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" }}{PARA 0 "" 0 "" {TEXT -1 134 "They ca n be read into a Maple session by commands similar to those that follo w, where each file path gives the location of the m-file." }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 102 "read \"C:\\\\Maple/procdrs/butcher .m\";\nread \"C:\\\\Maple/procdrs/roots.m\";\nread \"C:\\\\Maple/procd rs/intg.m\";" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}{PARA 0 "" 0 "" {TEXT -1 30 "#=============================" }}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 27 "Relations between the nodes" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 130 "For the Runge-Kutta schemes consi dered in this worksheet the stage order conditions for stage 4 togethe r with the condition that " }{XPPEDIT 18 0 "a[4,2]=0" "6#/&%\"aG6$\" \"%\"\"#\"\"!" }{TEXT -1 24 " imply that the nodes " }{XPPEDIT 18 0 "c[3]" "6#&%\"cG6#\"\"$" }{TEXT -1 7 " and " }{XPPEDIT 18 0 "c[4]" " 6#&%\"cG6#\"\"%" }{TEXT -1 22 " satisfy the relation" }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "c[3]=2/3" "6#/&%\"cG6#\"\"$*&\"\"# \"\"\"F'!\"\"" }{TEXT -1 1 " " }{XPPEDIT 18 0 "c[4]" "6#&%\"cG6#\"\"% " }{TEXT -1 2 ". " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 216 "SO_eqs := [op(StageOrderConditions(2,4..4,'e xpanded')),op(StageOrderConditions(3,4..4,'expanded'))];\nnode_eqs := \+ subs(a[4,2]=0,SO_eqs);\nsol := solve(\{op(node_eqs)\},indets(node_eqs) minus \{c[4]\}):\nc[3]=subs(sol,c[3]);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%'SO_eqsG7$/,&*&&%\"aG6$\"\"%\"\"#\"\"\"&%\"cG6#F-F.F.*&&F*6$F, \"\"$F.&F06#F5F.F.,$*&#F.F-F.*$)&F06#F,F-F.F.F./,&*&F)F.)F/F-F.F.*&F3F .)F6F-F.F.,$*&#F.F5F.*$)F=F5F.F.F." }}{PARA 11 "" 1 "" {XPPMATH 20 "6# >%)node_eqsG7$/*&&%\"aG6$\"\"%\"\"$\"\"\"&%\"cG6#F,F-,$*&#F-\"\"#F-*$) &F/6#F+F4F-F-F-/*&F(F-)F.F4F-,$*&#F-F,F-*$)F7F,F-F-F-" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/&%\"cG6#\"\"$,$*&#\"\"#F'\"\"\"&F%6#\"\"%F,F," }} }{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}}{PARA 0 "" 0 "" {TEXT -1 30 "#============================ =" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 30 "#=== ==========================" }}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 32 "con struction of a general scheme" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }} {PARA 0 "" 0 "" {TEXT -1 162 "See: A Parameter Study of Explicit Runge -Kutta Pairs of Orders 6(5), by Ch. Tsitouras,\n Applied Mathema tics Letters, Vol. 11, No. 1, pages 65 to 69, 1998. " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}} {PARA 0 "" 0 "" {TEXT -1 60 "#---------------------------------------- -------------------" }}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 48 "an alterna tive form for certain order conditions" }}{PARA 0 "" 0 "" {TEXT -1 0 " " }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 44 "ee: coefficients for the Sharp-Verner scheme" }} {PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 891 "# coefficients by Sharp and Verner 1994\nee := \{c[2]=1/12,c[3]=2 /15,c[4]=1/5,c[5]=8/15,c[6]=2/3,c[7]=19/20,c[8]=1,c[9]=1,\na[2,1]=1/12 ,a[3,1]=2/75,a[3,2]=8/75,\na[4,1]=1/20,a[4,2]=0,a[4,3]=3/20,\na[5,1]=8 8/135,a[5,2]=0,a[5,3]=-112/45,a[5,4]=64/27,\na[6,1]=-10891/11556,a[6,2 ]=0,a[6,3]=3880/963,\n a[6,4]=-8456/2889,a[6,5]=217/428,\na[7,1]=1718 911/4382720,a[7,2]=0,a[7,3]=-1000749/547840,\na[7,4]=819261/383488,a[7 ,5]=-671175/876544,a[7,6]=14535/14336,\na[8,1]=85153/203300,a[8,2]=0,a [8,3]=-6783/2140,\na[8,4]=10956/2675,a[8,5]=-38493/13375,a[8,6]=1152/4 25,a[8,7]=-7168/40375,\na[9,1]=53/912,a[9,2]=0,a[9,3]=0,a[9,4]=5/16,a[ 9,5]=27/112,a[9,6]=27/136,\na[9,7]=256/969,a[9,8]=-25/336,\nb[1]=53/91 2,b[2]=0,b[3]=0,b[4]=5/16,b[5]=27/112,b[6]=27/136,\nb[7]=256/969,b[8]= -25/336,\n`b*`[1]=617/10944,`b*`[2]=0,`b*`[3]=0,`b*`[4]=241/756,`b*`[5 ]=69/320,\n`b*`[6]=435/1904,`b*`[7]=10304/43605,`b*`[8]=0,`b*`[9]=-1/1 8\}:" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 " " {TEXT -1 142 "The algorithm presented by Tsitouras for the construct ion of a family of order 6 Runge-Kutta Pairs involves some alternative order conditions." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 5 "Let " }{XPPEDIT 18 0 "[A]" "6#7#%\"AG" }{TEXT -1 90 " be the 9 by 9 lower triangular matrix of linking coefficients from the B utcher tableau. " }}{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], b[8], 0];" "6#/7 #%\"bG7+&F%6#\"\"\"/&F%6#\"\"#\"\"!&F%6#\"\"$&F%6#\"\"%&F%6#\"\"&&F%6# \"\"'&F%6#\"\"(&F%6#\"\")F." }{TEXT -1 7 " and " }{XPPEDIT 18 0 "[`b *`] = [`b*`[1], `b*`[2] = 0, `b*`[3], `b*`[4], `b*`[5], `b*`[6], `b*`[ 7], `b*`[8], `b*`[9]];" "6#/7#%#b*G7+&F%6#\"\"\"/&F%6#\"\"#\"\"!&F%6# \"\"$&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#\"\"$" }{TEXT -1 3 ", " }{XPPEDIT 18 0 "c[4]" "6# &%\"cG6#\"\"%" }{TEXT -1 3 ", " }{XPPEDIT 18 0 "c[5]" "6#&%\"cG6#\"\" &" }{TEXT -1 3 ", " }{XPPEDIT 18 0 "c[6]" "6#&%\"cG6#\"\"'" }{TEXT -1 3 ", " }{XPPEDIT 18 0 "c[7]" "6#&%\"cG6#\"\"(" }{TEXT -1 3 ", " } {XPPEDIT 18 0 "c[8]=1" "6#/&%\"cG6#\"\")\"\"\"" }{TEXT -1 3 ", " } {XPPEDIT 18 0 "c[9]=1" "6#/&%\"cG6#\"\"*\"\"\"" }{TEXT -1 11 " and le t " }{TEXT 282 2 "Id" }{TEXT -1 32 " be the 9 by 9 identity matrix. " }}{PARA 0 "" 0 "" {TEXT -1 5 "Let " }{XPPEDIT 18 0 "[c]" "6#7#%\"cG " }{TEXT -1 38 " be the row vector whose entries are " }{XPPEDIT 18 0 "c[1]=0" "6#/&%\"cG6#\"\"\"\"\"!" }{TEXT -1 3 ", " }{XPPEDIT 18 0 " c[2]" "6#&%\"cG6#\"\"#" }{TEXT -1 3 ", " }{XPPEDIT 18 0 "c[3]" "6#&% \"cG6#\"\"$" }{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]" "6#&%\"cG6#\"\"(" }{TEXT -1 3 ", " }{XPPEDIT 18 0 "c[8]=1" "6#/&%\"cG6#\"\")\"\"\"" }{TEXT -1 3 ", " }{XPPEDIT 18 0 " c[9]=1" "6#/&%\"cG6#\"\"*\"\"\"" }{TEXT -1 2 ". " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 238 "A := matrix ([seq([seq(a[i,j],j=1..i-1),seq(0,j=i..9)],i=1..9)]):\nB := matrix([[s eq(b[i],i=1..9)]]):\n`B*` := matrix([[seq(`b*`[i],i=1..9)]]):\nId := l inalg[diag](1$9):\nC := linalg[diag](seq(c[i],i=1..9)):\nc_ := matrix( [seq([c[i]],i=1..9)]):" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 49 "#================================================" } }{PARA 0 "" 0 "" {TEXT -1 34 "(1) The first order condition is: " }} {PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "[b]^T*([C]-Id)*[A]*(C -c[4]*Id)*(C-c[5]*Id)*[c] = Int((x-1)*Int((t-c[4])*(t-c[5])*t,t = 0 .. x),x = 0 .. 1);" "6#/*.)7#%\"bG%\"TG\"\"\",&7#%\"CGF)%#IdG!\"\"F)7#% \"AGF),&F,F)*&&%\"cG6#\"\"%F)F-F)F.F),&F,F)*&&F46#\"\"&F)F-F)F.F)7#F4F )-%$IntG6$*&,&%\"xGF)F)F.F)-F>6$*(,&%\"tGF)&F46#F6F.F),&FGF)&F46#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[4])*(c[j-1]-c[5])*c[j-1],j = 2 .. 7) = -1/120+c[4]/60+c[ 5]/60-c[4]*c[5]/24;" "6#/-%$SumG6$**-F%6$*(&%\"bG6#%\"iG\"\"\",&&%\"cG 6#F.F/F/!\"\"F/&%\"aG6$F.,&%\"jGF/F/F4F//F.;F9\"\"(F/,&&F26#,&F9F/F/F4 F/&F26#\"\"%F4F/,&&F26#,&F9F/F/F4F/&F26#\"\"&F4F/&F26#,&F9F/F/F4F//F9; \"\"#F<,**&F/F/\"$?\"F4F4*&&F26#FCF/\"#gF4F/*&&F26#FJF/FWF4F/*(&F26#FC F/&F26#FJF/\"#CF4F4" }{TEXT -1 2 ". " }}{PARA 0 "" 0 "" {TEXT -1 1 " \+ " }}{PARA 0 "" 0 "" {TEXT -1 13 "We can omit " }{XPPEDIT 18 0 "j=2" " 6#/%\"jG\"\"#" }{TEXT -1 33 " in the outer summation because " } {XPPEDIT 18 0 "c[1]=0" "6#/&%\"cG6#\"\"\"\"\"!" }{TEXT -1 15 ", we can omit " }{XPPEDIT 18 0 "j=3" "6#/%\"jG\"\"$" }{TEXT -1 11 " because \+ " }{XPPEDIT 18 0 "a[i,2]=0" "6#/&%\"aG6$%\"iG\"\"#\"\"!" }{TEXT -1 8 ", for " }{XPPEDIT 18 0 "i=3" "6#/%\"iG\"\"$" }{TEXT -1 25 " . . 7, \+ and we can omit " }{XPPEDIT 18 0 "j=5" "6#/%\"jG\"\"&" }{TEXT -1 7 " \+ and " }{XPPEDIT 18 0 "j=6" "6#/%\"jG\"\"'" }{TEXT -1 34 " because o f obvious zero factors." }}{PARA 0 "" 0 "" {TEXT -1 13 "We can omit \+ " }{XPPEDIT 18 0 "j=4" "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,3],i=4..7)=0" "6#/-%$SumG6$*(&%\"bG6#%\"iG\"\"\",&&% \"cG6#F+F,F,!\"\"F,&%\"aG6$F+\"\"$F,/F+;\"\"%\"\"(\"\"!" }{TEXT -1 2 " . " }}{PARA 0 "" 0 "" {TEXT -1 24 "This gives the relation " }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "b[7]*(c[7]-1)*a[7,6]*(c[6]- c[4])*(c[6]-c[5])*c[6]=-1/120+c[4]/60+c[5]/60-c[4]*c[5]/24" "6#/*.&%\" bG6#\"\"(\"\"\",&&%\"cG6#F(F)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#F 8F)\"#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 61 "#-- ----------------------------------------------------------" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 62 "Int( (x-1)*Int((t-c[4])*(t-c[5])*t,t=0..x),x=0..1):\n%=value(%);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/-%$IntG6$*&,&%\"xG\"\"\"F*!\"\"F*-F%6$*(,&% \"tGF*&%\"cG6#\"\"%F+F*,&F0F*&F26#\"\"&F+F*F0F*/F0;\"\"!F)F*/F);F;F*,* #F*\"$?\"F+*&#F*\"#gF*F1F*F**&FBF*F6F*F**&#F*\"#CF**&F6F*F1F*F*F+" }}} {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 107 "We can u se the Sharp-Verner scheme to provide a numerical check for the order \+ condition in the matrix form." }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 159 "evalm(B &* (C-Id) &* A &* (C-c[4]*Id) &* (C-c[5]*Id) &* c_)[1,1 ]=int((x-1)*int((t-c[4])*(t-c[5])*t,t=0..x),x=0..1):\nsubs(\{c[1]=0,c[ 8]=1,c[9]=1\},%);\nsubs(ee,%);" }}{PARA 12 "" 1 "" {XPPMATH 20 "6#/,(* *,,*(&%\"bG6#\"\"$\"\"\",&&%\"cGF*F,F,!\"\"F,&%\"aG6$F+\"\"#F,F,*(&F)6 #\"\"%F,,&&F/F7F,F,F0F,&F26$F8F4F,F,*(&F)6#\"\"&F,,&F,F0&F/F?F,F,&F26$ F@F4F,F,*(&F)6#\"\"'F,,&F,F0&F/FGF,F,&F26$FHF4F,F,*(&F)6#\"\"(F,,&F,F0 &F/FOF,F,&F26$FPF4F,F,F,,&&F/6#F4F,F:F0F,,&FVF,FBF0F,FVF,F,**,**(F6F,F 9F,&F26$F8F+F,F,*(F>F,FAF,&F26$F@F+F,F,*(FFF,FIF,&F26$FHF+F,F,*(FNF,FQ F,&F26$FPF+F,F,F,,&F.F,F:F0F,,&F.F,FBF0F,F.F,F,*.FNF,FQF,&F26$FPFHF,,& F:F0FJF,F,,&FBF0FJF,F,FJF,F,,*#F,\"$?\"F0*&#F,\"#gF,F:F,F,*&F\\pF,FBF, F,*&#F,\"#CF,*&FBF,F:F,F,F0" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/#!\"\" \"%+=F$" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 80 "We can also make a numerical check of the order condition in the s ummation form." }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 151 "add(add(b [i]*(c[i]-1)*a[i,j-1],i=j..7)*(c[j-1]-c[4])*(c[j-1]-c[5])*c[j-1],j=2.. 7)=\n -1/120+c[4]/60+c[5]/60-c[4]*c[5]/24:\nsubs(\{c[1]=0\},%);\nsubs (ee,%);" }}{PARA 12 "" 1 "" {XPPMATH 20 "6#/,(**,,*(&%\"bG6#\"\"$\"\" \",&&%\"cGF*F,F,!\"\"F,&%\"aG6$F+\"\"#F,F,*(&F)6#\"\"%F,,&&F/F7F,F,F0F ,&F26$F8F4F,F,*(&F)6#\"\"&F,,&F,F0&F/F?F,F,&F26$F@F4F,F,*(&F)6#\"\"'F, ,&F,F0&F/FGF,F,&F26$FHF4F,F,*(&F)6#\"\"(F,,&F,F0&F/FOF,F,&F26$FPF4F,F, F,,&&F/6#F4F,F:F0F,,&FVF,FBF0F,FVF,F,**,**(F6F,F9F,&F26$F8F+F,F,*(F>F, FAF,&F26$F@F+F,F,*(FFF,FIF,&F26$FHF+F,F,*(FNF,FQF,&F26$FPF+F,F,F,,&F.F ,F:F0F,,&F.F,FBF0F,F.F,F,*.FNF,FQF,&F26$FPFHF,,&F:F0FJF,F,,&FBF0FJF,F, FJF,F,,*#F,\"$?\"F0*&#F,\"#gF,F:F,F,*&F\\pF,FBF,F,*&#F,\"#CF,*&FBF,F:F ,F,F0" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/#!\"\"\"%+=F$" }}}{PARA 0 " " 0 "" {TEXT -1 0 "" }{TEXT 281 0 "" }{TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 84 "After the omissions that we can make in the outer summati on we obtain the following." }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 132 "add(add(b[i]*(c[i]-1)*a[i,j-1],i=j..7)*(c[j-1]-c[4])*(c[j-1]-c[5] )*c[j-1],j=[7])=\n -1/120+c[4]/60+c[5]/60-c[4]*c[5]/24;\nsubs(ee,%); " }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/*.&%\"bG6#\"\"(\"\"\",&F)!\"\"&% \"cGF'F)F)&%\"aG6$F(\"\"'F),&&F-6#\"\"%F+&F-6#F1F)F),&&F-6#\"\"&F+F6F) F)F6F),*#F)\"$?\"F+*&#F)\"#gF)F3F)F)*&F@F)F9F)F)*&#F)\"#CF)*&F9F)F3F)F )F+" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/#!\"\"\"%+=F$" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 57 "#=================== =====================================" }}{PARA 0 "" 0 "" {TEXT -1 35 " (2) The second order condition is: " }}{PARA 256 "" 0 "" {TEXT -1 1 " \+ " }{XPPEDIT 18 0 "[`b*`]^T*[A]*(C-c[4]*Id)*(C-c[5]*Id)*[c] = Int(Int(( t-c[4])*(t-c[5])*t,t = 0 .. x),x = 0 .. 1);" "6#/*,)7#%#b*G%\"TG\"\"\" 7#%\"AGF),&%\"CGF)*&&%\"cG6#\"\"%F)%#IdGF)!\"\"F),&F-F)*&&F06#\"\"&F)F 3F)F4F)7#F0F)-%$IntG6$-F<6$*(,&%\"tGF)&F06#F2F4F),&FBF)&F06#F9F4F)FBF) /FB;\"\"!%\"xG/FK;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*`[i]*a[i,j-1],i = j .. 7)*(c[j-1]- c[4])*(c[j-1]-c[5])*c[j-1],j = 2 .. 9) = 1/20-c[4]/12-c[5]/12+c[4]*c[5 ]/6;" "6#/-%$SumG6$**-F%6$*&&%#b*G6#%\"iG\"\"\"&%\"aG6$F.,&%\"jGF/F/! \"\"F//F.;F4\"\"(F/,&&%\"cG6#,&F4F/F/F5F/&F;6#\"\"%F5F/,&&F;6#,&F4F/F/ F5F/&F;6#\"\"&F5F/&F;6#,&F4F/F/F5F//F4;\"\"#\"\"*,**&F/F/\"#?F5F/*&&F; 6#F@F/\"#7F5F5*&&F;6#FGF/FUF5F5*(&F;6#F@F/&F;6#FGF/\"\"'F5F/" }{TEXT -1 2 ". " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 13 "We can omit " }{XPPEDIT 18 0 "j=2" "6#/%\"jG\"\"#" }{TEXT -1 33 " in the outer summation because " }{XPPEDIT 18 0 "c[1]=0" "6#/&%\"cG6 #\"\"\"\"\"!" }{TEXT -1 15 ", we can omit " }{XPPEDIT 18 0 "j=3" "6#/ %\"jG\"\"$" }{TEXT -1 11 " because " }{XPPEDIT 18 0 "a[i,2]=0" "6#/& %\"aG6$%\"iG\"\"#\"\"!" }{TEXT -1 8 ", for " }{XPPEDIT 18 0 "i=3" "6 #/%\"iG\"\"$" }{TEXT -1 25 " . . 7, and we can omit " }{XPPEDIT 18 0 "j=5" "6#/%\"jG\"\"&" }{TEXT -1 7 " and " }{XPPEDIT 18 0 "j=6" "6#/% \"jG\"\"'" }{TEXT -1 34 " because of obvious zero factors." }}{PARA 0 "" 0 "" {TEXT -1 13 "We can omit " }{XPPEDIT 18 0 "j=4" "6#/%\"jG\" \"%" }{TEXT -1 11 " because " }}{PARA 256 "" 0 "" {TEXT -1 1 " " } {XPPEDIT 18 0 "Sum(`b*`[i]*a[i,3],i = 4 .. 9) = 0;" "6#/-%$SumG6$*&&%# b*G6#%\"iG\"\"\"&%\"aG6$F+\"\"$F,/F+;\"\"%\"\"*\"\"!" }{TEXT -1 2 ". \+ " }}{PARA 0 "" 0 "" {TEXT -1 24 "This gives the relation " }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "Sum(Sum(`b*`[i]*a[i,j-1],i \+ = j .. 7)*(c[j-1]-c[4])*(c[j-1]-c[5])*c[j-1],j = 7 .. 9) = 1/20-c[4]/1 2-c[5]/12+c[4]*c[5]/6;" "6#/-%$SumG6$**-F%6$*&&%#b*G6#%\"iG\"\"\"&%\"a G6$F.,&%\"jGF/F/!\"\"F//F.;F4\"\"(F/,&&%\"cG6#,&F4F/F/F5F/&F;6#\"\"%F5 F/,&&F;6#,&F4F/F/F5F/&F;6#\"\"&F5F/&F;6#,&F4F/F/F5F//F4;F8\"\"*,**&F/F /\"#?F5F/*&&F;6#F@F/\"#7F5F5*&&F;6#FGF/FTF5F5*(&F;6#F@F/&F;6#FGF/\"\"' F5F/" }{TEXT -1 2 ". " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 61 "#----------------------------------------------------- -------" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 56 "Int(Int((t-c[4])*(t-c[5])*t,t=0..x),x=0..1):\n%=value (%);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/-%$IntG6$-F%6$*(,&%\"tG\"\"\" &%\"cG6#\"\"%!\"\"F,,&F+F,&F.6#\"\"&F1F,F+F,/F+;\"\"!%\"xG/F9;F8F,,*#F ,\"#?F,*&#F,\"#7F,F-F,F1*&#F,FAF,F3F,F1*&#F,\"\"'F,*&F-F,F3F,F,F," }}} {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 107 "We can u se the Sharp-Verner scheme to provide a numerical check for the order \+ condition in the matrix form." }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 139 "evalm(`B*` &* A &* (C-c[4]*Id) &* (C-c[5]*Id) &* c_)[1,1]=int(i nt((t-c[4])*(t-c[5])*t,t=0..x),x=0..1):\nsubs(\{c[1]=0,c[8]=1\},%);\ns ubs(ee,%);" }}{PARA 12 "" 1 "" {XPPMATH 20 "6#/,,**,0*&&%#b*G6#\"\"$\" \"\"&%\"aG6$F+\"\"#F,F,*&&F)6#\"\"%F,&F.6$F4F0F,F,*&&F)6#\"\"&F,&F.6$F :F0F,F,*&&F)6#\"\"'F,&F.6$F@F0F,F,*&&F)6#\"\"(F,&F.6$FFF0F,F,*&&F)6#\" \")F,&F.6$FLF0F,F,*&&F)6#\"\"*F,&F.6$FRF0F,F,F,,&&%\"cG6#F0F,&FWF3!\" \"F,,&FVF,&FWF9FZF,FVF,F,**,.*&F2F,&F.6$F4F+F,F,*&F8F,&F.6$F:F+F,F,*&F >F,&F.6$F@F+F,F,*&FDF,&F.6$FFF+F,F,*&FJF,&F.6$FLF+F,F,*&FPF,&F.6$FRF+F ,F,F,,&&FWF*F,FYFZF,,&F\\pF,FfnFZF,F\\pF,F,**,(*&FDF,&F.6$FFF@F,F,*&FJ F,&F.6$FLF@F,F,*&FPF,&F.6$FRF@F,F,F,,&FYFZ&FWF?F,F,,&FfnFZFjpF,F,FjpF, F,**,&*&FJF,&F.6$FLFFF,F,*&FPF,&F.6$FRFFF,F,F,,&&FWFEF,FYFZF,,&FeqF,Ff nFZF,FeqF,F,**FPF,&F.6$FRFLF,,&F,F,FYFZF,,&F,F,FfnFZF,F,,*#F,\"#?F,*&# F,\"#7F,FYF,FZ*&#F,FarF,FfnF,FZ*&#F,F@F,*&FfnF,FYF,F,F," }}{PARA 11 " " 1 "" {XPPMATH 20 "6#/#\"\"\"\"$]\"F$" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 80 "We can also make a numerical check o f the order condition in the summation form." }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 152 "add(add(`b*`[i]*a[i,j-1],i=j..9)*(c[j-1]-c[4])*(c [j-1]-c[5])*c[j-1],j=[$2..9])=\n 1/20-c[4]/12-c[5]/12+c[4]*c[5]/6:\ns ubs(\{c[1]=0,c[8]=1\},%);\nsubs(ee,%);" }}{PARA 12 "" 1 "" {XPPMATH 20 "6#/,,**,0*&&%#b*G6#\"\"$\"\"\"&%\"aG6$F+\"\"#F,F,*&&F)6#\"\"%F,&F. 6$F4F0F,F,*&&F)6#\"\"&F,&F.6$F:F0F,F,*&&F)6#\"\"'F,&F.6$F@F0F,F,*&&F)6 #\"\"(F,&F.6$FFF0F,F,*&&F)6#\"\")F,&F.6$FLF0F,F,*&&F)6#\"\"*F,&F.6$FRF 0F,F,F,,&&%\"cG6#F0F,&FWF3!\"\"F,,&FVF,&FWF9FZF,FVF,F,**,.*&F2F,&F.6$F 4F+F,F,*&F8F,&F.6$F:F+F,F,*&F>F,&F.6$F@F+F,F,*&FDF,&F.6$FFF+F,F,*&FJF, &F.6$FLF+F,F,*&FPF,&F.6$FRF+F,F,F,,&&FWF*F,FYFZF,,&F\\pF,FfnFZF,F\\pF, F,**,(*&FDF,&F.6$FFF@F,F,*&FJF,&F.6$FLF@F,F,*&FPF,&F.6$FRF@F,F,F,,&FYF Z&FWF?F,F,,&FfnFZFjpF,F,FjpF,F,**,&*&FJF,&F.6$FLFFF,F,*&FPF,&F.6$FRFFF ,F,F,,&&FWFEF,FYFZF,,&FeqF,FfnFZF,FeqF,F,**FPF,&F.6$FRFLF,,&F,F,FYFZF, ,&F,F,FfnFZF,F,,*#F,\"#?F,*&#F,\"#7F,FYF,FZ*&#F,FarF,FfnF,FZ*&#F,F@F,* &FfnF,FYF,F,F," }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/#\"\"\"\"$]\"F$" }} }{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 84 "After th e omissions that we can make in the outer summation we obtain the foll owing." }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 124 "add(add(`b*`[i]*a [i,j-1],i=j..9)*(c[j-1]-c[4])*(c[j-1]-c[5])*c[j-1],j=7..9)=\n 1/20-c[ 4]/12-c[5]/12+c[4]*c[5]/6;\nsubs(ee,%);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/,(**,(*&&%#b*G6#\"\"(\"\"\"&%\"aG6$F+\"\"'F,F,*&&F)6#\"\")F,&F. 6$F4F0F,F,*&&F)6#\"\"*F,&F.6$F:F0F,F,F,,&&%\"cG6#\"\"%!\"\"&F?6#F0F,F, ,&&F?6#\"\"&FBFCF,F,FCF,F,**,&*&F2F,&F.6$F4F+F,F,*&F8F,&F.6$F:F+F,F,F, ,&&F?F*F,F>FBF,,&FRF,FFFBF,FRF,F,*,F8F,&F.6$F:F4F,,&&F?F3F,F>FBF,,&FXF ,FFFBF,FXF,F,,*#F,\"#?F,*&#F,\"#7F,F>F,FB*&#F,FinF,FFF,FB*&#F,F0F,*&FF F,F>F,F,F," }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/#\"\"\"\"$]\"F$" }}} {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 61 "#-------- ----------------------------------------------------" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}}{PARA 0 "" 0 "" {TEXT -1 60 "#----- ------------------------------------------------------" }}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 43 "Tsitouras' algorithm .. Sharp-Verner sche me" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 118 "In this subsection we illustrate the algorithm of Ch. Tsitouras by using it to construct a scheme of Sharp and Verner." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 161 "See: Completely Imbedded Runge-Kutta Pairs, by P. W. Sharp and J. H. Verner,\n SIAM Jou rnal on Numerical Analysis, Vol. 31, No. 4. (Aug., 1994), page 1185." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 22 "We spec ify the nodes: " }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "c[ 2] = 1/12;" "6#/&%\"cG6#\"\"#*&\"\"\"F)\"#7!\"\"" }{TEXT -1 3 ", " } {XPPEDIT 18 0 "c[4] = 1/5;" "6#/&%\"cG6#\"\"%*&\"\"\"F)\"\"&!\"\"" } {TEXT -1 3 ", " }{XPPEDIT 18 0 "c[5] = 8/15;" "6#/&%\"cG6#\"\"&*&\"\" )\"\"\"\"#:!\"\"" }{TEXT -1 3 ", " }{XPPEDIT 18 0 "c[6] = 2/3;" "6#/& %\"cG6#\"\"'*&\"\"#\"\"\"\"\"$!\"\"" }{TEXT -1 3 ", " }{XPPEDIT 18 0 "c[7] = 19/20;" "6#/&%\"cG6#\"\"(*&\"#>\"\"\"\"#?!\"\"" }{TEXT -1 3 ", " }{XPPEDIT 18 0 "c[8]=1" "6#/&%\"cG6#\"\")\"\"\"" }{TEXT -1 2 ", " }}{PARA 0 "" 0 "" {TEXT -1 31 "the zero linking coefficients: " }} {PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "a[4,2]=0" "6#/&%\"aG6 $\"\"%\"\"#\"\"!" }{TEXT -1 3 ", " }{XPPEDIT 18 0 "a[5,2]=0" "6#/&%\" aG6$\"\"&\"\"#\"\"!" }{TEXT -1 3 ", " }{XPPEDIT 18 0 "a[6,2]=0" "6#/& %\"aG6$\"\"'\"\"#\"\"!" }{TEXT -1 3 ", " }{XPPEDIT 18 0 "a[7,2]=0" "6 #/&%\"aG6$\"\"(\"\"#\"\"!" }{TEXT -1 3 ", " }{XPPEDIT 18 0 "a[8,2]=0 " "6#/&%\"aG6$\"\")\"\"#\"\"!" }{TEXT -1 2 ", " }}{PARA 0 "" 0 "" {TEXT -1 22 "and the zero weights: " }}{PARA 256 "" 0 "" {TEXT -1 1 " \+ " }{XPPEDIT 18 0 "b[2]=0" "6#/&%\"bG6#\"\"#\"\"!" }{TEXT -1 3 ", " } {XPPEDIT 18 0 "b[3]=0" "6#/&%\"bG6#\"\"$\"\"!" }{TEXT -1 3 ", " } {XPPEDIT 18 0 "`b*`[2]=0" "6#/&%#b*G6#\"\"#\"\"!" }{TEXT -1 3 ", " } {XPPEDIT 18 0 "`b*`[3]=0" "6#/&%#b*G6#\"\"$\"\"!" }{TEXT -1 3 ", " } {XPPEDIT 18 0 "`b*`[8]=0" "6#/&%#b*G6#\"\")\"\"!" }{TEXT -1 1 "." }} {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 90 "We shall \+ obtain expressions for all the coefficients of the scheme in terms of \+ the nodes " }{XPPEDIT 18 0 "c[2]" "6#&%\"cG6#\"\"#" }{TEXT -1 3 ", \+ " }{XPPEDIT 18 0 "c[4]" "6#&%\"cG6#\"\"%" }{TEXT -1 3 ", " }{XPPEDIT 18 0 "c[5]" "6#&%\"cG6#\"\"&" }{TEXT -1 3 ", " }{XPPEDIT 18 0 "c[6]" "6#&%\"cG6#\"\"'" }{TEXT -1 7 " and " }{XPPEDIT 18 0 "c[7]" "6#&%\"c G6#\"\"(" }{TEXT -1 1 "." }}{PARA 0 "" 0 "" {TEXT -1 10 "The node " } {XPPEDIT 18 0 "c[3]" "6#&%\"cG6#\"\"$" }{TEXT -1 27 " does not appear because " }{XPPEDIT 18 0 "c[3]=2/3" "6#/&%\"cG6#\"\"$*&\"\"#\"\"\"F' !\"\"" }{TEXT -1 1 " " }{XPPEDIT 18 0 "c[4]" "6#&%\"cG6#\"\"%" }{TEXT -1 2 ". " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 146 "e1 := \{c[2]=1/12,c[4]=1/5,c[5]=8/15,c[6]=2/3,c[7]=1 9/20,c[8]=1,c[9]=1,\n b[2]=0,b[3]=0,`b*`[2]=0,`b*`[3]=0,`b*`[8 ]=0,seq(a[i,2]=0,i=4..8)\}:" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 15 "" 0 "" {TEXT 283 6 "Step 1" }{TEXT -1 1 " " }}{PARA 0 "" 0 "" {TEXT -1 33 "We use the quadrature conditions:" }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "Sum(b[i],i=1..8)=1" "6#/-%$SumG6$&%\"bG 6#%\"iG/F*;\"\"\"\"\")F-" }{TEXT -1 15 ", " }{XPPEDIT 18 0 "Sum(b[i]*c[i]^(k-1),i = 2 .. 8) = 1/k;" "6#/-%$SumG6$*&&%\"bG6#%\"i G\"\"\")&%\"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 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 3 ", " }{XPPEDIT 18 0 "b[8]" "6#&%\"bG6#\"\")" }{TEXT -1 1 " ." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 142 "[add(b[i],i=1..8)=1,seq(add(b[i]*c[i]^(j-1),i=2..8)= 1/j,j=2..6)]:\ne2 := solve(\{op(subs(e1,%))\},\{seq(b[i],i=[1,$4..8]) \}):\ne3 := `union`(e1,e2):" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 49 "The weights of the order 6 scheme are as follow s." }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 31 "seq(b[i]=subs(e3,b[i]) ,i=1..8);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6*/&%\"bG6#\"\"\"#\"#`\"$7* /&F%6#\"\"#\"\"!/&F%6#\"\"$F//&F%6#\"\"%#\"\"&\"#;/&F%6#F9#\"#F\"$7\"/ &F%6#\"\"'#F?\"$O\"/&F%6#\"\"(#\"$c#\"$p*/&F%6#\"\")#!#D\"$O$" }}} {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 0 300 "e3 := \{a[8 ,2] = 0, c[9] = 1, c[8] = 1, a[5,2] = 0, a[4,2] = 0, a[6,2] = 0, a[7,2 ] = 0, b[3] = 0, b[2] = 0, `b*`[3] = 0, `b*`[2] = 0, b[1] = 53/912, b[ 8] = -25/336, b[7] = 256/969, b[6] = 27/136, `b*`[8] = 0, c[4] = 1/5, \+ c[2] = 1/12, c[6] = 2/3, b[5] = 27/112, b[4] = 5/16, c[5] = 8/15, c[7] = 19/20\}:" }}}{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 2" }{TEXT -1 1 " " }}{PARA 0 " " 0 "" {TEXT -1 35 "We use the stage-order conditions: " }}{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 3 ", " }}{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#*$&%\"cG6#\" \"%\"\"#" }{TEXT -1 15 ", " }{XPPEDIT 18 0 "Sum(a[4,j]*c[ j]^2,j = 2 .. 3) = 1/3;" "6#/-%$SumG6$*&&%\"aG6$\"\"%%\"jG\"\"\"*$&%\" cG6#F,\"\"#F-/F,;F2\"\"$*&F-F-F5!\"\"" }{TEXT -1 1 " " }{XPPEDIT 18 0 "c[4]^3;" "6#*$&%\"cG6#\"\"%\"\"$" }{TEXT -1 1 " " }}{PARA 0 "" 0 "" {TEXT -1 9 "to find " }{XPPEDIT 18 0 "c[3]" "6#&%\"cG6#\"\"$" }{TEXT -1 3 ", " }{XPPEDIT 18 0 "a[3,2]" "6#&%\"aG6$\"\"$\"\"#" }{TEXT -1 7 " and " }{XPPEDIT 18 0 "a[4,3];" "6#&%\"aG6$\"\"%\"\"$" }{TEXT -1 2 ". " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" } }{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 269 "[a[3,2]*c[2] = c[3]^2/2,se q(add(a[4,j]*c[j]^(k-1),j=2..3)=c[4]^k/k,k=[2,3])]:\neqns1 := subs(e3, %):\neqns1[1];\neqns1[2];\neqns1[3];\ne4 := solve(\{op(eqns1)\},\{a[3, 2],c[3],a[4,3]\}):\ne5 := `union`(e3,e4):\n``;\nc[3]=subs(e5,c[3]);\na [3,2]=subs(e5,a[3,2]);\na[4,3]=subs(e5,a[4,3]);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/,$*&#\"\"\"\"#7F'&%\"aG6$\"\"$\"\"#F'F',$*&#F'F-F'*$)& %\"cG6#F,F-F'F'F'" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/*&&%\"aG6$\"\"% \"\"$\"\"\"&%\"cG6#F)F*#F*\"#]" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/*&& %\"aG6$\"\"%\"\"$\"\"\")&%\"cG6#F)\"\"#F*#F*\"$v$" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%!G" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/&%\"cG6#\"\"$# \"\"#\"#:" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/&%\"aG6$\"\"$\"\"##\"\") \"#v" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/&%\"aG6$\"\"%\"\"$#F(\"#?" }} }{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 2 "e5" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 343 "e5 := \{a[8 ,2] = 0, c[9] = 1, c[8] = 1, a[5,2] = 0, a[4,2] = 0, a[6,2] = 0, a[7,2 ] = 0, b[3] = 0, b[2] = 0, `b*`[3] = 0, `b*`[2] = 0, a[3,2] = 8/75, c[ 3] = 2/15, a[4,3] = 3/20, b[1] = 53/912, b[8] = -25/336, b[7] = 256/96 9, b[6] = 27/136, `b*`[8] = 0, c[4] = 1/5, c[2] = 1/12, c[6] = 2/3, b[ 5] = 27/112, b[4] = 5/16, c[5] = 8/15, c[7] = 19/20\}:" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}} {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 15 "" 0 "" {TEXT -1 1 " " } {TEXT 285 6 "Step 3" }{TEXT -1 1 " " }}{PARA 0 "" 0 "" {TEXT -1 35 "We use the stage-order conditions: " }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "Sum(a[5,j]*c[j],j = 2 .. 4) = 1/2;" "6#/-%$SumG6$*&&% \"aG6$\"\"&%\"jG\"\"\"&%\"cG6#F,F-/F,;\"\"#\"\"%*&F-F-F3!\"\"" }{TEXT -1 1 " " }{XPPEDIT 18 0 "c[5]^2" "6#*$&%\"cG6#\"\"&\"\"#" }{TEXT -1 15 ", " }{XPPEDIT 18 0 "Sum(a[5,j]*c[j]^(2),j=2..4)=1/3" "6#/-%$SumG6$*&&%\"aG6$\"\"&%\"jG\"\"\"*$&%\"cG6#F,\"\"#F-/F,;F2\"\"%* &F-F-\"\"$!\"\"" }{TEXT -1 1 " " }{XPPEDIT 18 0 "c[5]^3" "6#*$&%\"cG6# \"\"&\"\"$" }{TEXT -1 1 " " }}{PARA 0 "" 0 "" {TEXT -1 9 "to find " } {XPPEDIT 18 0 "a[5,3]" "6#&%\"aG6$\"\"&\"\"$" }{TEXT -1 7 " and " } {XPPEDIT 18 0 "a[5,4]" "6#&%\"aG6$\"\"&\"\"%" }{TEXT -1 1 "." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 209 "[ seq(add(a[5,j]*c[j]^(k-1),j=2..4)=c[5]^k/k,k=[2,3])]:\neqns2 := subs(e 5,%):\neqns2[1];\neqns2[2];\ne6 := solve(\{op(eqns2)\},\{a[5,3],a[5,4] \}):\ne7 := `union`(e5,subs(e4,e6)):\n``;\nseq(a[5,j]=subs(e7,a[5,j]), j=[3,4]);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/,&*&#\"\"#\"#:\"\"\"&%\" aG6$\"\"&\"\"$F)F)*&#F)F-F)&F+6$F-\"\"%F)F)#\"#K\"$D#" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/,&*&#\"\"%\"$D#\"\"\"&%\"aG6$\"\"&\"\"$F)F)*&#F) \"#DF)&F+6$F-F'F)F)#\"$7&\"&D,\"" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%! G" }}{PARA 11 "" 1 "" {XPPMATH 20 "6$/&%\"aG6$\"\"&\"\"$#!$7\"\"#X/&F% 6$F'\"\"%#\"#k\"#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 0 377 "e7 := \{a[8,2] = 0, c[9] = 1, c[8] = 1, a[5,2] = 0, a[4,2] = \+ 0, a[6,2] = 0, a[7,2] = 0, b[3] = 0, b[2] = 0, `b*`[3] = 0, `b*`[2] = \+ 0, a[5,3] = -112/45, a[5,4] = 64/27, a[3,2] = 8/75, c[3] = 2/15, a[4,3 ] = 3/20, b[1] = 53/912, b[8] = -25/336, b[7] = 256/969, b[6] = 27/136 , `b*`[8] = 0, c[4] = 1/5, c[2] = 1/12, c[6] = 2/3, b[5] = 27/112, b[4 ] = 5/16, c[5] = 8/15, c[7] = 19/20\}:" }{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 286 6 "Step 4" }{TEXT -1 1 " " }}{PARA 0 "" 0 "" {TEXT -1 41 "We use the (column) simplifying condition" }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "Sum(b[i]*a[i,j],i=j..8)=b[j]*(1-c[j])" "6#/-%$Su mG6$*&&%\"bG6#%\"iG\"\"\"&%\"aG6$F+%\"jGF,/F+;F0\"\")*&&F)6#F0F,,&F,F, &%\"cG6#F0!\"\"F," }{TEXT -1 7 " for " }{XPPEDIT 18 0 "j=7" "6#/%\"j G\"\"(" }{TEXT -1 1 "," }}{PARA 0 "" 0 "" {TEXT -1 8 "that is," }} {PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "b[8]*a[8,7] = b[7]*(1 -c[7])" "6#/*&&%\"bG6#\"\")\"\"\"&%\"aG6$F(\"\"(F)*&&F&6#F-F),&F)F)&% \"cG6#F-!\"\"F)" }{TEXT -1 2 ", " }}{PARA 0 "" 0 "" {TEXT -1 9 "to fin d " }{XPPEDIT 18 0 "a[8,7]" "6#&%\"aG6$\"\")\"\"(" }{TEXT -1 1 "." }} {PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 113 "b[8]*a[8,7] = b[7]*(1-c[7]);\nsubs(e7,%);\ne8 := solve(\{%\},\{a[ 8,7]\}):\ne9 := `union`(e7,e8):\na[8,7]=subs(e9,a[8,7]);" }}{PARA 11 " " 1 "" {XPPMATH 20 "6#/*&&%\"bG6#\"\")\"\"\"&%\"aG6$F(\"\"(F)*&&F&6#F- F),&F)F)&%\"cGF0!\"\"F)" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/,$*&#\"#D \"$O$\"\"\"&%\"aG6$\"\")\"\"(F)!\"\"#\"#k\"%X[" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/&%\"aG6$\"\")\"\"(#!%or\"&v.%" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 2 "e9" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 399 "e9 := \{a[8,2] = 0, c[9] = \+ 1, c[8] = 1, a[5,2] = 0, a[4,2] = 0, a[6,2] = 0, a[7,2] = 0, b[3] = 0, b[2] = 0, `b*`[3] = 0, `b*`[2] = 0, a[5,3] = -112/45, a[5,4] = 64/27, a[8,7] = -7168/40375, a[3,2] = 8/75, c[3] = 2/15, a[4,3] = 3/20, b[1] = 53/912, b[8] = -25/336, b[7] = 256/969, b[6] = 27/136, `b*`[8] = 0, c[4] = 1/5, c[2] = 1/12, c[6] = 2/3, b[5] = 27/112, b[4] = 5/16, c[5] = 8/15, c[7] = 19/20\}:" }{TEXT -1 0 "" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}}{PARA 0 "" 0 " " {TEXT -1 0 "" }}{PARA 15 "" 0 "" {TEXT -1 1 " " }{TEXT 287 6 "Step 5 " }{TEXT -1 1 " " }}{PARA 0 "" 0 "" {TEXT -1 41 "We use the \"alternat ive\" order condition " }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "[b]^T*([C]-Id)*[A]*(C-c[4]*Id)*(C-c[5]*Id)*[c] = Int((x-1)*Int(( t-c[4])*(t-c[5])*t,t = 0 .. x),x = 0 .. 1);" "6#/*.)7#%\"bG%\"TG\"\"\" ,&7#%\"CGF)%#IdG!\"\"F)7#%\"AGF),&F,F)*&&%\"cG6#\"\"%F)F-F)F.F),&F,F)* &&F46#\"\"&F)F-F)F.F)7#F4F)-%$IntG6$*&,&%\"xGF)F)F.F)-F>6$*(,&%\"tGF)& F46#F6F.F),&FGF)&F46#F;F.F)FGF)/FG;\"\"!FBF)/FB;FOF)" }{TEXT -1 2 ". \+ " }}{PARA 0 "" 0 "" {TEXT -1 39 "This condition amounts to the relatio n " }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "b[7]*(c[7]-1)*a [7,6]*(c[6]-c[4])*(c[6]-c[5])*c[6]=-1/120+c[4]/60+c[5]/60-c[4]*c[5]/24 " "6#/*.&%\"bG6#\"\"(\"\"\",&&%\"cG6#F(F)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 27 "which can be used to fi nd " }{XPPEDIT 18 0 "a[7,6]" "6#&%\"aG6$\"\"(\"\"'" }{TEXT -1 2 ". " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 179 "b[7]*(c[7]-1)*a[7,6]*(c[6]-c[4])*(c[6]-c[5])*c[6] = -1/120+c[4] /60+c[5]/60-c[4]*c[5]/24;\nsubs(e9,%);\ne10 := solve(\{%\},a[7,6]):\ne 11 := `union`(e9,e10):\n``;\na[7,6]=subs(e11,a[7,6]);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/*.&%\"bG6#\"\"(\"\"\",&&%\"cGF'F)F)!\"\"F)&%\"aG6 $F(\"\"'F),&&F,6#F1F)&F,6#\"\"%F-F),&F3F)&F,6#\"\"&F-F)F3F),*#F)\"$?\" F-*&#F)\"#gF)F5F)F)*&F@F)F9F)F)*&#F)\"#CF)*&F5F)F9F)F)F-" }}{PARA 11 " " 1 "" {XPPMATH 20 "6#/,$*&#\"%#z\"\"(v.F$\"\"\"&%\"aG6$\"\"(\"\"'F)! \"\"#F/\"%+=" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%!G" }}{PARA 11 "" 1 " " {XPPMATH 20 "6#/&%\"aG6$\"\"(\"\"'#\"&NX\"\"&OV\"" }}}{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 0 422 "e11 := \{a[8,2] = 0, c [9] = 1, c[8] = 1, a[5,2] = 0, a[4,2] = 0, a[6,2] = 0, a[7,2] = 0, b[3 ] = 0, b[2] = 0, `b*`[3] = 0, `b*`[2] = 0, a[5,3] = -112/45, a[5,4] = \+ 64/27, a[8,7] = -7168/40375, a[7,6] = 14535/14336, a[3,2] = 8/75, c[3] = 2/15, a[4,3] = 3/20, b[1] = 53/912, b[8] = -25/336, b[7] = 256/969, b[6] = 27/136, `b*`[8] = 0, c[4] = 1/5, c[2] = 1/12, c[6] = 2/3, b[5] = 27/112, b[4] = 5/16, c[5] = 8/15, c[7] = 19/20\}:" }{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 288 6 "Step 6" }{TEXT -1 1 " " }}{PARA 0 "" 0 "" {TEXT -1 41 "We use the (column) simplifying condition" }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "Sum(b[i]*a[i,j],i=j..8)=b[j]*(1-c[j])" "6#/-%$SumG6$*&&%\"bG6#%\"iG\"\"\"&%\"aG6$F+%\"jGF,/F+;F0\"\")*&&F)6#F 0F,,&F,F,&%\"cG6#F0!\"\"F," }{TEXT -1 7 " for " }{XPPEDIT 18 0 "j = \+ 6;" "6#/%\"jG\"\"'" }{TEXT -1 1 "," }}{PARA 0 "" 0 "" {TEXT -1 8 "that is," }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "b[7]*a[7,6]+b [8]*a[8,6] = b[6]*(1-c[6])" "6#/,&*&&%\"bG6#\"\"(\"\"\"&%\"aG6$F)\"\"' F*F**&&F'6#\"\")F*&F,6$F2F.F*F**&&F'6#F.F*,&F*F*&%\"cG6#F.!\"\"F*" } {TEXT -1 2 ", " }}{PARA 0 "" 0 "" {TEXT -1 9 "to find " }{XPPEDIT 18 0 "a[8,6];" "6#&%\"aG6$\"\")\"\"'" }{TEXT -1 1 "." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 133 "add(b[i]*a[ i,6],i=7..8)=b[6]*(1-c[6]):\nsubs(e11,%);\ne12 := solve(\{%\},\{a[8,6] \}):\ne13 := `union`(e11,e12):\n``;\na[8,6]=subs(e13,a[8,6]);" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#/,&#\"#:\"#c\"\"\"*&#\"#D\"$O$F(&%\"aG 6$\"\")\"\"'F(!\"\"#\"\"*\"$O\"" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%!G " }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/&%\"aG6$\"\")\"\"'#\"%_6\"$D%" }} }{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 0 441 "e13 := \{a[ 8,2] = 0, c[9] = 1, c[8] = 1, a[5,2] = 0, a[4,2] = 0, a[6,2] = 0, a[7, 2] = 0, b[3] = 0, b[2] = 0, `b*`[3] = 0, `b*`[2] = 0, a[5,3] = -112/45 , a[5,4] = 64/27, a[8,7] = -7168/40375, a[7,6] = 14535/14336, a[3,2] = 8/75, c[3] = 2/15, a[4,3] = 3/20, b[1] = 53/912, b[8] = -25/336, b[7] = 256/969, b[6] = 27/136, `b*`[8] = 0, c[4] = 1/5, c[2] = 1/12, c[6] \+ = 2/3, b[5] = 27/112, b[4] = 5/16, a[8,6] = 1152/425, c[5] = 8/15, c[7 ] = 19/20\}:" }{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 289 6 "Step 7" }{TEXT -1 1 " " }} {PARA 0 "" 0 "" {TEXT -1 23 "We find the 6 weights " }{XPPEDIT 18 0 " `b*`[1];" "6#&%#b*G6#\"\"\"" }{TEXT -1 3 ", " }{XPPEDIT 18 0 "`b*`[4] ;" "6#&%#b*G6#\"\"%" }{TEXT -1 3 ", " }{XPPEDIT 18 0 "`b*`[5];" "6#&% #b*G6#\"\"&" }{TEXT -1 3 ", " }{XPPEDIT 18 0 "`b*`[6];" "6#&%#b*G6#\" \"'" }{TEXT -1 2 ", " }{XPPEDIT 18 0 "`b*`[7];" "6#&%#b*G6#\"\"(" } {TEXT -1 3 ", " }{XPPEDIT 18 0 "`b*`[9];" "6#&%#b*G6#\"\"*" }{TEXT -1 39 ", by using the 5 quadrature conditions:" }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "Sum(`b*`[i],i = 1 .. 9) = 1;" "6#/-%$Su mG6$&%#b*G6#%\"iG/F*;\"\"\"\"\"*F-" }{TEXT -1 15 ", " } {XPPEDIT 18 0 "Sum(`b*`[i]*c[i]^(k-1),i = 2 .. 9) = 1/k;" "6#/-%$SumG6 $*&&%#b*G6#%\"iG\"\"\")&%\"cG6#F+,&%\"kGF,F,!\"\"F,/F+;\"\"#\"\"**&F,F ,F2F3" }{TEXT -1 7 ", " }{XPPEDIT 18 0 "k = 2;" "6#/%\"kG\"\"#" } {TEXT -1 8 " . . 5, " }}{PARA 0 "" 0 "" {TEXT -1 48 "together with the \"alternative\" order condition " }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "[`b*`]^T*[A]*(C-c[4]*Id)*(C-c[5]*Id)*[c] = Int(Int((t- c[4])*(t-c[5])*t,t = 0 .. x),x = 0 .. 1);" "6#/*,)7#%#b*G%\"TG\"\"\"7# %\"AGF),&%\"CGF)*&&%\"cG6#\"\"%F)%#IdGF)!\"\"F),&F-F)*&&F06#\"\"&F)F3F )F4F)7#F0F)-%$IntG6$-F<6$*(,&%\"tGF)&F06#F2F4F),&FBF)&F06#F9F4F)FBF)/F B;\"\"!%\"xG/FK;FJF)" }{TEXT -1 2 ". " }}{PARA 0 "" 0 "" {TEXT -1 24 " This gives the relation " }}{PARA 256 "" 0 "" {TEXT -1 1 " " } {XPPEDIT 18 0 "Sum(Sum(`b*`[i]*a[i,j-1],i = j .. 7)*(c[j-1]-c[4])*(c[j -1]-c[5])*c[j-1],j = 7 .. 9) = -1/120+c[4]/60+c[5]/60-c[4]*c[5]/24;" " 6#/-%$SumG6$**-F%6$*&&%#b*G6#%\"iG\"\"\"&%\"aG6$F.,&%\"jGF/F/!\"\"F//F .;F4\"\"(F/,&&%\"cG6#,&F4F/F/F5F/&F;6#\"\"%F5F/,&&F;6#,&F4F/F/F5F/&F;6 #\"\"&F5F/&F;6#,&F4F/F/F5F//F4;F8\"\"*,**&F/F/\"$?\"F5F5*&&F;6#F@F/\"# gF5F/*&&F;6#FGF/FTF5F/*(&F;6#F@F/&F;6#FGF/\"#CF5F5" }{TEXT -1 2 ". " } }{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 41 "Expandin g the left-hand side gives . . . " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 76 "add(add(`b*`[i]*a[i,j-1],i=j ..9)*(c[j-1]-c[4])*(c[j-1]-c[5])*c[j-1],j=7..9);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#,(**,(*&&%#b*G6#\"\"(\"\"\"&%\"aG6$F*\"\"'F+F+*&&F(6#\" \")F+&F-6$F3F/F+F+*&&F(6#\"\"*F+&F-6$F9F/F+F+F+,&&%\"cG6#\"\"%!\"\"&F> 6#F/F+F+,&&F>6#\"\"&FAFBF+F+FBF+F+**,&*&F1F+&F-6$F3F*F+F+*&F7F+&F-6$F9 F*F+F+F+,&F=FA&F>F)F+F+,&FQF+FEFAF+FQF+F+*,F7F+&F-6$F9F3F+,&&F>F2F+F=F AF+,&FWF+FEFAF+FWF+F+" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT 260 4 "Note" }{TEXT -1 61 ": Since the last equation involv es the linking coefficients " }{XPPEDIT 18 0 "a[9,j]" "6#&%\"aG6$\"\" *%\"jG" }{TEXT -1 3 ", " }{XPPEDIT 18 0 "j=1" "6#/%\"jG\"\"\"" } {TEXT -1 43 " . . 8, we need to make the substitutions " }{XPPEDIT 18 0 "a[9,j]=b[j]" "6#/&%\"aG6$\"\"*%\"jG&%\"bG6#F(" }{TEXT -1 3 ", \+ " }{XPPEDIT 18 0 "j=1" "6#/%\"jG\"\"\"" }{TEXT -1 7 " . . 8." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 60 "First we set up the six equations for the six weights . . . " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 406 "`quad_eqs*` := [add(`b*`[i],i=1..9)=1,seq(add(`b*`[i]*c[i]^(j-1),i=2..9)=1/j,j=2. .5)]:\n`ord_eq*` := add(add(`b*`[i]*a[i,j-1],i=j..9)*(c[j-1]-c[4])*(c[ j-1]-c[5])*c[j-1],j=7..9)=\n 1/20-c[4]/12-c[5]/12+c[4] *c[5]/6:\nwt_eqs := [seq(a[9,i]=b[i],i=1..8)]:\n`eqns*` := simplify(su bs(e13,[op(`quad_eqs*`),subs(wt_eqs,`ord_eq*`)])):\nnops(`eqns*`);\nin dets(`eqns*`) minus \{c[4],c[5],c[6],c[7]\};\nnops(%);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#\"\"'" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#<(&%#b*G6 #\"\"'&F%6#\"\"*&F%6#\"\"(&F%6#\"\"&&F%6#\"\"%&F%6#\"\"\"" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#\"\"'" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }} {PARA 0 "" 0 "" {TEXT -1 30 " . . . and then we solve them." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 22 "in folevel[solve] := 4:" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 76 "e14 := solve(\{op(`eqns*`)\}):\ne15 := `union`(e13,e14):\ninfolevel[solve ] := 0:" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 49 "The weights of the order 5 scheme are as follows." }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 38 "seq(`b*`[i]=subs(e15,`b*`[i]),i=1..9);" } }{PARA 11 "" 1 "" {XPPMATH 20 "6+/&%#b*G6#\"\"\"#\"$<'\"&W4\"/&F%6#\" \"#\"\"!/&F%6#\"\"$F//&F%6#\"\"%#\"$T#\"$c(/&F%6#\"\"&#\"#p\"$?$/&F%6# \"\"'#\"$N%\"%/>/&F%6#\"\"(#\"&/.\"\"&0O%/&F%6#\"\")F//&F%6#\"\"*#!\" \"\"#=" }}}{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 0 559 "e15 := \{a[8,2] = 0, c[9] = 1, c[8] = 1, a[5,2] = 0, a[4,2] = 0, \+ a[6,2] = 0, a[7,2] = 0, b[3] = 0, b[2] = 0, `b*`[3] = 0, `b*`[2] = 0, \+ a[5,3] = -112/45, a[5,4] = 64/27, a[8,7] = -7168/40375, a[7,6] = 14535 /14336, a[3,2] = 8/75, c[3] = 2/15, a[4,3] = 3/20, `b*`[7] = 10304/436 05, b[1] = 53/912, b[8] = -25/336, b[7] = 256/969, b[6] = 27/136, `b*` [8] = 0, c[4] = 1/5, c[2] = 1/12, c[6] = 2/3, b[5] = 27/112, b[4] = 5/ 16, `b*`[5] = 69/320, `b*`[4] = 241/756, `b*`[6] = 435/1904, `b*`[1] = 617/10944, a[8,6] = 1152/425, c[5] = 8/15, c[7] = 19/20, `b*`[9] = -1 /18\}:" }{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 290 6 "Step 8" }{TEXT -1 1 " " }}{PARA 0 " " 0 "" {TEXT -1 22 "We use the relations: " }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "Sum(`b*`[i]*a[i,3],i=4..8)=0" "6#/-%$SumG6$*& &%#b*G6#%\"iG\"\"\"&%\"aG6$F+\"\"$F,/F+;\"\"%\"\")\"\"!" }{TEXT -1 6 " , " }{XPPEDIT 18 0 "Sum(b[i]*a[i,3],i=4..8)=b[3]*(1-c[3])" "6#/-%$ SumG6$*&&%\"bG6#%\"iG\"\"\"&%\"aG6$F+\"\"$F,/F+;\"\"%\"\")*&&F)6#F0F,, &F,F,&%\"cG6#F0!\"\"F," }{TEXT -1 6 ", " }{XPPEDIT 18 0 "Sum(b[i]* (c[i]-1)*a[i,3],i=4..7)=0" "6#/-%$SumG6$*(&%\"bG6#%\"iG\"\"\",&&%\"cG6 #F+F,F,!\"\"F,&%\"aG6$F+\"\"$F,/F+;\"\"%\"\"(\"\"!" }{TEXT -1 2 ", " } }{PARA 0 "" 0 "" {TEXT -1 9 "to find " }{XPPEDIT 18 0 "a[6,3]" "6#&% \"aG6$\"\"'\"\"$" }{TEXT -1 3 ", " }{XPPEDIT 18 0 "a[7,3]" "6#&%\"aG6 $\"\"(\"\"$" }{TEXT -1 7 " and " }{XPPEDIT 18 0 "a[8,3]" "6#&%\"aG6$ \"\")\"\"$" }{TEXT -1 1 "." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 272 "[add(`b*`[i]*a[i,3],i=4..8)=0,add( b[i]*a[i,3],i=4..8)=b[3]*(1-c[3]),add(b[i]*(c[i]-1)*a[i,3],i=4..7)=0]: \neqns3 := subs(e15,%):\neqns3[1];\neqns3[2];\neqns3[3];\ne16 := solve (\{op(eqns3)\},\{a[6,3],a[7,3],a[8,3]\}):\ne17 := `union`(e15,e16):\n` `;\nseq(a[i,3]=subs(e17,a[i,3]),i=6..8);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/,(#\"&>B\"\"&+_#!\"\"*&#\"$N%\"%/>\"\"\"&%\"aG6$\"\"'\"\"$F-F-* &#\"&/.\"\"&0O%F-&F/6$\"\"(F2F-F-\"\"!" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/,*#\"$x\"\"$?$!\"\"*&#\"#F\"$O\"\"\"\"&%\"aG6$\"\"'\"\"$F-F-*&# \"$c#\"$p*F-&F/6$\"\"(F2F-F-*&#\"#D\"$O$F-&F/6$\"\")F2F-F(\"\"!" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#/,(#\"#(*\"$+%\"\"\"*&#\"\"*\"$O\"F(&% \"aG6$\"\"'\"\"$F(!\"\"*&#\"#k\"%X[F(&F.6$\"\"(F1F(F2\"\"!" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%!G" }}{PARA 11 "" 1 "" {XPPMATH 20 "6%/&%\" aG6$\"\"'\"\"$#\"%!)Q\"$j*/&F%6$\"\"(F(#!(\\2+\"\"'Sya/&F%6$\"\")F(#!% $y'\"%S@" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 3 "e17" }} {PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 625 "e17 := \{a[8,2] = 0, c[9] = 1, c[8] = 1, a[5,2] = 0, a[4,2] = 0, \+ a[6,2] = 0, a[7,2] = 0, b[3] = 0, b[2] = 0, `b*`[3] = 0, `b*`[2] = 0, \+ a[5,3] = -112/45, a[5,4] = 64/27, a[8,7] = -7168/40375, a[7,6] = 14535 /14336, a[3,2] = 8/75, c[3] = 2/15, a[4,3] = 3/20, a[8,3] = -6783/2140 , a[6,3] = 3880/963, `b*`[7] = 10304/43605, b[1] = 53/912, a[7,3] = -1 000749/547840, b[8] = -25/336, b[7] = 256/969, b[6] = 27/136, `b*`[8] \+ = 0, c[4] = 1/5, c[2] = 1/12, c[6] = 2/3, b[5] = 27/112, b[4] = 5/16, \+ `b*`[5] = 69/320, `b*`[4] = 241/756, `b*`[6] = 435/1904, `b*`[1] = 617 /10944, a[8,6] = 1152/425, c[5] = 8/15, c[7] = 19/20, `b*`[9] = -1/18 \}:" }{TEXT -1 0 "" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}}{PARA 0 "" 0 "" {TEXT -1 0 "" }} {PARA 15 "" 0 "" {TEXT 291 6 "Step 9" }{TEXT -1 1 " " }}{PARA 0 "" 0 " " {TEXT -1 34 "We use the stage-order conditions:" }}{PARA 256 "" 0 " " {TEXT -1 1 " " }{XPPEDIT 18 0 "Sum(a[6,j]*c[j],j=2..5)=1/2" "6#/-%$S umG6$*&&%\"aG6$\"\"'%\"jG\"\"\"&%\"cG6#F,F-/F,;\"\"#\"\"&*&F-F-F3!\"\" " }{TEXT -1 1 " " }{XPPEDIT 18 0 "c[6]^2" "6#*$&%\"cG6#\"\"'\"\"#" } {TEXT -1 6 ", " }{XPPEDIT 18 0 "Sum(a[6,j]*c[j]^2,j=2..5)=1/3" "6# /-%$SumG6$*&&%\"aG6$\"\"'%\"jG\"\"\"*$&%\"cG6#F,\"\"#F-/F,;F2\"\"&*&F- F-\"\"$!\"\"" }{TEXT -1 1 " " }{XPPEDIT 18 0 "c[6]^3" "6#*$&%\"cG6#\" \"'\"\"$" }{TEXT -1 2 ", " }}{PARA 0 "" 0 "" {TEXT -1 9 "to find " } {XPPEDIT 18 0 "a[6,4]" "6#&%\"aG6$\"\"'\"\"%" }{TEXT -1 7 " and " } {XPPEDIT 18 0 "a[6,5]" "6#&%\"aG6$\"\"'\"\"&" }{TEXT -1 1 "." }}{PARA 0 "" 0 "" {TEXT -1 1 " " }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 217 " [seq(add(a[6,j]*c[j]^(k-1),j=2..5)=c[6]^k/k,k=[2,3])]:\neqns4 := subs( e17,%):\neqns4[1];\neqns4[2];\ne18 := solve(\{op(eqns4)\},\{a[6,4],a[6 ,5]\}):\ne19 := `union`(e17,e18):\n``;\na[6,4]=subs(e19,a[6,4]),a[6,5] =subs(e19,a[6,5]);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/,(#\"%_:\"%*)G \"\"\"*&#F(\"\"&F(&%\"aG6$\"\"'\"\"%F(F(*&#\"\")\"#:F(&F-6$F/F+F(F(#\" \"#\"\"*" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/,(#\"%/J\"&NL%\"\"\"*&#F( \"#DF(&%\"aG6$\"\"'\"\"%F(F(*&#\"#k\"$D#F(&F-6$F/\"\"&F(F(#\"\")\"#\") " }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%!G" }}{PARA 11 "" 1 "" {XPPMATH 20 "6$/&%\"aG6$\"\"'\"\"%#!%c%)\"%*)G/&F%6$F'\"\"&#\"$<#\"$G%" }}} {PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 3 "e19" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 664 "e19 := \{a[ 8,2] = 0, c[9] = 1, c[8] = 1, a[5,2] = 0, a[4,2] = 0, a[6,2] = 0, a[7, 2] = 0, b[3] = 0, b[2] = 0, `b*`[3] = 0, `b*`[2] = 0, a[5,3] = -112/45 , a[5,4] = 64/27, a[8,7] = -7168/40375, a[7,6] = 14535/14336, a[3,2] = 8/75, c[3] = 2/15, a[4,3] = 3/20, a[8,3] = -6783/2140, a[6,3] = 3880/ 963, `b*`[7] = 10304/43605, b[1] = 53/912, a[7,3] = -1000749/547840, b [8] = -25/336, b[7] = 256/969, b[6] = 27/136, `b*`[8] = 0, c[4] = 1/5, c[2] = 1/12, c[6] = 2/3, b[5] = 27/112, b[4] = 5/16, `b*`[5] = 69/320 , `b*`[4] = 241/756, `b*`[6] = 435/1904, `b*`[1] = 617/10944, a[8,6] = 1152/425, c[5] = 8/15, c[7] = 19/20, a[6,5] = 217/428, a[6,4] = -8456 /2889, `b*`[9] = -1/18\}:" }{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 292 7 "Step 10" }{TEXT -1 1 " " }}{PARA 0 "" 0 "" {TEXT -1 34 "We use the stage-order conditions:" }}{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 6 ", " }{XPPEDIT 18 0 "Sum(a[7,j] *c[j]^2,j = 2 .. 6) = 1/3;" "6#/-%$SumG6$*&&%\"aG6$\"\"(%\"jG\"\"\"*$& %\"cG6#F,\"\"#F-/F,;F2\"\"'*&F-F-\"\"$!\"\"" }{TEXT -1 1 " " } {XPPEDIT 18 0 "c[7]^3;" "6#*$&%\"cG6#\"\"(\"\"$" }{TEXT -1 2 ", " }} {PARA 0 "" 0 "" {TEXT -1 9 "to find " }{XPPEDIT 18 0 "a[7,4];" "6#&% \"aG6$\"\"(\"\"%" }{TEXT -1 7 " and " }{XPPEDIT 18 0 "a[7, 5];" "6#& %\"aG6$\"\"(\"\"&" }{TEXT -1 1 "." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 217 "[seq(add(a[7,j]*c[j]^(k-1), j=2..6)=c[7]^k/k,k=[2,3])]:\neqns5 := subs(e19,%):\neqns5[1];\neqns5[2 ];\ne20 := solve(\{op(eqns5)\},\{a[7,4],a[7,5]\}):\ne21 := `union`(e19 ,e20):\n``;\na[7,4]=subs(e21,a[7,4]),a[7,5]=subs(e21,a[7,5]);" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#/,(#\"(8-H)\")+W<>\"\"\"*&#F(\"\"&F(&% \"aG6$\"\"(\"\"%F(F(*&#\"\")\"#:F(&F-6$F/F+F(F(#\"$h$\"$+)" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/,(#\")8<8g\"*+!3Q9\"\"\"*&#F(\"#DF(&%\"aG6$ \"\"(\"\"%F(F(*&#\"#k\"$D#F(&F-6$F/\"\"&F(F(#\"%fo\"&+S#" }}{PARA 11 " " 1 "" {XPPMATH 20 "6#%!G" }}{PARA 11 "" 1 "" {XPPMATH 20 "6$/&%\"aG6$ \"\"(\"\"%#\"'h#>)\"')[$Q/&F%6$F'\"\"&#!'v6n\"'Wl()" }}}{PARA 0 "" 0 " " {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}} {SECT 1 {PARA 4 "" 0 "" {TEXT -1 3 "e21" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 664 "e19 := \{a[8,2] = 0, c [9] = 1, c[8] = 1, a[5,2] = 0, a[4,2] = 0, a[6,2] = 0, a[7,2] = 0, b[3 ] = 0, b[2] = 0, `b*`[3] = 0, `b*`[2] = 0, a[5,3] = -112/45, a[5,4] = \+ 64/27, a[8,7] = -7168/40375, a[7,6] = 14535/14336, a[3,2] = 8/75, c[3] = 2/15, a[4,3] = 3/20, a[8,3] = -6783/2140, a[6,3] = 3880/963, `b*`[7 ] = 10304/43605, b[1] = 53/912, a[7,3] = -1000749/547840, b[8] = -25/3 36, b[7] = 256/969, b[6] = 27/136, `b*`[8] = 0, c[4] = 1/5, c[2] = 1/1 2, c[6] = 2/3, b[5] = 27/112, b[4] = 5/16, `b*`[5] = 69/320, `b*`[4] = 241/756, `b*`[6] = 435/1904, `b*`[1] = 617/10944, a[8,6] = 1152/425, \+ c[5] = 8/15, c[7] = 19/20, a[6,5] = 217/428, a[6,4] = -8456/2889, `b*` [9] = -1/18\}:" }{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 293 7 "Step 11" }{TEXT -1 1 " " }} {PARA 0 "" 0 "" {TEXT -1 34 "We use the stage-order conditions:" }} {PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "Sum(a[8,j]*c[j],j = 2 .. 7) = 1/2;" "6#/-%$SumG6$*&&%\"aG6$\"\")%\"jG\"\"\"&%\"cG6#F,F-/F,; \"\"#\"\"(*&F-F-F3!\"\"" }{TEXT -1 1 " " }{XPPEDIT 18 0 "c[8]^2;" "6#* $&%\"cG6#\"\")\"\"#" }{TEXT -1 6 ", " }{XPPEDIT 18 0 "Sum(a[8,j]*c [j]^2,j = 2 .. 7) = 1/3;" "6#/-%$SumG6$*&&%\"aG6$\"\")%\"jG\"\"\"*$&% \"cG6#F,\"\"#F-/F,;F2\"\"(*&F-F-\"\"$!\"\"" }{TEXT -1 1 " " }{XPPEDIT 18 0 "c[8]^3;" "6#*$&%\"cG6#\"\")\"\"$" }{TEXT -1 2 ", " }}{PARA 0 "" 0 "" {TEXT -1 9 "to find " }{XPPEDIT 18 0 "a[8,4];" "6#&%\"aG6$\"\") \"\"%" }{TEXT -1 7 " and " }{XPPEDIT 18 0 "a[8,5];" "6#&%\"aG6$\"\") \"\"&" }{TEXT -1 1 "." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 217 "[seq(add(a[8,j]*c[j]^(k-1),j=2..7)=c[8]^ k/k,k=[2,3])]:\neqns6 := subs(e21,%):\neqns6[1];\neqns6[2];\ne22 := so lve(\{op(eqns6)\},\{a[8,4],a[8,5]\}):\ne23 := `union`(e21,e22):\n``;\n a[8,4]=subs(e23,a[8,4]),a[8,5]=subs(e23,a[8,5]);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/,(#\"'6E;\"']P8\"\"\"*&#F(\"\"&F(&%\"aG6$\"\")\"\"%F(F (*&#F/\"#:F(&F-6$F/F+F(F(#F(\"\"#" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/ ,(#\"'>7**\"(DJ+\"\"\"\"*&#F(\"#DF(&%\"aG6$\"\")\"\"%F(F(*&#\"#k\"$D#F (&F-6$F/\"\"&F(F(#F(\"\"$" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%!G" }} {PARA 11 "" 1 "" {XPPMATH 20 "6$/&%\"aG6$\"\")\"\"%#\"&c4\"\"%vE/&F%6$ F'\"\"&#!&$\\Q\"&vL\"" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 3 "e23" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 757 "e23 := \{a[8,2] = 0, c[9] = 1, c[8] = 1, a[5,2] = 0, a[4,2] = 0, a[6,2] = 0, a[7,2] = 0, b[3] = 0, b[2] = 0, `b*`[3] = 0, \+ `b*`[2] = 0, a[5,3] = -112/45, a[5,4] = 64/27, a[8,7] = -7168/40375, a [7,6] = 14535/14336, a[3,2] = 8/75, c[3] = 2/15, a[4,3] = 3/20, a[8,3] = -6783/2140, a[6,3] = 3880/963, `b*`[7] = 10304/43605, a[7,5] = -671 175/876544, a[7,4] = 819261/383488, b[1] = 53/912, a[7,3] = -1000749/5 47840, b[8] = -25/336, b[7] = 256/969, b[6] = 27/136, `b*`[8] = 0, c[4 ] = 1/5, c[2] = 1/12, c[6] = 2/3, b[5] = 27/112, b[4] = 5/16, `b*`[5] \+ = 69/320, `b*`[4] = 241/756, `b*`[6] = 435/1904, `b*`[1] = 617/10944, \+ a[8,6] = 1152/425, c[5] = 8/15, c[7] = 19/20, a[6,5] = 217/428, a[6,4] = -8456/2889, a[8,5] = -38493/13375, a[8,4] = 10956/2675, `b*`[9] = - 1/18\}:" }{TEXT -1 0 "" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}}{PARA 0 "" 0 "" {TEXT -1 0 " " }}{PARA 0 "" 0 "" {TEXT -1 31 "We use the row-sum conditions: " }} {PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "Sum(a[i,j],j=1..i-1)= c[i]" "6#/-%$SumG6$&%\"aG6$%\"iG%\"jG/F+;\"\"\",&F*F.F.!\"\"&%\"cG6#F* " }{TEXT -1 7 ", for " }{XPPEDIT 18 0 "i=2" "6#/%\"iG\"\"#" }{TEXT -1 8 " . . 8, " }}{PARA 0 "" 0 "" {TEXT -1 9 "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[8,1]" "6#&%\"aG6$\"\")\"\"\"" }{TEXT -1 2 ". " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 119 "[seq(add(a[ i,j],j=1..i-1)=c[i],i=2..8)]:\ne24 := solve(\{op(subs(e23,%))\},\{seq( a[i,1],i=2..8)\}):\ne25 := `union`(e23,e24):" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 60 "The linking coefficients \+ in the first column are as follows." }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 36 "seq(a[i,1]=subs(e25,a[i,1]),i=2..8);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6)/&%\"aG6$\"\"#\"\"\"#F(\"#7/&F%6$\"\"$F(#F'\"#v/&F %6$\"\"%F(#F(\"#?/&F%6$\"\"&F(#\"#))\"$N\"/&F%6$\"\"'F(#!&\"*3\"\"&c: \"/&F%6$\"\"(F(#\"(6*=<\"(?FQ%/&F%6$\"\")F(#\"&`^)\"'+L?" }}}{PARA 0 " " 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 22 "We use the equatio ns: " }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "a[9,i]=b[i]" "6#/&%\"aG6$\"\"*%\"iG&%\"bG6#F(" }{TEXT -1 3 ", " }{XPPEDIT 18 0 "i= 1" "6#/%\"iG\"\"\"" }{TEXT -1 8 " . . 8, " }}{PARA 0 "" 0 "" {TEXT -1 9 "to find " }{XPPEDIT 18 0 "a[9,i]" "6#&%\"aG6$\"\"*%\"iG" }{TEXT -1 2 ", " }{XPPEDIT 18 0 " i=1" "6#/%\"iG\"\"\"" }{TEXT -1 7 " . . 8. " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 116 "wt_eqs := [seq(a[9,i]=b[i],i=1..8)]:\ne26 := solve(\{op(subs( e25,%))\},\{seq(a[9,j],j=1..8)\}):\ne27 := `union`(e25,e26):\n" }}} {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 55 "The linki ng coefficients in the 9th row are as follows." }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 36 "seq(a[9,j]=subs(e27,a[9,j]),j=1..8);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6*/&%\"aG6$\"\"*\"\"\"#\"#`\"$7*/&F%6$F'\"\"# \"\"!/&F%6$F'\"\"$F0/&F%6$F'\"\"%#\"\"&\"#;/&F%6$F'F:#\"#F\"$7\"/&F%6$ F'\"\"'#F@\"$O\"/&F%6$F'\"\"(#\"$c#\"$p*/&F%6$F'\"\")#!#D\"$O$" }}} {PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 3 "e27" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1017 "e27 := \{a [8,2] = 0, c[9] = 1, c[8] = 1, a[5,2] = 0, a[4,2] = 0, a[6,2] = 0, a[7 ,2] = 0, a[9,3] = 0, a[9,2] = 0, b[3] = 0, b[2] = 0, `b*`[3] = 0, `b*` [2] = 0, a[7,1] = 1718911/4382720, a[8,1] = 85153/203300, a[5,3] = -11 2/45, a[5,4] = 64/27, a[6,1] = -10891/11556, a[5,1] = 88/135, a[8,7] = -7168/40375, a[7,6] = 14535/14336, a[4,1] = 1/20, a[3,1] = 2/75, a[3, 2] = 8/75, c[3] = 2/15, a[4,3] = 3/20, a[8,3] = -6783/2140, a[6,3] = 3 880/963, `b*`[7] = 10304/43605, a[7,5] = -671175/876544, a[7,4] = 8192 61/383488, b[1] = 53/912, a[7,3] = -1000749/547840, b[8] = -25/336, b[ 7] = 256/969, b[6] = 27/136, `b*`[8] = 0, c[4] = 1/5, c[2] = 1/12, c[6 ] = 2/3, b[5] = 27/112, a[9,1] = 53/912, a[9,4] = 5/16, a[9,5] = 27/11 2, a[9,6] = 27/136, a[9,7] = 256/969, a[9,8] = -25/336, b[4] = 5/16, a [2,1] = 1/12, `b*`[5] = 69/320, `b*`[4] = 241/756, `b*`[6] = 435/1904, `b*`[1] = 617/10944, a[8,6] = 1152/425, c[5] = 8/15, c[7] = 19/20, a[ 6,5] = 217/428, a[6,4] = -8456/2889, a[8,5] = -38493/13375, a[8,4] = 1 0956/2675, `b*`[9] = -1/18\}:" }{TEXT -1 0 "" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 49 "The Butcher tab leau for the scheme is as follows." }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 124 "subs(e27,matrix([seq([c[i],seq(a[i,j],j=1..i-1),``$( 10-i)],i=2..9),\n[`b`,seq(b[i],i=1..8),``],[`b*`,seq(`b*`[i],i=1..9)]] ));" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#K%'matrixG6#7,7,#\"\"\"\"#7F(%! GF+F+F+F+F+F+F+7,#\"\"#\"#:#F.\"#v#\"\")F1F+F+F+F+F+F+F+7,#F)\"\"&#F) \"#?\"\"!#\"\"$F8F+F+F+F+F+F+7,#F3F/#\"#))\"$N\"F9#!$7\"\"#X#\"#k\"#FF +F+F+F+F+7,#F.F;#!&\"*3\"\"&c:\"F9#\"%!)Q\"$j*#!%c%)\"%*)G#\"$<#\"$G%F +F+F+F+7,#\"#>F8#\"(6*=<\"(?FQ%F9#!(\\2+\"\"'Sya#\"'h#>)\"')[$Q#!'v6n \"'Wl()#\"&NX\"\"&OV\"F+F+F+7,F)#\"&`^)\"'+L?F9#!%$y'\"%S@#\"&c4\"\"%v E#!&$\\Q\"&vL\"#\"%_6\"$D%#!%or\"&v.%F+F+7,F)#\"#`\"$7*F9F9#F6\"#;#FF \"$7\"#FF\"$O\"#\"$c#\"$p*#!#D\"$O$F+7,%\"bGFepF9F9FhpFjpF\\qF^qFaqF+7 ,%#b*G#\"$<'\"&W4\"F9F9#\"$T#\"$c(#\"#p\"$?$#\"$N%\"%/>#\"&/.\"\"&0O%F 9#!\"\"\"#=Q(pprint06\"" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}}{PARA 0 "" 0 "" {TEXT -1 60 " #-----------------------------------------------------------" }}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 38 "Tsitouras' algorithm .. general scheme " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 129 "In t his subsection we use the algorithm of Ch. Tsitouras, as outlined in t he previous subsection, to construct a general scheme. " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 90 "We shall obtain expr essions for all the coefficients of the scheme in terms of the nodes \+ " }{XPPEDIT 18 0 "c[2]" "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 7 " and " }{XPPEDIT 18 0 "c[7]" "6#&%\"cG6#\"\"(" } {TEXT -1 1 "." }}{PARA 0 "" 0 "" {TEXT -1 10 "The node " }{XPPEDIT 18 0 "c[3]" "6#&%\"cG6#\"\"$" }{TEXT -1 27 " does not appear because \+ " }{XPPEDIT 18 0 "c[3]=2/3" "6#/&%\"cG6#\"\"$*&\"\"#\"\"\"F'!\"\"" } {TEXT -1 1 " " }{XPPEDIT 18 0 "c[4]" "6#&%\"cG6#\"\"%" }{TEXT -1 2 ". \+ " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT 260 4 "Note " }{TEXT -1 32 ": It is possible to also have " }{XPPEDIT 18 0 "`b*` [9]" "6#&%#b*G6#\"\"*" }{TEXT -1 123 " as a parameter but, since the \+ principal error norm and the stability polynomial for the order 6 sche me do not depend on " }{XPPEDIT 18 0 "`b*`[9]" "6#&%#b*G6#\"\"*" } {TEXT -1 116 ", it will be sufficient to obtain a solution that avoid s the occurrence of this extra parameter by requiring that " } {XPPEDIT 18 0 "`b*`[8]=0" "6#/&%#b*G6#\"\")\"\"!" }{TEXT -1 1 "." }} {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 52 "#-------- -------------------------------------------" }}{PARA 0 "" 0 "" {TEXT -1 22 "We specify the nodes: " }}{PARA 256 "" 0 "" {TEXT -1 1 " " } {XPPEDIT 18 0 "c[8]=1" "6#/&%\"cG6#\"\")\"\"\"" }{TEXT -1 3 ", " } {XPPEDIT 18 0 "c[9]=1" "6#/&%\"cG6#\"\"*\"\"\"" }{TEXT -1 1 "," }} {PARA 0 "" 0 "" {TEXT -1 31 "the zero linking coefficients: " }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "a[4,2]=0" "6#/&%\"aG6$\"\"% \"\"#\"\"!" }{TEXT -1 3 ", " }{XPPEDIT 18 0 "a[5,2]=0" "6#/&%\"aG6$\" \"&\"\"#\"\"!" }{TEXT -1 3 ", " }{XPPEDIT 18 0 "a[6,2]=0" "6#/&%\"aG6 $\"\"'\"\"#\"\"!" }{TEXT -1 3 ", " }{XPPEDIT 18 0 "a[7,2]=0" "6#/&%\" aG6$\"\"(\"\"#\"\"!" }{TEXT -1 3 ", " }{XPPEDIT 18 0 "a[8,2]=0" "6#/& %\"aG6$\"\")\"\"#\"\"!" }{TEXT -1 2 ", " }}{PARA 0 "" 0 "" {TEXT -1 22 "and the zero weights: " }}{PARA 256 "" 0 "" {TEXT -1 1 " " } {XPPEDIT 18 0 "b[2]=0" "6#/&%\"bG6#\"\"#\"\"!" }{TEXT -1 3 ", " } {XPPEDIT 18 0 "b[3]=0" "6#/&%\"bG6#\"\"$\"\"!" }{TEXT -1 3 ", " } {XPPEDIT 18 0 "`b*`[2]=0" "6#/&%#b*G6#\"\"#\"\"!" }{TEXT -1 3 ", " } {XPPEDIT 18 0 "`b*`[3]=0" "6#/&%#b*G6#\"\"$\"\"!" }{TEXT -1 3 ", " } {XPPEDIT 18 0 "`b*`[8]=0" "6#/&%#b*G6#\"\")\"\"!" }{TEXT -1 2 ". " }} {PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 87 "e1 := \{c[8]=1,c[9]=1,b[2]=0,b[3]=0,`b*`[2]=0,`b*`[3]=0,`b*`[8]=0, seq(a[i,2]=0,i=4..8)\}:" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 15 " " 0 "" {TEXT 270 6 "Step 1" }{TEXT -1 1 " " }}{PARA 0 "" 0 "" {TEXT -1 33 "We use the quadrature conditions:" }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "Sum(b[i],i=1..8)=1" "6#/-%$SumG6$&%\"bG6#%\"i G/F*;\"\"\"\"\")F-" }{TEXT -1 15 ", " }{XPPEDIT 18 0 "Sum (b[i]*c[i]^(k-1),i = 2 .. 8) = 1/k;" "6#/-%$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 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 3 " , " }{XPPEDIT 18 0 "b[8]" "6#&%\"bG6#\"\")" }{TEXT -1 1 "." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 142 "[ add(b[i],i=1..8)=1,seq(add(b[i]*c[i]^(j-1),i=2..8)=1/j,j=2..6)]:\ne2 : = solve(\{op(subs(e1,%))\},\{seq(b[i],i=[1,$4..8])\}):\ne3 := `union`( e1,e2):" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 62 "for ii to 8 do print(b[ii]=subs(e3,b[ii])); print(``) ; end do:" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/&%\"bG6#\"\"\",$*&#F'\"# gF'*,,B*,\"#IF'&%\"cG6#\"\"&F'&F16#\"\"'F'&F16#\"\"(F'&F16#\"\"%F'F'** \"#5F'F0F'F4F'F:F'!\"\"**F>F'F4F'F7F'F0F'F?*(F3F'F4F'F0F'F'**F>F'F4F'F 7F'F:F'F?*(F3F'F4F'F:F'F'*(F3F'F7F'F4F'F'*&\"\"$F'F4F'F?**F>F'F0F'F7F' F:F'F?*(F3F'F0F'F:F'F'*(F3F'F7F'F0F'F'*&FFF'F0F'F?*(F3F'F7F'F:F'F'*&FF F'F:F'F?\"\"#F'*&FFF'F7F'F?F'F0F?F4F?F7F?F:F?F'F'" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%!G" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/&%\"bG6#\"\"#\" \"!" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%!G" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/&%\"bG6#\"\"$\"\"!" }}{PARA 11 "" 1 "" {XPPMATH 20 "6# %!G" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/&%\"bG6#\"\"%,$*&#\"\"\"\"#gF+ *(,2\"\"#!\"\"*&\"\"$F+&%\"cG6#\"\"&F+F+*(F6F+&F46#\"\"(F+&F46#\"\"'F+ F0*&F2F+F8F+F+*&F2F+F;F+F+**\"#5F+F;F+F8F+F3F+F+*(F6F+F;F+F3F+F0*(F6F+ F8F+F3F+F0F+&F4F&F0,B*&F;F+)FDF2F+F0*$FGF+F0*$)FDF'F+F+*&F8F+)FDF/F+F+ *&F;F+FLF+F+*(F;F+F8F+FDF+F0*(F;F+F8F+FLF+F+*&F3F+FGF+F0*&F3F+FLF+F+*& F8F+FGF+F0*(F3F+F8F+FDF+F0*(F3F+F8F+FLF+F+*(F3F+F;F+FDF+F0*(F3F+F;F+FL F+F+*(F;F+F8F+F3F+F+**F3F+F;F+F8F+FDF+F0F0F+F+" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%!G" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/&%\"bG6#\"\"&,$ *&#\"\"\"\"#gF+**,2**\"#5F+&%\"cG6#\"\"'F+&F26#\"\"(F+&F26#\"\"%F+F+*( F'F+F1F+F8F+!\"\"*(F'F+F5F+F1F+F<*&\"\"$F+F1F+F+*(F'F+F5F+F8F+F<*&F?F+ F8F+F+\"\"#F<*&F?F+F5F+F+F+,&F8F<&F2F&F+FF+F1F+F;*(FBF+F5F+F8F+F+** F0F+F5F+F>F+F8F+F;*&\"#7F+F5F+F;*(FBF+F>F+F5F+F+*(FBF+F1F+F8F+F+**F0F+ F1F+F>F+F8F+F;*&FGF+F1F+F;*(FBF+F>F+F1F+F+*&FGF+F8F+F;*(FBF+F>F+F8F+F+ \"#5F+*&FGF+F>F+F;F+,&F8F+F+F;F;,&F+F;F1F+F;,&F+F;F5F+F;,&F+F;F>F+F;F+ F+" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%!G" }}}{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 1451 "e3 := \{c[8] = 1, b[3] = 0 , b[2] = 0, c[9] = 1, b[5] = -1/60*(10*c[6]*c[7]*c[4]-5*c[6]*c[4]-5*c[ 7]*c[6]+3*c[6]-5*c[7]*c[4]+3*c[4]-2+3*c[7])/(-c[4]+c[5])/c[5]/(-c[5]^3 +c[5]^2-c[7]*c[5]+c[7]*c[5]^2-c[6]*c[5]+c[6]*c[5]^2+c[7]*c[6]-c[6]*c[7 ]*c[5]), a[5,2] = 0, a[4,2] = 0,`b*`[8] = 0, `b*`[3] = 0, `b*`[2] = 0, a[8,2] = 0, a[7,2] = 0, a[6,2] = 0, b[1] = 1/60*(30*c[5]*c[6]*c[7]*c[ 4]-10*c[5]*c[6]*c[4]-10*c[6]*c[7]*c[5]+5*c[6]*c[5]-10*c[6]*c[7]*c[4]+5 *c[6]*c[4]+5*c[7]*c[6]-3*c[6]-10*c[5]*c[7]*c[4]+5*c[5]*c[4]+5*c[7]*c[5 ]-3*c[5]+5*c[7]*c[4]-3*c[4]+2-3*c[7])/c[5]/c[6]/c[7]/c[4], b[4] = 1/60 *(-2+3*c[5]-5*c[7]*c[6]+3*c[7]+3*c[6]+10*c[6]*c[7]*c[5]-5*c[6]*c[5]-5* c[7]*c[5])/c[4]/(-c[6]*c[4]^3-c[4]^3+c[4]^4+c[7]*c[4]^2+c[6]*c[4]^2-c[ 6]*c[7]*c[4]+c[6]*c[7]*c[4]^2-c[5]*c[4]^3+c[5]*c[4]^2-c[7]*c[4]^3-c[5] *c[7]*c[4]+c[5]*c[7]*c[4]^2-c[5]*c[6]*c[4]+c[5]*c[6]*c[4]^2+c[6]*c[7]* c[5]-c[5]*c[6]*c[7]*c[4]), b[6] = 1/60*(10*c[5]*c[7]*c[4]-5*c[5]*c[4]- 5*c[7]*c[5]+3*c[5]-5*c[7]*c[4]+3*c[4]-2+3*c[7])/(-c[4]+c[6])/(-c[5]+c[ 6])/c[6]/(c[6]^2-c[6]+c[7]-c[7]*c[6]), b[7] = -1/60*(10*c[5]*c[6]*c[4] -5*c[6]*c[5]-5*c[6]*c[4]+3*c[6]-5*c[5]*c[4]+3*c[5]+3*c[4]-2)/(c[4]-c[7 ])/(c[5]-c[7])/(c[6]-c[7])/c[7]/(-1+c[7]), b[8] = 1/60*(-20*c[5]*c[6]* c[4]+30*c[5]*c[6]*c[7]*c[4]+15*c[6]*c[5]-20*c[6]*c[7]*c[5]+15*c[6]*c[4 ]-20*c[6]*c[7]*c[4]-12*c[6]+15*c[7]*c[6]+15*c[5]*c[4]-20*c[5]*c[7]*c[4 ]-12*c[5]+15*c[7]*c[5]-12*c[4]+15*c[7]*c[4]+10-12*c[7])/(c[4]-1)/(-1+c [5])/(-1+c[6])/(-1+c[7])\}:" }{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 271 6 "Ste p 2" }{TEXT -1 1 " " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 " " {TEXT -1 9 "Because " }{XPPEDIT 18 0 "a[4,2]=0" "6#/&%\"aG6$\"\"%\" \"#\"\"!" }{TEXT -1 28 ", the stage-order equations " }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "a[4,2]*c[2]+a[4,3]*c[3] = 1/2" "6# /,&*&&%\"aG6$\"\"%\"\"#\"\"\"&%\"cG6#F*F+F+*&&F'6$F)\"\"$F+&F-6#F2F+F+ *&F+F+F*!\"\"" }{TEXT -1 1 " " }{XPPEDIT 18 0 "c[4]^2" "6#*$&%\"cG6#\" \"%\"\"#" }{TEXT -1 6 ", " }{XPPEDIT 18 0 "a[4,2]*c[2]^2+a[4,3]*c[ 3]^2 = 1/3" "6#/,&*&&%\"aG6$\"\"%\"\"#\"\"\"*$&%\"cG6#F*F*F+F+*&&F'6$F )\"\"$F+*$&F.6#F3F*F+F+*&F+F+F3!\"\"" }{TEXT -1 1 " " }{XPPEDIT 18 0 " c[4]^3" "6#*$&%\"cG6#\"\"%\"\"$" }{TEXT -1 1 " " }}{PARA 0 "" 0 "" {TEXT -1 7 "become " }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "a[4,3]*c[3] = 1/2" "6#/*&&%\"aG6$\"\"%\"\"$\"\"\"&%\"cG6#F)F**&F*F* \"\"#!\"\"" }{TEXT -1 1 " " }{XPPEDIT 18 0 "c[4]^2" "6#*$&%\"cG6#\"\"% \"\"#" }{TEXT -1 6 ", " }{XPPEDIT 18 0 "a[4,3]*c[3]^2 = 1/3" "6#/* &&%\"aG6$\"\"%\"\"$\"\"\"*$&%\"cG6#F)\"\"#F**&F*F*F)!\"\"" }{TEXT -1 1 " " }{XPPEDIT 18 0 "c[4]^3" "6#*$&%\"cG6#\"\"%\"\"$" }{TEXT -1 1 ". " }}{PARA 0 "" 0 "" {TEXT -1 57 "Dividing the second equation by the f irst equation gives " }}{PARA 256 "" 0 "" {TEXT -1 2 " " }{XPPEDIT 18 0 "c[3] = 2/3;" "6#/&%\"cG6#\"\"$*&\"\"#\"\"\"F'!\"\"" }{TEXT -1 1 " " }{XPPEDIT 18 0 "c[4]" "6#&%\"cG6#\"\"%" }{TEXT -1 2 ". " }}{PARA 0 "" 0 "" {TEXT -1 17 "Subsituting for " }{XPPEDIT 18 0 "c[3]" "6#&% \"cG6#\"\"$" }{TEXT -1 5 " in " }}{PARA 256 "" 0 "" {TEXT -1 2 " " } {XPPEDIT 18 0 "a[4,3]*c[3] = 1/2" "6#/*&&%\"aG6$\"\"%\"\"$\"\"\"&%\"cG 6#F)F**&F*F*\"\"#!\"\"" }{TEXT -1 1 " " }{XPPEDIT 18 0 "c[4]^2" "6#*$& %\"cG6#\"\"%\"\"#" }{TEXT -1 2 ", " }}{PARA 0 "" 0 "" {TEXT -1 6 "give s " }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "a[4,3]=3/4" "6# /&%\"aG6$\"\"%\"\"$*&F(\"\"\"F'!\"\"" }{TEXT -1 1 " " }{XPPEDIT 18 0 " c[4]" "6#&%\"cG6#\"\"%" }{TEXT -1 2 ". " }}{PARA 0 "" 0 "" {TEXT -1 17 "Subsituting for " }{XPPEDIT 18 0 "c[3]" "6#&%\"cG6#\"\"$" }{TEXT -1 30 " in the stage-order equation " }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "a[3,2]*c[2]=1/2" "6#/*&&%\"aG6$\"\"$\"\"#\"\"\"&% \"cG6#F)F**&F*F*F)!\"\"" }{TEXT -1 1 " " }{XPPEDIT 18 0 "c[3]^2" "6#*$ &%\"cG6#\"\"$\"\"#" }{TEXT -1 1 " " }}{PARA 0 "" 0 "" {TEXT -1 6 "give s " }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "a[3,2]=2/9" "6# /&%\"aG6$\"\"$\"\"#*&F(\"\"\"\"\"*!\"\"" }{TEXT -1 1 " " }{XPPEDIT 18 0 "c[4]^2/c[2]" "6#*&&%\"cG6#\"\"%\"\"#&F%6#F(!\"\"" }{TEXT -1 1 "." } }{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 24 "Thus we \+ can substitute " }{XPPEDIT 18 0 "c[3] = 2/3;" "6#/&%\"cG6#\"\"$*&\"\" #\"\"\"F'!\"\"" }{TEXT -1 1 " " }{XPPEDIT 18 0 "c[4]" "6#&%\"cG6#\"\"% " }{TEXT -1 20 " in the equations " }{XPPEDIT 18 0 "a[3,2]=c[3]^2/(2 *c[2])" "6#/&%\"aG6$\"\"$\"\"#*&&%\"cG6#F'F(*&F(\"\"\"&F+6#F(F.!\"\"" }{TEXT -1 7 " and " }{XPPEDIT 18 0 "a[4,3]=c[4]^2/(2*c[3])" "6#/&%\" aG6$\"\"%\"\"$*&&%\"cG6#F'\"\"#*&F-\"\"\"&F+6#F(F/!\"\"" }{TEXT -1 13 " to obtain " }{XPPEDIT 18 0 "a[3,2]" "6#&%\"aG6$\"\"$\"\"#" }{TEXT -1 7 " and " }{XPPEDIT 18 0 "a[4,3]" "6#&%\"aG6$\"\"%\"\"$" }{TEXT -1 15 " in terms of " }{XPPEDIT 18 0 "c[2]" "6#&%\"cG6#\"\"#" } {TEXT -1 7 " and " }{XPPEDIT 18 0 "c[4]" "6#&%\"cG6#\"\"%" }{TEXT -1 1 "." }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 102 "e4 := \{c[3]=2/3 *c[4]\}:\ne5 := `union`(e3,e4,subs(e4,\{a[4,3]=c[4]^2/(2*c[3]),a[3,2]= c[3]^2/(2*c[2])\})): " }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 46 "a[3,2]=subs(e5,a[3,2]),a[4,3]=subs( e5,a[4,3]);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6$/&%\"aG6$\"\"$\"\"#,$*& #F(\"\"*\"\"\"*&&%\"cG6#\"\"%F(&F06#F(!\"\"F-F-/&F%6$F2F',$*&#F'F2F-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 "e5" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1514 " e5 := \{c[8] = 1, b[3] = 0, b[2] = 0, c[9] = 1, b[5] = -1/60*(10*c[6]* c[7]*c[4]-5*c[6]*c[4]-5*c[7]*c[6]+3*c[6]-5*c[7]*c[4]+3*c[4]-2+3*c[7])/ (-c[4]+c[5])/c[5]/(-c[5]^3+c[5]^2-c[7]*c[5]+c[7]*c[5]^2-c[6]*c[5]+c[6] *c[5]^2+c[7]*c[6]-c[6]*c[7]*c[5]), a[5,2] = 0, a[4,2] = 0, `b*`[3] = 0 , `b*`[2] = 0, a[8,2] = 0, a[7,2] = 0, a[6,2] = 0, b[1] = 1/60*(30*c[5 ]*c[6]*c[7]*c[4]-10*c[5]*c[6]*c[4]-10*c[6]*c[7]*c[5]+5*c[6]*c[5]-10*c[ 6]*c[7]*c[4]+5*c[6]*c[4]+5*c[7]*c[6]-3*c[6]-10*c[5]*c[7]*c[4]+5*c[5]*c [4]+5*c[7]*c[5]-3*c[5]+5*c[7]*c[4]-3*c[4]+2-3*c[7])/c[5]/c[6]/c[7]/c[4 ], b[4] = 1/60*(-2+3*c[5]-5*c[7]*c[6]+3*c[7]+3*c[6]+10*c[6]*c[7]*c[5]- 5*c[6]*c[5]-5*c[7]*c[5])/c[4]/(-c[6]*c[4]^3-c[4]^3+c[4]^4+c[7]*c[4]^2+ c[6]*c[4]^2-c[6]*c[7]*c[4]+c[6]*c[7]*c[4]^2-c[5]*c[4]^3+c[5]*c[4]^2-c[ 7]*c[4]^3-c[5]*c[7]*c[4]+c[5]*c[7]*c[4]^2-c[5]*c[6]*c[4]+c[5]*c[6]*c[4 ]^2+c[6]*c[7]*c[5]-c[5]*c[6]*c[7]*c[4]), b[6] = 1/60*(10*c[5]*c[7]*c[4 ]-5*c[5]*c[4]-5*c[7]*c[5]+3*c[5]-5*c[7]*c[4]+3*c[4]-2+3*c[7])/(-c[4]+c [6])/(-c[5]+c[6])/c[6]/(c[6]^2-c[6]+c[7]-c[7]*c[6]), b[7] = -1/60*(10* c[5]*c[6]*c[4]-5*c[6]*c[5]-5*c[6]*c[4]+3*c[6]-5*c[5]*c[4]+3*c[5]+3*c[4 ]-2)/(c[4]-c[7])/(c[5]-c[7])/(c[6]-c[7])/c[7]/(-1+c[7]), `b*`[8] = 0, \+ b[8] = 1/60*(-20*c[5]*c[6]*c[4]+30*c[5]*c[6]*c[7]*c[4]+15*c[6]*c[5]-20 *c[6]*c[7]*c[5]+15*c[6]*c[4]-20*c[6]*c[7]*c[4]-12*c[6]+15*c[7]*c[6]+15 *c[5]*c[4]-20*c[5]*c[7]*c[4]-12*c[5]+15*c[7]*c[5]-12*c[4]+15*c[7]*c[4] +10-12*c[7])/(c[4]-1)/(-1+c[5])/(-1+c[6])/(-1+c[7]), c[3] = 2/3*c[4], \+ a[4,3] = 3/4*c[4], a[3,2] = 2/9*c[4]^2/c[2]\}:" }{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 272 6 "Step 3" }{TEXT -1 1 " " }}{PARA 0 "" 0 "" {TEXT -1 35 "We use the stage-order conditions: " }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{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#*$&%\"cG6#\"\"&\"\"#" } {TEXT -1 15 ", " }{XPPEDIT 18 0 "Sum(a[5,j]*c[j]^(2),j=2. .4)=1/3" "6#/-%$SumG6$*&&%\"aG6$\"\"&%\"jG\"\"\"*$&%\"cG6#F,\"\"#F-/F, ;F2\"\"%*&F-F-\"\"$!\"\"" }{TEXT -1 1 " " }{XPPEDIT 18 0 "c[5]^3" "6#* $&%\"cG6#\"\"&\"\"$" }{TEXT -1 1 " " }}{PARA 0 "" 0 "" {TEXT -1 9 "to \+ find " }{XPPEDIT 18 0 "a[5,3]" "6#&%\"aG6$\"\"&\"\"$" }{TEXT -1 7 " \+ and " }{XPPEDIT 18 0 "a[5,4]" "6#&%\"aG6$\"\"&\"\"%" }{TEXT -1 1 "." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 142 "[seq(add(a[5,j]*c[j]^(k-1),j=2..4)=c[5]^k/k,k=[2,3])];\ne6 := s olve(\{op(subs(e5,%))\},\{a[5,3],a[5,4]\}):\ne7 := `union`(e5,simplify (subs(e4,e6))):" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#7$/,(*&&%\"aG6$\"\" &\"\"#\"\"\"&%\"cG6#F+F,F,*&&F(6$F*\"\"$F,&F.6#F3F,F,*&&F(6$F*\"\"%F,& F.6#F9F,F,,$*&#F,F+F,*$)&F.6#F*F+F,F,F,/,(*&F'F,)F-F+F,F,*&F1F,)F4F+F, F,*&F7F,)F:F+F,F,,$*&#F,F3F,*$)FAF3F,F,F," }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 36 "seq(a[5,j]=subs(e7 ,a[5,j]),j=[3,4]);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6$/&%\"aG6$\"\"&\" \"$,$*&#F(\"\"%\"\"\"*(&%\"cG6#F'\"\"#,&*&F2F-F/F-F-*&F(F-&F06#F,F-!\" \"F-F6!\"#F-F8/&F%6$F'F,*(F/F2,&F6F8F/F-F-F6F9" }}}{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 1596 "e7 := \{c[8] = 1, b[3] = 0 , b[2] = 0, c[9] = 1, b[5] = -1/60*(10*c[6]*c[7]*c[4]-5*c[6]*c[4]-5*c[ 7]*c[6]+3*c[6]-5*c[7]*c[4]+3*c[4]-2+3*c[7])/(-c[4]+c[5])/c[5]/(-c[5]^3 +c[5]^2-c[7]*c[5]+c[7]*c[5]^2-c[6]*c[5]+c[6]*c[5]^2+c[7]*c[6]-c[6]*c[7 ]*c[5]), a[5,2] = 0, a[4,2] = 0, `b*`[3] = 0, `b*`[2] = 0, a[8,2] = 0, a[7,2] = 0, a[6,2] = 0, b[1] = 1/60*(30*c[5]*c[6]*c[7]*c[4]-10*c[5]*c [6]*c[4]-10*c[6]*c[7]*c[5]+5*c[6]*c[5]-10*c[6]*c[7]*c[4]+5*c[6]*c[4]+5 *c[7]*c[6]-3*c[6]-10*c[5]*c[7]*c[4]+5*c[5]*c[4]+5*c[7]*c[5]-3*c[5]+5*c [7]*c[4]-3*c[4]+2-3*c[7])/c[5]/c[6]/c[7]/c[4], b[4] = 1/60*(-2+3*c[5]- 5*c[7]*c[6]+3*c[7]+3*c[6]+10*c[6]*c[7]*c[5]-5*c[6]*c[5]-5*c[7]*c[5])/c [4]/(-c[6]*c[4]^3-c[4]^3+c[4]^4+c[7]*c[4]^2+c[6]*c[4]^2-c[6]*c[7]*c[4] +c[6]*c[7]*c[4]^2-c[5]*c[4]^3+c[5]*c[4]^2-c[7]*c[4]^3-c[5]*c[7]*c[4]+c [5]*c[7]*c[4]^2-c[5]*c[6]*c[4]+c[5]*c[6]*c[4]^2+c[6]*c[7]*c[5]-c[5]*c[ 6]*c[7]*c[4]), b[6] = 1/60*(10*c[5]*c[7]*c[4]-5*c[5]*c[4]-5*c[7]*c[5]+ 3*c[5]-5*c[7]*c[4]+3*c[4]-2+3*c[7])/(-c[4]+c[6])/(-c[5]+c[6])/c[6]/(c[ 6]^2-c[6]+c[7]-c[7]*c[6]), b[7] = -1/60*(10*c[5]*c[6]*c[4]-5*c[6]*c[5] -5*c[6]*c[4]+3*c[6]-5*c[5]*c[4]+3*c[5]+3*c[4]-2)/(c[4]-c[7])/(c[5]-c[7 ])/(c[6]-c[7])/c[7]/(-1+c[7]), `b*`[8] = 0, a[5,3] = -3/4*c[5]^2*(2*c[ 5]-3*c[4])/c[4]^2, b[8] = 1/60*(-20*c[5]*c[6]*c[4]+30*c[5]*c[6]*c[7]*c [4]+15*c[6]*c[5]-20*c[6]*c[7]*c[5]+15*c[6]*c[4]-20*c[6]*c[7]*c[4]-12*c [6]+15*c[7]*c[6]+15*c[5]*c[4]-20*c[5]*c[7]*c[4]-12*c[5]+15*c[7]*c[5]-1 2*c[4]+15*c[7]*c[4]+10-12*c[7])/(c[4]-1)/(-1+c[5])/(-1+c[6])/(-1+c[7]) , a[5,4] = c[5]^2*(-c[4]+c[5])/c[4]^2, c[3] = 2/3*c[4], a[4,3] = 3/4*c [4], a[3,2] = 2/9*c[4]^2/c[2]\}:" }{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 273 6 "Step 4" }{TEXT -1 1 " " }}{PARA 0 "" 0 "" {TEXT -1 41 "We use the ( column) simplifying condition" }}{PARA 256 "" 0 "" {TEXT -1 1 " " } {XPPEDIT 18 0 "Sum(b[i]*a[i,j],i=j..8)=b[j]*(1-c[j])" "6#/-%$SumG6$*&& %\"bG6#%\"iG\"\"\"&%\"aG6$F+%\"jGF,/F+;F0\"\")*&&F)6#F0F,,&F,F,&%\"cG6 #F0!\"\"F," }{TEXT -1 7 " for " }{XPPEDIT 18 0 "j=7" "6#/%\"jG\"\"( " }{TEXT -1 1 "," }}{PARA 0 "" 0 "" {TEXT -1 8 "that is," }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "b[8]*a[8,7] = b[7]*(1-c[7])" "6 #/*&&%\"bG6#\"\")\"\"\"&%\"aG6$F(\"\"(F)*&&F&6#F-F),&F)F)&%\"cG6#F-!\" \"F)" }{TEXT -1 2 ", " }}{PARA 0 "" 0 "" {TEXT -1 9 "to find " } {XPPEDIT 18 0 "a[8,7]" "6#&%\"aG6$\"\")\"\"(" }{TEXT -1 1 "." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 86 "b[ 8]*a[8,7] = b[7]*(1-c[7]);\ne8 := solve(\{subs(e7,%)\},\{a[8,7]\}):\ne 9 := `union`(e7,e8):" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/*&&%\"bG6#\" \")\"\"\"&%\"aG6$F(\"\"(F)*&&F&6#F-F),&F)F)&%\"cGF0!\"\"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 1976 "e9 := \{c[8] = 1, b [3] = 0, b[2] = 0, c[9] = 1, b[5] = -1/60*(10*c[6]*c[7]*c[4]-5*c[6]*c[ 4]-5*c[7]*c[6]+3*c[6]-5*c[7]*c[4]+3*c[4]-2+3*c[7])/(-c[4]+c[5])/c[5]/( -c[5]^3+c[5]^2-c[7]*c[5]+c[7]*c[5]^2-c[6]*c[5]+c[6]*c[5]^2+c[7]*c[6]-c [6]*c[7]*c[5]), a[5,2] = 0, a[4,2] = 0, `b*`[3] = 0, `b*`[2] = 0, a[8, 2] = 0, a[7,2] = 0, a[6,2] = 0, b[1] = 1/60*(30*c[5]*c[6]*c[7]*c[4]-10 *c[5]*c[6]*c[4]-10*c[6]*c[7]*c[5]+5*c[6]*c[5]-10*c[6]*c[7]*c[4]+5*c[6] *c[4]+5*c[7]*c[6]-3*c[6]-10*c[5]*c[7]*c[4]+5*c[5]*c[4]+5*c[7]*c[5]-3*c [5]+5*c[7]*c[4]-3*c[4]+2-3*c[7])/c[5]/c[6]/c[7]/c[4], b[4] = 1/60*(-2+ 3*c[5]-5*c[7]*c[6]+3*c[7]+3*c[6]+10*c[6]*c[7]*c[5]-5*c[6]*c[5]-5*c[7]* c[5])/c[4]/(-c[6]*c[4]^3-c[4]^3+c[4]^4+c[7]*c[4]^2+c[6]*c[4]^2-c[6]*c[ 7]*c[4]+c[6]*c[7]*c[4]^2-c[5]*c[4]^3+c[5]*c[4]^2-c[7]*c[4]^3-c[5]*c[7] *c[4]+c[5]*c[7]*c[4]^2-c[5]*c[6]*c[4]+c[5]*c[6]*c[4]^2+c[6]*c[7]*c[5]- c[5]*c[6]*c[7]*c[4]), b[6] = 1/60*(10*c[5]*c[7]*c[4]-5*c[5]*c[4]-5*c[7 ]*c[5]+3*c[5]-5*c[7]*c[4]+3*c[4]-2+3*c[7])/(-c[4]+c[6])/(-c[5]+c[6])/c [6]/(c[6]^2-c[6]+c[7]-c[7]*c[6]), a[8,7] = (10*c[5]*c[6]*c[4]-5*c[6]*c [5]-5*c[6]*c[4]+3*c[6]-5*c[5]*c[4]+3*c[5]+3*c[4]-2)/c[7]/(c[6]-c[7])/( c[5]-c[7])/(c[4]-c[7])/(-20*c[5]*c[6]*c[4]+30*c[5]*c[6]*c[7]*c[4]+15*c [6]*c[5]-20*c[6]*c[7]*c[5]+15*c[6]*c[4]-20*c[6]*c[7]*c[4]-12*c[6]+15*c [7]*c[6]+15*c[5]*c[4]-20*c[5]*c[7]*c[4]-12*c[5]+15*c[7]*c[5]-12*c[4]+1 5*c[7]*c[4]+10-12*c[7])*(c[4]-1)*(-1+c[5])*(-1+c[6])*(-1+c[7]), b[7] = -1/60*(10*c[5]*c[6]*c[4]-5*c[6]*c[5]-5*c[6]*c[4]+3*c[6]-5*c[5]*c[4]+3 *c[5]+3*c[4]-2)/(c[4]-c[7])/(c[5]-c[7])/(c[6]-c[7])/c[7]/(-1+c[7]), `b *`[8] = 0, a[5,3] = -3/4*c[5]^2*(2*c[5]-3*c[4])/c[4]^2, b[8] = 1/60*(- 20*c[5]*c[6]*c[4]+30*c[5]*c[6]*c[7]*c[4]+15*c[6]*c[5]-20*c[6]*c[7]*c[5 ]+15*c[6]*c[4]-20*c[6]*c[7]*c[4]-12*c[6]+15*c[7]*c[6]+15*c[5]*c[4]-20* c[5]*c[7]*c[4]-12*c[5]+15*c[7]*c[5]-12*c[4]+15*c[7]*c[4]+10-12*c[7])/( c[4]-1)/(-1+c[5])/(-1+c[6])/(-1+c[7]), a[5,4] = c[5]^2*(-c[4]+c[5])/c[ 4]^2, c[3] = 2/3*c[4], a[4,3] = 3/4*c[4], a[3,2] = 2/9*c[4]^2/c[2]\}: " }{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 274 6 "Step 5" }{TEXT -1 1 " " }} {PARA 0 "" 0 "" {TEXT -1 41 "We use the \"alternative\" order conditio n " }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "[b]^T*([C]-Id)* [A]*(C-c[4]*Id)*(C-c[5]*Id)*[c] = Int((x-1)*Int((t-c[4])*(t-c[5])*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)&F46# F;F.F)FGF)/FG;\"\"!FBF)/FB;FOF)" }{TEXT -1 2 ". " }}{PARA 0 "" 0 "" {TEXT -1 39 "This condition amounts to the relation " }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "b[7]*(c[7]-1)*a[7,6]*(c[6]-c[4])*( c[6]-c[5])*c[6]=-1/120+c[4]/60+c[5]/60-c[4]*c[5]/24" "6#/*.&%\"bG6#\" \"(\"\"\",&&%\"cG6#F(F)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 27 "which can be used to find " }{XPPEDIT 18 0 "a[7,6]" "6#&%\"aG6$\"\"(\"\"'" }{TEXT -1 2 ". " }}{PARA 0 "" 0 " " {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 147 "b[7]*(c[7 ]-1)*a[7,6]*(c[6]-c[4])*(c[6]-c[5])*c[6] = -1/120+c[4]/60+c[5]/60-c[4] *c[5]/24:\ne10 := solve(\{subs(e9,%)\},a[7,6]):\ne11 := `union`(e9,e10 ):" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 24 "a[7,6]=subs(e11,a[7,6]);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/&%\"aG6$\"\"(\"\"',$*&#\"\"\"\"\"#F,*4,**(\"\"&F,&%\"c G6#F1F,&F36#\"\"%F,F,F,F,*&F-F,F5F,!\"\"*&F-F,F2F,F9F,,&F5F,&F36#F'F9F ,,&F2F,F " 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 2172 "e11 := \{c[8] = 1, b[3] = \+ 0, b[2] = 0, c[9] = 1, b[5] = -1/60*(10*c[6]*c[7]*c[4]-5*c[6]*c[4]-5*c [7]*c[6]+3*c[6]-5*c[7]*c[4]+3*c[4]-2+3*c[7])/(-c[4]+c[5])/c[5]/(-c[5]^ 3+c[5]^2-c[7]*c[5]+c[7]*c[5]^2-c[6]*c[5]+c[6]*c[5]^2+c[7]*c[6]-c[6]*c[ 7]*c[5]), a[5,2] = 0, a[4,2] = 0, `b*`[3] = 0, `b*`[2] = 0, a[8,2] = 0 , a[7,2] = 0, a[6,2] = 0, b[1] = 1/60*(30*c[5]*c[6]*c[7]*c[4]-10*c[5]* c[6]*c[4]-10*c[6]*c[7]*c[5]+5*c[6]*c[5]-10*c[6]*c[7]*c[4]+5*c[6]*c[4]+ 5*c[7]*c[6]-3*c[6]-10*c[5]*c[7]*c[4]+5*c[5]*c[4]+5*c[7]*c[5]-3*c[5]+5* c[7]*c[4]-3*c[4]+2-3*c[7])/c[5]/c[6]/c[7]/c[4], b[4] = 1/60*(-2+3*c[5] -5*c[7]*c[6]+3*c[7]+3*c[6]+10*c[6]*c[7]*c[5]-5*c[6]*c[5]-5*c[7]*c[5])/ c[4]/(-c[6]*c[4]^3-c[4]^3+c[4]^4+c[7]*c[4]^2+c[6]*c[4]^2-c[6]*c[7]*c[4 ]+c[6]*c[7]*c[4]^2-c[5]*c[4]^3+c[5]*c[4]^2-c[7]*c[4]^3-c[5]*c[7]*c[4]+ c[5]*c[7]*c[4]^2-c[5]*c[6]*c[4]+c[5]*c[6]*c[4]^2+c[6]*c[7]*c[5]-c[5]*c [6]*c[7]*c[4]), b[6] = 1/60*(10*c[5]*c[7]*c[4]-5*c[5]*c[4]-5*c[7]*c[5] +3*c[5]-5*c[7]*c[4]+3*c[4]-2+3*c[7])/(-c[4]+c[6])/(-c[5]+c[6])/c[6]/(c [6]^2-c[6]+c[7]-c[7]*c[6]), a[8,7] = (10*c[5]*c[6]*c[4]-5*c[6]*c[5]-5* c[6]*c[4]+3*c[6]-5*c[5]*c[4]+3*c[5]+3*c[4]-2)/c[7]/(c[6]-c[7])/(c[5]-c [7])/(c[4]-c[7])/(-20*c[5]*c[6]*c[4]+30*c[5]*c[6]*c[7]*c[4]+15*c[6]*c[ 5]-20*c[6]*c[7]*c[5]+15*c[6]*c[4]-20*c[6]*c[7]*c[4]-12*c[6]+15*c[7]*c[ 6]+15*c[5]*c[4]-20*c[5]*c[7]*c[4]-12*c[5]+15*c[7]*c[5]-12*c[4]+15*c[7] *c[4]+10-12*c[7])*(c[4]-1)*(-1+c[5])*(-1+c[6])*(-1+c[7]), b[7] = -1/60 *(10*c[5]*c[6]*c[4]-5*c[6]*c[5]-5*c[6]*c[4]+3*c[6]-5*c[5]*c[4]+3*c[5]+ 3*c[4]-2)/(c[4]-c[7])/(c[5]-c[7])/(c[6]-c[7])/c[7]/(-1+c[7]), `b*`[8] \+ = 0, a[5,3] = -3/4*c[5]^2*(2*c[5]-3*c[4])/c[4]^2, b[8] = 1/60*(-20*c[5 ]*c[6]*c[4]+30*c[5]*c[6]*c[7]*c[4]+15*c[6]*c[5]-20*c[6]*c[7]*c[5]+15*c [6]*c[4]-20*c[6]*c[7]*c[4]-12*c[6]+15*c[7]*c[6]+15*c[5]*c[4]-20*c[5]*c [7]*c[4]-12*c[5]+15*c[7]*c[5]-12*c[4]+15*c[7]*c[4]+10-12*c[7])/(c[4]-1 )/(-1+c[5])/(-1+c[6])/(-1+c[7]), a[5,4] = c[5]^2*(-c[4]+c[5])/c[4]^2, \+ a[7,6] = 1/2*(5*c[5]*c[4]+1-2*c[4]-2*c[5])*(c[4]-c[7])*(c[5]-c[7])*(c[ 6]-c[7])*c[7]/(10*c[5]*c[6]*c[4]-5*c[6]*c[5]-5*c[6]*c[4]+3*c[6]-5*c[5] *c[4]+3*c[5]+3*c[4]-2)/(-c[4]+c[6])/(-c[5]+c[6])/c[6], c[3] = 2/3*c[4] , a[4,3] = 3/4*c[4], a[3,2] = 2/9*c[4]^2/c[2]\}:" }{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 275 6 "Step 6" }{TEXT -1 1 " " }}{PARA 0 "" 0 "" {TEXT -1 41 "We use the (column) simplifying condition" }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "Sum(b[i]*a[i,j],i=j..8)=b[j]*(1-c[j])" "6#/-%$SumG6$*&&%\"bG6#%\"iG\"\"\"&%\"aG6$F+%\"jGF,/F+;F0\"\")*&&F)6#F 0F,,&F,F,&%\"cG6#F0!\"\"F," }{TEXT -1 7 " for " }{XPPEDIT 18 0 "j = \+ 6;" "6#/%\"jG\"\"'" }{TEXT -1 1 "," }}{PARA 0 "" 0 "" {TEXT -1 8 "that is," }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "b[7]*a[7,6]+b [8]*a[8,6] = b[6]*(1-c[6])" "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 9 "to find " }{XPPEDIT 18 0 "a[8,6];" "6#&%\"aG6$\"\")\"\"'" }{TEXT -1 1 "." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 101 "add(b[i]*a[ i,6],i=7..8)=b[6]*(1-c[6]);\ne12 := solve(\{subs(e11,%)\},\{a[8,6]\}): \ne13 := `union`(e11,e12):" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/,&*&&% \"bG6#\"\"(\"\"\"&%\"aG6$F)\"\"'F*F**&&F'6#\"\")F*&F,6$F2F.F*F**&&F'6# F.F*,&&%\"cGF7!\"\"F*F*F*" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 24 "a[8,6]=subs(e13,a[8,6]);" }}{PARA 12 "" 1 "" {XPPMATH 20 "6#/&%\"aG6$\"\")\"\"',$*&#\"\"\"\"\"#F,*4,B** \"\"&F,&%\"cG6#F1F,&F36#F(F,&F36#\"\"%F,F,F5F,*(F-F,F5F,F7F,!\"\"*(F-F ,F5F,F2F,F;**\"#DF,F2F,&F36#\"\"(F,F7F,F,*&F(F,F2F,F,*&\"\"*F,F?F,F,*( \"#5F,)F?F-F,F7F,F,F9F;*&F(F,F7F,F,*&F(F,FGF,F;*(\"#9F,F?F,F7F,F;*(FFF ,F2F,F7F,F;*(FKF,F?F,F2F,F;**\"#?F,FGF,F2F,F7F,F;*(FFF,FGF,F2F,F,F,,&F 5F,F?F;F;,&F7F;F5F,F;,&F2F;F5F,F;F5F;,B**FOF,F2F,F5F,F7F,F;*,\"#IF,F2F ,F5F,F?F,F7F,F,*(\"#:F,F5F,F2F,F,**FOF,F5F,F?F,F2F,F;*(FYF,F5F,F7F,F,* *FOF,F5F,F?F,F7F,F;*&\"#7F,F5F,F;*(FYF,F?F,F5F,F,*(FYF,F2F,F7F,F,**FOF ,F2F,F?F,F7F,F;*&FhnF,F2F,F;*(FYF,F?F,F2F,F,*&FhnF,F7F,F;*(FYF,F?F,F7F ,F,FFF,*&FhnF,F?F,F;F;,&F7F,F,F;F,,&F,F;F2F,F,,&F,F;F5F,F,F,F," }}} {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 2656 "e13 := \{c [8] = 1, b[3] = 0, b[2] = 0, c[9] = 1, b[5] = -1/60*(10*c[6]*c[7]*c[4] -5*c[6]*c[4]-5*c[7]*c[6]+3*c[6]-5*c[7]*c[4]+3*c[4]-2+3*c[7])/(-c[4]+c[ 5])/c[5]/(-c[5]^3+c[5]^2-c[7]*c[5]+c[7]*c[5]^2-c[6]*c[5]+c[6]*c[5]^2+c [7]*c[6]-c[6]*c[7]*c[5]), a[5,2] = 0, a[4,2] = 0, `b*`[3] = 0, `b*`[2] = 0, a[8,2] = 0, a[7,2] = 0, a[6,2] = 0, b[1] = 1/60*(30*c[5]*c[6]*c[ 7]*c[4]-10*c[5]*c[6]*c[4]-10*c[6]*c[7]*c[5]+5*c[6]*c[5]-10*c[6]*c[7]*c [4]+5*c[6]*c[4]+5*c[7]*c[6]-3*c[6]-10*c[5]*c[7]*c[4]+5*c[5]*c[4]+5*c[7 ]*c[5]-3*c[5]+5*c[7]*c[4]-3*c[4]+2-3*c[7])/c[5]/c[6]/c[7]/c[4], b[4] = 1/60*(-2+3*c[5]-5*c[7]*c[6]+3*c[7]+3*c[6]+10*c[6]*c[7]*c[5]-5*c[6]*c[ 5]-5*c[7]*c[5])/c[4]/(-c[6]*c[4]^3-c[4]^3+c[4]^4+c[7]*c[4]^2+c[6]*c[4] ^2-c[6]*c[7]*c[4]+c[6]*c[7]*c[4]^2-c[5]*c[4]^3+c[5]*c[4]^2-c[7]*c[4]^3 -c[5]*c[7]*c[4]+c[5]*c[7]*c[4]^2-c[5]*c[6]*c[4]+c[5]*c[6]*c[4]^2+c[6]* c[7]*c[5]-c[5]*c[6]*c[7]*c[4]), b[6] = 1/60*(10*c[5]*c[7]*c[4]-5*c[5]* c[4]-5*c[7]*c[5]+3*c[5]-5*c[7]*c[4]+3*c[4]-2+3*c[7])/(-c[4]+c[6])/(-c[ 5]+c[6])/c[6]/(c[6]^2-c[6]+c[7]-c[7]*c[6]), a[8,7] = (10*c[5]*c[6]*c[4 ]-5*c[6]*c[5]-5*c[6]*c[4]+3*c[6]-5*c[5]*c[4]+3*c[5]+3*c[4]-2)/c[7]/(c[ 6]-c[7])/(c[5]-c[7])/(c[4]-c[7])/(-20*c[5]*c[6]*c[4]+30*c[5]*c[6]*c[7] *c[4]+15*c[6]*c[5]-20*c[6]*c[7]*c[5]+15*c[6]*c[4]-20*c[6]*c[7]*c[4]-12 *c[6]+15*c[7]*c[6]+15*c[5]*c[4]-20*c[5]*c[7]*c[4]-12*c[5]+15*c[7]*c[5] -12*c[4]+15*c[7]*c[4]+10-12*c[7])*(c[4]-1)*(-1+c[5])*(-1+c[6])*(-1+c[7 ]), b[7] = -1/60*(10*c[5]*c[6]*c[4]-5*c[6]*c[5]-5*c[6]*c[4]+3*c[6]-5*c [5]*c[4]+3*c[5]+3*c[4]-2)/(c[4]-c[7])/(c[5]-c[7])/(c[6]-c[7])/c[7]/(-1 +c[7]), `b*`[8] = 0, a[5,3] = -3/4*c[5]^2*(2*c[5]-3*c[4])/c[4]^2, a[8, 6] = 1/2*(5*c[5]*c[6]*c[4]+c[6]-2*c[6]*c[4]-2*c[6]*c[5]+25*c[5]*c[7]*c [4]+6*c[5]+9*c[7]+10*c[7]^2*c[4]-4+6*c[4]-6*c[7]^2-14*c[7]*c[4]-10*c[5 ]*c[4]-14*c[7]*c[5]-20*c[7]^2*c[5]*c[4]+10*c[7]^2*c[5])/(c[6]-c[7])/(- c[4]+c[6])/(-c[5]+c[6])/c[6]/(-20*c[5]*c[6]*c[4]+30*c[5]*c[6]*c[7]*c[4 ]+15*c[6]*c[5]-20*c[6]*c[7]*c[5]+15*c[6]*c[4]-20*c[6]*c[7]*c[4]-12*c[6 ]+15*c[7]*c[6]+15*c[5]*c[4]-20*c[5]*c[7]*c[4]-12*c[5]+15*c[7]*c[5]-12* c[4]+15*c[7]*c[4]+10-12*c[7])*(c[4]-1)*(-1+c[5])*(-1+c[6]), b[8] = 1/6 0*(-20*c[5]*c[6]*c[4]+30*c[5]*c[6]*c[7]*c[4]+15*c[6]*c[5]-20*c[6]*c[7] *c[5]+15*c[6]*c[4]-20*c[6]*c[7]*c[4]-12*c[6]+15*c[7]*c[6]+15*c[5]*c[4] -20*c[5]*c[7]*c[4]-12*c[5]+15*c[7]*c[5]-12*c[4]+15*c[7]*c[4]+10-12*c[7 ])/(c[4]-1)/(-1+c[5])/(-1+c[6])/(-1+c[7]), a[5,4] = c[5]^2*(-c[4]+c[5] )/c[4]^2, a[7,6] = 1/2*(5*c[5]*c[4]+1-2*c[4]-2*c[5])*(c[4]-c[7])*(c[5] -c[7])*(c[6]-c[7])*c[7]/(10*c[5]*c[6]*c[4]-5*c[6]*c[5]-5*c[6]*c[4]+3*c [6]-5*c[5]*c[4]+3*c[5]+3*c[4]-2)/(-c[4]+c[6])/(-c[5]+c[6])/c[6], c[3] \+ = 2/3*c[4], a[4,3] = 3/4*c[4], a[3,2] = 2/9*c[4]^2/c[2]\}:" }{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 276 6 "Step 7" }{TEXT -1 1 " " }}{PARA 0 "" 0 "" {TEXT -1 23 " We find the 6 weights " }{XPPEDIT 18 0 "`b*`[1];" "6#&%#b*G6#\"\"\"" }{TEXT -1 3 ", " }{XPPEDIT 18 0 "`b*`[4];" "6#&%#b*G6#\"\"%" }{TEXT -1 3 ", " }{XPPEDIT 18 0 "`b*`[5];" "6#&%#b*G6#\"\"&" }{TEXT -1 3 ", \+ " }{XPPEDIT 18 0 "`b*`[6];" "6#&%#b*G6#\"\"'" }{TEXT -1 2 ", " } {XPPEDIT 18 0 "`b*`[7];" "6#&%#b*G6#\"\"(" }{TEXT -1 3 ", " } {XPPEDIT 18 0 "`b*`[9];" "6#&%#b*G6#\"\"*" }{TEXT -1 39 ", by using th e 5 quadrature conditions:" }}{PARA 256 "" 0 "" {TEXT -1 1 " " } {XPPEDIT 18 0 "Sum(`b*`[i],i = 1 .. 9) = 1;" "6#/-%$SumG6$&%#b*G6#%\"i G/F*;\"\"\"\"\"*F-" }{TEXT -1 15 ", " }{XPPEDIT 18 0 "Sum (`b*`[i]*c[i]^(k-1),i = 2 .. 9) = 1/k;" "6#/-%$SumG6$*&&%#b*G6#%\"iG\" \"\")&%\"cG6#F+,&%\"kGF,F,!\"\"F,/F+;\"\"#\"\"**&F,F,F2F3" }{TEXT -1 7 ", " }{XPPEDIT 18 0 "k = 2;" "6#/%\"kG\"\"#" }{TEXT -1 8 " . . \+ 5, " }}{PARA 0 "" 0 "" {TEXT -1 48 "together with the \"alternative\" \+ order condition " }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "[ `b*`]^T*[A]*(C-c[4]*Id)*(C-c[5]*Id)*[c] = Int(Int((t-c[4])*(t-c[5])*t, t = 0 .. x),x = 0 .. 1);" "6#/*,)7#%#b*G%\"TG\"\"\"7#%\"AGF),&%\"CGF)* &&%\"cG6#\"\"%F)%#IdGF)!\"\"F),&F-F)*&&F06#\"\"&F)F3F)F4F)7#F0F)-%$Int G6$-F<6$*(,&%\"tGF)&F06#F2F4F),&FBF)&F06#F9F4F)FBF)/FB;\"\"!%\"xG/FK;F JF)" }{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]*a[i,j-1],i = j .. 7)*(c[j-1]-c[4])*(c[j-1]-c[5])*c[ j-1],j = 2 .. 9) = -1/120+c[4]/60+c[5]/60-c[4]*c[5]/24;" "6#/-%$SumG6$ **-F%6$*&&%#b*G6#%\"iG\"\"\"&%\"aG6$F.,&%\"jGF/F/!\"\"F//F.;F4\"\"(F/, &&%\"cG6#,&F4F/F/F5F/&F;6#\"\"%F5F/,&&F;6#,&F4F/F/F5F/&F;6#\"\"&F5F/&F ;6#,&F4F/F/F5F//F4;\"\"#\"\"*,**&F/F/\"$?\"F5F5*&&F;6#F@F/\"#gF5F/*&&F ;6#FGF/FUF5F/*(&F;6#F@F/&F;6#FGF/\"#CF5F5" }{TEXT -1 2 ". " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 13 "We can omit " } {XPPEDIT 18 0 "j=2" "6#/%\"jG\"\"#" }{TEXT -1 33 " in the outer summa tion because " }{XPPEDIT 18 0 "c[1]=0" "6#/&%\"cG6#\"\"\"\"\"!" } {TEXT -1 15 ", we can omit " }{XPPEDIT 18 0 "j=3" "6#/%\"jG\"\"$" } {TEXT -1 11 " because " }{XPPEDIT 18 0 "a[i,2]=0" "6#/&%\"aG6$%\"iG \"\"#\"\"!" }{TEXT -1 8 ", for " }{XPPEDIT 18 0 "i=3" "6#/%\"iG\"\"$ " }{TEXT -1 25 " . . 7, and we can omit " }{XPPEDIT 18 0 "j=5" "6#/% \"jG\"\"&" }{TEXT -1 7 " and " }{XPPEDIT 18 0 "j=6" "6#/%\"jG\"\"'" }{TEXT -1 34 " because of obvious zero factors." }}{PARA 0 "" 0 "" {TEXT -1 13 "We can omit " }{XPPEDIT 18 0 "j=4" "6#/%\"jG\"\"%" } {TEXT -1 11 " because " }}{PARA 256 "" 0 "" {TEXT -1 1 " " } {XPPEDIT 18 0 "Sum(`b*`[i]*a[i,3],i = 4 .. 9) = 0;" "6#/-%$SumG6$*&&%# b*G6#%\"iG\"\"\"&%\"aG6$F+\"\"$F,/F+;\"\"%\"\"*\"\"!" }{TEXT -1 2 ". \+ " }}{PARA 0 "" 0 "" {TEXT -1 24 "This gives the relation " }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "Sum(Sum(`b*`[i]*a[i,j-1],i \+ = j .. 7)*(c[j-1]-c[4])*(c[j-1]-c[5])*c[j-1],j = 7 .. 9) = -1/120+c[4] /60+c[5]/60-c[4]*c[5]/24;" "6#/-%$SumG6$**-F%6$*&&%#b*G6#%\"iG\"\"\"&% \"aG6$F.,&%\"jGF/F/!\"\"F//F.;F4\"\"(F/,&&%\"cG6#,&F4F/F/F5F/&F;6#\"\" %F5F/,&&F;6#,&F4F/F/F5F/&F;6#\"\"&F5F/&F;6#,&F4F/F/F5F//F4;F8\"\"*,**& F/F/\"$?\"F5F5*&&F;6#F@F/\"#gF5F/*&&F;6#FGF/FTF5F/*(&F;6#F@F/&F;6#FGF/ \"#CF5F5" }{TEXT -1 2 ". " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 " " 0 "" {TEXT -1 41 "Expanding the left-hand side gives . . . " }} {PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 76 "add(add(`b*`[i]*a[i,j-1],i=j..9)*(c[j-1]-c[4])*(c[j-1]-c[5])*c[j-1 ],j=7..9);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#,(**,(*&&%#b*G6#\"\"(\" \"\"&%\"aG6$F*\"\"'F+F+*&&F(6#\"\")F+&F-6$F3F/F+F+*&&F(6#\"\"*F+&F-6$F 9F/F+F+F+,&&%\"cG6#\"\"%!\"\"&F>6#F/F+F+,&&F>6#\"\"&FAFBF+F+FBF+F+**,& *&F1F+&F-6$F3F*F+F+*&F7F+&F-6$F9F*F+F+F+,&F=FA&F>F)F+F+,&FQF+FEFAF+FQF +F+*,F7F+&F-6$F9F3F+,&&F>F2F+F=FAF+,&FWF+FEFAF+FWF+F+" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT 260 4 "Note" }{TEXT -1 61 " : Since the last equation involves the linking coefficients " } {XPPEDIT 18 0 "a[9,j]" "6#&%\"aG6$\"\"*%\"jG" }{TEXT -1 3 ", " } {XPPEDIT 18 0 "j=1" "6#/%\"jG\"\"\"" }{TEXT -1 43 " . . 8, we need to \+ make the substitutions " }{XPPEDIT 18 0 "a[9,j]=b[j]" "6#/&%\"aG6$\" \"*%\"jG&%\"bG6#F(" }{TEXT -1 3 ", " }{XPPEDIT 18 0 "j=1" "6#/%\"jG\" \"\"" }{TEXT -1 7 " . . 8." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 60 "First we set up the six equations for the six wei ghts . . . " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 " " {MPLTEXT 1 0 406 "`quad_eqs*` := [add(`b*`[i],i=1..9)=1,seq(add(`b*` [i]*c[i]^(j-1),i=2..9)=1/j,j=2..5)]:\n`ord_eq*` := add(add(`b*`[i]*a[i ,j-1],i=j..9)*(c[j-1]-c[4])*(c[j-1]-c[5])*c[j-1],j=7..9)=\n \+ 1/20-c[4]/12-c[5]/12+c[4]*c[5]/6:\nwt_eqs := [seq(a[9,i]=b[i],i= 1..8)]:\n`eqns*` := simplify(subs(e13,[op(`quad_eqs*`),subs(wt_eqs,`or d_eq*`)])):\nnops(`eqns*`);\nindets(`eqns*`) minus \{c[4],c[5],c[6],c[ 7]\};\nnops(%);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#\"\"'" }}{PARA 11 " " 1 "" {XPPMATH 20 "6#<(&%#b*G6#\"\"\"&F%6#\"\"%&F%6#\"\"&&F%6#\"\"'&F %6#\"\"(&F%6#\"\"*" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#\"\"'" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 30 " . . . and then we solve them." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> \+ " 0 "" {MPLTEXT 1 0 22 "infolevel[solve] := 4:" }}}{EXCHG {PARA 0 "> \+ " 0 "" {MPLTEXT 1 0 120 "e14 := solve(\{op(`eqns*`)\},indets(`eqns*`) \+ minus \{c[4],c[5],c[6],c[7]\}):\ne15 := `union`(e13,e14):\ninfolevel[s olve] := 0:" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 9 "Examples:" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 26 "`b*`[9]=s ubs(e15,`b*`[9]);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/&%#b*G6#\"\"*,$* &#\"\"\"\"#5F+*&,>**\"#]F+)&%\"cG6#\"\"&\"\"#F+&F36#\"\"'F+)&F36#\"\"% F6F+F+**\"#SF+F1F+F7F+F;F+!\"\"*(F,F+F7F+F1F+F+**F?F+F2F+F7F+F:F+F@** \"#NF+F2F+F7F+F;F+F+*(F,F+F7F+F2F+F@*(F,F+F7F+F:F+F+*(F,F+F7F+F;F+F@*& \"\"$F+F7F+F+*(F5F+F1F+F;F+F@*(F5F+F2F+F:F+F@*(F,F+F2F+F;F+F+F2F@F;F@F +,>*&FIF+F7F+F+**\"#GF+F2F+F7F+F:F+F@**FPF+F1F+F7F+F;F+F@*(\"\")F+F7F+ F1F+F+*(F'F+F7F+F;F+F@*(F'F+F7F+F2F+F@**FPF+F2F+F7F+F;F+F+**\"#IF+F1F+ F7F+F:F+F+*(FSF+F7F+F:F+F+*(F=F+F1F+F;F+F@*(F=F+F2F+F:F+F@*(F'F+F2F+F; F+F+F2F@F;F@F@F+F+" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 26 "`b*`[4]=subs(e15,`b*`[4]);" }}{PARA 12 "" 1 "" {XPPMATH 20 "6#/&%#b*G6#\"\"%,$*&#\"\"\"\"#gF+*,,\\s*(\"#IF+)&%\" cG6#\"\"&\"\"$F+&F3F&F+!\"\"*(\"#UF+&F36#\"\"(F+)&F36#\"\"'\"\"#F+F8** \"#*)F+F2F+F;F+F7F+F+*&\"#=F+F?F+F8*&FAF+F2F+F+*&FAF+F7F+F+*(\"#FF+F;F +F?F+F+*&\"\"*F+)F2FBF+F8**\"$!=F+F;F+F>F+F1F+F+**\"$+\"F+F;F+F?F+F1F+ F8*&FJF+F>F+F+*,\"$'=F+F2F+F?F+F;F+F7F+F+*(\"#(*F+F?F+FMF+F8*(\"#mF+F? F+F2F+F+*(F,F+F?F+F1F+F+*(FQF+F>F+F1F+F8*(FLF+F;F+F2F+F8*(\"#9F+F;F+FM F+F+**FQF+F>F+F;F+)F7FBF+F8**\"$?\"F+F>F+F;F+F7F+F+*(F,F+F>F+FinF+F+*( FLF+F;F+F7F+F8**\"$.\"F+F?F+F;F+F2F+F8*(\"#OF+F?F+FinF+F8**\"#hF+F?F+F ;F+F7F+F8**F,F+F?F+F;F+FinF+F+**F0F+F2F+F;F+FinF+F8**\"$E\"F+F2F+F?F+F inF+F+*(FFF+F2F+FinF+F+**\"$7\"F+F2F+F?F+F7F+F8*(\"#dF+F2F+F7F+F8*(\"# RF+F?F+F7F+F+**\"$!GF+F>F+FinF+F2F+F8**\"$I$F+F>F+F7F+F2F+F+*,\"$!QF+F MF+F?F+F;F+FinF+F+*,\"$g&F+F>F+F;F+F7F+F2F+F8**\"$#>F+F>F+F;F+F2F+F+** \"$5$F+F>F+F;F+FMF+F8*,\"$I#F+F?F+F2F+F;F+FinF+F8**\"$c\"F+F?F+F;F+FMF +F+*(FOF+F>F+FMF+F+*,\"$+&F+F>F+FinF+F;F+F2F+F+*,\"$]*F+F>F+FMF+F;F+F7 F+F+*,\"$+*F+F>F+FMF+F;F+FinF+F8*,\"$+$F+F?F+F1F+F;F+FinF+F8*,\"$+'F+F >F+F1F+F;F+F7F+F8*(\"$<\"F+F>F+F2F+F8**\"$I\"F+F1F+F?F+F7F+F8**\"#]F+F 1F+F;F+F7F+F+*(\"#$*F+FMF+F7F+F+*,F\\qF+F1F+F?F+F;F+F7F+F+**F_rF+FMF+F ;F+FinF+F+**\"$!>F+FMF+F?F+FinF+F8**\"$]\"F+F1F+F?F+FinF+F+**\"$q%F+F> F+FMF+FinF+F+**\"$?$F+F>F+F1F+F7F+F+**FgqF+F>F+F1F+FinF+F8*(F0F+FMF+Fi nF+F8**\"$I&F+F>F+F7F+FMF+F8*(\"#vF+F>F+F7F+F8**F^qF+FMF+F?F+F7F+F+**F grF+FMF+F;F+F7F+F8*,\"$q#F+FMF+F?F+F;F+F7F+F8*,FiqF+F>F+F;F+F1F+FinF+F +F+,Z**FLF+F2F+F;F+F7F+F8*(F6F+F;F+F?F+F8*$FinF+F8*,\"#GF+F2F+F?F+F;F+ F7F+F8*&F;F+F2F+F+*&F;F+F7F+F+**FLF+F?F+F;F+F2F+F+*(FLF+F?F+FinF+F8*( \"\")F+F?F+)F7F6F+F+**FLF+F?F+F;F+F7F+F+**FbtF+F?F+F;F+FinF+F8*(F'F+F2 F+FctF+F8**F'F+F2F+F;F+FinF+F+**F\\tF+F2F+F?F+FinF+F+*(FLF+F2F+FinF+F+ **FLF+F2F+F?F+F7F+F8*&F2F+F7F+F8*(F6F+F?F+F7F+F+*,F0F+FMF+F?F+F;F+FinF +F8*,F\\tF+F?F+F2F+F;F+FinF+F+**FbtF+F?F+F;F+FMF+F8**F\\tF+F?F+F2F+Fct F+F8**F0F+FMF+F?F+FctF+F+**F\\tF+FMF+F?F+FinF+F8*(F'F+FMF+FinF+F8**Fbt F+FMF+F?F+F7F+F+**F'F+FMF+F;F+F7F+F+*,F\\tF+FMF+F?F+F;F+F7F+F+F8,&F7F8 F?F+F8,&F7F8F2F+F8F7F8F+F8" }}}{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 12989 "e15 := \{c[8] = 1, b[3] = 0, b[2] = 0, c[9] = 1, b [5] = -1/60*(10*c[6]*c[7]*c[4]-5*c[6]*c[4]-5*c[7]*c[6]+3*c[6]-5*c[7]*c [4]+3*c[4]-2+3*c[7])/(-c[4]+c[5])/c[5]/(-c[5]^3+c[5]^2-c[7]*c[5]+c[7]* c[5]^2-c[6]*c[5]+c[6]*c[5]^2+c[7]*c[6]-c[6]*c[7]*c[5]), a[5,2] = 0, a[ 4,2] = 0, `b*`[3] = 0, `b*`[2] = 0, a[8,2] = 0, a[7,2] = 0, a[6,2] = 0 , b[1] = 1/60*(30*c[5]*c[6]*c[7]*c[4]-10*c[5]*c[6]*c[4]-10*c[6]*c[7]*c [5]+5*c[6]*c[5]-10*c[6]*c[7]*c[4]+5*c[6]*c[4]+5*c[7]*c[6]-3*c[6]-10*c[ 5]*c[7]*c[4]+5*c[5]*c[4]+5*c[7]*c[5]-3*c[5]+5*c[7]*c[4]-3*c[4]+2-3*c[7 ])/c[5]/c[6]/c[7]/c[4], b[4] = 1/60*(-2+3*c[5]-5*c[7]*c[6]+3*c[7]+3*c[ 6]+10*c[6]*c[7]*c[5]-5*c[6]*c[5]-5*c[7]*c[5])/c[4]/(-c[6]*c[4]^3-c[4]^ 3+c[4]^4+c[7]*c[4]^2+c[6]*c[4]^2-c[6]*c[7]*c[4]+c[6]*c[7]*c[4]^2-c[5]* c[4]^3+c[5]*c[4]^2-c[7]*c[4]^3-c[5]*c[7]*c[4]+c[5]*c[7]*c[4]^2-c[5]*c[ 6]*c[4]+c[5]*c[6]*c[4]^2+c[6]*c[7]*c[5]-c[5]*c[6]*c[7]*c[4]), b[6] = 1 /60*(10*c[5]*c[7]*c[4]-5*c[5]*c[4]-5*c[7]*c[5]+3*c[5]-5*c[7]*c[4]+3*c[ 4]-2+3*c[7])/(-c[4]+c[6])/(-c[5]+c[6])/c[6]/(c[6]^2-c[6]+c[7]-c[7]*c[6 ]), `b*`[9] = 1/10*(50*c[5]^2*c[6]*c[4]^2-40*c[5]^2*c[6]*c[4]+10*c[6]* c[5]^2-40*c[5]*c[6]*c[4]^2+35*c[5]*c[6]*c[4]-10*c[6]*c[5]+10*c[6]*c[4] ^2-10*c[6]*c[4]+3*c[6]-5*c[5]^2*c[4]-5*c[5]*c[4]^2+10*c[5]*c[4]-c[5]-c [4])/(3*c[6]-28*c[5]*c[6]*c[4]^2-28*c[5]^2*c[6]*c[4]+8*c[6]*c[5]^2-9*c [6]*c[4]-9*c[6]*c[5]+28*c[5]*c[6]*c[4]+30*c[5]^2*c[6]*c[4]^2+8*c[6]*c[ 4]^2-4*c[5]^2*c[4]-4*c[5]*c[4]^2+9*c[5]*c[4]-c[5]-c[4]), `b*`[6] = -1/ 60*(-30*c[5]^3*c[4]+103*c[5]*c[7]*c[4]-18*c[6]+6*c[5]+6*c[4]+27*c[7]*c [6]-9*c[5]^2-9*c[4]^2-100*c[7]*c[6]*c[5]^3+522*c[5]*c[6]*c[7]*c[4]-111 *c[6]*c[5]^2+75*c[6]*c[5]+60*c[6]*c[5]^3-9*c[7]*c[5]+600*c[6]*c[7]*c[5 ]^3*c[4]^3+14*c[7]*c[5]^2-100*c[7]*c[6]*c[4]^3+14*c[7]*c[4]^2-9*c[7]*c [4]-117*c[6]*c[7]*c[5]-111*c[6]*c[4]^2+60*c[6]*c[4]^3-117*c[6]*c[7]*c[ 4]+180*c[6]*c[7]*c[4]^2-30*c[5]*c[4]^3-174*c[5]*c[7]*c[4]^2+486*c[5]*c [6]*c[4]^2+107*c[5]*c[4]^2-318*c[5]*c[6]*c[4]-66*c[5]*c[4]+75*c[6]*c[4 ]+500*c[5]*c[7]*c[6]*c[4]^3+1420*c[5]^2*c[6]*c[7]*c[4]^2-840*c[6]*c[5] *c[7]*c[4]^2+180*c[6]*c[7]*c[5]^2-900*c[6]*c[5]^2*c[7]*c[4]^3-900*c[6] *c[5]^3*c[7]*c[4]^2+50*c[7]*c[5]*c[4]^3-280*c[6]*c[5]*c[4]^3-280*c[5]^ 3*c[6]*c[4]+50*c[5]^3*c[7]*c[4]+107*c[5]^2*c[4]+500*c[5]^3*c[6]*c[7]*c [4]+50*c[5]^3*c[4]^2+470*c[5]^2*c[6]*c[4]^3+310*c[5]^2*c[7]*c[4]^2-770 *c[5]^2*c[6]*c[4]^2-90*c[5]^2*c[7]*c[4]^3-90*c[5]^3*c[7]*c[4]^2+470*c[ 5]^3*c[6]*c[4]^2-300*c[6]*c[5]^3*c[4]^3-180*c[5]^2*c[4]^2+486*c[5]^2*c [6]*c[4]-174*c[5]^2*c[7]*c[4]-840*c[5]^2*c[6]*c[7]*c[4]+50*c[5]^2*c[4] ^3)/(-9*c[5]*c[7]*c[4]-3*c[7]*c[6]+3*c[6]^2-28*c[5]*c[6]*c[7]*c[4]-c[6 ]*c[5]+c[7]*c[5]+8*c[6]^2*c[4]^2+c[7]*c[4]+9*c[6]*c[7]*c[5]+9*c[6]*c[7 ]*c[4]-8*c[6]*c[7]*c[4]^2+4*c[5]*c[7]*c[4]^2-4*c[5]*c[6]*c[4]^2+9*c[5] *c[6]*c[4]-c[6]*c[4]-28*c[6]^2*c[4]^2*c[5]+28*c[6]^2*c[4]*c[5]-30*c[5] ^2*c[6]*c[7]*c[4]^2+28*c[6]*c[5]*c[7]*c[4]^2-8*c[6]*c[7]*c[5]^2+8*c[6] ^2*c[5]^2-9*c[6]^2*c[5]+30*c[6]^2*c[5]^2*c[4]^2-28*c[6]^2*c[4]*c[5]^2- 9*c[6]^2*c[4]-4*c[5]^2*c[6]*c[4]+4*c[5]^2*c[7]*c[4]+28*c[5]^2*c[6]*c[7 ]*c[4])/(c[5]*c[4]-c[6]*c[4]-c[6]*c[5]+c[6]^2)/c[6], a[8,7] = (10*c[5] *c[6]*c[4]-5*c[6]*c[5]-5*c[6]*c[4]+3*c[6]-5*c[5]*c[4]+3*c[5]+3*c[4]-2) /c[7]/(c[6]-c[7])/(c[5]-c[7])/(c[4]-c[7])/(-20*c[5]*c[6]*c[4]+30*c[5]* c[6]*c[7]*c[4]+15*c[6]*c[5]-20*c[6]*c[7]*c[5]+15*c[6]*c[4]-20*c[6]*c[7 ]*c[4]-12*c[6]+15*c[7]*c[6]+15*c[5]*c[4]-20*c[5]*c[7]*c[4]-12*c[5]+15* c[7]*c[5]-12*c[4]+15*c[7]*c[4]+10-12*c[7])*(c[4]-1)*(-1+c[5])*(-1+c[6] )*(-1+c[7]), b[7] = -1/60*(10*c[5]*c[6]*c[4]-5*c[6]*c[5]-5*c[6]*c[4]+3 *c[6]-5*c[5]*c[4]+3*c[5]+3*c[4]-2)/(c[4]-c[7])/(c[5]-c[7])/(c[6]-c[7]) /c[7]/(-1+c[7]), `b*`[8] = 0, a[5,3] = -3/4*c[5]^2*(2*c[5]-3*c[4])/c[4 ]^2, `b*`[4] = -1/60*(-30*c[5]^3*c[4]-42*c[7]*c[6]^2+89*c[5]*c[7]*c[4] -18*c[6]+6*c[5]+6*c[4]+27*c[7]*c[6]-9*c[5]^2+180*c[7]*c[6]^2*c[5]^3-10 0*c[7]*c[6]*c[5]^3+27*c[6]^2+186*c[5]*c[6]*c[7]*c[4]-97*c[6]*c[5]^2+66 *c[6]*c[5]+60*c[6]*c[5]^3-100*c[6]^2*c[5]^3-9*c[7]*c[5]+14*c[7]*c[5]^2 -100*c[6]^2*c[7]*c[4]^2+120*c[6]^2*c[7]*c[4]+60*c[6]^2*c[4]^2-9*c[7]*c [4]-103*c[6]*c[7]*c[5]-36*c[6]*c[4]^2-61*c[6]*c[7]*c[4]+60*c[6]*c[7]*c [4]^2-30*c[5]*c[7]*c[4]^2+126*c[5]*c[6]*c[4]^2+18*c[5]*c[4]^2-112*c[5] *c[6]*c[4]-57*c[5]*c[4]+39*c[6]*c[4]-280*c[6]^2*c[4]^2*c[5]+330*c[6]^2 *c[4]*c[5]+380*c[5]^2*c[6]*c[7]*c[4]^2-560*c[6]^2*c[7]*c[4]*c[5]+192*c [6]^2*c[7]*c[5]-310*c[6]^2*c[7]*c[5]^2-230*c[6]*c[5]*c[7]*c[4]^2+156*c [6]*c[7]*c[5]^2+180*c[6]^2*c[5]^2+500*c[6]^2*c[4]^2*c[7]*c[5]+950*c[6] ^2*c[5]^2*c[7]*c[4]-900*c[6]^2*c[5]^2*c[7]*c[4]^2-300*c[6]*c[5]^3*c[7] *c[4]^2-600*c[6]^2*c[5]^3*c[7]*c[4]-117*c[6]^2*c[5]-130*c[5]^3*c[6]*c[ 4]+50*c[5]^3*c[7]*c[4]+93*c[5]^2*c[4]+230*c[5]^3*c[6]*c[7]*c[4]+50*c[5 ]^2*c[7]*c[4]^2-190*c[5]^2*c[6]*c[4]^2+150*c[5]^3*c[6]*c[4]^2+470*c[6] ^2*c[5]^2*c[4]^2+320*c[6]^2*c[5]^3*c[4]-300*c[6]^2*c[5]^3*c[4]^2-30*c[ 5]^2*c[4]^2-530*c[6]^2*c[4]*c[5]^2-75*c[6]^2*c[4]+156*c[5]^2*c[6]*c[4] -150*c[5]^2*c[7]*c[4]-270*c[5]^2*c[6]*c[7]*c[4]+600*c[6]^2*c[7]*c[5]^3 *c[4]^2)/(-9*c[5]*c[7]*c[4]-3*c[7]*c[6]-c[4]^2-28*c[5]*c[6]*c[7]*c[4]+ c[7]*c[5]+c[7]*c[4]+9*c[6]*c[7]*c[5]-9*c[6]*c[4]^2+8*c[6]*c[4]^3+9*c[6 ]*c[7]*c[4]-8*c[6]*c[7]*c[4]^2-4*c[5]*c[4]^3+4*c[5]*c[7]*c[4]^2+28*c[5 ]*c[6]*c[4]^2+9*c[5]*c[4]^2-9*c[5]*c[6]*c[4]-c[5]*c[4]+3*c[6]*c[4]-30* c[5]^2*c[6]*c[7]*c[4]^2+28*c[6]*c[5]*c[7]*c[4]^2-8*c[6]*c[7]*c[5]^2-28 *c[6]*c[5]*c[4]^3+30*c[5]^2*c[6]*c[4]^3-28*c[5]^2*c[6]*c[4]^2-4*c[5]^2 *c[4]^2+8*c[5]^2*c[6]*c[4]+4*c[5]^2*c[7]*c[4]+28*c[5]^2*c[6]*c[7]*c[4] )/(-c[4]+c[6])/(-c[4]+c[5])/c[4], a[8,6] = 1/2*(5*c[5]*c[6]*c[4]+c[6]- 2*c[6]*c[4]-2*c[6]*c[5]+25*c[5]*c[7]*c[4]+6*c[5]+9*c[7]+10*c[7]^2*c[4] -4+6*c[4]-6*c[7]^2-14*c[7]*c[4]-10*c[5]*c[4]-14*c[7]*c[5]-20*c[7]^2*c[ 5]*c[4]+10*c[7]^2*c[5])/(c[6]-c[7])/(-c[4]+c[6])/(-c[5]+c[6])/c[6]/(-2 0*c[5]*c[6]*c[4]+30*c[5]*c[6]*c[7]*c[4]+15*c[6]*c[5]-20*c[6]*c[7]*c[5] +15*c[6]*c[4]-20*c[6]*c[7]*c[4]-12*c[6]+15*c[7]*c[6]+15*c[5]*c[4]-20*c [5]*c[7]*c[4]-12*c[5]+15*c[7]*c[5]-12*c[4]+15*c[7]*c[4]+10-12*c[7])*(c [4]-1)*(-1+c[5])*(-1+c[6]), b[8] = 1/60*(-20*c[5]*c[6]*c[4]+30*c[5]*c[ 6]*c[7]*c[4]+15*c[6]*c[5]-20*c[6]*c[7]*c[5]+15*c[6]*c[4]-20*c[6]*c[7]* c[4]-12*c[6]+15*c[7]*c[6]+15*c[5]*c[4]-20*c[5]*c[7]*c[4]-12*c[5]+15*c[ 7]*c[5]-12*c[4]+15*c[7]*c[4]+10-12*c[7])/(c[4]-1)/(-1+c[5])/(-1+c[6])/ (-1+c[7]), `b*`[5] = 1/60*(-42*c[7]*c[6]^2+89*c[5]*c[7]*c[4]-18*c[6]+6 *c[5]+6*c[4]+27*c[7]*c[6]-9*c[4]^2+27*c[6]^2+186*c[5]*c[6]*c[7]*c[4]-3 6*c[6]*c[5]^2+39*c[6]*c[5]-9*c[7]*c[5]-100*c[6]^2*c[4]^3+600*c[6]^2*c[ 7]*c[5]^2*c[4]^3-100*c[7]*c[6]*c[4]^3-310*c[6]^2*c[7]*c[4]^2+192*c[6]^ 2*c[7]*c[4]+180*c[6]^2*c[4]^2+14*c[7]*c[4]^2-9*c[7]*c[4]-61*c[6]*c[7]* c[5]-97*c[6]*c[4]^2+60*c[6]*c[4]^3-103*c[6]*c[7]*c[4]+156*c[6]*c[7]*c[ 4]^2-30*c[5]*c[4]^3-150*c[5]*c[7]*c[4]^2+156*c[5]*c[6]*c[4]^2+93*c[5]* c[4]^2-112*c[5]*c[6]*c[4]-57*c[5]*c[4]+66*c[6]*c[4]+230*c[5]*c[7]*c[6] *c[4]^3-530*c[6]^2*c[4]^2*c[5]+330*c[6]^2*c[4]*c[5]+380*c[5]^2*c[6]*c[ 7]*c[4]^2+320*c[6]^2*c[5]*c[4]^3-560*c[6]^2*c[7]*c[4]*c[5]-600*c[7]*c[ 6]^2*c[5]*c[4]^3+120*c[6]^2*c[7]*c[5]-100*c[6]^2*c[7]*c[5]^2-270*c[6]* c[5]*c[7]*c[4]^2+60*c[6]*c[7]*c[5]^2+60*c[6]^2*c[5]^2+950*c[6]^2*c[4]^ 2*c[7]*c[5]+500*c[6]^2*c[5]^2*c[7]*c[4]-900*c[6]^2*c[5]^2*c[7]*c[4]^2- 300*c[6]*c[5]^2*c[7]*c[4]^3-75*c[6]^2*c[5]+50*c[7]*c[5]*c[4]^3-130*c[6 ]*c[5]*c[4]^3+18*c[5]^2*c[4]+150*c[5]^2*c[6]*c[4]^3+50*c[5]^2*c[7]*c[4 ]^2-190*c[5]^2*c[6]*c[4]^2-300*c[6]^2*c[5]^2*c[4]^3+470*c[6]^2*c[5]^2* c[4]^2-30*c[5]^2*c[4]^2-280*c[6]^2*c[4]*c[5]^2+180*c[7]*c[6]^2*c[4]^3- 117*c[6]^2*c[4]+126*c[5]^2*c[6]*c[4]-30*c[5]^2*c[7]*c[4]-230*c[5]^2*c[ 6]*c[7]*c[4])/(-4*c[5]^3*c[4]-9*c[5]*c[7]*c[4]-3*c[7]*c[6]-c[5]^2-28*c [5]*c[6]*c[7]*c[4]-9*c[6]*c[5]^2+3*c[6]*c[5]+8*c[6]*c[5]^3+c[7]*c[5]+c [7]*c[4]+9*c[6]*c[7]*c[5]+9*c[6]*c[7]*c[4]-8*c[6]*c[7]*c[4]^2+4*c[5]*c [7]*c[4]^2+8*c[5]*c[6]*c[4]^2-9*c[5]*c[6]*c[4]-c[5]*c[4]-30*c[5]^2*c[6 ]*c[7]*c[4]^2+28*c[6]*c[5]*c[7]*c[4]^2-8*c[6]*c[7]*c[5]^2-28*c[5]^3*c[ 6]*c[4]+9*c[5]^2*c[4]-28*c[5]^2*c[6]*c[4]^2+30*c[5]^3*c[6]*c[4]^2-4*c[ 5]^2*c[4]^2+28*c[5]^2*c[6]*c[4]+4*c[5]^2*c[7]*c[4]+28*c[5]^2*c[6]*c[7] *c[4])/(-c[5]+c[6])/(-c[4]+c[5])/c[5], a[5,4] = c[5]^2*(-c[4]+c[5])/c[ 4]^2, a[7,6] = 1/2*(5*c[5]*c[4]+1-2*c[4]-2*c[5])*(c[4]-c[7])*(c[5]-c[7 ])*(c[6]-c[7])*c[7]/(10*c[5]*c[6]*c[4]-5*c[6]*c[5]-5*c[6]*c[4]+3*c[6]- 5*c[5]*c[4]+3*c[5]+3*c[4]-2)/(-c[4]+c[6])/(-c[5]+c[6])/c[6], c[3] = 2/ 3*c[4], a[4,3] = 3/4*c[4], a[3,2] = 2/9*c[4]^2/c[2], `b*`[7] = 1/60*(- 30*c[5]^3*c[4]-18*c[6]+6*c[5]+6*c[4]-9*c[5]^2-9*c[4]^2+27*c[6]^2-97*c[ 6]*c[5]^2+66*c[6]*c[5]+60*c[6]*c[5]^3-100*c[6]^2*c[5]^3-100*c[6]^2*c[4 ]^3+180*c[6]^2*c[4]^2-97*c[6]*c[4]^2+60*c[6]*c[4]^3-30*c[5]*c[4]^3+312 *c[5]*c[6]*c[4]^2+107*c[5]*c[4]^2-215*c[5]*c[6]*c[4]-66*c[5]*c[4]+66*c [6]*c[4]-840*c[6]^2*c[4]^2*c[5]+522*c[6]^2*c[4]*c[5]+500*c[6]^2*c[5]*c [4]^3+180*c[6]^2*c[5]^2-117*c[6]^2*c[5]-230*c[6]*c[5]*c[4]^3-230*c[5]^ 3*c[6]*c[4]+107*c[5]^2*c[4]+50*c[5]^3*c[4]^2+380*c[5]^2*c[6]*c[4]^3-46 0*c[5]^2*c[6]*c[4]^2+380*c[5]^3*c[6]*c[4]^2+600*c[5]^3*c[6]^2*c[4]^3-9 00*c[6]^2*c[5]^2*c[4]^3+1420*c[6]^2*c[5]^2*c[4]^2-300*c[6]*c[5]^3*c[4] ^3+500*c[6]^2*c[5]^3*c[4]-900*c[6]^2*c[5]^3*c[4]^2-180*c[5]^2*c[4]^2-8 40*c[6]^2*c[4]*c[5]^2-117*c[6]^2*c[4]+312*c[5]^2*c[6]*c[4]+50*c[5]^2*c [4]^3)/c[7]/(-3*c[7]^3*c[6]+3*c[7]^2*c[6]^2-9*c[6]^2*c[7]^2*c[5]-8*c[7 ]*c[6]^2*c[5]^3+8*c[7]^2*c[6]*c[5]^3-c[5]*c[6]*c[7]*c[4]-c[7]^2*c[4]^2 -30*c[6]^2*c[7]*c[5]^2*c[4]^3-28*c[6]^2*c[7]^2*c[5]^2*c[4]+30*c[6]^2*c [7]^2*c[5]^2*c[4]^2-30*c[6]*c[7]*c[5]^3*c[4]^3+9*c[6]^2*c[7]*c[4]^2-3* c[6]^2*c[7]*c[4]+2*c[6]*c[7]^2*c[4]-9*c[6]*c[7]^2*c[4]^2-2*c[7]^2*c[5] *c[4]-c[7]^2*c[5]^2+c[6]*c[7]*c[4]^2+8*c[7]^2*c[6]*c[4]^3-4*c[5]*c[7]^ 2*c[4]^3+c[5]*c[7]*c[4]^2-c[5]*c[6]*c[4]^2+32*c[6]*c[5]*c[7]^2*c[4]^2- 9*c[7]^2*c[5]*c[6]*c[4]-4*c[5]*c[7]*c[6]*c[4]^3-9*c[6]^2*c[4]^2*c[5]+3 *c[6]^2*c[4]*c[5]+c[7]^3*c[4]+9*c[7]^3*c[4]*c[6]-9*c[7]^3*c[4]*c[5]-20 *c[5]^2*c[6]*c[7]*c[4]^2+8*c[6]^2*c[5]*c[4]^3+28*c[7]^3*c[4]^2*c[6]*c[ 5]+18*c[6]^2*c[7]*c[4]*c[5]+28*c[7]*c[6]^2*c[5]*c[4]^3-3*c[6]^2*c[7]*c [5]+9*c[6]^2*c[7]*c[5]^2+c[6]*c[7]*c[5]^2-8*c[6]*c[7]^3*c[5]^2-9*c[6]* c[7]^2*c[5]^2+8*c[6]^2*c[7]^2*c[5]^2+2*c[6]*c[7]^2*c[5]-36*c[6]^2*c[4] ^2*c[7]*c[5]-28*c[6]^2*c[7]^2*c[4]^2*c[5]-28*c[7]^3*c[4]*c[6]*c[5]-36* c[6]^2*c[5]^2*c[7]*c[4]+56*c[6]^2*c[5]^2*c[7]*c[4]^2+28*c[6]*c[5]^2*c[ 7]*c[4]^3+28*c[6]*c[5]^3*c[7]*c[4]^2+28*c[6]^2*c[5]^3*c[7]*c[4]+9*c[6] *c[7]^3*c[5]+9*c[7]^2*c[5]*c[4]^2+28*c[7]^3*c[5]^2*c[6]*c[4]+32*c[7]^2 *c[5]^2*c[6]*c[4]+8*c[6]^2*c[7]^2*c[4]^2-9*c[6]^2*c[7]^2*c[4]-4*c[5]^3 *c[6]*c[7]*c[4]+28*c[6]^2*c[7]^2*c[4]*c[5]-4*c[5]^2*c[6]*c[4]^3-9*c[5] ^2*c[7]*c[4]^2+9*c[5]^2*c[6]*c[4]^2+4*c[5]^2*c[7]*c[4]^3+4*c[5]^3*c[7] *c[4]^2-4*c[5]^3*c[6]*c[4]^2+30*c[5]^3*c[6]^2*c[4]^3-28*c[6]^2*c[5]^2* c[4]^3+28*c[6]^2*c[5]^2*c[4]^2+8*c[6]^2*c[5]^3*c[4]-28*c[6]^2*c[5]^3*c [4]^2-9*c[6]^2*c[4]*c[5]^2-8*c[7]*c[6]^2*c[4]^3+30*c[7]^2*c[5]^2*c[6]* c[4]^3-30*c[7]^3*c[5]^2*c[6]*c[4]^2+30*c[7]^2*c[5]^3*c[6]*c[4]^2-8*c[7 ]^3*c[4]^2*c[6]+4*c[7]^3*c[4]^2*c[5]+c[7]^3*c[5]-28*c[7]^2*c[6]*c[5]*c [4]^3-56*c[7]^2*c[5]^2*c[6]*c[4]^2-28*c[7]^2*c[5]^3*c[6]*c[4]-c[5]^2*c [6]*c[4]+c[5]^2*c[7]*c[4]+4*c[7]^3*c[5]^2*c[4]+9*c[7]^2*c[5]^2*c[4]-30 *c[6]^2*c[7]*c[5]^3*c[4]^2-8*c[7]^2*c[5]^2*c[4]^2-4*c[7]^2*c[5]^3*c[4] ), `b*`[1] = 1/60*(30*c[5]^3*c[4]+42*c[7]*c[6]^2-103*c[5]*c[7]*c[4]+18 *c[6]-6*c[5]-6*c[4]-27*c[7]*c[6]+9*c[5]^2+9*c[4]^2-180*c[7]*c[6]^2*c[5 ]^3+100*c[7]*c[6]*c[5]^3-27*c[6]^2-354*c[5]*c[6]*c[7]*c[4]+97*c[6]*c[5 ]^2-66*c[6]*c[5]-60*c[6]*c[5]^3+100*c[6]^2*c[5]^3+9*c[7]*c[5]+100*c[6] ^2*c[4]^3-2040*c[6]^2*c[7]*c[5]^2*c[4]^3-600*c[6]*c[7]*c[5]^3*c[4]^3-1 4*c[7]*c[5]^2+100*c[7]*c[6]*c[4]^3+310*c[6]^2*c[7]*c[4]^2-192*c[6]^2*c [7]*c[4]-180*c[6]^2*c[4]^2-14*c[7]*c[4]^2+9*c[7]*c[4]+103*c[6]*c[7]*c[ 5]+97*c[6]*c[4]^2-60*c[6]*c[4]^3+103*c[6]*c[7]*c[4]-156*c[6]*c[7]*c[4] ^2+30*c[5]*c[4]^3+174*c[5]*c[7]*c[4]^2-312*c[5]*c[6]*c[4]^2-107*c[5]*c [4]^2+215*c[5]*c[6]*c[4]+66*c[5]*c[4]-66*c[6]*c[4]-410*c[5]*c[7]*c[6]* c[4]^3+840*c[6]^2*c[4]^2*c[5]-522*c[6]^2*c[4]*c[5]-760*c[5]^2*c[6]*c[7 ]*c[4]^2-500*c[6]^2*c[5]*c[4]^3+932*c[6]^2*c[7]*c[4]*c[5]+1020*c[7]*c[ 6]^2*c[5]*c[4]^3-192*c[6]^2*c[7]*c[5]+310*c[6]^2*c[7]*c[5]^2+526*c[6]* c[5]*c[7]*c[4]^2-156*c[6]*c[7]*c[5]^2-180*c[6]^2*c[5]^2-1610*c[6]^2*c[ 4]^2*c[7]*c[5]-1610*c[6]^2*c[5]^2*c[7]*c[4]+2970*c[6]^2*c[5]^2*c[7]*c[ 4]^2+690*c[6]*c[5]^2*c[7]*c[4]^3+690*c[6]*c[5]^3*c[7]*c[4]^2+1020*c[6] ^2*c[5]^3*c[7]*c[4]+117*c[6]^2*c[5]-50*c[7]*c[5]*c[4]^3+230*c[6]*c[5]* c[4]^3+230*c[5]^3*c[6]*c[4]-50*c[5]^3*c[7]*c[4]-107*c[5]^2*c[4]-410*c[ 5]^3*c[6]*c[7]*c[4]-50*c[5]^3*c[4]^2-380*c[5]^2*c[6]*c[4]^3-310*c[5]^2 *c[7]*c[4]^2+460*c[5]^2*c[6]*c[4]^2+90*c[5]^2*c[7]*c[4]^3+90*c[5]^3*c[ 7]*c[4]^2-380*c[5]^3*c[6]*c[4]^2-600*c[5]^3*c[6]^2*c[4]^3+900*c[6]^2*c [5]^2*c[4]^3-1420*c[6]^2*c[5]^2*c[4]^2+300*c[6]*c[5]^3*c[4]^3-500*c[6] ^2*c[5]^3*c[4]+900*c[6]^2*c[5]^3*c[4]^2+180*c[5]^2*c[4]^2+840*c[6]^2*c [4]*c[5]^2-180*c[7]*c[6]^2*c[4]^3+117*c[6]^2*c[4]+1500*c[5]^3*c[6]^2*c [7]*c[4]^3-312*c[5]^2*c[6]*c[4]+174*c[5]^2*c[7]*c[4]+526*c[5]^2*c[6]*c [7]*c[4]-2040*c[6]^2*c[7]*c[5]^3*c[4]^2-50*c[5]^2*c[4]^3)/(3*c[6]-28*c [5]*c[6]*c[4]^2-28*c[5]^2*c[6]*c[4]+8*c[6]*c[5]^2-9*c[6]*c[4]-9*c[6]*c [5]+28*c[5]*c[6]*c[4]+30*c[5]^2*c[6]*c[4]^2+8*c[6]*c[4]^2-4*c[5]^2*c[4 ]-4*c[5]*c[4]^2+9*c[5]*c[4]-c[5]-c[4])/c[5]/c[6]/c[7]/c[4]\}:" }}} {PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 15 "" 0 "" {TEXT 277 6 "Step 8" }{TEXT -1 1 " " }}{PARA 0 "" 0 "" {TEXT -1 22 "We use the r elations: " }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "Sum(`b* `[i]*a[i,3],i=4..8)=0" "6#/-%$SumG6$*&&%#b*G6#%\"iG\"\"\"&%\"aG6$F+\" \"$F,/F+;\"\"%\"\")\"\"!" }{TEXT -1 6 ", " }{XPPEDIT 18 0 "Sum(b[i ]*a[i,3],i=4..8)=b[3]*(1-c[3])" "6#/-%$SumG6$*&&%\"bG6#%\"iG\"\"\"&%\" aG6$F+\"\"$F,/F+;\"\"%\"\")*&&F)6#F0F,,&F,F,&%\"cG6#F0!\"\"F," }{TEXT -1 6 ", " }{XPPEDIT 18 0 "Sum(b[i]*(c[i]-1)*a[i,3],i=4..7)=0" "6#/ -%$SumG6$*(&%\"bG6#%\"iG\"\"\",&&%\"cG6#F+F,F,!\"\"F,&%\"aG6$F+\"\"$F, /F+;\"\"%\"\"(\"\"!" }{TEXT -1 2 ", " }}{PARA 0 "" 0 "" {TEXT -1 9 "to find " }{XPPEDIT 18 0 "a[6,3]" "6#&%\"aG6$\"\"'\"\"$" }{TEXT -1 3 ", " }{XPPEDIT 18 0 "a[7,3]" "6#&%\"aG6$\"\"(\"\"$" }{TEXT -1 7 " and \+ " }{XPPEDIT 18 0 "a[8,3]" "6#&%\"aG6$\"\")\"\"$" }{TEXT -1 1 "." }} {PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 136 "[add(`b*`[i]*a[i,3],i=4..8)=0,add(b[i]*a[i,3],i=4..8)=b[3]*(1-c[3 ]),add(b[i]*(c[i]-1)*a[i,3],i=4..7)=0]:\neqns2 := simplify(subs(e15,%) ):" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 64 "nops(eqns2);\nindets(eqns2) minus \{c[4],c[5],c[6],c[ 7]\};\nnops(%);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#\"\"$" }}{PARA 11 " " 1 "" {XPPMATH 20 "6#<%&%\"aG6$\"\"'\"\"$&F%6$\"\"(F(&F%6$\"\")F(" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#\"\"$" }}}{PARA 0 "" 0 "" {TEXT -1 0 " " }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 93 "e16 := solve(\{op(eqns2) \},indets(eqns2) minus \{c[4],c[5],c[6],c[7]\}):\ne17 := `union`(e15,e 16):" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 8 "E xample:" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 24 "a[6,3]=subs(e17,a [6,3]);" }}{PARA 12 "" 1 "" {XPPMATH 20 "6#/&%\"aG6$\"\"'\"\"$,$*&#F( \"\"%\"\"\"**&%\"cG6#F,!\"#&F06#F'F-,R**\"#gF-)F/F(F-&F06#\"\"&F-)F3F( F-F-**\"#!*F-F9F-FF-FHF-F9F-F8F-F-**FPF-FHF-F?F-F9F-FA**F,F-FH F-F/F-F9F-F-*(F,F-FHF-F?F-FA**\"#IF-F3F-FIF-F8F-F-**F>F-FLF-F3F-FTF-F- **\"$!