_p$\",77FGB\"F-7$Fdp$\"+.xFE(*F-7$F ip$\"+4\\>P>F-7$F^q$!,'4#4k`\"F-7$Fcq$!,X!3*oE%F-7$Fhq$!,W9]eE%F-7$F]r $!,=cp7F-7$Fet$!-+jMn N8F-7$Fjt$!-X\"32aL\"F-7$F_u$!-i\">-7M\"F-7$Fdu$!-#>$[Qn8F-7$Fiu$!-\") )p.\"e9F-7$F^v$!-rfn`X:F-7$Fcv$!.*efHF-7$Fbbl$!-S6s4+BF-7$Fgbl$!-pc+8,@F-7$F \\cl$!-Q\\L3H?F-7$Facl$!-8Gt-H?F-7$Ffcl$!-)zIr*G?F-7$F[dl$!-,\\#f)G?F- 7$F`dl$!-WSpuG?F-7$Fedl$!-]_F=G?F-7$Fjdl$!-rc8bD?F-7$F_el$!-_\"ypy+#F- 7$Fdel$!-)p^'yB>F-7$Fiel$!-&f_@)4=F-7$F^fl$!-TL*y**f\"F-7$Fcfl$!-0%)4' HB\"F-7$Fhfl$!,yomG='F-7$F]gl$\"-O3_\"ey\"F-7$Fbgl$\"-d)Q2E&F-7$Fijl$\"-$p1wz`'F-7$F^[m$ \"-g7@ms&)F-7$Fc[m$\"-D)=.(y**F-7$Fh[m$\".-%fGKr6F-7$F]\\m$\".qg?sAF\" F-7$Fb\\m$\".MG'*)>J8F-7$Fg\\m$\".eA^e6L\"F-7$F\\]m$\".kI1=6L\"F-7$Fa] m$\".()Q'4$3L\"F-7$Ff]m$\".h96_1L\"F-7$F[^m$\".O>o&yJ8F-7$F`^m$\".Q(yE KS8F-7$Fe^m$\".%4*>\\=N\"F-7$Fj^m$\"/'*RBs6t8Fgv7$F__m$\"/U8_&e+T\"Fgv 7$Fd_m$\"/!ol%z7r9Fgv7$Fi_m$\"/^&3cWqd\"Fgv7$F^`m$\"/I'4'o\"Ru\"Fgv7$F c`m$\"/&pl9_!e=Fgv7$Fh`m$\"/R9pOu(*>Fgv7$F]am$\"0qo\\;Wu;#Faam7$Fcam$ \"5>2]o4M0;sBFham-Fjam6&F\\bmF]bmF\\jnFcbm-Ffbm6#%Hscheme~with~c[5]=c[ 6]=3/4~and~b[5]=b[6]G-%%FONTG6$%*HELVETICAG\"\"*-%+AXESLABELSG6$Q\"x6 \"Q!Fbas-%&TITLEG6#%Uerror~curves~for~7~stage~order~6~Runge-Kutta~meth odsG-%%VIEWG6$;F(Fcam%(DEFAULTG" 1 2 0 1 10 0 2 9 1 4 2 1.000000 45.000000 45.000000 0 1 "Butcher's scheme A" "scheme with simple nodes " "scheme with a relatively large stability region" "Butcher's scheme \+ B with c[5]=c[6]=1/2 and b[5]=b[6]" "scheme with c[5]=c[6]=3/4 and b[5 ]=b[6]" }}}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 47 "Test 12 of 7 stage, order 6 Runge-Kutta methods" }}{PARA 0 "" 0 "" {TEXT -1 0 " " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 256 "" 0 "" {TEXT -1 1 " " } {XPPEDIT 18 0 "dy/dx=exp(-x)/(x-1)^2" "6#/*&%#dyG\"\"\"%#dxG!\"\"*&-%$ expG6#,$%\"xGF(F&*$,&F.F&F&F(\"\"#F(" }{TEXT -1 2 " " }{XPPEDIT 18 0 "5*y*sin^7*7*x;" "6#*,\"\"&\"\"\"%\"yGF%%$sinG\"\"(F(F%%\"xGF%" } {TEXT -1 5 ", " }{XPPEDIT 18 0 "y(0) = 1;" "6#/-%\"yG6#\"\"!\"\"\" " }{TEXT -1 1 " " }}{PARA 0 "" 0 "" {TEXT -1 10 "Solution: " }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "y = exp(16/49+5/3136*cos*49 *x-cos*35*x/64+5/64*cos*21*x-25/64*cos*7*x);" "6#/%\"yG-%$expG6#,,*&\" #;\"\"\"\"#\\!\"\"F+*,\"\"&F+\"%OJF-%$cosGF+F,F+%\"xGF+F+**F1F+\"#NF+F 2F+\"#kF-F-*,F/F+F5F-F1F+\"#@F+F2F+F+*,\"#DF+F5F-F1F+\"\"(F+F2F+F-" } {TEXT -1 2 " " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 213 "de := diff(y(x),x)=5*y(x)*sin(7*x)^7;\nic := y( 0)=1;\ndsolve(\{de,ic\},y(x)):\ny(x)=combine((numer(rhs(%))/convert(de nom(rhs(%)),exp)));\nr := unapply(rhs(%),x):\nplot(r(x),x=0..5,font=[H ELVETICA,9],labels=[`x`,`y(x)`]);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#> %#deG/-%%diffG6$-%\"yG6#%\"xGF,,$*(\"\"&\"\"\"F)F0)-%$sinG6#,$*&\"\"(F 0F,F0F0F7F0F0" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%#icG/-%\"yG6#\"\"! \"\"\"" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/-%\"yG6#%\"xG-%$expG6#,,*&# \"#D\"#k\"\"\"-%$cosG6#,$*&\"\"(F0F'F0F0F0!\"\"*&#F0F/F0-F26#,$*&\"#NF 0F'F0F0F0F7*&#\"\"&\"%OJF0-F26#,$*&\"#\\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$\"33RW!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!eZ8'p.63>F27$$\"3rLLL$eI8k$FJ$\"3%[F&)3\"\\`>>F27$$ \"3_L$3-8>bx$FJ$\"3IprJEM*3#>F27$$\"3*QL$3xwq4RFJ$\"3rrv?(f28#>F27$$\" 3EM$eRA'*Q/%FJ$\"3X^'fP/+9#>F27$$\"33ML$3x%3yTFJ$\"3cOdy3HT@>F27$$\"3h +]PfyG7ZFJ$\"3VH&=@l89#>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$$\"3p LL$eRh;i'FJ$\"3)z%[YI??k9F27$$\"31nm\"zWo)\\nFJ$\"31\")Q9\"48QP\"F27$$ \"3W++++b2yoFJ$\"3k#Re\"fbk*G\"F27$$\"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$$\" 3[nm\"H2S3>)FJ$\"36f+OZ/c+5F27$$\"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\"F 27$$\"3!pTN@nfF3\"F2$\"3O\\W6vrpd6F27$$\"3E+]7GTt%4\"F2$\"3&p\"fsjqH;7 F27$$\"3i$e9Te3n5\"F2$\"3e&f$*[!\\V&G\"F27$$\"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$\"3vn0o ImOE;F27$$\"3[ek.aD&G;\"F2$\"3q#GT!4W()p;F27$$\"3C+voaFfp6F2$\"3Hrm)p6 m.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)zK=>F27$$\"3-++DJ12]9F2$\"32T7QjDP8>F27$$\"3/++]P [6j9F2$\"3+dVO')[G.>F27$$\"3W$ek`h0o[\"F2$\"3I$zM;9\\?'=F27$$\"3%o;HKR '\\5:F2$\"3-heuX3wwd\"F2$\"3yrf8;r3y8F27$$\"3#f*[o=f+z:F2$\"3lUh \"F2$\"3=x^]S+%)R6F27$$\"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$$\"3\"p m;a)))G=F2$\"3:l7r\"p]]+\"F27$$\"3%\\ilAZ9R\">F2$\"3^hCaoQo55F27$$\"3K Le9;0?E>F2$\"3:4Zh8k[?5F27$$\"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$$\"3*=/wP !Ry:?F2$\"3/M$G\"4m+g8F27$$\"3^P4YsP_A?F2$\"3$R*>L\"f'e19F27$$\"3dLe9T OEH?F2$\"3%\\u#Gw;Va9F27$$\"3kH2$)4N+O?F2$\"3\\;?B!3#z-:F27$$\"3EDc^yL uU?F2$\"37f,s,Z'3b\"F27$$\"3)3_+sC$[\\?F2$\"3C:ye37$yf\"F27$$\"3'pT&)e 6Bi0#F2$\"3VLP%*y0!Hk\"F27$$\"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$$\"3?+](=xpe =#F2$\"3.\"p)[NLI@>F27$$\"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$$\"3$p;a8d3AM#F2$\" 3/F6dAB#f\">F27$$\"31gc(z*f*=F27$$\"3um;zpSS\"R#F2$ \"3->e1#==?%=F27$$\"3uC\"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$$\"3AL3_D1l_DF 2$\"3/w0[YW]J5F27$$\"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$$\"3om\" zWi^bv#F2$\"3tr+Z;M5+5F27$$\"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$$\"3k\"H2 L7ST$GF2$\"3>]6Z^*QK.\"F27$$\"3)om\"H2 \"34'GF2$\"3c89;GxR$4\"F27$$\"3=](=<1#HuGF2$\"3H$eWMp\"*>9\"F27$$\"3YL e9;gn()GF2$\"3LgFbL%HW?\"F27$$\"3w;Hdq*f5!HF2$\"337;\"\\lV-G\"F27$$\"3 0+++DRW9HF2$\"3/%p0(oZBn8F27$$\"3-D19>2*4#HF2$\"3.aKDn)*p79F27$$\"3X]7 G8v`FHF2$\"39c[>!pr#f9F27$$\"3Uv=U2V3MHF2$\"3FI^JzcD1:F27$$\"3S+Dc,6jS HF2$\"3)fI*>\"*G\"Hb\"F27$$\"3O]P%)*oCP&HF2$\"3kq@\"z(y\"F27$$\"3C+v oaa+$*HF2$\"3](oH/*41IF2$\"3/1YYU*QP(=F27$$\"3:++DJ E>>IF2$\"3w5uUgAb'*=F27$$\"3Fv=<^8'=.$F2$\"3=[IQQI\\4>F27$$\"3S]P4r+`W IF2$\"3bz\"Rx1kj\">F27$$\"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$$\"3+++v=S2L KF2$\"39dX:D?6=>F27$$\"3Om\"zpo_$eKF2$\"3#zYTO,TK!>F27$$\"3;L$3_NJOG$F 2$\"3=J`$Q^]x&=F27$$\"3y;HK*oqiH$F2$\"3h#pBbd4n\"=F27$$\"3'**\\PM-5*3L F2$\"3uG&fHPn7w\"F27$$\"39$3_vN\\:K$F2$\"3Y(o1g2R=p\"F27$$\"3Jmmm\"p)= MLF2$\"3896'o\"f'3h\"F27$$\"3&)\\i:&GO4M$F2$\"3=*)4prhIk:F27$$\"3%H$ek yQoZLF2$\"3IceZ)o3k^\"F27$$\"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$$ \"3GLLeR%p\")Q$F2$\"3F&[%Q!RZWC\"F27$$\"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$\"3bMcTbf 795F27$$\"3mm\"H#oZ1\"\\$F2$\"38K^#3O0J+\"F27$$\"35+5F2 7$$\"35L$e*[$z*RNF2$\"3:J>88\">++\"F27$$\"3%o;Hd!fX$f$F2$\"3+/++++++5F 27$$\"3e++]iC$pk$F2$\"3>3i#3WY++\"F27$$\"3)*e*)f!y6&fOF2$\"3/sYmoa@+5F 27$$\"3%p\"zp)4\"4sOF2$\"3/b\">zh_2+\"F27$$\"3!\\(oz;/n%o$F2$\"3FA+oZ! R@+\"F27$$\"3ILe*[t\\sp$F2$\"3]TEK8J>05F27$$\"36]P4r$3Cs$F2$\"3cyY&>9K :-\"F27$$\"3[m;H2qcZPF2$\"3yA^kd)oL1\"F27$$\"3KL3F>grgPF2$\"3kx1i!p\" \\+6F27$$\"3s***\\7.lQx$F2$\"3WMY`\\)G.:\"F27$$\"3em\"HK/9qy$F2$\"3j!G cKQeN@\"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$$\"36nm;z5YEQF2$\"3%GUA'*pv(o9F27$$\"3J]i:&eNI$QF2$\"3eJ %\\YU_f^\"F27$$\"3^Le9\"45'RQF2$\"3][!)RoHki:F27$$\"3s;a8(f%=YQF2$\"3Z k-ND-43;F27$$\"3O+]7.\"fF&QF2$\"3]5UW.;d^;F27$$\"3i3_vlYhlQF2$\"3S)o\\ Z'>`GF27$$\"3Ymm;/OgbRF2$\"3e5rkQQq>>F27$$\"3*G$e*[$zV4SF2$\"3^Ar&38 89#>F27$$\"3w**\\ilAFjSF2$\"3Q')**40OT@>F27$$\"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$$\"3w3F>RL3GTF2$\"3A(\\vy/K(=>F27$$\"3Iz%\\lWkX8%F2$\"38-([e tep\">F27$$\"3t]i!RbX59%F2$\"38M:lS7J9>F27$$\"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?puckU3g 9F27$$\"3'Rek.9g(fUF2$\"30z2N6y#HT\"F27$$\"3SfRs3!)QmUF2$\"3E%zqA:-pO \"F27$$\"3)RL$3xe,tUF2$\"3Hzu(eD`EK\"F27$$\"3kv$4'\\=;'G%F2$\"32(*p'*e u>U7F27$$\"3G#[Z%F2$\"3S)\\h/++++\"F27$$\"3TM$3_5,-`%F2$\"3 y/++\"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$$\"37+]iSjE!z%F2$\"3&)4L&Q*oyW=F27$$\"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$$\"3a+]P40O\"*[F2$\"3L:$**HR(R@>F27$$\"3s+voa-oX\\F2$\"3%e \"e*Rr89#>F27$$\"\"&F)$\"3n\\kX'z.7#>F2-%'COLOURG6&%$RGBG$\"#5!\"\"F(F (-%%FONTG6$%*HELVETICAG\"\"*-%+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 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 0 752 "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 := [`Butcher's scheme A `,`scheme with simple nodes`,`scheme with a relatively large stability region`,`Butcher's scheme B with `*(c[5]=1/2,c[6]=1/2,b[5]=b[6]),`sch eme with `*(c[5]=3/4,c[6]=3/4,b[5]=b[6])]:\nerrs := []:\nDigits := 30: \nfor ct to 5 do\n Rn_RK6_||ct := RK6_||ct(R(x,y),x,y,x0,y0,hh,numst eps,false);\n sm := 0: numpts := nops(Rn_RK6_||ct):\n for ii to nu mpts do\n sm := sm+(Rn_RK6_||ct[ii,2]-r(Rn_RK6_||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,$*(\"\"&\"\"\"%\"yGF ,)-%$sinG6#,$*&\"\"(F,%\"xGF,F,F4F,F,7$%0initial~point:~G-%!G6$\"\"!F, 7$%/step~width:~~~G$F,!\"#7$%1no.~of~steps:~~~G\"$+&Q)pprint556\"" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#%!G" }}{PARA 11 "" 1 "" {XPPMATH 20 "6 #K%'matrixG6#7'7$%3Butcher's~scheme~AG$\"+VuCM**!#?7$%9scheme~with~sim ple~nodesG$\"+'oVM$QF+7$%Pscheme~with~a~relatively~large~stability~reg ionG$\"+7$*&%9Butcher's~scheme~B~with~G\"\"\"6%/&%\"cG6#\"\"&# F8\"\"#/&F<6#\"\"'F?/&%\"bGF=&FGFCF8$\"+ZDeNBF47$*&%-scheme~with~GF86% /F;#\"\"$\"\"%/FBFPFEF8$\"+CIDjfF+Q)pprint566\"" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 30 "The following code constr ucts " }{TEXT 260 20 "numerical procedures" }{TEXT -1 56 " for solutio ns 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 p oint where " }{XPPEDIT 18 0 "x = 4.999;" "6#/%\"xG-%&FloatG6$\"%**\\! \"$" }{TEXT -1 16 " is also given." }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 683 "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: `,numst eps]]);``;\nmthds := [`Butcher's scheme A`,`scheme with simple nodes`, `scheme with a relatively large stability region`,`Butcher's scheme B \+ with `*(c[5]=1/2,c[6]=1/2,b[5]=b[6]),`scheme with `*(c[5]=3/4,c[6]=3/4 ,b[5]=b[6])]:\nerrs := []:\nDigits := 30:\nfor ct to 5 do\n rn_RK6_| |ct := RK6_||ct(R(x,y),x,y,x0,y0,hh,numsteps,true);\nend do:\nxx := 4. 999: rxx := evalf(r(xx)):\nfor ct to 5 do\n errs := [op(errs),abs(rn _RK6_||ct(xx)-rxx)];\nend do:\nDigits := 10:\nlinalg[transpose]([mthds ,evalf(errs)]);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#K%'matrixG6#7&7$%0s lope~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)pprint576\"" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%!G " }}{PARA 11 "" 1 "" {XPPMATH 20 "6#K%'matrixG6#7'7$%3Butcher's~scheme ~AG$\"+HR!Q@#!#>7$%9scheme~with~simple~nodesG$\"+gIN&=#!#?7$%Pscheme~w ith~a~relatively~large~stability~regionG$\"+PTN+BF+7$*&%9Butcher's~sch eme~B~with~G\"\"\"6%/&%\"cG6#\"\"&#F8\"\"#/&F<6#\"\"'F?/&%\"bGF=&FGFCF 8$\"+xy(o(RF+7$*&%-scheme~with~GF86%/F;#\"\"$\"\"%/FBFPFEF8$\"+!R)>O6F +Q)pprint586\"" }}}{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 usin g the special procedure " }{TEXT 0 5 "NCint" }{TEXT -1 98 " to perfo rm numerical integration by the 7 point Newton-Cotes method over 200 e qual subintervals." }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 436 "mthds := [`Butcher's scheme A`,`scheme with simple nodes`,`scheme with a re latively large stability region`,`Butcher's scheme B with `*(c[5]=1/2, c[6]=1/2,b[5]=b[6]),`scheme with `*(c[5]=3/4,c[6]=3/4,b[5]=b[6])]:\ner rs := []:\nDigits := 20:\nfor ct to 5 do\n sm := NCint((r(x)-'rn_RK6 _||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]([m thds,evalf(errs)]);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#K%'matrixG6#7'7 $%3Butcher's~scheme~AG$\"+i'[_*)*!#?7$%9scheme~with~simple~nodesG$\"+3 [eTQF+7$%Pscheme~with~a~relatively~large~stability~regionG$\"+hpy_9!#> 7$*&%9Butcher's~scheme~B~with~G\"\"\"6%/&%\"cG6#\"\"&#F8\"\"#/&F<6#\" \"'F?/&%\"bGF=&FGFCF8$\"+?)4EL#F47$*&%-scheme~with~GF86%/F;#\"\"$\"\"% /FBFPFEF8$\"+U[-_fF+Q)pprint596\"" }}}{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 0 530 "evalf[20](plot([r(x)-'rn_RK6_1'(x),r(x)-'r n_RK6_2'(x),r(x)-'rn_RK6_3'(x),r(x)-'rn_RK6_4'(x),\nr(x)-'rn_RK6_5'(x) ],x=0..5,numpoints=100,font=[HELVETICA,9],\ncolor=[COLOR(RGB,.95,.2,0) ,COLOR(RGB,.45,0,.95),COLOR(RGB,0,.25,1),COLOR(RGB,0,.75,.2),COLOR(RGB ,.95,.45,0)],\nlegend=[`Butcher's scheme A`,`scheme with simple nodes` ,`scheme with a relatively large stability region`,`Butcher's scheme B with c[5]=c[6]=1/2 and b[5]=b[6]`,`scheme with c[5]=c[6]=3/4 and b[5] =b[6]`],title=`error curves for 7 stage order 6 Runge-Kutta methods`)) ;" }}{PARA 13 "" 1 "" {GLPLOT2D 977 637 637 {PLOTDATA 2 "6+-%'CURVESG6 %7c]m7$$\"\"!F)F(7$$\"5#HHHHHHz?k#!#@$!)hl<8!#>7$$\"5&eeeeeeeTG&F-$!)_ 4gIF07$$\"5]^^^^^w,$e(F-$\"(a<^\"F07$$\"5;<<<<AFC$!+xDd+6F07 $$\"5zyyyyGcu%G#FC$!+9m'y.\"F07$$\"5\"3333$=v\\\\BFC$!*DtLW)F07$$\"5$G GGGyS\\UT#FC$!*JSz*eF07$$\"5%QQQ)e_`iYCFC$!*e35)fF07$$\"5&[[[[tH,!zCFC $!*]f,\"eF07$$\"5ONN&G(p#*=&\\#FC$!*3,z>&F07$$\"5'eee3@Cx8^#FC$!*529)[ F07$$\"5OOO')[9_cFDFC$!*zFi\"\\F07$$\"5(ooooo=`Pa#FC$!*M(*=&\\F07$$\"5 onnnnn(>Qm#FC$!*c)*\\(fF07$$\"5[[[[[[j)Qy#FC$!*/Dp5*F07$$\"5)))))))))) Q'>R%GFC$!+#ppC6\"F07$$\"5HHHHHHH&R!HFC$!+>H:Z8F07$$\"5qpppp>i)R'HFC$! +T?1r8F07$$\"5555555&>S-$FC$!+Ts/3:F07$$\"50000000='3$FC$!+m#ef`\"F07$ $\"5++++++:M[JFC$!+Oo1n:F07$$\"5utttt[<)Q;$FC$!+@KVo:F07$$\"5[ZZZZ(*>U zJFC$!+:n\"\\c\"F07$$\"5A@@@@YA'\\>$FC$!+[N*ea\"F07$$\"5&\\\\\\\\\\\\- 0@$FC$!+%4HU`\"F07$$\"5UUUUU#*HeTKFC$!+CN)z`\"F07$$\"5!**)*)*)*)*[jEF$ FC$!+4$\\@`\"F07$$\"5!)zzzzza)pR$FC$!+2)[AO\"F07$$\"5ppppppuI@NFC$!+4! )4<7F07$$\"5GHHHHHaXyPFC$!*4:v&)*F07$$\"5)))))))))))Q.c.%FC$!*P![C&)F0 7$$\"5rrrrrrm#>H%FC$!*puU!))F07$$\"5aaaaaa*\\#[XFC$!*@$y:\"*F07$$\"5nn nn\"[FC$!*VU1k)F07$$\"5KLLLLe*RP%\\FC$!*7^ 1^)F07$$\"5ffffff*pb2&FC$!*KNre)F07$$\"5NLLLLL=!yI&FC$!+CyW:5F07$$\"52 22222P.SbFC$!+\"zI\"G7F07$$\"5+++++]FvqcFC$!+EMIa8F07$$\"5!HHHHHzr9!eF C$!+dY\\P:F07$$\"5:;;;m`::MeFC$!+hT;K:F07$$\"5SRRRR98$o'eFC$!+QX%*G:F0 7$$\"5+,,,w%>rJ)eFC$!+pagM:F07$$\"5giii7v5^**eFC$!+#[,fb\"F07$$\"5?CCC \\b4&e\"fFC$!+4gL`:F07$$\"5&eeeee$3>KfFC$!+WKX\\:F07$$\"5IKKKKd.b(*fFC $!+>ek!\\\"F07$$\"5yyyyyy)4H1'FC$!+^W1c9F07$$\"5qsssssx`GhFC$!+Be?t7F0 7$$\"5lmmmmmc;%>'FC$!+&oo#p5F07$$\"5ggggggNzfiFC$!*#=R:%*F07$$\"5baaaa a9UDjFC$!*B8@(fF07$$\"5SUUUUUsnckFC$!*:p(*)GF07$$\"5IIIIIII$ze'FC$!*R+ %z>F07$$\"5+.....BU9nFC$!*6!eI\\F07$$\"5vvvvvv:\"4%oFC$!*Jz))R(F07$$\" 5!RRRRR*Q`soFC$!*)o#))\\(F07$$\"5577777i:/pFC$!*&3;7!*F07$$\"5IIIIII&y d$pFC$!*3#pz))F07$$\"5][[[[[3SnpFC$!*,W^y)F07$$\"5qmmmmmJ-**pFC$!*QHc) ))F07$$\"5&[[[[[[X1.(FC$!*c]Tx)F07$$\"50.....yEiqFC$!*mxji)F07$$\"5@@@ @@@,*Q4(FC$!*'z@kyF07$$\"5vtttt)\\a(3sFC$!*By'[`F07$$\"5DEEEEw)=OK(FC$ !*0iBJ$F07$$\"5!)yyyy`K[QuFC$!*RuZi\"F07$$\"5JJJJJJwM`vFC$!)b;NEF07$$ \"5!44444M$FC7 $$\"5EFFFFF#z35*FC$\"*g86;$F07$$\"5XZZZZAm,?#*FC$\"*yi51$F07$$\"5lnnnn k%HF07$$\"5!zyyyGTT\"3(*FC$\"*?U=G$F07$$\"5qnnnnHOF07$$\"5]ZZZZAmcp**FC$\"*Fpzi$F07$$ \"5CCCuOxWA+5F0$\"*g[3`$F07$$\"5uttt[#H#\\.5F0$\"*jLS`$F07$$\"5\\[[t/+ i705F0$\"*0MI`$F07$$\"5CBBtg2,w15F0$\"*+8^^$F07$$\"5)zzHn^,%R35F0$\"*e [&GMF07$$\"5tssssAz-55F0$\"*'*3$RJF07$$\"5&[[[[[y@\\.\"F0$\"*\\E6a\"F0 7$$\"5(pppppk:)f5F0$!*IC,8#F07$$\"5EEEEw8F!G2\"F0$!*^\"y)>%F07$$\"5cbb bb!y*y&3\"F0$!*'oOwjF07$$\"5????&RJ$G#4\"F0$!*)eMb!)F07$$\"5&[[[[t%ox) 4\"F0$!*%o@6&)F07$$\"5=<)Gv\")F07 $$\"53333e&4!pB6F0$!*<#z2gF07$$\"5-----xihN6F0$!*RC0,$F07$$\"5]]]+QAyf Q6F0$!*m\\U?#F07$$\"5**)*)*)RxOz:9\"F0$!)91NfF07$$\"5[ZZ(*484cW6F0$!)n UrfF07$$\"5'ffff%eCaZ6F0$!)o#ex%F07$$\"5???&R6BL!\\6F0$!)]c#)=F07$$\"5 WWW%>Q+C0:\"F0$\")TEg>F07$$\"5ooo$*\\wZ,_6F0$\")U!G(>F07$$\"5$HHHz\"\\ b]`6F0$\")B/!)>F07$$\"5=<<#f=K'*\\:\"F0$\")d5G>F07$$\"5UTT\"RX4([c6F0$ \")q0X:F07$$\"5mll!>s'y(z:\"F0$!'9a\")F07$$\"5!**)*)*)*)R'o%f6F0$!)!fG L&F07$$\"5DDDDvPELs6F0$!*3G_I$F07$$\"5gggggNm>&=\"F0$!*sP02'F07$$\"5'f fffMjg!)>\"F0$!*OO=\\)F07$$\"5JJJJJJY#4@\"F0$!*L\\'z&*F07$$\"588888)y[ wB\"F0$!*%*y\"*)zF07$$\"5&\\\\\\\\\\%HPk7F0$!*FT#oXF07$$\"5------kj(G \"F0$!*Cn6&GF07$$\"544444f)**3J\"F0$!*+*3\"Q#F07$$\"52222dWCYB8F0$!*tr 5c#F07$$\"500000I]-O8F0$!*%)3Sx#F07$$\"5....`:we[8F0$!*)*pE#HF07$$\"5, ,,,,,-:h8F0$!*,$G>GF07$$\"511111\"eGTP\"F0$!*8d-i#F07$$\"566666hp5(Q\" F0$!*pI;T#F07$$\"5kjjj8^hf$R\"F0$!*::gK#F07$$\"5;;;;;T`3+9F0$!*39F0$!*E;\"*\\#F07$$\"5%\\\\\\*>E\">)49F0$!*f#e!QgFF07$$\"5(pppppMd%Q9F0$!*,P[_% F07$$\"5tsssss4&QY\"F0$!*%R)Gi(F07$$\"5aaaa/#zNhZ\"F0$!*[T^q)F07$$\"5O OOOO61U)[\"F0$!*b`*H%*F07$$\"5====oIaq+:F0$!*^URe)F07$$\"5+++++]-*H^\" F0$!*%*QCR'F07$$\"5UUUUn[.\")>:F0$!*s')Hl$F07$$\"5&[[[[tWIm_\"F0$!*_< \"4IF07$$\"5GFFF-Y0XL:F0$\")h??2%4tg:F0$\"*9)z\\;F0 7$$\"5RRRRRR5bn:F0$\"*mSp@\"F07$$\"5))))))))Q,#zOd\"F0$!*^CP8\"F07$$\" 5QQQQQjt!)z:F0$!*\"pP[MF07$$\"5)yyyy`_Nfe\"F0$!*&3A?PF07$$\"5PPPPP(oj? f\"F0$!*k:6/&F07$$\"5OOOOO6+K/;F0$!*>V?n%F07$$\"5NNNNNNjd;;F0$!*;\"flI F07$$\"5FFFFFxDmH;F0$\")8^o?F07$$\"5>>>>>>)[Fk\"F0$\"*/(f/@F07$$\"5666 66h]$el\"F0$\"*z!)yM$F07$$\"5......8#*o;F0$\"*(\\$)4[F07$$\"5qqqq?$Qz2 o\"F0$\"*jS%ygF07$$\"5QQQQQjuj#p\"F0$\"*3/jc'F07$$\"5AAAAZ.lc)p\"F0$\" *N>vj'F07$$\"51111cVb\\/F0$\"*V&o+nF07$$\"5#GGGySTN;\">F0$\"*\"z-ZiF07$$\"5999999%oy\">F0$\"* D$>LhF07$$\"5[[[[)4;>4$>F0$\"*2#=#f%F07$$\"5#GGGGy!*pR%>F0$\"*p(*R`$F0 7$$\"5;<<F0$\"*;EU<#F07$$\"5^^^^^,92q>F0$!*d/!e5F07$$\"5MMMM%=# 3k#)>F0$!*(y9!*HF07$$\"5<<<<F0$!*$4\"o_%F07$$\"5)yyy`sf_$)*>F0$ !*16al%F07$$\"5eeeeL_\\\\,?F0$!*Po&pZF07$$\"5HHHzT2tj/?F0$!*,(pO[F07$$ \"5++++]i'zx+#F0$!*!o%fo%F07$$\"5UTTTmsV19?F0$!*bjN^$F07$$\"5$GGGGG3\\ .-#F0$!):-lvF07$$\"5cbbbbIJ?L?F0$\"*LFpU#F07$$\"5GGGGGyr0Y?F0$\"*$\\pT [F07$$\"5YYYY@!pq#\\?F0$\"*wON6&F07$$\"5kkkk9-U[_?F0$\"*bJtH&F07$$\"5t ttB6e44a?F0$\"*_-+L&F07$$\"5#GGGySr(pb?F0$\"*7`2M&F07$$\"5#>>>W+Z/t0#F 0$\"*&pHX_F07$$\"5,,,,,E7\"*e?F0$\"*&3.'y%F07$$\"5QPPP()\\#Q`1#F0$\"*[ m#>SF07$$\"5uttttt_wr?F0$\"*q5Ai\"F07$$\"5'ffffM)G^%3#F0$!)`9z&)F07$$ \"5=====$\\gs4#F0$!*5Fqx#F07$$\"5'ff4(RWR&))4#F0$!*nQD0$F07$$\"5uttBh& RZ/5#F0$!*Gj**\\$F07$$\"5_^^w#o%3/-@F0$!*4z#3NF07$$\"5HHHH/)HMO5#F0$!* :Jj^$F07$$\"5%[[[t/?@o5#F0$!*\"4\"R`$F07$$\"5SSSS!H53+6#F0$!*@/(zMF07$ $\"5'fffM`+&>8@F0$!*(\\8\"\\$F07$$\"5^^^^w2>Q;@F0$!*EOZZ$F07$$\"5122d> 5)o&>@F0$!*$)QQ+$F07$$\"5iiiii7dvA@F0$!*T3(=GF07$$\"5YYYYYrJ=Y@F0$!)]e IYF07$$\"5IIIIII1hp@F0$\"*'=JKDF07$$\"566666')e.$=#F0$\"*QlGQ$F07$$\"5 #>>>>>9hk>#F0$\"*7Q.v$F07$$\"5KKKK#)pP<.AF0$\"*<.(\\QF07$$\"5sssss(R') )4AF0$\"*g_ww$F07$$\"5#HHHz;rUK@#F0$\"*V$*)fPF07$$\"57888jD!*f;AF0$\"* @O)\\PF07$$\"5KLLLeR`&*>AF0$\"*8))*pNF07$$\"5``````;JBAF0$\"*TMPc$F07$ $\"5PPPPPi$>`B#F0$\"*i1,E$F07$$\"500000!yM$fAF0$\"*H3SM$F07$$\"5,,,,E2nLiAF0$\"*& >U*R$F07$$\"5(ppppWjQ`E#F0$\"*I!)=S$F07$$\"5$HHHz;cS$oAF0$\"*:TqX$F07$ $\"5*))))))))))[U8F#F0$\"*7fog$F07$$\"5oooo$*\\QuxAF0$\"**>!)QOF07$$\" 5[[[[)4@XTG#F0$\"*i2,z$F07$$\"5GGGG.sla!H#F0$\"*)>_ZQF07$$\"533333Lz%p H#F0$\"*\"))\\QQF07$$\"5onnnH&F07$$\"5#=====to:P#F0$!*[ll&GF07$$\" 52222K))=PyBF0$!*$\\*y/$F07$$\"5KKKK#[/v^Q#F0$!*iy`]$F07$$\"5&\\\\\\uI iw&)Q#F0$!*=\\M]$F07$$\"5edddK,#y>R#F0$!*vTBY$F07$$\"5*))))Q^/**yOR#F0 $!*@77X$F07$$\"5???qdz(z`R#F0$!*Y:2V$F07$$\"5_^^Eqo03(R#F0$!*)fs^LF07$ $\"5$GGGGyN\"y)R#F0$!*)R\"z-$F07$$\"5MLLL$3n(Q7CF0$!(F3C(F07$$\"5%QQQQ Q)R*fU#F0$\"*@.o6$F07$$\"5kjjj8^gxPCF0$\"*0h!4kF07$$\"5WVVVV=\"e&\\CF0 $\"*6DP*yF07$$\"5CBBBt&=S8Y#F0$\"**Q@_cF07$$\"5.....`A7tCF0$\"*tqg_#F0 7$$\"5!)zzzz/]\\'[#F0$!)%4=/\"F07$$\"5ccccccx')*\\#F0$!)*>e*oF07$$\"5L LLLL30C8DF0$\")`!**)*)F07$$\"555555gKhEDF0$\"*.>y\"GF07$$\"54444fr5!)Q DF0$\"*jmR(\\F07$$\"533333$))))4b#F0$\"**)*4NpF07$$\"52222d%pwJc#F0$\" *O-\\!zF07$$\"5111111XOvDF0$\"*?%[q()F07$$\"5%RRRRk0'p)e#F0$\"*rNjg*F0 7$$\"5#====ogF?g#F0$\"+!fB++\"F07$$\"5wvvv+#Q$p3EF0$\"*7>n'**F07$$\"5q ppp>d\"f`h#F0$\"*hwd!**F07$$\"5kjjjQK\\-AEF0$\"*U,Sq*F07$$\"5ddddd22pG EF0$\"*ZLJi*F07$$\"5]]]]DW@WMEF0$\"*wiB\\*F07$$\"5VVVV$4e$>SEF0$\"*d#) fN*F07$$\"5OOOOh<]%fk#F0$\"*J2WN*F07$$\"5HHHHHakp^EF0$\"*7YGK*F07$$\"5 ::::lF$*>jEF0$\"*u'=!Q*F07$$\"5,,,,,,AquEF0$\"*pOf[*F07$$\"511111JE!yo #F0$\"+0\"Rsf*FC7$$\"566666hI!4q#F0$\"+.nUC'*FC7$$\"5;;;;;\"\\.Sr#F0$ \"*dHaa*F07$$\"5@@@@@@R5FFF0$\"*vxvU*F07$$\"5!**)*)*)*)*Ge(RFF0$\"*Diw K*F07$$\"5eeeeeeET_FF0$\"*n.xS*F07$$\"5FFFFFFq1lFF0$\"*(pG%f*F07$$\"5' fffffR@xx#F0$\"*VZZ')*F07$$\"5mlll!pWWSy#F0$\"+'e*y-5F07$$\"5NNNN&y\\n .z#F0$\"+&>qh+\"F07$$\"5???qKB!HNz#F0$\"+X\"F07$$\"5uuuuuC67FHF0$\"*OLZ(\\F07$$\"5^^^^wKO mLHF0$\"*lXJ&zF07$$\"5GGGGySh?SHF0$\"*tu2***F07$$\"50000!)['[n%HF0$\"+ 5!*pG5F07$$\"5#====o:\"H`HF0$\"+a%zB/\"F07$$\"5,,,E#QyE\\&HF0$\"+zZ*z/ \"F07$$\"5???q#3Til&HF0$\"+\\FmY5F07$$\"5SRR9$y.)>eHF0$\"+\"e&*y,\"F07 $$\"5feee$[mL)fHF0$\"*+@A2*F07$$\"5)pppW)=\\5jHF0$\"*6JA(*)F07$$\"5ONN N&G[k'F07$$\"5*))))))))) )=h%zHF0$\"*'=gJ^F07$$\"5YXXX&z:K>*HF0$\"*FRC&HF07$$\"5-----FJS/IF0$\" *dvQU#F07$$\"5eeee3'4uo,$F0$\"*G03v#F07$$\"5:::::l]MHIF0$\"*$3PRUF07$$ \"5cbbbb!G&fbIF0$\"*87!4sF07$$\"5'fffff\\X=3$F0$\"*dK*p'*F07$$\"5)zzzH <%e\"[3$F0$\"*$RLs'*F07$$\"5++++](='y(3$F0$\"*`Lms*F07$$\"5,,,^Qg8F*3$ F0$\"*XV-&)*F07$$\"5----FLlv!4$F0$\"*G-l'**F07$$\"5...`:1$F0$\"+$)3#3+\"F07$$\" 5onn$F0$\"*Um=)**F07$$\"5a```Gs;w+KF0$\"*ij-!)*F07$$\"5SRR*o7,1 R?$F0$\"*L3%*z*F07$$\"5DDDDD].02KF0$\"*8,Bw*F07$$\"5oooo=1xi>KF0$\"*>D yx)F07$$\"577777i]?KKF0$\"*))\\lk(F07$$\"5srrrrY&)HdKF0$\"*i^q6&F07$$ \"5JJJJJJ?R#G$F0$\"*^()**R#F07$$\"5HHHHz\"fW*)G$F0$\"*\"\\8dCF07$$\"5F FFFF_r\\&H$F0$\"*].We#F07$$\"5DDDDv7(\\?I$F0$\"*RL#\\QF07$$\"5BBBBBtAg 3LF0$\"*\"=SCYF07$$\"5@@@@rL[::LF0$\"*C&*)3iF07$$\"5>>>>>%R2oTMF07$$\"5N NNNNgp\"yS$F0$\"*kKyk$F07$$\"5!3333Lfy,U$F0$\"*;Xqt'F07$$\"5EEEEEE-aKM F0$\"*/bk?)F07$$\"5\\\\\\\\\\\\w$)fMF0$\"+a7yJ6F07$$\"5ssssss]8([$F0$ \"+\\%QIG\"F07$$\"5xxxx_'zz')[$F0$\"+L\"3JH\"F07$$\"5#GGGG._C-\\$F0$\" +npI:8F07$$\"5(yyyGTCp<\\$F0$\"+la&[J\"F07$$\"5#HHHHz'RJ$\\$F0$\"+:2X9 8F07$$\"5-...`:MS'\\$F0$\"+uPH98F07$$\"578888jG\\*\\$F0$\"+[Vb>8F07$$ \"5KLLLLe$F07$$\"5nnnn<0c]%z$F0 $\"*B'H=KF07$$\"5[[[[[tm<'z$F0$\"*4/VD$F07$$\"5HHHHzTx%yz$F0$\"*qYbN$F 07$$\"5555555)=&*z$F0$\"*nR3t$F07$$\"5LLLLL$3.i!QF0$\"*F;.:%F07$$\"5cc cccct)G\"QF0$\"*\"[#z]'F07$$\"5zzzzzH;d>QF0$\"*l%yg!*F07$$\"5-.....fDE QF0$\"+a'oX.\"F07$$\"5EEEEEw,%H$QF0$\"+v(HnM\"F07$$\"5\\\\\\\\\\\\WiRQ F0$\"+F;/%\\\"F07$$\"5yxxxFSPdUQF0$\"+k@5W:F07$$\"511111JI_XQF0$\"+OlE j:F07$$\"5????Xww*p%QF0$\"+jo%)p:F07$$\"5MMMM%=Ks%[QF0$\"+Jwkm:F07$$\" 5[[[[Bnp%*\\QF0$\"+#)H8N:F07$$\"5iiiii7;U^QF0$\"+'=&yT:F07$$\"5!4444M! 4PaQF0$\"+A^$*e:F07$$\"5>>>>>%>?t&QF0$\"+O0Q`:F07$$\"5MLLLeR[zeQF0$\"+ //.+:F07$$\"5[ZZZ(\\[p-'QF0$\"+>SA$Q\"F07$$\"5ihhhOITuhQF0$\"+A[E!R\"F 07$$\"5wvvvvv(=K'QF0$\"+0,2(R\"F07$$\"5*)))))))))Qf,vQF0$\"+:faz6F07$$ \"5------J\"o)QF0$\"*NE(*y*F07$$\"5mllllS)=O*QF0$\"*\"3B'o)F07$$\"5HHH HHzXU+RF0$\"*UF\"4%)F07$$\"55666h[u#Q!RF0$\"*)H\"[W)F07$$\"5#HHHHzJIs! RF0$\"*E'))4&)F07$$\"5$QQQ)e_<$*3RF0$\"*)**))f')F07$$\"5uuuuC(=L1\"RF0 $\"***\\d!*)F07$$\"5llll!>iMB\"RF0$\"*mX!>*)F07$$\"5ccccccg.9RF0$\"*^b G$*)F07$$\"5%QQQQQ`Zw#RF0$\"+Q*H*45F07$$\"5666666!f7%RF0$\"+VP$=C\"F07 $$\"5yyyyyGc#e'RF0$\"+,(QyY\"F07$$\"5YYYYYYAR!*RF0$\"+Rwa<;F07$$\"5+,, ,^Qfa-SF0$\"+(yFyh\"F07$$\"5bbbbbI'*p9SF0$\"+AbQ-;F07$$\"55555gAL&o-%F 0$\"+Y(H,e\"F07$$\"5kkkkk9q+RSF0$\"+?\">9c\"F07$$\"5OOOOOh_9lSF0$\"+LO *\\e\"F07$$\"5333333NG\"4%F0$\"+$e-dh\"F07$$\"5RRRRR*[Fv6%F0$\"+3$QZ_ \"F07$$\"5qqqqqq9xVTF0$\"+D6\"fB\"F07$$\"5#GGGGyg\\w;%F0$\"*&4?g'*F07$ $\"5&\\\\\\\\\\uF:>%F0$\"*v(4`')F07$$\"5#HHHHa]1U?%F0$\"+XIj+5F07$$\"5 !4444fE&)o@%F0$\"+?:(\\D\"F07$$\"5*)*)*)*[hkCKA%F0$\"+cphL:F07$$\"5))) )))))QEScHUF0$\"+'o$30nF07 $$\"577777P6c=VF0$\"**p,X%*F07$$\"588888jlQXVF0$\"+'yo)*H\"F07$$\"5!** )*)*)*[,)foVF0$\"+r]c5:F07$$\"5mmmmmm%4=R%F0$\"+rW%4k\"F07$$\"5)yyyyGm o%=WF0$\"+Mb!yh\"F07$$\"544444fy7XWF0$\"+*Rgbd\"F07$$\"53333e?JR^WF0$ \"+2p*pd\"F07$$\"522222#QewX%F0$\"+tCCy:F07$$\"51111cVO#RY%F0$\"+h&yce \"F07$$\"5000000*)=qWF0$\"+lti'f\"F07$$\"5.....G%>F[%F0$\",6g&*[g\"FC7 $$\"5,,,,,^*\\_\\%F0$\",5-\"*pg\"FC7$$\"5mlll!ppa9]%F0$\"+6^;-;F07$$\" 5IIII!GWfw]%F0$\"+lmp+;F07$$\"5&\\\\\\*p)=kQ^%F0$\"+!*zE#f\"F07$$\"5gf fffM*o+_%F0$\"+^fg\"e\"F07$$\"5*)))))))QE%yC`%F0$\"+^FNv:F07$$\"5===== =z)[a%F0$\"+g!*4y:F07$$\"5#=====G2tb%F0$\"+PU#Rf\"F07$$\"5XXXXXXmspXF0 $\"+w5uK;F07$$\"5EFFFFFj$fd%F0$\"+Bu'zj\"F07$$\"5344444g9#e%F0$\"+s$\\ 8l\"F07$$\"5!44444pb$)e%F0$\"+Z(*>^;F07$$\"5ssssss`c%f%F0$\"+DGhY;F07$ $\"5EFFFF-$**)>YF0$\"+otgx9F07$$\"5\"=====BL_k%F0$\"+[6a!H\"F07$$\"5EF FFF-+meYF0$\"+\")>(>7\"F07$$\"5ssssssn3sYF0$\"*hE`l)F07$$\"5XXXX&z:+)y YF0$\"*;:`B)F07$$\"5=====VN^&o%F0$\"*R88a(F07$$\"5aaaazN-())o%F0$\"*C' e5uF07$$\"5!4444%GpA#p%F0$\"*f?WE(F07$$\"5EFFF-@Oe&p%F0$\"*be%zF07$$\"5CCCC\\Ie40ZF0$\"*0_b$))F07$$\"5%[[ [[tM^7r%F0$\"+LPan6F07$$\"5XXXX?koSF07$$\"5[[[[[[W=[ZF0$\"+S4a -@F07$$\"51222d>qLgZF0$\"+i:!R)=F07$$\"5lllll!f*[sZF0$\"++5h#p\"F07$$ \"5CCCCuh@k%y%F0$\"+N/DN:F07$$\"5#GGGGGt%z'z%F0$\"+`c(eX\"F07$$\"5zzzz zHlxA[F0$\"+oXPI;F07$$\"5wwwwwE$e([[F0$\"+ZFQO>F07$$\"5jjjjjjNou[F0$\" +fX>w@F07$$\"5]]]]]+)31!\\F0$\"+R#GGB#F07$$\"5#GGGG._VA\"\\F0$\"+#en^@ #F07$$\"59::::S#yQ#\\F0$\"+$oHU>#F07$$\"5YZZZ(*fH^N\\F0$\"+IMVz@F07$$ \"5zzzzzzw9Z\\F0$\"+MN#z<#F07$$\"5%[[[[)f2Og\\F0$\"+\")p_2AF07$$\"5!