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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#\"\"$" }{TEXT -1 3 ", " }{XPPEDIT 18 0 "c[4]" "6# &%\"cG6#\"\"%" }{TEXT -1 3 ", " }{XPPEDIT 18 0 "c[5]" "6#&%\"cG6#\"\" &" }{TEXT -1 3 ", " }{XPPEDIT 18 0 "c[6]" "6#&%\"cG6#\"\"'" }{TEXT -1 3 ", " }{XPPEDIT 18 0 "c[7]" "6#&%\"cG6#\"\"(" }{TEXT -1 3 ", " } {XPPEDIT 18 0 "c[8]=1" "6#/&%\"cG6#\"\")\"\"\"" }{TEXT -1 3 ", " } {XPPEDIT 18 0 "c[9]=1" "6#/&%\"cG6#\"\"*\"\"\"" }{TEXT -1 11 " and le t " }{TEXT 281 2 "Id" }{TEXT -1 32 " be the 9 by 9 identity matrix. " }}{PARA 0 "" 0 "" {TEXT -1 5 "Let " }{XPPEDIT 18 0 "[c]" "6#7#%\"cG " }{TEXT -1 38 " be the row vector whose entries are " }{XPPEDIT 18 0 "c[1]=0" "6#/&%\"cG6#\"\"\"\"\"!" }{TEXT -1 3 ", " }{XPPEDIT 18 0 " c[2]" "6#&%\"cG6#\"\"#" }{TEXT -1 3 ", " }{XPPEDIT 18 0 "c[3]" "6#&% \"cG6#\"\"$" }{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 9 " in the " }{TEXT 260 15 "outer summation " }{TEXT -1 9 " 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 \"\"$" }{TEXT -1 25 " . . 7, and we can omit " }{XPPEDIT 18 0 "j=5" " 6#/%\"jG\"\"&" }{TEXT -1 7 " and " }{XPPEDIT 18 0 "j=6" "6#/%\"jG\" \"'" }{TEXT -1 34 " because of obvious zero factors." }}{PARA 0 "" 0 "" {TEXT -1 13 "We can omit " }{XPPEDIT 18 0 "j=4" "6#/%\"jG\"\"%" } {TEXT -1 30 " because (it turns out that) " }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "Sum(b[i]*(c[i]-1)*a[i,3],i=4..7)=0" "6# /-%$SumG6$*(&%\"bG6#%\"iG\"\"\",&&%\"cG6#F+F,F,!\"\"F,&%\"aG6$F+\"\"$F ,/F+;\"\"%\"\"(\"\"!" }{TEXT -1 2 ". " }}{PARA 0 "" 0 "" {TEXT -1 24 " This gives the relation " }}{PARA 256 "" 0 "" {TEXT -1 1 " " } {XPPEDIT 18 0 "b[7]*(c[7]-1)*a[7,6]*(c[6]-c[4])*(c[6]-c[5])*c[6]=-1/12 0+c[4]/60+c[5]/60-c[4]*c[5]/24" "6#/*.&%\"bG6#\"\"(\"\"\",&&%\"cG6#F(F )F)!\"\"F)&%\"aG6$F(\"\"'F),&&F,6#F2F)&F,6#\"\"%F.F),&&F,6#F2F)&F,6#\" \"&F.F)&F,6#F2F),**&F)F)\"$?\"F.F.*&&F,6#F8F)\"#gF.F)*&&F,6#F>F)FGF.F) *(&F,6#F8F)&F,6#F>F)\"#CF.F." }{TEXT -1 1 "." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 61 "#------------------------------ ------------------------------" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 62 "Int((x-1)*Int((t-c[4])*(t-c[ 5])*t,t=0..x),x=0..1):\n%=value(%);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6 #/-%$IntG6$*&,&%\"xG\"\"\"F*!\"\"F*-F%6$*(,&%\"tGF*&%\"cG6#\"\"%F+F*,& F0F*&F26#\"\"&F+F*F0F*/F0;\"\"!F)F*/F);F;F*,*#F*\"$?\"F+*&#F*\"#gF*F1F *F**&FBF*F6F*F**&#F*\"#CF**&F6F*F1F*F*F+" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 107 "We can use the Sharp-Verner schem e to provide a numerical check for the order condition in the matrix f orm." }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 159 "evalm(B &* (C-Id) & * A &* (C-c[4]*Id) &* (C-c[5]*Id) &* c_)[1,1]=int((x-1)*int((t-c[4])*( t-c[5])*t,t=0..x),x=0..1):\nsubs(\{c[1]=0,c[8]=1,c[9]=1\},%);\nsubs(ee ,%);" }}{PARA 12 "" 1 "" {XPPMATH 20 "6#/,(**,,*(&%\"bG6#\"\"$\"\"\",& &%\"cGF*F,F,!\"\"F,&%\"aG6$F+\"\"#F,F,*(&F)6#\"\"%F,,&&F/F7F,F,F0F,&F2 6$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$FPF+F,F,F,,&F.F,F:F 0F,,&F.F,FBF0F,F.F,F,*.FNF,FQF,&F26$FPFHF,,&F:F0FJF,F,,&FBF0FJF,F,FJF, F,,*#F,\"$?\"F0*&#F,\"#gF,F:F,F,*&F\\pF,FBF,F,*&#F,\"#CF,*&FBF,F:F,F,F 0" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/#!\"\"\"%+=F$" }}}{PARA 0 "" 0 " " {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 80 "We can also make a nume rical check of the order condition in the summation 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,FI F,&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$" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" } {TEXT 280 0 "" }{TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 84 "After the omissions that we can make in the outer summation we obtain the follo wing." }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 132 "add(add(b[i]*(c[i] -1)*a[i,j-1],i=j..7)*(c[j-1]-c[4])*(c[j-1]-c[5])*c[j-1],j=[7])=\n -1/ 120+c[4]/60+c[5]/60-c[4]*c[5]/24;\nsubs(ee,%);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/*.&%\"bG6#\"\"(\"\"\",&F)!\"\"&%\"cGF'F)F)&%\"aG6$F(\" \"'F),&&F-6#\"\"%F+&F-6#F1F)F),&&F-6#\"\"&F+F6F)F)F6F),*#F)\"$?\"F+*&# F)\"#gF)F3F)F)*&F@F)F9F)F)*&#F)\"#CF)*&F9F)F3F)F)F+" }}{PARA 11 "" 1 " " {XPPMATH 20 "6#/#!\"\"\"%+=F$" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 57 "#========================================================" }} {PARA 0 "" 0 "" {TEXT -1 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)%#Id GF)!\"\"F),&F-F)*&&F06#\"\"&F)F3F)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 o f 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]/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;F8\"\"*,**&F/F/\"#?F5F/*&&F;6#F@F/\"#7F5F5*&&F;6#FGF/F TF5F5*(&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$*(,&%\"tG\"\"\"&%\"cG6#\"\"%!\"\"F,,&F+F,&F.6#\"\"&F1F,F +F,/F+;\"\"!%\"xG/F9;F8F,,*#F,\"#?F,*&#F,\"#7F,F-F,F1*&#F,FAF,F3F,F1*& #F,\"\"'F,*&F-F,F3F,F,F," }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 " " 0 "" {TEXT -1 107 "We can use the Sharp-Verner scheme to provide a n umerical check for the order condition in the matrix form." }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 139 "evalm(`B*` &* A &* (C-c[4]*Id) &* \+ (C-c[5]*Id) &* c_)[1,1]=int(int((t-c[4])*(t-c[5])*t,t=0..x),x=0..1):\n subs(\{c[1]=0,c[8]=1\},%);\nsubs(ee,%);" }}{PARA 12 "" 1 "" {XPPMATH 20 "6#/,,**,0*&&%#b*G6#\"\"$\"\"\"&%\"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$" }} }{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 80 "We can a lso make a numerical check of the order condition in the summation for m." }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 149 "add(add(`b*`[i]*a[i,j -1],i=j..9)*(c[j-1]-c[4])*(c[j-1]-c[5])*c[j-1],j=2..9)=\n 1/20-c[4]/1 2-c[5]/12+c[4]*c[5]/6:\nsubs(\{c[1]=0,c[8]=1\},%);\nsubs(ee,%);" }} {PARA 12 "" 1 "" {XPPMATH 20 "6#/,,**,0*&&%#b*G6#\"\"$\"\"\"&%\"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$FFF0F,F,*&&F)6#\"\")F,&F.6$FLF0 F,F,*&&F)6#\"\"*F,&F.6$FRF0F,F,F,,&&%\"cG6#F0F,&FWF3!\"\"F,,&FVF,&FWF9 FZF,FVF,F,**,.*&F2F,&F.6$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$FRF+F,F,F,,&&FWF*F,F YFZF,,&F\\pF,FfnFZF,F\\pF,F,**,(*&FDF,&F.6$FFF@F,F,*&FJF,&F.6$FLF@F,F, *&FPF,&F.6$FRF@F,F,F,,&FYFZ&FWF?F,F,,&FfnFZFjpF,F,FjpF,F,**,&*&FJF,&F. 6$FLFFF,F,*&FPF,&F.6$FRFFF,F,F,,&&FWFEF,FYFZF,,&FeqF,FfnFZF,FeqF,F,**F PF,&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$" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 84 "After the omissions that we can make in the outer su mmation we obtain the following." }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 124 "add(add(`b*`[i]*a[i,j-1],i=j..9)*(c[j-1]-c[4])*(c[j-1]-c[5])* c[j-1],j=7..9)=\n 1/20-c[4]/12-c[5]/12+c[4]*c[5]/6;\nsubs(ee,%);" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#/,(**,(*&&%#b*G6#\"\"(\"\"\"&%\"aG6$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$F4 F+F,F,*&F8F,&F.6$F:F+F,F,F,,&&F?F*F,F>FBF,,&FRF,FFFBF,FRF,F,*,F8F,&F.6 $F:F4F,,&&F?F3F,F>FBF,,&FXF,FFFBF,FXF,F,,*#F,\"#?F,*&#F,\"#7F,F>F,FB*& #F,FinF,FFF,FB*&#F,F0F,*&FFF,F>F,F,F," }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/#\"\"\"\"$]\"F$" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 " " {TEXT -1 61 "#------------------------------------------------------ ------" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}}{PARA 0 "" 0 "" {TEXT -1 60 "#--------------------------------------------------- --------" }}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 43 "Tsitouras' algorithm \+ .. Sharp-Verner scheme" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 118 "In this subsection we illustrate the algorithm of C h. Tsitouras by using it to construct a scheme of Sharp and Verner." } }{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 161 "See: Co mpletely Imbedded Runge-Kutta Pairs, by P. W. Sharp and J. H. Verner, \n SIAM Journal on Numerical Analysis, Vol. 31, No. 4. (Aug., 19 94), page 1185." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 22 "We specify the nodes: " }}{PARA 256 "" 0 "" {TEXT -1 1 " \+ " }{XPPEDIT 18 0 "c[2] = 1/12;" "6#/&%\"cG6#\"\"#*&\"\"\"F)\"#7!\"\"" }{TEXT -1 3 ", " }{XPPEDIT 18 0 "c[4] = 1/5;" "6#/&%\"cG6#\"\"%*&\"\" \"F)\"\"&!\"\"" }{TEXT -1 3 ", " }{XPPEDIT 18 0 "c[5] = 8/15;" "6#/&% \"cG6#\"\"&*&\"\")\"\"\"\"#:!\"\"" }{TEXT -1 3 ", " }{XPPEDIT 18 0 "c [6] = 2/3;" "6#/&%\"cG6#\"\"'*&\"\"#\"\"\"\"\"$!\"\"" }{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 coe fficients: " }}{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#\"\"$\"\"!" }{TEXT -1 3 ", " }{XPPEDIT 18 0 "`b*`[8]=0" "6#/&%#b*G6#\"\")\"\"!" }{TEXT -1 1 "." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 90 "W e shall obtain expressions for all the coefficients of the scheme in t erms of the nodes " }{XPPEDIT 18 0 "c[2]" "6#&%\"cG6#\"\"#" }{TEXT -1 3 ", " }{XPPEDIT 18 0 "c[4]" "6#&%\"cG6#\"\"%" }{TEXT -1 3 ", " } {XPPEDIT 18 0 "c[5]" "6#&%\"cG6#\"\"&" }{TEXT -1 3 ", " }{XPPEDIT 18 0 "c[6]" "6#&%\"cG6#\"\"'" }{TEXT -1 7 " and " }{XPPEDIT 18 0 "c[7] " "6#&%\"cG6#\"\"(" }{TEXT -1 1 "." }}{PARA 0 "" 0 "" {TEXT -1 10 "The node " }{XPPEDIT 18 0 "c[3]" "6#&%\"cG6#\"\"$" }{TEXT -1 27 " does \+ not appear because " }{XPPEDIT 18 0 "c[3]=2/3" "6#/&%\"cG6#\"\"$*&\" \"#\"\"\"F'!\"\"" }{TEXT -1 1 " " }{XPPEDIT 18 0 "c[4]" "6#&%\"cG6#\" \"%" }{TEXT -1 2 ". " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 146 "e1 := \{c[2]=1/12,c[4]=1/5,c[5]=8/15,c[6 ]=2/3,c[7]=19/20,c[8]=1,c[9]=1,\n b[2]=0,b[3]=0,`b*`[2]=0,`b*` [3]=0,`b*`[8]=0,seq(a[i,2]=0,i=4..8)\}:" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 15 "" 0 "" {TEXT 282 6 "Step 1" }{TEXT -1 1 " " }}{PARA 0 "" 0 "" {TEXT -1 33 "We use the quadrature conditions:" }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "Sum(b[i],i=1..8)=1" "6#/-%$SumG 6$&%\"bG6#%\"iG/F*;\"\"\"\"\")F-" }{TEXT -1 15 ", " } {XPPEDIT 18 0 "Sum(b[i]*c[i]^(k-1),i = 2 .. 8) = 1/k;" "6#/-%$SumG6$*& &%\"bG6#%\"iG\"\"\")&%\"cG6#F+,&%\"kGF,F,!\"\"F,/F+;\"\"#\"\")*&F,F,F2 F3" }{TEXT -1 7 ", " }{XPPEDIT 18 0 "k = 2;" "6#/%\"kG\"\"#" } {TEXT -1 8 " . . 6, " }}{PARA 0 "" 0 "" {TEXT -1 21 "to find the weigh ts " }{XPPEDIT 18 0 "b[1]" "6#&%\"bG6#\"\"\"" }{TEXT -1 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 \+ follows." }}{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 283 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#/*&&%\"aG 6$\"\"$\"\"#\"\"\"&%\"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#* $&%\"cG6#\"\"%\"\"#" }{TEXT -1 15 ", " }{XPPEDIT 18 0 "Su m(a[4,j]*c[j]^2,j = 2 .. 3) = 1/3;" "6#/-%$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#&%\"cG 6#\"\"$" }{TEXT -1 3 ", " }{XPPEDIT 18 0 "a[3,2]" "6#&%\"aG6$\"\"$\" \"#" }{TEXT -1 7 " and " }{XPPEDIT 18 0 "a[4,3];" "6#&%\"aG6$\"\"%\" \"$" }{TEXT -1 2 ". " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 " " {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 269 "[a[3,2]*c [2] = c[3]^2/2,seq(add(a[4,j]*c[j]^(k-1),j=2..3)=c[4]^k/k,k=[2,3])]:\n eqns1 := subs(e3,%):\neqns1[1];\neqns1[2];\neqns1[3];\ne4 := solve(\{o p(eqns1)\},\{a[3,2],c[3],a[4,3]\}):\ne5 := `union`(e3,e4):\n``;\nc[3]= subs(e5,c[3]);\na[3,2]=subs(e5,a[3,2]);\na[4,3]=subs(e5,a[4,3]);" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#/,$*&#\"\"\"\"#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#\"\"$#\"\"#\"#:" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/&%\"aG 6$\"\"$\"\"##\"\")\"#v" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/&%\"aG6$\" \"%\"\"$#F(\"#?" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> \+ " 0 "" {MPLTEXT 1 0 1 ";" }}}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 2 "e5" } }{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 343 "e5 := \{a[8,2] = 0, c[9] = 1, c[8] = 1, a[5,2] = 0, a[4,2] = 0, a[6,2] = 0, a[7,2] = 0, b[3] = 0, b[2] = 0, `b*`[3] = 0, `b*`[2] = 0, a[3,2] = 8/75, c[3] = 2/15, a[4,3] = 3/20, b[1] = 53/912, b[8] = -25/ 336, b[7] = 256/969, b[6] = 27/136, `b*`[8] = 0, c[4] = 1/5, c[2] = 1/ 12, c[6] = 2/3, b[5] = 27/112, b[4] = 5/16, c[5] = 8/15, c[7] = 19/20 \}:" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 15 "" 0 " " {TEXT -1 1 " " }{TEXT 284 6 "Step 3" }{TEXT -1 1 " " }}{PARA 0 "" 0 "" {TEXT -1 35 "We use the stage-order conditions: " }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "Sum(a[5,j]*c[j],j = 2 .. 4) = 1/2;" "6#/-%$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$\"\"&\"\"%" }{TEXT -1 1 "." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 209 "[seq(add(a[5,j]*c[j]^(k-1),j=2..4)=c[5]^k/k,k=[2,3]) ]:\neqns2 := subs(e5,%):\neqns2[1];\neqns2[2];\ne6 := solve(\{op(eqns2 )\},\{a[5,3],a[5,4]\}):\ne7 := `union`(e5,subs(e4,e6)):\n``;\nseq(a[5, j]=subs(e7,a[5,j]),j=[3,4]);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/,&*&# \"\"#\"#:\"\"\"&%\"aG6$\"\"&\"\"$F)F)*&#F)F-F)&F+6$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] = 256/9 69, 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 285 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$\"\") \"\"(" }{TEXT -1 1 "." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 113 "b[8]*a[8,7] = b[7]*(1-c[7]);\nsubs(e7,%) ;\ne8 := solve(\{%\},\{a[8,7]\}):\ne9 := `union`(e7,e8):\na[8,7]=subs( e9,a[8,7]);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/*&&%\"bG6#\"\")\"\"\"& %\"aG6$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$\"\")\"\"(#!%or\"&v.%" } }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 2 "e9" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 399 "e9 := \{a[8 ,2] = 0, c[9] = 1, c[8] = 1, a[5,2] = 0, a[4,2] = 0, a[6,2] = 0, a[7,2 ] = 0, b[3] = 0, b[2] = 0, `b*`[3] = 0, `b*`[2] = 0, a[5,3] = -112/45, a[5,4] = 64/27, a[8,7] = -7168/40375, a[3,2] = 8/75, c[3] = 2/15, a[4 ,3] = 3/20, b[1] = 53/912, b[8] = -25/336, b[7] = 256/969, b[6] = 27/1 36, `b*`[8] = 0, c[4] = 1/5, c[2] = 1/12, c[6] = 2/3, b[5] = 27/112, b [4] = 5/16, c[5] = 8/15, c[7] = 19/20\}:" }{TEXT -1 0 "" }}}{PARA 0 " " 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}} }{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 15 "" 0 "" {TEXT -1 1 " " } {TEXT 286 6 "Step 5" }{TEXT -1 1 " " }}{PARA 0 "" 0 "" {TEXT -1 41 "We use the \"alternative\" order condition " }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "[b]^T*([C]-Id)*[A]*(C-c[4]*Id)*(C-c[5]*Id)*[c ] = Int((x-1)*Int((t-c[4])*(t-c[5])*t,t = 0 .. x),x = 0 .. 1);" "6#/*. )7#%\"bG%\"TG\"\"\",&7#%\"CGF)%#IdG!\"\"F)7#%\"AGF),&F,F)*&&%\"cG6#\" \"%F)F-F)F.F),&F,F)*&&F46#\"\"&F)F-F)F.F)7#F4F)-%$IntG6$*&,&%\"xGF)F)F .F)-F>6$*(,&%\"tGF)&F46#F6F.F),&FGF)&F46#F;F.F)FGF)/FG;\"\"!FBF)/FB;FO F)" }{TEXT -1 2 ". " }}{PARA 0 "" 0 "" {TEXT -1 39 "This condition amo unts 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#F8 F)&F,6#F>F)\"#CF.F." }{TEXT -1 1 "," }}{PARA 0 "" 0 "" {TEXT -1 27 "wh ich can be used to find " }{XPPEDIT 18 0 "a[7,6]" "6#&%\"aG6$\"\"(\" \"'" }{TEXT -1 2 ". " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 179 "b[7]*(c[7]-1)*a[7,6]*(c[6]-c[4])*(c[6]-c [5])*c[6] = -1/120+c[4]/60+c[5]/60-c[4]*c[5]/24;\nsubs(e9,%);\ne10 := \+ solve(\{%\},a[7,6]):\ne11 := `union`(e9,e10):\n``;\na[7,6]=subs(e11,a[ 7,6]);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/*.&%\"bG6#\"\"(\"\"\",&&%\" cGF'F)F)!\"\"F)&%\"aG6$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$\"\"(\"\"'#\"&NX\"\"&OV \"" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 3 "e11" }} {PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 422 "e11 := \{a[8,2] = 0, c[9] = 1, c[8] = 1, a[5,2] = 0, a[4,2] = 0, \+ a[6,2] = 0, a[7,2] = 0, b[3] = 0, b[2] = 0, `b*`[3] = 0, `b*`[2] = 0, \+ a[5,3] = -112/45, a[5,4] = 64/27, a[8,7] = -7168/40375, a[7,6] = 14535 /14336, a[3,2] = 8/75, c[3] = 2/15, a[4,3] = 3/20, b[1] = 53/912, b[8] = -25/336, b[7] = 256/969, b[6] = 27/136, `b*`[8] = 0, c[4] = 1/5, c[ 2] = 1/12, c[6] = 2/3, b[5] = 27/112, b[4] = 5/16, c[5] = 8/15, c[7] = 19/20\}:" }{TEXT -1 0 "" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}}{PARA 0 "" 0 "" {TEXT -1 0 " " }}{PARA 15 "" 0 "" {TEXT -1 1 " " }{TEXT 287 6 "Step 6" }{TEXT -1 1 " " }}{PARA 0 "" 0 "" {TEXT -1 41 "We use the (column) simplifying con dition" }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "Sum(b[i]*a[ i,j],i=j..8)=b[j]*(1-c[j])" "6#/-%$SumG6$*&&%\"bG6#%\"iG\"\"\"&%\"aG6$ F+%\"jGF,/F+;F0\"\")*&&F)6#F0F,,&F,F,&%\"cG6#F0!\"\"F," }{TEXT -1 7 " \+ for " }{XPPEDIT 18 0 "j = 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$\"\")\"\"'" } {TEXT -1 1 "." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 133 "add(b[i]*a[i,6],i=7..8)=b[6]*(1-c[6]):\nsubs(e1 1,%);\ne12 := solve(\{%\},\{a[8,6]\}):\ne13 := `union`(e11,e12):\n``; \na[8,6]=subs(e13,a[8,6]);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/,&#\"#: \"#c\"\"\"*&#\"#D\"$O$F(&%\"aG6$\"\")\"\"'F(!\"\"#\"\"*\"$O\"" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#%!G" }}{PARA 11 "" 1 "" {XPPMATH 20 "6 #/&%\"aG6$\"\")\"\"'#\"%_6\"$D%" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}{SECT 1 {PARA 4 "" 0 " " {TEXT -1 3 "e13" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 441 "e13 := \{a[8,2] = 0, c[9] = 1, c[8] = 1, a[5 ,2] = 0, a[4,2] = 0, a[6,2] = 0, a[7,2] = 0, b[3] = 0, b[2] = 0, `b*`[ 3] = 0, `b*`[2] = 0, a[5,3] = -112/45, a[5,4] = 64/27, a[8,7] = -7168/ 40375, a[7,6] = 14535/14336, a[3,2] = 8/75, c[3] = 2/15, a[4,3] = 3/20 , b[1] = 53/912, b[8] = -25/336, b[7] = 256/969, b[6] = 27/136, `b*`[8 ] = 0, c[4] = 1/5, c[2] = 1/12, c[6] = 2/3, b[5] = 27/112, b[4] = 5/16 , a[8,6] = 1152/425, c[5] = 8/15, c[7] = 19/20\}:" }{TEXT -1 0 "" }}} {PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 15 "" 0 "" {TEXT 288 6 "Step 7" }{TEXT -1 1 " " }}{PARA 0 "" 0 "" {TEXT -1 23 "We find the \+ 6 weights " }{XPPEDIT 18 0 "`b*`[1];" "6#&%#b*G6#\"\"\"" }{TEXT -1 3 ", " }{XPPEDIT 18 0 "`b*`[4];" "6#&%#b*G6#\"\"%" }{TEXT -1 3 ", " } {XPPEDIT 18 0 "`b*`[5];" "6#&%#b*G6#\"\"&" }{TEXT -1 3 ", " } {XPPEDIT 18 0 "`b*`[6];" "6#&%#b*G6#\"\"'" }{TEXT -1 2 ", " }{XPPEDIT 18 0 "`b*`[7];" "6#&%#b*G6#\"\"(" }{TEXT -1 3 ", " }{XPPEDIT 18 0 "`b *`[9];" "6#&%#b*G6#\"\"*" }{TEXT -1 39 ", by using the 5 quadrature co nditions:" }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "Sum(`b*` [i],i = 1 .. 9) = 1;" "6#/-%$SumG6$&%#b*G6#%\"iG/F*;\"\"\"\"\"*F-" } {TEXT -1 15 ", " }{XPPEDIT 18 0 "Sum(`b*`[i]*c[i]^(k-1),i = 2 .. 9) = 1/k;" "6#/-%$SumG6$*&&%#b*G6#%\"iG\"\"\")&%\"cG6#F+,&%\"k GF,F,!\"\"F,/F+;\"\"#\"\"**&F,F,F2F3" }{TEXT -1 7 ", " }{XPPEDIT 18 0 "k = 2;" "6#/%\"kG\"\"#" }{TEXT -1 8 " . . 5, " }}{PARA 0 "" 0 " " {TEXT -1 48 "together with the \"alternative\" order condition " }} {PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "[`b*`]^T*[A]*(C-c[4]* Id)*(C-c[5]*Id)*[c] = Int(Int((t-c[4])*(t-c[5])*t,t = 0 .. x),x = 0 .. 1);" "6#/*,)7#%#b*G%\"TG\"\"\"7#%\"AGF),&%\"CGF)*&&%\"cG6#\"\"%F)%#Id GF)!\"\"F),&F-F)*&&F06#\"\"&F)F3F)F4F)7#F0F)-%$IntG6$-F<6$*(,&%\"tGF)& F06#F2F4F),&FBF)&F06#F9F4F)FBF)/FB;\"\"!%\"xG/FK;FJF)" }{TEXT -1 2 ". \+ " }}{PARA 0 "" 0 "" {TEXT -1 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$\" \"*%\"jG&%\"bG6#F(" }{TEXT -1 3 ", " }{XPPEDIT 18 0 "j=1" "6#/%\"jG\" \"\"" }{TEXT -1 7 " . . 8." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 60 "First we set up the six equations for the six wei ghts . . . " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 " " {MPLTEXT 1 0 406 "`quad_eqs*` := [add(`b*`[i],i=1..9)=1,seq(add(`b*` [i]*c[i]^(j-1),i=2..9)=1/j,j=2..5)]:\n`ord_eq*` := add(add(`b*`[i]*a[i ,j-1],i=j..9)*(c[j-1]-c[4])*(c[j-1]-c[5])*c[j-1],j=7..9)=\n \+ 1/20-c[4]/12-c[5]/12+c[4]*c[5]/6:\nwt_eqs := [seq(a[9,i]=b[i],i= 1..8)]:\n`eqns*` := simplify(subs(e13,[op(`quad_eqs*`),subs(wt_eqs,`or d_eq*`)])):\nnops(`eqns*`);\nindets(`eqns*`) minus \{c[4],c[5],c[6],c[ 7]\};\nnops(%);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#\"\"'" }}{PARA 11 " " 1 "" {XPPMATH 20 "6#<(&%#b*G6#\"\"'&F%6#\"\"*&F%6#\"\"(&F%6#\"\"&&F% 6#\"\"%&F%6#\"\"\"" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#\"\"'" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 30 " . . . and then we solve them." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> \+ " 0 "" {MPLTEXT 1 0 22 "infolevel[solve] := 4:" }}}{EXCHG {PARA 0 "> \+ " 0 "" {MPLTEXT 1 0 76 "e14 := solve(\{op(`eqns*`)\}):\ne15 := `union` (e13,e14):\ninfolevel[solve] := 0:" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" } }{PARA 0 "" 0 "" {TEXT -1 49 "The weights of the order 5 scheme are as follows." }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 38 "seq(`b*`[i]=sub s(e15,`b*`[i]),i=1..9);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6+/&%#b*G6#\" \"\"#\"$<'\"&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] = 10304/43605, 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 289 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];\ne qns3[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] = -7 168/40375, 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, `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 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#/-%$SumG6$*&&%\"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$\"\"'\"\"&" } {TEXT -1 1 "." }}{PARA 0 "" 0 "" {TEXT -1 1 " " }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 217 "[seq(add(a[6,j]*c[j]^(k-1),j=2..5)=c[6]^k/k,k=[ 2,3])]:\neqns4 := subs(e17,%):\neqns4[1];\neqns4[2];\ne18 := solve(\{o p(eqns4)\},\{a[6,4],a[6,5]\}):\ne19 := `union`(e17,e18):\n``;\na[6,4]= subs(e19,a[6,4]),a[6,5]=subs(e19,a[6,5]);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/,(#\"%_:\"%*)G\"\"\"*&#F(\"\"&F(&%\"aG6$\"\"'\"\"%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 291 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#\"\"(\"\"$" }{TEXT -1 2 ", " }}{PARA 0 "" 0 "" {TEXT -1 9 "to fi nd " }{XPPEDIT 18 0 "a[7,4];" "6#&%\"aG6$\"\"(\"\"%" }{TEXT -1 7 " a nd " }{XPPEDIT 18 0 "a[7, 5];" "6#&%\"aG6$\"\"(\"\"&" }{TEXT -1 1 ". " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 217 "[seq(add(a[7,j]*c[j]^(k-1),j=2..6)=c[7]^k/k,k=[2,3])]:\neqns5 := subs(e19,%):\neqns5[1];\neqns5[2];\ne20 := solve(\{op(eqns5)\},\{a [7,4],a[7,5]\}):\ne21 := `union`(e19,e20):\n``;\na[7,4]=subs(e21,a[7,4 ]),a[7,5]=subs(e21,a[7,5]);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/,(#\"( 8-H)\")+W<>\"\"\"*&#F(\"\"&F(&%\"aG6$\"\"(\"\"%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] = -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, 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 11 " }{TEXT -1 1 " " }}{PARA 0 "" 0 "" {TEXT -1 34 "We use the stage-orde r 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$\"\")\"\"&" }{TEXT -1 1 "." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 217 "[seq(add(a[ 8,j]*c[j]^(k-1),j=2..7)=c[8]^k/k,k=[2,3])]:\neqns6 := subs(e21,%):\neq ns6[1];\neqns6[2];\ne22 := solve(\{op(eqns6)\},\{a[8,4],a[8,5]\}):\ne2 3 := `union`(e21,e22):\n``;\na[8,4]=subs(e23,a[8,4]),a[8,5]=subs(e23,a [8,5]);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/,(#\"'6E;\"']P8\"\"\"*&#F( \"\"&F(&%\"aG6$\"\")\"\"%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'\"\"&#!&$\\Q\"&vL\"" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}} {SECT 1 {PARA 4 "" 0 "" {TEXT -1 3 "e23" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 757 "e23 := \{a[8,2] = 0, c [9] = 1, c[8] = 1, a[5,2] = 0, a[4,2] = 0, a[6,2] = 0, a[7,2] = 0, b[3 ] = 0, b[2] = 0, `b*`[3] = 0, `b*`[2] = 0, a[5,3] = -112/45, a[5,4] = \+ 64/27, a[8,7] = -7168/40375, a[7,6] = 14535/14336, a[3,2] = 8/75, c[3] = 2/15, a[4,3] = 3/20, a[8,3] = -6783/2140, a[6,3] = 3880/963, `b*`[7 ] = 10304/43605, a[7,5] = -671175/876544, a[7,4] = 819261/383488, b[1] = 53/912, a[7,3] = -1000749/547840, b[8] = -25/336, b[7] = 256/969, b [6] = 27/136, `b*`[8] = 0, c[4] = 1/5, c[2] = 1/12, c[6] = 2/3, b[5] = 27/112, b[4] = 5/16, `b*`[5] = 69/320, `b*`[4] = 241/756, `b*`[6] = 4 35/1904, `b*`[1] = 617/10944, a[8,6] = 1152/425, c[5] = 8/15, c[7] = 1 9/20, a[6,5] = 217/428, a[6,4] = -8456/2889, a[8,5] = -38493/13375, a[ 8,4] = 10956/2675, `b*`[9] = -1/18\}:" }{TEXT -1 0 "" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}} {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 31 "We use th e row-sum conditions: " }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "Sum(a[i,j],j=1..i-1)=c[i]" "6#/-%$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$\"\")\"\"\"" }{TEXT -1 2 " . " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 119 "[seq(add(a[i,j],j=1..i-1)=c[i],i=2..8)]:\ne24 := sol ve(\{op(subs(e23,%))\},\{seq(a[i,1],i=2..8)\}):\ne25 := `union`(e23,e2 4):" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 60 "T he linking coefficients in the first column are as follows." }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 36 "seq(a[i,1]=subs(e25,a[i,1]),i=2..8) ;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6)/&%\"aG6$\"\"#\"\"\"#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$\"\")F(#\"&`^)\"'+ L?" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 22 "We use the equations: " }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "a[9,i]=b[i]" "6#/&%\"aG6$\"\"*%\"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$\"\" *%\"iG" }{TEXT -1 2 ", " }{XPPEDIT 18 0 " i=1" "6#/%\"iG\"\"\"" } {TEXT -1 7 " . . 8." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 116 "wt_eqs := [seq(a[9,i]=b[i],i=1..8)]:\ne26 \+ := solve(\{op(subs(e25,%))\},\{seq(a[9,j],j=1..8)\}):\ne27 := `union`( e25,e26):\n" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 55 "The linking coefficients in the 9th row are as follows." }} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 36 "seq(a[9,j]=subs(e27,a[9,j]), j=1..8);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6*/&%\"aG6$\"\"*\"\"\"#\"#` \"$7*/&F%6$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$" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 " " {MPLTEXT 1 0 1 ";" }}}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 3 "e27" }} {PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1017 "e27 := \{a[8,2] = 0, c[9] = 1, c[8] = 1, a[5,2] = 0, a[4,2] = 0, a[6,2] = 0, a[7,2] = 0, a[9,3] = 0, a[9,2] = 0, b[3] = 0, b[2] = 0, ` b*`[3] = 0, `b*`[2] = 0, a[7,1] = 1718911/4382720, a[8,1] = 85153/2033 00, a[5,3] = -112/45, a[5,4] = 64/27, a[6,1] = -10891/11556, a[5,1] = \+ 88/135, a[8,7] = -7168/40375, a[7,6] = 14535/14336, a[4,1] = 1/20, a[3 ,1] = 2/75, a[3,2] = 8/75, c[3] = 2/15, a[4,3] = 3/20, a[8,3] = -6783/ 2140, a[6,3] = 3880/963, `b*`[7] = 10304/43605, a[7,5] = -671175/87654 4, a[7,4] = 819261/383488, b[1] = 53/912, a[7,3] = -1000749/547840, b[ 8] = -25/336, b[7] = 256/969, b[6] = 27/136, `b*`[8] = 0, c[4] = 1/5, \+ c[2] = 1/12, c[6] = 2/3, b[5] = 27/112, a[9,1] = 53/912, a[9,4] = 5/16 , a[9,5] = 27/112, a[9,6] = 27/136, a[9,7] = 256/969, a[9,8] = -25/336 , b[4] = 5/16, a[2,1] = 1/12, `b*`[5] = 69/320, `b*`[4] = 241/756, `b* `[6] = 435/1904, `b*`[1] = 617/10944, a[8,6] = 1152/425, c[5] = 8/15, \+ c[7] = 19/20, a[6,5] = 217/428, a[6,4] = -8456/2889, a[8,5] = -38493/1 3375, a[8,4] = 10956/2675, `b*`[9] = -1/18\}:" }{TEXT -1 0 "" }}} {PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 49 "The Butcher tableau for the scheme is as follows." }}{EXCHG {PARA 0 " > " 0 "" {MPLTEXT 1 0 124 "subs(e27,matrix([seq([c[i],seq(a[i,j],j=1.. i-1),``$(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\"\"%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(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 this subsection we use the algorithm of Ch. Tsitouras, as outl ined in the previous subsection, to construct a general scheme. " }} {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 "" }}{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, sin ce the principal error norm and the stability polynomial for the order 6 scheme do not depend on " }{XPPEDIT 18 0 "`b*`[9]" "6#&%#b*G6#\"\" *" }{TEXT -1 116 ", it will be sufficient to obtain a solution that a voids 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#\"\"$\"\"!" }{TEXT -1 3 ", " } {XPPEDIT 18 0 "`b*`[8]=0" "6#/&%#b*G6#\"\")\"\"!" }{TEXT -1 2 ". " }} {PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 87 "e1 := \{c[8]=1,c[9]=1,b[2]=0,b[3]=0,`b*`[2]=0,`b*`[3]=0,`b*`[8]=0, seq(a[i,2]=0,i=4..8)\}:" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 15 " " 0 "" {TEXT 269 6 "Step 1" }{TEXT -1 1 " " }}{PARA 0 "" 0 "" {TEXT -1 33 "We use the quadrature conditions:" }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "Sum(b[i],i=1..8)=1" "6#/-%$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 270 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 271 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 272 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 273 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 274 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 275 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 276 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 277 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 278 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 279 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 293 23 "______________ _________" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT 258 84 "Schemes that have linking coeffic ients with maximum magnitude no greater than 18.35" }{TEXT -1 1 " " } }{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 256 "" 0 "" {TEXT -1 1 " " } {XPPEDIT 18 0 "matrix([[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, -10891/11556, 0, 3880/963, -8456/2889, 217/428, ``, ``, \+ ``, ``], [19/20, 1718911/4382720, 0, -1000749/547840, 819261/383488, - 671175/876544, 14535/14336, ``, ``, ``], [1, 85153/203300, 0, -6783/21 40, 10956/2675, -38493/13375, 1152/425, -7168/40375, ``, ``], [1, 53/9 12, 0, 0, 5/16, 27/112, 27/136, 256/969, -25/336, ``], [``, 617/10944, 0, 0, 241/756, 69/320, 435/1904, 10304/43605, 0, -1/18]])" "6#-%'matr ixG6#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)\" ')[$QF+,$*&\"'v6nF)\"'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)\"$7*F+F;F;*&F 8F)\"#;F+*&FIF)FEF+*&FIF)\"$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 weights 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 sc heme" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 35 "coefficient s of the combined scheme" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 891 "# 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 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 123 "subs(ee,mat rix([seq([c[i],seq(a[i,j],j=1..i-1),``$(10-i)],i=2..9),\n[`b`,seq(b[i] ,i=1..8),``],[`b*`,seq(`b*`[i],i=1..9)]]));" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#K%'matrixG6#7,7,#\"\"\"\"#7F(%!GF+F+F+F+F+F+F+7,#\"\"# \"#:#F.\"#v#\"\")F1F+F+F+F+F+F+F+7,#F)\"\"&#F)\"#?\"\"!#\"\"$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)pprint226\" " }}}{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$" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 114 "We can calculate the principal error norm of t he order 6 scheme, that is, the 2-norm of the principal error terms." }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 131 "errterms6_8 := PrincipalE rrorTerms(6,8,'expanded'):\nevalf(evalf[14](sqrt(add(subs(ee,errterms6 _8[i])^2,i=1.. nops(errterms6_8)))));" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#$\"+-L'f%z!#9" }}}{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#$\"+;.zC>!#7" }}} {PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}}{PARA 0 "" 0 "" {TEXT -1 52 "#------------------------------ ---------------------" }}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 43 "consecut ive construction of the two schemes" }}{PARA 0 "" 0 "" {TEXT -1 0 "" } }{PARA 0 "" 0 "" {TEXT -1 104 "First we construct the 8 stage order 6 \+ scheme by making use of a selection of \"simple\" order conditions." } }{PARA 0 "" 0 "" {TEXT -1 119 "We incorporate the stage-order equation s to ensure that stage 2 has stage-order 2 and stages 3 to 8 have stag e-order 3." }}{PARA 0 "" 0 "" {TEXT -1 48 "We also incorporate the sim plifying conditions: " }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "Sum(b[i]*a[i,j],i = j+1 .. 8) = b[j]*(1-c[j]);" "6#/-%$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 = 6;" "6#/%\"jG\"\"'" }{TEXT -1 7 " and " } {XPPEDIT 18 0 "j=7" "6#/%\"jG\"\"(" }{TEXT -1 9 " ( with " }{XPPEDIT 18 0 "c[1]=0" "6#/&%\"cG6#\"\"\"\"\"!" }{TEXT -1 3 " )." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 32 "The scheme is constr ucted with " }{XPPEDIT 18 0 "a[6,5];" "6#&%\"aG6$\"\"'\"\"&" }{TEXT -1 17 " as a parameter. " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 " " 0 "" {TEXT -1 78 "The 8 simple order conditions used are given (in a breviated form) as follows. " }}{PARA 0 "" 0 "" {TEXT -1 78 "These inc lude the 6 quadrature conditions and two additional order conditions. " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 178 "SO6 := SimpleOrderConditions(6):\n[seq([i,SO6[i]],i=[1,2,4,8, 16,24,29,32])]:\nlinalg[augment](linalg[delcols](%,2..2),matrix([[` ` ]$(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(\"\"$7%\"\")F)/*&F,F()F2F :F(#F(F57%\"#;F)/*&F,F()F2F5F(#F(\"\"&7%\"#CF)/*(F,F(F2F(-%!G6#*&%\"aG F(-FM6#*&F8F(FPF(F(F(#F(\"#s7%\"#HF)/*(F,F(F2F(-FM6#*&F?F(FPF(F(#F(FI7 %\"#KF)/*&F,F()F2FGF(#F(\"\"'Q)pprint666\"" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 388 "SO6_8 := SimpleOr derConditions(6,8,'expanded'):\ncdns := [seq(SO6_8[i],i=[1,2,4,8,16,24 ,29,32])]:\nSO_eqs := [op(RowSumConditions(8,'expanded')),op(StageOrde rConditions(2,8,'expanded')),\n op(StageOrderConditions( 3,4..8,'expanded'))]:\nsimp_eqs := [add(b[i]*a[i,1],i=1+1..8)=b[1],seq (add(b[i]*a[i,j],i=j+1..8)=b[j]*(1-c[j]),j=[6,7])]:\ncdns := [op(cdns) ,op(simp_eqs),op(SO_eqs)]:" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 22 "We specify the nodes: " }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "c[2]=1/12" "6#/&%\"cG6#\"\"#*&\"\"\"F) \"#7!\"\"" }{TEXT -1 3 ", " }{XPPEDIT 18 0 "c[4]=1/5" "6#/&%\"cG6#\" \"%*&\"\"\"F)\"\"&!\"\"" }{TEXT -1 3 ", " }{XPPEDIT 18 0 "c[5]=8/15" "6#/&%\"cG6#\"\"&*&\"\")\"\"\"\"#:!\"\"" }{TEXT -1 3 ", " }{XPPEDIT 18 0 "c[6]=2/3" "6#/&%\"cG6#\"\"'*&\"\"#\"\"\"\"\"$!\"\"" }{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 coe fficients: " }}{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$\"\")\"\"#\"\"!" }{TEXT -1 2 ", " }}{PARA 0 "" 0 "" {TEXT -1 20 "and the zero weights" }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "b[2]=0" "6#/&%\"bG6#\"\"#\"\"!" }{TEXT -1 3 ", " }{XPPEDIT 18 0 "b[3]=0" "6#/&%\"bG6#\"\"$\"\"!" }{TEXT -1 1 "." }}{PARA 0 "" 0 "" {TEXT -1 2 " " }}{PARA 0 "" 0 "" {TEXT -1 63 "We have 29 equations and 29 unknowns, excluding the parameter " } {XPPEDIT 18 0 "a[6,5]" "6#&%\"aG6$\"\"'\"\"&" }{TEXT -1 1 "." }} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 177 "e1 := \{c[2]=1/12,c[4]=1/5, c[5]=8/15,c[6]=2/3,c[7]=19/20,c[8]=1,\n seq(a[i,2]=0,i=4..8),b[2 ]=0,b[3]=0\}:\neqns := subs(e1,cdns):\nnops(%);\nindets(eqns) minus \{ a[6,5]\};\nnops(%);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#\"#H" }}{PARA 11 "" 1 "" {XPPMATH 20 "6# F'&F%6$F>F9&F%6$\"\"'F+&F%6$FEF'&F/6#FE&F/6#F3&F/6#\"\")&F%6$FEF9&F%6$ F3F'&F%6$F3F9&F%6$FNF'&F%6$FNF+&F/6#F9&F/6#F>&F%6$FNF9&F%6$FNF>&F%6$FN FE&F%6$FNF3&F%6$F3F>&F%6$F3FE" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#\"#H " }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 22 "infolevel[solve] := 4:" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 97 "e2 := solve(\{op(eqns)\},indets(eqns) minus \{a[6,5] \}):\ninfolevel[solve] := 0:\ne3 := `union`(e1,e2):" }}}{PARA 0 "" 0 " " {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}} {SECT 1 {PARA 4 "" 0 "" {TEXT -1 3 "e3 " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 821 "e3 := \{a[5,3] = -112/ 45, a[8,2] = 0, a[7,2] = 0, b[2] = 0, b[3] = 0, c[8] = 1, a[4,2] = 0, \+ a[5,2] = 0, a[6,2] = 0, a[8,5] = 1197/250-378/25*a[6,5], c[2] = 1/12, \+ c[4] = 1/5, c[5] = 8/15, c[6] = 2/3, c[7] = 19/20, a[2,1] = 1/12, a[8, 6] = 1152/425, a[4,1] = 1/20, a[7,3] = 250743/5120-12825/128*a[6,5], a [8,3] = 3003/20-1512/5*a[6,5], a[4,3] = 3/20, b[8] = -25/336, a[5,1] = 88/135, a[6,3] = 20*a[6,5]-55/9, a[8,1] = -72031/1900+378/5*a[6,5], a [3,2] = 8/75, a[6,1] = -5*a[6,5]+43/27, b[7] = 256/969, b[5] = 27/112, a[7,6] = 14535/14336, b[4] = 5/16, a[8,4] = -2964/25+6048/25*a[6,5], \+ a[5,4] = 64/27, c[3] = 2/15, a[6,4] = -16*a[6,5]+140/27, b[6] = 27/136 , a[7,1] = -504127/40960+12825/512*a[6,5], a[7,4] = -137997/3584+2565/ 32*a[6,5], a[7,5] = 14535/8192-2565/512*a[6,5], a[3,1] = 2/75, a[8,7] \+ = -7168/40375, b[1] = 53/912\}:" }{TEXT -1 0 "" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 91 "su bs(e3,matrix([seq([c[i],seq(a[i,j],j=1..i-1),``$(9-i)],i=2..8),\n[``,s eq(b[i],i=1..8)]]));" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#K%'matrixG6#7* 7+#\"\"\"\"#7F(%!GF+F+F+F+F+F+7+#\"\"#\"#:#F.\"#v#\"\")F1F+F+F+F+F+F+7 +#F)\"\"&#F)\"#?\"\"!#\"\"$F8F+F+F+F+F+7+#F3F/#\"#))\"$N\"F9#!$7\"\"#X #\"#k\"#FF+F+F+F+7+#F.F;,&*&F6F)&%\"aG6$\"\"'F6F)!\"\"#\"#VFFF)F9,&*&F 8F)FKF)F)#\"#b\"\"*FO,&*&\"#;F)FKF)FO#\"$S\"FFF)FKF+F+F+7+#\"#>F8,&*&# \"&DG\"\"$7&F)FKF)F)#\"'FT]\"&g4%FOF9,&*&#F\\o\"$G\"F)FKF)FO#\"'V2D\"% ?^F),&*&#\"%lD\"#KF)FKF)F)#\"'(*z8\"%%e$FO,&*&#F[pF]oF)FKF)FO#\"&NX\" \"%#>)F)#Fdp\"&OV\"F+F+7+F),&*&#\"$y$F6F)FKF)F)#\"&J?(\"%+>FOF9,&*&#\" %7:F6F)FKF)FO#\"%.IF8F),&*&#\"%[g\"#DF)FKF)F)#\"%kHFjqFO,&*&#F\\qFjqF) FKF)FO#\"%(>\"\"$]#F)#\"%_6\"$D%#!%or\"&v.%F+7+F+#\"#`\"$7*F9F9#F6FY#F F\"$7\"#FF\"$O\"#\"$c#\"$p*#!#D\"$O$Q)pprint676\"" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 104 "We use a selection of 7 \+ \"simple\" order conditions as given (in abreviated form) in the follo wing table. " }}{PARA 0 "" 0 "" {TEXT -1 78 "These include the 5 quadr ature conditions and two additional order conditions." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 194 "`SO5*` : = 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(\"\"$7%\"\")F)/*&F,F()F2F :F(#F(F57%\"#7F)/*&F,F(-%!G6#*&%\"aGF(-FF6#*&F8F(FIF(F(F(#F(\"#g7%\"#: F)/*&F,F(-FF6#*&F?F(FIF(F(#F(\"#?7%\"#;F)/*&F,F()F2F5F(#F(\"\"&Q)pprin t186\"" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 125 "The weights of the order 6 scheme provide the linking coefficient s for the 9th stage of the embedded order 5 scheme so that: " }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "a[9,i]=b[i]" "6#/&%\"aG6$\" \"*%\"iG&%\"bG6#F(" }{TEXT -1 3 ", " }{XPPEDIT 18 0 "i=1" "6#/%\"iG\" \"\"" }{TEXT -1 9 " . . . 8." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 192 "`SO5_9*` := subs(b=`b*`,Sim pleOrderConditions(5,9,'expanded')):\n`cdns*` := [seq(`SO5_9*`[i],i=[1 ,2,4,8,12,15,16])]:\nwt_eqns := [seq(b[i]=a[9,i],i=1..8)]:\n`cdns*` := [op(`cdns*`),op(wt_eqns)]: " }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }} {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#\"\"$\"\"!" }{TEXT -1 7 " and " } {XPPEDIT 18 0 "`b*`[8] = 0;" "6#/&%#b*G6#\"\")\"\"!" }{TEXT -1 2 ". " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 67 "e4 := \{c[9]=1,`b*`[2]=0,`b*`[3]=0,`b*`[8]=0\}:\ne5 := `union`(e 3,e4):" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 38 "We have 26 equations and 26 unknowns. " }}{EXCHG {PARA 0 "> " 0 " " {MPLTEXT 1 0 111 "eqns2 := [op(remove(_U->type(rhs(_U),realcons),e5) ),op(subs(e5,`cdns*`))]:\nnops(eqns2);\nindets(eqns2);\nnops(%);" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#\"#E" }}{PARA 11 "" 1 "" {XPPMATH 20 " 6#<<&%\"aG6$\"\"(\"\"\"&F%6$\"\"'F(&F%6$F+\"\"$&F%6$F+\"\"&&F%6$F+\"\" %&F%6$\"\"*F(&F%6$F7\"\"#&F%6$F7F.&F%6$F7F4&F%6$F7F1&F%6$F7F+&%#b*G6#F 4&FD6#F1&F%6$F'F.&F%6$F'F4&F%6$\"\")F.&F%6$FNF(&FD6#F(&F%6$F7F'&F%6$F7 FN&F%6$FNF4&F%6$FNF1&F%6$F'F1&FD6#F+&FD6#F'&FD6#F7" }}{PARA 11 "" 1 " " {XPPMATH 20 "6#\"#E" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 22 "infolevel[solve] := 4:" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 109 "e6 := solve(\{op(eqns2)\}):\ne7 := subs(a[6,5]=subs(e6,a[6,5]),e5):\ne8 := `union`(e6,e7):\ninfolevel[so lve] := 0:" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 " " {MPLTEXT 1 0 1 ";" }}}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 2 "e8" }} {PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1016 "e8 := \{b[5] = 27/112, a[8,6] = 1152/425, `b*`[5] = 69/320, c[3] = 2/15, 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] = 53/91 2, `b*`[1] = 617/10944, b[8] = -25/336, a[9,7] = 256/969, a[8,5] = -38 493/13375, 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] = 145 35/14336, 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 "s ubs(e8,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\"\"%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#\"$<'\"&W4\"F9 F9#\"$T#\"$c(#\"#p\"$?$#\"$N%\"%/>#\"&/.\"\"&0O%F9#!\"\"\"#=Q)pprint68 6\"" }}}{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(e8,RK6_8eqs)):\nmap(u->`if`(lhs(u)=rhs(u),0,1),%); \nnops(%);\nsimplify(subs(e8,`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 $" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#\"#<" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 44 "simultaneous construction of the two sche mes" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 119 "W e incorporate the stage-order equations to ensure that stage 2 has sta ge-order 2 and stages 3 to 8 have stage-order 3." }}{PARA 0 "" 0 "" {TEXT -1 48 "We also incorporate the simplifying conditions: " }} {PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "Sum(b[i]*a[i,j],i = j +1 .. 8) = b[j]*(1-c[j]);" "6#/-%$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 abreviat ed form) as follows. " }}{PARA 0 "" 0 "" {TEXT -1 76 "These include th e 6 quadrature conditions and 2 additional order conditions." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 178 "S O6 := SimpleOrderConditions(6):\n[seq([i,SO6[i]],i=[1,2,4,8,16,24,29,3 2])]:\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(\"\"$7%\"\")F)/*&F,F()F2F:F(#F(F5 7%\"#;F)/*&F,F()F2F5F(#F(\"\"&7%\"#CF)/*(F,F(F2F(-%!G6#*&%\"aGF(-FM6#* &F8F(FPF(F(F(#F(\"#s7%\"#HF)/*(F,F(F2F(-FM6#*&F?F(FPF(F(#F(FI7%\"#KF)/ *&F,F()F2FGF(#F(\"\"'Q)pprint586\"" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" } }{PARA 0 "" 0 "" {TEXT -1 136 "For the embedded order 5 scheme we use \+ a selection of 7 \"simple\" order conditions as given (in abreviated f orm) in the following table. " }}{PARA 0 "" 0 "" {TEXT -1 78 "These in clude the 5 quadrature conditions and two additional order conditions. " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 194 "`SO5*` := subs(b=`b*`,SimpleOrderConditions(5)):\n[seq([i,`SO 5*`[i]],i=[1,2,4,8,12,15,16])]:\nlinalg[augment](linalg[delcols](%,2.. 2),matrix([[` `]$(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(\"\"$7% \"\")F)/*&F,F()F2F:F(#F(F57%\"#7F)/*&F,F(-%!G6#*&%\"aGF(-FF6#*&F8F(FIF (F(F(#F(\"#g7%\"#:F)/*&F,F(-FF6#*&F?F(FIF(F(#F(\"#?7%\"#;F)/*&F,F()F2F 5F(#F(\"\"&Q)pprint186\"" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 582 "SO6_8 := SimpleOrderConditions(6,8 ,'expanded'):\nSO_eqs := [op(RowSumConditions(8,'expanded')),op(StageO rderConditions(2,8,'expanded')),\n op(StageOrderConditio ns(3,4..8,'expanded'))]:\n`SO5_9*` := subs(b=`b*`,SimpleOrderCondition s(5,9,'expanded')):\nord_cdns := [seq(SO6_8[i],i=[1,2,4,8,16,24,29,32] )]:\n`ord_cdns*` := [seq(`SO5_9*`[i],i=[1,2,4,8,12,15,16])]:\nsimp_eqs := [add(b[i]*a[i,1],i=1+1..8)=b[1],seq(add(b[i]*a[i,j],i=j+1..8)=b[j] *(1-c[j]),j=[6,7])]:\nwt_eqns := [seq(b[i]=a[9,i],i=1..8)]:\ncdns := [ op(ord_cdns),op(simp_eqs),op(SO_eqs),op(wt_eqns),op(`ord_cdns*`)]:" }} }{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 22 "We speci fy the nodes: " }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "c[2 ]=1/12" "6#/&%\"cG6#\"\"#*&\"\"\"F)\"#7!\"\"" }{TEXT -1 3 ", " } {XPPEDIT 18 0 "c[4]=1/5" "6#/&%\"cG6#\"\"%*&\"\"\"F)\"\"&!\"\"" } {TEXT -1 3 ", " }{XPPEDIT 18 0 "c[5]=8/15" "6#/&%\"cG6#\"\"&*&\"\")\" \"\"\"#:!\"\"" }{TEXT -1 3 ", " }{XPPEDIT 18 0 "c[6]=2/3" "6#/&%\"cG6 #\"\"'*&\"\"#\"\"\"\"\"$!\"\"" }{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 6 "" 1 "" {TEXT -1 53 "solve/rec2: solving for line ar equation in a[2,1]" }}{PARA 6 "" 1 "" {TEXT -1 53 "solve/rec2: \+ solving for linear equation in a[9,5]" }}{PARA 6 "" 1 "" {TEXT -1 53 "solve/rec2: solving for linear equation in a[5,1]" }}{PARA 6 " " 1 "" {TEXT -1 53 "solve/rec2: solving for linear equation in a[6 ,1]" }}{PARA 6 "" 1 "" {TEXT -1 53 "solve/rec2: solving for linear e quation in a[3,2]" }}{PARA 6 "" 1 "" {TEXT -1 53 "solve/rec2: solv ing for linear equation in a[6,4]" }}{PARA 6 "" 1 "" {TEXT -1 54 "so lve/rec2: solving for linear equation in `b*`[4]" }}{PARA 6 "" 1 " " {TEXT -1 53 "solve/rec2: solving for linear equation in a[9,6]" }}{PARA 6 "" 1 "" {TEXT -1 53 "solve/rec2: solving for linear equati on in a[7,1]" }}{PARA 6 "" 1 "" {TEXT -1 53 "solve/rec2: solving f or linear equation in a[9,1]" }}{PARA 6 "" 1 "" {TEXT -1 53 "solve/r ec2: solving for linear equation in a[8,4]" }}{PARA 6 "" 1 "" {TEXT -1 51 "solve/rec2: solving for linear equation in b[4]" }} {PARA 6 "" 1 "" {TEXT -1 54 "solve/rec2: solving for linear equation in `b*`[5]" }}{PARA 6 "" 1 "" {TEXT -1 51 "solve/rec2: solving fo r linear equation in b[1]" }}{PARA 6 "" 1 "" {TEXT -1 53 "solve/rec2 : solving for linear equation in a[9,7]" }}{PARA 6 "" 1 "" {TEXT -1 54 "solve/rec2: solving for linear equation in `b*`[1]" }} {PARA 6 "" 1 "" {TEXT -1 53 "solve/rec2: solving for linear equation in a[7,4]" }}{PARA 6 "" 1 "" {TEXT -1 53 "solve/rec2: solving for linear equation in a[4,1]" }}{PARA 6 "" 1 "" {TEXT -1 53 "solve/rec 2: solving for linear equation in a[5,4]" }}{PARA 6 "" 1 "" {TEXT -1 53 "solve/rec2: solving for linear equation in a[3,1]" }}{PARA 6 "" 1 "" {TEXT -1 54 "solve/rec2: solving for linear equation in \+ `b*`[6]" }}{PARA 6 "" 1 "" {TEXT -1 54 "solve/rec2: solving for line ar equation in `b*`[7]" }}{PARA 6 "" 1 "" {TEXT -1 51 "solve/rec2: \+ solving for linear equation in b[5]" }}{PARA 6 "" 1 "" {TEXT -1 53 "solve/rec2: solving for linear equation in a[9,4]" }}{PARA 6 "" 1 "" {TEXT -1 53 "solve/rec2: solving for linear equation in a[9,8 ]" }}{PARA 6 "" 1 "" {TEXT -1 53 "solve/rec2: solving for linear equ ation in a[8,1]" }}{PARA 6 "" 1 "" {TEXT -1 51 "solve/rec2: solvin g for linear equation in b[7]" }}{PARA 6 "" 1 "" {TEXT -1 51 "solve/ rec2: solving for linear equation in b[6]" }}{PARA 6 "" 1 "" {TEXT -1 53 "solve/rec2: solving for linear equation in a[8,5]" }} {PARA 6 "" 1 "" {TEXT -1 53 "solve/rec2: solving for linear equation in a[7,5]" }}{PARA 6 "" 1 "" {TEXT -1 53 "solve/rec2: solving for linear equation in a[6,5]" }}{PARA 6 "" 1 "" {TEXT -1 82 "Polynomia lSystem: 4550483 [8, 5000018428, a[8,6]] 11 11 9323 0 3 \+ 0" }}{PARA 6 "" 1 "" {TEXT -1 83 "PolynomialSystem: 2504962 [6, \+ 1000063000, a[8,7]] 10 10 8898 1 11 0" }}{PARA 6 "" 1 "" {TEXT -1 77 "PolynomialSystem: 734427 [8, 100003729, b[8]] 9 9 7802 1 11 0" }}{PARA 6 "" 1 "" {TEXT -1 80 "PolynomialSystem: 1522399 [7, 1000098500, a[7,6]] 8 8 5316 0 3 0" }} {PARA 6 "" 1 "" {TEXT -1 80 "PolynomialSystem: 1703683 [5, 1000199 680, a[4,3]] 7 7 7843 0 3 0" }}{PARA 6 "" 1 "" {TEXT -1 81 "PolynomialSystem: 5015306 [1, 5000114466, a[5,3]] 6 6 18 100 0 3 0" }}{PARA 6 "" 1 "" {TEXT -1 82 "PolynomialSystem: 68 84650 [1, 5500178852, a[6,3]] 5 5 13965 2 27 0" }}{PARA 6 "" 1 "" {TEXT -1 77 "PolynomialSystem: 512937 [4, 100004024, c[3 ]] 4 4 2572 3 44 1" }}{PARA 6 "" 1 "" {TEXT -1 78 "Polynom ialSystem: 418197 [1, 1000015645, a[7,3]] 3 3 326 0 3 \+ 1" }}{PARA 6 "" 1 "" {TEXT -1 78 "PolynomialSystem: 657391 [2, 550 0012222, a[8,3]] 2 2 147 0 3 1" }}{PARA 6 "" 1 "" {TEXT -1 77 "PolynomialSystem: 50551 [1, 100000568, `b*`[9]] 1 1 2 1 1 23 2" }}{PARA 6 "" 1 "" {TEXT -1 52 "PolynomialSystem: -10 [] 0 0 3 0 3 2" }}{PARA 6 "" 1 "" {TEXT -1 78 "Polynomi alSystem: 250525 [2, 100000726, `b*`[9]] 2 3 120 0 3 1 " }}{PARA 6 "" 1 "" {TEXT -1 79 "PolynomialSystem: 1529424 [6, 100 015428, c[3]] 5 6 13884 2 27 0" }}{PARA 6 "" 1 "" {TEXT -1 67 "PolynomialSystem: 1 solutions found, now doing backsubstitu tion" }}{PARA 6 "" 1 "" {TEXT -1 41 "backsubs: backsubstitution of \+ `b*`[9]" }}{PARA 6 "" 1 "" {TEXT -1 40 "backsubs: backsubstitution \+ of a[8,3]" }}{PARA 6 "" 1 "" {TEXT -1 40 "backsubs: backsubstituti on of a[7,3]" }}{PARA 6 "" 1 "" {TEXT -1 38 "backsubs: backsubstit ution of c[3]" }}{PARA 6 "" 1 "" {TEXT -1 40 "backsubs: backsubsti tution of a[6,3]" }}{PARA 6 "" 1 "" {TEXT -1 40 "backsubs: backsub stitution of a[5,3]" }}{PARA 6 "" 1 "" {TEXT -1 40 "backsubs: back substitution of a[4,3]" }}{PARA 6 "" 1 "" {TEXT -1 40 "backsubs: b acksubstitution of a[7,6]" }}{PARA 6 "" 1 "" {TEXT -1 38 "backsubs: \+ backsubstitution of b[8]" }}{PARA 6 "" 1 "" {TEXT -1 40 "backsubs: backsubstitution of a[8,7]" }}{PARA 6 "" 1 "" {TEXT -1 40 "backsu bs: backsubstitution of a[8,6]" }}}{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] = 2/15, 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] = 109 56/2675, a[4,3] = 3/20, `b*`[7] = 10304/43605, b[4] = 5/16, a[6,5] = 2 17/428, a[5,1] = 88/135, a[3,2] = 8/75, a[9,5] = 27/112, a[6,4] = -845 6/2889, 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] = 2 56/969, a[8,5] = -38493/13375, 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] = 14535/14336, 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,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)]]) );" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#K%'matrixG6#7,7,#\"\"\"\"#7F(%!G F+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(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(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 o f 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 0 891 "# coefficients by Sharp a nd 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]=1718911/4382720,a[7,2]=0,a[7,3]=-1 000749/547840,\na[7,4]=819261/383488,a[7,5]=-671175/876544,a[7,6]=1453 5/14336,\na[8,1]=85153/203300,a[8,2]=0,a[8,3]=-6783/2140,\na[8,4]=1095 6/2675,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]=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 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$\"\"&F3F9" }{TEXT -1 1 "." }} {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 77 "Dormand a nd Prince have suggested that as well as attempting to ensure that " }{XPPEDIT 18 0 "A[7];" "6#&%\"AG6#\"\"(" }{TEXT -1 73 " is a minimum, if the embedded scheme is to be used for error control, " }{XPPEDIT 18 0 "B[7];" "6#&%\"BG6#\"\"(" }{TEXT -1 7 " and " }{XPPEDIT 18 0 "C [7];" "6#&%\"CG6#\"\"(" }{TEXT -1 27 " should be chosen so that " }} {PARA 0 "" 0 "" {TEXT -1 68 "they are \"similar in magnitude\" and als o not differ too much from 1." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 186 "errterms6_8 := PrincipalErr orTerms(6,8,'expanded'):\n`errterms6_9*` :=subs(b=`b*`,PrincipalErrorT erms(6,9,'expanded')):\n`errterms5_9*` := subs(b=`b*`,PrincipalErrorTe rms(5,9,'expanded')):" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 326 "snmB := sqrt(add(evalf(subs(ee,`er rterms6_9*`[i]))^2,i=1..nops(`errterms6_9*`))):\nsdnB := sqrt(add(eval f(subs(ee,`errterms5_9*`[i]))^2,i=1..nops(`errterms5_9*`))):\nsnmC := \+ sqrt(add((evalf(subs(ee,`errterms6_9*`[i])-subs(ee,errterms6_8[i])))^2 ,i=1..nops(errterms6_8))):\n'B[7]'= evalf[8](snmB/sdnB);\n'C[7]'= eval f[8](snmC/sdnB);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/&%\"BG6#\"\"($\") w**o7!\"(" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/&%\"CG6#\"\"($\")\"G\"o7 !\"(" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}}{PARA 0 "" 0 "" {TEXT -1 52 "#----------------- ----------------------------------" }}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 27 "absolute stability regions " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}{SECT 1 {PARA 4 "" 0 " " {TEXT -1 36 "coefficients for the combined scheme" }}{PARA 0 "" 0 " " {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 884 "# coeffic ients by Sharp and Verner 1994\nee := \{c[2]=1/12,c[3]=2/15,c[4]=1/5,c [5]=8/15,c[6]=2/3,c[7]=19/20,c[8]=1,\na[2,1]=1/12,a[3,1]=2/75,a[3,2]=8 /75,\na[4,1]=1/20,a[4,2]=0,a[4,3]=3/20,\na[5,1]=88/135,a[5,2]=0,a[5,3] =-112/45,a[5,4]=64/27,\na[6,1]=-10891/11556,a[6,2]=0,a[6,3]=3880/963, \n a[6,4]=-8456/2889,a[6,5]=217/428,\na[7,1]=1718911/4382720,a[7,2]=0 ,a[7,3]=-1000749/547840,\na[7,4]=819261/383488,a[7,5]=-671175/876544,a [7,6]=14535/14336,\na[8,1]=85153/203300,a[8,2]=0,a[8,3]=-6783/2140,\na [8,4]=10956/2675,a[8,5]=-38493/13375,a[8,6]=1152/425,a[8,7]=-7168/4037 5,\na[9,1]=53/912,a[9,2]=0,a[9,3]=0,a[9,4]=5/16,a[9,5]=27/112,a[9,6]=2 7/136,\na[9,7]=256/969,a[9,8]=-25/336,\nb[1]=53/912,b[2]=0,b[3]=0,b[4] =5/16,b[5]=27/112,b[6]=27/136,\nb[7]=256/969,b[8]=-25/336,\n`b*`[1]=61 7/10944,`b*`[2]=0,`b*`[3]=0,`b*`[4]=241/756,`b*`[5]=69/320,\n`b*`[6]=4 35/1904,`b*`[7]=10304/43605,`b*`[8]=0,`b*`[9]=-1/18\}:" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}} {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 77 "The stabi lity function R for the 8 stage, order 6 scheme is given as follows." }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 75 "subs(ee,StabilityFunction( 6,8,'expanded')):\nR := unapply(%,z):\n'R(z)'=R(z);" }}{PARA 11 "" 1 " " {XPPMATH 20 "6#/-%\"RG6#%\"zG,4\"\"\"F)F'F)*&#F)\"\"#F)*$)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\"(+XW\"F)*$)F'\"\")F)F)F)" }}}{PARA 0 "" 0 "" {TEXT -1 0 " " }}{PARA 0 "" 0 "" {TEXT -1 124 "We can find the point where the boun dary of the stability region intersects the negative real axis by solv ing the equation: " }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "R(z) = 1;" "6#/-%\"RG6#%\"zG\"\"\"" }{TEXT -1 2 ". " }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 28 "z0 := newton(R(z)=1,z=-4.5);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%#z0G$!+#\\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=b lack):\np3 := plot([[z0,0],[z0,1]],linestyle=3,color=COLOR(RGB,0,.5,0) ):\nplots[display]([p1,p2,p3],view=[-5.19..0.49,-.07..1.47],font=[HELV ETICA,9]);" }}{PARA 13 "" 1 "" {GLPLOT2D 360 253 253 {PLOTDATA 2 "6+-% 'CURVESG6$7Z7$$!3Q++++++!>&!#<$\"3u'3,UrIZk$F*7$$!3QML3T![!f^F*$\"3MWL I!3!zfMF*7$$!3Ynm;#3'4G^F*$\"3m%\\kz\"GA$G$F*7$$!3a++DBT9(4&F*$\"3q['z qI*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$$!3 K+++\"HZkk$F*$\"32p\\;Gs&Q#>F]p7$$!3;++gW:!z_$F*$\"3)**)p]vehR:F]p7$$! 3hLL)*\\1D?MF*$\"3w'R#Gpckx7F]p7$$!3'ommSKVAH$F*$\"3Gt$QDn[K0\"F]p7$$! 3/nmEGV!Q=$F*$\"3p7$$!39++0(*RmdIF*$\"3qJ_Ws@i!G)F`s7$$!39 nmEI%3g%HF*$\"3%fo9V7Kq%yF`s7$$!3-++0xX]BGF*$\"3&>iRLH.Nt(F`s7$$!3*)** *\\\"R>&oq#F*$\"32;YGid7?zF`s7$$!3gmm;\\r8&e#F*$\"3YG%Q7i/jP)F`s7$$!3y mmrw\\OtCF*$\"3N*HOTf)=.!*F`s7$$!3SLL$))e.GN#F*$\"3Muk#>iBS))*F`s7$$!3 nLL)**=uvA#F*$\"3P;Q8$4r85\"F]p7$$!3K++:I;c=@F*$\"3AEH/!)RS<7F]p7$$!31 LL.z]#3+#F*$\"3U+qnl5uh8F]p7$$!3M++?,<>z=F*$\"3sZ#*)pHuF`\"F]p7$$!3;++ !4<(>g0>$*RBF]p7$$!3H++q 9zA<:F*$\"3EgIei.=%>#F]p7$$!3EnmEY;O-9F*$\"3EM&fs2%ogCF]p7$$!3#)*****p Q<(z7F*$\"3aL^*3u>9y#F]p7$$!3)RL$efMeo6F*$\"3>!y))eS*>3JF]p7$$!3I****f AZ3Z5F*$\"3l9(*4l8l4NF]p7$$!3xqm;(zQwK*F]p$\"3+IrC:(ef3MWF]p7$$!3'3nmm0VV'pF]p$\"3)H\\>7T%f$)\\F]p7$$!3 P)***\\iqATdF]p$\"3i;.dA#))>j&F]p7$$!3aFLL*)4AjXF]p$\"3-8Ajat4OjF]p7$$ !33LLLO'R&eLF]p$\"3C(QUPwus9(F]p7$$!3Uim;`O$Q;#F]p$\"3))Q$=?qkU0)F]p7$ $!3?*****>$H-m5F]p$\"3iayIQ4$)))*)F]p7$$\"3v*QLLU?>#>F`s$\"3[-s.#30%>5 F*7$$\"3%ymmY^avJ\"F]p$\"3@(G;qHH39\"F*7$$\"3E0+]HcU\\G3Z%F*Fi]l-%'SYMBOLG6#%'CIRCLEG-F]]l6&F_]lFd]lFd]lFd]l-%&STYLEG6 #%&POINTG-F$6&F_al-Fdal6#%&CROSSGFgalFial-F$6&F_al-Fdal6#%(DIAMONDGFga lFial-F$6%7$7$FaalFc]lF`al-%&COLORG6&F_]lFc]l$\"\"&!\"\"Fc]l-%*LINESTY LEG6#\"\"$-%+AXESLABELSG6%Q\"z6\"Q!Ficl-%%FONTG6#%(DEFAULTG-F\\dl6$%*H ELVETICAG\"\"*-%%VIEWG6$;$!$>&!\"#$\"#\\Fidl;$!\"(Fidl$\"$Z\"Fidl" 1 2 0 1 10 0 2 9 1 4 2 1.000000 45.000000 45.000000 0 0 "Curve 1" "Curve 2" "Curve 3" "Curve 4" "Curve 5" "Curve 6" }}}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 50 "The following picture shows the stability region. " }}{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 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,.45,.05,.45)):\np2 := plots[polygonplot]([seq ([pts[i-1],pts[i],[-2.25,0]],i=2..nops(pts))],\n style=patchn ogrid,color=COLOR(RGB,.85,.1,.85)):\npts := []: z0 := 2+4.75*I:\nfor c t 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(zz)]]:\nend do:\np3 := plot(pts, color=COLOR(RGB,.45,.05,.45)):\np4 := plots[polygonplot]([seq([pts[i-1 ],pts[i],[1.87,4.73]],i=2..nops(pts))],\n style=patchnogrid,c olor=COLOR(RGB,.85,.1,.85)):\npts := []: z0 := 2-4.75*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(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=[`R e(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******\\F2(>#FP7$$\"3'******p))G=W\"!#?$\"3))******=IC_BFP7$$\"39+ ++8nu=!)f!GFP7$$!3/+++J')y@KF_o$\"3=+++oeK]HFP7$$! 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In this way we see that there is " }{TEXT 260 53 "no largest inte rval on the nonnegative imaginary axis" }{TEXT -1 65 " that contains t he 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 the 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+1/720*z^6+559/28 89000*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)]]:\nend do:\nplot(p ts,color=COLOR(RGB,.85,0,.85),thickness=2,font=[HELVETICA,9]);\nDigits := 10:" }}{PARA 13 "" 1 "" {GLPLOT2D 316 302 302 {PLOTDATA 2 "6(-%'CU RVESG6#7fq7$$\"\"!F)F(7$$!:v\\(>e)ep(*zHkR#!#E$\":?Dy$Qy(*e`EfTJF-7$$! :]\\Sdc*GI,9?mRF-$\":%oi'>soxrI&=$G'F-7$$!:fIB?bW3WC8SK&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$$!:q*y$y>2#R`Xbv6F?$\":W-'pR1EL\"QLu#GF?7$$!: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\"*pPF?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$$!:Oku$[EC6Es8fCF?$\":2#\\!4 I]'4Dd!*Q(*F?7$$!:!*HF5X&3A4h)oV#F?$\":Z4R4x3\"[[dI05!#C7$$!:j=uEtQqmH \"foBF?$\":J)f))p+2'Qx?n.\"Ffu7$$!:B$QvT:6AU7\\J@F?$\":3T*Rx@<[Ac8o5Ff u7$$\":-7-)p57ve)y]K#F?$\":.\\#p>'>eQE]&*4\"Ffu7$$\":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$$\":]3\"y)z'ek8a Y$\\$F?$\":6$fe#z9ZhGkAQ\"Ffu7$$\":=%3:X@uB(*)R6e$F?$\":)*Q*R\"H>;mYtO T\"Ffu7$$\":q]vl*\\oufK]mOF?$\":<,-Be0yFZ\"3X9Ffu7$$\":M9&zP9(3EmN)\\P F?$\":owFi;sAh5)[w9Ffu7$$\":[!H\"\\'o,V@#[8$QF?$\":>]4A<+\")e8$*y]\"Ff u7$$\":QL>2^2<^D-7\"RF?$\":w\\PF%)4MVH'HR:Ffu7$$\":y&oz`,K7]@_*)RF?$\" :Slu:=f'erspq:Ffu7$$\":r4RfchvT?/k1%F?$\":(pK3BMpw4d4-;Ffu7$$\":Q46%3C tI]G#>9%F?$\":`RMH7'4i'>\"\\L;Ffu7$$\":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$$\":i<963)=?(=+\"H[F?$\":%ygtw([3V)z8Z>Ffu7$$\":\"pQg9 m0ann3\"*[F?$\":'z-o=B+N-)\\%y>Ffu7$$\":'ym^m!=&F?$\":S3%>F.(y.JMZ8#Ffu7$$\":qJ#4y')Q1l%RTB&F?$\":Dd a2P9eA->f;#Ffu7$$\":_\"Q1kGpF!3!)fG&F?$\":Pia2\"eRp\\F2(>#Ffu7$$\":8&z *=&zSK(3)4O`F?$\":d1K@(**>*y(>>GAFfu7$$\":o,-cg-?Av'R%Q&F?$\":*G?j@7HT &*GFfAFfu7$$\":1]:U0:%\\unwIaF?$\":@!3FVz<*RQ6.H#Ffu7$$\":qcgK3r5pP%3v aF?$\":PS,ez$[A5II@BFfu7$$\":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$$\":p(Rs v2`^7%y,m&F?$\":]%fkTev'\\V!QvCFfu7$$\":gxzo#4>\\@0P)o&F?$\":$GZq=],i< (zf]#Ffu7$$\":gL@!HE+idQ#Hr&F?$\":Vo+ynbd!G@\\ODFfu7$$\":)y0u&G)eK@!>M t&F?$\":2\"pc&4FgIu5pc#Ffu7$$\":+vf)GoAJ2@M\\dF?$\":Mc+tkdF?$\":,yC Alu$H_zYp&F?$\":e<`/]A%)>8'ewFFfu7$$\":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$$!:zz@1$Ffu7$$!:`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$$!:bxvZ;7f0BA3e'F?$\":U+'z*QKw ?Y*HqJFfu7$$!:yK/.\\(yeIU-)o'F?$\":c2'p$=!Qu[Vp'>$Ffu7$$!:wnB*>Ua\"48z ))y'F?$\":yulj0d+(GA#GA$Ffu-%*THICKNESSG6#\"\"#-%%FONTG6$%*HELVETICAG \"\"*-%&COLORG6&%$RGBG$\"#&)!\"#F(Fa]m-%+AXESLABELSG6$Q!6\"Fg]m-%%VIEW G6$%(DEFAULTGF\\^m" 1 2 0 1 10 2 2 6 1 4 2 1.000000 45.000000 45.000000 0 0 "Curve 1" }}}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 " " 0 "" {TEXT -1 108 "The relevant intersection points of the boundary \+ curve with the imaginary axis can be determined as follows." }}{PARA 0 "" 0 "" {TEXT -1 87 "First we look for points on the boundary curve \+ either side of each intersection point. " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 199 "Digits := 15:\nz0 := 1.1*I:\nfor ct from 33 to 36 do\n newton(R(z)=exp(ct*Pi/100*I),z=z0 );\nend do;``;\nz0 := 2.95*I:\nfor ct from 93 to 96 do\n newton(R(z) =exp(ct*Pi/100*I),z=z0);\nend do;\nDigits := 10:" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#^$$!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#^$$!02/Qsd([5!#<$\" 0fDDF9'yH!#9" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 108 "Then we apply the bisection method to calculate the para meter value associated with each intersection point." }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 328 "Digits := 15:\nreal_part := proc(u)\n \+ Re(newton(R(z)=exp(u*Pi*I),z=1.1*I))\nend proc:\nu0 := bisect('real_pa rt'(u),u=0.33..0.36);\nnewton(R(z)=exp(u0*Pi*I),z=1.1*I);``;\nreal_par t := proc(u)\n Re(newton(R(z)=exp(u*Pi*I),z=2.9*I))\nend proc:\nu0 : = bisect('real_part'(u),u=0.93..0.96);\nnewton(R(z)=exp(u0*Pi*I),z=2.9 *I);\nDigits := 10:" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%#u0G$\"0Z6QFx EV$!#:" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#^#$\"0Bw_4,%y5!#9" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%!G" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%#u0 G$\"0QomVm)\\%*!#:" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#^$$\"0)f8C8$[N$! #H$\"0?$>7m1OH!#9" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 4 "The " }{TEXT 260 73 "stability region intersects the noneg ative 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 func tion 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*`,Stabilit yFunction(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 real axis by solving \+ the equation: " }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "`R* `(z) = -1;" "6#/-%#R*G6#%\"zG,$\"\"\"!\"\"" }{TEXT -1 2 ". " }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 33 "z_0 := newton(`R*`(z)=-1,z=-3.5);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%$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,c olor=[red,blue]):\np_2 := plot([[[z_0,-1]]$3],style=point,symbol=[circ le,cross,diamond],color=black):\np_3 := plot([[z_0,0],[z_0,-1]],linest yle=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$$!3uLLL\\([B*QF*$!3=#4 yl=$R(R#F*7$$!3-++!Q$e')\\QF*$!3\"4S*f07'y?#F*7$$!3vmmE=HQ2QF*$!3P>'Rp Zp5.#F*7$$!3cLL$\\&*H=r$F*$!3_$y6jP]in\"F*7$$!3cLL`/Ok:OF*$!310n\"on&o s8F*7$$!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*$!3g6SV)*R1%[$FU7$$!3[LL $4x,i'HF*$!3Y\\7G%*4f!o#FU7$$!3!)****RaUdpGF*$!35/+Z7<[D>FU7$$!33+++w* \\Dx#F*$!3o9tO$om+I\"FU7$$!3')****f0\"\\!zEF*$!3pH\"pnJI!3!)