=F-FLF-F3F-F8F-FA**\"#aF-FLF-F3F-F?F-F-**\"#7F-FLF-F3F-F/F-FA** \"#=F-F3F-F9F-F8F-F-**F'F-F9F-F3F-F?F-F-*(F(F-FLF-F?F-F-*(F(F-F9F-F8F- FAF-,2*(\"#5F-FIF-F?F-F-*(FcoF-FLF-F8F-F-*(FZF-FLF-F?F-FA*(F'F-FLF-F/F -F-*$FLF-FA*(F'F-F9F-F?F-F-*&F9F-F/F-F-*$F?F-FAFAF-F-" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}} {SECT 1 {PARA 4 "" 0 "" {TEXT -1 3 "e17" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 19527 "e17 := \{c[8] = 1, b [3] = 0, b[2] = 0, c[9] = 1, b[5] = -1/60*(10*c[6]*c[7]*c[4]-5*c[6]*c[ 4]-5*c[7]*c[6]+3*c[6]-5*c[7]*c[4]+3*c[4]-2+3*c[7])/(-c[4]+c[5])/c[5]/( -c[5]^3+c[5]^2-c[7]*c[5]+c[7]*c[5]^2-c[6]*c[5]+c[6]*c[5]^2+c[7]*c[6]-c [6]*c[7]*c[5]), a[5,2] = 0, a[4,2] = 0, `b*`[3] = 0, `b*`[2] = 0, a[8, 2] = 0, a[7,2] = 0, a[6,2] = 0, b[1] = 1/60*(30*c[5]*c[6]*c[7]*c[4]-10 *c[5]*c[6]*c[4]-10*c[6]*c[7]*c[5]+5*c[6]*c[5]-10*c[6]*c[7]*c[4]+5*c[6] *c[4]+5*c[7]*c[6]-3*c[6]-10*c[5]*c[7]*c[4]+5*c[5]*c[4]+5*c[7]*c[5]-3*c [5]+5*c[7]*c[4]-3*c[4]+2-3*c[7])/c[5]/c[6]/c[7]/c[4], b[4] = 1/60*(-2+ 3*c[5]-5*c[7]*c[6]+3*c[7]+3*c[6]+10*c[6]*c[7]*c[5]-5*c[6]*c[5]-5*c[7]* c[5])/c[4]/(-c[6]*c[4]^3-c[4]^3+c[4]^4+c[7]*c[4]^2+c[6]*c[4]^2-c[6]*c[ 7]*c[4]+c[6]*c[7]*c[4]^2-c[5]*c[4]^3+c[5]*c[4]^2-c[7]*c[4]^3-c[5]*c[7] *c[4]+c[5]*c[7]*c[4]^2-c[5]*c[6]*c[4]+c[5]*c[6]*c[4]^2+c[6]*c[7]*c[5]- c[5]*c[6]*c[7]*c[4]), b[6] = 1/60*(10*c[5]*c[7]*c[4]-5*c[5]*c[4]-5*c[7 ]*c[5]+3*c[5]-5*c[7]*c[4]+3*c[4]-2+3*c[7])/(-c[4]+c[6])/(-c[5]+c[6])/c [6]/(c[6]^2-c[6]+c[7]-c[7]*c[6]), `b*`[9] = 1/10*(50*c[5]^2*c[6]*c[4]^ 2-40*c[5]^2*c[6]*c[4]+10*c[6]*c[5]^2-40*c[5]*c[6]*c[4]^2+35*c[5]*c[6]* c[4]-10*c[6]*c[5]+10*c[6]*c[4]^2-10*c[6]*c[4]+3*c[6]-5*c[5]^2*c[4]-5*c [5]*c[4]^2+10*c[5]*c[4]-c[5]-c[4])/(3*c[6]-28*c[5]*c[6]*c[4]^2-28*c[5] ^2*c[6]*c[4]+8*c[6]*c[5]^2-9*c[6]*c[4]-9*c[6]*c[5]+28*c[5]*c[6]*c[4]+3 0*c[5]^2*c[6]*c[4]^2+8*c[6]*c[4]^2-4*c[5]^2*c[4]-4*c[5]*c[4]^2+9*c[5]* c[4]-c[5]-c[4]), `b*`[6] = -1/60*(-30*c[5]^3*c[4]+103*c[5]*c[7]*c[4]-1 8*c[6]+6*c[5]+6*c[4]+27*c[7]*c[6]-9*c[5]^2-9*c[4]^2-100*c[7]*c[6]*c[5] ^3+522*c[5]*c[6]*c[7]*c[4]-111*c[6]*c[5]^2+75*c[6]*c[5]+60*c[6]*c[5]^3 -9*c[7]*c[5]+600*c[6]*c[7]*c[5]^3*c[4]^3+14*c[7]*c[5]^2-100*c[7]*c[6]* c[4]^3+14*c[7]*c[4]^2-9*c[7]*c[4]-117*c[6]*c[7]*c[5]-111*c[6]*c[4]^2+6 0*c[6]*c[4]^3-117*c[6]*c[7]*c[4]+180*c[6]*c[7]*c[4]^2-30*c[5]*c[4]^3-1 74*c[5]*c[7]*c[4]^2+486*c[5]*c[6]*c[4]^2+107*c[5]*c[4]^2-318*c[5]*c[6] *c[4]-66*c[5]*c[4]+75*c[6]*c[4]+500*c[5]*c[7]*c[6]*c[4]^3+1420*c[5]^2* c[6]*c[7]*c[4]^2-840*c[6]*c[5]*c[7]*c[4]^2+180*c[6]*c[7]*c[5]^2-900*c[ 6]*c[5]^2*c[7]*c[4]^3-900*c[6]*c[5]^3*c[7]*c[4]^2+50*c[7]*c[5]*c[4]^3- 280*c[6]*c[5]*c[4]^3-280*c[5]^3*c[6]*c[4]+50*c[5]^3*c[7]*c[4]+107*c[5] ^2*c[4]+500*c[5]^3*c[6]*c[7]*c[4]+50*c[5]^3*c[4]^2+470*c[5]^2*c[6]*c[4 ]^3+310*c[5]^2*c[7]*c[4]^2-770*c[5]^2*c[6]*c[4]^2-90*c[5]^2*c[7]*c[4]^ 3-90*c[5]^3*c[7]*c[4]^2+470*c[5]^3*c[6]*c[4]^2-300*c[6]*c[5]^3*c[4]^3- 180*c[5]^2*c[4]^2+486*c[5]^2*c[6]*c[4]-174*c[5]^2*c[7]*c[4]-840*c[5]^2 *c[6]*c[7]*c[4]+50*c[5]^2*c[4]^3)/(-9*c[5]*c[7]*c[4]-3*c[7]*c[6]+3*c[6 ]^2-28*c[5]*c[6]*c[7]*c[4]-c[6]*c[5]+c[7]*c[5]+8*c[6]^2*c[4]^2+c[7]*c[ 4]+9*c[6]*c[7]*c[5]+9*c[6]*c[7]*c[4]-8*c[6]*c[7]*c[4]^2+4*c[5]*c[7]*c[ 4]^2-4*c[5]*c[6]*c[4]^2+9*c[5]*c[6]*c[4]-c[6]*c[4]-28*c[6]^2*c[4]^2*c[ 5]+28*c[6]^2*c[4]*c[5]-30*c[5]^2*c[6]*c[7]*c[4]^2+28*c[6]*c[5]*c[7]*c[ 4]^2-8*c[6]*c[7]*c[5]^2+8*c[6]^2*c[5]^2-9*c[6]^2*c[5]+30*c[6]^2*c[5]^2 *c[4]^2-28*c[6]^2*c[4]*c[5]^2-9*c[6]^2*c[4]-4*c[5]^2*c[6]*c[4]+4*c[5]^ 2*c[7]*c[4]+28*c[5]^2*c[6]*c[7]*c[4])/(c[5]*c[4]-c[6]*c[4]-c[6]*c[5]+c [6]^2)/c[6], a[8,7] = (10*c[5]*c[6]*c[4]-5*c[6]*c[5]-5*c[6]*c[4]+3*c[6 ]-5*c[5]*c[4]+3*c[5]+3*c[4]-2)/c[7]/(c[6]-c[7])/(c[5]-c[7])/(c[4]-c[7] )/(-20*c[5]*c[6]*c[4]+30*c[5]*c[6]*c[7]*c[4]+15*c[6]*c[5]-20*c[6]*c[7] *c[5]+15*c[6]*c[4]-20*c[6]*c[7]*c[4]-12*c[6]+15*c[7]*c[6]+15*c[5]*c[4] -20*c[5]*c[7]*c[4]-12*c[5]+15*c[7]*c[5]-12*c[4]+15*c[7]*c[4]+10-12*c[7 ])*(c[4]-1)*(-1+c[5])*(-1+c[6])*(-1+c[7]), a[6,3] = 3/4*1/c[4]^2*c[6]* (60*c[4]^3*c[5]*c[6]^3-90*c[5]*c[6]^3*c[4]^2+36*c[6]^3*c[4]*c[5]-6*c[6 ]^3*c[5]+3*c[6]^3*c[4]-20*c[6]^2*c[5]^3*c[4]^2-80*c[6]^2*c[5]^2*c[4]^3 +150*c[6]^2*c[5]^2*c[4]^2-48*c[6]^2*c[4]*c[5]^2+8*c[6]^2*c[5]^2-60*c[5 ]*c[6]^2*c[4]^4+90*c[6]^2*c[5]*c[4]^3-48*c[6]^2*c[4]^2*c[5]+4*c[6]^2*c [4]*c[5]-4*c[6]^2*c[4]^2+30*c[6]*c[5]^3*c[4]^3+90*c[5]^2*c[6]*c[4]^4-1 80*c[5]^2*c[6]*c[4]^3+54*c[5]^2*c[6]*c[4]^2-12*c[5]^2*c[6]*c[4]+18*c[6 ]*c[5]*c[4]^3+6*c[5]*c[6]*c[4]^2+3*c[5]^2*c[4]^2-3*c[5]*c[4]^3)/(10*c[ 5]^3*c[4]^2+10*c[5]^2*c[4]^3-30*c[5]^2*c[4]^2+6*c[5]^2*c[4]-c[5]^2+6*c [5]*c[4]^2+c[5]*c[4]-c[4]^2), b[7] = -1/60*(10*c[5]*c[6]*c[4]-5*c[6]*c [5]-5*c[6]*c[4]+3*c[6]-5*c[5]*c[4]+3*c[5]+3*c[4]-2)/(c[4]-c[7])/(c[5]- c[7])/(c[6]-c[7])/c[7]/(-1+c[7]), `b*`[8] = 0, a[5,3] = -3/4*c[5]^2*(2 *c[5]-3*c[4])/c[4]^2, `b*`[4] = -1/60*(-30*c[5]^3*c[4]-42*c[7]*c[6]^2+ 89*c[5]*c[7]*c[4]-18*c[6]+6*c[5]+6*c[4]+27*c[7]*c[6]-9*c[5]^2+180*c[7] *c[6]^2*c[5]^3-100*c[7]*c[6]*c[5]^3+27*c[6]^2+186*c[5]*c[6]*c[7]*c[4]- 97*c[6]*c[5]^2+66*c[6]*c[5]+60*c[6]*c[5]^3-100*c[6]^2*c[5]^3-9*c[7]*c[ 5]+14*c[7]*c[5]^2-100*c[6]^2*c[7]*c[4]^2+120*c[6]^2*c[7]*c[4]+60*c[6]^ 2*c[4]^2-9*c[7]*c[4]-103*c[6]*c[7]*c[5]-36*c[6]*c[4]^2-61*c[6]*c[7]*c[ 4]+60*c[6]*c[7]*c[4]^2-30*c[5]*c[7]*c[4]^2+126*c[5]*c[6]*c[4]^2+18*c[5 ]*c[4]^2-112*c[5]*c[6]*c[4]-57*c[5]*c[4]+39*c[6]*c[4]-280*c[6]^2*c[4]^ 2*c[5]+330*c[6]^2*c[4]*c[5]+380*c[5]^2*c[6]*c[7]*c[4]^2-560*c[6]^2*c[7 ]*c[4]*c[5]+192*c[6]^2*c[7]*c[5]-310*c[6]^2*c[7]*c[5]^2-230*c[6]*c[5]* c[7]*c[4]^2+156*c[6]*c[7]*c[5]^2+180*c[6]^2*c[5]^2+500*c[6]^2*c[4]^2*c [7]*c[5]+950*c[6]^2*c[5]^2*c[7]*c[4]-900*c[6]^2*c[5]^2*c[7]*c[4]^2-300 *c[6]*c[5]^3*c[7]*c[4]^2-600*c[6]^2*c[5]^3*c[7]*c[4]-117*c[6]^2*c[5]-1 30*c[5]^3*c[6]*c[4]+50*c[5]^3*c[7]*c[4]+93*c[5]^2*c[4]+230*c[5]^3*c[6] *c[7]*c[4]+50*c[5]^2*c[7]*c[4]^2-190*c[5]^2*c[6]*c[4]^2+150*c[5]^3*c[6 ]*c[4]^2+470*c[6]^2*c[5]^2*c[4]^2+320*c[6]^2*c[5]^3*c[4]-300*c[6]^2*c[ 5]^3*c[4]^2-30*c[5]^2*c[4]^2-530*c[6]^2*c[4]*c[5]^2-75*c[6]^2*c[4]+156 *c[5]^2*c[6]*c[4]-150*c[5]^2*c[7]*c[4]-270*c[5]^2*c[6]*c[7]*c[4]+600*c [6]^2*c[7]*c[5]^3*c[4]^2)/(-9*c[5]*c[7]*c[4]-3*c[7]*c[6]-c[4]^2-28*c[5 ]*c[6]*c[7]*c[4]+c[7]*c[5]+c[7]*c[4]+9*c[6]*c[7]*c[5]-9*c[6]*c[4]^2+8* c[6]*c[4]^3+9*c[6]*c[7]*c[4]-8*c[6]*c[7]*c[4]^2-4*c[5]*c[4]^3+4*c[5]*c [7]*c[4]^2+28*c[5]*c[6]*c[4]^2+9*c[5]*c[4]^2-9*c[5]*c[6]*c[4]-c[5]*c[4 ]+3*c[6]*c[4]-30*c[5]^2*c[6]*c[7]*c[4]^2+28*c[6]*c[5]*c[7]*c[4]^2-8*c[ 6]*c[7]*c[5]^2-28*c[6]*c[5]*c[4]^3+30*c[5]^2*c[6]*c[4]^3-28*c[5]^2*c[6 ]*c[4]^2-4*c[5]^2*c[4]^2+8*c[5]^2*c[6]*c[4]+4*c[5]^2*c[7]*c[4]+28*c[5] ^2*c[6]*c[7]*c[4])/(-c[4]+c[6])/(-c[4]+c[5])/c[4], a[8,6] = 1/2*(5*c[5 ]*c[6]*c[4]+c[6]-2*c[6]*c[4]-2*c[6]*c[5]+25*c[5]*c[7]*c[4]+6*c[5]+9*c[ 7]+10*c[7]^2*c[4]-4+6*c[4]-6*c[7]^2-14*c[7]*c[4]-10*c[5]*c[4]-14*c[7]* c[5]-20*c[7]^2*c[5]*c[4]+10*c[7]^2*c[5])/(c[6]-c[7])/(-c[4]+c[6])/(-c[ 5]+c[6])/c[6]/(-20*c[5]*c[6]*c[4]+30*c[5]*c[6]*c[7]*c[4]+15*c[6]*c[5]- 20*c[6]*c[7]*c[5]+15*c[6]*c[4]-20*c[6]*c[7]*c[4]-12*c[6]+15*c[7]*c[6]+ 15*c[5]*c[4]-20*c[5]*c[7]*c[4]-12*c[5]+15*c[7]*c[5]-12*c[4]+15*c[7]*c[ 4]+10-12*c[7])*(c[4]-1)*(-1+c[5])*(-1+c[6]), b[8] = 1/60*(-20*c[5]*c[6 ]*c[4]+30*c[5]*c[6]*c[7]*c[4]+15*c[6]*c[5]-20*c[6]*c[7]*c[5]+15*c[6]*c [4]-20*c[6]*c[7]*c[4]-12*c[6]+15*c[7]*c[6]+15*c[5]*c[4]-20*c[5]*c[7]*c [4]-12*c[5]+15*c[7]*c[5]-12*c[4]+15*c[7]*c[4]+10-12*c[7])/(c[4]-1)/(-1 +c[5])/(-1+c[6])/(-1+c[7]), `b*`[5] = 1/60*(-42*c[7]*c[6]^2+89*c[5]*c[ 7]*c[4]-18*c[6]+6*c[5]+6*c[4]+27*c[7]*c[6]-9*c[4]^2+27*c[6]^2+186*c[5] *c[6]*c[7]*c[4]-36*c[6]*c[5]^2+39*c[6]*c[5]-9*c[7]*c[5]-100*c[6]^2*c[4 ]^3+600*c[6]^2*c[7]*c[5]^2*c[4]^3-100*c[7]*c[6]*c[4]^3-310*c[6]^2*c[7] *c[4]^2+192*c[6]^2*c[7]*c[4]+180*c[6]^2*c[4]^2+14*c[7]*c[4]^2-9*c[7]*c [4]-61*c[6]*c[7]*c[5]-97*c[6]*c[4]^2+60*c[6]*c[4]^3-103*c[6]*c[7]*c[4] +156*c[6]*c[7]*c[4]^2-30*c[5]*c[4]^3-150*c[5]*c[7]*c[4]^2+156*c[5]*c[6 ]*c[4]^2+93*c[5]*c[4]^2-112*c[5]*c[6]*c[4]-57*c[5]*c[4]+66*c[6]*c[4]+2 30*c[5]*c[7]*c[6]*c[4]^3-530*c[6]^2*c[4]^2*c[5]+330*c[6]^2*c[4]*c[5]+3 80*c[5]^2*c[6]*c[7]*c[4]^2+320*c[6]^2*c[5]*c[4]^3-560*c[6]^2*c[7]*c[4] *c[5]-600*c[7]*c[6]^2*c[5]*c[4]^3+120*c[6]^2*c[7]*c[5]-100*c[6]^2*c[7] *c[5]^2-270*c[6]*c[5]*c[7]*c[4]^2+60*c[6]*c[7]*c[5]^2+60*c[6]^2*c[5]^2 +950*c[6]^2*c[4]^2*c[7]*c[5]+500*c[6]^2*c[5]^2*c[7]*c[4]-900*c[6]^2*c[ 5]^2*c[7]*c[4]^2-300*c[6]*c[5]^2*c[7]*c[4]^3-75*c[6]^2*c[5]+50*c[7]*c[ 5]*c[4]^3-130*c[6]*c[5]*c[4]^3+18*c[5]^2*c[4]+150*c[5]^2*c[6]*c[4]^3+5 0*c[5]^2*c[7]*c[4]^2-190*c[5]^2*c[6]*c[4]^2-300*c[6]^2*c[5]^2*c[4]^3+4 70*c[6]^2*c[5]^2*c[4]^2-30*c[5]^2*c[4]^2-280*c[6]^2*c[4]*c[5]^2+180*c[ 7]*c[6]^2*c[4]^3-117*c[6]^2*c[4]+126*c[5]^2*c[6]*c[4]-30*c[5]^2*c[7]*c [4]-230*c[5]^2*c[6]*c[7]*c[4])/(-4*c[5]^3*c[4]-9*c[5]*c[7]*c[4]-3*c[7] *c[6]-c[5]^2-28*c[5]*c[6]*c[7]*c[4]-9*c[6]*c[5]^2+3*c[6]*c[5]+8*c[6]*c [5]^3+c[7]*c[5]+c[7]*c[4]+9*c[6]*c[7]*c[5]+9*c[6]*c[7]*c[4]-8*c[6]*c[7 ]*c[4]^2+4*c[5]*c[7]*c[4]^2+8*c[5]*c[6]*c[4]^2-9*c[5]*c[6]*c[4]-c[5]*c [4]-30*c[5]^2*c[6]*c[7]*c[4]^2+28*c[6]*c[5]*c[7]*c[4]^2-8*c[6]*c[7]*c[ 5]^2-28*c[5]^3*c[6]*c[4]+9*c[5]^2*c[4]-28*c[5]^2*c[6]*c[4]^2+30*c[5]^3 *c[6]*c[4]^2-4*c[5]^2*c[4]^2+28*c[5]^2*c[6]*c[4]+4*c[5]^2*c[7]*c[4]+28 *c[5]^2*c[6]*c[7]*c[4])/(-c[5]+c[6])/(-c[4]+c[5])/c[5], a[5,4] = c[5]^ 2*(-c[4]+c[5])/c[4]^2, a[7,6] = 1/2*(5*c[5]*c[4]+1-2*c[4]-2*c[5])*(c[4 ]-c[7])*(c[5]-c[7])*(c[6]-c[7])*c[7]/(10*c[5]*c[6]*c[4]-5*c[6]*c[5]-5* c[6]*c[4]+3*c[6]-5*c[5]*c[4]+3*c[5]+3*c[4]-2)/(-c[4]+c[6])/(-c[5]+c[6] )/c[6], c[3] = 2/3*c[4], a[4,3] = 3/4*c[4], a[3,2] = 2/9*c[4]^2/c[2], \+ `b*`[7] = 1/60*(-30*c[5]^3*c[4]-18*c[6]+6*c[5]+6*c[4]-9*c[5]^2-9*c[4]^ 2+27*c[6]^2-97*c[6]*c[5]^2+66*c[6]*c[5]+60*c[6]*c[5]^3-100*c[6]^2*c[5] ^3-100*c[6]^2*c[4]^3+180*c[6]^2*c[4]^2-97*c[6]*c[4]^2+60*c[6]*c[4]^3-3 0*c[5]*c[4]^3+312*c[5]*c[6]*c[4]^2+107*c[5]*c[4]^2-215*c[5]*c[6]*c[4]- 66*c[5]*c[4]+66*c[6]*c[4]-840*c[6]^2*c[4]^2*c[5]+522*c[6]^2*c[4]*c[5]+ 500*c[6]^2*c[5]*c[4]^3+180*c[6]^2*c[5]^2-117*c[6]^2*c[5]-230*c[6]*c[5] *c[4]^3-230*c[5]^3*c[6]*c[4]+107*c[5]^2*c[4]+50*c[5]^3*c[4]^2+380*c[5] ^2*c[6]*c[4]^3-460*c[5]^2*c[6]*c[4]^2+380*c[5]^3*c[6]*c[4]^2+600*c[5]^ 3*c[6]^2*c[4]^3-900*c[6]^2*c[5]^2*c[4]^3+1420*c[6]^2*c[5]^2*c[4]^2-300 *c[6]*c[5]^3*c[4]^3+500*c[6]^2*c[5]^3*c[4]-900*c[6]^2*c[5]^3*c[4]^2-18 0*c[5]^2*c[4]^2-840*c[6]^2*c[4]*c[5]^2-117*c[6]^2*c[4]+312*c[5]^2*c[6] *c[4]+50*c[5]^2*c[4]^3)/c[7]/(-3*c[7]^3*c[6]+3*c[7]^2*c[6]^2-9*c[6]^2* c[7]^2*c[5]-8*c[7]*c[6]^2*c[5]^3+8*c[7]^2*c[6]*c[5]^3-c[5]*c[6]*c[7]*c [4]-c[7]^2*c[4]^2-30*c[6]^2*c[7]*c[5]^2*c[4]^3-28*c[6]^2*c[7]^2*c[5]^2 *c[4]+30*c[6]^2*c[7]^2*c[5]^2*c[4]^2-30*c[6]*c[7]*c[5]^3*c[4]^3+9*c[6] ^2*c[7]*c[4]^2-3*c[6]^2*c[7]*c[4]+2*c[6]*c[7]^2*c[4]-9*c[6]*c[7]^2*c[4 ]^2-2*c[7]^2*c[5]*c[4]-c[7]^2*c[5]^2+c[6]*c[7]*c[4]^2+8*c[7]^2*c[6]*c[ 4]^3-4*c[5]*c[7]^2*c[4]^3+c[5]*c[7]*c[4]^2-c[5]*c[6]*c[4]^2+32*c[6]*c[ 5]*c[7]^2*c[4]^2-9*c[7]^2*c[5]*c[6]*c[4]-4*c[5]*c[7]*c[6]*c[4]^3-9*c[6 ]^2*c[4]^2*c[5]+3*c[6]^2*c[4]*c[5]+c[7]^3*c[4]+9*c[7]^3*c[4]*c[6]-9*c[ 7]^3*c[4]*c[5]-20*c[5]^2*c[6]*c[7]*c[4]^2+8*c[6]^2*c[5]*c[4]^3+28*c[7] ^3*c[4]^2*c[6]*c[5]+18*c[6]^2*c[7]*c[4]*c[5]+28*c[7]*c[6]^2*c[5]*c[4]^ 3-3*c[6]^2*c[7]*c[5]+9*c[6]^2*c[7]*c[5]^2+c[6]*c[7]*c[5]^2-8*c[6]*c[7] ^3*c[5]^2-9*c[6]*c[7]^2*c[5]^2+8*c[6]^2*c[7]^2*c[5]^2+2*c[6]*c[7]^2*c[ 5]-36*c[6]^2*c[4]^2*c[7]*c[5]-28*c[6]^2*c[7]^2*c[4]^2*c[5]-28*c[7]^3*c [4]*c[6]*c[5]-36*c[6]^2*c[5]^2*c[7]*c[4]+56*c[6]^2*c[5]^2*c[7]*c[4]^2+ 28*c[6]*c[5]^2*c[7]*c[4]^3+28*c[6]*c[5]^3*c[7]*c[4]^2+28*c[6]^2*c[5]^3 *c[7]*c[4]+9*c[6]*c[7]^3*c[5]+9*c[7]^2*c[5]*c[4]^2+28*c[7]^3*c[5]^2*c[ 6]*c[4]+32*c[7]^2*c[5]^2*c[6]*c[4]+8*c[6]^2*c[7]^2*c[4]^2-9*c[6]^2*c[7 ]^2*c[4]-4*c[5]^3*c[6]*c[7]*c[4]+28*c[6]^2*c[7]^2*c[4]*c[5]-4*c[5]^2*c [6]*c[4]^3-9*c[5]^2*c[7]*c[4]^2+9*c[5]^2*c[6]*c[4]^2+4*c[5]^2*c[7]*c[4 ]^3+4*c[5]^3*c[7]*c[4]^2-4*c[5]^3*c[6]*c[4]^2+30*c[5]^3*c[6]^2*c[4]^3- 28*c[6]^2*c[5]^2*c[4]^3+28*c[6]^2*c[5]^2*c[4]^2+8*c[6]^2*c[5]^3*c[4]-2 8*c[6]^2*c[5]^3*c[4]^2-9*c[6]^2*c[4]*c[5]^2-8*c[7]*c[6]^2*c[4]^3+30*c[ 7]^2*c[5]^2*c[6]*c[4]^3-30*c[7]^3*c[5]^2*c[6]*c[4]^2+30*c[7]^2*c[5]^3* c[6]*c[4]^2-8*c[7]^3*c[4]^2*c[6]+4*c[7]^3*c[4]^2*c[5]+c[7]^3*c[5]-28*c [7]^2*c[6]*c[5]*c[4]^3-56*c[7]^2*c[5]^2*c[6]*c[4]^2-28*c[7]^2*c[5]^3*c [6]*c[4]-c[5]^2*c[6]*c[4]+c[5]^2*c[7]*c[4]+4*c[7]^3*c[5]^2*c[4]+9*c[7] ^2*c[5]^2*c[4]-30*c[6]^2*c[7]*c[5]^3*c[4]^2-8*c[7]^2*c[5]^2*c[4]^2-4*c [7]^2*c[5]^3*c[4]), a[7,3] = 3/4*1/c[4]^2*c[7]*(-480*c[5]^2*c[6]*c[4]^ 4+15*c[6]*c[5]*c[4]^4+300*c[4]^2*c[5]^2*c[7]^3+60*c[7]^3*c[5]*c[4]^3-5 10*c[7]^3*c[5]^2*c[4]^3+54*c[7]*c[5]*c[4]^4+450*c[5]^2*c[4]^4*c[7]^2+1 50*c[4]^3*c[7]^3*c[5]^3-150*c[7]^2*c[5]^2*c[4]^5-450*c[4]^5*c[6]*c[5]^ 3-3*c[7]^3*c[4]^2+90*c[5]^2*c[4]^5*c[7]+120*c[5]^3*c[4]^4*c[7]+150*c[7 ]^3*c[5]^2*c[4]^4+100*c[5]^4*c[4]^2*c[7]^2*c[6]-180*c[6]*c[4]^5*c[5]*c [7]+30*c[6]*c[5]^2*c[4]^4*c[7]+60*c[5]^4*c[4]^3*c[6]-150*c[5]^4*c[4]^4 *c[6]-200*c[7]^2*c[4]^4*c[5]^3+300*c[6]*c[4]^5*c[5]*c[7]^2-50*c[7]^2*c [5]^4*c[4]^3+24*c[7]*c[6]*c[5]^3-40*c[7]^2*c[6]*c[5]^3-6*c[5]*c[6]*c[7 ]*c[4]+360*c[7]*c[4]^4*c[6]*c[5]+150*c[7]^2*c[5]^3*c[4]^3+2*c[7]^2*c[4 ]^2+6*c[5]^2*c[7]^3+330*c[6]*c[7]*c[5]^3*c[4]^3-450*c[7]^2*c[6]*c[5]*c [4]^4-12*c[7]*c[6]*c[4]^3-6*c[6]*c[7]^2*c[4]-2*c[7]^2*c[5]*c[4]-4*c[7] ^2*c[5]^2+6*c[6]*c[7]*c[4]^2+20*c[7]^2*c[6]*c[4]^3-10*c[5]*c[7]^2*c[4] ^3+6*c[5]*c[4]^3-6*c[5]*c[7]*c[4]^2-6*c[5]*c[6]*c[4]^2+150*c[6]*c[5]*c [7]^2*c[4]^2-72*c[7]^2*c[5]*c[6]*c[4]-270*c[5]*c[7]*c[6]*c[4]^3+9*c[7] ^3*c[4]*c[6]+3*c[7]^3*c[4]*c[5]-306*c[5]^2*c[6]*c[7]*c[4]^2+900*c[4]^5 *c[6]*c[7]*c[5]^3+600*c[5]^2*c[6]*c[7]^3*c[4]^4-420*c[7]^3*c[4]^2*c[6] *c[5]+90*c[6]*c[5]*c[7]*c[4]^2-12*c[6]*c[7]*c[5]^2+30*c[6]*c[7]^3*c[5] ^2+12*c[6]*c[7]^2*c[5]+123*c[7]^3*c[4]*c[6]*c[5]+360*c[6]*c[5]^2*c[7]* c[4]^3+360*c[6]*c[5]^3*c[7]*c[4]^2-18*c[6]*c[7]^3*c[5]+30*c[5]^4*c[4]^ 3*c[7]-80*c[7]^2*c[5]^2*c[4]^3+24*c[7]^2*c[5]*c[4]^2-30*c[7]*c[5]*c[4] ^3+3*c[6]*c[5]*c[4]^3-240*c[7]^3*c[5]^2*c[6]*c[4]-24*c[5]^3*c[6]*c[4]- 12*c[5]^3*c[7]*c[4]-144*c[5]^3*c[6]*c[7]*c[4]+9*c[5]^3*c[4]^2+294*c[5] ^2*c[6]*c[4]^3-42*c[5]^2*c[7]*c[4]^2-75*c[5]^2*c[6]*c[4]^2+252*c[5]^2* c[7]*c[4]^3+18*c[5]^3*c[7]*c[4]^2+189*c[5]^3*c[6]*c[4]^2-360*c[5]^2*c[ 4]^4*c[7]-720*c[6]*c[5]^3*c[4]^3-120*c[5]^3*c[7]*c[4]^3-60*c[5]^4*c[7] *c[4]^2*c[6]-6*c[5]^2*c[4]^2-780*c[7]^2*c[5]^2*c[6]*c[4]^3+810*c[7]^3* c[5]^2*c[6]*c[4]^2-1230*c[7]^2*c[5]^3*c[6]*c[4]^2-15*c[7]^3*c[4]^2*c[6 ]-21*c[7]^3*c[4]^2*c[5]+20*c[7]^2*c[6]*c[5]*c[4]^3+220*c[7]^2*c[5]^2*c [6]*c[4]^2+15*c[4]^4*c[5]^2+320*c[7]^2*c[5]^3*c[6]*c[4]+1850*c[7]^2*c[ 5]^3*c[6]*c[4]^3-60*c[5]^3*c[7]^3*c[4]^2+12*c[5]^2*c[6]*c[4]+12*c[5]^2 *c[7]*c[4]-66*c[7]^3*c[5]^2*c[4]+24*c[7]^2*c[5]^2*c[4]+72*c[5]^2*c[6]* c[7]*c[4]-110*c[7]^2*c[5]^2*c[4]^2+20*c[7]^2*c[5]^3*c[4]-9*c[4]^4*c[5] -60*c[7]^2*c[4]^4*c[5]-600*c[5]^2*c[6]*c[4]^5*c[7]^2-1200*c[7]^3*c[5]^ 2*c[4]^3*c[6]+630*c[7]^3*c[6]*c[5]*c[4]^3+300*c[5]^4*c[4]^4*c[6]*c[7]- 1800*c[5]^3*c[4]^4*c[6]*c[7]-15*c[5]^3*c[4]^3-300*c[7]^3*c[4]^4*c[5]*c [6]-20*c[7]^2*c[5]^3*c[4]^2-800*c[5]^3*c[4]^4*c[7]^2*c[6]-200*c[5]^4*c [4]^3*c[7]^2*c[6]+1150*c[5]^2*c[4]^4*c[7]^2*c[6]+180*c[5]^2*c[6]*c[4]^ 5+1140*c[4]^4*c[6]*c[5]^3)/(10*c[5]^3*c[4]^2+10*c[5]^2*c[4]^3-30*c[5]^ 2*c[4]^2+6*c[5]^2*c[4]-c[5]^2+6*c[5]*c[4]^2+c[5]*c[4]-c[4]^2)/(10*c[5] *c[6]*c[4]-5*c[6]*c[5]-5*c[6]*c[4]+3*c[6]-5*c[5]*c[4]+3*c[5]+3*c[4]-2) , `b*`[1] = 1/60*(30*c[5]^3*c[4]+42*c[7]*c[6]^2-103*c[5]*c[7]*c[4]+18* c[6]-6*c[5]-6*c[4]-27*c[7]*c[6]+9*c[5]^2+9*c[4]^2-180*c[7]*c[6]^2*c[5] ^3+100*c[7]*c[6]*c[5]^3-27*c[6]^2-354*c[5]*c[6]*c[7]*c[4]+97*c[6]*c[5] ^2-66*c[6]*c[5]-60*c[6]*c[5]^3+100*c[6]^2*c[5]^3+9*c[7]*c[5]+100*c[6]^ 2*c[4]^3-2040*c[6]^2*c[7]*c[5]^2*c[4]^3-600*c[6]*c[7]*c[5]^3*c[4]^3-14 *c[7]*c[5]^2+100*c[7]*c[6]*c[4]^3+310*c[6]^2*c[7]*c[4]^2-192*c[6]^2*c[ 7]*c[4]-180*c[6]^2*c[4]^2-14*c[7]*c[4]^2+9*c[7]*c[4]+103*c[6]*c[7]*c[5 ]+97*c[6]*c[4]^2-60*c[6]*c[4]^3+103*c[6]*c[7]*c[4]-156*c[6]*c[7]*c[4]^ 2+30*c[5]*c[4]^3+174*c[5]*c[7]*c[4]^2-312*c[5]*c[6]*c[4]^2-107*c[5]*c[ 4]^2+215*c[5]*c[6]*c[4]+66*c[5]*c[4]-66*c[6]*c[4]-410*c[5]*c[7]*c[6]*c [4]^3+840*c[6]^2*c[4]^2*c[5]-522*c[6]^2*c[4]*c[5]-760*c[5]^2*c[6]*c[7] *c[4]^2-500*c[6]^2*c[5]*c[4]^3+932*c[6]^2*c[7]*c[4]*c[5]+1020*c[7]*c[6 ]^2*c[5]*c[4]^3-192*c[6]^2*c[7]*c[5]+310*c[6]^2*c[7]*c[5]^2+526*c[6]*c [5]*c[7]*c[4]^2-156*c[6]*c[7]*c[5]^2-180*c[6]^2*c[5]^2-1610*c[6]^2*c[4 ]^2*c[7]*c[5]-1610*c[6]^2*c[5]^2*c[7]*c[4]+2970*c[6]^2*c[5]^2*c[7]*c[4 ]^2+690*c[6]*c[5]^2*c[7]*c[4]^3+690*c[6]*c[5]^3*c[7]*c[4]^2+1020*c[6]^ 2*c[5]^3*c[7]*c[4]+117*c[6]^2*c[5]-50*c[7]*c[5]*c[4]^3+230*c[6]*c[5]*c [4]^3+230*c[5]^3*c[6]*c[4]-50*c[5]^3*c[7]*c[4]-107*c[5]^2*c[4]-410*c[5 ]^3*c[6]*c[7]*c[4]-50*c[5]^3*c[4]^2-380*c[5]^2*c[6]*c[4]^3-310*c[5]^2* c[7]*c[4]^2+460*c[5]^2*c[6]*c[4]^2+90*c[5]^2*c[7]*c[4]^3+90*c[5]^3*c[7 ]*c[4]^2-380*c[5]^3*c[6]*c[4]^2-600*c[5]^3*c[6]^2*c[4]^3+900*c[6]^2*c[ 5]^2*c[4]^3-1420*c[6]^2*c[5]^2*c[4]^2+300*c[6]*c[5]^3*c[4]^3-500*c[6]^ 2*c[5]^3*c[4]+900*c[6]^2*c[5]^3*c[4]^2+180*c[5]^2*c[4]^2+840*c[6]^2*c[ 4]*c[5]^2-180*c[7]*c[6]^2*c[4]^3+117*c[6]^2*c[4]+1500*c[5]^3*c[6]^2*c[ 7]*c[4]^3-312*c[5]^2*c[6]*c[4]+174*c[5]^2*c[7]*c[4]+526*c[5]^2*c[6]*c[ 7]*c[4]-2040*c[6]^2*c[7]*c[5]^3*c[4]^2-50*c[5]^2*c[4]^3)/(3*c[6]-28*c[ 5]*c[6]*c[4]^2-28*c[5]^2*c[6]*c[4]+8*c[6]*c[5]^2-9*c[6]*c[4]-9*c[6]*c[ 5]+28*c[5]*c[6]*c[4]+30*c[5]^2*c[6]*c[4]^2+8*c[6]*c[4]^2-4*c[5]^2*c[4] -4*c[5]*c[4]^2+9*c[5]*c[4]-c[5]-c[4])/c[5]/c[6]/c[7]/c[4], a[8,3] = 3/ 4*(-1560*c[5]^2*c[6]*c[4]^4+585*c[6]*c[5]*c[4]^4-45*c[7]*c[5]*c[4]^4-1 2*c[5]^3*c[4]-450*c[4]^5*c[6]*c[5]^3+3*c[5]*c[7]*c[4]-150*c[5]^2*c[4]^ 5*c[7]-200*c[5]^3*c[4]^4*c[7]+300*c[6]*c[4]^5*c[5]*c[7]+2860*c[6]*c[5] ^2*c[4]^4*c[7]+160*c[5]^4*c[4]^3*c[6]-150*c[5]^4*c[4]^4*c[6]-4*c[5]^2+ 2*c[4]^2-40*c[7]*c[6]*c[5]^3+132*c[5]*c[6]*c[7]*c[4]-900*c[6]*c[5]^2*c [4]^5*c[7]-30*c[6]*c[5]^2+12*c[6]*c[5]-960*c[7]*c[4]^4*c[6]*c[5]+24*c[ 6]*c[5]^3+3140*c[6]*c[7]*c[5]^3*c[4]^3+6*c[7]*c[5]^2+20*c[7]*c[6]*c[4] ^3-3*c[7]*c[4]^2-18*c[6]*c[7]*c[5]+15*c[6]*c[4]^2-12*c[6]*c[4]^3+9*c[6 ]*c[7]*c[4]-24*c[6]*c[7]*c[4]^2-54*c[5]*c[4]^3-12*c[5]*c[7]*c[4]^2+348 *c[5]*c[6]*c[4]^2+9*c[5]*c[4]^2-87*c[5]*c[6]*c[4]-2*c[5]*c[4]-6*c[6]*c [4]+1035*c[5]*c[7]*c[6]*c[4]^3+1593*c[5]^2*c[6]*c[7]*c[4]^2+900*c[4]^5 *c[6]*c[7]*c[5]^3-534*c[6]*c[5]*c[7]*c[4]^2+48*c[6]*c[7]*c[5]^2-60*c[6 ]*c[5]^4*c[4]^2-3060*c[6]*c[5]^2*c[7]*c[4]^3-1560*c[6]*c[5]^3*c[7]*c[4 ]^2-50*c[5]^4*c[4]^3*c[7]+80*c[7]*c[5]*c[4]^3-652*c[6]*c[5]*c[4]^3-208 *c[5]^3*c[6]*c[4]+20*c[5]^3*c[7]*c[4]+54*c[5]^2*c[4]+360*c[5]^3*c[6]*c [7]*c[4]+82*c[5]^3*c[4]^2+1734*c[5]^2*c[6]*c[4]^3+388*c[5]^2*c[7]*c[4] ^2-944*c[5]^2*c[6]*c[4]^2-900*c[5]^2*c[7]*c[4]^3-135*c[5]^3*c[7]*c[4]^ 2+879*c[5]^3*c[6]*c[4]^2+660*c[5]^2*c[4]^4*c[7]-1680*c[6]*c[5]^3*c[4]^ 3+420*c[5]^3*c[7]*c[4]^3+100*c[5]^4*c[7]*c[4]^2*c[6]-246*c[5]^2*c[4]^2 -405*c[4]^4*c[5]^2+30*c[5]^4*c[4]^3+240*c[5]^2*c[6]*c[4]-84*c[5]^2*c[7 ]*c[4]-396*c[5]^2*c[6]*c[7]*c[4]+562*c[5]^2*c[4]^3+30*c[4]^4*c[5]+90*c [4]^5*c[5]^2+300*c[5]^4*c[4]^4*c[6]*c[7]-3000*c[5]^3*c[4]^4*c[6]*c[7]- 300*c[5]^4*c[4]^3*c[6]*c[7]-255*c[5]^3*c[4]^3+480*c[5]^2*c[6]*c[4]^5+1 540*c[4]^4*c[6]*c[5]^3-180*c[6]*c[5]*c[4]^5+120*c[5]^3*c[4]^4)/c[4]^2/ (150*c[5]^2*c[6]*c[4]^4-87*c[5]^3*c[4]-12*c[5]*c[7]*c[4]-200*c[5]^3*c[ 4]^4*c[7]-200*c[6]*c[5]^2*c[4]^4*c[7]-200*c[5]^4*c[4]^3*c[6]-10*c[5]^2 -10*c[4]^2+12*c[5]^3+12*c[4]^3+20*c[7]*c[6]*c[5]^3+15*c[5]*c[6]*c[7]*c [4]+12*c[6]*c[5]^2-15*c[6]*c[5]^3-1300*c[6]*c[7]*c[5]^3*c[4]^3+150*c[5 ]^4*c[7]*c[4]^2+12*c[7]*c[5]^2+20*c[7]*c[6]*c[4]^3+12*c[7]*c[4]^2-15*c [7]*c[4]^3+12*c[6]*c[4]^2-15*c[6]*c[4]^3-15*c[6]*c[7]*c[4]^2-87*c[5]*c [4]^3-72*c[5]*c[7]*c[4]^2-72*c[5]*c[6]*c[4]^2+60*c[5]*c[4]^2-12*c[5]*c [6]*c[4]+10*c[5]*c[4]-150*c[5]*c[7]*c[6]*c[4]^3-15*c[7]*c[5]^3-660*c[5 ]^2*c[6]*c[7]*c[4]^2+90*c[6]*c[5]*c[7]*c[4]^2-15*c[6]*c[7]*c[5]^2+150* c[6]*c[5]^4*c[4]^2+930*c[6]*c[5]^2*c[7]*c[4]^3+930*c[6]*c[5]^3*c[7]*c[ 4]^2-200*c[5]^4*c[4]^3*c[7]+110*c[7]*c[5]*c[4]^3+110*c[6]*c[5]*c[4]^3+ 110*c[5]^3*c[6]*c[4]+110*c[5]^3*c[7]*c[4]+60*c[5]^2*c[4]-150*c[5]^3*c[ 6]*c[7]*c[4]+550*c[5]^3*c[4]^2-690*c[5]^2*c[6]*c[4]^3+520*c[5]^2*c[7]* c[4]^2+520*c[5]^2*c[6]*c[4]^2-690*c[5]^2*c[7]*c[4]^3-690*c[5]^3*c[7]*c [4]^2-690*c[5]^3*c[6]*c[4]^2+150*c[5]^2*c[4]^4*c[7]+900*c[6]*c[5]^3*c[ 4]^3+900*c[5]^3*c[7]*c[4]^3-200*c[5]^4*c[7]*c[4]^2*c[6]-429*c[5]^2*c[4 ]^2-120*c[5]^4*c[4]^2-120*c[4]^4*c[5]^2+150*c[5]^4*c[4]^3-72*c[5]^2*c[ 6]*c[4]-72*c[5]^2*c[7]*c[4]+90*c[5]^2*c[6]*c[7]*c[4]+550*c[5]^2*c[4]^3 +300*c[5]^3*c[4]^4*c[6]*c[7]+300*c[5]^4*c[4]^3*c[6]*c[7]-690*c[5]^3*c[ 4]^3-200*c[4]^4*c[6]*c[5]^3+150*c[5]^3*c[4]^4)\}:" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 15 "" 0 "" {TEXT 278 6 "Step 9" } {TEXT -1 1 " " }}{PARA 0 "" 0 "" {TEXT -1 34 "We use the stage-order c onditions:" }}{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#*$&%\"cG6#\"\"'\"\"#" }{TEXT -1 6 ", " }{XPPEDIT 18 0 "Sum(a[ 6,j]*c[j]^2,j=2..5)=1/3" "6#/-%$SumG6$*&&%\"aG6$\"\"'%\"jG\"\"\"*$&%\" cG6#F,\"\"#F-/F,;F2\"\"&*&F-F-\"\"$!\"\"" }{TEXT -1 1 " " }{XPPEDIT 18 0 "c[6]^3" "6#*$&%\"cG6#\"\"'\"\"$" }{TEXT -1 2 ", " }}{PARA 0 "" 0 "" {TEXT -1 9 "to find " }{XPPEDIT 18 0 "a[6,4]" "6#&%\"aG6$\"\"'\" \"%" }{TEXT -1 7 " and " }{XPPEDIT 18 0 "a[6,5]" "6#&%\"aG6$\"\"'\" \"&" }{TEXT -1 1 "." }}{PARA 0 "" 0 "" {TEXT -1 1 " " }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 128 "[seq(add(a[6,j]*c[j]^(k-1),j=2..5)=c[6]^ k/k,k=[2,3])];\ne18 := solve(\{op(subs(e17,%))\},\{a[6,4],a[6,5]\}):\n e19 := `union`(e17,e18):" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#7$/,**&&% \"aG6$\"\"'\"\"#\"\"\"&%\"cG6#F+F,F,*&&F(6$F*\"\"$F,&F.6#F3F,F,*&&F(6$ F*\"\"%F,&F.6#F9F,F,*&&F(6$F*\"\"&F,&F.6#F?F,F,,$*&#F,F+F,*$)&F.6#F*F+ F,F,F,/,**&F'F,)F-F+F,F,*&F1F,)F4F+F,F,*&F7F,)F:F+F,F,*&F=F,)F@F+F,F,, $*&#F,F3F,*$)FGF3F,F,F," }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 15 " " 0 "" {TEXT 279 7 "Step 10" }{TEXT -1 1 " " }}{PARA 0 "" 0 "" {TEXT -1 34 "We use the stage-order conditions:" }}{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 6 ", " }{XPPEDIT 18 0 "Sum(a[7,j]*c[j]^2,j = 2 .. 6) = 1/ 3;" "6#/-%$SumG6$*&&%\"aG6$\"\"(%\"jG\"\"\"*$&%\"cG6#F,\"\"#F-/F,;F2\" \"'*&F-F-\"\"$!\"\"" }{TEXT -1 1 " " }{XPPEDIT 18 0 "c[7]^3;" "6#*$&% \"cG6#\"\"(\"\"$" }{TEXT -1 2 ", " }}{PARA 0 "" 0 "" {TEXT -1 9 "to fi nd " }{XPPEDIT 18 0 "a[7,4];" "6#&%\"aG6$\"\"(\"\"%" }{TEXT -1 7 " a nd " }{XPPEDIT 18 0 "a[7, 5];" "6#&%\"aG6$\"\"(\"\"&" }{TEXT -1 1 ". " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 128 "[seq(add(a[7,j]*c[j]^(k-1),j=2..6)=c[7]^k/k,k=[2,3])];\ne20 : = solve(\{op(subs(e19,%))\},\{a[7,4],a[7,5]\}):\ne21 := `union`(e19,e2 0):" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#7$/,,*&&%\"aG6$\"\"(\"\"#\"\"\" &%\"cG6#F+F,F,*&&F(6$F*\"\"$F,&F.6#F3F,F,*&&F(6$F*\"\"%F,&F.6#F9F,F,*& &F(6$F*\"\"&F,&F.6#F?F,F,*&&F(6$F*\"\"'F,&F.6#FEF,F,,$*&#F,F+F,*$)&F.6 #F*F+F,F,F,/,,*&F'F,)F-F+F,F,*&F1F,)F4F+F,F,*&F7F,)F:F+F,F,*&F=F,)F@F+ F,F,*&FCF,)FFF+F,F,,$*&#F,F3F,*$)FMF3F,F,F," }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 15 "" 0 "" {TEXT 280 7 "Step 11" }{TEXT -1 1 " " }} {PARA 0 "" 0 "" {TEXT -1 34 "We use the stage-order conditions:" }} {PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "Sum(a[8,j]*c[j],j = 2 .. 7) = 1/2;" "6#/-%$SumG6$*&&%\"aG6$\"\")%\"jG\"\"\"&%\"cG6#F,F-/F,; \"\"#\"\"(*&F-F-F3!\"\"" }{TEXT -1 1 " " }{XPPEDIT 18 0 "c[8]^2;" "6#* $&%\"cG6#\"\")\"\"#" }{TEXT -1 6 ", " }{XPPEDIT 18 0 "Sum(a[8,j]*c [j]^2,j = 2 .. 7) = 1/3;" "6#/-%$SumG6$*&&%\"aG6$\"\")%\"jG\"\"\"*$&% \"cG6#F,\"\"#F-/F,;F2\"\"(*&F-F-\"\"$!\"\"" }{TEXT -1 1 " " }{XPPEDIT 18 0 "c[8]^3;" "6#*$&%\"cG6#\"\")\"\"$" }{TEXT -1 2 ", " }}{PARA 0 "" 0 "" {TEXT -1 9 "to find " }{XPPEDIT 18 0 "a[8,4];" "6#&%\"aG6$\"\") \"\"%" }{TEXT -1 7 " and " }{XPPEDIT 18 0 "a[8,5];" "6#&%\"aG6$\"\") \"\"&" }{TEXT -1 1 "." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 128 "[seq(add(a[8,j]*c[j]^(k-1),j=2..7)=c[8]^ k/k,k=[2,3])];\ne22 := solve(\{op(subs(e21,%))\},\{a[8,4],a[8,5]\}):\n e23 := `union`(e21,e22):" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#7$/,.*&&% \"aG6$\"\")\"\"#\"\"\"&%\"cG6#F+F,F,*&&F(6$F*\"\"$F,&F.6#F3F,F,*&&F(6$ F*\"\"%F,&F.6#F9F,F,*&&F(6$F*\"\"&F,&F.6#F?F,F,*&&F(6$F*\"\"'F,&F.6#FE F,F,*&&F(6$F*\"\"(F,&F.6#FKF,F,,$*&#F,F+F,*$)&F.6#F*F+F,F,F,/,.*&F'F,) F-F+F,F,*&F1F,)F4F+F,F,*&F7F,)F:F+F,F,*&F=F,)F@F+F,F,*&FCF,)FFF+F,F,*& FIF,)FLF+F,F,,$*&#F,F3F,*$)FSF3F,F,F," }}}{PARA 0 "" 0 "" {TEXT -1 0 " " }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 3 "e23" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 54194 "e23 := \{c[8] = 1, b[3] = 0, b[2] = 0, c[9] = 1, b[5] = -1/60*(10*c[6]*c[7]*c[4]-5*c[6]*c[4]-5*c[7]*c[6]+3*c [6]-5*c[7]*c[4]+3*c[4]-2+3*c[7])/(-c[4]+c[5])/c[5]/(-c[5]^3+c[5]^2-c[7 ]*c[5]+c[7]*c[5]^2-c[6]*c[5]+c[6]*c[5]^2+c[7]*c[6]-c[6]*c[7]*c[5]), a[ 5,2] = 0, a[4,2] = 0, `b*`[3] = 0, `b*`[2] = 0, a[8,2] = 0, a[7,2] = 0 , a[6,2] = 0, a[6,5] = 1/2*c[6]*(30*c[6]^2*c[5]^2*c[4]^2-2*c[6]^2*c[4] ^2+c[6]^3*c[4]-12*c[6]^2*c[4]*c[5]^2-3*c[5]^2*c[6]*c[4]+c[5]*c[6]*c[4] ^2+c[6]*c[4]^3-12*c[6]^2*c[4]^2*c[5]-30*c[5]*c[6]^3*c[4]^2-20*c[5]*c[6 ]^2*c[4]^4+30*c[6]^2*c[5]*c[4]^3-c[5]*c[4]^3+20*c[4]^3*c[5]*c[6]^3-20* c[6]^2*c[5]^2*c[4]^3+c[5]^2*c[4]^2-30*c[5]^2*c[6]*c[4]^3+12*c[5]^2*c[6 ]*c[4]^2+2*c[6]^2*c[4]*c[5]+20*c[5]^2*c[6]*c[4]^4+2*c[6]^2*c[5]^2-2*c[ 6]^3*c[5]+12*c[6]^3*c[4]*c[5])/c[5]/(c[4]^3-2*c[5]*c[4]^2+30*c[5]^2*c[ 4]^3-6*c[5]*c[4]^3-10*c[4]^4*c[5]^2-c[5]^3+10*c[5]^4*c[4]^2-30*c[5]^3* c[4]^2+6*c[5]^3*c[4]+2*c[5]^2*c[4]), b[1] = 1/60*(30*c[5]*c[6]*c[7]*c[ 4]-10*c[5]*c[6]*c[4]-10*c[6]*c[7]*c[5]+5*c[6]*c[5]-10*c[6]*c[7]*c[4]+5 *c[6]*c[4]+5*c[7]*c[6]-3*c[6]-10*c[5]*c[7]*c[4]+5*c[5]*c[4]+5*c[7]*c[5 ]-3*c[5]+5*c[7]*c[4]-3*c[4]+2-3*c[7])/c[5]/c[6]/c[7]/c[4], b[4] = 1/60 *(-2+3*c[5]-5*c[7]*c[6]+3*c[7]+3*c[6]+10*c[6]*c[7]*c[5]-5*c[6]*c[5]-5* c[7]*c[5])/c[4]/(-c[6]*c[4]^3-c[4]^3+c[4]^4+c[7]*c[4]^2+c[6]*c[4]^2-c[ 6]*c[7]*c[4]+c[6]*c[7]*c[4]^2-c[5]*c[4]^3+c[5]*c[4]^2-c[7]*c[4]^3-c[5] *c[7]*c[4]+c[5]*c[7]*c[4]^2-c[5]*c[6]*c[4]+c[5]*c[6]*c[4]^2+c[6]*c[7]* c[5]-c[5]*c[6]*c[7]*c[4]), b[6] = 1/60*(10*c[5]*c[7]*c[4]-5*c[5]*c[4]- 5*c[7]*c[5]+3*c[5]-5*c[7]*c[4]+3*c[4]-2+3*c[7])/(-c[4]+c[6])/(-c[5]+c[ 6])/c[6]/(c[6]^2-c[6]+c[7]-c[7]*c[6]), `b*`[9] = 1/10*(50*c[5]^2*c[6]* c[4]^2-40*c[5]^2*c[6]*c[4]+10*c[6]*c[5]^2-40*c[5]*c[6]*c[4]^2+35*c[5]* c[6]*c[4]-10*c[6]*c[5]+10*c[6]*c[4]^2-10*c[6]*c[4]+3*c[6]-5*c[5]^2*c[4 ]-5*c[5]*c[4]^2+10*c[5]*c[4]-c[5]-c[4])/(3*c[6]-28*c[5]*c[6]*c[4]^2-28 *c[5]^2*c[6]*c[4]+8*c[6]*c[5]^2-9*c[6]*c[4]-9*c[6]*c[5]+28*c[5]*c[6]*c [4]+30*c[5]^2*c[6]*c[4]^2+8*c[6]*c[4]^2-4*c[5]^2*c[4]-4*c[5]*c[4]^2+9* c[5]*c[4]-c[5]-c[4]), a[7,5] = -1/2*c[7]*(-17*c[5]^2*c[6]*c[4]^4+5*c[6 ]*c[5]*c[4]^4+42*c[4]^2*c[5]^2*c[7]^3+17*c[7]^3*c[5]*c[4]^3-80*c[7]^3* c[5]^2*c[4]^3+c[7]*c[5]*c[4]^4+80*c[5]^2*c[4]^4*c[7]^2-60*c[4]^5*c[6]^ 2*c[5]^2+20*c[4]^3*c[7]^3*c[5]^3-20*c[7]^2*c[5]^2*c[4]^5+40*c[4]^5*c[6 ]*c[5]^3-5*c[5]*c[6]^2*c[4]^4+c[7]^3*c[4]^2-c[7]^2*c[4]^3-80*c[5]^3*c[ 4]^4*c[7]+20*c[7]^3*c[5]^2*c[4]^4-200*c[6]^2*c[4]^5*c[7]*c[5]^3-290*c[ 5]^4*c[4]^2*c[7]^2*c[6]+20*c[6]^2*c[7]*c[5]^4*c[4]^2+40*c[6]*c[5]^2*c[ 4]^4*c[7]-140*c[5]^4*c[4]^3*c[6]-100*c[5]^4*c[4]^5*c[6]+190*c[5]^4*c[4 ]^4*c[6]-40*c[7]^2*c[4]^4*c[5]^3+20*c[5]^4*c[4]^4*c[7]+2*c[4]^4*c[7]^2 -20*c[7]^2*c[5]^4*c[4]^3-4*c[6]^2*c[7]^2*c[5]-8*c[7]*c[6]^2*c[5]^3-3*c [7]*c[6]*c[5]^3+2*c[7]^2*c[6]*c[5]^3+60*c[7]*c[6]^2*c[5]*c[4]^5-40*c[6 ]*c[5]^2*c[4]^5*c[7]+17*c[7]*c[4]^4*c[6]*c[5]+80*c[7]^2*c[5]^3*c[4]^3+ 5*c[7]*c[4]^4*c[6]^2+c[5]^2*c[7]^3-260*c[6]^2*c[7]*c[5]^2*c[4]^3-12*c[ 6]^2*c[7]^2*c[5]^2*c[4]+20*c[6]^2*c[7]^2*c[5]^2*c[4]^2-40*c[6]*c[7]*c[ 5]^3*c[4]^3-41*c[6]^2*c[7]^3*c[4]*c[5]+20*c[7]^2*c[6]*c[5]*c[4]^4+3*c[ 7]*c[6]*c[4]^3-2*c[6]^2*c[7]*c[4]^2-3*c[6]*c[7]^2*c[4]^2+4*c[7]^2*c[6] *c[4]^3+6*c[5]*c[7]^2*c[4]^3+4*c[6]*c[5]*c[7]^2*c[4]^2+c[7]^2*c[5]*c[6 ]*c[4]-11*c[5]*c[7]*c[6]*c[4]^3+2*c[6]^2*c[4]^2*c[5]-c[7]^3*c[4]*c[5]+ 17*c[5]^2*c[6]*c[7]*c[4]^2-200*c[5]^4*c[7]^2*c[4]^4*c[6]-50*c[4]^5*c[6 ]*c[7]*c[5]^3-150*c[5]^2*c[6]*c[7]^3*c[4]^4-c[6]^2*c[5]*c[4]^3+2*c[7]^ 3*c[4]^2*c[6]*c[5]+2*c[6]^2*c[7]*c[4]*c[5]+108*c[7]*c[6]^2*c[5]*c[4]^3 +4*c[6]^2*c[7]*c[5]^2+2*c[6]*c[5]*c[7]*c[4]^2-10*c[6]^2*c[7]^3*c[5]^2+ 6*c[5]^4*c[7]*c[6]-8*c[6]*c[7]^3*c[5]^2+3*c[6]*c[7]^2*c[5]^2+2*c[6]^2* c[7]^2*c[5]^2-27*c[6]^2*c[4]^2*c[7]*c[5]-62*c[6]^2*c[7]^2*c[4]^2*c[5]+ 46*c[6]*c[5]^4*c[4]^2+20*c[4]^5*c[7]*c[5]^3+2*c[7]^3*c[4]*c[6]*c[5]-27 *c[6]^2*c[5]^2*c[7]*c[4]+120*c[6]^2*c[5]^2*c[7]*c[4]^2-35*c[6]*c[5]^2* c[7]*c[4]^3-37*c[6]*c[5]^3*c[7]*c[4]^2+53*c[6]^2*c[5]^3*c[7]*c[4]-54*c [7]^2*c[5]^2*c[4]^3+2*c[7]^2*c[5]*c[4]^2-c[7]*c[5]*c[4]^3-3*c[6]*c[5]* c[4]^3-2*c[4]^3*c[7]^3+63*c[7]^3*c[5]^2*c[6]*c[4]-16*c[7]^2*c[5]^2*c[6 ]*c[4]-5*c[6]*c[7]*c[4]^4+2*c[6]^2*c[7]^2*c[4]^2+2*c[6]^2*c[7]^2*c[4]- 10*c[5]^4*c[7]^2*c[6]+3*c[5]^3*c[6]*c[4]+3*c[5]^3*c[7]*c[4]+20*c[5]^3* c[6]*c[7]*c[4]-2*c[5]^3*c[4]^2+22*c[6]^2*c[7]^2*c[4]*c[5]+7*c[5]^2*c[6 ]*c[4]^3-c[5]^2*c[7]*c[4]^2+c[5]^2*c[6]*c[4]^2-4*c[5]^2*c[7]*c[4]^3-8* c[5]^3*c[7]*c[4]^2-22*c[5]^3*c[6]*c[4]^2+240*c[5]^3*c[6]^2*c[4]^3+100* c[4]^3*c[7]^2*c[5]^2*c[6]^2+150*c[4]^5*c[5]^3*c[6]^2-3*c[7]*c[5]^4*c[4 ]+5*c[7]^2*c[5]^4*c[4]+12*c[5]^2*c[4]^4*c[7]-98*c[6]^2*c[5]^2*c[4]^3+2 5*c[6]^2*c[5]^2*c[4]^2+54*c[6]*c[5]^3*c[4]^3+8*c[6]^2*c[5]^3*c[4]-63*c [6]^2*c[5]^3*c[4]^2+42*c[5]^3*c[7]*c[4]^3+60*c[5]^4*c[7]*c[4]^2*c[6]+2 00*c[5]^3*c[7]^3*c[4]^4*c[6]-4*c[6]^2*c[4]*c[5]^2+c[7]*c[6]^2*c[4]^3+2 00*c[7]^2*c[5]^2*c[6]*c[4]^3-240*c[7]^3*c[5]^2*c[6]*c[4]^2+180*c[7]^2* c[5]^3*c[6]*c[4]^2+3*c[5]^4*c[4]^2-80*c[5]^3*c[7]^3*c[6]*c[4]+c[7]^3*c [4]^2*c[6]-7*c[7]^3*c[4]^2*c[5]+20*c[4]^3*c[7]^2*c[5]*c[6]^2-19*c[7]^2 *c[6]*c[5]*c[4]^3-11*c[7]^2*c[5]^2*c[6]*c[4]^2-3*c[4]^4*c[5]^2-27*c[7] ^2*c[5]^3*c[6]*c[4]-480*c[7]^2*c[5]^3*c[6]*c[4]^3-380*c[5]^3*c[6]^2*c[ 4]^4-5*c[5]^4*c[4]^3-10*c[6]^2*c[4]^3*c[7]^2+130*c[5]^3*c[6]^2*c[7]*c[ 4]^3-5*c[7]^3*c[5]^2*c[4]-2*c[7]^2*c[5]^2*c[4]-4*c[5]^2*c[6]*c[7]*c[4] -160*c[6]^2*c[7]*c[5]^3*c[4]^2-80*c[6]^2*c[7]^2*c[4]*c[5]^3+2*c[5]^2*c [4]^3+12*c[7]^2*c[5]^2*c[4]^2+8*c[7]^2*c[5]^3*c[4]+400*c[7]^3*c[5]^2*c [6]^2*c[4]^3-17*c[7]^2*c[4]^4*c[5]+250*c[7]^2*c[5]^3*c[6]^2*c[4]^2-100 *c[6]^2*c[4]^5*c[5]*c[7]^2-200*c[7]^2*c[4]^5*c[6]*c[5]^3+150*c[5]^2*c[ 6]*c[4]^5*c[7]^2+380*c[7]^3*c[5]^2*c[4]^3*c[6]-20*c[7]^3*c[6]*c[5]*c[4 ]^3-210*c[7]^3*c[6]^2*c[5]*c[4]^3-6*c[5]^4*c[4]*c[6]+80*c[5]^4*c[7]^2* c[4]*c[6]+10*c[6]*c[7]^3*c[5]^3-36*c[5]^4*c[4]*c[7]*c[6]-50*c[4]^5*c[6 ]^2*c[7]*c[5]^2+200*c[4]^5*c[6]^2*c[7]^2*c[5]^2+160*c[6]^2*c[4]^4*c[5] ^2-450*c[7]^3*c[5]^3*c[4]^3*c[6]+5*c[6]^2*c[7]^3*c[4]^2-c[5]^3*c[7]^2+ 6*c[6]^2*c[7]^3*c[5]-160*c[7]*c[6]^2*c[5]*c[4]^4+200*c[5]^3*c[4]^4*c[6 ]^2*c[7]-350*c[5]^3*c[4]^3*c[6]^2*c[7]^2-350*c[5]^4*c[4]^4*c[6]*c[7]+2 30*c[5]^2*c[4]^4*c[7]*c[6]^2+150*c[5]^3*c[4]^4*c[6]*c[7]+100*c[5]^4*c[ 4]^3*c[6]*c[7]-3*c[6]^2*c[7]^3*c[4]-42*c[7]^2*c[5]^3*c[4]^2+50*c[5]^4* c[4]^4*c[6]^2-20*c[5]^4*c[4]^3*c[6]^2+500*c[5]^3*c[4]^4*c[7]^2*c[6]+45 0*c[5]^4*c[4]^3*c[7]^2*c[6]-340*c[5]^2*c[4]^4*c[7]^2*c[6]-50*c[5]^4*c[ 4]^3*c[6]^2*c[7]+150*c[7]^2*c[4]^4*c[6]^2*c[5]+200*c[7]^2*c[4]^4*c[5]^ 3*c[6]^2-350*c[7]^2*c[4]^4*c[6]^2*c[5]^2+10*c[7]^2*c[6]^2*c[5]^3-60*c[ 4]^4*c[6]*c[5]^3+100*c[7]^3*c[6]^2*c[5]*c[4]^4+200*c[5]^4*c[4]^5*c[6]* c[7]-200*c[7]^3*c[4]^4*c[6]^2*c[5]^2+5*c[5]^3*c[4]^4+80*c[6]^2*c[7]^3* c[5]^2*c[4]-270*c[6]^2*c[7]^3*c[5]^2*c[4]^2+290*c[5]^3*c[7]^3*c[4]^2*c [6]+140*c[6]^2*c[5]*c[7]^3*c[4]^2-5*c[7]^3*c[4]*c[5]^3)/c[5]/(100*c[5] ^2*c[6]*c[4]^4-18*c[6]*c[5]*c[4]^4+4*c[5]^3*c[4]+50*c[4]^5*c[6]^2*c[5] ^2+40*c[5]*c[6]^2*c[4]^4+3*c[5]^5-5*c[4]^4*c[6]^2+180*c[5]^4*c[4]^3*c[ 6]+100*c[5]^4*c[4]^5*c[6]-350*c[5]^4*c[4]^4*c[6]+140*c[5]^5*c[4]^2-180 *c[5]^5*c[4]^3-2*c[5]^4+30*c[4]^5*c[5]^3+300*c[5]^5*c[4]^3*c[6]+2*c[6] *c[5]^3-50*c[5]^5*c[4]^2*c[6]^2-40*c[5]^4*c[6]^2*c[4]+240*c[5]^4*c[6]^ 2*c[4]^2-3*c[6]^2*c[5]^3+3*c[6]^2*c[4]^3+5*c[5]^4*c[6]^2-2*c[6]*c[4]^3 +2*c[5]*c[4]^3+4*c[5]*c[6]*c[4]^2-6*c[6]^2*c[4]^2*c[5]-50*c[5]^4*c[4]^ 5-13*c[6]^2*c[5]*c[4]^3-40*c[6]*c[5]^4*c[4]^2+6*c[6]*c[5]*c[4]^3-15*c[ 5]^3*c[6]*c[4]-55*c[5]^2*c[6]*c[4]^3+6*c[5]^2*c[6]*c[4]^2+68*c[5]^3*c[ 6]*c[4]^2-100*c[4]^5*c[5]^3*c[6]^2+100*c[6]^2*c[5]^2*c[4]^3-100*c[6]*c [5]^3*c[4]^3+13*c[6]^2*c[5]^3*c[4]-100*c[6]^2*c[5]^3*c[4]^2-4*c[5]^2*c [4]^2+6*c[6]^2*c[4]*c[5]^2-68*c[5]^4*c[4]^2+23*c[4]^4*c[5]^2+350*c[5]^ 3*c[6]^2*c[4]^4-4*c[5]^2*c[6]*c[4]-9*c[5]^2*c[4]^3-3*c[4]^4*c[5]-5*c[5 ]^5*c[6]-30*c[5]^6*c[4]^2+50*c[5]^6*c[6]*c[4]^2+180*c[5]^4*c[4]^4+10*c [5]^4*c[4]*c[6]-240*c[6]^2*c[4]^4*c[5]^2+3*c[6]*c[4]^4-100*c[5]^6*c[4] ^3*c[6]+68*c[5]^3*c[4]^3-23*c[5]^5*c[4]-210*c[5]^5*c[6]*c[4]^2-350*c[5 ]^4*c[4]^3*c[6]^2-30*c[5]^2*c[6]*c[4]^5+50*c[5]^6*c[4]^3+60*c[4]^4*c[6 ]*c[5]^3+100*c[5]^5*c[6]^2*c[4]^3+9*c[5]^4*c[4]-140*c[5]^3*c[4]^4+40*c [5]^5*c[4]*c[6]), `b*`[6] = -1/60*(-30*c[5]^3*c[4]+103*c[5]*c[7]*c[4]- 18*c[6]+6*c[5]+6*c[4]+27*c[7]*c[6]-9*c[5]^2-9*c[4]^2-100*c[7]*c[6]*c[5 ]^3+522*c[5]*c[6]*c[7]*c[4]-111*c[6]*c[5]^2+75*c[6]*c[5]+60*c[6]*c[5]^ 3-9*c[7]*c[5]+600*c[6]*c[7]*c[5]^3*c[4]^3+14*c[7]*c[5]^2-100*c[7]*c[6] *c[4]^3+14*c[7]*c[4]^2-9*c[7]*c[4]-117*c[6]*c[7]*c[5]-111*c[6]*c[4]^2+ 60*c[6]*c[4]^3-117*c[6]*c[7]*c[4]+180*c[6]*c[7]*c[4]^2-30*c[5]*c[4]^3- 174*c[5]*c[7]*c[4]^2+486*c[5]*c[6]*c[4]^2+107*c[5]*c[4]^2-318*c[5]*c[6 ]*c[4]-66*c[5]*c[4]+75*c[6]*c[4]+500*c[5]*c[7]*c[6]*c[4]^3+1420*c[5]^2 *c[6]*c[7]*c[4]^2-840*c[6]*c[5]*c[7]*c[4]^2+180*c[6]*c[7]*c[5]^2-900*c [6]*c[5]^2*c[7]*c[4]^3-900*c[6]*c[5]^3*c[7]*c[4]^2+50*c[7]*c[5]*c[4]^3 -280*c[6]*c[5]*c[4]^3-280*c[5]^3*c[6]*c[4]+50*c[5]^3*c[7]*c[4]+107*c[5 ]^2*c[4]+500*c[5]^3*c[6]*c[7]*c[4]+50*c[5]^3*c[4]^2+470*c[5]^2*c[6]*c[ 4]^3+310*c[5]^2*c[7]*c[4]^2-770*c[5]^2*c[6]*c[4]^2-90*c[5]^2*c[7]*c[4] ^3-90*c[5]^3*c[7]*c[4]^2+470*c[5]^3*c[6]*c[4]^2-300*c[6]*c[5]^3*c[4]^3 -180*c[5]^2*c[4]^2+486*c[5]^2*c[6]*c[4]-174*c[5]^2*c[7]*c[4]-840*c[5]^ 2*c[6]*c[7]*c[4]+50*c[5]^2*c[4]^3)/(-9*c[5]*c[7]*c[4]-3*c[7]*c[6]+3*c[ 6]^2-28*c[5]*c[6]*c[7]*c[4]-c[6]*c[5]+c[7]*c[5]+8*c[6]^2*c[4]^2+c[7]*c [4]+9*c[6]*c[7]*c[5]+9*c[6]*c[7]*c[4]-8*c[6]*c[7]*c[4]^2+4*c[5]*c[7]*c [4]^2-4*c[5]*c[6]*c[4]^2+9*c[5]*c[6]*c[4]-c[6]*c[4]-28*c[6]^2*c[4]^2*c [5]+28*c[6]^2*c[4]*c[5]-30*c[5]^2*c[6]*c[7]*c[4]^2+28*c[6]*c[5]*c[7]*c [4]^2-8*c[6]*c[7]*c[5]^2+8*c[6]^2*c[5]^2-9*c[6]^2*c[5]+30*c[6]^2*c[5]^ 2*c[4]^2-28*c[6]^2*c[4]*c[5]^2-9*c[6]^2*c[4]-4*c[5]^2*c[6]*c[4]+4*c[5] ^2*c[7]*c[4]+28*c[5]^2*c[6]*c[7]*c[4])/(c[5]*c[4]-c[6]*c[4]-c[6]*c[5]+ c[6]^2)/c[6], a[8,7] = (10*c[5]*c[6]*c[4]-5*c[6]*c[5]-5*c[6]*c[4]+3*c[ 6]-5*c[5]*c[4]+3*c[5]+3*c[4]-2)/c[7]/(c[6]-c[7])/(c[5]-c[7])/(c[4]-c[7 ])/(-20*c[5]*c[6]*c[4]+30*c[5]*c[6]*c[7]*c[4]+15*c[6]*c[5]-20*c[6]*c[7 ]*c[5]+15*c[6]*c[4]-20*c[6]*c[7]*c[4]-12*c[6]+15*c[7]*c[6]+15*c[5]*c[4 ]-20*c[5]*c[7]*c[4]-12*c[5]+15*c[7]*c[5]-12*c[4]+15*c[7]*c[4]+10-12*c[ 7])*(c[4]-1)*(-1+c[5])*(-1+c[6])*(-1+c[7]), a[6,3] = 3/4*1/c[4]^2*c[6] *(60*c[4]^3*c[5]*c[6]^3-90*c[5]*c[6]^3*c[4]^2+36*c[6]^3*c[4]*c[5]-6*c[ 6]^3*c[5]+3*c[6]^3*c[4]-20*c[6]^2*c[5]^3*c[4]^2-80*c[6]^2*c[5]^2*c[4]^ 3+150*c[6]^2*c[5]^2*c[4]^2-48*c[6]^2*c[4]*c[5]^2+8*c[6]^2*c[5]^2-60*c[ 5]*c[6]^2*c[4]^4+90*c[6]^2*c[5]*c[4]^3-48*c[6]^2*c[4]^2*c[5]+4*c[6]^2* c[4]*c[5]-4*c[6]^2*c[4]^2+30*c[6]*c[5]^3*c[4]^3+90*c[5]^2*c[6]*c[4]^4- 180*c[5]^2*c[6]*c[4]^3+54*c[5]^2*c[6]*c[4]^2-12*c[5]^2*c[6]*c[4]+18*c[ 6]*c[5]*c[4]^3+6*c[5]*c[6]*c[4]^2+3*c[5]^2*c[4]^2-3*c[5]*c[4]^3)/(10*c [5]^3*c[4]^2+10*c[5]^2*c[4]^3-30*c[5]^2*c[4]^2+6*c[5]^2*c[4]-c[5]^2+6* c[5]*c[4]^2+c[5]*c[4]-c[4]^2), b[7] = -1/60*(10*c[5]*c[6]*c[4]-5*c[6]* c[5]-5*c[6]*c[4]+3*c[6]-5*c[5]*c[4]+3*c[5]+3*c[4]-2)/(c[4]-c[7])/(c[5] -c[7])/(c[6]-c[7])/c[7]/(-1+c[7]), `b*`[8] = 0, a[5,3] = -3/4*c[5]^2*( 2*c[5]-3*c[4])/c[4]^2, `b*`[4] = -1/60*(-30*c[5]^3*c[4]-42*c[7]*c[6]^2 +89*c[5]*c[7]*c[4]-18*c[6]+6*c[5]+6*c[4]+27*c[7]*c[6]-9*c[5]^2+180*c[7 ]*c[6]^2*c[5]^3-100*c[7]*c[6]*c[5]^3+27*c[6]^2+186*c[5]*c[6]*c[7]*c[4] -97*c[6]*c[5]^2+66*c[6]*c[5]+60*c[6]*c[5]^3-100*c[6]^2*c[5]^3-9*c[7]*c [5]+14*c[7]*c[5]^2-100*c[6]^2*c[7]*c[4]^2+120*c[6]^2*c[7]*c[4]+60*c[6] ^2*c[4]^2-9*c[7]*c[4]-103*c[6]*c[7]*c[5]-36*c[6]*c[4]^2-61*c[6]*c[7]*c [4]+60*c[6]*c[7]*c[4]^2-30*c[5]*c[7]*c[4]^2+126*c[5]*c[6]*c[4]^2+18*c[ 5]*c[4]^2-112*c[5]*c[6]*c[4]-57*c[5]*c[4]+39*c[6]*c[4]-280*c[6]^2*c[4] ^2*c[5]+330*c[6]^2*c[4]*c[5]+380*c[5]^2*c[6]*c[7]*c[4]^2-560*c[6]^2*c[ 7]*c[4]*c[5]+192*c[6]^2*c[7]*c[5]-310*c[6]^2*c[7]*c[5]^2-230*c[6]*c[5] *c[7]*c[4]^2+156*c[6]*c[7]*c[5]^2+180*c[6]^2*c[5]^2+500*c[6]^2*c[4]^2* c[7]*c[5]+950*c[6]^2*c[5]^2*c[7]*c[4]-900*c[6]^2*c[5]^2*c[7]*c[4]^2-30 0*c[6]*c[5]^3*c[7]*c[4]^2-600*c[6]^2*c[5]^3*c[7]*c[4]-117*c[6]^2*c[5]- 130*c[5]^3*c[6]*c[4]+50*c[5]^3*c[7]*c[4]+93*c[5]^2*c[4]+230*c[5]^3*c[6 ]*c[7]*c[4]+50*c[5]^2*c[7]*c[4]^2-190*c[5]^2*c[6]*c[4]^2+150*c[5]^3*c[ 6]*c[4]^2+470*c[6]^2*c[5]^2*c[4]^2+320*c[6]^2*c[5]^3*c[4]-300*c[6]^2*c [5]^3*c[4]^2-30*c[5]^2*c[4]^2-530*c[6]^2*c[4]*c[5]^2-75*c[6]^2*c[4]+15 6*c[5]^2*c[6]*c[4]-150*c[5]^2*c[7]*c[4]-270*c[5]^2*c[6]*c[7]*c[4]+600* c[6]^2*c[7]*c[5]^3*c[4]^2)/(-9*c[5]*c[7]*c[4]-3*c[7]*c[6]-c[4]^2-28*c[ 5]*c[6]*c[7]*c[4]+c[7]*c[5]+c[7]*c[4]+9*c[6]*c[7]*c[5]-9*c[6]*c[4]^2+8 *c[6]*c[4]^3+9*c[6]*c[7]*c[4]-8*c[6]*c[7]*c[4]^2-4*c[5]*c[4]^3+4*c[5]* c[7]*c[4]^2+28*c[5]*c[6]*c[4]^2+9*c[5]*c[4]^2-9*c[5]*c[6]*c[4]-c[5]*c[ 4]+3*c[6]*c[4]-30*c[5]^2*c[6]*c[7]*c[4]^2+28*c[6]*c[5]*c[7]*c[4]^2-8*c [6]*c[7]*c[5]^2-28*c[6]*c[5]*c[4]^3+30*c[5]^2*c[6]*c[4]^3-28*c[5]^2*c[ 6]*c[4]^2-4*c[5]^2*c[4]^2+8*c[5]^2*c[6]*c[4]+4*c[5]^2*c[7]*c[4]+28*c[5 ]^2*c[6]*c[7]*c[4])/(-c[4]+c[6])/(-c[4]+c[5])/c[4], a[8,6] = 1/2*(5*c[ 5]*c[6]*c[4]+c[6]-2*c[6]*c[4]-2*c[6]*c[5]+25*c[5]*c[7]*c[4]+6*c[5]+9*c [7]+10*c[7]^2*c[4]-4+6*c[4]-6*c[7]^2-14*c[7]*c[4]-10*c[5]*c[4]-14*c[7] *c[5]-20*c[7]^2*c[5]*c[4]+10*c[7]^2*c[5])/(c[6]-c[7])/(-c[4]+c[6])/(-c [5]+c[6])/c[6]/(-20*c[5]*c[6]*c[4]+30*c[5]*c[6]*c[7]*c[4]+15*c[6]*c[5] -20*c[6]*c[7]*c[5]+15*c[6]*c[4]-20*c[6]*c[7]*c[4]-12*c[6]+15*c[7]*c[6] +15*c[5]*c[4]-20*c[5]*c[7]*c[4]-12*c[5]+15*c[7]*c[5]-12*c[4]+15*c[7]*c [4]+10-12*c[7])*(c[4]-1)*(-1+c[5])*(-1+c[6]), b[8] = 1/60*(-20*c[5]*c[ 6]*c[4]+30*c[5]*c[6]*c[7]*c[4]+15*c[6]*c[5]-20*c[6]*c[7]*c[5]+15*c[6]* c[4]-20*c[6]*c[7]*c[4]-12*c[6]+15*c[7]*c[6]+15*c[5]*c[4]-20*c[5]*c[7]* c[4]-12*c[5]+15*c[7]*c[5]-12*c[4]+15*c[7]*c[4]+10-12*c[7])/(c[4]-1)/(- 1+c[5])/(-1+c[6])/(-1+c[7]), `b*`[5] = 1/60*(-42*c[7]*c[6]^2+89*c[5]*c [7]*c[4]-18*c[6]+6*c[5]+6*c[4]+27*c[7]*c[6]-9*c[4]^2+27*c[6]^2+186*c[5 ]*c[6]*c[7]*c[4]-36*c[6]*c[5]^2+39*c[6]*c[5]-9*c[7]*c[5]-100*c[6]^2*c[ 4]^3+600*c[6]^2*c[7]*c[5]^2*c[4]^3-100*c[7]*c[6]*c[4]^3-310*c[6]^2*c[7 ]*c[4]^2+192*c[6]^2*c[7]*c[4]+180*c[6]^2*c[4]^2+14*c[7]*c[4]^2-9*c[7]* c[4]-61*c[6]*c[7]*c[5]-97*c[6]*c[4]^2+60*c[6]*c[4]^3-103*c[6]*c[7]*c[4 ]+156*c[6]*c[7]*c[4]^2-30*c[5]*c[4]^3-150*c[5]*c[7]*c[4]^2+156*c[5]*c[ 6]*c[4]^2+93*c[5]*c[4]^2-112*c[5]*c[6]*c[4]-57*c[5]*c[4]+66*c[6]*c[4]+ 230*c[5]*c[7]*c[6]*c[4]^3-530*c[6]^2*c[4]^2*c[5]+330*c[6]^2*c[4]*c[5]+ 380*c[5]^2*c[6]*c[7]*c[4]^2+320*c[6]^2*c[5]*c[4]^3-560*c[6]^2*c[7]*c[4 ]*c[5]-600*c[7]*c[6]^2*c[5]*c[4]^3+120*c[6]^2*c[7]*c[5]-100*c[6]^2*c[7 ]*c[5]^2-270*c[6]*c[5]*c[7]*c[4]^2+60*c[6]*c[7]*c[5]^2+60*c[6]^2*c[5]^ 2+950*c[6]^2*c[4]^2*c[7]*c[5]+500*c[6]^2*c[5]^2*c[7]*c[4]-900*c[6]^2*c [5]^2*c[7]*c[4]^2-300*c[6]*c[5]^2*c[7]*c[4]^3-75*c[6]^2*c[5]+50*c[7]*c [5]*c[4]^3-130*c[6]*c[5]*c[4]^3+18*c[5]^2*c[4]+150*c[5]^2*c[6]*c[4]^3+ 50*c[5]^2*c[7]*c[4]^2-190*c[5]^2*c[6]*c[4]^2-300*c[6]^2*c[5]^2*c[4]^3+ 470*c[6]^2*c[5]^2*c[4]^2-30*c[5]^2*c[4]^2-280*c[6]^2*c[4]*c[5]^2+180*c [7]*c[6]^2*c[4]^3-117*c[6]^2*c[4]+126*c[5]^2*c[6]*c[4]-30*c[5]^2*c[7]* c[4]-230*c[5]^2*c[6]*c[7]*c[4])/(-4*c[5]^3*c[4]-9*c[5]*c[7]*c[4]-3*c[7 ]*c[6]-c[5]^2-28*c[5]*c[6]*c[7]*c[4]-9*c[6]*c[5]^2+3*c[6]*c[5]+8*c[6]* c[5]^3+c[7]*c[5]+c[7]*c[4]+9*c[6]*c[7]*c[5]+9*c[6]*c[7]*c[4]-8*c[6]*c[ 7]*c[4]^2+4*c[5]*c[7]*c[4]^2+8*c[5]*c[6]*c[4]^2-9*c[5]*c[6]*c[4]-c[5]* c[4]-30*c[5]^2*c[6]*c[7]*c[4]^2+28*c[6]*c[5]*c[7]*c[4]^2-8*c[6]*c[7]*c [5]^2-28*c[5]^3*c[6]*c[4]+9*c[5]^2*c[4]-28*c[5]^2*c[6]*c[4]^2+30*c[5]^ 3*c[6]*c[4]^2-4*c[5]^2*c[4]^2+28*c[5]^2*c[6]*c[4]+4*c[5]^2*c[7]*c[4]+2 8*c[5]^2*c[6]*c[7]*c[4])/(-c[5]+c[6])/(-c[4]+c[5])/c[5], a[5,4] = c[5] ^2*(-c[4]+c[5])/c[4]^2, a[7,6] = 1/2*(5*c[5]*c[4]+1-2*c[4]-2*c[5])*(c[ 4]-c[7])*(c[5]-c[7])*(c[6]-c[7])*c[7]/(10*c[5]*c[6]*c[4]-5*c[6]*c[5]-5 *c[6]*c[4]+3*c[6]-5*c[5]*c[4]+3*c[5]+3*c[4]-2)/(-c[4]+c[6])/(-c[5]+c[6 ])/c[6], c[3] = 2/3*c[4], a[4,3] = 3/4*c[4], a[3,2] = 2/9*c[4]^2/c[2], a[7,4] = -1/2*c[7]*(-68*c[5]^2*c[6]*c[4]^4-7*c[6]*c[5]*c[4]^4+7*c[4]^ 2*c[5]^2*c[7]^3-1400*c[5]^4*c[4]^4*c[6]^2*c[7]+11*c[7]^3*c[5]*c[4]^3-7 2*c[7]^3*c[5]^2*c[4]^3-3*c[7]*c[5]*c[4]^4-58*c[5]^2*c[4]^4*c[7]^2-120* c[6]*c[4]^7*c[7]*c[5]+320*c[4]^5*c[6]^2*c[5]^2-200*c[4]^3*c[7]^3*c[5]^ 3-100*c[4]^5*c[6]*c[5]^3-2*c[5]*c[6]^2*c[4]^4-600*c[4]^5*c[7]^3*c[6]*c [5]^3-200*c[5]^5*c[4]^3*c[7]^2*c[6]^2+400*c[5]^2*c[4]^6*c[6]^2*c[7]^2- 21*c[4]^5*c[5]*c[7]+120*c[5]^2*c[4]^5*c[7]-600*c[5]^3*c[4]^6*c[6]^2*c[ 7]-198*c[5]^3*c[4]^4*c[7]-280*c[7]^2*c[4]^5*c[5]^3+240*c[7]^3*c[5]^2*c [4]^4+1250*c[6]^2*c[4]^5*c[7]*c[5]^3+120*c[7]*c[5]^2*c[4]^6*c[6]-340*c [5]^4*c[4]^2*c[7]^2*c[6]+400*c[6]^2*c[7]*c[5]^4*c[4]^2-240*c[5]^2*c[4] ^6*c[7]-300*c[6]*c[5]*c[7]^2*c[4]^6-228*c[6]*c[4]^5*c[5]*c[7]-119*c[6] *c[5]^2*c[4]^4*c[7]-200*c[5]*c[7]^3*c[4]^6*c[6]-600*c[5]^4*c[7]*c[4]^6 *c[6]-120*c[4]^6*c[6]^2*c[5]^2+600*c[5]^2*c[7]^2*c[4]^6*c[6]+100*c[5]^ 2*c[7]^3*c[4]^6-187*c[5]^4*c[4]^3*c[6]-910*c[5]^4*c[4]^5*c[6]-320*c[5] ^2*c[6]*c[4]^6+620*c[5]^4*c[4]^4*c[6]-100*c[5]^4*c[7]^3*c[4]^4-40*c[7] ^2*c[4]^4*c[5]^3+40*c[7]^3*c[4]^5*c[5]+10*c[6]*c[7]^2*c[4]^5-10*c[6]*c [7]^3*c[4]^4+100*c[5]^4*c[4]^5*c[7]^2+80*c[5]^4*c[4]^4*c[7]+8*c[4]^3*c [7]^3*c[6]-2*c[7]^2*c[6]*c[4]^4+280*c[7]^3*c[5]^3*c[4]^4+80*c[6]*c[4]^ 5*c[5]*c[7]^2+c[4]^4*c[7]^2-360*c[5]^2*c[7]^3*c[4]^5-80*c[7]^2*c[5]^4* c[4]^3+40*c[7]^3*c[5]^4*c[4]^3-1200*c[5]^3*c[7]*c[4]^6*c[6]+10*c[6]^2* c[7]^3*c[4]^3-10*c[6]^2*c[4]^5*c[5]-12*c[7]*c[6]^2*c[5]^3-4*c[7]^2*c[6 ]*c[5]^3-80*c[5]^4*c[4]^5*c[7]-240*c[7]*c[6]^2*c[5]*c[4]^5-3*c[5]^4*c[ 4]^2*c[7]^2+20*c[5]^5*c[7]^2*c[4]^3+180*c[6]*c[5]^2*c[4]^5*c[7]-60*c[5 ]^4*c[7]^3*c[4]^2*c[6]+400*c[5]^2*c[7]^3*c[4]^6*c[6]-25*c[4]^5*c[5]^3+ 200*c[5]^5*c[4]^4*c[7]^2*c[6]+60*c[5]^2*c[4]^7*c[7]+100*c[7]*c[4]^4*c[ 6]*c[5]-40*c[5]*c[4]^6*c[7]^2-150*c[4]^4*c[6]^2*c[5]^5+50*c[5]^5*c[7]* c[4]^4*c[6]+174*c[7]^2*c[5]^3*c[4]^3+8*c[7]*c[4]^4*c[6]^2+600*c[5]^3*c [7]*c[4]^7*c[6]-60*c[5]^5*c[4]^2*c[6]^2*c[7]+200*c[5]^5*c[4]^4*c[6]^2* c[7]-40*c[5]^5*c[4]^4*c[6]-200*c[5]^5*c[7]*c[4]^5*c[6]-149*c[6]^2*c[5] ^4*c[7]*c[4]-24*c[5]^4*c[6]^2*c[4]+189*c[5]^4*c[6]^2*c[4]^2+300*c[5]^3 *c[4]^6*c[6]^2-20*c[5]^5*c[4]^4*c[7]-5*c[4]^5*c[5]*c[7]^2+600*c[4]^6*c [6]*c[5]^3-84*c[6]^2*c[7]*c[5]^2*c[4]^3-74*c[6]^2*c[7]^2*c[5]^2*c[4]+1 62*c[6]^2*c[7]^2*c[5]^2*c[4]^2+311*c[6]*c[7]*c[5]^3*c[4]^3+21*c[6]^2*c [7]^3*c[4]*c[5]-40*c[7]^2*c[6]^2*c[5]^4+11*c[5]^4*c[7]*c[4]^2+107*c[7] ^2*c[6]*c[5]*c[4]^4+100*c[7]^2*c[5]^5*c[6]^2*c[4]^2+320*c[5]^4*c[4]*c[ 7]^2*c[6]^2-10*c[6]^2*c[4]^4*c[7]^2+300*c[5]^2*c[4]^6*c[7]^2-3*c[7]^2* c[6]*c[4]^3-2*c[5]*c[7]^2*c[4]^3+15*c[6]*c[5]*c[7]^2*c[4]^2-8*c[5]*c[7 ]*c[6]*c[4]^3-14*c[5]^2*c[6]*c[7]*c[4]^2-2000*c[5]^4*c[7]^2*c[4]^4*c[6 ]+200*c[4]^5*c[6]*c[7]*c[5]^3+250*c[5]^2*c[6]*c[7]^3*c[4]^4+4*c[6]^2*c [5]*c[4]^3-26*c[7]^3*c[4]^2*c[6]*c[5]-69*c[7]*c[6]^2*c[5]*c[4]^3-18*c[ 6]^2*c[7]^3*c[5]^2+12*c[6]^2*c[7]^2*c[5]^2+10*c[6]^2*c[4]^2*c[7]*c[5]+ 50*c[6]^2*c[7]^2*c[4]^2*c[5]+31*c[6]*c[5]^4*c[4]^2+24*c[7]*c[6]^2*c[5] ^4+240*c[4]^5*c[7]*c[5]^3+2*c[6]^2*c[5]^2*c[7]*c[4]+39*c[6]^2*c[5]^2*c [7]*c[4]^2-17*c[6]*c[5]^2*c[7]*c[4]^3-73*c[6]*c[5]^3*c[7]*c[4]^2+59*c[ 6]^2*c[5]^3*c[7]*c[4]-12*c[5]^4*c[4]^3*c[7]-12*c[7]^2*c[5]^2*c[4]^3+40 *c[5]^4*c[7]^2*c[4]^4-c[4]^3*c[7]^3+17*c[7]^3*c[5]^2*c[6]*c[4]-9*c[7]^ 2*c[5]^2*c[6]*c[4]+3*c[6]*c[7]*c[4]^4+4*c[4]^5*c[5]+4*c[6]^2*c[7]^2*c[ 4]^2-150*c[5]^5*c[4]^3*c[7]^2*c[6]+21*c[5]^3*c[6]*c[7]*c[4]-14*c[6]^2* c[7]^2*c[4]*c[5]+23*c[5]^2*c[6]*c[4]^3+13*c[5]^2*c[7]*c[4]^3-11*c[5]^3 *c[7]*c[4]^2-17*c[5]^3*c[6]*c[4]^2+168*c[5]^3*c[6]^2*c[4]^3+50*c[5]^5* c[4]^3*c[6]^2*c[7]-220*c[4]^3*c[7]^2*c[5]^2*c[6]^2-6*c[6]*c[7]*c[4]^5- 580*c[4]^5*c[5]^3*c[6]^2-31*c[7]^3*c[4]^4*c[5]-2*c[7]^2*c[5]^4*c[4]-4* c[5]^2*c[4]^4*c[7]+53*c[6]^2*c[5]^2*c[4]^3-14*c[6]^2*c[5]^2*c[4]^2+47* c[6]*c[5]^3*c[4]^3+12*c[6]^2*c[5]^3*c[4]-59*c[6]^2*c[5]^3*c[4]^2+28*c[ 5]^3*c[7]*c[4]^3+127*c[5]^4*c[7]*c[4]^2*c[6]+1250*c[5]^3*c[7]^3*c[4]^4 *c[6]+10*c[6]*c[5]*c[4]^6+240*c[5]*c[4]^6*c[7]*c[6]-4*c[7]*c[6]^2*c[4] ^3+11*c[7]^2*c[5]^2*c[6]*c[4]^3-80*c[7]^3*c[5]^2*c[6]*c[4]^2-121*c[7]^ 2*c[5]^3*c[6]*c[4]^2-96*c[5]^3*c[7]^3*c[6]*c[4]+c[7]^3*c[4]^2*c[5]-68* c[4]^3*c[7]^2*c[5]*c[6]^2-80*c[7]^2*c[6]*c[5]*c[4]^3+68*c[7]^2*c[5]^2* c[6]*c[4]^2-10*c[4]^4*c[5]^2+20*c[7]^2*c[5]^3*c[6]*c[4]-240*c[7]^2*c[5 ]^3*c[6]*c[4]^3-9*c[5]^4*c[4]^3-2*c[6]^2*c[4]^3*c[7]^2+260*c[5]^3*c[6] ^2*c[7]*c[4]^3+49*c[5]^3*c[7]^3*c[4]^2-c[7]^3*c[5]^2*c[4]-228*c[6]^2*c [7]*c[5]^3*c[4]^2+30*c[6]^2*c[7]^2*c[4]*c[5]^3-2*c[7]^2*c[5]^2*c[4]^2+ 5*c[7]^2*c[5]^3*c[4]+40*c[5]^5*c[4]^3*c[6]*c[7]+36*c[4]^6*c[7]*c[5]+90 *c[7]^3*c[5]^2*c[6]^2*c[4]^3+16*c[7]^2*c[4]^4*c[5]-20*c[7]^2*c[5]^3*c[ 6]^2*c[4]^2+300*c[6]^2*c[4]^5*c[5]*c[7]^2+250*c[7]^2*c[4]^5*c[6]*c[5]^ 3-490*c[5]^2*c[6]*c[4]^5*c[7]^2+120*c[7]^3*c[5]^2*c[4]^3*c[6]+810*c[7] ^3*c[5]^3*c[6]^2*c[4]^2+111*c[7]^3*c[6]*c[5]*c[4]^3+280*c[7]^3*c[6]^2* c[5]*c[4]^3-240*c[7]^3*c[5]^3*c[6]^2*c[4]+15*c[5]^4*c[4]^4+60*c[5]^4*c [7]^2*c[4]*c[6]+6*c[6]*c[7]^3*c[5]^3-31*c[5]^4*c[4]*c[7]*c[6]-200*c[4] ^5*c[6]^2*c[7]*c[5]^2-500*c[4]^5*c[6]^2*c[7]^2*c[5]^2+9*c[4]^5*c[5]^2- 181*c[6]^2*c[4]^4*c[5]^2-1280*c[7]^3*c[5]^3*c[4]^3*c[6]-600*c[5]^2*c[4 ]^5*c[7]^3*c[6]-6*c[4]^6*c[5]-6*c[6]^2*c[7]^3*c[4]^2+100*c[5]^5*c[4]^5 *c[6]+120*c[5]*c[7]*c[4]^6*c[6]^2+420*c[6]*c[4]^5*c[7]^3*c[5]+600*c[5] ^4*c[4]^5*c[6]^2*c[7]+175*c[7]*c[6]^2*c[5]*c[4]^4-430*c[5]^3*c[4]^4*c[ 6]^2*c[7]+200*c[5]^3*c[4]^3*c[6]^2*c[7]^2-450*c[5]^4*c[4]^4*c[6]*c[7]+ 160*c[5]^2*c[4]^4*c[7]*c[6]^2-200*c[5]^3*c[4]^4*c[6]*c[7]-280*c[5]^4*c [4]^3*c[6]*c[7]+6*c[5]^3*c[4]^3-100*c[5]^2*c[4]^7*c[7]^2-1200*c[7]^3*c [5]^3*c[6]^2*c[4]^3-320*c[7]^3*c[4]^4*c[5]*c[6]-30*c[7]^2*c[5]^3*c[4]^ 2-300*c[6]*c[5]^3*c[4]^7-350*c[5]^4*c[4]^5*c[6]^2+600*c[6]^2*c[7]^3*c[ 5]^3*c[4]^4+600*c[6]*c[7]^2*c[5]^4*c[4]^5+200*c[6]^2*c[7]^3*c[4]^5*c[5 ]+200*c[7]^2*c[4]^7*c[6]*c[5]+1100*c[5]^4*c[4]^4*c[6]^2-720*c[5]^4*c[4 ]^3*c[6]^2+1750*c[7]^2*c[5]^4*c[6]^2*c[4]^3-1230*c[7]^2*c[5]^4*c[6]^2* c[4]^2+150*c[5]^4*c[7]^3*c[4]^3*c[6]+300*c[5]^4*c[4]^6*c[6]+480*c[5]^3 *c[4]^4*c[7]^2*c[6]+1420*c[5]^4*c[4]^3*c[7]^2*c[6]+20*c[5]^2*c[4]^4*c[ 7]^2*c[6]+130*c[5]^4*c[4]^3*c[6]^2*c[7]-40*c[7]^2*c[4]^4*c[6]^2*c[5]-3 50*c[7]^2*c[4]^4*c[5]^3*c[6]^2+230*c[7]^2*c[4]^4*c[6]^2*c[5]^2+10*c[4] ^6*c[5]^2+188*c[5]^2*c[6]*c[4]^5-400*c[6]^2*c[7]^3*c[5]^2*c[4]^5-600*c [6]^2*c[7]^2*c[5]^4*c[4]^4+120*c[5]^2*c[4]^7*c[6]-94*c[4]^4*c[6]*c[5]^ 3+6*c[6]*c[5]*c[4]^5+60*c[5]^5*c[6]^2*c[4]^3-200*c[6]^2*c[4]^6*c[5]*c[ 7]^2-420*c[7]^3*c[6]^2*c[5]*c[4]^4+1550*c[5]^4*c[4]^5*c[6]*c[7]+500*c[ 7]^3*c[4]^4*c[6]^2*c[5]^2-400*c[5]^2*c[4]^7*c[6]*c[7]^2+6*c[5]^3*c[4]^ 4+103*c[6]^2*c[7]^3*c[5]^2*c[4]-260*c[6]^2*c[7]^3*c[5]^2*c[4]^2+540*c[ 5]^3*c[7]^3*c[4]^2*c[6]-97*c[6]^2*c[5]*c[7]^3*c[4]^2+30*c[6]^2*c[7]^3* c[5]^3-4*c[7]^3*c[4]*c[5]^3)/(-240*c[5]^2*c[6]*c[4]^4+40*c[6]*c[5]*c[4 ]^4-9*c[5]^3*c[4]-100*c[4]^5*c[6]*c[5]^3-350*c[5]^4*c[4]^3*c[6]+30*c[5 ]^5*c[4]^2+2*c[5]^3-2*c[4]^3-50*c[5]^5*c[4]^3-3*c[5]^4+3*c[4]^4+50*c[4 ]^5*c[5]^3+100*c[5]^5*c[4]^3*c[6]-3*c[6]*c[5]^3+3*c[6]*c[4]^3+9*c[5]*c [4]^3-6*c[5]*c[6]*c[4]^2+4*c[5]*c[4]^2+240*c[6]*c[5]^4*c[4]^2+5*c[6]*c [5]^4-13*c[6]*c[5]*c[4]^3+13*c[5]^3*c[6]*c[4]-4*c[5]^2*c[4]+68*c[5]^3* c[4]^2+100*c[5]^2*c[6]*c[4]^3-100*c[5]^3*c[6]*c[4]^2-140*c[5]^4*c[4]^2 +140*c[4]^4*c[5]^2+180*c[5]^4*c[4]^3+6*c[5]^2*c[6]*c[4]-68*c[5]^2*c[4] ^3-23*c[4]^4*c[5]-40*c[5]^4*c[4]*c[6]-30*c[4]^5*c[5]^2-5*c[6]*c[4]^4-5 0*c[5]^5*c[6]*c[4]^2+50*c[5]^2*c[6]*c[4]^5+350*c[4]^4*c[6]*c[5]^3+23*c [5]^4*c[4]-180*c[5]^3*c[4]^4)/(-c[4]+c[6])/c[4]^2, `b*`[7] = 1/60*(-30 *c[5]^3*c[4]-18*c[6]+6*c[5]+6*c[4]-9*c[5]^2-9*c[4]^2+27*c[6]^2-97*c[6] *c[5]^2+66*c[6]*c[5]+60*c[6]*c[5]^3-100*c[6]^2*c[5]^3-100*c[6]^2*c[4]^ 3+180*c[6]^2*c[4]^2-97*c[6]*c[4]^2+60*c[6]*c[4]^3-30*c[5]*c[4]^3+312*c [5]*c[6]*c[4]^2+107*c[5]*c[4]^2-215*c[5]*c[6]*c[4]-66*c[5]*c[4]+66*c[6 ]*c[4]-840*c[6]^2*c[4]^2*c[5]+522*c[6]^2*c[4]*c[5]+500*c[6]^2*c[5]*c[4 ]^3+180*c[6]^2*c[5]^2-117*c[6]^2*c[5]-230*c[6]*c[5]*c[4]^3-230*c[5]^3* c[6]*c[4]+107*c[5]^2*c[4]+50*c[5]^3*c[4]^2+380*c[5]^2*c[6]*c[4]^3-460* c[5]^2*c[6]*c[4]^2+380*c[5]^3*c[6]*c[4]^2+600*c[5]^3*c[6]^2*c[4]^3-900 *c[6]^2*c[5]^2*c[4]^3+1420*c[6]^2*c[5]^2*c[4]^2-300*c[6]*c[5]^3*c[4]^3 +500*c[6]^2*c[5]^3*c[4]-900*c[6]^2*c[5]^3*c[4]^2-180*c[5]^2*c[4]^2-840 *c[6]^2*c[4]*c[5]^2-117*c[6]^2*c[4]+312*c[5]^2*c[6]*c[4]+50*c[5]^2*c[4 ]^3)/c[7]/(-3*c[7]^3*c[6]+3*c[7]^2*c[6]^2-9*c[6]^2*c[7]^2*c[5]-8*c[7]* c[6]^2*c[5]^3+8*c[7]^2*c[6]*c[5]^3-c[5]*c[6]*c[7]*c[4]-c[7]^2*c[4]^2-3 0*c[6]^2*c[7]*c[5]^2*c[4]^3-28*c[6]^2*c[7]^2*c[5]^2*c[4]+30*c[6]^2*c[7 ]^2*c[5]^2*c[4]^2-30*c[6]*c[7]*c[5]^3*c[4]^3+9*c[6]^2*c[7]*c[4]^2-3*c[ 6]^2*c[7]*c[4]+2*c[6]*c[7]^2*c[4]-9*c[6]*c[7]^2*c[4]^2-2*c[7]^2*c[5]*c [4]-c[7]^2*c[5]^2+c[6]*c[7]*c[4]^2+8*c[7]^2*c[6]*c[4]^3-4*c[5]*c[7]^2* c[4]^3+c[5]*c[7]*c[4]^2-c[5]*c[6]*c[4]^2+32*c[6]*c[5]*c[7]^2*c[4]^2-9* c[7]^2*c[5]*c[6]*c[4]-4*c[5]*c[7]*c[6]*c[4]^3-9*c[6]^2*c[4]^2*c[5]+3*c [6]^2*c[4]*c[5]+c[7]^3*c[4]+9*c[7]^3*c[4]*c[6]-9*c[7]^3*c[4]*c[5]-20*c [5]^2*c[6]*c[7]*c[4]^2+8*c[6]^2*c[5]*c[4]^3+28*c[7]^3*c[4]^2*c[6]*c[5] +18*c[6]^2*c[7]*c[4]*c[5]+28*c[7]*c[6]^2*c[5]*c[4]^3-3*c[6]^2*c[7]*c[5 ]+9*c[6]^2*c[7]*c[5]^2+c[6]*c[7]*c[5]^2-8*c[6]*c[7]^3*c[5]^2-9*c[6]*c[ 7]^2*c[5]^2+8*c[6]^2*c[7]^2*c[5]^2+2*c[6]*c[7]^2*c[5]-36*c[6]^2*c[4]^2 *c[7]*c[5]-28*c[6]^2*c[7]^2*c[4]^2*c[5]-28*c[7]^3*c[4]*c[6]*c[5]-36*c[ 6]^2*c[5]^2*c[7]*c[4]+56*c[6]^2*c[5]^2*c[7]*c[4]^2+28*c[6]*c[5]^2*c[7] *c[4]^3+28*c[6]*c[5]^3*c[7]*c[4]^2+28*c[6]^2*c[5]^3*c[7]*c[4]+9*c[6]*c [7]^3*c[5]+9*c[7]^2*c[5]*c[4]^2+28*c[7]^3*c[5]^2*c[6]*c[4]+32*c[7]^2*c [5]^2*c[6]*c[4]+8*c[6]^2*c[7]^2*c[4]^2-9*c[6]^2*c[7]^2*c[4]-4*c[5]^3*c [6]*c[7]*c[4]+28*c[6]^2*c[7]^2*c[4]*c[5]-4*c[5]^2*c[6]*c[4]^3-9*c[5]^2 *c[7]*c[4]^2+9*c[5]^2*c[6]*c[4]^2+4*c[5]^2*c[7]*c[4]^3+4*c[5]^3*c[7]*c [4]^2-4*c[5]^3*c[6]*c[4]^2+30*c[5]^3*c[6]^2*c[4]^3-28*c[6]^2*c[5]^2*c[ 4]^3+28*c[6]^2*c[5]^2*c[4]^2+8*c[6]^2*c[5]^3*c[4]-28*c[6]^2*c[5]^3*c[4 ]^2-9*c[6]^2*c[4]*c[5]^2-8*c[7]*c[6]^2*c[4]^3+30*c[7]^2*c[5]^2*c[6]*c[ 4]^3-30*c[7]^3*c[5]^2*c[6]*c[4]^2+30*c[7]^2*c[5]^3*c[6]*c[4]^2-8*c[7]^ 3*c[4]^2*c[6]+4*c[7]^3*c[4]^2*c[5]+c[7]^3*c[5]-28*c[7]^2*c[6]*c[5]*c[4 ]^3-56*c[7]^2*c[5]^2*c[6]*c[4]^2-28*c[7]^2*c[5]^3*c[6]*c[4]-c[5]^2*c[6 ]*c[4]+c[5]^2*c[7]*c[4]+4*c[7]^3*c[5]^2*c[4]+9*c[7]^2*c[5]^2*c[4]-30*c [6]^2*c[7]*c[5]^3*c[4]^2-8*c[7]^2*c[5]^2*c[4]^2-4*c[7]^2*c[5]^3*c[4]), a[6,4] = -1/2*(120*c[5]^2*c[6]*c[4]^4-12*c[6]*c[5]*c[4]^4-90*c[5]^2*c [6]^3*c[4]^2-60*c[5]*c[6]^2*c[4]^4-2*c[6]^3*c[4]^2+20*c[5]^4*c[4]^3*c[ 6]+40*c[6]^2*c[4]^5*c[5]+36*c[4]*c[5]^2*c[6]^3-20*c[5]^4*c[6]^2*c[4]^2 +8*c[6]^2*c[5]^3+2*c[6]^2*c[4]^3-24*c[5]*c[6]^3*c[4]^2-6*c[6]^2*c[4]^2 *c[5]+36*c[6]^2*c[5]*c[4]^3+60*c[4]^3*c[5]*c[6]^3-3*c[6]*c[5]*c[4]^3-1 1*c[5]^3*c[6]*c[4]+3*c[5]^3*c[4]^2-24*c[5]^2*c[6]*c[4]^3+13*c[5]^2*c[6 ]*c[4]^2+48*c[5]^3*c[6]*c[4]^2-60*c[5]^3*c[6]^2*c[4]^3-30*c[6]^2*c[5]^ 2*c[4]^3-12*c[6]^2*c[5]^2*c[4]^2-150*c[6]*c[5]^3*c[4]^3-48*c[6]^2*c[5] ^3*c[4]+150*c[6]^2*c[5]^3*c[4]^2-2*c[6]^2*c[4]*c[5]^2-5*c[5]^2*c[4]^3+ 2*c[4]^4*c[5]-6*c[5]^2*c[6]^3+60*c[6]^3*c[4]^3*c[5]^2+7*c[6]^3*c[4]*c[ 5]-60*c[5]^2*c[6]*c[4]^5-40*c[6]^3*c[4]^4*c[5]+60*c[4]^4*c[6]*c[5]^3)* c[6]/(c[4]^3-2*c[5]*c[4]^2+30*c[5]^2*c[4]^3-6*c[5]*c[4]^3-10*c[4]^4*c[ 5]^2-c[5]^3+10*c[5]^4*c[4]^2-30*c[5]^3*c[4]^2+6*c[5]^3*c[4]+2*c[5]^2*c [4])/c[4]^2, a[7,3] = 3/4*1/c[4]^2*c[7]*(-480*c[5]^2*c[6]*c[4]^4+15*c[ 6]*c[5]*c[4]^4+300*c[4]^2*c[5]^2*c[7]^3+60*c[7]^3*c[5]*c[4]^3-510*c[7] ^3*c[5]^2*c[4]^3+54*c[7]*c[5]*c[4]^4+450*c[5]^2*c[4]^4*c[7]^2+150*c[4] ^3*c[7]^3*c[5]^3-150*c[7]^2*c[5]^2*c[4]^5-450*c[4]^5*c[6]*c[5]^3-3*c[7 ]^3*c[4]^2+90*c[5]^2*c[4]^5*c[7]+120*c[5]^3*c[4]^4*c[7]+150*c[7]^3*c[5 ]^2*c[4]^4+100*c[5]^4*c[4]^2*c[7]^2*c[6]-180*c[6]*c[4]^5*c[5]*c[7]+30* c[6]*c[5]^2*c[4]^4*c[7]+60*c[5]^4*c[4]^3*c[6]-150*c[5]^4*c[4]^4*c[6]-2 00*c[7]^2*c[4]^4*c[5]^3+300*c[6]*c[4]^5*c[5]*c[7]^2-50*c[7]^2*c[5]^4*c [4]^3+24*c[7]*c[6]*c[5]^3-40*c[7]^2*c[6]*c[5]^3-6*c[5]*c[6]*c[7]*c[4]+ 360*c[7]*c[4]^4*c[6]*c[5]+150*c[7]^2*c[5]^3*c[4]^3+2*c[7]^2*c[4]^2+6*c [5]^2*c[7]^3+330*c[6]*c[7]*c[5]^3*c[4]^3-450*c[7]^2*c[6]*c[5]*c[4]^4-1 2*c[7]*c[6]*c[4]^3-6*c[6]*c[7]^2*c[4]-2*c[7]^2*c[5]*c[4]-4*c[7]^2*c[5] ^2+6*c[6]*c[7]*c[4]^2+20*c[7]^2*c[6]*c[4]^3-10*c[5]*c[7]^2*c[4]^3+6*c[ 5]*c[4]^3-6*c[5]*c[7]*c[4]^2-6*c[5]*c[6]*c[4]^2+150*c[6]*c[5]*c[7]^2*c [4]^2-72*c[7]^2*c[5]*c[6]*c[4]-270*c[5]*c[7]*c[6]*c[4]^3+9*c[7]^3*c[4] *c[6]+3*c[7]^3*c[4]*c[5]-306*c[5]^2*c[6]*c[7]*c[4]^2+900*c[4]^5*c[6]*c [7]*c[5]^3+600*c[5]^2*c[6]*c[7]^3*c[4]^4-420*c[7]^3*c[4]^2*c[6]*c[5]+9 0*c[6]*c[5]*c[7]*c[4]^2-12*c[6]*c[7]*c[5]^2+30*c[6]*c[7]^3*c[5]^2+12*c [6]*c[7]^2*c[5]+123*c[7]^3*c[4]*c[6]*c[5]+360*c[6]*c[5]^2*c[7]*c[4]^3+ 360*c[6]*c[5]^3*c[7]*c[4]^2-18*c[6]*c[7]^3*c[5]+30*c[5]^4*c[4]^3*c[7]- 80*c[7]^2*c[5]^2*c[4]^3+24*c[7]^2*c[5]*c[4]^2-30*c[7]*c[5]*c[4]^3+3*c[ 6]*c[5]*c[4]^3-240*c[7]^3*c[5]^2*c[6]*c[4]-24*c[5]^3*c[6]*c[4]-12*c[5] ^3*c[7]*c[4]-144*c[5]^3*c[6]*c[7]*c[4]+9*c[5]^3*c[4]^2+294*c[5]^2*c[6] *c[4]^3-42*c[5]^2*c[7]*c[4]^2-75*c[5]^2*c[6]*c[4]^2+252*c[5]^2*c[7]*c[ 4]^3+18*c[5]^3*c[7]*c[4]^2+189*c[5]^3*c[6]*c[4]^2-360*c[5]^2*c[4]^4*c[ 7]-720*c[6]*c[5]^3*c[4]^3-120*c[5]^3*c[7]*c[4]^3-60*c[5]^4*c[7]*c[4]^2 *c[6]-6*c[5]^2*c[4]^2-780*c[7]^2*c[5]^2*c[6]*c[4]^3+810*c[7]^3*c[5]^2* c[6]*c[4]^2-1230*c[7]^2*c[5]^3*c[6]*c[4]^2-15*c[7]^3*c[4]^2*c[6]-21*c[ 7]^3*c[4]^2*c[5]+20*c[7]^2*c[6]*c[5]*c[4]^3+220*c[7]^2*c[5]^2*c[6]*c[4 ]^2+15*c[4]^4*c[5]^2+320*c[7]^2*c[5]^3*c[6]*c[4]+1850*c[7]^2*c[5]^3*c[ 6]*c[4]^3-60*c[5]^3*c[7]^3*c[4]^2+12*c[5]^2*c[6]*c[4]+12*c[5]^2*c[7]*c [4]-66*c[7]^3*c[5]^2*c[4]+24*c[7]^2*c[5]^2*c[4]+72*c[5]^2*c[6]*c[7]*c[ 4]-110*c[7]^2*c[5]^2*c[4]^2+20*c[7]^2*c[5]^3*c[4]-9*c[4]^4*c[5]-60*c[7 ]^2*c[4]^4*c[5]-600*c[5]^2*c[6]*c[4]^5*c[7]^2-1200*c[7]^3*c[5]^2*c[4]^ 3*c[6]+630*c[7]^3*c[6]*c[5]*c[4]^3+300*c[5]^4*c[4]^4*c[6]*c[7]-1800*c[ 5]^3*c[4]^4*c[6]*c[7]-15*c[5]^3*c[4]^3-300*c[7]^3*c[4]^4*c[5]*c[6]-20* c[7]^2*c[5]^3*c[4]^2-800*c[5]^3*c[4]^4*c[7]^2*c[6]-200*c[5]^4*c[4]^3*c [7]^2*c[6]+1150*c[5]^2*c[4]^4*c[7]^2*c[6]+180*c[5]^2*c[6]*c[4]^5+1140* c[4]^4*c[6]*c[5]^3)/(10*c[5]^3*c[4]^2+10*c[5]^2*c[4]^3-30*c[5]^2*c[4]^ 2+6*c[5]^2*c[4]-c[5]^2+6*c[5]*c[4]^2+c[5]*c[4]-c[4]^2)/(10*c[5]*c[6]*c [4]-5*c[6]*c[5]-5*c[6]*c[4]+3*c[6]-5*c[5]*c[4]+3*c[5]+3*c[4]-2), `b*`[ 1] = 1/60*(30*c[5]^3*c[4]+42*c[7]*c[6]^2-103*c[5]*c[7]*c[4]+18*c[6]-6* c[5]-6*c[4]-27*c[7]*c[6]+9*c[5]^2+9*c[4]^2-180*c[7]*c[6]^2*c[5]^3+100* c[7]*c[6]*c[5]^3-27*c[6]^2-354*c[5]*c[6]*c[7]*c[4]+97*c[6]*c[5]^2-66*c [6]*c[5]-60*c[6]*c[5]^3+100*c[6]^2*c[5]^3+9*c[7]*c[5]+100*c[6]^2*c[4]^ 3-2040*c[6]^2*c[7]*c[5]^2*c[4]^3-600*c[6]*c[7]*c[5]^3*c[4]^3-14*c[7]*c [5]^2+100*c[7]*c[6]*c[4]^3+310*c[6]^2*c[7]*c[4]^2-192*c[6]^2*c[7]*c[4] -180*c[6]^2*c[4]^2-14*c[7]*c[4]^2+9*c[7]*c[4]+103*c[6]*c[7]*c[5]+97*c[ 6]*c[4]^2-60*c[6]*c[4]^3+103*c[6]*c[7]*c[4]-156*c[6]*c[7]*c[4]^2+30*c[ 5]*c[4]^3+174*c[5]*c[7]*c[4]^2-312*c[5]*c[6]*c[4]^2-107*c[5]*c[4]^2+21 5*c[5]*c[6]*c[4]+66*c[5]*c[4]-66*c[6]*c[4]-410*c[5]*c[7]*c[6]*c[4]^3+8 40*c[6]^2*c[4]^2*c[5]-522*c[6]^2*c[4]*c[5]-760*c[5]^2*c[6]*c[7]*c[4]^2 -500*c[6]^2*c[5]*c[4]^3+932*c[6]^2*c[7]*c[4]*c[5]+1020*c[7]*c[6]^2*c[5 ]*c[4]^3-192*c[6]^2*c[7]*c[5]+310*c[6]^2*c[7]*c[5]^2+526*c[6]*c[5]*c[7 ]*c[4]^2-156*c[6]*c[7]*c[5]^2-180*c[6]^2*c[5]^2-1610*c[6]^2*c[4]^2*c[7 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*c[4]^2+9*c[5]*c[4]-c[5]-c[4])/c[5]/c[6]/c[7]/c[4], a[8,4] = -1/2*(734 *c[5]^2*c[6]*c[4]^4-203*c[6]*c[5]*c[4]^4+3190*c[5]^4*c[4]^4*c[6]^2*c[7 ]-93*c[7]*c[5]*c[4]^4-636*c[5]^2*c[4]^4*c[7]^2-10*c[5]^3*c[4]-520*c[6] *c[4]^7*c[7]*c[5]-481*c[4]^5*c[6]^2*c[5]^2+645*c[7]^2*c[5]^2*c[4]^5-57 4*c[4]^5*c[6]*c[5]^3-172*c[5]*c[6]^2*c[4]^4+6*c[7]^2*c[4]^3+100*c[5]^5 *c[4]^3*c[7]^2*c[6]^2+60*c[4]^8*c[5]^2-600*c[5]^2*c[4]^6*c[6]^2*c[7]^2 +9*c[4]^5*c[5]*c[7]-164*c[5]^2*c[4]^5*c[7]+900*c[5]^3*c[4]^6*c[6]^2*c[ 7]+600*c[5]^3*c[4]^6*c[6]^2*c[7]^2-1104*c[5]^3*c[4]^4*c[7]-18*c[4]^4*c [6]^2+90*c[7]^2*c[4]^5*c[5]^3+1380*c[6]^2*c[4]^5*c[7]*c[5]^3-750*c[7]* c[5]^2*c[4]^6*c[6]+675*c[5]^4*c[4]^2*c[7]^2*c[6]+8*c[6]^2*c[4]^5-339*c [6]^2*c[7]*c[5]^4*c[4]^2-315*c[5]^2*c[4]^6*c[7]+640*c[6]*c[5]*c[7]^2*c [4]^6+46*c[6]*c[4]^5*c[5]*c[7]-1014*c[6]*c[5]^2*c[4]^4*c[7]+2100*c[5]^ 4*c[7]*c[4]^6*c[6]-390*c[6]^2*c[4]^6*c[5]+860*c[4]^6*c[6]^2*c[5]^2+140 *c[5]^5*c[4]^3*c[7]-1740*c[5]^2*c[7]^2*c[4]^6*c[6]-613*c[5]^4*c[4]^3*c 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[4]^4*c[6]^2-200*c[5]^5*c[7]*c[4]^6*c[6]-900*c[5]^3*c[7]*c[4]^7*c[6]+6 0*c[5]^5*c[4]^2*c[6]^2*c[7]-450*c[5]^5*c[4]^4*c[6]^2*c[7]-170*c[5]^5*c [4]^4*c[6]+200*c[5]^5*c[7]*c[4]^5*c[6]+143*c[6]^2*c[5]^4*c[7]*c[4]+34* c[5]^4*c[6]^2*c[4]-283*c[5]^4*c[6]^2*c[4]^2-530*c[5]^3*c[4]^6*c[6]^2+9 *c[4]^4*c[7]+600*c[5]^4*c[7]^2*c[4]^6*c[6]-140*c[5]^5*c[4]^4*c[7]-55*c [4]^5*c[5]*c[7]^2+10*c[6]^2*c[4]^3+100*c[5]^5*c[4]^4*c[7]^2+140*c[4]^6 *c[6]*c[5]^3-967*c[6]^2*c[7]*c[5]^2*c[4]^3-90*c[6]^2*c[7]^2*c[5]^2*c[4 ]+225*c[6]^2*c[7]^2*c[5]^2*c[4]^2-3082*c[6]*c[7]*c[5]^3*c[4]^3+185*c[5 ]^3*c[4]^6+40*c[7]^2*c[6]^2*c[5]^4+89*c[5]^4*c[7]*c[4]^2+284*c[7]^2*c[ 6]*c[5]*c[4]^4-100*c[7]^2*c[5]^5*c[6]^2*c[4]^2-340*c[5]^4*c[4]*c[7]^2* c[6]^2+200*c[5]^5*c[7]*c[4]^5*c[6]^2+7*c[6]*c[4]^5+27*c[7]*c[6]*c[4]^3 -4*c[6]^2*c[7]*c[4]^2-440*c[5]^2*c[4]^6*c[7]^2+120*c[4]^7*c[6]^2*c[5]- 9*c[7]*c[4]^3-10*c[6]*c[4]^3-22*c[7]^2*c[6]*c[4]^3-49*c[5]*c[7]^2*c[4] ^3-29*c[5]*c[4]^3+9*c[5]*c[7]*c[4]^2+10*c[5]*c[6]*c[4]^2-4*c[5]*c[4]^2 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6]+4*c[5]^2*c[4]-135*c[5]^3*c[6]*c[7]*c[4]+71*c[5]^3*c[4]^2-21*c[6]^2* c[7]^2*c[4]*c[5]-480*c[5]^2*c[6]*c[4]^3+42*c[5]^2*c[7]*c[4]^2+37*c[5]^ 2*c[6]*c[4]^2-480*c[5]^2*c[7]*c[4]^3-104*c[5]^3*c[7]*c[4]^2-97*c[5]^3* c[6]*c[4]^2-1536*c[5]^3*c[6]^2*c[4]^3+190*c[5]^5*c[4]^3*c[6]^2*c[7]+55 7*c[4]^3*c[7]^2*c[5]^2*c[6]^2-10*c[6]*c[7]*c[4]^5-730*c[4]^5*c[5]^3*c[ 6]^2-14*c[7]*c[5]^4*c[4]+10*c[7]^2*c[5]^4*c[4]+646*c[5]^2*c[4]^4*c[7]+ 498*c[6]^2*c[5]^2*c[4]^3-95*c[6]^2*c[5]^2*c[4]^2+744*c[6]*c[5]^3*c[4]^ 3-46*c[6]^2*c[5]^3*c[4]+334*c[6]^2*c[5]^3*c[4]^2+854*c[5]^3*c[7]*c[4]^ 3-490*c[5]^4*c[7]*c[4]^2*c[6]-30*c[7]*c[5]*c[4]^7-19*c[5]^2*c[4]^2+18* c[6]^2*c[4]*c[5]^2-472*c[6]*c[5]*c[4]^6+360*c[5]*c[4]^6*c[7]*c[6]-10*c [7]*c[6]^2*c[4]^3-811*c[7]^2*c[5]^2*c[6]*c[4]^3-993*c[7]^2*c[5]^3*c[6] *c[4]^2-44*c[5]^4*c[4]^2-451*c[4]^3*c[7]^2*c[5]*c[6]^2+10*c[6]*c[7]*c[ 4]^6-14*c[7]^2*c[6]*c[5]*c[4]^3-20*c[5]^5*c[4]^5+118*c[7]^2*c[5]^2*c[6 ]*c[4]^2-249*c[4]^4*c[5]^2+170*c[7]^2*c[5]^3*c[6]*c[4]+3829*c[7]^2*c[5 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9*c[6]*c[4]^4+600*c[5]^4*c[7]*c[4]^6*c[6]^2-2350*c[5]^4*c[4]^5*c[6]^2* c[7]-600*c[5]^3*c[7]^2*c[6]*c[4]^7-204*c[7]*c[6]^2*c[5]*c[4]^4-3630*c[ 5]^3*c[4]^4*c[6]^2*c[7]+740*c[5]^3*c[4]^3*c[6]^2*c[7]^2-2120*c[5]^4*c[ 4]^4*c[6]*c[7]+1460*c[5]^2*c[4]^4*c[7]*c[6]^2+2387*c[5]^3*c[4]^4*c[6]* c[7]+2548*c[5]^4*c[4]^3*c[6]*c[7]-498*c[5]^3*c[4]^3+100*c[5]^2*c[4]^7* c[7]^2+26*c[7]^2*c[5]^3*c[4]^2+600*c[6]*c[5]^3*c[4]^7+100*c[6]*c[5]^5* c[4]^6+300*c[6]*c[5]^4*c[4]^7-600*c[5]^4*c[7]*c[4]^7*c[6]+1600*c[5]^4* c[4]^5*c[6]^2-1500*c[6]^2*c[7]^2*c[5]^3*c[4]^5-2700*c[6]*c[7]^2*c[5]^4 *c[4]^5-200*c[7]^2*c[4]^7*c[6]*c[5]-2430*c[5]^4*c[4]^4*c[6]^2+1429*c[5 ]^4*c[4]^3*c[6]^2-2010*c[7]^2*c[5]^4*c[6]^2*c[4]^3+1410*c[7]^2*c[5]^4* c[6]^2*c[4]^2-1160*c[5]^4*c[4]^6*c[6]-4450*c[5]^3*c[4]^4*c[7]^2*c[6]-3 280*c[5]^4*c[4]^3*c[7]^2*c[6]+334*c[5]^2*c[4]^4*c[7]^2*c[6]-1220*c[5]^ 4*c[4]^3*c[6]^2*c[7]+810*c[7]^2*c[4]^4*c[6]^2*c[5]+1030*c[7]^2*c[4]^4* c[5]^3*c[6]^2-1850*c[7]^2*c[4]^4*c[6]^2*c[5]^2+354*c[4]^6*c[5]^2-629*c [5]^2*c[6]*c[4]^5-300*c[4]^8*c[6]*c[5]^3-600*c[5]^3*c[7]*c[6]^2*c[4]^7 +1300*c[6]^2*c[7]^2*c[5]^4*c[4]^4-200*c[6]^2*c[7]*c[4]^7*c[5]+100*c[5] ^4*c[4]^6*c[7]-320*c[5]^2*c[6]^2*c[4]^7-940*c[5]^2*c[4]^7*c[6]-48*c[7] ^2*c[6]^2*c[5]^3-535*c[4]^4*c[6]*c[5]^3+306*c[6]*c[5]*c[4]^5-160*c[5]^ 5*c[6]^2*c[4]^3+600*c[5]^2*c[7]*c[4]^7*c[6]^2+200*c[6]^2*c[4]^6*c[5]*c [7]^2-1510*c[5]^4*c[4]^5*c[6]*c[7]+6*c[5]^4*c[4]+600*c[5]^2*c[4]^7*c[6 ]*c[7]^2+814*c[5]^3*c[4]^4)/(690*c[5]^2*c[6]*c[4]^4-110*c[6]*c[5]*c[4] ^4-110*c[7]*c[5]*c[4]^4+48*c[5]^3*c[4]+200*c[4]^5*c[6]*c[5]^3-150*c[5] ^2*c[4]^5*c[7]-750*c[5]^3*c[4]^4*c[7]-930*c[6]*c[5]^2*c[4]^4*c[7]-200* c[5]^5*c[4]^3*c[7]+750*c[5]^4*c[4]^3*c[6]-120*c[5]^5*c[4]^2-10*c[5]^3+ 10*c[4]^3+150*c[5]^5*c[4]^3-15*c[7]*c[6]*c[5]^3+12*c[5]^4-12*c[4]^4+20 0*c[6]*c[5]^2*c[4]^5*c[7]-150*c[4]^5*c[5]^3-200*c[5]^5*c[4]^3*c[6]+150 *c[7]*c[4]^4*c[6]*c[5]+12*c[6]*c[5]^3+15*c[4]^4*c[7]-690*c[5]^4*c[7]*c [4]^2-15*c[5]^4*c[7]+150*c[5]^5*c[7]*c[4]^2+15*c[7]*c[6]*c[4]^3-12*c[7 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*c[7]-1100*c[5]^4*c[4]^3*c[6]*c[7]+150*c[5]^5*c[6]*c[4]^2-150*c[5]^2*c [6]*c[4]^5-750*c[4]^4*c[6]*c[5]^3-87*c[5]^4*c[4]+570*c[5]^3*c[4]^4)/(c [6]*c[4]-c[4]^2-c[7]*c[6]+c[7]*c[4])/c[4]^2, a[8,3] = 3/4*(-1560*c[5]^ 2*c[6]*c[4]^4+585*c[6]*c[5]*c[4]^4-45*c[7]*c[5]*c[4]^4-12*c[5]^3*c[4]- 450*c[4]^5*c[6]*c[5]^3+3*c[5]*c[7]*c[4]-150*c[5]^2*c[4]^5*c[7]-200*c[5 ]^3*c[4]^4*c[7]+300*c[6]*c[4]^5*c[5]*c[7]+2860*c[6]*c[5]^2*c[4]^4*c[7] +160*c[5]^4*c[4]^3*c[6]-150*c[5]^4*c[4]^4*c[6]-4*c[5]^2+2*c[4]^2-40*c[ 7]*c[6]*c[5]^3+132*c[5]*c[6]*c[7]*c[4]-900*c[6]*c[5]^2*c[4]^5*c[7]-30* c[6]*c[5]^2+12*c[6]*c[5]-960*c[7]*c[4]^4*c[6]*c[5]+24*c[6]*c[5]^3+3140 *c[6]*c[7]*c[5]^3*c[4]^3+6*c[7]*c[5]^2+20*c[7]*c[6]*c[4]^3-3*c[7]*c[4] ^2-18*c[6]*c[7]*c[5]+15*c[6]*c[4]^2-12*c[6]*c[4]^3+9*c[6]*c[7]*c[4]-24 *c[6]*c[7]*c[4]^2-54*c[5]*c[4]^3-12*c[5]*c[7]*c[4]^2+348*c[5]*c[6]*c[4 ]^2+9*c[5]*c[4]^2-87*c[5]*c[6]*c[4]-2*c[5]*c[4]-6*c[6]*c[4]+1035*c[5]* c[7]*c[6]*c[4]^3+1593*c[5]^2*c[6]*c[7]*c[4]^2+900*c[4]^5*c[6]*c[7]*c[5 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c[6]*c[5]^2*c[4]^4*c[7]-200*c[5]^4*c[4]^3*c[6]-10*c[5]^2-10*c[4]^2+12* c[5]^3+12*c[4]^3+20*c[7]*c[6]*c[5]^3+15*c[5]*c[6]*c[7]*c[4]+12*c[6]*c[ 5]^2-15*c[6]*c[5]^3-1300*c[6]*c[7]*c[5]^3*c[4]^3+150*c[5]^4*c[7]*c[4]^ 2+12*c[7]*c[5]^2+20*c[7]*c[6]*c[4]^3+12*c[7]*c[4]^2-15*c[7]*c[4]^3+12* c[6]*c[4]^2-15*c[6]*c[4]^3-15*c[6]*c[7]*c[4]^2-87*c[5]*c[4]^3-72*c[5]* c[7]*c[4]^2-72*c[5]*c[6]*c[4]^2+60*c[5]*c[4]^2-12*c[5]*c[6]*c[4]+10*c[ 5]*c[4]-150*c[5]*c[7]*c[6]*c[4]^3-15*c[7]*c[5]^3-660*c[5]^2*c[6]*c[7]* c[4]^2+90*c[6]*c[5]*c[7]*c[4]^2-15*c[6]*c[7]*c[5]^2+150*c[6]*c[5]^4*c[ 4]^2+930*c[6]*c[5]^2*c[7]*c[4]^3+930*c[6]*c[5]^3*c[7]*c[4]^2-200*c[5]^ 4*c[4]^3*c[7]+110*c[7]*c[5]*c[4]^3+110*c[6]*c[5]*c[4]^3+110*c[5]^3*c[6 ]*c[4]+110*c[5]^3*c[7]*c[4]+60*c[5]^2*c[4]-150*c[5]^3*c[6]*c[7]*c[4]+5 50*c[5]^3*c[4]^2-690*c[5]^2*c[6]*c[4]^3+520*c[5]^2*c[7]*c[4]^2+520*c[5 ]^2*c[6]*c[4]^2-690*c[5]^2*c[7]*c[4]^3-690*c[5]^3*c[7]*c[4]^2-690*c[5] ^3*c[6]*c[4]^2+150*c[5]^2*c[4]^4*c[7]+900*c[6]*c[5]^3*c[4]^3+900*c[5]^ 3*c[7]*c[4]^3-200*c[5]^4*c[7]*c[4]^2*c[6]-429*c[5]^2*c[4]^2-120*c[5]^4 *c[4]^2-120*c[4]^4*c[5]^2+150*c[5]^4*c[4]^3-72*c[5]^2*c[6]*c[4]-72*c[5 ]^2*c[7]*c[4]+90*c[5]^2*c[6]*c[7]*c[4]+550*c[5]^2*c[4]^3+300*c[5]^3*c[ 4]^4*c[6]*c[7]+300*c[5]^4*c[4]^3*c[6]*c[7]-690*c[5]^3*c[4]^3-200*c[4]^ 4*c[6]*c[5]^3+150*c[5]^3*c[4]^4), a[8,5] = -1/2*(893*c[5]^2*c[6]*c[4]^ 4-122*c[6]*c[5]*c[4]^4+850*c[5]^4*c[4]^4*c[6]^2*c[7]-101*c[7]*c[5]*c[4 ]^4-535*c[5]^2*c[4]^4*c[7]^2-44*c[5]^3*c[4]+160*c[4]^5*c[6]^2*c[5]^2+1 00*c[7]^2*c[5]^2*c[4]^5+170*c[4]^5*c[6]*c[5]^3+75*c[5]*c[6]^2*c[4]^4+1 6*c[7]^2*c[4]^3-9*c[5]*c[7]*c[4]-140*c[5]^2*c[4]^5*c[7]-780*c[5]^3*c[4 ]^4*c[7]-10*c[4]^4*c[6]^2-100*c[7]^2*c[4]^5*c[5]^3+450*c[6]^2*c[4]^5*c [7]*c[5]^3+510*c[5]^4*c[4]^2*c[7]^2*c[6]+490*c[6]^2*c[7]*c[5]^4*c[4]^2 -2224*c[6]*c[5]^2*c[4]^4*c[7]+10*c[5]^5*c[4]^3*c[7]-146*c[5]^4*c[4]^3* c[6]+90*c[5]^4*c[4]^5*c[6]+40*c[5]^4*c[4]^4*c[6]+570*c[7]^2*c[4]^4*c[5 ]^3+130*c[5]^4*c[4]^4*c[7]+25*c[7]^2*c[6]*c[4]^4+18*c[5]^5*c[4]^2-4*c[ 5]^2-4*c[4]^2-10*c[4]^4*c[7]^2+150*c[7]^2*c[5]^4*c[4]^3+5*c[5]^3-6*c[6 ]^2*c[7]^2*c[5]+10*c[4]^3-35*c[5]^5*c[4]^3-24*c[7]*c[6]^2*c[5]^3+27*c[ 7]*c[6]*c[5]^3-32*c[7]^2*c[6]*c[5]^3+23*c[5]*c[6]*c[7]*c[4]-c[5]^4-6*c [4]^4-60*c[7]*c[6]^2*c[5]*c[4]^5-45*c[5]^4*c[4]^2*c[7]^2+450*c[6]*c[5] ^2*c[4]^5*c[7]+10*c[6]*c[5]^2-80*c[4]^5*c[5]^3-260*c[5]^5*c[4]^3*c[6]+ 270*c[7]*c[4]^4*c[6]*c[5]+20*c[5]^5*c[4]^4-9*c[6]*c[5]^3+50*c[4]^4*c[6 ]^2*c[5]^5-600*c[5]^5*c[7]*c[4]^4*c[6]-760*c[7]^2*c[5]^3*c[4]^3+25*c[7 ]*c[4]^4*c[6]^2+240*c[5]^5*c[4]^4*c[6]+200*c[5]^5*c[7]*c[4]^5*c[6]+20* c[5]^5*c[4]^2*c[6]^2-90*c[6]^2*c[5]^4*c[7]*c[4]+71*c[5]^4*c[6]^2*c[4]- 383*c[5]^4*c[6]^2*c[4]^2+18*c[6]^2*c[5]^3+14*c[4]^4*c[7]-6*c[7]^2*c[4] ^2+16*c[6]^2*c[4]^3-1433*c[6]^2*c[7]*c[5]^2*c[4]^3-57*c[6]^2*c[7]^2*c[ 5]^2*c[4]+c[6]^2*c[7]^2*c[5]^2*c[4]^2-3185*c[6]*c[7]*c[5]^3*c[4]^3+100 *c[5]^4*c[7]*c[4]^2-195*c[7]^2*c[6]*c[5]*c[4]^4-15*c[5]^5*c[7]*c[4]^2- 20*c[6]^2*c[4]^4*c[7]^2+9*c[7]*c[5]^2+57*c[7]*c[6]*c[4]^3+19*c[6]^2*c[ 7]*c[4]^2-2*c[6]^2*c[7]*c[4]+5*c[5]^5*c[7]*c[4]+13*c[6]*c[7]^2*c[4]^2- 6*c[6]^2*c[4]^2+9*c[7]*c[4]^2-8*c[5]^4*c[6]^2+6*c[7]^2*c[5]*c[4]-6*c[7 ]^2*c[5]^2-23*c[7]*c[4]^3+10*c[6]*c[4]^2-26*c[6]*c[4]^3-21*c[6]*c[7]*c [4]^2-38*c[7]^2*c[6]*c[4]^3-110*c[5]*c[7]^2*c[4]^3-67*c[5]*c[4]^3-42*c [5]*c[7]*c[4]^2-49*c[5]*c[6]*c[4]^2+19*c[5]*c[4]^2-10*c[5]*c[6]*c[4]+4 *c[5]*c[4]-58*c[6]*c[5]*c[7]^2*c[4]^2-16*c[7]^2*c[5]*c[6]*c[4]-390*c[5 ]*c[7]*c[6]*c[4]^3+23*c[6]^2*c[4]^2*c[5]+8*c[6]^2*c[4]*c[5]-9*c[7]*c[5 ]^3+20*c[5]^4*c[4]^5-1308*c[5]^2*c[6]*c[7]*c[4]^2+900*c[5]^4*c[7]^2*c[ 4]^4*c[6]-450*c[4]^5*c[6]*c[7]*c[5]^3-106*c[6]^2*c[5]*c[4]^3-46*c[6]^2 *c[7]*c[4]*c[5]+25*c[7]*c[6]^2*c[5]*c[4]^3+4*c[6]^2*c[7]*c[5]+10*c[6]^ 2*c[7]*c[5]^2+97*c[6]*c[5]*c[7]*c[4]^2+10*c[5]^4*c[7]*c[6]-27*c[6]*c[7 ]*c[5]^2-10*c[6]^2*c[5]^2+22*c[6]*c[7]^2*c[5]^2+6*c[6]^2*c[7]^2*c[5]^2 +62*c[6]^2*c[4]^2*c[7]*c[5]-133*c[6]^2*c[7]^2*c[4]^2*c[5]-35*c[6]*c[5] ^4*c[4]^2+10*c[7]*c[6]^2*c[5]^4-7*c[6]*c[5]^4+140*c[4]^5*c[7]*c[5]^3-7 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c[4]^3-24*c[6]^2*c[5]^3*c[4]+285*c[5]^3*c[7]*c[4]^3+270*c[5]^4*c[7]*c[ 4]^2*c[6]+12*c[7]*c[6]^2*c[4]^3+340*c[7]^2*c[5]^2*c[6]*c[4]^3-410*c[7] ^2*c[5]^3*c[6]*c[4]^2-200*c[5]^5*c[4]^5*c[7]+70*c[4]^3*c[7]^2*c[5]*c[6 ]^2-42*c[7]^2*c[6]*c[5]*c[4]^3+150*c[5]^5*c[4]^5-30*c[7]^2*c[5]^2*c[6] *c[4]^2+12*c[4]^4*c[5]^2+87*c[7]^2*c[5]^3*c[6]*c[4]+510*c[7]^2*c[5]^3* c[6]*c[4]^3+690*c[5]^3*c[6]^2*c[4]^4-342*c[5]^4*c[4]^3-15*c[6]^2*c[4]^ 3*c[7]^2+510*c[5]^3*c[6]^2*c[7]*c[4]^3-20*c[5]^2*c[6]*c[7]*c[4]-410*c[ 6]^2*c[7]*c[5]^3*c[4]^2-70*c[6]^2*c[7]^2*c[4]*c[5]^3-10*c[5]^2*c[4]^3+ 24*c[7]^2*c[5]^2*c[4]^2-24*c[7]^2*c[5]^3*c[4]-200*c[7]^2*c[5]^6*c[4]^2 *c[6]+110*c[7]^2*c[4]*c[5]^5+1350*c[5]^5*c[4]^3*c[6]*c[7]+15*c[7]^2*c[ 4]^4*c[5]+300*c[6]^2*c[7]*c[5]^6*c[4]^3+510*c[7]^2*c[5]^3*c[6]^2*c[4]^ 2-550*c[5]^6*c[4]^2-600*c[7]*c[5]^6*c[4]^3+20*c[7]*c[6]^2*c[5]^5+570*c [5]^6*c[6]*c[4]^2+150*c[5]^2*c[6]*c[4]^5*c[7]^2+550*c[5]^4*c[4]^4+120* c[5]^7*c[4]^2+72*c[5]^4*c[4]*c[6]-40*c[5]^4*c[7]^2*c[4]*c[6]-57*c[5]^4 *c[4]*c[7]*c[6]+150*c[6]^2*c[5]^6*c[4]^2+150*c[4]^5*c[6]^2*c[7]*c[5]^2 -200*c[4]^5*c[6]^2*c[7]^2*c[5]^2-150*c[5]^7*c[7]*c[4]^2-110*c[6]^2*c[4 ]^4*c[5]^2+780*c[5]^5*c[4]^2*c[7]^2*c[6]-200*c[5]^5*c[4]^5*c[6]-600*c[ 5]^6*c[4]^3*c[6]-200*c[6]^2*c[7]*c[5]^6*c[4]^2-300*c[5]^4*c[4]^5*c[6]^ 2*c[7]+90*c[7]*c[6]^2*c[5]*c[4]^4+200*c[6]*c[5]^7*c[7]*c[4]^2-300*c[6] *c[5]^7*c[7]*c[4]^3-180*c[5]^3*c[4]^4*c[6]^2*c[7]-570*c[5]^4*c[4]^4*c[ 6]*c[7]-540*c[5]^2*c[4]^4*c[7]*c[6]^2+660*c[5]^3*c[4]^4*c[6]*c[7]+60*c [5]^4*c[4]^3*c[6]*c[7]+48*c[5]^3*c[4]^3-15*c[6]^2*c[5]^5-200*c[5]^6*c[ 6]^2*c[4]^3-15*c[5]^5*c[6]*c[7]-48*c[5]^5*c[4]+140*c[5]^5*c[6]*c[4]^2+ 200*c[5]^4*c[4]^5*c[6]^2+300*c[6]^2*c[7]^2*c[5]^3*c[4]^5-300*c[6]*c[7] ^2*c[5]^4*c[4]^5+700*c[7]*c[5]^6*c[4]^3*c[6]-750*c[5]^4*c[4]^4*c[6]^2+ 20*c[5]^5*c[7]^2*c[6]+1100*c[7]^2*c[5]^4*c[6]^2*c[4]^3-930*c[7]^2*c[5] ^4*c[6]^2*c[4]^2+300*c[7]^2*c[5]^6*c[4]^3*c[6]-20*c[5]^6*c[7]*c[6]-180 *c[5]^3*c[4]^4*c[7]^2*c[6]-750*c[5]^4*c[4]^3*c[7]^2*c[6]-540*c[5]^2*c[ 4]^4*c[7]^2*c[6]-750*c[5]^4*c[4]^3*c[6]^2*c[7]-150*c[7]^2*c[4]^4*c[6]^ 2*c[5]-1100*c[7]^2*c[4]^4*c[5]^3*c[6]^2+930*c[7]^2*c[4]^4*c[6]^2*c[5]^ 2+150*c[5]^6*c[4]*c[7]*c[6]-110*c[5]^6*c[4]*c[6]-150*c[5]^7*c[6]*c[4]^ 2+570*c[5]^6*c[4]^3-150*c[5]^7*c[4]^3+15*c[7]^2*c[6]^2*c[5]^3-440*c[4] ^4*c[6]*c[5]^3+750*c[5]^5*c[6]^2*c[4]^3-200*c[7]^2*c[5]^6*c[4]^3+12*c[ 5]^4*c[7]^2+200*c[5]^4*c[4]^5*c[6]*c[7]-20*c[5]^4*c[4]-87*c[5]^3*c[4]^ 4-30*c[5]^5*c[4]*c[6]+110*c[5]^5*c[6]^2*c[4])\}:" }}}{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 31 "We use the row- sum conditions: " }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "S um(a[i,j],j=1..i-1)=c[i]" "6#/-%$SumG6$&%\"aG6$%\"iG%\"jG/F+;\"\"\",&F *F.F.!\"\"&%\"cG6#F*" }{TEXT -1 7 ", for " }{XPPEDIT 18 0 "i=2" "6#/% \"iG\"\"#" }{TEXT -1 8 " . . 8, " }}{PARA 0 "" 0 "" {TEXT -1 9 "to fin d " }{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[8,1]" "6#&%\"aG6$\"\")\"\"\"" }{TEXT -1 2 ". " } }{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 119 "[seq(add(a[i,j],j=1..i-1)=c[i],i=2..8)]:\ne24 := solve(\{op(sub s(e23,%))\},\{seq(a[i,1],i=2..8)\}):\ne25 := `union`(e23,e24):" }}} {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 22 "We use th e equations: " }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "a[9, i]=b[i]" "6#/&%\"aG6$\"\"*%\"iG&%\"bG6#F(" }{TEXT -1 3 ", " } {XPPEDIT 18 0 "i=1" "6#/%\"iG\"\"\"" }{TEXT -1 8 " . . 8, " }}{PARA 0 "" 0 "" {TEXT -1 9 "to find " }{XPPEDIT 18 0 "a[9,i]" "6#&%\"aG6$\"\" *%\"iG" }{TEXT -1 2 ", " }{XPPEDIT 18 0 " i=1" "6#/%\"iG\"\"\"" } {TEXT -1 7 " . . 8." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 116 "wt_eqs := [seq(a[9,i]=b[i],i=1..8)]:\ne26 \+ := solve(\{op(subs(e25,%))\},\{seq(a[9,j],j=1..8)\}):\ne27 := `union`( e25,e26):\n" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 3 "e27" }} {PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 71948 "e27 := \{c[8] = 1, b[3] = 0, b[2] = 0, c[9] = 1, b[5] = -1/60*( 10*c[6]*c[7]*c[4]-5*c[6]*c[4]-5*c[7]*c[6]+3*c[6]-5*c[7]*c[4]+3*c[4]-2+ 3*c[7])/(-c[4]+c[5])/c[5]/(-c[5]^3+c[5]^2-c[7]*c[5]+c[7]*c[5]^2-c[6]*c [5]+c[6]*c[5]^2+c[7]*c[6]-c[6]*c[7]*c[5]), a[5,2] = 0, a[4,2] = 0, `b* `[3] = 0, `b*`[2] = 0, a[8,2] = 0, a[7,2] = 0, a[6,2] = 0, a[6,5] = 1/ 2*c[6]*(30*c[6]^2*c[5]^2*c[4]^2-2*c[6]^2*c[4]^2+c[6]^3*c[4]-12*c[6]^2* c[4]*c[5]^2-3*c[5]^2*c[6]*c[4]+c[5]*c[6]*c[4]^2+c[6]*c[4]^3-12*c[6]^2* c[4]^2*c[5]-30*c[5]*c[6]^3*c[4]^2-20*c[5]*c[6]^2*c[4]^4+30*c[6]^2*c[5] *c[4]^3-c[5]*c[4]^3+20*c[4]^3*c[5]*c[6]^3-20*c[6]^2*c[5]^2*c[4]^3+c[5] ^2*c[4]^2-30*c[5]^2*c[6]*c[4]^3+12*c[5]^2*c[6]*c[4]^2+2*c[6]^2*c[4]*c[ 5]+20*c[5]^2*c[6]*c[4]^4+2*c[6]^2*c[5]^2-2*c[6]^3*c[5]+12*c[6]^3*c[4]* c[5])/c[5]/(c[4]^3-2*c[5]*c[4]^2+30*c[5]^2*c[4]^3-6*c[5]*c[4]^3-10*c[4 ]^4*c[5]^2-c[5]^3+10*c[5]^4*c[4]^2-30*c[5]^3*c[4]^2+6*c[5]^3*c[4]+2*c[ 5]^2*c[4]), b[1] = 1/60*(30*c[5]*c[6]*c[7]*c[4]-10*c[5]*c[6]*c[4]-10*c [6]*c[7]*c[5]+5*c[6]*c[5]-10*c[6]*c[7]*c[4]+5*c[6]*c[4]+5*c[7]*c[6]-3* c[6]-10*c[5]*c[7]*c[4]+5*c[5]*c[4]+5*c[7]*c[5]-3*c[5]+5*c[7]*c[4]-3*c[ 4]+2-3*c[7])/c[5]/c[6]/c[7]/c[4], a[6,1] = -1/4*c[6]*(60*c[5]^2*c[6]*c [4]^4-90*c[5]^2*c[6]^3*c[4]^2-60*c[5]*c[6]^2*c[4]^4-2*c[6]^3*c[4]^2+50 *c[5]^4*c[4]^3*c[6]+40*c[6]^2*c[4]^5*c[5]-40*c[4]^5*c[5]^3+36*c[4]*c[5 ]^2*c[6]^3-20*c[5]^4*c[6]^2*c[4]^2+8*c[6]^2*c[5]^3+4*c[6]^2*c[4]^3-24* c[5]*c[6]^3*c[4]^2-8*c[6]^2*c[4]^2*c[5]+24*c[6]^2*c[5]*c[4]^3+60*c[4]^ 3*c[5]*c[6]^3-4*c[6]*c[5]*c[4]^3-14*c[5]^3*c[6]*c[4]+7*c[5]^3*c[4]^2+6 *c[5]^2*c[6]*c[4]^3+14*c[5]^2*c[6]*c[4]^2+66*c[5]^3*c[6]*c[4]^2-80*c[5 ]^3*c[6]^2*c[4]^3+30*c[6]^2*c[5]^2*c[4]^3-24*c[6]^2*c[5]^2*c[4]^2-240* c[6]*c[5]^3*c[4]^3-48*c[6]^2*c[5]^3*c[4]+150*c[6]^2*c[5]^3*c[4]^2-24*c [4]^4*c[5]^2-9*c[5]^2*c[4]^3+6*c[4]^4*c[5]-40*c[5]^4*c[4]^4-20*c[6]^2* c[4]^4*c[5]^2-6*c[5]^2*c[6]^3-2*c[6]*c[4]^4-24*c[5]^3*c[4]^3+60*c[6]^3 *c[4]^3*c[5]^2+7*c[6]^3*c[4]*c[5]-40*c[5]^2*c[6]*c[4]^5-40*c[6]^3*c[4] ^4*c[5]+110*c[4]^4*c[6]*c[5]^3+120*c[5]^3*c[4]^4)/c[5]/c[4]^2/(10*c[5] ^3*c[4]^2+10*c[5]^2*c[4]^3-30*c[5]^2*c[4]^2+6*c[5]^2*c[4]-c[5]^2+6*c[5 ]*c[4]^2+c[5]*c[4]-c[4]^2), b[4] = 1/60*(-2+3*c[5]-5*c[7]*c[6]+3*c[7]+ 3*c[6]+10*c[6]*c[7]*c[5]-5*c[6]*c[5]-5*c[7]*c[5])/c[4]/(-c[6]*c[4]^3-c [4]^3+c[4]^4+c[7]*c[4]^2+c[6]*c[4]^2-c[6]*c[7]*c[4]+c[6]*c[7]*c[4]^2-c [5]*c[4]^3+c[5]*c[4]^2-c[7]*c[4]^3-c[5]*c[7]*c[4]+c[5]*c[7]*c[4]^2-c[5 ]*c[6]*c[4]+c[5]*c[6]*c[4]^2+c[6]*c[7]*c[5]-c[5]*c[6]*c[7]*c[4]), b[6] = 1/60*(10*c[5]*c[7]*c[4]-5*c[5]*c[4]-5*c[7]*c[5]+3*c[5]-5*c[7]*c[4]+ 3*c[4]-2+3*c[7])/(-c[4]+c[6])/(-c[5]+c[6])/c[6]/(c[6]^2-c[6]+c[7]-c[7] *c[6]), `b*`[9] = 1/10*(50*c[5]^2*c[6]*c[4]^2-40*c[5]^2*c[6]*c[4]+10*c [6]*c[5]^2-40*c[5]*c[6]*c[4]^2+35*c[5]*c[6]*c[4]-10*c[6]*c[5]+10*c[6]* c[4]^2-10*c[6]*c[4]+3*c[6]-5*c[5]^2*c[4]-5*c[5]*c[4]^2+10*c[5]*c[4]-c[ 5]-c[4])/(3*c[6]-28*c[5]*c[6]*c[4]^2-28*c[5]^2*c[6]*c[4]+8*c[6]*c[5]^2 -9*c[6]*c[4]-9*c[6]*c[5]+28*c[5]*c[6]*c[4]+30*c[5]^2*c[6]*c[4]^2+8*c[6 ]*c[4]^2-4*c[5]^2*c[4]-4*c[5]*c[4]^2+9*c[5]*c[4]-c[5]-c[4]), a[7,5] = \+ -1/2*c[7]*(-17*c[5]^2*c[6]*c[4]^4+5*c[6]*c[5]*c[4]^4+42*c[4]^2*c[5]^2* c[7]^3+17*c[7]^3*c[5]*c[4]^3-80*c[7]^3*c[5]^2*c[4]^3+c[7]*c[5]*c[4]^4+ 80*c[5]^2*c[4]^4*c[7]^2-60*c[4]^5*c[6]^2*c[5]^2+20*c[4]^3*c[7]^3*c[5]^ 3-20*c[7]^2*c[5]^2*c[4]^5+40*c[4]^5*c[6]*c[5]^3-5*c[5]*c[6]^2*c[4]^4+c [7]^3*c[4]^2-c[7]^2*c[4]^3-80*c[5]^3*c[4]^4*c[7]+20*c[7]^3*c[5]^2*c[4] ^4-200*c[6]^2*c[4]^5*c[7]*c[5]^3-290*c[5]^4*c[4]^2*c[7]^2*c[6]+20*c[6] ^2*c[7]*c[5]^4*c[4]^2+40*c[6]*c[5]^2*c[4]^4*c[7]-140*c[5]^4*c[4]^3*c[6 ]-100*c[5]^4*c[4]^5*c[6]+190*c[5]^4*c[4]^4*c[6]-40*c[7]^2*c[4]^4*c[5]^ 3+20*c[5]^4*c[4]^4*c[7]+2*c[4]^4*c[7]^2-20*c[7]^2*c[5]^4*c[4]^3-4*c[6] ^2*c[7]^2*c[5]-8*c[7]*c[6]^2*c[5]^3-3*c[7]*c[6]*c[5]^3+2*c[7]^2*c[6]*c [5]^3+60*c[7]*c[6]^2*c[5]*c[4]^5-40*c[6]*c[5]^2*c[4]^5*c[7]+17*c[7]*c[ 4]^4*c[6]*c[5]+80*c[7]^2*c[5]^3*c[4]^3+5*c[7]*c[4]^4*c[6]^2+c[5]^2*c[7 ]^3-260*c[6]^2*c[7]*c[5]^2*c[4]^3-12*c[6]^2*c[7]^2*c[5]^2*c[4]+20*c[6] ^2*c[7]^2*c[5]^2*c[4]^2-40*c[6]*c[7]*c[5]^3*c[4]^3-41*c[6]^2*c[7]^3*c[ 4]*c[5]+20*c[7]^2*c[6]*c[5]*c[4]^4+3*c[7]*c[6]*c[4]^3-2*c[6]^2*c[7]*c[ 4]^2-3*c[6]*c[7]^2*c[4]^2+4*c[7]^2*c[6]*c[4]^3+6*c[5]*c[7]^2*c[4]^3+4* c[6]*c[5]*c[7]^2*c[4]^2+c[7]^2*c[5]*c[6]*c[4]-11*c[5]*c[7]*c[6]*c[4]^3 +2*c[6]^2*c[4]^2*c[5]-c[7]^3*c[4]*c[5]+17*c[5]^2*c[6]*c[7]*c[4]^2-200* c[5]^4*c[7]^2*c[4]^4*c[6]-50*c[4]^5*c[6]*c[7]*c[5]^3-150*c[5]^2*c[6]*c [7]^3*c[4]^4-c[6]^2*c[5]*c[4]^3+2*c[7]^3*c[4]^2*c[6]*c[5]+2*c[6]^2*c[7 ]*c[4]*c[5]+108*c[7]*c[6]^2*c[5]*c[4]^3+4*c[6]^2*c[7]*c[5]^2+2*c[6]*c[ 5]*c[7]*c[4]^2-10*c[6]^2*c[7]^3*c[5]^2+6*c[5]^4*c[7]*c[6]-8*c[6]*c[7]^ 3*c[5]^2+3*c[6]*c[7]^2*c[5]^2+2*c[6]^2*c[7]^2*c[5]^2-27*c[6]^2*c[4]^2* c[7]*c[5]-62*c[6]^2*c[7]^2*c[4]^2*c[5]+46*c[6]*c[5]^4*c[4]^2+20*c[4]^5 *c[7]*c[5]^3+2*c[7]^3*c[4]*c[6]*c[5]-27*c[6]^2*c[5]^2*c[7]*c[4]+120*c[ 6]^2*c[5]^2*c[7]*c[4]^2-35*c[6]*c[5]^2*c[7]*c[4]^3-37*c[6]*c[5]^3*c[7] *c[4]^2+53*c[6]^2*c[5]^3*c[7]*c[4]-54*c[7]^2*c[5]^2*c[4]^3+2*c[7]^2*c[ 5]*c[4]^2-c[7]*c[5]*c[4]^3-3*c[6]*c[5]*c[4]^3-2*c[4]^3*c[7]^3+63*c[7]^ 3*c[5]^2*c[6]*c[4]-16*c[7]^2*c[5]^2*c[6]*c[4]-5*c[6]*c[7]*c[4]^4+2*c[6 ]^2*c[7]^2*c[4]^2+2*c[6]^2*c[7]^2*c[4]-10*c[5]^4*c[7]^2*c[6]+3*c[5]^3* c[6]*c[4]+3*c[5]^3*c[7]*c[4]+20*c[5]^3*c[6]*c[7]*c[4]-2*c[5]^3*c[4]^2+ 22*c[6]^2*c[7]^2*c[4]*c[5]+7*c[5]^2*c[6]*c[4]^3-c[5]^2*c[7]*c[4]^2+c[5 ]^2*c[6]*c[4]^2-4*c[5]^2*c[7]*c[4]^3-8*c[5]^3*c[7]*c[4]^2-22*c[5]^3*c[ 6]*c[4]^2+240*c[5]^3*c[6]^2*c[4]^3+100*c[4]^3*c[7]^2*c[5]^2*c[6]^2+150 *c[4]^5*c[5]^3*c[6]^2-3*c[7]*c[5]^4*c[4]+5*c[7]^2*c[5]^4*c[4]+12*c[5]^ 2*c[4]^4*c[7]-98*c[6]^2*c[5]^2*c[4]^3+25*c[6]^2*c[5]^2*c[4]^2+54*c[6]* c[5]^3*c[4]^3+8*c[6]^2*c[5]^3*c[4]-63*c[6]^2*c[5]^3*c[4]^2+42*c[5]^3*c [7]*c[4]^3+60*c[5]^4*c[7]*c[4]^2*c[6]+200*c[5]^3*c[7]^3*c[4]^4*c[6]-4* c[6]^2*c[4]*c[5]^2+c[7]*c[6]^2*c[4]^3+200*c[7]^2*c[5]^2*c[6]*c[4]^3-24 0*c[7]^3*c[5]^2*c[6]*c[4]^2+180*c[7]^2*c[5]^3*c[6]*c[4]^2+3*c[5]^4*c[4 ]^2-80*c[5]^3*c[7]^3*c[6]*c[4]+c[7]^3*c[4]^2*c[6]-7*c[7]^3*c[4]^2*c[5] +20*c[4]^3*c[7]^2*c[5]*c[6]^2-19*c[7]^2*c[6]*c[5]*c[4]^3-11*c[7]^2*c[5 ]^2*c[6]*c[4]^2-3*c[4]^4*c[5]^2-27*c[7]^2*c[5]^3*c[6]*c[4]-480*c[7]^2* c[5]^3*c[6]*c[4]^3-380*c[5]^3*c[6]^2*c[4]^4-5*c[5]^4*c[4]^3-10*c[6]^2* c[4]^3*c[7]^2+130*c[5]^3*c[6]^2*c[7]*c[4]^3-5*c[7]^3*c[5]^2*c[4]-2*c[7 ]^2*c[5]^2*c[4]-4*c[5]^2*c[6]*c[7]*c[4]-160*c[6]^2*c[7]*c[5]^3*c[4]^2- 80*c[6]^2*c[7]^2*c[4]*c[5]^3+2*c[5]^2*c[4]^3+12*c[7]^2*c[5]^2*c[4]^2+8 *c[7]^2*c[5]^3*c[4]+400*c[7]^3*c[5]^2*c[6]^2*c[4]^3-17*c[7]^2*c[4]^4*c [5]+250*c[7]^2*c[5]^3*c[6]^2*c[4]^2-100*c[6]^2*c[4]^5*c[5]*c[7]^2-200* c[7]^2*c[4]^5*c[6]*c[5]^3+150*c[5]^2*c[6]*c[4]^5*c[7]^2+380*c[7]^3*c[5 ]^2*c[4]^3*c[6]-20*c[7]^3*c[6]*c[5]*c[4]^3-210*c[7]^3*c[6]^2*c[5]*c[4] ^3-6*c[5]^4*c[4]*c[6]+80*c[5]^4*c[7]^2*c[4]*c[6]+10*c[6]*c[7]^3*c[5]^3 -36*c[5]^4*c[4]*c[7]*c[6]-50*c[4]^5*c[6]^2*c[7]*c[5]^2+200*c[4]^5*c[6] ^2*c[7]^2*c[5]^2+160*c[6]^2*c[4]^4*c[5]^2-450*c[7]^3*c[5]^3*c[4]^3*c[6 ]+5*c[6]^2*c[7]^3*c[4]^2-c[5]^3*c[7]^2+6*c[6]^2*c[7]^3*c[5]-160*c[7]*c [6]^2*c[5]*c[4]^4+200*c[5]^3*c[4]^4*c[6]^2*c[7]-350*c[5]^3*c[4]^3*c[6] ^2*c[7]^2-350*c[5]^4*c[4]^4*c[6]*c[7]+230*c[5]^2*c[4]^4*c[7]*c[6]^2+15 0*c[5]^3*c[4]^4*c[6]*c[7]+100*c[5]^4*c[4]^3*c[6]*c[7]-3*c[6]^2*c[7]^3* c[4]-42*c[7]^2*c[5]^3*c[4]^2+50*c[5]^4*c[4]^4*c[6]^2-20*c[5]^4*c[4]^3* c[6]^2+500*c[5]^3*c[4]^4*c[7]^2*c[6]+450*c[5]^4*c[4]^3*c[7]^2*c[6]-340 *c[5]^2*c[4]^4*c[7]^2*c[6]-50*c[5]^4*c[4]^3*c[6]^2*c[7]+150*c[7]^2*c[4 ]^4*c[6]^2*c[5]+200*c[7]^2*c[4]^4*c[5]^3*c[6]^2-350*c[7]^2*c[4]^4*c[6] ^2*c[5]^2+10*c[7]^2*c[6]^2*c[5]^3-60*c[4]^4*c[6]*c[5]^3+100*c[7]^3*c[6 ]^2*c[5]*c[4]^4+200*c[5]^4*c[4]^5*c[6]*c[7]-200*c[7]^3*c[4]^4*c[6]^2*c [5]^2+5*c[5]^3*c[4]^4+80*c[6]^2*c[7]^3*c[5]^2*c[4]-270*c[6]^2*c[7]^3*c [5]^2*c[4]^2+290*c[5]^3*c[7]^3*c[4]^2*c[6]+140*c[6]^2*c[5]*c[7]^3*c[4] ^2-5*c[7]^3*c[4]*c[5]^3)/c[5]/(100*c[5]^2*c[6]*c[4]^4-18*c[6]*c[5]*c[4 ]^4+4*c[5]^3*c[4]+50*c[4]^5*c[6]^2*c[5]^2+40*c[5]*c[6]^2*c[4]^4+3*c[5] ^5-5*c[4]^4*c[6]^2+180*c[5]^4*c[4]^3*c[6]+100*c[5]^4*c[4]^5*c[6]-350*c [5]^4*c[4]^4*c[6]+140*c[5]^5*c[4]^2-180*c[5]^5*c[4]^3-2*c[5]^4+30*c[4] ^5*c[5]^3+300*c[5]^5*c[4]^3*c[6]+2*c[6]*c[5]^3-50*c[5]^5*c[4]^2*c[6]^2 -40*c[5]^4*c[6]^2*c[4]+240*c[5]^4*c[6]^2*c[4]^2-3*c[6]^2*c[5]^3+3*c[6] ^2*c[4]^3+5*c[5]^4*c[6]^2-2*c[6]*c[4]^3+2*c[5]*c[4]^3+4*c[5]*c[6]*c[4] ^2-6*c[6]^2*c[4]^2*c[5]-50*c[5]^4*c[4]^5-13*c[6]^2*c[5]*c[4]^3-40*c[6] *c[5]^4*c[4]^2+6*c[6]*c[5]*c[4]^3-15*c[5]^3*c[6]*c[4]-55*c[5]^2*c[6]*c [4]^3+6*c[5]^2*c[6]*c[4]^2+68*c[5]^3*c[6]*c[4]^2-100*c[4]^5*c[5]^3*c[6 ]^2+100*c[6]^2*c[5]^2*c[4]^3-100*c[6]*c[5]^3*c[4]^3+13*c[6]^2*c[5]^3*c [4]-100*c[6]^2*c[5]^3*c[4]^2-4*c[5]^2*c[4]^2+6*c[6]^2*c[4]*c[5]^2-68*c [5]^4*c[4]^2+23*c[4]^4*c[5]^2+350*c[5]^3*c[6]^2*c[4]^4-4*c[5]^2*c[6]*c [4]-9*c[5]^2*c[4]^3-3*c[4]^4*c[5]-5*c[5]^5*c[6]-30*c[5]^6*c[4]^2+50*c[ 5]^6*c[6]*c[4]^2+180*c[5]^4*c[4]^4+10*c[5]^4*c[4]*c[6]-240*c[6]^2*c[4] ^4*c[5]^2+3*c[6]*c[4]^4-100*c[5]^6*c[4]^3*c[6]+68*c[5]^3*c[4]^3-23*c[5 ]^5*c[4]-210*c[5]^5*c[6]*c[4]^2-350*c[5]^4*c[4]^3*c[6]^2-30*c[5]^2*c[6 ]*c[4]^5+50*c[5]^6*c[4]^3+60*c[4]^4*c[6]*c[5]^3+100*c[5]^5*c[6]^2*c[4] ^3+9*c[5]^4*c[4]-140*c[5]^3*c[4]^4+40*c[5]^5*c[4]*c[6]), `b*`[6] = -1/ 60*(-30*c[5]^3*c[4]+103*c[5]*c[7]*c[4]-18*c[6]+6*c[5]+6*c[4]+27*c[7]*c [6]-9*c[5]^2-9*c[4]^2-100*c[7]*c[6]*c[5]^3+522*c[5]*c[6]*c[7]*c[4]-111 *c[6]*c[5]^2+75*c[6]*c[5]+60*c[6]*c[5]^3-9*c[7]*c[5]+600*c[6]*c[7]*c[5 ]^3*c[4]^3+14*c[7]*c[5]^2-100*c[7]*c[6]*c[4]^3+14*c[7]*c[4]^2-9*c[7]*c [4]-117*c[6]*c[7]*c[5]-111*c[6]*c[4]^2+60*c[6]*c[4]^3-117*c[6]*c[7]*c[ 4]+180*c[6]*c[7]*c[4]^2-30*c[5]*c[4]^3-174*c[5]*c[7]*c[4]^2+486*c[5]*c [6]*c[4]^2+107*c[5]*c[4]^2-318*c[5]*c[6]*c[4]-66*c[5]*c[4]+75*c[6]*c[4 ]+500*c[5]*c[7]*c[6]*c[4]^3+1420*c[5]^2*c[6]*c[7]*c[4]^2-840*c[6]*c[5] *c[7]*c[4]^2+180*c[6]*c[7]*c[5]^2-900*c[6]*c[5]^2*c[7]*c[4]^3-900*c[6] *c[5]^3*c[7]*c[4]^2+50*c[7]*c[5]*c[4]^3-280*c[6]*c[5]*c[4]^3-280*c[5]^ 3*c[6]*c[4]+50*c[5]^3*c[7]*c[4]+107*c[5]^2*c[4]+500*c[5]^3*c[6]*c[7]*c [4]+50*c[5]^3*c[4]^2+470*c[5]^2*c[6]*c[4]^3+310*c[5]^2*c[7]*c[4]^2-770 *c[5]^2*c[6]*c[4]^2-90*c[5]^2*c[7]*c[4]^3-90*c[5]^3*c[7]*c[4]^2+470*c[ 5]^3*c[6]*c[4]^2-300*c[6]*c[5]^3*c[4]^3-180*c[5]^2*c[4]^2+486*c[5]^2*c [6]*c[4]-174*c[5]^2*c[7]*c[4]-840*c[5]^2*c[6]*c[7]*c[4]+50*c[5]^2*c[4] ^3)/(-9*c[5]*c[7]*c[4]-3*c[7]*c[6]+3*c[6]^2-28*c[5]*c[6]*c[7]*c[4]-c[6 ]*c[5]+c[7]*c[5]+8*c[6]^2*c[4]^2+c[7]*c[4]+9*c[6]*c[7]*c[5]+9*c[6]*c[7 ]*c[4]-8*c[6]*c[7]*c[4]^2+4*c[5]*c[7]*c[4]^2-4*c[5]*c[6]*c[4]^2+9*c[5] *c[6]*c[4]-c[6]*c[4]-28*c[6]^2*c[4]^2*c[5]+28*c[6]^2*c[4]*c[5]-30*c[5] ^2*c[6]*c[7]*c[4]^2+28*c[6]*c[5]*c[7]*c[4]^2-8*c[6]*c[7]*c[5]^2+8*c[6] ^2*c[5]^2-9*c[6]^2*c[5]+30*c[6]^2*c[5]^2*c[4]^2-28*c[6]^2*c[4]*c[5]^2- 9*c[6]^2*c[4]-4*c[5]^2*c[6]*c[4]+4*c[5]^2*c[7]*c[4]+28*c[5]^2*c[6]*c[7 ]*c[4])/(c[5]*c[4]-c[6]*c[4]-c[6]*c[5]+c[6]^2)/c[6], a[8,7] = (10*c[5] *c[6]*c[4]-5*c[6]*c[5]-5*c[6]*c[4]+3*c[6]-5*c[5]*c[4]+3*c[5]+3*c[4]-2) /c[7]/(c[6]-c[7])/(c[5]-c[7])/(c[4]-c[7])/(-20*c[5]*c[6]*c[4]+30*c[5]* c[6]*c[7]*c[4]+15*c[6]*c[5]-20*c[6]*c[7]*c[5]+15*c[6]*c[4]-20*c[6]*c[7 ]*c[4]-12*c[6]+15*c[7]*c[6]+15*c[5]*c[4]-20*c[5]*c[7]*c[4]-12*c[5]+15* c[7]*c[5]-12*c[4]+15*c[7]*c[4]+10-12*c[7])*(c[4]-1)*(-1+c[5])*(-1+c[6] )*(-1+c[7]), a[6,3] = 3/4*1/c[4]^2*c[6]*(60*c[4]^3*c[5]*c[6]^3-90*c[5] *c[6]^3*c[4]^2+36*c[6]^3*c[4]*c[5]-6*c[6]^3*c[5]+3*c[6]^3*c[4]-20*c[6] ^2*c[5]^3*c[4]^2-80*c[6]^2*c[5]^2*c[4]^3+150*c[6]^2*c[5]^2*c[4]^2-48*c [6]^2*c[4]*c[5]^2+8*c[6]^2*c[5]^2-60*c[5]*c[6]^2*c[4]^4+90*c[6]^2*c[5] *c[4]^3-48*c[6]^2*c[4]^2*c[5]+4*c[6]^2*c[4]*c[5]-4*c[6]^2*c[4]^2+30*c[ 6]*c[5]^3*c[4]^3+90*c[5]^2*c[6]*c[4]^4-180*c[5]^2*c[6]*c[4]^3+54*c[5]^ 2*c[6]*c[4]^2-12*c[5]^2*c[6]*c[4]+18*c[6]*c[5]*c[4]^3+6*c[5]*c[6]*c[4] ^2+3*c[5]^2*c[4]^2-3*c[5]*c[4]^3)/(10*c[5]^3*c[4]^2+10*c[5]^2*c[4]^3-3 0*c[5]^2*c[4]^2+6*c[5]^2*c[4]-c[5]^2+6*c[5]*c[4]^2+c[5]*c[4]-c[4]^2), \+ b[7] = -1/60*(10*c[5]*c[6]*c[4]-5*c[6]*c[5]-5*c[6]*c[4]+3*c[6]-5*c[5]* c[4]+3*c[5]+3*c[4]-2)/(c[4]-c[7])/(c[5]-c[7])/(c[6]-c[7])/c[7]/(-1+c[7 ]), a[7,1] = 1/4*c[7]*(-51*c[5]^2*c[6]*c[4]^4+14*c[6]*c[5]*c[4]^4+12*c [4]^2*c[5]^2*c[7]^3+2700*c[5]^4*c[4]^4*c[6]^2*c[7]+12*c[7]^3*c[5]*c[4] ^3-98*c[7]^3*c[5]^2*c[4]^3-2*c[7]*c[5]*c[4]^4+132*c[5]^2*c[4]^4*c[7]^2 -480*c[4]^5*c[6]^2*c[5]^2+200*c[4]^3*c[7]^3*c[5]^3-200*c[7]^2*c[5]^2*c [4]^5-760*c[4]^5*c[6]*c[5]^3-10*c[5]*c[6]^2*c[4]^4+200*c[5]^5*c[4]^3*c [7]^2*c[6]^2-400*c[5]^2*c[4]^6*c[6]^2*c[7]^2+4*c[4]^5*c[5]*c[7]-34*c[5 ]^2*c[4]^5*c[7]+400*c[5]^3*c[4]^6*c[6]^2*c[7]-98*c[5]^3*c[4]^4*c[7]+42 0*c[7]^2*c[4]^5*c[5]^3+200*c[7]^3*c[5]^2*c[4]^4-300*c[6]^2*c[4]^5*c[7] *c[5]^3-40*c[7]*c[5]^2*c[4]^6*c[6]+20*c[5]^4*c[4]^2*c[7]^2*c[6]-280*c[ 6]^2*c[7]*c[5]^4*c[4]^2-44*c[6]*c[4]^5*c[5]*c[7]-246*c[6]*c[5]^2*c[4]^ 4*c[7]+120*c[4]^6*c[6]^2*c[5]^2-100*c[5]^2*c[7]^2*c[4]^6*c[6]+141*c[5] ^4*c[4]^3*c[6]+1220*c[5]^4*c[4]^5*c[6]+400*c[5]^5*c[4]^5*c[6]^2-760*c[ 5]^4*c[4]^4*c[6]+100*c[5]^4*c[7]^3*c[4]^4-400*c[7]^2*c[4]^4*c[5]^3-200 *c[5]^4*c[4]^5*c[7]^2+200*c[5]^4*c[4]^4*c[7]-2*c[4]^3*c[7]^3*c[6]-8*c[ 7]^2*c[6]*c[4]^4-380*c[7]^3*c[5]^3*c[4]^4-40*c[6]*c[4]^5*c[5]*c[7]^2+2 *c[4]^4*c[7]^2-40*c[5]^2*c[7]^3*c[4]^5-200*c[7]^2*c[5]^4*c[4]^3-40*c[7 ]^3*c[5]^4*c[4]^3+200*c[5]^3*c[7]*c[4]^6*c[6]-10*c[6]^2*c[7]^3*c[4]^3+ 30*c[6]^2*c[4]^5*c[5]+12*c[7]*c[6]^2*c[5]^3+4*c[7]^2*c[6]*c[5]^3+400*c [5]^4*c[4]^6*c[6]^2-40*c[5]^3*c[7]*c[4]^6-380*c[5]^4*c[4]^5*c[7]+320*c [7]*c[6]^2*c[5]*c[4]^5+100*c[4]^6*c[6]^2*c[7]*c[5]^2+34*c[5]^4*c[4]^2* c[7]^2+40*c[5]^5*c[7]^2*c[4]^3+240*c[6]*c[5]^2*c[4]^5*c[7]+60*c[5]^4*c [7]^3*c[4]^2*c[6]+26*c[7]*c[4]^4*c[6]*c[5]-50*c[4]^4*c[6]^2*c[5]^5+200 *c[5]^5*c[7]*c[4]^4*c[6]+132*c[7]^2*c[5]^3*c[4]^3-2*c[7]*c[4]^4*c[6]^2 +60*c[5]^5*c[4]^2*c[6]^2*c[7]-500*c[5]^5*c[4]^4*c[6]^2*c[7]+160*c[5]^5 *c[4]^4*c[6]+134*c[6]^2*c[5]^4*c[7]*c[4]+24*c[5]^4*c[6]^2*c[4]-169*c[5 ]^4*c[6]^2*c[4]^2-500*c[5]^3*c[4]^6*c[6]^2-40*c[5]^5*c[4]^4*c[7]+34*c[ 4]^5*c[5]*c[7]^2-100*c[5]^3*c[7]^2*c[4]^6-100*c[5]^5*c[4]^4*c[7]^2+160 *c[4]^6*c[6]*c[5]^3-6*c[6]^2*c[7]*c[5]^2*c[4]^3+68*c[6]^2*c[7]^2*c[5]^ 2*c[4]-126*c[6]^2*c[7]^2*c[5]^2*c[4]^2-600*c[6]*c[7]*c[5]^3*c[4]^3-21* c[6]^2*c[7]^3*c[4]*c[5]+40*c[7]^2*c[6]^2*c[5]^4+4*c[5]^4*c[7]*c[4]^2+5 8*c[7]^2*c[6]*c[5]*c[4]^4-100*c[7]^2*c[5]^5*c[6]^2*c[4]^2-320*c[5]^4*c [4]*c[7]^2*c[6]^2+20*c[6]^2*c[4]^4*c[7]^2+40*c[5]^2*c[4]^6*c[7]^2+6*c[ 7]^2*c[6]*c[4]^3-8*c[6]*c[5]*c[7]^2*c[4]^2-4*c[5]*c[7]*c[6]*c[4]^3+14* c[5]^2*c[6]*c[7]*c[4]^2-910*c[4]^5*c[6]*c[7]*c[5]^3-340*c[5]^2*c[6]*c[ 7]^3*c[4]^4-4*c[6]^2*c[5]*c[4]^3+5*c[7]^3*c[4]^2*c[6]*c[5]+72*c[7]*c[6 ]^2*c[5]*c[4]^3+18*c[6]^2*c[7]^3*c[5]^2-12*c[6]^2*c[7]^2*c[5]^2-10*c[6 ]^2*c[4]^2*c[7]*c[5]-44*c[6]^2*c[7]^2*c[4]^2*c[5]-25*c[6]*c[5]^4*c[4]^ 2-24*c[7]*c[6]^2*c[5]^4+200*c[4]^5*c[7]*c[5]^3-2*c[6]^2*c[5]^2*c[7]*c[ 4]-42*c[6]^2*c[5]^2*c[7]*c[4]^2+74*c[6]*c[5]^2*c[7]*c[4]^3+80*c[6]*c[5 ]^3*c[7]*c[4]^2-50*c[6]^2*c[5]^3*c[7]*c[4]-34*c[5]^4*c[4]^3*c[7]-24*c[ 7]^2*c[5]^2*c[4]^3+420*c[5]^4*c[7]^2*c[4]^4-2*c[4]^3*c[7]^3+c[7]^3*c[5 ]^2*c[6]*c[4]+4*c[7]^2*c[5]^2*c[6]*c[4]-6*c[6]*c[7]*c[4]^4-4*c[6]^2*c[ 7]^2*c[4]^2+50*c[5]^5*c[4]^3*c[7]^2*c[6]-18*c[5]^3*c[6]*c[7]*c[4]+14*c [6]^2*c[7]^2*c[4]*c[5]-20*c[5]^2*c[6]*c[4]^3+2*c[5]^2*c[7]*c[4]^3-2*c[ 5]^3*c[7]*c[4]^2+16*c[5]^3*c[6]*c[4]^2-96*c[5]^3*c[6]^2*c[4]^3+100*c[5 ]^5*c[4]^3*c[6]^2*c[7]-60*c[4]^3*c[7]^2*c[5]^2*c[6]^2+10*c[6]*c[7]*c[4 ]^5+1540*c[4]^5*c[5]^3*c[6]^2-34*c[7]^3*c[4]^4*c[5]-4*c[7]^2*c[5]^4*c[ 4]+12*c[5]^2*c[4]^4*c[7]-41*c[6]^2*c[5]^2*c[4]^3+14*c[6]^2*c[5]^2*c[4] ^2-54*c[6]*c[5]^3*c[4]^3-12*c[6]^2*c[5]^3*c[4]+47*c[6]^2*c[5]^3*c[4]^2 +12*c[5]^3*c[7]*c[4]^3-98*c[5]^4*c[7]*c[4]^2*c[6]-50*c[5]^3*c[7]^3*c[4 ]^4*c[6]+4*c[7]*c[6]^2*c[4]^3+156*c[7]^2*c[5]^2*c[6]*c[4]^3-23*c[7]^3* c[5]^2*c[6]*c[4]^2+188*c[7]^2*c[5]^3*c[6]*c[4]^2+66*c[5]^3*c[7]^3*c[6] *c[4]+2*c[7]^3*c[4]^2*c[5]+100*c[5]^5*c[4]^5*c[7]+104*c[4]^3*c[7]^2*c[ 5]*c[6]^2-12*c[7]^2*c[6]*c[5]*c[4]^3-36*c[7]^2*c[5]^2*c[6]*c[4]^2-32*c [7]^2*c[5]^3*c[6]*c[4]-320*c[7]^2*c[5]^3*c[6]*c[4]^3-560*c[5]^3*c[6]^2 *c[4]^4-4*c[6]^2*c[4]^3*c[7]^2+240*c[5]^3*c[6]^2*c[7]*c[4]^3-34*c[5]^3 *c[7]^3*c[4]^2-2*c[7]^3*c[5]^2*c[4]+164*c[6]^2*c[7]*c[5]^3*c[4]^2-20*c [6]^2*c[7]^2*c[4]*c[5]^3+2*c[7]^2*c[5]^3*c[4]-130*c[5]^5*c[4]^3*c[6]*c [7]-90*c[7]^3*c[5]^2*c[6]^2*c[4]^3-16*c[7]^2*c[4]^4*c[5]-60*c[7]^2*c[5 ]^3*c[6]^2*c[4]^2-300*c[6]^2*c[4]^5*c[5]*c[7]^2-150*c[7]^2*c[4]^5*c[6] *c[5]^3+380*c[5]^2*c[6]*c[4]^5*c[7]^2+140*c[7]^3*c[5]^2*c[4]^3*c[6]-81 0*c[7]^3*c[5]^3*c[6]^2*c[4]^2-14*c[7]^3*c[6]*c[5]*c[4]^3-280*c[7]^3*c[ 6]^2*c[5]*c[4]^3+240*c[7]^3*c[5]^3*c[6]^2*c[4]-20*c[5]^4*c[7]^2*c[4]*c [6]-6*c[6]*c[7]^3*c[5]^3+22*c[5]^4*c[4]*c[7]*c[6]-280*c[4]^5*c[6]^2*c[ 7]*c[5]^2+400*c[4]^5*c[6]^2*c[7]^2*c[5]^2+253*c[6]^2*c[4]^4*c[5]^2+470 *c[7]^3*c[5]^3*c[4]^3*c[6]+100*c[5]^2*c[4]^5*c[7]^3*c[6]+6*c[6]^2*c[7] ^3*c[4]^2-300*c[5]^5*c[4]^5*c[6]-120*c[5]*c[7]*c[4]^6*c[6]^2-10*c[6]^2 *c[7]*c[4]^5-1100*c[5]^4*c[4]^5*c[6]^2*c[7]-226*c[7]*c[6]^2*c[5]*c[4]^ 4-770*c[5]^3*c[4]^4*c[6]^2*c[7]+280*c[5]^3*c[4]^3*c[6]^2*c[7]^2-1000*c [5]^4*c[4]^4*c[6]*c[7]+240*c[5]^2*c[4]^4*c[7]*c[6]^2+1240*c[5]^3*c[4]^ 4*c[6]*c[7]+600*c[5]^4*c[4]^3*c[6]*c[7]+1200*c[7]^3*c[5]^3*c[6]^2*c[4] ^3+40*c[7]^3*c[4]^4*c[5]*c[6]-16*c[7]^2*c[5]^3*c[4]^2-1250*c[5]^4*c[4] ^5*c[6]^2-600*c[6]^2*c[7]^3*c[5]^3*c[4]^4+200*c[6]^2*c[7]^2*c[5]^3*c[4 ]^5-200*c[6]^2*c[7]^3*c[4]^5*c[5]-140*c[5]^4*c[4]^4*c[6]^2+560*c[5]^4* c[4]^3*c[6]^2-1850*c[7]^2*c[5]^4*c[6]^2*c[4]^3+1230*c[7]^2*c[5]^4*c[6] ^2*c[4]^2-150*c[5]^4*c[7]^3*c[4]^3*c[6]-300*c[5]^4*c[4]^6*c[6]+310*c[5 ]^3*c[4]^4*c[7]^2*c[6]-70*c[5]^4*c[4]^3*c[7]^2*c[6]-380*c[5]^2*c[4]^4* c[7]^2*c[6]-850*c[5]^4*c[4]^3*c[6]^2*c[7]-40*c[7]^2*c[4]^4*c[6]^2*c[5] -450*c[7]^2*c[4]^4*c[5]^3*c[6]^2+250*c[7]^2*c[4]^4*c[6]^2*c[5]^2-4*c[7 ]^2*c[4]^5+136*c[5]^2*c[6]*c[4]^5+400*c[6]^2*c[7]^3*c[5]^2*c[4]^5+800* c[6]^2*c[7]^2*c[5]^4*c[4]^4+4*c[7]^3*c[4]^4+100*c[5]^4*c[4]^6*c[7]+437 *c[4]^4*c[6]*c[5]^3-22*c[6]*c[5]*c[4]^5-60*c[5]^5*c[6]^2*c[4]^3+200*c[ 6]^2*c[4]^6*c[5]*c[7]^2+420*c[7]^3*c[6]^2*c[5]*c[4]^4+400*c[5]^4*c[4]^ 5*c[6]*c[7]-500*c[7]^3*c[4]^4*c[6]^2*c[5]^2+100*c[5]^3*c[7]^3*c[4]^5-1 03*c[6]^2*c[7]^3*c[5]^2*c[4]+260*c[6]^2*c[7]^3*c[5]^2*c[4]^2-300*c[5]^ 3*c[7]^3*c[4]^2*c[6]+97*c[6]^2*c[5]*c[7]^3*c[4]^2-30*c[6]^2*c[7]^3*c[5 ]^3+4*c[7]^3*c[4]*c[5]^3)/c[6]/(-50*c[5]^2*c[6]*c[4]^4+23*c[5]^3*c[4]+ 100*c[5]^4*c[4]^3*c[6]+2*c[5]^2+2*c[4]^2-3*c[5]^3-3*c[4]^3-3*c[6]*c[5] ^2+5*c[6]*c[5]^3-3*c[6]*c[4]^2+5*c[6]*c[4]^3+23*c[5]*c[4]^3+18*c[5]*c[ 6]*c[4]^2-12*c[5]*c[4]^2+3*c[5]*c[6]*c[4]-2*c[5]*c[4]-50*c[6]*c[5]^4*c [4]^2-40*c[6]*c[5]*c[4]^3-40*c[5]^3*c[6]*c[4]-12*c[5]^2*c[4]-140*c[5]^ 3*c[4]^2+240*c[5]^2*c[6]*c[4]^3-140*c[5]^2*c[6]*c[4]^2+240*c[5]^3*c[6] *c[4]^2-400*c[6]*c[5]^3*c[4]^3+91*c[5]^2*c[4]^2+30*c[5]^4*c[4]^2+30*c[ 4]^4*c[5]^2-50*c[5]^4*c[4]^3+18*c[5]^2*c[6]*c[4]-140*c[5]^2*c[4]^3+210 *c[5]^3*c[4]^3+100*c[4]^4*c[6]*c[5]^3-50*c[5]^3*c[4]^4)/c[5]/c[4]^2, ` b*`[8] = 0, a[5,3] = -3/4*c[5]^2*(2*c[5]-3*c[4])/c[4]^2, `b*`[4] = -1/ 60*(-30*c[5]^3*c[4]-42*c[7]*c[6]^2+89*c[5]*c[7]*c[4]-18*c[6]+6*c[5]+6* c[4]+27*c[7]*c[6]-9*c[5]^2+180*c[7]*c[6]^2*c[5]^3-100*c[7]*c[6]*c[5]^3 +27*c[6]^2+186*c[5]*c[6]*c[7]*c[4]-97*c[6]*c[5]^2+66*c[6]*c[5]+60*c[6] *c[5]^3-100*c[6]^2*c[5]^3-9*c[7]*c[5]+14*c[7]*c[5]^2-100*c[6]^2*c[7]*c [4]^2+120*c[6]^2*c[7]*c[4]+60*c[6]^2*c[4]^2-9*c[7]*c[4]-103*c[6]*c[7]* c[5]-36*c[6]*c[4]^2-61*c[6]*c[7]*c[4]+60*c[6]*c[7]*c[4]^2-30*c[5]*c[7] *c[4]^2+126*c[5]*c[6]*c[4]^2+18*c[5]*c[4]^2-112*c[5]*c[6]*c[4]-57*c[5] *c[4]+39*c[6]*c[4]-280*c[6]^2*c[4]^2*c[5]+330*c[6]^2*c[4]*c[5]+380*c[5 ]^2*c[6]*c[7]*c[4]^2-560*c[6]^2*c[7]*c[4]*c[5]+192*c[6]^2*c[7]*c[5]-31 0*c[6]^2*c[7]*c[5]^2-230*c[6]*c[5]*c[7]*c[4]^2+156*c[6]*c[7]*c[5]^2+18 0*c[6]^2*c[5]^2+500*c[6]^2*c[4]^2*c[7]*c[5]+950*c[6]^2*c[5]^2*c[7]*c[4 ]-900*c[6]^2*c[5]^2*c[7]*c[4]^2-300*c[6]*c[5]^3*c[7]*c[4]^2-600*c[6]^2 *c[5]^3*c[7]*c[4]-117*c[6]^2*c[5]-130*c[5]^3*c[6]*c[4]+50*c[5]^3*c[7]* c[4]+93*c[5]^2*c[4]+230*c[5]^3*c[6]*c[7]*c[4]+50*c[5]^2*c[7]*c[4]^2-19 0*c[5]^2*c[6]*c[4]^2+150*c[5]^3*c[6]*c[4]^2+470*c[6]^2*c[5]^2*c[4]^2+3 20*c[6]^2*c[5]^3*c[4]-300*c[6]^2*c[5]^3*c[4]^2-30*c[5]^2*c[4]^2-530*c[ 6]^2*c[4]*c[5]^2-75*c[6]^2*c[4]+156*c[5]^2*c[6]*c[4]-150*c[5]^2*c[7]*c [4]-270*c[5]^2*c[6]*c[7]*c[4]+600*c[6]^2*c[7]*c[5]^3*c[4]^2)/(-9*c[5]* c[7]*c[4]-3*c[7]*c[6]-c[4]^2-28*c[5]*c[6]*c[7]*c[4]+c[7]*c[5]+c[7]*c[4 ]+9*c[6]*c[7]*c[5]-9*c[6]*c[4]^2+8*c[6]*c[4]^3+9*c[6]*c[7]*c[4]-8*c[6] *c[7]*c[4]^2-4*c[5]*c[4]^3+4*c[5]*c[7]*c[4]^2+28*c[5]*c[6]*c[4]^2+9*c[ 5]*c[4]^2-9*c[5]*c[6]*c[4]-c[5]*c[4]+3*c[6]*c[4]-30*c[5]^2*c[6]*c[7]*c [4]^2+28*c[6]*c[5]*c[7]*c[4]^2-8*c[6]*c[7]*c[5]^2-28*c[6]*c[5]*c[4]^3+ 30*c[5]^2*c[6]*c[4]^3-28*c[5]^2*c[6]*c[4]^2-4*c[5]^2*c[4]^2+8*c[5]^2*c [6]*c[4]+4*c[5]^2*c[7]*c[4]+28*c[5]^2*c[6]*c[7]*c[4])/(-c[4]+c[6])/(-c [4]+c[5])/c[4], a[8,6] = 1/2*(5*c[5]*c[6]*c[4]+c[6]-2*c[6]*c[4]-2*c[6] *c[5]+25*c[5]*c[7]*c[4]+6*c[5]+9*c[7]+10*c[7]^2*c[4]-4+6*c[4]-6*c[7]^2 -14*c[7]*c[4]-10*c[5]*c[4]-14*c[7]*c[5]-20*c[7]^2*c[5]*c[4]+10*c[7]^2* c[5])/(c[6]-c[7])/(-c[4]+c[6])/(-c[5]+c[6])/c[6]/(-20*c[5]*c[6]*c[4]+3 0*c[5]*c[6]*c[7]*c[4]+15*c[6]*c[5]-20*c[6]*c[7]*c[5]+15*c[6]*c[4]-20*c [6]*c[7]*c[4]-12*c[6]+15*c[7]*c[6]+15*c[5]*c[4]-20*c[5]*c[7]*c[4]-12*c [5]+15*c[7]*c[5]-12*c[4]+15*c[7]*c[4]+10-12*c[7])*(c[4]-1)*(-1+c[5])*( -1+c[6]), b[8] = 1/60*(-20*c[5]*c[6]*c[4]+30*c[5]*c[6]*c[7]*c[4]+15*c[ 6]*c[5]-20*c[6]*c[7]*c[5]+15*c[6]*c[4]-20*c[6]*c[7]*c[4]-12*c[6]+15*c[ 7]*c[6]+15*c[5]*c[4]-20*c[5]*c[7]*c[4]-12*c[5]+15*c[7]*c[5]-12*c[4]+15 *c[7]*c[4]+10-12*c[7])/(c[4]-1)/(-1+c[5])/(-1+c[6])/(-1+c[7]), a[5,1] \+ = 1/4*c[5]*(2*c[5]^2-5*c[5]*c[4]+4*c[4]^2)/c[4]^2, `b*`[5] = 1/60*(-42 *c[7]*c[6]^2+89*c[5]*c[7]*c[4]-18*c[6]+6*c[5]+6*c[4]+27*c[7]*c[6]-9*c[ 4]^2+27*c[6]^2+186*c[5]*c[6]*c[7]*c[4]-36*c[6]*c[5]^2+39*c[6]*c[5]-9*c [7]*c[5]-100*c[6]^2*c[4]^3+600*c[6]^2*c[7]*c[5]^2*c[4]^3-100*c[7]*c[6] *c[4]^3-310*c[6]^2*c[7]*c[4]^2+192*c[6]^2*c[7]*c[4]+180*c[6]^2*c[4]^2+ 14*c[7]*c[4]^2-9*c[7]*c[4]-61*c[6]*c[7]*c[5]-97*c[6]*c[4]^2+60*c[6]*c[ 4]^3-103*c[6]*c[7]*c[4]+156*c[6]*c[7]*c[4]^2-30*c[5]*c[4]^3-150*c[5]*c [7]*c[4]^2+156*c[5]*c[6]*c[4]^2+93*c[5]*c[4]^2-112*c[5]*c[6]*c[4]-57*c [5]*c[4]+66*c[6]*c[4]+230*c[5]*c[7]*c[6]*c[4]^3-530*c[6]^2*c[4]^2*c[5] +330*c[6]^2*c[4]*c[5]+380*c[5]^2*c[6]*c[7]*c[4]^2+320*c[6]^2*c[5]*c[4] ^3-560*c[6]^2*c[7]*c[4]*c[5]-600*c[7]*c[6]^2*c[5]*c[4]^3+120*c[6]^2*c[ 7]*c[5]-100*c[6]^2*c[7]*c[5]^2-270*c[6]*c[5]*c[7]*c[4]^2+60*c[6]*c[7]* c[5]^2+60*c[6]^2*c[5]^2+950*c[6]^2*c[4]^2*c[7]*c[5]+500*c[6]^2*c[5]^2* c[7]*c[4]-900*c[6]^2*c[5]^2*c[7]*c[4]^2-300*c[6]*c[5]^2*c[7]*c[4]^3-75 *c[6]^2*c[5]+50*c[7]*c[5]*c[4]^3-130*c[6]*c[5]*c[4]^3+18*c[5]^2*c[4]+1 50*c[5]^2*c[6]*c[4]^3+50*c[5]^2*c[7]*c[4]^2-190*c[5]^2*c[6]*c[4]^2-300 *c[6]^2*c[5]^2*c[4]^3+470*c[6]^2*c[5]^2*c[4]^2-30*c[5]^2*c[4]^2-280*c[ 6]^2*c[4]*c[5]^2+180*c[7]*c[6]^2*c[4]^3-117*c[6]^2*c[4]+126*c[5]^2*c[6 ]*c[4]-30*c[5]^2*c[7]*c[4]-230*c[5]^2*c[6]*c[7]*c[4])/(-4*c[5]^3*c[4]- 9*c[5]*c[7]*c[4]-3*c[7]*c[6]-c[5]^2-28*c[5]*c[6]*c[7]*c[4]-9*c[6]*c[5] ^2+3*c[6]*c[5]+8*c[6]*c[5]^3+c[7]*c[5]+c[7]*c[4]+9*c[6]*c[7]*c[5]+9*c[ 6]*c[7]*c[4]-8*c[6]*c[7]*c[4]^2+4*c[5]*c[7]*c[4]^2+8*c[5]*c[6]*c[4]^2- 9*c[5]*c[6]*c[4]-c[5]*c[4]-30*c[5]^2*c[6]*c[7]*c[4]^2+28*c[6]*c[5]*c[7 ]*c[4]^2-8*c[6]*c[7]*c[5]^2-28*c[5]^3*c[6]*c[4]+9*c[5]^2*c[4]-28*c[5]^ 2*c[6]*c[4]^2+30*c[5]^3*c[6]*c[4]^2-4*c[5]^2*c[4]^2+28*c[5]^2*c[6]*c[4 ]+4*c[5]^2*c[7]*c[4]+28*c[5]^2*c[6]*c[7]*c[4])/(-c[5]+c[6])/(-c[4]+c[5 ])/c[5], a[5,4] = c[5]^2*(-c[4]+c[5])/c[4]^2, a[7,6] = 1/2*(5*c[5]*c[4 ]+1-2*c[4]-2*c[5])*(c[4]-c[7])*(c[5]-c[7])*(c[6]-c[7])*c[7]/(10*c[5]*c [6]*c[4]-5*c[6]*c[5]-5*c[6]*c[4]+3*c[6]-5*c[5]*c[4]+3*c[5]+3*c[4]-2)/( -c[4]+c[6])/(-c[5]+c[6])/c[6], c[3] = 2/3*c[4], a[4,3] = 3/4*c[4], a[3 ,2] = 2/9*c[4]^2/c[2], a[7,4] = -1/2*c[7]*(-68*c[5]^2*c[6]*c[4]^4-7*c[ 6]*c[5]*c[4]^4+7*c[4]^2*c[5]^2*c[7]^3-1400*c[5]^4*c[4]^4*c[6]^2*c[7]+1 1*c[7]^3*c[5]*c[4]^3-72*c[7]^3*c[5]^2*c[4]^3-3*c[7]*c[5]*c[4]^4-58*c[5 ]^2*c[4]^4*c[7]^2-120*c[6]*c[4]^7*c[7]*c[5]+320*c[4]^5*c[6]^2*c[5]^2-2 00*c[4]^3*c[7]^3*c[5]^3-100*c[4]^5*c[6]*c[5]^3-2*c[5]*c[6]^2*c[4]^4-60 0*c[4]^5*c[7]^3*c[6]*c[5]^3-200*c[5]^5*c[4]^3*c[7]^2*c[6]^2+400*c[5]^2 *c[4]^6*c[6]^2*c[7]^2-21*c[4]^5*c[5]*c[7]+120*c[5]^2*c[4]^5*c[7]-600*c [5]^3*c[4]^6*c[6]^2*c[7]-198*c[5]^3*c[4]^4*c[7]-280*c[7]^2*c[4]^5*c[5] ^3+240*c[7]^3*c[5]^2*c[4]^4+1250*c[6]^2*c[4]^5*c[7]*c[5]^3+120*c[7]*c[ 5]^2*c[4]^6*c[6]-340*c[5]^4*c[4]^2*c[7]^2*c[6]+400*c[6]^2*c[7]*c[5]^4* c[4]^2-240*c[5]^2*c[4]^6*c[7]-300*c[6]*c[5]*c[7]^2*c[4]^6-228*c[6]*c[4 ]^5*c[5]*c[7]-119*c[6]*c[5]^2*c[4]^4*c[7]-200*c[5]*c[7]^3*c[4]^6*c[6]- 600*c[5]^4*c[7]*c[4]^6*c[6]-120*c[4]^6*c[6]^2*c[5]^2+600*c[5]^2*c[7]^2 *c[4]^6*c[6]+100*c[5]^2*c[7]^3*c[4]^6-187*c[5]^4*c[4]^3*c[6]-910*c[5]^ 4*c[4]^5*c[6]-320*c[5]^2*c[6]*c[4]^6+620*c[5]^4*c[4]^4*c[6]-100*c[5]^4 *c[7]^3*c[4]^4-40*c[7]^2*c[4]^4*c[5]^3+40*c[7]^3*c[4]^5*c[5]+10*c[6]*c [7]^2*c[4]^5-10*c[6]*c[7]^3*c[4]^4+100*c[5]^4*c[4]^5*c[7]^2+80*c[5]^4* c[4]^4*c[7]+8*c[4]^3*c[7]^3*c[6]-2*c[7]^2*c[6]*c[4]^4+280*c[7]^3*c[5]^ 3*c[4]^4+80*c[6]*c[4]^5*c[5]*c[7]^2+c[4]^4*c[7]^2-360*c[5]^2*c[7]^3*c[ 4]^5-80*c[7]^2*c[5]^4*c[4]^3+40*c[7]^3*c[5]^4*c[4]^3-1200*c[5]^3*c[7]* c[4]^6*c[6]+10*c[6]^2*c[7]^3*c[4]^3-10*c[6]^2*c[4]^5*c[5]-12*c[7]*c[6] ^2*c[5]^3-4*c[7]^2*c[6]*c[5]^3-80*c[5]^4*c[4]^5*c[7]-240*c[7]*c[6]^2*c [5]*c[4]^5-3*c[5]^4*c[4]^2*c[7]^2+20*c[5]^5*c[7]^2*c[4]^3+180*c[6]*c[5 ]^2*c[4]^5*c[7]-60*c[5]^4*c[7]^3*c[4]^2*c[6]+400*c[5]^2*c[7]^3*c[4]^6* c[6]-25*c[4]^5*c[5]^3+200*c[5]^5*c[4]^4*c[7]^2*c[6]+60*c[5]^2*c[4]^7*c [7]+100*c[7]*c[4]^4*c[6]*c[5]-40*c[5]*c[4]^6*c[7]^2-150*c[4]^4*c[6]^2* c[5]^5+50*c[5]^5*c[7]*c[4]^4*c[6]+174*c[7]^2*c[5]^3*c[4]^3+8*c[7]*c[4] ^4*c[6]^2+600*c[5]^3*c[7]*c[4]^7*c[6]-60*c[5]^5*c[4]^2*c[6]^2*c[7]+200 *c[5]^5*c[4]^4*c[6]^2*c[7]-40*c[5]^5*c[4]^4*c[6]-200*c[5]^5*c[7]*c[4]^ 5*c[6]-149*c[6]^2*c[5]^4*c[7]*c[4]-24*c[5]^4*c[6]^2*c[4]+189*c[5]^4*c[ 6]^2*c[4]^2+300*c[5]^3*c[4]^6*c[6]^2-20*c[5]^5*c[4]^4*c[7]-5*c[4]^5*c[ 5]*c[7]^2+600*c[4]^6*c[6]*c[5]^3-84*c[6]^2*c[7]*c[5]^2*c[4]^3-74*c[6]^ 2*c[7]^2*c[5]^2*c[4]+162*c[6]^2*c[7]^2*c[5]^2*c[4]^2+311*c[6]*c[7]*c[5 ]^3*c[4]^3+21*c[6]^2*c[7]^3*c[4]*c[5]-40*c[7]^2*c[6]^2*c[5]^4+11*c[5]^ 4*c[7]*c[4]^2+107*c[7]^2*c[6]*c[5]*c[4]^4+100*c[7]^2*c[5]^5*c[6]^2*c[4 ]^2+320*c[5]^4*c[4]*c[7]^2*c[6]^2-10*c[6]^2*c[4]^4*c[7]^2+300*c[5]^2*c [4]^6*c[7]^2-3*c[7]^2*c[6]*c[4]^3-2*c[5]*c[7]^2*c[4]^3+15*c[6]*c[5]*c[ 7]^2*c[4]^2-8*c[5]*c[7]*c[6]*c[4]^3-14*c[5]^2*c[6]*c[7]*c[4]^2-2000*c[ 5]^4*c[7]^2*c[4]^4*c[6]+200*c[4]^5*c[6]*c[7]*c[5]^3+250*c[5]^2*c[6]*c[ 7]^3*c[4]^4+4*c[6]^2*c[5]*c[4]^3-26*c[7]^3*c[4]^2*c[6]*c[5]-69*c[7]*c[ 6]^2*c[5]*c[4]^3-18*c[6]^2*c[7]^3*c[5]^2+12*c[6]^2*c[7]^2*c[5]^2+10*c[ 6]^2*c[4]^2*c[7]*c[5]+50*c[6]^2*c[7]^2*c[4]^2*c[5]+31*c[6]*c[5]^4*c[4] ^2+24*c[7]*c[6]^2*c[5]^4+240*c[4]^5*c[7]*c[5]^3+2*c[6]^2*c[5]^2*c[7]*c [4]+39*c[6]^2*c[5]^2*c[7]*c[4]^2-17*c[6]*c[5]^2*c[7]*c[4]^3-73*c[6]*c[ 5]^3*c[7]*c[4]^2+59*c[6]^2*c[5]^3*c[7]*c[4]-12*c[5]^4*c[4]^3*c[7]-12*c [7]^2*c[5]^2*c[4]^3+40*c[5]^4*c[7]^2*c[4]^4-c[4]^3*c[7]^3+17*c[7]^3*c[ 5]^2*c[6]*c[4]-9*c[7]^2*c[5]^2*c[6]*c[4]+3*c[6]*c[7]*c[4]^4+4*c[4]^5*c [5]+4*c[6]^2*c[7]^2*c[4]^2-150*c[5]^5*c[4]^3*c[7]^2*c[6]+21*c[5]^3*c[6 ]*c[7]*c[4]-14*c[6]^2*c[7]^2*c[4]*c[5]+23*c[5]^2*c[6]*c[4]^3+13*c[5]^2 *c[7]*c[4]^3-11*c[5]^3*c[7]*c[4]^2-17*c[5]^3*c[6]*c[4]^2+168*c[5]^3*c[ 6]^2*c[4]^3+50*c[5]^5*c[4]^3*c[6]^2*c[7]-220*c[4]^3*c[7]^2*c[5]^2*c[6] ^2-6*c[6]*c[7]*c[4]^5-580*c[4]^5*c[5]^3*c[6]^2-31*c[7]^3*c[4]^4*c[5]-2 *c[7]^2*c[5]^4*c[4]-4*c[5]^2*c[4]^4*c[7]+53*c[6]^2*c[5]^2*c[4]^3-14*c[ 6]^2*c[5]^2*c[4]^2+47*c[6]*c[5]^3*c[4]^3+12*c[6]^2*c[5]^3*c[4]-59*c[6] ^2*c[5]^3*c[4]^2+28*c[5]^3*c[7]*c[4]^3+127*c[5]^4*c[7]*c[4]^2*c[6]+125 0*c[5]^3*c[7]^3*c[4]^4*c[6]+10*c[6]*c[5]*c[4]^6+240*c[5]*c[4]^6*c[7]*c [6]-4*c[7]*c[6]^2*c[4]^3+11*c[7]^2*c[5]^2*c[6]*c[4]^3-80*c[7]^3*c[5]^2 *c[6]*c[4]^2-121*c[7]^2*c[5]^3*c[6]*c[4]^2-96*c[5]^3*c[7]^3*c[6]*c[4]+ c[7]^3*c[4]^2*c[5]-68*c[4]^3*c[7]^2*c[5]*c[6]^2-80*c[7]^2*c[6]*c[5]*c[ 4]^3+68*c[7]^2*c[5]^2*c[6]*c[4]^2-10*c[4]^4*c[5]^2+20*c[7]^2*c[5]^3*c[ 6]*c[4]-240*c[7]^2*c[5]^3*c[6]*c[4]^3-9*c[5]^4*c[4]^3-2*c[6]^2*c[4]^3* c[7]^2+260*c[5]^3*c[6]^2*c[7]*c[4]^3+49*c[5]^3*c[7]^3*c[4]^2-c[7]^3*c[ 5]^2*c[4]-228*c[6]^2*c[7]*c[5]^3*c[4]^2+30*c[6]^2*c[7]^2*c[4]*c[5]^3-2 *c[7]^2*c[5]^2*c[4]^2+5*c[7]^2*c[5]^3*c[4]+40*c[5]^5*c[4]^3*c[6]*c[7]+ 36*c[4]^6*c[7]*c[5]+90*c[7]^3*c[5]^2*c[6]^2*c[4]^3+16*c[7]^2*c[4]^4*c[ 5]-20*c[7]^2*c[5]^3*c[6]^2*c[4]^2+300*c[6]^2*c[4]^5*c[5]*c[7]^2+250*c[ 7]^2*c[4]^5*c[6]*c[5]^3-490*c[5]^2*c[6]*c[4]^5*c[7]^2+120*c[7]^3*c[5]^ 2*c[4]^3*c[6]+810*c[7]^3*c[5]^3*c[6]^2*c[4]^2+111*c[7]^3*c[6]*c[5]*c[4 ]^3+280*c[7]^3*c[6]^2*c[5]*c[4]^3-240*c[7]^3*c[5]^3*c[6]^2*c[4]+15*c[5 ]^4*c[4]^4+60*c[5]^4*c[7]^2*c[4]*c[6]+6*c[6]*c[7]^3*c[5]^3-31*c[5]^4*c [4]*c[7]*c[6]-200*c[4]^5*c[6]^2*c[7]*c[5]^2-500*c[4]^5*c[6]^2*c[7]^2*c [5]^2+9*c[4]^5*c[5]^2-181*c[6]^2*c[4]^4*c[5]^2-1280*c[7]^3*c[5]^3*c[4] ^3*c[6]-600*c[5]^2*c[4]^5*c[7]^3*c[6]-6*c[4]^6*c[5]-6*c[6]^2*c[7]^3*c[ 4]^2+100*c[5]^5*c[4]^5*c[6]+120*c[5]*c[7]*c[4]^6*c[6]^2+420*c[6]*c[4]^ 5*c[7]^3*c[5]+600*c[5]^4*c[4]^5*c[6]^2*c[7]+175*c[7]*c[6]^2*c[5]*c[4]^ 4-430*c[5]^3*c[4]^4*c[6]^2*c[7]+200*c[5]^3*c[4]^3*c[6]^2*c[7]^2-450*c[ 5]^4*c[4]^4*c[6]*c[7]+160*c[5]^2*c[4]^4*c[7]*c[6]^2-200*c[5]^3*c[4]^4* c[6]*c[7]-280*c[5]^4*c[4]^3*c[6]*c[7]+6*c[5]^3*c[4]^3-100*c[5]^2*c[4]^ 7*c[7]^2-1200*c[7]^3*c[5]^3*c[6]^2*c[4]^3-320*c[7]^3*c[4]^4*c[5]*c[6]- 30*c[7]^2*c[5]^3*c[4]^2-300*c[6]*c[5]^3*c[4]^7-350*c[5]^4*c[4]^5*c[6]^ 2+600*c[6]^2*c[7]^3*c[5]^3*c[4]^4+600*c[6]*c[7]^2*c[5]^4*c[4]^5+200*c[ 6]^2*c[7]^3*c[4]^5*c[5]+200*c[7]^2*c[4]^7*c[6]*c[5]+1100*c[5]^4*c[4]^4 *c[6]^2-720*c[5]^4*c[4]^3*c[6]^2+1750*c[7]^2*c[5]^4*c[6]^2*c[4]^3-1230 *c[7]^2*c[5]^4*c[6]^2*c[4]^2+150*c[5]^4*c[7]^3*c[4]^3*c[6]+300*c[5]^4* c[4]^6*c[6]+480*c[5]^3*c[4]^4*c[7]^2*c[6]+1420*c[5]^4*c[4]^3*c[7]^2*c[ 6]+20*c[5]^2*c[4]^4*c[7]^2*c[6]+130*c[5]^4*c[4]^3*c[6]^2*c[7]-40*c[7]^ 2*c[4]^4*c[6]^2*c[5]-350*c[7]^2*c[4]^4*c[5]^3*c[6]^2+230*c[7]^2*c[4]^4 *c[6]^2*c[5]^2+10*c[4]^6*c[5]^2+188*c[5]^2*c[6]*c[4]^5-400*c[6]^2*c[7] ^3*c[5]^2*c[4]^5-600*c[6]^2*c[7]^2*c[5]^4*c[4]^4+120*c[5]^2*c[4]^7*c[6 ]-94*c[4]^4*c[6]*c[5]^3+6*c[6]*c[5]*c[4]^5+60*c[5]^5*c[6]^2*c[4]^3-200 *c[6]^2*c[4]^6*c[5]*c[7]^2-420*c[7]^3*c[6]^2*c[5]*c[4]^4+1550*c[5]^4*c [4]^5*c[6]*c[7]+500*c[7]^3*c[4]^4*c[6]^2*c[5]^2-400*c[5]^2*c[4]^7*c[6] *c[7]^2+6*c[5]^3*c[4]^4+103*c[6]^2*c[7]^3*c[5]^2*c[4]-260*c[6]^2*c[7]^ 3*c[5]^2*c[4]^2+540*c[5]^3*c[7]^3*c[4]^2*c[6]-97*c[6]^2*c[5]*c[7]^3*c[ 4]^2+30*c[6]^2*c[7]^3*c[5]^3-4*c[7]^3*c[4]*c[5]^3)/(-240*c[5]^2*c[6]*c [4]^4+40*c[6]*c[5]*c[4]^4-9*c[5]^3*c[4]-100*c[4]^5*c[6]*c[5]^3-350*c[5 ]^4*c[4]^3*c[6]+30*c[5]^5*c[4]^2+2*c[5]^3-2*c[4]^3-50*c[5]^5*c[4]^3-3* c[5]^4+3*c[4]^4+50*c[4]^5*c[5]^3+100*c[5]^5*c[4]^3*c[6]-3*c[6]*c[5]^3+ 3*c[6]*c[4]^3+9*c[5]*c[4]^3-6*c[5]*c[6]*c[4]^2+4*c[5]*c[4]^2+240*c[6]* c[5]^4*c[4]^2+5*c[6]*c[5]^4-13*c[6]*c[5]*c[4]^3+13*c[5]^3*c[6]*c[4]-4* c[5]^2*c[4]+68*c[5]^3*c[4]^2+100*c[5]^2*c[6]*c[4]^3-100*c[5]^3*c[6]*c[ 4]^2-140*c[5]^4*c[4]^2+140*c[4]^4*c[5]^2+180*c[5]^4*c[4]^3+6*c[5]^2*c[ 6]*c[4]-68*c[5]^2*c[4]^3-23*c[4]^4*c[5]-40*c[5]^4*c[4]*c[6]-30*c[4]^5* c[5]^2-5*c[6]*c[4]^4-50*c[5]^5*c[6]*c[4]^2+50*c[5]^2*c[6]*c[4]^5+350*c [4]^4*c[6]*c[5]^3+23*c[5]^4*c[4]-180*c[5]^3*c[4]^4)/(-c[4]+c[6])/c[4]^ 2, `b*`[7] = 1/60*(-30*c[5]^3*c[4]-18*c[6]+6*c[5]+6*c[4]-9*c[5]^2-9*c[ 4]^2+27*c[6]^2-97*c[6]*c[5]^2+66*c[6]*c[5]+60*c[6]*c[5]^3-100*c[6]^2*c [5]^3-100*c[6]^2*c[4]^3+180*c[6]^2*c[4]^2-97*c[6]*c[4]^2+60*c[6]*c[4]^ 3-30*c[5]*c[4]^3+312*c[5]*c[6]*c[4]^2+107*c[5]*c[4]^2-215*c[5]*c[6]*c[ 4]-66*c[5]*c[4]+66*c[6]*c[4]-840*c[6]^2*c[4]^2*c[5]+522*c[6]^2*c[4]*c[ 5]+500*c[6]^2*c[5]*c[4]^3+180*c[6]^2*c[5]^2-117*c[6]^2*c[5]-230*c[6]*c [5]*c[4]^3-230*c[5]^3*c[6]*c[4]+107*c[5]^2*c[4]+50*c[5]^3*c[4]^2+380*c [5]^2*c[6]*c[4]^3-460*c[5]^2*c[6]*c[4]^2+380*c[5]^3*c[6]*c[4]^2+600*c[ 5]^3*c[6]^2*c[4]^3-900*c[6]^2*c[5]^2*c[4]^3+1420*c[6]^2*c[5]^2*c[4]^2- 300*c[6]*c[5]^3*c[4]^3+500*c[6]^2*c[5]^3*c[4]-900*c[6]^2*c[5]^3*c[4]^2 -180*c[5]^2*c[4]^2-840*c[6]^2*c[4]*c[5]^2-117*c[6]^2*c[4]+312*c[5]^2*c [6]*c[4]+50*c[5]^2*c[4]^3)/c[7]/(-3*c[7]^3*c[6]+3*c[7]^2*c[6]^2-9*c[6] ^2*c[7]^2*c[5]-8*c[7]*c[6]^2*c[5]^3+8*c[7]^2*c[6]*c[5]^3-c[5]*c[6]*c[7 ]*c[4]-c[7]^2*c[4]^2-30*c[6]^2*c[7]*c[5]^2*c[4]^3-28*c[6]^2*c[7]^2*c[5 ]^2*c[4]+30*c[6]^2*c[7]^2*c[5]^2*c[4]^2-30*c[6]*c[7]*c[5]^3*c[4]^3+9*c [6]^2*c[7]*c[4]^2-3*c[6]^2*c[7]*c[4]+2*c[6]*c[7]^2*c[4]-9*c[6]*c[7]^2* c[4]^2-2*c[7]^2*c[5]*c[4]-c[7]^2*c[5]^2+c[6]*c[7]*c[4]^2+8*c[7]^2*c[6] *c[4]^3-4*c[5]*c[7]^2*c[4]^3+c[5]*c[7]*c[4]^2-c[5]*c[6]*c[4]^2+32*c[6] *c[5]*c[7]^2*c[4]^2-9*c[7]^2*c[5]*c[6]*c[4]-4*c[5]*c[7]*c[6]*c[4]^3-9* c[6]^2*c[4]^2*c[5]+3*c[6]^2*c[4]*c[5]+c[7]^3*c[4]+9*c[7]^3*c[4]*c[6]-9 *c[7]^3*c[4]*c[5]-20*c[5]^2*c[6]*c[7]*c[4]^2+8*c[6]^2*c[5]*c[4]^3+28*c [7]^3*c[4]^2*c[6]*c[5]+18*c[6]^2*c[7]*c[4]*c[5]+28*c[7]*c[6]^2*c[5]*c[ 4]^3-3*c[6]^2*c[7]*c[5]+9*c[6]^2*c[7]*c[5]^2+c[6]*c[7]*c[5]^2-8*c[6]*c [7]^3*c[5]^2-9*c[6]*c[7]^2*c[5]^2+8*c[6]^2*c[7]^2*c[5]^2+2*c[6]*c[7]^2 *c[5]-36*c[6]^2*c[4]^2*c[7]*c[5]-28*c[6]^2*c[7]^2*c[4]^2*c[5]-28*c[7]^ 3*c[4]*c[6]*c[5]-36*c[6]^2*c[5]^2*c[7]*c[4]+56*c[6]^2*c[5]^2*c[7]*c[4] ^2+28*c[6]*c[5]^2*c[7]*c[4]^3+28*c[6]*c[5]^3*c[7]*c[4]^2+28*c[6]^2*c[5 ]^3*c[7]*c[4]+9*c[6]*c[7]^3*c[5]+9*c[7]^2*c[5]*c[4]^2+28*c[7]^3*c[5]^2 *c[6]*c[4]+32*c[7]^2*c[5]^2*c[6]*c[4]+8*c[6]^2*c[7]^2*c[4]^2-9*c[6]^2* c[7]^2*c[4]-4*c[5]^3*c[6]*c[7]*c[4]+28*c[6]^2*c[7]^2*c[4]*c[5]-4*c[5]^ 2*c[6]*c[4]^3-9*c[5]^2*c[7]*c[4]^2+9*c[5]^2*c[6]*c[4]^2+4*c[5]^2*c[7]* c[4]^3+4*c[5]^3*c[7]*c[4]^2-4*c[5]^3*c[6]*c[4]^2+30*c[5]^3*c[6]^2*c[4] ^3-28*c[6]^2*c[5]^2*c[4]^3+28*c[6]^2*c[5]^2*c[4]^2+8*c[6]^2*c[5]^3*c[4 ]-28*c[6]^2*c[5]^3*c[4]^2-9*c[6]^2*c[4]*c[5]^2-8*c[7]*c[6]^2*c[4]^3+30 *c[7]^2*c[5]^2*c[6]*c[4]^3-30*c[7]^3*c[5]^2*c[6]*c[4]^2+30*c[7]^2*c[5] ^3*c[6]*c[4]^2-8*c[7]^3*c[4]^2*c[6]+4*c[7]^3*c[4]^2*c[5]+c[7]^3*c[5]-2 8*c[7]^2*c[6]*c[5]*c[4]^3-56*c[7]^2*c[5]^2*c[6]*c[4]^2-28*c[7]^2*c[5]^ 3*c[6]*c[4]-c[5]^2*c[6]*c[4]+c[5]^2*c[7]*c[4]+4*c[7]^3*c[5]^2*c[4]+9*c [7]^2*c[5]^2*c[4]-30*c[6]^2*c[7]*c[5]^3*c[4]^2-8*c[7]^2*c[5]^2*c[4]^2- 4*c[7]^2*c[5]^3*c[4]), a[2,1] = c[2], a[6,4] = -1/2*(120*c[5]^2*c[6]*c [4]^4-12*c[6]*c[5]*c[4]^4-90*c[5]^2*c[6]^3*c[4]^2-60*c[5]*c[6]^2*c[4]^ 4-2*c[6]^3*c[4]^2+20*c[5]^4*c[4]^3*c[6]+40*c[6]^2*c[4]^5*c[5]+36*c[4]* c[5]^2*c[6]^3-20*c[5]^4*c[6]^2*c[4]^2+8*c[6]^2*c[5]^3+2*c[6]^2*c[4]^3- 24*c[5]*c[6]^3*c[4]^2-6*c[6]^2*c[4]^2*c[5]+36*c[6]^2*c[5]*c[4]^3+60*c[ 4]^3*c[5]*c[6]^3-3*c[6]*c[5]*c[4]^3-11*c[5]^3*c[6]*c[4]+3*c[5]^3*c[4]^ 2-24*c[5]^2*c[6]*c[4]^3+13*c[5]^2*c[6]*c[4]^2+48*c[5]^3*c[6]*c[4]^2-60 *c[5]^3*c[6]^2*c[4]^3-30*c[6]^2*c[5]^2*c[4]^3-12*c[6]^2*c[5]^2*c[4]^2- 150*c[6]*c[5]^3*c[4]^3-48*c[6]^2*c[5]^3*c[4]+150*c[6]^2*c[5]^3*c[4]^2- 2*c[6]^2*c[4]*c[5]^2-5*c[5]^2*c[4]^3+2*c[4]^4*c[5]-6*c[5]^2*c[6]^3+60* c[6]^3*c[4]^3*c[5]^2+7*c[6]^3*c[4]*c[5]-60*c[5]^2*c[6]*c[4]^5-40*c[6]^ 3*c[4]^4*c[5]+60*c[4]^4*c[6]*c[5]^3)*c[6]/(c[4]^3-2*c[5]*c[4]^2+30*c[5 ]^2*c[4]^3-6*c[5]*c[4]^3-10*c[4]^4*c[5]^2-c[5]^3+10*c[5]^4*c[4]^2-30*c [5]^3*c[4]^2+6*c[5]^3*c[4]+2*c[5]^2*c[4])/c[4]^2, a[3,1] = -2/9*c[4]*( c[4]-3*c[2])/c[2], a[4,1] = 1/4*c[4], a[7,3] = 3/4*1/c[4]^2*c[7]*(-480 *c[5]^2*c[6]*c[4]^4+15*c[6]*c[5]*c[4]^4+300*c[4]^2*c[5]^2*c[7]^3+60*c[ 7]^3*c[5]*c[4]^3-510*c[7]^3*c[5]^2*c[4]^3+54*c[7]*c[5]*c[4]^4+450*c[5] ^2*c[4]^4*c[7]^2+150*c[4]^3*c[7]^3*c[5]^3-150*c[7]^2*c[5]^2*c[4]^5-450 *c[4]^5*c[6]*c[5]^3-3*c[7]^3*c[4]^2+90*c[5]^2*c[4]^5*c[7]+120*c[5]^3*c [4]^4*c[7]+150*c[7]^3*c[5]^2*c[4]^4+100*c[5]^4*c[4]^2*c[7]^2*c[6]-180* c[6]*c[4]^5*c[5]*c[7]+30*c[6]*c[5]^2*c[4]^4*c[7]+60*c[5]^4*c[4]^3*c[6] -150*c[5]^4*c[4]^4*c[6]-200*c[7]^2*c[4]^4*c[5]^3+300*c[6]*c[4]^5*c[5]* c[7]^2-50*c[7]^2*c[5]^4*c[4]^3+24*c[7]*c[6]*c[5]^3-40*c[7]^2*c[6]*c[5] ^3-6*c[5]*c[6]*c[7]*c[4]+360*c[7]*c[4]^4*c[6]*c[5]+150*c[7]^2*c[5]^3*c [4]^3+2*c[7]^2*c[4]^2+6*c[5]^2*c[7]^3+330*c[6]*c[7]*c[5]^3*c[4]^3-450* c[7]^2*c[6]*c[5]*c[4]^4-12*c[7]*c[6]*c[4]^3-6*c[6]*c[7]^2*c[4]-2*c[7]^ 2*c[5]*c[4]-4*c[7]^2*c[5]^2+6*c[6]*c[7]*c[4]^2+20*c[7]^2*c[6]*c[4]^3-1 0*c[5]*c[7]^2*c[4]^3+6*c[5]*c[4]^3-6*c[5]*c[7]*c[4]^2-6*c[5]*c[6]*c[4] ^2+150*c[6]*c[5]*c[7]^2*c[4]^2-72*c[7]^2*c[5]*c[6]*c[4]-270*c[5]*c[7]* c[6]*c[4]^3+9*c[7]^3*c[4]*c[6]+3*c[7]^3*c[4]*c[5]-306*c[5]^2*c[6]*c[7] *c[4]^2+900*c[4]^5*c[6]*c[7]*c[5]^3+600*c[5]^2*c[6]*c[7]^3*c[4]^4-420* c[7]^3*c[4]^2*c[6]*c[5]+90*c[6]*c[5]*c[7]*c[4]^2-12*c[6]*c[7]*c[5]^2+3 0*c[6]*c[7]^3*c[5]^2+12*c[6]*c[7]^2*c[5]+123*c[7]^3*c[4]*c[6]*c[5]+360 *c[6]*c[5]^2*c[7]*c[4]^3+360*c[6]*c[5]^3*c[7]*c[4]^2-18*c[6]*c[7]^3*c[ 5]+30*c[5]^4*c[4]^3*c[7]-80*c[7]^2*c[5]^2*c[4]^3+24*c[7]^2*c[5]*c[4]^2 -30*c[7]*c[5]*c[4]^3+3*c[6]*c[5]*c[4]^3-240*c[7]^3*c[5]^2*c[6]*c[4]-24 *c[5]^3*c[6]*c[4]-12*c[5]^3*c[7]*c[4]-144*c[5]^3*c[6]*c[7]*c[4]+9*c[5] ^3*c[4]^2+294*c[5]^2*c[6]*c[4]^3-42*c[5]^2*c[7]*c[4]^2-75*c[5]^2*c[6]* c[4]^2+252*c[5]^2*c[7]*c[4]^3+18*c[5]^3*c[7]*c[4]^2+189*c[5]^3*c[6]*c[ 4]^2-360*c[5]^2*c[4]^4*c[7]-720*c[6]*c[5]^3*c[4]^3-120*c[5]^3*c[7]*c[4 ]^3-60*c[5]^4*c[7]*c[4]^2*c[6]-6*c[5]^2*c[4]^2-780*c[7]^2*c[5]^2*c[6]* c[4]^3+810*c[7]^3*c[5]^2*c[6]*c[4]^2-1230*c[7]^2*c[5]^3*c[6]*c[4]^2-15 *c[7]^3*c[4]^2*c[6]-21*c[7]^3*c[4]^2*c[5]+20*c[7]^2*c[6]*c[5]*c[4]^3+2 20*c[7]^2*c[5]^2*c[6]*c[4]^2+15*c[4]^4*c[5]^2+320*c[7]^2*c[5]^3*c[6]*c [4]+1850*c[7]^2*c[5]^3*c[6]*c[4]^3-60*c[5]^3*c[7]^3*c[4]^2+12*c[5]^2*c [6]*c[4]+12*c[5]^2*c[7]*c[4]-66*c[7]^3*c[5]^2*c[4]+24*c[7]^2*c[5]^2*c[ 4]+72*c[5]^2*c[6]*c[7]*c[4]-110*c[7]^2*c[5]^2*c[4]^2+20*c[7]^2*c[5]^3* c[4]-9*c[4]^4*c[5]-60*c[7]^2*c[4]^4*c[5]-600*c[5]^2*c[6]*c[4]^5*c[7]^2 -1200*c[7]^3*c[5]^2*c[4]^3*c[6]+630*c[7]^3*c[6]*c[5]*c[4]^3+300*c[5]^4 *c[4]^4*c[6]*c[7]-1800*c[5]^3*c[4]^4*c[6]*c[7]-15*c[5]^3*c[4]^3-300*c[ 7]^3*c[4]^4*c[5]*c[6]-20*c[7]^2*c[5]^3*c[4]^2-800*c[5]^3*c[4]^4*c[7]^2 *c[6]-200*c[5]^4*c[4]^3*c[7]^2*c[6]+1150*c[5]^2*c[4]^4*c[7]^2*c[6]+180 *c[5]^2*c[6]*c[4]^5+1140*c[4]^4*c[6]*c[5]^3)/(10*c[5]^3*c[4]^2+10*c[5] ^2*c[4]^3-30*c[5]^2*c[4]^2+6*c[5]^2*c[4]-c[5]^2+6*c[5]*c[4]^2+c[5]*c[4 ]-c[4]^2)/(10*c[5]*c[6]*c[4]-5*c[6]*c[5]-5*c[6]*c[4]+3*c[6]-5*c[5]*c[4 ]+3*c[5]+3*c[4]-2), `b*`[1] = 1/60*(30*c[5]^3*c[4]+42*c[7]*c[6]^2-103* c[5]*c[7]*c[4]+18*c[6]-6*c[5]-6*c[4]-27*c[7]*c[6]+9*c[5]^2+9*c[4]^2-18 0*c[7]*c[6]^2*c[5]^3+100*c[7]*c[6]*c[5]^3-27*c[6]^2-354*c[5]*c[6]*c[7] *c[4]+97*c[6]*c[5]^2-66*c[6]*c[5]-60*c[6]*c[5]^3+100*c[6]^2*c[5]^3+9*c [7]*c[5]+100*c[6]^2*c[4]^3-2040*c[6]^2*c[7]*c[5]^2*c[4]^3-600*c[6]*c[7 ]*c[5]^3*c[4]^3-14*c[7]*c[5]^2+100*c[7]*c[6]*c[4]^3+310*c[6]^2*c[7]*c[ 4]^2-192*c[6]^2*c[7]*c[4]-180*c[6]^2*c[4]^2-14*c[7]*c[4]^2+9*c[7]*c[4] +103*c[6]*c[7]*c[5]+97*c[6]*c[4]^2-60*c[6]*c[4]^3+103*c[6]*c[7]*c[4]-1 56*c[6]*c[7]*c[4]^2+30*c[5]*c[4]^3+174*c[5]*c[7]*c[4]^2-312*c[5]*c[6]* c[4]^2-107*c[5]*c[4]^2+215*c[5]*c[6]*c[4]+66*c[5]*c[4]-66*c[6]*c[4]-41 0*c[5]*c[7]*c[6]*c[4]^3+840*c[6]^2*c[4]^2*c[5]-522*c[6]^2*c[4]*c[5]-76 0*c[5]^2*c[6]*c[7]*c[4]^2-500*c[6]^2*c[5]*c[4]^3+932*c[6]^2*c[7]*c[4]* c[5]+1020*c[7]*c[6]^2*c[5]*c[4]^3-192*c[6]^2*c[7]*c[5]+310*c[6]^2*c[7] *c[5]^2+526*c[6]*c[5]*c[7]*c[4]^2-156*c[6]*c[7]*c[5]^2-180*c[6]^2*c[5] ^2-1610*c[6]^2*c[4]^2*c[7]*c[5]-1610*c[6]^2*c[5]^2*c[7]*c[4]+2970*c[6] ^2*c[5]^2*c[7]*c[4]^2+690*c[6]*c[5]^2*c[7]*c[4]^3+690*c[6]*c[5]^3*c[7] *c[4]^2+1020*c[6]^2*c[5]^3*c[7]*c[4]+117*c[6]^2*c[5]-50*c[7]*c[5]*c[4] ^3+230*c[6]*c[5]*c[4]^3+230*c[5]^3*c[6]*c[4]-50*c[5]^3*c[7]*c[4]-107*c [5]^2*c[4]-410*c[5]^3*c[6]*c[7]*c[4]-50*c[5]^3*c[4]^2-380*c[5]^2*c[6]* c[4]^3-310*c[5]^2*c[7]*c[4]^2+460*c[5]^2*c[6]*c[4]^2+90*c[5]^2*c[7]*c[ 4]^3+90*c[5]^3*c[7]*c[4]^2-380*c[5]^3*c[6]*c[4]^2-600*c[5]^3*c[6]^2*c[ 4]^3+900*c[6]^2*c[5]^2*c[4]^3-1420*c[6]^2*c[5]^2*c[4]^2+300*c[6]*c[5]^ 3*c[4]^3-500*c[6]^2*c[5]^3*c[4]+900*c[6]^2*c[5]^3*c[4]^2+180*c[5]^2*c[ 4]^2+840*c[6]^2*c[4]*c[5]^2-180*c[7]*c[6]^2*c[4]^3+117*c[6]^2*c[4]+150 0*c[5]^3*c[6]^2*c[7]*c[4]^3-312*c[5]^2*c[6]*c[4]+174*c[5]^2*c[7]*c[4]+ 526*c[5]^2*c[6]*c[7]*c[4]-2040*c[6]^2*c[7]*c[5]^3*c[4]^2-50*c[5]^2*c[4 ]^3)/(3*c[6]-28*c[5]*c[6]*c[4]^2-28*c[5]^2*c[6]*c[4]+8*c[6]*c[5]^2-9*c [6]*c[4]-9*c[6]*c[5]+28*c[5]*c[6]*c[4]+30*c[5]^2*c[6]*c[4]^2+8*c[6]*c[ 4]^2-4*c[5]^2*c[4]-4*c[5]*c[4]^2+9*c[5]*c[4]-c[5]-c[4])/c[5]/c[6]/c[7] /c[4], a[8,4] = -1/2*(734*c[5]^2*c[6]*c[4]^4-203*c[6]*c[5]*c[4]^4+3190 *c[5]^4*c[4]^4*c[6]^2*c[7]-93*c[7]*c[5]*c[4]^4-636*c[5]^2*c[4]^4*c[7]^ 2-10*c[5]^3*c[4]-520*c[6]*c[4]^7*c[7]*c[5]-481*c[4]^5*c[6]^2*c[5]^2+64 5*c[7]^2*c[5]^2*c[4]^5-574*c[4]^5*c[6]*c[5]^3-172*c[5]*c[6]^2*c[4]^4+6 *c[7]^2*c[4]^3+100*c[5]^5*c[4]^3*c[7]^2*c[6]^2+60*c[4]^8*c[5]^2-600*c[ 5]^2*c[4]^6*c[6]^2*c[7]^2+9*c[4]^5*c[5]*c[7]-164*c[5]^2*c[4]^5*c[7]+90 0*c[5]^3*c[4]^6*c[6]^2*c[7]+600*c[5]^3*c[4]^6*c[6]^2*c[7]^2-1104*c[5]^ 3*c[4]^4*c[7]-18*c[4]^4*c[6]^2+90*c[7]^2*c[4]^5*c[5]^3+1380*c[6]^2*c[4 ]^5*c[7]*c[5]^3-750*c[7]*c[5]^2*c[4]^6*c[6]+675*c[5]^4*c[4]^2*c[7]^2*c [6]+8*c[6]^2*c[4]^5-339*c[6]^2*c[7]*c[5]^4*c[4]^2-315*c[5]^2*c[4]^6*c[ 7]+640*c[6]*c[5]*c[7]^2*c[4]^6+46*c[6]*c[4]^5*c[5]*c[7]-1014*c[6]*c[5] ^2*c[4]^4*c[7]+2100*c[5]^4*c[7]*c[4]^6*c[6]-390*c[6]^2*c[4]^6*c[5]+860 *c[4]^6*c[6]^2*c[5]^2+140*c[5]^5*c[4]^3*c[7]-1740*c[5]^2*c[7]^2*c[4]^6 *c[6]-613*c[5]^4*c[4]^3*c[6]-100*c[4]^8*c[5]^2*c[7]+1140*c[5]^4*c[4]^5 *c[6]+968*c[5]^2*c[6]*c[4]^6-150*c[5]^5*c[4]^5*c[6]^2+263*c[5]^4*c[4]^ 4*c[6]+250*c[7]^2*c[4]^4*c[5]^3-120*c[6]*c[5]*c[4]^8-10*c[6]*c[7]^2*c[ 4]^5-6*c[6]*c[4]^6+780*c[5]^4*c[4]^4*c[7]+32*c[7]^2*c[6]*c[4]^4+1200*c [5]^3*c[7]^2*c[4]^6*c[6]-750*c[6]*c[4]^5*c[5]*c[7]^2-6*c[4]^4*c[7]^2-4 00*c[5]^4*c[7]^2*c[4]^5*c[6]^2+415*c[7]^2*c[5]^4*c[4]^3-990*c[5]^3*c[7 ]*c[4]^6*c[6]+433*c[6]^2*c[4]^5*c[5]+4*c[4]^3-60*c[5]^5*c[4]^3+30*c[7] *c[6]^2*c[5]^3+4*c[7]*c[6]*c[5]^3-6*c[7]^2*c[6]*c[5]^3-350*c[5]^4*c[4] ^6*c[6]^2-5*c[4]^4-250*c[5]^3*c[7]*c[4]^6-270*c[5]^4*c[4]^5*c[7]-330*c [7]*c[6]^2*c[5]*c[4]^5-1320*c[4]^6*c[6]^2*c[7]*c[5]^2-55*c[5]^4*c[4]^2 *c[7]^2-100*c[5]^5*c[7]^2*c[4]^3-96*c[6]*c[5]^2*c[4]^5*c[7]-562*c[4]^5 *c[5]^3-600*c[5]^5*c[4]^4*c[7]^2*c[6]+380*c[5]^2*c[4]^7*c[7]+160*c[5]^ 5*c[4]^3*c[6]+62*c[7]*c[4]^4*c[6]*c[5]+80*c[5]^5*c[4]^4+30*c[5]*c[4]^6 *c[7]^2+310*c[4]^4*c[6]^2*c[5]^5+450*c[5]^5*c[7]*c[4]^4*c[6]-356*c[7]^ 2*c[5]^3*c[4]^3+24*c[7]*c[4]^4*c[6]^2-200*c[5]^5*c[7]*c[4]^6*c[6]-900* c[5]^3*c[7]*c[4]^7*c[6]+60*c[5]^5*c[4]^2*c[6]^2*c[7]-450*c[5]^5*c[4]^4 *c[6]^2*c[7]-170*c[5]^5*c[4]^4*c[6]+200*c[5]^5*c[7]*c[4]^5*c[6]+143*c[ 6]^2*c[5]^4*c[7]*c[4]+34*c[5]^4*c[6]^2*c[4]-283*c[5]^4*c[6]^2*c[4]^2-5 30*c[5]^3*c[4]^6*c[6]^2+9*c[4]^4*c[7]+600*c[5]^4*c[7]^2*c[4]^6*c[6]-14 0*c[5]^5*c[4]^4*c[7]-55*c[4]^5*c[5]*c[7]^2+10*c[6]^2*c[4]^3+100*c[5]^5 *c[4]^4*c[7]^2+140*c[4]^6*c[6]*c[5]^3-967*c[6]^2*c[7]*c[5]^2*c[4]^3-90 *c[6]^2*c[7]^2*c[5]^2*c[4]+225*c[6]^2*c[7]^2*c[5]^2*c[4]^2-3082*c[6]*c [7]*c[5]^3*c[4]^3+185*c[5]^3*c[4]^6+40*c[7]^2*c[6]^2*c[5]^4+89*c[5]^4* c[7]*c[4]^2+284*c[7]^2*c[6]*c[5]*c[4]^4-100*c[7]^2*c[5]^5*c[6]^2*c[4]^ 2-340*c[5]^4*c[4]*c[7]^2*c[6]^2+200*c[5]^5*c[7]*c[4]^5*c[6]^2+7*c[6]*c [4]^5+27*c[7]*c[6]*c[4]^3-4*c[6]^2*c[7]*c[4]^2-440*c[5]^2*c[4]^6*c[7]^ 2+120*c[4]^7*c[6]^2*c[5]-9*c[7]*c[4]^3-10*c[6]*c[4]^3-22*c[7]^2*c[6]*c [4]^3-49*c[5]*c[7]^2*c[4]^3-29*c[5]*c[4]^3+9*c[5]*c[7]*c[4]^2+10*c[5]* c[6]*c[4]^2-4*c[5]*c[4]^2+40*c[6]*c[5]*c[7]^2*c[4]^2-109*c[5]*c[7]*c[6 ]*c[4]^3-20*c[6]^2*c[4]^2*c[5]+285*c[5]^4*c[4]^5-121*c[5]^2*c[6]*c[7]* c[4]^2+4790*c[5]^4*c[7]^2*c[4]^4*c[6]+1410*c[4]^5*c[6]*c[7]*c[5]^3+29* c[6]^2*c[5]*c[4]^3+14*c[6]^2*c[7]*c[4]*c[5]+238*c[7]*c[6]^2*c[5]*c[4]^ 3-12*c[6]^2*c[7]*c[5]^2-39*c[6]*c[5]*c[7]*c[4]^2+18*c[6]^2*c[7]^2*c[5] ^2-38*c[6]^2*c[4]^2*c[7]*c[5]+102*c[6]^2*c[7]^2*c[4]^2*c[5]+86*c[6]*c[ 5]^4*c[4]^2-24*c[7]*c[6]^2*c[5]^4+585*c[4]^5*c[7]*c[5]^3+c[4]^5+35*c[6 ]^2*c[5]^2*c[7]*c[4]-26*c[6]^2*c[5]^2*c[7]*c[4]^2+1128*c[6]*c[5]^2*c[7 ]*c[4]^3+706*c[6]*c[5]^3*c[7]*c[4]^2-138*c[6]^2*c[5]^3*c[7]*c[4]-685*c [5]^4*c[4]^3*c[7]+354*c[7]^2*c[5]^2*c[4]^3-370*c[5]^4*c[7]^2*c[4]^4-6* c[7]^2*c[5]*c[4]^2+70*c[7]*c[5]*c[4]^3+89*c[6]*c[5]*c[4]^3+200*c[4]^8* c[5]*c[6]*c[7]+390*c[6]*c[4]^7*c[5]-31*c[7]^2*c[5]^2*c[6]*c[4]-27*c[6] *c[7]*c[4]^4+10*c[4]^5*c[5]-80*c[5]^4*c[4]^6+200*c[5]^5*c[4]^5*c[7]^2* c[6]+6*c[6]^2*c[7]^2*c[4]^2+22*c[5]^3*c[6]*c[4]+19*c[5]^3*c[7]*c[4]+40 0*c[5]^5*c[4]^3*c[7]^2*c[6]+4*c[5]^2*c[4]-135*c[5]^3*c[6]*c[7]*c[4]+71 *c[5]^3*c[4]^2-21*c[6]^2*c[7]^2*c[4]*c[5]-480*c[5]^2*c[6]*c[4]^3+42*c[ 5]^2*c[7]*c[4]^2+37*c[5]^2*c[6]*c[4]^2-480*c[5]^2*c[7]*c[4]^3-104*c[5] ^3*c[7]*c[4]^2-97*c[5]^3*c[6]*c[4]^2-1536*c[5]^3*c[6]^2*c[4]^3+190*c[5 ]^5*c[4]^3*c[6]^2*c[7]+557*c[4]^3*c[7]^2*c[5]^2*c[6]^2-10*c[6]*c[7]*c[ 4]^5-730*c[4]^5*c[5]^3*c[6]^2-14*c[7]*c[5]^4*c[4]+10*c[7]^2*c[5]^4*c[4 ]+646*c[5]^2*c[4]^4*c[7]+498*c[6]^2*c[5]^2*c[4]^3-95*c[6]^2*c[5]^2*c[4 ]^2+744*c[6]*c[5]^3*c[4]^3-46*c[6]^2*c[5]^3*c[4]+334*c[6]^2*c[5]^3*c[4 ]^2+854*c[5]^3*c[7]*c[4]^3-490*c[5]^4*c[7]*c[4]^2*c[6]-30*c[7]*c[5]*c[ 4]^7-19*c[5]^2*c[4]^2+18*c[6]^2*c[4]*c[5]^2-472*c[6]*c[5]*c[4]^6+360*c [5]*c[4]^6*c[7]*c[6]-10*c[7]*c[6]^2*c[4]^3-811*c[7]^2*c[5]^2*c[6]*c[4] ^3-993*c[7]^2*c[5]^3*c[6]*c[4]^2-44*c[5]^4*c[4]^2-451*c[4]^3*c[7]^2*c[ 5]*c[6]^2+10*c[6]*c[7]*c[4]^6-14*c[7]^2*c[6]*c[5]*c[4]^3-20*c[5]^5*c[4 ]^5+118*c[7]^2*c[5]^2*c[6]*c[4]^2-249*c[4]^4*c[5]^2+170*c[7]^2*c[5]^3* c[6]*c[4]+3829*c[7]^2*c[5]^3*c[6]*c[4]^3+2208*c[5]^3*c[6]^2*c[4]^4+339 *c[5]^4*c[4]^3-6*c[6]^2*c[4]^3*c[7]^2+1871*c[5]^3*c[6]^2*c[7]*c[4]^3-1 0*c[5]^2*c[6]*c[4]-9*c[5]^2*c[7]*c[4]+6*c[7]^2*c[5]^2*c[4]+33*c[5]^2*c [6]*c[7]*c[4]+600*c[7]*c[4]^8*c[6]*c[5]^3+187*c[6]^2*c[7]*c[5]^3*c[4]^ 2+341*c[6]^2*c[7]^2*c[4]*c[5]^3+202*c[5]^2*c[4]^3-29*c[7]^2*c[5]^2*c[4 ]^2-10*c[7]^2*c[5]^3*c[4]+40*c[4]^4*c[5]-270*c[4]^7*c[5]^2-450*c[5]^5* c[4]^3*c[6]*c[7]+35*c[4]^6*c[7]*c[5]+80*c[7]^2*c[4]^4*c[5]-1163*c[7]^2 *c[5]^3*c[6]^2*c[4]^2+20*c[5]*c[4]^7-640*c[6]^2*c[4]^5*c[5]*c[7]^2+850 *c[7]^2*c[4]^5*c[6]*c[5]^3+1530*c[5]^2*c[6]*c[4]^5*c[7]^2-506*c[5]^4*c [4]^4+1420*c[5]^2*c[4]^7*c[6]*c[7]-16*c[5]^4*c[4]*c[6]-85*c[5]^4*c[7]^ 2*c[4]*c[6]+72*c[5]^4*c[4]*c[7]*c[6]+230*c[4]^5*c[6]^2*c[7]*c[5]^2+174 0*c[4]^5*c[6]^2*c[7]^2*c[5]^2-600*c[6]*c[4]^8*c[5]^2*c[7]-82*c[4]^5*c[ 5]^2-480*c[6]^2*c[4]^4*c[5]^2+320*c[4]^8*c[5]^2*c[6]-37*c[4]^6*c[5]+30 0*c[5]^3*c[4]^7*c[6]^2-90*c[5]^5*c[4]^5*c[6]+520*c[5]*c[7]*c[4]^6*c[6] ^2-10*c[6]^2*c[7]*c[4]^5+9*c[6]*c[4]^4+600*c[5]^4*c[7]*c[4]^6*c[6]^2-2 350*c[5]^4*c[4]^5*c[6]^2*c[7]-600*c[5]^3*c[7]^2*c[6]*c[4]^7-204*c[7]*c [6]^2*c[5]*c[4]^4-3630*c[5]^3*c[4]^4*c[6]^2*c[7]+740*c[5]^3*c[4]^3*c[6 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[5]^2*c[4]^7*c[6]-48*c[7]^2*c[6]^2*c[5]^3-535*c[4]^4*c[6]*c[5]^3+306*c [6]*c[5]*c[4]^5-160*c[5]^5*c[6]^2*c[4]^3+600*c[5]^2*c[7]*c[4]^7*c[6]^2 +200*c[6]^2*c[4]^6*c[5]*c[7]^2-1510*c[5]^4*c[4]^5*c[6]*c[7]+6*c[5]^4*c [4]+600*c[5]^2*c[4]^7*c[6]*c[7]^2+814*c[5]^3*c[4]^4)/(690*c[5]^2*c[6]* c[4]^4-110*c[6]*c[5]*c[4]^4-110*c[7]*c[5]*c[4]^4+48*c[5]^3*c[4]+200*c[ 4]^5*c[6]*c[5]^3-150*c[5]^2*c[4]^5*c[7]-750*c[5]^3*c[4]^4*c[7]-930*c[6 ]*c[5]^2*c[4]^4*c[7]-200*c[5]^5*c[4]^3*c[7]+750*c[5]^4*c[4]^3*c[6]-120 *c[5]^5*c[4]^2-10*c[5]^3+10*c[4]^3+150*c[5]^5*c[4]^3-15*c[7]*c[6]*c[5] ^3+12*c[5]^4-12*c[4]^4+200*c[6]*c[5]^2*c[4]^5*c[7]-150*c[4]^5*c[5]^3-2 00*c[5]^5*c[4]^3*c[6]+150*c[7]*c[4]^4*c[6]*c[5]+12*c[6]*c[5]^3+15*c[4] ^4*c[7]-690*c[5]^4*c[7]*c[4]^2-15*c[5]^4*c[7]+150*c[5]^5*c[7]*c[4]^2+1 5*c[7]*c[6]*c[4]^3-12*c[7]*c[4]^3-12*c[6]*c[4]^3-48*c[5]*c[4]^3+24*c[5 ]*c[7]*c[4]^2+24*c[5]*c[6]*c[4]^2-20*c[5]*c[4]^2-70*c[5]*c[7]*c[6]*c[4 ]^3+12*c[7]*c[5]^3-300*c[4]^5*c[6]*c[7]*c[5]^3-30*c[6]*c[5]*c[7]*c[4]^ 2+20*c[5]^4*c[7]*c[6]-690*c[6]*c[5]^4*c[4]^2-15*c[6]*c[5]^4+200*c[4]^5 *c[7]*c[5]^3+510*c[6]*c[5]^2*c[7]*c[4]^3-510*c[6]*c[5]^3*c[7]*c[4]^2+7 50*c[5]^4*c[4]^3*c[7]+57*c[7]*c[5]*c[4]^3+57*c[6]*c[5]*c[4]^3-20*c[6]* c[7]*c[4]^4-57*c[5]^3*c[6]*c[4]-57*c[5]^3*c[7]*c[4]+20*c[5]^2*c[4]-200 *c[5]^5*c[4]^2*c[6]*c[7]+70*c[5]^3*c[6]*c[7]*c[4]-342*c[5]^3*c[4]^2-41 0*c[5]^2*c[6]*c[4]^3-410*c[5]^2*c[7]*c[4]^3+410*c[5]^3*c[7]*c[4]^2+410 *c[5]^3*c[6]*c[4]^2+110*c[7]*c[5]^4*c[4]+690*c[5]^2*c[4]^4*c[7]+930*c[ 5]^4*c[7]*c[4]^2*c[6]+550*c[5]^4*c[4]^2-550*c[4]^4*c[5]^2-570*c[5]^4*c [4]^3-24*c[5]^2*c[6]*c[4]-24*c[5]^2*c[7]*c[4]+30*c[5]^2*c[6]*c[7]*c[4] +342*c[5]^2*c[4]^3+87*c[4]^4*c[5]+300*c[5]^5*c[4]^3*c[6]*c[7]+110*c[5] ^4*c[4]*c[6]-150*c[5]^4*c[4]*c[7]*c[6]+120*c[4]^5*c[5]^2+15*c[6]*c[4]^ 4+1100*c[5]^3*c[4]^4*c[6]*c[7]-1100*c[5]^4*c[4]^3*c[6]*c[7]+150*c[5]^5 *c[6]*c[4]^2-150*c[5]^2*c[6]*c[4]^5-750*c[4]^4*c[6]*c[5]^3-87*c[5]^4*c [4]+570*c[5]^3*c[4]^4)/(c[6]*c[4]-c[4]^2-c[7]*c[6]+c[7]*c[4])/c[4]^2, \+ a[8,1] = 1/4*(-2816*c[5]^2*c[6]*c[4]^4+424*c[6]*c[5]*c[4]^4-9720*c[5]^ 4*c[4]^4*c[6]^2*c[7]+372*c[7]*c[5]*c[4]^4+1752*c[5]^2*c[4]^4*c[7]^2+20 *c[5]^3*c[4]-1320*c[4]^5*c[6]^2*c[5]^2-1320*c[7]^2*c[5]^2*c[4]^5-4880* c[4]^5*c[6]*c[5]^3-264*c[5]*c[6]^2*c[4]^4-12*c[7]^2*c[4]^3+700*c[5]^5* c[4]^3*c[7]^2*c[6]^2-200*c[5]^2*c[4]^6*c[6]^2*c[7]^2-246*c[4]^5*c[5]*c [7]+1818*c[5]^2*c[4]^5*c[7]+1300*c[5]^3*c[4]^6*c[6]^2*c[7]-800*c[5]^3* c[4]^6*c[6]^2*c[7]^2+5526*c[5]^3*c[4]^4*c[7]+32*c[4]^4*c[6]^2+3160*c[7 ]^2*c[4]^5*c[5]^3-7740*c[6]^2*c[4]^5*c[7]*c[5]^3+780*c[7]*c[5]^2*c[4]^ 6*c[6]-325*c[5]^4*c[4]^2*c[7]^2*c[6]-20*c[6]^2*c[4]^5-1339*c[6]^2*c[7] *c[5]^4*c[4]^2-280*c[5]^2*c[4]^6*c[7]+668*c[6]*c[4]^5*c[5]*c[7]+6574*c [6]*c[5]^2*c[4]^4*c[7]+1500*c[5]^4*c[7]*c[4]^6*c[6]+200*c[4]^6*c[6]^2* c[5]^2-280*c[5]^5*c[4]^3*c[7]-600*c[5]^2*c[7]^2*c[4]^6*c[6]+2092*c[5]^ 4*c[4]^3*c[6]+3640*c[5]^4*c[4]^5*c[6]-320*c[5]^2*c[6]*c[4]^6+400*c[5]^ 5*c[4]^5*c[6]^2-4880*c[5]^4*c[4]^4*c[6]-4080*c[7]^2*c[4]^4*c[5]^3+50*c [6]*c[7]^2*c[4]^5-2400*c[5]^4*c[4]^5*c[7]^2-4160*c[5]^4*c[4]^4*c[7]-76 *c[7]^2*c[6]*c[4]^4+1600*c[5]^3*c[7]^2*c[4]^6*c[6]-490*c[6]*c[4]^5*c[5 ]*c[7]^2+32*c[4]^4*c[7]^2-7100*c[5]^4*c[7]^2*c[4]^5*c[6]^2-1320*c[7]^2 *c[5]^4*c[4]^3-2160*c[5]^3*c[7]*c[4]^6*c[6]+180*c[6]^2*c[4]^5*c[5]-8*c [4]^3+120*c[5]^5*c[4]^3+30*c[7]*c[6]^2*c[5]^3+4*c[7]*c[6]*c[5]^3-6*c[7 ]^2*c[6]*c[5]^3+400*c[5]^4*c[4]^6*c[6]^2+20*c[4]^4+780*c[5]^3*c[7]*c[4 ]^6+3060*c[5]^4*c[4]^5*c[7]-70*c[7]*c[6]^2*c[5]*c[4]^5-180*c[4]^6*c[6] ^2*c[7]*c[5]^2+180*c[5]^4*c[4]^2*c[7]^2+200*c[5]^5*c[7]^2*c[4]^3-5016* c[6]*c[5]^2*c[4]^5*c[7]+1720*c[4]^5*c[5]^3+1600*c[5]^5*c[4]^4*c[7]^2*c [6]-320*c[5]^5*c[4]^3*c[6]-972*c[7]*c[4]^4*c[6]*c[5]-320*c[5]^5*c[4]^4 -600*c[4]^4*c[6]^2*c[5]^5-2160*c[5]^5*c[7]*c[4]^4*c[6]+1752*c[7]^2*c[5 ]^3*c[4]^3-84*c[7]*c[4]^4*c[6]^2+60*c[5]^5*c[4]^2*c[6]^2*c[7]+1750*c[5 ]^5*c[4]^4*c[6]^2*c[7]+920*c[5]^5*c[4]^4*c[6]+1500*c[5]^5*c[7]*c[4]^5* c[6]+258*c[6]^2*c[5]^4*c[7]*c[4]-20*c[5]^4*c[6]^2*c[4]+180*c[5]^4*c[6] ^2*c[4]^2+400*c[5]^5*c[4]^5*c[7]^2-600*c[5]^3*c[4]^6*c[6]^2-46*c[4]^4* c[7]-1200*c[5]^4*c[7]^2*c[4]^6*c[6]+780*c[5]^5*c[4]^4*c[7]+180*c[4]^5* c[5]*c[7]^2-12*c[6]^2*c[4]^3-600*c[5]^3*c[7]^2*c[4]^6-600*c[5]^5*c[4]^ 4*c[7]^2+920*c[4]^6*c[6]*c[5]^3+1648*c[6]^2*c[7]*c[5]^2*c[4]^3-120*c[6 ]^2*c[7]^2*c[5]^2*c[4]+390*c[6]^2*c[7]^2*c[5]^2*c[4]^2+5722*c[6]*c[7]* c[5]^3*c[4]^3-320*c[5]^3*c[4]^6+40*c[7]^2*c[6]^2*c[5]^4-246*c[5]^4*c[7 ]*c[4]^2+686*c[7]^2*c[6]*c[5]*c[4]^4-100*c[7]^2*c[5]^5*c[6]^2*c[4]^2-4 00*c[5]^4*c[4]*c[7]^2*c[6]^2-1200*c[5]^5*c[7]*c[4]^5*c[6]^2+70*c[6]^2* c[4]^4*c[7]^2+32*c[6]*c[4]^5-42*c[7]*c[6]*c[4]^3-4*c[6]^2*c[7]*c[4]^2+ 200*c[5]^2*c[4]^6*c[7]^2+18*c[7]*c[4]^3+20*c[6]*c[4]^3+26*c[7]^2*c[6]* c[4]^3+72*c[5]*c[7]^2*c[4]^3+48*c[5]*c[4]^3-18*c[5]*c[7]*c[4]^2-20*c[5 ]*c[6]*c[4]^2+8*c[5]*c[4]^2-23*c[6]*c[5]*c[7]^2*c[4]^2+266*c[5]*c[7]*c [6]*c[4]^3+12*c[6]^2*c[4]^2*c[5]-1240*c[5]^4*c[4]^5+287*c[5]^2*c[6]*c[ 7]*c[4]^2-8120*c[5]^4*c[7]^2*c[4]^4*c[6]+11330*c[4]^5*c[6]*c[7]*c[5]^3 +72*c[6]^2*c[5]*c[4]^3+14*c[6]^2*c[7]*c[4]*c[5]+76*c[7]*c[6]^2*c[5]*c[ 4]^3-12*c[6]^2*c[7]*c[5]^2+40*c[6]*c[5]*c[7]*c[4]^2+18*c[6]^2*c[7]^2*c [5]^2-101*c[6]^2*c[4]^2*c[7]*c[5]+132*c[6]^2*c[7]^2*c[4]^2*c[5]-284*c[ 6]*c[5]^4*c[4]^2-24*c[7]*c[6]^2*c[5]^4-4160*c[4]^5*c[7]*c[5]^3-12*c[4] ^5+95*c[6]^2*c[5]^2*c[7]*c[4]-356*c[6]^2*c[5]^2*c[7]*c[4]^2-2598*c[6]* c[5]^2*c[7]*c[4]^3-698*c[6]*c[5]^3*c[7]*c[4]^2-300*c[6]^2*c[5]^3*c[7]* c[4]+1818*c[5]^4*c[4]^3*c[7]-692*c[7]^2*c[5]^2*c[4]^3+3160*c[5]^4*c[7] ^2*c[4]^4+12*c[7]^2*c[5]*c[4]^2-108*c[7]*c[5]*c[4]^3-120*c[6]*c[5]*c[4 ]^3+29*c[7]^2*c[5]^2*c[6]*c[4]+114*c[6]*c[7]*c[4]^4+104*c[4]^5*c[5]+20 0*c[5]^4*c[4]^6-1200*c[5]^5*c[4]^5*c[7]^2*c[6]+6*c[6]^2*c[7]^2*c[4]^2- 52*c[5]^3*c[6]*c[4]-46*c[5]^3*c[7]*c[4]-450*c[5]^5*c[4]^3*c[7]^2*c[6]- 8*c[5]^2*c[4]+60*c[5]^3*c[6]*c[7]*c[4]-160*c[5]^3*c[4]^2-21*c[6]^2*c[7 ]^2*c[4]*c[5]+1144*c[5]^2*c[6]*c[4]^3-108*c[5]^2*c[7]*c[4]^2-120*c[5]^ 2*c[6]*c[4]^2+1024*c[5]^2*c[7]*c[4]^3+372*c[5]^3*c[7]*c[4]^2+424*c[5]^ 3*c[6]*c[4]^2+1752*c[5]^3*c[6]^2*c[4]^3-660*c[5]^5*c[4]^3*c[6]^2*c[7]- 1153*c[4]^3*c[7]^2*c[5]^2*c[6]^2-72*c[6]*c[7]*c[4]^5+3160*c[4]^5*c[5]^ 3*c[6]^2+28*c[7]*c[5]^4*c[4]-20*c[7]^2*c[5]^4*c[4]-2472*c[5]^2*c[4]^4* c[7]-692*c[6]^2*c[5]^2*c[4]^3+72*c[6]^2*c[5]^2*c[4]^2-2816*c[6]*c[5]^3 *c[4]^3+32*c[6]^2*c[5]^3*c[4]-264*c[6]^2*c[5]^3*c[4]^2-2472*c[5]^3*c[7 ]*c[4]^3+566*c[5]^4*c[7]*c[4]^2*c[6]+48*c[5]^2*c[4]^2+1200*c[6]^2*c[4] ^5*c[5]^5*c[7]^2-12*c[6]^2*c[4]*c[5]^2+38*c[7]*c[6]^2*c[4]^3+1804*c[7] ^2*c[5]^2*c[6]*c[4]^3+258*c[7]^2*c[5]^3*c[6]*c[4]^2+104*c[5]^4*c[4]^2- 500*c[5]^5*c[4]^5*c[7]-256*c[4]^3*c[7]^2*c[5]*c[6]^2-176*c[7]^2*c[6]*c [5]*c[4]^3+200*c[5]^5*c[4]^5-212*c[7]^2*c[5]^2*c[6]*c[4]^2+1064*c[4]^4 *c[5]^2+8*c[7]^2*c[5]^3*c[6]*c[4]-3410*c[7]^2*c[5]^3*c[6]*c[4]^3-4080* c[5]^3*c[6]^2*c[4]^4-772*c[5]^4*c[4]^3+1200*c[7]^2*c[6]^2*c[4]^6*c[5]^ 4-36*c[6]^2*c[4]^3*c[7]^2-5368*c[5]^3*c[6]^2*c[7]*c[4]^3+20*c[5]^2*c[6 ]*c[4]+18*c[5]^2*c[7]*c[4]-12*c[7]^2*c[5]^2*c[4]-44*c[5]^2*c[6]*c[7]*c [4]+400*c[5]^4*c[4]^6*c[7]^2+1438*c[6]^2*c[7]*c[5]^3*c[4]^2+426*c[6]^2 *c[7]^2*c[4]*c[5]^3-452*c[5]^2*c[4]^3+72*c[7]^2*c[5]^2*c[4]^2+32*c[7]^ 2*c[5]^3*c[4]-160*c[4]^4*c[5]+690*c[5]^5*c[4]^3*c[6]*c[7]-264*c[7]^2*c [4]^4*c[5]-1833*c[7]^2*c[5]^3*c[6]^2*c[4]^2-260*c[6]^2*c[4]^5*c[5]*c[7 ]^2-8270*c[7]^2*c[4]^5*c[6]*c[5]^3+3780*c[5]^2*c[6]*c[4]^5*c[7]^2+1720 *c[5]^4*c[4]^4+32*c[5]^4*c[4]*c[6]+30*c[5]^4*c[7]^2*c[4]*c[6]-60*c[5]^ 4*c[4]*c[7]*c[6]+2480*c[4]^5*c[6]^2*c[7]*c[5]^2-1120*c[4]^5*c[6]^2*c[7 ]^2*c[5]^2-772*c[4]^5*c[5]^2-40*c[6]^2*c[7]^2*c[4]^5+1752*c[6]^2*c[4]^ 4*c[5]^2-1700*c[7]^2*c[6]^2*c[5]^5*c[4]^4-600*c[5]^5*c[4]^5*c[6]-120*c [5]*c[7]*c[4]^6*c[6]^2+50*c[6]^2*c[7]*c[4]^5-52*c[6]*c[4]^4-1200*c[5]^ 4*c[7]*c[4]^6*c[6]^2+7150*c[5]^4*c[4]^5*c[6]^2*c[7]+198*c[7]*c[6]^2*c[ 5]*c[4]^4+10560*c[5]^3*c[4]^4*c[6]^2*c[7]+5240*c[5]^3*c[4]^3*c[6]^2*c[ 7]^2+11240*c[5]^4*c[4]^4*c[6]*c[7]-3649*c[5]^2*c[4]^4*c[7]*c[6]^2-1420 7*c[5]^3*c[4]^4*c[6]*c[7]-4491*c[5]^4*c[4]^3*c[6]*c[7]+1064*c[5]^3*c[4 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+12*c[4]^3+20*c[7]*c[6]*c[5]^3+15*c[5]*c[6]*c[7]*c[4]+12*c[6]*c[5]^2-1 5*c[6]*c[5]^3-1300*c[6]*c[7]*c[5]^3*c[4]^3+150*c[5]^4*c[7]*c[4]^2+12*c [7]*c[5]^2+20*c[7]*c[6]*c[4]^3+12*c[7]*c[4]^2-15*c[7]*c[4]^3+12*c[6]*c [4]^2-15*c[6]*c[4]^3-15*c[6]*c[7]*c[4]^2-87*c[5]*c[4]^3-72*c[5]*c[7]*c [4]^2-72*c[5]*c[6]*c[4]^2+60*c[5]*c[4]^2-12*c[5]*c[6]*c[4]+10*c[5]*c[4 ]-150*c[5]*c[7]*c[6]*c[4]^3-15*c[7]*c[5]^3-660*c[5]^2*c[6]*c[7]*c[4]^2 +90*c[6]*c[5]*c[7]*c[4]^2-15*c[6]*c[7]*c[5]^2+150*c[6]*c[5]^4*c[4]^2+9 30*c[6]*c[5]^2*c[7]*c[4]^3+930*c[6]*c[5]^3*c[7]*c[4]^2-200*c[5]^4*c[4] ^3*c[7]+110*c[7]*c[5]*c[4]^3+110*c[6]*c[5]*c[4]^3+110*c[5]^3*c[6]*c[4] +110*c[5]^3*c[7]*c[4]+60*c[5]^2*c[4]-150*c[5]^3*c[6]*c[7]*c[4]+550*c[5 ]^3*c[4]^2-690*c[5]^2*c[6]*c[4]^3+520*c[5]^2*c[7]*c[4]^2+520*c[5]^2*c[ 6]*c[4]^2-690*c[5]^2*c[7]*c[4]^3-690*c[5]^3*c[7]*c[4]^2-690*c[5]^3*c[6 ]*c[4]^2+150*c[5]^2*c[4]^4*c[7]+900*c[6]*c[5]^3*c[4]^3+900*c[5]^3*c[7] *c[4]^3-200*c[5]^4*c[7]*c[4]^2*c[6]-429*c[5]^2*c[4]^2-120*c[5]^4*c[4]^ 2-120*c[4]^4*c[5]^2+150*c[5]^4*c[4]^3-72*c[5]^2*c[6]*c[4]-72*c[5]^2*c[ 7]*c[4]+90*c[5]^2*c[6]*c[7]*c[4]+550*c[5]^2*c[4]^3+300*c[5]^3*c[4]^4*c [6]*c[7]+300*c[5]^4*c[4]^3*c[6]*c[7]-690*c[5]^3*c[4]^3-200*c[4]^4*c[6] *c[5]^3+150*c[5]^3*c[4]^4), a[8,3] = 3/4*(-1560*c[5]^2*c[6]*c[4]^4+585 *c[6]*c[5]*c[4]^4-45*c[7]*c[5]*c[4]^4-12*c[5]^3*c[4]-450*c[4]^5*c[6]*c [5]^3+3*c[5]*c[7]*c[4]-150*c[5]^2*c[4]^5*c[7]-200*c[5]^3*c[4]^4*c[7]+3 00*c[6]*c[4]^5*c[5]*c[7]+2860*c[6]*c[5]^2*c[4]^4*c[7]+160*c[5]^4*c[4]^ 3*c[6]-150*c[5]^4*c[4]^4*c[6]-4*c[5]^2+2*c[4]^2-40*c[7]*c[6]*c[5]^3+13 2*c[5]*c[6]*c[7]*c[4]-900*c[6]*c[5]^2*c[4]^5*c[7]-30*c[6]*c[5]^2+12*c[ 6]*c[5]-960*c[7]*c[4]^4*c[6]*c[5]+24*c[6]*c[5]^3+3140*c[6]*c[7]*c[5]^3 *c[4]^3+6*c[7]*c[5]^2+20*c[7]*c[6]*c[4]^3-3*c[7]*c[4]^2-18*c[6]*c[7]*c [5]+15*c[6]*c[4]^2-12*c[6]*c[4]^3+9*c[6]*c[7]*c[4]-24*c[6]*c[7]*c[4]^2 -54*c[5]*c[4]^3-12*c[5]*c[7]*c[4]^2+348*c[5]*c[6]*c[4]^2+9*c[5]*c[4]^2 -87*c[5]*c[6]*c[4]-2*c[5]*c[4]-6*c[6]*c[4]+1035*c[5]*c[7]*c[6]*c[4]^3+ 1593*c[5]^2*c[6]*c[7]*c[4]^2+900*c[4]^5*c[6]*c[7]*c[5]^3-534*c[6]*c[5] *c[7]*c[4]^2+48*c[6]*c[7]*c[5]^2-60*c[6]*c[5]^4*c[4]^2-3060*c[6]*c[5]^ 2*c[7]*c[4]^3-1560*c[6]*c[5]^3*c[7]*c[4]^2-50*c[5]^4*c[4]^3*c[7]+80*c[ 7]*c[5]*c[4]^3-652*c[6]*c[5]*c[4]^3-208*c[5]^3*c[6]*c[4]+20*c[5]^3*c[7 ]*c[4]+54*c[5]^2*c[4]+360*c[5]^3*c[6]*c[7]*c[4]+82*c[5]^3*c[4]^2+1734* c[5]^2*c[6]*c[4]^3+388*c[5]^2*c[7]*c[4]^2-944*c[5]^2*c[6]*c[4]^2-900*c [5]^2*c[7]*c[4]^3-135*c[5]^3*c[7]*c[4]^2+879*c[5]^3*c[6]*c[4]^2+660*c[ 5]^2*c[4]^4*c[7]-1680*c[6]*c[5]^3*c[4]^3+420*c[5]^3*c[7]*c[4]^3+100*c[ 5]^4*c[7]*c[4]^2*c[6]-246*c[5]^2*c[4]^2-405*c[4]^4*c[5]^2+30*c[5]^4*c[ 4]^3+240*c[5]^2*c[6]*c[4]-84*c[5]^2*c[7]*c[4]-396*c[5]^2*c[6]*c[7]*c[4 ]+562*c[5]^2*c[4]^3+30*c[4]^4*c[5]+90*c[4]^5*c[5]^2+300*c[5]^4*c[4]^4* c[6]*c[7]-3000*c[5]^3*c[4]^4*c[6]*c[7]-300*c[5]^4*c[4]^3*c[6]*c[7]-255 *c[5]^3*c[4]^3+480*c[5]^2*c[6]*c[4]^5+1540*c[4]^4*c[6]*c[5]^3-180*c[6] *c[5]*c[4]^5+120*c[5]^3*c[4]^4)/c[4]^2/(150*c[5]^2*c[6]*c[4]^4-87*c[5] ^3*c[4]-12*c[5]*c[7]*c[4]-200*c[5]^3*c[4]^4*c[7]-200*c[6]*c[5]^2*c[4]^ 4*c[7]-200*c[5]^4*c[4]^3*c[6]-10*c[5]^2-10*c[4]^2+12*c[5]^3+12*c[4]^3+ 20*c[7]*c[6]*c[5]^3+15*c[5]*c[6]*c[7]*c[4]+12*c[6]*c[5]^2-15*c[6]*c[5] ^3-1300*c[6]*c[7]*c[5]^3*c[4]^3+150*c[5]^4*c[7]*c[4]^2+12*c[7]*c[5]^2+ 20*c[7]*c[6]*c[4]^3+12*c[7]*c[4]^2-15*c[7]*c[4]^3+12*c[6]*c[4]^2-15*c[ 6]*c[4]^3-15*c[6]*c[7]*c[4]^2-87*c[5]*c[4]^3-72*c[5]*c[7]*c[4]^2-72*c[ 5]*c[6]*c[4]^2+60*c[5]*c[4]^2-12*c[5]*c[6]*c[4]+10*c[5]*c[4]-150*c[5]* c[7]*c[6]*c[4]^3-15*c[7]*c[5]^3-660*c[5]^2*c[6]*c[7]*c[4]^2+90*c[6]*c[ 5]*c[7]*c[4]^2-15*c[6]*c[7]*c[5]^2+150*c[6]*c[5]^4*c[4]^2+930*c[6]*c[5 ]^2*c[7]*c[4]^3+930*c[6]*c[5]^3*c[7]*c[4]^2-200*c[5]^4*c[4]^3*c[7]+110 *c[7]*c[5]*c[4]^3+110*c[6]*c[5]*c[4]^3+110*c[5]^3*c[6]*c[4]+110*c[5]^3 *c[7]*c[4]+60*c[5]^2*c[4]-150*c[5]^3*c[6]*c[7]*c[4]+550*c[5]^3*c[4]^2- 690*c[5]^2*c[6]*c[4]^3+520*c[5]^2*c[7]*c[4]^2+520*c[5]^2*c[6]*c[4]^2-6 90*c[5]^2*c[7]*c[4]^3-690*c[5]^3*c[7]*c[4]^2-690*c[5]^3*c[6]*c[4]^2+15 0*c[5]^2*c[4]^4*c[7]+900*c[6]*c[5]^3*c[4]^3+900*c[5]^3*c[7]*c[4]^3-200 *c[5]^4*c[7]*c[4]^2*c[6]-429*c[5]^2*c[4]^2-120*c[5]^4*c[4]^2-120*c[4]^ 4*c[5]^2+150*c[5]^4*c[4]^3-72*c[5]^2*c[6]*c[4]-72*c[5]^2*c[7]*c[4]+90* c[5]^2*c[6]*c[7]*c[4]+550*c[5]^2*c[4]^3+300*c[5]^3*c[4]^4*c[6]*c[7]+30 0*c[5]^4*c[4]^3*c[6]*c[7]-690*c[5]^3*c[4]^3-200*c[4]^4*c[6]*c[5]^3+150 *c[5]^3*c[4]^4), a[8,5] = -1/2*(893*c[5]^2*c[6]*c[4]^4-122*c[6]*c[5]*c [4]^4+850*c[5]^4*c[4]^4*c[6]^2*c[7]-101*c[7]*c[5]*c[4]^4-535*c[5]^2*c[ 4]^4*c[7]^2-44*c[5]^3*c[4]+160*c[4]^5*c[6]^2*c[5]^2+100*c[7]^2*c[5]^2* c[4]^5+170*c[4]^5*c[6]*c[5]^3+75*c[5]*c[6]^2*c[4]^4+16*c[7]^2*c[4]^3-9 *c[5]*c[7]*c[4]-140*c[5]^2*c[4]^5*c[7]-780*c[5]^3*c[4]^4*c[7]-10*c[4]^ 4*c[6]^2-100*c[7]^2*c[4]^5*c[5]^3+450*c[6]^2*c[4]^5*c[7]*c[5]^3+510*c[ 5]^4*c[4]^2*c[7]^2*c[6]+490*c[6]^2*c[7]*c[5]^4*c[4]^2-2224*c[6]*c[5]^2 *c[4]^4*c[7]+10*c[5]^5*c[4]^3*c[7]-146*c[5]^4*c[4]^3*c[6]+90*c[5]^4*c[ 4]^5*c[6]+40*c[5]^4*c[4]^4*c[6]+570*c[7]^2*c[4]^4*c[5]^3+130*c[5]^4*c[ 4]^4*c[7]+25*c[7]^2*c[6]*c[4]^4+18*c[5]^5*c[4]^2-4*c[5]^2-4*c[4]^2-10* c[4]^4*c[7]^2+150*c[7]^2*c[5]^4*c[4]^3+5*c[5]^3-6*c[6]^2*c[7]^2*c[5]+1 0*c[4]^3-35*c[5]^5*c[4]^3-24*c[7]*c[6]^2*c[5]^3+27*c[7]*c[6]*c[5]^3-32 *c[7]^2*c[6]*c[5]^3+23*c[5]*c[6]*c[7]*c[4]-c[5]^4-6*c[4]^4-60*c[7]*c[6 ]^2*c[5]*c[4]^5-45*c[5]^4*c[4]^2*c[7]^2+450*c[6]*c[5]^2*c[4]^5*c[7]+10 *c[6]*c[5]^2-80*c[4]^5*c[5]^3-260*c[5]^5*c[4]^3*c[6]+270*c[7]*c[4]^4*c [6]*c[5]+20*c[5]^5*c[4]^4-9*c[6]*c[5]^3+50*c[4]^4*c[6]^2*c[5]^5-600*c[ 5]^5*c[7]*c[4]^4*c[6]-760*c[7]^2*c[5]^3*c[4]^3+25*c[7]*c[4]^4*c[6]^2+2 40*c[5]^5*c[4]^4*c[6]+200*c[5]^5*c[7]*c[4]^5*c[6]+20*c[5]^5*c[4]^2*c[6 ]^2-90*c[6]^2*c[5]^4*c[7]*c[4]+71*c[5]^4*c[6]^2*c[4]-383*c[5]^4*c[6]^2 *c[4]^2+18*c[6]^2*c[5]^3+14*c[4]^4*c[7]-6*c[7]^2*c[4]^2+16*c[6]^2*c[4] ^3-1433*c[6]^2*c[7]*c[5]^2*c[4]^3-57*c[6]^2*c[7]^2*c[5]^2*c[4]+c[6]^2* c[7]^2*c[5]^2*c[4]^2-3185*c[6]*c[7]*c[5]^3*c[4]^3+100*c[5]^4*c[7]*c[4] ^2-195*c[7]^2*c[6]*c[5]*c[4]^4-15*c[5]^5*c[7]*c[4]^2-20*c[6]^2*c[4]^4* 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4]^3-152*c[6]^2*c[5]^3*c[4]+772*c[6]^2*c[5]^3*c[4]^2+1076*c[5]^3*c[7]* c[4]^3+120*c[5]^4*c[7]*c[4]^2*c[6]-202*c[5]^2*c[4]^2+73*c[6]^2*c[4]*c[ 5]^2-42*c[7]*c[6]^2*c[4]^3-2311*c[7]^2*c[5]^2*c[6]*c[4]^3-1492*c[7]^2* c[5]^3*c[6]*c[4]^2-110*c[5]^4*c[4]^2+155*c[4]^3*c[7]^2*c[5]*c[6]^2+269 *c[7]^2*c[6]*c[5]*c[4]^3+1027*c[7]^2*c[5]^2*c[6]*c[4]^2-327*c[4]^4*c[5 ]^2+284*c[7]^2*c[5]^3*c[6]*c[4]+3210*c[7]^2*c[5]^3*c[6]*c[4]^3+1310*c[ 5]^3*c[6]^2*c[4]^4+242*c[5]^4*c[4]^3+35*c[6]^2*c[4]^3*c[7]^2+2510*c[5] ^3*c[6]^2*c[7]*c[4]^3-73*c[5]^2*c[6]*c[4]-66*c[5]^2*c[7]*c[4]+43*c[7]^ 2*c[5]^2*c[4]+211*c[5]^2*c[6]*c[7]*c[4]-1154*c[6]^2*c[7]*c[5]^3*c[4]^2 +444*c[5]^2*c[4]^3-306*c[7]^2*c[5]^2*c[4]^2-44*c[7]^2*c[5]^3*c[4]+44*c [4]^4*c[5]+670*c[5]^5*c[4]^3*c[6]*c[7]+75*c[7]^2*c[4]^4*c[5]+250*c[7]^ 2*c[5]^3*c[6]^2*c[4]^2+6*c[5]^5*c[6]+100*c[6]^2*c[4]^5*c[5]*c[7]^2+600 *c[7]^2*c[4]^5*c[6]*c[5]^3-400*c[5]^2*c[6]*c[4]^5*c[7]^2-165*c[5]^4*c[ 4]^4+58*c[5]^4*c[4]*c[6]-90*c[5]^4*c[7]^2*c[4]*c[6]-90*c[5]^4*c[4]*c[7 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5]^3*c[4]^3-15*c[7]*c[4]^4*c[6]^2-15*c[5]^5*c[7]^2+780*c[5]^5*c[4]^2*c [6]^2*c[7]+750*c[5]^5*c[4]^4*c[6]+300*c[5]^5*c[7]*c[4]^5*c[6]-690*c[5] ^5*c[4]^2*c[6]^2-40*c[6]^2*c[5]^4*c[7]*c[4]-690*c[5]^5*c[4]^2*c[7]^2-5 7*c[5]^4*c[6]^2*c[4]+410*c[5]^4*c[6]^2*c[4]^2+750*c[5]^5*c[4]^4*c[7]+3 40*c[6]^2*c[7]*c[5]^2*c[4]^3-30*c[6]^2*c[7]^2*c[5]^2*c[4]-750*c[6]*c[7 ]*c[5]^3*c[4]^3-20*c[7]^2*c[6]^2*c[5]^4-342*c[5]^4*c[7]*c[4]^2+90*c[7] ^2*c[6]*c[5]*c[4]^4+200*c[7]^2*c[5]^5*c[6]^2*c[4]^2-10*c[5]^4*c[7]+140 *c[5]^5*c[7]*c[4]^2+150*c[5]^4*c[4]*c[7]^2*c[6]^2+20*c[6]^2*c[4]^4*c[7 ]^2-10*c[7]*c[6]*c[4]^3-30*c[5]^5*c[7]*c[4]+12*c[5]^4*c[6]^2+12*c[7]^2 *c[6]*c[4]^3-12*c[5]*c[7]^2*c[4]^3-24*c[6]*c[5]*c[7]^2*c[4]^2+24*c[5]* c[7]*c[6]*c[4]^3-12*c[5]^6+150*c[7]^2*c[5]^6*c[4]^2-120*c[5]^4*c[4]^5+ 48*c[5]^2*c[6]*c[7]*c[4]^2+1100*c[5]^4*c[7]^2*c[4]^4*c[6]-150*c[4]^5*c [6]*c[7]*c[5]^3-12*c[6]^2*c[5]*c[4]^3-42*c[7]*c[6]^2*c[5]*c[4]^3+20*c[ 6]*c[5]*c[7]*c[4]^2+12*c[5]^4*c[7]*c[6]-24*c[6]^2*c[4]^2*c[7]*c[5]+30* c[6]^2*c[7]^2*c[4]^2*c[5]-342*c[6]*c[5]^4*c[4]^2-10*c[6]*c[5]^4+120*c[ 4]^5*c[7]*c[5]^3+24*c[6]^2*c[5]^2*c[7]*c[4]-30*c[6]^2*c[5]^2*c[7]*c[4] ^2-243*c[6]*c[5]^2*c[7]*c[4]^3+372*c[6]*c[5]^3*c[7]*c[4]^2+87*c[6]^2*c [5]^3*c[7]*c[4]+410*c[5]^4*c[4]^3*c[7]+57*c[7]^2*c[5]^2*c[4]^3-750*c[5 ]^4*c[7]^2*c[4]^4+10*c[7]*c[5]*c[4]^3+10*c[6]*c[5]*c[4]^3+24*c[7]^2*c[ 5]^2*c[6]*c[4]+12*c[6]*c[7]*c[4]^4+200*c[7]*c[5]^7*c[4]^3-110*c[5]^6*c [7]*c[4]+15*c[6]*c[5]^6+20*c[5]^3*c[6]*c[4]+20*c[5]^3*c[7]*c[4]+150*c[ 7]*c[6]*c[5]^5*c[4]-900*c[5]^5*c[4]^3*c[7]^2*c[6]-750*c[5]^5*c[4]^2*c[ 6]*c[7]-96*c[5]^3*c[6]*c[7]*c[4]+200*c[6]*c[5]^7*c[4]^3+20*c[5]^3*c[4] ^2-36*c[5]^2*c[6]*c[4]^3-20*c[5]^2*c[7]*c[4]^2-20*c[5]^2*c[6]*c[4]^2-3 6*c[5]^2*c[7]*c[4]^3-24*c[5]^3*c[7]*c[4]^2-24*c[5]^3*c[6]*c[4]^2-410*c [5]^3*c[6]^2*c[4]^3-900*c[5]^5*c[4]^3*c[6]^2*c[7]-510*c[4]^3*c[7]^2*c[ 5]^2*c[6]^2-630*c[7]*c[5]^6*c[6]*c[4]^2-150*c[4]^5*c[5]^3*c[6]^2+72*c[ 7]*c[5]^4*c[4]-57*c[7]^2*c[5]^4*c[4]+72*c[5]^2*c[4]^4*c[7]+57*c[6]^2*c [5]^2*c[4]^3+24*c[6]^2*c[5]^2*c[4]^2+285*c[6]*c[5]^3*c[4]^3-24*c[6]^2* c[5]^3*c[4]+285*c[5]^3*c[7]*c[4]^3+270*c[5]^4*c[7]*c[4]^2*c[6]+12*c[7] *c[6]^2*c[4]^3+340*c[7]^2*c[5]^2*c[6]*c[4]^3-410*c[7]^2*c[5]^3*c[6]*c[ 4]^2-200*c[5]^5*c[4]^5*c[7]+70*c[4]^3*c[7]^2*c[5]*c[6]^2-42*c[7]^2*c[6 ]*c[5]*c[4]^3+150*c[5]^5*c[4]^5-30*c[7]^2*c[5]^2*c[6]*c[4]^2+12*c[4]^4 *c[5]^2+87*c[7]^2*c[5]^3*c[6]*c[4]+510*c[7]^2*c[5]^3*c[6]*c[4]^3+690*c [5]^3*c[6]^2*c[4]^4-342*c[5]^4*c[4]^3-15*c[6]^2*c[4]^3*c[7]^2+510*c[5] ^3*c[6]^2*c[7]*c[4]^3-20*c[5]^2*c[6]*c[7]*c[4]-410*c[6]^2*c[7]*c[5]^3* c[4]^2-70*c[6]^2*c[7]^2*c[4]*c[5]^3-10*c[5]^2*c[4]^3+24*c[7]^2*c[5]^2* c[4]^2-24*c[7]^2*c[5]^3*c[4]-200*c[7]^2*c[5]^6*c[4]^2*c[6]+110*c[7]^2* c[4]*c[5]^5+1350*c[5]^5*c[4]^3*c[6]*c[7]+15*c[7]^2*c[4]^4*c[5]+300*c[6 ]^2*c[7]*c[5]^6*c[4]^3+510*c[7]^2*c[5]^3*c[6]^2*c[4]^2-550*c[5]^6*c[4] ^2-600*c[7]*c[5]^6*c[4]^3+20*c[7]*c[6]^2*c[5]^5+570*c[5]^6*c[6]*c[4]^2 +150*c[5]^2*c[6]*c[4]^5*c[7]^2+550*c[5]^4*c[4]^4+120*c[5]^7*c[4]^2+72* c[5]^4*c[4]*c[6]-40*c[5]^4*c[7]^2*c[4]*c[6]-57*c[5]^4*c[4]*c[7]*c[6]+1 50*c[6]^2*c[5]^6*c[4]^2+150*c[4]^5*c[6]^2*c[7]*c[5]^2-200*c[4]^5*c[6]^ 2*c[7]^2*c[5]^2-150*c[5]^7*c[7]*c[4]^2-110*c[6]^2*c[4]^4*c[5]^2+780*c[ 5]^5*c[4]^2*c[7]^2*c[6]-200*c[5]^5*c[4]^5*c[6]-600*c[5]^6*c[4]^3*c[6]- 200*c[6]^2*c[7]*c[5]^6*c[4]^2-300*c[5]^4*c[4]^5*c[6]^2*c[7]+90*c[7]*c[ 6]^2*c[5]*c[4]^4+200*c[6]*c[5]^7*c[7]*c[4]^2-300*c[6]*c[5]^7*c[7]*c[4] ^3-180*c[5]^3*c[4]^4*c[6]^2*c[7]-570*c[5]^4*c[4]^4*c[6]*c[7]-540*c[5]^ 2*c[4]^4*c[7]*c[6]^2+660*c[5]^3*c[4]^4*c[6]*c[7]+60*c[5]^4*c[4]^3*c[6] *c[7]+48*c[5]^3*c[4]^3-15*c[6]^2*c[5]^5-200*c[5]^6*c[6]^2*c[4]^3-15*c[ 5]^5*c[6]*c[7]-48*c[5]^5*c[4]+140*c[5]^5*c[6]*c[4]^2+200*c[5]^4*c[4]^5 *c[6]^2+300*c[6]^2*c[7]^2*c[5]^3*c[4]^5-300*c[6]*c[7]^2*c[5]^4*c[4]^5+ 700*c[7]*c[5]^6*c[4]^3*c[6]-750*c[5]^4*c[4]^4*c[6]^2+20*c[5]^5*c[7]^2* c[6]+1100*c[7]^2*c[5]^4*c[6]^2*c[4]^3-930*c[7]^2*c[5]^4*c[6]^2*c[4]^2+ 300*c[7]^2*c[5]^6*c[4]^3*c[6]-20*c[5]^6*c[7]*c[6]-180*c[5]^3*c[4]^4*c[ 7]^2*c[6]-750*c[5]^4*c[4]^3*c[7]^2*c[6]-540*c[5]^2*c[4]^4*c[7]^2*c[6]- 750*c[5]^4*c[4]^3*c[6]^2*c[7]-150*c[7]^2*c[4]^4*c[6]^2*c[5]-1100*c[7]^ 2*c[4]^4*c[5]^3*c[6]^2+930*c[7]^2*c[4]^4*c[6]^2*c[5]^2+150*c[5]^6*c[4] *c[7]*c[6]-110*c[5]^6*c[4]*c[6]-150*c[5]^7*c[6]*c[4]^2+570*c[5]^6*c[4] ^3-150*c[5]^7*c[4]^3+15*c[7]^2*c[6]^2*c[5]^3-440*c[4]^4*c[6]*c[5]^3+75 0*c[5]^5*c[6]^2*c[4]^3-200*c[7]^2*c[5]^6*c[4]^3+12*c[5]^4*c[7]^2+200*c [5]^4*c[4]^5*c[6]*c[7]-20*c[5]^4*c[4]-87*c[5]^3*c[4]^4-30*c[5]^5*c[4]* c[6]+110*c[5]^5*c[6]^2*c[4]), a[9,7] = -1/60*(10*c[5]*c[6]*c[4]-5*c[6] *c[5]-5*c[6]*c[4]+3*c[6]-5*c[5]*c[4]+3*c[5]+3*c[4]-2)/c[7]/(-c[5]*c[6] *c[4]+c[5]*c[6]*c[7]*c[4]+c[5]*c[7]*c[4]-c[7]^2*c[5]*c[4]+c[6]*c[7]*c[ 4]-c[6]*c[7]^2*c[4]-c[7]^2*c[4]+c[7]^3*c[4]+c[6]*c[7]*c[5]-c[6]*c[7]^2 *c[5]-c[7]^2*c[5]+c[7]^3*c[5]-c[7]^2*c[6]+c[7]^3*c[6]+c[7]^3-c[7]^4), \+ a[9,2] = 0, a[9,3] = 0, a[9,5] = -1/60*(10*c[6]*c[7]*c[4]-5*c[6]*c[4]- 5*c[7]*c[6]+3*c[6]-5*c[7]*c[4]+3*c[4]-2+3*c[7])/c[5]/(c[5]^3*c[4]-c[5] ^2*c[4]+c[5]*c[7]*c[4]-c[5]^2*c[7]*c[4]+c[5]*c[6]*c[4]-c[5]^2*c[6]*c[4 ]-c[6]*c[7]*c[4]+c[5]*c[6]*c[7]*c[4]-c[5]^4+c[5]^3-c[7]*c[5]^2+c[7]*c[ 5]^3-c[6]*c[5]^2+c[6]*c[5]^3+c[6]*c[7]*c[5]-c[6]*c[7]*c[5]^2), a[9,8] \+ = 1/60*(-20*c[5]*c[6]*c[4]+30*c[5]*c[6]*c[7]*c[4]+15*c[6]*c[5]-20*c[6] *c[7]*c[5]+15*c[6]*c[4]-20*c[6]*c[7]*c[4]-12*c[6]+15*c[7]*c[6]+15*c[5] *c[4]-20*c[5]*c[7]*c[4]-12*c[5]+15*c[7]*c[5]-12*c[4]+15*c[7]*c[4]+10-1 2*c[7])/(-c[4]+c[7]*c[4]+c[6]*c[4]-c[6]*c[7]*c[4]+c[5]*c[4]-c[5]*c[7]* c[4]-c[5]*c[6]*c[4]+c[5]*c[6]*c[7]*c[4]+1-c[7]-c[6]+c[7]*c[6]-c[5]+c[7 ]*c[5]+c[6]*c[5]-c[6]*c[7]*c[5]), a[9,1] = 1/60*(30*c[5]*c[6]*c[7]*c[4 ]-10*c[5]*c[6]*c[4]-10*c[6]*c[7]*c[5]+5*c[6]*c[5]-10*c[6]*c[7]*c[4]+5* c[6]*c[4]+5*c[7]*c[6]-3*c[6]-10*c[5]*c[7]*c[4]+5*c[5]*c[4]+5*c[7]*c[5] -3*c[5]+5*c[7]*c[4]-3*c[4]+2-3*c[7])/c[5]/c[6]/c[7]/c[4], a[9,4] = 1/6 0*(-2+3*c[5]-5*c[7]*c[6]+3*c[7]+3*c[6]+10*c[6]*c[7]*c[5]-5*c[6]*c[5]-5 *c[7]*c[5])/c[4]/(-c[6]*c[4]^3-c[4]^3+c[4]^4+c[7]*c[4]^2+c[6]*c[4]^2-c [6]*c[7]*c[4]+c[6]*c[7]*c[4]^2-c[5]*c[4]^3+c[5]*c[4]^2-c[7]*c[4]^3-c[5 ]*c[7]*c[4]+c[5]*c[7]*c[4]^2-c[5]*c[6]*c[4]+c[5]*c[6]*c[4]^2+c[6]*c[7] *c[5]-c[5]*c[6]*c[7]*c[4]), a[9,6] = 1/60*(10*c[5]*c[7]*c[4]-5*c[5]*c[ 4]-5*c[7]*c[5]+3*c[5]-5*c[7]*c[4]+3*c[4]-2+3*c[7])/c[6]/(c[6]^2*c[4]*c [5]-c[5]*c[6]*c[4]+c[5]*c[7]*c[4]-c[5]*c[6]*c[7]*c[4]-c[6]^3*c[4]+c[6] ^2*c[4]-c[6]*c[7]*c[4]+c[6]^2*c[7]*c[4]-c[6]^3*c[5]+c[6]^2*c[5]-c[6]*c [7]*c[5]+c[6]^2*c[7]*c[5]+c[6]^4-c[6]^3+c[7]*c[6]^2-c[6]^3*c[7])\}:" } }}{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 49 "a[7,6]=subs( e27,a[7,6]);\na[6,5]=subs(e27,a[6,5]);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/&%\"aG6$\"\"(\"\"',$*&#\"\"\"\"\"#F,*4,**(\"\"&F,&%\"cG6#F1F,&F 36#\"\"%F,F,F,F,*&F-F,F5F,!