** )*)*)*)RQdt\\F0$\"+UUbFAF07$$\"5UUUU#*z.=!)\\F0$\"+%\\axB#F07$$\"5&\\ \\\\\\*>py')\\F0$\"+'y\"RPAF07$$\"5A@@@'**=!4!*\\F0$\"+\"HZ\"HAF07$$\" 5[ZZZ(*fMR$*\\F0$\"+dc4HAF07$$\"5hgg5)\\4X]*\\F0$\"+#[f)GAF07$$\"5uttt )*Hnp'*\\F0$\"+TnIFAF07$$\"5(ooo$*\\O[$)*\\F0$\"+1lE?AF07$$\"\"&F)$\"+ vZ&f>#F0-%&COLORG6&%$RGBG$\"#&*!\"#$\"\"#!\"\"F(-%'LEGENDG6#%3Butcher' s~scheme~AG-F$6%7_dnF'7$F+$!)jyJ?F07$F2$!)gHxYF07$F7$\"(g=\"oF07$F<$\" *=!yw5F07$$\"5YXXX&zN?G0\"FC$\"*v_(e7F07$$\"5>>>>>WIX<6FC$\"*EvyN\"F07 $$\"51111J(Qp(\\6FC$\"*#4Lf8F07$$\"5#HHHH/t&3#=\"FC$\"*[cDL\"F07$$\"5O OO'))>!RC)>\"FC$\"*v(yT7F07$$\"5zzzzat?S97FC$\"*'G!\\A\"F07$$\"5ABBt5X -cI7FC$\"*=AiA\"F07$FA$\"*#))RE7F07$$\"599999*y$)fP\"FC$\")eVt%)F07$FG $\"(N\">DF07$FQ$!)o'4M*F07$Fen$!*n')f<#F07$F_o$!*2[!*)>F07$Fio$!)`piTF 07$$\"5!)zzzat:7/dP*F07$$\"5!**)*)*)*['G+%*HFC$\"*.IW)zF07$F_t$\"*.A^<(F07$Fdt$\"* 8;GQ'F07$Fit$\"*kXb&[F07$Fcu$\"*=y0]%F07$F]v$\"*\\'**[IF07$$\"5oooooVF /EKFC$\"*%*>H0$F07$Fbv$\"*OmT0$F07$$\"5;;;;;TK7dKFC$\"*I#HPIF07$Fgv$\" *ucn%HF07$$\"5&[[[[[[C[L$FC$\"*0sC5#F07$F\\w$\"*\"eOK?F07$$\"5FFFFFxf1 GMFC$\"*aw%G?F07$$\"5uuuuuuk9fMFC$\"*W9F/#F07$$\"5AAAAAspA!\\$FC$\"*ay YJ#F07$Faw$\"*YI8k#F07$$\"5[\\\\\\\\\\9))\\OFC$\"*1)zgOF07$Ffw$\"*N\"y [\\F07$$\"5344444%Hq!RFC$\"*\"H*RU'F07$F[x$\"*,k[u'F07$F`x$\"*PF$3jF07 $Fex$\"*uw<$eF07$F_y$\"*Wzy,'F07$Fiy$\"*XuEc'F07$$\"5ijjjQdu'[%[FC$\"* f!ejlF07$$\"5>???qd\\#y([FC$\"*:s2g'F07$$\"5[[[)fyq.V*[FC$\"*\"GL)p'F0 7$$\"5wwww,eCy5\\FC$\"*6t9w'F07$$\"5/00b<37EF\\FC$\"*1n9w'F07$F^z$\"*f @:w'F07$$\"5XYYY'*e\\l4]FC$\"*e@*)p'F07$Fcz$\"*KUsk'F07$$\"5XYYYY'*eo \">&FC$\"*$)[+)eF07$Fhz$\"*$>C-XF07$$\"5?????qx\"RU&FC$\"*sD\\R$F07$F] [l$\"*g$*\\X#F07$$\"5IIII!yY8Fd&FC$\"*z\"H%R#F07$$\"5b````GKR0cFC$\"*4 )[#)>F07$$\"5!onnn#*)H2QcFC$\"*gx%z>F07$Fb[l$\"*&G4*)>F07$$\"5XYYYYrA6 OdFC$\"*'3(fB#F07$Fg[l$\"*LgmQ$F07$$\"5]aaaHt;\"y\"eFC$\"*lO4Q$F07$F\\ \\l$\"*q/eP$F07$$\"5!yxxFSV\"\\]eFC$\"*I\\OQ$F07$Fa\\l$\"*ccNZ$F07$Ff \\l$\"*F&F07$F`]l$\"*QU#G`F07$Fe]l$\"*x!z:`F07$$\" 5]ZZZA;2`[fFC$\"*51iJ&F07$$\"55444f'fq['fFC$\"*T$[.aF07$$\"5qqqq&pZ57) fFC$\"*C#))ReF07$Fj]l$\"**\\N?tF07$$\"5bbbb0=,BIgFC$\"**GnlwF07$F_^l$ \"*^*3/xF07$$\"5vvvvvDQs&4'FC$\"*qMKD*F07$Fd^l$\"*&f*=x*F07$$\"5?@@@@Y Z%\\9'FC$\"*SZ_t*F07$$\"5qpppp>^b'FC$\"*H66y&F07$F]`l$\"*Y\"oZXF 07$$\"5SRRRR*=WPg'FC$\"*X_6x#F07$$\"5][[[[[`b>mFC$\"*8N)\\FF07$$\"5gdd dd2lONmFC$\"*:\"pEFF07$$\"5lmmmmmw<^mFC$\"*CG@o#F07$$\"5qvvvvD)))pm'FC $\"*O5<_#F07$$\"5![[[[[)**z#o'FC$\"*j*4\\>F07$$\"5!RRRRR96')p'FC$\")=u ZCF07$Fb`l$!&r/%F07$$\"5SRRRRRpmxnFC$!)6Z!Q$F07$Fg`l$!*S#=i=F07$F\\al$ !*Ho$\\>F07$Faal$!*'f\")=GF07$Ffal$!*7Fvx#F07$F[bl$!*&G%)fFF07$$\"5gdd dd2?@$)pFC$!*Plb!GF07$F`bl$!*rdy&HF07$$\"5!eddddKM[,(FC$!*n)Q`HF07$Feb l$!*CdK$HF07$Fjbl$!*9t#*)GF07$F_cl$!*W\">QFF07$Ficl$!*>,ok\"F07$Fcdl$! )1IosF07$$\"556666'[K**o(FC$!)Xgc5F07$Fhdl$\")'GSp$F07$$\"5!3333$o(4[* yFC$\")-*GB%F07$$\"5qqqqq&>-J'zFC$\")D^lVF07$$\"5gggg5BYRJ!)FC$\")J=KG F07$F]el$!(jf$fF07$Fbel$!)@+'R*F07$Fgel$!*=V=Q\"F07$F\\fl$!*el_C\"F07$ Fafl$!*E&*40\"F07$$\"5511c=4y^[))FC$!*dW,0\"F07$$\"5]ZZZs.>Ml))FC$!*k2 T/\"F07$$\"5!*))))QE)*f;#)))FC$!**42=5F07$$\"5IIII!G4!**)*))FC$!)1=b$* F07$$\"55888)=GQE$*)FC$!)+\"QF*F07$Fffl$!)4,[#*F07$$\"5ghhh6\\GeL!*FC$ !)u*\\**)F07$F[gl$!)t'\\y*F07$F`gl$!*1fB8\"F07$Fegl$!*'R;&H\"F07$$\"5q sssZm$Q*o$*FC$!*wtHI\"F07$$\"5vxxxF:Fs)R*FC$!*ED*y8F07$$\"5!GGGyS12&G% *FC$!*a3hQ\"F07$Fjgl$!*BkdQ\"F07$$\"5&zzzz/6gy^*FC$!*c$eO8F07$F_hl$!*; 6FI\"F07$Fihl$!)clDGF07$Fa[m$\")MMHWF07$Ff[m$!)TaO6F07$F[\\m$!*9-NW\"F 07$Fe\\m$!*$*Rcg#F07$Fc^m$!*j')=*HF07$$\"56666'[+Fx6\"F0$!*NXL)GF07$Fh ^m$!*(*z>'=F07$$\"50000I'=`'H6F0$!)`%fV#F07$F]_m$\")?;zGF07$$\"5EEE,q \\q5P6F0$\")x6:\\F07$Fb_m$\"*6**o>\"F07$$\"5uuu*f]f)3S6F0$\"*g'G2IF07$ Fg_m$\"*-c%GIF07$F\\`m$\"*\\^B3$F07$Fa`m$\"*=$=0NF07$Ff`m$\"*V]\"3XF07 $F[am$\"*-J=%fF07$F`am$\"*Q$=!)fF07$Feam$\"*jv'>gF07$F_bm$\"*9B>?'F07$ Fibm$\"*(yR,wF07$$\"5eddd#)Q1!f;\"F0$\"*'*GYY)F07$F^cm$\"*b\"R5%*F07$$ \"5#HHHzmjk(y6F0$\"*O_([$*F07$Fccm$\"*y8\"G!*F07$$\"5WWW%p]j7%)=\"F0$ \"*[l)o&)F07$$\"5GGGG`M'G;>\"F0$\"*h-7F(F07$$\"5???XEMmB$>\"F0$\"*/LWH (F07$$\"5777i*RjW[>\"F0$\"*+zMI(F07$$\"5///zsLEX'>\"F0$\"*!o\"zA(F07$F hcm$\"*Kl:#oF07$$\"5)yyG\">L'o'*>\"F0$\"*`^tV&F07$$\"5!)zzH#Hjw7?\"F0$ \"*w-6$\\F07$$\"5srrYlKY)G?\"F0$\"*:2J%\\F07$$\"5kjjjQKE\\/7F0$\"*\"yQ Y\\F07$$\"5cbb!=@j+h?\"F0$\"*FE6*[F07$$\"5[ZZ(\\=j3x?\"F0$\"*))>Pf%F07 $$\"5SRR9eJmJ47F0$\"*3Q@c$F07$F]dm$\"*+!))pEF07$$\"5AAAAs4nGC7F0$\"*V[ >/\"F07$Fbdm$\")f8PIF07$$\"5////am3,^7F0$\"*+U49\"F07$Fgdm$\"*\")*>.AF 07$$\"5[[[[[tY+w7F0$\"*!)*)fK$F07$F\\em$\"*q'R7VF07$$\"5zyyyGmAX$H\"F0 $\"*ZCI\"[F07$$\"5cbbbbI\"o#*H\"F0$\"*dr#e\\F07$$\"5vuuCi'4A2I\"F0$\"* \"oRZ]F07$$\"5%RRR*oig<-8F0$\"*c2u/&F07$$\"5888jvG+j.8F0$\"*#HUZ]F07$$ \"5KKKK#[*R308F0$\"*2;t/&F07$$\"5qqqq&p#>*zI\"F0$\"*&>GO]F07$Faem$\"*@ sn'\\F07$Ffem$\"*e5&*o%F07$F[fm$\"*/NFO%F07$$\"5aaa/Uwc;R8F0$\"*'[4ZUF 07$$\"5////zAjIU8F0$\"*\"\\d]TF07$$\"5a``.;ppWX8F0$\"*>)[\\TF07$F`fm$ \"*)yfNTF07$$\"5___-!>EG#\\F07$$\"5CCCuhe#H()Q\"F0$\"*3=\"o\\F07$$\"5QPPP7c:N!R\"F0$ \"*apH0&F07$$\"5^]]+j`Q(>R\"F0$\"*beH0&F07$Fdgm$\"*-\"\\F07$F^hm$\"*J8<\"\\F07$Fchm$\"*(zl))[F 07$F]im$\"*,wkZ%F07$Fgim$\"*xqMT%F07$$\"544444M0wD9F0$\"*Mq\\b$F07$F\\ jm$\"*$>ku@F07$$\"5&[[[[)fT:^9F0$\")p5jdF07$Fajm$\")\\?3EF07$$\"5YXX?9 vlQl9F0$\")R#*REF07$$\"5===obx@#pY\"F0$\")2ziGF07$$\"5\"44fr*zxXo9F0$ \")\"ok'QF07$$\"5kjjjQ#Q$**p9F0$\")UuJtF07$$\"5444f@(ekIZ\"F0$\")KbTtF 07$Ffjm$\")t(er(F07$$\"5XXXXq,#yA[\"F0$\"*%o+/@F07$F[[n$\"*%ow=FF07$$ \"5444%yP@c**[\"F0$\"*,Zi;%F07$$\"5#===$>;=\\\"\\\"F0$\"*$R[AUF07$$\"5 aaazg=u-$\\\"F0$\"*MSC@%F07$$\"5FFFF-@Ic%\\\"F0$\"*_F4@%F07$$\"5+++vVB ')4'\\\"F0$\"*%R!oE%F07$$\"5sssA&eAMw\\\"F0$\"*&\\'>b%F07$$\"5XXXqEG)p \"*\\\"F0$\"*g*p=bF07$F`[n$\"*t+Bj'F07$$\"5\"44f'4L5C-:F0$\"*1E3h'F07$ $\"5kjj8^Nmx.:F0$\"*a()4f'F07$$\"5OOOh#zB7`]\"F0$\"*e2Ff'F07$$\"54444M Sy%o]\"F0$\"*7;/q'F07$$\"5#==obFW$Q3:F0$\"*(**\\erF07$$\"5aaa/*4:F 0$\"*Evk_)F07$$\"5FFF_eZYX6:F0$\"*@X$3')F07$Fe[n$\"*'*zEd)F07$$\"5ggg& oYF&p9:F0$\"*])[Q&)F07$$\"5@@@rL*H+k^\"F0$\"*3\"G[&)F07$$\"5#==o0SK0\" =:F0$\"*>2Dt)F07$Fj[n$\"*0y=S*F07$$\"5...GMt`^@:F0$\"*1X@[*F07$$\"5kjj 8,)R?K_\"F0$\"*'GsG%*F07$$\"5CCC*zEUD\\_\"F0$\"*v5VP*F07$F_\\n$\"*#)=T J*F07$$\"5111co'\\S+`\"F0$\"*e!R*)))F07$Fd\\n$\"*V*3t()F07$$\"5)yyG\"p qb:N:F0$\"*873q)F07$$\"5\\[[)f`fgo`\"F0$\"*YJVa)F07$$\"5544%G+il&Q:F0$ \"*3Pj.)F07$Fi\\n$\"*c*H+oF07$F^]n$\"*6V$)p'F07$Fc]n$\"*PaeO'F07$Fh]n$ \"*=#)3Z&F07$F]^n$\"*-kB#QF07$Fb^n$\"*ky=z$F07$Fg^n$\"*iutv$F07$$\"59: :Srmehb:F0$\"*QN%zOF07$$\"5vvvDQ\"*3Kd:F0$\"*\\z3P$F07$$\"5OOO60;f-f:F 0$\"*>1\\C#F07$F\\_n$\")!*3UtF07$$\"5ddd#)QlfVi:F0$\")dHzsF07$$\"5===o 0!*49k:F0$\")#\\T;(F07$$\"5yyy`s9g%ec\"F0$\")y,]lF07$Fa_n$\")d25NF07$$ \"5wwwE*)zI3p:F0$!)^J8eF07$$\"59999R?^hq:F0$!*ZSty\"F07$$\"5^^^,*3;Z@d \"F0$!*IWPx\"F07$Ff_n$!*XM>w\"F07$$\"5jjjjQ#GVnd\"F0$!*h%G[=F07$F[`n$! *Z%G&=$F07$F``n$!*=K+K$F07$Fe`n$!*1I=,%F07$$\"5'oooo$\\=>)f\"F0$!*5$=Y RF07$Fj`n$!*+3\\$RF07$$\"5'eeeeL<[/h\"F0$!*<:yd$F07$F_an$!*w;qZ$F07$Fi an$!*M/o,#F07$Fcbn$!)R^Y$*F07$$\"5'ooo=JM][n\"F0$!)_TxsF07$Fhbn$!)(30' \\F07$$\"5aaaaHB%3no\"F0$!)9V7\\F07$F]cn$!)h&Gv%F07$Fgcn$!)f!4&oF07$Fa dn$!*T>b3\"F07$Ffdn$!*Pwtw\"F07$F[en$!*K*[!>#F07$$\"5RRRR9eOw[nFF07$Fifn$!*#[\"p!=F07$$\"59999R&*\\q)z\"F0$!*g&e.=F07 $F^gn$!*Z'y0=F07$$\"5)zzHUX#)Gj!=F0$!*+o(3=F07$$\"5uuuCP!f`y!=F0$!*8MA #=F07$$\"5^^^E?c$y$4=F0$!*P0$o=F07$$\"5GGGG.AJ!4\"=F0$!*'Qy1>F07$$\"5# ===$p`E&R\"=F0$!*w6q!>F07$Fcgn$!*QB)>>F07$F]hn$!*x*\\)>#F07$Fghn$!*Wx^ G#F07$Fain$!*amF>#F07$F[jn$!*)*\\([=F07$Fejn$!*BN=2\"F07$F_[o$!)c,@oF0 7$Fd[o$!)Is,cF07$Fi[o$!))31n%F07$F^\\o$!)Xkq[F07$Fc\\o$!)bd(H&F07$F]]o $!*5)z7;F07$Fg]o$!*PP$HJF07$F\\^o$!*yq@u$F07$Fa^o$!*0b:I%F07$Ff^o$!*#4 4qVF07$F[_o$!*hU?U%F07$F`_o$!*\\&e'[%F07$Fe_o$!*W`zX%F07$Fj_o$!*_;\"RS F07$F_`o$!*7%plEF07$$\"5#>>>%zQe&>-#F0$!*Mjro#F07$$\"5-,,,w%fiN-#F0$!* xhsq#F07$$\"5655gs]$p^-#F0$!*0G*3FF07$$\"5?>>>p1hxE?F0$!*K_^g#F07$$\"5 HGGyliGQG?F0$!*o,s5#F07$$\"5QPPPi='*)*H?F0$!)TqTZF07$$\"5ZYY'*eujfJ?F0 $!)+d1YF07$Fd`o$!))*=LYF07$$\"5lkk9_'))4[.#F0$!)n$G^%F07$$\"5uttt[UmTO ?F0$!)\\neMF07$$\"5$GGG`%)RB!Q?F0$\")%R4G\"F07$$\"5#>>>>W:I'R?F0$\"*ik Fp\"F07$$\"55555NmO%G/#F0$\"*-*G!Q#F07$Fi`o$\"*=7x\\#F07$F^ao$\"*!**eP SF07$Fcao$\"*na))=&F07$F]bo$\"*mAqH&F07$Fgbo$\"*!4IehF07$F\\co$\"*B'p+ tF07$Faco$\"*b==(yF07$$\"5&[[[)fy!R\"y?F0$\"*6_J!yF07$Ffco$\"*_Tl4(F07 $$\"5utt[nMj5'3#F0$\"*C$HoqF07$$\"5_^^,*ey*p(3#F0$\"*6(3KoF07$$\"5IHHa 5PKH*3#F0$\"*.%*p$fF07$$\"52222K)o')34#F0$\"*V2x4&F07$$\"5%[[)f`R,[#4# F0$\"*q(e8^F07$$\"5iii7v!ftS4#F0$\"*6QX7&F07$$\"5SSSl'>/nc4#F0$\"*d[u4 &F07$F[do$\"*.5Q*[F07$F`do$\"*wz!4TF07$Fedo$\"*%)R'*o#F07$Fjdo$\"*d#*f p#F07$F_eo$\"*%>9+FF07$$\"5122#e#\\xA0@F0$\"*_RJo#F07$Fdeo$\"*Bpkb#F07 $$\"5iii()o^YT3@F0$\"*Xu#[?F07$Fieo$\")!>7A&F07$Fcfo$\")r>lXF07$F]go$! )`c*>*F07$$\"5eeeeLxDhG@F0$!*+f18\"F07$$\"5aaaa/U%pW8#F0$!*kqdX\"F07$$ \"5```GAeO$f8#F0$!*;+LX\"F07$$\"5___-SuyRP@F0$!*+bWV\"F07$$\"5^^^wd!4i )Q@F0$!*o9SN\"F07$$\"5]]]]v1jKS@F0$!*')3l<\"F07$$\"5[[[)4\"RZDV@F0$!*1 Xr<\"F07$Fbgo$!*Is?:\"F07$$\"5UUUUYN\"F07$$\"5MMMMflPvj@F0$\")44,vF07$Fggo$\"*zfCc\"F07$F\\ho$\"*s r(pEF07$Faho$\"*X],>$F07$$\"5---_*))HR\")>#F0$\"*cDNA$F07$$\"57777(eX< )*>#F0$\"*z-fJ$F07$$\"5AAAs%Gh&\\,AF0$\"*x!\\JLF07$Ffho$\"*5y9L$F07$$ \"5UUU#*zE>&[?#F0$\"*K;5L$F07$$\"5____x$3Il?#F0$\"*%f_ELF07$$\"5iii7vS #3#3AF0$\"*^X:I$F07$F[io$\"*$F07$Feio$\"*3D_<$F0 7$$\"5ABBtg#=x#=AF0$\"*HlC6$F07$Fjio$\"*!Ga)*GF07$$\"5UVV$fl\\L;A#F0$ \"*2j!*)GF07$F_jo$\"*xA*))GF07$Fdjo$\"*>v!zDF07$Fijo$\"*el>S#F07$Fc[p$ \"*)HRaDF07$Fg\\p$\"*]M&eHF07$$\"5yyyGTpJauAF0$\"*aj(fHF07$F\\]p$\"*') Gv+$F07$$\"5jjj))>!>W$zAF0$\"*b'\\NJF07$$\"5eee3YIX%4G#F0$\"*)e2RKF07$ $\"5```Gsq[a#G#F0$\"*2'3RKF07$Fa]p$\"*c4%RKF07$$\"5QQQ)3:*eM(G#F0$\"* \"QC`KF07$Ff]p$\"*)[_FLF07$$\"5===ob_su$H#F0$\"*/#QFLF07$F[^p$\"*W*=9L F07$F`^p$\"*i`zh#F07$Fe^p$\"*^ibh\"F07$$\"5!4444f@2[L#F0$\")!pk?&F07$F j^p$!)C(H)fF07$$\"5OOOO'Q(z=`BF0$!*VI6F\"F07$$\"5=====$*[JfBF0$!*G'=q8 F07$$\"54444%GNyBO#F0$!*l\"y39F07$$\"5++++]7=WlBF0$!*nN\"*R\"F07$$\"5 \"444f@F0&oBF0$!*(p#=?\"F07$F__p$!)x2(4(F07$Fd_p$!)XqhEF07$Fi_p$\"))\\ u+*F07$F^`p$\"***GAk\"F07$Fc`p$\"*?[N9$F07$Fh`p$\"*)eCOJF07$F]ap$\"*Ok k:$F07$Fbap$\"*^OaL$F07$Fgap$\"*eC35%F07$$\"5999R&p9#[+CF0$\"*+J!fbF07 $$\"5YXX&zg$H=-CF0$\"*:7z`&F07$$\"5xww^?DP)QS#F0$\"*X&z=bF07$$\"53333L 9Xe0CF0$\"*w\\#HbF07$$\"5rqq?e#4')*3CF0$\"*h=WR'F07$F\\bp$\"*%e:XtF07$ $\"5feeeLF3>>CF0$\"*2B:n(F07$Fabp$\"*r[w!yF07$$\"5uttt[<])=V#F0$\"*LnR (pF07$Ffbp$\"*uY4['F07$$\"5766'[&4)[#RCF0$\"*V[\">cF07$$\"5fee3'zc@2W# F0$\"*+'>4ZF07$$\"5111JPEV>UCF0$\"*=N'yYF07$$\"5a```y%3nOW#F0$\"*tRck% F07$$\"5-,,w>V)R^W#F0$\"*Tu;f%F07$$\"5\\[[)4;g7mW#F0$\"*,AaV%F07$$\"5' ff4A+O&3[CF0$\"*ji4$RF07$F[cp$\"*y+k[#F07$$\"5RQQ)e_j.DX#F0$\"*CS$*o\" F07$$\"5MLLL3_\"\\aX#F0$\"*VMUi\"F07$$\"5#33e&\\5>#pX#F0$\"*3ZKV\"F07$ $\"5HGGy!*oYReCF0$\")+-YzF07$$\"5wvv+KFu')fCF0$!)?\\A)*F07$F`cp$!*CO$H 7F07$$\"5sqqX9WH\"GY#F0$!*9I2A\"F07$$\"5>==ob-dGkCF0$!*\">4=7F07$$\"5m ll!p4YedY#F0$!*wWTD\"F07$$\"59888Q>7BnCF0$!*bfIW\"F07$$\"5hggNzxRqoCF0 $!*j:+3#F07$$\"5333e?On7Z$F0 7$Fecp$!*t%pp8F07$Fbf p$!**3ow8F07$$\"5\"444MS%*G?f#F0$!*9FZS\"F07$$\"5)yyyG;$=O&f#F0$!*\"zR 19F07$$\"5&[[[B#>Zp)f#F0$!*1?`\\\"F07$Fgfp$!**=4h;F07$Fagp$!*?4f2#F07$ F[hp$!*g=\\m#F07$F`hp$!*7r!**GF07$Fehp$!*,y_7$F07$Fjhp$!*K^v7$F07$F_ip $!*x4#zJF07$Fdip$!*az64$F07$Fiip$!*cy'GHF07$$\"5GFFF_3t(zn#F0$!*y>n*GF 07$$\"5a```.;CD\"o#F0$!*&pNuFF07$$\"5!)zzzaBv_%o#F0$!*/QRx#F07$F^jp$!* Av!eFF07$$\"5eeee3YGN%p#F0$!*[<$F07$$\"522222#)=#Hu#F0$!*S\\_<$F07$$ \"5CCCCCua3YFF0$!*8$HsJF07$$\"5TTTTTm!\\#\\FF0$!*<\\G5$F07$Fg[q$!*F(\\ VIF07$F\\\\q$!*!=gPFF07$Fa\\q$!*hy=C#F07$F[]q$!*aH%4:F07$Fi^q$!*&)=;N \"F07$$\"5WWWWp]mL4GF0$!*CesR\"F07$F^_q$!*DC8W\"F07$$\"5%QQQ)e_F)>#GF0 $!*B)yIP:\"F07$F^dq$! *s;.;\"F07$$\"5qqq&p(f#*HNHF0$!*+&H F0$\"*\\9rE%F07$$\"5jiiP\")Hbl^HF0$\"*V/fH%F07$F]eq$\"**)4^K%F07$Fafq$ \"*N-Jy&F07$F[gq$\"*cr%*3'F07$$\"5uttB'oUZ'pHF0$\"*u'*f?'F07$F`gq$\"*v [*fiF07$$\"5JJJc(yIaX(HF0$\"*cS_G'F07$$\"5]]]+)[$**=wHF0$\"*BV!ziF07$$ \"5qppW)=cDy(HF0$\"*c:u7'F07$Fegq$\"*_3E[&F07$$\"5'ff4A+1?5)HF0$\"*<84 0&F07$$\"5/..`:J*yD)HF0$\"*F[&p]F07$$\"5655&)G-y8%)HF0$\"*\")fN3&F07$$ \"5=<<'f+$F0$\")0EBSF07$ $\"5;;;mGp3_2IF0$\")Vu:/uz!4IF0$!)Z[kfF07$$\"5IIIIb6'Q1, $F0$!*9M$*e\"F07$$\"5PPPio#[(>7IF0$!*=fQNcP,$F0$!*:ab f\"F07$$\"5^^^E&\\A:`,$F0$!*)=F8;F07$Fdhq$!*qcrp\"F07$$\"5'ooo=1e4J-$F 0$!*E_d$GF07$Fihq$!*O/&fIF07$$\"5ONNN&G%=\"F07 $Fa[r$\"*5![E:F07$Ff[r$\"*XGue\"F07$F[\\r$\"*!y(ef\"F07$F`\\r$\"*&*HVO \"F07$Fe\\r$\")s**R!*F07$$\"5aaaaa/m$f8$F0$\")Q(p'zF07$Fj\\r$\")SqnmF0 7$$\"512222dx1\\JF0$\")D$[)pF07$F_]r$\")QKLuF07$Fd]r$\"*pr$=6F07$Fi]r$ \"*ak9c\"F07$F^^r$\"*b@Md\"F07$Fc^r$\"*h#*He\"F07$F]_r$\"*([([F\"F07$F g_r$\"*D]8A\"F07$F\\`r$!)F`(G\"F07$Fa`r$!*[(*zX\"F07$$\"5#>>>>W!=vWKF0 $!*+IZY#F07$Ff`r$!*Nz13$F07$$\"5ihhh'y\">djKF0$!*pC:-$F07$$\"5_^^^,*GX )pKF0$!*H0z>#F07$$\"5ZYY'*eu>)HF$F0$!*#[-\"4#F07$$\"5UTTT;g'=hF$F0$!*r k=.#F07$$\"5OOO'QdMb#zKF0$!*lXT0\"F07$F[ar$!)*p+)GF07$F`ar$\")nT\")zF0 7$Fear$\"*eHO6#F07$$\"5EEEE^KMx)H$F0$\"*Hra.$F07$Fjar$\"*V0%\\WF07$$\" 5uuuC(G&yo.LF0$\"*Z6LV%F07$$\"5CCCC*H*fK0LF0$\"*w[mV%F07$$\"5uttB6LT'p I$F0$\"*q%HVXF07$F_br$\"*wlO+&F07$Fdbr$\"*>Q%3gF07$Fibr$\"*]MML'F07$F^ cr$\"*%p5UfF07$Fccr$\"*<0l*\\F07$$\"5?>>W+-8KOLF0$\"*ki@*[F07$Fhcr$\"* #QB`XF07$$\"5GFF-rv)Q$RLF0$\"*)GhCNF07$F]dr$\"*')fzh#F07$Fbdr$\"*Lb(yD F07$Fgdr$\"*B_'QBF07$$\"5_^^w#of\"R[LF0$\"*E$>*o\"F07$$\"5cbb0o$Q+*\\L F0$!)F$py\"F07$$\"5gffM`q\"49N$F0$!)!yvt$F07$F\\er$!)Aa:PF07$$\"5srr@4 Jb$fN$F0$!)g`%R%F07$Faer$!*>%[^;F07$$\"5%QQ)3l\"*=YgLF0$!*reF8$F07$$\" 5)yyy.&y1(>O$F0$!*OM$4JF07$$\"5#>>pc`YzMO$F0$!*GMu3$F07$Ffer$!*Y703$F0 7$$\"5///a\"f#e+oLF0$!*XC9[$F07$F[fr$!*X$Q%4&F07$$\"5???qKt4/uLF0$!*Yr 7-&F07$F`fr$!*;dU1&F07$$\"5KKKd)QLn&yLF0$!*q#4``F07$$\"5OOO'Q27w+Q$F0$ !*[o18'F07$$\"5SSS:f2\\e\"Q$F0$!*[+j3'F07$Fefr$!*0&yUgF07$Figr$!*$zT5d F07$Fchr$!*ZV:<%F07$Fhhr$!*;<+z#F07$F]ir$!*N]5E#F07$Fe[s$!*/lh2$F07$F_ \\s$!*.SF07$$\"5!333e?J&)Ga$F0$!*')y(\\ SF07$$\"5///a\"4&[/YNF0$!*`Q60%F07$Fd\\s$!*X+=1%F07$$\"5]]]+jGRO_NF0$! *H4]1%F07$$\"5uttt[nM_bNF0$!*=(fjSF07$$\"5(pppWj+$oeNF0$!*'=PDSF07$Fi \\s$!*](>_RF07$F^]s$!*MpUy$F07$Fc]s$!*0naj$F07$F]^s$!*+`Nx$F07$Fg^s$!* J6G1%F07$F\\_s$!*.4Ka$F07$Fa_s$!*$*G#eEF07$$\"5XXX&zgQM*)o$F0$!*?-%=DF 07$$\"5????X^F3#p$F0$!*nc/N#F07$$\"5%\\\\\\Co6J_p$F0$!*.d2N#F07$$\"5pp pp>#[z$)p$F0$!*t!==BF07$$\"5WWW%pv%y_,PF0$!*6A1D#F07$$\"5=>>>%H@wYq$F0 $!*$=;aAF07$$\"5ccc\"GcR]iq$F0$!*GK)eAF07$$\"5$RRR9$yX#yq$F0$!*_:kF#F0 7$$\"5IJJ1+h()R4PF0$!*/&)*RBF07$Ff_s$!*/,KR#F07$$\"5nnnn<0kcBPF0$!***G hu#F07$F[`s$!*1(*\\C$F07$Fe`s$!*yVa#[F07$F_as$!*_ETC'F07$Fdas$!*d:(zkF 07$Fias$!*P\\'RmF07$F^bs$!*)Ro!p'F07$Fcbs$!*,mut'F07$Fhbs$!*%*Qhv'F07$ F]cs$!*PsMm'F07$$\"5srrrrY4'G!QF0$!*&=Z\"o'F07$Fbcs$!*te*fnF07$$\"5%\\ \\\\\\*>_a4QF0$!*+Y];'F07$Fgcs$!*'HlkfF07$F\\ds$!*;\"ogZF07$Fads$!*7'4 #H%F07$$\"5$QQQQ8(p#z#QF0$!*QGH\"RF07$$\"5kkkkkR!)fHQF0$!*I)Q&[#F07$$ \"5XXXX&z5p7$QF0$!*i/5%=F07$Ffds$!*&[7b=F07$$\"52222dW7hMQF0$!*62v&=F0 7$$\"5)yyyyGJ#GOQF0$!*+`Ox\"F07$$\"5oooo=\"Q`z$QF0$!*HM>K\"F07$F[es$\" )DVzEF07$F`es$\")!pjT*F07$Fees$\")puJ**F07$F_fs$\"**)R4s\"F07$Fifs$\"* mtQL$F07$Fcgs$\"*F\"HjNF07$Fghs$\"*.P`n%F07$$\"5////am!oh'QF0$\"*\"zL> ZF07$$\"5KKKKKdt6pQF0$\"**[TRYF07$$\"5YYYYr-?fqQF0$\"*&pm$[%F07$$\"5gg gg5[m1sQF0$\"*17D]%F07$$\"5uuuu\\$HTN(QF0$\"*4D+_%F07$F\\is$\"*FY!GXF0 7$$\"5YXXXX?X\"4)QF0$\"*&yi\")HF07$Fais$\"*y*)='GF07$$\"5$HHHzm`9&))QF 0$\"*hGmE#F07$$\"5%QQQQ8(f@!*QF0$\")dlnlF07$$\"5vuuu*fS<>*QF0$\")k,(e' F07$Ffis$\")!Gde'F07$$\"5dcccJv-K&*QF0$\")WrCjF07$$\"5[ZZZ(*4<-(*QF0$ \")-tmXF07$$\"5QQQQjWJs)*QF0$!)R\\UFF07$F[js$!*tbFw\"F07$$\"5????&R,E@ !RF0$!*(QhmF07$Fjjs$!*Ue*[EF07$F_[t$!*QI7m$F07$Fd[t$!*/Nnm$F07$Fi[t$!*(*3Sn$F 07$$\"5QQQQ)e#*Qu\"RF0$!*W@C#QF07$$\"5?????&zT3#RF0$!*aPip%F07$$\"5--- -_kYCCRF0$!*w_eq%F07$F^\\t$!*c+Yu%F07$$\"5vuuu\\o*[$HRF0$!*g.N![F07$$ \"5mlll:./0JRF0$!*z9J$[F07$$\"5dccc\"y$=vKRF0$!*b0f$[F07$$\"5[ZZZZsKXM RF0$!*;yq$[F07$$\"5IHHHzTh&y$RF0$!*.0hu%F07$Fc\\t$!*;S0E%F07$$\"5%\\\\ \\\\*>Ba`RF0$!*+h0F$F07$Fh\\t$!*US#Q@F07$$\"5iiiiiP*3\"yRF0$!*fF*45F07 $F]]t$!)[f\"[\"F07$Fg]t$!)l.))QF07$Fa^t$!*$ex;5F07$Ff^t$!)69ElF07$F[_t $!)8\\%z\"F07$$\"5utttt)\\0W5%F0$!)!)QudF07$F`_t$!*$4#)e9F07$$\"5AAAAs %[)3CTF0$!*Tz#QBF07$$\"5/0000![\\18%F0$!*tsvV$F07$$\"5(yyyy`Z5s8%F0$!* XfN`$F07$Fe_t$!*3!3rVF07$$\"5tttt)\\+T(\\TF0$!*&3swZF07$$\"5wwwwER0rbT F0$!*)o\"y$[F07$$\"5zzzzat+ohTF0$!*n-be%F07$Fj_t$!*``lS%F07$$\"5&eee3@ 9>O<%F0$!*pp,T$F07$$\"5))))))))Qw')ezTF0$!*[M7$>F07$$\"5SSS!HNWtD=%F0$ !*k&e79F07$$\"5#>>>p1@eb=%F0$!*1*zm8F07$$\"5WVV$4y(Ha)=%F0$!)F4(*eF07$ F_`t$\"*t(y_5F07$$\"5qpp%4+f7J>%F0$\"*PA,0\"F07$$\"5WWW%p]V(p%>%F0$\"* f$=e5F07$$\"5>>>%H,G#G'>%F0$\"*SX\\8\"F07$$\"5%RRR*=Dr'y>%F0$\"*0D+[\" F07$$\"5ooo$\\-(>X*>%F0$\"*6ieg#F07$$\"5VVV$4`\"o.,UF0$\"*K?DM$F07$$\" 5===$p.m@E?%F0$\"*UF#HLF07$Fd`t$\"*:U$>LF07$$\"5UUU#\\b>wt?%F0$\"*vL'y MF07$$\"5\">>>pc)ea5UF0$\"*NE*HZF07$$\"5STT\"*yvbr8UF0$\"*L(e$o%F07$Fi `t$\"*uq$eYF07$$\"5SSS!Hg&\\0?UF0$\"*pOBp%F07$F^at$\"*1\"ePYF07$$\"5QR R*oiL%REUF0$\"*gxB`%F07$Fcat$\"*+@Ka$F07$$\"5QQQ)3lrLFB%F0$\"*T^;6$F07 $Fhat$\"*1->,$F07$$\"5OPP([n4t!RUF0$\"*g[$)y\"F07$F]bt$\")C#zD&F07$$\" 5QQQQj%4O`C%F0$\")%>y#[F07$$\"5*)*)*)*)R-%H%[UF0$!)Nk2PF07$$\"5kll:Gcg (*\\UF0$!*7lwO#F07$$\"5STTT;5F_^UF0$!**zk)R#F07$$\"5;<jw#F07$$\"5???qdzfDfUF0$!*\\7+(QF07$$\"5&ffffMj-3E%F0$!*f$3X\\ F07$$\"5qrr@M(G\\BE%F0$!*9/r!\\F07$$\"5YZZZATf*QE%F0$!*5]@([F07$$\"5AB Bt5&fUaE%F0$!*E#Rh[F07$Fgbt$!*7Q4'\\F07$F\\ct$!*X^6b'F07$Fact$!*X8N.(F 07$Ffct$!*8vW?(F07$F[dt$!*2,!)H(F07$F`dt$!*&)Q-\"fF07$Fedt$!*LRSX%F07$ $\"5_^^^,*G#*pN%F0$!*Fki$RF07$Fjdt$!*Gv$RMF07$$\"54444%y(3SuVF0$!*:q0D $F07$$\"5GGGGySP?!Q%F0$!*wg2;$F07$$\"5)yyy`A<0JQ%F0$!*%[xdJF07$$\"5ZZZ Zs.m+'Q%F0$!*jw!eJF07$$\"5122d>N!3*)Q%F0$!*<$e8KF07$F_et$!*9T'*H$F07$F det$!*'p,>>WvIrJ_%F0$!*IRR([F07$$\"5CCCC \\!oti_%F0$!*e*Gy[F07$$\"5ccc1W`gPHXF0$!*.Uo$\\F07$F`it$!*7%Hp\\F07$$ \"5a```GsJoQXF0$!*>mt&\\F07$Feit$!*;P_#\\F07$F_jt$!*W_$oRF07$Fc[u$!*d@ n?$F07$Fh[u$!*`c&HPF07$F]\\u$!*#*fg$[F07$Fb\\u$!*e%3JcF07$Fg\\u$!*&Q;r mF07$Fa]u$!*u3*euF07$Fe^u$!*1()e(yF07$$\"5yyy.&G>z/q%F0$!*8bHo(F07$$\" 5%RRRk?2=?q%F0$!*.G5u(F07$$\"5444%y7&pb.ZF0$!*z_\"*z(F07$Fj^u$!*M`)\\y F07$$\"5aaa/#*)et\"3ZF0$!*T79p(F07$F__u$!*4$=WoF07$$\"5*****\\ilA!z7ZF 0$!*=)z'*oF07$$\"59::lx0\"HVr%F0$!*r(yWpF07$$\"5III0*\\)z'er%F0$!*ggi& pF07$Fd_u$!*rt\"3oF07$$\"5ggg&=MuX*=ZF0$!*I?H:'F07$$\"5wvvDjAY[?ZF0$!* !3kN]F07$$\"5\"44fY=]B?s%F0$!*!f(R2&F07$Fi_u$!*Ub06&F07$$\"5@@@YFg75DZ F0$!*))*)p7&F07$$\"5OOO')[R,kEZF0$!*9kf.&F07$$\"5^^^Eq=!z\"GZF0$!*N;:c %F07$$\"5mmmm\"z*yrHZF0$!*`)p@IF07$$\"5'pppWjl&zKZF0$!*'Q`gDF07$F^`u$! *rB%GDF07$$\"5edd2?t6&*QZF0$!*,#\\P8F07$$\"5)yyyG;$*G?u%F0$\")Ro.:F07$ $\"5...G%3\"ycVZF0$\")z0J:F07$$\"5===o0!p1^u%F0$\")!*RO$F07$$\"5rrr@Mi^PjZF0$\"*3:(GKF07$$\"5OOOO60LTmZF0$\"*?Q9C$F07$ $\"5+,,^)yW^%pZF0$\"*0+i(GF07$F]au$\"***y;m#F07$$\"5%\\\\\\*>wecyZF0$ \"*)RR$4#F07$Fbau$\")8&GV)F07$$\"5ccc\"GJBhhy%F0$\")_3exF07$$\"5)))))) Q^/.o(y%F0$\")z$\\w%F07$$\"5?@@'**eP*>*y%F0$!)\"yc.&F07$$\"5````GZ%=2z %F0$!**>]'f\"F07$$\"5'ee3r'=vB#z%F0$!*T$[+;F07$$\"5===o0!fcPz%F0$!*ORn g\"F07$$\"5]]]DWhcF&z%F0$!*EC`j\"F07$Fgau$!*$*pjm,[F0$! *?Rw'RF07$$\"512222#=!H.[F0$!*.rc(RF07$$\"5=>>>p1z`1[F0$!*6C63%F07$$\" 5IJJJJJcy4[F0$!*FccX&F07$$\"5abbbb!3\"G;[F0$!*rC&\\dF07$F\\bu$!*S2$>lF 07$$\"5/////z>FH[F0$!*y!o2lF07$$\"5GGGGGGuwN[F0$!*PP$fkF07$$\"5_____xG EU[F0$!*2Cyw&F07$Fabu$!*ApFO&F07$$\"5[[[[)fjR_&[F0$!*#3-9ZF07$$\"5???? ?X4sh[F0$!*R@pi$F07$$\"5#>>>>WD-#o[F0$!*HI0R$F07$Ffbu$!*Lx!)p#F07$$\"5 12222#=Yw)[F0$!*6hj/#F07$F[cu$!*6IE\">F07$Fecu$!*%))=1DF07$F_du$!*@4Yv #F07$Fddu$!*A!y)H#F07$Fidu$!*)pa\"*>F07$$\"5...`!\\ZD_(\\F0$!*T/&*)>F0 7$$\"5;;;;\"*4r(o(\\F0$!*[Fy(>F07$$\"5HHHz\"\\uG&y\\F0$!*U()[$>F07$F^e u$!*uX_$=F07$$\"5oooo$*\\O[$)\\F0$!*2^_$=F07$Fceu$!*Gh,%=F07$F]fu$!*UL J'>F07$Fagu$!*g)GYCF0-Ffgu6&Fhgu$\"#XF[huF(Figu-F`hu6#%9scheme~with~si mple~nodesG-F$6%7]\\nF'7$F+$!)i5E?F07$F2$!)2j1ZF07$F7$\"(J*zlF07$F<$\" *FIP5\"F07$Fciu$\"*uJSI\"F07$Fhiu$\"*ep.Y\"F07$Fbju$\"*L)[f9F07$FA$\"* !*pFV\"F07$Fi[v$\"*5pCD\"F07$FG$\")peG&*F07$FL$\")(p;,*F07$FQ$\")8b]*) F07$$\"5!)zzzzHZbI=FC$\")2t&*))F07$FV$\")LSI%*F07$$\"5EDDDD+k9G>FC$\") #\\Op*F07$$\"532222dpng>FC$\")?23**F07$$\"5!*)))))))Q^2K*>FC$\"*'>e&= \"F07$Fen$\"*8XXL\"F07$Fjn$\"*-1l\"=F07$F_o$\"*>3hQ#F07$Fdo$\"*+a;H%F0 7$Fio$\"*y&)HK&F07$F]]v$\"*LJ;V(F07$F^p$\"*:Gvd(F07$Fe]v$\"*yYZ!yF07$F j]v$\"*#p5)e)F07$F_^v$\"+cfF,6F07$Fcp$\"+VRce6F07$Fhp$\"+jsHy6F07$F]q$ \"+3bpr7F07$Fbq$\"+x1s)[\"F07$Fgq$\"+J,-<;F07$F\\r$\"+1reG;F07$Far$\"+ Ut3T;F07$Fi_v$\"+5!*=f;F07$F^`v$\"+#y#4.ACBF07$F_t$\"+!o\"fN@F07$Fit$\"+\"o!p3>F07$Fgv$ \"+t0C3&3=h\"F07$F\\w$\"+iG.)f\"F07$Fgev$\"+v^x'f\"F07$F \\fv$\"+WYE*f\"F07$Fafv$\"+(plTi\"F07$Faw$\"+I9Ja;F07$Ffw$\"+MH)[)=F07 $F[x$\"+ddCm?F07$F`x$\"+EA#G-#F07$Fex$\"+odku>F07$F_y$\"+V)[J*>F07$Fiy $\"+EX$y/#F07$Fehv$\"+K[#z/#F07$Fjhv$\"+m/o^?F07$F_iv$\"+lYch?F07$Fdiv $\"+'*H(z1#F07$Fiiv$\"+]+(z1#F07$F^z$\"+XG(z1#F07$Fajv$\"+%zX>1#F07$Fc z$\"+([Sn0#F07$Fhz$\"+F07$F_^l$\"+OK\"[?#F07$Fd^l$\"+qC=.CF07 $Fi^l$\"+OdEZCF07$Fcbw$\"+b8]xCF07$Fhbw$\"+:%GZY#F07$F]cw$\"+\\\\k^CF0 7$F^_l$\"+O@(zV#F07$Fecw$\"+'*G/!R#F07$Fc_l$\"+/YKTBF07$F]dw$\"+Eryr@F 07$Fh_l$\"+4*o-+#F07$Fedw$\"+=Cr>=F07$Fjdw$\"+.RA0;F07$F_ew$\"+1fhu:F0 7$F]`l$\"+6`$eR\"F07$F\\fw$\"+29*G9\"F07$Fffw$\"+7Ey@6F07$F[gw$\"+Fk%R 4\"F07$F`gw$\"+)3]*45F07$Fegw$\"*EH=s(F07$Fb`l$\"*'oZItF07$F]hw$\"*cvl h'F07$Fg`l$\"*D0k>%F07$F[bl$\"*.m#HAF07$F_cl$\"*map;\"F07$$\"5]ZZZ(*4B K^rFC$\"*89/0\"F07$Fdcl$\"*?lu,\"F07$$\"5++++](o'=msFC$\"*HV;+\"F07$Fi cl$\"*a)>Y5F07$F^dl$\"*\"F07$F][x$\"*V_([9F07$ Fhdl$\"*D51p\"F07$Fe[x$\"*0D-p\"F07$Fj[x$\"*)Q0y;F07$F_\\x$\"*sC))[\"F 07$F]el$\"*%*Q&H6F07$Fbel$\")RSoBF07$Fgel$!)!yh5#F07$F\\fl$!(St^(F07$F afl$\")ED*>\"F07$Fb^x$\")?\\lBF07$Fffl$\"*`_SZ#FC7$F__x$\"*0%fPFFC7$F[ gl$\")2!G&>F07$F`gl$\"($[8TF07$Fegl$!)7#oA\"F07$Fjgl$!):!o9#F07$F_hl$! )E%yK\"F07$Fihl$\")Z(\\'*)F07$Fa[m$\"*VSpu\"F07$Ff[m$\"*F\"F07$F`\\m$\"*4\"F07$Fe\\m$\"*(p(>B\"F07$F_]m$\"*=(*Qb \"F07$Fc^m$\"*[>Q0$F07$Fibx$\"*Z6FZ$F07$Fh^m$\"*vO9P&F07$$\"5ccc1%4krm 7\"F0$\"*ihfi&F07$Facx$\"*Ag^(zF07$$\"5a``.mJZjK6F0$\"*Dyxz)F07$F]_m$ \"*XlJ**)F07$Ficx$\"*M,?M*F07$Fb_m$\"+bE'*R5F07$Fadx$\"+;e>-8F07$Fg_m$ \"+u.O68F07$F\\`m$\"+`wCJ8F07$Fa`m$\"+1ZI.9F07$Ff`m$\"+_mR_:F07$F[am$ \"+?W'Qw\"F07$F`am$\"+kWCvk5CF07$Fifx$\"+z)*\\`CF0 7$Fccm$\"+h1bxCF07$Ffgx$\"+C%Q5M#F07$Fhcm$\"+@iS9BF07$F\\jx$\"+NV'o8#F 07$F]dm$\"+\\\"G@!>F07$Fc[y$\"+oiuO.#F07$F`fm$\"+cf^I?F07$Fi`y$\"+m2.H?F07$F^ay$\"+E>6H?F07$$\"5xww^ XJ#RkN\"F0$\"+I[nH?F07$Fcay$\"+G(4B.#F07$$\"5EEE^#y()z&f8F0$\"+..dT?F0 7$Fefm$\"+b*Qj/#F07$Fjfm$\"+3!4q2#F07$F_gm$\"+^PA4@F07$Faby$\"+(>9R6#F 07$Ffby$\"+*4tD7#F07$F[cy$\"+C&oD7#F07$Fdgm$\"+9+cA@F07$Fccy$\"+M=\\A@ F07$Fhcy$\"+I6%>7#F07$F]dy$\"+\"RD!>@F07$Figm$\"+wPn3@F07$F^hm$\"+&4>' 3@F07$Fchm$\"+1WF1@F07$F]im$\"+c87l?F07$Fgim$\"+?&f(e?F07$F\\jm$\"+`Jr L=F07$Fajm$\"+H/g\\;F07$Ffjm$\"+F,'zq\"F07$F[[n$\"+]ex<>F07$Fdiy$\"+8( Q\\2#F07$F`[n$\"+PfzDBF07$F[[z$\"+\\([#=BF07$F`[z$\"+O)z1J#F07$Fe[z$\" +;Y<0BF07$Fj[z$\"+l*e0J#F07$F_\\z$\"+(3KBN#F07$Fd\\z$\"+F`u)[#F07$Fi\\ z$\"+F07$Fc]n$\"+' QJ-\">F07$Fh]n$\"+O?nzquZ:F07$Fb^n$\"+AEUN:F07$Fg^n$\"+- U_A:F07$Fabz$\"+(***p.:F07$Ffbz$\"+M.p`9F07$F[cz$\"+H<*QH\"F07$F\\_n$ \"+8)eJ3\"F07$Fhcz$\"+V@Ik5F07$Fa_n$\"*DRs(**F07$Fedz$\"*7PMi)F07$Fjdz $\"*8\\f*oF07$F_ez$\"*tqK%oF07$Ff_n$\"*$ye)y'F07$Fgez$\"*8z'>lF07$F[`n $\"*f(>-WF07$F``n$\"*Q!>FRF07$Fe`n$\"*tKTQ#F07$Fj`n$\"*za)*e\"F07$F_an $\"*Hz7L\"F07$Fdan$\"*'\\s\\8F07$Fian$\"*9j7T\"F07$F^bn$\"*i9m^\"F07$F cbn$\"*`K&pF07$Fchz$\"*T&Q!)>F07$F ]cn$\"*SRZ$>F07$Fgcn$\"*,u&*o\"F07$Fadn$\"*(\\Zt7F07$Ffdn$\")E)*GeF07$ F[en$\")X=a:F07$F`en$\"(Fs\"fF07$Feen$\")TYR:F07$Fjen$\")L%3p#F07$F_fn $\")[HLVF07$Fdfn$\")O`#R%F07$Fifn$\"*)yu&Q&FC7$F][[l$\"*PUCU&FC7$F^gn$ \")Z(pS&F07$Fd\\[l$\")_M+WF07$Fcgn$\")UcpUF07$Fhgn$\")F`lFF07$F]hn$\") #z*o9F07$Fbhn$\")r_)>\"F07$Fghn$\"(JM'eF07$F\\in$\"(v\\J*F07$Fain$\")k 4,:F07$Ffin$\")m0>TF07$F[jn$\")'eE&\\F07$F_[o$\"*xgtp\"F07$Fc\\o$\"*uo e,#F07$Fh\\o$\"*w&y[F0$\"*lLl_\"F07$F\\^o$\"*>g\\b\"F07$$\"5wvvv +Kb#*))>F0$\"*v$3b;F07$Fa^o$\"*=?**y\"F07$F[_o$\"*t,IS#F07$Fe_o$\"*wi6 p#F07$Fj_o$\"*wf*3RF07$F_`o$\"*w9=Q'F07$Fba[l$\"*vIt['F07$F\\b[l$\"*)f ?-oF07$Fab[l$\"*Y@Sg(F07$Ffb[l$\"+JU(Q+\"F07$F[c[l$\"++0G95F07$Fd`o$\" ++iOA5F07$Fcc[l$\"+\"37D.\"F07$Fhc[l$\"+>pwb5F07$F]d[l$\"+_0%48\"F07$F bd[l$\"+!3Z.O\"F07$Fgd[l$\"+QlNu9F07$Fi`o$\"+\\)[v]\"F07$F^ao$\"+sXDX< F07$Fcao$\"+V!zy#>F07$F]bo$\"+`(z#f>F07$Fgbo$\"+j*4\"3@F07$F\\co$\"+)p wtJ#F07$Faco$\"+dj-%[#F07$$\"5IHHz;w@&\\2#F0$\"+D7y2DF07$Faf[l$\"+.h7> DF07$$\"5iiiP\")HDtz?F0$\"+8:_\"\\#F07$$\"5SSS!H5)fK\"3#F0$\"++/V*[#F0 7$$\"5===VCK%>H3#F0$\"+8yL*\\#F07$Ffco$\"+i?J3DF07$F^g[l$\"+]&yl\\#F07 $Fhg[l$\"+-IsUBF07$F]h[l$\"+[32]BF07$Fbh[l$\"+#=vmN#F07$Fgh[l$\"+A51fB F07$F[do$\"+NbnUBF07$F_eo$\"+N7&*H@F07$Fieo$\"+\\Wq/>F07$F^fo$\"+@l*4 \">F07$Fcfo$\"+`=z3>F07$Fhfo$\"+#3Azz\"F07$F]go$\"+q&[$fF07$F\\ho$\"+7me3@F07$F aho$\"+q\"G8;#F07$F__\\l$\"+6wtk@F07$Fd_\\l$\"+t6Bu@F07$Fi_\\l$\"+%*G% e<#F07$Ffho$\"+iT%e<#F07$Ff`\\l$\"+5xNv@F07$F[io$\"+0=7j@F07$F`io$\"+u ?