!#>7$$!3?L Ltd89%f#F*$!3+WZ\")Ra_%>%Fhp7$$!3_mm1Ly<$\\#F*$!3?'f'f^J$4z$!#?7$$!3ym mE%[[wS#F*$\"3#GS3$3-alBFhp7$$!3')****z=v:3BF*$\"3k1-+so9D^Fhp7$$!3?nm Ec64?AF*$\"3Q8/Ke*)\\usFhp7$$!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=++?z c;$4\"F*$\"3yPE&pJ;-N$FU7$$!3FnmEKpc-5F*$\"3/XHkqRtoOFU7$$!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$$\"3s'****R;!fERF U$\"3;q\\DfP\"4[\"F*7$$\"3!***************[FU$\"3#HsJ&=zJK;F*-%'COLOUR G6&%$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$F^sFa\\l7$F csFa\\l7$FhsFa\\l7$F]tFa\\l7$FbtFa\\l7$FgtFa\\l7$F\\uFa\\l7$FauFa\\l7$ FfuFa\\l7$F[vFa\\l7$F`vFa\\l7$FevFa\\l7$FjvFa\\l7$F_wFa\\l7$FdwFa\\l7$ FiwFa\\l7$F^xFa\\l7$FcxFa\\l7$FhxFa\\l7$F]yFa\\l7$FbyFa\\l7$FgyFa\\l7$ F\\zFa\\l7$FazFa\\l7$FfzFa\\l7$F[[lFa\\l7$F`[lFa\\l-Fe[l6&Fg[lF[\\lF[ \\lFh[l-F$6&7#7$$!3%)******oz.qMF*Fa\\l-%'SYMBOLG6#%'CIRCLEG-Fe[l6&Fg[ lF\\\\lF\\\\lF\\\\l-%&STYLEG6#%&POINTG-F$6&Fg_l-F\\`l6#%&CROSSGF_`lFa` l-F$6&Fg_l-F\\`l6#%(DIAMONDGF_`lFa`l-F$6%7$7$Fi_lF[\\lFh_l-%&COLORG6&F g[lF[\\l$\"\"&Fb\\lF[\\l-%*LINESTYLEG6#\"\"$-%+AXESLABELSG6%Q\"z6\"Q!F `bl-%%FONTG6#%(DEFAULTG-Fcbl6$%*HELVETICAG\"\"*-%%VIEWG6$;$!$*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+31903/104004000*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 z0 := zz :\n pts := [op(pts),[Re(zz),Im(zz)]]:\nend do:\np_1 := plot(pts,colo r=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,color=CO LOR(RGB,.65,0,.65)):\npts := []: z0 := 1.8+4*I:\nfor ct from 0 to 50 d o\n zz := newton(`R*`(z)=exp(ct*Pi/25*I),z=z0):\n z0 := zz:\n pt s := [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],[1.7 3,3.87]],i=2..nops(pts))],\n style=patchnogrid,color=COLOR(RG B,.65,0,.65)):\npts := []: z0 := 1.8-4*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(zz)]]:\nend do:\np_5 := plot(pts,color=COLOR(RGB,.3 3,0,.33)):\np_6 := plots[polygonplot]([seq([pts[i-1],pts[i],[1.73,-3.8 7]],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]]],col or=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 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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]=- 8456/2889,a[6,5]=217/428,\na[7,1]=1718911/4382720,a[7,2]=0,a[7,3]=-100 0749/547840,\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]=1152/425,a[8,7]=-7168/40375,\na[9,1]=5 3/912,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]=2 7/112,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]=-1/18\}:" }}}{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$F9F'\"\"!/&F%6$F9 F.#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$\"\"(F(#\"( 6*=<\"(?FQ%/&F%6$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#FY\"$7\"/&F%6$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 "#=============================" }} {SECT 1 {PARA 4 "" 0 "" {TEXT -1 13 "scheme with " }{XPPEDIT 18 0 "c[ 7] = 24/25;" "6#/&%\"cG6#\"\"(*&\"#C\"\"\"\"#D!\"\"" }{TEXT -1 1 " " } }{PARA 257 "" 0 "" {TEXT -1 33 "The scheme constructed here has " } {XPPEDIT 18 0 "c[6] = 27/40;" "6#/&%\"cG6#\"\"'*&\"#F\"\"\"\"#S!\"\"" }{TEXT -1 7 " and " }{XPPEDIT 18 0 "c[7] = 24/25;" "6#/&%\"cG6#\"\"( *&\"#C\"\"\"\"#D!\"\"" }{TEXT -1 1 "." }}{PARA 0 "" 0 "" {TEXT -1 6 "W ith " }{XPPEDIT 18 0 "c[6]" "6#&%\"cG6#\"\"'" }{TEXT -1 7 " and " } {XPPEDIT 18 0 "c[7]" "6#&%\"cG6#\"\"(" }{TEXT -1 39 " having these fi xed values 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[5]" "6#&%\"cG6#\"\"&" }{TEXT -1 50 " are c hosen to minimize the principal error norm." }}{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 scheme" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1879 "ee := \{c[ 2]=33/577,\nc[3]=173/1200,\nc[4]=173/800,\nc[5]=260/463,\nc[6]=27/40, \nc[7]=24/25,\nc[8]=1,\nc[9]=1,\n\na[2,1]=33/577,\na[3,1]=-3567433/950 40000,\na[3,2]=17269033/95040000,\na[4,1]=173/3200,\na[4,2]=0,\na[4,3] =519/3200,\na[5,1]=1877748548260/2970538457863,\na[5,2]=0,\na[5,3]=-71 26513680000/2970538457863,\na[5,4]=6916886080000/2970538457863,\na[6,1 ]=-65244579431964655428477/69653246199915888640000,\na[6,2]=0,\na[6,3] =2694429616783931433/669742751922268160,\na[6,4]=-62138409816102019904 637/21415191928402504983040,\na[6,5]=145922219908058645309313/29766179 4320406364160000,\na[7,1]=3890754282260741440349323/301023695697054704 8875000,\na[7,2]=0,\na[7,3]=-45910927628476367208/8576173666582755125, \na[7,4]=5986037178684297350750710688/1207688208130690660530130125,\na [7,5]=-3106990980562910309356750843/3003073489062493679278875000,\na[7 ,6]=2999575140352/2729866330125,\na[8,1]=3914147173803753655861/144460 6034961051585792,\na[8,2]=0,\na[8,3]=-2276990272364785725/192922814498 003684,\na[8,4]=24219185248932516188948175/2263933167309766272933307, \na[8,5]=-4801005224579354456654139837/1661667052909362562177244416,\n a[8,6]=232855940800/97230777633,\na[8,7]=-57039375/516949856,\na[9,1]= 7466867/116588160,\na[9,2]=0,\na[9,3]=0,\na[9,4]=3227648000000/9720313 182027,\na[9,5]=179724568369012721/784946257231456320,\na[9,6]=2343680 00/1186672113,\na[9,7]=16390625/58886016,\na[9,8]=-14011/138852,\n\nb[ 1]=7466867/116588160,\nb[2]=0,\nb[3]=0,\nb[4]=3227648000000/9720313182 027,\nb[5]=179724568369012721/784946257231456320,\nb[6]=234368000/1186 672113,\nb[7]=16390625/58886016,\nb[8]=-14011/138852,\n\n`b*`[1]=33846 82763489201/53914632194067840,\n`b*`[2]=0,\n`b*`[3]=0,\n`b*`[4]=721434 77722071040000/214048979481496847913,\n`b*`[5]=24566463552585428091992 041/120995959197297938812618560,\n`b*`[6]=125185218908556800/548760616 063863687,\n`b*`[7]=2889850698904375/11670449557585536,\n`b*`[8]=-2285 6669025961/321051233221740,\n`b*`[9]=-1/135\}:" }}}{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 64 "The Butcher tab leau in exact and approximate form is as follows." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 488 "subs(ee,mat rix([[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],se q(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)],[``$4,`b*`[9]]]));" } }{PARA 11 "" 1 "" {XPPMATH 20 "6#K%'matrixG6#747'#\"#L\"$x&F(%!GF+F+7' #\"$t\"\"%+7#!(Luc$\")++/&*#\")L!ps\"F2F+F+7'#F.\"$+)#F.\"%+K\"\"!#\"$ >&F9F+7'#\"$g#\"$j%#\".g#[&[x(=\".jyXQ0(HF:#!.++o8l7(FC#\".++3')o\"pFC 7'#\"#F\"#S#!8x%Galk>VzXCl\"8++k))e\"**>YKlpF:#\"4L9$Ry;'HWp#\"3g\"oA# >vU(p'#!8PY!*>?5;)4%Q@'\"8SI)\\]-%G>>:9#7'F+F+F+F+#\"98$4`ke!3*>A#f9\" 9++;kjS?VzhwH7'#\"#C\"#D#\":B$\\.WT2E#Ga2*Q\":+]()[qaqp&pB5IF:#!53sOw% Gw#4\"f%\"4D^v#emO# \";v\"[*)=;D$*[_=>U#\":2L$HFm(4t;LRE#7'F+F+#!=P)RTlcWNzXA05![\"=;WCx@c i$4H0n;m\"#\"-+3%f&GB\",LwxIs*#!)v$Rq&\"*c)\\p^7'Fho#\"(noY(\"*g\")e; \"F:F:#\".+++[wA$\".F?=8.s*7'F+#\"3@F,p$oXsz\"\"3?jXJsDY\\y#\"*+!oVB\" +8@n'=\"#\")D1R;\");g))e#!&6S\"\"'_)Q\"7'F+%=_________________________ ___GFarFarFar7'%\"bGF]qF:F:F`qFcq7'%#b*G#\"1,#*[jFo%Q$\"2Sy1%>KY\"R&F: F:#\"5++/r?sxM9s\"68z%o\\\"[z*[S@7'F+#\";T?*>4Gae_NYmX#\" ff*47#\"3+ob3*=_=D\"\"3(ojQ1;1w[&#\"1vV!*)p])*)G\"2Obed&\\/n6#!/hf-pm& G#\"0S " 0 "" {MPLTEXT 1 0 136 "subs(ee,mat rix([seq([c[i],seq(a[i,j],j=1..i-1),``$(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,$\")uB>d!\"*F(%!GF+F+F+F+F+F+F+ 7,$\")nmT9!\")$!)Ah`PF*$\")z-<=F/F+F+F+F+F+F+F+7,$\")+]i@F/$\")+D1aF*$ \"\"!F:$\")](=i\"F/F+F+F+F+F+F+7,$\")3b:cF/$\")'R7K'F/F9$!)Y1*R#!\"($ \")d\\GBFDF+F+F+F+F+7,$\")++]nF/$!)^0n$*F/F9$\")93BSFD$!)Ug,HFD$\")DG- \\F/F+F+F+F+7,$\")+++'*F/$\")w]#H\"FDF9$!)8J``FD$\")\"3m&\\FD$!)PgM5FD $\")$*z)4\"FDF+F+F+7,$\"\"\"F:$\")6\\4FFDF9$!)(f-=\"!\"'$\")Oyp5F`o$!) .F*)GFD$\")*y[R#FD$!)JQ.6F/F+F+7,Fjn$\")9[/kF*F9F9$\")&=0K$F/$\")F/$\")&\\My#F/$!)+145F/F+7,%\"bGFjoF9F9F\\pF^pF`pFbpFdpF+7,% #b*G$\")c&yF'F*F9F9$\")\">/P$F/$\")SNI?F/$\")bB\"G#F/$\")?@wCF/$!)9K>r F*$!)uS2u!#5Q)pprint196\"" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 146 "RK6_8eqs := [op(RowSumConditions(8 ,'expanded')),op(OrderConditions(6,8,'expanded'))]:\n`RK5_9eqs*` := su bs(b=`b*`,OrderConditions(5,9,'expanded')):" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 7 "Check: " }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 148 "simplify(subs(ee,RK6_8eqs)):\nmap(u->`if`(lhs(u )=rhs(u),0,1),%);\nnops(%);\nsimplify(subs(ee,`RK5_9eqs*`)):\nmap(u->` if`(lhs(u)=rhs(u),0,1),%);\nnops(%);" }{TEXT -1 0 "" }}{PARA 11 "" 1 " " {XPPMATH 20 "6#7N\"\"!F$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$" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 114 "We can calculate the principal erro r norm of the order 6 scheme, that is, the 2-norm of the principal err or terms." }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 131 "errterms6_8 := PrincipalErrorTerms(6,8,'expanded'):\nevalf(evalf[14](sqrt(add(subs(e e,errterms6_8[i])^2,i=1.. nops(errterms6_8)))));" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#$\"+#R!RFO!#9" }}}{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#$ \"+ " 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(\"\"$7%\"\")F)/*&F,F ()F2F:F(#F(F57%\"#7F)/*&F,F(-%!G6#*&%\"aGF(-FF6#*&F8F(FIF(F(F(#F(\"#g7 %\"#:F)/*&F,F(-FF6#*&F?F(FIF(F(#F(\"#?7%\"#;F)/*&F,F()F2F5F(#F(\"\"&Q) pprint186\"" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 582 "SO6_8 := SimpleOrderConditions(6,8,'expanded'):\n SO_eqs := [op(RowSumConditions(8,'expanded')),op(StageOrderConditions( 2,8,'expanded')),\n op(StageOrderConditions(3,4..8,'expa nded'))]:\n`SO5_9*` := subs(b=`b*`,SimpleOrderConditions(5,9,'expanded ')):\nord_cdns := [seq(SO6_8[i],i=[1,2,4,8,16,24,29,32])]:\n`ord_cdns* ` := [seq(`SO5_9*`[i],i=[1,2,4,8,12,15,16])]:\nsimp_eqs := [add(b[i]*a [i,1],i=1+1..8)=b[1],seq(add(b[i]*a[i,j],i=j+1..8)=b[j]*(1-c[j]),j=[6, 7])]:\nwt_eqns := [seq(b[i]=a[9,i],i=1..8)]:\ncdns := [op(ord_cdns),op (simp_eqs),op(SO_eqs),op(wt_eqns),op(`ord_cdns*`)]:" }}}{PARA 0 "" 0 " " {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 22 "We specify the nodes: \+ " }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "c[2] = 33/577;" " 6#/&%\"cG6#\"\"#*&\"#L\"\"\"\"$x&!\"\"" }{TEXT -1 3 ", " }{XPPEDIT 18 0 "c[4] = 173/800;" "6#/&%\"cG6#\"\"%*&\"$t\"\"\"\"\"$+)!\"\"" } {TEXT -1 3 ", " }{XPPEDIT 18 0 "c[5] = 260/463;" "6#/&%\"cG6#\"\"&*& \"$g#\"\"\"\"$j%!\"\"" }{TEXT -1 3 ", " }{XPPEDIT 18 0 "c[6] = 27/40; " "6#/&%\"cG6#\"\"'*&\"#F\"\"\"\"#S!\"\"" }{TEXT -1 3 ", " }{XPPEDIT 18 0 "c[7] = 24/25;" "6#/&%\"cG6#\"\"(*&\"#C\"\"\"\"#D!\"\"" }{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*`[9] = -1/135;" "6#/&%#b*G6#\" \"*,$*&\"\"\"F*\"$N\"!\"\"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 0 215 "e1 := \{c[2 ]=33/577,c[4]=173/800,c[5]=260/463,c[6]=27/40,c[7]=24/25,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]=-1/135\}:\neqns := subs(e1,cdns):\nnops(%);\nindets(eqns);\nnops(%) ;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#\"#W" }}{PARA 12 "" 1 "" {XPPMATH 20 "6#&F<6$F*F'&F<6 $F*F:&F<6$\"\"*F'&F<6$FIF>&F<6$FIF:&F<6$FIF*&F<6$FIF-&F<6$FIF0&F<6$FIF 3&F<6$FIF6&%#b*GF)&FYF,&FYF/&FYF2&FYF5&F<6$F-F:&F<6$F-F*&F<6$F0F'&F<6$ F0F:&F<6$F0F*&F<6$F0F-&F<6$F3F'&F<6$F3F:&F<6$F3F*&F<6$F3F-&F<6$F3F0&F< 6$F6F'&F<6$F-F'&FYF&&F<6$F6F:&F<6$F6F*&F<6$F6F-&F<6$F6F0&F<6$F6F3" }} {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 1998 "e3 := \{a[4,1] = 173/3200, a[7,1] = 3890754282260741440349323/3010236956970547048875000, `b*`[8] = -22856669025961/321051233221740, a[2,1] = 33/577, b[8] = -14011/138 852, a[8,1] = 3914147173803753655861/1444606034961051585792, a[4,2] = \+ 0, a[5,2] = 0, a[6,2] = 0, a[7,2] = 0, `b*`[7] = 2889850698904375/1167 0449557585536, a[8,2] = 0, a[9,5] = 179724568369012721/784946257231456 320, b[4] = 3227648000000/9720313182027, `b*`[6] = 125185218908556800/ 548760616063863687, `b*`[3] = 0, a[9,2] = 0, a[9,3] = 0, a[8,5] = -480 1005224579354456654139837/1661667052909362562177244416, a[9,4] = 32276 48000000/9720313182027, `b*`[1] = 3384682763489201/53914632194067840, \+ a[9,8] = -14011/138852, b[1] = 7466867/116588160, a[5,4] = 69168860800 00/2970538457863, `b*`[4] = 72143477722071040000/214048979481496847913 , c[8] = 1, c[9] = 1, b[2] = 0, b[3] = 0, `b*`[2] = 0, b[6] = 23436800 0/1186672113, b[5] = 179724568369012721/784946257231456320, a[3,1] = - 3567433/95040000, a[5,1] = 1877748548260/2970538457863, a[8,4] = 24219 185248932516188948175/2263933167309766272933307, a[9,1] = 7466867/1165 88160, c[5] = 260/463, c[6] = 27/40, c[7] = 24/25, c[4] = 173/800, c[2 ] = 33/577, `b*`[5] = 24566463552585428091992041/120995959197297938812 618560, a[8,3] = -2276990272364785725/192922814498003684, `b*`[9] = -1 /135, a[7,3] = -45910927628476367208/8576173666582755125, a[9,6] = 234 368000/1186672113, a[7,5] = -3106990980562910309356750843/300307348906 2493679278875000, a[3,2] = 17269033/95040000, a[6,1] = -65244579431964 655428477/69653246199915888640000, b[7] = 16390625/58886016, a[7,4] = \+ 5986037178684297350750710688/1207688208130690660530130125, a[6,3] = 26 94429616783931433/669742751922268160, a[5,3] = -7126513680000/29705384 57863, c[3] = 173/1200, a[9,7] = 16390625/58886016, a[4,3] = 519/3200, a[7,6] = 2999575140352/2729866330125, a[8,7] = -57039375/516949856, a [6,4] = -62138409816102019904637/21415191928402504983040, a[8,6] = 232 855940800/97230777633, a[6,5] = 145922219908058645309313/2976617943204 06364160000\}:" }{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)],[``$4,`b*`[9]]]));" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#K%'matrixG6#747'#\"#L\"$x&F(%!GF+F+7'# \"$t\"\"%+7#!(Luc$\")++/&*#\")L!ps\"F2F+F+7'#F.\"$+)#F.\"%+K\"\"!#\"$> &F9F+7'#\"$g#\"$j%#\".g#[&[x(=\".jyXQ0(HF:#!.++o8l7(FC#\".++3')o\"pFC7 '#\"#F\"#S#!8x%Galk>VzXCl\"8++k))e\"**>YKlpF:#\"4L9$Ry;'HWp#\"3g\"oA#> vU(p'#!8PY!*>?5;)4%Q@'\"8SI)\\]-%G>>:9#7'F+F+F+F+#\"98$4`ke!3*>A#f9\"9 ++;kjS?VzhwH7'#\"#C\"#D#\":B$\\.WT2E#Ga2*Q\":+]()[qaqp&pB5IF:#!53sOw%G w#4\"f%\"4D^v#emO# \";v\"[*)=;D$*[_=>U#\":2L$HFm(4t;LRE#7'F+F+#!=P)RTlcWNzXA05![\"=;WCx@c i$4H0n;m\"#\"-+3%f&GB\",LwxIs*#!)v$Rq&\"*c)\\p^7'Fho#\"(noY(\"*g\")e; \"F:F:#\".+++[wA$\".F?=8.s*7'F+#\"3@F,p$oXsz\"\"3?jXJsDY\\y#\"*+!oVB\" +8@n'=\"#\")D1R;\");g))e#!&6S\"\"'_)Q\"7'F+%=_________________________ ___GFarFarFar7'%\"bGF]qF:F:F`qFcq7'%#b*G#\"1,#*[jFo%Q$\"2Sy1%>KY\"R&F: F:#\"5++/r?sxM9s\"68z%o\\\"[z*[S@7'F+#\";T?*>4Gae_NYmX#\" ff*47#\"3+ob3*=_=D\"\"3(ojQ1;1w[&#\"1vV!*)p])*)G\"2Obed&\\/n6#!/hf-pm& G#\"0S " 0 "" {MPLTEXT 1 0 136 "subs(e3,mat rix([seq([c[i],seq(a[i,j],j=1..i-1),``$(10-i)],i=2..9),\n[`b`,seq(b[i] ,i=1..8),``],[`b*`,seq(`b*`[i],i=1..9)]])):\nevalf[8](%);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#K%'matrixG6#7,7,$\")uB>d!\"*F(%!GF+F+F+F+F+F+F+ 7,$\")nmT9!\")$!)Ah`PF*$\")z-<=F/F+F+F+F+F+F+F+7,$\")+]i@F/$\")+D1aF*$ \"\"!F:$\")](=i\"F/F+F+F+F+F+F+7,$\")3b:cF/$\")'R7K'F/F9$!)Y1*R#!\"($ \")d\\GBFDF+F+F+F+F+7,$\")++]nF/$!)^0n$*F/F9$\")93BSFD$!)Ug,HFD$\")DG- \\F/F+F+F+F+7,$\")+++'*F/$\")w]#H\"FDF9$!)8J``FD$\")\"3m&\\FD$!)PgM5FD $\")$*z)4\"FDF+F+F+7,$\"\"\"F:$\")6\\4FFDF9$!)(f-=\"!\"'$\")Oyp5F`o$!) .F*)GFD$\")*y[R#FD$!)JQ.6F/F+F+7,Fjn$\")9[/kF*F9F9$\")&=0K$F/$\")F/$\")&\\My#F/$!)+145F/F+7,%\"bGFjoF9F9F\\pF^pF`pFbpFdpF+7,% #b*G$\")c&yF'F*F9F9$\")\">/P$F/$\")SNI?F/$\")bB\"G#F/$\")?@wCF/$!)9K>r F*$!)uS2u!#5Q)pprint176\"" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 146 "RK6_8eqs := [op(RowSumConditions(8 ,'expanded')),op(OrderConditions(6,8,'expanded'))]:\n`RK5_9eqs*` := su bs(b=`b*`,OrderConditions(5,9,'expanded')):" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 7 "Check: " }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 148 "simplify(subs(e3,RK6_8eqs)):\nmap(u->`if`(lhs(u )=rhs(u),0,1),%);\nnops(%);\nsimplify(subs(e3,`RK5_9eqs*`)):\nmap(u->` if`(lhs(u)=rhs(u),0,1),%);\nnops(%);" }{TEXT -1 0 "" }}{PARA 11 "" 1 " " {XPPMATH 20 "6#7N\"\"!F$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 28 "determination of 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[5] " "6#&%\"cG6#\"\"&" }{TEXT -1 1 " " }}{PARA 4 "" 0 "" {TEXT -1 0 "" }} {PARA 0 "" 0 "" {TEXT -1 79 "In this section we obtain the nodes of a \+ modification of a scheme by Tsitouras." }}{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*(3 0*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]+2 8*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]+1 90*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]^2-27*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]^4-10 0*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]), 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]+1 32*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]^2-91 0*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+20 0*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]-4 1*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]^3-1 2*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+2 2*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+42 0*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 +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]+1 5*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]+15 6*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+95 0*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]^3-10 0*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]-1 98*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]+20 0*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+2 40*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+1 1*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]^3-49 0*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]^2-30 0*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]+20 0*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+10 3*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]^3-2 3*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+18 0*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]^3-1 0*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]-20 0*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]+3 60*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+3 60*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]-8 0*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+84 0*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+23 0*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[8,4] = -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]-7 50*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]^5-132 0*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]^2-530*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]^ 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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]^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]-32 80*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 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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]+7 0*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 ]+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]+150 0*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]+17 20*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- 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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]^2-14207*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]+2 860*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]-4 29*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]+49 0*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+1 4*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]^3-70*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[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 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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] /(72*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 ]+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] 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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]+150*c[6]^2*c[5]^6*c[4]^2+15 0*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]-20 0*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 33 "#=================== =============" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 22 "We specify the nodes " }{XPPEDIT 18 0 "c[6] = 27/40;" "6 #/&%\"cG6#\"\"'*&\"#F\"\"\"\"#S!\"\"" }{TEXT -1 7 " and " }{XPPEDIT 18 0 "c[7] = 24/25;" "6#/&%\"cG6#\"\"(*&\"#C\"\"\"\"#D!\"\"" }{TEXT -1 27 " and determine 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[5]" "6#&%\"cG6#\"\"&" }{TEXT -1 65 " that minimize the principal errror norm (subject to the nodes " }{XPPEDIT 18 0 "c[6]" "6#&%\"cG6#\"\"'" }{TEXT -1 7 " and " } {XPPEDIT 18 0 "c[7]" "6#&%\"cG6#\"\"(" }{TEXT -1 19 " remaining fixed )." }}{PARA 0 "" 0 "" {TEXT -1 85 "We use the general solution to obt ain expressions for the coefficients in terms of " }{XPPEDIT 18 0 "c[ 2]" "6#&%\"cG6#\"\"#" }{TEXT -1 3 ", " }{XPPEDIT 18 0 "c[4]" "6#&%\"c G6#\"\"%" }{TEXT -1 7 " and " }{XPPEDIT 18 0 "c[5]" "6#&%\"cG6#\"\"& " }{TEXT -1 2 ". " }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 71 "eA := \+ \{c[6]=27/40,c[7]=24/25\}:\neB := `union`(eA,simplify(subs(eA,eG))):" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 2 "eB" }}{PARA 0 "" 0 " " {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 14902 "eB := \+ \{`b*`[2] = 0, c[9] = 1, a[8,7] = -203125/1368*(70*c[5]*c[4]-15*c[5]-1 5*c[4]+1)*(c[4]-1)*(-1+c[5])/(25*c[5]-24)/(25*c[4]-24)/(348*c[5]*c[4]- 87*c[5]-87*c[4]+20), `b*`[5] = 1/20*(77849*c[4]+32975*c[5]-24300*c[5]^ 2+711850*c[5]*c[4]^2+434700*c[5]^2*c[4]^3+217800*c[5]^2*c[4]-282930*c[ 5]*c[4]-180380*c[4]^2-580350*c[5]^2*c[4]^2-498000*c[5]*c[4]^3+118260*c [4]^3-9639)/(6792*c[4]+8817*c[5]+5400*c[5]^3-12259*c[5]^2+27384*c[5]*c [4]^2+20250*c[5]^3*c[4]^2-22900*c[5]^3*c[4]+49884*c[5]^2*c[4]-33859*c[ 5]*c[4]-5184*c[4]^2-42340*c[5]^2*c[4]^2-1944)/(40*c[5]-27)/(c[4]-c[5]) /c[5], b[4] = 1/12*(261*c[5]-67)/c[4]/(-2635*c[4]^3+1000*c[4]^4+2283*c [4]^2-648*c[4]-1000*c[5]*c[4]^3+2635*c[5]*c[4]^2-2283*c[5]*c[4]+648*c[ 5]), c[8] = 1, b[3] = 0, b[2] = 0, `b*`[4] = -1/20*(32975*c[4]+77849*c [5]+118260*c[5]^3-180380*c[5]^2+217800*c[5]*c[4]^2+434700*c[5]^3*c[4]^ 2-498000*c[5]^3*c[4]+711850*c[5]^2*c[4]-282930*c[5]*c[4]-24300*c[4]^2- 580350*c[5]^2*c[4]^2-9639)/(8817*c[4]+6792*c[5]-5184*c[5]^2+49884*c[5] *c[4]^2+20250*c[5]^2*c[4]^3+27384*c[5]^2*c[4]-33859*c[5]*c[4]-12259*c[ 4]^2-42340*c[5]^2*c[4]^2-22900*c[5]*c[4]^3+5400*c[4]^3-1944)/(40*c[4]- 27)/(c[4]-c[5])/c[4], `b*`[1] = 1/38880*(-233547*c[4]-233547*c[5]-3547 80*c[5]^3+541140*c[5]^2-5248040*c[5]*c[4]^2-10208540*c[5]^2*c[4]^3-102 08540*c[5]^3*c[4]^2+3596220*c[5]^3*c[4]-5248040*c[5]^2*c[4]+2106942*c[ 5]*c[4]+541140*c[4]^2+7857000*c[5]^3*c[4]^3+14100420*c[5]^2*c[4]^2+359 6220*c[5]*c[4]^3-354780*c[4]^3+28917)/(-283*c[4]-283*c[5]+216*c[5]^2-9 16*c[5]*c[4]^2-916*c[5]^2*c[4]+1116*c[5]*c[4]+216*c[4]^2+810*c[5]^2*c[ 4]^2+81)/c[5]/c[4], a[8,4] = -1/10*(99668400*c[5]^5*c[4]^3-1465776315* c[5]^4*c[4]^3+548597150*c[5]^3*c[4]^6-1747285880*c[5]^3*c[4]^4-1706396 100*c[4]^6*c[5]^2-15746400*c[5]^5*c[4]^2+470794250*c[4]^6*c[5]-2329647 600*c[5]^4*c[4]^5+192402250*c[5]^5*c[4]^5+186300000*c[4]^8*c[5]^3+1432 070735*c[4]^5*c[5]^2-8173872*c[5]^3+2309472*c[5]^2-214224250*c[5]^5*c[ 4]^4-478044935*c[4]^5*c[5]+22568108*c[5]*c[4]^2+7705498*c[5]^2*c[4]^3- 525743280*c[4]^4*c[5]^2+466030165*c[5]^4*c[4]^2-484966642*c[5]^3*c[4]^ 2+94014914*c[5]^3*c[4]-18041990*c[5]^2*c[4]-2694384*c[5]*c[4]+77869175 0*c[4]^5*c[5]^3+276794860*c[4]^4*c[5]+2519348650*c[5]^4*c[4]^4+769824* c[4]^2+1310023080*c[5]^3*c[4]^3-7704800*c[4]^5+6298560*c[5]^4+8892310* c[4]^4-79972710*c[5]^4*c[4]+54387665*c[5]^2*c[4]^2-98286899*c[5]*c[4]^ 3-4387334*c[4]^3-208800000*c[4]^8*c[5]^2+1070019250*c[5]^4*c[4]^6+9625 08000*c[4]^7*c[5]^2-239731000*c[5]*c[4]^7+2430000*c[4]^6+48600000*c[5] *c[4]^8-62100000*c[5]^5*c[4]^6-186300000*c[4]^7*c[5]^4-677200500*c[4]^ 7*c[5]^3)/(-3480*c[5]^5*c[4]^3+11310*c[5]^4*c[4]^3-11310*c[5]^3*c[4]^4 +870*c[5]^5*c[4]^2-870*c[4]^5*c[5]^2+20*c[5]^3+40*c[5]*c[4]^2-426*c[5] ^2*c[4]^3+4898*c[4]^4*c[5]^2-4898*c[5]^4*c[4]^2+426*c[5]^3*c[4]^2-33*c [5]^3*c[4]-40*c[5]^2*c[4]+3480*c[4]^5*c[5]^3-870*c[4]^4*c[5]-87*c[5]^4 +87*c[4]^4+870*c[5]^4*c[4]+33*c[5]*c[4]^3-20*c[4]^3)/(-1635*c[4]+1000* c[4]^2+648)/c[4]^2, a[6,4] = -27/5120000*(-864000*c[5]^4*c[4]^3-259200 0*c[5]^3*c[4]^4+2592000*c[4]^5*c[5]^2-233280*c[5]^3+118098*c[5]^2-1166 400*c[4]^5*c[5]+647352*c[5]*c[4]^2+1050620*c[5]^2*c[4]^3-5184000*c[4]^ 4*c[5]^2+583200*c[5]^4*c[4]^2-6639600*c[5]^3*c[4]^2+1874880*c[5]^3*c[4 ]-650268*c[5]^2*c[4]-137781*c[5]*c[4]+2927320*c[4]^4*c[5]+39366*c[4]^2 +8229600*c[5]^3*c[4]^3+1559790*c[5]^2*c[4]^2-2101140*c[5]*c[4]^3-58320 *c[4]^3)/(-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] = 54/390625*(-21082500*c[5]^4*c[4]^3-20076750 0*c[5]^3*c[4]^4+342144*c[4]-63247500*c[4]^5*c[5]^2-684288*c[5]-1944000 *c[5]^3+2450400*c[5]^2+14580000*c[4]^5*c[5]-21971190*c[5]*c[4]^2-21940 6650*c[5]^2*c[4]^3+205412600*c[4]^4*c[5]^2+4860000*c[5]^4*c[4]^2-93654 075*c[5]^3*c[4]^2+20097000*c[5]^3*c[4]-23609460*c[5]^2*c[4]+5330928*c[ 5]*c[4]+58218750*c[4]^5*c[5]^3-47530425*c[4]^4*c[5]+19406250*c[5]^4*c[ 4]^4-1225200*c[4]^2+212383000*c[5]^3*c[4]^3+101958085*c[5]^2*c[4]^2+49 335155*c[5]*c[4]^3+972000*c[4]^3)/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)/(70*c[5]*c[4]-15*c[5]-15*c[4]+1), a[6,1] = 27/10240000*(-2160000 *c[5]^4*c[4]^3-12432000*c[5]^3*c[4]^4+1728000*c[4]^5*c[5]^2-233280*c[5 ]^3+118098*c[5]^2-1166400*c[4]^5*c[5]+705672*c[5]*c[4]^2-1738980*c[5]^ 2*c[4]^3-472800*c[4]^4*c[5]^2+583200*c[5]^4*c[4]^2-7673200*c[5]^3*c[4] ^2+2004480*c[5]^3*c[4]-708588*c[5]^2*c[4]-137781*c[5]*c[4]+2560000*c[4 ]^5*c[5]^3+2152920*c[4]^4*c[5]+2560000*c[5]^4*c[4]^4+39366*c[4]^2+1423 6800*c[5]^3*c[4]^3+86400*c[4]^4+1866510*c[5]^2*c[4]^2-1708020*c[5]*c[4 ]^3-116640*c[4]^3)/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), a[8 ,6] = -83200/1539*(5285*c[5]*c[4]+1073-2130*c[4]-2130*c[5])*(c[4]-1)*( -1+c[5])/(40*c[4]-27)/(40*c[5]-27)/(348*c[5]*c[4]-87*c[5]-87*c[4]+20), a[5,3] = 3/4*c[5]^2*(-2*c[5]+3*c[4])/c[4]^2, a[7,4] = -12/390625*(-20 93332500*c[5]^5*c[4]^3+26751339000*c[5]^4*c[4]^3-12458812500*c[5]^3*c[ 4]^6+19641069600*c[5]^3*c[4]^4+18293173000*c[4]^6*c[5]^2+393660000*c[5 ]^5*c[4]^2-4589754000*c[4]^6*c[5]+22043231250*c[5]^4*c[4]^5-1552500000 *c[5]^5*c[4]^5-23169434400*c[4]^5*c[5]^2+198482400*c[5]^3-55427328*c[5 ]^2+3243806250*c[5]^5*c[4]^4+6714425350*c[4]^5*c[5]-476086032*c[5]*c[4 ]^2-1198933635*c[5]^2*c[4]^3+11735508875*c[4]^4*c[5]^2-9793290075*c[5] ^4*c[4]^2+9627246285*c[5]^3*c[4]^2-2077172460*c[5]^3*c[4]+374772528*c[ 5]^2*c[4]+64665216*c[5]*c[4]+2228535500*c[4]^5*c[5]^3-4762358370*c[4]^ 4*c[5]-36004612500*c[5]^4*c[4]^4-18475776*c[4]^2-21841162800*c[5]^3*c[ 4]^3+58320000*c[4]^5-157464000*c[5]^4-126259200*c[4]^4+1841382000*c[5] ^4*c[4]-907502310*c[5]^2*c[4]^2+1879649100*c[5]*c[4]^3+86706720*c[4]^3 -4657500000*c[5]^4*c[4]^6-5059800000*c[4]^7*c[5]^2+1166400000*c[5]*c[4 ]^7+4657500000*c[4]^7*c[5]^3)/(-700*c[5]^5*c[4]^3+2250*c[5]^4*c[4]^3-2 250*c[5]^3*c[4]^4+150*c[5]^5*c[4]^2-150*c[4]^5*c[5]^2+c[5]^3+2*c[5]*c[ 4]^2+20*c[5]^2*c[4]^3+880*c[4]^4*c[5]^2-880*c[5]^4*c[4]^2-20*c[5]^3*c[ 4]^2+9*c[5]^3*c[4]-2*c[5]^2*c[4]+700*c[4]^5*c[5]^3-160*c[4]^4*c[5]-15* c[5]^4+15*c[4]^4+160*c[5]^4*c[4]-9*c[5]*c[4]^3-c[4]^3)/(40*c[4]-27)/c[ 4]^2, `b*`[8] = 0, a[5,1] = 1/4*c[5]*(2*c[5]^2-5*c[5]*c[4]+4*c[4]^2)/c [4]^2, a[7,2] = 0, a[6,2] = 0, a[4,2] = 0, `b*`[3] = 0, a[8,2] = 0, a[ 5,2] = 0, c[3] = 2/3*c[4], a[4,3] = 3/4*c[4], a[3,2] = 2/9*c[4]^2/c[2] , b[1] = 1/7776*(1618*c[5]*c[4]-261*c[5]-261*c[4]+67)/c[5]/c[4], a[9,8 ] = 1/156*(348*c[5]*c[4]-87*c[5]-87*c[4]+20)/(-c[4]+c[5]*c[4]+1-c[5]), c[7] = 24/25, a[2,1] = c[2], a[3,1] = -2/9*c[4]*(c[4]-3*c[2])/c[2], a [4,1] = 1/4*c[4], `b*`[6] = 25600/4617*(-27831*c[4]-27831*c[5]-24300*c [5]^3+46155*c[5]^2-276310*c[5]*c[4]^2-302350*c[5]^2*c[4]^3-302350*c[5] ^3*c[4]^2+153000*c[5]^3*c[4]-276310*c[5]^2*c[4]+156486*c[5]*c[4]+46155 *c[4]^2+186300*c[5]^3*c[4]^3+518010*c[5]^2*c[4]^2+153000*c[5]*c[4]^3-2 4300*c[4]^3+5346)/(-283*c[4]-283*c[5]+216*c[5]^2-916*c[5]*c[4]^2-916*c [5]^2*c[4]+1116*c[5]*c[4]+216*c[4]^2+810*c[5]^2*c[4]^2+81)/(1600*c[5]* c[4]-1080*c[4]-1080*c[5]+729), `b*`[7] = -15625/16416*(-4413*c[4]-4413 *c[5]-8100*c[5]^3+12060*c[5]^2-104200*c[5]*c[4]^2-165700*c[5]^2*c[4]^3 -165700*c[5]^3*c[4]^2+68100*c[5]^3*c[4]-104200*c[5]^2*c[4]+42738*c[5]* c[4]+12060*c[4]^2+113400*c[5]^3*c[4]^3+250380*c[5]^2*c[4]^2+68100*c[5] *c[4]^3-8100*c[4]^3+243)/(-211608*c[4]-211608*c[5]-129600*c[5]^3+29421 6*c[5]^2-1503691*c[5]*c[4]^2-1058500*c[5]^2*c[4]^3-1058500*c[5]^3*c[4] ^2+684600*c[5]^3*c[4]-1503691*c[5]^2*c[4]+1033041*c[5]*c[4]+294216*c[4 ]^2+506250*c[5]^3*c[4]^3+2263260*c[5]^2*c[4]^2+684600*c[5]*c[4]^3-1296 00*c[4]^3+46656), a[9,2] = 0, a[9,3] = 0, a[8,3] = 3/20*(-104400*c[5]^ 4*c[4]^3-976500*c[5]^3*c[4]^4+1782*c[4]-313200*c[4]^5*c[5]^2-3564*c[5] -9720*c[5]^3+12614*c[5]^2+72900*c[4]^5*c[5]-113652*c[5]*c[4]^2-1114430 *c[5]^2*c[4]^3+1028880*c[4]^4*c[5]^2+24300*c[5]^4*c[4]^2-465155*c[5]^3 *c[4]^2+100080*c[5]^3*c[4]-121248*c[5]^2*c[4]+27691*c[5]*c[4]+279450*c [4]^5*c[5]^3-240405*c[4]^4*c[5]+93150*c[5]^4*c[4]^4-6307*c[4]^2+104892 0*c[5]^3*c[4]^3+521544*c[5]^2*c[4]^2+253380*c[5]*c[4]^3+4860*c[4]^3)/c [4]^2/(3480*c[5]^4*c[4]^3+3480*c[5]^3*c[4]^4+87*c[5]^3-20*c[5]^2+120*c [5]*c[4]^2+4898*c[5]^2*c[4]^3-870*c[4]^4*c[5]^2-870*c[5]^4*c[4]^2+4898 *c[5]^3*c[4]^2-870*c[5]^3*c[4]+120*c[5]^2*c[4]+20*c[5]*c[4]-20*c[4]^2- 12180*c[5]^3*c[4]^3-1296*c[5]^2*c[4]^2-870*c[5]*c[4]^3+87*c[4]^3), b[5 ] = -1/12*(261*c[4]-67)/(c[4]-c[5])/c[5]/(1000*c[5]^3-2635*c[5]^2+2283 *c[5]-648), a[9,4] = 1/12*(261*c[5]-67)/c[4]/(-2635*c[4]^3+1000*c[4]^4 +2283*c[4]^2-648*c[4]-1000*c[5]*c[4]^3+2635*c[5]*c[4]^2-2283*c[5]*c[4] +648*c[5]), b[8] = 1/156*(348*c[5]*c[4]-87*c[5]-87*c[4]+20)/(c[4]-1)/( -1+c[5]), b[7] = -390625/16416*(70*c[5]*c[4]-15*c[5]-15*c[4]+1)/(25*c[ 4]-24)/(25*c[5]-24), a[9,1] = 1/7776*(1618*c[5]*c[4]-261*c[5]-261*c[4] +67)/c[5]/c[4], c[6] = 27/40, `b*`[9] = 1/10*(1350*c[5]^2*c[4]^2-1280* c[5]^2*c[4]+270*c[5]^2-1280*c[5]*c[4]^2+1345*c[5]*c[4]-310*c[5]+270*c[ 4]^2-310*c[4]+81)/(-283*c[4]-283*c[5]+216*c[5]^2-916*c[5]*c[4]^2-916*c [5]^2*c[4]+1116*c[5]*c[4]+216*c[4]^2+810*c[5]^2*c[4]^2+81), a[6,3] = 2 43/10240000*(1463660*c[5]*c[4]^3-970650*c[5]*c[4]^2+275076*c[5]*c[4]-3 9366*c[5]+19683*c[4]-194400*c[5]^3*c[4]^2-3369600*c[5]^2*c[4]^3+229960 0*c[5]^2*c[4]^2-639360*c[5]^2*c[4]+77760*c[5]^2-583200*c[4]^4*c[5]-388 80*c[4]^2+432000*c[5]^3*c[4]^3+1296000*c[4]^4*c[5]^2)/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[6] = 1024000/60021*(115*c[5]*c[4]-45*c[5] -45*c[4]+22)/(40*c[5]-27)/(40*c[4]-27), a[7,1] = 2/1171875*(700927500* c[5]^5*c[4]^3-6624683000*c[5]^4*c[4]^3+451562500*c[5]^3*c[4]^6-4668410 200*c[5]^3*c[4]^4-1006755000*c[4]^6*c[5]^2-131220000*c[5]^5*c[4]^2+262 440000*c[4]^6*c[5]-4309093750*c[5]^4*c[4]^5+631250000*c[5]^5*c[4]^5+23 25944800*c[4]^5*c[5]^2-66160800*c[5]^3+18475776*c[5]^2-1120343750*c[5] ^5*c[4]^4-716890150*c[4]^5*c[5]+133587504*c[5]*c[4]^2+211854445*c[5]^2 *c[4]^3-1662339225*c[4]^4*c[5]^2+2667317525*c[5]^4*c[4]^2-2540651995*c [5]^3*c[4]^2+618640620*c[5]^3*c[4]-104378256*c[5]^2*c[4]-21555072*c[5] *c[4]+1049807500*c[4]^5*c[5]^3+738503090*c[4]^4*c[5]+7997177500*c[5]^4 *c[4]^4+6158592*c[4]^2+5241865200*c[5]^3*c[4]^3-13170000*c[4]^5+524880 00*c[5]^4+33673200*c[4]^4-555789000*c[5]^4*c[4]+195554250*c[5]^2*c[4]^ 2-394423020*c[5]*c[4]^3-26604000*c[4]^3+631250000*c[5]^4*c[4]^6)/(700* c[5]^4*c[4]^3+700*c[5]^3*c[4]^4+15*c[5]^3-c[5]^2+6*c[5]*c[4]^2+880*c[5 ]^2*c[4]^3-150*c[4]^4*c[5]^2-150*c[5]^4*c[4]^2+880*c[5]^3*c[4]^2-160*c [5]^3*c[4]+6*c[5]^2*c[4]+c[5]*c[4]-c[4]^2-2400*c[5]^3*c[4]^3-140*c[5]^ 2*c[4]^2-160*c[5]*c[4]^3+15*c[4]^3)/c[5]/c[4]^2, a[8,1] = 1/3240*(2076 9450*c[5]^5*c[4]^3-195860650*c[5]^4*c[4]^3+12952450*c[5]^3*c[4]^6-1409 75060*c[5]^3*c[4]^4-29968950*c[4]^6*c[5]^2-3936600*c[5]^5*c[4]^2+78732 00*c[4]^6*c[5]-122466100*c[5]^4*c[4]^5+17599700*c[5]^5*c[4]^5+70749230 *c[4]^5*c[5]^2-2043468*c[5]^3+577368*c[5]^2-32318450*c[5]^5*c[4]^4-219 26935*c[4]^5*c[5]+4152197*c[5]*c[4]^2+6403627*c[5]^2*c[4]^3-51244960*c [4]^4*c[5]^2+79391915*c[5]^4*c[4]^2-78194398*c[5]^3*c[4]^2+19081736*c[ 5]^3*c[4]-3263465*c[5]^2*c[4]-673596*c[5]*c[4]+31003870*c[4]^5*c[5]^3+ 22852130*c[4]^4*c[5]+232645270*c[5]^4*c[4]^4+192456*c[4]^2+160468790*c [5]^3*c[4]^3-385665*c[4]^5+1574640*c[5]^4+1014200*c[4]^4-16598625*c[5] ^4*c[4]+6155850*c[5]^2*c[4]^2-12222726*c[5]*c[4]^3-820991*c[4]^3+17599 700*c[5]^4*c[4]^6)/c[5]/c[4]^2/(3480*c[5]^4*c[4]^3+3480*c[5]^3*c[4]^4+ 87*c[5]^3-20*c[5]^2+120*c[5]*c[4]^2+4898*c[5]^2*c[4]^3-870*c[4]^4*c[5] ^2-870*c[5]^4*c[4]^2+4898*c[5]^3*c[4]^2-870*c[5]^3*c[4]+120*c[5]^2*c[4 ]+20*c[5]*c[4]-20*c[4]^2-12180*c[5]^3*c[4]^3-1296*c[5]^2*c[4]^2-870*c[ 5]*c[4]^3+87*c[4]^3), a[7,5] = -12/390625*(4895537500*c[5]^4*c[4]^3+94 20966500*c[5]^3*c[4]^4+1552500000*c[5]^4*c[4]^5-9237888*c[4]+209333250 0*c[4]^5*c[5]^2+18475776*c[5]+126259200*c[5]^3-86706720*c[5]^2-3936600 00*c[4]^5*c[5]+503281530*c[5]*c[4]^2+6396719750*c[5]^2*c[4]^3-60196242 00*c[4]^4*c[5]^2-2468130000*c[5]^4*c[4]^2+5105897425*c[5]^3*c[4]^2-126 3072600*c[5]^3*c[4]+809717940*c[5]^2*c[4]-137074896*c[5]*c[4]-32438062 50*c[4]^5*c[5]^3+1078991475*c[4]^4*c[5]-4524618750*c[5]^4*c[4]^4+39906 000*c[4]^2-10145466000*c[5]^3*c[4]^3-58320000*c[5]^4+19755000*c[4]^4+6 02910000*c[5]^4*c[4]-3194585295*c[5]^2*c[4]^2-1069433385*c[5]*c[4]^3-5 0509800*c[4]^3)/c[5]/(108900*c[5]^5*c[4]^3-60750*c[5]^4*c[4]^3+95950*c [5]^3*c[4]^4-39250*c[5]^5*c[4]^2+28000*c[5]^4*c[4]^5+6000*c[5]^6*c[4]^ 2+4050*c[4]^5*c[5]^2-27*c[5]^3-54*c[5]*c[4]^2-900*c[5]^2*c[4]^3-30160* c[4]^4*c[5]^2+22960*c[5]^4*c[4]^2+540*c[5]^3*c[4]^2-323*c[5]^3*c[4]+54 *c[5]^2*c[4]-24900*c[4]^5*c[5]^3+4920*c[4]^4*c[5]-90000*c[5]^4*c[4]^4+ 800*c[5]^3*c[4]^3+445*c[5]^4+6400*c[5]^5*c[4]-405*c[4]^4-28000*c[5]^6* c[4]^3-3960*c[5]^4*c[4]+80*c[5]^2*c[4]^2+203*c[5]*c[4]^3+27*c[4]^3-600 *c[5]^5), a[6,5] = -27/5120000*(1457200*c[5]^2*c[4]^2-58320*c[4]^2+196 83*c[4]-479520*c[5]^2*c[4]-897210*c[5]*c[4]^2+43200*c[4]^3-583200*c[4] ^4*c[5]+1204460*c[5]*c[4]^3-1879200*c[5]^2*c[4]^3+294516*c[5]*c[4]+864 000*c[4]^4*c[5]^2+58320*c[5]^2-39366*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]), a[8,5] = -1/10*(2013 66250*c[5]^5*c[4]^3-620051950*c[5]^4*c[4]^3-629823850*c[5]^3*c[4]^4-10 2037500*c[5]^5*c[4]^2-192402250*c[5]^4*c[4]^5+384912*c[4]+62100000*c[5 ]^5*c[4]^5-99668400*c[4]^5*c[5]^2-769824*c[5]-8892310*c[5]^3+4387334*c [5]^2-184018750*c[5]^5*c[4]^4+15746400*c[4]^5*c[5]-19142297*c[5]*c[4]^ 2-307603140*c[5]^2*c[4]^3+289224135*c[4]^4*c[5]^2+313755025*c[5]^4*c[4 ]^2-343893660*c[5]^3*c[4]^2+86187520*c[5]^3*c[4]-39300343*c[5]^2*c[4]+ 5289736*c[5]*c[4]+214224250*c[4]^5*c[5]^3-43186405*c[4]^4*c[5]+5685762 00*c[5]^4*c[4]^4-1641982*c[4]^2+682198050*c[5]^3*c[4]^3+7704800*c[5]^4 +25020000*c[5]^5*c[4]-771330*c[4]^4-77581825*c[5]^4*c[4]+152960414*c[5 ]^2*c[4]^2+42062390*c[5]*c[4]^3+2028400*c[4]^3-2430000*c[5]^5)/c[5]/(- 20746890*c[5]^5*c[4]^3-3480000*c[5]^7*c[4]^3+6902880*c[5]^4*c[4]^3-162 07110*c[5]^3*c[4]^4+8997990*c[5]^5*c[4]^2-6559800*c[5]^4*c[4]^5+348000 0*c[5]^5*c[4]^5-6320450*c[5]^6*c[4]^2-563760*c[4]^5*c[5]^2+12960*c[5]^ 3-11310000*c[5]^5*c[4]^4+25920*c[5]*c[4]^2-350003*c[5]^2*c[4]^3+468335 4*c[4]^4*c[5]^2-3870414*c[5]^4*c[4]^2+316048*c[5]^3*c[4]^2+44016*c[5]^ 3*c[4]-25920*c[5]^2*c[4]+3677490*c[4]^5*c[5]^3-706005*c[4]^4*c[5]+2338 9850*c[5]^4*c[4]^4+729510*c[5]^3*c[4]^3-89076*c[5]^4-1455450*c[5]^5*c[ 4]+56376*c[4]^4+16999800*c[5]^6*c[4]^3+577715*c[5]^4*c[4]-65400*c[5]^2 *c[4]^2+54084*c[5]*c[4]^3-12960*c[4]^3-87000*c[5]^6+162245*c[5]^5+8700 00*c[5]^6*c[4]+870000*c[5]^7*c[4]^2), a[7,6] = -38912/1875*(5*c[5]*c[4 ]+1-2*c[4]-2*c[5])*(25*c[4]-24)*(25*c[5]-24)/(70*c[5]*c[4]-15*c[5]-15* c[4]+1)/(40*c[4]-27)/(40*c[5]-27), a[9,6] = 1024000/60021*(115*c[5]*c[ 4]-45*c[5]-45*c[4]+22)/(1600*c[5]*c[4]-1080*c[4]-1080*c[5]+729), a[9,7 ] = -390625/16416*(70*c[5]*c[4]-15*c[5]-15*c[4]+1)/(625*c[5]*c[4]-600* c[4]-600*c[5]+576), a[9,5] = -1/12*(261*c[4]-67)/c[5]/(1000*c[5]^3*c[4 ]-2635*c[5]^2*c[4]+2283*c[5]*c[4]-648*c[4]-1000*c[5]^4+2635*c[5]^3-228 3*c[5]^2+648*c[5]), a[5,4] = -c[5]^2*(c[4]-c[5])/c[4]^2\}:" }}}{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 98 "A len gthy computation gives an expression for the square of the principal e rror norm in terms of " }{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[5]" "6#&%\"cG6#\"\"&" }{TEXT -1 2 ". " }} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 261 "errterms6_8 := PrincipalErr orTerms(6,8,'expanded'):\nsm := 0:\nfor ct to nops(errterms6_8) do\n \+ print(ct);\n sm := sm+(simplify(subs(eB,errterms6_8[ct])))^2;\nend \+ do:\nsm := simplify(sm):\nprin_err_norm_sqrd := unapply(%,c[2],c[4],c[ 5]):\nprin_err_norm_sqrd(u,v,w);" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}{SECT 1 {PARA 4 "" 0 " " {TEXT -1 18 "prin_err_norm_sqrd" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 4269 "prin_err_norm_sqrd := (u,v ,w)->1/29393280000000000*(-18543889677600*v*u^2*w^2-17063300072923200* w^3*v^6-160530646225950*w^6*v-2444797630800000*u^2*v^4*w^5+24179619310 245000*v^4*u^2*w^4-3457174934430000*v^5*u^2*w^2-68631458679000*v^3*u^2 *w+386331034950*u^2*v^2-191834127120000*w*v^7+882165816000000*v^10*w^2 -3967050427200000*v^9*w^2+4568692132565640*v^5*w^3-468125211352770*w^5 *v^2+53985763395588000*w^5*v^6+29786772127766000*v^6*w^6+5286963150000 *w^5*u-5843256214723200*w^6*v^3+128774092167000*v*u^2*w^3+151262595288 00*v^2*u^2*w-31391922681000000*u^2*v^7*w^4+6864627042000000*u^2*v^8*w^ 4-35810863536000000*w^4*v^9-28486427661612000*w^6*v^5-254343789882000* w^7*v^4+13349490052500*w^8*v^4+455054623109640*w^5*v^3-38546928000000* u^2*v^2*w^5+350065800000000*u^2*v^6*w^6+3100276620000000*u^2*v^7*w^5-7 892568558000000*u^2*v^6*w^5+18692387998200000*u^2*v^7*w^3-229703040000 000*u^2*v^5*w^6+6400467234000000*u^2*v^5*w^5+54549930443400000*u^2*v^6 *w^4+37685088000000*u^2*v^4*w^6-1394750826000000*w^6*v^6*u-85065989400 0000*v^9*w^2*u-9152836056000000*v^9*w^4*u+116185301401050*v^2*u^2*w^2- 1002382483914000*v^2*u^2*w^3+79304447250000*u^2*v^6*w-1545324139800*u^ 2*w*v-209712471255000*u^2*v^5*w+4602424115205000*u^2*v^6*w^2-497651620 000500*v^3*u^2*w^2+50740815557700000*w^4*v^6*u+1759331840525*v^6+25421 3699589*v^4-1334713011120*v^5+1303635443589*w^4-6709381971120*w^5+8640 093840525*w^6+815645149200*w^3*u+47236978366452500*v^8*w^4+98577911250 00*u^2*w^4-5873905196909670*w^4*v^4-21737211267882000*w^4*v^7+27391038 921540000*v^5*u^2*w^3+1562353014096000*v^4*u^2*w^2-47418175101150000*u ^2*v^5*w^4-182066790150000*u^2*v*w^4+5580497916000000*v^9*w^3*u+140127 3270000000*w^6*v^5*u+3825370054800000*v^8*w^2*u-14111597685780000*v^4* u^2*w^3+4684882015737000*v^3*u^2*w^3-31834696127850000*u^2*v^6*w^3+172 345278325500*u^2*v^4*w+637994920500000*u^2*v^8*w^2+11614796986160100*w ^4*v^5+42964786518480*w^3*v^2-3876052230378*w^3*v-1681945938378*w*v^3+ 4187500157367*w^2*v^2-4185373437000000*u^2*v^8*w^3+1551467318310000*u^ 2*v^2*w^4+473447494800000*u^2*v^3*w^5-2869027541100000*u^2*v^7*w^2-276 8791863600*u*v*w^2+839367751959000*u*v^4*w^2+1158010562781000*u*v^3*w^ 3+34165646424000*u*v^2*w^2-1325145279928770*v^5*w^2+165225934270050*v^ 6*w+3439300591125450*v^6*w^2-24608220559179600*w^5*v^5-578718302400000 0*w^3*v^10+344516983312764*v^4*w^2-150217526394436*w^4*v^2+95675553915 3000*w^7*v^5+24701505494400000*v^9*w^3-63975311124047000*w^5*v^7-77342 27877180000*v^3*u^2*w^4+4987428790120100*w^5*v^4-5719791995806800*w^4* v^6-41064468943200000*v^8*w^3-1539347684805000*w^7*v^6+371471818900350 00*w^5*v^8+16701761806452*v^4*w-2669898010500*w^7*v^2+408661796970000* w^7*v^7-6075675790010500*v^7*w^2-136136399801152*w^3*v^3-1553698109680 5000*w^6*v^7-71450626368850*v^5*w-380538308884950*w^3*v^4+729030456000 00*w*v^8+6816923949360000*v^8*w^2+1436678484643050*w^4*v^3-40782257460 0*u*v^3-104454097965000*w^8*v^5-7805752920000*u^2*w^3-4153212792000*w^ 4*u-2473690119948*w^4*v+1290575309373450*w^6*v^2-55535792263920*w^2*v^ 3+9491829984000000*w^4*v^10-8310504384000000*v^9*w^5-2643481575000*v^5 *u+2076606396000*v^4*u+2464447781250*u^2*v^4-1951438230000*u^2*v^3+348 99296255976000*w^3*v^7+1545324139800*u^2*w^2+204330898485000*w^8*v^6+3 6910207656000*w^7*v^3+16276477615153200*w^6*v^4+3329644474485000*w^6*v ^8+94345385619150*w^5*v-115480819311300*u*v^2*w^3-208194866951400*u*v^ 3*w^2+87790197235500*u*v^4*w+1996129793700*u*v^2*w+194117350338000*w^4 *u*v^3-3791223636480000*w^5*u*v^3+808016029740000*w^5*u*v^2-1033663680 0000*w^6*u*v^2-5883059204712000*v^4*w^3*u+18284718709620000*v^5*u*w^3- 35783531388450000*v^6*u*w^3-95402626200000*v^7*u*w+10713996589230000*w ^5*u*v^4-6263993747340000*v^7*u*w^2+252015126660000*v^6*u*w+4981944902 220000*v^6*u*w^2+127428260400000*w^6*u*v^3-19774969538400*u*v^3*w-9788 6891340000*w^5*u*v+56152752822000*w^4*v*u-2438154088398000*v^5*u*w^2-1 7750837105100000*w^5*v^5*u+15052800369600000*w^5*v^6*u-646639315344000 00*w^4*v^7*u-590709672000000*w^5*v^8*u+4057325873280000*w^4*u*v^4-2751 98556888000*w^4*u*v^2+94617525600*w^3*u*v-212541872454000*v^5*u*w+4161 5483788800000*w^3*v^7*u-3942738828000000*v^7*w^5*u+333293940000000*w^6 *v^7*u-21114311674500000*w^4*v^5*u+39719732346000000*v^8*w^4*u-2460122 0418600000*v^8*w^3*u-613747335600000*w^6*v^4*u)/(10*w^3*v^2+10*w^2*v^3 -30*w^2*v^2+6*w^2*v-w^2+6*w*v^2+v*w-v^2)^2:" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 124 "Using a one dimensi onal minimization procedure and cycling around the nodes gives very sl ow convergence towards the minimum." }}{PARA 0 "" 0 "" {TEXT -1 19 "We use the values " }{XPPEDIT 18 0 "c[2] = 1/16;" "6#/&%\"cG6#\"\"#*&\" \"\"F)\"#;!\"\"" }{TEXT -1 3 ", " }{XPPEDIT 18 0 "c[4] = 1/5;" "6#/&% \"cG6#\"\"%*&\"\"\"F)\"\"&!\"\"" }{TEXT -1 7 " and " }{XPPEDIT 18 0 "c[5] = 8/15;" "6#/&%\"cG6#\"\"&*&\"\")\"\"\"\"#:!\"\"" }{TEXT -1 22 " as starting values. " }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 486 "D igits := 30:\nc_2 := 1/16: c_4 := 1/5: c_5 := 8/15:\nfor ct to 120000 \+ do\n c_2 := findmin(prin_err_norm_sqrd(c2,c_4,c_5),c2=\{0.03,c_2,0.1 \},convergence=location)[1];\n c_4 := findmin(prin_err_norm_sqrd(c_2 ,c4,c_5),c4=\{0.2,c_4,0.23\},convergence=location)[1];\n mn := findm in(prin_err_norm_sqrd(c_2,c_4,c5),c5=\{0.5,c_5,0.63\},convergence=loca tion); \n c_5 := mn[1]:\n if `mod`(ct,1000)=0 then\n print(c[ 2]=c_2,c[4]=c_4,c[5]=c_5);\n print(mn[2]);\n end if;\nend do:\n Digits := 10:" }}{PARA 11 "" 1 "" {XPPMATH 20 "6%/&%\"cG6#\"\"#$\"?]( \\+g!)>07D#)>Ioa&!#J/&F%6#\"\"%$\"?gS\\NN#\\w_taAM25#!#I/&F%6#\"\"&$\" ?n#GM'))pI'*Rp>!)y)\\&F1" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#$\"?/PeV:6 ,%pKi!*R1X\"!#Q" }}{PARA 11 "" 1 "" {XPPMATH 20 "6%/&%\"cG6#\"\"#$\"?] X,QPu+ouU%>geh&!#J/&F%6#\"\"%$\"??`q'>fJiQ^BW&*e7#!#I/&F%6#\"\"&$\"?'4 #**3OD&HR'4]^XXbF1" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#$\"?ozZ1$4CKuzLZ `ZO\"!#Q" }}{PARA 11 "" 1 "" {XPPMATH 20 "6%/&%\"cG6#\"\"#$\"?(4ylw:# \\)y5-F$[]c!#J/&F%6#\"\"%$\"?nj.4Cv\"=&3hP(yz8#!#I/&F%6#\"\"&$\"?B+,K \"f20ddf_.$obF1" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#$\"?KlBs#HT.C&H5&= \"Q8!#Q" }}{PARA 11 "" 1 "" {XPPMATH 20 "6%/&%\"cG6#\"\"#$\"?%QD)oN&oq )eEIZIrc!#J/&F%6#\"\"%$\"?Pz#z&)pj[-PoQo]9#!#I/&F%6#\"\"&$\"?S>9&HaDY] +VEV=e&F1" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#$\"??n:O^o.>p1^.=F8!#Q" } }{PARA 259 "" 0 "" {TEXT -1 1 ":" }}{PARA 259 "" 0 "" {TEXT -1 1 ":" } }{PARA 11 "" 1 "" {XPPMATH 20 "6%/&%\"cG6#\"\"#$\"?z71MERBX4oB*RUs&!#J /&F%6#\"\"%$\"?M%QZ&eXJ-EGPF\\i@!#I/&F%6#\"\"&$\"?Y#QuY=+qLD'yaa:cF1" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#$\"?8*))pV\\Rn&)Hnf#z:8!#Q" }}{PARA 11 "" 1 "" {XPPMATH 20 "6%/&%\"cG6#\"\"#$\"?%zI(o7SQA5oB*RUs&!#J/&F%6# \"\"%$\"?ndkrU^3FEGPF\\i@!#I/&F%6#\"\"&$\"?dl)p^XW`QD'yaa:cF1" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#$\"?M:MgB&Rn&)Hnf#z:8!#Q" }}{PARA 11 " " 1 "" {XPPMATH 20 "6%/&%\"cG6#\"\"#$\"?SN*)yJL/%3\"oB*RUs&!#J/&F%6#\" \"%$\"?!*H'Rf/#)ok#GPF\\i@!#I/&F%6#\"\"&$\"?()>\"=LRqiEGPF\\i@!#I/&F%6#\"\"&$\"?(>W(Hg4'[XD'yaa:cF1" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#$\"?^rKS_'Rn&)Hnf#z:8!#Q" }}}{PARA 0 " " 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 78 "The following grap hs give a visual check that we have found a (local) minimum." }} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 382 "c_2 := .57242399237e-1: pp \+ := .13157925967e-8:\np1 := evalf[30](plot(prin_err_norm_sqrd(c[2],.216 24927373,.56155454786),c[2]=0.052..0.0625,\n color=COLOR(RGB,.5,0,. 9))):\np2 := plot([[[c_2,pp]]$4],style=point,symbol=[circle$2,diamond, cross],symbolsize=[12,10$3],\n color=[black,red$3]):\nplots[ display]([p1,p2],font=[HELVETICA,9],view=[0.052..0.0625,1.3157e-9..1.3 1636e-9]);" }}{PARA 13 "" 1 "" {GLPLOT2D 367 347 347 {PLOTDATA 2 "6*-% 'CURVESG6$7S7$$\"#_!\"$$\"?9.Xl_F1$\"?u$p6!\\Gili>D3B;8F-7$$\"?++++++++v=!y RxG&F1$\"?o&HKuA'>KwCR))=;8F-7$$\"?+++++++]i!*4j<5`F1$\"?yy(3&G5+\"GV; :\\hJ\"F-7$$\"?+++++++]7.Xy(4L&F1$\"?D+Q,1/@uwjEU6;8F-7$$\"?+++++++]7y Hm^_`F1$\"?rD6k8!e18#Qg*zgJ\"F-7$$\"?+++++++]7yr?zu`F1$\"?)z'4%*yTMN4) =bYgJ\"F-7$$\"?+++++++]i:wg*pR&F1$\"?NmaF1$\"?$)olc.zgA]5-`)fJ\"F-7$$\"?++++++++v$*4F&*RaF1$\"?1pxpTzA Wo>t1'fJ\"F-7$$\"?++++++++]7%y*fiaF1$\"?.-pw3djk_*4'\\$fJ\"F-7$$\"?+++ +++++]P\\)R`[&F1$\"?>:%eJ\"F-7$$\"?+++++++]7`J==%f&F1$\"?La\\`@mzP_Qux#eJ\"F -7$$\"?+++++++++D&QA[h&F1$\"?gv:_O69Oi\"[\\-dF1$\"?')pHW)=j4Z5ug$z:8F-7$$\"?++++++++v$4t\"[CdF1$\"?q#o!R-Oo !\\^of#z:8F-7$$\"?+++++++]7yJ:jZdF1$\"?*4^Ai\"pxh_lMPz:8F-7$$\"?++++++ +](o9]$yndF1$\"?rGZ-b?X09')Rlz:8F-7$$\"?++++++++D\"3=[&*y&F1$\"?ieh'zS z<&)RoY,eJ\"F-7$$\"?+++++++++DCK.7eF1$\"?S7ASjb\\YH1E'3eJ\"F-7$$\"?+++ +++++Dc_/.MeF1$\"?WziIAu+bd)[m\")4PUeJ\"F-7$$\"? +++++++++DD'z,!fF1$\"?C1%fl]H$HJ9\"RSTK3=D\"[#*eJ\"F-7 $$\"?++++++++]7hF1$\"?e)RTaXflK@$pt6;8F-7$$\"?+++++++++D/ErRhF1$\"?YbrB EF%e?=Ee^hJ\"F-7$$\"?++++++++v$*f=(H;'F1$\"?M@)yf<+%e([;!H>;8F-7$$\"?+ ++++++++]:_x$='F1$\"?#[1a]B?6/.UwJiJ\"F-7$$\"?+++++++]7`Jf&f?'F1$\"?*H ^g\"z2@med#=viJ\"F-7$$\"?+++++++](opq&=FiF1$\"?'Gj^\\:)4YI5c'=jJ\"F-7$ $\"$D'!\"%$\"?x-4R)yV:!*eSYnjJ\"F--%&COLORG6&%$RGBG$\"\"&!\"\"$\"\"!Fb [l$\"\"*F`[l-F$6&7#7$$\"3C++qB*RUs&!#>$\"3.++q'f#z:8!#E-%'COLOURG6&F][ lFb[lFb[lFb[l-%'SYMBOLG6$%'CIRCLEG\"#7-%&STYLEG6#%&POINTG-F$6&Fg[l-F` \\l6&F][l$\"*++++\"!\")Fa[lFa[l-Fc\\l6$Fe\\l\"#5Fg\\l-F$6&Fg[lF]]l-Fc \\l6$%(DIAMONDGFd]lFg\\l-F$6&Fg[lF]]l-Fc\\l6$%&CROSSGFd]lFg\\l-%%FONTG 6$%*HELVETICAGFd[l-%+AXESLABELSG6%Q%c[2]6\"Q!Fg^l-F`^l6#%(DEFAULTG-%%V IEWG6$;F(Fez;$\"&dJ\"!#8$\"'O;8!#9" 1 2 0 1 10 0 2 9 1 4 2 1.000000 45.000000 45.000000 0 0 "Curve 1" "Curve 2" "Curve 3" "Curve 4" "Curve 5" }}}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 395 "c_4 := .21624927373: pp := .13157925967e-8:\np1 := e valf[30](plot(prin_err_norm_sqrd(.57242399237e-1,c[4],.56155454786),c[ 4]=0.2162435..0.216255,\n color=COLOR(RGB,0,.7,.2))):\np2 := plot([ [[c_4,pp]]$4],style=point,symbol=[circle$2,diamond,cross],symbolsize=[ 12,10$3],\n color=[black,cyan$3]):\nplots[display]([p1,p2],f ont=[HELVETICA,9],view=[0.2162435..0.216255,1.3157e-9..1.31636e-9]);" }}{PARA 13 "" 1 "" {GLPLOT2D 400 369 369 {PLOTDATA 2 "6*-%'CURVESG6$7S 7$$\"(NC;#!\"($\"?F]sy&*z'fNl%Q;O;8!#Q7$$\"?nmmmmm;HFn1vVi@!#I$\"?$=HY Vve7I?wQ8jJ\"F-7$$\"?LLLLL$3xE:xoRC;#F1$\"?-u&>nb%)))*f`PJF;8F-7$$\"?n mmmmmT5d_S@Wi@F1$\"?nX$\\C;#F1$\"?'))4lP]Fq(*QzI9hJ\"F-7$$\"?+++++]P4B?/.=' fJ\"F-7$$\"?++++++]P(G4wjC;#F1$\"?`^ym0\\NJ?r%>OfJ\"F-7$$\"?++++++]7t] ^iYi@F1$\"?EryXPq')G@L!f7fJ\"F-7$$\"?++++++]i7k^'oC;#F1$\"?%3Pz>&\\\\$ =cR%=*eJ\"F-7$$\"?nmmmm;/EG>J3Zi@F1$\"?Itu*QhhA9XkqueJ\"F-7$$\"?LLLLLL $eW!*GUtC;#F1$\"?Ck)zr$)=u@@!Qk&eJ\"F-7$$\"?LLLLLLL$G)R=cZi@F1$\"?X-qX _m%fi*HcF%eJ\"F-7$$\"?+++++]PMeHs\"yC;#F1$\"?r$*o2A4TQ5i7*GeJ\"F-7$$\" ?LLLLLLL3z#HV![i@F1$\"?nT$[u(R*4\"G==&=eJ\"F-7$$\"?+++++]P%e*>8H[i@F1$ \"?M&y#*3!>mwp%Q74eJ\"F-7$$\"?+++++]7`Z+v_[i@F1$\"?(z(=iKDkiC*R8-eJ\"F -7$$\"?LLLLLL3x1IRx[i@F1$\"?KE>Bp27GVrvoz:8F-7$$\"?LLLLL$3_g$H-+\\i@F1 $\"?tBBp&QTxs^#yQz:8F-7$$\"?nmmmmm\"z>PKW#\\i@F1$\"?wx@7v*[i6W:h#z:8F- 7$$\"?nmmmm;/w:py\\\\i@F1$\"?zh'\\Ek-KVVxX$z:8F-7$$\"?+++++]il?\"e=(\\ i@F1$\"?e4*yJzSo-y())fz:8F-7$$\"?nmmmmmTgEcp&*\\i@F1$\"?W#pIx@.>wz#*f+ eJ\"F-7$$\"?+++++++v-@K?]i@F1$\"?m1%\\$)Rd3P..T2eJ\"F-7$$\"?++++++v=0V TW]i@F1$\"?:CDhD=$=/d&)3;eJ\"F-7$$\"?+++++]P%)\\Zsn]i@F1$\"?-wP8'o\"Qt /2$QEeJ\"F-7$$\"?++++++DJCqg$4D;#F1$\"?y'3FOXc^hh7+SeJ\"F-7$$\"?LLLLLL L3*Rjo6D;#F1$\"?hqtf]&euy8D?aeJ\"F-7$$\"?++++++]7VXpT^i@F1$\"?e>)3aP\\ U!yD>9(eJ\"F-7$$\"?nmmmm;/E]_>k^i@F1$\"?#z:.`l!\\h!=H&)))eJ\"F-7$$\"?+ +++++]PmWz)=D;#F1$\"?UUZD,t\"*e%4t!*4fJ\"F-7$$\"?LLLLL$3x;TS>@D;#F1$\" ?7lr9hG&)p4S@;$fJ\"F-7$$\"?+++++](=PfMhBD;#F1$\"?3\"4H=$RyDlq#HcfJ\"F- 7$$\"?LLLLLL$e*G))yf_i@F1$\"?(=o+#4SH\"*=$[O#)fJ\"F-7$$\"?+++++]P4@Eb% GD;#F1$\"?R)H0eXp8k+#G<,;8F-7$$\"?nmmmmmm;'4.%3`i@F1$\"?FCyO5i+i*R'4?/ ;8F-7$$\"?nmmmmm\"HF5$prwr)R3]2;8F-7$$\"?LLLLL$3x6C#) pND;#F1$\"?,(G*yHu^38!Qw4hJ\"F-7$$\"?+++++++vU!4#z`i@F1$\"?8Sos0ryS\\U $[VhJ\"F-7$$\"?LLLLLLeklMo/ai@F1$\"?-b9M+8&\\ov5B%=;8F-7$$\"?nmmmmmm;$ 4ouUD;#F1$\"?G$o3EfFe$*G%yDA;8F-7$$\"?+++++]PMe7w^ai@F1$\"?R.c^E'=o$p7 WaE;8F-7$$\"?+++++]i:A)=jQ3jJ\"F-7$$\"'bi@!\"'$ \"?cn)*\\c<'>_A " 0 "" {MPLTEXT 1 0 389 "c_5 := .56155454786: pp := .13157925967e-8:\np1 := evalf[30](plot (prin_err_norm_sqrd(.57242399237e-1,.21624927373,c[5]),c[5]=0.5615433. .0.5615658,\n color=COLOR(RGB,0.6,.2,.2))):\np2 := plot([[[c_5,pp] ]$4],style=point,symbol=[circle$2,diamond,cross],symbolsize=[12,10$3], color=[black,green$3]):\nplots[display]([p1,p2],font=[HELVETICA,9],vie w=[0.5615433..0.5615658,1.3157e-9..1.31636e-9]);" }}{PARA 13 "" 1 "" {GLPLOT2D 389 359 359 {PLOTDATA 2 "6*-%'CURVESG6$7S7$$\"(Lah&!\"($\"?/ 73Hm;YzV^gIO;8!#Q7$$\"?++++++](ofV!zV:c!#I$\"?*G'H2\\6%)GXsTVJ;8F-7$$ \"?+++++]7Gohr@W:cF1$\"?fDOUp,D\"QeYstiJ\"F-7$$\"?++++++D\"y$fqpW:cF1$ \"?]>i0#p!o^?\"o,IiJ\"F-7$$\"?++++++vo9Q,=X:cF1$\"?(yV#**okEwYBD\")=;8 F-7$$\"?+++++]il#4#4mX:cF1$\"?HJ-MdE'y34+`[hJ\"F-7$$\"?+++++]7y5om5Y:c F1$\"?+yyG60Tp*)p)o8hJ\"F-7$$\"?+++++]7`\\8#olah&F1$\"?')*ptgInuA_l]zg J\"F-7$$\"?+++++]7`RWb/Z:cF1$\"?#)QW*>mJaC\"p\"=YgJ\"F-7$$\"?+++++]i!* [W8_Z:cF1$\"?pS6Q:zUWDg8],;8F-7$$\"?++++++](oLw5![:cF1$\"?XYVr-!45SM!) 3&)fJ\"F-7$$\"?++++++vVyV=W[:cF1$\"?9yL5*zR_M8@_gfJ\"F-7$$\"?++++++]7B Qr#*[:cF1$\"?!G3PET0ok;]([$fJ\"F-7$$\"?++++++]PMDWT\\:cF1$\"?d3tk>QJ\" =+AE6fJ\"F-7$$\"?++++++](on,%))\\:cF1$\"?s(=m&>R3eJ \"F-7$$\"?+++++]PM(RTOJbh&F1$\"?4(f#z:8F-7$$\"?+++++]7`Rh \\.b:cF1$\"?Z?.:XM5]E#Qm$z:8F-7$$\"?+++++](=<$*ymabh&F1$\"?2ObLM\\^qj< %R'z:8F-7$$\"?++++++DJIvJ$fbh&F1$\"?)oJx33ek*\\;E7!eJ\"F-7$$\"?+++++++ Dm(*\\Tc:cF1$\"?s\\%e8K$)H^17F3eJ\"F-7$$\"?++++++D1%oO')obh&F1$\"?D1zx >gdfo\">=)RxvdVt:%eJ\"F-7$$\"?+++++++D6\\QIe:cF1$\"? prFEu'RQ,[***f&eJ\"F-7$$\"?++++++]P%en*ye:cF1$\"?kJ*e-DmO'3CaM(eJ\"F-7 $$\"?+++++]7.22*H#f:cF1$\"?PPirTRlk%*4*4\"*eJ\"F-7$$\"?++++++]73'>6(f: cF1$\"?v%RAy\"Q7WxSyB\"fJ\"F-7$$\"?+++++]7G`^S;g:cF1$\"?;$4f(3>3,r'pHM fJ\"F-7$$\"?+++++]iS'fTP1ch&F1$\"?'Rv\"HI:*4^HIDZeh:cF1$\"?lL]szwL3 @U%)\\,;8F-7$$\"?+++++++]#\\O^?ch&F1$\"?vjcG4e2QrHJa/;8F-7$$\"?++++++v oWr&GDch&F1$\"?UR'y,nfk8IOeygJ\"F-7$$\"?+++++]7yXE=+j:cF1$\"?mH3vT/)p> ZTZ8hJ\"F-7$$\"?+++++++Dm)pOMch&F1$\"?p7B/kC)z'=Y,t9;8F-7$$\"?++++++vV G6^$Rch&F1$\"?5Q2fp%=]$4]^\")=;8F-7$$\"?+++++++]Z(*3Qk:cF1$\"?SFhd8ll` h$*plA;8F-7$$\"?+++++]7G`)>c[ch&F1$\"?&H>4!Rf^?Fa)[piJ\"F-7$$\"?+++++] (=#HA6Jl:cF1$\"?wm[)y`lR#[zdCJ;8F-7$$\"(ech&F*$\"?T)3$)**=lQ/A\\pgjJ\" F--%&COLORG6&%$RGBG$\"\"'!\"\"$\"\"#F_[lF`[l-F$6&7#7$$\"3J++gyaa:c!#=$ \"3.++q'f#z:8!#E-%'COLOURG6&F\\[l\"\"!F_\\lF_\\l-%'SYMBOLG6$%'CIRCLEG \"#7-%&STYLEG6#%&POINTG-F$6&Fd[l-F]\\l6&F\\[l$F_\\lF_\\l$\"*++++\"!\") F]]l-Fa\\l6$Fc\\l\"#5Fe\\l-F$6&Fd[lF[]l-Fa\\l6$%(DIAMONDGFc]lFe\\l-F$6 &Fd[lF[]l-Fa\\l6$%&CROSSGFc]lFe\\l-%%FONTG6$%*HELVETICAG\"\"*-%+AXESLA BELSG6%Q%c[5]6\"Q!Fg^l-F_^l6#%(DEFAULTG-%%VIEWG6$;F(Fez;$\"&dJ\"!#8$\" 'O;8!#9" 1 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5B3F030498321128103F03049A4B8B85A43F0304A021305DE63F0304A9B5C98C3D3F03 04B7070254D43F0304C814A5ACC73F0304DCDDBAFB493F0304F5633A64C53F030511A2 CB0F443F0305319C2005573F0306757FF6E2A13F03063B67F6CECE3F03060517E698A3 3F0305D28DC304A03F0305A3C9B5F4D13F030578CACF8DB53F03055191DC246A3F0305 2E1C484F753F03050E6B8E80273F0304F27EAE6E873F0304DA5382F0403F0304C5EB96 FF2D3F0304B5453586FF3F0304A8606156513F03049F3C1383023F030499D86981833F 030498336FC93D3F03049A4DF5951B3F0304A0270A82693F0304A9BE341FBF3F0304B7 121D275D3F0304C822DA094D3F0304DCEF20DA6E3F0304F576EE6CFA3F030511B9FB9D 973F0306B346028AAD3F03067568503C093F03063B514924D33F03060501C299113F03 05D278AA1E7A3F0305A3B60E5AD03F030578B858723F3F0305518064512A3F03052E0C 2A320C3F03050E5D302AD03F0304F271DAB4E13F0304DA4880B68D3F0304C5E1B12576 3F0304B53CB6D74B3F0304A85A5E1BE33F03049F382FB52D3F030499D6C142483F0304 9834A6B8383F03049A517D50713F0304A02D3D06A53F0304A9C69671D13F0304B71D52 B83C3F0304C830BCFE3B3F0304DD009DFA323F0304F58BE77D3A3F0306F4D2F20DF13F 0306B32DC25FB63F03067550267F5E3F03063B3A79A3813F030604EB3ACB2C3F0305D2 634C98F63F0305A3A186F1023F030578A53FC9593F0305516DB78D7C3F03052DFC5784 703F03050E4E0603DB3F0304F2641C48293F0304DA3D275B203F0304C5D8590F7F3F03 04B535FC293F3F0304A8549B54883F03049F3578E41F3F030499D69774283F03049835 F00E383F03049A5536940A3F0304A0337A83253F0304A9CFF7C3A23F0304B7290B8C58 3F0304C83FE8B2673F0304DD12B9C5FA3F03073A27E8C6F93F0306F4BAD0476F3F0306 B3156412883F03067538B84CDF3F03063B243EE84F3F030604D5CFC1E13F0305D24EEC D3223F0305A38EAC12753F03057892F2815C3F0305515D9C63793F03052DECB5242F3F 03050E4084A4533F0304F257BCF1773F0304DA31E755C33F0304C5CF64235F3F0304B5 2EDB2B0E3F0304A84FFDC4283F03049F31FB2CED3F030499D4A351F83F03049836DB72 E43F03049A59861CCE3F0304A03A42C7393F0304A9D8F3FA173F0304B7347E679F3F03 04C84E14E5363F03078345936F4D3F03073A0FFE0E893F0306F4A280C9993F0306B2FD B5A9BE3F03067521B419C43F03063B0D4EB2343F030604C0812B163F0305D23A2964BA 3F0305A37AAB7BF63F030578810F9AAD3F0305514C677C5B3F03052DDD5E440B3F0305 0E3290BBC53F0304F24BD020693F0304DA271DC7593F0304C5C62BA13F3F0304B527A9 259A3F0304A84AEA89AD3F03049F2F49A6793F030499D4585A6D3F03049838FBDC093F 03049A5D87692B3F0304A040B15CF33F0304A9E2318F813F0304B741B9CAA83F0307D0 2CBBC8543F0307832D0B30153F030739F71B5A813F0306F48A5D01003F0306B2E684AD A23F0306750A91E4583F03063AF7C52D7A3F030604ABBBD9983F0305D225EB62E33F03 05A367B090163F0305786FAE6F243F0305513C347FE63F03052DCDA4374D3F03050E24 701F063F0304F23F1B86413F0304DA1CA00E993F0304C5BD717C993F0304B52110C1DD 3F0304A8468471C53F03049F2CB9CF7E3F030499D4711ED93F0304983B0040B23F0304 9A61ECADD43F0304A04770AE863F0304A9EB7A53763F030820DD9D6C1E3F0307D0140F DFF53F0307831458FEE03F030739DEB9DF7E3F0306F4732B4AED3F0306B2CFBDBB293F 030674F42B62313F03063AE20676E73F030604966FCF713F0305D21278CE383F0305A3 54DA14DD3F0305785DD2910A3F0305512B8AD20B3F03052DBF1305303F03050E1707D8 223F0304F232F0C08F3F0304DA12FDCAC93F0304C5B554CDC03F0304B51A8E2C763F03 04A8425EF71B3F03049F2A5F101B3F030499D45679223F0304983D22C30F3F03049A66 8775FB3F0304A04F3ABC443F03087558FDDD583F030820C54009453F0307CFFBC60DC0 3F030782FC280AB93F030739C780F31A3F0306F45B4F763F3F0306B2B8C933753F0306 74DE1F72B53F03063ACC0AC41B3F030604821609A73F0305D1FF3A2C013F0305A34262 58273F0305784C68F0123F0305511B47CE5F3F03052DB00F27583F03050E0999BA253F 0304F2273893F23F0304DA08C65BFF3F0304C5AD3E4F823F0304B51453D89B3F0304A8 3D7A1FF23F03049F286BD95C3F030499D3F1220B3F0304983F065EB93F03049A6B1DA6 1B3F0308CDA0C492483F03087540C592203F030820AD0A5BE53F0307CFE3BB562F3F03 0782E4ED1F4F3F030739AF8296B23F0306F4444E750B3F0306B2A236FC2B3F030674C8 B420133F03063AB7053A363F0306046D5889113F0305D1EB4E13BD3F0305A32FE45B04 3F0305783BD69E863F0305510C0AC6853F03052DA1B41B6B3F03050DFC96631D3F0304 F21C5BE13A3F0304D9FF7DE1253F0304C5A58365903F0304B50E7535AB3F0304A839ED 35C33F03049F268AE2E23F030499D3BD2A3D3F03049842976E37-%&COLORG6&%$RGBG$ \"#X!\"#$\"\"!F>$\"#&*F<-%%FONTG6$%*HELVETICAG\"\"*-%*AXESSTYLEG6#%$BO XG-%+AXESLABELSG6%%%c[4]G%%c[5]G%!G-%+PROJECTIONG6%$!#yF>$\"#cF>\"\"\" " 1 2 0 1 10 0 2 1 1 2 2 1.000000 56.000000 -78.000000 1 0 "Curve 1" } }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 36 "#----- ------------------------------" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 151 "nds := [c[2]=.57242399237e- 1,c[4]=.21624927373,c[5]=.56155454786]:\nevalf[10](%);\nfor dgt from 7 by -1 to 4 do\n map(convert,nds,rational,dgt);\nend do;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#7%/&%\"cG6#\"\"#$\"+C*RUs&!#6/&F&6#\"\"%$\"+ PF\\i@!#5/&F&6#\"\"&$\"+zaa:cF2" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#7%/ &%\"cG6#\"\"##\"$:\"\"%4?/&F&6#\"\"%#\"$a$\"%P;/&F&6#\"\"&#\"$g#\"$j% " }}{PARA 11 "" 1 "" {XPPMATH 20 "6#7%/&%\"cG6#\"\"##\"#m\"%`6/&F&6#\" \"%#\"$t\"\"$+)/&F&6#\"\"&#\"$g#\"$j%" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#7%/&%\"cG6#\"\"##\"#<\"$(H/&F&6#\"\"%#\"$t\"\"$+)/&F&6#\"\"&#\"#t\" $I\"" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#7%/&%\"cG6#\"\"##\"#:\"$i#/&F& 6#\"\"%#\"\")\"#P/&F&6#\"\"&#\"#K\"#d" }}}{PARA 0 "" 0 "" {TEXT -1 0 " " }}{PARA 0 "" 0 "" {TEXT -1 56 "The minimum value for the principal e rror norm is . . . " }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 86 "evalf [25](prin_err_norm_sqrd(.5724239924e-1,.2162492737,.5615545479)):\neva lf(sqrt(%));" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#$\"+cbQFO!#9" }}} {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 9 "Setting \+ " }{XPPEDIT 18 0 "c[4] = 173/800;" "6#/&%\"cG6#\"\"%*&\"$t\"\"\"\"\"$+ )!\"\"" }{TEXT -1 7 " and " }{XPPEDIT 18 0 "c[5] = 260/463;" "6#/&% \"cG6#\"\"&*&\"$g#\"\"\"\"$j%!\"\"" }{TEXT -1 46 " the principal erro r norm is a minimum when " }{XPPEDIT 18 0 "c[2] = 33/577;" "6#/&%\"cG 6#\"\"#*&\"#L\"\"\"\"$x&!\"\"" }{TEXT -1 19 " (approximately). " }} {PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 113 "mn := evalf[20](findmin(prin_err_norm_sqrd(c2,173/800,260/463),c2 =0.04..0.08)):\nc[2]=mn[1];convert(%,rational,5);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/&%\"cG6#\"\"#$\"5/hGJ&3\"e5>d!#@" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/&%\"cG6#\"\"##\"#L\"$x&" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 6 "When " }{XPPEDIT 18 0 "c[2] = 33/5 77;" "6#/&%\"cG6#\"\"#*&\"#L\"\"\"\"$x&!\"\"" }{TEXT -1 3 ", " } {XPPEDIT 18 0 "c[4] = 173/800;" "6#/&%\"cG6#\"\"%*&\"$t\"\"\"\"\"$+)! \"\"" }{TEXT -1 7 " and " }{XPPEDIT 18 0 "c[5] = 260/463;" "6#/&%\"c G6#\"\"&*&\"$g#\"\"\"\"$j%!\"\"" }{TEXT -1 66 ", the principal error \+ norm is given (approximately) as follows. " }}{EXCHG {PARA 0 "> " 0 " " {MPLTEXT 1 0 70 "evalf[20](prin_err_norm_sqrd(33/577,173/800,260/463 )):\nevalf(sqrt(%));" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#$\"+\"R!RFO!#9 " }}}{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 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 comb ined scheme" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 " " {MPLTEXT 1 0 1879 "ee := \{c[2]=33/577,\nc[3]=173/1200,\nc[4]=173/80 0,\nc[5]=260/463,\nc[6]=27/40,\nc[7]=24/25,\nc[8]=1,\nc[9]=1,\n\na[2,1 ]=33/577,\na[3,1]=-3567433/95040000,\na[3,2]=17269033/95040000,\na[4,1 ]=173/3200,\na[4,2]=0,\na[4,3]=519/3200,\na[5,1]=1877748548260/2970538 457863,\na[5,2]=0,\na[5,3]=-7126513680000/2970538457863,\na[5,4]=69168 86080000/2970538457863,\na[6,1]=-65244579431964655428477/6965324619991 5888640000,\na[6,2]=0,\na[6,3]=2694429616783931433/669742751922268160, \na[6,4]=-62138409816102019904637/21415191928402504983040,\na[6,5]=145 922219908058645309313/297661794320406364160000,\na[7,1]=38907542822607 41440349323/3010236956970547048875000,\na[7,2]=0,\na[7,3]=-45910927628 476367208/8576173666582755125,\na[7,4]=5986037178684297350750710688/12 07688208130690660530130125,\na[7,5]=-3106990980562910309356750843/3003 073489062493679278875000,\na[7,6]=2999575140352/2729866330125,\na[8,1] =3914147173803753655861/1444606034961051585792,\na[8,2]=0,\na[8,3]=-22 76990272364785725/192922814498003684,\na[8,4]=242191852489325161889481 75/2263933167309766272933307,\na[8,5]=-4801005224579354456654139837/16 61667052909362562177244416,\na[8,6]=232855940800/97230777633,\na[8,7]= -57039375/516949856,\na[9,1]=7466867/116588160,\na[9,2]=0,\na[9,3]=0, \na[9,4]=3227648000000/9720313182027,\na[9,5]=179724568369012721/78494 6257231456320,\na[9,6]=234368000/1186672113,\na[9,7]=16390625/58886016 ,\na[9,8]=-14011/138852,\n\nb[1]=7466867/116588160,\nb[2]=0,\nb[3]=0, \nb[4]=3227648000000/9720313182027,\nb[5]=179724568369012721/784946257 231456320,\nb[6]=234368000/1186672113,\nb[7]=16390625/58886016,\nb[8]= -14011/138852,\n\n`b*`[1]=3384682763489201/53914632194067840,\n`b*`[2] =0,\n`b*`[3]=0,\n`b*`[4]=72143477722071040000/214048979481496847913,\n `b*`[5]=24566463552585428091992041/120995959197297938812618560,\n`b*`[ 6]=125185218908556800/548760616063863687,\n`b*`[7]=2889850698904375/11 670449557585536,\n`b*`[8]=-22856669025961/321051233221740,\n`b*`[9]=-1 /135\}:" }}}{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 compone nts are the principal error terms of the embedded 9 stage, order 5 sch eme (the error terms of order 6) and let " }{XPPEDIT 18 0 "`T*`[6,9]; " "6#&%#T*G6$\"\"'\"\"*" }{TEXT -1 99 " denote the vector whose compo nents are the error terms of order 7 of the embedded order 5 scheme." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 39 "Denote \+ the 2-norm of these vectors by " }{XPPEDIT 18 0 "abs(abs(T[6,8]));" " 6#-%$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#\"\"(-%$absG6#-F)6#&%\"TG6$\"\"'\"\")" }{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$\"\"&F3F9 " }{TEXT -1 1 "." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 77 "Dormand and Prince have suggested that as well as attempt ing to ensure that " }{XPPEDIT 18 0 "A[7];" "6#&%\"AG6#\"\"(" }{TEXT -1 73 " is a minimum, if the embedded scheme is to be used for error \+ control, " }{XPPEDIT 18 0 "B[7];" "6#&%\"BG6#\"\"(" }{TEXT -1 7 " an d " }{XPPEDIT 18 0 "C[7];" "6#&%\"CG6#\"\"(" }{TEXT -1 27 " should b e chosen so that " }}{PARA 0 "" 0 "" {TEXT -1 68 "they are \"similar i n magnitude\" and also not differ too much from 1." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 186 "errterms6_8 := PrincipalErrorTerms(6,8,'expanded'):\n`errterms6_9*` :=subs(b=`b*` ,PrincipalErrorTerms(6,9,'expanded')):\n`errterms5_9*` := subs(b=`b*`, PrincipalErrorTerms(5,9,'expanded')):" }}}{PARA 0 "" 0 "" {TEXT -1 0 " " }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 326 "snmB := sqrt(add(evalf( subs(ee,`errterms6_9*`[i]))^2,i=1..nops(`errterms6_9*`))):\nsdnB := sq rt(add(evalf(subs(ee,`errterms5_9*`[i]))^2,i=1..nops(`errterms5_9*`))) :\nsnmC := sqrt(add((evalf(subs(ee,`errterms6_9*`[i])-subs(ee,errterms 6_8[i])))^2,i=1..nops(errterms6_8))):\n'B[7]'= evalf[8](snmB/sdnB);\n' C[7]'= evalf[8](snmC/sdnB);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/&%\"BG 6#\"\"($\")s&3I\"!\"(" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/&%\"CG6#\"\" ($\")r\"pJ\"!