\"\"*&F-F,F2F,F9F,,&F5F,&F36#F'F9F,,&F2F,F< F9F,,&&F36#F(F,F**FAF,F6F,F/F,F:F,F>*(F7F,F/F,F9F,F,*&F/F,)F:FAF,F,**F CF,F5F,F9F,F7F,F>**F4F,F7F,F@F,F9F,F>**\"#?F,F7F,F5F,)F:F**F4F,F5 F,F7F,FGF,F,*&F7F,FGF,F>**FKF,FGF,F7F,F@F,F,**FKF,F5F,F6F,FGF,F>*&F6F, F9F,F,**F4F,F6F,F/F,FGF,F>**FCF,F6F,F/F,F9F,F,**F-F,F5F,F:F,F7F,F,**FK F,F6F,F/F,FLF,F,*(F-F,F5F,F6F,F,*(F-F,F@F,F7F,F>**FCF,F@F,F:F,F7F,F,F, F7F>,6*$FGF,F,*(F-F,F7F,F9F,F>*(F4F,F6F,FGF,F,*(F'F,F7F,FGF,F>*(\"#5F, FLF,F6F,F>*$)F7FAF,F>*(FinF,)F7F*(F'F,F[oF,F :F,F,*(F-F,F6F,F:F,F,F>F,F," }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}{SECT 1 {PARA 4 "" 0 " " {TEXT -1 44 "ee: coefficients for the Sharp-Verner scheme" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 891 "# coefficients by Sharp and Verner 1994\nee := \{c[2]=1/12,c[3]=2/15,c[ 4]=1/5,c[5]=8/15,c[6]=2/3,c[7]=19/20,c[8]=1,c[9]=1,\na[2,1]=1/12,a[3,1 ]=2/75,a[3,2]=8/75,\na[4,1]=1/20,a[4,2]=0,a[4,3]=3/20,\na[5,1]=88/135, a[5,2]=0,a[5,3]=-112/45,a[5,4]=64/27,\na[6,1]=-10891/11556,a[6,2]=0,a[ 6,3]=3880/963,\n a[6,4]=-8456/2889,a[6,5]=217/428,\na[7,1]=1718911/43 82720,a[7,2]=0,a[7,3]=-1000749/547840,\na[7,4]=819261/383488,a[7,5]=-6 71175/876544,a[7,6]=14535/14336,\na[8,1]=85153/203300,a[8,2]=0,a[8,3]= -6783/2140,\na[8,4]=10956/2675,a[8,5]=-38493/13375,a[8,6]=1152/425,a[8 ,7]=-7168/40375,\na[9,1]=53/912,a[9,2]=0,a[9,3]=0,a[9,4]=5/16,a[9,5]=2 7/112,a[9,6]=27/136,\na[9,7]=256/969,a[9,8]=-25/336,\nb[1]=53/912,b[2] =0,b[3]=0,b[4]=5/16,b[5]=27/112,b[6]=27/136,\nb[7]=256/969,b[8]=-25/33 6,\n`b*`[1]=617/10944,`b*`[2]=0,`b*`[3]=0,`b*`[4]=241/756,`b*`[5]=69/3 20,\n`b*`[6]=435/1904,`b*`[7]=10304/43605,`b*`[8]=0,`b*`[9]=-1/18\}:" }}}{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 6 "Check:" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 113 "eA := \{c[2 ]=1/12,c[4]=1/5,c[5]=8/15,c[6]=2/3,c[7]=19/20\}:\neB := `union`(eA,sim plify(subs(eA,e27))):\nevalb(ee=eB);" }}{PARA 11 "" 1 "" {XPPMATH 20 " 6#%%trueG" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 " " {MPLTEXT 1 0 1 ";" }}}}{PARA 0 "" 0 "" {TEXT -1 60 "#--------------- --------------------------------------------" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}}{PARA 0 "" 0 "" {TEXT 294 23 "______________ _________" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 30 "#=============================" }}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 26 "Sharp-Verner (1994) scheme" }}{PARA 257 "" 0 "" {TEXT -1 0 "" } }{PARA 0 "" 0 "" {TEXT -1 161 "See: Completely Imbedded Runge-Kutta Pa irs, by P. W. Sharp and J. H. Verner,\n SIAM Journal on Numeric al Analysis, Vol. 31, No. 4. (Aug., 1994), page 1185." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "ma trix([[1/12, 1/12, ``, ``, ``, ``, ``, ``, ``, ``], [2/15, 2/75, 8/75, ``, ``, ``, ``, ``, ``, ``], [1/5, 1/20, 0, 3/20, ``, ``, ``, ``, ``, ``], [8/15, 88/135, 0, -112/45, 64/27, ``, ``, ``, ``, ``], [2/3, -10 891/11556, 0, 3880/963, -8456/2889, 217/428, ``, ``, ``, ``], [19/20, \+ 1718911/4382720, 0, -1000749/547840, 819261/383488, -671175/876544, 14 535/14336, ``, ``, ``], [1, 85153/203300, 0, -6783/2140, 10956/2675, - 38493/13375, 1152/425, -7168/40375, ``, ``], [1, 53/912, 0, 0, 5/16, 2 7/112, 27/136, 256/969, -25/336, ``], [``, 617/10944, 0, 0, 241/756, 6 9/320, 435/1904, 10304/43605, 0, -1/18]])" "6#-%'matrixG6#7+7,*&\"\"\" F)\"#7!\"\"*&F)F)F*F+%!GF-F-F-F-F-F-F-7,*&\"\"#F)\"#:F+*&F0F)\"#vF+*& \"\")F)F3F+F-F-F-F-F-F-F-7,*&F)F)\"\"&F+*&F)F)\"#?F+\"\"!*&\"\"$F)F:F+ F-F-F-F-F-F-7,*&F5F)F1F+*&\"#))F)\"$N\"F+F;,$*&\"$7\"F)\"#XF+F+*&\"#kF )\"#FF+F-F-F-F-F-7,*&F0F)F=F+,$*&\"&\"*3\"F)\"&c:\"F+F+F;*&\"%!)QF)\"$ j*F+,$*&\"%c%)F)\"%*)GF+F+*&\"$<#F)\"$G%F+F-F-F-F-7,*&\"#>F)F:F+*&\"(6 *=)F)\"')[$QF+,$*&\"'v6 nF)\"'Wl()F+F+*&\"&NX\"F)\"&OV\"F+F-F-F-7,F)*&\"&`^)F)\"'+L?F+F;,$*&\" %$y'F)\"%S@F+F+*&\"&c4\"F)\"%vEF+,$*&\"&$\\QF)\"&vL\"F+F+*&\"%_6F)\"$D %F+,$*&\"%orF)\"&v.%F+F+F-F-7,F)*&\"#`F)\"$7*F+F;F;*&F8F)\"#;F+*&FIF)F EF+*&FIF)\"$O\"F+*&\"$c#F)\"$p*F+,$*&\"#DF)\"$O$F+F+F-7,F-*&\"$<'F)\"& W4\"F+F;F;*&\"$T#F)\"$c(F+*&\"#pF)\"$?$F+*&\"$N%F)\"%/>F+*&\"&/.\"F)\" &0O%F+F;,$*&F)F)\"#=F+F+" }{TEXT -1 1 " " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 63 " The last-but-one row gives the we ights for the order 6 scheme." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}{PARA 0 "" 0 "" {TEXT -1 52 "#---------------------------------------------------" }}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 28 "checking the combined scheme" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 35 "coefficients of the combined s cheme" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 891 "# coefficients by Sharp and Verner 1994\nee := \{c[2 ]=1/12,c[3]=2/15,c[4]=1/5,c[5]=8/15,c[6]=2/3,c[7]=19/20,c[8]=1,c[9]=1, \na[2,1]=1/12,a[3,1]=2/75,a[3,2]=8/75,\na[4,1]=1/20,a[4,2]=0,a[4,3]=3/ 20,\na[5,1]=88/135,a[5,2]=0,a[5,3]=-112/45,a[5,4]=64/27,\na[6,1]=-1089 1/11556,a[6,2]=0,a[6,3]=3880/963,\n a[6,4]=-8456/2889,a[6,5]=217/428, \na[7,1]=1718911/4382720,a[7,2]=0,a[7,3]=-1000749/547840,\na[7,4]=8192 61/383488,a[7,5]=-671175/876544,a[7,6]=14535/14336,\na[8,1]=85153/2033 00,a[8,2]=0,a[8,3]=-6783/2140,\na[8,4]=10956/2675,a[8,5]=-38493/13375, a[8,6]=1152/425,a[8,7]=-7168/40375,\na[9,1]=53/912,a[9,2]=0,a[9,3]=0,a [9,4]=5/16,a[9,5]=27/112,a[9,6]=27/136,\na[9,7]=256/969,a[9,8]=-25/336 ,\nb[1]=53/912,b[2]=0,b[3]=0,b[4]=5/16,b[5]=27/112,b[6]=27/136,\nb[7]= 256/969,b[8]=-25/336,\n`b*`[1]=617/10944,`b*`[2]=0,`b*`[3]=0,`b*`[4]=2 41/756,`b*`[5]=69/320,\n`b*`[6]=435/1904,`b*`[7]=10304/43605,`b*`[8]=0 ,`b*`[9]=-1/18\}:" }}}{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 0 123 "subs(ee,matrix([seq([c[i],s eq(a[i,j],j=1..i-1),``$(10-i)],i=2..9),\n[`b`,seq(b[i],i=1..8),``],[`b *`,seq(`b*`[i],i=1..9)]]));" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#K%'matr ixG6#7,7,#\"\"\"\"#7F(%!GF+F+F+F+F+F+F+7,#\"\"#\"#:#F.\"#v#\"\")F1F+F+ F+F+F+F+F+7,#F)\"\"&#F)\"#?\"\"!#\"\"$F8F+F+F+F+F+F+7,#F3F/#\"#))\"$N \"F9#!$7\"\"#X#\"#k\"#FF+F+F+F+F+7,#F.F;#!&\"*3\"\"&c:\"F9#\"%!)Q\"$j* #!%c%)\"%*)G#\"$<#\"$G%F+F+F+F+7,#\"#>F8#\"(6*=<\"(?FQ%F9#!(\\2+\"\"'S ya#\"'h#>)\"')[$Q#!'v6n\"'Wl()#\"&NX\"\"&OV\"F+F+F+7,F)#\"&`^)\"'+L?F9 #!%$y'\"%S@#\"&c4\"\"%vE#!&$\\Q\"&vL\"#\"%_6\"$D%#!%or\"&v.%F+F+7,F)# \"#`\"$7*F9F9#F6\"#;#FF\"$7\"#FF\"$O\"#\"$c#\"$p*#!#D\"$O$F+7,%\"bGFep F9F9FhpFjpF\\qF^qFaqF+7,%#b*G#\"$<'\"&W4\"F9F9#\"$T#\"$c(#\"#p\"$?$#\" $N%\"%/>#\"&/.\"\"&0O%F9#!\"\"\"#=Q)pprint226\"" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 146 "RK6_8eqs := [op(RowSumConditions(8,'expanded')),op(OrderConditions(6,8,'expanded' ))]:\n`RK5_9eqs*` := subs(b=`b*`,OrderConditions(5,9,'expanded')):" }} }{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 7 "Check: " }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 148 "simplify(subs(ee,RK6_8eqs )):\nmap(u->`if`(lhs(u)=rhs(u),0,1),%);\nnops(%);\nsimplify(subs(ee,`R K5_9eqs*`)):\nmap(u->`if`(lhs(u)=rhs(u),0,1),%);\nnops(%);" }{TEXT -1 0 "" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#7N\"\"!F$F$F$F$F$F$F$F$F$F$F$F$ F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#\"#W" }}{PARA 11 "" 1 "" {XPPMATH 20 " 6#73\"\"!F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#\"#<" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 " " {TEXT -1 86 "Next we set-up stage-order condtions to check for stage -orders from 2 to 5 inclusive. " }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 85 "for ct from 2 to 5 do\n so||ct||_8 := StageOrderConditions(c t,8,'expanded');\nend do:" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 34 "Stages 4 to 8 have stage-order 3. " }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 173 "[seq([seq(expand(subs(ee,so||i||_8 [j])),i=2..5)],j=1..6)]:\nmap(proc(L) local i; for i to nops(L) do if \+ not evalb(L[i]) then break end if end do; i end proc,%):\nsimplify(%); " }}{PARA 11 "" 1 "" {XPPMATH 20 "6#7(\"\"#\"\"$F%F%F%F%" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 53 "None of the princ ipal error conditions are satisfied." }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 123 "RK6_8err_eqs := PrincipalErrorConditions(6,8,'expand ed'):\nexpand(subs(ee,RK6_8err_eqs)):\nmap(u->`if`(lhs(u)=rhs(u),0,1), %);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#7R\"\"\"F$F$F$F$F$F$F$F$F$F$F$F $F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F $" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 114 "We can calculate the principal error norm of the order 6 scheme, that is , the 2-norm of the principal error terms." }}{EXCHG {PARA 0 "> " 0 " " {MPLTEXT 1 0 131 "errterms6_8 := PrincipalErrorTerms(6,8,'expanded') :\nevalf(evalf[14](sqrt(add(subs(ee,errterms6_8[i])^2,i=1.. nops(errte rms6_8)))));" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#$\"+-L'f%z!#9" }}} {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 79 "The 2 nor m of the principal error of the order 5 embedded scheme is as follows. " }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 153 "`errterms5_9*` := subs( b=`b*`,PrincipalErrorTerms(5,9,'expanded')):\nevalf(evalf[14](sqrt(add (subs(ee,`errterms5_9*`[i])^2,i=1.. nops(`errterms5_9*`)))));" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#$\"+;.zC>!#7" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}}{PARA 0 "" 0 "" {TEXT -1 52 "#---------------------------------------------- -----" }}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 44 "simultaneous constructio n of the two schemes" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 " " {TEXT -1 119 "We incorporate the stage-order equations to ensure tha t stage 2 has stage-order 2 and stages 3 to 8 have stage-order 3." }} {PARA 0 "" 0 "" {TEXT -1 48 "We also incorporate the simplifying condi tions: " }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "Sum(b[i]*a [i,j],i = j+1 .. 8) = b[j]*(1-c[j]);" "6#/-%$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=1" "6#/%\"jG\"\"\"" }{TEXT -1 10 " ( with " }{XPPEDIT 18 0 "c[1 ]=0" "6#/&%\"cG6#\"\"\"\"\"!" }{TEXT -1 5 " ), " }{XPPEDIT 18 0 "j = \+ 6;" "6#/%\"jG\"\"'" }{TEXT -1 7 " and " }{XPPEDIT 18 0 "j=7" "6#/%\" jG\"\"(" }{TEXT -1 1 "." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 78 "The 8 simple order conditions used are given (in abr eviated form) as follows. " }}{PARA 0 "" 0 "" {TEXT -1 76 "These inclu de the 6 quadrature conditions and 2 additional order conditions." }} {PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 178 "SO6 := SimpleOrderConditions(6):\n[seq([i,SO6[i]],i=[1,2,4,8,16,2 4,29,32])]:\nlinalg[augment](linalg[delcols](%,2..2),matrix([[` `]$(l inalg[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(\"\"$7%\"\")F)/*&F,F()F2F :F(#F(F57%\"#;F)/*&F,F()F2F5F(#F(\"\"&7%\"#CF)/*(F,F(F2F(-%!G6#*&%\"aG F(-FM6#*&F8F(FPF(F(F(#F(\"#s7%\"#HF)/*(F,F(F2F(-FM6#*&F?F(FPF(F(#F(FI7 %\"#KF)/*&F,F()F2FGF(#F(\"\"'Q)pprint586\"" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 136 "For the embedded order 5 schem e we use a selection of 7 \"simple\" order conditions as given (in abr eviated form) in the following table. " }}{PARA 0 "" 0 "" {TEXT -1 78 "These include the 5 quadrature conditions and two additional order co nditions." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 194 "`SO5*` := subs(b=`b*`,SimpleOrderConditions(5)):\n[s eq([i,`SO5*`[i]],i=[1,2,4,8,12,15,16])]:\nlinalg[augment](linalg[delco ls](%,2..2),matrix([[` `]$(linalg[rowdim](%))]),linalg[delcols](%,1.. 1));" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#K%'matrixG6#7)7%\"\"\"%#~~G/*& %#b*GF(%\"eGF(F(7%\"\"#F)/*&F,F(%\"cGF(#F(F/7%\"\"%F)/*&F,F()F2F/F(#F( \"\"$7%\"\")F)/*&F,F()F2F:F(#F(F57%\"#7F)/*&F,F(-%!G6#*&%\"aGF(-FF6#*& F8F(FIF(F(F(#F(\"#g7%\"#:F)/*&F,F(-FF6#*&F?F(FIF(F(#F(\"#?7%\"#;F)/*&F ,F()F2F5F(#F(\"\"&Q)pprint186\"" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 582 "SO6_8 := SimpleOrderConditi ons(6,8,'expanded'):\nSO_eqs := [op(RowSumConditions(8,'expanded')),op (StageOrderConditions(2,8,'expanded')),\n op(StageOrderC onditions(3,4..8,'expanded'))]:\n`SO5_9*` := subs(b=`b*`,SimpleOrderCo nditions(5,9,'expanded')):\nord_cdns := [seq(SO6_8[i],i=[1,2,4,8,16,24 ,29,32])]:\n`ord_cdns*` := [seq(`SO5_9*`[i],i=[1,2,4,8,12,15,16])]:\ns imp_eqs := [add(b[i]*a[i,1],i=1+1..8)=b[1],seq(add(b[i]*a[i,j],i=j+1.. 8)=b[j]*(1-c[j]),j=[6,7])]:\nwt_eqns := [seq(b[i]=a[9,i],i=1..8)]:\ncd ns := [op(ord_cdns),op(simp_eqs),op(SO_eqs),op(wt_eqns),op(`ord_cdns*` )]:" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 22 "W e specify the nodes: " }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "c[2]=1/12" "6#/&%\"cG6#\"\"#*&\"\"\"F)\"#7!\"\"" }{TEXT -1 3 ", \+ " }{XPPEDIT 18 0 "c[4]=1/5" "6#/&%\"cG6#\"\"%*&\"\"\"F)\"\"&!\"\"" } {TEXT -1 3 ", " }{XPPEDIT 18 0 "c[5]=8/15" "6#/&%\"cG6#\"\"&*&\"\")\" \"\"\"#:!\"\"" }{TEXT -1 3 ", " }{XPPEDIT 18 0 "c[6]=2/3" "6#/&%\"cG6 #\"\"'*&\"\"#\"\"\"\"\"$!\"\"" }{TEXT -1 3 ", " }{XPPEDIT 18 0 "c[7]= 19/20" "6#/&%\"cG6#\"\"(*&\"#>\"\"\"\"#?!\"\"" }{TEXT -1 3 ", " } {XPPEDIT 18 0 "c[8]=1" "6#/&%\"cG6#\"\")\"\"\"" }{TEXT -1 2 ", " }} {PARA 0 "" 0 "" {TEXT -1 31 "the zero linking coefficients: " }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "a[4,2]=0" "6#/&%\"aG6$\"\"% \"\"#\"\"!" }{TEXT -1 3 ", " }{XPPEDIT 18 0 "a[5,2]=0" "6#/&%\"aG6$\" \"&\"\"#\"\"!" }{TEXT -1 3 ", " }{XPPEDIT 18 0 "a[6,2]=0" "6#/&%\"aG6 $\"\"'\"\"#\"\"!" }{TEXT -1 3 ", " }{XPPEDIT 18 0 "a[7,2]=0" "6#/&%\" aG6$\"\"(\"\"#\"\"!" }{TEXT -1 3 ", " }{XPPEDIT 18 0 "a[8,2]=0" "6#/& %\"aG6$\"\")\"\"#\"\"!" }{TEXT -1 2 ", " }}{PARA 0 "" 0 "" {TEXT -1 22 "and the zero weights: " }}{PARA 256 "" 0 "" {TEXT -1 1 " " } {XPPEDIT 18 0 "b[2]=0" "6#/&%\"bG6#\"\"#\"\"!" }{TEXT -1 3 ", " } {XPPEDIT 18 0 "b[3]=0" "6#/&%\"bG6#\"\"$\"\"!" }{TEXT -1 1 "." }} {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 125 "The weig hts of the order 6 scheme provide the linking coefficients for the 9th stage of the embedded order 5 scheme so that: " }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "a[9,i]=b[i]" "6#/&%\"aG6$\"\"*%\"iG&%\" bG6#F(" }{TEXT -1 3 ", " }{XPPEDIT 18 0 "i=1" "6#/%\"iG\"\"\"" } {TEXT -1 9 " . . . 8." }}{PARA 0 "" 0 "" {TEXT -1 22 "We also specify \+ that " }{XPPEDIT 18 0 "c[9] = 1;" "6#/&%\"cG6#\"\"*\"\"\"" }{TEXT -1 3 ", " }{XPPEDIT 18 0 "`b*`[2] = 0;" "6#/&%#b*G6#\"\"#\"\"!" }{TEXT -1 3 ", " }{XPPEDIT 18 0 "`b*`[3] = 0;" "6#/&%#b*G6#\"\"$\"\"!" } {TEXT -1 7 " and " }{XPPEDIT 18 0 "`b*`[8] = 0;" "6#/&%#b*G6#\"\")\" \"!" }{TEXT -1 2 ". " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 " " {TEXT -1 37 "We have 44 equations and 44 unknowns." }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 199 "e1 := \{c[2]=1/12,c[4]=1/5,c[5]=8/15,c[6 ]=2/3,c[7]=19/20,c[8]=1,c[9]=1,\n seq(a[i,2]=0,i=4..8),b[2]=0,b[ 3]=0,`b*`[2]=0,`b*`[3]=0,`b*`[8]=0\}:\neqns := subs(e1,cdns):\nnops(%) ;\nindets(eqns);\nnops(%);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#\"#W" }} {PARA 12 "" 1 "" {XPPMATH 20 "6#F'&F%6$F>F9&F%6$\"\"'F+&F%6$FEF'&F/6#FE&F/6#F3&F/6#\"\")&F%6$FEF >&F%6$FEF9&F%6$\"\"*F+&F%6$FUF(&F%6$FUF'&F%6$FUF9&F%6$FUF>&F%6$FUFE&%# b*G6#F9&F[o6#F>&F%6$F3F'&F%6$F3F9&F%6$FNF'&F%6$FNF+&F[oF0&F%6$FUF3&F%6 $FUFN&F/F\\o&F/F^o&F%6$FNF9&F%6$FNF>&F%6$FNFE&F%6$FNF3&F%6$F3F>&F%6$F3 FE&F[oFI&F[oFK&F[o6#FU" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#\"#W" }}} {PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 22 "infolevel[solve] := 4:" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 69 "e2 := solve(\{op(eqns)\}):\ninfolevel[solve] := 0:\ne3 := `union`( e1,e2):" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 2 "e3" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1016 " e3 := \{b[5] = 27/112, a[8,6] = 1152/425, `b*`[5] = 69/320, c[3] = 2/1 5, a[7,5] = -671175/876544, c[9] = 1, `b*`[2] = 0, `b*`[3] = 0, b[6] = 27/136, b[3] = 0, `b*`[8] = 0, a[8,4] = 10956/2675, a[4,3] = 3/20, `b *`[7] = 10304/43605, b[4] = 5/16, a[6,5] = 217/428, a[5,1] = 88/135, a [3,2] = 8/75, a[9,5] = 27/112, a[6,4] = -8456/2889, a[5,4] = 64/27, `b *`[6] = 435/1904, a[8,2] = 0, a[5,3] = -112/45, `b*`[9] = -1/18, a[7,4 ] = 819261/383488, a[8,3] = -6783/2140, a[8,7] = -7168/40375, `b*`[4] \+ = 241/756, a[9,1] = 53/912, a[9,3] = 0, a[9,2] = 0, b[1] = 53/912, `b* `[1] = 617/10944, b[8] = -25/336, a[9,7] = 256/969, a[8,5] = -38493/13 375, b[7] = 256/969, c[5] = 8/15, c[6] = 2/3, a[9,8] = -25/336, c[4] = 1/5, a[7,1] = 1718911/4382720, a[9,6] = 27/136, a[6,2] = 0, a[9,4] = \+ 5/16, a[4,1] = 1/20, a[3,1] = 2/75, a[6,3] = 3880/963, b[2] = 0, a[5,2 ] = 0, c[2] = 1/12, a[7,3] = -1000749/547840, c[7] = 19/20, a[2,1] = 1 /12, a[4,2] = 0, a[7,2] = 0, a[8,1] = 85153/203300, a[7,6] = 14535/143 36, c[8] = 1, a[6,1] = -10891/11556\}:" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 123 "subs(e3,mat rix([seq([c[i],seq(a[i,j],j=1..i-1),``$(10-i)],i=2..9),\n[`b`,seq(b[i] ,i=1..8),``],[`b*`,seq(`b*`[i],i=1..9)]]));" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#K%'matrixG6#7,7,#\"\"\"\"#7F(%!GF+F+F+F+F+F+F+7,#\"\"# \"#:#F.\"#v#\"\")F1F+F+F+F+F+F+F+7,#F)\"\"&#F)\"#?\"\"!#\"\"$F8F+F+F+F +F+F+7,#F3F/#\"#))\"$N\"F9#!$7\"\"#X#\"#k\"#FF+F+F+F+F+7,#F.F;#!&\"*3 \"\"&c:\"F9#\"%!)Q\"$j*#!%c%)\"%*)G#\"$<#\"$G%F+F+F+F+7,#\"#>F8#\"(6*= <\"(?FQ%F9#!(\\2+\"\"'Sya#\"'h#>)\"')[$Q#!'v6n\"'Wl()#\"&NX\"\"&OV\"F+ F+F+7,F)#\"&`^)\"'+L?F9#!%$y'\"%S@#\"&c4\"\"%vE#!&$\\Q\"&vL\"#\"%_6\"$ D%#!%or\"&v.%F+F+7,F)#\"#`\"$7*F9F9#F6\"#;#FF\"$7\"#FF\"$O\"#\"$c#\"$p *#!#D\"$O$F+7,%\"bGFepF9F9FhpFjpF\\qF^qFaqF+7,%#b*G#\"$<'\"&W4\"F9F9# \"$T#\"$c(#\"#p\"$?$#\"$N%\"%/>#\"&/.\"\"&0O%F9#!\"\"\"#=Q)pprint576\" " }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 146 "RK6_8eqs := [op(RowSumConditions(8,'expanded')),op(O rderConditions(6,8,'expanded'))]:\n`RK5_9eqs*` := subs(b=`b*`,OrderCon ditions(5,9,'expanded')):" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 7 "Check: " }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 148 "simplify(subs(e3,RK6_8eqs)):\nmap(u->`if`(lhs(u)=rhs(u),0,1),%); \nnops(%);\nsimplify(subs(e3,`RK5_9eqs*`)):\nmap(u->`if`(lhs(u)=rhs(u) ,0,1),%);\nnops(%);" }{TEXT -1 0 "" }}{PARA 11 "" 1 "" {XPPMATH 20 "6# 7N\"\"!F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$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#\"#W" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#73\"\"!F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F $" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#\"#<" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}}{PARA 0 "" 0 "" {TEXT -1 52 "#--------------------------------------------------- " }}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 38 "characteristics of the embedd ed scheme" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 35 "coefficient s of the combined scheme" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 891 "# coefficients by Sharp and Verner 1994\nee := \{c[2]=1/12,c[3]=2/15,c[4]=1/5,c[5]=8/15,c[6]=2/3,c[7]=19 /20,c[8]=1,c[9]=1,\na[2,1]=1/12,a[3,1]=2/75,a[3,2]=8/75,\na[4,1]=1/20, a[4,2]=0,a[4,3]=3/20,\na[5,1]=88/135,a[5,2]=0,a[5,3]=-112/45,a[5,4]=64 /27,\na[6,1]=-10891/11556,a[6,2]=0,a[6,3]=3880/963,\n a[6,4]=-8456/28 89,a[6,5]=217/428,\na[7,1]=1718911/4382720,a[7,2]=0,a[7,3]=-1000749/54 7840,\na[7,4]=819261/383488,a[7,5]=-671175/876544,a[7,6]=14535/14336, \na[8,1]=85153/203300,a[8,2]=0,a[8,3]=-6783/2140,\na[8,4]=10956/2675,a [8,5]=-38493/13375,a[8,6]=1152/425,a[8,7]=-7168/40375,\na[9,1]=53/912, a[9,2]=0,a[9,3]=0,a[9,4]=5/16,a[9,5]=27/112,a[9,6]=27/136,\na[9,7]=256 /969,a[9,8]=-25/336,\nb[1]=53/912,b[2]=0,b[3]=0,b[4]=5/16,b[5]=27/112, b[6]=27/136,\nb[7]=256/969,b[8]=-25/336,\n`b*`[1]=617/10944,`b*`[2]=0, `b*`[3]=0,`b*`[4]=241/756,`b*`[5]=69/320,\n`b*`[6]=435/1904,`b*`[7]=10 304/43605,`b*`[8]=0,`b*`[9]=-1/18\}:" }}}{PARA 0 "" 0 "" {TEXT -1 0 " " }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 5 "Let " }{XPPEDIT 18 0 "T[6 , 8];" "6#&%\"TG6$\"\"'\"\")" }{TEXT -1 127 " denote the vector whose components are the principal error terms of the 8 stage, order 6 sche me (the error terms of order 7)." }}{PARA 0 "" 0 "" {TEXT -1 5 "Let \+ " }{XPPEDIT 18 0 "`T*`[5,9];" "6#&%#T*G6$\"\"&\"\"*" }{TEXT -1 145 " \+ denote the vector whose components are the principal error terms of th e embedded 9 stage, order 5 scheme (the error terms of order 6) and le t " }{XPPEDIT 18 0 "`T*`[6,9];" "6#&%#T*G6$\"\"'\"\"*" }{TEXT -1 99 " denote the vector whose components are the error terms of order 7 of the embedded order 5 scheme." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }} {PARA 0 "" 0 "" {TEXT -1 39 "Denote the 2-norm of these vectors by " }{XPPEDIT 18 0 "abs(abs(T[6,8]));" "6#-%$absG6#-F$6#&%\"TG6$\"\"'\"\") " }{TEXT -1 3 ", " }{XPPEDIT 18 0 "abs(abs(`T*`[5,9]));" "6#-%$absG6# -F$6#&%#T*G6$\"\"&\"\"*" }{TEXT -1 7 " and " }{XPPEDIT 18 0 "abs(abs (`T*`[6,9]));" "6#-%$absG6#-F$6#&%#T*G6$\"\"'\"\"*" }{TEXT -1 15 " re spectively." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 8 "Define: " }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "A[7 ] = abs(abs(T[6, 8]));" "6#/&%\"AG6#\"\"(-%$absG6#-F)6#&%\"TG6$\"\"'\" \")" }{TEXT -1 6 ", " }{XPPEDIT 18 0 "B[7] = abs(abs(`T*`[6,9]))/a bs(abs(`T*`[5,9]));" "6#/&%\"BG6#\"\"(*&-%$absG6#-F*6#&%#T*G6$\"\"'\" \"*\"\"\"-F*6#-F*6#&F/6$\"\"&F2!\"\"" }{TEXT -1 6 ", " }{XPPEDIT 18 0 "C[7] = abs(abs(`T*`[6,9]-T[6, 8]))/abs(abs(`T*`[5, 9]));" "6#/&% \"CG6#\"\"(*&-%$absG6#-F*6#,&&%#T*G6$\"\"'\"\"*\"\"\"&%\"TG6$F2\"\")! \"\"F4-F*6#-F*6#&F06$\"\"&F3F9" }{TEXT -1 1 "." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 77 "Dormand and Prince have s uggested that as well as attempting to ensure that " }{XPPEDIT 18 0 " A[7];" "6#&%\"AG6#\"\"(" }{TEXT -1 73 " is a minimum, if the embedded scheme is to be used for error control, " }{XPPEDIT 18 0 "B[7];" "6# &%\"BG6#\"\"(" }{TEXT -1 7 " and " }{XPPEDIT 18 0 "C[7];" "6#&%\"CG6 #\"\"(" }{TEXT -1 27 " should be chosen so that " }}{PARA 0 "" 0 "" {TEXT -1 68 "they are \"similar in magnitude\" and also not differ too much from 1." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 186 "errterms6_8 := PrincipalErrorTerms(6,8,'expande d'):\n`errterms6_9*` :=subs(b=`b*`,PrincipalErrorTerms(6,9,'expanded') ):\n`errterms5_9*` := subs(b=`b*`,PrincipalErrorTerms(5,9,'expanded')) :" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 326 "snmB := sqrt(add(evalf(subs(ee,`errterms6_9*`[i]))^2 ,i=1..nops(`errterms6_9*`))):\nsdnB := sqrt(add(evalf(subs(ee,`errterm s5_9*`[i]))^2,i=1..nops(`errterms5_9*`))):\nsnmC := sqrt(add((evalf(su bs(ee,`errterms6_9*`[i])-subs(ee,errterms6_8[i])))^2,i=1..nops(errterm s6_8))):\n'B[7]'= evalf[8](snmB/sdnB);\n'C[7]'= evalf[8](snmC/sdnB);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/&%\"BG6#\"\"($\")w**o7!\"(" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#/&%\"CG6#\"\"($\")\"G\"o7!\"(" }}} {PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}}{PARA 0 "" 0 "" {TEXT -1 52 "#------------------------------ ---------------------" }}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 27 "absolute stability regions " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 36 "c oefficients for the combined scheme" }}{PARA 0 "" 0 "" {TEXT -1 0 "" } }{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 884 "# coefficients by Sharp an d Verner 1994\nee := \{c[2]=1/12,c[3]=2/15,c[4]=1/5,c[5]=8/15,c[6]=2/3 ,c[7]=19/20,c[8]=1,\na[2,1]=1/12,a[3,1]=2/75,a[3,2]=8/75,\na[4,1]=1/20 ,a[4,2]=0,a[4,3]=3/20,\na[5,1]=88/135,a[5,2]=0,a[5,3]=-112/45,a[5,4]=6 4/27,\na[6,1]=-10891/11556,a[6,2]=0,a[6,3]=3880/963,\n a[6,4]=-8456/2 889,a[6,5]=217/428,\na[7,1]=1718911/4382720,a[7,2]=0,a[7,3]=-1000749/5 47840,\na[7,4]=819261/383488,a[7,5]=-671175/876544,a[7,6]=14535/14336, \na[8,1]=85153/203300,a[8,2]=0,a[8,3]=-6783/2140,\na[8,4]=10956/2675,a [8,5]=-38493/13375,a[8,6]=1152/425,a[8,7]=-7168/40375,\na[9,1]=53/912, a[9,2]=0,a[9,3]=0,a[9,4]=5/16,a[9,5]=27/112,a[9,6]=27/136,\na[9,7]=256 /969,a[9,8]=-25/336,\nb[1]=53/912,b[2]=0,b[3]=0,b[4]=5/16,b[5]=27/112, b[6]=27/136,\nb[7]=256/969,b[8]=-25/336,\n`b*`[1]=617/10944,`b*`[2]=0, `b*`[3]=0,`b*`[4]=241/756,`b*`[5]=69/320,\n`b*`[6]=435/1904,`b*`[7]=10 304/43605,`b*`[8]=0,`b*`[9]=-1/18\}:" }}}{PARA 0 "" 0 "" {TEXT -1 0 " " }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 77 "The stability function R \+ for the 8 stage, order 6 scheme is given as follows." }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 75 "subs(ee,StabilityFunction(6,8,'expanded') ):\nR := unapply(%,z):\n'R(z)'=R(z);" }}{PARA 11 "" 1 "" {XPPMATH 20 " 6#/-%\"RG6#%\"zG,4\"\"\"F)F'F)*&#F)\"\"#F)*$)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&\"(+!*)GF)*$)F'\"\"(F)F)F)*&#\"#J\"(+X W\"F)*$)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 sta bility region intersects the negative real axis by solving the equatio n: " }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "R(z) = 1;" "6# /-%\"RG6#%\"zG\"\"\"" }{TEXT -1 2 ". " }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 28 "z0 := newton(R(z)=1,z=-4.5);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%#z0G$!+#\\G3Z%!\"*" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 303 "z0 := newton(R(z)=1,z=-4. 5):\np1 := plot([R(z),1],z=-5.19..0.49,color=[red,blue]):\np2 := plot( [[[z0,1]]$3],style=point,symbol=[circle,cross,diamond],color=black):\n p3 := plot([[z0,0],[z0,1]],linestyle=3,color=COLOR(RGB,0,.5,0)):\nplot s[display]([p1,p2,p3],view=[-5.19..0.49,-.07..1.47],font=[HELVETICA,9] );" }}{PARA 13 "" 1 "" {GLPLOT2D 403 237 237 {PLOTDATA 2 "6+-%'CURVESG 6$7Z7$$!3Q++++++!>&!#<$\"3u'3,UrIZk$F*7$$!3QML3T![!f^F*$\"3MWLI!3!zfMF *7$$!3Ynm;#3'4G^F*$\"3m%\\kz\"GA$G$F*7$$!3a++DBT9(4&F*$\"3q['zqI*p9JF* 7$$!3kLLLk@>m]F*$\"370Rnu4!R&HF*7$$!3E+]U'*)HB,&F*$\"3'Q!3ZM2f\"p#F*7$ $!3!pm;&GwYe\\F*$\"3#3>)>$***>]CF*7$$!3s+](\\(Q*y*[F*$\"3D:*eI:9>?#F*7 $$!3nLLV@,KP[F*$\"3;QTnRhIw>F*7$$!3'RLLd%[MwZF*$\"3cEn$e\"F*7$$!3E+]<*4%oaYF*$\"3wmq%oihcT\"F*7$$!3;nmJG ')*Rf%F*$\"3o+u^il%QE\"F*7$$!3uLLyGAZ\"[%F*$\"3go;OD+l?5F*7$$!3%3+])fw &\\O%F*$\"3\"4m;e]&fV\")!#=7$$!3$QL$)f7eWC%F*$\"3tB\\G%oc)=kF]p7$$!3A+ +lN]MCTF*$\"3kFtqwClV]F]p7$$!3ummYeRz+SF*$\"3'[4K#y=dCRF]p7$$!3_LLV-,( >*QF*$\"3N\"*z*[hiP9$F]p7$$!35++S:-YpPF*$\"3;)RRmBqJX#F]p7$$!3K+++\"HZ kk$F*$\"32p\\;Gs&Q#>F]p7$$!3;++gW:!z_$F*$\"3)**)p]vehR:F]p7$$!3hLL)*\\ 1D?MF*$\"3w'R#Gpckx7F]p7$$!3'ommSKVAH$F*$\"3Gt$QDn[K0\"F]p7$$!3/nmEGV! 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" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 " " {MPLTEXT 1 0 1316 "R := z -> 1+z+1/2*z^2+1/6*z^3+1/24*z^4+1/120*z^5+ 1/720*z^6+559/2889000*z^7+31/1444500*z^8:\npts := []:\nz0 := 0:\nfor c t from 0 to 240 do\n zz := newton(R(z)=exp(ct*Pi/20*I),z=z0):\n z0 := zz:\n pts := [op(pts),[Re(zz),Im(zz)]]:\nend do:\np1 := plot(pts ,color=COLOR(RGB,.45,.05,.45)):\np2 := plots[polygonplot]([seq([pts[i- 1],pts[i],[-2.25,0]],i=2..nops(pts))],\n style=patchnogrid,co lor=COLOR(RGB,.85,.1,.85)):\npts := []: z0 := 2+4.75*I:\nfor ct from 0 to 60 do\n zz := newton(R(z)=exp(ct*Pi/30*I),z=z0):\n z0 := zz:\n pts := [op(pts),[Re(zz),Im(zz)]]:\nend do:\np3 := plot(pts,color=CO LOR(RGB,.45,.05,.45)):\np4 := plots[polygonplot]([seq([pts[i-1],pts[i] ,[1.87,4.73]],i=2..nops(pts))],\n style=patchnogrid,color=COL OR(RGB,.85,.1,.85)):\npts := []: z0 := 2-4.75*I:\nfor ct from 0 to 60 \+ do\n zz := newton(R(z)=exp(ct*Pi/30*I),z=z0):\n z0 := zz:\n pts \+ := [op(pts),[Re(zz),Im(zz)]]:\nend do:\np5 := plot(pts,color=COLOR(RGB ,.45,.05,.45)):\np6 := plots[polygonplot]([seq([pts[i-1],pts[i],[1.87, -4.73]],i=2..nops(pts))],\n style=patchnogrid,color=COLOR(RGB ,.85,.1,.85)):\np7 := plot([[[-5.19,0],[2.29,0]],[[0,-5.19],[0,5.19]]] ,color=black,linestyle=3):\nplots[display]([p||(1..7)],view=[-5.19..2. 29,-5.19..5.19],font=[HELVETICA,9],\n labels=[`Re(z)`,`Im (z)`],axes=boxed,scaling=constrained);" }}{PARA 13 "" 1 "" {GLPLOT2D 381 565 565 {PLOTDATA 2 "6/-%'CURVESG6$7]z7$$\"\"!F)F(7$F($\"3++++Fjzq :!#=7$$!3/+++*>N\"[8!#F$\"3&)******QEfTJF-7$$!3'******\\R3'*)H!#E$\"3C +++Y()Q7ZF-7$$!3-+++&yj[O#!#D$\"37+++YO=$G'F-7$$!3[+++x5MH'*F=$\"3W+++ xT(R&yF-7$$!3++++Qh^\")>!#C$\"3b*****Ro_ZU*F-7$$\"3'******f#4`t5FH$\"3 ))*****RE]&*4\"!#<7$$\"3<+++KT6EE!#B$\"35+++%))=mD\"FP7$$\"30+++Z>KU7! #A$\"3'******pYtOT\"FP7$$\"37+++i%>]2%FZ$\"33+++sspq:FP7$$\"3++++.n\\% 3\"!#@$\"3'******z&[lF'\\+![#F_o$\"3)******H@vW)=FP7$ $\"3<+++:yk.]F_o$\"3')******>_-T?FP7$$\"3*******>8lG+*F_o$\"3y******\\ 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" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 148 "We can distort the bound ary curve horizontally by taking the 11th root of the real part of poi nts along the curve. In this way we see that there is " }{TEXT 260 53 "no largest interval on the nonnegative imaginary axis" }{TEXT -1 65 " that contains the origin and lies inside the stability region. " }} {PARA 0 "" 0 "" {TEXT -1 119 "However the stability region intersects \+ the nonnegative imaginary axis in an interval that does not contain th e origin." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 343 "R := z -> 1+z+1/2*z^2+1/6*z^3+1/24*z^4+1/120*z^5+1/7 20*z^6+559/2889000*z^7+31/1444500*z^8:\nDigits := 25:\npts := []: z0 : = 0:\nfor ct from 0 to 105 do\n zz := newton(R(z)=exp(ct*Pi/100*I),z =z0):\n z0 := zz:\n pts := [op(pts),[surd(Re(zz),11),Im(zz)]]:\nen d do:\nplot(pts,color=COLOR(RGB,.85,0,.85),thickness=2,font=[HELVETICA ,9]);\nDigits := 10:" }}{PARA 13 "" 1 "" {GLPLOT2D 316 302 302 {PLOTDATA 2 "6(-%'CURVESG6#7fq7$$\"\"!F)F(7$$!:v\\(>e)ep(*zHkR#!#E$\": ?Dy$Qy(*e`EfTJF-7$$!:]\\Sdc*GI,9?mRF-$\":%oi'>soxrI&=$G'F-7$$!:fIB?bW3 WC8SK&F-$\":\"e=g\"**)ptgzxC%*F-7$$!:-&)f6Uj/:&zselF-$\":S&)R!z:E*oW`B#)*F-$\":9N@xC:6j&[6*>#F?7$$! :^JBU&onJ()G%33\"F?$\":>T,QVZG)>TF8DF?7$$!:q*y$y>2#R`Xbv6F?$\":W-'pR1E L\"QLu#GF?7$$!:9a$*z([*3)p`ym7F?$\":lyaT'ydEREfTJF?7$$!:s$e$*Q7%Qd%3%[ N\"F?$\":r&G+\\eOC\"*=vbMF?7$$!:%fUF3T\")48w%*R9F?$\":amAL+\"oxL6\"*pP F?7$$!:^&H8^eJ')zFFA:F?$\":x#yhkDQ%>OqS3%F?7$$!:vCt5Kuc#fk$>g\"F?$\":s &\\P\\I`,p&H#)R%F?7$$!:\"=t/.3'Q(3*>!z;F?$\":T<(['[QGgu)Q7ZF?7$$!:cP7x ??6GlqNv\"F?$\":NVGr$33J\")yaE]F?7$$!:nj,)*)y3Am`gD=F?$\":pr:XTaa*fpqS `F?7$$!:t![_mb\\DD46&*=F?$\"::YN2\"*GOK'f'[l&F?7$$!:O3[$)og*fvP/i>F?$ \":Esmtm:#)z'[-pfF?7$$!:2sa`KaodaFj-#F?$\":9/t4..rek$=$G'F?7$$!:\\jJ%oEEUtf'F?7$$!:$H`SBaBvX*[k9#F?$\":t\\8^,4ztn+:\" pF?7$$!:\"pq\"yHwk3Q:>?#F?$\":ziq2G*=IT)ecA(F?7$$!:#z8En@QNyJ'RD#F?$\" :>v_icWlpp;)RvF?7$$!:Jca\"yIds&46AI#F?$\":yKP+./;pJW#F?$ \":XD=c&)f5\"*F?7$$!:^-qdQ)RhjrIeCF?$\":e'eS)pxpVo_ZU*F?7$$!:Oku$ [EC6Es8fCF?$\":2#\\!4I]'4Dd!*Q(*F?7$$!:!*HF5X&3A4h)oV#F?$\":Z4R4x3\"[[ dI05!#C7$$!:j=uEtQqmH\"foBF?$\":J)f))p+2'Qx?n.\"Ffu7$$!:B$QvT:6AU7\\J@ F?$\":3T*Rx@<[Ac8o5Ffu7$$\":-7-)p57ve)y]K#F?$\":.\\#p>'>eQE]&*4\"Ffu7$ $\":DzsX[1\\L.'=yDF?$\":RK0O.vACmk48\"Ffu7$$\"::s167:!fnccWFF?$\":W')* Gz4jiw(yB;\"Ffu7$$\":Y[/qE=![!\\0-)GF?$\":8!ep'z/zyb#z$>\"Ffu7$$\":cHA `L#=5fOw**HF?$\":`jQ?J#*4$\\f?D7Ffu7$$\":$oHL^Gf:xgJ4JF?$\":/DYG*H-K%) )=mD\"Ffu7$$\":`4Q@y$3HY)o>@$F?$\":#*H8Ffu7$$\":*eq_$)z$*o+35.MF?$\":!Q`B[nA\"=5a3N\" Ffu7$$\":]3\"y)z'ek8aY$\\$F?$\":6$fe#z9ZhGkAQ\"Ffu7$$\":=%3:X@uB(*)R6e $F?$\":)*Q*R\"H>;mYtOT\"Ffu7$$\":q]vl*\\oufK]mOF?$\":<,-Be0yFZ\"3X9Ffu 7$$\":M9&zP9(3EmN)\\PF?$\":owFi;sAh5)[w9Ffu7$$\":[!H\"\\'o,V@#[8$QF?$ \":>]4A<+\")e8$*y]\"Ffu7$$\":QL>2^2<^D-7\"RF?$\":w\\PF%)4MVH'HR:Ffu7$$ \":y&oz`,K7]@_*)RF?$\":Slu:=f'erspq:Ffu7$$\":r4RfchvT?/k1%F?$\":(pK3BM pw4d4-;Ffu7$$\":Q46%3CtI]G#>9%F?$\":`RMH7'4i'>\"\\L;Ffu7$$\":F6yc[*oDA ^8;UF?$\":(pa;!R:m%eK)[m\"Ffu7$$\":CKr-Ps%[2R3*G%F?$\":i#Q&*Q`vZ_8F'p \"Ffu7$$\":03`K@\"o!RM+3O%F?$\":i6ZyxZ!Ge[lFFfu7$$\":i<963)=?(=+\"H[F?$\":%ygtw( [3V)z8Z>Ffu7$$\":\"pQg9m0ann3\"*[F?$\":'z-o=B+N-)\\%y>Ffu7$$\":'ym^m!=&F?$\":S3%>F.(y.JMZ8#Ffu7$$\": qJ#4y')Q1l%RTB&F?$\":Dda2P9eA->f;#Ffu7$$\":_\"Q1kGpF!3!)fG&F?$\":Pia2 \"eRp\\F2(>#Ffu7$$\":8&z*=&zSK(3)4O`F?$\":d1K@(**>*y(>>GAFfu7$$\":o,-c g-?Av'R%Q&F?$\":*G?j@7HT&*GFfAFfu7$$\":1]:U0:%\\unwIaF?$\":@!3FVz<*RQ6 .H#Ffu7$$\":qcgK3r5pP%3vaF?$\":PS,ez$[A5II@BFfu7$$\":7p8.3q8^\"*3s^&F? $\":\"*fF+ocp%>IC_BFfu7$$\":$oL:3s?gf)zpb&F?$\":l/]tE(3IKj7$Q#Ffu7$$\" :T\"o6Q&z'pWH@%f&F?$\":4\"o675/&QaZRT#Ffu7$$\":eI,ng?-[>&pGcF?$\":\"ov $y@`Ej#4qWCFfu7$$\":p(Rsv2`^7%y,m&F?$\":]%fkTev'\\V!QvCFfu7$$\":gxzo#4 >\\@0P)o&F?$\":$GZq=],i<(zf]#Ffu7$$\":gL@!HE+idQ#Hr&F?$\":Vo+ynbd!G@\\ ODFfu7$$\":)y0u&G)eK@!>Mt&F?$\":2\"pc&4FgIu5pc#Ffu7$$\":+vf)GoAJ2@M\\d F?$\":Mc+tkdF?$\":,yCAlu$H_zYp&F?$\":e<`/]A%)>8'ewFFfu7 $$\":i==&RshI>0ITcF?$\":2@ILV_^@=!)f!GFfu7$$\":!>#Q]^ha7Dd:c&F?$\":pSp [-p9y.5_$GFfu7$$\":n%=8@1aX#))z%RaF?$\":J3h)4mHQmvEkGFfu7$$\":CRK[,5n. \">BN_F?$\":(\\R0%>!fy\"pWJ*GFfu7$$\":6A%=0jH>\"yq[x%F?$\":TDn.QwD7UL= #HFfu7$$!:BD5$>Ha,3U`9[F?$\":b:?1Htg$oeK]HFfu7$$!:b%)[$=Xr&HrX)f`F?$\" :\"*4ipPfDDF9'yHFfu7$$!:;!ev+8,ox;/ccF?$\":v2w^,!ew85p1IFfu7$$!:u1hEJ* fs&y*oteF?$\":tN]'QlCkB'[X.$Ffu7$$!:zz@1$F fu7$$!:`hLl\\jW#*y?\\?'F?$\":&)e9t)36nptd*3$Ffu7$$!:M([')fqy)ej\"fTjF? $\":&3]8p!QS'oVt;JFfu7$$!:%3;I9*Qxo?lfY'F?$\":UPU-ugf&eRkVJFfu7$$!:bxv Z;7f0BA3e'F?$\":U+'z*QKw?Y*HqJFfu7$$!:yK/.\\(yeIU-)o'F?$\":c2'p$=!Qu[V p'>$Ffu7$$!:wnB*>Ua\"48z))y'F?$\":yulj0d+(GA#GA$Ffu-%*THICKNESSG6#\"\" #-%%FONTG6$%*HELVETICAG\"\"*-%&COLORG6&%$RGBG$\"#&)!\"#F(Fa]m-%+AXESLA BELSG6$Q!6\"Fg]m-%%VIEWG6$%(DEFAULTGF\\^m" 1 2 0 1 10 2 2 6 1 4 2 1.000000 45.000000 45.000000 0 0 "Curve 1" }}}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 108 "The relevant intersection poin ts of the boundary curve with the imaginary axis can be determined as \+ follows." }}{PARA 0 "" 0 "" {TEXT -1 87 "First we look for points on t he boundary curve either side of each intersection point. " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 199 "Dig its := 15:\nz0 := 1.1*I:\nfor ct from 33 to 36 do\n newton(R(z)=exp( ct*Pi/100*I),z=z0);\nend do;``;\nz0 := 2.95*I:\nfor ct from 93 to 96 d o\n newton(R(z)=exp(ct*Pi/100*I),z=z0);\nend do;\nDigits := 10:" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#^$$!0Vr7v5kJ\"!#@$\"0qgQx?n.\"!#9" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#^$$!0n%3;O!f7%!#A$\"0s\"[Ac8o5!#9" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#^$$\"0+!zD4`t5!#@$\"0?eQE]&*4\"!#9" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#^$$\"0XEf6naM$!#@$\"0vACmk48\"!#9" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#%!G" }}{PARA 11 "" 1 "" {XPPMATH 20 "6 #^$$\"0#H)3(*fk4)!#=$\"0!fy\"pWJ*G!#9" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#^$$\"0?3l'4cTH!#=$\"0wD7UL=#H!#9" }}{PARA 11 "" 1 "" {XPPMATH 20 "6 #^$$!0I52j)y@K!#=$\"0tg$oeK]H!#9" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#^$ $!02/Qsd([5!#<$\"0fDDF9'yH!#9" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }} {PARA 0 "" 0 "" {TEXT -1 108 "Then we apply the bisection method to ca lculate the parameter value associated with each intersection point." }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 328 "Digits := 15:\nreal_part \+ := proc(u)\n Re(newton(R(z)=exp(u*Pi*I),z=1.1*I))\nend proc:\nu0 := \+ bisect('real_part'(u),u=0.33..0.36);\nnewton(R(z)=exp(u0*Pi*I),z=1.1*I );``;\nreal_part := proc(u)\n Re(newton(R(z)=exp(u*Pi*I),z=2.9*I))\n end proc:\nu0 := bisect('real_part'(u),u=0.93..0.96);\nnewton(R(z)=exp (u0*Pi*I),z=2.9*I);\nDigits := 10:" }}{PARA 11 "" 1 "" {XPPMATH 20 "6# >%#u0G$\"0Z6QFxEV$!#:" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#^#$\"0Bw_4,%y 5!#9" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%!G" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%#u0G$\"0QomVm)\\%*!#:" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#^$$\"0)f8C8$[N$!#H$\"0?$>7m1OH!#9" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 4 "The " }{TEXT 260 73 "stability regi on intersects the nonegative imaginary axis in the interval" }{TEXT -1 3 " " }{XPPEDIT 18 0 "[1.0784, 2.9361];" "6#7$-%&FloatG6$\"&%y5! \"%-F%6$\"&h$HF(" }{TEXT -1 18 " (approximately)." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 37 "#------------------------ ------------" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 80 "The stability function R* for the 8 stage, order 9 scheme is \+ given as follows." }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 97 "subs(ee ,subs(b=`b*`,StabilityFunction(5,9,'expanded'))):\n`R*` := unapply(%,z ):\n'`R*`(z)'=`R*`(z);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/-%#R*G6#%\" zG,6\"\"\"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)*&#\"&j_\"\")+/ S5F)*$)F'F1F)F)F)*&#\"&.>$\"*+S+/\"F)*$)F'\"\"(F)F)F)*&#\"$f&\")+?+_F) *$)F'\"\")F)F)!\"\"*&#\"#J\")+5+EF)*$)F'\"\"*F)F)FU" }}}{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 rea l axis by solving the equation: " }}{PARA 256 "" 0 "" {TEXT -1 1 " " } {XPPEDIT 18 0 "`R*`(z) = -1;" "6#/-%#R*G6#%\"zG,$\"\"\"!\"\"" }{TEXT -1 2 ". " }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 33 "z_0 := newton(`R *`(z)=-1,z=-3.5);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%$z_0G$!+pz.qM! \"*" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 325 "z_0 := newton(`R*`(z)=-1,z=-3.45):\np_1 := plot([`R* `(z),-1],z=-3.99..0.49,color=[red,blue]):\np_2 := plot([[[z_0,-1]]$3], style=point,symbol=[circle,cross,diamond],color=black):\np_3 := plot([ [z_0,0],[z_0,-1]],linestyle=3,color=COLOR(RGB,0,.5,0)):\nplots[display ]([p_1,p_2,p_3],view=[-3.99..0.49,-1.57..1.47],font=[HELVETICA,9]);" } }{PARA 13 "" 1 "" {GLPLOT2D 398 284 284 {PLOTDATA 2 "6+-%'CURVESG6$7U7 $$!3A++++++!*R!#<$!3wYkHzad&)GF*7$$!3wmmmuVj#F*7$$!3u LLL\\([B*QF*$!3=#4yl=$R(R#F*7$$!3-++!Q$e')\\QF*$!3\"4S*f07'y?#F*7$$!3v mmE=HQ2QF*$!3P>'RpZp5.#F*7$$!3cLL$\\&*H=r$F*$!3_$y6jP]in\"F*7$$!3cLL`/ Ok:OF*$!310n\"on&os8F*7$$!3\"pmm5T9*>NF*$!3=gvj59)p6\"F*7$$!3zLL`%>h6V $F*$!37m5q'GV$e\"*!#=7$$!30++gzBERLF*$!3Q&Hw2\\37R(FU7$$!3mLLt$\\?UC$F *$!3]V'3P3L^&eFU7$$!33++S3M[\\JF*$!3id9v#4.kd%FU7$$!3immYrY._IF*$!3g6S V)*R1%[$FU7$$!3[LL$4x,i'HF*$!3Y\\7G%*4f!o#FU7$$!3!)****RaUdpGF*$!35/+Z 7<[D>FU7$$!33+++w*\\Dx#F*$!3o9tO$om+I\"FU7$$!3')****f0\"\\!zEF*$!3pH\" pnJI!3!)!#>7$$!3?LLtd89%f#F*$!3+WZ\")Ra_%>%Fhp7$$!3_mm1Ly<$\\#F*$!3?'f 'f^J$4z$!#?7$$!3ymmE%[[wS#F*$\"3#GS3$3-alBFhp7$$!3')****z=v:3BF*$\"3k1 -+so9D^Fhp7$$!3?nmEc64?AF*$\"3Q8/Ke*)\\usFhp7$$!30++!))RoM7#F*$\"3#))z ]#*)ya2%*Fhp7$$!3$)****R%og9.#F*$\"3#z**py[2%H6FU7$$!3\"ommmGga$>F*$\" 3Vvbyw%=!=8FU7$$!3#ommM&>IZ=F*$\"3S?\\'\\\"F*$\"3@]*[8 !)Gt-$FU7$$!3=++?zc;$4\"F*$\"3yPE&pJ;-N$FU7$$!3FnmEKpc-5F*$\"3/XHkqRto OFU7$$!31,++?VLe!*FU$\"3$R&[.`^zTSFU7$$!3_PLLL]y\"=)FU$\"3`)GFhp$\"3ymcuS1j>5F*7$$\"3'oLLtN*z'=\"FU$\"3F!*yYl%4g7\"F*7 $$\"3Ikmm%>4W2#FU$\"3N_9bA]_I7F*7$$\"3(\\++?T'y?IFU$\"3=T%3+onEN\"F*7$ $\"3s'****R;!fERFU$\"3;q\\DfP\"4[\"F*7$$\"3!***************[FU$\"3#HsJ &=zJK;F*-%'COLOURG6&%$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$F_qFa\\l7$FeqFa\\l7$FjqFa\\l7$F_rFa\\l7$FdrFa\\l7$FirFa \\l7$F^sFa\\l7$FcsFa\\l7$FhsFa\\l7$F]tFa\\l7$FbtFa\\l7$FgtFa\\l7$F\\uF a\\l7$FauFa\\l7$FfuFa\\l7$F[vFa\\l7$F`vFa\\l7$FevFa\\l7$FjvFa\\l7$F_wF a\\l7$FdwFa\\l7$FiwFa\\l7$F^xFa\\l7$FcxFa\\l7$FhxFa\\l7$F]yFa\\l7$FbyF a\\l7$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$$!3%)******oz.qMF*Fa\\l-%'SYMBOLG6#%'CI RCLEG-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[\\lF h_l-%&COLORG6&Fg[lF[\\l$\"\"&Fb\\lF[\\l-%*LINESTYLEG6#\"\"$-%+AXESLABE LSG6%Q\"z6\"Q!F`bl-%%FONTG6#%(DEFAULTG-Fcbl6$%*HELVETICAG\"\"*-%%VIEWG 6$;$!$*R!\"#$\"#\\F`cl;$!$d\"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 82 "The following picture shows the stability region for the 9 stage, order 5 scheme. " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1365 "`R*` := z -> 1+z+1/2*z^2+1 /6*z^3+1/24*z^4+1/120*z^5+\n 15263/10400400*z^6+31903/1040040 00*z^7-559/52002000*z^8-31/26001000*z^9:\npts := []: z0 := 0:\nfor ct \+ from 0 to 200 do\n zz := newton(`R*`(z)=exp(ct*Pi/20*I),z=z0):\n z 0 := zz:\n pts := [op(pts),[Re(zz),Im(zz)]]:\nend do:\np_1 := plot(p ts,color=COLOR(RGB,.33,0,.33)):\np_2 := plots[polygonplot]([seq([pts[i -1],pts[i],[-1.75,0]],i=2..nops(pts))],\n style=patchnogrid,c olor=COLOR(RGB,.65,0,.65)):\npts := []: z0 := 1.8+4*I:\nfor ct from 0 \+ to 60 do\n zz := newton(`R*`(z)=exp(ct*Pi/30*I),z=z0):\n z0 := zz: \n pts := [op(pts),[Re(zz),Im(zz)]]:\nend do:\np_3 := plot(pts,color =COLOR(RGB,.33,0,.33)):\np_4 := plots[polygonplot]([seq([pts[i-1],pts[ i],[1.73,3.87]],i=2..nops(pts))],\n style=patchnogrid,color=C OLOR(RGB,.65,0,.65)):\npts := []: z0 := 1.8-4*I:\nfor ct from 0 to 60 \+ do\n zz := newton(`R*`(z)=exp(ct*Pi/30*I),z=z0):\n z0 := zz:\n p ts := [op(pts),[Re(zz),Im(zz)]]:\nend do:\np_5 := plot(pts,color=COLOR (RGB,.33,0,.33)):\np_6 := plots[polygonplot]([seq([pts[i-1],pts[i],[1. 73,-3.87]],i=2..nops(pts))],\n style=patchnogrid,color=COLOR( RGB,.65,0,.65)):\np_7 := plot([[[-3.99,0],[2.19,0]],[[0,-4.39],[0,4.39 ]]],color=black,linestyle=3):\nplots[display]([p_||(1..7)],view=[-3.99 ..2.19,-4.39..4.39],font=[HELVETICA,9],\n labels=[`Re(z)`,` Im(z)`],axes=boxed,scaling=constrained);" }}{PARA 13 "" 1 "" {GLPLOT2D 390 543 543 {PLOTDATA 2 "6/-%'CURVESG6$7ew7$$\"\"!F)F(7$$\"3 &******R$3'*>7!#E$\"3++++Fjzq:!#=7$$\"3k+++K\"\\m_)!#D$\"31+++AFfTJF07 $$\"3*******\\Dv,5\"!#B$\"3z*******p*Q7ZF07$$\"3#******4s!p3rF:$\"3!** ****f;'=$G'F07$$\"32+++pCB6J!#A$\"3;+++s\"fR&yF07$$\"3'******f6_/0\"!# @$\"3m*****\\d(eC%*F07$$\"3$)*****Hoeo#HFK$\"3'******4\"eY*4\"!#<7$$\" 3#******Rb%)R*pFK$\"35+++`zIc7FS7$$\"35+++MBMl9!#?$\"3/+++J]u79FS7$$\" 34+++!3B2s#Ffn$\"3++++#)=Ko:FS7$$\"3O+++/Uq![%Ffn$\"3'******\\j)GAv&G%R$Ffn$\"3*******4odoU#FS7$$!30+++=$ 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z%FganF]abl7%Fe^cl7$F^``l$!+^l;%y%FganF]abl7%Fi^cl7$Fh_`l$!+Q#fnx%Fgan F]abl7%F]_cl7$Fb_`l$!+pe)*oZFganF]abl7%Fa_cl7$Fg^`l$!+G=*3w%FganF]abl7 %Fe_clFg`blF]ablFhj^lF[^r-F$6%7$7$$!3Q++++++!>&FSF(7$$\"3/++++++!H#FSF (-%'COLOURG6&F_^nF)F)F)-%*LINESTYLEG6#\"\"$-F$6%7$7$F(F]`cl7$F($\"3Q++ ++++!>&FSFb`clFe`cl-%*AXESSTYLEG6#%$BOXG-%%FONTG6$%*HELVETICAG\"\"*-%( SCALINGG6#%,CONSTRAINEDG-%+AXESLABELSG6%%&Re(z)G%&Im(z)G-Feacl6#%(DEFA ULTG-%%VIEWG6$;$!$>&Fb^n$\"$H#Fb^n;Fibcl$\"$>&Fb^n" 1 2 0 1 10 0 2 9 1 2 1 1.000000 45.000000 45.000000 0 0 "Curve 1" "Curve 2" "Curve 3" " Curve 4" "Curve 5" "Curve 6" "Curve 7" "Curve 8" "Curve 9" "Curve 10" "Curve 11" "Curve 12" "Curve 13" "Curve 14" }}}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}}{PARA 0 "" 0 "" {TEXT -1 52 "#--------------------------------------------------- " }}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 36 "coefficients for the combined scheme" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 35 "coefficient s of the combined scheme" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 891 "# coefficients by Sharp and Verner 1994\nee := \{c[2]=1/12,c[3]=2/15,c[4]=1/5,c[5]=8/15,c[6]=2/3,c[7]=19 /20,c[8]=1,c[9]=1,\na[2,1]=1/12,a[3,1]=2/75,a[3,2]=8/75,\na[4,1]=1/20, a[4,2]=0,a[4,3]=3/20,\na[5,1]=88/135,a[5,2]=0,a[5,3]=-112/45,a[5,4]=64 /27,\na[6,1]=-10891/11556,a[6,2]=0,a[6,3]=3880/963,\n a[6,4]=-8456/28 89,a[6,5]=217/428,\na[7,1]=1718911/4382720,a[7,2]=0,a[7,3]=-1000749/54 7840,\na[7,4]=819261/383488,a[7,5]=-671175/876544,a[7,6]=14535/14336, \na[8,1]=85153/203300,a[8,2]=0,a[8,3]=-6783/2140,\na[8,4]=10956/2675,a [8,5]=-38493/13375,a[8,6]=1152/425,a[8,7]=-7168/40375,\na[9,1]=53/912, a[9,2]=0,a[9,3]=0,a[9,4]=5/16,a[9,5]=27/112,a[9,6]=27/136,\na[9,7]=256 /969,a[9,8]=-25/336,\nb[1]=53/912,b[2]=0,b[3]=0,b[4]=5/16,b[5]=27/112, b[6]=27/136,\nb[7]=256/969,b[8]=-25/336,\n`b*`[1]=617/10944,`b*`[2]=0, `b*`[3]=0,`b*`[4]=241/756,`b*`[5]=69/320,\n`b*`[6]=435/1904,`b*`[7]=10 304/43605,`b*`[8]=0,`b*`[9]=-1/18\}:" }}}{PARA 0 "" 0 "" {TEXT -1 0 " " }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT 259 5 "nodes" }{TEXT -1 2 ": " } }{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 31 "seq(c[i]=subs(ee,c[i]),i=2. .9);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6*/&%\"cG6#\"\"##\"\"\"\"#7/&F%6 #\"\"$#F'\"#:/&F%6#\"\"%#F)\"\"&/&F%6#F6#\"\")F0/&F%6#\"\"'#F'F./&F%6# \"\"(#\"#>\"#?/&F%6#F;F)/&F%6#\"\"*F)" }}}{PARA 0 "" 0 "" {TEXT -1 0 " " }}{PARA 0 "" 0 "" {TEXT 259 51 "linking coefficients for the 9 stage order 5 scheme" }{TEXT -1 1 ":" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 49 "seq(seq(a[i,j]=subs(ee,a[i,j]),j=1..i-1),i=1..9);" }}{PARA 12 "" 1 "" {XPPMATH 20 "6F/&%\"aG6$\"\"#\"\"\"#F(\"#7/&F%6$\"\"$F(#F'\"#v /&F%6$F.F'#\"\")F0/&F%6$\"\"%F(#F(\"#?/&F%6$F9F'\"\"!/&F%6$F9F.#F.F;/& F%6$\"\"&F(#\"#))\"$N\"/&F%6$FGF'F?/&F%6$FGF.#!$7\"\"#X/&F%6$FGF9#\"#k \"#F/&F%6$\"\"'F(#!&\"*3\"\"&c:\"/&F%6$FgnF'F?/&F%6$FgnF.#\"%!)Q\"$j*/ &F%6$FgnF9#!%c%)\"%*)G/&F%6$FgnFG#\"$<#\"$G%/&F%6$\"\"(F(#\"(6*=<\"(?F Q%/&F%6$FcpF'F?/&F%6$FcpF.#!(\\2+\"\"'Sya/&F%6$FcpF9#\"'h#>)\"')[$Q/&F %6$FcpFG#!'v6n\"'Wl()/&F%6$FcpFgn#\"&NX\"\"&OV\"/&F%6$F5F(#\"&`^)\"'+L ?/&F%6$F5F'F?/&F%6$F5F.#!%$y'\"%S@/&F%6$F5F9#\"&c4\"\"%vE/&F%6$F5FG#!& $\\Q\"&vL\"/&F%6$F5Fgn#\"%_6\"$D%/&F%6$F5Fcp#!%or\"&v.%/&F%6$\"\"*F(# \"#`\"$7*/&F%6$F\\uF'F?/&F%6$F\\uF.F?/&F%6$F\\uF9#FG\"#;/&F%6$F\\uFG#F Y\"$7\"/&F%6$F\\uFgn#FY\"$O\"/&F%6$F\\uFcp#\"$c#\"$p*/&F%6$F\\uF5#!#D \"$O$" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT 259 38 "weights for the 8 stage order 6 scheme" }{TEXT -1 1 ":" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 31 "seq(b[i]=subs(ee,b[i]),i=1..8);" }} {PARA 11 "" 1 "" {XPPMATH 20 "6*/&%\"bG6#\"\"\"#\"#`\"$7*/&F%6#\"\"#\" \"!/&F%6#\"\"$F//&F%6#\"\"%#\"\"&\"#;/&F%6#F9#\"#F\"$7\"/&F%6#\"\"'#F? \"$O\"/&F%6#\"\"(#\"$c#\"$p*/&F%6#\"\")#!#D\"$O$" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT 259 38 "weights for the 9 stage \+ order 5 scheme" }{TEXT -1 1 ":" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 37 "seq(`b*`[i]=subs(ee,`b*`[i]),i=1..9);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6+/&%#b*G6#\"\"\"#\"$<'\"&W4\"/&F%6#\"\"#\"\"!/&F%6#\"\"$ F//&F%6#\"\"%#\"$T#\"$c(/&F%6#\"\"&#\"#p\"$?$/&F%6#\"\"'#\"$N%\"%/>/&F %6#\"\"(#\"&/.\"\"&0O%/&F%6#\"\")F//&F%6#\"\"*#!\"\"\"#=" }}}{PARA 0 " " 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}} }{PARA 0 "" 0 "" {TEXT -1 52 "#--------------------------------------- ------------" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}}{PARA 0 "" 0 "" {TEXT -1 30 "#=============================" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT 258 58 "Schemes designed to h ave specific stability characterstics" }}{PARA 0 "" 0 "" {TEXT -1 59 " #==========================================================" }}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 44 "a scheme with a wide real stability int erval" }}{PARA 257 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 85 "This scheme is constructed to have a stability polynomial suggeste d by Ch. Tsitouras." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 " " {TEXT -1 162 "See: A Parameter Study of Explicit Runge-Kutta Pairs o f Orders 6(5), by Ch. Tsitouras,\n Applied Mathematics Letters, \+ Vol. 11, No. 1, pages 65 to 69, 1998. " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}{PARA 0 "" 0 "" {TEXT -1 52 "#---------------------------------------------------" }} {SECT 1 {PARA 4 "" 0 "" {TEXT -1 28 "checking the combined scheme" }} {PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 35 "coefficients of the comb ined scheme" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 " " {MPLTEXT 1 0 1931 "ee := \{c[2]=1/192,\nc[3]=56/435,\nc[4]=28/145,\n c[5]=56/169,\nc[6]=37/58,\nc[7]=121/158,\nc[8]=1,\nc[9]=1,\n\na[2,1]=1 /192,\na[3,1]=-30744/21025,\na[3,2]=100352/63075,\na[4,1]=7/145,\na[4, 2]=0,\na[4,3]=21/145,\na[5,1]=523516/4826809,\na[5,2]=0,\na[5,3]=-8891 40/4826809,\na[5,4]=1965040/4826809,\na[6,1]=41044746763290003/5173977 9980736512,\na[6,2]=0,\na[6,3]=-57685603608510645/24977824818286592,\n a[6,4]=83145940696701395/53969942910940672,\na[6,5]=537728352095420589 35/87647187287367651328,\na[7,1]=-19737371728514492920457623/384824669 72140721924237312,\na[7,2]=0,\na[7,3]=27466253014073155807766505/14560 933448918110998360064,\na[7,4]=-117670134102158631127838015/1229878843 09611901825434112,\na[7,5]=-19483500956399449871589744537/481311655153 988159050791915520,\na[7,6]=913336658253963141/2343933323194077385,\na [8,1]=2484813954171031615807/2538960546022287823744,\na[8,2]=0,\na[8,3 ]=-229147524517236387495/87335252197326853888,\na[8,4]=130548415875501 708716128185/106468788692037442075798624,\na[8,5]=11103534575830173429 908962617/8186576761928888116508540672,\na[8,6]=-10679027011796052/131 81450824860817,\na[8,7]=2238377869046599644/2564667687663709133,\na[9, 1]=397933/5470080,\na[9,2]=0,\na[9,3]=0,\na[9,4]=5450042155625/3220528 4983872,\na[9,5]=1677324270326783/6406236767145600,\na[9,6]=1792009578 46/1323774725175,\na[9,7]=77461817788426/272279942939565,\na[9,8]=4704 9059/616363020,\n\nb[1]=397933/5470080,\nb[2]=0,\nb[3]=0,\nb[4]=545004 2155625/32205284983872,\nb[5]=1677324270326783/6406236767145600,\nb[6] =179200957846/1323774725175,\nb[7]=77461817788426/272279942939565,\nb[ 8]=47049059/616363020,\n\n`b*`[1]=58931524118239849/837231220422643200 ,\n`b*`[2]=0,\n`b*`[3]=0,\n`b*`[4]=180077138918196243205225/9858455470 89116780242176,\n`b*`[5]=239006278908625056478598819/98051608511643687 3474624000,\n`b*`[6]=4448038142979447869029/28944615297423551638500,\n `b*`[7]=1616804268713313415249069/5953458735623079359484300,\n`b*`[8]= 713536679628437051/13476908236860104400,\n`b*`[9]=1/40\}:" }}}{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 0 488 "subs(ee,matrix([[c[2],a[2,1],``$3],\n [c[3],a[3,1],a[3,2],``$2 ],\n [c[4],seq(a[4,i],i=1..3),``],\n [c[5],seq(a[5,i],i=1..4)],\n [ c[6],seq(a[6,i],i=1..4)],[``$4,a[6,5]],\n [c[7],seq(a[7,i],i=1..4)],[ ``$3,a[7,5],a[7,6]],\n [c[8],seq(a[8,i],i=1..4)],[``$2,seq(a[8,i],i=5 ..7)],\n [c[9],seq(a[9,i],i=1..4)],[``,seq(a[9,i],i=5..8)],\n [``,`_ ___________________________`$4],\n [`b`,seq(b[i],i=1..4)],[``,seq(b[i ],i=5..8)],\n [`b*`,seq(`b*`[i],i=1..4)],[``,seq(`b*`[i],i=5..8)],[`` $4,`b*`[9]]]));" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#K%'matrixG6#747'#\" \"\"\"$#>F(%!GF+F+7'#\"#c\"$N%#!&W2$\"&D5##\"'_.5\"&vI'F+F+7'#\"#G\"$X \"#\"\"(F9\"\"!#\"#@F9F+7'#F.\"$p\"#\"';N_\"(4o#[F<#!'S\"*))FD#\"(S]'> FD7'#\"#P\"#e#\"2.+HjnuW5%\"27lt!)*z(R<&F<#!2X1^3Og&od\"2#f'G=[#y(\\## \"2&R,npSf9$)\"2s1%4\"H%*pR&7'F+F+F+F+#\"5N*e?a4_$Gx`\"5G8lntG(=Zw)7'# \"$@\"\"$e\"#!;BwX?H\\9&G\";7tBC>sS@(pY#[QF<#\";0lw2e:tS,`iYF\";k +O)*46=*[M$4c9#!<:!Qy7J'e@5M,n<\"\"<7TVD=!>h4V)y)H77'F+F+F+#!>PXu*er) \\%*Rc4]$[>\"??b\">z]!f\"))R:b;J\"[#\"3TJ'RDemL8*\"4&QxS>BL$RM#7'F)#\" 72ehJ5__L()#\"<&=G hr3<]veT[08\"w wl=)#!2_gz6q-z1\"\"2<3'[#3X\"=8#\"4W'*fY!pyPQA\"4L\"4Pm(onYc#7'F)#\"'L zR\"(!3qaF=*Qr2!=\"9w@C!y;\"*3Zb%e)*7'F+#\"<>)) fyk0D'3*yi+R#\"<+SiuM(oV;^3;0)*#\"7H!pyWzH9Q![W\"8+&Q;bBuH:Y%*G#\":p! \\_T8LroU!oh\"\":+V[f$zIiN(eM&f#\"3^qVG'zm`8(\"5+W5goB3pZ87'F+F+F+F+#F )\"#SQ(pprint06\"" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 " > " 0 "" {MPLTEXT 1 0 136 "subs(ee,matrix([seq([c[i],seq(a[i,j],j=1..i -1),``$(10-i)],i=2..9),\n[`b`,seq(b[i],i=1..8),``],[`b*`,seq(`b*`[i],i =1..9)]])):\nevalf[8](%);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#K%'matrix G6#7,7,$\")LL3_!#5F(%!GF+F+F+F+F+F+F+7,$\")jN(G\"!\")$!)#fAY\"!\"($\") [*4f\"F2F+F+F+F+F+F+F+7,$\")X.J>F/$\")ieF[!\"*$\"\"!F<$\")fF[9F/F+F+F+ F+F+F+7,$\")&4OJ$F/$\")1g%3\"F/F;$!)m3U=F/$\")a4rSF/F+F+F+F+F+7,$\").J zjF/$\")(=H$zF/F;$!)FZ4BF2$\")sfS:F2$\")n9NhF/F+F+F+F+7,$\")yAewF/$!)e #*G^F/F;$\")vH')=F2$!)+in&*F/$!)6+[SF:$\")$)f'*QF/F+F+F+7,$\"\"\"F<$\" )st'y*F/F;$!)!pPi#F2$\")h;E7F2$\"))4jN\"F2$!)mb,\")F/$\"),vF()F/F+F+7, F[o$\")*>ZF(F:F;F;$\")>G#p\"F/$\")xE=EF/$\")>r`8F/$\")H$\\%GF/$\")eLLw F:F+7,%\"bGFjoF;F;F\\pF^pF`pFbpFdpF+7,%#b*G$\")*e)QqF:F;F;$\")jiE=F/$ \")fbPCF/$\")7uO:F/$\")%Rdr#F/$\")K^%H&F:$\")+++DF:Q(pprint16\"" }}} {PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 146 "RK6_8eqs := [op(RowSumConditions(8,'expanded')),op(OrderCondition s(6,8,'expanded'))]:\n`RK5_9eqs*` := subs(b=`b*`,OrderConditions(5,9,' expanded')):" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 7 "Check: " }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 148 "simp lify(subs(ee,RK6_8eqs)):\nmap(u->`if`(lhs(u)=rhs(u),0,1),%);\nnops(%); \nsimplify(subs(ee,`RK5_9eqs*`)):\nmap(u->`if`(lhs(u)=rhs(u),0,1),%); \nnops(%);" }{TEXT -1 0 "" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#7N\"\"!F$ F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$ F$F$F$F$F$F$F$" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#\"#W" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#73\"\"!F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#\"#<" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }} {PARA 0 "" 0 "" {TEXT -1 86 "Next we set-up stage-order condtions to c heck for stage-orders from 2 to 5 inclusive. " }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 85 "for ct from 2 to 5 do\n so||ct||_8 := StageOrd erConditions(ct,8,'expanded');\nend do:" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 34 "Stages 4 to 8 have stage-order 3. \+ " }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 173 "[seq([seq(expand(subs(e e,so||i||_8[j])),i=2..5)],j=1..6)]:\nmap(proc(L) local i; for i to nop s(L) do if not evalb(L[i]) then break end if end do; i end proc,%):\ns implify(%); " }}{PARA 11 "" 1 "" {XPPMATH 20 "6#7(\"\"#\"\"$F%F%F%F%" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 53 "None o f the principal error conditions are satisfied." }}{EXCHG {PARA 0 "> \+ " 0 "" {MPLTEXT 1 0 123 "RK6_8err_eqs := PrincipalErrorConditions(6,8, 'expanded'):\nexpand(subs(ee,RK6_8err_eqs)):\nmap(u->`if`(lhs(u)=rhs(u ),0,1),%);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#7R\"\"\"F$F$F$F$F$F$F$F$ F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$ F$F$F$F$" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 114 "We can calculate 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 0 131 "errterms6_8 := PrincipalErrorTerms(6,8,'expa nded'):\nevalf(evalf[14](sqrt(add(subs(ee,errterms6_8[i])^2,i=1.. nops (errterms6_8)))));" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#$\"+g:6OH!#8" }} }{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 79 "The 2 no rm of the principal error of the order 5 embedded scheme is as follows ." }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 153 "`errterms5_9*` := subs (b=`b*`,PrincipalErrorTerms(5,9,'expanded')):\nevalf(evalf[14](sqrt(ad d(subs(ee,`errterms5_9*`[i])^2,i=1.. nops(`errterms5_9*`)))));" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#$\"+lc^;A!#8" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}}{PARA 0 "" 0 "" {TEXT -1 52 "#---------------------------------------------- -----" }}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 44 "simultaneous constructio n of the two schemes" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 " " {TEXT -1 119 "We incorporate the stage-order equations to ensure tha t stage 2 has stage-order 2 and stages 3 to 8 have stage-order 3." }} {PARA 0 "" 0 "" {TEXT -1 48 "We also incorporate the simplifying condi tions: " }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "Sum(b[i]*a [i,j],i = j+1 .. 8) = b[j]*(1-c[j]);" "6#/-%$SumG6$*&&%\"bG6#%\"iG\"\" \"&%\"aG6$F+%\"jGF,/F+;,&F0F,F,F,\"\")*&&F)6#F0F,,&F,F,&%\"cG6#F0!\"\" F," }{TEXT -1 1 " " }}{PARA 0 "" 0 "" {TEXT -1 5 "for " }{XPPEDIT 18 0 "j=1" "6#/%\"jG\"\"\"" }{TEXT -1 10 " ( with " }{XPPEDIT 18 0 "c[1 ]=0" "6#/&%\"cG6#\"\"\"\"\"!" }{TEXT -1 5 " ), " }{XPPEDIT 18 0 "j = \+ 6;" "6#/%\"jG\"\"'" }{TEXT -1 7 " and " }{XPPEDIT 18 0 "j=7" "6#/%\" jG\"\"(" }{TEXT -1 1 "." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 78 "The 8 simple order conditions used are given (in abr eviated form) as follows. " }}{PARA 0 "" 0 "" {TEXT -1 78 "These inclu de the 6 quadrature conditions and two additional order conditions." } }{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 178 "SO6 := SimpleOrderConditions(6):\n[seq([i,SO6[i]],i=[1,2,4,8,16 ,24,29,32])]:\nlinalg[augment](linalg[delcols](%,2..2),matrix([[` `]$ (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(\"\"$7%\"\")F)/*&F,F()F2F :F(#F(F57%\"#;F)/*&F,F()F2F5F(#F(\"\"&7%\"#CF)/*(F,F(F2F(-%!G6#*&%\"aG F(-FM6#*&F8F(FPF(F(F(#F(\"#s7%\"#HF)/*(F,F(F2F(-FM6#*&F?F(FPF(F(#F(FI7 %\"#KF)/*&F,F()F2FGF(#F(\"\"'Q)pprint596\"" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 136 "For the embedded order 5 schem e we use a selection of 7 \"simple\" order conditions as given (in abr eviated form) in the following table. " }}{PARA 0 "" 0 "" {TEXT -1 78 "These include the 5 quadrature conditions and two additional order co nditions." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 194 "`SO5*` := subs(b=`b*`,SimpleOrderConditions(5)):\n[s eq([i,`SO5*`[i]],i=[1,2,4,8,12,15,16])]:\nlinalg[augment](linalg[delco ls](%,2..2),matrix([[` `]$(linalg[rowdim](%))]),linalg[delcols](%,1.. 1));" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#K%'matrixG6#7)7%\"\"\"%#~~G/*& %#b*GF(%\"eGF(F(7%\"\"#F)/*&F,F(%\"cGF(#F(F/7%\"\"%F)/*&F,F()F2F/F(#F( \"\"$7%\"\")F)/*&F,F()F2F:F(#F(F57%\"#7F)/*&F,F(-%!G6#*&%\"aGF(-FF6#*& F8F(FIF(F(F(#F(\"#g7%\"#:F)/*&F,F(-FF6#*&F?F(FIF(F(#F(\"#?7%\"#;F)/*&F ,F()F2F5F(#F(\"\"&Q)pprint186\"" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 582 "SO6_8 := SimpleOrderConditi ons(6,8,'expanded'):\nSO_eqs := [op(RowSumConditions(8,'expanded')),op (StageOrderConditions(2,8,'expanded')),\n op(StageOrderC onditions(3,4..8,'expanded'))]:\n`SO5_9*` := subs(b=`b*`,SimpleOrderCo nditions(5,9,'expanded')):\nord_cdns := [seq(SO6_8[i],i=[1,2,4,8,16,24 ,29,32])]:\n`ord_cdns*` := [seq(`SO5_9*`[i],i=[1,2,4,8,12,15,16])]:\ns imp_eqs := [add(b[i]*a[i,1],i=1+1..8)=b[1],seq(add(b[i]*a[i,j],i=j+1.. 8)=b[j]*(1-c[j]),j=[6,7])]:\nwt_eqns := [seq(b[i]=a[9,i],i=1..8)]:\ncd ns := [op(ord_cdns),op(simp_eqs),op(SO_eqs),op(wt_eqns),op(`ord_cdns*` )]:" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 22 "W e specify the nodes: " }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "c[2] = 1/192;" "6#/&%\"cG6#\"\"#*&\"\"\"F)\"$#>!\"\"" }{TEXT -1 3 ", " }{XPPEDIT 18 0 "c[4] = 28/145;" "6#/&%\"cG6#\"\"%*&\"#G\"\"\" \"$X\"!\"\"" }{TEXT -1 3 ", " }{XPPEDIT 18 0 "c[5] = 56/169;" "6#/&% \"cG6#\"\"&*&\"#c\"\"\"\"$p\"!\"\"" }{TEXT -1 3 ", " }{XPPEDIT 18 0 " c[6] = 37/58;" "6#/&%\"cG6#\"\"'*&\"#P\"\"\"\"#e!\"\"" }{TEXT -1 3 ", \+ " }{XPPEDIT 18 0 "c[7] = 121/158;" "6#/&%\"cG6#\"\"(*&\"$@\"\"\"\"\"$ e\"!\"\"" }{TEXT -1 3 ", " }{XPPEDIT 18 0 "c[8]=1" "6#/&%\"cG6#\"\") \"\"\"" }{TEXT -1 2 ", " }}{PARA 0 "" 0 "" {TEXT -1 31 "the zero linki ng coefficients: " }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 " a[4,2]=0" "6#/&%\"aG6$\"\"%\"\"#\"\"!" }{TEXT -1 3 ", " }{XPPEDIT 18 0 "a[5,2]=0" "6#/&%\"aG6$\"\"&\"\"#\"\"!" }{TEXT -1 3 ", " }{XPPEDIT 18 0 "a[6,2]=0" "6#/&%\"aG6$\"\"'\"\"#\"\"!" }{TEXT -1 3 ", " } {XPPEDIT 18 0 "a[7,2]=0" "6#/&%\"aG6$\"\"(\"\"#\"\"!" }{TEXT -1 3 ", \+ " }{XPPEDIT 18 0 "a[8,2]=0" "6#/&%\"aG6$\"\")\"\"#\"\"!" }{TEXT -1 2 " , " }}{PARA 0 "" 0 "" {TEXT -1 22 "and the zero weights: " }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "b[2]=0" "6#/&%\"bG6#\"\"#\" \"!" }{TEXT -1 3 ", " }{XPPEDIT 18 0 "b[3]=0" "6#/&%\"bG6#\"\"$\"\"! " }{TEXT -1 1 "." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT 260 4 "Note" }{TEXT -1 89 ": Calculations relating to the choice of nodes are performed in the following subsection." }}{PARA 0 "" 0 " " {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 125 "The weights of the ord er 6 scheme provide the linking coefficients for the 9th stage of the \+ embedded order 5 scheme so that: " }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "a[9,i]=b[i]" "6#/&%\"aG6$\"\"*%\"iG&%\"bG6#F(" }{TEXT -1 3 ", " }{XPPEDIT 18 0 "i=1" "6#/%\"iG\"\"\"" }{TEXT -1 9 " . . . 8 ." }}{PARA 0 "" 0 "" {TEXT -1 22 "We also specify that " }{XPPEDIT 18 0 "c[9] = 1;" "6#/&%\"cG6#\"\"*\"\"\"" }{TEXT -1 3 ", " }{XPPEDIT 18 0 "`b*`[2] = 0;" "6#/&%#b*G6#\"\"#\"\"!" }{TEXT -1 3 ", " } {XPPEDIT 18 0 "`b*`[3] = 0;" "6#/&%#b*G6#\"\"$\"\"!" }{TEXT -1 7 " an d " }{XPPEDIT 18 0 "`b*`[9] = 1/40;" "6#/&%#b*G6#\"\"**&\"\"\"F)\"#S! \"\"" }{TEXT -1 2 ". " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 37 "We have 44 equations and 44 unknowns." }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 212 "e1 := \{c[2]=1/192,c[4]=28/145,c[5]=56/1 69,c[6]=37/58,c[7]=121/158,c[8]=1,c[9]=1,\n seq(a[i,2]=0,i=4..8) ,b[2]=0,b[3]=0,`b*`[2]=0,`b*`[3]=0,`b*`[9]=1/40\}:\neqns := subs(e1,cd ns):\nnops(%);\nindets(eqns);\nnops(%);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#\"#W" }}{PARA 12 "" 1 "" {XPPMATH 20 "6#FL&F%6#FL&F%6#F4&F%6#FC&F/6$\"\"*FL &F/6$FgoF>&F/6$FgoF1&F/6$FgoF4&F/6$FgoFC&F/6$FgoF'&F/6$FgoF-&F/6$FgoF* &%#b*GFbo&FgpFdo&FgpF&&FgpF,&FgpF)&FgpF`o" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#\"#W" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 22 "infolevel[solve] := 4:" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 69 "e2 := solve(\{op(eqns)\}):\ninfolevel[sol ve] := 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 2050 "e3 := \{`b*`[5] = 23900627890862505647859881 9/980516085116436873474624000, c[8] = 1, c[9] = 1, a[9,2] = 0, a[9,3] \+ = 0, c[2] = 1/192, `b*`[4] = 180077138918196243205225/9858455470891167 80242176, b[5] = 1677324270326783/6406236767145600, c[4] = 28/145, c[5 ] = 56/169, b[2] = 0, b[3] = 0, `b*`[2] = 0, `b*`[3] = 0, a[4,2] = 0, \+ a[5,2] = 0, a[6,2] = 0, a[7,2] = 0, a[8,2] = 0, a[7,4] = -117670134102 158631127838015/122987884309611901825434112, a[9,1] = 397933/5470080, \+ a[7,6] = 913336658253963141/2343933323194077385, b[7] = 77461817788426 /272279942939565, a[9,8] = 47049059/616363020, a[6,3] = -5768560360851 0645/24977824818286592, a[8,7] = 2238377869046599644/25646676876637091 33, b[8] = 47049059/616363020, a[8,6] = -10679027011796052/13181450824 860817, a[8,4] = 130548415875501708716128185/1064687886920374420757986 24, c[7] = 121/158, a[7,3] = 27466253014073155807766505/14560933448918 110998360064, a[5,4] = 1965040/4826809, `b*`[9] = 1/40, b[1] = 397933/ 5470080, a[8,1] = 2484813954171031615807/2538960546022287823744, a[7,5 ] = -19483500956399449871589744537/481311655153988159050791915520, a[5 ,3] = -889140/4826809, a[6,4] = 83145940696701395/53969942910940672, a [4,3] = 21/145, a[3,2] = 100352/63075, c[3] = 56/435, a[2,1] = 1/192, \+ `b*`[1] = 58931524118239849/837231220422643200, a[5,1] = 523516/482680 9, a[8,3] = -229147524517236387495/87335252197326853888, a[9,5] = 1677 324270326783/6406236767145600, b[4] = 5450042155625/32205284983872, a[ 6,5] = 53772835209542058935/87647187287367651328, `b*`[6] = 4448038142 979447869029/28944615297423551638500, a[6,1] = 41044746763290003/51739 779980736512, a[9,4] = 5450042155625/32205284983872, b[6] = 1792009578 46/1323774725175, c[6] = 37/58, a[9,6] = 179200957846/1323774725175, a [9,7] = 77461817788426/272279942939565, a[8,5] = 111035345758301734299 08962617/8186576761928888116508540672, a[7,1] = -197373717285144929204 57623/38482466972140721924237312, `b*`[8] = 713536679628437051/1347690 8236860104400, `b*`[7] = 1616804268713313415249069/5953458735623079359 484300, a[4,1] = 7/145, a[3,1] = -30744/21025\}:" }{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 0 488 "subs(e3,matrix([[c[2],a[2,1],``$3],\n [c[3],a[3,1], a[3,2],``$2],\n [c[4],seq(a[4,i],i=1..3),``],\n [c[5],seq(a[5,i],i=1 ..4)],\n [c[6],seq(a[6,i],i=1..4)],[``$4,a[6,5]],\n [c[7],seq(a[7,i] ,i=1..4)],[``$3,a[7,5],a[7,6]],\n [c[8],seq(a[8,i],i=1..4)],[``$2,seq (a[8,i],i=5..7)],\n [c[9],seq(a[9,i],i=1..4)],[``,seq(a[9,i],i=5..8)] ,\n [``,`____________________________`$4],\n [`b`,seq(b[i],i=1..4)], [``,seq(b[i],i=5..8)],\n [`b*`,seq(`b*`[i],i=1..4)],[``,seq(`b*`[i],i =5..8)],[``$4,`b*`[9]]]));" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#K%'matri xG6#747'#\"\"\"\"$#>F(%!GF+F+7'#\"#c\"$N%#!&W2$\"&D5##\"'_.5\"&vI'F+F+ 7'#\"#G\"$X\"#\"\"(F9\"\"!#\"#@F9F+7'#F.\"$p\"#\"';N_\"(4o#[F<#!'S\"*) )FD#\"(S]'>FD7'#\"#P\"#e#\"2.+HjnuW5%\"27lt!)*z(R<&F<#!2X1^3Og&od\"2#f 'G=[#y(\\##\"2&R,npSf9$)\"2s1%4\"H%*pR&7'F+F+F+F+#\"5N*e?a4_$Gx`\"5G8l ntG(=Zw)7'#\"$@\"\"$e\"#!;BwX?H\\9&G\";7tBC>sS@(pY#[QF<#\";0lw2e: tS,`iYF\";k+O)*46=*[M$4c9#!<:!Qy7J'e@5M,n<\"\"<7TVD=!>h4V)y)H77'F+F+F+ #!>PXu*er)\\%*Rc4]$[>\"??b\">z]!f\"))R:b;J\"[#\"3TJ'RDemL8*\"4&QxS>BL$ RM#7'F)#\"72ehJ5__ L()#\"<&=Ghr3<]veT[08\"wwl=)#!2_gz6q-z1\"\"2<3'[#3X\"=8#\"4W'*fY!pyPQA\"4L\"4Pm(onYc #7'F)#\"'LzR\"(!3qaF=*Qr2!=\"9w@C!y;\"*3Zb%e)*7 'F+#\"<>))fyk0D'3*yi+R#\"<+SiuM(oV;^3;0)*#\"7H!pyWzH9Q![W\"8+&Q;bBuH:Y %*G#\":p!\\_T8LroU!oh\"\":+V[f$zIiN(eM&f#\"3^qVG'zm`8(\"5+W5goB3pZ87'F +F+F+F+#F)\"#SQ(pprint86\"" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 136 "subs(e3,matrix([seq([c[i],seq(a[i, j],j=1..i-1),``$(10-i)],i=2..9),\n[`b`,seq(b[i],i=1..8),``],[`b*`,seq( `b*`[i],i=1..9)]])):\nevalf[8](%);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6# K%'matrixG6#7,7,$\")LL3_!#5F(%!GF+F+F+F+F+F+F+7,$\")jN(G\"!\")$!)#fAY \"!\"($\")[*4f\"F2F+F+F+F+F+F+F+7,$\")X.J>F/$\")ieF[!\"*$\"\"!F<$\")fF [9F/F+F+F+F+F+F+7,$\")&4OJ$F/$\")1g%3\"F/F;$!)m3U=F/$\")a4rSF/F+F+F+F+ F+7,$\").JzjF/$\")(=H$zF/F;$!)FZ4BF2$\")sfS:F2$\")n9NhF/F+F+F+F+7,$\") yAewF/$!)e#*G^F/F;$\")vH')=F2$!)+in&*F/$!)6+[SF:$\")$)f'*QF/F+F+F+7,$ \"\"\"F<$\")st'y*F/F;$!)!pPi#F2$\")h;E7F2$\"))4jN\"F2$!)mb,\")F/$\"),v F()F/F+F+7,F[o$\")*>ZF(F:F;F;$\")>G#p\"F/$\")xE=EF/$\")>r`8F/$\")H$\\% GF/$\")eLLwF:F+7,%\"bGFjoF;F;F\\pF^pF`pFbpFdpF+7,%#b*G$\")*e)QqF:F;F;$ \")jiE=F/$\")fbPCF/$\")7uO:F/$\")%Rdr#F/$\")K^%H&F:$\")+++DF:Q)pprint1 06\"" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 146 "RK6_8eqs := [op(RowSumConditions(8,'expanded')),op(O rderConditions(6,8,'expanded'))]:\n`RK5_9eqs*` := subs(b=`b*`,OrderCon ditions(5,9,'expanded')):" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 7 "Check: " }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 148 "simplify(subs(e3,RK6_8eqs)):\nmap(u->`if`(lhs(u)=rhs(u),0,1),%); \nnops(%);\nsimplify(subs(e3,`RK5_9eqs*`)):\nmap(u->`if`(lhs(u)=rhs(u) ,0,1),%);\nnops(%);" }{TEXT -1 0 "" }}{PARA 11 "" 1 "" {XPPMATH 20 "6# 7N\"\"!F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$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#\"#W" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#73\"\"!F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F $" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#\"#<" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 21 "determining the nodes" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 22 "eG .. general solution" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 71947 "eG := \+ \{c[8] = 1, b[3] = 0, b[2] = 0, c[9] = 1, b[5] = -1/60*(10*c[6]*c[7]*c [4]-5*c[6]*c[4]-5*c[7]*c[6]+3*c[6]-5*c[7]*c[4]+3*c[4]-2+3*c[7])/(-c[4] +c[5])/c[5]/(-c[5]^3+c[5]^2-c[7]*c[5]+c[7]*c[5]^2-c[6]*c[5]+c[6]*c[5]^ 2+c[7]*c[6]-c[6]*c[7]*c[5]), a[5,2] = 0, a[4,2] = 0, `b*`[3] = 0, `b*` [2] = 0, a[8,2] = 0, a[7,2] = 0, a[6,2] = 0, a[6,5] = 1/2*c[6]*(30*c[6 ]^2*c[5]^2*c[4]^2-2*c[6]^2*c[4]^2+c[6]^3*c[4]-12*c[6]^2*c[4]*c[5]^2-3* c[5]^2*c[6]*c[4]+c[5]*c[6]*c[4]^2+c[6]*c[4]^3-12*c[6]^2*c[4]^2*c[5]-30 *c[5]*c[6]^3*c[4]^2-20*c[5]*c[6]^2*c[4]^4+30*c[6]^2*c[5]*c[4]^3-c[5]*c [4]^3+20*c[4]^3*c[5]*c[6]^3-20*c[6]^2*c[5]^2*c[4]^3+c[5]^2*c[4]^2-30*c [5]^2*c[6]*c[4]^3+12*c[5]^2*c[6]*c[4]^2+2*c[6]^2*c[4]*c[5]+20*c[5]^2*c [6]*c[4]^4+2*c[6]^2*c[5]^2-2*c[6]^3*c[5]+12*c[6]^3*c[4]*c[5])/c[5]/(c[ 4]^3-2*c[5]*c[4]^2+30*c[5]^2*c[4]^3-6*c[5]*c[4]^3-10*c[4]^4*c[5]^2-c[5 ]^3+10*c[5]^4*c[4]^2-30*c[5]^3*c[4]^2+6*c[5]^3*c[4]+2*c[5]^2*c[4]), b[ 1] = 1/60*(30*c[5]*c[6]*c[7]*c[4]-10*c[5]*c[6]*c[4]-10*c[6]*c[7]*c[5]+ 5*c[6]*c[5]-10*c[6]*c[7]*c[4]+5*c[6]*c[4]+5*c[7]*c[6]-3*c[6]-10*c[5]*c [7]*c[4]+5*c[5]*c[4]+5*c[7]*c[5]-3*c[5]+5*c[7]*c[4]-3*c[4]+2-3*c[7])/c [5]/c[6]/c[7]/c[4], a[6,1] = -1/4*c[6]*(60*c[5]^2*c[6]*c[4]^4-90*c[5]^ 2*c[6]^3*c[4]^2-60*c[5]*c[6]^2*c[4]^4-2*c[6]^3*c[4]^2+50*c[5]^4*c[4]^3 *c[6]+40*c[6]^2*c[4]^5*c[5]-40*c[4]^5*c[5]^3+36*c[4]*c[5]^2*c[6]^3-20* c[5]^4*c[6]^2*c[4]^2+8*c[6]^2*c[5]^3+4*c[6]^2*c[4]^3-24*c[5]*c[6]^3*c[ 4]^2-8*c[6]^2*c[4]^2*c[5]+24*c[6]^2*c[5]*c[4]^3+60*c[4]^3*c[5]*c[6]^3- 4*c[6]*c[5]*c[4]^3-14*c[5]^3*c[6]*c[4]+7*c[5]^3*c[4]^2+6*c[5]^2*c[6]*c [4]^3+14*c[5]^2*c[6]*c[4]^2+66*c[5]^3*c[6]*c[4]^2-80*c[5]^3*c[6]^2*c[4 ]^3+30*c[6]^2*c[5]^2*c[4]^3-24*c[6]^2*c[5]^2*c[4]^2-240*c[6]*c[5]^3*c[ 4]^3-48*c[6]^2*c[5]^3*c[4]+150*c[6]^2*c[5]^3*c[4]^2-24*c[4]^4*c[5]^2-9 *c[5]^2*c[4]^3+6*c[4]^4*c[5]-40*c[5]^4*c[4]^4-20*c[6]^2*c[4]^4*c[5]^2- 6*c[5]^2*c[6]^3-2*c[6]*c[4]^4-24*c[5]^3*c[4]^3+60*c[6]^3*c[4]^3*c[5]^2 +7*c[6]^3*c[4]*c[5]-40*c[5]^2*c[6]*c[4]^5-40*c[6]^3*c[4]^4*c[5]+110*c[ 4]^4*c[6]*c[5]^3+120*c[5]^3*c[4]^4)/c[5]/c[4]^2/(10*c[5]^3*c[4]^2+10*c [5]^2*c[4]^3-30*c[5]^2*c[4]^2+6*c[5]^2*c[4]-c[5]^2+6*c[5]*c[4]^2+c[5]* c[4]-c[4]^2), b[4] = 1/60*(-2+3*c[5]-5*c[7]*c[6]+3*c[7]+3*c[6]+10*c[6] *c[7]*c[5]-5*c[6]*c[5]-5*c[7]*c[5])/c[4]/(-c[6]*c[4]^3-c[4]^3+c[4]^4+c [7]*c[4]^2+c[6]*c[4]^2-c[6]*c[7]*c[4]+c[6]*c[7]*c[4]^2-c[5]*c[4]^3+c[5 ]*c[4]^2-c[7]*c[4]^3-c[5]*c[7]*c[4]+c[5]*c[7]*c[4]^2-c[5]*c[6]*c[4]+c[ 5]*c[6]*c[4]^2+c[6]*c[7]*c[5]-c[5]*c[6]*c[7]*c[4]), b[6] = 1/60*(10*c[ 5]*c[7]*c[4]-5*c[5]*c[4]-5*c[7]*c[5]+3*c[5]-5*c[7]*c[4]+3*c[4]-2+3*c[7 ])/(-c[4]+c[6])/(-c[5]+c[6])/c[6]/(c[6]^2-c[6]+c[7]-c[7]*c[6]), `b*`[9 ] = 1/10*(50*c[5]^2*c[6]*c[4]^2-40*c[5]^2*c[6]*c[4]+10*c[6]*c[5]^2-40* c[5]*c[6]*c[4]^2+35*c[5]*c[6]*c[4]-10*c[6]*c[5]+10*c[6]*c[4]^2-10*c[6] *c[4]+3*c[6]-5*c[5]^2*c[4]-5*c[5]*c[4]^2+10*c[5]*c[4]-c[5]-c[4])/(3*c[ 6]-28*c[5]*c[6]*c[4]^2-28*c[5]^2*c[6]*c[4]+8*c[6]*c[5]^2-9*c[6]*c[4]-9 *c[6]*c[5]+28*c[5]*c[6]*c[4]+30*c[5]^2*c[6]*c[4]^2+8*c[6]*c[4]^2-4*c[5 ]^2*c[4]-4*c[5]*c[4]^2+9*c[5]*c[4]-c[5]-c[4]), a[7,5] = -1/2*c[7]*(-17 *c[5]^2*c[6]*c[4]^4+5*c[6]*c[5]*c[4]^4+42*c[4]^2*c[5]^2*c[7]^3+17*c[7] ^3*c[5]*c[4]^3-80*c[7]^3*c[5]^2*c[4]^3+c[7]*c[5]*c[4]^4+80*c[5]^2*c[4] ^4*c[7]^2-60*c[4]^5*c[6]^2*c[5]^2+20*c[4]^3*c[7]^3*c[5]^3-20*c[7]^2*c[ 5]^2*c[4]^5+40*c[4]^5*c[6]*c[5]^3-5*c[5]*c[6]^2*c[4]^4+c[7]^3*c[4]^2-c [7]^2*c[4]^3-80*c[5]^3*c[4]^4*c[7]+20*c[7]^3*c[5]^2*c[4]^4-200*c[6]^2* c[4]^5*c[7]*c[5]^3-290*c[5]^4*c[4]^2*c[7]^2*c[6]+20*c[6]^2*c[7]*c[5]^4 *c[4]^2+40*c[6]*c[5]^2*c[4]^4*c[7]-140*c[5]^4*c[4]^3*c[6]-100*c[5]^4*c [4]^5*c[6]+190*c[5]^4*c[4]^4*c[6]-40*c[7]^2*c[4]^4*c[5]^3+20*c[5]^4*c[ 4]^4*c[7]+2*c[4]^4*c[7]^2-20*c[7]^2*c[5]^4*c[4]^3-4*c[6]^2*c[7]^2*c[5] -8*c[7]*c[6]^2*c[5]^3-3*c[7]*c[6]*c[5]^3+2*c[7]^2*c[6]*c[5]^3+60*c[7]* c[6]^2*c[5]*c[4]^5-40*c[6]*c[5]^2*c[4]^5*c[7]+17*c[7]*c[4]^4*c[6]*c[5] +80*c[7]^2*c[5]^3*c[4]^3+5*c[7]*c[4]^4*c[6]^2+c[5]^2*c[7]^3-260*c[6]^2 *c[7]*c[5]^2*c[4]^3-12*c[6]^2*c[7]^2*c[5]^2*c[4]+20*c[6]^2*c[7]^2*c[5] ^2*c[4]^2-40*c[6]*c[7]*c[5]^3*c[4]^3-41*c[6]^2*c[7]^3*c[4]*c[5]+20*c[7 ]^2*c[6]*c[5]*c[4]^4+3*c[7]*c[6]*c[4]^3-2*c[6]^2*c[7]*c[4]^2-3*c[6]*c[ 7]^2*c[4]^2+4*c[7]^2*c[6]*c[4]^3+6*c[5]*c[7]^2*c[4]^3+4*c[6]*c[5]*c[7] ^2*c[4]^2+c[7]^2*c[5]*c[6]*c[4]-11*c[5]*c[7]*c[6]*c[4]^3+2*c[6]^2*c[4] ^2*c[5]-c[7]^3*c[4]*c[5]+17*c[5]^2*c[6]*c[7]*c[4]^2-200*c[5]^4*c[7]^2* c[4]^4*c[6]-50*c[4]^5*c[6]*c[7]*c[5]^3-150*c[5]^2*c[6]*c[7]^3*c[4]^4-c [6]^2*c[5]*c[4]^3+2*c[7]^3*c[4]^2*c[6]*c[5]+2*c[6]^2*c[7]*c[4]*c[5]+10 8*c[7]*c[6]^2*c[5]*c[4]^3+4*c[6]^2*c[7]*c[5]^2+2*c[6]*c[5]*c[7]*c[4]^2 -10*c[6]^2*c[7]^3*c[5]^2+6*c[5]^4*c[7]*c[6]-8*c[6]*c[7]^3*c[5]^2+3*c[6 ]*c[7]^2*c[5]^2+2*c[6]^2*c[7]^2*c[5]^2-27*c[6]^2*c[4]^2*c[7]*c[5]-62*c [6]^2*c[7]^2*c[4]^2*c[5]+46*c[6]*c[5]^4*c[4]^2+20*c[4]^5*c[7]*c[5]^3+2 *c[7]^3*c[4]*c[6]*c[5]-27*c[6]^2*c[5]^2*c[7]*c[4]+120*c[6]^2*c[5]^2*c[ 7]*c[4]^2-35*c[6]*c[5]^2*c[7]*c[4]^3-37*c[6]*c[5]^3*c[7]*c[4]^2+53*c[6 ]^2*c[5]^3*c[7]*c[4]-54*c[7]^2*c[5]^2*c[4]^3+2*c[7]^2*c[5]*c[4]^2-c[7] *c[5]*c[4]^3-3*c[6]*c[5]*c[4]^3-2*c[4]^3*c[7]^3+63*c[7]^3*c[5]^2*c[6]* c[4]-16*c[7]^2*c[5]^2*c[6]*c[4]-5*c[6]*c[7]*c[4]^4+2*c[6]^2*c[7]^2*c[4 ]^2+2*c[6]^2*c[7]^2*c[4]-10*c[5]^4*c[7]^2*c[6]+3*c[5]^3*c[6]*c[4]+3*c[ 5]^3*c[7]*c[4]+20*c[5]^3*c[6]*c[7]*c[4]-2*c[5]^3*c[4]^2+22*c[6]^2*c[7] ^2*c[4]*c[5]+7*c[5]^2*c[6]*c[4]^3-c[5]^2*c[7]*c[4]^2+c[5]^2*c[6]*c[4]^ 2-4*c[5]^2*c[7]*c[4]^3-8*c[5]^3*c[7]*c[4]^2-22*c[5]^3*c[6]*c[4]^2+240* c[5]^3*c[6]^2*c[4]^3+100*c[4]^3*c[7]^2*c[5]^2*c[6]^2+150*c[4]^5*c[5]^3 *c[6]^2-3*c[7]*c[5]^4*c[4]+5*c[7]^2*c[5]^4*c[4]+12*c[5]^2*c[4]^4*c[7]- 98*c[6]^2*c[5]^2*c[4]^3+25*c[6]^2*c[5]^2*c[4]^2+54*c[6]*c[5]^3*c[4]^3+ 8*c[6]^2*c[5]^3*c[4]-63*c[6]^2*c[5]^3*c[4]^2+42*c[5]^3*c[7]*c[4]^3+60* c[5]^4*c[7]*c[4]^2*c[6]+200*c[5]^3*c[7]^3*c[4]^4*c[6]-4*c[6]^2*c[4]*c[ 5]^2+c[7]*c[6]^2*c[4]^3+200*c[7]^2*c[5]^2*c[6]*c[4]^3-240*c[7]^3*c[5]^ 2*c[6]*c[4]^2+180*c[7]^2*c[5]^3*c[6]*c[4]^2+3*c[5]^4*c[4]^2-80*c[5]^3* c[7]^3*c[6]*c[4]+c[7]^3*c[4]^2*c[6]-7*c[7]^3*c[4]^2*c[5]+20*c[4]^3*c[7 ]^2*c[5]*c[6]^2-19*c[7]^2*c[6]*c[5]*c[4]^3-11*c[7]^2*c[5]^2*c[6]*c[4]^ 2-3*c[4]^4*c[5]^2-27*c[7]^2*c[5]^3*c[6]*c[4]-480*c[7]^2*c[5]^3*c[6]*c[ 4]^3-380*c[5]^3*c[6]^2*c[4]^4-5*c[5]^4*c[4]^3-10*c[6]^2*c[4]^3*c[7]^2+ 130*c[5]^3*c[6]^2*c[7]*c[4]^3-5*c[7]^3*c[5]^2*c[4]-2*c[7]^2*c[5]^2*c[4 ]-4*c[5]^2*c[6]*c[7]*c[4]-160*c[6]^2*c[7]*c[5]^3*c[4]^2-80*c[6]^2*c[7] ^2*c[4]*c[5]^3+2*c[5]^2*c[4]^3+12*c[7]^2*c[5]^2*c[4]^2+8*c[7]^2*c[5]^3 *c[4]+400*c[7]^3*c[5]^2*c[6]^2*c[4]^3-17*c[7]^2*c[4]^4*c[5]+250*c[7]^2 *c[5]^3*c[6]^2*c[4]^2-100*c[6]^2*c[4]^5*c[5]*c[7]^2-200*c[7]^2*c[4]^5* c[6]*c[5]^3+150*c[5]^2*c[6]*c[4]^5*c[7]^2+380*c[7]^3*c[5]^2*c[4]^3*c[6 ]-20*c[7]^3*c[6]*c[5]*c[4]^3-210*c[7]^3*c[6]^2*c[5]*c[4]^3-6*c[5]^4*c[ 4]*c[6]+80*c[5]^4*c[7]^2*c[4]*c[6]+10*c[6]*c[7]^3*c[5]^3-36*c[5]^4*c[4 ]*c[7]*c[6]-50*c[4]^5*c[6]^2*c[7]*c[5]^2+200*c[4]^5*c[6]^2*c[7]^2*c[5] ^2+160*c[6]^2*c[4]^4*c[5]^2-450*c[7]^3*c[5]^3*c[4]^3*c[6]+5*c[6]^2*c[7 ]^3*c[4]^2-c[5]^3*c[7]^2+6*c[6]^2*c[7]^3*c[5]-160*c[7]*c[6]^2*c[5]*c[4 ]^4+200*c[5]^3*c[4]^4*c[6]^2*c[7]-350*c[5]^3*c[4]^3*c[6]^2*c[7]^2-350* c[5]^4*c[4]^4*c[6]*c[7]+230*c[5]^2*c[4]^4*c[7]*c[6]^2+150*c[5]^3*c[4]^ 4*c[6]*c[7]+100*c[5]^4*c[4]^3*c[6]*c[7]-3*c[6]^2*c[7]^3*c[4]-42*c[7]^2 *c[5]^3*c[4]^2+50*c[5]^4*c[4]^4*c[6]^2-20*c[5]^4*c[4]^3*c[6]^2+500*c[5 ]^3*c[4]^4*c[7]^2*c[6]+450*c[5]^4*c[4]^3*c[7]^2*c[6]-340*c[5]^2*c[4]^4 *c[7]^2*c[6]-50*c[5]^4*c[4]^3*c[6]^2*c[7]+150*c[7]^2*c[4]^4*c[6]^2*c[5 ]+200*c[7]^2*c[4]^4*c[5]^3*c[6]^2-350*c[7]^2*c[4]^4*c[6]^2*c[5]^2+10*c [7]^2*c[6]^2*c[5]^3-60*c[4]^4*c[6]*c[5]^3+100*c[7]^3*c[6]^2*c[5]*c[4]^ 4+200*c[5]^4*c[4]^5*c[6]*c[7]-200*c[7]^3*c[4]^4*c[6]^2*c[5]^2+5*c[5]^3 *c[4]^4+80*c[6]^2*c[7]^3*c[5]^2*c[4]-270*c[6]^2*c[7]^3*c[5]^2*c[4]^2+2 90*c[5]^3*c[7]^3*c[4]^2*c[6]+140*c[6]^2*c[5]*c[7]^3*c[4]^2-5*c[7]^3*c[ 4]*c[5]^3)/c[5]/(100*c[5]^2*c[6]*c[4]^4-18*c[6]*c[5]*c[4]^4+4*c[5]^3*c [4]+50*c[4]^5*c[6]^2*c[5]^2+40*c[5]*c[6]^2*c[4]^4+3*c[5]^5-5*c[4]^4*c[ 6]^2+180*c[5]^4*c[4]^3*c[6]+100*c[5]^4*c[4]^5*c[6]-350*c[5]^4*c[4]^4*c [6]+140*c[5]^5*c[4]^2-180*c[5]^5*c[4]^3-2*c[5]^4+30*c[4]^5*c[5]^3+300* c[5]^5*c[4]^3*c[6]+2*c[6]*c[5]^3-50*c[5]^5*c[4]^2*c[6]^2-40*c[5]^4*c[6 ]^2*c[4]+240*c[5]^4*c[6]^2*c[4]^2-3*c[6]^2*c[5]^3+3*c[6]^2*c[4]^3+5*c[ 5]^4*c[6]^2-2*c[6]*c[4]^3+2*c[5]*c[4]^3+4*c[5]*c[6]*c[4]^2-6*c[6]^2*c[ 4]^2*c[5]-50*c[5]^4*c[4]^5-13*c[6]^2*c[5]*c[4]^3-40*c[6]*c[5]^4*c[4]^2 +6*c[6]*c[5]*c[4]^3-15*c[5]^3*c[6]*c[4]-55*c[5]^2*c[6]*c[4]^3+6*c[5]^2 *c[6]*c[4]^2+68*c[5]^3*c[6]*c[4]^2-100*c[4]^5*c[5]^3*c[6]^2+100*c[6]^2 *c[5]^2*c[4]^3-100*c[6]*c[5]^3*c[4]^3+13*c[6]^2*c[5]^3*c[4]-100*c[6]^2 *c[5]^3*c[4]^2-4*c[5]^2*c[4]^2+6*c[6]^2*c[4]*c[5]^2-68*c[5]^4*c[4]^2+2 3*c[4]^4*c[5]^2+350*c[5]^3*c[6]^2*c[4]^4-4*c[5]^2*c[6]*c[4]-9*c[5]^2*c [4]^3-3*c[4]^4*c[5]-5*c[5]^5*c[6]-30*c[5]^6*c[4]^2+50*c[5]^6*c[6]*c[4] ^2+180*c[5]^4*c[4]^4+10*c[5]^4*c[4]*c[6]-240*c[6]^2*c[4]^4*c[5]^2+3*c[ 6]*c[4]^4-100*c[5]^6*c[4]^3*c[6]+68*c[5]^3*c[4]^3-23*c[5]^5*c[4]-210*c [5]^5*c[6]*c[4]^2-350*c[5]^4*c[4]^3*c[6]^2-30*c[5]^2*c[6]*c[4]^5+50*c[ 5]^6*c[4]^3+60*c[4]^4*c[6]*c[5]^3+100*c[5]^5*c[6]^2*c[4]^3+9*c[5]^4*c[ 4]-140*c[5]^3*c[4]^4+40*c[5]^5*c[4]*c[6]), `b*`[6] = -1/60*(-30*c[5]^3 *c[4]+103*c[5]*c[7]*c[4]-18*c[6]+6*c[5]+6*c[4]+27*c[7]*c[6]-9*c[5]^2-9 *c[4]^2-100*c[7]*c[6]*c[5]^3+522*c[5]*c[6]*c[7]*c[4]-111*c[6]*c[5]^2+7 5*c[6]*c[5]+60*c[6]*c[5]^3-9*c[7]*c[5]+600*c[6]*c[7]*c[5]^3*c[4]^3+14* c[7]*c[5]^2-100*c[7]*c[6]*c[4]^3+14*c[7]*c[4]^2-9*c[7]*c[4]-117*c[6]*c [7]*c[5]-111*c[6]*c[4]^2+60*c[6]*c[4]^3-117*c[6]*c[7]*c[4]+180*c[6]*c[ 7]*c[4]^2-30*c[5]*c[4]^3-174*c[5]*c[7]*c[4]^2+486*c[5]*c[6]*c[4]^2+107 *c[5]*c[4]^2-318*c[5]*c[6]*c[4]-66*c[5]*c[4]+75*c[6]*c[4]+500*c[5]*c[7 ]*c[6]*c[4]^3+1420*c[5]^2*c[6]*c[7]*c[4]^2-840*c[6]*c[5]*c[7]*c[4]^2+1 80*c[6]*c[7]*c[5]^2-900*c[6]*c[5]^2*c[7]*c[4]^3-900*c[6]*c[5]^3*c[7]*c [4]^2+50*c[7]*c[5]*c[4]^3-280*c[6]*c[5]*c[4]^3-280*c[5]^3*c[6]*c[4]+50 *c[5]^3*c[7]*c[4]+107*c[5]^2*c[4]+500*c[5]^3*c[6]*c[7]*c[4]+50*c[5]^3* c[4]^2+470*c[5]^2*c[6]*c[4]^3+310*c[5]^2*c[7]*c[4]^2-770*c[5]^2*c[6]*c [4]^2-90*c[5]^2*c[7]*c[4]^3-90*c[5]^3*c[7]*c[4]^2+470*c[5]^3*c[6]*c[4] ^2-300*c[6]*c[5]^3*c[4]^3-180*c[5]^2*c[4]^2+486*c[5]^2*c[6]*c[4]-174*c [5]^2*c[7]*c[4]-840*c[5]^2*c[6]*c[7]*c[4]+50*c[5]^2*c[4]^3)/(-9*c[5]*c [7]*c[4]-3*c[7]*c[6]+3*c[6]^2-28*c[5]*c[6]*c[7]*c[4]-c[6]*c[5]+c[7]*c[ 5]+8*c[6]^2*c[4]^2+c[7]*c[4]+9*c[6]*c[7]*c[5]+9*c[6]*c[7]*c[4]-8*c[6]* c[7]*c[4]^2+4*c[5]*c[7]*c[4]^2-4*c[5]*c[6]*c[4]^2+9*c[5]*c[6]*c[4]-c[6 ]*c[4]-28*c[6]^2*c[4]^2*c[5]+28*c[6]^2*c[4]*c[5]-30*c[5]^2*c[6]*c[7]*c [4]^2+28*c[6]*c[5]*c[7]*c[4]^2-8*c[6]*c[7]*c[5]^2+8*c[6]^2*c[5]^2-9*c[ 6]^2*c[5]+30*c[6]^2*c[5]^2*c[4]^2-28*c[6]^2*c[4]*c[5]^2-9*c[6]^2*c[4]- 4*c[5]^2*c[6]*c[4]+4*c[5]^2*c[7]*c[4]+28*c[5]^2*c[6]*c[7]*c[4])/(c[5]* c[4]-c[6]*c[4]-c[6]*c[5]+c[6]^2)/c[6], a[8,7] = (10*c[5]*c[6]*c[4]-5*c [6]*c[5]-5*c[6]*c[4]+3*c[6]-5*c[5]*c[4]+3*c[5]+3*c[4]-2)/c[7]/(c[6]-c[ 7])/(c[5]-c[7])/(c[4]-c[7])/(-20*c[5]*c[6]*c[4]+30*c[5]*c[6]*c[7]*c[4] +15*c[6]*c[5]-20*c[6]*c[7]*c[5]+15*c[6]*c[4]-20*c[6]*c[7]*c[4]-12*c[6] +15*c[7]*c[6]+15*c[5]*c[4]-20*c[5]*c[7]*c[4]-12*c[5]+15*c[7]*c[5]-12*c [4]+15*c[7]*c[4]+10-12*c[7])*(c[4]-1)*(-1+c[5])*(-1+c[6])*(-1+c[7]), a [6,3] = 3/4*1/c[4]^2*c[6]*(60*c[4]^3*c[5]*c[6]^3-90*c[5]*c[6]^3*c[4]^2 +36*c[6]^3*c[4]*c[5]-6*c[6]^3*c[5]+3*c[6]^3*c[4]-20*c[6]^2*c[5]^3*c[4] ^2-80*c[6]^2*c[5]^2*c[4]^3+150*c[6]^2*c[5]^2*c[4]^2-48*c[6]^2*c[4]*c[5 ]^2+8*c[6]^2*c[5]^2-60*c[5]*c[6]^2*c[4]^4+90*c[6]^2*c[5]*c[4]^3-48*c[6 ]^2*c[4]^2*c[5]+4*c[6]^2*c[4]*c[5]-4*c[6]^2*c[4]^2+30*c[6]*c[5]^3*c[4] ^3+90*c[5]^2*c[6]*c[4]^4-180*c[5]^2*c[6]*c[4]^3+54*c[5]^2*c[6]*c[4]^2- 12*c[5]^2*c[6]*c[4]+18*c[6]*c[5]*c[4]^3+6*c[5]*c[6]*c[4]^2+3*c[5]^2*c[ 4]^2-3*c[5]*c[4]^3)/(10*c[5]^3*c[4]^2+10*c[5]^2*c[4]^3-30*c[5]^2*c[4]^ 2+6*c[5]^2*c[4]-c[5]^2+6*c[5]*c[4]^2+c[5]*c[4]-c[4]^2), b[7] = -1/60*( 10*c[5]*c[6]*c[4]-5*c[6]*c[5]-5*c[6]*c[4]+3*c[6]-5*c[5]*c[4]+3*c[5]+3* c[4]-2)/(c[4]-c[7])/(c[5]-c[7])/(c[6]-c[7])/c[7]/(-1+c[7]), a[7,1] = 1 /4*c[7]*(-51*c[5]^2*c[6]*c[4]^4+14*c[6]*c[5]*c[4]^4+12*c[4]^2*c[5]^2*c [7]^3+2700*c[5]^4*c[4]^4*c[6]^2*c[7]+12*c[7]^3*c[5]*c[4]^3-98*c[7]^3*c [5]^2*c[4]^3-2*c[7]*c[5]*c[4]^4+132*c[5]^2*c[4]^4*c[7]^2-480*c[4]^5*c[ 6]^2*c[5]^2+200*c[4]^3*c[7]^3*c[5]^3-200*c[7]^2*c[5]^2*c[4]^5-760*c[4] ^5*c[6]*c[5]^3-10*c[5]*c[6]^2*c[4]^4+200*c[5]^5*c[4]^3*c[7]^2*c[6]^2-4 00*c[5]^2*c[4]^6*c[6]^2*c[7]^2+4*c[4]^5*c[5]*c[7]-34*c[5]^2*c[4]^5*c[7 ]+400*c[5]^3*c[4]^6*c[6]^2*c[7]-98*c[5]^3*c[4]^4*c[7]+420*c[7]^2*c[4]^ 5*c[5]^3+200*c[7]^3*c[5]^2*c[4]^4-300*c[6]^2*c[4]^5*c[7]*c[5]^3-40*c[7 ]*c[5]^2*c[4]^6*c[6]+20*c[5]^4*c[4]^2*c[7]^2*c[6]-280*c[6]^2*c[7]*c[5] ^4*c[4]^2-44*c[6]*c[4]^5*c[5]*c[7]-246*c[6]*c[5]^2*c[4]^4*c[7]+120*c[4 ]^6*c[6]^2*c[5]^2-100*c[5]^2*c[7]^2*c[4]^6*c[6]+141*c[5]^4*c[4]^3*c[6] +1220*c[5]^4*c[4]^5*c[6]+400*c[5]^5*c[4]^5*c[6]^2-760*c[5]^4*c[4]^4*c[ 6]+100*c[5]^4*c[7]^3*c[4]^4-400*c[7]^2*c[4]^4*c[5]^3-200*c[5]^4*c[4]^5 *c[7]^2+200*c[5]^4*c[4]^4*c[7]-2*c[4]^3*c[7]^3*c[6]-8*c[7]^2*c[6]*c[4] ^4-380*c[7]^3*c[5]^3*c[4]^4-40*c[6]*c[4]^5*c[5]*c[7]^2+2*c[4]^4*c[7]^2 -40*c[5]^2*c[7]^3*c[4]^5-200*c[7]^2*c[5]^4*c[4]^3-40*c[7]^3*c[5]^4*c[4 ]^3+200*c[5]^3*c[7]*c[4]^6*c[6]-10*c[6]^2*c[7]^3*c[4]^3+30*c[6]^2*c[4] ^5*c[5]+12*c[7]*c[6]^2*c[5]^3+4*c[7]^2*c[6]*c[5]^3+400*c[5]^4*c[4]^6*c [6]^2-40*c[5]^3*c[7]*c[4]^6-380*c[5]^4*c[4]^5*c[7]+320*c[7]*c[6]^2*c[5 ]*c[4]^5+100*c[4]^6*c[6]^2*c[7]*c[5]^2+34*c[5]^4*c[4]^2*c[7]^2+40*c[5] ^5*c[7]^2*c[4]^3+240*c[6]*c[5]^2*c[4]^5*c[7]+60*c[5]^4*c[7]^3*c[4]^2*c [6]+26*c[7]*c[4]^4*c[6]*c[5]-50*c[4]^4*c[6]^2*c[5]^5+200*c[5]^5*c[7]*c [4]^4*c[6]+132*c[7]^2*c[5]^3*c[4]^3-2*c[7]*c[4]^4*c[6]^2+60*c[5]^5*c[4 ]^2*c[6]^2*c[7]-500*c[5]^5*c[4]^4*c[6]^2*c[7]+160*c[5]^5*c[4]^4*c[6]+1 34*c[6]^2*c[5]^4*c[7]*c[4]+24*c[5]^4*c[6]^2*c[4]-169*c[5]^4*c[6]^2*c[4 ]^2-500*c[5]^3*c[4]^6*c[6]^2-40*c[5]^5*c[4]^4*c[7]+34*c[4]^5*c[5]*c[7] ^2-100*c[5]^3*c[7]^2*c[4]^6-100*c[5]^5*c[4]^4*c[7]^2+160*c[4]^6*c[6]*c [5]^3-6*c[6]^2*c[7]*c[5]^2*c[4]^3+68*c[6]^2*c[7]^2*c[5]^2*c[4]-126*c[6 ]^2*c[7]^2*c[5]^2*c[4]^2-600*c[6]*c[7]*c[5]^3*c[4]^3-21*c[6]^2*c[7]^3* c[4]*c[5]+40*c[7]^2*c[6]^2*c[5]^4+4*c[5]^4*c[7]*c[4]^2+58*c[7]^2*c[6]* c[5]*c[4]^4-100*c[7]^2*c[5]^5*c[6]^2*c[4]^2-320*c[5]^4*c[4]*c[7]^2*c[6 ]^2+20*c[6]^2*c[4]^4*c[7]^2+40*c[5]^2*c[4]^6*c[7]^2+6*c[7]^2*c[6]*c[4] ^3-8*c[6]*c[5]*c[7]^2*c[4]^2-4*c[5]*c[7]*c[6]*c[4]^3+14*c[5]^2*c[6]*c[ 7]*c[4]^2-910*c[4]^5*c[6]*c[7]*c[5]^3-340*c[5]^2*c[6]*c[7]^3*c[4]^4-4* c[6]^2*c[5]*c[4]^3+5*c[7]^3*c[4]^2*c[6]*c[5]+72*c[7]*c[6]^2*c[5]*c[4]^ 3+18*c[6]^2*c[7]^3*c[5]^2-12*c[6]^2*c[7]^2*c[5]^2-10*c[6]^2*c[4]^2*c[7 ]*c[5]-44*c[6]^2*c[7]^2*c[4]^2*c[5]-25*c[6]*c[5]^4*c[4]^2-24*c[7]*c[6] ^2*c[5]^4+200*c[4]^5*c[7]*c[5]^3-2*c[6]^2*c[5]^2*c[7]*c[4]-42*c[6]^2*c [5]^2*c[7]*c[4]^2+74*c[6]*c[5]^2*c[7]*c[4]^3+80*c[6]*c[5]^3*c[7]*c[4]^ 2-50*c[6]^2*c[5]^3*c[7]*c[4]-34*c[5]^4*c[4]^3*c[7]-24*c[7]^2*c[5]^2*c[ 4]^3+420*c[5]^4*c[7]^2*c[4]^4-2*c[4]^3*c[7]^3+c[7]^3*c[5]^2*c[6]*c[4]+ 4*c[7]^2*c[5]^2*c[6]*c[4]-6*c[6]*c[7]*c[4]^4-4*c[6]^2*c[7]^2*c[4]^2+50 *c[5]^5*c[4]^3*c[7]^2*c[6]-18*c[5]^3*c[6]*c[7]*c[4]+14*c[6]^2*c[7]^2*c [4]*c[5]-20*c[5]^2*c[6]*c[4]^3+2*c[5]^2*c[7]*c[4]^3-2*c[5]^3*c[7]*c[4] ^2+16*c[5]^3*c[6]*c[4]^2-96*c[5]^3*c[6]^2*c[4]^3+100*c[5]^5*c[4]^3*c[6 ]^2*c[7]-60*c[4]^3*c[7]^2*c[5]^2*c[6]^2+10*c[6]*c[7]*c[4]^5+1540*c[4]^ 5*c[5]^3*c[6]^2-34*c[7]^3*c[4]^4*c[5]-4*c[7]^2*c[5]^4*c[4]+12*c[5]^2*c [4]^4*c[7]-41*c[6]^2*c[5]^2*c[4]^3+14*c[6]^2*c[5]^2*c[4]^2-54*c[6]*c[5 ]^3*c[4]^3-12*c[6]^2*c[5]^3*c[4]+47*c[6]^2*c[5]^3*c[4]^2+12*c[5]^3*c[7 ]*c[4]^3-98*c[5]^4*c[7]*c[4]^2*c[6]-50*c[5]^3*c[7]^3*c[4]^4*c[6]+4*c[7 ]*c[6]^2*c[4]^3+156*c[7]^2*c[5]^2*c[6]*c[4]^3-23*c[7]^3*c[5]^2*c[6]*c[ 4]^2+188*c[7]^2*c[5]^3*c[6]*c[4]^2+66*c[5]^3*c[7]^3*c[6]*c[4]+2*c[7]^3 *c[4]^2*c[5]+100*c[5]^5*c[4]^5*c[7]+104*c[4]^3*c[7]^2*c[5]*c[6]^2-12*c [7]^2*c[6]*c[5]*c[4]^3-36*c[7]^2*c[5]^2*c[6]*c[4]^2-32*c[7]^2*c[5]^3*c [6]*c[4]-320*c[7]^2*c[5]^3*c[6]*c[4]^3-560*c[5]^3*c[6]^2*c[4]^4-4*c[6] ^2*c[4]^3*c[7]^2+240*c[5]^3*c[6]^2*c[7]*c[4]^3-34*c[5]^3*c[7]^3*c[4]^2 -2*c[7]^3*c[5]^2*c[4]+164*c[6]^2*c[7]*c[5]^3*c[4]^2-20*c[6]^2*c[7]^2*c [4]*c[5]^3+2*c[7]^2*c[5]^3*c[4]-130*c[5]^5*c[4]^3*c[6]*c[7]-90*c[7]^3* c[5]^2*c[6]^2*c[4]^3-16*c[7]^2*c[4]^4*c[5]-60*c[7]^2*c[5]^3*c[6]^2*c[4 ]^2-300*c[6]^2*c[4]^5*c[5]*c[7]^2-150*c[7]^2*c[4]^5*c[6]*c[5]^3+380*c[ 5]^2*c[6]*c[4]^5*c[7]^2+140*c[7]^3*c[5]^2*c[4]^3*c[6]-810*c[7]^3*c[5]^ 3*c[6]^2*c[4]^2-14*c[7]^3*c[6]*c[5]*c[4]^3-280*c[7]^3*c[6]^2*c[5]*c[4] ^3+240*c[7]^3*c[5]^3*c[6]^2*c[4]-20*c[5]^4*c[7]^2*c[4]*c[6]-6*c[6]*c[7 ]^3*c[5]^3+22*c[5]^4*c[4]*c[7]*c[6]-280*c[4]^5*c[6]^2*c[7]*c[5]^2+400* c[4]^5*c[6]^2*c[7]^2*c[5]^2+253*c[6]^2*c[4]^4*c[5]^2+470*c[7]^3*c[5]^3 *c[4]^3*c[6]+100*c[5]^2*c[4]^5*c[7]^3*c[6]+6*c[6]^2*c[7]^3*c[4]^2-300* c[5]^5*c[4]^5*c[6]-120*c[5]*c[7]*c[4]^6*c[6]^2-10*c[6]^2*c[7]*c[4]^5-1 100*c[5]^4*c[4]^5*c[6]^2*c[7]-226*c[7]*c[6]^2*c[5]*c[4]^4-770*c[5]^3*c [4]^4*c[6]^2*c[7]+280*c[5]^3*c[4]^3*c[6]^2*c[7]^2-1000*c[5]^4*c[4]^4*c [6]*c[7]+240*c[5]^2*c[4]^4*c[7]*c[6]^2+1240*c[5]^3*c[4]^4*c[6]*c[7]+60 0*c[5]^4*c[4]^3*c[6]*c[7]+1200*c[7]^3*c[5]^3*c[6]^2*c[4]^3+40*c[7]^3*c [4]^4*c[5]*c[6]-16*c[7]^2*c[5]^3*c[4]^2-1250*c[5]^4*c[4]^5*c[6]^2-600* c[6]^2*c[7]^3*c[5]^3*c[4]^4+200*c[6]^2*c[7]^2*c[5]^3*c[4]^5-200*c[6]^2 *c[7]^3*c[4]^5*c[5]-140*c[5]^4*c[4]^4*c[6]^2+560*c[5]^4*c[4]^3*c[6]^2- 1850*c[7]^2*c[5]^4*c[6]^2*c[4]^3+1230*c[7]^2*c[5]^4*c[6]^2*c[4]^2-150* c[5]^4*c[7]^3*c[4]^3*c[6]-300*c[5]^4*c[4]^6*c[6]+310*c[5]^3*c[4]^4*c[7 ]^2*c[6]-70*c[5]^4*c[4]^3*c[7]^2*c[6]-380*c[5]^2*c[4]^4*c[7]^2*c[6]-85 0*c[5]^4*c[4]^3*c[6]^2*c[7]-40*c[7]^2*c[4]^4*c[6]^2*c[5]-450*c[7]^2*c[ 4]^4*c[5]^3*c[6]^2+250*c[7]^2*c[4]^4*c[6]^2*c[5]^2-4*c[7]^2*c[4]^5+136 *c[5]^2*c[6]*c[4]^5+400*c[6]^2*c[7]^3*c[5]^2*c[4]^5+800*c[6]^2*c[7]^2* c[5]^4*c[4]^4+4*c[7]^3*c[4]^4+100*c[5]^4*c[4]^6*c[7]+437*c[4]^4*c[6]*c [5]^3-22*c[6]*c[5]*c[4]^5-60*c[5]^5*c[6]^2*c[4]^3+200*c[6]^2*c[4]^6*c[ 5]*c[7]^2+420*c[7]^3*c[6]^2*c[5]*c[4]^4+400*c[5]^4*c[4]^5*c[6]*c[7]-50 0*c[7]^3*c[4]^4*c[6]^2*c[5]^2+100*c[5]^3*c[7]^3*c[4]^5-103*c[6]^2*c[7] ^3*c[5]^2*c[4]+260*c[6]^2*c[7]^3*c[5]^2*c[4]^2-300*c[5]^3*c[7]^3*c[4]^ 2*c[6]+97*c[6]^2*c[5]*c[7]^3*c[4]^2-30*c[6]^2*c[7]^3*c[5]^3+4*c[7]^3*c [4]*c[5]^3)/c[6]/(-50*c[5]^2*c[6]*c[4]^4+23*c[5]^3*c[4]+100*c[5]^4*c[4 ]^3*c[6]+2*c[5]^2+2*c[4]^2-3*c[5]^3-3*c[4]^3-3*c[6]*c[5]^2+5*c[6]*c[5] ^3-3*c[6]*c[4]^2+5*c[6]*c[4]^3+23*c[5]*c[4]^3+18*c[5]*c[6]*c[4]^2-12*c [5]*c[4]^2+3*c[5]*c[6]*c[4]-2*c[5]*c[4]-50*c[6]*c[5]^4*c[4]^2-40*c[6]* c[5]*c[4]^3-40*c[5]^3*c[6]*c[4]-12*c[5]^2*c[4]-140*c[5]^3*c[4]^2+240*c [5]^2*c[6]*c[4]^3-140*c[5]^2*c[6]*c[4]^2+240*c[5]^3*c[6]*c[4]^2-400*c[ 6]*c[5]^3*c[4]^3+91*c[5]^2*c[4]^2+30*c[5]^4*c[4]^2+30*c[4]^4*c[5]^2-50 *c[5]^4*c[4]^3+18*c[5]^2*c[6]*c[4]-140*c[5]^2*c[4]^3+210*c[5]^3*c[4]^3 +100*c[4]^4*c[6]*c[5]^3-50*c[5]^3*c[4]^4)/c[5]/c[4]^2, `b*`[8] = 0, a[ 5,3] = -3/4*c[5]^2*(2*c[5]-3*c[4])/c[4]^2, `b*`[4] = -1/60*(-30*c[5]^3 *c[4]-42*c[7]*c[6]^2+89*c[5]*c[7]*c[4]-18*c[6]+6*c[5]+6*c[4]+27*c[7]*c [6]-9*c[5]^2+180*c[7]*c[6]^2*c[5]^3-100*c[7]*c[6]*c[5]^3+27*c[6]^2+186 *c[5]*c[6]*c[7]*c[4]-97*c[6]*c[5]^2+66*c[6]*c[5]+60*c[6]*c[5]^3-100*c[ 6]^2*c[5]^3-9*c[7]*c[5]+14*c[7]*c[5]^2-100*c[6]^2*c[7]*c[4]^2+120*c[6] ^2*c[7]*c[4]+60*c[6]^2*c[4]^2-9*c[7]*c[4]-103*c[6]*c[7]*c[5]-36*c[6]*c [4]^2-61*c[6]*c[7]*c[4]+60*c[6]*c[7]*c[4]^2-30*c[5]*c[7]*c[4]^2+126*c[ 5]*c[6]*c[4]^2+18*c[5]*c[4]^2-112*c[5]*c[6]*c[4]-57*c[5]*c[4]+39*c[6]* c[4]-280*c[6]^2*c[4]^2*c[5]+330*c[6]^2*c[4]*c[5]+380*c[5]^2*c[6]*c[7]* c[4]^2-560*c[6]^2*c[7]*c[4]*c[5]+192*c[6]^2*c[7]*c[5]-310*c[6]^2*c[7]* c[5]^2-230*c[6]*c[5]*c[7]*c[4]^2+156*c[6]*c[7]*c[5]^2+180*c[6]^2*c[5]^ 2+500*c[6]^2*c[4]^2*c[7]*c[5]+950*c[6]^2*c[5]^2*c[7]*c[4]-900*c[6]^2*c [5]^2*c[7]*c[4]^2-300*c[6]*c[5]^3*c[7]*c[4]^2-600*c[6]^2*c[5]^3*c[7]*c [4]-117*c[6]^2*c[5]-130*c[5]^3*c[6]*c[4]+50*c[5]^3*c[7]*c[4]+93*c[5]^2 *c[4]+230*c[5]^3*c[6]*c[7]*c[4]+50*c[5]^2*c[7]*c[4]^2-190*c[5]^2*c[6]* c[4]^2+150*c[5]^3*c[6]*c[4]^2+470*c[6]^2*c[5]^2*c[4]^2+320*c[6]^2*c[5] ^3*c[4]-300*c[6]^2*c[5]^3*c[4]^2-30*c[5]^2*c[4]^2-530*c[6]^2*c[4]*c[5] ^2-75*c[6]^2*c[4]+156*c[5]^2*c[6]*c[4]-150*c[5]^2*c[7]*c[4]-270*c[5]^2 *c[6]*c[7]*c[4]+600*c[6]^2*c[7]*c[5]^3*c[4]^2)/(-9*c[5]*c[7]*c[4]-3*c[ 7]*c[6]-c[4]^2-28*c[5]*c[6]*c[7]*c[4]+c[7]*c[5]+c[7]*c[4]+9*c[6]*c[7]* c[5]-9*c[6]*c[4]^2+8*c[6]*c[4]^3+9*c[6]*c[7]*c[4]-8*c[6]*c[7]*c[4]^2-4 *c[5]*c[4]^3+4*c[5]*c[7]*c[4]^2+28*c[5]*c[6]*c[4]^2+9*c[5]*c[4]^2-9*c[ 5]*c[6]*c[4]-c[5]*c[4]+3*c[6]*c[4]-30*c[5]^2*c[6]*c[7]*c[4]^2+28*c[6]* c[5]*c[7]*c[4]^2-8*c[6]*c[7]*c[5]^2-28*c[6]*c[5]*c[4]^3+30*c[5]^2*c[6] *c[4]^3-28*c[5]^2*c[6]*c[4]^2-4*c[5]^2*c[4]^2+8*c[5]^2*c[6]*c[4]+4*c[5 ]^2*c[7]*c[4]+28*c[5]^2*c[6]*c[7]*c[4])/(-c[4]+c[6])/(-c[4]+c[5])/c[4] , a[8,6] = 1/2*(5*c[5]*c[6]*c[4]+c[6]-2*c[6]*c[4]-2*c[6]*c[5]+25*c[5]* c[7]*c[4]+6*c[5]+9*c[7]+10*c[7]^2*c[4]-4+6*c[4]-6*c[7]^2-14*c[7]*c[4]- 10*c[5]*c[4]-14*c[7]*c[5]-20*c[7]^2*c[5]*c[4]+10*c[7]^2*c[5])/(c[6]-c[ 7])/(-c[4]+c[6])/(-c[5]+c[6])/c[6]/(-20*c[5]*c[6]*c[4]+30*c[5]*c[6]*c[ 7]*c[4]+15*c[6]*c[5]-20*c[6]*c[7]*c[5]+15*c[6]*c[4]-20*c[6]*c[7]*c[4]- 12*c[6]+15*c[7]*c[6]+15*c[5]*c[4]-20*c[5]*c[7]*c[4]-12*c[5]+15*c[7]*c[ 5]-12*c[4]+15*c[7]*c[4]+10-12*c[7])*(c[4]-1)*(-1+c[5])*(-1+c[6]), b[8] = 1/60*(-20*c[5]*c[6]*c[4]+30*c[5]*c[6]*c[7]*c[4]+15*c[6]*c[5]-20*c[6 ]*c[7]*c[5]+15*c[6]*c[4]-20*c[6]*c[7]*c[4]-12*c[6]+15*c[7]*c[6]+15*c[5 ]*c[4]-20*c[5]*c[7]*c[4]-12*c[5]+15*c[7]*c[5]-12*c[4]+15*c[7]*c[4]+10- 12*c[7])/(c[4]-1)/(-1+c[5])/(-1+c[6])/(-1+c[7]), a[5,1] = 1/4*c[5]*(2* c[5]^2-5*c[5]*c[4]+4*c[4]^2)/c[4]^2, `b*`[5] = 1/60*(-42*c[7]*c[6]^2+8 9*c[5]*c[7]*c[4]-18*c[6]+6*c[5]+6*c[4]+27*c[7]*c[6]-9*c[4]^2+27*c[6]^2 +186*c[5]*c[6]*c[7]*c[4]-36*c[6]*c[5]^2+39*c[6]*c[5]-9*c[7]*c[5]-100*c [6]^2*c[4]^3+600*c[6]^2*c[7]*c[5]^2*c[4]^3-100*c[7]*c[6]*c[4]^3-310*c[ 6]^2*c[7]*c[4]^2+192*c[6]^2*c[7]*c[4]+180*c[6]^2*c[4]^2+14*c[7]*c[4]^2 -9*c[7]*c[4]-61*c[6]*c[7]*c[5]-97*c[6]*c[4]^2+60*c[6]*c[4]^3-103*c[6]* c[7]*c[4]+156*c[6]*c[7]*c[4]^2-30*c[5]*c[4]^3-150*c[5]*c[7]*c[4]^2+156 *c[5]*c[6]*c[4]^2+93*c[5]*c[4]^2-112*c[5]*c[6]*c[4]-57*c[5]*c[4]+66*c[ 6]*c[4]+230*c[5]*c[7]*c[6]*c[4]^3-530*c[6]^2*c[4]^2*c[5]+330*c[6]^2*c[ 4]*c[5]+380*c[5]^2*c[6]*c[7]*c[4]^2+320*c[6]^2*c[5]*c[4]^3-560*c[6]^2* c[7]*c[4]*c[5]-600*c[7]*c[6]^2*c[5]*c[4]^3+120*c[6]^2*c[7]*c[5]-100*c[ 6]^2*c[7]*c[5]^2-270*c[6]*c[5]*c[7]*c[4]^2+60*c[6]*c[7]*c[5]^2+60*c[6] ^2*c[5]^2+950*c[6]^2*c[4]^2*c[7]*c[5]+500*c[6]^2*c[5]^2*c[7]*c[4]-900* c[6]^2*c[5]^2*c[7]*c[4]^2-300*c[6]*c[5]^2*c[7]*c[4]^3-75*c[6]^2*c[5]+5 0*c[7]*c[5]*c[4]^3-130*c[6]*c[5]*c[4]^3+18*c[5]^2*c[4]+150*c[5]^2*c[6] *c[4]^3+50*c[5]^2*c[7]*c[4]^2-190*c[5]^2*c[6]*c[4]^2-300*c[6]^2*c[5]^2 *c[4]^3+470*c[6]^2*c[5]^2*c[4]^2-30*c[5]^2*c[4]^2-280*c[6]^2*c[4]*c[5] ^2+180*c[7]*c[6]^2*c[4]^3-117*c[6]^2*c[4]+126*c[5]^2*c[6]*c[4]-30*c[5] ^2*c[7]*c[4]-230*c[5]^2*c[6]*c[7]*c[4])/(-4*c[5]^3*c[4]-9*c[5]*c[7]*c[ 4]-3*c[7]*c[6]-c[5]^2-28*c[5]*c[6]*c[7]*c[4]-9*c[6]*c[5]^2+3*c[6]*c[5] +8*c[6]*c[5]^3+c[7]*c[5]+c[7]*c[4]+9*c[6]*c[7]*c[5]+9*c[6]*c[7]*c[4]-8 *c[6]*c[7]*c[4]^2+4*c[5]*c[7]*c[4]^2+8*c[5]*c[6]*c[4]^2-9*c[5]*c[6]*c[ 4]-c[5]*c[4]-30*c[5]^2*c[6]*c[7]*c[4]^2+28*c[6]*c[5]*c[7]*c[4]^2-8*c[6 ]*c[7]*c[5]^2-28*c[5]^3*c[6]*c[4]+9*c[5]^2*c[4]-28*c[5]^2*c[6]*c[4]^2+ 30*c[5]^3*c[6]*c[4]^2-4*c[5]^2*c[4]^2+28*c[5]^2*c[6]*c[4]+4*c[5]^2*c[7 ]*c[4]+28*c[5]^2*c[6]*c[7]*c[4])/(-c[5]+c[6])/(-c[4]+c[5])/c[5], a[5,4 ] = c[5]^2*(-c[4]+c[5])/c[4]^2, a[7,6] = 1/2*(5*c[5]*c[4]+1-2*c[4]-2*c [5])*(c[4]-c[7])*(c[5]-c[7])*(c[6]-c[7])*c[7]/(10*c[5]*c[6]*c[4]-5*c[6 ]*c[5]-5*c[6]*c[4]+3*c[6]-5*c[5]*c[4]+3*c[5]+3*c[4]-2)/(-c[4]+c[6])/(- c[5]+c[6])/c[6], c[3] = 2/3*c[4], a[4,3] = 3/4*c[4], a[3,2] = 2/9*c[4] ^2/c[2], a[7,4] = -1/2*c[7]*(-68*c[5]^2*c[6]*c[4]^4-7*c[6]*c[5]*c[4]^4 +7*c[4]^2*c[5]^2*c[7]^3-1400*c[5]^4*c[4]^4*c[6]^2*c[7]+11*c[7]^3*c[5]* c[4]^3-72*c[7]^3*c[5]^2*c[4]^3-3*c[7]*c[5]*c[4]^4-58*c[5]^2*c[4]^4*c[7 ]^2-120*c[6]*c[4]^7*c[7]*c[5]+320*c[4]^5*c[6]^2*c[5]^2-200*c[4]^3*c[7] ^3*c[5]^3-100*c[4]^5*c[6]*c[5]^3-2*c[5]*c[6]^2*c[4]^4-600*c[4]^5*c[7]^ 3*c[6]*c[5]^3-200*c[5]^5*c[4]^3*c[7]^2*c[6]^2+400*c[5]^2*c[4]^6*c[6]^2 *c[7]^2-21*c[4]^5*c[5]*c[7]+120*c[5]^2*c[4]^5*c[7]-600*c[5]^3*c[4]^6*c [6]^2*c[7]-198*c[5]^3*c[4]^4*c[7]-280*c[7]^2*c[4]^5*c[5]^3+240*c[7]^3* c[5]^2*c[4]^4+1250*c[6]^2*c[4]^5*c[7]*c[5]^3+120*c[7]*c[5]^2*c[4]^6*c[ 6]-340*c[5]^4*c[4]^2*c[7]^2*c[6]+400*c[6]^2*c[7]*c[5]^4*c[4]^2-240*c[5 ]^2*c[4]^6*c[7]-300*c[6]*c[5]*c[7]^2*c[4]^6-228*c[6]*c[4]^5*c[5]*c[7]- 119*c[6]*c[5]^2*c[4]^4*c[7]-200*c[5]*c[7]^3*c[4]^6*c[6]-600*c[5]^4*c[7 ]*c[4]^6*c[6]-120*c[4]^6*c[6]^2*c[5]^2+600*c[5]^2*c[7]^2*c[4]^6*c[6]+1 00*c[5]^2*c[7]^3*c[4]^6-187*c[5]^4*c[4]^3*c[6]-910*c[5]^4*c[4]^5*c[6]- 320*c[5]^2*c[6]*c[4]^6+620*c[5]^4*c[4]^4*c[6]-100*c[5]^4*c[7]^3*c[4]^4 -40*c[7]^2*c[4]^4*c[5]^3+40*c[7]^3*c[4]^5*c[5]+10*c[6]*c[7]^2*c[4]^5-1 0*c[6]*c[7]^3*c[4]^4+100*c[5]^4*c[4]^5*c[7]^2+80*c[5]^4*c[4]^4*c[7]+8* c[4]^3*c[7]^3*c[6]-2*c[7]^2*c[6]*c[4]^4+280*c[7]^3*c[5]^3*c[4]^4+80*c[ 6]*c[4]^5*c[5]*c[7]^2+c[4]^4*c[7]^2-360*c[5]^2*c[7]^3*c[4]^5-80*c[7]^2 *c[5]^4*c[4]^3+40*c[7]^3*c[5]^4*c[4]^3-1200*c[5]^3*c[7]*c[4]^6*c[6]+10 *c[6]^2*c[7]^3*c[4]^3-10*c[6]^2*c[4]^5*c[5]-12*c[7]*c[6]^2*c[5]^3-4*c[ 7]^2*c[6]*c[5]^3-80*c[5]^4*c[4]^5*c[7]-240*c[7]*c[6]^2*c[5]*c[4]^5-3*c [5]^4*c[4]^2*c[7]^2+20*c[5]^5*c[7]^2*c[4]^3+180*c[6]*c[5]^2*c[4]^5*c[7 ]-60*c[5]^4*c[7]^3*c[4]^2*c[6]+400*c[5]^2*c[7]^3*c[4]^6*c[6]-25*c[4]^5 *c[5]^3+200*c[5]^5*c[4]^4*c[7]^2*c[6]+60*c[5]^2*c[4]^7*c[7]+100*c[7]*c [4]^4*c[6]*c[5]-40*c[5]*c[4]^6*c[7]^2-150*c[4]^4*c[6]^2*c[5]^5+50*c[5] ^5*c[7]*c[4]^4*c[6]+174*c[7]^2*c[5]^3*c[4]^3+8*c[7]*c[4]^4*c[6]^2+600* c[5]^3*c[7]*c[4]^7*c[6]-60*c[5]^5*c[4]^2*c[6]^2*c[7]+200*c[5]^5*c[4]^4 *c[6]^2*c[7]-40*c[5]^5*c[4]^4*c[6]-200*c[5]^5*c[7]*c[4]^5*c[6]-149*c[6 ]^2*c[5]^4*c[7]*c[4]-24*c[5]^4*c[6]^2*c[4]+189*c[5]^4*c[6]^2*c[4]^2+30 0*c[5]^3*c[4]^6*c[6]^2-20*c[5]^5*c[4]^4*c[7]-5*c[4]^5*c[5]*c[7]^2+600* c[4]^6*c[6]*c[5]^3-84*c[6]^2*c[7]*c[5]^2*c[4]^3-74*c[6]^2*c[7]^2*c[5]^ 2*c[4]+162*c[6]^2*c[7]^2*c[5]^2*c[4]^2+311*c[6]*c[7]*c[5]^3*c[4]^3+21* c[6]^2*c[7]^3*c[4]*c[5]-40*c[7]^2*c[6]^2*c[5]^4+11*c[5]^4*c[7]*c[4]^2+ 107*c[7]^2*c[6]*c[5]*c[4]^4+100*c[7]^2*c[5]^5*c[6]^2*c[4]^2+320*c[5]^4 *c[4]*c[7]^2*c[6]^2-10*c[6]^2*c[4]^4*c[7]^2+300*c[5]^2*c[4]^6*c[7]^2-3 *c[7]^2*c[6]*c[4]^3-2*c[5]*c[7]^2*c[4]^3+15*c[6]*c[5]*c[7]^2*c[4]^2-8* c[5]*c[7]*c[6]*c[4]^3-14*c[5]^2*c[6]*c[7]*c[4]^2-2000*c[5]^4*c[7]^2*c[ 4]^4*c[6]+200*c[4]^5*c[6]*c[7]*c[5]^3+250*c[5]^2*c[6]*c[7]^3*c[4]^4+4* c[6]^2*c[5]*c[4]^3-26*c[7]^3*c[4]^2*c[6]*c[5]-69*c[7]*c[6]^2*c[5]*c[4] ^3-18*c[6]^2*c[7]^3*c[5]^2+12*c[6]^2*c[7]^2*c[5]^2+10*c[6]^2*c[4]^2*c[ 7]*c[5]+50*c[6]^2*c[7]^2*c[4]^2*c[5]+31*c[6]*c[5]^4*c[4]^2+24*c[7]*c[6 ]^2*c[5]^4+240*c[4]^5*c[7]*c[5]^3+2*c[6]^2*c[5]^2*c[7]*c[4]+39*c[6]^2* c[5]^2*c[7]*c[4]^2-17*c[6]*c[5]^2*c[7]*c[4]^3-73*c[6]*c[5]^3*c[7]*c[4] ^2+59*c[6]^2*c[5]^3*c[7]*c[4]-12*c[5]^4*c[4]^3*c[7]-12*c[7]^2*c[5]^2*c [4]^3+40*c[5]^4*c[7]^2*c[4]^4-c[4]^3*c[7]^3+17*c[7]^3*c[5]^2*c[6]*c[4] -9*c[7]^2*c[5]^2*c[6]*c[4]+3*c[6]*c[7]*c[4]^4+4*c[4]^5*c[5]+4*c[6]^2*c [7]^2*c[4]^2-150*c[5]^5*c[4]^3*c[7]^2*c[6]+21*c[5]^3*c[6]*c[7]*c[4]-14 *c[6]^2*c[7]^2*c[4]*c[5]+23*c[5]^2*c[6]*c[4]^3+13*c[5]^2*c[7]*c[4]^3-1 1*c[5]^3*c[7]*c[4]^2-17*c[5]^3*c[6]*c[4]^2+168*c[5]^3*c[6]^2*c[4]^3+50 *c[5]^5*c[4]^3*c[6]^2*c[7]-220*c[4]^3*c[7]^2*c[5]^2*c[6]^2-6*c[6]*c[7] *c[4]^5-580*c[4]^5*c[5]^3*c[6]^2-31*c[7]^3*c[4]^4*c[5]-2*c[7]^2*c[5]^4 *c[4]-4*c[5]^2*c[4]^4*c[7]+53*c[6]^2*c[5]^2*c[4]^3-14*c[6]^2*c[5]^2*c[ 4]^2+47*c[6]*c[5]^3*c[4]^3+12*c[6]^2*c[5]^3*c[4]-59*c[6]^2*c[5]^3*c[4] ^2+28*c[5]^3*c[7]*c[4]^3+127*c[5]^4*c[7]*c[4]^2*c[6]+1250*c[5]^3*c[7]^ 3*c[4]^4*c[6]+10*c[6]*c[5]*c[4]^6+240*c[5]*c[4]^6*c[7]*c[6]-4*c[7]*c[6 ]^2*c[4]^3+11*c[7]^2*c[5]^2*c[6]*c[4]^3-80*c[7]^3*c[5]^2*c[6]*c[4]^2-1 21*c[7]^2*c[5]^3*c[6]*c[4]^2-96*c[5]^3*c[7]^3*c[6]*c[4]+c[7]^3*c[4]^2* c[5]-68*c[4]^3*c[7]^2*c[5]*c[6]^2-80*c[7]^2*c[6]*c[5]*c[4]^3+68*c[7]^2 *c[5]^2*c[6]*c[4]^2-10*c[4]^4*c[5]^2+20*c[7]^2*c[5]^3*c[6]*c[4]-240*c[ 7]^2*c[5]^3*c[6]*c[4]^3-9*c[5]^4*c[4]^3-2*c[6]^2*c[4]^3*c[7]^2+260*c[5 ]^3*c[6]^2*c[7]*c[4]^3+49*c[5]^3*c[7]^3*c[4]^2-c[7]^3*c[5]^2*c[4]-228* c[6]^2*c[7]*c[5]^3*c[4]^2+30*c[6]^2*c[7]^2*c[4]*c[5]^3-2*c[7]^2*c[5]^2 *c[4]^2+5*c[7]^2*c[5]^3*c[4]+40*c[5]^5*c[4]^3*c[6]*c[7]+36*c[4]^6*c[7] *c[5]+90*c[7]^3*c[5]^2*c[6]^2*c[4]^3+16*c[7]^2*c[4]^4*c[5]-20*c[7]^2*c [5]^3*c[6]^2*c[4]^2+300*c[6]^2*c[4]^5*c[5]*c[7]^2+250*c[7]^2*c[4]^5*c[ 6]*c[5]^3-490*c[5]^2*c[6]*c[4]^5*c[7]^2+120*c[7]^3*c[5]^2*c[4]^3*c[6]+ 810*c[7]^3*c[5]^3*c[6]^2*c[4]^2+111*c[7]^3*c[6]*c[5]*c[4]^3+280*c[7]^3 *c[6]^2*c[5]*c[4]^3-240*c[7]^3*c[5]^3*c[6]^2*c[4]+15*c[5]^4*c[4]^4+60* c[5]^4*c[7]^2*c[4]*c[6]+6*c[6]*c[7]^3*c[5]^3-31*c[5]^4*c[4]*c[7]*c[6]- 200*c[4]^5*c[6]^2*c[7]*c[5]^2-500*c[4]^5*c[6]^2*c[7]^2*c[5]^2+9*c[4]^5 *c[5]^2-181*c[6]^2*c[4]^4*c[5]^2-1280*c[7]^3*c[5]^3*c[4]^3*c[6]-600*c[ 5]^2*c[4]^5*c[7]^3*c[6]-6*c[4]^6*c[5]-6*c[6]^2*c[7]^3*c[4]^2+100*c[5]^ 5*c[4]^5*c[6]+120*c[5]*c[7]*c[4]^6*c[6]^2+420*c[6]*c[4]^5*c[7]^3*c[5]+ 600*c[5]^4*c[4]^5*c[6]^2*c[7]+175*c[7]*c[6]^2*c[5]*c[4]^4-430*c[5]^3*c [4]^4*c[6]^2*c[7]+200*c[5]^3*c[4]^3*c[6]^2*c[7]^2-450*c[5]^4*c[4]^4*c[ 6]*c[7]+160*c[5]^2*c[4]^4*c[7]*c[6]^2-200*c[5]^3*c[4]^4*c[6]*c[7]-280* c[5]^4*c[4]^3*c[6]*c[7]+6*c[5]^3*c[4]^3-100*c[5]^2*c[4]^7*c[7]^2-1200* c[7]^3*c[5]^3*c[6]^2*c[4]^3-320*c[7]^3*c[4]^4*c[5]*c[6]-30*c[7]^2*c[5] ^3*c[4]^2-300*c[6]*c[5]^3*c[4]^7-350*c[5]^4*c[4]^5*c[6]^2+600*c[6]^2*c [7]^3*c[5]^3*c[4]^4+600*c[6]*c[7]^2*c[5]^4*c[4]^5+200*c[6]^2*c[7]^3*c[ 4]^5*c[5]+200*c[7]^2*c[4]^7*c[6]*c[5]+1100*c[5]^4*c[4]^4*c[6]^2-720*c[ 5]^4*c[4]^3*c[6]^2+1750*c[7]^2*c[5]^4*c[6]^2*c[4]^3-1230*c[7]^2*c[5]^4 *c[6]^2*c[4]^2+150*c[5]^4*c[7]^3*c[4]^3*c[6]+300*c[5]^4*c[4]^6*c[6]+48 0*c[5]^3*c[4]^4*c[7]^2*c[6]+1420*c[5]^4*c[4]^3*c[7]^2*c[6]+20*c[5]^2*c [4]^4*c[7]^2*c[6]+130*c[5]^4*c[4]^3*c[6]^2*c[7]-40*c[7]^2*c[4]^4*c[6]^ 2*c[5]-350*c[7]^2*c[4]^4*c[5]^3*c[6]^2+230*c[7]^2*c[4]^4*c[6]^2*c[5]^2 +10*c[4]^6*c[5]^2+188*c[5]^2*c[6]*c[4]^5-400*c[6]^2*c[7]^3*c[5]^2*c[4] ^5-600*c[6]^2*c[7]^2*c[5]^4*c[4]^4+120*c[5]^2*c[4]^7*c[6]-94*c[4]^4*c[ 6]*c[5]^3+6*c[6]*c[5]*c[4]^5+60*c[5]^5*c[6]^2*c[4]^3-200*c[6]^2*c[4]^6 *c[5]*c[7]^2-420*c[7]^3*c[6]^2*c[5]*c[4]^4+1550*c[5]^4*c[4]^5*c[6]*c[7 ]+500*c[7]^3*c[4]^4*c[6]^2*c[5]^2-400*c[5]^2*c[4]^7*c[6]*c[7]^2+6*c[5] ^3*c[4]^4+103*c[6]^2*c[7]^3*c[5]^2*c[4]-260*c[6]^2*c[7]^3*c[5]^2*c[4]^ 2+540*c[5]^3*c[7]^3*c[4]^2*c[6]-97*c[6]^2*c[5]*c[7]^3*c[4]^2+30*c[6]^2 *c[7]^3*c[5]^3-4*c[7]^3*c[4]*c[5]^3)/(-240*c[5]^2*c[6]*c[4]^4+40*c[6]* c[5]*c[4]^4-9*c[5]^3*c[4]-100*c[4]^5*c[6]*c[5]^3-350*c[5]^4*c[4]^3*c[6 ]+30*c[5]^5*c[4]^2+2*c[5]^3-2*c[4]^3-50*c[5]^5*c[4]^3-3*c[5]^4+3*c[4]^ 4+50*c[4]^5*c[5]^3+100*c[5]^5*c[4]^3*c[6]-3*c[6]*c[5]^3+3*c[6]*c[4]^3+ 9*c[5]*c[4]^3-6*c[5]*c[6]*c[4]^2+4*c[5]*c[4]^2+240*c[6]*c[5]^4*c[4]^2+ 5*c[6]*c[5]^4-13*c[6]*c[5]*c[4]^3+13*c[5]^3*c[6]*c[4]-4*c[5]^2*c[4]+68 *c[5]^3*c[4]^2+100*c[5]^2*c[6]*c[4]^3-100*c[5]^3*c[6]*c[4]^2-140*c[5]^ 4*c[4]^2+140*c[4]^4*c[5]^2+180*c[5]^4*c[4]^3+6*c[5]^2*c[6]*c[4]-68*c[5 ]^2*c[4]^3-23*c[4]^4*c[5]-40*c[5]^4*c[4]*c[6]-30*c[4]^5*c[5]^2-5*c[6]* c[4]^4-50*c[5]^5*c[6]*c[4]^2+50*c[5]^2*c[6]*c[4]^5+350*c[4]^4*c[6]*c[5 ]^3+23*c[5]^4*c[4]-180*c[5]^3*c[4]^4)/(-c[4]+c[6])/c[4]^2, `b*`[7] = 1 /60*(-30*c[5]^3*c[4]-18*c[6]+6*c[5]+6*c[4]-9*c[5]^2-9*c[4]^2+27*c[6]^2 -97*c[6]*c[5]^2+66*c[6]*c[5]+60*c[6]*c[5]^3-100*c[6]^2*c[5]^3-100*c[6] ^2*c[4]^3+180*c[6]^2*c[4]^2-97*c[6]*c[4]^2+60*c[6]*c[4]^3-30*c[5]*c[4] ^3+312*c[5]*c[6]*c[4]^2+107*c[5]*c[4]^2-215*c[5]*c[6]*c[4]-66*c[5]*c[4 ]+66*c[6]*c[4]-840*c[6]^2*c[4]^2*c[5]+522*c[6]^2*c[4]*c[5]+500*c[6]^2* c[5]*c[4]^3+180*c[6]^2*c[5]^2-117*c[6]^2*c[5]-230*c[6]*c[5]*c[4]^3-230 *c[5]^3*c[6]*c[4]+107*c[5]^2*c[4]+50*c[5]^3*c[4]^2+380*c[5]^2*c[6]*c[4 ]^3-460*c[5]^2*c[6]*c[4]^2+380*c[5]^3*c[6]*c[4]^2+600*c[5]^3*c[6]^2*c[ 4]^3-900*c[6]^2*c[5]^2*c[4]^3+1420*c[6]^2*c[5]^2*c[4]^2-300*c[6]*c[5]^ 3*c[4]^3+500*c[6]^2*c[5]^3*c[4]-900*c[6]^2*c[5]^3*c[4]^2-180*c[5]^2*c[ 4]^2-840*c[6]^2*c[4]*c[5]^2-117*c[6]^2*c[4]+312*c[5]^2*c[6]*c[4]+50*c[ 5]^2*c[4]^3)/c[7]/(-3*c[7]^3*c[6]+3*c[7]^2*c[6]^2-9*c[6]^2*c[7]^2*c[5] -8*c[7]*c[6]^2*c[5]^3+8*c[7]^2*c[6]*c[5]^3-c[5]*c[6]*c[7]*c[4]-c[7]^2* c[4]^2-30*c[6]^2*c[7]*c[5]^2*c[4]^3-28*c[6]^2*c[7]^2*c[5]^2*c[4]+30*c[ 6]^2*c[7]^2*c[5]^2*c[4]^2-30*c[6]*c[7]*c[5]^3*c[4]^3+9*c[6]^2*c[7]*c[4 ]^2-3*c[6]^2*c[7]*c[4]+2*c[6]*c[7]^2*c[4]-9*c[6]*c[7]^2*c[4]^2-2*c[7]^ 2*c[5]*c[4]-c[7]^2*c[5]^2+c[6]*c[7]*c[4]^2+8*c[7]^2*c[6]*c[4]^3-4*c[5] *c[7]^2*c[4]^3+c[5]*c[7]*c[4]^2-c[5]*c[6]*c[4]^2+32*c[6]*c[5]*c[7]^2*c [4]^2-9*c[7]^2*c[5]*c[6]*c[4]-4*c[5]*c[7]*c[6]*c[4]^3-9*c[6]^2*c[4]^2* c[5]+3*c[6]^2*c[4]*c[5]+c[7]^3*c[4]+9*c[7]^3*c[4]*c[6]-9*c[7]^3*c[4]*c [5]-20*c[5]^2*c[6]*c[7]*c[4]^2+8*c[6]^2*c[5]*c[4]^3+28*c[7]^3*c[4]^2*c [6]*c[5]+18*c[6]^2*c[7]*c[4]*c[5]+28*c[7]*c[6]^2*c[5]*c[4]^3-3*c[6]^2* c[7]*c[5]+9*c[6]^2*c[7]*c[5]^2+c[6]*c[7]*c[5]^2-8*c[6]*c[7]^3*c[5]^2-9 *c[6]*c[7]^2*c[5]^2+8*c[6]^2*c[7]^2*c[5]^2+2*c[6]*c[7]^2*c[5]-36*c[6]^ 2*c[4]^2*c[7]*c[5]-28*c[6]^2*c[7]^2*c[4]^2*c[5]-28*c[7]^3*c[4]*c[6]*c[ 5]-36*c[6]^2*c[5]^2*c[7]*c[4]+56*c[6]^2*c[5]^2*c[7]*c[4]^2+28*c[6]*c[5 ]^2*c[7]*c[4]^3+28*c[6]*c[5]^3*c[7]*c[4]^2+28*c[6]^2*c[5]^3*c[7]*c[4]+ 9*c[6]*c[7]^3*c[5]+9*c[7]^2*c[5]*c[4]^2+28*c[7]^3*c[5]^2*c[6]*c[4]+32* c[7]^2*c[5]^2*c[6]*c[4]+8*c[6]^2*c[7]^2*c[4]^2-9*c[6]^2*c[7]^2*c[4]-4* c[5]^3*c[6]*c[7]*c[4]+28*c[6]^2*c[7]^2*c[4]*c[5]-4*c[5]^2*c[6]*c[4]^3- 9*c[5]^2*c[7]*c[4]^2+9*c[5]^2*c[6]*c[4]^2+4*c[5]^2*c[7]*c[4]^3+4*c[5]^ 3*c[7]*c[4]^2-4*c[5]^3*c[6]*c[4]^2+30*c[5]^3*c[6]^2*c[4]^3-28*c[6]^2*c [5]^2*c[4]^3+28*c[6]^2*c[5]^2*c[4]^2+8*c[6]^2*c[5]^3*c[4]-28*c[6]^2*c[ 5]^3*c[4]^2-9*c[6]^2*c[4]*c[5]^2-8*c[7]*c[6]^2*c[4]^3+30*c[7]^2*c[5]^2 *c[6]*c[4]^3-30*c[7]^3*c[5]^2*c[6]*c[4]^2+30*c[7]^2*c[5]^3*c[6]*c[4]^2 -8*c[7]^3*c[4]^2*c[6]+4*c[7]^3*c[4]^2*c[5]+c[7]^3*c[5]-28*c[7]^2*c[6]* c[5]*c[4]^3-56*c[7]^2*c[5]^2*c[6]*c[4]^2-28*c[7]^2*c[5]^3*c[6]*c[4]-c[ 5]^2*c[6]*c[4]+c[5]^2*c[7]*c[4]+4*c[7]^3*c[5]^2*c[4]+9*c[7]^2*c[5]^2*c [4]-30*c[6]^2*c[7]*c[5]^3*c[4]^2-8*c[7]^2*c[5]^2*c[4]^2-4*c[7]^2*c[5]^ 3*c[4]), a[2,1] = c[2], a[6,4] = -1/2*(120*c[5]^2*c[6]*c[4]^4-12*c[6]* c[5]*c[4]^4-90*c[5]^2*c[6]^3*c[4]^2-60*c[5]*c[6]^2*c[4]^4-2*c[6]^3*c[4 ]^2+20*c[5]^4*c[4]^3*c[6]+40*c[6]^2*c[4]^5*c[5]+36*c[4]*c[5]^2*c[6]^3- 20*c[5]^4*c[6]^2*c[4]^2+8*c[6]^2*c[5]^3+2*c[6]^2*c[4]^3-24*c[5]*c[6]^3 *c[4]^2-6*c[6]^2*c[4]^2*c[5]+36*c[6]^2*c[5]*c[4]^3+60*c[4]^3*c[5]*c[6] ^3-3*c[6]*c[5]*c[4]^3-11*c[5]^3*c[6]*c[4]+3*c[5]^3*c[4]^2-24*c[5]^2*c[ 6]*c[4]^3+13*c[5]^2*c[6]*c[4]^2+48*c[5]^3*c[6]*c[4]^2-60*c[5]^3*c[6]^2 *c[4]^3-30*c[6]^2*c[5]^2*c[4]^3-12*c[6]^2*c[5]^2*c[4]^2-150*c[6]*c[5]^ 3*c[4]^3-48*c[6]^2*c[5]^3*c[4]+150*c[6]^2*c[5]^3*c[4]^2-2*c[6]^2*c[4]* c[5]^2-5*c[5]^2*c[4]^3+2*c[4]^4*c[5]-6*c[5]^2*c[6]^3+60*c[6]^3*c[4]^3* c[5]^2+7*c[6]^3*c[4]*c[5]-60*c[5]^2*c[6]*c[4]^5-40*c[6]^3*c[4]^4*c[5]+ 60*c[4]^4*c[6]*c[5]^3)*c[6]/(c[4]^3-2*c[5]*c[4]^2+30*c[5]^2*c[4]^3-6*c [5]*c[4]^3-10*c[4]^4*c[5]^2-c[5]^3+10*c[5]^4*c[4]^2-30*c[5]^3*c[4]^2+6 *c[5]^3*c[4]+2*c[5]^2*c[4])/c[4]^2, a[3,1] = -2/9*c[4]*(c[4]-3*c[2])/c [2], a[4,1] = 1/4*c[4], a[7,3] = 3/4*1/c[4]^2*c[7]*(-480*c[5]^2*c[6]*c [4]^4+15*c[6]*c[5]*c[4]^4+300*c[4]^2*c[5]^2*c[7]^3+60*c[7]^3*c[5]*c[4] ^3-510*c[7]^3*c[5]^2*c[4]^3+54*c[7]*c[5]*c[4]^4+450*c[5]^2*c[4]^4*c[7] ^2+150*c[4]^3*c[7]^3*c[5]^3-150*c[7]^2*c[5]^2*c[4]^5-450*c[4]^5*c[6]*c [5]^3-3*c[7]^3*c[4]^2+90*c[5]^2*c[4]^5*c[7]+120*c[5]^3*c[4]^4*c[7]+150 *c[7]^3*c[5]^2*c[4]^4+100*c[5]^4*c[4]^2*c[7]^2*c[6]-180*c[6]*c[4]^5*c[ 5]*c[7]+30*c[6]*c[5]^2*c[4]^4*c[7]+60*c[5]^4*c[4]^3*c[6]-150*c[5]^4*c[ 4]^4*c[6]-200*c[7]^2*c[4]^4*c[5]^3+300*c[6]*c[4]^5*c[5]*c[7]^2-50*c[7] ^2*c[5]^4*c[4]^3+24*c[7]*c[6]*c[5]^3-40*c[7]^2*c[6]*c[5]^3-6*c[5]*c[6] *c[7]*c[4]+360*c[7]*c[4]^4*c[6]*c[5]+150*c[7]^2*c[5]^3*c[4]^3+2*c[7]^2 *c[4]^2+6*c[5]^2*c[7]^3+330*c[6]*c[7]*c[5]^3*c[4]^3-450*c[7]^2*c[6]*c[ 5]*c[4]^4-12*c[7]*c[6]*c[4]^3-6*c[6]*c[7]^2*c[4]-2*c[7]^2*c[5]*c[4]-4* c[7]^2*c[5]^2+6*c[6]*c[7]*c[4]^2+20*c[7]^2*c[6]*c[4]^3-10*c[5]*c[7]^2* c[4]^3+6*c[5]*c[4]^3-6*c[5]*c[7]*c[4]^2-6*c[5]*c[6]*c[4]^2+150*c[6]*c[ 5]*c[7]^2*c[4]^2-72*c[7]^2*c[5]*c[6]*c[4]-270*c[5]*c[7]*c[6]*c[4]^3+9* c[7]^3*c[4]*c[6]+3*c[7]^3*c[4]*c[5]-306*c[5]^2*c[6]*c[7]*c[4]^2+900*c[ 4]^5*c[6]*c[7]*c[5]^3+600*c[5]^2*c[6]*c[7]^3*c[4]^4-420*c[7]^3*c[4]^2* c[6]*c[5]+90*c[6]*c[5]*c[7]*c[4]^2-12*c[6]*c[7]*c[5]^2+30*c[6]*c[7]^3* c[5]^2+12*c[6]*c[7]^2*c[5]+123*c[7]^3*c[4]*c[6]*c[5]+360*c[6]*c[5]^2*c [7]*c[4]^3+360*c[6]*c[5]^3*c[7]*c[4]^2-18*c[6]*c[7]^3*c[5]+30*c[5]^4*c [4]^3*c[7]-80*c[7]^2*c[5]^2*c[4]^3+24*c[7]^2*c[5]*c[4]^2-30*c[7]*c[5]* c[4]^3+3*c[6]*c[5]*c[4]^3-240*c[7]^3*c[5]^2*c[6]*c[4]-24*c[5]^3*c[6]*c [4]-12*c[5]^3*c[7]*c[4]-144*c[5]^3*c[6]*c[7]*c[4]+9*c[5]^3*c[4]^2+294* c[5]^2*c[6]*c[4]^3-42*c[5]^2*c[7]*c[4]^2-75*c[5]^2*c[6]*c[4]^2+252*c[5 ]^2*c[7]*c[4]^3+18*c[5]^3*c[7]*c[4]^2+189*c[5]^3*c[6]*c[4]^2-360*c[5]^ 2*c[4]^4*c[7]-720*c[6]*c[5]^3*c[4]^3-120*c[5]^3*c[7]*c[4]^3-60*c[5]^4* c[7]*c[4]^2*c[6]-6*c[5]^2*c[4]^2-780*c[7]^2*c[5]^2*c[6]*c[4]^3+810*c[7 ]^3*c[5]^2*c[6]*c[4]^2-1230*c[7]^2*c[5]^3*c[6]*c[4]^2-15*c[7]^3*c[4]^2 *c[6]-21*c[7]^3*c[4]^2*c[5]+20*c[7]^2*c[6]*c[5]*c[4]^3+220*c[7]^2*c[5] ^2*c[6]*c[4]^2+15*c[4]^4*c[5]^2+320*c[7]^2*c[5]^3*c[6]*c[4]+1850*c[7]^ 2*c[5]^3*c[6]*c[4]^3-60*c[5]^3*c[7]^3*c[4]^2+12*c[5]^2*c[6]*c[4]+12*c[ 5]^2*c[7]*c[4]-66*c[7]^3*c[5]^2*c[4]+24*c[7]^2*c[5]^2*c[4]+72*c[5]^2*c [6]*c[7]*c[4]-110*c[7]^2*c[5]^2*c[4]^2+20*c[7]^2*c[5]^3*c[4]-9*c[4]^4* c[5]-60*c[7]^2*c[4]^4*c[5]-600*c[5]^2*c[6]*c[4]^5*c[7]^2-1200*c[7]^3*c [5]^2*c[4]^3*c[6]+630*c[7]^3*c[6]*c[5]*c[4]^3+300*c[5]^4*c[4]^4*c[6]*c [7]-1800*c[5]^3*c[4]^4*c[6]*c[7]-15*c[5]^3*c[4]^3-300*c[7]^3*c[4]^4*c[ 5]*c[6]-20*c[7]^2*c[5]^3*c[4]^2-800*c[5]^3*c[4]^4*c[7]^2*c[6]-200*c[5] ^4*c[4]^3*c[7]^2*c[6]+1150*c[5]^2*c[4]^4*c[7]^2*c[6]+180*c[5]^2*c[6]*c [4]^5+1140*c[4]^4*c[6]*c[5]^3)/(10*c[5]^3*c[4]^2+10*c[5]^2*c[4]^3-30*c [5]^2*c[4]^2+6*c[5]^2*c[4]-c[5]^2+6*c[5]*c[4]^2+c[5]*c[4]-c[4]^2)/(10* c[5]*c[6]*c[4]-5*c[6]*c[5]-5*c[6]*c[4]+3*c[6]-5*c[5]*c[4]+3*c[5]+3*c[4 ]-2), `b*`[1] = 1/60*(30*c[5]^3*c[4]+42*c[7]*c[6]^2-103*c[5]*c[7]*c[4] +18*c[6]-6*c[5]-6*c[4]-27*c[7]*c[6]+9*c[5]^2+9*c[4]^2-180*c[7]*c[6]^2* c[5]^3+100*c[7]*c[6]*c[5]^3-27*c[6]^2-354*c[5]*c[6]*c[7]*c[4]+97*c[6]* c[5]^2-66*c[6]*c[5]-60*c[6]*c[5]^3+100*c[6]^2*c[5]^3+9*c[7]*c[5]+100*c [6]^2*c[4]^3-2040*c[6]^2*c[7]*c[5]^2*c[4]^3-600*c[6]*c[7]*c[5]^3*c[4]^ 3-14*c[7]*c[5]^2+100*c[7]*c[6]*c[4]^3+310*c[6]^2*c[7]*c[4]^2-192*c[6]^ 2*c[7]*c[4]-180*c[6]^2*c[4]^2-14*c[7]*c[4]^2+9*c[7]*c[4]+103*c[6]*c[7] *c[5]+97*c[6]*c[4]^2-60*c[6]*c[4]^3+103*c[6]*c[7]*c[4]-156*c[6]*c[7]*c [4]^2+30*c[5]*c[4]^3+174*c[5]*c[7]*c[4]^2-312*c[5]*c[6]*c[4]^2-107*c[5 ]*c[4]^2+215*c[5]*c[6]*c[4]+66*c[5]*c[4]-66*c[6]*c[4]-410*c[5]*c[7]*c[ 6]*c[4]^3+840*c[6]^2*c[4]^2*c[5]-522*c[6]^2*c[4]*c[5]-760*c[5]^2*c[6]* c[7]*c[4]^2-500*c[6]^2*c[5]*c[4]^3+932*c[6]^2*c[7]*c[4]*c[5]+1020*c[7] *c[6]^2*c[5]*c[4]^3-192*c[6]^2*c[7]*c[5]+310*c[6]^2*c[7]*c[5]^2+526*c[ 6]*c[5]*c[7]*c[4]^2-156*c[6]*c[7]*c[5]^2-180*c[6]^2*c[5]^2-1610*c[6]^2 *c[4]^2*c[7]*c[5]-1610*c[6]^2*c[5]^2*c[7]*c[4]+2970*c[6]^2*c[5]^2*c[7] *c[4]^2+690*c[6]*c[5]^2*c[7]*c[4]^3+690*c[6]*c[5]^3*c[7]*c[4]^2+1020*c [6]^2*c[5]^3*c[7]*c[4]+117*c[6]^2*c[5]-50*c[7]*c[5]*c[4]^3+230*c[6]*c[ 5]*c[4]^3+230*c[5]^3*c[6]*c[4]-50*c[5]^3*c[7]*c[4]-107*c[5]^2*c[4]-410 *c[5]^3*c[6]*c[7]*c[4]-50*c[5]^3*c[4]^2-380*c[5]^2*c[6]*c[4]^3-310*c[5 ]^2*c[7]*c[4]^2+460*c[5]^2*c[6]*c[4]^2+90*c[5]^2*c[7]*c[4]^3+90*c[5]^3 *c[7]*c[4]^2-380*c[5]^3*c[6]*c[4]^2-600*c[5]^3*c[6]^2*c[4]^3+900*c[6]^ 2*c[5]^2*c[4]^3-1420*c[6]^2*c[5]^2*c[4]^2+300*c[6]*c[5]^3*c[4]^3-500*c [6]^2*c[5]^3*c[4]+900*c[6]^2*c[5]^3*c[4]^2+180*c[5]^2*c[4]^2+840*c[6]^ 2*c[4]*c[5]^2-180*c[7]*c[6]^2*c[4]^3+117*c[6]^2*c[4]+1500*c[5]^3*c[6]^ 2*c[7]*c[4]^3-312*c[5]^2*c[6]*c[4]+174*c[5]^2*c[7]*c[4]+526*c[5]^2*c[6 ]*c[7]*c[4]-2040*c[6]^2*c[7]*c[5]^3*c[4]^2-50*c[5]^2*c[4]^3)/(3*c[6]-2 8*c[5]*c[6]*c[4]^2-28*c[5]^2*c[6]*c[4]+8*c[6]*c[5]^2-9*c[6]*c[4]-9*c[6 ]*c[5]+28*c[5]*c[6]*c[4]+30*c[5]^2*c[6]*c[4]^2+8*c[6]*c[4]^2-4*c[5]^2* c[4]-4*c[5]*c[4]^2+9*c[5]*c[4]-c[5]-c[4])/c[5]/c[6]/c[7]/c[4], a[8,4] \+ = -1/2*(734*c[5]^2*c[6]*c[4]^4-203*c[6]*c[5]*c[4]^4+3190*c[5]^4*c[4]^4 *c[6]^2*c[7]-93*c[7]*c[5]*c[4]^4-636*c[5]^2*c[4]^4*c[7]^2-10*c[5]^3*c[ 4]-520*c[6]*c[4]^7*c[7]*c[5]-481*c[4]^5*c[6]^2*c[5]^2+645*c[7]^2*c[5]^ 2*c[4]^5-574*c[4]^5*c[6]*c[5]^3-172*c[5]*c[6]^2*c[4]^4+6*c[7]^2*c[4]^3 +100*c[5]^5*c[4]^3*c[7]^2*c[6]^2+60*c[4]^8*c[5]^2-600*c[5]^2*c[4]^6*c[ 6]^2*c[7]^2+9*c[4]^5*c[5]*c[7]-164*c[5]^2*c[4]^5*c[7]+900*c[5]^3*c[4]^ 6*c[6]^2*c[7]+600*c[5]^3*c[4]^6*c[6]^2*c[7]^2-1104*c[5]^3*c[4]^4*c[7]- 18*c[4]^4*c[6]^2+90*c[7]^2*c[4]^5*c[5]^3+1380*c[6]^2*c[4]^5*c[7]*c[5]^ 3-750*c[7]*c[5]^2*c[4]^6*c[6]+675*c[5]^4*c[4]^2*c[7]^2*c[6]+8*c[6]^2*c [4]^5-339*c[6]^2*c[7]*c[5]^4*c[4]^2-315*c[5]^2*c[4]^6*c[7]+640*c[6]*c[ 5]*c[7]^2*c[4]^6+46*c[6]*c[4]^5*c[5]*c[7]-1014*c[6]*c[5]^2*c[4]^4*c[7] +2100*c[5]^4*c[7]*c[4]^6*c[6]-390*c[6]^2*c[4]^6*c[5]+860*c[4]^6*c[6]^2 *c[5]^2+140*c[5]^5*c[4]^3*c[7]-1740*c[5]^2*c[7]^2*c[4]^6*c[6]-613*c[5] ^4*c[4]^3*c[6]-100*c[4]^8*c[5]^2*c[7]+1140*c[5]^4*c[4]^5*c[6]+968*c[5] ^2*c[6]*c[4]^6-150*c[5]^5*c[4]^5*c[6]^2+263*c[5]^4*c[4]^4*c[6]+250*c[7 ]^2*c[4]^4*c[5]^3-120*c[6]*c[5]*c[4]^8-10*c[6]*c[7]^2*c[4]^5-6*c[6]*c[ 4]^6+780*c[5]^4*c[4]^4*c[7]+32*c[7]^2*c[6]*c[4]^4+1200*c[5]^3*c[7]^2*c [4]^6*c[6]-750*c[6]*c[4]^5*c[5]*c[7]^2-6*c[4]^4*c[7]^2-400*c[5]^4*c[7] ^2*c[4]^5*c[6]^2+415*c[7]^2*c[5]^4*c[4]^3-990*c[5]^3*c[7]*c[4]^6*c[6]+ 433*c[6]^2*c[4]^5*c[5]+4*c[4]^3-60*c[5]^5*c[4]^3+30*c[7]*c[6]^2*c[5]^3 +4*c[7]*c[6]*c[5]^3-6*c[7]^2*c[6]*c[5]^3-350*c[5]^4*c[4]^6*c[6]^2-5*c[ 4]^4-250*c[5]^3*c[7]*c[4]^6-270*c[5]^4*c[4]^5*c[7]-330*c[7]*c[6]^2*c[5 ]*c[4]^5-1320*c[4]^6*c[6]^2*c[7]*c[5]^2-55*c[5]^4*c[4]^2*c[7]^2-100*c[ 5]^5*c[7]^2*c[4]^3-96*c[6]*c[5]^2*c[4]^5*c[7]-562*c[4]^5*c[5]^3-600*c[ 5]^5*c[4]^4*c[7]^2*c[6]+380*c[5]^2*c[4]^7*c[7]+160*c[5]^5*c[4]^3*c[6]+ 62*c[7]*c[4]^4*c[6]*c[5]+80*c[5]^5*c[4]^4+30*c[5]*c[4]^6*c[7]^2+310*c[ 4]^4*c[6]^2*c[5]^5+450*c[5]^5*c[7]*c[4]^4*c[6]-356*c[7]^2*c[5]^3*c[4]^ 3+24*c[7]*c[4]^4*c[6]^2-200*c[5]^5*c[7]*c[4]^6*c[6]-900*c[5]^3*c[7]*c[ 4]^7*c[6]+60*c[5]^5*c[4]^2*c[6]^2*c[7]-450*c[5]^5*c[4]^4*c[6]^2*c[7]-1 70*c[5]^5*c[4]^4*c[6]+200*c[5]^5*c[7]*c[4]^5*c[6]+143*c[6]^2*c[5]^4*c[ 7]*c[4]+34*c[5]^4*c[6]^2*c[4]-283*c[5]^4*c[6]^2*c[4]^2-530*c[5]^3*c[4] ^6*c[6]^2+9*c[4]^4*c[7]+600*c[5]^4*c[7]^2*c[4]^6*c[6]-140*c[5]^5*c[4]^ 4*c[7]-55*c[4]^5*c[5]*c[7]^2+10*c[6]^2*c[4]^3+100*c[5]^5*c[4]^4*c[7]^2 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7]+35*c[4]^6*c[7]*c[5]+80*c[7]^2*c[4]^4*c[5]-1163*c[7]^2*c[5]^3*c[6]^2 *c[4]^2+20*c[5]*c[4]^7-640*c[6]^2*c[4]^5*c[5]*c[7]^2+850*c[7]^2*c[4]^5 *c[6]*c[5]^3+1530*c[5]^2*c[6]*c[4]^5*c[7]^2-506*c[5]^4*c[4]^4+1420*c[5 ]^2*c[4]^7*c[6]*c[7]-16*c[5]^4*c[4]*c[6]-85*c[5]^4*c[7]^2*c[4]*c[6]+72 *c[5]^4*c[4]*c[7]*c[6]+230*c[4]^5*c[6]^2*c[7]*c[5]^2+1740*c[4]^5*c[6]^ 2*c[7]^2*c[5]^2-600*c[6]*c[4]^8*c[5]^2*c[7]-82*c[4]^5*c[5]^2-480*c[6]^ 2*c[4]^4*c[5]^2+320*c[4]^8*c[5]^2*c[6]-37*c[4]^6*c[5]+300*c[5]^3*c[4]^ 7*c[6]^2-90*c[5]^5*c[4]^5*c[6]+520*c[5]*c[7]*c[4]^6*c[6]^2-10*c[6]^2*c [7]*c[4]^5+9*c[6]*c[4]^4+600*c[5]^4*c[7]*c[4]^6*c[6]^2-2350*c[5]^4*c[4 ]^5*c[6]^2*c[7]-600*c[5]^3*c[7]^2*c[6]*c[4]^7-204*c[7]*c[6]^2*c[5]*c[4 ]^4-3630*c[5]^3*c[4]^4*c[6]^2*c[7]+740*c[5]^3*c[4]^3*c[6]^2*c[7]^2-212 0*c[5]^4*c[4]^4*c[6]*c[7]+1460*c[5]^2*c[4]^4*c[7]*c[6]^2+2387*c[5]^3*c [4]^4*c[6]*c[7]+2548*c[5]^4*c[4]^3*c[6]*c[7]-498*c[5]^3*c[4]^3+100*c[5 ]^2*c[4]^7*c[7]^2+26*c[7]^2*c[5]^3*c[4]^2+600*c[6]*c[5]^3*c[4]^7+100*c 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5]^3-264*c[5]*c[6]^2*c[4]^4-12*c[7]^2*c[4]^3+700*c[5]^5*c[4]^3*c[7]^2* c[6]^2-200*c[5]^2*c[4]^6*c[6]^2*c[7]^2-246*c[4]^5*c[5]*c[7]+1818*c[5]^ 2*c[4]^5*c[7]+1300*c[5]^3*c[4]^6*c[6]^2*c[7]-800*c[5]^3*c[4]^6*c[6]^2* c[7]^2+5526*c[5]^3*c[4]^4*c[7]+32*c[4]^4*c[6]^2+3160*c[7]^2*c[4]^5*c[5 ]^3-7740*c[6]^2*c[4]^5*c[7]*c[5]^3+780*c[7]*c[5]^2*c[4]^6*c[6]-325*c[5 ]^4*c[4]^2*c[7]^2*c[6]-20*c[6]^2*c[4]^5-1339*c[6]^2*c[7]*c[5]^4*c[4]^2 -280*c[5]^2*c[4]^6*c[7]+668*c[6]*c[4]^5*c[5]*c[7]+6574*c[6]*c[5]^2*c[4 ]^4*c[7]+1500*c[5]^4*c[7]*c[4]^6*c[6]+200*c[4]^6*c[6]^2*c[5]^2-280*c[5 ]^5*c[4]^3*c[7]-600*c[5]^2*c[7]^2*c[4]^6*c[6]+2092*c[5]^4*c[4]^3*c[6]+ 3640*c[5]^4*c[4]^5*c[6]-320*c[5]^2*c[6]*c[4]^6+400*c[5]^5*c[4]^5*c[6]^ 2-4880*c[5]^4*c[4]^4*c[6]-4080*c[7]^2*c[4]^4*c[5]^3+50*c[6]*c[7]^2*c[4 ]^5-2400*c[5]^4*c[4]^5*c[7]^2-4160*c[5]^4*c[4]^4*c[7]-76*c[7]^2*c[6]*c [4]^4+1600*c[5]^3*c[7]^2*c[4]^6*c[6]-490*c[6]*c[4]^5*c[5]*c[7]^2+32*c[ 4]^4*c[7]^2-7100*c[5]^4*c[7]^2*c[4]^5*c[6]^2-1320*c[7]^2*c[5]^4*c[4]^3 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2*c[5]^2*c[4]^3+72*c[6]^2*c[5]^2*c[4]^2-2816*c[6]*c[5]^3*c[4]^3+32*c[6 ]^2*c[5]^3*c[4]-264*c[6]^2*c[5]^3*c[4]^2-2472*c[5]^3*c[7]*c[4]^3+566*c [5]^4*c[7]*c[4]^2*c[6]+48*c[5]^2*c[4]^2+1200*c[6]^2*c[4]^5*c[5]^5*c[7] ^2-12*c[6]^2*c[4]*c[5]^2+38*c[7]*c[6]^2*c[4]^3+1804*c[7]^2*c[5]^2*c[6] *c[4]^3+258*c[7]^2*c[5]^3*c[6]*c[4]^2+104*c[5]^4*c[4]^2-500*c[5]^5*c[4 ]^5*c[7]-256*c[4]^3*c[7]^2*c[5]*c[6]^2-176*c[7]^2*c[6]*c[5]*c[4]^3+200 *c[5]^5*c[4]^5-212*c[7]^2*c[5]^2*c[6]*c[4]^2+1064*c[4]^4*c[5]^2+8*c[7] ^2*c[5]^3*c[6]*c[4]-3410*c[7]^2*c[5]^3*c[6]*c[4]^3-4080*c[5]^3*c[6]^2* c[4]^4-772*c[5]^4*c[4]^3+1200*c[7]^2*c[6]^2*c[4]^6*c[5]^4-36*c[6]^2*c[ 4]^3*c[7]^2-5368*c[5]^3*c[6]^2*c[7]*c[4]^3+20*c[5]^2*c[6]*c[4]+18*c[5] ^2*c[7]*c[4]-12*c[7]^2*c[5]^2*c[4]-44*c[5]^2*c[6]*c[7]*c[4]+400*c[5]^4 *c[4]^6*c[7]^2+1438*c[6]^2*c[7]*c[5]^3*c[4]^2+426*c[6]^2*c[7]^2*c[4]*c [5]^3-452*c[5]^2*c[4]^3+72*c[7]^2*c[5]^2*c[4]^2+32*c[7]^2*c[5]^3*c[4]- 160*c[4]^4*c[5]+690*c[5]^5*c[4]^3*c[6]*c[7]-264*c[7]^2*c[4]^4*c[5]-183 3*c[7]^2*c[5]^3*c[6]^2*c[4]^2-260*c[6]^2*c[4]^5*c[5]*c[7]^2-8270*c[7]^ 2*c[4]^5*c[6]*c[5]^3+3780*c[5]^2*c[6]*c[4]^5*c[7]^2+1720*c[5]^4*c[4]^4 +32*c[5]^4*c[4]*c[6]+30*c[5]^4*c[7]^2*c[4]*c[6]-60*c[5]^4*c[4]*c[7]*c[ 6]+2480*c[4]^5*c[6]^2*c[7]*c[5]^2-1120*c[4]^5*c[6]^2*c[7]^2*c[5]^2-772 *c[4]^5*c[5]^2-40*c[6]^2*c[7]^2*c[4]^5+1752*c[6]^2*c[4]^4*c[5]^2-1700* c[7]^2*c[6]^2*c[5]^5*c[4]^4-600*c[5]^5*c[4]^5*c[6]-120*c[5]*c[7]*c[4]^ 6*c[6]^2+50*c[6]^2*c[7]*c[4]^5-52*c[6]*c[4]^4-1200*c[5]^4*c[7]*c[4]^6* c[6]^2+7150*c[5]^4*c[4]^5*c[6]^2*c[7]+198*c[7]*c[6]^2*c[5]*c[4]^4+1056 0*c[5]^3*c[4]^4*c[6]^2*c[7]+5240*c[5]^3*c[4]^3*c[6]^2*c[7]^2+11240*c[5 ]^4*c[4]^4*c[6]*c[7]-3649*c[5]^2*c[4]^4*c[7]*c[6]^2-14207*c[5]^3*c[4]^ 4*c[6]*c[7]-4491*c[5]^4*c[4]^3*c[6]*c[7]+1064*c[5]^3*c[4]^3-264*c[7]^2 *c[5]^3*c[4]^2-2400*c[5]^4*c[4]^5*c[6]^2+6020*c[6]^2*c[7]^2*c[5]^3*c[4 ]^5+6800*c[6]*c[7]^2*c[5]^4*c[4]^5+3160*c[5]^4*c[4]^4*c[6]^2-1320*c[5] ^4*c[4]^3*c[6]^2-5800*c[7]^2*c[5]^4*c[6]^2*c[4]^3+1940*c[7]^2*c[5]^4*c [6]^2*c[4]^2-600*c[5]^4*c[4]^6*c[6]+28*c[4]^5*c[7]+9740*c[5]^3*c[4]^4* c[7]^2*c[6]+2920*c[5]^4*c[4]^3*c[7]^2*c[6]-4775*c[5]^2*c[4]^4*c[7]^2*c [6]+4980*c[5]^4*c[4]^3*c[6]^2*c[7]+210*c[7]^2*c[4]^4*c[6]^2*c[5]-8860* c[7]^2*c[4]^4*c[5]^3*c[6]^2+2140*c[7]^2*c[4]^4*c[6]^2*c[5]^2+120*c[4]^ 6*c[5]^2-20*c[7]^2*c[4]^5+2092*c[5]^2*c[6]*c[4]^5+9920*c[6]^2*c[7]^2*c [5]^4*c[4]^4-500*c[5]^4*c[4]^6*c[7]-48*c[7]^2*c[6]^2*c[5]^3+6404*c[4]^ 4*c[6]*c[5]^3-284*c[6]*c[5]*c[4]^5+200*c[5]^5*c[6]^2*c[4]^3+200*c[6]^2 *c[4]^6*c[5]*c[7]^2-8820*c[5]^4*c[4]^5*c[6]*c[7]-12*c[5]^4*c[4]-2324*c [5]^3*c[4]^4)/c[6]/c[5]/c[7]/c[4]^2/(150*c[5]^2*c[6]*c[4]^4-87*c[5]^3* c[4]-12*c[5]*c[7]*c[4]-200*c[5]^3*c[4]^4*c[7]-200*c[6]*c[5]^2*c[4]^4*c [7]-200*c[5]^4*c[4]^3*c[6]-10*c[5]^2-10*c[4]^2+12*c[5]^3+12*c[4]^3+20* c[7]*c[6]*c[5]^3+15*c[5]*c[6]*c[7]*c[4]+12*c[6]*c[5]^2-15*c[6]*c[5]^3- 1300*c[6]*c[7]*c[5]^3*c[4]^3+150*c[5]^4*c[7]*c[4]^2+12*c[7]*c[5]^2+20* c[7]*c[6]*c[4]^3+12*c[7]*c[4]^2-15*c[7]*c[4]^3+12*c[6]*c[4]^2-15*c[6]* c[4]^3-15*c[6]*c[7]*c[4]^2-87*c[5]*c[4]^3-72*c[5]*c[7]*c[4]^2-72*c[5]* c[6]*c[4]^2+60*c[5]*c[4]^2-12*c[5]*c[6]*c[4]+10*c[5]*c[4]-150*c[5]*c[7 ]*c[6]*c[4]^3-15*c[7]*c[5]^3-660*c[5]^2*c[6]*c[7]*c[4]^2+90*c[6]*c[5]* c[7]*c[4]^2-15*c[6]*c[7]*c[5]^2+150*c[6]*c[5]^4*c[4]^2+930*c[6]*c[5]^2 *c[7]*c[4]^3+930*c[6]*c[5]^3*c[7]*c[4]^2-200*c[5]^4*c[4]^3*c[7]+110*c[ 7]*c[5]*c[4]^3+110*c[6]*c[5]*c[4]^3+110*c[5]^3*c[6]*c[4]+110*c[5]^3*c[ 7]*c[4]+60*c[5]^2*c[4]-150*c[5]^3*c[6]*c[7]*c[4]+550*c[5]^3*c[4]^2-690 *c[5]^2*c[6]*c[4]^3+520*c[5]^2*c[7]*c[4]^2+520*c[5]^2*c[6]*c[4]^2-690* c[5]^2*c[7]*c[4]^3-690*c[5]^3*c[7]*c[4]^2-690*c[5]^3*c[6]*c[4]^2+150*c [5]^2*c[4]^4*c[7]+900*c[6]*c[5]^3*c[4]^3+900*c[5]^3*c[7]*c[4]^3-200*c[ 5]^4*c[7]*c[4]^2*c[6]-429*c[5]^2*c[4]^2-120*c[5]^4*c[4]^2-120*c[4]^4*c [5]^2+150*c[5]^4*c[4]^3-72*c[5]^2*c[6]*c[4]-72*c[5]^2*c[7]*c[4]+90*c[5 ]^2*c[6]*c[7]*c[4]+550*c[5]^2*c[4]^3+300*c[5]^3*c[4]^4*c[6]*c[7]+300*c [5]^4*c[4]^3*c[6]*c[7]-690*c[5]^3*c[4]^3-200*c[4]^4*c[6]*c[5]^3+150*c[ 5]^3*c[4]^4), a[8,3] = 3/4*(-1560*c[5]^2*c[6]*c[4]^4+585*c[6]*c[5]*c[4 ]^4-45*c[7]*c[5]*c[4]^4-12*c[5]^3*c[4]-450*c[4]^5*c[6]*c[5]^3+3*c[5]*c [7]*c[4]-150*c[5]^2*c[4]^5*c[7]-200*c[5]^3*c[4]^4*c[7]+300*c[6]*c[4]^5 *c[5]*c[7]+2860*c[6]*c[5]^2*c[4]^4*c[7]+160*c[5]^4*c[4]^3*c[6]-150*c[5 ]^4*c[4]^4*c[6]-4*c[5]^2+2*c[4]^2-40*c[7]*c[6]*c[5]^3+132*c[5]*c[6]*c[ 7]*c[4]-900*c[6]*c[5]^2*c[4]^5*c[7]-30*c[6]*c[5]^2+12*c[6]*c[5]-960*c[ 7]*c[4]^4*c[6]*c[5]+24*c[6]*c[5]^3+3140*c[6]*c[7]*c[5]^3*c[4]^3+6*c[7] *c[5]^2+20*c[7]*c[6]*c[4]^3-3*c[7]*c[4]^2-18*c[6]*c[7]*c[5]+15*c[6]*c[ 4]^2-12*c[6]*c[4]^3+9*c[6]*c[7]*c[4]-24*c[6]*c[7]*c[4]^2-54*c[5]*c[4]^ 3-12*c[5]*c[7]*c[4]^2+348*c[5]*c[6]*c[4]^2+9*c[5]*c[4]^2-87*c[5]*c[6]* c[4]-2*c[5]*c[4]-6*c[6]*c[4]+1035*c[5]*c[7]*c[6]*c[4]^3+1593*c[5]^2*c[ 6]*c[7]*c[4]^2+900*c[4]^5*c[6]*c[7]*c[5]^3-534*c[6]*c[5]*c[7]*c[4]^2+4 8*c[6]*c[7]*c[5]^2-60*c[6]*c[5]^4*c[4]^2-3060*c[6]*c[5]^2*c[7]*c[4]^3- 1560*c[6]*c[5]^3*c[7]*c[4]^2-50*c[5]^4*c[4]^3*c[7]+80*c[7]*c[5]*c[4]^3 -652*c[6]*c[5]*c[4]^3-208*c[5]^3*c[6]*c[4]+20*c[5]^3*c[7]*c[4]+54*c[5] ^2*c[4]+360*c[5]^3*c[6]*c[7]*c[4]+82*c[5]^3*c[4]^2+1734*c[5]^2*c[6]*c[ 4]^3+388*c[5]^2*c[7]*c[4]^2-944*c[5]^2*c[6]*c[4]^2-900*c[5]^2*c[7]*c[4 ]^3-135*c[5]^3*c[7]*c[4]^2+879*c[5]^3*c[6]*c[4]^2+660*c[5]^2*c[4]^4*c[ 7]-1680*c[6]*c[5]^3*c[4]^3+420*c[5]^3*c[7]*c[4]^3+100*c[5]^4*c[7]*c[4] ^2*c[6]-246*c[5]^2*c[4]^2-405*c[4]^4*c[5]^2+30*c[5]^4*c[4]^3+240*c[5]^ 2*c[6]*c[4]-84*c[5]^2*c[7]*c[4]-396*c[5]^2*c[6]*c[7]*c[4]+562*c[5]^2*c [4]^3+30*c[4]^4*c[5]+90*c[4]^5*c[5]^2+300*c[5]^4*c[4]^4*c[6]*c[7]-3000 *c[5]^3*c[4]^4*c[6]*c[7]-300*c[5]^4*c[4]^3*c[6]*c[7]-255*c[5]^3*c[4]^3 +480*c[5]^2*c[6]*c[4]^5+1540*c[4]^4*c[6]*c[5]^3-180*c[6]*c[5]*c[4]^5+1 20*c[5]^3*c[4]^4)/c[4]^2/(150*c[5]^2*c[6]*c[4]^4-87*c[5]^3*c[4]-12*c[5 ]*c[7]*c[4]-200*c[5]^3*c[4]^4*c[7]-200*c[6]*c[5]^2*c[4]^4*c[7]-200*c[5 ]^4*c[4]^3*c[6]-10*c[5]^2-10*c[4]^2+12*c[5]^3+12*c[4]^3+20*c[7]*c[6]*c [5]^3+15*c[5]*c[6]*c[7]*c[4]+12*c[6]*c[5]^2-15*c[6]*c[5]^3-1300*c[6]*c [7]*c[5]^3*c[4]^3+150*c[5]^4*c[7]*c[4]^2+12*c[7]*c[5]^2+20*c[7]*c[6]*c [4]^3+12*c[7]*c[4]^2-15*c[7]*c[4]^3+12*c[6]*c[4]^2-15*c[6]*c[4]^3-15*c [6]*c[7]*c[4]^2-87*c[5]*c[4]^3-72*c[5]*c[7]*c[4]^2-72*c[5]*c[6]*c[4]^2 +60*c[5]*c[4]^2-12*c[5]*c[6]*c[4]+10*c[5]*c[4]-150*c[5]*c[7]*c[6]*c[4] ^3-15*c[7]*c[5]^3-660*c[5]^2*c[6]*c[7]*c[4]^2+90*c[6]*c[5]*c[7]*c[4]^2 -15*c[6]*c[7]*c[5]^2+150*c[6]*c[5]^4*c[4]^2+930*c[6]*c[5]^2*c[7]*c[4]^ 3+930*c[6]*c[5]^3*c[7]*c[4]^2-200*c[5]^4*c[4]^3*c[7]+110*c[7]*c[5]*c[4 ]^3+110*c[6]*c[5]*c[4]^3+110*c[5]^3*c[6]*c[4]+110*c[5]^3*c[7]*c[4]+60* c[5]^2*c[4]-150*c[5]^3*c[6]*c[7]*c[4]+550*c[5]^3*c[4]^2-690*c[5]^2*c[6 ]*c[4]^3+520*c[5]^2*c[7]*c[4]^2+520*c[5]^2*c[6]*c[4]^2-690*c[5]^2*c[7] *c[4]^3-690*c[5]^3*c[7]*c[4]^2-690*c[5]^3*c[6]*c[4]^2+150*c[5]^2*c[4]^ 4*c[7]+900*c[6]*c[5]^3*c[4]^3+900*c[5]^3*c[7]*c[4]^3-200*c[5]^4*c[7]*c [4]^2*c[6]-429*c[5]^2*c[4]^2-120*c[5]^4*c[4]^2-120*c[4]^4*c[5]^2+150*c [5]^4*c[4]^3-72*c[5]^2*c[6]*c[4]-72*c[5]^2*c[7]*c[4]+90*c[5]^2*c[6]*c[ 7]*c[4]+550*c[5]^2*c[4]^3+300*c[5]^3*c[4]^4*c[6]*c[7]+300*c[5]^4*c[4]^ 3*c[6]*c[7]-690*c[5]^3*c[4]^3-200*c[4]^4*c[6]*c[5]^3+150*c[5]^3*c[4]^4 ), a[8,5] = -1/2*(893*c[5]^2*c[6]*c[4]^4-122*c[6]*c[5]*c[4]^4+850*c[5] ^4*c[4]^4*c[6]^2*c[7]-101*c[7]*c[5]*c[4]^4-535*c[5]^2*c[4]^4*c[7]^2-44 *c[5]^3*c[4]+160*c[4]^5*c[6]^2*c[5]^2+100*c[7]^2*c[5]^2*c[4]^5+170*c[4 ]^5*c[6]*c[5]^3+75*c[5]*c[6]^2*c[4]^4+16*c[7]^2*c[4]^3-9*c[5]*c[7]*c[4 ]-140*c[5]^2*c[4]^5*c[7]-780*c[5]^3*c[4]^4*c[7]-10*c[4]^4*c[6]^2-100*c [7]^2*c[4]^5*c[5]^3+450*c[6]^2*c[4]^5*c[7]*c[5]^3+510*c[5]^4*c[4]^2*c[ 7]^2*c[6]+490*c[6]^2*c[7]*c[5]^4*c[4]^2-2224*c[6]*c[5]^2*c[4]^4*c[7]+1 0*c[5]^5*c[4]^3*c[7]-146*c[5]^4*c[4]^3*c[6]+90*c[5]^4*c[4]^5*c[6]+40*c [5]^4*c[4]^4*c[6]+570*c[7]^2*c[4]^4*c[5]^3+130*c[5]^4*c[4]^4*c[7]+25*c [7]^2*c[6]*c[4]^4+18*c[5]^5*c[4]^2-4*c[5]^2-4*c[4]^2-10*c[4]^4*c[7]^2+ 150*c[7]^2*c[5]^4*c[4]^3+5*c[5]^3-6*c[6]^2*c[7]^2*c[5]+10*c[4]^3-35*c[ 5]^5*c[4]^3-24*c[7]*c[6]^2*c[5]^3+27*c[7]*c[6]*c[5]^3-32*c[7]^2*c[6]*c 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6]*c[4]^2-26*c[6]*c[4]^3-21*c[6]*c[7]*c[4]^2-38*c[7]^2*c[6]*c[4]^3-110 *c[5]*c[7]^2*c[4]^3-67*c[5]*c[4]^3-42*c[5]*c[7]*c[4]^2-49*c[5]*c[6]*c[ 4]^2+19*c[5]*c[4]^2-10*c[5]*c[6]*c[4]+4*c[5]*c[4]-58*c[6]*c[5]*c[7]^2* c[4]^2-16*c[7]^2*c[5]*c[6]*c[4]-390*c[5]*c[7]*c[6]*c[4]^3+23*c[6]^2*c[ 4]^2*c[5]+8*c[6]^2*c[4]*c[5]-9*c[7]*c[5]^3+20*c[5]^4*c[4]^5-1308*c[5]^ 2*c[6]*c[7]*c[4]^2+900*c[5]^4*c[7]^2*c[4]^4*c[6]-450*c[4]^5*c[6]*c[7]* c[5]^3-106*c[6]^2*c[5]*c[4]^3-46*c[6]^2*c[7]*c[4]*c[5]+25*c[7]*c[6]^2* c[5]*c[4]^3+4*c[6]^2*c[7]*c[5]+10*c[6]^2*c[7]*c[5]^2+97*c[6]*c[5]*c[7] *c[4]^2+10*c[5]^4*c[7]*c[6]-27*c[6]*c[7]*c[5]^2-10*c[6]^2*c[5]^2+22*c[ 6]*c[7]^2*c[5]^2+6*c[6]^2*c[7]^2*c[5]^2+62*c[6]^2*c[4]^2*c[7]*c[5]-133 *c[6]^2*c[7]^2*c[4]^2*c[5]-35*c[6]*c[5]^4*c[4]^2+10*c[7]*c[6]^2*c[5]^4 -7*c[6]*c[5]^4+140*c[4]^5*c[7]*c[5]^3-70*c[6]^2*c[5]^2*c[7]*c[4]+583*c [6]^2*c[5]^2*c[7]*c[4]^2+2898*c[6]*c[5]^2*c[7]*c[4]^3+1462*c[6]*c[5]^3 *c[7]*c[4]^2+208*c[6]^2*c[5]^3*c[7]*c[4]-225*c[5]^4*c[4]^3*c[7]+704*c[ 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5*c[4]^5*c[7]+70*c[4]^3*c[7]^2*c[5]*c[6]^2-42*c[7]^2*c[6]*c[5]*c[4]^3+ 150*c[5]^5*c[4]^5-30*c[7]^2*c[5]^2*c[6]*c[4]^2+12*c[4]^4*c[5]^2+87*c[7 ]^2*c[5]^3*c[6]*c[4]+510*c[7]^2*c[5]^3*c[6]*c[4]^3+690*c[5]^3*c[6]^2*c [4]^4-342*c[5]^4*c[4]^3-15*c[6]^2*c[4]^3*c[7]^2+510*c[5]^3*c[6]^2*c[7] *c[4]^3-20*c[5]^2*c[6]*c[7]*c[4]-410*c[6]^2*c[7]*c[5]^3*c[4]^2-70*c[6] ^2*c[7]^2*c[4]*c[5]^3-10*c[5]^2*c[4]^3+24*c[7]^2*c[5]^2*c[4]^2-24*c[7] ^2*c[5]^3*c[4]-200*c[7]^2*c[5]^6*c[4]^2*c[6]+110*c[7]^2*c[4]*c[5]^5+13 50*c[5]^5*c[4]^3*c[6]*c[7]+15*c[7]^2*c[4]^4*c[5]+300*c[6]^2*c[7]*c[5]^ 6*c[4]^3+510*c[7]^2*c[5]^3*c[6]^2*c[4]^2-550*c[5]^6*c[4]^2-600*c[7]*c[ 5]^6*c[4]^3+20*c[7]*c[6]^2*c[5]^5+570*c[5]^6*c[6]*c[4]^2+150*c[5]^2*c[ 6]*c[4]^5*c[7]^2+550*c[5]^4*c[4]^4+120*c[5]^7*c[4]^2+72*c[5]^4*c[4]*c[ 6]-40*c[5]^4*c[7]^2*c[4]*c[6]-57*c[5]^4*c[4]*c[7]*c[6]+150*c[6]^2*c[5] ^6*c[4]^2+150*c[4]^5*c[6]^2*c[7]*c[5]^2-200*c[4]^5*c[6]^2*c[7]^2*c[5]^ 2-150*c[5]^7*c[7]*c[4]^2-110*c[6]^2*c[4]^4*c[5]^2+780*c[5]^5*c[4]^2*c[ 7]^2*c[6]-200*c[5]^5*c[4]^5*c[6]-600*c[5]^6*c[4]^3*c[6]-200*c[6]^2*c[7 ]*c[5]^6*c[4]^2-300*c[5]^4*c[4]^5*c[6]^2*c[7]+90*c[7]*c[6]^2*c[5]*c[4] ^4+200*c[6]*c[5]^7*c[7]*c[4]^2-300*c[6]*c[5]^7*c[7]*c[4]^3-180*c[5]^3* c[4]^4*c[6]^2*c[7]-570*c[5]^4*c[4]^4*c[6]*c[7]-540*c[5]^2*c[4]^4*c[7]* c[6]^2+660*c[5]^3*c[4]^4*c[6]*c[7]+60*c[5]^4*c[4]^3*c[6]*c[7]+48*c[5]^ 3*c[4]^3-15*c[6]^2*c[5]^5-200*c[5]^6*c[6]^2*c[4]^3-15*c[5]^5*c[6]*c[7] -48*c[5]^5*c[4]+140*c[5]^5*c[6]*c[4]^2+200*c[5]^4*c[4]^5*c[6]^2+300*c[ 6]^2*c[7]^2*c[5]^3*c[4]^5-300*c[6]*c[7]^2*c[5]^4*c[4]^5+700*c[7]*c[5]^ 6*c[4]^3*c[6]-750*c[5]^4*c[4]^4*c[6]^2+20*c[5]^5*c[7]^2*c[6]+1100*c[7] ^2*c[5]^4*c[6]^2*c[4]^3-930*c[7]^2*c[5]^4*c[6]^2*c[4]^2+300*c[7]^2*c[5 ]^6*c[4]^3*c[6]-20*c[5]^6*c[7]*c[6]-180*c[5]^3*c[4]^4*c[7]^2*c[6]-750* c[5]^4*c[4]^3*c[7]^2*c[6]-540*c[5]^2*c[4]^4*c[7]^2*c[6]-750*c[5]^4*c[4 ]^3*c[6]^2*c[7]-150*c[7]^2*c[4]^4*c[6]^2*c[5]-1100*c[7]^2*c[4]^4*c[5]^ 3*c[6]^2+930*c[7]^2*c[4]^4*c[6]^2*c[5]^2+150*c[5]^6*c[4]*c[7]*c[6]-110 *c[5]^6*c[4]*c[6]-150*c[5]^7*c[6]*c[4]^2+570*c[5]^6*c[4]^3-150*c[5]^7* c[4]^3+15*c[7]^2*c[6]^2*c[5]^3-440*c[4]^4*c[6]*c[5]^3+750*c[5]^5*c[6]^ 2*c[4]^3-200*c[7]^2*c[5]^6*c[4]^3+12*c[5]^4*c[7]^2+200*c[5]^4*c[4]^5*c [6]*c[7]-20*c[5]^4*c[4]-87*c[5]^3*c[4]^4-30*c[5]^5*c[4]*c[6]+110*c[5]^ 5*c[6]^2*c[4]), a[9,7] = -1/60*(10*c[5]*c[6]*c[4]-5*c[6]*c[5]-5*c[6]*c [4]+3*c[6]-5*c[5]*c[4]+3*c[5]+3*c[4]-2)/c[7]/(-c[5]*c[6]*c[4]+c[5]*c[6 ]*c[7]*c[4]+c[5]*c[7]*c[4]-c[7]^2*c[5]*c[4]+c[6]*c[7]*c[4]-c[6]*c[7]^2 *c[4]-c[7]^2*c[4]+c[7]^3*c[4]+c[6]*c[7]*c[5]-c[6]*c[7]^2*c[5]-c[7]^2*c [5]+c[7]^3*c[5]-c[7]^2*c[6]+c[7]^3*c[6]+c[7]^3-c[7]^4), a[9,2] = 0, a[ 9,3] = 0, a[9,5] = -1/60*(10*c[6]*c[7]*c[4]-5*c[6]*c[4]-5*c[7]*c[6]+3* c[6]-5*c[7]*c[4]+3*c[4]-2+3*c[7])/c[5]/(c[5]^3*c[4]-c[5]^2*c[4]+c[5]*c [7]*c[4]-c[5]^2*c[7]*c[4]+c[5]*c[6]*c[4]-c[5]^2*c[6]*c[4]-c[6]*c[7]*c[ 4]+c[5]*c[6]*c[7]*c[4]-c[5]^4+c[5]^3-c[7]*c[5]^2+c[7]*c[5]^3-c[6]*c[5] ^2+c[6]*c[5]^3+c[6]*c[7]*c[5]-c[6]*c[7]*c[5]^2), a[9,8] = 1/60*(-20*c[ 5]*c[6]*c[4]+30*c[5]*c[6]*c[7]*c[4]+15*c[6]*c[5]-20*c[6]*c[7]*c[5]+15* c[6]*c[4]-20*c[6]*c[7]*c[4]-12*c[6]+15*c[7]*c[6]+15*c[5]*c[4]-20*c[5]* c[7]*c[4]-12*c[5]+15*c[7]*c[5]-12*c[4]+15*c[7]*c[4]+10-12*c[7])/(-c[4] +c[7]*c[4]+c[6]*c[4]-c[6]*c[7]*c[4]+c[5]*c[4]-c[5]*c[7]*c[4]-c[5]*c[6] *c[4]+c[5]*c[6]*c[7]*c[4]+1-c[7]-c[6]+c[7]*c[6]-c[5]+c[7]*c[5]+c[6]*c[ 5]-c[6]*c[7]*c[5]), a[9,1] = 1/60*(30*c[5]*c[6]*c[7]*c[4]-10*c[5]*c[6] *c[4]-10*c[6]*c[7]*c[5]+5*c[6]*c[5]-10*c[6]*c[7]*c[4]+5*c[6]*c[4]+5*c[ 7]*c[6]-3*c[6]-10*c[5]*c[7]*c[4]+5*c[5]*c[4]+5*c[7]*c[5]-3*c[5]+5*c[7] *c[4]-3*c[4]+2-3*c[7])/c[5]/c[6]/c[7]/c[4], a[9,4] = 1/60*(-2+3*c[5]-5 *c[7]*c[6]+3*c[7]+3*c[6]+10*c[6]*c[7]*c[5]-5*c[6]*c[5]-5*c[7]*c[5])/c[ 4]/(-c[6]*c[4]^3-c[4]^3+c[4]^4+c[7]*c[4]^2+c[6]*c[4]^2-c[6]*c[7]*c[4]+ c[6]*c[7]*c[4]^2-c[5]*c[4]^3+c[5]*c[4]^2-c[7]*c[4]^3-c[5]*c[7]*c[4]+c[ 5]*c[7]*c[4]^2-c[5]*c[6]*c[4]+c[5]*c[6]*c[4]^2+c[6]*c[7]*c[5]-c[5]*c[6 ]*c[7]*c[4]), a[9,6] = 1/60*(10*c[5]*c[7]*c[4]-5*c[5]*c[4]-5*c[7]*c[5] +3*c[5]-5*c[7]*c[4]+3*c[4]-2+3*c[7])/c[6]/(c[6]^2*c[4]*c[5]-c[5]*c[6]* c[4]+c[5]*c[7]*c[4]-c[5]*c[6]*c[7]*c[4]-c[6]^3*c[4]+c[6]^2*c[4]-c[6]*c [7]*c[4]+c[6]^2*c[7]*c[4]-c[6]^3*c[5]+c[6]^2*c[5]-c[6]*c[7]*c[5]+c[6]^ 2*c[7]*c[5]+c[6]^4-c[6]^3+c[7]*c[6]^2-c[6]^3*c[7])\}:" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}} {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 5 "Let " } {XPPEDIT 18 0 "t[7]" "6#&%\"tG6#\"\"(" }{TEXT -1 7 " and " } {XPPEDIT 18 0 "t[8]" "6#&%\"tG6#\"\")" }{TEXT -1 29 " denote the coeff icients of " }{XPPEDIT 18 0 "z^7" "6#*$%\"zG\"\"(" }{TEXT -1 7 " and " }{XPPEDIT 18 0 "z^8" "6#*$%\"zG\"\")" }{TEXT -1 43 " respectively in the stability polynomial." }}{PARA 0 "" 0 "" {TEXT -1 29 "Followin g Tsitouras, we set " }{XPPEDIT 18 0 "t[7] = 1/6331;" "6#/&%\"tG6#\" \"(*&\"\"\"F)\"%Jj!\"\"" }{TEXT -1 7 " and " }{XPPEDIT 18 0 "t[8] = \+ 1/128550;" "6#/&%\"tG6#\"\")*&\"\"\"F)\"']&G\"!\"\"" }{TEXT -1 37 " t o obtain two equations involving " }{XPPEDIT 18 0 "c[4]" "6#&%\"cG6# \"\"%" }{TEXT -1 3 ", " }{XPPEDIT 18 0 "c[5]" "6#&%\"cG6#\"\"&" } {TEXT -1 7 " and " }{XPPEDIT 18 0 "c[6]" "6#&%\"cG6#\"\"'" }{TEXT -1 2 ". " }}{PARA 0 "" 0 "" {TEXT -1 52 "Note that the stability polyn omial only depends on " }{XPPEDIT 18 0 "c[4]" "6#&%\"cG6#\"\"%" } {TEXT -1 3 ", " }{XPPEDIT 18 0 "c[5]" "6#&%\"cG6#\"\"&" }{TEXT -1 7 " and " }{XPPEDIT 18 0 "c[6]" "6#&%\"cG6#\"\"'" }{TEXT -1 2 ". " }} {PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 142 "Rz := StabilityFunction(6,8,'expanded'):\neqA := simplify(subs(eG ,coeff(Rz,z^7)))=1/6331:%;\neqB := simplify(subs(eG,coeff(Rz,z^8)))=1/ 128550:%;" }}{PARA 12 "" 1 "" {XPPMATH 20 "6#/,$*&#\"\"\"\"%S9F'*&,hn* (\"\"#F'&%\"cG6#\"\"'F')&F.6#\"\"&F,F'F'*(\"\"%F'F-F')&F.6#F6\"\"$F'! \"\"*(\"#5F'F2F'F7F'F;*(\"#?F'F1F')F8F,F'F;*(\"#SF')F8F4F'F1F'F'**\"#! )F'F-F'F2F'FCF'F;**\"#IF'F2F'F-F'F@F'F'*&F2F'F@F'F'*(\"#**F'F1F'F7F'F' *(\"$S\"F')F8F6F'F1F'F;*(F4F')F2F:F'F@F'F;*(F,F'FOF'F8F'F'*&F1F'F8F'F' **\"#)*F'F-F'F2F'F7F'F;**\"#MF'FOF'F-F'F8F'F'**F0F'F1F'F-F'F8F'F;**\"$ g\"F'F1F'F-F'F7F'F'**\"#>F'F1F'F-F'F@F'F;**\"$S$F'F1F'F-F'FMF'F;**F4F' F2F'F-F'F8F'F;*(\"#9F'FMF'F2F'F'*(F?F'FOF'F7F'F;*(FBF'FOF'FMF'F'**\"$? \"F'FOF'F-F'F@F'F;**\"$!>F'F-F'FOF'F7F'F'**\"$+#F'F1F'F-F'FCF'F'**\"$+ \"F'FMF'F-F'FOF'F;**FXF'F-F'F2F'FMF'F'*(F,F'F-F'F@F'F'*(F6F'F-F'FOF'F; F',2*(F=F'FOF'F@F'F'*(F=F'F1F'F7F'F'*(FGF'F1F'F@F'F;*(F0F'F1F'F8F'F'*$ F1F'F;*(F0F'F2F'F@F'F'*&F2F'F8F'F'*$F@F'F;F;F'F;#F'\"%Jj" }}{PARA 11 " " 1 "" {XPPMATH 20 "6#/,$*&#\"\"\"\"%S9F'*,&%\"cG6#\"\"%F'&F+6#\"\"&F' ,0**\"#IF'F.F'&F+6#\"\"'F')F*\"\"#F'!\"\"**\"#7F'F.F'F4F'F*F'F'**\"#?F 'F4F'F.F')F*\"\"$F'F'*(F8F'F4F'F.F'F9*&F.F'F*F'F'*$F7F'F9*&F4F'F*F'F'F ',**(F0F'F.F'F*F'F'F'F'*&F8F'F*F'F9*&F8F'F.F'F9F',2*(\"#5F')F.F?F'F7F' F'*(FJF')F.F8F'F>F'F'*(F3F'FMF'F7F'F9*(F6F'FMF'F*F'F'*$FMF'F9*(F6F'F.F 'F7F'F'FAF'FBF9F9F'F'#F'\"']&G\"" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }} {PARA 0 "" 0 "" {TEXT -1 55 "We can solve the two equations to obtain \+ formulas for " }{XPPEDIT 18 0 "c[5]" "6#&%\"cG6#\"\"&" }{TEXT -1 7 " \+ and " }{XPPEDIT 18 0 "c[6]" "6#&%\"cG6#\"\"'" }{TEXT -1 15 " in ter ms of " }{XPPEDIT 18 0 "c[4]" "6#&%\"cG6#\"\"%" }{TEXT -1 1 "." }} {PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 96 "sol := solve(\{eqA,eqB\},\{c[5],c[6]\}):\nc[5]=subs(sol,c[5]);eqC \+ := %:\nc[6]=subs(sol,c[6]);eqD := %:" }}{PARA 11 "" 1 "" {XPPMATH 20 " 6#/&%\"cG6#\"\"&,$*(\"'))QI\"\"\"&F%6#\"\"%F+,**&\")qmDaF+)F,\"\"$F+F+ *&\")N$Gr#F+)F,\"\"#F+!\"\"*&\"(+_3$F+F,F+F+\"'W>:F8F8F8" }}{PARA 12 " " 1 "" {XPPMATH 20 "6#/&%\"cG6#\"\"',$*&#\"#C\")N$Gr#\"\"\"**&F%6#\"\" %!\"\",2*&\"2w*e0W&QlT\"F-F/F-F-*&\"5:z![\"f/T7\"\\%F-)F/F1F-F2*&\"3!3 'y)Q<@kC$F-)F/\"\"#F-F2*&\"4?$oXGm\\&f%\\F-)F/\"\"$F-F-*&\"6]1cn!)*HQ' *4AF-)F/\"\"&F-F-\"0[+t%*4\\Q%F2*&\"6].Tl@Bmo#*Q&F-)F/F'F-F2*&\"6+jnF< 'e\\$))3&F-)F/\"\"(F-F-F-,,*&\")v\"=[$F-F;F-F-*&\"*SL^3\"F-F?F-F2*&\"( ko*RF-F/F-F2\"'W>:F-*&FQF-F8F-F-F2,**&\")5*y\"[F-F?F-F-*&\")&p6!=F-F;F -F2*&\"'c9cF-F/F-F2\"'KeXF-F2F-F2" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }} {PARA 0 "" 0 "" {TEXT -1 33 "#================================" }} {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 24 "The follo wing procedure " }{TEXT 0 13 "prin_err_norm" }{TEXT -1 54 " calculates the principal error norm given the nodes " }{XPPEDIT 18 0 "c[2]" "6# &%\"cG6#\"\"#" }{TEXT -1 3 ", " }{XPPEDIT 18 0 "c[4]" "6#&%\"cG6#\"\" %" }{TEXT -1 7 " and " }{XPPEDIT 18 0 "c[7]" "6#&%\"cG6#\"\"(" } {TEXT -1 87 " by using the general solution together with the precedi ng two equations to determine " }{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 1 ";" }}}{SECT 1 {PARA 4 "" 0 "" {TEXT 0 13 "prin_er r_norm" }{TEXT -1 1 " " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 51 "errterms6_8 := PrincipalErrorTerms( 6,8,'expanded'):" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 889 "prin_err_norm := proc(c2,c4,c7)\n local c2 r,c4r,c5r,c6r,c7r,e1,sm,ct;\n global e2;\n \n c2r := convert(c2,r ational,Digits);\n c4r := convert(c4,rational,Digits); \n c5r := - 303888*c4r/(54256670*c4r^3-27128335*c4r^2+3085200*c4r-151944);\n c6r := -24/27128335*1/c4r*(14165385440558976*c4r-44911241045914807915*c4 r^4-324642117388786080*c4r^2+4945954966284568320*c4r^3+220996382998067 560650*c4r^5-438490994730048-538926866232165410350*c4r^6+5088834958617 27676300*c4r^7)/(34818175*c4r^2-108513340*c4r^3-3996864*c4r+151944+108 513340*c4r^4)/(48178910*c4r^3-18011695*c4r^2-561456*c4r+455832);\n c 7r := convert(c7,rational,Digits); \n\n e1 := \{c[2]=c2r,c[4]=c4r,c[ 5]=c5r,c[6]=c6r,c[7]=c7r,c[8]=1,c[9]=1\};\n e2 := `union`(e1,simplif y(subs(e1,eG))):\n sm := 0;\n for ct to nops(errterms6_8) do\n \+ sm := sm+(simplify(subs(e2,errterms6_8[ct])))^2;\n end do;\n eva lf(sqrt(sm));\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 31 "To avoid a negative value for " } {XPPEDIT 18 0 "c[2]" "6#&%\"cG6#\"\"#" }{TEXT -1 10 ", we set " } {XPPEDIT 18 0 "c[2]=1/192" "6#/&%\"cG6#\"\"#*&\"\"\"F)\"$#>!\"\"" } {TEXT -1 57 " and minimize the principal error norm with respect to \+ " }{XPPEDIT 18 0 "c[4]" "6#&%\"cG6#\"\"%" }{TEXT -1 7 " and " } {XPPEDIT 18 0 "c[7]" "6#&%\"cG6#\"\"(" }{TEXT -1 29 " using the start ing values " }{XPPEDIT 18 0 "c[4]=39/200" "6#/&%\"cG6#\"\"%*&\"#R\"\" \"\"$+#!\"\"" }{TEXT -1 7 " and " }{XPPEDIT 18 0 "c[7]=3/4" "6#/&%\" cG6#\"\"(*&\"\"$\"\"\"\"\"%!\"\"" }{TEXT -1 1 "." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 50 "We use the one dimensiona l minimization procedure " }{TEXT 0 7 "findmin" }{TEXT -1 51 " and min imize with resect to each node alternately." }}{PARA 0 "" 0 "" {TEXT -1 2 " " }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 292 "Digits := 15:\n c_2 := 1/192: c_4 := .195: c_7 := .75:\nfor ii to 10 do \n c_4 := op (1,findmin('prin_err_norm'(c_2,c4,c_7),c4=\{0.18,c_4,0.21\}));\n mn \+ := findmin('prin_err_norm'(c_2,c_4,c7),c7=\{0.7,c_7,0.82\});\n c_7 : = mn[1]:\n print(c[4]=c_4,c[7]=c_7);\n print(mn[2]);\nend do:\nDig its := 10:" }}{PARA 11 "" 1 "" {XPPMATH 20 "6$/&%\"cG6#\"\"%$\"0.%)Q4i ?$>!#:/&F%6#\"\"($\"0WhOizVl(F*" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#$\" 0oqS3Yn$H!#=" }}{PARA 11 "" 1 "" {XPPMATH 20 "6$/&%\"cG6#\"\"%$\"0:G)4 ()=J>!#:/&F%6#\"\"($\"0=!#:/&F%6#\"\"($\"0'=AC=udwF*" }}{PARA 11 "" 1 "" {XPPMATH 20 "6# $\"0V:-Y^j$H!#=" }}{PARA 11 "" 1 "" {XPPMATH 20 "6$/&%\"cG6#\"\"%$\"0$ *41]r6$>!#:/&F%6#\"\"($\"0b#f.IudwF*" }}{PARA 11 "" 1 "" {XPPMATH 20 " 6#$\"0lj,Y^j$H!#=" }}{PARA 11 "" 1 "" {XPPMATH 20 "6$/&%\"cG6#\"\"%$\" 0W(e%\\r6$>!#:/&F%6#\"\"($\"0Mde-Vxl(F*" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#$\"0jj,Y^j$H!#=" }}{PARA 11 "" 1 "" {XPPMATH 20 "6$/&%\"cG6#\"\" %$\"0vtW\\r6$>!#:/&F%6#\"\"($\"0lxi-Vxl(F*" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#$\"0jj,Y^j$H!#=" }}{PARA 11 "" 1 "" {XPPMATH 20 "6$/&% \"cG6#\"\"%$\"0grW\\r6$>!#:/&F%6#\"\"($\"0f&GEIudwF*" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#$\"0jj,Y^j$H!#=" }}{PARA 11 "" 1 "" {XPPMATH 20 "6$/ &%\"cG6#\"\"%$\"0crW\\r6$>!#:/&F%6#\"\"($\"0u&GEIudwF*" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#$\"0jj,Y^j$H!#=" }}{PARA 11 "" 1 "" {XPPMATH 20 "6 $/&%\"cG6#\"\"%$\"0crW\\r6$>!#:/&F%6#\"\"($\"0u&GEIudwF*" }}{PARA 11 " " 1 "" {XPPMATH 20 "6#$\"0jj,Y^j$H!#=" }}{PARA 11 "" 1 "" {XPPMATH 20 "6$/&%\"cG6#\"\"%$\"0crW\\r6$>!#:/&F%6#\"\"($\"0u&GEIudwF*" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#$\"0jj,Y^j$H!#=" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 65 "We consider various rational ap proximations for the values found." }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 138 "nds := [c[4]=.193117149447156,c[7]=.765774302628574] :\nevalf[10](%);\nfor dgt from 6 by -1 to 3 do\n map(convert,nds,rat ional,dgt);\nend do;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#7$/&%\"cG6#\" \"%$\"+%\\r6$>!#5/&F&6#\"\"($\"+EIudwF+" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#7$/&%\"cG6#\"\"%#\"$,\"\"$B&/&F&6#\"\"(#\"$,)\"%Y5" }}{PARA 11 " " 1 "" {XPPMATH 20 "6#7$/&%\"cG6#\"\"%#\"#G\"$X\"/&F&6#\"\"(#\"#&)\"$6 \"" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#7$/&%\"cG6#\"\"%#\"#6\"#d/&F&6# \"\"(#\"#O\"#Z" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#7$/&%\"cG6#\"\"%#\" \"&\"#E/&F&6#\"\"(#\"#5\"#8" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 79 "#---------------------------------------------- --------------------------------" }}{PARA 0 "" 0 "" {TEXT -1 44 "Tsito uras mentions a scheme with the nodes " }{XPPEDIT 18 0 "c[2] = -25/20 4;" "6#/&%\"cG6#\"\"#,$*&\"#D\"\"\"\"$/#!\"\"F-" }{TEXT -1 3 ", " } {XPPEDIT 18 0 "c[4] = 620/261;" "6#/&%\"cG6#\"\"%*&\"$?'\"\"\"\"$h#!\" \"" }{TEXT -1 3 ", " }{XPPEDIT 18 0 "c[7] = 452/385;" "6#/&%\"cG6#\" \"(*&\"$_%\"\"\"\"$&Q!\"\"" }{TEXT -1 116 ", but there must be a (non- obvious) misprint in these values, since the corresponding principal e rror norm is . . . " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 39 "prin_err_norm(-25/204,620/261,452/385);" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#$\"0rZ&HfCB!\"\"" }{TEXT -1 4 " , " }{XPPEDIT 18 0 "c[4] = \+ 28/145;" "6#/&%\"cG6#\"\"%*&\"#G\"\"\"\"$X\"!\"\"" }{TEXT -1 7 " and \+ " }{XPPEDIT 18 0 "c[7]=85/111" "6#/&%\"cG6#\"\"(*&\"#&)\"\"\"\"$6\"! \"\"" }{TEXT -1 43 " gives the following principal error norm." }} {PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 35 "prin_err_norm(1/192,28/145,85/111);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#$\"0Pt[(3OOH!#=" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 29 "A slightly better value for " }{XPPEDIT 18 0 "c[7] " "6#&%\"cG6#\"\"(" }{TEXT -1 11 " is . . . " }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 115 "mn := findmin('prin_err_norm'(1/192,28/145,c7), c7=0.75..0.78,convergence=location):\nc[7]=convert(mn[1],rational,5); " }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/&%\"cG6#\"\"(#\"$@\"\"$e\"" }}} {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 9 "Setting \+ " }{XPPEDIT 18 0 "c[2] = 1/192;" "6#/&%\"cG6#\"\"#*&\"\"\"F)\"$#>!\"\" " }{TEXT -1 4 " , " }{XPPEDIT 18 0 "c[4] = 28/145;" "6#/&%\"cG6#\"\"% *&\"#G\"\"\"\"$X\"!\"\"" }{TEXT -1 7 " and " }{XPPEDIT 18 0 "c[7] = \+ 121/158;" "6#/&%\"cG6#\"\"(*&\"$@\"\"\"\"\"$e\"!\"\"" }{TEXT -1 43 " \+ gives the following principal error norm." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 36 "prin_err_norm(1/192,2 8/145,121/158);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#$\"0(RsM1OOH!#=" }} }{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 24 "Using th e formulas for " }{XPPEDIT 18 0 "c[5]" "6#&%\"cG6#\"\"&" }{TEXT -1 7 " and " }{XPPEDIT 18 0 "c[6]" "6#&%\"cG6#\"\"'" }{TEXT -1 15 " in t erms of " }{XPPEDIT 18 0 "c[4]" "6#&%\"cG6#\"\"%" }{TEXT -1 58 " woul d suggest the followng rational values for the nodes " }{XPPEDIT 18 0 " c[5]" "6#&%\"cG6#\"\"&" }{TEXT -1 7 " and " }{XPPEDIT 18 0 "c[6]" "6#&%\"cG6#\"\"'" }{TEXT -1 1 "." }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 105 "evalf(subs(c[4]=28/145,eqC)):\nconvert(%,rational,5);\nevalf( subs(c[4]=28/145,eqD)):\nconvert(%,rational,5);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/&%\"cG6#\"\"&#\"#c\"$p\"" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/&%\"cG6#\"\"'#\"#P\"#e" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }} {PARA 0 "" 0 "" {TEXT -1 155 "Taking these values gives a principal er ror norm of approximately 0.00019024 nd the maximum magnitude of the \+ linking coefficients is approximately 4.478." }}{PARA 0 "" 0 "" {TEXT -1 61 "The real stability interval is approximately [ -4.31360, 0]." }}{PARA 0 "" 0 "" {TEXT -1 53 "(See the abreviated calculations \+ in a later section.)" }}{PARA 0 "" 0 "" {TEXT -1 51 "#---------------- ----------------------------------" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}}{PARA 0 "" 0 "" {TEXT -1 52 "#----------------- ----------------------------------" }}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 38 "characteristics of the embedded scheme" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 35 "coefficients of the combined scheme" }} {PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1931 "ee := \{c[2]=1/192,\nc[3]=56/435,\nc[4]=28/145,\nc[5]=56/169,\nc [6]=37/58,\nc[7]=121/158,\nc[8]=1,\nc[9]=1,\n\na[2,1]=1/192,\na[3,1]=- 30744/21025,\na[3,2]=100352/63075,\na[4,1]=7/145,\na[4,2]=0,\na[4,3]=2 1/145,\na[5,1]=523516/4826809,\na[5,2]=0,\na[5,3]=-889140/4826809,\na[ 5,4]=1965040/4826809,\na[6,1]=41044746763290003/51739779980736512,\na[ 6,2]=0,\na[6,3]=-57685603608510645/24977824818286592,\na[6,4]=83145940 696701395/53969942910940672,\na[6,5]=53772835209542058935/876471872873 67651328,\na[7,1]=-19737371728514492920457623/384824669721407219242373 12,\na[7,2]=0,\na[7,3]=27466253014073155807766505/14560933448918110998 360064,\na[7,4]=-117670134102158631127838015/1229878843096119018254341 12,\na[7,5]=-19483500956399449871589744537/481311655153988159050791915 520,\na[7,6]=913336658253963141/2343933323194077385,\na[8,1]=248481395 4171031615807/2538960546022287823744,\na[8,2]=0,\na[8,3]=-229147524517 236387495/87335252197326853888,\na[8,4]=130548415875501708716128185/10 6468788692037442075798624,\na[8,5]=11103534575830173429908962617/81865 76761928888116508540672,\na[8,6]=-10679027011796052/13181450824860817, \na[8,7]=2238377869046599644/2564667687663709133,\na[9,1]=397933/54700 80,\na[9,2]=0,\na[9,3]=0,\na[9,4]=5450042155625/32205284983872,\na[9,5 ]=1677324270326783/6406236767145600,\na[9,6]=179200957846/132377472517 5,\na[9,7]=77461817788426/272279942939565,\na[9,8]=47049059/616363020, \n\nb[1]=397933/5470080,\nb[2]=0,\nb[3]=0,\nb[4]=5450042155625/3220528 4983872,\nb[5]=1677324270326783/6406236767145600,\nb[6]=179200957846/1 323774725175,\nb[7]=77461817788426/272279942939565,\nb[8]=47049059/616 363020,\n\n`b*`[1]=58931524118239849/837231220422643200,\n`b*`[2]=0,\n `b*`[3]=0,\n`b*`[4]=180077138918196243205225/985845547089116780242176, \n`b*`[5]=239006278908625056478598819/980516085116436873474624000,\n`b *`[6]=4448038142979447869029/28944615297423551638500,\n`b*`[7]=1616804 268713313415249069/5953458735623079359484300,\n`b*`[8]=713536679628437 051/13476908236860104400,\n`b*`[9]=1/40\}:" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 5 "Let " }{XPPEDIT 18 0 "T[6, 8];" "6#&%\"TG6$\"\"'\"\")" }{TEXT -1 127 " denote the vector whose components are the principal error terms of the 8 stage, order \+ 6 scheme (the error terms of order 7)." }}{PARA 0 "" 0 "" {TEXT -1 5 " Let " }{XPPEDIT 18 0 "`T*`[5,9];" "6#&%#T*G6$\"\"&\"\"*" }{TEXT -1 145 " denote the vector whose components are the principal error term s of the embedded 9 stage, order 5 scheme (the error terms of order 6) and let " }{XPPEDIT 18 0 "`T*`[6,9];" "6#&%#T*G6$\"\"'\"\"*" }{TEXT -1 99 " denote the vector whose components are the error terms of ord er 7 of the embedded order 5 scheme." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 39 "Denote the 2-norm of these vectors by \+ " }{XPPEDIT 18 0 "abs(abs(T[6,8]));" "6#-%$absG6#-F$6#&%\"TG6$\"\"'\" \")" }{TEXT -1 3 ", " }{XPPEDIT 18 0 "abs(abs(`T*`[5,9]));" "6#-%$abs G6#-F$6#&%#T*G6$\"\"&\"\"*" }{TEXT -1 7 " and " }{XPPEDIT 18 0 "abs( abs(`T*`[6,9]));" "6#-%$absG6#-F$6#&%#T*G6$\"\"'\"\"*" }{TEXT -1 15 " \+ respectively." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 8 "Define: " }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "A[7] = abs(abs(T[6, 8]));" "6#/&%\"AG6#\"\"(-%$absG6#-F)6#&%\"TG6$ \"\"'\"\")" }{TEXT -1 6 ", " }{XPPEDIT 18 0 "B[7] = abs(abs(`T*`[6 ,9]))/abs(abs(`T*`[5,9]));" "6#/&%\"BG6#\"\"(*&-%$absG6#-F*6#&%#T*G6$ \"\"'\"\"*\"\"\"-F*6#-F*6#&F/6$\"\"&F2!\"\"" }{TEXT -1 6 ", " } {XPPEDIT 18 0 "C[7] = abs(abs(`T*`[6,9]-T[6, 8]))/abs(abs(`T*`[5, 9])) ;" "6#/&%\"CG6#\"\"(*&-%$absG6#-F*6#,&&%#T*G6$\"\"'\"\"*\"\"\"&%\"TG6$ F2\"\")!\"\"F4-F*6#-F*6#&F06$\"\"&F3F9" }{TEXT -1 1 "." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 77 "Dormand and Prince h ave suggested that as well as attempting to ensure that " }{XPPEDIT 18 0 "A[7];" "6#&%\"AG6#\"\"(" }{TEXT -1 73 " is a minimum, if the em bedded scheme is to be used for error control, " }{XPPEDIT 18 0 "B[7] ;" "6#&%\"BG6#\"\"(" }{TEXT -1 7 " and " }{XPPEDIT 18 0 "C[7];" "6#& %\"CG6#\"\"(" }{TEXT -1 27 " should be chosen so that " }}{PARA 0 "" 0 "" {TEXT -1 68 "they are \"similar in magnitude\" and also not diffe r too much from 1." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 " > " 0 "" {MPLTEXT 1 0 186 "errterms6_8 := PrincipalErrorTerms(6,8,'exp anded'):\n`errterms6_9*` :=subs(b=`b*`,PrincipalErrorTerms(6,9,'expand ed')):\n`errterms5_9*` := subs(b=`b*`,PrincipalErrorTerms(5,9,'expande d')):" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 326 "snmB := sqrt(add(evalf(subs(ee,`errterms6_9*`[i]))^2 ,i=1..nops(`errterms6_9*`))):\nsdnB := sqrt(add(evalf(subs(ee,`errterm s5_9*`[i]))^2,i=1..nops(`errterms5_9*`))):\nsnmC := sqrt(add((evalf(su bs(ee,`errterms6_9*`[i])-subs(ee,errterms6_8[i])))^2,i=1..nops(errterm s6_8))):\n'B[7]'= evalf[8](snmB/sdnB);\n'C[7]'= evalf[8](snmC/sdnB);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/&%\"BG6#\"\"($\")hhE8!\"(" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/&%\"CG6#\"\"($\")(R)G8!\"(" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}} {PARA 0 "" 0 "" {TEXT -1 52 "#---------------------------------------- -----------" }}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 27 "absolute stability regions " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 35 "coefficient s of the combined scheme" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1931 "ee := \{c[2]=1/192,\nc[3]=56/435, \nc[4]=28/145,\nc[5]=56/169,\nc[6]=37/58,\nc[7]=121/158,\nc[8]=1,\nc[9 ]=1,\n\na[2,1]=1/192,\na[3,1]=-30744/21025,\na[3,2]=100352/63075,\na[4 ,1]=7/145,\na[4,2]=0,\na[4,3]=21/145,\na[5,1]=523516/4826809,\na[5,2]= 0,\na[5,3]=-889140/4826809,\na[5,4]=1965040/4826809,\na[6,1]=410447467 63290003/51739779980736512,\na[6,2]=0,\na[6,3]=-57685603608510645/2497 7824818286592,\na[6,4]=83145940696701395/53969942910940672,\na[6,5]=53 772835209542058935/87647187287367651328,\na[7,1]=-19737371728514492920 457623/38482466972140721924237312,\na[7,2]=0,\na[7,3]=2746625301407315 5807766505/14560933448918110998360064,\na[7,4]=-1176701341021586311278 38015/122987884309611901825434112,\na[7,5]=-19483500956399449871589744 537/481311655153988159050791915520,\na[7,6]=913336658253963141/2343933 323194077385,\na[8,1]=2484813954171031615807/2538960546022287823744,\n a[8,2]=0,\na[8,3]=-229147524517236387495/87335252197326853888,\na[8,4] =130548415875501708716128185/106468788692037442075798624,\na[8,5]=1110 3534575830173429908962617/8186576761928888116508540672,\na[8,6]=-10679 027011796052/13181450824860817,\na[8,7]=2238377869046599644/2564667687 663709133,\na[9,1]=397933/5470080,\na[9,2]=0,\na[9,3]=0,\na[9,4]=54500 42155625/32205284983872,\na[9,5]=1677324270326783/6406236767145600,\na [9,6]=179200957846/1323774725175,\na[9,7]=77461817788426/2722799429395 65,\na[9,8]=47049059/616363020,\n\nb[1]=397933/5470080,\nb[2]=0,\nb[3] =0,\nb[4]=5450042155625/32205284983872,\nb[5]=1677324270326783/6406236 767145600,\nb[6]=179200957846/1323774725175,\nb[7]=77461817788426/2722 79942939565,\nb[8]=47049059/616363020,\n\n`b*`[1]=58931524118239849/83 7231220422643200,\n`b*`[2]=0,\n`b*`[3]=0,\n`b*`[4]=1800771389181962432 05225/985845547089116780242176,\n`b*`[5]=239006278908625056478598819/9 80516085116436873474624000,\n`b*`[6]=4448038142979447869029/2894461529 7423551638500,\n`b*`[7]=1616804268713313415249069/59534587356230793594 84300,\n`b*`[8]=713536679628437051/13476908236860104400,\n`b*`[9]=1/40 \}:" }}}{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 8 stage, order 6 schem e is given as follows." }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 75 "su bs(ee,StabilityFunction(6,8,'expanded')):\nR := unapply(%,z):\n'R(z)'= R(z);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/-%\"RG6#%\"zG,4\"\"\"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)*$)F'F1F)F)F)*&#\") \"*>7U\"-&3N3om#F)*$)F'\"\"(F)F)F)*&#\"*<'4O`\"/+!z=2v&oF)*$)F'\"\")F) F)F)" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT 260 4 " Note" }{TEXT -1 63 ": The scheme has been constructed so that the coef ficients of " }{XPPEDIT 18 0 "z^6" "6#*$%\"zG\"\"'" }{TEXT -1 7 " an d " }{XPPEDIT 18 0 "z^7" "6#*$%\"zG\"\"(" }{TEXT -1 30 " are approxi mately equal to " }{XPPEDIT 18 0 "1/6331;" "6#*&\"\"\"F$\"%Jj!\"\"" } {TEXT -1 7 " and " }{XPPEDIT 18 0 "1/128550;" "6#*&\"\"\"F$\"']&G\"! \"\"" }{TEXT -1 15 " respectively." }}{PARA 0 "" 0 "" {TEXT -1 0 "" } }{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 57 "266680835085/42121991,68575 071879000/533609617;\nevalf(%);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6$#\" -&3N3om#\")\"*>7U#\"/+!z=2v&o\"*<'4O`" }}{PARA 11 "" 1 "" {XPPMATH 20 "6$$\"+`X:Jj!\"'$\"+mo6&G\"!\"%" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }} {PARA 0 "" 0 "" {TEXT -1 124 "We can find the point where the boundary of the stability region intersects the negative real axis by solving \+ the equation: " }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "R(z ) = 1;" "6#/-%\"RG6#%\"zG\"\"\"" }{TEXT -1 2 ". " }}{EXCHG {PARA 0 "> \+ " 0 "" {MPLTEXT 1 0 28 "z0 := newton(R(z)=1,z=-8.1);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%#z0G$!+'R!H*3)!\"*" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 303 "z0 := newton(R(z)=1,z= -8.1):\np1 := plot([R(z),1],z=-8.69..0.49,color=[red,blue]):\np2 := pl ot([[[z0,1]]$3],style=point,symbol=[circle,cross,diamond],color=black) :\np3 := plot([[z0,0],[z0,1]],linestyle=3,color=COLOR(RGB,0,.5,0)):\np lots[display]([p1,p2,p3],view=[-8.69..0.49,-.57..1.47],font=[HELVETICA ,9]);" }}{PARA 13 "" 1 "" {GLPLOT2D 485 267 267 {PLOTDATA 2 "6+-%'CURV ESG6$7in7$$!3]*************o)!#<$\"3f\"p\\PgEzT&F*7$$!31*\\(=Jb(*R')F* $\"3-ZzLLe^k[F*7$$!3Q+]Pi5&**e)F*$\"3iw&p`%zx^VF*7$$!3&**\\iNfE*R&)F*$ \"3C3XnU[SxQF*7$$!3]***\\Z7-**[)F*$\"3pBpuX5>RMF*7$$!3;Dc\">5wjW)F*$\" 3Qdyd.,o&3$F*7$$!3/\\73z+&GS)F*$\"3'pbf2_>mv#F*7$$!3#H(oCcSKf$)F*$\"3i f`7=St]CF*7$$!3d)\\7M.)z:$)F*$\"3#***)33&)*zm@F*7$$!3k[(oN!*)*y@)F*$\" 3VH7qNsQ-;F*7$$!3r)*\\st(***>\")F*$\"34NgrE.pI6F*7$$!3q)****[q^9-)F*$ \"31ROdn)H/Q(!#=7$$!3f**\\2OO!H#zF*$\"3_&4$RRd4uTF^o7$$!36](=!\\R#[#yF *$\"3%)[V_teU/;F^o7$$!3v*\\i>EWns(F*$!31iE\"p_pAH%!#>7$$!3c*\\7?Tz[a(F *$!3b!*3V#RS<.$F^o7$$!3o*\\iH:C2X(F*$!3M@qZ4E8$*QF^o7$$!3!)*\\7R*)olN( F*$!31xS#z\\wy[%F^o7$$!3M+DJ!R$>fsF*$!3[Xul'\\%3o[F^o7$$!3)**\\7n)y\"= ;(F*$!3/k!p4Tt40&F^o7$$!3q*\\(enYvkqF*$!3%\\OR9ZLO2&F^o7$$!3W*\\i%[9pn pF*$!3=:&zvOf*o\\F^o7$$!3')***\\ba3!onF*$!3m*4fX$=XnWF^o7$$!3o**\\(ftF @f'F*$!3\\S5j>#)*)HQF^o7$$!3E***\\;gFTR'F*$!3[su(y0=)>IF^o7$$!3g***\\( fWJ&>'F*$!3.$3dqp9D?#F^o7$$!3U***\\$e6s.gF*$!3cfhp/3:s9F^o7$$!3\"*)\\7 d3O(HeF*$!3qN]QY#3_())F^p7$$!3W***\\h+^Gi&F*$!3imwiFY!G3$F^p7$$!3a**** f1AfZaF*$\"3[H\\?:0$*yz!#?7$$!3k*\\7'eXsV_F*$\"3M)p/v*)*HwTF^p7$$!3S** **4YrEj]F*$\"3;.^s&e0dC'F^p7$$!3;*\\7Oww_'[F*$\"3AWT$>!zVowF^p7$$!3;* \\P))4Vnn%F*$\"3C[v.QMJb$)F^p7$$!3I**\\7)*z-![%F*$\"3%=UXA#)*yZ&)F^p7$ $!3%))\\iy7#Q*H%F*$\"3'R9fjLs]R)F^p7$$!3W**\\P_6`/TF*$\"3r8108E4 M!)[)F^p7$$!3Q***\\dD7-P#F*$\"3'=L]%*37`()*F^p7$$!3!**\\7`\"zf!>#F*$\" 3y[L5]=P_6F^o7$$!3C***\\))*>B%*>F*$\"3c-NRdJz!Q\"F^o7$$!3h*\\7Y(pY4=F* $\"31r#R3n%R[;F^o7$$!3B*\\imGMjh\"F*$\"3ePI%\\/r<*>F^o7$$!3!****\\Fy6v U\"F*$\"3oW@GVof,CF^o7$$!3b*\\7.7K)H7F*$\"3'>sqeQ0W#HF^o7$$!3$))***f5J WR5F*$\"3u;9k#f\\o`$F^o7$$!33*)*\\n\\GuW)F^o$\"3y$)egmRw'H%F^o7$$!3*=) \\7_![l^'F^o$\"3(H\\hjFT=@&F^o7$$!3E\")***pXpAu%F^o$\"3:Qek6IQ**))F^p$\"3av!y+=7&[\" *F^o7$$\"3k,](Q,!H\\5F^o$\"3RIpBa " 0 "" {MPLTEXT 1 0 1697 "R := z -> 1+z+1/2*z^2+1/6*z^3+1/24*z^4+1/120*z^5+1/720*z^6+\n \+ 42121991/266680835085*z^7+533609617/68575071879000*z^8:\npts := []: z0 := 0: tt := 0: \nwhile tt<=241/20 do\n zz := newton(`R`(z)=e xp(tt*Pi*I),z=z0):\n z0 := zz:\n if (7/20<=tt and tt<=27/20) or (2 13/20<=tt and tt<=233/20) then\n hh := 1/40\n else \n hh : = 1/20\n end if;\n tt := tt+hh;\n pts := [op(pts),[Re(zz),Im(zz) ]]:\nend do:\np1 := plot(pts,color=COLOR(RGB,.1,.23,.5)):\np2 := plots [polygonplot]([seq([pts[i-1],pts[i],[-4,0]],i=2..nops(pts))],\n \+ style=patchnogrid,color=COLOR(RGB,.2,.45,1)):\npts := []: z0 := 1.3 +4.5*I: tt := 0: \nwhile tt<=61/30 do\n zz := newton(R(z)=exp(tt*Pi* I),z=z0):\n z0 := zz:\n if (13/30<=tt and tt<=53/30) then\n h h := 1/60\n else \n hh := 1/30\n end if;\n tt := tt+hh;\n \+ pts := [op(pts),[Re(zz),Im(zz)]]:\nend do:\np3 := plot(pts,color=COLO R(RGB,.1,.23,.5)):\np4 := plots[polygonplot]([seq([pts[i-1],pts[i],[1. 16,4.38]],i=2..nops(pts))],\n style=patchnogrid,color=COLOR(R GB,.2,.45,1)):\npts := []: z0 := 1.3-4.5*I: tt := 0: \nwhile tt<=61/30 do\n zz := newton(R(z)=exp(tt*Pi*I),z=z0):\n z0 := zz:\n if (3/ 10<=tt and tt<=8/5) then\n hh := 1/60\n else \n hh := 1/30 \n end if;\n tt := tt+hh;\n pts := [op(pts),[Re(zz),Im(zz)]]:\ne nd do:\np5 := plot(pts,color=COLOR(RGB,.1,.23,.5)):\np6 := plots[polyg onplot]([seq([pts[i-1],pts[i],[1.16,-4.38]],i=2..nops(pts))],\n \+ style=patchnogrid,color=COLOR(RGB,.2,.45,1)):\np7 := plot([[[-8.79, 0],[1.99,0]],[[0,-5.19],[0,5.19]]],color=black,linestyle=3):\nplots[di splay]([p||(1..7)],view=[-8.79..1.99,-5.19..5.19],font=[HELVETICA,9], \n labels=[`Re(z)`,`Im(z)`],axes=boxed,scaling=constraine d);" }}{PARA 13 "" 1 "" {GLPLOT2D 625 529 529 {PLOTDATA 2 "6/-%'CURVES G6$7f\\l7$$\"\"!F)F(7$F($\"3#******fK'zq:!#=7$$!32+++bTsmp?m&!#D$\"3@+++hpQ7ZF-7$$!3&)*****\\Eu[i&!#C$\" 31+++w0<$G'F-7$$!37+++(=h&GL!#B$\"3/+++WQ\"R&yF-7$$!3)******>Nh\"=9!#A $\"3Z+++9eaC%*F-7$$!31+++b@V7[FI$\"3#******\\(G\\*4\"!#<7$$!3J+++VlP-$ )FI$\"3!******H*f*z<\"FQ7$$!37+++d![7Q\"!#@$\"3-+++_O[c7FQ7$$!38+++]-^ DAFZ$\"31+++n:&\\L\"FQ7$$!3%******f%[1&[$FZ$\"36+++E]R89FQ7$$!3)****** >\"e$)>`FZ$\"3-+++&44=\\\"FQ7$$!3i*****4I?b$zFZ$\"31+++L*)=q:FQ7$$!35+ ++&GC#f6!#?$\"3%******z?I&[;FQ7$$!35+++\"p#Rh;F^p$\"3.+++H'Hos\"FQ7$$! 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" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 148 "We can distort the boundary curve horizontally by taking the 11th roo t of the real part of points along the curve. In this way we see that \+ there is " }{TEXT 260 53 "no largest interval on the nonnegative imagi nary axis" }{TEXT -1 65 " that contains the origin and lies inside the stability region. " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 377 "R := z -> 1+z+1/2*z^2+1/6*z^3+1/24*z^4+1 /120*z^5+1/720*z^6+\n 42121991/266680835085*z^7+533609617/685 75071879000*z^8:\nDigits := 25:\npts := []: z0 := 0:\nfor ct from 0 to 120 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,.95),thickness=2,font=[HELVETICA,9]);\nDigits := 10:" } }{PARA 13 "" 1 "" {GLPLOT2D 222 299 299 {PLOTDATA 2 "6(-%'CURVESG6#7er 7$$\"\"!F)F(7$$!:4E&*3hA&>7stiI!#E$\":o0CW5n*e`EfTJF-7$$!:+n'fe#pp!o*R .2&F-$\":M]FZC&R;2`=$G'F-7$$!:Gzdr.n*=!HA#4oF-$\":)*o%\\=_C]gzxC%*F-7$ $!:$f$Rg)ewhMbq$R)F-$\":;J$>Z^hBhqjc7!#D7$$!:@kx1R&QI:mSs)*F-$\":AtIr) *eVeK'zq:F?7$$!:sS\"H`;[(p%y=F6F?$\":V)*)fm]1v)eb\\)=F?7$$!:'R'4(*RUAU Fn3E\"F?$\":jtzyf'*>v%[6*>#F?7$$!:p4Z!\\%zq+T+%*Q\"F?$\":)[ng`E4[(4uK^ #F?7$$!:PKDRwnY$RBf8:F?$\":\"pqbDclYILVFGF?7$$!:joQ/uPi$fh/M;F?$\":8uS VsHdJ`#fTJF?7$$!:FGrAMNtsnG7v\"F?$\":U(pW^$\\))\\o^dX$F?7$$!:#=([Oa[R. q5b'=F?$\":gC(yS^Neb2\"*pPF?7$$!:Ioj)3i?2[I>x>F?$\":TOY_/g!o,(pS3%F?7$ $!:A)R<&y\\7j0Cl3#F?$\":g&))4g#)p_j%G#)R%F?7$$!:ky:F`8'*[w6P>#F?$\":H? 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" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 32 "z_0 := newton(`R*`( z)=1,z=-8.1);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%$z_0G$!+:d0e!)!\"* " }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 319 "z_0 := newton(`R*`(z)=1,z=-8.1):\np_1 := plot([`R*`( z),1],z=-8.69..0.49,color=[red,blue]):\np_2 := plot([[[z_0,1]]$3],styl e=point,symbol=[circle,cross,diamond],color=black):\np_3 := plot([[z_0 ,0],[z_0,1]],linestyle=3,color=COLOR(RGB,0,.5,0)):\nplots[display]([p_ 1,p_2,p_3],view=[-8.69..0.49,-.07..1.47],font=[HELVETICA,9]);" }} {PARA 13 "" 1 "" {GLPLOT2D 449 255 255 {PLOTDATA 2 "6+-%'CURVESG6$7hn7 $$!3]*************o)!#<$\"3Gq3Gb9xjHF*7$$!31*\\(=Jb(*R')F*$\"3%Q(e$oct ut#F*7$$!3Q+]Pi5&**e)F*$\"3gnII&p#zDDF*7$$!3&**\\iNfE*R&)F*$\"3QKJKcD( zK#F*7$$!3]***\\Z7-**[)F*$\"3t[/GuoGV@F*7$$!3;Dc\">5wjW)F*$\"3uv%GNYNF *>F*7$$!3/\\73z+&GS)F*$\"3!e=ZAzd6&=F*7$$!3#H(oCcSKf$)F*$\"3Nm.@$*z7=< F*7$$!3d)\\7M.)z:$)F*$\"3!3fRqjNKf\"F*7$$!3k[(oN!*)*y@)F*$\"3M18L3(Q+M \"F*7$$!3r)*\\st(***>\")F*$\"3'*pj$>(4!=7\"F*7$$!3q)****[q^9-)F*$\"3e% )enm9)\\L*!#=7$$!3f**\\2OO!H#zF*$\"3=!3ETK=2t(F^o7$$!36](=!\\R#[#yF*$ \"3#*p;ky.EWns(F*$\"3VcO$4<@&R_F^o7$$!3c*\\7?Tz[a(F*$ \"33SK&*GQ\"\\g$F^o7$$!3!)*\\7R*)olN(F*$\"3EPp7#QZUV#F^o7$$!3)**\\7n)y \"=;(F*$\"3&oD[+,j3k\"F^o7$$!3W*\\i%[9pnpF*$\"3#Q\\)oE37b6F^o7$$!3')** *\\ba3!onF*$\"3QC#oO*)z\\v)!#>7$$!3o**\\(ftF@f'F*$\"3urWuoewDvFgq7$$!3 E***\\;gFTR'F*$\"3Md=KU$y>+(Fgq7$$!3g***\\(fWJ&>'F*$\"3m(Q10G=n)pFgq7$ $!3U***\\$e6s.gF*$\"3+QRn$Rg\"yrFgq7$$!3\"*)\\7d3O(HeF*$\"39V4RZ'foP(F gq7$$!3W***\\h+^Gi&F*$\"3gk:VWc?IvFgq7$$!3a****f1AfZaF*$\"3$eR)\\9c`Tv Fgq7$$!3k*\\7'eXsV_F*$\"368].J@'3S(Fgq7$$!3S****4YrEj]F*$\"3%H!f@/]#)[ rFgq7$$!3;*\\7Oww_'[F*$\"3K/&))3w#QknFgq7$$!3;*\\P))4Vnn%F*$\"3-13YzZg JjFgq7$$!3I**\\7)*z-![%F*$\"3e:76&QZh&eFgq7$$!3%))\\iy7#Q*H%F*$\"3(>N \"G#)Q'zV&Fgq7$$!3W**\\P_6`/TF*$\"3GJR^G\"*fZ]Fgq7$$!3j*\\72vN@!RF*$\" 3U\\oY+sz\\ZFgq7$$!36*\\(ee*\\fs$F*$\"3$)G3x!4/'4YFgq7$$!3e**\\KwWmNNF *$\"3Per2Z7a4YFgq7$$!3\"*****p`43RLF*$\"3xH&GZ.5.![Fgq7$$!3)***\\-HBwY JF*$\"3wOa&*=;'*)>&Fgq7$$!3!)*\\73<%ogHF*$\"3;Dl\\\"4)>2eFgq7$$!3]**\\ d,h2aFF*$\"33R+u*o)eonFgq7$$!3S++5i&H%oDF*$\"3`7K8imR?zFgq7$$!3Q***\\d D7-P#F*$\"3i/ylkS^#\\*Fgq7$$!3!**\\7`\"zf!>#F*$\"3a=\\M*Q3n7\"F^o7$$!3 C***\\))*>B%*>F*$\"3o/z#4cu]O\"F^o7$$!3h*\\7Y(pY4=F*$\"3;x#)fZ,0R;F^o7 $$!3B*\\imGMjh\"F*$\"3m4xs^gs')>F^o7$$!3!****\\Fy6vU\"F*$\"3y)3#)\\Ko! *R#F^o7$$!3b*\\7.7K)H7F*$\"3,CmJWfJBHF^o7$$!3$))***f5JWR5F*$\"3KQi7PPV ONF^o7$$!33*)*\\n\\GuW)F^o$\"3ms'\\J!*QmH%F^o7$$!3*=)\\7_![l^'F^o$\"3o B/z#)Q\"=@&F^o7$$!3E\")***pXpAu%F^o$\"3B:2?5vlBiF^o7$$!3+$)*\\(ffu3FF^ o$\"3#4zuNE@ri(F^o7$$!3]t****>IQ**))Fgq$\"3#zT$))z@^[\"*F^o7$$\"3k,](Q ,!H\\5F^o$\"3qXL=aF^o$\"3%Q-zw^P'=7F*7$$\"3v,]7_ *y`!HF^o$\"3'p " 0 "" {MPLTEXT 1 0 1847 "`R*` := z -> 1+z+1/2*z^2+1/6*z^3+1/24*z^4+1/120*z^5 +\n 944100953333055801679/699723966496841608644000*z^6+\n 4280 86926242079513313/2798895865987366434576000*z^7+\n 980936041767266 25091/10495859497452624129660000*z^8+\n 533609617/2743002875160000 *z^9:\npts := []: z0 := 0: tt := 0: \nwhile tt<=241/20 do\n zz := ne wton(`R*`(z)=exp(tt*Pi*I),z=z0):\n z0 := zz:\n if (13/20<=tt and t t<=29/20) or (211/20<=tt and tt<=227/20) then\n hh := 1/40\n el se \n hh := 1/20\n end if;\n tt := tt+hh;\n pts := [op(pts) ,[Re(zz),Im(zz)]]:\nend do:\np_1 := plot(pts,color=COLOR(RGB,0,.18,.4) ):\np_2 := plots[polygonplot]([seq([pts[i-1],pts[i],[-4,0]],i=2..nops( pts))],\n style=patchnogrid,color=COLOR(RGB,0,.35,.8)):\npts \+ := []: z0 := 1.25+4.6*I: tt := 0: \nwhile tt<=61/30 do\n zz := newto n(`R*`(z)=exp(tt*Pi*I),z=z0):\n z0 := zz:\n if (13/30<=tt and tt<= 53/30) then\n hh := 1/60\n else \n hh := 1/30\n end if; \n tt := tt+hh;\n pts := [op(pts),[Re(zz),Im(zz)]]:\nend do:\np_3 \+ := plot(pts,color=COLOR(RGB,0,.18,.4)):\np_4 := plots[polygonplot]([se q([pts[i-1],pts[i],[1.02,4.51]],i=2..nops(pts))],\n style=pat chnogrid,color=COLOR(RGB,0,.35,.8)):\npts := []: z0 := 1.25-4.6*I: tt \+ := 0: \nwhile tt<=61/30 do\n zz := newton(`R*`(z)=exp(tt*Pi*I),z=z0) :\n z0 := zz:\n if (3/10<=tt and tt<=8/5) then\n hh := 1/60\n else \n hh := 1/30\n end if;\n tt := tt+hh;\n pts := [op (pts),[Re(zz),Im(zz)]]:\nend do:\np_5 := plot(pts,color=COLOR(RGB,0,.1 8,.4)):\np_6 := plots[polygonplot]([seq([pts[i-1],pts[i],[1.02,-4.51]] ,i=2..nops(pts))],\n style=patchnogrid,color=COLOR(RGB,0,.35, .8)):\np_7 := plot([[[-8.79,0],[1.79,0]],[[0,-5.19],[0,5.19]]],color=b lack,linestyle=3):\nplots[display]([p_||(1..7)],view=[-8.79..1.79,-5.1 9..5.19],font=[HELVETICA,9],\n labels=[`Re(z)`,`Im(z)`],axe s=boxed,scaling=constrained);" }}{PARA 13 "" 1 "" {GLPLOT2D 681 553 553 {PLOTDATA 2 "6/-%'CURVESG6$7^\\l7$$\"\"!F)F(7$$!3=+++G@Z#*f!#F$\"3 ++++Fjzq:!#=7$$!38+++(yvs!R!#D$\"3C+++PEfTJF07$$!3=+++&3jhe%!#C$\"31++ +<()Q7ZF07$$!3A+++m8(=o#!#B$\"3q*****pa$=$G'F07$$!3/+++d/ct5!#A$\"3I++ +^X(R&yF07$$!3m*****f8[_Q$FF$\"38+++t#eZU*F07$$!3U*****z?rX0*FF$\"3))* ****pf`&*4\"!#<7$$!3)******>\"ojX@!#@$\"3/+++\"*Hjc7FS7$$!36+++'Go2j%F W$\"3#******p%=s89FS7$$!3S+++H3Zt#*FW$\"3%******>/S3d\"FS7$$!3#******> ([0Y+++Y(**>7#FS7$$ !3C*****f#=E)z)Fao$\"36+++n-;,AFS7$$!3.+++'\\Sb6\"!#>$\"3!)*****>dN0G# FS7$$!3,+++^A`;H=#F[q$\"3********ohe?DFS7$$!3#******psa)*p#F[q$\"3?+++x x^,EFS7$$!3'******pmKYK$F[q$\"3'******z\"o2$o#FS7$$!3++++2!>%zSF[q$\"3 ++++u[QlFFS7$$!3!)******Qs/#*\\F[q$\"35+++MDd[GFS7$$!3A+++*H;z4'F[q$\" 3?+++(fqF$HFS7$$!3!******>.kCW(F[q$\"3.+++C<4=IFS7$$!3%*******>;5%3*F[ q$\"3%)*****H\"[f/JFS7$$!3)******zK#p46F0$\"39+++?+A#>$FS7$$!3-+++#G6r N\"F0$\"3-+++c+o!G$FS7$$!3'******\\iR1m\"F0$\"3')*****p'[IpLFS7$$!3!** *****Hn9H?F0$\"3#******4S()oX$FS7$$!3++++]IkmCF0$\"34+++K!y;a$FS7$$!31 +++XzUoHF0$\"3*)*****>$[r@OFS7$$!39+++pOp?NF0$\"3')*****R(\\V&p$FS7$$! 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YF^goF^b\\mF\\eflF[^u-%+AXESLABELSG6%%&Re(z)G%&Im(z)G-%%FONTG6#%(DEFAU LTG-%(SCALINGG6#%,CONSTRAINEDG-Fe]^m6$%*HELVETICAG\"\"*-%*AXESSTYLEG6# %$BOXG-%%VIEWG6$;$!$z)Fico$\"$*>Fico;$!$>&Fico$\"$>&Fico" 1 2 0 1 10 0 2 9 1 2 1 1.000000 45.000000 45.000000 0 0 "Curve 1" "Curve 2" "Curv e 3" "Curve 4" "Curve 5" "Curve 6" "Curve 7" "Curve 8" "Curve 9" "Curv e 10" "Curve 11" "Curve 12" "Curve 13" "Curve 14" }}}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}}{PARA 0 "" 0 "" {TEXT -1 52 "#---------------------------------------------- -----" }}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 36 "coefficients for the com bined scheme" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 35 "coeffici ents of the combined scheme" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1931 "ee := \{c[2]=1/192,\nc[3]=56/435, \nc[4]=28/145,\nc[5]=56/169,\nc[6]=37/58,\nc[7]=121/158,\nc[8]=1,\nc[9 ]=1,\n\na[2,1]=1/192,\na[3,1]=-30744/21025,\na[3,2]=100352/63075,\na[4 ,1]=7/145,\na[4,2]=0,\na[4,3]=21/145,\na[5,1]=523516/4826809,\na[5,2]= 0,\na[5,3]=-889140/4826809,\na[5,4]=1965040/4826809,\na[6,1]=410447467 63290003/51739779980736512,\na[6,2]=0,\na[6,3]=-57685603608510645/2497 7824818286592,\na[6,4]=83145940696701395/53969942910940672,\na[6,5]=53 772835209542058935/87647187287367651328,\na[7,1]=-19737371728514492920 457623/38482466972140721924237312,\na[7,2]=0,\na[7,3]=2746625301407315 5807766505/14560933448918110998360064,\na[7,4]=-1176701341021586311278 38015/122987884309611901825434112,\na[7,5]=-19483500956399449871589744 537/481311655153988159050791915520,\na[7,6]=913336658253963141/2343933 323194077385,\na[8,1]=2484813954171031615807/2538960546022287823744,\n a[8,2]=0,\na[8,3]=-229147524517236387495/87335252197326853888,\na[8,4] =130548415875501708716128185/106468788692037442075798624,\na[8,5]=1110 3534575830173429908962617/8186576761928888116508540672,\na[8,6]=-10679 027011796052/13181450824860817,\na[8,7]=2238377869046599644/2564667687 663709133,\na[9,1]=397933/5470080,\na[9,2]=0,\na[9,3]=0,\na[9,4]=54500 42155625/32205284983872,\na[9,5]=1677324270326783/6406236767145600,\na [9,6]=179200957846/1323774725175,\na[9,7]=77461817788426/2722799429395 65,\na[9,8]=47049059/616363020,\n\nb[1]=397933/5470080,\nb[2]=0,\nb[3] =0,\nb[4]=5450042155625/32205284983872,\nb[5]=1677324270326783/6406236 767145600,\nb[6]=179200957846/1323774725175,\nb[7]=77461817788426/2722 79942939565,\nb[8]=47049059/616363020,\n\n`b*`[1]=58931524118239849/83 7231220422643200,\n`b*`[2]=0,\n`b*`[3]=0,\n`b*`[4]=1800771389181962432 05225/985845547089116780242176,\n`b*`[5]=239006278908625056478598819/9 80516085116436873474624000,\n`b*`[6]=4448038142979447869029/2894461529 7423551638500,\n`b*`[7]=1616804268713313415249069/59534587356230793594 84300,\n`b*`[8]=713536679628437051/13476908236860104400,\n`b*`[9]=1/40 \}:" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 " " {TEXT 259 5 "nodes" }{TEXT -1 1 ":" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 31 "seq(c[i]=subs(ee,c[i]),i=2..9);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6*/&%\"cG6#\"\"##\"\"\"\"$#>/&F%6#\"\"$#\"#c\"$N%/&F%6#\" \"%#\"#G\"$X\"/&F%6#\"\"&#F0\"$p\"/&F%6#\"\"'#\"#P\"#e/&F%6#\"\"(#\"$@ \"\"$e\"/&F%6#\"\")F)/&F%6#\"\"*F)" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" } }{PARA 0 "" 0 "" {TEXT 259 51 "linking coefficients for the 9 stage or der 5 scheme" }{TEXT -1 1 ":" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 49 "seq(seq(a[i,j]=subs(ee,a[i,j]),j=1..i-1),i=1..9);" }}{PARA 12 "" 1 "" {XPPMATH 20 "6F/&%\"aG6$\"\"#\"\"\"#F(\"$#>/&F%6$\"\"$F(#!&W2$\"& D5#/&F%6$F.F'#\"'_.5\"&vI'/&F%6$\"\"%F(#\"\"(\"$X\"/&F%6$F;F'\"\"!/&F% 6$F;F.#\"#@F>/&F%6$\"\"&F(#\"';N_\"(4o#[/&F%6$FKF'FB/&F%6$FKF.#!'S\"*) )FN/&F%6$FKF;#\"(S]'>FN/&F%6$\"\"'F(#\"2.+HjnuW5%\"27lt!)*z(R<&/&F%6$F inF'FB/&F%6$FinF.#!2X1^3Og&od\"2#f'G=[#y(\\#/&F%6$FinF;#\"2&R,npSf9$) \"2s1%4\"H%*pR&/&F%6$FinFK#\"5N*e?a4_$Gx`\"5G8lntG(=Zw)/&F%6$F=F(#!;Bw X?H\\9&G\";7tBC>sS@(pY#[Q/&F%6$F=F'FB/&F%6$F=F.#\";0lw2e:tS,`iYF \";k+O)*46=*[M$4c9/&F%6$F=F;#!<:!Qy7J'e@5M,n<\"\"<7TVD=!>h4V)y)H7/&F%6 $F=FK#!>PXu*er)\\%*Rc4]$[>\"??b\">z]!f\"))R:b;J\"[/&F%6$F=Fin#\"3TJ'RD emL8*\"4&QxS>BL$RM#/&F%6$\"\")F(#\"72ehJ5__L()/&F%6$FfrF;#\"<&=Ghr 3<]veT[08\"wwl=)/&F%6$FfrFin#!2_gz6q-z1\"\"2<3'[#3X\"=8/&F%6$FfrF=#\"4W'*fY!pyP QA\"4L\"4Pm(onYc#/&F%6$\"\"*F(#\"'LzR\"(!3qa/&F%6$F^uF'FB/&F%6$F^uF.FB /&F%6$F^uF;#\".Dc:U+X&\"/sQ)\\G0A$/&F%6$F^uFK#\"1$yE.FCtn\"\"1+c9nnB1k /&F%6$F^uFin#\"-Yy&4?z\"\".v^suPK\"/&F%6$F^uF=#\"/E%)y<=Yx\"0l&RH%*zAF /&F%6$F^uFfr#\")f!\\q%\"*?IO;'" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }} {PARA 0 "" 0 "" {TEXT 259 38 "weights for the 8 stage order 6 scheme" }{TEXT -1 1 ":" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 31 "seq(b[i]=s ubs(ee,b[i]),i=1..8);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6*/&%\"bG6#\"\" \"#\"'LzR\"(!3qa/&F%6#\"\"#\"\"!/&F%6#\"\"$F//&F%6#\"\"%#\".Dc:U+X&\"/ sQ)\\G0A$/&F%6#\"\"&#\"1$yE.FCtn\"\"1+c9nnB1k/&F%6#\"\"'#\"-Yy&4?z\"\" .v^suPK\"/&F%6#\"\"(#\"/E%)y<=Yx\"0l&RH%*zAF/&F%6#\"\")#\")f!\\q%\"*?I O;'" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT 259 38 " weights for the 9 stage order 5 scheme" }{TEXT -1 1 ":" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 37 "seq(`b*`[i]=subs(ee,`b*`[i]),i=1..9 );" }}{PARA 12 "" 1 "" {XPPMATH 20 "6+/&%#b*G6#\"\"\"#\"2\\)R#=T_J*e\" 3+KkA/AJs$)/&F%6#\"\"#\"\"!/&F%6#\"\"$F//&F%6#\"\"%#\"9D_?Vi>=*Qr2!=\" 9w@C!y;\"*3Zb%e)*/&F%6#\"\"&#\"<>))fyk0D'3*yi+R#\"<+SiuM(oV;^3;0)*/&F% 6#\"\"'#\"7H!pyWzH9Q![W\"8+&Q;bBuH:Y%*G/&F%6#\"\"(#\":p!\\_T8LroU!oh\" \":+V[f$zIiN(eM&f/&F%6#\"\")#\"3^qVG'zm`8(\"5+W5goB3pZ8/&F%6#\"\"*#F' \"#S" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}}{PARA 0 "" 0 "" {TEXT -1 52 "#----------------- ----------------------------------" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}}{PARA 0 "" 0 "" {TEXT -1 59 "#================= =========================================" }}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 46 "a scheme with a large imaginary axis inclusion" }}{PARA 257 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 161 "See: A Param eter Study of Explicit Runge-Kutta Pairs of Orders 6(5), by Ch. Tsitou ras,\n Applied Mathematics Letters, Vol. 11, No. 1, pages 65 to 69, 1998." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 257 "" 0 "" {TEXT -1 98 "The scheme constructed here is a minor modification of a scheme presented in the preceding paper. " }}{PARA 0 "" 0 "" {TEXT -1 0 "" } }{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}{PARA 0 "" 0 "" {TEXT -1 52 "#---------------------------------------------------" }}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 28 "checking the combined scheme" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 35 "coefficients of the combined s cheme" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1757 "ee := \{c[2]=14/145,\nc[3]=59/321,\nc[4]=59/214,\nc [5]=47/60,\nc[6]=53/75,\nc[7]=4/35,\nc[8]=1,\nc[9]=1,\n\na[2,1]=14/145 ,\na[3,1]=25547/2885148,\na[3,2]=504745/2885148,\na[4,1]=59/856,\na[4, 2]=0,\na[4,3]=177/856,\na[5,1]=109313963/93987000,\na[5,2]=0,\na[5,3]= -280562881/62658000,\na[5,4]=770307017/187974000,\na[6,1]=259491568474 60933/19448380820547506250,\na[6,2]=0,\na[6,3]=13843311346643833/13793 1778869131250,\na[6,4]=322308298919703822734/674279501001748115625,\na [6,5]=8562169944665744/67437276532035625,\na[7,1]=-1124425445878279929 044/5080674836868609224225,\na[7,2]=0,\na[7,3]=10208986940897432574/28 55457555847472065,\na[7,4]=-1389076296899785384285714176/3218458025953 21532838393475,\na[7,5]=-8341061397510669205248/2889892848216596175835 ,\na[7,6]=7781543704350/1963746281231,\na[8,1]=-4662888761921915772773 /8195396815968383745983,\na[8,2]=0,\na[8,3]=2931920524405086861/940000 781782231318,\na[8,4]=-182277776762433747385270230/7416489655164903191 4481339,\na[8,5]=-146526017349731441994000/935639633573123866302727,\n a[8,6]=3595419767671875/3518327954949241,\na[8,7]=1418176375/335006407 22,\na[9,1]=577999/7054512,\na[9,2]=0,\na[9,3]=0,\na[9,4]=341100580906 24/83078929543805,\na[9,5]=1110240000/12869442287,\na[9,6]=77530078125 /230762131864,\na[9,7]=203784875/52405236624,\na[9,8]=1341697/16490760 ,\n\nb[1]=577999/7054512,\nb[2]=0,\nb[3]=0,\nb[4]=34110058090624/83078 929543805,\nb[5]=1110240000/12869442287,\nb[6]=77530078125/23076213186 4,\nb[7]=203784875/52405236624,\nb[8]=1341697/16490760,\n\n`b*`[1]=463 127996257/5946086987136,\n`b*`[2]=0,\n`b*`[3]=0,\n`b*`[4]=126224639819 736010009/315113992065678492180,\n`b*`[5]=624913261131500/888880796265 0317,\n`b*`[6]=450569541695078125/1275082620954224192,\n`b*`[7]=115589 702589080125/7818298262438511744,\n`b*`[8]=12589356280823/273360468363 840,\n`b*`[9]=1/27\}:" }}}{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 0 488 "subs(ee,matrix([[c[2],a [2,1],``$3],\n [c[3],a[3,1],a[3,2],``$2],\n [c[4],seq(a[4,i],i=1..3) ,``],\n [c[5],seq(a[5,i],i=1..4)],\n [c[6],seq(a[6,i],i=1..4)],[``$4 ,a[6,5]],\n [c[7],seq(a[7,i],i=1..4)],[``$3,a[7,5],a[7,6]],\n [c[8], seq(a[8,i],i=1..4)],[``$2,seq(a[8,i],i=5..7)],\n [c[9],seq(a[9,i],i=1 ..4)],[``,seq(a[9,i],i=5..8)],\n [``,`____________________________`$4 ],\n [`b`,seq(b[i],i=1..4)],[``,seq(b[i],i=5..8)],\n [`b*`,seq(`b*`[ i],i=1..4)],[``,seq(`b*`[i],i=5..8)],[``$4,`b*`[9]]]));" }}{PARA 11 " " 1 "" {XPPMATH 20 "6#K%'matrixG6#747'#\"#9\"$X\"F(%!GF+F+7'#\"#f\"$@$ #\"&Zb#\"([^)G#\"'XZ]F2F+F+7'#F.\"$9##F.\"$c)\"\"!#\"$x\"F9F+7'#\"#Z\" #g#\"*jRJ4\"\")+q)R*F:#!*\")Gc!G\")+!eE'#\"*F:#\"2LQkY8JVQ\"\"3]78p)y<$z8#\"6MF#Qq>*)H3 BK\"6Dc6[<+,&zUn7'F+F+F+F+#\"1WdmW*p@c)\"2Dc.KlFPu'7'#\"\"%\"#N#!7W!H* z#yeWDW7\"\"7DUA4'oo$[n!3&F:#\"5uDV(*3%p)*3-\"\"4l?ZZebda&G#!=wTr&G%Q& y**oHw!*Q\"\"7'\"\"\"#!7tFx:>#>w))Gm%\"7$)fu$Qof\"oR&>)F:#\"4ho 30W_?>$H\"3=8B#y\"y++%*#!.\\;b'*[;u7'F+F+#!9+S* >WJ(\\t,El9\"9FFImQ7tNjRc$*#\"1v=nn(>af$\"1T#\\\\&zK=N#\"+vj<=9\",A2k+ N$7'Fjo#\"'**zd\"(7X0(F:F:#\"/C14e+6M\"/0QaH*yI)7'F+#\"+++C56\",(GU%pG \"#\",D\"y+`x\"-k=8i2B#\"*v[y.#\",CmB0C&#\"((pT8\")g2\\;7'F+%=________ ____________________GFcrFcrFcr7'%\"bGF_qF:F:FbqFeq7'%#b*G#\"-di*z7j%\" .Or)p3YfF:F:#\"64+,O(>)RYAE\"\"6!=#\\yc1#*R6:$7'F+#\"0+:8hK\"\\i\"1<.l iz!)))))#\"3D\"y]pT&p0X\"4#>CU&4i#3v7#\"3D,3*e-(*e:\"\"4W<^QCE)H=y#\"/ B3Gc$*e7\"0SQOo/Ot#7'F+F+F+F+#Fjo\"#FQ(pprint66\"" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 136 "subs(ee,mat rix([seq([c[i],seq(a[i,j],j=1..i-1),``$(10-i)],i=2..9),\n[`b`,seq(b[i] ,i=1..8),``],[`b*`,seq(`b*`[i],i=1..9)]])):\nevalf[8](%);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#K%'matrixG6#7,7,$\")C98AF/F:$\")VDvNFC$!)$ofJ%FC$!)u G')GFC$\")9giRFCF+F+F+7,$\"\"\"F;$!)Pk*o&F/F:$\")=1>JFC$!)ltdCFC$!)=0m :F/$\"):\">-\"FC$\")8GLUF*F+F+7,F[o$\")OK$>)F*F:F:$\")7u0TF/$\")t%pi)F *$\"),ufLF/$\")dj))QF2$\")K0O\")F*F+7,%\"bGFjoF:F:F\\pF^pF`pFbpFdpF+7, %#b*G$\")hy)y(F*F:F:$\")=o0SF/$\")#Q..(F*$\"))\\O`$F/$\")4Xy9F*$\")\\S 0YF*$\")Pq.PF*Q(pprint76\"" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 146 "RK6_8eqs := [op(RowSumConditions(8 ,'expanded')),op(OrderConditions(6,8,'expanded'))]:\n`RK5_9eqs*` := su bs(b=`b*`,OrderConditions(5,9,'expanded')):" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 7 "Check: " }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 148 "simplify(subs(ee,RK6_8eqs)):\nmap(u->`if`(lhs(u )=rhs(u),0,1),%);\nnops(%);\nsimplify(subs(ee,`RK5_9eqs*`)):\nmap(u->` if`(lhs(u)=rhs(u),0,1),%);\nnops(%);" }{TEXT -1 0 "" }}{PARA 11 "" 1 " " {XPPMATH 20 "6#7N\"\"!F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$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#\"#W" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#73\"\"!F$F$F$F$F$F$F$F$F $F$F$F$F$F$F$F$" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#\"#<" }}}{PARA 0 " " 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 86 "Next we set-up sta ge-order condtions to check for stage-orders from 2 to 5 inclusive. " }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 85 "for ct from 2 to 5 do\n \+ so||ct||_8 := StageOrderConditions(ct,8,'expanded');\nend do:" }}} {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 34 "Stages 4 \+ to 8 have stage-order 3. " }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 173 "[seq([seq(expand(subs(ee,so||i||_8[j])),i=2..5)],j=1..6)]:\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 53 "None of the principal error conditions are sati sfied." }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 123 "RK6_8err_eqs := P rincipalErrorConditions(6,8,'expanded'):\nexpand(subs(ee,RK6_8err_eqs) ):\nmap(u->`if`(lhs(u)=rhs(u),0,1),%);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#7R\"\"\"F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$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 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 114 "We can calculate the principal erro r norm of the order 6 scheme, that is, the 2-norm of the principal err or terms." }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 131 "errterms6_8 := PrincipalErrorTerms(6,8,'expanded'):\nevalf(evalf[14](sqrt(add(subs(e e,errterms6_8[i])^2,i=1.. nops(errterms6_8)))));" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#$\"+#))oB!>!#8" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }} {PARA 0 "" 0 "" {TEXT -1 79 "The 2 norm of the principal error of the \+ order 5 embedded scheme is as follows." }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 153 "`errterms5_9*` := subs(b=`b*`,PrincipalErrorTerms(5, 9,'expanded')):\nevalf(evalf[14](sqrt(add(subs(ee,`errterms5_9*`[i])^2 ,i=1.. nops(`errterms5_9*`)))));" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#$ \"+khH?j!#8" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}}{PARA 0 "" 0 "" {TEXT -1 52 "#-------------- -------------------------------------" }}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 44 "simultaneous construction of the two schemes" }}{PARA 0 " " 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 119 "We incorporate th e stage-order equations to ensure that stage 2 has stage-order 2 and s tages 3 to 8 have stage-order 3." }}{PARA 0 "" 0 "" {TEXT -1 48 "We al so incorporate the simplifying conditions: " }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "Sum(b[i]*a[i,j],i = j+1 .. 8) = 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=1" "6#/%\"jG\"\"\"" } {TEXT -1 10 " ( with " }{XPPEDIT 18 0 "c[1]=0" "6#/&%\"cG6#\"\"\"\" \"!" }{TEXT -1 5 " ), " }{XPPEDIT 18 0 "j = 6;" "6#/%\"jG\"\"'" } {TEXT -1 7 " and " }{XPPEDIT 18 0 "j=7" "6#/%\"jG\"\"(" }{TEXT -1 1 "." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 78 "The 8 simple order conditions used are given (in abreviated form) as foll ows. " }}{PARA 0 "" 0 "" {TEXT -1 78 "These include the 6 quadrature c onditions and two additional order conditions." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 178 "SO6 := Simp leOrderConditions(6):\n[seq([i,SO6[i]],i=[1,2,4,8,16,24,29,32])]:\nlin alg[augment](linalg[delcols](%,2..2),matrix([[` `]$(linalg[rowdim](%) )]),linalg[delcols](%,1..1));" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#K%'ma trixG6#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%\"#;F)/*&F ,F()F2F5F(#F(\"\"&7%\"#CF)/*(F,F(F2F(-%!G6#*&%\"aGF(-FM6#*&F8F(FPF(F(F (#F(\"#s7%\"#HF)/*(F,F(F2F(-FM6#*&F?F(FPF(F(#F(FI7%\"#KF)/*&F,F()F2FGF (#F(\"\"'Q)pprint596\"" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 136 "For the embedded order 5 scheme we use a selection \+ of 7 \"simple\" order conditions as given (in abreviated form) in the \+ following table. " }}{PARA 0 "" 0 "" {TEXT -1 78 "These include the 5 \+ quadrature conditions and two additional order conditions." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 194 "`SO 5*` := subs(b=`b*`,SimpleOrderConditions(5)):\n[seq([i,`SO5*`[i]],i=[1 ,2,4,8,12,15,16])]:\nlinalg[augment](linalg[delcols](%,2..2),matrix([[ ` `]$(linalg[rowdim](%))]),linalg[delcols](%,1..1));" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#K%'matrixG6#7)7%\"\"\"%#~~G/*&%#b*GF(%\"eGF(F(7%\" \"#F)/*&F,F(%\"cGF(#F(F/7%\"\"%F)/*&F,F()F2F/F(#F(\"\"$7%\"\")F)/*&F,F ()F2F:F(#F(F57%\"#7F)/*&F,F(-%!G6#*&%\"aGF(-FF6#*&F8F(FIF(F(F(#F(\"#g7 %\"#:F)/*&F,F(-FF6#*&F?F(FIF(F(#F(\"#?7%\"#;F)/*&F,F()F2F5F(#F(\"\"&Q) pprint186\"" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 582 "SO6_8 := SimpleOrderConditions(6,8,'expanded'):\n SO_eqs := [op(RowSumConditions(8,'expanded')),op(StageOrderConditions( 2,8,'expanded')),\n op(StageOrderConditions(3,4..8,'expa nded'))]:\n`SO5_9*` := subs(b=`b*`,SimpleOrderConditions(5,9,'expanded ')):\nord_cdns := [seq(SO6_8[i],i=[1,2,4,8,16,24,29,32])]:\n`ord_cdns* ` := [seq(`SO5_9*`[i],i=[1,2,4,8,12,15,16])]:\nsimp_eqs := [add(b[i]*a [i,1],i=1+1..8)=b[1],seq(add(b[i]*a[i,j],i=j+1..8)=b[j]*(1-c[j]),j=[6, 7])]:\nwt_eqns := [seq(b[i]=a[9,i],i=1..8)]:\ncdns := [op(ord_cdns),op (simp_eqs),op(SO_eqs),op(wt_eqns),op(`ord_cdns*`)]:" }}}{PARA 0 "" 0 " " {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 22 "We specify the nodes: \+ " }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "c[2] = 14/145;" " 6#/&%\"cG6#\"\"#*&\"#9\"\"\"\"$X\"!\"\"" }{TEXT -1 3 ", " }{XPPEDIT 18 0 "c[4] = 59/214;" "6#/&%\"cG6#\"\"%*&\"#f\"\"\"\"$9#!\"\"" }{TEXT -1 3 ", " }{XPPEDIT 18 0 "c[5] = 47/60;" "6#/&%\"cG6#\"\"&*&\"#Z\"\" \"\"#g!\"\"" }{TEXT -1 3 ", " }{XPPEDIT 18 0 "c[6] = 53/75;" "6#/&%\" cG6#\"\"'*&\"#`\"\"\"\"#v!\"\"" }{TEXT -1 3 ", " }{XPPEDIT 18 0 "c[7] = 4/35;" "6#/&%\"cG6#\"\"(*&\"\"%\"\"\"\"#N!\"\"" }{TEXT -1 3 ", " } {XPPEDIT 18 0 "c[8]=1" "6#/&%\"cG6#\"\")\"\"\"" }{TEXT -1 2 ", " }} {PARA 0 "" 0 "" {TEXT -1 31 "the zero linking coefficients: " }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "a[4,2]=0" "6#/&%\"aG6$\"\"% \"\"#\"\"!" }{TEXT -1 3 ", " }{XPPEDIT 18 0 "a[5,2]=0" "6#/&%\"aG6$\" \"&\"\"#\"\"!" }{TEXT -1 3 ", " }{XPPEDIT 18 0 "a[6,2]=0" "6#/&%\"aG6 $\"\"'\"\"#\"\"!" }{TEXT -1 3 ", " }{XPPEDIT 18 0 "a[7,2]=0" "6#/&%\" aG6$\"\"(\"\"#\"\"!" }{TEXT -1 3 ", " }{XPPEDIT 18 0 "a[8,2]=0" "6#/& %\"aG6$\"\")\"\"#\"\"!" }{TEXT -1 2 ", " }}{PARA 0 "" 0 "" {TEXT -1 22 "and the zero weights: " }}{PARA 256 "" 0 "" {TEXT -1 1 " " } {XPPEDIT 18 0 "b[2]=0" "6#/&%\"bG6#\"\"#\"\"!" }{TEXT -1 3 ", " } {XPPEDIT 18 0 "b[3]=0" "6#/&%\"bG6#\"\"$\"\"!" }{TEXT -1 1 "." }} {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT 260 4 "Note" } {TEXT -1 89 ": Calculations relating to the choice of nodes are perfor med in the following subsection." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }} {PARA 0 "" 0 "" {TEXT -1 125 "The weights of the order 6 scheme provid e the linking coefficients for the 9th stage of the embedded order 5 s cheme so that: " }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "a[ 9,i]=b[i]" "6#/&%\"aG6$\"\"*%\"iG&%\"bG6#F(" }{TEXT -1 3 ", " } {XPPEDIT 18 0 "i=1" "6#/%\"iG\"\"\"" }{TEXT -1 9 " . . . 8." }}{PARA 0 "" 0 "" {TEXT -1 22 "We also specify that " }{XPPEDIT 18 0 "c[9] = \+ 1;" "6#/&%\"cG6#\"\"*\"\"\"" }{TEXT -1 3 ", " }{XPPEDIT 18 0 "`b*`[2] = 0;" "6#/&%#b*G6#\"\"#\"\"!" }{TEXT -1 3 ", " }{XPPEDIT 18 0 "`b*`[ 3] = 0;" "6#/&%#b*G6#\"\"$\"\"!" }{TEXT -1 7 " and " }{XPPEDIT 18 0 "`b*`[9] = 1/27;" "6#/&%#b*G6#\"\"**&\"\"\"F)\"#F!\"\"" }{TEXT -1 2 ". " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 37 "We h ave 44 equations and 44 unknowns." }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 209 "e1 := \{c[2]=14/145,c[4]=59/214,c[5]=47/60,c[6]=53/7 5,c[7]=4/35,c[8]=1,c[9]=1,\n seq(a[i,2]=0,i=4..8),b[2]=0,b[3]=0, `b*`[2]=0,`b*`[3]=0,`b*`[9]=1/27\}:\neqns := subs(e1,cdns):\nnops(%); \nindets(eqns);\nnops(%);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#\"#W" }} {PARA 12 "" 1 "" {XPPMATH 20 "6#FL&F%6#FL&F%6#F4&F%6#FC&F/6$\"\"*FL&F/6$FgoF>&F/ 6$FgoF1&F/6$FgoF4&F/6$FgoFC&F/6$FgoF'&F/6$FgoF-&F/6$FgoF*&%#b*GFbo&Fgp Fdo&FgpF&&FgpF,&FgpF)&FgpF`o" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#\"#W" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{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 := `uni on`(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 1876 "e3 := \{a[6,1] = 25949156847460933/19448380820547506250, `b*`[7] = 115589702589080125/7818298262438511744, a[8,3] = 293192052440508686 1/940000781782231318, c[8] = 1, c[9] = 1, c[4] = 59/214, c[5] = 47/60, c[6] = 53/75, c[7] = 4/35, c[2] = 14/145, a[9,2] = 0, a[9,3] = 0, `b* `[8] = 12589356280823/273360468363840, a[9,6] = 77530078125/2307621318 64, a[6,3] = 13843311346643833/137931778869131250, a[7,1] = -112442544 5878279929044/5080674836868609224225, b[4] = 34110058090624/8307892954 3805, a[9,1] = 577999/7054512, a[6,4] = 322308298919703822734/67427950 1001748115625, a[8,1] = -4662888761921915772773/8195396815968383745983 , a[5,4] = 770307017/187974000, `b*`[1] = 463127996257/5946086987136, \+ a[7,3] = 10208986940897432574/2855457555847472065, a[9,7] = 203784875/ 52405236624, a[5,3] = -280562881/62658000, b[2] = 0, b[3] = 0, `b*`[2] = 0, `b*`[3] = 0, `b*`[9] = 1/27, a[4,2] = 0, a[5,2] = 0, a[6,2] = 0, a[7,2] = 0, a[8,2] = 0, a[7,4] = -1389076296899785384285714176/321845 802595321532838393475, a[2,1] = 14/145, a[9,5] = 1110240000/1286944228 7, c[3] = 59/321, a[7,6] = 7781543704350/1963746281231, `b*`[5] = 6249 13261131500/8888807962650317, a[6,5] = 8562169944665744/67437276532035 625, b[6] = 77530078125/230762131864, `b*`[6] = 450569541695078125/127 5082620954224192, a[9,8] = 1341697/16490760, a[4,3] = 177/856, a[8,6] \+ = 3595419767671875/3518327954949241, a[9,4] = 34110058090624/830789295 43805, b[7] = 203784875/52405236624, a[8,7] = 1418176375/33500640722, \+ a[8,4] = -182277776762433747385270230/74164896551649031914481339, a[8, 5] = -146526017349731441994000/935639633573123866302727, a[7,5] = -834 1061397510669205248/2889892848216596175835, b[1] = 577999/7054512, b[5 ] = 1110240000/12869442287, `b*`[4] = 126224639819736010009/3151139920 65678492180, a[3,1] = 25547/2885148, b[8] = 1341697/16490760, a[3,2] = 504745/2885148, a[5,1] = 109313963/93987000, a[4,1] = 59/856\}:" } {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 0 488 "subs(e3,matrix([[c[2],a[2,1],``$3] ,\n [c[3],a[3,1],a[3,2],``$2],\n [c[4],seq(a[4,i],i=1..3),``],\n [c [5],seq(a[5,i],i=1..4)],\n [c[6],seq(a[6,i],i=1..4)],[``$4,a[6,5]],\n [c[7],seq(a[7,i],i=1..4)],[``$3,a[7,5],a[7,6]],\n [c[8],seq(a[8,i], i=1..4)],[``$2,seq(a[8,i],i=5..7)],\n [c[9],seq(a[9,i],i=1..4)],[``,s eq(a[9,i],i=5..8)],\n [``,`____________________________`$4],\n [`b`, seq(b[i],i=1..4)],[``,seq(b[i],i=5..8)],\n [`b*`,seq(`b*`[i],i=1..4)] ,[``,seq(`b*`[i],i=5..8)],[``$4,`b*`[9]]]));" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#K%'matrixG6#747'#\"#9\"$X\"F(%!GF+F+7'#\"#f\"$@$#\"&Zb# \"([^)G#\"'XZ]F2F+F+7'#F.\"$9##F.\"$c)\"\"!#\"$x\"F9F+7'#\"#Z\"#g#\"*j RJ4\"\")+q)R*F:#!*\")Gc!G\")+!eE'#\"*F:#\"2LQkY8JVQ\"\"3]78p)y<$z8#\"6MF#Qq>*)H3BK\"6Dc6 [<+,&zUn7'F+F+F+F+#\"1WdmW*p@c)\"2Dc.KlFPu'7'#\"\"%\"#N#!7W!H*z#yeWDW7 \"\"7DUA4'oo$[n!3&F:#\"5uDV(*3%p)*3-\"\"4l?ZZebda&G#!=wTr&G%Q&y**oHw!* Q\"\"7'\"\"\"#!7tFx:>#>w))Gm%\"7$)fu$Qof\"oR&>)F:#\"4ho30W_?>$H \"3=8B#y\"y++%*#!.\\;b'*[;u7'F+F+#!9+S*>WJ(\\t, El9\"9FFImQ7tNjRc$*#\"1v=nn(>af$\"1T#\\\\&zK=N#\"+vj<=9\",A2k+N$7'Fjo# \"'**zd\"(7X0(F:F:#\"/C14e+6M\"/0QaH*yI)7'F+#\"+++C56\",(GU%pG\"#\",D \"y+`x\"-k=8i2B#\"*v[y.#\",CmB0C&#\"((pT8\")g2\\;7'F+%=_______________ _____________GFcrFcrFcr7'%\"bGF_qF:F:FbqFeq7'%#b*G#\"-di*z7j%\".Or)p3Y fF:F:#\"64+,O(>)RYAE\"\"6!=#\\yc1#*R6:$7'F+#\"0+:8hK\"\\i\"1<.liz!)))) )#\"3D\"y]pT&p0X\"4#>CU&4i#3v7#\"3D,3*e-(*e:\"\"4W<^QCE)H=y#\"/B3Gc$*e 7\"0SQOo/Ot#7'F+F+F+F+#Fjo\"#FQ(pprint46\"" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 133 "subs(e3,matrix([s eq([c[i],seq(a[i,j],j=1..i-1),``$(10-i)],i=2..9),\n[`b`,seq(b[i],i=1.. 8),``],[`b*`,seq(`b*`[i],i=1..9)]])):\nevalf(%);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#K%'matrixG6#7,7,$\"+9C>98AF/F:$\"+8VDvNFC$!+[$of J%FC$!+GuG')GFC$\"+q8giRFCF+F+F+7,$\"\"\"F;$!+qOk*o&F/F:$\"+%zh!>JFC$! +ZltdCFC$!+W=0m:F/$\"+\"\\6>-\"FC$\"+[8GLUF*F+F+7,F[o$\"+\\OK$>)F*F:F: $\"+k6u0TF/$\"+qs%pi)F*$\"+r+ufLF/$\"+>dj))QF2$\"+fJ0O\")F*F+7,%\"bGFj oF:F:F\\pF^pF`pFbpFdpF+7,%#b*G$\"+#4'y)y(F*F:F:$\"+[=o0SF/$\"+$>Q..(F* $\"+v(\\O`$F/$\"+?4Xy9F*$\"+b[S0YF*$\"+/Pq.PF*Q(pprint56\"" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 146 "R K6_8eqs := [op(RowSumConditions(8,'expanded')),op(OrderConditions(6,8, 'expanded'))]:\n`RK5_9eqs*` := subs(b=`b*`,OrderConditions(5,9,'expand ed')):" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 7 "Check: " }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 148 "simplify(subs(e 3,RK6_8eqs)):\nmap(u->`if`(lhs(u)=rhs(u),0,1),%);\nnops(%);\nsimplify( subs(e3,`RK5_9eqs*`)):\nmap(u->`if`(lhs(u)=rhs(u),0,1),%);\nnops(%);" }{TEXT -1 0 "" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#7N\"\"!F$F$F$F$F$F$F$ F$F$F$F$F$F$F$F$F$F$F$F$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#\"#W" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#73\"\"!F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#\"#<" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 21 "determining the nodes" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}{SECT 1 {PARA 4 "" 0 " " {TEXT -1 22 "eG .. general solution" }}{PARA 0 "" 0 "" {TEXT -1 0 " " }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 71947 "eG := \{c[8] = 1, b[3 ] = 0, b[2] = 0, c[9] = 1, b[5] = -1/60*(10*c[6]*c[7]*c[4]-5*c[6]*c[4] -5*c[7]*c[6]+3*c[6]-5*c[7]*c[4]+3*c[4]-2+3*c[7])/(-c[4]+c[5])/c[5]/(-c [5]^3+c[5]^2-c[7]*c[5]+c[7]*c[5]^2-c[6]*c[5]+c[6]*c[5]^2+c[7]*c[6]-c[6 ]*c[7]*c[5]), a[5,2] = 0, a[4,2] = 0, `b*`[3] = 0, `b*`[2] = 0, a[8,2] = 0, a[7,2] = 0, a[6,2] = 0, a[6,5] = 1/2*c[6]*(30*c[6]^2*c[5]^2*c[4] ^2-2*c[6]^2*c[4]^2+c[6]^3*c[4]-12*c[6]^2*c[4]*c[5]^2-3*c[5]^2*c[6]*c[4 ]+c[5]*c[6]*c[4]^2+c[6]*c[4]^3-12*c[6]^2*c[4]^2*c[5]-30*c[5]*c[6]^3*c[ 4]^2-20*c[5]*c[6]^2*c[4]^4+30*c[6]^2*c[5]*c[4]^3-c[5]*c[4]^3+20*c[4]^3 *c[5]*c[6]^3-20*c[6]^2*c[5]^2*c[4]^3+c[5]^2*c[4]^2-30*c[5]^2*c[6]*c[4] ^3+12*c[5]^2*c[6]*c[4]^2+2*c[6]^2*c[4]*c[5]+20*c[5]^2*c[6]*c[4]^4+2*c[ 6]^2*c[5]^2-2*c[6]^3*c[5]+12*c[6]^3*c[4]*c[5])/c[5]/(c[4]^3-2*c[5]*c[4 ]^2+30*c[5]^2*c[4]^3-6*c[5]*c[4]^3-10*c[4]^4*c[5]^2-c[5]^3+10*c[5]^4*c [4]^2-30*c[5]^3*c[4]^2+6*c[5]^3*c[4]+2*c[5]^2*c[4]), b[1] = 1/60*(30*c [5]*c[6]*c[7]*c[4]-10*c[5]*c[6]*c[4]-10*c[6]*c[7]*c[5]+5*c[6]*c[5]-10* c[6]*c[7]*c[4]+5*c[6]*c[4]+5*c[7]*c[6]-3*c[6]-10*c[5]*c[7]*c[4]+5*c[5] *c[4]+5*c[7]*c[5]-3*c[5]+5*c[7]*c[4]-3*c[4]+2-3*c[7])/c[5]/c[6]/c[7]/c [4], a[6,1] = -1/4*c[6]*(60*c[5]^2*c[6]*c[4]^4-90*c[5]^2*c[6]^3*c[4]^2 -60*c[5]*c[6]^2*c[4]^4-2*c[6]^3*c[4]^2+50*c[5]^4*c[4]^3*c[6]+40*c[6]^2 *c[4]^5*c[5]-40*c[4]^5*c[5]^3+36*c[4]*c[5]^2*c[6]^3-20*c[5]^4*c[6]^2*c [4]^2+8*c[6]^2*c[5]^3+4*c[6]^2*c[4]^3-24*c[5]*c[6]^3*c[4]^2-8*c[6]^2*c [4]^2*c[5]+24*c[6]^2*c[5]*c[4]^3+60*c[4]^3*c[5]*c[6]^3-4*c[6]*c[5]*c[4 ]^3-14*c[5]^3*c[6]*c[4]+7*c[5]^3*c[4]^2+6*c[5]^2*c[6]*c[4]^3+14*c[5]^2 *c[6]*c[4]^2+66*c[5]^3*c[6]*c[4]^2-80*c[5]^3*c[6]^2*c[4]^3+30*c[6]^2*c [5]^2*c[4]^3-24*c[6]^2*c[5]^2*c[4]^2-240*c[6]*c[5]^3*c[4]^3-48*c[6]^2* c[5]^3*c[4]+150*c[6]^2*c[5]^3*c[4]^2-24*c[4]^4*c[5]^2-9*c[5]^2*c[4]^3+ 6*c[4]^4*c[5]-40*c[5]^4*c[4]^4-20*c[6]^2*c[4]^4*c[5]^2-6*c[5]^2*c[6]^3 -2*c[6]*c[4]^4-24*c[5]^3*c[4]^3+60*c[6]^3*c[4]^3*c[5]^2+7*c[6]^3*c[4]* c[5]-40*c[5]^2*c[6]*c[4]^5-40*c[6]^3*c[4]^4*c[5]+110*c[4]^4*c[6]*c[5]^ 3+120*c[5]^3*c[4]^4)/c[5]/c[4]^2/(10*c[5]^3*c[4]^2+10*c[5]^2*c[4]^3-30 *c[5]^2*c[4]^2+6*c[5]^2*c[4]-c[5]^2+6*c[5]*c[4]^2+c[5]*c[4]-c[4]^2), b [4] = 1/60*(-2+3*c[5]-5*c[7]*c[6]+3*c[7]+3*c[6]+10*c[6]*c[7]*c[5]-5*c[ 6]*c[5]-5*c[7]*c[5])/c[4]/(-c[6]*c[4]^3-c[4]^3+c[4]^4+c[7]*c[4]^2+c[6] *c[4]^2-c[6]*c[7]*c[4]+c[6]*c[7]*c[4]^2-c[5]*c[4]^3+c[5]*c[4]^2-c[7]*c [4]^3-c[5]*c[7]*c[4]+c[5]*c[7]*c[4]^2-c[5]*c[6]*c[4]+c[5]*c[6]*c[4]^2+ c[6]*c[7]*c[5]-c[5]*c[6]*c[7]*c[4]), b[6] = 1/60*(10*c[5]*c[7]*c[4]-5* c[5]*c[4]-5*c[7]*c[5]+3*c[5]-5*c[7]*c[4]+3*c[4]-2+3*c[7])/(-c[4]+c[6]) /(-c[5]+c[6])/c[6]/(c[6]^2-c[6]+c[7]-c[7]*c[6]), `b*`[9] = 1/10*(50*c[ 5]^2*c[6]*c[4]^2-40*c[5]^2*c[6]*c[4]+10*c[6]*c[5]^2-40*c[5]*c[6]*c[4]^ 2+35*c[5]*c[6]*c[4]-10*c[6]*c[5]+10*c[6]*c[4]^2-10*c[6]*c[4]+3*c[6]-5* c[5]^2*c[4]-5*c[5]*c[4]^2+10*c[5]*c[4]-c[5]-c[4])/(3*c[6]-28*c[5]*c[6] *c[4]^2-28*c[5]^2*c[6]*c[4]+8*c[6]*c[5]^2-9*c[6]*c[4]-9*c[6]*c[5]+28*c [5]*c[6]*c[4]+30*c[5]^2*c[6]*c[4]^2+8*c[6]*c[4]^2-4*c[5]^2*c[4]-4*c[5] *c[4]^2+9*c[5]*c[4]-c[5]-c[4]), a[7,5] = -1/2*c[7]*(-17*c[5]^2*c[6]*c[ 4]^4+5*c[6]*c[5]*c[4]^4+42*c[4]^2*c[5]^2*c[7]^3+17*c[7]^3*c[5]*c[4]^3- 80*c[7]^3*c[5]^2*c[4]^3+c[7]*c[5]*c[4]^4+80*c[5]^2*c[4]^4*c[7]^2-60*c[ 4]^5*c[6]^2*c[5]^2+20*c[4]^3*c[7]^3*c[5]^3-20*c[7]^2*c[5]^2*c[4]^5+40* c[4]^5*c[6]*c[5]^3-5*c[5]*c[6]^2*c[4]^4+c[7]^3*c[4]^2-c[7]^2*c[4]^3-80 *c[5]^3*c[4]^4*c[7]+20*c[7]^3*c[5]^2*c[4]^4-200*c[6]^2*c[4]^5*c[7]*c[5 ]^3-290*c[5]^4*c[4]^2*c[7]^2*c[6]+20*c[6]^2*c[7]*c[5]^4*c[4]^2+40*c[6] *c[5]^2*c[4]^4*c[7]-140*c[5]^4*c[4]^3*c[6]-100*c[5]^4*c[4]^5*c[6]+190* c[5]^4*c[4]^4*c[6]-40*c[7]^2*c[4]^4*c[5]^3+20*c[5]^4*c[4]^4*c[7]+2*c[4 ]^4*c[7]^2-20*c[7]^2*c[5]^4*c[4]^3-4*c[6]^2*c[7]^2*c[5]-8*c[7]*c[6]^2* c[5]^3-3*c[7]*c[6]*c[5]^3+2*c[7]^2*c[6]*c[5]^3+60*c[7]*c[6]^2*c[5]*c[4 ]^5-40*c[6]*c[5]^2*c[4]^5*c[7]+17*c[7]*c[4]^4*c[6]*c[5]+80*c[7]^2*c[5] ^3*c[4]^3+5*c[7]*c[4]^4*c[6]^2+c[5]^2*c[7]^3-260*c[6]^2*c[7]*c[5]^2*c[ 4]^3-12*c[6]^2*c[7]^2*c[5]^2*c[4]+20*c[6]^2*c[7]^2*c[5]^2*c[4]^2-40*c[ 6]*c[7]*c[5]^3*c[4]^3-41*c[6]^2*c[7]^3*c[4]*c[5]+20*c[7]^2*c[6]*c[5]*c [4]^4+3*c[7]*c[6]*c[4]^3-2*c[6]^2*c[7]*c[4]^2-3*c[6]*c[7]^2*c[4]^2+4*c [7]^2*c[6]*c[4]^3+6*c[5]*c[7]^2*c[4]^3+4*c[6]*c[5]*c[7]^2*c[4]^2+c[7]^ 2*c[5]*c[6]*c[4]-11*c[5]*c[7]*c[6]*c[4]^3+2*c[6]^2*c[4]^2*c[5]-c[7]^3* c[4]*c[5]+17*c[5]^2*c[6]*c[7]*c[4]^2-200*c[5]^4*c[7]^2*c[4]^4*c[6]-50* c[4]^5*c[6]*c[7]*c[5]^3-150*c[5]^2*c[6]*c[7]^3*c[4]^4-c[6]^2*c[5]*c[4] ^3+2*c[7]^3*c[4]^2*c[6]*c[5]+2*c[6]^2*c[7]*c[4]*c[5]+108*c[7]*c[6]^2*c [5]*c[4]^3+4*c[6]^2*c[7]*c[5]^2+2*c[6]*c[5]*c[7]*c[4]^2-10*c[6]^2*c[7] ^3*c[5]^2+6*c[5]^4*c[7]*c[6]-8*c[6]*c[7]^3*c[5]^2+3*c[6]*c[7]^2*c[5]^2 +2*c[6]^2*c[7]^2*c[5]^2-27*c[6]^2*c[4]^2*c[7]*c[5]-62*c[6]^2*c[7]^2*c[ 4]^2*c[5]+46*c[6]*c[5]^4*c[4]^2+20*c[4]^5*c[7]*c[5]^3+2*c[7]^3*c[4]*c[ 6]*c[5]-27*c[6]^2*c[5]^2*c[7]*c[4]+120*c[6]^2*c[5]^2*c[7]*c[4]^2-35*c[ 6]*c[5]^2*c[7]*c[4]^3-37*c[6]*c[5]^3*c[7]*c[4]^2+53*c[6]^2*c[5]^3*c[7] *c[4]-54*c[7]^2*c[5]^2*c[4]^3+2*c[7]^2*c[5]*c[4]^2-c[7]*c[5]*c[4]^3-3* c[6]*c[5]*c[4]^3-2*c[4]^3*c[7]^3+63*c[7]^3*c[5]^2*c[6]*c[4]-16*c[7]^2* c[5]^2*c[6]*c[4]-5*c[6]*c[7]*c[4]^4+2*c[6]^2*c[7]^2*c[4]^2+2*c[6]^2*c[ 7]^2*c[4]-10*c[5]^4*c[7]^2*c[6]+3*c[5]^3*c[6]*c[4]+3*c[5]^3*c[7]*c[4]+ 20*c[5]^3*c[6]*c[7]*c[4]-2*c[5]^3*c[4]^2+22*c[6]^2*c[7]^2*c[4]*c[5]+7* c[5]^2*c[6]*c[4]^3-c[5]^2*c[7]*c[4]^2+c[5]^2*c[6]*c[4]^2-4*c[5]^2*c[7] *c[4]^3-8*c[5]^3*c[7]*c[4]^2-22*c[5]^3*c[6]*c[4]^2+240*c[5]^3*c[6]^2*c [4]^3+100*c[4]^3*c[7]^2*c[5]^2*c[6]^2+150*c[4]^5*c[5]^3*c[6]^2-3*c[7]* c[5]^4*c[4]+5*c[7]^2*c[5]^4*c[4]+12*c[5]^2*c[4]^4*c[7]-98*c[6]^2*c[5]^ 2*c[4]^3+25*c[6]^2*c[5]^2*c[4]^2+54*c[6]*c[5]^3*c[4]^3+8*c[6]^2*c[5]^3 *c[4]-63*c[6]^2*c[5]^3*c[4]^2+42*c[5]^3*c[7]*c[4]^3+60*c[5]^4*c[7]*c[4 ]^2*c[6]+200*c[5]^3*c[7]^3*c[4]^4*c[6]-4*c[6]^2*c[4]*c[5]^2+c[7]*c[6]^ 2*c[4]^3+200*c[7]^2*c[5]^2*c[6]*c[4]^3-240*c[7]^3*c[5]^2*c[6]*c[4]^2+1 80*c[7]^2*c[5]^3*c[6]*c[4]^2+3*c[5]^4*c[4]^2-80*c[5]^3*c[7]^3*c[6]*c[4 ]+c[7]^3*c[4]^2*c[6]-7*c[7]^3*c[4]^2*c[5]+20*c[4]^3*c[7]^2*c[5]*c[6]^2 -19*c[7]^2*c[6]*c[5]*c[4]^3-11*c[7]^2*c[5]^2*c[6]*c[4]^2-3*c[4]^4*c[5] ^2-27*c[7]^2*c[5]^3*c[6]*c[4]-480*c[7]^2*c[5]^3*c[6]*c[4]^3-380*c[5]^3 *c[6]^2*c[4]^4-5*c[5]^4*c[4]^3-10*c[6]^2*c[4]^3*c[7]^2+130*c[5]^3*c[6] ^2*c[7]*c[4]^3-5*c[7]^3*c[5]^2*c[4]-2*c[7]^2*c[5]^2*c[4]-4*c[5]^2*c[6] *c[7]*c[4]-160*c[6]^2*c[7]*c[5]^3*c[4]^2-80*c[6]^2*c[7]^2*c[4]*c[5]^3+ 2*c[5]^2*c[4]^3+12*c[7]^2*c[5]^2*c[4]^2+8*c[7]^2*c[5]^3*c[4]+400*c[7]^ 3*c[5]^2*c[6]^2*c[4]^3-17*c[7]^2*c[4]^4*c[5]+250*c[7]^2*c[5]^3*c[6]^2* c[4]^2-100*c[6]^2*c[4]^5*c[5]*c[7]^2-200*c[7]^2*c[4]^5*c[6]*c[5]^3+150 *c[5]^2*c[6]*c[4]^5*c[7]^2+380*c[7]^3*c[5]^2*c[4]^3*c[6]-20*c[7]^3*c[6 ]*c[5]*c[4]^3-210*c[7]^3*c[6]^2*c[5]*c[4]^3-6*c[5]^4*c[4]*c[6]+80*c[5] ^4*c[7]^2*c[4]*c[6]+10*c[6]*c[7]^3*c[5]^3-36*c[5]^4*c[4]*c[7]*c[6]-50* c[4]^5*c[6]^2*c[7]*c[5]^2+200*c[4]^5*c[6]^2*c[7]^2*c[5]^2+160*c[6]^2*c [4]^4*c[5]^2-450*c[7]^3*c[5]^3*c[4]^3*c[6]+5*c[6]^2*c[7]^3*c[4]^2-c[5] ^3*c[7]^2+6*c[6]^2*c[7]^3*c[5]-160*c[7]*c[6]^2*c[5]*c[4]^4+200*c[5]^3* c[4]^4*c[6]^2*c[7]-350*c[5]^3*c[4]^3*c[6]^2*c[7]^2-350*c[5]^4*c[4]^4*c [6]*c[7]+230*c[5]^2*c[4]^4*c[7]*c[6]^2+150*c[5]^3*c[4]^4*c[6]*c[7]+100 *c[5]^4*c[4]^3*c[6]*c[7]-3*c[6]^2*c[7]^3*c[4]-42*c[7]^2*c[5]^3*c[4]^2+ 50*c[5]^4*c[4]^4*c[6]^2-20*c[5]^4*c[4]^3*c[6]^2+500*c[5]^3*c[4]^4*c[7] ^2*c[6]+450*c[5]^4*c[4]^3*c[7]^2*c[6]-340*c[5]^2*c[4]^4*c[7]^2*c[6]-50 *c[5]^4*c[4]^3*c[6]^2*c[7]+150*c[7]^2*c[4]^4*c[6]^2*c[5]+200*c[7]^2*c[ 4]^4*c[5]^3*c[6]^2-350*c[7]^2*c[4]^4*c[6]^2*c[5]^2+10*c[7]^2*c[6]^2*c[ 5]^3-60*c[4]^4*c[6]*c[5]^3+100*c[7]^3*c[6]^2*c[5]*c[4]^4+200*c[5]^4*c[ 4]^5*c[6]*c[7]-200*c[7]^3*c[4]^4*c[6]^2*c[5]^2+5*c[5]^3*c[4]^4+80*c[6] ^2*c[7]^3*c[5]^2*c[4]-270*c[6]^2*c[7]^3*c[5]^2*c[4]^2+290*c[5]^3*c[7]^ 3*c[4]^2*c[6]+140*c[6]^2*c[5]*c[7]^3*c[4]^2-5*c[7]^3*c[4]*c[5]^3)/c[5] /(100*c[5]^2*c[6]*c[4]^4-18*c[6]*c[5]*c[4]^4+4*c[5]^3*c[4]+50*c[4]^5*c [6]^2*c[5]^2+40*c[5]*c[6]^2*c[4]^4+3*c[5]^5-5*c[4]^4*c[6]^2+180*c[5]^4 *c[4]^3*c[6]+100*c[5]^4*c[4]^5*c[6]-350*c[5]^4*c[4]^4*c[6]+140*c[5]^5* c[4]^2-180*c[5]^5*c[4]^3-2*c[5]^4+30*c[4]^5*c[5]^3+300*c[5]^5*c[4]^3*c [6]+2*c[6]*c[5]^3-50*c[5]^5*c[4]^2*c[6]^2-40*c[5]^4*c[6]^2*c[4]+240*c[ 5]^4*c[6]^2*c[4]^2-3*c[6]^2*c[5]^3+3*c[6]^2*c[4]^3+5*c[5]^4*c[6]^2-2*c [6]*c[4]^3+2*c[5]*c[4]^3+4*c[5]*c[6]*c[4]^2-6*c[6]^2*c[4]^2*c[5]-50*c[ 5]^4*c[4]^5-13*c[6]^2*c[5]*c[4]^3-40*c[6]*c[5]^4*c[4]^2+6*c[6]*c[5]*c[ 4]^3-15*c[5]^3*c[6]*c[4]-55*c[5]^2*c[6]*c[4]^3+6*c[5]^2*c[6]*c[4]^2+68 *c[5]^3*c[6]*c[4]^2-100*c[4]^5*c[5]^3*c[6]^2+100*c[6]^2*c[5]^2*c[4]^3- 100*c[6]*c[5]^3*c[4]^3+13*c[6]^2*c[5]^3*c[4]-100*c[6]^2*c[5]^3*c[4]^2- 4*c[5]^2*c[4]^2+6*c[6]^2*c[4]*c[5]^2-68*c[5]^4*c[4]^2+23*c[4]^4*c[5]^2 +350*c[5]^3*c[6]^2*c[4]^4-4*c[5]^2*c[6]*c[4]-9*c[5]^2*c[4]^3-3*c[4]^4* c[5]-5*c[5]^5*c[6]-30*c[5]^6*c[4]^2+50*c[5]^6*c[6]*c[4]^2+180*c[5]^4*c [4]^4+10*c[5]^4*c[4]*c[6]-240*c[6]^2*c[4]^4*c[5]^2+3*c[6]*c[4]^4-100*c [5]^6*c[4]^3*c[6]+68*c[5]^3*c[4]^3-23*c[5]^5*c[4]-210*c[5]^5*c[6]*c[4] ^2-350*c[5]^4*c[4]^3*c[6]^2-30*c[5]^2*c[6]*c[4]^5+50*c[5]^6*c[4]^3+60* c[4]^4*c[6]*c[5]^3+100*c[5]^5*c[6]^2*c[4]^3+9*c[5]^4*c[4]-140*c[5]^3*c [4]^4+40*c[5]^5*c[4]*c[6]), `b*`[6] = -1/60*(-30*c[5]^3*c[4]+103*c[5]* c[7]*c[4]-18*c[6]+6*c[5]+6*c[4]+27*c[7]*c[6]-9*c[5]^2-9*c[4]^2-100*c[7 ]*c[6]*c[5]^3+522*c[5]*c[6]*c[7]*c[4]-111*c[6]*c[5]^2+75*c[6]*c[5]+60* c[6]*c[5]^3-9*c[7]*c[5]+600*c[6]*c[7]*c[5]^3*c[4]^3+14*c[7]*c[5]^2-100 *c[7]*c[6]*c[4]^3+14*c[7]*c[4]^2-9*c[7]*c[4]-117*c[6]*c[7]*c[5]-111*c[ 6]*c[4]^2+60*c[6]*c[4]^3-117*c[6]*c[7]*c[4]+180*c[6]*c[7]*c[4]^2-30*c[ 5]*c[4]^3-174*c[5]*c[7]*c[4]^2+486*c[5]*c[6]*c[4]^2+107*c[5]*c[4]^2-31 8*c[5]*c[6]*c[4]-66*c[5]*c[4]+75*c[6]*c[4]+500*c[5]*c[7]*c[6]*c[4]^3+1 420*c[5]^2*c[6]*c[7]*c[4]^2-840*c[6]*c[5]*c[7]*c[4]^2+180*c[6]*c[7]*c[ 5]^2-900*c[6]*c[5]^2*c[7]*c[4]^3-900*c[6]*c[5]^3*c[7]*c[4]^2+50*c[7]*c [5]*c[4]^3-280*c[6]*c[5]*c[4]^3-280*c[5]^3*c[6]*c[4]+50*c[5]^3*c[7]*c[ 4]+107*c[5]^2*c[4]+500*c[5]^3*c[6]*c[7]*c[4]+50*c[5]^3*c[4]^2+470*c[5] ^2*c[6]*c[4]^3+310*c[5]^2*c[7]*c[4]^2-770*c[5]^2*c[6]*c[4]^2-90*c[5]^2 *c[7]*c[4]^3-90*c[5]^3*c[7]*c[4]^2+470*c[5]^3*c[6]*c[4]^2-300*c[6]*c[5 ]^3*c[4]^3-180*c[5]^2*c[4]^2+486*c[5]^2*c[6]*c[4]-174*c[5]^2*c[7]*c[4] -840*c[5]^2*c[6]*c[7]*c[4]+50*c[5]^2*c[4]^3)/(-9*c[5]*c[7]*c[4]-3*c[7] *c[6]+3*c[6]^2-28*c[5]*c[6]*c[7]*c[4]-c[6]*c[5]+c[7]*c[5]+8*c[6]^2*c[4 ]^2+c[7]*c[4]+9*c[6]*c[7]*c[5]+9*c[6]*c[7]*c[4]-8*c[6]*c[7]*c[4]^2+4*c [5]*c[7]*c[4]^2-4*c[5]*c[6]*c[4]^2+9*c[5]*c[6]*c[4]-c[6]*c[4]-28*c[6]^ 2*c[4]^2*c[5]+28*c[6]^2*c[4]*c[5]-30*c[5]^2*c[6]*c[7]*c[4]^2+28*c[6]*c [5]*c[7]*c[4]^2-8*c[6]*c[7]*c[5]^2+8*c[6]^2*c[5]^2-9*c[6]^2*c[5]+30*c[ 6]^2*c[5]^2*c[4]^2-28*c[6]^2*c[4]*c[5]^2-9*c[6]^2*c[4]-4*c[5]^2*c[6]*c [4]+4*c[5]^2*c[7]*c[4]+28*c[5]^2*c[6]*c[7]*c[4])/(c[5]*c[4]-c[6]*c[4]- c[6]*c[5]+c[6]^2)/c[6], a[8,7] = (10*c[5]*c[6]*c[4]-5*c[6]*c[5]-5*c[6] *c[4]+3*c[6]-5*c[5]*c[4]+3*c[5]+3*c[4]-2)/c[7]/(c[6]-c[7])/(c[5]-c[7]) /(c[4]-c[7])/(-20*c[5]*c[6]*c[4]+30*c[5]*c[6]*c[7]*c[4]+15*c[6]*c[5]-2 0*c[6]*c[7]*c[5]+15*c[6]*c[4]-20*c[6]*c[7]*c[4]-12*c[6]+15*c[7]*c[6]+1 5*c[5]*c[4]-20*c[5]*c[7]*c[4]-12*c[5]+15*c[7]*c[5]-12*c[4]+15*c[7]*c[4 ]+10-12*c[7])*(c[4]-1)*(-1+c[5])*(-1+c[6])*(-1+c[7]), a[6,3] = 3/4*1/c [4]^2*c[6]*(60*c[4]^3*c[5]*c[6]^3-90*c[5]*c[6]^3*c[4]^2+36*c[6]^3*c[4] *c[5]-6*c[6]^3*c[5]+3*c[6]^3*c[4]-20*c[6]^2*c[5]^3*c[4]^2-80*c[6]^2*c[ 5]^2*c[4]^3+150*c[6]^2*c[5]^2*c[4]^2-48*c[6]^2*c[4]*c[5]^2+8*c[6]^2*c[ 5]^2-60*c[5]*c[6]^2*c[4]^4+90*c[6]^2*c[5]*c[4]^3-48*c[6]^2*c[4]^2*c[5] +4*c[6]^2*c[4]*c[5]-4*c[6]^2*c[4]^2+30*c[6]*c[5]^3*c[4]^3+90*c[5]^2*c[ 6]*c[4]^4-180*c[5]^2*c[6]*c[4]^3+54*c[5]^2*c[6]*c[4]^2-12*c[5]^2*c[6]* c[4]+18*c[6]*c[5]*c[4]^3+6*c[5]*c[6]*c[4]^2+3*c[5]^2*c[4]^2-3*c[5]*c[4 ]^3)/(10*c[5]^3*c[4]^2+10*c[5]^2*c[4]^3-30*c[5]^2*c[4]^2+6*c[5]^2*c[4] -c[5]^2+6*c[5]*c[4]^2+c[5]*c[4]-c[4]^2), b[7] = -1/60*(10*c[5]*c[6]*c[ 4]-5*c[6]*c[5]-5*c[6]*c[4]+3*c[6]-5*c[5]*c[4]+3*c[5]+3*c[4]-2)/(c[4]-c [7])/(c[5]-c[7])/(c[6]-c[7])/c[7]/(-1+c[7]), a[7,1] = 1/4*c[7]*(-51*c[ 5]^2*c[6]*c[4]^4+14*c[6]*c[5]*c[4]^4+12*c[4]^2*c[5]^2*c[7]^3+2700*c[5] ^4*c[4]^4*c[6]^2*c[7]+12*c[7]^3*c[5]*c[4]^3-98*c[7]^3*c[5]^2*c[4]^3-2* c[7]*c[5]*c[4]^4+132*c[5]^2*c[4]^4*c[7]^2-480*c[4]^5*c[6]^2*c[5]^2+200 *c[4]^3*c[7]^3*c[5]^3-200*c[7]^2*c[5]^2*c[4]^5-760*c[4]^5*c[6]*c[5]^3- 10*c[5]*c[6]^2*c[4]^4+200*c[5]^5*c[4]^3*c[7]^2*c[6]^2-400*c[5]^2*c[4]^ 6*c[6]^2*c[7]^2+4*c[4]^5*c[5]*c[7]-34*c[5]^2*c[4]^5*c[7]+400*c[5]^3*c[ 4]^6*c[6]^2*c[7]-98*c[5]^3*c[4]^4*c[7]+420*c[7]^2*c[4]^5*c[5]^3+200*c[ 7]^3*c[5]^2*c[4]^4-300*c[6]^2*c[4]^5*c[7]*c[5]^3-40*c[7]*c[5]^2*c[4]^6 *c[6]+20*c[5]^4*c[4]^2*c[7]^2*c[6]-280*c[6]^2*c[7]*c[5]^4*c[4]^2-44*c[ 6]*c[4]^5*c[5]*c[7]-246*c[6]*c[5]^2*c[4]^4*c[7]+120*c[4]^6*c[6]^2*c[5] ^2-100*c[5]^2*c[7]^2*c[4]^6*c[6]+141*c[5]^4*c[4]^3*c[6]+1220*c[5]^4*c[ 4]^5*c[6]+400*c[5]^5*c[4]^5*c[6]^2-760*c[5]^4*c[4]^4*c[6]+100*c[5]^4*c [7]^3*c[4]^4-400*c[7]^2*c[4]^4*c[5]^3-200*c[5]^4*c[4]^5*c[7]^2+200*c[5 ]^4*c[4]^4*c[7]-2*c[4]^3*c[7]^3*c[6]-8*c[7]^2*c[6]*c[4]^4-380*c[7]^3*c [5]^3*c[4]^4-40*c[6]*c[4]^5*c[5]*c[7]^2+2*c[4]^4*c[7]^2-40*c[5]^2*c[7] ^3*c[4]^5-200*c[7]^2*c[5]^4*c[4]^3-40*c[7]^3*c[5]^4*c[4]^3+200*c[5]^3* c[7]*c[4]^6*c[6]-10*c[6]^2*c[7]^3*c[4]^3+30*c[6]^2*c[4]^5*c[5]+12*c[7] *c[6]^2*c[5]^3+4*c[7]^2*c[6]*c[5]^3+400*c[5]^4*c[4]^6*c[6]^2-40*c[5]^3 *c[7]*c[4]^6-380*c[5]^4*c[4]^5*c[7]+320*c[7]*c[6]^2*c[5]*c[4]^5+100*c[ 4]^6*c[6]^2*c[7]*c[5]^2+34*c[5]^4*c[4]^2*c[7]^2+40*c[5]^5*c[7]^2*c[4]^ 3+240*c[6]*c[5]^2*c[4]^5*c[7]+60*c[5]^4*c[7]^3*c[4]^2*c[6]+26*c[7]*c[4 ]^4*c[6]*c[5]-50*c[4]^4*c[6]^2*c[5]^5+200*c[5]^5*c[7]*c[4]^4*c[6]+132* c[7]^2*c[5]^3*c[4]^3-2*c[7]*c[4]^4*c[6]^2+60*c[5]^5*c[4]^2*c[6]^2*c[7] -500*c[5]^5*c[4]^4*c[6]^2*c[7]+160*c[5]^5*c[4]^4*c[6]+134*c[6]^2*c[5]^ 4*c[7]*c[4]+24*c[5]^4*c[6]^2*c[4]-169*c[5]^4*c[6]^2*c[4]^2-500*c[5]^3* c[4]^6*c[6]^2-40*c[5]^5*c[4]^4*c[7]+34*c[4]^5*c[5]*c[7]^2-100*c[5]^3*c [7]^2*c[4]^6-100*c[5]^5*c[4]^4*c[7]^2+160*c[4]^6*c[6]*c[5]^3-6*c[6]^2* c[7]*c[5]^2*c[4]^3+68*c[6]^2*c[7]^2*c[5]^2*c[4]-126*c[6]^2*c[7]^2*c[5] ^2*c[4]^2-600*c[6]*c[7]*c[5]^3*c[4]^3-21*c[6]^2*c[7]^3*c[4]*c[5]+40*c[ 7]^2*c[6]^2*c[5]^4+4*c[5]^4*c[7]*c[4]^2+58*c[7]^2*c[6]*c[5]*c[4]^4-100 *c[7]^2*c[5]^5*c[6]^2*c[4]^2-320*c[5]^4*c[4]*c[7]^2*c[6]^2+20*c[6]^2*c [4]^4*c[7]^2+40*c[5]^2*c[4]^6*c[7]^2+6*c[7]^2*c[6]*c[4]^3-8*c[6]*c[5]* c[7]^2*c[4]^2-4*c[5]*c[7]*c[6]*c[4]^3+14*c[5]^2*c[6]*c[7]*c[4]^2-910*c [4]^5*c[6]*c[7]*c[5]^3-340*c[5]^2*c[6]*c[7]^3*c[4]^4-4*c[6]^2*c[5]*c[4 ]^3+5*c[7]^3*c[4]^2*c[6]*c[5]+72*c[7]*c[6]^2*c[5]*c[4]^3+18*c[6]^2*c[7 ]^3*c[5]^2-12*c[6]^2*c[7]^2*c[5]^2-10*c[6]^2*c[4]^2*c[7]*c[5]-44*c[6]^ 2*c[7]^2*c[4]^2*c[5]-25*c[6]*c[5]^4*c[4]^2-24*c[7]*c[6]^2*c[5]^4+200*c [4]^5*c[7]*c[5]^3-2*c[6]^2*c[5]^2*c[7]*c[4]-42*c[6]^2*c[5]^2*c[7]*c[4] ^2+74*c[6]*c[5]^2*c[7]*c[4]^3+80*c[6]*c[5]^3*c[7]*c[4]^2-50*c[6]^2*c[5 ]^3*c[7]*c[4]-34*c[5]^4*c[4]^3*c[7]-24*c[7]^2*c[5]^2*c[4]^3+420*c[5]^4 *c[7]^2*c[4]^4-2*c[4]^3*c[7]^3+c[7]^3*c[5]^2*c[6]*c[4]+4*c[7]^2*c[5]^2 *c[6]*c[4]-6*c[6]*c[7]*c[4]^4-4*c[6]^2*c[7]^2*c[4]^2+50*c[5]^5*c[4]^3* c[7]^2*c[6]-18*c[5]^3*c[6]*c[7]*c[4]+14*c[6]^2*c[7]^2*c[4]*c[5]-20*c[5 ]^2*c[6]*c[4]^3+2*c[5]^2*c[7]*c[4]^3-2*c[5]^3*c[7]*c[4]^2+16*c[5]^3*c[ 6]*c[4]^2-96*c[5]^3*c[6]^2*c[4]^3+100*c[5]^5*c[4]^3*c[6]^2*c[7]-60*c[4 ]^3*c[7]^2*c[5]^2*c[6]^2+10*c[6]*c[7]*c[4]^5+1540*c[4]^5*c[5]^3*c[6]^2 -34*c[7]^3*c[4]^4*c[5]-4*c[7]^2*c[5]^4*c[4]+12*c[5]^2*c[4]^4*c[7]-41*c [6]^2*c[5]^2*c[4]^3+14*c[6]^2*c[5]^2*c[4]^2-54*c[6]*c[5]^3*c[4]^3-12*c [6]^2*c[5]^3*c[4]+47*c[6]^2*c[5]^3*c[4]^2+12*c[5]^3*c[7]*c[4]^3-98*c[5 ]^4*c[7]*c[4]^2*c[6]-50*c[5]^3*c[7]^3*c[4]^4*c[6]+4*c[7]*c[6]^2*c[4]^3 +156*c[7]^2*c[5]^2*c[6]*c[4]^3-23*c[7]^3*c[5]^2*c[6]*c[4]^2+188*c[7]^2 *c[5]^3*c[6]*c[4]^2+66*c[5]^3*c[7]^3*c[6]*c[4]+2*c[7]^3*c[4]^2*c[5]+10 0*c[5]^5*c[4]^5*c[7]+104*c[4]^3*c[7]^2*c[5]*c[6]^2-12*c[7]^2*c[6]*c[5] *c[4]^3-36*c[7]^2*c[5]^2*c[6]*c[4]^2-32*c[7]^2*c[5]^3*c[6]*c[4]-320*c[ 7]^2*c[5]^3*c[6]*c[4]^3-560*c[5]^3*c[6]^2*c[4]^4-4*c[6]^2*c[4]^3*c[7]^ 2+240*c[5]^3*c[6]^2*c[7]*c[4]^3-34*c[5]^3*c[7]^3*c[4]^2-2*c[7]^3*c[5]^ 2*c[4]+164*c[6]^2*c[7]*c[5]^3*c[4]^2-20*c[6]^2*c[7]^2*c[4]*c[5]^3+2*c[ 7]^2*c[5]^3*c[4]-130*c[5]^5*c[4]^3*c[6]*c[7]-90*c[7]^3*c[5]^2*c[6]^2*c [4]^3-16*c[7]^2*c[4]^4*c[5]-60*c[7]^2*c[5]^3*c[6]^2*c[4]^2-300*c[6]^2* c[4]^5*c[5]*c[7]^2-150*c[7]^2*c[4]^5*c[6]*c[5]^3+380*c[5]^2*c[6]*c[4]^ 5*c[7]^2+140*c[7]^3*c[5]^2*c[4]^3*c[6]-810*c[7]^3*c[5]^3*c[6]^2*c[4]^2 -14*c[7]^3*c[6]*c[5]*c[4]^3-280*c[7]^3*c[6]^2*c[5]*c[4]^3+240*c[7]^3*c [5]^3*c[6]^2*c[4]-20*c[5]^4*c[7]^2*c[4]*c[6]-6*c[6]*c[7]^3*c[5]^3+22*c [5]^4*c[4]*c[7]*c[6]-280*c[4]^5*c[6]^2*c[7]*c[5]^2+400*c[4]^5*c[6]^2*c [7]^2*c[5]^2+253*c[6]^2*c[4]^4*c[5]^2+470*c[7]^3*c[5]^3*c[4]^3*c[6]+10 0*c[5]^2*c[4]^5*c[7]^3*c[6]+6*c[6]^2*c[7]^3*c[4]^2-300*c[5]^5*c[4]^5*c [6]-120*c[5]*c[7]*c[4]^6*c[6]^2-10*c[6]^2*c[7]*c[4]^5-1100*c[5]^4*c[4] ^5*c[6]^2*c[7]-226*c[7]*c[6]^2*c[5]*c[4]^4-770*c[5]^3*c[4]^4*c[6]^2*c[ 7]+280*c[5]^3*c[4]^3*c[6]^2*c[7]^2-1000*c[5]^4*c[4]^4*c[6]*c[7]+240*c[ 5]^2*c[4]^4*c[7]*c[6]^2+1240*c[5]^3*c[4]^4*c[6]*c[7]+600*c[5]^4*c[4]^3 *c[6]*c[7]+1200*c[7]^3*c[5]^3*c[6]^2*c[4]^3+40*c[7]^3*c[4]^4*c[5]*c[6] -16*c[7]^2*c[5]^3*c[4]^2-1250*c[5]^4*c[4]^5*c[6]^2-600*c[6]^2*c[7]^3*c [5]^3*c[4]^4+200*c[6]^2*c[7]^2*c[5]^3*c[4]^5-200*c[6]^2*c[7]^3*c[4]^5* c[5]-140*c[5]^4*c[4]^4*c[6]^2+560*c[5]^4*c[4]^3*c[6]^2-1850*c[7]^2*c[5 ]^4*c[6]^2*c[4]^3+1230*c[7]^2*c[5]^4*c[6]^2*c[4]^2-150*c[5]^4*c[7]^3*c [4]^3*c[6]-300*c[5]^4*c[4]^6*c[6]+310*c[5]^3*c[4]^4*c[7]^2*c[6]-70*c[5 ]^4*c[4]^3*c[7]^2*c[6]-380*c[5]^2*c[4]^4*c[7]^2*c[6]-850*c[5]^4*c[4]^3 *c[6]^2*c[7]-40*c[7]^2*c[4]^4*c[6]^2*c[5]-450*c[7]^2*c[4]^4*c[5]^3*c[6 ]^2+250*c[7]^2*c[4]^4*c[6]^2*c[5]^2-4*c[7]^2*c[4]^5+136*c[5]^2*c[6]*c[ 4]^5+400*c[6]^2*c[7]^3*c[5]^2*c[4]^5+800*c[6]^2*c[7]^2*c[5]^4*c[4]^4+4 *c[7]^3*c[4]^4+100*c[5]^4*c[4]^6*c[7]+437*c[4]^4*c[6]*c[5]^3-22*c[6]*c [5]*c[4]^5-60*c[5]^5*c[6]^2*c[4]^3+200*c[6]^2*c[4]^6*c[5]*c[7]^2+420*c [7]^3*c[6]^2*c[5]*c[4]^4+400*c[5]^4*c[4]^5*c[6]*c[7]-500*c[7]^3*c[4]^4 *c[6]^2*c[5]^2+100*c[5]^3*c[7]^3*c[4]^5-103*c[6]^2*c[7]^3*c[5]^2*c[4]+ 260*c[6]^2*c[7]^3*c[5]^2*c[4]^2-300*c[5]^3*c[7]^3*c[4]^2*c[6]+97*c[6]^ 2*c[5]*c[7]^3*c[4]^2-30*c[6]^2*c[7]^3*c[5]^3+4*c[7]^3*c[4]*c[5]^3)/c[6 ]/(-50*c[5]^2*c[6]*c[4]^4+23*c[5]^3*c[4]+100*c[5]^4*c[4]^3*c[6]+2*c[5] ^2+2*c[4]^2-3*c[5]^3-3*c[4]^3-3*c[6]*c[5]^2+5*c[6]*c[5]^3-3*c[6]*c[4]^ 2+5*c[6]*c[4]^3+23*c[5]*c[4]^3+18*c[5]*c[6]*c[4]^2-12*c[5]*c[4]^2+3*c[ 5]*c[6]*c[4]-2*c[5]*c[4]-50*c[6]*c[5]^4*c[4]^2-40*c[6]*c[5]*c[4]^3-40* c[5]^3*c[6]*c[4]-12*c[5]^2*c[4]-140*c[5]^3*c[4]^2+240*c[5]^2*c[6]*c[4] ^3-140*c[5]^2*c[6]*c[4]^2+240*c[5]^3*c[6]*c[4]^2-400*c[6]*c[5]^3*c[4]^ 3+91*c[5]^2*c[4]^2+30*c[5]^4*c[4]^2+30*c[4]^4*c[5]^2-50*c[5]^4*c[4]^3+ 18*c[5]^2*c[6]*c[4]-140*c[5]^2*c[4]^3+210*c[5]^3*c[4]^3+100*c[4]^4*c[6 ]*c[5]^3-50*c[5]^3*c[4]^4)/c[5]/c[4]^2, `b*`[8] = 0, a[5,3] = -3/4*c[5 ]^2*(2*c[5]-3*c[4])/c[4]^2, `b*`[4] = -1/60*(-30*c[5]^3*c[4]-42*c[7]*c [6]^2+89*c[5]*c[7]*c[4]-18*c[6]+6*c[5]+6*c[4]+27*c[7]*c[6]-9*c[5]^2+18 0*c[7]*c[6]^2*c[5]^3-100*c[7]*c[6]*c[5]^3+27*c[6]^2+186*c[5]*c[6]*c[7] *c[4]-97*c[6]*c[5]^2+66*c[6]*c[5]+60*c[6]*c[5]^3-100*c[6]^2*c[5]^3-9*c [7]*c[5]+14*c[7]*c[5]^2-100*c[6]^2*c[7]*c[4]^2+120*c[6]^2*c[7]*c[4]+60 *c[6]^2*c[4]^2-9*c[7]*c[4]-103*c[6]*c[7]*c[5]-36*c[6]*c[4]^2-61*c[6]*c [7]*c[4]+60*c[6]*c[7]*c[4]^2-30*c[5]*c[7]*c[4]^2+126*c[5]*c[6]*c[4]^2+ 18*c[5]*c[4]^2-112*c[5]*c[6]*c[4]-57*c[5]*c[4]+39*c[6]*c[4]-280*c[6]^2 *c[4]^2*c[5]+330*c[6]^2*c[4]*c[5]+380*c[5]^2*c[6]*c[7]*c[4]^2-560*c[6] ^2*c[7]*c[4]*c[5]+192*c[6]^2*c[7]*c[5]-310*c[6]^2*c[7]*c[5]^2-230*c[6] *c[5]*c[7]*c[4]^2+156*c[6]*c[7]*c[5]^2+180*c[6]^2*c[5]^2+500*c[6]^2*c[ 4]^2*c[7]*c[5]+950*c[6]^2*c[5]^2*c[7]*c[4]-900*c[6]^2*c[5]^2*c[7]*c[4] ^2-300*c[6]*c[5]^3*c[7]*c[4]^2-600*c[6]^2*c[5]^3*c[7]*c[4]-117*c[6]^2* c[5]-130*c[5]^3*c[6]*c[4]+50*c[5]^3*c[7]*c[4]+93*c[5]^2*c[4]+230*c[5]^ 3*c[6]*c[7]*c[4]+50*c[5]^2*c[7]*c[4]^2-190*c[5]^2*c[6]*c[4]^2+150*c[5] ^3*c[6]*c[4]^2+470*c[6]^2*c[5]^2*c[4]^2+320*c[6]^2*c[5]^3*c[4]-300*c[6 ]^2*c[5]^3*c[4]^2-30*c[5]^2*c[4]^2-530*c[6]^2*c[4]*c[5]^2-75*c[6]^2*c[ 4]+156*c[5]^2*c[6]*c[4]-150*c[5]^2*c[7]*c[4]-270*c[5]^2*c[6]*c[7]*c[4] +600*c[6]^2*c[7]*c[5]^3*c[4]^2)/(-9*c[5]*c[7]*c[4]-3*c[7]*c[6]-c[4]^2- 28*c[5]*c[6]*c[7]*c[4]+c[7]*c[5]+c[7]*c[4]+9*c[6]*c[7]*c[5]-9*c[6]*c[4 ]^2+8*c[6]*c[4]^3+9*c[6]*c[7]*c[4]-8*c[6]*c[7]*c[4]^2-4*c[5]*c[4]^3+4* c[5]*c[7]*c[4]^2+28*c[5]*c[6]*c[4]^2+9*c[5]*c[4]^2-9*c[5]*c[6]*c[4]-c[ 5]*c[4]+3*c[6]*c[4]-30*c[5]^2*c[6]*c[7]*c[4]^2+28*c[6]*c[5]*c[7]*c[4]^ 2-8*c[6]*c[7]*c[5]^2-28*c[6]*c[5]*c[4]^3+30*c[5]^2*c[6]*c[4]^3-28*c[5] ^2*c[6]*c[4]^2-4*c[5]^2*c[4]^2+8*c[5]^2*c[6]*c[4]+4*c[5]^2*c[7]*c[4]+2 8*c[5]^2*c[6]*c[7]*c[4])/(-c[4]+c[6])/(-c[4]+c[5])/c[4], a[8,6] = 1/2* (5*c[5]*c[6]*c[4]+c[6]-2*c[6]*c[4]-2*c[6]*c[5]+25*c[5]*c[7]*c[4]+6*c[5 ]+9*c[7]+10*c[7]^2*c[4]-4+6*c[4]-6*c[7]^2-14*c[7]*c[4]-10*c[5]*c[4]-14 *c[7]*c[5]-20*c[7]^2*c[5]*c[4]+10*c[7]^2*c[5])/(c[6]-c[7])/(-c[4]+c[6] )/(-c[5]+c[6])/c[6]/(-20*c[5]*c[6]*c[4]+30*c[5]*c[6]*c[7]*c[4]+15*c[6] *c[5]-20*c[6]*c[7]*c[5]+15*c[6]*c[4]-20*c[6]*c[7]*c[4]-12*c[6]+15*c[7] *c[6]+15*c[5]*c[4]-20*c[5]*c[7]*c[4]-12*c[5]+15*c[7]*c[5]-12*c[4]+15*c [7]*c[4]+10-12*c[7])*(c[4]-1)*(-1+c[5])*(-1+c[6]), b[8] = 1/60*(-20*c[ 5]*c[6]*c[4]+30*c[5]*c[6]*c[7]*c[4]+15*c[6]*c[5]-20*c[6]*c[7]*c[5]+15* c[6]*c[4]-20*c[6]*c[7]*c[4]-12*c[6]+15*c[7]*c[6]+15*c[5]*c[4]-20*c[5]* c[7]*c[4]-12*c[5]+15*c[7]*c[5]-12*c[4]+15*c[7]*c[4]+10-12*c[7])/(c[4]- 1)/(-1+c[5])/(-1+c[6])/(-1+c[7]), a[5,1] = 1/4*c[5]*(2*c[5]^2-5*c[5]*c [4]+4*c[4]^2)/c[4]^2, `b*`[5] = 1/60*(-42*c[7]*c[6]^2+89*c[5]*c[7]*c[4 ]-18*c[6]+6*c[5]+6*c[4]+27*c[7]*c[6]-9*c[4]^2+27*c[6]^2+186*c[5]*c[6]* c[7]*c[4]-36*c[6]*c[5]^2+39*c[6]*c[5]-9*c[7]*c[5]-100*c[6]^2*c[4]^3+60 0*c[6]^2*c[7]*c[5]^2*c[4]^3-100*c[7]*c[6]*c[4]^3-310*c[6]^2*c[7]*c[4]^ 2+192*c[6]^2*c[7]*c[4]+180*c[6]^2*c[4]^2+14*c[7]*c[4]^2-9*c[7]*c[4]-61 *c[6]*c[7]*c[5]-97*c[6]*c[4]^2+60*c[6]*c[4]^3-103*c[6]*c[7]*c[4]+156*c [6]*c[7]*c[4]^2-30*c[5]*c[4]^3-150*c[5]*c[7]*c[4]^2+156*c[5]*c[6]*c[4] ^2+93*c[5]*c[4]^2-112*c[5]*c[6]*c[4]-57*c[5]*c[4]+66*c[6]*c[4]+230*c[5 ]*c[7]*c[6]*c[4]^3-530*c[6]^2*c[4]^2*c[5]+330*c[6]^2*c[4]*c[5]+380*c[5 ]^2*c[6]*c[7]*c[4]^2+320*c[6]^2*c[5]*c[4]^3-560*c[6]^2*c[7]*c[4]*c[5]- 600*c[7]*c[6]^2*c[5]*c[4]^3+120*c[6]^2*c[7]*c[5]-100*c[6]^2*c[7]*c[5]^ 2-270*c[6]*c[5]*c[7]*c[4]^2+60*c[6]*c[7]*c[5]^2+60*c[6]^2*c[5]^2+950*c [6]^2*c[4]^2*c[7]*c[5]+500*c[6]^2*c[5]^2*c[7]*c[4]-900*c[6]^2*c[5]^2*c [7]*c[4]^2-300*c[6]*c[5]^2*c[7]*c[4]^3-75*c[6]^2*c[5]+50*c[7]*c[5]*c[4 ]^3-130*c[6]*c[5]*c[4]^3+18*c[5]^2*c[4]+150*c[5]^2*c[6]*c[4]^3+50*c[5] ^2*c[7]*c[4]^2-190*c[5]^2*c[6]*c[4]^2-300*c[6]^2*c[5]^2*c[4]^3+470*c[6 ]^2*c[5]^2*c[4]^2-30*c[5]^2*c[4]^2-280*c[6]^2*c[4]*c[5]^2+180*c[7]*c[6 ]^2*c[4]^3-117*c[6]^2*c[4]+126*c[5]^2*c[6]*c[4]-30*c[5]^2*c[7]*c[4]-23 0*c[5]^2*c[6]*c[7]*c[4])/(-4*c[5]^3*c[4]-9*c[5]*c[7]*c[4]-3*c[7]*c[6]- c[5]^2-28*c[5]*c[6]*c[7]*c[4]-9*c[6]*c[5]^2+3*c[6]*c[5]+8*c[6]*c[5]^3+ c[7]*c[5]+c[7]*c[4]+9*c[6]*c[7]*c[5]+9*c[6]*c[7]*c[4]-8*c[6]*c[7]*c[4] ^2+4*c[5]*c[7]*c[4]^2+8*c[5]*c[6]*c[4]^2-9*c[5]*c[6]*c[4]-c[5]*c[4]-30 *c[5]^2*c[6]*c[7]*c[4]^2+28*c[6]*c[5]*c[7]*c[4]^2-8*c[6]*c[7]*c[5]^2-2 8*c[5]^3*c[6]*c[4]+9*c[5]^2*c[4]-28*c[5]^2*c[6]*c[4]^2+30*c[5]^3*c[6]* c[4]^2-4*c[5]^2*c[4]^2+28*c[5]^2*c[6]*c[4]+4*c[5]^2*c[7]*c[4]+28*c[5]^ 2*c[6]*c[7]*c[4])/(-c[5]+c[6])/(-c[4]+c[5])/c[5], a[5,4] = c[5]^2*(-c[ 4]+c[5])/c[4]^2, a[7,6] = 1/2*(5*c[5]*c[4]+1-2*c[4]-2*c[5])*(c[4]-c[7] )*(c[5]-c[7])*(c[6]-c[7])*c[7]/(10*c[5]*c[6]*c[4]-5*c[6]*c[5]-5*c[6]*c [4]+3*c[6]-5*c[5]*c[4]+3*c[5]+3*c[4]-2)/(-c[4]+c[6])/(-c[5]+c[6])/c[6] , c[3] = 2/3*c[4], a[4,3] = 3/4*c[4], a[3,2] = 2/9*c[4]^2/c[2], a[7,4] = -1/2*c[7]*(-68*c[5]^2*c[6]*c[4]^4-7*c[6]*c[5]*c[4]^4+7*c[4]^2*c[5]^ 2*c[7]^3-1400*c[5]^4*c[4]^4*c[6]^2*c[7]+11*c[7]^3*c[5]*c[4]^3-72*c[7]^ 3*c[5]^2*c[4]^3-3*c[7]*c[5]*c[4]^4-58*c[5]^2*c[4]^4*c[7]^2-120*c[6]*c[ 4]^7*c[7]*c[5]+320*c[4]^5*c[6]^2*c[5]^2-200*c[4]^3*c[7]^3*c[5]^3-100*c [4]^5*c[6]*c[5]^3-2*c[5]*c[6]^2*c[4]^4-600*c[4]^5*c[7]^3*c[6]*c[5]^3-2 00*c[5]^5*c[4]^3*c[7]^2*c[6]^2+400*c[5]^2*c[4]^6*c[6]^2*c[7]^2-21*c[4] ^5*c[5]*c[7]+120*c[5]^2*c[4]^5*c[7]-600*c[5]^3*c[4]^6*c[6]^2*c[7]-198* c[5]^3*c[4]^4*c[7]-280*c[7]^2*c[4]^5*c[5]^3+240*c[7]^3*c[5]^2*c[4]^4+1 250*c[6]^2*c[4]^5*c[7]*c[5]^3+120*c[7]*c[5]^2*c[4]^6*c[6]-340*c[5]^4*c [4]^2*c[7]^2*c[6]+400*c[6]^2*c[7]*c[5]^4*c[4]^2-240*c[5]^2*c[4]^6*c[7] -300*c[6]*c[5]*c[7]^2*c[4]^6-228*c[6]*c[4]^5*c[5]*c[7]-119*c[6]*c[5]^2 *c[4]^4*c[7]-200*c[5]*c[7]^3*c[4]^6*c[6]-600*c[5]^4*c[7]*c[4]^6*c[6]-1 20*c[4]^6*c[6]^2*c[5]^2+600*c[5]^2*c[7]^2*c[4]^6*c[6]+100*c[5]^2*c[7]^ 3*c[4]^6-187*c[5]^4*c[4]^3*c[6]-910*c[5]^4*c[4]^5*c[6]-320*c[5]^2*c[6] *c[4]^6+620*c[5]^4*c[4]^4*c[6]-100*c[5]^4*c[7]^3*c[4]^4-40*c[7]^2*c[4] ^4*c[5]^3+40*c[7]^3*c[4]^5*c[5]+10*c[6]*c[7]^2*c[4]^5-10*c[6]*c[7]^3*c [4]^4+100*c[5]^4*c[4]^5*c[7]^2+80*c[5]^4*c[4]^4*c[7]+8*c[4]^3*c[7]^3*c [6]-2*c[7]^2*c[6]*c[4]^4+280*c[7]^3*c[5]^3*c[4]^4+80*c[6]*c[4]^5*c[5]* c[7]^2+c[4]^4*c[7]^2-360*c[5]^2*c[7]^3*c[4]^5-80*c[7]^2*c[5]^4*c[4]^3+ 40*c[7]^3*c[5]^4*c[4]^3-1200*c[5]^3*c[7]*c[4]^6*c[6]+10*c[6]^2*c[7]^3* c[4]^3-10*c[6]^2*c[4]^5*c[5]-12*c[7]*c[6]^2*c[5]^3-4*c[7]^2*c[6]*c[5]^ 3-80*c[5]^4*c[4]^5*c[7]-240*c[7]*c[6]^2*c[5]*c[4]^5-3*c[5]^4*c[4]^2*c[ 7]^2+20*c[5]^5*c[7]^2*c[4]^3+180*c[6]*c[5]^2*c[4]^5*c[7]-60*c[5]^4*c[7 ]^3*c[4]^2*c[6]+400*c[5]^2*c[7]^3*c[4]^6*c[6]-25*c[4]^5*c[5]^3+200*c[5 ]^5*c[4]^4*c[7]^2*c[6]+60*c[5]^2*c[4]^7*c[7]+100*c[7]*c[4]^4*c[6]*c[5] -40*c[5]*c[4]^6*c[7]^2-150*c[4]^4*c[6]^2*c[5]^5+50*c[5]^5*c[7]*c[4]^4* c[6]+174*c[7]^2*c[5]^3*c[4]^3+8*c[7]*c[4]^4*c[6]^2+600*c[5]^3*c[7]*c[4 ]^7*c[6]-60*c[5]^5*c[4]^2*c[6]^2*c[7]+200*c[5]^5*c[4]^4*c[6]^2*c[7]-40 *c[5]^5*c[4]^4*c[6]-200*c[5]^5*c[7]*c[4]^5*c[6]-149*c[6]^2*c[5]^4*c[7] *c[4]-24*c[5]^4*c[6]^2*c[4]+189*c[5]^4*c[6]^2*c[4]^2+300*c[5]^3*c[4]^6 *c[6]^2-20*c[5]^5*c[4]^4*c[7]-5*c[4]^5*c[5]*c[7]^2+600*c[4]^6*c[6]*c[5 ]^3-84*c[6]^2*c[7]*c[5]^2*c[4]^3-74*c[6]^2*c[7]^2*c[5]^2*c[4]+162*c[6] ^2*c[7]^2*c[5]^2*c[4]^2+311*c[6]*c[7]*c[5]^3*c[4]^3+21*c[6]^2*c[7]^3*c [4]*c[5]-40*c[7]^2*c[6]^2*c[5]^4+11*c[5]^4*c[7]*c[4]^2+107*c[7]^2*c[6] *c[5]*c[4]^4+100*c[7]^2*c[5]^5*c[6]^2*c[4]^2+320*c[5]^4*c[4]*c[7]^2*c[ 6]^2-10*c[6]^2*c[4]^4*c[7]^2+300*c[5]^2*c[4]^6*c[7]^2-3*c[7]^2*c[6]*c[ 4]^3-2*c[5]*c[7]^2*c[4]^3+15*c[6]*c[5]*c[7]^2*c[4]^2-8*c[5]*c[7]*c[6]* c[4]^3-14*c[5]^2*c[6]*c[7]*c[4]^2-2000*c[5]^4*c[7]^2*c[4]^4*c[6]+200*c [4]^5*c[6]*c[7]*c[5]^3+250*c[5]^2*c[6]*c[7]^3*c[4]^4+4*c[6]^2*c[5]*c[4 ]^3-26*c[7]^3*c[4]^2*c[6]*c[5]-69*c[7]*c[6]^2*c[5]*c[4]^3-18*c[6]^2*c[ 7]^3*c[5]^2+12*c[6]^2*c[7]^2*c[5]^2+10*c[6]^2*c[4]^2*c[7]*c[5]+50*c[6] ^2*c[7]^2*c[4]^2*c[5]+31*c[6]*c[5]^4*c[4]^2+24*c[7]*c[6]^2*c[5]^4+240* c[4]^5*c[7]*c[5]^3+2*c[6]^2*c[5]^2*c[7]*c[4]+39*c[6]^2*c[5]^2*c[7]*c[4 ]^2-17*c[6]*c[5]^2*c[7]*c[4]^3-73*c[6]*c[5]^3*c[7]*c[4]^2+59*c[6]^2*c[ 5]^3*c[7]*c[4]-12*c[5]^4*c[4]^3*c[7]-12*c[7]^2*c[5]^2*c[4]^3+40*c[5]^4 *c[7]^2*c[4]^4-c[4]^3*c[7]^3+17*c[7]^3*c[5]^2*c[6]*c[4]-9*c[7]^2*c[5]^ 2*c[6]*c[4]+3*c[6]*c[7]*c[4]^4+4*c[4]^5*c[5]+4*c[6]^2*c[7]^2*c[4]^2-15 0*c[5]^5*c[4]^3*c[7]^2*c[6]+21*c[5]^3*c[6]*c[7]*c[4]-14*c[6]^2*c[7]^2* c[4]*c[5]+23*c[5]^2*c[6]*c[4]^3+13*c[5]^2*c[7]*c[4]^3-11*c[5]^3*c[7]*c [4]^2-17*c[5]^3*c[6]*c[4]^2+168*c[5]^3*c[6]^2*c[4]^3+50*c[5]^5*c[4]^3* c[6]^2*c[7]-220*c[4]^3*c[7]^2*c[5]^2*c[6]^2-6*c[6]*c[7]*c[4]^5-580*c[4 ]^5*c[5]^3*c[6]^2-31*c[7]^3*c[4]^4*c[5]-2*c[7]^2*c[5]^4*c[4]-4*c[5]^2* c[4]^4*c[7]+53*c[6]^2*c[5]^2*c[4]^3-14*c[6]^2*c[5]^2*c[4]^2+47*c[6]*c[ 5]^3*c[4]^3+12*c[6]^2*c[5]^3*c[4]-59*c[6]^2*c[5]^3*c[4]^2+28*c[5]^3*c[ 7]*c[4]^3+127*c[5]^4*c[7]*c[4]^2*c[6]+1250*c[5]^3*c[7]^3*c[4]^4*c[6]+1 0*c[6]*c[5]*c[4]^6+240*c[5]*c[4]^6*c[7]*c[6]-4*c[7]*c[6]^2*c[4]^3+11*c [7]^2*c[5]^2*c[6]*c[4]^3-80*c[7]^3*c[5]^2*c[6]*c[4]^2-121*c[7]^2*c[5]^ 3*c[6]*c[4]^2-96*c[5]^3*c[7]^3*c[6]*c[4]+c[7]^3*c[4]^2*c[5]-68*c[4]^3* c[7]^2*c[5]*c[6]^2-80*c[7]^2*c[6]*c[5]*c[4]^3+68*c[7]^2*c[5]^2*c[6]*c[ 4]^2-10*c[4]^4*c[5]^2+20*c[7]^2*c[5]^3*c[6]*c[4]-240*c[7]^2*c[5]^3*c[6 ]*c[4]^3-9*c[5]^4*c[4]^3-2*c[6]^2*c[4]^3*c[7]^2+260*c[5]^3*c[6]^2*c[7] *c[4]^3+49*c[5]^3*c[7]^3*c[4]^2-c[7]^3*c[5]^2*c[4]-228*c[6]^2*c[7]*c[5 ]^3*c[4]^2+30*c[6]^2*c[7]^2*c[4]*c[5]^3-2*c[7]^2*c[5]^2*c[4]^2+5*c[7]^ 2*c[5]^3*c[4]+40*c[5]^5*c[4]^3*c[6]*c[7]+36*c[4]^6*c[7]*c[5]+90*c[7]^3 *c[5]^2*c[6]^2*c[4]^3+16*c[7]^2*c[4]^4*c[5]-20*c[7]^2*c[5]^3*c[6]^2*c[ 4]^2+300*c[6]^2*c[4]^5*c[5]*c[7]^2+250*c[7]^2*c[4]^5*c[6]*c[5]^3-490*c [5]^2*c[6]*c[4]^5*c[7]^2+120*c[7]^3*c[5]^2*c[4]^3*c[6]+810*c[7]^3*c[5] ^3*c[6]^2*c[4]^2+111*c[7]^3*c[6]*c[5]*c[4]^3+280*c[7]^3*c[6]^2*c[5]*c[ 4]^3-240*c[7]^3*c[5]^3*c[6]^2*c[4]+15*c[5]^4*c[4]^4+60*c[5]^4*c[7]^2*c [4]*c[6]+6*c[6]*c[7]^3*c[5]^3-31*c[5]^4*c[4]*c[7]*c[6]-200*c[4]^5*c[6] ^2*c[7]*c[5]^2-500*c[4]^5*c[6]^2*c[7]^2*c[5]^2+9*c[4]^5*c[5]^2-181*c[6 ]^2*c[4]^4*c[5]^2-1280*c[7]^3*c[5]^3*c[4]^3*c[6]-600*c[5]^2*c[4]^5*c[7 ]^3*c[6]-6*c[4]^6*c[5]-6*c[6]^2*c[7]^3*c[4]^2+100*c[5]^5*c[4]^5*c[6]+1 20*c[5]*c[7]*c[4]^6*c[6]^2+420*c[6]*c[4]^5*c[7]^3*c[5]+600*c[5]^4*c[4] ^5*c[6]^2*c[7]+175*c[7]*c[6]^2*c[5]*c[4]^4-430*c[5]^3*c[4]^4*c[6]^2*c[ 7]+200*c[5]^3*c[4]^3*c[6]^2*c[7]^2-450*c[5]^4*c[4]^4*c[6]*c[7]+160*c[5 ]^2*c[4]^4*c[7]*c[6]^2-200*c[5]^3*c[4]^4*c[6]*c[7]-280*c[5]^4*c[4]^3*c [6]*c[7]+6*c[5]^3*c[4]^3-100*c[5]^2*c[4]^7*c[7]^2-1200*c[7]^3*c[5]^3*c [6]^2*c[4]^3-320*c[7]^3*c[4]^4*c[5]*c[6]-30*c[7]^2*c[5]^3*c[4]^2-300*c [6]*c[5]^3*c[4]^7-350*c[5]^4*c[4]^5*c[6]^2+600*c[6]^2*c[7]^3*c[5]^3*c[ 4]^4+600*c[6]*c[7]^2*c[5]^4*c[4]^5+200*c[6]^2*c[7]^3*c[4]^5*c[5]+200*c [7]^2*c[4]^7*c[6]*c[5]+1100*c[5]^4*c[4]^4*c[6]^2-720*c[5]^4*c[4]^3*c[6 ]^2+1750*c[7]^2*c[5]^4*c[6]^2*c[4]^3-1230*c[7]^2*c[5]^4*c[6]^2*c[4]^2+ 150*c[5]^4*c[7]^3*c[4]^3*c[6]+300*c[5]^4*c[4]^6*c[6]+480*c[5]^3*c[4]^4 *c[7]^2*c[6]+1420*c[5]^4*c[4]^3*c[7]^2*c[6]+20*c[5]^2*c[4]^4*c[7]^2*c[ 6]+130*c[5]^4*c[4]^3*c[6]^2*c[7]-40*c[7]^2*c[4]^4*c[6]^2*c[5]-350*c[7] ^2*c[4]^4*c[5]^3*c[6]^2+230*c[7]^2*c[4]^4*c[6]^2*c[5]^2+10*c[4]^6*c[5] ^2+188*c[5]^2*c[6]*c[4]^5-400*c[6]^2*c[7]^3*c[5]^2*c[4]^5-600*c[6]^2*c [7]^2*c[5]^4*c[4]^4+120*c[5]^2*c[4]^7*c[6]-94*c[4]^4*c[6]*c[5]^3+6*c[6 ]*c[5]*c[4]^5+60*c[5]^5*c[6]^2*c[4]^3-200*c[6]^2*c[4]^6*c[5]*c[7]^2-42 0*c[7]^3*c[6]^2*c[5]*c[4]^4+1550*c[5]^4*c[4]^5*c[6]*c[7]+500*c[7]^3*c[ 4]^4*c[6]^2*c[5]^2-400*c[5]^2*c[4]^7*c[6]*c[7]^2+6*c[5]^3*c[4]^4+103*c [6]^2*c[7]^3*c[5]^2*c[4]-260*c[6]^2*c[7]^3*c[5]^2*c[4]^2+540*c[5]^3*c[ 7]^3*c[4]^2*c[6]-97*c[6]^2*c[5]*c[7]^3*c[4]^2+30*c[6]^2*c[7]^3*c[5]^3- 4*c[7]^3*c[4]*c[5]^3)/(-240*c[5]^2*c[6]*c[4]^4+40*c[6]*c[5]*c[4]^4-9*c [5]^3*c[4]-100*c[4]^5*c[6]*c[5]^3-350*c[5]^4*c[4]^3*c[6]+30*c[5]^5*c[4 ]^2+2*c[5]^3-2*c[4]^3-50*c[5]^5*c[4]^3-3*c[5]^4+3*c[4]^4+50*c[4]^5*c[5 ]^3+100*c[5]^5*c[4]^3*c[6]-3*c[6]*c[5]^3+3*c[6]*c[4]^3+9*c[5]*c[4]^3-6 *c[5]*c[6]*c[4]^2+4*c[5]*c[4]^2+240*c[6]*c[5]^4*c[4]^2+5*c[6]*c[5]^4-1 3*c[6]*c[5]*c[4]^3+13*c[5]^3*c[6]*c[4]-4*c[5]^2*c[4]+68*c[5]^3*c[4]^2+ 100*c[5]^2*c[6]*c[4]^3-100*c[5]^3*c[6]*c[4]^2-140*c[5]^4*c[4]^2+140*c[ 4]^4*c[5]^2+180*c[5]^4*c[4]^3+6*c[5]^2*c[6]*c[4]-68*c[5]^2*c[4]^3-23*c [4]^4*c[5]-40*c[5]^4*c[4]*c[6]-30*c[4]^5*c[5]^2-5*c[6]*c[4]^4-50*c[5]^ 5*c[6]*c[4]^2+50*c[5]^2*c[6]*c[4]^5+350*c[4]^4*c[6]*c[5]^3+23*c[5]^4*c [4]-180*c[5]^3*c[4]^4)/(-c[4]+c[6])/c[4]^2, `b*`[7] = 1/60*(-30*c[5]^3 *c[4]-18*c[6]+6*c[5]+6*c[4]-9*c[5]^2-9*c[4]^2+27*c[6]^2-97*c[6]*c[5]^2 +66*c[6]*c[5]+60*c[6]*c[5]^3-100*c[6]^2*c[5]^3-100*c[6]^2*c[4]^3+180*c [6]^2*c[4]^2-97*c[6]*c[4]^2+60*c[6]*c[4]^3-30*c[5]*c[4]^3+312*c[5]*c[6 ]*c[4]^2+107*c[5]*c[4]^2-215*c[5]*c[6]*c[4]-66*c[5]*c[4]+66*c[6]*c[4]- 840*c[6]^2*c[4]^2*c[5]+522*c[6]^2*c[4]*c[5]+500*c[6]^2*c[5]*c[4]^3+180 *c[6]^2*c[5]^2-117*c[6]^2*c[5]-230*c[6]*c[5]*c[4]^3-230*c[5]^3*c[6]*c[ 4]+107*c[5]^2*c[4]+50*c[5]^3*c[4]^2+380*c[5]^2*c[6]*c[4]^3-460*c[5]^2* c[6]*c[4]^2+380*c[5]^3*c[6]*c[4]^2+600*c[5]^3*c[6]^2*c[4]^3-900*c[6]^2 *c[5]^2*c[4]^3+1420*c[6]^2*c[5]^2*c[4]^2-300*c[6]*c[5]^3*c[4]^3+500*c[ 6]^2*c[5]^3*c[4]-900*c[6]^2*c[5]^3*c[4]^2-180*c[5]^2*c[4]^2-840*c[6]^2 *c[4]*c[5]^2-117*c[6]^2*c[4]+312*c[5]^2*c[6]*c[4]+50*c[5]^2*c[4]^3)/c[ 7]/(-3*c[7]^3*c[6]+3*c[7]^2*c[6]^2-9*c[6]^2*c[7]^2*c[5]-8*c[7]*c[6]^2* c[5]^3+8*c[7]^2*c[6]*c[5]^3-c[5]*c[6]*c[7]*c[4]-c[7]^2*c[4]^2-30*c[6]^ 2*c[7]*c[5]^2*c[4]^3-28*c[6]^2*c[7]^2*c[5]^2*c[4]+30*c[6]^2*c[7]^2*c[5 ]^2*c[4]^2-30*c[6]*c[7]*c[5]^3*c[4]^3+9*c[6]^2*c[7]*c[4]^2-3*c[6]^2*c[ 7]*c[4]+2*c[6]*c[7]^2*c[4]-9*c[6]*c[7]^2*c[4]^2-2*c[7]^2*c[5]*c[4]-c[7 ]^2*c[5]^2+c[6]*c[7]*c[4]^2+8*c[7]^2*c[6]*c[4]^3-4*c[5]*c[7]^2*c[4]^3+ c[5]*c[7]*c[4]^2-c[5]*c[6]*c[4]^2+32*c[6]*c[5]*c[7]^2*c[4]^2-9*c[7]^2* c[5]*c[6]*c[4]-4*c[5]*c[7]*c[6]*c[4]^3-9*c[6]^2*c[4]^2*c[5]+3*c[6]^2*c [4]*c[5]+c[7]^3*c[4]+9*c[7]^3*c[4]*c[6]-9*c[7]^3*c[4]*c[5]-20*c[5]^2*c [6]*c[7]*c[4]^2+8*c[6]^2*c[5]*c[4]^3+28*c[7]^3*c[4]^2*c[6]*c[5]+18*c[6 ]^2*c[7]*c[4]*c[5]+28*c[7]*c[6]^2*c[5]*c[4]^3-3*c[6]^2*c[7]*c[5]+9*c[6 ]^2*c[7]*c[5]^2+c[6]*c[7]*c[5]^2-8*c[6]*c[7]^3*c[5]^2-9*c[6]*c[7]^2*c[ 5]^2+8*c[6]^2*c[7]^2*c[5]^2+2*c[6]*c[7]^2*c[5]-36*c[6]^2*c[4]^2*c[7]*c [5]-28*c[6]^2*c[7]^2*c[4]^2*c[5]-28*c[7]^3*c[4]*c[6]*c[5]-36*c[6]^2*c[ 5]^2*c[7]*c[4]+56*c[6]^2*c[5]^2*c[7]*c[4]^2+28*c[6]*c[5]^2*c[7]*c[4]^3 +28*c[6]*c[5]^3*c[7]*c[4]^2+28*c[6]^2*c[5]^3*c[7]*c[4]+9*c[6]*c[7]^3*c [5]+9*c[7]^2*c[5]*c[4]^2+28*c[7]^3*c[5]^2*c[6]*c[4]+32*c[7]^2*c[5]^2*c [6]*c[4]+8*c[6]^2*c[7]^2*c[4]^2-9*c[6]^2*c[7]^2*c[4]-4*c[5]^3*c[6]*c[7 ]*c[4]+28*c[6]^2*c[7]^2*c[4]*c[5]-4*c[5]^2*c[6]*c[4]^3-9*c[5]^2*c[7]*c [4]^2+9*c[5]^2*c[6]*c[4]^2+4*c[5]^2*c[7]*c[4]^3+4*c[5]^3*c[7]*c[4]^2-4 *c[5]^3*c[6]*c[4]^2+30*c[5]^3*c[6]^2*c[4]^3-28*c[6]^2*c[5]^2*c[4]^3+28 *c[6]^2*c[5]^2*c[4]^2+8*c[6]^2*c[5]^3*c[4]-28*c[6]^2*c[5]^3*c[4]^2-9*c [6]^2*c[4]*c[5]^2-8*c[7]*c[6]^2*c[4]^3+30*c[7]^2*c[5]^2*c[6]*c[4]^3-30 *c[7]^3*c[5]^2*c[6]*c[4]^2+30*c[7]^2*c[5]^3*c[6]*c[4]^2-8*c[7]^3*c[4]^ 2*c[6]+4*c[7]^3*c[4]^2*c[5]+c[7]^3*c[5]-28*c[7]^2*c[6]*c[5]*c[4]^3-56* c[7]^2*c[5]^2*c[6]*c[4]^2-28*c[7]^2*c[5]^3*c[6]*c[4]-c[5]^2*c[6]*c[4]+ c[5]^2*c[7]*c[4]+4*c[7]^3*c[5]^2*c[4]+9*c[7]^2*c[5]^2*c[4]-30*c[6]^2*c [7]*c[5]^3*c[4]^2-8*c[7]^2*c[5]^2*c[4]^2-4*c[7]^2*c[5]^3*c[4]), a[2,1] = c[2], a[6,4] = -1/2*(120*c[5]^2*c[6]*c[4]^4-12*c[6]*c[5]*c[4]^4-90* c[5]^2*c[6]^3*c[4]^2-60*c[5]*c[6]^2*c[4]^4-2*c[6]^3*c[4]^2+20*c[5]^4*c [4]^3*c[6]+40*c[6]^2*c[4]^5*c[5]+36*c[4]*c[5]^2*c[6]^3-20*c[5]^4*c[6]^ 2*c[4]^2+8*c[6]^2*c[5]^3+2*c[6]^2*c[4]^3-24*c[5]*c[6]^3*c[4]^2-6*c[6]^ 2*c[4]^2*c[5]+36*c[6]^2*c[5]*c[4]^3+60*c[4]^3*c[5]*c[6]^3-3*c[6]*c[5]* c[4]^3-11*c[5]^3*c[6]*c[4]+3*c[5]^3*c[4]^2-24*c[5]^2*c[6]*c[4]^3+13*c[ 5]^2*c[6]*c[4]^2+48*c[5]^3*c[6]*c[4]^2-60*c[5]^3*c[6]^2*c[4]^3-30*c[6] ^2*c[5]^2*c[4]^3-12*c[6]^2*c[5]^2*c[4]^2-150*c[6]*c[5]^3*c[4]^3-48*c[6 ]^2*c[5]^3*c[4]+150*c[6]^2*c[5]^3*c[4]^2-2*c[6]^2*c[4]*c[5]^2-5*c[5]^2 *c[4]^3+2*c[4]^4*c[5]-6*c[5]^2*c[6]^3+60*c[6]^3*c[4]^3*c[5]^2+7*c[6]^3 *c[4]*c[5]-60*c[5]^2*c[6]*c[4]^5-40*c[6]^3*c[4]^4*c[5]+60*c[4]^4*c[6]* c[5]^3)*c[6]/(c[4]^3-2*c[5]*c[4]^2+30*c[5]^2*c[4]^3-6*c[5]*c[4]^3-10*c [4]^4*c[5]^2-c[5]^3+10*c[5]^4*c[4]^2-30*c[5]^3*c[4]^2+6*c[5]^3*c[4]+2* c[5]^2*c[4])/c[4]^2, a[3,1] = -2/9*c[4]*(c[4]-3*c[2])/c[2], a[4,1] = 1 /4*c[4], a[7,3] = 3/4*1/c[4]^2*c[7]*(-480*c[5]^2*c[6]*c[4]^4+15*c[6]*c [5]*c[4]^4+300*c[4]^2*c[5]^2*c[7]^3+60*c[7]^3*c[5]*c[4]^3-510*c[7]^3*c [5]^2*c[4]^3+54*c[7]*c[5]*c[4]^4+450*c[5]^2*c[4]^4*c[7]^2+150*c[4]^3*c [7]^3*c[5]^3-150*c[7]^2*c[5]^2*c[4]^5-450*c[4]^5*c[6]*c[5]^3-3*c[7]^3* c[4]^2+90*c[5]^2*c[4]^5*c[7]+120*c[5]^3*c[4]^4*c[7]+150*c[7]^3*c[5]^2* c[4]^4+100*c[5]^4*c[4]^2*c[7]^2*c[6]-180*c[6]*c[4]^5*c[5]*c[7]+30*c[6] *c[5]^2*c[4]^4*c[7]+60*c[5]^4*c[4]^3*c[6]-150*c[5]^4*c[4]^4*c[6]-200*c [7]^2*c[4]^4*c[5]^3+300*c[6]*c[4]^5*c[5]*c[7]^2-50*c[7]^2*c[5]^4*c[4]^ 3+24*c[7]*c[6]*c[5]^3-40*c[7]^2*c[6]*c[5]^3-6*c[5]*c[6]*c[7]*c[4]+360* c[7]*c[4]^4*c[6]*c[5]+150*c[7]^2*c[5]^3*c[4]^3+2*c[7]^2*c[4]^2+6*c[5]^ 2*c[7]^3+330*c[6]*c[7]*c[5]^3*c[4]^3-450*c[7]^2*c[6]*c[5]*c[4]^4-12*c[ 7]*c[6]*c[4]^3-6*c[6]*c[7]^2*c[4]-2*c[7]^2*c[5]*c[4]-4*c[7]^2*c[5]^2+6 *c[6]*c[7]*c[4]^2+20*c[7]^2*c[6]*c[4]^3-10*c[5]*c[7]^2*c[4]^3+6*c[5]*c [4]^3-6*c[5]*c[7]*c[4]^2-6*c[5]*c[6]*c[4]^2+150*c[6]*c[5]*c[7]^2*c[4]^ 2-72*c[7]^2*c[5]*c[6]*c[4]-270*c[5]*c[7]*c[6]*c[4]^3+9*c[7]^3*c[4]*c[6 ]+3*c[7]^3*c[4]*c[5]-306*c[5]^2*c[6]*c[7]*c[4]^2+900*c[4]^5*c[6]*c[7]* c[5]^3+600*c[5]^2*c[6]*c[7]^3*c[4]^4-420*c[7]^3*c[4]^2*c[6]*c[5]+90*c[ 6]*c[5]*c[7]*c[4]^2-12*c[6]*c[7]*c[5]^2+30*c[6]*c[7]^3*c[5]^2+12*c[6]* c[7]^2*c[5]+123*c[7]^3*c[4]*c[6]*c[5]+360*c[6]*c[5]^2*c[7]*c[4]^3+360* c[6]*c[5]^3*c[7]*c[4]^2-18*c[6]*c[7]^3*c[5]+30*c[5]^4*c[4]^3*c[7]-80*c [7]^2*c[5]^2*c[4]^3+24*c[7]^2*c[5]*c[4]^2-30*c[7]*c[5]*c[4]^3+3*c[6]*c [5]*c[4]^3-240*c[7]^3*c[5]^2*c[6]*c[4]-24*c[5]^3*c[6]*c[4]-12*c[5]^3*c [7]*c[4]-144*c[5]^3*c[6]*c[7]*c[4]+9*c[5]^3*c[4]^2+294*c[5]^2*c[6]*c[4 ]^3-42*c[5]^2*c[7]*c[4]^2-75*c[5]^2*c[6]*c[4]^2+252*c[5]^2*c[7]*c[4]^3 +18*c[5]^3*c[7]*c[4]^2+189*c[5]^3*c[6]*c[4]^2-360*c[5]^2*c[4]^4*c[7]-7 20*c[6]*c[5]^3*c[4]^3-120*c[5]^3*c[7]*c[4]^3-60*c[5]^4*c[7]*c[4]^2*c[6 ]-6*c[5]^2*c[4]^2-780*c[7]^2*c[5]^2*c[6]*c[4]^3+810*c[7]^3*c[5]^2*c[6] *c[4]^2-1230*c[7]^2*c[5]^3*c[6]*c[4]^2-15*c[7]^3*c[4]^2*c[6]-21*c[7]^3 *c[4]^2*c[5]+20*c[7]^2*c[6]*c[5]*c[4]^3+220*c[7]^2*c[5]^2*c[6]*c[4]^2+ 15*c[4]^4*c[5]^2+320*c[7]^2*c[5]^3*c[6]*c[4]+1850*c[7]^2*c[5]^3*c[6]*c [4]^3-60*c[5]^3*c[7]^3*c[4]^2+12*c[5]^2*c[6]*c[4]+12*c[5]^2*c[7]*c[4]- 66*c[7]^3*c[5]^2*c[4]+24*c[7]^2*c[5]^2*c[4]+72*c[5]^2*c[6]*c[7]*c[4]-1 10*c[7]^2*c[5]^2*c[4]^2+20*c[7]^2*c[5]^3*c[4]-9*c[4]^4*c[5]-60*c[7]^2* c[4]^4*c[5]-600*c[5]^2*c[6]*c[4]^5*c[7]^2-1200*c[7]^3*c[5]^2*c[4]^3*c[ 6]+630*c[7]^3*c[6]*c[5]*c[4]^3+300*c[5]^4*c[4]^4*c[6]*c[7]-1800*c[5]^3 *c[4]^4*c[6]*c[7]-15*c[5]^3*c[4]^3-300*c[7]^3*c[4]^4*c[5]*c[6]-20*c[7] ^2*c[5]^3*c[4]^2-800*c[5]^3*c[4]^4*c[7]^2*c[6]-200*c[5]^4*c[4]^3*c[7]^ 2*c[6]+1150*c[5]^2*c[4]^4*c[7]^2*c[6]+180*c[5]^2*c[6]*c[4]^5+1140*c[4] ^4*c[6]*c[5]^3)/(10*c[5]^3*c[4]^2+10*c[5]^2*c[4]^3-30*c[5]^2*c[4]^2+6* c[5]^2*c[4]-c[5]^2+6*c[5]*c[4]^2+c[5]*c[4]-c[4]^2)/(10*c[5]*c[6]*c[4]- 5*c[6]*c[5]-5*c[6]*c[4]+3*c[6]-5*c[5]*c[4]+3*c[5]+3*c[4]-2), `b*`[1] = 1/60*(30*c[5]^3*c[4]+42*c[7]*c[6]^2-103*c[5]*c[7]*c[4]+18*c[6]-6*c[5] -6*c[4]-27*c[7]*c[6]+9*c[5]^2+9*c[4]^2-180*c[7]*c[6]^2*c[5]^3+100*c[7] *c[6]*c[5]^3-27*c[6]^2-354*c[5]*c[6]*c[7]*c[4]+97*c[6]*c[5]^2-66*c[6]* c[5]-60*c[6]*c[5]^3+100*c[6]^2*c[5]^3+9*c[7]*c[5]+100*c[6]^2*c[4]^3-20 40*c[6]^2*c[7]*c[5]^2*c[4]^3-600*c[6]*c[7]*c[5]^3*c[4]^3-14*c[7]*c[5]^ 2+100*c[7]*c[6]*c[4]^3+310*c[6]^2*c[7]*c[4]^2-192*c[6]^2*c[7]*c[4]-180 *c[6]^2*c[4]^2-14*c[7]*c[4]^2+9*c[7]*c[4]+103*c[6]*c[7]*c[5]+97*c[6]*c [4]^2-60*c[6]*c[4]^3+103*c[6]*c[7]*c[4]-156*c[6]*c[7]*c[4]^2+30*c[5]*c [4]^3+174*c[5]*c[7]*c[4]^2-312*c[5]*c[6]*c[4]^2-107*c[5]*c[4]^2+215*c[ 5]*c[6]*c[4]+66*c[5]*c[4]-66*c[6]*c[4]-410*c[5]*c[7]*c[6]*c[4]^3+840*c [6]^2*c[4]^2*c[5]-522*c[6]^2*c[4]*c[5]-760*c[5]^2*c[6]*c[7]*c[4]^2-500 *c[6]^2*c[5]*c[4]^3+932*c[6]^2*c[7]*c[4]*c[5]+1020*c[7]*c[6]^2*c[5]*c[ 4]^3-192*c[6]^2*c[7]*c[5]+310*c[6]^2*c[7]*c[5]^2+526*c[6]*c[5]*c[7]*c[ 4]^2-156*c[6]*c[7]*c[5]^2-180*c[6]^2*c[5]^2-1610*c[6]^2*c[4]^2*c[7]*c[ 5]-1610*c[6]^2*c[5]^2*c[7]*c[4]+2970*c[6]^2*c[5]^2*c[7]*c[4]^2+690*c[6 ]*c[5]^2*c[7]*c[4]^3+690*c[6]*c[5]^3*c[7]*c[4]^2+1020*c[6]^2*c[5]^3*c[ 7]*c[4]+117*c[6]^2*c[5]-50*c[7]*c[5]*c[4]^3+230*c[6]*c[5]*c[4]^3+230*c [5]^3*c[6]*c[4]-50*c[5]^3*c[7]*c[4]-107*c[5]^2*c[4]-410*c[5]^3*c[6]*c[ 7]*c[4]-50*c[5]^3*c[4]^2-380*c[5]^2*c[6]*c[4]^3-310*c[5]^2*c[7]*c[4]^2 +460*c[5]^2*c[6]*c[4]^2+90*c[5]^2*c[7]*c[4]^3+90*c[5]^3*c[7]*c[4]^2-38 0*c[5]^3*c[6]*c[4]^2-600*c[5]^3*c[6]^2*c[4]^3+900*c[6]^2*c[5]^2*c[4]^3 -1420*c[6]^2*c[5]^2*c[4]^2+300*c[6]*c[5]^3*c[4]^3-500*c[6]^2*c[5]^3*c[ 4]+900*c[6]^2*c[5]^3*c[4]^2+180*c[5]^2*c[4]^2+840*c[6]^2*c[4]*c[5]^2-1 80*c[7]*c[6]^2*c[4]^3+117*c[6]^2*c[4]+1500*c[5]^3*c[6]^2*c[7]*c[4]^3-3 12*c[5]^2*c[6]*c[4]+174*c[5]^2*c[7]*c[4]+526*c[5]^2*c[6]*c[7]*c[4]-204 0*c[6]^2*c[7]*c[5]^3*c[4]^2-50*c[5]^2*c[4]^3)/(3*c[6]-28*c[5]*c[6]*c[4 ]^2-28*c[5]^2*c[6]*c[4]+8*c[6]*c[5]^2-9*c[6]*c[4]-9*c[6]*c[5]+28*c[5]* c[6]*c[4]+30*c[5]^2*c[6]*c[4]^2+8*c[6]*c[4]^2-4*c[5]^2*c[4]-4*c[5]*c[4 ]^2+9*c[5]*c[4]-c[5]-c[4])/c[5]/c[6]/c[7]/c[4], a[8,4] = -1/2*(734*c[5 ]^2*c[6]*c[4]^4-203*c[6]*c[5]*c[4]^4+3190*c[5]^4*c[4]^4*c[6]^2*c[7]-93 *c[7]*c[5]*c[4]^4-636*c[5]^2*c[4]^4*c[7]^2-10*c[5]^3*c[4]-520*c[6]*c[4 ]^7*c[7]*c[5]-481*c[4]^5*c[6]^2*c[5]^2+645*c[7]^2*c[5]^2*c[4]^5-574*c[ 4]^5*c[6]*c[5]^3-172*c[5]*c[6]^2*c[4]^4+6*c[7]^2*c[4]^3+100*c[5]^5*c[4 ]^3*c[7]^2*c[6]^2+60*c[4]^8*c[5]^2-600*c[5]^2*c[4]^6*c[6]^2*c[7]^2+9*c [4]^5*c[5]*c[7]-164*c[5]^2*c[4]^5*c[7]+900*c[5]^3*c[4]^6*c[6]^2*c[7]+6 00*c[5]^3*c[4]^6*c[6]^2*c[7]^2-1104*c[5]^3*c[4]^4*c[7]-18*c[4]^4*c[6]^ 2+90*c[7]^2*c[4]^5*c[5]^3+1380*c[6]^2*c[4]^5*c[7]*c[5]^3-750*c[7]*c[5] ^2*c[4]^6*c[6]+675*c[5]^4*c[4]^2*c[7]^2*c[6]+8*c[6]^2*c[4]^5-339*c[6]^ 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5]*c[4]^3+89*c[6]*c[5]*c[4]^3+200*c[4]^8*c[5]*c[6]*c[7]+390*c[6]*c[4]^ 7*c[5]-31*c[7]^2*c[5]^2*c[6]*c[4]-27*c[6]*c[7]*c[4]^4+10*c[4]^5*c[5]-8 0*c[5]^4*c[4]^6+200*c[5]^5*c[4]^5*c[7]^2*c[6]+6*c[6]^2*c[7]^2*c[4]^2+2 2*c[5]^3*c[6]*c[4]+19*c[5]^3*c[7]*c[4]+400*c[5]^5*c[4]^3*c[7]^2*c[6]+4 *c[5]^2*c[4]-135*c[5]^3*c[6]*c[7]*c[4]+71*c[5]^3*c[4]^2-21*c[6]^2*c[7] ^2*c[4]*c[5]-480*c[5]^2*c[6]*c[4]^3+42*c[5]^2*c[7]*c[4]^2+37*c[5]^2*c[ 6]*c[4]^2-480*c[5]^2*c[7]*c[4]^3-104*c[5]^3*c[7]*c[4]^2-97*c[5]^3*c[6] *c[4]^2-1536*c[5]^3*c[6]^2*c[4]^3+190*c[5]^5*c[4]^3*c[6]^2*c[7]+557*c[ 4]^3*c[7]^2*c[5]^2*c[6]^2-10*c[6]*c[7]*c[4]^5-730*c[4]^5*c[5]^3*c[6]^2 -14*c[7]*c[5]^4*c[4]+10*c[7]^2*c[5]^4*c[4]+646*c[5]^2*c[4]^4*c[7]+498* c[6]^2*c[5]^2*c[4]^3-95*c[6]^2*c[5]^2*c[4]^2+744*c[6]*c[5]^3*c[4]^3-46 *c[6]^2*c[5]^3*c[4]+334*c[6]^2*c[5]^3*c[4]^2+854*c[5]^3*c[7]*c[4]^3-49 0*c[5]^4*c[7]*c[4]^2*c[6]-30*c[7]*c[5]*c[4]^7-19*c[5]^2*c[4]^2+18*c[6] ^2*c[4]*c[5]^2-472*c[6]*c[5]*c[4]^6+360*c[5]*c[4]^6*c[7]*c[6]-10*c[7]* c[6]^2*c[4]^3-811*c[7]^2*c[5]^2*c[6]*c[4]^3-993*c[7]^2*c[5]^3*c[6]*c[4 ]^2-44*c[5]^4*c[4]^2-451*c[4]^3*c[7]^2*c[5]*c[6]^2+10*c[6]*c[7]*c[4]^6 -14*c[7]^2*c[6]*c[5]*c[4]^3-20*c[5]^5*c[4]^5+118*c[7]^2*c[5]^2*c[6]*c[ 4]^2-249*c[4]^4*c[5]^2+170*c[7]^2*c[5]^3*c[6]*c[4]+3829*c[7]^2*c[5]^3* c[6]*c[4]^3+2208*c[5]^3*c[6]^2*c[4]^4+339*c[5]^4*c[4]^3-6*c[6]^2*c[4]^ 3*c[7]^2+1871*c[5]^3*c[6]^2*c[7]*c[4]^3-10*c[5]^2*c[6]*c[4]-9*c[5]^2*c [7]*c[4]+6*c[7]^2*c[5]^2*c[4]+33*c[5]^2*c[6]*c[7]*c[4]+600*c[7]*c[4]^8 *c[6]*c[5]^3+187*c[6]^2*c[7]*c[5]^3*c[4]^2+341*c[6]^2*c[7]^2*c[4]*c[5] ^3+202*c[5]^2*c[4]^3-29*c[7]^2*c[5]^2*c[4]^2-10*c[7]^2*c[5]^3*c[4]+40* c[4]^4*c[5]-270*c[4]^7*c[5]^2-450*c[5]^5*c[4]^3*c[6]*c[7]+35*c[4]^6*c[ 7]*c[5]+80*c[7]^2*c[4]^4*c[5]-1163*c[7]^2*c[5]^3*c[6]^2*c[4]^2+20*c[5] *c[4]^7-640*c[6]^2*c[4]^5*c[5]*c[7]^2+850*c[7]^2*c[4]^5*c[6]*c[5]^3+15 30*c[5]^2*c[6]*c[4]^5*c[7]^2-506*c[5]^4*c[4]^4+1420*c[5]^2*c[4]^7*c[6] *c[7]-16*c[5]^4*c[4]*c[6]-85*c[5]^4*c[7]^2*c[4]*c[6]+72*c[5]^4*c[4]*c[ 7]*c[6]+230*c[4]^5*c[6]^2*c[7]*c[5]^2+1740*c[4]^5*c[6]^2*c[7]^2*c[5]^2 -600*c[6]*c[4]^8*c[5]^2*c[7]-82*c[4]^5*c[5]^2-480*c[6]^2*c[4]^4*c[5]^2 +320*c[4]^8*c[5]^2*c[6]-37*c[4]^6*c[5]+300*c[5]^3*c[4]^7*c[6]^2-90*c[5 ]^5*c[4]^5*c[6]+520*c[5]*c[7]*c[4]^6*c[6]^2-10*c[6]^2*c[7]*c[4]^5+9*c[ 6]*c[4]^4+600*c[5]^4*c[7]*c[4]^6*c[6]^2-2350*c[5]^4*c[4]^5*c[6]^2*c[7] -600*c[5]^3*c[7]^2*c[6]*c[4]^7-204*c[7]*c[6]^2*c[5]*c[4]^4-3630*c[5]^3 *c[4]^4*c[6]^2*c[7]+740*c[5]^3*c[4]^3*c[6]^2*c[7]^2-2120*c[5]^4*c[4]^4 *c[6]*c[7]+1460*c[5]^2*c[4]^4*c[7]*c[6]^2+2387*c[5]^3*c[4]^4*c[6]*c[7] +2548*c[5]^4*c[4]^3*c[6]*c[7]-498*c[5]^3*c[4]^3+100*c[5]^2*c[4]^7*c[7] ^2+26*c[7]^2*c[5]^3*c[4]^2+600*c[6]*c[5]^3*c[4]^7+100*c[6]*c[5]^5*c[4] ^6+300*c[6]*c[5]^4*c[4]^7-600*c[5]^4*c[7]*c[4]^7*c[6]+1600*c[5]^4*c[4] ^5*c[6]^2-1500*c[6]^2*c[7]^2*c[5]^3*c[4]^5-2700*c[6]*c[7]^2*c[5]^4*c[4 ]^5-200*c[7]^2*c[4]^7*c[6]*c[5]-2430*c[5]^4*c[4]^4*c[6]^2+1429*c[5]^4* c[4]^3*c[6]^2-2010*c[7]^2*c[5]^4*c[6]^2*c[4]^3+1410*c[7]^2*c[5]^4*c[6] 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1200*c[5]^5*c[7]*c[4]^5*c[6]^2+70*c[6]^2*c[4]^4*c[7]^2+32*c[6]*c[4]^5- 42*c[7]*c[6]*c[4]^3-4*c[6]^2*c[7]*c[4]^2+200*c[5]^2*c[4]^6*c[7]^2+18*c [7]*c[4]^3+20*c[6]*c[4]^3+26*c[7]^2*c[6]*c[4]^3+72*c[5]*c[7]^2*c[4]^3+ 48*c[5]*c[4]^3-18*c[5]*c[7]*c[4]^2-20*c[5]*c[6]*c[4]^2+8*c[5]*c[4]^2-2 3*c[6]*c[5]*c[7]^2*c[4]^2+266*c[5]*c[7]*c[6]*c[4]^3+12*c[6]^2*c[4]^2*c [5]-1240*c[5]^4*c[4]^5+287*c[5]^2*c[6]*c[7]*c[4]^2-8120*c[5]^4*c[7]^2* c[4]^4*c[6]+11330*c[4]^5*c[6]*c[7]*c[5]^3+72*c[6]^2*c[5]*c[4]^3+14*c[6 ]^2*c[7]*c[4]*c[5]+76*c[7]*c[6]^2*c[5]*c[4]^3-12*c[6]^2*c[7]*c[5]^2+40 *c[6]*c[5]*c[7]*c[4]^2+18*c[6]^2*c[7]^2*c[5]^2-101*c[6]^2*c[4]^2*c[7]* c[5]+132*c[6]^2*c[7]^2*c[4]^2*c[5]-284*c[6]*c[5]^4*c[4]^2-24*c[7]*c[6] ^2*c[5]^4-4160*c[4]^5*c[7]*c[5]^3-12*c[4]^5+95*c[6]^2*c[5]^2*c[7]*c[4] -356*c[6]^2*c[5]^2*c[7]*c[4]^2-2598*c[6]*c[5]^2*c[7]*c[4]^3-698*c[6]*c [5]^3*c[7]*c[4]^2-300*c[6]^2*c[5]^3*c[7]*c[4]+1818*c[5]^4*c[4]^3*c[7]- 692*c[7]^2*c[5]^2*c[4]^3+3160*c[5]^4*c[7]^2*c[4]^4+12*c[7]^2*c[5]*c[4] ^2-108*c[7]*c[5]*c[4]^3-120*c[6]*c[5]*c[4]^3+29*c[7]^2*c[5]^2*c[6]*c[4 ]+114*c[6]*c[7]*c[4]^4+104*c[4]^5*c[5]+200*c[5]^4*c[4]^6-1200*c[5]^5*c [4]^5*c[7]^2*c[6]+6*c[6]^2*c[7]^2*c[4]^2-52*c[5]^3*c[6]*c[4]-46*c[5]^3 *c[7]*c[4]-450*c[5]^5*c[4]^3*c[7]^2*c[6]-8*c[5]^2*c[4]+60*c[5]^3*c[6]* c[7]*c[4]-160*c[5]^3*c[4]^2-21*c[6]^2*c[7]^2*c[4]*c[5]+1144*c[5]^2*c[6 ]*c[4]^3-108*c[5]^2*c[7]*c[4]^2-120*c[5]^2*c[6]*c[4]^2+1024*c[5]^2*c[7 ]*c[4]^3+372*c[5]^3*c[7]*c[4]^2+424*c[5]^3*c[6]*c[4]^2+1752*c[5]^3*c[6 ]^2*c[4]^3-660*c[5]^5*c[4]^3*c[6]^2*c[7]-1153*c[4]^3*c[7]^2*c[5]^2*c[6 ]^2-72*c[6]*c[7]*c[4]^5+3160*c[4]^5*c[5]^3*c[6]^2+28*c[7]*c[5]^4*c[4]- 20*c[7]^2*c[5]^4*c[4]-2472*c[5]^2*c[4]^4*c[7]-692*c[6]^2*c[5]^2*c[4]^3 +72*c[6]^2*c[5]^2*c[4]^2-2816*c[6]*c[5]^3*c[4]^3+32*c[6]^2*c[5]^3*c[4] -264*c[6]^2*c[5]^3*c[4]^2-2472*c[5]^3*c[7]*c[4]^3+566*c[5]^4*c[7]*c[4] ^2*c[6]+48*c[5]^2*c[4]^2+1200*c[6]^2*c[4]^5*c[5]^5*c[7]^2-12*c[6]^2*c[ 4]*c[5]^2+38*c[7]*c[6]^2*c[4]^3+1804*c[7]^2*c[5]^2*c[6]*c[4]^3+258*c[7 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40*c[6]^2*c[7]^2*c[4]^5+1752*c[6]^2*c[4]^4*c[5]^2-1700*c[7]^2*c[6]^2*c [5]^5*c[4]^4-600*c[5]^5*c[4]^5*c[6]-120*c[5]*c[7]*c[4]^6*c[6]^2+50*c[6 ]^2*c[7]*c[4]^5-52*c[6]*c[4]^4-1200*c[5]^4*c[7]*c[4]^6*c[6]^2+7150*c[5 ]^4*c[4]^5*c[6]^2*c[7]+198*c[7]*c[6]^2*c[5]*c[4]^4+10560*c[5]^3*c[4]^4 *c[6]^2*c[7]+5240*c[5]^3*c[4]^3*c[6]^2*c[7]^2+11240*c[5]^4*c[4]^4*c[6] *c[7]-3649*c[5]^2*c[4]^4*c[7]*c[6]^2-14207*c[5]^3*c[4]^4*c[6]*c[7]-449 1*c[5]^4*c[4]^3*c[6]*c[7]+1064*c[5]^3*c[4]^3-264*c[7]^2*c[5]^3*c[4]^2- 2400*c[5]^4*c[4]^5*c[6]^2+6020*c[6]^2*c[7]^2*c[5]^3*c[4]^5+6800*c[6]*c [7]^2*c[5]^4*c[4]^5+3160*c[5]^4*c[4]^4*c[6]^2-1320*c[5]^4*c[4]^3*c[6]^ 2-5800*c[7]^2*c[5]^4*c[6]^2*c[4]^3+1940*c[7]^2*c[5]^4*c[6]^2*c[4]^2-60 0*c[5]^4*c[4]^6*c[6]+28*c[4]^5*c[7]+9740*c[5]^3*c[4]^4*c[7]^2*c[6]+292 0*c[5]^4*c[4]^3*c[7]^2*c[6]-4775*c[5]^2*c[4]^4*c[7]^2*c[6]+4980*c[5]^4 *c[4]^3*c[6]^2*c[7]+210*c[7]^2*c[4]^4*c[6]^2*c[5]-8860*c[7]^2*c[4]^4*c [5]^3*c[6]^2+2140*c[7]^2*c[4]^4*c[6]^2*c[5]^2+120*c[4]^6*c[5]^2-20*c[7 ]^2*c[4]^5+2092*c[5]^2*c[6]*c[4]^5+9920*c[6]^2*c[7]^2*c[5]^4*c[4]^4-50 0*c[5]^4*c[4]^6*c[7]-48*c[7]^2*c[6]^2*c[5]^3+6404*c[4]^4*c[6]*c[5]^3-2 84*c[6]*c[5]*c[4]^5+200*c[5]^5*c[6]^2*c[4]^3+200*c[6]^2*c[4]^6*c[5]*c[ 7]^2-8820*c[5]^4*c[4]^5*c[6]*c[7]-12*c[5]^4*c[4]-2324*c[5]^3*c[4]^4)/c [6]/c[5]/c[7]/c[4]^2/(150*c[5]^2*c[6]*c[4]^4-87*c[5]^3*c[4]-12*c[5]*c[ 7]*c[4]-200*c[5]^3*c[4]^4*c[7]-200*c[6]*c[5]^2*c[4]^4*c[7]-200*c[5]^4* c[4]^3*c[6]-10*c[5]^2-10*c[4]^2+12*c[5]^3+12*c[4]^3+20*c[7]*c[6]*c[5]^ 3+15*c[5]*c[6]*c[7]*c[4]+12*c[6]*c[5]^2-15*c[6]*c[5]^3-1300*c[6]*c[7]* c[5]^3*c[4]^3+150*c[5]^4*c[7]*c[4]^2+12*c[7]*c[5]^2+20*c[7]*c[6]*c[4]^ 3+12*c[7]*c[4]^2-15*c[7]*c[4]^3+12*c[6]*c[4]^2-15*c[6]*c[4]^3-15*c[6]* c[7]*c[4]^2-87*c[5]*c[4]^3-72*c[5]*c[7]*c[4]^2-72*c[5]*c[6]*c[4]^2+60* c[5]*c[4]^2-12*c[5]*c[6]*c[4]+10*c[5]*c[4]-150*c[5]*c[7]*c[6]*c[4]^3-1 5*c[7]*c[5]^3-660*c[5]^2*c[6]*c[7]*c[4]^2+90*c[6]*c[5]*c[7]*c[4]^2-15* c[6]*c[7]*c[5]^2+150*c[6]*c[5]^4*c[4]^2+930*c[6]*c[5]^2*c[7]*c[4]^3+93 0*c[6]*c[5]^3*c[7]*c[4]^2-200*c[5]^4*c[4]^3*c[7]+110*c[7]*c[5]*c[4]^3+ 110*c[6]*c[5]*c[4]^3+110*c[5]^3*c[6]*c[4]+110*c[5]^3*c[7]*c[4]+60*c[5] ^2*c[4]-150*c[5]^3*c[6]*c[7]*c[4]+550*c[5]^3*c[4]^2-690*c[5]^2*c[6]*c[ 4]^3+520*c[5]^2*c[7]*c[4]^2+520*c[5]^2*c[6]*c[4]^2-690*c[5]^2*c[7]*c[4 ]^3-690*c[5]^3*c[7]*c[4]^2-690*c[5]^3*c[6]*c[4]^2+150*c[5]^2*c[4]^4*c[ 7]+900*c[6]*c[5]^3*c[4]^3+900*c[5]^3*c[7]*c[4]^3-200*c[5]^4*c[7]*c[4]^ 2*c[6]-429*c[5]^2*c[4]^2-120*c[5]^4*c[4]^2-120*c[4]^4*c[5]^2+150*c[5]^ 4*c[4]^3-72*c[5]^2*c[6]*c[4]-72*c[5]^2*c[7]*c[4]+90*c[5]^2*c[6]*c[7]*c [4]+550*c[5]^2*c[4]^3+300*c[5]^3*c[4]^4*c[6]*c[7]+300*c[5]^4*c[4]^3*c[ 6]*c[7]-690*c[5]^3*c[4]^3-200*c[4]^4*c[6]*c[5]^3+150*c[5]^3*c[4]^4), a [8,3] = 3/4*(-1560*c[5]^2*c[6]*c[4]^4+585*c[6]*c[5]*c[4]^4-45*c[7]*c[5 ]*c[4]^4-12*c[5]^3*c[4]-450*c[4]^5*c[6]*c[5]^3+3*c[5]*c[7]*c[4]-150*c[ 5]^2*c[4]^5*c[7]-200*c[5]^3*c[4]^4*c[7]+300*c[6]*c[4]^5*c[5]*c[7]+2860 *c[6]*c[5]^2*c[4]^4*c[7]+160*c[5]^4*c[4]^3*c[6]-150*c[5]^4*c[4]^4*c[6] -4*c[5]^2+2*c[4]^2-40*c[7]*c[6]*c[5]^3+132*c[5]*c[6]*c[7]*c[4]-900*c[6 ]*c[5]^2*c[4]^5*c[7]-30*c[6]*c[5]^2+12*c[6]*c[5]-960*c[7]*c[4]^4*c[6]* c[5]+24*c[6]*c[5]^3+3140*c[6]*c[7]*c[5]^3*c[4]^3+6*c[7]*c[5]^2+20*c[7] *c[6]*c[4]^3-3*c[7]*c[4]^2-18*c[6]*c[7]*c[5]+15*c[6]*c[4]^2-12*c[6]*c[ 4]^3+9*c[6]*c[7]*c[4]-24*c[6]*c[7]*c[4]^2-54*c[5]*c[4]^3-12*c[5]*c[7]* c[4]^2+348*c[5]*c[6]*c[4]^2+9*c[5]*c[4]^2-87*c[5]*c[6]*c[4]-2*c[5]*c[4 ]-6*c[6]*c[4]+1035*c[5]*c[7]*c[6]*c[4]^3+1593*c[5]^2*c[6]*c[7]*c[4]^2+ 900*c[4]^5*c[6]*c[7]*c[5]^3-534*c[6]*c[5]*c[7]*c[4]^2+48*c[6]*c[7]*c[5 ]^2-60*c[6]*c[5]^4*c[4]^2-3060*c[6]*c[5]^2*c[7]*c[4]^3-1560*c[6]*c[5]^ 3*c[7]*c[4]^2-50*c[5]^4*c[4]^3*c[7]+80*c[7]*c[5]*c[4]^3-652*c[6]*c[5]* c[4]^3-208*c[5]^3*c[6]*c[4]+20*c[5]^3*c[7]*c[4]+54*c[5]^2*c[4]+360*c[5 ]^3*c[6]*c[7]*c[4]+82*c[5]^3*c[4]^2+1734*c[5]^2*c[6]*c[4]^3+388*c[5]^2 *c[7]*c[4]^2-944*c[5]^2*c[6]*c[4]^2-900*c[5]^2*c[7]*c[4]^3-135*c[5]^3* c[7]*c[4]^2+879*c[5]^3*c[6]*c[4]^2+660*c[5]^2*c[4]^4*c[7]-1680*c[6]*c[ 5]^3*c[4]^3+420*c[5]^3*c[7]*c[4]^3+100*c[5]^4*c[7]*c[4]^2*c[6]-246*c[5 ]^2*c[4]^2-405*c[4]^4*c[5]^2+30*c[5]^4*c[4]^3+240*c[5]^2*c[6]*c[4]-84* c[5]^2*c[7]*c[4]-396*c[5]^2*c[6]*c[7]*c[4]+562*c[5]^2*c[4]^3+30*c[4]^4 *c[5]+90*c[4]^5*c[5]^2+300*c[5]^4*c[4]^4*c[6]*c[7]-3000*c[5]^3*c[4]^4* c[6]*c[7]-300*c[5]^4*c[4]^3*c[6]*c[7]-255*c[5]^3*c[4]^3+480*c[5]^2*c[6 ]*c[4]^5+1540*c[4]^4*c[6]*c[5]^3-180*c[6]*c[5]*c[4]^5+120*c[5]^3*c[4]^ 4)/c[4]^2/(150*c[5]^2*c[6]*c[4]^4-87*c[5]^3*c[4]-12*c[5]*c[7]*c[4]-200 *c[5]^3*c[4]^4*c[7]-200*c[6]*c[5]^2*c[4]^4*c[7]-200*c[5]^4*c[4]^3*c[6] -10*c[5]^2-10*c[4]^2+12*c[5]^3+12*c[4]^3+20*c[7]*c[6]*c[5]^3+15*c[5]*c [6]*c[7]*c[4]+12*c[6]*c[5]^2-15*c[6]*c[5]^3-1300*c[6]*c[7]*c[5]^3*c[4] ^3+150*c[5]^4*c[7]*c[4]^2+12*c[7]*c[5]^2+20*c[7]*c[6]*c[4]^3+12*c[7]*c [4]^2-15*c[7]*c[4]^3+12*c[6]*c[4]^2-15*c[6]*c[4]^3-15*c[6]*c[7]*c[4]^2 -87*c[5]*c[4]^3-72*c[5]*c[7]*c[4]^2-72*c[5]*c[6]*c[4]^2+60*c[5]*c[4]^2 -12*c[5]*c[6]*c[4]+10*c[5]*c[4]-150*c[5]*c[7]*c[6]*c[4]^3-15*c[7]*c[5] ^3-660*c[5]^2*c[6]*c[7]*c[4]^2+90*c[6]*c[5]*c[7]*c[4]^2-15*c[6]*c[7]*c [5]^2+150*c[6]*c[5]^4*c[4]^2+930*c[6]*c[5]^2*c[7]*c[4]^3+930*c[6]*c[5] ^3*c[7]*c[4]^2-200*c[5]^4*c[4]^3*c[7]+110*c[7]*c[5]*c[4]^3+110*c[6]*c[ 5]*c[4]^3+110*c[5]^3*c[6]*c[4]+110*c[5]^3*c[7]*c[4]+60*c[5]^2*c[4]-150 *c[5]^3*c[6]*c[7]*c[4]+550*c[5]^3*c[4]^2-690*c[5]^2*c[6]*c[4]^3+520*c[ 5]^2*c[7]*c[4]^2+520*c[5]^2*c[6]*c[4]^2-690*c[5]^2*c[7]*c[4]^3-690*c[5 ]^3*c[7]*c[4]^2-690*c[5]^3*c[6]*c[4]^2+150*c[5]^2*c[4]^4*c[7]+900*c[6] *c[5]^3*c[4]^3+900*c[5]^3*c[7]*c[4]^3-200*c[5]^4*c[7]*c[4]^2*c[6]-429* c[5]^2*c[4]^2-120*c[5]^4*c[4]^2-120*c[4]^4*c[5]^2+150*c[5]^4*c[4]^3-72 *c[5]^2*c[6]*c[4]-72*c[5]^2*c[7]*c[4]+90*c[5]^2*c[6]*c[7]*c[4]+550*c[5 ]^2*c[4]^3+300*c[5]^3*c[4]^4*c[6]*c[7]+300*c[5]^4*c[4]^3*c[6]*c[7]-690 *c[5]^3*c[4]^3-200*c[4]^4*c[6]*c[5]^3+150*c[5]^3*c[4]^4), a[8,5] = -1/ 2*(893*c[5]^2*c[6]*c[4]^4-122*c[6]*c[5]*c[4]^4+850*c[5]^4*c[4]^4*c[6]^ 2*c[7]-101*c[7]*c[5]*c[4]^4-535*c[5]^2*c[4]^4*c[7]^2-44*c[5]^3*c[4]+16 0*c[4]^5*c[6]^2*c[5]^2+100*c[7]^2*c[5]^2*c[4]^5+170*c[4]^5*c[6]*c[5]^3 +75*c[5]*c[6]^2*c[4]^4+16*c[7]^2*c[4]^3-9*c[5]*c[7]*c[4]-140*c[5]^2*c[ 4]^5*c[7]-780*c[5]^3*c[4]^4*c[7]-10*c[4]^4*c[6]^2-100*c[7]^2*c[4]^5*c[ 5]^3+450*c[6]^2*c[4]^5*c[7]*c[5]^3+510*c[5]^4*c[4]^2*c[7]^2*c[6]+490*c [6]^2*c[7]*c[5]^4*c[4]^2-2224*c[6]*c[5]^2*c[4]^4*c[7]+10*c[5]^5*c[4]^3 *c[7]-146*c[5]^4*c[4]^3*c[6]+90*c[5]^4*c[4]^5*c[6]+40*c[5]^4*c[4]^4*c[ 6]+570*c[7]^2*c[4]^4*c[5]^3+130*c[5]^4*c[4]^4*c[7]+25*c[7]^2*c[6]*c[4] ^4+18*c[5]^5*c[4]^2-4*c[5]^2-4*c[4]^2-10*c[4]^4*c[7]^2+150*c[7]^2*c[5] ^4*c[4]^3+5*c[5]^3-6*c[6]^2*c[7]^2*c[5]+10*c[4]^3-35*c[5]^5*c[4]^3-24* c[7]*c[6]^2*c[5]^3+27*c[7]*c[6]*c[5]^3-32*c[7]^2*c[6]*c[5]^3+23*c[5]*c [6]*c[7]*c[4]-c[5]^4-6*c[4]^4-60*c[7]*c[6]^2*c[5]*c[4]^5-45*c[5]^4*c[4 ]^2*c[7]^2+450*c[6]*c[5]^2*c[4]^5*c[7]+10*c[6]*c[5]^2-80*c[4]^5*c[5]^3 -260*c[5]^5*c[4]^3*c[6]+270*c[7]*c[4]^4*c[6]*c[5]+20*c[5]^5*c[4]^4-9*c [6]*c[5]^3+50*c[4]^4*c[6]^2*c[5]^5-600*c[5]^5*c[7]*c[4]^4*c[6]-760*c[7 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5*c[4]^3*c[7]^2*c[6]-750*c[5]^5*c[4]^2*c[6]*c[7]-96*c[5]^3*c[6]*c[7]*c [4]+200*c[6]*c[5]^7*c[4]^3+20*c[5]^3*c[4]^2-36*c[5]^2*c[6]*c[4]^3-20*c [5]^2*c[7]*c[4]^2-20*c[5]^2*c[6]*c[4]^2-36*c[5]^2*c[7]*c[4]^3-24*c[5]^ 3*c[7]*c[4]^2-24*c[5]^3*c[6]*c[4]^2-410*c[5]^3*c[6]^2*c[4]^3-900*c[5]^ 5*c[4]^3*c[6]^2*c[7]-510*c[4]^3*c[7]^2*c[5]^2*c[6]^2-630*c[7]*c[5]^6*c [6]*c[4]^2-150*c[4]^5*c[5]^3*c[6]^2+72*c[7]*c[5]^4*c[4]-57*c[7]^2*c[5] ^4*c[4]+72*c[5]^2*c[4]^4*c[7]+57*c[6]^2*c[5]^2*c[4]^3+24*c[6]^2*c[5]^2 *c[4]^2+285*c[6]*c[5]^3*c[4]^3-24*c[6]^2*c[5]^3*c[4]+285*c[5]^3*c[7]*c [4]^3+270*c[5]^4*c[7]*c[4]^2*c[6]+12*c[7]*c[6]^2*c[4]^3+340*c[7]^2*c[5 ]^2*c[6]*c[4]^3-410*c[7]^2*c[5]^3*c[6]*c[4]^2-200*c[5]^5*c[4]^5*c[7]+7 0*c[4]^3*c[7]^2*c[5]*c[6]^2-42*c[7]^2*c[6]*c[5]*c[4]^3+150*c[5]^5*c[4] ^5-30*c[7]^2*c[5]^2*c[6]*c[4]^2+12*c[4]^4*c[5]^2+87*c[7]^2*c[5]^3*c[6] *c[4]+510*c[7]^2*c[5]^3*c[6]*c[4]^3+690*c[5]^3*c[6]^2*c[4]^4-342*c[5]^ 4*c[4]^3-15*c[6]^2*c[4]^3*c[7]^2+510*c[5]^3*c[6]^2*c[7]*c[4]^3-20*c[5] ^2*c[6]*c[7]*c[4]-410*c[6]^2*c[7]*c[5]^3*c[4]^2-70*c[6]^2*c[7]^2*c[4]* c[5]^3-10*c[5]^2*c[4]^3+24*c[7]^2*c[5]^2*c[4]^2-24*c[7]^2*c[5]^3*c[4]- 200*c[7]^2*c[5]^6*c[4]^2*c[6]+110*c[7]^2*c[4]*c[5]^5+1350*c[5]^5*c[4]^ 3*c[6]*c[7]+15*c[7]^2*c[4]^4*c[5]+300*c[6]^2*c[7]*c[5]^6*c[4]^3+510*c[ 7]^2*c[5]^3*c[6]^2*c[4]^2-550*c[5]^6*c[4]^2-600*c[7]*c[5]^6*c[4]^3+20* c[7]*c[6]^2*c[5]^5+570*c[5]^6*c[6]*c[4]^2+150*c[5]^2*c[6]*c[4]^5*c[7]^ 2+550*c[5]^4*c[4]^4+120*c[5]^7*c[4]^2+72*c[5]^4*c[4]*c[6]-40*c[5]^4*c[ 7]^2*c[4]*c[6]-57*c[5]^4*c[4]*c[7]*c[6]+150*c[6]^2*c[5]^6*c[4]^2+150*c [4]^5*c[6]^2*c[7]*c[5]^2-200*c[4]^5*c[6]^2*c[7]^2*c[5]^2-150*c[5]^7*c[ 7]*c[4]^2-110*c[6]^2*c[4]^4*c[5]^2+780*c[5]^5*c[4]^2*c[7]^2*c[6]-200*c [5]^5*c[4]^5*c[6]-600*c[5]^6*c[4]^3*c[6]-200*c[6]^2*c[7]*c[5]^6*c[4]^2 -300*c[5]^4*c[4]^5*c[6]^2*c[7]+90*c[7]*c[6]^2*c[5]*c[4]^4+200*c[6]*c[5 ]^7*c[7]*c[4]^2-300*c[6]*c[5]^7*c[7]*c[4]^3-180*c[5]^3*c[4]^4*c[6]^2*c [7]-570*c[5]^4*c[4]^4*c[6]*c[7]-540*c[5]^2*c[4]^4*c[7]*c[6]^2+660*c[5] ^3*c[4]^4*c[6]*c[7]+60*c[5]^4*c[4]^3*c[6]*c[7]+48*c[5]^3*c[4]^3-15*c[6 ]^2*c[5]^5-200*c[5]^6*c[6]^2*c[4]^3-15*c[5]^5*c[6]*c[7]-48*c[5]^5*c[4] +140*c[5]^5*c[6]*c[4]^2+200*c[5]^4*c[4]^5*c[6]^2+300*c[6]^2*c[7]^2*c[5 ]^3*c[4]^5-300*c[6]*c[7]^2*c[5]^4*c[4]^5+700*c[7]*c[5]^6*c[4]^3*c[6]-7 50*c[5]^4*c[4]^4*c[6]^2+20*c[5]^5*c[7]^2*c[6]+1100*c[7]^2*c[5]^4*c[6]^ 2*c[4]^3-930*c[7]^2*c[5]^4*c[6]^2*c[4]^2+300*c[7]^2*c[5]^6*c[4]^3*c[6] -20*c[5]^6*c[7]*c[6]-180*c[5]^3*c[4]^4*c[7]^2*c[6]-750*c[5]^4*c[4]^3*c [7]^2*c[6]-540*c[5]^2*c[4]^4*c[7]^2*c[6]-750*c[5]^4*c[4]^3*c[6]^2*c[7] -150*c[7]^2*c[4]^4*c[6]^2*c[5]-1100*c[7]^2*c[4]^4*c[5]^3*c[6]^2+930*c[ 7]^2*c[4]^4*c[6]^2*c[5]^2+150*c[5]^6*c[4]*c[7]*c[6]-110*c[5]^6*c[4]*c[ 6]-150*c[5]^7*c[6]*c[4]^2+570*c[5]^6*c[4]^3-150*c[5]^7*c[4]^3+15*c[7]^ 2*c[6]^2*c[5]^3-440*c[4]^4*c[6]*c[5]^3+750*c[5]^5*c[6]^2*c[4]^3-200*c[ 7]^2*c[5]^6*c[4]^3+12*c[5]^4*c[7]^2+200*c[5]^4*c[4]^5*c[6]*c[7]-20*c[5 ]^4*c[4]-87*c[5]^3*c[4]^4-30*c[5]^5*c[4]*c[6]+110*c[5]^5*c[6]^2*c[4]), a[9,7] = -1/60*(10*c[5]*c[6]*c[4]-5*c[6]*c[5]-5*c[6]*c[4]+3*c[6]-5*c[ 5]*c[4]+3*c[5]+3*c[4]-2)/c[7]/(-c[5]*c[6]*c[4]+c[5]*c[6]*c[7]*c[4]+c[5 ]*c[7]*c[4]-c[7]^2*c[5]*c[4]+c[6]*c[7]*c[4]-c[6]*c[7]^2*c[4]-c[7]^2*c[ 4]+c[7]^3*c[4]+c[6]*c[7]*c[5]-c[6]*c[7]^2*c[5]-c[7]^2*c[5]+c[7]^3*c[5] -c[7]^2*c[6]+c[7]^3*c[6]+c[7]^3-c[7]^4), a[9,2] = 0, a[9,3] = 0, a[9,5 ] = -1/60*(10*c[6]*c[7]*c[4]-5*c[6]*c[4]-5*c[7]*c[6]+3*c[6]-5*c[7]*c[4 ]+3*c[4]-2+3*c[7])/c[5]/(c[5]^3*c[4]-c[5]^2*c[4]+c[5]*c[7]*c[4]-c[5]^2 *c[7]*c[4]+c[5]*c[6]*c[4]-c[5]^2*c[6]*c[4]-c[6]*c[7]*c[4]+c[5]*c[6]*c[ 7]*c[4]-c[5]^4+c[5]^3-c[7]*c[5]^2+c[7]*c[5]^3-c[6]*c[5]^2+c[6]*c[5]^3+ c[6]*c[7]*c[5]-c[6]*c[7]*c[5]^2), a[9,8] = 1/60*(-20*c[5]*c[6]*c[4]+30 *c[5]*c[6]*c[7]*c[4]+15*c[6]*c[5]-20*c[6]*c[7]*c[5]+15*c[6]*c[4]-20*c[ 6]*c[7]*c[4]-12*c[6]+15*c[7]*c[6]+15*c[5]*c[4]-20*c[5]*c[7]*c[4]-12*c[ 5]+15*c[7]*c[5]-12*c[4]+15*c[7]*c[4]+10-12*c[7])/(-c[4]+c[7]*c[4]+c[6] *c[4]-c[6]*c[7]*c[4]+c[5]*c[4]-c[5]*c[7]*c[4]-c[5]*c[6]*c[4]+c[5]*c[6] *c[7]*c[4]+1-c[7]-c[6]+c[7]*c[6]-c[5]+c[7]*c[5]+c[6]*c[5]-c[6]*c[7]*c[ 5]), a[9,1] = 1/60*(30*c[5]*c[6]*c[7]*c[4]-10*c[5]*c[6]*c[4]-10*c[6]*c [7]*c[5]+5*c[6]*c[5]-10*c[6]*c[7]*c[4]+5*c[6]*c[4]+5*c[7]*c[6]-3*c[6]- 10*c[5]*c[7]*c[4]+5*c[5]*c[4]+5*c[7]*c[5]-3*c[5]+5*c[7]*c[4]-3*c[4]+2- 3*c[7])/c[5]/c[6]/c[7]/c[4], a[9,4] = 1/60*(-2+3*c[5]-5*c[7]*c[6]+3*c[ 7]+3*c[6]+10*c[6]*c[7]*c[5]-5*c[6]*c[5]-5*c[7]*c[5])/c[4]/(-c[6]*c[4]^ 3-c[4]^3+c[4]^4+c[7]*c[4]^2+c[6]*c[4]^2-c[6]*c[7]*c[4]+c[6]*c[7]*c[4]^ 2-c[5]*c[4]^3+c[5]*c[4]^2-c[7]*c[4]^3-c[5]*c[7]*c[4]+c[5]*c[7]*c[4]^2- c[5]*c[6]*c[4]+c[5]*c[6]*c[4]^2+c[6]*c[7]*c[5]-c[5]*c[6]*c[7]*c[4]), a [9,6] = 1/60*(10*c[5]*c[7]*c[4]-5*c[5]*c[4]-5*c[7]*c[5]+3*c[5]-5*c[7]* c[4]+3*c[4]-2+3*c[7])/c[6]/(c[6]^2*c[4]*c[5]-c[5]*c[6]*c[4]+c[5]*c[7]* c[4]-c[5]*c[6]*c[7]*c[4]-c[6]^3*c[4]+c[6]^2*c[4]-c[6]*c[7]*c[4]+c[6]^2 *c[7]*c[4]-c[6]^3*c[5]+c[6]^2*c[5]-c[6]*c[7]*c[5]+c[6]^2*c[7]*c[5]+c[6 ]^4-c[6]^3+c[7]*c[6]^2-c[6]^3*c[7])\}:" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 5 "Let " }{XPPEDIT 18 0 "t[7 ]" "6#&%\"tG6#\"\"(" }{TEXT -1 7 " and " }{XPPEDIT 18 0 "t[8]" "6#&% \"tG6#\"\")" }{TEXT -1 29 " denote the coefficients of " }{XPPEDIT 18 0 "z^7" "6#*$%\"zG\"\"(" }{TEXT -1 7 " and " }{XPPEDIT 18 0 "z^8 " "6#*$%\"zG\"\")" }{TEXT -1 43 " respectively in the stability polyn omial." }}{PARA 0 "" 0 "" {TEXT -1 29 "Following Tsitouras, we set " }{XPPEDIT 18 0 "t[7] = 1/7!;" "6#/&%\"tG6#\"\"(*&\"\"\"F)-%*factorialG 6#F'!\"\"" }{TEXT -1 7 " and " }{XPPEDIT 18 0 "t[8] = 1/8!;" "6#/&% \"tG6#\"\")*&\"\"\"F)-%*factorialG6#F'!\"\"" }{TEXT -1 37 " to obtain two equations involving " }{XPPEDIT 18 0 "c[4]" "6#&%\"cG6#\"\"%" } {TEXT -1 3 ", " }{XPPEDIT 18 0 "c[5]" "6#&%\"cG6#\"\"&" }{TEXT -1 7 " and " }{XPPEDIT 18 0 "c[6]" "6#&%\"cG6#\"\"'" }{TEXT -1 2 ". " }} {PARA 0 "" 0 "" {TEXT -1 52 "Note that the stability polynomial only d epends on " }{XPPEDIT 18 0 "c[4]" "6#&%\"cG6#\"\"%" }{TEXT -1 3 ", \+ " }{XPPEDIT 18 0 "c[5]" "6#&%\"cG6#\"\"&" }{TEXT -1 7 " and " } {XPPEDIT 18 0 "c[6]" "6#&%\"cG6#\"\"'" }{TEXT -1 2 ". " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 136 "Rz := \+ StabilityFunction(6,8,'expanded'):\neqA := simplify(subs(eG,coeff(Rz,z ^7)))=1/7!:%;\neqB := simplify(subs(eG,coeff(Rz,z^8)))=1/8!:%;" }} {PARA 12 "" 1 "" {XPPMATH 20 "6#/,$*&#\"\"\"\"%S9F'*&,hn*(\"\"#F'&%\"c G6#\"\"'F')&F.6#\"\"%F,F'F'*(F4F'F-F')&F.6#\"\"&\"\"$F'!\"\"**\"#!)F'F -F'F7F')F2F9F'F;**\"$g\"F')F7F,F'F-F')F2F:F'F'**\"#>F'FAF'F-F'F1F'F;** \"$S$F'FAF'F-F')F2F4F'F;**F9F'F7F'F-F'F2F'F;*(\"#SF'F>F'FAF'F'**\"#)*F 'F-F'F7F'FBF'F;**\"#MF'F6F'F-F'F2F'F'**\"$?\"F'F6F'F-F'F1F'F;*(FJF'F6F 'FGF'F'**\"$!>F'F-F'F6F'FBF'F'**\"$+#F'FAF'F-F'F>F'F'*(\"#?F'F6F'FBF'F ;*(\"#9F'FGF'F7F'F'*(F,F'F-F'FAF'F'*(F4F'F-F'FBF'F;*(\"#5F'F7F'FBF'F;* (FWF'FAF'F1F'F;**\"$+\"F'FGF'F-F'F6F'F;**\"#IF'F7F'F-F'F1F'F'*(F,F'F6F 'F2F'F'*&FAF'F2F'F'*(F9F'F6F'F1F'F;**F@F'F-F'F7F'FGF'F'*(\"$S\"F'FGF'F AF'F;*(\"#**F'FAF'FBF'F'*&F7F'F1F'F'**F0F'FAF'F-F'F2F'F;F',2*(FgnF'F6F 'F1F'F'*(FgnF'FAF'FBF'F'*(F\\oF'FAF'F1F'F;*(F0F'FAF'F2F'F'*$FAF'F;*(F0 F'F7F'F1F'F'*&F7F'F2F'F'*$F1F'F;F;F'F;#F'\"%S]" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/,$*&#\"\"\"\"%S9F'*,&%\"cG6#\"\"%F'&F+6#\"\"&F',0**\"# IF'F.F'&F+6#\"\"'F')F*\"\"#F'!\"\"**\"#7F'F.F'F4F'F*F'F'**\"#?F'F4F'F. F')F*\"\"$F'F'*(F8F'F4F'F.F'F9*&F.F'F*F'F'*$F7F'F9*&F4F'F*F'F'F',**(F0 F'F.F'F*F'F'F'F'*&F8F'F*F'F9*&F8F'F.F'F9F',2*(\"#5F')F.F?F'F7F'F'*(FJF ')F.F8F'F>F'F'*(F3F'FMF'F7F'F9*(F6F'FMF'F*F'F'*$FMF'F9*(F6F'F.F'F7F'F' FAF'FBF9F9F'F'#F'\"&?.%" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 " " 0 "" {TEXT -1 55 "We can solve the two equations to obtain formulas \+ for " }{XPPEDIT 18 0 "c[5]" "6#&%\"cG6#\"\"&" }{TEXT -1 7 " and " } {XPPEDIT 18 0 "c[6]" "6#&%\"cG6#\"\"'" }{TEXT -1 15 " in terms of " }{XPPEDIT 18 0 "c[4]" "6#&%\"cG6#\"\"%" }{TEXT -1 1 "." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 96 "sol := \+ solve(\{eqA,eqB\},\{c[5],c[6]\}):\nc[5]=subs(sol,c[5]);eqC := %:\nc[6] =subs(sol,c[6]);eqD := %:" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/&%\"cG6# \"\"&,$*(\"\"#\"\"\"&F%6#\"\"%F+,**&\"$7\"F+)F,\"\"$F+F+*&\"#cF+)F,F*F +!\"\"*&\"\")F+F,F+F+F+F7F7F7" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/&%\" cG6#\"\"',$*&#\"\"\"\"#cF+**&F%6#\"\"%!\"\",2\"\"$F1*&\"#gF+F.F+F+*&\" 'c9@F+)F.F'F+F1*&\"'g\")=F+)F.\"\"(F+F+*&\"$#zF+)F.\"\"#F+F1*&\"%;mF+) F.F3F+F+*&\"'g(4\"F+)F.\"\"&F+F+*&\"&KS$F+)F.F0F+F1F+,,*&\"##)F+F?F+F+ *&\"$C#F+FCF+F1*&\"#9F+F.F+F1F+F+*&FOF+FJF+F+F1,**&F0F+F?F+F+*&\"#;F+F .F+F1*&\"#sF+FCF+F+F3F+F1F+F1" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }} {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 24 "The follo wing procedure " }{TEXT 0 13 "prin_err_norm" }{TEXT -1 54 " calculates the principal error norm given the nodes " }{XPPEDIT 18 0 "c[2]" "6# &%\"cG6#\"\"#" }{TEXT -1 3 ", " }{XPPEDIT 18 0 "c[4]" "6#&%\"cG6#\"\" %" }{TEXT -1 7 " and " }{XPPEDIT 18 0 "c[7]" "6#&%\"cG6#\"\"(" } {TEXT -1 87 " by using the general solution together with the precedi ng two equations to determine " }{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 1 ";" }}}{SECT 1 {PARA 4 "" 0 "" {TEXT 0 13 "prin_er r_norm" }{TEXT -1 1 " " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 51 "errterms6_8 := PrincipalErrorTerms( 6,8,'expanded'):" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 695 "prin_err_norm := proc(c2,c4,c7)\n local c2 r,c4r,c5r,c6r,c7r,e1,sm,ct;\n global e2;\n \n c2r := convert(c2,r ational,Digits);\n c4r := convert(c4,rational,Digits); \n c5r := - 2*c4r/(112*c4r^3-56*c4r^2+8*c4r-1);\n c6r := -1/56*1/c4r*(-3+60*c4r+ 109760*c4r^5-792*c4r^2+6616*c4r^3-34032*c4r^4+188160*c4r^7-211456*c4r^ 6)/\n (82*c4r^2-224*c4r^3-14*c4r+1+224*c4r^4)/(3+72*c4r^3+4*c4r^ 2-16*c4r);\n c7r := convert(c7,rational,Digits); \n\n e1 := \{c[2] =c2r,c[4]=c4r,c[5]=c5r,c[6]=c6r,c[7]=c7r,c[8]=1,c[9]=1\};\n\n e2 := \+ `union`(e1,simplify(subs(e1,eG))):\n sm := 0;\n for ct to nops(err terms6_8) do\n sm := sm+(simplify(subs(e2,errterms6_8[ct])))^2;\n end do;\n evalf(sqrt(sm));\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 29 "We can determine val ues for " }{XPPEDIT 18 0 "c[2]" "6#&%\"cG6#\"\"#" }{TEXT -1 3 ", " } {XPPEDIT 18 0 "c[4]" "6#&%\"cG6#\"\"%" }{TEXT -1 7 " and " } {XPPEDIT 18 0 "c[7]" "6#&%\"cG6#\"\"(" }{TEXT -1 121 " that make the \+ principal error norm a minimum by a simple cycling method using a one \+ dimensional minimization procedure." }}{PARA 0 "" 0 "" {TEXT -1 0 "" } }{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 379 "Digits := 15:\nc_2 := .1: \+ c_4 := .275: c_7 := .12:\nfor ii to 40 do \n c_2 := op(1,findmin('pr in_err_norm'(c2,c_4,c_7),c2=\{0.09,c_2,0.14\}));\n c_4 := op(1,findm in('prin_err_norm'(c_2,c4,c_7),c4=\{0.26,c_4,0.29\}));\n mn := findm in('prin_err_norm'(c_2,c_4,c7),c7=\{0.07,c_7,0.13\},convergence=locati on);\n c_7 := mn[1]:\n print(c[2]=c_2,c[4]=c_4,c[7]=c_7);\n prin t(mn[2]);\nend do:" }}{PARA 11 "" 1 "" {XPPMATH 20 "6%/&%\"cG6#\"\"#$ \"0-++++++*!#;/&F%6#\"\"%$\"0@=*zD,dF!#:/&F%6#\"\"($\"0\\%4\"*HSE7F1" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#$\"0\"3f%HuD!>!#=" }}{PARA 11 "" 1 " " {XPPMATH 20 "6%/&%\"cG6#\"\"#$\"0i*[XBv%f*!#;/&F%6#\"\"%$\"0Jf?/erv# !#:/&F%6#\"\"($\"0'\\)pCQJ@\"F1" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#$\" 0$H)GfvB!>!#=" }}{PARA 11 "" 1 "" {XPPMATH 20 "6%/&%\"cG6#\"\"#$\"0+.1 U-%\\'*!#;/&F%6#\"\"%$\"0AP?!o9dF!#:/&F%6#\"\"($\"0n1B!*\\]NP->!#=" }}{PARA 11 "" 1 "" {XPPMATH 20 "6%/&%\"cG6#\"\"#$\"02Qo#[wa'*!#;/&F%6#\"\"%$\"0jFksHrv#!# :/&F%6#\"\"($\"06.2aiQ>\"F1" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#$\"0:_E 3tB!>!#=" }}{PARA 11 "" 1 "" {XPPMATH 20 "6%/&%\"cG6#\"\"#$\"0t(Hi1ib' *!#;/&F%6#\"\"%$\"0B!=4u6dF!#:/&F%6#\"\"($\"0#zGM8`)=\"F1" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#$\"04?#fGP->!#=" }}{PARA 11 "" 1 "" {XPPMATH 20 "6%/&%\"cG6#\"\"#$\"0Tc+eafl*!#;/&F%6#\"\"%$\"0?\\204rv#!#:/&F%6#\" \"($\"03o*o.%\\=\"F1" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#$\"0]EqvsB!>!# =" }}{PARA 11 "" 1 "" {XPPMATH 20 "6%/&%\"cG6#\"\"#$\"0m0?Pdhl*!#;/&F% 6#\"\"%$\"0\\d!#=" }}{PARA 11 "" 1 "" {XPPMATH 20 "6%/& %\"cG6#\"\"#$\"0c\\qb\"Hc'*!#;/&F%6#\"\"%$\"0(*)pM'*4dF!#:/&F%6#\"\"($ \"0?_*3n*3=\"F1" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#$\"0#yq*osB!>!#=" } }{PARA 259 "" 0 "" {TEXT -1 1 ":" }}{PARA 259 "" 0 "" {TEXT -1 1 ":" } }{PARA 11 "" 1 "" {XPPMATH 20 "6%/&%\"cG6#\"\"#$\"0oojvlll*!#;/&F%6#\" \"%$\"0#*=C&=4dF!#:/&F%6#\"\"($\"0-]4s_v<\"F1" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#$\"0.gBnsB!>!#=" }}{PARA 11 "" 1 "" {XPPMATH 20 "6%/&% \"cG6#\"\"#$\"0$*ypvlll*!#;/&F%6#\"\"%$\"0b,C&=4dF!#:/&F%6#\"\"($\"0\" R[8Fbx6F1" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#$\"0.gBnsB!>!#=" }}{PARA 11 "" 1 "" {XPPMATH 20 "6%/&%\"cG6#\"\"#$\"0y*Qddcc'*!#;/&F%6#\"\"%$\" 0')*Q_=4dF!#:/&F%6#\"\"($\"0Gf%3Fbx6F1" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#$\"0.gBnsB!>!#=" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 65 "We consider various rational approximations for the \+ values found." }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 163 "nds := [c[ 2]=.965656575738978e-1,c[4]=.275709185238986,c[7]=.117755270845928]:\n evalf[10](%);\nfor dgt from 6 by -1 to 3 do\n map(convert,nds,ration al,dgt);\nend do;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#7%/&%\"cG6#\"\"#$ \"+ddcc'*!#6/&F&6#\"\"%$\"+_=4dF!#5/&F&6#\"\"($\"+3Fbx6F2" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#7%/&%\"cG6#\"\"##\"#X\"$m%/&F&6#\"\"%#\"$E\"\"$ d%/&F&6#\"\"(#\"$G\"\"%(3\"" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#7%/&%\" cG6#\"\"##\"#X\"$m%/&F&6#\"\"%#\"#f\"$9#/&F&6#\"\"(#\"#j\"$N&" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#7%/&%\"cG6#\"\"##\"#9\"$X\"/&F&6#\"\"% #\"\")\"#H/&F&6#\"\"(#F(\"#<" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#7%/&% \"cG6#\"\"##\"\"$\"#J/&F&6#\"\"%#\"\")\"#H/&F&6#\"\"(#F(\"#<" }}} {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 79 "#-------- ---------------------------------------------------------------------- " }}{PARA 0 "" 0 "" {TEXT -1 44 "Tsitouras mentions a scheme with the \+ nodes " }{XPPEDIT 18 0 "c[2] = 3/31" "6#/&%\"cG6#\"\"#*&\"\"$\"\"\"\" #J!\"\"" }{TEXT -1 3 ", " }{XPPEDIT 18 0 "c[4] = 8/29" "6#/&%\"cG6#\" \"%*&\"\")\"\"\"\"#H!\"\"" }{TEXT -1 3 ", " }{XPPEDIT 18 0 "c[7] = 2/ 17" "6#/&%\"cG6#\"\"(*&\"\"#\"\"\"\"# " 0 "" {MPLTEXT 1 0 30 "prin_err_norm(3/31,8/29,2/17);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#$\"+Q$zV!>!#8" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 " " {TEXT -1 24 "Using the formulas for " }{XPPEDIT 18 0 "c[5]" "6#&%\" cG6#\"\"&" }{TEXT -1 7 " and " }{XPPEDIT 18 0 "c[6]" "6#&%\"cG6#\"\" '" }{TEXT -1 15 " in terms of " }{XPPEDIT 18 0 "c[4]" "6#&%\"cG6#\" \"%" }{TEXT -1 59 " would suggest the following rational values for th e nodes " }{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 0 101 "evalf(subs(c[4]=8/29,eqC)):\nconve rt(%,rational,5);\nevalf(subs(c[4]=8/29,eqD)):\nconvert(%,rational,5); " }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/&%\"cG6#\"\"&#\"#S\"#^" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/&%\"cG6#\"\"'#\"$9\"\"$h\"" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 156 "Taking these values gives a principal error norm of approximately 0.00019045 and the max imum magnitude of the linking coefficients is approximately 4.493." } }{PARA 0 "" 0 "" {TEXT -1 61 "The real stability interval is approxima tely [ -4.31346, 0]." }}{PARA 0 "" 0 "" {TEXT -1 53 "(See the abrevia ted calculations in a later section.)" }}{PARA 0 "" 0 "" {TEXT -1 79 " #--------------------------------------------------------------------- ---------" }}{PARA 0 "" 0 "" {TEXT -1 11 "If we set " }{XPPEDIT 18 0 "c[2]=14/145" "6#/&%\"cG6#\"\"#*&\"#9\"\"\"\"$X\"!\"\"" }{TEXT -1 7 " \+ and " }{XPPEDIT 18 0 "c[4]=59/214" "6#/&%\"cG6#\"\"%*&\"#f\"\"\"\"$9 #!\"\"" }{TEXT -1 26 " then a good choice for " }{XPPEDIT 18 0 "c[7] " "6#&%\"cG6#\"\"(" }{TEXT -1 6 " is " }{XPPEDIT 18 0 "c[7]=4/35" "6 #/&%\"cG6#\"\"(*&\"\"%\"\"\"\"#N!\"\"" }{TEXT -1 1 "." }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 122 "mn := findmin('prin_err_norm'(14/145,59/ 214,c7),c7=0.1..0.13,convergence=location):\nc[4]=convert(mn[1],ration al,4);\nmn[2];" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/&%\"cG6#\"\"%#F'\"# N" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#$\"+RYP->!#8" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 24 "Using the formulas for \+ " }{XPPEDIT 18 0 "c[5]" "6#&%\"cG6#\"\"&" }{TEXT -1 7 " and " } {XPPEDIT 18 0 "c[6]" "6#&%\"cG6#\"\"'" }{TEXT -1 15 " in terms of " }{XPPEDIT 18 0 "c[4]" "6#&%\"cG6#\"\"%" }{TEXT -1 59 " would suggest t he following rational values for the nodes " }{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 0 105 "e valf(subs(c[4]=59/214,eqC)):\nconvert(%,rational,5);\nevalf(subs(c[4]= 59/214,eqD)):\nconvert(%,rational,5);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/&%\"cG6#\"\"&#\"#Z\"#g" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/&%\"cG6 #\"\"'#\"#`\"#v" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 155 "Taking these values gives a principal error norm of appr oximately 0.00019024 nd the maximum magnitude of the linking coeffici ents is approximately 4.478." }}{PARA 0 "" 0 "" {TEXT -1 61 "The real stability interval is approximately [ -4.31360, 0]." }}{PARA 0 "" 0 "" {TEXT -1 53 "(See the abreviated calculations in a later section.) " }}{PARA 0 "" 0 "" {TEXT -1 50 "#------------------------------------ -------------" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}} {PARA 0 "" 0 "" {TEXT -1 52 "#---------------------------------------- -----------" }}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 38 "characteristics of the embedded scheme" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 35 "coefficients of the combined scheme" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1757 "ee := \{c[2]=14/145,\nc[ 3]=59/321,\nc[4]=59/214,\nc[5]=47/60,\nc[6]=53/75,\nc[7]=4/35,\nc[8]=1 ,\nc[9]=1,\n\na[2,1]=14/145,\na[3,1]=25547/2885148,\na[3,2]=504745/288 5148,\na[4,1]=59/856,\na[4,2]=0,\na[4,3]=177/856,\na[5,1]=109313963/93 987000,\na[5,2]=0,\na[5,3]=-280562881/62658000,\na[5,4]=770307017/1879 74000,\na[6,1]=25949156847460933/19448380820547506250,\na[6,2]=0,\na[6 ,3]=13843311346643833/137931778869131250,\na[6,4]=32230829891970382273 4/674279501001748115625,\na[6,5]=8562169944665744/67437276532035625,\n a[7,1]=-1124425445878279929044/5080674836868609224225,\na[7,2]=0,\na[7 ,3]=10208986940897432574/2855457555847472065,\na[7,4]=-138907629689978 5384285714176/321845802595321532838393475,\na[7,5]=-834106139751066920 5248/2889892848216596175835,\na[7,6]=7781543704350/1963746281231,\na[8 ,1]=-4662888761921915772773/8195396815968383745983,\na[8,2]=0,\na[8,3] =2931920524405086861/940000781782231318,\na[8,4]=-18227777676243374738 5270230/74164896551649031914481339,\na[8,5]=-146526017349731441994000/ 935639633573123866302727,\na[8,6]=3595419767671875/3518327954949241,\n a[8,7]=1418176375/33500640722,\na[9,1]=577999/7054512,\na[9,2]=0,\na[9 ,3]=0,\na[9,4]=34110058090624/83078929543805,\na[9,5]=1110240000/12869 442287,\na[9,6]=77530078125/230762131864,\na[9,7]=203784875/5240523662 4,\na[9,8]=1341697/16490760,\n\nb[1]=577999/7054512,\nb[2]=0,\nb[3]=0, \nb[4]=34110058090624/83078929543805,\nb[5]=1110240000/12869442287,\nb [6]=77530078125/230762131864,\nb[7]=203784875/52405236624,\nb[8]=13416 97/16490760,\n\n`b*`[1]=463127996257/5946086987136,\n`b*`[2]=0,\n`b*`[ 3]=0,\n`b*`[4]=126224639819736010009/315113992065678492180,\n`b*`[5]=6 24913261131500/8888807962650317,\n`b*`[6]=450569541695078125/127508262 0954224192,\n`b*`[7]=115589702589080125/7818298262438511744,\n`b*`[8]= 12589356280823/273360468363840,\n`b*`[9]=1/27\}:" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 5 "Let " } {XPPEDIT 18 0 "T[6, 8];" "6#&%\"TG6$\"\"'\"\")" }{TEXT -1 127 " denot e the vector whose components are the principal error terms of the 8 s tage, order 6 scheme (the error terms of order 7)." }}{PARA 0 "" 0 "" {TEXT -1 5 "Let " }{XPPEDIT 18 0 "`T*`[5,9];" "6#&%#T*G6$\"\"&\"\"*" }{TEXT -1 145 " denote the vector whose components are the principal \+ error terms of the embedded 9 stage, order 5 scheme (the error terms o f order 6) and let " }{XPPEDIT 18 0 "`T*`[6,9];" "6#&%#T*G6$\"\"'\"\" *" }{TEXT -1 99 " denote the vector whose components are the error te rms of order 7 of the embedded order 5 scheme." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 39 "Denote the 2-norm of thes e vectors by " }{XPPEDIT 18 0 "abs(abs(T[6,8]));" "6#-%$absG6#-F$6#&% \"TG6$\"\"'\"\")" }{TEXT -1 3 ", " }{XPPEDIT 18 0 "abs(abs(`T*`[5,9]) );" "6#-%$absG6#-F$6#&%#T*G6$\"\"&\"\"*" }{TEXT -1 7 " and " } {XPPEDIT 18 0 "abs(abs(`T*`[6,9]));" "6#-%$absG6#-F$6#&%#T*G6$\"\"'\" \"*" }{TEXT -1 15 " respectively." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }} {PARA 0 "" 0 "" {TEXT -1 8 "Define: " }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "A[7] = abs(abs(T[6, 8]));" "6#/&%\"AG6#\"\"(-%$absG 6#-F)6#&%\"TG6$\"\"'\"\")" }{TEXT -1 6 ", " }{XPPEDIT 18 0 "B[7] = abs(abs(`T*`[6,9]))/abs(abs(`T*`[5,9]));" "6#/&%\"BG6#\"\"(*&-%$absG6 #-F*6#&%#T*G6$\"\"'\"\"*\"\"\"-F*6#-F*6#&F/6$\"\"&F2!\"\"" }{TEXT -1 6 ", " }{XPPEDIT 18 0 "C[7] = abs(abs(`T*`[6,9]-T[6, 8]))/abs(abs( `T*`[5, 9]));" "6#/&%\"CG6#\"\"(*&-%$absG6#-F*6#,&&%#T*G6$\"\"'\"\"*\" \"\"&%\"TG6$F2\"\")!\"\"F4-F*6#-F*6#&F06$\"\"&F3F9" }{TEXT -1 1 "." }} {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 77 "Dormand a nd Prince have suggested that as well as attempting to ensure that " }{XPPEDIT 18 0 "A[7];" "6#&%\"AG6#\"\"(" }{TEXT -1 73 " is a minimum, if the embedded scheme is to be used for error control, " }{XPPEDIT 18 0 "B[7];" "6#&%\"BG6#\"\"(" }{TEXT -1 7 " and " }{XPPEDIT 18 0 "C [7];" "6#&%\"CG6#\"\"(" }{TEXT -1 27 " should be chosen so that " }} {PARA 0 "" 0 "" {TEXT -1 68 "they are \"similar in magnitude\" and als o not differ too much from 1." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 186 "errterms6_8 := PrincipalErr orTerms(6,8,'expanded'):\n`errterms6_9*` :=subs(b=`b*`,PrincipalErrorT erms(6,9,'expanded')):\n`errterms5_9*` := subs(b=`b*`,PrincipalErrorTe rms(5,9,'expanded')):" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 326 "snmB := sqrt(add(evalf(subs(ee,`er rterms6_9*`[i]))^2,i=1..nops(`errterms6_9*`))):\nsdnB := sqrt(add(eval f(subs(ee,`errterms5_9*`[i]))^2,i=1..nops(`errterms5_9*`))):\nsnmC := \+ sqrt(add((evalf(subs(ee,`errterms6_9*`[i])-subs(ee,errterms6_8[i])))^2 ,i=1..nops(errterms6_8))):\n'B[7]'= evalf[8](snmB/sdnB);\n'C[7]'= eval f[8](snmC/sdnB);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/&%\"BG6#\"\"($\") `![I\"!\"(" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/&%\"CG6#\"\"($\")J$pF\" !\"(" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}}{PARA 0 "" 0 "" {TEXT -1 52 "#----------------- ----------------------------------" }}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 27 "absolute stability regions " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}{SECT 1 {PARA 4 "" 0 " " {TEXT -1 35 "coefficients of the combined scheme" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1757 "ee := \{c[ 2]=14/145,\nc[3]=59/321,\nc[4]=59/214,\nc[5]=47/60,\nc[6]=53/75,\nc[7] =4/35,\nc[8]=1,\nc[9]=1,\n\na[2,1]=14/145,\na[3,1]=25547/2885148,\na[3 ,2]=504745/2885148,\na[4,1]=59/856,\na[4,2]=0,\na[4,3]=177/856,\na[5,1 ]=109313963/93987000,\na[5,2]=0,\na[5,3]=-280562881/62658000,\na[5,4]= 770307017/187974000,\na[6,1]=25949156847460933/19448380820547506250,\n a[6,2]=0,\na[6,3]=13843311346643833/137931778869131250,\na[6,4]=322308 298919703822734/674279501001748115625,\na[6,5]=8562169944665744/674372 76532035625,\na[7,1]=-1124425445878279929044/5080674836868609224225,\n a[7,2]=0,\na[7,3]=10208986940897432574/2855457555847472065,\na[7,4]=-1 389076296899785384285714176/321845802595321532838393475,\na[7,5]=-8341 061397510669205248/2889892848216596175835,\na[7,6]=7781543704350/19637 46281231,\na[8,1]=-4662888761921915772773/8195396815968383745983,\na[8 ,2]=0,\na[8,3]=2931920524405086861/940000781782231318,\na[8,4]=-182277 776762433747385270230/74164896551649031914481339,\na[8,5]=-14652601734 9731441994000/935639633573123866302727,\na[8,6]=3595419767671875/35183 27954949241,\na[8,7]=1418176375/33500640722,\na[9,1]=577999/7054512,\n a[9,2]=0,\na[9,3]=0,\na[9,4]=34110058090624/83078929543805,\na[9,5]=11 10240000/12869442287,\na[9,6]=77530078125/230762131864,\na[9,7]=203784 875/52405236624,\na[9,8]=1341697/16490760,\n\nb[1]=577999/7054512,\nb[ 2]=0,\nb[3]=0,\nb[4]=34110058090624/83078929543805,\nb[5]=1110240000/1 2869442287,\nb[6]=77530078125/230762131864,\nb[7]=203784875/5240523662 4,\nb[8]=1341697/16490760,\n\n`b*`[1]=463127996257/5946086987136,\n`b* `[2]=0,\n`b*`[3]=0,\n`b*`[4]=126224639819736010009/3151139920656784921 80,\n`b*`[5]=624913261131500/8888807962650317,\n`b*`[6]=45056954169507 8125/1275082620954224192,\n`b*`[7]=115589702589080125/7818298262438511 744,\n`b*`[8]=12589356280823/273360468363840,\n`b*`[9]=1/27\}:" }}} {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 8 stage, order 6 scheme is given as \+ follows." }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 75 "subs(ee,Stabilit yFunction(6,8,'expanded')):\nR := unapply(%,z):\n'R(z)'=R(z);" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#/-%\"RG6#%\"zG,4\"\"\"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)*&#\"0<$o#))=kX$ \"4+gXyS*Q/UT')\"4+?\"p:)y(3%[$F)*$)F'\" \")F)F)F)" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT 260 4 "Note" }{TEXT -1 63 ": The scheme has been constructed so that t he coefficients of " }{XPPEDIT 18 0 "z^6" "6#*$%\"zG\"\"'" }{TEXT -1 7 " and " }{XPPEDIT 18 0 "z^7" "6#*$%\"zG\"\"(" }{TEXT -1 30 " are \+ approximately equal to " }{XPPEDIT 18 0 "1/7!" "6#*&\"\"\"F$-%*factor ialG6#\"\"(!\"\"" }{TEXT -1 7 " and " }{XPPEDIT 18 0 "1/8!" "6#*&\" \"\"F$-%*factorialG6#\"\")!\"\"" }{TEXT -1 15 " respectively." }} {PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 51 "1/7!,345641888268317/1742043894078456000;\nevalf(%);" }}{PARA 11 " " 1 "" {XPPMATH 20 "6$#\"\"\"\"%S]#\"0<$o#))=kX$\"4+gXyS*Q/U<" }} {PARA 11 "" 1 "" {XPPMATH 20 "6$$\"+%)p7%)>!#8$\"+()p6%)>F%" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 50 "1/ 8!,86411937200369/3484087788156912000;\nevalf(%);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6$#\"\"\"\"&?.%#\"/p.?P>T')\"4+?\"p:)y(3%[$" }}{PARA 11 " " 1 "" {XPPMATH 20 "6$$\"+I(e,[#!#9$\"+'G)=![#F%" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 124 "We can find the point wh ere the boundary of the stability region intersects the negative real \+ axis by solving the equation: " }}{PARA 256 "" 0 "" {TEXT -1 1 " " } {XPPEDIT 18 0 "R(z) = 1;" "6#/-%\"RG6#%\"zG\"\"\"" }{TEXT -1 2 ". " }} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 28 "z0 := newton(R(z)=1,z=-4.3); " }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%#z0G$!+#>&f8V!\"*" }}}{PARA 0 " " 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 303 "z0 : = newton(R(z)=1,z=-4.3):\np1 := plot([R(z),1],z=-5.09..0.49,color=[red ,blue]):\np2 := plot([[[z0,1]]$3],style=point,symbol=[circle,cross,dia mond],color=black):\np3 := plot([[z0,0],[z0,1]],linestyle=3,color=COLO R(RGB,0,.5,0)):\nplots[display]([p1,p2,p3],view=[-5.09..0.49,-.07..1.4 7],font=[HELVETICA,9]);" }}{PARA 13 "" 1 "" {GLPLOT2D 360 253 253 {PLOTDATA 2 "6+-%'CURVESG6$7Z7$$!3')*************3&!#<$\"3g%e/WoYo9%F* 7$$!3_*\\P*pHff]F*$\"35prt;?&z$RF*7$$!32+]()Rf=H]F*$\"3IArx;GUQPF*7$$! 3i+D\")4*y()*\\F*$\"3!=')G)Q[*ya$F*7$$!3G++vz=Po\\F*$\"3iUKDoe,mLF*7$$ !35]i?\"*yX:\\F*$\"3+rh7At7pIF*7$$!3#**\\iE!Rai[F*$\"3U#3l\">>p&z#F*7$ $!3P\\PW)eOI![F*$\"34Xp%f7+U^#F*7$$!3r**\\Au#HNu%F*$\"3gd&z!*[!=eAF*7$ $!3u*****o]FOo%F*$\"32(=!pTq:C?F*7$$!3x**\\dRdsBYF*$\"3:'y&fq'**>\"=F* 7$$!3#)\\P*[n3Tc%F*$\"3j`=\"o0)z?;F*7$$!3*)*\\7-h\"\\/XF*$\"35SyntD'yW \"F*7$$!3s*\\i#4j%RR%F*$\"3)\\(>!o$oXq6F*7$$!35+D;`I[zUF*$\"3Cw[E4k:Z$ *!#=7$$!3W*\\i**)\\5hTF*$\"3'e_maRnBP(F]p7$$!3!**\\7nc1J/%F*$\"3gO1e0/ g%z&F]p7$$!3'****\\XoI<#RF*$\"3U)oL'>(Qz]%F]p7$$!3++]ZTF#[\"QF*$\"3Fi] iz#Q#3OF]p7$$!3s***\\'=(pWp$F*$\"3DO'4=d(e5GF]p7$$!3#)***\\Z^AOd$F*$\" 3W^!48A'R(>#F]p7$$!3))***\\8%Q;dMF*$\"3KBW$=.=1v\"F]p7$$!3#**\\i*3#39N $F*$\"37WZS^odW9F]p7$$!3t***\\J`acA$F*$\"3E)\\8a>F0=\"F]p7$$!3l****fuY 7>JF*$\"38Dra+@wD5F]p7$$!3q*\\iQ70_*HF*$\"31(p4xLQg4*!#>7$$!3+++5C`^&) GF*$\"3Oyocy8GM&)Fes7$$!3q*\\i)G#o^w#F*$\"3u'G9F*p$yK)Fes7$$!3W*\\(eM$ p0l#F*$\"3)oGUXe8>Y)Fes7$$!3]**\\i5u*4`#F*$\"3a)4u#Gyn\"*))Fes7$$!3T* \\7\"eI>@CF*$\"3&eA7\\0#o8&*Fes7$$!3n**\\()HUv-BF*$\"36sML51[S5F]p7$$! 3y*\\iRdH(z@F*$\"3['o+nMzd:\"F]p7$$!3o*\\P$\\ijs?F*$\"3YJiU7Xgu7F]p7$$ !3S**\\#[_sp&>F*$\"3(QVrwQMDU\"F]p7$$!3y****pz0[P=F*$\"3a5%)*prFxf\"F] p7$$!3')**\\_B5e?_r2*F]p$\" 3OBj<<,YMSF]p7$$!3*)**\\74%3K!zF]p$\"3!z`N&R^*p`%F]p7$$!3E****\\xPYbnF ]p$\"3yHfS0Qy)3&F]p7$$!3)4+Dc^\")Qb&F]p$\"3)p2E`!\\\\QdF]p7$$!3)e****f )\\h'R%F]p$\"3$Gq!RoYaUkF]p7$$!3F)**\\<\"G98KF]p$\"3=Z`Q+I&>D(F]p7$$!3 'G*\\i%Qq%R?F]p$\"3YUet/c0b\")F]p7$$!3xr****pJ()4'*Fes$\"35wp*HEVP3*F] p7$$\"3k<+]_)f2v#Fes$\"3@w5_D%*)y-\"F*7$$\"3)3++!Qdi!Q\"F]p$\"3g*49dQZ ![6F*7$$\"3o4]PhBPfDF]p$\"3si*z]lr;H\"F*7$$\"3]/]i%G$e(o$F]p$\"3-^i**= \"QfW\"F*7$$\"3!***************[F]p$\"3)H,K_@;Bj\"F*-%'COLOURG6&%$RGBG $\"*++++\"!\")$\"\"!Fd]lFc]l-F$6$7S7$F($\"\"\"Fd]l7$F=Fi]l7$FGFi]l7$FQ Fi]l7$FenFi]l7$F_oFi]l7$FdoFi]l7$FioFi]l7$F_pFi]l7$FdpFi]l7$FipFi]l7$F ^qFi]l7$FcqFi]l7$FhqFi]l7$F]rFi]l7$FbrFi]l7$FgrFi]l7$F\\sFi]l7$FasFi]l 7$FgsFi]l7$F\\tFi]l7$FatFi]l7$FftFi]l7$F[uFi]l7$F`uFi]l7$FeuFi]l7$FjuF i]l7$F_vFi]l7$FdvFi]l7$FivFi]l7$F^wFi]l7$FcwFi]l7$FhwFi]l7$F]xFi]l7$Fb xFi]l7$FgxFi]l7$F\\yFi]l7$FayFi]l7$FfyFi]l7$F[zFi]l7$F`zFi]l7$FezFi]l7 $FjzFi]l7$F_[lFi]l7$Fd[lFi]l7$Fi[lFi]l7$F^\\lFi]l7$Fc\\lFi]l7$Fh\\lFi] l-F]]l6&F_]lFc]lFc]lF`]l-F$6&7#7$$!3>+++#>&f8VF*Fi]l-%'SYMBOLG6#%'CIRC LEG-F]]l6&F_]lFd]lFd]lFd]l-%&STYLEG6#%&POINTG-F$6&F_al-Fdal6#%&CROSSGF galFial-F$6&F_al-Fdal6#%(DIAMONDGFgalFial-F$6%7$7$FaalFc]lF`al-%&COLOR G6&F_]lFc]l$\"\"&!\"\"Fc]l-%*LINESTYLEG6#\"\"$-%%FONTG6$%*HELVETICAG\" \"*-%+AXESLABELSG6%Q\"z6\"Q!F^dl-Ffcl6#%(DEFAULTG-%%VIEWG6$;$!$4&!\"#$ \"#\\Fidl;$!\"(Fidl$\"$Z\"Fidl" 1 2 0 1 10 0 2 9 1 4 2 1.000000 45.000000 45.000000 0 0 "Curve 1" "Curve 2" "Curve 3" "Curve 4" "Curve 5" "Curve 6" }}}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 50 "The following picture shows the stability region. " }} {PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1372 "R := z -> 1+z+1/2*z^2+1/6*z^3+1/24*z^4+1/120*z^5+1/720*z^6+\n \+ 345641888268317/1742043894078456000*z^7+86411937200369/3484087788156 912000*z^8:\npts := []: z0 := 0:\nfor ct from 0 to 240 do\n zz := ne wton(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,.48,.05,.13)): \np2 := plots[polygonplot]([seq([pts[i-1],pts[i],[-2.15,0]],i=2..nops( pts))],\n style=patchnogrid,color=COLOR(RGB,.95,.1,.25)):\npt s := []: z0 := 2.1+4.7*I:\nfor ct from 0 to 40 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:\np3 := plot(pts,color=COLOR(RGB,.48,.05,.13)):\np4 := \+ plots[polygonplot]([seq([pts[i-1],pts[i],[2.03,4.71]],i=2..nops(pts))] ,\n style=patchnogrid,color=COLOR(RGB,.95,.1,.25)):\npts := [ ]: z0 := 2.1-4.7*I:\nfor ct from 0 to 40 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:\np5 := plot(pts,color=COLOR(RGB,.48,.05,.13)):\np6 := plots[ polygonplot]([seq([pts[i-1],pts[i],[2.03,-4.71]],i=2..nops(pts))],\n \+ style=patchnogrid,color=COLOR(RGB,.95,.1,.25)):\np7 := plot([[ [-5.19,0],[2.39,0]],[[0,-5.19],[0,5.19]]],color=black,linestyle=3):\np lots[display]([p||(1..7)],view=[-5.19..2.39,-5.19..5.19],font=[HELVETI CA,9],\n labels=[`Re(z)`,`Im(z)`],axes=boxed,scaling=cons trained);" }}{PARA 13 "" 1 "" {GLPLOT2D 381 565 565 {PLOTDATA 2 "6/-%' CURVESG6$7]z7$$\"\"!F)F(7$F($\"3++++Fjzq:!#=7$F($\"3)******Rl#fTJF-7$$ \"3/+++]RW&H\"!#E$\"3!)******3!*Q7ZF-7$$\"39+++[HHZA!#D$\"3I+++hc=$G'F -7$$\"34+++sBhG?!#C$\"3C+++5S)R&yF-7$$\"3/+++%H*)z?\"!#B$\"3,+++5')yC% *F-7$$\"39+++RI$QQ&FF$\"3#******R'4c*4\"!#<7$$\"3,+++!)y)f$>!#A$\"3-++ +ihkc7FN7$$\"3++++xtz$*eFR$\"3++++vZt89FN7$$\"3/+++,f4p:!#@$\"3)****** \\$4#3d\"FN7$$\"3')*****4fSit$Fgn$\"3*)*****Ry!)ys\"FN7$$\"3O+++'z[F3) 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"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 "[-4.4314, 0];" "6#7$,$-%&FloatG6$\"&9V%!\"%!\"\"\"\"!" }{TEXT -1 2 ". " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 260 "We can distort the boundary curve horizontally by taking the 11th root of the real part of points along the curve. In this way we see t hat the largest interval on the nonnegative imaginary axis that contai ns the origin and lies inside the stability region is " }{XPPEDIT 18 0 "[0, 3.4];" "6#7$\"\"!-%&FloatG6$\"#M!\"\"" }{TEXT -1 18 " approxim ately. " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 395 "R := z -> 1+z+1/2*z^2+1/6*z^3+1/24*z^4+1/120*z^5+1/7 20*z^6+\n 345641888268317/1742043894078456000*z^7+86411937200369/3 484087788156912000*z^8:\nDigits := 25:\npts := []: z0 := 0:\nfor ct fr om 0 to 120 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,.9,0,.2),thickness=2,font=[HELVETICA,9]);\nDigits := \+ 10:" }}{PARA 13 "" 1 "" {GLPLOT2D 261 276 276 {PLOTDATA 2 "6(-%'CURVES G6#7er7$$\"\"!F)F(7$$\":!p')))>(*3Wi[\"HC\"!#E$\":?plVKz*e`EfTJF-7$$\" :iXG(pOdeH5InCF-$\":U&Q2H!fzrI&=$G'F-7$$\":')\\ND2JA()4_Kf$F-$\":'3I#f U`p2'zxC%*F-7$$\":X!>.WA)*Q&et$yYF-$\":%yP9OEhVhqjc7!#D7$$\":)HgN&odRY 3mit&F-$\":\\5VP;#fb\\)=F? 7$$\":Ll!\\\"**)\\A8.(Rz(F-$\":CJ5n86Xv&[6*>#F?7$$\":(z')y^z7]lQJ+))F- $\":`Kxo4lyH7uK^#F?7$$\":)><@')4(yhw$Q%z*F-$\":;REy6OP&)QLu#GF?7$$\":z ^aym'\\'f*ovx5F?$\":pgZ./`vVl#fTJF?7$$\":G[hWwW6u3%4v6F?$\":e%p9;ily?> vbMF?7$$\":T?\\v&HW8,Z`r7F?$\":;\\88.r#H)=6*pPF?7$$\":qks2Lkd@4ZrO\"F? 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" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 113 "Digits := 15:\nz0 := 3.4*I:\nfor ct from 112 to 115 \+ do\n newton(R(z)=exp(ct*Pi/100*I),z=z0);\nend do;\nDigits := 10:" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#^$$\"0O+M\\4>*G!#<$\"0!*e/#zuqL!#9" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#^$$\"0z6w@f0p#!#=$\"0U>Y=;IR$!#9" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#^$$!0g,@N!H\"e#!#<$\"0h!z3%e\\T$!#9" } }{PARA 11 "" 1 "" {XPPMATH 20 "6#^$$!0^:P(Qkgc!#<$\"0&eLgNdOM!#9" }}} {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 107 "Then we \+ apply the bisection method to calculate the parameter value associated with the intersection point." }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 176 "real_part := proc(u)\n Re(newton(R(z)=exp(u*Pi*I),z=3.4*I))\n end proc:\nDigits := 15:\nu0 := bisect('real_part'(u),u=1.12..1.15);\n newton(R(z)=exp(u0*Pi*I),z=3.4*I);\nDigits := 10:" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%#u0G$\"0iD%*Hz48\"!#9" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#^#$\"03CVSz^R$!#9" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 " " 0 "" {TEXT -1 4 "The " }{TEXT 260 112 "largest interval on the nonne gative imaginary axis that contains the origin and lies inside the sta bility region" }{TEXT -1 5 " is " }{XPPEDIT 18 0 "[0, 3.3952];" "6#7$ \"\"!-%&FloatG6$\"&_R$!\"%" }{TEXT -1 18 " (approximately)." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 37 "#-------------- ----------------------" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 80 "The stability function R* for the 9 stage, order 5 scheme is given as follows." }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 97 "subs(ee,subs(b=`b*`,StabilityFunction(5,9,'expanded'))):\n`R*` := \+ unapply(%,z):\n'`R*`(z)'=`R*`(z);" }}{PARA 12 "" 1 "" {XPPMATH 20 "6#/ -%#R*G6#%\"zG,6\"\"\"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)*&# \"8@HoNm]d,B&*z$\";+O(oil2pZKU*)f#F)*$)F'F1F)F)F)*&#\"8(oZRh'ej4t2h'\" <+gtoil2pZKU*)f#F)*$)F'\"\"(F)F)F)*&#\"8\"zj)31Wz.1<6\"\"<+?ZPDJ:Q&\\Y )y>&F)*$)F'\"\")F)F)F)*&#\"/p.?P>T')\"5+SiO-Gq.2%*F)*$)F'\"\"*F)F)F)" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 124 "We ca n find the point where the boundary of the stability region intersects the negative real axis by solving the equation: " }}{PARA 256 "" 0 " " {TEXT -1 1 " " }{XPPEDIT 18 0 "`R*`(z) = -1;" "6#/-%#R*G6#%\"zG,$\" \"\"!\"\"" }{TEXT -1 2 ". " }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 33 "z_0 := newton(`R*`(z)=-1,z=-4.3);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%$z_0G$!+Pg_KV!\"*" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 324 "z_0 := newton(`R*`(z)=-1,z=-4.3): \np_1 := plot([`R*`(z),-1],z=-4.89..0.49,color=[red,blue]):\np_2 := pl ot([[[z_0,-1]]$3],style=point,symbol=[circle,cross,diamond],color=blac k):\np_3 := plot([[z_0,0],[z_0,-1]],linestyle=3,color=COLOR(RGB,0,.5,0 )):\nplots[display]([p_1,p_2,p_3],view=[-4.89..0.49,-1.57..1.47],font= [HELVETICA,9]);" }}{PARA 13 "" 1 "" {GLPLOT2D 398 284 284 {PLOTDATA 2 "6+-%'CURVESG6$7S7$$!3o*************)[!#<$!3aVO*4P\"4#R#F*7$$!3mKLe58t sZF*$!3$Gu7![8n6?F*7$$!3'f;a4X'pqYF*$!3N2LWa&*=B(pzS9F*7$$!3iK$e'y!Q/W%F*$!3?q%G\"F*7$$!3AmT+uvZDVF*$!3A&4 YEcF:))*!#=7$$!3_L3AqW*)=UF*$!3m_G6rPvC#)FF7$$!3`*\\(yRQ`3TF*$!3AUuY!* *=Kv'FF7$$!3!H$3#zr)R%*RF*$!31W4'4nO(faFF7$$!3u*\\P)G'H1)QF*$!3QK!G0m$F*$!3G\"yNe,.Lt#FF7$ 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iKLLzHSjSFF$\"3!4$)Ra'o$3m'FF7$$!3mGLei\"\\B#HFF$\"3cI9vO;$fY(FF7$$!3g i;aZQu!z\"FF$\"3%)[\"RD5P/O)FF7$$!3FJ****p3;4vFcp$\"3YP%*e3]ew#*FF7$$ \"3DiL$3rQ%3WFcp$\"3MZ!\\;aq]/\"F*7$$\"3&pmmY=on]\"FF$\"3?z+sg3ii6F*7$ $\"3u+]7DeEVEFF$\"3ith\"za`DI\"F*7$$\"3D)*\\(33R5t$FF$\"3k@\"GG[NAX\"F *7$$\"3!***************[FF$\"35\"o=hf " 0 "" {MPLTEXT 1 0 1519 " `R*` := z -> 1+z+1/2*z^2+1/6*z^3+1/24*z^4+1/120*z^5+\n 37995230157 506635682921/25989423247690765626873600*z^6+\n 6610773096358661394 7687/259894232476907656268736000*z^7+\n 11117060379440608863791/51 9788464953815312537472000*z^8+\n 86411937200369/940703702802366240 00*z^9:\npts := []: z0 := 0:\nfor ct from 0 to 200 do\n zz := newton (`R*`(z)=exp(ct*Pi/20*I),z=z0):\n z0 := zz:\n pts := [op(pts),[Re( zz),Im(zz)]]:\nend do:\np_1 := plot(pts,color=COLOR(RGB,.43,0,.08)):\n p_2 := plots[polygonplot]([seq([pts[i-1],pts[i],[-2.15,0]],i=2..nops(p ts))],\n style=patchnogrid,color=COLOR(RGB,.85,0,.15)):\npts \+ := []: z0 := 2.2+4.45*I:\nfor ct from 0 to 50 do\n zz := newton(`R*` (z)=exp(ct*Pi/25*I),z=z0):\n z0 := zz:\n pts := [op(pts),[Re(zz),I m(zz)]]:\nend do:\np_3 := plot(pts,color=COLOR(RGB,.43,0,.08)):\np_4 : = plots[polygonplot]([seq([pts[i-1],pts[i],[2.08,4.43]],i=2..nops(pts) )],\n style=patchnogrid,color=COLOR(RGB,.85,0,.15)):\npts := \+ []: z0 := 2.2-4.45*I:\nfor ct from 0 to 50 do\n zz := newton(`R*`(z) =exp(ct*Pi/25*I),z=z0):\n z0 := zz:\n pts := [op(pts),[Re(zz),Im(z z)]]:\nend do:\np_5 := plot(pts,color=COLOR(RGB,.43,0,.08)):\np_6 := p lots[polygonplot]([seq([pts[i-1],pts[i],[2.08,-4.43]],i=2..nops(pts))] ,\n style=patchnogrid,color=COLOR(RGB,.85,0,.15)):\np_7 := pl ot([[[-5.09,0],[2.49,0]],[[0,-4.89],[0,4.89]]],color=black,linestyle=3 ):\nplots[display]([p_||(1..7)],view=[-5.09..2.49,-4.89..4.89],font=[H ELVETICA,9],\n labels=[`Re(z)`,`Im(z)`],axes=boxed,scaling= constrained);" }}{PARA 13 "" 1 "" {GLPLOT2D 407 552 552 {PLOTDATA 2 "6 /-%'CURVESG6$7ew7$$\"\"!F)F(7$$\"3-+++op!f5\"!#E$\"3#******fK'zq:!#=7$ 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TFf^alFi^al-%%FONTG6$%*HELVETICAG\"\"*-%+AXESLABELSG6%%&Re(z)G%&Im(z)G -Fe_al6#%(DEFAULTG-%*AXESSTYLEG6#%$BOXG-%(SCALINGG6#%,CONSTRAINEDG-%%V IEWG6$;$!$>&Fc^n$\"$R#Fc^n;F]aal$\"$>&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" "Curve 9" "Curve 10" "Curve 11" "Curve 12" "Curve 13" "Curve 14" }}}}{PARA 0 "" 0 "" {TEXT -1 0 " " }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}}{PARA 0 "" 0 "" {TEXT -1 52 "#---------------------------------------------------" }} {SECT 1 {PARA 4 "" 0 "" {TEXT -1 36 "coefficients for the combined sch eme" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 35 "coefficient s of the combined scheme" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1757 "ee := \{c[2]=14/145,\nc[3]=59/321 ,\nc[4]=59/214,\nc[5]=47/60,\nc[6]=53/75,\nc[7]=4/35,\nc[8]=1,\nc[9]=1 ,\n\na[2,1]=14/145,\na[3,1]=25547/2885148,\na[3,2]=504745/2885148,\na[ 4,1]=59/856,\na[4,2]=0,\na[4,3]=177/856,\na[5,1]=109313963/93987000,\n a[5,2]=0,\na[5,3]=-280562881/62658000,\na[5,4]=770307017/187974000,\na [6,1]=25949156847460933/19448380820547506250,\na[6,2]=0,\na[6,3]=13843 311346643833/137931778869131250,\na[6,4]=322308298919703822734/6742795 01001748115625,\na[6,5]=8562169944665744/67437276532035625,\na[7,1]=-1 124425445878279929044/5080674836868609224225,\na[7,2]=0,\na[7,3]=10208 986940897432574/2855457555847472065,\na[7,4]=-138907629689978538428571 4176/321845802595321532838393475,\na[7,5]=-8341061397510669205248/2889 892848216596175835,\na[7,6]=7781543704350/1963746281231,\na[8,1]=-4662 888761921915772773/8195396815968383745983,\na[8,2]=0,\na[8,3]=29319205 24405086861/940000781782231318,\na[8,4]=-182277776762433747385270230/7 4164896551649031914481339,\na[8,5]=-146526017349731441994000/935639633 573123866302727,\na[8,6]=3595419767671875/3518327954949241,\na[8,7]=14 18176375/33500640722,\na[9,1]=577999/7054512,\na[9,2]=0,\na[9,3]=0,\na [9,4]=34110058090624/83078929543805,\na[9,5]=1110240000/12869442287,\n a[9,6]=77530078125/230762131864,\na[9,7]=203784875/52405236624,\na[9,8 ]=1341697/16490760,\n\nb[1]=577999/7054512,\nb[2]=0,\nb[3]=0,\nb[4]=34 110058090624/83078929543805,\nb[5]=1110240000/12869442287,\nb[6]=77530 078125/230762131864,\nb[7]=203784875/52405236624,\nb[8]=1341697/164907 60,\n\n`b*`[1]=463127996257/5946086987136,\n`b*`[2]=0,\n`b*`[3]=0,\n`b *`[4]=126224639819736010009/315113992065678492180,\n`b*`[5]=6249132611 31500/8888807962650317,\n`b*`[6]=450569541695078125/127508262095422419 2,\n`b*`[7]=115589702589080125/7818298262438511744,\n`b*`[8]=125893562 80823/273360468363840,\n`b*`[9]=1/27\}:" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}}{PARA 0 "" 0 " " {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT 259 5 "nodes" }{TEXT -1 1 ":" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 31 "seq(c[i]=subs(ee,c[i]),i=2 ..9);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6*/&%\"cG6#\"\"##\"#9\"$X\"/&F% 6#\"\"$#\"#f\"$@$/&F%6#\"\"%#F0\"$9#/&F%6#\"\"&#\"#Z\"#g/&F%6#\"\"'#\" #`\"#v/&F%6#\"\"(#F5\"#N/&F%6#\"\")\"\"\"/&F%6#\"\"*FP" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT 259 51 "linking coefficient s for the 9 stage order 5 scheme" }{TEXT -1 1 ":" }}{EXCHG {PARA 0 "> \+ " 0 "" {MPLTEXT 1 0 49 "seq(seq(a[i,j]=subs(ee,a[i,j]),j=1..i-1),i=1.. 9);" }}{PARA 12 "" 1 "" {XPPMATH 20 "6F/&%\"aG6$\"\"#\"\"\"#\"#9\"$X\" /&F%6$\"\"$F(#\"&Zb#\"([^)G/&F%6$F/F'#\"'XZ]F2/&F%6$\"\"%F(#\"#f\"$c)/ &F%6$F;F'\"\"!/&F%6$F;F/#\"$x\"F>/&F%6$\"\"&F(#\"*jRJ4\"\")+q)R*/&F%6$ FKF'FB/&F%6$FKF/#!*\")Gc!G\")+!eE'/&F%6$FKF;#\"*/&F%6$F[oF'FB/&F%6$F[oF/#\"2LQkY8JVQ \"\"3]78p)y<$z8/&F%6$F[oF;#\"6MF#Qq>*)H3BK\"6Dc6[<+,&zUn/&F%6$F[oFK#\" 1WdmW*p@c)\"2Dc.KlFPu'/&F%6$\"\"(F(#!7W!H*z#yeWDW7\"\"7DUA4'oo$[n!3&/& F%6$FgpF'FB/&F%6$FgpF/#\"5uDV(*3%p)*3-\"\"4l?ZZebda&G/&F%6$FgpF;#!=wTr &G%Q&y**oHw!*Q\"\"/&F%6$\"\")F(#!7tFx:>#>w))Gm%\" 7$)fu$Qof\"oR&>)/&F%6$FirF'FB/&F%6$FirF/#\"4ho30W_?>$H\"3=8B#y\"y++%*/ &F%6$FirF;#!.\\;b'*[;u/&F%6$FirFK#!9+S*>WJ(\\t, El9\"9FFImQ7tNjRc$*/&F%6$FirF[o#\"1v=nn(>af$\"1T#\\\\&zK=N/&F%6$FirFgp #\"+vj<=9\",A2k+N$/&F%6$\"\"*F(#\"'**zd\"(7X0(/&F%6$FauF'FB/&F%6$FauF/ FB/&F%6$FauF;#\"/C14e+6M\"/0QaH*yI)/&F%6$FauFK#\"+++C56\",(GU%pG\"/&F% 6$FauF[o#\",D\"y+`x\"-k=8i2B/&F%6$FauFgp#\"*v[y.#\",CmB0C&/&F%6$FauFir #\"((pT8\")g2\\;" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT 259 38 "weights for the 8 stage order 6 scheme" }{TEXT -1 1 ":" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 31 "seq(b[i]=subs(ee,b[i]),i=1 ..8);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6*/&%\"bG6#\"\"\"#\"'**zd\"(7X0 (/&F%6#\"\"#\"\"!/&F%6#\"\"$F//&F%6#\"\"%#\"/C14e+6M\"/0QaH*yI)/&F%6# \"\"&#\"+++C56\",(GU%pG\"/&F%6#\"\"'#\",D\"y+`x\"-k=8i2B/&F%6#\"\"(#\" *v[y.#\",CmB0C&/&F%6#\"\")#\"((pT8\")g2\\;" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT 259 38 "weights for the 9 stage order \+ 5 scheme" }{TEXT -1 1 ":" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 37 " seq(`b*`[i]=subs(ee,`b*`[i]),i=1..9);" }}{PARA 12 "" 1 "" {XPPMATH 20 "6+/&%#b*G6#\"\"\"#\"-di*z7j%\".Or)p3Yf/&F%6#\"\"#\"\"!/&F%6#\"\"$F//& F%6#\"\"%#\"64+,O(>)RYAE\"\"6!=#\\yc1#*R6:$/&F%6#\"\"&#\"0+:8hK\"\\i\" 1<.liz!)))))/&F%6#\"\"'#\"3D\"y]pT&p0X\"4#>CU&4i#3v7/&F%6#\"\"(#\"3D,3 *e-(*e:\"\"4W<^QCE)H=y/&F%6#\"\")#\"/B3Gc$*e7\"0SQOo/Ot#/&F%6#\"\"*#F' \"#F" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}}{PARA 0 "" 0 "" {TEXT -1 52 "#----------------- ----------------------------------" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}}{PARA 0 "" 0 "" {TEXT -1 59 "#================= =========================================" }}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 82 "a scheme with a quite large stability region and non-zero imaginary axis inclusion" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 " ;" }}}{PARA 0 "" 0 "" {TEXT -1 52 "#---------------------------------- -----------------" }}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 28 "checking the combined scheme" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> \+ " 0 "" {MPLTEXT 1 0 1 ";" }}}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 35 "coef ficients of the combined scheme" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1743 "ee := \{c[2]=7/71,\nc[3]=4 6/255,\nc[4]=23/85,\nc[5]=27/29,\nc[6]=61/90,\nc[7]=18/77,\nc[8]=1,\nc [9]=1,\n\na[2,1]=7/71,\na[3,1]=6992/455175,\na[3,2]=75118/455175,\na[4 ,1]=23/340,\na[4,2]=0,\na[4,3]=69/340,\na[5,1]=125814087/51607124,\na[ 5,2]=0,\na[5,3]=-481282155/51607124,\na[5,4]=100879020/12901781,\na[6, 1]=3323091540925544113/29573952540280488000,\na[6,2]=0,\na[6,3]=-10703 826229567511/24340701679243200,\na[6,4]=169241467001919447401/17831998 0502135683200,\na[6,5]=2551748271602005217/45506989353097008000,\na[7, 1]=-376385546372076652128/6002133498919339474385,\na[7,2]=0,\na[7,3]=1 9305251563909761378/19679126225965047457,\na[7,4]=-8122491009809811364 010481/9979717849963846796488754, a[7,5]=-155386566948917103529551/230 1833170083782096614930,\na[7,6]=107267076044076240/544838019860392951, \na[8,1]=-1972640388605306005321/257739734717483367780,\na[8,2]=0, a[8 ,3]=12354606914500963/386396118220908,\na[8,4]=-5685687973257841249168 4/1967664304821917372209,\na[8,5]=-17589460594924790303414/92934689150 952840583755,\na[8,6]=370576933768080/119264931607213,\na[8,7]=1150428 860219177/432917708558301,\na[9,1]=636077/8182296,\na[9,2]=0,\na[9,3]= 0,\na[9,4]=24315051125/87140294626,\na[9,5]=30418033967/271431025272, \na[9,6]=862843671000/2241531443039,\na[9,7]=1670085824869/14714276549 256,\na[9,8]=13887/424328,\n\nb[1]=636077/8182296,\nb[2]=0,\nb[3]=0,\n b[4]=24315051125/87140294626,\nb[5]=30418033967/271431025272,\nb[6]=86 2843671000/2241531443039,\nb[7]=1670085824869/14714276549256,\nb[8]=13 887/424328,\n\n`b*`[1]=231122308093/3481256020752,\n`b*`[2]=0,\n`b*`[3 ]=0,\n`b*`[4]=4006504071740375/84742592073525056,\n`b*`[5]=23826240438 664855/461934347497102656,\n`b*`[6]=1723971130382250/4007085927807811, \n`b*`[7]=123485516897451247/368256795835268016,\n`b*`[8]=63672019983/ 6138204944224,\n`b*`[9]=1/17\}:" }}}{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 0 488 "subs(ee,matrix([[ c[2],a[2,1],``$3],\n [c[3],a[3,1],a[3,2],``$2],\n [c[4],seq(a[4,i],i =1..3),``],\n [c[5],seq(a[5,i],i=1..4)],\n [c[6],seq(a[6,i],i=1..4)] ,[``$4,a[6,5]],\n [c[7],seq(a[7,i],i=1..4)],[``$3,a[7,5],a[7,6]],\n \+ [c[8],seq(a[8,i],i=1..4)],[``$2,seq(a[8,i],i=5..7)],\n [c[9],seq(a[9, i],i=1..4)],[``,seq(a[9,i],i=5..8)],\n [``,`_________________________ ___`$4],\n [`b`,seq(b[i],i=1..4)],[``,seq(b[i],i=5..8)],\n [`b*`,seq (`b*`[i],i=1..4)],[``,seq(`b*`[i],i=5..8)],[``$4,`b*`[9]]]));" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#K%'matrixG6#747'#\"\"(\"#rF(%!GF+F+7'# \"#Y\"$b##\"%#*p\"'v^X#\"&=^(F2F+F+7'#\"#B\"#&)#F7\"$S$\"\"!#\"#pF:F+7 '#\"#F\"#H#\"*(39e7\")Crg^F;#!*b@G\"[FD#\"*?!z35\")\"y,H\"7'#\"#h\"#!* #\"48TaD4a\"4BL\"5+!)[!GSD&RdHF;#!26vcHi#Qq5\"2+KCz;qSV##\"6,uW>>+n9Cp \"\"6+KoN@]!)*>$y\"7'F+F+F+F+#\"4<_+-;F[*)\\L@+'F;#\"5y8w4Rc^_I>\"5du/lfAE\"z'>#!:\"[5S O6)4)45\\A\")\":a()['zYQ'*\\yrz**7'F+F+F+#!9^&HN5<*[pc'Qb\"\":I\\h'4#y $3qJ$=I##\"3Si2Wg2ns5\"3^HRg)>!Q[a7'\"\"\"#!7@`+1`g)QSE(>\"6!ynL[#[Ikw'>7'F+F+#! 89MI!zC\\fg%*e<\"8bPeSG&4:*oMH*#\"0!3oP$pdq$\"08sgJ\\E>\"#\"1x\">-')G/ :\"\"0,$e&3x\"HV7'Fjo#\"'xgj\"('H#=)F;F;#\",D60:V#\",EYHSr)7'F+#\",nR. =/$\"-s_-J9F#\"-+5nVG')\".RIWJ:C##\".p[#e3q;\"/c#\\lF9Z\"#\"&()Q\"\"'G VU7'F+%=____________________________GFcrFcrFcr7'%\"bGF_qF;F;FbqFeq7'%# b*G#\"-$43B7J#\"._2-c7[$F;F;#\"1v.urS]1S\"2c]_t?fUZ)7'F+#\"2b[mQ/CEQ# \"3cE5(\\ZV$>Y#\"1]AQI6(Rs\"\"16y!y#f32S#\"3Z7X(*o^&[B\"\"3;!o_$ezc#o$ #\",$)*>?nj\".CU%\\?Qh7'F+F+F+F+#Fjo\"# " 0 "" {MPLTEXT 1 0 136 "subs(ee, matrix([seq([c[i],seq(a[i,j],j=1..i-1),``$(10-i)],i=2..9),\n[`b`,seq(b [i],i=1..8),``],[`b*`,seq(`b*`[i],i=1..9)]])):\nevalf[8](%);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#K%'matrixG6#7,7,$\")\\:f)*!\"*F(%!GF+F+F+F+F +F+F+7,$\");#R!=!\")$\")D6O:F*$\").J];F/F+F+F+F+F+F+F+7,$\")C)eq#F/$\" )fqknF*$\"\"!F:$\")=TH?F/F+F+F+F+F+F+7,$\")[M5$*F/$\")4#zV#!\"(F9$!)g) eK*FB$\")'***=yFBF+F+F+F+F+7,$\")yxxnF/$\")\\lB6F/F9$!)6](R%F/$\")k)3 \\*F/$\")mP2cF*F+F+F+F+7,$\")BmPBF/$!)E'3F'F*F9$\")Y,5)*F/$!)')**Q\")F /$!)tb]nF*$\")%)yo>F/F+F+F+7,$\"\"\"F:$!)Qh`wFBF9$\")UR(>$!\"'$!)?c*)G F`o$!)!pE*=F/$\")V<2JFB$\")YQdEFBF+F+7,Fjn$\")0#Qx(F*F9F9$\")QL!z#F/$ \")Vl?6F/$\")!\\$\\QF/$\")0,N6F/$\")TqsKF*F+7,%\"bGFjoF9F9F\\pF^pF`pFb pFdpF+7,%#b*G$\")%\\!RmF*F9F9$\")<&ys%F*$\")z#z:&F*$\")kI-VF/$\")[C`LF /$\")oIP5F*$\")HN#)eF*Q(pprint36\"" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" } }{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 146 "RK6_8eqs := [op(RowSumCond itions(8,'expanded')),op(OrderConditions(6,8,'expanded'))]:\n`RK5_9eqs *` := subs(b=`b*`,OrderConditions(5,9,'expanded')):" }}}{PARA 0 "" 0 " " {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 7 "Check: " }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 148 "simplify(subs(ee,RK6_8eqs)):\nmap(u->`if `(lhs(u)=rhs(u),0,1),%);\nnops(%);\nsimplify(subs(ee,`RK5_9eqs*`)):\nm ap(u->`if`(lhs(u)=rhs(u),0,1),%);\nnops(%);" }{TEXT -1 0 "" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#7N\"\"!F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F $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#\"#W" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#73\"\"!F$F$F$F$ F$F$F$F$F$F$F$F$F$F$F$F$" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#\"#<" }}} {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 86 "Next we s et-up stage-order condtions to check for stage-orders from 2 to 5 incl usive. " }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 85 "for ct from 2 to \+ 5 do\n so||ct||_8 := StageOrderConditions(ct,8,'expanded');\nend do: " }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 34 "Stag es 4 to 8 have stage-order 3. " }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 173 "[seq([seq(expand(subs(ee,so||i||_8[j])),i=2..5)],j=1..6)]:\nmap (proc(L) local i; for i to nops(L) do if not evalb(L[i]) then break en d if end do; i end proc,%):\nsimplify(%); " }}{PARA 11 "" 1 "" {XPPMATH 20 "6#7(\"\"#\"\"$F%F%F%F%" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 53 "None of the principal error conditions \+ are satisfied." }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 123 "RK6_8err_ eqs := PrincipalErrorConditions(6,8,'expanded'):\nexpand(subs(ee,RK6_8 err_eqs)):\nmap(u->`if`(lhs(u)=rhs(u),0,1),%);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#7R\"\"\"F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$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 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 0 131 "err terms6_8 := PrincipalErrorTerms(6,8,'expanded'):\nevalf(evalf[14](sqrt (add(subs(ee,errterms6_8[i])^2,i=1.. nops(errterms6_8)))));" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#$\"+SI#3!>!#8" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 79 "The 2 norm of the principal error \+ of the order 5 embedded scheme is as follows." }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 153 "`errterms5_9*` := subs(b=`b*`,PrincipalErrorTer ms(5,9,'expanded')):\nevalf(evalf[14](sqrt(add(subs(ee,`errterms5_9*`[ i])^2,i=1.. nops(`errterms5_9*`)))));" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#$\"+a!#8" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> \+ " 0 "" {MPLTEXT 1 0 1 ";" }}}}{PARA 0 "" 0 "" {TEXT -1 52 "#---------- -----------------------------------------" }}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 44 "simultaneous construction of the two schemes" }}{PARA 0 " " 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 119 "We incorporate th e stage-order equations to ensure that stage 2 has stage-order 2 and s tages 3 to 8 have stage-order 3." }}{PARA 0 "" 0 "" {TEXT -1 48 "We al so incorporate the simplifying conditions: " }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "Sum(b[i]*a[i,j],i = j+1 .. 8) = 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=1" "6#/%\"jG\"\"\"" } {TEXT -1 10 " ( with " }{XPPEDIT 18 0 "c[1]=0" "6#/&%\"cG6#\"\"\"\" \"!" }{TEXT -1 5 " ), " }{XPPEDIT 18 0 "j = 6;" "6#/%\"jG\"\"'" } {TEXT -1 7 " and " }{XPPEDIT 18 0 "j=7" "6#/%\"jG\"\"(" }{TEXT -1 1 "." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 78 "The 8 simple order conditions used are given (in abreviated form) as foll ows. " }}{PARA 0 "" 0 "" {TEXT -1 78 "These include the 6 quadrature c onditions and two additional order conditions." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 178 "SO6 := Simp leOrderConditions(6):\n[seq([i,SO6[i]],i=[1,2,4,8,16,24,29,32])]:\nlin alg[augment](linalg[delcols](%,2..2),matrix([[` `]$(linalg[rowdim](%) )]),linalg[delcols](%,1..1));" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#K%'ma trixG6#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%\"#;F)/*&F ,F()F2F5F(#F(\"\"&7%\"#CF)/*(F,F(F2F(-%!G6#*&%\"aGF(-FM6#*&F8F(FPF(F(F (#F(\"#s7%\"#HF)/*(F,F(F2F(-FM6#*&F?F(FPF(F(#F(FI7%\"#KF)/*&F,F()F2FGF (#F(\"\"'Q)pprint596\"" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 136 "For the embedded order 5 scheme we use a selection \+ of 7 \"simple\" order conditions as given (in abreviated form) in the \+ following table. " }}{PARA 0 "" 0 "" {TEXT -1 78 "These include the 5 \+ quadrature conditions and two additional order conditions." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 194 "`SO 5*` := subs(b=`b*`,SimpleOrderConditions(5)):\n[seq([i,`SO5*`[i]],i=[1 ,2,4,8,12,15,16])]:\nlinalg[augment](linalg[delcols](%,2..2),matrix([[ ` `]$(linalg[rowdim](%))]),linalg[delcols](%,1..1));" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#K%'matrixG6#7)7%\"\"\"%#~~G/*&%#b*GF(%\"eGF(F(7%\" \"#F)/*&F,F(%\"cGF(#F(F/7%\"\"%F)/*&F,F()F2F/F(#F(\"\"$7%\"\")F)/*&F,F ()F2F:F(#F(F57%\"#7F)/*&F,F(-%!G6#*&%\"aGF(-FF6#*&F8F(FIF(F(F(#F(\"#g7 %\"#:F)/*&F,F(-FF6#*&F?F(FIF(F(#F(\"#?7%\"#;F)/*&F,F()F2F5F(#F(\"\"&Q) pprint186\"" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 582 "SO6_8 := SimpleOrderConditions(6,8,'expanded'):\n SO_eqs := [op(RowSumConditions(8,'expanded')),op(StageOrderConditions( 2,8,'expanded')),\n op(StageOrderConditions(3,4..8,'expa nded'))]:\n`SO5_9*` := subs(b=`b*`,SimpleOrderConditions(5,9,'expanded ')):\nord_cdns := [seq(SO6_8[i],i=[1,2,4,8,16,24,29,32])]:\n`ord_cdns* ` := [seq(`SO5_9*`[i],i=[1,2,4,8,12,15,16])]:\nsimp_eqs := [add(b[i]*a [i,1],i=1+1..8)=b[1],seq(add(b[i]*a[i,j],i=j+1..8)=b[j]*(1-c[j]),j=[6, 7])]:\nwt_eqns := [seq(b[i]=a[9,i],i=1..8)]:\ncdns := [op(ord_cdns),op (simp_eqs),op(SO_eqs),op(wt_eqns),op(`ord_cdns*`)]:" }}}{PARA 0 "" 0 " " {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 22 "We specify the nodes: \+ " }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "c[2] = 7/71;" "6# /&%\"cG6#\"\"#*&\"\"(\"\"\"\"#r!\"\"" }{TEXT -1 3 ", " }{XPPEDIT 18 0 "c[4] = 23/85;" "6#/&%\"cG6#\"\"%*&\"#B\"\"\"\"#&)!\"\"" }{TEXT -1 3 ", " }{XPPEDIT 18 0 "c[5] = 27/29;" "6#/&%\"cG6#\"\"&*&\"#F\"\"\"\" #H!\"\"" }{TEXT -1 3 ", " }{XPPEDIT 18 0 "c[6] = 61/90;" "6#/&%\"cG6# \"\"'*&\"#h\"\"\"\"#!*!\"\"" }{TEXT -1 3 ", " }{XPPEDIT 18 0 "c[7] = \+ 18/77;" "6#/&%\"cG6#\"\"(*&\"#=\"\"\"\"#x!\"\"" }{TEXT -1 3 ", " } {XPPEDIT 18 0 "c[8]=1" "6#/&%\"cG6#\"\")\"\"\"" }{TEXT -1 2 ", " }} {PARA 0 "" 0 "" {TEXT -1 31 "the zero linking coefficients: " }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "a[4,2]=0" "6#/&%\"aG6$\"\"% \"\"#\"\"!" }{TEXT -1 3 ", " }{XPPEDIT 18 0 "a[5,2]=0" "6#/&%\"aG6$\" \"&\"\"#\"\"!" }{TEXT -1 3 ", " }{XPPEDIT 18 0 "a[6,2]=0" "6#/&%\"aG6 $\"\"'\"\"#\"\"!" }{TEXT -1 3 ", " }{XPPEDIT 18 0 "a[7,2]=0" "6#/&%\" aG6$\"\"(\"\"#\"\"!" }{TEXT -1 3 ", " }{XPPEDIT 18 0 "a[8,2]=0" "6#/& %\"aG6$\"\")\"\"#\"\"!" }{TEXT -1 2 ", " }}{PARA 0 "" 0 "" {TEXT -1 22 "and the zero weights: " }}{PARA 256 "" 0 "" {TEXT -1 1 " " } {XPPEDIT 18 0 "b[2]=0" "6#/&%\"bG6#\"\"#\"\"!" }{TEXT -1 3 ", " } {XPPEDIT 18 0 "b[3]=0" "6#/&%\"bG6#\"\"$\"\"!" }{TEXT -1 1 "." }} {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT 260 4 "Note" } {TEXT -1 89 ": Calculations relating to the choice of nodes are perfor med in the following subsection." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }} {PARA 0 "" 0 "" {TEXT -1 125 "The weights of the order 6 scheme provid e the linking coefficients for the 9th stage of the embedded order 5 s cheme so that: " }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "a[ 9,i]=b[i]" "6#/&%\"aG6$\"\"*%\"iG&%\"bG6#F(" }{TEXT -1 3 ", " } {XPPEDIT 18 0 "i=1" "6#/%\"iG\"\"\"" }{TEXT -1 9 " . . . 8." }}{PARA 0 "" 0 "" {TEXT -1 22 "We also specify that " }{XPPEDIT 18 0 "c[9] = \+ 1;" "6#/&%\"cG6#\"\"*\"\"\"" }{TEXT -1 3 ", " }{XPPEDIT 18 0 "`b*`[2] = 0;" "6#/&%#b*G6#\"\"#\"\"!" }{TEXT -1 3 ", " }{XPPEDIT 18 0 "`b*`[ 3] = 0;" "6#/&%#b*G6#\"\"$\"\"!" }{TEXT -1 7 " and " }{XPPEDIT 18 0 "`b*`[9] = 1/17;" "6#/&%#b*G6#\"\"**&\"\"\"F)\"# " 0 "" {MPLTEXT 1 0 207 "e1 := \{c[2]=7/71,c[4]=23/85,c[5]=27/29,c[6]=61/90,c [7]=18/77,c[8]=1,c[9]=1,\n seq(a[i,2]=0,i=4..8),b[2]=0,b[3]=0,`b *`[2]=0,`b*`[3]=0,`b*`[9]=1/17\}:\neqns := subs(e1,cdns):\nnops(%);\ni ndets(eqns);\nnops(%);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#\"#W" }} {PARA 12 "" 1 "" {XPPMATH 20 "6# " 0 "" {MPLTEXT 1 0 22 "info level[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 1862 "e3 := \{ c[5] = 27/29, c[4] = 23/85, c[2] = 7/71, `b*`[9] = 1/17, `b*`[3] = 0, \+ `b*`[2] = 0, b[3] = 0, b[2] = 0, c[9] = 1, c[8] = 1, c[7] = 18/77, c[6 ] = 61/90, a[8,2] = 0, a[7,2] = 0, a[6,2] = 0, a[5,2] = 0, a[4,2] = 0, a[6,1] = 3323091540925544113/29573952540280488000, a[7,5] = -15538656 6948917103529551/2301833170083782096614930, a[8,7] = 1150428860219177/ 432917708558301, a[9,1] = 636077/8182296, `b*`[6] = 1723971130382250/4 007085927807811, b[7] = 1670085824869/14714276549256, `b*`[5] = 238262 40438664855/461934347497102656, a[8,1] = -1972640388605306005321/25773 9734717483367780, a[8,3] = 12354606914500963/386396118220908, a[8,6] = 370576933768080/119264931607213, a[2,1] = 7/71, a[3,1] = 6992/455175, b[5] = 30418033967/271431025272, a[9,8] = 13887/424328, a[6,5] = 2551 748271602005217/45506989353097008000, a[9,3] = 0, a[9,2] = 0, a[7,4] = -8122491009809811364010481/9979717849963846796488754, a[4,1] = 23/340 , a[9,7] = 1670085824869/14714276549256, `b*`[4] = 4006504071740375/84 742592073525056, a[9,6] = 862843671000/2241531443039, b[4] = 243150511 25/87140294626, `b*`[8] = 63672019983/6138204944224, a[9,4] = 24315051 125/87140294626, a[5,3] = -481282155/51607124, `b*`[7] = 1234855168974 51247/368256795835268016, a[7,6] = 107267076044076240/5448380198603929 51, a[4,3] = 69/340, a[8,5] = -17589460594924790303414/929346891509528 40583755, a[9,5] = 30418033967/271431025272, a[5,1] = 125814087/516071 24, a[3,2] = 75118/455175, b[8] = 13887/424328, a[7,1] = -376385546372 076652128/6002133498919339474385, a[7,3] = 19305251563909761378/196791 26225965047457, a[5,4] = 100879020/12901781, b[1] = 636077/8182296, a[ 6,4] = 169241467001919447401/178319980502135683200, c[3] = 46/255, a[8 ,4] = -56856879732578412491684/1967664304821917372209, `b*`[1] = 23112 2308093/3481256020752, a[6,3] = -10703826229567511/24340701679243200, \+ b[6] = 862843671000/2241531443039\}:" }{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 0 488 "subs(e3,matrix([[c[2],a[2,1],``$3],\n [c[3],a[3,1],a[3,2],``$2], \n [c[4],seq(a[4,i],i=1..3),``],\n [c[5],seq(a[5,i],i=1..4)],\n [c[ 6],seq(a[6,i],i=1..4)],[``$4,a[6,5]],\n [c[7],seq(a[7,i],i=1..4)],[`` $3,a[7,5],a[7,6]],\n [c[8],seq(a[8,i],i=1..4)],[``$2,seq(a[8,i],i=5.. 7)],\n [c[9],seq(a[9,i],i=1..4)],[``,seq(a[9,i],i=5..8)],\n [``,`___ _________________________`$4],\n [`b`,seq(b[i],i=1..4)],[``,seq(b[i], i=5..8)],\n [`b*`,seq(`b*`[i],i=1..4)],[``,seq(`b*`[i],i=5..8)],[``$4 ,`b*`[9]]]));" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#K%'matrixG6#747'#\"\" (\"#rF(%!GF+F+7'#\"#Y\"$b##\"%#*p\"'v^X#\"&=^(F2F+F+7'#\"#B\"#&)#F7\"$ S$\"\"!#\"#pF:F+7'#\"#F\"#H#\"*(39e7\")Crg^F;#!*b@G\"[FD#\"*?!z35\")\" y,H\"7'#\"#h\"#!*#\"48TaD4a\"4BL\"5+!)[!GSD&RdHF;#!26vcHi#Qq5\"2+KCz;q SV##\"6,uW>>+n9Cp\"\"6+KoN@]!)*>$y\"7'F+F+F+F+#\"4<_+-;F[*)\\L@+'F;#\"5y8w4Rc^_I>\"5du/ lfAE\"z'>#!:\"[5SO6)4)45\\A\")\":a()['zYQ'*\\yrz**7'F+F+F+#!9^&HN5<*[p c'Qb\"\":I\\h'4#y$3qJ$=I##\"3Si2Wg2ns5\"3^HRg)>!Q[a7'\"\"\"#!7@`+1`g)Q SE(>\"6!ynL[#[Ikw'>7'F+F+#!89MI!zC\\fg%*e<\"8bPeSG&4:*oMH*#\"0!3oP$pdq$\"08sgJ\\ E>\"#\"1x\">-')G/:\"\"0,$e&3x\"HV7'Fjo#\"'xgj\"('H#=)F;F;#\",D60:V#\", EYHSr)7'F+#\",nR.=/$\"-s_-J9F#\"-+5nVG')\".RIWJ:C##\".p[#e3q;\"/c#\\lF 9Z\"#\"&()Q\"\"'GVU7'F+%=____________________________GFcrFcrFcr7'%\"bG F_qF;F;FbqFeq7'%#b*G#\"-$43B7J#\"._2-c7[$F;F;#\"1v.urS]1S\"2c]_t?fUZ)7 'F+#\"2b[mQ/CEQ#\"3cE5(\\ZV$>Y#\"1]AQI6(Rs\"\"16y!y#f32S#\"3Z7X(*o^&[B \"\"3;!o_$ezc#o$#\",$)*>?nj\".CU%\\?Qh7'F+F+F+F+#Fjo\"# " 0 "" {MPLTEXT 1 0 136 "subs(e3,matrix([seq([c[i],seq(a[i,j],j=1..i-1),``$(10-i)],i=2 ..9),\n[`b`,seq(b[i],i=1..8),``],[`b*`,seq(`b*`[i],i=1..9)]])):\nevalf [8](%);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#K%'matrixG6#7,7,$\")\\:f)*! \"*F(%!GF+F+F+F+F+F+F+7,$\");#R!=!\")$\")D6O:F*$\").J];F/F+F+F+F+F+F+F +7,$\")C)eq#F/$\")fqknF*$\"\"!F:$\")=TH?F/F+F+F+F+F+F+7,$\")[M5$*F/$\" )4#zV#!\"(F9$!)g)eK*FB$\")'***=yFBF+F+F+F+F+7,$\")yxxnF/$\")\\lB6F/F9$ !)6](R%F/$\")k)3\\*F/$\")mP2cF*F+F+F+F+7,$\")BmPBF/$!)E'3F'F*F9$\")Y,5 )*F/$!)')**Q\")F/$!)tb]nF*$\")%)yo>F/F+F+F+7,$\"\"\"F:$!)Qh`wFBF9$\")U R(>$!\"'$!)?c*)GF`o$!)!pE*=F/$\")V<2JFB$\")YQdEFBF+F+7,Fjn$\")0#Qx(F*F 9F9$\")QL!z#F/$\")Vl?6F/$\")!\\$\\QF/$\")0,N6F/$\")TqsKF*F+7,%\"bGFjoF 9F9F\\pF^pF`pFbpFdpF+7,%#b*G$\")%\\!RmF*F9F9$\")<&ys%F*$\")z#z:&F*$\") kI-VF/$\")[C`LF/$\")oIP5F*$\")HN#)eF*Q(pprint16\"" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 146 "RK6_8eqs := [op(RowSumConditions(8,'expanded')),op(OrderConditions(6,8,'expanded' ))]:\n`RK5_9eqs*` := subs(b=`b*`,OrderConditions(5,9,'expanded')):" }} }{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 7 "Check: " }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 148 "simplify(subs(e3,RK6_8eqs )):\nmap(u->`if`(lhs(u)=rhs(u),0,1),%);\nnops(%);\nsimplify(subs(e3,`R K5_9eqs*`)):\nmap(u->`if`(lhs(u)=rhs(u),0,1),%);\nnops(%);" }{TEXT -1 0 "" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#7N\"\"!F$F$F$F$F$F$F$F$F$F$F$F$ F$F$F$F$F$F$F$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#\"#W" }}{PARA 11 "" 1 "" {XPPMATH 20 " 6#73\"\"!F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#\"#<" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 21 "determining the nodes" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 22 "eG .. general solution" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 71947 "eG := \{c[8] = 1, b[3] = \+ 0, b[2] = 0, c[9] = 1, b[5] = -1/60*(10*c[6]*c[7]*c[4]-5*c[6]*c[4]-5*c [7]*c[6]+3*c[6]-5*c[7]*c[4]+3*c[4]-2+3*c[7])/(-c[4]+c[5])/c[5]/(-c[5]^ 3+c[5]^2-c[7]*c[5]+c[7]*c[5]^2-c[6]*c[5]+c[6]*c[5]^2+c[7]*c[6]-c[6]*c[ 7]*c[5]), a[5,2] = 0, a[4,2] = 0, `b*`[3] = 0, `b*`[2] = 0, a[8,2] = 0 , a[7,2] = 0, a[6,2] = 0, a[6,5] = 1/2*c[6]*(30*c[6]^2*c[5]^2*c[4]^2-2 *c[6]^2*c[4]^2+c[6]^3*c[4]-12*c[6]^2*c[4]*c[5]^2-3*c[5]^2*c[6]*c[4]+c[ 5]*c[6]*c[4]^2+c[6]*c[4]^3-12*c[6]^2*c[4]^2*c[5]-30*c[5]*c[6]^3*c[4]^2 -20*c[5]*c[6]^2*c[4]^4+30*c[6]^2*c[5]*c[4]^3-c[5]*c[4]^3+20*c[4]^3*c[5 ]*c[6]^3-20*c[6]^2*c[5]^2*c[4]^3+c[5]^2*c[4]^2-30*c[5]^2*c[6]*c[4]^3+1 2*c[5]^2*c[6]*c[4]^2+2*c[6]^2*c[4]*c[5]+20*c[5]^2*c[6]*c[4]^4+2*c[6]^2 *c[5]^2-2*c[6]^3*c[5]+12*c[6]^3*c[4]*c[5])/c[5]/(c[4]^3-2*c[5]*c[4]^2+ 30*c[5]^2*c[4]^3-6*c[5]*c[4]^3-10*c[4]^4*c[5]^2-c[5]^3+10*c[5]^4*c[4]^ 2-30*c[5]^3*c[4]^2+6*c[5]^3*c[4]+2*c[5]^2*c[4]), b[1] = 1/60*(30*c[5]* c[6]*c[7]*c[4]-10*c[5]*c[6]*c[4]-10*c[6]*c[7]*c[5]+5*c[6]*c[5]-10*c[6] *c[7]*c[4]+5*c[6]*c[4]+5*c[7]*c[6]-3*c[6]-10*c[5]*c[7]*c[4]+5*c[5]*c[4 ]+5*c[7]*c[5]-3*c[5]+5*c[7]*c[4]-3*c[4]+2-3*c[7])/c[5]/c[6]/c[7]/c[4], a[6,1] = -1/4*c[6]*(60*c[5]^2*c[6]*c[4]^4-90*c[5]^2*c[6]^3*c[4]^2-60* c[5]*c[6]^2*c[4]^4-2*c[6]^3*c[4]^2+50*c[5]^4*c[4]^3*c[6]+40*c[6]^2*c[4 ]^5*c[5]-40*c[4]^5*c[5]^3+36*c[4]*c[5]^2*c[6]^3-20*c[5]^4*c[6]^2*c[4]^ 2+8*c[6]^2*c[5]^3+4*c[6]^2*c[4]^3-24*c[5]*c[6]^3*c[4]^2-8*c[6]^2*c[4]^ 2*c[5]+24*c[6]^2*c[5]*c[4]^3+60*c[4]^3*c[5]*c[6]^3-4*c[6]*c[5]*c[4]^3- 14*c[5]^3*c[6]*c[4]+7*c[5]^3*c[4]^2+6*c[5]^2*c[6]*c[4]^3+14*c[5]^2*c[6 ]*c[4]^2+66*c[5]^3*c[6]*c[4]^2-80*c[5]^3*c[6]^2*c[4]^3+30*c[6]^2*c[5]^ 2*c[4]^3-24*c[6]^2*c[5]^2*c[4]^2-240*c[6]*c[5]^3*c[4]^3-48*c[6]^2*c[5] ^3*c[4]+150*c[6]^2*c[5]^3*c[4]^2-24*c[4]^4*c[5]^2-9*c[5]^2*c[4]^3+6*c[ 4]^4*c[5]-40*c[5]^4*c[4]^4-20*c[6]^2*c[4]^4*c[5]^2-6*c[5]^2*c[6]^3-2*c [6]*c[4]^4-24*c[5]^3*c[4]^3+60*c[6]^3*c[4]^3*c[5]^2+7*c[6]^3*c[4]*c[5] -40*c[5]^2*c[6]*c[4]^5-40*c[6]^3*c[4]^4*c[5]+110*c[4]^4*c[6]*c[5]^3+12 0*c[5]^3*c[4]^4)/c[5]/c[4]^2/(10*c[5]^3*c[4]^2+10*c[5]^2*c[4]^3-30*c[5 ]^2*c[4]^2+6*c[5]^2*c[4]-c[5]^2+6*c[5]*c[4]^2+c[5]*c[4]-c[4]^2), b[4] \+ = 1/60*(-2+3*c[5]-5*c[7]*c[6]+3*c[7]+3*c[6]+10*c[6]*c[7]*c[5]-5*c[6]*c [5]-5*c[7]*c[5])/c[4]/(-c[6]*c[4]^3-c[4]^3+c[4]^4+c[7]*c[4]^2+c[6]*c[4 ]^2-c[6]*c[7]*c[4]+c[6]*c[7]*c[4]^2-c[5]*c[4]^3+c[5]*c[4]^2-c[7]*c[4]^ 3-c[5]*c[7]*c[4]+c[5]*c[7]*c[4]^2-c[5]*c[6]*c[4]+c[5]*c[6]*c[4]^2+c[6] *c[7]*c[5]-c[5]*c[6]*c[7]*c[4]), b[6] = 1/60*(10*c[5]*c[7]*c[4]-5*c[5] *c[4]-5*c[7]*c[5]+3*c[5]-5*c[7]*c[4]+3*c[4]-2+3*c[7])/(-c[4]+c[6])/(-c [5]+c[6])/c[6]/(c[6]^2-c[6]+c[7]-c[7]*c[6]), `b*`[9] = 1/10*(50*c[5]^2 *c[6]*c[4]^2-40*c[5]^2*c[6]*c[4]+10*c[6]*c[5]^2-40*c[5]*c[6]*c[4]^2+35 *c[5]*c[6]*c[4]-10*c[6]*c[5]+10*c[6]*c[4]^2-10*c[6]*c[4]+3*c[6]-5*c[5] ^2*c[4]-5*c[5]*c[4]^2+10*c[5]*c[4]-c[5]-c[4])/(3*c[6]-28*c[5]*c[6]*c[4 ]^2-28*c[5]^2*c[6]*c[4]+8*c[6]*c[5]^2-9*c[6]*c[4]-9*c[6]*c[5]+28*c[5]* c[6]*c[4]+30*c[5]^2*c[6]*c[4]^2+8*c[6]*c[4]^2-4*c[5]^2*c[4]-4*c[5]*c[4 ]^2+9*c[5]*c[4]-c[5]-c[4]), a[7,5] = -1/2*c[7]*(-17*c[5]^2*c[6]*c[4]^4 +5*c[6]*c[5]*c[4]^4+42*c[4]^2*c[5]^2*c[7]^3+17*c[7]^3*c[5]*c[4]^3-80*c [7]^3*c[5]^2*c[4]^3+c[7]*c[5]*c[4]^4+80*c[5]^2*c[4]^4*c[7]^2-60*c[4]^5 *c[6]^2*c[5]^2+20*c[4]^3*c[7]^3*c[5]^3-20*c[7]^2*c[5]^2*c[4]^5+40*c[4] ^5*c[6]*c[5]^3-5*c[5]*c[6]^2*c[4]^4+c[7]^3*c[4]^2-c[7]^2*c[4]^3-80*c[5 ]^3*c[4]^4*c[7]+20*c[7]^3*c[5]^2*c[4]^4-200*c[6]^2*c[4]^5*c[7]*c[5]^3- 290*c[5]^4*c[4]^2*c[7]^2*c[6]+20*c[6]^2*c[7]*c[5]^4*c[4]^2+40*c[6]*c[5 ]^2*c[4]^4*c[7]-140*c[5]^4*c[4]^3*c[6]-100*c[5]^4*c[4]^5*c[6]+190*c[5] ^4*c[4]^4*c[6]-40*c[7]^2*c[4]^4*c[5]^3+20*c[5]^4*c[4]^4*c[7]+2*c[4]^4* c[7]^2-20*c[7]^2*c[5]^4*c[4]^3-4*c[6]^2*c[7]^2*c[5]-8*c[7]*c[6]^2*c[5] ^3-3*c[7]*c[6]*c[5]^3+2*c[7]^2*c[6]*c[5]^3+60*c[7]*c[6]^2*c[5]*c[4]^5- 40*c[6]*c[5]^2*c[4]^5*c[7]+17*c[7]*c[4]^4*c[6]*c[5]+80*c[7]^2*c[5]^3*c [4]^3+5*c[7]*c[4]^4*c[6]^2+c[5]^2*c[7]^3-260*c[6]^2*c[7]*c[5]^2*c[4]^3 -12*c[6]^2*c[7]^2*c[5]^2*c[4]+20*c[6]^2*c[7]^2*c[5]^2*c[4]^2-40*c[6]*c [7]*c[5]^3*c[4]^3-41*c[6]^2*c[7]^3*c[4]*c[5]+20*c[7]^2*c[6]*c[5]*c[4]^ 4+3*c[7]*c[6]*c[4]^3-2*c[6]^2*c[7]*c[4]^2-3*c[6]*c[7]^2*c[4]^2+4*c[7]^ 2*c[6]*c[4]^3+6*c[5]*c[7]^2*c[4]^3+4*c[6]*c[5]*c[7]^2*c[4]^2+c[7]^2*c[ 5]*c[6]*c[4]-11*c[5]*c[7]*c[6]*c[4]^3+2*c[6]^2*c[4]^2*c[5]-c[7]^3*c[4] *c[5]+17*c[5]^2*c[6]*c[7]*c[4]^2-200*c[5]^4*c[7]^2*c[4]^4*c[6]-50*c[4] ^5*c[6]*c[7]*c[5]^3-150*c[5]^2*c[6]*c[7]^3*c[4]^4-c[6]^2*c[5]*c[4]^3+2 *c[7]^3*c[4]^2*c[6]*c[5]+2*c[6]^2*c[7]*c[4]*c[5]+108*c[7]*c[6]^2*c[5]* c[4]^3+4*c[6]^2*c[7]*c[5]^2+2*c[6]*c[5]*c[7]*c[4]^2-10*c[6]^2*c[7]^3*c [5]^2+6*c[5]^4*c[7]*c[6]-8*c[6]*c[7]^3*c[5]^2+3*c[6]*c[7]^2*c[5]^2+2*c [6]^2*c[7]^2*c[5]^2-27*c[6]^2*c[4]^2*c[7]*c[5]-62*c[6]^2*c[7]^2*c[4]^2 *c[5]+46*c[6]*c[5]^4*c[4]^2+20*c[4]^5*c[7]*c[5]^3+2*c[7]^3*c[4]*c[6]*c [5]-27*c[6]^2*c[5]^2*c[7]*c[4]+120*c[6]^2*c[5]^2*c[7]*c[4]^2-35*c[6]*c [5]^2*c[7]*c[4]^3-37*c[6]*c[5]^3*c[7]*c[4]^2+53*c[6]^2*c[5]^3*c[7]*c[4 ]-54*c[7]^2*c[5]^2*c[4]^3+2*c[7]^2*c[5]*c[4]^2-c[7]*c[5]*c[4]^3-3*c[6] *c[5]*c[4]^3-2*c[4]^3*c[7]^3+63*c[7]^3*c[5]^2*c[6]*c[4]-16*c[7]^2*c[5] ^2*c[6]*c[4]-5*c[6]*c[7]*c[4]^4+2*c[6]^2*c[7]^2*c[4]^2+2*c[6]^2*c[7]^2 *c[4]-10*c[5]^4*c[7]^2*c[6]+3*c[5]^3*c[6]*c[4]+3*c[5]^3*c[7]*c[4]+20*c [5]^3*c[6]*c[7]*c[4]-2*c[5]^3*c[4]^2+22*c[6]^2*c[7]^2*c[4]*c[5]+7*c[5] ^2*c[6]*c[4]^3-c[5]^2*c[7]*c[4]^2+c[5]^2*c[6]*c[4]^2-4*c[5]^2*c[7]*c[4 ]^3-8*c[5]^3*c[7]*c[4]^2-22*c[5]^3*c[6]*c[4]^2+240*c[5]^3*c[6]^2*c[4]^ 3+100*c[4]^3*c[7]^2*c[5]^2*c[6]^2+150*c[4]^5*c[5]^3*c[6]^2-3*c[7]*c[5] ^4*c[4]+5*c[7]^2*c[5]^4*c[4]+12*c[5]^2*c[4]^4*c[7]-98*c[6]^2*c[5]^2*c[ 4]^3+25*c[6]^2*c[5]^2*c[4]^2+54*c[6]*c[5]^3*c[4]^3+8*c[6]^2*c[5]^3*c[4 ]-63*c[6]^2*c[5]^3*c[4]^2+42*c[5]^3*c[7]*c[4]^3+60*c[5]^4*c[7]*c[4]^2* c[6]+200*c[5]^3*c[7]^3*c[4]^4*c[6]-4*c[6]^2*c[4]*c[5]^2+c[7]*c[6]^2*c[ 4]^3+200*c[7]^2*c[5]^2*c[6]*c[4]^3-240*c[7]^3*c[5]^2*c[6]*c[4]^2+180*c [7]^2*c[5]^3*c[6]*c[4]^2+3*c[5]^4*c[4]^2-80*c[5]^3*c[7]^3*c[6]*c[4]+c[ 7]^3*c[4]^2*c[6]-7*c[7]^3*c[4]^2*c[5]+20*c[4]^3*c[7]^2*c[5]*c[6]^2-19* c[7]^2*c[6]*c[5]*c[4]^3-11*c[7]^2*c[5]^2*c[6]*c[4]^2-3*c[4]^4*c[5]^2-2 7*c[7]^2*c[5]^3*c[6]*c[4]-480*c[7]^2*c[5]^3*c[6]*c[4]^3-380*c[5]^3*c[6 ]^2*c[4]^4-5*c[5]^4*c[4]^3-10*c[6]^2*c[4]^3*c[7]^2+130*c[5]^3*c[6]^2*c [7]*c[4]^3-5*c[7]^3*c[5]^2*c[4]-2*c[7]^2*c[5]^2*c[4]-4*c[5]^2*c[6]*c[7 ]*c[4]-160*c[6]^2*c[7]*c[5]^3*c[4]^2-80*c[6]^2*c[7]^2*c[4]*c[5]^3+2*c[ 5]^2*c[4]^3+12*c[7]^2*c[5]^2*c[4]^2+8*c[7]^2*c[5]^3*c[4]+400*c[7]^3*c[ 5]^2*c[6]^2*c[4]^3-17*c[7]^2*c[4]^4*c[5]+250*c[7]^2*c[5]^3*c[6]^2*c[4] ^2-100*c[6]^2*c[4]^5*c[5]*c[7]^2-200*c[7]^2*c[4]^5*c[6]*c[5]^3+150*c[5 ]^2*c[6]*c[4]^5*c[7]^2+380*c[7]^3*c[5]^2*c[4]^3*c[6]-20*c[7]^3*c[6]*c[ 5]*c[4]^3-210*c[7]^3*c[6]^2*c[5]*c[4]^3-6*c[5]^4*c[4]*c[6]+80*c[5]^4*c [7]^2*c[4]*c[6]+10*c[6]*c[7]^3*c[5]^3-36*c[5]^4*c[4]*c[7]*c[6]-50*c[4] ^5*c[6]^2*c[7]*c[5]^2+200*c[4]^5*c[6]^2*c[7]^2*c[5]^2+160*c[6]^2*c[4]^ 4*c[5]^2-450*c[7]^3*c[5]^3*c[4]^3*c[6]+5*c[6]^2*c[7]^3*c[4]^2-c[5]^3*c [7]^2+6*c[6]^2*c[7]^3*c[5]-160*c[7]*c[6]^2*c[5]*c[4]^4+200*c[5]^3*c[4] ^4*c[6]^2*c[7]-350*c[5]^3*c[4]^3*c[6]^2*c[7]^2-350*c[5]^4*c[4]^4*c[6]* c[7]+230*c[5]^2*c[4]^4*c[7]*c[6]^2+150*c[5]^3*c[4]^4*c[6]*c[7]+100*c[5 ]^4*c[4]^3*c[6]*c[7]-3*c[6]^2*c[7]^3*c[4]-42*c[7]^2*c[5]^3*c[4]^2+50*c [5]^4*c[4]^4*c[6]^2-20*c[5]^4*c[4]^3*c[6]^2+500*c[5]^3*c[4]^4*c[7]^2*c [6]+450*c[5]^4*c[4]^3*c[7]^2*c[6]-340*c[5]^2*c[4]^4*c[7]^2*c[6]-50*c[5 ]^4*c[4]^3*c[6]^2*c[7]+150*c[7]^2*c[4]^4*c[6]^2*c[5]+200*c[7]^2*c[4]^4 *c[5]^3*c[6]^2-350*c[7]^2*c[4]^4*c[6]^2*c[5]^2+10*c[7]^2*c[6]^2*c[5]^3 -60*c[4]^4*c[6]*c[5]^3+100*c[7]^3*c[6]^2*c[5]*c[4]^4+200*c[5]^4*c[4]^5 *c[6]*c[7]-200*c[7]^3*c[4]^4*c[6]^2*c[5]^2+5*c[5]^3*c[4]^4+80*c[6]^2*c [7]^3*c[5]^2*c[4]-270*c[6]^2*c[7]^3*c[5]^2*c[4]^2+290*c[5]^3*c[7]^3*c[ 4]^2*c[6]+140*c[6]^2*c[5]*c[7]^3*c[4]^2-5*c[7]^3*c[4]*c[5]^3)/c[5]/(10 0*c[5]^2*c[6]*c[4]^4-18*c[6]*c[5]*c[4]^4+4*c[5]^3*c[4]+50*c[4]^5*c[6]^ 2*c[5]^2+40*c[5]*c[6]^2*c[4]^4+3*c[5]^5-5*c[4]^4*c[6]^2+180*c[5]^4*c[4 ]^3*c[6]+100*c[5]^4*c[4]^5*c[6]-350*c[5]^4*c[4]^4*c[6]+140*c[5]^5*c[4] ^2-180*c[5]^5*c[4]^3-2*c[5]^4+30*c[4]^5*c[5]^3+300*c[5]^5*c[4]^3*c[6]+ 2*c[6]*c[5]^3-50*c[5]^5*c[4]^2*c[6]^2-40*c[5]^4*c[6]^2*c[4]+240*c[5]^4 *c[6]^2*c[4]^2-3*c[6]^2*c[5]^3+3*c[6]^2*c[4]^3+5*c[5]^4*c[6]^2-2*c[6]* c[4]^3+2*c[5]*c[4]^3+4*c[5]*c[6]*c[4]^2-6*c[6]^2*c[4]^2*c[5]-50*c[5]^4 *c[4]^5-13*c[6]^2*c[5]*c[4]^3-40*c[6]*c[5]^4*c[4]^2+6*c[6]*c[5]*c[4]^3 -15*c[5]^3*c[6]*c[4]-55*c[5]^2*c[6]*c[4]^3+6*c[5]^2*c[6]*c[4]^2+68*c[5 ]^3*c[6]*c[4]^2-100*c[4]^5*c[5]^3*c[6]^2+100*c[6]^2*c[5]^2*c[4]^3-100* c[6]*c[5]^3*c[4]^3+13*c[6]^2*c[5]^3*c[4]-100*c[6]^2*c[5]^3*c[4]^2-4*c[ 5]^2*c[4]^2+6*c[6]^2*c[4]*c[5]^2-68*c[5]^4*c[4]^2+23*c[4]^4*c[5]^2+350 *c[5]^3*c[6]^2*c[4]^4-4*c[5]^2*c[6]*c[4]-9*c[5]^2*c[4]^3-3*c[4]^4*c[5] -5*c[5]^5*c[6]-30*c[5]^6*c[4]^2+50*c[5]^6*c[6]*c[4]^2+180*c[5]^4*c[4]^ 4+10*c[5]^4*c[4]*c[6]-240*c[6]^2*c[4]^4*c[5]^2+3*c[6]*c[4]^4-100*c[5]^ 6*c[4]^3*c[6]+68*c[5]^3*c[4]^3-23*c[5]^5*c[4]-210*c[5]^5*c[6]*c[4]^2-3 50*c[5]^4*c[4]^3*c[6]^2-30*c[5]^2*c[6]*c[4]^5+50*c[5]^6*c[4]^3+60*c[4] ^4*c[6]*c[5]^3+100*c[5]^5*c[6]^2*c[4]^3+9*c[5]^4*c[4]-140*c[5]^3*c[4]^ 4+40*c[5]^5*c[4]*c[6]), `b*`[6] = -1/60*(-30*c[5]^3*c[4]+103*c[5]*c[7] *c[4]-18*c[6]+6*c[5]+6*c[4]+27*c[7]*c[6]-9*c[5]^2-9*c[4]^2-100*c[7]*c[ 6]*c[5]^3+522*c[5]*c[6]*c[7]*c[4]-111*c[6]*c[5]^2+75*c[6]*c[5]+60*c[6] *c[5]^3-9*c[7]*c[5]+600*c[6]*c[7]*c[5]^3*c[4]^3+14*c[7]*c[5]^2-100*c[7 ]*c[6]*c[4]^3+14*c[7]*c[4]^2-9*c[7]*c[4]-117*c[6]*c[7]*c[5]-111*c[6]*c [4]^2+60*c[6]*c[4]^3-117*c[6]*c[7]*c[4]+180*c[6]*c[7]*c[4]^2-30*c[5]*c [4]^3-174*c[5]*c[7]*c[4]^2+486*c[5]*c[6]*c[4]^2+107*c[5]*c[4]^2-318*c[ 5]*c[6]*c[4]-66*c[5]*c[4]+75*c[6]*c[4]+500*c[5]*c[7]*c[6]*c[4]^3+1420* c[5]^2*c[6]*c[7]*c[4]^2-840*c[6]*c[5]*c[7]*c[4]^2+180*c[6]*c[7]*c[5]^2 -900*c[6]*c[5]^2*c[7]*c[4]^3-900*c[6]*c[5]^3*c[7]*c[4]^2+50*c[7]*c[5]* c[4]^3-280*c[6]*c[5]*c[4]^3-280*c[5]^3*c[6]*c[4]+50*c[5]^3*c[7]*c[4]+1 07*c[5]^2*c[4]+500*c[5]^3*c[6]*c[7]*c[4]+50*c[5]^3*c[4]^2+470*c[5]^2*c [6]*c[4]^3+310*c[5]^2*c[7]*c[4]^2-770*c[5]^2*c[6]*c[4]^2-90*c[5]^2*c[7 ]*c[4]^3-90*c[5]^3*c[7]*c[4]^2+470*c[5]^3*c[6]*c[4]^2-300*c[6]*c[5]^3* c[4]^3-180*c[5]^2*c[4]^2+486*c[5]^2*c[6]*c[4]-174*c[5]^2*c[7]*c[4]-840 *c[5]^2*c[6]*c[7]*c[4]+50*c[5]^2*c[4]^3)/(-9*c[5]*c[7]*c[4]-3*c[7]*c[6 ]+3*c[6]^2-28*c[5]*c[6]*c[7]*c[4]-c[6]*c[5]+c[7]*c[5]+8*c[6]^2*c[4]^2+ c[7]*c[4]+9*c[6]*c[7]*c[5]+9*c[6]*c[7]*c[4]-8*c[6]*c[7]*c[4]^2+4*c[5]* c[7]*c[4]^2-4*c[5]*c[6]*c[4]^2+9*c[5]*c[6]*c[4]-c[6]*c[4]-28*c[6]^2*c[ 4]^2*c[5]+28*c[6]^2*c[4]*c[5]-30*c[5]^2*c[6]*c[7]*c[4]^2+28*c[6]*c[5]* c[7]*c[4]^2-8*c[6]*c[7]*c[5]^2+8*c[6]^2*c[5]^2-9*c[6]^2*c[5]+30*c[6]^2 *c[5]^2*c[4]^2-28*c[6]^2*c[4]*c[5]^2-9*c[6]^2*c[4]-4*c[5]^2*c[6]*c[4]+ 4*c[5]^2*c[7]*c[4]+28*c[5]^2*c[6]*c[7]*c[4])/(c[5]*c[4]-c[6]*c[4]-c[6] *c[5]+c[6]^2)/c[6], a[8,7] = (10*c[5]*c[6]*c[4]-5*c[6]*c[5]-5*c[6]*c[4 ]+3*c[6]-5*c[5]*c[4]+3*c[5]+3*c[4]-2)/c[7]/(c[6]-c[7])/(c[5]-c[7])/(c[ 4]-c[7])/(-20*c[5]*c[6]*c[4]+30*c[5]*c[6]*c[7]*c[4]+15*c[6]*c[5]-20*c[ 6]*c[7]*c[5]+15*c[6]*c[4]-20*c[6]*c[7]*c[4]-12*c[6]+15*c[7]*c[6]+15*c[ 5]*c[4]-20*c[5]*c[7]*c[4]-12*c[5]+15*c[7]*c[5]-12*c[4]+15*c[7]*c[4]+10 -12*c[7])*(c[4]-1)*(-1+c[5])*(-1+c[6])*(-1+c[7]), a[6,3] = 3/4*1/c[4]^ 2*c[6]*(60*c[4]^3*c[5]*c[6]^3-90*c[5]*c[6]^3*c[4]^2+36*c[6]^3*c[4]*c[5 ]-6*c[6]^3*c[5]+3*c[6]^3*c[4]-20*c[6]^2*c[5]^3*c[4]^2-80*c[6]^2*c[5]^2 *c[4]^3+150*c[6]^2*c[5]^2*c[4]^2-48*c[6]^2*c[4]*c[5]^2+8*c[6]^2*c[5]^2 -60*c[5]*c[6]^2*c[4]^4+90*c[6]^2*c[5]*c[4]^3-48*c[6]^2*c[4]^2*c[5]+4*c [6]^2*c[4]*c[5]-4*c[6]^2*c[4]^2+30*c[6]*c[5]^3*c[4]^3+90*c[5]^2*c[6]*c [4]^4-180*c[5]^2*c[6]*c[4]^3+54*c[5]^2*c[6]*c[4]^2-12*c[5]^2*c[6]*c[4] +18*c[6]*c[5]*c[4]^3+6*c[5]*c[6]*c[4]^2+3*c[5]^2*c[4]^2-3*c[5]*c[4]^3) /(10*c[5]^3*c[4]^2+10*c[5]^2*c[4]^3-30*c[5]^2*c[4]^2+6*c[5]^2*c[4]-c[5 ]^2+6*c[5]*c[4]^2+c[5]*c[4]-c[4]^2), b[7] = -1/60*(10*c[5]*c[6]*c[4]-5 *c[6]*c[5]-5*c[6]*c[4]+3*c[6]-5*c[5]*c[4]+3*c[5]+3*c[4]-2)/(c[4]-c[7]) /(c[5]-c[7])/(c[6]-c[7])/c[7]/(-1+c[7]), a[7,1] = 1/4*c[7]*(-51*c[5]^2 *c[6]*c[4]^4+14*c[6]*c[5]*c[4]^4+12*c[4]^2*c[5]^2*c[7]^3+2700*c[5]^4*c [4]^4*c[6]^2*c[7]+12*c[7]^3*c[5]*c[4]^3-98*c[7]^3*c[5]^2*c[4]^3-2*c[7] *c[5]*c[4]^4+132*c[5]^2*c[4]^4*c[7]^2-480*c[4]^5*c[6]^2*c[5]^2+200*c[4 ]^3*c[7]^3*c[5]^3-200*c[7]^2*c[5]^2*c[4]^5-760*c[4]^5*c[6]*c[5]^3-10*c [5]*c[6]^2*c[4]^4+200*c[5]^5*c[4]^3*c[7]^2*c[6]^2-400*c[5]^2*c[4]^6*c[ 6]^2*c[7]^2+4*c[4]^5*c[5]*c[7]-34*c[5]^2*c[4]^5*c[7]+400*c[5]^3*c[4]^6 *c[6]^2*c[7]-98*c[5]^3*c[4]^4*c[7]+420*c[7]^2*c[4]^5*c[5]^3+200*c[7]^3 *c[5]^2*c[4]^4-300*c[6]^2*c[4]^5*c[7]*c[5]^3-40*c[7]*c[5]^2*c[4]^6*c[6 ]+20*c[5]^4*c[4]^2*c[7]^2*c[6]-280*c[6]^2*c[7]*c[5]^4*c[4]^2-44*c[6]*c [4]^5*c[5]*c[7]-246*c[6]*c[5]^2*c[4]^4*c[7]+120*c[4]^6*c[6]^2*c[5]^2-1 00*c[5]^2*c[7]^2*c[4]^6*c[6]+141*c[5]^4*c[4]^3*c[6]+1220*c[5]^4*c[4]^5 *c[6]+400*c[5]^5*c[4]^5*c[6]^2-760*c[5]^4*c[4]^4*c[6]+100*c[5]^4*c[7]^ 3*c[4]^4-400*c[7]^2*c[4]^4*c[5]^3-200*c[5]^4*c[4]^5*c[7]^2+200*c[5]^4* c[4]^4*c[7]-2*c[4]^3*c[7]^3*c[6]-8*c[7]^2*c[6]*c[4]^4-380*c[7]^3*c[5]^ 3*c[4]^4-40*c[6]*c[4]^5*c[5]*c[7]^2+2*c[4]^4*c[7]^2-40*c[5]^2*c[7]^3*c [4]^5-200*c[7]^2*c[5]^4*c[4]^3-40*c[7]^3*c[5]^4*c[4]^3+200*c[5]^3*c[7] *c[4]^6*c[6]-10*c[6]^2*c[7]^3*c[4]^3+30*c[6]^2*c[4]^5*c[5]+12*c[7]*c[6 ]^2*c[5]^3+4*c[7]^2*c[6]*c[5]^3+400*c[5]^4*c[4]^6*c[6]^2-40*c[5]^3*c[7 ]*c[4]^6-380*c[5]^4*c[4]^5*c[7]+320*c[7]*c[6]^2*c[5]*c[4]^5+100*c[4]^6 *c[6]^2*c[7]*c[5]^2+34*c[5]^4*c[4]^2*c[7]^2+40*c[5]^5*c[7]^2*c[4]^3+24 0*c[6]*c[5]^2*c[4]^5*c[7]+60*c[5]^4*c[7]^3*c[4]^2*c[6]+26*c[7]*c[4]^4* c[6]*c[5]-50*c[4]^4*c[6]^2*c[5]^5+200*c[5]^5*c[7]*c[4]^4*c[6]+132*c[7] ^2*c[5]^3*c[4]^3-2*c[7]*c[4]^4*c[6]^2+60*c[5]^5*c[4]^2*c[6]^2*c[7]-500 *c[5]^5*c[4]^4*c[6]^2*c[7]+160*c[5]^5*c[4]^4*c[6]+134*c[6]^2*c[5]^4*c[ 7]*c[4]+24*c[5]^4*c[6]^2*c[4]-169*c[5]^4*c[6]^2*c[4]^2-500*c[5]^3*c[4] ^6*c[6]^2-40*c[5]^5*c[4]^4*c[7]+34*c[4]^5*c[5]*c[7]^2-100*c[5]^3*c[7]^ 2*c[4]^6-100*c[5]^5*c[4]^4*c[7]^2+160*c[4]^6*c[6]*c[5]^3-6*c[6]^2*c[7] *c[5]^2*c[4]^3+68*c[6]^2*c[7]^2*c[5]^2*c[4]-126*c[6]^2*c[7]^2*c[5]^2*c [4]^2-600*c[6]*c[7]*c[5]^3*c[4]^3-21*c[6]^2*c[7]^3*c[4]*c[5]+40*c[7]^2 *c[6]^2*c[5]^4+4*c[5]^4*c[7]*c[4]^2+58*c[7]^2*c[6]*c[5]*c[4]^4-100*c[7 ]^2*c[5]^5*c[6]^2*c[4]^2-320*c[5]^4*c[4]*c[7]^2*c[6]^2+20*c[6]^2*c[4]^ 4*c[7]^2+40*c[5]^2*c[4]^6*c[7]^2+6*c[7]^2*c[6]*c[4]^3-8*c[6]*c[5]*c[7] ^2*c[4]^2-4*c[5]*c[7]*c[6]*c[4]^3+14*c[5]^2*c[6]*c[7]*c[4]^2-910*c[4]^ 5*c[6]*c[7]*c[5]^3-340*c[5]^2*c[6]*c[7]^3*c[4]^4-4*c[6]^2*c[5]*c[4]^3+ 5*c[7]^3*c[4]^2*c[6]*c[5]+72*c[7]*c[6]^2*c[5]*c[4]^3+18*c[6]^2*c[7]^3* c[5]^2-12*c[6]^2*c[7]^2*c[5]^2-10*c[6]^2*c[4]^2*c[7]*c[5]-44*c[6]^2*c[ 7]^2*c[4]^2*c[5]-25*c[6]*c[5]^4*c[4]^2-24*c[7]*c[6]^2*c[5]^4+200*c[4]^ 5*c[7]*c[5]^3-2*c[6]^2*c[5]^2*c[7]*c[4]-42*c[6]^2*c[5]^2*c[7]*c[4]^2+7 4*c[6]*c[5]^2*c[7]*c[4]^3+80*c[6]*c[5]^3*c[7]*c[4]^2-50*c[6]^2*c[5]^3* c[7]*c[4]-34*c[5]^4*c[4]^3*c[7]-24*c[7]^2*c[5]^2*c[4]^3+420*c[5]^4*c[7 ]^2*c[4]^4-2*c[4]^3*c[7]^3+c[7]^3*c[5]^2*c[6]*c[4]+4*c[7]^2*c[5]^2*c[6 ]*c[4]-6*c[6]*c[7]*c[4]^4-4*c[6]^2*c[7]^2*c[4]^2+50*c[5]^5*c[4]^3*c[7] ^2*c[6]-18*c[5]^3*c[6]*c[7]*c[4]+14*c[6]^2*c[7]^2*c[4]*c[5]-20*c[5]^2* c[6]*c[4]^3+2*c[5]^2*c[7]*c[4]^3-2*c[5]^3*c[7]*c[4]^2+16*c[5]^3*c[6]*c [4]^2-96*c[5]^3*c[6]^2*c[4]^3+100*c[5]^5*c[4]^3*c[6]^2*c[7]-60*c[4]^3* c[7]^2*c[5]^2*c[6]^2+10*c[6]*c[7]*c[4]^5+1540*c[4]^5*c[5]^3*c[6]^2-34* c[7]^3*c[4]^4*c[5]-4*c[7]^2*c[5]^4*c[4]+12*c[5]^2*c[4]^4*c[7]-41*c[6]^ 2*c[5]^2*c[4]^3+14*c[6]^2*c[5]^2*c[4]^2-54*c[6]*c[5]^3*c[4]^3-12*c[6]^ 2*c[5]^3*c[4]+47*c[6]^2*c[5]^3*c[4]^2+12*c[5]^3*c[7]*c[4]^3-98*c[5]^4* c[7]*c[4]^2*c[6]-50*c[5]^3*c[7]^3*c[4]^4*c[6]+4*c[7]*c[6]^2*c[4]^3+156 *c[7]^2*c[5]^2*c[6]*c[4]^3-23*c[7]^3*c[5]^2*c[6]*c[4]^2+188*c[7]^2*c[5 ]^3*c[6]*c[4]^2+66*c[5]^3*c[7]^3*c[6]*c[4]+2*c[7]^3*c[4]^2*c[5]+100*c[ 5]^5*c[4]^5*c[7]+104*c[4]^3*c[7]^2*c[5]*c[6]^2-12*c[7]^2*c[6]*c[5]*c[4 ]^3-36*c[7]^2*c[5]^2*c[6]*c[4]^2-32*c[7]^2*c[5]^3*c[6]*c[4]-320*c[7]^2 *c[5]^3*c[6]*c[4]^3-560*c[5]^3*c[6]^2*c[4]^4-4*c[6]^2*c[4]^3*c[7]^2+24 0*c[5]^3*c[6]^2*c[7]*c[4]^3-34*c[5]^3*c[7]^3*c[4]^2-2*c[7]^3*c[5]^2*c[ 4]+164*c[6]^2*c[7]*c[5]^3*c[4]^2-20*c[6]^2*c[7]^2*c[4]*c[5]^3+2*c[7]^2 *c[5]^3*c[4]-130*c[5]^5*c[4]^3*c[6]*c[7]-90*c[7]^3*c[5]^2*c[6]^2*c[4]^ 3-16*c[7]^2*c[4]^4*c[5]-60*c[7]^2*c[5]^3*c[6]^2*c[4]^2-300*c[6]^2*c[4] ^5*c[5]*c[7]^2-150*c[7]^2*c[4]^5*c[6]*c[5]^3+380*c[5]^2*c[6]*c[4]^5*c[ 7]^2+140*c[7]^3*c[5]^2*c[4]^3*c[6]-810*c[7]^3*c[5]^3*c[6]^2*c[4]^2-14* c[7]^3*c[6]*c[5]*c[4]^3-280*c[7]^3*c[6]^2*c[5]*c[4]^3+240*c[7]^3*c[5]^ 3*c[6]^2*c[4]-20*c[5]^4*c[7]^2*c[4]*c[6]-6*c[6]*c[7]^3*c[5]^3+22*c[5]^ 4*c[4]*c[7]*c[6]-280*c[4]^5*c[6]^2*c[7]*c[5]^2+400*c[4]^5*c[6]^2*c[7]^ 2*c[5]^2+253*c[6]^2*c[4]^4*c[5]^2+470*c[7]^3*c[5]^3*c[4]^3*c[6]+100*c[ 5]^2*c[4]^5*c[7]^3*c[6]+6*c[6]^2*c[7]^3*c[4]^2-300*c[5]^5*c[4]^5*c[6]- 120*c[5]*c[7]*c[4]^6*c[6]^2-10*c[6]^2*c[7]*c[4]^5-1100*c[5]^4*c[4]^5*c [6]^2*c[7]-226*c[7]*c[6]^2*c[5]*c[4]^4-770*c[5]^3*c[4]^4*c[6]^2*c[7]+2 80*c[5]^3*c[4]^3*c[6]^2*c[7]^2-1000*c[5]^4*c[4]^4*c[6]*c[7]+240*c[5]^2 *c[4]^4*c[7]*c[6]^2+1240*c[5]^3*c[4]^4*c[6]*c[7]+600*c[5]^4*c[4]^3*c[6 ]*c[7]+1200*c[7]^3*c[5]^3*c[6]^2*c[4]^3+40*c[7]^3*c[4]^4*c[5]*c[6]-16* c[7]^2*c[5]^3*c[4]^2-1250*c[5]^4*c[4]^5*c[6]^2-600*c[6]^2*c[7]^3*c[5]^ 3*c[4]^4+200*c[6]^2*c[7]^2*c[5]^3*c[4]^5-200*c[6]^2*c[7]^3*c[4]^5*c[5] -140*c[5]^4*c[4]^4*c[6]^2+560*c[5]^4*c[4]^3*c[6]^2-1850*c[7]^2*c[5]^4* c[6]^2*c[4]^3+1230*c[7]^2*c[5]^4*c[6]^2*c[4]^2-150*c[5]^4*c[7]^3*c[4]^ 3*c[6]-300*c[5]^4*c[4]^6*c[6]+310*c[5]^3*c[4]^4*c[7]^2*c[6]-70*c[5]^4* c[4]^3*c[7]^2*c[6]-380*c[5]^2*c[4]^4*c[7]^2*c[6]-850*c[5]^4*c[4]^3*c[6 ]^2*c[7]-40*c[7]^2*c[4]^4*c[6]^2*c[5]-450*c[7]^2*c[4]^4*c[5]^3*c[6]^2+ 250*c[7]^2*c[4]^4*c[6]^2*c[5]^2-4*c[7]^2*c[4]^5+136*c[5]^2*c[6]*c[4]^5 +400*c[6]^2*c[7]^3*c[5]^2*c[4]^5+800*c[6]^2*c[7]^2*c[5]^4*c[4]^4+4*c[7 ]^3*c[4]^4+100*c[5]^4*c[4]^6*c[7]+437*c[4]^4*c[6]*c[5]^3-22*c[6]*c[5]* c[4]^5-60*c[5]^5*c[6]^2*c[4]^3+200*c[6]^2*c[4]^6*c[5]*c[7]^2+420*c[7]^ 3*c[6]^2*c[5]*c[4]^4+400*c[5]^4*c[4]^5*c[6]*c[7]-500*c[7]^3*c[4]^4*c[6 ]^2*c[5]^2+100*c[5]^3*c[7]^3*c[4]^5-103*c[6]^2*c[7]^3*c[5]^2*c[4]+260* c[6]^2*c[7]^3*c[5]^2*c[4]^2-300*c[5]^3*c[7]^3*c[4]^2*c[6]+97*c[6]^2*c[ 5]*c[7]^3*c[4]^2-30*c[6]^2*c[7]^3*c[5]^3+4*c[7]^3*c[4]*c[5]^3)/c[6]/(- 50*c[5]^2*c[6]*c[4]^4+23*c[5]^3*c[4]+100*c[5]^4*c[4]^3*c[6]+2*c[5]^2+2 *c[4]^2-3*c[5]^3-3*c[4]^3-3*c[6]*c[5]^2+5*c[6]*c[5]^3-3*c[6]*c[4]^2+5* c[6]*c[4]^3+23*c[5]*c[4]^3+18*c[5]*c[6]*c[4]^2-12*c[5]*c[4]^2+3*c[5]*c [6]*c[4]-2*c[5]*c[4]-50*c[6]*c[5]^4*c[4]^2-40*c[6]*c[5]*c[4]^3-40*c[5] ^3*c[6]*c[4]-12*c[5]^2*c[4]-140*c[5]^3*c[4]^2+240*c[5]^2*c[6]*c[4]^3-1 40*c[5]^2*c[6]*c[4]^2+240*c[5]^3*c[6]*c[4]^2-400*c[6]*c[5]^3*c[4]^3+91 *c[5]^2*c[4]^2+30*c[5]^4*c[4]^2+30*c[4]^4*c[5]^2-50*c[5]^4*c[4]^3+18*c [5]^2*c[6]*c[4]-140*c[5]^2*c[4]^3+210*c[5]^3*c[4]^3+100*c[4]^4*c[6]*c[ 5]^3-50*c[5]^3*c[4]^4)/c[5]/c[4]^2, `b*`[8] = 0, a[5,3] = -3/4*c[5]^2* (2*c[5]-3*c[4])/c[4]^2, `b*`[4] = -1/60*(-30*c[5]^3*c[4]-42*c[7]*c[6]^ 2+89*c[5]*c[7]*c[4]-18*c[6]+6*c[5]+6*c[4]+27*c[7]*c[6]-9*c[5]^2+180*c[ 7]*c[6]^2*c[5]^3-100*c[7]*c[6]*c[5]^3+27*c[6]^2+186*c[5]*c[6]*c[7]*c[4 ]-97*c[6]*c[5]^2+66*c[6]*c[5]+60*c[6]*c[5]^3-100*c[6]^2*c[5]^3-9*c[7]* c[5]+14*c[7]*c[5]^2-100*c[6]^2*c[7]*c[4]^2+120*c[6]^2*c[7]*c[4]+60*c[6 ]^2*c[4]^2-9*c[7]*c[4]-103*c[6]*c[7]*c[5]-36*c[6]*c[4]^2-61*c[6]*c[7]* c[4]+60*c[6]*c[7]*c[4]^2-30*c[5]*c[7]*c[4]^2+126*c[5]*c[6]*c[4]^2+18*c [5]*c[4]^2-112*c[5]*c[6]*c[4]-57*c[5]*c[4]+39*c[6]*c[4]-280*c[6]^2*c[4 ]^2*c[5]+330*c[6]^2*c[4]*c[5]+380*c[5]^2*c[6]*c[7]*c[4]^2-560*c[6]^2*c [7]*c[4]*c[5]+192*c[6]^2*c[7]*c[5]-310*c[6]^2*c[7]*c[5]^2-230*c[6]*c[5 ]*c[7]*c[4]^2+156*c[6]*c[7]*c[5]^2+180*c[6]^2*c[5]^2+500*c[6]^2*c[4]^2 *c[7]*c[5]+950*c[6]^2*c[5]^2*c[7]*c[4]-900*c[6]^2*c[5]^2*c[7]*c[4]^2-3 00*c[6]*c[5]^3*c[7]*c[4]^2-600*c[6]^2*c[5]^3*c[7]*c[4]-117*c[6]^2*c[5] -130*c[5]^3*c[6]*c[4]+50*c[5]^3*c[7]*c[4]+93*c[5]^2*c[4]+230*c[5]^3*c[ 6]*c[7]*c[4]+50*c[5]^2*c[7]*c[4]^2-190*c[5]^2*c[6]*c[4]^2+150*c[5]^3*c [6]*c[4]^2+470*c[6]^2*c[5]^2*c[4]^2+320*c[6]^2*c[5]^3*c[4]-300*c[6]^2* c[5]^3*c[4]^2-30*c[5]^2*c[4]^2-530*c[6]^2*c[4]*c[5]^2-75*c[6]^2*c[4]+1 56*c[5]^2*c[6]*c[4]-150*c[5]^2*c[7]*c[4]-270*c[5]^2*c[6]*c[7]*c[4]+600 *c[6]^2*c[7]*c[5]^3*c[4]^2)/(-9*c[5]*c[7]*c[4]-3*c[7]*c[6]-c[4]^2-28*c [5]*c[6]*c[7]*c[4]+c[7]*c[5]+c[7]*c[4]+9*c[6]*c[7]*c[5]-9*c[6]*c[4]^2+ 8*c[6]*c[4]^3+9*c[6]*c[7]*c[4]-8*c[6]*c[7]*c[4]^2-4*c[5]*c[4]^3+4*c[5] *c[7]*c[4]^2+28*c[5]*c[6]*c[4]^2+9*c[5]*c[4]^2-9*c[5]*c[6]*c[4]-c[5]*c [4]+3*c[6]*c[4]-30*c[5]^2*c[6]*c[7]*c[4]^2+28*c[6]*c[5]*c[7]*c[4]^2-8* c[6]*c[7]*c[5]^2-28*c[6]*c[5]*c[4]^3+30*c[5]^2*c[6]*c[4]^3-28*c[5]^2*c [6]*c[4]^2-4*c[5]^2*c[4]^2+8*c[5]^2*c[6]*c[4]+4*c[5]^2*c[7]*c[4]+28*c[ 5]^2*c[6]*c[7]*c[4])/(-c[4]+c[6])/(-c[4]+c[5])/c[4], a[8,6] = 1/2*(5*c [5]*c[6]*c[4]+c[6]-2*c[6]*c[4]-2*c[6]*c[5]+25*c[5]*c[7]*c[4]+6*c[5]+9* c[7]+10*c[7]^2*c[4]-4+6*c[4]-6*c[7]^2-14*c[7]*c[4]-10*c[5]*c[4]-14*c[7 ]*c[5]-20*c[7]^2*c[5]*c[4]+10*c[7]^2*c[5])/(c[6]-c[7])/(-c[4]+c[6])/(- c[5]+c[6])/c[6]/(-20*c[5]*c[6]*c[4]+30*c[5]*c[6]*c[7]*c[4]+15*c[6]*c[5 ]-20*c[6]*c[7]*c[5]+15*c[6]*c[4]-20*c[6]*c[7]*c[4]-12*c[6]+15*c[7]*c[6 ]+15*c[5]*c[4]-20*c[5]*c[7]*c[4]-12*c[5]+15*c[7]*c[5]-12*c[4]+15*c[7]* c[4]+10-12*c[7])*(c[4]-1)*(-1+c[5])*(-1+c[6]), b[8] = 1/60*(-20*c[5]*c [6]*c[4]+30*c[5]*c[6]*c[7]*c[4]+15*c[6]*c[5]-20*c[6]*c[7]*c[5]+15*c[6] *c[4]-20*c[6]*c[7]*c[4]-12*c[6]+15*c[7]*c[6]+15*c[5]*c[4]-20*c[5]*c[7] *c[4]-12*c[5]+15*c[7]*c[5]-12*c[4]+15*c[7]*c[4]+10-12*c[7])/(c[4]-1)/( -1+c[5])/(-1+c[6])/(-1+c[7]), a[5,1] = 1/4*c[5]*(2*c[5]^2-5*c[5]*c[4]+ 4*c[4]^2)/c[4]^2, `b*`[5] = 1/60*(-42*c[7]*c[6]^2+89*c[5]*c[7]*c[4]-18 *c[6]+6*c[5]+6*c[4]+27*c[7]*c[6]-9*c[4]^2+27*c[6]^2+186*c[5]*c[6]*c[7] *c[4]-36*c[6]*c[5]^2+39*c[6]*c[5]-9*c[7]*c[5]-100*c[6]^2*c[4]^3+600*c[ 6]^2*c[7]*c[5]^2*c[4]^3-100*c[7]*c[6]*c[4]^3-310*c[6]^2*c[7]*c[4]^2+19 2*c[6]^2*c[7]*c[4]+180*c[6]^2*c[4]^2+14*c[7]*c[4]^2-9*c[7]*c[4]-61*c[6 ]*c[7]*c[5]-97*c[6]*c[4]^2+60*c[6]*c[4]^3-103*c[6]*c[7]*c[4]+156*c[6]* c[7]*c[4]^2-30*c[5]*c[4]^3-150*c[5]*c[7]*c[4]^2+156*c[5]*c[6]*c[4]^2+9 3*c[5]*c[4]^2-112*c[5]*c[6]*c[4]-57*c[5]*c[4]+66*c[6]*c[4]+230*c[5]*c[ 7]*c[6]*c[4]^3-530*c[6]^2*c[4]^2*c[5]+330*c[6]^2*c[4]*c[5]+380*c[5]^2* c[6]*c[7]*c[4]^2+320*c[6]^2*c[5]*c[4]^3-560*c[6]^2*c[7]*c[4]*c[5]-600* c[7]*c[6]^2*c[5]*c[4]^3+120*c[6]^2*c[7]*c[5]-100*c[6]^2*c[7]*c[5]^2-27 0*c[6]*c[5]*c[7]*c[4]^2+60*c[6]*c[7]*c[5]^2+60*c[6]^2*c[5]^2+950*c[6]^ 2*c[4]^2*c[7]*c[5]+500*c[6]^2*c[5]^2*c[7]*c[4]-900*c[6]^2*c[5]^2*c[7]* c[4]^2-300*c[6]*c[5]^2*c[7]*c[4]^3-75*c[6]^2*c[5]+50*c[7]*c[5]*c[4]^3- 130*c[6]*c[5]*c[4]^3+18*c[5]^2*c[4]+150*c[5]^2*c[6]*c[4]^3+50*c[5]^2*c [7]*c[4]^2-190*c[5]^2*c[6]*c[4]^2-300*c[6]^2*c[5]^2*c[4]^3+470*c[6]^2* c[5]^2*c[4]^2-30*c[5]^2*c[4]^2-280*c[6]^2*c[4]*c[5]^2+180*c[7]*c[6]^2* c[4]^3-117*c[6]^2*c[4]+126*c[5]^2*c[6]*c[4]-30*c[5]^2*c[7]*c[4]-230*c[ 5]^2*c[6]*c[7]*c[4])/(-4*c[5]^3*c[4]-9*c[5]*c[7]*c[4]-3*c[7]*c[6]-c[5] ^2-28*c[5]*c[6]*c[7]*c[4]-9*c[6]*c[5]^2+3*c[6]*c[5]+8*c[6]*c[5]^3+c[7] *c[5]+c[7]*c[4]+9*c[6]*c[7]*c[5]+9*c[6]*c[7]*c[4]-8*c[6]*c[7]*c[4]^2+4 *c[5]*c[7]*c[4]^2+8*c[5]*c[6]*c[4]^2-9*c[5]*c[6]*c[4]-c[5]*c[4]-30*c[5 ]^2*c[6]*c[7]*c[4]^2+28*c[6]*c[5]*c[7]*c[4]^2-8*c[6]*c[7]*c[5]^2-28*c[ 5]^3*c[6]*c[4]+9*c[5]^2*c[4]-28*c[5]^2*c[6]*c[4]^2+30*c[5]^3*c[6]*c[4] ^2-4*c[5]^2*c[4]^2+28*c[5]^2*c[6]*c[4]+4*c[5]^2*c[7]*c[4]+28*c[5]^2*c[ 6]*c[7]*c[4])/(-c[5]+c[6])/(-c[4]+c[5])/c[5], a[5,4] = c[5]^2*(-c[4]+c [5])/c[4]^2, a[7,6] = 1/2*(5*c[5]*c[4]+1-2*c[4]-2*c[5])*(c[4]-c[7])*(c [5]-c[7])*(c[6]-c[7])*c[7]/(10*c[5]*c[6]*c[4]-5*c[6]*c[5]-5*c[6]*c[4]+ 3*c[6]-5*c[5]*c[4]+3*c[5]+3*c[4]-2)/(-c[4]+c[6])/(-c[5]+c[6])/c[6], c[ 3] = 2/3*c[4], a[4,3] = 3/4*c[4], a[3,2] = 2/9*c[4]^2/c[2], a[7,4] = - 1/2*c[7]*(-68*c[5]^2*c[6]*c[4]^4-7*c[6]*c[5]*c[4]^4+7*c[4]^2*c[5]^2*c[ 7]^3-1400*c[5]^4*c[4]^4*c[6]^2*c[7]+11*c[7]^3*c[5]*c[4]^3-72*c[7]^3*c[ 5]^2*c[4]^3-3*c[7]*c[5]*c[4]^4-58*c[5]^2*c[4]^4*c[7]^2-120*c[6]*c[4]^7 *c[7]*c[5]+320*c[4]^5*c[6]^2*c[5]^2-200*c[4]^3*c[7]^3*c[5]^3-100*c[4]^ 5*c[6]*c[5]^3-2*c[5]*c[6]^2*c[4]^4-600*c[4]^5*c[7]^3*c[6]*c[5]^3-200*c [5]^5*c[4]^3*c[7]^2*c[6]^2+400*c[5]^2*c[4]^6*c[6]^2*c[7]^2-21*c[4]^5*c [5]*c[7]+120*c[5]^2*c[4]^5*c[7]-600*c[5]^3*c[4]^6*c[6]^2*c[7]-198*c[5] ^3*c[4]^4*c[7]-280*c[7]^2*c[4]^5*c[5]^3+240*c[7]^3*c[5]^2*c[4]^4+1250* c[6]^2*c[4]^5*c[7]*c[5]^3+120*c[7]*c[5]^2*c[4]^6*c[6]-340*c[5]^4*c[4]^ 2*c[7]^2*c[6]+400*c[6]^2*c[7]*c[5]^4*c[4]^2-240*c[5]^2*c[4]^6*c[7]-300 *c[6]*c[5]*c[7]^2*c[4]^6-228*c[6]*c[4]^5*c[5]*c[7]-119*c[6]*c[5]^2*c[4 ]^4*c[7]-200*c[5]*c[7]^3*c[4]^6*c[6]-600*c[5]^4*c[7]*c[4]^6*c[6]-120*c [4]^6*c[6]^2*c[5]^2+600*c[5]^2*c[7]^2*c[4]^6*c[6]+100*c[5]^2*c[7]^3*c[ 4]^6-187*c[5]^4*c[4]^3*c[6]-910*c[5]^4*c[4]^5*c[6]-320*c[5]^2*c[6]*c[4 ]^6+620*c[5]^4*c[4]^4*c[6]-100*c[5]^4*c[7]^3*c[4]^4-40*c[7]^2*c[4]^4*c [5]^3+40*c[7]^3*c[4]^5*c[5]+10*c[6]*c[7]^2*c[4]^5-10*c[6]*c[7]^3*c[4]^ 4+100*c[5]^4*c[4]^5*c[7]^2+80*c[5]^4*c[4]^4*c[7]+8*c[4]^3*c[7]^3*c[6]- 2*c[7]^2*c[6]*c[4]^4+280*c[7]^3*c[5]^3*c[4]^4+80*c[6]*c[4]^5*c[5]*c[7] ^2+c[4]^4*c[7]^2-360*c[5]^2*c[7]^3*c[4]^5-80*c[7]^2*c[5]^4*c[4]^3+40*c [7]^3*c[5]^4*c[4]^3-1200*c[5]^3*c[7]*c[4]^6*c[6]+10*c[6]^2*c[7]^3*c[4] ^3-10*c[6]^2*c[4]^5*c[5]-12*c[7]*c[6]^2*c[5]^3-4*c[7]^2*c[6]*c[5]^3-80 *c[5]^4*c[4]^5*c[7]-240*c[7]*c[6]^2*c[5]*c[4]^5-3*c[5]^4*c[4]^2*c[7]^2 +20*c[5]^5*c[7]^2*c[4]^3+180*c[6]*c[5]^2*c[4]^5*c[7]-60*c[5]^4*c[7]^3* c[4]^2*c[6]+400*c[5]^2*c[7]^3*c[4]^6*c[6]-25*c[4]^5*c[5]^3+200*c[5]^5* c[4]^4*c[7]^2*c[6]+60*c[5]^2*c[4]^7*c[7]+100*c[7]*c[4]^4*c[6]*c[5]-40* c[5]*c[4]^6*c[7]^2-150*c[4]^4*c[6]^2*c[5]^5+50*c[5]^5*c[7]*c[4]^4*c[6] +174*c[7]^2*c[5]^3*c[4]^3+8*c[7]*c[4]^4*c[6]^2+600*c[5]^3*c[7]*c[4]^7* c[6]-60*c[5]^5*c[4]^2*c[6]^2*c[7]+200*c[5]^5*c[4]^4*c[6]^2*c[7]-40*c[5 ]^5*c[4]^4*c[6]-200*c[5]^5*c[7]*c[4]^5*c[6]-149*c[6]^2*c[5]^4*c[7]*c[4 ]-24*c[5]^4*c[6]^2*c[4]+189*c[5]^4*c[6]^2*c[4]^2+300*c[5]^3*c[4]^6*c[6 ]^2-20*c[5]^5*c[4]^4*c[7]-5*c[4]^5*c[5]*c[7]^2+600*c[4]^6*c[6]*c[5]^3- 84*c[6]^2*c[7]*c[5]^2*c[4]^3-74*c[6]^2*c[7]^2*c[5]^2*c[4]+162*c[6]^2*c [7]^2*c[5]^2*c[4]^2+311*c[6]*c[7]*c[5]^3*c[4]^3+21*c[6]^2*c[7]^3*c[4]* c[5]-40*c[7]^2*c[6]^2*c[5]^4+11*c[5]^4*c[7]*c[4]^2+107*c[7]^2*c[6]*c[5 ]*c[4]^4+100*c[7]^2*c[5]^5*c[6]^2*c[4]^2+320*c[5]^4*c[4]*c[7]^2*c[6]^2 -10*c[6]^2*c[4]^4*c[7]^2+300*c[5]^2*c[4]^6*c[7]^2-3*c[7]^2*c[6]*c[4]^3 -2*c[5]*c[7]^2*c[4]^3+15*c[6]*c[5]*c[7]^2*c[4]^2-8*c[5]*c[7]*c[6]*c[4] ^3-14*c[5]^2*c[6]*c[7]*c[4]^2-2000*c[5]^4*c[7]^2*c[4]^4*c[6]+200*c[4]^ 5*c[6]*c[7]*c[5]^3+250*c[5]^2*c[6]*c[7]^3*c[4]^4+4*c[6]^2*c[5]*c[4]^3- 26*c[7]^3*c[4]^2*c[6]*c[5]-69*c[7]*c[6]^2*c[5]*c[4]^3-18*c[6]^2*c[7]^3 *c[5]^2+12*c[6]^2*c[7]^2*c[5]^2+10*c[6]^2*c[4]^2*c[7]*c[5]+50*c[6]^2*c [7]^2*c[4]^2*c[5]+31*c[6]*c[5]^4*c[4]^2+24*c[7]*c[6]^2*c[5]^4+240*c[4] ^5*c[7]*c[5]^3+2*c[6]^2*c[5]^2*c[7]*c[4]+39*c[6]^2*c[5]^2*c[7]*c[4]^2- 17*c[6]*c[5]^2*c[7]*c[4]^3-73*c[6]*c[5]^3*c[7]*c[4]^2+59*c[6]^2*c[5]^3 *c[7]*c[4]-12*c[5]^4*c[4]^3*c[7]-12*c[7]^2*c[5]^2*c[4]^3+40*c[5]^4*c[7 ]^2*c[4]^4-c[4]^3*c[7]^3+17*c[7]^3*c[5]^2*c[6]*c[4]-9*c[7]^2*c[5]^2*c[ 6]*c[4]+3*c[6]*c[7]*c[4]^4+4*c[4]^5*c[5]+4*c[6]^2*c[7]^2*c[4]^2-150*c[ 5]^5*c[4]^3*c[7]^2*c[6]+21*c[5]^3*c[6]*c[7]*c[4]-14*c[6]^2*c[7]^2*c[4] *c[5]+23*c[5]^2*c[6]*c[4]^3+13*c[5]^2*c[7]*c[4]^3-11*c[5]^3*c[7]*c[4]^ 2-17*c[5]^3*c[6]*c[4]^2+168*c[5]^3*c[6]^2*c[4]^3+50*c[5]^5*c[4]^3*c[6] ^2*c[7]-220*c[4]^3*c[7]^2*c[5]^2*c[6]^2-6*c[6]*c[7]*c[4]^5-580*c[4]^5* c[5]^3*c[6]^2-31*c[7]^3*c[4]^4*c[5]-2*c[7]^2*c[5]^4*c[4]-4*c[5]^2*c[4] ^4*c[7]+53*c[6]^2*c[5]^2*c[4]^3-14*c[6]^2*c[5]^2*c[4]^2+47*c[6]*c[5]^3 *c[4]^3+12*c[6]^2*c[5]^3*c[4]-59*c[6]^2*c[5]^3*c[4]^2+28*c[5]^3*c[7]*c [4]^3+127*c[5]^4*c[7]*c[4]^2*c[6]+1250*c[5]^3*c[7]^3*c[4]^4*c[6]+10*c[ 6]*c[5]*c[4]^6+240*c[5]*c[4]^6*c[7]*c[6]-4*c[7]*c[6]^2*c[4]^3+11*c[7]^ 2*c[5]^2*c[6]*c[4]^3-80*c[7]^3*c[5]^2*c[6]*c[4]^2-121*c[7]^2*c[5]^3*c[ 6]*c[4]^2-96*c[5]^3*c[7]^3*c[6]*c[4]+c[7]^3*c[4]^2*c[5]-68*c[4]^3*c[7] ^2*c[5]*c[6]^2-80*c[7]^2*c[6]*c[5]*c[4]^3+68*c[7]^2*c[5]^2*c[6]*c[4]^2 -10*c[4]^4*c[5]^2+20*c[7]^2*c[5]^3*c[6]*c[4]-240*c[7]^2*c[5]^3*c[6]*c[ 4]^3-9*c[5]^4*c[4]^3-2*c[6]^2*c[4]^3*c[7]^2+260*c[5]^3*c[6]^2*c[7]*c[4 ]^3+49*c[5]^3*c[7]^3*c[4]^2-c[7]^3*c[5]^2*c[4]-228*c[6]^2*c[7]*c[5]^3* c[4]^2+30*c[6]^2*c[7]^2*c[4]*c[5]^3-2*c[7]^2*c[5]^2*c[4]^2+5*c[7]^2*c[ 5]^3*c[4]+40*c[5]^5*c[4]^3*c[6]*c[7]+36*c[4]^6*c[7]*c[5]+90*c[7]^3*c[5 ]^2*c[6]^2*c[4]^3+16*c[7]^2*c[4]^4*c[5]-20*c[7]^2*c[5]^3*c[6]^2*c[4]^2 +300*c[6]^2*c[4]^5*c[5]*c[7]^2+250*c[7]^2*c[4]^5*c[6]*c[5]^3-490*c[5]^ 2*c[6]*c[4]^5*c[7]^2+120*c[7]^3*c[5]^2*c[4]^3*c[6]+810*c[7]^3*c[5]^3*c [6]^2*c[4]^2+111*c[7]^3*c[6]*c[5]*c[4]^3+280*c[7]^3*c[6]^2*c[5]*c[4]^3 -240*c[7]^3*c[5]^3*c[6]^2*c[4]+15*c[5]^4*c[4]^4+60*c[5]^4*c[7]^2*c[4]* c[6]+6*c[6]*c[7]^3*c[5]^3-31*c[5]^4*c[4]*c[7]*c[6]-200*c[4]^5*c[6]^2*c [7]*c[5]^2-500*c[4]^5*c[6]^2*c[7]^2*c[5]^2+9*c[4]^5*c[5]^2-181*c[6]^2* c[4]^4*c[5]^2-1280*c[7]^3*c[5]^3*c[4]^3*c[6]-600*c[5]^2*c[4]^5*c[7]^3* c[6]-6*c[4]^6*c[5]-6*c[6]^2*c[7]^3*c[4]^2+100*c[5]^5*c[4]^5*c[6]+120*c [5]*c[7]*c[4]^6*c[6]^2+420*c[6]*c[4]^5*c[7]^3*c[5]+600*c[5]^4*c[4]^5*c [6]^2*c[7]+175*c[7]*c[6]^2*c[5]*c[4]^4-430*c[5]^3*c[4]^4*c[6]^2*c[7]+2 00*c[5]^3*c[4]^3*c[6]^2*c[7]^2-450*c[5]^4*c[4]^4*c[6]*c[7]+160*c[5]^2* c[4]^4*c[7]*c[6]^2-200*c[5]^3*c[4]^4*c[6]*c[7]-280*c[5]^4*c[4]^3*c[6]* c[7]+6*c[5]^3*c[4]^3-100*c[5]^2*c[4]^7*c[7]^2-1200*c[7]^3*c[5]^3*c[6]^ 2*c[4]^3-320*c[7]^3*c[4]^4*c[5]*c[6]-30*c[7]^2*c[5]^3*c[4]^2-300*c[6]* c[5]^3*c[4]^7-350*c[5]^4*c[4]^5*c[6]^2+600*c[6]^2*c[7]^3*c[5]^3*c[4]^4 +600*c[6]*c[7]^2*c[5]^4*c[4]^5+200*c[6]^2*c[7]^3*c[4]^5*c[5]+200*c[7]^ 2*c[4]^7*c[6]*c[5]+1100*c[5]^4*c[4]^4*c[6]^2-720*c[5]^4*c[4]^3*c[6]^2+ 1750*c[7]^2*c[5]^4*c[6]^2*c[4]^3-1230*c[7]^2*c[5]^4*c[6]^2*c[4]^2+150* c[5]^4*c[7]^3*c[4]^3*c[6]+300*c[5]^4*c[4]^6*c[6]+480*c[5]^3*c[4]^4*c[7 ]^2*c[6]+1420*c[5]^4*c[4]^3*c[7]^2*c[6]+20*c[5]^2*c[4]^4*c[7]^2*c[6]+1 30*c[5]^4*c[4]^3*c[6]^2*c[7]-40*c[7]^2*c[4]^4*c[6]^2*c[5]-350*c[7]^2*c [4]^4*c[5]^3*c[6]^2+230*c[7]^2*c[4]^4*c[6]^2*c[5]^2+10*c[4]^6*c[5]^2+1 88*c[5]^2*c[6]*c[4]^5-400*c[6]^2*c[7]^3*c[5]^2*c[4]^5-600*c[6]^2*c[7]^ 2*c[5]^4*c[4]^4+120*c[5]^2*c[4]^7*c[6]-94*c[4]^4*c[6]*c[5]^3+6*c[6]*c[ 5]*c[4]^5+60*c[5]^5*c[6]^2*c[4]^3-200*c[6]^2*c[4]^6*c[5]*c[7]^2-420*c[ 7]^3*c[6]^2*c[5]*c[4]^4+1550*c[5]^4*c[4]^5*c[6]*c[7]+500*c[7]^3*c[4]^4 *c[6]^2*c[5]^2-400*c[5]^2*c[4]^7*c[6]*c[7]^2+6*c[5]^3*c[4]^4+103*c[6]^ 2*c[7]^3*c[5]^2*c[4]-260*c[6]^2*c[7]^3*c[5]^2*c[4]^2+540*c[5]^3*c[7]^3 *c[4]^2*c[6]-97*c[6]^2*c[5]*c[7]^3*c[4]^2+30*c[6]^2*c[7]^3*c[5]^3-4*c[ 7]^3*c[4]*c[5]^3)/(-240*c[5]^2*c[6]*c[4]^4+40*c[6]*c[5]*c[4]^4-9*c[5]^ 3*c[4]-100*c[4]^5*c[6]*c[5]^3-350*c[5]^4*c[4]^3*c[6]+30*c[5]^5*c[4]^2+ 2*c[5]^3-2*c[4]^3-50*c[5]^5*c[4]^3-3*c[5]^4+3*c[4]^4+50*c[4]^5*c[5]^3+ 100*c[5]^5*c[4]^3*c[6]-3*c[6]*c[5]^3+3*c[6]*c[4]^3+9*c[5]*c[4]^3-6*c[5 ]*c[6]*c[4]^2+4*c[5]*c[4]^2+240*c[6]*c[5]^4*c[4]^2+5*c[6]*c[5]^4-13*c[ 6]*c[5]*c[4]^3+13*c[5]^3*c[6]*c[4]-4*c[5]^2*c[4]+68*c[5]^3*c[4]^2+100* c[5]^2*c[6]*c[4]^3-100*c[5]^3*c[6]*c[4]^2-140*c[5]^4*c[4]^2+140*c[4]^4 *c[5]^2+180*c[5]^4*c[4]^3+6*c[5]^2*c[6]*c[4]-68*c[5]^2*c[4]^3-23*c[4]^ 4*c[5]-40*c[5]^4*c[4]*c[6]-30*c[4]^5*c[5]^2-5*c[6]*c[4]^4-50*c[5]^5*c[ 6]*c[4]^2+50*c[5]^2*c[6]*c[4]^5+350*c[4]^4*c[6]*c[5]^3+23*c[5]^4*c[4]- 180*c[5]^3*c[4]^4)/(-c[4]+c[6])/c[4]^2, `b*`[7] = 1/60*(-30*c[5]^3*c[4 ]-18*c[6]+6*c[5]+6*c[4]-9*c[5]^2-9*c[4]^2+27*c[6]^2-97*c[6]*c[5]^2+66* c[6]*c[5]+60*c[6]*c[5]^3-100*c[6]^2*c[5]^3-100*c[6]^2*c[4]^3+180*c[6]^ 2*c[4]^2-97*c[6]*c[4]^2+60*c[6]*c[4]^3-30*c[5]*c[4]^3+312*c[5]*c[6]*c[ 4]^2+107*c[5]*c[4]^2-215*c[5]*c[6]*c[4]-66*c[5]*c[4]+66*c[6]*c[4]-840* c[6]^2*c[4]^2*c[5]+522*c[6]^2*c[4]*c[5]+500*c[6]^2*c[5]*c[4]^3+180*c[6 ]^2*c[5]^2-117*c[6]^2*c[5]-230*c[6]*c[5]*c[4]^3-230*c[5]^3*c[6]*c[4]+1 07*c[5]^2*c[4]+50*c[5]^3*c[4]^2+380*c[5]^2*c[6]*c[4]^3-460*c[5]^2*c[6] *c[4]^2+380*c[5]^3*c[6]*c[4]^2+600*c[5]^3*c[6]^2*c[4]^3-900*c[6]^2*c[5 ]^2*c[4]^3+1420*c[6]^2*c[5]^2*c[4]^2-300*c[6]*c[5]^3*c[4]^3+500*c[6]^2 *c[5]^3*c[4]-900*c[6]^2*c[5]^3*c[4]^2-180*c[5]^2*c[4]^2-840*c[6]^2*c[4 ]*c[5]^2-117*c[6]^2*c[4]+312*c[5]^2*c[6]*c[4]+50*c[5]^2*c[4]^3)/c[7]/( -3*c[7]^3*c[6]+3*c[7]^2*c[6]^2-9*c[6]^2*c[7]^2*c[5]-8*c[7]*c[6]^2*c[5] ^3+8*c[7]^2*c[6]*c[5]^3-c[5]*c[6]*c[7]*c[4]-c[7]^2*c[4]^2-30*c[6]^2*c[ 7]*c[5]^2*c[4]^3-28*c[6]^2*c[7]^2*c[5]^2*c[4]+30*c[6]^2*c[7]^2*c[5]^2* c[4]^2-30*c[6]*c[7]*c[5]^3*c[4]^3+9*c[6]^2*c[7]*c[4]^2-3*c[6]^2*c[7]*c [4]+2*c[6]*c[7]^2*c[4]-9*c[6]*c[7]^2*c[4]^2-2*c[7]^2*c[5]*c[4]-c[7]^2* c[5]^2+c[6]*c[7]*c[4]^2+8*c[7]^2*c[6]*c[4]^3-4*c[5]*c[7]^2*c[4]^3+c[5] *c[7]*c[4]^2-c[5]*c[6]*c[4]^2+32*c[6]*c[5]*c[7]^2*c[4]^2-9*c[7]^2*c[5] *c[6]*c[4]-4*c[5]*c[7]*c[6]*c[4]^3-9*c[6]^2*c[4]^2*c[5]+3*c[6]^2*c[4]* c[5]+c[7]^3*c[4]+9*c[7]^3*c[4]*c[6]-9*c[7]^3*c[4]*c[5]-20*c[5]^2*c[6]* c[7]*c[4]^2+8*c[6]^2*c[5]*c[4]^3+28*c[7]^3*c[4]^2*c[6]*c[5]+18*c[6]^2* c[7]*c[4]*c[5]+28*c[7]*c[6]^2*c[5]*c[4]^3-3*c[6]^2*c[7]*c[5]+9*c[6]^2* c[7]*c[5]^2+c[6]*c[7]*c[5]^2-8*c[6]*c[7]^3*c[5]^2-9*c[6]*c[7]^2*c[5]^2 +8*c[6]^2*c[7]^2*c[5]^2+2*c[6]*c[7]^2*c[5]-36*c[6]^2*c[4]^2*c[7]*c[5]- 28*c[6]^2*c[7]^2*c[4]^2*c[5]-28*c[7]^3*c[4]*c[6]*c[5]-36*c[6]^2*c[5]^2 *c[7]*c[4]+56*c[6]^2*c[5]^2*c[7]*c[4]^2+28*c[6]*c[5]^2*c[7]*c[4]^3+28* c[6]*c[5]^3*c[7]*c[4]^2+28*c[6]^2*c[5]^3*c[7]*c[4]+9*c[6]*c[7]^3*c[5]+ 9*c[7]^2*c[5]*c[4]^2+28*c[7]^3*c[5]^2*c[6]*c[4]+32*c[7]^2*c[5]^2*c[6]* c[4]+8*c[6]^2*c[7]^2*c[4]^2-9*c[6]^2*c[7]^2*c[4]-4*c[5]^3*c[6]*c[7]*c[ 4]+28*c[6]^2*c[7]^2*c[4]*c[5]-4*c[5]^2*c[6]*c[4]^3-9*c[5]^2*c[7]*c[4]^ 2+9*c[5]^2*c[6]*c[4]^2+4*c[5]^2*c[7]*c[4]^3+4*c[5]^3*c[7]*c[4]^2-4*c[5 ]^3*c[6]*c[4]^2+30*c[5]^3*c[6]^2*c[4]^3-28*c[6]^2*c[5]^2*c[4]^3+28*c[6 ]^2*c[5]^2*c[4]^2+8*c[6]^2*c[5]^3*c[4]-28*c[6]^2*c[5]^3*c[4]^2-9*c[6]^ 2*c[4]*c[5]^2-8*c[7]*c[6]^2*c[4]^3+30*c[7]^2*c[5]^2*c[6]*c[4]^3-30*c[7 ]^3*c[5]^2*c[6]*c[4]^2+30*c[7]^2*c[5]^3*c[6]*c[4]^2-8*c[7]^3*c[4]^2*c[ 6]+4*c[7]^3*c[4]^2*c[5]+c[7]^3*c[5]-28*c[7]^2*c[6]*c[5]*c[4]^3-56*c[7] ^2*c[5]^2*c[6]*c[4]^2-28*c[7]^2*c[5]^3*c[6]*c[4]-c[5]^2*c[6]*c[4]+c[5] ^2*c[7]*c[4]+4*c[7]^3*c[5]^2*c[4]+9*c[7]^2*c[5]^2*c[4]-30*c[6]^2*c[7]* c[5]^3*c[4]^2-8*c[7]^2*c[5]^2*c[4]^2-4*c[7]^2*c[5]^3*c[4]), a[2,1] = c [2], a[6,4] = -1/2*(120*c[5]^2*c[6]*c[4]^4-12*c[6]*c[5]*c[4]^4-90*c[5] ^2*c[6]^3*c[4]^2-60*c[5]*c[6]^2*c[4]^4-2*c[6]^3*c[4]^2+20*c[5]^4*c[4]^ 3*c[6]+40*c[6]^2*c[4]^5*c[5]+36*c[4]*c[5]^2*c[6]^3-20*c[5]^4*c[6]^2*c[ 4]^2+8*c[6]^2*c[5]^3+2*c[6]^2*c[4]^3-24*c[5]*c[6]^3*c[4]^2-6*c[6]^2*c[ 4]^2*c[5]+36*c[6]^2*c[5]*c[4]^3+60*c[4]^3*c[5]*c[6]^3-3*c[6]*c[5]*c[4] ^3-11*c[5]^3*c[6]*c[4]+3*c[5]^3*c[4]^2-24*c[5]^2*c[6]*c[4]^3+13*c[5]^2 *c[6]*c[4]^2+48*c[5]^3*c[6]*c[4]^2-60*c[5]^3*c[6]^2*c[4]^3-30*c[6]^2*c [5]^2*c[4]^3-12*c[6]^2*c[5]^2*c[4]^2-150*c[6]*c[5]^3*c[4]^3-48*c[6]^2* c[5]^3*c[4]+150*c[6]^2*c[5]^3*c[4]^2-2*c[6]^2*c[4]*c[5]^2-5*c[5]^2*c[4 ]^3+2*c[4]^4*c[5]-6*c[5]^2*c[6]^3+60*c[6]^3*c[4]^3*c[5]^2+7*c[6]^3*c[4 ]*c[5]-60*c[5]^2*c[6]*c[4]^5-40*c[6]^3*c[4]^4*c[5]+60*c[4]^4*c[6]*c[5] ^3)*c[6]/(c[4]^3-2*c[5]*c[4]^2+30*c[5]^2*c[4]^3-6*c[5]*c[4]^3-10*c[4]^ 4*c[5]^2-c[5]^3+10*c[5]^4*c[4]^2-30*c[5]^3*c[4]^2+6*c[5]^3*c[4]+2*c[5] ^2*c[4])/c[4]^2, a[3,1] = -2/9*c[4]*(c[4]-3*c[2])/c[2], a[4,1] = 1/4*c [4], a[7,3] = 3/4*1/c[4]^2*c[7]*(-480*c[5]^2*c[6]*c[4]^4+15*c[6]*c[5]* c[4]^4+300*c[4]^2*c[5]^2*c[7]^3+60*c[7]^3*c[5]*c[4]^3-510*c[7]^3*c[5]^ 2*c[4]^3+54*c[7]*c[5]*c[4]^4+450*c[5]^2*c[4]^4*c[7]^2+150*c[4]^3*c[7]^ 3*c[5]^3-150*c[7]^2*c[5]^2*c[4]^5-450*c[4]^5*c[6]*c[5]^3-3*c[7]^3*c[4] ^2+90*c[5]^2*c[4]^5*c[7]+120*c[5]^3*c[4]^4*c[7]+150*c[7]^3*c[5]^2*c[4] ^4+100*c[5]^4*c[4]^2*c[7]^2*c[6]-180*c[6]*c[4]^5*c[5]*c[7]+30*c[6]*c[5 ]^2*c[4]^4*c[7]+60*c[5]^4*c[4]^3*c[6]-150*c[5]^4*c[4]^4*c[6]-200*c[7]^ 2*c[4]^4*c[5]^3+300*c[6]*c[4]^5*c[5]*c[7]^2-50*c[7]^2*c[5]^4*c[4]^3+24 *c[7]*c[6]*c[5]^3-40*c[7]^2*c[6]*c[5]^3-6*c[5]*c[6]*c[7]*c[4]+360*c[7] *c[4]^4*c[6]*c[5]+150*c[7]^2*c[5]^3*c[4]^3+2*c[7]^2*c[4]^2+6*c[5]^2*c[ 7]^3+330*c[6]*c[7]*c[5]^3*c[4]^3-450*c[7]^2*c[6]*c[5]*c[4]^4-12*c[7]*c [6]*c[4]^3-6*c[6]*c[7]^2*c[4]-2*c[7]^2*c[5]*c[4]-4*c[7]^2*c[5]^2+6*c[6 ]*c[7]*c[4]^2+20*c[7]^2*c[6]*c[4]^3-10*c[5]*c[7]^2*c[4]^3+6*c[5]*c[4]^ 3-6*c[5]*c[7]*c[4]^2-6*c[5]*c[6]*c[4]^2+150*c[6]*c[5]*c[7]^2*c[4]^2-72 *c[7]^2*c[5]*c[6]*c[4]-270*c[5]*c[7]*c[6]*c[4]^3+9*c[7]^3*c[4]*c[6]+3* c[7]^3*c[4]*c[5]-306*c[5]^2*c[6]*c[7]*c[4]^2+900*c[4]^5*c[6]*c[7]*c[5] ^3+600*c[5]^2*c[6]*c[7]^3*c[4]^4-420*c[7]^3*c[4]^2*c[6]*c[5]+90*c[6]*c [5]*c[7]*c[4]^2-12*c[6]*c[7]*c[5]^2+30*c[6]*c[7]^3*c[5]^2+12*c[6]*c[7] ^2*c[5]+123*c[7]^3*c[4]*c[6]*c[5]+360*c[6]*c[5]^2*c[7]*c[4]^3+360*c[6] *c[5]^3*c[7]*c[4]^2-18*c[6]*c[7]^3*c[5]+30*c[5]^4*c[4]^3*c[7]-80*c[7]^ 2*c[5]^2*c[4]^3+24*c[7]^2*c[5]*c[4]^2-30*c[7]*c[5]*c[4]^3+3*c[6]*c[5]* c[4]^3-240*c[7]^3*c[5]^2*c[6]*c[4]-24*c[5]^3*c[6]*c[4]-12*c[5]^3*c[7]* c[4]-144*c[5]^3*c[6]*c[7]*c[4]+9*c[5]^3*c[4]^2+294*c[5]^2*c[6]*c[4]^3- 42*c[5]^2*c[7]*c[4]^2-75*c[5]^2*c[6]*c[4]^2+252*c[5]^2*c[7]*c[4]^3+18* c[5]^3*c[7]*c[4]^2+189*c[5]^3*c[6]*c[4]^2-360*c[5]^2*c[4]^4*c[7]-720*c [6]*c[5]^3*c[4]^3-120*c[5]^3*c[7]*c[4]^3-60*c[5]^4*c[7]*c[4]^2*c[6]-6* c[5]^2*c[4]^2-780*c[7]^2*c[5]^2*c[6]*c[4]^3+810*c[7]^3*c[5]^2*c[6]*c[4 ]^2-1230*c[7]^2*c[5]^3*c[6]*c[4]^2-15*c[7]^3*c[4]^2*c[6]-21*c[7]^3*c[4 ]^2*c[5]+20*c[7]^2*c[6]*c[5]*c[4]^3+220*c[7]^2*c[5]^2*c[6]*c[4]^2+15*c [4]^4*c[5]^2+320*c[7]^2*c[5]^3*c[6]*c[4]+1850*c[7]^2*c[5]^3*c[6]*c[4]^ 3-60*c[5]^3*c[7]^3*c[4]^2+12*c[5]^2*c[6]*c[4]+12*c[5]^2*c[7]*c[4]-66*c [7]^3*c[5]^2*c[4]+24*c[7]^2*c[5]^2*c[4]+72*c[5]^2*c[6]*c[7]*c[4]-110*c [7]^2*c[5]^2*c[4]^2+20*c[7]^2*c[5]^3*c[4]-9*c[4]^4*c[5]-60*c[7]^2*c[4] ^4*c[5]-600*c[5]^2*c[6]*c[4]^5*c[7]^2-1200*c[7]^3*c[5]^2*c[4]^3*c[6]+6 30*c[7]^3*c[6]*c[5]*c[4]^3+300*c[5]^4*c[4]^4*c[6]*c[7]-1800*c[5]^3*c[4 ]^4*c[6]*c[7]-15*c[5]^3*c[4]^3-300*c[7]^3*c[4]^4*c[5]*c[6]-20*c[7]^2*c [5]^3*c[4]^2-800*c[5]^3*c[4]^4*c[7]^2*c[6]-200*c[5]^4*c[4]^3*c[7]^2*c[ 6]+1150*c[5]^2*c[4]^4*c[7]^2*c[6]+180*c[5]^2*c[6]*c[4]^5+1140*c[4]^4*c [6]*c[5]^3)/(10*c[5]^3*c[4]^2+10*c[5]^2*c[4]^3-30*c[5]^2*c[4]^2+6*c[5] ^2*c[4]-c[5]^2+6*c[5]*c[4]^2+c[5]*c[4]-c[4]^2)/(10*c[5]*c[6]*c[4]-5*c[ 6]*c[5]-5*c[6]*c[4]+3*c[6]-5*c[5]*c[4]+3*c[5]+3*c[4]-2), `b*`[1] = 1/6 0*(30*c[5]^3*c[4]+42*c[7]*c[6]^2-103*c[5]*c[7]*c[4]+18*c[6]-6*c[5]-6*c [4]-27*c[7]*c[6]+9*c[5]^2+9*c[4]^2-180*c[7]*c[6]^2*c[5]^3+100*c[7]*c[6 ]*c[5]^3-27*c[6]^2-354*c[5]*c[6]*c[7]*c[4]+97*c[6]*c[5]^2-66*c[6]*c[5] -60*c[6]*c[5]^3+100*c[6]^2*c[5]^3+9*c[7]*c[5]+100*c[6]^2*c[4]^3-2040*c [6]^2*c[7]*c[5]^2*c[4]^3-600*c[6]*c[7]*c[5]^3*c[4]^3-14*c[7]*c[5]^2+10 0*c[7]*c[6]*c[4]^3+310*c[6]^2*c[7]*c[4]^2-192*c[6]^2*c[7]*c[4]-180*c[6 ]^2*c[4]^2-14*c[7]*c[4]^2+9*c[7]*c[4]+103*c[6]*c[7]*c[5]+97*c[6]*c[4]^ 2-60*c[6]*c[4]^3+103*c[6]*c[7]*c[4]-156*c[6]*c[7]*c[4]^2+30*c[5]*c[4]^ 3+174*c[5]*c[7]*c[4]^2-312*c[5]*c[6]*c[4]^2-107*c[5]*c[4]^2+215*c[5]*c [6]*c[4]+66*c[5]*c[4]-66*c[6]*c[4]-410*c[5]*c[7]*c[6]*c[4]^3+840*c[6]^ 2*c[4]^2*c[5]-522*c[6]^2*c[4]*c[5]-760*c[5]^2*c[6]*c[7]*c[4]^2-500*c[6 ]^2*c[5]*c[4]^3+932*c[6]^2*c[7]*c[4]*c[5]+1020*c[7]*c[6]^2*c[5]*c[4]^3 -192*c[6]^2*c[7]*c[5]+310*c[6]^2*c[7]*c[5]^2+526*c[6]*c[5]*c[7]*c[4]^2 -156*c[6]*c[7]*c[5]^2-180*c[6]^2*c[5]^2-1610*c[6]^2*c[4]^2*c[7]*c[5]-1 610*c[6]^2*c[5]^2*c[7]*c[4]+2970*c[6]^2*c[5]^2*c[7]*c[4]^2+690*c[6]*c[ 5]^2*c[7]*c[4]^3+690*c[6]*c[5]^3*c[7]*c[4]^2+1020*c[6]^2*c[5]^3*c[7]*c [4]+117*c[6]^2*c[5]-50*c[7]*c[5]*c[4]^3+230*c[6]*c[5]*c[4]^3+230*c[5]^ 3*c[6]*c[4]-50*c[5]^3*c[7]*c[4]-107*c[5]^2*c[4]-410*c[5]^3*c[6]*c[7]*c [4]-50*c[5]^3*c[4]^2-380*c[5]^2*c[6]*c[4]^3-310*c[5]^2*c[7]*c[4]^2+460 *c[5]^2*c[6]*c[4]^2+90*c[5]^2*c[7]*c[4]^3+90*c[5]^3*c[7]*c[4]^2-380*c[ 5]^3*c[6]*c[4]^2-600*c[5]^3*c[6]^2*c[4]^3+900*c[6]^2*c[5]^2*c[4]^3-142 0*c[6]^2*c[5]^2*c[4]^2+300*c[6]*c[5]^3*c[4]^3-500*c[6]^2*c[5]^3*c[4]+9 00*c[6]^2*c[5]^3*c[4]^2+180*c[5]^2*c[4]^2+840*c[6]^2*c[4]*c[5]^2-180*c [7]*c[6]^2*c[4]^3+117*c[6]^2*c[4]+1500*c[5]^3*c[6]^2*c[7]*c[4]^3-312*c [5]^2*c[6]*c[4]+174*c[5]^2*c[7]*c[4]+526*c[5]^2*c[6]*c[7]*c[4]-2040*c[ 6]^2*c[7]*c[5]^3*c[4]^2-50*c[5]^2*c[4]^3)/(3*c[6]-28*c[5]*c[6]*c[4]^2- 28*c[5]^2*c[6]*c[4]+8*c[6]*c[5]^2-9*c[6]*c[4]-9*c[6]*c[5]+28*c[5]*c[6] *c[4]+30*c[5]^2*c[6]*c[4]^2+8*c[6]*c[4]^2-4*c[5]^2*c[4]-4*c[5]*c[4]^2+ 9*c[5]*c[4]-c[5]-c[4])/c[5]/c[6]/c[7]/c[4], a[8,4] = -1/2*(734*c[5]^2* c[6]*c[4]^4-203*c[6]*c[5]*c[4]^4+3190*c[5]^4*c[4]^4*c[6]^2*c[7]-93*c[7 ]*c[5]*c[4]^4-636*c[5]^2*c[4]^4*c[7]^2-10*c[5]^3*c[4]-520*c[6]*c[4]^7* c[7]*c[5]-481*c[4]^5*c[6]^2*c[5]^2+645*c[7]^2*c[5]^2*c[4]^5-574*c[4]^5 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6]+780*c[5]^5*c[4]^4*c[7]+180*c[4]^5*c[5]*c[7]^2-12*c[6]^2*c[4]^3-600* c[5]^3*c[7]^2*c[4]^6-600*c[5]^5*c[4]^4*c[7]^2+920*c[4]^6*c[6]*c[5]^3+1 648*c[6]^2*c[7]*c[5]^2*c[4]^3-120*c[6]^2*c[7]^2*c[5]^2*c[4]+390*c[6]^2 *c[7]^2*c[5]^2*c[4]^2+5722*c[6]*c[7]*c[5]^3*c[4]^3-320*c[5]^3*c[4]^6+4 0*c[7]^2*c[6]^2*c[5]^4-246*c[5]^4*c[7]*c[4]^2+686*c[7]^2*c[6]*c[5]*c[4 ]^4-100*c[7]^2*c[5]^5*c[6]^2*c[4]^2-400*c[5]^4*c[4]*c[7]^2*c[6]^2-1200 *c[5]^5*c[7]*c[4]^5*c[6]^2+70*c[6]^2*c[4]^4*c[7]^2+32*c[6]*c[4]^5-42*c [7]*c[6]*c[4]^3-4*c[6]^2*c[7]*c[4]^2+200*c[5]^2*c[4]^6*c[7]^2+18*c[7]* c[4]^3+20*c[6]*c[4]^3+26*c[7]^2*c[6]*c[4]^3+72*c[5]*c[7]^2*c[4]^3+48*c [5]*c[4]^3-18*c[5]*c[7]*c[4]^2-20*c[5]*c[6]*c[4]^2+8*c[5]*c[4]^2-23*c[ 6]*c[5]*c[7]^2*c[4]^2+266*c[5]*c[7]*c[6]*c[4]^3+12*c[6]^2*c[4]^2*c[5]- 1240*c[5]^4*c[4]^5+287*c[5]^2*c[6]*c[7]*c[4]^2-8120*c[5]^4*c[7]^2*c[4] ^4*c[6]+11330*c[4]^5*c[6]*c[7]*c[5]^3+72*c[6]^2*c[5]*c[4]^3+14*c[6]^2* c[7]*c[4]*c[5]+76*c[7]*c[6]^2*c[5]*c[4]^3-12*c[6]^2*c[7]*c[5]^2+40*c[6 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4]^3+72*c[7]^2*c[5]^2*c[4]^2+32*c[7]^2*c[5]^3*c[4]-160*c[4]^4*c[5]+690 *c[5]^5*c[4]^3*c[6]*c[7]-264*c[7]^2*c[4]^4*c[5]-1833*c[7]^2*c[5]^3*c[6 ]^2*c[4]^2-260*c[6]^2*c[4]^5*c[5]*c[7]^2-8270*c[7]^2*c[4]^5*c[6]*c[5]^ 3+3780*c[5]^2*c[6]*c[4]^5*c[7]^2+1720*c[5]^4*c[4]^4+32*c[5]^4*c[4]*c[6 ]+30*c[5]^4*c[7]^2*c[4]*c[6]-60*c[5]^4*c[4]*c[7]*c[6]+2480*c[4]^5*c[6] ^2*c[7]*c[5]^2-1120*c[4]^5*c[6]^2*c[7]^2*c[5]^2-772*c[4]^5*c[5]^2-40*c [6]^2*c[7]^2*c[4]^5+1752*c[6]^2*c[4]^4*c[5]^2-1700*c[7]^2*c[6]^2*c[5]^ 5*c[4]^4-600*c[5]^5*c[4]^5*c[6]-120*c[5]*c[7]*c[4]^6*c[6]^2+50*c[6]^2* c[7]*c[4]^5-52*c[6]*c[4]^4-1200*c[5]^4*c[7]*c[4]^6*c[6]^2+7150*c[5]^4* c[4]^5*c[6]^2*c[7]+198*c[7]*c[6]^2*c[5]*c[4]^4+10560*c[5]^3*c[4]^4*c[6 ]^2*c[7]+5240*c[5]^3*c[4]^3*c[6]^2*c[7]^2+11240*c[5]^4*c[4]^4*c[6]*c[7 ]-3649*c[5]^2*c[4]^4*c[7]*c[6]^2-14207*c[5]^3*c[4]^4*c[6]*c[7]-4491*c[ 5]^4*c[4]^3*c[6]*c[7]+1064*c[5]^3*c[4]^3-264*c[7]^2*c[5]^3*c[4]^2-2400 *c[5]^4*c[4]^5*c[6]^2+6020*c[6]^2*c[7]^2*c[5]^3*c[4]^5+6800*c[6]*c[7]^ 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^3*c[4]^3+150*c[5]^4*c[7]*c[4]^2+12*c[7]*c[5]^2+20*c[7]*c[6]*c[4]^3+12 *c[7]*c[4]^2-15*c[7]*c[4]^3+12*c[6]*c[4]^2-15*c[6]*c[4]^3-15*c[6]*c[7] *c[4]^2-87*c[5]*c[4]^3-72*c[5]*c[7]*c[4]^2-72*c[5]*c[6]*c[4]^2+60*c[5] *c[4]^2-12*c[5]*c[6]*c[4]+10*c[5]*c[4]-150*c[5]*c[7]*c[6]*c[4]^3-15*c[ 7]*c[5]^3-660*c[5]^2*c[6]*c[7]*c[4]^2+90*c[6]*c[5]*c[7]*c[4]^2-15*c[6] *c[7]*c[5]^2+150*c[6]*c[5]^4*c[4]^2+930*c[6]*c[5]^2*c[7]*c[4]^3+930*c[ 6]*c[5]^3*c[7]*c[4]^2-200*c[5]^4*c[4]^3*c[7]+110*c[7]*c[5]*c[4]^3+110* c[6]*c[5]*c[4]^3+110*c[5]^3*c[6]*c[4]+110*c[5]^3*c[7]*c[4]+60*c[5]^2*c [4]-150*c[5]^3*c[6]*c[7]*c[4]+550*c[5]^3*c[4]^2-690*c[5]^2*c[6]*c[4]^3 +520*c[5]^2*c[7]*c[4]^2+520*c[5]^2*c[6]*c[4]^2-690*c[5]^2*c[7]*c[4]^3- 690*c[5]^3*c[7]*c[4]^2-690*c[5]^3*c[6]*c[4]^2+150*c[5]^2*c[4]^4*c[7]+9 00*c[6]*c[5]^3*c[4]^3+900*c[5]^3*c[7]*c[4]^3-200*c[5]^4*c[7]*c[4]^2*c[ 6]-429*c[5]^2*c[4]^2-120*c[5]^4*c[4]^2-120*c[4]^4*c[5]^2+150*c[5]^4*c[ 4]^3-72*c[5]^2*c[6]*c[4]-72*c[5]^2*c[7]*c[4]+90*c[5]^2*c[6]*c[7]*c[4]+ 550*c[5]^2*c[4]^3+300*c[5]^3*c[4]^4*c[6]*c[7]+300*c[5]^4*c[4]^3*c[6]*c [7]-690*c[5]^3*c[4]^3-200*c[4]^4*c[6]*c[5]^3+150*c[5]^3*c[4]^4), a[8,3 ] = 3/4*(-1560*c[5]^2*c[6]*c[4]^4+585*c[6]*c[5]*c[4]^4-45*c[7]*c[5]*c[ 4]^4-12*c[5]^3*c[4]-450*c[4]^5*c[6]*c[5]^3+3*c[5]*c[7]*c[4]-150*c[5]^2 *c[4]^5*c[7]-200*c[5]^3*c[4]^4*c[7]+300*c[6]*c[4]^5*c[5]*c[7]+2860*c[6 ]*c[5]^2*c[4]^4*c[7]+160*c[5]^4*c[4]^3*c[6]-150*c[5]^4*c[4]^4*c[6]-4*c [5]^2+2*c[4]^2-40*c[7]*c[6]*c[5]^3+132*c[5]*c[6]*c[7]*c[4]-900*c[6]*c[ 5]^2*c[4]^5*c[7]-30*c[6]*c[5]^2+12*c[6]*c[5]-960*c[7]*c[4]^4*c[6]*c[5] +24*c[6]*c[5]^3+3140*c[6]*c[7]*c[5]^3*c[4]^3+6*c[7]*c[5]^2+20*c[7]*c[6 ]*c[4]^3-3*c[7]*c[4]^2-18*c[6]*c[7]*c[5]+15*c[6]*c[4]^2-12*c[6]*c[4]^3 +9*c[6]*c[7]*c[4]-24*c[6]*c[7]*c[4]^2-54*c[5]*c[4]^3-12*c[5]*c[7]*c[4] ^2+348*c[5]*c[6]*c[4]^2+9*c[5]*c[4]^2-87*c[5]*c[6]*c[4]-2*c[5]*c[4]-6* c[6]*c[4]+1035*c[5]*c[7]*c[6]*c[4]^3+1593*c[5]^2*c[6]*c[7]*c[4]^2+900* c[4]^5*c[6]*c[7]*c[5]^3-534*c[6]*c[5]*c[7]*c[4]^2+48*c[6]*c[7]*c[5]^2- 60*c[6]*c[5]^4*c[4]^2-3060*c[6]*c[5]^2*c[7]*c[4]^3-1560*c[6]*c[5]^3*c[ 7]*c[4]^2-50*c[5]^4*c[4]^3*c[7]+80*c[7]*c[5]*c[4]^3-652*c[6]*c[5]*c[4] ^3-208*c[5]^3*c[6]*c[4]+20*c[5]^3*c[7]*c[4]+54*c[5]^2*c[4]+360*c[5]^3* c[6]*c[7]*c[4]+82*c[5]^3*c[4]^2+1734*c[5]^2*c[6]*c[4]^3+388*c[5]^2*c[7 ]*c[4]^2-944*c[5]^2*c[6]*c[4]^2-900*c[5]^2*c[7]*c[4]^3-135*c[5]^3*c[7] *c[4]^2+879*c[5]^3*c[6]*c[4]^2+660*c[5]^2*c[4]^4*c[7]-1680*c[6]*c[5]^3 *c[4]^3+420*c[5]^3*c[7]*c[4]^3+100*c[5]^4*c[7]*c[4]^2*c[6]-246*c[5]^2* c[4]^2-405*c[4]^4*c[5]^2+30*c[5]^4*c[4]^3+240*c[5]^2*c[6]*c[4]-84*c[5] ^2*c[7]*c[4]-396*c[5]^2*c[6]*c[7]*c[4]+562*c[5]^2*c[4]^3+30*c[4]^4*c[5 ]+90*c[4]^5*c[5]^2+300*c[5]^4*c[4]^4*c[6]*c[7]-3000*c[5]^3*c[4]^4*c[6] *c[7]-300*c[5]^4*c[4]^3*c[6]*c[7]-255*c[5]^3*c[4]^3+480*c[5]^2*c[6]*c[ 4]^5+1540*c[4]^4*c[6]*c[5]^3-180*c[6]*c[5]*c[4]^5+120*c[5]^3*c[4]^4)/c [4]^2/(150*c[5]^2*c[6]*c[4]^4-87*c[5]^3*c[4]-12*c[5]*c[7]*c[4]-200*c[5 ]^3*c[4]^4*c[7]-200*c[6]*c[5]^2*c[4]^4*c[7]-200*c[5]^4*c[4]^3*c[6]-10* c[5]^2-10*c[4]^2+12*c[5]^3+12*c[4]^3+20*c[7]*c[6]*c[5]^3+15*c[5]*c[6]* c[7]*c[4]+12*c[6]*c[5]^2-15*c[6]*c[5]^3-1300*c[6]*c[7]*c[5]^3*c[4]^3+1 50*c[5]^4*c[7]*c[4]^2+12*c[7]*c[5]^2+20*c[7]*c[6]*c[4]^3+12*c[7]*c[4]^ 2-15*c[7]*c[4]^3+12*c[6]*c[4]^2-15*c[6]*c[4]^3-15*c[6]*c[7]*c[4]^2-87* c[5]*c[4]^3-72*c[5]*c[7]*c[4]^2-72*c[5]*c[6]*c[4]^2+60*c[5]*c[4]^2-12* c[5]*c[6]*c[4]+10*c[5]*c[4]-150*c[5]*c[7]*c[6]*c[4]^3-15*c[7]*c[5]^3-6 60*c[5]^2*c[6]*c[7]*c[4]^2+90*c[6]*c[5]*c[7]*c[4]^2-15*c[6]*c[7]*c[5]^ 2+150*c[6]*c[5]^4*c[4]^2+930*c[6]*c[5]^2*c[7]*c[4]^3+930*c[6]*c[5]^3*c [7]*c[4]^2-200*c[5]^4*c[4]^3*c[7]+110*c[7]*c[5]*c[4]^3+110*c[6]*c[5]*c [4]^3+110*c[5]^3*c[6]*c[4]+110*c[5]^3*c[7]*c[4]+60*c[5]^2*c[4]-150*c[5 ]^3*c[6]*c[7]*c[4]+550*c[5]^3*c[4]^2-690*c[5]^2*c[6]*c[4]^3+520*c[5]^2 *c[7]*c[4]^2+520*c[5]^2*c[6]*c[4]^2-690*c[5]^2*c[7]*c[4]^3-690*c[5]^3* c[7]*c[4]^2-690*c[5]^3*c[6]*c[4]^2+150*c[5]^2*c[4]^4*c[7]+900*c[6]*c[5 ]^3*c[4]^3+900*c[5]^3*c[7]*c[4]^3-200*c[5]^4*c[7]*c[4]^2*c[6]-429*c[5] ^2*c[4]^2-120*c[5]^4*c[4]^2-120*c[4]^4*c[5]^2+150*c[5]^4*c[4]^3-72*c[5 ]^2*c[6]*c[4]-72*c[5]^2*c[7]*c[4]+90*c[5]^2*c[6]*c[7]*c[4]+550*c[5]^2* c[4]^3+300*c[5]^3*c[4]^4*c[6]*c[7]+300*c[5]^4*c[4]^3*c[6]*c[7]-690*c[5 ]^3*c[4]^3-200*c[4]^4*c[6]*c[5]^3+150*c[5]^3*c[4]^4), a[8,5] = -1/2*(8 93*c[5]^2*c[6]*c[4]^4-122*c[6]*c[5]*c[4]^4+850*c[5]^4*c[4]^4*c[6]^2*c[ 7]-101*c[7]*c[5]*c[4]^4-535*c[5]^2*c[4]^4*c[7]^2-44*c[5]^3*c[4]+160*c[ 4]^5*c[6]^2*c[5]^2+100*c[7]^2*c[5]^2*c[4]^5+170*c[4]^5*c[6]*c[5]^3+75* c[5]*c[6]^2*c[4]^4+16*c[7]^2*c[4]^3-9*c[5]*c[7]*c[4]-140*c[5]^2*c[4]^5 *c[7]-780*c[5]^3*c[4]^4*c[7]-10*c[4]^4*c[6]^2-100*c[7]^2*c[4]^5*c[5]^3 +450*c[6]^2*c[4]^5*c[7]*c[5]^3+510*c[5]^4*c[4]^2*c[7]^2*c[6]+490*c[6]^ 2*c[7]*c[5]^4*c[4]^2-2224*c[6]*c[5]^2*c[4]^4*c[7]+10*c[5]^5*c[4]^3*c[7 ]-146*c[5]^4*c[4]^3*c[6]+90*c[5]^4*c[4]^5*c[6]+40*c[5]^4*c[4]^4*c[6]+5 70*c[7]^2*c[4]^4*c[5]^3+130*c[5]^4*c[4]^4*c[7]+25*c[7]^2*c[6]*c[4]^4+1 8*c[5]^5*c[4]^2-4*c[5]^2-4*c[4]^2-10*c[4]^4*c[7]^2+150*c[7]^2*c[5]^4*c [4]^3+5*c[5]^3-6*c[6]^2*c[7]^2*c[5]+10*c[4]^3-35*c[5]^5*c[4]^3-24*c[7] *c[6]^2*c[5]^3+27*c[7]*c[6]*c[5]^3-32*c[7]^2*c[6]*c[5]^3+23*c[5]*c[6]* c[7]*c[4]-c[5]^4-6*c[4]^4-60*c[7]*c[6]^2*c[5]*c[4]^5-45*c[5]^4*c[4]^2* c[7]^2+450*c[6]*c[5]^2*c[4]^5*c[7]+10*c[6]*c[5]^2-80*c[4]^5*c[5]^3-260 *c[5]^5*c[4]^3*c[6]+270*c[7]*c[4]^4*c[6]*c[5]+20*c[5]^5*c[4]^4-9*c[6]* c[5]^3+50*c[4]^4*c[6]^2*c[5]^5-600*c[5]^5*c[7]*c[4]^4*c[6]-760*c[7]^2* c[5]^3*c[4]^3+25*c[7]*c[4]^4*c[6]^2+240*c[5]^5*c[4]^4*c[6]+200*c[5]^5* c[7]*c[4]^5*c[6]+20*c[5]^5*c[4]^2*c[6]^2-90*c[6]^2*c[5]^4*c[7]*c[4]+71 *c[5]^4*c[6]^2*c[4]-383*c[5]^4*c[6]^2*c[4]^2+18*c[6]^2*c[5]^3+14*c[4]^ 4*c[7]-6*c[7]^2*c[4]^2+16*c[6]^2*c[4]^3-1433*c[6]^2*c[7]*c[5]^2*c[4]^3 -57*c[6]^2*c[7]^2*c[5]^2*c[4]+c[6]^2*c[7]^2*c[5]^2*c[4]^2-3185*c[6]*c[ 7]*c[5]^3*c[4]^3+100*c[5]^4*c[7]*c[4]^2-195*c[7]^2*c[6]*c[5]*c[4]^4-15 *c[5]^5*c[7]*c[4]^2-20*c[6]^2*c[4]^4*c[7]^2+9*c[7]*c[5]^2+57*c[7]*c[6] *c[4]^3+19*c[6]^2*c[7]*c[4]^2-2*c[6]^2*c[7]*c[4]+5*c[5]^5*c[7]*c[4]+13 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5]^2*c[6]*c[4]^4-12*c[6]*c[5]*c[4]^4+1100*c[5]^4*c[4]^4*c[6]^2*c[7]-12 *c[7]*c[5]*c[4]^4-110*c[5]^2*c[4]^4*c[7]^2+120*c[4]^5*c[6]*c[5]^3+15*c [5]*c[6]^2*c[4]^4-300*c[5]^5*c[4]^3*c[7]^2*c[6]^2+10*c[5]^5-440*c[5]^3 *c[4]^4*c[7]-150*c[7]^2*c[4]^5*c[5]^3+180*c[5]^4*c[4]^2*c[7]^2*c[6]+18 0*c[6]^2*c[7]*c[5]^4*c[4]^2+350*c[6]*c[5]^2*c[4]^4*c[7]-570*c[5]^5*c[4 ]^3*c[7]-150*c[5]^5*c[4]*c[6]*c[7]^2-150*c[5]^5*c[4]*c[6]^2*c[7]+410*c [5]^4*c[4]^3*c[6]-120*c[5]^4*c[4]^4*c[6]+690*c[7]^2*c[4]^4*c[5]^3+200* c[5]^4*c[4]^5*c[7]^2-120*c[5]^4*c[4]^4*c[7]-15*c[7]^2*c[6]*c[4]^4+342* c[5]^5*c[4]^2-12*c[7]*c[6]^2*c[5]^3+10*c[7]*c[6]*c[5]^3-12*c[7]^2*c[6] *c[5]^3+15*c[5]^6*c[7]+410*c[5]^4*c[4]^2*c[7]^2+750*c[5]^5*c[7]^2*c[4] ^3-120*c[6]*c[5]^2*c[4]^5*c[7]+570*c[5]^6*c[4]^2*c[7]-570*c[5]^5*c[4]^ 3*c[6]-57*c[7]*c[4]^4*c[6]*c[5]+87*c[5]^6*c[4]-570*c[5]^5*c[4]^4-1100* c[5]^5*c[7]*c[4]^4*c[6]-410*c[7]^2*c[5]^3*c[4]^3-15*c[7]*c[4]^4*c[6]^2 -15*c[5]^5*c[7]^2+780*c[5]^5*c[4]^2*c[6]^2*c[7]+750*c[5]^5*c[4]^4*c[6] +300*c[5]^5*c[7]*c[4]^5*c[6]-690*c[5]^5*c[4]^2*c[6]^2-40*c[6]^2*c[5]^4 *c[7]*c[4]-690*c[5]^5*c[4]^2*c[7]^2-57*c[5]^4*c[6]^2*c[4]+410*c[5]^4*c [6]^2*c[4]^2+750*c[5]^5*c[4]^4*c[7]+340*c[6]^2*c[7]*c[5]^2*c[4]^3-30*c [6]^2*c[7]^2*c[5]^2*c[4]-750*c[6]*c[7]*c[5]^3*c[4]^3-20*c[7]^2*c[6]^2* c[5]^4-342*c[5]^4*c[7]*c[4]^2+90*c[7]^2*c[6]*c[5]*c[4]^4+200*c[7]^2*c[ 5]^5*c[6]^2*c[4]^2-10*c[5]^4*c[7]+140*c[5]^5*c[7]*c[4]^2+150*c[5]^4*c[ 4]*c[7]^2*c[6]^2+20*c[6]^2*c[4]^4*c[7]^2-10*c[7]*c[6]*c[4]^3-30*c[5]^5 *c[7]*c[4]+12*c[5]^4*c[6]^2+12*c[7]^2*c[6]*c[4]^3-12*c[5]*c[7]^2*c[4]^ 3-24*c[6]*c[5]*c[7]^2*c[4]^2+24*c[5]*c[7]*c[6]*c[4]^3-12*c[5]^6+150*c[ 7]^2*c[5]^6*c[4]^2-120*c[5]^4*c[4]^5+48*c[5]^2*c[6]*c[7]*c[4]^2+1100*c [5]^4*c[7]^2*c[4]^4*c[6]-150*c[4]^5*c[6]*c[7]*c[5]^3-12*c[6]^2*c[5]*c[ 4]^3-42*c[7]*c[6]^2*c[5]*c[4]^3+20*c[6]*c[5]*c[7]*c[4]^2+12*c[5]^4*c[7 ]*c[6]-24*c[6]^2*c[4]^2*c[7]*c[5]+30*c[6]^2*c[7]^2*c[4]^2*c[5]-342*c[6 ]*c[5]^4*c[4]^2-10*c[6]*c[5]^4+120*c[4]^5*c[7]*c[5]^3+24*c[6]^2*c[5]^2 *c[7]*c[4]-30*c[6]^2*c[5]^2*c[7]*c[4]^2-243*c[6]*c[5]^2*c[7]*c[4]^3+37 2*c[6]*c[5]^3*c[7]*c[4]^2+87*c[6]^2*c[5]^3*c[7]*c[4]+410*c[5]^4*c[4]^3 *c[7]+57*c[7]^2*c[5]^2*c[4]^3-750*c[5]^4*c[7]^2*c[4]^4+10*c[7]*c[5]*c[ 4]^3+10*c[6]*c[5]*c[4]^3+24*c[7]^2*c[5]^2*c[6]*c[4]+12*c[6]*c[7]*c[4]^ 4+200*c[7]*c[5]^7*c[4]^3-110*c[5]^6*c[7]*c[4]+15*c[6]*c[5]^6+20*c[5]^3 *c[6]*c[4]+20*c[5]^3*c[7]*c[4]+150*c[7]*c[6]*c[5]^5*c[4]-900*c[5]^5*c[ 4]^3*c[7]^2*c[6]-750*c[5]^5*c[4]^2*c[6]*c[7]-96*c[5]^3*c[6]*c[7]*c[4]+ 200*c[6]*c[5]^7*c[4]^3+20*c[5]^3*c[4]^2-36*c[5]^2*c[6]*c[4]^3-20*c[5]^ 2*c[7]*c[4]^2-20*c[5]^2*c[6]*c[4]^2-36*c[5]^2*c[7]*c[4]^3-24*c[5]^3*c[ 7]*c[4]^2-24*c[5]^3*c[6]*c[4]^2-410*c[5]^3*c[6]^2*c[4]^3-900*c[5]^5*c[ 4]^3*c[6]^2*c[7]-510*c[4]^3*c[7]^2*c[5]^2*c[6]^2-630*c[7]*c[5]^6*c[6]* c[4]^2-150*c[4]^5*c[5]^3*c[6]^2+72*c[7]*c[5]^4*c[4]-57*c[7]^2*c[5]^4*c [4]+72*c[5]^2*c[4]^4*c[7]+57*c[6]^2*c[5]^2*c[4]^3+24*c[6]^2*c[5]^2*c[4 ]^2+285*c[6]*c[5]^3*c[4]^3-24*c[6]^2*c[5]^3*c[4]+285*c[5]^3*c[7]*c[4]^ 3+270*c[5]^4*c[7]*c[4]^2*c[6]+12*c[7]*c[6]^2*c[4]^3+340*c[7]^2*c[5]^2* c[6]*c[4]^3-410*c[7]^2*c[5]^3*c[6]*c[4]^2-200*c[5]^5*c[4]^5*c[7]+70*c[ 4]^3*c[7]^2*c[5]*c[6]^2-42*c[7]^2*c[6]*c[5]*c[4]^3+150*c[5]^5*c[4]^5-3 0*c[7]^2*c[5]^2*c[6]*c[4]^2+12*c[4]^4*c[5]^2+87*c[7]^2*c[5]^3*c[6]*c[4 ]+510*c[7]^2*c[5]^3*c[6]*c[4]^3+690*c[5]^3*c[6]^2*c[4]^4-342*c[5]^4*c[ 4]^3-15*c[6]^2*c[4]^3*c[7]^2+510*c[5]^3*c[6]^2*c[7]*c[4]^3-20*c[5]^2*c [6]*c[7]*c[4]-410*c[6]^2*c[7]*c[5]^3*c[4]^2-70*c[6]^2*c[7]^2*c[4]*c[5] ^3-10*c[5]^2*c[4]^3+24*c[7]^2*c[5]^2*c[4]^2-24*c[7]^2*c[5]^3*c[4]-200* c[7]^2*c[5]^6*c[4]^2*c[6]+110*c[7]^2*c[4]*c[5]^5+1350*c[5]^5*c[4]^3*c[ 6]*c[7]+15*c[7]^2*c[4]^4*c[5]+300*c[6]^2*c[7]*c[5]^6*c[4]^3+510*c[7]^2 *c[5]^3*c[6]^2*c[4]^2-550*c[5]^6*c[4]^2-600*c[7]*c[5]^6*c[4]^3+20*c[7] *c[6]^2*c[5]^5+570*c[5]^6*c[6]*c[4]^2+150*c[5]^2*c[6]*c[4]^5*c[7]^2+55 0*c[5]^4*c[4]^4+120*c[5]^7*c[4]^2+72*c[5]^4*c[4]*c[6]-40*c[5]^4*c[7]^2 *c[4]*c[6]-57*c[5]^4*c[4]*c[7]*c[6]+150*c[6]^2*c[5]^6*c[4]^2+150*c[4]^ 5*c[6]^2*c[7]*c[5]^2-200*c[4]^5*c[6]^2*c[7]^2*c[5]^2-150*c[5]^7*c[7]*c [4]^2-110*c[6]^2*c[4]^4*c[5]^2+780*c[5]^5*c[4]^2*c[7]^2*c[6]-200*c[5]^ 5*c[4]^5*c[6]-600*c[5]^6*c[4]^3*c[6]-200*c[6]^2*c[7]*c[5]^6*c[4]^2-300 *c[5]^4*c[4]^5*c[6]^2*c[7]+90*c[7]*c[6]^2*c[5]*c[4]^4+200*c[6]*c[5]^7* c[7]*c[4]^2-300*c[6]*c[5]^7*c[7]*c[4]^3-180*c[5]^3*c[4]^4*c[6]^2*c[7]- 570*c[5]^4*c[4]^4*c[6]*c[7]-540*c[5]^2*c[4]^4*c[7]*c[6]^2+660*c[5]^3*c [4]^4*c[6]*c[7]+60*c[5]^4*c[4]^3*c[6]*c[7]+48*c[5]^3*c[4]^3-15*c[6]^2* c[5]^5-200*c[5]^6*c[6]^2*c[4]^3-15*c[5]^5*c[6]*c[7]-48*c[5]^5*c[4]+140 *c[5]^5*c[6]*c[4]^2+200*c[5]^4*c[4]^5*c[6]^2+300*c[6]^2*c[7]^2*c[5]^3* c[4]^5-300*c[6]*c[7]^2*c[5]^4*c[4]^5+700*c[7]*c[5]^6*c[4]^3*c[6]-750*c [5]^4*c[4]^4*c[6]^2+20*c[5]^5*c[7]^2*c[6]+1100*c[7]^2*c[5]^4*c[6]^2*c[ 4]^3-930*c[7]^2*c[5]^4*c[6]^2*c[4]^2+300*c[7]^2*c[5]^6*c[4]^3*c[6]-20* c[5]^6*c[7]*c[6]-180*c[5]^3*c[4]^4*c[7]^2*c[6]-750*c[5]^4*c[4]^3*c[7]^ 2*c[6]-540*c[5]^2*c[4]^4*c[7]^2*c[6]-750*c[5]^4*c[4]^3*c[6]^2*c[7]-150 *c[7]^2*c[4]^4*c[6]^2*c[5]-1100*c[7]^2*c[4]^4*c[5]^3*c[6]^2+930*c[7]^2 *c[4]^4*c[6]^2*c[5]^2+150*c[5]^6*c[4]*c[7]*c[6]-110*c[5]^6*c[4]*c[6]-1 50*c[5]^7*c[6]*c[4]^2+570*c[5]^6*c[4]^3-150*c[5]^7*c[4]^3+15*c[7]^2*c[ 6]^2*c[5]^3-440*c[4]^4*c[6]*c[5]^3+750*c[5]^5*c[6]^2*c[4]^3-200*c[7]^2 *c[5]^6*c[4]^3+12*c[5]^4*c[7]^2+200*c[5]^4*c[4]^5*c[6]*c[7]-20*c[5]^4* c[4]-87*c[5]^3*c[4]^4-30*c[5]^5*c[4]*c[6]+110*c[5]^5*c[6]^2*c[4]), a[9 ,7] = -1/60*(10*c[5]*c[6]*c[4]-5*c[6]*c[5]-5*c[6]*c[4]+3*c[6]-5*c[5]*c [4]+3*c[5]+3*c[4]-2)/c[7]/(-c[5]*c[6]*c[4]+c[5]*c[6]*c[7]*c[4]+c[5]*c[ 7]*c[4]-c[7]^2*c[5]*c[4]+c[6]*c[7]*c[4]-c[6]*c[7]^2*c[4]-c[7]^2*c[4]+c [7]^3*c[4]+c[6]*c[7]*c[5]-c[6]*c[7]^2*c[5]-c[7]^2*c[5]+c[7]^3*c[5]-c[7 ]^2*c[6]+c[7]^3*c[6]+c[7]^3-c[7]^4), a[9,2] = 0, a[9,3] = 0, a[9,5] = \+ -1/60*(10*c[6]*c[7]*c[4]-5*c[6]*c[4]-5*c[7]*c[6]+3*c[6]-5*c[7]*c[4]+3* c[4]-2+3*c[7])/c[5]/(c[5]^3*c[4]-c[5]^2*c[4]+c[5]*c[7]*c[4]-c[5]^2*c[7 ]*c[4]+c[5]*c[6]*c[4]-c[5]^2*c[6]*c[4]-c[6]*c[7]*c[4]+c[5]*c[6]*c[7]*c [4]-c[5]^4+c[5]^3-c[7]*c[5]^2+c[7]*c[5]^3-c[6]*c[5]^2+c[6]*c[5]^3+c[6] *c[7]*c[5]-c[6]*c[7]*c[5]^2), a[9,8] = 1/60*(-20*c[5]*c[6]*c[4]+30*c[5 ]*c[6]*c[7]*c[4]+15*c[6]*c[5]-20*c[6]*c[7]*c[5]+15*c[6]*c[4]-20*c[6]*c [7]*c[4]-12*c[6]+15*c[7]*c[6]+15*c[5]*c[4]-20*c[5]*c[7]*c[4]-12*c[5]+1 5*c[7]*c[5]-12*c[4]+15*c[7]*c[4]+10-12*c[7])/(-c[4]+c[7]*c[4]+c[6]*c[4 ]-c[6]*c[7]*c[4]+c[5]*c[4]-c[5]*c[7]*c[4]-c[5]*c[6]*c[4]+c[5]*c[6]*c[7 ]*c[4]+1-c[7]-c[6]+c[7]*c[6]-c[5]+c[7]*c[5]+c[6]*c[5]-c[6]*c[7]*c[5]), a[9,1] = 1/60*(30*c[5]*c[6]*c[7]*c[4]-10*c[5]*c[6]*c[4]-10*c[6]*c[7]* c[5]+5*c[6]*c[5]-10*c[6]*c[7]*c[4]+5*c[6]*c[4]+5*c[7]*c[6]-3*c[6]-10*c [5]*c[7]*c[4]+5*c[5]*c[4]+5*c[7]*c[5]-3*c[5]+5*c[7]*c[4]-3*c[4]+2-3*c[ 7])/c[5]/c[6]/c[7]/c[4], a[9,4] = 1/60*(-2+3*c[5]-5*c[7]*c[6]+3*c[7]+3 *c[6]+10*c[6]*c[7]*c[5]-5*c[6]*c[5]-5*c[7]*c[5])/c[4]/(-c[6]*c[4]^3-c[ 4]^3+c[4]^4+c[7]*c[4]^2+c[6]*c[4]^2-c[6]*c[7]*c[4]+c[6]*c[7]*c[4]^2-c[ 5]*c[4]^3+c[5]*c[4]^2-c[7]*c[4]^3-c[5]*c[7]*c[4]+c[5]*c[7]*c[4]^2-c[5] *c[6]*c[4]+c[5]*c[6]*c[4]^2+c[6]*c[7]*c[5]-c[5]*c[6]*c[7]*c[4]), a[9,6 ] = 1/60*(10*c[5]*c[7]*c[4]-5*c[5]*c[4]-5*c[7]*c[5]+3*c[5]-5*c[7]*c[4] +3*c[4]-2+3*c[7])/c[6]/(c[6]^2*c[4]*c[5]-c[5]*c[6]*c[4]+c[5]*c[7]*c[4] -c[5]*c[6]*c[7]*c[4]-c[6]^3*c[4]+c[6]^2*c[4]-c[6]*c[7]*c[4]+c[6]^2*c[7 ]*c[4]-c[6]^3*c[5]+c[6]^2*c[5]-c[6]*c[7]*c[5]+c[6]^2*c[7]*c[5]+c[6]^4- c[6]^3+c[7]*c[6]^2-c[6]^3*c[7])\}:" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" } }{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 5 "Let " }{XPPEDIT 18 0 "t[7 ]" "6#&%\"tG6#\"\"(" }{TEXT -1 7 " and " }{XPPEDIT 18 0 "t[8]" "6#&% \"tG6#\"\")" }{TEXT -1 29 " denote the coefficients of " }{XPPEDIT 18 0 "z^7" "6#*$%\"zG\"\"(" }{TEXT -1 7 " and " }{XPPEDIT 18 0 "z^8 " "6#*$%\"zG\"\")" }{TEXT -1 43 " respectively in the stability polyn omial." }}{PARA 0 "" 0 "" {TEXT -1 8 "We set " }{XPPEDIT 18 0 "t[7] = 17/(16*`.`*7!);" "6#/&%\"tG6#\"\"(*&\"#<\"\"\"*(\"#;F*%\".GF*-%*facto rialG6#F'F*!\"\"" }{XPPEDIT 18 0 "`` = 17/80640;" "6#/%!G*&\"#<\"\"\" \"&S1)!\"\"" }{TEXT -1 7 " and " }{XPPEDIT 18 0 "t[8] = 1/8!;" "6#/& %\"tG6#\"\")*&\"\"\"F)-%*factorialG6#F'!\"\"" }{XPPEDIT 18 0 "`` = 1/4 0320;" "6#/%!G*&\"\"\"F&\"&?.%!\"\"" }{TEXT -1 37 " to obtain two equ ations involving " }{XPPEDIT 18 0 "c[4]" "6#&%\"cG6#\"\"%" }{TEXT -1 3 ", " }{XPPEDIT 18 0 "c[5]" "6#&%\"cG6#\"\"&" }{TEXT -1 7 " and " }{XPPEDIT 18 0 "c[6]" "6#&%\"cG6#\"\"'" }{TEXT -1 2 ". " }}{PARA 0 "" 0 "" {TEXT -1 52 "Note that the stability polynomial only depends on \+ " }{XPPEDIT 18 0 "c[4]" "6#&%\"cG6#\"\"%" }{TEXT -1 3 ", " }{XPPEDIT 18 0 "c[5]" "6#&%\"cG6#\"\"&" }{TEXT -1 7 " and " }{XPPEDIT 18 0 "c[ 6]" "6#&%\"cG6#\"\"'" }{TEXT -1 2 ". " }}{PARA 0 "" 0 "" {TEXT -1 0 " " }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 143 "Rz := StabilityFunction (6,8,'expanded'):\neqA := simplify(subs(eG,coeff(Rz,z^7)))=17/80640:%; \neqB := simplify(subs(eG,coeff(Rz,z^8)))=1/40320:%;" }}{PARA 12 "" 1 "" {XPPMATH 20 "6#/,$*&#\"\"\"\"%S9F'*&,hn**\"#IF'&%\"cG6#\"\"&F'&F.6# \"\"'F')&F.6#\"\"%\"\"#F'!\"\"*&)F-F8F'F5F'F9*(F8F')F-\"\"$F'F5F'F9*(F 0F'F=F'F4F'F'*(\"#?F'F;F'F4F'F'*(\"$S\"F')F5F7F'F;F'F'*(\"#**F'F;F')F5 F>F'F9*&F-F'F4F'F9*(\"#SF'F=F'FDF'F9*(FAF'F=F'FGF'F'*(\"#5F'F-F'FGF'F' *(\"#9F'FDF'F-F'F9*(F8F'F1F'F;F'F9*(F8F'F1F'F4F'F9*(F7F'F1F'FGF'F'*(FJ F')F5F0F'F;F'F9**F0F'F-F'F1F'F5F'F'**\"$S$F'F;F'F1F'FDF'F'**\"#>F'F;F' F1F'F4F'F'**\"$g\"F'F;F'F1F'FGF'F9**F3F'F;F'F1F'F5F'F'**\"$+\"F'FDF'F1 F'F=F'F'**\"$+#F'F;F'F1F'FTF'F9**\"$!>F'F1F'F=F'FGF'F9**\"$?\"F'F=F'F1 F'F4F'F'**\"#MF'F=F'F1F'F5F'F9**\"#)*F'F1F'F-F'FGF'F'**FenF'F1F'F-F'FD F'F9*(F7F'F1F'F=F'F'**\"#!)F'F1F'F-F'FTF'F'F',2*(FMF'F=F'F4F'F'*(FMF'F ;F'FGF'F'*(F,F'F;F'F4F'F9*(F3F'F;F'F5F'F'*$F;F'F9*(F3F'F-F'F4F'F'*&F-F 'F5F'F'*$F4F'F9F9F'F'#\"#<\"&S1)" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/, $*&#\"\"\"\"%S9F'*,&%\"cG6#\"\"%F'&F+6#\"\"&F',0**\"#7F'F.F'&F+6#\"\"' F'F*F'F'*(\"\"#F'F4F'F.F'!\"\"**\"#IF'F.F'F4F')F*F8F'F9*&F.F'F*F'F'** \"#?F'F4F'F.F')F*\"\"$F'F'*$F " 0 "" {MPLTEXT 1 0 96 "sol := \+ solve(\{eqA,eqB\},\{c[5],c[6]\}):\nc[5]=subs(sol,c[5]);eqC := %:\nc[6] =subs(sol,c[6]);eqD := %:" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/&%\"cG6# \"\"&,$*(\"\"%\"\"\"&F%6#F*F+,**&\"$C#F+)F,\"\"$F+F+*&\"$7\"F+)F,\"\"# F+!\"\"*&\"# " 0 "" {MPLTEXT 1 0 1 ";" }}}{SECT 1 {PARA 4 "" 0 "" {TEXT 0 13 "pr in_err_norm" }{TEXT -1 1 " " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 51 "errterms6_8 := PrincipalErrorTerms( 6,8,'expanded'):" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 701 "prin_err_norm := proc(c2,c4,c7)\n local c2 r,c4r,c5r,c6r,c7r,e1,sm,ct;\n global e2;\n \n c2r := convert(c2,r ational,Digits);\n c4r := convert(c4,rational,Digits); \n c5r := - 4*c4r/(224*c4r^3-112*c4r^2+17*c4r-2);\n c6r := -1/112*1/c4r*(-24+4 92*c4r-6594*c4r^2+55465*c4r^3-285184*c4r^4+913920*c4r^5+1555456*c4r^7- 1741824*c4r^6)/(166*c4r^2-448*c4r^3-29*c4r+2+448*c4r^4)/(8*c4r^2+144*c 4r^3-31*c4r+6);\n c7r := convert(c7,rational,Digits); \n\n e1 := \+ \{c[2]=c2r,c[4]=c4r,c[5]=c5r,c[6]=c6r,c[7]=c7r,c[8]=1,c[9]=1\};\n\n \+ e2 := `union`(e1,simplify(subs(e1,eG))):\n sm := 0;\n for ct to no ps(errterms6_8) do\n sm := sm+(simplify(subs(e2,errterms6_8[ct])) )^2;\n end do;\n evalf(sqrt(sm));\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 29 "We can determin e values for " }{XPPEDIT 18 0 "c[2]" "6#&%\"cG6#\"\"#" }{TEXT -1 3 ", " }{XPPEDIT 18 0 "c[4]" "6#&%\"cG6#\"\"%" }{TEXT -1 7 " and " } {XPPEDIT 18 0 "c[7]" "6#&%\"cG6#\"\"(" }{TEXT -1 121 " that make the \+ principal error norm a minimum by a simple cycling method using a one \+ dimensional minimization procedure." }}{PARA 0 "" 0 "" {TEXT -1 0 "" } }{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 367 "Digits := 15:\nc_2 := .1: \+ c_4 := .27: c_7 := .3:\nfor ii to 30 do \n c_2 := op(1,findmin('prin _err_norm'(c2,c_4,c_7),c2=\{0.01,c_2,0.16\}));\n c_4 := op(1,findmin ('prin_err_norm'(c_2,c4,c_7),c4=\{0.2,c_4,0.35\}));\n mn := findmin( 'prin_err_norm'(c_2,c_4,c7),c7=\{0.2,c_7,0.4\});\n c_7 := mn[1]:\n \+ print(c[2]=c_2,c[4]=c_4,c[7]=c_7);\n print(mn[2]);\nend do:\nDigits := 10:" }}{PARA 11 "" 1 "" {XPPMATH 20 "6%/&%\"cG6#\"\"#$\"0&\\E9!*4* *))!#;/&F%6#\"\"%$\"0Rynp/&3F!#:/&F%6#\"\"($\"0ak*))>YVEF1" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#$\"0w9D!=m->!#=" }}{PARA 11 "" 1 "" {XPPMATH 20 "6%/&%\"cG6#\"\"#$\"0*>1#=8*[)*!#;/&F%6#\"\"%$\"0/gO$=F2F! #:/&F%6#\"\"($\"0N'=.'pyW#F1" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#$\"0Oa *z(\\4!>!#=" }}{PARA 11 "" 1 "" {XPPMATH 20 "6%/&%\"cG6#\"\"#$\"0tH#Gx ;0**!#;/&F%6#\"\"%$\"0'*=q!RF1F!#:/&F%6#\"\"($\"0&H(\\'f#QP#F1" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#$\"0$o^.(p2!>!#=" }}{PARA 11 "" 1 "" {XPPMATH 20 "6%/&%\"cG6#\"\"#$\"0Vpl/CA))*!#;/&F%6#\"\"%$\"0G^Hfpeq#!# :/&F%6#\"\"($\"0G\"*zTg)[BF1" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#$\"0[p zXU2!>!#=" }}{PARA 260 "" 0 "" {TEXT -1 1 ":" }}{PARA 260 "" 0 "" {TEXT -1 1 ":" }}{PARA 11 "" 1 "" {XPPMATH 20 "6%/&%\"cG6#\"\"#$\"0xC3 L(=h)*!#;/&F%6#\"\"%$\"0**H8Zjcq#!#:/&F%6#\"\"($\"032Xu$>PBF1" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#$\"0dfUsQ2!>!#=" }}{PARA 11 "" 1 "" {XPPMATH 20 "6%/&%\"cG6#\"\"#$\"0oC3L(=h)*!#;/&F%6#\"\"%$\"0**H8Zjcq#! #:/&F%6#\"\"($\"012Xu$>PBF1" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#$\"0dfU sQ2!>!#=" }}{PARA 11 "" 1 "" {XPPMATH 20 "6%/&%\"cG6#\"\"#$\"0qC3L(=h) *!#;/&F%6#\"\"%$\"0**H8Zjcq#!#:/&F%6#\"\"($\"032Xu$>PBF1" }}{PARA 11 " " 1 "" {XPPMATH 20 "6#$\"0dfUsQ2!>!#=" }}}{PARA 0 "" 0 "" {TEXT -1 0 " " }}{PARA 0 "" 0 "" {TEXT -1 65 "We consider various rational approxim ations for the values found." }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 163 "nds := [c[2]=.986118733082470e-1,c[4]=.270566347132999,c[7]=.2337 19374450708]:\nevalf[10](%);\nfor dgt from 6 by -1 to 3 do\n map(con vert,nds,rational,dgt);\nend do;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#7% /&%\"cG6#\"\"#$\"+Jt=h)*!#6/&F&6#\"\"%$\"+rMm0F!#5/&F&6#\"\"($\"+XP>PB F2" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#7%/&%\"cG6#\"\"##\"#r\"$?(/&F&6# \"\"%#\"$[\"\"$Z&/&F&6#\"\"(#\"$i#\"%@6" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#7%/&%\"cG6#\"\"##\"#k\"$\\'/&F&6#\"\"%#\"#B\"#&)/&F&6#\"\"(#\"#h \"$h#" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#7%/&%\"cG6#\"\"##\"\"(\"#r/&F &6#\"\"%#\"#B\"#&)/&F&6#F*#\"#=\"#x" }}{PARA 11 "" 1 "" {XPPMATH 20 "6 #7%/&%\"cG6#\"\"##\"\"(\"#r/&F&6#\"\"%#\"\"$\"#6/&F&6#F*#F/\"#<" }}} {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 21 "Choosing \+ the values " }{XPPEDIT 18 0 "c[2] = 7/71;" "6#/&%\"cG6#\"\"#*&\"\"(\" \"\"\"#r!\"\"" }{TEXT -1 3 ", " }{XPPEDIT 18 0 "c[4] = 23/85;" "6#/&% \"cG6#\"\"%*&\"#B\"\"\"\"#&)!\"\"" }{TEXT -1 7 " and " }{XPPEDIT 18 0 "c[7] = 18/77;" "6#/&%\"cG6#\"\"(*&\"#=\"\"\"\"#x!\"\"" }{TEXT -1 44 " gives the following principal error norm. " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 32 "prin_err_nor m(7/71,23/85,18/77);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#$\"+lZu+>!#8" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }} {PARA 0 "" 0 "" {TEXT -1 24 "Using the formulas for " }{XPPEDIT 18 0 "c[5]" "6#&%\"cG6#\"\"&" }{TEXT -1 7 " and " }{XPPEDIT 18 0 "c[6]" " 6#&%\"cG6#\"\"'" }{TEXT -1 15 " in terms of " }{XPPEDIT 18 0 "c[4]" "6#&%\"cG6#\"\"%" }{TEXT -1 58 " would suggest the followng rational v alues for the nodes " }{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 0 113 "c_4 := 23/85:\nevalf(su bs(c[4]=c_4,eqC)):\nconvert(%,rational,5);\nevalf(subs(c[4]=c_4,eqD)): \nconvert(%,rational,5);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/&%\"cG6# \"\"&#\"#F\"#H" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/&%\"cG6#\"\"'#\"#h \"#!*" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 156 "Taking these values gives a principal error norm of approximately 0.00019008 and the maximum magnitude of the linking coefficients is \+ approximately 31.97." }}{PARA 0 "" 0 "" {TEXT -1 60 "The real stabili ty interval is approximately [ -4.5144, 0]." }}{PARA 0 "" 0 "" {TEXT -1 53 "(See the abreviated calculations in a later section.)" }}{PARA 0 "" 0 "" {TEXT -1 50 "#---------------------------------------------- ---" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}}{PARA 0 "" 0 " " {TEXT -1 52 "#---------------------------------------------------" } }{SECT 1 {PARA 4 "" 0 "" {TEXT -1 38 "characteristics of the embedded \+ scheme" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 35 "coefficient s of the combined scheme" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1743 "ee := \{c[2]=7/71,\nc[3]=46/255, \nc[4]=23/85,\nc[5]=27/29,\nc[6]=61/90,\nc[7]=18/77,\nc[8]=1,\nc[9]=1, \n\na[2,1]=7/71,\na[3,1]=6992/455175,\na[3,2]=75118/455175,\na[4,1]=23 /340,\na[4,2]=0,\na[4,3]=69/340,\na[5,1]=125814087/51607124,\na[5,2]=0 ,\na[5,3]=-481282155/51607124,\na[5,4]=100879020/12901781,\na[6,1]=332 3091540925544113/29573952540280488000,\na[6,2]=0,\na[6,3]=-10703826229 567511/24340701679243200,\na[6,4]=169241467001919447401/17831998050213 5683200,\na[6,5]=2551748271602005217/45506989353097008000,\na[7,1]=-37 6385546372076652128/6002133498919339474385,\na[7,2]=0,\na[7,3]=1930525 1563909761378/19679126225965047457,\na[7,4]=-8122491009809811364010481 /9979717849963846796488754, a[7,5]=-155386566948917103529551/230183317 0083782096614930,\na[7,6]=107267076044076240/544838019860392951,\na[8, 1]=-1972640388605306005321/257739734717483367780,\na[8,2]=0, a[8,3]=12 354606914500963/386396118220908,\na[8,4]=-56856879732578412491684/1967 664304821917372209,\na[8,5]=-17589460594924790303414/92934689150952840 583755,\na[8,6]=370576933768080/119264931607213,\na[8,7]=1150428860219 177/432917708558301,\na[9,1]=636077/8182296,\na[9,2]=0,\na[9,3]=0,\na[ 9,4]=24315051125/87140294626,\na[9,5]=30418033967/271431025272,\na[9,6 ]=862843671000/2241531443039,\na[9,7]=1670085824869/14714276549256,\na [9,8]=13887/424328,\n\nb[1]=636077/8182296,\nb[2]=0,\nb[3]=0,\nb[4]=24 315051125/87140294626,\nb[5]=30418033967/271431025272,\nb[6]=862843671 000/2241531443039,\nb[7]=1670085824869/14714276549256,\nb[8]=13887/424 328,\n\n`b*`[1]=231122308093/3481256020752,\n`b*`[2]=0,\n`b*`[3]=0,\n` b*`[4]=4006504071740375/84742592073525056,\n`b*`[5]=23826240438664855/ 461934347497102656,\n`b*`[6]=1723971130382250/4007085927807811,\n`b*`[ 7]=123485516897451247/368256795835268016,\n`b*`[8]=63672019983/6138204 944224,\n`b*`[9]=1/17\}:" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}}{PARA 0 "" 0 "" {TEXT -1 0 " " }}{PARA 0 "" 0 "" {TEXT -1 5 "Let " }{XPPEDIT 18 0 "T[6, 8];" "6#&% \"TG6$\"\"'\"\")" }{TEXT -1 127 " denote the vector whose components \+ are the principal error terms of the 8 stage, order 6 scheme (the erro r terms of order 7)." }}{PARA 0 "" 0 "" {TEXT -1 5 "Let " }{XPPEDIT 18 0 "`T*`[5,9];" "6#&%#T*G6$\"\"&\"\"*" }{TEXT -1 145 " denote the v ector whose components are the principal error terms of the embedded 9 stage, order 5 scheme (the error terms of order 6) and let " } {XPPEDIT 18 0 "`T*`[6,9];" "6#&%#T*G6$\"\"'\"\"*" }{TEXT -1 99 " deno te the vector whose components are the error terms of order 7 of the e mbedded order 5 scheme." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 39 "Denote the 2-norm of these vectors by " }{XPPEDIT 18 0 "abs(abs(T[6,8]));" "6#-%$absG6#-F$6#&%\"TG6$\"\"'\"\")" }{TEXT -1 3 ", " }{XPPEDIT 18 0 "abs(abs(`T*`[5,9]));" "6#-%$absG6#-F$6#&%#T *G6$\"\"&\"\"*" }{TEXT -1 7 " and " }{XPPEDIT 18 0 "abs(abs(`T*`[6,9 ]));" "6#-%$absG6#-F$6#&%#T*G6$\"\"'\"\"*" }{TEXT -1 15 " respectivel y." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 8 "Defi ne: " }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "A[7] = abs(ab s(T[6, 8]));" "6#/&%\"AG6#\"\"(-%$absG6#-F)6#&%\"TG6$\"\"'\"\")" } {TEXT -1 6 ", " }{XPPEDIT 18 0 "B[7] = abs(abs(`T*`[6,9]))/abs(abs (`T*`[5,9]));" "6#/&%\"BG6#\"\"(*&-%$absG6#-F*6#&%#T*G6$\"\"'\"\"*\"\" \"-F*6#-F*6#&F/6$\"\"&F2!\"\"" }{TEXT -1 6 ", " }{XPPEDIT 18 0 "C[ 7] = abs(abs(`T*`[6,9]-T[6, 8]))/abs(abs(`T*`[5, 9]));" "6#/&%\"CG6#\" \"(*&-%$absG6#-F*6#,&&%#T*G6$\"\"'\"\"*\"\"\"&%\"TG6$F2\"\")!\"\"F4-F* 6#-F*6#&F06$\"\"&F3F9" }{TEXT -1 1 "." }}{PARA 0 "" 0 "" {TEXT -1 0 " " }}{PARA 0 "" 0 "" {TEXT -1 77 "Dormand and Prince have suggested tha t as well as attempting to ensure that " }{XPPEDIT 18 0 "A[7];" "6#&% \"AG6#\"\"(" }{TEXT -1 73 " is a minimum, if the embedded scheme is t o be used for error control, " }{XPPEDIT 18 0 "B[7];" "6#&%\"BG6#\"\" (" }{TEXT -1 7 " and " }{XPPEDIT 18 0 "C[7];" "6#&%\"CG6#\"\"(" } {TEXT -1 27 " should be chosen so that " }}{PARA 0 "" 0 "" {TEXT -1 68 "they are \"similar in magnitude\" and also not differ too much fro m 1." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 186 "errterms6_8 := PrincipalErrorTerms(6,8,'expanded'): \n`errterms6_9*` :=subs(b=`b*`,PrincipalErrorTerms(6,9,'expanded')):\n `errterms5_9*` := subs(b=`b*`,PrincipalErrorTerms(5,9,'expanded')):" } }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 326 "snmB := sqrt(add(evalf(subs(ee,`errterms6_9*`[i]))^2,i=1..nops( `errterms6_9*`))):\nsdnB := sqrt(add(evalf(subs(ee,`errterms5_9*`[i])) ^2,i=1..nops(`errterms5_9*`))):\nsnmC := sqrt(add((evalf(subs(ee,`errt erms6_9*`[i])-subs(ee,errterms6_8[i])))^2,i=1..nops(errterms6_8))):\n' B[7]'= evalf[8](snmB/sdnB);\n'C[7]'= evalf[8](snmC/sdnB);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/&%\"BG6#\"\"($\")2^C " 0 "" {MPLTEXT 1 0 1 ";" }}}}{PARA 0 "" 0 "" {TEXT -1 52 "#--------------------------------------------------- " }}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 27 "absolute stability regions " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 35 "coefficients of the co mbined scheme" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1743 "ee := \{c[2]=7/71,\nc[3]=46/255,\nc[4]=23/85, \nc[5]=27/29,\nc[6]=61/90,\nc[7]=18/77,\nc[8]=1,\nc[9]=1,\n\na[2,1]=7/ 71,\na[3,1]=6992/455175,\na[3,2]=75118/455175,\na[4,1]=23/340,\na[4,2] =0,\na[4,3]=69/340,\na[5,1]=125814087/51607124,\na[5,2]=0,\na[5,3]=-48 1282155/51607124,\na[5,4]=100879020/12901781,\na[6,1]=3323091540925544 113/29573952540280488000,\na[6,2]=0,\na[6,3]=-10703826229567511/243407 01679243200,\na[6,4]=169241467001919447401/178319980502135683200,\na[6 ,5]=2551748271602005217/45506989353097008000,\na[7,1]=-376385546372076 652128/6002133498919339474385,\na[7,2]=0,\na[7,3]=19305251563909761378 /19679126225965047457,\na[7,4]=-8122491009809811364010481/997971784996 3846796488754, a[7,5]=-155386566948917103529551/2301833170083782096614 930,\na[7,6]=107267076044076240/544838019860392951,\na[8,1]=-197264038 8605306005321/257739734717483367780,\na[8,2]=0, a[8,3]=123546069145009 63/386396118220908,\na[8,4]=-56856879732578412491684/19676643048219173 72209,\na[8,5]=-17589460594924790303414/92934689150952840583755,\na[8, 6]=370576933768080/119264931607213,\na[8,7]=1150428860219177/432917708 558301,\na[9,1]=636077/8182296,\na[9,2]=0,\na[9,3]=0,\na[9,4]=24315051 125/87140294626,\na[9,5]=30418033967/271431025272,\na[9,6]=86284367100 0/2241531443039,\na[9,7]=1670085824869/14714276549256,\na[9,8]=13887/4 24328,\n\nb[1]=636077/8182296,\nb[2]=0,\nb[3]=0,\nb[4]=24315051125/871 40294626,\nb[5]=30418033967/271431025272,\nb[6]=862843671000/224153144 3039,\nb[7]=1670085824869/14714276549256,\nb[8]=13887/424328,\n\n`b*`[ 1]=231122308093/3481256020752,\n`b*`[2]=0,\n`b*`[3]=0,\n`b*`[4]=400650 4071740375/84742592073525056,\n`b*`[5]=23826240438664855/4619343474971 02656,\n`b*`[6]=1723971130382250/4007085927807811,\n`b*`[7]=1234855168 97451247/368256795835268016,\n`b*`[8]=63672019983/6138204944224,\n`b*` [9]=1/17\}:" }}}{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 8 stage, order 6 sc heme is given as follows." }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 75 "subs(ee,StabilityFunction(6,8,'expanded')):\nR := unapply(%,z):\n'R(z )'=R(z);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/-%\"RG6#%\"zG,4\"\"\"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)*&# \".zL?9xH\"\"1++#)QYKchF)*$)F'\"\"(F)F)F)*&#\"+([G?l&\"0++mB?,G#F)*$)F '\"\")F)F)F)" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT 260 4 "Note" }{TEXT -1 63 ": The scheme has been constructed so \+ that the coefficients of " }{XPPEDIT 18 0 "z^6" "6#*$%\"zG\"\"'" } {TEXT -1 7 " and " }{XPPEDIT 18 0 "z^7" "6#*$%\"zG\"\"(" }{TEXT -1 30 " are approximately equal to " }{XPPEDIT 18 0 "17/(16*`.`*7!);" " 6#*&\"#<\"\"\"*(\"#;F%%\".GF%-%*factorialG6#\"\"(F%!\"\"" }{TEXT -1 7 " and " }{XPPEDIT 18 0 "1/8!;" "6#*&\"\"\"F$-%*factorialG6#\"\")!\" \"" }{TEXT -1 15 " respectively." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 52 "17/(16*7!),1297714203379/615 6324638820000;\nevalf(%);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6$#\"#<\"&S 1)#\".zL?9xH\"\"1++#)QYKch" }}{PARA 11 "" 1 "" {XPPMATH 20 "6$$\"+@\\8 3@!#8$\"+,m$z5#F%" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 " > " 0 "" {MPLTEXT 1 0 42 "1/8!,5652028487/228012023660000;\nevalf(%); " }}{PARA 11 "" 1 "" {XPPMATH 20 "6$#\"\"\"\"&?.%#\"+([G?l&\"0++mB?,G# " }}{PARA 11 "" 1 "" {XPPMATH 20 "6$$\"+I(e,[#!#9$\"+T\"H)yCF%" }}} {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 124 "We can f ind the point where the boundary of the stability region intersects th e negative real axis by solving the equation: " }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "R(z) = 1;" "6#/-%\"RG6#%\"zG\"\"\"" } {TEXT -1 2 ". " }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 28 "z0 := newt on(R(z)=1,z=-4.5);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%#z0G$!+(zQW^%! \"*" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 303 "z0 := newton(R(z)=1,z=-4.5):\np1 := plot([R(z),1],z= -4.99..0.49,color=[red,blue]):\np2 := plot([[[z0,1]]$3],style=point,sy mbol=[circle,cross,diamond],color=black):\np3 := plot([[z0,0],[z0,1]], linestyle=3,color=COLOR(RGB,0,.5,0)):\nplots[display]([p1,p2,p3],view= [-4.99..0.49,-.07..1.47],font=[HELVETICA,9]);" }}{PARA 13 "" 1 "" {GLPLOT2D 440 227 227 {PLOTDATA 2 "6+-%'CURVESG6$7Z7$$!3A++++++!*\\!#< $\"3h@fP'R2a`#F*7$$!3Wm;z)*y8g\\F*$\"3_;[&oj(f)R#F*7$$!3cLLe(zv-$\\F*$ \"3s_N8Z;JoAF*7$$!3o+]P'p8/!\\F*$\"3U#=E!RAGW@F*7$$!3#pmm^f^0([F*$\"30 a+3d9DE?F*7$$!3#3]()f)ee=[F*$\"3!GZ@_LAW$=F*7$$!3%QL3oIz\"3ZF*$\"3c?ri!fj!z9F*7$$!3imm,F%Q(\\YF*$\"3U)Q!4 \"*4x;8F*7$$!3Gnm1o,\"4f%F*$\"3CV0%>\\5&p6F*7$$!33nm64>3KXF*$\"30%yG#R %Rq.\"F*7$$!3I+Dh]K`tWF*$\"31aSRh#zh=*!#=7$$!3\\L$3@f%)\\T%F*$\"3(zUH* H]%F*$\"3Av!Gd )=bR]F^o7$$!3%omTR&=vxSF*$\"3oGpy$>f)*)QF^o7$$!3/+]x(4o='RF*$\"3+\"G6m u'G%*HF^o7$$!3OF*$\"3Gzv5S#=()R\"F^o7$$!3?++]Qxz+NF*$\"3QfXRp (4B5\"F^o7$$!3/++5QhU'Q$F*$\"3K#e@\"GCG=!*!#>7$$!3mm;%zwlDG$F*$\"3-!ob ETYmx(Far7$$!3[LLBUd1fJF*$\"3aD,:)*)[5(oFar7$$!3eLL$4-XW0$F*$\"3qE_7S` !G\\'Far7$$!3;+]n]iuKHF*$\"3vL>9_YE5kFar7$$!3ILL$z@A]#GF*$\"3qfm\"\\5N $)f'Far7$$!3!)**\\n!)=$oq#F*$\"3V:g:#=Cr.(Far7$$!3')**\\-InG%f#F*$\"3W L4\"R)[\"zk(Far7$$!3ILL3sw&oZ#F*$\"3Gs&3g*Hpi%)Far7$$!3#HL3&R6-pBF*$\" 3CF2AW()>f$*Far7$$!3#om;4([q_AF*$\"3o611K#*4[5F^o7$$!3xm;%z&\\)=8#F*$ \"3ihG'[e**==\"F^o7$$!3$***\\_o3rE?F*$\"3Q]LHpzV88F^o7$$!3hmmhq*>J\">F *$\"3%\\\\0isvCZ\"F^o7$$!35++?e%pdz\"F*$\"3EnAAfK0d;F^o7$$!3+++:w['4o \"F*$\"3'4_)QJ;wf=F^o7$$!3-+]()Rb))p:F*$\"3wg9[lD:z?F^o7$$!3#)***\\a(3 bY9F*$\"3m!Q8\"e1$GN#F^o7$$!3cLL$R>HdL\"F*$\"3f#['3J:0HEF^o7$$!3z**** \\%R.u@\"F*$\"3okb'4TI'fHF^o7$$!3pm;aLE=56F*$\"35?/07Oz%H$F^o7$$!3E%** **4@?'H**F^o$\"3;yVudDo/PF^o7$$!3MML3+cmE))F^o$\"3T-z@:*Rn8%F^o7$$!3%H **\\(H-wtwF^o$\"3DWgUP*yAk%F^o7$$!3cOLL)\\%eYlF^o$\"3!G1T,F*='>&F^o7$$ !3g.+vof`m`F^o$\"3wbRm.[,ZeF^o7$$!3Dkmm#)*3+B%F^o$\"3?Sy2e,y]lF^o7$$!3 Yjm;()funIF^o$\"3'\\nK3Lk\"etF^o7$$!3HBL3;r5:>F^o$\"33;d_-y5d#)F^o7$$! 3_^****>q^f&)Far$\"37#[*4fxlz\"*F^o7$$\"3OMnm\"G*fzNFar$\"3\\$QAGQWk. \"F*7$$\"3\"RLL8'ppV9F^o$\"3]T@mV6Jb6F*7$$\"3@0+D$4>8g#F^o$\"3U_T5w65( H\"F*7$$\"3Q,+v#=6$4PF^o$\"3gH: \+ " 0 "" {MPLTEXT 1 0 1368 "R := z -> 1+z+1/2*z^2+1/6*z^3+1/24*z^4+1/120 *z^5+1/720*z^6+\n 1297714203379/6156324638820000*z^7+56520284 87/228012023660000*z^8:\npts := []: z0 := 0:\nfor ct from 0 to 240 do \n zz := newton(R(z)=exp(ct*Pi/20*I),z=z0):\n z0 := zz:\n pts := [op(pts),[Re(zz),Im(zz)]]:\nend do:\np1 := plot(pts,color=COLOR(RGB,. 48,.23,.05)):\np2 := plots[polygonplot]([seq([pts[i-1],pts[i],[-2.25,0 ]],i=2..nops(pts))],\n style=patchnogrid,color=COLOR(RGB,.95, .45,.1)):\npts := []: z0 := 2.15+4.65*I:\nfor ct from 0 to 40 do\n z z := newton(R(z)=exp(ct*Pi/20*I),z=z0):\n z0 := zz:\n pts := [op(p ts),[Re(zz),Im(zz)]]:\nend do:\np3 := plot(pts,color=COLOR(RGB,.48,.23 ,.05)):\np4 := plots[polygonplot]([seq([pts[i-1],pts[i],[2.07,4.63]],i =2..nops(pts))],\n style=patchnogrid,color=COLOR(RGB,.95,.45, .1)):\npts := []: z0 := 2.15-4.65*I:\nfor ct from 0 to 40 do\n zz := newton(R(z)=exp(ct*Pi/20*I),z=z0):\n z0 := zz:\n pts := [op(pts), [Re(zz),Im(zz)]]:\nend do:\np5 := plot(pts,color=COLOR(RGB,.48,.23,.05 )):\np6 := plots[polygonplot]([seq([pts[i-1],pts[i],[2.07,-4.63]],i=2. .nops(pts))],\n style=patchnogrid,color=COLOR(RGB,.95,.45,.1) ):\np7 := plot([[[-5.19,0],[2.49,0]],[[0,-5.19],[0,5.19]]],color=black ,linestyle=3):\nplots[display]([p||(1..7)],view=[-5.19..2.49,-5.19..5. 19],font=[HELVETICA,9],\n labels=[`Re(z)`,`Im(z)`],axes=b oxed,scaling=constrained);" }}{PARA 13 "" 1 "" {GLPLOT2D 625 529 529 {PLOTDATA 2 "6/-%'CURVESG6$7]z7$$\"\"!F)F(7$F($\"3++++Fjzq:!#=7$$\"35+ 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"6#7$,$-%&FloatG6$\"&W^%!\"%! \"\"\"\"!" }{TEXT -1 2 ". " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 260 "We can distort the boundary curve horizontally b y taking the 11th root of the real part of points along the curve. In \+ this way we see that the largest interval on the nonnegative imaginary axis that contains the origin and lies inside the stability region is " }{XPPEDIT 18 0 "[0, 3.3];" "6#7$\"\"!-%&FloatG6$\"#L!\"\"" }{TEXT -1 18 " approximately. " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 383 "R := z -> 1+z+1/2*z^2+1/6*z^3+1/24 *z^4+1/120*z^5+1/720*z^6+\n 1297714203379/6156324638820000*z^7+565 2028487/228012023660000*z^8:\nDigits := 25:\npts := []: z0 := 0:\nfor \+ ct from 0 to 120 do\n zz := newton(R(z)=exp(ct*Pi/100*I),z=z0):\n \+ z0 := zz:\n pts := [op(pts),[surd(Re(zz),11),Im(zz)]]:\nend do:\nplo t(pts,color=COLOR(RGB,.85,.4,0),thickness=2,font=[HELVETICA,9]);\nDigi ts := 10:" }}{PARA 13 "" 1 "" {GLPLOT2D 261 276 276 {PLOTDATA 2 "6(-%' CURVESG6#7er7$$\"\"!F)F(7$$\":ke-*e_1%)>sL!*G!#E$\":xsJA1$)*e`EfTJF-7$ $\":x8JH-=8&3Z,&y%F-$\":!*o+gwO%=2`=$G'F-7$$\":IrX'>&=BtqEiU'F-$\":Kd# G1e4&3'zxC%*F-7$$\":&*>;'*\\q8*Q?!>#zF-$\":JDSbO\"p\\hqjc7!#D7$$\":yI) RYH-2>!fzJ*F-$\":m*3W3A]3Fjzq:F?7$$\":6HvSc,V2.\\R1\"F?$\":>u_L%\\->$f b\\)=F?7$$\":x0l*eAsk/Z@!>\"F?$\":_Jn7Pf]0'[6*>#F?7$$\":2_&[G`-E(3a;J \"F?$\":VbAM-Zu08uK^#F?7$$\":^[(RY&yo!)[H!H9F?$\":ZXll8`7dSLu#GF?7$$\" :&pmadPfrCB\"Ha\"F?$\":>,?O\\NQ**o#fTJF?7$$\":P\\5;$\\'yc7XPl\"F?$\":t ED+#z_M*)>vbMF?7$$\":Uj\"Gb%3sQbz=w\"F?$\":o:M0E3jGJ6*pPF?7$$\":,5/#[( zhY&4gn=F?$\":E#f#)R=S4t12%3%F?7$$\":JtQFl$p8u[9r>F?$\":I%y&)[mGE(3I#) R%F?7$$\":3UZ`EYJ+c3F2#F?$\":2vR!o/,Fy&*Q7ZF?7$$\":x#R8!)y[%=?fC<#F?$ \":IL(>`t4$e<\\l-&F?7$$\":,w0ip*\\O!)*R0F#F?$\":QAdU#egn<*32M&F?7$$\": **H&>Xj?G.Y2nBF?$\":.#eA(>)4y])o[l&F?7$$\":9\"GF=i8go2YGM-%3A#QnDsF?7$$\":&3e$)*e'pVm)=!>HF?$\":1JY 5#3WzYl$)RvF?7$$\":_h#>cm\\PEm72IF?$\":f5VPG>&oN,+ayF?7$$\":))*oxdYg\\!=$F?$\":5<3!*=i^s^IB[ )F?7$$\":F5nHneS6aHeE$F?$\":#psOB%**e,k(\\'z)F?7$$\":V3)eZb%)o%G<.N$F? 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)fX![')yxoF?$\":0s%=]0C.AR$fn#Ffu7$$\":KLv>#HPM*)ev-pF?$\":J\")*4Ng&4$ Qz^/FFfu7$$\":F!f)*ooIJtw>DpF?$\":gjF)\\o^83y(Gt#Ffu7$$\":#*>pLdwg[B$) \\%pF?$\":2SK>Q^\"3N\\+hFFfu7$$\":O4@C0:.4co>'pF?$\":qu?sdDC<;\"*))y#F fu7$$\":E,W'oSm%3:$*f(pF?$\":\"G[+iOdd-)Gl\"GFfu7$$\":G0f:mD(GWe(o)pF? $\":BN'\\-Q=An1\"R%GFfu7$$\":nl09\">I9;%4W*pF?$\":Hi%G[Z)3O2I5(GFfu7$$ \":1_2_o2SH^`$)*pF?$\":W)[/h;Fz^3)y*GFfu7$$\"::*y%f&[N,6`U)*pF?$\":ng+ B#)*QzQtXCHFfu7$$\":6k2FKqql(pG%*pF?$\":)3NJ#pM![lVv]HFfu7$$\":oqC1\"4 NkE&Gb)pF?$\":vqp@/(zhOswwHFfu7$$\":v`s/0=)H)*QkrpF?$\":>g`w>!fP0<\\-I Ffu7$$\":M>G,))=,ao'*>&pF?$\":S()=*oF?$\":$fJZ@#f$p8$)*y2$Ffu7$ $\":g^vb`L)*o&R*)[oF?$\":'f#[Q@4VXeMC5$Ffu7$$\":)=+VyK*=+tYZz'F?$\":xE sQvpDc$omEJFfu7$$\":I&p(o([6!Gx\\ks'F?$\":#[B$*3GI\"Q$Gf]JFfu7$$\":<+X L#ej%R(fB= v>$Ffu7$$\":u.8(p6,rQ6=qjF?$\":z\"z(RJ!3C(>90A$Ffu7$$\":O4e-&3%eQ15&Qh F?$\":DPrK)Hp')zp>VKFfu7$$\":a_w.Bd`yI!43dF?$\":,oRBxD!f`_clKFfu7$$!:2 J&ymw-]oh1j^F?$\":j&=*3b\"*G=#yh(G$Ffu7$$!:!e[?YfCM1IW7gF?$\":Y$=IH'pM [h`$4LFfu7$$!:(GB=.4(zcQucO'F?$\":D0wEr&4@'or2L$Ffu7$$!:DaM!e&p+#Q**G/ mF?$\":#>Ex(z=DC@r=N$Ffu7$$!:Tj:nCKiw[v!*y'F?$\":YKH'**[v)[\\^EP$Ffu7$ $!:U6&G(y1Sgr%3UpF?$\":_(pFL/\"GT'>6$R$Ffu7$$!:JZ.ta.3\\))3R2(F?$\"::h 9P$))fe&=_KT$Ffu7$$!:fB#od[5:73[!>(F?$\":#pVlIp;_m=2LMFfu7$$!:;0a1ZH.- ^qaH(F?$\":4c#4s%)odj3d_MFfu7$$!:\\+*fb%*QaLEJ\"R(F?$\":#*yM(p`N`!>\\< Z$Ffu7$$!:*>:Rd4hPu)3(zuF?$\":G2(Q0C2'e-21\\$Ffu-%%FONTG6$%*HELVETICAG \"\"*-%+AXESLABELSG6$Q!6\"Fgam-%*THICKNESSG6#\"\"#-%&COLORG6&%$RGBG$\" #&)!\"#$\"\"%!\"\"F(-%%VIEWG6$%(DEFAULTGFjbm" 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 poin t 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 0 113 "Digits := 15:\nz0 := 3.3*I:\nfor ct from 108 to 111 \+ do\n newton(R(z)=exp(ct*Pi/100*I),z=z0);\nend do;\nDigits := 10:" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#^$$\"0DN&)*GHjY!#<$\"0$p')zp>VK!#9" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#^$$\"0'\\L&=Qg4#!#<$\"0E!f`_clK!#9" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#^$$!0(30An**\\p!#=$\"0#*G=#yh(G$!#9" } }{PARA 11 "" 1 "" {XPPMATH 20 "6#^$$!06$30gf6P!#<$\"0qM[h`$4L!#9" }}} {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 107 "Then we \+ apply the bisection method to calculate the parameter value associated with the intersection point." }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 176 "real_part := proc(u)\n Re(newton(R(z)=exp(u*Pi*I),z=3.3*I))\n end proc:\nDigits := 15:\nu0 := bisect('real_part'(u),u=1.08..1.11);\n newton(R(z)=exp(u0*Pi*I),z=3.3*I);\nDigits := 10:" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%#u0G$\"0R'>JPe(4\"!#9" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#^#$\"0NUNJ=BG$!#9" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 " " 0 "" {TEXT -1 4 "The " }{TEXT 260 112 "largest interval on the nonne gative imaginary axis that contains the origin and lies inside the sta bility region" }{TEXT -1 5 " is " }{XPPEDIT 18 0 "[0, 3.2823];" "6#7$ \"\"!-%&FloatG6$\"&BG$!\"%" }{TEXT -1 18 " (approximately)." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 37 "#-------------- ----------------------" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 80 "The stability function R* for the 9 stage, order 5 scheme is given as follows." }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 97 "subs(ee,subs(b=`b*`,StabilityFunction(5,9,'expanded'))):\n`R*` := \+ unapply(%,z):\n'`R*`(z)'=`R*`(z);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/ -%#R*G6#%\"zG,6\"\"\"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)*&# \"5P$orM]\\xLk#\"8+?\"p6vc\"*=6\"y\"F)*$)F'F1F)F)F)*&#\"5tCL0H " 0 "" {MPLTEXT 1 0 33 "z_0 := new ton(`R*`(z)=-1,z=-4.5);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%$z_0G$!+? ax=X!\"*" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 324 "z_0 := newton(`R*`(z)=-1,z=-4.5):\np_1 := plot([`R*` (z),-1],z=-4.99..0.49,color=[red,blue]):\np_2 := plot([[[z_0,-1]]$3],s tyle=point,symbol=[circle,cross,diamond],color=black):\np_3 := plot([[ z_0,0],[z_0,-1]],linestyle=3,color=COLOR(RGB,0,.5,0)):\nplots[display] ([p_1,p_2,p_3],view=[-4.99..0.49,-1.47..1.47],font=[HELVETICA,9]);" }} {PARA 13 "" 1 "" {GLPLOT2D 366 248 248 {PLOTDATA 2 "6+-%'CURVESG6$7U7$ $!3A++++++!*\\!#<$!3Y$>?/mH'3BF*7$$!3cLLe(zv-$\\F*$!3Y&R$H--Y(3#F*7$$! 3#pmm^f^0([F*$!37aQNh$G[)=F*7$$!3#3]()f)ee=[F*$!3m723fd`AV]F\"F*7$$!33nm64>3KXF *$!3C,UZJwUD5F*7$$!3\\L$3@f%)\\T%F*$!3kR5$)z21#>)!#=7$$!3jm;u*Q?kI%F*$ !34W](H#=k+mFP7$$!3E+]ZY%3S>%F*$!3_Wp/(ek#H_FP7$$!3%omTR&=vxSF*$!3+A?P &Hc;1%FP7$$!3/+]x(4o='RF*$!3GQ\"e!**Q#36$FP7$$!3*zt J#FP7$$!3[mm^!Qvwt$F*$!3MT2%3](QZOF*$!3i[>\"o%RrC7 FP7$$!3?++]Qxz+NF*$!3*)3eQ#*fHN!)!#>7$$!3/++5QhU'Q$F*$!3_RU<@AvlZFcp7$ $!3mm;%zwlDG$F*$!36E,=MgYLBFcp7$$!3[LLBUd1fJF*$\"36MBa\">xFs$!#@7$$!3e LL$4-XW0$F*$\"3\\I%\\(>40.ZG?F-(Fcp7$$!3ILL3sw&oZ#F*$\"3;='3=hZ->)Fcp7 $$!3#HL3&R6-pBF*$\"3QKBd\"pd`G*Fcp7$$!3#om;4([q_AF*$\"3QZG)31yG0\"FP7$ $!3xm;%z&\\)=8#F*$\"3#oygNF&e#>\"FP7$$!3$***\\_o3rE?F*$\"37_5&e#zvD8FP 7$$!3hmmhq*>J\">F*$\"3e(3j#Q[T%[\"FP7$$!35++?e%pdz\"F*$\"3y-AEI*ztm\"F P7$$!3+++:w['4o\"F*$\"3#RC$Q&=c!o=FP7$$!3-+]()Rb))p:F*$\"3'RSYl`wa3#FP 7$$!3#)***\\a(3bY9F*$\"3E%G0!e'GsN#FP7$$!3cLL$R>HdL\"F*$\"3s\\`^ia0KEF P7$$!3z****\\%R.u@\"F*$\"3['3#zFI^hHFP7$$!3pm;aLE=56F*$\"3#RXHenbfH$FP 7$$!3E%****4@?'H**FP$\"3d+>/;&>`q$FP7$$!3MML3+cmE))FP$\"3YZ[SSE2PTFP7$ $!3%H**\\(H-wtwFP$\"3._?_!\\ICk%FP7$$!3cOLL)\\%eYlFP$\"37`=z30D'>&FP7$ $!3g.+vof`m`FP$\"3EX!)eJU.ZeFP7$$!3Dkmm#)*3+B%FP$\"3EJl<-]y]lFP7$$!3Yj m;()funIFP$\"3/.Cvh];etFP7$$!3HBL3;r5:>FP$\"3%=fUs%y5d#)FP7$$!3_^****> q^f&)Fcp$\"3'))4m%fxlz\"*FP7$$\"3OMnm\"G*fzNFcp$\"3x&eAGQWk.\"F*7$$\"3 \"RLL8'ppV9FP$\"3!fm`X96`:\"F*7$$\"3@0+D$4>8g#FP$\"3#\\A\"Q275(H\"F*7$ $\"3Q,+v#=6$4PFP$\"3w=H/NN3\\9F*7$$\"3!***************[FP$\"31G%=3wax=XF*Fa\\l-%'SYMBOLG6#%'CIRCLE G-Fe[l6&Fg[lF\\\\lF\\\\lF\\\\l-%&STYLEG6#%&POINTG-F$6&Fg_l-F\\`l6#%&CR OSSGF_`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$%*HEL VETICAG\"\"*-%+AXESLABELSG6%Q\"z6\"Q!Febl-F]bl6#%(DEFAULTG-%%VIEWG6$;$ !$*\\!\"#$\"#\\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 82 "The following picture shows the stability region for the 9 stage, order 5 scheme. " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1484 "`R*` := z -> 1+z+1/2*z^2+1 /6*z^3+1/24*z^4+1/120*z^5+\n 26433774950347168337/1781111891567511 6912000*z^6+\n 10412134172905332473/44527797289187792280000*z^7+\n 1803950400787159537/89055594578375584560000*z^8+\n 5652028487 /3876204402220000*z^9:\npts := []: z0 := 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:\np_1 := plot(pts,color=COLOR(RGB ,.43,.2,0)):\np_2 := plots[polygonplot]([seq([pts[i-1],pts[i],[-2.25,0 ]],i=2..nops(pts))],\n style=patchnogrid,color=COLOR(RGB,.85, .4,0)):\npts := []: z0 := 2.1+4.65*I:\nfor ct from 0 to 50 do\n zz : = newton(`R*`(z)=exp(ct*Pi/25*I),z=z0):\n z0 := zz:\n pts := [op(p ts),[Re(zz),Im(zz)]]:\nend do:\np_3 := plot(pts,color=COLOR(RGB,.43,.2 ,0)):\np_4 := plots[polygonplot]([seq([pts[i-1],pts[i],[1.98,4.61]],i= 2..nops(pts))],\n style=patchnogrid,color=COLOR(RGB,.85,.4,0) ):\npts := []: z0 := 2.1-4.65*I:\nfor ct from 0 to 50 do\n zz := new ton(`R*`(z)=exp(ct*Pi/25*I),z=z0):\n z0 := zz:\n pts := [op(pts),[ Re(zz),Im(zz)]]:\nend do:\np_5 := plot(pts,color=COLOR(RGB,.43,.2,0)): \np_6 := plots[polygonplot]([seq([pts[i-1],pts[i],[1.98,-4.61]],i=2..n ops(pts))],\n style=patchnogrid,color=COLOR(RGB,.85,.4,0)):\n p_7 := plot([[[-5.19,0],[2.49,0]],[[0,-5.19],[0,5.19]]],color=black,li nestyle=3):\nplots[display]([p_||(1..7)],view=[-5.19..2.49,-5.19..5.19 ],font=[HELVETICA,9],\n labels=[`Re(z)`,`Im(z)`],axes=boxed ,scaling=constrained);" }}{PARA 13 "" 1 "" {GLPLOT2D 681 553 553 {PLOTDATA 2 "6/-%'CURVESG6$7ew7$$\"\"!F)F(7$$\"3)*******[,iF9!#E$\"35+ ++Djzq:!#=7$$\"35+++0>bz!*!#D$\"3$)*****4Z#fTJF07$$\"3)******z^DD-\"!# B$\"3*******z$eQ7ZF07$$\"3w*****pc=Ok&F:$\"3Q+++p9;$G'F07$$\"3#******4 5Ep4#!#A$\"3y*****pGmQ&yF07$$\"35+++l4TIgFE$\"3^*****H;eVU*F07$$\"3!** *****4oqU9!#@$\"3%******>.K%*4\"!#<7$$\"3;+++Qvw))HFP$\"3++++0MJc7FS7$ 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}} {PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1743 "ee := \{c[2]=7/71,\nc[3]=46/255,\nc[4]=23/85,\nc[5]=27/29,\nc[6] =61/90,\nc[7]=18/77,\nc[8]=1,\nc[9]=1,\n\na[2,1]=7/71,\na[3,1]=6992/45 5175,\na[3,2]=75118/455175,\na[4,1]=23/340,\na[4,2]=0,\na[4,3]=69/340, \na[5,1]=125814087/51607124,\na[5,2]=0,\na[5,3]=-481282155/51607124,\n a[5,4]=100879020/12901781,\na[6,1]=3323091540925544113/295739525402804 88000,\na[6,2]=0,\na[6,3]=-10703826229567511/24340701679243200,\na[6,4 ]=169241467001919447401/178319980502135683200,\na[6,5]=255174827160200 5217/45506989353097008000,\na[7,1]=-376385546372076652128/600213349891 9339474385,\na[7,2]=0,\na[7,3]=19305251563909761378/196791262259650474 57,\na[7,4]=-8122491009809811364010481/9979717849963846796488754, a[7, 5]=-155386566948917103529551/2301833170083782096614930,\na[7,6]=107267 076044076240/544838019860392951,\na[8,1]=-1972640388605306005321/25773 9734717483367780,\na[8,2]=0, a[8,3]=12354606914500963/386396118220908, \na[8,4]=-56856879732578412491684/1967664304821917372209,\na[8,5]=-175 89460594924790303414/92934689150952840583755,\na[8,6]=370576933768080/ 119264931607213,\na[8,7]=1150428860219177/432917708558301,\na[9,1]=636 077/8182296,\na[9,2]=0,\na[9,3]=0,\na[9,4]=24315051125/87140294626,\na [9,5]=30418033967/271431025272,\na[9,6]=862843671000/2241531443039,\na [9,7]=1670085824869/14714276549256,\na[9,8]=13887/424328,\n\nb[1]=6360 77/8182296,\nb[2]=0,\nb[3]=0,\nb[4]=24315051125/87140294626,\nb[5]=304 18033967/271431025272,\nb[6]=862843671000/2241531443039,\nb[7]=1670085 824869/14714276549256,\nb[8]=13887/424328,\n\n`b*`[1]=231122308093/348 1256020752,\n`b*`[2]=0,\n`b*`[3]=0,\n`b*`[4]=4006504071740375/84742592 073525056,\n`b*`[5]=23826240438664855/461934347497102656,\n`b*`[6]=172 3971130382250/4007085927807811,\n`b*`[7]=123485516897451247/3682567958 35268016,\n`b*`[8]=63672019983/6138204944224,\n`b*`[9]=1/17\}:" }}} {PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT 259 5 "nodes" }{TEXT -1 1 ":" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 31 "se q(c[i]=subs(ee,c[i]),i=2..9);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6*/&%\" cG6#\"\"##\"\"(\"#r/&F%6#\"\"$#\"#Y\"$b#/&F%6#\"\"%#\"#B\"#&)/&F%6#\" \"&#\"#F\"#H/&F%6#\"\"'#\"#h\"#!*/&F%6#F)#\"#=\"#x/&F%6#\"\")\"\"\"/&F %6#\"\"*FQ" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT 259 51 "linking coefficients for the 9 stage order 5 scheme" }{TEXT -1 1 ":" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 49 "seq(seq(a[i,j]=su bs(ee,a[i,j]),j=1..i-1),i=1..9);" }}{PARA 12 "" 1 "" {XPPMATH 20 "6F/& %\"aG6$\"\"#\"\"\"#\"\"(\"#r/&F%6$\"\"$F(#\"%#*p\"'v^X/&F%6$F/F'#\"&=^ (F2/&F%6$\"\"%F(#\"#B\"$S$/&F%6$F;F'\"\"!/&F%6$F;F/#\"#pF>/&F%6$\"\"&F (#\"*(39e7\")Crg^/&F%6$FKF'FB/&F%6$FKF/#!*b@G\"[FN/&F%6$FKF;#\"*?!z35 \")\"y,H\"/&F%6$\"\"'F(#\"48TaD4a\"4BL\"5+!)[!GSD&RdH/&F%6$FjnF'FB/&F% 6$FjnF/#!26vcHi#Qq5\"2+KCz;qSV#/&F%6$FjnF;#\"6,uW>>+n9Cp\"\"6+KoN@]!)* 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.RIWJ:C#/&F%6#\"\"(#\".p[#e3q;\"/c#\\lF9Z\"/&F%6#\"\")#\"&()Q\"\"'GVU " }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT 259 38 "wei ghts for the 9 stage order 5 scheme" }{TEXT -1 1 ":" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 37 "seq(`b*`[i]=subs(ee,`b*`[i]),i=1..9);" }} {PARA 12 "" 1 "" {XPPMATH 20 "6+/&%#b*G6#\"\"\"#\"-$43B7J#\"._2-c7[$/& F%6#\"\"#\"\"!/&F%6#\"\"$F//&F%6#\"\"%#\"1v.urS]1S\"2c]_t?fUZ)/&F%6#\" \"&#\"2b[mQ/CEQ#\"3cE5(\\ZV$>Y/&F%6#\"\"'#\"1]AQI6(Rs\"\"16y!y#f32S/&F %6#\"\"(#\"3Z7X(*o^&[B\"\"3;!o_$ezc#o$/&F%6#\"\")#\",$)*>?nj\".CU%\\?Q h/&F%6#\"\"*#F'\"#<" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}}{PARA 0 "" 0 "" {TEXT -1 52 "#----- ----------------------------------------------" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}}{PARA 0 "" 0 "" {TEXT -1 59 "#============ ==============================================" }}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 44 "a scheme with a quite large stability region" }} {PARA 257 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 81 "The sch eme constructed to have a stability polynomial suggested by Ch. Tsitou ras." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 162 " See: A Parameter Study of Explicit Runge-Kutta Pairs of Orders 6(5), b y Ch. Tsitouras,\n Applied Mathematics Letters, Vol. 11, No. 1, \+ pages 65 to 69, 1998. " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}{PARA 0 "" 0 "" {TEXT -1 52 "# ---------------------------------------------------" }}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 28 "checking the combined scheme" }}{PARA 0 "" 0 " " {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}} {SECT 1 {PARA 4 "" 0 "" {TEXT -1 35 "coefficients of the combined sche me" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1880 "ee := \{c[2]=1/60,\nc[3]=4/45,\nc[4]=2/15,\nc[5]=18 0/389,\nc[6]=177/262,\nc[7]=125/126,\nc[8]=1,\nc[9]=1,\n\na[2,1]=1/60, \na[3,1]=-4/27,\na[3,2]=32/135,\na[4,1]=1/30,\na[4,2]=0,\na[4,3]=1/10, \na[5,1]=73104030/58863869,\na[5,2]=0,\na[5,3]=-279389250/58863869,\na [5,4]=233523000/58863869,\na[6,1]=-107221818666951807169/2561961567822 3628160,\na[6,2]=0,\na[6,3]=328927584642593578425/20495692542578902528 ,\na[6,4]=-3717245049788716341975/307755633334661333272,\na[6,5]=87713 642192227304821351/98481802667091626647040,\na[7,1]=223899211079219075 995406125/20066238404586623311945344,\na[7,2]=0,\na[7,3]=-273812933690 66082540625/622049813755756227072,\na[7,4]=2493079698743912733446875/7 1342417962706618257992,\na[7,5]=-2021984654539866742602494375/88907803 8644174940599689728,\na[7,6]=999354901744231797500/8462741215004819472 87,\na[8,1]=15677672759360359499/1267222156368417000,\na[8,2]=0,\na[8, 3]=-414775576646775/8485270768736,\na[8,4]=45606172048348599823275/117 5113514395365856852,\na[8,5]=-321038818302974478087638483/123915646917 936012537205920,\na[8,6]=8496514166784579485/6535296976627611297,\na[8 ,7]=-857445886276749/93424049779207625,\na[9,1]=10645627/477900000,\na [9,2]=0,\na[9,3]=0,\na[9,4]=4717085625/17726544496,\na[9,5]=7334570581 73503507/2441724811071084000,\na[9,6]=3157306853548189/136247861366536 50,\na[9,7]=15203797026579/11457084981250,\na[9,8]=-407713/355300,\n\n b[1]=10645627/477900000,\nb[2]=0,\nb[3]=0,\nb[4]=4717085625/1772654449 6,\nb[5]=733457058173503507/2441724811071084000,\nb[6]=315730685354818 9/13624786136653650,\nb[7]=15203797026579/11457084981250,\nb[8]=-40771 3/355300,\n`b*`[1]=15760837607929/1669012981875000,\n`b*`[2]=0,\n`b*`[ 3]=0,\n`b*`[4]=1099642053438585/3714479960779514,\n`b*`[5]=20710911423 05288598181339/8527454295551205104475000,\n`b*`[6]=1134425723844512936 90689/380664489430348936942500,\n`b*`[7]=824482795143047119317/9603025 01853174687500,\n`b*`[8]=-5198669774391/7445076113750,\n`b*`[9]=-1/150 \}:" }}}{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 0 488 "subs(ee,matrix([[c[2],a[2,1],``$3],\n [c[ 3],a[3,1],a[3,2],``$2],\n [c[4],seq(a[4,i],i=1..3),``],\n [c[5],seq( a[5,i],i=1..4)],\n [c[6],seq(a[6,i],i=1..4)],[``$4,a[6,5]],\n [c[7], seq(a[7,i],i=1..4)],[``$3,a[7,5],a[7,6]],\n [c[8],seq(a[8,i],i=1..4)] ,[``$2,seq(a[8,i],i=5..7)],\n [c[9],seq(a[9,i],i=1..4)],[``,seq(a[9,i ],i=5..8)],\n [``,`____________________________`$4],\n [`b`,seq(b[i] ,i=1..4)],[``,seq(b[i],i=5..8)],\n [`b*`,seq(`b*`[i],i=1..4)],[``,seq (`b*`[i],i=5..8)],[``$4,`b*`[9]]]));" }}{PARA 11 "" 1 "" {XPPMATH 20 " 6#K%'matrixG6#747'#\"\"\"\"#gF(%!GF+F+7'#\"\"%\"#X#!\"%\"#F#\"#K\"$N\" F+F+7'#\"\"#\"#:#F)\"#I\"\"!#F)\"#5F+7'#\"$!=\"$*Q#\")IS5t\")pQ')eF<#! *]#*Qz#FE#\"*+I_L#FE7'#\"$x\"\"$i##!6pr!=&pm==A2\"\"5g\"GOAych>c#F<#\" 6D%yNfUYeF*G$\"5GD!*yDa#p&\\?#!7v>M;()y\\]CUOr()\"8SqkE;4nE!=[)*7'#\"$D\"\"$E\"#\"#z5@**QA\"; W`%>JBme/%Qi1?F<#!8D1a#3m!pLH\"QF\"6sqAcdv8)\\?i#\":voWLF\"Ru)pzI\\#\" 8#*zD=mqizTU8(7'F+F+F+#!=vV\\-Eum)RXl%)>-#\"[+:7ui%)7'F)#\"5*\\f.OfFnxc\"\"4+qToj:AsE\"F<#!0vnkw bx9%\".O(o2F&[)#\"8vK#)*f[$[?<1c%\"7_o&eO&R9N6v67'F+F+#!<$[Qw3yW(HI=)Q 5K\"r/V^z#[C)\"6+vouJ&=]-.'*#!.\"Ru(p')>&\".]P6w]W(7'F+F+F+F+#!\"\"\"$ ]\"Q(pprint46\"" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> \+ " 0 "" {MPLTEXT 1 0 136 "subs(ee,matrix([seq([c[i],seq(a[i,j],j=1..i-1 ),``$(10-i)],i=2..9),\n[`b`,seq(b[i],i=1..8),``],[`b*`,seq(`b*`[i],i=1 ..9)]])):\nevalf[8](%);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#K%'matrixG6 #7,7,$\")nmm;!\"*F(%!GF+F+F+F+F+F+F+7,$\")*)))))))F*$!):[\"[\"!\")$\") /PqBF1F+F+F+F+F+F+F+7,$\")LLL8F1$\")LLLLF*$\"\"!F:$\")+++5F1F+F+F+F+F+ F+7,$\")%\\si%F1$\")o\">C\"!\"(F9$!)BOYZFB$\")//'HF1$\")KtGCF1$\")'>,)HF1$\")rl&e)F1$!)`p#)pF1$!)nmmmFioQ(pprint56 \"" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 146 "RK6_8eqs := [op(RowSumConditions(8,'expanded')),op(O rderConditions(6,8,'expanded'))]:\n`RK5_9eqs*` := subs(b=`b*`,OrderCon ditions(5,9,'expanded')):" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 7 "Check: " }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 148 "simplify(subs(ee,RK6_8eqs)):\nmap(u->`if`(lhs(u)=rhs(u),0,1),%); \nnops(%);\nsimplify(subs(ee,`RK5_9eqs*`)):\nmap(u->`if`(lhs(u)=rhs(u) ,0,1),%);\nnops(%);" }{TEXT -1 0 "" }}{PARA 11 "" 1 "" {XPPMATH 20 "6# 7N\"\"!F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$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#\"#W" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#73\"\"!F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F $" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#\"#<" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 86 "Next we set-up stage-order cond tions to check for stage-orders from 2 to 5 inclusive. " }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 85 "for ct from 2 to 5 do\n so||ct||_ 8 := StageOrderConditions(ct,8,'expanded');\nend do:" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 34 "Stages 4 to 8 have sta ge-order 3. " }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 173 "[seq([seq(e xpand(subs(ee,so||i||_8[j])),i=2..5)],j=1..6)]:\nmap(proc(L) local i; \+ for i to nops(L) do if not evalb(L[i]) then break end if end do; i end proc,%):\nsimplify(%); " }}{PARA 11 "" 1 "" {XPPMATH 20 "6#7(\"\"#\" \"$F%F%F%F%" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 53 "None of the principal error conditions are satisfied." }} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 123 "RK6_8err_eqs := PrincipalEr rorConditions(6,8,'expanded'):\nexpand(subs(ee,RK6_8err_eqs)):\nmap(u- >`if`(lhs(u)=rhs(u),0,1),%);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#7R\"\" \"F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$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 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 114 "We can calculate the principal error norm of t he order 6 scheme, that is, the 2-norm of the principal error terms." }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 131 "errterms6_8 := PrincipalE rrorTerms(6,8,'expanded'):\nevalf(evalf[14](sqrt(add(subs(ee,errterms6 _8[i])^2,i=1.. nops(errterms6_8)))));" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#$\"+U!oSW\"!#8" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 79 "The 2 norm of the principal error of the order 5 embedded scheme is as follows." }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 153 "` errterms5_9*` := subs(b=`b*`,PrincipalErrorTerms(5,9,'expanded')):\nev alf(evalf[14](sqrt(add(subs(ee,`errterms5_9*`[i])^2,i=1.. nops(`errter ms5_9*`)))));" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#$\"+S\"=3D'!#8" }}} {PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}}{PARA 0 "" 0 "" {TEXT -1 52 "#------------------------------ ---------------------" }}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 44 "simultan eous construction of the two schemes" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 119 "We incorporate the stage-order equatio ns to ensure that stage 2 has stage-order 2 and stages 3 to 8 have sta ge-order 3." }}{PARA 0 "" 0 "" {TEXT -1 48 "We also incorporate the si mplifying conditions: " }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "Sum(b[i]*a[i,j],i = j+1 .. 8) = 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=1" "6#/%\"jG\"\"\"" }{TEXT -1 10 " ( with " } {XPPEDIT 18 0 "c[1]=0" "6#/&%\"cG6#\"\"\"\"\"!" }{TEXT -1 5 " ), " } {XPPEDIT 18 0 "j = 6;" "6#/%\"jG\"\"'" }{TEXT -1 7 " and " } {XPPEDIT 18 0 "j=7" "6#/%\"jG\"\"(" }{TEXT -1 1 "." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 78 "The 8 simple order condit ions used are given (in abreviated form) as follows. " }}{PARA 0 "" 0 "" {TEXT -1 78 "These include the 6 quadrature conditions and two addi tional order conditions." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 178 "SO6 := SimpleOrderConditions(6):\n [seq([i,SO6[i]],i=[1,2,4,8,16,24,29,32])]:\nlinalg[augment](linalg[del cols](%,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%\"#;F)/*&F,F()F2F5F(#F(\"\"&7%\"# CF)/*(F,F(F2F(-%!G6#*&%\"aGF(-FM6#*&F8F(FPF(F(F(#F(\"#s7%\"#HF)/*(F,F( F2F(-FM6#*&F?F(FPF(F(#F(FI7%\"#KF)/*&F,F()F2FGF(#F(\"\"'Q)pprint596\" " }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 136 "For the embedded order 5 scheme we use a selection of 7 \"simple\" order \+ conditions as given (in abreviated form) in the following table. " }} {PARA 0 "" 0 "" {TEXT -1 78 "These include the 5 quadrature conditions and two additional order conditions." }}{PARA 0 "" 0 "" {TEXT -1 0 " " }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 194 "`SO5*` := subs(b=`b*`,S impleOrderConditions(5)):\n[seq([i,`SO5*`[i]],i=[1,2,4,8,12,15,16])]: \nlinalg[augment](linalg[delcols](%,2..2),matrix([[` `]$(linalg[rowdi m](%))]),linalg[delcols](%,1..1));" }}{PARA 11 "" 1 "" {XPPMATH 20 "6# K%'matrixG6#7)7%\"\"\"%#~~G/*&%#b*GF(%\"eGF(F(7%\"\"#F)/*&F,F(%\"cGF(# F(F/7%\"\"%F)/*&F,F()F2F/F(#F(\"\"$7%\"\")F)/*&F,F()F2F:F(#F(F57%\"#7F )/*&F,F(-%!G6#*&%\"aGF(-FF6#*&F8F(FIF(F(F(#F(\"#g7%\"#:F)/*&F,F(-FF6#* &F?F(FIF(F(#F(\"#?7%\"#;F)/*&F,F()F2F5F(#F(\"\"&Q)pprint186\"" }}} {PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 582 "SO6_8 := SimpleOrderConditions(6,8,'expanded'):\nSO_eqs := [op(Ro wSumConditions(8,'expanded')),op(StageOrderConditions(2,8,'expanded')) ,\n op(StageOrderConditions(3,4..8,'expanded'))]:\n`SO5_ 9*` := subs(b=`b*`,SimpleOrderConditions(5,9,'expanded')):\nord_cdns : = [seq(SO6_8[i],i=[1,2,4,8,16,24,29,32])]:\n`ord_cdns*` := [seq(`SO5_9 *`[i],i=[1,2,4,8,12,15,16])]:\nsimp_eqs := [add(b[i]*a[i,1],i=1+1..8)= b[1],seq(add(b[i]*a[i,j],i=j+1..8)=b[j]*(1-c[j]),j=[6,7])]:\nwt_eqns : = [seq(b[i]=a[9,i],i=1..8)]:\ncdns := [op(ord_cdns),op(simp_eqs),op(SO _eqs),op(wt_eqns),op(`ord_cdns*`)]:" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 22 "We specify the nodes: " }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "c[2] = 1/60;" "6#/&%\"cG6#\"\"#*& \"\"\"F)\"#g!\"\"" }{TEXT -1 3 ", " }{XPPEDIT 18 0 "c[4] = 2/15;" "6# /&%\"cG6#\"\"%*&\"\"#\"\"\"\"#:!\"\"" }{TEXT -1 3 ", " }{XPPEDIT 18 0 "c[5] = 180/389;" "6#/&%\"cG6#\"\"&*&\"$!=\"\"\"\"$*Q!\"\"" }{TEXT -1 3 ", " }{XPPEDIT 18 0 "c[6] = 177/262;" "6#/&%\"cG6#\"\"'*&\"$x\" \"\"\"\"$i#!\"\"" }{TEXT -1 3 ", " }{XPPEDIT 18 0 "c[7] = 125/126;" " 6#/&%\"cG6#\"\"(*&\"$D\"\"\"\"\"$E\"!\"\"" }{TEXT -1 3 ", " } {XPPEDIT 18 0 "c[8]=1" "6#/&%\"cG6#\"\")\"\"\"" }{TEXT -1 2 ", " }} {PARA 0 "" 0 "" {TEXT -1 31 "the zero linking coefficients: " }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "a[4,2]=0" "6#/&%\"aG6$\"\"% \"\"#\"\"!" }{TEXT -1 3 ", " }{XPPEDIT 18 0 "a[5,2]=0" "6#/&%\"aG6$\" \"&\"\"#\"\"!" }{TEXT -1 3 ", " }{XPPEDIT 18 0 "a[6,2]=0" "6#/&%\"aG6 $\"\"'\"\"#\"\"!" }{TEXT -1 3 ", " }{XPPEDIT 18 0 "a[7,2]=0" "6#/&%\" aG6$\"\"(\"\"#\"\"!" }{TEXT -1 3 ", " }{XPPEDIT 18 0 "a[8,2]=0" "6#/& %\"aG6$\"\")\"\"#\"\"!" }{TEXT -1 2 ", " }}{PARA 0 "" 0 "" {TEXT -1 22 "and the zero weights: " }}{PARA 256 "" 0 "" {TEXT -1 1 " " } {XPPEDIT 18 0 "b[2]=0" "6#/&%\"bG6#\"\"#\"\"!" }{TEXT -1 3 ", " } {XPPEDIT 18 0 "b[3]=0" "6#/&%\"bG6#\"\"$\"\"!" }{TEXT -1 1 "." }} {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT 260 4 "Note" } {TEXT -1 89 ": Calculations relating to the choice of nodes are perfor med in the following subsection." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }} {PARA 0 "" 0 "" {TEXT -1 125 "The weights of the order 6 scheme provid e the linking coefficients for the 9th stage of the embedded order 5 s cheme so that: " }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "a[ 9,i]=b[i]" "6#/&%\"aG6$\"\"*%\"iG&%\"bG6#F(" }{TEXT -1 3 ", " } {XPPEDIT 18 0 "i=1" "6#/%\"iG\"\"\"" }{TEXT -1 9 " . . . 8." }}{PARA 0 "" 0 "" {TEXT -1 22 "We also specify that " }{XPPEDIT 18 0 "c[9] = \+ 1;" "6#/&%\"cG6#\"\"*\"\"\"" }{TEXT -1 3 ", " }{XPPEDIT 18 0 "`b*`[2] = 0;" "6#/&%#b*G6#\"\"#\"\"!" }{TEXT -1 3 ", " }{XPPEDIT 18 0 "`b*`[ 3] = 0;" "6#/&%#b*G6#\"\"$\"\"!" }{TEXT -1 7 " and " }{XPPEDIT 18 0 "`b*`[9] = -1/150;" "6#/&%#b*G6#\"\"*,$*&\"\"\"F*\"$]\"!\"\"F," } {TEXT -1 2 ". " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 37 "We have 44 equations and 44 unknowns." }}{EXCHG {PARA 0 " > " 0 "" {MPLTEXT 1 0 214 "e1 := \{c[2]=1/60,c[4]=2/15,c[5]=180/389,c[ 6]=177/262,c[7]=125/126,c[8]=1,c[9]=1,\n seq(a[i,2]=0,i=4..8),b[ 2]=0,b[3]=0,`b*`[2]=0,`b*`[3]=0,`b*`[9]=-1/150\}:\neqns := subs(e1,cdn s):\nnops(%);\nindets(eqns);\nnops(%);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#\"#W" }}{PARA 12 "" 1 "" {XPPMATH 20 "6# " 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 2000 "e3 := \{b[2] = 0, b[3] = 0, b[4] = 4717085625/17726544496, c[8] \+ = 1, c[9] = 1, a[5,2] = 0, a[4,2] = 0, `b*`[3] = 0, `b*`[2] = 0, a[9,3 ] = 0, a[9,2] = 0, a[8,2] = 0, a[7,2] = 0, a[6,2] = 0, b[5] = 73345705 8173503507/2441724811071084000, b[8] = -407713/355300, a[5,1] = 731040 30/58863869, a[8,1] = 15677672759360359499/1267222156368417000, b[6] = 3157306853548189/13624786136653650, a[9,5] = 733457058173503507/24417 24811071084000, a[4,3] = 1/10, a[8,6] = 8496514166784579485/6535296976 627611297, a[9,6] = 3157306853548189/13624786136653650, `b*`[7] = 8244 82795143047119317/960302501853174687500, `b*`[5] = 2071091142305288598 181339/8527454295551205104475000, a[8,5] = -32103881830297447808763848 3/123915646917936012537205920, a[7,1] = 223899211079219075995406125/20 066238404586623311945344, c[4] = 2/15, c[5] = 180/389, c[6] = 177/262, c[7] = 125/126, `b*`[9] = -1/150, a[3,1] = -4/27, a[9,4] = 4717085625 /17726544496, a[6,5] = 87713642192227304821351/98481802667091626647040 , a[7,6] = 999354901744231797500/846274121500481947287, a[8,7] = -8574 45886276749/93424049779207625, a[9,7] = 15203797026579/11457084981250, a[7,3] = -27381293369066082540625/622049813755756227072, a[7,5] = -20 21984654539866742602494375/889078038644174940599689728, c[3] = 4/45, a [8,3] = -414775576646775/8485270768736, a[6,3] = 328927584642593578425 /20495692542578902528, a[5,3] = -279389250/58863869, a[4,1] = 1/30, a[ 2,1] = 1/60, b[7] = 15203797026579/11457084981250, `b*`[6] = 113442572 384451293690689/380664489430348936942500, a[6,4] = -371724504978871634 1975/307755633334661333272, `b*`[4] = 1099642053438585/371447996077951 4, `b*`[1] = 15760837607929/1669012981875000, a[9,1] = 10645627/477900 000, a[5,4] = 233523000/58863869, a[3,2] = 32/135, a[7,4] = 2493079698 743912733446875/71342417962706618257992, a[9,8] = -407713/355300, b[1] = 10645627/477900000, a[8,4] = 45606172048348599823275/11751135143953 65856852, c[2] = 1/60, a[6,1] = -107221818666951807169/256196156782236 28160, `b*`[8] = -5198669774391/7445076113750\}:" }{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 0 488 "subs(e3,matrix([[c[2],a[2,1],``$3],\n [c[3],a[3,1], a[3,2],``$2],\n [c[4],seq(a[4,i],i=1..3),``],\n [c[5],seq(a[5,i],i=1 ..4)],\n [c[6],seq(a[6,i],i=1..4)],[``$4,a[6,5]],\n [c[7],seq(a[7,i] ,i=1..4)],[``$3,a[7,5],a[7,6]],\n [c[8],seq(a[8,i],i=1..4)],[``$2,seq (a[8,i],i=5..7)],\n [c[9],seq(a[9,i],i=1..4)],[``,seq(a[9,i],i=5..8)] ,\n [``,`____________________________`$4],\n [`b`,seq(b[i],i=1..4)], [``,seq(b[i],i=5..8)],\n [`b*`,seq(`b*`[i],i=1..4)],[``,seq(`b*`[i],i =5..8)],[``$4,`b*`[9]]]));" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#K%'matri xG6#747'#\"\"\"\"#gF(%!GF+F+7'#\"\"%\"#X#!\"%\"#F#\"#K\"$N\"F+F+7'#\" \"#\"#:#F)\"#I\"\"!#F)\"#5F+7'#\"$!=\"$*Q#\")IS5t\")pQ')eF<#!*]#*Qz#FE #\"*+I_L#FE7'#\"$x\"\"$i##!6pr!=&pm==A2\"\"5g\"GOAych>c#F<#\"6D%yNfUYe F*G$\"5GD!*yDa#p&\\?#!7v>M;()y\\]CUOr()\"8SqkE;4nE!=[)*7'#\"$D\"\"$E\"#\"#z5@**QA\";W`%>JBme/ %Qi1?F<#!8D1a#3m!pLH\"QF\"6sqAcdv8)\\?i#\":voWLF\"Ru)pzI\\#\"8#*zD=mqi zTU8(7'F+F+F+#!=vV\\-Eum)RXl%)>-#\"[+:7ui%)7'F)#\"5*\\f.OfFnxc\"\"4+qToj:AsE\"F<#!0vnkwbx9%\".O( o2F&[)#\"8vK#)*f[$[?<1c%\"7_o&eO&R9N6v67'F+F+#!<$[Qw3yW(HI=)Q5K\"r/V^ z#[C)\"6+vouJ&=]-.'*#!.\"Ru(p')>&\".]P6w]W(7'F+F+F+F+#!\"\"\"$]\"Q(ppr int26\"" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 136 "subs(e3,matrix([seq([c[i],seq(a[i,j],j=1..i-1),``$(1 0-i)],i=2..9),\n[`b`,seq(b[i],i=1..8),``],[`b*`,seq(`b*`[i],i=1..9)]]) ):\nevalf[8](%);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#K%'matrixG6#7,7,$ \")nmm;!\"*F(%!GF+F+F+F+F+F+F+7,$\")*)))))))F*$!):[\"[\"!\")$\")/PqBF1 F+F+F+F+F+F+F+7,$\")LLL8F1$\")LLLLF*$\"\"!F:$\")+++5F1F+F+F+F+F+F+7,$ \")%\\si%F1$\")o\">C\"!\"(F9$!)BOYZFB$\")//' HF1$\")KtGCF1$\")'>,)HF1$\")rl&e)F1$!)`p#)pF1$!)nmmmFioQ(pprint36\"" } }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 146 "RK6_8eqs := [op(RowSumConditions(8,'expanded')),op(OrderConditi ons(6,8,'expanded'))]:\n`RK5_9eqs*` := subs(b=`b*`,OrderConditions(5,9 ,'expanded')):" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 7 "Check: " }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 148 "simp lify(subs(e3,RK6_8eqs)):\nmap(u->`if`(lhs(u)=rhs(u),0,1),%);\nnops(%); \nsimplify(subs(e3,`RK5_9eqs*`)):\nmap(u->`if`(lhs(u)=rhs(u),0,1),%); \nnops(%);" }{TEXT -1 0 "" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#7N\"\"!F$ F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$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#\"#W" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#73\"\"!F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#\"#<" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}}{SECT 1 {PARA 4 "" 0 " " {TEXT -1 21 "determining the nodes" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 22 "eG .. general solution" }}{PARA 0 "" 0 "" {TEXT -1 0 " " }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 71947 "eG := \{c[8] = 1, b[3 ] = 0, b[2] = 0, c[9] = 1, b[5] = -1/60*(10*c[6]*c[7]*c[4]-5*c[6]*c[4] -5*c[7]*c[6]+3*c[6]-5*c[7]*c[4]+3*c[4]-2+3*c[7])/(-c[4]+c[5])/c[5]/(-c [5]^3+c[5]^2-c[7]*c[5]+c[7]*c[5]^2-c[6]*c[5]+c[6]*c[5]^2+c[7]*c[6]-c[6 ]*c[7]*c[5]), a[5,2] = 0, a[4,2] = 0, `b*`[3] = 0, `b*`[2] = 0, a[8,2] = 0, a[7,2] = 0, a[6,2] = 0, a[6,5] = 1/2*c[6]*(30*c[6]^2*c[5]^2*c[4] ^2-2*c[6]^2*c[4]^2+c[6]^3*c[4]-12*c[6]^2*c[4]*c[5]^2-3*c[5]^2*c[6]*c[4 ]+c[5]*c[6]*c[4]^2+c[6]*c[4]^3-12*c[6]^2*c[4]^2*c[5]-30*c[5]*c[6]^3*c[ 4]^2-20*c[5]*c[6]^2*c[4]^4+30*c[6]^2*c[5]*c[4]^3-c[5]*c[4]^3+20*c[4]^3 *c[5]*c[6]^3-20*c[6]^2*c[5]^2*c[4]^3+c[5]^2*c[4]^2-30*c[5]^2*c[6]*c[4] ^3+12*c[5]^2*c[6]*c[4]^2+2*c[6]^2*c[4]*c[5]+20*c[5]^2*c[6]*c[4]^4+2*c[ 6]^2*c[5]^2-2*c[6]^3*c[5]+12*c[6]^3*c[4]*c[5])/c[5]/(c[4]^3-2*c[5]*c[4 ]^2+30*c[5]^2*c[4]^3-6*c[5]*c[4]^3-10*c[4]^4*c[5]^2-c[5]^3+10*c[5]^4*c [4]^2-30*c[5]^3*c[4]^2+6*c[5]^3*c[4]+2*c[5]^2*c[4]), b[1] = 1/60*(30*c [5]*c[6]*c[7]*c[4]-10*c[5]*c[6]*c[4]-10*c[6]*c[7]*c[5]+5*c[6]*c[5]-10* c[6]*c[7]*c[4]+5*c[6]*c[4]+5*c[7]*c[6]-3*c[6]-10*c[5]*c[7]*c[4]+5*c[5] *c[4]+5*c[7]*c[5]-3*c[5]+5*c[7]*c[4]-3*c[4]+2-3*c[7])/c[5]/c[6]/c[7]/c [4], a[6,1] = -1/4*c[6]*(60*c[5]^2*c[6]*c[4]^4-90*c[5]^2*c[6]^3*c[4]^2 -60*c[5]*c[6]^2*c[4]^4-2*c[6]^3*c[4]^2+50*c[5]^4*c[4]^3*c[6]+40*c[6]^2 *c[4]^5*c[5]-40*c[4]^5*c[5]^3+36*c[4]*c[5]^2*c[6]^3-20*c[5]^4*c[6]^2*c [4]^2+8*c[6]^2*c[5]^3+4*c[6]^2*c[4]^3-24*c[5]*c[6]^3*c[4]^2-8*c[6]^2*c [4]^2*c[5]+24*c[6]^2*c[5]*c[4]^3+60*c[4]^3*c[5]*c[6]^3-4*c[6]*c[5]*c[4 ]^3-14*c[5]^3*c[6]*c[4]+7*c[5]^3*c[4]^2+6*c[5]^2*c[6]*c[4]^3+14*c[5]^2 *c[6]*c[4]^2+66*c[5]^3*c[6]*c[4]^2-80*c[5]^3*c[6]^2*c[4]^3+30*c[6]^2*c [5]^2*c[4]^3-24*c[6]^2*c[5]^2*c[4]^2-240*c[6]*c[5]^3*c[4]^3-48*c[6]^2* c[5]^3*c[4]+150*c[6]^2*c[5]^3*c[4]^2-24*c[4]^4*c[5]^2-9*c[5]^2*c[4]^3+ 6*c[4]^4*c[5]-40*c[5]^4*c[4]^4-20*c[6]^2*c[4]^4*c[5]^2-6*c[5]^2*c[6]^3 -2*c[6]*c[4]^4-24*c[5]^3*c[4]^3+60*c[6]^3*c[4]^3*c[5]^2+7*c[6]^3*c[4]* c[5]-40*c[5]^2*c[6]*c[4]^5-40*c[6]^3*c[4]^4*c[5]+110*c[4]^4*c[6]*c[5]^ 3+120*c[5]^3*c[4]^4)/c[5]/c[4]^2/(10*c[5]^3*c[4]^2+10*c[5]^2*c[4]^3-30 *c[5]^2*c[4]^2+6*c[5]^2*c[4]-c[5]^2+6*c[5]*c[4]^2+c[5]*c[4]-c[4]^2), b [4] = 1/60*(-2+3*c[5]-5*c[7]*c[6]+3*c[7]+3*c[6]+10*c[6]*c[7]*c[5]-5*c[ 6]*c[5]-5*c[7]*c[5])/c[4]/(-c[6]*c[4]^3-c[4]^3+c[4]^4+c[7]*c[4]^2+c[6] *c[4]^2-c[6]*c[7]*c[4]+c[6]*c[7]*c[4]^2-c[5]*c[4]^3+c[5]*c[4]^2-c[7]*c [4]^3-c[5]*c[7]*c[4]+c[5]*c[7]*c[4]^2-c[5]*c[6]*c[4]+c[5]*c[6]*c[4]^2+ c[6]*c[7]*c[5]-c[5]*c[6]*c[7]*c[4]), b[6] = 1/60*(10*c[5]*c[7]*c[4]-5* c[5]*c[4]-5*c[7]*c[5]+3*c[5]-5*c[7]*c[4]+3*c[4]-2+3*c[7])/(-c[4]+c[6]) /(-c[5]+c[6])/c[6]/(c[6]^2-c[6]+c[7]-c[7]*c[6]), `b*`[9] = 1/10*(50*c[ 5]^2*c[6]*c[4]^2-40*c[5]^2*c[6]*c[4]+10*c[6]*c[5]^2-40*c[5]*c[6]*c[4]^ 2+35*c[5]*c[6]*c[4]-10*c[6]*c[5]+10*c[6]*c[4]^2-10*c[6]*c[4]+3*c[6]-5* c[5]^2*c[4]-5*c[5]*c[4]^2+10*c[5]*c[4]-c[5]-c[4])/(3*c[6]-28*c[5]*c[6] *c[4]^2-28*c[5]^2*c[6]*c[4]+8*c[6]*c[5]^2-9*c[6]*c[4]-9*c[6]*c[5]+28*c [5]*c[6]*c[4]+30*c[5]^2*c[6]*c[4]^2+8*c[6]*c[4]^2-4*c[5]^2*c[4]-4*c[5] *c[4]^2+9*c[5]*c[4]-c[5]-c[4]), a[7,5] = -1/2*c[7]*(-17*c[5]^2*c[6]*c[ 4]^4+5*c[6]*c[5]*c[4]^4+42*c[4]^2*c[5]^2*c[7]^3+17*c[7]^3*c[5]*c[4]^3- 80*c[7]^3*c[5]^2*c[4]^3+c[7]*c[5]*c[4]^4+80*c[5]^2*c[4]^4*c[7]^2-60*c[ 4]^5*c[6]^2*c[5]^2+20*c[4]^3*c[7]^3*c[5]^3-20*c[7]^2*c[5]^2*c[4]^5+40* c[4]^5*c[6]*c[5]^3-5*c[5]*c[6]^2*c[4]^4+c[7]^3*c[4]^2-c[7]^2*c[4]^3-80 *c[5]^3*c[4]^4*c[7]+20*c[7]^3*c[5]^2*c[4]^4-200*c[6]^2*c[4]^5*c[7]*c[5 ]^3-290*c[5]^4*c[4]^2*c[7]^2*c[6]+20*c[6]^2*c[7]*c[5]^4*c[4]^2+40*c[6] *c[5]^2*c[4]^4*c[7]-140*c[5]^4*c[4]^3*c[6]-100*c[5]^4*c[4]^5*c[6]+190* c[5]^4*c[4]^4*c[6]-40*c[7]^2*c[4]^4*c[5]^3+20*c[5]^4*c[4]^4*c[7]+2*c[4 ]^4*c[7]^2-20*c[7]^2*c[5]^4*c[4]^3-4*c[6]^2*c[7]^2*c[5]-8*c[7]*c[6]^2* c[5]^3-3*c[7]*c[6]*c[5]^3+2*c[7]^2*c[6]*c[5]^3+60*c[7]*c[6]^2*c[5]*c[4 ]^5-40*c[6]*c[5]^2*c[4]^5*c[7]+17*c[7]*c[4]^4*c[6]*c[5]+80*c[7]^2*c[5] ^3*c[4]^3+5*c[7]*c[4]^4*c[6]^2+c[5]^2*c[7]^3-260*c[6]^2*c[7]*c[5]^2*c[ 4]^3-12*c[6]^2*c[7]^2*c[5]^2*c[4]+20*c[6]^2*c[7]^2*c[5]^2*c[4]^2-40*c[ 6]*c[7]*c[5]^3*c[4]^3-41*c[6]^2*c[7]^3*c[4]*c[5]+20*c[7]^2*c[6]*c[5]*c [4]^4+3*c[7]*c[6]*c[4]^3-2*c[6]^2*c[7]*c[4]^2-3*c[6]*c[7]^2*c[4]^2+4*c [7]^2*c[6]*c[4]^3+6*c[5]*c[7]^2*c[4]^3+4*c[6]*c[5]*c[7]^2*c[4]^2+c[7]^ 2*c[5]*c[6]*c[4]-11*c[5]*c[7]*c[6]*c[4]^3+2*c[6]^2*c[4]^2*c[5]-c[7]^3* c[4]*c[5]+17*c[5]^2*c[6]*c[7]*c[4]^2-200*c[5]^4*c[7]^2*c[4]^4*c[6]-50* c[4]^5*c[6]*c[7]*c[5]^3-150*c[5]^2*c[6]*c[7]^3*c[4]^4-c[6]^2*c[5]*c[4] ^3+2*c[7]^3*c[4]^2*c[6]*c[5]+2*c[6]^2*c[7]*c[4]*c[5]+108*c[7]*c[6]^2*c [5]*c[4]^3+4*c[6]^2*c[7]*c[5]^2+2*c[6]*c[5]*c[7]*c[4]^2-10*c[6]^2*c[7] ^3*c[5]^2+6*c[5]^4*c[7]*c[6]-8*c[6]*c[7]^3*c[5]^2+3*c[6]*c[7]^2*c[5]^2 +2*c[6]^2*c[7]^2*c[5]^2-27*c[6]^2*c[4]^2*c[7]*c[5]-62*c[6]^2*c[7]^2*c[ 4]^2*c[5]+46*c[6]*c[5]^4*c[4]^2+20*c[4]^5*c[7]*c[5]^3+2*c[7]^3*c[4]*c[ 6]*c[5]-27*c[6]^2*c[5]^2*c[7]*c[4]+120*c[6]^2*c[5]^2*c[7]*c[4]^2-35*c[ 6]*c[5]^2*c[7]*c[4]^3-37*c[6]*c[5]^3*c[7]*c[4]^2+53*c[6]^2*c[5]^3*c[7] *c[4]-54*c[7]^2*c[5]^2*c[4]^3+2*c[7]^2*c[5]*c[4]^2-c[7]*c[5]*c[4]^3-3* c[6]*c[5]*c[4]^3-2*c[4]^3*c[7]^3+63*c[7]^3*c[5]^2*c[6]*c[4]-16*c[7]^2* c[5]^2*c[6]*c[4]-5*c[6]*c[7]*c[4]^4+2*c[6]^2*c[7]^2*c[4]^2+2*c[6]^2*c[ 7]^2*c[4]-10*c[5]^4*c[7]^2*c[6]+3*c[5]^3*c[6]*c[4]+3*c[5]^3*c[7]*c[4]+ 20*c[5]^3*c[6]*c[7]*c[4]-2*c[5]^3*c[4]^2+22*c[6]^2*c[7]^2*c[4]*c[5]+7* c[5]^2*c[6]*c[4]^3-c[5]^2*c[7]*c[4]^2+c[5]^2*c[6]*c[4]^2-4*c[5]^2*c[7] *c[4]^3-8*c[5]^3*c[7]*c[4]^2-22*c[5]^3*c[6]*c[4]^2+240*c[5]^3*c[6]^2*c [4]^3+100*c[4]^3*c[7]^2*c[5]^2*c[6]^2+150*c[4]^5*c[5]^3*c[6]^2-3*c[7]* c[5]^4*c[4]+5*c[7]^2*c[5]^4*c[4]+12*c[5]^2*c[4]^4*c[7]-98*c[6]^2*c[5]^ 2*c[4]^3+25*c[6]^2*c[5]^2*c[4]^2+54*c[6]*c[5]^3*c[4]^3+8*c[6]^2*c[5]^3 *c[4]-63*c[6]^2*c[5]^3*c[4]^2+42*c[5]^3*c[7]*c[4]^3+60*c[5]^4*c[7]*c[4 ]^2*c[6]+200*c[5]^3*c[7]^3*c[4]^4*c[6]-4*c[6]^2*c[4]*c[5]^2+c[7]*c[6]^ 2*c[4]^3+200*c[7]^2*c[5]^2*c[6]*c[4]^3-240*c[7]^3*c[5]^2*c[6]*c[4]^2+1 80*c[7]^2*c[5]^3*c[6]*c[4]^2+3*c[5]^4*c[4]^2-80*c[5]^3*c[7]^3*c[6]*c[4 ]+c[7]^3*c[4]^2*c[6]-7*c[7]^3*c[4]^2*c[5]+20*c[4]^3*c[7]^2*c[5]*c[6]^2 -19*c[7]^2*c[6]*c[5]*c[4]^3-11*c[7]^2*c[5]^2*c[6]*c[4]^2-3*c[4]^4*c[5] ^2-27*c[7]^2*c[5]^3*c[6]*c[4]-480*c[7]^2*c[5]^3*c[6]*c[4]^3-380*c[5]^3 *c[6]^2*c[4]^4-5*c[5]^4*c[4]^3-10*c[6]^2*c[4]^3*c[7]^2+130*c[5]^3*c[6] ^2*c[7]*c[4]^3-5*c[7]^3*c[5]^2*c[4]-2*c[7]^2*c[5]^2*c[4]-4*c[5]^2*c[6] *c[7]*c[4]-160*c[6]^2*c[7]*c[5]^3*c[4]^2-80*c[6]^2*c[7]^2*c[4]*c[5]^3+ 2*c[5]^2*c[4]^3+12*c[7]^2*c[5]^2*c[4]^2+8*c[7]^2*c[5]^3*c[4]+400*c[7]^ 3*c[5]^2*c[6]^2*c[4]^3-17*c[7]^2*c[4]^4*c[5]+250*c[7]^2*c[5]^3*c[6]^2* c[4]^2-100*c[6]^2*c[4]^5*c[5]*c[7]^2-200*c[7]^2*c[4]^5*c[6]*c[5]^3+150 *c[5]^2*c[6]*c[4]^5*c[7]^2+380*c[7]^3*c[5]^2*c[4]^3*c[6]-20*c[7]^3*c[6 ]*c[5]*c[4]^3-210*c[7]^3*c[6]^2*c[5]*c[4]^3-6*c[5]^4*c[4]*c[6]+80*c[5] ^4*c[7]^2*c[4]*c[6]+10*c[6]*c[7]^3*c[5]^3-36*c[5]^4*c[4]*c[7]*c[6]-50* c[4]^5*c[6]^2*c[7]*c[5]^2+200*c[4]^5*c[6]^2*c[7]^2*c[5]^2+160*c[6]^2*c [4]^4*c[5]^2-450*c[7]^3*c[5]^3*c[4]^3*c[6]+5*c[6]^2*c[7]^3*c[4]^2-c[5] ^3*c[7]^2+6*c[6]^2*c[7]^3*c[5]-160*c[7]*c[6]^2*c[5]*c[4]^4+200*c[5]^3* c[4]^4*c[6]^2*c[7]-350*c[5]^3*c[4]^3*c[6]^2*c[7]^2-350*c[5]^4*c[4]^4*c [6]*c[7]+230*c[5]^2*c[4]^4*c[7]*c[6]^2+150*c[5]^3*c[4]^4*c[6]*c[7]+100 *c[5]^4*c[4]^3*c[6]*c[7]-3*c[6]^2*c[7]^3*c[4]-42*c[7]^2*c[5]^3*c[4]^2+ 50*c[5]^4*c[4]^4*c[6]^2-20*c[5]^4*c[4]^3*c[6]^2+500*c[5]^3*c[4]^4*c[7] ^2*c[6]+450*c[5]^4*c[4]^3*c[7]^2*c[6]-340*c[5]^2*c[4]^4*c[7]^2*c[6]-50 *c[5]^4*c[4]^3*c[6]^2*c[7]+150*c[7]^2*c[4]^4*c[6]^2*c[5]+200*c[7]^2*c[ 4]^4*c[5]^3*c[6]^2-350*c[7]^2*c[4]^4*c[6]^2*c[5]^2+10*c[7]^2*c[6]^2*c[ 5]^3-60*c[4]^4*c[6]*c[5]^3+100*c[7]^3*c[6]^2*c[5]*c[4]^4+200*c[5]^4*c[ 4]^5*c[6]*c[7]-200*c[7]^3*c[4]^4*c[6]^2*c[5]^2+5*c[5]^3*c[4]^4+80*c[6] ^2*c[7]^3*c[5]^2*c[4]-270*c[6]^2*c[7]^3*c[5]^2*c[4]^2+290*c[5]^3*c[7]^ 3*c[4]^2*c[6]+140*c[6]^2*c[5]*c[7]^3*c[4]^2-5*c[7]^3*c[4]*c[5]^3)/c[5] /(100*c[5]^2*c[6]*c[4]^4-18*c[6]*c[5]*c[4]^4+4*c[5]^3*c[4]+50*c[4]^5*c [6]^2*c[5]^2+40*c[5]*c[6]^2*c[4]^4+3*c[5]^5-5*c[4]^4*c[6]^2+180*c[5]^4 *c[4]^3*c[6]+100*c[5]^4*c[4]^5*c[6]-350*c[5]^4*c[4]^4*c[6]+140*c[5]^5* c[4]^2-180*c[5]^5*c[4]^3-2*c[5]^4+30*c[4]^5*c[5]^3+300*c[5]^5*c[4]^3*c [6]+2*c[6]*c[5]^3-50*c[5]^5*c[4]^2*c[6]^2-40*c[5]^4*c[6]^2*c[4]+240*c[ 5]^4*c[6]^2*c[4]^2-3*c[6]^2*c[5]^3+3*c[6]^2*c[4]^3+5*c[5]^4*c[6]^2-2*c [6]*c[4]^3+2*c[5]*c[4]^3+4*c[5]*c[6]*c[4]^2-6*c[6]^2*c[4]^2*c[5]-50*c[ 5]^4*c[4]^5-13*c[6]^2*c[5]*c[4]^3-40*c[6]*c[5]^4*c[4]^2+6*c[6]*c[5]*c[ 4]^3-15*c[5]^3*c[6]*c[4]-55*c[5]^2*c[6]*c[4]^3+6*c[5]^2*c[6]*c[4]^2+68 *c[5]^3*c[6]*c[4]^2-100*c[4]^5*c[5]^3*c[6]^2+100*c[6]^2*c[5]^2*c[4]^3- 100*c[6]*c[5]^3*c[4]^3+13*c[6]^2*c[5]^3*c[4]-100*c[6]^2*c[5]^3*c[4]^2- 4*c[5]^2*c[4]^2+6*c[6]^2*c[4]*c[5]^2-68*c[5]^4*c[4]^2+23*c[4]^4*c[5]^2 +350*c[5]^3*c[6]^2*c[4]^4-4*c[5]^2*c[6]*c[4]-9*c[5]^2*c[4]^3-3*c[4]^4* c[5]-5*c[5]^5*c[6]-30*c[5]^6*c[4]^2+50*c[5]^6*c[6]*c[4]^2+180*c[5]^4*c [4]^4+10*c[5]^4*c[4]*c[6]-240*c[6]^2*c[4]^4*c[5]^2+3*c[6]*c[4]^4-100*c [5]^6*c[4]^3*c[6]+68*c[5]^3*c[4]^3-23*c[5]^5*c[4]-210*c[5]^5*c[6]*c[4] ^2-350*c[5]^4*c[4]^3*c[6]^2-30*c[5]^2*c[6]*c[4]^5+50*c[5]^6*c[4]^3+60* c[4]^4*c[6]*c[5]^3+100*c[5]^5*c[6]^2*c[4]^3+9*c[5]^4*c[4]-140*c[5]^3*c [4]^4+40*c[5]^5*c[4]*c[6]), `b*`[6] = -1/60*(-30*c[5]^3*c[4]+103*c[5]* c[7]*c[4]-18*c[6]+6*c[5]+6*c[4]+27*c[7]*c[6]-9*c[5]^2-9*c[4]^2-100*c[7 ]*c[6]*c[5]^3+522*c[5]*c[6]*c[7]*c[4]-111*c[6]*c[5]^2+75*c[6]*c[5]+60* c[6]*c[5]^3-9*c[7]*c[5]+600*c[6]*c[7]*c[5]^3*c[4]^3+14*c[7]*c[5]^2-100 *c[7]*c[6]*c[4]^3+14*c[7]*c[4]^2-9*c[7]*c[4]-117*c[6]*c[7]*c[5]-111*c[ 6]*c[4]^2+60*c[6]*c[4]^3-117*c[6]*c[7]*c[4]+180*c[6]*c[7]*c[4]^2-30*c[ 5]*c[4]^3-174*c[5]*c[7]*c[4]^2+486*c[5]*c[6]*c[4]^2+107*c[5]*c[4]^2-31 8*c[5]*c[6]*c[4]-66*c[5]*c[4]+75*c[6]*c[4]+500*c[5]*c[7]*c[6]*c[4]^3+1 420*c[5]^2*c[6]*c[7]*c[4]^2-840*c[6]*c[5]*c[7]*c[4]^2+180*c[6]*c[7]*c[ 5]^2-900*c[6]*c[5]^2*c[7]*c[4]^3-900*c[6]*c[5]^3*c[7]*c[4]^2+50*c[7]*c [5]*c[4]^3-280*c[6]*c[5]*c[4]^3-280*c[5]^3*c[6]*c[4]+50*c[5]^3*c[7]*c[ 4]+107*c[5]^2*c[4]+500*c[5]^3*c[6]*c[7]*c[4]+50*c[5]^3*c[4]^2+470*c[5] ^2*c[6]*c[4]^3+310*c[5]^2*c[7]*c[4]^2-770*c[5]^2*c[6]*c[4]^2-90*c[5]^2 *c[7]*c[4]^3-90*c[5]^3*c[7]*c[4]^2+470*c[5]^3*c[6]*c[4]^2-300*c[6]*c[5 ]^3*c[4]^3-180*c[5]^2*c[4]^2+486*c[5]^2*c[6]*c[4]-174*c[5]^2*c[7]*c[4] -840*c[5]^2*c[6]*c[7]*c[4]+50*c[5]^2*c[4]^3)/(-9*c[5]*c[7]*c[4]-3*c[7] *c[6]+3*c[6]^2-28*c[5]*c[6]*c[7]*c[4]-c[6]*c[5]+c[7]*c[5]+8*c[6]^2*c[4 ]^2+c[7]*c[4]+9*c[6]*c[7]*c[5]+9*c[6]*c[7]*c[4]-8*c[6]*c[7]*c[4]^2+4*c [5]*c[7]*c[4]^2-4*c[5]*c[6]*c[4]^2+9*c[5]*c[6]*c[4]-c[6]*c[4]-28*c[6]^ 2*c[4]^2*c[5]+28*c[6]^2*c[4]*c[5]-30*c[5]^2*c[6]*c[7]*c[4]^2+28*c[6]*c [5]*c[7]*c[4]^2-8*c[6]*c[7]*c[5]^2+8*c[6]^2*c[5]^2-9*c[6]^2*c[5]+30*c[ 6]^2*c[5]^2*c[4]^2-28*c[6]^2*c[4]*c[5]^2-9*c[6]^2*c[4]-4*c[5]^2*c[6]*c [4]+4*c[5]^2*c[7]*c[4]+28*c[5]^2*c[6]*c[7]*c[4])/(c[5]*c[4]-c[6]*c[4]- c[6]*c[5]+c[6]^2)/c[6], a[8,7] = (10*c[5]*c[6]*c[4]-5*c[6]*c[5]-5*c[6] *c[4]+3*c[6]-5*c[5]*c[4]+3*c[5]+3*c[4]-2)/c[7]/(c[6]-c[7])/(c[5]-c[7]) /(c[4]-c[7])/(-20*c[5]*c[6]*c[4]+30*c[5]*c[6]*c[7]*c[4]+15*c[6]*c[5]-2 0*c[6]*c[7]*c[5]+15*c[6]*c[4]-20*c[6]*c[7]*c[4]-12*c[6]+15*c[7]*c[6]+1 5*c[5]*c[4]-20*c[5]*c[7]*c[4]-12*c[5]+15*c[7]*c[5]-12*c[4]+15*c[7]*c[4 ]+10-12*c[7])*(c[4]-1)*(-1+c[5])*(-1+c[6])*(-1+c[7]), a[6,3] = 3/4*1/c [4]^2*c[6]*(60*c[4]^3*c[5]*c[6]^3-90*c[5]*c[6]^3*c[4]^2+36*c[6]^3*c[4] *c[5]-6*c[6]^3*c[5]+3*c[6]^3*c[4]-20*c[6]^2*c[5]^3*c[4]^2-80*c[6]^2*c[ 5]^2*c[4]^3+150*c[6]^2*c[5]^2*c[4]^2-48*c[6]^2*c[4]*c[5]^2+8*c[6]^2*c[ 5]^2-60*c[5]*c[6]^2*c[4]^4+90*c[6]^2*c[5]*c[4]^3-48*c[6]^2*c[4]^2*c[5] +4*c[6]^2*c[4]*c[5]-4*c[6]^2*c[4]^2+30*c[6]*c[5]^3*c[4]^3+90*c[5]^2*c[ 6]*c[4]^4-180*c[5]^2*c[6]*c[4]^3+54*c[5]^2*c[6]*c[4]^2-12*c[5]^2*c[6]* c[4]+18*c[6]*c[5]*c[4]^3+6*c[5]*c[6]*c[4]^2+3*c[5]^2*c[4]^2-3*c[5]*c[4 ]^3)/(10*c[5]^3*c[4]^2+10*c[5]^2*c[4]^3-30*c[5]^2*c[4]^2+6*c[5]^2*c[4] -c[5]^2+6*c[5]*c[4]^2+c[5]*c[4]-c[4]^2), b[7] = -1/60*(10*c[5]*c[6]*c[ 4]-5*c[6]*c[5]-5*c[6]*c[4]+3*c[6]-5*c[5]*c[4]+3*c[5]+3*c[4]-2)/(c[4]-c [7])/(c[5]-c[7])/(c[6]-c[7])/c[7]/(-1+c[7]), a[7,1] = 1/4*c[7]*(-51*c[ 5]^2*c[6]*c[4]^4+14*c[6]*c[5]*c[4]^4+12*c[4]^2*c[5]^2*c[7]^3+2700*c[5] ^4*c[4]^4*c[6]^2*c[7]+12*c[7]^3*c[5]*c[4]^3-98*c[7]^3*c[5]^2*c[4]^3-2* c[7]*c[5]*c[4]^4+132*c[5]^2*c[4]^4*c[7]^2-480*c[4]^5*c[6]^2*c[5]^2+200 *c[4]^3*c[7]^3*c[5]^3-200*c[7]^2*c[5]^2*c[4]^5-760*c[4]^5*c[6]*c[5]^3- 10*c[5]*c[6]^2*c[4]^4+200*c[5]^5*c[4]^3*c[7]^2*c[6]^2-400*c[5]^2*c[4]^ 6*c[6]^2*c[7]^2+4*c[4]^5*c[5]*c[7]-34*c[5]^2*c[4]^5*c[7]+400*c[5]^3*c[ 4]^6*c[6]^2*c[7]-98*c[5]^3*c[4]^4*c[7]+420*c[7]^2*c[4]^5*c[5]^3+200*c[ 7]^3*c[5]^2*c[4]^4-300*c[6]^2*c[4]^5*c[7]*c[5]^3-40*c[7]*c[5]^2*c[4]^6 *c[6]+20*c[5]^4*c[4]^2*c[7]^2*c[6]-280*c[6]^2*c[7]*c[5]^4*c[4]^2-44*c[ 6]*c[4]^5*c[5]*c[7]-246*c[6]*c[5]^2*c[4]^4*c[7]+120*c[4]^6*c[6]^2*c[5] ^2-100*c[5]^2*c[7]^2*c[4]^6*c[6]+141*c[5]^4*c[4]^3*c[6]+1220*c[5]^4*c[ 4]^5*c[6]+400*c[5]^5*c[4]^5*c[6]^2-760*c[5]^4*c[4]^4*c[6]+100*c[5]^4*c [7]^3*c[4]^4-400*c[7]^2*c[4]^4*c[5]^3-200*c[5]^4*c[4]^5*c[7]^2+200*c[5 ]^4*c[4]^4*c[7]-2*c[4]^3*c[7]^3*c[6]-8*c[7]^2*c[6]*c[4]^4-380*c[7]^3*c [5]^3*c[4]^4-40*c[6]*c[4]^5*c[5]*c[7]^2+2*c[4]^4*c[7]^2-40*c[5]^2*c[7] ^3*c[4]^5-200*c[7]^2*c[5]^4*c[4]^3-40*c[7]^3*c[5]^4*c[4]^3+200*c[5]^3* c[7]*c[4]^6*c[6]-10*c[6]^2*c[7]^3*c[4]^3+30*c[6]^2*c[4]^5*c[5]+12*c[7] *c[6]^2*c[5]^3+4*c[7]^2*c[6]*c[5]^3+400*c[5]^4*c[4]^6*c[6]^2-40*c[5]^3 *c[7]*c[4]^6-380*c[5]^4*c[4]^5*c[7]+320*c[7]*c[6]^2*c[5]*c[4]^5+100*c[ 4]^6*c[6]^2*c[7]*c[5]^2+34*c[5]^4*c[4]^2*c[7]^2+40*c[5]^5*c[7]^2*c[4]^ 3+240*c[6]*c[5]^2*c[4]^5*c[7]+60*c[5]^4*c[7]^3*c[4]^2*c[6]+26*c[7]*c[4 ]^4*c[6]*c[5]-50*c[4]^4*c[6]^2*c[5]^5+200*c[5]^5*c[7]*c[4]^4*c[6]+132* c[7]^2*c[5]^3*c[4]^3-2*c[7]*c[4]^4*c[6]^2+60*c[5]^5*c[4]^2*c[6]^2*c[7] -500*c[5]^5*c[4]^4*c[6]^2*c[7]+160*c[5]^5*c[4]^4*c[6]+134*c[6]^2*c[5]^ 4*c[7]*c[4]+24*c[5]^4*c[6]^2*c[4]-169*c[5]^4*c[6]^2*c[4]^2-500*c[5]^3* c[4]^6*c[6]^2-40*c[5]^5*c[4]^4*c[7]+34*c[4]^5*c[5]*c[7]^2-100*c[5]^3*c [7]^2*c[4]^6-100*c[5]^5*c[4]^4*c[7]^2+160*c[4]^6*c[6]*c[5]^3-6*c[6]^2* c[7]*c[5]^2*c[4]^3+68*c[6]^2*c[7]^2*c[5]^2*c[4]-126*c[6]^2*c[7]^2*c[5] ^2*c[4]^2-600*c[6]*c[7]*c[5]^3*c[4]^3-21*c[6]^2*c[7]^3*c[4]*c[5]+40*c[ 7]^2*c[6]^2*c[5]^4+4*c[5]^4*c[7]*c[4]^2+58*c[7]^2*c[6]*c[5]*c[4]^4-100 *c[7]^2*c[5]^5*c[6]^2*c[4]^2-320*c[5]^4*c[4]*c[7]^2*c[6]^2+20*c[6]^2*c [4]^4*c[7]^2+40*c[5]^2*c[4]^6*c[7]^2+6*c[7]^2*c[6]*c[4]^3-8*c[6]*c[5]* c[7]^2*c[4]^2-4*c[5]*c[7]*c[6]*c[4]^3+14*c[5]^2*c[6]*c[7]*c[4]^2-910*c [4]^5*c[6]*c[7]*c[5]^3-340*c[5]^2*c[6]*c[7]^3*c[4]^4-4*c[6]^2*c[5]*c[4 ]^3+5*c[7]^3*c[4]^2*c[6]*c[5]+72*c[7]*c[6]^2*c[5]*c[4]^3+18*c[6]^2*c[7 ]^3*c[5]^2-12*c[6]^2*c[7]^2*c[5]^2-10*c[6]^2*c[4]^2*c[7]*c[5]-44*c[6]^ 2*c[7]^2*c[4]^2*c[5]-25*c[6]*c[5]^4*c[4]^2-24*c[7]*c[6]^2*c[5]^4+200*c [4]^5*c[7]*c[5]^3-2*c[6]^2*c[5]^2*c[7]*c[4]-42*c[6]^2*c[5]^2*c[7]*c[4] ^2+74*c[6]*c[5]^2*c[7]*c[4]^3+80*c[6]*c[5]^3*c[7]*c[4]^2-50*c[6]^2*c[5 ]^3*c[7]*c[4]-34*c[5]^4*c[4]^3*c[7]-24*c[7]^2*c[5]^2*c[4]^3+420*c[5]^4 *c[7]^2*c[4]^4-2*c[4]^3*c[7]^3+c[7]^3*c[5]^2*c[6]*c[4]+4*c[7]^2*c[5]^2 *c[6]*c[4]-6*c[6]*c[7]*c[4]^4-4*c[6]^2*c[7]^2*c[4]^2+50*c[5]^5*c[4]^3* c[7]^2*c[6]-18*c[5]^3*c[6]*c[7]*c[4]+14*c[6]^2*c[7]^2*c[4]*c[5]-20*c[5 ]^2*c[6]*c[4]^3+2*c[5]^2*c[7]*c[4]^3-2*c[5]^3*c[7]*c[4]^2+16*c[5]^3*c[ 6]*c[4]^2-96*c[5]^3*c[6]^2*c[4]^3+100*c[5]^5*c[4]^3*c[6]^2*c[7]-60*c[4 ]^3*c[7]^2*c[5]^2*c[6]^2+10*c[6]*c[7]*c[4]^5+1540*c[4]^5*c[5]^3*c[6]^2 -34*c[7]^3*c[4]^4*c[5]-4*c[7]^2*c[5]^4*c[4]+12*c[5]^2*c[4]^4*c[7]-41*c [6]^2*c[5]^2*c[4]^3+14*c[6]^2*c[5]^2*c[4]^2-54*c[6]*c[5]^3*c[4]^3-12*c [6]^2*c[5]^3*c[4]+47*c[6]^2*c[5]^3*c[4]^2+12*c[5]^3*c[7]*c[4]^3-98*c[5 ]^4*c[7]*c[4]^2*c[6]-50*c[5]^3*c[7]^3*c[4]^4*c[6]+4*c[7]*c[6]^2*c[4]^3 +156*c[7]^2*c[5]^2*c[6]*c[4]^3-23*c[7]^3*c[5]^2*c[6]*c[4]^2+188*c[7]^2 *c[5]^3*c[6]*c[4]^2+66*c[5]^3*c[7]^3*c[6]*c[4]+2*c[7]^3*c[4]^2*c[5]+10 0*c[5]^5*c[4]^5*c[7]+104*c[4]^3*c[7]^2*c[5]*c[6]^2-12*c[7]^2*c[6]*c[5] *c[4]^3-36*c[7]^2*c[5]^2*c[6]*c[4]^2-32*c[7]^2*c[5]^3*c[6]*c[4]-320*c[ 7]^2*c[5]^3*c[6]*c[4]^3-560*c[5]^3*c[6]^2*c[4]^4-4*c[6]^2*c[4]^3*c[7]^ 2+240*c[5]^3*c[6]^2*c[7]*c[4]^3-34*c[5]^3*c[7]^3*c[4]^2-2*c[7]^3*c[5]^ 2*c[4]+164*c[6]^2*c[7]*c[5]^3*c[4]^2-20*c[6]^2*c[7]^2*c[4]*c[5]^3+2*c[ 7]^2*c[5]^3*c[4]-130*c[5]^5*c[4]^3*c[6]*c[7]-90*c[7]^3*c[5]^2*c[6]^2*c [4]^3-16*c[7]^2*c[4]^4*c[5]-60*c[7]^2*c[5]^3*c[6]^2*c[4]^2-300*c[6]^2* c[4]^5*c[5]*c[7]^2-150*c[7]^2*c[4]^5*c[6]*c[5]^3+380*c[5]^2*c[6]*c[4]^ 5*c[7]^2+140*c[7]^3*c[5]^2*c[4]^3*c[6]-810*c[7]^3*c[5]^3*c[6]^2*c[4]^2 -14*c[7]^3*c[6]*c[5]*c[4]^3-280*c[7]^3*c[6]^2*c[5]*c[4]^3+240*c[7]^3*c [5]^3*c[6]^2*c[4]-20*c[5]^4*c[7]^2*c[4]*c[6]-6*c[6]*c[7]^3*c[5]^3+22*c [5]^4*c[4]*c[7]*c[6]-280*c[4]^5*c[6]^2*c[7]*c[5]^2+400*c[4]^5*c[6]^2*c [7]^2*c[5]^2+253*c[6]^2*c[4]^4*c[5]^2+470*c[7]^3*c[5]^3*c[4]^3*c[6]+10 0*c[5]^2*c[4]^5*c[7]^3*c[6]+6*c[6]^2*c[7]^3*c[4]^2-300*c[5]^5*c[4]^5*c [6]-120*c[5]*c[7]*c[4]^6*c[6]^2-10*c[6]^2*c[7]*c[4]^5-1100*c[5]^4*c[4] ^5*c[6]^2*c[7]-226*c[7]*c[6]^2*c[5]*c[4]^4-770*c[5]^3*c[4]^4*c[6]^2*c[ 7]+280*c[5]^3*c[4]^3*c[6]^2*c[7]^2-1000*c[5]^4*c[4]^4*c[6]*c[7]+240*c[ 5]^2*c[4]^4*c[7]*c[6]^2+1240*c[5]^3*c[4]^4*c[6]*c[7]+600*c[5]^4*c[4]^3 *c[6]*c[7]+1200*c[7]^3*c[5]^3*c[6]^2*c[4]^3+40*c[7]^3*c[4]^4*c[5]*c[6] -16*c[7]^2*c[5]^3*c[4]^2-1250*c[5]^4*c[4]^5*c[6]^2-600*c[6]^2*c[7]^3*c [5]^3*c[4]^4+200*c[6]^2*c[7]^2*c[5]^3*c[4]^5-200*c[6]^2*c[7]^3*c[4]^5* c[5]-140*c[5]^4*c[4]^4*c[6]^2+560*c[5]^4*c[4]^3*c[6]^2-1850*c[7]^2*c[5 ]^4*c[6]^2*c[4]^3+1230*c[7]^2*c[5]^4*c[6]^2*c[4]^2-150*c[5]^4*c[7]^3*c [4]^3*c[6]-300*c[5]^4*c[4]^6*c[6]+310*c[5]^3*c[4]^4*c[7]^2*c[6]-70*c[5 ]^4*c[4]^3*c[7]^2*c[6]-380*c[5]^2*c[4]^4*c[7]^2*c[6]-850*c[5]^4*c[4]^3 *c[6]^2*c[7]-40*c[7]^2*c[4]^4*c[6]^2*c[5]-450*c[7]^2*c[4]^4*c[5]^3*c[6 ]^2+250*c[7]^2*c[4]^4*c[6]^2*c[5]^2-4*c[7]^2*c[4]^5+136*c[5]^2*c[6]*c[ 4]^5+400*c[6]^2*c[7]^3*c[5]^2*c[4]^5+800*c[6]^2*c[7]^2*c[5]^4*c[4]^4+4 *c[7]^3*c[4]^4+100*c[5]^4*c[4]^6*c[7]+437*c[4]^4*c[6]*c[5]^3-22*c[6]*c [5]*c[4]^5-60*c[5]^5*c[6]^2*c[4]^3+200*c[6]^2*c[4]^6*c[5]*c[7]^2+420*c [7]^3*c[6]^2*c[5]*c[4]^4+400*c[5]^4*c[4]^5*c[6]*c[7]-500*c[7]^3*c[4]^4 *c[6]^2*c[5]^2+100*c[5]^3*c[7]^3*c[4]^5-103*c[6]^2*c[7]^3*c[5]^2*c[4]+ 260*c[6]^2*c[7]^3*c[5]^2*c[4]^2-300*c[5]^3*c[7]^3*c[4]^2*c[6]+97*c[6]^ 2*c[5]*c[7]^3*c[4]^2-30*c[6]^2*c[7]^3*c[5]^3+4*c[7]^3*c[4]*c[5]^3)/c[6 ]/(-50*c[5]^2*c[6]*c[4]^4+23*c[5]^3*c[4]+100*c[5]^4*c[4]^3*c[6]+2*c[5] ^2+2*c[4]^2-3*c[5]^3-3*c[4]^3-3*c[6]*c[5]^2+5*c[6]*c[5]^3-3*c[6]*c[4]^ 2+5*c[6]*c[4]^3+23*c[5]*c[4]^3+18*c[5]*c[6]*c[4]^2-12*c[5]*c[4]^2+3*c[ 5]*c[6]*c[4]-2*c[5]*c[4]-50*c[6]*c[5]^4*c[4]^2-40*c[6]*c[5]*c[4]^3-40* c[5]^3*c[6]*c[4]-12*c[5]^2*c[4]-140*c[5]^3*c[4]^2+240*c[5]^2*c[6]*c[4] ^3-140*c[5]^2*c[6]*c[4]^2+240*c[5]^3*c[6]*c[4]^2-400*c[6]*c[5]^3*c[4]^ 3+91*c[5]^2*c[4]^2+30*c[5]^4*c[4]^2+30*c[4]^4*c[5]^2-50*c[5]^4*c[4]^3+ 18*c[5]^2*c[6]*c[4]-140*c[5]^2*c[4]^3+210*c[5]^3*c[4]^3+100*c[4]^4*c[6 ]*c[5]^3-50*c[5]^3*c[4]^4)/c[5]/c[4]^2, `b*`[8] = 0, a[5,3] = -3/4*c[5 ]^2*(2*c[5]-3*c[4])/c[4]^2, `b*`[4] = -1/60*(-30*c[5]^3*c[4]-42*c[7]*c [6]^2+89*c[5]*c[7]*c[4]-18*c[6]+6*c[5]+6*c[4]+27*c[7]*c[6]-9*c[5]^2+18 0*c[7]*c[6]^2*c[5]^3-100*c[7]*c[6]*c[5]^3+27*c[6]^2+186*c[5]*c[6]*c[7] *c[4]-97*c[6]*c[5]^2+66*c[6]*c[5]+60*c[6]*c[5]^3-100*c[6]^2*c[5]^3-9*c [7]*c[5]+14*c[7]*c[5]^2-100*c[6]^2*c[7]*c[4]^2+120*c[6]^2*c[7]*c[4]+60 *c[6]^2*c[4]^2-9*c[7]*c[4]-103*c[6]*c[7]*c[5]-36*c[6]*c[4]^2-61*c[6]*c [7]*c[4]+60*c[6]*c[7]*c[4]^2-30*c[5]*c[7]*c[4]^2+126*c[5]*c[6]*c[4]^2+ 18*c[5]*c[4]^2-112*c[5]*c[6]*c[4]-57*c[5]*c[4]+39*c[6]*c[4]-280*c[6]^2 *c[4]^2*c[5]+330*c[6]^2*c[4]*c[5]+380*c[5]^2*c[6]*c[7]*c[4]^2-560*c[6] ^2*c[7]*c[4]*c[5]+192*c[6]^2*c[7]*c[5]-310*c[6]^2*c[7]*c[5]^2-230*c[6] *c[5]*c[7]*c[4]^2+156*c[6]*c[7]*c[5]^2+180*c[6]^2*c[5]^2+500*c[6]^2*c[ 4]^2*c[7]*c[5]+950*c[6]^2*c[5]^2*c[7]*c[4]-900*c[6]^2*c[5]^2*c[7]*c[4] ^2-300*c[6]*c[5]^3*c[7]*c[4]^2-600*c[6]^2*c[5]^3*c[7]*c[4]-117*c[6]^2* c[5]-130*c[5]^3*c[6]*c[4]+50*c[5]^3*c[7]*c[4]+93*c[5]^2*c[4]+230*c[5]^ 3*c[6]*c[7]*c[4]+50*c[5]^2*c[7]*c[4]^2-190*c[5]^2*c[6]*c[4]^2+150*c[5] ^3*c[6]*c[4]^2+470*c[6]^2*c[5]^2*c[4]^2+320*c[6]^2*c[5]^3*c[4]-300*c[6 ]^2*c[5]^3*c[4]^2-30*c[5]^2*c[4]^2-530*c[6]^2*c[4]*c[5]^2-75*c[6]^2*c[ 4]+156*c[5]^2*c[6]*c[4]-150*c[5]^2*c[7]*c[4]-270*c[5]^2*c[6]*c[7]*c[4] +600*c[6]^2*c[7]*c[5]^3*c[4]^2)/(-9*c[5]*c[7]*c[4]-3*c[7]*c[6]-c[4]^2- 28*c[5]*c[6]*c[7]*c[4]+c[7]*c[5]+c[7]*c[4]+9*c[6]*c[7]*c[5]-9*c[6]*c[4 ]^2+8*c[6]*c[4]^3+9*c[6]*c[7]*c[4]-8*c[6]*c[7]*c[4]^2-4*c[5]*c[4]^3+4* c[5]*c[7]*c[4]^2+28*c[5]*c[6]*c[4]^2+9*c[5]*c[4]^2-9*c[5]*c[6]*c[4]-c[ 5]*c[4]+3*c[6]*c[4]-30*c[5]^2*c[6]*c[7]*c[4]^2+28*c[6]*c[5]*c[7]*c[4]^ 2-8*c[6]*c[7]*c[5]^2-28*c[6]*c[5]*c[4]^3+30*c[5]^2*c[6]*c[4]^3-28*c[5] ^2*c[6]*c[4]^2-4*c[5]^2*c[4]^2+8*c[5]^2*c[6]*c[4]+4*c[5]^2*c[7]*c[4]+2 8*c[5]^2*c[6]*c[7]*c[4])/(-c[4]+c[6])/(-c[4]+c[5])/c[4], a[8,6] = 1/2* (5*c[5]*c[6]*c[4]+c[6]-2*c[6]*c[4]-2*c[6]*c[5]+25*c[5]*c[7]*c[4]+6*c[5 ]+9*c[7]+10*c[7]^2*c[4]-4+6*c[4]-6*c[7]^2-14*c[7]*c[4]-10*c[5]*c[4]-14 *c[7]*c[5]-20*c[7]^2*c[5]*c[4]+10*c[7]^2*c[5])/(c[6]-c[7])/(-c[4]+c[6] )/(-c[5]+c[6])/c[6]/(-20*c[5]*c[6]*c[4]+30*c[5]*c[6]*c[7]*c[4]+15*c[6] *c[5]-20*c[6]*c[7]*c[5]+15*c[6]*c[4]-20*c[6]*c[7]*c[4]-12*c[6]+15*c[7] *c[6]+15*c[5]*c[4]-20*c[5]*c[7]*c[4]-12*c[5]+15*c[7]*c[5]-12*c[4]+15*c [7]*c[4]+10-12*c[7])*(c[4]-1)*(-1+c[5])*(-1+c[6]), b[8] = 1/60*(-20*c[ 5]*c[6]*c[4]+30*c[5]*c[6]*c[7]*c[4]+15*c[6]*c[5]-20*c[6]*c[7]*c[5]+15* c[6]*c[4]-20*c[6]*c[7]*c[4]-12*c[6]+15*c[7]*c[6]+15*c[5]*c[4]-20*c[5]* c[7]*c[4]-12*c[5]+15*c[7]*c[5]-12*c[4]+15*c[7]*c[4]+10-12*c[7])/(c[4]- 1)/(-1+c[5])/(-1+c[6])/(-1+c[7]), a[5,1] = 1/4*c[5]*(2*c[5]^2-5*c[5]*c [4]+4*c[4]^2)/c[4]^2, `b*`[5] = 1/60*(-42*c[7]*c[6]^2+89*c[5]*c[7]*c[4 ]-18*c[6]+6*c[5]+6*c[4]+27*c[7]*c[6]-9*c[4]^2+27*c[6]^2+186*c[5]*c[6]* c[7]*c[4]-36*c[6]*c[5]^2+39*c[6]*c[5]-9*c[7]*c[5]-100*c[6]^2*c[4]^3+60 0*c[6]^2*c[7]*c[5]^2*c[4]^3-100*c[7]*c[6]*c[4]^3-310*c[6]^2*c[7]*c[4]^ 2+192*c[6]^2*c[7]*c[4]+180*c[6]^2*c[4]^2+14*c[7]*c[4]^2-9*c[7]*c[4]-61 *c[6]*c[7]*c[5]-97*c[6]*c[4]^2+60*c[6]*c[4]^3-103*c[6]*c[7]*c[4]+156*c [6]*c[7]*c[4]^2-30*c[5]*c[4]^3-150*c[5]*c[7]*c[4]^2+156*c[5]*c[6]*c[4] ^2+93*c[5]*c[4]^2-112*c[5]*c[6]*c[4]-57*c[5]*c[4]+66*c[6]*c[4]+230*c[5 ]*c[7]*c[6]*c[4]^3-530*c[6]^2*c[4]^2*c[5]+330*c[6]^2*c[4]*c[5]+380*c[5 ]^2*c[6]*c[7]*c[4]^2+320*c[6]^2*c[5]*c[4]^3-560*c[6]^2*c[7]*c[4]*c[5]- 600*c[7]*c[6]^2*c[5]*c[4]^3+120*c[6]^2*c[7]*c[5]-100*c[6]^2*c[7]*c[5]^ 2-270*c[6]*c[5]*c[7]*c[4]^2+60*c[6]*c[7]*c[5]^2+60*c[6]^2*c[5]^2+950*c [6]^2*c[4]^2*c[7]*c[5]+500*c[6]^2*c[5]^2*c[7]*c[4]-900*c[6]^2*c[5]^2*c [7]*c[4]^2-300*c[6]*c[5]^2*c[7]*c[4]^3-75*c[6]^2*c[5]+50*c[7]*c[5]*c[4 ]^3-130*c[6]*c[5]*c[4]^3+18*c[5]^2*c[4]+150*c[5]^2*c[6]*c[4]^3+50*c[5] ^2*c[7]*c[4]^2-190*c[5]^2*c[6]*c[4]^2-300*c[6]^2*c[5]^2*c[4]^3+470*c[6 ]^2*c[5]^2*c[4]^2-30*c[5]^2*c[4]^2-280*c[6]^2*c[4]*c[5]^2+180*c[7]*c[6 ]^2*c[4]^3-117*c[6]^2*c[4]+126*c[5]^2*c[6]*c[4]-30*c[5]^2*c[7]*c[4]-23 0*c[5]^2*c[6]*c[7]*c[4])/(-4*c[5]^3*c[4]-9*c[5]*c[7]*c[4]-3*c[7]*c[6]- c[5]^2-28*c[5]*c[6]*c[7]*c[4]-9*c[6]*c[5]^2+3*c[6]*c[5]+8*c[6]*c[5]^3+ c[7]*c[5]+c[7]*c[4]+9*c[6]*c[7]*c[5]+9*c[6]*c[7]*c[4]-8*c[6]*c[7]*c[4] ^2+4*c[5]*c[7]*c[4]^2+8*c[5]*c[6]*c[4]^2-9*c[5]*c[6]*c[4]-c[5]*c[4]-30 *c[5]^2*c[6]*c[7]*c[4]^2+28*c[6]*c[5]*c[7]*c[4]^2-8*c[6]*c[7]*c[5]^2-2 8*c[5]^3*c[6]*c[4]+9*c[5]^2*c[4]-28*c[5]^2*c[6]*c[4]^2+30*c[5]^3*c[6]* c[4]^2-4*c[5]^2*c[4]^2+28*c[5]^2*c[6]*c[4]+4*c[5]^2*c[7]*c[4]+28*c[5]^ 2*c[6]*c[7]*c[4])/(-c[5]+c[6])/(-c[4]+c[5])/c[5], a[5,4] = c[5]^2*(-c[ 4]+c[5])/c[4]^2, a[7,6] = 1/2*(5*c[5]*c[4]+1-2*c[4]-2*c[5])*(c[4]-c[7] )*(c[5]-c[7])*(c[6]-c[7])*c[7]/(10*c[5]*c[6]*c[4]-5*c[6]*c[5]-5*c[6]*c [4]+3*c[6]-5*c[5]*c[4]+3*c[5]+3*c[4]-2)/(-c[4]+c[6])/(-c[5]+c[6])/c[6] , c[3] = 2/3*c[4], a[4,3] = 3/4*c[4], a[3,2] = 2/9*c[4]^2/c[2], a[7,4] = -1/2*c[7]*(-68*c[5]^2*c[6]*c[4]^4-7*c[6]*c[5]*c[4]^4+7*c[4]^2*c[5]^ 2*c[7]^3-1400*c[5]^4*c[4]^4*c[6]^2*c[7]+11*c[7]^3*c[5]*c[4]^3-72*c[7]^ 3*c[5]^2*c[4]^3-3*c[7]*c[5]*c[4]^4-58*c[5]^2*c[4]^4*c[7]^2-120*c[6]*c[ 4]^7*c[7]*c[5]+320*c[4]^5*c[6]^2*c[5]^2-200*c[4]^3*c[7]^3*c[5]^3-100*c [4]^5*c[6]*c[5]^3-2*c[5]*c[6]^2*c[4]^4-600*c[4]^5*c[7]^3*c[6]*c[5]^3-2 00*c[5]^5*c[4]^3*c[7]^2*c[6]^2+400*c[5]^2*c[4]^6*c[6]^2*c[7]^2-21*c[4] ^5*c[5]*c[7]+120*c[5]^2*c[4]^5*c[7]-600*c[5]^3*c[4]^6*c[6]^2*c[7]-198* c[5]^3*c[4]^4*c[7]-280*c[7]^2*c[4]^5*c[5]^3+240*c[7]^3*c[5]^2*c[4]^4+1 250*c[6]^2*c[4]^5*c[7]*c[5]^3+120*c[7]*c[5]^2*c[4]^6*c[6]-340*c[5]^4*c [4]^2*c[7]^2*c[6]+400*c[6]^2*c[7]*c[5]^4*c[4]^2-240*c[5]^2*c[4]^6*c[7] -300*c[6]*c[5]*c[7]^2*c[4]^6-228*c[6]*c[4]^5*c[5]*c[7]-119*c[6]*c[5]^2 *c[4]^4*c[7]-200*c[5]*c[7]^3*c[4]^6*c[6]-600*c[5]^4*c[7]*c[4]^6*c[6]-1 20*c[4]^6*c[6]^2*c[5]^2+600*c[5]^2*c[7]^2*c[4]^6*c[6]+100*c[5]^2*c[7]^ 3*c[4]^6-187*c[5]^4*c[4]^3*c[6]-910*c[5]^4*c[4]^5*c[6]-320*c[5]^2*c[6] *c[4]^6+620*c[5]^4*c[4]^4*c[6]-100*c[5]^4*c[7]^3*c[4]^4-40*c[7]^2*c[4] ^4*c[5]^3+40*c[7]^3*c[4]^5*c[5]+10*c[6]*c[7]^2*c[4]^5-10*c[6]*c[7]^3*c [4]^4+100*c[5]^4*c[4]^5*c[7]^2+80*c[5]^4*c[4]^4*c[7]+8*c[4]^3*c[7]^3*c [6]-2*c[7]^2*c[6]*c[4]^4+280*c[7]^3*c[5]^3*c[4]^4+80*c[6]*c[4]^5*c[5]* c[7]^2+c[4]^4*c[7]^2-360*c[5]^2*c[7]^3*c[4]^5-80*c[7]^2*c[5]^4*c[4]^3+ 40*c[7]^3*c[5]^4*c[4]^3-1200*c[5]^3*c[7]*c[4]^6*c[6]+10*c[6]^2*c[7]^3* c[4]^3-10*c[6]^2*c[4]^5*c[5]-12*c[7]*c[6]^2*c[5]^3-4*c[7]^2*c[6]*c[5]^ 3-80*c[5]^4*c[4]^5*c[7]-240*c[7]*c[6]^2*c[5]*c[4]^5-3*c[5]^4*c[4]^2*c[ 7]^2+20*c[5]^5*c[7]^2*c[4]^3+180*c[6]*c[5]^2*c[4]^5*c[7]-60*c[5]^4*c[7 ]^3*c[4]^2*c[6]+400*c[5]^2*c[7]^3*c[4]^6*c[6]-25*c[4]^5*c[5]^3+200*c[5 ]^5*c[4]^4*c[7]^2*c[6]+60*c[5]^2*c[4]^7*c[7]+100*c[7]*c[4]^4*c[6]*c[5] -40*c[5]*c[4]^6*c[7]^2-150*c[4]^4*c[6]^2*c[5]^5+50*c[5]^5*c[7]*c[4]^4* c[6]+174*c[7]^2*c[5]^3*c[4]^3+8*c[7]*c[4]^4*c[6]^2+600*c[5]^3*c[7]*c[4 ]^7*c[6]-60*c[5]^5*c[4]^2*c[6]^2*c[7]+200*c[5]^5*c[4]^4*c[6]^2*c[7]-40 *c[5]^5*c[4]^4*c[6]-200*c[5]^5*c[7]*c[4]^5*c[6]-149*c[6]^2*c[5]^4*c[7] *c[4]-24*c[5]^4*c[6]^2*c[4]+189*c[5]^4*c[6]^2*c[4]^2+300*c[5]^3*c[4]^6 *c[6]^2-20*c[5]^5*c[4]^4*c[7]-5*c[4]^5*c[5]*c[7]^2+600*c[4]^6*c[6]*c[5 ]^3-84*c[6]^2*c[7]*c[5]^2*c[4]^3-74*c[6]^2*c[7]^2*c[5]^2*c[4]+162*c[6] ^2*c[7]^2*c[5]^2*c[4]^2+311*c[6]*c[7]*c[5]^3*c[4]^3+21*c[6]^2*c[7]^3*c [4]*c[5]-40*c[7]^2*c[6]^2*c[5]^4+11*c[5]^4*c[7]*c[4]^2+107*c[7]^2*c[6] *c[5]*c[4]^4+100*c[7]^2*c[5]^5*c[6]^2*c[4]^2+320*c[5]^4*c[4]*c[7]^2*c[ 6]^2-10*c[6]^2*c[4]^4*c[7]^2+300*c[5]^2*c[4]^6*c[7]^2-3*c[7]^2*c[6]*c[ 4]^3-2*c[5]*c[7]^2*c[4]^3+15*c[6]*c[5]*c[7]^2*c[4]^2-8*c[5]*c[7]*c[6]* c[4]^3-14*c[5]^2*c[6]*c[7]*c[4]^2-2000*c[5]^4*c[7]^2*c[4]^4*c[6]+200*c [4]^5*c[6]*c[7]*c[5]^3+250*c[5]^2*c[6]*c[7]^3*c[4]^4+4*c[6]^2*c[5]*c[4 ]^3-26*c[7]^3*c[4]^2*c[6]*c[5]-69*c[7]*c[6]^2*c[5]*c[4]^3-18*c[6]^2*c[ 7]^3*c[5]^2+12*c[6]^2*c[7]^2*c[5]^2+10*c[6]^2*c[4]^2*c[7]*c[5]+50*c[6] ^2*c[7]^2*c[4]^2*c[5]+31*c[6]*c[5]^4*c[4]^2+24*c[7]*c[6]^2*c[5]^4+240* c[4]^5*c[7]*c[5]^3+2*c[6]^2*c[5]^2*c[7]*c[4]+39*c[6]^2*c[5]^2*c[7]*c[4 ]^2-17*c[6]*c[5]^2*c[7]*c[4]^3-73*c[6]*c[5]^3*c[7]*c[4]^2+59*c[6]^2*c[ 5]^3*c[7]*c[4]-12*c[5]^4*c[4]^3*c[7]-12*c[7]^2*c[5]^2*c[4]^3+40*c[5]^4 *c[7]^2*c[4]^4-c[4]^3*c[7]^3+17*c[7]^3*c[5]^2*c[6]*c[4]-9*c[7]^2*c[5]^ 2*c[6]*c[4]+3*c[6]*c[7]*c[4]^4+4*c[4]^5*c[5]+4*c[6]^2*c[7]^2*c[4]^2-15 0*c[5]^5*c[4]^3*c[7]^2*c[6]+21*c[5]^3*c[6]*c[7]*c[4]-14*c[6]^2*c[7]^2* c[4]*c[5]+23*c[5]^2*c[6]*c[4]^3+13*c[5]^2*c[7]*c[4]^3-11*c[5]^3*c[7]*c [4]^2-17*c[5]^3*c[6]*c[4]^2+168*c[5]^3*c[6]^2*c[4]^3+50*c[5]^5*c[4]^3* c[6]^2*c[7]-220*c[4]^3*c[7]^2*c[5]^2*c[6]^2-6*c[6]*c[7]*c[4]^5-580*c[4 ]^5*c[5]^3*c[6]^2-31*c[7]^3*c[4]^4*c[5]-2*c[7]^2*c[5]^4*c[4]-4*c[5]^2* c[4]^4*c[7]+53*c[6]^2*c[5]^2*c[4]^3-14*c[6]^2*c[5]^2*c[4]^2+47*c[6]*c[ 5]^3*c[4]^3+12*c[6]^2*c[5]^3*c[4]-59*c[6]^2*c[5]^3*c[4]^2+28*c[5]^3*c[ 7]*c[4]^3+127*c[5]^4*c[7]*c[4]^2*c[6]+1250*c[5]^3*c[7]^3*c[4]^4*c[6]+1 0*c[6]*c[5]*c[4]^6+240*c[5]*c[4]^6*c[7]*c[6]-4*c[7]*c[6]^2*c[4]^3+11*c [7]^2*c[5]^2*c[6]*c[4]^3-80*c[7]^3*c[5]^2*c[6]*c[4]^2-121*c[7]^2*c[5]^ 3*c[6]*c[4]^2-96*c[5]^3*c[7]^3*c[6]*c[4]+c[7]^3*c[4]^2*c[5]-68*c[4]^3* c[7]^2*c[5]*c[6]^2-80*c[7]^2*c[6]*c[5]*c[4]^3+68*c[7]^2*c[5]^2*c[6]*c[ 4]^2-10*c[4]^4*c[5]^2+20*c[7]^2*c[5]^3*c[6]*c[4]-240*c[7]^2*c[5]^3*c[6 ]*c[4]^3-9*c[5]^4*c[4]^3-2*c[6]^2*c[4]^3*c[7]^2+260*c[5]^3*c[6]^2*c[7] *c[4]^3+49*c[5]^3*c[7]^3*c[4]^2-c[7]^3*c[5]^2*c[4]-228*c[6]^2*c[7]*c[5 ]^3*c[4]^2+30*c[6]^2*c[7]^2*c[4]*c[5]^3-2*c[7]^2*c[5]^2*c[4]^2+5*c[7]^ 2*c[5]^3*c[4]+40*c[5]^5*c[4]^3*c[6]*c[7]+36*c[4]^6*c[7]*c[5]+90*c[7]^3 *c[5]^2*c[6]^2*c[4]^3+16*c[7]^2*c[4]^4*c[5]-20*c[7]^2*c[5]^3*c[6]^2*c[ 4]^2+300*c[6]^2*c[4]^5*c[5]*c[7]^2+250*c[7]^2*c[4]^5*c[6]*c[5]^3-490*c [5]^2*c[6]*c[4]^5*c[7]^2+120*c[7]^3*c[5]^2*c[4]^3*c[6]+810*c[7]^3*c[5] ^3*c[6]^2*c[4]^2+111*c[7]^3*c[6]*c[5]*c[4]^3+280*c[7]^3*c[6]^2*c[5]*c[ 4]^3-240*c[7]^3*c[5]^3*c[6]^2*c[4]+15*c[5]^4*c[4]^4+60*c[5]^4*c[7]^2*c [4]*c[6]+6*c[6]*c[7]^3*c[5]^3-31*c[5]^4*c[4]*c[7]*c[6]-200*c[4]^5*c[6] ^2*c[7]*c[5]^2-500*c[4]^5*c[6]^2*c[7]^2*c[5]^2+9*c[4]^5*c[5]^2-181*c[6 ]^2*c[4]^4*c[5]^2-1280*c[7]^3*c[5]^3*c[4]^3*c[6]-600*c[5]^2*c[4]^5*c[7 ]^3*c[6]-6*c[4]^6*c[5]-6*c[6]^2*c[7]^3*c[4]^2+100*c[5]^5*c[4]^5*c[6]+1 20*c[5]*c[7]*c[4]^6*c[6]^2+420*c[6]*c[4]^5*c[7]^3*c[5]+600*c[5]^4*c[4] ^5*c[6]^2*c[7]+175*c[7]*c[6]^2*c[5]*c[4]^4-430*c[5]^3*c[4]^4*c[6]^2*c[ 7]+200*c[5]^3*c[4]^3*c[6]^2*c[7]^2-450*c[5]^4*c[4]^4*c[6]*c[7]+160*c[5 ]^2*c[4]^4*c[7]*c[6]^2-200*c[5]^3*c[4]^4*c[6]*c[7]-280*c[5]^4*c[4]^3*c [6]*c[7]+6*c[5]^3*c[4]^3-100*c[5]^2*c[4]^7*c[7]^2-1200*c[7]^3*c[5]^3*c [6]^2*c[4]^3-320*c[7]^3*c[4]^4*c[5]*c[6]-30*c[7]^2*c[5]^3*c[4]^2-300*c [6]*c[5]^3*c[4]^7-350*c[5]^4*c[4]^5*c[6]^2+600*c[6]^2*c[7]^3*c[5]^3*c[ 4]^4+600*c[6]*c[7]^2*c[5]^4*c[4]^5+200*c[6]^2*c[7]^3*c[4]^5*c[5]+200*c [7]^2*c[4]^7*c[6]*c[5]+1100*c[5]^4*c[4]^4*c[6]^2-720*c[5]^4*c[4]^3*c[6 ]^2+1750*c[7]^2*c[5]^4*c[6]^2*c[4]^3-1230*c[7]^2*c[5]^4*c[6]^2*c[4]^2+ 150*c[5]^4*c[7]^3*c[4]^3*c[6]+300*c[5]^4*c[4]^6*c[6]+480*c[5]^3*c[4]^4 *c[7]^2*c[6]+1420*c[5]^4*c[4]^3*c[7]^2*c[6]+20*c[5]^2*c[4]^4*c[7]^2*c[ 6]+130*c[5]^4*c[4]^3*c[6]^2*c[7]-40*c[7]^2*c[4]^4*c[6]^2*c[5]-350*c[7] ^2*c[4]^4*c[5]^3*c[6]^2+230*c[7]^2*c[4]^4*c[6]^2*c[5]^2+10*c[4]^6*c[5] ^2+188*c[5]^2*c[6]*c[4]^5-400*c[6]^2*c[7]^3*c[5]^2*c[4]^5-600*c[6]^2*c [7]^2*c[5]^4*c[4]^4+120*c[5]^2*c[4]^7*c[6]-94*c[4]^4*c[6]*c[5]^3+6*c[6 ]*c[5]*c[4]^5+60*c[5]^5*c[6]^2*c[4]^3-200*c[6]^2*c[4]^6*c[5]*c[7]^2-42 0*c[7]^3*c[6]^2*c[5]*c[4]^4+1550*c[5]^4*c[4]^5*c[6]*c[7]+500*c[7]^3*c[ 4]^4*c[6]^2*c[5]^2-400*c[5]^2*c[4]^7*c[6]*c[7]^2+6*c[5]^3*c[4]^4+103*c [6]^2*c[7]^3*c[5]^2*c[4]-260*c[6]^2*c[7]^3*c[5]^2*c[4]^2+540*c[5]^3*c[ 7]^3*c[4]^2*c[6]-97*c[6]^2*c[5]*c[7]^3*c[4]^2+30*c[6]^2*c[7]^3*c[5]^3- 4*c[7]^3*c[4]*c[5]^3)/(-240*c[5]^2*c[6]*c[4]^4+40*c[6]*c[5]*c[4]^4-9*c [5]^3*c[4]-100*c[4]^5*c[6]*c[5]^3-350*c[5]^4*c[4]^3*c[6]+30*c[5]^5*c[4 ]^2+2*c[5]^3-2*c[4]^3-50*c[5]^5*c[4]^3-3*c[5]^4+3*c[4]^4+50*c[4]^5*c[5 ]^3+100*c[5]^5*c[4]^3*c[6]-3*c[6]*c[5]^3+3*c[6]*c[4]^3+9*c[5]*c[4]^3-6 *c[5]*c[6]*c[4]^2+4*c[5]*c[4]^2+240*c[6]*c[5]^4*c[4]^2+5*c[6]*c[5]^4-1 3*c[6]*c[5]*c[4]^3+13*c[5]^3*c[6]*c[4]-4*c[5]^2*c[4]+68*c[5]^3*c[4]^2+ 100*c[5]^2*c[6]*c[4]^3-100*c[5]^3*c[6]*c[4]^2-140*c[5]^4*c[4]^2+140*c[ 4]^4*c[5]^2+180*c[5]^4*c[4]^3+6*c[5]^2*c[6]*c[4]-68*c[5]^2*c[4]^3-23*c [4]^4*c[5]-40*c[5]^4*c[4]*c[6]-30*c[4]^5*c[5]^2-5*c[6]*c[4]^4-50*c[5]^ 5*c[6]*c[4]^2+50*c[5]^2*c[6]*c[4]^5+350*c[4]^4*c[6]*c[5]^3+23*c[5]^4*c [4]-180*c[5]^3*c[4]^4)/(-c[4]+c[6])/c[4]^2, `b*`[7] = 1/60*(-30*c[5]^3 *c[4]-18*c[6]+6*c[5]+6*c[4]-9*c[5]^2-9*c[4]^2+27*c[6]^2-97*c[6]*c[5]^2 +66*c[6]*c[5]+60*c[6]*c[5]^3-100*c[6]^2*c[5]^3-100*c[6]^2*c[4]^3+180*c [6]^2*c[4]^2-97*c[6]*c[4]^2+60*c[6]*c[4]^3-30*c[5]*c[4]^3+312*c[5]*c[6 ]*c[4]^2+107*c[5]*c[4]^2-215*c[5]*c[6]*c[4]-66*c[5]*c[4]+66*c[6]*c[4]- 840*c[6]^2*c[4]^2*c[5]+522*c[6]^2*c[4]*c[5]+500*c[6]^2*c[5]*c[4]^3+180 *c[6]^2*c[5]^2-117*c[6]^2*c[5]-230*c[6]*c[5]*c[4]^3-230*c[5]^3*c[6]*c[ 4]+107*c[5]^2*c[4]+50*c[5]^3*c[4]^2+380*c[5]^2*c[6]*c[4]^3-460*c[5]^2* c[6]*c[4]^2+380*c[5]^3*c[6]*c[4]^2+600*c[5]^3*c[6]^2*c[4]^3-900*c[6]^2 *c[5]^2*c[4]^3+1420*c[6]^2*c[5]^2*c[4]^2-300*c[6]*c[5]^3*c[4]^3+500*c[ 6]^2*c[5]^3*c[4]-900*c[6]^2*c[5]^3*c[4]^2-180*c[5]^2*c[4]^2-840*c[6]^2 *c[4]*c[5]^2-117*c[6]^2*c[4]+312*c[5]^2*c[6]*c[4]+50*c[5]^2*c[4]^3)/c[ 7]/(-3*c[7]^3*c[6]+3*c[7]^2*c[6]^2-9*c[6]^2*c[7]^2*c[5]-8*c[7]*c[6]^2* c[5]^3+8*c[7]^2*c[6]*c[5]^3-c[5]*c[6]*c[7]*c[4]-c[7]^2*c[4]^2-30*c[6]^ 2*c[7]*c[5]^2*c[4]^3-28*c[6]^2*c[7]^2*c[5]^2*c[4]+30*c[6]^2*c[7]^2*c[5 ]^2*c[4]^2-30*c[6]*c[7]*c[5]^3*c[4]^3+9*c[6]^2*c[7]*c[4]^2-3*c[6]^2*c[ 7]*c[4]+2*c[6]*c[7]^2*c[4]-9*c[6]*c[7]^2*c[4]^2-2*c[7]^2*c[5]*c[4]-c[7 ]^2*c[5]^2+c[6]*c[7]*c[4]^2+8*c[7]^2*c[6]*c[4]^3-4*c[5]*c[7]^2*c[4]^3+ c[5]*c[7]*c[4]^2-c[5]*c[6]*c[4]^2+32*c[6]*c[5]*c[7]^2*c[4]^2-9*c[7]^2* c[5]*c[6]*c[4]-4*c[5]*c[7]*c[6]*c[4]^3-9*c[6]^2*c[4]^2*c[5]+3*c[6]^2*c [4]*c[5]+c[7]^3*c[4]+9*c[7]^3*c[4]*c[6]-9*c[7]^3*c[4]*c[5]-20*c[5]^2*c [6]*c[7]*c[4]^2+8*c[6]^2*c[5]*c[4]^3+28*c[7]^3*c[4]^2*c[6]*c[5]+18*c[6 ]^2*c[7]*c[4]*c[5]+28*c[7]*c[6]^2*c[5]*c[4]^3-3*c[6]^2*c[7]*c[5]+9*c[6 ]^2*c[7]*c[5]^2+c[6]*c[7]*c[5]^2-8*c[6]*c[7]^3*c[5]^2-9*c[6]*c[7]^2*c[ 5]^2+8*c[6]^2*c[7]^2*c[5]^2+2*c[6]*c[7]^2*c[5]-36*c[6]^2*c[4]^2*c[7]*c [5]-28*c[6]^2*c[7]^2*c[4]^2*c[5]-28*c[7]^3*c[4]*c[6]*c[5]-36*c[6]^2*c[ 5]^2*c[7]*c[4]+56*c[6]^2*c[5]^2*c[7]*c[4]^2+28*c[6]*c[5]^2*c[7]*c[4]^3 +28*c[6]*c[5]^3*c[7]*c[4]^2+28*c[6]^2*c[5]^3*c[7]*c[4]+9*c[6]*c[7]^3*c [5]+9*c[7]^2*c[5]*c[4]^2+28*c[7]^3*c[5]^2*c[6]*c[4]+32*c[7]^2*c[5]^2*c [6]*c[4]+8*c[6]^2*c[7]^2*c[4]^2-9*c[6]^2*c[7]^2*c[4]-4*c[5]^3*c[6]*c[7 ]*c[4]+28*c[6]^2*c[7]^2*c[4]*c[5]-4*c[5]^2*c[6]*c[4]^3-9*c[5]^2*c[7]*c [4]^2+9*c[5]^2*c[6]*c[4]^2+4*c[5]^2*c[7]*c[4]^3+4*c[5]^3*c[7]*c[4]^2-4 *c[5]^3*c[6]*c[4]^2+30*c[5]^3*c[6]^2*c[4]^3-28*c[6]^2*c[5]^2*c[4]^3+28 *c[6]^2*c[5]^2*c[4]^2+8*c[6]^2*c[5]^3*c[4]-28*c[6]^2*c[5]^3*c[4]^2-9*c [6]^2*c[4]*c[5]^2-8*c[7]*c[6]^2*c[4]^3+30*c[7]^2*c[5]^2*c[6]*c[4]^3-30 *c[7]^3*c[5]^2*c[6]*c[4]^2+30*c[7]^2*c[5]^3*c[6]*c[4]^2-8*c[7]^3*c[4]^ 2*c[6]+4*c[7]^3*c[4]^2*c[5]+c[7]^3*c[5]-28*c[7]^2*c[6]*c[5]*c[4]^3-56* c[7]^2*c[5]^2*c[6]*c[4]^2-28*c[7]^2*c[5]^3*c[6]*c[4]-c[5]^2*c[6]*c[4]+ c[5]^2*c[7]*c[4]+4*c[7]^3*c[5]^2*c[4]+9*c[7]^2*c[5]^2*c[4]-30*c[6]^2*c [7]*c[5]^3*c[4]^2-8*c[7]^2*c[5]^2*c[4]^2-4*c[7]^2*c[5]^3*c[4]), a[2,1] = c[2], a[6,4] = -1/2*(120*c[5]^2*c[6]*c[4]^4-12*c[6]*c[5]*c[4]^4-90* c[5]^2*c[6]^3*c[4]^2-60*c[5]*c[6]^2*c[4]^4-2*c[6]^3*c[4]^2+20*c[5]^4*c [4]^3*c[6]+40*c[6]^2*c[4]^5*c[5]+36*c[4]*c[5]^2*c[6]^3-20*c[5]^4*c[6]^ 2*c[4]^2+8*c[6]^2*c[5]^3+2*c[6]^2*c[4]^3-24*c[5]*c[6]^3*c[4]^2-6*c[6]^ 2*c[4]^2*c[5]+36*c[6]^2*c[5]*c[4]^3+60*c[4]^3*c[5]*c[6]^3-3*c[6]*c[5]* c[4]^3-11*c[5]^3*c[6]*c[4]+3*c[5]^3*c[4]^2-24*c[5]^2*c[6]*c[4]^3+13*c[ 5]^2*c[6]*c[4]^2+48*c[5]^3*c[6]*c[4]^2-60*c[5]^3*c[6]^2*c[4]^3-30*c[6] ^2*c[5]^2*c[4]^3-12*c[6]^2*c[5]^2*c[4]^2-150*c[6]*c[5]^3*c[4]^3-48*c[6 ]^2*c[5]^3*c[4]+150*c[6]^2*c[5]^3*c[4]^2-2*c[6]^2*c[4]*c[5]^2-5*c[5]^2 *c[4]^3+2*c[4]^4*c[5]-6*c[5]^2*c[6]^3+60*c[6]^3*c[4]^3*c[5]^2+7*c[6]^3 *c[4]*c[5]-60*c[5]^2*c[6]*c[4]^5-40*c[6]^3*c[4]^4*c[5]+60*c[4]^4*c[6]* c[5]^3)*c[6]/(c[4]^3-2*c[5]*c[4]^2+30*c[5]^2*c[4]^3-6*c[5]*c[4]^3-10*c [4]^4*c[5]^2-c[5]^3+10*c[5]^4*c[4]^2-30*c[5]^3*c[4]^2+6*c[5]^3*c[4]+2* c[5]^2*c[4])/c[4]^2, a[3,1] = -2/9*c[4]*(c[4]-3*c[2])/c[2], a[4,1] = 1 /4*c[4], a[7,3] = 3/4*1/c[4]^2*c[7]*(-480*c[5]^2*c[6]*c[4]^4+15*c[6]*c [5]*c[4]^4+300*c[4]^2*c[5]^2*c[7]^3+60*c[7]^3*c[5]*c[4]^3-510*c[7]^3*c [5]^2*c[4]^3+54*c[7]*c[5]*c[4]^4+450*c[5]^2*c[4]^4*c[7]^2+150*c[4]^3*c [7]^3*c[5]^3-150*c[7]^2*c[5]^2*c[4]^5-450*c[4]^5*c[6]*c[5]^3-3*c[7]^3* c[4]^2+90*c[5]^2*c[4]^5*c[7]+120*c[5]^3*c[4]^4*c[7]+150*c[7]^3*c[5]^2* c[4]^4+100*c[5]^4*c[4]^2*c[7]^2*c[6]-180*c[6]*c[4]^5*c[5]*c[7]+30*c[6] *c[5]^2*c[4]^4*c[7]+60*c[5]^4*c[4]^3*c[6]-150*c[5]^4*c[4]^4*c[6]-200*c [7]^2*c[4]^4*c[5]^3+300*c[6]*c[4]^5*c[5]*c[7]^2-50*c[7]^2*c[5]^4*c[4]^ 3+24*c[7]*c[6]*c[5]^3-40*c[7]^2*c[6]*c[5]^3-6*c[5]*c[6]*c[7]*c[4]+360* c[7]*c[4]^4*c[6]*c[5]+150*c[7]^2*c[5]^3*c[4]^3+2*c[7]^2*c[4]^2+6*c[5]^ 2*c[7]^3+330*c[6]*c[7]*c[5]^3*c[4]^3-450*c[7]^2*c[6]*c[5]*c[4]^4-12*c[ 7]*c[6]*c[4]^3-6*c[6]*c[7]^2*c[4]-2*c[7]^2*c[5]*c[4]-4*c[7]^2*c[5]^2+6 *c[6]*c[7]*c[4]^2+20*c[7]^2*c[6]*c[4]^3-10*c[5]*c[7]^2*c[4]^3+6*c[5]*c [4]^3-6*c[5]*c[7]*c[4]^2-6*c[5]*c[6]*c[4]^2+150*c[6]*c[5]*c[7]^2*c[4]^ 2-72*c[7]^2*c[5]*c[6]*c[4]-270*c[5]*c[7]*c[6]*c[4]^3+9*c[7]^3*c[4]*c[6 ]+3*c[7]^3*c[4]*c[5]-306*c[5]^2*c[6]*c[7]*c[4]^2+900*c[4]^5*c[6]*c[7]* c[5]^3+600*c[5]^2*c[6]*c[7]^3*c[4]^4-420*c[7]^3*c[4]^2*c[6]*c[5]+90*c[ 6]*c[5]*c[7]*c[4]^2-12*c[6]*c[7]*c[5]^2+30*c[6]*c[7]^3*c[5]^2+12*c[6]* c[7]^2*c[5]+123*c[7]^3*c[4]*c[6]*c[5]+360*c[6]*c[5]^2*c[7]*c[4]^3+360* c[6]*c[5]^3*c[7]*c[4]^2-18*c[6]*c[7]^3*c[5]+30*c[5]^4*c[4]^3*c[7]-80*c [7]^2*c[5]^2*c[4]^3+24*c[7]^2*c[5]*c[4]^2-30*c[7]*c[5]*c[4]^3+3*c[6]*c [5]*c[4]^3-240*c[7]^3*c[5]^2*c[6]*c[4]-24*c[5]^3*c[6]*c[4]-12*c[5]^3*c [7]*c[4]-144*c[5]^3*c[6]*c[7]*c[4]+9*c[5]^3*c[4]^2+294*c[5]^2*c[6]*c[4 ]^3-42*c[5]^2*c[7]*c[4]^2-75*c[5]^2*c[6]*c[4]^2+252*c[5]^2*c[7]*c[4]^3 +18*c[5]^3*c[7]*c[4]^2+189*c[5]^3*c[6]*c[4]^2-360*c[5]^2*c[4]^4*c[7]-7 20*c[6]*c[5]^3*c[4]^3-120*c[5]^3*c[7]*c[4]^3-60*c[5]^4*c[7]*c[4]^2*c[6 ]-6*c[5]^2*c[4]^2-780*c[7]^2*c[5]^2*c[6]*c[4]^3+810*c[7]^3*c[5]^2*c[6] *c[4]^2-1230*c[7]^2*c[5]^3*c[6]*c[4]^2-15*c[7]^3*c[4]^2*c[6]-21*c[7]^3 *c[4]^2*c[5]+20*c[7]^2*c[6]*c[5]*c[4]^3+220*c[7]^2*c[5]^2*c[6]*c[4]^2+ 15*c[4]^4*c[5]^2+320*c[7]^2*c[5]^3*c[6]*c[4]+1850*c[7]^2*c[5]^3*c[6]*c [4]^3-60*c[5]^3*c[7]^3*c[4]^2+12*c[5]^2*c[6]*c[4]+12*c[5]^2*c[7]*c[4]- 66*c[7]^3*c[5]^2*c[4]+24*c[7]^2*c[5]^2*c[4]+72*c[5]^2*c[6]*c[7]*c[4]-1 10*c[7]^2*c[5]^2*c[4]^2+20*c[7]^2*c[5]^3*c[4]-9*c[4]^4*c[5]-60*c[7]^2* c[4]^4*c[5]-600*c[5]^2*c[6]*c[4]^5*c[7]^2-1200*c[7]^3*c[5]^2*c[4]^3*c[ 6]+630*c[7]^3*c[6]*c[5]*c[4]^3+300*c[5]^4*c[4]^4*c[6]*c[7]-1800*c[5]^3 *c[4]^4*c[6]*c[7]-15*c[5]^3*c[4]^3-300*c[7]^3*c[4]^4*c[5]*c[6]-20*c[7] ^2*c[5]^3*c[4]^2-800*c[5]^3*c[4]^4*c[7]^2*c[6]-200*c[5]^4*c[4]^3*c[7]^ 2*c[6]+1150*c[5]^2*c[4]^4*c[7]^2*c[6]+180*c[5]^2*c[6]*c[4]^5+1140*c[4] ^4*c[6]*c[5]^3)/(10*c[5]^3*c[4]^2+10*c[5]^2*c[4]^3-30*c[5]^2*c[4]^2+6* c[5]^2*c[4]-c[5]^2+6*c[5]*c[4]^2+c[5]*c[4]-c[4]^2)/(10*c[5]*c[6]*c[4]- 5*c[6]*c[5]-5*c[6]*c[4]+3*c[6]-5*c[5]*c[4]+3*c[5]+3*c[4]-2), `b*`[1] = 1/60*(30*c[5]^3*c[4]+42*c[7]*c[6]^2-103*c[5]*c[7]*c[4]+18*c[6]-6*c[5] -6*c[4]-27*c[7]*c[6]+9*c[5]^2+9*c[4]^2-180*c[7]*c[6]^2*c[5]^3+100*c[7] *c[6]*c[5]^3-27*c[6]^2-354*c[5]*c[6]*c[7]*c[4]+97*c[6]*c[5]^2-66*c[6]* c[5]-60*c[6]*c[5]^3+100*c[6]^2*c[5]^3+9*c[7]*c[5]+100*c[6]^2*c[4]^3-20 40*c[6]^2*c[7]*c[5]^2*c[4]^3-600*c[6]*c[7]*c[5]^3*c[4]^3-14*c[7]*c[5]^ 2+100*c[7]*c[6]*c[4]^3+310*c[6]^2*c[7]*c[4]^2-192*c[6]^2*c[7]*c[4]-180 *c[6]^2*c[4]^2-14*c[7]*c[4]^2+9*c[7]*c[4]+103*c[6]*c[7]*c[5]+97*c[6]*c [4]^2-60*c[6]*c[4]^3+103*c[6]*c[7]*c[4]-156*c[6]*c[7]*c[4]^2+30*c[5]*c [4]^3+174*c[5]*c[7]*c[4]^2-312*c[5]*c[6]*c[4]^2-107*c[5]*c[4]^2+215*c[ 5]*c[6]*c[4]+66*c[5]*c[4]-66*c[6]*c[4]-410*c[5]*c[7]*c[6]*c[4]^3+840*c [6]^2*c[4]^2*c[5]-522*c[6]^2*c[4]*c[5]-760*c[5]^2*c[6]*c[7]*c[4]^2-500 *c[6]^2*c[5]*c[4]^3+932*c[6]^2*c[7]*c[4]*c[5]+1020*c[7]*c[6]^2*c[5]*c[ 4]^3-192*c[6]^2*c[7]*c[5]+310*c[6]^2*c[7]*c[5]^2+526*c[6]*c[5]*c[7]*c[ 4]^2-156*c[6]*c[7]*c[5]^2-180*c[6]^2*c[5]^2-1610*c[6]^2*c[4]^2*c[7]*c[ 5]-1610*c[6]^2*c[5]^2*c[7]*c[4]+2970*c[6]^2*c[5]^2*c[7]*c[4]^2+690*c[6 ]*c[5]^2*c[7]*c[4]^3+690*c[6]*c[5]^3*c[7]*c[4]^2+1020*c[6]^2*c[5]^3*c[ 7]*c[4]+117*c[6]^2*c[5]-50*c[7]*c[5]*c[4]^3+230*c[6]*c[5]*c[4]^3+230*c [5]^3*c[6]*c[4]-50*c[5]^3*c[7]*c[4]-107*c[5]^2*c[4]-410*c[5]^3*c[6]*c[ 7]*c[4]-50*c[5]^3*c[4]^2-380*c[5]^2*c[6]*c[4]^3-310*c[5]^2*c[7]*c[4]^2 +460*c[5]^2*c[6]*c[4]^2+90*c[5]^2*c[7]*c[4]^3+90*c[5]^3*c[7]*c[4]^2-38 0*c[5]^3*c[6]*c[4]^2-600*c[5]^3*c[6]^2*c[4]^3+900*c[6]^2*c[5]^2*c[4]^3 -1420*c[6]^2*c[5]^2*c[4]^2+300*c[6]*c[5]^3*c[4]^3-500*c[6]^2*c[5]^3*c[ 4]+900*c[6]^2*c[5]^3*c[4]^2+180*c[5]^2*c[4]^2+840*c[6]^2*c[4]*c[5]^2-1 80*c[7]*c[6]^2*c[4]^3+117*c[6]^2*c[4]+1500*c[5]^3*c[6]^2*c[7]*c[4]^3-3 12*c[5]^2*c[6]*c[4]+174*c[5]^2*c[7]*c[4]+526*c[5]^2*c[6]*c[7]*c[4]-204 0*c[6]^2*c[7]*c[5]^3*c[4]^2-50*c[5]^2*c[4]^3)/(3*c[6]-28*c[5]*c[6]*c[4 ]^2-28*c[5]^2*c[6]*c[4]+8*c[6]*c[5]^2-9*c[6]*c[4]-9*c[6]*c[5]+28*c[5]* c[6]*c[4]+30*c[5]^2*c[6]*c[4]^2+8*c[6]*c[4]^2-4*c[5]^2*c[4]-4*c[5]*c[4 ]^2+9*c[5]*c[4]-c[5]-c[4])/c[5]/c[6]/c[7]/c[4], a[8,4] = -1/2*(734*c[5 ]^2*c[6]*c[4]^4-203*c[6]*c[5]*c[4]^4+3190*c[5]^4*c[4]^4*c[6]^2*c[7]-93 *c[7]*c[5]*c[4]^4-636*c[5]^2*c[4]^4*c[7]^2-10*c[5]^3*c[4]-520*c[6]*c[4 ]^7*c[7]*c[5]-481*c[4]^5*c[6]^2*c[5]^2+645*c[7]^2*c[5]^2*c[4]^5-574*c[ 4]^5*c[6]*c[5]^3-172*c[5]*c[6]^2*c[4]^4+6*c[7]^2*c[4]^3+100*c[5]^5*c[4 ]^3*c[7]^2*c[6]^2+60*c[4]^8*c[5]^2-600*c[5]^2*c[4]^6*c[6]^2*c[7]^2+9*c [4]^5*c[5]*c[7]-164*c[5]^2*c[4]^5*c[7]+900*c[5]^3*c[4]^6*c[6]^2*c[7]+6 00*c[5]^3*c[4]^6*c[6]^2*c[7]^2-1104*c[5]^3*c[4]^4*c[7]-18*c[4]^4*c[6]^ 2+90*c[7]^2*c[4]^5*c[5]^3+1380*c[6]^2*c[4]^5*c[7]*c[5]^3-750*c[7]*c[5] ^2*c[4]^6*c[6]+675*c[5]^4*c[4]^2*c[7]^2*c[6]+8*c[6]^2*c[4]^5-339*c[6]^ 2*c[7]*c[5]^4*c[4]^2-315*c[5]^2*c[4]^6*c[7]+640*c[6]*c[5]*c[7]^2*c[4]^ 6+46*c[6]*c[4]^5*c[5]*c[7]-1014*c[6]*c[5]^2*c[4]^4*c[7]+2100*c[5]^4*c[ 7]*c[4]^6*c[6]-390*c[6]^2*c[4]^6*c[5]+860*c[4]^6*c[6]^2*c[5]^2+140*c[5 ]^5*c[4]^3*c[7]-1740*c[5]^2*c[7]^2*c[4]^6*c[6]-613*c[5]^4*c[4]^3*c[6]- 100*c[4]^8*c[5]^2*c[7]+1140*c[5]^4*c[4]^5*c[6]+968*c[5]^2*c[6]*c[4]^6- 150*c[5]^5*c[4]^5*c[6]^2+263*c[5]^4*c[4]^4*c[6]+250*c[7]^2*c[4]^4*c[5] ^3-120*c[6]*c[5]*c[4]^8-10*c[6]*c[7]^2*c[4]^5-6*c[6]*c[4]^6+780*c[5]^4 *c[4]^4*c[7]+32*c[7]^2*c[6]*c[4]^4+1200*c[5]^3*c[7]^2*c[4]^6*c[6]-750* 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6]*c[5]*c[7]*c[4]^2+18*c[6]^2*c[7]^2*c[5]^2-38*c[6]^2*c[4]^2*c[7]*c[5] +102*c[6]^2*c[7]^2*c[4]^2*c[5]+86*c[6]*c[5]^4*c[4]^2-24*c[7]*c[6]^2*c[ 5]^4+585*c[4]^5*c[7]*c[5]^3+c[4]^5+35*c[6]^2*c[5]^2*c[7]*c[4]-26*c[6]^ 2*c[5]^2*c[7]*c[4]^2+1128*c[6]*c[5]^2*c[7]*c[4]^3+706*c[6]*c[5]^3*c[7] *c[4]^2-138*c[6]^2*c[5]^3*c[7]*c[4]-685*c[5]^4*c[4]^3*c[7]+354*c[7]^2* c[5]^2*c[4]^3-370*c[5]^4*c[7]^2*c[4]^4-6*c[7]^2*c[5]*c[4]^2+70*c[7]*c[ 5]*c[4]^3+89*c[6]*c[5]*c[4]^3+200*c[4]^8*c[5]*c[6]*c[7]+390*c[6]*c[4]^ 7*c[5]-31*c[7]^2*c[5]^2*c[6]*c[4]-27*c[6]*c[7]*c[4]^4+10*c[4]^5*c[5]-8 0*c[5]^4*c[4]^6+200*c[5]^5*c[4]^5*c[7]^2*c[6]+6*c[6]^2*c[7]^2*c[4]^2+2 2*c[5]^3*c[6]*c[4]+19*c[5]^3*c[7]*c[4]+400*c[5]^5*c[4]^3*c[7]^2*c[6]+4 *c[5]^2*c[4]-135*c[5]^3*c[6]*c[7]*c[4]+71*c[5]^3*c[4]^2-21*c[6]^2*c[7] ^2*c[4]*c[5]-480*c[5]^2*c[6]*c[4]^3+42*c[5]^2*c[7]*c[4]^2+37*c[5]^2*c[ 6]*c[4]^2-480*c[5]^2*c[7]*c[4]^3-104*c[5]^3*c[7]*c[4]^2-97*c[5]^3*c[6] *c[4]^2-1536*c[5]^3*c[6]^2*c[4]^3+190*c[5]^5*c[4]^3*c[6]^2*c[7]+557*c[ 4]^3*c[7]^2*c[5]^2*c[6]^2-10*c[6]*c[7]*c[4]^5-730*c[4]^5*c[5]^3*c[6]^2 -14*c[7]*c[5]^4*c[4]+10*c[7]^2*c[5]^4*c[4]+646*c[5]^2*c[4]^4*c[7]+498* c[6]^2*c[5]^2*c[4]^3-95*c[6]^2*c[5]^2*c[4]^2+744*c[6]*c[5]^3*c[4]^3-46 *c[6]^2*c[5]^3*c[4]+334*c[6]^2*c[5]^3*c[4]^2+854*c[5]^3*c[7]*c[4]^3-49 0*c[5]^4*c[7]*c[4]^2*c[6]-30*c[7]*c[5]*c[4]^7-19*c[5]^2*c[4]^2+18*c[6] ^2*c[4]*c[5]^2-472*c[6]*c[5]*c[4]^6+360*c[5]*c[4]^6*c[7]*c[6]-10*c[7]* c[6]^2*c[4]^3-811*c[7]^2*c[5]^2*c[6]*c[4]^3-993*c[7]^2*c[5]^3*c[6]*c[4 ]^2-44*c[5]^4*c[4]^2-451*c[4]^3*c[7]^2*c[5]*c[6]^2+10*c[6]*c[7]*c[4]^6 -14*c[7]^2*c[6]*c[5]*c[4]^3-20*c[5]^5*c[4]^5+118*c[7]^2*c[5]^2*c[6]*c[ 4]^2-249*c[4]^4*c[5]^2+170*c[7]^2*c[5]^3*c[6]*c[4]+3829*c[7]^2*c[5]^3* c[6]*c[4]^3+2208*c[5]^3*c[6]^2*c[4]^4+339*c[5]^4*c[4]^3-6*c[6]^2*c[4]^ 3*c[7]^2+1871*c[5]^3*c[6]^2*c[7]*c[4]^3-10*c[5]^2*c[6]*c[4]-9*c[5]^2*c [7]*c[4]+6*c[7]^2*c[5]^2*c[4]+33*c[5]^2*c[6]*c[7]*c[4]+600*c[7]*c[4]^8 *c[6]*c[5]^3+187*c[6]^2*c[7]*c[5]^3*c[4]^2+341*c[6]^2*c[7]^2*c[4]*c[5] 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6]^2*c[4]^3+600*c[5]^2*c[7]*c[4]^7*c[6]^2+200*c[6]^2*c[4]^6*c[5]*c[7]^ 2-1510*c[5]^4*c[4]^5*c[6]*c[7]+6*c[5]^4*c[4]+600*c[5]^2*c[4]^7*c[6]*c[ 7]^2+814*c[5]^3*c[4]^4)/(690*c[5]^2*c[6]*c[4]^4-110*c[6]*c[5]*c[4]^4-1 10*c[7]*c[5]*c[4]^4+48*c[5]^3*c[4]+200*c[4]^5*c[6]*c[5]^3-150*c[5]^2*c [4]^5*c[7]-750*c[5]^3*c[4]^4*c[7]-930*c[6]*c[5]^2*c[4]^4*c[7]-200*c[5] ^5*c[4]^3*c[7]+750*c[5]^4*c[4]^3*c[6]-120*c[5]^5*c[4]^2-10*c[5]^3+10*c [4]^3+150*c[5]^5*c[4]^3-15*c[7]*c[6]*c[5]^3+12*c[5]^4-12*c[4]^4+200*c[ 6]*c[5]^2*c[4]^5*c[7]-150*c[4]^5*c[5]^3-200*c[5]^5*c[4]^3*c[6]+150*c[7 ]*c[4]^4*c[6]*c[5]+12*c[6]*c[5]^3+15*c[4]^4*c[7]-690*c[5]^4*c[7]*c[4]^ 2-15*c[5]^4*c[7]+150*c[5]^5*c[7]*c[4]^2+15*c[7]*c[6]*c[4]^3-12*c[7]*c[ 4]^3-12*c[6]*c[4]^3-48*c[5]*c[4]^3+24*c[5]*c[7]*c[4]^2+24*c[5]*c[6]*c[ 4]^2-20*c[5]*c[4]^2-70*c[5]*c[7]*c[6]*c[4]^3+12*c[7]*c[5]^3-300*c[4]^5 *c[6]*c[7]*c[5]^3-30*c[6]*c[5]*c[7]*c[4]^2+20*c[5]^4*c[7]*c[6]-690*c[6 ]*c[5]^4*c[4]^2-15*c[6]*c[5]^4+200*c[4]^5*c[7]*c[5]^3+510*c[6]*c[5]^2* c[7]*c[4]^3-510*c[6]*c[5]^3*c[7]*c[4]^2+750*c[5]^4*c[4]^3*c[7]+57*c[7] *c[5]*c[4]^3+57*c[6]*c[5]*c[4]^3-20*c[6]*c[7]*c[4]^4-57*c[5]^3*c[6]*c[ 4]-57*c[5]^3*c[7]*c[4]+20*c[5]^2*c[4]-200*c[5]^5*c[4]^2*c[6]*c[7]+70*c [5]^3*c[6]*c[7]*c[4]-342*c[5]^3*c[4]^2-410*c[5]^2*c[6]*c[4]^3-410*c[5] ^2*c[7]*c[4]^3+410*c[5]^3*c[7]*c[4]^2+410*c[5]^3*c[6]*c[4]^2+110*c[7]* c[5]^4*c[4]+690*c[5]^2*c[4]^4*c[7]+930*c[5]^4*c[7]*c[4]^2*c[6]+550*c[5 ]^4*c[4]^2-550*c[4]^4*c[5]^2-570*c[5]^4*c[4]^3-24*c[5]^2*c[6]*c[4]-24* c[5]^2*c[7]*c[4]+30*c[5]^2*c[6]*c[7]*c[4]+342*c[5]^2*c[4]^3+87*c[4]^4* c[5]+300*c[5]^5*c[4]^3*c[6]*c[7]+110*c[5]^4*c[4]*c[6]-150*c[5]^4*c[4]* c[7]*c[6]+120*c[4]^5*c[5]^2+15*c[6]*c[4]^4+1100*c[5]^3*c[4]^4*c[6]*c[7 ]-1100*c[5]^4*c[4]^3*c[6]*c[7]+150*c[5]^5*c[6]*c[4]^2-150*c[5]^2*c[6]* c[4]^5-750*c[4]^4*c[6]*c[5]^3-87*c[5]^4*c[4]+570*c[5]^3*c[4]^4)/(c[6]* c[4]-c[4]^2-c[7]*c[6]+c[7]*c[4])/c[4]^2, a[8,1] = 1/4*(-2816*c[5]^2*c[ 6]*c[4]^4+424*c[6]*c[5]*c[4]^4-9720*c[5]^4*c[4]^4*c[6]^2*c[7]+372*c[7] *c[5]*c[4]^4+1752*c[5]^2*c[4]^4*c[7]^2+20*c[5]^3*c[4]-1320*c[4]^5*c[6] ^2*c[5]^2-1320*c[7]^2*c[5]^2*c[4]^5-4880*c[4]^5*c[6]*c[5]^3-264*c[5]*c [6]^2*c[4]^4-12*c[7]^2*c[4]^3+700*c[5]^5*c[4]^3*c[7]^2*c[6]^2-200*c[5] ^2*c[4]^6*c[6]^2*c[7]^2-246*c[4]^5*c[5]*c[7]+1818*c[5]^2*c[4]^5*c[7]+1 300*c[5]^3*c[4]^6*c[6]^2*c[7]-800*c[5]^3*c[4]^6*c[6]^2*c[7]^2+5526*c[5 ]^3*c[4]^4*c[7]+32*c[4]^4*c[6]^2+3160*c[7]^2*c[4]^5*c[5]^3-7740*c[6]^2 *c[4]^5*c[7]*c[5]^3+780*c[7]*c[5]^2*c[4]^6*c[6]-325*c[5]^4*c[4]^2*c[7] ^2*c[6]-20*c[6]^2*c[4]^5-1339*c[6]^2*c[7]*c[5]^4*c[4]^2-280*c[5]^2*c[4 ]^6*c[7]+668*c[6]*c[4]^5*c[5]*c[7]+6574*c[6]*c[5]^2*c[4]^4*c[7]+1500*c [5]^4*c[7]*c[4]^6*c[6]+200*c[4]^6*c[6]^2*c[5]^2-280*c[5]^5*c[4]^3*c[7] -600*c[5]^2*c[7]^2*c[4]^6*c[6]+2092*c[5]^4*c[4]^3*c[6]+3640*c[5]^4*c[4 ]^5*c[6]-320*c[5]^2*c[6]*c[4]^6+400*c[5]^5*c[4]^5*c[6]^2-4880*c[5]^4*c [4]^4*c[6]-4080*c[7]^2*c[4]^4*c[5]^3+50*c[6]*c[7]^2*c[4]^5-2400*c[5]^4 *c[4]^5*c[7]^2-4160*c[5]^4*c[4]^4*c[7]-76*c[7]^2*c[6]*c[4]^4+1600*c[5] 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6*c[6]+780*c[5]^5*c[4]^4*c[7]+180*c[4]^5*c[5]*c[7]^2-12*c[6]^2*c[4]^3- 600*c[5]^3*c[7]^2*c[4]^6-600*c[5]^5*c[4]^4*c[7]^2+920*c[4]^6*c[6]*c[5] ^3+1648*c[6]^2*c[7]*c[5]^2*c[4]^3-120*c[6]^2*c[7]^2*c[5]^2*c[4]+390*c[ 6]^2*c[7]^2*c[5]^2*c[4]^2+5722*c[6]*c[7]*c[5]^3*c[4]^3-320*c[5]^3*c[4] ^6+40*c[7]^2*c[6]^2*c[5]^4-246*c[5]^4*c[7]*c[4]^2+686*c[7]^2*c[6]*c[5] *c[4]^4-100*c[7]^2*c[5]^5*c[6]^2*c[4]^2-400*c[5]^4*c[4]*c[7]^2*c[6]^2- 1200*c[5]^5*c[7]*c[4]^5*c[6]^2+70*c[6]^2*c[4]^4*c[7]^2+32*c[6]*c[4]^5- 42*c[7]*c[6]*c[4]^3-4*c[6]^2*c[7]*c[4]^2+200*c[5]^2*c[4]^6*c[7]^2+18*c [7]*c[4]^3+20*c[6]*c[4]^3+26*c[7]^2*c[6]*c[4]^3+72*c[5]*c[7]^2*c[4]^3+ 48*c[5]*c[4]^3-18*c[5]*c[7]*c[4]^2-20*c[5]*c[6]*c[4]^2+8*c[5]*c[4]^2-2 3*c[6]*c[5]*c[7]^2*c[4]^2+266*c[5]*c[7]*c[6]*c[4]^3+12*c[6]^2*c[4]^2*c [5]-1240*c[5]^4*c[4]^5+287*c[5]^2*c[6]*c[7]*c[4]^2-8120*c[5]^4*c[7]^2* c[4]^4*c[6]+11330*c[4]^5*c[6]*c[7]*c[5]^3+72*c[6]^2*c[5]*c[4]^3+14*c[6 ]^2*c[7]*c[4]*c[5]+76*c[7]*c[6]^2*c[5]*c[4]^3-12*c[6]^2*c[7]*c[5]^2+40 *c[6]*c[5]*c[7]*c[4]^2+18*c[6]^2*c[7]^2*c[5]^2-101*c[6]^2*c[4]^2*c[7]* c[5]+132*c[6]^2*c[7]^2*c[4]^2*c[5]-284*c[6]*c[5]^4*c[4]^2-24*c[7]*c[6] ^2*c[5]^4-4160*c[4]^5*c[7]*c[5]^3-12*c[4]^5+95*c[6]^2*c[5]^2*c[7]*c[4] -356*c[6]^2*c[5]^2*c[7]*c[4]^2-2598*c[6]*c[5]^2*c[7]*c[4]^3-698*c[6]*c [5]^3*c[7]*c[4]^2-300*c[6]^2*c[5]^3*c[7]*c[4]+1818*c[5]^4*c[4]^3*c[7]- 692*c[7]^2*c[5]^2*c[4]^3+3160*c[5]^4*c[7]^2*c[4]^4+12*c[7]^2*c[5]*c[4] ^2-108*c[7]*c[5]*c[4]^3-120*c[6]*c[5]*c[4]^3+29*c[7]^2*c[5]^2*c[6]*c[4 ]+114*c[6]*c[7]*c[4]^4+104*c[4]^5*c[5]+200*c[5]^4*c[4]^6-1200*c[5]^5*c [4]^5*c[7]^2*c[6]+6*c[6]^2*c[7]^2*c[4]^2-52*c[5]^3*c[6]*c[4]-46*c[5]^3 *c[7]*c[4]-450*c[5]^5*c[4]^3*c[7]^2*c[6]-8*c[5]^2*c[4]+60*c[5]^3*c[6]* c[7]*c[4]-160*c[5]^3*c[4]^2-21*c[6]^2*c[7]^2*c[4]*c[5]+1144*c[5]^2*c[6 ]*c[4]^3-108*c[5]^2*c[7]*c[4]^2-120*c[5]^2*c[6]*c[4]^2+1024*c[5]^2*c[7 ]*c[4]^3+372*c[5]^3*c[7]*c[4]^2+424*c[5]^3*c[6]*c[4]^2+1752*c[5]^3*c[6 ]^2*c[4]^3-660*c[5]^5*c[4]^3*c[6]^2*c[7]-1153*c[4]^3*c[7]^2*c[5]^2*c[6 ]^2-72*c[6]*c[7]*c[4]^5+3160*c[4]^5*c[5]^3*c[6]^2+28*c[7]*c[5]^4*c[4]- 20*c[7]^2*c[5]^4*c[4]-2472*c[5]^2*c[4]^4*c[7]-692*c[6]^2*c[5]^2*c[4]^3 +72*c[6]^2*c[5]^2*c[4]^2-2816*c[6]*c[5]^3*c[4]^3+32*c[6]^2*c[5]^3*c[4] -264*c[6]^2*c[5]^3*c[4]^2-2472*c[5]^3*c[7]*c[4]^3+566*c[5]^4*c[7]*c[4] ^2*c[6]+48*c[5]^2*c[4]^2+1200*c[6]^2*c[4]^5*c[5]^5*c[7]^2-12*c[6]^2*c[ 4]*c[5]^2+38*c[7]*c[6]^2*c[4]^3+1804*c[7]^2*c[5]^2*c[6]*c[4]^3+258*c[7 ]^2*c[5]^3*c[6]*c[4]^2+104*c[5]^4*c[4]^2-500*c[5]^5*c[4]^5*c[7]-256*c[ 4]^3*c[7]^2*c[5]*c[6]^2-176*c[7]^2*c[6]*c[5]*c[4]^3+200*c[5]^5*c[4]^5- 212*c[7]^2*c[5]^2*c[6]*c[4]^2+1064*c[4]^4*c[5]^2+8*c[7]^2*c[5]^3*c[6]* c[4]-3410*c[7]^2*c[5]^3*c[6]*c[4]^3-4080*c[5]^3*c[6]^2*c[4]^4-772*c[5] ^4*c[4]^3+1200*c[7]^2*c[6]^2*c[4]^6*c[5]^4-36*c[6]^2*c[4]^3*c[7]^2-536 8*c[5]^3*c[6]^2*c[7]*c[4]^3+20*c[5]^2*c[6]*c[4]+18*c[5]^2*c[7]*c[4]-12 *c[7]^2*c[5]^2*c[4]-44*c[5]^2*c[6]*c[7]*c[4]+400*c[5]^4*c[4]^6*c[7]^2+ 1438*c[6]^2*c[7]*c[5]^3*c[4]^2+426*c[6]^2*c[7]^2*c[4]*c[5]^3-452*c[5]^ 2*c[4]^3+72*c[7]^2*c[5]^2*c[4]^2+32*c[7]^2*c[5]^3*c[4]-160*c[4]^4*c[5] +690*c[5]^5*c[4]^3*c[6]*c[7]-264*c[7]^2*c[4]^4*c[5]-1833*c[7]^2*c[5]^3 *c[6]^2*c[4]^2-260*c[6]^2*c[4]^5*c[5]*c[7]^2-8270*c[7]^2*c[4]^5*c[6]*c [5]^3+3780*c[5]^2*c[6]*c[4]^5*c[7]^2+1720*c[5]^4*c[4]^4+32*c[5]^4*c[4] *c[6]+30*c[5]^4*c[7]^2*c[4]*c[6]-60*c[5]^4*c[4]*c[7]*c[6]+2480*c[4]^5* c[6]^2*c[7]*c[5]^2-1120*c[4]^5*c[6]^2*c[7]^2*c[5]^2-772*c[4]^5*c[5]^2- 40*c[6]^2*c[7]^2*c[4]^5+1752*c[6]^2*c[4]^4*c[5]^2-1700*c[7]^2*c[6]^2*c [5]^5*c[4]^4-600*c[5]^5*c[4]^5*c[6]-120*c[5]*c[7]*c[4]^6*c[6]^2+50*c[6 ]^2*c[7]*c[4]^5-52*c[6]*c[4]^4-1200*c[5]^4*c[7]*c[4]^6*c[6]^2+7150*c[5 ]^4*c[4]^5*c[6]^2*c[7]+198*c[7]*c[6]^2*c[5]*c[4]^4+10560*c[5]^3*c[4]^4 *c[6]^2*c[7]+5240*c[5]^3*c[4]^3*c[6]^2*c[7]^2+11240*c[5]^4*c[4]^4*c[6] *c[7]-3649*c[5]^2*c[4]^4*c[7]*c[6]^2-14207*c[5]^3*c[4]^4*c[6]*c[7]-449 1*c[5]^4*c[4]^3*c[6]*c[7]+1064*c[5]^3*c[4]^3-264*c[7]^2*c[5]^3*c[4]^2- 2400*c[5]^4*c[4]^5*c[6]^2+6020*c[6]^2*c[7]^2*c[5]^3*c[4]^5+6800*c[6]*c [7]^2*c[5]^4*c[4]^5+3160*c[5]^4*c[4]^4*c[6]^2-1320*c[5]^4*c[4]^3*c[6]^ 2-5800*c[7]^2*c[5]^4*c[6]^2*c[4]^3+1940*c[7]^2*c[5]^4*c[6]^2*c[4]^2-60 0*c[5]^4*c[4]^6*c[6]+28*c[4]^5*c[7]+9740*c[5]^3*c[4]^4*c[7]^2*c[6]+292 0*c[5]^4*c[4]^3*c[7]^2*c[6]-4775*c[5]^2*c[4]^4*c[7]^2*c[6]+4980*c[5]^4 *c[4]^3*c[6]^2*c[7]+210*c[7]^2*c[4]^4*c[6]^2*c[5]-8860*c[7]^2*c[4]^4*c [5]^3*c[6]^2+2140*c[7]^2*c[4]^4*c[6]^2*c[5]^2+120*c[4]^6*c[5]^2-20*c[7 ]^2*c[4]^5+2092*c[5]^2*c[6]*c[4]^5+9920*c[6]^2*c[7]^2*c[5]^4*c[4]^4-50 0*c[5]^4*c[4]^6*c[7]-48*c[7]^2*c[6]^2*c[5]^3+6404*c[4]^4*c[6]*c[5]^3-2 84*c[6]*c[5]*c[4]^5+200*c[5]^5*c[6]^2*c[4]^3+200*c[6]^2*c[4]^6*c[5]*c[ 7]^2-8820*c[5]^4*c[4]^5*c[6]*c[7]-12*c[5]^4*c[4]-2324*c[5]^3*c[4]^4)/c [6]/c[5]/c[7]/c[4]^2/(150*c[5]^2*c[6]*c[4]^4-87*c[5]^3*c[4]-12*c[5]*c[ 7]*c[4]-200*c[5]^3*c[4]^4*c[7]-200*c[6]*c[5]^2*c[4]^4*c[7]-200*c[5]^4* c[4]^3*c[6]-10*c[5]^2-10*c[4]^2+12*c[5]^3+12*c[4]^3+20*c[7]*c[6]*c[5]^ 3+15*c[5]*c[6]*c[7]*c[4]+12*c[6]*c[5]^2-15*c[6]*c[5]^3-1300*c[6]*c[7]* c[5]^3*c[4]^3+150*c[5]^4*c[7]*c[4]^2+12*c[7]*c[5]^2+20*c[7]*c[6]*c[4]^ 3+12*c[7]*c[4]^2-15*c[7]*c[4]^3+12*c[6]*c[4]^2-15*c[6]*c[4]^3-15*c[6]* c[7]*c[4]^2-87*c[5]*c[4]^3-72*c[5]*c[7]*c[4]^2-72*c[5]*c[6]*c[4]^2+60* c[5]*c[4]^2-12*c[5]*c[6]*c[4]+10*c[5]*c[4]-150*c[5]*c[7]*c[6]*c[4]^3-1 5*c[7]*c[5]^3-660*c[5]^2*c[6]*c[7]*c[4]^2+90*c[6]*c[5]*c[7]*c[4]^2-15* c[6]*c[7]*c[5]^2+150*c[6]*c[5]^4*c[4]^2+930*c[6]*c[5]^2*c[7]*c[4]^3+93 0*c[6]*c[5]^3*c[7]*c[4]^2-200*c[5]^4*c[4]^3*c[7]+110*c[7]*c[5]*c[4]^3+ 110*c[6]*c[5]*c[4]^3+110*c[5]^3*c[6]*c[4]+110*c[5]^3*c[7]*c[4]+60*c[5] ^2*c[4]-150*c[5]^3*c[6]*c[7]*c[4]+550*c[5]^3*c[4]^2-690*c[5]^2*c[6]*c[ 4]^3+520*c[5]^2*c[7]*c[4]^2+520*c[5]^2*c[6]*c[4]^2-690*c[5]^2*c[7]*c[4 ]^3-690*c[5]^3*c[7]*c[4]^2-690*c[5]^3*c[6]*c[4]^2+150*c[5]^2*c[4]^4*c[ 7]+900*c[6]*c[5]^3*c[4]^3+900*c[5]^3*c[7]*c[4]^3-200*c[5]^4*c[7]*c[4]^ 2*c[6]-429*c[5]^2*c[4]^2-120*c[5]^4*c[4]^2-120*c[4]^4*c[5]^2+150*c[5]^ 4*c[4]^3-72*c[5]^2*c[6]*c[4]-72*c[5]^2*c[7]*c[4]+90*c[5]^2*c[6]*c[7]*c [4]+550*c[5]^2*c[4]^3+300*c[5]^3*c[4]^4*c[6]*c[7]+300*c[5]^4*c[4]^3*c[ 6]*c[7]-690*c[5]^3*c[4]^3-200*c[4]^4*c[6]*c[5]^3+150*c[5]^3*c[4]^4), a [8,3] = 3/4*(-1560*c[5]^2*c[6]*c[4]^4+585*c[6]*c[5]*c[4]^4-45*c[7]*c[5 ]*c[4]^4-12*c[5]^3*c[4]-450*c[4]^5*c[6]*c[5]^3+3*c[5]*c[7]*c[4]-150*c[ 5]^2*c[4]^5*c[7]-200*c[5]^3*c[4]^4*c[7]+300*c[6]*c[4]^5*c[5]*c[7]+2860 *c[6]*c[5]^2*c[4]^4*c[7]+160*c[5]^4*c[4]^3*c[6]-150*c[5]^4*c[4]^4*c[6] -4*c[5]^2+2*c[4]^2-40*c[7]*c[6]*c[5]^3+132*c[5]*c[6]*c[7]*c[4]-900*c[6 ]*c[5]^2*c[4]^5*c[7]-30*c[6]*c[5]^2+12*c[6]*c[5]-960*c[7]*c[4]^4*c[6]* c[5]+24*c[6]*c[5]^3+3140*c[6]*c[7]*c[5]^3*c[4]^3+6*c[7]*c[5]^2+20*c[7] *c[6]*c[4]^3-3*c[7]*c[4]^2-18*c[6]*c[7]*c[5]+15*c[6]*c[4]^2-12*c[6]*c[ 4]^3+9*c[6]*c[7]*c[4]-24*c[6]*c[7]*c[4]^2-54*c[5]*c[4]^3-12*c[5]*c[7]* c[4]^2+348*c[5]*c[6]*c[4]^2+9*c[5]*c[4]^2-87*c[5]*c[6]*c[4]-2*c[5]*c[4 ]-6*c[6]*c[4]+1035*c[5]*c[7]*c[6]*c[4]^3+1593*c[5]^2*c[6]*c[7]*c[4]^2+ 900*c[4]^5*c[6]*c[7]*c[5]^3-534*c[6]*c[5]*c[7]*c[4]^2+48*c[6]*c[7]*c[5 ]^2-60*c[6]*c[5]^4*c[4]^2-3060*c[6]*c[5]^2*c[7]*c[4]^3-1560*c[6]*c[5]^ 3*c[7]*c[4]^2-50*c[5]^4*c[4]^3*c[7]+80*c[7]*c[5]*c[4]^3-652*c[6]*c[5]* c[4]^3-208*c[5]^3*c[6]*c[4]+20*c[5]^3*c[7]*c[4]+54*c[5]^2*c[4]+360*c[5 ]^3*c[6]*c[7]*c[4]+82*c[5]^3*c[4]^2+1734*c[5]^2*c[6]*c[4]^3+388*c[5]^2 *c[7]*c[4]^2-944*c[5]^2*c[6]*c[4]^2-900*c[5]^2*c[7]*c[4]^3-135*c[5]^3* c[7]*c[4]^2+879*c[5]^3*c[6]*c[4]^2+660*c[5]^2*c[4]^4*c[7]-1680*c[6]*c[ 5]^3*c[4]^3+420*c[5]^3*c[7]*c[4]^3+100*c[5]^4*c[7]*c[4]^2*c[6]-246*c[5 ]^2*c[4]^2-405*c[4]^4*c[5]^2+30*c[5]^4*c[4]^3+240*c[5]^2*c[6]*c[4]-84* c[5]^2*c[7]*c[4]-396*c[5]^2*c[6]*c[7]*c[4]+562*c[5]^2*c[4]^3+30*c[4]^4 *c[5]+90*c[4]^5*c[5]^2+300*c[5]^4*c[4]^4*c[6]*c[7]-3000*c[5]^3*c[4]^4* c[6]*c[7]-300*c[5]^4*c[4]^3*c[6]*c[7]-255*c[5]^3*c[4]^3+480*c[5]^2*c[6 ]*c[4]^5+1540*c[4]^4*c[6]*c[5]^3-180*c[6]*c[5]*c[4]^5+120*c[5]^3*c[4]^ 4)/c[4]^2/(150*c[5]^2*c[6]*c[4]^4-87*c[5]^3*c[4]-12*c[5]*c[7]*c[4]-200 *c[5]^3*c[4]^4*c[7]-200*c[6]*c[5]^2*c[4]^4*c[7]-200*c[5]^4*c[4]^3*c[6] -10*c[5]^2-10*c[4]^2+12*c[5]^3+12*c[4]^3+20*c[7]*c[6]*c[5]^3+15*c[5]*c [6]*c[7]*c[4]+12*c[6]*c[5]^2-15*c[6]*c[5]^3-1300*c[6]*c[7]*c[5]^3*c[4] ^3+150*c[5]^4*c[7]*c[4]^2+12*c[7]*c[5]^2+20*c[7]*c[6]*c[4]^3+12*c[7]*c [4]^2-15*c[7]*c[4]^3+12*c[6]*c[4]^2-15*c[6]*c[4]^3-15*c[6]*c[7]*c[4]^2 -87*c[5]*c[4]^3-72*c[5]*c[7]*c[4]^2-72*c[5]*c[6]*c[4]^2+60*c[5]*c[4]^2 -12*c[5]*c[6]*c[4]+10*c[5]*c[4]-150*c[5]*c[7]*c[6]*c[4]^3-15*c[7]*c[5] ^3-660*c[5]^2*c[6]*c[7]*c[4]^2+90*c[6]*c[5]*c[7]*c[4]^2-15*c[6]*c[7]*c [5]^2+150*c[6]*c[5]^4*c[4]^2+930*c[6]*c[5]^2*c[7]*c[4]^3+930*c[6]*c[5] ^3*c[7]*c[4]^2-200*c[5]^4*c[4]^3*c[7]+110*c[7]*c[5]*c[4]^3+110*c[6]*c[ 5]*c[4]^3+110*c[5]^3*c[6]*c[4]+110*c[5]^3*c[7]*c[4]+60*c[5]^2*c[4]-150 *c[5]^3*c[6]*c[7]*c[4]+550*c[5]^3*c[4]^2-690*c[5]^2*c[6]*c[4]^3+520*c[ 5]^2*c[7]*c[4]^2+520*c[5]^2*c[6]*c[4]^2-690*c[5]^2*c[7]*c[4]^3-690*c[5 ]^3*c[7]*c[4]^2-690*c[5]^3*c[6]*c[4]^2+150*c[5]^2*c[4]^4*c[7]+900*c[6] *c[5]^3*c[4]^3+900*c[5]^3*c[7]*c[4]^3-200*c[5]^4*c[7]*c[4]^2*c[6]-429* c[5]^2*c[4]^2-120*c[5]^4*c[4]^2-120*c[4]^4*c[5]^2+150*c[5]^4*c[4]^3-72 *c[5]^2*c[6]*c[4]-72*c[5]^2*c[7]*c[4]+90*c[5]^2*c[6]*c[7]*c[4]+550*c[5 ]^2*c[4]^3+300*c[5]^3*c[4]^4*c[6]*c[7]+300*c[5]^4*c[4]^3*c[6]*c[7]-690 *c[5]^3*c[4]^3-200*c[4]^4*c[6]*c[5]^3+150*c[5]^3*c[4]^4), a[8,5] = -1/ 2*(893*c[5]^2*c[6]*c[4]^4-122*c[6]*c[5]*c[4]^4+850*c[5]^4*c[4]^4*c[6]^ 2*c[7]-101*c[7]*c[5]*c[4]^4-535*c[5]^2*c[4]^4*c[7]^2-44*c[5]^3*c[4]+16 0*c[4]^5*c[6]^2*c[5]^2+100*c[7]^2*c[5]^2*c[4]^5+170*c[4]^5*c[6]*c[5]^3 +75*c[5]*c[6]^2*c[4]^4+16*c[7]^2*c[4]^3-9*c[5]*c[7]*c[4]-140*c[5]^2*c[ 4]^5*c[7]-780*c[5]^3*c[4]^4*c[7]-10*c[4]^4*c[6]^2-100*c[7]^2*c[4]^5*c[ 5]^3+450*c[6]^2*c[4]^5*c[7]*c[5]^3+510*c[5]^4*c[4]^2*c[7]^2*c[6]+490*c [6]^2*c[7]*c[5]^4*c[4]^2-2224*c[6]*c[5]^2*c[4]^4*c[7]+10*c[5]^5*c[4]^3 *c[7]-146*c[5]^4*c[4]^3*c[6]+90*c[5]^4*c[4]^5*c[6]+40*c[5]^4*c[4]^4*c[ 6]+570*c[7]^2*c[4]^4*c[5]^3+130*c[5]^4*c[4]^4*c[7]+25*c[7]^2*c[6]*c[4] ^4+18*c[5]^5*c[4]^2-4*c[5]^2-4*c[4]^2-10*c[4]^4*c[7]^2+150*c[7]^2*c[5] 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7]*c[4]^2+2898*c[6]*c[5]^2*c[7]*c[4]^3+1462*c[6]*c[5]^3*c[7]*c[4]^2+20 8*c[6]^2*c[5]^3*c[7]*c[4]-225*c[5]^4*c[4]^3*c[7]+704*c[7]^2*c[5]^2*c[4 ]^3-100*c[5]^4*c[7]^2*c[4]^4+29*c[7]^2*c[5]*c[4]^2+152*c[7]*c[5]*c[4]^ 3+181*c[6]*c[5]*c[4]^3-178*c[7]^2*c[5]^2*c[6]*c[4]-36*c[6]*c[7]*c[4]^4 -18*c[6]^2*c[7]^2*c[4]^2+3*c[6]^2*c[7]^2*c[4]+10*c[5]^4*c[7]^2*c[6]+77 *c[5]^3*c[6]*c[4]+75*c[5]^3*c[7]*c[4]+90*c[7]*c[6]*c[5]^5*c[4]+29*c[5] ^2*c[4]-350*c[5]^5*c[4]^2*c[6]*c[7]-234*c[5]^3*c[6]*c[7]*c[4]+279*c[5] ^3*c[4]^2+54*c[6]^2*c[7]^2*c[4]*c[5]-1192*c[5]^2*c[6]*c[4]^3+450*c[5]^ 2*c[7]*c[4]^2+522*c[5]^2*c[6]*c[4]^2-990*c[5]^2*c[7]*c[4]^3-502*c[5]^3 *c[7]*c[4]^2-614*c[5]^3*c[6]*c[4]^2-1638*c[5]^3*c[6]^2*c[4]^3+260*c[4] ^3*c[7]^2*c[5]^2*c[6]^2-310*c[4]^5*c[5]^3*c[6]^2-5*c[7]*c[5]^4*c[4]-5* c[7]^2*c[5]^4*c[4]+737*c[5]^2*c[4]^4*c[7]+948*c[6]^2*c[5]^2*c[4]^3-426 *c[6]^2*c[5]^2*c[4]^2+1443*c[6]*c[5]^3*c[4]^3-152*c[6]^2*c[5]^3*c[4]+7 72*c[6]^2*c[5]^3*c[4]^2+1076*c[5]^3*c[7]*c[4]^3+120*c[5]^4*c[7]*c[4]^2 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4]^4*c[5]^2-100*c[5]^5*c[4]^5*c[6]+6*c[5]^3*c[7]^2+16*c[6]*c[4]^4-200* c[5]^4*c[4]^5*c[6]^2*c[7]+15*c[7]*c[6]^2*c[5]*c[4]^4-1990*c[5]^3*c[4]^ 4*c[6]^2*c[7]-750*c[5]^3*c[4]^3*c[6]^2*c[7]^2+210*c[5]^4*c[4]^4*c[6]*c [7]+1100*c[5]^2*c[4]^4*c[7]*c[6]^2+2380*c[5]^3*c[4]^4*c[6]*c[7]-50*c[5 ]^4*c[4]^3*c[6]*c[7]-594*c[5]^3*c[4]^3-10*c[5]^5*c[6]*c[7]-3*c[5]^5*c[ 4]+328*c[7]^2*c[5]^3*c[4]^2+166*c[5]^5*c[6]*c[4]^2+150*c[5]^4*c[4]^5*c [6]^2-200*c[6]*c[7]^2*c[5]^4*c[4]^5-680*c[5]^4*c[4]^4*c[6]^2+850*c[5]^ 4*c[4]^3*c[6]^2+300*c[7]^2*c[5]^4*c[6]^2*c[4]^3-100*c[7]^2*c[5]^4*c[6] ^2*c[4]^2-2570*c[5]^3*c[4]^4*c[7]^2*c[6]-1130*c[5]^4*c[4]^3*c[7]^2*c[6 ]+1840*c[5]^2*c[4]^4*c[7]^2*c[6]-1060*c[5]^4*c[4]^3*c[6]^2*c[7]-170*c[ 7]^2*c[4]^4*c[6]^2*c[5]+500*c[7]^2*c[4]^4*c[5]^3*c[6]^2-110*c[7]^2*c[4 ]^4*c[6]^2*c[5]^2-160*c[5]^2*c[6]*c[4]^5-200*c[6]^2*c[7]^2*c[5]^4*c[4] ^4-1067*c[4]^4*c[6]*c[5]^3-70*c[5]^5*c[6]^2*c[4]^3-200*c[5]^4*c[4]^5*c [6]*c[7]+14*c[5]^4*c[4]+434*c[5]^3*c[4]^4-52*c[5]^5*c[4]*c[6])/c[5]/(7 2*c[5]^2*c[6]*c[4]^4-12*c[6]*c[5]*c[4]^4+1100*c[5]^4*c[4]^4*c[6]^2*c[7 ]-12*c[7]*c[5]*c[4]^4-110*c[5]^2*c[4]^4*c[7]^2+120*c[4]^5*c[6]*c[5]^3+ 15*c[5]*c[6]^2*c[4]^4-300*c[5]^5*c[4]^3*c[7]^2*c[6]^2+10*c[5]^5-440*c[ 5]^3*c[4]^4*c[7]-150*c[7]^2*c[4]^5*c[5]^3+180*c[5]^4*c[4]^2*c[7]^2*c[6 ]+180*c[6]^2*c[7]*c[5]^4*c[4]^2+350*c[6]*c[5]^2*c[4]^4*c[7]-570*c[5]^5 *c[4]^3*c[7]-150*c[5]^5*c[4]*c[6]*c[7]^2-150*c[5]^5*c[4]*c[6]^2*c[7]+4 10*c[5]^4*c[4]^3*c[6]-120*c[5]^4*c[4]^4*c[6]+690*c[7]^2*c[4]^4*c[5]^3+ 200*c[5]^4*c[4]^5*c[7]^2-120*c[5]^4*c[4]^4*c[7]-15*c[7]^2*c[6]*c[4]^4+ 342*c[5]^5*c[4]^2-12*c[7]*c[6]^2*c[5]^3+10*c[7]*c[6]*c[5]^3-12*c[7]^2* c[6]*c[5]^3+15*c[5]^6*c[7]+410*c[5]^4*c[4]^2*c[7]^2+750*c[5]^5*c[7]^2* c[4]^3-120*c[6]*c[5]^2*c[4]^5*c[7]+570*c[5]^6*c[4]^2*c[7]-570*c[5]^5*c [4]^3*c[6]-57*c[7]*c[4]^4*c[6]*c[5]+87*c[5]^6*c[4]-570*c[5]^5*c[4]^4-1 100*c[5]^5*c[7]*c[4]^4*c[6]-410*c[7]^2*c[5]^3*c[4]^3-15*c[7]*c[4]^4*c[ 6]^2-15*c[5]^5*c[7]^2+780*c[5]^5*c[4]^2*c[6]^2*c[7]+750*c[5]^5*c[4]^4* c[6]+300*c[5]^5*c[7]*c[4]^5*c[6]-690*c[5]^5*c[4]^2*c[6]^2-40*c[6]^2*c[ 5]^4*c[7]*c[4]-690*c[5]^5*c[4]^2*c[7]^2-57*c[5]^4*c[6]^2*c[4]+410*c[5] ^4*c[6]^2*c[4]^2+750*c[5]^5*c[4]^4*c[7]+340*c[6]^2*c[7]*c[5]^2*c[4]^3- 30*c[6]^2*c[7]^2*c[5]^2*c[4]-750*c[6]*c[7]*c[5]^3*c[4]^3-20*c[7]^2*c[6 ]^2*c[5]^4-342*c[5]^4*c[7]*c[4]^2+90*c[7]^2*c[6]*c[5]*c[4]^4+200*c[7]^ 2*c[5]^5*c[6]^2*c[4]^2-10*c[5]^4*c[7]+140*c[5]^5*c[7]*c[4]^2+150*c[5]^ 4*c[4]*c[7]^2*c[6]^2+20*c[6]^2*c[4]^4*c[7]^2-10*c[7]*c[6]*c[4]^3-30*c[ 5]^5*c[7]*c[4]+12*c[5]^4*c[6]^2+12*c[7]^2*c[6]*c[4]^3-12*c[5]*c[7]^2*c [4]^3-24*c[6]*c[5]*c[7]^2*c[4]^2+24*c[5]*c[7]*c[6]*c[4]^3-12*c[5]^6+15 0*c[7]^2*c[5]^6*c[4]^2-120*c[5]^4*c[4]^5+48*c[5]^2*c[6]*c[7]*c[4]^2+11 00*c[5]^4*c[7]^2*c[4]^4*c[6]-150*c[4]^5*c[6]*c[7]*c[5]^3-12*c[6]^2*c[5 ]*c[4]^3-42*c[7]*c[6]^2*c[5]*c[4]^3+20*c[6]*c[5]*c[7]*c[4]^2+12*c[5]^4 *c[7]*c[6]-24*c[6]^2*c[4]^2*c[7]*c[5]+30*c[6]^2*c[7]^2*c[4]^2*c[5]-342 *c[6]*c[5]^4*c[4]^2-10*c[6]*c[5]^4+120*c[4]^5*c[7]*c[5]^3+24*c[6]^2*c[ 5]^2*c[7]*c[4]-30*c[6]^2*c[5]^2*c[7]*c[4]^2-243*c[6]*c[5]^2*c[7]*c[4]^ 3+372*c[6]*c[5]^3*c[7]*c[4]^2+87*c[6]^2*c[5]^3*c[7]*c[4]+410*c[5]^4*c[ 4]^3*c[7]+57*c[7]^2*c[5]^2*c[4]^3-750*c[5]^4*c[7]^2*c[4]^4+10*c[7]*c[5 ]*c[4]^3+10*c[6]*c[5]*c[4]^3+24*c[7]^2*c[5]^2*c[6]*c[4]+12*c[6]*c[7]*c [4]^4+200*c[7]*c[5]^7*c[4]^3-110*c[5]^6*c[7]*c[4]+15*c[6]*c[5]^6+20*c[ 5]^3*c[6]*c[4]+20*c[5]^3*c[7]*c[4]+150*c[7]*c[6]*c[5]^5*c[4]-900*c[5]^ 5*c[4]^3*c[7]^2*c[6]-750*c[5]^5*c[4]^2*c[6]*c[7]-96*c[5]^3*c[6]*c[7]*c [4]+200*c[6]*c[5]^7*c[4]^3+20*c[5]^3*c[4]^2-36*c[5]^2*c[6]*c[4]^3-20*c [5]^2*c[7]*c[4]^2-20*c[5]^2*c[6]*c[4]^2-36*c[5]^2*c[7]*c[4]^3-24*c[5]^ 3*c[7]*c[4]^2-24*c[5]^3*c[6]*c[4]^2-410*c[5]^3*c[6]^2*c[4]^3-900*c[5]^ 5*c[4]^3*c[6]^2*c[7]-510*c[4]^3*c[7]^2*c[5]^2*c[6]^2-630*c[7]*c[5]^6*c [6]*c[4]^2-150*c[4]^5*c[5]^3*c[6]^2+72*c[7]*c[5]^4*c[4]-57*c[7]^2*c[5] ^4*c[4]+72*c[5]^2*c[4]^4*c[7]+57*c[6]^2*c[5]^2*c[4]^3+24*c[6]^2*c[5]^2 *c[4]^2+285*c[6]*c[5]^3*c[4]^3-24*c[6]^2*c[5]^3*c[4]+285*c[5]^3*c[7]*c [4]^3+270*c[5]^4*c[7]*c[4]^2*c[6]+12*c[7]*c[6]^2*c[4]^3+340*c[7]^2*c[5 ]^2*c[6]*c[4]^3-410*c[7]^2*c[5]^3*c[6]*c[4]^2-200*c[5]^5*c[4]^5*c[7]+7 0*c[4]^3*c[7]^2*c[5]*c[6]^2-42*c[7]^2*c[6]*c[5]*c[4]^3+150*c[5]^5*c[4] ^5-30*c[7]^2*c[5]^2*c[6]*c[4]^2+12*c[4]^4*c[5]^2+87*c[7]^2*c[5]^3*c[6] *c[4]+510*c[7]^2*c[5]^3*c[6]*c[4]^3+690*c[5]^3*c[6]^2*c[4]^4-342*c[5]^ 4*c[4]^3-15*c[6]^2*c[4]^3*c[7]^2+510*c[5]^3*c[6]^2*c[7]*c[4]^3-20*c[5] ^2*c[6]*c[7]*c[4]-410*c[6]^2*c[7]*c[5]^3*c[4]^2-70*c[6]^2*c[7]^2*c[4]* c[5]^3-10*c[5]^2*c[4]^3+24*c[7]^2*c[5]^2*c[4]^2-24*c[7]^2*c[5]^3*c[4]- 200*c[7]^2*c[5]^6*c[4]^2*c[6]+110*c[7]^2*c[4]*c[5]^5+1350*c[5]^5*c[4]^ 3*c[6]*c[7]+15*c[7]^2*c[4]^4*c[5]+300*c[6]^2*c[7]*c[5]^6*c[4]^3+510*c[ 7]^2*c[5]^3*c[6]^2*c[4]^2-550*c[5]^6*c[4]^2-600*c[7]*c[5]^6*c[4]^3+20* c[7]*c[6]^2*c[5]^5+570*c[5]^6*c[6]*c[4]^2+150*c[5]^2*c[6]*c[4]^5*c[7]^ 2+550*c[5]^4*c[4]^4+120*c[5]^7*c[4]^2+72*c[5]^4*c[4]*c[6]-40*c[5]^4*c[ 7]^2*c[4]*c[6]-57*c[5]^4*c[4]*c[7]*c[6]+150*c[6]^2*c[5]^6*c[4]^2+150*c [4]^5*c[6]^2*c[7]*c[5]^2-200*c[4]^5*c[6]^2*c[7]^2*c[5]^2-150*c[5]^7*c[ 7]*c[4]^2-110*c[6]^2*c[4]^4*c[5]^2+780*c[5]^5*c[4]^2*c[7]^2*c[6]-200*c [5]^5*c[4]^5*c[6]-600*c[5]^6*c[4]^3*c[6]-200*c[6]^2*c[7]*c[5]^6*c[4]^2 -300*c[5]^4*c[4]^5*c[6]^2*c[7]+90*c[7]*c[6]^2*c[5]*c[4]^4+200*c[6]*c[5 ]^7*c[7]*c[4]^2-300*c[6]*c[5]^7*c[7]*c[4]^3-180*c[5]^3*c[4]^4*c[6]^2*c [7]-570*c[5]^4*c[4]^4*c[6]*c[7]-540*c[5]^2*c[4]^4*c[7]*c[6]^2+660*c[5] ^3*c[4]^4*c[6]*c[7]+60*c[5]^4*c[4]^3*c[6]*c[7]+48*c[5]^3*c[4]^3-15*c[6 ]^2*c[5]^5-200*c[5]^6*c[6]^2*c[4]^3-15*c[5]^5*c[6]*c[7]-48*c[5]^5*c[4] +140*c[5]^5*c[6]*c[4]^2+200*c[5]^4*c[4]^5*c[6]^2+300*c[6]^2*c[7]^2*c[5 ]^3*c[4]^5-300*c[6]*c[7]^2*c[5]^4*c[4]^5+700*c[7]*c[5]^6*c[4]^3*c[6]-7 50*c[5]^4*c[4]^4*c[6]^2+20*c[5]^5*c[7]^2*c[6]+1100*c[7]^2*c[5]^4*c[6]^ 2*c[4]^3-930*c[7]^2*c[5]^4*c[6]^2*c[4]^2+300*c[7]^2*c[5]^6*c[4]^3*c[6] -20*c[5]^6*c[7]*c[6]-180*c[5]^3*c[4]^4*c[7]^2*c[6]-750*c[5]^4*c[4]^3*c [7]^2*c[6]-540*c[5]^2*c[4]^4*c[7]^2*c[6]-750*c[5]^4*c[4]^3*c[6]^2*c[7] -150*c[7]^2*c[4]^4*c[6]^2*c[5]-1100*c[7]^2*c[4]^4*c[5]^3*c[6]^2+930*c[ 7]^2*c[4]^4*c[6]^2*c[5]^2+150*c[5]^6*c[4]*c[7]*c[6]-110*c[5]^6*c[4]*c[ 6]-150*c[5]^7*c[6]*c[4]^2+570*c[5]^6*c[4]^3-150*c[5]^7*c[4]^3+15*c[7]^ 2*c[6]^2*c[5]^3-440*c[4]^4*c[6]*c[5]^3+750*c[5]^5*c[6]^2*c[4]^3-200*c[ 7]^2*c[5]^6*c[4]^3+12*c[5]^4*c[7]^2+200*c[5]^4*c[4]^5*c[6]*c[7]-20*c[5 ]^4*c[4]-87*c[5]^3*c[4]^4-30*c[5]^5*c[4]*c[6]+110*c[5]^5*c[6]^2*c[4]), a[9,7] = -1/60*(10*c[5]*c[6]*c[4]-5*c[6]*c[5]-5*c[6]*c[4]+3*c[6]-5*c[ 5]*c[4]+3*c[5]+3*c[4]-2)/c[7]/(-c[5]*c[6]*c[4]+c[5]*c[6]*c[7]*c[4]+c[5 ]*c[7]*c[4]-c[7]^2*c[5]*c[4]+c[6]*c[7]*c[4]-c[6]*c[7]^2*c[4]-c[7]^2*c[ 4]+c[7]^3*c[4]+c[6]*c[7]*c[5]-c[6]*c[7]^2*c[5]-c[7]^2*c[5]+c[7]^3*c[5] -c[7]^2*c[6]+c[7]^3*c[6]+c[7]^3-c[7]^4), a[9,2] = 0, a[9,3] = 0, a[9,5 ] = -1/60*(10*c[6]*c[7]*c[4]-5*c[6]*c[4]-5*c[7]*c[6]+3*c[6]-5*c[7]*c[4 ]+3*c[4]-2+3*c[7])/c[5]/(c[5]^3*c[4]-c[5]^2*c[4]+c[5]*c[7]*c[4]-c[5]^2 *c[7]*c[4]+c[5]*c[6]*c[4]-c[5]^2*c[6]*c[4]-c[6]*c[7]*c[4]+c[5]*c[6]*c[ 7]*c[4]-c[5]^4+c[5]^3-c[7]*c[5]^2+c[7]*c[5]^3-c[6]*c[5]^2+c[6]*c[5]^3+ c[6]*c[7]*c[5]-c[6]*c[7]*c[5]^2), a[9,8] = 1/60*(-20*c[5]*c[6]*c[4]+30 *c[5]*c[6]*c[7]*c[4]+15*c[6]*c[5]-20*c[6]*c[7]*c[5]+15*c[6]*c[4]-20*c[ 6]*c[7]*c[4]-12*c[6]+15*c[7]*c[6]+15*c[5]*c[4]-20*c[5]*c[7]*c[4]-12*c[ 5]+15*c[7]*c[5]-12*c[4]+15*c[7]*c[4]+10-12*c[7])/(-c[4]+c[7]*c[4]+c[6] *c[4]-c[6]*c[7]*c[4]+c[5]*c[4]-c[5]*c[7]*c[4]-c[5]*c[6]*c[4]+c[5]*c[6] *c[7]*c[4]+1-c[7]-c[6]+c[7]*c[6]-c[5]+c[7]*c[5]+c[6]*c[5]-c[6]*c[7]*c[ 5]), a[9,1] = 1/60*(30*c[5]*c[6]*c[7]*c[4]-10*c[5]*c[6]*c[4]-10*c[6]*c [7]*c[5]+5*c[6]*c[5]-10*c[6]*c[7]*c[4]+5*c[6]*c[4]+5*c[7]*c[6]-3*c[6]- 10*c[5]*c[7]*c[4]+5*c[5]*c[4]+5*c[7]*c[5]-3*c[5]+5*c[7]*c[4]-3*c[4]+2- 3*c[7])/c[5]/c[6]/c[7]/c[4], a[9,4] = 1/60*(-2+3*c[5]-5*c[7]*c[6]+3*c[ 7]+3*c[6]+10*c[6]*c[7]*c[5]-5*c[6]*c[5]-5*c[7]*c[5])/c[4]/(-c[6]*c[4]^ 3-c[4]^3+c[4]^4+c[7]*c[4]^2+c[6]*c[4]^2-c[6]*c[7]*c[4]+c[6]*c[7]*c[4]^ 2-c[5]*c[4]^3+c[5]*c[4]^2-c[7]*c[4]^3-c[5]*c[7]*c[4]+c[5]*c[7]*c[4]^2- c[5]*c[6]*c[4]+c[5]*c[6]*c[4]^2+c[6]*c[7]*c[5]-c[5]*c[6]*c[7]*c[4]), a [9,6] = 1/60*(10*c[5]*c[7]*c[4]-5*c[5]*c[4]-5*c[7]*c[5]+3*c[5]-5*c[7]* c[4]+3*c[4]-2+3*c[7])/c[6]/(c[6]^2*c[4]*c[5]-c[5]*c[6]*c[4]+c[5]*c[7]* c[4]-c[5]*c[6]*c[7]*c[4]-c[6]^3*c[4]+c[6]^2*c[4]-c[6]*c[7]*c[4]+c[6]^2 *c[7]*c[4]-c[6]^3*c[5]+c[6]^2*c[5]-c[6]*c[7]*c[5]+c[6]^2*c[7]*c[5]+c[6 ]^4-c[6]^3+c[7]*c[6]^2-c[6]^3*c[7])\}:" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 5 "Let " }{XPPEDIT 18 0 "t[7 ]" "6#&%\"tG6#\"\"(" }{TEXT -1 7 " and " }{XPPEDIT 18 0 "t[8]" "6#&% \"tG6#\"\")" }{TEXT -1 29 " denote the coefficients of " }{XPPEDIT 18 0 "z^7" "6#*$%\"zG\"\"(" }{TEXT -1 7 " and " }{XPPEDIT 18 0 "z^8 " "6#*$%\"zG\"\")" }{TEXT -1 43 " respectively in the stability polyn omial." }}{PARA 0 "" 0 "" {TEXT -1 29 "Following Tsitouras, we set " }{XPPEDIT 18 0 "t[7] = 77/(80*`.`*7!);" "6#/&%\"tG6#\"\"(*&\"#x\"\"\"* (\"#!)F*%\".GF*-%*factorialG6#F'F*!\"\"" }{XPPEDIT 18 0 " ``=11/57600 " "6#/%!G*&\"#6\"\"\"\"&+w&!\"\"" }{TEXT -1 7 " and " }{XPPEDIT 18 0 "t[8] = 7/(10*`.`*8!);" "6#/&%\"tG6#\"\")*&\"\"(\"\"\"*(\"#5F*%\".GF *-%*factorialG6#F'F*!\"\"" }{XPPEDIT 18 0 "`` = 1/57600;" "6#/%!G*&\" \"\"F&\"&+w&!\"\"" }{TEXT -1 37 " to obtain two equations involving \+ " }{XPPEDIT 18 0 "c[4]" "6#&%\"cG6#\"\"%" }{TEXT -1 3 ", " }{XPPEDIT 18 0 "c[5]" "6#&%\"cG6#\"\"&" }{TEXT -1 7 " and " }{XPPEDIT 18 0 "c[ 6]" "6#&%\"cG6#\"\"'" }{TEXT -1 2 ". " }}{PARA 0 "" 0 "" {TEXT -1 52 " Note that the stability polynomial only depends on " }{XPPEDIT 18 0 " c[4]" "6#&%\"cG6#\"\"%" }{TEXT -1 3 ", " }{XPPEDIT 18 0 "c[5]" "6#&% \"cG6#\"\"&" }{TEXT -1 7 " and " }{XPPEDIT 18 0 "c[6]" "6#&%\"cG6#\" \"'" }{TEXT -1 2 ". " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 143 "Rz := StabilityFunction(6,8,'expanded'): \neqA := simplify(subs(eG,coeff(Rz,z^7)))=11/57600:%;\neqB := simplify (subs(eG,coeff(Rz,z^8)))=1/57600:%;" }}{PARA 12 "" 1 "" {XPPMATH 20 "6 #/,$*&#\"\"\"\"%S9F'*&,hn**\"#IF'&%\"cG6#\"\"&F'&F.6#\"\"'F')&F.6#\"\" %\"\"#F'!\"\"*&)F-F8F'F5F'F9*(F8F')F-\"\"$F'F5F'F9*(F0F'F=F'F4F'F'*(\" #?F'F;F'F4F'F'*(\"$S\"F')F5F7F'F;F'F'*(\"#**F'F;F')F5F>F'F9*&F-F'F4F'F 9*(\"#SF'F=F'FDF'F9*(FAF'F=F'FGF'F'*(\"#5F'F-F'FGF'F'*(\"#9F'FDF'F-F'F 9*(F8F'F1F'F;F'F9*(F8F'F1F'F4F'F9*(F7F'F1F'FGF'F'*(FJF')F5F0F'F;F'F9** F0F'F-F'F1F'F5F'F'**\"$S$F'F;F'F1F'FDF'F'**\"#>F'F;F'F1F'F4F'F'**\"$g \"F'F;F'F1F'FGF'F9**F3F'F;F'F1F'F5F'F'**\"$+\"F'FDF'F1F'F=F'F'**\"$+#F 'F;F'F1F'FTF'F9**\"$!>F'F1F'F=F'FGF'F9**\"$?\"F'F=F'F1F'F4F'F'**\"#MF' F=F'F1F'F5F'F9**\"#)*F'F1F'F-F'FGF'F'**FenF'F1F'F-F'FDF'F9*(F7F'F1F'F= F'F'**\"#!)F'F1F'F-F'FTF'F'F',2*(FMF'F=F'F4F'F'*(FMF'F;F'FGF'F'*(F,F'F ;F'F4F'F9*(F3F'F;F'F5F'F'*$F;F'F9*(F3F'F-F'F4F'F'*&F-F'F5F'F'*$F4F'F9F 9F'F'#\"#6\"&+w&" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/,$*&#\"\"\"\"%S9F '*,&%\"cG6#\"\"%F'&F+6#\"\"&F',0**\"#7F'F.F'&F+6#\"\"'F'F*F'F'*(\"\"#F 'F4F'F.F'!\"\"**\"#IF'F.F'F4F')F*F8F'F9*&F.F'F*F'F'**\"#?F'F4F'F.F')F* \"\"$F'F'*$F " 0 "" {MPLTEXT 1 0 96 "sol := solve(\{eqA,eqB\},\{c[5],c[6 ]\}):\nc[5]=subs(sol,c[5]);eqC := %:\nc[6]=subs(sol,c[6]);eqD := %:" } }{PARA 11 "" 1 "" {XPPMATH 20 "6#/&%\"cG6#\"\"&,$*(\"\"#\"\"\"&F%6#\" \"%F+,**&\"$g\"F+)F,\"\"$F+F+*&\"#!)F+)F,F*F+!\"\"*&\"#6F+F,F+F+F+F7F7 F7" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/&%\"cG6#\"\"',$*&#\"\"\"\"#!)F+ **&F%6#\"\"%!\"\",2\"\"$F1*&\"#pF+F.F+F+*&\"%*3\"F+)F.\"\"#F+F1*&\"&$y 5F+)F.F3F+F+*&\"&+a'F+)F.F0F+F1*&\"'g " 0 "" {MPLTEXT 1 0 1 ";" }}} {SECT 1 {PARA 4 "" 0 "" {TEXT 0 13 "prin_err_norm" }{TEXT -1 1 " " }} {PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 51 "errterms6_8 := PrincipalErrorTerms(6,8,'expanded'):" }}}{PARA 0 " " 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 704 "prin _err_norm := proc(c2,c4,c7)\n local c2r,c4r,c5r,c6r,c7r,e1,sm,ct;\n \+ global e2;\n \n c2r := convert(c2,rational,Digits);\n c4r := co nvert(c4,rational,Digits); \n c5r := -2*c4r/(160*c4r^3-80*c4r^2+11* c4r-1);\n c6r := -1/80*1/c4r*(-3-65400*c4r^4+10783*c4r^3-1089*c4r^2 +69*c4r+241760*c4r^5+460800*c4r^7-505600*c4r^6)/\n (112*c4r^ 2-320*c4r^3-17*c4r+1+320*c4r^4)/(120*c4r^3-20*c4r^2-13*c4r+3);\n c7r := convert(c7,rational,Digits);\n e1 := \{c[2]=c2r,c[4]=c4r,c[5]=c5 r,c[6]=c6r,c[7]=c7r,c[8]=1,c[9]=1\};\n e2 := `union`(e1,simplify(sub s(e1,eG))):\n sm := 0;\n for ct to nops(errterms6_8) do\n sm \+ := sm+(simplify(subs(e2,errterms6_8[ct])))^2;\n end do;\n evalf(sq rt(sm));\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 11 "The nodes " }{XPPEDIT 18 0 "c[2]=-5/18" "6#/&%\"cG6#\"\"#,$*&\"\"&\"\"\"\"#=!\"\"F-" }{TEXT -1 3 ", " } {XPPEDIT 18 0 "c[4]=2/15" "6#/&%\"cG6#\"\"%*&\"\"#\"\"\"\"#:!\"\"" } {TEXT -1 7 " and " }{XPPEDIT 18 0 "c[7]=125/126" "6#/&%\"cG6#\"\"(*& \"$D\"\"\"\"\"$E\"!\"\"" }{TEXT -1 78 " suggested by Tsitouras give th e following value for the principal error norm." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 34 "prin_err_nor m(-5/18,2/15,125/126);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#$\"+OQcW9!#8 " }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 59 "The \+ maximum magnitude of the linking coefficients is . . . " }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 59 "evalf(max(seq(seq(subs(e2,abs(a[i,j ])),j=1..i-1),i=2..9)));" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#$\"+n7)))) [!\")" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 30 "The corresponding values for " }{XPPEDIT 18 0 "c[5]" "6#&%\"cG6#\"\" &" }{TEXT -1 7 " and " }{XPPEDIT 18 0 "c[6]" "6#&%\"cG6#\"\"'" } {TEXT -1 12 " are . . . " }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 38 "c[5]=subs(e2,c[5]),c[6]=subs(e2,c[6]);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6$/&%\"cG6#\"\"&#\"$!=\"$*Q/&F%6#\"\"'#\"(R'[R\"(K[%e" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 29 "Tsitouras quotes \+ the values " }{XPPEDIT 18 0 "c[5]=180/389" "6#/&%\"cG6#\"\"&*&\"$!=\" \"\"\"$*Q!\"\"" }{TEXT -1 7 " and " }{XPPEDIT 18 0 "c[6]=163679/2422 80" "6#/&%\"cG6#\"\"'*&\"'zO;\"\"\"\"'!GU#!\"\"" }{TEXT -1 18 ". This \+ value for " }{XPPEDIT 18 0 "c[6]" "6#&%\"cG6#\"\"'" }{TEXT -1 62 " i s a 12 digit rational approximation for the previous value." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 38 "co nvert(3948639/5844832.,rational,12);" }}{PARA 11 "" 1 "" {XPPMATH 20 " 6##\"'zO;\"'!GU#" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 11 "If we set " }{XPPEDIT 18 0 "c[4]=2/15" "6#/&%\"cG6#\"\"% *&\"\"#\"\"\"\"#:!\"\"" }{TEXT -1 7 " and " }{XPPEDIT 18 0 "c[7]=125 /126" "6#/&%\"cG6#\"\"(*&\"$D\"\"\"\"\"$E\"!\"\"" }{TEXT -1 46 ", the \+ principal error norm is a minimum when " }{XPPEDIT 18 0 "c[2]" "6#&% \"cG6#\"\"#" }{TEXT -1 1 " " }{TEXT 266 1 "~" }{TEXT -1 1 " " } {XPPEDIT 18 0 "-0" "6#,$\"\"!!\"\"" }{TEXT -1 5 ".042." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 94 "evalf[1 5](findmin('prin_err_norm'(c2,2/15,125/126),c2=\{-0.3,-5/18,0.3\},conv ergence=location));" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#7$$!0:q$)4Av?%! #;$\"0\\B:wZRW\"!#=" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 " " {TEXT -1 35 "Taking a small positive value for " }{XPPEDIT 18 0 "c[ 2]" "6#&%\"cG6#\"\"#" }{TEXT -1 7 ", say " }{XPPEDIT 18 0 "c[2]=1/60 " "6#/&%\"cG6#\"\"#*&\"\"\"F)\"#g!\"\"" }{TEXT -1 76 ", gives a value for the principal error norm which is close to the minimum." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 33 "pr in_err_norm(1/60,2/15,125/126);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#$\" +Mg)RW\"!#8" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 72 "The maximum magnitude of the linking coefficients is the same a s before." }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 59 "evalf(max(seq(s eq(subs(e2,abs(a[i,j])),j=1..i-1),i=2..9)));" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#$\"+n7))))[!\")" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 833 "cc1 := -.4207522098e-1: nm1 := .14439477615e-3:\ncc2 := -5/18: nm2 := .1444563836e-3:\ncc3 := 1/6 0: nm3 := .144398603407233e-3:\nst := .14438e-3: \np1 := plot([[[cc1,n m1]]$4],style=point,symbol=[circle$2,diamond,cross],color=[black,green $3],symbolsize=[12,10$3]):\np2 := plot([[cc1,st],[cc1,nm1]],color=blac k,linestyle=3):\np3 := plot([[[cc2,nm2]]$4],style=point,symbol=[circle $2,diamond,cross],color=[black,cyan$3],symbolsize=[12,10$3]):\np4 := p lot([[cc2,st],[cc2,nm2]],color=black,linestyle=3):\np5 := plot([[[cc3, nm3]]$4],style=point,symbol=[circle$2,diamond,cross],color=[black,cora l$3],symbolsize=[12,10$3]):\np6 := plot([[cc3,st],[cc3,nm3]],color=red ,linestyle=3):\np7 := plot('prin_err_norm'(c2,2/15,125/126),c2=-0.4..0 .3,color=COLOR(RGB,.5,0,.9)):\nplots[display]([p||(1..7)],font=[HELVET ICA,9],view=[-0.4..0.3,.14438e-3..0.14453e-3]);" }}{PARA 13 "" 1 "" {GLPLOT2D 673 440 440 {PLOTDATA 2 "65-%'CURVESG6&7#7$$!+)4Av?%!#6$\",: wZRW\"!#9-%'COLOURG6&%$RGBG\"\"!F2F2-%'SYMBOLG6$%'CIRCLEG\"#7-%&STYLEG 6#%&POINTG-F$6&F&-F/6&F1$F2F2$\"*++++\"!\")F@-F46$F6\"#5F8-F$6&F&F>-F4 6$%(DIAMONDGFFF8-F$6&F&F>-F46$%&CROSSGFFF8-F$6%7$7$F($\"&QW\"FCF'F.-%* LINESTYLEG6#\"\"$-F$6&7#7$$!0yxxxxxx#!#:$\"+OQcW9!#8F.F3F8-F$6&Fgn-F/6 &F1F@FAFAFDF8-F$6&FgnFaoFIF8-F$6&FgnFaoFNF8-F$6%7$7$FinFUFhnF.FW-F$6&7 #7$$\"0nmmmmmm\"!#;$\"0LsS.')RW\"!#=F.F3F8-F$6&F]p-F/6&F1FA$\")AR!)\\F CF@FDF8-F$6&F]pFgpFIF8-F$6&F]pFgpFNF8-F$6%7$7$F_pFUF^p-F/6&F1FAF@F@FW- F$6$7S7$$!\"%!\"\"$\"0b[)>!o`W\"Fdp7$$!0LL3#*>u%QF[o$\"0:Ebc\\_W\"Fdp7 $$!0n\"z43m9PF[o$\"0@W$*o]^W\"Fdp7$$!0L$e/$f`c$F[o$\"0;a?:W]W\"Fdp7$$! 0M$3K\"o]T$F[o$\"0!*f8*=%\\W\"Fdp7$$!0n\"Hn7\\lKF[o$\"0c793X[W\"Fdp7$$ !0MekO9o7$F[o$\"0FAZ$GF[o$\"0Fjb&RfW9Fdp7$$!0+Dc#Gp'o#F[o$\"0\">8]r^W9Fdp7$$!0nmTF[o$\"0I?Wo2 UW\"Fdp7$$!0Me9(3(*==F[o$\"0O0$*ekTW\"Fdp7$$!0nmTO:7m\"F[o$\"0FSXU=TW \"Fdp7$$!0nmmvvv_\"F[o$\"0Z)oMO3W9Fdp7$$!0,DJ7@@P\"F[o$\"0AW\"\\\"[SW \"Fdp7$$!0nm;VByR*F[o$\"03U8kxRW\"Fdp7$$!/n;zp\"y*yF[o$\"0C,k'G'RW\"Fdp7$ $!/n\"H-V._'F[o$\"052Yp`RW\"Fdp7$$!/M$3F^X.&F[o$\"0**o,_[RW\"Fdp7$$!/M e97B\"\\$F[o$\"0921L[RW\"Fdp7$$!/,volwZ@F[o$\"0>ksY_RW\"Fdp7$$!.M$e%zy 'pF[o$\"0%oBK9'RW\"Fdp7$$\".****\\\\@-)F[o$\"0Sh&)fvRW\"Fdp7$$\"/**\\P opoAF[o$\"0JZ3G%*RW\"Fdp7$$\"/*\\(oMf(o$F[o$\"0(GZ(*o,W9Fdp7$$\"/**\\i i.j_F[o$\"0Y95CZSW\"Fdp7$$\"/KL$oT'ymF[o$\"0t\\9?zSW\"Fdp7$$\"/***\\i- ,>)F[o$\"0M1=B=TW\"Fdp7$$\"/mT&)3rf&*F[o$\"0IvU(z:W9Fdp7$$\"0***\\Zaq0 6F[o$\"0\\XE=1UW\"Fdp7$$\"0K3-\"QfY7F[o$\"0\\h?3cUW\"Fdp7$$\"0*\\PWF'Q R\"F[o$\"0yP_%HJW9Fdp7$$\"0KLe/Xy`\"F[o$\"0@er=tVW\"Fdp7$$\"0*\\(=<\"e )o\"F[o$\"0WYz F[o$\"0K3\\3)eW9Fdp7$$\"0K3-7d%H@F[o$\"0Ihj$*oYW\"Fdp7$$\"0****\\p]ZE# F[o$\"0nNQYZZW\"Fdp7$$\"0K$e*R7)>CF[o$\"0*fl]C%[W\"Fdp7$$\"0lmmV,&eDF[ o$\"0/\"R=>$\\W\"Fdp7$$\"0)\\(o(GP1FF[o$\"0\\P++K]W\"Fdp7$$\"0)\\78Z!z %GF[o$\"0DiQKK^W\"Fdp7$$FZF[r$\"0#)yn2X_W\"Fdp-%&COLORG6&F1$\"\"&F[rF@ $\"\"*F[r-%%FONTG6$%*HELVETICAGFcal-%+AXESLABELSG6%Q!6\"F[bl-Feal6#%(D EFAULTG-%%VIEWG6$;FiqFj`l;FU$\"&`W\"FC" 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" "Curve 8" "Curve 9" "Curve 10" "Curve 11" "Curve 12" "Curve 13" "Curve 14" "Curve 15" "Curve 16" }}}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 51 "We could use a \+ simpler rational approximation for " }{XPPEDIT 18 0 "c[6]" "6#&%\"cG6 #\"\"'" }{TEXT -1 10 ", namely " }{XPPEDIT 18 0 "c[6]=177/262" "6#/&% \"cG6#\"\"'*&\"$x\"\"\"\"\"$i#!\"\"" }{TEXT -1 1 "." }}{PARA 0 "" 0 " " {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 37 "convert(39 48639/5844832.,rational,5);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6##\"$x\" \"$i#" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 19 "Taking the values " }{XPPEDIT 18 0 "c[2] = 1/60" "6#/&%\"cG6#\"\"#*& \"\"\"F)\"#g!\"\"" }{TEXT -1 3 ", " }{XPPEDIT 18 0 "c[4] = 2/15" "6#/ &%\"cG6#\"\"%*&\"\"#\"\"\"\"#:!\"\"" }{TEXT -1 3 ", " }{XPPEDIT 18 0 "c[5] = 180/389" "6#/&%\"cG6#\"\"&*&\"$!=\"\"\"\"$*Q!\"\"" }{TEXT -1 3 ", " }{XPPEDIT 18 0 "c[6] = 177/262" "6#/&%\"cG6#\"\"'*&\"$x\"\"\" \"\"$i#!\"\"" }{TEXT -1 3 ", " }{XPPEDIT 18 0 "c[7] = 125/126" "6#/&% \"cG6#\"\"(*&\"$D\"\"\"\"\"$E\"!\"\"" }{TEXT -1 138 " gives a princip al error norm of approximately 0.00014441 and the maximum magnitude o f the linking coefficients is approximately 48.89." }}{PARA 0 "" 0 " " {TEXT -1 60 "The real stability interval is approximately [ -4.8903 , 0]." }}{PARA 0 "" 0 "" {TEXT -1 53 "(See the abreviated calculations in a later section.)" }}{PARA 0 "" 0 "" {TEXT -1 49 "#--------------- ---------------------------------" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}}{PARA 0 "" 0 "" {TEXT -1 52 "#----------------- ----------------------------------" }}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 38 "characteristics of the embedded scheme" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 35 "coefficients of the combined scheme" }} {PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1880 "ee := \{c[2]=1/60,\nc[3]=4/45,\nc[4]=2/15,\nc[5]=180/389,\nc[6]= 177/262,\nc[7]=125/126,\nc[8]=1,\nc[9]=1,\n\na[2,1]=1/60,\na[3,1]=-4/2 7,\na[3,2]=32/135,\na[4,1]=1/30,\na[4,2]=0,\na[4,3]=1/10,\na[5,1]=7310 4030/58863869,\na[5,2]=0,\na[5,3]=-279389250/58863869,\na[5,4]=2335230 00/58863869,\na[6,1]=-107221818666951807169/25619615678223628160,\na[6 ,2]=0,\na[6,3]=328927584642593578425/20495692542578902528,\na[6,4]=-37 17245049788716341975/307755633334661333272,\na[6,5]=877136421922273048 21351/98481802667091626647040,\na[7,1]=223899211079219075995406125/200 66238404586623311945344,\na[7,2]=0,\na[7,3]=-27381293369066082540625/6 22049813755756227072,\na[7,4]=2493079698743912733446875/71342417962706 618257992,\na[7,5]=-2021984654539866742602494375/889078038644174940599 689728,\na[7,6]=999354901744231797500/846274121500481947287,\na[8,1]=1 5677672759360359499/1267222156368417000,\na[8,2]=0,\na[8,3]=-414775576 646775/8485270768736,\na[8,4]=45606172048348599823275/1175113514395365 856852,\na[8,5]=-321038818302974478087638483/1239156469179360125372059 20,\na[8,6]=8496514166784579485/6535296976627611297,\na[8,7]=-85744588 6276749/93424049779207625,\na[9,1]=10645627/477900000,\na[9,2]=0,\na[9 ,3]=0,\na[9,4]=4717085625/17726544496,\na[9,5]=733457058173503507/2441 724811071084000,\na[9,6]=3157306853548189/13624786136653650,\na[9,7]=1 5203797026579/11457084981250,\na[9,8]=-407713/355300,\n\nb[1]=10645627 /477900000,\nb[2]=0,\nb[3]=0,\nb[4]=4717085625/17726544496,\nb[5]=7334 57058173503507/2441724811071084000,\nb[6]=3157306853548189/13624786136 653650,\nb[7]=15203797026579/11457084981250,\nb[8]=-407713/355300,\n`b *`[1]=15760837607929/1669012981875000,\n`b*`[2]=0,\n`b*`[3]=0,\n`b*`[4 ]=1099642053438585/3714479960779514,\n`b*`[5]=207109114230528859818133 9/8527454295551205104475000,\n`b*`[6]=113442572384451293690689/3806644 89430348936942500,\n`b*`[7]=824482795143047119317/96030250185317468750 0,\n`b*`[8]=-5198669774391/7445076113750,\n`b*`[9]=-1/150\}:" }}} {PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 5 " Let " }{XPPEDIT 18 0 "T[6, 8];" "6#&%\"TG6$\"\"'\"\")" }{TEXT -1 127 " denote the vector whose components are the principal error terms of the 8 stage, order 6 scheme (the error terms of order 7)." }}{PARA 0 "" 0 "" {TEXT -1 5 "Let " }{XPPEDIT 18 0 "`T*`[5,9];" "6#&%#T*G6$\"\" &\"\"*" }{TEXT -1 145 " denote the vector whose components are the pr incipal error terms of the embedded 9 stage, order 5 scheme (the error terms of order 6) and let " }{XPPEDIT 18 0 "`T*`[6,9];" "6#&%#T*G6$ \"\"'\"\"*" }{TEXT -1 99 " denote the vector whose components are the error terms of order 7 of the embedded order 5 scheme." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 39 "Denote the 2-norm of these vectors by " }{XPPEDIT 18 0 "abs(abs(T[6,8]));" "6#-%$absG6#-F $6#&%\"TG6$\"\"'\"\")" }{TEXT -1 3 ", " }{XPPEDIT 18 0 "abs(abs(`T*`[ 5,9]));" "6#-%$absG6#-F$6#&%#T*G6$\"\"&\"\"*" }{TEXT -1 7 " and " } {XPPEDIT 18 0 "abs(abs(`T*`[6,9]));" "6#-%$absG6#-F$6#&%#T*G6$\"\"'\" \"*" }{TEXT -1 15 " respectively." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }} {PARA 0 "" 0 "" {TEXT -1 8 "Define: " }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "A[7] = abs(abs(T[6, 8]));" "6#/&%\"AG6#\"\"(-%$absG 6#-F)6#&%\"TG6$\"\"'\"\")" }{TEXT -1 6 ", " }{XPPEDIT 18 0 "B[7] = abs(abs(`T*`[6,9]))/abs(abs(`T*`[5,9]));" "6#/&%\"BG6#\"\"(*&-%$absG6 #-F*6#&%#T*G6$\"\"'\"\"*\"\"\"-F*6#-F*6#&F/6$\"\"&F2!\"\"" }{TEXT -1 6 ", " }{XPPEDIT 18 0 "C[7] = abs(abs(`T*`[6,9]-T[6, 8]))/abs(abs( `T*`[5, 9]));" "6#/&%\"CG6#\"\"(*&-%$absG6#-F*6#,&&%#T*G6$\"\"'\"\"*\" \"\"&%\"TG6$F2\"\")!\"\"F4-F*6#-F*6#&F06$\"\"&F3F9" }{TEXT -1 1 "." }} {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 77 "Dormand a nd Prince have suggested that as well as attempting to ensure that " }{XPPEDIT 18 0 "A[7];" "6#&%\"AG6#\"\"(" }{TEXT -1 73 " is a minimum, if the embedded scheme is to be used for error control, " }{XPPEDIT 18 0 "B[7];" "6#&%\"BG6#\"\"(" }{TEXT -1 7 " and " }{XPPEDIT 18 0 "C [7];" "6#&%\"CG6#\"\"(" }{TEXT -1 27 " should be chosen so that " }} {PARA 0 "" 0 "" {TEXT -1 68 "they are \"similar in magnitude\" and als o not differ too much from 1." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 186 "errterms6_8 := PrincipalErr orTerms(6,8,'expanded'):\n`errterms6_9*` :=subs(b=`b*`,PrincipalErrorT erms(6,9,'expanded')):\n`errterms5_9*` := subs(b=`b*`,PrincipalErrorTe rms(5,9,'expanded')):" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 326 "snmB := sqrt(add(evalf(subs(ee,`er rterms6_9*`[i]))^2,i=1..nops(`errterms6_9*`))):\nsdnB := sqrt(add(eval f(subs(ee,`errterms5_9*`[i]))^2,i=1..nops(`errterms5_9*`))):\nsnmC := \+ sqrt(add((evalf(subs(ee,`errterms6_9*`[i])-subs(ee,errterms6_8[i])))^2 ,i=1..nops(errterms6_8))):\n'B[7]'= evalf[8](snmB/sdnB);\n'C[7]'= eval f[8](snmC/sdnB);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/&%\"BG6#\"\"($\") LM[8!\"(" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/&%\"CG6#\"\"($\")[J%Q\"! \"(" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}}{PARA 0 "" 0 "" {TEXT -1 52 "#----------------- ----------------------------------" }}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 27 "absolute stability regions " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}{SECT 1 {PARA 4 "" 0 " " {TEXT -1 35 "coefficients of the combined scheme" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1880 "ee := \{c[ 2]=1/60,\nc[3]=4/45,\nc[4]=2/15,\nc[5]=180/389,\nc[6]=177/262,\nc[7]=1 25/126,\nc[8]=1,\nc[9]=1,\n\na[2,1]=1/60,\na[3,1]=-4/27,\na[3,2]=32/13 5,\na[4,1]=1/30,\na[4,2]=0,\na[4,3]=1/10,\na[5,1]=73104030/58863869,\n a[5,2]=0,\na[5,3]=-279389250/58863869,\na[5,4]=233523000/58863869,\na[ 6,1]=-107221818666951807169/25619615678223628160,\na[6,2]=0,\na[6,3]=3 28927584642593578425/20495692542578902528,\na[6,4]=-371724504978871634 1975/307755633334661333272,\na[6,5]=87713642192227304821351/9848180266 7091626647040,\na[7,1]=223899211079219075995406125/2006623840458662331 1945344,\na[7,2]=0,\na[7,3]=-27381293369066082540625/62204981375575622 7072,\na[7,4]=2493079698743912733446875/71342417962706618257992,\na[7, 5]=-2021984654539866742602494375/889078038644174940599689728,\na[7,6]= 999354901744231797500/846274121500481947287,\na[8,1]=15677672759360359 499/1267222156368417000,\na[8,2]=0,\na[8,3]=-414775576646775/848527076 8736,\na[8,4]=45606172048348599823275/1175113514395365856852,\na[8,5]= -321038818302974478087638483/123915646917936012537205920,\na[8,6]=8496 514166784579485/6535296976627611297,\na[8,7]=-857445886276749/93424049 779207625,\na[9,1]=10645627/477900000,\na[9,2]=0,\na[9,3]=0,\na[9,4]=4 717085625/17726544496,\na[9,5]=733457058173503507/2441724811071084000, \na[9,6]=3157306853548189/13624786136653650,\na[9,7]=15203797026579/11 457084981250,\na[9,8]=-407713/355300,\n\nb[1]=10645627/477900000,\nb[2 ]=0,\nb[3]=0,\nb[4]=4717085625/17726544496,\nb[5]=733457058173503507/2 441724811071084000,\nb[6]=3157306853548189/13624786136653650,\nb[7]=15 203797026579/11457084981250,\nb[8]=-407713/355300,\n`b*`[1]=1576083760 7929/1669012981875000,\n`b*`[2]=0,\n`b*`[3]=0,\n`b*`[4]=10996420534385 85/3714479960779514,\n`b*`[5]=2071091142305288598181339/85274542955512 05104475000,\n`b*`[6]=113442572384451293690689/38066448943034893694250 0,\n`b*`[7]=824482795143047119317/960302501853174687500,\n`b*`[8]=-519 8669774391/7445076113750,\n`b*`[9]=-1/150\}:" }}}{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 8 stage, order 6 scheme is given as follows." }} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 75 "subs(ee,StabilityFunction(6, 8,'expanded')):\nR := unapply(%,z):\n'R(z)'=R(z);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/-%\"RG6#%\"zG,4\"\"\"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)*&#\"+8M!H6\"\".+%3VeFeF)*$)F'\" \"(F)F)F)*&#\"+f_*G6\"\"/+CRFM5kF)*$)F'\"\")F)F)F)" }}}{PARA 0 "" 0 " " {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT 260 4 "Note" }{TEXT -1 63 ": T he scheme has been constructed so that the coefficients of " } {XPPEDIT 18 0 "z^6" "6#*$%\"zG\"\"'" }{TEXT -1 7 " and " }{XPPEDIT 18 0 "z^7" "6#*$%\"zG\"\"(" }{TEXT -1 30 " are approximately equal to " }{XPPEDIT 18 0 "11/57600;" "6#*&\"#6\"\"\"\"&+w&!\"\"" }{TEXT -1 7 " and " }{XPPEDIT 18 0 "1/57600;" "6#*&\"\"\"F$\"&+w&!\"\"" } {TEXT -1 15 " respectively." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 61 "1112903413/5827584308400,111 2895259/64103427392400;\nevalf(%);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6$ #\"+8M!H6\"\".+%3VeFe#\"+f_*G6\"\"/+CRFM5k" }}{PARA 11 "" 1 "" {XPPMATH 20 "6$$\"+Qlr4>!#8$\"+CK4O " 0 "" {MPLTEXT 1 0 27 "11/57600,1/57600;\nev alf(%);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6$#\"#6\"&+w&#\"\"\"F%" }} {PARA 11 "" 1 "" {XPPMATH 20 "6$$\"+AAs4>!#8$\"+666O " 0 "" {MPLTEXT 1 0 28 "z0 := newton(R(z)=1,z=- 4.9);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%#z0G$!+pDI!*[!\"*" }}} {PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 303 "z0 := newton(R(z)=1,z=-4.9):\np1 := plot([R(z),1],z=-5.29..0.49,c olor=[red,blue]):\np2 := plot([[[z0,1]]$3],style=point,symbol=[circle, cross,diamond],color=black):\np3 := plot([[z0,0],[z0,1]],linestyle=3,c olor=COLOR(RGB,0,.5,0)):\nplots[display]([p1,p2,p3],view=[-5.29..0.49, -.07..1.47],font=[HELVETICA,9]);" }}{PARA 13 "" 1 "" {GLPLOT2D 396 256 256 {PLOTDATA 2 "6+-%'CURVESG6$7Z7$$!3/++++++!H&!#<$\"3kaD,ooh\"3# F*7$$!3Wn\"HA6.&e_F*$\"3G@y(o:t)o>F*7$$!3'RLeWA1qA&F*$\"3;GC4RXeh=F*7$ $!3Y+voO$4b>&F*$\"3Yj]qa@_f?4^F*$\"3/Le?aT([]\"F*7$$!3)H$3Pa8Ra]F*$\"3q/B#3XM2O\"F*7$$!3K]i] h6v#*\\F*$\"3I!R0AZjM@\"F*7$$!3wm;ko46J\\F*$\"3)=a3m^<13\"F*7$$!3Gnmc% =i!p[F*$\"3y%)[Dh8:-'*!#=7$$!3#pm\"\\+M,2[F*$\"3z00#)o'FZ7$$!3%o;/$ [\")**oXF*$\"3z\"zr?xL&=`FZ7$$!3!)*\\PlEK/X%F*$\"3Ex0p\">@t<%FZ7$$!3Wm T+i7\"yK%F*$\"3H#>)H.?NUKFZ7$$!3j*\\(e/Ne0UF*$\"35bPZ.ox8DFZ7$$!3aLLQK s&)zSF*$\"3\"H%*y%\\_KO>FZ7$$!3gm;Rju6pRF*$\"3;'4Agm!3X:FZ7$$!3c***\\@ r]W%QF*$\"3h*f&p(G_3@\"FZ7$$!3#****\\s1s#>PF*$\"3=\"Q[Ms&>j'*!#>7$$!3a ***\\yCR')f$F*$\"3#f$z;#fOU*zF\\r7$$!3UmT+\"4$4*[$F*$\"3o5hmg=\\ZpF\\r 7$$!3cLL)\\6K)eLF*$\"3CH,!=aXx;'F\\r7$$!35LL$>)R[[KF*$\"3bxo]l>yEeF\\r 7$$!37+vBqG7?JF*$\"3\\].2u(3zs&F\\r7$$!3SLLVO:]1IF*$\"3;W(fuB\"yheF\\r 7$$!3!**\\P_#4%=)GF*$\"3%>IY/R[.@'F\\r7$$!3!**\\7PaMJw#F*$\"3DH[GpB79n F\\r7$$!3CL$3x)oFREF*$\"3pvJH\"eWBS(F\\r7$$!3DL3K&*o`DDF*$\"3gm,1u#3P< )F\\r7$$!3pm;zZH&GS#F*$\"31L$Rr-iV:*F\\r7$$!3omT+1)=aF#F*$\"31#y;4UbT. \"FZ7$$!31+D'4,([k@F*$\"3)GQpg)[N_6FZ7$$!3Hm;CLwnW?F*$\"312`^dw&oH\"FZ 7$$!3,++qAG!4#>F*$\"3E4T\"F*$\"3o.;8zqp=IFZ7$$!3o***\\L2YT2\"F*$\"3Agj+i9$fT$FZ7$$!3)* Q$3dRD\"y&*FZ$\"3r%=D>.Lt$QFZ7$$!3-'*\\(yw/@O)FZ$\"3IG%*H3Y]LVFZ7$$!3e LL$eLAK<(FZ$\"3%HlnHYx0)[FZ7$$!3j/]P4EdGfFZ$\"3a'or:*yXFbFZ7$$!3=fmm#* p#)HZFZ$\"3W+&Gwv39B'FZ7$$!3)ym;4YOR]$FZ$\"3K31W*\\2T/(FZ7$$!3+K$3<#p> )G#FZ$\"3;kv)e]>Z&zFZ7$$!3M#****pae5<\"FZ$\"3+G[v8.\"\\*))FZ7$$\"3/tm; %*43$4\"F\\r$\"3#pS9&p2*4,\"F*7$$\"3![LL8H$[a7FZ$\"3'>\\C/fcO8\"F*7$$ \"3%3+Dw*)yaZ#FZ$\"3Mu)[9q!)3G\"F*7$$\"3*=+v$)[FTk$FZ$\"35-r(RIo'R9F*7 $$\"3!***************[FZ$\"3F_M/ShJK;F*-%'COLOURG6&%$RGBG$\"*++++\"!\" )$\"\"!Fd]lFc]l-F$6$7S7$F($\"\"\"Fd]l7$F=Fi]l7$FGFi]l7$FQFi]l7$FfnFi]l 7$F`oFi]l7$FeoFi]l7$FjoFi]l7$F_pFi]l7$FdpFi]l7$FipFi]l7$F^qFi]l7$FcqFi ]l7$FhqFi]l7$F^rFi]l7$FcrFi]l7$FhrFi]l7$F]sFi]l7$FbsFi]l7$FgsFi]l7$F\\ tFi]l7$FatFi]l7$FftFi]l7$F[uFi]l7$F`uFi]l7$FeuFi]l7$FjuFi]l7$F_vFi]l7$ FdvFi]l7$FivFi]l7$F^wFi]l7$FcwFi]l7$FhwFi]l7$F]xFi]l7$FbxFi]l7$FgxFi]l 7$F\\yFi]l7$FayFi]l7$FfyFi]l7$F[zFi]l7$F`zFi]l7$FezFi]l7$FjzFi]l7$F_[l Fi]l7$Fd[lFi]l7$Fi[lFi]l7$F^\\lFi]l7$Fc\\lFi]l7$Fh\\lFi]l-F]]l6&F_]lFc ]lFc]lF`]l-F$6&7#7$$!39+++pDI!*[F*Fi]l-%'SYMBOLG6#%'CIRCLEG-F]]l6&F_]l Fd]lFd]lFd]l-%&STYLEG6#%&POINTG-F$6&F_al-Fdal6#%&CROSSGFgalFial-F$6&F_ al-Fdal6#%(DIAMONDGFgalFial-F$6%7$7$FaalFc]lF`al-%&COLORG6&F_]lFc]l$\" \"&!\"\"Fc]l-%*LINESTYLEG6#\"\"$-%+AXESLABELSG6%Q\"z6\"Q!Ficl-%%FONTG6 #%(DEFAULTG-F\\dl6$%*HELVETICAG\"\"*-%%VIEWG6$;$!$H&!\"#$\"#\\Fidl;$! \"(Fidl$\"$Z\"Fidl" 1 2 0 1 10 0 2 9 1 4 2 1.000000 45.000000 45.000000 0 0 "Curve 1" "Curve 2" "Curve 3" "Curve 4" "Curve 5" "Curve 6" }}}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 50 "T he following picture shows the stability region. " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1710 "R := z -> \+ 1+z+1/2*z^2+1/6*z^3+1/24*z^4+1/120*z^5+1/720*z^6+\n 111290341 3/5827584308400*z^7+1112895259/64103427392400*z^8:\npts := []: z0 := 0 : tt := 0: \nwhile tt<=241/20 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 (213/20<=tt an d tt<=233/20) then\n hh := 1/40\n else \n hh := 1/20\n e nd if;\n tt := tt+hh;\n pts := [op(pts),[Re(zz),Im(zz)]]:\nend do: \np1 := plot(pts,color=COLOR(RGB,.05,.4,.13)):\np2 := plots[polygonplo t]([seq([pts[i-1],pts[i],[-2.45,0]],i=2..nops(pts))],\n style =patchnogrid,color=COLOR(RGB,.1,.8,.25)):\npts := []: z0 := 1.8+4.7*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,. 05,.4,.13)):\np4 := plots[polygonplot]([seq([pts[i-1],pts[i],[1.68,4.6 8]],i=2..nops(pts))],\n style=patchnogrid,color=COLOR(RGB,.1, .8,.25)):\npts := []: z0 := 1.8-4.7*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<=t t 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,.05,.4,.13)):\np6 := plots[polygonpl ot]([seq([pts[i-1],pts[i],[1.68,-4.68]],i=2..nops(pts))],\n s tyle=patchnogrid,color=COLOR(RGB,.1,.8,.25)):\np7 := plot([[[-5.49,0], [2.09,0]],[[0,-5.19],[0,5.19]]],color=black,linestyle=3):\nplots[displ ay]([p||(1..7)],view=[-5.49..2.09,-5.19..5.19],font=[HELVETICA,9],\n \+ labels=[`Re(z)`,`Im(z)`],axes=boxed,scaling=constrained); " }}{PARA 13 "" 1 "" {GLPLOT2D 625 529 529 {PLOTDATA 2 "6/-%'CURVESG6$ 7f\\l7$$\"\"!F)F(7$F($\"3++++Fjzq:!#=7$F($\"3u*****4j#fTJF-7$F($\"3C++ +&e)Q7ZF-7$$!3++++l?YeO!#F$\"3o*****>F#=$G'F-7$$!3)******\\LdD<&!#E$\" 3q*****RpmR&yF-7$$!3G+++\\DK5X!#D$\"3_+++]AsC%*F-7$$!33+++(*Hl/G!#C$\" 3%******fBS&*4\"!#<7$$!3]+++wJ#GM'FI$\"3#******Hio!y6FL7$$!3++++>z*)e8 !#B$\"33+++%z!fc7FL7$$!35+++LvnvFFU$\"32+++#*R5N8FL7$$!3V+++'[@TV&FU$ \"3.+++pWg89FL7$$!3(******4r9U-\"!#A$\"31+++3s3#\\\"FL7$$!3%******R&QT l=F_o$\"3'*******)eX0d\"FL7$$!3K+++2-]$H$F_o$\"3#******R*4(*[;FL7$$!3' ******f\\K@l&F_o$\"3&*******eBNFY#Fip$\"33+++\"4u?' >FL7$$!31+++H#3#[QFip$\"3A+++\"[,,/#FL7$$!3m*****RFG\")*eFip$\"3')**** *>Juz6#FL7$$!39+++E'HR())Fip$\"39+++Adl&>#FL7$$!34+++\"Q0=J\"!#?$\"3/+ ++fH5tAFL7$$!3,+++))p(o!>Fcr$\"3#******p?n-N#FL7$$!3\"*******oqcFFFcr$ \"3++++$e$4FCFL7$$!3.+++4&y8%QFcr$\"3'******zG@N]#FL7$$!3()******etZH` Fcr$\"35+++jQ[zDFL7$$!3-+++P+N(G(Fcr$\"31+++_%4\\l#FL7$$!3!3++!3K#[#)* Fcr$\"3')******f4sHFFL7$$!3$******>6GlI\"!#>$\"36+++Pf$Q!GFL7$$!3:+++L OR9)Q]?AFgt$\"31+++Mih\\HFL7$$!3/+++.W zRGFgt$\"3!*******4I3@IFL7$$!3w*****\\$4?(e$Fgt$\"3#)*****H2`94$FL7$$! 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" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 148 "We can distort the boundary curve horizontally by taking the 11th root of the real part of points along the curve. In this way we see t hat there is " }{TEXT 260 53 "no largest interval on the nonnegative i maginary axis" }{TEXT -1 65 " that contains the origin and lies inside the stability region. " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 382 "R := z -> 1+z+1/2*z^2+1/6*z^3+1/24 *z^4+1/120*z^5+1/720*z^6+\n 1112903413/5827584308400*z^7+1112 895259/64103427392400*z^8:\nDigits := 25:\npts := []: z0 := 0:\nfor ct from 0 to 120 do\n zz := newton(R(z)=exp(ct*Pi/100*I),z=z0):\n z0 := zz:\n pts := [op(pts),[surd(Re(zz),11),Im(zz)]]:\nend do:\nplot( pts,color=COLOR(RGB,0,.75,.25),thickness=2,font=[HELVETICA,9]);\nDigit s := 10:" }}{PARA 13 "" 1 "" {GLPLOT2D 222 299 299 {PLOTDATA 2 "6(-%'C URVESG6#7er7$$\"\"!F)F(7$$!:nl&GpzL=Z=AE6!#E$\":-1Xh2x*e`EfTJF-7$$!:)y 'Rk4t_5ss]'=F-$\":W,N9$3n<2`=$G'F-7$$!:*HJ>z]Pft3D3DF-$\":;_F\"4s,sgzx C%*F-7$$!:V32\"z!H$=qyN.JF-$\":W!Hg1?!*Rhqjc7!#D7$$!:9w_l%>:\\e(*)on$F -$\":ppn>Jv=mK'zq:F?7$$!:(=E%*\\s!=ag/'[UF-$\":\"z/G(e#=_\"fb\\)=F?7$$ !:R4b(f*H*ok(4H$[F-$\":!)3Py')H]c&[6*>#F?7$$!:\\=lX'\\lYl&=!QaF-$\":`# e.l(e>\"=TF8DF?7$$!:Okf\\V+DDD%RmgF-$\":pq>$3G$otPLu#GF?7$$!:OE!4Iz$o0 'pq;nF-$\":/BT*)\\XA3j#fTJF?7$$!:5>_dT5$H\"fmfQ(F-$\":^PkP$**)eW(=vbMF ?7$$!:)\\@u!yiwDCG42)F-$\":#)3)\\OvKD-6\"*pPF?7$$!:dVO8#3_j`$G(o()F-$ \":^M`OQt%\\0.2%3%F?7$$!:Ib()38\"\\['>`qZ*F-$\":P#fOpj(\\>ZH#)R%F?7$$! :rv?jbWl4F5%>5F?$\":vb+Fa7b[e)Q7ZF?7$$!:\"p&o;;O@@J\\=4\"F?$\":UuCS$z# ='*R&)Q#e\"\\;\"F?$\":3X())3<;!Hb12M&F?7$$!:jD3$ybv`2 o_Q7F?$\":iB65L]k*R`'[l&F?7$$!:9&)HG8(Q\"F?$\":I;3pt#p2sA=$G'F?7$$!:5v/Tz6e29K?Y\"F?$\":>+kgl qEtFStf'F?7$$!:;Qgol;>%4)ys`\"F?$\":a\")ogxHZL&y\\6pF?7$$!:a@Lw;f'pHD% Gh\"F?$\":$)>i_&GE<#)[lDsF?7$$!:RSq\"HP[]asp)o\"F?$\":W+xP2D)=?7\")RvF ?7$$!:/vODCH?!o-#[w\"F?$\":/Vi0or,QpmR&yF?7$$!:/bXRQh,L\\\">T=F?$\":^C MuuT[X4@\"o\")F?7$$!:v:Ur=aI%pHz<>F?$\":g)3@J(\\uNF?$\":?C/T!ePVNcU'z)F?7$$!:\"zq@STx#G5B;2#F?$\":ws'>EAF?$\":$ *4(*yzQ,F^m)Q(*F?7$$!:!eC]rWOhU+t.BF?$\":J$=emIz\"et+`+\"!#C7$$!:**H'= >p+/9OT\"Q#F?$\":Mw]4t!zi8WrO5Ffu7$$!:s'o)[(3S&3IO#fCF?$\":x&*G8b,:%4w 7o5Ffu7$$!:vR1gXxi/+)=PDF?$\":2R-$[b2MO-a*4\"Ffu7$$!:15@;.6K#Q!f_h#F?$ \":6')em_48\"Ffu7$$!:Z%>IPb&yX5SMp#F?$\":g6IrM))>'pLOi6Ffu7$$!:.6 nVW^\\-?A_mqIOx$>\"Ffu7$$!:GaM\\w2^!)e'4]GF?$\":*R7\"\\M c>(GG=D7Ffu7$$!:S#z?)f&[l\\ZbGHF?$\":BbK*y6=<%z!fc7Ffu7$$!:IO+x?Z(RY$) 32IF?$\":GDJU:\">!\\L(*zG\"Ffu7$$!:\"RBO.@L?$=*o&3$F?$\":@_NTii(eLAS>8 Ffu7$$!:c\"exPv(H\\<\\V;$F?$\":v$z+UVE0X_!3N\"Ffu7$$!:R^E\"*GcmXJgIC$F ?$\":B=Qa@>dL41AQ\"Ffu7$$!:HP`hXuEGk9=K$F?$\":\">@'[Qtk'oWg89Ffu7$$!:O #GG6mBv=Ug+MF?$\":q]o$H[!yQ-+]W\"Ffu7$$!:%e^%Q`bG54@%zMF?$\":$Q[>+8&R5 P#Rw9Ffu7$$!:P*f1;_%f(psDeNF?$\":T?*Qrf@\\x5y2:Ffu7$$!:'3T)Hm$3^)p/rj$ F?$\":+KoO=@+=ml\"R:Ffu7$$!:X_vok#*pkAbfr$F?$\":xUx[?GI%*eX0d\"Ffu7$$! :!p\\R=)H91e+[z$F?$\":#f.pKes>o-#>g\"Ffu7$$!:8w9#[V!*GL;Ffu7$$!:3^Ym\"Ffu7$$!:\\x]MhpG /X?7.%F?$\":[E_1z.iF#f+'p\"Ffu7$$!:@*[\\b&*)=o&*e*4TF?$\":f]>EFzj$fBNF KCWF?$\":5@i6gb0znME&=Ffu7$$!:@Itr`2v4S9F]%F?$\":BPS2\"*p\"HMM# R)=Ffu7$$!:+%=NAcHo?h+\"e%F?$\":1+tpo8fcq'>:>Ffu7$$!:Z6(etY[N?c=fYF?$ \":WH'zD6YQ8HXY>Ffu7$$!:6XAq+xSq+Tst%F?$\":MaP`AXT^N!px>Ffu7$$!:\"*4yx OEZB#*f^\"[F?$\":laCj2*3?Ffu7$$!:;[pUmy*yB&HH*[F?$\":6FNf0mm7[,, /#Ffu7$$!:?DxDj84)ok`q\\F?$\":Beq`G)y/'4r72#Ffu7$$!:%*33$)fed=!p'z/&F? 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$\":q$fkFXBU]GFuJFfu7$$!:Zv**R!HZ]i\"3wi(F?$\":5-\"QPYvnvxW,KFfu7$$!:A -N&pap%4\"p0&o(F?$\":]RdS,Si\\H\"RGKFfu7$$!:cCF)fB2qn_kTxF?$\":v.O>8E( Q=O4bKFfu7$$!:$4T[@/?+E-O(z(F?$\":W?elgw.XpW:G$Ffu7$$!:x*oMrjzl-$*=_yF ?$\":?&>z/W.w[Ut2LFfu7$$!:#\\j:4o\"pDh?h!zF?$\":*[;LW`[(=$=lLLFfu7$$!: D/v8\\Q2o&H9fzF?$\":=#fy44f.yoGfLFfu7$$!:I(\\biV1'H#fC6!)F?$\":qfG1(>! fCxGYQ$Ffu7$$!:lZIhWl$p&**>C1)F?$\":[U&yY\"39u#pm4MFfu7$$!:j#p@[SPuuml 7\")F?$\":2;)F?$\":X*R\">6d#)zI!zeMF fu7$$!:$f#Qt%o'[#e'*G5#)F?$\":$)3BCu.nwEbG[$Ffu7$$!:9bR/T3$zJ^nd#)F?$ \":>joF%[._Thd1NFfu7$$!:g(4`<0u8x<5/$)F?$\":E#=&*fz$zI!Q%*HNFfu7$$!:&* z5y8,Cp$zc\\$)F?$\":5$)HpmjO1p\\Hb$Ffu7$$!:8VWifginhtSR)F?$\":QNRB(z'f #*)eevNFfu7$$!:LS!o(Ht7^a?wV)F?$\":A`%z\"eX@V?Xyf$Ffu-%+AXESLABELSG6$Q !6\"Fbam-%%FONTG6$%*HELVETICAG\"\"*-%&COLORG6&%$RGBGF($\"#v!\"#$\"#DF_ bm-%*THICKNESSG6#\"\"#-%%VIEWG6$%(DEFAULTGFibm" 1 2 0 1 10 2 2 6 1 4 2 1.000000 46.000000 45.000000 0 0 "Curve 1" }}}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 37 "#------------------------ ------------" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 80 "The stability function R* for the 9 stage, order 5 scheme is \+ given as follows." }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 97 "subs(ee ,subs(b=`b*`,StabilityFunction(5,9,'expanded'))):\n`R*` := unapply(%,z ):\n'`R*`(z)'=`R*`(z);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/-%#R*G6#%\" zG,6\"\"\"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)*&#\"0$fHY_HeE \"3+q-!==BV'=F)*$)F'F1F)F)F)*&#\"28%=Ihf*=6$\"6++eBs(yb " 0 "" {MPLTEXT 1 0 33 "z_0 := newton(`R*`(z)=-1,z=-4.9);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%$z_0G$!+moz<\\!\"*" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 324 "z_0 := newton(`R*`(z)=-1,z=-4.9):\np_1 := plot([`R*`(z),-1],z=-5.39..0.49,co lor=[red,blue]):\np_2 := plot([[[z_0,-1]]$3],style=point,symbol=[circl e,cross,diamond],color=black):\np_3 := plot([[z_0,0],[z_0,-1]],linesty le=3,color=COLOR(RGB,0,.5,0)):\nplots[display]([p_1,p_2,p_3],view=[-5. 39..0.49,-1.47..1.47],font=[HELVETICA,9]);" }}{PARA 13 "" 1 "" {GLPLOT2D 363 274 274 {PLOTDATA 2 "6+-%'CURVESG6$7S7$$!3o************* Q&!#<$!3$frEWbk&R>F*7$$!3[****\\LF$=E&F*$!3&R8Tz^Pgj\"F*7$$!3'***\\A!3 :.:&F*$!3W\"Gs%G(yHS\"F*7$$!3%)***\\e\"=!\\-&F*$!3I@:0,6_s6F*7$$!3g*** \\4Bd')*[F*$!3#oG%F*$!3(op(f=iB1MFA7$$!3 W****H10#*eTF*$!3\"[A&>3.jIEFA7$$!3o***\\V#[EYSF*$!3Wke!*=r>f?FA7$$!3Z *****)37W>RF*$!3m#**oX!3@B:FA7$$!3a****\\Vo4#z$F*$!3Il9T1a3#3\"FA7$$!3 Q****4^pPpOF*$!3bfkicP%yL(!#>7$$!3o**\\-Kb$zb$F*$!3o=oT()Gi;ZF^p7$$!3P *****e!4UDMF*$!3!*oAf8B5Y@F^p7$$!3g****fNO;8LF*$!3-]`)e&**H9M!#?7$$!3m **\\UVGF*$\"3]&Q()[gk%*>&F^p7$$!3 W***\\ii;Mp#F*$\"3tS:u;?6HjF^p7$$!3u**\\#R\")3xd#F*$\"3'R+Lk9->O(F^p7$ $!3)****\\nI-HX#F*$\"3>>#[!\\cU7&)F^p7$$!3p**\\-AMEBBF*$\"3+Ro.y'4*)y* F^p7$$!3#)**\\x\"R7/@#F*$\"3O)>JYq$))*4\"FA7$$!3_***\\u=I&)3#F*$\"3Zq( *)yqERC\"FA7$$!39++?WRhi>F*$\"3-z8#oj*\\59FA7$$!3()***\\cYH%R=F*$\"3Sh 4POw9%f\"FA7$$!3p**\\i[@C?UT%z\"FA7$$!3')***\\R&\\!ze\"F* $\"3%yhG,@Xm/#FA7$$!33++g)4%**o9F*$\"3(p;BYu)z.BFA7$$!3%)****\\z8.U8F* $\"3![&e\"H.-Yh#FA7$$!3&***\\i&G%)pA\"F*$\"3O9:mVfoKHFA7$$!3<****4Cu?, 6F*$\"3!H>>v9B_K$FA7$$!3K)**\\U*>hG)*FA$\"3dugDT#FA$\"3C,9.#p/k&yFA7$$!3Q% ****>;%4w7FA$\"3Xjk0H5(>!))FA7$$\"3U^/+]cTUEF^q$\"3h;t!*4fk-5F*7$$\"3w ,++o?T\">\"FA$\"3+oGr!)*Gl7\"F*7$$\"3J0+vl@`LCFA$\"39DX:'3>bF\"F*7$$\" 3-0+D!f*RAOFA$\"3aHx*[oVlV\"F*7$$\"3!***************[FA$\"3k6i0EmJK;F* -%'COLOURG6&%$RGBG$\"*++++\"!\")$\"\"!Fb[lFa[l-F$6$7S7$F($!\"\"Fb[l7$F .Fg[l7$F3Fg[l7$F8Fg[l7$F=Fg[l7$FCFg[l7$FHFg[l7$FMFg[l7$FRFg[l7$FWFg[l7 $FfnFg[l7$F[oFg[l7$F`oFg[l7$FeoFg[l7$FjoFg[l7$F`pFg[l7$FepFg[l7$FjpFg[ l7$F`qFg[l7$FeqFg[l7$FjqFg[l7$F_rFg[l7$FdrFg[l7$FirFg[l7$F^sFg[l7$FcsF g[l7$FhsFg[l7$F]tFg[l7$FbtFg[l7$FgtFg[l7$F\\uFg[l7$FauFg[l7$FfuFg[l7$F [vFg[l7$F`vFg[l7$FevFg[l7$FjvFg[l7$F_wFg[l7$FdwFg[l7$FiwFg[l7$F^xFg[l7 $FcxFg[l7$FhxFg[l7$F]yFg[l7$FbyFg[l7$FgyFg[l7$F\\zFg[l7$FazFg[l7$FfzFg [l-F[[l6&F][lFa[lFa[lF^[l-F$6&7#7$$!3x*****f'oz<\\F*Fg[l-%'SYMBOLG6#%' CIRCLEG-F[[l6&F][lFb[lFb[lFb[l-%&STYLEG6#%&POINTG-F$6&F]_l-Fb_l6#%&CRO SSGFe_lFg_l-F$6&F]_l-Fb_l6#%(DIAMONDGFe_lFg_l-F$6%7$7$F__lFa[lF^_l-%&C OLORG6&F][lFa[l$\"\"&Fh[lFa[l-%*LINESTYLEG6#\"\"$-%+AXESLABELSG6%Q\"z6 \"Q!Ffal-%%FONTG6#%(DEFAULTG-Fial6$%*HELVETICAG\"\"*-%%VIEWG6$;$!$R&! \"#$\"#\\Ffbl;$!$Z\"Ffbl$\"$Z\"Ffbl" 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 82 "The following picture shows the stability region for the \+ 9 stage, order 5 scheme. " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1797 "`R*` := z -> 1+z+1/2*z^2+1/6*z^3+ 1/24*z^4+1/120*z^5+\n 265829524629593/186432318180027000*z^6+31118 959613018413/162817557877223580000*z^7+\n 29952403326058547/322378 7645969026884000*z^8-1112895259/9615514108860000*z^9:\npts := []: z0 : = 0: tt := 0: \nwhile tt<=201/20 do\n zz := newton(`R*`(z)=exp(tt*Pi *I),z=z0):\n z0 := zz:\n if (13/20<=tt and tt<=29/20) or (171/20<= tt and tt<=187/20) 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)]]:\ne nd do:\np_1 := plot(pts,color=COLOR(RGB,0,.35,.1)):\np_2 := plots[poly gonplot]([seq([pts[i-1],pts[i],[-4,0]],i=2..nops(pts))],\n st yle=patchnogrid,color=COLOR(RGB,0,.7,.2)):\npts := []: z0 := 1.6+4.5*I : tt := 0:\nwhile tt<=61/30 do\n zz := newton(`R*`(z)=exp(tt*Pi*I),z =z0):\n z0 := zz:\n if (1/2<=tt and tt<=7/5) then\n hh := 1/5 0\n else \n hh := 1/25\n end if;\n tt := tt+hh;\n pts := \+ [op(pts),[Re(zz),Im(zz)]]:\nend do:\np_3 := plot(pts,color=COLOR(RGB,0 ,.18,.4)):\np_4 := plots[polygonplot]([seq([pts[i-1],pts[i],[1.47,4.38 ]],i=2..nops(pts))],\n style=patchnogrid,color=COLOR(RGB,0,.7 ,.2)):\npts := []: z0 := 1.6-4.5*I: tt := 0:\nwhile tt<=61/30 do\n z z := newton(`R*`(z)=exp(tt*Pi*I),z=z0):\n z0 := zz:\n if (3/5<=tt \+ and tt<=3/2) then\n hh := 1/50\n else \n hh := 1/25\n en d if;\n tt := tt+hh;\n pts := [op(pts),[Re(zz),Im(zz)]]:\nend do: \np_5 := plot(pts,color=COLOR(RGB,0,.18,.4)):\np_6 := plots[polygonplo t]([seq([pts[i-1],pts[i],[1.47,-4.38]],i=2..nops(pts))],\n st yle=patchnogrid,color=COLOR(RGB,0,.7,.2)):\np_7 := plot([[[-5.49,0],[2 .09,0]],[[0,-5.19],[0,5.19]]],color=black,linestyle=3):\nplots[display ]([p_||(1..7)],view=[-5.49..2.09,-5.19..5.19],font=[HELVETICA,9],\n \+ labels=[`Re(z)`,`Im(z)`],axes=boxed,scaling=constrained);" }} {PARA 13 "" 1 "" {GLPLOT2D 681 553 553 {PLOTDATA 2 "6/-%'CURVESG6$7fy7 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4,1]=1/30,\na[4,2]=0,\na[4,3]=1/10,\na[5,1]=73104030/58863869,\na[5,2] =0,\na[5,3]=-279389250/58863869,\na[5,4]=233523000/58863869,\na[6,1]=- 107221818666951807169/25619615678223628160,\na[6,2]=0,\na[6,3]=3289275 84642593578425/20495692542578902528,\na[6,4]=-3717245049788716341975/3 07755633334661333272,\na[6,5]=87713642192227304821351/9848180266709162 6647040,\na[7,1]=223899211079219075995406125/2006623840458662331194534 4,\na[7,2]=0,\na[7,3]=-27381293369066082540625/622049813755756227072, \na[7,4]=2493079698743912733446875/71342417962706618257992,\na[7,5]=-2 021984654539866742602494375/889078038644174940599689728,\na[7,6]=99935 4901744231797500/846274121500481947287,\na[8,1]=15677672759360359499/1 267222156368417000,\na[8,2]=0,\na[8,3]=-414775576646775/8485270768736, \na[8,4]=45606172048348599823275/1175113514395365856852,\na[8,5]=-3210 38818302974478087638483/123915646917936012537205920,\na[8,6]=849651416 6784579485/6535296976627611297,\na[8,7]=-857445886276749/9342404977920 7625,\na[9,1]=10645627/477900000,\na[9,2]=0,\na[9,3]=0,\na[9,4]=471708 5625/17726544496,\na[9,5]=733457058173503507/2441724811071084000,\na[9 ,6]=3157306853548189/13624786136653650,\na[9,7]=15203797026579/1145708 4981250,\na[9,8]=-407713/355300,\n\nb[1]=10645627/477900000,\nb[2]=0, \nb[3]=0,\nb[4]=4717085625/17726544496,\nb[5]=733457058173503507/24417 24811071084000,\nb[6]=3157306853548189/13624786136653650,\nb[7]=152037 97026579/11457084981250,\nb[8]=-407713/355300,\n`b*`[1]=15760837607929 /1669012981875000,\n`b*`[2]=0,\n`b*`[3]=0,\n`b*`[4]=1099642053438585/3 714479960779514,\n`b*`[5]=2071091142305288598181339/852745429555120510 4475000,\n`b*`[6]=113442572384451293690689/380664489430348936942500,\n `b*`[7]=824482795143047119317/960302501853174687500,\n`b*`[8]=-5198669 774391/7445076113750,\n`b*`[9]=-1/150\}:" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}}{PARA 0 "" 0 " " {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT 259 5 "nodes" }{TEXT -1 1 ":" }}{EXCHG 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"#----------------- ----------------------------------" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}}{PARA 0 "" 0 "" {TEXT -1 59 "#================= =========================================" }}{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 0 770 "SO6_8 := SimpleOrderConditions(6,8,'expanded'):\nSO_eqs := [op(Ro wSumConditions(8,'expanded')),op(StageOrderConditions(2,8,'expanded')) ,\n op(StageOrderConditions(3,4..8,'expanded'))]:\n`SO5_ 9*` := subs(b=`b*`,SimpleOrderConditions(5,9,'expanded')):\nord_cdns : = [seq(SO6_8[i],i=[1,2,4,8,16,24,29,32])]:\n`ord_cdns*` := [seq(`SO5_9 *`[i],i=[1,2,4,8,12,15,16])]:\nsimp_eqs := [add(b[i]*a[i,1],i=1+1..8)= b[1],seq(add(b[i]*a[i,j],i=j+1..8)=b[j]*(1-c[j]),j=[6,7])]:\nwt_eqns : = [seq(b[i]=a[9,i],i=1..8)]:\ncdns := [op(ord_cdns),op(simp_eqs),op(SO _eqs),op(wt_eqns),op(`ord_cdns*`)]:\n\nerrterms6_8 := PrincipalErrorTe rms(6,8,'expanded'):\n`errterms6_9*` :=subs(b=`b*`,PrincipalErrorTerms (6,9,'expanded')):\n`errterms5_9*` := subs(b=`b*`,PrincipalErrorTerms( 5,9,'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 1 " " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1985 "calc_RKcoeffs := proc()\n local eqns,sm,ct,Rz,stb6,stb5,nmB,snmB,dnB,sdnB,nmC,snmC,B _7,C_7,nrm;\n global e1,e2,e3;\n\n e1 := \{c[2]=c_2,c[4]=c_4,c[5]= c_5,c[6]=c_6,c[7]=c_7,c[8]=1,c[9]=1,seq(a[i,2]=0,i=4..8),\n b [2]=0,b[3]=0,`b*`[2]=0,`b*`[3]=0,`b*`[bs[1]]=bs[2]\};\n eqns := subs (e1,cdns):\n e2 := solve(\{op(eqns)\});\n e3 := `union`(e1,e2);\n \+ Digits := 14;\n sm := 0;\n for ct to nops(errterms6_8) do\n \+ sm := sm+(evalf(subs(e3,errterms6_8[ct])))^2;\n end do;\n Rz := s ubs(e3,StabilityFunction(6,8,'expanded'));\n stb6 := max(fsolve(Rz=1 ,z=-9..-1e-7),fsolve(Rz=-1,z=-9..-1e-7));\n stb6 := evalf[8](stb6); \n Rz := subs(e3,subs(b=`b*`,StabilityFunction(5,9,'expanded')));\n \+ stb5 := evalf[8](max(fsolve(Rz=1,z=-9..-1e-7),fsolve(Rz=-1,z=-9..-1e -7)));\n stb5 := evalf[8](stb5);\n nmB := 0;\n for ct to nops(`e rrterms6_9*`) do\n nmB := nmB+evalf(subs(e3,`errterms6_9*`[ct]))^ 2;\n end do:\n snmB := sqrt(nmB);\n dnB := 0;\n for ct to nops (`errterms5_9*`) do\n dnB := dnB+evalf(subs(e3,`errterms5_9*`[ct] ))^2;\n end do;\n sdnB := sqrt(dnB);\n nmC := 0;\n for ct to n ops(errterms6_8) do\n nmC := nmC+(evalf(subs(e3,`errterms6_9*`[ct ]))-evalf(subs(e3,errterms6_8[ct])))^2;\n end do;\n snmC := sqrt(s implify(nmC));\n B_7 := evalf[8](snmB/sdnB);\n C_7 := evalf[8](snm C/sdnB);\n print(`nodes:`,c[2]=c_2,c[3]=subs(e3,c[3]),c[4]=c_4,c[5]= c_5,c[6]=c_6,c[7]=c_7);\n print(`order 6 weights:`,seq(b[i]=evalf[6] (subs(e3,b[i])),i=[1,$4..8]));\n print(`order 5 weights:`,seq(`b*`[i ]=evalf[6](subs(e3,`b*`[i])),i=[1,$4..9]));\n nrm := evalf(max(seq(s eq(subs(e3,abs(a[i,j])),j=1..i-1),i=2..9)));\n print(infinity*`-norm of linking coeffs`=evalf[10](nrm));\n print(`2-norm of principal er ror of order 6 scheme` = evalf[10](sqrt(sm)));\n print(`2-norm of pr incipal error of order 5 scheme` = evalf[10](sdnB));\n print(`order \+ 6 stability interval` = [stb6,0]);\n print(`order 5 stability interv al` = [stb5,0]);\n print('B[7]'=B_7,'C[7]'=C_7);\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 41 "#========================================" }}{PARA 0 "" 0 "" {TEXT -1 19 "Sharp-Verner scheme" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 93 "c_2 := 1/12: c_4 := 1/5: c_5 := 8/15: c_6 := 2/3: c_7 := 19/20: bs := [8,0]:\ncalc_RKcoeffs();" }}{PARA 11 "" 1 "" {XPPMATH 20 "6)%'node s:G/&%\"cG6#\"\"##\"\"\"\"#7/&F&6#\"\"$#F(\"#:/&F&6#\"\"%#F*\"\"&/&F&6 #F7#\"\")F1/&F&6#\"\"'#F(F//&F&6#\"\"(#\"#>\"#?" }}{PARA 11 "" 1 "" {XPPMATH 20 "6)%1order~6~weights:G/&%\"bG6#\"\"\"$\"'S6e!\"(/&F&6#\"\" %$\"'+DJ!\"'/&F&6#\"\"&$\"'r5CF2/&F&6#\"\"'$\"'H&)>F2/&F&6#\"\"($\"'!> k#F2/&F&6#\"\")$!'[SuF+" }}{PARA 11 "" 1 "" {XPPMATH 20 "6*%1order~5~w eights:G/&%#b*G6#\"\"\"$\"'zPc!\"(/&F&6#\"\"%$\"'$y=$!\"'/&F&6#\"\"&$ \"'Dc@F2/&F&6#\"\"'$\"'m%G#F2/&F&6#\"\"($\"'.jBF2/&F&6#\"\")$\"\"!FJ/& F&6#\"\"*$!'cbbF+" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/*&%)infinityG\" \"\"%8-norm~of~linking~coeffsGF&$\"+N4q&4%!\"*" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/%L2-norm~of~principal~error~of~order~6~schemeG$\"+-L'f %z!#9" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/%L2-norm~of~principal~error~ of~order~5~schemeG$\"+:.zC>!#7" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/%;o rder~6~stability~intervalG7$$!)&G3Z%!\"(\"\"!" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/%;order~5~stability~intervalG7$$!)!Q+Z$!\"(\"\"!" }} {PARA 11 "" 1 "" {XPPMATH 20 "6$/&%\"BG6#\"\"($\")w**o7!\"(/&%\"CGF&$ \")\"G\"o7F*" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 46 "#=============================================" }}{PARA 0 "" 0 "" {TEXT -1 141 "A scheme which has a similar stability functio n to that of a scheme of Tsitouras chosen so that the scheme has a wid e real stability interval" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 106 "c_2 := 1/192: c_4 := 28/145: c_5 := 56/169: c_6 := 37/58: c_7 := \+ 121/158: bs := [9,1/40]:\ncalc_RKcoeffs();" }}{PARA 11 "" 1 "" {XPPMATH 20 "6)%'nodes:G/&%\"cG6#\"\"##\"\"\"\"$#>/&F&6#\"\"$#\"#c\"$N %/&F&6#\"\"%#\"#G\"$X\"/&F&6#\"\"&#F1\"$p\"/&F&6#\"\"'#\"#P\"#e/&F&6# \"\"(#\"$@\"\"$e\"" }}{PARA 11 "" 1 "" {XPPMATH 20 "6)%1order~6~weight s:G/&%\"bG6#\"\"\"$\"'sus!\"(/&F&6#\"\"%$\"'G#p\"!\"'/&F&6#\"\"&$\"'F= EF2/&F&6#\"\"'$\"'r`8F2/&F&6#\"\"($\"'$\\%GF2/&F&6#\"\")$\"'MLwF+" }} {PARA 11 "" 1 "" {XPPMATH 20 "6*%1order~5~weights:G/&%#b*G6#\"\"\"$\"' ')Qq!\"(/&F&6#\"\"%$\"'jE=!\"'/&F&6#\"\"&$\"'cPCF2/&F&6#\"\"'$\"'uO:F2 /&F&6#\"\"($\"'u:FF2/&F&6#\"\")$\"'^%H&F+/&F&6#\"\"*$\"'++DF+" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#/*&%)infinityG\"\"\"%8-norm~of~linking ~coeffsGF&$\"00zy'*oPi#!#9" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/%L2-nor m~of~principal~error~of~order~6~schemeG$\"0KD,c6h$H!#=" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/%L2-norm~of~principal~error~of~order~5~schemeG$\" 0%yDlc^;A!#=" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/%;order~6~stability~i ntervalG7$$!)/H*3)!\"(\"\"!" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/%;orde r~5~stability~intervalG7$$!)d0e!)!\"(\"\"!" }}{PARA 11 "" 1 "" {XPPMATH 20 "6$/&%\"BG6#\"\"($\")hhE8!\"(/&%\"CGF&$\")(R)G8F*" }}} {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 46 "#======== =====================================" }}{PARA 0 "" 0 "" {TEXT -1 64 " Tsitouras' scheme with a large imaginary axis inclusion (nearly)" }} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 101 "c_2 := 3/31: c_4 := 8/29: c _5 := 40/51: c_6 := 114/161: c_7 := 2/17: bs := [9,1/20]:\ncalc_RKcoef fs();" }}{PARA 11 "" 1 "" {XPPMATH 20 "6)%'nodes:G/&%\"cG6#\"\"##\"\"$ \"#J/&F&6#F*#\"#;\"#()/&F&6#\"\"%#\"\")\"#H/&F&6#\"\"&#\"#S\"#^/&F&6# \"\"'#\"$9\"\"$h\"/&F&6#\"\"(#F(\"#<" }}{PARA 11 "" 1 "" {XPPMATH 20 " 6)%1order~6~weights:G/&%\"bG6#\"\"\"$\"'z+#)!\"(/&F&6#\"\"%$\"'d2T!\"' /&F&6#\"\"&$\"'kjzF+/&F&6#\"\"'$\"'*=U$F2/&F&6#\"\"($\"'.TR!\")/&F&6# \"\")$\"'!p9)F+" }}{PARA 11 "" 1 "" {XPPMATH 20 "6*%1order~5~weights:G /&%#b*G6#\"\"\"$\"'3rw!\"(/&F&6#\"\"%$\"'6qR!\"'/&F&6#\"\"&$\"'rzdF+/& F&6#\"\"'$\"'IfOF2/&F&6#\"\"($\"'(4(=F+/&F&6#\"\")$\"'A%Q$F+/&F&6#\"\" *$\"'++]F+" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/*&%)infinityG\"\"\"%8-n orm~of~linking~coeffsGF&$\"+ifh#\\%!\"*" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/%L2-norm~of~principal~error~of~order~6~schemeG$\"+F*=X!>!#8" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#/%L2-norm~of~principal~error~of~order~ 5~schemeG$\"+h!H$>')!#8" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/%;order~6~ stability~intervalG7$$!)wX8V!\"(\"\"!" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/%;order~5~stability~intervalG7$$!)WlSS!\"(\"\"!" }}{PARA 11 "" 1 " " {XPPMATH 20 "6$/&%\"BG6#\"\"($\")0=%H\"!\"(/&%\"CGF&$\")Whz7F*" }}} {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 71 "modificat ion of Tsitouras' scheme with a large imaginary axis inclusion" }} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 103 "c_2 := 14/145: c_4 := 59/21 4: c_5 := 47/60: c_6 := 53/75: c_7 := 4/35: bs := [9,1/27]:\ncalc_RKco effs();" }}{PARA 11 "" 1 "" {XPPMATH 20 "6)%'nodes:G/&%\"cG6#\"\"##\"# 9\"$X\"/&F&6#\"\"$#\"#f\"$@$/&F&6#\"\"%#F1\"$9#/&F&6#\"\"&#\"#Z\"#g/&F &6#\"\"'#\"#`\"#v/&F&6#\"\"(#F6\"#N" }}{PARA 11 "" 1 "" {XPPMATH 20 "6 )%1order~6~weights:G/&%\"bG6#\"\"\"$\"'K$>)!\"(/&F&6#\"\"%$\"'u0T!\"'/ &F&6#\"\"&$\"'&pi)F+/&F&6#\"\"'$\"'ufLF2/&F&6#\"\"($\"'k))Q!\")/&F&6# \"\")$\"'0O\")F+" }}{PARA 11 "" 1 "" {XPPMATH 20 "6*%1order~5~weights: G/&%#b*G6#\"\"\"$\"'z)y(!\"(/&F&6#\"\"%$\"'o0S!\"'/&F&6#\"\"&$\"'MIqF+ /&F&6#\"\"'$\"'lLNF2/&F&6#\"\"($\"'Xy9F+/&F&6#\"\")$\"'S0YF+/&F&6#\"\" *$\"'q.PF+" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/*&%)infinityG\"\"\"%8-n orm~of~linking~coeffsGF&$\"0e;X]'oxW!#9" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/%L2-norm~of~principal~error~of~order~6~schemeG$\"0b*z\"))oB!>!# =" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/%L2-norm~of~principal~error~of~o rder~5~schemeG$\"0=2W;'H?j!#=" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/%;or der~6~stability~intervalG7$$!)_f8V!\"(\"\"!" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/%;order~5~stability~intervalG7$$!)g_KV!\"(\"\"!" }} {PARA 11 "" 1 "" {XPPMATH 20 "6$/&%\"BG6#\"\"($\")`![I\"!\"(/&%\"CGF&$ \")J$pF\"F*" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 54 "#=====================================================" }} {PARA 0 "" 0 "" {TEXT -1 80 "scheme with a quite large stability regio n and non-zero imaginary axis inclusion" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 101 "c_2 := 7/71: c_4 := 23/85: c_5 := 27/29: c_6 := 61/9 0: c_7 := 18/77: bs := [9,1/17]:\ncalc_RKcoeffs();" }}{PARA 11 "" 1 " " {XPPMATH 20 "6)%'nodes:G/&%\"cG6#\"\"##\"\"(\"#r/&F&6#\"\"$#\"#Y\"$b #/&F&6#\"\"%#\"#B\"#&)/&F&6#\"\"&#\"#F\"#H/&F&6#\"\"'#\"#h\"#!*/&F&6#F *#\"#=\"#x" }}{PARA 11 "" 1 "" {XPPMATH 20 "6)%1order~6~weights:G/&%\" bG6#\"\"\"$\"'#Qx(!\"(/&F&6#\"\"%$\"'L!z#!\"'/&F&6#\"\"&$\"'l?6F2/&F&6 #\"\"'$\"'N\\QF2/&F&6#\"\"($\"',N6F2/&F&6#\"\")$\"'qsKF+" }}{PARA 11 " " 1 "" {XPPMATH 20 "6*%1order~5~weights:G/&%#b*G6#\"\"\"$\"'0Rm!\"(/&F &6#\"\"%$\"'&ys%F+/&F&6#\"\"&$\"'$z:&F+/&F&6#\"\"'$\"'J-V!\"'/&F&6#\" \"($\"'C`LF>/&F&6#\"\")$\"'JP5F+/&F&6#\"\"*$\"'N#)eF+" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/*&%)infinityG\"\"\"%8-norm~of~linking~coeffsGF&$ \"+fTR(>$!\")" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/%L2-norm~of~principa l~error~of~order~6~schemeG$\"+SI#3!>!#8" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/%L2-norm~of~principal~error~of~order~5~schemeG$\"+;K[>a!#8" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#/%;order~6~stability~intervalG7$$!))QW ^%!\"(\"\"!" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/%;order~5~stability~in tervalG7$$!)ax=X!\"(\"\"!" }}{PARA 11 "" 1 "" {XPPMATH 20 "6$/&%\"BG6# \"\"($\")2^C " 0 "" {MPLTEXT 1 0 115 "c_2 := -5/18: c_4 := 2/15: c_5 := 180/389: c_6 := 3948639/5844832 : c_7 := 125/126: bs := [9,1/20]:\ncalc_RKcoeffs();" }}{PARA 11 "" 1 " " {XPPMATH 20 "6)%'nodes:G/&%\"cG6#\"\"##!\"&\"#=/&F&6#\"\"$#\"\"%\"#X /&F&6#F1#F(\"#:/&F&6#\"\"&#\"$!=\"$*Q/&F&6#\"\"'#\"(R'[R\"(K[%e/&F&6# \"\"(#\"$D\"\"$E\"" }}{PARA 11 "" 1 "" {XPPMATH 20 "6)%1order~6~weight s:G/&%\"bG6#\"\"\"$\"'iFA!\"(/&F&6#\"\"%$\"'-hE!\"'/&F&6#\"\"&$\"'!R+$ F2/&F&6#\"\"'$\"'ItF+/&F&6#\"\"'$!';cEF+ /&F&6#\"\"($\"'GU[!\"&/&F&6#\"\")$!'i=XFE/&F&6#\"\"*$\"'++]F2" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#/*&%)infinityG\"\"\"%8-norm~of~linking ~coeffsGF&$\"+n7))))[!\")" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/%L2-norm ~of~principal~error~of~order~6~schemeG$\"+OQcW9!#8" }}{PARA 11 "" 1 " " {XPPMATH 20 "6#/%L2-norm~of~principal~error~of~order~5~schemeG$\"+JL #QA'!#7" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/%;order~6~stability~interv alG7$$!)=H!*[!\"(\"\"!" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/%;order~5~s tability~intervalG7$$!) " 0 "" {MPLTEXT 1 0 108 "c_2 := 1/60: c_4 := 2/1 5: c_5 := 180/389: c_6 := 177/262: c_7 := 125/126: bs := [9,-1/150]:\n calc_RKcoeffs();" }}{PARA 11 "" 1 "" {XPPMATH 20 "6)%'nodes:G/&%\"cG6# \"\"##\"\"\"\"#g/&F&6#\"\"$#\"\"%\"#X/&F&6#F1#F(\"#:/&F&6#\"\"&#\"$!= \"$*Q/&F&6#\"\"'#\"$x\"\"$i#/&F&6#\"\"(#\"$D\"\"$E\"" }}{PARA 11 "" 1 "" {XPPMATH 20 "6)%1order~6~weights:G/&%\"bG6#\"\"\"$\"'eFA!\"(/&F&6# \"\"%$\"'.hE!\"'/&F&6#\"\"&$\"'&Q+$F2/&F&6#\"\"'$\"'L " 0 "" {MPLTEXT 1 0 1 ";" }}}}{PARA 0 "" 0 "" {TEXT -1 23 " #======================" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 35 "#==================================" }}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 36 "Test-bed procedures for the examples" }} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}{SECT 1 {PARA 4 "" 0 " " {TEXT -1 9 "RK6_8step" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 2923 "rk6step := proc(x_rk6step::realco ns)\n local c2,c3,c4,c5,c6,c7,c8,a21,a31,a32,a41,a42,a43,a51,a52,a53 ,a54,\n a61,a62,a63,a64,a65,a71,a72,a73,a74,a75,a76,a81,a82,a83,a84, a85,a86,a87,\n f1,f2,f3,f4,f5,f6,f7,f8,b1,b2,b3,b4,b5,b6,b7,b8,\n \+ xk,yk,t,jF,jM,jS,n,h,data,fn,xx,ys,saveDigits;\n options `Copyright \+ 2004 by Peter Stone`;\n \n data := SOLN_;\n\n saveDigits := Digi ts;\n Digits := max(trunc(evalhf(Digits)),Digits+5);\n\n # procedu re 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 error \"evaluation of slope field fa iled at %1\",evalf([X_,Y_],saveDigits);\n end if;\n val;\n \+ end proc;\n\n xx := evalf(x_rk6step);\n n := nops(data);\n\n \+ if (data[1,1]data[n,1] or xxdata[1,1])) then\n \+ error \"independent variable is outside 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_; c8 := c8_;\n a2 1 := c2; a31 := a31_; a32 := a32_; a41 := a41_; a42 := a42_; a43 := a4 3_;\n 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 a81 := a81_; a82 := a82_; a83 := a83_; a84 := a84_; a85 := a85_; \+ a86 := a86_; a87 := a87_;\n b1 := b1_; b2 := b2_; b3 := b3_; b4 := b 4_; b5 := b5_; b6 := b6_; b7 := b7_; b8 := b8_;\n # Perform a binary search for the 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 else jS := jM end if;\n end do;\n if jM = n then j F := n-1; jS := n end if;\n else\n while jS-jF> 1 do\n j M := trunc((jF+jS)/2);\n if xx<=data[jM,1] then jF := jM else j S := jM end if;\n end do;\n if jM = n then jF := n-1; jS := \+ n end if;\n end if;\n \n # Get the data needed from the list.\n \+ xk := data[jF,1];\n yk := data[jF,2];\n\n # Do one step with step -size ..\n h := xx-xk;\n f1 := fn(xk,yk);\n t := a21*f1;\n f2 \+ := fn(xk + c2*h,yk + t*h);\n t := a31*f1 + a32*f2;\n f3 := fn(xk + c3*h,yk + t*h);\n t := a41*f1 + a42*f2 + a43*f3;\n f4 := fn(xk + \+ c4*h,yk + t*h);\n t := a51*f1 + a52*f2 + a53*f3 + a54*f4;\n f5 := \+ fn(xk + c5*h,yk + t*h);\n t := a61*f1 + a62*f2 + a63*f3 + a64*f4 + a 65*f5;\n f6 := fn(xk + c6*h,yk + t*h);\n t := a71*f1 + a72*f2 + a7 3*f3 + a74*f4 + a75*f5 + a76*f6;\n f7 := fn(xk + c7*h,yk + t*h);\n \+ t := a81*f1 + a82*f2 + a83*f3 + a84*f4 + a85*f5 + a86*f6 + a87*f7;\n \+ f8 := fn(xk + c8*h,yk + t*h);\n \n ys := yk + (b1*f1 + b2*f2 + b3*f3 + b4*f4 + b5*f5 + b6*f6 + b7*f7 + b8*f8)*h;\n\n evalf[saveD igits](ys);\nend proc: # of rk6step" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 26 "RK6_1 Sharp-Verner scheme" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 3400 "RK6_1 := proc(fx y,x,y,xx,yy,h,stps,bb)\n local c2,c3,c4,c5,c6,c7,c8,a21,a31,a32,a41, a42,a43,a51,\n a52,a53,a54,a61,a62,a63,a64,a65,a71,a72,a73,a74,a75,a 76,\n a81,a82,a83,a84,a85,a86,a87,b1,b2,b3,b4,b5,b6,b7,b8,\n f1,f2 ,f3,f4,f5,f6,f7,f8,t,k,fn,xk,yk,soln,eqns,A,saveDigits;\n saveDigits := Digits;\n Digits := max(trunc(evalhf(Digits)),Digits+5);\n\n f n := unapply(fxy,x,y);\n\n A := matrix([[1/12,1/12,0,0,0,0,0,0,0],\n [2/15,2/75,8/75,0,0,0,0,0,0],\n [1/5,1/ 20,0,3/20,0,0,0,0,0],\n [8/15,88/135,0,-112/45,64/27,0, 0,0,0],\n [2/3,-10891/11556,0,3880/963,-8456/2889,217/4 28,0,0,0],\n [19/20,1718911/4382720,0,-1000749/547840,8 19261/383488,-671175/876544,\n 14535/14336,0,0] ,\n [1,85153/203300,0,-6783/2140,10956/2675,-38493/1337 5,1152/425,-7168/40375,0],\n [0,53/912,0,0,5/16,27/112, 27/136,256/969,-25/336]]);\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 := e valf(A[5,1]);\n c7 := evalf(A[6,1]);\n c8 := evalf(A[7,1]);\n a2 1 := 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 := ev alf(A[6,6]);\n a76 := evalf(A[6,7]);\n a81 := evalf(A[7,2]);\n a 82 := evalf(A[7,3]);\n a83 := evalf(A[7,4]);\n a84 := evalf(A[7,5] );\n a85 := evalf(A[7,6]);\n a86 := evalf(A[7,7]);\n a87 := eval f(A[7,8]);\n b1 := evalf(A[8,2]);\n b2 := evalf(A[8,3]);\n b3 := evalf(A[8,4]);\n b4 := evalf(A[8,5]);\n b5 := evalf(A[8,6]);\n \+ b6 := evalf(A[8,7]);\n b7 := evalf(A[8,8]);\n b8 := evalf(A[8,9]); \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 + a5 3*f3 + a54*f4;\n f5 := fn(xk + c5*h,yk + t*h);\n t := a61*f1 + a62*f2 + a63*f3 + a64*f4 + a65*f5;\n f6 := fn(xk + c6*h,yk + t *h);\n t := a71*f1 + a72*f2 + a73*f3 + a74*f4 + a75*f5 + a76*f6; \n f7 := fn(xk + c7*h,yk + t*h);\n t := a81*f1 + a82*f2 + a8 3*f3 + a84*f4 + a85*f5 + a86*f6 + a87*f7;\n f8 := fn(xk + c8*h,yk + t*h);\n\n yk := yk + (b1*f1 + b2*f2 + b3*f3 + b4*f4 + b5*f5 + \+ b6*f6 + b7*f7 + b8*f8)*h;\n xk := xk + h:\n soln := soln,[xk ,yk];\n end do;\n if bb=true then\n eqns := \{SOLN_=[soln],FX Y_=fxy,X_=x,Y_=y,c2_=c2,c3_=c3,\n c4_=c4,c5_=c5,c6_=c6,c7_=c7, c8_=c8,a31_=a31,a32_=a32,a41_=a41,\n a42_=a42,a43_=a43,a51_=a5 1,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 a81_=a81,a82_=a82,a83_=a83,a84_=a84,a85_=a85,a86_ =a86,a87_=a87,\n b1_=b1,b2_=b2,b3_=b3,b4_=b4,b5_=b5,b6_=b6,b7_ =b7,b8_=b8\};\n return subs(eqns,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 50 "RK6_2 scheme with a wide rea l stability interval " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 4175 "RK6_2 := proc(fxy,x,y,xx,yy,h,stps,bb) \n local c2,c3,c4,c5,c6,c7,c8,a21,a31,a32,a41,a42,a43,a51,\n a52,a 53,a54,a61,a62,a63,a64,a65,a71,a72,a73,a74,a75,a76,\n a81,a82,a83,a8 4,a85,a86,a87,b1,b2,b3,b4,b5,b6,b7,b8,\n f1,f2,f3,f4,f5,f6,f7,f8,t,k ,fn,xk,yk,soln,eqns,A,saveDigits;\n saveDigits := Digits;\n Digits := max(trunc(evalhf(Digits)),Digits+5);\n\n fn := unapply(fxy,x,y); \n\n A := matrix([[1/192,1/192,0,0,0,0,0,0,0],\n [56/435,-30 744/21025,100352/63075,0,0,0,0,0,0],\n [28/145,7/145,0,21/145, 0,0,0,0,0],\n [56/169,523516/4826809,0,-889140/4826809,1965040 /4826809,0,0,0,0],\n [37/58,41044746763290003/5173977998073651 2,0,-57685603608510645/24977824818286592,\n 8314594069670 1395/53969942910940672,53772835209542058935/87647187287367651328,0,0,0 ],\n [121/158,-19737371728514492920457623/38482466972140721924 237312,0,\n 27466253014073155807766505/145609334489181109 98360064,\n -117670134102158631127838015/12298788430961190 1825434112,\n -19483500956399449871589744537/4813116551539 88159050791915520,\n 913336658253963141/23439333231940773 85,0,0],\n [1,2484813954171031615807/2538960546022287823744,0, \n -229147524517236387495/87335252197326853888,\n \+ 130548415875501708716128185/106468788692037442075798624,\n \+ 11103534575830173429908962617/8186576761928888116508540672,\n \+ -10679027011796052/13181450824860817,2238377869046599644/25 64667687663709133,0],\n [0,397933/5470080,0,0,5450042155625/32 205284983872,\n 1677324270326783/6406236767145600,1792009 57846/1323774725175,\n 77461817788426/272279942939565,470 49059/616363020]]);\n\n c2 := evalf(A[1,1]);\n c3 := evalf(A[2,1]) ;\n c4 := evalf(A[3,1]);\n c5 := evalf(A[4,1]);\n c6 := evalf(A[ 5,1]);\n c7 := evalf(A[6,1]);\n c8 := evalf(A[7,1]);\n 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 a81 := evalf(A[7,2]);\n a82 := \+ evalf(A[7,3]);\n a83 := evalf(A[7,4]);\n a84 := evalf(A[7,5]);\n \+ a85 := evalf(A[7,6]);\n a86 := evalf(A[7,7]);\n a87 := evalf(A[7, 8]);\n b1 := evalf(A[8,2]);\n b2 := evalf(A[8,3]);\n b3 := evalf (A[8,4]);\n b4 := evalf(A[8,5]);\n b5 := evalf(A[8,6]);\n b6 := \+ evalf(A[8,7]);\n b7 := evalf(A[8,8]);\n b8 := evalf(A[8,9]);\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 t := a81*f1 + a82*f2 + a83*f3 + a 84*f4 + a85*f5 + a86*f6 + a87*f7;\n f8 := fn(xk + c8*h,yk + t*h); \n\n yk := yk + (b1*f1 + b2*f2 + b3*f3 + b4*f4 + b5*f5 + b6*f6 + \+ b7*f7 + b8*f8)*h;\n xk := xk + h:\n soln := soln,[xk,yk];\n \+ end do;\n if bb=true then\n eqns := \{SOLN_=[soln],FXY_=fxy,X _=x,Y_=y,c2_=c2,c3_=c3,\n c4_=c4,c5_=c5,c6_=c6,c7_=c7,c8_=c8,a 31_=a31,a32_=a32,a41_=a41,\n a42_=a42,a43_=a43,a51_=a51,a52_=a 52,a53_=a53,a54_=a54,\n a61_=a61,a62_=a62,a63_=a63,a64_=a64,a6 5_=a65,\n a71_=a71,a72_=a72,a73_=a73,a74_=a74,a75_=a75,a76_=a7 6,\n a81_=a81,a82_=a82,a83_=a83,a84_=a84,a85_=a85,a86_=a86,a87 _=a87,\n b1_=b1,b2_=b2,b3_=b3,b4_=b4,b5_=b5,b6_=b6,b7_=b7,b8_= b8\};\n return subs(eqns,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 51 "RK6_3 scheme with a large imagin ary axis inclusion" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 " > " 0 "" {MPLTEXT 1 0 4105 "RK6_3 := proc(fxy,x,y,xx,yy,h,stps,bb)\n \+ local c2,c3,c4,c5,c6,c7,c8,a21,a31,a32,a41,a42,a43,a51,\n a52,a53,a 54,a61,a62,a63,a64,a65,a71,a72,a73,a74,a75,a76,\n a81,a82,a83,a84,a8 5,a86,a87,b1,b2,b3,b4,b5,b6,b7,b8,\n f1,f2,f3,f4,f5,f6,f7,f8,t,k,fn, xk,yk,soln,eqns,A,saveDigits;\n saveDigits := Digits;\n Digits := \+ max(trunc(evalhf(Digits)),Digits+5);\n\n fn := unapply(fxy,x,y);\n\n A := matrix([[14/145,14/145,0,0,0,0,0,0,0],\n [59/321,2554 7/2885148,504745/2885148,0,0,0,0,0,0],\n [59/214,59/856,0,177 /856,0,0,0,0,0],\n [47/60,109313963/93987000,0,-280562881/626 58000,770307017/187974000,0,0,0,0],\n [53/75,2594915684746093 3/19448380820547506250,0,\n 13843311346643833/137931778 869131250,322308298919703822734/674279501001748115625,\n \+ 8562169944665744/67437276532035625,0,0,0],\n [4/35,-1124425 445878279929044/5080674836868609224225,0,\n 10208986940 897432574/2855457555847472065,\n -1389076296899785384285 714176/321845802595321532838393475,\n -83410613975106692 05248/2889892848216596175835,7781543704350/1963746281231,0,0],\n \+ [1,-4662888761921915772773/8195396815968383745983,0,\n \+ 2931920524405086861/940000781782231318,\n -182277776 762433747385270230/74164896551649031914481339,\n -146526 017349731441994000/935639633573123866302727,\n 35954197 67671875/3518327954949241,1418176375/33500640722,0],\n [0,577 999/7054512,0,0,34110058090624/83078929543805,1110240000/12869442287, \n 77530078125/230762131864,203784875/52405236624,13416 97/16490760]]);\n\n c2 := evalf(A[1,1]);\n c3 := evalf(A[2,1]);\n \+ c4 := evalf(A[3,1]);\n c5 := evalf(A[4,1]);\n c6 := evalf(A[5,1] );\n c7 := evalf(A[6,1]);\n c8 := evalf(A[7,1]);\n 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 := e valf(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 := eva lf(A[5,6]);\n a71 := evalf(A[6,2]);\n a72 := evalf(A[6,3]);\n a7 3 := evalf(A[6,4]);\n a74 := evalf(A[6,5]);\n a75 := evalf(A[6,6]) ;\n a76 := evalf(A[6,7]);\n a81 := evalf(A[7,2]);\n a82 := evalf (A[7,3]);\n a83 := evalf(A[7,4]);\n a84 := evalf(A[7,5]);\n a85 \+ := evalf(A[7,6]);\n a86 := evalf(A[7,7]);\n a87 := evalf(A[7,8]); \n b1 := evalf(A[8,2]);\n b2 := evalf(A[8,3]);\n b3 := evalf(A[8 ,4]);\n b4 := evalf(A[8,5]);\n b5 := evalf(A[8,6]);\n b6 := eval f(A[8,7]);\n b7 := evalf(A[8,8]);\n b8 := evalf(A[8,9]);\n\n xk \+ := evalf(xx);\n yk := evalf(yy);\n soln := [xk,yk]; \n for k fro m 1 to stps do\n f1 := fn(xk,yk);\n t := a21*f1;\n f2 : = fn(xk + c2*h,yk + t*h);\n t := a31*f1 + a32*f2;\n f3 := fn (xk + c3*h,yk + t*h);\n t := a41*f1 + a42*f2 + a43*f3;\n f4 \+ := fn(xk + c4*h,yk + t*h);\n t := a51*f1 + a52*f2 + a53*f3 + a54* f4;\n f5 := fn(xk + c5*h,yk + t*h);\n t := a61*f1 + a62*f2 + a63*f3 + a64*f4 + a65*f5;\n f6 := fn(xk + c6*h,yk + t*h);\n \+ t := a71*f1 + a72*f2 + a73*f3 + a74*f4 + a75*f5 + a76*f6;\n f7 : = fn(xk + c7*h,yk + t*h);\n t := a81*f1 + a82*f2 + a83*f3 + a84*f 4 + a85*f5 + a86*f6 + a87*f7;\n f8 := fn(xk + c8*h,yk + t*h);\n\n yk := yk + (b1*f1 + b2*f2 + b3*f3 + b4*f4 + b5*f5 + b6*f6 + b7*f 7 + b8*f8)*h;\n xk := xk + h:\n soln := soln,[xk,yk];\n en d do;\n if bb=true then\n eqns := \{SOLN_=[soln],FXY_=fxy,X_=x, Y_=y,c2_=c2,c3_=c3,\n c4_=c4,c5_=c5,c6_=c6,c7_=c7,c8_=c8,a31_= a31,a32_=a32,a41_=a41,\n a42_=a42,a43_=a43,a51_=a51,a52_=a52,a 53_=a53,a54_=a54,\n a61_=a61,a62_=a62,a63_=a63,a64_=a64,a65_=a 65,\n a71_=a71,a72_=a72,a73_=a73,a74_=a74,a75_=a75,a76_=a76,\n a81_=a81,a82_=a82,a83_=a83,a84_=a84,a85_=a85,a86_=a86,a87_=a8 7,\n b1_=b1,b2_=b2,b3_=b3,b4_=b4,b5_=b5,b6_=b6,b7_=b7,b8_=b8\} ;\n return subs(eqns,eval(rk6step)); \n else\n return eval f[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 87 "RK6_4 scheme with a quite large stabil ity region and non-zero imaginary axis inclusion" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 4105 "RK6_4 := p roc(fxy,x,y,xx,yy,h,stps,bb)\n local c2,c3,c4,c5,c6,c7,c8,a21,a31,a3 2,a41,a42,a43,a51,\n a52,a53,a54,a61,a62,a63,a64,a65,a71,a72,a73,a74 ,a75,a76,\n a81,a82,a83,a84,a85,a86,a87,b1,b2,b3,b4,b5,b6,b7,b8,\n \+ f1,f2,f3,f4,f5,f6,f7,f8,t,k,fn,xk,yk,soln,eqns,A,saveDigits;\n save Digits := Digits;\n Digits := max(trunc(evalhf(Digits)),Digits+5);\n \n fn := unapply(fxy,x,y);\n\n A := matrix([[7/71,7/71,0,0,0,0,0,0 ,0],\n [46/255,6992/455175,75118/455175,0,0,0,0,0,0],\n \+ [23/85,23/340,0,69/340,0,0,0,0,0],\n [27/29,125814087/516 07124,0,-481282155/51607124,100879020/12901781,0,0,0,0],\n [6 1/90,3323091540925544113/29573952540280488000,0,\n -1070 3826229567511/24340701679243200,169241467001919447401/1783199805021356 83200,\n 2551748271602005217/45506989353097008000,0,0,0 ],\n [18/77,-376385546372076652128/6002133498919339474385,0, \n 19305251563909761378/19679126225965047457,\n \+ -8122491009809811364010481/9979717849963846796488754,\n \+ -155386566948917103529551/2301833170083782096614930,\n \+ 107267076044076240/544838019860392951,0,0],\n [1,-1972 640388605306005321/257739734717483367780,0,12354606914500963/386396118 220908,\n -56856879732578412491684/196766430482191737220 9,\n -17589460594924790303414/92934689150952840583755,\n 370576933768080/119264931607213,1150428860219177/43291 7708558301,0],\n [0,636077/8182296,0,0,24315051125/8714029462 6,30418033967/271431025272,\n 862843671000/224153144303 9,1670085824869/14714276549256,13887/424328]]);\n\n c2 := evalf(A[1, 1]);\n c3 := evalf(A[2,1]);\n c4 := evalf(A[3,1]);\n c5 := evalf (A[4,1]);\n c6 := evalf(A[5,1]);\n c7 := evalf(A[6,1]);\n c8 := \+ evalf(A[7,1]);\n a21 := c2;\n a31 := evalf(A[2,2]);\n a32 := eva lf(A[2,3]);\n a41 := evalf(A[3,2]);\n a42 := evalf(A[3,3]);\n a4 3 := evalf(A[3,4]);\n a51 := evalf(A[4,2]);\n a52 := evalf(A[4,3]) ;\n a53 := evalf(A[4,4]);\n a54 := evalf(A[4,5]);\n a61 := evalf (A[5,2]);\n a62 := evalf(A[5,3]);\n a63 := evalf(A[5,4]);\n a64 \+ := evalf(A[5,5]);\n a65 := evalf(A[5,6]);\n a71 := evalf(A[6,2]); \n a72 := evalf(A[6,3]);\n a73 := evalf(A[6,4]);\n a74 := evalf( A[6,5]);\n a75 := evalf(A[6,6]);\n a76 := evalf(A[6,7]);\n a81 : = evalf(A[7,2]);\n a82 := evalf(A[7,3]);\n a83 := evalf(A[7,4]);\n a84 := evalf(A[7,5]);\n a85 := evalf(A[7,6]);\n a86 := evalf(A[ 7,7]);\n a87 := evalf(A[7,8]);\n b1 := evalf(A[8,2]);\n b2 := ev alf(A[8,3]);\n b3 := evalf(A[8,4]);\n b4 := evalf(A[8,5]);\n b5 \+ := evalf(A[8,6]);\n b6 := evalf(A[8,7]);\n b7 := evalf(A[8,8]);\n \+ b8 := evalf(A[8,9]);\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 t := \+ a81*f1 + a82*f2 + a83*f3 + a84*f4 + a85*f5 + a86*f6 + a87*f7;\n f 8 := fn(xk + c8*h,yk + t*h);\n\n yk := yk + (b1*f1 + b2*f2 + b3*f 3 + b4*f4 + b5*f5 + b6*f6 + b7*f7 + b8*f8)*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,c 5_=c5,c6_=c6,c7_=c7,c8_=c8,a31_=a31,a32_=a32,a41_=a41,\n a42_= a42,a43_=a43,a51_=a51,a52_=a52,a53_=a53,a54_=a54,\n a61_=a61,a 62_=a62,a63_=a63,a64_=a64,a65_=a65,\n a71_=a71,a72_=a72,a73_=a 73,a74_=a74,a75_=a75,a76_=a76,\n a81_=a81,a82_=a82,a83_=a83,a8 4_=a84,a85_=a85,a86_=a86,a87_=a87,\n b1_=b1,b2_=b2,b3_=b3,b4_= b4,b5_=b5,b6_=b6,b7_=b7,b8_=b8\};\n return subs(eqns,eval(rk6step )); \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 ";" }}}}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 49 "RK6_5 sch eme with a quite large stability region" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 4161 "RK6_5 := proc(fxy,x,y ,xx,yy,h,stps,bb)\n local c2,c3,c4,c5,c6,c7,c8,a21,a31,a32,a41,a42,a 43,a51,\n a52,a53,a54,a61,a62,a63,a64,a65,a71,a72,a73,a74,a75,a76,\n a81,a82,a83,a84,a85,a86,a87,b1,b2,b3,b4,b5,b6,b7,b8,\n f1,f2,f3,f 4,f5,f6,f7,f8,t,k,fn,xk,yk,soln,eqns,A,saveDigits;\n saveDigits := D igits;\n Digits := max(trunc(evalhf(Digits)),Digits+5);\n\n fn := \+ unapply(fxy,x,y);\n\n A := matrix([[1/60,1/60,0,0,0,0,0,0,0],\n \+ [4/45,-4/27,32/135,0,0,0,0,0,0],\n [2/15,1/30,0,1/10,0,0 ,0,0,0],\n [180/389,73104030/58863869,0,-279389250/58863869,2 33523000/58863869,0,0,0,0],\n [177/262,-107221818666951807169 /25619615678223628160,0,\n 328927584642593578425/2049569 2542578902528,-3717245049788716341975/307755633334661333272,\n \+ 87713642192227304821351/98481802667091626647040,0,0,0],\n \+ [125/126,223899211079219075995406125/20066238404586623311945344,0, \n -27381293369066082540625/622049813755756227072,\n \+ 2493079698743912733446875/71342417962706618257992,\n \+ -2021984654539866742602494375/889078038644174940599689728,\n \+ 999354901744231797500/846274121500481947287,0,0],\n \+ [1,15677672759360359499/1267222156368417000,0,-414775576646775/84852 70768736,\n 45606172048348599823275/11751135143953658568 52,\n -321038818302974478087638483/1239156469179360125372 05920,\n 8496514166784579485/6535296976627611297,-857445 886276749/93424049779207625,0],\n [0,10645627/477900000,0,0,4 717085625/17726544496,733457058173503507/2441724811071084000,\n \+ 3157306853548189/13624786136653650,15203797026579/114570849812 50,-407713/355300]]);\n\n c2 := evalf(A[1,1]);\n c3 := evalf(A[2,1 ]);\n c4 := evalf(A[3,1]);\n c5 := evalf(A[4,1]);\n c6 := evalf( A[5,1]);\n c7 := evalf(A[6,1]);\n c8 := evalf(A[7,1]);\n a21 := \+ c2;\n a31 := evalf(A[2,2]);\n a32 := evalf(A[2,3]);\n a41 := eva lf(A[3,2]);\n a42 := evalf(A[3,3]);\n a43 := evalf(A[3,4]);\n a5 1 := evalf(A[4,2]);\n a52 := evalf(A[4,3]);\n a53 := evalf(A[4,4]) ;\n a54 := evalf(A[4,5]);\n a61 := evalf(A[5,2]);\n a62 := evalf (A[5,3]);\n a63 := evalf(A[5,4]);\n a64 := evalf(A[5,5]);\n a65 \+ := evalf(A[5,6]);\n a71 := evalf(A[6,2]);\n a72 := evalf(A[6,3]); \n a73 := evalf(A[6,4]);\n a74 := evalf(A[6,5]);\n a75 := evalf( A[6,6]);\n a76 := evalf(A[6,7]);\n a81 := evalf(A[7,2]);\n a82 : = evalf(A[7,3]);\n a83 := evalf(A[7,4]);\n a84 := evalf(A[7,5]);\n a85 := evalf(A[7,6]);\n a86 := evalf(A[7,7]);\n a87 := evalf(A[ 7,8]);\n b1 := evalf(A[8,2]);\n b2 := evalf(A[8,3]);\n b3 := eva lf(A[8,4]);\n b4 := evalf(A[8,5]);\n b5 := evalf(A[8,6]);\n b6 : = evalf(A[8,7]);\n b7 := evalf(A[8,8]);\n b8 := evalf(A[8,9]);\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 + a6 2*f2 + a63*f3 + a64*f4 + a65*f5;\n f6 := fn(xk + c6*h,yk + t*h); \n t := a71*f1 + a72*f2 + a73*f3 + a74*f4 + a75*f5 + a76*f6;\n \+ f7 := fn(xk + c7*h,yk + t*h);\n t := a81*f1 + a82*f2 + a83*f3 \+ + a84*f4 + a85*f5 + a86*f6 + a87*f7;\n f8 := fn(xk + c8*h,yk + t* h);\n\n yk := yk + (b1*f1 + b2*f2 + b3*f3 + b4*f4 + b5*f5 + b6*f6 + b7*f7 + b8*f8)*h;\n xk := xk + h:\n soln := soln,[xk,yk]; \n end do;\n if bb=true then\n eqns := \{SOLN_=[soln],FXY_=fx y,X_=x,Y_=y,c2_=c2,c3_=c3,\n c4_=c4,c5_=c5,c6_=c6,c7_=c7,c8_=c 8,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 a81_=a81,a82_=a82,a83_=a83,a84_=a84,a85_=a85,a86_=a86, a87_=a87,\n b1_=b1,b2_=b2,b3_=b3,b4_=b4,b5_=b5,b6_=b6,b7_=b7,b 8_=b8\};\n return subs(eqns,eval(rk6step)); \n else\n retu rn evalf[saveDigits]([soln]);\n end if;\nend 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 "These tests do not \+ make use of the embedded order 4 method for error correction." }} {SECT 1 {PARA 4 "" 0 "" {TEXT -1 48 "Test 1 of 8 stage, order 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)*sin(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)*exp(-x)*y(x); \nic := y(0)=1;\ndsolve(\{de,ic\},y(x)):\ny(x)=simplify(numer(rhs(%))/ convert(denom(rhs(%)),exp));\nf := unapply(rhs(%),x):\nplot(f(x),x=0.. 5,0..1.45,font=[HELVETICA,9],labels=[`x`,`y(x)`]);" }}{PARA 11 "" 1 " " {XPPMATH 20 "6#>%#deG/-%%diffG6$-%\"yG6#%\"xGF,,$*,\"#7\"\"\"F,F0-%$ cosG6#,$*&\"\"%F0F,F0F0F0-%$expG6#,$F,!\"\"F0F)F0F0" }}{PARA 11 "" 1 " " {XPPMATH 20 "6#>%#icG/-%\"yG6#\"\"!\"\"\"" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/-%\"yG6#%\"xG-%$expG6#,,*&#\"#7\"#<\"\"\"*(F'F0-%$cosG 6#,$*&\"\"%F0F'F0F0F0-F)6#,$F'!\"\"F0F0F;*&#\"$!=\"$*GF0*&F8F0F2F0F0F0 *&#\"#[F/F0*(F8F0-%$sinGF4F0F'F0F0F0*&#\"#'*F?F0*&F8F0FEF0F0F0#F>F?F; " }}{PARA 13 "" 1 "" {GLPLOT2D 577 286 286 {PLOTDATA 2 "6&-%'CURVESG6$ 7er7$$\"\"!F)$\"\"\"F)7$$\"3gmmTN@Ki8!#>$\"3Fk>e\"G.6+\"!#<7$$\"3ALL$3 FWYs#F/$\"3!H*fm:2P/5F27$$\"3%)***\\iSmp3%F/$\"3Qn()\\Dat45F27$$\"3Wmm mT&)G\\aF/$\"34$Q7t`Dr,\"F27$$\"3m****\\7G$R<)F/$\"3S2-*\\9jw.\"F27$$ \"3GLLL3x&)*3\"!#=$\"3U([#>C\\El5F27$$\"3))**\\i!R(*Rc\"FJ$\"3>&=^@[0u 7\"F27$$\"3umm\"H2P\"Q?FJ$\"3k\\#o#G?)=?\"F27$$\"3!***\\PMnNrDFJ$\"3s_ j<)f!R*G\"F27$$\"3MLL$eRwX5$FJ$\"37'\\4u:c`O\"F27$$\"3_LLe*[`HP$FJ$\"3 [!