#>;#F07$Feio$\"+R;Pg@F07$Fia\\l$\"+rI3a@F07$Fjio$\"+W![E8#F07$Fab\\l$ \"+)=)pJ@F07$F_jo$\"+=ToJ@F07$Fdjo$\"+Ko\\+@F07$Fijo$\"+g=`#3#F07$Fc[p $\"+\"4'e(4#F07$Fg\\p$\"+5))3Q@F07$Fec\\l$\"+\"*>@Q@F07$F\\]p$\"+*p?I9 #F07$F]d\\l$\"+86#f:#F07$Fbd\\l$\"+36Pm@F07$Fgd\\l$\"+44Pm@F07$Fa]p$\" +^>Sm@F07$Ff]p$\"+>FZv@F07$F[^p$\"+ww6u@F07$F`^p$\"+6(*e/@F07$Fe^p$\"+ mF$Q+#F07$Fef\\l$\"+\\kJ$*=F07$Fj^p$\"+!e#e!y\"F07$$\"5+++]P%y#f[BF0$ \"+`g\"=w\"F07$$\"5XXXX?9X7]BF0$\"+K5K=F07$Fc`p$\"+,'zA>#F07$F gap$\"+!*eitAF07$F`[]l$\"++#HCS#F07$F\\bp$\"+eZ``DF07$F]\\]l$\"+(GUI`# F07$Fabp$\"+I'R6\\#F07$Fe\\]l$\"+o_i5BF07$Ffbp$\"+E\")p2AF07$F]]]l$\"+ )**[j2#F07$Fb]]l$\"+r^NR>F07$Fg]]l$\"+*p'yE>F07$F\\^]l$\"+97#Q\">F07$F a^]l$\"+!*H!z*=F07$Ff^]l$\"+oK\"z'=F07$F[_]l$\"+ZWc!z\"F07$F[cp$\"+)pa le\"F07$Fc_]l$\"+8)f=Y\"F07$Fh_]l$\"+#G6cV\"F07$F]`]l$\"+d--,9F07$Fb`] l$\"+1*)418F07$Fg`]l$\"+av]e5F07$F`cp$\"+zdq;5F07$Fda]l$\"+l\")*4+\"F0 7$F^b]l$\"*s@g_*F07$Fcb]l$\"**[tb&)F07$Fhb]l$\"*WZE]'F07$F]c]l$\"*`$) \\X'F07$Fecp$\"*4:qS'F07$$\"5UTTT\"*G'3)zCF0$\"*Za(GUF07$Fjcp$\"*linv$ F07$$\"5====o!Q\"=$\\#F0$\"*Hq#fCF07$F_dp$\"*Wn%**=F07$$\"5%\\\\\\\\C8 al]#F0$\"**fe'z\"F07$Fddp$\"*51Ok\"F07$$\"5srrr@%)o#*>DF0$\"*#)zVk\"F0 7$Fidp$\"*O0!=;F07$Fcep$\"*fl^%=F07$F]fp$\"*C@\"*=#F07$Fgfp$\"*-1W)=F0 7$F[hp$\")+X%e)F07$F`hp$\")P>?iF07$Fehp$\")CWJRF07$Fjhp$\")&4y!RF07$F_ ip$\")MQuLF07$Fdip$\")_e[UF07$Fiip$\")q&z(eF07$Fch]l$\")k`*>'F07$Fhh]l $\")T5KuF07$F]i]l$\")fKOuF07$F^jp$\"*>/of(FC7$Fei]l$\"*E%)GE)FC7$Fcjp$ \")eRB!)F07$Fhjp$\")&\\y!oF07$F][q$\")&4.*\\F07$Fhj]l$\")DgJPF07$Fb[^l $\")x=JPF07$Fg[^l$\")(p;q$F07$Fb[q$\")xr[MF07$F_\\^l$\")!pJT$F07$Fd\\^ l$\")[)HW$F07$Fi\\^l$\")bWLTF07$Fg[q$\")u^BZF07$F\\\\q$\")V\"fz(F07$Fa \\q$\"*v&y#G\"F07$F[]q$\"*t(e`?F07$Fi^q$\"*mhFF#F07$F]^^l$\"*n&)4H#F07 $F^_q$\"*0)*))H#F07$$\"5**)*)*[6F7#)=GF0$\"*/!f]AF07$Fe^^l$\"*%e4r@F07 $$\"5ooo=1yU9DGF0$\"*\"3+w@F07$Fc_q$\"*0@48#F07$$\"5'ooo=Jf0Y$GF0$\"*J y(**>F07$Fh_q$\"*Ve0&=F07$$\"5````Gs^?ZGF0$\"*@J@'=F07$F]`q$\"*%*o9!=F 07$Fb`q$\"*HQ$==F07$Fg`q$\"*h_;(=F07$Faaq$\"*!zb!>#F07$F_cq$\"*\"=mSIF 07$Fi_^l$\"*`[a$QF07$Fdcq$\"*1`&Q[F07$$\"5,--_ROH/=HF0$\"*y\"fTaF07$Fa `^l$\"*BB')e(F07$$\"5$QQQj>1&4CHF0$\"*'\\>5xF07$Ficq$\"*QN>9)F07$Fi`^l $\"*q>dF*F07$F^a^l$\"+idH_6F07$Fca^l$\"+**zWh6F07$F^dq$\"+`n$4<\"F07$F [b^l$\"+*R1W=\"F07$F`b^l$\"+n*=)>7F07$Feb^l$\"+RN([L\"F07$Fcdq$\"+==O6 ;F07$Fbc^l$\"+d&3bj\"F07$Fhdq$\"+#HXIo\"F07$F_d^l$\"+\\`oz F07$Fihq$\"+JWGk-y\"z\"F07$F^iq$\"+v$ff)=F07$Fi^_l$\"+` 2cN?F07$Fciq$\"+T`_x@F07$F]jq$\"+\"RFd=#F07$Fa[r$\"+rIW?AF07$Ff[r$\"+i $3oA#F07$F[\\r$\"+CZtFAF07$F`\\r$\"+[Rm/AF07$Fe\\r$\"+MmSe@F07$Fc`_l$ \"+apeZ@F07$Fj\\r$\"+n[OM@F07$F[a_l$\"+KaYP@F07$F_]r$\"+f$p=9#F07$Fd]r $\"+%R=%z@F07$Fi]r$\"+H*fSA#F07$F^^r$\"+*z'HDAF07$Fc^r$\"+b0VEAF07$$\" 5vuu*4Q;Xg>$F0$\"+u$F0$\"+1+F5 AF07$F]_r$\"+_!Qd>#F07$Fb_r$\"+E(*e&>#F07$Fg_r$\"+3,B!>#F07$F\\`r$\"+; Tva?F07$Fa`r$\"+Sw(4#>F07$F[c_l$\"+%QA\">=F07$Ff`r$\"+Xn)yv\"F07$Fhc_l $\"+\\F)f'=F07$F[ar$\"+'pvH1#F07$Fear$\"+ghK*G#F07$F_br$\"+x8MKDF07$$ \"5sssAN8/C5LF0$\"+G[$)GEF07$$\"5AAAAZ`&y=J$F0$\"+O@?;EF07$$\"5srr@f$p ;NJ$F0$\"+-3E.EF07$Fdbr$\"+C?N!f#F07$$\"5????&R6J%=LF0$\"+w[$*oDF07$Fi br$\"+AVZVDF07$F^cr$\"+xEPSCF07$Fccr$\"+Qh6ZAF07$F[h_l$\"+mZIAAF07$Fhc r$\"+`+Dk@F07$Fch_l$\"+YD35?F07$F]dr$\"+iD&R(=F07$Fbdr$\"+()e&z%=F07$F gdr$\"+%e`Vz\"F07$Fai_l$\"+QH&fp\"F07$Ffi_l$\"+%\\KVV\"F07$F[j_l$\"+%o kwR\"F07$F\\er$\"+$GGuQ\"F07$Fcj_l$\"+^L7d8F07$Faer$\"+([G?<\"F07$F`[` l$\"*Kv_a*F07$Ffer$\"*#oex$*F07$F]\\`l$\"*x]ng)F07$F[fr$\"*_'>GhF07$F` fr$\"*VI4w&F07$Fefr$\"*_Pn!QF07$F_gr$\"*A9Jb#F07$Figr$\"*F&)y-#F07$F^h r$\"*r_Y&>F07$Fchr$\"*l-\"o>F07$Fhhr$\"*l;#GBF07$F]ir$\"*$f:FDF07$Fe[s $\"*y*=F;F07$F_\\s$\")uov\")F07$$\"5'ff4sP+Y\"QNF0$\")%eV!yF07$Fa_`l$ \")g*Rs'F07$$\"5=>>%HEa08a$F0$\")3d6kF07$Ff_`l$\")ZU6kF07$F[``l$\")3F( R'F07$Fd\\s$\")y6$G'F07$Fh``l$\")WFeiF07$Fi\\s$\")A%*otF07$F^]s$\")Qfa !*F07$Fc]s$\"+L%Hb0\"FC7$$\"5$GGGG`>XNf$F0$\"+abd26FC7$Fh]s$\"*%*[)e5F 07$$\"5)zzzHUq%=.OF0$\"*@!od5F07$$\"5OOOO'Q(yR1OF0$\"*rx@0\"F07$$\"5uu uu\\V5h4OF0$\")ENP&*F07$F]^s$\")^/#>*F07$Fb^s$\")!\\nZ(F07$Fg^s$\")gcu iF07$F\\_s$\"*!3wZ6F07$Fa_s$\"*J[,1#F07$Fgb`l$\"*U0X@#F07$F\\c`l$\"*r5 ?S#F07$Fac`l$\"*O]jS#F07$Ffc`l$\"*ty&eCF07$F`d`l$\"*lPnd#F07$Ff_s$\"*^ cYb#F07$Fge`l$\"*9[fT#F07$F[`s$\"*aL([AF07$$\"5yyyyGm'3Cu$F0$\"*>o_7#F 07$F``s$\"*A4B7#F07$$\"5-...`li!\\v$F0$\"*?,W5#F07$Fe`s$\"*\"e<-@F07$$ \"5EFFFxkQSnPF0$\"*7-49#F07$Fj`s$\"*,&*R;#F07$$\"5]^^^,k9!*zPF0$\"**er _AF07$F_as$\"*UpQK#F07$F]cs$\"*v!G&R$F07$Fgcs$\"*_PDt&F07$F\\ds$\"*3s# >!)F07$Fads$\"*jXJ@*F07$F]i`l$\"*+rb()*F07$Fbi`l$\"+7(*f-7F07$Fgi`l$\" +%\\(e18F07$Ffds$\"+QJ2<8F07$F_j`l$\"+F%o\"H8F07$Fdj`l$\"+*GTLN\"F07$F ij`l$\"+]KYH9F07$F[es$\"+oeTn;F07$F`es$\"+%e&y%y\"F07$Fees$\"+/%QL\"=F 07$Fjes$\"+(y!\\[=F07$F_fs$\"+h([6%>F07$Fdfs$\"+\"G>m<#F07$Fifs$\"+$R \")>?#F07$Fcgs$\"+#=7zF#F07$Fghs$\"+$)\\h6DF07$Fh\\al$\"+))f%zd#F07$F \\is$\"+F0gIEF07$$\"5=<<4a#F07$$\"5utttB6Q'Q)QF0$\"+0kz[DF07$$\"5)yyyGmX Q`)QF0$\"+$GOTb#F07$Fais$\"+oho[DF07$Ffis$\"+yBK`BF07$F[js$\"+h[J:@F07 $F[aal$\"+t%*))>@F07$F`js$\"+`(RR7#F07$Fcaal$\"+*y$pC@F07$Fejs$\"+D^?4 @F07$Fjjs$\"+j,;Q?F07$F_[t$\"+'3k#G>F07$Fd[t$\"+%QI6$>F07$Fi[t$\"+[6cL >F07$Fgbal$\"+;0&=#>F07$F\\cal$\"+'>tC#=F07$Facal$\"+WatD=F07$F^\\t$\" +$o4L#=F07$Fical$\"+$oVF\"=F07$F^dal$\"+o-[1=F07$Fcdal$\"+]Ga2=F07$Fhd al$\"+^8i3=F07$F]eal$\"+?>S==F07$Fc\\t$\"+J9*H'=F07$Fh\\t$\"+iMGx?F07$ F]]t$\"+\"Gf%yAF07$Fg]t$\"+h@uaAF07$Fa^t$\"+WSZ\">#F07$Ff^t$\"+gnnFAF0 7$F[_t$\"+a F07$Fe_t$\"+ZMc`=F07$F[ial$\"+))*=o\"=F07$F`ial$\"+gJT1=F07$Feial$\"+a 9cT=F07$Fj_t$\"+FDZ_=F07$F]jal$\"+\\Q4i>F07$Fbjal$\"+Div<@F07$F\\[bl$ \"+Ybai@F07$F_`t$\"+SMB5CF07$Fd`t$\"+-Nz&e#F07$Fi`t$\"+K4.?EF07$F^at$ \"+)\\U%GDF07$Fcat$\"+!=nMJ#F07$Fa`bl$\"+T**oCAF07$Fhat$\"+g?L'=#F07$F i`bl$\"+90\">*>F07$F]bt$\"+,ui$z\"F07$Faabl$\"+=b.jMG\"F07$Fbcbl$\"+s.=E7F07$Fgcbl$\"+votj5F07$F \\dbl$\"*(\\Fb!*F07$Ffdbl$\"*ij@\"*)F07$Fgbt$\"*k'4T&)F07$$\"5uuuC(G!f `oUF0$\"*&e&3x(F07$$\"5]]]]vcD3qUF0$\"*q%GpeF07$$\"5EEEwj5#H;F%F0$\"*2 sW#eF07$F\\ct$\"*-D&zdF07$$\"5_```Gs\"piF%F0$\"*zI(>cF07$Fact$\"*gZ;[% F07$Ffct$\"*G!)*=PF07$F[dt$\"*YmQw#F07$$\"5OOOOhn?W)H%F0$\"*`dme#F07$$ \"5ihhh6C%[^I%F0$\"*gksK#F07$$\"5(ooo=1ya=J%F0$\"*)))HbAF07$F`dt$\"*8? sA#F07$$\"5iiii7]Q(>L%F0$\"*#3>vAF07$Fedt$\"*9-$QBF07$Fefbl$\"*([o$\\# F07$Fjdt$\"*=-Lr#F07$F]gbl$\"*$)>D!GF07$Fbgbl$\"*h`\\!GF07$Fggbl$\"*V` A!GF07$F\\hbl$\"*XBez#F07$Fahbl$\"*Bm*>FF07$F_et$\"*b%*4h#F07$Fdet$\"* a)[b;F07$Fiet$\")\"Q7**)F07$F]gt$\"*BM8A\"F07$Fggt$\"+)Qo=Q\"FC7$F\\ht $\"*E.yI\"F07$Faht$\"*Iz^G\"F07$Ffht$\"*2J`:\"F07$F[it$\")Lw2**F07$F`i t$\")dDS*)F07$Feit$\")()zq$*F07$F_jt$\"*z+W!>F07$Fc[u$\"*+X@u#F07$$\"5 OOOO6b))*3g%F0$\"*w>I'GF07$$\"5********\\PBB2YF0$\"*IJw'GF07$$\"5!===$ py!*R5YF0$\"*lsu!GF07$$\"5ijjj))>ec8YF0$\"*ngS\"GF07$$\"5`aaH[!>\\^h%F 0$\"*&=H;GF07$$\"5WXX&z5cKnh%F0$\"*nk5\"GF07$$\"5NOOhnJfJ=YF0$\"*\"3=x FF07$Fh[u$\"*8&*)fEF07$$\"5344fYVg1BYF0$\"*QYhl#F07$$\"5!444fYyKii%F0$ \"*Yr\"fEF07$$\"5sssA&e_*RHYF0$\"*8L1b#F07$$\"5aaaa/nicKYF0$\"*\"4H%\\ #F07$$\"5====V\\(**)QYF0$\"*Q(*)fCF07$F]\\u$\"*RmlT#F07$Fb\\u$\"*-q7U# F07$Fg\\u$\"*Ut$)[#F07$Fa]u$\"*&oo9FF07$Fe^u$\"*J)R)y$F07$Fj^u$\"*9p[` %F07$F__u$\"*1_'>nF07$F\\_cl$\"*+*)>$oF07$Fd_u$\"*`]eH(F07$Fi_cl$\"*eX jQ)F07$F^`cl$\"+_+\"\\,\"F07$Fc`cl$\"+1([E-\"F07$Fi_u$\"+ilmI5F07$F[ac l$\"+()*R:/\"F07$F`acl$\"+&eov1\"F07$Feacl$\"+$RVx9\"F07$Fjacl$\"+a3qy 8F07$F_bcl$\"+S#>.Z\"F07$F^`u$\"+6.&4]\"F07$Fgbcl$\"+,)\\ip\"F07$F\\cc l$\"+C)fZ$>F07$Ffccl$\"+99@j>F07$Fc`u$\"+5XEm?F07$$\"5!33eq)>Nq\\ZF0$ \"+F8feAF07$$\"5788jD\"fA7v%F0$\"+2xuJBF07$$\"5WXX?ki;u_ZF0$\"+*R+`M#F 07$Fcdcl$\"+](H\"fBF07$$\"5UUU#*zw))HdZF0$\"+Ls\\0CF07$Fh`u$\"+(G+(yDF 07$F`ecl$\"+x%)\\HEF07$F]au$\"+2:HgEF07$Fbau$\"+M)fNc#F07$Fgau$\"+FK&H M#F07$Fjicl$\"+S=GB@F07$Fdjcl$\"+dP_r>F07$Fijcl$\"+,_Z]>F07$F\\bu$\"+C 'f_'=F07$Fa[dl$\"+wu1m=F07$Ff[dl$\"+LZ&4(=F07$F[\\dl$\"+'yuv$>F07$Fabu $\"+(H(Hz>F07$Fh\\dl$\"+3-g`@F07$Ffbu$\"+EVrZAF07$$\"5[\\\\\\C=U#z([F0 $\"+tcCgAF07$$\"5MNNN&G([;\")[F0$\"+o\"p\"4BF07$$\"5?@@@YFbS%)[F0$\"+9 yJ4BF07$Fe]dl$\"+[es8BF07$$\"5******\\P4lE*)[F0$\"+ID#[K#F07$$\"5#HHHz m$o)3*[F0$\"+w$=XL#F07$$\"5&eee$)R;2D*[F0$\"+')Q_MBF07$$\"5yyyyG\"\\FT *[F0$\"+g3`MBF07$$\"5kkkk*e9ot*[F0$\"+Pi:MBF07$F[cu$\"+\")GtFBF07$Fecu $\"+\"3-#oAF07$F_du$\"+R*4HC#F07$Fddu$\"+WuZ)G#F07$Fidu$\"+v.S>BF07$Fi ^dl$\"+$y/'>BF07$F^_dl$\"+LOy?BF07$Fc_dl$\"+o&Q^K#F07$F^eu$\"+'=!HNBF0 7$F[`dl$\"+N\"y_L#F07$Fceu$\"+!HzZL#F07$F]fu$\"+M9kABF07$Fagu$\"+&RAVF #F0-Ffgu6&FhguF($\"#DF[hu$\"\"\"F)-F`hu6#%Pscheme~with~a~relatively~la rge~stability~regionG-F$6%7e\\nF'7$F+$!)3e*[$F07$F2$!)C$Q3)F07$F7$\"(n hw*F07$F<$\"*%oz)p\"F07$$\"5;::lFVeM/5FC$\"*pnS%>F07$$\"5feee$[,/0-\"F C$\"*#y%[%>F07$$\"5---_R'=im.\"FC$\"**=$e%>F07$Fciu$\"*-6![>F07$$\"5KK KK2,n8&3\"FC$\"*yMw'>F07$Fhiu$\"*]>4)>F07$F]ju$\"*\"*R<)>F07$Fbju$\"*S FB!>F07$Fgju$\"*!=4p;F07$F\\[v$\"*(40E;F07$Fa[v$\"*?_xi\"F07$FA$\"*Z-r i\"F07$Fi[v$\")g)=q*F07$FG$\")c]i>F07$$\"5CDDDDv-Jq:FC$\")!pm$>F07$FL$ \")gK%*=F07$$\"5_____-DV+cJ#F07$F_o$\"*izoG$F07$Fdo$\"*'=+okF07$Fio $\"*U*y0\")F07$F]]v$\"+^G!p9\"F07$F^p$\"+#e\\$p6F07$Fe]v$\"+**)4Q?\"F0 7$Fj]v$\"+*f\\8K\"F07$F_^v$\"+\"[H?o\"F07$Fcp$\"+`sDnF07$Fbq$\"+%f_gA#F07$Fgq$\"+2oV,CF07$F\\r$\"+0og=CF0 7$Far$\"+V'epV#F07$F^`v$\"+4w#)>DF07$Fh`v$\"+y;ZBHF07$F]av$\"+,i?eHF07 $Ffr$\"+$\\LH+$F07$Feav$\"+X\"z.G$F07$F[s$\"+.-r[LF07$Fes$\"+kAfwKF07$ F_t$\"+iKX\"3$F07$Fit$\"+y&R&fGF07$Fgv$\"+![(p!p#F07$$\"5kjjjjQP?)G$FC $\"+:,KrEF07$$\"5QPPPP()Ru.LFC$\"+04VPEF07$$\"576666OUG>LFC$\"+qXxREF0 7$F_ev$\"+)[q>k#F07$$\"5eeeeeLZO]LFC$\"+\"GDUk#F07$$\"5KKKKK#)\\!fO$FC $\"+?#>zk#F07$$\"511111J_W\"Q$FC$\"+98yeEF07$F\\w$\"+.%eVp#F07$F\\fv$ \"+]Ut:FF07$Faw$\"+_.$)oGF07$Fifv$\"+-TwxIF07$Ffw$\"+'ev$=LF07$Fagv$\" +(z#4#e$F07$F[x$\"+v&3'QOF07$F`x$\"+%3))Qc$F07$Fex$\"+3WQ\"[$F07$F_y$ \"+x4H8NF07$Fiy$\"+?i$F07$Fhz$\"+wb1TKF07$F][l$\"+3K\\MGF07$Fi[w$\"+Fp X;GF07$F^\\w$\"+0ZR)p#F07$Fc\\w$\"+))Q>%p#F07$Fb[l$\"+LpQ&o#F07$$\"5gh hhOIE4(o&FC$\"+Qs%Hn#F07$$\"5?BBBt5DV.dFC$\"+3+8bEF07$$\"5![[[)4\"Rs(> dFC$\"+$H#3_EF07$F[]w$\"+P`')[EF07$$\"55333$=:_Cv&FC$\"+8p0YEF07$$\"5q ppp>K?zodFC$\"+7'prk#F07$$\"5IJJJc7>8&y&FC$\"+;bilEF07$Fg[l$\"++\"H*GF F07$Fe]l$\"+#z2S*GF07$F_^l$\"+t4U3JF07$Fd^l$\"+62h5LF07$Fi^l$\"+D?>`LF 07$F^_l$\"+N&*oKLF07$Fc_l$\"+5!eq@$F07$F]dw$\"+1y&>+$F07$Fh_l$\"+d0.\" y#F07$Fedw$\"+@'3)QDF07$Fjdw$\"+D/wZAF07$F_ew$\"+_%[[?#F07$F]`l$\"+TDN [>F07$F\\fw$\"+#Rq4e\"F07$Fffw$\"+!Q]8b\"F07$F[gw$\"+84J5:F07$F`gw$\"+ 9&=LQ\"F07$Fegw$\"+;gB>5F07$Fb`l$\"*H+Eg*F07$F]hw$\"*Ou>a)F07$Fg`l$\"* Avn![F07$F[bl$\"*fz>\">F07$F_cl$\")())zX&F07$Fdcl$\")!G%3MF07$Ficl$\") $e8N#F07$F^dl$\").^!p\"F07$Fcdl$\")DwgJF07$F][x$\"*OSu/\"F07$Fhdl$\"*! \\#G)>F07$Fe[x$\"*yBw?#F07$Fj[x$\"*JgzH#F07$$\"5llllS4%[s*zFC$\"*#G/FA F07$F_\\x$\"*PW^?#F07$$\"5533e&*HxY[!)FC$\"*9h@?#F07$$\"5bbbb!o$3al!)F C$\"*$e+#=#F07$$\"5+..`lVRh#3)FC$\"*(3$\\3#F07$F]el$\"*Emmt\"F07$Fbel$ \")A)[A$F07$Fgel$!)wL!G%F07$F\\fl$!)W%[$>F07$Fafl$\");\"eT\"F07$Fh]x$ \"))zZ`\"F07$Fb^x$\")vx8MF07$Fg^x$\"*\\#paNFC7$Fffl$\"*dL&*f$FC7$F__x$ \"*6%HXSFC7$F[gl$\")z[\"p#F07$F`gl$\"'YuTF07$Fegl$!)[MpFF07$Fjgl$!)7pV VF07$F_hl$!)d[VHF07$Fihl$\"*Z;%39F07$Fa[m$\"*0J\"RBF07$Ff[m$\"*ho_K\"F 07$F[\\m$\")oDFlF07$F`\\m$\")HvV()F07$Fe\\m$\"*Mu%=6F07$F_]m$\"*sP`x\" F07$Fc^m$\"*,Y9N%F07$Fibx$\"*kqK-&F07$Fh^m$\"*'G=k\")F07$Facx$\"+q`tD7 F07$F]_m$\"+e]I%Q\"F07$Fb_m$\"+a!z;f\"F07$Fg_m$\"+@Ke()>F07$F\\`m$\"+b 7]BF07$F[am$\"+F#G7g#F07$F`am$\"+G )4!=EF07$Feam$\"+NE\"\\j#F07$F_bm$\"+?KU(o#F07$Fibm$\"+orojHF07$Fafx$ \"+PCRhJF07$F^cm$\"+CXH'R$F07$Fifx$\"+*px*[MF07$Fccm$\"+wJKtMF07$Ffgx$ \"+zzD7F0$\"+AID-GF07$$\"5\\\\\\*>,>Y4A\"F0$\"+$)\\TTFF07$$\"5'ee3@*\\khA7F 0$\"+Xu%\\u#F07$Fc[y$\"+Yq1[FF07$$\"5eeeL_pp&fA\"F0$\"+b;p\\FF07$$\"5& \\\\\\C$HsiF7F0$\"+!y&\\YFF07$$\"5KJJc7*[(HH7F0$\"+q6TKFF07$$\"5onnn#* [x'4B\"F0$\"+9U1BFF07$$\"5SSS!H&o#3VB\"F0$\"+,6bFFF07$Fbdm$\"+]q,UFF07 $F[\\y$\"+.j(R*HF07$Fgdm$\"+'=[o?$F07$F\\em$\"+f[1#f$F07$Faem$\"+L!p\" 3PF07$Ffem$\"+bBYgOF07$F[fm$\"+:g1/OF07$Fg_y$\"+O_0%e$F07$F\\`y$\"+$)Q LnNF07$Fa`y$\"+q]9nNF07$F`fm$\"+capkNF07$Fi`y$\"+05CiNF07$F^ay$\"+62Qi NF07$Fgbfl$\"+X-NjNF07$Fcay$\"+b0*yc$F07$F_cfl$\"+hP&Qe$F07$Fefm$\"+t` 2#f$F07$Fjfm$\"+'y%yWOF07$F_gm$\"+_,/+PF07$Faby$\"+)f[!3PF07$Ffby$\"+2 s!Gs$F07$F[cy$\"+\"=*zAPF07$Fdgm$\"+%=%yAPF07$Fccy$\"+cTmAPF07$Fhcy$\" +U&)p@PF07$F]dy$\"+A\")f;PF07$Figm$\"+\"*e^)p$F07$F^hm$\"+N-U)p$F07$Fc hm$\"+8pL%p$F07$F]im$\"+b**fAOF07$Fgim$\"+RY]6OF07$Faey$\"+*eS!fMF07$F \\jm$\"+,()=/KF07$Fiey$\"+0rEvGF07$Fajm$\"+P!38v#F07$F`gy$\"+A+DSFF07$ Ffjm$\"+%[9&HFF07$F]hy$\"+(R.O$GF07$F[[n$\"+;E\")yGF07$F`[n$\"+SY^iKF0 7$Fe[n$\"+\")*))4T$F07$Fa]z$\"+)3[`R$F07$Ff]z$\"+<'4RQ$F07$F[^z$\"+37z *Q$F07$Fj[n$\"+#)>%GW$F07$Fh^z$\"+,'ouT$F07$F_\\n$\"+Z&=TP$F07$Fd\\n$ \"+6*43?$F07$Fi\\n$\"+1su$z#F07$F^]n$\"+]2P_FF07$Fc]n$\"+Sc$ym#F07$Fh] n$\"+BYc*[#F07$F]^n$\"+BOwp@F07$Fb^n$\"+(>'[_@F07$Fg^n$\"+IPRM@F07$Fab z$\"+A(zx5#F07$Ffbz$\"+s'[d.#F07$F[cz$\"+Zpx,=F07$F\\_n$\"+\"G!f!\\\"F 07$Fhcz$\"+M=`k9F07$Fa_n$\"+AZ5l8F07$Fedz$\"+mN8d6F07$Fjdz$\"*+YY!*)F0 7$F_ez$\"*%fgO))F07$Ff_n$\"*-'Gl()F07$Fgez$\"*hB:P)F07$F[`n$\"**[M(3&F 07$F``n$\"*^F%*R%F07$Fe`n$\"*#p!p8#F07$Fj`n$\"*IUg8\"F07$F_an$\")C)4c) F07$$\"5JJJJJc%>Ji\"F0$\")#oSq(F07$Fdan$\")6einF07$$\"5BBBBB)p0ij\"F0$ \")u*4V'F07$Fian$\")vf2hF07$$\"5:::::S>H\\;F0$\")%QCD(F07$F^bn$\")`RE# )F07$$\"522222#=yBm\"F0$\"*T>9J\"F07$Fcbn$\"*98uf\"F07$Fhbn$\"*\"pt2DF 07$F]cn$\"*H)>HFF07$Fgcn$\"*(GEHDF07$Fadn$\"*`m&\\>F07$Ffdn$\")Zu_%)F0 7$F[en$\")\"Q9K\"F07$F`en$!(.g'HF07$Feen$\")_1W8F07$F_fn$\")6rUhF07$Fi fn$\"*-lT%zFC7$F][[l$\"*SB[+)FC7$F^gn$\")L\\tzF07$Fd\\[l$\")NQRiF07$Fc gn$\")ts9gF07$Fhgn$\")%*RIMF07$F]hn$\")3\\17F07$Fbhn$\"(MsU(F07$Fghn$! (o6.$F07$F\\in$\"(gF(GF07$Fain$\").:i7F07$Ffin$\")`Y+dF07$F[jn$\")#f** 3(F07$Fejn$\"*d_3*>F07$F_[o$\"*NDce#F07$Fd[o$\"*@/It#F07$Fi[o$\"*)RRUG F07$F^\\o$\"*T=-o#F07$Fc\\o$\"*$on+EF07$Fh\\o$\"*Emmg\"F07$F]]o$\"*p\\ a<\"F07$$\"5*******\\7G&\\]>F0$\"*B%*y-\"F07$Fb]o$\"*;u$[5F07$$\"5MMMM 4Ggaj>F0$\"*K>&f6F07$Fg]o$\"*gE\"y8F07$F\\^o$\"*dD%R;F07$Fa^o$\"*\"\\9 2@F07$F[_o$\"*[80=$F07$Fe_o$\"*(\\TROF07$Fj_o$\"*$p)4q&F07$F_`o$\"*cHv u*F07$Fba[l$\"*5)y3**F07$F\\b[l$\"+(yk'R5F07$Fab[l$\"+\"QwS;\"F07$Ffb[ l$\"+E.VS:F07$F[c[l$\"+h%>kb\"F07$Fd`o$\"+-%>)o:F07$Fcc[l$\"+QxB%e\"F0 7$Fhc[l$\"+Teo=;F07$F]d[l$\"+&Qixs\"F07$Fbd[l$\"+*RRi0#F07$Fgd[l$\"+-m 3?AF07$Fi`o$\"+=_4oAF07$F^ao$\"+-@P$e#F07$Fcao$\"+fdGCGF07$F]bo$\"+(4) yoGF07$Fgbo$\"+$p?>0$F07$F\\co$\"++dy4LF07$Faco$\"+71L%\\$F07$F_[hl$\" +<3lFNF07$Faf[l$\"+PJ'=a$F07$Fg[hl$\"+'3g2]$F07$F\\\\hl$\"+;2U(\\$F07$ Fa\\hl$\"+;kM6NF07$Ffco$\"+s\"*4CNF07$F^g[l$\"+cxg9NF07$Fhg[l$\"+rw2RL F07$Fbh[l$\"+_e7fLF07$F[do$\"+cwV[LF07$F_eo$\"+`byDJF07$Fieo$\"+qcO4HF 07$F^fo$\"+^1->HF07$Fcfo$\"+eU*4#HF07$Fhfo$\"+s#)[MGF07$F]go$\"+9F12GF 07$Fa[\\l$\"+(4Z=\"GF07$Ff[\\l$\"+,)>\">GF07$F`\\\\l$\"+lfNLGF07$Fj\\ \\l$\"+o^qKHF07$F_]\\l$\"+KR,NHF07$Fbgo$\"+A(oF%HF07$F\\^\\l$\"+)*)[W; $F07$Fggo$\"+#[gb[$F07$F\\ho$\"+BN%[o$F07$Faho$\"+(G\\mx$F07$F__\\l$\" +Yw`#y$F07$Fd_\\l$\"+<^*))z$F07$Fi_\\l$\"+5#o;!QF07$Ffho$\"+\"\\q;!QF0 7$Fa`\\l$\"+Caf,QF07$Ff`\\l$\"+&eL3!QF07$F[a\\l$\"+#Rwlz$F07$F[io$\"+k kszPF07$F`io$\"+,flxPF07$Feio$\"+*3#*\\x$F07$Fia\\l$\"++X=kPF07$Fjio$ \"+%HQts$F07$Fab\\l$\"+)Q0ds$F07$F_jo$\"+T7oDPF07$Fdjo$\"+cb2$F07$F ]g\\l$\"+bJ.:HF07$Fbg\\l$\"+8'4H&GF07$$\"5kjj8,Bm%3O#F0$\"+#zh#>GF07$F gg\\l$\"+ltz;GF07$$\"5aaa/n#35RO#F0$\"+z*\\T\"GF07$F\\h\\l$\"+`?O6GF07 $Fah\\l$\"+otn6GF07$F__p$\"+,HsFGF07$Fgap$\"+JX+JKF07$Fabp$\"+SNV?MF07 $Fe\\]l$\"+4N!4?$F07$Ffbp$\"+TDWkIF07$Fb]]l$\"+$*HJ;FF07$F\\^]l$\"+C>d !o#F07$Ff^]l$\"+Q51F07$Fb`]l$\"+C<#=#=F07$Fg`]l$\"+/1g^9F07$F`cp$ \"+$40(*Q\"F07$Fda]l$\"+Du2o8F07$F^b]l$\"+vnw&H\"F07$Fcb]l$\"+X6%za#F0$\"*voK5\"F07$$\"5qqqXk1a Y\\DF0$\"*y23C\"F07$Fcep$\"*5brL\"F07$Fhep$\"**pLy=F07$F]fp$\"*=sa_#F0 7$Fgd]l$\"*GJvq#F07$F\\e]l$\"*dg(3IF07$Fae]l$\"*2dx+$F07$Fbfp$\"*W(4hI F07$Fie]l$\"*-0A8$F07$F^f]l$\"*C6t7$F07$Fcf]l$\"*Xkz-$F07$Fgfp$\"*=Ao$ GF07$Fagp$\"*#3V9AF07$F[hp$\"*KgND\"F07$F`hp$\")fw7')F07$Fehp$\")\"H;w %F07$Fjhp$\")l!=s%F07$F_ip$\")'p=$QF07$Fdip$\")p!HM&F07$Fiip$\")G9V\") F07$$\"5999kwa(Rjn#F0$\")'*4W#)F07$Fch]l$\")*R[p)F07$$\"5TSS!zA'[hzEF0 $\"*4/w,\"F07$Fhh]l$\"*'y)33\"F07$F]i]l$\"*_6;3\"F07$F^jp$\"+)>!346FC7 $Fei]l$\"++'4HA\"FC7$Fcjp$\"+xr3\"=\"FC7$Fhjp$\")9v=(*F07$F][q$\")4S(f 'F07$Fcj]l$\")\"f'4fF07$Fhj]l$\")3wRWF07$F][^l$\")LvRWF07$Fb[^l$\")e/R WF07$Fg[^l$\"):])Q%F07$Fb[q$\")s%z&RF07$F_\\^l$\")jy(*QF07$Fd\\^l$\")Z f[RF07$Fi\\^l$\")\\jF^F07$Fg[q$\")pyNhF07$F\\\\q$\"*iiE8\"F07$Fa\\q$\" *-H!f>F07$Ff\\q$\"*Q!=oDF07$F[]q$\"*R(*G4$F07$Fe]q$\"*I(G5JF07$Fi^q$\" *5$*GD$F07$F^_q$\"*]\"\\**HF07$Fc_q$\"*\\IJK#F07$Fegil$\"*hR#4>F07$Fh_ q$\"*q0a`\"F07$F]hil$\"*\")=*R:F07$F]`q$\"*.5^[\"F07$Fb`q$\"*X]Go\"F07 $Fg`q$\"*CY4.#F07$Faaq$\"*OW,u#F07$F_cq$\"*&p]yTF07$Fi_^l$\"*l)H#\\&F0 7$Fdcq$\"*(e(3<(F07$Fgiil$\"*&evR\")F07$Fa`^l$\"+\"o7>;\"F07$F_jil$\"+ Ou`!=\"F07$Ficq$\"+GJwY7F07$Fi`^l$\"+aGu>9F07$F^a^l$\"+Ys'3w\"F07$Fca^ l$\"+ZF&[x\"F07$F^dq$\"+,\\K*y\"F07$F[b^l$\"+m?]4=F07$F`b^l$\"+32!3'=F 07$Feb^l$\"+F[!Q-#F07$Fcdq$\"+YD_5CF07$Fbc^l$\"+:dgYCF07$Fhdq$\"+4d18D F07$F_d^l$\"+!=#fREF07$Fdd^l$\"+6')*p)HF07$Fid^l$\"+h\"[r+$F07$F]eq$\" +n:=FIF07$Fafq$\"+J#)3eLF07$F[gq$\"+NPspMF07$Fge^l$\"+ik!Gb$F07$F`gq$ \"+6))p)f$F07$F_f^l$\"+9G9:OF07$Fdf^l$\"+`#4#GOF07$Fif^l$\"+$e?mi$F07$ Fegq$\"+@M([d$F07$Fjgq$\"+E7EpLF07$F_hq$\"+[V'*\\JF07$Fdhq$\"+4ijfHF07 $Fihq$\"+p()**)*GF07$Fa^_l$\"+Z,V`IF07$F^iq$\"+OHXcKF07$Fi^_l$\"+3FBQN F07$Fciq$\"+gM9\"z$F07$F]jq$\"+mqS0QF07$Fa[r$\"+P1mlQF07$Ff[r$\"+:kgwQ F07$F[\\r$\"+Cz=yQF07$F`\\r$\"+.PYQQF07$Fe\\r$\"+ez/fPF07$Fc`_l$\"+$)z \\SPF07$Fj\\r$\"+0!=zr$F07$F[a_l$\"+kDIBPF07$F_]r$\"+'GP4t$F07$Fd]r$\" +j&)\\&z$F07$Fi]r$\"+YS0sQF07$F^^r$\"+'pCT(QF07$Fc^r$\"+rp'e(QF07$Fh^r $\"+,G4pQF07$F]_r$\"+&G^C#QF07$Fb_r$\"+6F>AQF07$Fg_r$\"+BT#G\"QF07$F\\ `r$\"+A![Ld$F07$Fa`r$\"+\"=1oK$F07$F[c_l$\"+-/$)>JF07$Ff`r$\"+nR#p%HF0 7$$\"5?>>W]*)o')eKF0$\"+w%*)R#HF07$$\"5nmm;HK_VgKF0$\"+,qo*)GF07$$\"59 99*y]d.?E$F0$\"+'Heo)GF07$Fcc_l$\"+ok%Q)GF07$$\"5544Mlg-9lKF0$\"+3*y3) GF07$$\"5dcc1W.'3nE$F0$\"+StbzGF07$$\"5///zAYpFoKF0$\"+KMd')GF07$Fhc_l $\"+B0_AHF07$F]d_l$\"+\\'>4#HF07$Fbd_l$\"+=mG;HF07$Fgd_l$\"+pMz.IF07$F [ar$\"+$eX&pIF07$Fear$\"+j&*\\pKF07$F_br$\"+(*R.0NF07$F_fjl$\"+Jo37OF0 7$Fdfjl$\"+j\"HZf$F07$Fifjl$\"+uq%pd$F07$Fdbr$\"+5fAfNF07$Fagjl$\"+d&G 9`$F07$Fibr$\"+K*H<]$F07$F^cr$\"+hs$)oLF07$Fccr$\"+O5;EJF07$Fhcr$\"+x# *[;IF07$F]dr$\"+:=\\KEF07$Fbdr$\"+!R$*ff#F07$Fgdr$\"+d:1@DF07$Fai_l$\" +RPL#Q#F07$Ffi_l$\"+4lC4?F07$F[j_l$\"+R&*)p&>F07$F\\er$\"+`AlU>F07$Fcj _l$\"+2hO**=F07$Faer$\"+=\\OA;F07$F`[`l$\"+5&RSH\"F07$Ffer$\"+$*3\"3F \"F07$F]\\`l$\"+\"*pK`6F07$F[fr$\"*3!y#o(F07$F`fr$\"*xyV;(F07$Fefr$\"* \"[b-UF07$F_gr$\"*`mC[#F07$Figr$\"*7fA'=F07$F^hr$\"*Kp>i\"F07$Fchr$\"* (4I%[\"F07$Fhhr$\"*y1.R#F07$F]ir$\"*\\_7\\$F07$Fe[s$\"*Ci9Y#F07$F_\\s$ \"*,mZ8\"F07$F]][m$\"*9&=s5F07$Fa_`l$\")%p=!*)F07$Fe][m$\")jsv$)F07$Ff _`l$\")\")[v$)F07$F[``l$\")'*z^$)F07$Fd\\s$\"))4o;)F07$Fh``l$\")Y^L\") F07$Fi\\s$\"*lK[+\"F07$F^]s$\"*#yP%H\"F07$Fc]s$\"+Qai^:FC7$F__[m$\"+@` OS;FC7$Fh]s$\"*wPgb\"F07$Fg_[m$\"*o:Sb\"F07$F\\`[m$\"*wcXa\"F07$Fa`[m$ \"*.Q_P\"F07$F]^s$\"*LMeJ\"F07$Fb^s$\"*$4N@5F07$Fg^s$\")_ua\")F07$F\\_ s$\"*Gjap\"F07$Fa_s$\"*]!3VJF07$Fgb`l$\"*z?OM$F07$F\\c`l$\"*,!Q\"e$F07 $Fac`l$\"*z')fe$F07$Ffc`l$\"*;>2h$F07$F[d`l$\"*R!)*QOF07$F`d`l$\"*nnGk $F07$Fed`l$\"*\\Ghj$F07$Fjd`l$\"*GGHf$F07$F_e`l$\"*Kp8V$F07$Ff_s$\"*wV ^I$F07$Fge`l$\"*q*e^FF07$F[`s$\"*_Ph@#F07$$\"5ssssZmUGRPF0$\"*^n4/#F07 $F]c[m$\"*t;)G>F07$$\"5%[[[)4mI`XPF0$\"*zx&R>F07$F``s$\"*1*=Q>F07$$\"5 'ppp>d'=y^PF0$\"*T\\h%>F07$Fec[m$\"*p&fi>F07$$\"53444Ml1.ePF0$\"*$Rl5? F07$Fe`s$\"*5!*G;#F07$Fj`s$\"*6^RX#F07$F_as$\"*Eje$GF07$F]cs$\"*@%)op% F07$Fgcs$\"*LWpe)F07$F\\ds$\"+`3FB7F07$Fads$\"+A\"=7T\"F07$F]i`l$\"+fM +6:F07$Fbi`l$\"+Z-AK=F07$Fgi`l$\"+3Z8()>F07$Ffds$\"+_Y2.?F07$F_j`l$\"+ ;OG@?F07$Fdj`l$\"+%=Wj0#F07$Fij`l$\"+0^.j@F07$F[es$\"+*=*4!\\#F07$F`es $\"+-1j_EF07$Fees$\"+B&4Rp#F07$F_fs$\"+t'R&fGF07$Fifs$\"+)o/w=$F07$Fcg s$\"+g;O*G$F07$Fghs$\"+'ym@c$F07$Fh\\al$\"+O$[*QOF07$F\\is$\"+0L!*)p$F 07$$\"5/...G%e!\\wQF0$\"+!*\\.2PF07$F_i[m$\"+)**=fp$F07$$\"5KJJJ1v)R%z QF0$\"+MbOHOF07$F_^al$\"+_x^#e$F07$Fgi[m$\"+C%QWf$F07$F\\j[m$\"+U3m0OF 07$Faj[m$\"+(oWRh$F07$Fais$\"+p)3+h$F07$Ffis$\"+qyy*R$F07$F[js$\"+XZ,g JF07$F`js$\"+X70tJF07$Fejs$\"+Yb`jJF07$Fjjs$\"+,BO*4$F07$F_[t$\"+STy** HF07$Fd[t$\"+F![U+$F07$Fi[t$\"+5RF0$\"+hHlxHF07$F\\cal$\"+b\"y'\\ HF07$Facal$\"+XQ@bHF07$F^\\t$\"+6u(e'HF07$Fhdal$\"+Xvx?IF07$Fc\\t$\"+j rAvJF07$Feeal$\"+H,T#Q$F07$Fh\\t$\"+:$)y'f$F07$F]fal$\"+E2Y+QF07$F]]t$ \"+fiv_RF07$Fg]t$\"+1*y=\"RF07$Fa^t$\"+K$yL!QF07$Ff^t$\"+I8\"e'QF07$F[ _t$\"+T]oZRF07$Fagal$\"+m\"\\#yQF07$F`_t$\"+P#)o@PF07$F^hal$\"+&embN$F 07$Fe_t$\"+rJ^bJF07$F`ial$\"+otm1IF07$Fj_t$\"+SU&3'HF07$Fbjal$\"+[#o@: $F07$F_`t$\"+r@y>MF07$Fi[bl$\"+C#o*3MF07$F^\\bl$\"+j-(*)R$F07$Fc\\bl$ \"+!eMkR$F07$Fh\\bl$\"+2ZGCMF07$F]]bl$\"+p!f9a$F07$Fb]bl$\"+;Wc8OF07$F g]bl$\"+\\1/*f$F07$Fd`t$\"+%GzWe$F07$$\"5nnn#*[]8z0UF0$\"+k$F07$Fi`t$\"+A9\"Qg$F07$F^at$\"+V!3F\\$F07$Fcat$\"+3aa@KF07$Fa `bl$\"+g/G2JF07$Fhat$\"+VGaaIF07$Fi`bl$\"+5$HSz#F07$F]bt$\"+#)4DDDF07$ Faabl$\"+rY?#[#F07$Ffabl$\"+=l*>G#F07$F[bbl$\"+a1bv=F07$F`bbl$\"+6i*>& =F07$Febbl$\"+Q<*y$=F07$Fbbt$\"+\">g:#=F07$F]cbl$\"+\"Rr5z\"F07$Fbcbl$ \"+3(y_q\"F07$Fgcbl$\"+(*='zX\"F07$F\\dbl$\"+!H/h@\"F07$Ffdbl$\"+,Az'> \"F07$Fgbt$\"+:QqT6F07$Fge\\m$\"+GrZA5F07$F\\f\\m$\"*2G3E(F07$Faf\\m$ \"*a)Q0sF07$F\\ct$\"*:a&\\rF07$Fif\\m$\"*()H%GpF07$Fact$\"*Z;L@&F07$Ff ct$\"*$>C0TF07$F[dt$\"*?(G9GF07$F\\h\\m$\"*&)esJ#F07$F`dt$\"*AWF5#F07$ $\"5PPPPi$\\n_K%F0$\"*x=U+#F07$Fih\\m$\"*'**R4>F07$$\"5)yyyGm?!oQVF0$ \"*Fzg#>F07$Fedt$\"*\"QE.?F07$Fjdt$\"*FPvA$F07$F_et$\"*XG?'QF07$Fdet$ \"*a5:Z#F07$Fiet$\"*S\")e@\"F07$F]gt$\"*XT8x\"F07$Fggt$\"+zp!e/#FC7$F \\ht$\"*7,!=>F07$Faht$\"*u=\"z=F07$Ffht$\"*3$)el\"F07$F[it$\"*(y[t8F07 $F`it$\"*Y!*y?\"F07$Feit$\"*gQ=G\"F07$F_jt$\"*DH\"))GF07$Fc[u$\"*E6#\\ SF07$Fh[u$\"*d$ohIF07$F]\\u$\"*RCZN#F07$Fg\\u$\"*Z!3!)HF07$Fe^u$\"*L9H F&F07$Fj^u$\"*-'fJlF07$F__u$\"+MXG95F07$F\\_cl$\"+p`IJ5F07$Fd_u$\"+F#3 S5\"F07$Fi_cl$\"+mW%fF\"F07$F^`cl$\"+E9Z`:F07$Fc`cl$\"++lJl:F07$Fi_u$ \"+Z>ex:F07$F[acl$\"+9E6%f\"F07$F`acl$\"+XN5L;F07$Feacl$\"+]'p;v\"F07$ Fjacl$\"+U\\')*3#F07$F_bcl$\"+t+YCAF07$F^`u$\"+gdXpAF07$Fgbcl$\"+Oc!f` #F07$F\\ccl$\"+bLIdGF07$Ffccl$\"+Znj)*GF07$Fc`u$\"+3SLJIF07$Fcdcl$\"+[ 5N(R$F07$Fh`u$\"+dzU[OF07$F`ecl$\"+)=w\">PF07$F]au$\"+>\"pWu$F07$Fbau$ \"+(*zWVOF07$Fgau$\"+N;\\SF07$F\\[^m$\"+2IYOSF07$Fa[^m$\"+UDZOSF07$Ff[^m$\"+ 1Z[OSF07$F[\\^m$\"+)zQe.%F07$F[cu$\"+U(=Z-%F07$Fecu$\"+()*QC#RF07$F_du $\"+/UDzQF07$Fddu$\"+?fldRF07$Fidu$\"+#[32,%F07$Fi^dl$\"+3&e5,%F07$F^_ dl$\"+v_28SF07$Fc_dl$\"+Yf^?SF07$F^eu$\"+cB$y.%F07$F[`dl$\"+49\"y.%F07 $Fceu$\"+uW$p.%F07$F]fu$\"+dbq:SF07$Fagu$\"+m**\\JRF0-Ffgu6&FhguF($\"# vF[huF\\hu-F`hu6#%TButcher's~scheme~B~with~c[5]=c[6]=1/2~and~b[5]=b[6] G-F$6%7[fmF'7$F+$!)Qt_-!Q7F07$Fbju$\"*h`(47F07$ Fgju$\"*=zY6\"F07$F\\[v$\"*S3o4\"F07$Fa[v$\"*fxz4\"F07$FA$\"*A()y4\"F0 7$Fi[v$\")kO*p'F07$FG$!),C>QF07$FQ$!*.V:x\"F07$Fen$!*VGcF$F07$F_o$!*x= r8$F07$Fio$!*E#[h>F07$F^p$!*))oJ,\"F07$Fcp$\")Z&4$eF07$F]q$\")hYo))F07 $Far$\"*'*y53#F07$Fh`v$\"*tNw)HF07$Ffr$\"*s.s1$F07$$\"5)yyyyGTOQp#FC$ \"*sL,:$F07$Feav$\"*/;\")=$F07$$\"5====o!Qh)QFFC$\"*JJL?$F07$$\"5GGGGG .(pQv#FC$\"*[3C@$F07$$\"5QQQQ)e-y)oFFC$\"*av1>$F07$F[s$\"*j-q0$F07$F`s $\"*)e1qDF07$Fes$\"*$e`,8F07$Fcbv$\"*Jc%48F07$Fjs$\"*^^4D\"F07$F[cv$\" )G_*3$F07$F_t$!)tmsCF07$Fit$!*Yv6v\"F07$Fgv$!*%p>JHF07$Fei^m$!*ASV5$F0 7$Fji^m$!*!\\_1MF07$F_j^m$!*Z`&4MF07$F_ev$!*\\YCT$F07$F\\[_m$!***>$>MF 07$F\\w$!*x[KQ$F07$Fgev$!*B3(pLF07$F\\fv$!*qO7O$F07$Fafv$!*]I07$F07$Fa w$!*xPU$GF07$Fifv$!*m'=()>F07$Ffw$!)za*4*F07$Fagv$\")+'oR$F07$F[x$\")! f\\8'F07$F`x$\")OEjBF07$Fex$!)q#4u\"F07$F_y$!(x*[8F07$Fiy$\")n5hXF07$F ehv$\")L&*oXF07$Fjhv$\");`*)[F07$F_iv$\")c9IdF07$Fdiv$\")u8uiF07$Fiiv$ \")O#F07$F][l$!*:&\\rHF07$Fi[w$!*a*R@IF07$ F^\\w$!*+*R4MF07$Fc\\w$!*r9VS$F07$Fb[l$!*4:0S$F07$F[]w$!*%ywLLF07$Fg[l $!*VZHq#F07$Fe]l$!*F,wa\"F07$F_^l$!(*f\\XF07$Fd^l$\"*6rXK\"F07$Fi^l$\" *A+l.#F07$F^_l$\"*(estBF07$Fc_l$\"*aOZw#F07$F]dw$\"*)))*e`#F07$Fh_l$\" *7xSE#F07$Fjdw$\"*:%e$4\"F07$F]`l$\")9:VRF07$Fgew$!)LsAhF07$F\\fw$!)3) e2'F07$Fafw$!)*Q3/'F07$Fffw$!)i/WhF07$F`gw$!*%)H\\1\"F07$Fb`l$!*%3kdBF 07$F]hw$!*^MTa#F07$Fg`l$!*Muun$F07$F\\al$!*SmHs$F07$Faal$!*P$QvWF07$Ff al$!*7G'4WF07$F[bl$!*?Q%oVF07$Faiw$!*Q%R.WF07$F`bl$!*(*=oa%F07$Fiiw$!* c(eJXF07$Febl$!*&Hr+XF07$Fjbl$!*B$pNWF07$F_cl$!*r$=]UF07$Ficl$!*A$QfFF 07$Fcdl$!*fOZT\"F07$F][x$!)vGJiF07$Fhdl$!('yZ')F07$Fe[x$!(c#)*GF07$Fj[ x$!(haO\"F07$F_\\x$!)?!)o:F07$F]el$!)cj*p%F07$Fbel$!*0O8D\"F07$Fgel$!* LNlj\"F07$F\\fl$!*qc)=:F07$Fafl$!*9>9N\"F07$Fc]x$!**fo]8F07$Fh]x$!*0%[ X8F07$F]^x$!*V`IK\"F07$Fb^x$!*y_>D\"F07$Fg^x$!*\\X\\C\"F07$Fffl$!*CDFC \"F07$F__x$!*S95A\"F07$F[gl$!*E#>*G\"F07$F`gl$!