\"(" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}}{PARA 0 "" 0 "" {TEXT -1 52 "#--------- ------------------------------------------" }}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 26 "absolute stability regions" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}{SECT 1 {PARA 4 " " 0 "" {TEXT -1 35 "coefficients of the combined scheme" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1879 "ee := \{c[2]=33/577,\nc[3]=173/1200,\nc[4]=173/800,\nc[5]=260/463,\nc[6]=27 /40,\nc[7]=24/25,\nc[8]=1,\nc[9]=1,\n\na[2,1]=33/577,\na[3,1]=-3567433 /95040000,\na[3,2]=17269033/95040000,\na[4,1]=173/3200,\na[4,2]=0,\na[ 4,3]=519/3200,\na[5,1]=1877748548260/2970538457863,\na[5,2]=0,\na[5,3] =-7126513680000/2970538457863,\na[5,4]=6916886080000/2970538457863,\na [6,1]=-65244579431964655428477/69653246199915888640000,\na[6,2]=0,\na[ 6,3]=2694429616783931433/669742751922268160,\na[6,4]=-6213840981610201 9904637/21415191928402504983040,\na[6,5]=145922219908058645309313/2976 61794320406364160000,\na[7,1]=3890754282260741440349323/30102369569705 47048875000,\na[7,2]=0,\na[7,3]=-45910927628476367208/8576173666582755 125,\na[7,4]=5986037178684297350750710688/1207688208130690660530130125 ,\na[7,5]=-3106990980562910309356750843/3003073489062493679278875000, \na[7,6]=2999575140352/2729866330125,\na[8,1]=3914147173803753655861/1 444606034961051585792,\na[8,2]=0,\na[8,3]=-2276990272364785725/1929228 14498003684,\na[8,4]=24219185248932516188948175/2263933167309766272933 307,\na[8,5]=-4801005224579354456654139837/166166705290936256217724441 6,\na[8,6]=232855940800/97230777633,\na[8,7]=-57039375/516949856,\na[9 ,1]=7466867/116588160,\na[9,2]=0,\na[9,3]=0,\na[9,4]=3227648000000/972 0313182027,\na[9,5]=179724568369012721/784946257231456320,\na[9,6]=234 368000/1186672113,\na[9,7]=16390625/58886016,\na[9,8]=-14011/138852,\n \nb[1]=7466867/116588160,\nb[2]=0,\nb[3]=0,\nb[4]=3227648000000/972031 3182027,\nb[5]=179724568369012721/784946257231456320,\nb[6]=234368000/ 1186672113,\nb[7]=16390625/58886016,\nb[8]=-14011/138852,\n\n`b*`[1]=3 384682763489201/53914632194067840,\n`b*`[2]=0,\n`b*`[3]=0,\n`b*`[4]=72 143477722071040000/214048979481496847913,\n`b*`[5]=2456646355258542809 1992041/120995959197297938812618560,\n`b*`[6]=125185218908556800/54876 0616063863687,\n`b*`[7]=2889850698904375/11670449557585536,\n`b*`[8]=- 22856669025961/321051233221740,\n`b*`[9]=-1/135\}:" }}}{PARA 0 "" 0 " " {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}} {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 77 "The stabi lity function R for the 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)*&#\"3h-M'ze-(f7\"6+++_B<2$yWkF )*$)F'\"\"(F)F)F)*&#\"126G46u]X\"6++g0<:#\\VL>F)*$)F'\"\")F)F)F)" }}} {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 124 "We can f ind the point where the boundary of the stability region intersects th e negative real axis by solving the equation: " }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "R(z) = 1;" "6#/-%\"RG6#%\"zG\"\"\"" } {TEXT -1 2 ". " }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 28 "z0 := newt on(R(z)=1,z=-4.4);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%#z0G$!+RU\\]V! \"*" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 303 "z0 := newton(R(z)=1,z=-4.4):\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 360 253 253 {PLOTDATA 2 "6+-%'CURVESG6$7Z7$$!3A++++++!*\\!#< $\"3&ROIz\"3ZF*$\"3QtC:Q-Fj>F*7$$!3imm,F%Q(\\YF*$\"3 '*4?09j(Qw\"F*7$$!3Gnm1o,\"4f%F*$\"3M)4na)Rl\"e\"F*7$$!33nm64>3KXF*$\" 3oP*4T:'\\;9F*7$$!3I+Dh]K`tWF*$\"3qO*=hG)on7F*7$$!3\\L$3@f%)\\T%F*$\"3 pu_*>\"=8L6F*7$$!3jm;u*Q?kI%F*$\"3sA'QUw#yt\"*!#=7$$!3E+]ZY%3S>%F*$\"3 B(QC>.z9M(Fho7$$!3%omTR&=vxSF*$\"3kz'4loqy!eFho7$$!3/+]x(4o='RF*$\"3'* pq,sb*Re%Fho7$$!3OF*$\"3)f&peXh'zF#Fho7$$!3?++]Qxz+NF*$\"3DUE mht63=Fho7$$!3/++5QhU'Q$F*$\"3#>%zZ'))=&o9Fho7$$!3mm;%zwlDG$F*$\"3Cyfv $pp'Q7Fho7$$!3[LLBUd1fJF*$\"3$=.[A&oUW5Fho7$$!3eLL$4-XW0$F*$\"39d5K)>V ![$*!#>7$$!3;+]n]iuKHF*$\"3%yGLu6&p'e)F`s7$$!3ILL$z@A]#GF*$\"3#pSKJKUD H)F`s7$$!3!)**\\n!)=$oq#F*$\"3vQ-yY=-4$)F`s7$$!3')**\\-InG%f#F*$\"3;x= 8F@(Qg)F`s7$$!3ILL3sw&oZ#F*$\"3+pry3:Ki\"*F`s7$$!3#HL3&R6-pBF*$\"3MP%o %y2J\">F*$ \"3kV<]9Ch%[\"Fho7$$!35++?e%pdz\"F*$\"3u$)[`gQ$\\m\"Fho7$$!3+++:w['4o \"F*$\"3-O#))4yyZ'=Fho7$$!3-+]()Rb))p:F*$\"3;k:.SUH#3#Fho7$$!3#)***\\a (3bY9F*$\"3e(=Tw$HiaBFho7$$!3cLL$R>HdL\"F*$\"3=N3MNz3IEFho7$$!3z****\\ %R.u@\"F*$\"3'>ZIyqx,'HFho7$$!3pm;aLE=56F*$\"3Q*fU0W$3&H$Fho7$$!3E%*** *4@?'H**Fho$\"3*)Qq;lm\"[q$Fho7$$!3MML3+cmE))Fho$\"3Pa/.3$*zOTFho7$$!3 %H**\\(H-wtwFho$\"3q?!Q)e9IUYFho7$$!3cOLL)\\%eYlFho$\"3a4#3)\\n>'>&Fho 7$$!3g.+vof`m`Fho$\"3'RDBJo;q%eFho7$$!3Dkmm#)*3+B%Fho$\"3k\"*yz;0y]lFh o7$$!3Yjm;()funIFho$\"3K5x0pV;etFho7$$!3HBL3;r5:>Fho$\"3#4w^R!y5d#)Fho 7$$!3_^****>q^f&)F`s$\"3-3Y5fxlz\"*Fho7$$\"3OMnm\"G*fzNF`s$\"3Q#QAGQWk .\"F*7$$\"3\"RLL8'ppV9Fho$\"3gi=kV6Jb6F*7$$\"3@0+D$4>8g#Fho$\"3oK?%[<, rH\"F*7$$\"3Q,+v#=6$4PFho$\"3!3,44D$3\\9F*7$$\"3!***************[Fho$ \"3Bk^,\">;Bj\"F*-%'COLOURG6&%$RGBG$\"*++++\"!\")$\"\"!Fd]lFc]l-F$6$7S 7$F($\"\"\"Fd]l7$F=Fi]l7$FGFi]l7$FQFi]l7$FenFi]l7$F_oFi]l7$FdoFi]l7$Fj oFi]l7$F_pFi]l7$FdpFi]l7$FipFi]l7$F^qFi]l7$FcqFi]l7$FhqFi]l7$F]rFi]l7$ FbrFi]l7$FgrFi]l7$F\\sFi]l7$FbsFi]l7$FgsFi]l7$F\\tFi]l7$FatFi]l7$FftFi ]l7$F[uFi]l7$F`uFi]l7$FeuFi]l7$FjuFi]l7$F_vFi]l7$FdvFi]l7$FivFi]l7$F^w Fi]l7$FcwFi]l7$FhwFi]l7$F]xFi]l7$FbxFi]l7$FgxFi]l7$F\\yFi]l7$FayFi]l7$ FfyFi]l7$F[zFi]l7$F`zFi]l7$FezFi]l7$FjzFi]l7$F_[lFi]l7$Fd[lFi]l7$Fi[lF i]l7$F^\\lFi]l7$Fc\\lFi]l7$Fh\\lFi]l-F]]l6&F_]lFc]lFc]lF`]l-F$6&7#7$$! 3m******QU\\]VF*Fi]l-%'SYMBOLG6#%'CIRCLEG-F]]l6&F_]lFd]lFd]lFd]l-%&STY LEG6#%&POINTG-F$6&F_al-Fdal6#%&CROSSGFgalFial-F$6&F_al-Fdal6#%(DIAMOND GFgalFial-F$6%7$7$FaalFc]lF`al-%&COLORG6&F_]lFc]l$\"\"&!\"\"Fc]l-%*LIN ESTYLEG6#\"\"$-%%FONTG6$%*HELVETICAG\"\"*-%+AXESLABELSG6%Q\"z6\"Q!F^dl -Ffcl6#%(DEFAULTG-%%VIEWG6$;$!$*\\!\"#$\"#\\Fidl;$!\"(Fidl$\"$Z\"Fidl " 1 2 0 1 10 0 2 9 1 4 2 1.000000 45.000000 45.000000 0 0 "Curve 1" "C urve 2" "Curve 3" "Curve 4" "Curve 5" "Curve 6" }}}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 50 "The following picture sho ws the stability region. " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1381 "R := z -> 1+z+1/2*z^2+1/6*z^3+1/2 4*z^4+1/120*z^5+1/720*z^6+\n 125970258796340261/644478307172352000 000*z^7+4550741109281107/193343492151705600000*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,.23,.13,.48)):\np2 := plots[polygonplot]([se q([pts[i-1],pts[i],[-2.2,0]],i=2..nops(pts))],\n style=patchn ogrid,color=COLOR(RGB,.45,.25,.95)):\npts := []: z0 := 2+4.75*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(zz)]]:\nend do:\np3 := plot(pts ,color=COLOR(RGB,.23,.13,.48)):\np4 := plots[polygonplot]([seq([pts[i- 1],pts[i],[1.97,4.73]],i=2..nops(pts))],\n style=patchnogrid, color=COLOR(RGB,.45,.25,.95)):\npts := []: z0 := 2-4.75*I:\nfor ct fro m 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(zz)]]:\nend do:\np5 := plot(pts,color =COLOR(RGB,.23,.13,.48)):\np6 := plots[polygonplot]([seq([pts[i-1],pts [i],[1.97,-4.73]],i=2..nops(pts))],\n style=patchnogrid,color =COLOR(RGB,.45,.25,.95)):\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))*****zC_[Q\"!#F$\"3()*****fk#fTJF-7$$!3!)*****4^ Zt(G!#E$\"3:+++f))Q7ZF-7$$!37+++f+ks>!#D$\"33+++eX=$G'F-7$$!35+++\"[B- B&F=$\"3++++o)yR&yF-7$$\"3m+++O?6(Q*F=$\"3[+++$yqZU*F-7$$\"3-+++h\"[6] \"!#B$\"33+++`fb*4\"!#<7$$\"36+++T#R+!zFM$\"31+++sTjc7FP7$$\"31+++F&o' =H!#A$\"3*******zk4PT\"FP7$$\"3O+++N7BA()FY$\"3#******z9u2d\"FP7$$\"3. +++x$ftB#!#@$\"36+++FK!ys\"FP7$$\"39+++4o$f3&F^o$\"3)******4uOZ)=FP7$$ 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}{TEXT -1 89 " (or stability interval) is the intersection of the stability region with the real line." }}{PARA 0 "" 0 "" {TEXT -1 59 "For this scheme the stability interval is (approximately) " } {XPPEDIT 18 0 "[-4.3505, 0];" "6#7$,$-%&FloatG6$\"&0N%!\"%!\"\"\"\"!" }{TEXT -1 2 ". " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 148 "We can distort the boundary curve horizontally by taking the 11th root of the real part of points along the curve. In this way we see that there is " }{TEXT 260 53 "no largest interval on the nonn egative imaginary axis" }{TEXT -1 65 " that contains the origin and li es inside the stability region. " }}{PARA 0 "" 0 "" {TEXT -1 119 "How ever the stability region intersects the nonnegative imaginary axis in an interval that does not contain the origin." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 406 "R := z -> 1 +z+1/2*z^2+1/6*z^3+1/24*z^4+1/120*z^5+1/720*z^6+\n 125970258796340 261/644478307172352000000*z^7+4550741109281107/193343492151705600000*z ^8:\nDigits := 25:\npts := []: z0 := 0:\nfor ct from 0 to 120 do\n z z := newton(R(z)=exp(ct*Pi/100*I),z=z0):\n z0 := zz:\n pts := [op( pts),[surd(Re(zz),11),Im(zz)]]:\nend do:\nplot(pts,color=COLOR(RGB,.45 ,0,.95),thickness=2,font=[HELVETICA,9]);\nDigits := 10:" }}{PARA 13 " " 1 "" {GLPLOT2D 261 276 276 {PLOTDATA 2 "6(-%'CURVESG6#7er7$$\"\"!F)F (7$$!:pM&\\()4%>s&4#3T#!#E$\":XzwJVy*e`EfTJF-7$$!:a*)p``z_)Qjj*)RF-$\" :gQb5)\\%yrI&=$G'F-7$$!:xgJ?u\\!*oU1YN&F-$\":S]r))\\0]2'zxC%*F-7$$!:#R r(fdn_p90\\f'F-$\":t%38mS:Uhqjc7!#D7$$!:8sd_*e)yoOC![xF-$\":w7-\"fcpsE jzq:F?7$$!:hb)zIu>bXx4M))F-$\":cu3i358>fb\\)=F?7$$!:.!>ry%ea@Rv`')*F-$ \":([Fu#>r8o&[6*>#F?7$$!:G$G8nj&[qw()\\3\"F?$\":`,Z$)*3#=67uK^#F?7$$!: y\\:>V\"=\\X)3$z6F?$\":.&*)\\xj'*H%QLu#GF?7$$!:`;'***4W;!*p$))p7F?$\": R&f`\"yOGbk#fTJF?7$$!:nlZhRktTPzpN\"F?$\":Ps(Qs\\Rd.>vbMF?7$$!:f;ai&Rg !z>w2W\"F?$\":u^O4-`+n:6*pPF?7$$!:BJ#p0z!f'o]Q@:F?$\":MSKd/%3c-/2%3%F? 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" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 200 "Digits := 15:\nz0 := 0.9*I:\nfor ct from 27 to 30 do \n newton(R(z)=exp(ct*Pi/100*I),z=z0);\nend do;``;\nz0 := 3.3*I:\nfo r ct from 107 to 110 do\n newton(R(z)=exp(ct*Pi/100*I),z=z0);\nend d o;\nDigits := 10:" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#^$$!0kjMjD#eV!#A$ \"0!ySnfH#[)!#:" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#^$$!0/bt(=s_@!#A$\" 0$\"0J%yM,5!H$!#9" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#^$$!03AQjL'\\?!#<$\"0*=5.wK9L!#9" }}{PARA 11 "" 1 " " {XPPMATH 20 "6#^$$!0n1+L'eFV!#<$\"0.pA@U#QL!#9" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 108 "Then we apply the bisect ion method to calculate the parameter value associated with each inter section point." }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 328 "Digits := 15:\nreal_part := proc(u)\n Re(newton(R(z)=exp(u*Pi*I),z=0.9*I))\ne nd proc:\nu0 := bisect('real_part'(u),u=0.27..0.30);\nnewton(R(z)=exp( u0*Pi*I),z=0.9*I);``;\nreal_part := proc(u)\n Re(newton(R(z)=exp(u*P i*I),z=3.3*I))\nend proc:\nu0 := bisect('real_part'(u),u=1.07..1.10); \nnewton(R(z)=exp(u0*Pi*I),z=3.3*I);\nDigits := 10:" }}{PARA 11 "" 1 " " {XPPMATH 20 "6#>%#u0G$\"02,z#fmdG!#:" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#^#$\"089Gl;w(*)!#:" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%!G" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#>%#u0G$\"0!\\DJ92!3\"!#9" }}{PARA 11 " " 1 "" {XPPMATH 20 "6#^$$\"0,,%eQ1&y$!#H$\"0&fB'Hu-H$!#9" }}}{PARA 0 " " 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 4 "The " }{TEXT 260 73 "stability region intersects the nonegative imaginary axis in the i nterval" }{TEXT -1 39 " [ 0.8978, 3.2903 ] (approximately)." }} {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 37 "#-------- ----------------------------" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 80 "The stability function R* for the 9 stage, or der 5 scheme is given as follows." }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 97 "subs(ee,subs(b=`b*`,StabilityFunction(5,9,'expanded') )):\n`R*` := unapply(%,z):\n'`R*`(z)'=`R*`(z);" }}{PARA 12 "" 1 "" {XPPMATH 20 "6#/-%#R*G6#%\"zG,6\"\"\"F)F'F)*&#F)\"\"#F)*$)F'F,F)F)F)*& #F)\"\"'F)*$)F'\"\"$F)F)F)*&#F)\"#CF)*$)F'\"\"%F)F)F)*&#F)\"$?\"F)*$)F '\"\"&F)F)F)*&#\";ty]@gWVdnwG7]\">+++K_Ds\\mK&oIHNF)*$)F'F1F)F)F)*&#\" =**plJk(**e')=,5U;$\"A+++g\"))pX*o7C/m8T8F)*$)F'\"\"(F)F)F)*&#\"<(fVR4 )zcpNzgH.#FJF)*$)F'\"\")F)F)F)*&#\"126G46u]X\"8+++c-[S9P,h#F)*$)F'\"\" *F)F)!\"\"" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 124 "We can find the point where the boundary of the stability regi on intersects the negative real axis by solving the equation: " }} {PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "`R*`(z) = -1;" "6#/-% #R*G6#%\"zG,$\"\"\"!\"\"" }{TEXT -1 2 ". " }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 33 "z_0 := newton(`R*`(z)=-1,z=-4.4);" }}{PARA 11 "" 1 " " {XPPMATH 20 "6#>%$z_0G$!+jd#zP%!\"*" }}}{PARA 0 "" 0 "" {TEXT -1 0 " " }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 324 "z_0 := newton(`R*`(z)=- 1,z=-4.4):\np_1 := plot([`R*`(z),-1],z=-5.09..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=[-5.09..0.49,-1.57.. 1.47],font=[HELVETICA,9]);" }}{PARA 13 "" 1 "" {GLPLOT2D 398 284 284 {PLOTDATA 2 "6+-%'CURVESG6$7S7$$!3')*************3&!#<$!3Ex$*f$pH_d#F* 7$$!3G++vz=Po\\F*$!3ek'GW2vhA#F*7$$!3#**\\iE!Rai[F*$!3')oS0k)p8&>F*7$$ !3r**\\Au#HNu%F*$!3/K*)Rh!HHn\"F*7$$!3x**\\dRdsBYF*$!3EnQ;s6`B9F*7$$!3 *)*\\7-h\"\\/XF*$!3FRyTiw./7F*7$$!3s*\\i#4j%RR%F*$!3?\\'>X$fEC5F*7$$!3 5+D;`I[zUF*$!3I&)*>fT_Fg)!#=7$$!3W*\\i**)\\5hTF*$!3+2[r0KXBrFP7$$!3!** \\7nc1J/%F*$!3UOZBdfQZeFP7$$!3'****\\XoI<#RF*$!3o@ua\"yj+s%FP7$$!3++]Z TF#[\"QF*$!3yJeJ[F$f'QFP7$$!3s***\\'=(pWp$F*$!3))\\/$FP7$$!3#)** *\\Z^AOd$F*$!3M5i@;%GHM#FP7$$!3))***\\8%Q;dMF*$!3W^Td9*GYx\"FP7$$!3#** \\i*3#39N$F*$!3)oW&pVOhN8FP7$$!3t***\\J`acA$F*$!3%*y@t^b=Y*)!#>7$$!3l* ***fuY7>JF*$!3'GRfNQs+z&Fhp7$$!3q*\\iQ70_*HF*$!3gbw\"zearm#Fhp7$$!3+++ 5C`^&)GF*$!3MQ\"G;#Qu:I!#?7$$!3q*\\i)G#o^w#F*$\"3mgThJE`d>Fhp7$$!3W*\\ (eM$p0l#F*$\"3'4'QP\\QuhQFhp7$$!3]**\\i5u*4`#F*$\"3WfTa`&\\$ocFhp7$$!3 T*\\7\"eI>@CF*$\"3b+:1&oO]A(Fhp7$$!3n**\\()HUv-BF*$\"3g[rGG^Y_))Fhp7$$ !3y*\\iRdH(z@F*$\"3Qba<)RAW0\"FP7$$!3o*\\P$\\ijs?F*$\"3,v1u_X:17FP7$$! 3S**\\#[_sp&>F*$\"3qy%4jte)y8FP7$$!3y****pz0[P=F*$\"3siMDCC7r:FP7$$!3' )**\\_B5e?FP7$$!3 i**\\2&R*)=[\"F*$\"3Y#yS0q\\\"oAFP7$$!3'*****4?a/p8F*$\"3W%))[oWB9a#FP 7$$!3O***\\2Rg&[7F*$\"3ar8%\\G5#oGFP7$$!36+DcYIQR6F*$\"39CIeo^s*>$FP7$ $!3#*)**\\=PB+-\"F*$\"3r9UxVbq0OFP7$$!3c-]i)>_r2*FP$\"3,*35'o4SMSFP7$$ !3*)**\\74%3K!zFP$\"3W>=y#o%)p`%FP7$$!3E****\\xPYbnFP$\"3#yM;:L&y)3&FP 7$$!3)4+Dc^\")Qb&FP$\"3m`CP6r\\QdFP7$$!3)e****f)\\h'R%FP$\"3s$Q$4/caUk FP7$$!3F)**\\<\"G98KFP$\"3a/hF,K&>D(FP7$$!3'G*\\i%Qq%R?FP$\"3#QTm9ib]: )FP7$$!3xr****pJ()4'*Fhp$\"3s2W@jKu$3*FP7$$\"3k<+]_)f2v#Fhp$\"3>!4@bU* )y-\"F*7$$\"3)3++!Qdi!Q\"FP$\"3s#[lfQZ![6F*7$$\"3o4]PhBPfDFP$\"3mM%)Rm ;n\"H\"F*7$$\"3]/]i%G$e(o$FP$\"35Zv " 0 "" {MPLTEXT 1 0 1534 "`R*` := z -> 1+z+1/2*z^2+1 /6*z^3+1/24*z^4+1/120*z^5+\n 50122876675743446021507873/3529306853 2664972255232000000*z^6+\n 3164210011886589976431656999/1341136604 2412689456988160000000*z^7+\n 203296079356956798093943597/13411366 042412689456988160000000*z^8-\n 4550741109281107/26101371440480256 000000*z^9:\npts := []: z0 := 0:\nfor ct from 0 to 200 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:\np_1 := plot(pts,color=COLOR(RGB,.2,0,.4)): \np_2 := plots[polygonplot]([seq([pts[i-1],pts[i],[-2.2,0]],i=2..nops( pts))],\n style=patchnogrid,color=COLOR(RGB,.4,0,.8)):\npts : = []: z0 := 1.9+4.4*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( zz)]]:\nend do:\np_3 := plot(pts,color=COLOR(RGB,.2,0,.4)):\np_4 := pl ots[polygonplot]([seq([pts[i-1],pts[i],[1.85,4.37]],i=2..nops(pts))], \n style=patchnogrid,color=COLOR(RGB,.4,0,.8)):\npts := []: z 0 := 1.9-4.4*I:\nfor ct from 0 to 50 do\n zz := newton(`R*`(z)=exp(c t*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,.2,0,.4)):\np_6 := plots[po lygonplot]([seq([pts[i-1],pts[i],[1.85,-4.37]],i=2..nops(pts))],\n \+ style=patchnogrid,color=COLOR(RGB,.4,0,.8)):\np_7 := plot([[[-5. 09,0],[2.19,0]],[[0,-4.99],[0,4.99]]],color=black,linestyle=3):\nplots [display]([p_||(1..7)],view=[-5.09..2.19,-4.99..4.99],font=[HELVETICA, 9],\n labels=[`Re(z)`,`Im(z)`],axes=boxed,scaling=constrain ed);" }}{PARA 13 "" 1 "" {GLPLOT2D 407 552 552 {PLOTDATA 2 "6/-%'CURVE SG6$7ew7$$\"\"!F)F(7$$\"3J+++%p)>=[!#F$\"3++++Fjzq:!#=7$$\"3$******Rr# o,L!#D$\"3')*****fm#fTJF07$$\"3y*****pnsg:%!#C$\"3=+++u!*Q7ZF07$$\"3)) *****fsIVi#!#B$\"3C+++@Z=$G'F07$$\"3-+++P#=!H6!#A$\"3e*****zArR&yF07$$ \"3')*****>:9ix$FF$\"3]+++^(3ZU*F07$$\"3/+++VvV^5!#@$\"33+++Vl_*4\"!#< 7$$\"3$*******[&=b`#FQ$\"3/+++F%HlD\"FT7$$\"3q*****\\27[U&FQ$\"32+++6? 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l$!+jc[8[FjanF[]al7%F[gal7$F^c_l$!+(3L#4[FjanF[]al7%F_gal7$Fhb_l$!+5P8 /[FjanF[]al7%Fcgal7$Fbb_l$!+0oC)z%FjanF[]al7%Fggal7$F\\b_l$!+ZSj\"z%Fj anF[]al7%F[hal7$Ffa_l$!+\\(fVy%FjanF[]al7%F_hal7$F`a_l$!+$>\"\\wZFjanF []al7%Fchal7$Fj`_l$!+f!*4oZFjanF[]al7%Fghal7$F_`_l$!+UzDfZFjanF[]al7%F [ialFe\\alF[]alFa_^lF^^r-F$6%7$7$$!3Q++++++!>&FTF(7$$\"3/++++++!H#FTF( -%'COLOURG6&F`^nF)F)F)-%*LINESTYLEG6#\"\"$-F$6%7$7$F(Fcial7$F($\"3Q+++ +++!>&FTFhialF[jal-%%FONTG6$%*HELVETICAG\"\"*-%*AXESSTYLEG6#%$BOXG-%+A XESLABELSG6%%&Re(z)G%&Im(z)G-Fgjal6#%(DEFAULTG-%(SCALINGG6#%,CONSTRAIN EDG-%%VIEWG6$;$!$>&F__s$\"$H#F__s;F_\\bl$\"$>&F__s" 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 1879 "ee := \{c[2]=33/577,\nc[3]=173/12 00,\nc[4]=173/800,\nc[5]=260/463,\nc[6]=27/40,\nc[7]=24/25,\nc[8]=1,\n c[9]=1,\n\na[2,1]=33/577,\na[3,1]=-3567433/95040000,\na[3,2]=17269033/ 95040000,\na[4,1]=173/3200,\na[4,2]=0,\na[4,3]=519/3200,\na[5,1]=18777 48548260/2970538457863,\na[5,2]=0,\na[5,3]=-7126513680000/297053845786 3,\na[5,4]=6916886080000/2970538457863,\na[6,1]=-652445794319646554284 77/69653246199915888640000,\na[6,2]=0,\na[6,3]=2694429616783931433/669 742751922268160,\na[6,4]=-62138409816102019904637/21415191928402504983 040,\na[6,5]=145922219908058645309313/297661794320406364160000,\na[7,1 ]=3890754282260741440349323/3010236956970547048875000,\na[7,2]=0,\na[7 ,3]=-45910927628476367208/8576173666582755125,\na[7,4]=598603717868429 7350750710688/1207688208130690660530130125,\na[7,5]=-31069909805629103 09356750843/3003073489062493679278875000,\na[7,6]=2999575140352/272986 6330125,\na[8,1]=3914147173803753655861/1444606034961051585792,\na[8,2 ]=0,\na[8,3]=-2276990272364785725/192922814498003684,\na[8,4]=24219185 248932516188948175/2263933167309766272933307,\na[8,5]=-480100522457935 4456654139837/1661667052909362562177244416,\na[8,6]=232855940800/97230 777633,\na[8,7]=-57039375/516949856,\na[9,1]=7466867/116588160,\na[9,2 ]=0,\na[9,3]=0,\na[9,4]=3227648000000/9720313182027,\na[9,5]=179724568 369012721/784946257231456320,\na[9,6]=234368000/1186672113,\na[9,7]=16 390625/58886016,\na[9,8]=-14011/138852,\n\nb[1]=7466867/116588160,\nb[ 2]=0,\nb[3]=0,\nb[4]=3227648000000/9720313182027,\nb[5]=17972456836901 2721/784946257231456320,\nb[6]=234368000/1186672113,\nb[7]=16390625/58 886016,\nb[8]=-14011/138852,\n\n`b*`[1]=3384682763489201/5391463219406 7840,\n`b*`[2]=0,\n`b*`[3]=0,\n`b*`[4]=72143477722071040000/2140489794 81496847913,\n`b*`[5]=24566463552585428091992041/120995959197297938812 618560,\n`b*`[6]=125185218908556800/548760616063863687,\n`b*`[7]=28898 50698904375/11670449557585536,\n`b*`[8]=-22856669025961/32105123322174 0,\n`b*`[9]=-1/135\}:" }}}{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#\"\"##\"#L\"$x&/&F%6#\"\"$#\" $t\"\"%+7/&F%6#\"\"%#F0\"$+)/&F%6#\"\"&#\"$g#\"$j%/&F%6#\"\"'#\"#F\"#S /&F%6#\"\"(#\"#C\"#D/&F%6#\"\")\"\"\"/&F%6#\"\"*FQ" }}}{PARA 0 "" 0 " " {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT 259 51 "linking coefficients f or 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$\"\"#\"\"\"#\"#L\"$x&/&F %6$\"\"$F(#!(Luc$\")++/&*/&F%6$F/F'#\")L!ps\"F2/&F%6$\"\"%F(#\"$t\"\"% +K/&F%6$F;F'\"\"!/&F%6$F;F/#\"$>&F>/&F%6$\"\"&F(#\".g#[&[x(=\".jyXQ0(H /&F%6$FKF'FB/&F%6$FKF/#!.++o8l7(FN/&F%6$FKF;#\".++3')o\"pFN/&F%6$\"\"' F(#!8x%Galk>VzXCl\"8++k))e\"**>YKlp/&F%6$FinF'FB/&F%6$FinF/#\"4L9$Ry;' HWp#\"3g\"oA#>vU(p'/&F%6$FinF;#!8PY!*>?5;)4%Q@'\"8SI)\\]-%G>>:9#/&F%6$ FinFK#\"98$4`ke!3*>A#f9\"9++;kjS?VzhwH/&F%6$\"\"(F(#\":B$\\.WT2E#Ga2*Q \":+]()[qaqp&pB5I/&F%6$FepF'FB/&F%6$FepF/#!53sOw%Gw#4\"f%\"4D^v#emO/&F%6$FgrF;#\";v\"[*)=;D$*[_=>U#\":2L$HF m(4t;LRE#/&F%6$FgrFK#!=P)RTlcWNzXA05![\"=;WCx@ci$4H0n;m\"/&F%6$FgrFin# \"-+3%f&GB\",LwxIs*/&F%6$FgrFep#!)v$Rq&\"*c)\\p^/&F%6$\"\"*F(#\"(noY( \"*g\")e;\"/&F%6$F_uF'FB/&F%6$F_uF/FB/&F%6$F_uF;#\".+++[wA$\".F?=8.s*/ &F%6$F_uFK#\"3@F,p$oXsz\"\"3?jXJsDY\\y/&F%6$F_uFin#\"*+!oVB\"+8@n'=\"/ &F%6$F_uFep#\")D1R;\");g))e/&F%6$F_uFgr#!&6S\"\"'_)Q\"" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT 259 38 "weights for the 8 s tage 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#\"\"\"#\"(noY(\"*g\")e;\"/&F%6#\"\"#\"\"!/&F%6 #\"\"$F//&F%6#\"\"%#\".+++[wA$\".F?=8.s*/&F%6#\"\"&#\"3@F,p$oXsz\"\"3? jXJsDY\\y/&F%6#\"\"'#\"*+!oVB\"+8@n'=\"/&F%6#\"\"(#\")D1R;\");g))e/&F% 6#\"\")#!&6S\"\"'_)Q\"" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT 259 38 "weights for the 9 stage order 5 scheme" }{TEXT -1 1 ":" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 37 "seq(`b*`[i]=subs(ee, `b*`[i]),i=1..9);" }}{PARA 12 "" 1 "" {XPPMATH 20 "6+/&%#b*G6#\"\"\"# \"1,#*[jFo%Q$\"2Sy1%>KY\"R&/&F%6#\"\"#\"\"!/&F%6#\"\"$F//&F%6#\"\"%#\" 5++/r?sxM9s\"68z%o\\\"[z*[S@/&F%6#\"\"&#\";T?*>4Gae_NYmX#\"ff*47/&F%6#\"\"'#\"3+ob3*=_=D\"\"3(ojQ1;1w[&/&F%6#\"\"(#\"1vV!*)p])* )G\"2Obed&\\/n6/&F%6#\"\")#!/hf-pm&G#\"0S " 0 "" {MPLTEXT 1 0 1 ";" }}}}{PARA 0 "" 0 "" {TEXT -1 52 "#----------------- ----------------------------------" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}}{PARA 0 "" 0 "" {TEXT -1 33 "#================= ===============" }}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 13 "scheme with \+ " }{XPPEDIT 18 0 "c[7] = 35/36;" "6#/&%\"cG6#\"\"(*&\"#N\"\"\"\"#O!\" \"" }{TEXT -1 1 " " }}{PARA 257 "" 0 "" {TEXT -1 33 "The scheme constr ucted here has " }{XPPEDIT 18 0 "c[6] = 97/140;" "6#/&%\"cG6#\"\"'*& \"#(*\"\"\"\"$S\"!\"\"" }{TEXT -1 7 " and " }{XPPEDIT 18 0 "c[7] = 3 5/36;" "6#/&%\"cG6#\"\"(*&\"#N\"\"\"\"#O!\"\"" }{TEXT -1 1 "." }} {PARA 0 "" 0 "" {TEXT -1 6 "With " }{XPPEDIT 18 0 "c[6]" "6#&%\"cG6# \"\"'" }{TEXT -1 7 " and " }{XPPEDIT 18 0 "c[7]" "6#&%\"cG6#\"\"(" } {TEXT -1 39 " having these fixed values 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[5]" "6#&%\"cG6#\" \"&" }{TEXT -1 50 " are chosen to minimize the principal error norm. " }}{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 1734 "ee := \{c[2]=7/95,\nc[3]=35/243,\nc[4]=35/162,\nc [5]=31/55,\nc[6]=97/140,\nc[7]=35/36,\nc[8]=1,\nc[9]=1,\n\na[2,1]=7/95 ,\na[3,1]=385/118098,\na[3,2]=16625/118098,\na[4,1]=35/648,\na[4,2]=0, \na[4,3]=35/216,\na[5,1]=262364129/407618750,\na[5,2]=0,\na[5,3]=-9969 09687/407618750,\na[5,4]=482147154/203809375,\na[6,1]=-369745979892278 00011/27927480760535300000,\na[6,2]=0,\na[6,3]=40382359908551133219/72 07091809170400000,\na[6,4]=-5837069082360096304407/1395022708312545550 000,\na[6,5]=13508218613909220883/22593673904425227520,\na[7,1]=756727 539023617977739/325483930565422777440,\na[7,2]=0,\na[7,3]=-11344125743 485787/1187841741251840,\na[7,4]=93403161519631272506393/1118867565006 9067742640,\na[7,5]=-3811592140111638097686625/33088119234517402722408 96,\na[7,6]=102601928198000/102454959175119,\na[8,1]=25079250578935479 0081/66169955021507254750,\na[8,2]=0,\na[8,3]=-349400598312667239/2200 5305959929250,\na[8,4]=17728479258264420469702254/12897128609419953407 62625,\na[8,5]=-61925787315598406758085/27764881218432626442221,\na[8, 6]=43795320281930/26932915012729,\na[8,7]=-96493150422/1790131071575, \na[9,1]=1414477/22101450,\na[9,2]=0,\na[9,3]=0,\na[9,4]=6047117272638 /18236245267975,\na[9,5]=31062711625/123650103496,\na[9,6]=27376502125 /148102904499,\na[9,7]=26504253/76308925,\na[9,8]=-23459/131064,\n\nb[ 1]=1414477/22101450,\nb[2]=0,\nb[3]=0,\nb[4]=6047117272638/18236245267 975,\nb[5]=31062711625/123650103496,\nb[6]=27376502125/148102904499,\n b[7]=26504253/76308925,\nb[8]=-23459/131064,\n\n`b*`[1]=9659334433759/ 154907736963000,\n`b*`[2]=0,\n`b*`[3]=0,\n`b*`[4]=21597935405438536443 /63908374454260348250,\n`b*`[5]=230190597481295825/1039987387676705088 ,\n`b*`[6]=8367042728056775/37747068053062584,\n`b*`[7]=28150317302067 /97244486689000,\n`b*`[8]=-77493123437/612413141440,\n`b*`[9]=-1/160\} :" }}}{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 64 "The Butcher tableau in exact and approximate form is as follows." }}{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)],[``$4,`b*`[9]]]));" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#K%'matri xG6#747'#\"\"(\"#&*F(%!GF+F+7'#\"#N\"$V##\"$&Q\"')4=\"#\"&Dm\"F2F+F+7' #F.\"$i\"#F.\"$['\"\"!#F.\"$;#F+7'#\"#J\"#b#\"*HTOi#\"*](=wSF:#!*(o4p* *FC#\"*ar9#[\"*v$4Q?7'#\"#(*\"$S\"#!56+!yA*)zfup$\"5++IN0w![Fz#F:#\"5> K8^&3*fBQS\"4++Sq\"4=42s#!72WI'4gB3pq$e\"7++bXDJ3F-&R\"7'F+F+F+F+#\"5$ )3A4Rh=#3N\"\"5?vADW!Rn$fA7'#F.\"#O#\"6Rx(zhB!Rvsc(\"6SuxAacIR[D$F:#!2 (y&[Vd7W8\"\"1S=DT<%y=\"#\"8$R1DFJ'>:;.M*\"8SEun!p+lv')=67'F+F+F+#!:Dm o(4Q;6S@f6Q\":'*3Cs-u^M#>\")3L#\"0+!)>G>g-\"\"0>^%4'Gr*G\"7'F+F+#!8&3enS)f:tyD>'\"8@AWEEV=7)[wF#\"/I> G?`zV\"/HF,:H$p##!,A/:$\\'*\".v:2J,z\"7'Fho#\"(xWT\"\")]95AF:F:#\".QEF I#\"4)30nn(Q()*R5#\"1vn0GF/n$)\"2%eiI0oquP#\" /n?I<.:G\"/+!*o'[Ws*#!,PM7$\\x\"-S998Ch7'F+F+F+F+#!\"\"\"$g\"Q(pprint9 6\"" }}}{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),``$(1 0-i)],i=2..9),\n[`b`,seq(b[i],i=1..8),``],[`b*`,seq(`b*`[i],i=1..9)]]) ):\nevalf[8](%);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#K%'matrixG6#7,7,$ \")6Uot!\"*F(%!GF+F+F+F+F+F+F+7,$\")#H.W\"!\")$\")W+gK!#5$\")#HxS\"F/F +F+F+F+F+F+F+7,$\")Q\\g@F/$\")YB,aF*$\"\"!F;$\")/P?;F/F+F+F+F+F+F+7,$ \")OOOcF/$\")z]OkF/F:$!)9pXC!\"($\")qnlBFEF+F+F+F+F+7,$\")9dGpF/$!)/&R K\"FEF:$\")A9.cFE$!)3@%=%FE$\")9wyfF/F+F+F+F+7,$\")AAA(*F/$\")1$\\K#FE F:$!)'*>]&*FE$\")(3![$)FE$!)>&>:\"FE$\")XV,5FEF+F+F+7,$\"\"\"F;$\")m7! z$FEF:$!);!ye\"!\"'$\")ngu8Fao$!)QOIAFE$\")\"*3E;FE$!)TG!R&F*F+F+7,F[o $\")&G**R'F*F:F:$\")\"))fJ$F/$\")g97DF/$\")%y%[=F/$\")NGtMF/$!)*)))*y \"F/F+7,%\"bGF[pF:F:F]pF_pFapFcpFepF+7,%#b*G$\").aNiF*F:F:$\")d^zLF/$ \")!)R8AF/$\")tg;AF/$\")$)z%*GF/$!)LPl7F/$!)++]iF2Q)pprint106\"" }}} {PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 146 "RK6_8eqs := [op(RowSumConditions(8,'expanded')),op(OrderCondition s(6,8,'expanded'))]:\n`RK5_9eqs*` := subs(b=`b*`,OrderConditions(5,9,' expanded')):" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 7 "Check: " }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 148 "simp lify(subs(ee,RK6_8eqs)):\nmap(u->`if`(lhs(u)=rhs(u),0,1),%);\nnops(%); \nsimplify(subs(ee,`RK5_9eqs*`)):\nmap(u->`if`(lhs(u)=rhs(u),0,1),%); \nnops(%);" }{TEXT -1 0 "" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#7N\"\"!F$ 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#$\"+6OWXE!#9" }} }{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 79 "The 2 no rm of the principal error of the order 5 embedded scheme is as follows ." }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 153 "`errterms5_9*` := subs (b=`b*`,PrincipalErrorTerms(5,9,'expanded')):\nevalf(evalf[14](sqrt(ad d(subs(ee,`errterms5_9*`[i])^2,i=1.. nops(`errterms5_9*`)))));" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#$\"+1\"Q0t&!#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( \"\"$7%\"\")F)/*&F,F()F2F:F(#F(F57%\"#7F)/*&F,F(-%!G6#*&%\"aGF(-FF6#*& F8F(FIF(F(F(#F(\"#g7%\"#:F)/*&F,F(-FF6#*&F?F(FIF(F(#F(\"#?7%\"#;F)/*&F ,F()F2F5F(#F(\"\"&Q)pprint186\"" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 582 "SO6_8 := SimpleOrderConditi ons(6,8,'expanded'):\nSO_eqs := [op(RowSumConditions(8,'expanded')),op (StageOrderConditions(2,8,'expanded')),\n op(StageOrderC onditions(3,4..8,'expanded'))]:\n`SO5_9*` := subs(b=`b*`,SimpleOrderCo nditions(5,9,'expanded')):\nord_cdns := [seq(SO6_8[i],i=[1,2,4,8,16,24 ,29,32])]:\n`ord_cdns*` := [seq(`SO5_9*`[i],i=[1,2,4,8,12,15,16])]:\ns imp_eqs := [add(b[i]*a[i,1],i=1+1..8)=b[1],seq(add(b[i]*a[i,j],i=j+1.. 8)=b[j]*(1-c[j]),j=[6,7])]:\nwt_eqns := [seq(b[i]=a[9,i],i=1..8)]:\ncd ns := [op(ord_cdns),op(simp_eqs),op(SO_eqs),op(wt_eqns),op(`ord_cdns*` )]:" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 22 "W e specify the nodes: " }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "c[2] = 7/95;" "6#/&%\"cG6#\"\"#*&\"\"(\"\"\"\"#&*!\"\"" }{TEXT -1 3 ", " }{XPPEDIT 18 0 "c[4] = 35/162;" "6#/&%\"cG6#\"\"%*&\"#N\"\" \"\"$i\"!\"\"" }{TEXT -1 3 ", " }{XPPEDIT 18 0 "c[5] = 31/55;" "6#/&% \"cG6#\"\"&*&\"#J\"\"\"\"#b!\"\"" }{TEXT -1 3 ", " }{XPPEDIT 18 0 "c[ 6] = 97/140;" "6#/&%\"cG6#\"\"'*&\"#(*\"\"\"\"$S\"!\"\"" }{TEXT -1 3 " , " }{XPPEDIT 18 0 "c[7] = 35/36;" "6#/&%\"cG6#\"\"(*&\"#N\"\"\"\"#O! \"\"" }{TEXT -1 3 ", " }{XPPEDIT 18 0 "c[8]=1" "6#/&%\"cG6#\"\")\"\" \"" }{TEXT -1 2 ", " }}{PARA 0 "" 0 "" {TEXT -1 31 "the zero linking c oefficients: " }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "a[4, 2]=0" "6#/&%\"aG6$\"\"%\"\"#\"\"!" }{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#\"\"$\"\"!" } {TEXT -1 1 "." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 125 "The weights of the order 6 scheme provide the linking co efficients for the 9th stage of the embedded order 5 scheme so that: \+ " }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "a[9,i]=b[i]" "6#/ &%\"aG6$\"\"*%\"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# \"\"$\"\"!" }{TEXT -1 7 " and " }{XPPEDIT 18 0 "`b*`[9] = -1/160;" " 6#/&%#b*G6#\"\"*,$*&\"\"\"F*\"$g\"!\"\"F," }{TEXT -1 2 ". " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 37 "We have 44 equati ons and 44 unknowns." }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 211 "e1 \+ := \{c[2]=7/95,c[4]=35/162,c[5]=31/55,c[6]=97/140,c[7]=35/36,c[8]=1,c[ 9]=1,\n seq(a[i,2]=0,i=4..8),b[2]=0,b[3]=0,`b*`[2]=0,`b*`[3]=0,` b*`[9]=-1/160\}:\neqns := subs(e1,cdns):\nnops(%);\nindets(eqns);\nnop s(%);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#\"#W" }}{PARA 12 "" 1 "" {XPPMATH 20 "6#&F<6$F*F'&F<6 $F*F:&F<6$\"\"*F'&F<6$FIF>&F<6$FIF:&F<6$FIF*&F<6$FIF-&F<6$FIF0&F<6$FIF 3&F<6$FIF6&%#b*GF)&FYF,&FYF/&FYF2&FYF5&F<6$F-F:&F<6$F-F*&F<6$F0F'&F<6$ F0F:&F<6$F0F*&F<6$F0F-&F<6$F3F'&F<6$F3F:&F<6$F3F*&F<6$F3F-&F<6$F3F0&F< 6$F6F'&F<6$F-F'&FYF&&F<6$F6F:&F<6$F6F*&F<6$F6F-&F<6$F6F0&F<6$F6F3" }} {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 1853 "e3 := \{a[9,4] = 604711727 2638/18236245267975, a[7,4] = 93403161519631272506393/1118867565006906 7742640, a[7,1] = 756727539023617977739/325483930565422777440, `b*`[8] = -77493123437/612413141440, a[6,4] = -5837069082360096304407/1395022 708312545550000, b[1] = 1414477/22101450, a[2,1] = 7/95, `b*`[4] = 215 97935405438536443/63908374454260348250, a[4,2] = 0, a[5,2] = 0, a[6,2] = 0, a[7,2] = 0, a[8,2] = 0, b[8] = -23459/131064, a[8,4] = 177284792 58264420469702254/1289712860941995340762625, a[5,4] = 482147154/203809 375, a[7,3] = -11344125743485787/1187841741251840, a[8,3] = -349400598 312667239/22005305959929250, a[9,1] = 1414477/22101450, `b*`[3] = 0, a [9,2] = 0, a[9,3] = 0, a[6,3] = 40382359908551133219/72070918091704000 00, a[5,3] = -996909687/407618750, `b*`[7] = 28150317302067/9724448668 9000, a[4,1] = 35/648, `b*`[1] = 9659334433759/154907736963000, a[8,5] = -61925787315598406758085/27764881218432626442221, c[8] = 1, c[9] = \+ 1, b[2] = 0, b[3] = 0, `b*`[2] = 0, c[2] = 7/95, c[4] = 35/162, c[5] = 31/55, a[9,7] = 26504253/76308925, a[8,1] = 250792505789354790081/661 69955021507254750, c[6] = 97/140, c[7] = 35/36, a[5,1] = 262364129/407 618750, `b*`[9] = -1/160, a[4,3] = 35/216, a[9,5] = 31062711625/123650 103496, a[9,8] = -23459/131064, a[3,2] = 16625/118098, `b*`[5] = 23019 0597481295825/1039987387676705088, a[7,6] = 102601928198000/1024549591 75119, b[4] = 6047117272638/18236245267975, a[8,7] = -96493150422/1790 131071575, a[8,6] = 43795320281930/26932915012729, a[9,6] = 2737650212 5/148102904499, c[3] = 35/243, b[5] = 31062711625/123650103496, a[3,1] = 385/118098, b[7] = 26504253/76308925, a[6,5] = 13508218613909220883 /22593673904425227520, b[6] = 27376502125/148102904499, a[7,5] = -3811 592140111638097686625/3308811923451740272240896, `b*`[6] = 83670427280 56775/37747068053062584, a[6,1] = -36974597989227800011/27927480760535 300000\}:" }{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)],[``$4,`b*`[9]]]));" }}{PARA 11 " " 1 "" {XPPMATH 20 "6#K%'matrixG6#747'#\"\"(\"#&*F(%!GF+F+7'#\"#N\"$V# #\"$&Q\"')4=\"#\"&Dm\"F2F+F+7'#F.\"$i\"#F.\"$['\"\"!#F.\"$;#F+7'#\"#J \"#b#\"*HTOi#\"*](=wSF:#!*(o4p**FC#\"*ar9#[\"*v$4Q?7'#\"#(*\"$S\"#!56+ !yA*)zfup$\"5++IN0w![Fz#F:#\"5>K8^&3*fBQS\"4++Sq\"4=42s#!72WI'4gB3pq$e \"7++bXDJ3F-&R\"7'F+F+F+F+#\"5$)3A4Rh=#3N\"\"5?vADW!Rn$fA7'#F.\"#O#\"6 Rx(zhB!Rvsc(\"6SuxAacIR[D$F:#!2(y&[Vd7W8\"\"1S=DT<%y=\"#\"8$R1DFJ'>:;. M*\"8SEun!p+lv')=67'F+F+F+#!:Dmo(4Q;6S@f6Q\":'*3Cs-u^M#>\")3L#\"0+!)>G >g-\"\"0>^%4'Gr*G\"7'F+F+#!8&3en S)f:tyD>'\"8@AWEEV=7)[wF#\"/I>G?`zV\"/HF,:H$p##!,A/:$\\'*\".v:2J,z\"7' Fho#\"(xWT\"\")]95AF:F:#\".QEFI#\"4)30nn(Q()* R5#\"1vn0GF/n$)\"2%eiI0oquP#\"/n?I<.:G\"/+!*o'[Ws*#!,PM7$\\x\"-S998Ch7 'F+F+F+F+#!\"\"\"$g\"Q(pprint76\"" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 136 "subs(e3,matrix([seq([c[i],s eq(a[i,j],j=1..i-1),``$(10-i)],i=2..9),\n[`b`,seq(b[i],i=1..8),``],[`b *`,seq(`b*`[i],i=1..9)]])):\nevalf[8](%);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#K%'matrixG6#7,7,$\")6Uot!\"*F(%!GF+F+F+F+F+F+F+7,$\")#H .W\"!\")$\")W+gK!#5$\")#HxS\"F/F+F+F+F+F+F+F+7,$\")Q\\g@F/$\")YB,aF*$ \"\"!F;$\")/P?;F/F+F+F+F+F+F+7,$\")OOOcF/$\")z]OkF/F:$!)9pXC!\"($\")qn lBFEF+F+F+F+F+7,$\")9dGpF/$!)/&RK\"FEF:$\")A9.cFE$!)3@%=%FE$\")9wyfF/F +F+F+F+7,$\")AAA(*F/$\")1$\\K#FEF:$!)'*>]&*FE$\")(3![$)FE$!)>&>:\"FE$ \")XV,5FEF+F+F+7,$\"\"\"F;$\")m7!z$FEF:$!);!ye\"!\"'$\")ngu8Fao$!)QOIA FE$\")\"*3E;FE$!)TG!R&F*F+F+7,F[o$\")&G**R'F*F:F:$\")\"))fJ$F/$\")g97D F/$\")%y%[=F/$\")NGtMF/$!)*)))*y\"F/F+7,%\"bGF[pF:F:F]pF_pFapFcpFepF+7 ,%#b*G$\").aNiF*F:F:$\")d^zLF/$\")!)R8AF/$\")tg;AF/$\")$)z%*GF/$!)LPl7 F/$!)++]iF2Q(pprint86\"" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 146 "RK6_8eqs := [op(RowSumConditions(8 ,'expanded')),op(OrderConditions(6,8,'expanded'))]:\n`RK5_9eqs*` := su bs(b=`b*`,OrderConditions(5,9,'expanded')):" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 7 "Check: " }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 148 "simplify(subs(e3,RK6_8eqs)):\nmap(u->`if`(lhs(u )=rhs(u),0,1),%);\nnops(%);\nsimplify(subs(e3,`RK5_9eqs*`)):\nmap(u->` if`(lhs(u)=rhs(u),0,1),%);\nnops(%);" }{TEXT -1 0 "" }}{PARA 11 "" 1 " " {XPPMATH 20 "6#7N\"\"!F$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 28 "determination of 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[5] " "6#&%\"cG6#\"\"&" }{TEXT -1 1 " " }}{PARA 4 "" 0 "" {TEXT -1 0 "" }} {PARA 0 "" 0 "" {TEXT -1 79 "In this section we obtain the nodes of a \+ modification of a scheme by Tsitouras." }}{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*(3 0*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]+2 8*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]+1 90*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]^2-27*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]^4-10 0*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]), 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]+1 32*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]^2-91 0*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+20 0*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]-4 1*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]^3-1 2*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+2 2*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+42 0*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 +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]+1 5*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]+15 6*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+95 0*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]^3-10 0*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]-1 98*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]+20 0*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+2 40*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+1 1*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]^3-49 0*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]^2-30 0*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]+20 0*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+10 3*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]^3-2 3*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+18 0*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]^3-1 0*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]-20 0*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]+3 60*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+3 60*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]-8 0*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+84 0*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+23 0*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[8,4] = -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]-7 50*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] 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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]+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]+4 98*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 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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+1 0*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]+7 0*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 ]+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]+150 0*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]+17 20*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]^2-400*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+1 8*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]+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- 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 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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]+2 860*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]-4 29*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]+49 0*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+1 4*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]^3-70*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[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 ]+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+1 027*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]-9 0*c[5]^4*c[7]^2*c[4]*c[6]-90*c[5]^4*c[4]*c[7]*c[6]-190*c[4]^5*c[6]^2*c [7]*c[5]^2-100*c[4]^5*c[6]^2*c[7]^2*c[5]^2+60*c[4]^5*c[5]^2-745*c[6]^2 *c[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]^4-2 00*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 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*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 ]+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+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]^3-2 0*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]^5-30*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+15 0*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]-20 0*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 33 "#=================== =============" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 22 "We specify the nodes " }{XPPEDIT 18 0 "c[6] = 97/140;" " 6#/&%\"cG6#\"\"'*&\"#(*\"\"\"\"$S\"!\"\"" }{TEXT -1 7 " and " } {XPPEDIT 18 0 "c[7] = 35/36;" "6#/&%\"cG6#\"\"(*&\"#N\"\"\"\"#O!\"\"" }{TEXT -1 27 " and determine 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[5]" "6#&%\"cG6#\"\"&" } {TEXT -1 65 " that minimize the principal errror norm (subject to the nodes " }{XPPEDIT 18 0 "c[6]" "6#&%\"cG6#\"\"'" }{TEXT -1 7 " and \+ " }{XPPEDIT 18 0 "c[7]" "6#&%\"cG6#\"\"(" }{TEXT -1 19 " remaining fi xed)." }}{PARA 0 "" 0 "" {TEXT -1 85 "We use the general solution to \+ obtain expressions for the coefficients in terms of " }{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[5]" "6#&%\"cG6#\" \"&" }{TEXT -1 2 ". " }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 72 "eA : = \{c[6]=97/140,c[7]=35/36\}:\neB := `union`(eA,simplify(subs(eA,eG))) :" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 2 "eB" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 15909 "eB := \{c[9] = 1, c[8] = 1, a[5,3] = 3/4*c[5]^2*(-2*c[5]+3*c[4])/c[4] ^2, `b*`[3] = 0, a[4,1] = 1/4*c[4], a[3,1] = -2/9*c[4]*(c[4]-3*c[2])/c [2], `b*`[2] = 0, `b*`[5] = 1/30*(-15150360*c[5]*c[4]^3-18162210*c[5]^ 2*c[4]^2+2483776*c[4]+1123060*c[5]-5569145*c[4]^2-844870*c[5]^2+360355 0*c[4]^3-9075720*c[5]*c[4]+7107730*c[5]^2*c[4]+21990305*c[5]*c[4]^2+13 356900*c[5]^2*c[4]^3-326211)/(-219786*c[5]^2*c[4]^2+35455*c[4]+45931*c [5]-27160*c[4]^2-63628*c[5]^2+27936*c[5]^3-175628*c[5]*c[4]-117936*c[5 ]^3*c[4]+257796*c[5]^2*c[4]+104760*c[5]^3*c[4]^2+142596*c[5]*c[4]^2-10 185)/(140*c[5]-97)/(c[4]-c[5])/c[5], b[2] = 0, b[3] = 0, a[2,1] = c[2] , `b*`[1] = 1/101850*(35488350*c[5]*c[4]^3+137541135*c[5]^2*c[4]^2-248 3776*c[4]-2483776*c[5]+5569145*c[4]^2+5569145*c[5]^2-3603550*c[4]^3-36 03550*c[5]^3+76111050*c[5]^3*c[4]^3+21436126*c[5]*c[4]+326211+35488350 *c[5]^3*c[4]-52163475*c[5]^2*c[4]-99154200*c[5]^3*c[4]^2-52163475*c[5] *c[4]^2-99154200*c[5]^2*c[4]^3)/(291-3276*c[5]*c[4]^2-3276*c[5]^2*c[4] +776*c[5]^2-1013*c[4]-1013*c[5]+3976*c[5]*c[4]+2910*c[5]^2*c[4]^2+776* c[4]^2)/c[5]/c[4], a[9,2] = 0, a[9,3] = 0, b[5] = -1/60*(7110*c[4]-187 9)/(c[4]-c[5])/c[5]/(5040*c[5]^3-13432*c[5]^2+11787*c[5]-3395), b[7] = -39366/1925*(270*c[5]*c[4]-65*c[5]-65*c[4]+11)/(36*c[4]-35)/(36*c[5]- 35), b[6] = 21008750/137643*(170*c[5]*c[4]-67*c[5]-67*c[4]+33)/(140*c[ 4]-97)/(140*c[5]-97), `b*`[9] = 1/10*(4850*c[5]^2*c[4]^2-4580*c[5]^2*c [4]+970*c[5]^2-4580*c[5]*c[4]^2+4795*c[5]*c[4]-1110*c[5]+970*c[4]^2-11 10*c[4]+291)/(291-3276*c[5]*c[4]^2-3276*c[5]^2*c[4]+776*c[5]^2-1013*c[ 4]-1013*c[5]+3976*c[5]*c[4]+2910*c[5]^2*c[4]^2+776*c[4]^2), a[6,4] = - 97/768320000*(-96478140*c[5]*c[4]^3+73232090*c[5]^2*c[4]^2+1825346*c[4 ]^2+5476038*c[5]^2-2634520*c[4]^3-10538080*c[5]^3-38024000*c[5]^4*c[4] ^3-114072000*c[5]^3*c[4]^4+364215600*c[5]^3*c[4]^3+132868920*c[4]^4*c[ 5]-6388711*c[5]*c[4]+84141680*c[5]^3*c[4]-30221708*c[5]^2*c[4]-2281440 00*c[4]^4*c[5]^2+26345200*c[5]^4*c[4]^2-297078600*c[5]^3*c[4]^2+298077 12*c[5]*c[4]^2+44106220*c[5]^2*c[4]^3+114072000*c[4]^5*c[5]^2-52690400 *c[4]^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])/c[4]^2, a[9,5] = -1/60*(7110*c[4]-1879)/c[5]/(5040*c[5] ^3*c[4]-13432*c[5]^2*c[4]+11787*c[5]*c[4]-3395*c[4]-5040*c[5]^4+13432* c[5]^3-11787*c[5]^2+3395*c[5]), a[8,1] = 1/13580*(-1379191952*c[5]*c[4 ]^3+705562118*c[5]^2*c[4]^2+112669078*c[4]^4+176512840*c[5]^4+21734790 *c[4]^2+65204370*c[5]^2-91704548*c[4]^3-228958800*c[5]^3-1853538000*c[ 5]^4*c[4]-3586290300*c[5]^5*c[4]^4+1452229200*c[5]^3*c[4]^6-1362009490 0*c[5]^4*c[4]^5-441282100*c[5]^5*c[4]^2+2306397300*c[5]^5*c[4]^3+19625 59600*c[5]^4*c[4]^6-21763835720*c[5]^4*c[4]^3+1962559600*c[5]^5*c[4]^5 -15559009500*c[5]^3*c[4]^4+17785460128*c[5]^3*c[4]^3+2575125394*c[4]^4 *c[5]+25823191040*c[5]^4*c[4]^4+3389913740*c[4]^5*c[5]^3-76071765*c[5] *c[4]+2128579338*c[5]^3*c[4]-368451368*c[5]^2*c[4]-5694943812*c[4]^4*c [5]^2-3331815000*c[4]^6*c[5]^2+8849336340*c[5]^4*c[4]^2-8703556881*c[5 ]^3*c[4]^2+467001428*c[5]*c[4]^2+692679527*c[5]^2*c[4]^3+7862971280*c[ 4]^5*c[5]^2+882564200*c[4]^6*c[5]-2463048100*c[4]^5*c[5]-42699320*c[4] ^5)/c[5]/c[4]^2/(-24610*c[5]*c[4]^3-39020*c[5]^2*c[4]^2-621*c[4]^2-621 *c[5]^2+2500*c[4]^3+2500*c[5]^3+96100*c[5]^4*c[4]^3+96100*c[5]^3*c[4]^ 4-338300*c[5]^3*c[4]^3+621*c[5]*c[4]-24610*c[5]^3*c[4]+3726*c[5]^2*c[4 ]-25000*c[4]^4*c[5]^2-25000*c[5]^4*c[4]^2+138870*c[5]^3*c[4]^2+3726*c[ 5]*c[4]^2+138870*c[5]^2*c[4]^3), b[1] = 1/203700*(43130*c[5]*c[4]-7110 *c[5]-7110*c[4]+1879)/c[5]/c[4], b[4] = 1/60*(-1879+7110*c[5])/c[4]/(- 13432*c[4]^3+5040*c[4]^4+11787*c[4]^2-3395*c[4]-5040*c[5]*c[4]^3+13432 *c[5]*c[4]^2-11787*c[5]*c[4]+3395*c[5]), a[8,3] = 3/4*(1356881*c[5]*c[ 4]^3+2773147*c[5]^2*c[4]^2+9603*c[4]-19206*c[5]-33720*c[4]^2+67440*c[5 ]^2+25996*c[4]^3-51992*c[5]^3-553580*c[5]^4*c[4]^3-5182520*c[5]^3*c[4] ^4+5566540*c[5]^3*c[4]^3-1285680*c[4]^4*c[5]+494700*c[5]^4*c[4]^4+1484 100*c[4]^5*c[5]^3+148956*c[5]*c[4]+533384*c[5]^3*c[4]-645780*c[5]^2*c[ 4]+5454980*c[4]^4*c[5]^2+129980*c[5]^4*c[4]^2-2474952*c[5]^3*c[4]^2-61 1154*c[5]*c[4]^2-5911092*c[5]^2*c[4]^3-1660740*c[4]^5*c[5]^2+389940*c[ 4]^5*c[5])/c[4]^2/(-24610*c[5]*c[4]^3-39020*c[5]^2*c[4]^2-621*c[4]^2-6 21*c[5]^2+2500*c[4]^3+2500*c[5]^3+96100*c[5]^4*c[4]^3+96100*c[5]^3*c[4 ]^4-338300*c[5]^3*c[4]^3+621*c[5]*c[4]-24610*c[5]^3*c[4]+3726*c[5]^2*c [4]-25000*c[4]^4*c[5]^2-25000*c[5]^4*c[4]^2+138870*c[5]^3*c[4]^2+3726* c[5]*c[4]^2+138870*c[5]^2*c[4]^3), a[6,3] = 291/1536640000*(199303380* c[5]*c[4]^3-133961850*c[5]*c[4]^2+38125268*c[5]*c[4]-5476038*c[5]+2738 019*c[4]-26345200*c[5]^3*c[4]^2-447596800*c[5]^2*c[4]^3+308485800*c[5] ^2*c[4]^2-86042880*c[5]^2*c[4]+10538080*c[5]^2-79035600*c[4]^4*c[5]-52 69040*c[4]^2+57036000*c[5]^3*c[4]^3+171108000*c[4]^4*c[5]^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)/c[4]^2, a[7,2] = 0, a[9,7] = -39366/1925*(27 0*c[5]*c[4]-65*c[5]-65*c[4]+11)/(1296*c[5]*c[4]-1260*c[4]-1260*c[5]+12 25), a[7,3] = 35/746496*(559444262*c[5]*c[4]^3+1146562330*c[5]^2*c[4]^ 2+3921225*c[4]-7842450*c[5]-13881035*c[4]^2+27762070*c[5]^2+10918320*c [4]^3-21836640*c[5]^3-234063360*c[5]^4*c[4]^3-2218515840*c[5]^3*c[4]^4 +2357904320*c[5]^3*c[4]^3-535692260*c[4]^4*c[5]+213710400*c[5]^4*c[4]^ 4+641131200*c[4]^5*c[5]^3+60935735*c[5]*c[4]+224645184*c[5]^3*c[4]-266 270792*c[5]^2*c[4]+2288294280*c[4]^4*c[5]^2+54591600*c[5]^4*c[4]^2-104 4263864*c[5]^3*c[4]^2-250737464*c[5]*c[4]^2-2455881064*c[5]^2*c[4]^3-7 02190080*c[4]^5*c[5]^2+163774800*c[4]^5*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)/(270*c[5]*c[4]-65*c[5]-65*c[4]+11), a[5,4] = -c[5]^2 *(c[4]-c[5])/c[4]^2, `b*`[4] = -1/30*(-18162210*c[5]^2*c[4]^2+1123060* c[4]+2483776*c[5]-844870*c[4]^2-5569145*c[5]^2+3603550*c[5]^3-9075720* c[5]*c[4]-15150360*c[5]^3*c[4]+21990305*c[5]^2*c[4]+13356900*c[5]^3*c[ 4]^2+7107730*c[5]*c[4]^2-326211)/(-117936*c[5]*c[4]^3-219786*c[5]^2*c[ 4]^2+45931*c[4]+35455*c[5]-63628*c[4]^2-27160*c[5]^2+27936*c[4]^3-1756 28*c[5]*c[4]+142596*c[5]^2*c[4]+257796*c[5]*c[4]^2+104760*c[5]^2*c[4]^ 3-10185)/(140*c[4]-97)/(c[4]-c[5])/c[4], `b*`[8] = 0, a[4,2] = 0, a[5, 2] = 0, a[6,2] = 0, a[8,2] = 0, a[7,4] = -35/3359232*(76352513556*c[5] *c[4]^3-37493769796*c[5]^2*c[4]^2-5042885680*c[4]^4-6354462240*c[5]^4- 760717650*c[4]^2-2282152950*c[5]^2+3518021990*c[4]^3+8078762370*c[5]^3 +73760400144*c[5]^4*c[4]+126960883200*c[5]^5*c[4]^4-483412924800*c[5]^ 3*c[4]^6+855285580800*c[5]^4*c[4]^5+15886155600*c[5]^5*c[4]^2-83213537 760*c[5]^5*c[4]^3-179516736000*c[5]^4*c[4]^6+1057904062160*c[5]^4*c[4] ^3-59838912000*c[5]^5*c[4]^5-196613222400*c[4]^7*c[5]^2+45856944000*c[ 5]*c[4]^7+775984692280*c[5]^3*c[4]^4-871330979568*c[5]^3*c[4]^3-192181 525468*c[4]^4*c[5]-1410331700000*c[5]^4*c[4]^4+87582355680*c[4]^5*c[5] ^3+2662511775*c[5]*c[4]-83943480392*c[5]^3*c[4]+15367444505*c[5]^2*c[4 ]+465972983036*c[4]^4*c[5]^2+715804681920*c[4]^6*c[5]^2-390181793064*c [5]^4*c[4]^2+386930507534*c[5]^3*c[4]^2-19430872475*c[5]*c[4]^2-463257 21858*c[5]^2*c[4]^3-914127894080*c[4]^5*c[5]^2-181766144000*c[4]^6*c[5 ]+268431273240*c[4]^5*c[5]+179516736000*c[5]^3*c[4]^7+2292847200*c[4]^ 5)/(c[5]*c[4]^3+65*c[4]^4-65*c[5]^4-11*c[4]^3+11*c[5]^3+660*c[5]^4*c[4 ]+650*c[5]^5*c[4]^2-2700*c[5]^5*c[4]^3+8750*c[5]^4*c[4]^3-8750*c[5]^3* c[4]^4-660*c[4]^4*c[5]+2700*c[4]^5*c[5]^3-c[5]^3*c[4]-22*c[5]^2*c[4]+3 680*c[4]^4*c[5]^2-3680*c[5]^4*c[4]^2+180*c[5]^3*c[4]^2+22*c[5]*c[4]^2- 180*c[5]^2*c[4]^3-650*c[4]^5*c[5]^2)/(140*c[4]-97)/c[4]^2, a[9,8] = 1/ 2580*(9610*c[5]*c[4]-2500*c[5]-2500*c[4]+621)/(-c[4]+c[5]*c[4]+1-c[5]) , a[9,4] = 1/60*(-1879+7110*c[5])/c[4]/(-13432*c[4]^3+5040*c[4]^4+1178 7*c[4]^2-3395*c[4]-5040*c[5]*c[4]^3+13432*c[5]*c[4]^2-11787*c[5]*c[4]+ 3395*c[5]), a[7,5] = -35/3359232*(-43880090374*c[5]*c[4]^3-12887496777 0*c[5]^2*c[4]^2-380358825*c[4]+760717650*c[5]+782737200*c[4]^4-2292847 200*c[5]^4+1619302195*c[4]^2-3518021990*c[5]^2-2024158360*c[4]^3+50428 85680*c[5]^3+23587744320*c[5]^4*c[4]+59838912000*c[5]^4*c[4]^5+1904127 66720*c[5]^4*c[4]^3+370674951680*c[5]^3*c[4]^4-400988397920*c[5]^3*c[4 ]^3+43912180580*c[4]^4*c[5]-175259433600*c[5]^4*c[4]^4-126960883200*c[ 4]^5*c[5]^3-5634636595*c[5]*c[4]-50211339408*c[5]^3*c[4]+32719095804*c [5]^2*c[4]-240715314120*c[4]^4*c[5]^2-96291004320*c[5]^4*c[4]^2+202464 679088*c[5]^3*c[4]^2+20744261428*c[5]*c[4]^2+257143912368*c[5]^2*c[4]^ 3+83213537760*c[4]^5*c[5]^2-15886155600*c[4]^5*c[5])/c[5]/(-1637*c[5]* c[4]^3+3080*c[5]^2*c[4]^2-6305*c[4]^4+7845*c[5]^4+1067*c[4]^3-1067*c[5 ]^3-64160*c[5]^4*c[4]-378000*c[5]^6*c[4]^3+92400*c[5]^5*c[4]+91000*c[5 ]^6*c[4]^2+378000*c[5]^4*c[4]^5-578250*c[5]^5*c[4]^2+1486900*c[5]^5*c[ 4]^3-848750*c[5]^4*c[4]^3+1363950*c[5]^3*c[4]^4-25200*c[5]^3*c[4]^3+73 120*c[4]^4*c[5]-1225000*c[5]^4*c[4]^4-352900*c[4]^5*c[5]^3-2983*c[5]^3 *c[4]+2134*c[5]^2*c[4]-449360*c[4]^4*c[5]^2+382160*c[5]^4*c[4]^2-17460 *c[5]^3*c[4]^2-2134*c[5]*c[4]^2+17600*c[5]^2*c[4]^3+63050*c[4]^5*c[5]^ 2-9100*c[5]^5), c[7] = 35/36, a[5,1] = 1/4*c[5]*(2*c[5]^2-5*c[5]*c[4]+ 4*c[4]^2)/c[4]^2, a[8,5] = -1/2*(1188410971*c[5]*c[4]^3+4254299841*c[5 ]^2*c[4]^2+10867395*c[4]-21734790*c[5]-21349660*c[4]^4+209864528*c[5]^ 4-45852274*c[4]^2+122352710*c[5]^2+56334539*c[4]^3-244972528*c[5]^3-21 05495216*c[5]^4*c[4]+672063840*c[5]^5*c[4]-4924821600*c[5]^5*c[4]^4-51 99215200*c[5]^4*c[4]^5-2736563040*c[5]^5*c[4]^2+5392638720*c[5]^5*c[4] ^3-16775019580*c[5]^4*c[4]^3+1662192000*c[5]^5*c[4]^5-17216165420*c[5] ^3*c[4]^4+18673648658*c[5]^3*c[4]^3-1214082410*c[4]^4*c[5]+15367291680 *c[5]^4*c[4]^4+5852440760*c[4]^5*c[5]^3+149398222*c[5]*c[4]+2366232752 *c[5]^3*c[4]-1093066993*c[5]^2*c[4]+8009127410*c[4]^4*c[5]^2+850257378 8*c[5]^4*c[4]^2-9431184222*c[5]^3*c[4]^2-543274093*c[5]*c[4]^2-8536013 308*c[5]^2*c[4]^3-2756699660*c[4]^5*c[5]^2+441282100*c[4]^5*c[5]-65509 920*c[5]^5)/c[5]/(9373702*c[5]*c[4]^3-10422864*c[5]^2*c[4]^2+8487500*c [4]^4-13698932*c[5]^4-2108295*c[4]^3+2108295*c[5]^3-484344000*c[5]^7*c [4]^3+87579862*c[5]^4*c[4]+126000000*c[5]^7*c[4]^2+2385503200*c[5]^6*c [4]^3-212706160*c[5]^5*c[4]-1579032000*c[5]^5*c[4]^4-909704800*c[5]^6* c[4]^2+124034400*c[5]^6*c[4]-932471200*c[5]^4*c[4]^5+1322898440*c[5]^5 *c[4]^2-2955473100*c[5]^5*c[4]^3+991027100*c[5]^4*c[4]^3+484344000*c[5 ]^5*c[4]^5-2353084940*c[5]^3*c[4]^4+127107760*c[5]^3*c[4]^3-104530950* c[4]^4*c[5]+3329118400*c[5]^4*c[4]^4+536059500*c[4]^5*c[5]^3+6260594*c [5]^3*c[4]-4216590*c[5]^2*c[4]+690590770*c[4]^4*c[5]^2-592392370*c[5]^ 4*c[4]^2+55181630*c[5]^3*c[4]^2+4216590*c[5]*c[4]^2-62340382*c[5]^2*c[ 4]^3-84875000*c[4]^5*c[5]^2+24109840*c[5]^5-12600000*c[5]^6), c[6] = 9 7/140, a[6,5] = -97/768320000*(65076200*c[5]^2*c[4]^2-2634520*c[4]^2+9 12673*c[4]-21510720*c[5]^2*c[4]-41286110*c[5]*c[4]^2+1901200*c[4]^3-26 345200*c[4]^4*c[5]+55027260*c[5]*c[4]^3-83381200*c[5]^2*c[4]^3+1358659 6*c[5]*c[4]+38024000*c[4]^4*c[5]^2+2634520*c[5]^2-1825346*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*`[6] = 300125/6402*(813540*c[5]*c[4]^3+2743860*c[5]^2*c[4]^2-14917 5*c[4]-149175*c[5]+246728*c[4]^2+246728*c[5]^2-129980*c[4]^3-129980*c[ 5]^3+989400*c[5]^3*c[4]^3+28809+833794*c[5]*c[4]+813540*c[5]^3*c[4]-14 68008*c[5]^2*c[4]-1603260*c[5]^3*c[4]^2-1468008*c[5]*c[4]^2-1603260*c[ 5]^2*c[4]^3)/(291-3276*c[5]*c[4]^2-3276*c[5]^2*c[4]+776*c[5]^2-1013*c[ 4]-1013*c[5]+3976*c[5]*c[4]+2910*c[5]^2*c[4]^2+776*c[4]^2)/(19600*c[5] *c[4]-13580*c[4]-13580*c[5]+9409), a[7,6] = -264110000/636417*(5*c[5]* c[4]+1-2*c[4]-2*c[5])*(36*c[4]-35)*(36*c[5]-35)/(270*c[5]*c[4]-65*c[5] -65*c[4]+11)/(140*c[4]-97)/(140*c[5]-97), c[3] = 2/3*c[4], a[4,3] = 3/ 4*c[4], a[3,2] = 2/9*c[4]^2/c[2], b[8] = 1/2580*(9610*c[5]*c[4]-2500*c [5]-2500*c[4]+621)/(c[4]-1)/(-1+c[5]), a[9,1] = 1/203700*(43130*c[5]*c [4]-7110*c[5]-7110*c[4]+1879)/c[5]/c[4], `b*`[7] = -2187/3850*(993100* c[5]*c[4]^3+3585980*c[5]^2*c[4]^2-86973*c[4]-86973*c[5]+9603+199960*c[ 4]^2+199960*c[5]^2-126100*c[4]^3-126100*c[5]^3+1571400*c[5]^3*c[4]^3+6 98198*c[5]*c[4]+993100*c[5]^3*c[4]-1569400*c[5]^2*c[4]-2327700*c[5]^3* c[4]^2-1569400*c[5]*c[4]^2-2327700*c[5]^2*c[4]^3)/(5133456*c[5]*c[4]^3 +16973166*c[5]^2*c[4]^2-1607585*c[4]-1607585*c[5]+2226980*c[4]^2+22269 80*c[5]^2-977760*c[4]^3-977760*c[5]^3+356475+3771360*c[5]^3*c[4]^3+780 0496*c[5]*c[4]+5133456*c[5]^3*c[4]-11313468*c[5]^2*c[4]-7912296*c[5]^3 *c[4]^2-11313468*c[5]*c[4]^2-7912296*c[5]^2*c[4]^3), a[7,1] = 35/65169 1008*(-48546197536*c[5]*c[4]^3+24582169976*c[5]^2*c[4]^2+4048316720*c[ 4]^4+6354462240*c[5]^4+760717650*c[4]^2+2282152950*c[5]^2-3238604390*c [4]^3-8078762370*c[5]^3-66937222944*c[5]^4*c[4]-132771758400*c[5]^5*c[ 4]^4+53797420800*c[5]^3*c[4]^6-509544158400*c[5]^4*c[4]^5-15886155600* c[5]^5*c[4]^2+83767181760*c[5]^5*c[4]^3+74423577600*c[5]^4*c[4]^6-7916 16822080*c[5]^4*c[4]^3+74423577600*c[5]^5*c[4]^5-558190640280*c[5]^3*c [4]^4+631602983496*c[5]^3*c[4]^3+90720981740*c[4]^4*c[5]+950191840640* c[5]^4*c[4]^4+123957263360*c[4]^5*c[5]^3-2662511775*c[5]*c[4]+75178654 952*c[5]^3*c[4]-12892140905*c[5]^2*c[4]-201025695268*c[4]^4*c[5]^2-120 570131520*c[4]^6*c[5]^2+320331834024*c[5]^4*c[4]^2-307879677422*c[5]^3 *c[4]^2+16406597375*c[5]*c[4]^2+24767106946*c[5]^2*c[4]^3+280213188160 *c[4]^5*c[5]^2+31772311200*c[4]^6*c[5]-87473248840*c[4]^5*c[5]-1565474 400*c[4]^5)/(-660*c[5]*c[4]^3-840*c[5]^2*c[4]^2-11*c[4]^2-11*c[5]^2+65 *c[4]^3+65*c[5]^3+2700*c[5]^4*c[4]^3+2700*c[5]^3*c[4]^4-9400*c[5]^3*c[ 4]^3+11*c[5]*c[4]-660*c[5]^3*c[4]+66*c[5]^2*c[4]-650*c[4]^4*c[5]^2-650 *c[5]^4*c[4]^2+3680*c[5]^3*c[4]^2+66*c[5]*c[4]^2+3680*c[5]^2*c[4]^3)/c [5]/c[4]^2, a[6,1] = 97/1536640000*(-78769820*c[5]*c[4]^3+87138010*c[5 ]^2*c[4]^2+3802400*c[4]^4+1825346*c[4]^2+5476038*c[5]^2-5269040*c[4]^3 -10538080*c[5]^3-95060000*c[5]^4*c[4]^3-538412000*c[5]^3*c[4]^4+627524 800*c[5]^3*c[4]^3+99078520*c[4]^4*c[5]+109760000*c[5]^4*c[4]^4+1097600 00*c[4]^5*c[5]^3-6388711*c[5]*c[4]+89845280*c[5]^3*c[4]-32856228*c[5]^ 2*c[4]-21870800*c[4]^4*c[5]^2+26345200*c[5]^4*c[4]^2-342276200*c[5]^3* c[4]^2+32442232*c[5]*c[4]^2-80989380*c[5]^2*c[4]^3+76048000*c[4]^5*c[5 ]^2-52690400*c[4]^5*c[5])/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), a[9,6] = 21008750/137643*(170*c[5]*c[4]-67*c[5]-67*c[4]+33)/(19600 *c[5]*c[4]-13580*c[4]-13580*c[5]+9409), a[8,6] = -3687250/1067*(25730* c[5]*c[4]+5181-10327*c[4]-10327*c[5])*(c[4]-1)*(-1+c[5])/(140*c[4]-97) /(140*c[5]-97)/(9610*c[5]*c[4]-2500*c[5]-2500*c[4]+621), a[8,7] = -564 246/385*(270*c[5]*c[4]-65*c[5]-65*c[4]+11)*(c[4]-1)*(-1+c[5])/(36*c[5] -35)/(36*c[4]-35)/(9610*c[5]*c[4]-2500*c[5]-2500*c[4]+621), a[8,4] = - 1/2*(-2739057579*c[5]*c[4]^3+1528762665*c[5]^2*c[4]^2+244972528*c[4]^4 +176512840*c[5]^4+21734790*c[4]^2+65204370*c[5]^2-122352710*c[4]^3-228 958800*c[5]^3-5580086400*c[4]^8*c[5]^2-2225805884*c[5]^4*c[4]+65509920 *c[4]^6-5852440760*c[5]^5*c[4]^4+15134726280*c[5]^3*c[4]^6-63077943400 *c[5]^4*c[4]^5-441282100*c[5]^5*c[4]^2+2756699660*c[5]^5*c[4]^3+287811 04800*c[5]^4*c[4]^6-40301700634*c[5]^4*c[4]^3+5199215200*c[5]^5*c[4]^5 +25873088320*c[4]^7*c[5]^2-6501469120*c[5]*c[4]^7-47423890938*c[5]^3*c [4]^4+35929350356*c[5]^3*c[4]^3+7672964442*c[4]^4*c[5]+68734626820*c[5 ]^4*c[4]^4-1662192000*c[5]^5*c[4]^6+20666212460*c[4]^5*c[5]^3-76071765 *c[5]*c[4]+2613788621*c[5]^3*c[4]-506550698*c[5]^2*c[4]-14385172050*c[ 4]^4*c[5]^2-46171622120*c[4]^6*c[5]^2+12899781458*c[5]^4*c[4]^2-134016 07179*c[5]^3*c[4]^2+631963550*c[5]*c[4]^2+166854717*c[5]^2*c[4]^3+3900 9521196*c[4]^5*c[5]^2+12858656680*c[4]^6*c[5]-13157184608*c[4]^5*c[5]+ 1310198400*c[4]^8*c[5]+4986576000*c[4]^8*c[5]^3-18276196800*c[5]^3*c[4 ]^7-4986576000*c[5]^4*c[4]^7-209864528*c[4]^5)/(1226*c[5]*c[4]^3+2500* c[4]^4-2500*c[5]^4-621*c[4]^3+621*c[5]^3+24610*c[5]^4*c[4]+25000*c[5]^ 5*c[4]^2-96100*c[5]^5*c[4]^3+313300*c[5]^4*c[4]^3-313300*c[5]^3*c[4]^4 -24610*c[4]^4*c[5]+96100*c[4]^5*c[5]^3-1226*c[5]^3*c[4]-1242*c[5]^2*c[ 4]+138870*c[4]^4*c[5]^2-138870*c[5]^4*c[4]^2+14410*c[5]^3*c[4]^2+1242* c[5]*c[4]^2-14410*c[5]^2*c[4]^3-25000*c[4]^5*c[5]^2)/(-8392*c[4]+5040* c[4]^2+3395)/c[4]^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 0 "" 0 "" {TEXT -1 98 "A lengthy computation giv es an expression for the square of the principal error norm in terms o f " }{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[5]" "6#&%\"cG6#\"\"&" }{TEXT -1 2 ". " }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 261 "errterms6_8 := PrincipalErrorTerms (6,8,'expanded'):\nsm := 0:\nfor ct to nops(errterms6_8) do\n print( ct);\n sm := sm+(simplify(subs(eB,errterms6_8[ct])))^2;\nend do:\nsm := simplify(sm):\nprin_err_norm_sqrd := unapply(%,c[2],c[4],c[5]):\np rin_err_norm_sqrd(u,v,w);" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 18 "prin_err_norm_sqrd" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 4128 "prin_err_norm_sqrd := (u,v,w)->1/ 3950456832000000*(-1008953225158*w^5+39176587901*v^4+264721907341*v^6- 203340185158*v^5+1285510755341*w^6+5102386458396520*w^3*v^7+5110211802 70956*v^6*w^2-22041806722648*w^4*v^2-2318807273674000*w^6*v^7+63536385 6303*w^2*v^2-9349238053641600*w^5*v^7+4443007053098600*w^6*v^6-5189413 369600000*v^9*w^4+31607043210000*w^8*v^6+3625337592320000*w^3*v^9-8513 29113600000*w^3*v^10+1382846976000000*w^4*v^10+497069147210000*w^6*v^8 -1229638272000000*w^5*v^9+131030486720000*v^10*w^2-588390980800000*v^9 *w^2+5976001052520*w^7*v^3+1010549836464000*v^8*w^2+6479727286442*w^3* v^2-8341150671958*w^2*v^3-197462718290632*v^5*w^2+123037477200*w^3*u-6 20413560000*w^4*u+24717481359328*v^6*w+5771596021200000*w^4*v^8*u-2021 40338400000*w^6*v^6*u-610319955600000*w^5*v^7*u-556829151948000*w^5*u* v^3+8386349115600*w^4*u*v+37356389460000*v^6*u*w+49403844000000*w^6*v^ 7*u-88730691840000*w^6*v^4*u+202503331800000*w^6*v^5*u-361277839332000 0*v^8*w^3*u-9377727053400000*w^4*v^7*u-73794888000000*w^5*v^8*u+184623 44040000*w^6*u*v^3-1503306000000*w^6*u*v^2-14433890104800*w^5*u*v+8209 24502400000*v^9*w^3*u-10236286167300*u^2*v^3*w+692216797444200*u^2*v^3 *w^3+2230275036240000*w^5*v^6*u-515681538750000*u^2*v^5*w^2-1413838864 8000*v^7*u*w+567377017200000*v^8*w^2*u-1333459584000000*v^9*w^4*u-3102 6735315000*u^2*v^5*w-126350826480000*v^9*w^2*u+25612441914600*u^2*v^4* w-2080971447444000*u^2*v^4*w^3-231839641800*u^2*v*w-74741888727000*u^2 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v^4-20803556060646*w^3*v^3+2520698744746*v^4*w-868550378389704*w^4*v^4 +198104491901*w^4+7826920712052000*w^5*v^6+664434450151204*v^5*w^3+152 092884118400*w^7*v^5-40933485808000*w^7*v^4-587600615002*w^3*v+5149053 0096552*v^4*w^2-403212917654*w^4*v-440049810820*w^7*v^2-86500362441344 0*w^6*v^3+14176017931716*w^5*v+695000565496980*w^5*v^4+191063527888556 *w^6*v^2-2490705677637440*w^3*v^6-23802159197472*w^6*v+173215853753698 0*w^4*v^5-71023377175432*w^5*v^2+77753563290404*w^5*v^3-35295469815492 80*w^5*v^5+6785809540814100*w^4*v^8-924313560004120*w^4*v^6+6321408642 0000*w^7*v^7-3036320974048000*w^4*v^7-4238246715708000*w^6*v^5+2413641 249019880*w^6*v^4-10713846584284*v^5*w)/(10*w^3*v^2+10*w^2*v^3-30*w^2* v^2+6*w^2*v-w^2+6*w*v^2+w*v-v^2)^2:" }}}{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 54 "Preliminary investigation s suggested that the values " }{XPPEDIT 18 0 "c[2] = 25/403;" "6#/&% \"cG6#\"\"#*&\"#D\"\"\"\"$.%!\"\"" }{TEXT -1 3 ", " }{XPPEDIT 18 0 "c [4] = 65/326;" "6#/&%\"cG6#\"\"%*&\"#l\"\"\"\"$E$!\"\"" }{TEXT -1 7 " \+ and " }{XPPEDIT 18 0 "c[5] = 47/88;" "6#/&%\"cG6#\"\"&*&\"#Z\"\"\"\" #))!\"\"" }{TEXT -1 89 " give a value for the (square of the) princip al error norm that is close to the minimum." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 124 "Using a one dimensional minimi zation procedure and cycling around the nodes gives very slow converge nce towards the minimum." }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 494 "Digits := 30:\nc_2 := 25/403: c_4 := 65/326: c_5 := 47/88:\nfor ct to 100000 do\n c_2 := findmin(prin_err_norm_sqrd(c2,c_4,c_5),c2=\{0.05 ,c_2,0.13\},convergence=location)[1];\n c_4 := findmin(prin_err_norm _sqrd(c_2,c4,c_5),c4=\{0.19,c_4,0.22\},convergence=location)[1];\n m n := findmin(prin_err_norm_sqrd(c_2,c_4,c5),c5=\{0.5,c_5,0.63\},conver gence=location); \n c_5 := mn[1]:\n if `mod`(ct,1000)=0 then\n \+ print(c[2]=c_2,c[4]=c_4,c[5]=c_5);\n print(mn[2]);\n end if; \nend do:\nDigits := 10:" }}{PARA 11 "" 1 "" {XPPMATH 20 "6%/&%\"cG6# \"\"#$\"?k'Q)G&[(*H=jr53`'p!#J/&F%6#\"\"%$\"?Z5()=N@*\\Lz)=gb)4#!#I/&F %6#\"\"&$\"?d(R*z6n%)[`I$zz4_&F1" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#$ \"?=9Z.GuRc'Q4Fd-C)!#R" }}{PARA 11 "" 1 "" {XPPMATH 20 "6%/&%\"cG6#\" \"#$\"?:@_eLf\"fs'f1G$\\2(!#J/&F%6#\"\"%$\"?9xSaV')442U)HLf7#!#I/&F%6# \"\"&$\"?kPvs760rd)>X$)4d&F1" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#$\"?hi lQ1#o=h&)G-@8R(!#R" }}{PARA 11 "" 1 "" {XPPMATH 20 "6%/&%\"cG6#\"\"#$ \"?Fw5k@p4!3>TlBl7(!#J/&F%6#\"\"%$\"?H\\9mcbD'*=tnJnQ@!#I/&F%6#\"\"&$ \"?Rt.RD<4S=NHIu%f&F1" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#$\"?%Qwq_`R/) oK'eD3:(!#R" }}{PARA 259 "" 0 "" {TEXT -1 1 ":" }}{PARA 259 "" 0 "" {TEXT -1 1 ":" }}{PARA 11 "" 1 "" {XPPMATH 20 "6%/&%\"cG6#\"\"#$\"?BP \\7*[PjM$)eEwb@(!#J/&F%6#\"\"%$\"?itf3;&yk_T&GuQg@!#I/&F%6#\"\"&$\"?PU !\\!*)\\%)yFw?C(fj&F1" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#$\"?JiXX2D1b$ >f2sw)p!#R" }}{PARA 11 "" 1 "" {XPPMATH 20 "6%/&%\"cG6#\"\"#$\"?'3hKXF i7N$)eEwb@(!#J/&F%6#\"\"%$\"?`:m!Rrow_T&GuQg@!#I/&F%6#\"\"&$\"?*\\94zV I6yi2Usfj&F1" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#$\"?_NN%R!=1b$>f2sw)p! #R" }}{PARA 11 "" 1 "" {XPPMATH 20 "6%/&%\"cG6#\"\"#$\"?f)ot>*p*[N$)eE wb@(!#J/&F%6#\"\"%$\"?%*o67BqaG:aGuQg@!#I/&F%6#\"\"&$\"?ejL=!*p\"Gyi2U sfj&F1" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#$\"?>syvB?1b$>f2sw)p!#R" }}} {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 78 "The follo wing graphs give a visual check that we have found a (local) minimum. " }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 383 "c_2 := .72155762659e-1: pp := .69876720759e-9:\np1 := evalf[30](plot(prin_err_norm_sqrd(c[2], .216038742854,.56359724208),c[2]=0.065..0.0793,\n color=COLOR(RGB,. 5,0,.9))):\np2 := plot([[[c_2,pp]]$4],style=point,symbol=[circle$2,dia mond,cross],symbolsize=[12,10$3],\n color=[black,red$3]):\np lots[display]([p1,p2],font=[HELVETICA,9],view=[0.065..0.0793,6.986e-10 ..6.9966e-10]);" }}{PARA 13 "" 1 "" {GLPLOT2D 367 347 347 {PLOTDATA 2 "6*-%'CURVESG6$7S7$$\"#l!\"$$\"?xk_olvF7$>wDZm*p!#R7$$\"?LLLLLLLL$e/$* p6`'!#J$\"?,'RqI'pac/r$Q#)e*pF-7$$\"?nmmmmmm;aG?2HelF1$\"?T%>wJ\"\\]WT z!eW_*pF-7$$\"?LLLLLLLL3#\\)3z)e'F1$\"?*R_>(*fN9S'>4\"eX*pF-7$$\"?LLLL LLLLeWCK\\>mF1$\"?N\"*H&)yDAyuIB+!R*pF-7$$\"?nmmmmmm;/6T'\\+l'F1$\"?( \\Qw?=iLsi[(yF$*pF-7$$\"?LLLLLLL$3_3Nz$ymF1$\"?CqrVvf'HWEUIIF*pF-7$$\" ?