\\'y0#yNR\"F27$$\"3rLLL$eI8k$FJ$\"3N\"Ha_9o@T\"F27$$\"3_L$3-8>bx$FJ$ \"3@))>@pAD<9F27$$\"3*QL$3xwq4RFJ$\"3a@g!fsi#>9F27$$\"3EM$eRA'*Q/%FJ$ \"3^DvP/8/=9F27$$\"33ML$3x%3yTFJ$\"3bF0p:\"oMT\"F27$$\"3h+]PfyG7ZFJ$\" 3e=U+Y19h8F27$$\"3emm\"z%4\\Y_FJ$\"3Yii#4W6uD\"F27$$\"3'QLL3FGT\\&FJ$ \"3c!QStI8]>\"F27$$\"32++v$flWv*FJ7$$\"3I++vVVX$\\'FJ$ \"3w/21T*\\F&*)FJ7$$\"31nm\"zWo)\\nFJ$\"3E>3;k'H:;)FJ7$$\"3%QL$3_DG1qF J$\"31le1yn9(R(FJ7$$\"3]***\\il'pisFJ$\"3E!)4GzFfsmFJ7$$\"3+MLe*[!)y_( FJ$\"3CJpN=**=vfFJ7$$\"3Qnm\"HKkIz(FJ$\"3'oU:>LtrL&FJ7$$\"3!3+]i:[#e!) 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ArFJ7$$\"3e++]iC$pk$F2$\"3ma\\oRiHQqFJ7$$\"3ILe*[t\\sp$F2$\"3'e9/wG(3M oFJ7$$\"3[m;H2qcZPF2$\"3CYQ8S*3be'FJ7$$\"3O+]7.\"fF&QF2$\"3**Q8E[N&3+' FJ7$$\"3Ymm;/OgbRF2$\"3kN#z0%oN^aFJ7$$\"3w**\\ilAFjSF2$\"3[i8#)*p//*\\ FJ7$$\"3ym\"zW7@^6%F2$\"3>C%QCunR#[FJ7$$\"3yLLL$)*pp;%F2$\"3g*yCm#3E'p %FJ7$$\"3)QL3-$H**>UF2$\"3$*o:W?mr0YFJ7$$\"3)RL$3xe,tUF2$\"3!\\Bp&*))o Xb%FJ7$$\"3h+v=n(*fDVF2$\"3kIpK$)H$3a%FJ7$$\"3Cn;HdO=yVF2$\"3u&G6!oNOh XFJ7$$\"3MMe9\"z-lU%F2$\"3kC\">#=Lu2YFJ7$$\"3a+++D>#[Z%F2$\"3w_(eqj7vn %FJ7$$\"3SnmT&G!e&e%F2$\"3W>T$>g**p!\\FJ7$$\"3#RLLL)Qk%o%F2$\"3'yDBP_q :;&FJ7$$\"37+]iSjE!z%F2$\"3J;fP@m(pV&FJ7$$\"3a+]P40O\"*[F2$\"3!>+$=fU- gcFJ7$$\"\"&F)$\"3h(Q0fOqh\"eFJ-%'COLOURG6&%$RGBG$\"#5!\"\"F(F(-%%FONT G6$%*HELVETICAG\"\"*-%+AXESLABELSG6$%\"xG%%y(x)G-%%VIEWG6$;F(F]am;F($ \"$X\"!\"#" 1 2 0 1 10 0 2 9 1 4 2 1.000000 45.000000 45.000000 0 0 "C urve 1" }}}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 32 "The following code constructs a " }{TEXT 260 17 "discrete solution " }{TEXT -1 44 " based on each of the methods and gives the " }{TEXT 260 22 "root mean square error" }{TEXT -1 18 " of each solution." }} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 819 "F := (x,y) -> 12*x*cos(4*x) *exp(-x)*y: hh := 0.01: numsteps := 500: x0 := 0: y0 := 1:\nmatrix([[` slope field: `,F(x,y)],[`initial point: `,``(x0,y0)],[`step width: \+ `,hh],\n[`no. of steps: `,numsteps]]);``;\nmthds := [`Sharp-Verner \+ scheme`,`scheme with a wide real stability interval`,`scheme with a la rge imaginary axis inclusion`,`scheme with a quite large stability reg ion and non-zero imaginary axis inclusion`,`scheme with a quite large \+ stability region`]: errs := []:\nDigits := 25:\nfor ct to 5 do\n Fn_ RK6_||ct := RK6_||ct(F(x,y),x,y,x0,y0,hh,numsteps,false);\n sm := 0: numpts := nops(Fn_RK6_||ct):\n for ii to numpts do\n sm := sm+ (Fn_RK6_||ct[ii,2]-f(Fn_RK6_||ct[ii,1]))^2;\n end do:\n errs := [o p(errs),sqrt(sm/numpts)];\nend do:\nDigits := 10:\nlinalg[transpose](c onvert([mthds,evalf(errs)],matrix));" }}{PARA 11 "" 1 "" {XPPMATH 20 " 6#K%'matrixG6#7&7$%0slope~field:~~~G,$*,\"#7\"\"\"%\"xGF,-%$cosG6#,$*& \"\"%F,F-F,F,F,-%$expG6#,$F-!\"\"F,%\"yGF,F,7$%0initial~point:~G-%!G6$ \"\"!F,7$%/step~width:~~~G$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$%4Sharp-Verner~schemeG$\"+[%Qxi$!#B7$%K scheme~with~a~wide~real~stability~intervalG$\"+ " 0 "" {MPLTEXT 1 0 750 "F := (x,y) -> 12*x*cos(4*x) *exp(-x)*y: hh := 0.01: numsteps := 500: x0 := 0: y0 := 1:\nmatrix([[` slope field: `,F(x,y)],[`initial point: `,``(x0,y0)],[`step width: \+ `,hh],\n[`no. of steps: `,numsteps]]);``;\nmthds := [`Sharp-Verner \+ scheme`,`scheme with a wide real stability interval`,`scheme with a la rge imaginary axis inclusion`,`scheme with a quite large stability reg ion and non-zero imaginary axis inclusion`,`scheme with a quite large \+ stability region`]: errs := []:\nDigits := 25:\nfor ct to 5 do\n fn_ RK6_||ct := RK6_||ct(F(x,y),x,y,x0,y0,hh,numsteps,true);\nend do:\nxx \+ := 4.999: fxx := evalf(f(xx)):\nfor ct to 5 do\n errs := [op(errs),a bs(fn_RK6_||ct(xx)-fxx)];\nend do:\nDigits := 10:\nlinalg[transpose](c onvert([mthds,evalf(errs)],matrix));" }}{PARA 11 "" 1 "" {XPPMATH 20 " 6#K%'matrixG6#7&7$%0slope~field:~~~G,$*,\"#7\"\"\"%\"xGF,-%$cosG6#,$*& \"\"%F,F-F,F,F,-%$expG6#,$F-!\"\"F,%\"yGF,F,7$%0initial~point:~G-%!G6$ \"\"!F,7$%/step~width:~~~G$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$%4Sharp-Verner~schemeG$\"+<)HNU$!#B7$%K scheme~with~a~wide~real~stability~intervalG$\"+*p!e*\\#!#A7$%Mscheme~w ith~a~large~imaginary~axis~inclusionG$\"+*4Ujf$F+7$%[pscheme~with~a~qu ite~large~stability~region~and~non-zero~imaginary~axis~inclusionG$\"+H ()R$\\#F+7$%Kscheme~with~a~quite~large~stability~regionG$\"+m7'eI$F+Q) 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 "[0, 5];" "6#7$\"\"!\"\"&" }{TEXT -1 82 " of each Runge-Kutta method is estimated as follows using the spe cial procedure " }{TEXT 0 5 "NCint" }{TEXT -1 98 " to perform numeri cal integration by the 7 point Newton-Cotes method over 200 equal subi ntervals." }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 494 "mthds := [`Sha rp-Verner scheme`,`scheme with a wide real stability interval`,`scheme with a large imaginary axis inclusion`,`scheme with a quite large sta bility region and non-zero imaginary axis inclusion`,`scheme with a qu ite large stability region`]: errs := []:\nDigits := 20:\nfor ct to 5 \+ do\n sm := NCint((f(x)-'fn_RK6_||ct'(x))^2,x=0..5,adaptive=false,num points=7,factor=200);\n errs := [op(errs),sqrt(sm/5)];\nend do:\nDig its := 10:\nlinalg[transpose](convert([mthds,evalf(errs)],matrix));" } }{PARA 11 "" 1 "" {XPPMATH 20 "6#K%'matrixG6#7'7$%4Sharp-Verner~scheme G$\"+\")RcIO!#B7$%Kscheme~with~a~wide~real~stability~intervalG$\"+Kj:< B!#A7$%Mscheme~with~a~large~imaginary~axis~inclusionG$\"+s&QX@&F+7$%[p scheme~with~a~quite~large~stability~region~and~non-zero~imaginary~axis ~inclusionG$\"+-ZPHaF+7$%Kscheme~with~a~quite~large~stability~regionG$ \"+V&y'eZF+Q)pprint196\"" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 " " 0 "" {TEXT -1 92 "The following error graphs are constructed using t he numerical procedures for the solutions." }}{EXCHG {PARA 0 "> " 0 " " {MPLTEXT 1 0 557 "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 ,font=[HELVETICA,9],\ncolor=[COLOR(RGB,.9,0,.9),COLOR(RGB,0,.15,1),COL OR(RGB,.95,0,.2),COLOR(RGB,.95,.45,0),COLOR(RGB,0,.75,.2)],\nlegend=[` Sharp-Verner scheme`,`scheme with a wide real stability interval`,`sch eme with a large imaginary axis inclusion`,`scheme with a quite large \+ stability region and non-zero imag axis inclusion`,`scheme with a quit e large stability region`],title=`error curves for 8 stage order 6 Run ge-Kutta methods`));" }}{PARA 13 "" 1 "" {GLPLOT2D 980 585 585 {PLOTDATA 2 "6+-%'CURVESG6%7ct7$$\"\"!F)F(7$$\"5NLLL$3FWYs#!#@$\"&ee#! #>7$$\"5qmmmmT&)G\\aF-$\"&R)zF07$$\"50+++]7G$R<)F-$\"'Bw:F07$$\"5MLLLL 3x&)*3\"!#?$\"'s)Q#F07$$\"5nmm\"z%\\v#pK\"F>$\"'UJLF07$$\"5+++]i!R(*Rc \"F>$\"'awTF07$$\"5MLL3xJs1,=F>$\"'ug`F07$$\"5nmmm\"H2P\"Q?F>$\"'f$='F 07$$\"5MLLek.pu/BF>$\"'MEsF07$$\"5+++]PMnNrDF>$\"'E>zF07$$\"5nmmT5ll'z $GF>$\"'\"eh)F07$$\"5MLLL$eRwX5$F>$\"'SD\"*F07$$\"5MLL$3F%\\wQKF>$\"'] #G*F07$$\"5MLLLe*[`HP$F>$\"']2%*F07$$\"5MLL3-jx/SMF>$\"'ee%*F07$$\"5ML L$ek.Ur]$F>$\"'=#[*F07$$\"5MLLe*)4jBuNF>$\"'`1&*F07$$\"5MLLLL$eI8k$F>$ \"'/(\\*F07$$\"5MLLL3xwq4RF>$\"'91#*F07$$\"5NLLL$3x%3yTF>$\"'1Z')F07$$ \"5-+](oHaN;J%F>$\"'a\\zF07$$\"5ommT5:j=XWF>$\"''\\V(F07$$\"5NL$eRs3P( yXF>$\"'=SnF07$$\"5-++]PfyG7ZF>$\"'&Hb&F07$$\"5om;/^J'Qe%[F>$\"')3y%F0 7$$\"5NLLek.%*Qz\\F>$\"'YEQF07$$\"5++]7yv,%H6&F>$\"'HpBF07$$\"5ommm\"z 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" }}{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 soluti on." }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 822 "G := (x,y) -> x/y: h h := 0.05: numsteps := 200: x0 := 0: y0 := 1:\nmatrix([[`slope field: \+ `,G(x,y)],[`initial point: `,``(x0,y0)],[`step width: `,hh],\n[`no . of steps: `,numsteps]]);``;\nmthds := [`Sharp-Verner scheme`,`sche me with a wide real stability interval`,`scheme with a large imaginary axis inclusion`,`scheme with a quite large stability region and non-z ero imaginary axis inclusion`,`scheme with a quite large stability reg ion`]: errs := []:\nDigits := 20:\ng := x -> sqrt(1+x^2):\nfor ct to 5 do\n Gn_RK6_||ct := RK6_||ct(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/numpts)];\nend do:\nDigits := 10:\nlinalg[t ranspose](convert([mthds,evalf(errs)],matrix));" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#K%'matrixG6#7&7$%0slope~field:~~~G*&%\"xG\"\"\"%\"yG!\" \"7$%0initial~point:~G-%!G6$\"\"!F+7$%/step~width:~~~G$\"\"&!\"#7$%1no .~of~steps:~~~G\"$+#Q)pprint226\"" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#% !G" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#K%'matrixG6#7'7$%4Sharp-Verner~s chemeG$\"+6 " 0 "" {MPLTEXT 1 0 752 "G := (x,y) -> \+ x/y: hh := 0.05: numsteps := 200: x0 := 0: y0 := 1:\nmatrix([[`slope f ield: `,G(x,y)],[`initial point: `,``(x0,y0)],[`step width: `,hh], \n[`no. of steps: `,numsteps]]);``;\nmthds := [`Sharp-Verner scheme` ,`scheme with a wide real stability interval`,`scheme with a large ima ginary axis inclusion`,`scheme with a quite large stability region and non-zero imaginary axis inclusion`,`scheme with a quite large stabili ty region`]: errs := []:\nDigits := 25:\nfor ct to 5 do\n gn_RK6_||c t := RK6_||ct(G(x,y),x,y,x0,y0,hh,numsteps,true);\nend do:\ng := x -> \+ sqrt(1+x^2):\nxx := 9.99: gxx := evalf(g(xx)):\nfor ct to 5 do\n err s := [op(errs),abs(gn_RK6_||ct(xx)-gxx)];\nend do:\nDigits := 10:\nlin alg[transpose](convert([mthds,evalf(errs)],matrix));" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#K%'matrixG6#7&7$%0slope~field:~~~G*&%\"xG\"\"\"%\"yG !\"\"7$%0initial~point:~G-%!G6$\"\"!F+7$%/step~width:~~~G$\"\"&!\"#7$% 1no.~of~steps:~~~G\"$+#Q)pprint246\"" }}{PARA 11 "" 1 "" {XPPMATH 20 " 6#%!G" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#K%'matrixG6#7'7$%4Sharp-Verne r~schemeG$\"*rT=d$!#B7$%Kscheme~with~a~wide~real~stability~intervalG$ \"+V!p/\"QF+7$%Mscheme~with~a~large~imaginary~axis~inclusionG$\"**G%oQ $F+7$%[pscheme~with~a~quite~large~stability~region~and~non-zero~imagin ary~axis~inclusionG$\"*6LT9'F+7$%Kscheme~with~a~quite~large~stability~ regionG$\")4EDeF+Q)pprint256\"" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }} {PARA 0 "" 0 "" {TEXT -1 4 "The " }{TEXT 260 22 "root mean square erro r" }{TEXT -1 20 " over the interval " }{XPPEDIT 18 0 "[0, 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 met hod over 100 equal subintervals." }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 520 "mthds := [`Sharp-Verner scheme`,`scheme with a wide real stab ility interval`,`scheme with a large imaginary axis inclusion`,\n`sche me with a quite large stability region and non-zero imaginary axis inc lusion`,`scheme with a quite large stability region`]: errs := []:\nDi gits := 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,adaptive=false,numpoints=7,factor=100) ;\n errs := [op(errs),sqrt(sm/10)];\nend do:\nDigits := 10:\nlinalg[ transpose](convert([mthds,evalf(errs)],matrix));" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#K%'matrixG6#7'7$%4Sharp-Verner~schemeG$\"+Lw!>6\"!#B7$% Kscheme~with~a~wide~real~stability~intervalG$\"+rK#)[9!#A7$%Mscheme~wi th~a~large~imaginary~axis~inclusionG$\"+4$pms\"F+7$%[pscheme~with~a~qu ite~large~stability~region~and~non-zero~imaginary~axis~inclusionG$\"+J VCa=F+7$%Kscheme~with~a~quite~large~stability~regionG$\"+X)[gz\"F+Q)pp rint266\"" }}}{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 0 558 "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=[HELVE TICA,9],\ncolor=[COLOR(RGB,.9,0,.9),COLOR(RGB,0,.15,1),COLOR(RGB,.95,0 ,.2),COLOR(RGB,.95,.45,0),COLOR(RGB,0,.75,.2)],\nlegend=[`Sharp-Verner scheme`,`scheme with a wide real stability interval`,`scheme with a l arge imaginary axis inclusion`,`scheme with a quite large stability re gion and non-zero imag axis inclusion`,`scheme with a quite large stab ility region`],title=`error curves for 8 stage order 6 Runge-Kutta met hods`));" }}{PARA 13 "" 1 "" {GLPLOT2D 973 558 558 {PLOTDATA 2 "6+-%'C URVESG6%7bp7$$\"\"!F)F(7$$\"5lmmmmT&)G\\a!#@$!&]F#!#>7$$\"5LLLLL3x&)*3 \"!#?$!&E#pF07$$\"5mmmT5:Q(z:\"F4$!&!>pF07$$\"5+++](=#**3E7F4$!&I$pF07 $$\"5LLLekGg?%H\"F4$!&b/(F07$$\"5mmmmTN@Ki8F4$!&J_(F07$$\"5+++v=U#Q/V \"F4$!&Q0*F07$$\"5LLL$e*[Vb)\\\"F4$!'R98F07$$\"5mmm\"HdXqmc\"F4$!'LE8F 07$$\"5++++]ilyM;F4$!')\\K\"F07$$\"5LLLLe*)4D2>F4$!'!\\[\"F07$$\"5mmmm m;arz@F4$!'**G?F07$$\"5+++D\"yD&y;CF4$!'96AF07$$\"5LLL$e*)4bQl#F4$!'%[ q#F07$$\"5mmmT5S\\#4*GF4$!'y$z#F07$$\"5++++D\"y%*z7$F4$!'jjKF07$$\"5mm m;ajW8-OF4$!'oVOF07$$\"5LLLL$e9ui2%F4$!''e\"QF07$$\"5mm;z>h!z&4UF4$!'- )z$F07$$\"5+++DcwR)GM%F4$!'Y#y$F07$$\"5LL$3F>*))=wWF4$!'*Ry$F07$$\"5mm m;H2Q\\4YF4$!'0tPF07$$\"5LLL3-QO5w[F4$!'%\\r$F07$$\"5++++voMrU^F4$!'P[ NF07$$\"5NLL$3-8Lfn&F4$!'A#=$F07$$\"5mmmmm\"z_\"4iF4$!'\")>FF07$$\"5lm 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" }}{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 0 822 "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 := [`Sharp-Verner scheme`,`scheme with a wide real stability i nterval`,`scheme with a large imaginary axis inclusion`,`scheme with a quite large stability region and non-zero imaginary axis inclusion`,` scheme with a quite large stability region`]: errs := []:\nDigits := 2 0:\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: numpts := nops(Hn_R K6_||ct):\n for ii to numpts do\n sm := sm+(Hn_RK6_||ct[ii,2]-h (Hn_RK6_||ct[ii,1]))^2;\n end do:\n errs := [op(errs),sqrt(sm/nump ts)];\nend do:\nDigits := 10:\nlinalg[transpose](convert([mthds,evalf( errs)],matrix));" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#K%'matrixG6#7&7$%0 slope~field:~~~G,$*&%\"xG\"\"\"%\"yGF,!\"\"7$%0initial~point:~G-%!G6$ \"\"!F,7$%/step~width:~~~G$F,F.7$%1no.~of~steps:~~~G\"$+\"Q)pprint276 \"" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%!G" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#K%'matrixG6#7'7$%4Sharp-Verner~schemeG$\"+T*R8o\"!#?7$% Kscheme~with~a~wide~real~stability~intervalG$\"+OG2IzF+7$%Mscheme~with ~a~large~imaginary~axis~inclusionG$\"+GpA9rF+7$%[pscheme~with~a~quite~ large~stability~region~and~non-zero~imaginary~axis~inclusionG$\"+2Ms;g F+7$%Kscheme~with~a~quite~large~stability~regionG$\"+O\")HE6F+Q)pprint 286\"" }}}{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 "9.99;" "6#-%&F loatG6$\"$***!\"#" }{TEXT -1 16 " is also given." }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 752 "H := (x,y) -> -x*y: hh := 0.1: numsteps := 1 00: x0 := 0: y0 := 1:\nmatrix([[`slope field: `,H(x,y)],[`initial po int: `,``(x0,y0)],[`step width: `,hh],\n[`no. of steps: `,numsteps ]]);``;\nmthds := [`Sharp-Verner scheme`,`scheme with a wide real stab ility interval`,`scheme with a large imaginary axis inclusion`,`scheme with a quite large stability region and non-zero imaginary axis inclu sion`,`scheme with a quite large stability region`]: errs := []:\nDigi ts := 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](convert([mthds, evalf(errs)],matrix));" }}{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)pprin t296\"" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%!G" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#K%'matrixG6#7'7$%4Sharp-Verner~schemeG$\"+c_gME!#N7$%Ks cheme~with~a~wide~real~stability~intervalG$\"+h4pO:!#M7$%Mscheme~with~ a~large~imaginary~axis~inclusionG$\"+t'4;;#F+7$%[pscheme~with~a~quite~ large~stability~region~and~non-zero~imaginary~axis~inclusionG$\"+2#3E2 &F+7$%Kscheme~with~a~quite~large~stability~regionG$\"+`l`)p\"F+Q)pprin t306\"" }}}{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 interval [0, 0.5] of each Runge-Kutta method is estimated as follow s 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 0 519 " mthds := [`Sharp-Verner scheme`,`scheme with a wide real stability int erval`,`scheme with a large imaginary axis inclusion`,\n`scheme with a quite large stability region and non-zero imaginary axis inclusion`,` scheme with a quite large stability region`]: errs := []:\nDigits := 2 0:\nh := x -> exp(-x^2/2):\nfor ct to 5 do\n sm := NCint((h(x)-'hn_R K6_||ct'(x))^2,x=0..10,adaptive=false,numpoints=7,factor=50);\n errs := [op(errs),sqrt(sm/10)];\nend do:\nDigits := 10:\nlinalg[transpose] (convert([mthds,evalf(errs)],matrix));" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#K%'matrixG6#7'7$%4Sharp-Verner~schemeG$\"+)HJuc\"!#?7$%Kscheme~w ith~a~wide~real~stability~intervalG$\"+*[ZA_(F+7$%Mscheme~with~a~large ~imaginary~axis~inclusionG$\"+w0$[n'F+7$%[pscheme~with~a~quite~large~s tability~region~and~non-zero~imaginary~axis~inclusionG$\"+]]cxcF+7$%Ks cheme~with~a~quite~large~stability~regionG$\"+HT[%3\"F+Q)pprint316\"" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 92 "The fo llowing error graphs are constructed using the numerical procedures fo r the solutions." }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 552 "evalf[2 0](plot(['hn_RK6_1'(x)-h(x),'hn_RK6_2'(x)-h(x),'hn_RK6_3'(x)-h(x),\n'h n_RK6_4'(x)-h(x),'hn_RK6_5'(x)-h(x)],x=0..6,numpoints=100,\ncolor=[COL OR(RGB,.9,0,.9),COLOR(RGB,0,.15,1),COLOR(RGB,.95,0,.2),COLOR(RGB,.95,. 45,0),COLOR(RGB,0,.75,.2)],\nlegend=[`Sharp-Verner scheme`,`scheme wit h a wide real stability interval`,`scheme with a large imaginary axis \+ inclusion`,`scheme with a quite large stability region and non-zero im ag axis inclusion`,`scheme with a quite large stability region`],title =`error curves for 8 stage order 6 Runge-Kutta methods`));" }}{PARA 13 "" 1 "" {GLPLOT2D 959 562 562 {PLOTDATA 2 "6*-%'CURVESG6%7ecl7$$\" \"!F)F(7$$\"5_^^^^^^\\qJ!#@$\"%,e!#?7$$\"5.......*4M'F-$\"(N?[\"F07$$ \"5!======@'*4*F-$\")H9eEF07$$\"5111111_#e=\"F0$\")>qOcF07$$\"5.....`O 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G. Lether: Mathematics of Computation, Vol. 20, no. 95, (July 1966) \+ page 381. " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "dy/dx = -32*x*y*ln(2);" "6#/*&%#dyG\"\"\"%#dx G!\"\",$**\"#KF&%\"xGF&%\"yGF&-%#lnG6#\"\"#F&F(" }{TEXT -1 6 ", " }{XPPEDIT 18 0 "y(-1) = 1/8;" "6#/-%\"yG6#,$\"\"\"!\"\"*&F(F(\"\")F)" }{TEXT -1 1 " " }}{PARA 0 "" 0 "" {TEXT -1 10 "Solution: " }}{PARA 256 "" 0 "" {TEXT -1 2 " " }{XPPEDIT 18 0 "y = 2^(13-6*x^2);" "6#/%\" yG)\"\"#,&\"#8\"\"\"*&\"\"'F)*$%\"xGF&F)!\"\"" }{TEXT -1 2 ". " }} {PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 191 "de := diff(y(x),x)=-32*x*y(x)*ln(2);\nic := y(-1)=1/8;\ndsolve(\{ de,ic\},y(x)):\ny(x)=2^simplify(log[2](rhs(%)));\nk := unapply(rhs(%), x):\nplot(k(x),x=-1..1,font=[HELVETICA,9],labels=[`x`,`y(x)`]);" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#>%#deG/-%%diffG6$-%\"yG6#%\"xGF,,$**\" #K\"\"\"F,F0F)F0-%#lnG6#\"\"#F0!\"\"" }}{PARA 11 "" 1 "" {XPPMATH 20 " 6#>%#icG/-%\"yG6#!\"\"#\"\"\"\"\")" }}{PARA 11 "" 1 "" {XPPMATH 20 "6# /-%\"yG6#%\"xG)\"\"#,&\"#8\"\"\"*&\"#;F,)F'F)F,!\"\"" }}{PARA 13 "" 1 "" {GLPLOT2D 577 286 286 {PLOTDATA 2 "6&-%'CURVESG6$7eo7$$!\"\"\"\"!$ \"3+++++++]7!#=7$$!3ommm;p0k&*F-$\"3!Hg[[\"[$)=KF-7$$!3wKL$3$3(F-7$$!3mmmmT%p\"e()F-$\"3!=E-TWD`l\"!#<7$$!3:mmm\"4m(G$)F- $\"3M\"fONp()[t$F=7$$!3\"QLL3i.9!zF-$\"3A!e'4A%*Rg!)F=7$$!3\"ommT!R=0v F-$\"3%z2Mbncie\"!#;7$$!3u****\\P8#\\4(F-$\"3C>dT>$)H#3$FM7$$!3+nm;/si qmF-$\"3gp%*z`g)4*eFM7$$!3[++](y$pZiF-$\"3%R6L-Y$zz5!#:7$$!33LLL$yaE\" 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{MPLTEXT 1 0 830 "K := (x,y) -> -32*x*y(x)*ln(2): hh := 0.01 : numsteps := 200: x0 := -1: y0 := 1/8:\nmatrix([[`slope field: `,K( x,y)],[`initial point: `,``(x0,y0)],[`step width: `,hh],\n[`no. of s teps: `,numsteps]]);``;\nmthds := [`Sharp-Verner scheme`,`scheme wit h a wide real stability interval`,`scheme with a large imaginary axis \+ inclusion`,\n`scheme with a quite large stability region and non-zero \+ imaginary axis inclusion`,`scheme with a quite large stability region` ]: errs := []:\nDigits := 20:\nfor ct to 5 do\n Kn_RK6_||ct := RK6_| |ct(evalf(K(x,y)),x,y,x0,evalf(y0),hh,numsteps,false);\n sm := 0: nu mpts := nops(Kn_RK6_||ct):\n for ii to numpts do\n sm := sm+(Kn _RK6_||ct[ii,2]-k(Kn_RK6_||ct[ii,1]))^2;\n end do:\n errs := [op(e rrs),sqrt(sm/numpts)];\nend do:\nDigits := 10:\nlinalg[transpose](conv ert([mthds,evalf(errs)],matrix));" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#K %'matrixG6#7&7$%0slope~field:~~~G,$**\"#K\"\"\"%\"xGF,-%\"yG6#F-F,-%#l nG6#\"\"#F,!\"\"7$%0initial~point:~G-%!G6$F5#F,\"\")7$%/step~width:~~~ G$F,!\"#7$%1no.~of~steps:~~~G\"$+#Q)pprint326\"" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%!G" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#K%'matrixG6#7'7$ %4Sharp-Verner~schemeG$\"+p)z`g%!#:7$%Kscheme~with~a~wide~real~stabili ty~intervalG$\"+uL1TD!#97$%Mscheme~with~a~large~imaginary~axis~inclusi onG$\"+W5>:xF+7$%[pscheme~with~a~quite~large~stability~region~and~non- zero~imaginary~axis~inclusionG$\"+='Go]\"F+7$%Kscheme~with~a~quite~lar ge~stability~regionG$\"+S.p3]F+Q)pprint336\"" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 30 "The following code constr ucts " }{TEXT 260 20 "numerical procedures" }{TEXT -1 56 " for solutio ns based on each of the Runge-Kutta schemes." }}{PARA 0 "" 0 "" {TEXT -1 75 "The error in the value obtained by each of the methods at the p oint where " }{XPPEDIT 18 0 "x = 0;" "6#/%\"xG\"\"!" }{TEXT -1 20 ".9 95 is also given." }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 760 "K := \+ (x,y) -> -32*x*y(x)*ln(2): hh := 0.01: numsteps := 200: x0 := -1: y0 : = 1/8:\nmatrix([[`slope field: `,K(x,y)],[`initial point: `,``(x0,y0 )],[`step width: `,hh],\n[`no. of steps: `,numsteps]]);``;\nmthds \+ := [`Sharp-Verner scheme`,`scheme with a wide real stability interval` ,`scheme with a large imaginary axis inclusion`,`scheme with a quite l arge stability region and non-zero imaginary axis inclusion`,`scheme w ith a quite large stability region`]: errs := []:\nDigits := 20:\nfor \+ ct to 5 do\n kn_RK6_||ct := RK6_||ct(evalf(K(x,y)),x,y,x0,evalf(y0), hh,numsteps,true);\nend do:\nxx := 0.995: kxx := evalf(k(xx)):\nfor ct to 5 do\n errs := [op(errs),abs(kn_RK6_||ct(xx)-kxx)];\nend do:\nDi gits := 10:\nlinalg[transpose](convert([mthds,evalf(errs)],matrix));" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#K%'matrixG6#7&7$%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 )pprint346\"" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%!G" }}{PARA 11 "" 1 " " {XPPMATH 20 "6#K%'matrixG6#7'7$%4Sharp-Verner~schemeG$\"+P%4\")o%!#? 7$%Kscheme~with~a~wide~real~stability~intervalG$\"++i&Q<\"!#>7$%Mschem e~with~a~large~imaginary~axis~inclusionG$\"+WbEx#*F+7$%[pscheme~with~a ~quite~large~stability~region~and~non-zero~imaginary~axis~inclusionG$ \"+VEfoJF+7$%Kscheme~with~a~quite~large~stability~regionG$\"*w))3z&F+Q )pprint356\"" }}}{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 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 100 e qual subintervals." }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 495 "mthds := [`Sharp-Verner scheme`,`scheme with a wide real stability interval `,`scheme with a large imaginary axis inclusion`,`scheme with a quite \+ large stability region and non-zero imaginary axis inclusion`,`scheme \+ with a quite large stability region`]: errs := []:\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)];\nen d do:\nDigits := 10:\nlinalg[transpose](convert([mthds,evalf(errs)],ma trix));" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#K%'matrixG6#7'7$%4Sharp-Ver ner~schemeG$\"+o9)oh%!#:7$%Kscheme~with~a~wide~real~stability~interval G$\"+c\"3ua#!#97$%Mscheme~with~a~large~imaginary~axis~inclusionG$\"+:M XMxF+7$%[pscheme~with~a~quite~large~stability~region~and~non-zero~imag inary~axis~inclusionG$\"+)f#f5:F+7$%Kscheme~with~a~quite~large~stabili ty~regionG$\"+%p)>@]F+Q)pprint366\"" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 92 "The following error graphs are construc ted using the numerical procedures for the solutions." }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 558 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Fgp$\",Oo>`T$F`p7$F\\q$\",:m(*Q\"RF`p7$Faq$\",Avq$\\WF`p7$Ffq$\",b%H98 ^F`p7$F[r$\",pT;B\"eF`p7$F`r$\",?V)p^kF`p7$Fer$\",N`c/5(F`p7$Fjr$\",== W-\"yF`p7$F_s$\",([;)G])F`p7$Fds$\",Ls>:8*F`p7$Fis$\",F&zJ:(*F`p7$F^t$ \"-vUq8E5F`p7$Fct$\"-5$)R#G2\"F`p7$Fht$\"-#yRyv5\"F`p7$F^u$\"-'p/qO8\" F`p7$Fcu$\"-^mc!R9\"F`p7$Fhu$\"-4w3N^6F`p7$F]v$\"-pq3,a6F`p7$Fbv$\"-$y #*\\f:\"F`p7$Fgv$\"-$*\\6:\"F`p7$Ffx$\"-e# pFX9\"F`p7$F[y$\"-<@Y/M6F`p7$F`y$\"-mSm&)36F`p7$Fey$\"-`bcRv5F`p7$Fjy$ \"--`+)3.\"F`p7$F_z$\",#4.F)y*F`p7$Fdz$\",>Unc=*F`p7$Fiz$\",R')**G`)F` p7$F^[l$\",vHN9'yF`p7$Fc[l$\",(f)pE<(F`p7$Fh[l$\",8HvK]'F`p7$F]\\l$\", L!*fH%eF`p7$Fb\\l$\",/w7J8&F`p7$Fg\\l$\",Lw$HfWF`p7$F\\]l$\",X1MB*QF`p 7$Fa]l$\",)e#\\pO$F`p7$Ff]l$\",Hf*)oR#F`p7$F[^l$\",L;)H+XFfn7$F__l$\",35Gjq#Ffn 7$Fd_l$\",X0'H;:Ffn7$Fi_l$\",RHF=J)FL7$F^`l$\",/LD5G%FL7$Fc`l$\",+7')* *4#FL7$Fh`l$\"-f2W\\L5F<7$F]al$\",?\"yc$G%F<7$Fbal$\",2@D;w\"F<7$Fgal$ \",Kmp/&fF/7$F\\bl$\",YVqKa\"F/7$Fabl$!+Swfb6F/-Ffbl6&FhblF+$\"#vF^amF a_n-F\\cl6#%Kscheme~with~a~quite~large~stability~regionG-%%FONTG6$%*HE LVETICAGFjbl-%&TITLEG6#%Uerror~curves~for~8~stage~order~6~Runge-Kutta~ methodsG-%+AXESLABELSG6$Q\"x6\"Q!Ff\\p-%%VIEWG6$;F(Fabl%(DEFAULTG" 1 2 0 1 10 0 2 9 1 4 2 1.000000 45.000000 45.000000 0 1 "Sharp-Verner sc heme" "scheme with a wide real stability interval" "scheme with a larg e imaginary axis inclusion" "scheme with a quite large stability regio n and non-zero imag axis inclusion" "scheme with a quite large stabili ty region" }}}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 " " {MPLTEXT 1 0 1 ";" }}}}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 47 "Test 5 o f 8 stage, order 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 "de := diff(y(x),x)=16/((16*x+1)*y(x));\nic := y(0)=1;\ndsolve( \{de,ic\},y(x));\ns := unapply(rhs(%),x):\nplot(s(x),x=0..0.5,0..2.6,f ont=[HELVETICA,9],labels=[`x`,`y(x)`]);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%#deG/-%%diffG6$-%\"yG6#%\"xGF,,$*(\"#;\"\"\",&*&F/F0F,F0F0F0F0 !\"\"F)F3F0" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%#icG/-%\"yG6#\"\"!\" \"\"" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/-%\"yG6#%\"xG*$,&*&\"\"#\"\" \"-%#lnG6#,&*&\"#;F,F'F,F,F,F,F,F,F,F,#F,F+" }}{PARA 13 "" 1 "" {GLPLOT2D 503 318 318 {PLOTDATA 2 "6&-%'CURVESG6$7U7$$\"\"!F)$\"\"\"F) 7$$\"3WmmmT&)G\\a!#?$\"3EP,(Qyl.3\"!#<7$$\"3ILLL3x&)*3\"!#>$\"3?25A!pa &\\6F27$$\"3-+]i!R(*Rc\"F6$\"3oz*p77wF?\"F27$$\"3umm\"H2P\"Q?F6$\"3]_v ibZz]7F27$$\"3MLL$eRwX5$F6$\"3!Qb+fY'3W8F27$$\"3CLL$3x%3yTF6$\"31#\\\\ E7=EU\"F27$$\"3=mm\"z%4\\Y_F6$\"3s(e$4OUg*[\"F27$$\"3)HL$eR-/PiF6$\"3. fPtw=4W:F27$$\"3A***\\il'pisF6$\"3/07@a`R%f\"F27$$\"3`KLe*)>VB$)F6$\"3 K!\\`od36k\"F27$$\"3!))**\\7`l2Q*F6$\"3#HUv\"fmC$o\"F27$$\"3smm;/j$o/ \"!#=$\"3:'H!f>cuAjU6Fco$\"3K$o8QC!za=F27$$\" 3)*****\\P[6j9Fco$\"39iuo+OIZ=F27$$\"3KL$e*[z(yb\"Fco$\"3Q:]fA\\>F27$ $\"3))**\\iSj0x=Fco$\"3-5Hbh&QF%>F27$$\"3Wmmm\"pW`(>Fco$\"3So#znsrC'>F 27$$\"35+]i!f#=$3#Fco$\"3w)>Y)R!pI)>F27$$\"3/+](=xpe=#Fco$\"3?*eB@.[<+ #F27$$\"3smm\"H28IH#Fco$\"3/Fyh^(\\.-#F27$$\"3km;zpSS\"R#Fco$\"3)4US+% ypO?F27$$\"3GLL3_?`(\\#Fco$\"3#4Cj+a0O0#F27$$\"3#HLe*)>pxg#Fco$\"3ab\\ mG7Vq?F27$$\"3u**\\Pf4t.FFco$\"3Cx7m@=^%3#F27$$\"32LLe*Gst!GFco$\"3Q

>IFco $\"3&ocGC'[]F@F27$$\"3h**\\i!RU07$Fco$\"3HCH$Q\")f.9#F27$$\"3b***\\(=S 2LKFco$\"3C`wrWc9a@F27$$\"3Kmmm\"p)=MLFco$\"3;=S,IA7m@F27$$\"3!*****\\ (=]@W$Fco$\"3w4%eC\"p]y@F27$$\"35L$e*[$z*RNFco$\"3UyOr,.R*=#F27$$\"3#* ****\\iC$pk$Fco$\"3wIdFs1%4?#F27$$\"39m;H2qcZPFco$\"3Qbx\"QY%\\6AF27$$ \"3q**\\7.\"fF&QFco$\"3f+!e(oz@AAF27$$\"3Ymm;/OgbRFco$\"36qG(yA8CB#F27 $$\"3y**\\ilAFjSFco$\"3v.zLgjzUAF27$$\"3YLLL$)*pp;%Fco$\"3IImU*yHDD#F2 7$$\"3?LL3xe,tUFco$\"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;LLL$)Qk%o%Fco$\"35e8d5)>wH#F27$$\"37+]iSjE!z%Fco$\"3e%4h. zwhI#F27$$\"35+]P40O\"*[Fco$\"3Nwd2,K=9BF27$$\"3++++++++]Fco$\"3m'>()) [`fABF2-%'COLOURG6&%$RGBG$\"#5!\"\"F(F(-%%FONTG6$%*HELVETICAG\"\"*-%+A XESLABELSG6$%\"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 cod e constructs a " }{TEXT 260 17 "discrete solution" }{TEXT -1 44 " base d on each of the methods and gives the " }{TEXT 260 22 "root mean squa re error" }{TEXT -1 18 " of each solution." }}{EXCHG {PARA 0 "> " 0 " " {MPLTEXT 1 0 812 "S := (x,y) -> 16/((16*x+1)*y): hh := 0.005: numste ps := 100: x0 := 0: y0 := 1:\nmatrix([[`slope field: `,S(x,y)],[`ini tial point: `,``(x0,y0)],[`step width: `,hh],\n[`no. of steps: `,n umsteps]]);``;\nmthds := [`Sharp-Verner scheme`,`scheme with a wide re al stability interval`,`scheme with a large imaginary axis inclusion`, `scheme with a quite large stability region and non-zero imaginary axi s inclusion`,`scheme with a quite large stability region`]: errs := [] :\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: numpts := 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(errs),sqrt(sm/numpts)];\nen d do:\nDigits := 10:\nlinalg[transpose](convert([mthds,evalf(errs)],ma trix));" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#K%'matrixG6#7&7$%0slope~fie ld:~~~G,$*(\"#;\"\"\",&*&F+F,%\"xGF,F,F,F,!\"\"%\"yGF0F,7$%0initial~po int:~G-%!G6$\"\"!F,7$%/step~width:~~~G$\"\"&!\"$7$%1no.~of~steps:~~~G \"$+\"Q)pprint376\"" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%!G" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#K%'matrixG6#7'7$%4Sharp-Verner~schemeG$\"+/% *33V!#?7$%Kscheme~with~a~wide~real~stability~intervalG$\"+_v4-;!#>7$%M scheme~with~a~large~imaginary~axis~inclusionG$\"+a>4FcF+7$%[pscheme~wi th~a~quite~large~stability~region~and~non-zero~imaginary~axis~inclusio nG$\"+J&3l_$F+7$%Kscheme~with~a~quite~large~stability~regionG$\"+[RMG' )F+Q)pprint386\"" }}}{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 0 744 "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 := [`Sharp-Verner scheme`,`schem e with a wide real stability interval`,`scheme with a large imaginary \+ axis inclusion`,`scheme with a quite large stability region and non-ze ro imaginary axis inclusion`,`scheme with a quite large stability regi on`]: errs := []:\nDigits := 20:\nfor ct to 5 do\n sn_RK6_||ct := RK 6_||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 errs := [op(errs),abs(sn_RK6_||c t(xx)-sxx)];\nend do:\nDigits := 10:\nlinalg[transpose](convert([mthds ,evalf(errs)],matrix));" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#K%'matrixG6 #7&7$%0slope~field:~~~G,$*(\"#;\"\"\",&*&F+F,%\"xGF,F,F,F,!\"\"%\"yGF0 F,7$%0initial~point:~G-%!G6$\"\"!F,7$%/step~width:~~~G$\"\"&!\"$7$%1no .~of~steps:~~~G\"$+\"Q)pprint396\"" }}{PARA 11 "" 1 "" {XPPMATH 20 "6# %!G" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#K%'matrixG6#7'7$%4Sharp-Verner~ schemeG$\"*4V#fN!#>7$%Kscheme~with~a~wide~real~stability~intervalG$\"+ L.IB8F+7$%Mscheme~with~a~large~imaginary~axis~inclusionG$\"*kQ+n%F+7$% [pscheme~with~a~quite~large~stability~region~and~non-zero~imaginary~ax is~inclusionG$\"*&H`AHF+7$%Kscheme~with~a~quite~large~stability~region G$\"*qlk8(F+Q)pprint406\"" }}}{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 interval [0, 0.5] of each Runge-Kutta method is estimated as follows using the special procedure " }{TEXT 0 5 "NC int" }{TEXT -1 97 " to perform numerical integration by the 7 point N ewton-Cotes method over 50 equal subintervals." }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 497 "mthds := [`Sharp-Verner scheme`,`scheme with a \+ wide real stability interval`,`scheme with a large imaginary axis incl usion`,`scheme with a quite large stability region and non-zero imagin ary axis inclusion`,`scheme with a quite large stability region`]: err s := []:\nDigits := 20:\nfor ct to 5 do\n sm := NCint((s(x)-'sn_RK6_ ||ct'(x))^2,x=0..0.5,adaptive=false,numpoints=7,factor=50);\n errs : = [op(errs),sqrt(sm/0.5)];\nend do:\nDigits := 10:\nlinalg[transpose]( convert([mthds,evalf(errs)],matrix));" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#K%'matrixG6#7'7$%4Sharp-Verner~schemeG$\"+T_h1V!#?7$%Kscheme~with~a ~wide~real~stability~intervalG$\"+0_b,;!#>7$%Mscheme~with~a~large~imag inary~axis~inclusionG$\"+<]7DcF+7$%[pscheme~with~a~quite~large~stabili ty~region~and~non-zero~imaginary~axis~inclusionG$\"+?HCDNF+7$%Kscheme~ with~a~quite~large~stability~regionG$\"+(*oOD')F+Q)pprint416\"" }}} {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 0 559 "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=[HELVETICA,9],\ncolor= [COLOR(RGB,.9,0,.9),COLOR(RGB,0,.15,1),COLOR(RGB,.95,0,.2),COLOR(RGB,. 95,.45,0),COLOR(RGB,0,.75,.2)],\nlegend=[`Sharp-Verner scheme`,`scheme with a wide real stability interval`,`scheme with a large imaginary a xis inclusion`,`scheme with a quite large stability region and non-zer o imag axis inclusion`,`scheme with a quite large stability region`],t itle=`error curves for 8 stage order 6 Runge-Kutta methods`));" }} {PARA 13 "" 1 "" {GLPLOT2D 982 531 531 {PLOTDATA 2 "6+-%'CURVESG6%7^p7 $$\"\"!F)F(7$$\"5SLLL3x1h6o!#B$!$q'!#>7$$\"5ommmTN@Ki8!#A$!&E(yF07$$\" 5NLL3FpE!Hq\"F4$!'n)f$F07$$\"5-++]7.K[V?F4$!(@kB\"F07$$\"5omm\"zptjSQ# F4$!(o&)[$F07$$\"5NLLL$3FWYs#F4$!(_>_)F07$$\"5-++vo/[AlIF4$!)v'['=F07$ 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" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 169 "de := diff( y(x),x)=(1+2*(x+1)*sin(3*x))*exp(-y(x));\nic := y(0)=0;\ndsolve(\{de,i c\},y(x));\nu := unapply(rhs(%),x):\nplot(u(x),x=0..5,font=[HELVETICA, 9],labels=[`x`,`y(x)`]);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%#deG/-%% diffG6$-%\"yG6#%\"xGF,*&,&\"\"\"F/*(\"\"#F/,&F,F/F/F/F/-%$sinG6#,$*&\" \"$F/F,F/F/F/F/F/-%$expG6#,$F)!\"\"F/" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%#icG/-%\"yG6#\"\"!F)" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/-%\"yG6# %\"xG-%#lnG6#,,F'\"\"\"*&#\"\"#\"\"*F,-%$sinG6#,$*&\"\"$F,F'F,F,F,F,*& #F/F6F,*&F'F,-%$cosGF3F,F,!\"\"*&#F/F6F,F:F,F<#\"\"&F6F," }}{PARA 13 " " 1 "" {GLPLOT2D 503 318 318 {PLOTDATA 2 "6&-%'CURVESG6$7bp7$$\"\"!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 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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 0 824 "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 \+ := [`Sharp-Verner scheme`,`scheme with a wide real stability interval` ,`scheme with a large imaginary axis inclusion`,`scheme with a quite l arge stability region and non-zero imaginary axis inclusion`,`scheme w ith a quite large stability region`]: errs := []:\nDigits := 25:\nfor \+ ct to 5 do\n Un_RK6_||ct := RK6_||ct(U(x,y),x,y,x0,y0,hh,numsteps,fa lse);\n sm := 0: numpts := nops(Un_RK6_||ct):\n for ii to numpts d o\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:\nl inalg[transpose](convert([mthds,evalf(errs)],matrix));" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#K%'matrixG6#7&7$%0slope~field:~~~G*&,&\"\"\"F+*(\" \"#F+,&%\"xGF+F+F+F+-%$sinG6#,$*&\"\"$F+F/F+F+F+F+F+-%$expG6#,$%\"yG! \"\"F+7$%0initial~point:~G-%!G6$\"\"!FA7$%/step~width:~~~G$F+!\"#7$%1n o.~of~steps:~~~G\"$+&Q)pprint426\"" }}{PARA 11 "" 1 "" {XPPMATH 20 "6# %!G" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#K%'matrixG6#7'7$%4Sharp-Verner~ schemeG$\"+7Z2y`!#C7$%Kscheme~with~a~wide~real~stability~intervalG$\"+ NfHU7!#B7$%Mscheme~with~a~large~imaginary~axis~inclusionG$\"+*ykbY%F+7 $%[pscheme~with~a~quite~large~stability~region~and~non-zero~imaginary~ axis~inclusionG$\"+^!>8`)F+7$%Kscheme~with~a~quite~large~stability~reg ionG$\"+!*>@6[F+Q)pprint436\"" }}}{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 0 755 " 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 := [`Sharp-Verner scheme`,`scheme with a wide real stabi lity interval`,`scheme with a large imaginary axis inclusion`,`scheme \+ with a quite large stability region and non-zero imaginary axis inclus ion`,`scheme with a quite large stability region`]: errs := []:\nDigit s := 30:\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:\nDi gits := 10:\nlinalg[transpose](convert([mthds,evalf(errs)],matrix));" }}{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+-%$e xpG6#,$%\"yG!\"\"F+7$%0initial~point:~G-%!G6$\"\"!FA7$%/step~width:~~~ G$F+!\"#7$%1no.~of~steps:~~~G\"$+&Q)pprint466\"" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%!G" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#K%'matrixG6#7'7$ %4Sharp-Verner~schemeG$\"+v&)y(Q$!#C7$%Kscheme~with~a~wide~real~stabil ity~intervalG$\"+@\"*o\\[F+7$%Mscheme~with~a~large~imaginary~axis~incl usionG$\"+ba?R`!#D7$%[pscheme~with~a~quite~large~stability~region~and~ non-zero~imaginary~axis~inclusionG$\"+A=.)>'F47$%Kscheme~with~a~quite~ large~stability~regionG$\"+.^&\\q#F+Q)pprint476\"" }}}{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 "NCi nt" }{TEXT -1 98 " to perform numerical integration by the 7 point Ne wton-Cotes method over 200 equal subintervals." }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 494 "mthds := [`Sharp-Verner scheme`,`scheme with a \+ wide real stability interval`,`scheme with a large imaginary axis incl usion`,`scheme with a quite large stability region and non-zero imagin ary axis inclusion`,`scheme with a quite large stability region`]: err s := []:\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](con vert([mthds,evalf(errs)],matrix));" }}{PARA 11 "" 1 "" {XPPMATH 20 "6# K%'matrixG6#7'7$%4Sharp-Verner~schemeG$\"+@\"p9Q&!#C7$%Kscheme~with~a~ wide~real~stability~intervalG$\"+cPMQ7!#B7$%Mscheme~with~a~large~imagi nary~axis~inclusionG$\"+6H,hWF+7$%[pscheme~with~a~quite~large~stabilit y~region~and~non-zero~imaginary~axis~inclusionG$\"+)*\\'=`)F+7$%Kschem e~with~a~quite~large~stability~regionG$\"+I4B=[F+Q)pprint486\"" }}} {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 0 557 "evalf[20]( plot(['un_RK6_1'(x)-u(x),'un_RK6_2'(x)-u(x),'un_RK6_3'(x)-u(x),'un_RK6 _4'(x)-u(x),\n'un_RK6_5'(x)-u(x)],x=0..5,font=[HELVETICA,9],\ncolor=[C OLOR(RGB,.9,0,.9),COLOR(RGB,0,.15,1),COLOR(RGB,.95,0,.2),COLOR(RGB,.95 ,.45,0),COLOR(RGB,0,.75,.2)],\nlegend=[`Sharp-Verner scheme`,`scheme w ith a wide real stability interval`,`scheme with a large imaginary axi s inclusion`,`scheme with a quite large stability region and non-zero \+ imag axis inclusion`,`scheme with a quite large stability region`],tit le=`error curves for 8 stage order 6 Runge-Kutta methods`));" }}{PARA 13 "" 1 "" {GLPLOT2D 1016 586 586 {PLOTDATA 2 "6+-%'CURVESG6%7as7$$\" \"!F)F(7$$\"5qmmmmT&)G\\a!#@$!'?G[F-7$$\"5MLLLL3x&)*3\"!#?$!&#zwF37$$ \"5+++]i!R(*Rc\"F3$!&ps)F37$$\"5nmmm\"H2P\"Q?F3$!&pY*F37$$\"5MLLL$eRwX 5$F3$!':*=\"F37$$\"5NLLL$3x%3yTF3$!'O_9F37$$\"5ommm\"z%4\\Y_F3$!'`J;F3 7$$\"5NLLLeR-/PiF3$!&&e;!#>7$$\"5-+++DcmpisF3$!&$*f\"FT7$$\"5OLLLe*)>V B$)F3$!&r^\"FT7$$\"5.+++DJbw!Q*F3$!&;W\"FT7$$\"5nmmm;/j$o/\"FT$!&dR\"F T7$$\"5MLLL3_>jU6FT$!&_Q\"FT7$$\"5++++]i^Z]7FT$!&#*R\"FT7$$\"5++++](=h (e8FT$!&QT\"FT7$$\"5++++]7!Q4T\"FT$!&`Q\"FT7$$\"5++++]P[6j9FT$!&-K\"FT 7$$\"5nmm\"HKR'\\5:FT$!&)o6FT7$$\"5MLL$e*[z(yb\"FT$!%C&*FT7$$\"5+++Dc, 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" }}{PARA 0 "" 0 "" {TEXT -1 10 "Solutio n: " }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "y = 1/3" "6#/% \"yG*&\"\"\"F&\"\"$!\"\"" }{TEXT -1 1 " " }{XPPEDIT 18 0 "exp(-4/3*sin (3*x)+8/3*sin(3/2*x)*cos(3/2*x))+2/3" "6#,&-%$expG6#,&*(\"\"%\"\"\"\" \"$!\"\"-%$sinG6#*&F+F*%\"xGF*F*F,**\"\")F*F+F,-F.6#*(F+F*\"\"#F,F1F*F *-%$cosG6#*(F+F*F7F,F1F*F*F*F**&F7F*F+F,F*" }{TEXT -1 1 " " }{XPPEDIT 18 0 "exp(-4/3*sin(3*x)-x)" "6#-%$expG6#,&*(\"\"%\"\"\"\"\"$!\"\"-%$si nG6#*&F*F)%\"xGF)F)F+F0F+" }{TEXT -1 2 ". " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 181 "de := diff(y(x),x )=-(1+4*cos(3*x))*(y(x)-1/3);\nic := y(0)=1;\nsimplify(dsolve(\{de,ic \},y(x)));\nv := unapply(rhs(%),x):\nplot(v(x),x=0..5,0..1.1,font=[HEL VETICA,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$$\"\"!F)$\"\"\"F)7$$\"3gmm TN@Ki8!#>$\"3W+7cSy5h&*!#=7$$\"3ALL$3FWYs#F/$\"3KtP[t*Q;:*F27$$\"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,'pRB 0LX'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[hQO W]BpF27$$\"37++]i^Z]7Fgt$\"3IVnF)*yXIuF27$$\"35+++v\"=YI\"Fgt$\"3ahS!3 L%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)\\\"F gt$\"3p\\\\U!)G36))F27$$\"3%o;HKR'\\5:Fgt$\"3%G4&GMdV(z)F27$$\"3-]P4rr =M:Fgt$\"3Erd.MaCV()F27$$\"3UL$e*[z(yb\"Fgt$\"3m)))[\\1qQl)F27$$\"34+D c,#>Uh\"Fgt$\"3(fTb\\\\y3J)F27$$\"3wmm;a/cq;Fgt$\"3-!y\"yF27$$\" 3\"pm;a)))G=BtF27$$\"3%ommmJFgt$\"3%RlX>.MR=&F27$$\"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$$\"3Jmmm\"p)=MLFgt$\"3RsXuk([b#Q F27$$\"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-Tml[`MOF27$$\"3w**\\ilAFjSFgt$\" 3&zNMj#[Z%Fgt$ \"3ADU\\K%G5O$F27$$\"3SnmT&G!e&e%Fgt$\"35gRzc#\\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 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 0 819 "V := \+ (x,y) -> -(1+4*cos(3*x))*(y-1/3): hh := 0.02: numsteps := 250: x0 := 0 : y0 := 1:\nmatrix([[`slope field: `,V(x,y)],[`initial point: `,``(x 0,y0)],[`step width: `,hh],\n[`no. of steps: `,numsteps]]);``;\nmt hds := [`Sharp-Verner scheme`,`scheme with a wide real stability inter val`,`scheme with a large imaginary axis inclusion`,`scheme with a qui te large stability region and non-zero imaginary axis inclusion`,`sche me with a quite large stability region`]: errs := []:\nDigits := 30:\n for ct to 5 do\n Vn_RK6_||ct := RK6_||ct(V(x,y),x,y,x0,y0,hh,numstep s,false);\n sm := 0: numpts := nops(Vn_RK6_||ct):\n for ii to nump ts 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[transpose](convert([mthds,evalf(errs)],matrix));" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#K%'matrixG6#7&7$%0slope~field:~~~G,$*&,&\"\" \"F,*&\"\"%F,-%$cosG6#,$*&\"\"$F,%\"xGF,F,F,F,F,,&%\"yGF,#F,F4!\"\"F,F 97$%0initial~point:~G-%!G6$\"\"!F,7$%/step~width:~~~G$\"\"#!\"#7$%1no. ~of~steps:~~~G\"$]#Q)pprint496\"" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%! G" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#K%'matrixG6#7'7$%4Sharp-Verner~sc hemeG$\"+ITT+6!#@7$%Kscheme~with~a~wide~real~stability~intervalG$\"+vZ )oK#F+7$%Mscheme~with~a~large~imaginary~axis~inclusionG$\"+*[&RlHF+7$% [pscheme~with~a~quite~large~stability~region~and~non-zero~imaginary~ax is~inclusionG$\"+^mQE`F+7$%Kscheme~with~a~quite~large~stability~region G$\"+*=xmG)!#AQ)pprint506\"" }}}{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 th e Runge-Kutta schemes." }}{PARA 0 "" 0 "" {TEXT -1 75 "The error in th e 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 0 750 "V : = (x,y) -> -(1+4*cos(3*x))*(y-1/3): hh := 0.02: numsteps := 250: x0 := 0: y0 := 1:\nmatrix([[`slope field: `,V(x,y)],[`initial point: `,`` (x0,y0)],[`step width: `,hh],\n[`no. of steps: `,numsteps]]);``;\n mthds := [`Sharp-Verner scheme`,`scheme with a wide real stability int erval`,`scheme with a large imaginary axis inclusion`,`scheme with a q uite large stability region and non-zero imaginary axis inclusion`,`sc heme with a quite large stability region`]: errs := []:\nDigits := 30: \nfor ct to 5 do\n vn_RK6_||ct := RK6_||ct(V(x,y),x,y,x0,y0,hh,numst eps,true);\nend do:\nxx := 4.999: vxx := evalf(v(xx)):\nfor ct to 5 do \n errs := [op(errs),abs(vn_RK6_||ct(xx)-vxx)];\nend do:\nDigits := \+ 10:\nlinalg[transpose](convert([mthds,evalf(errs)],matrix));" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#K%'matrixG6#7&7$%0slope~field:~~~G,$*&,&\"\" \"F,*&\"\"%F,-%$cosG6#,$*&\"\"$F,%\"xGF,F,F,F,F,,&%\"yGF,#F,F4!\"\"F,F 97$%0initial~point:~G-%!G6$\"\"!F,7$%/step~width:~~~G$\"\"#!\"#7$%1no. ~of~steps:~~~G\"$]#Q)pprint516\"" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%! G" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#K%'matrixG6#7'7$%4Sharp-Verner~sc hemeG$\"+!4jYC#!#B7$%Kscheme~with~a~wide~real~stability~intervalG$\"+R ^y9G!#A7$%Mscheme~with~a~large~imaginary~axis~inclusionG$\"+z#H3\"=F+7 $%[pscheme~with~a~quite~large~stability~region~and~non-zero~imaginary~ axis~inclusionG$\"+L>(>Q'F+7$%Kscheme~with~a~quite~large~stability~reg ionG$\"+yEjQhF+Q)pprint526\"" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }} {PARA 0 "" 0 "" {TEXT -1 4 "The " }{TEXT 260 22 "root mean square erro r" }{TEXT -1 20 " over the interval " }{XPPEDIT 18 0 "[0, 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 met hod over 100 equal subintervals." }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 494 "mthds := [`Sharp-Verner scheme`,`scheme with a wide real stab ility interval`,`scheme with a large imaginary axis inclusion`,`scheme with a quite large stability region and non-zero imaginary axis inclu sion`,`scheme with a quite large stability region`]: errs := []:\nDigi ts := 20:\nfor ct to 5 do\n sm := NCint((v(x)-'vn_RK6_||ct'(x))^2,x= 0..5,adaptive=false,numpoints=7,factor=100);\n errs := [op(errs),sqr t(sm/5)];\nend do:\nDigits := 10:\nlinalg[transpose](convert([mthds,ev alf(errs)],matrix));" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#K%'matrixG6#7' 7$%4Sharp-Verner~schemeG$\"+(p**R5\"!#@7$%Kscheme~with~a~wide~real~sta bility~intervalG$\"+o\"GQH#F+7$%Mscheme~with~a~large~imaginary~axis~in clusionG$\"+$fo>'HF+7$%[pscheme~with~a~quite~large~stability~region~an d~non-zero~imaginary~axis~inclusionG$\"+h[0D`F+7$%Kscheme~with~a~quite ~large~stability~regionG$\"+7p7&G)!#AQ)pprint536\"" }}}{PARA 0 "" 0 " " {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 92 "The following error gra phs are constructed using the numerical procedures for the solutions. 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Jx[F07$$\"5nm;a)ehe%)o\"Fjt$\")\\_O[F07$$\"5nmT&QGkT9p\"Fjt$\")3p_ZF07 $Fhz$\")w%=\\%F07$$\"5nmmTgxnN1F07$F][l$\"(>UA'F07$Fb[l$!) #pE@#F07$Fg[l$!)m;:KF07$$\"5Mek`;'G([pZJ)F07$Ff\\l$\")]6O)*F07$F^hn$\"*%p X#4\"F07$Fchn$\"*!z!z?\"F07$$\"5n\"zp[25uc+#Fjt$\"*]P$)=\"F07$Fhhn$\"* !pmp6F07$$\"5+v$fL%pRT7?Fjt$\"*2(*f:\"F07$F]in$\"*'*QG;\"F07$Fbin$\"** )G@B\"F07$Fgin$\"*9MID\"F07$F\\jn$\"*a$\\K7F07$Fajn$\"**e'G@\"F07$Ffjn $\"*=D&37F07$F[]l$\"*c([17F07$F`]l$\"*P\"\\<5F07$Fe]l$\")D&p4)F07$F_^l $\")wB-gF07$Fi^l$\"))z)yWF07$F^_l$\")OPoMF07$Fc_l$\")6)*oGF07$F`]o$\") >(4^#F07$Fh_l$\")F!zI#F07$Fh]o$\")V)o?#F07$F]`l$\")$R!y@F07$Fb`l$\")*H N>#F07$Fh^o$\")lbZAF07$Fg`l$\")*))*eBF07$F`_o$\")+,]DF07$F\\al$\")?)o$ GF07$$\"5,++]7y#=o'HFjt$\")1ATKF07$Faal$\")ySoPF07$Ffal$\")FDh]F07$F[b l$\")/46nF07$F`bl$\")'3$H$)F07$Febl$\"*D;t2\"F07$Fjbl$\"*fTs=\"F07$F_c l$\"*G+**G\"F07$$\"5,](oa&=RmYNFjt$\"*RVPH\"F07$Fdcl$\"*%ph(H\"F07$$\" 5M$eRZx0L+c$Fjt$\"*G=AJ\"F07$Ficl$\"*G.SJ\"F07$F^dl$\"*()GHK\"F07$Fcdl $\"**G%=K\"F07$$\"5M$3_D`ZS,g$Fjt$\"*Dx3K\"F07$$\"5,+v=#\\/Dog$Fjt$\"* K[&=8F07$$\"5o;H#=Xh4Nh$Fjt$\"*h%=:8F07$Fhdl$\"*!\\M08F07$$\"5,]P4r`(y oi$Fjt$\"*,q4I\"F07$$\"5om\"H2LKjNj$Fjt$\"*'*z]H\"F07$$\"5,voa53c!pj$F jt$\"*_2!*G\"F07$$\"5M$ek.H*yCSOFjt$\"*GhiF\"F07$$\"5o\"H#=qx,fVOFjt$ \"*_*=t7F07$F]el$\"*&R%*p7F07$Fbel$\"*hpO9\"F07$Fgel$\"),o!e*F07$Febo$ \")9\"eZ(F07$F\\fl$\")zn*)fF07$F]co$\")DVF\\F07$Fafl$\")OxSVF07$Feco$ \")4j4TF07$Fffl$\")F07$$\"5-++v=n(*fDVFjt$\")u?C:F07$Fegl$\")'fk;\"F07$Ffdo$\"(!3\"H* F07$Fjgl$\"(O_m(F07$F^eo$\"(J%[kF07$F_hl$\"(A/s&F07$Fdhl$\"(nL0&F07$Fi hl$\"(KM([F07$F^il$\"(&*y;&F07$Fcil$\"(AW:'F0-Fhil6&FjilF($\"#vFefoFcd r-F_jl6#%Kscheme~with~a~quite~large~stability~regionG-%%FONTG6$%*HELVE TICAGF\\jl-%&TITLEG6#%Uerror~curves~for~8~stage~order~6~Runge-Kutta~me thodsG-%+AXESLABELSG6$Q\"x6\"Q!F[fw-%%VIEWG6$;F(Fcil%(DEFAULTG" 1 2 0 1 10 0 2 9 1 4 2 1.000000 45.000000 45.000000 0 1 "Sharp-Verner scheme " "scheme with a wide real stability interval" "scheme with a large im aginary axis inclusion" "scheme with a quite large stability region an d non-zero imag axis inclusion" "scheme with a quite large stability r egion" }}}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 47 "Test 8 of \+ 8 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+sqr t(64+9*x^8-324*x^6+2916*x^4))^(1/3);" "6#/-%$rhoG6#%\"xG),(*&\"#a\"\" \"*$F'\"\"#F,F,*&\"\"$F,*$F'\"\"%F,!\"\"-%%sqrtG6#,*\"#kF,*&\"\"*F,*$F '\"\")F,F,*&\"$C$F,*$F'\"\"'F,F3*&\"%;HF,*$F'F2F,F,F,*&F,F,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(rhs(%),x):\nplot(w(x), x=0..4,0..3.7,numpoints=75,font=[HELVETICA,9],labels=[`x`,`y(x)`]);" } }{PARA 11 "" 1 "" {XPPMATH 20 "6#>%#deG/-%%diffG6$-%\"yG6#%\"xGF,*(F, \"\"\",&\"\"*F.*$)F,\"\"#F.!\"\"F.,&F.F.*$)F)F3F.F.F4" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%#icG/-%\"yG6#\"\"!F)" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/-%\"yG6#%\"xG,&*&\"\"#!\"\",(*&\"\"$\"\"\")F'\"\"%F/F+ *&\"#aF/)F'F*F/F/*$,*\"#kF/*&\"\"*F/)F'\"\")F/F/*&\"$C$F/)F'\"\"'F/F+* &\"%;HF/F0F/F/#F/F*F/#F/F.F/*&F*F/F,#F+F.F+" }}{PARA 13 "" 1 "" {GLPLOT2D 503 318 318 {PLOTDATA 2 "6&-%'CURVESG6$7io7$$\"\"!F)F(7$$\"3 ()=*=*=*Qx#G!#>$\"3_LLtbH2)f$!#?7$$\"3uPy$y$yZbcF-$\"3ZF^'eEW*Q9F-7$$ \"3;_8N^$ye6)F-$\"3C$\"3aT8Yqv-h6F>7$$\"3oKCVKs3o@F>$\"3c?q**e5wz?F>7$$\"3$4\" 3\"3T.Ds#F>$\"3+#H`Y\")*G6KF>7$$\"3jy$y$y\"=lB$F>$\"3L\\!fpl0?S%F>7$$ 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Fho$\"3uI!*3pTpuMFho7$$\"3E#*=*=p]\"pNFho$\"3s<,a='3,U$Fho7$$\"37.Fq-Y #3i$Fho$\"3w<#3Q&zxhLFho7$$\"31wcnbdutOFho$\"3?^'pWBqHH$Fho7$$\"32dnv' *p'o+i*\\/FFFho7$$\"\"%F)$\"3CxC=rRoRDFho-%'COLOU RG6&%$RGBG$\"#5!\"\"F(F(-%%FONTG6$%*HELVETICAG\"\"*-%+AXESLABELSG6$%\" xG%%y(x)G-%%VIEWG6$;F(F`cl;F($\"#PFjcl" 1 2 0 1 10 0 2 9 1 4 2 1.000000 45.000000 45.000000 0 0 "Curve 1" }}}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 32 "The following code constructs a " }{TEXT 260 17 "discrete solution" }{TEXT -1 44 " based on each of t he methods and gives the " }{TEXT 260 22 "root mean square error" } {TEXT -1 18 " of each solution." }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 813 "W := (x,y) -> x*(9-x^2)/(1+y^2): hh := 0.01: numsteps := 400: x0 := 0: y0 := 0:\nmatrix([[`slope field: `,W(x,y)],[`initial point : `,``(x0,y0)],[`step width: `,hh],\n[`no. of steps: `,numsteps]]) ;``;\nmthds := [`Sharp-Verner scheme`,`scheme with a wide real stabili ty interval`,`scheme with a large imaginary axis inclusion`,`scheme wi th a quite large stability region and non-zero imaginary axis inclusio n`,`scheme with a quite large stability region`]: errs := []:\nDigits \+ := 30:\nfor ct to 5 do\n Wn_RK6_||ct := RK6_||ct(W(x,y),x,y,x0,y0,hh ,numsteps,false);\n sm := 0: numpts := nops(Wn_RK6_||ct):\n for ii to numpts do\n sm := sm+(Wn_RK6_||ct[ii,2]-w(Wn_RK6_||ct[ii,1])) ^2;\n end do:\n errs := [op(errs),sqrt(sm/numpts)];\nend do:\nDigi ts := 10:\nlinalg[transpose](convert([mthds,evalf(errs)],matrix));" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#K%'matrixG6#7&7$%0slope~field:~~~G*(% \"xG\"\"\",&\"\"*F+*$)F*\"\"#F+!\"\"F+,&F+F+*$)%\"yGF0F+F+F17$%0initia l~point:~G-%!G6$\"\"!F;7$%/step~width:~~~G$F+!\"#7$%1no.~of~steps:~~~G \"$+%Q)pprint546\"" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%!G" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#K%'matrixG6#7'7$%4Sharp-Verner~schemeG$\"+iNJd$ *!#C7$%Kscheme~with~a~wide~real~stability~intervalG$\"+f@#Qn)F+7$%Msch eme~with~a~large~imaginary~axis~inclusionG$\"+Q]Fg))F+7$%[pscheme~with ~a~quite~large~stability~region~and~non-zero~imaginary~axis~inclusionG $\"+$)\\sa6!#B7$%Kscheme~with~a~quite~large~stability~regionG$\"+ " 0 "" {MPLTEXT 1 0 744 "W := (x,y) -> x*(9-x ^2)/(1+y^2): hh := 0.01: numsteps := 400: x0 := 0: y0 := 0:\nmatrix([[ `slope field: `,W(x,y)],[`initial point: `,``(x0,y0)],[`step width: \+ `,hh],\n[`no. of steps: `,numsteps]]);``;\nmthds := [`Sharp-Verner scheme`,`scheme with a wide real stability interval`,`scheme with a l arge imaginary axis inclusion`,`scheme with a quite large stability re gion and non-zero imaginary axis inclusion`,`scheme with a quite large stability region`]: errs := []:\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]( convert([mthds,evalf(errs)],matrix));" }}{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$\"\"!F;7$%/ste p~width:~~~G$F+!\"#7$%1no.~of~steps:~~~G\"$+%Q)pprint566\"" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%!G" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#K%'ma trixG6#7'7$%4Sharp-Verner~schemeG$\"+22TZM!#C7$%Kscheme~with~a~wide~re al~stability~intervalG$\"+eYPwZ!#D7$%Mscheme~with~a~large~imaginary~ax is~inclusionG$\"+CL\"*36F+7$%[pscheme~with~a~quite~large~stability~reg ion~and~non-zero~imaginary~axis~inclusionG$\"+C#[K'GF+7$%Kscheme~with~ a~quite~large~stability~regionG$\"+$=uw1$F+Q)pprint576\"" }}}{PARA 0 " " 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 4 "The " }{TEXT 260 22 "root mean square error" }{TEXT -1 20 " over the interval " } {XPPEDIT 18 0 "[0,4]" "6#7$\"\"!\"\"%" }{TEXT -1 82 " of each Runge-K utta 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 0 494 "mthds := [`Sharp-Verner sch eme`,`scheme with a wide real stability interval`,`scheme with a large imaginary axis inclusion`,`scheme with a quite large stability region and non-zero imaginary axis inclusion`,`scheme with a quite large sta bility region`]: errs := []:\nDigits := 20:\nfor ct to 5 do\n sm := \+ NCint((w(x)-'wn_RK6_||ct'(x))^2,x=0..4,adaptive=false,numpoints=7,fact or=200);\n errs := [op(errs),sqrt(sm/4)];\nend do:\nDigits := 10:\nl inalg[transpose](convert([mthds,evalf(errs)],matrix));" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#K%'matrixG6#7'7$%4Sharp-Verner~schemeG$\"+?;$QK*!# C7$%Kscheme~with~a~wide~real~stability~intervalG$\"+X@,`\")F+7$%Mschem e~with~a~large~imaginary~axis~inclusionG$\"+\\TL5()F+7$%[pscheme~with~ a~quite~large~stability~region~and~non-zero~imaginary~axis~inclusionG$ \"+*)ykL6!#B7$%Kscheme~with~a~quite~large~stability~regionG$\"+ZR')z5F 8Q)pprint586\"" }}}{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 0 575 "evalf[20](plot([w(x)-'wn_RK6_1'(x),w(x)-'wn_RK6_2'(x ),w(x)-'wn_RK6_3'(x),\nw(x)-'wn_RK6_4'(x),w(x)-'wn_RK6_5'(x)],x=0..4,- 4.9e-14..4.9e-14,font=[HELVETICA,9],\ncolor=[COLOR(RGB,.9,0,.9),COLOR( RGB,0,.15,1),COLOR(RGB,.95,0,.2),COLOR(RGB,.95,.45,0),COLOR(RGB,0,.75, .2)],\nlegend=[`Sharp-Verner scheme`,`scheme with a wide real stabilit y interval`,`scheme with a large imaginary axis inclusion`,`scheme wit h a quite large stability region and non-zero imag axis inclusion`,`sc heme with a quite large stability region`],title=`error curves for 8 s tage order 6 Runge-Kutta methods`));" }}{PARA 13 "" 1 "" {GLPLOT2D 906 673 673 {PLOTDATA 2 "6+-%'CURVESG6%7eq7$$\"\"!F)F(7$$\"5mmmmm;arz@ !#@$\"'<'4\"!#A7$$\"5LLLLLL3VfVF-$\"'0(['F07$$\"5lmmmmT&)G\\aF-$\"'4&H 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RLFM7$Fit$\"(vb4$FM7$F]v$\"(&FM$\"(Sz+#FM7$Fb[l$\"(VK(>FM7$$\"5qmmmmTg-0cFM$\"'5=> F[\\l7$Fg[l$\"'jb=F[\\l7$Fb\\l$\"'0x:F[\\l7$F\\]l$\"'\"QL\"F[\\l7$Fa]l $\"'yO6F[\\l7$Ff]l$\"'a+5F[\\l7$F[^l$\"&sy)F[\\l7$F`^l$\"&\\\"yF[\\l7$ Fe^l$\"&I0(F[\\l7$Fj^l$\"&oZ'F[\\l7$F__l$\"&A!fF[\\l7$Fd_l$\"&0\\&F[\\ l7$Fi_l$\"&93&F[\\l7$F^`l$\"&.x%F[\\l7$Fc`l$\"&XZ%F[\\l7$Fh`l$\"&,B%F[ \\l7$F]al$\"&s+%F[\\l7$Fbal$\"&y#QF[\\l7$Fgal$\"&ul$F[\\l7$F\\bl$\"&E] $F[\\l7$Fabl$\"&UQ$F[\\l7$Ffbl$\"&;F$F[\\l7$F[cl$\"&.<$F[\\l7$F`cl$\"& Y3$F[\\l7$Fecl$\"&M,$F[\\l7$Fjcl$\"&s%HF[\\l7$F_dl$\"&%)*GF[\\l7$Fddl$ \"&x&GF[\\l7$Fidl$\"&1$GF[\\l7$F^el$\"&=\"GF[\\l7$Fcel$\"&\\!GF[\\l7$F hel$\"&%4GF[\\l7$F]fl$\"&q#GF[\\l7$Fbfl$\"&)fGF[\\l7$Fgfl$\"&*3HF[\\l7 $F\\gl$\"&)yHF[\\l7$Fagl$\"&@2$F[\\l7$Ffgl$\"&I=$F[\\l7$F[hl$\"&-N$F[ \\l7$F`hl$\"&[a$F[\\l7$Fjhl$\"&f$QF[\\l7$Fdil$\"&\"fSF[\\l7$F^jl$\"&:X %F[\\l7$Fcjl$\"&^\"[F[\\l7$Fhjl$\"&)=aF[\\l7$Fb[m$\"&me'F[\\l7$F\\\\m$ \"&Of*F[\\l-Fa\\m6&Fc\\mF($\"#vF]bnFdeo-Fh\\m6#%Kscheme~with~a~quite~l arge~stability~regionG-%%FONTG6$%*HELVETICAGFe\\m-%&TITLEG6#%Uerror~cu rves~for~8~stage~order~6~Runge-Kutta~methodsG-%+AXESLABELSG6$Q\"x6\"Q! Feiq-%%VIEWG6$;F(F\\\\m;$!#\\!#:$\"#\\F^jq" 1 2 0 1 10 0 2 9 1 4 2 1.000000 45.000000 45.000000 0 1 "Sharp-Verner scheme" "scheme with a \+ wide real stability interval" "scheme with a large imaginary axis incl usion" "scheme with a quite large stability region and non-zero imag a xis inclusion" "scheme with a quite large stability region" }}}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 48 "Test 9 of 8 stage, order 6 Ru nge-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 := unap ply(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/-%%diffG6$-%\"yG 6#%\"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$7Y7$$\"\"!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$\"3IyW%eHk>[*FJ 7$$\"3+++]Peui=FJ$\"3#4`!o2+#G**)FJ7$$\"37+++]$)z%=#FJ$\"3OGH4wwYu&)FJ 7$$\"3A++]i3&o]#FJ$\"3=1g%=M2W@)FJ7$$\"3%)***\\(oX*y9$FJ$\"31u2v$Q9&Gw FJ7$$\"3z***\\P9CAu$FJ$\"3=XIMTf7+sFJ7$$\"3!)***\\P*zhdVFJ$\"3P$G(zQ8# 4%oFJ7$$\"31++v$>fS*\\FJ$\"3X'3%RcqqPlFJ7$$\"3$)***\\(=$f%GcFJ$\"3mYY% G?7\"*G'FJ7$$\"3Q+++Dy,\"G'FJ$\"3?u@-35zxgFJ7$$\"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_FJ7$$\"33+]7j=_ 68F,$\"3'e4m\")R`oD&FJ7$$\"33++vVy!eP\"F,$\"3a@U-1/NZ_FJ7$$\"34+](=WU[ V\"F,$\"3Nrr*HO\"oU_FJ7$$\"3)****\\7B>&)\\\"F,$\"3'HX%)zwR1C&FJ7$$\"3) ***\\P>:mk:F,$\"3<^\"Q\"4\"y-C&FJ7$$\"3'***\\iv&QAi\"F,$\"3:*4?^OZ,C&F J7$$\"31++vtLU%o\"F,$\"3\"3gSMou)Q_FJ7$$\"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\\FJ7$$\"3')****\\ i@OtBF,$\"3m09))4iC_[FJ7$$\"3')**\\PfL'zV#F,$\"3%Gjf])o8tZFJ7$$\"3>+++ !*>=+DF,$\"3[G/4+_V#p%FJ7$$\"3-++DE&4Qc#F,$\"3!**R*=7x[1YFJ7$$\"3=+]P% >5pi#F,$\"3f7E:iH**=XFJ7$$\"39+++bJ*[o#F,$\"3cgVvc$ovV%FJ7$$\"33++Dr\" [8v#F,$\"3Ln\\jDQ5WVFJ7$$\"3++++Ijy5GF,$\"3OZ!)Q%zK7E%FJ7$$\"31+]P/)fT (GF,$\"3)*4_&egIW<%FJ7$$\"31+]i0j\"[$HF,$\"3qns]&)H\\$4%FJ7$$\"\"$F)$ \"3ntdq;jW4SFJ-%'COLOURG6&%$RGBG$\"#5!\"\"F(F(-%%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 "The follo wing 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 0 826 "M := (x,y) -> -(1+cos(2*x))*y^3: hh := 0.0 1: 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 := [`Sharp-Verner scheme`,`scheme with a wide real stability interval`,`scheme with a large imaginary a xis inclusion`,`scheme with a quite large stability region and non-zer o imaginary axis inclusion`,`scheme with a quite large stability regio n`]: errs := []:\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[ii,1]))^2;\n end do:\n errs := [op(errs), sqrt(sm/numpts)];\nend do:\nDigits := 10:\nlinalg[transpose](convert([ mthds,evalf(errs)],matrix));" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#K%'mat rixG6#7&7$%0slope~field:~~~G,$*&,&\"\"\"F,-%$cosG6#,$*&\"\"#F,%\"xGF,F ,F,F,)%\"yG\"\"$F,!\"\"7$%0initial~point:~G-%!G6$\"\"!*$F2#F,F27$%/ste p~width:~~~G$F,!\"#7$%1no.~of~steps:~~~G\"$+$Q)pprint596\"" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%!G" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#K%'ma trixG6#7'7$%4Sharp-Verner~schemeG$\"+P'[%*z&!#A7$%Kscheme~with~a~wide~ real~stability~intervalG$\"+8%*)H$[!#@7$%Mscheme~with~a~large~imaginar y~axis~inclusionG$\"+ " 0 "" {MPLTEXT 1 0 757 "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. o f steps: `,numsteps]]);``;\nmthds := [`Sharp-Verner scheme`,`scheme \+ with a wide real stability interval`,`scheme with a large imaginary ax is inclusion`,`scheme with a quite large stability region and non-zero imaginary axis inclusion`,`scheme with a quite large stability region `]: errs := []:\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)):\nfor ct to 5 do\n errs := [op(errs),abs(mn_RK6 _||ct(xx)-mxx)];\nend do:\nDigits := 10:\nlinalg[transpose](convert([m thds,evalf(errs)],matrix));" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#K%'matr ixG6#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)pprint616\"" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%!G" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#K%'matri xG6#7'7$%4Sharp-Verner~schemeG$\"+To(*H6!#A7$%Kscheme~with~a~wide~real ~stability~intervalG$\"+!>s?D*F+7$%Mscheme~with~a~large~imaginary~axis ~inclusionG$\"+tM#yz(!#B7$%[pscheme~with~a~quite~large~stability~regio n~and~non-zero~imaginary~axis~inclusionG$\"+^%4n5$F+7$%Kscheme~with~a~ quite~large~stability~regionG$\"+'e0Gd\"F+Q)pprint626\"" }}}{PARA 0 " " 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 4 "The " }{TEXT 260 22 "root mean square error" }{TEXT -1 20 " over the interval " } {XPPEDIT 18 0 "[0, 3];" "6#7$\"\"!\"\"$" }{TEXT -1 82 " of each Runge -Kutta method is estimated as follows using the special procedure " } {TEXT 0 5 "NCint" }{TEXT -1 98 " to perform numerical integration by \+ the 7 point Newton-Cotes method over 150 equal subintervals." }} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 494 "mthds := [`Sharp-Verner sch eme`,`scheme with a wide real stability interval`,`scheme with a large imaginary axis inclusion`,`scheme with a quite large stability region and non-zero imaginary axis inclusion`,`scheme with a quite large sta bility region`]: errs := []:\nDigits := 20:\nfor ct to 5 do\n sm := \+ NCint((m(x)-'mn_RK6_||ct'(x))^2,x=0..3,adaptive=false,numpoints=7,fact or=150);\n errs := [op(errs),sqrt(sm/3)];\nend do:\nDigits := 10:\nl inalg[transpose](convert([mthds,evalf(errs)],matrix));" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#K%'matrixG6#7'7$%4Sharp-Verner~schemeG$\"+k,%Qs&!# A7$%Kscheme~with~a~wide~real~stability~intervalG$\"+=M/pZ!#@7$%Mscheme ~with~a~large~imaginary~axis~inclusionG$\"+.oyGUF+7$%[pscheme~with~a~q uite~large~stability~region~and~non-zero~imaginary~axis~inclusionG$\"+ eHbA;F07$%Kscheme~with~a~quite~large~stability~regionG$\"+3!f.!zF+Q)pp rint636\"" }}}{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 0 559 "evalf[20](plot(['mn_RK6_1'(x)-m(x),'mn_RK6_2'(x)-m(x),'mn_RK6_3 '(x)-m(x),\n'mn_RK6_4'(x)-m(x),'mn_RK6_5'(x)-m(x)],x=0..0.5,font=[HELV ETICA,9],\ncolor=[COLOR(RGB,.9,0,.9),COLOR(RGB,0,.15,1),COLOR(RGB,.95, 0,.2),COLOR(RGB,.95,.45,0),COLOR(RGB,0,.75,.2)],\nlegend=[`Sharp-Verne r scheme`,`scheme with a wide real stability interval`,`scheme with a \+ large imaginary axis inclusion`,`scheme with a quite large stability r egion and non-zero imag axis inclusion`,`scheme with a quite large sta bility region`],title=`error curves for 8 stage order 6 Runge-Kutta me thods`));" }}{PARA 13 "" 1 "" {GLPLOT2D 975 500 500 {PLOTDATA 2 "6+-%' CURVESG6%7_q7$$\"\"!F)F(7$$\"5ommmTN@Ki8!#A$!#:!#>7$$\"5NLLL$3FWYs#F-$ !%p=F07$$\"5omm;aQ`!eS$F-$!%T()F07$$\"5-+++D1k'p3%F-$!&E2$F07$$\"5OLL$ eRZF\"oZF-$!&'p))F07$$\"5qmmmmT&)G\\aF-$!'\\;AF07$$\"50++]P4'\\/8'F-$! 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`s$!)L,tdF07$Fes$!)/pkgF07$Fjs$!),--fF07$F_t$!)89tdF07$Fdt$!)q&ow&F07$ Fit$!)cC)y&F07$F^u$!)'=Y&eF07$Fcu$!)m=\"*fF07$Fhu$!)&[0j'F07$F]v$!)tE' z'F07$Fbv$!)mjDmF07$Fgv$!):;(\\'F07$F\\w$!)>YBnF07$Faw$!)?:`pF07$Ffw$! )jX\"z'F07$F[x$!)$y(zmF07$F^cm$!)dX#o'F07$F`x$!)y'Gr'F07$F[dm$!)%3[y'F 07$Fex$!)y<Bd #Fa^l7$F`cl$!*6$R%[#Fa^l7$Fecl$!*pB`R#Fa^l7$Fjcl$!*l=#4BFa^l7$F_dl$!*d 8.B#Fa^l7$Fddl$!*Cw&e@Fa^l7$Fidl$!*X.Q3#Fa^l7$F^el$!*(zk??Fa^l7$Fcel$! *a+r&>Fa^l7$Fhel$!*P>F!>Fa^l7$F]fl$!*n-l%=Fa^l7$Fbfl$!*eokz\"Fa^l7$Fgf l$!*SHpu\"Fa^l7$F\\gl$!*=L5q\"Fa^l7$Fagl$!*')*[b;Fa^l7$Ffgl$!*%H)Qh\"F a^l7$F[hl$!*lsMd\"Fa^l7$F`hl$!*>$RN:Fa^l7$Fehl$!*#y/-:Fa^l7$Fjhl$!*@Tc Y\"Fa^l7$F_il$!*UGYV\"Fa^l7$Fdil$!*xlIS\"Fa^l7$Fiil$!*aMUP\"Fa^l7$F^jl $!*k\\YM\"Fa^l-Fdjl6&FfjlFjao$\"#XF^_nF(-Fjjl6#%foscheme~with~a~quite~ large~stability~region~and~non-zero~imag~axis~inclusionG-F$6%7_qF'7$F+ $F-F07$F2$!%cEF07$F7$!&yB\"F07$F<$!&YL%F07$FA$!'DY7F07$FF$!'N,JF07$FK$ !'#=\"pF07$FP$!(7?T\"F07$FU$!(1%)o#F07$FZ$!(u#H[F07$Fin$!([=N'F07$F^o$ !(46E)F07$Fco$!)\">L1\"F07$Fho$!)'4bN\"F07$F]p$!)pc7F07$Ffr$!)&f@4#F07$F[s$!)E!eI#F07$F`s$!);3=EF07$Fes$!)qQ [FF07$Fjs$!):ruEF07$F_t$!)e!)=EF07$Fdt$!)gQ=EF07$Fit$!)$**>j#F07$F^u$! )m*ym#F07$Fcu$!),sPFF07$Fhu$!)X)z/$F07$F]v$!)!ep7$F07$Fbv$!),c[IF07$Fg v$!)'>H*HF07$F\\w$!)+!*>JF07$Faw$!)3'*RKF07$Ffw$!)!f[;$F07$F[x$!)ka:$F07$F][l$!)y H5JF07$Fb[l$!)N$=6$F07$Fg[l$!)J4BJF07$F\\\\l$!)L4[JF07$Fa\\l$!))[?>$F0 7$Ff\\l$!)Z&G<$F07$F[]l$!)#o:9$F07$F`]l$!)'>4-$F07$Fe]l$!)_HxGF07$Fj]l $!)-SJFF07$F_^l$!)sf$e#F07$Fe^l$!)I\\iCF07$Fj^l$!)9*)HBF07$F__l$!*RHe? #Fa^l7$Fd_l$!*5H^4#Fa^l7$Fi_l$!*5\"*=+#Fa^l7$F^`l$!*?yy*=Fa^l7$Fc`l$!* E%f;=Fa^l7$Fh`l$!*TZzFa^l7$F`hl$!)WScxFa^l7$Fehl$!)t-*e(Fa ^l7$Fjhl$!)MI1uFa^l7$F_il$!).[]sFa^l7$Fdil$!)&Q>4(Fa^l7$Fiil$!)#))p%pF a^l7$F^jl$!)'=%)z'Fa^l-Fdjl6&FfjlF($\"#vF^_nF\\bo-Fjjl6#%Kscheme~with~ a~quite~large~stability~regionG-%%FONTG6$%*HELVETICAGFhjl-%&TITLEG6#%U error~curves~for~8~stage~order~6~Runge-Kutta~methodsG-%+AXESLABELSG6$Q \"x6\"Q!Fbiq-%%VIEWG6$;F(F^jl%(DEFAULTG" 1 2 0 1 10 0 2 9 1 4 2 1.000000 45.000000 45.000000 0 1 "Sharp-Verner scheme" "scheme with a \+ wide real stability interval" "scheme with a large imaginary axis incl usion" "scheme with a quite large stability region and non-zero imag a xis inclusion" "scheme with a quite large stability region" }}}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 47 "Test 10 of 8 stage, order 6 R unge-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;\ndsolve(\{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#,*#\"\"\"\"\"&F0 *&#F0\"\"#F0-F)6#,$*&,&-%$expG6#,$*&F4!\"\"F1F3F0F0F0F0F0,&F:F0F0F?F?F ?F0F0*&#\"\"$\"#9F0-%$sinG6#,$*&\"\"(F0F'F0F0F0F0*&#F0F1F0-%$cosG6#,$* &F1F0F'F0F0F0F?" }}{PARA 12 "" 1 "" {XPPMATH 20 "6#/-%\"yG6#%\"xG-%#ln G6#,$*&,*-%$expG6#,4#\"\"#\"\"&!\"\"*&#\"$#>\"\"(\"\"\"*&-%$sinGF&F:)- %$cosGF&\"\"'F:F:F5*&#\"$S#F9F:*&F$\"33uw,WSt45F,7$$\"3WmmmT&)G\\aF0$\"3 yUB%H69F5*!#=7$$\"3m****\\7G$R<)F0$\"3[G6@7@G;#)F87$$\"3GLLL3x&)*3\"F8 $\"3u_\"Hlv:eW(F87$$\"3))**\\i!R(*Rc\"F8$\"3aT]N\"zi(yjF87$$\"3umm\"H2 P\"Q?F8$\"3xQ:-NK+KcF87$$\"3YLek.pu/BF8$\"3:Vt%)yf.P`F87$$\"3!***\\PMn NrDF8$\"3\"zkU]7kD7&F87$$\"37$eR(\\;m/FF8$\"3#f>&4'\\tL/&F87$$\"3MmT5l l'z$GF8$\"3#3U'>%*z$=)\\F87$$\"37](o/[r7(HF8$\"3oQ2*>TRs$\\F87$$\"3MLL $eRwX5$F8$\"359>\\xg!*3\\F87$$\"3:L$3F%\\wQKF8$\"3wA2?_M<'*[F87$$\"3_L Le*[`HP$F8$\"3Y^)o(RAn)*[F87$$\"3*QLek.Ur]$F8$\"3K()))eR[\"e\"\\F87$$ \"3rLLL$eI8k$F8$\"35_F/H[-Z\\F87$$\"3*QL$3xwq4RF8$\"3y)RBYN;$\\]F87$$ \"33ML$3x%3yTF8$\"3F1fX,$G1?&F87$$\"3h+]PfyG7ZF8$\"3-%yxl/-Si&F87$$\"3 emm\"z%4\\Y_F8$\"3E(Q!R(Q;g:'F87$$\"32++v$flMLe*)>VB$)F8$\"3R-PW?nJGpF87$$\"3wmmTg()4_))F8$\"3nU!**oz*pDlF87$ $\"3Y++DJbw!Q*F8$\"35wIZl^+ohF87$$\"3=nT&)3\\m_'*F8$\"3:%=[TB`#HgF87$$ \"3+N$ekGkX#**F8$\"3;==#[laH$fF87$$\"3nTg_(R^g+\"F,$\"3:fYLpC*H!fF87$$ \"31]iSmjk>5F,$\"3?Z/nW4A')eF87$$\"3XekGN8CL5F,$\"3S5jkI7P$)eF87$$\"3% 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7$$\"3?]iSmTI-?F,$\"3]%GeCR0B@$F87$$\"3*=/wP!Ry:?F,$\"3OG\"o!ej/aKF87$ $\"3dLe9TOEH?F,$\"32!yA?()GSJ$F87$$\"3'pT&)e6Bi0#F,$\"3EaW;d/\"=\\$F87 $$\"3K+]i!f#=$3#F,$\"35[niL?f`PF87$$\"3/++D\"=EX8#F,$\"3)=7zi$ePDXF87$ $\"3?+](=xpe=#F,$\"3rT'oxRDBu&F87$$\"3$pTNrfbE@#F,$\"3I&**Rb6vOf'F87$$ \"3mLeRA9WRAF,$\"3Y7cI=h\\:wF87$$\"3S]ilZsAmAF,$\"3O*zb%4IS@))F87$$\"3 7nm\"H28IH#F,$\"3I'yDR:j=-\"F,7$$\"3!oTN@#3hF,7$$\"3WeRseStdCF,$\"3d+$en#[n??F,7$$\"3) oT5l0+5Z#F,$\"3%)[_;#*)R!\\?F,7$$\"35YOSbIjxCF,$\"3s$>URy8d0#F,7$$\"3I voHagE%[#F,$\"3+2\\y.l?d?F,7$$\"3_/,>`!**3\\#F,$\"3+HUb4&*\\`?F,7$$\"3 GLL3_?`(\\#F,$\"33zGRRomW?F,7$$\"3.$3-)Q84DDF,$\"3Q^^lOB%z&>F,7$$\"3AL 3_D1l_DF,$\"3aG*H8v%))4=F,7$$\"3I3-))o-VmDF,$\"3Y]!f;TM@s\"F,7$$\"3S$e RA\"*4-e#F,$\"3we!)pZ(f(H;F,7$$\"3[e*)fb&*)Rf#F,$\"3!o'>0Ar[N:F,7$$\"3 fL$e*)>pxg#F,$\"3(zC:0fj9W\"F,7$$\"3V+D1R'f:&H/%HG\"F,7$$ \"3%omm\"z+vbEF,$\"3;#=@:,\\a8\"F,7$$\"3AL3F>0uzEF,$\"3C!3Cbi?=+\"F,7$ $\"33+]Pf4t.FF,$\"3l3u3()HVJ))F87$$\"3Q$3F>HT'HFF,$\"3Xy2J2Y\"y#F,$\"3qi_f3vnxfF87$$\" 3uLLe*Gst!GF,$\"3eVA%fwpRK&F87$$\"3)om\"H2\"34'GF,$\"3Ar]3R$*[ONF87$$\"3C +voaa+$*HF,$\"3>.\"*[foPONF87$$\"3>](oH/*41IF,$\"3UC#)e$)41_NF87$$\"3: ++DJE>>IF,$\"3cpJ^K5P$e$F87$$\"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$F,$\"3P\\JG%fz/\"F,7$$\" 3C$ekyZ2mY$F,$\"3%zJ4cdg.0\"F,7$$\"3=vo/Bh$)yMF,$\"3i,C<-Qm\\5F,7$$\"3 mm\"H#oZ1\"\\$F,$\"3%43uAU\"*f/\"F,7$$\"36]Pfe?_:NF,$\"3]?:\"y&yYI5F,7 $$\"35L$e*[$z*RNF,$\"3WZcwFIo05F,7$$\"3%o;Hd!fX$f$F,$\"3\\R'zK#G91$*F8 7$$\"3e++]iC$pk$F,$\"3E?Ix(pVQY)F87$$\"3ILe*[t\\sp$F,$\"3-#z!\\!\\E.v( F87$$\"3[m;H2qcZPF,$\"3!Q'*ep\"3v-sF87$$\"3s***\\7.lQx$F,$\"3Il\"\\:&* zo*pF87$$\"3UL$3_0j,!QF,$\"3cWvs*pf#\\oF87$$\"3F+v=n?J8QF,$\"3KU^fg!Qu z'F87$$\"36nm;z5YEQF,$\"3-)RYLJ)=gnF87$$\"3^Le9\"45'RQF,$\"39oisVpNPnF 87$$\"3O+]7.\"fF&QF,$\"3VEh1S\\sGnF87$$\"3i3_vlYhlQF,$\"33cF9)GTPt'F87 $$\"3)oT&QG-ZyQF,$\"3aR=Q3Gt^nF87$$\"39Dc,\"zD8*QF,$\"3!zI/$G#3By'F87$ $\"3TLek`8=/RF,$\"33OP7dc+DoF87$$\"3$*\\i!*yC*)HRF,$\"3reG3V'=X%pF87$$ \"3Ymm;/OgbRF,$\"33DC(y#>%[5(F87$$\"3*G$e*[$zV4SF,$\"39]/NuwfQvF87$$\" 3w**\\ilAFjSF,$\"3D8xFXpB@!)F87$$\"3#G3_]p'>*3%F,$\"3qve-u54K#)F87$$\" 3ym\"zW7@^6%F,$\"3%eL`Rp%o0%)F87$$\"3w3F>RL3GTF,$\"3!)Q$yF^SJZ)F87$$\" 3t]i!RbX59%F,$\"3i)pQA'3/D&)F87$$\"3#=z>'ox+aTF,$\"37$4T=0@'f&)F87$$\" 3yLLL$)*pp;%F,$\"3/<(p'zRPv&)F87$$\"3!Q3_+sD-=%F,$\"3Ga;6C(p2d)F87$$\" 3#Q$3xc9[$>%F,$\"3X[ZFn2MW&)F87$$\"3'Qe*[$>Pn?%F,$\"3u\\G!HhJc\\)F87$$ \"3)QL3-$H**>UF,$\"3[:#3'R6kC%)F87$$\"3#R$ek.W]YUF,$\"3+Xc4Am3=#)F87$$ \"3)RL$3xe,tUF,$\"3%p?zi!3VLzF87$$\"3Cn;HdO=yVF,$\"3;#>X)HduejF87$$\"3 a+++D>#[Z%F,$\"3qyAO;Hx\")\\F87$$\"3TM$3_5,-`%F,$\"3'RSzCT^zW%F87$$\"3 SnmT&G!e&e%F,$\"3%=ER;G9D:%F87$$\"3/]i:NK'zf%F,$\"3*f`:i,h67%F87$$\"3f Le*[=Y.h%F,$\"3U'>WD]()H5%F87$$\"386but%F,$\"3')>+X\\xqwZF87$$\"37+ ]iSjE!z%F,$\"3<\"z**[W(=JcF87$$\"3y*\\7G))Rb\"[F,$\"3[YJ\"GK]h>'F87$$ \"3L+++DM\"3%[F,$\"3!**fZR-9n(oF87$$\"3)3](=np3m[F,$\"3gVt^2I*Ho(F87$$ \"3a+]P40O\"*[F,$\"3`9x`tb,B')F87$$\"3>]7.#Q?&=\\F,$\"3Ik*[g6ply*F87$$ \"3s+voa-oX\\F,$\"3)pQ=b([U56F,7$$\"3O]PMF,%G(\\F,$\"3_;`pzy]b7F,7$$\" \"&F)$\"3ftg')yo>49F,-%'COLOURG6&%$RGBG$\"#5!\"\"F(F(-%%FONTG6$%*HELVE TICAG\"\"*-%+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 follo wing 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 0 850 "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)],[`initi al point: `,``(x0,y0)],[`step width: `,hh],\n[`no. of steps: `,num steps]]);``;\nmthds := [`Sharp-Verner scheme`,`scheme with a wide real stability interval`,`scheme with a large imaginary axis inclusion`,`s cheme with a quite large stability region and non-zero imaginary axis \+ inclusion`,`scheme with a quite large stability region`]: errs := []: \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: numpts := nops(Pn_RK6_| |ct):\n for ii to numpts do\n sm := sm+(Pn_RK6_||ct[ii,2]-evalf (p(Pn_RK6_||ct[ii,1])))^2;\n end do:\n errs := [op(errs),sqrt(sm/n umpts)];\nend do:\nDigits := 10:\nlinalg[transpose](convert([mthds,eva lf(errs)],matrix));" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#K%'matrixG6#7&7 $%0slope~field:~~~G,$*&,&*&\"\"#\"\"\"-%$sinG6#,$*&\"\"&F.%\"xGF.F.F.F .*&\"\"$F.-%$cosG6#,$*&\"\"(F.F5F.F.F.F.F.-%%sinhG6#%\"yGF.!\"\"7$%0in itial~point:~G-%!G6$\"\"!,$*&F-FBF4#F.F-F.7$%/step~width:~~~G$F.!\"#7$ %1no.~of~steps:~~~G\"$+&Q(pprint66\"" }}{PARA 11 "" 1 "" {XPPMATH 20 " 6#%!G" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#K%'matrixG6#7'7$%4Sharp-Verne r~schemeG$\"+.'*z2?!#@7$%Kscheme~with~a~wide~real~stability~intervalG$ \"+#\\/<*zF+7$%Mscheme~with~a~large~imaginary~axis~inclusionG$\"+)GFTP #F+7$%[pscheme~with~a~quite~large~stability~region~and~non-zero~imagin ary~axis~inclusionG$\"+g8J#3%F+7$%Kscheme~with~a~quite~large~stability ~regionG$\"+'y>%48F+Q(pprint76\"" }}}{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 0 774 " 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 := [`Sharp-Verner scheme`,`scheme with a wid e real stability interval`,`scheme with a large imaginary axis inclusi on`,`scheme with a quite large stability region and non-zero imaginary axis inclusion`,`scheme with a quite large stability region`]: errs : = []:\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 := ev alf(p(xx)):\nfor ct to 5 do\n errs := [op(errs),abs(pn_RK6_||ct(xx)- pxx)];\nend do:\nDigits := 10:\nlinalg[transpose](convert([mthds,evalf (errs)],matrix));" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#K%'matrixG6#7&7$% 0slope~field:~~~G,$*&,&*&\"\"#\"\"\"-%$sinG6#,$*&\"\"&F.%\"xGF.F.F.F.* &\"\"$F.-%$cosG6#,$*&\"\"(F.F5F.F.F.F.F.-%%sinhG6#%\"yGF.!\"\"7$%0init ial~point:~G-%!G6$\"\"!,$*&F-FBF4#F.F-F.7$%/step~width:~~~G$F.!\"#7$%1 no.~of~steps:~~~G\"$+&Q)pprint666\"" }}{PARA 11 "" 1 "" {XPPMATH 20 "6 #%!G" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#K%'matrixG6#7'7$%4Sharp-Verner ~schemeG$\"+T^+W5!#@7$%Kscheme~with~a~wide~real~stability~intervalG$\" +wUpI8!#?7$%Mscheme~with~a~large~imaginary~axis~inclusionG$\"+4re3KF+7 $%[pscheme~with~a~quite~large~stability~region~and~non-zero~imaginary~ axis~inclusionG$\"+'[:y)fF+7$%Kscheme~with~a~quite~large~stability~reg ionG$\"+[oP]p!#AQ)pprint676\"" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }} {PARA 0 "" 0 "" {TEXT -1 4 "The " }{TEXT 260 22 "root mean square erro r" }{TEXT -1 20 " over the interval " }{XPPEDIT 18 0 "[0, 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 met hod over 200 equal subintervals." }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 494 "mthds := [`Sharp-Verner scheme`,`scheme with a wide real stab ility interval`,`scheme with a large imaginary axis inclusion`,`scheme with a quite large stability region and non-zero imaginary axis inclu sion`,`scheme with a quite large stability region`]: errs := []:\nDigi ts := 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(errs),sqr t(sm/5)];\nend do:\nDigits := 10:\nlinalg[transpose](convert([mthds,ev alf(errs)],matrix));" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#K%'matrixG6#7' 7$%4Sharp-Verner~schemeG$\"+yL`.?!#@7$%Kscheme~with~a~wide~real~stabil ity~intervalG$\"+Jo3czF+7$%Mscheme~with~a~large~imaginary~axis~inclusi onG$\"+%*Gj\"Q#F+7$%[pscheme~with~a~quite~large~stability~region~and~n on-zero~imaginary~axis~inclusionG$\"+LmYySF+7$%Kscheme~with~a~quite~la rge~stability~regionG$\"+0ke88F+Q)pprint686\"" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 92 "The following error graph s are constructed using the numerical procedures for the solutions." } }{EXCHG {PARA 0 "> " 0 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G. Lether: Mathematics of Computation, Vol. 20, no. 95, (Ju ly 1966) page 382. " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "dy/dx=exp(-x)/(x-1)^2" "6#/*&%#dyG\" \"\"%#dxG!\"\"*&-%$expG6#,$%\"xGF(F&*$,&F.F&F&F(\"\"#F(" }{TEXT -1 2 " " }{XPPEDIT 18 0 "cos(1/(x-1))-y" "6#,&-%$cosG6#*&\"\"\"F(,&%\"xGF(F (!\"\"F+F(%\"yGF+" }{TEXT -1 5 ", " }{XPPEDIT 18 0 "y(0)=sin*1" "6# /-%\"yG6#\"\"!*&%$sinG\"\"\"F*F*" }{TEXT -1 1 " " }}{PARA 0 "" 0 "" {TEXT -1 10 "Solution: " }}{PARA 256 "" 0 "" {TEXT -1 2 " " } {XPPEDIT 18 0 "y = -exp(-x)*sin(1/(x-1))" "6#/%\"yG,$*&-%$expG6#,$%\"x G!\"\"\"\"\"-%$sinG6#*&F-F-,&F+F-F-F,F,F-F," }{TEXT -1 1 " " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 185 "d e := diff(y(x),x)=exp(-x)/(x-1)^2*cos(1/(x-1))-y(x);\nic := y(0)=sin(1 );\ndsolve(\{de,ic\},y(x));\nq := unapply(rhs(%),x):\nplot(q(x),x=0..1 -1/(6*Pi),font=[HELVETICA,9],labels=[`x`,`y(x)`]);" }}{PARA 11 "" 1 " " {XPPMATH 20 "6#>%#deG/-%%diffG6$-%\"yG6#%\"xGF,,&*(-%$expG6#,$F,!\" \"\"\"\",&F,F4F4F3!\"#-%$cosG6#*&F4F4F5F3F4F4F)F3" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%#icG/-%\"yG6#\"\"!-%$sinG6#\"\"\"" }}{PARA 11 "" 1 " " {XPPMATH 20 "6#/-%\"yG6#%\"xG,$*&-%$expG6#,$F'!\"\"\"\"\"-%$sinG6#*& F/F/,&F'F/F/F.F.F/F." }}{PARA 13 "" 1 "" {GLPLOT2D 603 396 396 {PLOTDATA 2 "6&-%'CURVESG6$7[r7$$\"\"!F)$\"30l*y![)4ZT)!#=7$$\"3#>=\"* )>z2k?!#>$\"3m%p=Qu6DN)F,7$$\"3kMQU\"3@+'QF0$\"3m=O<,5$yO#)F,7$$\"38UQ!)p4'G\"zF0$\"3/Z^$zTlU<)F,7$$\" 3BY$*R0>JO**F0$\"36ty1)z*36\")F,7$$\"3wbXC%*4B\"=\"F,$\"3A;o(=P!Q^!)F, 7$$\"3M!)fxC(zaP\"F,$\"3E8^!ydw!))zF,7$$\"3kgswR?Pw:F,$\"3T8>lD8j?zF,7 $$\"3Qnlb!4?mx\"F,$\"3%p4\\s^P4&yF,7$$\"3OsvSC)*f#)>F,$\"3/$H(=wa6wxF, 7$$\"3)Rk+h)o-k@F,$\"3LR8d$>`qq(F,7$$\"3Q^Vo'yq#oBF,$\"3YB)Qc;#3DwF,7$ $\"3?0sMKLNtDF,$\"3,;%fG`C(F,7$$ \"3S+dSsVlWLF,$\"3&36sy[X09(F,7$$\"3EOur83&\\b$F,$\"37)QgTzpp+(F,7$$\" 3c#**Qwz)4TPF,$\"3M]xfz#>l(oF,7$$\"3wx#p)QELXRF,$\"3UR-VbS%zr'F,7$$\"3 \"yy;Gb6)RTF,$\"3%40quzD%\\lF,7$$\"3p2KM(*)HFM%F,$\"3W'4!o9@F_jF,7$$\" 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\\RTgHcmRGF,7$$\"3C\\b#*Q:tj#*F,$\"3M$o6)eV:lLF,7$$\"3oE3CP5fw#*F,$\"3 Ma)HpV]]I!H!G$*F,$\"3a%4t 07BP*GF,7$$\"3J=s\")G&))3M*F,$\"3f(3)H;[%o+#F,7$$\"3w&\\Kr-[PN*F,$\"33 n1kl[]%*F,$!3m (=[SoWqQ#F,7$$\"3%>saO,CmX*F,$!3GDS5eP#en\"F,7$$\"3AhB\"Gw`IY*F,$!3U$3 !Gg0_(o)F07$$\"3]++(>^$[p%*F,$!3V'=8$[D+C:!#C-%'COLOURG6&%$RGBG$\"#5! \"\"F(F(-%%FONTG6$%*HELVETICAG\"\"*-%+AXESLABELSG6$%\"xG%%y(x)G-%%VIEW G6$;F($\"+BN[p%*!#5%(DEFAULTG" 1 2 0 1 10 0 2 9 1 4 2 1.000000 45.000000 45.000000 0 0 "Curve 1" }}}}{PARA 0 "" 0 "" {TEXT -1 0 "" }} {PARA 0 "" 0 "" {TEXT -1 93 "The following code constructs a discrete \+ solution based on each of the methods and gives the " }{TEXT 260 22 "r oot mean square error" }{TEXT -1 18 " of each solution." }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 866 "Q := (x,y) -> exp(-x)/(x-1)^2*cos( 1/(x-1))-y: hh := 1/500-1/(3000*Pi): numsteps := 500: x0 := 0: y0 := s in(1):\nmatrix([[`slope field: `,Q(x,y)],[`initial point: `,``(x0,y0 )],[`step width: `,hh],\n[`no. of steps: `,numsteps]]);``;\nmthds \+ := [`Sharp-Verner scheme`,`scheme with a wide real stability interval` ,`scheme with a large imaginary axis inclusion`,`scheme with a quite l arge stability region and non-zero imaginary axis inclusion`,`scheme w ith a quite large stability region`]: errs := []:\nDigits := 30:\nfor \+ ct to 5 do\n Qn_RK6_||ct := RK6_||ct(Q(x,y),x,y,x0,evalf[33](y0),eva lf[33](hh),numsteps,false);\n sm := 0: numpts := nops(Qn_RK6_||ct): \n for ii to numpts do\n sm := sm+(Qn_RK6_||ct[ii,2]-q(Qn_RK6_| |ct[ii,1]))^2;\n end do:\n errs := [op(errs),sqrt(sm/numpts)];\nen d do:\nDigits := 10:\nlinalg[transpose](convert([mthds,evalf(errs)],ma trix));" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#K%'matrixG6#7&7$%0slope~fie ld:~~~G,&*(-%$expG6#,$%\"xG!\"\"\"\"\",&F/F1F1F0!\"#-%$cosG6#*&F1F1F2F 0F1F1%\"yGF07$%0initial~point:~G-%!G6$\"\"!-%$sinG6#F17$%/step~width:~ ~~G,&#F1\"$+&F1*&F1F1*&\"%+IF1%#PiGF1F0F07$%1no.~of~steps:~~~GFFQ)ppri nt696\"" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%!G" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#K%'matrixG6#7'7$%4Sharp-Verner~schemeG$\"+gvM4E!#>7$%Ks cheme~with~a~wide~real~stability~intervalG$\"+ln$oJ\"!#=7$%Mscheme~wit h~a~large~imaginary~axis~inclusionG$\"+;+^GlD\"F0Q)pp rint706\"" }}}{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 = 0;" "6#/%\"xG\"\"!" }{TEXT -1 21 ".9469 is also given." }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 790 "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):\n matrix([[`slope field: `,Q(x,y)],[`initial point: `,``(x0,y0)],[`ste p width: `,hh],\n[`no. of steps: `,numsteps]]);``;\nmthds := [`Sha rp-Verner scheme`,`scheme with a wide real stability interval`,`scheme with a large imaginary axis inclusion`,`scheme with a quite large sta bility region and non-zero imaginary axis inclusion`,`scheme with a qu ite large stability region`]: errs := []:\nDigits := 30:\nfor ct to 5 \+ do\n qn_RK6_||ct := RK6_||ct(Q(x,y),x,y,x0,evalf(y0),evalf(hh),numst eps,true);\nend do:\nxx := 0.9469: qxx := evalf(q(xx)):\nfor ct to 5 d o\n errs := [op(errs),abs(qn_RK6_||ct(xx)-qxx)];\nend do:\nDigits := 10:\nlinalg[transpose](convert([mthds,evalf(errs)],matrix));" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#K%'matrixG6#7&7$%0slope~field:~~~G,&*( -%$expG6#,$%\"xG!\"\"\"\"\",&F/F1F1F0!\"#-%$cosG6#*&F1F1F2F0F1F1%\"yGF 07$%0initial~point:~G-%!G6$\"\"!-%$sinG6#F17$%/step~width:~~~G,&#F1\"$ +&F1*&F1F1*&\"%+IF1%#PiGF1F0F07$%1no.~of~steps:~~~GFFQ)pprint716\"" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#%!G" }}{PARA 11 "" 1 "" {XPPMATH 20 "6 #K%'matrixG6#7'7$%4Sharp-Verner~schemeG$\"+1MwyW!#=7$%Kscheme~with~a~w ide~real~stability~intervalG$\"+,%[OI#!#<7$%Mscheme~with~a~large~imagi nary~axis~inclusionG$\"+bO%\\-$F07$%[pscheme~with~a~quite~large~stabil ity~region~and~non-zero~imaginary~axis~inclusionG$\"+*=.1,#F07$%Kschem e~with~a~quite~large~stability~regionG$\"+>o$=>#F0Q)pprint726\"" }}} {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 4 "The " } {TEXT 260 22 "root mean square error" }{TEXT -1 19 " over the interval " }{XPPEDIT 18 0 " [0, 1-1/(6*Pi)] " "6#7$\"\"!,&\"\"\"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 0 514 "mthds := [`Sharp-Verner scheme`,`scheme with a wide \+ real stability interval`,`scheme with a large imaginary axis inclusion `,`scheme with a quite large stability region and non-zero imaginary a xis inclusion`,`scheme with a quite large stability region`]: errs := \+ []:\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,numpoints=7,factor=200);\n err s := [op(errs),sqrt(sm/(1-1/(6*Pi)))];\nend do:\nDigits := 10:\nlinalg [transpose](convert([mthds,evalf(errs)],matrix));" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#K%'matrixG6#7'7$%4Sharp-Verner~schemeG$\"+&*=#yc\"!#>7$ %Kscheme~with~a~wide~real~stability~intervalG$\"+$R()>`(F+7$%Mscheme~w ith~a~large~imaginary~axis~inclusionG$\"+.skv)*F+7$%[pscheme~with~a~qu ite~large~stability~region~and~non-zero~imaginary~axis~inclusionG$\"+/ '\\%=lF+7$%Kscheme~with~a~quite~large~stability~regionG$\"+'=?sC(F+Q)p print736\"" }}}{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 0 577 "evalf[25](plot(['qn_RK6_1'(x)-q(x),'qn_RK6_2'(x)-q(x),'qn_RK6_3 '(x)-q(x),\n'qn_RK6_4'(x)-q(x),'qn_RK6_5'(x)-q(x)],x=0..0.7,-1.9e-18.. 1.9e-18,font=[HELVETICA,9],\ncolor=[COLOR(RGB,.9,0,.9),COLOR(RGB,0,.15 ,1),COLOR(RGB,.95,0,.2),COLOR(RGB,.95,.45,0),COLOR(RGB,0,.75,.2)],\nle gend=[`Sharp-Verner scheme`,`scheme with a wide real stability interva l`,`scheme with a large imaginary axis inclusion`,`scheme with a quite large stability region and non-zero imag axis inclusion`,`scheme with a quite large stability region`],title=`error curves for 8 stage orde r 6 Runge-Kutta methods`));" }}{PARA 13 "" 1 "" {GLPLOT2D 1006 578 578 {PLOTDATA 2 "6+-%'CURVESG6%7fn7$$\"\"!F)F(7$$\":mmmmmmm;z+e_\"!#E$ \"%r()!#D7$$\":LLLLLL$3->R`GF-$\"&ss\"F07$$\":mmmmmmmT&pSYVF-$\"&Lw#F0 7$$\":lmmmmmm\"z'=$\\eF-$\"&[#RF07$$\":KLLLLL$3Ft3XtF-$\"&#H_F07$$\":l mmmmm;aLc=t)F-$\"&Pr'F07$$\":++++++v=`xn,\"F0$\"&K?)F07$$\":mmmmmmT&y/ Gl6F0$\"'()35F07$$\":++++++vV<2LJ\"F0$\"'oB7F07$$\":LLLLLLLe#3dl9F0$\" '#pY\"F07$$\":mmmmmm;Ht%o*f\"F0$\"'J1F0$\"'QkBF07$$\":+++++++Dxg$[?F0$\"'viFF07$$\":mmm 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" 0 "" {MPLTEXT 1 0 1 ";" }}}}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 48 "Test 12 of 8 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 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" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 810 "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 := [`Sharp-Verner schem e`,`scheme with a wide real stability interval`,`scheme with a large i maginary axis inclusion`,`scheme with a quite large stability region a nd non-zero imaginary axis inclusion`,`scheme with a quite large stabi lity region`]: errs := []:\nDigits := 30:\nfor ct to 5 do\n Rn_RK6_| |ct := RK6_||ct(R(x,y),x,y,x0,y0,hh,numsteps,false);\n sm := 0: nump ts := nops(Rn_RK6_||ct):\n for ii to numpts do\n sm := sm+(Rn_R K6_||ct[ii,2]-r(Rn_RK6_||ct[ii,1]))^2;\n end do:\n errs := [op(err s),sqrt(sm/numpts)];\nend do:\nDigits := 10:\nlinalg[transpose](conver t([mthds,evalf(errs)],matrix));" }}{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)pprint746\"" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%!G" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#K%'matrixG6#7'7$ %4Sharp-Verner~schemeG$\"+*G%HU7!#>7$%Kscheme~with~a~wide~real~stabili ty~intervalG$\"+*f$)H-\"F+7$%Mscheme~with~a~large~imaginary~axis~inclu sionG$\"+4J>:8F+7$%[pscheme~with~a~quite~large~stability~region~and~no n-zero~imaginary~axis~inclusionG$\"+v@,`FF+7$%Kscheme~with~a~quite~lar ge~stability~regionG$\"+IvP)Q)!#?Q)pprint756\"" }}}{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 0 741 "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 := [`Sharp-Verner scheme`,`scheme with a wide real s tability interval`,`scheme with a large imaginary axis inclusion`,`sch eme with a quite large stability region and non-zero imaginary axis in clusion`,`scheme with a quite large stability region`]: errs := []:\nD igits := 25:\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)):\nfo r ct to 5 do\n errs := [op(errs),abs(rn_RK6_||ct(xx)-rxx)];\nend do: \nDigits := 10:\nlinalg[transpose](convert([mthds,evalf(errs)],matrix) );" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#K%'matrixG6#7&7$%0slope~field:~~ ~G,$*(\"\"&\"\"\"%\"yGF,)-%$sinG6#,$*&\"\"(F,%\"xGF,F,F4F,F,7$%0initia l~point:~G-%!G6$\"\"!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$%4Sharp-Verner~schemeG$\"+$y=sT# !#>7$%Kscheme~with~a~wide~real~stability~intervalG$\"+iVf*\\\"F+7$%Msc heme~with~a~large~imaginary~axis~inclusionG$\"+#>Z#*f#F+7$%[pscheme~wi th~a~quite~large~stability~region~and~non-zero~imaginary~axis~inclusio nG$\"+!)\\BFcF+7$%Kscheme~with~a~quite~large~stability~regionG$\"+Jtz: 9F+Q(pprint36\"" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 4 "The " }{TEXT 260 22 "root mean square error" }{TEXT -1 20 " over the interval " }{XPPEDIT 18 0 "[0, 5];" "6#7$\"\"!\"\"&" } {TEXT -1 82 " of each Runge-Kutta method is estimated as 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 0 494 "mthds := [`Sharp-Verner scheme`,`scheme with a wide real stability interval `,`scheme with a large imaginary axis inclusion`,`scheme with a quite \+ large stability region and non-zero imaginary axis inclusion`,`scheme \+ with a quite large stability region`]: errs := []:\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](convert([mthds,evalf(errs)],mat rix));" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#K%'matrixG6#7'7$%4Sharp-Vern er~schemeG$\"+Tv))R7!#>7$%Kscheme~with~a~wide~real~stability~intervalG $\"+@zIA5F+7$%Mscheme~with~a~large~imaginary~axis~inclusionG$\"+!fw@J \"F+7$%[pscheme~with~a~quite~large~stability~region~and~non-zero~imagi nary~axis~inclusionG$\"+3/#ou#F+7$%Kscheme~with~a~quite~large~stabilit y~regionG$\"+YiGw$)!#?Q)pprint786\"" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 92 "The following error graphs are construc ted using the numerical procedures for the solutions." }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 557 "evalf[20](plot(['rn_RK6_1'(x)-r(x),'rn_R K6_2'(x)-r(x),'rn_RK6_3'(x)-r(x),\n'rn_RK6_4'(x)-r(x),'rn_RK6_5'(x)-r( x)],x=0..5,font=[HELVETICA,9],\ncolor=[COLOR(RGB,.9,0,.9),COLOR(RGB,0, .15,1),COLOR(RGB,.95,0,.2),COLOR(RGB,.95,.45,0),COLOR(RGB,0,.75,.2)], \nlegend=[`Sharp-Verner scheme`,`scheme with a wide real stability int erval`,`scheme with a large imaginary axis inclusion`,`scheme with a q uite large stability region and non-zero imag axis inclusion`,`scheme \+ with a quite large stability region`],title=`error curves for 8 stage \+ order 6 Runge-Kutta methods`));" }}{PARA 13 "" 1 "" {GLPLOT2D 990 636 636 {PLOTDATA 2 "6+-%'CURVESG6%7a_l7$$\"\"!F)F(7$$\"5NLLL$3FWYs#!#@$!( >f3%!#>7$$\"5qmmmmT&)G\\aF-$!(Eg1(F07$$\"5SLLL3x1h6oF-$!($p$\\#F07$$\" 50+++]7G$R<)F-$\"((GXpF07$$\"5qmmm\"z%\\DO&*F-$\"(t1\\*F07$$\"5MLLLL3x &)*3\"!#?$\"(\">KEF07$$\"5nmm\"z%\\v#pK\"FH$!)=@\"R$F07$$\"5+++]i!R(*R c\"FH$!)O0EnF07$$\"5MLL3xJs1,=FH$!*,4lc\"F07$$\"5nmmm\"H2P\"Q?FH$!*Q/J 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%F0$!+%4l6X\"F07$$\"5-+v=+=F07$Ff\\q$!+Sh^C=F07$F\\j[l$!+:V3F=F07$F[]q$!+yTN B=F07$F`]q$!+K\\^-=F07$Fe]q$!+(4u%G]M&[F0$!+'*\\XK;F07 $Fj]q$!+$=sI`\"F07$$\"5-+]7GQPsy[F0$!+33vN9F07$F_^q$!+yf3&Q\"F07$Fd^q$ !+%['=C9F07$Fi^q$!+we$HY\"F07$F^_q$!+\"Q4NV\"F07$Fc_q$!+kc(pR\"F07$Ff \\\\l$!+b>Q'R\"F07$Fh_q$!+>P8(Q\"F07$F^]\\l$!+v\\b%Q\"F07$F]`q$!+S!Q[Q \"F07$Fb`q$!+X:['R\"F07$Fg`q$!+1**eQ9F0-F\\aq6&F^aqF($\"#vF_^\\lFbbcl- Fcaq6#%Kscheme~with~a~quite~large~stability~regionG-%+AXESLABELSG6$Q\" x6\"Q!Fej_m-%&TITLEG6#%Uerror~curves~for~8~stage~order~6~Runge-Kutta~m ethodsG-%%FONTG6$%*HELVETICAGF`aq-%%VIEWG6$;F(Fg`q%(DEFAULTG" 1 2 0 1 10 0 2 9 1 4 2 1.000000 45.000000 45.000000 0 1 "Sharp-Verner scheme" "scheme with a wide real stability interval" "scheme with a large imag inary axis inclusion" "scheme with a quite large stability region and \+ non-zero imag axis inclusion" "scheme with a quite large stability reg ion" }}}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 48 "Test 13 of 8 stage, order 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&%\"yG F&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 differ ential equation " }{XPPEDIT 18 0 "dy/dx = cos*x+2*y;" "6#/*&%#dyG\" \"\"%#dxG!\"\",&*&%$cosGF&%\"xGF&F&*&\"\"#F&%\"yGF&F&" }{TEXT -1 64 " \+ contains an exponential term, but with the initial condition " } {XPPEDIT 18 0 "y(0) = -2/5" "6#/-%\"yG6#\"\"!,$*&\"\"#\"\"\"\"\"&!\"\" F-" }{TEXT -1 23 " this term disappears." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 50 "de := diff(y(x),x)=co s(x)+2*y(x);\ndsolve(de,y(x));" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%#d eG/-%%diffG6$-%\"yG6#%\"xGF,,&-%$cosGF+\"\"\"*&\"\"#F0F)F0F0" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/-%\"yG6#%\"xG,(*&#\"\"#\"\"&\"\"\"-%$cosGF& F-!\"\"*&#F-F,F--%$sinGF&F-F-*&-%$expG6#,$*&F+F-F'F-F-F-%$_C1GF-F-" }} }{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 105 "Any sli ght 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,,&-%$c osGF+\"\"\"*&\"\"#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$$\"3ELLLLBxV5E$F,$!35;%fC]([[JF,7$$\"3MLLLLAKn\\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?LL L=\"\\g**FK7$$\"3\")*****\\[A4]\"FO$\"3Xgu?U;&er\"F ,7$$\"3wmmm'3Q\\n\"FO$\"3S\\4.g(y\\S#F,7$$\"3OLLLB6@G=FO$\"3e*[f2BGC&H F,7$$\"3&)******f-w+?FO$\"375@EVOJ&[$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! 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[`Sharp-Verner scheme`,`s cheme with a wide real stability interval`,`scheme with a large imagin ary axis inclusion`,`scheme with a quite large stability region and no n-zero imaginary axis inclusion`,`scheme with a quite large stability \+ region`]: errs := []:\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: nu mpts := 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(e rrs),sqrt(sm/numpts)];\nend do:\nDigits := 10:\nlinalg[transpose](conv ert([mthds,evalf(errs)],matrix));" }}{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(pprint46\"" }}{PARA 11 "" 1 "" {XPPMATH 20 " 6#%!G" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#K%'matrixG6#7'7$%4Sharp-Verne r~schemeG$\"+3=5F7$%Kscheme~with~a~quite~large~stabili ty~regionG$\"+[Q89\\F8Q(pprint56\"" }}}{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 = 7.999;" "6#/%\"xG-%&FloatG6$\"%**z!\"$" }{TEXT -1 16 " is also given." }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 747 "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 := [ `Sharp-Verner scheme`,`scheme with a wide real stability interval`,`sc heme with a large imaginary axis inclusion`,`scheme with a quite large stability region and non-zero imaginary axis inclusion`,`scheme with \+ a quite large stability region`]: errs := []:\nDigits := 20:\nfor ct t o 5 do\n en_RK6_||ct := RK6_||ct(E(x,y),x,y,x0,evalf(y0),hh,numsteps ,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](convert([mthds,evalf(errs)],matrix));" }}{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(pprint66\"" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#%!G" }}{PARA 11 "" 1 "" {XPPMATH 20 "6 #K%'matrixG6#7'7$%4Sharp-Verner~schemeG$\"+mheq&*!#=7$%Kscheme~with~a~ wide~real~stability~intervalG$\"+aMu[$*F+7$%Mscheme~with~a~large~imagi nary~axis~inclusionG$\"+^())GZ\"!#<7$%[pscheme~with~a~quite~large~stab ility~region~and~non-zero~imaginary~axis~inclusionG$\"+os*\\!\\F+7$%Ks cheme~with~a~quite~large~stability~regionG$\"+ba7BFF+Q(pprint76\"" }}} {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 4 "The " } {TEXT 260 22 "root mean square error" }{TEXT -1 20 " over the interval " }{XPPEDIT 18 0 "[0, 8];" "6#7$\"\"!\"\")" }{TEXT -1 82 " of each \+ Runge-Kutta method is estimated as follows using the special procedure " }{TEXT 0 5 "NCint" }{TEXT -1 98 " to perform numerical integratio n by the 7 point Newton-Cotes method over 200 equal subintervals." }} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 494 "mthds := [`Sharp-Verner sch eme`,`scheme with a wide real stability interval`,`scheme with a large imaginary axis inclusion`,`scheme with a quite large stability region and non-zero imaginary axis inclusion`,`scheme with a quite large sta bility region`]: errs := []:\nDigits := 20:\nfor ct to 5 do\n sm := \+ NCint((e(x)-'en_RK6_||ct'(x))^2,x=0..8,adaptive=false,numpoints=7,fact or=200);\n errs := [op(errs),sqrt(sm/8)];\nend do:\nDigits := 10:\nl inalg[transpose](convert([mthds,evalf(errs)],matrix));" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#K%'matrixG6#7'7$%4Sharp-Verner~schemeG$\"+:PC&p\"! #=7$%Kscheme~with~a~wide~real~stability~intervalG$\"+=$\\fl\"F+7$%Msch eme~with~a~large~imaginary~axis~inclusionG$\"+Qq$*3EF+7$%[pscheme~with ~a~quite~large~stability~region~and~non-zero~imaginary~axis~inclusionG $\"+_ID)o)!#>7$%Kscheme~with~a~quite~large~stability~regionG$\"+ix[B[F 8Q(pprint86\"" }}}{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 0 538 "evalf[20](plot([e(x)-'en_RK6_1'(x),e(x)-'en_RK6_2'(x ),e(x)-'en_RK6_3'(x),\ne(x)-'en_RK6_4'(x),e(x)-'en_RK6_5'(x)],x=0..2, \ncolor=[COLOR(RGB,.9,0,.9),COLOR(RGB,0,.15,1),COLOR(RGB,.95,0,.2),COL OR(RGB,.95,.45,0),COLOR(RGB,0,.75,.2)],\nlegend=[`Sharp-Verner scheme` ,`scheme with a wide real stability interval`,`scheme with a large ima ginary axis inclusion`,`scheme with a quite large stability region and non-zero imag axis inclusion`,`scheme with a quite large stability re gion`],title=`error 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0 2 " ; " }}}}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 48 "Test 14 of 8 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 "Sol ution: " }}{PARA 256 "" 0 "" {TEXT -1 2 " " }{XPPEDIT 18 0 "y=100/101 " "6#/%\"yG*&\"$+\"\"\"\"\"$,\"!\"\"" }{TEXT -1 1 " " }{XPPEDIT 18 0 " x*cos*x-990/10201" "6#,&*(%\"xG\"\"\"%$cosGF&F%F&F&*&\"$!**F&\"&,-\"! \"\"F+" }{TEXT -1 1 " " }{XPPEDIT 18 0 "cos*x+10/101" "6#,&*&%$cosG\" \"\"%\"xGF&F&*&\"#5F&\"$,\"!\"\"F&" }{TEXT -1 1 " " }{XPPEDIT 18 0 "x* sin*x-200/10201" "6#,&*(%\"xG\"\"\"%$sinGF&F%F&F&*&\"$+#F&\"&,-\"!\"\" F+" }{TEXT -1 1 " " }{XPPEDIT 18 0 "sin*x+(990/10201+sqrt(5))*exp(-10* x)" "6#,&*&%$sinG\"\"\"%\"xGF&F&*&,&*&\"$!**F&\"&,-\"!\"\"F&-%%sqrtG6# \"\"&F&F&-%$expG6#,$*&\"#5F&F'F&F-F&F&" }{TEXT -1 1 "." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 163 "de := \+ diff(y(x),x)=10*x*cos(x)-10*y(x);\nic := y(0)=sqrt(5);\ndsolve(\{de,ic \},y(x));\nb := unapply(rhs(%),x):\nplot(b(x),x=0..5,font=[HELVETICA,9 ],labels=[`x`,`y(x)`]);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%#deG/-%%d iffG6$-%\"yG6#%\"xGF,,&*(\"#5\"\"\"F,F0-%$cosGF+F0F0*&F/F0F)F0!\"\"" } }{PARA 11 "" 1 "" {XPPMATH 20 "6#>%#icG/-%\"yG6#\"\"!*$\"\"&#\"\"\"\" \"#" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/-%\"yG6#%\"xG,,*&#\"$+\"\"$,\" \"\"\"*&F'F--%$cosGF&F-F-F-*&#\"$!**\"&,-\"F-F/F-!\"\"*&#\"#5F,F-*&-%$ sinGF&F-F'F-F-F-*&#\"$+#F4F-F:F-F5*&-%$expG6#,$*&F8F-F'F-F5F-,&#F3F4F- *$\"\"&#F-\"\"#F-F-F-" }}{PARA 13 "" 1 "" {GLPLOT2D 703 312 312 {PLOTDATA 2 "6&-%'CURVESG6$7hn7$$\"\"!F)$\"3\")*y*\\xz1OA!#<7$$\"3ALL$ 3FWYs#!#>$\"3*pc)\\jF;12h\"H\"48F,7$$ \"3m****\\7G$R<)F0$\"3<_u(oLbK,\"F,7$$\"3GLLL3x&)*3\"!#=$\"3(**[ro!GyV 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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 0 832 "B := (x,y) -> 10*x*cos(x)-10*y: hh := 0.01: numsteps := 500: \+ x0 := 0: y0 := sqrt(5):\nmatrix([[`slope field: `,B(x,y)],[`initial \+ point: `,``(x0,y0)],[`step width: `,hh],\n[`no. of steps: `,numste ps]]);``;\nmthds := [`Sharp-Verner scheme`,`scheme with a wide real st ability interval`,`scheme with a large imaginary axis inclusion`,`sche me with a quite large stability region and non-zero imaginary axis inc lusion`,`scheme with a quite large stability region`]: errs := []:\nDi gits := 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]-evalf(b(B n_RK6_||ct[ii,1])))^2;\n end do:\n errs := [op(errs),sqrt(sm/numpt s)];\nend do:\nDigits := 10:\nlinalg[transpose](convert([mthds,evalf(e rrs)],matrix));" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#K%'matrixG6#7&7$%0s lope~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(pprint86\"" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%!G" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#K%'matrixG6#7'7$%4Sharp-V erner~schemeG$\"+#QX#o%)!#A7$%Kscheme~with~a~wide~real~stability~inter valG$\"+,!H[0(!#@7$%Mscheme~with~a~large~imaginary~axis~inclusionG$\"+ ;s-*o#F+7$%[pscheme~with~a~quite~large~stability~region~and~non-zero~i maginary~axis~inclusionG$\"+iFK`AF07$%Kscheme~with~a~quite~large~stabi lity~regionG$\"+rB:D7F0Q(pprint96\"" }}}{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 0 756 " B := (x,y) -> 10*x*cos(x)-10*y: hh := 0.01: numsteps := 500: x0 := 0: \+ y0 := sqrt(5):\nmatrix([[`slope field: `,B(x,y)],[`initial point: `, ``(x0,y0)],[`step width: `,hh],\n[`no. of steps: `,numsteps]]);``; \nmthds := [`Sharp-Verner scheme`,`scheme with a wide real stability i nterval`,`scheme with a large imaginary axis inclusion`,`scheme with a quite large stability region and non-zero imaginary axis inclusion`,` scheme with a quite large stability region`]: errs := []:\nDigits := 2 5:\nfor ct to 5 do\n bn_RK6_||ct := RK6_||ct(B(x,y),x,y,x0,evalf(y0) ,hh,numsteps,true);\nend do:\nxx := 4.999: bxx := evalf(b(xx)):\nfor c t to 5 do\n errs := [op(errs),abs(bn_RK6_||ct(xx)-bxx)];\nend do:\nD igits := 10:\nlinalg[transpose](convert([mthds,evalf(errs)],matrix)); " }}{PARA 11 "" 1 "" {XPPMATH 20 "6#K%'matrixG6#7&7$%0slope~field:~~~G ,&*(\"#5\"\"\"%\"xGF,-%$cosG6#F-F,F,*&F+F,%\"yGF,!\"\"7$%0initial~poin t:~G-%!G6$\"\"!*$\"\"&#F,\"\"#7$%/step~width:~~~G$F,!\"#7$%1no.~of~ste ps:~~~G\"$+&Q)pprint116\"" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%!G" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#K%'matrixG6#7'7$%4Sharp-Verner~schemeG $\"+$\\s,[(!#B7$%Kscheme~with~a~wide~real~stability~intervalG$\"+Zu%RG \"!#A7$%Mscheme~with~a~large~imaginary~axis~inclusionG$\"+<5dE=F07$%[p scheme~with~a~quite~large~stability~region~and~non-zero~imaginary~axis ~inclusionG$\"+9@%[d'F+7$%Kscheme~with~a~quite~large~stability~regionG $\"+Wc(=\"**!#CQ)pprint126\"" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }} {PARA 0 "" 0 "" {TEXT -1 4 "The " }{TEXT 260 22 "root mean square erro r" }{TEXT -1 20 " over the interval " }{XPPEDIT 18 0 "[0, 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 met hod over 200 equal subintervals." }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 494 "mthds := [`Sharp-Verner scheme`,`scheme with a wide real stab ility interval`,`scheme with a large imaginary axis inclusion`,`scheme with a quite large stability region and non-zero imaginary axis inclu sion`,`scheme with a quite large stability region`]: errs := []:\nDigi ts := 20:\nfor ct to 5 do\n sm := NCint((b(x)-'bn_RK6_||ct'(x))^2,x= 0..5,adaptive=false,numpoints=7,factor=200);\n errs := [op(errs),sqr t(sm/5)];\nend do:\nDigits := 10:\nlinalg[transpose](convert([mthds,ev alf(errs)],matrix));" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#K%'matrixG6#7' 7$%4Sharp-Verner~schemeG$\"+s-tn\")!#A7$%Kscheme~with~a~wide~real~stab ility~intervalG$\"+e?N/o!#@7$%Mscheme~with~a~large~imaginary~axis~incl usionG$\"+3Rv#f#F+7$%[pscheme~with~a~quite~large~stability~region~and~ non-zero~imaginary~axis~inclusionG$\"+pn?t@F07$%Kscheme~with~a~quite~l arge~stability~regionG$\"+f9v\"=\"F0Q)pprint136\"" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 92 "The following error graph s are constructed using the numerical procedures for the solutions." } }{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 555 "evalf[20](plot(['bn_RK6_1' (x)-b(x),'bn_RK6_2'(x)-b(x),'bn_RK6_3'(x)-b(x),\n'bn_RK6_4'(x)-b(x),'b n_RK6_5'(x)-b(x)],x=0..0.65,numpoints=100,\ncolor=[COLOR(RGB,.9,0,.9), COLOR(RGB,0,.15,1),COLOR(RGB,.95,0,.2),COLOR(RGB,.95,.45,0),COLOR(RGB, 0,.75,.2)],\nlegend=[`Sharp-Verner scheme`,`scheme with a wide real st ability interval`,`scheme with a large imaginary axis inclusion`,`sche me with a quite large stability region and non-zero imag axis inclusio n`,`scheme with a quite large stability region`],title=`error curves f or 8 stage order 6 Runge-Kutta methods`));" }}{PARA 13 "" 1 "" {GLPLOT2D 999 566 566 {PLOTDATA 2 "6*-%'CURVESG6%7b\\l7$$\"\"!F)F(7$$ \"5SSSSSS:N<7$$\"5\"333333.ZV$F-$\"%MjF07$$\"5+,,,,^)yLH%F -$\"&N+$F07$$\"5?@@@@@Y0_^F-$\"'Iq5F07$$\"5STTTT\"RI2,'F-$\"'\\JJF07$$ 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[7'F]hl7$F_hl$!59&45.nEQpO&F]hl7$Fdhl$!5;$=5/oYK]P%F]hl7$Fihl$!6$G-%\\ d9JEXP$F]il7$F_il$!6YO?)3(QumvC#F]il7$Fdil$!6))G-Gk3HCP&**Fhil7$Fjil$ \"62y/(44#G\"z5DFhil7$F_jl$\"7p\\YcL/.(z(\\:Fhil7$Fdjl$\"8(*>$z&>(*Qdn ,)GFhjl7$Fjjl$\"8to**=:eUJzUD%Fhjl7$F_[m$\"9a],zj=#Qk#4(f&F_hn7$Fd[m$ \"9S#oSK'**zV&3^s'F_hn7$Fi[m$\"9oK^^miwI>PJzF_hn7$F^\\m$\":2fmT;-aRODH D*Fjbo7$Fc\\m$\":1\"fLHLy.>d8v5F_hn7$Fa^m$\":#=7Q2yO$\\AA'37F_hn7$Fgbm $\":!HEk^dGm.**Rg7F_hn7$Fedm$\":[/QNf3@s%)3DR\"F_hn7$Feim$\":?H()**3nP zz%y[9F_hn7$Fa]n$\":%\\GWn&4oGV9F_hn7$Fagn$\" :\"pgb_m&)=m>?G8F_hn7$Ffgn$\":l^?*oi(z3i[&=7F_hn7$F[hn$\":;*Gh,pk+-OQ< 6F_hn7$Fahn$\":fA\"f)Q+S_dhs-\"F_hn-Fghn6&FihnF]in$\"#vF*Feer-F_in6#%K scheme~with~a~quite~large~stability~regionG-%+AXESLABELSG6$Q\"x6\"Q!Fj `u-%&TITLEG6#%Uerror~curves~for~8~stage~order~6~Runge-Kutta~methodsG-% %VIEWG6$;F(Fahn%(DEFAULTG" 1 2 0 1 10 0 2 9 1 4 2 1.000000 45.000000 45.000000 0 1 "Sharp-Verner scheme" "scheme with a wide real stability interval" "scheme with a large imaginary axis inclusion" "scheme with a quite large stability region and non-zero imag axis inclusion" "sch eme with a quite large stability region" }}}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 47 "Test 15 of 8 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 and Sedgwick,\n Siam Journal on Numerical Ana lysis, 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 "Solut ion: " }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "2*ln((x^2+y^ 2)/(x^2))+4*arctan(y/x)+4*ln*x-2*ln*2-Pi = 0;" "6#/,,*&\"\"#\"\"\"-%#l nG6#*&,&*$%\"xGF&F'*$%\"yGF&F'F'*$F.F&!\"\"F'F'*&\"\"%F'-%'arctanG6#*& F0F'F.F2F'F'*(F4F'F)F'F.F'F'*(F&F'F)F'F&F'F2%#PiGF2\"\"!" }{TEXT -1 2 ". " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 73 "de := diff(y(x),x)=(y(x)-x)/(y(x)+x);\nic := y(1)=1; \ndsolve(\{de,ic\},y(x));" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%#deG/-% %diffG6$-%\"yG6#%\"xGF,*&,&F)\"\"\"F,!\"\"F/,&F)F/F,F/F0" }}{PARA 11 " " 1 "" {XPPMATH 20 "6#>%#icG/-%\"yG6#\"\"\"F)" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/-%\"yG6#%\"xG-%'RootOfG6#,,*&\"\"#\"\"\"-%#lnG6#*&,&*$ )F'F-F.F.*$)%#_ZGF-F.F.F.F'!\"#F.!\"\"*&\"\"%F.-%'arctanG6#*&F8F.F'F:F .F:*&F " 0 " " {MPLTEXT 1 0 1 ";" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 " " {TEXT -1 41 "The solution can be given more simply as " }}{PARA 0 " " 0 "" {TEXT -1 0 "" }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 " ln(x^2+y^2)+2*arctan(y/x)=ln*2+Pi/2" "6#/,&-%#lnG6#,&*$%\"xG\"\"# \"\"\"*$%\"yGF+F,F,*&F+F,-%'arctanG6#*&F.F,F*!\"\"F,F,,&*&F&F,F+F,F,*& %#PiGF,F+F4F," }{TEXT -1 2 ". " }}{PARA 0 "" 0 "" {TEXT -1 18 "The sol ution (for " }{TEXT 269 1 "x" }{TEXT -1 47 " increasing) is the sectio n of the polar curve " }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "r=sqrt(2)*exp(Pi/4-theta)" "6#/%\"rG*&-%%sqrtG6#\"\"#\"\"\"-%$ex pG6#,&*&%#PiGF*\"\"%!\"\"F*%&thetaGF2F*" }{TEXT -1 5 ", " } {XPPEDIT 18 0 "-Pi/4<=theta" "6#1,$*&%#PiG\"\"\"\"\"%!\"\"F)%&thetaG" }{XPPEDIT 18 0 "``<=Pi/4" "6#1%!G*&%#PiG\"\"\"\"\"%!\"\"" }{TEXT -1 2 ". " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 7 "Che ck: " }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 60 "ln((x^2+y^2))+2*arct an(y/x)=ln(2)+Pi/2;\nimplicitdiff(%,y,x);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/,&-%#lnG6#,&*$)%\"xG\"\"#\"\"\"F-*$)%\"yGF,F-F-F-*&F,F --%'arctanG6#*&F0F-F+!\"\"F-F-,&-F&6#F,F-*&F,F6%#PiGF-F-" }}{PARA 11 " " 1 "" {XPPMATH 20 "6#,$*&,&%\"xG\"\"\"%\"yG!\"\"F',&F(F'F&F'F)F)" }}} {PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 471 "p1 := plot([sqrt(2)*exp(Pi/4-t),t,t=-Pi/4..Pi/4],coords=polar,thi ckness=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,symbol=[circle$2,diamond,cross],\n \+ symbolsize=[12,10$3],color=[black,green$3]):\nplots[di splay]([p1,p2,p3,p4],font=[HELVETICA,9],labels=[`x`,`y(x)`]);" }} {PARA 13 "" 1 "" {GLPLOT2D 567 520 520 {PLOTDATA 2 "6,-%'CURVESG6%7S7$ $\"3F_`'4Qx/\"[!#<$!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$$ \"3)>>'Q`Nt[YF*$!3pIdcz?N/uDVF*$!3Mk<%4i8_'>F*7$$\"3neCR^2f VUF*$!3U`V:L2,dAqSF *$!3]0%Rz+R[P\"F*7$$\"3CpGt*G2)))RF*$!3Vje1<#>j@\"F*7$$\"31&*>Xy@@*)QF *$!3KrYY*z8q.\"F*7$$\"3m&*QGM(*y-QF*$!3Z+5yDwaF*)!#=7$$\"3x^E')y`D+PF* $!3%[->)G'=aL(F\\q7$$\"3v7s?m@*zg$F*$!3c,0?p.L,gF\\q7$$\"3'*yXFYgX0NF* $!3H?*=wI4nh%F\\q7$$\"3W4K\"RWEoS$F*$!3%3mwkU\\IP$F\\q7$$\"3sJbIp3<.LF *$!3OMj)e)4)4:#F\\q7$$\"3$p#*\\.zXv?$F*$!3gM#f*yWs%4\"F\\q7$$\"368R(=G rT5$F*$!3oLM#4e`nS#!#?7$$\"3O349%H!z'*HF*$\"3?)\\7Z#>+:5F\\q7$$\"3_&4u xb.N!HF*$\"3?;dqg#**3'=F\\q7$$\"3Q!f))4wMJ!GF*$\"3Rn1jA(f_r#F\\q7$$\"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$$\" 3tj&H(oDW3:F*$\"3!)e5j3rW%R*F\\q7$$\"3Hm+#y!yfG9F*$\"3U,&*3jpEm&*F\\q7 $$\"3YmX)QzoqN\"F*$\"3]k9op*4mp*F\\q7$$\"3s5rrvvGx7F*$\"3P5fR'>0`\")*F \\q7$$\"3XZ9deE%z?\"F*$\"3Gc$*zYS?&*)*F\\q7$$\"30pa^\"yv$))F\\q$\"3]-mK&*3^l**F\\q7$$\"3eM?/ #4dlu(F\\q$\"3uEm+DD_n)*F\\q7$$\"3%*yjGB$o\\&oF\\q$\"3AaVWo?rO(*F\\q7$ $\"3[>li88H;gF\\q$\"3w@=8&*eNF\\q$\"3yN hC*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-c f?z3(F\\q7$$\"3f(3R5[t$3HFc_l$\"3f7r>seQEnF\\q7$$!37`UU,+,&Q)F`s$\"3yN -Jd*yHO'F\\q7$$!3$)3\"RF\\q$\"3#4c;J=K@(HF\\q7$$ !3iuBr]$3f1#F\\q$\"3Gm]`0-<5CF\\q7$$!3G4mSr>(e2#F\\q$\"30UK<,&yQ#>F\\q 7$$!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_l7$$!3Q^mSZRTp9Fc_l$!3_uFr$)*ycw $Fc_l7$$!3-[9 Nr!)oebDFc_l7$$\"3Wcg/C\"='=WF`s$!37)o9o#3(GD#Fc_l7$$\"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`s 7$$\"3/Xu8A%33['F`s$!3;Is^'z2(yxFehl7$$\"3)Q4tG5rBz&F`s$\"3c0'H(*oS@v% !#H-Fjz6&F\\[lFa[lFa[lFa[l-%*LINESTYLEGFd[l-F$6%7S7$$\"3W'4ORO![>WF*$! 3Ar[\"*HXwawF*7$$\"3VdsTJRpPWF*$!3M'>\"=)=2ge(F*7$$\"3'G4%*HMRJX%F*$!3 8,ZT))oTEvF*7$$\"3_L#[)Gh1qWF*$!3SWWx3PmfuF*7$$\"3_-)f6JNm[%F*$!3#o7AC d\\FR(F*7$$\"3akH:B>m-XF*$!3:r-1\\\\VEtF*7$$\"3C*o#*G!46\"Q)[RgXF*$!35r.'eb*[sqF*7$$\"3I*G!GQ%[pF*7$$\"3u*H#3E-h*f%F*$!3wdnf&y\"pUX'F*7$$\"3CLFg!*\\T'o%F*$!3WZZJr$p:R'F*7$$\"3 9!\\a\\())R&p%F*$!3JUA:p?9KjF*7$$\"3*yZODQ#Q/ZF*$!3eR\\wA`UqiF*7$$\"3# Gs'H4VG7ZF*$!3@Fo#z)3,9iF*7$$\"3zqGL?zV?ZF*$!3[?z6=2W`hF*7$$\"3,4vY0\\ ]GZF*$!3')*GM%4S$34'F*7$$\"3<`G9ym>NZF*$!3T9'Qi<\"fOgF*7$$\"3J@Tq\"p\" 3UZF*$!3!y4%3&*fFyfF*7$$\"3w'3pk%[#)[ZF*$!3<#*G&H-C$=fF*7$$\"31ni$>nh] v%F*$!3CB5%zdh*feF*7$$\"3G2i'*30wgZF*$!3$H7(43_w.eF*7$$\"3)Q\"Q9p_qmZF *$!3a]Y_qGoTdF*7$$\"3[]LaAnqrZF*$!3kKO%\\U\"='o&F*7$$\"3!zPO+/&pwZF*$! 3R2\")Hxq@FcF*7$$\"3oO**41X!4y%F*$!3Ayw?\\-0ubF*7$$\"3G3]Qj><&y%F*$!3t <&zB%>@;bF*7$$\"3?1\"fYJr))y%F*$!3Crd]bY1iaF*7$$\"3eA/XIST#z%F*$!3Z>]2 +%\\dS&F*7$$\"392#[Idgbz%F*$!3Y'QI)[?(4N&F*7$$\"3gZd(yDA&)z%F*$!3#*=D? 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\\o-y%F-$!+G<\")eSF-7$$\"+D\"Q.z%F-$!+!p\\]>%F-7$$\"+SHP&z%F-$!+M@kwUF -7$$\"+axS+[F-$!+ " 0 "" {MPLTEXT 1 0 877 "C := (x,y) -> (y-x)/(y+x): \+ hh := 0.01: numsteps := 375: x0 := 1: y0 := 1:\nmatrix([[`slope field: `,C(x,y)],[`initial point: `,``(x0,y0)],[`step width: `,hh],\n[`n o. of steps: `,numsteps]]);``;\nmthds := [`Sharp-Verner scheme`,`sch eme with a wide real stability interval`,`scheme with a large imaginar y axis inclusion`,`scheme with a quite large stability region and non- zero imaginary axis inclusion`,`scheme with a quite large stability re gion`]: errs := []: vals := []:\nDigits := 25:\nfor ct to 5 do\n Cn _RK6_||ct := RK6_||ct(C(x,y),x,y,x0,y0,hh,numsteps,false);\n sm := 0 : numpts := nops(Cn_RK6_||ct):\n for ii to numpts do\n if ct=1 \+ then vals := [op(vals),phi(Cn_RK6_||ct[ii,1])] end if;\n sm := sm +(Cn_RK6_||ct[ii,2]-vals[ii])^2;\n end do:\n errs := [op(errs),sqr t(sm/numpts)];\nend do:\nDigits := 10:\nlinalg[transpose](convert([mth ds,evalf(errs)],matrix));" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#K%'matrix G6#7&7$%0slope~field:~~~G*&,&%\"yG\"\"\"%\"xG!\"\"F,,&F-F,F+F,F.7$%0in itial~point:~G-%!G6$F,F,7$%/step~width:~~~G$F,!\"#7$%1no.~of~steps:~~~ G\"$v$Q)pprint146\"" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%!G" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#K%'matrixG6#7'7$%4Sharp-Verner~schemeG$\"+gg U_m!#A7$%Kscheme~with~a~wide~real~stability~intervalG$\"+DT2GD!#@7$%Ms cheme~with~a~large~imaginary~axis~inclusionG$\"+z3\\\"*o!#B7$%[pscheme ~with~a~quite~large~stability~region~and~non-zero~imaginary~axis~inclu sionG$\"+([MD8%F+7$%Kscheme~with~a~quite~large~stability~regionG$\"+e^ D)Q(F+Q)pprint156\"" }}}{PARA 0 "" 0 "" {TEXT -1 2 " " }}{PARA 0 "" 0 "" {TEXT -1 30 "The following code constructs " }{TEXT 260 20 "numer ical procedures" }{TEXT -1 56 " for solutions based on each of the Run ge-Kutta schemes." }}{PARA 0 "" 0 "" {TEXT -1 75 "The error in the val ue obtained by each of the methods at the point where " }{XPPEDIT 18 0 "x = 4.749;" "6#/%\"xG-%&FloatG6$\"%\\Z!\"$" }{TEXT -1 16 " is also given." }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 740 "C := (x,y) -> (y -x)/(y+x): hh := 0.01: numsteps := 375: x0 := 1: y0 := 1:\nmatrix([[`s lope field: `,C(x,y)],[`initial point: `,``(x0,y0)],[`step width: \+ `,hh],\n[`no. of steps: `,numsteps]]);``;\nmthds := [`Sharp-Verner s cheme`,`scheme with a wide real stability interval`,`scheme with a lar ge imaginary axis inclusion`,`scheme with a quite large stability regi on and non-zero imaginary axis inclusion`,`scheme with a quite large s tability region`]: errs := []:\nDigits := 30:\nfor ct to 5 do\n cn_R K6_||ct := RK6_||ct(C(x,y),x,y,x0,y0,hh,numsteps,true);\nend do:\nxx : = 4.749: cxx := evalf(phi(xx)):\nfor ct to 5 do\n errs := [op(errs), abs(cn_RK6_||ct(xx)-cxx)];\nend do:\nDigits := 10:\nlinalg[transpose]( convert([mthds,evalf(errs)],matrix));" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#K%'matrixG6#7&7$%0slope~field:~~~G*&,&%\"yG\"\"\"%\"xG!\"\"F,,&F-F, F+F,F.7$%0initial~point:~G-%!G6$F,F,7$%/step~width:~~~G$F,!\"#7$%1no.~ of~steps:~~~G\"$v$Q)pprint166\"" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%!G " }}{PARA 11 "" 1 "" {XPPMATH 20 "6#K%'matrixG6#7'7$%4Sharp-Verner~sch emeG$\"+=,Is\")!#@7$%Kscheme~with~a~wide~real~stability~intervalG$\"+7 =f'4$!#?7$%Mscheme~with~a~large~imaginary~axis~inclusionG$\"+M=W>l!#A7 $%[pscheme~with~a~quite~large~stability~region~and~non-zero~imaginary~ axis~inclusionG$\"+!3C)G_F+7$%Kscheme~with~a~quite~large~stability~reg ionG$\"+h=v&3*F+Q)pprint176\"" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }} {PARA 0 "" 0 "" {TEXT -1 4 "The " }{TEXT 260 22 "root mean square erro r" }{TEXT -1 20 " over the interval " }{XPPEDIT 18 0 "[1, 4.75];" "6# 7$\"\"\"-%&FloatG6$\"$v%!\"#" }{TEXT -1 82 " of each Runge-Kutta meth od is estimated as follows using the special procedure " }{TEXT 0 5 " NCint" }{TEXT -1 98 " to perform numerical integration by the 7 point Newton-Cotes method over 200 equal subintervals." }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 501 "mthds := [`Sharp-Verner scheme`,`scheme with a wide real stability interval`,`scheme with a large imaginary axis i nclusion`,`scheme with a quite large stability region and non-zero ima ginary axis inclusion`,`scheme with a quite large stability region`]: \+ errs := []:\nDigits := 20:\nfor ct to 5 do\n sm := NCint(('phi'(x)-' cn_RK6_||ct'(x))^2,x=1..4.75,adaptive=false,numpoints=7,factor=200);\n errs := [op(errs),sqrt(sm/5)];\nend do:\nDigits := 10:\nlinalg[tran spose](convert([mthds,evalf(errs)],matrix));" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#K%'matrixG6#7'7$%4Sharp-Verner~schemeG$\"+Uh[9K!#A7$%Ks cheme~with~a~wide~real~stability~intervalG$\"+Fm.37!#@7$%Mscheme~with~ a~large~imaginary~axis~inclusionG$\"+EKH%G#!#B7$%[pscheme~with~a~quite ~large~stability~region~and~non-zero~imaginary~axis~inclusionG$\"+*p6- ;#F+7$%Kscheme~with~a~quite~large~stability~regionG$\"+yj??NF+Q)pprint 186\"" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 92 "The following error graphs are constructed using the numerical proced ures for the solutions." }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 598 " evalf[30](plot(['cn_RK6_1'(x)-'phi'(x),'cn_RK6_2'(x)-'phi'(x),'cn_RK6_ 3'(x)-'phi'(x),\n'cn_RK6_4'(x)-'phi'(x),'cn_RK6_5'(x)-'phi'(x)],x=1..3 .75,-2.7e-17..5.5e-17,font=[HELVETICA,9],\ncolor=[COLOR(RGB,.9,0,.9),C OLOR(RGB,0,.15,1),COLOR(RGB,.95,0,.2),COLOR(RGB,.95,.45,0),COLOR(RGB,0 ,.75,.2)],\nlegend=[`Sharp-Verner scheme`,`scheme with a wide real sta bility interval`,`scheme with a large imaginary axis inclusion`,`schem e with a quite large stability region and non-zero imag axis inclusion `,`scheme with a quite large stability region`],title=`error curves fo r 8 stage order 6 Runge-Kutta methods`));" }}{PARA 13 "" 1 "" {GLPLOT2D 1041 597 597 {PLOTDATA 2 "6+-%'CURVESG6%7jn7$$\"\"\"\"\"!$F* F*7$$\"?LLLLLLLL$eR'>'F27$$\"?LLLLLLLLeRiYzH7F/$!-(= YkSh(F27$$\"?nmmmmmm;a8-qb)G\"F/$!-q\"*)*4b()F27$$\"?LLLLLLL$3xJ@PIM\" 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