*(z%=U\"F07$Fegl$!*?H@c \"F07$Fb`x$!*7QTj\"F07$Fjgl$!**Q+S;F07$F_ax$!*(4(of\"F07$F_hl$!*\\Stc \"F07$Fihl$!)1AlmF07$Fa[m$!'m8IF07$$\"5zyyyy`[ZA5F0$!)$3h1#F07$Ff[m$!) /skiF07$$\"5\"4444frot/\"F0$!*-#\\j7F07$F[\\m$!*sn=U#F07$Fe\\m$!*@Vf,% F07$Fc^m$!*w%QoXF07$Fh^m$!*\\>;x$F07$F]_m$!*'*RqU#F07$Fg_m$!)I/V*)F07$ Fa`m$!)\\qYsF07$Feam$\"))=2,%F07$Fibm$\")k)e)))F07$$\"5#==oI'Rm2h6F0$ \"*sNO4\"F07$$\"5uttBORYoi6F0$\"*FX.5\"F07$$\"5mllS4REHk6F0$\"*jRs5\"F 07$Fafx$\"*&om96F07$$\"5UTT\"*GQm6p6F0$\"*Z2s2\"F07$F^cm$\"*NJU+\"F07$ Fifx$\").s%*oF07$Fccm$\")P$[G\"F07$Ffgx$!*[XiE\"F07$Fhcm$!*,N'H;F07$F \\jx$!*2T_*GF07$F]dm$!*]e)HVF07$Fc[y$!*16.O&F07$Fbdm$!*ib\"odF07$F[\\y $!*vRm)\\F07$Fgdm$!*.jm5%F07$Fc\\y$!*R!3nJF07$F\\em$!*WgEL#F07$F[]y$!* ]Hi!>F07$F`]y$!*9tAy\"F07$Fd^y$!*E%H1t\"F07$Figm$!*+k=#=F07$F^hm$!*0x>#=F07$Fchm$!*i&\\T= F07$F]im$!*!Gj#>#F07$Fgim$!*1/gC#F07$Faey$!*[v'pHF07$F\\jm$!*Z%[F07$Fj[z$!*BT4$=F07$Fe[n$!)2/\\_F07$Fj[n$ \"):)HW$F07$F_\\n$\")Y<`ZF07$$\"5YXXq,saLG:F0$\")![HZ&F07$Fe_z$\")z1os F07$$\"5nmmTN@buJ:F0$\")zA@sF07$Fd\\n$\")h;srF07$Fb`z$\")7A(y'F07$Fi\\ n$\")+F8:F07$Fc]n$\"(Z@-%F07$Fg^n$!*V#*)G6F07$Ffbz$!*vZ!)H\"F07$F\\_n$ !*.'e@GF07$Fccz$!*Eo!)z#F07$Fhcz$!*i0\"yFF07$F]dz$!*\")e>z#F07$Fa_n$!* ,oN(HF07$Ff_n$!*^D#pVF07$F[`n$!*I#[NaF07$$\"5wvvD)QSR8e\"F0$!**='ec&F0 7$$\"58888QW9(Ge\"F0$!*dfW_&F07$$\"5]]]+)[[.We\"F0$!*c;i[&F07$F``n$!*( fIlaF07$$\"5iiiiP1'***)e\"F0$!*2h$QdF07$Fe`n$!*W@q.'F07$Fj`n$!*0P%feF0 7$F_an$!*8^^E&F07$Fdan$!*vJY1%F07$Fian$!*3\\>>$F07$F^bn$!*ua(=DF07$Fcb n$!*2:Uw\"F07$F[hz$!*;VW_\"F07$Fhbn$!*EB]F\"F07$Fchz$!*ok!o7F07$F]cn$! *=j\"[7F07$Fgcn$!*)3:U9F07$Fadn$!*P(>0=F07$Ffdn$!*!=B4CF07$F[en$!*bF(y FF07$Fgiz$!*o!*\\\"GF07$F`en$!*J+0'GF07$F_jz$!*&>%)4GF07$Feen$!*!p6yFF 07$F_fn$!*e*HQDF07$Fifn$!*:\\([CF07$F][[l$!*!H\"fW#F07$F^gn$!*DyyW#F07 $Fj[[l$!*rn?Y#F07$Fd\\[l$!*f4]`#F07$Fi\\[l$!*41_`#F07$Fcgn$!*e^ia#F07$ Fhgn$!*AHbn#F07$F]hn$!*'>T'y#F07$Fbhn$!*[6&4GF07$Fghn$!*(*G3'GF07$F\\i n$!*qw.$GF07$Fain$!*Mw,y#F07$Ffin$!*#og^DF07$F[jn$!*rT\"zCF07$Fejn$!*$ 4p)y\"F07$F_[o$!*#GzL9F07$Fd[o$!*?g1K\"F07$Fi[o$!*8nZB\"F07$F^\\o$!*C1 (f7F07$Fc\\o$!*]TzI\"F07$Fh\\o$!*?\"3$*>F07$F]]o$!*5^(3EF07$Fb]o$!*t+L P$F07$Fg]o$!*Xn*[ZF07$F\\^o$!*(pjQbF07$Fa^o$!*v$e.iF07$F[_o$!*OjpL'F07 $Fe_o$!*elcV'F07$Fj_o$!*66U6'F07$F_`o$!*F4`7&F07$Fd`o$!*aB*oQF07$Fi`o$ !*\"fyOCF07$Fcao$!*ip%Q8F07$Fgbo$!*[')H8\"F07$$\"5555g(>)z^g?F0$!)=B2* )F07$$\"5?>>>%ztC@1#F0$!)(R-'*)F07$$\"5HGGy!R\\JP1#F0$!)/y7!*F07$F\\co $!)JBy!*F07$$\"5cbbb!=w^&o?F0$!*y!R/5F07$Faco$!*XhXD\"F07$Faf[l$!*N.)3 :F07$Ffco$!*Z@]N#F07$Fhg[l$!*Nlh'QF07$F[do$!*_p&ySF07$F_eo$!*;qD`&F07$ Fieo$!*EeQ!pF07$Fcfo$!*JPt)pF07$F]go$!*[:G#yF07$Fa[\\l$!*M#f\\zF07$Ff[ \\l$!*EBf6)F07$F[\\\\l$!*!e_:\")F07$F`\\\\l$!*m(>&4)F07$Fe\\\\l$!*mMy+ )F07$Fj\\\\l$!*$\\>ByF07$F_]\\l$!*wa*GyF07$Fbgo$!*-4<\"yF07$F\\^\\l$!* 4***fpF07$Fggo$!*t]5a&F07$F\\ho$!*CDcg%F07$Faho$!*1o<%F07$Fia\\l$!*\"H$4B%F07$Fjio$!*Lk`T%F07$Fab\\l$! *&p`BWF07$F_jo$!**zlBWF07$Fdjo$!*V]1p%F07$Fijo$!*NBJ%[F07$Fc[p$!*;y9r% F07$Fg\\p$!*1bIO%F07$Fec\\l$!*:%*>O%F07$F\\]p$!*KF3K%F07$F]d\\l$!*Il0@ %F07$Fbd\\l$!*`F87%F07$Fgd\\l$!*9987%F07$Fa]p$!*)*G57%F07$Ff]p$!*mK\\/ %F07$F[^p$!*u3i0%F07$F`^p$!*EU)[YF07$Fe^p$!*>XH\\&F07$Fef\\l$!*/W5S'F0 7$Fj^p$!*)4A=tF07$F]g\\l$!*'[(p)yF07$Fbg\\l$!*lF:,)F07$Fgg\\l$!*tK`2)F 07$F\\h\\l$!**o!p0)F07$Fah\\l$!*+'3_zF07$F__p$!*%=!Rq(F07$Fd_p$!*gn:T( F07$Fi_p$!*8$45nF07$Fc`p$!*=_)e`F07$Fgap$!*\\kUp%F07$F`[]l$!*(HAIPF07$ F\\bp$!*yRbP#F07$F]\\]l$!*7L,&=F07$Fabp$!*kX9X\"F07$$\"5zyyGm+&R*GCF0$ !*T#[\"Q\"F07$Fe\\]l$!*1A+M\"F07$$\"5A@@'**exdLV#F0$!*$4NK8F07$$\"5poo =JM0$[V#F0$!*q#HF8F07$$\"5;;;Ts#H.jV#F0$!*`3*Q8F07$Ffbp$!*QY&=9F07$F\\ ^]l$!*\"*4h+#F07$F[cp$!*N^c)HF07$$\"5#44fYo(3.^CF0$!*1*okLF07$Fc_]l$!* *oCTLF07$$\"5'ee3rORwRX#F0$!*D<-K$F07$Fh_]l$!*_r$=LF07$Fb`]l$!*')[Ow$F 07$F`cp$!*!o#4(\\F07$F_a]l$!*7V\\$\\F07$Fda]l$!*-,K!\\F07$Fia]l$!*0\\( )*[F07$F^b]l$!*R#z.]F07$Fcb]l$!*\"*GeV&F07$Fhb]l$!*,(3\\kF07$F]c]l$!*, H=S'F07$Fecp$!*r!QbjF07$Fjcp$!*SvU=(F07$F_dp$!*3*4DtF07$Fddp$!*\"*HYd' F07$Fidp$!*XXlq&F07$F^ep$!*-mnq%F07$Fcep$!*`OCm$F07$Fhep$!*yc?4$F07$F] fp$!*\"HnsEF07$Fgd]l$!*\"))3#f#F07$F\\e]l$!*yJFZ#F07$Fae]l$!*1U$pCF07$ Fbfp$!*dtRZ#F07$Fie]l$!*yK')\\#F07$F^f]l$!*!)H$*\\#F07$Fcf]l$!*&yE\"e# F07$Fgfp$!*![HMFF07$Fagp$!*)Q+3JF07$F[hp$!*[h'GOF07$F`hp$!*&HlLQF07$Fe hp$!*#3QISF07$Fjhp$!*'\\LKSF07$F_ip$!*!*yr2%F07$Fdip$!*[#G,SF07$Fiip$! *8-7'QF07$Fch]l$!*igO$QF07$Fhh]l$!*tL#GPF07$F]i]l$!*vsys$F07$F^jp$!*56 Ur$F07$Fei]l$!*&\\ydOF07$Fcjp$!*Yq#zOF07$Fhjp$!*TKUy$F07$F][q$!*\"=CSR F07$Fcj]l$!*WCF07$F^_q$!*'HIXDF07$Fc_q$!*q\\C,$F07$Fh_ q$!*+XP-%F07$F]`q$!*MM5x%F07$Fb`q$!*&)=0i&F07$Fg`q$!*oTfq'F07$Faaq$!* \"*4mm(F07$F_cq$!*wSa$zF07$Fdcq$!*[HAk(F07$Ficq$!*Cb\"HmF07$F^dq$!*u3x N&F07$Fcdq$!*d@\\*RF07$Fbc^l$!*//M0%F07$Fhdq$!*0pz.%F07$F_d^l$!*H6$=QF 07$Fdd^l$!*#3BAJF07$Fid^l$!*-*GVJF07$F]eq$!*cGQ;$F07$Fbeq$!*rx3=$F07$F geq$!*zC6=$F07$F\\fq$!*A\"zIJF07$Fafq$!*;TZ(HF07$Fffq$!*CBM)HF07$F[gq$ !*1e=.$F07$F`gq$!*l6xf$F07$Fegq$!*\"p\\ZWF07$F`h^l$!*>'3M\\F07$Fjgq$!* ?q#3lF07$F\\j^l$!*!3g^pF07$F_hq$!*t-\"*=)F07$Fdhq$!*\"[0!e*F07$Fihq$!+ B!)\\P5F07$Fa^_l$!+y5N-5F07$F^iq$!*4\\bC*F07$Fi^_l$!*(Ht-!)F07$Fciq$!* JPL\"oF07$F]jq$!*0>]u'F07$Fa[r$!*SRHX'F07$Ff[r$!*Bp7S'F07$F[\\r$!*B8UR 'F07$F`\\r$!*skSf'F07$Fe\\r$!*'\\!3*pF07$Fc`_l$!*CVK3(F07$Fj\\r$!*[_]> (F07$F[a_l$!*oIw;(F07$F_]r$!*Mr)GrF07$Fd]r$!*>\"\\0oF07$Fi]r$!*ieOU'F0 7$F^^r$!*HQKT'F07$Fc^r$!*YJYS'F07$Fh^r$!*&RoPkF07$F]_r$!*<(fnmF07$Fb_r $!*NY!omF07$Fg_r$!*6]Dr'F07$F\\`r$!*3@9&yF07$Fa`r$!*>\"*Q&*)F07$F[c_l$ !*'4UF07$Fbdr$!*o(3aTF07$Fgdr$!*wt8?%F07$F\\er$!*q$f6bF07$Faer$ !*)f]whF07$Ffer$!*)3yhpF07$F[fr$!*.8iM)F07$$\"5;;;TZ'=KDP$F0$!*%*HRG)F 07$Fe\\`l$!*w/VA)F07$$\"5CCC*z,w\\bP$F0$!*i\"[!=)F07$F`fr$!*\"*3??)F07 $F]]`l$!*!)o?U)F07$Fb]`l$!*%f.s!*F07$Fg]`l$!*6!Q1!*F07$Fefr$!*i!*=%*)F 07$Fjfr$!*#>'Q'*)F07$F_gr$!*%yy!*))F07$Fdgr$!*[8\\c)F07$Figr$!*$=:)G)F 07$F^hr$!*t6s*pF07$Fchr$!*sz*[hF07$$\"5)yyyyy$*)=YMF0$!*pXKO&F07$Fhhr$ !*co9G%F07$$\"556666hj[tMF0$!*\\%4QQF07$F]ir$!*9bWF0 7$F_\\s$!*+l=7&F07$Fa_`l$!*6ZjC&F07$Ff_`l$!*.jIF&F07$F[``l$!*>PUF&F07$ Fd\\s$!*\\ONG&F07$Fc``l$!*iljG&F07$Fh``l$!*!o9&G&F07$F]a`l$!*\\*=__F07 $Fi\\s$!*<&4*=&F07$F^]s$!*,^V/&F07$Fc]s$!*'*eh\"\\F07$F]^s$!*'3UN]F07$ Fg^s$!*!>X%G&F07$F\\_s$!*T#pH[F07$Fa_s$!*mt1/%F07$Fgb`l$!*HvB\"RF07$F \\c`l$!*Qpxv$F07$Fac`l$!*b,%fPF07$Ffc`l$!*.>-t$F07$F[d`l$!*%=DoOF07$F` d`l$!*\"G$Qn$F07$Fjd`l$!*tM#*p$F07$Ff_s$!*fHP#QF07$Fge`l$!*YKeA%F07$F[ `s$!*\"*Q7%[F07$F``s$!*&)Q\"*y&F07$Fe`s$!*#z)>0(F07$Fj`s$!*?'HyzF07$F_ as$!*EMk*))F07$Fias$!*-#yW$*F07$F]cs$!*!47O%*F07$Fgg`l$!*LwA\\*F07$Fbc s$!*wW3i*F07$F_h`l$!*)zgr\"*F07$Fgcs$!*+wr1*F07$F\\ds$!*`)yW$)F07$Fads $!*2,%z\")F07$Ffds$!*3c%eoF07$F[es$!*Q#zefF07$Fees$!*W.$fdF07$Fifs$!*p a$G]F07$Fcgs$!*]ay6&F07$Fghs$!*cB4:&F07$F\\is$!*\"\\!z/'F07$Fais$!*!oR ?wF07$Ffis$!*l%=*=*F07$F[js$!+\\/Tx5F07$Fejs$!+(*G2)4\"F07$Fi[t$!+'3S[ ?\"F07$Fgbal$!+!pKc@\"F07$F\\cal$!+I/>m7F07$Facal$!+vjdo7F07$F^\\t$!++ ')*4F\"F07$F^dal$!+y'y)o7F07$Fhdal$!+W2;q7F07$F]eal$!+))=ci7F07$Fc\\t$ !+w)G(>7F07$Feeal$!+^J6Q6F07$Fh\\t$!+lo&Q/\"F07$F]fal$!*:Oo[*F07$F]]t$ !*7XIv)F07$Fg]t$!*O+:'*)F07$Fa^t$!*%ob-&*F07$Ff^t$!*]%H)=*F07$F[_t$!*! p\")z()F07$Fagal$!*ak&=\"*F07$F`_t$!*1F<')*F07$Figal$!+Agmf5F07$F^hal$ !+ywi]6F07$Fchal$!+O%4x:\"F07$Fe_t$!+O>iE7F07$$\"5YYYrFaQEXTF0$!+EhhE7 F07$$\"5AAAs%yBcn9%F0$!+H&y(G7F07$$\"5)zzH<9i[#[TF0$!+hNoP7F07$F[ial$! +xN&HE\"F07$$\"5CDDv7sds_TF0$!+SYgo7F07$F`ial$!+i6&oE\"F07$Feial$!+:(G #f7F07$Fj_t$!+S33X7F07$F]jal$!+RU\"o=\"F07$Fbjal$!+8Vr'4\"F07$F\\[bl$! +K>@c5F07$F_`t$!*-tn0*F07$Fh\\bl$!*(>tg')F07$Fd`t$!*$\\CktF07$Fd^bl$!* 'Ry7hF07$Fi`t$!*i,I%fF07$Fa_bl$!*bigU&F07$F^at$!*JEJO&F07$$\"5kkk*37\\ 4[A%F0$!*H+8L&F07$Fi_bl$!*qdXI&F07$$\"5899*G8=zzA%F0$!*?a.J&F07$Fcat$! *x5iW&F07$Fa`bl$!*PeGZ&F07$Fhat$!*%3!>U&F07$Fi`bl$!*4,Q'eF07$F]bt$!*?. 6M'F07$Fbbt$!*^i,g(F07$Fgbt$!*l(e(**)F07$F\\ct$!+>5&*35F07$Fact$!+\"[h \"R5F07$Ffct$!+m*yH/\"F07$F[dt$!+`2tV5F07$F\\h\\m$!*3\\=o*F07$F`dt$!*M KSd)F07$Fih\\m$!*n'olsF07$Fedt$!*x19['F07$Fefbl$!*lz1!eF07$Fjdt$!*luM@ &F07$F]gbl$!*I\"[/]F07$Fbgbl$!*Oer!\\F07$Fggbl$!*&4^-\\F07$F\\hbl$!*a? 7!\\F07$Fahbl$!*N;?&\\F07$F_et$!*E8;.&F07$Fdet$!*fnO&eF07$Fiet$!*PSr]' F07$F]gt$!*Jr!HiF07$Fggt$!*(3!H4'F07$Faht$!*Ynn<'F07$F[it$!*J&[HkF07$F [jbl$!*'[_HkF07$F`jbl$!*-tKV'F07$Fejbl$!*VLO['F07$F`it$!*3F:^'F07$F][c l$!*ZR5]'F07$Feit$!*[4HZ'F07$F_jt$!*i//j&F07$Fc[u$!*c'HU\\F07$Fh[u$!*? .![bF07$F]\\u$!*O%)))*pF07$Fb\\u$!*zTk8)F07$Fg\\u$!*3v)R&*F07$Fa]u$!+* e>B0\"F07$Fe^u$!+'fr%46F07$F__u$!+v\"z&[5F07$Fi_u$!*[3$G'*F07$F^`u$!*0 [P])F07$Fc`u$!*Da#>uF07$Fae]m$!*#e#[2(F07$Ffe]m$!*')3F*pF07$F[f]m$!*)y CLqF07$Fcdcl$!*w$zsqF07$Fcf]m$!*Mk'[rF07$Fh`u$!*4KmK(F07$F`ecl$!*a\\9] (F07$F]au$!*vYcY)F07$F]fcl$!*\\$e_!*F07$Fbau$!+Kul65F07$Fjfcl$!+%[X9/ \"F07$Fdgcl$!+l?e!=\"F07$F^hcl$!+F#*R'=\"F07$Fgau$!+\"G`6?\"F07$Fjicl$ !+s\"fTM\"F07$Fdjcl$!+8bLQ9F07$Fijcl$!+1!H4Y\"F07$F\\bu$!+)G!p0:F07$Fa [dl$!+'*)z5]\"F07$Ff[dl$!+-?!\\\\\"F07$F[\\dl$!+c\"ffV\"F07$Fabu$!+(Qp GS\"F07$Fh\\dl$!+hN=e7F07$Ffbu$!+G3()z6F07$Fe]dl$!+zUTC6F07$F[cu$!+O[6 86F07$Fecu$!+;bGk6F07$F_du$!+o@m&=\"F07$Fddu$!+&e^j9\"F07$Fidu$!+.b()> 6F07$Fi^dl$!+y()p>6F07$F^_dl$!+\"R\"p=6F07$Fc_dl$!+U%*)\\6\"F07$F^eu$! +t;S16F07$F[`dl$!+UxR16F07$Fceu$!+U#4o5\"F07$F]fu$!+)R#H<6F07$Fagu$!+0 xUe6F0-Ffgu6&FhguFiguFj`dlF(-F`hu6#%Hscheme~with~c[5]=c[6]=3/4~and~b[5 ]=b[6]G-%%FONTG6$%*HELVETICAG\"\"*-%+AXESLABELSG6$Q\"x6\"Q!Fhb`n-%&TIT LEG6#%Uerror~curves~for~7~stage~order~6~Runge-Kutta~methodsG-%%VIEWG6$ ;F(Fagu%(DEFAULTG" 1 2 0 1 10 0 2 9 1 4 2 1.000000 45.000000 45.000000 0 1 "Butcher's scheme A" "scheme with simple nodes" "scheme \+ with a relatively large stability region" "Butcher's scheme B with c[5 ]=c[6]=1/2 and b[5]=b[6]" "scheme with c[5]=c[6]=3/4 and b[5]=b[6]" }} }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 47 "Test 13 of 7 stage, o rder 6 Runge-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!\"\",&*& %$cosGF&%\"xGF&F&*&\"\"#F&%\"yGF&F&" }{TEXT -1 64 " contains an expon ential term, but with the initial condition " }{XPPEDIT 18 0 "y(0) = \+ -2/5" "6#/-%\"yG6#\"\"!,$*&\"\"#\"\"\"\"\"&!\"\"F-" }{TEXT -1 23 " th is term 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(d e,y(x));" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%#deG/-%%diffG6$-%\"yG6#% \"xGF,,&-%$cosGF+\"\"\"*&\"\"#F0F)F0F0" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/-%\"yG6#%\"xG,(*&#\"\"#\"\"&\"\"\"-%$cosGF&F-!\"\"*&#F-F,F--%$s inGF&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 numerical solution from the correct solution tends to become rapidly magnified." }}{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, font=[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 0 758 "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 := [`Butcher's scheme A`,`scheme with simple nodes`,`scheme wi th a relatively large stability region`,`Butcher's scheme B with `*(c[ 5]=1/2,c[6]=1/2,b[5]=b[6]),`scheme with `*(c[5]=3/4,c[6]=3/4,b[5]=b[6] )]:\nerrs := []:\nDigits := 20:\nfor ct to 5 do\n En_RK6_||ct := RK6 _||ct(E(x,y),x,y,x0,evalf(y0),hh,numsteps,false);\n sm := 0: numpts \+ := nops(En_RK6_||ct):\n for ii to numpts do\n sm := sm+(En_RK6_ ||ct[ii,2]-e(En_RK6_||ct[ii,1]))^2;\n end do:\n errs := [op(errs), sqrt(sm/numpts)];\nend do:\nDigits := 10:\nlinalg[transpose]([mthds,ev alf(errs)]);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#K%'matrixG6#7&7$%0slop e~field:~~~G,&-%$cosG6#%\"xG\"\"\"*&\"\"#F.%\"yGF.F.7$%0initial~point: ~G-%!G6$\"\"!#!\"#\"\"&7$%/step~width:~~~G$F0F97$%1no.~of~steps:~~~G\" $+%Q)pprint606\"" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%!G" }}{PARA 11 " " 1 "" {XPPMATH 20 "6#K%'matrixG6#7'7$%3Butcher's~scheme~AG$\"+\">1=z# !#;7$%9scheme~with~simple~nodesG$\"+En.\\@!#<7$%Pscheme~with~a~relativ ely~large~stability~regionG$\"+@jwX=!#=7$*&%9Butcher's~scheme~B~with~G \"\"\"6%/&%\"cG6#\"\"&#F9\"\"#/&F=6#\"\"'F@/&%\"bGF>&FHFDF9$\"+)**\\2L \"F+7$*&%-scheme~with~GF96%/F<#\"\"$\"\"%/FCFQFFF9$\"+2:6&)=F0Q)pprint 616\"" }}}{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 "The error in the value obtained by ea ch of the methods at the point where " }{XPPEDIT 18 0 "x = 7.999;" "6 #/%\"xG-%&FloatG6$\"%**z!\"$" }{TEXT -1 16 " is also given." }} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 689 "E := (x,y) -> cos(x)+2*y: h h := 0.02: numsteps := 400: x0 := 0: y0 := -2/5:\nmatrix([[`slope fiel d: `,E(x,y)],[`initial point: `,``(x0,y0)],[`step width: `,hh],\n[ `no. of steps: `,numsteps]]);``;\nmthds := [`Butcher's scheme A`,`sc heme with simple nodes`,`scheme with a relatively large stability regi on`,`Butcher's scheme B with `*(c[5]=1/2,c[6]=1/2,b[5]=b[6]),`scheme w ith `*(c[5]=3/4,c[6]=3/4,b[5]=b[6])]:\nerrs := []:\nDigits := 20:\nfor ct to 5 do\n en_RK6_||ct := RK6_||ct(E(x,y),x,y,x0,evalf(y0),hh,num steps,true);\nend do:\nxx := 7.999: exx := evalf(e(xx)):\nfor ct to 5 \+ do\n errs := [op(errs),abs(en_RK6_||ct(xx)-exx)];\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)pprint626\"" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%!G" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#K%'matrixG6 #7'7$%3Butcher's~scheme~AG$\"+$Rbqa\"!#:7$%9scheme~with~simple~nodesG$ \"+I-(3>\"!#;7$%Pscheme~with~a~relatively~large~stability~regionG$\"+H ]\"G-\"!#<7$*&%9Butcher's~scheme~B~with~G\"\"\"6%/&%\"cG6#\"\"&#F9\"\" #/&F=6#\"\"'F@/&%\"bGF>&FHFDF9$\"+eiButF07$*&%-scheme~with~GF96%/F<#\" \"$\"\"%/FCFQFFF9$\"+R$=Y/\"F0Q)pprint636\"" }}}{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 "[0, 8 ];" "6#7$\"\"!\"\")" }{TEXT -1 82 " of each Runge-Kutta method is est imated 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 0 436 "mthds := [`Butcher's scheme A`,`scheme with simple n odes`,`scheme with a relatively large stability region`,`Butcher's sch eme B with `*(c[5]=1/2,c[6]=1/2,b[5]=b[6]),`scheme with `*(c[5]=3/4,c[ 6]=3/4,b[5]=b[6])]:\nerrs := []:\nDigits := 20:\nfor ct to 5 do\n sm := NCint((e(x)-'en_RK6_||ct'(x))^2,x=0..8,adaptive=false,numpoints=7, factor=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$%3Butcher's~scheme~AG$\"+y&3.u#!#;7$%9s cheme~with~simple~nodesG$\"+GcR4@!#<7$%Pscheme~with~a~relatively~large ~stability~regionG$\"+.%><\"=!#=7$*&%9Butcher's~scheme~B~with~G\"\"\"6 %/&%\"cG6#\"\"&#F9\"\"#/&F=6#\"\"'F@/&%\"bGF>&FHFDF9$\"+4I?18F+7$*&%-s cheme~with~GF96%/F<#\"\"$\"\"%/FCFQFFF9$\"+q(Q.&=F0Q)pprint646\"" }}} {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 0 516 "evalf[20]( plot([e(x)-'en_RK6_1'(x),e(x)-'en_RK6_2'(x),e(x)-'en_RK6_3'(x),e(x)-'e n_RK6_4'(x),\ne(x)-'en_RK6_5'(x)],x=0..2,font=[HELVETICA,9],\ncolor=[C OLOR(RGB,.95,.2,0),COLOR(RGB,.45,0,.95),COLOR(RGB,0,.25,1),COLOR(RGB,0 ,.75,.2),COLOR(RGB,.95,.45,0)],\nlegend=[`Butcher's scheme A`,`scheme \+ with simple nodes`,`scheme with a relatively large stability region`,` Butcher's scheme B with c[5]=c[6]=1/2 and b[5]=b[6]`,`scheme with c[5] =c[6]=3/4 and b[5]=b[6]`],title=`error curves for 7 stage order 6 Rung e-Kutta methods`));" }}{PARA 13 "" 1 "" {GLPLOT2D 817 633 633 {PLOTDATA 2 "6+-%'CURVESG6%7S7$$\"\"!F)F(7$$\"5MLLLLL3VfV!#@$\"(v&3B!# ?7$$\"5nmmmm\"H[D:)F-$\"()4YZF07$$\"5LLLLLe0$=C\"F0$\"(%4*Q(F07$$\"5LL LLL3RBr;F0$\")uYB5F07$$\"5nmmm;zjf)4#F0$\")+NH8F07$$\"5MLLL$e4;[\\#F0$ \");mY;F07$$\"5++++]i'y]!HF0$\")h7!*>F07$$\"5MLLL$ezs$HLF0$\")/@mBF07$ $\"5++++]7iI_PF0$\")%*4uFF07$$\"5nmmmm;_M(=%F0$\")`E]KF07$$\"5MLLLL3y_ qXF0$\")7_aOF07$$\"5+++++]1!>+&F0$\")3\">?%F07$$\"5+++++]Z/NaF0$\")!>Y t%F07$$\"5+++++]$fC&eF0$\")2h\"H&F07$$\"5MLLL$ez6:B'F0$\")r2WeF07$$\"5 nmmmm;=C#o'F0$\")]%Q_'F07$$\"5nmmmmm#pS1(F0$\")GUgrF07$$\"5,+++]i`A3vF 0$\")')4PzF07$$\"5nmmmmm(y8!zF0$\")AQ(o)F07$$\"5,+++]i.tK$)F0$\")-Tk&* F07$$\"5,+++](3zMu)F0$\"*2T]O\"F-7$$\"5LLL Lezw5V5!#>$\"+(*GE&\\\"F-7$$\"5++++v$Q#\\\"3\"Fas$\"+>xB<;F-7$$\"5LLLL $e\"*[H7\"Fas$\",_51(eF-7$$\"5++++]_qn 27Fas$\"+?p\"Q3#F-7$$\"5++++Dcp@[7Fas$\"+!GA%eAF-7$$\"5++++]2'HKH\"Fas $\"+(\\c'oCF-7$$\"5nmmmmwanL8Fas$\"*9RJn#F07$$\"5+++++v+'oP\"Fas$\"*xk \"4HF07$$\"5LLLLeR<*fT\"Fas$\"*-k19$F07$$\"5+++++&)Hxe9Fas$\"*n0JT$F07 $$\"5nmmm\"H!o-*\\\"Fas$\"*#eI\"p$F07$$\"5++++DTO5T:Fas$\"*s)Q0SF07$$ \"5nmmmmT9C#e\"Fas$\"*5j\"RVF07$$\"5++++D1*3`i\"Fas$\"*['y=ZF07$$\"5LL LLL$*zym;Fas$\"*-&H:^F07$$\"5LLLL$3N1#4Fas$\"*PB8J )F07$$\"5++++v.Uac>Fas$\"*h2L**)F07$$\"\"#F)$\"*i!p%y*F0-%&COLORG6&%$R GBG$\"#&*!\"#$Fez!\"\"F(-%'LEGENDG6#%3Butcher's~scheme~AG-F$6%7SF'7$F+ $!'\"zs\"F07$F2$!'cbNF07$F7$!'aSbF07$F<$!'3\"o(F07$FA$!'^&)**F07$FF$!( uzB\"F07$FK$!(zu\\\"F07$FP$!(-?y\"F07$FU$!(;54#F07$FZ$!(CCX#F07$Fin$!( E$fFF07$F^o$!(bf<$F07$Fco$!(i8e$F07$Fho$!(:d+%F07$F]p$!(8sU%F07$Fbp$!( qd%\\F07$Fgp$!(9AV&F07$F\\q$!(Rc-'F07$Faq$!(z'*f'F07$Ffq$!(T3F(F07$F[r $!(3B'zF07$F`r$!)*z@u)F-7$Fer$!)`83&*F-7$Fjr$!*9I//\"F-7$F_s$!*(=TS6F- 7$Fes$!*s)4M7F-7$Fjs$!+N?uU8F^t7$F`t$!*$30k9F-7$Fet$!*#3]#f\"F-7$Fjt$! *[Bns\"F-7$F_u$!*o/$))=F-7$Fdu$!)/dX?F07$Fiu$!)V;FAF07$F^v$!)@D0CF07$F cv$!)n-:EF07$Fhv$!)5:HGF07$F]w$!)f)42$F07$Fbw$!)3)yK$F07$Fgw$!)5/?OF07 $F\\x$!)wDDRF07$Fax$!)B;kUF07$Ffx$!)T1HYF07$F[y$!)nC\"*\\F07$F`y$!)%oD W&F07$Fey$!)hE!)eF07$Fjy$!)a'fQ'F07$F_z$!)776pF07$Fdz$!)E#3_(F0-Fiz6&F [[l$\"#XF^[lF(F\\[l-Fb[l6#%9scheme~with~simple~nodesG-F$6%7SF'7$F+$!&Y d%F07$F2$!&V?*F07$F7$!'7-9F07$F<$!'w**=F07$FA$!')HT#F07$FF$!'XAHF07$FK $!'I_MF07$FP$!'j4SF07$FU$!'J(e%F07$FZ$!'A?_F07$Fin$!',`dF07$F^o$!'t2kF 07$Fco$!'h[qF07$Fho$!'q*o(F07$F]p$!'%))G)F07$Fbp$!'2I!*F07$Fgp$!'Ls'*F 07$F\\q$!()3Y5F07$Faq$!(1u6\"F07$Ffq$!(b)*>\"F07$F[r$!(?4G\"F07$F`r$!) %\\wO\"F-7$Fer$!)Zv_9F-7$Fjr$!)t=V:F-7$F_s$!)]JX;F-7$Fes$!)7oQF-7$Fet$!)\\%R4#F-7$Fjt$!)Hr?AF-7$F_u$!)$z^P#F -7$Fdu$!(C*=DF07$Fiu$!(a0o#F07$F^v$!(x0%GF07$Fcv$!(#3F07$Fbw$\"*Pe;8#F07$Fgw$\"*xDHJ#F07$F\\x$\"*AP>]#F07$Fax$\"*s< =r#F07$Ffx$\"*i*fPHF07$F[y$\"*%RCF07$F^o$!(;[y#F07$Fco$!(Z.9$F07$Fho$!(-D^$F07$F]p$!(k@)QF07$Fbp$ !(\\pL%F07$Fgp$!('ejZF07$F\\q$!(\\SG&F07$Faq$!(Dvy&F07$Ffq$!()=wjF07$F [r$!(rE)pF07$F`r$!)zqmwF-7$Fer$!)u`Q$)F-7$Fjr$!)NgC\"*F-7$F_s$!*Ce,+\" F-7$Fes$!*[NB3\"F-7$Fjs$!+S9jx6F^t7$F`t$!*8PSG\"F-7$Fet$!*s2nR\"F-7$Fj t$!*'HW9:F-7$F_u$!*Yvhl\"F-7$Fdu$!)Y7%z\"F07$Fiu$!)]T`>F07$F^v$!)&H'4@ F07$Fcv$!)Ek$H#F07$Fhv$!)'p9[#F07$F]w$!)gg$p#F07$Fbw$!)3&*=HF07$Fgw$!) %H_<$F07$F\\x$!)?'HW$F07$Fax$!)aCSPF07$Ffx$!)QLgSF07$F[y$!)(Q!yVF07$F` y$!)R$Rx%F07$Fey$!)+)y:&F07$Fjy$!)pZ,cF07$F_z$!)89igF07$Fdz$!)`(pf'F0- Fiz6&F[[lF\\[lFjdlF(-Fb[l6#%Hscheme~with~c[5]=c[6]=3/4~and~b[5]=b[6]G- %%FONTG6$%*HELVETICAG\"\"*-%+AXESLABELSG6$Q\"x6\"Q!Ffbn-%&TITLEG6#%Uer ror~curves~for~7~stage~order~6~Runge-Kutta~methodsG-%%VIEWG6$;F(Fdz%(D EFAULTG" 1 2 0 1 10 0 2 9 1 4 2 1.000000 45.000000 45.000000 0 1 "Butc her's scheme A" "scheme with simple nodes" "scheme with a relatively l arge stability region" "Butcher's scheme B with c[5]=c[6]=1/2 and b[5] =b[6]" "scheme with c[5]=c[6]=3/4 and b[5]=b[6]" }}}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 516 "evalf[20](p lot([e(x)-'en_RK6_1'(x),e(x)-'en_RK6_2'(x),e(x)-'en_RK6_3'(x),e(x)-'en _RK6_4'(x),\ne(x)-'en_RK6_5'(x)],x=2..8,font=[HELVETICA,9],\ncolor=[CO LOR(RGB,.95,.2,0),COLOR(RGB,.45,0,.95),COLOR(RGB,0,.25,1),COLOR(RGB,0, .75,.2),COLOR(RGB,.95,.45,0)],\nlegend=[`Butcher's scheme A`,`scheme w ith simple nodes`,`scheme with a relatively large stability region`,`B utcher's scheme B with c[5]=c[6]=1/2 and b[5]=b[6]`,`scheme with c[5]= c[6]=3/4 and b[5]=b[6]`],title=`error curves for 7 stage order 6 Runge -Kutta methods`));" }}{PARA 13 "" 1 "" {GLPLOT2D 813 648 648 {PLOTDATA 2 "6+-%'CURVESG6%7Z7$$\"\"#\"\"!$\"*i!p%y*!#?7$$\"5+++++DHyI @!#>$\"+.uYk7F-7$$\"5++++v[kdWAF1$\"+C=j!e\"F-7$$\"5++++]n\"\\DP#F1$\" +Cu-M?F-7$$\"5++++]s,P,DF1$\"+!z>Di#F-7$$\"5++++v8*y&HEF1$\"+r&R3Q$F-7 $$\"5++++vG[W[FF1$\"+kN%)zUF-7$$\"5++++v)fB:(GF1$\"+ud'fY&F-7$$\"5++++ vQ=\"))*HF1$\"+cR_TqF-7$$\"5++++vj=pDJF1$\"+BY;n!*F-7$$\"5+++++lN?cKF1 $\",gXij<\"F-7$$\"5++++]U$e6P$F1$\",#4jvz9F-7$$\"5+++++&>q0]$F1$\",ioL h\">F-7$$\"5+++++DM^IOF1$\",smQU[#F-7$$\"5+++++0ytbPF1$\",ht[2>$F-7$$ \"5++++vQNXpQF1$\",!RF=0SF-7$$\"5+++++XDn/SF1$\",#)[!e[_F-7$$\"5+++++! y?#>TF1$\"-L(\\,(*f'!#@7$$\"5++++v3wY_UF1$\".YRa(*\\h)!#A7$$\"5+++++IO TqVF1$\".VFN!p!4\"F^q7$$\"5++++v3\">)*\\%F1$\"-5n'zGT\"F-7$$\"5++++DEP /BYF1$\"-hEPw2=F-7$$\"5++++](o:;v%F1$\"-D$*p)yL#F-7$$\"5++++v$)[op[F1$ \"->s8igHF-7$$\"5++++]i%Qq*\\F1$\"-6a1]>QF-7$$\"5++++vQIKH^F1$\"-F%puj (\\F-7$$\"5++++D^rZW_F1$\"-*R;f_E'F-7$$\"5++++]Zn%)o`F1$\"-yk_oM!)F-7$ $\"5+++++5FL(\\&F1$\".[h!***)Q5F-7$$\"5++++]d6.BcF1$\".@#\\t%eL\"F-7$$ \"5++++vo3lWdF1$\".4*)*Hr.B#F-7$$\"5++ +++Ik-,gF1$\".%\\7],XGF-7$$\"5+++++D-eIhF1$\".\\xZ-lo$F-7$$\"5++++v=_( zC'F1$\".R-/=@m%F-7$$\"5+++++b*=jP'F1$\".PWF`k-'F-7$$\"5++++v3/3(\\'F1 $\".Ri%p!Gn(F-7$$\"5++++vB4JBmF1$\".p.`ij()*F-7$$\"5+++++DVsYnF1$\"/*> r\"H8k7F-7$$\"5++++v=n#f(oF1$\"/pa'Gzoj\"F-7$$\"5+++++!)RO+qF1$\"/!>^) 3V*4#F-7$$\"5++++]_!>w7(F1$\"/dg#\\3zq#F-7$$\"5++++v)Q?QD(F1$\"0#=H]xQ &[$F^q7$$\"5+++++5jyptF1$\"1XEV-=>&R%Fdq7$$\"5++++DE8COuF1$\"0$f!)*>c* >]F^q7$$\"5++++]Ujp-vF1$\"0b7.5HNt&F^q7$$\"5++++D,X8ivF1$\"0\\&frfFdkF ^q7$$\"5+++++gEd@wF1$\"/pHK=QssF-7$$\"5+++]PMh%\\o(F1$\"/ps(y'4b#)F-7$ $\"5++++v3'>$[xF1$\"/ZiJdgq$*F-7$$\"5+++++5h(*3yF1$\"0M'*\\(4#z0\"F-7$ $\"5++++D6EjpyF1$\"0'>JN(pV>\"F-7$$\"5+++vVeWA-zF1$\"0y#4'e;[F\"F-7$$ \"5+++]i0j\"[$zF1$\"0w'fW>og8F-7$$\"5+++D\"G:3u'zF1$\"0'Qol2L_9F-7$$\" \")F*$\"0*oT)f_,b\"F--%&COLORG6&%$RGBG$\"#&*!\"#$F)!\"\"$F*F*-%'LEGEND G6#%3Butcher's~scheme~AG-F$6%7Z7$F($!)E#3_(F-7$F/$!)y!Gs*F-7$F5$!*E]d@ \"F-7$F:$!*3X[c\"F-7$F?$!*_$*z,#F-7$FD$!*!\\$=g#F-7$FI$!**=*RH$F-7$FN$ !*\"p;2UF-7$FS$!*#39?aF-7$FX$!*0m&zpF-7$Fgn$!*Lj`0*F-7$F\\o$!+sF4R6F-7 $Fao$!+Ep,v9F-7$Ffo$!+%>RB\">F-7$F[p$!+d)*>cCF-7$F`p$!+6V8$3$F-7$Fep$! +y1FSSF-7$Fjp$!,O&eK!3&F^q7$F`q$!-Wp7kJmFdq7$Ffq$!,MfxeR)F^q7$F[r$!,nJ )f(3\"F-7$F`r$!,=\"oc\"R\"F-7$Fer$!,-zP'*z\"F-7$Fjr$!,/@(**yAF-7$F_s$! ,YvK,%HF-7$Fds$!,'pTlIQF-7$Fis$!,j]#zA[F-7$F^t$!,pON[='F-7$Fct$!,oY/r* zF-7$Fht$!-U\"G!HG5F-7$F]u$!-)o`g9J\"F-7$Fbu$!-sGT5=$Ho#F^q7$F^y$!0^%3ozE$Q$Fdq7$Fcy$!/Az$> !>kQF^q7$Fhy$!/7FlTZ8WF^q7$F]z$!/hA5(*eq\\F^q7$Fbz$!.beFI!)f&F-7$Fgz$! .&3!e!\\ajF-7$F\\[l$!.hx9rJ@(F-7$Fa[l$!.hb68N9)F-7$Ff[l$!.(4F*[Q>*F-7$ F[\\l$!.,mA,J\")*F-7$F`\\l$!/$)zLjSZ5F-7$Fe\\l$!/AqcW&z6\"F-7$Fj\\l$!/ 7d_VD$>\"F--F_]l6&Fa]l$\"#XFd]lFg]lFb]l-Fi]l6#%9scheme~with~simple~nod esG-F$6%7Z7$F($!(vr:(F-7$F/$!(y*=!*F-7$F5$!)h\"\\5\"F-7$F:$!)#4$*R\"F- 7$F?$!))f#z_j$F-7$FS$!)4vmYF-7$ FX$!)HG'*fF-7$Fgn$!)@SpxF-7$F\\o$!)CUm(*F-7$Fao$!*]nTE\"F-7$Ffo$!*-i)Q ;F-7$F[p$!*7H^5#F-7$F`p$!*P&*Gk#F-7$Fep$!*3XUY$F-7$Fjp$!+bB*pN%F^q7$F` q$!,Kn%y)o&Fdq7$Ffq$!+t$)p.sF^q7$F[r$!*zYOL*F-7$F`r$!++)>W>\"F-7$Fer$! +\\!))[a\"F-7$Fjr$!+`3hc>F-7$F_s$!+:fYCDF-7$Fds$!+vNN*G$F-7$Fis$!+c'H: 9%F-7$F^t$!+!y19J&F-7$Fct$!+(H\"*z'oF-7$Fht$!+67IJ))F-7$F]u$!,!*zYj7\" F-7$Fbu$!,*)Q9cZ\"F-7$Fgu$!,&H6#4)=F-7$F\\v$!,:ensV#F-7$Fav$!,D`\"H#3$ F-7$Ffv$!,E_7V)RF-7$F[w$!,dx&ys]F-7$F`w$!,5kX'HlF-7$Few$!,PutwN)F-7$Fj w$!-**[V?#3\"F-7$F_x$!-FjY,)Q\"F-7$Fdx$!-&f%>I!z\"F-7$Fix$!.ZJCAVI#F^q 7$F^y$!/@h`q#e!HFdq7$Fcy$!.N5k\"))=LF^q7$Fhy$!.')ze]1z$F^q7$F]z$!.Ps&f 9pUF^q7$Fbz$!-1K?/3[F-7$Fgz$!-Zt0vdaF-7$F\\[l$!-4(*RD&>'F-7$Fa[l$!-6te I%*pF-7$Ff[l$!-&p*zT'*yF-7$F[\\l$!-!H5\"GG%)F-7$F`\\l$!-K)=of**)F-7$Fe \\l$!-%*)e\"*=g*F-7$Fj\\l$!.'R&pi[-\"F--F_]l6&Fa]lFg]l$\"#DFd]l$\"\"\" F*-Fi]l6#%Pscheme~with~a~relatively~large~stability~regionG-F$6%7Z7$F( $\"*nP8s%F-7$F/$\"*cl@3'F-7$F5$\"*3'>%e(F-7$F:$\"*l%*3u*F-7$F?$\"+e;%Q D\"F-7$FD$\"+%\\@Zh\"F-7$FI$\"+!4[D/#F-7$FN$\"++5@2EF-7$FS$\"+roPdLF-7 $FX$\"+Sm6AVF-7$Fgn$\"+\"yF-7$Fep$\",Dp?8]#F-7$F jp$\"-$fn0`9$F^q7$F`q$\".\"y)ose5%Fdq7$Ffq$\"-_5(4$)>&F^q7$F[r$\",[W_S t'F-7$F`r$\",A>.jh)F-7$Fer$\"-K4!>V6\"F-7$Fjr$\"-LET:69F-7$F_s$\"-S)R^ 0#=F-7$Fds$\"-P0J*>P#F-7$Fis$\"-m>-O')HF-7$F^t$\"-*GR$yHQF-7$Fct$\"-#e 6-?&\\F-7$Fht$\"-MdOF-7$F`w$\".SU[&p2ZF-7$Few$\".1=g_c-'F-7$Fjw$\".MD9'R-yF- 7$F_x$\"/L*pv?2+\"F-7$Fdx$\"/82#Qf2H\"F-7$Fix$\"0)H[yZNh;F^q7$F^y$\"1F $e#\\W-&4#Fdq7$Fcy$\"0M!4Rj#GR#F^q7$Fhy$\"0M+[\"*fHt#F^q7$F]z$\"0ghh/V z2$F^q7$Fbz$\"/,j&ftkY$F-7$Fgz$\"/dGZr*[$RF-7$F\\[l$\"/QcM'=mY%F-7$Fa[ l$\"/#*Rj]rU]F-7$Ff[l$\"/1\\w`6$p&F-7$F[\\l$\"/?pxadwgF-7$F`\\l$\"/q'[ gje['F-7$Fe\\l$\"//*)=$>F#pF-7$Fj\\l$\"/zNr%***)Q(F--F_]l6&Fa]lFg]l$\" #vFd]lFe]l-Fi]l6#%TButcher's~scheme~B~with~c[5]=c[6]=1/2~and~b[5]=b[6] G-F$6%7Z7$F($!)`(pf'F-7$F/$!)8aG&)F-7$F5$!*rCk1\"F-7$F:$!*Y[EP\"F-7$F? $!*6[,x\"F-7$FD$!*\\*G#G#F-7$FI$!*bW%*)GF-7$FN$!*Gu/p$F-7$FS$!*r%[aZF- 7$FX$!*\\)RAhF-7$Fgn$!*(zEVzF-7$F\\o$!*D-?***F-7$Fao$!+;&pQH\"F-7$Ffo$ !+(4%[x;F-7$F[p$!+oHba@F-7$F`p$!+DK\\/FF-7$Fep$!+!*H3WNF-7$Fjp$!,5<3kX %F^q7$F`q$!-Ik_?F-7$F_s$!,e^_!zDF-7$Fds$!,z93-O$F-7$ Fis$!,#H5]IUF-7$F^t$!,L6q_U&F-7$Fct$!,ass\\,(F-7$Fht$!,]+]+-*F-7$F]u$! -u[))R]6F-7$Fbu$!-Nc@52:F-7$Fgu$!-hV+/@>F-7$F\\v$!-\"RdP#*[#F-7$Fav$!- !HU-![JF-7$Ffv$!-X%[U#pSF-7$F[w$!-&\\))44=&F-7$F`w$!-fxn\")omF-7$Few$! -(4K0e`)F-7$Fjw$!.>M3q_5\"F-7$F_x$!./'H))f<9F-7$Fdx$!.jv?h%G=F-7$Fix$! /f&feQMN#F^q7$F^y$!0%yw\\cwnHFdq7$Fcy$!/[Cl`i*Q$F^q7$Fhy$!/bMc7XrQF^q7 $F]z$!