+++++++](oV78xq'F1$\"?;hj$>R/s1u[(H>#*pF-7$$\"?LLLLLLL$3-\\:]!QnF1$\" ??3*)*eq7Ya9y**o;*pF-7$$\"?+++++++]P>%*)*GonF1$\"??5&f%)[9#eFK4)y6*pF- 7$$\"?nmmmmmmm;*H=&R*z'F1$\"?\"*p8%p+.B%pEO!32*pF-7$$\"?LLLLLLLLeHQFzE oF1$\"?KC9(3!>FP!GNZ@.*pF-7$$\"?++++++++]Z'*ejdoF1$\"?FubRU];JqR\"y<** )pF-7$$\"?++++++++]i*p0'))oF1$\"?,-Tp8XVvu0')fa*)pF-7$$\"?++++++++]_V3 X=pF1$\"?Ahr(od2)*[Uh]>#*)pF-7$$\"?LLLLLLL$3-M4`b%pF1$\"?q)fx4bz+Q)z$3 ]*))pF-7$$\"?nmmmmmmm;*)*G!yxpF1$\"?*)3BhP6[45TGKm))pF-7$$\"?nmmmmmmmm c_430qF1$\"?)QaUR72E-M)4([%))pF-7$$\"?+++++++](=M6Qo.(F1$\"?I>/\"))G!4 *\\tv/K#))pF-7$$\"?nmmmmmmmm\"=e[\\1(F1$\"?)30O_+#=uzoi(p!))pF-7$$\"?+ ++++++](=4A!z&4(F1$\"?8SY8w0z4hUzN#z)pF-7$$\"?+++++++]iva(e^7(F1$\"?G. 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g!e3^wdQJ*fDr&e*pF-7$$\"$$z!\"%$\"?f9L8Womr8yn$=m*pF--%&COLORG6&%$RGBG $\"\"&!\"\"$\"\"!Fb[l$\"\"*F`[l-F$6&7#7$$\"3c++!fEwb@(!#>$\"3=++!f2sw) p!#F-%'COLOURG6&F][lFb[lFb[lFb[l-%'SYMBOLG6$%'CIRCLEG\"#7-%&STYLEG6#%& POINTG-F$6&Fg[l-F`\\l6&F][l$\"*++++\"!\")Fa[lFa[l-Fc\\l6$Fe\\l\"#5Fg\\ l-F$6&Fg[lF]]l-Fc\\l6$%(DIAMONDGFd]lFg\\l-F$6&Fg[lF]]l-Fc\\l6$%&CROSSG Fd]lFg\\l-%%FONTG6$%*HELVETICAGFd[l-%+AXESLABELSG6%Q%c[2]6\"Q!Fg^l-F`^ l6#%(DEFAULTG-%%VIEWG6$;F(Fez;$\"%')p!#8$\"&m*p!#9" 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" }}}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 394 "c_4 := .216038742854: pp := .69876720759e- 9:\np1 := evalf[30](plot(prin_err_norm_sqrd(.72155762659e-1,c[4],.5635 9724208),c[4]=0.21603..0.2160475,\n color=COLOR(RGB,0,.7,.2))):\np2 := plot([[[c_4,pp]]$4],style=point,symbol=[circle$2,diamond,cross],sy mbolsize=[12,10$3],\n color=[black,cyan$3]):\nplots[display] ([p1,p2],font=[HELVETICA,9],view=[0.21603..0.2160475,6.986e-10..6.9966 e-10]);" }}{PARA 13 "" 1 "" {GLPLOT2D 400 369 369 {PLOTDATA 2 "6*-%'CU RVESG6$7S7$$\"&.;#!\"&$\"?aQ-3s#[(3rrvFg'*p!#R7$$\"?nmmmmm;z>]9QIg@!#I $\"?)G,Wlm&3!GPy.Ue*pF-7$$\"?LLLLL$3_vzM82.;#F1$\"?4*fCpSVoWJ-c2_*pF-7 $$\"?nmmmmmT&QaKg@ F1$\"?;9]G!zH>pVVHq@*pF-7$$\"?nmmmm;aj>,K\"H.;#F1$\"?2\"Gv$[S%p&))oD&[ ;*pF-7$$\"?+++++]Pf$zE$GLg@F1$\"?!>?Y**G>gw)HV.;\"*pF-7$$\"?LLLLLL$ekq #RmLg@F1$\"?!o$>F\"*[6ir/\\9p!*pF-7$$\"?nmmmmm\"HK=@**R.;#F1$\"?d'o!\\ sKz'Hg6S1.*pF-7$$\"?++++++](oImwV.;#F1$\"?Q(\\Ww:uk@6YF/**)pF-7$$\"?++ ++++]i:kcvMg@F1$\"?xG^)3D'zY;a?R`*)pF-7$$\"?++++++]7$>!47Ng@F1$\"?eOFP #3)\\BJOB(3#*)pF-7$$\"?nmmmm;a8#Gd_a.;#F1$\"?#*o&e`bo$=B1!RS*))pF-7$$ \"?LLLLLL$e*ehp%e.;#F1$\"?Wq_kVm7W))pF-7$$\"?+++++](=#>(ppl.;#F1$\"?!eTJLH&yN(QCmD#))p F-7$$\"?LLLLLLL3U1P\"p.;#F1$\"?2d%*)yrpV\"**QPV1))pF-7$$\"?+++++](=n!R 6HPg@F1$\"?!ffG8)=Zs%3X?>z)pF-7$$\"?+++++]i:?W0lPg@F1$\"?sK#>F\"Rg@F1$\"?Z9H4WG>&HcJT*o()pF-7$$\"?+++ ++]7yNeIYRg@F1$\"?I\"[;R0L/.(*\\'Ht()pF-7$$\"?nmmmmmTN,.e#)Rg@F1$\"?yB vIM7E0xhs(4y)pF-7$$\"?+++++++vt`0?Sg@F1$\"?HWa#*4*>zdnug@z)pF-7$$\"?++ ++++v$4U? 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"" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 36 "#-------- ---------------------------" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 152 "nds := [c[2]=.72155762659e-1,c[4]=.216038742854,c[5]=.56359724208 ]:\nevalf[10](%);\nfor dgt from 7 by -1 to 4 do\n map(convert,nds,ra tional,dgt);\nend do;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#7%/&%\"cG6#\" \"#$\"+mid:s!#6/&F&6#\"\"%$\"+HuQg@!#5/&F&6#\"\"&$\"+@C(fj&F2" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#7%/&%\"cG6#\"\"##\"$j\"\"%fA/&F&6#\"\" %#\"$D'\"%$*G/&F&6#\"\"&#\"%t:\"%\"z#" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#7%/&%\"cG6#\"\"##\"#y\"%\"3\"/&F&6#\"\"%#\"$K\"\"$6'/&F&6#\"\"&#\"$ d#\"$c%" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#7%/&%\"cG6#\"\"##\"#y\"%\"3 \"/&F&6#\"\"%#\"#N\"$i\"/&F&6#\"\"&#\"#J\"#b" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#7%/&%\"cG6#\"\"##\"\"(\"#(*/&F&6#\"\"%#\"\")\"#P/&F&6# \"\"&#\"#J\"#b" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 56 "The minimum value for the principal error norm is . . . \+ " }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 90 "evalf[25](prin_err_norm_ sqrd(.72155762659e-1,.216038742854,.56359724208)):\nevalf(sqrt(%));" } }{PARA 11 "" 1 "" {XPPMATH 20 "6#$\"+E0UVE!#9" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 9 "Setting " }{XPPEDIT 18 0 "c[4] = 35/162;" "6#/&%\"cG6#\"\"%*&\"#N\"\"\"\"$i\"!\"\"" }{TEXT -1 7 " and " }{XPPEDIT 18 0 "c[5] = 31/55;" "6#/&%\"cG6#\"\"&*&\"#J\"\" \"\"#b!\"\"" }{TEXT -1 46 " the principal error norm is a minimum whe n " }{XPPEDIT 18 0 "c[2] = 7/95;" "6#/&%\"cG6#\"\"#*&\"\"(\"\"\"\"#&* !\"\"" }{TEXT -1 19 " (approximately). " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 112 "mn := evalf[20](find min(prin_err_norm_sqrd(c2,35/162,31/55),c2=0.07..0.075)):\nc[2]=mn[1]; \nconvert(%,rational,4);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/&%\"cG6# \"\"#$\"5Jm5f4u1oqt!#@" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/&%\"cG6#\" \"##\"\"(\"#&*" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 6 "When " }{XPPEDIT 18 0 "c[2] = 7/95;" "6#/&%\"cG6#\"\"#*& \"\"(\"\"\"\"#&*!\"\"" }{TEXT -1 3 ", " }{XPPEDIT 18 0 "c[4] = 35/162 ;" "6#/&%\"cG6#\"\"%*&\"#N\"\"\"\"$i\"!\"\"" }{TEXT -1 7 " and " } {XPPEDIT 18 0 "c[5] = 31/55;" "6#/&%\"cG6#\"\"&*&\"#J\"\"\"\"#b!\"\"" }{TEXT -1 67 ", the principal error norm is given (approximately) as follows. " }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 65 "evalf[15](pri n_err_norm_sqrd(7/95,35/162,31/55)):\nevalf(sqrt(%));" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#$\"+6OWXE!#9" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }} {PARA 0 "" 0 "" {TEXT -1 54 "#---------------------------------------- -------------" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}} {PARA 0 "" 0 "" {TEXT -1 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 1734 "ee := \{c[2]=7/95,\nc[3] =35/243,\nc[4]=35/162,\nc[5]=31/55,\nc[6]=97/140,\nc[7]=35/36,\nc[8]=1 ,\nc[9]=1,\n\na[2,1]=7/95,\na[3,1]=385/118098,\na[3,2]=16625/118098,\n a[4,1]=35/648,\na[4,2]=0,\na[4,3]=35/216,\na[5,1]=262364129/407618750, \na[5,2]=0,\na[5,3]=-996909687/407618750,\na[5,4]=482147154/203809375, \na[6,1]=-36974597989227800011/27927480760535300000,\na[6,2]=0,\na[6,3 ]=40382359908551133219/7207091809170400000,\na[6,4]=-58370690823600963 04407/1395022708312545550000,\na[6,5]=13508218613909220883/22593673904 425227520,\na[7,1]=756727539023617977739/325483930565422777440,\na[7,2 ]=0,\na[7,3]=-11344125743485787/1187841741251840,\na[7,4]=934031615196 31272506393/11188675650069067742640,\na[7,5]=-381159214011163809768662 5/3308811923451740272240896,\na[7,6]=102601928198000/102454959175119, \na[8,1]=250792505789354790081/66169955021507254750,\na[8,2]=0,\na[8,3 ]=-349400598312667239/22005305959929250,\na[8,4]=177284792582644204697 02254/1289712860941995340762625,\na[8,5]=-61925787315598406758085/2776 4881218432626442221,\na[8,6]=43795320281930/26932915012729,\na[8,7]=-9 6493150422/1790131071575,\na[9,1]=1414477/22101450,\na[9,2]=0,\na[9,3] =0,\na[9,4]=6047117272638/18236245267975,\na[9,5]=31062711625/12365010 3496,\na[9,6]=27376502125/148102904499,\na[9,7]=26504253/76308925,\na[ 9,8]=-23459/131064,\n\nb[1]=1414477/22101450,\nb[2]=0,\nb[3]=0,\nb[4]= 6047117272638/18236245267975,\nb[5]=31062711625/123650103496,\nb[6]=27 376502125/148102904499,\nb[7]=26504253/76308925,\nb[8]=-23459/131064, \n\n`b*`[1]=9659334433759/154907736963000,\n`b*`[2]=0,\n`b*`[3]=0,\n`b *`[4]=21597935405438536443/63908374454260348250,\n`b*`[5]=230190597481 295825/1039987387676705088,\n`b*`[6]=8367042728056775/3774706805306258 4,\n`b*`[7]=28150317302067/97244486689000,\n`b*`[8]=-77493123437/61241 3141440,\n`b*`[9]=-1/160\}:" }}}{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 comp onents are the principal error terms of the 8 stage, order 6 scheme (t he error terms of order 7)." }}{PARA 0 "" 0 "" {TEXT -1 5 "Let " } {XPPEDIT 18 0 "`T*`[5,9];" "6#&%#T*G6$\"\"&\"\"*" }{TEXT -1 145 " den ote the vector whose components are the principal error terms of the e mbedded 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 " d enote the vector whose components are the error terms of order 7 of th e 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#\"\"($\")c(4L\"!\"(" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#/&%\"CG6#\"\"($\")7;R8!\"(" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}}{PARA 0 "" 0 "" {TEXT -1 52 "#------------------------------------ ---------------" }}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 26 "absolute stabi lity regions" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 35 "coeffici ents of the combined scheme" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1734 "ee := \{c[2]=7/95,\nc[3]=35/243, \nc[4]=35/162,\nc[5]=31/55,\nc[6]=97/140,\nc[7]=35/36,\nc[8]=1,\nc[9]= 1,\n\na[2,1]=7/95,\na[3,1]=385/118098,\na[3,2]=16625/118098,\na[4,1]=3 5/648,\na[4,2]=0,\na[4,3]=35/216,\na[5,1]=262364129/407618750,\na[5,2] =0,\na[5,3]=-996909687/407618750,\na[5,4]=482147154/203809375,\na[6,1] =-36974597989227800011/27927480760535300000,\na[6,2]=0,\na[6,3]=403823 59908551133219/7207091809170400000,\na[6,4]=-5837069082360096304407/13 95022708312545550000,\na[6,5]=13508218613909220883/2259367390442522752 0,\na[7,1]=756727539023617977739/325483930565422777440,\na[7,2]=0,\na[ 7,3]=-11344125743485787/1187841741251840,\na[7,4]=93403161519631272506 393/11188675650069067742640,\na[7,5]=-3811592140111638097686625/330881 1923451740272240896,\na[7,6]=102601928198000/102454959175119,\na[8,1]= 250792505789354790081/66169955021507254750,\na[8,2]=0,\na[8,3]=-349400 598312667239/22005305959929250,\na[8,4]=17728479258264420469702254/128 9712860941995340762625,\na[8,5]=-61925787315598406758085/2776488121843 2626442221,\na[8,6]=43795320281930/26932915012729,\na[8,7]=-9649315042 2/1790131071575,\na[9,1]=1414477/22101450,\na[9,2]=0,\na[9,3]=0,\na[9, 4]=6047117272638/18236245267975,\na[9,5]=31062711625/123650103496,\na[ 9,6]=27376502125/148102904499,\na[9,7]=26504253/76308925,\na[9,8]=-234 59/131064,\n\nb[1]=1414477/22101450,\nb[2]=0,\nb[3]=0,\nb[4]=604711727 2638/18236245267975,\nb[5]=31062711625/123650103496,\nb[6]=27376502125 /148102904499,\nb[7]=26504253/76308925,\nb[8]=-23459/131064,\n\n`b*`[1 ]=9659334433759/154907736963000,\n`b*`[2]=0,\n`b*`[3]=0,\n`b*`[4]=2159 7935405438536443/63908374454260348250,\n`b*`[5]=230190597481295825/103 9987387676705088,\n`b*`[6]=8367042728056775/37747068053062584,\n`b*`[7 ]=28150317302067/97244486689000,\n`b*`[8]=-77493123437/612413141440,\n `b*`[9]=-1/160\}:" }}}{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)\"#C F)*$)F'\"\"%F)F)F)*&#F)\"$?\"F)*$)F'\"\"&F)F)F)*&#F)\"$?(F)*$)F'F1F)F) F)*&#\"/ZN$fNBT\"\"2+!3A@sfMsF)*$)F'\"\"(F)F)F)*&#\"-FN[0x9\"1+'Hb3`2V '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 stabil ity 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=-4.4);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%#z0G$!+U4V&Q%!\"*" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" } }{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 303 "z0 := newton(R(z)=1,z=-4.4 ):\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):\np 3 := plot([[z0,0],[z0,1]],linestyle=3,color=COLOR(RGB,0,.5,0)):\nplots [display]([p1,p2,p3],view=[-5.19..0.49,-.07..1.47],font=[HELVETICA,9]) ;" }}{PARA 13 "" 1 "" {GLPLOT2D 360 253 253 {PLOTDATA 2 "6+-%'CURVESG6 $7Z7$$!3Q++++++!>&!#<$\"3Y1^'R&)\\LE%F*7$$!3QML3T![!f^F*$\"3g&[,Mn,&[S F*7$$!3Ynm;#3'4G^F*$\"3,vjo3$)HVQF*7$$!3a++DBT9(4&F*$\"33u%Q\\djtk$F*7 $$!3kLLLk@>m]F*$\"33wU&GwL.Y$F*7$$!3E+]U'*)HB,&F*$\"3#>WApFf]:$F*7$$!3 !pm;&GwYe\\F*$\"3wcKH\"HCR(GF*7$$!3s+](\\(Q*y*[F*$\"3'Q%4dx#HXe#F*7$$! 3nLLV@,KP[F*$\"365Rf&yS8K#F*7$$!3'RLLd%[MwZF*$\"3c5#R*='y23#F*7$$!3NLL .q&p`r%F*$\"3KkbmZHqi=F*7$$!3E+]<*4%oaYF*$\"3WC*3\"REe-'yvF]p7$$!3A++ lN]MCTF*$\"3?=bvWd]bfF]p7$$!3ummYeRz+SF*$\"3!y-Mj`)*4j%F]p7$$!3_LLV-,( >*QF*$\"3_J')p&3yRq$F]p7$$!35++S:-YpPF*$\"3;paG8Y.\")GF]p7$$!3K+++\"HZ kk$F*$\"3[CIz\"\\XrC#F]p7$$!3;++gW:!z_$F*$\"3y\"HA=>uRy\"F]p7$$!3hLL)* \\1D?MF*$\"3]oGNsPYl9F]p7$$!3'ommSKVAH$F*$\"3))p6uFA'*)=\"F]p7$$!3/nmE GV!Q=$F*$\"3.dK%4D\\_-\"F]p7$$!39++0(*RmdIF*$\"33Y7wE$f6+*!#>7$$!39nmE I%3g%HF*$\"3wD(4qDb*p$)Fes7$$!3-++0xX]BGF*$\"3Gw$*=9E^&4)Fes7$$!3*)*** \\\"R>&oq#F*$\"3k7))o,-$3<)Fes7$$!3gmm;\\r8&e#F*$\"3(Gr\"[z&=Qa)Fes7$$ !3ymmrw\\OtCF*$\"3\"y&y@T>e;\"*Fes7$$!3SLL$))e.GN#F*$\"3#=]fLa!pc**Fes 7$$!3nLL)**=uvA#F*$\"3`Q2HvB\"e5\"F]p7$$!3K++:I;c=@F*$\"3E/MRv\\@?7F]p 7$$!31LL.z]#3+#F*$\"3Mio>qpRj8F]p7$$!3M++?,<>z=F*$\"3gF]p7$$!3H ++q9zA<:F*$\"3#fJg99%G%>#F]p7$$!3EnmEY;O-9F*$\"33_\"f@TD2Y#F]p7$$!3#)* ****pQ<(z7F*$\"3q$y6z7J9y#F]p7$$!3)RL$efMeo6F*$\"3mK@pZ/?3JF]p7$$!3I** **fAZ3Z5F*$\"3#GZp*>$\\'4NF]p7$$!3xqm;(zQwK*F]p$\"3![2*zX(pY$RF]p7$$!3 &z***\\)ecE8)F]p$\"3xg&z.y%3MWF]p7$$!3'3nmm0VV'pF]p$\"3W#o-L(Qf$)\\F]p 7$$!3P)***\\iqATdF]p$\"3s#ema/))>j&F]p7$$!3aFLL*)4AjXF]p$\"3**>Y#=J(4O jF]p7$$!33LLLO'R&eLF]p$\"3q$peyvus9(F]p7$$!3Uim;`O$Q;#F]p$\"3W/sq,ZEa! )F]p7$$!3?*****>$H-m5F]p$\"3E.aIQ4$)))*)F]p7$$\"3v*QLLU?>#>Fes$\"3[-s. #30%>5F*7$$\"3%ymmY^avJ\"F]p$\"3g9w,(HH39\"F*7$$\"3E0+]HcU&!\"#$\"#\\Fidl;$!\"(Fidl$\"$Z\"Fidl " 1 2 0 1 10 0 2 9 1 4 2 1.000000 45.000000 45.000000 0 0 "Curve 1" "C urve 2" "Curve 3" "Curve 4" "Curve 5" "Curve 6" }}}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 50 "The following picture sho ws the stability region. " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1364 "R := z -> 1+z+1/2*z^2+1/6*z^3+1/2 4*z^4+1/120*z^5+1/720*z^6+\n 14123355933547/72345972212208000*z^7+ 147705483527/6430753085529600*z^8:\npts := []: z0 := 0:\nfor ct from 0 to 240 do\n zz := newton(R(z)=exp(ct*Pi/20*I),z=z0):\n z0 := zz: \n pts := [op(pts),[Re(zz),Im(zz)]]:\nend do:\np1 := plot(pts,color= COLOR(RGB,.48,.23,.08)):\np2 := plots[polygonplot]([seq([pts[i-1],pts[ i],[-2.2,0]],i=2..nops(pts))],\n style=patchnogrid,color=COLO R(RGB,.95,.45,.15)):\npts := []: z0 := 2+4.75*I:\nfor ct from 0 to 50 \+ do\n zz := newton(R(z)=exp(ct*Pi/25*I),z=z0):\n z0 := zz:\n pts \+ := [op(pts),[Re(zz),Im(zz)]]:\nend do:\np3 := plot(pts,color=COLOR(RGB ,.48,.23,.08)):\np4 := plots[polygonplot]([seq([pts[i-1],pts[i],[1.94, 4.73]],i=2..nops(pts))],\n style=patchnogrid,color=COLOR(RGB, .95,.45,.15)):\npts := []: z0 := 2-4.75*I:\nfor ct from 0 to 50 do\n \+ zz := newton(R(z)=exp(ct*Pi/25*I),z=z0):\n z0 := zz:\n pts := [op (pts),[Re(zz),Im(zz)]]:\nend do:\np5 := plot(pts,color=COLOR(RGB,.48,. 23,.08)):\np6 := plots[polygonplot]([seq([pts[i-1],pts[i],[1.94,-4.73] ],i=2..nops(pts))],\n style=patchnogrid,color=COLOR(RGB,.95,. 45,.15)):\np7 := plot([[[-5.19,0],[2.29,0]],[[0,-5.19],[0,5.19]]],colo r=black,linestyle=3):\nplots[display]([p||(1..7)],view=[-5.19..2.29,-5 .19..5.19],font=[HELVETICA,9],\n labels=[`Re(z)`,`Im(z)`] ,axes=boxed,scaling=constrained);" }}{PARA 13 "" 1 "" {GLPLOT2D 381 565 565 {PLOTDATA 2 "6/-%'CURVESG6$7]z7$$\"\"!F)F(7$F($\"3++++Fjzq:!#= 7$$!3)******4=Wp4\"!#F$\"3z*****\\k#fTJF-7$$!3%)*****pP<1@#!#E$\"3')** ***>%))Q7ZF-7$$!3%*******3z_.9!#D$\"3U+++,W=$G'F-7$$!33+++Pa,aDF=$\"3E +++sz(R&yF-7$$\"3\"******\\>y+t\"!#C$\"3N+++%)pwC%*F-7$$\"3)******\\@# )\\j\"!#B$\"3'******zla&*4\"!#<7$$\"3u*****R;sp&yFN$\"3++++T/jc7FQ7$$ \"37+++cWR\"z#!#A$\"3)******fE+PT\"FQ7$$\"3O******)e^8:)FZ$\"3!******> 4`2d\"FQ7$$\"3'******4fEk0#!#@$\"32+++O/wF#FQ7$$\"3')*****pGyE\"HF_p$\"3?+++h#GFN#FQ7$$ \"3!******>(zU'[%F_p$\"3z*****f\\Mk]#FQ7$$\"3')*****Rk2`B'F_p$\"3%**** **p+5xl#FQ7$$\"3r*****fg59k(F_p$\"3))*****>kza!GFQ7$$\"3d*****f#\\cHyF _p$\"3#)*****>l)f[HFQ7$$\"3g*****z\"*e(QcF_p$\"3-+++1S+'3$FQ7$$!3!**** **zEIh&=F_o$\"3,+++3i\"o@$FQ7$$!3/+++@r9v5!#>$\"3!******\\&GNSLFQ7$$!3 !******R'4>#o#Fhr$\"33+++nx3cMFQ7$$!3I+++rGGr[Fhr$\"3++++`$>Oc$FQ7$$!3 o*****RhP'HwFhr$\"35+++(e!oiOFQ7$$!3/+++Ws!44\"F-$\"33+++9W<`PFQ7$$!31 +++aO(RY\"F-$\"3A+++]s>NQFQ7$$!3%********>.W(=F-$\"3')*****p0F!4RFQ7$$ !3!******\\dGZJ#F-$\"3'******45z](RFQ7$$!3!*******\\GRyFF-$\"3v*****z8 [Q.%FQ7$$!33+++U#y*fKF-$\"3/+++c^&e3%FQ7$$!3E+++MJ=bPF-$\"3!******z(zg JTFQ7$$!3)*******[ingUF-$\"3I+++CqdrTFQ7$$!3!******pi9Rx%F-$\"3$****** pH$=1UFQ7$$!3')*****zHrHH&F-$\"3!)******[YzNUFQ7$$!3l*****R4\"R;eF-$\" 3?+++OpsgUFQ7$$!3W+++j#yIM'F-$\"3y*****zmZ7G%FQ7$$!3a*****p,8A(oF-$\"3 E+++P1e(H%FQ7$$!3^+++'=#>.uF-$\"3f*****pm5*4VFQ7$$!3s******H#*eNzF-$\" 3*******f.)Q=VFQ7$$!3K+++f(H\"p%)F-$\"3N+++*RKJK%FQ7$$!3R+++5)zO+*F-$ \"3G+++ahBCVFQ7$$!3u*****pt7oF%FQ7$$!3!******H()R WA\"FQ$\"39+++_Q#oD%FQ7$$!3#******\\sf'z7FQ$\"3r*****fo1LB%FQ7$$!3#*** ***4z*fN8FQ$\"3c*****zBZi?%FQ7$$!3!******4,0DR\"FQ$\"3u*****R%3lvTFQ7$ $!33+++WFo]9FQ$\"3R+++$4k:9%FQ7$$!33+++7Y]5:FQ$\"3j*****pr1T5%FQ7$$!3# ******>#[Rs:FQ$\"3Q+++8U^jSFQ7$$!30+++zMzO;FQ$\"3@+++?l=?SFQ7$$!3++++& H$3/+FZ(RFQ7$$!3/+++t&zWx\"FQ$\"3))*****fyTz#RFQ7$$!3'* *****pq6z%=FQ$\"3?+++pew!)QFQ7$$!3&******\\:cR#>FQ$\"3\")*****pL5T$QFQ 7$$!35+++$zw=+#FQ$\"3y*****R*=o)y$FQ7$$!3-+++X'o23#FQ$\"3=+++y4'[u$FQ7 $$!3!******H\"Qtf@FQ$\"3y******Q0p-PFQ7$$!3<+++u:,QAFQ$\"3%)*****zLb>m $FQ7$$!39+++0F.:BFQ$\"35+++8DHAOFQ7$$!3%******pA9/R#FQ$\"3/+++VhG$e$FQ 7$$!3!*******R]#RY#FQ$\"3<+++]'GXa$FQ7$$!36+++$\\Xa`#FQ$\"3()*****HG_c ]$FQ7$$!36+++8A$\\g#FQ$\"3/+++S0MmMFQ7$$!3#******HX\"RsEFQ$\"3++++`%Gj U$FQ7$$!3,+++Y/'yt#FQ$\"3-+++1jGFQ$\"3!)*****>(35+LFQ7$$!32+++$)R$H#HFQ$\"3;+++hHZbKFQ 7$$!3-+++'Q(3\")HFQ$\"3.+++'>#R4KFQ7$$!3.+++t1gPIFQ$\"3%)*****\\d$yhJF Q7$$!3))*****p3aD4$FQ$\"3$******\\a)e7JFQ7$$!37+++Y'Hg9$FQ$\"3))*****p >j<1$FQ7$$!3-+++h86)>$FQ$\"3&******zEx#4IFQ7$$!3)******\\W&))[KFQ$\"3' ******\\'Q6bHFQ7$$!37+++*HS%)H$FQ$\"3%******>#)p#**GFQ7$$!3)******H5mo M$FQ$\"32+++dlvTGFQ7$$!39+++-QD%R$FQ$\"3?+++v7g#y#FQ7$$!33+++mLpSMFQ$ \"3+++++$[=s#FQ7$$!3!******p-cfEFQ7$$!35+++vl1JNFQ $\"3)******zZ=ef#FQ7$$!3/+++%oZ^d$FQ$\"3$)*****z+B2`#FQ7$$!3@+++yoc=OF Q$\"3++++<0RkCFQ7$$!3!)*****4Uc8m$FQ$\"3!)*****\\=^pR#FQ7$$!37+++V\\_. 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" }} {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 148 "We can d istort the boundary curve horizontally by taking the 11th root of the \+ real part of points along the curve. In this way we see that there is \+ " }{TEXT 260 53 "no largest interval on the nonnegative imaginary axis " }{TEXT -1 65 " that contains the origin and lies inside the stabilit y region. " }}{PARA 0 "" 0 "" {TEXT -1 119 "However the stability reg ion intersects the nonnegative imaginary axis in an interval that does not contain the origin." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 389 "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 14123355933547/72345972212208000*z^7+1 47705483527/6430753085529600*z^8:\nDigits := 25:\npts := []: z0 := 0: \nfor ct from 0 to 107 do\n zz := newton(R(z)=exp(ct*Pi/100*I),z=z0) :\n z0 := zz:\n pts := [op(pts),[surd(Re(zz),11),Im(zz)]]:\nend do :\nplot(pts,color=COLOR(RGB,.85,.25,0),thickness=2,font=[HELVETICA,9]) ;\nDigits := 10:" }}{PARA 13 "" 1 "" {GLPLOT2D 261 276 276 {PLOTDATA 2 "6(-%'CURVESG6#7hq7$$\"\"!F)F(7$$!:*)=%omh6,?#))RO#!#E$\":L/0-Oy*e`E fTJF-7$$!:v?X2QF0;'*p>\"RF-$\":p7^%*eNyrI&=$G'F-7$$!:5*p\"))f9]ns)**\\ _F-$\":'yS6yN%[2'zxC%*F-7$$!:MT6,\"fSQ8-UlkF-$\":U1o%Q7.Uhqjc7!#D7$$!: &>aUQ=Z.h=!\\f(F-$\":`/Yal+@nK'zq:F?7$$!:l]6y`*o\")R`3e')F-$\":;XBdVP \"*=fb\\)=F?7$$!:OdOb\\Y;E-tom*F-$\":/oK6!)Q[n&[6*>#F?7$$!:2\")RP@(oIY 1!H1\"F?$\":atOr*\\x%47uK^#F?7$$!:eDZK&=$*Qp@(\\:\"F?$\":\"=BAsM.!RQLu #GF?7$$!:**Gi\")=2lfwgKC\"F?$\":0_A#*R&*oYk#fTJF?7$$!:ZEa'))yj`D%3!G8F ?$\":D!3>J$zY=!>vbMF?7$$!:+![1B'fYisn$49F?$\":;FPR#)4?M:6*pPF?7$$!:)R \"ej?g&ytsT([\"F?$\":O/=:/\")=mRqS3%F?7$$!:21dN2\"4ma!p@c\"F?$\":))*Qb 6\"yfxiH#)R%F?7$$!:'z9*Q_[Q4VpNj\"F?$\":t%)HN:$y3U))Q7ZF?7$$!:!QUS!=d) Q[S\\,22M& F?7$$!:0%G.W\"*yGJ=)f#=F?$\":q\")=JUjgtJm[l&F?7$$!:Z5%e;`F?$\":[(\\zs?nL,W=$G'F?7$$!:s))G :(yWe?H0x>F?$\":&pTr7/WqOLM(f'F?7$$!:*\\#R(fq&fxTJS,#F?$\":j%*pa3\\%R! =-:\"pF?7$$!:h*HL&R>Dq.!*3/#F?$\"::z@oa88X\"4mDsF?7$$!:z@YA'oJGmC/`?F? $\":k>)Glb()3>&>)RvF?7$$!:m<55_3a@l^0/#F?$\":*))Q'fSC>>(z(R&yF?7$$!:Z# )*[[4FRJKLt>F?$\":7!)p1D8O$[i8o\")F?7$$\":;HsznYm7U85%**F-$\":+.--D^S3 K%H#[)F?7$$\":(3a!Rq408.vl3#F?$\":*p]P[sPbe@X'z)F?7$$\":Faw^G3uu'3d\"G #F?$\":'*e%f*=-8ks416*F?7$$\":\")QKtsrcD%y;GCF?$\":ldMyyTES)pwC%*F?7$$ \":RNoiTSlE_[Wb#F?$\":hpb50\\(3%)Q#*Q(*F?7$$\":*)orD(R#3q_(RpEF?$\":lE %3G**H-P!3`+\"!#C7$$\":OMV?Z1f&[!\\qx#F?$\":NBEx:e=ujBn.\"Ffu7$$\":]\" >R%ze3uEf&zGF?$\":H&\\y(f\")37=R\"o5Ffu7$$\":+.9p$f!Q.$G@yHF?$\":J,%H] ^S(zla&*4\"Ffu7$$\":k9h$e!)>OjW$Q2$F?$\":Lk$G$)p;$F?$\":%*\\A6oIf[N&Qi6Ffu7$$\":?XT=4QI;Na!eKF?$\":#f!o*H'*z %y`+Q>\"Ffu7$$\":AjJHZz\\Z&fLZLF?$\":x4TL!H7_xb@D7Ffu7$$\":SdMu=x;R!R/ NMF?$\":J?)>h%*f-T/jc7Ffu7$$\":,iR)*4+1T2V8_$F?$\":za]%Hw5;(3X!)G\"Ffu 7$$\":9(\\)3\\oSbZhjg$F?$\":&Q#Q\\^HuWYf%>8Ffu7$$\":A^)pCFF*>v*>!p$F?$ \":)zlz)*eP-4N(3N\"Ffu7$$\":e]*Q*3!yKsx$Hx$F?$\":!)GBsMPi<9(G#Q\"Ffu7$ $\":p:p&R[[\"RJRY&QF?$\":Y1&=d:][ClZ\"Ffu7$$\":%p1$=L* QD*y')R4%F?$\":\"*3*o&*o(*eh_$z]\"Ffu7$$\":d3!)*GMj0;-'><%F?$\":OY:i*G O+\")[MR:Ffu7$$\":xbJ33sLmAn!\\UF?$\":VHfku!>&=4`2d\"Ffu7$$\":y7t)Rei6 #pA`K%F?$\":,yQ^z()*)[ff@g\"Ffu7$$\":#3^#fHkHf9P2S%F?$\":R6l:7Z)pNScL; Ffu7$$\":7>)*o(*)yq\"G<`Z%F?$\":[qN6c&H`'*f'\\m\"Ffu7$$\":\"G%eUxe4lLm !\\XF?$\":@d\")yT03t)\\O'p\"Ffu7$$\":UR5pr@gGQ%)>i%F?$\":=[QD.GwcVgxs \"Ffu7$$\":*3+/J6&HsaoSp%F?$\":v!f50I<;w;:fy')=$Qz`!z\"Ffu7$$\":,PnXY$Q#[!*4d$[F?$\":%=oLe$G&*QL=>#=Ffu7$$ \":zI9yG^y=!yC0\\F?$\":W=G75I*)H%=H`=Ffu7$$\":-Z%))pGM\")>L\"R(\\F?$\" :iY#*)z\\QP8g\">Ffu7$$ \":sq-Oggmd4g&3^F?$\":b&G-B[<:f&et%>Ffu7$$\":d[zQUzjU>,X<&F?$\":(RJf%f c%o]6py>Ffu7$$\":b=A<>cQ.Y*[R_F?$\":z>RK&[E(*G#4+,#Ffu7$$\":!*GO@;fE]( y\\.`F?$\":Xo***\\?D\\S1JT?Ffu7$$\":/Jc'4U(zrW'\\m`F?$\":Gc1W_S1#yHfs? 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" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 201 "Digits := 1 5:\nz0 := 0.83*I:\nfor ct from 25 to 28 do\n newton(R(z)=exp(ct*Pi/1 00*I),z=z0);\nend do;``;\nz0 := 3.2*I:\nfor ct from 103 to 106 do\n \+ newton(R(z)=exp(ct*Pi/100*I),z=z0);\nend do;\nDigits := 10:" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#^$$!0n&)HV:Sb#!#A$\"0C>>(z(R&y!#:" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#^$$!0)\\SA0(ow\"!#A$\"08O$[i8o\")!#:" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#^$$\"05sEu\\*p$*!#E$\"0^S3K%H#[)!#:" } }{PARA 11 "" 1 "" {XPPMATH 20 "6#^$$\"0[.7wzTE$!#A$\"0x`&e@X'z)!#:" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#%!G" }}{PARA 11 "" 1 "" {XPPMATH 20 "6 #^$$\"0_2U+Sql#!#<$\"09J6@I`;$!#9" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#^ $$\"0rs&*fzqK\"!#<$\"0xMh`;7>$!#9" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#^ $$!0[!\\o-8c=!#=$\"0VQv?;o@$!#9" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#^$$ !0&)*z_@6*)=!#<$\"0LKOOC@C$!#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 0 330 "Digits := 15:\nreal_part \+ := proc(u)\n Re(newton(R(z)=exp(u*Pi*I),z=0.83*I))\nend proc:\nu0 := bisect('real_part'(u),u=0.25..0.28);\nnewton(R(z)=exp(u0*Pi*I),z=0.83 *I);``;\nreal_part := proc(u)\n Re(newton(R(z)=exp(u*Pi*I),z=3.2*I)) \nend proc:\nu0 := bisect('real_part'(u),u=1.03..1.05);\nnewton(R(z)=e xp(u0*Pi*I),z=3.2*I);\nDigits := 10:" }}{PARA 11 "" 1 "" {XPPMATH 20 " 6#>%#u0G$\"0F+0@h**p#!#:" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#^#$\"0AR0Y s@[)!#:" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%!G" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%#u0G$\"0_1G!o$)[5!#9" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#^#$\"0.)f_L&Q@$!#9" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 4 "The " }{TEXT 260 73 "stability region intersects the \+ nonegative imaginary axis in the interval" }{TEXT -1 40 " [ 0.8482, \+ 3.21385 ] (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 f unction 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*`,Stabilit yFunction(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)*&#\"42Cu59(49id\"7gh6li\"yi[l0%F)*$ )F'F1F)F)F)*&#\"5rk?NR-pl2z\"9++of()o%)*=d/Q$F)*$)F'\"\"(F)F)F)*&#\"58 O&*Q)3F)*f/$\":++3eD83RJu#G?F)*$)F'\"\")F)F)F)*&#\"-FN[0x9\"4+gt%o$\\? *G5F)*$)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 s tability region intersects the negative real axis by solving the equat ion: " }}{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=-4.4);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%$z_0G$!+Qu(eR%!\"*" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 324 "z_0 := newt on(`R*`(z)=-1,z=-4.4):\np_1 := plot([`R*`(z),-1],z=-4.99..0.49,color=[ red,blue]):\np_2 := plot([[[z_0,-1]]$3],style=point,symbol=[circle,cro ss,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,-1.57..1.47],font=[HELVETICA,9]);" }}{PARA 13 "" 1 "" {GLPLOT2D 398 284 284 {PLOTDATA 2 "6+-%'CURVESG6$7S7$$!3A++++++!*\\!#<$!3!\\iAqH <(RAF*7$$!3#pmm^f^0([F*$!3'GH5!yR)*G>F*7$$!3%QL3opE!=)*R9F*7$$!33nm64>3KXF*$!3zLESrJ)4A\"F*7$ $!3\\L$3@f%)\\T%F*$!3hlI.u.,H5F*7$$!3jm;u*Q?kI%F*$!3W.DQLTIB()!#=7$$!3 E+]ZY%3S>%F*$!3IU>V%R2&)H(FK7$$!3%omTR&=vxSF*$!3HbCaS,*p,'FK7$$!3/+]x( 4o='RF*$!3p[tiC:S9\\FK7$$!3[URFK7$$!3[mm^!Qvwt$F*$ !3*p1!p_II2KFK7$$!3#)*****=Az%>OF*$!3O,7$$!3eLL$4-XW0$F*$!39\\G(=5!>^P Fhp7$$!3;+]n]iuKHF*$!3x.9F;!yR,\"Fhp7$$!3ILL$z@A]#GF*$\"3q\"y'*RnpB3\" Fhp7$$!3!)**\\n!)=$oq#F*$\"3b;a,b*pA6$Fhp7$$!3')**\\-InG%f#F*$\"3E=1Z1 Sd_[Fhp7$$!3ILL3sw&oZ#F*$\"3e)*=33fA'pge*Fhp7$$!3xm;%z&\\)=8#F*$\"3u,!*fuw &[7\"FK7$$!3$***\\_o3rE?F*$\"3CW)3v!>%eF\"FK7$$!3hmmhq*>J\">F*$\"3a1() 3ao@\\9FK7$$!35++?e%pdz\"F*$\"39*)yeLCUV;FK7$$!3+++:w['4o\"F*$\"3%Qjc_ yO?&=FK7$$!3-+]()Rb))p:F*$\"3%\\6#)yw?\\2#FK7$$!3#)***\\a(3bY9F*$\"3U` 7duQ#3N#FK7$$!3cLL$R>HdL\"F*$\"3&*R/%e**=\"GEFK7$$!3z****\\%R.u@\"F*$ \"3wW/9s$y#fHFK7$$!3pm;aLE=56F*$\"3oIi1f?o%H$FK7$$!3E%****4@?'H**FK$\" 3O(=Zw)Rn/PFK7$$!3MML3+cmE))FK$\"3;)3e?6dn8%FK7$$!3%H**\\(H-wtwFK$\"3G )RET`&HUYFK7$$!3cOLL)\\%eYlFK$\"3_E9'>&FK7$$!3g.+vof`m`FK$\"3_&*) z=r=q%eFK7$$!3Dkmm#)*3+B%FK$\"3G1mK18y]lFK7$$!3Yjm;()funIFK$\"3iEM;FX; etFK7$$!3HBL3;r5:>FK$\"3wBpt:y5d#)FK7$$!3_^****>q^f&)Fhp$\"3g_b@fxlz\" *FK7$$\"3OMnm\"G*fzNFhp$\"3m^C#GQWk.\"F*7$$\"3\"RLL8'ppV9FK$\"3iGj(R96 `:\"F*7$$\"3@0+D$4>8g#FK$\"3Y\"\\Vv=,rH\"F*7$$\"3Q,+v#=6$4PFK$\"3Ubb@n L3\\9F*7$$\"3!***************[FK$\"32P*G-'oJK;F*-%'COLOURG6&%$RGBG$\"* ++++\"!\")$\"\"!Fa[lF`[l-F$6$7S7$F($!\"\"Fa[l7$F.Ff[l7$F3Ff[l7$F8Ff[l7 $F=Ff[l7$FBFf[l7$FGFf[l7$FMFf[l7$FRFf[l7$FWFf[l7$FfnFf[l7$F[oFf[l7$F`o Ff[l7$FeoFf[l7$FjoFf[l7$F_pFf[l7$FdpFf[l7$FjpFf[l7$F_qFf[l7$FdqFf[l7$F iqFf[l7$F^rFf[l7$FcrFf[l7$FhrFf[l7$F]sFf[l7$FbsFf[l7$FgsFf[l7$F\\tFf[l 7$FatFf[l7$FftFf[l7$F[uFf[l7$F`uFf[l7$FeuFf[l7$FjuFf[l7$F_vFf[l7$FdvFf [l7$FivFf[l7$F^wFf[l7$FcwFf[l7$FhwFf[l7$F]xFf[l7$FbxFf[l7$FgxFf[l7$F\\ yFf[l7$FayFf[l7$FfyFf[l7$F[zFf[l7$F`zFf[l7$FezFf[l-Fjz6&F\\[lF`[lF`[lF ][l-F$6&7#7$$!3I+++Qu(eR%F*Ff[l-%'SYMBOLG6#%'CIRCLEG-Fjz6&F\\[lFa[lFa[ lFa[l-%&STYLEG6#%&POINTG-F$6&F\\_l-Fa_l6#%&CROSSGFd_lFf_l-F$6&F\\_l-Fa _l6#%(DIAMONDGFd_lFf_l-F$6%7$7$F^_lF`[lF]_l-%&COLORG6&F\\[lF`[l$\"\"&F g[lF`[l-%*LINESTYLEG6#\"\"$-%%FONTG6$%*HELVETICAG\"\"*-%+AXESLABELSG6% Q\"z6\"Q!Fjal-Fbal6#%(DEFAULTG-%%VIEWG6$;$!$*\\!\"#$\"#\\Febl;$!$d\"Fe bl$\"$Z\"Febl" 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 follo wing picture shows the stability region for the 9 stage, order 5 schem e. " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1488 "`R*` := z -> 1+z+1/2*z^2+1/6*z^3+1/24*z^4+1/120*z^5 +\n 5762140971410742407/4056548627816265116160*z^6+\n 79076569 023935206471/338045718984688759680000*z^7+\n 30459982708838953613/ 2028274313908132558080000*z^8-\n 147705483527/1028920493684736000* 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,.38,.1,0)):\np_2 \+ := plots[polygonplot]([seq([pts[i-1],pts[i],[-2.2,0]],i=2..nops(pts))] ,\n style=patchnogrid,color=COLOR(RGB,.75,.2,0)):\npts := []: z0 := 1.9+4.4*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(zz)]] :\nend do:\np_3 := plot(pts,color=COLOR(RGB,.38,.1,0)):\np_4 := plots[ polygonplot]([seq([pts[i-1],pts[i],[1.83,4.38]],i=2..nops(pts))],\n \+ style=patchnogrid,color=COLOR(RGB,.75,.2,0)):\npts := []: z0 := 1.9-4.4*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(zz)]]:\nend do:\np_5 := plot(pts,color=COLOR(RGB,.38,.1,0)):\np_6 := plots[polygo nplot]([seq([pts[i-1],pts[i],[1.83,-4.38]],i=2..nops(pts))],\n \+ style=patchnogrid,color=COLOR(RGB,.75,.2,0)):\np_7 := plot([[[-5.09, 0],[2.19,0]],[[0,-4.99],[0,4.99]]],color=black,linestyle=3):\nplots[di splay]([p_||(1..7)],view=[-5.09..2.19,-4.99..4.99],font=[HELVETICA,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$$\"3]*****f6#Q][!#F$\"3++++Fjzq:!#=7$$\"3?+++jFD3L!# D$\"3#*******fEfTJF07$$\"31+++XlaPT!#C$\"37+++m*)Q7ZF07$$\"37+++Fq@%f# !#B$\"3W+++sR=$G'F07$$\"3++++=GN36!#A$\"3)*******)*z'R&yF07$$\"3%***** *z-$>$o$FF$\"3_+++M')pC%*F07$$\"3,+++tqT>5!#@$\"3++++dS_*4\"!#<7$$\"3# ******4$*QWW#FQ$\"33+++([ClD\"FT7$$\"3*)*****Rrk3?&FQ$\"35+++%>%R89FT7 $$\"3y+++hB1d**FQ$\"3++++$G$)*p:FT7$$\"3'******4'oTE+++anx/QFao$\"3-++ +CXlM?FT7$$\"3M+++onbbYFao$\"3%)*****>Lze=#FT7$$\"3,+++g'Rmg%Fao$\"36+ ++GF#RL#FT7$$\"3:+++dO>,FFao$\"3))*****>&*)*yZ#FT7$$!37+++HZ3QAFao$\"3 !)*****p<')ph#FT7$$!31+++%pj\\9\"!#>$\"3'******H>^/v#FT7$$!3'*******Qv 8/EFeq$\"3'******z7:w(GFT7$$!3O+++,*H8o%Feq$\"3)******R>/y*HFT7$$!3i++ +2B*HT(Feq$\"3:+++FJM5JFT7$$!31+++nhsy5F0$\"3#)*****\\N=Y@$FT7$$!37+++ ZU\"\\Z\"F0$\"3,+++F^=5LFT7$$!31+++3rV@>F0$\"3%******f'y&oR$FT7$$!3,++ +evo3CF0$\"3?+++P[tuMFT7$$!3&******4v!QFHF0$\"34+++*)H:WNFT7$$!3A+++EF ZpMF0$\"3y*****Hu/cg$FT7$$!3>+++2t\\GSF0$\"39+++]LlfOFT7$$!3#******\\` 8&*f%F0$\"3!******>)z'oq$FT7$$!3'******Hxp)y^F0$\"3#******z@$yZPFT7$$! 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25/148102904499,\na[9,7]=26504253/76308925,\na[9,8]=-23459/131064,\n\n b[1]=1414477/22101450,\nb[2]=0,\nb[3]=0,\nb[4]=6047117272638/182362452 67975,\nb[5]=31062711625/123650103496,\nb[6]=27376502125/148102904499, \nb[7]=26504253/76308925,\nb[8]=-23459/131064,\n\n`b*`[1]=965933443375 9/154907736963000,\n`b*`[2]=0,\n`b*`[3]=0,\n`b*`[4]=215979354054385364 43/63908374454260348250,\n`b*`[5]=230190597481295825/10399873876767050 88,\n`b*`[6]=8367042728056775/37747068053062584,\n`b*`[7]=281503173020 67/97244486689000,\n`b*`[8]=-77493123437/612413141440,\n`b*`[9]=-1/160 \}:" }}}{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#\"\"$#\"#N\"$V#/&F%6#\" \"%#F0\"$i\"/&F%6#\"\"&#\"#J\"#b/&F%6#\"\"'#\"#(*\"$S\"/&F%6#F)#F0\"#O /&F%6#\"\")\"\"\"/&F%6#\"\"*FO" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }} {PARA 0 "" 0 "" {TEXT 259 51 "linking coefficients for the 9 stage ord er 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%6$\"\"$F(#\"$&Q \"')4=\"/&F%6$F/F'#\"&Dm\"F2/&F%6$\"\"%F(#\"#N\"$['/&F%6$F;F'\"\"!/&F% 6$F;F/#F=\"$;#/&F%6$\"\"&F(#\"*HTOi#\"*](=wS/&F%6$FKF'FB/&F%6$FKF/#!*( o4p**FN/&F%6$FKF;#\"*ar9#[\"*v$4Q?/&F%6$\"\"'F(#!56+!yA*)zfup$\"5++IN0 w![Fz#/&F%6$FjnF'FB/&F%6$FjnF/#\"5>K8^&3*fBQS\"4++Sq\"4=42s/&F%6$FjnF; #!72WI'4gB3pq$e\"7++bXDJ3F-&R\"/&F%6$FjnFK#\"5$)3A4Rh=#3N\"\"5?vADW!Rn $fA/&F%6$F*F(#\"6Rx(zhB!Rvsc(\"6SuxAacIR[D$/&F%6$F*F'FB/&F%6$F*F/#!2(y &[Vd7W8\"\"1S=DT<%y=\"/&F%6$F*F;#\"8$R1DFJ'>:;.M*\"8SEun!p+lv')=6/&F%6 $F*FK#!:Dmo(4Q;6S@f6Q\":'*3Cs-u^M#>\")3L/&F%6$F*Fjn#\"0+!)>G>g-\"\"0>^ %4'Gr*G\"/&F%6$FgrFK#!8&3enS)f:tyD>'\"8@AWEEV=7)[wF/&F%6$FgrFjn# \"/I>G?`zV\"/HF,:H$p#/&F%6$FgrF*#!,A/:$\\'*\".v:2J,z\"/&F%6$\"\"*F(#\" (xWT\"\")]95A/&F%6$F_uF'FB/&F%6$F_uF/FB/&F%6$F_uF;#\".QEF " 0 "" {MPLTEXT 1 0 31 "seq(b[i]=subs(ee,b[i]),i=1..8);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6*/&%\"bG6#\"\"\"#\"(xWT\"\")]95A/&F%6#\"\"#\"\"!/&F%6#\"\"$F//&F%6# \"\"%#\".QEF " 0 "" {MPLTEXT 1 0 37 "seq(`b*`[i]=subs(ee,`b*`[i]),i=1..9);" }} {PARA 12 "" 1 "" {XPPMATH 20 "6+/&%#b*G6#\"\"\"#\".fPVM$f'*\"0+I'pt2\\ :/&F%6#\"\"#\"\"!/&F%6#\"\"$F//&F%6#\"\"%#\"5Vk`QaSNzf@\"5]#[.EaWP3R'/ &F%6#\"\"&#\"3DeH\"[(f!>I#\"4)30nn(Q()*R5/&F%6#\"\"'#\"1vn0GF/n$)\"2%e iI0oquP/&F%6#\"\"(#\"/n?I<.:G\"/+!*o'[Ws*/&F%6#\"\")#!,PM7$\\x\"-S998C h/&F%6#\"\"*#!\"\"\"$g\"" }}}{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 33 "#===== ===========================" }}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 13 "sc heme with " }{XPPEDIT 18 0 "c[7] = 63/64;" "6#/&%\"cG6#\"\"(*&\"#j\" \"\"\"#k!\"\"" }{TEXT -1 1 " " }}{PARA 257 "" 0 "" {TEXT -1 33 "The sc heme constructed here has " }{XPPEDIT 18 0 "c[6] = 64/91;" "6#/&%\"cG 6#\"\"'*&\"#k\"\"\"\"#\"*!\"\"" }{TEXT -1 7 " and " }{XPPEDIT 18 0 " c[7] = 63/64;" "6#/&%\"cG6#\"\"(*&\"#j\"\"\"\"#k!\"\"" }{TEXT -1 1 ". " }}{PARA 0 "" 0 "" {TEXT -1 6 "With " }{XPPEDIT 18 0 "c[6]" "6#&%\"c G6#\"\"'" }{TEXT -1 7 " and " }{XPPEDIT 18 0 "c[7]" "6#&%\"cG6#\"\"( " }{TEXT -1 39 " having these fixed values 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[5]" "6#&%\"cG6# \"\"&" }{TEXT -1 50 " are chosen to minimize the principal error norm ." }}{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 2196 "ee := \{c[2]=19/220,\nc[3]=248/1719,\nc[4]=124/57 3,\nc[5]=159/281,\nc[6]=64/91,\nc[7]=63/64,\nc[8]=1,\nc[9]=1,\n\na[2,1 ]=19/220,\na[3,1]=1334488/56144259,\na[3,2]=6765440/56144259,\na[4,1]= 31/573,\na[4,2]=0,\na[4,3]=31/191,\na[5,1]=443985772809/682326636832, \na[5,2]=0,\na[5,3]=-1687953692799/682326636832,\na[5,4]=815026549419/ 341163318416,\na[6,1]=-17777250984185217125334224/11518294736403027809 706743,\na[6,2]=0,\na[6,3]=1412202709832277291195888/21732631578118920 3956731,\na[6,4]=-60026762217662324171682463968/1222743050479704818221 7556253,\na[6,5]=443591723399448798344389120/6743536074445822618704791 69,\na[7,1]=3116283851584560023118564368757831/95530192818476958207207 1268073472,\na[7,2]=0,\na[7,3]=-15069358907111564759839129615089/11265 35292670718846783102910464,\na[7,4]=4608655696349942937342097596091097 426937/402287173573717757962711842819464495104,\na[7,5]=-1041097287013 501701647909592093443697/767936864410813987103949029050941440,\na[7,6] =126705961361465452994391/126885461609550467563520,\na[8,1]=2688622152 256014375593487443/635826944384194229962077552,\na[8,2]=0,\na[8,3]=-16 5064430696291307962649/9442530342150159719648,\na[8,4]=856473441344050 61626206038309671468707/5745317881110570819440070776064368272,\na[8,5] =-1868050624284923482186332306552702/977068785600982269831702681478435 ,\na[8,6]=16842653200587834711/13166474974917410080,\na[8,7]=-60849112 3177947136/27366835598969517651,\na[9,1]=2465359/38465280,\na[9,2]=0, \na[9,3]=0,\na[9,4]=29999319743551167/90312244465616140,\na[9,5]=24405 21938223593/9364974686865720,\na[9,6]=1241823968749/6822515208960,\na[ 9,7]=177732994465792/327930128445465,\na[9,8]=-11266189/29580120,\n\nb [1]=2465359/38465280,\nb[2]=0,\nb[3]=0,\nb[4]=29999319743551167/903122 44465616140,\nb[5]=2440521938223593/9364974686865720,\nb[6]=1241823968 749/6822515208960,\nb[7]=177732994465792/327930128445465,\nb[8]=-11266 189/29580120,\n\n`b*`[1]=739837987028857/11890322590648320,\n`b*`[2]=0 ,\n`b*`[3]=0,\n`b*`[4]=1204992945826748851603162095/355106389081239848 5590445952,\n`b*`[5]=328936061938716185672737/143966065434173057741640 0,\n`b*`[6]=102717532147008407320237/461800408829724604569600,\n`b*`[7 ]=9268295603184461676544/21663580993679805137685,\n`b*`[8]=-9633553238 997677/35173968040963440,\n`b*`[9]=-1/156\}:" }}}{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 64 "The Butcher tab leau in exact and approximate form is as follows." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 576 "subs(ee,mat rix([[c[2],a[2,1],``$2],\n [c[3],a[3,1],a[3,2],``],\n [c[4],seq(a[4, i],i=1..3)],\n [c[5],seq(a[5,i],i=1..3)],[``$3,a[5,4]],\n [c[6],seq( a[6,i],i=1..3)],[``$2,a[6,4],a[6,5]],\n [c[7],seq(a[7,i],i=1..3)],[`` ,seq(a[7,i],i=4..6)],\n [c[8],seq(a[8,i],i=1..3)],[``,seq(a[8,i],i=4. .6)],[``$3,a[8,7]],\n [c[9],seq(a[9,i],i=1..3)],[``,seq(a[9,i],i=4..6 )],[``$2,a[9,7],a[9,8]],\n [``,`_____________________________________ `$3],\n [`b`,seq(b[i],i=1..3)],[``,seq(b[i],i=4..6)],[``$2,b[7],b[8]] ,\n [`b*`,seq(`b*`[i],i=1..3)],[``,seq(`b*`[i],i=4..6)],[``,seq(`b*`[ i],i=7..9)]]));" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#K%'matrixG6#787&#\" #>\"$?#F(%!GF+7&#\"$[#\"%><#\"()[M8\")fU9c#\"(Saw'F2F+7&#\"$C\"\"$t&# \"#JF8\"\"!#F:\"$\">7&#\"$f\"\"$\"G#\"-4Gx&)RW\"-KojEBoF;#!.*z#p`zo\"F D7&F+F+F+#\"->%\\l-:)\"-;%=L;T$7&#\"#k\"#\"*#!;CULDr@&=%)4Dxx\"\";Vnq4 y-.kt%H=:\"F;#\":))e>\"HxA$)4F?79\"9Jn&R?*=\"y:jK<#7&F+F+#!>oRY#orTKiw @in-g\">`ibIb*F;#!A*3:'H\"R)fZc6r!*e$p] \"\"@k/\"H5$yY)=2n#HNl7\"7&F+#\"IPpU(4\"4'f(4Ut$H%*\\jpb'3Y\"H/^\\k%>G %=rizv%Q Wp#ejF;#!9\\E'zI\"H'pIW1l\"\"7['>(f,:U.`U%*7&F+#\"G2(o9n4$Qg?E;10W8Wtk &)\"Fs#oV1w22S%>3d56)yJXd#!C-Fb1BL'=#[B\\GC10o=\"BN%y9o-<$)pA)4g&yoq(* #\"56Z$ye+KlUo\"\"5!35u\"\\(\\ZmJ\"7&F+F+F+#!3Or%z$***H\"2ShhlWC7.*#\"1$fB#Q>_SC \"1?d'oou\\O*#\".\\(oR#=C\"\".g*3_^Ao7&F+F+#\"0#zlW*Htx\"\"0laWG,$zK#! )*=m7\"\")?,eH7&F+%F_____________________________________GFerFer7&%\"b GF`qF;F;FcqF]r7&%#b*G#\"0d)Gq)z$)R(\"2?$[1fA.*=\"F;F;7&F+#\"=&4iJg^)[n #e%H*\\?\"\"=_fW!f&[)R73*Q1^N#\"9PFn&=;(Q>1O*G$\":+kTx0tTVlg'R9#\"9P-K 2%3q9Kvr-\"\"9+'pXgC(H)3/!=Y7&F+#\"7WlnhW=.cHo#*\"8&oP^!)zO*4ej;##!1xw **QKbL'*\"2SM'4/oR " 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,$\")OOO')!\"*F(%!GF+F+F+F+F+F+F +7,$\")#*pU9!\")$\")8*oP#F*$\"),,07F/F+F+F+F+F+F+F+7,$\")*[S;#F/$\")A7 5aF*$\"\"!F:$\")m.B;F/F+F+F+F+F+F+7,$\")IOecF/$\")*Qp]'F/F9$!)1#QZ#!\" ($\")I'*)Q#FDF+F+F+F+F+7,$\")q'H.(F/$!)ERV:FDF9$\")]2)\\'FD$!)')=4\\FD $\")&G!ylF/F+F+F+F+7,$\")+vV)*F/$\")J4iKFDF9$!)InP8!\"'$\")MhX6FY$!)pq b8FD$\")M&e)**F/F+F+F+7,$\"\"\"F:$\")RaGUFDF9$!)`4[FD$\")v?z7FD$!)=YBAF*F+F+7,F[o$\")*4$4kF*F9F9$\")Mt@LF/$\")-,1EF /$\")]=?=F/$\")M%)>aF/$!)Hq3QF/F+7,%\"bGFjoF9F9F\\pF^pF`pFbpFdpF+7,%#b *G$\")g=AiF*F9F9$\")%HLR$F/$\")m\"[G#F/$\")TGCAF/$\")UGyUF/$!)/$)QFF/$ !)kD5k!#5Q)pprint236\"" }}}{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#$\"+6<$ou\"!#9" }}}{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#$ \"+-0QSe!#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] = 19/220;" " 6#/&%\"cG6#\"\"#*&\"#>\"\"\"\"$?#!\"\"" }{TEXT -1 3 ", " }{XPPEDIT 18 0 "c[4] = 124/573;" "6#/&%\"cG6#\"\"%*&\"$C\"\"\"\"\"$t&!\"\"" } {TEXT -1 3 ", " }{XPPEDIT 18 0 "c[5] = 159/281;" "6#/&%\"cG6#\"\"&*& \"$f\"\"\"\"\"$\"G!\"\"" }{TEXT -1 3 ", " }{XPPEDIT 18 0 "c[6] = 64/9 1;" "6#/&%\"cG6#\"\"'*&\"#k\"\"\"\"#\"*!\"\"" }{TEXT -1 3 ", " } {XPPEDIT 18 0 "c[7] = 63/64;" "6#/&%\"cG6#\"\"(*&\"#j\"\"\"\"#k!\"\"" }{TEXT -1 3 ", " }{XPPEDIT 18 0 "c[8]=1" "6#/&%\"cG6#\"\")\"\"\"" } {TEXT -1 2 ", " }}{PARA 0 "" 0 "" {TEXT -1 31 "the zero linking coeffi cients: " }}{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 weights 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 speci fy 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] = -1/156;" "6#/&%#b*G6 #\"\"*,$*&\"\"\"F*\"$c\"!\"\"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 0 215 "e1 := \{c[2 ]=19/220,c[4]=124/573,c[5]=159/281,c[6]=64/91,c[7]=63/64,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]=-1/156\}:\neqns := subs(e1,cdns):\nnops(%);\nindets(eqns);\nnops(%) ;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#\"#W" }}{PARA 12 "" 1 "" {XPPMATH 20 "6#&F<6$F*F'&F<6 $F*F:&F<6$\"\"*F'&F<6$FIF>&F<6$FIF:&F<6$FIF*&F<6$FIF-&F<6$FIF0&F<6$FIF 3&F<6$FIF6&%#b*GF)&FYF,&FYF/&FYF2&FYF5&F<6$F-F:&F<6$F-F*&F<6$F0F'&F<6$ F0F:&F<6$F0F*&F<6$F0F-&F<6$F3F'&F<6$F3F:&F<6$F3F*&F<6$F3F-&F<6$F3F0&F< 6$F6F'&F<6$F-F'&FYF&&F<6$F6F:&F<6$F6F*&F<6$F6F-&F<6$F6F0&F<6$F6F3" }} {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 2315 "e3 := \{b[7] = 17773299446 5792/327930128445465, a[9,7] = 177732994465792/327930128445465, a[9,6] = 1241823968749/6822515208960, b[1] = 2465359/38465280, a[2,1] = 19/2 20, a[4,2] = 0, a[5,2] = 0, a[6,2] = 0, a[7,2] = 0, a[8,2] = 0, c[6] = 64/91, a[6,1] = -17777250984185217125334224/1151829473640302780970674 3, a[9,5] = 2440521938223593/9364974686865720, `b*`[3] = 0, a[9,4] = 2 9999319743551167/90312244465616140, a[9,2] = 0, a[9,3] = 0, a[9,1] = 2 465359/38465280, a[5,4] = 815026549419/341163318416, a[8,3] = -1650644 30696291307962649/9442530342150159719648, c[8] = 1, c[9] = 1, b[2] = 0 , b[3] = 0, `b*`[2] = 0, a[9,8] = -11266189/29580120, b[4] = 299993197 43551167/90312244465616140, a[7,3] = -15069358907111564759839129615089 /1126535292670718846783102910464, a[6,3] = 1412202709832277291195888/2 17326315781189203956731, `b*`[6] = 102717532147008407320237/4618004088 29724604569600, c[2] = 19/220, a[7,4] = 460865569634994293734209759609 1097426937/402287173573717757962711842819464495104, `b*`[7] = 92682956 03184461676544/21663580993679805137685, a[5,3] = -1687953692799/682326 636832, b[8] = -11266189/29580120, c[7] = 63/64, c[4] = 124/573, c[5] \+ = 159/281, a[8,4] = 85647344134405061626206038309671468707/57453178811 10570819440070776064368272, a[4,1] = 31/573, a[4,3] = 31/191, a[8,6] = 16842653200587834711/13166474974917410080, a[6,5] = 44359172339944879 8344389120/674353607444582261870479169, b[5] = 2440521938223593/936497 4686865720, `b*`[9] = -1/156, a[8,7] = -608491123177947136/27366835598 969517651, a[7,6] = 126705961361465452994391/126885461609550467563520, `b*`[4] = 1204992945826748851603162095/3551063890812398485590445952, \+ c[3] = 248/1719, a[7,1] = 3116283851584560023118564368757831/955301928 184769582072071268073472, a[7,5] = -1041097287013501701647909592093443 697/767936864410813987103949029050941440, a[3,2] = 6765440/56144259, ` b*`[8] = -9633553238997677/35173968040963440, a[8,1] = 268862215225601 4375593487443/635826944384194229962077552, a[6,4] = -60026762217662324 171682463968/12227430504797048182217556253, `b*`[1] = 739837987028857/ 11890322590648320, b[6] = 1241823968749/6822515208960, a[3,1] = 133448 8/56144259, `b*`[5] = 328936061938716185672737/14396606543417305774164 00, a[8,5] = -1868050624284923482186332306552702/977068785600982269831 702681478435, a[5,1] = 443985772809/682326636832\}:" }{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 0 576 "subs(e3,matrix([[c[2],a[2,1],``$2],\n [c[3],a[3,1], a[3,2],``],\n [c[4],seq(a[4,i],i=1..3)],\n [c[5],seq(a[5,i],i=1..3)] ,[``$3,a[5,4]],\n [c[6],seq(a[6,i],i=1..3)],[``$2,a[6,4],a[6,5]],\n \+ [c[7],seq(a[7,i],i=1..3)],[``,seq(a[7,i],i=4..6)],\n [c[8],seq(a[8,i] ,i=1..3)],[``,seq(a[8,i],i=4..6)],[``$3,a[8,7]],\n [c[9],seq(a[9,i],i =1..3)],[``,seq(a[9,i],i=4..6)],[``$2,a[9,7],a[9,8]],\n [``,`________ _____________________________`$3],\n [`b`,seq(b[i],i=1..3)],[``,seq(b [i],i=4..6)],[``$2,b[7],b[8]],\n [`b*`,seq(`b*`[i],i=1..3)],[``,seq(` b*`[i],i=4..6)],[``,seq(`b*`[i],i=7..9)]]));" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#K%'matrixG6#787&#\"#>\"$?#F(%!GF+7&#\"$[#\"%><#\"()[M8 \")fU9c#\"(Saw'F2F+7&#\"$C\"\"$t&#\"#JF8\"\"!#F:\"$\">7&#\"$f\"\"$\"G# \"-4Gx&)RW\"-KojEBoF;#!.*z#p`zo\"FD7&F+F+F+#\"->%\\l-:)\"-;%=L;T$7&#\" #k\"#\"*#!;CULDr@&=%)4Dxx\"\";Vnq4y-.kt%H=:\"F;#\":))e>\"HxA$)4F?79\"9 Jn&R?*=\"y:jK<#7&F+F+#!>oRY#orTKiw@in-g\">`ibIb*F;#!A*3:'H\"R)fZc6r!*e$p]\"\"@k/\"H5$yY)=2n#HNl7\"7&F+#\"IPpU (4\"4'f(4Ut$H%*\\jpb'3Y\"H/^\\k%>G%=rizv%QWp#ejF;#!9\\E'zI\"H'pIW1l\"\"7['>(f ,:U.`U%*7&F+#\"G2(o9n4$Qg?E;10W8Wtk&)\"Fs#oV1w22S%>3d56)yJXd#!C-Fb1BL' =#[B\\GC10o=\"BN%y9o-<$)pA)4g&yoq(*#\"56Z$ye+KlUo\"\"5!35u\"\\(\\ZmJ\" 7&F+F+F+#!3Or%z$***H\"2ShhlWC7.*#\"1$fB#Q>_SC\"1?d'oou\\O*#\".\\(oR#=C\"\".g*3_^ Ao7&F+F+#\"0#zlW*Htx\"\"0laWG,$zK#!)*=m7\"\")?,eH7&F+%F_______________ ______________________GFerFer7&%\"bGF`qF;F;FcqF]r7&%#b*G#\"0d)Gq)z$)R( \"2?$[1fA.*=\"F;F;7&F+#\"=&4iJg^)[n#e%H*\\?\"\"=_fW!f&[)R73*Q1^N#\"9PF n&=;(Q>1O*G$\":+kTx0tTVlg'R9#\"9P-K2%3q9Kvr-\"\"9+'pXgC(H)3/!=Y7&F+#\" 7WlnhW=.cHo#*\"8&oP^!)zO*4ej;##!1xw**QKbL'*\"2SM'4/oR " 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, $\")OOO')!\"*F(%!GF+F+F+F+F+F+F+7,$\")#*pU9!\")$\")8*oP#F*$\"),,07F/F+ F+F+F+F+F+F+7,$\")*[S;#F/$\")A75aF*$\"\"!F:$\")m.B;F/F+F+F+F+F+F+7,$\" )IOecF/$\")*Qp]'F/F9$!)1#QZ#!\"($\")I'*)Q#FDF+F+F+F+F+7,$\")q'H.(F/$!) ERV:FDF9$\")]2)\\'FD$!)')=4\\FD$\")&G!ylF/F+F+F+F+7,$\")+vV)*F/$\")J4i KFDF9$!)InP8!\"'$\")MhX6FY$!)pqb8FD$\")M&e)**F/F+F+F+7,$\"\"\"F:$\")Ra GUFDF9$!)`4[FD$\")v?z7FD$!)=YBAF*F+F+7,F[o$\" )*4$4kF*F9F9$\")Mt@LF/$\")-,1EF/$\")]=?=F/$\")M%)>aF/$!)Hq3QF/F+7,%\"b GFjoF9F9F\\pF^pF`pFbpFdpF+7,%#b*G$\")g=AiF*F9F9$\")%HLR$F/$\")m\"[G#F/ $\")TGCAF/$\")UGyUF/$!)/$)QFF/$!)kD5k!#5Q)pprint216\"" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 146 "RK6_8e qs := [op(RowSumConditions(8,'expanded')),op(OrderConditions(6,8,'expa nded'))]:\n`RK5_9eqs*` := subs(b=`b*`,OrderConditions(5,9,'expanded')) :" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 7 "Chec k: " }}{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 28 "determination of 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[5]" "6#&%\"cG6#\"\"&" } {TEXT -1 1 " " }}{PARA 4 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 79 "In this section we obtain the nodes of a modification of \+ a scheme by Tsitouras." }}{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 33 "#======================== ========" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 22 "We specify the nodes " }{XPPEDIT 18 0 "c[6] = 64/91;" "6#/&%\"cG6 #\"\"'*&\"#k\"\"\"\"#\"*!\"\"" }{TEXT -1 7 " and " }{XPPEDIT 18 0 "c [7] = 63/64;" "6#/&%\"cG6#\"\"(*&\"#j\"\"\"\"#k!\"\"" }{TEXT -1 27 " a nd determine 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[5]" "6#&%\"cG6#\"\"&" }{TEXT -1 65 " that \+ minimize the principal errror norm (subject to the nodes " }{XPPEDIT 18 0 "c[6]" "6#&%\"cG6#\"\"'" }{TEXT -1 7 " and " }{XPPEDIT 18 0 "c[ 7]" "6#&%\"cG6#\"\"(" }{TEXT -1 19 " remaining fixed)." }}{PARA 0 "" 0 "" {TEXT -1 85 "We use the general solution to obtain expressions f or the coefficients in terms of " }{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[5]" "6#&%\"cG6#\"\"&" }{TEXT -1 2 ". \+ " }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 71 "eA := \{c[6]=64/91,c[7]= 63/64\}:\neB := `union`(eA,simplify(subs(eA,eG))):" }}}{PARA 0 "" 0 " " {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}} {SECT 1 {PARA 4 "" 0 "" {TEXT -1 2 "eB" }}{PARA 0 "" 0 "" {TEXT -1 0 " " }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 15934 "eB := \{`b*`[7] = -41 94304/1546965*(-1059230*c[5]^3*c[4]^2+97823*c[5]^2-60160*c[5]^3-45162* c[4]+97823*c[4]^2-45162*c[5]+5760+460050*c[4]^3*c[5]-1059230*c[4]^3*c[ 5]^2+339406*c[4]*c[5]-737485*c[4]*c[5]^2+460050*c[5]^3*c[4]+710400*c[5 ]^3*c[4]^3-60160*c[4]^3+1646700*c[4]^2*c[5]^2-737485*c[4]^2*c[5])/(-16 572416*c[5]^3*c[4]^2+4721472*c[5]^2-2064384*c[5]^3-3421467*c[4]+472147 2*c[4]^2-3421467*c[5]+762048+10790144*c[4]^3*c[5]-16572416*c[4]^3*c[5] ^2+16528179*c[4]*c[5]-23881132*c[4]*c[5]^2+10790144*c[5]^3*c[4]+786432 0*c[5]^3*c[4]^3-2064384*c[4]^3+35701120*c[4]^2*c[5]^2-23881132*c[4]^2* c[5]), a[8,7] = -50331648/11459*(185*c[4]*c[5]-47*c[5]-47*c[4]+10)*(c[ 4]-1)*(-1+c[5])/(64*c[5]-63)/(64*c[4]-63)/(11740*c[4]*c[5]-3093*c[5]-3 093*c[4]+772), a[9,1] = 1/241920*(51790*c[4]*c[5]-8647*c[5]-8647*c[4]+ 2321)/c[4]/c[5], c[3] = 2/3*c[4], a[4,3] = 3/4*c[4], a[3,2] = 2/9*c[4] ^2/c[2], a[7,6] = -77716530801/2147483648*(5*c[4]*c[5]+1-2*c[4]-2*c[5] )*(64*c[4]-63)*(64*c[5]-63)/(185*c[4]*c[5]-47*c[5]-47*c[4]+10)/(91*c[4 ]-64)/(91*c[5]-64), a[4,2] = 0, a[5,2] = 0, a[6,2] = 0, a[7,2] = 0, a[ 8,2] = 0, b[7] = -268435456/1546965*(185*c[4]*c[5]-47*c[5]-47*c[4]+10) /(64*c[4]-63)/(64*c[5]-63), a[6,5] = -32/68574961*(18295459*c[4]^2*c[5 ]^2-745472*c[4]^2+262144*c[4]-6062784*c[4]*c[5]^2-11807168*c[4]^2*c[5] +529984*c[4]^3-7454720*c[4]^4*c[5]+15671389*c[4]^3*c[5]-23354240*c[4]^ 3*c[5]^2+3891200*c[4]*c[5]+10599680*c[4]^4*c[5]^2+745472*c[5]^2-524288 *c[5])/c[5]/(-c[4]^3+2*c[4]^2*c[5]-30*c[4]^3*c[5]^2+6*c[4]^3*c[5]+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[4]*c[5]^2), `b*`[9] = 1/10*(3200*c[4]^2*c[5]^2-3015*c[4]*c[5]^2+640* c[5]^2-3015*c[4]^2*c[5]+3150*c[4]*c[5]-731*c[5]+640*c[4]^2-731*c[4]+19 2)/(512*c[5]^2+192-667*c[4]+512*c[4]^2-667*c[5]+2611*c[4]*c[5]-2156*c[ 4]*c[5]^2+1920*c[4]^2*c[5]^2-2156*c[4]^2*c[5]), c[6] = 64/91, `b*`[8] \+ = 0, a[7,3] = 567/67108864*(-4023335652*c[5]^3*c[4]^2+108657738*c[5]^2 -84639744*c[5]^3+15494976*c[4]-54328869*c[4]^2-30989952*c[5]+812646400 *c[5]^4*c[4]^4-2084351488*c[4]^4*c[5]-898691200*c[5]^4*c[4]^3+24379392 00*c[4]^5*c[5]^3+2190353124*c[4]^3*c[5]-9514173970*c[4]^3*c[5]^2+63479 8080*c[5]*c[4]^5+211599360*c[5]^4*c[4]^2+240268581*c[4]*c[5]-103851102 2*c[4]*c[5]^2+867314944*c[5]^3*c[4]-2696073600*c[4]^5*c[5]^2-847064320 0*c[5]^3*c[4]^4+9050207290*c[5]^3*c[4]^3+42319872*c[4]^3+4460423796*c[ 4]^2*c[5]^2+8819680570*c[4]^4*c[5]^2-986993219*c[4]^2*c[5])/c[4]^2/(10 *c[5]^3*c[4]^2+10*c[4]^3*c[5]^2-30*c[4]^2*c[5]^2+6*c[4]*c[5]^2-c[5]^2+ 6*c[4]^2*c[5]+c[4]*c[5]-c[4]^2)/(185*c[4]*c[5]-47*c[5]-47*c[4]+10), b[ 4] = 1/60*(-2321+8647*c[5])/c[4]/(-15653*c[4]^3+5824*c[4]^4+13861*c[4] ^2-4032*c[4]-5824*c[4]^3*c[5]+15653*c[4]^2*c[5]-13861*c[4]*c[5]+4032*c [5]), a[9,7] = -268435456/1546965*(185*c[4]*c[5]-47*c[5]-47*c[4]+10)/( 4096*c[4]*c[5]-4032*c[4]-4032*c[5]+3969), a[5,4] = -c[5]^2*(c[4]-c[5]) /c[4]^2, `b*`[3] = 0, a[9,2] = 0, a[9,3] = 0, b[5] = -1/60*(8647*c[4]- 2321)/(c[4]-c[5])/c[5]/(5824*c[5]^3-15653*c[5]^2+13861*c[5]-4032), `b* `[6] = 68574961/6286080*(-1928450*c[5]^3*c[4]^2+35136+298950*c[5]^2-15 7440*c[5]^3-181197*c[4]+298950*c[4]^2-181197*c[5]+981050*c[4]^3*c[5]-1 928450*c[4]^3*c[5]^2+1008291*c[4]*c[5]-1770598*c[4]*c[5]^2+981050*c[5] ^3*c[4]+1190400*c[5]^3*c[4]^3-157440*c[4]^3+3300430*c[4]^2*c[5]^2-1770 598*c[4]^2*c[5])/(512*c[5]^2+192-667*c[4]+512*c[4]^2-667*c[5]+2611*c[4 ]*c[5]-2156*c[4]*c[5]^2+1920*c[4]^2*c[5]^2-2156*c[4]^2*c[5])/(8281*c[4 ]*c[5]-5824*c[4]-5824*c[5]+4096), a[8,5] = -1/2*(-13363607416*c[5]^3*c [4]^2+296006528*c[5]^4+175609130*c[5]^2-91693056*c[5]^5-348440746*c[5] ^3+15740928*c[4]-65959830*c[4]^2-31481856*c[5]+21524520380*c[5]^4*c[4] ^4-1750080621*c[4]^4*c[5]-23519702042*c[5]^4*c[4]^3+7503180160*c[5]^5* c[4]^3+8262633020*c[4]^5*c[5]^3+2310963200*c[5]^5*c[4]^5-2961798013*c[ 5]^4*c[4]+1719192194*c[4]^3*c[5]-12194176768*c[4]^3*c[5]^2-7280657920* c[4]^5*c[5]^4+634798080*c[5]*c[4]^5+11941631067*c[5]^4*c[4]^2+21647855 0*c[4]*c[5]-1565892493*c[4]*c[5]^2+3357335284*c[5]^3*c[4]-3927736380*c [4]^5*c[5]^2-24318633814*c[5]^3*c[4]^4+26410713672*c[5]^3*c[4]^3+93813 5744*c[5]^5*c[4]-30573882*c[4]^4+80792784*c[4]^3-3813630528*c[5]^5*c[4 ]^2-6846955520*c[4]^4*c[5]^5+6090473054*c[4]^2*c[5]^2+11421723457*c[4] ^4*c[5]^2-788906347*c[4]^2*c[5])/c[5]/(82527872*c[5]^3*c[4]^2-18013632 *c[5]^6-20058964*c[5]^4+34897225*c[5]^5+3112704*c[5]^3+180136320*c[5]^ 7*c[4]^2+4761397570*c[5]^4*c[4]^4-152562633*c[4]^4*c[5]+1438562048*c[5 ]^4*c[4]^3-4239141570*c[5]^5*c[4]^3+777367770*c[4]^5*c[5]^3+683737600* c[5]^5*c[4]^5+176455552*c[5]^6*c[4]+128296111*c[5]^4*c[4]+13793236*c[4 ]^3*c[5]-93158575*c[4]^3*c[5]^2-1334060920*c[4]^5*c[5]^4-868531702*c[5 ]^4*c[4]^2-6225408*c[4]*c[5]^2+8970728*c[5]^3*c[4]-124709760*c[4]^5*c[ 5]^2-683737600*c[5]^7*c[4]^3-3401503262*c[5]^3*c[4]^4+188224438*c[5]^3 *c[4]^3+3385273720*c[5]^6*c[4]^3-1299623770*c[5]^6*c[4]^2-306762178*c[ 5]^5*c[4]+12470976*c[4]^4-3112704*c[4]^3+1911195422*c[5]^5*c[4]^2-2231 349120*c[4]^4*c[5]^5-15175976*c[4]^2*c[5]^2+1005083074*c[4]^4*c[5]^2+6 225408*c[4]^2*c[5]), a[2,1] = c[2], a[8,1] = 1/448*(-346535362*c[5]^3* c[4]^2+7053312*c[5]^4+2623488*c[5]^2-9156896*c[5]^3+874496*c[4]^2+7803 8500*c[5]^4*c[4]^6+1024823670*c[5]^4*c[4]^4+103288446*c[4]^4*c[5]-8643 88746*c[5]^4*c[4]^3+35266560*c[4]^6*c[5]+91596530*c[5]^5*c[4]^3+134366 550*c[4]^5*c[5]^3+78038500*c[5]^5*c[4]^5-132236590*c[4]^6*c[5]^2-73862 837*c[5]^4*c[4]-55410318*c[4]^3*c[5]+26688131*c[4]^3*c[5]^2-540970520* c[4]^5*c[5]^4+57670090*c[4]^6*c[5]^3-98560835*c[5]*c[4]^5+352180141*c[ 5]^4*c[4]^2-3060736*c[4]*c[5]-14826915*c[4]*c[5]^2+84874472*c[5]^3*c[4 ]+311895734*c[4]^5*c[5]^2-616921052*c[5]^3*c[4]^4+706380726*c[5]^3*c[4 ]^3-1698549*c[4]^5+4488488*c[4]^4-3664435*c[4]^3-17633280*c[5]^5*c[4]^ 2-142317110*c[4]^4*c[5]^5+28714674*c[4]^2*c[5]^2-225646650*c[4]^4*c[5] ^2+18736835*c[4]^2*c[5])/c[5]/c[4]^2/(170950*c[5]^3*c[4]^2-772*c[5]^2+ 3093*c[5]^3-772*c[4]^2+117400*c[5]^4*c[4]^3-30298*c[4]^3*c[5]+170950*c [4]^3*c[5]^2-30930*c[5]^4*c[4]^2+772*c[4]*c[5]+4632*c[4]*c[5]^2-30298* c[5]^3*c[4]+117400*c[5]^3*c[4]^4-414060*c[5]^3*c[4]^3+3093*c[4]^3-4853 6*c[4]^2*c[5]^2-30930*c[4]^4*c[5]^2+4632*c[4]^2*c[5]), a[9,4] = 1/60*( -2321+8647*c[5])/c[4]/(-15653*c[4]^3+5824*c[4]^4+13861*c[4]^2-4032*c[4 ]-5824*c[4]^3*c[5]+15653*c[4]^2*c[5]-13861*c[4]*c[5]+4032*c[5]), b[6] \+ = 6240321451/169724160*(310*c[4]*c[5]-123*c[5]-123*c[4]+61)/(91*c[4]-6 4)/(91*c[5]-64), a[7,1] = 63/2147483648*(-395323037271*c[5]^3*c[4]^2+8 125415424*c[5]^4+2975035392*c[5]^2-10431142848*c[5]^3+991678464*c[4]^2 +92584140800*c[5]^4*c[4]^6+1195286075550*c[5]^4*c[4]^4+117552090665*c[ 4]^4*c[5]-1001572181708*c[5]^4*c[4]^3+40627077120*c[4]^6*c[5]+10604913 7920*c[5]^5*c[4]^3+156322465950*c[4]^5*c[5]^3+92584140800*c[5]^5*c[4]^ 5-152773927680*c[4]^6*c[5]^2-85239712896*c[5]^4*c[4]-63032492630*c[4]^ 3*c[5]+30619244233*c[4]^3*c[5]^2-636902396160*c[4]^5*c[5]^4+6746951872 0*c[4]^6*c[5]^3-112677766592*c[5]*c[4]^5+406960951232*c[5]^4*c[4]^2-34 70874624*c[4]*c[5]-16811318727*c[4]*c[5]^2+96737786958*c[5]^3*c[4]+357 224299892*c[4]^5*c[5]^2-709499994546*c[5]^3*c[4]^4+807630281340*c[5]^3 *c[4]^3-1977478272*c[4]^5+5166144270*c[4]^4-4176677799*c[4]^3-20313538 560*c[5]^5*c[4]^2-166572644480*c[4]^4*c[5]^5+32484424714*c[4]^2*c[5]^2 -257046329284*c[4]^4*c[5]^2+21292390791*c[4]^2*c[5])/(2620*c[5]^3*c[4] ^2-10*c[5]^2+47*c[5]^3-10*c[4]^2+1850*c[5]^4*c[4]^3-467*c[4]^3*c[5]+26 20*c[4]^3*c[5]^2-470*c[5]^4*c[4]^2+10*c[4]*c[5]+60*c[4]*c[5]^2-467*c[5 ]^3*c[4]+1850*c[5]^3*c[4]^4-6490*c[5]^3*c[4]^3+47*c[4]^3-679*c[4]^2*c[ 5]^2-470*c[4]^4*c[5]^2+60*c[4]^2*c[5])/c[5]/c[4]^2, c[8] = 1, c[9] = 1 , b[2] = 0, b[3] = 0, `b*`[2] = 0, a[7,5] = -63/33554432*(512983446528 *c[5]^3*c[4]^2-5776662528*c[5]^4-9074550681*c[5]^2+12858172992*c[5]^3- 991678464*c[4]+4176677799*c[4]^2+1983356928*c[5]-435379814400*c[5]^4*c [4]^4+113151153600*c[4]^4*c[5]+475023287040*c[5]^4*c[4]^3-317546086400 *c[4]^5*c[5]^3+59194244928*c[5]^4*c[4]-113853627049*c[4]^3*c[5]+657586 456912*c[4]^3*c[5]^2+147901644800*c[4]^5*c[5]^4-40627077120*c[5]*c[4]^ 5-240969408512*c[5]^4*c[4]^2-14666776551*c[4]*c[5]+84120317501*c[4]*c[ 5]^2-127549156867*c[5]^3*c[4]+210540552960*c[4]^5*c[5]^2+931999580160* c[5]^3*c[4]^4-1012712595060*c[5]^3*c[4]^3+1977478272*c[4]^4-5166144270 *c[4]^3-330675092506*c[4]^2*c[5]^2-612544554868*c[4]^4*c[5]^2+54036905 519*c[4]^2*c[5])/c[5]/(-13568*c[5]^3*c[4]^2+3918*c[5]^4-4277*c[5]^5-64 0*c[5]^3-547820*c[5]^4*c[4]^4+34165*c[4]^4*c[5]-385280*c[5]^4*c[4]^3+6 66220*c[5]^5*c[4]^3-161170*c[4]^5*c[5]^3-31071*c[5]^4*c[4]-1742*c[4]^3 *c[5]+14751*c[4]^3*c[5]^2+168350*c[4]^5*c[5]^4+186972*c[5]^4*c[4]^2+12 80*c[4]*c[5]^2-988*c[5]^3*c[4]+30080*c[4]^5*c[5]^2+623700*c[5]^3*c[4]^ 4-19292*c[5]^3*c[4]^3-168350*c[5]^6*c[4]^3+42770*c[5]^6*c[4]^2+42497*c [5]^5*c[4]-3008*c[4]^4+640*c[4]^3-268500*c[5]^5*c[4]^2+1820*c[4]^2*c[5 ]^2-210177*c[4]^4*c[5]^2-1280*c[4]^2*c[5]), a[9,6] = 6240321451/169724 160*(310*c[4]*c[5]-123*c[5]-123*c[4]+61)/(8281*c[4]*c[5]-5824*c[4]-582 4*c[5]+4096), a[4,1] = 1/4*c[4], a[3,1] = -2/9*c[4]*(c[4]-3*c[2])/c[2] , `b*`[4] = -1/60*(22022400*c[5]^3*c[4]^2-9192094*c[5]^2+5907200*c[5]^ 3+1944609*c[4]-1479936*c[4]^2+4167585*c[5]-15277681*c[4]*c[5]+36322918 *c[4]*c[5]^2-24823010*c[5]^3*c[4]-30243170*c[4]^2*c[5]^2+12087278*c[4] ^2*c[5]-562752)/(-12096-32256*c[5]^2+54309*c[4]-74944*c[4]^2+42021*c[5 ]-137984*c[4]^3*c[5]+122880*c[4]^3*c[5]^2-207181*c[4]*c[5]+168596*c[4] *c[5]^2+32768*c[4]^3-258944*c[4]^2*c[5]^2+302932*c[4]^2*c[5])/(91*c[4] -64)/(c[4]-c[5])/c[4], b[1] = 1/241920*(51790*c[4]*c[5]-8647*c[5]-8647 *c[4]+2321)/c[4]/c[5], a[7,4] = -63/33554432*(991016267063*c[5]^3*c[4] ^2-16250830848*c[5]^4-5950070784*c[5]^2+20862285696*c[5]^3-1983356928* c[4]^2-443704934400*c[5]^4*c[4]^6+115533250560*c[5]*c[4]^7-35308140109 40*c[5]^4*c[4]^4-494014793337*c[4]^4*c[5]+2667334727576*c[5]^4*c[4]^3- 460606125056*c[4]^6*c[5]-210540552960*c[5]^5*c[4]^3+212022942720*c[4]^ 5*c[5]^3-147901644800*c[5]^5*c[4]^5+443704934400*c[5]^3*c[4]^7+1796234 950460*c[4]^6*c[5]^2+187653641088*c[5]^4*c[4]+197230039213*c[4]^3*c[5] -114554716664*c[4]^3*c[5]^2+2125531296000*c[4]^5*c[5]^4-1199466086400* c[4]^6*c[5]^3+685172674008*c[5]*c[4]^5-490685395200*c[4]^7*c[5]^2-9890 10083264*c[5]^4*c[4]^2+6941749248*c[4]*c[5]+39984003801*c[4]*c[5]^2-21 5766119004*c[5]^3*c[4]-2308286082776*c[4]^5*c[5]^2+1967784038696*c[5]^ 3*c[4]^4-2220936003832*c[5]^3*c[4]^3+5776662528*c[4]^5-12858172992*c[4 ]^4+9074550681*c[4]^3+40627077120*c[5]^5*c[4]^2+317546086400*c[4]^4*c[ 5]^5-98204168111*c[4]^2*c[5]^2+1181858111120*c[4]^4*c[5]^2-50356190745 *c[4]^2*c[5])/(212*c[5]^3*c[4]^2-47*c[5]^4+10*c[5]^3-467*c[4]^4*c[5]+6 020*c[5]^4*c[4]^3-1850*c[5]^5*c[4]^3+1850*c[4]^5*c[5]^3+467*c[5]^4*c[4 ]+13*c[4]^3*c[5]-212*c[4]^3*c[5]^2-2620*c[5]^4*c[4]^2-20*c[4]*c[5]^2-1 3*c[5]^3*c[4]-470*c[4]^5*c[5]^2-6020*c[5]^3*c[4]^4+47*c[4]^4-10*c[4]^3 +470*c[5]^5*c[4]^2+2620*c[4]^4*c[5]^2+20*c[4]^2*c[5])/(91*c[4]-64)/c[4 ]^2, a[8,3] = 3/4*(-2985923*c[5]^3*c[4]^2+81758*c[5]^2-62976*c[5]^3+11 712*c[4]-40879*c[4]^2-23424*c[5]+595200*c[5]^4*c[4]^4-1557825*c[4]^4*c [5]-666170*c[5]^4*c[4]^3+1785600*c[4]^5*c[5]^3+1646672*c[4]^3*c[5]-712 2068*c[4]^3*c[5]^2+472320*c[5]*c[4]^5+157440*c[5]^4*c[4]^2+181423*c[4] *c[5]-780708*c[4]*c[5]^2+644324*c[5]^3*c[4]-1998510*c[4]^5*c[5]^2-6235 880*c[5]^3*c[4]^4+6701940*c[5]^3*c[4]^3+31488*c[4]^3+3348052*c[4]^2*c[ 5]^2+6566820*c[4]^4*c[5]^2-744060*c[4]^2*c[5])/c[4]^2/(170950*c[5]^3*c [4]^2-772*c[5]^2+3093*c[5]^3-772*c[4]^2+117400*c[5]^4*c[4]^3-30298*c[4 ]^3*c[5]+170950*c[4]^3*c[5]^2-30930*c[5]^4*c[4]^2+772*c[4]*c[5]+4632*c [4]*c[5]^2-30298*c[5]^3*c[4]+117400*c[5]^3*c[4]^4-414060*c[5]^3*c[4]^3 +3093*c[4]^3-48536*c[4]^2*c[5]^2-30930*c[4]^4*c[5]^2+4632*c[4]^2*c[5]) , a[6,1] = 16/68574961*(-96164341*c[5]^3*c[4]^2+1572864*c[5]^2-2981888 *c[5]^3+524288*c[4]^2+30142840*c[5]^4*c[4]^4+28328494*c[4]^4*c[5]-2649 9200*c[5]^4*c[4]^3+30142840*c[4]^5*c[5]^3-22554368*c[4]^3*c[5]-2330848 5*c[4]^3*c[5]^2-14909440*c[5]*c[4]^5+7454720*c[5]^4*c[4]^2-1835008*c[4 ]*c[5]-9437184*c[4]*c[5]^2+25311104*c[5]^3*c[4]+21199360*c[4]^5*c[5]^2 -148726760*c[5]^3*c[4]^4+175100744*c[5]^3*c[4]^3+1059968*c[4]^4-149094 4*c[4]^3+25118848*c[4]^2*c[5]^2-6258616*c[4]^4*c[5]^2+9273344*c[4]^2*c [5])/c[5]/c[4]^2/(10*c[5]^3*c[4]^2+10*c[4]^3*c[5]^2-30*c[4]^2*c[5]^2+6 *c[4]*c[5]^2-c[5]^2+6*c[4]^2*c[5]+c[4]*c[5]-c[4]^2), `b*`[5] = 1/60*(- 1479936*c[5]^2+4167585*c[4]-9192094*c[4]^2+1944609*c[5]-24823010*c[4]^ 3*c[5]+22022400*c[4]^3*c[5]^2-15277681*c[4]*c[5]+12087278*c[4]*c[5]^2+ 5907200*c[4]^3-30243170*c[4]^2*c[5]^2+36322918*c[4]^2*c[5]-562752)/(12 2880*c[5]^3*c[4]^2-12096-74944*c[5]^2+32768*c[5]^3+42021*c[4]-32256*c[ 4]^2+54309*c[5]-207181*c[4]*c[5]+302932*c[4]*c[5]^2-137984*c[5]^3*c[4] -258944*c[4]^2*c[5]^2+168596*c[4]^2*c[5])/(91*c[5]-64)/(c[4]-c[5])/c[5 ], `b*`[1] = 1/241920*(-158504650*c[5]^3*c[4]^2+9192094*c[5]^2-5907200 *c[5]^3-4167585*c[4]+9192094*c[4]^2-4167585*c[5]+57246690*c[4]^3*c[5]- 158504650*c[4]^3*c[5]^2+35156495*c[4]*c[5]-84486206*c[4]*c[5]^2+572466 90*c[5]^3*c[4]+121459200*c[5]^3*c[4]^3-5907200*c[4]^3+220432710*c[4]^2 *c[5]^2-84486206*c[4]^2*c[5]+562752)/(512*c[5]^2+192-667*c[4]+512*c[4] ^2-667*c[5]+2611*c[4]*c[5]-2156*c[4]*c[5]^2+1920*c[4]^2*c[5]^2-2156*c[ 4]^2*c[5])/c[5]/c[4], a[5,1] = 1/4*c[5]*(2*c[5]^2-5*c[4]*c[5]+4*c[4]^2 )/c[4]^2, b[8] = 1/1620*(11740*c[4]*c[5]-3093*c[5]-3093*c[4]+772)/(c[4 ]-1)/(-1+c[5]), a[5,3] = 3/4*c[5]^2*(-2*c[5]+3*c[4])/c[4]^2, a[9,8] = \+ 1/1620*(11740*c[4]*c[5]-3093*c[5]-3093*c[4]+772)/(-c[4]+c[4]*c[5]+1-c[ 5]), a[6,3] = 48/68574961*(56553879*c[4]^3*c[5]-38304384*c[4]^2*c[5]+1 0928128*c[4]*c[5]-1572864*c[5]+786432*c[4]-7454720*c[5]^3*c[4]^2-12521 6000*c[4]^3*c[5]^2+86790249*c[4]^2*c[5]^2-24251136*c[4]*c[5]^2+2981888 *c[5]^2-22364160*c[4]^4*c[5]-1490944*c[4]^2+15899520*c[5]^3*c[4]^3+476 98560*c[4]^4*c[5]^2)/(10*c[5]^3*c[4]^2+10*c[4]^3*c[5]^2-30*c[4]^2*c[5] ^2+6*c[4]*c[5]^2-c[5]^2+6*c[4]^2*c[5]+c[4]*c[5]-c[4]^2)/c[4]^2, c[7] = 63/64, a[8,4] = -1/2*(-19113709750*c[5]^3*c[4]^2+253919232*c[5]^4+944 45568*c[5]^2-329648256*c[5]^3+31481856*c[4]^2+40172724480*c[5]^4*c[4]^ 6-2310963200*c[5]^5*c[4]^6-9143470720*c[5]*c[4]^7+96982397406*c[5]^4*c [4]^4+10972509712*c[4]^4*c[5]-57192166905*c[5]^4*c[4]^3+18183694510*c[ 4]^6*c[5]+3927736380*c[5]^5*c[4]^3+28651494282*c[4]^5*c[5]^3+728065792 0*c[5]^5*c[4]^5-25566220800*c[5]^3*c[4]^7+1833861120*c[5]*c[4]^8-64838 454584*c[4]^6*c[5]^2-3186331590*c[5]^4*c[4]-3931799689*c[4]^3*c[5]+191 104978*c[4]^3*c[5]^2-88493091200*c[4]^5*c[5]^4+21484334720*c[4]^6*c[5] ^3-6932889600*c[5]^4*c[4]^7-18714686047*c[5]*c[4]^5+36147194820*c[4]^7 *c[5]^2+18395438177*c[5]^4*c[4]^2-110186496*c[4]*c[5]-730948074*c[4]*c [5]^2+3743661638*c[5]^3*c[4]-7759548160*c[4]^8*c[5]^2+55065578045*c[4] ^5*c[5]^2-66769309590*c[5]^3*c[4]^4+50966508156*c[5]^3*c[4]^3+69328896 00*c[4]^8*c[5]^3-296006528*c[4]^5+91693056*c[4]^6+348440746*c[4]^4-175 609130*c[4]^3-634798080*c[5]^5*c[4]^2-8262633020*c[4]^4*c[5]^5+2208224 875*c[4]^2*c[5]^2-20377597468*c[4]^4*c[5]^2+910077610*c[4]^2*c[5])/(18 238*c[5]^3*c[4]^2-3093*c[5]^4+772*c[5]^3-30298*c[4]^4*c[5]+383130*c[5] ^4*c[4]^3-117400*c[5]^5*c[4]^3+117400*c[4]^5*c[5]^3+30298*c[5]^4*c[4]+ 1539*c[4]^3*c[5]-18238*c[4]^3*c[5]^2-170950*c[5]^4*c[4]^2-1544*c[4]*c[ 5]^2-1539*c[5]^3*c[4]-30930*c[4]^5*c[5]^2-383130*c[5]^3*c[4]^4+3093*c[ 4]^4-772*c[4]^3+30930*c[5]^5*c[4]^2+170950*c[4]^4*c[5]^2+1544*c[4]^2*c [5])/(-9829*c[4]+5824*c[4]^2+4032)/c[4]^2, a[6,4] = -32/68574961*(-836 10345*c[5]^3*c[4]^2+1572864*c[5]^2-2981888*c[5]^3+524288*c[4]^2+377025 86*c[4]^4*c[5]-10599680*c[5]^4*c[4]^3-27557184*c[4]^3*c[5]+11940911*c[ 4]^3*c[5]^2-14909440*c[5]*c[4]^5+7454720*c[5]^4*c[4]^2-1835008*c[4]*c[ 5]-8691712*c[4]*c[5]^2+23721152*c[5]^3*c[4]+31799040*c[4]^5*c[5]^2-317 99040*c[5]^3*c[4]^4+101861760*c[5]^3*c[4]^3-745472*c[4]^3+21176000*c[4 ]^2*c[5]^2-63598080*c[4]^4*c[5]^2+8527872*c[4]^2*c[5])/(-c[4]^3+2*c[4] ^2*c[5]-30*c[4]^3*c[5]^2+6*c[4]^3*c[5]+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[4]*c[5]^2)/c[4]^2, a[9,5] = -1/60*(8647*c[4]-2321)/c[5]/(5824*c[5]^3*c[4]-15653*c[4]*c[5]^2+138 61*c[4]*c[5]-4032*c[4]-5824*c[5]^4+15653*c[5]^3-13861*c[5]^2+4032*c[5] ), a[8,6] = -20346417/104768*(233710*c[4]*c[5]+46833-93575*c[4]-93575* c[5])*(c[4]-1)*(-1+c[5])/(91*c[4]-64)/(91*c[5]-64)/(11740*c[4]*c[5]-30 93*c[5]-3093*c[4]+772)\}:" }}}{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 98 "A lengthy computation gives an expres sion for the square of the principal error norm in terms of " } {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[5] " "6#&%\"cG6#\"\"&" }{TEXT -1 2 ". " }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 261 "errterms6_8 := PrincipalErrorTerms(6,8,'expanded'): \nsm := 0:\nfor ct to nops(errterms6_8) do\n print(ct);\n sm := sm +(simplify(subs(eB,errterms6_8[ct])))^2;\nend do:\nsm := simplify(sm): \nprin_err_norm_sqrd := unapply(%,c[2],c[4],c[5]):\nprin_err_norm_sqrd (u,v,w);" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 18 "prin_err_no rm_sqrd" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 4400 "prin_err_norm_sqrd := (u,v,w)->1/142427136983040000 *(-47544734077920000*v^9*w^5+19295906334054300*w^6*v^8-945166446191520 0*w^7*v^6+56316087005400*w^4*u^2-10185195792456*w*v^3+2106400576477902 00*w^5*v^8+258649095450436700*w^4*v^8-134517287684065524*w^5*v^5-90018 466536475200*w^6*v^7-113819809838440020*w^4*v^7+196904316201927600*w^3 *v^7-359559977519372840*w^5*v^7-7726628000551590*v^5*w^2+1724943549896 85620*w^6*v^6-924956849474982*w^6*v+424250996736000*w*v^8+606341879201 9160*w^7*v^5+67653988931274756*w^4*v^5-327842188145172*w^2*v^3+2018517 507631344*v^4*w^2+19957045982745231*v^6*w^2+25106567947884*w^2*v^2-151 93083686400*v^5*u+30386167372800*w^5*u+2278156492800*v^2*u^2+124207023 7014300*w^8*v^6-666414346609800*w^8*v^5+1864510997895000*v^6*u^2*w^6-1 224766719625500*v^3*u*w^2-1034666110112400*w^4*u^2*v-116150895286200*v ^3*u*w-24079962411360000*v^7*w^5*u+7753357652040000*w^6*v^5*u-12348792 95618700*w*v^5*u-359452890364824000*w^4*v^7*u-139599660803472000*v^8*w ^3*u-180845685447381000*v^6*u^2*w^3-265329190139751000*v^5*u^2*w^4-201 74775802637400*v^5*u^2*w^2-51184489927860000*v^9*w^4*u+269005133326482 00*v^3*u^2*w^3+22082311673856000*v^8*w^2*u-5774968723575600*w^3*u^2*v^ 2+8777695955988600*w^4*u^2*v^2+744359117405400*w^3*u^2*v-1093515116544 00*v*u^2*w^2-9112625971200*v*u^2*w+1911111160140000*w^6*v^7*u+22134702 0719340000*v^8*w^4*u-1626364155865500*w^4*v^2*u-16561733755392000*v^7* u^2*w^2+21850531487465400*w^4*v^4*u+39352054659926400*v^8*w^2+12138074 459100*v^4*u-11326771987200*v^3*u^2-2424241094400*v^3*u-24276148918200 *w^4*u-37703714781534492*w^4*v^6-17877741844140*w^7*v^2-16426291683760 20*w^7*v^4+140035264911168000*v^9*w^3+53080211777040000*v^10*w^4-19869 4346395680000*v^9*w^4+2484140474028600*w^7*v^7+93588761541771108*w^6*v ^4-35140034059488940*v^7*w^2+299950113456922440*w^5*v^6-16447429196192 5560*w^6*v^5-1118116924243200*w*v^7-22900175069184000*v^9*w^2-27994418 10679590*w^5*v^2-33761677007782251*w^4*v^4+99241658600436*v^4*w+822175 1914980762*w^4*v^3+967540588324218*v^6*w+3250879924087422*w^5*v^3+2556 2850082442622*v^5*w^3+26039983791612356*w^5*v^4-16639957681164*w^4*v+9 112625971200*w^2*u^2+5100762562560000*v^10*w^2+4848482188800*w^3*u+893 88709220700*w^8*v^4+14079021751350*v^4*u^2+1414066094723700*w^4*v^3*u- 21513449747122200*w^5*v^3*u+709312376682000*w^6*v^3*u-57932050176000*w ^6*v^2*u-80745552432243000*v^4*u^2*w^3-2939810550979500*v^3*u^2*w^2-73 6623857700*w^3*v*u+88656071788800*v^2*u^2*w-4918592471040000*v^9*w^2*u +999553008122100*v^4*u^2*w+9188988417753750*v^4*u^2*w^2+68668080152355 0*v^2*u^2*w^2-400600467009900*v^3*u^2*w-23800122328320000*v^8*u^2*w^3- 13463148762882000*v^4*u^2*w^5+4599050226028200*w^5*v^2*u-1206499863596 400*v^5*u^2*w+16920399467790000*v^7*u^2*w^5+86336440067988000*w^5*v^6* u-2608420815720000*w^5*v^8*u-7741725476940000*w^6*v^6*u+14539424783400 00*v^6*u*w-36224008364415600*w^2*v^7*u-549791297280000*v^7*u*w-1426271 3362223100*w^2*v^5*u+28981882232350200*w^2*v^6*u+236105960098761000*w^ 3*v^7*u-116487381722583600*w^4*v^5*u-3401641219977000*w^6*v^4*u+104138 506067761800*w^3*v^5*u-559473498225000*w^5*v*u-203393972799403200*w^3* v^6*u+281478004324587000*w^4*v^6*u-100763967364695000*w^5*v^5*u+607123 41130404000*w^5*v^4*u-33561539998558200*w^3*v^4*u-45307087948800*w^3*u ^2+204702841980000*v^4*u^2*w^6-1235582007660000*v^5*u^2*w^6-1638534925 4400*v*u*w^2+38388367445895000*v^8*u^2*w^4-175197426136470000*v^7*u^2* w^4+106109727402924000*v^7*u^2*w^3-43061316471810000*v^6*u^2*w^5+36889 44353280000*v^8*u^2*w^2+135744885087352200*w^4*v^4*u^2-435800968212096 00*w^4*u^2*v^3+156143749626707400*w^3*v^5*u^2+2620696941954000*w^5*u^2 *v^3-656216771252400*v^2*u*w^3+11829036268800*v^2*u*w+2663602199080020 0*v^6*u^2*w^2+455792510592000*v^6*u^2*w-214736718924000*w^5*u^2*v^2+49 28921234056800*v^4*u*w^2+201226258022400*v^2*u*w^2+6609446239563000*v^ 3*u*w^3+304455552329223000*v^6*u^2*w^4+35074953799704000*v^5*u^2*w^5+5 12958881910600*v^4*u*w+240549320386800*w^7*v^3-1996817450783238*w^3*v^ 4-827108823620638*w^3*v^3-232203581071056000*v^8*w^3+328024208239200*w ^4*v*u-32908811120640000*v^10*w^3+255343012228428*w^3*v^2-335368755324 22644*w^6*v^3+7833774177828*w^4-8029307131572*v^5+50079811802607*w^6+1 0365408754607*v^6+1559431521828*v^4-39602434135572*w^5+741352856776523 1*w^6*v^2-23209631760456*w^3*v-96054274137925044*w^3*v^6-8527904954414 56*w^4*v^2-420237318662932*v^5*w+556005210399468*w^5*v+317334964377600 00*v^9*w^3*u)/(10*w^3*v^2+10*w^2*v^3-30*w^2*v^2+6*w^2*v-w^2+6*w*v^2+w* v-v^2)^2:" }}}{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 47 "Preliminary investigations suggest that, when " } {XPPEDIT 18 0 "c[2] = 20/237;" "6#/&%\"cG6#\"\"#*&\"#?