/l,=p9gVF^q7$Fbz$!.T&=4`5\\F-7$Fgz$!.,Ci*3ubF-7$F\\[l$!.qJ#[JFjF -7$Fa[l$!.T-i+M9(F-7$Ff[l$!.CWqVZ1)F-7$F[\\l$!.Yv/Xzg)F-7$F`\\l$!.'H\" yLx=*F-7$Fe\\l$!.R\")>ul!)*F-7$Fj\\l$!/Xwk'4n/\"F--F_]l6&Fa]lFb]lFihlF g]l-Fi]l6#%Hscheme~with~c[5]=c[6]=3/4~and~b[5]=b[6]G-%%FONTG6$%*HELVET ICAG\"\"*-%+AXESLABELSG6$Q\"x6\"Q!F][o-%&TITLEG6#%Uerror~curves~for~7~ stage~order~6~Runge-Kutta~methodsG-%%VIEWG6$;F(Fj\\l%(DEFAULTG" 1 2 0 1 10 0 2 9 1 4 2 1.000000 45.000000 45.000000 0 1 "Butcher's scheme A " "scheme with simple nodes" "scheme with a relatively large stability region" "Butcher's scheme B with c[5]=c[6]=1/2 and b[5]=b[6]" "scheme with c[5]=c[6]=3/4 and b[5]=b[6]" }}}}{PARA 0 "" 0 "" {TEXT -1 0 "" } }{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 2 " ;" }}}}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 47 "Test 14 of 7 stage, order 6 Runge-Kutta methods" }} {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 256 "" 0 "" {TEXT -1 3 " " } {XPPEDIT 18 0 "dy/dx = 10*x*cos*x-10*y;" "6#/*&%#dyG\"\"\"%#dxG!\"\",& **\"#5F&%\"xGF&%$cosGF&F,F&F&*&F+F&%\"yGF&F(" }{TEXT -1 8 ", " } {XPPEDIT 18 0 "y(0) = sqrt(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#,$* &\"#5F&F'F&F-F&F&" }{TEXT -1 1 "." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 163 "de := diff(y(x),x)=10*x*cos (x)-10*y(x);\nic := y(0)=sqrt(5);\ndsolve(\{de,ic\},y(x));\nb := unapp ly(rhs(%),x):\nplot(b(x),x=0..5,font=[HELVETICA,9],labels=[`x`,`y(x)`] );" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%#deG/-%%diffG6$-%\"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--%$cosG F&F-F-F-*&#\"$!**\"&,-\"F-F/F-!\"\"*&#\"#5F,F-*&-%$sinGF&F-F'F-F-F-*&# \"$+#F4F-F:F-F5*&-%$expG6#,$*&F8F-F'F-F5F-,&#F3F4F-*$\"\"&#F-\"\"#F-F- F-" }}{PARA 13 "" 1 "" {GLPLOT2D 703 312 312 {PLOTDATA 2 "6&-%'CURVESG 6$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$$\"3MmT5ll'z$GF@$ \"3/?Np5C\\bJF@7$$\"3MLL$eRwX5$F@$\"3)GTJ!oG0$3$F@7$$\"3rLLL$eI8k$F@$ \"3FyHM$p'GKJF@7$$\"33ML$3x%3yTF@$\"3n$**Q]`\"yRLF@7$$\"3emm\"z%4\\Y_F @$\"3%)*)G8T#p2%RF@7$$\"3`LLeR-/PiF@$\"3PZ.%R2Cm^%F@7$$\"3]***\\il'pis F@$\"31e'*fKlL9]F@7$$\"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$$\"3%ommmJF,$!3!\\F1 VDv$)p&F@7$$\"3K+]i!f#=$3#F,$!3V$*[LN4F4!)F@7$$\"3?+](=xpe=#F,$!3$[6ra l`?.\"F,7$$\"37nm\"H28IH#F,$!3s6ToLL*4G\"F,7$$\"3um;zpSS\"R#F,$!3/xdj' GyI^\"F,7$$\"3GLL3_?`(\\#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$$\"3m m\"H#oZ1\"\\$F,$!3)o4&)z%=-oKF,7$$\"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$$ \"3ILe*[t\\sp$F,$!3D1>x`HA5KF,7$$\"3[m;H2qcZPF,$!3/[q%\\V.-<$F,7$$\"3O +]7.\"fF&QF,$!3KL?tX&>E0$F,7$$\"3Ymm;/OgbRF,$!3KQEMNc$G*GF,7$$\"3w**\\ ilAFjSF,$!3/QR)44g!yEF,7$$\"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$%\"xG%%y(x)G-%%VIEW G6$;F(F\\^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 "d iscrete solution" }{TEXT -1 44 " based on each of the methods and give s the " }{TEXT 260 22 "root mean square error" }{TEXT -1 18 " of each \+ solution." }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 767 "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)],[`s tep width: `,hh],\n[`no. of steps: `,numsteps]]);``;\nmthds := [`B utcher's scheme A`,`scheme with simple nodes`,`scheme with a relativel y large stability region`,`Butcher's scheme B with `*(c[5]=1/2,c[6]=1/ 2,b[5]=b[6]),`scheme with `*(c[5]=3/4,c[6]=3/4,b[5]=b[6])]:\nerrs := [ ]:\nDigits := 20:\nfor ct to 5 do\n Bn_RK6_||ct := RK6_||ct(B(x,y),x ,y,x0,evalf(y0),hh,numsteps,false);\n sm := 0: numpts := nops(Bn_RK6 _||ct):\n for ii to numpts do\n sm := sm+(Bn_RK6_||ct[ii,2]-b(B n_RK6_||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\"$+&Q)pprint656\"" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%!G" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#K%'matrixG6#7'7$%3Butcher's~scheme~AG$ \"+LXoP;!#>7$%9scheme~with~simple~nodesG$\"+op%o0(!#?7$%Pscheme~with~a ~relatively~large~stability~regionG$\"+xDPeqF07$*&%9Butcher's~scheme~B ~with~G\"\"\"6%/&%\"cG6#\"\"&#F8\"\"#/&F<6#\"\"'F?/&%\"bGF=&FGFCF8$\"+ X8zK;F+7$*&%-scheme~with~GF86%/F;#\"\"$\"\"%/FBFPFEF8$\"+oI7SqF0Q)ppri nt666\"" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 30 "The following code constructs " }{TEXT 260 20 "numerical procedure s" }{TEXT -1 56 " for solutions based on each of the Runge-Kutta schem es." }}{PARA 0 "" 0 "" {TEXT -1 75 "The error in the value obtained by each of the methods at the point where " }{XPPEDIT 18 0 "x = 4.999; " "6#/%\"xG-%&FloatG6$\"%**\\!\"$" }{TEXT -1 16 " is also given." }} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 698 "B := (x,y) -> 10*x*cos(x)-1 0*y: hh := 0.01: numsteps := 500: x0 := 0: y0 := sqrt(5):\nmatrix([[`s lope field: `,B(x,y)],[`initial point: `,``(x0,y0)],[`step width: \+ `,hh],\n[`no. of steps: `,numsteps]]);``;\nmthds := [`Butcher's sche me A`,`scheme with simple nodes`,`scheme with a relatively large stabi lity region`,`Butcher's scheme B with `*(c[5]=1/2,c[6]=1/2,b[5]=b[6]), `scheme with `*(c[5]=3/4,c[6]=3/4,b[5]=b[6])]:\nerrs := []:\nDigits := 25:\nfor ct to 5 do\n bn_RK6_||ct := RK6_||ct(B(x,y),x,y,x0,evalf(y 0),hh,numsteps,true);\nend do:\nxx := 4.999: bxx := evalf(b(xx)):\nfor ct to 5 do\n errs := [op(errs),abs(bn_RK6_||ct(xx)-bxx)];\nend do: \nDigits := 10:\nlinalg[transpose]([mthds,evalf(errs)]);" }}{PARA 11 " " 1 "" {XPPMATH 20 "6#K%'matrixG6#7&7$%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) pprint676\"" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%!G" }}{PARA 11 "" 1 " " {XPPMATH 20 "6#K%'matrixG6#7'7$%3Butcher's~scheme~AG$\"+Os.!3\"!#?7$ %9scheme~with~simple~nodesG$\"+5EPch!#A7$%Pscheme~with~a~relatively~la rge~stability~regionG$\"+bMkeBF07$*&%9Butcher's~scheme~B~with~G\"\"\"6 %/&%\"cG6#\"\"&#F8\"\"#/&F<6#\"\"'F?/&%\"bGF=&FGFCF8$\"+x&)*yD)!#@7$*& %-scheme~with~GF86%/F;#\"\"$\"\"%/FBFQFEF8$\"+NC)zo&F0Q)pprint686\"" } }}{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$\"\"!\"\"&" }{TEXT -1 82 " of each \+ Runge-Kutta method is estimated as follows using the special procedure " }{TEXT 0 5 "NCint" }{TEXT -1 98 " to perform numerical integratio n by the 7 point Newton-Cotes method over 200 equal subintervals." }} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 436 "mthds := [`Butcher's scheme A`,`scheme with simple nodes`,`scheme with a relatively large stabili ty region`,`Butcher's scheme B with `*(c[5]=1/2,c[6]=1/2,b[5]=b[6]),`s cheme with `*(c[5]=3/4,c[6]=3/4,b[5]=b[6])]:\nerrs := []:\nDigits := 2 0:\nfor ct to 5 do\n sm := NCint((b(x)-'bn_RK6_||ct'(x))^2,x=0..5,ad aptive=false,numpoints=7,factor=200);\n errs := [op(errs),sqrt(sm/5) ];\nend do:\nDigits := 10:\nlinalg[transpose]([mthds,evalf(errs)]);" } }{PARA 11 "" 1 "" {XPPMATH 20 "6#K%'matrixG6#7'7$%3Butcher's~scheme~AG $\"+LiOn6!#>7$%9scheme~with~simple~nodesG$\"+;:5k@!#@7$%Pscheme~with~a ~relatively~large~stability~regionG$\"+: " 0 "" {MPLTEXT 1 0 533 "evalf[20](plot(['bn_RK6_1'(x)-b(x),'bn_RK6_2'(x)-b(x),'bn_RK6_3 '(x)-b(x),'bn_RK6_4'(x)-b(x),\n'bn_RK6_5'(x)-b(x)],x=0..0.65,numpoints =100,font=[HELVETICA,9],\ncolor=[COLOR(RGB,.95,.2,0),COLOR(RGB,.45,0,. 95),COLOR(RGB,0,.25,1),COLOR(RGB,0,.75,.2),COLOR(RGB,.95,.45,0)],\nleg end=[`Butcher's scheme A`,`scheme with simple nodes`,`scheme with a re latively large stability region`,`Butcher's scheme B with c[5]=c[6]=1/ 2 and b[5]=b[6]`,`scheme with c[5]=c[6]=3/4 and b[5]=b[6]`],title=`err or curves for 7 stage order 6 Runge-Kutta methods`));" }}{PARA 13 "" 1 "" {GLPLOT2D 938 575 575 {PLOTDATA 2 "6+-%'CURVESG6%7b\\l7$$\"\"!F)F (7$$\"5SSSSSS:N<7$$\"5\"333333.ZV$F-$\"')3q)F07$$\"5+,,,, ^)yLH%F-$\"(\"eZTF07$$\"5?@@@@@Y0_^F-$\")$zc[\"F07$$\"5STTTT\"RI2,'F-$ \")QKpVF07$$\"5ihhhhhhSpoF-$\"*L%H76F07$$\"5XXXX&zNNlh(F-$\"*i'p!H#F07 $$\"5IHHHHaXmj$)F-$\"*[8)3WF07$$\"55888j]Pz5\"*F-$\"*$onA!)F07$$\"5&pp ppp%H#z&)*F-$\"+L)GDR\"F07$$\"5HHHagC0Z/5!#@$\"+$Hr%Q:F07$$\"5*)))))Q^ a([J-\"F[o$\"+$Q+c`\"F07$$\"5[[[BU%)p#=/\"F[o$\"+J[tK:F07$$\"53333L9_] g5F[o$\"+HY()H:F07$$\"5FFFx9u;'y4\"F[o$\"+P.F07$$\"5555555R#o&>F[o$\"+ZkQ@C F07$$\"5GGGG.s(oz(>F[o$\"+%\\Aue#F07$$\"5YYYY'Rj8\"**>F[o$\"+*)>&ox#F0 7$$\"5kkkk*e\\e--#F[o$\"+5\\qzFF07$$\"5$GGGGyN.9/#F[o$\"+tM$Qx#F07$$\" 5?>>>p\"3$p$3#F[o$\"+7y7iFF07$$\"5cbbbb0G)f7#F[o$\"+:@Z]FF07$$\"5GGGGG `Ac5AF[o$\"+O%Hts#F07$$\"5,,,,,,<9&H#F[o$\"+uDe/FF07$$\"5utttt[6szBF[o $\"+wq*Ho#F07$$\"5YYYYY'f+VY#F[o$\"+)[Z[m#F07$$\"5>>>>>W+))[DF[o$\"+]N _bEF07$$\"5#>>>>>\\fMj#F[o$\"+-u,mEF07$$\"5aaaaH[pjdx$F07$$\"5SRRkq4CILIF[o$\"+M;ynPF07$$\"50000!QxYV0$F[o$ \"+;4')fPF07$$\"5OOO'))>]Nk4$F[o$\"+\"[pSu$F07$$\"5onnnOF[o$\"+#*3?$f$F07$$\"5333333)Q ^x$F[o$\"+0fB!p$F07$$\"588888j`AJRF[o$\"+*>Cn8%F07$$\"5%[[[[)4+jrRF[o$ \"+)QuCO%F07$$\"5ccccccY.7SF[o$\"+P\")>bXF07$$\"5GGGGG.$RC0%F[o$\"+^,$ o`%F07$$\"5+++++]R%G4%F[o$\"+$HO&=XF07$$\"5WVVVVVKltTF[o$\"+CV<#[%F07$ $\"5(ooooo`iWD%F[o$\"+6n;YWF07$$\"5uttttB63;WF[o$\"+Wu/xVF07$$\"5hgggg 5(*pxXF[o$\"+$*Q'oK%F07$$\"533333e?*>\"\\F[o$\"+W#)e/ZF07$$\"5cbbbb0WG Y_F[o$\"+!QMF.&F07$$\"5SRRRRRX*GT&F[o$\"+$RT9&\\F07$$\"5DBBBBtY]zbF[o$ \"+[9E))[F07$$\"5?::::S(4Gm&F[o$\"+2a()z[F07$$\"5522222[6YdF[o$\"+SBb2 \\F07$$\"5+**)*)*)R()>%HeF[o$\"+aKw**\\F07$$\"5\"44444%\\s7fF[o$\"+D`# 3?&F07$$\"5XWW%>8ppb&fF[o$\"+!f>rO&F07$$\"5+)zzHf7/'F[o$\"+74YxbF07$$\"50000bUR5%3'F[o$\"+lbh`bF07$$\"557 77PVMzphF[o$\"+TOB1bF07$$\"5?>>>>WH[biF[o$\"+3$4$faF07$$\"5NLLL$e%>'oU 'F[o$\"+,6no`F07$$\"5[ZZZZZ4C)f'F[o$\"+`w[)H&F07$$\"5NLLLL$QU,!pF[o$\" +)*y#G_&F07$$\"5?>>>>>Q/-sF[o$\"+)[(f$z&F07$$\"5+++++v&y>P(F[o$\"+2+s' p&F07$$\"5!33333L8>a(F[o$\"+\"y*[5cF07$$\"5?@@@r32)oi(F[o$\"+zqq\"e&F0 7$$\"5ghhhh'3[=r(F[o$\"+IwWwbF07$$\"5+---_ka\"oz(F[o$\"+o1$\\h&F07$$\" 5VUUUUUGy\")yF[o$\"+t*)eHdF07$$\"5baaaa/\"*4nzF[o$\"+bDErfF07$$\"5lmmm mm`T_!)F[o$\",YVj53'!#?7$$\"5!)yyyyG;tP\")F[o$\",OT.%HgF_gl7$$\"5!4444 4*y/B#)F[o$\",u***>yfF_gl7$$\"5::::::/o$R)F[o$\",++a!yeF_gl7$$\"5SRRRR RHJk&)F[o$\",54?-z&F_gl7$$\"5][[[[[]=$*))F[o$\",GD`T!fF_gl7$$\"5eddddd r0A#*F[o$\",Gcvb3'F_gl7$$\"5&eeee$[3Qr$*F[o$\",cdsf*fF_gl7$$\"5:9999RX q?&*F[o$\",#)H+I\"fF_gl7$$\"5IGGG`%Qm`f*F[o$\",`lj\"zeF_gl7$$\"5XUUU#* H#G+n*F[o$\",56&*o&eF_gl7$$\"5]\\\\*>E:ftq*F[o$\",]:;F&eF_gl7$$\"5gccc Jv+pW(*F[o$\",/YF^&eF_gl7$$\"5ljj8,)*4-#y*F[o$\",(p@.meF_gl7$$\"5rqqqq ?>N>)*F[o$\",Bwax)eF_gl7$$\"5gdddKw?83**F[o$\",=B,0+'F_gl7$$\"5XWWW%>B 7p***F[o$\",ZE9MC'F_gl7$$\"5888jvQ#p&35F_gl$\",KI!o,iF_gl7$$\"5#====VD Zu,\"F_gl$\",X9oo9'F_gl7$$\"5?>>>W&G._.\"F_gl$\",TOk!RgF_gl7$$\"5dcccc ;$fH0\"F_gl$\",HXC!RfF_gl7$$\"5yxxxFg,[g5F_gl$\",Zr:Z!fF_gl7$$\"5+**)* )*)R+,!o5F_gl$\",_j&*=)eF_gl7$$\"5gfff%eUh<2\"F_gl$\",-*eOxeF_gl7$$\"5 @???qZ=_v5F_gl$\",#>yLzeF_gl7$$\"5#333e&pAGz5F_gl$\",Oct'*)eF_gl7$$\"5 UTTTT\"pUI3\"F_gl$\",EZW1\"fF_gl7$$\"5jiii7NNc!4\"F_gl$\",zV8g*fF_gl7$ $\"5%QQQQ)yV3)4\"F_gl$\",Am\"RkhF_gl7$$\"5999kw*ek**4\"F_gl$\",\\t&)QA 'F_gl7$$\"5WWWWp+[%=5\"F_gl$\",>z5O@'F_gl7$$\"5uuuCi6]s.6F_gl$\",$p)Q> ?'F_gl7$$\"50000bA_g06F_gl$\",G())G!>'F_gl7$$\"5mlllSWcO46F_gl$\",&>X0 nhF_gl7$$\"5EEEEEmg786F_gl$\",Rl2R9'F_gl7$$\"5SSSS!H=?\"[6F_gl$\",uY&o NfF_gl7$$\"5baaaa*H9J=\"F_gl$\",)pOxoeF_gl7$$\"5'eeeL-Be3>\"F_gl$\",G7 Me%fF_gl7$$\"5=<<<#4;-')>\"F_gl$\",JZmQ5'F_gl7$$\"5%GGyli7uC?\"F_gl$\" ,TNc(GhF_gl7$$\"5\\[[)4;4Yj?\"F_gl$\",h!>20hF_gl7$$\"5999R&p0=-@\"F_gl $\",i.z93'F_gl7$$\"5!)zzzHA+497F_gl$\",juxz0'F_gl7$$\"5UUUUn$)ydH7F_gl $\",Nxm\\'fF_gl7$$\"500000Xd1X7F_gl$\",@c_\\(eF_gl7$$\"5uttBOk?c`7F_gl $\",!f?PHeF_gl7$$\"5UUUUn$Qe?E\"F_gl$\",zie1z&F_gl7$$\"5666h)Hqa0F\"F_ gl$\",_Bxcw&F_gl7$$\"5!)zzzHA50z7F_gl$\",3\\:iw&F_gl7$$\"5\\[[)4;MZvG \"F_gl$\",\"f-E6eF_gl7$$\"5=<<<#4mVgH\"F_gl$\",mV)))HfF_gl7$$\"5NM%o\\ 2un\")H\"F_gl$\",D8$pvfF_gl7$$\"5_^^wd?=H+8F_gl$\",x')\\)>gF_gl7$$\"5p o=cS+fT-8F_gl$\",hzwq+'F_gl7$$\"5'eeeL-)*RXI\"F_gl$\",t#3L%*fF_gl7$$\" 5???&*))R\")y38F_gl$\",E)*>*ofF_gl7$$\"5baaaa*HOIJ\"F_gl$\",G40=\\\"F_gl$\",b!p&)fbF_gl7$$\"5(oooo=B4t]\"F_gl$\",3#[ $\\k&F_gl7$$\"5zyyyGEcvS:F_gl$\",8D%*)faF_gl7$$\"5rqqqq???u:F_gl$\",QU X?K&F_gl7$$\"5!*)))))QEss:f\"F_gl$\",EHGLP&F_gl7$$\"53222dCM%*3;F_gl$ \",&[y1QaF_gl7$$\"5EDDD]ETJE;F_gl$\",LbZWM&F_gl7$$\"5WVVVVG[oV;F_gl$\" ,$*eDLD&F_gl7$$\"5/...`vg!)e;F_gl$\",l&Q3\"=&F_gl7$$\"5jiiiiAt#Rn\"F_g l$\",iQGK8&F_gl7$$\"5UUUU4mp\"F_gl$\",vgeUA&F_gl7$$\"5#====o\" )pTq\"F_gl$\",![pn^_F_gl7$$\"5cccc1RD$ot\"F_gl$\",NLrJ3&F_gl7$$\"5JJJJ Jh_\\pF_gl$\",G#Q\"z![F_gl7$$\"5GFFFx%zY\\$>F_gl$\",N\\=pl%F _gl7$$\"5+++++Dt)o'>F_gl$\",>>XS_%F_gl7$$\"5:::lF`Mvv>F_gl$\",156K]%F_ gl7$$\"5IIIIb\"e>Y)>F_gl$\",8m;>]%F_gl7$$\"5XXX&H)4d[$*>F_gl$\",B(yZ!e%F_gl7$$\"5!444fY4%3??F_gl$\",1Zo** \\%F_gl7$$\"5@@@@@^j\"y.#F_gl$\",Le-6U%F_gl7$$\"5!**)*)*)Rs&\\P0#F_gl$ \",2%[m`VF_gl7$$\"5eeeee$z#op?F_gl$\",`kZ')H%F_gl7$$\"5#HHHzTS\\w2#F_g l$\",^mzKG%F_gl7$$\"5FFFFx9gh&3#F_gl$\",x)z#RG%F_gl7$$\"5ihhhODEe$4#F_ gl$\",e)\\T5VF_gl7$$\"5'fffffB\\:5#F_gl$\",cM\"G`VF_gl7$$\"5&\\\\\\\\ \\Ytb8#F_gl$\",H(**y2UF_gl7$$\"5%RRRRRp(fp@F_gl$\",DG2;3%F_gl7$$\"5$HH HagW0t<#F_gl$\",!*3Tj1%F_gl7$$\"5#>>>p\")>8]=#F_gl$\",!elJkSF_gl7$$\"5 \"444%G]4s#>#F_gl$\",pctL3%F_gl7$$\"5!**)*)*)R-(G/?#F_gl$\",#e:.HTF_gl 7$$\"5)yyyGm?We@#F_gl$\",@(*oe1%F_gl7$$\"5'eeee3rf7B#F_gl$\",7y?P+%F_g l7$$\"5VUUUU#es')H#F_gl$\",v_v_*QF_gl7$$\"5?>>>p6xQIBF_gl$\",N]9%)y$F_ gl7$$\"5'ffff4%G5iBF_gl$\",SE3bn$F_gl7$$\"5cbbb0.2RqBF_gl$\",**Q\"y_OF _gl7$$\"5:::::l&y'yBF_gl$\",q8;!QOF_gl7$$\"5uuuuCFk'pQ#F_gl$\",_b&)oj$ F_gl7$$\"5MMMMM*Ga_R#F_gl$\",+40!eOF_gl7$$\"59999R?#)R*R#F_gl$\",QT_1o $F_gl7$$\"5%RRRR9:UNS#F_gl$\",vqX#G4hDF_gl$\",V[I'oKF_gl7$$\"5onnnn2QXEEF_gl$\",3wfx <$F_gl7$$\"5'eeeee&[H$p#F_gl$\",(*zO./$F_gl7$$\"5UTTTTECefFF_gl$\",Z\" H\\))GF_gl7$$\"5SRRRR>Q\\?GF_gl$\",$eQz7GF_gl7$$\"5gffff\\^I!*GF_gl$\" ,Gknpm#F_gl7$$\"5cbbbbN_u_HF_gl$\",Y.Q,b#F_gl7$$\"5YXXXX!R>$>IF_gl$\", `MuQY#F_gl7$$\"5POOOO^$RI3$F_gl$\",6SF9L#F_gl7$$\"5**)*)*)*)*y@z`JF_gl $\"-(4L^IA#F[o7$$\"5%RRRR*G*e]@$F_gl$\"-00!=Q:#F[o7$$\"588888Qsf%G$F_g l$\"-!Q=)3F?F[o7$$\"5)yyyyy&Q(zM$F_gl$\"-RfI>T>F[o7$$\"5&[[[[)>zH&R\"F[o7$$\"5cbbbbX&*HtQF_gl$\"-6&\\dmI\"F[o7$$\"5qpppp%e[\"QRF_gl$ \"-umG$*[7F[o7$$\"5vuuuuW\"*R1SF_gl$\"-\"pQp7>\"F[o7$$\"5xwwwwOB=oSF_g l$\"-Q%o#R@6F[o7$$\"5!**)*)*)*)HVYOTF_gl$\"-k_>)p1\"F[o7$$\"5wvvvv!em= ?%F_gl$\"-V^YF=5F[o7$$\"5rqqqqS'4rE%F_gl$\",oRE'\\&*F[o7$$\"5qpppppiDN VF_gl$\",;^(Ro!*F[o7$$\"5yxxxx-=-)R%F_gl$\",lbhuj)F[o7$$\"599999%H-BY% F_gl$\",NaP77)F[o7$$\"5baaaa%fvK`%F_gl$\",x]FUn(F[o7$$\"5lkkkk/h`(f%F_ gl$\",x/_$ysF[o7$$\"5)yyyyG^aKm%F_gl$\",B\"HWLoF[o7$$\"5FEEEET%)3IZF_g l$\",QgUbY'F[o7$$\"5#>>>>p#=_\"z%F_gl$\",(pp49hF[o7$$\"5nmmmmEy+d[F_gl $\",UCI]v&F[o7$$\"5tssssU`*>#\\F_gl$\",uvT&QaF[o7$$\"5NMMMM%y6:*\\F_gl $\",1Q4:4&F[o7$$\"5jiiiiKq&G0&F_gl$\",:7VO![F[o7$$\"5XWWWW9njB^F_gl$\" ,l[Xc\\%F[o7$$\"5TSSSS?*4v=&F_gl$\",h^M7A%F[o7$$\"5/////>\"42D&F_gl$\" ,O[M!oRF[o7$$\"5^]]]]g&o'=`F_gl$\",JYtfq$F[o7$$\"5#>>>>>\"H!pQ&F_gl$\" ,ziJuX$F[o7$$\"5WVVVVog)*[aF_gl$\"-&G,\\BC$F-7$$\"5$HHHHHi:\\^&F_gl$\" -`_*>c,$F-7$$\"5XWWWWCkDzbF_gl$\"-=9>;AGF-7$$\"522222KD+\\cF_gl$\"-24h q4EF-7$$\"5nmmmm1BN4dF_gl$\"-!3:06U#F-7$$\"5#====o@m'ydF_gl$\"--UT&4D# F-7$$\"5KJJJJO\\#Q%eF_gl$\"-hJ*RZ2#F-7$$\"5kjjjj$Ha$3fF_gl$\"-$)eKh&*= F-7$$\"5baaaa%)\\$H(fF_gl$\"-d..'4x\"F-7$$\"5POOOO,K!)QgF_gl$\"-F0Bq4; F-7$$\"5tssss2Ci3hF_gl$\"-4N/&*R9F-7$$\"5.....$yRE<'F_gl$\"-G;-cV8F-7$ $\"5onnnn_J$eB'F_gl$\"-&*G==.7F-7$$\"5!)zzzzCeQ.jF_gl$\"-*Gi/H0\"F-7$$ \"5mllllS9zqjF_gl$\",a1e]x*F-7$$\"5utttt$)>HJkF_gl$\",_h0F_)F-7$$\"#l! \"#$\",S2=([rF--%&COLORG6&%$RGBG$\"#&*F_bp$\"\"#!\"\"F(-%'LEGENDG6#%3B utcher's~scheme~AG-F$6%7cblF'7$F+$\"$K\"F07$F2$\"&Li\"F07$F7$\"&%3wF07 $F<$\"'&)yEF07$FA$\"'pTxF07$FF$\"(Zg$>F07$FK$\"(n\\#RF07$FP$\"(iYV(F07 $FU$\")\\9J8F07$FZ$\")*RGF#F07$Fin$\")18.DF07$F_o$\")'f%)\\#F07$Fdo$\" )sz$\\#F07$Fio$\")O9*[#F07$F^p$\")E')zCF07$Fcp$\")!=1Z#F07$Fhp$\")SF_C F07$F]q$\")>PMCF07$Fbq$\")5M.CF07$Fgq$\"))H4W#F07$F\\r$\")mjUDF07$Far$ \")bVlFF07$Ffr$\")L`%>$F07$F[s$\")?!f&RF07$F`s$\")zg>UF07$Fes$\")W!)=X F07$Fjs$\")Q+BXF07$F_t$\"),X8XF07$Fdt$\")JS%\\%F07$Fit$\")xVvWF07$F^u$ \")uyPWF07$Fcu$\"):$3S%F07$Fhu$\").(fO%F07$F]v$\")%fsL%F07$Fbv$\")[0CV F07$Fgv$\")K\"\\M%F07$F\\w$\")XCKWF07$Faw$\")WeRYF07$Ffw$\")ta\\]F07$F [x$\")f^$y&F07$F`x$\")&Rk.'F07$Fex$\")lLWhF07$Fjx$\")*>98'F07$F_y$\")- `=hF07$Fdy$\")B$G4'F07$Fiy$\")WCngF07$F^z$\")MW;gF07$Fcz$\")I]mfF07$Fh z$\")DAAfF07$F][l$\").0$)eF07$Fb[l$\")3qbeF07$Fg[l$\")b&G&eF07$F\\\\l$ \")G/@gF07$Fa\\l$\")xp\\nF07$Ff\\l$\")sh4rF07$F[]l$\"):'QT(F07$F`]l$\" )j'RQ(F07$Fe]l$\")?>atF07$Fj]l$\")A,&H(F07$F_^l$\")cUOsF07$Fd^l$\")sQC rF07$Fi^l$\")smXqF07$F^_l$\")n0vwF07$Fc_l$\")%zD>)F07$Fh_l$\")Ykg!)F07 $F]`l$\")3cgzF07$$\"5?>>>p1s:@cF[o$\")z)*\\zF07$Fb`l$\")@F]zF07$$\"5:6 66htAY/dF[o$\"),LlzF07$Fg`l$\")#>,+)F07$F\\al$\")P.b\")F07$Faal$\")f%G [)F07$Ffal$\")IJ\\()F07$F[bl$\")S$Q5*F07$F`bl$\")![83*F07$Febl$\")E_U! *F07$Fjbl$\")_Pl*)F07$F_cl$\")t)*))))F07$Fdcl$\")P&=u)F07$Ficl$\");dI' )F07$F^dl$\")$)G3!*F07$Fcdl$\")'oeV*F07$Fhdl$\")%f#y#*F07$F]el$\")^VR \"*F07$Fbel$\")8$[4*F07$Fgel$\")l!**3*F07$F\\fl$\"))Hq:*F07$Fafl$\")( \\lM*F07$Fffl$\")(Q]t*F07$F[gl$\"*wVt!**F_gl7$Fagl$\"*wyJ#)*F_gl7$Ffgl $\"*%GwR(*F_gl7$F[hl$\"*!y#od*F_gl7$F`hl$\"*KmbV*F_gl7$Fehl$\"*#*oIj*F _gl7$Fjhl$\"*vI'=**F_gl7$F_il$\"*#*HFx*F_gl7$Fdil$\"*SL&Q'*F_gl7$Fiil$ \"*1oZe*F_gl7$F^jl$\"*=S2b*F_gl7$Fcjl$\"*t0aa*F_gl7$Fhjl$\"*F!*4b*F_gl 7$F][m$\"*G'\\q&*F_gl7$Fb[m$\"*)3a2'*F_gl7$Fg[m$\"*C(*Hz*F_gl7$F\\\\m$ \"+%HC\"=5F_gl7$Fa\\m$\"+k+E65F_gl7$Ff\\m$\"+bBK-5F_gl7$F[]m$\"*iGv%)* F_gl7$F`]m$\"*r.bo*F_gl7$Fe]m$\"*hK4j*F_gl7$Fj]m$\"*oXff*F_gl7$F_^m$\" *e$***e*F_gl7$Fd^m$\"*>,[f*F_gl7$Fi^m$\"*p+Lh*F_gl7$F^_m$\"*H>!\\'*F_g l7$Fc_m$\"*B&o*y*F_gl7$Fh_m$\"+7.,15F_gl7$F]`m$\"+OeW:5F_gl7$Fb`m$\"+3 Nw85F_gl7$Fg`m$\"+5#f=,\"F_gl7$F\\am$\"+!\\e*45F_gl7$Faam$\"+dx;15F_gl 7$Ffam$\"+48R-5F_gl7$$\"5LLLLeCJiI6F_gl$\"*T%H])*F_gl7$F[bm$\"*Y4[o*F_ gl7$$\"5%RRRk?roo:\"F_gl$\"*IHEh*F_gl7$$\"5[ZZZATshl6F_gl$\"*nU(e&*F_g l7$$\"5-,,^QqdOu6F_gl$\"*dz-a*F_gl7$F`bm$\"*`o`e*F_gl7$Febm$\"*2,Fr*F_ gl7$Fjbm$\"*Non'**F_gl7$F_cm$\"+H@c+5F_gl7$Fdcm$\"*xap'**F_gl7$Ficm$\" *yP%G**F_gl7$F^dm$\"*Zq+*)*F_gl7$Fcdm$\"*P[#Q(*F_gl7$Fhdm$\"*#ek\"f*F_ gl7$F]em$\"*OOz^*F_gl7$Fbem$\"*pthX*F_gl7$Fgem$\"*s5yT*F_gl7$F\\fm$\"* VU>U*F_gl7$Fafm$\"*P0%)\\*F_gl7$Fffm$\"*i![\"p*F_gl7$F[gm$\"*[w[w*F_gl 7$F`gm$\"*tf^$)*F_gl7$Fegm$\"*@\"H9)*F_gl7$Fjgm$\"*-nMz*F_gl7$F_hm$\"* 3^>v*F_gl7$Fdhm$\"*g61r*F_gl7$Fihm$\"*GqWS*F_gl7$F^im$\"*+`nA*F_gl7$Fc im$\"*PlaU*F_gl7$Fhim$\"*=!=$[*F_gl7$F]jm$\"*PEXK*F_gl7$Fbjm$\"*,m5<*F _gl7$Fgjm$\"*K2#[!*F_gl7$F\\[n$\"*+;y)*)F_gl7$Fa[n$\"*\\7F5*F_gl7$Ff[n $\"*)F_gl7$F`an$\"*)H3t#)F_gl7$Fean$\"*+yY8)F_gl7$Fjan$\"*)f.**zF_gl7$F_b n$\"*(F_gl7$Fchn$\"*)3napF_gl7$Fhhn$\"*hF!\\nF _gl7$F]in$\"*oUhs'F_gl7$Fbin$\"*pLhs'F_gl7$Fgin$\"*B&yhnF_gl7$F\\jn$\" *Ka7%oF_gl7$Fajn$\"*b-mt'F_gl7$Ffjn$\"*k[Oj'F_gl7$F[[o$\"*lh5Z'F_gl7$F `[o$\"*>`VH'F_gl7$Fe[o$\"*bw\"3hF_gl7$Fj[o$\"*KF?2'F_gl7$F_\\o$\"*RD-0 'F_gl7$Fd\\o$\"*Q)[_gF_gl7$Fi\\o$\"*1lJ4'F_gl7$F^]o$\"*U#*R8'F_gl7$Fc] o$\"*%Gg>hF_gl7$Fh]o$\"*7'H%4'F_gl7$F]^o$\"*7%4pgF_gl7$Fb^o$\"*#f+>gF_ gl7$Fg^o$\"*&fOpfF_gl7$F\\_o$\"*`_#[dF_gl7$Fa_o$\"*G\"=oaF_gl7$Ff_o$\" *J2gL&F_gl7$F[`o$\"*tp47&F_gl7$$\"5kjjj8T'Qks#F_gl$\"*!3RJ]F_gl7$F``o$ \"*>kG([F_gl7$$\"5TSSS!H7Q+z#F_gl$\"*7+_\"[F_gl7$Fe`o$\"*[!ynZF_gl7$Fj `o$\"*b*eMXF_gl7$F_ao$\"*+GrM%F_gl7$Fdao$\"*@0eA%F_gl7$Fiao$\"*\\)[2SF _gl7$F^bo$\"+F#G)RQF[o7$$\"5'eeeLF_]9;$F_gl$\"+M0a8QF[o7$$\"5ssssZm)3 \"pJF_gl$\"+\"R*o!z$F[o7$$\"5fff4A5swwJF_gl$\"+g>rtPF[o7$$\"5YYYY'RbDW =$F_gl$\"+%37jw$F[o7$$\"5LLL$3x*Q3#>$F_gl$\"+ju(Qx$F[o7$$\"5????XTAu*> $F_gl$\"+au'R!QF[o7$$\"5222d>&e+u?$F_gl$\"+tcUxPF[o7$Fcbo$\"+&f2'[PF[o 7$$\"5a````$3G)\\KF_gl$\"+Irf@OF[o7$Fhbo$\"+q^cQNF[o7$F]co$\"+\\!4!3MF [o7$Fbco$\"+R&e'*H$F[o7$Fgco$\"+8&)QFJF[o7$F\\do$\"+XK#Q,$F[o7$$\"5(oo ooox&ohNF_gl$\"+6%)3oHF[o7$$\"5;;;;;ck8yNF_gl$\"+v&*pNHF[o7$$\"5YXXXXN re%f$F_gl$\"++8:UHF[o7$Fado$\"+y4]GHF[o7$$\"5=<<_*y$F_gl$\"+_6Q)e #F[o7$F`eo$\"+l0s*f#F[o7$$\"5OOOO'Q]y#RQF_gl$\"+0b!H^#F[o7$Feeo$\"+r/^ RCF[o7$$\"55444M0=^*)QF_gl$\"+B&QDV#F[o7$$\"5jiii7lSs0RF_gl$\"+4flUCF[ o7$$\"5;;;;\"\\KO>#RF_gl$\"+dlP.CF[o7$Fjeo$\"+CZ$[O#F[o7$$\"5'fff4ZA6_ &RF_gl$\"+A/;EBF[o7$$\"5AAAAskQFsRF_gl$\"+!z6ZH#F[o7$$\"5[[[[t/lL*)RF_ gl$\"+.10(G#F[o7$F_fo$\"+h1)eH#F[o7$$\"5wvvvvS2HPSF_gl$\"+o88EAF[o7$Fd fo$\"+[^Fk@F[o7$$\"510000NGD&3%F_gl$\"+]!>(\\@F[o7$$\"5MLLLLLLK-TF_gl$ \"+g#)op@F[o7$$\"5ihhhhJQR>TF_gl$\"+Qe'H8#F[o7$Fifo$\"+%oOp4#F[o7$$\"5 OOOOh#*[\"G:%F_gl$\"+V^#Q1#F[o7$$\"5$GGGG`Xl\"pTF_gl$\"+U$*pN?F[o7$$\" 5IHHH/=g^&=%F_gl$\"+25(Q-#F[o7$F^go$\"+j3pW?F[o7$$\"5CBBBt5\")[MUF_gl$ \"+sR6z>F[o7$Fcgo$\"+38Y?>F[o7$$\"5YXXX&zHYTG%F_gl$\"+Aux1>F[o7$$\"5?? ???bH=,VF_gl$\"+,]:H>F[o7$$\"5&\\\\\\\\Ch>#=VF_gl$\"+tvc'*=F[o7$Fhgo$ \"+mKek=F[o7$$\"5srrr'HlZ4N%F_gl$\"+J\"*>O=F[o7$$\"5utttBO!RmO%F_gl$\" +Q;W6=F[o7$$\"5wvvv]>/L#Q%F_gl$\"+\\\\R)z\"F[o7$F]ho$\"+:Lu;=F[o7$$\"5 'ffff%[?;IWF_gl$\"+8'p)oc/![%F_gl$\" +q0U)p\"F[o7$$\"5MMMMMW*)y(\\%F_gl$\"+m3!fr\"F[o7$$\"5WWWWWpA`:XF_gl$ \"+dm?'p\"F[o7$Fgho$\"+F#4km\"F[o7$F\\io$\"+P4pA;F[o7$$\"5EEEEw3`RIYF_ gl$\"+!R*R\"e\"F[o7$Faio$\"+B.:L:F[o7$$\"5[ZZZ(\\*H'*zYF_gl$\"+A?m>:F[ o7$$\"532222x9n'p%F_gl$\"+\"pS^`\"F[o7$$\"5ommm;f*zLr%F_gl$\"+l3JC:F[o 7$Ffio$\"+)Hp!*\\\"F[o7$F[jo$\"+ZdqY9F[o7$$\"5hggg&oK$*y![F_gl$\"+;qva 9F[o7$$\"5IHHHzE[EC[F_gl$\"+*zP6V\"F[o7$$\"5)zzzHnKO1%[F_gl$\"+7;-39F[ o7$F`jo$\"+A`P'Q\"F[o7$$\"5qpppp%e,&*)[F_gl$\"+)Gp=P\"F[o7$Fejo$\"+9[&F_gl$\",oI)>V5F-7$$\"5bbbbIMKV)\\&F_gl$\",36Nz1\"F-7$Fb]p$\",`Hyr0 \"F-7$$\"5qooooBg3ZbF_gl$\",lGRS-\"F-7$Fg]p$\",=v1R+\"F-7$$\"55555N^Hp 'f&F_gl$\",<2X[-\"F-7$$\"5vvvvDy%HTh&F_gl$\",:t]0-\"F-7$$\"5STTT;0gcJc F_gl$\",U3IH+\"F-7$F\\^p$\"+2Z4g)*F-7$$\"5&ppp>d(**3kcF_gl$\"+JgkO(*F- 7$$\"5&oooo$>u>>p'*e&\\z&F_gl$\"+L'owa*F-7$$\"5bccccwbC6eF_gl$\"+mdx)e*F-7$$ \"5&RRRRkDNv#eF_gl$\"+0)>RV*F-7$F[_p$\"+h`S$G*F-7$$\"5SRRRkvs&*feF_gl$ \"+PKx]\"*F-7$$\"5]ZZZ(\\h*3weF_gl$\"+](z;3*F-7$$\"5bbbbIa>A#*eF_gl$\" +3)\\\"3#*F-7$F`_p$\"+$=)=S$*F-7$$\"554444RYkSfF_gl$\"+;^eW!*F-7$Fe_p$ \"+d&Ro#))F-7$$\"5++++vQ?S*)fF_gl$\"+2*oy!*)F-7$$\"5XXXX&H4pe+'F_gl$\" +\\E::\"*F-7$$\"5!444fr9OB-'F_gl$\"++)*Hm*)F-7$Fj_p$\"+Fip?))F-7$$\"5X XXX&H+ei0'F_gl$\"+k$[(y')F-7$$\"5baaaa/GrtgF_gl$\"+o2!Qf)F-7$$\"5ljjj8 1w;\"4'F_gl$\"+6/]4()F-7$F_`p$\"+4o.p))F-7$$\"5!zyyy`4J19'F_gl$\"+='G3 f)F-7$Fd`p$\"+G'\\%)Q)F-7$$\"5?>>>WD\"Q%)='F_gl$\"+51]k%)F-7$$\"5NNNN& yYOU?'F_gl$\"+A!R1r)F-7$$\"5]^^^E5[.?iF_gl$\"+l]6u&)F-7$Fi`p$\"+&42-W) F-7$$\"5qqqqq?8s_iF_gl$\"+'ybdI)F-7$$\"5vtttt)[4'piF_gl$\"+gD;6#)F-7$$ \"5!onnnnl(\\'G'F_gl$\"+S9Bg#)F-7$F^ap$\"+*y!GV&)F-7$$\"5vssssK')3PjF_ gl$\"+nmxg#)F-7$Fcap$\"+af)\\/)F-7$$\"5qnnnUwl\"fQ'F_gl$\"+s7=F07$FF$\"(v()f%F07$FK$\"('>5%*F07$FP$\")yZ* z\"F07$FU$\")))G`KF07$FZ$\");55cF07$Fin$\")JM!>'F07$F_o$\")9zyhF07$Fdo $\")8EnhF07$Fio$\")FvbhF07$F^p$\")/!G8'F07$Fcp$\")w$*4hF07$Fhp$\")%fF07$Fgq$\")@EEgF07$F\\r$\")(='oiF07$Far $\")YT3oF07$Ffr$\")!zD'yF07$F[s$\")B9e(*F07$F`s$\"*N$)>/\"F07$Fes$\"*) HJ<6F07$Fjs$\"*Q<%=6F07$F_t$\"*+bg6\"F07$Fdt$\"*@X86\"F07$Fit$\"*abm5 \"F07$F^u$\"*oWt4\"F07$Fcu$\"*'z>)3\"F07$Fhu$\"*UQ&z5F07$F]v$\"*g:B2\" F07$Fbv$\"*na(o5F07$Fgv$\"*oUL2\"F07$F\\w$\"*9dS4\"F07$Faw$\"*o1V9\"F0 7$Ffw$\"*Iz^C\"F07$F[x$\"*3u#G9F07$F`x$\"*5p=\\\"F07$Fex$\"*d*4>:F07$F jx$\"*51f^\"F07$F_y$\"*K>F^\"F07$Fdy$\"**eO1:F07$Fiy$\"*dR+]\"F07$F^z$ \"*[yu[\"F07$Fcz$\"*j@^Z\"F07$Fhz$\"*wOTY\"F07$F][l$\"*nYVX\"F07$Fb[l$ \"*eZtW\"F07$Fg[l$\"*_+iW\"F07$F\\\\l$\"*!4A'[\"F07$Fa\\l$\"*\"R)fm\"F 07$Ff\\l$\"*=Zgv\"F07$F[]l$\"*ZRE$=F07$F`]l$\"*n\\_#=F07$Fe]l$\"*t*)y \"=F07$Fj]l$\"*zgK!=F07$F_^l$\"*+w()y\"F07$Fd^l$\"*D75w\"F07$Fi^l$\"*& o5TF07$F]`l$\"*8v o'>F07$Fb`l$\"*h?Q'>F07$Fg`l$\"**RSv>F07$F\\al$\"*9LH,#F07$Faal$\"*u.Q 4#F07$Ffal$\"*zQ-;#F07$F[bl$\"*Jf#\\AF07$F`bl$\"*UxPC#F07$Febl$\"*f%=M AF07$Fjbl$\"*CB^@#F07$F_cl$\"*NZi>#F07$Fdcl$\"**o#)f@F07$Ficl$\"*.x=8# F07$F^dl$\"*u.KA#F07$Fcdl$\"*9\"oIBF07$Fhdl$\"*ADzQBF_gl7$Fffm$\"+IAD'Q#F_gl7$F[gm$\" +gGZ/CF_gl7$F`gm$\"+*G\"*>U#F_gl7$Fegm$\"+VA&oT#F_gl7$Fjgm$\"+1Ts6CF_g l7$F_hm$\"+c/],CF_gl7$Fdhm$\"+Y-K\"R#F_gl7$Fihm$\"+#)R(eJ#F_gl7$F^im$ \"+'G()4F#F_gl7$Fcim$\"+LW->BF_gl7$Fhim$\"+,i[LBF_gl7$F]jm$\"+UWW%H#F_ gl7$Fbjm$\"+xRjcAF_gl7$Fgjm$\"+Hm9EAF_gl7$F\\[n$\"+M@c5AF_gl7$Fa[n$\"+ !RLyB#F_gl7$Ff[n$\"+;ucrAF_gl7$F[\\n$\"+J'>r>#F_gl7$F`\\n$\"+?%e?9#F_g l7$Fe\\n$\"+#ozH;#F_gl7$Fj\\n$\"+Y1h)=#F_gl7$F_]n$\"+hG$4:#F_gl7$Fd]n$ \"+))*yU6#F_gl7$Fi]n$\"+XLI&3#F_gl7$F^^n$\"+!Q7j1#F_gl7$Fc^n$\"+.tql?F _gl7$Fh^n$\"+J4.w?F_gl7$F]_n$\"+w]7.@F_gl7$Fb_n$\"+F2%R6#F_gl7$Fg_n$\" +v)=h/#F_gl7$F\\`n$\"+ocL*)>F_gl7$Fa`n$\"+]!4P)>F_gl7$Ff`n$\"+#4a!))>F _gl7$F[an$\"++xu3?F_gl7$F`an$\"+w8%p-#F_gl7$Fean$\"+s8.$*>F_gl7$Fjan$ \"+Uzyf>F_gl7$F_bn$\"+2v$*G>F_gl7$Fdbn$\"+Iln/>F_gl7$Fibn$\"+8+:**=F_g l7$F^cn$\"+JhU->F_gl7$Fccn$\"+b%f*>>F_gl7$Fhcn$\"+b\\;O>F_gl7$F]dn$\"+ \"Hi`(=F_gl7$Fbdn$\"+dK/A=F_gl7$Fgdn$\"+-$=Q\"=F_gl7$F\\en$\"+y<\\8=F_ gl7$Faen$\"+L8fE=F_gl7$Ffen$\"+(4-^%=F_gl7$F[fn$\"+IQn7=F_gl7$F`fn$\"+ \"z54y\"F_gl7$Fefn$\"+p!R;F_gl7$Ff jn$\"+#*y'Rh\"F_gl7$F[[o$\"+(Q\"*4d\"F_gl7$F`[o$\"+^g)y_\"F_gl7$Fe[o$ \"+O;X#[\"F_gl7$Fj[o$\"+N0Rt9F_gl7$F_\\o$\"+kafn9F_gl7$Fd\\o$\"+@BMn9F _gl7$Fi\\o$\"+kp/w9F_gl7$F^]o$\"+CyA&[\"F_gl7$Fc]o$\"+*)[j\"[\"F_gl7$F h]o$\"+-y]v9F_gl7$F]^o$\"+pgSp9F_gl7$Fb^o$\"+U*ysX\"F_gl7$Fg^o$\"+C#f_ W\"F_gl7$F\\_o$\"+^%\\*)Q\"F_gl7$Fa_o$\"+OA))>8F_gl7$Ff_o$\"+\"eNSG\"F _gl7$F[`o$\"+]pIH7F_gl7$F``o$\"+=I>o6F_gl7$Fe`o$\"+)f)fQ6F_gl7$Fj`o$\" +\"[2.3\"F_gl7$F_ao$\"+boVL5F_gl7$Fdao$\"*,hp***F_gl7$Fiao$\"*M*[k%*F_ gl7$F^bo$\"+Zq+L!*F[o7$Fcbo$\"+vg_l()F[o7$Fhbo$\"+!)*GbD)F[o7$F]co$\"+ >Q'[\"zF[o7$Fbco$\"+>BR1wF[o7$Fgco$\"+$GW()>(F[o7$F\\do$\"+D[))*)oF[o7 $Fado$\"+)[mTj'F[o7$Ffdo$\"+r@,RiF[o7$F[eo$\"+QyizfF[o7$F`eo$\"+X%3gw& F[o7$Feeo$\"+\"G(z-aF[o7$Fjeo$\"+%)GV\"=&F[o7$F_fo$\"+,2jj\\F[o7$Fdfo$ \"+)3HUn%F[o7$Fifo$\"+%*[bnWF[o7$F^go$\"+$*p6(G%F[o7$Fcgo$\"+oFAASF[o7 $Fhgo$\"+E/sTQF[o7$F]ho$\"+:ab\"o$F[o7$Fbho$\"+bjhlMF[o7$Fgho$\"+x7t*H $F[o7$F\\io$\"+dm*H:$F[o7$Faio$\"+$R+c'HF[o7$Ffio$\"+)\\6H$GF[o7$F[jo$ \"+d&e`p#F[o7$F`jo$\"+_!z2b#F[o7$Fejo$\"+u!z1W#F[o7$Fjjo$\"+'49FI#F[o7 $F_[p$\"+N1F(=#F[o7$Fd[p$\"+&pU+3#F[o7$Fi[p$\"+6>An>F[o7$F^\\p$\"+'pv, (=F[o7$Fc\\p$\"+\")>%Hy\"F[o7$Fh\\p$\"+HH0y;F[o7$F]]p$\",&oKI(f\"F-7$F b]p$\",`01c_\"F-7$Fg]p$\",=P;iV\"F-7$F\\^p$\",2iE9O\"F-7$Fa^p$\",!ey_2 8F-7$Ff^p$\",-%*oXA\"F-7$F[_p$\",hKOj;\"F-7$F`_p$\",$)RRd6\"F-7$Fe_p$ \",dT3$[5F-7$Fj_p$\"+F#p\\***F-7$F_`p$\"+4[*[^*F-7$Fd`p$\"+GSyW*)F-7$F i`p$\"+&z(\\b&)F-7$F^ap$\"+*G;%o\")F-7$Fcap$\"+aBq]wF-7$Fhap$\"+_QWXtF -7$F]bp$\"+Snv6qF--Fcbp6&FebpF($\"#DF_bp$\"\"\"F)-F\\cp6#%Pscheme~with ~a~relatively~large~stability~regionG-F$6%7b\\lF'F*7$F2$\"'#4q)F07$F7$ \"(/w9%F07$F<$\")\"*o&[\"F07$FA$\")uNpVF07$FF$\"*6/B6\"F07$FK$\"*(*=2H #F07$FP$\"*tg)3WF07$FU$\"*bqF-)F07$FZ$\"+[ka#R\"F07$Fin$\"+P5\\Q:F07$F _o$\"+\"4?c`\"F07$Fdo$\"+,XvK:F07$Fio$\"+jU*)H:F07$F^p$\"+)*)*=C:F07$F cp$\"+?x]=:F07$Fhp$\"+O(Gs]\"F07$F]q$\"+`i>'\\\"F07$Fbq$\"+bC\\w9F07$F gq$\"+=e1'\\\"F07$F\\r$\"+>c$[b\"F07$Far$\"+rEA(o\"F07$Ffr$\"+5J;[>F07 $F[s$\"+?+Z@CF07$F`s$\"+gs^(e#F07$Fes$\"+t&fpx#F07$Fjs$\"+^G\")zFF07$F _t$\"+&=TRx#F07$Fdt$\"+q]BiFF07$Fit$\"+?*y0v#F07$F^u$\"+X`VFFF07$Fcu$ \"+Gcl#F07$Fgv$\" +t^7mEF07$F\\w$\"+*H@ir#F07$Faw$\"+?u`RGF07$Ffw$\"+gRe*3$F07$F[x$\"+&H 4va$F07$F`x$\"+mkO2PF07$Fex$\"+$Rxfx$F07$Fjx$\"+X%R!oPF07$F_y$\"+&==,w $F07$Fdy$\"+qcKWPF07$Fiy$\"+I/gGPF07$F^z$\"+7dP(p$F07$Fcz$\"+))\\kmOF0 7$Fhz$\"+c3GROF07$F][l$\"+'4yZh$F07$Fb[l$\"+_P+(f$F07$Fg[l$\"+'y^Mf$F0 7$F\\\\l$\"+#H80p$F07$Fa\\l$\"+b$*4PTF07$Ff\\l$\"+gw*GO%F07$F[]l$\"+HH mbXF07$F`]l$\"+pIHPXF07$Fe]l$\"+Xt**=XF07$Fj]l$\"+m;j#[%F07$F_^l$\"+!R ?mW%F07$Fd^l$\"+^W\\xVF07$Fi^l$\"+#Q4tK%F07$F^_l$\"+T-<0ZF07$Fc_l$\"+K 'RM.&F07$Fh_l$\"+/d8_\\F07$F]`l$\"+,:&*))[F07$Fb`l$\"+*Hs0)[F07$Fg`l$ \"+d-F3\\F07$F\\al$\"+ay_+]F07$Faal$\"+#H!o,_F07$Ffal$\"+[k/o`F07$F[bl $\"+-(z>f&F07$F`bl$\"+rG[ybF07$Febl$\"+aJjabF07$Fjbl$\"+]DC2bF07$F_cl$ \"+P'4.Y&F07$Fdcl$\"+1blp`F07$Ficl$\"+)3m%*H&F07$F^dl$\"+Xj(R_&F07$Fcd l$\"+&4T\\z&F07$Fhdl$\"+b9/)p&F07$F]el$\"+Y`z6cF07$Fbel$\"+1R,$e&F07$F gel$\"++7xxbF07$F\\fl$\"+.,I;cF07$Fafl$\"+aa0JdF07$Fffl$\"+u@\"H(fF07$ F[gl$\",1e2G3'F_gl7$Fagl$\",EuK6.'F_gl7$Ffgl$\",%\\Y\"*zfF_gl7$F[hl$\" ,gXS(zeF_gl7$F`hl$\",\\f()=z&F_gl7$Fehl$\",#=(>g!fF_gl7$Fjhl$\",-M(p(3 'F_gl7$F_il$\",m]j!)*fF_gl7$Fdil$\",#ei1:fF_gl7$Fiil$\",:mC7)eF_gl7$F^ jl$\",*3H'*eeF_gl7$Fcjl$\",%*[%zaeF_gl7$Fhjl$\",E9Bs&eF_gl7$F][m$\",g, a\"oeF_gl7$Fb[m$\",^/9**)eF_gl7$Fg[m$\",U`8G+'F_gl7$F\\\\m$\",rx;gC'F_ gl7$Fa\\m$\",R'RF/iF_gl7$Ff\\m$\",R))Q%\\hF_gl7$F[]m$\",zK!fTgF_gl7$F` ]m$\",ax8:%fF_gl7$Fe]m$\",t?*>2fF_gl7$Fj]m$\",He(Q%)eF_gl7$F_^m$\",grq )zeF_gl7$Fd^m$\",?vj=)eF_gl7$Fi^m$\",ZBJA*eF_gl7$F^_m$\",9XZK\"fF_gl7$ Fc_m$\",*)pi()*fF_gl7$Fh_m$\",R;%RnhF_gl7$F]`m$\",h0upA'F_gl7$Fb`m$\", `8&p;iF_gl7$Fg`m$\",*=u,0iF_gl7$F\\am$\",%R;O$>'F_gl7$Faam$\",Jv:,<'F_ gl7$Ffam$\",\")Rdp9'F_gl7$F[bm$\",1(\\jQfF_gl7$F`bm$\",>+L=(eF_gl7$Feb m$\",/af!\\fF_gl7$Fjbm$\",0T%Q2hF_gl7$F_cm$\",NKOB8'F_gl7$Fdcm$\",2/Q' 3hF_gl7$Ficm$\",(*QJ]3'F_gl7$F^dm$\",FP;:1'F_gl7$Fcdm$\",;=^%ofF_gl7$F hdm$\",OW'QyeF_gl7$F]em$\",?W%yKeF_gl7$Fbem$\",nkhSz&F_gl7$Fgem$\",i!e 4pdF_gl7$F\\fm$\",#*f'ppdF_gl7$Fafm$\",\"Qm)[\"eF_gl7$Fffm$\",S(yzLfF_ gl7$F[gm$\",uq.(zfF_gl7$F`gm$\",2ceR-'F_gl7$Fegm$\",7xw6,'F_gl7$Fjgm$ \",I5A%)*fF_gl7$F_hm$\",[\"R*H(fF_gl7$Fdhm$\",cvtw%fF_gl7$Fihm$\",!GY% *fdF_gl7$F^im$\",O`+qk&F_gl7$Fcim$\",K$zjmdF_gl7$Fhim$\",\"3g+/eF_gl7$ F]jm$\",:6&*oq&F_gl7$Fbjm$\",Vc#y7cF_gl7$Fgjm$\",+!RiObF_gl7$F\\[n$\", 9F%3(\\&F_gl7$Fa[n$\",9x[Yc&F_gl7$Ff[n$\",N!o5]cF_gl7$F[\\n$\",,2)*[Y& F_gl7$F`\\n$\",3@9qK&F_gl7$Fe\\n$\",.')e'y`F_gl7$Fj\\n$\",ot%F_gl7$Fibn$\",l_HFs%F_gl7$F^cn$\",O6;1t%F_gl7$Fccn$ \",SQTWx%F_gl7$Fhcn$\",W=\"[:[F_gl7$F]dn$\",(3)[Um%F_gl7$Fbdn$\",g1C7` %F_gl7$Fgdn$\",uwQ/^%F_gl7$F\\en$\",;92$4XF_gl7$Faen$\",g()G>a%F_gl7$F fen$\",kZm')e%F_gl7$F[fn$\",C(*=!3XF_gl7$F`fn$\",r58!HWF_gl7$Fefn$\",' *)oYhVF_gl7$Fjfn$\",4Q?kI%F_gl7$F_gn$\",(3*=6H%F_gl7$Fdgn$\",*[E%>H%F_ gl7$Fign$\",sC#y=VF_gl7$F^hn$\",Q,9@O%F_gl7$Fchn$\",,AGj@%F_gl7$Fhhn$ \",(*4v**3%F_gl7$F]in$\",&*GsZ2%F_gl7$Fbin$\",5b:H2%F_gl7$Fgin$\",2+3B 4%F_gl7$F\\jn$\",!GU^QTF_gl7$Fajn$\",me1_2%F_gl7$Ffjn$\",]*f\"H,%F_gl7 $F[[o$\",d\">G0RF_gl7$F`[o$\",6]X#)z$F_gl7$Fe[o$\",?,*3&o$F_gl7$Fj[o$ \",-(QOiOF_gl7$F_\\o$\",$*Q$oZOF_gl7$Fd\\o$\",Dv\"yYOF_gl7$Fi\\o$\",%H wOoOF_gl7$F^]o$\",3!yP\"p$F_gl7$Fc]o$\",@h%\\#o$F_gl7$Fh]o$\",`=msm$F_ gl7$F]^o$\",Px+@l$F_gl7$Fb^o$\",3bf>i$F_gl7$Fg^o$\",d6%3#f$F_gl7$F\\_o $\",F&H2^MF_gl7$Fa_o$\",7\\_%zKF_gl7$Ff_o$\",0e^&*=$F_gl7$F[`o$\",\"pE [_IF_gl7$F``o$\",(Rwd+HF_gl7$Fe`o$\",p;kf#GF_gl7$Fj`o$\",)eG?!o#F_gl7$ F_ao$\",)>*\\Nc#F_gl7$Fdao$\",$H\"z$yCF_gl7$Fiao$\",J9,cM#F_gl7$F^bo$ \"-d4:tPAF[o7$Fcbo$\"-NCXrp@F[o7$Fhbo$\"-!Q.+E/#F[o7$F]co$\"-zd@Bd>F[o 7$Fbco$\"-4FR))y=F[o7$Fgco$\"-.l\\lxn F[o7$F[jo$\",nPY$pjF[o7$F`jo$\",#>%\\&3gF[o7$Fejo$\",k*Hs1dF[o7$Fjjo$ \",Ew\"4f`F[o7$F_[p$\",XV*3q]F[o7$Fd[p$\",l\"GyvZF[o7$Fi[p$\",@M)>(\\% F[o7$F^\\p$\",Y=]pC%F[o7$Fc\\p$\",h&3z**RF[o7$Fh\\p$\",*4R\"[u$F[o7$F] ]p$\"-&3c@N`$F-7$Fb]p$\"-`b/pALF-7$Fg]p$\"-=k44u#F-7$Ff^p$\"--xq5dDF-7$F[_p$\"-h92_!R#F-7$F`_p$\"-$[ (RcGAF-7$Fe_p$\"-di#>s3#F-7$Fj_p$\"-F2cPQ>F-7$F_`p$\"-4W&=Wy\"F-7$Fd`p $\"-G_))yq;F-7$Fi`p$\"-&>=)*Ra\"F-7$F^ap$\"-*Gv@1T\"F-7$Fcap$\"-a?=V:8 F-7$Fhap$\"-_>\" eF07$F_o$!(t5!eF07$Fdo$!([-z&F07$Fio$!(V%zdF07$F^p$!($*yv&F07$Fcp$!(Dk t&F07$Fhp$!(YPp&F07$F]q$!(\"e^cF07$Fbq$!(kmc&F07$Fgq$!(,?e&F07$F\\r$!( ')=u&F07$Far$!(Ok;'F07$Ffr$!(yd5(F07$F[s$!(wD)*)F07$F`s$!()pv'*F07$Fes $!)dM[5F07$Fjs$!)c()\\5F07$F_t$!)!ew/\"F07$Fdt$!)pBV5F07$Fit$!)W$)Q5F0 7$F^u$!)r3I5F07$Fcu$!)!Q9-\"F07$Fhu$!)%=I,\"F07$F]v$!)iK05F07$Fbv$!(+! )***F07$Fgv$!(#4****F07$F\\w$!)=*G,\"F07$Faw$!)\"RE0\"F07$Ffw$!)\\!R9 \"F07$F[x$!)/@G8F07$F`x$!)X\"fR\"F07$Fex$!)\\kD9F07$Fjx$!)xkA9F07$F_y$ !)ql>9F07$Fdy$!)Xp89F07$Fiy$!)pv29F07$F^z$!)#ffR\"F07$Fcz$!),H%Q\"F07$ F][l$!)Duj8F07$Fg[l$!)N4^8F07$F\\\\l$!)dHx8F07$Fa\\l$!)m%Ra\"F07$Fi^l$ !)'='H;F07$F^_l$!)dbdF07$Ffa l$!)(>E,#F07$F[bl$!)KK2@F07$F`bl$!)Jw-@F07$Febl$!)Ix$4#F07$Fjbl$!)&3f2 #F07$F_cl$!)??e?F07$Fdcl$!)\"yN-#F07$Ficl$!)*)3%*>F07$F^dl$!)$>k1#F07$ Fcdl$!)$pD=#F07$Fhdl$!)()*e9#F07$F]el$!))z<6#F07$Fbel$!)fk)4#F07$Fgel$ !)c,$4#F07$F\\fl$!):6.@F07$Fafl$!)#HP9#F07$Fffl$!)CYSAF07$F[gl$!*%zt)G #F_gl7$Fagl$!*M%HpAF_gl7$Ffgl$!*Y<+D#F_gl7$F[hl$!*!*)47AF_gl7$F`hl$!*0 *=x@F_gl7$Fehl$!*_$f3AF_gl7$Fjhl$!*s\")yG#F_gl7$F_il$!*qaSD#F_gl7$Fdil $!*m>=A#F_gl7$Fiil$!*&\\r2AF_gl7$F^jl$!*p*3(>#F_gl7$Fhjl$!*iYL>#F_gl7$ Fb[m$!*Y%Q-AF_gl7$Fg[m$!*e/NC#F_gl7$F\\\\m$!*)z[VBF_gl7$Fa\\m$!*kb%GBF _gl7$Ff\\m$!*BvyI#F_gl7$F[]m$!*!oInAF_gl7$F`]m$!*gV'GAF_gl7$Fe]m$!*4(R 9AF_gl7$Fj]m$!*fOO?#F_gl7$Fd^m$!*^P(*>#F_gl7$F^_m$!*II&3AF_gl7$Fc_m$!* tg&RAF_gl7$Fh_m$!*&=y2BF_gl7$F]`m$!*])3LBF_gl7$Fb`m$!*=-$HBF_gl7$Fg`m$ !*sE\\K#F_gl7$F\\am$!*Yf0K#F_gl7$Faam$!*d\\=J#F_gl7$Ffam$!*PsJI#F_gl7$ F[bm$!*2-XA#F_gl7$F`bm$!*^D,>#F_gl7$Febm$!*pOz@#F_gl7$Fjbm$!*9i>G#F_gl 7$F_cm$!*,-LH#F_gl7$Fdcm$!*hRWG#F_gl7$Ficm$!*W6cF#F_gl7$F^dm$!*RB#F_gl7$Fhdm$!*Hdz>#F_gl7$Fbem$!*h7V;#F_gl7$F\\fm$!*$)) z\\@F_gl7$Fafm$!*F7U;#F_gl7$Fffm$!*=z,@#F_gl7$F[gm$!*Go!HAF_gl7$F`gm$! *.ixC#F_gl7$Fegm$!*r#*HC#F_gl7$Fjgm$!*XL#QAF_gl7$F_hm$!*CX(GAF_gl7$Fdh m$!*G(H>AF_gl7$Fihm$!*!*)))[@F_gl7$F^im$!*Y7-5#F_gl7$Fcim$!*7tA9#F_gl7 $Fhim$!*#)3,;#F_gl7$F]jm$!*c]R7#F_gl7$Fbjm$!*x9')3#F_gl7$Fgjm$!*48(e?F _gl7$F\\[n$!*?K*R?F_gl7$Fa[n$!*Ki71#F_gl7$Ff[n$!*:ul4#F_gl7$F[\\n$!*Q& pF?F_gl7$F`\\n$!*Zr<(>F_gl7$Fe\\n$!*eBj)>F_gl7$Fj\\n$!*O+J,#F_gl7$F_]n $!*LN%y>F_gl7$Fd]n$!*G#\\W>F_gl7$Fi]n$!*/Rm\">F_gl7$F^^n$!*)pv&*=F_gl7 $Fc^n$!*98F*=F_gl7$Fh^n$!*)p2+>F_gl7$F]_n$!*L4^#>F_gl7$Fb_n$!*L]n$>F_g l7$Fg_n$!*!)RX(=F_gl7$F\\`n$!*aN#>=F_gl7$Fa`n$!*k>;\"=F_gl7$Ff`n$!*i') G\"=F_gl7$F[an$!*HO0$=F_gl7$F`an$!*<>'[=F_gl7$Fean$!*B\"p<=F_gl7$Fjan$ !*%RI(y\"F_gl7$F_bn$!*:F&e#F_gl$!*6<3Z\"F_gl7$F \\jn$!*&pDz9F_gl7$Fajn$!*HGmX\"F_gl7$Ffjn$!*SZVV\"F_gl7$F[[o$!*$))*RQ \"F_gl7$Fe[o$!*,ZWI\"F_gl7$Fg^o$!*)z.g7F_gl7$F\\_o$!*(p***>\"F_gl7$Fa_ o$!*v$HO6F_gl7$Ff_o$!*3V=4\"F_gl7$F[`o$!*RuO.\"F_gl7$F``o$!)bZu(*F_gl7 $Fe`o$!)'43P*F_gl7$Fj`o$!)1>(y)F_gl7$F_ao$!)!*[L$)F_gl7$Fdao$!)dq#)yF_ gl7$Fiao$!)#=iR(F_gl7$F^bo$!*BZM$pF[o7$Fcbo$!*0K6`'F[o7$Fhbo$!*]HP2'F[ o7$F]co$!*\"G)>p&F[o7$Fbco$!*6(H`_F[o7$Fgco$!*nzr#\\F[o7$F\\do$!*bEc`% F[o7$Fado$!*7!=ITF[o7$Ffdo$!*\\g#RQF[o7$F[eo$!*is>[$F[o7$F`eo$!*:=-5$F [o7$Feeo$!**yyqGF[o7$Fjeo$!*'oyEDF[o7$Fdhs$!*)HF![#F[o7$Fihs$!*?^rT#F[ o7$F^is$!*()fwH#F[o7$F_fo$!*\\7Z9#F[o7$Ffis$!*sx\"z?F[o7$Fdfo$!*7')z*> F[o7$F^js$!*gy)**=F[o7$Fcjs$!*+\"y5\"F[o7$$\"53333LpzivUF_gl$!*i@t: \"F[o7$Fj\\t$!*[\\E5\"F[o7$$\"5$GGGylikEH%F_gl$!*rAH,\"F[o7$F_]t$!)H:9 *)F[o7$Fd]t$!)xaj()F[o7$Fhgo$!)9F8')F[o7$F\\^t$!)zX_%)F[o7$Fa^t$!)7Bl \")F[o7$$\"5vuuC(ys%[uVF_gl$!)7r))yF[o7$Ff^t$!)h@MuF[o7$$\"5xwwE96h>>>pcRF[o7$F``t$!)tNv9F[o7$Fgho$!)VzZ9F[o7$$\"532222A2M \\XF_gl$!)NC,9F[o7$$\"5gffff\\eSlXF_gl$!)[*oA\"F[o7$$\"5'eeeeLTQMd%F_g l$!(_$*)**F[o7$$\"577777x4Z\"e%F_gl$!(r1w&F[o7$$\"5QQQQ)3a.&*e%F_gl$\" (9F[o7$F[at$\")g%[)=F[o7$Faio$\")tto>F[o7$$\"5on nn#Rv3;n%F_gl$\")A*>:#F[o7$Fcat$\")#Q7`#F[o7$$\"5GFFF-OsJ)o%F_gl$\")<: XKF[o7$Fhat$\")@l+XF[o7$F]bt$\")D,X^F[o7$Ffio$\")Qhg]F[o7$F[jo$\")Z@9n F[o7$$\"5444M!>q9cz%F_gl$\")8:'Q(F[o7$$\"5EEEw)od2(*z%F_gl$\")L\"oD)F[ o7$$\"5WVV=(=X+Q![F_gl$\")N='H)F[o7$Fhbt$\")wHi#)F[o7$$\"5'\\\\\\Co2zg \"[F_gl$\")2&\\>)F[o7$F]ct$\")*G$G\")F[o7$Fbct$\")sI.!)F[o7$F`jo$\")U. UzF[o7$Fjct$\")3ob$*F[o7$Fejo$\"*C8Y5\"F[o7$$\"598888`WPR\\F_gl$\"*=jh 3\"F[o7$$\"5a````jNvc\\F_gl$\"*pPW2\"F[o7$$\"5utttt=JWl\\F_gl$\"*/f!y5 F[o7$$\"5%RRRRRnKT(\\F_gl$\"*jJr4\"F[o7$$\"599999HA#G)\\F_gl$\"*J%fW6F [o7$Fjjo$\"*E$)3C\"F[o7$$\"57666OieM&*\\F_gl$\"*M)R18F[o7$$\"5)yyyy.%* z\"**\\F_gl$\"*#zl!R\"F[o7$$\"5lkkkR=S,.]F_gl$\"**yB29F[o7$$\"5STTTT'4 [o+&F_gl$\"*!G&=S\"F[o7$$\"5&\\\\\\\\CD;X,&F_gl$\"*$\\9\"R\"F[o7$$\"5] [[[[3W=A]F_gl$\"*kG0Q\"F[o7$$\"5bbbbb?2_P]F_gl$\"*hS*f8F[o7$F_[p$\"*Xf NM\"F[o7$$\"5b````toC)3&F_gl$\"*h@7Y\"F[o7$Fd[p$\"*lH8k\"F[o7$$\"5XVVV $f,0'R^F_gl$\"*:gfh\"F[o7$$\"5XUUUU&F_gl$\"*Qt y#=F[o7$$\"5IJJJ1?(4L?&F_gl$\"*(y\"3$>F[o7$$\"5vwwE*)>'47@&F_gl$\"*uCc \">F[o7$$\"5?AAAs>&4\">_F_gl$\"*7b0!>F[o7$$\"55888Q>$4\\B&F_gl$\"*$Q.r =F[o7$F^\\p$\"*cu^%=F[o7$Faet$\"*P(eQ=F[o7$Ffet$\"*O&\\9>F[o7$$\"5533e &*pP=$H&F_gl$\"*]cr-#F[o7$F[ft$\"*2K&y@F[o7$$\"5qpp>KIO<5`F_gl$\"*4/,; #F[o7$Fc\\p$\"*JN=9#F[o7$Fcft$\"*6Of5#F[o7$Fhft$\"*>m]2#F[o7$F]gt$\"*U L42#F[o7$Fh\\p$\"*\\qh;#F[o7$$\"5&eee$[1Lm%R&F_gl$\"*tshG#F[o7$$\"5!)z zz/,PU-aF_gl$\"*Oe.T#F[o7$$\"5vttBh&4%=5aF_gl$\"+cbs\"R#F-7$$\"5qnnn!\\&F_gl$\"+]%)> GCF-7$Fbht$\"+3.$ff#F-7$$\"5DCCuhGWn1bF_gl$\"+f4QAEF-7$Fb]p$\"+`!f3g#F -7$$\"5!3333L#3+JbF_gl$\"+/d[fDF-7$Fjht$\"+lm!4_#F-7$$\"5gccc1C7KF-7$Fh\\u$\"+0]InJF-7$F[_p$\"+h^gJ^KF-7$F`_p$\"+$oYcT$F-7$Fb^u$ \"+;$Q!3LF-7$Fe_p$\"+d$GwD$F-7$Fj^u$\"+2!4;Q$F-7$F__u$\"+\\*oUg$F-7$Fd _u$\"++uTXNF-7$Fj_p$\"+Fu<)[$F-7$F\\`u$\"+kaKOMF-7$Fa`u$\"+o^*)GMF-7$F f`u$\"+6\\$3e$F-7$F_`p$\"+411nPF-7$F^au$\"+=YM\\OF-7$Fd`p$\"+GUb*e$F-7 $Ffau$\"+5$p8q$F-7$F[bu$\"+AYb]RF-7$F`bu$\"+l&R'))QF-7$Fi`p$\"+&H$3GQF -7$Fhbu$\"+')fgpPF-7$F]cu$\"+g'))Gu$F-7$$\"5IDDDvsN0yiF_gl$\"+y,,kPF-7 $Fbcu$\"+S)RJ$QF-7$$\"5IGGGyS<%\\H'F_gl$\"+VpG#)RF-7$F^ap$\"+*ol[6%F-7 $Fjcu$\"+nQ,zRF-7$Fcap$\"+aA5&*QF-7$$\"5qmm;a3SNyjF_gl$\"+\"4*\\;RF-7$ Fbdu$\"+s9qxRF-7$$\"5qoo=JW\"zMR'F_gl$\"+Fh*=5%F-7$Fgdu$\"+#H#3!G%F-7$ F\\eu$\"+?U$e@%F-7$Fhap$\"+_yq_TF-7$F]bp$\"+S<%[V%F--Fcbp6&FebpFfbpFhe uF(-F\\cp6#%Hscheme~with~c[5]=c[6]=3/4~and~b[5]=b[6]G-%%FONTG6$%*HELVE TICAG\"\"*-%+AXESLABELSG6$Q\"x6\"Q!F\\_al-%&TITLEG6#%Uerror~curves~for ~7~stage~order~6~Runge-Kutta~methodsG-%%VIEWG6$;F(F]bp%(DEFAULTG" 1 2 0 1 10 0 2 9 1 4 2 1.000000 45.000000 45.000000 0 1 "Butcher's scheme \+ A" "scheme with simple nodes" "scheme with a relatively large stabilit y region" "Butcher's scheme B with c[5]=c[6]=1/2 and b[5]=b[6]" "schem e with c[5]=c[6]=3/4 and b[5]=b[6]" }}}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 519 "evalf[25](plot(['bn_RK6_1 '(x)-b(x),'bn_RK6_2'(x)-b(x),'bn_RK6_3'(x)-b(x),'bn_RK6_4'(x)-b(x),\n' bn_RK6_5'(x)-b(x)],x=0.65..5,font=[HELVETICA,9],\ncolor=[COLOR(RGB,.95 ,.2,0),COLOR(RGB,.45,0,.95),COLOR(RGB,0,.25,1),COLOR(RGB,0,.75,.2),COL OR(RGB,.95,.45,0)],\nlegend=[`Butcher's scheme A`,`scheme with simple \+ nodes`,`scheme with a relatively large stability region`,`Butcher's sc heme B with c[5]=c[6]=1/2 and b[5]=b[6]`,`scheme with c[5]=c[6]=3/4 an d b[5]=b[6]`],title=`error curves for 7 stage order 6 Runge-Kutta meth ods`));" }}{PARA 13 "" 1 "" {GLPLOT2D 880 564 564 {PLOTDATA 2 "6+-%'CU RVESG6%7]w7$$\"#l!\"#$\"1G8*Q3=([r!#F7$$\":+++++]i:0Wqt'!#D$\"1PH+^4'e /%F-7$$\":++++++DJ5)3upF1$\"1zs&3,+ET\"F-7$$\":+++++](oa@86sF1$!1]?') \\LTG:F-7$$\":++++++]i?w\"[uF1$!16'3'Q%ew<$F-7$$\":+++++vV)HxngyF1$!2m c%y5)4\")*e!#G7$$\":+++++]PMDzJF)F1$!2TF)=X,;$*zFH7$$\":+++++D1*e.3P() F1$!3_]hdBe*\\,\"FH7$$\":++++++vVY\")4?*F1$!3j.oiXv\"p?\"FH7$$\":+++++ ]Pf`pWV*F1$!3WSB.tSO;7FH7$$\":+++++++vgdzm*F1$!3sZ)=]X.'G7FH7$$\":++++ +D\"GV;q%y*F1$!3UO=nH!RPE\"FH7$$\":+++++]i!zcW,**F1$!3\"f(fS*>:IO\"FH7 $$\":++++]P%[r*==+\"!#C$!4i#p?HC2Ce8!#H7$$\":+++++]i]P$\\85Fio$!4>pb*z <:]_8F\\p7$$\":+++++voCrWk5\"Fio$!44'p:k,7\\:\"Fio $!4e1yk$QBi)\\\"F\\p7$$\":+++]iS;Vx7w:\"Fio$!4(37Y)))[p5[\"F\\p7$$\":+ +++]7.-$eIg6Fio$!4v[TM&fwgw:F\\p7$$\":+++]P%)*3')))*H;\"Fio$!4-z<0W^\\ Z`\"F\\p7$$\":++++Dcw>%>pl6Fio$!4Fftap/\")p\\\"F\\p7$$\":+++]7Gjy*\\Qo 6Fio$!4CO%4Dz>H+:F\\p7$$\":++++++]P0y5<\"Fio$!4&[g#=a*\\Ns:F\\p7$$\":+ +++](oHx-&==\"Fio$!4^dlc,\"Fio$!4HmPozTlDc \"F\\p7$$\":++++++vV^wP?\"Fio$!4$QoW+I0G^:F\\p7$$\":+++++D1z_I\\@\"Fio $!4\"4i\"*f#[q/a\"F\\p7$$\":+++++]P9a%3E7Fio$!4IU@w7,-A`\"F\\p7$$\":++ +++vo\\bQsB\"Fio$!5tg#QX[Jn:`\"!#I7$$\":++++Dc^$eq-S7Fio$!5'p``31;hMj \"Fav7$$\":++++]PMwa\"Fav7$$\":+++++++&oDR[7Fio$!5g_mv4#=p#\\:Fav7$$\":++++]i l_dpRD\"Fio$!5Y$fxc1S@_d\"Fav7$$\":+++++DJ?eY&f7Fio$!5*\\'=X5cyR0;Fav7 $$\":++++D19a3NBE\"Fio$!5t%HOFL4@`g\"Fav7$$\":++++](oz)eB^E\"Fio$!5e`, PCQ\\ki:Fav7$$\":++++voz@47zE\"Fio$!5F9*Gb<#fG\\:Fav7$$\":+++++]ibf+2F \"Fio$!5T,p:U)yvej\"Fav7$$\":++++DJX*)4*[t7Fio$!5s!H0PfX*)4f\"Fav7$$\" :++++]7GBgxiF\"Fio$!5Kdwf;%)3F`:Fav7$$\":++++v$4r0h1z7Fio$!5Q9zj/c%)H& e\"Fav7$$\":+++++v$44Y&=G\"Fio$!5)z%oz2PF#3i\"Fav7$$\":+++++vVe5!R$H\" Fio$!5,y@4x#pH&*f\"Fav7$$\":+++++v$f-c#\\I\"Fio$!5_i9C$pcU\"z:Fav7$$\" :+++++vV$*4hkJ\"Fio$!5*)*yT#Hb#eSc\"Fav7$$\":+++++v$4'f'*zK\"Fio$!5je@ KiU&)[o:Fav7$$\":+++++vVG4K&R8Fio$!5*zAQ*\\fE.G;Fav7$$\":+++++v$f*en5N \"Fio$!5!\\Gv,P'QT];Fav7$$\":+++++vVj3.EO\"Fio$!50vgzouJME;Fav7$$\":++ +++v$4$eQTP\"Fio$!5&GKFKh\\7Eg\"Fav7$$\":++++]Pf`3PcQ\"Fio$!5MM:-oBrs \"e\"Fav7$$\":++++++DweNrR\"Fio$!5(GC\"4a&y\"Qq:Fav7$$\":+++]iS\">8-,+ 9Fio$!5:$z5fCGq,n\"Fav7$$\":++++D\"yvQ[)GS\"Fio$!5n&G!)H\")ooGi\"Fav7$ $\":+++](=UKk%fdS\"Fio$!5w8ob?!)Q;!e\"Fav7$$\":++++]i!*)*3M'39Fio$!5\" 3D)3yGtr)e\"Fav7$$\":+++]7.dar3:T\"Fio$!5.+-wE*)3\"[k\"Fav7$$\":++++vV B5M$Q99Fio$!57i?6.y&)o)f\"Fav7$$\":+++]P%)*emzD<9Fio$!5#=sa*pV*3'p:Fav 7$$\":+++++Dc@fK,U\"Fio$!5gjLsNBevm;Fav7$$\":+++++](o'fHJW\"Fio$!5_GFh @'4?]h\"Fav7$$\":+++++v=7gEhY\"Fio$!6,yn&*QK-='o:!#J7$$\":++++](=#H4PM ^\"Fio$!6SjY$*)\\M8]\"f\"F\\al7$$\":++++++DYeZ2c\"Fio$!6HSz:!ytY\"=h\" F\\al7$$\":+++++vo9p=Cg\"Fio$!6()e%>y*)Qqdh:F\\al7$$\":+++++]7$)z*3W;F io$!6&[c^T5mN43:F\\al7$$\":++++++v8R8zt\"Fio$!7/\\,'3QH,y)*R\"!#K7$$\" :++++++]A\"y,&y\"Fio$!7d'pHG[Yv5@O\"Ffbl7$$\":++++++DJBA@$=Fio$!7faAF# *HvYkQ8Ffbl7$$\":+++++++?:>[&=Fio$!7y7]?.@EzZx7Ffbl7$$\":++++++v32;v(= Fio$!7%e-:0b*\\@))=H\"Ffbl7 $$\":+++++v$f+.>$)=Fio$!7KB]g))o>qyb7Ffbl7$$\":++++]7`aTFg)=Fio$!7d*y# puZ&)*QPA\"Ffbl7$$\":+++++]7.`k)))=Fio$!7k=l9%Gxz:IB\"Ffbl7$$\":++++]( =&H?\"Ffbl7$$\":++++++](*)H@+>F io$!7y^D\"*yYDhYj7Ffbl7$$\":+++++](=\\9c6>Fio$!7fm^vE3;eAJ7Ffbl7$$\":+ +++++D'3*4H#>Fio$!8<.b/:n5:x$*>\"!#L7$$\":+++++v$f:QN0?Fio$!8NQ=\\;#>= OWU5F]gl7$$\":++++++D^f(Q.@Fio$!7pXB`9tS$*Rd()F]gl7$$\":+++++++b1Nk=#F io$!8J+YPEPiPiV\"p!#M7$$\":+++++vVj,RIG#Fio$!89&45.Fxs$G-)[F]hl7$$\":+ +++++]<))\\&oBFio$!8;$=5/ydrXW2GF]hl7$$\":+++++vVQ&)oBY#Fio$!8$G-%\\dk vL,.>%!#N7$$\":+++++DJ:q1*>F]il7$$\":+++++]PMP@\\ k#Fio$\"::r(>d8fr+CZNV!#O7$$\":+++++v=2a@0t#Fio$\":6y/(44-Y1hvtqFhil7$ $\":+++++]7`)G&G#GFio$\":p\\YcL/&Qlj<5%*Fhil7$$\":+++++v$4.#f(=HFio$\" :?$z&>(*[lXo8y<\"F]il7$$\":+++++DcY$fC-IFio$\":o**=:e_ae]%yS9F]il7$$\" :+++++]P>*QT#4$Fio$\"::!zj=#[Mk_xYn\"F]il7$$\":++++++]Z@mb=$Fio$\":oSK '**zGJb^pW=F]il7$$\":+++++](=*e(pwKFio$\":8:lEmZL,2St.#F]il7$$\":+++++ vV)z=([O$Fio$\":nT;-afUcslKD#F]il7$$\":+++++]7jRuFY$Fio$\":fLHLy.ar`^O \\#F]il7$$\":++++v=#*phqPZ$Fio$\":58T(Hz`GY(4q[#F]il7$$\":++++](=nPow% [$Fio$\":E(z^`/)e#F]il7$$\":+++ +voHM>'4!e$Fio$\":zx*GR%)psc7=kFF]il7$$\":+++]il(f'RJIe$Fio$\":tPrb\"G %4:w!H%o#F]il7$$\":++++]il(*fmfe$Fio$\":uJ354usu0BSh#F]il7$$\":+++]PfL H!=!*)e$Fio$\":tIt)f'>E#f(4Ul#F]il7$$\":++++Dc,h+P=f$Fio$\":ww[w%esBf: j.1ES\"z#F]il7$$\":++++]7`/&)HHj$ Fio$\":@OeSb-9'48dcFF]il7$$\":++++++DJmqYk$Fio$\":Ek^dGc9svjIs#F]il7$$ \":+++]PfLWe4`l$Fio$\":1k*p?Xy+k]3?FF]il7$$\":++++v=Ud][fm$Fio$\":)4/S Cat&z3p)=FF]il7$$\":+++]7y]qU(ewOFio$\":bE=CYmtjhW5s#F]il7$$\":++++]Pf $[jA(o$Fio$\":B'[2(Qh^$\\x6HFF]il7$$\":++++Dcw4>/&3PFio$\":I%p'pe#)oFi \\$zFF]il7$$\":+++++v$fL?yHPFio$\":\"QNf3@k'zDx#o$)3?2 J#GF]il7$$\":++]i!*yedU9#zPFio$\":#\\+#pm2N**)QE'*GF]il7$$\":+++DJ?LN@ A@y$Fio$\":fDZa8tU]+;+%HF]il7$$\":++](=1x62KGF]il7$$\":+++v$f3jkLv$z$Fio $\":T(4q'R:a/U)f+HF]il7$$\":++++voz,%*o&*z$Fio$\":KK&Q>]P<$>*\\_HF]il7 $$\":++]i:5azswC!QFio$\":!eVf.^Z*fFm^%HF]il7$$\":+++Dc^Gd^%Q0QFio$\":R p;&>C/w1=ekGF]il7$$\":++](oHH].BH3QFio$\":uGl6c(yd%G()z&GF]il7$$\":+++ ]PMx74+7\"QFio$\":N$zF.\"H&oopJ!*HF]il7$$\":++]7yv^!zy59QFio$\":7$pbA% oHQ^$>0HF]il7$$\":+++v=Kf%GF]il7$$\":++]Pfe+ YXB*>QFio$\":zQ!Q&\\p2f/w+-$F]il7$$\":++++++vBCJG#QFio$\":t)**3nPGuZ%QFio$\":%z3&p,e\\(*GQv!HF]il7$$\":++++]Pf$psg mQFio$\":`88^i/5(p_eyGF]il7$$\":++]P%)*e/rKMpQFio$\":_e2F:qW5fI!pHF]il 7$$\":+++v=UKFFz?(QFio$\":$Q'y&Hy43a([++$F]il7$$\":++]7`%*=WF:[(QFio$ \":`Z!z.!onZaQ3#HF]il7$$\":+++](oa5w7bxQFio$\":vNs;RRNu\\/;)GF]il7$$\" :++](=#*>zxsG!)QFio$\":)[QqlKB'*Q11fIF]il7$$\":+++Dc^y%zK-$)QFio$\":uu &)p*efM2pcwHF]il7$$\":++]i!R];\"Gfd)QFio$\":'H8Ac9^b**Ro\")*pHF]il7$$\":+++++v =(>/!>+%Fio$\":15*R9`%QUo%pOIF]il7$$\":++++DJ&p)4pU-%Fio$\":Uv1N&H4H$y t`'HF]il7$$\":++++](=nxxjYSFio$\":(=UC%\\OV9()f#4HF]il7$$\":++++vV[mX1 !pSFio$\":Eh,?Qw;i\"4$3'HF]il7$$\":++++++Dc8v84%Fio$\":g&yU9/d,#ypb.$F ]il7$$\":+++]PfeD5%3.TFio$\":B7%*G#4yy9$p.)HF]il7$$\":++++v=#\\pIz9TFi o$\":HwCb[#o<8`GFHF]il7$$\":++]PfeD7J?x6%Fio$\":BPu>az0tXD!))GF]il7$$ \":+++vV)*eHbZ17%Fio$\":,PI*[mNuBGrWIF]il7$$\":++]7GQ#p%zuN7%Fio$\":Y \"o/Z$4!\\`+4dHF]il7$$\":+++]7yDk.-l7%Fio$\":$)oma4T&3\"Gsb)GF]il7$$\" :++](oz\"f\"y#H%HTFio$\":]wx,!)oxA%4SyHF]il7$$\":+++D\"yD*)>lNKTFio$\" :[eTUK_6Knk!))HF]il7$$\":++]il(fihPGNTFio$\":TY=/3^%R+=60HF]il7$$\":++ ++]PfL+6#QTFio$\":AQ'p1N(zu5'y*)GF]il7$$\":+++v=*\\TFio$\":LKOw]c'y8ZIMIF]il7$$\":++](=nj-7 s%G:%Fio$\":6/**>F<1#QP=iHF]il7$$\":+++DcwfPXud:%Fio$\":O!G-h?&>7n>F)G F]il7$$\":++]iS;$\\&p,(eTFio$\":LE$4o'Rk)yzh+HF]il7$$\":++++DcEs$*G;;% Fio$\":%40*o&4^')[-8#*HF]il7$$\":+++]i:gT!zLtTFio$\":nDUS(f:s2p%\\$HF] il7$$\":+++++v$4ro/&=%Fio$\":Ch!G&R%e(Qb=0)GF]il7$$\":+++++++b)QEvUFio $\":qtcj66p]*)Haz#F]il7$$\":+++++]iI\"Q_nVFio$\":JOAO^<`!p9&el#F]il7$$ \":+++++vV=y>!fWFio$\":[Rjc*=NCo6cfDF]il7$$\":++++++]Z2&4VXFio$\":a'4. )*Q4c^pUF]il7$$\":+++++vVjrJv\"[Fio$\":0#*oi(*QU' HlvdPO(F-7$$\" :++++](o/opUawF1$\"00$Rzjn7tF-7$FD$\"1MjP6gO$H(FH7$$\":++++]iS;\\Gp1)F 1$\"1vO/O:X0tFH7$FJ$\"1fA5A<(=N(FH7$FO$\"1[zpY&Rwr(FH7$FT$\"1Pc#=*\\.Q #)FH7$FY$\"1czT;A'\\1)FH7$Fhn$\"1G#Gp3y6&zFH7$$\":++++DJXkhVrp*F1$\"1l d-\"RK^O)FH7$$\":++++]i!RD'Hjs*F1$\"1EO<=GB!G)FH7$$\":++++v$fLMc^b(*F1 $\"1HQ5]Ptb!)FH7$F]o$\"1eVqAA) FH7$Fbo$\"14k;$)>z!f)FH7$Fgo$\"2Q2\")=hTy\\)F\\p7$F^p$\"2\"3.JB1q0%)F \\p7$$\":+++]7GQAH7^-\"Fio$\"2'zRB[:?D$)F\\p7$$\":++++D19%47tO5Fio$\"2 \\o!*)H2G%G)F\\p7$$\":++]7y+3(QfjR5Fio$\"2IT())GsHw')F\\p7$$\":+++DJ&> +o1aU5Fio$\"2_E8x?c(G')F\\p7$$\":++]P%)*eH(RXa/\"Fio$\"2)o;Rnl2%R)F\\p 7$$\":+++]P%)*eE,N[5Fio$\"238>\\4_>R)F\\p7$$\":+++vVtx^efT0\"Fio$\"2rE +8B+v`)F\\p7$$\":++++]ilP/p*f5Fio$\"2nCoEn4z#*)F\\p7$$\":++++v=<\"yo?$ 3\"Fio$\"2*yAIrO2]()F\\p7$Fcp$\"2\"Rqr\"yVIf)F\\p7$Fhp$\"2)H$*pI)fih)F \\p7$F]q$\"2)e$y?r3Uo)F\\p7$Fbq$\"2Q>&3hXcF))F\\p7$Fgq$\"2f#HjTLr\"4*F \\p7$F\\r$\"20kG%30v4\"*F\\p7$Far$\"2U$Rn0Q(R())F\\p7$Ffr$\"28ze6$*Qtw )F\\p7$F[s$\"2D^d()*3;@$*F\\p7$F`s$\"2)4ikrGot!*F\\p7$Fes$\"2tSCa*R(* \\))F\\p7$Fjs$\"2wjkO!p6l))F\\p7$F_t$\"2:&f%)o%zDG*F\\p7$Fdt$\"2\\U(>& GcEC*F\\p7$Fit$\"2rLI0\"fX,#*F\\p7$F^u$\"2<;B;=q[7*F\\p7$Fcu$\"24z0X_% )>0*F\\p7$Fhu$\"2qdZH=nY**)F\\p7$F]v$\"3FRPt1eS#)*)Fav7$Fcv$\"3/jWG=*3 Rd*Fav7$Fhv$\"3^*4]/Jb2J*Fav7$F]w$\"3WXIvb@^q!*Fav7$Fbw$\"3SZV&Qq'\\x! *Fav7$Fgw$\"3a1ChCO'eA*Fav7$F\\x$\"3,N@VPgO(R*Fav7$Fax$\"3F0(G^=GfR*Fa v7$Ffx$\"3UY=[*f-g9*Fav7$F[y$\"3t&3#GfZ,m!*Fav7$F`y$\"3f)4&=-I*)o&*Fav 7$Fey$\"3G4Fy#QBjI*Fav7$Fjy$\"3oU.#4V8_3*Fav7$F_z$\"3i&3+C8r$p#*Fav7$F dz$\"3-_TP'zsaZ*Fav7$Fiz$\"3*>#[^Ct3Y$*Fav7$F^[l$\"3[PN%Q'3SA#*Fav7$Fc [l$\"365sM3RyH\"*Fav7$Fh[l$\"3PT)*o(>I2:*Fav7$F]\\l$\"3,s'*Fav7$Fg\\l$\"3&\\#zNhxiw%*Fav7$F\\]l$\"3:xcCSDcN $*Fav7$Fa]l$\"3ml9NPQ<6#*Fav7$Ff]l$\"38dZ-NW;U\"*Fav7$F[^l$\"3&o?hoz2: s*Fav7$F`^l$\"3L9(\\p>yhW*Fav7$Fe^l$\"3C'=%G!H)Q(>*Fav7$Fj^l$\"3>\\Z<\"eK8*Fav7$F^`l$\"3SOE&3G_sp*Fav7$Fc`l$\"3[r-:'4JCR*Fav7$Fh`l$\"4*> A$=!Rt\"*=\"*F\\al7$F^al$\"4gO`n`ro[C*F\\al7$Fcal$\"4rf?NrublN*F\\al7$ Fhal$\"48T0P&eQ_g!*F\\al7$F]bl$\"4:N%=:6&*=Y()F\\al7$Fbbl$\"5'4&)R>,[& z4\")Ffbl7$Fhbl$\"5V..@I'3Wx@Z(Ffbl7 $Fadl$\"5ow\\fh!\\nLE(Ffbl7$Ffdl$\"5V5s5i)**=y2(Ffbl7$F[el$\"5O\"[`lzh v08(Ffbl7$F`el$\"5Zz1***eT9S#GN$pF]gl7$F_gl$\"6lh\"3&eVos\">gF]gl7$Fdgl$\"6JawY9#GV+Z] F]gl7$Figl$\"7p*RDOF5[nd(RF]hl7$F_hl$\"7'[!**oHG^EL#z#F]hl7$Fdhl$\"7%o \")*e>)3yY&*e\"F]hl7$Fihl$\"7\" F]il7$Fdil$!9))G-Gk3j$\\J8b#Fhil7$Fjil$!9$>_H!4z#yTG$QTFhil7$F_jl$!:0. NNkc4XqJ))[&!#P7$Fdjl$!:.!o?/G5\"3=_f&oF]ap7$Fijl$!:FJ+\"[=u0K#)>x$)F] ap7$F^[m$!:%\\)4i8y6&*pRss*F]ap7$Fc[m$!:=$fn.+A0n)p02\"Fhil7$Fh[m$!:n[ [LPBT4C)\\\"=\"Fhil7$F]\\m$!:M$e$yf/U/RgfI\"Fhil7$Fb\\m$!:4k1n;i\"RKGR W9Fhil7$Fg\\m$!:)*o)e-2#[O-e/W\"Fhil7$F\\]m$!:ZF?#e7e*fPXoV\"Fhil7$Fa] m$!:)HFf1L&*o0xqM9Fhil7$Ff]m$!:bP$G54xQ/FnO9Fhil7$F[^m$!:,3pQQI)4Ty7y9 Fhil7$F`^m$!:y=E>KmrQ%f0j:Fhil7$Fe^m$!:rORM2K\"ol'y\\a\"Fhil7$Fj^m$!:+ \\N`REr9k[s_\"Fhil7$F__m$!:Po'\\xyP5R*>!)\\\"Fhil7$Fd_m$!:5A-rg:**[]!* **f\"Fhil7$Fi_m$!:kA'GW=Y19*yqn5$R*o\"Fhil7$Fidm$!:#oRI1d[>e\"=(p;Fhil7$F^ em$!:4Q?uA<3*4gQK;Fhil7$Fcem$!:\"3&*zILKboS`u;Fhil7$Fhem$!:7WFbkoF9Wb* *p\"Fhil7$F]fm$!::))z$)eo=wr!f_;Fhil7$Fbfm$!:RkX`R!RwaXPP;Fhil7$Fgfm$! :&e-*H.Y:5#R2x;Fhil7$F\\gm$!:$on91)\\4@D.pq\"Fhil7$Fagm$!:1UcS'*[0N(fv -;_;Fhil7$F`hm$!:a m?s'*3r5?q(GX;Fh il7$F_im$!:77'>Y]+%R&='eu\"Fhil7$Fdim$!:r7+\"HB1J@_p/^YDT1o\"Fhil7$F^jm$!:tko)[PN`k'pOm\"Fhil7$Fcjm$!:x9CHZ)*[yN6er \"Fhil7$Fhjm$!:ph8Uq@=T0bQt\"Fhil7$F][n$!:sC&4i*>@;Ck!)o\"Fhil7$Fb[n$! :cUwK31D`*GEl;Fhil7$Fg[n$!:=^hHMn\")zUoyw\"Fhil7$F\\\\n$!:f_U,.TRWo$>? `$*f9$Gx\"Fhil7$Fe]n$!:#>+ia/KI*f 3es\"Fhil7$Fj]n$!:>^(**e!**[iQ'f%o\"Fhil7$F_^n$!:G0O+)[aJ_nN:6\"=(z#)=*G#))49mn\"Fhil7$Fjbn$!:&yh.L\\'=mY.wm\"F hil7$F_cn$!:-tE#\\,P'yOwBp\"Fhil7$Fdcn$!:zwOO#\\$)=lG(4v\"Fhil7$Ficn$! :*)e4+GF5:tv$4x*Qza_z&[j;Fhil7$Fcdn$!:rOn!>L!e*4fjt;F hil7$Fhdn$!:j!\\4J/z0ipbEZgl'H$G%>m\"Fhil7$Fgen$!:(HEVO))y1!fa>h\"Fhil7$F\\fn$!:%ojxj[iSXgVI:F hil7$Fafn$!::0mL/\"oH%olQZ\"Fhil7$Fffn$!:bM!p>5O]^`7\"R\"Fhil7$F[gn$!: @c<\"RJ]&*e%QjE\"Fhil7$F`gn$!:$RWZL9,'yULm5\"Fhil7$Fegn$!:%[z5tBIA4H1' [*F]ap7$Fjgn$!:3rQ)4`j`^%>%[zF]ap7$F_hn$!:u(39h**H5:&z&4jF]ap-Fehn6&Fg hn$\"#XF*F]inFhhn-F_in6#%9scheme~with~simple~nodesG-F$6%7bw7$F($\"0G-# onv6qF-7$$\":+++++D\"yD?_=mF1$\"0\\\"*HCx=P'F-7$F/$\"0P*f/,d%z&F-7$$\" :+++++vVt2mb&oF1$\"0PGdU#ov_F-7$F5$\"0zQ0d\">>[F-7$F:$\"0]d2&y$\\6%F-7 $F?$\"0*G(H7dKX$F-7$FD$\"1M.dsgaUEFH7$FJ$\"1f_l'RT15#FH7$FO$\"1[*GS'o8 TUUlYoN\"F\\p7$Fgq$\"2f#\\ e)fP9Z\"z6W\"F\\p7$Fdt$\"2 \\U_.b#GS9F\\p7$Fit$\"2rL(e`RfR9F\\p7$Fdz$\"3-_TBB>h]:Fav7$F\\]l$\"3:x 'HsL?1h\"Fav7$Fa]l$\"3ml9[?Y,*f\"Fav7$Ff]l$\"38dnV6\"fvf\"Fav7$F[^l$\" 3&o?.;/b\"3H;;Fav7$Fj^l$ \"3>\\()*3(*[$G;Fav7$F__l$\"3(**zNGV'>#p\"Fav7$Fd_l$\"3)y$RL[FxW;Fav7$ Fi_l$\"3=ys?J-%eh\"Fav7$F^`l$\"3SOc?u3+DA$))oW7Lm\"F\\al7$F^al$\"4gO`YV@a)R!Q6H&f=F\\al7$$\":++++DcwRaHFq\"Fio$\"5))\\6X%\\RU!H=Ffbl7$$ \":++++]P4J\\dWr\"Fio$\"53K(ps.UG$*z\"Ffbl7$$\":++++v=UAW&=EK**f)y\"Ffbl7$F] cl$\"5TXxsGt!