\"\"\"\"$P#!\"\" " }{TEXT -1 3 ", " }{XPPEDIT 18 0 "c[4] = 16/75;" "6#/&%\"cG6#\"\"%*& \"#;\"\"\"\"#v!\"\"" }{TEXT -1 7 " and " }{XPPEDIT 18 0 "c[5] = 23/4 1;" "6#/&%\"cG6#\"\"&*&\"#B\"\"\"\"#T!\"\"" }{TEXT -1 50 ", the princ ipal error norm is close to a minimum." }}{PARA 0 "" 0 "" {TEXT -1 158 "Taking these values as starting values cycling around the nodes w ith a one dimensional minimization procedure gives very slow convergen ce towards the minimum." }}{PARA 0 "" 0 "" {TEXT -1 2 ". " }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 493 "Digits := 30:\nc_2 := 13/153: c_4 \+ := 34/159: c_5 := 23/41:\nfor ct to 120000 do\n c_2 := findmin(prin_ err_norm_sqrd(c2,c_4,c_5),c2=\{0.03,c_2,0.13\},convergence=location)[1 ];\n c_4 := findmin(prin_err_norm_sqrd(c_2,c4,c_5),c4=\{0.2,c_4,0.23 \},convergence=location)[1];\n mn := findmin(prin_err_norm_sqrd(c_2, c_4,c5),c5=\{0.5,c_5,0.62\},convergence=location); \n c_5 := mn[1]: \n if `mod`(ct,1000)=0 then\n print(c[2]=c_2,c[4]=c_4,c[5]=c_5) ;\n print(mn[2]);\n end if;\nend do:\nDigits := 10:" }}{PARA 11 "" 1 "" {XPPMATH 20 "6%/&%\"cG6#\"\"#$\"?A/Lk%)\\*zDt8Z02a)!#J/&F%6 #\"\"%$\"?\"H,g:M/4XHt*yGZ@!#I/&F%6#\"\"&$\"?VG%>Pydp:989Tni&F1" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#$\"?&3_BS0)y&3UawMu9$!#R" }}{PARA 11 " " 1 "" {XPPMATH 20 "6%/&%\"cG6#\"\"#$\"?wEC(QLGXZCz)f_m&)!#J/&F%6#\"\" %$\"?W,%zek*3cHDNov_@!#I/&F%6#\"\"&$\"?\\6qEJ'pA4$*Q:%*pj&F1" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#$\"?0d2o9B6D54'H%G&4$!#R" }}{PARA 11 "" 1 " " {XPPMATH 20 "6%/&%\"cG6#\"\"#$\"?e=6Jvl7F#4JP%*He)!#J/&F%6#\"\"%$\"? !e%)GM>J*[7(RDdi:#!#I/&F%6#\"\"&$\"?f4WX`i%HOMa<(eVcF1" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#$\"?o))R+(R#R4)H:flB2$!#R" }}{PARA 11 "" 1 "" {XPPMATH 20 "6%/&%\"cG6#\"\"#$\"??3F%)o9zwJH^@%Rf)!#J/&F%6#\"\"%$\"?\" f\"[aC%)*p')!#J/&F%6#\"\"%$\"?Q )yuH\\V.-Q%=?/k@!#I/&F%6#\"\"&$\"?CIbPJD<3!fejN$ecF1" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#$\"?KX)30r[/ZUW[j80$!#R" }}{PARA 11 "" 1 "" {XPPMATH 20 "6%/&%\"cG6#\"\"#$\"?Ql(oETS(y$QQ5g%>')!#J/&F%6#\"\"%$\"?r wR>-PM?!Q%=?/k@!#I/&F%6#\"\"&$\"?7?G\\HH<3!fejN$ecF1" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#$\"?\"4.#=%H^/ZUW[j80$!#R" }}{PARA 11 "" 1 "" {XPPMATH 20 "6%/&%\"cG6#\"\"#$\"?7)yyAlT(y$QQ5g%>')!#J/&F%6#\"\"%$\"?t T)*o`QM?!Q%=?/k@!#I/&F%6#\"\"&$\"?!*)*[l')!#J/&F%6#\"\"%$\"? \\AqAkRM?!Q%=?/k@!#I/&F%6#\"\"&$\"?#eu!)yUt\"3!fejN$ecF1" }}{PARA 11 " " 1 "" {XPPMATH 20 "6#$\"?e%y\"en'[/ZUW[j80$!#R" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 78 "The following graphs give a visual check that we have found a (local) minimum." }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 380 "c_2 := .86194601038e-1: pp := .305136348 44e-9:\np1 := evalf[30](plot(prin_err_norm_sqrd(c[2],.21640420184,.565 83356359),c[2]=0.081..0.0914,\n color=COLOR(RGB,.5,0,.9))):\np2 := \+ plot([[[c_2,pp]]$4],style=point,symbol=[circle$2,diamond,cross],symbol size=[12,10$3],\n color=[black,red$3]):\nplots[display]([p1,p2] ,font=[HELVETICA,9],view=[0.081..0.0914,3.0505e-10..3.0556e-10]);" }} {PARA 13 "" 1 "" {GLPLOT2D 388 361 361 {PLOTDATA 2 "6*-%'CURVESG6$7S7$ $\"#\")!\"$$\"?1kZY\">QX*Q*z'pibI!#R7$$\"?LLLLLLLLLLS!pE7)!#J$\"?oIg,M ()[$RWR(HEbIF-7$$\"?nmmmmmmmm6^KRU\")F1$\"?$))o;BM7xI'Hw%f\\0$F-7$$\"? 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v!\"#$\"#&)F<$\"\"!F@-%%FONTG6$%*HELVETICAG\"\"*-%* AXESSTYLEG6#%$BOXG-%+AXESLABELSG6%%%c[4]G%%c[5]G%!G-%+PROJECTIONG6%$!# yF@$\"#cF@\"\"\"" 1 2 0 1 10 0 2 1 1 2 2 1.000000 56.000000 -78.000000 1 0 "Curve 1" }}}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 " " 0 "" {TEXT -1 36 "#-----------------------------------" }}{PARA 0 " " 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 113 "c_5 \+ := .56583356359: pp := .30513634844e-9:\np1 := evalf[30](plot(prin_err _norm_sqrd(.86194601038e-1,.21640420184," }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 151 "nds := [c[2]=.861946 01038e-1,c[4]=.21640420184,c[5]=.56583356359]:\nevalf[10](%);\nfor dgt from 6 by -1 to 4 do\n map(convert,nds,rational,dgt);\nend do;" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#7%/&%\"cG6#\"\"#$\"+/,Y>')!#6/&F&6#\" \"%$\"+=?/k@!#5/&F&6#\"\"&$\"+OcLecF2" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#7%/&%\"cG6#\"\"##\"$=\"\"%p8/&F&6#\"\"%#\"$C\"\"$t&/&F&6#\"\"&#\"$f \"\"$\"G" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#7%/&%\"cG6#\"\"##\"$=\"\"% p8/&F&6#\"\"%#\"#H\"$M\"/&F&6#\"\"&#\"#V\"#w" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#7%/&%\"cG6#\"\"##\"\"&\"#e/&F&6#\"\"%#\"\")\"#P/&F&6#F* #\"#I\"#`" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 56 "The minimum value for the principal error norm is . . . " }} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 86 "evalf[25](prin_err_norm_sqrd (.8619460104e-1,.2164042018,.5658335636)):\nevalf(sqrt(%));" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#$\"+S_\"ou\"!#9" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 9 "Setting " }{XPPEDIT 18 0 "c[4] \+ = 124/573;" "6#/&%\"cG6#\"\"%*&\"$C\"\"\"\"\"$t&!\"\"" }{TEXT -1 7 " \+ and " }{XPPEDIT 18 0 "c[5] = 159/281;" "6#/&%\"cG6#\"\"&*&\"$f\"\"\" \"\"$\"G!\"\"" }{TEXT -1 46 " the principal error norm is a minimum w hen " }{XPPEDIT 18 0 "c[2] = 19/220;" "6#/&%\"cG6#\"\"#*&\"#>\"\"\"\" $?#!\"\"" }{TEXT -1 19 " (approximately). " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 116 "mn := evalf[20](f indmin(prin_err_norm_sqrd(c2,124/573,159/281),c2=0.075..0.095)):\nc[2] =mn[1];\nconvert(%,rational,5);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/&% \"cG6#\"\"#$\"5\"Ru<>#=/qN')!#@" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/&% \"cG6#\"\"##\"#>\"$?#" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 6 "When " }{XPPEDIT 18 0 "c[2] = 19/220;" "6#/&%\"cG6# \"\"#*&\"#>\"\"\"\"$?#!\"\"" }{TEXT -1 3 ", " }{XPPEDIT 18 0 "c[4] = \+ 124/573;" "6#/&%\"cG6#\"\"%*&\"$C\"\"\"\"\"$t&!\"\"" }{TEXT -1 7 " an d " }{XPPEDIT 18 0 "c[5] = 159/281;" "6#/&%\"cG6#\"\"&*&\"$f\"\"\"\" \"$\"G!\"\"" }{TEXT -1 65 ", the principal error norm is given (appro ximately) as follows. " }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 70 "ev alf[20](prin_err_norm_sqrd(19/220,124/573,159/281)):\nevalf(sqrt(%)); " }}{PARA 11 "" 1 "" {XPPMATH 20 "6#$\"+6<$ou\"!#9" }}}{PARA 0 "" 0 " " {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 44 "#---------------------- ---------------------" }}{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 2196 "ee := \{c[2]=19/ 220,\nc[3]=248/1719,\nc[4]=124/573,\nc[5]=159/281,\nc[6]=64/91,\nc[7]= 63/64,\nc[8]=1,\nc[9]=1,\n\na[2,1]=19/220,\na[3,1]=1334488/56144259,\n a[3,2]=6765440/56144259,\na[4,1]=31/573,\na[4,2]=0,\na[4,3]=31/191,\na [5,1]=443985772809/682326636832,\na[5,2]=0,\na[5,3]=-1687953692799/682 326636832,\na[5,4]=815026549419/341163318416,\na[6,1]=-177772509841852 17125334224/11518294736403027809706743,\na[6,2]=0,\na[6,3]=14122027098 32277291195888/217326315781189203956731,\na[6,4]=-60026762217662324171 682463968/12227430504797048182217556253,\na[6,5]=443591723399448798344 389120/674353607444582261870479169,\na[7,1]=31162838515845600231185643 68757831/955301928184769582072071268073472,\na[7,2]=0,\na[7,3]=-150693 58907111564759839129615089/1126535292670718846783102910464,\na[7,4]=46 08655696349942937342097596091097426937/4022871735737177579627118428194 64495104,\na[7,5]=-1041097287013501701647909592093443697/7679368644108 13987103949029050941440,\na[7,6]=126705961361465452994391/126885461609 550467563520,\na[8,1]=2688622152256014375593487443/6358269443841942299 62077552,\na[8,2]=0,\na[8,3]=-165064430696291307962649/944253034215015 9719648,\na[8,4]=85647344134405061626206038309671468707/57453178811105 70819440070776064368272,\na[8,5]=-1868050624284923482186332306552702/9 77068785600982269831702681478435,\na[8,6]=16842653200587834711/1316647 4974917410080,\na[8,7]=-608491123177947136/27366835598969517651,\na[9, 1]=2465359/38465280,\na[9,2]=0,\na[9,3]=0,\na[9,4]=29999319743551167/9 0312244465616140,\na[9,5]=2440521938223593/9364974686865720,\na[9,6]=1 241823968749/6822515208960,\na[9,7]=177732994465792/327930128445465,\n a[9,8]=-11266189/29580120,\n\nb[1]=2465359/38465280,\nb[2]=0,\nb[3]=0, \nb[4]=29999319743551167/90312244465616140,\nb[5]=2440521938223593/936 4974686865720,\nb[6]=1241823968749/6822515208960,\nb[7]=17773299446579 2/327930128445465,\nb[8]=-11266189/29580120,\n\n`b*`[1]=73983798702885 7/11890322590648320,\n`b*`[2]=0,\n`b*`[3]=0,\n`b*`[4]=1204992945826748 851603162095/3551063890812398485590445952,\n`b*`[5]=328936061938716185 672737/1439660654341730577416400,\n`b*`[6]=102717532147008407320237/46 1800408829724604569600,\n`b*`[7]=9268295603184461676544/21663580993679 805137685,\n`b*`[8]=-9633553238997677/35173968040963440,\n`b*`[9]=-1/1 56\}:" }}}{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 compone nts are the principal error terms of the embedded 9 stage, order 5 sch eme (the error terms of order 6) and let " }{XPPEDIT 18 0 "`T*`[6,9]; " "6#&%#T*G6$\"\"'\"\"*" }{TEXT -1 99 " denote the vector whose compo nents 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#\"\"(-%$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 that as well as attempt ing 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 " an d " }{XPPEDIT 18 0 "C[7];" "6#&%\"CG6#\"\"(" }{TEXT -1 27 " should b e chosen so that " }}{PARA 0 "" 0 "" {TEXT -1 68 "they are \"similar i n 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,'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 := sq rt(add(evalf(subs(ee,`errterms5_9*`[i]))^2,i=1..nops(`errterms5_9*`))) :\nsnmC := sqrt(add((evalf(subs(ee,`errterms6_9*`[i])-subs(ee,errterms 6_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#/&%\"BG 6#\"\"($\")YtZ8!\"(" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/&%\"CG6#\"\"($ \")ts_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 26 "absolute stability regions" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}{SECT 1 {PARA 4 " " 0 "" {TEXT -1 35 "coefficients of the combined scheme" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 2196 "ee := \{c[2]=19/220,\nc[3]=248/1719,\nc[4]=124/573,\nc[5]=159/281,\nc[6]=64 /91,\nc[7]=63/64,\nc[8]=1,\nc[9]=1,\n\na[2,1]=19/220,\na[3,1]=1334488/ 56144259,\na[3,2]=6765440/56144259,\na[4,1]=31/573,\na[4,2]=0,\na[4,3] =31/191,\na[5,1]=443985772809/682326636832,\na[5,2]=0,\na[5,3]=-168795 3692799/682326636832,\na[5,4]=815026549419/341163318416,\na[6,1]=-1777 7250984185217125334224/11518294736403027809706743,\na[6,2]=0,\na[6,3]= 1412202709832277291195888/217326315781189203956731,\na[6,4]=-600267622 17662324171682463968/12227430504797048182217556253,\na[6,5]=4435917233 99448798344389120/674353607444582261870479169,\na[7,1]=311628385158456 0023118564368757831/955301928184769582072071268073472,\na[7,2]=0,\na[7 ,3]=-15069358907111564759839129615089/1126535292670718846783102910464, \na[7,4]=4608655696349942937342097596091097426937/40228717357371775796 2711842819464495104,\na[7,5]=-1041097287013501701647909592093443697/76 7936864410813987103949029050941440,\na[7,6]=126705961361465452994391/1 26885461609550467563520,\na[8,1]=2688622152256014375593487443/63582694 4384194229962077552,\na[8,2]=0,\na[8,3]=-165064430696291307962649/9442 530342150159719648,\na[8,4]=85647344134405061626206038309671468707/574 5317881110570819440070776064368272,\na[8,5]=-1868050624284923482186332 306552702/977068785600982269831702681478435,\na[8,6]=16842653200587834 711/13166474974917410080,\na[8,7]=-608491123177947136/2736683559896951 7651,\na[9,1]=2465359/38465280,\na[9,2]=0,\na[9,3]=0,\na[9,4]=29999319 743551167/90312244465616140,\na[9,5]=2440521938223593/9364974686865720 ,\na[9,6]=1241823968749/6822515208960,\na[9,7]=177732994465792/3279301 28445465,\na[9,8]=-11266189/29580120,\n\nb[1]=2465359/38465280,\nb[2]= 0,\nb[3]=0,\nb[4]=29999319743551167/90312244465616140,\nb[5]=244052193 8223593/9364974686865720,\nb[6]=1241823968749/6822515208960,\nb[7]=177 732994465792/327930128445465,\nb[8]=-11266189/29580120,\n\n`b*`[1]=739 837987028857/11890322590648320,\n`b*`[2]=0,\n`b*`[3]=0,\n`b*`[4]=12049 92945826748851603162095/3551063890812398485590445952,\n`b*`[5]=3289360 61938716185672737/1439660654341730577416400,\n`b*`[6]=1027175321470084 07320237/461800408829724604569600,\n`b*`[7]=9268295603184461676544/216 63580993679805137685,\n`b*`[8]=-9633553238997677/35173968040963440,\n` b*`[9]=-1/156\}:" }}}{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)\"#C F)*$)F'\"\"%F)F)F)*&#F)\"$?\"F)*$)F'\"\"&F)F)F)*&#F)\"$?(F)*$)F'F1F)F) F)*&#\"4H*Q$3OZt>\"p\"8S%)>RVft.Dra$F)*$)F'\"\"(F)F)F)*&#\"2@&=-PPn6A \"6!HL0ix.^7`)*F)*$)F'\"\")F)F)F)" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }} {PARA 0 "" 0 "" {TEXT -1 124 "We can find the point where the boundary of the stability region intersects the negative real axis by solving \+ the equation: " }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "R(z ) = 1;" "6#/-%\"RG6#%\"zG\"\"\"" }{TEXT -1 2 ". " }}{EXCHG {PARA 0 "> \+ " 0 "" {MPLTEXT 1 0 28 "z0 := newton(R(z)=1,z=-4.4);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%#z0G$!+`*>uT%!\"*" }}}{PARA 0 "" 0 "" {TEXT -1 0 " " }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 303 "z0 := newton(R(z)=1,z=- 4.4):\np1 := plot([R(z),1],z=-5.19..0.49,color=[red,blue]):\np2 := plo t([[[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)):\npl ots[display]([p1,p2,p3],view=[-5.19..0.49,-.07..1.47],font=[HELVETICA, 9]);" }}{PARA 13 "" 1 "" {GLPLOT2D 360 253 253 {PLOTDATA 2 "6+-%'CURVE SG6$7Z7$$!3Q++++++!>&!#<$\"3\\[-s0?![-%F*7$$!3QML3T![!f^F*$\"3)f11O738 #QF*7$$!3Ynm;#3'4G^F*$\"3UF&)=$\\$)pi$F*7$$!3a++DBT9(4&F*$\"3jZ()=+\"p 9W$F*7$$!3kLLLk@>m]F*$\"3;/.I')yTkKF*7$$!3E+]U'*)HB,&F*$\"3?4Za_D]vHF* 7$$!3!pm;&GwYe\\F*$\"3N!fw/=>&4FF*7$$!3s+](\\(Q*y*[F*$\"3>(=(f?q\"eV#F *7$$!3nLLV@,KP[F*$\"3om#QS@&*p=#F*7$$!3'RLLd%[MwZF*$\"3GX][nRlf>F*7$$! 3NLL.q&p`r%F*$\"3=#)4'H\"pk`*QF*$\"35&oa6gWvZ$F]p7$$!35++S:-YpPF*$\"3'**=(3])Rpq#F]p7$$!3K++ +\"HZkk$F*$\"3i'o;DfQY6#F]p7$$!3;++gW:!z_$F*$\"3grPdGc/$o\"F]p7$$!3hLL )*\\1D?MF*$\"3Kp3YB_H(Q\"F]p7$$!3'ommSKVAH$F*$\"3kJ42*p8>8\"F]p7$$!3/n mEGV!Q=$F*$\"36brDe:,?)*!#>7$$!39++0(*RmdIF*$\"39&)4l%**G&oq#F*$\"3i26ikR[e!)F`s7$$!3gmm;\\r8&e#F*$\"31qpZ,+Jn%)F`s7$$! 3ymmrw\\OtCF*$\"31dUWyztj!*F`s7$$!3SLL$))e.GN#F*$\"3U#f1LTd>#**F`s7$$! 3nLL)**=uvA#F*$\"3'[OyQ(=i.6F]p7$$!3K++:I;c=@F*$\"3N7OM'H#y=7F]p7$$!31 LL.z]#3+#F*$\"3ASy(\\x:DO\"F]p7$$!3M++?,<>z=F*$\"34ql!\\]uJ`\"F]p7$$!3 ;++!4<(>g$>F]p7$$!3 H++q9zA<:F*$\"3M+ixKQ?%>#F]p7$$!3EnmEY;O-9F*$\"3[\"p'fEbogCF]p7$$!3#)* ****pQ<(z7F*$\"3;5Fe4PT\"y#F]p7$$!3)RL$efMeo6F*$\"3!=L!p_H>3JF]p7$$!3I ****fAZ3Z5F*$\"3s!4R*>nk4NF]p7$$!3xqm;(zQwK*F]p$\"3[pf2b*oY$RF]p7$$!3& z***\\)ecE8)F]p$\"3DGkMCY3MWF]p7$$!3'3nmm0VV'pF]p$\"3/i))3pQf$)\\F]p7$ $!3P)***\\iqATdF]p$\"3>\\$3v0))>j&F]p7$$!3aFLL*)4AjXF]p$\"3oa?x;t4OjF] p7$$!33LLLO'R&eLF]p$\"3G>2ueZFZrF]p7$$!3Uim;`O$Q;#F]p$\"3;/$H-m5F]p$\"3KxeIQ4$)))*)F]p7$$\"3v*QLLU?>#>F`s$\"3[-s.#30% >5F*7$$\"3%ymmY^avJ\"F]p$\"3uT%F*Fi]l-%'SYMBOLG6#%'CIRCLEG-F]]l6&F_]lFd]lFd]lFd]l-%&STY LEG6#%&POINTG-F$6&F_al-Fdal6#%&CROSSGFgalFial-F$6&F_al-Fdal6#%(DIAMOND GFgalFial-F$6%7$7$FaalFc]lF`al-%&COLORG6&F_]lFc]l$\"\"&!\"\"Fc]l-%*LIN ESTYLEG6#\"\"$-%%FONTG6$%*HELVETICAG\"\"*-%+AXESLABELSG6%Q\"z6\"Q!F^dl -Ffcl6#%(DEFAULTG-%%VIEWG6$;$!$>&!\"#$\"#\\Fidl;$!\"(Fidl$\"$Z\"Fidl" 1 2 0 1 10 0 2 9 1 4 2 1.000000 45.000000 45.000000 0 0 "Curve 1" "Cur ve 2" "Curve 3" "Curve 4" "Curve 5" "Curve 6" }}}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 50 "The following picture sho ws the stability region. " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1385 "R := z -> 1+z+1/2*z^2+1/6*z^3+1/2 4*z^4+1/120*z^5+1/720*z^6+\n 6911973473608338929/35471250373594339 198440*z^7+22116737370218521/985312510377620533290*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:\np 1 := plot(pts,color=COLOR(RGB,.33,.43,.08)):\np2 := plots[polygonplot] ([seq([pts[i-1],pts[i],[-2.2,0]],i=2..nops(pts))],\n style=pa tchnogrid,color=COLOR(RGB,.65,.85,.15)):\npts := []: z0 := 2+4.75*I:\n for ct from 0 to 60 do\n zz := newton(R(z)=exp(ct*Pi/30*I),z=z0):\n \+ z0 := zz:\n pts := [op(pts),[Re(zz),Im(zz)]]:\nend do:\np3 := plot (pts,color=COLOR(RGB,.33,.43,.08)):\np4 := plots[polygonplot]([seq([pt s[i-1],pts[i],[1.92,4.73]],i=2..nops(pts))],\n style=patchnog rid,color=COLOR(RGB,.65,.85,.15)):\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,c olor=COLOR(RGB,.33,.43,.08)):\np6 := plots[polygonplot]([seq([pts[i-1] ,pts[i],[1.92,-4.73]],i=2..nops(pts))],\n style=patchnogrid,c olor=COLOR(RGB,.65,.85,.15)):\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 labe ls=[`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\\*****\\)z%=g*!#G$\"3=+++VEfTJF-7$$!3++++![S( =>!#E$\"3E+++?))Q7ZF-7$$!3%******\\'\\o\">\"!#D$\"3Y*****\\@%=$G'F-7$$ !3++++&*RLz=F=$\"3]******pp(R&yF-7$$\"3$******>3!zD+++:d>T?FQ7$$\"3-+++3Bn&[\"!#?$\" 3++++U,I(>#FQ7$$\"3!******f#Q&zY#F_p$\"3/+++%G%\\_BFQ7$$\"3.+++UhF8PF_ p$\"3y*****ziih]#FQ7$$\"3?+++\"R$zy\\F_p$\"33+++n_YdEFQ7$$\"3r*****H'f ')GdF_p$\"3()*****4k\"Q0GFQ7$$\"3++++,XJ$4&F_p$\"3?+++@&=)[HFQ7$$\"3-+ 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stability interval is (approximately) " }{XPPEDIT 18 0 "[-4.4174, 0];" "6#7$,$-%&FloatG6$\"&uT%!\"%!\"\"\"\"!" }{TEXT -1 2 ". " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 148 "We can distort the boundary curve horizontally by taking the 11th root of the real part of points along the curve. In this way we see t hat there is " }{TEXT 260 53 "no largest interval on the nonnegative i maginary axis" }{TEXT -1 65 " that contains the origin and lies inside the stability region. " }}{PARA 0 "" 0 "" {TEXT -1 119 "However the \+ stability region intersects the nonnegative imaginary axis in an inter val that does not contain the origin." }}{PARA 0 "" 0 "" {TEXT -1 0 " " }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 408 "R := z -> 1+z+1/2*z^2+1 /6*z^3+1/24*z^4+1/120*z^5+1/720*z^6+\n 6911973473608338929/3547125 0373594339198440*z^7+22116737370218521/985312510377620533290*z^8:\nDig its := 25:\npts := []: z0 := 0:\nfor ct from 0 to 107 do\n zz := new ton(R(z)=exp(ct*Pi/100*I),z=z0):\n z0 := zz:\n pts := [op(pts),[su rd(Re(zz),11),Im(zz)]]:\nend do:\nplot(pts,color=COLOR(RGB,.2,.8,0),th ickness=2,font=[HELVETICA,9]);\nDigits := 10:" }}{PARA 13 "" 1 "" {GLPLOT2D 261 276 276 {PLOTDATA 2 "6(-%'CURVESG6#7hq7$$\"\"!F)F(7$$!:R UK\\#o%yRiclL#!#E$\":%GT'=Dy*e`EfTJF-7$$!:)*RO(4%*G6_4`mQF-$\":\"R:f#o @yrI&=$G'F-7$$!:s&3Zh$*z'y+<*)=&F-$\":#)=88&[gugzxC%*F-7$$!:#>$o5\"\\s :#z?+R'F-$\":&[JsG7&=91PmD\"!#D7$$!:4WphFmS.9eg](F-$\":])y$Q#F?7$$!:J*z^2gRCH7H]5F?$\":?6.2^@227uK^#F?7$$!:vP#*G^&e TQ2=T6F?$\":kE4IBoVLQLu#GF?7$$!:?@Zt*GK:+:IG7F?$\":!>'*HIHk[VEfTJF?7$$ !:B%G[W)GjLH()=J\"F?$\":T'zs*fW-&**=vbMF?7$$!:HPVp0Z&f!H#3#R\"F?$\":q! 4Pkv(H!\\6\"*pPF?7$$!:ir?xx+vS'e&*o9F?$\":d#)o%)\\Q!y)QqS3%F?7$$!:**H4 ]'=%\\n>4Da\"F?$\":Jd+zIVAVhH#)R%F?7$$!:C\"pug\\]I'GvEh\"F?$\":K]k-!3[ %)>))Q7ZF?7$$!:Q_NI:k(e0CJz;F?$\":uBd3t%o!z*zaE]F?7$$!:'3sd/`E]@'*=UF?$\":0&o /5fTp9U=$G'F?7$$!:6`*>4tP:'3ti%>F?$\":(QF..'Q.y1Vtf'F?7$$!:7Q9713R*e&* Q!)>F?$\":bCEL$H50*z,:\"pF?7$$!:8j3h5pW$*[!G.?F?$\":P3)yYk?Z\"QgcA(F?7 $$!:X`:g0gTA0\"=4?F?$\":U%)[nl8^Qy=)RvF?7$$!:TTH(4`DZ(HMW)>F?$\":!Rr3: Jm)\\:))Ryy7%>wa#F?$\":@s\\L+U`^r=*Q(*F?7$$ \":#fFB2fTh2%H#eEF?$\":*eP,^j-yxtI05!#C7$$\":pNbvZyFl;nDw#F?$\"::<8fvH 8G!GsO5Ffu7$$\":$[@HK-UKaBPiGF?$\":7b>irk\")>8Q\"o5Ffu7$$\":&p64#p!o6z =reHF?$\":Qe)y%*fVrZLb*4\"Ffu7$$\":%>*>nXZG!\\\\G_IF?$\":n@Ox%)H@)G%o4 8\"Ffu7$$\":iSn*=TqvyGdVJF?$\":Ot`hQDl'\\LQi6Ffu7$$\":\"Hx:UIq)*)p>HB$ F?$\":'pM'z*RM;z!)z$>\"Ffu7$$\":wB)GUnH8h'y0K$F?$\":YW+sAG#fzD@D7Ffu7$ $\":)*Rnj)Rq1$eSnS$F?$\":p23vg0n^!oic7Ffu7$$\":>Or\"[B8H8>b\"\\$F?$\": [ynn&fbO+2/)G\"Ffu7$$\":UTpOTgVTnE^d$F?$\":r_D8Xliz>a%>8Ffu7$$\":#)G.n _a?_gavl$F?$\":\"[&)Gy2-u;s'3N\"Ffu7$$\":Z'>P!zq^E'o!*QPF?$\":i\"Q2MdZ $)e'zAQ\"Ffu7$$\"::5XC7[=t4S#>QF?$\":[SdeO:$o19p89Ffu7$$\":_-ktr?\\(G% *f)*QF?$\":\"4F\\nWX_>B5X9Ffu7$$\":4Xqh[NU'[1-xRF?$\":A-=z:W:%HA^w9Ffu 7$$\":Ex,\">B3Sv=`aSF?$\":D*>?P'H8n$4#z]\"Ffu7$$\":@*4'H&pNw4[:JTF?$\" :6cO'zCJ)\\?G$R:Ffu7$$\":_]2f:)eK&p0p?%F?$\":sUn_!=8GbPtq:Ffu7$$\":^U] $zT9+dgz\"G%F?$\":%p2q]<5xfs8-;Ffu7$$\":<_)QlOblpK$eN%F?$\":&HC!3'4WiM $QNj\"Ffu7$$\":S)=ADM6x%*4-HWF?$\":4jxdQ=p@`O\\m\"Ffu7$$\":/+U)=-)GH^f 8]%F?$\":)Hxm%fNRALJjp\"Ffu7$$\"::(Hs36xpsf%Gd%F?$\":g#*3;O)[pK@sF1j=#=Ffu7$$ \":@C6MTel;=8,&[F?$\":_P.=(eY_A)HK&=Ffu7$$\":g?.#)*p<\\F?$\":4Dk% >p]1]ze%)=Ffu7$$\":x%es,:)=W!=N$)\\F?$\":\")4v()*f>M-h$f\">Ffu7$$\":ty $)fP01q@c&[]F?$\":VV]BO=oTwss%>Ffu7$$\":Md,5;FH'>Yy7^F?$\":R,BD]OU6B'f y>Ffu7$$\":sVY')=b5Lq3g<&F?$\":QNw&obTr#f/*4?Ffu7$$\":4/(R()y9]*)p>Q_F ?$\":n)4Ho_Tu9d>T?Ffu7$$\":r`*=)\\/C\"eYJ*H&F?$\":q%*Rn#f2s@sYs?Ffu7$$ \":%H(\\%RT$\\yMB$f`F?$\":`EpB4_K)ykr.@Ffu7$$\":P!G!4*Rp\"p#4==aF?$\": I(\\E[vK8u0%\\8#Ffu7$$\":*3!z7<2;>BTeZ&F?$\":@p*3,U!y5IOh;#Ffu7$$\":+V @hmL#)3x`A`&F?$\":Xm'RLb)[<9+t>#Ffu7$$\":>\\m)\\Pht&Qjte&F?$\":3'Q+)eG I6DG%GAFfu7$$\":vR_N.0'F?$\":g<59g\\Xp)HkODFfu7$$\":(\\>*e\\G>6[)e&3'F?$\":#=$zcC3 mzd@qc#Ffu7$$\":!***Hdi_'**3sF=hF?$\":RukD&H+\"HP!H(f#Ffu7$$\":nx]1,)y D=f?[hF?$\":5$fTt[/9(4Tui#Ffu7$$\":+M7&>iuv5>:vhF?$\":9#p(y?$\\*pElul# Ffu7$$\":=P%)[>Jp6&*e))>'F?$\":8^zU&=h$4Bato#Ffu7$$\":<`bG,GWA()G!>iF? $\"::X#o#\\>mO@*4W^ZaL'GFfu7$$\":U&o!z$QVJ<4rRiF?$\":!*>+?TUX M@]?*GFfu7$$\":8=ZmKe)>mk9>iF?$\":F+@V')p&3Z4a?HFfu7$$\":\\P]&[v*)GG:$ z='F?$\":pT3n&p)y4_=)[HFfu7$$\":'>d*\\BUFp*zAVhF?$\"::)y(=U^>(=](o(HFf u7$$\":]+XPV*)38JW13'F?$\":0.jLyzU>y-Z+$Ffu7$$\":#oB%)p*[pr+-G*fF?$\": (HYw!37tOU%HKIFfu7$$\":X10Vc7C:E3d'eF?$\":T<#z*odAe$GkfIFfu7$$\":Tn3$Ffu7$$\":T*\\1ILmBO:*>G&F?$\":U7.j=OB3%He8 JFfu7$$!:PvN#z#REwd&[(z%F?$\":9XoITv%p1=;SJFfu7$$!:jyd%z&R:')z1\\f&F?$ \":\"yoM$RSE>xrk;$Ffu7$$!:\")RAIJJ56Ykr$fF?$\":R>;sJ[#=oq]#>$Ffu7$$!:* ))*\\3\"3#yITVQ<'F?$\":Mc(Hu'\\P?;i#=KFfu7$$!:T5(H9HRNJ7(3O'F?$\":;b') \\H^P5uJPC$Ffu7$$!:J-bv:!=RmM_=lF?$\":CHk7V6C/q5*oKFfu-%*THICKNESSG6# \"\"#-%%FONTG6$%*HELVETICAG\"\"*-%&COLORG6&%$RGBG$Fa]m!\"\"$\"\")F\\^m F(-%+AXESLABELSG6$Q!6\"Fb^m-%%VIEWG6$%(DEFAULTGFg^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 intersectio n points of the boundary curve with the imaginary axis can be determin ed as follows." }}{PARA 0 "" 0 "" {TEXT -1 87 "First we look for point s on the boundary curve either side of each intersection point. " }} {PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 201 "Digits := 15:\nz0 := 0.82*I:\nfor ct from 25 to 28 do\n newton( R(z)=exp(ct*Pi/100*I),z=z0);\nend do;``;\nz0 := 3.1*I:\nfor ct from 10 0 to 103 do\n newton(R(z)=exp(ct*Pi/100*I),z=z0);\nend do;\nDigits : = 10:" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#^$$!0?4;*RLz=!#A$\"0jw,(p(R&y !#:" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#^$$!0L-d'47@5!#A$\"0%\\'*)*[8o \")!#:" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#^$$\"012y67Vi(!#B$\"0>3C_#H# [)!#:" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#^$$\"0M*frwSVR!#A$\"0d(G&y\\k z)!#:" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%!G" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#^$$\"03V%G/rN>!#<$\"0jG7>Tn3$!#9" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#^$$\"0;)=N?UG*)!#=$\"0OB3%He8J!#9" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#^$$!0DK![D\\)4$!#=$\"0v%p1=;SJ!#9" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#^$$!01Yx/&f\"o\"!#<$\"0SE>xrk;$!#9" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 108 "Then we apply the bisect ion method to calculate the parameter value associated with each inter section point." }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 330 "Digits := 15:\nreal_part := proc(u)\n Re(newton(R(z)=exp(u*Pi*I),z=0.82*I))\n end proc:\nu0 := bisect('real_part'(u),u=0.25..0.28);\nnewton(R(z)=exp (u0*Pi*I),z=0.82*I);``;\nreal_part := proc(u)\n Re(newton(R(z)=exp(u *Pi*I),z=3.1*I))\nend proc:\nu0 := bisect('real_part'(u),u=1.00..1.03) ;\nnewton(R(z)=exp(u0*Pi*I),z=3.1*I);\nDigits := 10:" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%#u0G$\"0\"o'>RjYm#!#:" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#^#$\"0Z#e7*z7P)!#:" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#% !G" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%#u0G$\"0!3l!y]v,\"!#9" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#^#$\"0xPupwO8$!#9" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 4 "The " }{TEXT 260 73 "stabi lity region intersects the nonegative imaginary axis in the interval" }{TEXT -1 38 " [ 0.8371, 3.1337 ] (approximately)." }}{PARA 0 "" 0 " " {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 37 "#---------------------- --------------" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 80 "The stability function R* for the 9 stage, order 5 sche me is given as follows." }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 97 "s ubs(ee,subs(b=`b*`,StabilityFunction(5,9,'expanded'))):\n`R*` := unapp ly(%,z):\n'`R*`(z)'=`R*`(z);" }}{PARA 12 "" 1 "" {XPPMATH 20 "6#/-%#R* G6#%\"zG,6\"\"\"F)F'F)*&#F)\"\"#F)*$)F'F,F)F)F)*&#F)\"\"'F)*$)F'\"\"$F )F)F)*&#F)\"#CF)*$)F'\"\"%F)F)F)*&#F)\"$?\"F)*$)F'\"\"&F)F)F)*&#\"?*Gi \"**38N%\\Q&exM)e'\"B!33\\>BS5!=.#GZwqRYF)*$)F'F1F)F)F)*&#\">2e8*znkG!zh)\"Bkk#fb=K3WDc#y6mVR?F)*$)F'\"\")F)F)F)*&#\"2@&=-PPn6A\"9SK>.)3*=;v3P:F)*$ )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 stabilit y 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,$\"\"\"!\"\"" }{TEXT -1 2 ". " }}{EXCHG {PARA 0 "> " 0 " " {MPLTEXT 1 0 33 "z_0 := newton(`R*`(z)=-1,z=-4.4);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%$z_0G$!+J9\\>W!\"*" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 324 "z_0 := newton(`R*`(z)= -1,z=-4.4):\np_1 := plot([`R*`(z),-1],z=-4.99..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=[-4.99..0.49,-1.57.. 1.47],font=[HELVETICA,9]);" }}{PARA 13 "" 1 "" {GLPLOT2D 398 284 284 {PLOTDATA 2 "6+-%'CURVESG6$7S7$$!3A++++++!*\\!#<$!3=4EJM54g@F*7$$!3#pm m^f^0([F*$!3%=>t)Ro(4'=F*7$$!3%QL3o3KXF*$!3GDBPSpby6F*7$$!3\\L$3@f%)\\T% F*$!38U))y1\\1L**!#=7$$!3jm;u*Q?kI%F*$!3`i8,3`V?%)FF7$$!3E+]ZY%3S>%F*$ !3i;;3vcLWqFF7$$!3%omTR&=vxSF*$!3\"=tkGNJh!eFF7$$!3/+]x(4o='RF*$!3!*4? 5CBSSZFF7$$!3OF*$!3uPQ,aO#QS#FF7$$!3?++]Qxz+NF*$!3y(>Y\")4L@#=FF 7$$!3/++5QhU'Q$F*$!31]#40Hr&[8FF7$$!3mm;%zwlDG$F*$!3iP2MJ?+=)*!#>7$$!3 [LLBUd1fJF*$!30Cb1iBmhx7FF7$$!3hmmhq*>J\">F*$\"3='=gYd-/X\"FF7$$!35++?e%pd z\"F*$\"3oOt8$*H=W;FF7$$!3+++:w['4o\"F*$\"3s$Gv\"*[9D&=FF7$$!3-+]()Rb) )p:F*$\"3;yLNqa@v?FF7$$!3#)***\\a(3bY9F*$\"3[DZa!z))4N#FF7$$!3cLL$R>Hd L\"F*$\"3vVF:IC@GEFF7$$!3z****\\%R.u@\"F*$\"31d9!fVE$fHFF7$$!3pm;aLE=5 6F*$\"3OtrJ$y1ZH$FF7$$!3E%****4@?'H**FF$\"3*\\qQ0)\\o/PFF7$$!3MML3+cmE ))FF$\"36K[9['>&FF7$$!3g.+vof`m`FF$\"3-n)Ql\")=q%eFF7$$!3Dkmm#)*3+B%F F$\"3Y$o\\3K\"y]lFF7$$!3Yjm;()funIFF$\"3I')\\nFX;etFF7$$!3HBL3;r5:>FF$ \"3uU:n:y5d#)FF7$$!3_^****>q^f&)Fcp$\"39>V@fxlz\"*FF7$$\"3OMnm\"G*fzNF cp$\"3x]C#GQWk.\"F*7$$\"3\"RLL8'ppV9FF$\"3[S)pR96`:\"F*7$$\"3@0+D$4>8g #FF$\"3/^qD(=,rH\"F*7$$\"3Q,+v#=6$4PFF$\"3uZ7GkL3\\9F*7$$\"3!********* ******[FF$\"3P*=#eToJK;F*-%'COLOURG6&%$RGBG$\"*++++\"!\")$\"\"!Fb[lFa[ l-F$6$7S7$F($!\"\"Fb[l7$F.Fg[l7$F3Fg[l7$F8Fg[l7$F=Fg[l7$FBFg[l7$FHFg[l 7$FMFg[l7$FRFg[l7$FWFg[l7$FfnFg[l7$F[oFg[l7$F`oFg[l7$FeoFg[l7$FjoFg[l7 $F_pFg[l7$FepFg[l7$FjpFg[l7$F_qFg[l7$FeqFg[l7$FjqFg[l7$F_rFg[l7$FdrFg[ l7$FirFg[l7$F^sFg[l7$FcsFg[l7$FhsFg[l7$F]tFg[l7$FbtFg[l7$FgtFg[l7$F\\u Fg[l7$FauFg[l7$FfuFg[l7$F[vFg[l7$F`vFg[l7$FevFg[l7$FjvFg[l7$F_wFg[l7$F dwFg[l7$FiwFg[l7$F^xFg[l7$FcxFg[l7$FhxFg[l7$F]yFg[l7$FbyFg[l7$FgyFg[l7 $F\\zFg[l7$FazFg[l7$FfzFg[l-F[[l6&F][lFa[lFa[lF^[l-F$6&7#7$$!3(******4 V\"\\>WF*Fg[l-%'SYMBOLG6#%'CIRCLEG-F[[l6&F][lFb[lFb[lFb[l-%&STYLEG6#%& POINTG-F$6&F]_l-Fb_l6#%&CROSSGFe_lFg_l-F$6&F]_l-Fb_l6#%(DIAMONDGFe_lFg _l-F$6%7$7$F__lFa[lF^_l-%&COLORG6&F][lFa[l$\"\"&Fh[lFa[l-%*LINESTYLEG6 #\"\"$-%%FONTG6$%*HELVETICAG\"\"*-%+AXESLABELSG6%Q\"z6\"Q!F[bl-Fcal6#% (DEFAULTG-%%VIEWG6$;$!$*\\!\"#$\"#\\Ffbl;$!$d\"Ffbl$\"$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 st ability region for the 9 stage, order 5 scheme. " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1552 "`R*` := z \+ -> 1+z+1/2*z^2+1/6*z^3+1/24*z^4+1/120*z^5+\n 658834775853849435130 899162289/463970764728203180104023194908080*z^6+\n 861790286419436 71767799135807/371176611782562544083218555926464*z^7+\n 3037134775 4705998770250385/2039431932871222769688014043552*z^8-\n 2211673737 0218521/153708751618908803193240*z^9:\npts := []: z0 := 0:\nfor ct fro m 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,.07,.4,0)):\np_2 := plots[polygonplot]([seq([pts[i-1], pts[i],[-2.2,0]],i=2..nops(pts))],\n style=patchnogrid,color= COLOR(RGB,.13,.8,0)):\npts := []: z0 := 1.9+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(zz)]]:\nend do:\np_3 := plot(pts,color=CO LOR(RGB,.07,.4,0)):\np_4 := plots[polygonplot]([seq([pts[i-1],pts[i],[ 1.82,4.39]],i=2..nops(pts))],\n style=patchnogrid,color=COLOR (RGB,.13,.8,0)):\npts := []: z0 := 1.9-4.45*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(zz)]]:\nend do:\np_5 := plot(pts,color=COLOR(R GB,.07,.4,0)):\np_6 := plots[polygonplot]([seq([pts[i-1],pts[i],[1.82, -4.39]],i=2..nops(pts))],\n style=patchnogrid,color=COLOR(RGB ,.13,.8,0)):\np_7 := plot([[[-5.09,0],[2.19,0]],[[0,-4.99],[0,4.99]]], color=black,linestyle=3):\nplots[display]([p_||(1..7)],view=[-5.09..2. 19,-4.99..4.99],font=[HELVETICA,9],\n labels=[`Re(z)`,`Im(z )`],axes=boxed,scaling=constrained);" }}{PARA 13 "" 1 "" {GLPLOT2D 407 552 552 {PLOTDATA 2 "6/-%'CURVESG6$7ew7$$\"\"!F)F(7$$\"3u*****>YZe x%!#F$\"3++++Fjzq:!#=7$$\"3A+++J]&3D$!#D$\"3:+++cEfTJF07$$\"3E+++0(4V0 %!#C$\"3C+++2*)Q7ZF07$$\"3?+++&[RS`#!#B$\"3))*****fd$=$G'F07$$\"3'**** **fmF#z5!#A$\"39+++kj'R&yF07$$\"39+++fI_vNFF$\"3E+++aQpC%*F07$$\"3g+++ 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941440,\na[7,6]=126705961361465452994391/126885461609550467563520,\na[ 8,1]=2688622152256014375593487443/635826944384194229962077552,\na[8,2] =0,\na[8,3]=-165064430696291307962649/9442530342150159719648,\na[8,4]= 85647344134405061626206038309671468707/5745317881110570819440070776064 368272,\na[8,5]=-1868050624284923482186332306552702/977068785600982269 831702681478435,\na[8,6]=16842653200587834711/13166474974917410080,\na [8,7]=-608491123177947136/27366835598969517651,\na[9,1]=2465359/384652 80,\na[9,2]=0,\na[9,3]=0,\na[9,4]=29999319743551167/90312244465616140, \na[9,5]=2440521938223593/9364974686865720,\na[9,6]=1241823968749/6822 515208960,\na[9,7]=177732994465792/327930128445465,\na[9,8]=-11266189/ 29580120,\n\nb[1]=2465359/38465280,\nb[2]=0,\nb[3]=0,\nb[4]=2999931974 3551167/90312244465616140,\nb[5]=2440521938223593/9364974686865720,\nb [6]=1241823968749/6822515208960,\nb[7]=177732994465792/327930128445465 ,\nb[8]=-11266189/29580120,\n\n`b*`[1]=739837987028857/118903225906483 20,\n`b*`[2]=0,\n`b*`[3]=0,\n`b*`[4]=1204992945826748851603162095/3551 063890812398485590445952,\n`b*`[5]=328936061938716185672737/1439660654 341730577416400,\n`b*`[6]=102717532147008407320237/4618004088297246045 69600,\n`b*`[7]=9268295603184461676544/21663580993679805137685,\n`b*`[ 8]=-9633553238997677/35173968040963440,\n`b*`[9]=-1/156\}:" }}}{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#\" \"##\"#>\"$?#/&F%6#\"\"$#\"$[#\"%> " 0 "" {MPLTEXT 1 0 49 "seq(seq(a[i ,j]=subs(ee,a[i,j]),j=1..i-1),i=1..9);" }}{PARA 12 "" 1 "" {XPPMATH 20 "6F/&%\"aG6$\"\"#\"\"\"#\"#>\"$?#/&F%6$\"\"$F(#\"()[M8\")fU9c/&F%6$ F/F'#\"(Saw'F2/&F%6$\"\"%F(#\"#J\"$t&/&F%6$F;F'\"\"!/&F%6$F;F/#F=\"$\" >/&F%6$\"\"&F(#\"-4Gx&)RW\"-KojEBo/&F%6$FKF'FB/&F%6$FKF/#!.*z#p`zo\"FN /&F%6$FKF;#\"->%\\l-:)\"-;%=L;T$/&F%6$\"\"'F(#!;CULDr@&=%)4Dxx\"\";Vnq 4y-.kt%H=:\"/&F%6$FjnF'FB/&F%6$FjnF/#\":))e>\"HxA$)4F?79\"9Jn&R?*=\"y: jK<#/&F%6$FjnF;#!>oRY#orTKiw@in-g\">`ibIb*/&F%6$FfpF'FB/&F%6$FfpF/#!A*3:'H\"R)fZc6r!*e$p]\"\"@ k/\"H5$yY)=2n#HNl7\"/&F%6$FfpF;#\"IPpU(4\"4'f(4Ut$H%*\\jpb'3Y\"H/^\\k% >G%=rizv%QWp#ej/&F%6$FhrF'FB/&F%6$FhrF/#!9\\E'zI\" H'pIW1l\"\"7['>(f,:U.`U%*/&F%6$FhrF;#\"G2(o9n4$Qg?E;10W8Wtk&)\"Fs#oV1w 22S%>3d56)yJXd/&F%6$FhrFK#!C-Fb1BL'=#[B\\GC10o=\"BN%y9o-<$)pA)4g&yoq(* /&F%6$FhrFjn#\"56Z$ye+KlUo\"\"5!35u\"\\(\\ZmJ\"/&F%6$FhrFfp#!3Or%z$***H\"2ShhlWC7.*/&F%6$F`uFK#\"1$fB#Q>_SC\"1?d 'oou\\O*/&F%6$F`uFjn#\".\\(oR#=C\"\".g*3_^Ao/&F%6$F`uFfp#\"0#zlW*Htx\" \"0laWG,$zK/&F%6$F`uFhr#!)*=m7\"\")?,eH" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT 259 38 "weights for the 8 stage order 6 s cheme" }{TEXT -1 1 ":" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 31 "seq (b[i]=subs(ee,b[i]),i=1..8);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6*/&%\"b G6#\"\"\"#\"(f`Y#\")!Gl%Q/&F%6#\"\"#\"\"!/&F%6#\"\"$F//&F%6#\"\"%#\"2n 6bV(>$***H\"2ShhlWC7.*/&F%6#\"\"&#\"1$fB#Q>_SC\"1?d'oou\\O*/&F%6#\"\"' #\".\\(oR#=C\"\".g*3_^Ao/&F%6#\"\"(#\"0#zlW*Htx\"\"0laWG,$zK/&F%6#\"\" )#!)*=m7\"\")?,eH" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT 259 38 "weights for the 9 stage order 5 scheme" }{TEXT -1 1 ":" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 37 "seq(`b*`[i]=subs(ee,`b*`[i ]),i=1..9);" }}{PARA 12 "" 1 "" {XPPMATH 20 "6+/&%#b*G6#\"\"\"#\"0d)Gq )z$)R(\"2?$[1fA.*=\"/&F%6#\"\"#\"\"!/&F%6#\"\"$F//&F%6#\"\"%#\"=&4iJg^ )[n#e%H*\\?\"\"=_fW!f&[)R73*Q1^N/&F%6#\"\"&#\"9PFn&=;(Q>1O*G$\":+kTx0t TVlg'R9/&F%6#\"\"'#\"9P-K2%3q9Kvr-\"\"9+'pXgC(H)3/!=Y/&F%6#\"\"(#\"7Wl nhW=.cHo#*\"8&oP^!)zO*4ej;#/&F%6#\"\")#!1xw**QKbL'*\"2SM'4/oR " 0 "" {MPLTEXT 1 0 1 ";" }}}}{PARA 0 "" 0 "" {TEXT -1 52 "#-------- -------------------------------------------" }}{EXCHG {PARA 0 "> " 0 " " {MPLTEXT 1 0 1 ";" }}}}{PARA 0 "" 0 "" {TEXT -1 33 "#=============== =================" }}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 13 "scheme with \+ " }{XPPEDIT 18 0 "c[7] = 125/126;" "6#/&%\"cG6#\"\"(*&\"$D\"\"\"\"\"$ E\"!\"\"" }{TEXT -1 1 " " }}{PARA 257 "" 0 "" {TEXT -1 33 "The scheme \+ constructed here has " }{XPPEDIT 18 0 "c[6] = 91/129;" "6#/&%\"cG6#\" \"'*&\"#\"*\"\"\"\"$H\"!\"\"" }{TEXT -1 7 " and " }{XPPEDIT 18 0 "c[ 7] = 125/126;" "6#/&%\"cG6#\"\"(*&\"$D\"\"\"\"\"$E\"!\"\"" }{TEXT -1 1 "." }}{PARA 0 "" 0 "" {TEXT -1 6 "With " }{XPPEDIT 18 0 "c[6]" "6#& %\"cG6#\"\"'" }{TEXT -1 7 " and " }{XPPEDIT 18 0 "c[7]" "6#&%\"cG6# \"\"(" }{TEXT -1 39 " having these fixed values 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[5] " "6#&%\"cG6#\"\"&" }{TEXT -1 50 " are chosen to minimize the princip al error norm." }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}} {PARA 0 "" 0 "" {TEXT -1 52 "#---------------------------------------- -----------" }}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 28 "checking the combi ned scheme" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 " " {MPLTEXT 1 0 1 ";" }}}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 35 "coefficie nts of the combined scheme" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1967 "ee := \{c[2]=6/65,\nc[3]=23/159, \nc[4]=23/106,\nc[5]=80/141,\nc[6]=91/129,\nc[7]=125/126,\nc[8]=1,\nc[ 9]=1,\n\na[2,1]=6/65,\na[3,1]=9499/303372,\na[3,2]=34385/303372,\na[4, 1]=23/424,\na[4,2]=0,\na[4,3]=69/424,\na[5,1]=967715920/1482903909,\na [5,2]=0,\na[5,3]=-1226377600/494301303,\na[5,4]=3552780800/1482903909, \na[6,1]=-46804789259240403389999/30226267716384768392160,\na[6,2]=0, \na[6,3]=410550459697310264150/62971391075801600817,\na[6,4]=-48733588 56094925562114476/989343525191918950435887,\na[6,5]=731598505213969160 5041/11082753205258491358240,\na[7,1]=38140471378772278038787625/10601 915981402462257077264,\na[7,2]=0,\na[7,3]=-2006972716153785212305625/1 35921999761570028936888,\na[7,4]=22464333486903984266603841845000/1782 850965624830561723340121359,\na[7,5]=-3335402692827978879904464125/225 2537921258500779474698736,\na[7,6]=627869105822046875/6033274244855569 53,\na[8,1]=146733704741261404039925503/35929601425079440010660000,\na [8,2]=0,\na[8,3]=-332175497511735087990/19741539244549142863,\na[8,4]= 6401812090729586424603870720224/446765171619464803439374766203,\na[8,5 ]=-4265990808672245834945068273/2437965633092800464693941600,\na[8,6]= 330884756892468554/282253958465901877,\na[8,7]=-28749933344522124/3047 037469905514375,\na[9,1]=6205673/96600000,\na[9,2]=0,\na[9,3]=0,\na[9, 4]=43180775629715/129606111568587,\na[9,5]=10385429951981/398503440080 00,\na[9,6]=315039093349/1710630556040,\na[9,7]=1569042174483/15794058 53125,\na[9,8]=-9646151/11543640,\n\nb[1]=6205673/96600000,\nb[2]=0,\n b[3]=0,\nb[4]=43180775629715/129606111568587,\nb[5]=10385429951981/398 50344008000,\nb[6]=315039093349/1710630556040,\nb[7]=1569042174483/157 9405853125,\nb[8]=-9646151/11543640,\n\n`b*`[1]=5026843877900203/80720 277237600000,\n`b*`[2]=0,\n`b*`[3]=0,\n`b*`[4]=11352736352605242548485 /33323272041738045062256,\n`b*`[5]=581508736832735921107/2561499296336 591776000,\n`b*`[6]=323536238315689614389/1429426218785286161440,\n`b* `[7]=310826349095675205429/406084020815219450000,\n`b*`[8]=-4557126969 08507/742001768698080,\n`b*`[9]=-1/132\}:" }}}{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 64 "The Butcher tableau \+ in exact and approximate form is as follows." }}{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)],[``$4,`b*`[9]]]));" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#K%'matrixG6#747'#\"\"'\"#lF(%!GF+F+7'# \"#B\"$f\"#\"%*\\*\"'sLI#\"&&QMF2F+F+7'#F.\"$1\"#F.\"$C%\"\"!#\"#pF9F+ 7'#\"#!)\"$T\"#\"*?frn*\"+4R!H[\"F:#!++wPE7\"*.8I%\\#\"++3y_NFC7'#\"# \"*\"$H\"#!8****Q./Cf#*y/o%\"8g@RoZQ;xEE-$F:#\"6]TE5tpf/b5%\"5<3g,e2\" RrH'#!:wW6ib#\\4c)eL([\"9()eV]*=>>DNM*)*7'F+F+F+F+#\"7T]g\"pR@0&)fJ(\" 8S#e8\\e_?`F367'#\"$D\"\"$E\"#\";DwyQ!yAxy8ZS\"Q\";ks2dAY-9)f\">g5F:#! :DcI7_y`hrsp+#\"9))o$*G+dh(**>#f8#\"A+]%=%QgmU)R!p[LVYA\"@f87SLsh0$[il 4&Gy\"7'F+F+F+#!=DTY/*z)yz#Gp-aL$\"=O()pu%z2]e7#z`_A#\"3vo/Ae5pyi\"3`p b&[CuK.'7'\"\"\"#\"<.b#*RSSh7u/PtY\"\";++m5+Wz]U,'Hf$F:#!6!*z3N<^(\\v@ L\"5jG9\\XCR:u>#\"@C-sqQgCkeH247=S'\"?.iwu$RM![Y>;&*HaQ5\"/+!3SM])R#\"-\\L4R]J\".SgbI1r\"#\".$[u@/p:\".DJ&eSz:#!( ^hk*\")SOa67'F+%=____________________________GFbrFbrFbr7'%\"bGF^qF:F:F aqFdq7'%#b*G#\"1.-!z(Q%o-&\"2++gPsF?2)F:F:#\"8&[[DC0ENOFN6\"8cA1X!Q_\"3-%31%#!02&3pp7dX\"0!3)po<+U(7'F+F+F+F+#!\" \"\"$K\"Q(pprint56\"" }}}{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),``$(10-i)],i=2..9),\n[`b`,seq(b[i],i=1..8),``],[`b*`,seq(`b*`[i ],i=1..9)]])):\nevalf[8](%);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#K%'mat rixG6#7,7,$\")#p2B*!\"*F(%!GF+F+F+F+F+F+F+7,$\")4aY9!\")$\")$R68$F*$\" )pUL6F/F+F+F+F+F+F+F+7,$\")8\")p@F/$\")$GXU&F*$\"\"!F:$\")&eti\"F/F+F+ F+F+F+F+7,$\")*ePn&F/$\")o\"e_'F/F9$!)D.\"[#!\"($\")n#eR#FDF+F+F+F+F+7 ,$\")OEaqF/$!)1[[:FDF9$\")Yj>lFD$!)6&e#\\FD$\")VB,mF/F+F+F+F+7,$\")\\j ?**F/$\")u](f$FDF9$!)@cw9!\"'$\")J-g7FY$!)5t![\"FD$\")snS5FDF+F+F+7,$ \"\"\"F:$\")C#R3%FDF9$!)@i#o\"FY$\")b#HV\"FY$!)e\")\\ " 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$" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 114 "We can calculate the principal error norm of t he order 6 scheme, that is, the 2-norm of the principal error terms." }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 131 "errterms6_8 := PrincipalE rrorTerms(6,8,'expanded'):\nevalf(evalf[14](sqrt(add(subs(ee,errterms6 _8[i])^2,i=1.. nops(errterms6_8)))));" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#$\"+yRjT8!#9" }}}{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#$\"+7%4i'e!#8" }}} {PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}}{PARA 0 "" 0 "" {TEXT -1 52 "#------------------------------ ---------------------" }}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 44 "simultan eous construction of the two schemes" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 119 "We incorporate the stage-order equatio ns to ensure that stage 2 has stage-order 2 and stages 3 to 8 have sta ge-order 3." }}{PARA 0 "" 0 "" {TEXT -1 48 "We also incorporate the si mplifying conditions: " }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "Sum(b[i]*a[i,j],i = j+1 .. 8) = b[j]*(1-c[j]);" "6#/-%$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] = 6/65;" "6#/&%\"cG6#\"\"#*& \"\"'\"\"\"\"#l!\"\"" }{TEXT -1 3 ", " }{XPPEDIT 18 0 "c[4] = 23/106; " "6#/&%\"cG6#\"\"%*&\"#B\"\"\"\"$1\"!\"\"" }{TEXT -1 3 ", " } {XPPEDIT 18 0 "c[5] = 80/141;" "6#/&%\"cG6#\"\"&*&\"#!)\"\"\"\"$T\"!\" \"" }{TEXT -1 3 ", " }{XPPEDIT 18 0 "c[6] = 91/129;" "6#/&%\"cG6#\"\" '*&\"#\"*\"\"\"\"$H\"!\"\"" }{TEXT -1 3 ", " }{XPPEDIT 18 0 "c[7] = 1 25/126;" "6#/&%\"cG6#\"\"(*&\"$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 -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*`[9] = -1/132;" "6#/&%#b*G6#\" \"*,$*&\"\"\"F*\"$K\"!\"\"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 ]=6/65,c[4]=23/106,c[5]=80/141,c[6]=91/129,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 ]=-1/132\}:\neqns := subs(e1,cdns):\nnops(%);\nindets(eqns);\nnops(%); " }}{PARA 11 "" 1 "" {XPPMATH 20 "6#\"#W" }}{PARA 12 "" 1 "" {XPPMATH 20 "6#&F<6$F*F'&F<6$F*F:&F< 6$\"\"*F'&F<6$FIF>&F<6$FIF:&F<6$FIF*&F<6$FIF-&F<6$FIF0&F<6$FIF3&F<6$FI F6&%#b*GF)&FYF,&FYF/&FYF2&FYF5&F<6$F-F:&F<6$F-F*&F<6$F0F'&F<6$F0F:&F<6 $F0F*&F<6$F0F-&F<6$F3F'&F<6$F3F:&F<6$F3F*&F<6$F3F-&F<6$F3F0&F<6$F6F'&F <6$F-F'&FYF&&F<6$F6F:&F<6$F6F*&F<6$F6F-&F<6$F6F0&F<6$F6F3" }}{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)\}):\n infolevel[solve] := 0:\ne3 := `union`(e1,e2):" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 2 "e3" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 2086 "e3 := \{`b*`[4] = 11352736 352605242548485/33323272041738045062256, a[4,2] = 0, a[5,2] = 0, a[6,2 ] = 0, a[7,2] = 0, a[8,2] = 0, a[8,1] = 146733704741261404039925503/35 929601425079440010660000, a[7,3] = -2006972716153785212305625/13592199 9761570028936888, `b*`[3] = 0, a[9,2] = 0, a[9,3] = 0, a[9,7] = 156904 2174483/1579405853125, a[9,1] = 6205673/96600000, a[9,6] = 31503909334 9/1710630556040, a[7,5] = -3335402692827978879904464125/22525379212585 00779474698736, a[9,5] = 10385429951981/39850344008000, a[6,5] = 73159 85052139691605041/11082753205258491358240, `b*`[5] = 58150873683273592 1107/2561499296336591776000, a[2,1] = 6/65, b[1] = 6205673/96600000, a [5,3] = -1226377600/494301303, a[8,7] = -28749933344522124/30470374699 05514375, c[4] = 23/106, c[5] = 80/141, c[6] = 91/129, c[7] = 125/126, b[6] = 315039093349/1710630556040, a[6,4] = -487335885609492556211447 6/989343525191918950435887, c[8] = 1, c[9] = 1, b[2] = 0, b[3] = 0, `b *`[2] = 0, c[2] = 6/65, `b*`[9] = -1/132, c[3] = 23/159, a[5,4] = 3552 780800/1482903909, a[6,1] = -46804789259240403389999/30226267716384768 392160, a[9,8] = -9646151/11543640, a[8,3] = -332175497511735087990/19 741539244549142863, a[8,5] = -4265990808672245834945068273/24379656330 92800464693941600, a[5,1] = 967715920/1482903909, `b*`[6] = 3235362383 15689614389/1429426218785286161440, b[7] = 1569042174483/1579405853125 , `b*`[8] = -455712696908507/742001768698080, b[5] = 10385429951981/39 850344008000, a[7,1] = 38140471378772278038787625/10601915981402462257 077264, a[4,1] = 23/424, b[8] = -9646151/11543640, a[4,3] = 69/424, `b *`[1] = 5026843877900203/80720277237600000, a[8,6] = 33088475689246855 4/282253958465901877, a[6,3] = 410550459697310264150/62971391075801600 817, `b*`[7] = 310826349095675205429/406084020815219450000, a[7,6] = 6 27869105822046875/603327424485556953, a[7,4] = 22464333486903984266603 841845000/1782850965624830561723340121359, a[3,1] = 9499/303372, a[9,4 ] = 43180775629715/129606111568587, b[4] = 43180775629715/129606111568 587, a[8,4] = 6401812090729586424603870720224/446765171619464803439374 766203, a[3,2] = 34385/303372\}:" }{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 "s ubs(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)],[``$4,`b*`[ 9]]]));" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#K%'matrixG6#747'#\"\"'\"#lF (%!GF+F+7'#\"#B\"$f\"#\"%*\\*\"'sLI#\"&&QMF2F+F+7'#F.\"$1\"#F.\"$C%\" \"!#\"#pF9F+7'#\"#!)\"$T\"#\"*?frn*\"+4R!H[\"F:#!++wPE7\"*.8I%\\#\"++3 y_NFC7'#\"#\"*\"$H\"#!8****Q./Cf#*y/o%\"8g@RoZQ;xEE-$F:#\"6]TE5tpf/b5% \"5<3g,e2\"RrH'#!:wW6ib#\\4c)eL([\"9()eV]*=>>DNM*)*7'F+F+F+F+#\"7T]g\" pR@0&)fJ(\"8S#e8\\e_?`F367'#\"$D\"\"$E\"#\";DwyQ!yAxy8ZS\"Q\";ks2dAY-9 )f\">g5F:#!:DcI7_y`hrsp+#\"9))o$*G+dh(**>#f8#\"A+]%=%QgmU)R!p[LVYA\"@f 87SLsh0$[il4&Gy\"7'F+F+F+#!=DTY/*z)yz#Gp-aL$\"=O()pu%z2]e7#z`_A#\"3vo/ Ae5pyi\"3`pb&[CuK.'7'\"\"\"#\"<.b#*RSSh7u/PtY\"\";++m5+Wz]U,'Hf$F:#!6! *z3N<^(\\v@L\"5jG9\\XCR:u>#\"@C-sqQgCkeH247=S'\"?.iwu$RM![Y>;&*HaQ5\"/+!3SM])R#\"-\\L4R]J\".SgbI1r\"#\".$[u@/p:\" .DJ&eSz:#!(^hk*\")SOa67'F+%=____________________________GFbrFbrFbr7'% \"bGF^qF:F:FaqFdq7'%#b*G#\"1.-!z(Q%o-&\"2++gPsF?2)F:F:#\"8&[[DC0ENOFN6 \"8cA1X!Q_\"3-%31%#!02&3pp7dX\"0!3)po<+U(7' F+F+F+F+#!\"\"\"$K\"Q(pprint36\"" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 136 "subs(e3,matrix([seq([c[i],s eq(a[i,j],j=1..i-1),``$(10-i)],i=2..9),\n[`b`,seq(b[i],i=1..8),``],[`b *`,seq(`b*`[i],i=1..9)]])):\nevalf[8](%);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#K%'matrixG6#7,7,$\")#p2B*!\"*F(%!GF+F+F+F+F+F+F+7,$\")4 aY9!\")$\")$R68$F*$\")pUL6F/F+F+F+F+F+F+F+7,$\")8\")p@F/$\")$GXU&F*$\" \"!F:$\")&eti\"F/F+F+F+F+F+F+7,$\")*ePn&F/$\")o\"e_'F/F9$!)D.\"[#!\"($ \")n#eR#FDF+F+F+F+F+7,$\")OEaqF/$!)1[[:FDF9$\")Yj>lFD$!)6&e#\\FD$\")VB ,mF/F+F+F+F+7,$\")\\j?**F/$\")u](f$FDF9$!)@cw9!\"'$\")J-g7FY$!)5t![\"F D$\")snS5FDF+F+F+7,$\"\"\"F:$\")C#R3%FDF9$!)@i#o\"FY$\")b#HV\"FY$!)e\" )\\ " 0 "" {MPLTEXT 1 0 146 "RK6_8eqs := [op(RowSumConditions(8 ,'expanded')),op(OrderConditions(6,8,'expanded'))]:\n`RK5_9eqs*` := su bs(b=`b*`,OrderConditions(5,9,'expanded')):" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 7 "Check: " }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 148 "simplify(subs(e3,RK6_8eqs)):\nmap(u->`if`(lhs(u )=rhs(u),0,1),%);\nnops(%);\nsimplify(subs(e3,`RK5_9eqs*`)):\nmap(u->` if`(lhs(u)=rhs(u),0,1),%);\nnops(%);" }{TEXT -1 0 "" }}{PARA 11 "" 1 " " {XPPMATH 20 "6#7N\"\"!F$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 28 "determination of 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[5] " "6#&%\"cG6#\"\"&" }{TEXT -1 1 " " }}{PARA 4 "" 0 "" {TEXT -1 0 "" }} {PARA 0 "" 0 "" {TEXT -1 79 "In this section we obtain the nodes of a \+ modification of a scheme by Tsitouras." }}{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*(3 0*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]+2 8*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]+1 90*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]^2-27*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]^4-10 0*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]), 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]+1 32*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]^2-91 0*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+20 0*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]-4 1*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]^3-1 2*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+2 2*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+42 0*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 +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]+1 5*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]+15 6*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+95 0*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]^3-10 0*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]-1 98*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]+20 0*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+2 40*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+1 1*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]^3-49 0*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]^2-30 0*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]+20 0*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+10 3*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]^3-2 3*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+18 0*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]^3-1 0*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]-20 0*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]+3 60*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+3 60*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]-8 0*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+84 0*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+23 0*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[8,4] = -1/2*(734* 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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]+150 0*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* 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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 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*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]+2 860*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]-4 29*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]+49 0*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 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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]^3-70*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[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] 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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] /(72*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 ]+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+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]^3-2 0*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]^5-30*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+15 0*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]-20 0*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 33 "#=================== =============" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 22 "We specify the nodes " }{XPPEDIT 18 0 "c[6] = 91/129;" " 6#/&%\"cG6#\"\"'*&\"#\"*\"\"\"\"$H\"!