*4H=Ffbl7$Fbcl$\"5A()\\p;*4V^x\"Ffbl7$Fgcl$\"5;u\\oQm#pEu \"Ffbl7$Fadl$\"5ow\\>t^))H\"z\"Ffbl7$F[el$\"5O\"[`xgN:ew\"Ffbl7$F`el$ \"5Zz1RTb>S8=Ffbl7$Feel$\"5k!*y5a\\4Mj(>Z$ H+G.l\"F]gl7$Fdgl$\"6JawYNOjgLh\"F]gl7$$\":+++++]7.L6\\9#Fio$\"6ofJ4S) Q#*yU:F]gl7$Figl$\"7p*RDO()e(3ht9F]hl7$F_hl$\"7'[!**oH-#3pfO\"F]hl7$Fd hl$\"7%o\")*e>f]o%=>\"F]hl7$Fihl$\"8I5F]il7$F_il$\"7ajz6H^g #HNL)F]il7$Fdil$\"87r(>d8H%*GUJfFhil7$Fjil$\"82y/(44#**oYe!QFhil7$F_jl $\"8p\\YcL/@p]$e7Fhil7$Fdjl$!9.!o?/G5i!\\z#R\"F]ap7$Fijl$!9FJ+\"[=um(H !H&RF]ap7$F^[m$!:Y\\)4i8yh\"\\>(Gm!#Q7$Fc[m$!:g<$fn.+-=PM`!*F`fr7$Fh[m $!:u'[[LPBzr_Ge6F]ap7$F]\\m$!:TLe$yf/dbKt=F]ap7$F`^m$!:#y=E>KmGyi_>?F]ap7$Fe^m$!:6n$RM2KKX7p5?F]ap7$ Fj^m$!:$**[N`REO9k)=+#F]ap7$F^`m$!:xDo\"**3f)p.:w*>F]ap7$Fbam$!:dh`zj/ 7F8w\"=?F]ap7$Fgam$!:7c3-#[G5W#F]ap7$Fa bm$!:0zjTfW2@g\"px@F]ap7$Ffbm$!:rtN[Ur`B2:^;#F]ap7$F`cm$!:W,f*fvX\\42V *=#F]ap7$Fjcm$!:Xw8DHh[%z2(eA#F]ap7$F_dm$!:9q0LITFWURsH#F]ap7$Fddm$!:b >Y19*y*oZ>/X#F]ap7$$\":+++]iS\"p%=89u$Fio$!:MXDgza0Drh*pCF]ap7$Fidm$!: jf)Qd#F]ap7$Fagm$!:h?k0k*[Tra 4rDF]ap7$Ffgm$!:-hI$[!edkG34]#F]ap7$F[hm$!:?ErM)QCjs`l)\\#F]ap7$F`hm$! :Zl1An*3&>fXYi#F]ap7$Fehm$!:*zoIWx:'=)[&*\\DF]ap7$Fjhm$!:k'z.m)e^WH5!* \\#F]ap7$F_im$!:@@h>Y]5*[qHkEF]ap7$Fdim$!:3F,5HBm\"*=V@g#F]ap7$Fiim$!: k07\\I)>Co\"QJf#F]ap7$F^jm$!:MZ'o)[P&)p!f*\\f#F]ap7$Fcjm$!:pZT#HZ)4![K b%o#F]ap7$Fhjm$!:#oh8Uq@ZI]#yr#F]ap7$F][n$!:*3/Rhk#F]ap7$Fb[n$! :fDkF$31Mi\")Q7EF]ap7$Fg[n$!:'=^hHMn\"eOFcy#F]ap7$F\\\\n$!:#f_U,.TgN*4 0r#F]ap7$Fa\\n$!:P.nyPa)H1\"*=VEF]ap7$Ff\\n$!:h&4M(H7$[keYcEF]ap7$F[]n $!:/yz#RF]#*Rb=\\FF]ap7$F`]n$!:^rbK/>BuxSq$GF]ap7$Fe]n$!:@>+ia/#*[^L,z #F]ap7$Fj]n$!:&>^(**e!*f/#>t^FF]ap7$F_^n$!:u_g.!)[k+bnq$GF]ap7$Fd^n$!: n$**3g&okq7\")p$HF]ap7$Fi^n$!:odC$\\Yq8_zQ'*GF]ap7$F^_n$!:r7ybd]`N\\<. (GF]ap7$Fc_n$!:FuQ)*zh.D^th&HF]ap7$Fh_n$!:US9sbeHtL0$oIF]ap7$F]`n$!:\\ w(e5x!>wp#HFIF]ap7$Fb`n$!:[rBvW^P.KU!))HF]ap7$Ffan$!:Ox*Qz_(fh6+$F ]ap7$Fcdn$!:-nt1>L]wSv^-$F]ap7$Fhdn$!:K1\\4J/>emt,8$F]ap7$F]en$!:SLudf -9#f3e&3$F]ap7$Fben$!:$f(Q>Zg:%y*=N/$F]ap7$$\":+++]7`%*y*RK'>%Fio$!:lu FB3vFuZ&>AIF]ap7$$\":++++DJ&pC6g2UFio$!:r)oL!*y+\"e;,R-$F]ap7$$\":+++] P4'\\^#y)=UFio$!:1>$4\">*o+#QyE3$F]ap7$$\":++++](oHy`:IUFio$!:(f)zmbdb %f)e%GKF]ap7$$\":++++vV)*=j4FD%Fio$!:qi()e4S@\\3J!eJF]ap7$Fgen$!:oHEVO ))=8Nq44$F]ap7$F\\fn$!:Sojxj[iZ`CA3$F]ap7$Fafn$!:^^gOV5)*o08P8$F]ap7$F ffn$!:_X.p>5c(=4%G7$F]ap7$$\":++++DJXk/S^b%Fio$!:hyzgP1HpKka0$F]ap7$$ \":++++]iS\"=]=nXFio$!:nj0'\\Y]8Gh&G,$F]ap7$$\":++++v$f$)*)*H#zXFio$!: &>I$yYf\"yXWP%3$F]ap7$$\":+++++DJ:'\\F\"f%Fio$!:dS'*pq7k#y_8SJF]ap7$$ \":++++DcEK$*>Lg%Fio$!:Zf%Hm!G58\" y8B;IF]ap7$$\":++++v=6U IYFio$!:7B\"z'QZ[rw>q7$F]ap7$$\":+++]P%[YiBVLYFio$!:5$oI\\7&)))p$>W.$F ]ap7$$\":+++v=n)Q0OWOYFio$!:pe?RhLfO+wt&HF]ap7$F[gn$!:8iv6RJkrr/40$F]a p7$$\":+++v$f3))R\"[@k%Fio$!:(oO`M`*)*oQ%eiIF]ap7$$\":+++](=#\\9VT[k%F io$!:'[m+eD>;&)3&H)HF]ap7$$\":+++D\"yv,BZ`ZYFio$!:Ar%pIBE%\\'*H6%HF]ap 7$$\":++++v$fe9!G-l%Fio$!::%G*)y@TR/wS5JF]ap7$$\":+++voHahI@Hl%Fio$!:6 iomm(RfhQ!y-$F]ap7$$\":+++]ilAxf9cl%Fio$!:s\"3F#=KrM)QB_HF]ap7$$\":+++ Dc,\"H*)yIeYFio$!:&>Q@IngHV\"Gj%HF]ap7$$\":++++]Pf3=,5m%Fio$!:'>r))G(R ?#=ucuIF]ap7$$\":++++D\"G8ZVxrYFio$!:6i\"4(>.bI7r'QIF]ap7$$\":+++++D1M ^ZDo%Fio$!:bW\"QDM65w*RF+$F]ap7$$\":++++]7`f%Q4/ZFio$!:L7f\"*p]^]B?7$H F]ap7$F`gn$!:JRWZL9v8aMM'GF]ap7$Fegn$!:%[z5tB?'=l\"4+FF]ap7$Fjgn$!:3rQ )4`.j(=)=\\DF]ap7$F_hn$!:u(39h***zNI[bW#F]ap-Fehn6&FghnF]in$\"#DF*$\" \"\"Fahn-F_in6#%Pscheme~with~a~relatively~large~stability~regionG-F$6% 7_x7$F($\"2G(4qrU/&3\"F-7$F\\dq$\"1\\mtqK/0#*F-7$F/$\"1Pn&>fB$>xF-7$Fd dq$\"1P&4ZM&GrjF-7$F5$\"1z07,\\t.^F-7$F:$\"1]T5%RT<]#F-7$F?$\"0*Q1<4dX !)F-7$FD$!2mEeR&fhspI]cr!HZ)F\\p7$Fcp$!34 'R=+7RWB*F\\p7$Fhp$!3-n4Pknl.$*F\\p7$F]q$!37k&euY[-U*F\\p7$Fbq$!3i!oxh O)4>'*F\\p7$Fgq$!3T21$oS>;&**F\\p7$F\\r$!3&fLO$**>az**F\\p7$Far$!3e1'= L[Y:s*F\\p7$Ffr$!3(3Kk\\&[()4'*F\\p7$F[s$!4v[xoS0*GC5F\\p7$F`s$!3-z*z2 8]4(**F\\p7$Fes$!3Ff3#)R-'es*F\\p7$Fjs$!3CO'=S.Q9v*F\\p7$F_t$!4&[!)*z[ w5I-\"F\\p7$Fdt$!4^dx!*\\nx8-\"F\\p7$Fit$!4HmQ]*fZT>5F\\p7$F^u$!4$Q))[ 6!H`L,\"F\\p7$Fcu$!4\"47ed**\\a25F\\p7$Fhu$!4IUD#R_ZO.5F\\p7$F]v$!5tgU &fGw(>/5Fav7$Fcv$!5'p`#G`cz2s5Fav7$Fhv$!5\\+*zN^t5E/\"Fav7$F]w$!5ca4s& oEed,\"Fav7$Fbw$!5g_w\"H&RQ;<5Fav7$Fgw$!5Y$fl\"*yR')\\.\"Fav7$F\\x$!5* \\'y-'30Uc0\"Fav7$Fax$!5t%H0Zrm&)e0\"Fav7$Ffx$!5e`\"\\$o@g#y-\"Fav7$F[ y$!5F9R&[U6Y#>5Fav7$F`y$!5T,p?P\"[-r2\"Fav7$Fey$!5s!HE\"4])[v/\"Fav7$F jy$!5Kd1%[#)=bF-\"Fav7$F_z$!5Q9zgJ_hNW5Fav7$Fdz$!5)z%=],_vFo5Fav7$Fiz$ !5,y^\\*[_&Hb5Fav7$F^[l$!5_iu%y_!=(G/\"Fav7$Fc[l$!5*)*yS**[BXR.\"Fav7$ Fh[l$!5je\"GulU@!Q5Fav7$F]\\l$!5*zAyemQR*y5Fav7$Fb\\l$!5!\\G([\">%>/&4 \"Fav7$Fg\\l$!50v]=&[nX+3\"Fav7$F\\]l$!5&GK%oxI2Cl5Fav7$Fa]l$!5MMb+2su I_5Fav7$Ff]l$!5(GC81KQad/\"Fav7$F[^l$!5:$z'=))f_386Fav7$F`^l$!5n&G@H&f 7c\"3\"Fav7$Fe^l$!5w8yAD0/7`5Fav7$Fj^l$!5\"3DF(=Bd9f5Fav7$F__l$!5.+s@, U:9(4\"Fav7$Fd_l$!57i!y\"4$Rzj1\"Fav7$Fi_l$!5#=s!3L603Z5Fav7$F^`l$!5gj 87Bdlu76Fav7$Fc`l$!5_GPZuOn2!3\"Fav7$Fh`l$!6,ynH6c.*)30\"F\\al7$F^al$! 6SjYEh5WB32\"F\\al7$Fcal$!6HSz\">eOTH*3\"F\\al7$Fhal$!6()e%4[*H+$>f5F \\al7$F]bl$!6&[cTS_[jyE5F\\al7$Ff^r$!6+`e_l\"))G+25F\\al7$F[_r$!5?C(QQ eU)*)*\\%)F]gl7$F_gl$!7N Q=\\SySle`uF]gl7$Fdgl$!7pXB`q^w4cAkF]gl7$Figl$!8J+YPE0(*>d&H_F]hl7$F_h l$!89&45.2h!z6#\\RF]hl7$Fdhl$!8;$=5/)p8aHIf#F]hl7$Fihl$!9$G-%\\dM([sj0 1\"F]il7$F_il$\"8ajz6Hht7l\\8&F]il7$Fdil$\":7r(>d8Rn!p&e#3#Fhil7$Fjil$ \":2y/(44iD2dhiQFhil7$F_jl$\":p\\YcL/Bg9A*RaFhil7$Fdjl$\":+Kz&>(*o]T'* [WqFhil7$Fijl$\":(o**=:e#\\](*)='z)Fhil7$F^[m$\"::!zj=#[%*=tJ'Q5F]il7$ Fc[m$\":oSK'**ztjAe2e6F]il7$Fh[m$\":8:lEm&o2v\" F]il7$Fe^m$\":L1cEz'yZ*\\!yJ$QD?4DM%*ejS==F ]il7$Fabm$\":@OeSbUYm'e&oz\"F]il7$Ffbm$\":Ek^dGcU`&z8wy< F]il7$Fjcm$\":B'[2(QJ=F]il7$Fddm$ \":\"QNf3@bzAMk:>F]il7$Fidm$\":Kgp$H9]_4Q(e*=F]il7$F^em$\":>'zDx#=ccVu f&=F]il7$Fcem$\":#\\+#pmZ3rJuY!>F]il7$Fhem$\":fDZa88_S^>R$>F]il7$F]fm$ \":=,i69V5uni+)=F]il7$Fbfm$\":cVl/'4*R5rUJ'=F]il7$Fgfm$\":T(4q'RXM*4z5 4>F]il7$F\\gm$\":KK&Q>]Xi?l6W>F]il7$Fagm$\":!eVf.^b%\\jq&R>F]il7$Ffgm$ \":Rp;&>CXszd^')=F]il7$F[hm$\":uGl6c2UzcoC)=F]il7$F`hm$\":N$zF.\"zYIJ] /(>F]il7$Fehm$\":7$pbA%yoa*4O9>F]il7$Fjhm$\":.iR8T=\"[f[Sv=F]il7$F_im$ \":zQ!Q&\\>91_e6*>F]il7$Fdim$\":t)**3nPX^i'QU%>F]il7$Fiim$\":%z3&p,o( \\U%R#>>F]il7$F^jm$\":`88^i9\"**=kP->F]il7$Fcjm$\":_e2F:??!yg!G'>F]il7 $Fhjm$\":$Q'y&Hy)4UEBP)>F]il7$F][n$\":`Z!z.!Q*3'*QNJ>F]il7$Fb[n$\":vNs ;RH?`&ec0>F]il7$Fg[n$\":)[QqlKOXbl!R-#F]il7$F\\\\n$\":uu&)p*eL(**4G$p> F]il7$Fa\\n$\":'H8Ac9mX&)*\\-#>F]il7$Ff\\n$\":0fEq(oI\")e^CF>F]il7$F[] n$\":A?2E(\\y2J[F#)>F]il7$F`]n$\":HWn&4oTP)>F]il7$Fj]n$\":)[-5%4+a3]]r$>F]il7$F_^n$\":ZR'*>^&e\\rutv>F]il7$ Fd^n$\":15*R9`,\"\\[QI-#F]il7$Fi^n$\":Uv1N&H^JP&Gy(>F]il7$F^_n$\":(=UC %\\E[0fRF%>F]il7$Fc_n$\":Eh,?QmPQ$[-!)>F]il7$Fh_n$\":g&yU9/BQk[.L?F]il 7$F]`n$\":B7%*G#4`i<\\E(*>F]il7$Fb`n$\":HwCb[-q$yE)G'>F]il7$Fg`n$\":BP u>apcy'3vO>F]il7$F\\an$\":,PI*[mr1ic&G/#F]il7$Faan$\":Y\"o/Z$***\\&*e1 %)>F]il7$Ffan$\":$)oma4rw!\\+9O>F]il7$F[bn$\":]wx,!)e:Tr!>**>F]il7$F`b n$\":[eTUKUS<_rg+#F]il7$Febn$\":TY=/3@H:4%R]>F]il7$Fjbn$\":AQ'p1Nf@)p. /%>F]il7$F_cn$\":qKx])Haml2-q>F]il7$Fdcn$\":LKOw]OST:h&R?F]il7$Ficn$\" :6/**>FZb)fn9\"*>F]il7$F^dn$\":O!G-h?\">-3ex$>F]il7$Fcdn$\":LE$4o'\\Z \"f#Q-&>F]il7$Fhdn$\":%40*o&4G'pwED,#F]il7$F]en$\":nDUS(f#*[A4Iv>F]il7 $Fben$\":Ch!G&Rk9Um+*R>F]il7$$\":++]7`%Rz_h'y=%Fio$\":mQl`&z?ePRv>>F]i l7$$\":+++D19%\\Mao!>%Fio$\"::6WF1cs%Qa*3-#F]il7$$\":++]PfL%>;Z]$>%Fio $\":TKlcxP$QE\"Q['>F]il7$Fhcs$\":Esw\"\\AG_D)=x\">F]il7$$\":++]ils%fzK 9*>%Fio$\":=&eu$pp_JZ*3e>F]il7$$\":+++v=#\\HhD'>?%Fio$\":iO\"pER*[F+!> \"*>F]il7$$\":++](=<^*H%=y/UFio$\":bsCmr0+v10p$>F]il7$F]ds$\":7j'4@*Rp PD&f4>F]il7$Fbds$\":\"o!*33Ju$G)fuN>F]il7$Fgds$\":9?LWUC9\\tVM,#F]il7$ F\\es$\":Q7T!*fGYec7A&>F]il7$Fgen$\":qtcj6J:kpVP*=F]il7$F\\fn$\":JOAO^ d^MW05\"=F]il7$Fafn$\":[Rjc*=c]ALee\"F]il7$Fjgn$\":Hh,pkud-J\"4B5F]il7$F_hn$\":B\"f)Q+!>vn 3C([)Fhil-Fehn6&FghnF]in$\"#vF*Fjhn-F_in6#%TButcher's~scheme~B~with~c[ 5]=c[6]=1/2~and~b[5]=b[6]G-F$6%7fw7$F($\"0GIa#=%[V%F-7$F?$\"0*QO?X*3L& F-7$FJ$\"1f-AFPNDgFH7$FO$\"1[\\nn%)p#e'FH7$FT$\"1P'Gc(=n2sFH7$FY$\"1c* oD'4V2rFH7$Fhn$\"1G_A)=]-0(FH7$F[]o$\"1lP'RX]=V(FH7$F`]o$\"1E;)G'>%)ft FH7$Fe]o$\"1H)\\)G1`grFH7$F]o$\"1e$4oula=(FH7$F]^o$\"1bE!)*[2/I(FH7$Fb o$\"14%o/e_-n(FH7$Fgo$\"2Q2'e%\\\\?g(F\\p7$F^p$\"2\"3V5!*e'H`(F\\p7$F[ _o$\"2'z*Rl%G%HZ(F\\p7$F`_o$\"2\\o9Mq!fZuF\\p7$Fe_o$\"2ITG,M#\\2yF\\p7 $Fj_o$\"2_Eu3gWrw(F\\p7$F_`o$\"2)om>Dt)fb(F\\p7$Fd`o$\"238'eXjRcvF\\p7 $Fi`o$\"2rE:CB/Wp(F\\p7$F^ao$\"2nCK%R<-b!)F\\p7$Fcao$\"2*y-'f`X,\"zF\\ p7$Fcp$\"2\"R!4*HU'4y(F\\p7$Fhp$\"2)H.*Gg:x!yF\\p7$F]q$\"2)eL3iTyuyF\\ p7$Fbq$\"2Q>*>vpE5!)F\\p7$Fgq$\"2f#H%*)>idD)F\\p7$F\\r$\"20kIB02NF)F\\ p7$Far$\"2U$*Q$=PPf!)F\\p7$Ffr$\"28z]-wLG'zF\\p7$F[s$\"2D^GUZ;%p%)F\\p 7$F`s$\"2)4sav>bW#)F\\p7$Fes$\"2tS(yl2HT!)F\\p7$Fjs$\"2wj$*)4Rtb!)F\\p 7$F_t$\"2:&pTN3%zV)F\\p7$Fdt$\"2\\U#e'[N\\S)F\\p7$Fit$\"2rLykk'[q$)F\\ p7$F^u$\"2<;o@_zNI)F\\p7$Fcu$\"24zB^z#yR#)F\\p7$Fhu$\"2qdJY%4$**=)F\\p 7$F]v$\"3FR2H5G&4=)Fav7$Fcv$\"3/j/o(*H#=s)Fav7$Fhv$\"3^*4r!y(*3#[)Fav7 $F]w$\"3WXqy#)Q+w:G_)Fav7$F^[l$\"3[P&*>p0R6%)Fav7$Fc[l$\"365-)=5d\"G$)Fav7$Fh [l$\"3PT[1WU[[$)Fav7$F]\\l$\"3,sd?Ij_i')Fav7$Fb\\l$\"35:P$3Yt+y)Fav7$F g\\l$\"3&\\#p!pq:.l)Fav7$F\\]l$\"3:xE?&y5D_)Fav7$Fa]l$\"3mlMA\"yK)4%)F av7$Ff]l$\"38d\\ZW07+V%)Fav7$F__l$\"3(**z61Lb ;u)Fav7$Fd_l$\"3)y$R,^<['\\)Fav7$Fi_l$\"3=yULJ37T$)Fav7$F^`l$\"3SO;t>- A$3CRr@+xp%G0&oFfbl7$Fadl$\"5ow\\HB_&*3fmFfbl7$Ff dl$\"5V5sS(*e-%*)['Ffbl7$F[el$\"5O\"[`sKH2u`'Ffbl7$F`el$\"5Zz1R1Q;R!o' Ffbl7$Feel$\"5k!*y!p#o/r&\\'Ffbl7$Fjel$\"58Yl#f^VS#yjFfbl7$F_fl$\"5A[u G1\"y!e*p'Ffbl7$Fdfl$\"5TL[Wf4w^GlFfbl7$Fifl$\"6$o\\az'*)eQ&fjF]gl7$F_ gl$\"6lh\"3&*)oLln_&F]gl7$Fdgl$\"6JawYkuMlFk%F]gl7$Figl$\"7p*RDOn')=Nd m$F]hl7$F_hl$\"7'[!**oHt#H&G)e#F]hl7$Fdhl$\"7%o\")*e>oM&4/\\\"F]hl7$Fi hl$\"7_H!4ziGN1QPFhil7$F_jl$!:0.NNkcHRaIO(\\F]ap7$Fdjl$!:.!o ?/G58%)oVCiF]ap7$Fijl$!:FJ+\"[=ukjaU:wF]ap7$F^[m$!:%\\)4i8yJiAC8&))F]a p7$Fc[m$!:wJfn.+-%yf4\\(*F]ap7$Fh[m$!:n[[LPB?6I'fw5Fhil7$F]\\m$!:M$e$y f/aF%ym!>\"Fhil7$Fb\\m$!:4k1n;iCcDxvJ\"Fhil7$Fg\\m$!:)*o)e-2A#R<_SJ\"F hil7$F\\]m$!:ZF?#e7Q&f2=3J\"Fhil7$Fa]m$!:)HFf1L02cX#*38Fhil7$Ff]m$!:bP $G54d%=an2J\"Fhil7$F[^m$!:,3pQQIXJe-([8Fhil7$F`^m$!:y=E>Km\"*zSKkU\"Fh il7$Fe^m$!:rORM2K\\^J***49Fhil7$Fj^m$!:+\\N`REiqdzQR\"Fhil7$F__m$!:Po' \\xyF>\\ZS\"Fhil7$Fbam$!:;O&zj/iss=? %Q\"Fhil7$Fgam$!:i&3-#[Gw)HNZV9Fhil7$F\\bm$!:Dc&392o&[IaYZ\"Fhil7$Fabm $!:!zjTfWZ-7rBc9Fhil7$Ffbm$!:Pd$[UrBv\\*3&Q9Fhil7$F[cm$!:Nf.Izab\">f!p V\"Fhil7$F`cm$!:9!f*fvX`rACiV\"Fhil7$Fecm$!:^MY19*yj/3*)Ga \"Fhil7$Fidm$!:#oRI1dGZ:'z]_\"Fhil7$F^em$!:4Q?uA<,`K&3\"\\\"Fhil7$Fcem $!:\"3&*zILiJOvdH:Fhil7$Fhem$!:7WFbko'4Qz)Gb\"Fhil7$F]fm$!::))z$)eoMzo :'4:Fhil7$Fbfm$!:RkX`R!ps@hn&\\\"Fhil7$Fgfm$!:&e-*H.YPWNZ?`\"Fhil7$F\\ gm$!:$on91)\\Bmw+$f:Fhil7$Fagm$!:1UcS'*[*o7Cdb:Fhil7$Ffgm$!:51L[!e(HFz !)H^\"Fhil7$F[hm$!:h7Z$)QClm-/$4:Fhil7$F`hm$!:am?s'*3%\\)*>Sz:Fhil7$Fe hm$!:!)oIWx:F\\iOW`\"Fhil7$Fjhm$!:mz.m)e6d$GAI]\"Fhil7$F_im$!:77'>Y]5$ y*z1&f\"Fhil7$Fdim$!:r7+\"HBENW*pub\"Fhil7$Fiim$!:c?\"\\I)>B0)QhN:Fhil 7$F^jm$!:tko)[PX6G*4-_\"Fhil7$Fcjm$!:x9CHZ)fQ5:'yc\"Fhil7$Fhjm$!:ph8Uq @a(o3V%e\"Fhil7$F][n$!:sC&4i*>'*eI#eU:Fhil7$Fb[n$!:cUwK31q!)R5<_\"Fhil 7$Fg[n$!:=^hHMn%)p/ybh\"Fhil7$F\\\\n$!:f_U,.Tz)e*4?d\"Fhil7$Fa\\n$!:Lq 'yPa)f,OsF`\"Fhil7$Ff\\n$!:c4M(H7LF.v)y`\"Fhil7$F[]n$!:!yz#RF]seV01e\" Fhil7$F`]n$!::dDV!>tG#[4.i\"Fhil7$Fe]n$!:#>+ia/#frCfud\"Fhil7$Fj]n$!:> ^(**e!*RUy))*)R:Fhil7$F_^n$!:G0O+)[uXZe7o:Fhil7$Fd^n$!:P**3g&oM+#*>`.; Fhil7$Fi^n$!:xXK\\YqH6fCec\"Fhil7$F^_n$!:F\"ybd]Vju52O:Fhil7$Fc_n$!:V( Q)*zhL_r#=Jc\"Fhil7$Fh_n$!:/W@d&ez$e=aFg\"Fhil7$F]`n$!:lxe5x!fe$R(et:F hil7$Fb`n$!::P_Z9v'>$>Eba\"Fhil7$Fg`n$!:nFc-e/[u\"\\jC:Fhil7$F\\an$!:! *H'p5N8HT#=vg\"Fhil7$Faan$!:Y&=`HlI!eTB:Fh il7$F[bn$!:(\\BA)*>JAkuJs:Fhil7$F`bn$!:<:%ednn:y&)ex:Fhil7$Febn$!:'f` \"e>*GTdswL:Fhil7$Fjbn$!:&yh.L\\wF:r[D:Fhil7$F_cn$!:-tE#\\,F&z\"*f#[:F hil7$Fdcn$!:zwOO#\\.\"p\"\\#>g\"Fhil7$Ficn$!:*)e4+GFuJ3#)Qc\"Fhil7$F^d n$!:T'>x*QzX9W()=_\"Fhil7$Fcdn$!:rOn!>L!e\\vW6`\"Fhil7$Fhdn$!:j!\\4J/4 FK9nz:Fhil7$F]en$!:MVx&f-%QIVf%\\:Fhil7$Fben$!:f(Q>ZgvC`jo?:Fhil7$Fgen $!:(HEVO))=DfFdv9Fhil7$F\\fn$!:%ojxj[i3F?e,9Fhil7$Fafn$!::0mL/\"yK``]] 8Fhil7$Fffn$!:bM!p>5')eGI[v7Fhil7$F[gn$!:@c<\"RJ91(zp>;\"Fhil7$F`gn$!: $RWZL9nO8NL;5Fhil7$Fegn$!:%[z5tB?+)*)3Gs)F]ap7$Fjgn$!:3rQ)4`L_]7\"=K(F ]ap7$F_hn$!:u(39h**>9cB&4$eF]ap-Fehn6&FghnFhhnF`cqF]in-F_in6#%Hscheme~ with~c[5]=c[6]=3/4~and~b[5]=b[6]G-%%FONTG6$%*HELVETICAG\"\"*-%+AXESLAB ELSG6$Q\"x6\"Q!Fj^y-%&TITLEG6#%Uerror~curves~for~7~stage~order~6~Runge -Kutta~methodsG-%%VIEWG6$;F(F_hn%(DEFAULTG" 1 2 0 1 10 0 2 9 1 4 2 1.000000 45.000000 45.000000 0 1 "Butcher's scheme A" "scheme with sim ple nodes" "scheme with a relatively large stability region" "Butcher' s scheme B with c[5]=c[6]=1/2 and b[5]=b[6]" "scheme with c[5]=c[6]=3/ 4 and b[5]=b[6]" }}}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 47 "Te st 15 of 7 stage, order 6 Runge-Kutta methods" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 198 "See: Comparing Numerical Methods for Ordinary Differential Equations, Hull, Enright, Fellen an d Sedgwick,\n Siam Journal on Numerical Analysis, 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 "dy/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*arc tan(y/x)+4*ln*x-2*ln*2-Pi = 0;" "6#/,,*&\"\"#\"\"\"-%#lnG6#*&,&*$%\"xG F&F'*$%\"yGF&F'F'*$F.F&!\"\"F'F'*&\"\"%F'-%'arctanG6#*&F0F'F.F2F'F'*(F 4F'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 solut ion 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,*&%#PiGF,F+F4F," }{TEXT -1 2 ". " }}{PARA 0 "" 0 "" {TEXT -1 18 "The solution (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-t heta)" "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#*&F0F-F+!\"\"F-F-,&-F&6#F,F- *&F,F6%#PiGF-F-" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#,$*&,&%\"yG!\"\"%\" xG\"\"\"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=red):\np2 := plot([sqrt( 2)*exp(Pi/4-t),t,t=Pi/4..2*Pi],coords=polar,color=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,symb ol=[circle$2,diamond,cross],\n symbolsize=[12,10$3],c olor=[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*$!3pIdcz?N/uD VF*$!3Mk<%4i8_'>F*7$$\"3neCR^2fVUF*$!3U`V:L2,dAqSF*$!3]0%Rz+R[P\"F*7$$\"3CpGt*G2)))RF*$!3 Vje1<#>j@\"F*7$$\"31&*>Xy@@*)QF*$!3KrYY*z8q.\"F*7$$\"3m&*QGM(*y-QF*$!3 Z+5yDwaF*)!#=7$$\"3x^E')y`D+PF*$!3%[->)G'=aL(F\\q7$$\"3v7s?m@*zg$F*$!3 c,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'*H F*$\"3?)\\7Z#>+:5F\\q7$$\"3_&4uxb.N!HF*$\"3?;dqg#**3'=F\\q7$$\"3Q!f))4 wMJ!GF*$\"3Rn1jA(f_r#F\\q7$$\"3`xr$)p%\\+q#F*$\"37>6@EK]NNF\\q7$$\"3;# Rcs]s**f#F*$\"3]^E27l!)yUF\\q7$$\"3Y0BF*$\"3I%H$\\j$zL='F \\q7$$\"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$$\"3tj&H(oDW3:F*$\"3!)e5j3rW%R*F\\q7$$\"3H m+#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%*yj GB$o\\&oF\\q$\"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$$\"3f(3R5[t$3HFc_l$\"3f7r>se QEnF\\q7$$!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$$!3G 4mSr>(e2#F\\q$\"30UK<,&yQ#>F\\q7$$!3#o3.QX91/#F\\q$\"3P$*fjX!f&H:F\\q7 $$!3#y(eD!G(*f&>F\\q$\"3UEc#G.d\"=6F\\q7$$!3=qle`Y)o&=F\\q$\"37JSjs:(p ;)Fc_l7$$!30utN:SE> B.UUNGWgF`s7$$!3zWz&))>R47\"F\\q$!3Unq>(fHx'=Fc_l7$$!3_y,&)zs9)3t![ Fc_l$!3_$o;W4rzI%Fc_l7$$!3E\\<<**)*)Rz$Fc_l$!3e#*))[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_l 7$$!3Q^mSZRTp9Fc_l$!3_uFr$)*ycw$Fc_l7$$!3-[9Nr!)oebDFc_l7$$\"3Wcg/C\"='=WF`s$!37)o9 o#3(GD#Fc_l7$$\"33dhy:jnDiF`s$!3?g29(Qlz%>Fc_l7$$\"3-K8`c=f([(F`s$!35q &z[NHem\"Fc_l7$$\"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$!36Hbzi()GSn;F`s7$$\"3/Xu8A%33['F`s$!3;Is^'z2(yxFehl7$$ \"3)Q4tG5rBz&F`s$\"3c0'H(*oS@v%!#H-Fjz6&F\\[lFa[lFa[lFa[l-%*LINESTYLEG Fd[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*$!3SWWx3P mfuF*7$$\"3_-)f6JNm[%F*$!3#o7ACd\\FR(F*7$$\"3akH:B>m-XF*$!3:r-1\\\\VEt F*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$$\"3CLF g!*\\T'o%F*$!3WZZJr$p:R'F*7$$\"39!\\a\\())R&p%F*$!3JUA:p?9KjF*7$$\"3*y ZODQ#Q/ZF*$!3eR\\wA`UqiF*7$$\"3#Gs'H4VG7ZF*$!3@Fo#z)3,9iF*7$$\"3zqGL?z V?ZF*$!3[?z6=2W`hF*7$$\"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$$\"3w'3pk%[#)[Z F*$!3<#*G&H-C$=fF*7$$\"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$$\"3oO**41X!4y%F*$!3Ay w?\\-0ubF*7$$\"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$$\"3ZJ;\"pbc5![F*$!34Ww)4ym# R_F*7$$\"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$$\"\"\"Fa[lFj\\n7$ %*undefinedGF]]nF_]m-%'SYMBOLG6$%'CIRCLEG\"#7-%&STYLEG6#%&POINTG-F$6&F h\\n-Fjz6&F\\[lF`[lF][lF`[l-F_]n6$Fa]n\"#5Fc]n-F$6&Fh\\nFi]n-F_]n6$%(D IAMONDGF]^nFc]n-F$6&Fh\\nFi]n-F_]n6$%&CROSSGF]^nFc]n-%%FONTG6$%*HELVET ICAG\"\"*-%+AXESLABELSG6%%\"xG%%y(x)G-Fi^n6#%(DEFAULTG-%%VIEWG6$Fd_nFd _n" 1 2 0 1 10 0 2 9 1 4 2 1.000000 45.000000 45.000000 0 0 "Curve 1" "Curve 2" "Curve 3" "Curve 4" "Curve 5" "Curve 6" "Curve 7" }}}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 29 "The following p rocedure uses " }{TEXT 0 6 "fsolve" }{TEXT -1 23 " to solve the equati on " }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 " ln(x^2+y^2)+2 *arctan(y/x)=ln*2+Pi/2" "6#/,&-%#lnG6#,&*$%\"xG\"\"#\"\"\"*$%\"yGF+F,F ,*&F+F,-%'arctanG6#*&F.F,F*!\"\"F,F,,&*&F&F,F+F,F,*&%#PiGF,F+F4F," } {TEXT -1 1 " " }}{PARA 0 "" 0 "" {TEXT -1 4 "for " }{TEXT 265 1 "y" } {TEXT -1 25 " numerically in terms of " }{TEXT 266 1 "x" }{TEXT -1 1 " ." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 209 "phi := proc(x) local y;\n fsolve(ln(x^2+y^2 )+2*arctan(y/x)=ln(2)+Pi/2,y=-x..7/2-x);\n end proc:\nuu := eval f(exp(Pi/2)):\nplot('phi'(x),x=1..uu,numpoints=100,font=[HELVETICA,9], labels=[`x`,`y(x)`]);" }}{PARA 13 "" 1 "" {GLPLOT2D 515 404 404 {PLOTDATA 2 "6&-%'CURVESG6$7jq7$$\"\"\"\"\"!$\"+++++5!\"*7$$\"+M.FS5F- $\"+!Hsf***!#57$$\"+N$4`2\"F-$\"+]6*f)**F37$$\"+eVr96F-$\"+Zyon**F37$$ \"+#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-$\"+ig9X'*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$$\"+6kKP?F-$\"+KCbovF3 7$$\"+q%*)o2#F-$\"+LT&QQ(F37$$\"+0Uf:@F-$\"+Okc'>(F37$$\"+EI/`@F-$\"+7 w54qF37$$\"+cNi%>#F-$\"+Snr$z'F37$$\"+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$$\"+.DQ,DF-$\"+iJ/i\\F37$$\"+(*))pRDF-$\"+qG<-ZF37$$\"+VH))yDF-$ \"+%H()*GWF37$$\"+EDuLBF37$$\"+#GbZ)GF-$\"+E =zC?F37$$\"+O0_DHF-$\"+Z[&em\"F37$$\"+SOniHF-$\"+<&>.L\"F37$$\"+(H8L+$ F-$\"+L)p\"R&*!#67$$\"+[%y$QIF-$\"+H`(4@'F]\\l7$$\"+iNJyIF-$\"+j(QwK#F ]\\l7$$\"+6*))o6$F-$!+[(['>:F]\\l7$$\"+f!Ra:$F-$!+RiBhaF]\\l7$$\"+)QZQ >$F-$!+d^2([*F]\\l7$$\"+oduIKF-$!+O[*\\M\"F37$$\"+%*QjqKF-$!+%Q!>%y\"F 37$$\"+r,l3LF-$!+B!>N@#F37$$\"+70m[LF-$!+?8/xEF37$$\"+0&z[Q$F-$!+_aG2J F37$$\"+t#3\\U$F-$!+e$R\\f$F37$$\"+d)[KY$F-$!+\")RRuSF37$$\"+>h\\,NF-$ !+)*o;lXF37$$\"+LbWTNF-$!+6&)e\"4&F37$$\"+,/CyNF-$!+4mR*e&F37$$\"+zL#f h$F-$!+w:g7hF37$$\"+l)Hvl$F-$!+Zam1nF37$$\"+I6?&p$F-$!+q5-gsF37$$\"+aq sLPF-$!+g\\%=%yF37$$\"+Fp!Hx$F-$!+)QD3X)F37$$\"+a3#*3QF-$!+'*3rE!*F37$ $\"+M0JZQF-$!+?;We'*F37$$\"+*)zS&)QF-$!+6MXI5F-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-$!+)[X5D#F-7$$\"+C&QOY%F-$!+d+zqBF- 7$$\"+([(\\,XF-$!+HVk)\\#F-7$$\"+L76SXF-$!+'GZ)QEF-7$$\"+d6/\"e%F-$!+$ pm1!GF-7$$\"+!*)p&=YF-$!+?m5kHF-7$$\"+obhbYF-$!+9%4S9$F-7$$\"+(y;_p%F- $!+/&R[O$F-7$$\"+*zJZt%F-$!+`i>JOF-7$$\"+M`Y_ZF-$!+El5vPF-7$$\"+o))>qZ F-$!+iFfWRF-7$$\"+'\\o-y%F-$!+G<\")eSF-7$$\"+D\"Q.z%F-$!+!p\\]>%F-7$$ \"+SHP&z%F-$!+M@kwUF-7$$\"+axS+[F-$!+ " 0 "" {MPLTEXT 1 0 819 "C := \+ (x,y) -> (y-x)/(y+x): hh := 0.01: numsteps := 375: x0 := 1: y0 := 1:\n matrix([[`slope field: `,C(x,y)],[`initial point: `,``(x0,y0)],[`ste p width: `,hh],\n[`no. of steps: `,numsteps]]);``;\nmthds := [`But cher's scheme A`,`scheme with simple nodes`,`scheme with a relatively \+ large stability region`,`Butcher's scheme B with `*(c[5]=1/2,c[6]=1/2, b[5]=b[6]),`scheme with `*(c[5]=3/4,c[6]=3/4,b[5]=b[6])]:\nerrs := []: vals := []:\nDigits := 25:\nfor ct to 5 do\n Cn_RK6_||ct := RK6_|| ct(C(x,y),x,y,x0,y0,hh,numsteps,false);\n sm := 0: numpts := nops(Cn _RK6_||ct):\n for ii to numpts do\n if ct=1 then vals := [op(va ls),phi(Cn_RK6_||ct[ii,1])] end if;\n sm := sm+(Cn_RK6_||ct[ii,2] -vals[ii])^2;\n end do:\n errs := [op(errs),sqrt(sm/numpts)];\nend do:\nDigits := 10:\nlinalg[transpose]([mthds,evalf(errs)]);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#K%'matrixG6#7&7$%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$Q)pprint706\"" }}{PARA 11 " " 1 "" {XPPMATH 20 "6#%!G" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#K%'matrix G6#7'7$%3Butcher's~scheme~AG$\"+Q9&HU\"!#?7$%9scheme~with~simple~nodes G$\"+5l(\\G#!#@7$%Pscheme~with~a~relatively~large~stability~regionG$\" +mS&*RDF07$*&%9Butcher's~scheme~B~with~G\"\"\"6%/&%\"cG6#\"\"&#F8\"\"# /&F<6#\"\"'F?/&%\"bGF=&FGFCF8$\"+2ZB?DF07$*&%-scheme~with~GF86%/F;#\" \"$\"\"%/FBFPFEF8$\"+\"\\^X5\"F0Q)pprint716\"" }}}{PARA 0 "" 0 "" {TEXT -1 2 " " }}{PARA 0 "" 0 "" {TEXT -1 30 "The following code cons tructs " }{TEXT 260 20 "numerical procedures" }{TEXT -1 56 " for solut ions 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 = 4.749;" "6#/%\"xG-%&FloatG6$\" %\\Z!\"$" }{TEXT -1 16 " is also given." }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 682 "C := (x,y) -> (y-x)/(y+x): hh := 0.01: numsteps := 3 75: x0 := 1: y0 := 1:\nmatrix([[`slope field: `,C(x,y)],[`initial po int: `,``(x0,y0)],[`step width: `,hh],\n[`no. of steps: `,numsteps ]]);``;\nmthds := [`Butcher's scheme A`,`scheme with simple nodes`,`sc heme with a relatively large stability region`,`Butcher's scheme B wit h `*(c[5]=1/2,c[6]=1/2,b[5]=b[6]),`scheme with `*(c[5]=3/4,c[6]=3/4,b[ 5]=b[6])]:\nerrs := []:\nDigits := 30:\nfor ct to 5 do\n cn_RK6_||ct := RK6_||ct(C(x,y),x,y,x0,y0,hh,numsteps,true);\nend do:\nxx := 4.749 : cxx := evalf(phi(xx)):\nfor ct to 5 do\n errs := [op(errs),abs(cn_ RK6_||ct(xx)-cxx)];\nend do:\nDigits := 10:\nlinalg[transpose]([mthds, evalf(errs)]);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#K%'matrixG6#7&7$%0sl ope~field:~~~G*&,&%\"yG\"\"\"%\"xG!\"\"F,,&F-F,F+F,F.7$%0initial~point :~G-%!G6$F,F,7$%/step~width:~~~G$F,!\"#7$%1no.~of~steps:~~~G\"$v$Q)ppr int726\"" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%!G" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#K%'matrixG6#7'7$%3Butcher's~scheme~AG$\"+`5'4x\"!#>7$%9 scheme~with~simple~nodesG$\"+Wk+)z#!#?7$%Pscheme~with~a~relatively~lar ge~stability~regionG$\"+I./6JF07$*&%9Butcher's~scheme~B~with~G\"\"\"6% /&%\"cG6#\"\"&#F8\"\"#/&F<6#\"\"'F?/&%\"bGF=&FGFCF8$\"+@QYdHF07$*&%-sc heme~with~GF86%/F;#\"\"$\"\"%/FBFPFEF8$\"+F5_^8F0Q)pprint736\"" }}} {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 4 "The " } {TEXT 260 22 "root mean square error" }{TEXT -1 20 " over the interval " }{XPPEDIT 18 0 "[1, 4.75];" "6#7$\"\"\"-%&FloatG6$\"$v%!\"#" } {TEXT -1 82 " of each Runge-Kutta method is estimated as follows usin g the special procedure " }{TEXT 0 5 "NCint" }{TEXT -1 98 " to perfo rm numerical integration by the 7 point Newton-Cotes method over 200 e qual subintervals." }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 443 "mthds := [`Butcher's scheme A`,`scheme with simple nodes`,`scheme with a re latively large stability region`,`Butcher's scheme B with `*(c[5]=1/2, c[6]=1/2,b[5]=b[6]),`scheme with `*(c[5]=3/4,c[6]=3/4,b[5]=b[6])]:\ner rs := []:\nDigits := 20:\nfor ct to 5 do\n sm := NCint(('phi'(x)-'cn _RK6_||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[transp ose]([mthds,evalf(errs)]);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#K%'matri xG6#7'7$%3Butcher's~scheme~AG$\"+\\v`Hu!#@7$%9scheme~with~simple~nodes G$\"+t\\i*3\"F+7$%Pscheme~with~a~relatively~large~stability~regionG$\" +Jbl67F+7$*&%9Butcher's~scheme~B~with~G\"\"\"6%/&%\"cG6#\"\"&#F7\"\"#/ &F;6#\"\"'F>/&%\"bGF<&FFFBF7$\"+%RkX1\"F+7$*&%-scheme~with~GF76%/F:#\" \"$\"\"%/FAFOFDF7$\"+%owZC&!#AQ)pprint746\"" }}}{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 0 557 "evalf[30](plot(['cn_RK6_1'( x)-'phi'(x),'cn_RK6_2'(x)-'phi'(x),'cn_RK6_3'(x)-'phi'(x),'cn_RK6_4'(x )-'phi'(x),\n'cn_RK6_5'(x)-'phi'(x)],x=1..3.75,-3.9e-16..3.5e-16,font= [HELVETICA,9],\ncolor=[COLOR(RGB,.95,.2,0),COLOR(RGB,.45,0,.95),COLOR( RGB,0,.25,1),COLOR(RGB,0,.75,.2),COLOR(RGB,.95,.45,0)],\nlegend=[`Butc her's scheme A`,`scheme with simple nodes`,`scheme with a relatively l arge stability region`,`Butcher's scheme B with c[5]=c[6]=1/2 and b[5] =b[6]`,`scheme with c[5]=c[6]=3/4 and b[5]=b[6]`],title=`error curves \+ for 7 stage order 6 Runge-Kutta methods`));" }}{PARA 13 "" 1 "" {GLPLOT2D 893 735 735 {PLOTDATA 2 "6+-%'CURVESG6%7V7$$\"\"\"\"\"!$F*F* 7$$\"?LLLLLLLL$eR(G$o&=F/$!0YH.L oW5\"F27$$\"?nmmmmmmm;z\\#3)=>F/$!0ii9$G0&>\"F27$$\"?nmmmmmmmm;C&48(>F /$!03-eK-jF\"F27$$\"?