\"\"" }{TEXT -1 7 " and " } {XPPEDIT 18 0 "c[7] = 125/126;" "6#/&%\"cG6#\"\"(*&\"$D\"\"\"\"\"$E\"! \"\"" }{TEXT -1 27 " and determine 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[5]" "6#&%\"cG6#\"\"&" }{TEXT -1 65 " that minimize the principal errror norm (subject to th e nodes " }{XPPEDIT 18 0 "c[6]" "6#&%\"cG6#\"\"'" }{TEXT -1 7 " and \+ " }{XPPEDIT 18 0 "c[7]" "6#&%\"cG6#\"\"(" }{TEXT -1 19 " remaining f ixed)." }}{PARA 0 "" 0 "" {TEXT -1 85 "We use the general solution to obtain expressions for the coefficients in terms of " }{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[5]" "6#&%\"cG6# \"\"&" }{TEXT -1 2 ". " }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 74 "eA := \{c[6]=91/129,c[7]=125/126\}:\neB := `union`(eA,simplify(subs(eA,e G))):" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 2 "eB" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 16480 "eB := \{c[8] = 1, a[8,2] = 0, `b*`[2] = 0, `b*`[3] = 0, a[4,2] = 0, a [5,2] = 0, c[9] = 1, b[2] = 0, b[3] = 0, a[7,2] = 0, a[6,2] = 0, `b*`[ 8] = 0, a[5,3] = -3/4*c[5]^2*(2*c[5]-3*c[4])/c[4]^2, a[2,1] = c[2], a[ 3,1] = -2/9*c[4]*(c[4]-3*c[2])/c[2], a[4,1] = 1/4*c[4], 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], a[5,1] = 1/4*c[5]*(2*c[5]^2-5*c[5]*c[4]+4*c[4]^2)/c[4]^ 2, a[9,2] = 0, a[9,3] = 0, a[9,5] = 1/60*(24557*c[4]-6610)/c[5]/(-1625 4*c[5]^3*c[4]+43845*c[5]^2*c[4]-38966*c[5]*c[4]+11375*c[4]+16254*c[5]^ 4-43845*c[5]^3+38966*c[5]^2-11375*c[5]), `b*`[4] = 1/60*(23991240*c[5] ^3-37388372*c[5]^2+17006868*c[5]+7994592*c[4]-123537570*c[5]^2*c[4]^2+ 147894592*c[5]^2*c[4]-6113744*c[4]^2+49618972*c[5]*c[4]^2+89707800*c[5 ]^3*c[4]^2-100870380*c[5]^3*c[4]-62459575*c[5]*c[4]-2308215)/(-91000*c [5]^2+118500*c[5]+152898*c[4]-727314*c[5]^2*c[4]^2-386064*c[5]*c[4]^3+ 91728*c[4]^3+474728*c[5]^2*c[4]-210448*c[4]^2+850334*c[5]*c[4]^2+34398 0*c[5]^2*c[4]^3-583073*c[5]*c[4]-34125)/(129*c[4]-91)/(-c[4]+c[5])/c[4 ], b[8] = 1/2280*(33240*c[5]*c[4]-8683*c[5]-8683*c[4]+2073)/(c[4]-1)/( -1+c[5]), `b*`[5] = -1/60*(-6113744*c[5]^2+7994592*c[5]+17006868*c[4]- 123537570*c[5]^2*c[4]^2-100870380*c[5]*c[4]^3+23991240*c[4]^3+49618972 *c[5]^2*c[4]-37388372*c[4]^2+147894592*c[5]*c[4]^2+89707800*c[5]^2*c[4 ]^3-62459575*c[5]*c[4]-2308215)/(91728*c[5]^3-210448*c[5]^2+152898*c[5 ]+118500*c[4]-727314*c[5]^2*c[4]^2+850334*c[5]^2*c[4]-91000*c[4]^2+474 728*c[5]*c[4]^2+343980*c[5]^3*c[4]^2-386064*c[5]^3*c[4]-583073*c[5]*c[ 4]-34125)/(129*c[5]-91)/(-c[4]+c[5])/c[5], b[4] = -1/60*(-6610+24557*c [5])/c[4]/(43845*c[4]^3-16254*c[4]^4-38966*c[4]^2+11375*c[4]+16254*c[5 ]*c[4]^3-43845*c[5]*c[4]^2+38966*c[5]*c[4]-11375*c[5]), `b*`[1] = 1/68 2500*(-23991240*c[5]^3+37388372*c[5]^2-17006868*c[5]-17006868*c[4]+890 451300*c[5]^2*c[4]^2+231593700*c[5]*c[4]^3-23991240*c[4]^3-342146740*c [5]^2*c[4]+37388372*c[4]^2-342146740*c[5]*c[4]^2-639791220*c[5]^2*c[4] ^3-639791220*c[5]^3*c[4]^2+231593700*c[5]^3*c[4]+142771009*c[5]*c[4]+4 89953100*c[5]^3*c[4]^3+2308215)/(273-3064*c[5]*c[4]^2-3064*c[5]^2*c[4] +728*c[5]^2-948*c[4]-948*c[5]+3709*c[5]*c[4]+2730*c[5]^2*c[4]^2+728*c[ 4]^2)/c[5]/c[4], a[6,1] = 91/1107691524*(-8545992*c[5]^3+4521426*c[5]^ 2+3028662*c[4]^4+72258732*c[5]^2*c[4]^2-64794912*c[5]*c[4]^3-4272996*c [4]^3-75716550*c[5]^4*c[4]^3-27128556*c[5]^2*c[4]+1507142*c[4]^2+26631 696*c[5]*c[4]^2-67027515*c[5]^2*c[4]^3-17974344*c[4]^4*c[5]^2-27521001 9*c[5]^3*c[4]^2+72476586*c[5]^3*c[4]-5274997*c[5]*c[4]+85867560*c[4]^5 *c[5]^3+81357646*c[4]^4*c[5]+85867560*c[5]^4*c[4]^4+500419896*c[5]^3*c [4]^3-424179090*c[5]^3*c[4]^4+60573240*c[4]^5*c[5]^2-42729960*c[4]^5*c [5]+21364980*c[5]^4*c[4]^2)/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), a[6,4] = -91/553845762*(8545992*c[5]^3-4521426*c[5]^2-60954075*c [5]^2*c[4]^2+79128231*c[5]*c[4]^3+2136498*c[4]^3+30286620*c[5]^4*c[4]^ 3+24992058*c[5]^2*c[4]-1507142*c[4]^2-24495198*c[5]*c[4]^2-33910599*c[ 5]^2*c[4]^3+181719720*c[4]^4*c[5]^2+239365305*c[5]^3*c[4]^2-67933593*c [5]^3*c[4]+5274997*c[5]*c[4]-108116374*c[4]^4*c[5]-291244590*c[5]^3*c[ 4]^3+90859860*c[5]^3*c[4]^4-90859860*c[4]^5*c[5]^2+42729960*c[4]^5*c[5 ]-21364980*c[5]^4*c[4]^2)/(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[7,1] = 125/91745244864*(-333132891 000*c[5]^3+95490281250*c[5]^2+257721282000*c[5]^4+164063313000*c[4]^4+ 1049516831059*c[5]^2*c[4]^2-2017658076374*c[5]*c[4]^3-4822588106280*c[ 4]^6*c[5]^2-133409015750*c[4]^3-62423865000*c[4]^5-19917976165920*c[5] ^4*c[4]^5-2698116908256*c[5]^4*c[4]-5225842069560*c[5]^5*c[4]^4-315966 72372066*c[5]^4*c[4]^3-644303205000*c[5]^5*c[4]^2+3347652028140*c[5]^5 *c[4]^3+1288606410000*c[4]^6*c[5]-539784257750*c[5]^2*c[4]+31830093750 *c[4]^2+681936586250*c[5]*c[4]^2+952442566824*c[5]^2*c[4]^3-8160901028 596*c[4]^4*c[5]^2-12594642577344*c[5]^3*c[4]^2+3085067185878*c[5]^3*c[ 4]-111405328125*c[5]*c[4]+4943151138600*c[4]^5*c[5]^3+3757925648286*c[ 4]^4*c[5]+37566654289020*c[5]^4*c[4]^4+25683365550618*c[5]^3*c[4]^3-22 496020236042*c[5]^3*c[4]^4+11322525685124*c[4]^5*c[5]^2+2110136835240* c[5]^3*c[4]^6-3590093761828*c[4]^5*c[5]+12865851972036*c[5]^4*c[4]^2+2 883727753200*c[5]^4*c[4]^6+2883727753200*c[5]^5*c[4]^5)/(68*c[5]^3-15* c[5]^2-1001*c[5]^2*c[4]^2-673*c[5]*c[4]^3+68*c[4]^3+2650*c[5]^4*c[4]^3 +90*c[5]^2*c[4]-15*c[4]^2+90*c[5]*c[4]^2+3780*c[5]^2*c[4]^3-680*c[4]^4 *c[5]^2+3780*c[5]^3*c[4]^2-673*c[5]^3*c[4]+15*c[5]*c[4]-9310*c[5]^3*c[ 4]^3+2650*c[5]^3*c[4]^4-680*c[5]^4*c[4]^2)/c[5]/c[4]^2, a[6,3] = 91/12 3076836*(54058187*c[5]*c[4]^3-36670452*c[5]*c[4]^2+10467184*c[5]*c[4]- 1507142*c[5]+753571*c[4]-7121660*c[5]^3*c[4]^2-119346500*c[5]^2*c[4]^3 +82817097*c[5]^2*c[4]^2-23149308*c[5]^2*c[4]+2848664*c[5]^2-21364980*c [4]^4*c[5]-1424332*c[4]^2+15143310*c[5]^3*c[4]^3+45429930*c[4]^4*c[5]^ 2)/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*`[6] = 30769209/28264 60*(-449540*c[5]^3+854238*c[5]^2-518526*c[5]-518526*c[4]+9396710*c[5]^ 2*c[4]^2+2795650*c[5]*c[4]^3-449540*c[4]^3-5049096*c[5]^2*c[4]+854238* c[4]^2-5049096*c[5]*c[4]^2-5487030*c[5]^2*c[4]^3-5487030*c[5]^3*c[4]^2 +2795650*c[5]^3*c[4]+2879673*c[5]*c[4]+3385200*c[5]^3*c[4]^3+100737)/( 273-3064*c[5]*c[4]^2-3064*c[5]^2*c[4]+728*c[5]^2-948*c[4]-948*c[5]+370 9*c[5]*c[4]+2730*c[5]^2*c[4]^2+728*c[4]^2)/(16641*c[5]*c[4]-11739*c[4] -11739*c[5]+8281), `b*`[7] = -7001316/970625*(-123760*c[5]^3+202128*c[ 5]^2-94257*c[5]-94257*c[4]+3363700*c[5]^2*c[4]^2+941300*c[5]*c[4]^3-12 3760*c[4]^3-1512885*c[5]^2*c[4]+202128*c[4]^2-1512885*c[5]*c[4]^2-2160 030*c[5]^2*c[4]^3-2160030*c[5]^3*c[4]^2+941300*c[5]^3*c[4]+700491*c[5] *c[4]+1446900*c[5]^3*c[4]^3+12285)/(-11466000*c[5]^3+26306000*c[5]^2-1 9112250*c[5]-19112250*c[4]+198056334*c[5]^2*c[4]^2+59815728*c[5]*c[4]^ 3-11466000*c[4]^3-132808198*c[5]^2*c[4]+26306000*c[4]^2-132808198*c[5] *c[4]^2-91641564*c[5]^2*c[4]^3-91641564*c[5]^3*c[4]^2+59815728*c[5]^3* c[4]+92149273*c[5]*c[4]+43341480*c[5]^3*c[4]^3+4265625), a[7,4] = -125 /504094752*(-333132891000*c[5]^3+95490281250*c[5]^2+257721282000*c[5]^ 4+204412582500*c[4]^4+1581888358309*c[5]^2*c[4]^2-3154107182624*c[5]*c [4]^3-28338307412460*c[4]^6*c[5]^2-144983718875*c[4]^3-91335289500*c[4 ]^5-33303236294160*c[5]^4*c[4]^5-2969822268756*c[5]^4*c[4]-49892175320 40*c[5]^5*c[4]^4-42066260201718*c[5]^4*c[4]^3-644303205000*c[5]^5*c[4] ^2+3324646948140*c[5]^5*c[4]^3+7302967071456*c[4]^6*c[5]-641602523375* c[5]^2*c[4]+31830093750*c[4]^2+806315742500*c[5]*c[4]^2+1787896530360* c[5]^2*c[4]^3-18728170819102*c[4]^4*c[5]^2-15781709984733*c[5]^3*c[4]^ 2+3440076186378*c[5]^3*c[4]-111405328125*c[5]*c[4]-3186623660220*c[4]^ 5*c[5]^3+7883573870985*c[4]^4*c[5]+55512312454980*c[5]^4*c[4]^4+353131 01948088*c[5]^3*c[4]^3-31269203039130*c[5]^3*c[4]^4+36518379418004*c[4 ]^5*c[5]^2+18756458884800*c[5]^3*c[4]^6-10899858434938*c[4]^5*c[5]+156 30620557482*c[5]^4*c[4]^2+6932903140800*c[5]^4*c[4]^6+2310967713600*c[ 5]^5*c[4]^5+7721325841320*c[4]^7*c[5]^2-1826705790000*c[5]*c[4]^7-6932 903140800*c[5]^3*c[4]^7)/(-15*c[5]^3+68*c[5]^4-68*c[4]^4-22*c[5]*c[4]^ 3+15*c[4]^3-673*c[5]^4*c[4]-8630*c[5]^4*c[4]^3-680*c[5]^5*c[4]^2+2650* c[5]^5*c[4]^3+30*c[5]^2*c[4]-30*c[5]*c[4]^2+328*c[5]^2*c[4]^3-3780*c[4 ]^4*c[5]^2-328*c[5]^3*c[4]^2+22*c[5]^3*c[4]-2650*c[4]^5*c[5]^3+673*c[4 ]^4*c[5]+8630*c[5]^3*c[4]^4+680*c[4]^5*c[5]^2+3780*c[5]^4*c[4]^2)/(129 *c[4]-91)/c[4]^2, a[9,7] = -882165816/970625*(265*c[5]*c[4]-68*c[5]-68 *c[4]+15)/(15876*c[5]*c[4]-15750*c[4]-15750*c[5]+15625), b[5] = 1/60*( 24557*c[4]-6610)/(-c[4]+c[5])/c[5]/(16254*c[5]^3-43845*c[5]^2+38966*c[ 5]-11375), a[8,7] = -3192600096/194125*(265*c[5]*c[4]-68*c[5]-68*c[4]+ 15)*(c[4]-1)*(-1+c[5])/(126*c[5]-125)/(126*c[4]-125)/(33240*c[5]*c[4]- 8683*c[5]-8683*c[4]+2073), a[8,6] = -81574182/141323*(462535*c[5]*c[4] +92550-185057*c[4]-185057*c[5])*(c[4]-1)*(-1+c[5])/(129*c[4]-91)/(129* c[5]-91)/(33240*c[5]*c[4]-8683*c[5]-8683*c[4]+2073), a[9,1] = 1/682500 *(146610*c[5]*c[4]-24557*c[5]-24557*c[4]+6610)/c[5]/c[4], a[8,3] = 3/4 *(-179816*c[5]^3+233754*c[5]^2-67158*c[5]+33579*c[4]+9554487*c[5]^2*c[ 4]^2+4709577*c[5]*c[4]^3+89908*c[4]^3-1896570*c[5]^4*c[4]^3-2229444*c[ 5]^2*c[4]-116877*c[4]^2-2131296*c[5]*c[4]^2-20303208*c[5]^2*c[4]^3+187 05170*c[4]^4*c[5]^2-8510433*c[5]^3*c[4]^2+1837524*c[5]^3*c[4]+519825*c [5]*c[4]+5077800*c[4]^5*c[5]^3-4450395*c[4]^4*c[5]+1692600*c[5]^4*c[4] ^4+19082350*c[5]^3*c[4]^3-17741880*c[5]^3*c[4]^4-5689710*c[4]^5*c[5]^2 +1348620*c[4]^5*c[5]+449540*c[5]^4*c[4]^2)/c[4]^2/(8683*c[5]^3-2073*c[ 5]^2-133146*c[5]^2*c[4]^2-85338*c[5]*c[4]^3+8683*c[4]^3+332400*c[5]^4* c[4]^3+12438*c[5]^2*c[4]-2073*c[4]^2+12438*c[5]*c[4]^2+480660*c[5]^2*c [4]^3-86830*c[4]^4*c[5]^2+480660*c[5]^3*c[4]^2-85338*c[5]^3*c[4]+2073* c[5]*c[4]-1170860*c[5]^3*c[4]^3+332400*c[5]^3*c[4]^4-86830*c[5]^4*c[4] ^2), c[6] = 91/129, a[9,8] = 1/2280*(33240*c[5]*c[4]-8683*c[5]-8683*c[ 4]+2073)/(-c[4]+c[5]*c[4]+1-c[5]), b[1] = 1/682500*(146610*c[5]*c[4]-2 4557*c[5]-24557*c[4]+6610)/c[5]/c[4], a[6,5] = 91/553845762*(52366131* c[5]^2*c[4]^2-2136498*c[4]^2+753571*c[4]-17361981*c[5]^2*c[4]-33911787 *c[5]*c[4]^2+1514331*c[4]^3-21364980*c[4]^4*c[5]+44972201*c[5]*c[4]^3- 66794910*c[5]^2*c[4]^3+11179350*c[5]*c[4]+30286620*c[4]^4*c[5]^2+21364 98*c[5]^2-1507142*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*`[9] = 1/10*(4550*c[5]^2*c[4]^2-4285* c[5]^2*c[4]+910*c[5]^2-4285*c[5]*c[4]^2+4475*c[5]*c[4]-1039*c[5]+910*c [4]^2-1039*c[4]+273)/(273-3064*c[5]*c[4]^2-3064*c[5]^2*c[4]+728*c[5]^2 -948*c[4]-948*c[5]+3709*c[5]*c[4]+2730*c[5]^2*c[4]^2+728*c[4]^2), a[8, 4] = -1/2*(2658951750*c[5]^3-763922250*c[5]^2-2045407000*c[5]^4-279638 4138*c[4]^4-17847973344*c[5]^2*c[4]^2+31690934044*c[5]*c[4]^3+51759220 0074*c[4]^6*c[5]^2+1414862766*c[4]^3+2366844438*c[4]^5+705338993380*c[ 5]^4*c[4]^5+25616735299*c[5]^4*c[4]+66028174740*c[5]^5*c[4]^4+45816273 6909*c[5]^4*c[4]^3+5113517500*c[5]^5*c[4]^2-31512707320*c[5]^5*c[4]^3- 145609503260*c[4]^6*c[5]+5904224643*c[5]^2*c[4]-254640750*c[4]^2-73441 57956*c[5]*c[4]^2-1370095700*c[5]^2*c[4]^3+163316689688*c[4]^4*c[5]^2+ 153604709318*c[5]^3*c[4]^2-30135380297*c[5]^3*c[4]+891242625*c[5]*c[4] -227807334942*c[4]^5*c[5]^3-88303072416*c[4]^4*c[5]-774983847556*c[5]^ 4*c[4]^4-408687105857*c[5]^3*c[4]^3+533790104768*c[5]^3*c[4]^4-4406239 34571*c[4]^5*c[5]^2-171848038740*c[5]^3*c[4]^6+150257206783*c[4]^5*c[5 ]-147663584952*c[5]^4*c[4]^2+61653697560*c[4]^8*c[5]^2-319448666880*c[ 5]^4*c[4]^6-57969998520*c[5]^5*c[4]^5-287860886100*c[4]^7*c[5]^2+73030 996500*c[5]*c[4]^7-730682316*c[4]^6+18341013600*c[5]^5*c[4]^6-14613646 320*c[5]*c[4]^8-55023040800*c[4]^8*c[5]^3+55023040800*c[5]^4*c[4]^7+20 3447134800*c[5]^3*c[4]^7)/(-2073*c[5]^3+8683*c[5]^4-8683*c[4]^4-3755*c [5]*c[4]^3+2073*c[4]^3-85338*c[5]^4*c[4]-1084030*c[5]^4*c[4]^3-86830*c [5]^5*c[4]^2+332400*c[5]^5*c[4]^3+4146*c[5]^2*c[4]-4146*c[5]*c[4]^2+47 808*c[5]^2*c[4]^3-480660*c[4]^4*c[5]^2-47808*c[5]^3*c[4]^2+3755*c[5]^3 *c[4]-332400*c[4]^5*c[5]^3+85338*c[4]^4*c[5]+1084030*c[5]^3*c[4]^4+868 30*c[4]^5*c[5]^2+480660*c[5]^4*c[4]^2)/(-27591*c[4]+16254*c[4]^2+11375 )/c[4]^2, b[7] = -882165816/970625*(265*c[5]*c[4]-68*c[5]-68*c[4]+15)/ (126*c[4]-125)/(126*c[5]-125), a[9,4] = -1/60*(-6610+24557*c[5])/c[4]/ (43845*c[4]^3-16254*c[4]^4-38966*c[4]^2+11375*c[4]+16254*c[5]*c[4]^3-4 3845*c[5]*c[4]^2+38966*c[5]*c[4]-11375*c[5]), a[8,5] = 1/2*(-279638413 8*c[5]^3+1414862766*c[5]^2-254640750*c[5]+127320375*c[4]+2366844438*c[ 5]^4-246403457*c[4]^4+48996573070*c[5]^2*c[4]^2+13882747526*c[5]*c[4]^ 3+651348317*c[4]^3-57969998520*c[5]^4*c[4]^5-23654505213*c[5]^4*c[4]-5 4347834520*c[5]^5*c[4]^4-187440898422*c[5]^4*c[4]^3-30320676918*c[5]^5 *c[4]^2+59591401380*c[5]^5*c[4]^3-12606236970*c[5]^2*c[4]-532265235*c[ 4]^2-6382937944*c[5]*c[4]^2-97978089271*c[5]^2*c[4]^3+91685597725*c[4] ^4*c[5]^2-107034229850*c[5]^3*c[4]^2+26915359362*c[5]^3*c[4]+175128367 2*c[5]*c[4]+66028174740*c[4]^5*c[5]^3-14109970004*c[4]^4*c[5]+17142502 0840*c[5]^4*c[4]^4+211293490470*c[5]^3*c[4]^3-194406410584*c[5]^3*c[4] ^4-31512707320*c[4]^5*c[5]^2+7466778774*c[5]^5*c[4]-730682316*c[5]^5+5 113517500*c[4]^5*c[5]+95273536877*c[5]^4*c[4]^2+18341013600*c[5]^5*c[4 ]^5)/c[5]/(-23580375*c[5]^3+155965268*c[5]^4-98769125*c[4]^4+114392286 *c[5]^2*c[4]^2-99909268*c[5]*c[4]^3+23580375*c[4]^3+10582583220*c[5]^4 *c[4]^5+10208374170*c[5]^6*c[4]^2-26791072020*c[5]^6*c[4]^3-1006934871 *c[5]^4*c[4]+17619823620*c[5]^5*c[4]^4-11553770018*c[5]^4*c[4]^3-15026 652542*c[5]^5*c[4]^2+33690521730*c[5]^5*c[4]^3+5402829600*c[5]^7*c[4]^ 3+47160750*c[5]^2*c[4]-47160750*c[5]*c[4]^2+681114747*c[5]^2*c[4]^3-79 63201740*c[4]^4*c[5]^2-611205084*c[5]^3*c[4]^2-71679161*c[5]^3*c[4]-61 76776530*c[4]^5*c[5]^3+1210292403*c[4]^4*c[5]-37722119370*c[5]^4*c[4]^ 4-1380104298*c[5]^3*c[4]^3+26979815162*c[5]^3*c[4]^4+987691250*c[4]^5* c[5]^2+2415594528*c[5]^5*c[4]-273267195*c[5]^5-1411334820*c[5]^7*c[4]^ 2+6786578028*c[5]^4*c[4]^2+141133482*c[5]^6-5402829600*c[5]^5*c[4]^5-1 387083852*c[5]^6*c[4]), a[7,5] = 125/504094752*(204412582500*c[5]^3-14 4983718875*c[5]^2+31830093750*c[5]-15915046875*c[4]-91335289500*c[5]^4 +31211932500*c[4]^4-5266866226986*c[5]^2*c[4]^2-1820460486123*c[5]*c[4 ]^3-82031656500*c[4]^3+2310967713600*c[5]^4*c[4]^5+934078029066*c[5]^4 *c[4]+7467421763340*c[5]^4*c[4]^3+1341749649376*c[5]^2*c[4]+6670450787 5*c[4]^2+865860651187*c[5]*c[4]^2+10452687727292*c[5]^2*c[4]^3-9707627 699026*c[4]^4*c[5]^2+8126531994882*c[5]^3*c[4]^2-2023947071535*c[5]^3* c[4]-235124179750*c[5]*c[4]-4989217532040*c[4]^5*c[5]^3+1802395500290* c[4]^4*c[5]-6825451568040*c[5]^4*c[4]^4-16010745168654*c[5]^3*c[4]^3+1 4693253828960*c[5]^3*c[4]^4+3324646948140*c[4]^5*c[5]^2-644303205000*c [4]^5*c[5]-3795743883708*c[5]^4*c[4]^2)/c[5]/(1365*c[5]^3-8123*c[5]^4+ 6188*c[4]^4-3870*c[5]^2*c[4]^2+3937*c[5]*c[4]^3-1365*c[4]^3-341850*c[5 ]^4*c[4]^5-87720*c[5]^6*c[4]^2+341850*c[5]^6*c[4]^3+64081*c[5]^4*c[4]+ 785330*c[5]^4*c[4]^3+549500*c[5]^5*c[4]^2-1354420*c[5]^5*c[4]^3-2730*c [5]^2*c[4]+2730*c[5]*c[4]^2-32686*c[5]^2*c[4]^3+430797*c[4]^4*c[5]^2+2 9848*c[5]^3*c[4]^2+1868*c[5]^3*c[4]+328870*c[4]^5*c[5]^3-70015*c[4]^4* c[5]+1113270*c[5]^4*c[4]^4+42312*c[5]^3*c[4]^3-1272950*c[5]^3*c[4]^4-6 1880*c[4]^5*c[5]^2-86817*c[5]^5*c[4]+8772*c[5]^5-386292*c[5]^4*c[4]^2) , a[8,1] = 1/45500*(-2658951750*c[5]^3+763922250*c[5]^2+2045407000*c[5 ]^4+1302696634*c[4]^4+8406154320*c[5]^2*c[4]^2-16115752180*c[5]*c[4]^3 -38218898360*c[4]^6*c[5]^2-1064530470*c[4]^3-492806914*c[4]^5-15635211 4280*c[5]^4*c[4]^5-21394642414*c[5]^4*c[4]-41143673840*c[5]^5*c[4]^4-2 50034979034*c[5]^4*c[4]^3-5113517500*c[5]^5*c[4]^2+26501552890*c[5]^5* c[4]^3+10227035000*c[4]^6*c[5]-4318588095*c[5]^2*c[4]+254640750*c[4]^2 +5449695345*c[5]*c[4]^2+7574007107*c[5]^2*c[4]^3-65136981563*c[4]^4*c[ 5]^2-100453619283*c[5]^3*c[4]^2+24617215134*c[5]^3*c[4]-891242625*c[5] *c[4]+39039652910*c[4]^5*c[5]^3+30008405592*c[4]^4*c[5]+296293354160*c [5]^4*c[4]^4+204617310062*c[5]^3*c[4]^3-178694725858*c[5]^3*c[4]^4+901 24727216*c[4]^5*c[5]^2+16616301160*c[5]^3*c[4]^6-28602952382*c[4]^5*c[ 5]+101952051743*c[5]^4*c[4]^2+22548997200*c[5]^4*c[4]^6+22548997200*c[ 5]^5*c[4]^5)/c[5]/c[4]^2/(8683*c[5]^3-2073*c[5]^2-133146*c[5]^2*c[4]^2 -85338*c[5]*c[4]^3+8683*c[4]^3+332400*c[5]^4*c[4]^3+12438*c[5]^2*c[4]- 2073*c[4]^2+12438*c[5]*c[4]^2+480660*c[5]^2*c[4]^3-86830*c[4]^4*c[5]^2 +480660*c[5]^3*c[4]^2-85338*c[5]^3*c[4]+2073*c[5]*c[4]-1170860*c[5]^3* c[4]^3+332400*c[5]^3*c[4]^4-86830*c[5]^4*c[4]^2), a[7,3] = 125/1120210 56*(-944034000*c[5]^3+1220267000*c[5]^2-349781250*c[5]+174890625*c[4]+ 49938442842*c[5]^2*c[4]^2+24596164974*c[5]*c[4]^3+472017000*c[4]^3-997 5873180*c[5]^4*c[4]^3-11643681886*c[5]^2*c[4]-610133500*c[4]^2-1111530 6307*c[5]*c[4]^2-106264771082*c[5]^2*c[4]^3+98167009210*c[4]^4*c[5]^2- 44737205880*c[5]^3*c[4]^2+9654553272*c[5]^3*c[4]+2708821000*c[5]*c[4]+ 26871717600*c[4]^5*c[5]^3-23311475432*c[4]^4*c[5]+8957239200*c[5]^4*c[ 4]^4+100422151890*c[5]^3*c[4]^3-93646927920*c[5]^3*c[4]^4-29927619540* c[4]^5*c[5]^2+7080255000*c[4]^5*c[5]+2360085000*c[5]^4*c[4]^2)/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)/(265*c[5]*c[4]-68*c[5]-68*c[4]+15), b[6] = 3969227961/107405480*(620*c[5]*c[4]-247*c[5]-247*c[4]+123)/(12 9*c[4]-91)/(129*c[5]-91), a[9,6] = 3969227961/107405480*(620*c[5]*c[4] -247*c[5]-247*c[4]+123)/(16641*c[5]*c[4]-11739*c[4]-11739*c[5]+8281), \+ c[7] = 125/126, a[7,6] = -15434296375/566328672*(5*c[5]*c[4]+1-2*c[4]- 2*c[5])*(126*c[4]-125)*(126*c[5]-125)/(265*c[5]*c[4]-68*c[5]-68*c[4]+1 5)/(129*c[4]-91)/(129*c[5]-91)\}:" }}}{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 98 "A lengthy computation gives an \+ expression for the square of the principal error norm in terms of " } {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[5] " "6#&%\"cG6#\"\"&" }{TEXT -1 2 ". " }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 261 "errterms6_8 := PrincipalErrorTerms(6,8,'expanded'): \nsm := 0:\nfor ct to nops(errterms6_8) do\n print(ct);\n sm := sm +(simplify(subs(eB,errterms6_8[ct])))^2;\nend do:\nsm := simplify(sm): \nprin_err_norm_sqrd := unapply(%,c[2],c[4],c[5]):\nprin_err_norm_sqrd (u,v,w);" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 18 "prin_err_no rm_sqrd" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 4304 "prin_err_norm_sqrd := (u,v,w)->1/41087220088320000* (5985882089045329*v^6*w^2-884006550865800*v^3*u^2*w^2-2413505604053340 0*v^4*u^2*w^3-120202863918300*v^3*u^2*w+567772676820000*w^6*v^7*u-2739 522076200*v*u^2*w-32874264914400*v*u^2*w^2-15205553073180000*v^9*w^4*u -6050959109879400*v^5*u^2*w^2-360889289632800*v^5*u^2*w-14698967119200 00*v^9*w^2*u+26622198930600*v^2*u^2*w-272947222800*w^3*v*u+20652585746 4000*v^2*u^2*w^2+2760523365006450*v^4*u^2*w^2+299528118275400*v^4*u^2* w+222931570863600*w^3*u^2*v+2618430538415400*w^4*u^2*v^2+2210853015081 329*w^6*v^2+20257026952236760*w^4*v^5-98491704970260*w^2*v^3-697740040 5138*w^3*v+606164618664180*v^4*w^2+76773375666540*w^3*v^2+984883804802 534*w^5*v^3+7623466248997734*v^5*w^3-9992134522826468*w^6*v^3+77188813 33699160*w^5*v^4-5071447026168*w^4*v-839148872556784*w^5*v^2-100989324 49223209*w^4*v^4+29796538417032*v^4*w+71440255658800*w^7*v^3+245874459 6087740*w^4*v^3+289907110539860*v^6*w+27860245910641444*w^6*v^4-334454 542233600*w*v^7-10530355657876000*v^7*w^2-11404428312971356*w^4*v^6+89 193277178690760*w^5*v^6-48920186219421240*w^6*v^5-28667196699136868*w^ 3*v^6+11777982983171200*v^8*w^2-254798376198220*w^4*v^2-5318631556000* w^7*v^2-126065017413298*v^5*w+166574633447502*w^5*v-589804221701060*w^ 3*v^4-69253475091280000*v^8*w^3-249526845611226*w^3*v^3-23191069059967 84*v^5*w^2+51260836965352260*w^6*v^6-106881242304191920*w^5*v^7-276107 944576940*w^6*v+126786365888000*w*v^8+1797463900704080*w^7*v^5+7681372 3811908000*w^4*v^8-39989456518287732*w^5*v^5-26741127500591400*w^6*v^7 -33689253068495780*w^4*v^7+62593258355266000*w^5*v^8+58757670169345200 *w^3*v^7-487550282719780*w^7*v^4+734446318769800*w^7*v^7+4174217606406 4000*v^9*w^3+15768721705520000*v^10*w^4-59029812802400000*v^9*w^4-6847 449702912000*v^9*w^2-197642331614000*w^8*v^5+5730857954504900*w^6*v^8- 2796265951151400*w^7*v^6+16829503747200*w^4*u^2-3061853133138*w*v^3+14 964898697000*w^6-13580088631200*w^3*u^2+2739522076200*w^2*u^2+23550074 78769*w^4+1524337330880000*v^10*w^2+1457604930600*w^3*u-2406060960660* v^5-9805469011520000*v^10*w^3+26593157780000*w^8*v^4+3096473977000*v^6 +367223159384900*w^8*v^6-14124895347360000*v^9*w^5+468735818769*v^4+42 07375936800*v^4*u^2-7276446626400*w^4*u+3638223313200*v^4*u+6848805190 50*v^2*u^2-3395022157800*v^3*u^2-728802465300*v^3*u+7547600004507*w^2* v^2-4540340422800*v^5*u-11869725744660*w^5+9080680845600*w^5*u-1728536 063464800*w^3*u^2*v^2+46631538558592200*w^3*v^5*u^2-12991034925306000* w^4*u^2*v^3+40437610651254600*w^4*v^4*u^2+3556168434000*v^2*u*w-196099 103892600*v^2*u*w^3+781960193712000*w^5*u^2*v^3-64169649576000*w^5*u^2 *v^2+136210212684000*v^6*u^2*w+7973776096572600*v^6*u^2*w^2+1976248400 124600*v^3*u*w^3+60474256479000*v^2*u*w^2+1481046402695400*v^4*u*w^2-3 4876438191000*v^3*u*w-308881655013600*w^4*u^2*v-368104795947000*v^3*u* w^2+553840520085000*v^6*u^2*w^6-78983988898131000*v^5*u^2*w^4-53955364 178097000*v^6*u^2*w^3+153894022950600*v^4*u*w+10441576069092000*v^5*u^ 2*w^5+90560124734409000*v^6*u^2*w^4-4925929472100*v*u*w^2-368117225100 000*v^5*u^2*w^6+61168654980000*v^4*u^2*w^6+31633240230972000*v^7*u^2*w ^3-52076679920730000*v^7*u^2*w^4+11404164804885000*v^8*u^2*w^4+5026361 150970000*v^7*u^2*w^5+1102422533940000*v^8*u^2*w^2-12802369024170000*v ^6*u^2*w^5+1371795333460200*w^5*v^2*u-4012544000250000*v^4*u^2*w^5-709 1455267260000*v^8*u^2*w^3-4952173445856000*v^7*u^2*w^2-487807194834000 *w^4*v^2*u+65785665886920000*v^8*w^4*u+9455273689680000*v^9*w^3*u+6602 897927808000*v^8*w^2*u+8046506364183000*v^3*u^2*w^3+98288667658800*w^4 *v*u+2307208651020000*w^6*v^5*u-7153708510080000*v^7*w^5*u-23008351093 20000*w^6*v^6*u-774492048360000*w^5*v^8*u+25666072723944000*w^5*v^6*u- 164301644052000*v^7*u*w-10843346249668800*w^2*v^7*u+434905224656400*v^ 6*u*w-41615825013936000*v^8*w^3*u-106883800188912000*w^4*v^7*u-3700104 59300400*w*v^5*u+70435383815403000*w^3*v^7*u+8689849698882600*w^2*v^6* u-4282311737839500*w^2*v^5*u+31115743922489400*w^3*v^5*u-1013583639951 000*w^6*v^4*u-34648389909136800*w^4*v^5*u+83733371585631000*w^4*v^6*u- 60729365390178600*w^3*v^6*u-167021833636800*w^5*v*u-10032892441536600* w^3*v^4*u+18079926255369000*w^5*v^4*u-29981754783315000*w^5*v^5*u-6411 818018118600*w^5*v^3*u+432373453089300*w^4*v^3*u+6483848711236200*w^4* v^4*u-17311972860000*w^6*v^2*u+211644787572000*w^6*v^3*u)/(10*w^3*v^2+ 10*w^2*v^3-30*w^2*v^2+6*w^2*v-w^2+6*w*v^2+w*v-v^2)^2:" }}}{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 70 "c[2] = 26/291, c[4] = 22/103, c[5] = 216/385, c[6] = 91/129;\neval f(%);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6&/&%\"cG6#\"\"##\"#E\"$\"H/&F% 6#\"\"%#\"#A\"$.\"/&F%6#\"\"&#\"$;#\"$&Q/&F%6#\"\"'#\"#\"*\"$H\"" }} {PARA 11 "" 1 "" {XPPMATH 20 "6&/&%\"cG6#\"\"#$\"+/zqM*)!#6/&F%6#\"\"% $\"+IB#f8#!#5/&F%6#\"\"&$\"+5'*Q5cF1/&F%6#\"\"'$\"+mNEaqF1" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 46 "Preliminary inv estigations suggest that when " }{XPPEDIT 18 0 "c[2] = 13/144;" "6#/& %\"cG6#\"\"#*&\"#8\"\"\"\"$W\"!\"\"" }{TEXT -1 3 ", " }{XPPEDIT 18 0 "c[4] = 25/117;" "6#/&%\"cG6#\"\"%*&\"#D\"\"\"\"$<\"!\"\"" }{TEXT -1 7 " and " }{XPPEDIT 18 0 "c[5] = 23/41;" "6#/&%\"cG6#\"\"&*&\"#B\"\" \"\"#T!\"\"" }{TEXT -1 49 " the principal error norm is close to a mi nimum." }}{PARA 0 "" 0 "" {TEXT -1 158 "Taking these values as startin g values cycling around the nodes with a one dimensional minimization \+ procedure gives very slow convergence towards the minimum." }}{PARA 0 "" 0 "" {TEXT -1 1 "." }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 493 "Di gits := 30:\nc_2 := 13/144: c_4 := 25/117: c_5 := 23/41:\nfor ct to 12 0000 do\n c_2 := findmin(prin_err_norm_sqrd(c2,c_4,c_5),c2=\{0.03,c_ 2,0.13\},convergence=location)[1];\n c_4 := findmin(prin_err_norm_sq rd(c_2,c4,c_5),c4=\{0.2,c_4,0.23\},convergence=location)[1];\n mn := findmin(prin_err_norm_sqrd(c_2,c_4,c5),c5=\{0.5,c_5,0.62\},convergenc e=location); \n c_5 := mn[1]:\n if `mod`(ct,1000)=0 then\n pr int(c[2]=c_2,c[4]=c_4,c[5]=c_5);\n print(mn[2]);\n end if;\nend do:\nDigits := 10:" }}{PARA 11 "" 1 "" {XPPMATH 20 "6%/&%\"cG6#\"\"#$ \"?\\BD.2Bd3<1'erA3*!#J/&F%6#\"\"%$\"?w%y>E'[B'es<.Kz9#!#I/&F%6#\"\"&$ \"?)podZtS+D&zJHkKcF1" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#$\"?te]b*Gx/s 0SyPH'>!#R" }}{PARA 11 "" 1 "" {XPPMATH 20 "6%/&%\"cG6#\"\"#$\"?TnawUn N,t^=#Ho6*!#J/&F%6#\"\"%$\"?#y'43(z8/'4FK\"p]:#!#I/&F%6#\"\"&$\"?*ROyA \"zmF]+r<%fk&F1" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#$\"?\"oq;sY=\"HM9yt vt=!#R" }}{PARA 11 "" 1 "" {XPPMATH 20 "6%/&%\"cG6#\"\"#$\"?Rk\"=8?))> 'QI2*\\$Q\"*!#J/&F%6#\"\"%$\"?j(R[$*R_Xoay1_&f@!#I/&F%6#\"\"&$\"?a9%o$ p!*='3L73XVl&F1" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#$\"?fWs=t5#4Ejsvc_$ =!#R" }}{PARA 11 "" 1 "" {XPPMATH 20 "6%/&%\"cG6#\"\"#$\"?O/JXe8+ixEl \\]_\"*!#J/&F%6#\"\"%$\"?;\">c'Q9()>@F0q^i@!#I/&F%6#\"\"&$\"?S%Q!RI6W* p()\\@C*fcF1" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#$\"?\\H'3\\y*H&4QfRrt \"=!#R" }}{PARA 259 "" 0 "" {TEXT -1 1 ":" }}{PARA 259 "" 0 "" {TEXT -1 1 ":" }}{PARA 11 "" 1 "" {XPPMATH 20 "6%/&%\"cG6#\"\"#$\"?Nk7lH-7?S I%Q]g=*!#J/&F%6#\"\"%$\"?!oV#4UB#)38Qogfp@!#I/&F%6#\"\"&$\"?bzB`oXf8! \\c*QJtcF1" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#$\"?[LM;?USzd[+:d*z\"!#R " }}{PARA 11 "" 1 "" {XPPMATH 20 "6%/&%\"cG6#\"\"#$\"?*3>*QIr7?SI%Q]g= *!#J/&F%6#\"\"%$\"?t&y'*owB)38Qogfp@!#I/&F%6#\"\"&$\"?1dUGtsf8!\\c*QJt cF1" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#$\"?&GKlQS.%zd[+:d*z\"!#R" }} {PARA 11 "" 1 "" {XPPMATH 20 "6%/&%\"cG6#\"\"#$\"?>p+76:8?SI%Q]g=*!#J/ &F%6#\"\"%$\"?.]UM:[#)38Qogfp@!#I/&F%6#\"\"&$\"?<*)*QOE*f8!\\c*QJtcF1 " }}{PARA 11 "" 1 "" {XPPMATH 20 "6#$\"?a.GBDpSzd[+:d*z\"!#R" }}{PARA 11 "" 1 "" {XPPMATH 20 "6%/&%\"cG6#\"\"#$\"?!HZwPbO,-/VQ]g=*!#J/&F%6# \"\"%$\"?0J#RweD)38Qogfp@!#I/&F%6#\"\"&$\"?TC-))H2g8!\\c*QJtcF1" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#$\"?qK&=J95%zd[+:d*z\"!#R" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 78 "The following gra phs give a visual check that we have found a (local) minimum." }} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 380 "c_2 := .91860503843e-1: pp \+ := .17995715005e-9:\np1 := evalf[30](plot(prin_err_norm_sqrd(c[2],.216 95960684,.567331389565),c[2]=0.087..0.0967,\n color=COLOR(RGB,.5,0, .9))):\np2 := plot([[[c_2,pp]]$4],style=point,symbol=[circle$2,diamond ,cross],symbolsize=[12,10$3],\n color=[black,red$3]):\nplots[di splay]([p1,p2],font=[HELVETICA,9],view=[0.087..0.0967,1.799e-10..1.803 3e-10]);" }}{PARA 13 "" 1 "" {GLPLOT2D 388 361 361 {PLOTDATA 2 "6*-%'C URVESG6$7S7$$\"#()!\"$$\"?eHNz*4j/TBGl#G.=!#R7$$\"?nmmmmmmm;aRK9@()!#J $\"?U)4+p&R*p4rU!o'H!=F-7$$\"?LLLLLLL$e9#f)R&R()F1$\"?uelAH:T7yz7Mq-=F -7$$\"?nmmmmmmm\"z?yG-w)F1$\"?::C!=F-7$$\"?nmmmmmmmTbW[0\"y )F1$\"?Ay,))F1$\"?$H[l.3(367:96*= !=F-7$$\"?nmmmmmm;zk!e)*4#))F1$\"?FU^>g8v'H5Z\"\\m,=F-7$$\"?+++++++]78 :j*3%))F1$\"?'3LX[)HGY+NCHW,=F-7$$\"?nmmmmmm;zf!eu9'))F1$\"?PZjp![t4Om fUE7!=F-7$$\"?+++++++]iI^o)>)))F1$\"?T2PHIDPbxJiQ-,=F-7$$\"?LLLLLLLL$3 IC'3.*)F1$\"?DlS07$\\$)*\\$GHH3!=F-7$$\"?nmmmmmmmTq)fq;#*)F1$\"?!=J>-' zxUo?,&p1!=F-7$$\"?++++++++]_\"=#fU*)F1$\"?hCUOVyuu9_(f-0!=F-7$$\"?+++ +++++]Pq'*fj*)F1$\"?'=*f5[3k0PJX)[.!=F-7$$\"?++++++++]ZyU%Q)*)F1$\"?mg FRmdK!o+cz8-!=F-7$$\"?nmmmmmm;z4A$GA+*F1$\"?*=RrT%f.BWA8B5+=F-7$$\"?LL LLLLLL$3\"G()3C!*F1$\"?ajqmjNy;G!4+=F-7$$\"?nmmmmmm;zp$*fv'Q*F1$\" ?\"H)fUVIyqP1*H/-!=F-7$$\"?++++++++]s(*[]2%*F1$\"?];rh\\0\")G7;**=M+=F -7$$\"?LLLLLLL$e9%**z-F%*F1$\"?#*=/s1\"\\lUA5s$[+=F-7$$\"?+++++++]i+m_ VZ%*F1$\"?]GQeat]WDXmZk+=F-7$$\"?LLLLLLLL$3#*4(Qn%*F1$\"?+b55mgzl_@n[ \"3!=F-7$$\"?+++++++]7`>[F)[*F1$\"?iQcI9W`/mOZj+,=F-7$$\"?mmmmmmmmmww@ R3&*F1$\"?i-9i)o9Fm8Ss.7!=F-7$$\"?mmmmmmmmT:!3l*G&*F1$\"?XdhyKcn@CM?(= 9!=F-7$$\"?LLLLLLL$e9&HwO\\&*F1$\"?:^z#G.'z!y1\"p]k,=F-7$$\"?+++++++++ X`a6o&*F1$\"?)y1(*H4N#>I@!fk=!=F-7$$\"?LLLLLLLL3PvDg*e*F1$\"?j(ops-l#) )[0n(H@!=F-7$$\"?mmmmmmmmmO84#)3'*F1$\"?Q/lf^%H-MLLBzB!=F-7$$\"?++++++ +]73q;JH'*F1$\"?cL?;48?()RG)*zl-=F-7$$\"?+++++++](=)QR#*['*F1$\"??1)4c Lw[G`r;PH!=F-7$$\"$n*!\"%$\"?IM!4'4b1;xSV1D.=F--%&COLORG6&%$RGBG$\"\"& !\"\"$\"\"!Fb[l$\"\"*F`[l-F$6&7#7$$\"3#*****H%Q]g=*!#>$\"31++]+:d*z\"! #F-%'COLOURG6&F][lFb[lFb[lFb[l-%'SYMBOLG6$%'CIRCLEG\"#7-%&STYLEG6#%&PO INTG-F$6&Fg[l-F`\\l6&F][l$\"*++++\"!\")Fa[lFa[l-Fc\\l6$Fe\\l\"#5Fg\\l- F$6&Fg[lF]]l-Fc\\l6$%(DIAMONDGFd]lFg\\l-F$6&Fg[lF]]l-Fc\\l6$%&CROSSGFd ]lFg\\l-%%FONTG6$%*HELVETICAGFd[l-%+AXESLABELSG6%Q%c[2]6\"Q!Fg^l-F`^l6 #%(DEFAULTG-%%VIEWG6$;F(Fez;$\"%*z\"!#8$\"&L!=!#9" 1 2 0 1 10 0 2 9 1 4 2 1.000000 45.000000 45.000000 0 0 "Curve 1" "Curve 2" "Curve 3" "Cu rve 4" "Curve 5" }}}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 396 "c_4 := .21695960684: pp := .17995715005e-9: \np1 := evalf[30](plot(prin_err_norm_sqrd(.91860503843e-1,c[4],.567331 389565),c[4]=0.216954..0.2169652,\n color=COLOR(RGB,0,.7,.2))):\np2 := plot([[[c_4,pp]]$4],style=point,symbol=[circle$2,diamond,cross],sy mbolsize=[12,10$3],\n color=[black,cyan$3]):\nplots[display] ([p1,p2],font=[HELVETICA,9],view=[0.216954..0.2169652,1.799e-10..1.803 2e-10]);" }}{PARA 13 "" 1 "" {GLPLOT2D 368 346 346 {PLOTDATA 2 "6*-%'C URVESG6$7S7$$\"'ap@!\"'$\"?&G!GB*[QZvVN*RB.=!#R7$$\"?nmmmmmmm7GTCap@!# I$\"?A?kiv'py0-dPAH!=F-7$$\"?LLLLLLLVqUlXap@F1$\"?jD5q^FUfm7#[iE!=F-7$ $\"?nmmmmmmE6Dapap@F1$\"?U$QFPfr.Frxq#Q-=F-7$$\"?nmmmmmm'))4*e$\\&p@F1 $\"?a`35,-#=&y\\^W6-=F-7$$\"?LLLLLLLB(R@v^&p@F1$\"?NZ^ge5O['**ezg=!=F- 7$$\"?nmmmmmmO,(4(Rbp@F1$\"?u572oPH\")Ql/vj,=F-7$$\"?+++++++50Woibp@F1 $\"?(G@(>%)fi#>`*[$=9!=F-7$$\"?nmmmmmmcw[W'e&p@F1$\"?'e(3=avp!)HF(f/7! =F-7$$\"?+++++++!z9H,h&p@F1$\"?2)ov$>b#Rm\\')e/5!=F-7$$\"?LLLLLLL8K8\\ Mcp@F1$\"?dV=qv[@*)=MhC\"3!=F-7$$\"?nmmmmmmEd&\\fl&p@F1$\"?wZo#z`\\jTC #oYl+=F-7$$\"?+++++++SOk5!o&p@F1$\"?P#z,w=,9/^v%)*[+=F-7$$\"?++++++++1 DO/dp@F1$\"?@J*=[)=$[U9N,Q.!=F-7$$\"?+++++++gBxtFdp@F1$\"?],(pA=#Q)32A l/-!=F-7$$\"?nmmmmmmcgY'*[dp@F1$\"?3;SE<$oSz>/d%4+=F-7$$\"?LLLLLLLtTb? udp@F1$\"?xo+&>\"y=p8oKt(**z\"F-7$$\"?LLLLLLL$*yye&z&p@F1$\"?(4UHQ9+-: ;7j*))*z\"F-7$$\"?+++++++I?1Y?ep@F1$\"?wNvTmQAmRZ@5!)*z\"F-7$$\"?LLLLL LL$4@xC%ep@F1$\"?h_,ct'[=VS;iM(*z\"F-7$$\"?+++++++I+Hjmep@F1$\"?u!)plv (zlzDpxu'*z\"F-7$$\"?+++++++!*G[j*)ep@F1$\"?Y\"p\"Q.n.TN9W/j*z\"F-7$$ \"?LLLLLLLLG\\j8fp@F1$\"?Cw&G`bjC8P4N(f*z\"F-7$$\"?LLLLLLLj6XnNfp@F1$ \"?%zRdyS))*z \"F-7$$\"?LLLLLLL$pEeo9'p@F1$\"?u?azKoU=\"*e2n(**z\"F-7$$\"?++++++++U; /rhp@F1$\"?70*eI-!)y.gA())3+=F-7$$\"?nmmmmmm;u`&H>'p@F1$\"?J=wv&4RsgYO M--!=F-7$$\"?+++++++grG\"p@'p@F1$\"?YBKCb]p;>yv#R.!=F-7$$\"?LLLLLLLj4] XRip@F1$\"?&\\\"=,=F-7$$\"?nmmmmmmYcb:djp@F1$\"?]lQ[q^P\"\\&=I4T,=F-7$$\"?LLLLL LLBRJr!Q'p@F1$\"?8/+(H)e0umrRij,=F-7$$\"?+++++++?6,O-kp@F1$\"?[')\\!)[ kwpSEsZ&=!=F-7$$\"?LLLLLLL$R)*prU'p@F1$\"?h\"z]pT+j'R*3y=@!=F-7$$\"?nm mmmmm')H-O\\kp@F1$\"?9@%Qgo$)QlKVIZ'p@F1$\"?iX !*=I>%p5u/xWE!=F-7$$\"?+++++++5aZm&\\'p@F1$\"?^l^ " 0 "" {MPLTEXT 1 0 388 "c_5 := .567331389565: pp := .17995715005e-9:\np1 := \+ evalf[30](plot(prin_err_norm_sqrd(.91860503843e-1,.21695960684,c[5]),c [5]=0.567321..0.5673418,\n color=COLOR(RGB,0.6,.2,.2))):\np2 := pl ot([[[c_5,pp]]$4],style=point,symbol=[circle$2,diamond,cross],symbolsi ze=[12,10$3],color=[black,green$3]):\nplots[display]([p1,p2],font=[HEL VETICA,9],view=[0.567321..0.5673418,1.799e-10..1.8031e-10]);" }}{PARA 13 "" 1 "" {GLPLOT2D 389 359 359 {PLOTDATA 2 "6*-%'CURVESG6$7S7$$\"'@t c!\"'$\"?c^Bq&[/*ynV[e2.=!#R7$$\"?nmmmmmmm!3Q`9Kn&!#I$\"?\">*)>(4GB+;P _kx-=F-7$$\"?LLLLLLLB-ly%=Kn&F1$\"?YU86j65*RV/%o_-=F-7$$\"?nmmmmmm1y.: HAtcF1$\"?KmP]-DaIfkI#eA!=F-7$$\"?nmmmmmmYE$3QFKn&F1$\"?0?wykKy9.L!y+? !=F-7$$\"?LLLLLLLVBSD=BtcF1$\"?v$RNbc1\"olKTuv,=F-7$$\"?nmmmmmm'R(3YfB tcF1$\"?c_v$HvT([MQ=La,=F-7$$\"?+++++++!4=G@SKn&F1$\"?!Hs+su+k:lxCL8!= F-7$$\"?nmmmmmmwqZDYCtcF1$\"?\\_1;!>r$3B_[%G6!=F-7$$\"?+++++++5Y)R-\\K n&F1$\"?z[d)e*om%R(>3p$4!=F-7$$\"?LLLLLLL`-R[NDtcF1$\"?OJkO&>:_Uv*4Iv+ =F-7$$\"?nmmmmmm1#*[LvDtcF1$\"?Of**)HY\"RF0la?g+=F-7$$\"?+++++++gnw>?E tcF1$\"?F,&o%)GdN:x^YW/!=F-7$$\"?++++++++aYClEtcF1$\"?xE,&G$)f6$G#yQ*H +=F-7$$\"?+++++++Ssdl3FtcF1$\"?V\"=5\"fJZI5$H0s,!=F-7$$\"?nmmmmmmwEs2[ FtcF1$\"?gDFsAD^S**yDq1+=F-7$$\"?LLLLLLL$*[J&\\zKn&F1$\"?'\\Y**>1!>Qrw t_&**z\"F-7$$\"?LLLLLLLt.KmMGtcF1$\"?'*>Dq!G#['Q5wwr)*z\"F-7$$\"?+++++ ++qPa&3)GtcF1$\"?f>n\")y-*oVX&4vy*z\"F-7$$\"?LLLLLLLt\"RV<#HtcF1$\"?m% 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ls=[`c[4]`,`c[5]`,``],\n orientation=[-78,56],color=COLOR(RGB,0,.75 ,85),font=[HELVETICA,9]);\nDigits := 10:" }}{PARA 13 "" 1 "" {GLPLOT3D 525 430 430 {PLOTDATA 3 "6'-%%GRIDG6&;$\"'ap@!\"'$\"(_'p@!\" (;$\"'@tcF)$\"(=Mn&F,X,%)anythingG6\"6\"[gl'!%\"!!#\\bm\":\":3EEC30FCB 40BB4F93EEC3185D9CA67C43EEC32279CE65D8E3EEC32E1FD2AEF4E3EEC33B4EE577BA A3EEC34A070D8634D3EEC35A47A3C3D1E3EEC36C110AD5DF43EEC37F6224AA15A3EEC3 943B0BE07163EEC3AA9B29E03AA3EEC3C28249DB3BC3EEC3DBEFE320D403EEC3F6E36C 11BF23EEC4135CE43566C3EEC4315B50ECB093EEC450DEADE5FEF3EEC471E66F5CE1E3 EEC49471D8CACC23EEC4B880A9AFE743EEC4DE121A553363EEC50526182F5B93EEC52D BB7D124C33EEC557D233CA2BF3EEC5836978E29AB3EEC308A1B19B9753EEC30FA1AE20 3E63EEC3182B5AC81FD3EEC3223F4ABBFA83EEC32DDC983221C3EEC33B033EEC9A03EE C349B2D55EE0D3EEC359EB5996AFB3EEC36BAC2CF54453EEC37EF51D0E8123EEC393C5 89216DB3EEC3AA1D6F7708B3EEC3C1FC5694A173EEC3DB61E6D73573EEC3F64D2B569E 93EEC412BE74AADF03EEC430B54DB69243EEC4503046745553EEC4713001DAB603EEC4 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5193EEC2FC9DECDA8703EEC2FE7B1FDC6C53EEC301E3E89E2463EEC306D74C7A16E3EE C30D55D8313243EEC3155EE1FC9A53EEC31EF2205F8283EEC32A0FA0A665D3EEC336B7 47160D83EEC344E7E8B9A363EEC354A2376B8A13EEC365E587638D03EEC378B128EDCE 43EEC3718D8CCD2863EEC35F4530B0B313EEC34E893E0839B3EEC33F59BA6C0743EEC3 31B6286A04A3EEC3259F38950FF3EEC31B149A6652A3EEC31216211C2693EEC30AA407 E565C3EEC304BDDA627F53EEC30063B8AB0473EEC2FD95812B87B3EEC2FC52CEDC5833 EEC2FC9C356D7303EEC2FE7091FBDE03EEC301D05B3E6073EEC306BB29469DB3EEC30D 310ED93003EEC315316E93E963EEC31EBC3D836CE3EEC329D148A0AA83EEC3367004E3 0A53EEC344987F2063A3EEC3544A5856AFD3EEC365854D989193EEC385DE92006CF3EE C372009CFE7E53EEC35FAF283F89A3EEC34EEA03B9A913EEC33FB1470E4BC3EEC33204 EC5F08F3EEC325E4FC8D09D3EEC31B516D4C4313EEC3124A01E30C03EEC30ACE98B3F9 B3EEC304DF7DC70243EEC3007CA78F6143EEC2FDA57F2B7BE3EEC2FC5A2CC64863EEC2 FC9A83E99F23EEC2FE6649B660B3EEC301BD44F46F93EEC3069F803FEDC3EEC30D0C83 688043EEC3150465D7DC73EEC31E866D9E2E63EEC329931A937273EEC33629699A9283 EEC3444975E82FC3EEC353F2D3437093EEC39BCE8254FD73EEC3865B23439393EEC372 7412D1D813EEC36019556B0343EEC34F4B39FFD083EEC340094C380B03EEC33253BFF4 8E43EEC3262B0A3967A3EEC31B8E303F82C3EEC3127DE4049CC3EEC30AF9B29A54C3EE C30501B6D45D73EEC3009598ABA2B3EEC2FDB59A3AAE43EEC2FC6190559403EEC2FC99 3E8E4AF3EEC2FE5C130A3933EEC301AA4AEF7C03EEC30683E0DB0CB3EEC30CE8423FF1 53EEC314D78C2F2D43EEC31E511D778353EEC3295524A7B9A3EEC335E2FE4755B3EEC3 43FA9649B1B3EEC3B35D8AD786B3EEC39C54D5A0E413EEC386D841AFBB83EEC372E837 43D7B3EEC360845299D923EEC34FAD0408D793EEC34061CF9642A3EEC332A37B314383 EEC3267160BAE923EEC31BCB8DBC66A3EEC312B219F446E3EEC30B253A272F63EEC305 243D9C50B3EEC300AF4883A3E3EEC2FDC6965719C3EEC2FC69786EEC63EEC2FC982C9E 7523EEC2FE5262625BD3EEC3019802B97603EEC30668D54E8B83EEC30CC49E991BD3EE C314AB2A7DFE43EEC31E1C595E0723EEC32917908C34B3EEC3359D124F79A3EEC3CC8B FA0CADD3EEC3B3EDB1AD3193EEC39CDB70AA76C3EEC387559624E0F3EEC3735C3BC47C A3EEC360EF4053AF53EEC3500EBFCCADD3EEC340BABE417273EEC332F2F693D343EEC3 26B7E494C5C3EEC31C0965A1D063EEC312E6A77F83A3EEC30B50B9EDF543EEC30546CB 28C203EEC300C9226BFA13EEC2FDD75B45CF83EEC2FC71894A5D63EEC2FC9777C3BA63 EEC2FE4910FBD9A3EEC30185EAC84733EEC3064E1B5CEA93EEC30CA127F713D3EEC314 7EF0600723EEC31DE768139CD3EEC328DA703D9BC3EEC3E75969D8A743EEC3CD256008 C1B3EEC3B47DF64EAD43EEC39D6251941FB3EEC387D36E4E7613EEC373D0D38357E3EE C3615ABCCEC643EEC35070E78C3CA3EEC34113C7C332E3EEC3334312A5C6C3EEC326FE EEE7A463EEC31C4718C69233EEC3131BE4CD8943EEC30B7CD6D60EB3EEC30569F063CE 73EEC300E37E84A573EEC2FDE8E8C29573EEC2FC7A45989403EEC2FC975ABD5593EEC2 FE3FC8C9C0B3EEC30173EF33B213EEC306336763E783EEC30C7DEB967CA3EEC3145320 232E33EEC31DB2F97DC373EEC403C5C9B28F83EEC3E7FC93DEBC63EEC3CDBF71A8D4A3 EEC3B50E9A232D93EEC39DE9EFBDF033EEC38851D3C8D1E3EEC37445E02E3243EEC361 C6A507FEB3EEC350D3C9151AA3EEC3416D64737893EEC33393B12A82B3EEC327461689 F4A3EEC31C859D8DE6C3EEC3135131D72693EEC30BA92B835ED3EEC3058D7D0DC383EE C300FDF875A383EEC2FDFA8BD68EB3EEC2FC82E3AFF8D3EEC2FC970BF628A3EEC2FE37 1B910A13EEC3016293A0FA13EEC3061953A49D93EEC30C5B390B6073EEC31427B6F866 E3EEC421D16456FE33EEC404729E542793EEC3E8A0666E9193EEC3CE59F43B9E73EEC3 B59F9D25DB13EEC39E71DE5185C3EEC388D07512F6A3EEC374BB390FB833EEC36232BF 458443EEC35136C25AB8E3EEC341C7511C6E13EEC333E47DCF1063EEC3278E15454F13 EEC31CC43F137D93EEC31386F2191163EEC30BD626DF0D63EEC305B139500BA3EEC301 18BDBA4163EEC2FE0CA06A5913EEC2FC8C26BF7BD3EEC2FC979855B6B3EEC2FE2ECFA8 B763EEC3015182E25D23EEC305FF8CF2B0E3EEC30C388FAB366-%&COLORG6&%$RGBG$ \"\"!F;$\"#v!\"#$\"#&)F;-%%FONTG6$%*HELVETICAG\"\"*-%*AXESSTYLEG6#%$BO XG-%+AXESLABELSG6%%%c[4]G%%c[5]G%!G-%+PROJECTIONG6%$!#yF;$\"#cF;\"\"\" " 1 2 0 1 10 0 2 1 1 2 2 1.000000 56.000000 -78.000000 1 0 "Curve 1" } }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 36 "#----- ------------------------------" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 152 "nds := [c[2]=.91860503843e- 1,c[4]=.21695960684,c[5]=.567331389565]:\nevalf[10](%);\nfor dgt from \+ 7 by -1 to 4 do\n map(convert,nds,rational,dgt);\nend do;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#7%/&%\"cG6#\"\"#$\"+%Q]g=*!#6/&F&6#\"\"%$\"+ ogfp@!#5/&F&6#\"\"&$\"+'*QJtcF2" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#7%/ &%\"cG6#\"\"##\"#z\"$g)/&F&6#\"\"%#\"$%G\"%48/&F&6#\"\"&#\"$b%\"$-)" } }{PARA 11 "" 1 "" {XPPMATH 20 "6#7%/&%\"cG6#\"\"##\"#z\"$g)/&F&6#\"\"% #\"#()\"$,%/&F&6#\"\"&#\"$)>\"$\\$" }}{PARA 11 "" 1 "" {XPPMATH 20 "6# 7%/&%\"cG6#\"\"##\"#N\"$\"Q/&F&6#\"\"%#\"#B\"$1\"/&F&6#\"\"&#\"#f\"$/ \"" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#7%/&%\"cG6#\"\"##\"\"*\"#)*/&F&6 #\"\"%#\"#=\"#$)/&F&6#\"\"&#\"#@\"#P" }}}{PARA 0 "" 0 "" {TEXT -1 0 " " }}{PARA 0 "" 0 "" {TEXT -1 56 "The minimum value for the principal e rror norm is . . . " }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 90 "evalf [25](prin_err_norm_sqrd(.91860503843e-1,.21695960684,.567331389565)): \nevalf(sqrt(%));" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#$\"+%3\"[T8!#9" } }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 9 "Setting \+ " }{XPPEDIT 18 0 "c[2] = 9/98;" "6#/&%\"cG6#\"\"#*&\"\"*\"\"\"\"#)*! \"\"" }{TEXT -1 7 " and " }{XPPEDIT 18 0 "c[4] = 23/106;" "6#/&%\"cG 6#\"\"%*&\"#B\"\"\"\"$1\"!\"\"" }{TEXT -1 46 " the principal error no rm is a minimum when " }{XPPEDIT 18 0 "c[5] = 80/141;" "6#/&%\"cG6#\" \"&*&\"#!)\"\"\"\"$T\"!\"\"" }{TEXT -1 19 " (approximately). " }} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 112 "mn := evalf[20](findmin(pri n_err_norm_sqrd(9/98,23/106,c5),c5=0.567..0.568)):\nc[5]=mn[1];\nconve rt(%,rational,6);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/&%\"cG6#\"\"&$\" 5<:ytU\\Bstc!#?" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/&%\"cG6#\"\"&#\"#! )\"$T\"" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 9 "Setting " }{XPPEDIT 18 0 "c[4] = 23/106;" "6#/&%\"cG6#\"\"%*&\"#B \"\"\"\"$1\"!\"\"" }{TEXT -1 7 " and " }{XPPEDIT 18 0 "c[5] = 80/141 ;" "6#/&%\"cG6#\"\"&*&\"#!)\"\"\"\"$T\"!\"\"" }{TEXT -1 46 " the prin cipal error norm is a minimum when " }{XPPEDIT 18 0 "c[2] = 6/65;" "6 #/&%\"cG6#\"\"#*&\"\"'\"\"\"\"#l!\"\"" }{TEXT -1 19 " (approximately). " }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 111 "mn := evalf[20](findm in(prin_err_norm_sqrd(c2,23/106,80/141),c2=0.08..0.1)):\nc[2]=mn[1];\n convert(%,rational,4);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/&%\"cG6#\" \"#$\"5Tvme#RHYWB*!#@" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/&%\"cG6#\"\" ##\"\"'\"#l" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 6 "When " }{XPPEDIT 18 0 "c[2] = 6/65;" "6#/&%\"cG6#\"\"#*&\"\"'\" \"\"\"#l!\"\"" }{TEXT -1 3 ", " }{XPPEDIT 18 0 "c[4] = 23/106;" "6#/& %\"cG6#\"\"%*&\"#B\"\"\"\"$1\"!\"\"" }{TEXT -1 7 " and " }{XPPEDIT 18 0 "c[5] = 80/141;" "6#/&%\"cG6#\"\"&*&\"#!)\"\"\"\"$T\"!\"\"" } {TEXT -1 65 ", the principal error norm is given (approximately) as f ollows. " }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 66 "evalf[20](prin_e rr_norm_sqrd(6/65,23/106,80/141)):\nevalf(sqrt(%));" }}{PARA 11 "" 1 " " {XPPMATH 20 "6#$\"+yRjT8!#9" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }} {PARA 0 "" 0 "" {TEXT -1 54 "#---------------------------------------- -------------" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}} {PARA 0 "" 0 "" {TEXT -1 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 1967 "ee := \{c[2]=6/65,\nc[3] =23/159,\nc[4]=23/106,\nc[5]=80/141,\nc[6]=91/129,\nc[7]=125/126,\nc[8 ]=1,\nc[9]=1,\n\na[2,1]=6/65,\na[3,1]=9499/303372,\na[3,2]=34385/30337 2,\na[4,1]=23/424,\na[4,2]=0,\na[4,3]=69/424,\na[5,1]=967715920/148290 3909,\na[5,2]=0,\na[5,3]=-1226377600/494301303,\na[5,4]=3552780800/148 2903909,\na[6,1]=-46804789259240403389999/30226267716384768392160,\na[ 6,2]=0,\na[6,3]=410550459697310264150/62971391075801600817,\na[6,4]=-4 873358856094925562114476/989343525191918950435887,\na[6,5]=73159850521 39691605041/11082753205258491358240,\na[7,1]=3814047137877227803878762 5/10601915981402462257077264,\na[7,2]=0,\na[7,3]=-20069727161537852123 05625/135921999761570028936888,\na[7,4]=224643334869039842666038418450 00/1782850965624830561723340121359,\na[7,5]=-3335402692827978879904464 125/2252537921258500779474698736,\na[7,6]=627869105822046875/603327424 485556953,\na[8,1]=146733704741261404039925503/35929601425079440010660 000,\na[8,2]=0,\na[8,3]=-332175497511735087990/19741539244549142863,\n a[8,4]=6401812090729586424603870720224/446765171619464803439374766203, \na[8,5]=-4265990808672245834945068273/2437965633092800464693941600,\n a[8,6]=330884756892468554/282253958465901877,\na[8,7]=-287499333445221 24/3047037469905514375,\na[9,1]=6205673/96600000,\na[9,2]=0,\na[9,3]=0 ,\na[9,4]=43180775629715/129606111568587,\na[9,5]=10385429951981/39850 344008000,\na[9,6]=315039093349/1710630556040,\na[9,7]=1569042174483/1 579405853125,\na[9,8]=-9646151/11543640,\n\nb[1]=6205673/96600000,\nb[ 2]=0,\nb[3]=0,\nb[4]=43180775629715/129606111568587,\nb[5]=10385429951 981/39850344008000,\nb[6]=315039093349/1710630556040,\nb[7]=1569042174 483/1579405853125,\nb[8]=-9646151/11543640,\n\n`b*`[1]=502684387790020 3/80720277237600000,\n`b*`[2]=0,\n`b*`[3]=0,\n`b*`[4]=1135273635260524 2548485/33323272041738045062256,\n`b*`[5]=581508736832735921107/256149 9296336591776000,\n`b*`[6]=323536238315689614389/142942621878528616144 0,\n`b*`[7]=310826349095675205429/406084020815219450000,\n`b*`[8]=-455 712696908507/742001768698080,\n`b*`[9]=-1/132\}:" }}}{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#\"\"($\") #QLN\"!\"(" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/&%\"CG6#\"\"($\")QHc8! \"(" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}}{PARA 0 "" 0 "" {TEXT -1 52 "#----------------- ----------------------------------" }}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 26 "absolute stability regions" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}{SECT 1 {PARA 4 "" 0 " " {TEXT -1 35 "coefficients of the combined scheme" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1967 "ee := \{c[ 2]=6/65,\nc[3]=23/159,\nc[4]=23/106,\nc[5]=80/141,\nc[6]=91/129,\nc[7] =125/126,\nc[8]=1,\nc[9]=1,\n\na[2,1]=6/65,\na[3,1]=9499/303372,\na[3, 2]=34385/303372,\na[4,1]=23/424,\na[4,2]=0,\na[4,3]=69/424,\na[5,1]=96 7715920/1482903909,\na[5,2]=0,\na[5,3]=-1226377600/494301303,\na[5,4]= 3552780800/1482903909,\na[6,1]=-46804789259240403389999/30226267716384 768392160,\na[6,2]=0,\na[6,3]=410550459697310264150/629713910758016008 17,\na[6,4]=-4873358856094925562114476/989343525191918950435887,\na[6, 5]=7315985052139691605041/11082753205258491358240,\na[7,1]=38140471378 772278038787625/10601915981402462257077264,\na[7,2]=0,\na[7,3]=-200697 2716153785212305625/135921999761570028936888,\na[7,4]=2246433348690398 4266603841845000/1782850965624830561723340121359,\na[7,5]=-33354026928 27978879904464125/2252537921258500779474698736,\na[7,6]=62786910582204 6875/603327424485556953,\na[8,1]=146733704741261404039925503/359296014 25079440010660000,\na[8,2]=0,\na[8,3]=-332175497511735087990/197415392 44549142863,\na[8,4]=6401812090729586424603870720224/44676517161946480 3439374766203,\na[8,5]=-4265990808672245834945068273/24379656330928004 64693941600,\na[8,6]=330884756892468554/282253958465901877,\na[8,7]=-2 8749933344522124/3047037469905514375,\na[9,1]=6205673/96600000,\na[9,2 ]=0,\na[9,3]=0,\na[9,4]=43180775629715/129606111568587,\na[9,5]=103854 29951981/39850344008000,\na[9,6]=315039093349/1710630556040,\na[9,7]=1 569042174483/1579405853125,\na[9,8]=-9646151/11543640,\n\nb[1]=6205673 /96600000,\nb[2]=0,\nb[3]=0,\nb[4]=43180775629715/129606111568587,\nb[ 5]=10385429951981/39850344008000,\nb[6]=315039093349/1710630556040,\nb [7]=1569042174483/1579405853125,\nb[8]=-9646151/11543640,\n\n`b*`[1]=5 026843877900203/80720277237600000,\n`b*`[2]=0,\n`b*`[3]=0,\n`b*`[4]=11 352736352605242548485/33323272041738045062256,\n`b*`[5]=58150873683273 5921107/2561499296336591776000,\n`b*`[6]=323536238315689614389/1429426 218785286161440,\n`b*`[7]=310826349095675205429/406084020815219450000, \n`b*`[8]=-455712696908507/742001768698080,\n`b*`[9]=-1/132\}:" }}} {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)*&#\"/^^\")fYj> \"3Cgom**zN45F)*$)F'\"\"(F)F)F)*&#\"-vR=Gvb\"21:n\"**\\RBDF)*$)F'\"\") F)F)F)" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 124 "We can find the point where the boundary of the stability region \+ intersects the negative real axis by solving the equation: " }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "R(z) = 1;" "6#/-%\"RG6#%\"z G\"\"\"" }{TEXT -1 2 ". " }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 28 " z0 := newton(R(z)=1,z=-4.4);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%#z0G $!+*3D&QW!\"*" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 303 "z0 := newton(R(z)=1,z=-4.4):\np1 := plot([R(z), 1],z=-5.19..0.49,color=[red,blue]):\np2 := plot([[[z0,1]]$3],style=poi nt,symbol=[circle,cross,diamond],color=black):\np3 := plot([[z0,0],[z0 ,1]],linestyle=3,color=COLOR(RGB,0,.5,0)):\nplots[display]([p1,p2,p3], view=[-5.19..0.49,-.07..1.47],font=[HELVETICA,9]);" }}{PARA 13 "" 1 " " {GLPLOT2D 360 253 253 {PLOTDATA 2 "6+-%'CURVESG6$7Z7$$!3Q++++++!>&!# <$\"3Q9g0A(eM(QF*7$$!3QML3T![!f^F*$\"33W^v!*3BxOF*7$$!3Ynm;#3'4G^F*$\" 3C%o7?Wl)*[$F*7$$!3a++DBT9(4&F*$\"3bi!p,`95J$F*7$$!3kLLLk@>m]F*$\"35B% \\\\#>MSJF*7$$!3E+]U'*)HB,&F*$\"3)[mm!)Ry='GF*7$$!3!pm;&GwYe\\F*$\"3_J =]v(obg#F*7$$!3s+](\\(Q*y*[F*$\"3HuR')G&y=M#F*7$$!3nLLV@,KP[F*$\"3q&3K G<:A5#F*7$$!3'RLLd%[MwZF*$\"3arFCo\\H$)=F*7$$!3NLL.q&p`r%F*$\"3!fdN2Ro \\o\"F*7$$!3E+]<*4%oaYF*$\"3\\**>Z@jN1:F*7$$!3;nmJG')*Rf%F*$\"3J;:S(Q? \\M\"F*7$$!3uLLyGAZ\"[%F*$\"3ikFgD_@'3\"F*7$$!3%3+])fw&\\O%F*$\"35nk\" ocZlm)!#=7$$!3$QL$)f7eWC%F*$\"3)GMaES(zHoF]p7$$!3A++lN]MCTF*$\"35jc%RI LVO&F]p7$$!3ummYeRz+SF*$\"3oD+O0J\"4<%F]p7$$!3_LLV-,(>*QF*$\"3U>0YN1`P LF]p7$$!35++S:-YpPF*$\"3k0N%GZv'*f#F]p7$$!3K+++\"HZkk$F*$\"3aF!oyBfN %f&*!#>7$$!39++0(*RmdIF*$\"3\"f9f9Gsk])F`s7$$!39nmEI%3g%HF*$\"3O5%e)>% yu+)F`s7$$!3-++0xX]BGF*$\"3*G!oGk.eTyF`s7$$!3*)***\\\"R>&oq#F*$\"3)3XJ V7%p#*zF`s7$$!3gmm;\\r8&e#F*$\"3wQYSW5%HU)F`s7$$!3ymmrw\\OtCF*$\"3b#HZ O:/M.*F`s7$$!3SLL$))e.GN#F*$\"3nIJJE8F-**F`s7$$!3nLL)**=uvA#F*$\"3([.c BX*R-6F]p7$$!3K++:I;c=@F*$\"3x5$*yVa*z@\"F]p7$$!31LL.z]#3+#F*$\"39+F:p ;/i8F]p7$$!3M++?,<>z=F*$\"3HTDg-V!H`\"F]p7$$!3;++!4<(>g$>F]p7$$!3H++q9zA<:F*$\"3i!*)*)z)p; %>#F]p7$$!3EnmEY;O-9F*$\"39&ebbgo1Y#F]p7$$!3#)*****pQ<(z7F*$\"3Cj:v>sS \"y#F]p7$$!3)RL$efMeo6F*$\"31OZ+\"o!>3JF]p7$$!3I****fAZ3Z5F*$\"3woXyci k4NF]p7$$!3xqm;(zQwK*F]p$\"3'zvRh**oY$RF]p7$$!3&z***\\)ecE8)F]p$\"3uL` 1RZ3MWF]p7$$!3'3nmm0VV'pF]p$\"3m$)H^SRf$)\\F]p7$$!3P)***\\iqATdF]p$\"3 u)HU[3))>j&F]p7$$!3aFLL*)4AjXF]p$\"3PX!eRK(4OjF]p7$$!33LLLO'R&eLF]p$\" 3/TeyfZFZrF]p7$$!3Uim;`O$Q;#F]p$\"3'f?>=qkU0)F]p7$$!3?*****>$H-m5F]p$ \"3]UjIQ4$)))*)F]p7$$\"3v*QLLU?>#>F`s$\"3[-s.#30%>5F*7$$\"3%ymmY^avJ\" F]p$\"3Fdq,(HH39\"F*7$$\"3E0+]HcU))zhJK;F*-%'COL OURG6&%$RGBG$\"*++++\"!\")$\"\"!Fd]lFc]l-F$6$7S7$F($\"\"\"Fd]l7$F=Fi]l 7$FGFi]l7$FQFi]l7$FenFi]l7$F_oFi]l7$FdoFi]l7$FioFi]l7$F_pFi]l7$FdpFi]l 7$FipFi]l7$F^qFi]l7$FcqFi]l7$FhqFi]l7$F]rFi]l7$FbrFi]l7$FgrFi]l7$F\\sF i]l7$FbsFi]l7$FgsFi]l7$F\\tFi]l7$FatFi]l7$FftFi]l7$F[uFi]l7$F`uFi]l7$F euFi]l7$FjuFi]l7$F_vFi]l7$FdvFi]l7$FivFi]l7$F^wFi]l7$FcwFi]l7$FhwFi]l7 $F]xFi]l7$FbxFi]l7$FgxFi]l7$F\\yFi]l7$FayFi]l7$FfyFi]l7$F[zFi]l7$F`zFi ]l7$FezFi]l7$FjzFi]l7$F_[lFi]l7$Fd[lFi]l7$Fi[lFi]l7$F^\\lFi]l7$Fc\\lFi ]l7$Fh\\lFi]l-F]]l6&F_]lFc]lFc]lF`]l-F$6&7#7$$!3k******)3D&QWF*Fi]l-%' SYMBOLG6#%'CIRCLEG-F]]l6&F_]lFd]lFd]lFd]l-%&STYLEG6#%&POINTG-F$6&F_al- Fdal6#%&CROSSGFgalFial-F$6&F_al-Fdal6#%(DIAMONDGFgalFial-F$6%7$7$FaalF c]lF`al-%&COLORG6&F_]lFc]l$\"\"&!\"\"Fc]l-%*LINESTYLEG6#\"\"$-%%FONTG6 $%*HELVETICAG\"\"*-%+AXESLABELSG6%Q\"z6\"Q!F^dl-Ffcl6#%(DEFAULTG-%%VIE WG6$;$!$>&!\"#$\"#\\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 1366 "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 \+ 19634659815151/100935799966686024*z^7+557528183975/252339499916715 06*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,.08,.23,.48)):\np2 := plots[polygonplot]([seq([pts[i-1],pts[i],[-2.2,0]],i=2..nops(pts)) ],\n style=patchnogrid,color=COLOR(RGB,.15,.45,.95)):\npts := []: z0 := 2+4.75*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(zz)]] :\nend do:\np3 := plot(pts,color=COLOR(RGB,.08,.23,.48)):\np4 := plots [polygonplot]([seq([pts[i-1],pts[i],[1.90,4.73]],i=2..nops(pts))],\n \+ style=patchnogrid,color=COLOR(RGB,.15,.45,.95)):\npts := []: z 0 := 2-4.75*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(zz)]]:\nend do:\np5 := plot(pts,color=COLOR(RGB,.08,.23,.48)):\np6 := plots[polyg onplot]([seq([pts[i-1],pts[i],[1.90,-4.73]],i=2..nops(pts))],\n \+ style=patchnogrid,color=COLOR(RGB,.15,.45,.95)):\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=constrai ned);" }}{PARA 13 "" 1 "" {GLPLOT2D 381 565 565 {PLOTDATA 2 "6/-%'CURV ESG6$7]z7$$\"\"!F)F(7$F($\"3++++Fjzq:!#=7$$!3O+++1Cd_&*!#G$\"35+++UEfT JF-7$$!3))*****>m?4%>!#E$\"3z*****4!))Q7ZF-7$$!33+++I*Q7E\"!#D$\"3'*** ****fS=$G'F-7$$!39+++&zAFk#F=$\"3=+++th(R&yF-7$$\"38+++)>wMA\"!#C$\"3] ******)))fZU*F-7$$\"3/+++L\"\\BG\"!#B$\"3%******>Q_&*4\"!#<7$$\"3!)*** **z`DHF'FN$\"35+++6Uic7FQ7$$\"30+++v\"yiB#!#A$\"3********p_o89FQ7$$\"3 $)*****zr3M^'FZ$\"3!******>s?2d\"FQ7$$\"3$******40nAj\"!#@$\"3/+++#*pp F(>#FQ7$$\"3*)*****fGpK9#F_p$ 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" }}{PARA 0 "" 0 "" {TEXT -1 59 "For this scheme the stability interva l is (approximately) " }{XPPEDIT 18 0 "[-4.4563, 0];" "6#7$,$-%&Float G6$\"&jX%!\"%!\"\"\"\"!" }{TEXT -1 2 ". " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 148 "We can distort the boundary curve horizontally by taking the 11th root of the real part of points along the curve. In this way we see that there is " }{TEXT 260 53 "no large st interval on the nonnegative imaginary axis" }{TEXT -1 65 " that con tains the origin and lies inside the stability region. " }}{PARA 0 " " 0 "" {TEXT -1 119 "However the stability region intersects the nonne gative imaginary axis in an interval that does not contain the origin. " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 391 "R := z -> 1+z+1/2*z^2+1/6*z^3+1/24*z^4+1/120*z^5+1/720*z^6+\n 19634659815151/100935799966686024*z^7+557528183975/25233949991671 506*z^8:\nDigits := 25:\npts := []: z0 := 0:\nfor ct from 0 to 107 do \n zz := newton(R(z)=exp(ct*Pi/100*I),z=z0):\n z0 := zz:\n pts : = [op(pts),[surd(Re(zz),11),Im(zz)]]:\nend do:\nplot(pts,color=COLOR(R GB,0,.45,.95),thickness=2,font=[HELVETICA,9]);\nDigits := 10:" }} {PARA 13 "" 1 "" {GLPLOT2D 261 276 276 {PLOTDATA 2 "6(-%'CURVESG6#7hq7 $$\"\"!F)F(7$$!:N[b>b?*3jt`LB!#E$\":8&)41:y*e`EfTJF-7$$!:i'QE%e?.y\\<; 'QF-$\":,2[4q3yrI&=$G'F-7$$!:ZVzW#)[$310^#=&F-$\":zPs\"[CQugzxC%*F-7$$ !:9N\")z1fHX9gCQ'F-$\":iKYu)RoThqjc7!#D7$$!:1D)fAI?\\Jao(\\(F-$\":*)\\ Fd(QVqEjzq:F?7$$!:6L)\\vK1R`4gZ&)F-$\":-d+VG#F?7$$!:&H*zov;`7J^%\\5F?$\":E9B:]r([?TF8DF? 7$$!:kEYZ8([s5%R/9\"F?$\":Guc%*3vQGQLu#GF?7$$!:,D&Go\"QOL\">sF7F?$\":P 'Q/cT2UUEfTJF?7$$!:\"\\ZX\"3C*G^Oa68F?$\":@'*4F@R/u*=vbMF?7$$!:7R%4Kd% 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u7$$\":*p'Hp%[S#zPU!)4'F?$\":PY%)o\\@%GqP/)*e2'F?$\":7autH`Sj;vM'GFfu7$$\":DR$Gc E+SPa\"[/'F?$\":#\\202*=S=<@A*GFfu7$$\":%pX1())>X>(=M**fF?$\":Psx3Z?.I hn2#HFfu7$$\":Qd+ewk!*y9ZW$fF?$\":e&RDzH!H([k5\\HFfu7$$\":\"et_BR.4[$3 9%eF?$\":+HVT(='*3P)Hs(HFfu7$$\":#)G)f4f\"4.@NGq&F?$\":>C8V:lVH:I^+$Ff u7$$\":t9@MS/pFd6XZ&F?$\":$y]F@.(=-++G.$Ffu7$$\":+5.R?b$4Q*4:&\\F?$\": iQv+\\H?!>ABgIFfu7$$!:h8&f6mu3$Ffu7$$!:s())G -J(Q1up;i&F?$\":I.B8y)\\2AjN9JFfu7$$!:1#4()H*)\\MYdxFR#yG\"y%f$>$Ffu7$$!:$)47f.0&)4i9HY'F?$\":j+e4EbZZ\"RY>KFfu 7$$!:%)yS(e$Q1vT=qf'F?$\":EQY=8B!Ho;0XKFfu7$$!:s " 0 "" {MPLTEXT 1 0 201 "Digits := 15:\nz0 := 0.85*I:\nfor ct fro m 26 to 29 do\n newton(R(z)=exp(ct*Pi/100*I),z=z0);\nend do;``;\nz0 \+ := 3.05*I:\nfor ct from 98 to 101 do\n newton(R(z)=exp(ct*Pi/100*I), z=z0);\nend do;\nDigits := 10:" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#^$$! 0'z%o!*R5<#!#A$\"0ZdS$Q8o\")!#:" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#^$$ !0Pl]e$4!R*!#B$\"0E\"p96H#[)!#:" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#^$$ \"0:$>&z1nY\"!#A$\"0jCD%zW'z)!#:" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#^$ $\"0`cBSc#3c!#A$\"0i5LB/16*!#:" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%!G " }}{PARA 11 "" 1 "" {XPPMATH 20 "6#^$$\"0J$GL#HPK\"!#<$\"0q=-++G.$!#9 " }}{PARA 11 "" 1 "" {XPPMATH 20 "6#^$$\"07=]KgkQ%!#=$\"0H?!>ABgI!#9" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#^$$!0rH/Dt7!f!#=$\"0JMz))>u3$!#9" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#^$$!0>I(*)RAs " 0 "" {MPLTEXT 1 0 332 "Digits := 15:\nreal_part := proc(u)\n Re(newton(R(z)=exp(u*Pi *I),z=0.58*I))\nend proc:\nu0 := bisect('real_part'(u),u=0.26..0.29); \nnewton(R(z)=exp(u0*Pi*I),z=0.85*I);``;\nreal_part := proc(u)\n Re( newton(R(z)=exp(u*Pi*I),z=3.05*I))\nend proc:\nu0 := bisect('real_part '(u),u=0.98..1.01);\nnewton(R(z)=exp(u0*Pi*I),z=3.05*I);\nDigits := 10 :" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%#u0G$\"0,$3mZNYF!#:" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#^#$\"0Zx%Qy\"zi)!#:" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%!G" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%#u0G$\"0$RcuzT W**!#:" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#^$$\"0W7(=eQ-T!#H$\"0A'G@!RB 2$!#9" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 4 " The " }{TEXT 260 73 "stability region intersects the nonegative imagin ary axis in the interval" }{TEXT -1 38 " [ 0.8628, 3.0723 ] (approxi mately)." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 37 "#------------------------------------" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 80 "The stability function R* for th e 9 stage, order 5 scheme is given as follows." }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 97 "subs(ee,subs(b=`b*`,StabilityFunction(5,9,'expan ded'))):\n`R*` := unapply(%,z):\n'`R*`(z)'=`R*`(z);" }}{PARA 12 "" 1 " " {XPPMATH 20 "6#/-%#R*G6#%\"zG,6\"\"\"F)F'F)*&#F)\"\"#F)*$)F'F,F)F)F) *&#F)\"\"'F)*$)F'\"\"$F)F)F)*&#F)\"#CF)*$)F'\"\"%F)F)F)*&#F)\"$?\"F)*$ )F'\"\"&F)F)F)*&#\"82zHl&QS\\'pKX$\";![()zV>4@lp!)HV#F)*$)F'F1F)F)F)*& #\"7\\..!pr\"HhO#4%\";SaW49%zgl]S%p " 0 " " {MPLTEXT 1 0 33 "z_0 := newton(`R*`(z)=-1,z=-4.4);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%$z_0G$!+In " 0 "" {MPLTEXT 1 0 324 "z_0 := newton(`R*`(z)=- 1,z=-4.4):\np_1 := plot([`R*`(z),-1],z=-4.99..0.49,color=[red,blue]): \np_2 := plot([[[z_0,-1]]$3],style=point,symbol=[circle,cross,diamond] ,color=black):\np_3 := plot([[z_0,0],[z_0,-1]],linestyle=3,color=COLOR (RGB,0,.5,0)):\nplots[display]([p_1,p_2,p_3],view=[-4.99..0.49,-1.57.. 1.47],font=[HELVETICA,9]);" }}{PARA 13 "" 1 "" {GLPLOT2D 398 284 284 {PLOTDATA 2 "6+-%'CURVESG6$7S7$$!3A++++++!*\\!#<$!3Eg)e^'>U/@F*7$$!3#p mm^f^0([F*$!3m)fx%\\)3]\"=F*7$$!3%QL3o3KXF*$!3'oB\"yQXU_6F*7$$!3\\L$3@ f%)\\T%F*$!3e8I')=+!*>(*!#=7$$!3jm;u*Q?kI%F*$!3$[79JQ'oW#)FF7$$!3E+]ZY %3S>%F*$!3Rg\\weE1,pFF7$$!3%omTR&=vxSF*$!3#)[ft#>I2p&FF7$$!3/+]x(4o='R F*$!3ooS=g%*)yk%FF7$$!3hts$FF7$$!3[mm^!Qvwt$F*$!3[ x;Y66:IIFF7$$!3#)*****=Az%>OF*$!3QU@dzgMdBFF7$$!3?++]Qxz+NF*$!3+\"4^_p @gy\"FF7$$!3/++5QhU'Q$F*$!3)>xEOHo/K\"FF7$$!3mm;%zwlDG$F*$!3+2yqSit&f* !#>7$$!3[LLBUd1fJF*$!3CJ&)yF*R0&fFcp7$$!3eLL$4-XW0$F*$!3K;lT0Y#4K$Fcp7 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:= newton(`R*`(z)=exp( ct*Pi/20*I),z=z0):\n z0 := zz:\n pts := [op(pts),[Re(zz),Im(zz)]]: \nend do:\np_1 := plot(pts,color=COLOR(RGB,0,.13,.5)):\np_2 := plots[p olygonplot]([seq([pts[i-1],pts[i],[-2.2,0]],i=2..nops(pts))],\n \+ style=patchnogrid,color=COLOR(RGB,0,.25,1)):\npts := []: z0 := 1.9+ 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(zz)]]:\nend do: \np_3 := plot(pts,color=COLOR(RGB,0,.13,.5)):\np_4 := plots[polygonplo t]([seq([pts[i-1],pts[i],[1.81,4.39]],i=2..nops(pts))],\n sty le=patchnogrid,color=COLOR(RGB,0,.25,1)):\npts := []: z0 := 1.9-4.45*I :\nfor ct from 0 to 50 do\n zz := newton(`R*`(z)=exp(ct*Pi/25*I),z=z 0):\n z0 := zz:\n pts := [op(pts),[Re(zz),Im(zz)]]:\nend do:\np_5 \+ := plot(pts,color=COLOR(RGB,0,.13,.5)):\np_6 := plots[polygonplot]([se q([pts[i-1],pts[i],[1.81,-4.39]],i=2..nops(pts))],\n style=pa tchnogrid,color=COLOR(RGB,0,.25,1)):\np_7 := plot([[[-5.09,0],[2.19,0] 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oefficients of the combined scheme" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1967 "ee := \{c[2]=6/65,\nc[3]=2 3/159,\nc[4]=23/106,\nc[5]=80/141,\nc[6]=91/129,\nc[7]=125/126,\nc[8]= 1,\nc[9]=1,\n\na[2,1]=6/65,\na[3,1]=9499/303372,\na[3,2]=34385/303372, \na[4,1]=23/424,\na[4,2]=0,\na[4,3]=69/424,\na[5,1]=967715920/14829039 09,\na[5,2]=0,\na[5,3]=-1226377600/494301303,\na[5,4]=3552780800/14829 03909,\na[6,1]=-46804789259240403389999/30226267716384768392160,\na[6, 2]=0,\na[6,3]=410550459697310264150/62971391075801600817,\na[6,4]=-487 3358856094925562114476/989343525191918950435887,\na[6,5]=7315985052139 691605041/11082753205258491358240,\na[7,1]=38140471378772278038787625/ 10601915981402462257077264,\na[7,2]=0,\na[7,3]=-2006972716153785212305 625/135921999761570028936888,\na[7,4]=22464333486903984266603841845000 /1782850965624830561723340121359,\na[7,5]=-333540269282797887990446412 5/2252537921258500779474698736,\na[7,6]=627869105822046875/60332742448 5556953,\na[8,1]=146733704741261404039925503/3592960142507944001066000 0,\na[8,2]=0,\na[8,3]=-332175497511735087990/19741539244549142863,\na[ 8,4]=6401812090729586424603870720224/446765171619464803439374766203,\n a[8,5]=-4265990808672245834945068273/2437965633092800464693941600,\na[ 8,6]=330884756892468554/282253958465901877,\na[8,7]=-28749933344522124 /3047037469905514375,\na[9,1]=6205673/96600000,\na[9,2]=0,\na[9,3]=0, \na[9,4]=43180775629715/129606111568587,\na[9,5]=10385429951981/398503 44008000,\na[9,6]=315039093349/1710630556040,\na[9,7]=1569042174483/15 79405853125,\na[9,8]=-9646151/11543640,\n\nb[1]=6205673/96600000,\nb[2 ]=0,\nb[3]=0,\nb[4]=43180775629715/129606111568587,\nb[5]=103854299519 81/39850344008000,\nb[6]=315039093349/1710630556040,\nb[7]=15690421744 83/1579405853125,\nb[8]=-9646151/11543640,\n\n`b*`[1]=5026843877900203 /80720277237600000,\n`b*`[2]=0,\n`b*`[3]=0,\n`b*`[4]=11352736352605242 548485/33323272041738045062256,\n`b*`[5]=581508736832735921107/2561499 296336591776000,\n`b*`[6]=323536238315689614389/1429426218785286161440 ,\n`b*`[7]=310826349095675205429/406084020815219450000,\n`b*`[8]=-4557 12696908507/742001768698080,\n`b*`[9]=-1/132\}:" }}}{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#\"\"##\"\"' \"#l/&F%6#\"\"$#\"#B\"$f\"/&F%6#\"\"%#F0\"$1\"/&F%6#\"\"&#\"#!)\"$T\"/ &F%6#F)#\"#\"*\"$H\"/&F%6#\"\"(#\"$D\"\"$E\"/&F%6#\"\")\"\"\"/&F%6#\" \"*FP" }}}{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 $\"\"#\"\"\"#\"\"'\"#l/&F%6$\"\"$F(#\"%*\\*\"'sLI/&F%6$F/F'#\"&&QMF2/& F%6$\"\"%F(#\"#B\"$C%/&F%6$F;F'\"\"!/&F%6$F;F/#\"#pF>/&F%6$\"\"&F(#\"* ?frn*\"+4R!H[\"/&F%6$FKF'FB/&F%6$FKF/#!++wPE7\"*.8I%\\/&F%6$FKF;#\"++3 y_NFN/&F%6$F*F(#!8****Q./Cf#*y/o%\"8g@RoZQ;xEE-$/&F%6$F*F'FB/&F%6$F*F/ #\"6]TE5tpf/b5%\"5<3g,e2\"RrH'/&F%6$F*F;#!:wW6ib#\\4c)eL([\"9()eV]*=>> DNM*)*/&F%6$F*FK#\"7T]g\"pR@0&)fJ(\"8S#e8\\e_?`F36/&F%6$\"\"(F(#\";Dwy Q!yAxy8ZS\"Q\";ks2dAY-9)f\">g5/&F%6$FepF'FB/&F%6$FepF/#!:DcI7_y`hrsp+# \"9))o$*G+dh(**>#f8/&F%6$FepF;#\"A+]%=%QgmU)R!p[LVYA\"@f87SLsh0$[il4&G y\"/&F%6$FepFK#!=DTY/*z)yz#Gp-aL$\"=O()pu%z2]e7#z`_A/&F%6$FepF*#\"3vo/ Ae5pyi\"3`pb&[CuK.'/&F%6$\"\")F(#\"<.b#*RSSh7u/PtY\"\";++m5+Wz]U,'Hf$/ &F%6$FgrF'FB/&F%6$FgrF/#!6!*z3N<^(\\v@L\"5jG9\\XCR:u>/&F%6$FgrF;#\"@C- sqQgCkeH247=S'\"?.iwu$RM![Y>;&*HaQ5\" /+!3SM])R/&F%6$F_uF*#\"-\\L4R]J\".SgbI1r\"/&F%6$F_uFep#\".$[u@/p:\".DJ &eSz:/&F%6$F_uFgr#!(^hk*\")SOa6" }}}{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#\"\" \"#\"(tc?'\")++g'*/&F%6#\"\"#\"\"!/&F%6#\"\"$F//&F%6#\"\"%#\"/:(Hcx!=V \"0(eo:61'H\"/&F%6#\"\"&#\"/\")>&*HaQ5\"/+!3SM])R/&F%6#\"\"'#\"-\\L4R] J\".SgbI1r\"/&F%6#\"\"(#\".$[u@/p:\".DJ&eSz:/&F%6#\"\")#!(^hk*\")SOa6 " }}}{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#\"\"\"#\"1.-!z(Q%o-&\"2++gPsF ?2)/&F%6#\"\"#\"\"!/&F%6#\"\"$F//&F%6#\"\"%#\"8&[[DC0ENOFN6\"8cA1X!Q_\"3-%31%/&F%6# \"\")#!02&3pp7dX\"0!3)po<+U(/&F%6#\"\"*#!\"\"\"$K\"" }}}{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 33 "#================================" }}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 25 "Papakostas' scheme with " }{XPPEDIT 18 0 "c[7] = 199/200;" "6#/&%\"cG6#\"\"(*&\"$*>\"\"\"\"$+#!\"\"" }{TEXT -1 1 " " }}{PARA 257 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 211 "See: On Phase-Fitted modified Runge-Kutta Pairs of order 6(5), by Ch. Tsitouras and I. Th. Famelis,\n International Conference of Numerical Analysis and Applied Mathematics, Crete, (2006) Extended Ab stract." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 257 "" 0 "" {TEXT -1 151 "The scheme constructed here is a minor modification of the scheme presented in the preceding paper (It was originally constructed by S. N. Papakostas). " }}{PARA 257 "" 0 "" {TEXT -1 237 "The order 6 scheme is exactly the same (except that all the linking coefficents are give n in exact rational form) but the embedded order 5 scheme is altered s o that it has similar stability charactersitics to those of 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 36 "coefficients for the combined scheme" }}{PARA 0 "" 0 " " {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1841 "ee := \{ c[2]=17/183,\nc[3]=12/83,\nc[4]=18/83,\nc[5]=71/125,\nc[6]=42/59,\nc[7 ]=199/200,\nc[8]=1,\nc[9]=1,\n\na[2,1]=17/183,\na[3,1]=3756/117113,\na [3,2]=13176/117113,\na[4,1]=9/166,\na[4,2]=0,\na[4,3]=27/166,\na[5,1]= 207751751/316406250,\na[5,2]=0,\na[5,3]=-526769377/210937500,\na[5,4]= 1524242129/632812500,\na[6,1]=-4970082682619223281/2887511529739311186 ,\na[6,2]=0,\na[6,3]=97919278033879057/13556392158400522,\na[6,4]=-407 131674007930877068/74078904949579652469,\na[6,5]=123760185520426875000 0/1753200750473385108433,\na[7,1]=176597685527535385020980411/42773485 015591331328000000,\na[7,2]=0,\na[7,3]=-6793162515552646891859/4016289 67282547712000,\na[7,4]=12704926019361287204873446554247/8866594026530 54716778496000000,\na[7,5]=-50728836334509259632278125/326575917180086 85915971584,\na[7,6]=51536223982796190703/51293749413888000000,\na[8,1 ]=299033520572337573523/66918720793812357519,\na[8,2]=0,\na[8,3]=-1655 0269823961899/902146153892364,\na[8,4]=49920346343238033627496282/3215 735869387500624775563,\na[8,5]=-1686432488955761721093750/978844996793 357447730403,\na[8,6]=161901609084039/149698803705724,\na[8,7]=-305146 137600000/54760341991955873,\na[9,1]=24503/381483,\na[9,2]=0,\na[9,3]= 0,\na[9,4]=1366847103121/4106349847584,\na[9,5]=20339599609375/7593391 3767768,\na[9,6]=35031290651/194765546144,\na[9,7]=16620160000000/1100 1207123543,\na[9,8]=-14933/11016,\n\nb[1]=24503/381483,\nb[2]=0,\nb[3] =0,\nb[4]=1366847103121/4106349847584,\nb[5]=20339599609375/7593391376 7768,\nb[6]=35031290651/194765546144,\nb[7]=16620160000000/11001207123 543,\nb[8]=-14933/11016,\n\n`b*`[1]=61010485298317/979331468960880,\n` b*`[2]=0,\n`b*`[3]=0,\n`b*`[4]=320207313882553286621/94122281340699239 5200,\n`b*`[5]=6845867841119140625/29008216787127405534,\n`b*`[6]=1241 09197949158875473/562495660250110816320,\n`b*`[7]=19339714537078400000 /16810691577722216811,\n`b*`[8]=-211029377951/210416202900,\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 64 "The Butcher tableau in exact and approximate form is as follows." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 576 "subs(ee,matrix([[c[2],a[2,1],``$2],\n [c[3],a[3,1], a[3,2],``],\n [c[4],seq(a[4,i],i=1..3)],\n [c[5],seq(a[5,i],i=1..3)] ,[``$3,a[5,4]],\n [c[6],seq(a[6,i],i=1..3)],[``$2,a[6,4],a[6,5]],\n \+ [c[7],seq(a[7,i],i=1..3)],[``,seq(a[7,i],i=4..6)],\n [c[8],seq(a[8,i] ,i=1..3)],[``,seq(a[8,i],i=4..6)],[``$3,a[8,7]],\n [c[9],seq(a[9,i],i =1..3)],[``,seq(a[9,i],i=4..6)],[``$2,a[9,7],a[9,8]],\n [``,`________ _____________________________`$3],\n [`b`,seq(b[i],i=1..3)],[``,seq(b [i],i=4..6)],[``$2,b[7],b[8]],\n [`b*`,seq(`b*`[i],i=1..3)],[``,seq(` b*`[i],i=4..6)],[``,seq(`b*`[i],i=7..9)]]));" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#K%'matrixG6#787&#\"#<\"$$=F(%!GF+7&#\"#7\"#$)#\"%cP\"'8 r6#\"&wJ\"F2F+7&#\"#=F/#\"\"*\"$m\"\"\"!#\"#FF:7&#\"#r\"$D\"#\"*^Eo#3q\\\"4'=6$R(H:^()GF;#\"2d!zQ.y#>z*\"2A0Se@RcN\"7&F+F+#!6oq(3$z+u ;82%\"5pClz&\\\\!*yS(#\"7++voU?b=gP7\"7L%3^Qt/v+Kv\"7&#\"$*>\"$+##\"<6 /)4-&QNv_&o(fw\"\";+++G8L\"f:][tF%F;#!7f=*ok_b^iJz'\"6+?rZDGn*G;S7&F+# \"AZUbYM([?(Gh$>g#\\q7\"?+++'\\ynraIl-%fm))#!;D\"yAjf#4XLO)G2&\";%erf \"fo3!='z#)RAO:&\"5+++))QT\\PH^7&\"\"\"#\"6BNdPBd?N.*H\"5> vN7Qz?(=p'F;#!2**='R#)p-b;\"0kB*Q:Y@!*7&F+#\";#G'\\FO.QKMY.#*\\\":jbxC 1](Qpet:K#!:]P4@;\"0Cdq.))p\\ \"7&F+F+F+#!0++gPh90$\"2te&>*>MgZ&7&F\\p#\"&.X#\"'$[\"QF;F;7&F+#\".@J5 ZoO\"\".%eZ)\\j5%#\"/v$4'*fR.#\"/oxw8R$f(#\",^1HJ]$\"-WhalZ>7&F+F+#\"/ +++g,i;\"/VN727+6#!&L\\\"\"&;5\"7&F+%F________________________________ _____GFgrFgr7&%\"bGFbqF;F;FeqF_r7&%#b*G#\"/<$)H&[55'\"0!)3'*o9Lz*F;F;7 &F+#\"6@mG`D)QJ2-K\"6+_R#*pS8GAT*#\"4D19>6%y'e%o\"5MbSFry;#3!H#\"6ta() e\"\\z>4T7\"6?j\"36]-m&\\i&7&F+#\"5++Syq`9(R$>\"56o@Axd\"p5o\"#!-^zPH5 @\"-+H?;/@#!\"\"\"$]\"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),``$(10-i)],i=2..9),\n[`b`,seq(b[i],i=1..8),``],[` b*`,seq(`b*`[i],i=1..9)]])):\nevalf[8](%);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#K%'matrixG6#7,7,$\")vh*G*!\"*F(%!GF+F+F+F+F+F+F+7,$\")J yX9!\")$\")*er?$F*$\")s1D6F/F+F+F+F+F+F+F+7,$\")Zno@F/$\")no@aF*$\"\"! F:$\")g]E;F/F+F+F+F+F+F+7,$\")++!o&F/$\")8)fc'F/F9$!)qF(\\#!\"($\")*y' 3CFDF+F+F+F+F+7,$\")Tk=rF/$!)RB@f\\&FD$\")&*4fqF/F +F+F+F+7,$\")++]**F/$\")>nGTFDF9$!)DS\"p\"!\"'$\")\")*GV\"FY$!)^N`:FD$ \")ss/5FDF+F+F+7,$\"\"\"F:$\")ygoWFDF9$!)UaM=FY$\")rP_:FY$!)+)Gs\"FD$ \")d^\"3\"FD$!)GRsb!#5F+F+7,F[o$\")44BkF*F9F9$\")#='GLF/$\")BfyEF/$\") !R')z\"F/$\")yv5:FD$!)Pdb8FDF+7,%\"bGF[pF9F9F]pF_pFapFcpFepF+7,%#b*G$ \")&4)HiF*F9F9$\")_.-MF/$\")a(*fBF/$\")GS1AF/$\")8W]6FD$!)T\"H+\"FD$!) nmmmFioQ(pprint86\"" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 146 "RK6_8eqs := [op(RowSumConditions(8,'expa nded')),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*`)):\nmap(u->`if`(l hs(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$" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 114 "We can calculate the principal erro r norm of the order 6 scheme, that is, the 2-norm of the principal err or terms." }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 131 "errterms6_8 := PrincipalErrorTerms(6,8,'expanded'):\nevalf(evalf[14](sqrt(add(subs(e e,errterms6_8[i])^2,i=1.. nops(errterms6_8)))));" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#$\"+O7OJ7!#9" }}}{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,'expand ed')):\nevalf(evalf[14](sqrt(add(subs(ee,`errterms5_9*`[i])^2,i=1.. no ps(`errterms5_9*`)))));" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#$\"+U**zfd! #8" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}}{PARA 0 "" 0 "" {TEXT -1 52 "#----------------- ----------------------------------" }}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 44 "simultaneous construction of the two schemes" }}{PARA 0 "" 0 " " {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 119 "We incorporate the sta ge-order equations to ensure that stage 2 has stage-order 2 and stages 3 to 8 have stage-order 3." }}{PARA 0 "" 0 "" {TEXT -1 48 "We also in corporate the simplifying conditions: " }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "Sum(b[i]*a[i,j],i = j+1 .. 8) = b[j]*(1-c[j]);" "6#/-%$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 " an d " }{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 follows. " }}{PARA 0 "" 0 "" {TEXT -1 78 "These include the 6 quadrature conditions and t wo additional order conditions." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 178 "SO6 := SimpleOrderCondition s(6):\n[seq([i,SO6[i]],i=[1,2,4,8,16,24,29,32])]:\nlinalg[augment](lin alg[delcols](%,2..2),matrix([[` `]$(linalg[rowdim](%))]),linalg[delco ls](%,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(\"\"$7%\"\")F)/*&F,F()F2F:F(#F(F57%\"#;F)/*&F,F()F2F5F(#F(\" \"&7%\"#CF)/*(F,F(F2F(-%!G6#*&%\"aGF(-FM6#*&F8F(FPF(F(F(#F(\"#s7%\"#HF )/*(F,F(F2F(-FM6#*&F?F(FPF(F(#F(FI7%\"#KF)/*&F,F()F2FGF(#F(\"\"'Q)ppri nt596\"" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 136 "For the embedded order 5 scheme we use a selection of 7 \"simple \" order conditions as given (in abreviated form) in the following tab le. " }}{PARA 0 "" 0 "" {TEXT -1 78 "These include the 5 quadrature co nditions and two additional order conditions." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 194 "`SO5*` := s ubs(b=`b*`,SimpleOrderConditions(5)):\n[seq([i,`SO5*`[i]],i=[1,2,4,8,1 2,15,16])]:\nlinalg[augment](linalg[delcols](%,2..2),matrix([[` `]$(l inalg[rowdim](%))]),linalg[delcols](%,1..1));" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#K%'matrixG6#7)7%\"\"\"%#~~G/*&%#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)pprin t186\"" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 582 "SO6_8 := SimpleOrderConditions(6,8,'expanded'):\nSO_ eqs := [op(RowSumConditions(8,'expanded')),op(StageOrderConditions(2,8 ,'expanded')),\n op(StageOrderConditions(3,4..8,'expande d'))]:\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(si mp_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] = 17/183;" "6#/ &%\"cG6#\"\"#*&\"#<\"\"\"\"$$=!