+++++++]PM()4QK?F/$!0#z*435+P\"F27$$\"?nmmmmmmmmT !eRk3#F/$!0I(3'RwSX\"F27$$\"?+++++++]P%[U]d9#F/$!0dVu!QKa:F27$$\"?++++ +++]7`u$GA?#F/$!0k^b>@ql\"F27$$\"?nmmmmmmmT5!>d6E#F/$!0:p(RTNnF2 7$$\"?LLLLLLL$3F%fIFMCF/$!0t&RF3cA@F27$$\"?+++++++]ilF?0([#F/$!0p>)zNt UAF27$$\"?LLLLLLLL3FfZ0WDF/$!0yj]6.cQ#F27$$\"?+++++++++veT%Hg#F/$!0\"o [>V7UDF27$$\"?++++++++v=ZfbgEF/$!0s\"RR^M3FF27$$\"?+++++++]P%[J)H;FF/$ !0&4DFU2qGF27$$\"?++++++++DJ52>yFF/$!0Ve+38M2$F27$$\"?nmmmmmmmmT!y.Q$G F/$!0[4z?wZF$F27$$\"?++++++++]7.E=$*GF/$!0k(*[X'H1NF27$$\"?LLLLLLL$3F> k))p%HF/$!0'pd#=%yFPF27$$\"?++++++++]PaG\"e+$F/$!1-&z'HLF,S!#J7$$\"?nm mmmmm;/,a=;hIF/$!1:\"3$3?<&H%Fgv7$$\"?+++++++](=n]JF/$!1u\\@r&)H9YF gv7$$\"?nmmmmmmm;H#)>evJF/$!1OfQS[Em\\Fgv7$$\"?+++++++]P4Y(*zMKF/$!0i^ 'ez&pQ&F27$$\"?LLLLLLLLL$3\\L=H$F/$!0&fr(3,e&eF27$$\"?LLLLLLLLeRK(e,N$ F/$!0Ii#z4h(R'F27$$\"?nmmmmmm;/^65+3MF/$!0Yk-QW8(pF27$$\"?+++++++++ve? :hMF/$!0Rtm5/[k(F27$$\"?nmmmmmmm\"zp:p?_$F/$!0(R(**o7'*\\)F27$$\"?LLLL LLLLL$e8ald$F/$!05\\,TR)o$*F27$$\"?mmmmmmmT&)e6.g0OF/$!02$[.J'Q#**F27$ $\"?+++++++]PM([YYj$F/$!1pI\\avt`5F27$$\"?+++++++++v$QZCm$F/$!1>&H5&** [?6F27$$\"?+++++++]i:!G[-p$F/$!1_([/$pb%>\"F27$$\"?+++++++D\"y+9C,s$F/ $!1b$[;tm#z7F27$$\"$v$!\"#$!1$\".89^e[_%F27$FC$\".;C4$QJ _F27$FH$\".;qaJ>'eF27$FM$\".)Rg>7JkF27$FR$\".G2KQx%pF27$FW$\".Jjn\")[W (F27$Ffn$\".;[zcL#zF27$F[o$\".(Qwa\\D$)F27$F`o$\".bDD%Rg()F27$Feo$\".y $\\t3)=*F27$Fjo$\"./j!G\"zf*F27$F_p$\".8Y@;='**F27$Fdp$\"/,If69S5F27$F ip$\"/sV#Guz2\"F27$F^q$\"/+$))[o<7\"F27$Fcq$\"/:Szy%4;\"F27$Fhq$\"/T6B fb07F27$F]r$\"/_bdba\\7F27$Fbr$\"/#>OQrlH\"F27$Fgr$\"/oLd-mS8F27$F\\s$ \"/v+G%e1R\"F27$Fas$\"/O/UM`W9F27$Ffs$\"/b#G3(G$\\\"F27$F[t$\"/j8N0z[: F27$F`t$\"/7!)='\\)3;F27$Fet$\"/QG&G64n\"F27$Fjt$\"/%3bS!4LF27$F^v$\"/LkZ#\\e.#F27$ Fcv$\"0%*[AN()p7#Fgv7$Fiv$\"0$=Dq&*\\?AFgv7$F^w$\"01!Hbp#F27$Fbx$\"/Ivm>@[GF27 $Fgx$\"/\"eY;H3,$F27$F\\y$\"/2*[!>'o=$F27$Fay$\"/-g&oz#3MF27$Ffy$\"/#) *Qj=!HOF27$F`z$\"/v)*yCX3RF27$Fjz$\"/*>h#QeFUF27$Fd[l$\"/\\%GW\\?i%F2- Fj[l6&F\\\\l$\"#XFf[lF+F]\\l-Fc\\l6#%9scheme~with~simple~nodesG-F$6%7S F'7$F-$\"./-MC7c\"F27$F4$\".$\".C`>Xxh%F 27$FC$\"._i>a;N&F27$FH$\".Xu*[(=,'F27$FM$\".vs5*\\7mF27$FR$\".hR&o[hrF 27$FW$\".JW//Vp(F27$Ffn$\".q7fx5@)F27$F[o$\".O&)*[Y[')F27$F`o$\".9yr:V 7*F27$Feo$\".#=EnN&f*F27$Fjo$\"/NT>?)\\+\"F27$F_p$\"/^kFgWX5F27$Fdp$\" /[.!*em%4\"F27$Fip$\"/([yV.t8\"F27$F^q$\"/TFuqm'=\"F27$Fcq$\"/rZ'*=(3B \"F27$Fhq$\"/*G7.v:G\"F27$F]r$\"/L!QT)*=L\"F27$Fbr$\"/RKqNw&Q\"F27$Fgr $\"/nn^\"[hV\"F27$F\\s$\"/j-_;!Q\\\"F27$Fas$\"/Sto.+c:F27$Ffs$\"/C^A\" pBh\"F27$F[t$\"/Nen].x;F27$F`t$\"/i=j83g#*=F27$F_u$\"/pj:*=*z>F27$Fdu$\"/'zX'p>2O#Fgv7$Fiv$\"0-\")Q*ostCFgv7$F^w$\" 0(y&=(p'yf#Fgv7$Fcw$\"0KscG85t#Fgv7$Fhw$\"/TGgoP&)GF27$F]x$\"/o,dww^IF 27$Fbx$\"/Nd[\"o+C$F27$Fgx$\"/'=xNk+W$F27$F\\y$\"/oh`.!*fOF27$Fay$\"/_ +d#)HORF27$Ffy$\"/KbFlJ7UF27$F`z$\"/Lga>7lXF27$Fjz$\"/#[KbP5(\\F27$Fd[ l$\"/[EIsptaF2-Fj[l6&F\\\\lF+$\"#DFf[lF(-Fc\\l6#%Pscheme~with~a~relati vely~large~stability~regionG-F$6%7VF'7$F-$\".j4&\\_;LF27$F4$\".[*yG$>u &F27$F9$\".w6OGpG)F27$F>$\"/<_W?a_5F27$FC$\"/g^DYZZ7F27$FH$\"/`f+$oPV \"F27$FM$\"/8C1g#Hh\"F27$FR$\"/@138o%y\"F27$FW$\"/%[.:<3'>F27$Ffn$\"/x TpX)*R@F27$F[o$\"/$p;0>\")H#F27$F`o$\"/CqZ\"fiZ#F27$Feo$\"/7%*3OGfEF27 $Fjo$\"/w4o4nUGF27$F_p$\"/C#QpE'3IF27$Fdp$\"/N%>c!3=KF27$Fip$\"/V/&=[e S$F27$F^q$\"/+@s76COF27$Fcq$\"/AV)Rt5#QF27$Fhq$\"/HP=xjbSF27$F]r$\"/mE ,R>'H%F27$Fbr$\"/WzCF9cXF27$Fgr$\"/U\"[@Mwz%F27$F\\s$\"/X?rXE'3&F27$Fa s$\"/:M,`d+aF27$Ffs$\"/o&o5`$)o&F27$F[t$\"/`+/,(3.'F27$F`t$\"/RfE&)p2k F27$Fet$\"/!=@uW\"4oF27$Fjt$\"/'*4\\)3@?(F27$F_u$\"/Eu)eNop(F27$Fdu$\" /9vu6C)=)F27$Fiu$\"/fBXBIb()F27$F^v$\"/IR\\$o(*H*F27$Fcv$\"0zQnIRP(**F gv7$Fiv$\"1&RsOM7+2\"Fgv7$F^w$\"1ZRF+P:\\6Fgv7$Fcw$\"1(y$=PEnO7Fgv7$Fh w$\"0a[`)*o:M\"F27$F]x$\"0=g%f\"F27$Fgx$\"0DHR LR'QF27$Fay$\"0IpT9QM7#F27$Ffy$\"0:.W%H^UBF27$F[ z$\"0=8;^tC[#F27$F`z$\"0%>b!yorj#F27$Fez$\"0O=P*QW0GF27$Fjz$\"0d)oj&R? *HF27$F_[l$\"03YMP@_?$F27$Fd[l$\"0Smn&yoVMF2-Fj[l6&F\\\\lF+$\"#vFf[lF_ \\l-Fc\\l6#%TButcher's~scheme~B~with~c[5]=c[6]=1/2~and~b[5]=b[6]G-F$6% 7SF'7$F-$\".\\7(=iz8F27$F4$\".-3)e4PBF27$F9$\".,FE`aG$F27$F>$\".k]d)Qm SF27$FC$\".Wy]^Yq%F27$FH$\".RBugXF&F27$FM$\".8Z(3#))y&F27$FR$\".A,(**3 biF27$FW$\".Rpp#3.nF27$Ffn$\".eJMsK8(F27$F[o$\".t!pv(Q\\(F27$F`o$\".:D 3KF)yF27$Feo$\".%4(e8QE)F27$Fjo$\".Cg]Gvi)F27$F_p$\".#oZ8\\\\*)F27$Fdp $\".cDVGmL*F27$Fip$\".yPCG\"o'*F27$F^q$\"/*=%Rm205F27$Fcq$\"/PQ\\g?R5F 27$Fhq$\"/jCmK#y2\"F27$F]r$\"/Z'H(el:6F27$Fbr$\"/a\"G0Qf:\"F27$Fgr$\"/ ))of%\\O>\"F27$F\\s$\"/%o?EDgB\"F27$Fas$\"/mz*f&[\"G\"F27$Ffs$\"/kmQ:W A8F27$F[t$\"/%y)HPgo8F27$F`t$\"/ZCLuC=9F27$Fet$\"/yn5p2p9F27$Fjt$\"/(Q Kd%3?:F27$F_u$\"/'Qk8R*z:F27$Fdu$\"/Q\\<6oO;F27$Fiu$\"/tL$yv/q\"F27$F^ v$\"/G^3ODhg='4>$=Fgv7$Fiv$\"0RF,Fgv7$F^w$\"0x+_,>8 )>Fgv7$Fcw$\"0\"fYR4Vj?Fgv7$Fhw$\"/L6XD9c@F27$F]x$\"/f51u4`AF27$Fbx$\" /rmR)o0O#F27$Fgx$\"/!RIsX[Z#F27$F\\y$\"/94'*o(Gf#F27$Fay$\"/:I='e+u#F2 7$Ffy$\"/+Mmry%)GF27$F`z$\"/bg;y(41$F27$Fjz$\"/ooqL?bKF27$Fd[l$\"/*)y7 %R8\\$F2-Fj[l6&F\\\\lF]\\lF[flF+-Fc\\l6#%Hscheme~with~c[5]=c[6]=3/4~an d~b[5]=b[6]G-%%FONTG6$%*HELVETICAG\"\"*-%+AXESLABELSG6$Q\"x6\"Q!F^dn-% &TITLEG6#%Uerror~curves~for~7~stage~order~6~Runge-Kutta~methodsG-%%VIE WG6$;F(Fd[l;$!#R!#<$\"#NF[en" 1 2 0 1 10 0 2 9 1 4 2 1.000000 45.000000 45.000000 0 1 "Butcher's scheme A" "scheme with simple nodes " "scheme with a relatively large stability region" "Butcher's scheme \+ B with c[5]=c[6]=1/2 and b[5]=b[6]" "scheme with c[5]=c[6]=3/4 and b[5 ]=b[6]" }}}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 559 "evalf[30](plot(['cn_RK6_1'(x)-'phi'(x),'cn_RK6_2'(x) -'phi'(x),'cn_RK6_3'(x)-'phi'(x),'cn_RK6_4'(x)-'phi'(x),\n'cn_RK6_5'(x )-'phi'(x)],x=3.75..4.6,-8.9e-14..8.9e-14,font=[HELVETICA,9],\ncolor=[ COLOR(RGB,.95,.2,0),COLOR(RGB,.45,0,.95),COLOR(RGB,0,.25,1),COLOR(RGB, 0,.75,.2),COLOR(RGB,.95,.45,0)],\nlegend=[`Butcher's scheme A`,`scheme with simple nodes`,`scheme with a relatively large stability region`, `Butcher's scheme B with c[5]=c[6]=1/2 and b[5]=b[6]`,`scheme with c[5 ]=c[6]=3/4 and b[5]=b[6]`],title=`error curves for 7 stage order 6 Run ge-Kutta methods`));" }}{PARA 13 "" 1 "" {GLPLOT2D 958 662 662 {PLOTDATA 2 "6+-%'CURVESG6%7[o7$$\"$v$!\"#$!1RQF1$!1LXP1#>Rs\"F-7$$\"?nmmmmmm;H2%oHg&QF1 $!1S\"y;%fz+=F-7$$\"?+++++++]i:VeYtQF1$!0A+)zlB.>F17$$\"?nmmmmmm;H#QM) \\\"*QF1$!0,sh.Ox,#F17$$\"?+++++++]7.9IZ4RF1$!0bo'pn%*G@F17$$\"?LLLLLL LL$3<2d&***[F17$$\"?LLLLLLLLeRAA\")RTF1$!0Gty\"oUu`F17$$\"?LLLLLL L$ek=pQl:%F1$!0GgN-5Hu&F17$$\"?nmmmmmmmT&)[/euTF1$!09NZsz+O'F17$$\"?nm mmmmm;H#Qw?L>%F1$!06!=rka(4(F17$$\"?+++++++]P4jUj4UF1$!0e`F]Ko&yF17$$ \"?nmmmmmmm\"H#*G`sA%F1$!0!QjbTil&)F17$$\"?+++++++++DnaXXUF1$!0-7cUJ\" ['*F17$$\"?++++++++DJZFEjUF1$!1b\"RaTur4\"F17$$\"?+++++++]iS1A\\!G%F1$ !1J,XXA0c7F17$$\"?++++++++v=$eA'*H%F1$!1x')H;brK9F17$$\"?LLLLLLLLLex? \"oJ%F1$!19(>$=r'>f\"F17$$\"?++++++++](=`l^L%F1$!1$3>nT$=b=F17$$\"?nmm mmmm;HK*['z^VF1$!1r/eR'>>=#F17$$\"?++++++++]i=&y*pVF1$!1'z*3!G?lf#F17$ $\"?LLLLLLL$eR7R'3(Q%F1$!152=t'H'3HF17$$\"?+++++++]7`Z!p\\S%F1$!1%G1$* *y?)\\$F17$$\"?LLLLLLLL$3Fh_CU%F1$!1sDj1+v!G%F17$$\"?+++++++]i:&Gc2W%F 1$!1JC3.$4*>`F17$$\"?mmmmmmmmm;(*[QeWF1$!1$=bCA!)*[iF17$$\"?mmmmmmmmT5 *p7kZ%F1$!1E(zI^C*\\xF17$$\"?LLLLLLL$eR2^%F1$!2nl!QbY068F17$$\"?LLLLLLLL3_['[&HXF1$!2qZX%e/96 $F17$$\"?+++ ++++++D#GQHd%F1$!2O()[*o#R3c$F17$$\"?+++++++v=U%yMsd%F1$!2Bq%yU*=+n$F1 7$$\"?+++++++]Pf'GJ:e%F1$!2'))*p9di)*H%F17$$\"?++++++DJ&pd()RQe%F1$!2> %=Fr/!)HVF17$$\"?++++++]7`%\\Y[he%F1$!2j'3@!pd\\Q%F17$$\"?++++++v$4@T0 d%)e%F1$!2A`^yH'pTYF17$$\"?+++++++voHVcw!f%F1$!2kN`6pp/D&F17$$\"?+++++ +DcEZKU2$f%F1$!2I7rIz&R(G&F17$$\"?++++++]P%[;#GQ&f%F1$!2\"ea-js#zL&F17 $$\"?+++++]7GjB;r`'f%F1$!2-da*y?^'R&F17$$\"?++++++v=U#3T\"p(f%F1$!2:KN #4H=IbF17$$\"?+++++]P4@T0d%))f%F1$!23MPTdAG$eF17$$\"#Y!\"\"$!2^ez$R\"R nZ'F1-%&COLORG6&%$RGBG$\"#&*F*$\"\"#F]_l$\"\"!Fi_l-%'LEGENDG6#%3Butche r's~scheme~AG-F$6%7eo7$F($\"/\\%GW\\?i%F-7$F/$\"/&F-7$FD$\"/ns*Qz&[`F-7$FI$\" /Z89\")p.bF-7$FN$\".OAelrp&F17$FS$\".Z)oHL4fF17$FX$\".S@1\\o6'F17$Fgn$ \".;_2p%HjF17$F\\o$\".]_&=_hlF17$Fao$\".[xum*RoF17$Ffo$\".E$o%*3TrF17$ F[p$\".+kcpDS(F17$F`p$\".CoLi)*o(F17$Fep$\".kjWyY1)F17$Fjp$\".T'=zYV%) F17$F_q$\".3xQ,a\"))F17$Fdq$\".dwGPv=*F17$Fiq$\".bJuFHn*F17$F^r$\"/?Fa 0#)=5F17$Fcr$\"/IOim(H2\"F17$Fhr$\"/5TxK)f6\"F17$F]s$\"/aGH!3x<\"F17$F bs$\"/]cc#zZC\"F17$Fgs$\"/%)*p\"o918F17$F\\t$\"/CfYTVT;F17$F_v$\"/HsqpQy:F17$Fdv$\"/k&*[DYC9F17 $Fiv$\"/'>)\\6B#H\"F17$F^w$\".pCN&G7))F17$Fcw$\".=%*fpgQ\"F17$Fhw$!/:# Q1+/7\"F17$F]x$!/s_=Ub*R#F17$Fbx$!/9X**yP/\\F17$Fgx$!/(f-A8%3%*F17$F\\ y$!0b!y#e#*>q\"F17$Fay$!0@7]?Gu\"GF17$Ffy$!0pE>-=[D$F17$F[z$!0*=THv#o6 %F17$F`z$!0t.CP&z4aF17$Fez$!0)oJ.2rlrF17$Fjz$!0I#Hyjq9!)F17$F_[l$!0'Rp k`/N&*F17$$\"?++++++voHz2C,uXF1$!0sNDX0zc*F17$$\"?++++++]PfLLl3vXF1$!0 $\\c!3dmh*F17$$\"?++++++D1*y)e1;wXF1$!0;A.*z%fr*F17$Fd[l$!0nFo)3cY**F1 7$$\"?++++++]7y]NIQzXF1$!1M9O[=xf6F17$Fi[l$!1d>/O_/w7F17$$\"?+++++]iS; =\"e&o#e%F1$!1AbDdHO!G\"F17$F^\\l$!1A9sls4&G\"F17$$\"?+++++](=Ud.<%*\\ e%F1$!1OUK\"z9>H\"F17$Fc\\l$!14X$zW\\hI\"F17$$\"?+++++]7.K`fFI(e%F1$!1 C.Q`&47M\"F17$Fh\\l$!1SJAhR&eU\"F17$$\"?+++++]P%)*3([8h*e%F1$!1!)*pv-c _h\"F17$F]]l$!1)3!e3`,AF17$Ff^l$!1FWrM(*>-?F17$$\"?+++++voagq_GU*f%F1$!1![F=$=0T@F17$ F[_l$!1=A]9XJUBF1-Fa_l6&Fc_l$\"#XF*Fh_lFd_l-F[`l6#%9scheme~with~simple ~nodesG-F$6%7eo7$F($\"/[EIsptaF-7$F/$\"/Gz/yRCcF-7$F5$\"/>p#*QL(y&F-7$ F:$\"/iL]Or()fF-7$F?$\"/6r2\"eE?'F-7$FD$\"/UF[Fh.kF-7$FI$\"/c%>&=M-mF- 7$FN$\".dJu;G&oF17$FS$\".1[j2y7(F17$FX$\".PuysdR(F17$Fgn$\"..$oEfpwF17 $F\\o$\".%f9ivrzF17$Fao$\".ar*)Q\\L)F17$Ffo$\".\")e2'fG()F17$F[p$\".b) >f#z1*F17$F`p$\".hQL3UW*F17$Fep$\".4(R(>q$**F17$Fjp$\"/u:]k!R/\"F17$F_ q$\"/e\"Rr*z#4\"F17$Fdq$\"/%>q8D?9\"F17$Fiq$\"/m&ev]o?\"F17$F^r$\"/Lyg -?w7F17$Fcr$\"/'py\\$R\\8F17$Fhr$\"/M^6C<29F17$F]s$\"/fTVL'>\\\"F17$Fb s$\"/xkvPF&e\"F17$Fgs$\"/CJa%y@n\"F17$F\\t$\"/$>l@WEv\"F17$Fat$\"/,S/x F17$F[u$\"/R>hLcf?F17$F`u$\"/iMy.;]@F17$Feu$\"/ 0PCtm=AF17$Fju$\"/F6jo()pAF17$F_v$\"/^08KIeAF17$Fdv$\"/3'z.Tu:#F17$Fiv $\"/Um6ldi?F17$F^w$\"/'z%[\")4'p\"F17$Fcw$\".Ro94\\\")*F17$Fhw$!.1&fn1 AGF17$F]x$!/g+$=2%)e\"F17$Fbx$!/1'QL0@?%F17$Fgx$!/!Qzr`H)*)F17$F\\y$!0 0\\xtUmr\"F17$Fay$!0\\#*))od]#HF17$Ffy$!0Y*o]tm*R$F17$F[z$!0./%z9KRVF1 7$F`z$!03To+QLv&F17$Fez$!0I%R&y[*ywF17$Fjz$!0&Q\\XN\">h)F17$F_[l$!1(H% z>TPG5F17$F[jl$!1I]!**)*G>.\"F17$F`jl$!1ZvT+#Rs.\"F17$Fejl$!1gX@f18[5F 17$Fd[l$!1Tm!eZ>N2\"F17$F][m$!1H[A$G)eb7F17$Fi[l$!1P$=[x7PQ\"F17$Fe[m$ !1Jy))\\cR)Q\"F17$F^\\l$!19wqX!RNR\"F17$F]\\m$!1e%eVI()4S\"F17$Fc\\l$! 1b:jL*RmT\"F17$Fe\\m$!1`X\"F17$Fh\\l$!1&o;gs,)[:F17$F]]m$!1Nq,%) 4-eI\"*)=F17$Fg]l$!1Zg\\Od&*4>F1 7$F\\^l$!1T_EjVXR>F17$Fa^l$!1@FM$GFE,#F17$Fa^m$!17qT%*>W\"3#F17$Ff^l$! 1BXpqPK&=#F17$Fi^m$!1e;*32D*QBF17$F[_l$!1S>sO0bhDF1-Fa_l6&Fc_lFh_l$\"# DF*$\"\"\"Fi_l-F[`l6#%Pscheme~with~a~relatively~large~stability~region G-F$6%7in7$F($\"0Smn&yoVMF-7$F/$\"0qX2uh^d$F-7$F5$\"0b@&44kFPF-7$F:$\" 0\")e3$zb>RF-7$F?$\"0'=5\"HI>F-7$FD$\"0sUBel&>VF-7$FI$\"0(HsnIO6XF-7 $FN$\"/1aCRMmZF17$FS$\"/)pPPn10&F17$FX$\"/::DNEE`F17$Fgn$\"/0ST='og&F1 7$F\\o$\"/-g#47`$fF17$Fao$\"/x%eWR\"QjF17$Ffo$\"/.-`:K&y'F17$F[p$\"/9m +%*HirF17$F`p$\"/4te%[Jg(F17$Fep$\"/:m%y*f*>)F17$Fjp$\"/OOLl\"3%))F17$ F_q$\"/GtWs#*\\%*F17$Fdq$\"0'p=t?\"*35F17$Fiq$\"06v2)R^)4\"F17$F^r$\"0 !fAI\"R(*>\"F17$Fcr$\"0aCp(zj58F17$Fhr$\"02iwFTmR\"F17$F]s$\"0tqI%=!*Q :F17$Fbs$\"0lp?G#G2Q8.AP(F17$Fcw$\"0#psq'\\(3))F1 7$Fhw$\"1hYvvk;l5F17$F]x$\"1*)48&=0kA\"F17$Fbx$\"1W1v.pMw9F17$Fgx$\"12 ^&=m*))G=F17$F\\y$\"1;Oc$)e<&H#F17$Fay$\"1h[-XuxdGF17$Ffy$\"1'fJ+zak1$ F17$F[z$\"1AvUwlK>MF17$F`z$\"13'e+k2h)QF17$Fez$\"1N7Ky'F17$Fb]l$\"1)p;0+L3 $oF17$Fg]l$\"1C=6Zv=\"*oF17$F\\^l$\"1!oyt/*)3&pF17$Fa^l$\"1ig\"QO>V2(F 17$Ff^l$\"1uPZ'p\\cL(F17$F[_l$\"1un#\\:OU'yF1-Fa_l6&Fc_lFh_l$\"#vF*Ff_ l-F[`l6#%TButcher's~scheme~B~with~c[5]=c[6]=1/2~and~b[5]=b[6]G-F$6%7eo 7$F($\"/*)y7%R8\\$F-7$F/$\"/RdT9DlNF-7$F5$\"/VUS.FROF-7$F:$\"/1`#o%QGP F-7$F?$\"/e&)y3/BQF-7$FD$\"/7wWax9RF-7$FI$\"/RucwY/SF-7$FN$\".J*z6U6TF 17$FS$\".mqJz!GUF17$FX$\".05/&F17$Fep$\".AWk3mR&F17$Fjp$\".[+^\\Ef&F17$F_q$\".r:pQOz&F17$Fd q$\".HHLx>*fF17$Fiq$\".#=^)yaC'F17$F^r$\".%40L.8lF17$Fcr$\".(z\"o;tz'F 17$Fhr$\".ho\\26.(F17$F]s$\".U=MNgN(F17$Fbs$\".l8pDBr(F17$Fgs$\".<6Ow; /)F17$F\\t$\".H;D)yd$)F17$Fat$\".0eZ;\\u)F17$Fft$\".qk7x\\8*F17$F[u$\" ..JgA,]*F17$F`u$\".#Rr]:Q)*F17$Feu$\"/!o([T$*45F17$Fju$\"/]nQM8F5F17$F _v$\"/@16\\!)=5F17$Fdv$\".!Q0Cr`(*F17$Fiv$\".=@,y@P*F17$F^w$\".Tn#3M)) yF17$Fcw$\".:aNk,-&F17$Fhw$!,pyQT#GF17$F]x$!.ikb%QK_F17$Fbx$!/s'=EHdc \"F17$Fgx$!/L$)3BXwMF17$F\\y$!/$flq(RcnF17$Fay$!0e0%Rl_h6F17$Ffy$!0[W$ ='GEN\"F17$F[z$!0n165S:t\"F17$F`z$!0t:&=%=JI#F17$Fez$!0OAn-:M3$F17$Fjz $!01]ayDFY$F17$F_[l$!0QAYx;<9%F17$F[jl$!0x%QtZ1cTF17$F`jl$!0\"*))\\#Rf xTF17$Fejl$!0L0%fN$>A%F17$Fd[l$!0F+.;gaK%F17$F][m$!0>(GDVmn]F17$Fi[l$! 08)G(\\$Q*e&F17$Fe[m$!01JDW-$3cF17$F^\\l$!0$)Gxc5\"HcF17$F]\\m$!09@'3L PfcF17$Fc\\l$!00G%R#fKs&F17$Fe\\m$!0nQYJ99)eF17$Fh\\l$!0)H7*4!pjiF17$F ]]m$!0\">0.q!ojf\")F17$Fa^m$!0DJPbu=W)F17$F f^l$!0Yut._y'))F17$Fi^m$!0Juu9Mv\\*F17$F[_l$!1UE*\\O))4/\"F1-Fa_l6&Fc_ lFd_lFb_mFh_l-F[`l6#%Hscheme~with~c[5]=c[6]=3/4~and~b[5]=b[6]G-%%FONTG 6$%*HELVETICAG\"\"*-%+AXESLABELSG6$Q\"x6\"Q!F[io-%&TITLEG6#%Uerror~cur ves~for~7~stage~order~6~Runge-Kutta~methodsG-%%VIEWG6$;F(F[_l;$!#*)!#: $\"#*)Fhio" 1 2 0 1 10 0 2 9 1 4 2 1.000000 45.000000 45.000000 0 1 "B utcher's scheme A" "scheme with simple nodes" "scheme with a relativel y large stability region" "Butcher's scheme B with c[5]=c[6]=1/2 and b [5]=b[6]" "scheme with c[5]=c[6]=3/4 and b[5]=b[6]" }}}}{PARA 0 "" 0 " " {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 559 "evalf[30] (plot(['cn_RK6_1'(x)-'phi'(x),'cn_RK6_2'(x)-'phi'(x),'cn_RK6_3'(x)-'ph i'(x),'cn_RK6_4'(x)-'phi'(x),\n'cn_RK6_5'(x)-'phi'(x)],x=4.6..4.75,-2. 9e-11..1.5e-12,font=[HELVETICA,9],\ncolor=[COLOR(RGB,.95,.2,0),COLOR(R GB,.45,0,.95),COLOR(RGB,0,.25,1),COLOR(RGB,0,.75,.2),COLOR(RGB,.95,.45 ,0)],\nlegend=[`Butcher's scheme A`,`scheme with simple nodes`,`scheme with a relatively large stability region`,`Butcher's scheme B with c[ 5]=c[6]=1/2 and b[5]=b[6]`,`scheme with c[5]=c[6]=3/4 and b[5]=b[6]`], title=`error curves for 7 stage order 6 Runge-Kutta methods`));" }} {PARA 13 "" 1 "" {GLPLOT2D 952 675 675 {PLOTDATA 2 "6+-%'CURVESG6%7dq7 $$\"#Y!\"\"$!2^ez$R\"RnZ'!#H7$$\"?++++++++]7t&pKg%F-$!2v(Q->U+WlF-7$$ \"?+++++++](=7T9hg%F-$!2Kj\\yR(>Z _(F-7$$\"?++++++++DJaU`7YF-$!2tM(zLkQm\")F-7$$\"?+++++++]P%GZRdh%F-$!2 n%\\l/&o.H)F-7$$\"?+++++++](=276(=YF-$!2dO%\\8;XV!*F-7$$\"?+++++++](o* *3)y@YF-$!3iBJwigsJ5F-7$$\"?+++++++](ofHq\\i%F-$!3yb%H:[F\\/\"F-7$$\"? +++++++]Pf'HU\"GYF-$!3M6:Vr%[S6\"F-7$$\"?++++++++]7*309j%F-$!3V\"H$>y[ /B8F-7$$\"?++++++++Dce*yUj%F-$!3qw]kk2rP8F-7$$\"?++++++++]([D9vj%F-$!3 ?G,0)Hl%F-$!3`xmZG<@.BF- 7$$\"?+++++++](=-p6jl%F-$!3ol)*H(>I8O#F-7$$\"?+++++++++vS.EfYF-$!3s^'z &yr5sFF-7$$\"?+++++++](=xZ&\\iYF-$!3zjD.o=)*4JF-7$$\"?+++++++]i:$4wbm% F-$!3*G/6\"o%F-$!3%o,/ \"[7LigF-7$$\"?++++++++voo6A%o%F-$!3rxHgQcafhF-7$$\"?+++++++](=KCFeo%F -$!3nu2y3ApbiF-7$$\"?+++++++++v*\\lF-7$$\"?++++++ +]PM[X+*o%F-$!3?5h%=(zUouF-7$$\"?++++++++v$*ydd!p%F-$!3AlNnm\")*F-7$$\"?++ +++++]ilS*3&)p%F-$!4ZAr**3x8p/\"F-7$$\"?+++++++DJq+tE*p%F-$!4,x;+3dtq: \"F-7$$\"?+++++++++vgc-+ZF-$!4v#=jYNoqH8F-7$$\"?++++++++v=$3X;q%F-$!4q c(z(Qk9t/ZF-$!4.s?AAj]'o8F-7$$\"?+++++++](o/Q*>1ZF-$!4ady\"fQsN*R\"F-7$$\"?+ +++++]7.#)G:+2ZF-$!4ji^+GoymV\"F-7$$\"?+++++++v=F-7$$\"?++++++]P4'Hti, r%F-$!4s^;jjKiM4#F-7$$\"?++++++D1R]7,a5ZF-$!4?2\"GjE%G#)4#F-7$$\"?++++ +++vo/#\\<4r%F-$!4viHY<&=-.@F-7$$\"?++++++]7G8^An6ZF-$!4\"f!pkS<&p7@F- 7$$\"?+++++++](=-,FCr%F-$!4aOHbXHDD7#F-7$$\"?+++++++]il\"*[+9ZF-$!4Mj) H^ph3X@F-7$$\"?+++++++]P4tFe:ZF-$!4L\\R`$H17$QH#F-7$$\"?++++ ++](=nVx'*yr%F-$!4(33$z,7q=U#F-7$$\"?++++++++]73\"o'=ZF-$!40#G-\\P*)*o k#F-7$$\"?++++++]Pf3BcZ>ZF-$!4QzkjsR\"[_IF-7$$\"?+++++++vo/QJG?ZF-$!47 jte<8C:Y$F-7$$\"?++++++]7y+`14@ZF-$!4.k'*p8Gu,[$F-7$$\"?+++++++](oz;)* =s%F-$!4?T$4]0s4*\\$F-7$$\"?+++++++vVtLOXBZF-$!4Tak(*\\ b5Lx\"oK!RF-7$$\"?++++++++DJw/>GZF-$!4VS;%\\ElX9UF-7$$\"?++++++]iSmM#z *GZF-$!4Cw))44*[')fZF-7$$\"?+++++++Dc,$*zwHZF-$!4o(*>nQOSto&F-7$$\"?++ +++DJq&4yem)HZF-$!4OD+Ee$Q))ReF-7$$\"?+++++]i:Ng#=l*HZF-$!41GO(\\Hz6-g F-7$$\"?+++++v$4Y(RxP1IZF-$!4Q8/='zpmkgF-7$$\"?++++++D19>sB;IZF-$!4h$z +(G>s!pgF-7$$\"?+++++](oHzJ$*Re@#eAhF-7$$\"?+++++++vVt$3&zKZF-$!4C_!G/EiM!>'F-7 $$\"?+++++++++vdYCMZF-$!42i@];BH%piF-7$$\"?+++++++DJXX`2NZF-$!4do^kF-7$$\"?+++++++v$f3sOnt%F- $!4L\"3[9;@RmmF-7$$\"?++++++++Dc3ucPZF-$!4'=&[%zKq'*zqF-7$$\"?+++++++v oa&Q5$QZF-$!4)=H]cp0T]xF-7$$\"?+++++++]7`iL0RZF-$!40>L&4*RFI*))F-7$$\" ?+++++++Dc^RjzRZF-$!5?cEI`')ywx5F-7$$\"?+++++++++];$R0u%F-$!5-y%f8i>OB :\"F-7$$\"?+++++++v$fLlB@u%F-$!5\\)*[xT5$)*y;\"F-7$$\"?+++++++](=-*zqV ZF-$!5%[bLlP.&)[=\"F-7$$\"?+++++++vV['>mWu%F-$!5c%=,'e:))H&>\"F-7$$\"? +++++++++v-WAXZF-$!5T#f9zAe&R57F-7$$\"?+++++++Dc,4E)fu%F-$!5G+o9U&QkdB \"F-7$$\"?+++++++]7G:3uYZF-$!5Q`]'3?Jt;G\"F-7$$\"?++++++]P4Y6cbZZF-$!5 E[&[MIM2UP\"F-7$$\"?+++++++D1k2/P[ZF-$!5(38+[$f/,Z:F-7$$\"?++++++vo/t0 yx[ZF-$!5=g0E2E*Q/o\"F-7$$\"?++++++]7.#Q?&=\\ZF-$!5\"H+$*)*y)3od=F-7$$ \"?+++++]PM_'G!*)Q\\ZF-$!5(o1]J_A.k'>F-7$$\"?++++++Dc,\">g#f\\ZF-$!5)) R#>IT#F-7$F9$!1$y=8c6n*GF-7$F>$!1:P(*\\Z&4D$F-7$FC$!1o`cpBV2LF-7$FH $!17H\"y;_vu$F-7$FM$!1K1Imo6:XF-7$FR$!1zhSp\\QwXF-7$FW$!1M?Gagn+]F-7$F fn$!1C2EKnafjF-7$F[o$!1B@rZ4rJkF-7$F`o$!11*eIq&4BoF-7$Feo$!1,#Ra'3*>3* F-7$Fjo$!1kaHaN'[>*F-7$F_p$!1<_2#oaH^*F-7$Fdp$!2exxha$3>8F-7$Fip$!2D\\ )y\"[=VL\"F-7$F^q$!2g)*)H/\\!QP\"F-7$Fcq$!2)Rj>tzi/F-7$F]r$!2$Qkic'=)=?F-7$Fbr$!2(R2\"zp7&*R#F-7$Fgr$!2\\5i`(\\i(*HF-7$F \\s$!2[a\">#Q*o]IF-7$Fas$!2-iuo(*Q?Y$F-7$Ffs$!22(e&\\m+[m%F-7$F[t$!2Q1 #3j;uSZF-7$Fet$!29Q!z$4XS6&F-7$F_u$!2Er2G\"G;*[(F-7$Fiu$!2SAp%Q`66wF-7 $F^v$!2tMTxCEKs(F-7$Fcv$!2Sotcfz%)3)F-7$Fhv$!2hyO@mF%)\\)F-7$F]w$!2iFZ 4JvsA*F-7$Fbw$!3!pb&yrv'y/\"F-7$Fgw$!3$o(4xZ[vY7F-7$F\\x$!3!\\&z%e!R,e 7F-7$Fax$!3R1Oc.izp7F-7$Ffx$!3YO]t!\\4SG\"F-7$F[y$!366>P@1i<8F-7$F`y$! 3V1KQHv(4O\"F-7$Fey$!3_\"4`kj'GV9F-7$Fjy$!3\\0(=Pt/Yf\"F-7$F_z$!3D&G1M gd?'=F-7$Fdz$!3ed#eOr-s/#F-7$Fiz$!3$[36%4$pl<#F-7$F^[l$!3?O!phhC:=#F-7 $Fc[l$!3eVi6T$3l=#F-7$Fh[l$!3tb,6Hjc'>#F-7$F]\\l$!3!e%*GKz&z1AF-7$Fb\\ l$!3:'Rhv]G1B#F-7$Fg\\l$!3ezJZRR([F#F-7$F\\]l$!3$)*4!oFqTBBF-7$Fa]l$!3 sHK&p)zW9CF-7$Ff]l$!32L(Rx$Q6$e#F-7$F[^l$!3Cm6_d5$e)GF-7$F`^l$!31k-Tq! o9W$F-7$Fe^l$!3%[gc)*yyz+%F-7$Fj^l$!3i,PC1KdHSF-7$F__l$!3jO+kEj[^SF-7$ Fd_l$!3uaE&\\L1h4%F-7$Fi_l$!3oD![esj=;%F-7$F^`l$!3r\"H6T&Q#)GUF-7$Fc`l $!3@d<:j2;bVF-7$Fh`l$!3(R(R;v#yzf%F-7$F]al$!3,08R9]+a]F-7$Fbal$!3g=O)e ig,(eF-7$Fgal$!3%yEFb&o\"RG(F-7$F\\bl$!3,Zf`dEd=vF-7$Fabl$!3$e6^j0s'ox F-7$Ffbl$!3``\"R*HK`kyF-7$F[cl$!3%fw+$=eCqyF-7$F`cl$!3gP5S-Gq\")yF-7$F ecl$!3J5M;AC?$*yF-7$Fjcl$!3J#o$em4L;zF-7$F_dl$!3yeZScpjRzF-7$Fddl$!3FT ?LX1hF!)F-7$Fidl$!3N>^y,uPK\")F-7$F^el$!3G'z<7<_mA)F-7$Fcel$!3uI2aeT<& R)F-7$Fhel$!3CIs!pc9qs)F-7$F]fl$!3289'[l'f'Q*F-7$Fbfl$!4CGu2O#=G[5F-7$ Fgfl$!4`'*)H!)H&o*Q7F-7$F\\gl$!4gon#)fPT%f:F-7$Fagl$!4D/_1&)H(H'o\"F-7 $Ffgl$!46&pof%4t!4o$F-7$Fg[m$!4K&>A/D-R&*RF--F]\\m6&F_\\m$\"#XFi[mFd\\mF`\\m-Fg\\m 6#%9scheme~with~simple~nodesG-F$6%7aq7$F($!1S>sO0bhDF-7$F/$!1\\U$RXh#) e#F-7$F4$!1j*30i(4QEF-7$F9$!1#)zGV/'\\<$F-7$F>$!1>NGw6,nNF-7$FC$!1RV/] fFHOF-7$FH$!1iIJ)*3Bv$=;5F-7$F_p$!2Qp8Ny2:0\"F-7$Fdp$!2Tw^ -Fk-Y\"F-7$Fip$!2')fxrmIrZ\"F-7$F^q$!2(4cv'e()4_\"F-7$Fcq$!2DTz=G<*))= F-7$Fhq$!2?V$pWET$>#F-7$F]r$!2F.G]m4\"QAF-7$Fbr$!2MJ`)))QshEF-7$Fgr$!2 H\\)znn!pK$F-7$F\\s$!2PNN'*e^eQ$F-7$Fas$!2/C%)>ynP%QF-7$Ffs$!2h#)H;j.> =&F-7$F[t$!2d#f#ox*Gm_F-7$Fet$!2Whi?\"[(>o&F-7$F_u$!2W?f#)Hg]K)F-7$Fiu $!28aM#)QR1Y)F-7$F^v$!2El.sNIae)F-7$Fcv$!2t4paAtA**)F-7$Fhv$!2z$z5C2!) [%*F-7$F]w$!35xEYbD-E5F-7$Fbw$!3DPWD\">'Hl6F-7$Fgw$!3I82%*)pvlQ\"F-7$F \\x$!3NmU8[u4*R\"F-7$Fax$!3o#p+WM-AT\"F-7$Ffx$!3zr&ppSA!G9F-7$F[y$!3h2 Rs!4jaY\"F-7$F`y$!3n&R.OidP^\"F-7$Fey$!3xep$)*o9ag\"F-7$Fjy$!3RBG0>'fQ x\"F-7$F_z$!3l%>io>v92#F-7$Fdz$!3*=)[5lzWxAF-7$Fiz$!3#[RN9(*[8U#F-7$F^ [l$!3bys4e:'oU#F-7$Fc[l$!3qqWFfdSKCF-7$Fh[l$!3A7)[a!\\fVCF-7$F]\\l$!3W K26t](\\X#F-7$Fb\\l$!3&pw2<9'\\\"[#F-7$Fg\\l$!3iK!y-#[xIDF-7$F\\]l$!3D h-_H\"\\[e#F-7$Fa]l$!3/-[zP2B'o#F-7$Ff]l$!3RE5$o\\7S(GF-7$F[^l$!3ST0Mo N\"4@$F-7$F`^l$!3yQ$>E]c!HQF-7$Fe^l$!3$z(GV#z0\"fWF-7$Fj^l$!3o.\"G#>38 $[%F-7$F__l$!3L^)fR`5v]%F-7$Fd_l$!3P;J9[*erb%F-7$Fi_l$!3[gSKnZPIYF-7$F ^`l$!3I%*>W.`&\\q%F-7$Fc`l$!3QB&G*H+lX[F-7$Fh`l$!3uAZkcM(f6&F-7$F]al$! 3k%psJw`Mi&F-7$Fbal$!3f;zI-2KJlF-7$Fgal$!3zOY!3(yI.\")F-7$F\\bl$!3#fnE -'QU')F-7$Ffbl$!3@ZO%49e([()F-7$F[cl$!3V]*Ho+8^ v)F-7$F`cl$!3g;\"[@7eyw)F-7$Fecl$!3fO0zm1l!y)F-7$Fjcl$!3yH$)4,'zj!))F- 7$F_dl$!30l*e:&fIK))F-7$Fddl$!3j=yVWS&\\A7[wY!*F-7$F^ el$!3CR38w$H<:*F-7$Fcel$!3z2BLvm6F-7$Fgfl$!4\"yN7!pL)=y8F-7$F\\gl$!46N+ 'GOSLMF-7$F[hl$!4kvu*GcAY G>F-7$F`hl$!4\\x#)R?Nsg%>F-7$Fehl$!4(ylP7Bnss>F-7$Fjhl$!4k7]Q%>?l>?F-7 $F_il$!4_95#3M0'z5#F-7$Fdil$!4*z'yd'oZw\"H#F-7$Fiil$!43s<&\\'))=Wk#F-7 $F^jl$!4t$f#fb\">X@HF-7$Fcjl$!4ls%e]xgo$H$F-7$Fhjl$!4i+Y^G'G*Q_$F-7$F] [m$!44+4?/qN()y$F-7$Fb[m$!4r%)\\@3>*o#4%F-7$Fg[m$!4hC3pFc32W%F--F]\\m6 &F_\\mFd\\m$\"#DFi[m$\"\"\"Fe\\m-Fg\\m6#%Pscheme~with~a~relatively~lar ge~stability~regionG-F$6%7ip7$F($\"1un#\\:OU'yF-7$F/$\"1T9,'pOd%zF-7$F 4$\"1O-$\"1&[#p/?pH#*F-7$FC$\"1b/tMMUb $*F-7$FH$\"1!o^+e/I*)*F-7$FM$\"27V=&=(G?2\"F-7$FR$\"266z)*zm]3\"F-7$FW $\"2.S(pHVCI6F-7$Ffn$\"2Pn0&pTVV7F-7$F[o$\"2#>fbUs!pD\"F-7$F`o$\"2TrG[ 93QH\"F-7$Feo$\"2pPGM%=vI9F-7$Fjo$\"24FWgtE\"[9F-7$F_p$\"2NLlLCPaZ\"F- 7$Fdp$\"2u5XC:^;i\"F-7$Fip$\"29y;]>1.k\"F-7$F^q$\"2Kk;r>W+n\"F-7$Fcq$ \"29#eKnpp^u)*>o;F-7$Fet$\"2.0GUXB>i\"F-7 $F_u$\"1eHG9(eim(F-7$Fiu$\"1ZkV7\\P\"y(F-7$Fcv$\"1%f#psM]8mF-7$Fgw$!2U 8v1SNhj\"F-7$Fax$!2/#[abd'pm\"F-7$Ffx$!2Z%z!*o`E'p\"F-7$F[y$!2QsN9]hq# =F-7$F`y$!2zP8B&>_Z?F-7$Fey$!24,W'*HDT_#F-7$Fjy$!2j.-M7]-\\$F-7$F_z$!2 oG03j$oR`F-7$F]\\l$!2MU7g#[/XxF-7$Fb\\l$!2$\\ss+IyKyF-7$Fg\\l$!2)[^(4# z**[!)F-7$F\\]l$!2qB))\\[(eW$)F-7$Fa]l$!2y*z3ID_m*)F-7$Ff]l$!3n2jaI\"f C-\"F-7$F[^l$!3))[X(=^-\\E\"F-7$F`^l$!3e$pBIbK'Q\"H#F-7$ Fi_l$!3gZ&=o4t=L#F-7$F^`l$!31q.>*oa)zBF-7$Fc`l$!3z)Rc0/L,[#F-7$$\"?+++ +++]7y+gu*ps%F-$!3K:%)Q#y(3mDF-7$Fh`l$!3zH22VaQ*o#F-7$$\"?++++++]P4@/G zFZF-$!3k,;&*H&HV'GF-7$F]al$!35d,@R8j4JF-7$Fbal$!3p.A!f4F/\\%>nF-7$F]fl$!3$*G#)exzm(R(F-7$Fbfl$!3Lsru%[gre)F-7$Fgfl$!4Yp rP$H))Hv5F-7$F\\gl$!4k2@K@x4[X\"F-7$Fagl$!4Th-0WGgmg\"F-7$Ffgl$!4mQo)Q ftNG;F-7$F[hl$!4BX2\\v]$>_;F-7$F`hl$!4cTR$)RXuum\"F-7$Fehl$!4!)))\\Nvq '[\"p\"F-7$Fjhl$!4x=*[r[(Rht\"F-7$F_il$!4F#z\"y]A\"4D=F-7$Fdil$!4N'zoA Hvd??F-7$Fiil$!4`4K)3k*R[T#F-7$F^jl$!4_N^t,ML[t#F-7$Fcjl$!4d\\lfVEDU<$ F-7$Fhjl$!4-]S/:!G<]MF-7$F][m$!4x'o]5s'34x$F-7$Fb[m$!4RqT)G#oLF9%F-7$F g[m$!4pTsCD@)psXF--F]\\m6&F_\\mFd\\m$\"#vFi[mFb\\m-Fg\\m6#%TButcher's~ scheme~B~with~c[5]=c[6]=1/2~and~b[5]=b[6]G-F$6%7aq7$F($!1UE*\\O))4/\"F -7$F/$!1&ej(*eX=0\"F-7$F4$!1$e2d5eA2\"F-7$F9$!1em*)*=(3$H\"F-7$F>$!1p \")QBX2(z9F-7$FH$!1qw;&*\\2\"o\"F-7$FM$!17%)*zOr>.#F- 7$FR$!1\"H)3E*z'f?F-7$FW$!1=[8cPbaAF-7$Ffn$!1\\NfsZpyGF-7$F[o$!1K7;%=F 9\"HF-7$F`o$!1?\\r`4x\"4$F-7$Feo$!16,L_y4MTF-7$Fjo$!1iSgal_&=%F-7$F_p$ !1%QB[2]CL%F-7$Fdp$!1'zSp*3BPgF-7$Fip$!1S8A'**pp5'F-7$F^q$!1BvvbRr*G'F -7$Fcq$!1^%H%>-=IyF-7$Fhq$!1M=t43?/\"*F-7$F]r$!1$Q*[*>p1H*F-7$Fbr$!2:B &[BM526F-7$Fgr$!2/I>()\\\\mQ\"F-7$F\\s$!2`$G:b:G69F-7$Fas$!2wwCk+^Wg\" F-7$Ffs$!2Yh$[4xGp@F-7$F[t$!2`^ka#)\\Y?#F-7$Fet$!2xQa\"R*o0Q#F-7$F_u$! 2D_2XKY7]$F-7$Fiu$!2!*ebm&pGeNF-7$F^v$!2T;0.%e06OF-7$Fcv$!2c78$GZ.%y$F -7$Fhv$!20vm(ofSyRF-7$F]w$!2jPgAz2SK%F-7$Fbw$!2/D#Q1nJ<\\F-7$Fgw$!2[VV djq+'eF-7$F\\x$!2YlX#=8*H\"fF-7$Fax$!2%H[PJRRofF-7$Ffx$!2I]KQN![NgF-7$ F[y$!2BHsg$49&>'F-7$F`y$!2h8astr:S'F-7$Fey$!2(e2*eEuOz'F-7$Fjy$!2asXQ8 %e9vF-7$F_z$!2Uc%Hr'[')y)F-7$Fdz$!2VJ%z%H>0n*F-7$Fiz$!3&p$4!4)*e'G5F-7 $F^[l$!3=z+&p)3+J5F-7$Fc[l$!33g2F-7$Fj^l$!376'eC+S^\">F -7$F__l$!3<*>@='[bD>F-7$Fd_l$!3\\9_dKNxY>F-7$Fi_l$!37/qvBzF-7$F^`l$ !3!p#fx'3)H5?F-7$Fc`l$!3$QWs)G`,r?F-7$Fh`l$!3sa*)*3o8y=#F-7$F]al$!37/Z 3oxB2CF-7$Fbal$!3a*>Cexu**z#F-7$Fgal$!3PJdbl.G![$F-7$F\\bl$!3)**>b!3s> $f$F-7$Fabl$!3z6D(*paa8PF-7$Ffbl$!3c:MZ/CmfPF-7$F[cl$!3`'QLNK$RiPF-7$F `cl$!3(3hjsOqyw$F-7$Fecl$!3+4;;$znLx$F-7$Fjcl$!3!*Gw]iWU%y$F-7$F_dl$!3 ]hn()*)fc&z$F-7$Fddl$!3$4j@`&ziPQF-7$Fidl$!3'[R$f`?#)F-7$F[hl$!3z9*Q(\\*e5M) F-7$F`hl$!3n***\\\\L\\uT)F-7$Fehl$!3baF#)4fQL&)F-7$Fjhl$!3L!)Hg=)**zt) F-7$F_il$!3FN^1rVtB\"*F-7$Fdil$!3OQ-2fFdF**F-7$Fiil$!4S$4.X$4Er9\"F-7$ F^jl$!43cytlCh%o7F-7$Fcjl$!4ZMF--F]\\m6&F_ \\mF`\\mF^`nFd\\m-Fg\\m6#%Hscheme~with~c[5]=c[6]=3/4~and~b[5]=b[6]G-%% FONTG6$%*HELVETICAG\"\"*-%+AXESLABELSG6$Q\"x6\"Q!Fcjq-%&TITLEG6#%Uerro r~curves~for~7~stage~order~6~Runge-Kutta~methodsG-%%VIEWG6$;F(Fg[m;$F- !#7$\"#:!#8" 1 2 0 1 10 0 2 9 1 4 2 1.000000 45.000000 45.000000 0 1 " Butcher's scheme A" "scheme with simple nodes" "scheme with a relative ly large stability region" "Butcher's scheme B with c[5]=c[6]=1/2 and \+ b[5]=b[6]" "scheme with c[5]=c[6]=3/4 and b[5]=b[6]" }}}}{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 34 "#=================================" }}{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 }