\"\"" }{TEXT -1 3 ", " }{XPPEDIT 18 0 "c[4] = 18/83;" "6#/&%\"cG6#\"\"%*&\"#=\"\"\"\"#$)!\"\"" }{TEXT -1 3 " , " }{XPPEDIT 18 0 "c[5] = 71/125;" "6#/&%\"cG6#\"\"&*&\"#r\"\"\"\"$D \"!\"\"" }{TEXT -1 3 ", " }{XPPEDIT 18 0 "c[6] = 42/59;" "6#/&%\"cG6# \"\"'*&\"#U\"\"\"\"#f!\"\"" }{TEXT -1 3 ", " }{XPPEDIT 18 0 "c[7] = 1 99/200;" "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 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]=17/183,c[4]=18/83,c[5]=71/125, c[6]=42/59,c[7]=199/200,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:\ne 3 := `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 1960 "e3 := \{`b*`[2] = 0, `b*`[3] = 0, b[2] = 0, c[9] = \+ 1, c[8] = 1, a[5,4] = 1524242129/632812500, c[3] = 12/83, b[4] = 13668 47103121/4106349847584, b[8] = -14933/11016, a[9,8] = -14933/11016, a[ 6,1] = -4970082682619223281/2887511529739311186, a[8,2] = 0, c[5] = 71 /125, c[7] = 199/200, a[5,2] = 0, a[6,2] = 0, a[7,2] = 0, a[4,2] = 0, \+ a[9,7] = 16620160000000/11001207123543, a[9,5] = 20339599609375/759339 13767768, a[8,5] = -1686432488955761721093750/978844996793357447730403 , a[9,6] = 35031290651/194765546144, `b*`[1] = 61010485298317/97933146 8960880, a[8,3] = -16550269823961899/902146153892364, b[1] = 24503/381 483, b[3] = 0, a[7,3] = -6793162515552646891859/401628967282547712000, a[4,1] = 9/166, b[6] = 35031290651/194765546144, b[7] = 1662016000000 0/11001207123543, a[7,6] = 51536223982796190703/51293749413888000000, \+ a[9,4] = 1366847103121/4106349847584, a[6,3] = 97919278033879057/13556 392158400522, a[5,1] = 207751751/316406250, a[2,1] = 17/183, a[5,3] = \+ -526769377/210937500, a[9,1] = 24503/381483, a[8,7] = -305146137600000 /54760341991955873, `b*`[4] = 320207313882553286621/941222813406992395 200, a[8,1] = 299033520572337573523/66918720793812357519, b[5] = 20339 599609375/75933913767768, `b*`[7] = 19339714537078400000/1681069157772 2216811, a[8,6] = 161901609084039/149698803705724, a[7,4] = 1270492601 9361287204873446554247/886659402653054716778496000000, a[6,4] = -40713 1674007930877068/74078904949579652469, c[2] = 17/183, c[4] = 18/83, c[ 6] = 42/59, a[9,2] = 0, a[9,3] = 0, a[3,2] = 13176/117113, a[3,1] = 37 56/117113, `b*`[9] = -1/150, `b*`[5] = 6845867841119140625/29008216787 127405534, a[4,3] = 27/166, a[7,1] = 176597685527535385020980411/42773 485015591331328000000, `b*`[8] = -211029377951/210416202900, a[7,5] = \+ -50728836334509259632278125/32657591718008685915971584, a[6,5] = 12376 01855204268750000/1753200750473385108433, a[8,4] = 4992034634323803362 7496282/3215735869387500624775563, `b*`[6] = 124109197949158875473/562 495660250110816320\}:" }}{PARA 12 "" 1 "" {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 0 576 "subs(e3,matrix([[c[2],a[2,1],``$2],\n [c[3],a[3,1],a[3,2],``],\n [c[4],seq(a[4,i],i=1..3)],\n [c[5],seq(a[5,i],i=1..3)],[``$3,a[5,4] ],\n [c[6],seq(a[6,i],i=1..3)],[``$2,a[6,4],a[6,5]],\n [c[7],seq(a[7 ,i],i=1..3)],[``,seq(a[7,i],i=4..6)],\n [c[8],seq(a[8,i],i=1..3)],[`` ,seq(a[8,i],i=4..6)],[``$3,a[8,7]],\n [c[9],seq(a[9,i],i=1..3)],[``,s eq(a[9,i],i=4..6)],[``$2,a[9,7],a[9,8]],\n [``,`_____________________ ________________`$3],\n [`b`,seq(b[i],i=1..3)],[``,seq(b[i],i=4..6)], [``$2,b[7],b[8]],\n [`b*`,seq(`b*`[i],i=1..3)],[``,seq(`b*`[i],i=4..6 )],[``,seq(`b*`[i],i=7..9)]]));" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#K%' matrixG6#787&#\"#<\"$$=F(%!GF+7&#\"#7\"#$)#\"%cP\"'8r6#\"&wJ\"F2F+7&# \"#=F/#\"\"*\"$m\"\"\"!#\"#FF:7&#\"#r\"$D\"#\"*^Eo#3q\\\"4'=6$R( H:^()GF;#\"2d!zQ.y#>z*\"2A0Se@RcN\"7&F+F+#!6oq(3$z+u;82%\"5pClz&\\\\!* yS(#\"7++voU?b=gP7\"7L%3^Qt/v+Kv\"7&#\"$*>\"$+##\"<6/)4-&QNv_&o(fw\"\" ;+++G8L\"f:][tF%F;#!7f=*ok_b^iJz'\"6+?rZDGn*G;S7&F+#\"AZUbYM([?(Gh$>g# \\q7\"?+++'\\ynraIl-%fm))#!;D\"yAjf#4XLO)G2&\";%erf\"fo3!= 'z#)RAO:&\"5+++))QT\\PH^7&\"\"\"#\"6BNdPBd?N.*H\"5>vN7Qz?(=p'F;#!2**=' R#)p-b;\"0kB*Q:Y@!*7&F+#\";#G'\\FO.QKMY.#*\\\":jbxC1](Qpet:K#!:]P4@;\"0Cdq.))p\\\"7&F+F+F+#!0++gPh9 0$\"2te&>*>MgZ&7&F\\p#\"&.X#\"'$[\"QF;F;7&F+#\".@J5ZoO\"\".%eZ)\\j5%# \"/v$4'*fR.#\"/oxw8R$f(#\",^1HJ]$\"-WhalZ>7&F+F+#\"/+++g,i;\"/VN727+6# !&L\\\"\"&;5\"7&F+%F_____________________________________GFgrFgr7&%\"b GFbqF;F;FeqF_r7&%#b*G#\"/<$)H&[55'\"0!)3'*o9Lz*F;F;7&F+#\"6@mG`D)QJ2-K \"6+_R#*pS8GAT*#\"4D19>6%y'e%o\"5MbSFry;#3!H#\"6ta()e\"\\z>4T7\"6?j\"3 6]-m&\\i&7&F+#\"5++Syq`9(R$>\"56o@Axd\"p5o\"#!-^zPH5@\"-+H?;/@#!\"\"\" $]\"Q(pprint36\"" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 133 "subs(e3,matrix([seq([c[i],seq(a[i,j],j=1..i- 1),``$(10-i)],i=2..9),\n[`b`,seq(b[i],i=1..8),``],[`b*`,seq(`b*`[i],i= 1..9)]])):\nevalf(%);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#K%'matrixG6#7 ,7,$\"+'[<'*G*!#6F(%!GF+F+F+F+F+F+F+7,$\"+LJyX9!#5$\"+)*)er?$F*$\"+Vs1 D6F/F+F+F+F+F+F+F+7,$\"+*pu'o@F/$\"+Zno@aF*$\"\"!F:$\"+Cg]E;F/F+F+F+F+ F+F+7,$\"++++!o&F/$\"+m7)fc'F/F9$!+ZqF(\\#!\"*$\"+?*y'3CFDF+F+F+F+F+7, $\"+oSk=rF/$!+$)QB@f\\&FD$\"+1&*4fqF/F+F+F+F+7, $\"++++]**F/$\"+(*=nGTFDF9$!+IDS\"p\"!\")$\"+K\")*GV\"FY$!+!4bLb\"FD$ \"+wrs/5FDF+F+F+7,$\"\"\"F:$\"+VygoWFDF9$!+(=WX$=FY$\"+sqP_:FY$!+@+)Gs \"FD$\"+N?S$F/$\"+9a(*fBF/$\"+kFS1AF/$\"+S8 W]6FD$!+1T\"H+\"FD$!+nmmmmFioQ(pprint16\"" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 146 "RK6_8eqs := [op(R owSumConditions(8,'expanded')),op(OrderConditions(6,8,'expanded'))]:\n `RK5_9eqs*` := subs(b=`b*`,OrderConditions(5,9,'expanded')):" }}} {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 7 "Check: " } }{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 148 "simplify(subs(e3,RK6_8eqs) ):\nmap(u->`if`(lhs(u)=rhs(u),0,1),%);\nnops(%);\nsimplify(subs(e3,`RK 5_9eqs*`)):\nmap(u->`if`(lhs(u)=rhs(u),0,1),%);\nnops(%);" }{TEXT -1 0 "" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#7N\"\"!F$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 28 "determination of 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[5]" "6#&%\"cG6#\"\"&" }{TEXT -1 1 " \+ " }}{PARA 4 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 22 "eG .. general soluti on" }}{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+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]^3-1 1*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+38 0*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]+15 0*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+10 0*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]^2-3 0*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]^2-8 40*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]), 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]+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]-6 0*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]^3-2 80*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]^5-103*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]-10 3*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]-1 2*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]^3-1 00*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+1 20*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+6 0*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]+12 6*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]+1 89*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]+2 1*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+1 68*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+42 0*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+1 550*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+6 0*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+2 8*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]-27 0*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]^2-31 2*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+30 0*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+1 80*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]^2-5 0*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 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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+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 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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]^2-247 2*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+2 0*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 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^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]-4 775*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+5 20*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]^2-69 0*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]^3-5 34*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]^2-306 0*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]+2 0*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]^4-5 35*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+1 30*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 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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]^2-4 2*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]), 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 33 "#================================" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 22 "We specify the nodes " } {XPPEDIT 18 0 "c[6] = 42/59;" "6#/&%\"cG6#\"\"'*&\"#U\"\"\"\"#f!\"\"" }{TEXT -1 7 " and " }{XPPEDIT 18 0 "c[7] = 199/200;" "6#/&%\"cG6#\" \"(*&\"$*>\"\"\"\"$+#!\"\"" }{TEXT -1 27 " and determine 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[ 5]" "6#&%\"cG6#\"\"&" }{TEXT -1 65 " that minimize the principal errr or norm (subject to the nodes " }{XPPEDIT 18 0 "c[6]" "6#&%\"cG6#\"\" '" }{TEXT -1 7 " and " }{XPPEDIT 18 0 "c[7]" "6#&%\"cG6#\"\"(" } {TEXT -1 19 " remaining fixed)." }}{PARA 0 "" 0 "" {TEXT -1 85 "We u se the general solution to obtain expressions for the coefficients in \+ terms of " }{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[5]" "6#&%\"cG6#\"\"&" }{TEXT -1 2 ". " }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 73 "eA := \{c[6]=42/59,c[7]=199/200\}: \neB := `union`(eA,simplify(subs(eA,eG))):" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 2 "eB" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 16278 "eB := \{a[9,5] = -1/60*(1 8275*c[4]-4967)/c[5]/(11800*c[5]^3*c[4]-31941*c[5]^2*c[4]+28499*c[5]*c [4]-8358*c[4]-11800*c[5]^4+31941*c[5]^3-28499*c[5]^2+8358*c[5]), a[8,1 ] = 1/16716*(8146104082*c[4]^4*c[5]-48224617900*c[5]^3*c[4]^4-57952844 40*c[5]^4*c[4]-67638491180*c[5]^4*c[4]^3-1386592200*c[5]^5*c[4]^2-4233 7584200*c[5]^4*c[4]^5-7758616940*c[4]^5*c[5]+10511585090*c[4]^5*c[5]^3 +24373477120*c[4]^5*c[5]^2+2773184400*c[4]^6*c[5]-288212224*c[4]^3+691 54092*c[4]^2-4376228940*c[5]*c[4]^3+2291750028*c[5]^2*c[4]^2-720584970 *c[5]^3+207462276*c[5]^2+352215472*c[4]^4+554636880*c[5]^4-133157340*c [4]^5+1478353813*c[5]*c[4]^2+2032562984*c[5]^2*c[4]^3-17614274189*c[4] ^4*c[5]^2+27602681095*c[5]^4*c[4]^2-27172054136*c[5]^3*c[4]^2+66628899 40*c[5]^3*c[4]-1172693395*c[5]^2*c[4]-242039322*c[5]*c[4]+80147724740* c[5]^4*c[4]^4+55280294446*c[5]^3*c[4]^3+7167337950*c[5]^5*c[4]^3-10339 955700*c[4]^6*c[5]^2-11130211300*c[5]^5*c[4]^4+4508442500*c[5]^3*c[4]^ 6+6113177800*c[5]^4*c[4]^6+6113177800*c[5]^5*c[4]^5)/c[5]/c[4]^2/(2492 00*c[5]^3*c[4]^4+249200*c[5]^4*c[4]^3+6645*c[4]^3-1678*c[4]^2-64790*c[ 5]*c[4]^3-105160*c[5]^2*c[4]^2+6645*c[5]^3-1678*c[5]^2+10068*c[5]*c[4] ^2+365650*c[5]^2*c[4]^3-66450*c[4]^4*c[5]^2-66450*c[5]^4*c[4]^2+365650 *c[5]^3*c[4]^2-64790*c[5]^3*c[4]+10068*c[5]^2*c[4]+1678*c[5]*c[4]-8805 00*c[5]^3*c[4]^3), a[4,2] = 0, `b*`[6] = 12117361/8419320*(-331800*c[4 ]^3-382755*c[4]+630214*c[4]^2+2060050*c[5]*c[4]^3+6916070*c[5]^2*c[4]^ 2-382755*c[5]-331800*c[5]^3+630214*c[5]^2-3718654*c[5]*c[4]^2-4040890* c[5]^2*c[4]^3-4040890*c[5]^3*c[4]^2+2060050*c[5]^3*c[4]-3718654*c[5]^2 *c[4]+2122199*c[5]*c[4]+2494800*c[5]^3*c[4]^3+74466)/(126-1412*c[5]*c[ 4]^2-1412*c[5]^2*c[4]+336*c[5]^2-437*c[4]-437*c[5]+1707*c[5]*c[4]+1260 *c[5]^2*c[4]^2+336*c[4]^2)/(3481*c[5]*c[4]-2478*c[4]-2478*c[5]+1764), \+ a[8,5] = -1/2*(3996955000*c[5]^5*c[4]-7657623815*c[4]^4*c[5]-104641886 290*c[5]^3*c[4]^4-12700553425*c[5]^4*c[4]-100576479790*c[5]^4*c[4]^3-1 6225257000*c[5]^5*c[4]^2-31102975600*c[5]^4*c[4]^5+2773184400*c[4]^5*c [5]+35539650700*c[4]^5*c[5]^3-17022739500*c[4]^5*c[5]^2+352215472*c[4] ^3+69154092*c[4]-288212224*c[4]^2+7544001892*c[5]*c[4]^3+26498640284*c [5]^2*c[4]^2-138308184*c[5]-1507902610*c[5]^3+765666014*c[5]^2-1331573 40*c[4]^4+1272068780*c[5]^4-391524000*c[5]^5-3472668987*c[5]*c[4]^2-52 960988194*c[5]^2*c[4]^3+49536560145*c[4]^4*c[5]^2+51138792735*c[5]^4*c [4]^2-57651294808*c[5]^3*c[4]^2+14500168388*c[5]^3*c[4]-6817138749*c[5 ]^2*c[4]+951414694*c[5]*c[4]+91969147300*c[5]^4*c[4]^4+113761264620*c[ 5]^3*c[4]^3+31879986000*c[5]^5*c[4]^3-29073040000*c[5]^5*c[4]^4+981288 0000*c[5]^5*c[4]^5)/c[5]/(-5653039450*c[5]^6*c[4]^2-1345326790*c[5]^5* c[4]-675351765*c[4]^4*c[5]-14932908550*c[5]^3*c[4]^4+14624927200*c[5]^ 6*c[4]^3+570856663*c[5]^4*c[4]+6327463900*c[5]^4*c[4]^3+8396311750*c[5 ]^5*c[4]^2+764522000*c[5]^6*c[4]-5803247200*c[5]^4*c[4]^5+3421183050*c [4]^5*c[5]^3-555389100*c[4]^5*c[5]^2-14024724*c[4]^3+62406032*c[5]*c[4 ]^3-67593196*c[5]^2*c[4]^2+14024724*c[5]^3+55538910*c[4]^4-89335508*c[ 5]^4+153637345*c[5]^5-78411000*c[5]^6+28049448*c[5]*c[4]^2-426155503*c [5]^2*c[4]^3+4439449090*c[4]^4*c[5]^2-3869194870*c[5]^4*c[4]^2+3770132 60*c[5]^3*c[4]^2+38983762*c[5]^3*c[4]-28049448*c[5]^2*c[4]+20710451050 *c[5]^4*c[4]^4+853483570*c[5]^3*c[4]^3-18478594650*c[5]^5*c[4]^3-96057 90000*c[5]^5*c[4]^4+2940560000*c[5]^5*c[4]^5+784110000*c[5]^7*c[4]^2-2 940560000*c[5]^7*c[4]^3), a[7,1] = 199/134400000000*(1622916359777*c[4 ]^4*c[5]-9640340513460*c[5]^3*c[4]^4-1159706076400*c[5]^4*c[4]-1355283 2452530*c[5]^4*c[4]^3-277318440000*c[5]^5*c[4]^2-8526862020000*c[5]^4* c[4]^5-1547937460200*c[4]^5*c[5]+2109477367950*c[4]^5*c[5]^3+486553482 4920*c[4]^5*c[5]^2+554636880000*c[4]^6*c[5]-57448191421*c[4]^3+1376166 4308*c[4]^2-871730756580*c[5]*c[4]^3+455727376491*c[5]^2*c[4]^2-143603 871306*c[5]^3+41284992924*c[5]^2+70426172038*c[4]^4+110927376000*c[5]^ 4-26723232400*c[4]^5+294390098770*c[5]*c[4]^2+405958221489*c[5]^2*c[4] ^3-3510117856640*c[4]^4*c[5]^2+5525940933060*c[5]^4*c[4]^2-54175485601 03*c[5]^3*c[4]^2+1328063907784*c[5]^3*c[4]-233356213312*c[5]^2*c[4]-48 165825078*c[5]*c[4]+16093276777950*c[5]^4*c[4]^4+11029715034370*c[5]^3 *c[4]^3+1435778794000*c[5]^5*c[4]^3-2069860616000*c[4]^6*c[5]^2-223743 8010000*c[5]^5*c[4]^4+906009990000*c[5]^3*c[4]^6+1234238000000*c[5]^4* c[4]^6+1234238000000*c[5]^5*c[4]^5)/(1250*c[5]^3*c[4]^4+1250*c[5]^4*c[ 4]^3+33*c[4]^3-8*c[4]^2-323*c[5]*c[4]^3-511*c[5]^2*c[4]^2+33*c[5]^3-8* c[5]^2+48*c[5]*c[4]^2+1820*c[5]^2*c[4]^3-330*c[4]^4*c[5]^2-330*c[5]^4* c[4]^2+1820*c[5]^3*c[4]^2-323*c[5]^3*c[4]+48*c[5]^2*c[4]+8*c[5]*c[4]-4 410*c[5]^3*c[4]^3)/c[5]/c[4]^2, a[9,8] = 1/1020*(24920*c[5]*c[4]-6645* c[5]-6645*c[4]+1678)/(-c[4]+c[5]*c[4]+1-c[5]), a[9,1] = 1/501480*(1083 30*c[5]*c[4]-18275*c[5]-18275*c[4]+4967)/c[5]/c[4], a[6,1] = 21/242347 22*(7975806*c[4]^4*c[5]-40727700*c[5]^3*c[4]^4-7310100*c[5]^4*c[4]^3-4 163040*c[4]^5*c[5]+8215160*c[4]^5*c[5]^3+5848080*c[4]^5*c[5]^2-416304* c[4]^3+148176*c[4]^2-6358296*c[5]*c[4]^3+7118916*c[5]^2*c[4]^2-832608* c[5]^3+444528*c[5]^2+292404*c[4]^4+2610720*c[5]*c[4]^2-6596361*c[5]^2* c[4]^3-1761504*c[4]^4*c[5]^2+2081520*c[5]^4*c[4]^2-26698385*c[5]^3*c[4 ]^2+7042476*c[5]^3*c[4]-2667168*c[5]^2*c[4]-518616*c[5]*c[4]+8215160*c [5]^4*c[4]^4+48343656*c[5]^3*c[4]^3)/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), a[6,4] = -21/12117361*(10551746*c[4]^4*c[5]-8772120*c[5 ]^3*c[4]^4-2924040*c[5]^4*c[4]^3-4163040*c[4]^5*c[5]+8772120*c[4]^5*c[ 5]^2-208152*c[4]^3+148176*c[4]^2-7753410*c[5]*c[4]^3+6016206*c[5]^2*c[ 4]^2-832608*c[5]^3+444528*c[5]^2+2402568*c[5]*c[4]^2+3212743*c[5]^2*c[ 4]^3-17544240*c[4]^4*c[5]^2+2081520*c[5]^4*c[4]^2-23245233*c[5]^3*c[4] ^2+6603870*c[5]^3*c[4]-2459016*c[5]^2*c[4]-518616*c[5]*c[4]+28174860*c [5]^3*c[4]^3)/(-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[9,6] = 714924299/143128440*(990*c[5]*c[4]-395 *c[5]-395*c[4]+197)/(3481*c[5]*c[4]-2478*c[4]-2478*c[5]+1764), a[9,4] \+ = 1/60*(-4967+18275*c[5])/c[4]/(-31941*c[4]^3+11800*c[4]^4+28499*c[4]^ 2-8358*c[4]-11800*c[5]*c[4]^3+31941*c[5]*c[4]^2-28499*c[5]*c[4]+8358*c [5]), a[7,3] = 597/6400000000*(-130517075400*c[4]^4*c[5]-521956280000* c[5]^3*c[4]^4-55645070000*c[5]^4*c[4]^3+39616920000*c[4]^5*c[5]+149688 000000*c[4]^5*c[5]^3-166935210000*c[4]^5*c[5]^2+2641128000*c[4]^3+9829 76022*c[4]-3419139793*c[4]^2+137882420000*c[5]*c[4]^3+279306330280*c[5 ]^2*c[4]^2-1965952044*c[5]-5282256000*c[5]^3+6838279586*c[5]^2-6241060 5321*c[5]*c[4]^2-593670029510*c[5]^2*c[4]^3+547920565950*c[4]^4*c[5]^2 +13205640000*c[5]^4*c[4]^2-249839968460*c[5]^3*c[4]^2+53951564000*c[5] ^3*c[4]-65169767226*c[5]^2*c[4]+15214852057*c[5]*c[4]+49896000000*c[5] ^4*c[4]^4+560205411150*c[5]^3*c[4]^3)/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)/(125*c[5]*c[4]-33*c[5]-33*c[4]+8), b[8] = 1/1020*(24920*c[5] *c[4]-6645*c[5]-6645*c[4]+1678)/(c[4]-1)/(-1+c[5]), b[5] = -1/60*(1827 5*c[4]-4967)/(c[4]-c[5])/c[5]/(11800*c[5]^3-31941*c[5]^2+28499*c[5]-83 58), `b*`[1] = 1/501480*(824418-8330280*c[4]^3-5982075*c[4]+13031062*c [4]^2+79714750*c[5]*c[4]^3+305377310*c[5]^2*c[4]^2-5982075*c[5]-833028 0*c[5]^3+13031062*c[5]^2-118021282*c[5]*c[4]^2-219132550*c[5]^2*c[4]^3 -219132550*c[5]^3*c[4]^2+79714750*c[5]^3*c[4]-118021282*c[5]^2*c[4]+49 591907*c[5]*c[4]+167680800*c[5]^3*c[4]^3)/(126-1412*c[5]*c[4]^2-1412*c [5]^2*c[4]+336*c[5]^2-437*c[4]-437*c[5]+1707*c[5]*c[4]+1260*c[5]^2*c[4 ]^2+336*c[4]^2)/c[5]/c[4], a[9,7] = -16000000000/1994577*(125*c[5]*c[4 ]-33*c[5]-33*c[4]+8)/(40000*c[5]*c[4]-39800*c[4]-39800*c[5]+39601), c[ 3] = 2/3*c[4], a[4,3] = 3/4*c[4], c[8] = 1, b[3] = 0, b[2] = 0, c[9] = 1, `b*`[2] = 0, a[5,2] = 0, `b*`[3] = 0, a[8,2] = 0, a[7,2] = 0, a[6, 2] = 0, `b*`[8] = 0, a[5,1] = 1/4*c[5]*(2*c[5]^2-5*c[5]*c[4]+4*c[4]^2) /c[4]^2, a[3,2] = 2/9*c[4]^2/c[2], a[2,1] = c[2], a[3,1] = -2/9*c[4]*( c[4]-3*c[2])/c[2], a[4,1] = 1/4*c[4], a[9,2] = 0, a[9,3] = 0, c[6] = 4 2/59, c[7] = 199/200, a[8,4] = -1/2*(47700652872*c[4]^4*c[5]-286548976 330*c[5]^3*c[4]^4-13864210380*c[5]^4*c[4]-247061558665*c[5]^4*c[4]^3-2 773184400*c[5]^5*c[4]^2-378637306200*c[5]^4*c[4]^5-80997372315*c[4]^5* c[5]+121371840590*c[4]^5*c[5]^3+236859538345*c[4]^5*c[5]^2+78317424150 *c[4]^6*c[5]-765666014*c[4]^3+138308184*c[4]^2-17146390825*c[5]*c[4]^3 +9678026355*c[5]^2*c[4]^2-1441169940*c[5]^3+414924552*c[5]^2+150790261 0*c[4]^4+1109273760*c[5]^4-1272068780*c[4]^5+391524000*c[4]^6+39791823 62*c[5]*c[4]^2+669396364*c[5]^2*c[4]^3-87920158864*c[4]^4*c[5]^2+79787 475835*c[5]^4*c[4]^2-82915492080*c[5]^3*c[4]^2+16296682974*c[5]^3*c[4] -3201214892*c[5]^2*c[4]-484078644*c[5]*c[4]+416950658850*c[5]^4*c[4]^4 +220100515586*c[5]^3*c[4]^3+17022739500*c[5]^5*c[4]^3-277678978760*c[4 ]^6*c[5]^2-32956220000*c[4]^8*c[5]^2-35539650700*c[5]^5*c[4]^4+9282300 6000*c[5]^3*c[4]^6+171154306800*c[5]^4*c[4]^6+31102975600*c[5]^5*c[4]^ 5+154134686900*c[4]^7*c[5]^2-39199897600*c[5]*c[4]^7-109125046800*c[5] ^3*c[4]^7+7830480000*c[5]*c[4]^8-9812880000*c[5]^5*c[4]^6+29438640000* c[4]^8*c[5]^3-29438640000*c[5]^4*c[4]^7)/(-64790*c[4]^4*c[5]-814050*c[ 5]^3*c[4]^4+64790*c[5]^4*c[4]+814050*c[5]^4*c[4]^3+66450*c[5]^5*c[4]^2 +249200*c[4]^5*c[5]^3-66450*c[4]^5*c[5]^2-1678*c[4]^3+3423*c[5]*c[4]^3 +1678*c[5]^3+6645*c[4]^4-6645*c[5]^4+3356*c[5]*c[4]^2-40370*c[5]^2*c[4 ]^3+365650*c[4]^4*c[5]^2-365650*c[5]^4*c[4]^2+40370*c[5]^3*c[4]^2-3423 *c[5]^3*c[4]-3356*c[5]^2*c[4]-249200*c[5]^5*c[4]^3)/(-20141*c[4]+11800 *c[4]^2+8358)/c[4]^2, a[7,4] = -199/3200000000*(-6776076072449*c[4]^4* c[5]+26694408497420*c[5]^3*c[4]^4+2550895559600*c[5]^4*c[4]+3599139198 9060*c[5]^4*c[4]^3+554636880000*c[5]^5*c[4]^2+28335462180000*c[5]^4*c[ 4]^5+9343115154200*c[4]^5*c[5]+2753347170800*c[4]^5*c[5]^3-31194932734 440*c[4]^5*c[5]^2-6242945392400*c[4]^6*c[5]+124819403417*c[4]^3-275233 28616*c[4]^2+2716268163331*c[5]*c[4]^3-1367896813287*c[5]^2*c[4]^2+287 207742612*c[5]^3-82569985848*c[5]^2-175169706220*c[4]^4-221854752000*c [5]^4+77913276000*c[4]^5-695362526945*c[5]*c[4]^2-1519958511488*c[5]^2 *c[4]^3+16031073783460*c[4]^4*c[5]^2-13403797235920*c[5]^4*c[4]^2+1355 2403110271*c[5]^3*c[4]^2-2959235171028*c[5]^3*c[4]+553962894263*c[5]^2 *c[4]+96331650156*c[5]*c[4]-47361389943900*c[5]^4*c[4]^4-3025227192436 0*c[5]^3*c[4]^3-2848993376000*c[5]^5*c[4]^3+24148040780700*c[4]^6*c[5] ^2+4255847920000*c[5]^5*c[4]^4-15966720000000*c[5]^3*c[4]^6-5887728000 000*c[5]^4*c[4]^6-1962576000000*c[5]^5*c[4]^5-6566118260000*c[4]^7*c[5 ]^2+1558265520000*c[5]*c[4]^7+5887728000000*c[5]^3*c[4]^7)/(-323*c[4]^ 4*c[5]-4080*c[5]^3*c[4]^4+323*c[5]^4*c[4]+4080*c[5]^4*c[4]^3+330*c[5]^ 5*c[4]^2+1250*c[4]^5*c[5]^3-330*c[4]^5*c[5]^2-8*c[4]^3+15*c[5]*c[4]^3+ 8*c[5]^3+33*c[4]^4-33*c[5]^4+16*c[5]*c[4]^2-188*c[5]^2*c[4]^3+1820*c[4 ]^4*c[5]^2-1820*c[5]^4*c[4]^2+188*c[5]^3*c[4]^2-15*c[5]^3*c[4]-16*c[5] ^2*c[4]-1250*c[5]^5*c[4]^3)/(59*c[4]-42)/c[4]^2, a[7,5] = -199/3200000 000*(1554615340200*c[4]^4*c[5]+12547749776400*c[5]^3*c[4]^4+7957855690 00*c[5]^4*c[4]+6353986696000*c[5]^4*c[4]^3+1962576000000*c[5]^4*c[4]^5 -554636880000*c[4]^5*c[5]-4255847920000*c[4]^5*c[5]^3+2848993376000*c[ 4]^5*c[5]^2-70426172038*c[4]^3-13761664308*c[4]+57448191421*c[4]^2-157 3345196177*c[5]*c[4]^3-4528372755202*c[5]^2*c[4]^2+27523328616*c[5]+17 5169706220*c[5]^3-124819403417*c[5]^2+26723232400*c[4]^4-77913276000*c [5]^4+749206323051*c[5]*c[4]^2+8979739172440*c[5]^2*c[4]^3-83296882411 20*c[4]^4*c[5]^2-3231844524000*c[5]^4*c[4]^2+6951740140980*c[5]^3*c[4] ^2-1732269366955*c[5]^3*c[4]+1153938421971*c[5]^2*c[4]-203256301351*c[ 5]*c[4]-5802623920000*c[5]^4*c[4]^4-13686389594520*c[5]^3*c[4]^3)/c[5] /(19470*c[5]^6*c[4]^2+19057*c[5]^5*c[4]+15513*c[4]^4*c[5]+278740*c[5]^ 3*c[4]^4-73750*c[5]^6*c[4]^3-14451*c[5]^4*c[4]-171360*c[5]^4*c[4]^3-12 1240*c[5]^5*c[4]^2+73750*c[5]^4*c[4]^5-71970*c[4]^5*c[5]^3+13860*c[4]^ 5*c[5]^2+336*c[4]^3-1102*c[5]*c[4]^3+944*c[5]^2*c[4]^2-336*c[5]^3-1386 *c[4]^4+1858*c[5]^4-1947*c[5]^5-672*c[5]*c[4]^2+8781*c[5]^2*c[4]^3-954 97*c[4]^4*c[5]^2+87532*c[5]^4*c[4]^2-7896*c[5]^3*c[4]^2-314*c[5]^3*c[4 ]+672*c[5]^2*c[4]-240720*c[5]^4*c[4]^4-11092*c[5]^3*c[4]^3+293220*c[5] ^5*c[4]^3), b[4] = 1/60*(-4967+18275*c[5])/c[4]/(-31941*c[4]^3+11800*c [4]^4+28499*c[4]^2-8358*c[4]-11800*c[5]*c[4]^3+31941*c[5]*c[4]^2-28499 *c[5]*c[4]+8358*c[5]), a[6,5] = -21/12117361*(5082083*c[5]^2*c[4]^2-20 8152*c[4]^2+74088*c[4]-1687518*c[5]^2*c[4]-3325350*c[5]*c[4]^2+146202* c[4]^3-2081520*c[4]^4*c[5]+4398661*c[5]*c[4]^3-6467580*c[5]^2*c[4]^3+1 097208*c[5]*c[4]+2924040*c[4]^4*c[5]^2+208152*c[5]^2-148176*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[6] = 714924299/143128440*(990*c[5]*c[4]-395*c[5]-395*c[4]+197)/(5 9*c[4]-42)/(59*c[5]-42), a[8,3] = 3/4*(-3284025*c[4]^4*c[5]-13070200*c [5]^3*c[4]^4-1396450*c[5]^4*c[4]^3+995400*c[4]^5*c[5]+3742200*c[4]^5*c [5]^3-4189350*c[4]^5*c[5]^2+66360*c[4]^3+24822*c[4]-86215*c[4]^2+34758 10*c[5]*c[4]^3+7037402*c[5]^2*c[4]^2-49644*c[5]-132720*c[5]^3+172430*c [5]^2-1574664*c[5]*c[4]^2-14945180*c[5]^2*c[4]^3+13769940*c[4]^4*c[5]^ 2+331800*c[5]^4*c[4]^2-6272315*c[5]^3*c[4]^2+1354900*c[5]^3*c[4]-16428 12*c[5]^2*c[4]+384079*c[5]*c[4]+1247400*c[5]^4*c[4]^4+14054340*c[5]^3* c[4]^3)/c[4]^2/(249200*c[5]^3*c[4]^4+249200*c[5]^4*c[4]^3+6645*c[4]^3- 1678*c[4]^2-64790*c[5]*c[4]^3-105160*c[5]^2*c[4]^2+6645*c[5]^3-1678*c[ 5]^2+10068*c[5]*c[4]^2+365650*c[5]^2*c[4]^3-66450*c[4]^4*c[5]^2-66450* c[5]^4*c[4]^2+365650*c[5]^3*c[4]^2-64790*c[5]^3*c[4]+10068*c[5]^2*c[4] +1678*c[5]*c[4]-880500*c[5]^3*c[4]^3), b[1] = 1/501480*(108330*c[5]*c[ 4]-18275*c[5]-18275*c[4]+4967)/c[5]/c[4], a[5,3] = 3/4*c[5]^2*(-2*c[5] +3*c[4])/c[4]^2, a[5,4] = -c[5]^2*(c[4]-c[5])/c[4]^2, `b*`[9] = 1/10*( 2100*c[5]^2*c[4]^2-1975*c[5]^2*c[4]+420*c[5]^2-1975*c[5]*c[4]^2+2060*c [5]*c[4]-479*c[5]+420*c[4]^2-479*c[4]+126)/(126-1412*c[5]*c[4]^2-1412* c[5]^2*c[4]+336*c[5]^2-437*c[4]-437*c[5]+1707*c[5]*c[4]+1260*c[5]^2*c[ 4]^2+336*c[4]^2), a[8,7] = -27200000000/664859*(125*c[5]*c[4]-33*c[5]- 33*c[4]+8)*(c[4]-1)*(-1+c[5])/(200*c[5]-199)/(200*c[4]-199)/(24920*c[5 ]*c[4]-6645*c[5]-6645*c[4]+1678), a[6,3] = 189/24234722*(5275873*c[5]* c[4]^3-3595452*c[5]*c[4]^2+1027824*c[5]*c[4]-148176*c[5]+74088*c[4]-69 3840*c[5]^3*c[4]^2-11547480*c[5]^2*c[4]^3+8040815*c[5]^2*c[4]^2-225002 4*c[5]^2*c[4]+277536*c[5]^2-2081520*c[4]^4*c[5]-138768*c[4]^2+1462020* c[5]^3*c[4]^3+4386060*c[4]^4*c[5]^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 )/c[4]^2, b[7] = -16000000000/1994577*(125*c[5]*c[4]-33*c[5]-33*c[4]+8 )/(200*c[4]-199)/(200*c[5]-199), `b*`[4] = -1/60*(-824418+2855007*c[4] -2189880*c[4]^2-43144090*c[5]^2*c[4]^2+5982075*c[5]+8330280*c[5]^3-130 31062*c[5]^2+17511570*c[5]*c[4]^2+31185000*c[5]^3*c[4]^2-34985590*c[5] ^3*c[4]+51509610*c[5]^2*c[4]-21974077*c[5]*c[4])/(67200*c[4]^3+112163* c[4]-154264*c[4]^2-282400*c[5]*c[4]^3-533140*c[5]^2*c[4]^2+86963*c[5]- 66864*c[5]^2+622388*c[5]*c[4]^2+252000*c[5]^2*c[4]^3+348188*c[5]^2*c[4 ]-427093*c[5]*c[4]-25074)/(59*c[4]-42)/(c[4]-c[5])/c[4], a[8,6] = -349 1443/140322*(1612090*c[5]*c[4]+322477-644895*c[4]-644895*c[5])*(c[4]-1 )*(-1+c[5])/(59*c[4]-42)/(59*c[5]-42)/(24920*c[5]*c[4]-6645*c[5]-6645* c[4]+1678), `b*`[5] = 1/60*(-824418+8330280*c[4]^3+5982075*c[4]-130310 62*c[4]^2-34985590*c[5]*c[4]^3-43144090*c[5]^2*c[4]^2+2855007*c[5]-218 9880*c[5]^2+51509610*c[5]*c[4]^2+31185000*c[5]^2*c[4]^3+17511570*c[5]^ 2*c[4]-21974077*c[5]*c[4])/(86963*c[4]-66864*c[4]^2-533140*c[5]^2*c[4] ^2+112163*c[5]+67200*c[5]^3-154264*c[5]^2+348188*c[5]*c[4]^2+252000*c[ 5]^3*c[4]^2-282400*c[5]^3*c[4]+622388*c[5]^2*c[4]-427093*c[5]*c[4]-250 74)/(59*c[5]-42)/(c[4]-c[5])/c[5], a[7,6] = -136548076561/134400000000 *(5*c[5]*c[4]+1-2*c[4]-2*c[5])*(200*c[4]-199)*(200*c[5]-199)/(125*c[5] *c[4]-33*c[5]-33*c[4]+8)/(59*c[4]-42)/(59*c[5]-42), `b*`[7] = -8000000 0/1994577*(-27720*c[4]^3-21954*c[4]+45825*c[4]^2+207630*c[5]*c[4]^3+73 8420*c[5]^2*c[4]^2-21954*c[5]-27720*c[5]^3+45825*c[5]^2-336157*c[5]*c[ 4]^2-471910*c[5]^2*c[4]^3-471910*c[5]^3*c[4]^2+207630*c[5]^3*c[4]-3361 57*c[5]^2*c[4]+158292*c[5]*c[4]+315000*c[5]^3*c[4]^3+3024)/(-13372800* c[4]^3-22320437*c[4]+30698536*c[4]^2+69637600*c[5]*c[4]^3+230572460*c[ 5]^2*c[4]^2-22320437*c[5]-13372800*c[5]^3+30698536*c[5]^2-154708012*c[ 5]*c[4]^2-106628000*c[5]^2*c[4]^3-106628000*c[5]^3*c[4]^2+69637600*c[5 ]^3*c[4]-154708012*c[5]^2*c[4]+107424107*c[5]*c[4]+50400000*c[5]^3*c[4 ]^3+4989726)\}:" }}}{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 98 "A lengthy computation gives an expression for the square of the principal error norm in terms of " }{XPPEDIT 18 0 "c[2 ]" "6#&%\"cG6#\"\"#" }{TEXT -1 3 ", " }{XPPEDIT 18 0 "c[4]" "6#&%\"cG 6#\"\"%" }{TEXT -1 7 " and " }{XPPEDIT 18 0 "c[5]" "6#&%\"cG6#\"\"& " }{TEXT -1 2 ". " }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 261 "errter ms6_8 := PrincipalErrorTerms(6,8,'expanded'):\nsm := 0:\nfor ct to nop s(errterms6_8) do\n print(ct);\n sm := sm+(simplify(subs(eB,errter ms6_8[ct])))^2;\nend do:\nsm := simplify(sm):\nprin_err_norm_sqrd := u napply(%,c[2],c[4],c[5]):\nprin_err_norm_sqrd(u,v,w);" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}} {SECT 1 {PARA 4 "" 0 "" {TEXT -1 18 "prin_err_norm_sqrd" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 4642 "prin_ err_norm_sqrd := (u,v,w)->1/4092720307200000000*(-7069785241691760000* v^8*w^3-1078840647324178420*v^7*w^2-34359462863040000*w*v^7-5656015776 18420*w^7*v^2+156350277216000000*v^10*w^2+5993350032915275520*w^3*v^7+ 13025259576000000*w*v^8+9076500623530680600*w^5*v^6-12963159167154496* v^5*w-3381480243960462300*w^4*v^7+1608138205254000000*v^10*w^4-1002861 243216000000*v^10*w^3+1206755392951920000*v^8*w^2-26009384960935480*w^ 4*v^2-2922120513285797340*w^3*v^6-1197961432847225220*w^4*v^6+75560225 35763520*w^7*v^3+773223557761961180*w^5*v^4+105169183760593314*w^5*v^3 +613511619515197281*v^6*w^2-4052267100731065980*w^5*v^5-25798160913382 606*w^3*v^3+7907224084395408*w^3*v^2-720050105072592*w^3*v-10337245772 10563109*w^4*v^4-2751367858843170000*w^6*v^7+7792205693843452100*w^4*v ^8-10130818623565392*w^2*v^3+62245996374927720*v^4*w^2-502655941194467 9400*w^6*v^5-1448654315220000000*v^9*w^5-6007792745589600000*v^9*w^4+6 404764849212135000*w^5*v^8+4265103947731200000*v^9*w^3-208120653502650 00*w^8*v^5+38290387208062500*w^8*v^6-701874742464000000*v^9*w^2+285867 0224177430780*w^6*v^4-10910444530391606600*w^5*v^7+5272755766050685100 *w^6*v^6+2828007888092100*w^8*v^4+589739644238062500*w^6*v^8+765807744 16125000*w^7*v^7-316865907992592*w*v^3+9225734526485505000*u^2*v^6*w^4 -8048764940980365000*u^2*v^5*w^4-725283577683000000*u^2*v^8*w^3+779153 126646888*w^2*v^2+55784186348625000*u^2*v^6*w^6+44564943516660000*v^6* w*u+509425370039250000*u^2*v^7*w^5+3233128422056400000*u^2*v^7*w^3+116 3028523442625000*u^2*v^8*w^4-17089553113605000*w^5*u*v-530683822501245 0000*u^2*v^7*w^4-655360357463379000*w^5*u*v^3-507605840532000000*u^2*v ^7*w^2+21513928387770000*w^6*u*v^3+10095029512833600*w^4*v*u+113074754 058000000*u^2*v^8*w^2-1025398566317410200*v^4*w^3*u+21238130636852850* u^2*w^2*v^2-3379458418761600*u^2*w^2*v+79375504119786000*u^2*w^5*v^3-1 761593498280000*w^6*u*v^2-177005778809050800*u^2*w^3*v^2+6215789920284 000*u*v^2*w^2+366023177994600*u*v^2*w+46110383160021900*w^4*v^3*u+2673 22362031405800*u^2*w^4*v^2+823525988755062600*u^2*w^3*v^3+140268925331 955000*w^5*u*v^2-35664504290100*w^3*u*v+4122869096278737000*u^2*w^4*v^ 4+22841523231294600*u^2*w^3*v+15798684716924400*u*v^4*w-65197572685560 00*u^2*w^5*v^2-3583691177484600*u*v^3*w-50220018495402300*w^4*v^2*u-13 25346087917376000*u^2*w^4*v^3-37822631967845100*u*v^3*w^2-506833945443 000*u*v*w^2+152122332500816400*u*v^4*w^2+201981435702791400*u*v^3*w^3- 20023682178241800*u*v^2*w^3-2468761668979767000*u^2*w^3*v^4-3155903205 6963600*u^2*w^4*v+2733995415954600*u^2*v^2*w+4767284104470531000*u^2*v ^5*w^3+30707881852605300*u^2*v^4*w+283674531580231050*u^2*v^4*w^2+8174 21209405605000*u^2*v^6*w^2-36954364856220000*u^2*v^5*w-909021594411819 00*u^2*v^3*w^2-12335867516076300*u^2*v^3*w-150766338744000000*v^9*w^2* u-621052146163827000*u^2*v^5*w^2-1550704697923500000*v^9*w^4*u+1394986 5653220000*u^2*v^6*w+6701705588875500000*w^4*v^8*u-745081796100600000* w^5*v^7*u-234133589510100000*w^6*v^6*u-16838227372680000*v^7*w*u+96704 4770244000000*v^9*w^3*u-3519124771133934000*w^4*v^5*u+2633273578121580 000*w^5*v^6*u+654989875384473000*w^4*v^4*u-3067014340384245000*w^5*v^5 *u-73689934437000000*w^5*v^8*u+58240083982500000*w^6*v^7*u-43958303870 6348100*v^5*w^2*u+7195745962302975000*v^7*w^3*u+3179576272509999000*v^ 5*w^3*u+676807787376000000*v^8*w^2*u-1111408874261910000*v^7*w^2*u-379 52742633900300*v^5*w*u-6204580077802302000*v^6*w^3*u+89133278216636700 0*v^6*w^2*u+1847998942652178000*w^5*v^4*u-4253070661686000000*v^8*w^3* u+8516452141038465000*v^6*w^4*u-10879055506266660000*v^7*w^4*u-1029852 81122535000*w^6*v^4*u+234465605564940000*w^6*v^5*u-281621534896800*u^2 *w*v-5513684678552295000*u^2*v^6*w^3-1296940927618350000*u^2*v^6*w^5+6 174883868580000*u^2*v^4*w^6+48672594355896*v^4+70405383724200*u^2*v^2+ 373679837981100*v^4*u-75070803515400*v^3*u+430243607187450*u^2*v^4-406 987892998470000*u^2*v^4*w^5-37119291570900000*u^2*v^5*w^6+105833085736 2660000*u^2*v^5*w^5-248948398395792*v^5-1392354942098400*u^2*w^3+92999 1043548000*w^5*u+3070508215911840*v^4*w-59240198700946998*w^3*v^4-3480 88735524600*u^2*v^3-464995521774000*v^5*u+281621534896800*u^2*w^2+1720 974428749800*u^2*w^4+319160618312921*v^6-1024398177529853340*w^6*v^3-5 34254478017760*w^4*v-747359675962200*w^4*u+150141607030800*w^3*u+22661 1842589831681*w^6*v^2+17147793063803904*w^5*v+251426147259073002*w^4*v ^3-86671683388864206*w^5*v^2+243248247067896*w^4-28309889951793702*w^6 *v+2079368348649537180*w^4*v^5+29794872533522298*v^6*w-514768784848623 00*w^7*v^4-1222164990855792*w^5+189036893207273400*w^7*v^5-29217851929 9170000*w^7*v^6+776089127020436514*v^5*w^3-237910641565024206*v^5*w^2+ 1535943466320921*w^6)/(10*w^3*v^2+10*w^2*v^3-30*w^2*v^2+6*w^2*v-w^2+6* w*v^2+w*v-v^2)^2:" }}}{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 12 "The values " }{XPPEDIT 18 0 "c[2] = 17/183;" " 6#/&%\"cG6#\"\"#*&\"#<\"\"\"\"$$=!\"\"" }{TEXT -1 3 ", " }{XPPEDIT 18 0 "c[4] = 18/83;" "6#/&%\"cG6#\"\"%*&\"#=\"\"\"\"#$)!\"\"" }{TEXT -1 7 " and " }{XPPEDIT 18 0 "c[5] = 71/125;" "6#/&%\"cG6#\"\"&*&\"#r \"\"\"\"$D\"!\"\"" }{TEXT -1 108 " of Papakostas' scheme already give a value for the principal error norm that is close to a minimum (with " }{XPPEDIT 18 0 "c[6]" "6#&%\"cG6#\"\"'" }{TEXT -1 7 " and " } {XPPEDIT 18 0 "c[7]" "6#&%\"cG6#\"\"(" }{TEXT -1 9 " fixed)." }} {PARA 0 "" 0 "" {TEXT -1 97 "These values can be used as starting valu es to minimize the (square of the) principal error norm." }}{PARA 0 " " 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 119 "Using a one dimen sional minimization procedure and cycling around the nodes gives slow \+ convergence towards the minimum." }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 494 "Digits := 30:\nc_2 := 17/183: c_4 := 18/83: c_5 := 71/125:\nf or ct to 100000 do\n c_2 := findmin(prin_err_norm_sqrd(c2,c_4,c_5),c 2=\{0.05,c_2,0.13\},convergence=location)[1];\n c_4 := findmin(prin_ err_norm_sqrd(c_2,c4,c_5),c4=\{0.19,c_4,0.25\},convergence=location)[1 ];\n mn := findmin(prin_err_norm_sqrd(c_2,c_4,c5),c5=\{0.5,c_5,0.64 \},convergence=location); \n c_5 := mn[1]:\n if `mod`(ct,1000)=0 t hen\n print(c[2]=c_2,c[4]=c_4,c[5]=c_5);\n print(mn[2]);\n \+ end if;\nend do:\nDigits := 10:" }}{PARA 11 "" 1 "" {XPPMATH 20 "6%/& %\"cG6#\"\"#$\"?lK6%4afw6*[]\"!#R" }}{PARA 11 "" 1 "" {XPPMATH 20 "6%/&%\"cG6#\"\"#$ \"?&Gjlty'y9\\3\"G\\_F*!#J/&F%6#\"\"%$\"?E'))>92\"4-\"\\>k!4m@!#I/&F%6 #\"\"&$\"?$[&*4!\\\"ez\"R+-r'\\n&F1" }}{PARA 11 "" 1 "" {XPPMATH 20 "6 #$\"?))Qfq:)oJ2OHY0T]\"!#R" }}{PARA 11 "" 1 "" {XPPMATH 20 "6%/&%\"cG6 #\"\"#$\"?,_@![znU#RQ%yPJF*!#J/&F%6#\"\"%$\"?t')H>YFYqD)f@[c;#!#I/&F%6 #\"\"&$\"?]g*p(Hp7*y!eyK8ucF1" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#$\"?? Y8R&><74tuZ2P]\"!#R" }}{PARA 259 "" 0 "" {TEXT -1 1 ":" }}{PARA 259 " " 0 "" {TEXT -1 1 ":" }}{PARA 11 "" 1 "" {XPPMATH 20 "6%/&%\"cG6#\"\"# $\"?d%yc$*G\"=!Rb#G=+o#*!#J/&F%6#\"\"%$\"?hG\\<&GZYDSAXtX;#!#I/&F%6#\" \"&$\"?\"3j'Q7*>KvbRH4@n&F1" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#$\"?N(= w$oObwt`)32L]\"!#R" }}{PARA 11 "" 1 "" {XPPMATH 20 "6%/&%\"cG6#\"\"#$ \"?gXtmp1 " 0 "" {MPLTEXT 1 0 393 "c_2 := .9268001828 26e-1: pp := .1503307088537e-9:\np1 := evalf[30](plot(prin_err_norm_sq rd(c[2],.216457345224,.567210929396),c[2]=0.09..0.0954,\n co lor=COLOR(RGB,.5,0,.9))):\np2 := plot([[[c_2,pp]]$4],style=point,symbo l=[circle$2,diamond,cross],symbolsize=[12,10$3],\n color=[bl ack,red$3]):\nplots[display]([p1,p2],font=[HELVETICA,9],view=[0.09..0. 0954,1.5032e-10..1.5044e-10]);" }}{PARA 13 "" 1 "" {GLPLOT2D 367 347 347 {PLOTDATA 2 "6*-%'CURVESG6$7S7$$\"\"*!\"#$\"?bedfpH63&3*>))R/:!#R7 $$\"?+++++++++Dj/x6!*!#J$\"?q(\\v\"\\Y/)4>&H]I/:F-7$$\"?++++++++vQ!)=, A!*F1$\"?eVu1:LK&zb&\\oA/:F-7$$\"?++++++++]2D%HN.*F1$\"?agL!Gq>NX]stUT ]\"F-7$$\"?++++++++]_:L7X!*F1$\"?L,B+$ymh\"\\AS@1/:F-7$$\"?++++++++vB- @mc!*F1$\"?pu]@_.+$)=Y$)f)R]\"F-7$$\"?++++++++veM+On!*F1$\"?$R'H;o240s C#**=R]\"F-7$$\"?++++++++v)Q7P%y!*F1$\"?5mPs^)GaB&*HH`Q]\"F-7$$\"?++++ ++++v[lI*)*3*F1$\"?#Q[\\?^')R[)Qq#*y.:F-7$$\"?++++++++vtnAJ,\"*F1$\"?L oT#QLDt=aNUHP]\"F-7$$\"?+++++++++&3KeI6*F1$\"?kpP4DweZ@i**>n.:F-7$$\"? ++++++++]#3D/M7*F1$\"?WohSOfZ\")*3Z*[i.:F-7$$\"?+++++++++b<80N\"*F1$\" ?))=*>n=gy%HCfdd.:F-7$$\"?+++++++++D3iuY\"*F1$\"?0ZG3SV2\"468dIN]\"F-7 $$\"?+++++++++X-k,e\"*F1$\"?g;.3;\\N^TTe4\\.:F-7$$\"?++++++++v[=3Do\"* F1$\"?N,w&)oq4PuSJ$eM]\"F-7$$\"?+++++++++0H0U!=*F1$\"?,M8!>:%o5`,zOU.: F-7$$\"?+++++++++?q)H2>*F1$\"?6+UhN')\\S)yj%yR.:F-7$$\"?++++++++vy%3AF ?*F1$\"?q'*['z=iB#\\,i=P.:F-7$$\"?+++++++++qEsL8#*F1$\"?'y!>(*HsdBD74D N.:F-7$$\"?++++++++vy>P)\\A*F1$\"?+2%HV*fgSd!o@NL]\"F-7$$\"?++++++++DO NR2O#*F1$\"?mGYzk*)4.v:$eAL]\"F-7$$\"?++++++++](=TXwC*F1$\"?JnqhF+hrZ7 (Q8L]\"F-7$$\"?++++++++v`R;Fe#*F1$\"?8[m:KE*H)>hF&3L]\"F-7$$\"?+++++++ +]ihMtp#*F1$\"?l2Q@g/1SZ6MrI.:F-7$$\"?++++++++v[t!R;G*F1$\"??K'eu&eP9] K:*4L]\"F-7$$\"?++++++++DhVH+#H*F1$\"?)[L$pf0x@(zW%eJ.:F-7$$\"?+++++++ +]F2i>.$*F1$\"?d![h(f\"eaG!y:fK.:F-7$$\"?+++++++++!R%*fZJ*F1$\"?<\\$*F1$\"?? WfS5)f<@6WB2M]\"F-7$$\"?+++++++++qyB4g$*F1$\"?=\"3him2>]3P*fV.:F-7$$\" ?+++++++++D?Avr$*F1$\"?F\\2BIN%4\\]AqqM]\"F-7$$\"?++++++++vopxJ#Q*F1$ \"?N6y4%)eZnvjAd].:F-7$$\"?+++++++++&fqoQR*F1$\"?\"3I:$o%\\W$fA#*ya.:F -7$$\"?++++++++vyOst/%*F1$\"?*=`D)y;v4sAu7f.:F-7$$\"?++++++++v8$)z4;%* F1$\"?!zi8/5*Q#*\\^f/k.:F-7$$\"?+++++++++D*=0sU*F1$\"?h,5KDd5$))[.M#p. :F-7$$\"?++++++++vo/M$)Q%*F1$\"?g'p+B?2$fK?t1v.:F-7$$\"?+++++++++?eF.] %*F1$\"?SaxF_q%)oieS2\"Q]\"F-7$$\"?++++++++]s9d[h%*F1$\"?7iHIH?V&*ob6h (Q]\"F-7$$\"?++++++++v)\\$Q%GZ*F1$\"?EUM?F4*G4/:F-7$$\"?+++++++++SR:%f]*F1$\"?kk(G`I=p)*[XjnT]\"F-7$$\"?++++ ++++vyk([t^*F1$\"?jY\")pnuWHgWC@D/:F-7$$\"?++++++++D,NpEG&*F1$\"?Mm?^q pZ$HnqpOV]\"F-7$$\"$a*!\"%$\"?T/7yn6ZWxa@;V/:F--%&COLORG6&%$RGBG$\"\"& !\"\"$\"\"!Fb[l$F)F`[l-F$6&7#7$$\"3?++EG=+o#*!#>$\"30+q`)32L]\"!#F-%'C OLOURG6&F][lFb[lFb[lFb[l-%'SYMBOLG6$%'CIRCLEG\"#7-%&STYLEG6#%&POINTG-F $6&Ff[l-F_\\l6&F][l$\"*++++\"!\")Fa[lFa[l-Fb\\l6$Fd\\l\"#5Ff\\l-F$6&Ff 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],view=[0.5672045..0.5672175,1.5032e-10..1.50446e-10]);" }}{PARA 13 " " 1 "" {GLPLOT2D 368 366 366 {PLOTDATA 2 "6*-%'CURVESG6$7S7$$\"(X?n&! \"($\"?\"*HFdd@7VqJF#eW]\"!#R7$$\"?nmmmmmmT+jLy/sc!#I$\"?-v4NpYiv;Ea*e V]\"F-7$$\"?LLLLLLe*Qc\"*H]?n&F1$\"?Xu\\\"4O%Hdkj@iF/:F-7$$\"?nmmmmm;H ')*=2`?n&F1$\"?,:og&oI4lQ5B(=/:F-7$$\"?nmmmmm;//-je0scF1$\"??=K3Rj')RL G\")>5/:F-7$$\"?LLLLLLekk(3ke?n&F1$\"?O1BwBmnvBK]9-/:F-7$$\"?nmmmmm\"H i/j@h?n&F1$\"?LClo%**RJ$QwJ1&R]\"F-7$$\"?++++++D18,$)Q1scF1$\"?o6YD*Q2 VSq.?\")Q]\"F-7$$\"?nmmmmm\"Hy#p-Y/SzxV#)e.:F-7$$\"?+++++++v3zF.3scF1$\"?IJm Vw1X+[\"yiSN]\"F-7$$\"?+++++++vd)4/$3scF1$\"?6n1HB)=gZ7&>*)\\.:F-7$$\" ?nmmmmm\"HnE[]&3scF1$\"?.P^[\"z?8]erfkM]\"F-7$$\"?LLLLLLL3=dM%)3scF1$ \"?<\"*>/[YvFJQ&=GM]\"F-7$$\"?LLLLLLLL-X;44scF1$\"?N`4v_D*[Tt*z5S.:F-7 $$\"?++++++Dc[Y.Q4scF1$\"?WLePG#4)e$od'QP.:F-7$$\"?LLLLLLL$)>'*ej4scF1 $\"?![\"y`s%)4,8z[ON.:F-7$$\"?++++++Dctui\"*4scF1$\"?gDyAeX%y5Z0lNL]\" F-7$$\"?++++++voShK=5scF1$\"?Qc*>)p0GP'4(zDK.:F-7$$\"?LLLLLL$e*)R$=Y5s 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{MPLTEXT 1 0 154 "nds := [c[2]=.926800182826e-1,c[4]=.216457345224,c[5]=.56721092 9396]:\nevalf[10](%);\nfor dgt from 6 by -1 to 4 do\n map(convert,nd s,rational,dgt);\nend do;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#7%/&%\"cG 6#\"\"#$\"+G=+o#*!#6/&F&6#\"\"%$\"+_Mdk@!#5/&F&6#\"\"&$\"+%H4@n&F2" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#7%/&%\"cG6#\"\"##\"$Q\"\"%*[\"/&F&6#\" \"%#\"$@\"\"$f&/&F&6#\"\"&#\"$t\"\"$0$" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#7%/&%\"cG6#\"\"##\"#>\"$0#/&F&6#\"\"%#\"#]\"$J#/&F&6#\"\"&#\"#Q \"#n" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#7%/&%\"cG6#\"\"##\"\"&\"#a/&F& 6#\"\"%#\"#@\"#(*/&F&6#F*#\"#<\"#I" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" } }{PARA 0 "" 0 "" {TEXT -1 56 "The minimum value for the principal erro r norm is . . . " }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 92 "evalf[25 ](prin_err_norm_sqrd(.926800182826e-1,.216457345224,.567210929396)):\n evalf(sqrt(%));" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#$\"+TU4E7!#9" }}} {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 66 "#-------- ---------------------------------------------------------" }}{PARA 0 " " 0 "" {TEXT -1 6 "When " }{XPPEDIT 18 0 "c[2] = 19/205;" "6#/&%\"cG6 #\"\"#*&\"#>\"\"\"\"$0#!\"\"" }{TEXT -1 3 ", " }{XPPEDIT 18 0 "c[4] = 121/559;" "6#/&%\"cG6#\"\"%*&\"$@\"\"\"\"\"$f&!\"\"" }{TEXT -1 7 " a nd " }{XPPEDIT 18 0 "c[5] = 173/305;" "6#/&%\"cG6#\"\"&*&\"$t\"\"\"\" \"$0$!\"\"" }{TEXT -1 67 ", the principal error norm is given (approx imately) as follows. " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 70 "evalf[15](prin_err_norm_sqrd(19/205 ,121/559,173/305)):\nevalf(sqrt(%));" }}{PARA 11 "" 1 "" {XPPMATH 20 " 6#$\"+th5E7!#9" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 124 "The maximum magnitude of the linking coefficients is app roximately 18.65 and the real stability interval is approximately " }{XPPEDIT 18 0 " [-4.4487, 0]" "6#7$,$-%&FloatG6$\"&([W!\"%!\"\"\"\"! " }{TEXT -1 2 ". " }}{PARA 0 "" 0 "" {TEXT -1 53 "(See the abreviated \+ calculations in a later section.)" }}{PARA 0 "" 0 "" {TEXT -1 66 "#--- --------------------------------------------------------------" }} {PARA 0 "" 0 "" {TEXT -1 6 "When " }{XPPEDIT 18 0 "c[2] = 19/205;" "6 #/&%\"cG6#\"\"#*&\"#>\"\"\"\"$0#!\"\"" }{TEXT -1 3 ", " }{XPPEDIT 18 0 "c[4] = 50/231;" "6#/&%\"cG6#\"\"%*&\"#]\"\"\"\"$J#!\"\"" }{TEXT -1 7 " and " }{XPPEDIT 18 0 "c[5] = 173/305;" "6#/&%\"cG6#\"\"&*&\"$t\" \"\"\"\"$0$!\"\"" }{TEXT -1 67 ", the principal error norm is given ( approximately) as follows. " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 69 "evalf[15](prin_err_norm_sqrd (19/205,50/231,173/305)):\nevalf(sqrt(%));" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#$\"+QP%)G7!#9" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }} {PARA 0 "" 0 "" {TEXT -1 124 "The maximum magnitude of the linking coe fficients is approximately 18.54 and the real stability interval is ap proximately " }{XPPEDIT 18 0 "[-4.4502, 0];" "6#7$,$-%&FloatG6$\"&-X %!\"%!\"\"\"\"!" }{TEXT -1 2 ". " }}{PARA 0 "" 0 "" {TEXT -1 53 "(See \+ the abreviated calculations in a later section.)" }}{PARA 0 "" 0 "" {TEXT -1 66 "#-------------------------------------------------------- ---------" }}{PARA 0 "" 0 "" {TEXT -1 6 "When " }{XPPEDIT 18 0 "c[2] \+ = 17/183;" "6#/&%\"cG6#\"\"#*&\"#<\"\"\"\"$$=!\"\"" }{TEXT -1 3 ", " }{XPPEDIT 18 0 "c[4] = 18/83;" "6#/&%\"cG6#\"\"%*&\"#=\"\"\"\"#$)!\"\" " }{TEXT -1 7 " and " }{XPPEDIT 18 0 "c[5] = 71/125;" "6#/&%\"cG6#\" \"&*&\"#r\"\"\"\"$D\"!\"\"" }{TEXT -1 106 " (the values used by Papak ostas' scheme) the principal error norm is given (approximately) as f ollows. " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 67 "evalf[15](prin_err_norm_sqrd(17/183,18/83,71/125)):\n evalf(sqrt(%));" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#$\"+O7OJ7!#9" }}} {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 124 "The maxi mum magnitude of the linking coefficients is approximately 18.35 and t he real stability interval is approximately " }{XPPEDIT 18 0 "[-4.45 00, 0];" "6#7$,$-%&FloatG6$\"&+X%!\"%!\"\"\"\"!" }{TEXT -1 2 ". " }} {PARA 0 "" 0 "" {TEXT -1 53 "(See the abreviated calculations in a lat er section.)" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT 258 10 "Conclusion" }{TEXT -1 106 ": There is little, if anything, to \+ be gained by changing the nodes to either of the previous alternatives ." }}{PARA 0 "" 0 "" {TEXT -1 66 "#----------------------------------- ------------------------------" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}}{PARA 0 "" 0 "" {TEXT -1 52 "#---------------------------- -----------------------" }}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 38 "charac teristics of the embedded scheme" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}{SECT 1 {PARA 4 "" 0 " " {TEXT -1 36 "coefficients for the combined scheme" }}{PARA 0 "" 0 " " {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1841 "ee := \{ c[2]=17/183,\nc[3]=12/83,\nc[4]=18/83,\nc[5]=71/125,\nc[6]=42/59,\nc[7 ]=199/200,\nc[8]=1,\nc[9]=1,\n\na[2,1]=17/183,\na[3,1]=3756/117113,\na [3,2]=13176/117113,\na[4,1]=9/166,\na[4,2]=0,\na[4,3]=27/166,\na[5,1]= 207751751/316406250,\na[5,2]=0,\na[5,3]=-526769377/210937500,\na[5,4]= 1524242129/632812500,\na[6,1]=-4970082682619223281/2887511529739311186 ,\na[6,2]=0,\na[6,3]=97919278033879057/13556392158400522,\na[6,4]=-407 131674007930877068/74078904949579652469,\na[6,5]=123760185520426875000 0/1753200750473385108433,\na[7,1]=176597685527535385020980411/42773485 015591331328000000,\na[7,2]=0,\na[7,3]=-6793162515552646891859/4016289 67282547712000,\na[7,4]=12704926019361287204873446554247/8866594026530 54716778496000000,\na[7,5]=-50728836334509259632278125/326575917180086 85915971584,\na[7,6]=51536223982796190703/51293749413888000000,\na[8,1 ]=299033520572337573523/66918720793812357519,\na[8,2]=0,\na[8,3]=-1655 0269823961899/902146153892364,\na[8,4]=49920346343238033627496282/3215 735869387500624775563,\na[8,5]=-1686432488955761721093750/978844996793 357447730403,\na[8,6]=161901609084039/149698803705724,\na[8,7]=-305146 137600000/54760341991955873,\na[9,1]=24503/381483,\na[9,2]=0,\na[9,3]= 0,\na[9,4]=1366847103121/4106349847584,\na[9,5]=20339599609375/7593391 3767768,\na[9,6]=35031290651/194765546144,\na[9,7]=16620160000000/1100 1207123543,\na[9,8]=-14933/11016,\n\nb[1]=24503/381483,\nb[2]=0,\nb[3] =0,\nb[4]=1366847103121/4106349847584,\nb[5]=20339599609375/7593391376 7768,\nb[6]=35031290651/194765546144,\nb[7]=16620160000000/11001207123 543,\nb[8]=-14933/11016,\n\n`b*`[1]=61010485298317/979331468960880,\n` b*`[2]=0,\n`b*`[3]=0,\n`b*`[4]=320207313882553286621/94122281340699239 5200,\n`b*`[5]=6845867841119140625/29008216787127405534,\n`b*`[6]=1241 09197949158875473/562495660250110816320,\n`b*`[7]=19339714537078400000 /16810691577722216811,\n`b*`[8]=-211029377951/210416202900,\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 compone nts are the principal error terms of the embedded 9 stage, order 5 sch eme (the error terms of order 6) and let " }{XPPEDIT 18 0 "`T*`[6,9]; " "6#&%#T*G6$\"\"'\"\"*" }{TEXT -1 99 " denote the vector whose compo nents are the error terms of order 7 of the embedded order 5 scheme." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 39 "Denote \+ the 2-norm of these vectors by " }{XPPEDIT 18 0 "abs(abs(T[6,8]));" " 6#-%$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#\"\"(-%$absG6#-F)6#&%\"TG6$\"\"'\"\")" }{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$\"\"&F3F9 " }{TEXT -1 1 "." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 77 "Dormand and Prince have suggested that as well as attempt ing to ensure that " }{XPPEDIT 18 0 "A[7];" "6#&%\"AG6#\"\"(" }{TEXT -1 73 " is a minimum, if the embedded scheme is to be used for error \+ control, " }{XPPEDIT 18 0 "B[7];" "6#&%\"BG6#\"\"(" }{TEXT -1 7 " an d " }{XPPEDIT 18 0 "C[7];" "6#&%\"CG6#\"\"(" }{TEXT -1 27 " should b e chosen so that " }}{PARA 0 "" 0 "" {TEXT -1 68 "they are \"similar i n magnitude\" and also not differ too much from 1." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 186 "errterms6_8 := PrincipalErrorTerms(6,8,'expanded'):\n`errterms6_9*` :=subs(b=`b*` ,PrincipalErrorTerms(6,9,'expanded')):\n`errterms5_9*` := subs(b=`b*`, PrincipalErrorTerms(5,9,'expanded')):" }}}{PARA 0 "" 0 "" {TEXT -1 0 " " }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 326 "snmB := sqrt(add(evalf( subs(ee,`errterms6_9*`[i]))^2,i=1..nops(`errterms6_9*`))):\nsdnB := sq rt(add(evalf(subs(ee,`errterms5_9*`[i]))^2,i=1..nops(`errterms5_9*`))) :\nsnmC := sqrt(add((evalf(subs(ee,`errterms6_9*`[i])-subs(ee,errterms 6_8[i])))^2,i=1..nops(errterms6_8))):\n'B[7]'= evalf[8](snmB/sdnB);\n' C[7]'= evalf[8](snmC/sdnB);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/&%\"BG 6#\"\"($\")&G=O\"!\"(" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/&%\"CG6#\"\" ($\")`gi8!\"(" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}}{PARA 0 "" 0 "" {TEXT -1 52 "#------------ ---------------------------------------" }}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 27 "absolute stability regions " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 36 "coefficients for the combined scheme" }}{PARA 0 " " 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1841 "ee \+ := \{c[2]=17/183,\nc[3]=12/83,\nc[4]=18/83,\nc[5]=71/125,\nc[6]=42/59, \nc[7]=199/200,\nc[8]=1,\nc[9]=1,\n\na[2,1]=17/183,\na[3,1]=3756/11711 3,\na[3,2]=13176/117113,\na[4,1]=9/166,\na[4,2]=0,\na[4,3]=27/166,\na[ 5,1]=207751751/316406250,\na[5,2]=0,\na[5,3]=-526769377/210937500,\na[ 5,4]=1524242129/632812500,\na[6,1]=-4970082682619223281/28875115297393 11186,\na[6,2]=0,\na[6,3]=97919278033879057/13556392158400522,\na[6,4] =-407131674007930877068/74078904949579652469,\na[6,5]=1237601855204268 750000/1753200750473385108433,\na[7,1]=176597685527535385020980411/427 73485015591331328000000,\na[7,2]=0,\na[7,3]=-6793162515552646891859/40 1628967282547712000,\na[7,4]=12704926019361287204873446554247/88665940 2653054716778496000000,\na[7,5]=-50728836334509259632278125/3265759171 8008685915971584,\na[7,6]=51536223982796190703/51293749413888000000,\n a[8,1]=299033520572337573523/66918720793812357519,\na[8,2]=0,\na[8,3]= -16550269823961899/902146153892364,\na[8,4]=49920346343238033627496282 /3215735869387500624775563,\na[8,5]=-1686432488955761721093750/9788449 96793357447730403,\na[8,6]=161901609084039/149698803705724,\na[8,7]=-3 05146137600000/54760341991955873,\na[9,1]=24503/381483,\na[9,2]=0,\na[ 9,3]=0,\na[9,4]=1366847103121/4106349847584,\na[9,5]=20339599609375/75 933913767768,\na[9,6]=35031290651/194765546144,\na[9,7]=16620160000000 /11001207123543,\na[9,8]=-14933/11016,\n\nb[1]=24503/381483,\nb[2]=0, \nb[3]=0,\nb[4]=1366847103121/4106349847584,\nb[5]=20339599609375/7593 3913767768,\nb[6]=35031290651/194765546144,\nb[7]=16620160000000/11001 207123543,\nb[8]=-14933/11016,\n\n`b*`[1]=61010485298317/9793314689608 80,\n`b*`[2]=0,\n`b*`[3]=0,\n`b*`[4]=320207313882553286621/94122281340 6992395200,\n`b*`[5]=6845867841119140625/29008216787127405534,\n`b*`[6 ]=124109197949158875473/562495660250110816320,\n`b*`[7]=19339714537078 400000/16810691577722216811,\n`b*`[8]=-211029377951/210416202900,\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 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)*$)F'F1F)F)F)* &#\"/fT)fNeI&\"3+7UGk8KGFF)*$)F'\"\"(F)F)F)*&#\"-H5`5v**\"2+-2QF-sa%F) *$)F'\"\")F)F)F)" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 124 "We can find the point where the boundary of the stabilit y region intersects the negative real axis by solving the equation: " }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "R(z) = 1;" "6#/-%\" RG6#%\"zG\"\"\"" }{TEXT -1 2 ". " }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 28 "z0 := newton(R(z)=1,z=-4.4);" }}{PARA 11 "" 1 "" {XPPMATH 20 " 6#>%#z0G$!+UT)*\\W!\"*" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 303 "z0 := newton(R(z)=1,z=-4.4):\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):\np3 := plo t([[z0,0],[z0,1]],linestyle=3,color=COLOR(RGB,0,.5,0)):\nplots[display ]([p1,p2,p3],view=[-5.19..0.49,-.07..1.47],font=[HELVETICA,9]);" }} {PARA 13 "" 1 "" {GLPLOT2D 360 253 253 {PLOTDATA 2 "6+-%'CURVESG6$7Z7$ $!3Q++++++!>&!#<$\"3?OpPxZ&fz$F*7$$!3QML3T![!f^F*$\"3;5A$zazLg$F*7$$!3 Ynm;#3'4G^F*$\"30KnX$>:&>MF*7$$!3a++DBT9(4&F*$\"3sA$4oG>SC$F*7$$!3kLLL k@>m]F*$\"3)Hm#y`3cwIF*7$$!3E+]U'*)HB,&F*$\"3T7;p%))pL!GF*7$$!3!pm;&Gw Ye\\F*$\"3A>Ffau%>b#F*7$$!3s+](\\(Q*y*[F*$\"3#)Q!G*)GBLH#F*7$$!3nLLV@, KP[F*$\"3)[`t()=-$e?F*7$$!3'RLLd%[MwZF*$\"3XBPLyQ#\\;F*7$$!3E+]<*4%oaYF*$\"3?,5vhl*QF*$\"3 WL4\"H-S=E$F]p7$$!35++S:-YpPF*$\"3p$*>p)Gb7a#F]p7$$!3K+++\"HZkk$F*$\"3 s*\\M#*Gm'))>F]p7$$!3;++gW:!z_$F*$\"3%fiqmj+se\"F]p7$$!3hLL)*\\1D?MF*$ \"3@`Ej!zWJJ\"F]p7$$!3'ommSKVAH$F*$\"3AB))*[Auy2\"F]p7$$!3/nmEGV!Q=$F* $\"3x!3^5P!)3T*!#>7$$!39++0(*RmdIF*$\"3MR'>iC(\\*R)F`s7$$!39nmEI%3g%HF *$\"3'eXeuc;%GzF`s7$$!3-++0xX]BGF*$\"3)*HmG>Mh&y(F`s7$$!3*)***\\\"R>&o q#F*$\"3)e%f%fg'*H&zF`s7$$!3gmm;\\r8&e#F*$\"37$)*QUp`cR)F`s7$$!3ymmrw \\OtCF*$\"30Ny=$[vV,*F`s7$$!3SLL$))e.GN#F*$\"3C!**4fh<'*))*F`s7$$!3nLL )**=uvA#F*$\"33\"=)[!H!f,6F]p7$$!3K++:I;c=@F*$\"3**3DyT)eu@\"F]p7$$!31 LL.z]#3+#F*$\"3I(p**p(eqh8F]p7$$!3M++?,<>z=F*$\"3**)*R@!p.F`\"F]p7$$!3 ;++!4<(>gF]p7$$!3 H++q9zA<:F*$\"3+:R5yE8%>#F]p7$$!3EnmEY;O-9F*$\"35l#QEw]1Y#F]p7$$!3#)** ***pQ<(z7F*$\"3%p*fY.*)R\"y#F]p7$$!3)RL$efMeo6F*$\"3?%3E\"*z'=3JF]p7$$ !3I****fAZ3Z5F*$\"3s+!y3sW'4NF]p7$$!3xqm;(zQwK*F]p$\"3oJ%eIUoY$RF]p7$$ !3&z***\\)ecE8)F]p$\"3CS#oQc%3MWF]p7$$!3'3nmm0VV'pF]p$\"3CVp(f*Qf$)\\F ]p7$$!3P)***\\iqATdF]p$\"3%zR\"Gx!))>j&F]p7$$!3aFLL*)4AjXF]p$\"3e0x?Bt 4OjF]p7$$!33LLLO'R&eLF]p$\"3i`&*yfZFZrF]p7$$!3Uim;`O$Q;#F]p$\"3-eN#=qk U0)F]p7$$!3?*****>$H-m5F]p$\"3C+kIQ4$)))*)F]p7$$\"3v*QLLU?>#>F`s$\"3[- s.#30%>5F*7$$\"3%ymmY^avJ\"F]p$\"3(f+9%)*\\WF*Fi]l-%'SYMBOLG6#%'CIRCLEG-F]]l6&F_]lFd]lFd]lFd]l- %&STYLEG6#%&POINTG-F$6&F_al-Fdal6#%&CROSSGFgalFial-F$6&F_al-Fdal6#%(DI AMONDGFgalFial-F$6%7$7$FaalFc]lF`al-%&COLORG6&F_]lFc]l$\"\"&!\"\"Fc]l- %*LINESTYLEG6#\"\"$-%+AXESLABELSG6%Q\"z6\"Q!Ficl-%%FONTG6#%(DEFAULTG-F \\dl6$%*HELVETICAG\"\"*-%%VIEWG6$;$!$>&!\"#$\"#\\Fidl;$!\"(Fidl$\"$Z\" Fidl" 1 2 0 1 10 0 2 9 1 4 2 1.000000 45.000000 45.000000 0 0 "Curve 1 " "Curve 2" "Curve 3" "Curve 4" "Curve 5" "Curve 6" }}}}{PARA 0 "" 0 " " {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 50 "The following picture s hows the stability region. " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1361 "R := z -> 1+z+1/2*z^2+1/6*z^3+1/2 4*z^4+1/120*z^5+1/720*z^6+\n 53058355984159/272832136428421200*z^7 +997510531029/45472022738070200*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,.05,.13)):\np2 := plots[polygonplot]([seq([pts[i-1],pts [i],[-2.2,0]],i=2..nops(pts))],\n style=patchnogrid,color=COL OR(RGB,.95,.1,.25)):\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=COLOR(RGB ,.48,.05,.13)):\np4 := plots[polygonplot]([seq([pts[i-1],pts[i],[1.9,4 .72]],i=2..nops(pts))],\n style=patchnogrid,color=COLOR(RGB,. 95,.1,.25)):\npts := []: z0 := 2-4.75*I:\nfor ct from 0 to 60 do\n z z := newton(R(z)=exp(ct*Pi/30*I),z=z0):\n z0 := zz:\n pts := [op(p ts),[Re(zz),Im(zz)]]:\nend do:\np5 := plot(pts,color=COLOR(RGB,.48,.05 ,.13)):\np6 := plots[polygonplot]([seq([pts[i-1],pts[i],[1.9,-4.72]],i =2..nops(pts))],\n style=patchnogrid,color=COLOR(RGB,.95,.1,. 25)):\np7 := plot([[[-5.19,0],[2.29,0]],[[0,-5.19],[0,5.19]]],color=bl ack,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)`],axe s=boxed,scaling=constrained);" }}{PARA 13 "" 1 "" {GLPLOT2D 381 565 565 {PLOTDATA 2 "6/-%'CURVESG6$7]z7$$\"\"!F)F(7$F($\"3++++Fjzq:!#=7$$! 3*******H='HJ')!#G$\"35+++UEfTJF-7$$!3&******\\Z&GDS=$G'F-7$$!3%)******3)H%H%f(R&yF-7$$\"33+++N888:!#C$\"3')*****p))eZU*F-7$$\"32+++>OAU8! #B$\"33+++N?b*4\"!#<7$$\"3p*****4%4#HK'FN$\"3#******p>BmD\"FQ7$$\"39++ +#4*o:A!#A$\"3%******Hp#o89FQ7$$\"3Q*****fK!R'Q'FZ$\"3#******z)[rq:FQ7 $$\"33+++AJ=)e\"!#@$\"31+++U]oF#FQ7$$\"3#)*****zp%[C?F_p$\"3%******H!eF_BFQ7$$\"31+++t_JZHF_p$\"3 -+++kI\"f]#FQ7$$\"3(******4[\\%RPF_p$\"3))*****>k[sl#FQ7$$\"3-+++j!fl% 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}{TEXT 260 30 "interval of absolute stability" }{TEXT -1 89 " \+ (or stability interval) is the intersection of the stability region wi th the real line." }}{PARA 0 "" 0 "" {TEXT -1 59 "For this scheme the \+ stability interval is (approximately) " }{XPPEDIT 18 0 "[-4.4500, 0]; " "6#7$,$-%&FloatG6$\"&+X%!\"%!\"\"\"\"!" }{TEXT -1 2 ". " }}{PARA 0 " " 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 148 "We can distort th e boundary curve horizontally by taking the 11th root of the real part of points along the curve. In this way we see that there is " }{TEXT 260 53 "no largest interval on the nonnegative imaginary axis" }{TEXT -1 65 " that contains the origin and lies inside the stability region. " }}{PARA 0 "" 0 "" {TEXT -1 119 "However the stability region inter sects the nonnegative imaginary axis in an interval that does not cont ain the origin." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> \+ " 0 "" {MPLTEXT 1 0 389 "R := z -> 1+z+1/2*z^2+1/6*z^3+1/24*z^4+1/120* z^5+1/720*z^6+\n 53058355984159/272832136428421200*z^7+99751053102 9/45472022738070200*z^8:\nDigits := 25:\npts := []: z0 := 0:\nfor ct f rom 0 to 107 do\n zz := newton(R(z)=exp(ct*Pi/100*I),z=z0):\n z0 : = zz:\n pts := [op(pts),[surd(Re(zz),11),Im(zz)]]:\nend do:\nplot(pt s,color=COLOR(RGB,.9,0,.2),thickness=2,font=[HELVETICA,9]);\nDigits := 10:" }}{PARA 13 "" 1 "" {GLPLOT2D 261 276 276 {PLOTDATA 2 "6(-%'CURVE SG6#7hq7$$\"\"!F)F(7$$!:6]f***fnj?./9B!#E$\":faaV8y*e`EfTJF-7$$!:etu2) 4_F+(p#HQF-$\":\"y(=`g1yrI&=$G'F-7$$!:'420>*f!o;$4*Q^F-$\":)[*os@YV2'z xC%*F-7$$!:<249l#)*RRJVGjF-$\":JfB+Sc;91PmD\"!#D7$$!:Y!31p-D8O!3PV(F-$ \":,tT!4'*HqEjzq:F?7$$!:^A*[Sgfp%3MRZ)F-$\":'4D]:Vi#=fb\\)=F?7$$!:^QX! e)4R9$Qtg%*F-$\":TeGQbral&[6*>#F?7$$!:<*z4x`y>7a;S5F?$\":1mszr`[/7uK^# F?7$$!:HY_y,8\"F?$\":>'HYY7ju#QLu#GF?7$$!:Z&3]/q?`f*fk@\"F?$\":2 2T?KJ?Ak#fTJF?7$$!:F=YcJDD))yT#*H\"F?$\":D\"fX\"4])*p*=vbMF?7$$!:jV=_` !Qj;F?$\":]8#z1W;GgzaE]F?7$$!:7-7% )f)35G?]D9.A[[1H9m[l& F?7$$!:(z@Z4emmt')[P=F?$\":k$fF]kV#*H^-pfF?7$$!:+KFb2WP!pJ&f)=F?$\":o% )=(f.PM?S=$G'F?7$$!:=?E!p0<#*Gv2G>F?$\":#f\"GU403')yUtf'F?7$$!:m%HXG]Q `o-;i>F?$\":Uf^T:RVUS,:\"pF?7$$!:%R\\+GVo4CkJ&)>F?$\":RA&4&H*>F?$\":6wS3(G>#p-=)RvF?7$$!:/g9v*[1(Ho%\\p>F?$\":l *yOetb?V^FjAF?$\":%fsH(Q/+4Z.16*F?7$$\":&R9eeLHdo4x)R#F?$\" :wc:7k_Ln))eZU*F?7$$\":]5-f&)yb%GI/=DF?$\":v\\.0L)H])[8*Q(*F?7$$\":U*f C96(f8(HxFEF?$\":lSS*>GTs7nI05!#C7$$\":-)3YIO;$Hda6t#F?$\":,w!=N(y^F'> sO5Ffu7$$\":')4_&)>XXB\\f*HGF?$\":^>@HAm>#yq8o5Ffu7$$\":>4k@cdj**fp_#H F?$\":F\"4lrS7hM?b*4\"Ffu7$$\":_lotFnKSb\"za1k5si>$F?$\": aRqG2pL,j&z$>\"Ffu7$$\":%**fwx$)\\$>^IGG$F?$\":RW'ohiZ>*f4_A\"Ffu7$$\" ::N#e'QL6J^wyO$F?$\":9:'>D+(ot>BmD\"Ffu7$$\":l[<%[d(>$ohb^MF?$\":*HuF? zqTbj.)G\"Ffu7$$\":Jd,u#p8q8Ffu7$$\":25!G$4 2D_&4=;$p#o89Ffu7$$\":f/Ab1'Gb/&yE&QF ?$\":%\\Mv0-*fe1#4X9Ffu7$$\":&p;.-[t#o*H&)HRF?$\":Q/Z*49DF9-]w9Ffu7$$ \":m6e,0R#ydN41SF?$\":#*py`W9R`\"p!z]\"Ffu7$$\":VX;n^G\\+??93%F?$\":>R ]`bW/!3>JR:Ffu7$$\":-.>I5**\\kw.q%G=P?I%F?$\":d)\\AFx -tLL^L;Ffu7$$\":ia&Rm.45Yl!QP%F?$\":?YxbIAl%F? $\":y1erg&yRl9W!z\"Ffu7$$\":z6$[i_;N1k@>ZF?$\":^&*e!eb\"*)=y4=#=Ffu7$$ \":y^.DK`&*HXHby%F?$\":#>..WMnSG,<`=Ffu7$$\":[t![NC@#\\>y3&[F?$\":*=DD J//508_%)=Ffu7$$\":u#fVLa*[LyQ_\"\\F?$\":M`&oPXjzf>'e\">Ffu7$$\":utau2 +g'[Tey\\F?$\":oV*\\])[fNd!>Z>Ffu7$$\":2r@n)3*G[&Q)3/&F?$\":t5dChEQcW0 &y>Ffu7$$\":qLWaF?$\":U'3uP<'opTOr>#Ffu7$$\":RJX``S())eCp'\\&F?$\":C5W4 %pBnqIDGAFfu7$$\":bD)QTH\")pAB`ZbF?$\":(zk3Btac!))H$fAFfu7$$\":r#Qa,-O `*funf&F?$\":G+T=1?-@Zi.H#Ffu7$$\":7[t*eqWp$yGVk&F?$\":@>\"\\PtZ0]&G&eF?$\":+# [#o))\\IUHT`Z#Ffu7$$\":*Q%\\sk\\^^\")Hx)eF?$\":i]$yJf,zjI\"f]#Ffu7$$\" :7#*fu4!Gf5H\"*>fF?$\":!ywDFu!HQL$RODFfu7$$\":;Ch%H_r&oyu\"\\fF?$\":Ym %fF_d.`ZxmDFfu7$$\":jn:H`=%\\)Q\\_(fF?$\":A$zAd2%=#3(\\qf#Ffu7$$\":l,= aP?=%[[#y*fF?$\":K#*)H9f&pKN5si#Ffu7$$\":Q(4uA['fZ!4`;gF?$\":%fvu;5'[< k[sl#Ffu7$$\":![zE>N)=rY@4.'F?$\":(zTDCIMbij:(o#Ffu7$$\":kF;BH`dx\"*\\ //'F?$\":w,BI-Mtc3]OmYlu#Ffu7$$ \":[E?W\"o6,;!)*>/'F?$\":(*))p'[N'['[B,wFFfu7$$\":q0)Q-,u?^r*>.'F?$\": t]xzdVViv8`!GFfu7$$\":PK5x^+zWltG,'F?$\":usG3w3CHWUW$GFfu7$$\":]sii** \\:8\"yU#)fF?$\":(**H8(R?4c.!RjGFfu7$$\":&HLv#*>$4s%))QPfF?$\":S_07^v* )eE[@*GFfu7$$\":EFd\"=]Ik6FbseF?$\":v8bvG+#e+!42#HFfu7$$\":#z(z)*G]7tM 9)ydF?$\":YeI5awqzEk!\\HFfu7$$\":wj:h#*4'\\BfvPcF?$\":a,aF?$\":k*[41Z*e_WF^+$Ffu7$$\":ogxMh!)H@rD>#[F?$\":8 4E>ZdTuQ?G.$Ffu7$$!:T!3TP[<%*f&*46^F?$\":x'*>'omO#\\%zFgIFfu7$$!:d2YM^ ^RZxsag&F?$\":f.pYI:_j<$\\(3$Ffu7$$!:;(f^QG^foAI\"*eF?$\":Yp[KF?w!e$fW 6$Ffu7$$!:CSDZPn/kt]Q5'F?$\":4%Q$**4B=!z*p69$Ffu7$$!:oPp35hG^d2zF'F?$ \":Pq+k,*))3L(=w;$Ffu7$$!:C2e(eW1[Gs)yU'F?$\":`wz:f&G-5&*z$>$Ffu7$$!:: Q\">iO@U)o#>hlF?$\":EKk$=(QUbQ1(>KFfu7$$!:I,))Q>@CT,U@o'F?$\":/gkq/DW< hL`C$Ffu7$$!:\\D`,,;v!R$yMz'F?$\":^4.0A*[b3cnqKFfu-%*THICKNESSG6#\"\"# -%&COLORG6&%$RGBG$\"\"*!\"\"F($Fa]mFh]m-%+AXESLABELSG6$Q!6\"F]^m-%%FON TG6$%*HELVETICAGFg]m-%%VIEWG6$%(DEFAULTGFf^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 0 201 "Dig its := 15:\nz0 := 0.85*I:\nfor ct from 25 to 28 do\n newton(R(z)=exp (ct*Pi/100*I),z=z0);\nend do;``;\nz0 := 3.05*I:\nfor ct from 97 to 100 do\n newton(R(z)=exp(ct*Pi/100*I),z=z0);\nend do;\nDigits := 10:" } }{PARA 11 "" 1 "" {XPPMATH 20 "6#^$$!0i.e!)H%H " 0 "" {MPLTEXT 1 0 331 "Digits := 15:\nreal_part \+ := proc(u)\n Re(newton(R(z)=exp(u*Pi*I),z=0.85*I))\nend proc:\nu0 := bisect('real_part'(u),u=0.25..0.28);\nnewton(R(z)=exp(u0*Pi*I),z=0.85 *I);``;\nreal_part := proc(u)\n Re(newton(R(z)=exp(u*Pi*I),z=3.05*I) )\nend proc:\nu0 := bisect('real_part'(u),u=0.97..1.0);\nnewton(R(z)=e xp(u0*Pi*I),z=3.05*I);\nDigits := 10:" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%#u0G$\"0D+&pwzpE!#:" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#^#$\"0mmo% zS(Q)!#:" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%!G" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%#u0G$\"0^xva;i$)*!#:" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#^$$\"0p))\\Oj09(!#H$\"0!*zE7#zUI!#9" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 4 "The " }{TEXT 260 73 "stability region intersects the nonegative imaginary axis in the interval" }{TEXT -1 3 " " }{XPPEDIT 18 0 "[0*.8387, 3.0428];" "6#7$*&\"\"!\"\"\"-%&Float G6$\"%(Q)!\"%F&-F(6$\"&G/$F+" }{TEXT -1 18 " (approximately)." }} {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 37 "#-------- ----------------------------" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 80 "The stability function R* for the 9 stage, or der 5 scheme is given as follows." }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 97 "subs(ee,subs(b=`b*`,StabilityFunction(5,9,'expanded') )):\n`R*` := unapply(%,z):\n'`R*`(z)'=`R*`(z);" }}{PARA 12 "" 1 "" {XPPMATH 20 "6#/-%#R*G6#%\"zG,6\"\"\"F)F'F)*&#F)\"\"#F)*$)F'F,F)F)F)*& #F)\"\"'F)*$)F'\"\"$F)F)F)*&#F)\"#CF)*$)F'\"\"%F)F)F)*&#F)\"$?\"F)*$)F '\"\"&F)F)F)*&#\"7\"\\7HX)yj.$*36\"9]2J\"fBr+hMq\"yF)*$)F'F1F)F)F)*&# \"8JBs+o;L'Q+O9\";++'[I())p0)oFOD'F)*$)F'\"\"(F)F)F)*&#\"6J)>UOeP%=H6$ \";++i,\"Hm&oiDa%3#F)*$)F'\"\")F)F)F)*&#\"-V.^.DL\"4++^.p8,OF#F)*$)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 stabilit y 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,$\"\"\"!\"\"" }{TEXT -1 2 ". " }}{EXCHG {PARA 0 "> " 0 " " {MPLTEXT 1 0 33 "z_0 := newton(`R*`(z)=-1,z=-4.5);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%$z_0G$!+Gk]xW!\"*" }}}{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=-5.09..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=[-5.09..0.49,-1.57.. 1.47],font=[HELVETICA,9]);" }}{PARA 13 "" 1 "" {GLPLOT2D 398 284 284 {PLOTDATA 2 "6+-%'CURVESG6$7S7$$!3')*************3&!#<$!3attk%p>.@#F*7 $$!3G++vz=Po\\F*$!3yF)p_s]S\">F*7$$!3#**\\iE!Rai[F*$!3w!3e/N[*z;F*7$$! 3r**\\Au#HNu%F*$!3\"yU-Kv))>W\"F*7$$!3x**\\dRdsBYF*$!35kj.)4+#G7F*7$$! 3*)*\\7-h\"\\/XF*$!3aAe(ecK&R5F*7$$!3s*\\i#4j%RR%F*$!3u\"=7e@;k%))!#=7 $$!35+D;`I[zUF*$!36$*)3&>X[IuFK7$$!3W*\\i**)\\5hTF*$!3wq\"GruK1:'FK7$$ !3!**\\7nc1J/%F*$!33FK7$$!3))***\\8%Q;dMF*$!3]p5tS^6&\\\"FK7$$ !3#**\\i*3#39N$F*$!3ONa%[+E$46FK7$$!3t***\\J`acA$F*$!3@A\")GmUv-s!#>7$ $!3l****fuY7>JF*$!3e%>K.S^VS%Fhp7$$!3q*\\iQ70_*HF*$!3*oAtQ1+xh\"Fhp7$$ !3+++5C`^&)GF*$\"3)fTT2^M.OJ%Fhp7$$!3]**\\i5u*4`#F*$\"39)*4@,]^'*fFh p7$$!3T*\\7\"eI>@CF*$\"3YVSMEI=muFhp7$$!3n**\\()HUv-BF*$\"3Fy#yWH:D-*F hp7$$!3y*\\iRdH(z@F*$\"3oDu9\\9,m5FK7$$!3o*\\P$\\ijs?F*$\"3>Np8[&)H97F K7$$!3S**\\#[_sp&>F*$\"3%zYpDa+VQ\"FK7$$!3y****pz0[P=F*$\"354Qw+Whu:FK 7$$!3')**\\_B5e?V!y*> FK7$$!3i**\\2&R*)=[\"F*$\"3)y+$**)))4*oAFK7$$!3'*****4?a/p8F*$\"3?X*>y Eb=a#FK7$$!3O***\\2Rg&[7F*$\"3#\\H0BIL%oGFK7$$!36+DcYIQR6F*$\"3C'*>uo/ %)*>$FK7$$!3#*)**\\=PB+-\"F*$\"3X(fCp?ddg$FK7$$!3c-]i)>_r2*FK$\"32:-q& )HUMSFK7$$!3*)**\\74%3K!zFK$\"3+I(ylf#*p`%FK7$$!3E****\\xPYbnFK$\"3?$o #4rxy)3&FK7$$!3)4+Dc^\")Qb&FK$\"3i:W8`w\\QdFK7$$!3)e****f)\\h'R%FK$\"3 !es]noXDW'FK7$$!3F)**\\<\"G98KFK$\"3i!)R*e?`>D(FK7$$!3'G*\\i%Qq%R?FK$ \"39JhC@c0b\")FK7$$!3xr****pJ()4'*Fhp$\"3A(p1KEVP3*FK7$$\"3k<+]_)f2v#F hp$\"3J*3@bU*)y-\"F*7$$\"3)3++!Qdi!Q\"FK$\"3Yf%[fQZ![6F*7$$\"3o4]PhBPf DFK$\"3Q\"z&\\l;n\"H\"F*7$$\"3]/]i%G$e(o$FK$\"3_j^F>#QfW\"F*7$$\"3!*** ************[FK$\"3y!y__!oJK;F*-%'COLOURG6&%$RGBG$\"*++++\"!\")$\"\"!F b[lFa[l-F$6$7S7$F($!\"\"Fb[l7$F.Fg[l7$F3Fg[l7$F8Fg[l7$F=Fg[l7$FBFg[l7$ FGFg[l7$FMFg[l7$FRFg[l7$FWFg[l7$FfnFg[l7$F[oFg[l7$F`oFg[l7$FeoFg[l7$Fj oFg[l7$F_pFg[l7$FdpFg[l7$FjpFg[l7$F_qFg[l7$FdqFg[l7$FjqFg[l7$F_rFg[l7$ FdrFg[l7$FirFg[l7$F^sFg[l7$FcsFg[l7$FhsFg[l7$F]tFg[l7$FbtFg[l7$FgtFg[l 7$F\\uFg[l7$FauFg[l7$FfuFg[l7$F[vFg[l7$F`vFg[l7$FevFg[l7$FjvFg[l7$F_wF g[l7$FdwFg[l7$FiwFg[l7$F^xFg[l7$FcxFg[l7$FhxFg[l7$F]yFg[l7$FbyFg[l7$Fg yFg[l7$F\\zFg[l7$FazFg[l7$FfzFg[l-F[[l6&F][lFa[lFa[lF^[l-F$6&7#7$$!3)) *****zU1vZ%F*Fg[l-%'SYMBOLG6#%'CIRCLEG-F[[l6&F][lFb[lFb[lFb[l-%&STYLEG 6#%&POINTG-F$6&F]_l-Fb_l6#%&CROSSGFe_lFg_l-F$6&F]_l-Fb_l6#%(DIAMONDGFe _lFg_l-F$6%7$7$F__lFa[lF^_l-%&COLORG6&F][lFa[l$\"\"&Fh[lFa[l-%*LINESTY LEG6#\"\"$-%+AXESLABELSG6%Q\"z6\"Q!Ffal-%%FONTG6#%(DEFAULTG-Fial6$%*HE LVETICAG\"\"*-%%VIEWG6$;$!$4&!\"#$\"#\\Ffbl;$!$d\"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 1500 "`R*` := \+ z -> 1+z+1/2*z^2+1/6*z^3+1/24*z^4+1/120*z^5+\n 1108930363788452912 491/781703461007123591310750*z^6+\n 14360038633166800722331/625362 76880569887304860000*z^7+\n 311291843758364219831/2084542562685662 9101620000*z^8-332503510343/2273601136903510000*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)):\np_2 := plots[polygonplot] ([seq([pts[i-1],pts[i],[-2.2,0]],i=2..nops(pts))],\n style=pa tchnogrid,color=COLOR(RGB,.85,0,.15)):\npts := []: z0 := 1.9+4.45*I:\n for 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(zz)]]:\nend do:\np_3 := \+ plot(pts,color=COLOR(RGB,.43,0,.08)):\np_4 := plots[polygonplot]([seq( [pts[i-1],pts[i],[1.82,4.4]],i=2..nops(pts))],\n style=patchn ogrid,color=COLOR(RGB,.85,0,.15)):\npts := []: z0 := 1.9-4.45*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(zz)]]:\nend do:\np_5 := plot (pts,color=COLOR(RGB,.43,0,.08)):\np_6 := plots[polygonplot]([seq([pts [i-1],pts[i],[1.82,-4.4]],i=2..nops(pts))],\n style=patchnogr id,color=COLOR(RGB,.85,0,.15)):\np_7 := plot([[[-5.09,0],[2.19,0]],[[0 ,-4.99],[0,4.99]]],color=black,linestyle=3):\nplots[display]([p_||(1.. 7)],view=[-5.09..2.19,-4.99..4.99],font=[HELVETICA,9],\n la bels=[`Re(z)`,`Im(z)`],axes=boxed,scaling=constrained);" }}{PARA 13 " " 1 "" {GLPLOT2D 407 552 552 {PLOTDATA 2 "6/-%'CURVESG6$7ew7$$\"\"!F)F (7$$\"3y*****>eg3c%!#F$\"3++++Fjzq:!#=7$$\"3;+++pc3+J!#D$\"3!******Hl# fTJF07$$\"3?+++4ydeQ!#C$\"3:+++f))Q7ZF07$$\"3%******4KpjS#!#B$\"3A+++& G$=$G'F07$$\"3-+++!>(fA5!#A$\"3%******>QlR&yF07$$\"3y******[@!3Q$FF$\" 3<+++G?pC%*F07$$\"3(******fOA8J*FF$\"3)*******GI_*4\"!#<7$$\"35+++no$> 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4" "Curve 5" "Curve 6" "Curve 7" "Curve 8" "Curve 9" "Curve 1 0" "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 36 "coeffici ents for the combined scheme" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1841 "ee := \{c[2]=17/183,\nc[3] =12/83,\nc[4]=18/83,\nc[5]=71/125,\nc[6]=42/59,\nc[7]=199/200,\nc[8]=1 ,\nc[9]=1,\n\na[2,1]=17/183,\na[3,1]=3756/117113,\na[3,2]=13176/117113 ,\na[4,1]=9/166,\na[4,2]=0,\na[4,3]=27/166,\na[5,1]=207751751/31640625 0,\na[5,2]=0,\na[5,3]=-526769377/210937500,\na[5,4]=1524242129/6328125 00,\na[6,1]=-4970082682619223281/2887511529739311186,\na[6,2]=0,\na[6, 3]=97919278033879057/13556392158400522,\na[6,4]=-407131674007930877068 /74078904949579652469,\na[6,5]=1237601855204268750000/1753200750473385 108433,\na[7,1]=176597685527535385020980411/42773485015591331328000000 ,\na[7,2]=0,\na[7,3]=-6793162515552646891859/401628967282547712000,\na [7,4]=12704926019361287204873446554247/886659402653054716778496000000, \na[7,5]=-50728836334509259632278125/32657591718008685915971584,\na[7, 6]=51536223982796190703/51293749413888000000,\na[8,1]=2990335205723375 73523/66918720793812357519,\na[8,2]=0,\na[8,3]=-16550269823961899/9021 46153892364,\na[8,4]=49920346343238033627496282/3215735869387500624775 563,\na[8,5]=-1686432488955761721093750/978844996793357447730403,\na[8 ,6]=161901609084039/149698803705724,\na[8,7]=-305146137600000/54760341 991955873,\na[9,1]=24503/381483,\na[9,2]=0,\na[9,3]=0,\na[9,4]=1366847 103121/4106349847584,\na[9,5]=20339599609375/75933913767768,\na[9,6]=3 5031290651/194765546144,\na[9,7]=16620160000000/11001207123543,\na[9,8 ]=-14933/11016,\n\nb[1]=24503/381483,\nb[2]=0,\nb[3]=0,\nb[4]=13668471 03121/4106349847584,\nb[5]=20339599609375/75933913767768,\nb[6]=350312 90651/194765546144,\nb[7]=16620160000000/11001207123543,\nb[8]=-14933/ 11016,\n\n`b*`[1]=61010485298317/979331468960880,\n`b*`[2]=0,\n`b*`[3] =0,\n`b*`[4]=320207313882553286621/941222813406992395200,\n`b*`[5]=684 5867841119140625/29008216787127405534,\n`b*`[6]=124109197949158875473/ 562495660250110816320,\n`b*`[7]=19339714537078400000/16810691577722216 811,\n`b*`[8]=-211029377951/210416202900,\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 {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#\"\"##\"#<\"$$=/&F%6#\"\"$#\"#7\"#$)/&F%6#\"\"%#\"#=F1/&F%6#\"\"&# \"#r\"$D\"/&F%6#\"\"'#\"#U\"#f/&F%6#\"\"(#\"$*>\"$+#/&F%6#\"\")\"\"\"/ &F%6#\"\"*FQ" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT 259 51 "linking coefficients for the 9 stage order 5 scheme" } {TEXT -1 1 ":" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 49 "seq(seq(a[i ,j]=subs(ee,a[i,j]),j=1..i-1),i=1..9);" }}{PARA 12 "" 1 "" {XPPMATH 20 "6F/&%\"aG6$\"\"#\"\"\"#\"#<\"$$=/&F%6$\"\"$F(#\"%cP\"'8r6/&F%6$F/F '#\"&wJ\"F2/&F%6$\"\"%F(#\"\"*\"$m\"/&F%6$F;F'\"\"!/&F%6$F;F/#\"#FF>/& F%6$\"\"&F(#\"*^Eo#3q\\\"4'=6$R(H:^()G/& F%6$F[oF'FB/&F%6$F[oF/#\"2d!zQ.y#>z*\"2A0Se@RcN\"/&F%6$F[oF;#!6oq(3$z+ u;82%\"5pClz&\\\\!*yS(/&F%6$F[oFK#\"7++voU?b=gP7\"7L%3^Qt/v+Kv\"/&F%6$ \"\"(F(#\"<6/)4-&QNv_&o(fw\"\";+++G8L\"f:][tF%/&F%6$FgpF'FB/&F%6$FgpF/ #!7f=*ok_b^iJz'\"6+?rZDGn*G;S/&F%6$FgpF;#\"AZUbYM([?(Gh$>g#\\q7\"?+++' \\ynraIl-%fm))/&F%6$FgpFK#!;D\"yAjf#4XLO)G2&\";%erf\"fo3!='z#)RAO:&\"5+++))QT\\PH^/&F%6$\"\")F(#\"6BNdPBd?N.*H\"5>vN 7Qz?(=p'/&F%6$FirF'FB/&F%6$FirF/#!2**='R#)p-b;\"0kB*Q:Y@!*/&F%6$FirF;# \";#G'\\FO.QKMY.#*\\\":jbxC1](Qpet:K/&F%6$FirFK#!:]P4@;\"0Cdq.))p\\\"/&F%6$FirFgp#!0++g Ph90$\"2te&>*>MgZ&/&F%6$F=F(#\"&.X#\"'$[\"Q/&F%6$F=F'FB/&F%6$F=F/FB/&F %6$F=F;#\".@J5ZoO\"\".%eZ)\\j5%/&F%6$F=FK#\"/v$4'*fR.#\"/oxw8R$f(/&F%6 $F=F[o#\",^1HJ]$\"-WhalZ>/&F%6$F=Fgp#\"/+++g,i;\"/VN727+6/&F%6$F=Fir#! &L\\\"\"&;5\"" }}}{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#\"\"\"#\"&.X#\"'$[\" Q/&F%6#\"\"#\"\"!/&F%6#\"\"$F//&F%6#\"\"%#\".@J5ZoO\"\".%eZ)\\j5%/&F%6 #\"\"&#\"/v$4'*fR.#\"/oxw8R$f(/&F%6#\"\"'#\",^1HJ]$\"-WhalZ>/&F%6#\"\" (#\"/+++g,i;\"/VN727+6/&F%6#\"\")#!&L\\\"\"&;5\"" }}}{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#\"\"\"#\"/<$)H&[55'\"0!)3'*o9Lz*/&F%6#\"\"#\" \"!/&F%6#\"\"$F//&F%6#\"\"%#\"6@mG`D)QJ2-K\"6+_R#*pS8GAT*/&F%6#\"\"&# \"4D19>6%y'e%o\"5MbSFry;#3!H/&F%6#\"\"'#\"6ta()e\"\\z>4T7\"6?j\"36]-m& \\i&/&F%6#\"\"(#\"5++Syq`9(R$>\"56o@Axd\"p5o\"/&F%6#\"\")#!-^zPH5@\"-+ H?;/@/&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 33 " #================================" }}{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 "S O6_8 := SimpleOrderConditions(6,8,'expanded'):\nSO_eqs := [op(RowSumCo nditions(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],s eq(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 := PrincipalErrorTerms(6, 8,'expanded'):\n`errterms6_9*` :=subs(b=`b*`,PrincipalErrorTerms(6,9,' expanded')):\n`errterms5_9*` := subs(b=`b*`,PrincipalErrorTerms(5,9,'e xpanded')):" }}}{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 1879 "calc_RKcoeffs := proc()\n local eqns,sm,ct,Rz,stb6,stb5,nmB,snmB,dnB,sdnB,nmC,snmC,B_7,C_7;\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 sm := 0;\n \+ for ct to nops(errterms6_8) do\n sm := sm+(evalf(subs(e3,errter ms6_8[ct])))^2;\n end do;\n Rz := subs(e3,StabilityFunction(6,8,'e xpanded'));\n stb6 := evalf[8](max(fsolve(Rz=1,z=-9..-1e-7),fsolve(R z=-1,z=-9..-1e-7)));\n Rz := subs(e3,subs(b=`b*`,StabilityFunction(5 ,9,'expanded')));\n stb5 := evalf[8](max(fsolve(Rz=1,z=-9..-1e-7),fs olve(Rz=-1,z=-9..-1e-7)));\n nmB := 0;\n for ct to nops(`errterms6 _9*`) do\n nmB := nmB+evalf(subs(e3,`errterms6_9*`[ct]))^2;\n e nd do:\n snmB := sqrt(nmB);\n dnB := 0;\n for ct to nops(`errter ms5_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 nops(errt erms6_8) do\n nmC := nmC+(evalf(subs(e3,`errterms6_9*`[ct]))-eval f(subs(e3,errterms6_8[ct])))^2;\n end do;\n snmC := sqrt(simplify( nmC));\n B_7 := evalf[8](snmB/sdnB);\n C_7 := evalf[8](snmC/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 print(infinity*`-norm of linki ng coeffs`=evalf(max(seq(seq(subs(e3,abs(a[i,j])),j=1..i-1),i=2..9)))) ;\n print(`2-norm of principal error of order 6 scheme` = evalf(sqrt (sm)));\n print(`2-norm of principal error of order 5 scheme` = sdnB );\n print(`order 6 stability interval` = [stb6,0]);\n print(`orde r 5 stability interval` = [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)%'nodes: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~weights: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~erro r~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#/%;order~6~stability~intervalG7$$!)&G3Z%!\"( \"\"!" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/%;order~5~stability~interval G7$$!)!Q+Z$!\"(\"\"!" }}{PARA 11 "" 1 "" {XPPMATH 20 "6$/&%\"BG6#\"\"( $\")w**o7!\"(/&%\"CGF&$\")\"G\"o7F*" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 40 "#-------------------------------------- -" }}{PARA 0 "" 0 "" {TEXT -1 28 "modified Sharp-Verner scheme" }} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 99 "c_2 := 1/16: c_4 := 1/5: c_5 := 8/15: c_6 := 27/40: c_7 := 24/25: bs := [8,-1/13]:\ncalc_RKcoeffs( );" }}{PARA 11 "" 1 "" {XPPMATH 20 "6)%'nodes:G/&%\"cG6#\"\"##\"\"\"\" #;/&F&6#\"\"$#F(\"#:/&F&6#\"\"%#F*\"\"&/&F&6#F7#\"\")F1/&F&6#\"\"'#\"# F\"#S/&F&6#\"\"(#\"#C\"#D" }}{PARA 11 "" 1 "" {XPPMATH 20 "6)%1order~6 ~weights:G/&%\"bG6#\"\"\"$\"'a4e!\"(/&F&6#\"\"%$\"'+DJ!\"'/&F&6#\"\"&$ \"'YfCF2/&F&6#\"\"'$\"'92?F2/&F&6#\"\"($\"'UuHF2/&F&6#\"\")$!')p9\"F2 " }}{PARA 11 "" 1 "" {XPPMATH 20 "6*%1order~5~weights:G/&%#b*G6#\"\"\" $\"'\">f&!\"(/&F&6#\"\"%$\"'8.K!\"'/&F&6#\"\"&$\"'\\f@F2/&F&6#\"\"'$\" 'okBF2/&F&6#\"\"($\"'a]DF2/&F&6#\"\")$!'J#p(F+/&F&6#\"\"*$!'c!y'!\")" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/*&%)infinityG\"\"\"%8-norm~of~linki ng~coeffsGF&$\"+-0(3`\"!\")" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/%L2-no rm~of~principal~error~of~order~6~schemeG$\"+5=Q(=&!#9" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/%L2-norm~of~principal~error~of~order~5~schemeG$\" +FTuf`!#8" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/%;order~6~stability~inte rvalG7$$!)4y=W!\"(\"\"!" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/%;order~5~ stability~intervalG7$$!)1#[T%!\"(\"\"!" }}{PARA 11 "" 1 "" {XPPMATH 20 "6$/&%\"BG6#\"\"($\"))o8I\"!\"(/&%\"CGF&$\")1-?8F*" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 37 "#------------------- -----------------" }}{PARA 0 "" 0 "" {TEXT -1 15 "a scheme with " } {XPPEDIT 18 0 "c[7]=24/25" "6#/&%\"cG6#\"\"(*&\"#C\"\"\"\"#D!\"\"" } {TEXT -1 1 " " }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 109 "c_2 := 33/ 577: c_4 := 173/800: c_5 := 260/463: c_6 := 27/40: c_7 := 24/25: bs := [9,-1/135]:\ncalc_RKcoeffs();" }}{PARA 11 "" 1 "" {XPPMATH 20 "6)%'no des:G/&%\"cG6#\"\"##\"#L\"$x&/&F&6#\"\"$#\"$t\"\"%+7/&F&6#\"\"%#F1\"$+ )/&F&6#\"\"&#\"$g#\"$j%/&F&6#\"\"'#\"#F\"#S/&F&6#\"\"(#\"#C\"#D" }} {PARA 11 "" 1 "" {XPPMATH 20 "6)%1order~6~weights:G/&%\"bG6#\"\"\"$\"' [/k!\"(/&F&6#\"\"%$\"'_?L!\"'/&F&6#\"\"&$\"'k*G#F2/&F&6#\"\"'$\"'+v>F2 /&F&6#\"\"($\"'X$y#F2/&F&6#\"\")$!'145F2" }}{PARA 11 "" 1 "" {XPPMATH 20 "6*%1order~5~weights:G/&%#b*G6#\"\"\"$\"''yF'!\"(/&F&6#\"\"%$\"'UqL !\"'/&F&6#\"\"&$\"'NI?F2/&F&6#\"\"'$\"'C\"G#F2/&F&6#\"\"($\"'@wCF2/&F& 6#\"\")$!'K>rF+/&F&6#\"\"*$!'T2u!\")" }}{PARA 11 "" 1 "" {XPPMATH 20 " 6#/*&%)infinityG\"\"\"%8-norm~of~linking~coeffsGF&$\"+@(f-=\"!\")" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#/%L2-norm~of~principal~error~of~order~ 6~schemeG$\"+!R!RFO!#9" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/%L2-norm~of ~principal~error~of~order~5~schemeG$\"+:vqZc!#8" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/%;order~6~stability~intervalG7$$!)U\\]V!\"(\"\"!" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#/%;order~5~stability~intervalG7$$!)e#z P%!\"(\"\"!" }}{PARA 11 "" 1 "" {XPPMATH 20 "6$/&%\"BG6#\"\"($\")s&3I \"!\"(/&%\"CGF&$\")r\"pJ\"F*" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }} {PARA 0 "" 0 "" {TEXT -1 31 "#------------------------------" }}{PARA 0 "" 0 "" {TEXT -1 15 "a scheme with " }{XPPEDIT 18 0 "c[7]=35/36" "6 #/&%\"cG6#\"\"(*&\"#N\"\"\"\"#O!\"\"" }{TEXT -1 1 " " }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 105 "c_2 := 7/95: c_4 := 35/162: c_5 := 31/55 : c_6 := 97/140: c_7 := 35/36: bs := [9,-1/160]:\ncalc_RKcoeffs();" }} {PARA 11 "" 1 "" {XPPMATH 20 "6)%'nodes:G/&%\"cG6#\"\"##\"\"(\"#&*/&F& 6#\"\"$#\"#N\"$V#/&F&6#\"\"%#F1\"$i\"/&F&6#\"\"&#\"#J\"#b/&F&6#\"\"'# \"#(*\"$S\"/&F&6#F*#F1\"#O" }}{PARA 11 "" 1 "" {XPPMATH 20 "6)%1order~ 6~weights:G/&%\"bG6#\"\"\"$\"'$**R'!\"(/&F&6#\"\"%$\"'*fJ$!\"'/&F&6#\" \"&$\"':7DF2/&F&6#\"\"'$\"'[[=F2/&F&6#\"\"($\"'GtMF2/&F&6#\"\")$!'*)*y \"F2" }}{PARA 11 "" 1 "" {XPPMATH 20 "6*%1order~5~weights:G/&%#b*G6#\" \"\"$\"'aNi!\"(/&F&6#\"\"%$\"'_zL!\"'/&F&6#\"\"&$\"'S8AF2/&F&6#\"\"'$ \"'h;AF2/&F&6#\"\"($\"'![*GF2/&F&6#\"\")$!'Pl7F2/&F&6#\"\"*$!'+]i!\") " }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/*&%)infinityG\"\"\"%8-norm~of~lin king~coeffsGF&$\"+#f,ye\"!\")" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/%L2- norm~of~principal~error~of~order~6~schemeG$\"+4OWXE!#9" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/%L2-norm~of~principal~error~of~order~5~schemeG$\" +1\"Q0t&!#8" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/%;order~6~stability~in tervalG7$$!)4V&Q%!\"(\"\"!" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/%;order ~5~stability~intervalG7$$!)u(eR%!\"(\"\"!" }}{PARA 11 "" 1 "" {XPPMATH 20 "6$/&%\"BG6#\"\"($\")c(4L\"!\"(/&%\"CGF&$\")7;R8F*" }}} {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 31 "#-------- ----------------------" }}{PARA 0 "" 0 "" {TEXT -1 15 "a scheme with \+ " }{XPPEDIT 18 0 "c[7] = 63/64;" "6#/&%\"cG6#\"\"(*&\"#j\"\"\"\"#k!\" \"" }{TEXT -1 1 " " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 " > " 0 "" {MPLTEXT 1 0 109 "c_2 := 19/220: c_4 := 124/573: c_5 := 159/2 81: c_6 := 64/91: c_7 := 63/64: bs := [9,-1/156]:\ncalc_RKcoeffs();" } }{PARA 11 "" 1 "" {XPPMATH 20 "6)%'nodes:G/&%\"cG6#\"\"##\"#>\"$?#/&F& 6#\"\"$#\"$[#\"%>aF2/&F&6#\"\") $!'q3QF2" }}{PARA 11 "" 1 "" {XPPMATH 20 "6*%1order~5~weights:G/&%#b*G 6#\"\"\"$\"'>Ai!\"(/&F&6#\"\"%$\"'L$R$!\"'/&F&6#\"\"&$\"'#[G#F2/&F&6# \"\"'$\"'GCAF2/&F&6#\"\"($\"'GyUF2/&F&6#\"\")$!'$)QFF2/&F&6#\"\"*$!'E5 k!\")" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/*&%)infinityG\"\"\"%8-norm~o f~linking~coeffsGF&$\"+<`4[W!\"(\"\"!" }}{PARA 11 "" 1 "" {XPPMATH 20 "6$/&%\"BG6#\"\"($\")YtZ8!\"(/&%\"CGF&$\")ts_8F*" }}} {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 57 "#-------- ------------------------------------------------" }}{PARA 0 "" 0 "" {TEXT -1 15 "a scheme with " }{XPPEDIT 18 0 "c[7] = 125/126;" "6#/&% \"cG6#\"\"(*&\"$D\"\"\"\"\"$E\"!\"\"" }{TEXT -1 1 " " }}{PARA 0 "" 0 " " {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 108 "c_2 := 6/ 65: c_4 := 23/106: c_5 := 80/141: c_6 := 91/129: c_7 := 125/126: bs := [9,-1/132]:\ncalc_RKcoeffs();" }}{PARA 11 "" 1 "" {XPPMATH 20 "6)%'no des:G/&%\"cG6#\"\"##\"\"'\"#l/&F&6#\"\"$#\"#B\"$f\"/&F&6#\"\"%#F1\"$1 \"/&F&6#\"\"&#\"#!)\"$T\"/&F&6#F*#\"#\"*\"$H\"/&F&6#\"\"(#\"$D\"\"$E\" " }}{PARA 11 "" 1 "" {XPPMATH 20 "6)%1order~6~weights:G/&%\"bG6#\"\"\" $\"'4Ck!\"(/&F&6#\"\"%$\"'pJL!\"'/&F&6#\"\"&$\"'61EF2/&F&6#\"\"'$\"'lT =F2/&F&6#\"\"($\"'QM**F2/&F&6#\"\")$!'Dc$)F2" }}{PARA 11 "" 1 "" {XPPMATH 20 "6*%1order~5~weights:G/&%#b*G6#\"\"\"$\"'\\Fi!\"(/&F&6#\" \"%$\"'&oS$!\"'/&F&6#\"\"&$\"'>qAF2/&F&6#\"\"'$\"'SjAF2/&F&6#\"\"($\"' CawF2/&F&6#\"\")$!'nThF2/&F&6#\"\"*$!'wvv!\")" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/*&%)infinityG\"\"\"%8-norm~of~linking~coeffsGF&$\"+w?i #o\"!\")" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/%L2-norm~of~principal~err or~of~order~6~schemeG$\"+yRjT8!#9" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/ %L2-norm~of~principal~error~of~order~5~schemeG$\"+7%4i'e!#8" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/%;order~6~stability~intervalG7$$!)^_QW!\"( \"\"!" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/%;order~5~stability~interval G7$$!)n\"\"\"\"$+ #!\"\"" }{TEXT -1 1 " " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 106 "c_2 := 17/183: c_4 := 18/83: c_5 : = 71/125: c_6 := 42/59: c_7 := 199/200: bs := [9,1/20]:\ncalc_RKcoeffs ();" }}{PARA 11 "" 1 "" {XPPMATH 20 "6)%'nodes:G/&%\"cG6#\"\"##\"#<\"$ $=/&F&6#\"\"$#\"#7\"#$)/&F&6#\"\"%#\"#=F2/&F&6#\"\"&#\"#r\"$D\"/&F&6# \"\"'#\"#U\"#f/&F&6#\"\"(#\"$*>\"$+#" }}{PARA 11 "" 1 "" {XPPMATH 20 " 6)%1order~6~weights:G/&%\"bG6#\"\"\"$\"'4Bk!\"(/&F&6#\"\"%$\"'iGL!\"'/ &F&6#\"\"&$\"'fyEF2/&F&6#\"\"'$\"'k)z\"F2/&F&6#\"\"($\"'w5:!\"&/&F&6# \"\")$!'db8FE" }}{PARA 11 "" 1 "" {XPPMATH 20 "6*%1order~5~weights:G/& %#b*G6#\"\"\"$\"'qsy!\"(/&F&6#\"\"%$\"'*zx#!\"'/&F&6#\"\"&$\"'Ao]F2/&F &6#\"\"'$!'ff7F2/&F&6#\"\"($\"'88U!\"&/&F&6#\"\")$!'_+SFE/&F&6#\"\"*$ \"'++]F+" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/*&%)infinityG\"\"\"%8-nor m~of~linking~coeffsGF&$\"+(=WX$=!\")" }}{PARA 11 "" 1 "" {XPPMATH 20 " 6#/%L2-norm~of~principal~error~of~order~6~schemeG$\"+P7OJ7!#9" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#/%L2-norm~of~principal~error~of~order~ 5~schemeG$\"+d*\\)>V!#7" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/%;order~6~ stability~intervalG7$$!)T)*\\W!\"(\"\"!" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/%;order~5~stability~intervalG7$$!)k)e3$!\"(\"\"!" }}{PARA 11 " " 1 "" {XPPMATH 20 "6$/&%\"BG6#\"\"($\")Tti8!\"(/&%\"CGF&$\")`gi8F*" } }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 58 "Papakos tas' scheme with a modified embedded order 5 scheme" }}{PARA 0 "" 0 " " {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 108 "c_2 := 17 /183: c_4 := 18/83: c_5 := 71/125: c_6 := 42/59: c_7 := 199/200: bs := [9,-1/150]:\ncalc_RKcoeffs();" }}{PARA 11 "" 1 "" {XPPMATH 20 "6)%'no des:G/&%\"cG6#\"\"##\"#<\"$$=/&F&6#\"\"$#\"#7\"#$)/&F&6#\"\"%#\"#=F2/& F&6#\"\"&#\"#r\"$D\"/&F&6#\"\"'#\"#U\"#f/&F&6#\"\"(#\"$*>\"$+#" }} {PARA 11 "" 1 "" {XPPMATH 20 "6)%1order~6~weights:G/&%\"bG6#\"\"\"$\"' 4Bk!\"(/&F&6#\"\"%$\"'iGL!\"'/&F&6#\"\"&$\"'fyEF2/&F&6#\"\"'$\"'k)z\"F 2/&F&6#\"\"($\"'w5:!\"&/&F&6#\"\")$!'db8FE" }}{PARA 11 "" 1 "" {XPPMATH 20 "6*%1order~5~weights:G/&%#b*G6#\"\"\"$\"'\")Hi!\"(/&F&6#\" \"%$\"'/-M!\"'/&F&6#\"\"&$\"')*fBF2/&F&6#\"\"'$\"'S1AF2/&F&6#\"\"($\"' W]6!\"&/&F&6#\"\")$!'\"H+\"FE/&F&6#\"\"*$!'nmm!\")" }}{PARA 11 "" 1 " " {XPPMATH 20 "6#/*&%)infinityG\"\"\"%8-norm~of~linking~coeffsGF&$\"+( =WX$=!\")" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/%L2-norm~of~principal~er ror~of~order~6~schemeG$\"+P7OJ7!#9" }}{PARA 11 "" 1 "" {XPPMATH 20 "6# /%L2-norm~of~principal~error~of~order~5~schemeG$\"+U**zfd!#8" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/%;order~6~stability~intervalG7$$!)T)*\\W!\" (\"\"!" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/%;order~5~stability~interva lG7$$!)k]xW!\"(\"\"!" }}{PARA 11 "" 1 "" {XPPMATH 20 "6$/&%\"BG6#\"\"( $\")&G=O\"!\"(/&%\"CGF&$\")`gi8F*" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }} {PARA 0 "" 0 "" {TEXT -1 57 "#---------------------------------------- ----------------" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}} {PARA 0 "" 0 "" {TEXT -1 23 "#======================" }}{PARA 0 "" 0 " " {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 35 "#====================== ============" }}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 37 "Test-bed procedur es 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::realcons)\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,a 75,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 := Digits;\n Digits := max(trunc(evalhf(Digits)),D igits+5);\n\n # procedure to evaluate the slope field\n fn := proc (X_,Y_)\n local val; \n val := traperror(evalf(FXY_));\n \+ if val=lasterror or not type(val,numeric) then\n error \"eva luation of slope field failed 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 i nterpolation interval: %1\",evalf(data[1,1])..evalf(data[n,1]);\n en d if;\n\n c2 := c2_; c3 := c3_; c4 := c4_; c5 := c5_; c6 := c6_; c7 \+ := c7_; c8 := c8_;\n a21 := c2; a31 := a31_; a32 := a32_; a41 := a41 _; a42 := a42_; a43 := a43_;\n a51 := a51_; a52 := a52_; a53 := a53_ ; a54 := a54_;\n a61 := a61_; a62 := a62_; a63 := a63_; a64 := a64_; a65 := a65_;\n a71 := a71_; a72 := a72_; a73 := a73_; a74 := a74_; \+ a75 := a75_; a76 := a76_;\n a81 := a81_; a82 := a82_; a83 := a83_; a 84 := 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 # 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 jF := n-1; jS := n end if;\n else\n whi le jS-jF> 1 do\n jM := trunc((jF+jS)/2);\n if xx<=data[j M,1] then jF := jM else jS := jM end if;\n end do;\n if jM = n then jF := n-1; jS := n end if;\n end if;\n \n # Get the data \+ needed from the list.\n xk := 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 + a 43*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 + 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[saveDigits](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 33 "RK6_1 Sharp-Verner sch eme with " }{XPPEDIT 18 0 "c[7]=19/20" "6#/&%\"cG6#\"\"(*&\"#>\"\"\" \"#?!\"\"" }{TEXT -1 1 " " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 3400 "RK6_1 := proc(fxy,x,y,xx,yy,h,stp s,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,a76,\n a81,a82,a 83,a84,a85,a86,a87,b1,b2,b3,b4,b5,b6,b7,b8,\n f1,f2,f3,f4,f5,f6,f7,f 8,t,k,fn,xk,yk,soln,eqns,A,saveDigits;\n saveDigits := Digits;\n D igits := max(trunc(evalhf(Digits)),Digits+5);\n\n fn := 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,-8456/2889,217/428,0,0,0],\n \+ [19/20,1718911/4382720,0,-1000749/547840,819261/383488,-671 175/876544,\n 14535/14336,0,0],\n \+ [1,85153/203300,0,-6783/2140,10956/2675,-38493/13375,1152/425,-7168/ 40375,0],\n [0,53/912,0,0,5/16,27/112,27/136,256/969,-2 5/336]]);\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 := 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 := ev alf(A[7,6]);\n a86 := evalf(A[7,7]);\n a87 := evalf(A[7,8]);\n b 1 := 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 := ev alf(xx);\n yk := evalf(yy);\n soln := [xk,yk]; \n for k from 1 t o stps do\n f1 := fn(xk,yk);\n t := a21*f1;\n f2 := fn( xk + c2*h,yk + t*h);\n t := a31*f1 + a32*f2;\n f3 := fn(xk + c3*h,yk + t*h);\n t := a41*f1 + a42*f2 + a43*f3;\n f4 := fn (xk + c4*h,yk + t*h);\n t := a51*f1 + a52*f2 + a53*f3 + a54*f4;\n f5 := fn(xk + c5*h,yk + t*h);\n t := a61*f1 + a62*f2 + a63* f3 + a64*f4 + a65*f5;\n f6 := fn(xk + c6*h,yk + t*h);\n t := a71*f1 + a72*f2 + a73*f3 + a74*f4 + a75*f5 + a76*f6;\n f7 := fn( xk + c7*h,yk + t*h);\n t := a81*f1 + a82*f2 + a83*f3 + a84*f4 + a 85*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 + b 8*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,a31_=a31,a 32_=a32,a41_=a41,\n a42_=a42,a43_=a43,a51_=a51,a52_=a52,a53_=a 53,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[sav eDigits]([soln]);\n end if;\nend proc: " }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 20 "RK6_2 scheme with " }{XPPEDIT 18 0 "c[7 ] = 24/25;" "6#/&%\"cG6#\"\"(*&\"#C\"\"\"\"#D!\"\"" }{TEXT -1 1 " " }} {PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 4216 "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,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,save Digits;\n saveDigits := Digits;\n Digits := max(trunc(evalhf(Digit s)),Digits+5);\n\n fn := unapply(fxy,x,y);\n\n A := matrix([[33/57 7,33/577,0,0,0,0,0,0,0],\n [173/1200,-3567433/95040000,172690 33/95040000,0,0,0,0,0,0],\n [173/800,173/3200,0,519/3200,0,0, 0,0,0],\n [260/463,1877748548260/2970538457863,0,-71265136800 00/2970538457863,\n 6916886080000/2970538457863,0,0,0,0] ,\n [27/40,-65244579431964655428477/69653246199915888640000,0 ,\n 2694429616783931433/669742751922268160,\n \+ -62138409816102019904637/21415191928402504983040,\n 1 45922219908058645309313/297661794320406364160000,0,0,0],\n [2 4/25,3890754282260741440349323/3010236956970547048875000,0,\n \+ -45910927628476367208/8576173666582755125,\n 598603 7178684297350750710688/1207688208130690660530130125,\n -3 106990980562910309356750843/3003073489062493679278875000,2999575140352 /2729866330125,0,0],\n [1,3914147173803753655861/144460603496 1051585792,0,\n -2276990272364785725/192922814498003684, \n 24219185248932516188948175/2263933167309766272933307, \n -4801005224579354456654139837/166166705290936256217724 4416,\n 232855940800/97230777633,-57039375/516949856,0], \n [0,7466867/116588160,0,0,3227648000000/9720313182027,17972 4568369012721/784946257231456320,\n 234368000/1186672113 ,16390625/58886016,-14011/138852]]);\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 := ev alf(A[7,5]);\n a85 := evalf(A[7,6]);\n a86 := evalf(A[7,7]);\n a 87 := 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 := eval f(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 + a5 2*f2 + a53*f3 + a54*f4;\n f5 := fn(xk + c5*h,yk + t*h);\n t \+ := a61*f1 + a62*f2 + a63*f3 + a64*f4 + a65*f5;\n f6 := fn(xk + c6 *h,yk + t*h);\n t := a71*f1 + a72*f2 + a73*f3 + a74*f4 + a75*f5 + a76*f6;\n f7 := fn(xk + c7*h,yk + t*h);\n t := a81*f1 + a82 *f2 + a83*f3 + a84*f4 + a85*f5 + a86*f6 + a87*f7;\n f8 := fn(xk + c8*h,yk + t*h);\n\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,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,b8_=b8\};\n return subs(eqns,eval(rk6step)); \n els e\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 20 "RK6_3 scheme with \+ " }{XPPEDIT 18 0 "c[7] = 35/36;" "6#/&%\"cG6#\"\"(*&\"#N\"\"\"\"#O!\" \"" }{TEXT -1 1 " " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 " > " 0 "" {MPLTEXT 1 0 4128 "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([[7/95,7/95,0,0,0,0,0,0,0],\n [35/243,385/1180 98,16625/118098,0,0,0,0,0,0],\n [35/162,35/648,0,35/216,0,0,0 ,0,0],\n [31/55,262364129/407618750,0,-996909687/407618750,48 2147154/203809375,0,0,0,0],\n [97/140,-36974597989227800011/2 7927480760535300000,0,\n 40382359908551133219/720709180 9170400000,\n -5837069082360096304407/139502270831254555 0000,\n 13508218613909220883/22593673904425227520,0,0,0 ],\n [35/36,756727539023617977739/325483930565422777440,0,\n \+ -11344125743485787/1187841741251840,\n 93 403161519631272506393/11188675650069067742640,\n -381159 2140111638097686625/3308811923451740272240896,\n 102601 928198000/102454959175119,0,0],\n [1,250792505789354790081/66 169955021507254750,0,\n -349400598312667239/220053059599 29250,\n 17728479258264420469702254/1289712860941995340 762625,\n -61925787315598406758085/277648812184326264422 21,\n 43795320281930/26932915012729,-96493150422/179013 1071575,0],\n [0,1414477/22101450,0,0,6047117272638/182362452 67975,31062711625/123650103496,\n 27376502125/148102904 499,26504253/76308925,-23459/131064]]);\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 := eva lf(A[3,4]);\n a51 := evalf(A[4,2]);\n a52 := evalf(A[4,3]);\n a5 3 := 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 + 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*f 4 + 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 := \{SOL N_=[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,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,a 85_=a85,a86_=a86,a87_=a87,\n b1_=b1,b2_=b2,b3_=b3,b4_=b4,b5_=b 5,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 20 "RK6_4 sch eme with " }{XPPEDIT 18 0 "c[7] = 63/64;" "6#/&%\"cG6#\"\"(*&\"#j\"\" \"\"#k!\"\"" }{TEXT -1 1 " " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 4526 "RK6_4 := proc(fxy,x,y,xx,yy,h,stp s,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,a76,\n a81,a82,a 83,a84,a85,a86,a87,b1,b2,b3,b4,b5,b6,b7,b8,\n f1,f2,f3,f4,f5,f6,f7,f 8,t,k,fn,xk,yk,soln,eqns,A,saveDigits;\n saveDigits := Digits;\n D igits := max(trunc(evalhf(Digits)),Digits+5);\n\n fn := unapply(fxy, x,y);\n\n A := matrix([[19/220,19/220,0,0,0,0,0,0,0],\n [24 8/1719,1334488/56144259,6765440/56144259,0,0,0,0,0,0],\n [124 /573,31/573,0,31/191,0,0,0,0,0],\n [159/281,443985772809/6823 26636832,0,-1687953692799/682326636832,\n 815026549419 /341163318416,0,0,0,0],\n [64/91,-17777250984185217125334224/ 11518294736403027809706743,0,\n 1412202709832277291195 888/217326315781189203956731,\n -6002676221766232417168 2463968/12227430504797048182217556253,\n 4435917233994 48798344389120/674353607444582261870479169,0,0,0],\n [63/64,3 116283851584560023118564368757831/955301928184769582072071268073472,0, \n -15069358907111564759839129615089/112653529267071884 6783102910464,\n 4608655696349942937342097596091097426 937/402287173573717757962711842819464495104,\n -1041097 287013501701647909592093443697/767936864410813987103949029050941440,\n 126705961361465452994391/126885461609550467563520,0,0 ],\n [1,2688622152256014375593487443/635826944384194229962077 552,0,\n -165064430696291307962649/94425303421501597196 48,\n 85647344134405061626206038309671468707/574531788 1110570819440070776064368272,\n -1868050624284923482186 332306552702/977068785600982269831702681478435,\n 1684 2653200587834711/13166474974917410080,-608491123177947136/273668355989 69517651,0],\n [0,2465359/38465280,0,0,29999319743551167/9031 2244465616140,\n 2440521938223593/9364974686865720,124 1823968749/6822515208960,\n 177732994465792/3279301284 45465,-11266189/29580120]]);\n\n c2 := evalf(A[1,1]);\n c3 := eval f(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 := ev alf(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 + a 53*f3 + a54*f4;\n f5 := fn(xk + c5*h,yk + t*h);\n t := a61*f 1 + 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 20 "RK6_5 scheme with " } {XPPEDIT 18 0 "c[7] = 125/126;" "6#/&%\"cG6#\"\"(*&\"$D\"\"\"\"\"$E\"! \"\"" }{TEXT -1 1 " " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 4290 "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,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([[6/65,6/65,0,0,0,0,0,0,0],\n [23/159,9499 /303372,34385/303372,0,0,0,0,0,0],\n [23/106,23/424,0,69/424, 0,0,0,0,0],\n [80/141,967715920/1482903909,0,-1226377600/4943 01303,3552780800/1482903909,0,0,0,0],\n [91/129,-468047892592 40403389999/30226267716384768392160,0,\n 41055045969731 0264150/62971391075801600817,\n -48733588560949255621144 76/989343525191918950435887,\n 7315985052139691605041/1 1082753205258491358240,0,0,0],\n [125/126,3814047137877227803 8787625/10601915981402462257077264,0,\n -200697271615378 5212305625/135921999761570028936888,\n 2246433348690398 4266603841845000/1782850965624830561723340121359,\n -333 5402692827978879904464125/2252537921258500779474698736,\n \+ 627869105822046875/603327424485556953,0,0],\n [1,146733704 741261404039925503/35929601425079440010660000,0,\n -3321 75497511735087990/19741539244549142863,\n 6401812090729 586424603870720224/446765171619464803439374766203,\n -42 65990808672245834945068273/2437965633092800464693941600,\n \+ 330884756892468554/282253958465901877,-28749933344522124/304703746 9905514375,0],\n [0,6205673/96600000,0,0,43180775629715/12960 6111568587,10385429951981/39850344008000,\n 31503909334 9/1710630556040,1569042174483/1579405853125,-9646151/11543640]]);\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 := eva lf(A[3,3]);\n a43 := evalf(A[3,4]);\n a51 := evalf(A[4,2]);\n a5 2 := 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 := eval f(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 + a 65*f5;\n f6 := fn(xk + c6*h,yk + t*h);\n t := a71*f1 + a72*f 2 + a73*f3 + a74*f4 + a75*f5 + a76*f6;\n f7 := fn(xk + c7*h,yk + \+ t*h);\n t := a81*f1 + a82*f2 + a83*f3 + a84*f4 + a85*f5 + a86*f6 \+ + a87*f7;\n f8 := fn(xk + c8*h,yk + t*h);\n\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,a31_=a31,a32_=a32,a41_=a 41,\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,a 82_=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 sub s(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 32 "RK6_6 Papakostas' scheme with " }{XPPEDIT 18 0 "c[7] \+ = 199/200;" "6#/&%\"cG6#\"\"(*&\"$*>\"\"\"\"$+#!\"\"" }{TEXT -1 1 " " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 4164 "RK6_6 := proc(fxy,x,y,xx,yy,h,stps,bb)\n local c2,c3,c4,c5,c 6,c7,c8,a21,a31,a32,a41,a42,a43,a51,\n a52,a53,a54,a61,a62,a63,a64,a 65,a71,a72,a73,a74,a75,a76,\n a81,a82,a83,a84,a85,a86,a87,b1,b2,b3,b 4,b5,b6,b7,b8,\n f1,f2,f3,f4,f5,f6,f7,f8,t,k,fn,xk,yk,soln,eqns,A,sa veDigits;\n saveDigits := Digits;\n Digits := max(trunc(evalhf(Dig its)),Digits+5);\n\n fn := unapply(fxy,x,y);\n\n A := matrix([[17/ 183,17/183,0,0,0,0,0,0,0],\n [12/83,3756/117113,13176/117113, 0,0,0,0,0,0],\n [18/83,9/166,0,27/166,0,0,0,0,0],\n \+ [71/125,207751751/316406250,0,-526769377/210937500,1524242129/63281250 0,0,0,0,0],\n [42/59,-4970082682619223281/2887511529739311186 ,0,97919278033879057/13556392158400522,\n -4071316740079 30877068/74078904949579652469,\n 1237601855204268750000 /1753200750473385108433,0,0,0],\n [199/200,176597685527535385 020980411/42773485015591331328000000,0,\n -6793162515552 646891859/401628967282547712000,\n 12704926019361287204 873446554247/886659402653054716778496000000,\n -50728836 334509259632278125/32657591718008685915971584,\n 515362 23982796190703/51293749413888000000,0,0],\n [1,29903352057233 7573523/66918720793812357519,0,-16550269823961899/902146153892364,\n \+ 49920346343238033627496282/3215735869387500624775563,\n \+ -1686432488955761721093750/978844996793357447730403,\n \+ 161901609084039/149698803705724,-305146137600000/5476034 1991955873,0],\n [0,24503/381483,0,0,1366847103121/4106349847 584,20339599609375/75933913767768,\n 35031290651/194765 546144,16620160000000/11001207123543,-14933/11016]]);\n\n c2 := eval f(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 := ev alf(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(x k,yk);\n t := a21*f1;\n f2 := fn(xk + c2*h,yk + t*h);\n \+ t := a31*f1 + a32*f2;\n f3 := fn(xk + c3*h,yk + t*h);\n t : = a41*f1 + a42*f2 + a43*f3;\n f4 := fn(xk + c4*h,yk + t*h);\n \+ t := a51*f1 + a52*f2 + a53*f3 + a54*f4;\n f5 := fn(xk + c5*h,yk + t*h);\n t := a61*f1 + a62*f2 + a63*f3 + a64*f4 + a65*f5;\n \+ f6 := fn(xk + c6*h,yk + t*h);\n t := a71*f1 + a72*f2 + a73*f3 + a74*f4 + a75*f5 + a76*f6;\n f7 := fn(xk + c7*h,yk + t*h);\n \+ t := a81*f1 + a82*f2 + a83*f3 + a84*f4 + a85*f5 + a86*f6 + a87*f7;\n \+ f8 := fn(xk + c8*h,yk + t*h);\n\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,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_=b 3,b4_=b4,b5_=b5,b6_=b6,b7_=b7,b8_=b8\};\n return subs(eqns,eval(r k6step)); \n else\n return evalf[saveDigits]([soln]);\n end i f;\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 47 "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 := unapp ly(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-%$expG6#,,*&#\"#7 \"#<\"\"\"*(F'F0-%$cosG6#,$*&\"\"%F0F'F0F0F0-F)6#,$F'!\"\"F0F0F;*&#\"$ !=\"$*GF0*&F8F0F2F0F0F0*&#\"#[F/F0*(F8F0-%$sinGF4F0F'F0F0F0*&#\"#'*F?F 0*&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$$\"3ALL$3FWYs#F/$\"3!H*fm:2P/5F27$$\"3%)***\\iSmp3%F /$\"3Qn()\\Dat45F27$$\"3WmmmT&)G\\aF/$\"34$Q7t`Dr,\"F27$$\"3m****\\7G$ R<)F/$\"3S2-*\\9jw.\"F27$$\"3GLLL3x&)*3\"!#=$\"3U([#>C\\El5F27$$\"3))* *\\i!R(*Rc\"FJ$\"3>&=^@[0u7\"F27$$\"3umm\"H2P\"Q?FJ$\"3k\\#o#G?)=?\"F2 7$$\"3!***\\PMnNrDFJ$\"3s_j<)f!R*G\"F27$$\"3MLL$eRwX5$FJ$\"37'\\4u:c`O \"F27$$\"3_LLe*[`HP$FJ$\"3[!\\'y0#yNR\"F27$$\"3rLLL$eI8k$FJ$\"3N\"Ha_9 o@T\"F27$$\"3_L$3-8>bx$FJ$\"3@))>@pAD<9F27$$\"3*QL$3xwq4RFJ$\"3a@g!fsi #>9F27$$\"3EM$eRA'*Q/%FJ$\"3^DvP/8/=9F27$$\"33ML$3x%3yTFJ$\"3bF0p:\"oM T\"F27$$\"3h+]PfyG7ZFJ$\"3e=U+Y19h8F27$$\"3emm\"z%4\\Y_FJ$\"3Yii#4W6uD 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%p\")Q$F2$\"3#o,C;(=8foFJ7$$\"3B++](=]@W$F2$\"3#G%=QV$\\;4(FJ7$$\"3C$e kyZ2mY$F2$\"3u,muc\"4C(FJ7$$\"3h Tgx.2vFNF2$\"3/^M\"Q[;lC(FJ7$$\"35L$e*[$z*RNF2$\"3=wJ%fi2nC(FJ7$$\"3)* \\PMFwrmNF2$\"3R[i&\\xl(GsFJ7$$\"3%o;Hd!fX$f$F2$\"3IEKi0hy'=(FJ7$$\"3r $e9T=%>?OF2$\"3(>gS`&3dArFJ7$$\"3e++]iC$pk$F2$\"3ma\\oRiHQqFJ7$$\"3ILe *[t\\sp$F2$\"3'e9/wG(3MoFJ7$$\"3[m;H2qcZPF2$\"3CYQ8S*3be'FJ7$$\"3O+]7. \"fF&QF2$\"3**Q8E[N&3+'FJ7$$\"3Ymm;/OgbRF2$\"3kN#z0%oN^aFJ7$$\"3w**\\i lAFjSF2$\"3[i8#)*p//*\\FJ7$$\"3ym\"zW7@^6%F2$\"3>C%QCunR#[FJ7$$\"3yLLL $)*pp;%F2$\"3g*yCm#3E'p%FJ7$$\"3)QL3-$H**>UF2$\"3$*o:W?mr0YFJ7$$\"3)RL $3xe,tUF2$\"3!\\Bp&*))oXb%FJ7$$\"3h+v=n(*fDVF2$\"3kIpK$)H$3a%FJ7$$\"3C n;HdO=yVF2$\"3u&G6!oNOhXFJ7$$\"3MMe9\"z-lU%F2$\"3kC\">#=Lu2YFJ7$$\"3a+ ++D>#[Z%F2$\"3w_(eqj7vn%FJ7$$\"3SnmT&G!e&e%F2$\"3W>T$>g**p!\\FJ7$$\"3# RLLL)Qk%o%F2$\"3'yDBP_q:;&FJ7$$\"37+]iSjE!z%F2$\"3J;fP@m(pV&FJ7$$\"3a+ ]P40O\"*[F2$\"3!>+$=fU-gcFJ7$$\"\"&F)$\"3h(Q0fOqh\"eFJ-%'COLOURG6&%$RG BG$\"#5!\"\"F(F(-%%FONTG6$%*HELVETICAG\"\"*-%+AXESLABELSG6$%\"xG%%y(x) G-%%VIEWG6$;F(F]am;F($\"$X\"!\"#" 1 2 0 1 10 0 2 9 1 4 2 1.000000 45.000000 45.000000 0 0 "Curve 1" }}}}{PARA 0 "" 0 "" {TEXT -1 0 "" }} {PARA 0 "" 0 "" {TEXT -1 32 "The following code constructs a " }{TEXT 260 17 "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 0 770 "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: `,``(x 0,y0)],[`step width: `,hh],\n[`no. of steps: `,numsteps]]);``;\nmt hds := [`Sharp-Verner scheme with `*``(c[7]=19/20),`scheme with `*``(c [7]=24/25),`scheme with `*``(c[7]=35/36),`scheme with `*``(c[7]=63/64) ,`scheme with `*``(c[7]=125/126),`Papakostas' scheme with `*``(c[7]=19 9/200)]: errs := []:\nDigits := 25:\nfor ct to 6 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_||c t[ii,2]-f(Fn_RK6_||ct[ii,1]))^2;\n end do:\n errs := [op(errs),sqr t(sm/numpts)];\nend do:\nDigits := 10:\nlinalg[transpose]([mthds,evalf (errs)]);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#K%'matrixG6#7&7$%0slope~f ield:~~~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(pprint46\"" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%!G" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#K%'matrixG6#7(7$ *&%:Sharp-Verner~scheme~with~G\"\"\"-%!G6#/&%\"cG6#\"\"(#\"#>\"#?F*$\" +[%Qxi$!#B7$*&%-scheme~with~GF*-F,6#/F/#\"#C\"#DF*$\"+\"yBGB$F87$*&F;F *-F,6#/F/#\"#N\"#OF*$\"+1xFj@F87$*&F;F*-F,6#/F/#\"#j\"#kF*$\"+AnGD$*!# C7$*&F;F*-F,6#/F/#\"$D\"\"$E\"F*$\"+;%yk9$FX7$*&%9Papakostas'~scheme~w ith~GF*-F,6#/F/#\"$*>\"$+#F*$\"+pP/M\\FXQ(pprint56\"" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 30 "The following code con structs " }{TEXT 260 20 "numerical procedures" }{TEXT -1 56 " for solu tions based on each of the Runge-Kutta schemes." }}{PARA 0 "" 0 "" {TEXT -1 75 "The error in the value obtained by each of the methods at the point where " }{XPPEDIT 18 0 "x = 4.999;" "6#/%\"xG-%&FloatG6$\" %**\\!\"$" }{TEXT -1 16 " is also given." }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 701 "F := (x,y) -> 12*x*cos(4*x)*exp(-x)*y: hh := 0.01: n umsteps := 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 with `*``(c[7]=19/ 20),`scheme with `*``(c[7]=24/25),`scheme with `*``(c[7]=35/36),`schem e with `*``(c[7]=63/64),`scheme with `*``(c[7]=125/126),`Papakostas' s cheme with `*``(c[7]=199/200)]: errs := []:\nDigits := 25:\nfor ct to \+ 6 do\n fn_RK6_||ct := RK6_||ct(F(x,y),x,y,x0,y0,hh,numsteps,true);\n end do:\nxx := 4.999: fxx := evalf(f(xx)):\nfor ct to 6 do\n errs := [op(errs),abs(fn_RK6_||ct(xx)-fxx)];\nend do:\nDigits := 10:\nlinalg[ transpose]([mthds,evalf(errs)]);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#K% 'matrixG6#7&7$%0slope~field:~~~G,$*,\"#7\"\"\"%\"xGF,-%$cosG6#,$*&\"\" %F,F-F,F,F,-%$expG6#,$F-!\"\"F,%\"yGF,F,7$%0initial~point:~G-%!G6$\"\" !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$*&%:Sharp-Verner~scheme~with~G\"\"\"-%!G6#/&%\"cG6# \"\"(#\"#>\"#?F*$\"+<)HNU$!#B7$*&%-scheme~with~GF*-F,6#/F/#\"#C\"#DF*$ \"+TX(o_$F87$*&F;F*-F,6#/F/#\"#N\"#OF*$\"+y@$\\H#F87$*&F;F*-F,6#/F/#\" #j\"#kF*$\"+m_B2')!#C7$*&F;F*-F,6#/F/#\"$D\"\"$E\"F*$\"+'[k_>\"FX7$*&% 9Papakostas'~scheme~with~GF*-F,6#/F/#\"$*>\"$+#F*$\"+)=V,R&FXQ(pprint7 6\"" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 4 "Th e " }{TEXT 260 22 "root mean square error" }{TEXT -1 20 " over the int erval " }{XPPEDIT 18 0 "[0, 5];" "6#7$\"\"!\"\"&" }{TEXT -1 82 " of \+ each Runge-Kutta method is estimated as follows using the special proc edure " }{TEXT 0 5 "NCint" }{TEXT -1 98 " to perform numerical integ ration by the 7 point Newton-Cotes method over 200 equal subintervals. " }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 445 "mthds := [`Sharp-Verner scheme with `*``(c[7]=19/20),`scheme with `*``(c[7]=24/25),`scheme wi th `*``(c[7]=35/36),`scheme with `*``(c[7]=63/64),`scheme with `*``(c[ 7]=125/126),`Papakostas' scheme with `*``(c[7]=199/200)]: errs := []: \nDigits := 20:\nfor ct to 6 do\n sm := NCint((f(x)-'fn_RK6_||ct'(x) )^2,x=0..5,adaptive=false,numpoints=7,factor=200);\n errs := [op(err s),sqrt(sm/5)];\nend do:\nDigits := 10:\nlinalg[transpose]([mthds,eval f(errs)]);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#K%'matrixG6#7(7$*&%:Shar p-Verner~scheme~with~G\"\"\"-%!G6#/&%\"cG6#\"\"(#\"#>\"#?F*$\"+\")RcIO !#B7$*&%-scheme~with~GF*-F,6#/F/#\"#C\"#DF*$\"+s_pJKF87$*&F;F*-F,6#/F/ #\"#N\"#OF*$\"+haDi@F87$*&F;F*-F,6#/F/#\"#j\"#kF*$\"+!z6FJ*!#C7$*&F;F* -F,6#/F/#\"$D\"\"$E\"F*$\"+Ab(*GJFX7$*&%9Papakostas'~scheme~with~GF*-F ,6#/F/#\"$*>\"$+#F*$\"+5$zv#\\FXQ(pprint86\"" }}}{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 0 541 "evalf[20](plot([f(x)-'fn_R K6_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),f(x)-'fn_RK6_6'(x)],x=0..5,font=[HELVETICA,9],\ncolor =[COLOR(RGB,.95,0,.95),COLOR(RGB,.5,0,.95),COLOR(RGB,.95,.45,0),COLOR( RGB,.2,.85,0),\nCOLOR(RGB,0,.3,.95),COLOR(RGB,.95,0,.2)],legend=[`Shar p-Verner scheme with c[7]=19/20`,`scheme with c[7]=24/25`,`scheme with c[7]=35/36`,`scheme with c[7]=63/64`,`scheme with c[7]=125/126`,`Papa kostas' scheme with c[7]=199/200`],title=`error curves for 8 stage ord er 6 Runge-Kutta methods`));" }}{PARA 13 "" 1 "" {GLPLOT2D 995 628 628 {PLOTDATA 2 "6,-%'CURVESG6%7ct7$$\"\"!F)F(7$$\"5NLLL$3FWYs#!#@$\"& ee#!#>7$$\"5qmmmmT&)G\\aF-$\"&R)zF07$$\"50+++]7G$R<)F-$\"'Bw:F07$$\"5M LLLL3x&)*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 $='F07$$\"5MLLek.pu/BF>$\"'MEsF07$$\"5+++]PMnNrDF>$\"'E>zF07$$\"5nmmT5 ll'z$GF>$\"'\"eh)F07$$\"5MLLL$eRwX5$F>$\"'SD\"*F07$$\"5MLL$3F%\\wQKF>$ \"']#G*F07$$\"5MLLLe*[`HP$F>$\"']2%*F07$$\"5MLL3-jx/SMF>$\"'ee%*F07$$ \"5MLL$ek.Ur]$F>$\"'=#[*F07$$\"5MLLe*)4jBuNF>$\"'`1&*F07$$\"5MLLLL$eI8 k$F>$\"'/(\\*F07$$\"5MLLL3xwq4RF>$\"'91#*F07$$\"5NLLL$3x%3yTF>$\"'1Z') F07$$\"5-+](oHaN;J%F>$\"'a\\zF07$$\"5ommT5:j=XWF>$\"''\\V(F07$$\"5NL$e Rs3P(yXF>$\"'=SnF07$$\"5-++]PfyG7ZF>$\"'&Hb&F07$$\"5om;/^J'Qe%[F>$\"') 3y%F07$$\"5NLLek.%*Qz\\F>$\"'YEQF07$$\"5++]7yv,%H6&F>$\"'HpBF07$$\"5om mm\"z%4\\Y_F>$\"'%)H:F07$$\"5+++]P4'4.P&F>$\"&zh'F07$$\"5NLLL$3FGT\\&F >$!&!y^F07$$\"5qmm;HKp%zh&F>$!'AM9F07$$\"5++++v$fl$!'(=-#F07$$\"5 qmmmm;HS*)fF>$!''>9$F07$$\"5NLLLeR-/PiF>$!(;]\"QF>7$$\"5qmmmm\"HZ_O'F> $!(g$=RF>7$$\"5++++vVVX$\\'F>$!(OE.%F>7$$\"5qmm;zpybdlF>$!(?E+%F>7$$\" 5NLLL$eRh;i'F>$!(zf-%F>7$$\"5+++](=#\\w&o'F>$!(:J'RF>7$$\"5qmmm\"zWo) \\nF>$!(&oGRF>7$$\"5NLLL3_DG1qF>$!(D'QPF>7$$\"5-+++DcmpisF>$!(&z0NF>7$ $\"5qmmm\"Hd)G&R(F>$!(b(yMF>7$$\"5NLLLe*[!)y_(F>$!(#o\\MF>7$$\"5qmmm\" zWwTf(F>$!(D>X$F>7$$\"5++++D1CZgwF>$!(SET$F>7$$\"5NLLLek$ons(F>$!(9>[$ F>7$$\"5qmmm\"HKkIz(F>$!(5t\\$F>7$$\"5SLLL3-B@EyF>$!(?N_$F>7$$\"50+++D \"Gg$fyF>$!(h'yMF>7$$\"5qmmmTg#3D*yF>$!(A:a$F>7$$\"5SLLLeRilDzF>$!(myd $F>7$$\"5qmmm\"z>_>*zF>$!(:df$F>7$$\"50+++Dc\"[#e!)F>$!(jUf$F>7$$\"5qm mm\"H2S3>)F>$!(`)=PF>7$$\"5OLLLe*)>VB$)F>$!((\\TQF>7$$\"50++v$fL:&*Q)F >$!(mZ$QF>7$$\"5qmm;H#o)fb%)F>$!(7#[QF>7$$\"50+](oaNS')[)F>$!($y!)QF>7 $$\"5SLLekG?o@&)F>$!(/f%RF>7$$\"5qm;H#=qBZb)F>$!(QV*QF>7$$\"50++++v`w( e)F>$!(8h\"RF>7$$\"5qmmTN@([Ql)F>$!(Rt#RF>7$$\"5SLL$3x1K*>()F>$!(_r*RF >7$$\"50++D19a,'y)F>$!(yi%RF>7$$\"5qmmmTg()4_))F>$!(Rj%RF>7$$\"5+++]7` aE%)*)F>$!(fQ\"RF>7$$\"5NLLL$e9Kk6*F>$!(eM*QF>7$$\"5qmm;aQ))f[#*F>$!(u sx$F>7$$\"5.+++DJbw!Q*F>$!(wvl$F>7$$\"5NLL$ekGkX#**F>$!(z#*)HF>7$$\"5n mmm;/j$o/\"F0$!(XTF#F>7$$\"5+++]7GTt%4\"F0$!(O&[7$$\"5MLLL3_>jU6F0$ !(q,W\"F>7$$\"5nm;aQ`B6c6F0$!(r_S\"F>7$$\"5+++voaFfp6F0$!(,lP\"F>7$$\" 5ML$e*)f:tI=\"F0$!(G;P\"F>7$$\"5nmm;HdNb'>\"F0$!(37Q\"F>7$$\"5MLLe*)fV ^B7F0$!(K&e9F>7$$\"5++++]i^Z]7F0$!(\"*)3;F>7$$\"5+++++v\"=YI\"F0$!(A3- #F>7$$\"5++++](=h(e8F0$!(I'oDF>7$$\"5++++]7!Q4T\"F0$!(>g1$F>7$$\"5++++ ]P[6j9F0$!(vUS$F>7$$\"5nmm\"HKR'\\5:F0$!(n$=NF>7$$\"5MLL$e*[z(yb\"F0$! 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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 squar e error" }{TEXT -1 18 " of each solution." }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 773 "G := (x,y) -> x/y: hh := 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 with `*``(c[7]=19/20),`scheme with `* ``(c[7]=24/25),`scheme with `*``(c[7]=35/36),`scheme with `*``(c[7]=63 /64),`scheme with `*``(c[7]=125/126),`Papakostas' scheme with `*``(c[7 ]=199/200)]: errs := []:\nDigits := 20:\ng := x -> sqrt(1+x^2):\nfor c t to 6 do\n Gn_RK6_||ct := RK6_||ct(G(x,y),x,y,x0,y0,hh,numsteps,fal se);\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 d o:\n errs := [op(errs),sqrt(sm/numpts)];\nend do:\nDigits := 10:\nli nalg[transpose]([mthds,evalf(errs)]);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#K%'matrixG6#7&7$%0slope~field:~~~G*&%\"xG\"\"\"%\"yG!\"\"7$%0initia l~point:~G-%!G6$\"\"!F+7$%/step~width:~~~G$\"\"&!\"#7$%1no.~of~steps:~ ~~G\"$+#Q(pprint96\"" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%!G" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#K%'matrixG6#7(7$*&%:Sharp-Verner~scheme~with ~G\"\"\"-%!G6#/&%\"cG6#\"\"(#\"#>\"#?F*$\"+6\"$+#F*$\"+ UD*))*HFDQ)pprint106\"" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{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 "9.99;" "6#-%&FloatG6$\"$***!\"#" }{TEXT -1 16 " is also given." }} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 703 "G := (x,y) -> x/y: hh := 0. 05: numsteps := 200: x0 := 0: y0 := 1:\nmatrix([[`slope field: `,G(x ,y)],[`initial point: `,``(x0,y0)],[`step width: `,hh],\n[`no. of st eps: `,numsteps]]);``;\nmthds := [`Sharp-Verner scheme with `*``(c[7 ]=19/20),`scheme with `*``(c[7]=24/25),`scheme with `*``(c[7]=35/36),` scheme with `*``(c[7]=63/64),`scheme with `*``(c[7]=125/126),`Papakost as' scheme with `*``(c[7]=199/200)]: errs := []:\nDigits := 25:\nfor c t to 6 do\n gn_RK6_||ct := RK6_||ct(G(x,y),x,y,x0,y0,hh,numsteps,tru e);\nend do:\ng := x -> sqrt(1+x^2):\nxx := 9.99: gxx := evalf(g(xx)): \nfor ct to 6 do\n errs := [op(errs),abs(gn_RK6_||ct(xx)-gxx)];\nend do:\nDigits := 10:\nlinalg[transpose]([mthds,evalf(errs)]);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#K%'matrixG6#7&7$%0slope~field:~~~G*&%\"xG\" \"\"%\"yG!\"\"7$%0initial~point:~G-%!G6$\"\"!F+7$%/step~width:~~~G$\" \"&!\"#7$%1no.~of~steps:~~~G\"$+#Q)pprint116\"" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%!G" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#K%'matrixG6#7(7$ *&%:Sharp-Verner~scheme~with~G\"\"\"-%!G6#/&%\"cG6#\"\"(#\"#>\"#?F*$\" *rT=d$!#B7$*&%-scheme~with~GF*-F,6#/F/#\"#C\"#DF*$\"*&)*>(3$F87$*&F;F* -F,6#/F/#\"#N\"#OF*$\"*>yh@#F87$*&F;F*-F,6#/F/#\"#j\"#kF*$\"*\"Rv!e\"F 87$*&F;F*-F,6#/F/#\"$D\"\"$E\"F*$\"*r)e/7F87$*&%9Papakostas'~scheme~wi th~GF*-F,6#/F/#\"$*>\"$+#F*$\"*v.\"e5F8Q)pprint126\"" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 4 "The " }{TEXT 260 22 "ro ot mean square error" }{TEXT -1 20 " over the interval " }{XPPEDIT 18 0 "[0, 10]" "6#7$\"\"!\"#5" }{TEXT -1 82 " of each Runge-Kutta met hod is estimated as follows using the special procedure " }{TEXT 0 5 "NCint" }{TEXT -1 98 " to perform numerical integration by the 7 poin t Newton-Cotes method over 100 equal subintervals." }}{EXCHG {PARA 0 " > " 0 "" {MPLTEXT 1 0 470 "mthds := [`Sharp-Verner scheme with `*``(c[ 7]=19/20),`scheme with `*``(c[7]=24/25),`scheme with `*``(c[7]=35/36), `scheme with `*``(c[7]=63/64),`scheme with `*``(c[7]=125/126),`Papakos tas' scheme with `*``(c[7]=199/200)]: errs := []:\nDigits := 20:\ng := x -> sqrt(1+x^2):\nfor ct to 6 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]([mthds ,evalf(errs)]);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#K%'matrixG6#7(7$*&% :Sharp-Verner~scheme~with~G\"\"\"-%!G6#/&%\"cG6#\"\"(#\"#>\"#?F*$\"+Lw !>6\"!#B7$*&%-scheme~with~GF*-F,6#/F/#\"#C\"#DF*$\"+$4-c/)!#C7$*&F;F*- F,6#/F/#\"#N\"#OF*$\"+>l:idFD7$*&F;F*-F,6#/F/#\"#j\"#kF*$\"+]KY6UFD7$* &F;F*-F,6#/F/#\"$D\"\"$E\"F*$\"+a98OLFD7$*&%9Papakostas'~scheme~with~G F*-F,6#/F/#\"$*>\"$+#F*$\"+]S`!*HFDQ)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 0 542 "evalf[20](plot([g(x)-'gn_R K6_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),g(x)-'gn_RK6_6'(x)],x=0..10,font=[HELVETICA,9],\ncolo r=[COLOR(RGB,.95,0,.95),COLOR(RGB,.5,0,.95),COLOR(RGB,.95,.45,0),COLOR (RGB,.2,.85,0),\nCOLOR(RGB,0,.3,.95),COLOR(RGB,.95,0,.2)],legend=[`Sha rp-Verner scheme with c[7]=19/20`,`scheme with c[7]=24/25`,`scheme wit h c[7]=35/36`,`scheme with c[7]=63/64`,`scheme with c[7]=125/126`,`Pap akostas' scheme with c[7]=199/200`],title=`error curves for 8 stage or der 6 Runge-Kutta methods`));" }}{PARA 13 "" 1 "" {GLPLOT2D 1097 790 790 {PLOTDATA 2 "6,-%'CURVESG6%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$$\"5 mmm\"HdXqmc\"F4$!'LE8F07$$\"5++++]ilyM;F4$!')\\K\"F07$$\"5LLLLe*)4D2>F 4$!'!\\[\"F07$$\"5mmmmm;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% 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$eF07$F_y$\"&-Q&F07$Fiy$\"&M&\\F07$F^z$\"&!eXF07$Fcz$\"&vC%F07$Fhz$\"& )QRF07$F][l$\"&em$F07$Fb[l$\"&OV$F07$Fg[l$\"&aC$F07$F\\\\l$\"&c/$F07$F a\\l$\"&N*GF07$Ff\\l$\"&Ut#F07$F[]l$\"&mg#F07$F`]l$\"&$zCF07$Fe]l$\"&) oBF07$Fj]l$\"&LE#F07$F_^l$\"&U<#F07$Fd^l$\"&`3#F07$Fi^l$\"&0+#F07$F^_l $\"&<$>F07$Fc_l$\"&F'=F07$Fh_l$\"&jz\"F07$F]`l$\"&dt\"F07$Fb`l$\"&1o\" F07$Fg`l$\"&Oi\"F07$F\\al$\"&bd\"F07$Faal$\"&r_\"F07$Ffal$\"&d[\"F07$F [bl$\"&IW\"F07$F`bl$\"&\\S\"F07$Febl$\"&sO\"F07$Fjbl$\"&AL\"F07$F_cl$ \"&uH\"F07$Fdcl$\"&cE\"F07$Ficl$\"&ZB\"F07$F^dl$\"&a?\"F07$Fcdl$\"&(z6 F07$Fhdl$\"&::\"F07$F]el$\"&v7\"F07$Fbel$\"&H5\"F07$Fgel$\"&.3\"F07$F \\fl$\"%d5F`fl-Fbfl6&FdflFeflF(F_ip-Fifl6#%EPapakostas'~scheme~with~c[ 7]=199/200G-%%FONTG6$%*HELVETICAG\"\"*-%+AXESLABELSG6$Q\"x6\"Q!Ffgr-%& TITLEG6#%Uerror~curves~for~8~stage~order~6~Runge-Kutta~methodsG-%%VIEW G6$;F(F\\fl%(DEFAULTG" 1 2 0 1 10 0 2 9 1 4 2 1.000000 45.000000 45.000000 0 1 "Sharp-Verner scheme with c[7]=19/20" "scheme with c[7]= 24/25" "scheme with c[7]=35/36" "scheme with c[7]=63/64" "scheme with \+ c[7]=125/126" "Papakostas' scheme with c[7]=199/200" }}}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}} {SECT 1 {PARA 4 "" 0 "" {TEXT -1 47 "Test 3 of 8 stage, order 6 Runge- Kutta methods " }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "dy/ dx = -x*y;" "6#/*&%#dyG\"\"\"%#dxG!\"\",$*&%\"xGF&%\"yGF&F(" }{TEXT -1 11 ", " }{XPPEDIT 18 0 "y(0)=1" "6#/-%\"yG6#\"\"!\"\"\"" } {TEXT -1 1 " " }}{PARA 0 "" 0 "" {TEXT -1 10 "Solution: " }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "y = exp(-x^2/2);" "6#/%\"yG-%$e xpG6#,$*&%\"xG\"\"#F+!\"\"F," }{TEXT -1 2 ". " }}{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 0 773 "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 with `*``(c[7]=19/20),`scheme with `* ``(c[7]=24/25),`scheme with `*``(c[7]=35/36),`scheme with `*``(c[7]=63 /64),`scheme with `*``(c[7]=125/126),`Papakostas' scheme with `*``(c[7 ]=199/200)]: errs := []:\nDigits := 20:\nh := x -> exp(-x^2/2):\nfor c t to 6 do\n Hn_RK6_||ct := RK6_||ct(H(x,y),x,y,x0,y0,hh,numsteps,fal se);\n sm := 0: numpts := nops(Hn_RK6_||ct):\n for ii to numpts do \n sm := sm+(Hn_RK6_||ct[ii,2]-h(Hn_RK6_||ct[ii,1]))^2;\n end d o:\n errs := [op(errs),sqrt(sm/numpts)];\nend do:\nDigits := 10:\nli nalg[transpose]([mthds,evalf(errs)]);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#K%'matrixG6#7&7$%0slope~field:~~~G,$*&%\"xG\"\"\"%\"yGF,!\"\"7$%0in itial~point:~G-%!G6$\"\"!F,7$%/step~width:~~~G$F,F.7$%1no.~of~steps:~~ ~G\"$+\"Q)pprint146\"" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%!G" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#K%'matrixG6#7(7$*&%:Sharp-Verner~scheme~with ~G\"\"\"-%!G6#/&%\"cG6#\"\"(#\"#>\"#?F*$\"+T*R8o\"!#?7$*&%-scheme~with ~GF*-F,6#/F/#\"#C\"#DF*$\"+$\\5nU$F87$*&F;F*-F,6#/F/#\"#N\"#OF*$\"+/pg JLF87$*&F;F*-F,6#/F/#\"#j\"#kF*$\"+61FjJF87$*&F;F*-F,6#/F/#\"$D\"\"$E \"F*$\"+[&G!yHF87$*&%9Papakostas'~scheme~with~GF*-F,6#/F/#\"$*>\"$+#F* $\"+N))*\\#GF8Q)pprint156\"" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 30 "The following code constructs " }{TEXT 260 20 " numerical procedures" }{TEXT -1 56 " for solutions based on each of th e Runge-Kutta schemes." }}{PARA 0 "" 0 "" {TEXT -1 75 "The error in th e value obtained by each of the methods at the point where " } {XPPEDIT 18 0 "9.99;" "6#-%&FloatG6$\"$***!\"#" }{TEXT -1 16 " is als o given." }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 703 "H := (x,y) -> - x*y: hh := 0.1: numsteps := 100: x0 := 0: y0 := 1:\nmatrix([[`slope fi eld: `,H(x,y)],[`initial point: `,``(x0,y0)],[`step width: `,hh], \n[`no. of steps: `,numsteps]]);``;\nmthds := [`Sharp-Verner scheme \+ with `*``(c[7]=19/20),`scheme with `*``(c[7]=24/25),`scheme with `*``( c[7]=35/36),`scheme with `*``(c[7]=63/64),`scheme with `*``(c[7]=125/1 26),`Papakostas' scheme with `*``(c[7]=199/200)]: errs := []:\nDigits \+ := 20:\nfor ct to 6 do\n hn_RK6_||ct := RK6_||ct(H(x,y),x,y,x0,y0,hh ,numsteps,true);\nend do:\nh := x -> exp(-x^2/2):\nxx := 9.99: hxx := \+ evalf(h(xx)):\nfor ct to 6 do\n errs := [op(errs),abs(hn_RK6_||ct(xx )-hxx)];\nend do:\nDigits := 10:\nlinalg[transpose]([mthds,evalf(errs) ]);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#K%'matrixG6#7&7$%0slope~field:~ ~~G,$*&%\"xG\"\"\"%\"yGF,!\"\"7$%0initial~point:~G-%!G6$\"\"!F,7$%/ste p~width:~~~G$F,F.7$%1no.~of~steps:~~~G\"$+\"Q)pprint166\"" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%!G" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#K%'matri xG6#7(7$*&%:Sharp-Verner~scheme~with~G\"\"\"-%!G6#/&%\"cG6#\"\"(#\"#> \"#?F*$\"+c_gME!#N7$*&%-scheme~with~GF*-F,6#/F/#\"#C\"#DF*$\"+9fj)p#F8 7$*&F;F*-F,6#/F/#\"#N\"#OF*$\"+\\NMMDF87$*&F;F*-F,6#/F/#\"#j\"#kF*$\"+ Es2]CF87$*&F;F*-F,6#/F/#\"$D\"\"$E\"F*$\"+Kz3SCF87$*&%9Papakostas'~sch eme~with~GF*-F,6#/F/#\"$*>\"$+#F*$\"+/HBtBF8Q)pprint176\"" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 4 "The " }{TEXT 260 22 "root mean square error" }{TEXT -1 110 " over the interval [0, 0.5 ] of each Runge-Kutta method is estimated as follows using the specia l procedure " }{TEXT 0 5 "NCint" }{TEXT -1 97 " to perform numerical integration by the 7 point Newton-Cotes method over 50 equal subinter vals." }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 469 "mthds := [`Sharp-V erner scheme with `*``(c[7]=19/20),`scheme with `*``(c[7]=24/25),`sche me with `*``(c[7]=35/36),`scheme with `*``(c[7]=63/64),`scheme with `* ``(c[7]=125/126),`Papakostas' scheme with `*``(c[7]=199/200)]: errs := []:\nDigits := 20:\nh := x -> exp(-x^2/2):\nfor ct to 6 do\n sm := \+ NCint((h(x)-'hn_RK6_||ct'(x))^2,x=0..10,adaptive=false,numpoints=7,fac tor=50);\n errs := [op(errs),sqrt(sm/10)];\nend do:\nDigits := 10:\n linalg[transpose]([mthds,evalf(errs)]);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#K%'matrixG6#7(7$*&%:Sharp-Verner~scheme~with~G\"\"\"-%!G6#/&%\"c G6#\"\"(#\"#>\"#?F*$\"+)HJuc\"!#?7$*&%-scheme~with~GF*-F,6#/F/#\"#C\"# DF*$\"+z#p(4KF87$*&F;F*-F,6#/F/#\"#N\"#OF*$\"+!)\\y?JF87$*&F;F*-F,6#/F /#\"#j\"#kF*$\"+K9[iHF87$*&F;F*-F,6#/F/#\"$D\"\"$E\"F*$\"+&eRxy#F87$*& %9Papakostas'~scheme~with~GF*-F,6#/F/#\"$*>\"$+#F*$\"+hCrVEF8Q)pprint1 86\"" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 92 " The following error graphs are constructed using the numerical procedu res for the solutions." }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 555 "e valf[20](plot(['hn_RK6_1'(x)-h(x),'hn_RK6_2'(x)-h(x),'hn_RK6_3'(x)-h(x ),\n'hn_RK6_4'(x)-h(x),'hn_RK6_5'(x)-h(x),'hn_RK6_6'(x)-h(x)],x=0..6,n umpoints=100,font=[HELVETICA,9],\ncolor=[COLOR(RGB,.95,0,.95),COLOR(RG B,.5,0,.95),COLOR(RGB,.95,.45,0),COLOR(RGB,.2,.85,0),\nCOLOR(RGB,0,.3, .95),COLOR(RGB,.95,0,.2)],legend=[`Sharp-Verner scheme with c[7]=19/20 `,`scheme with c[7]=24/25`,`scheme with c[7]=35/36`,`scheme with c[7]= 63/64`,`scheme with c[7]=125/126`,`Papakostas' scheme with c[7]=199/20 0`],title=`error curves for 8 stage order 6 Runge-Kutta methods`));" } }{PARA 13 "" 1 "" {GLPLOT2D 1085 681 681 {PLOTDATA 2 "6,-%'CURVESG6%7e cl7$$\"\"!F)F(7$$\"5_^^^^^^\\qJ!#@$\"%,e!#?7$$\"5.......*4M'F-$\"(N?[ \"F07$$\"5!======@'*4*F-$\")H9eEF07$$\"5111111_#e=\"F0$\")>qOcF07$$\"5 .....`O%4M\"F0$\")Y%4j&F07$$\"5++++++@1'\\\"F0$\")9>$p&F07$$\"5[[[[[B8 it:F0$\")Z-OeF07$$\"5(pppppa!=^;F0$\")[SxhF07$$\"5YXXXXq(R(GF0$\"*kN4d\"F07$$\" 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x6\"Q!F]es-%&TITLEG6#%Uerror~curves~for~8~stage~order~6~Runge-Kutta~me thodsG-%%VIEWG6$;F(Fg]m%(DEFAULTG" 1 2 0 1 10 0 2 9 1 4 2 1.000000 45.000000 45.000000 0 1 "Sharp-Verner scheme with c[7]=19/20" "scheme \+ with c[7]=24/25" "scheme with c[7]=35/36" "scheme with c[7]=63/64" "sc heme with c[7]=125/126" "Papakostas' scheme with c[7]=199/200" }}}} {PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 563 "evalf[20](plot(['hn_RK6_1'(x)/h(x)-1,'hn_RK6_2'(x)/h(x)-1,'hn_RK6 _3'(x)/h(x)-1,'hn_RK6_4'(x)/h(x)-1,\n'hn_RK6_5'(x)/h(x)-1,'hn_RK6_6'(x )/h(x)-1],x=0..10,font=[HELVETICA,9],\ncolor=[COLOR(RGB,.95,0,.95),COL OR(RGB,.5,0,.95),COLOR(RGB,.95,.45,0),COLOR(RGB,.2,.85,0),\nCOLOR(RGB, 0,.3,.95),COLOR(RGB,.95,0,.2)],legend=[`Sharp-Verner scheme with c[7]= 19/20`,`scheme with c[7]=24/25`,`scheme with c[7]=35/36`,`scheme with \+ c[7]=63/64`,`scheme with c[7]=125/126`,`Papakostas' scheme with c[7]=1 99/200`],title=`relative error curves for 8 stage order 6 Runge-Kutta \+ 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yV[p=5\"F-7$F]em$\"1C;nj=*G5\"F-7$Fbem$\"18x UbPdc%F-7$F9$\"0$fGw$Ryc%F-7$F>$\"0\"=*f.')ed%F-7$FC$\"0+]T#*>`e%F-7$F H$\"0!4n)*\\g,YF-7$FM$\"0/_LAU)GYF-7$FR$\"07K*3;=tYF-7$FW$\"0PoZScA&[F -7$Ffn$\"0qZ4=rgM]F-7$F`o$\"0*yF'G?Y.&F-7$Feo$\"0eY +d\"oN]F-7$Fjo$\"0?TIYp(R]F-7$F_p$\"0m#R.l-a]F-7$Fdp$\"030/5!yp]F-7$Fi p$\"0#R.D*3h4&F-7$F^q$\"0JcMeQ*Q^F-7$Fcq$\"0Ix\"*z)*p?&F-7$Fhq$\"0dW1- :nJ&F-7$F]r$\"06^Z)RU'[&F-7$Fbr$\"0\"\\8cm:8bF-7$Fgr$\"07Y\"ft&F-7$Fbw$ \"0:0_v&*Q%eF-7$Fgw$\"0'o?!fpd*fF-7$F\\x$\"0`v/vk#3hF-7$Fax$\"0mPAvk#3 hF-7$Ffx$\"01?B#[E3hF-7$F[y$\"0\"f%)eOF3hF-7$F`y$\"0*H9xPU3hF-7$Fey$\" 0xxN(*G#4hF-7$Fjy$\"0iH?yeC6'F-7$F_z$\"0(eo-69(\\:'4iF-7$Fc[l$ \"0U.NE<=F'F-7$F]bn$\"03Y>D$pmjF-7$Fh[l$\"0Q'4k(G!4lF-7$F]\\l$\"0Bh'[/ T9mF-7$Fb\\l$\"0&oF=VF?nF-7$Fg\\l$\"0>&G=VF?nF-7$F\\]l$\"0T8*=VF?nF-7$ Fa]l$\"0h[8Qu-s'F-7$Ff]l$\"0NvZKv-s'F-7$F[^l$\"04UaH0.s'F-7$F`^l$\"0TZ `\"[f?nF-7$Fe^l$\"0rxtLRBs'F-7$Fj^l$\"06J!f=pHnF-7$F__l$\"0@*3J@sPnF-7 $Fd_l$\"0/m?l`8v'F-7$Fi_l$\"0eK7e3Qx'F-7$F^`l$\"0Ns`9c)4oF-7$Fc`l$\"0+ 3*)[s3(pF-7$Fh`l$\"0/Us9'pgtF-7$F]al$\"0UuQ_ " 0 "" {MPLTEXT 1 0 1 ";" }}}}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 47 "Test 4 of 8 stage, orde r 6 Runge-Kutta methods " }}{PARA 0 "" 0 "" {TEXT -1 81 "F. G. Lether: Mathematics of Computation, Vol. 20, no. 95, (July 1966) page 381. " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 256 "" 0 "" {TEXT -1 1 " " } {XPPEDIT 18 0 "dy/dx = -32*x*y*ln(2);" "6#/*&%#dyG\"\"\"%#dxG!\"\",$** \"#KF&%\"xGF&%\"yGF&-%#lnG6#\"\"#F&F(" }{TEXT -1 6 ", " }{XPPEDIT 18 0 "y(-1) = 1/8;" "6#/-%\"yG6#,$\"\"\"!\"\"*&F(F(\"\")F)" }{TEXT -1 1 " " }}{PARA 0 "" 0 "" {TEXT -1 10 "Solution: " }}{PARA 256 "" 0 "" {TEXT -1 2 " " }{XPPEDIT 18 0 "y = 2^(13-6*x^2);" "6#/%\"yG)\"\"#,&\" #8\"\"\"*&\"\"'F)*$%\"xGF&F)!\"\"" }{TEXT -1 2 ". " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 191 "de := diff( y(x),x)=-32*x*y(x)*ln(2);\nic := y(-1)=1/8;\ndsolve(\{de,ic\},y(x)):\n y(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=0vF-$\"3%z2Mbncie\"! #;7$$!3u****\\P8#\\4(F-$\"3C>dT>$)H#3$FM7$$!3+nm;/siqmF-$\"3gp%*z`g)4* eFM7$$!3[++](y$pZiF-$\"3%R6L-Y$zz5!#:7$$!33LLL$yaE\"eF-$\"3xvp\"p)==K> 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$\"3s*)***\\PQ#\\\")Fft$\"3!QM&=wHL5wFap7$$\"3ilm\"z\\1A-\"F-$\"3#*[#H (\\2i&H(Fap7$$\"3GKLLe\"*[H7F-$\"3))\\\\;@heFpFap7$$\"3ylm;HCjV9F-$\"3 )e+$\\9-Y,lFap7$$\"3I*******pvxl\"F-$\"3S%z:5s)zRgFap7$$\"3g)***\\7JFn =F-$\"31))p(30[[c&Fap7$$\"3#z****\\_qn2#F-$\"3ae5F\"zuv2&Fap7$$\"3=)** \\P/q%zAF-$\"3ZUhzOe!Rg%Fap7$$\"3U)***\\i&p@[#F-$\"3r&f%4uLbOTFap7$$\" 3L)**\\(=GB2FF-$\"3WV]5@%**Rj$Fap7$$\"3B)****\\2'HKHF-$\"3ul]=$GLo:$Fa p7$$\"3uJL$3UDX8$F-$\"3sKZjodBbFFap7$$\"3ElmmmZvOLF-$\"3!>\\-t_7IQ#Fap 7$$\"3i******\\2goPF-$\"3Q>G9F7l&p\"Fap7$$\"3UKL$eR<*fTF-$\"3?\"Fap7$$\"3m******\\)Hxe%F-$\"3V-?C_;$p$zFgn7$$\"3ckm;H!o-*\\F-$\"31 MiF2c]v^Fgn7$$\"3y)***\\7k.6aF-$\"3#pB[/J``=$Fgn7$$\"3#emmmT9C#eF-$\"3 &*=.D]9+3>Fgn7$$\"33****\\i!*3`iF-$\"3%HX+j$our5Fgn7$$\"3%QLLL$*zym'F- $\"3!o4*yfd(\\\"fFM7$$\"3wKLL3N1#4(F-$\"3!\\\\K5**)='4$FM7$$\"3Nmm;HYt 7vF-$\"3%o[)olFVm:FM7$$\"3Y*******p(G**yF-$\"3)3H-pcT.4)F=7$$\"3]mmmT6 KU$)F-$\"35omE\\#[Ck$F=7$$\"3fKLLLbdQ()F-$\"3TxwT%Qu%>ei< " 0 "" {MPLTEXT 1 0 780 "K := (x,y) -> -32*x*y(x)*ln(2): hh := 0.01: numsteps := 200: \+ x0 := -1: y0 := 1/8:\nmatrix([[`slope field: `,K(x,y)],[`initial poi nt: `,``(x0,y0)],[`step width: `,hh],\n[`no. of steps: `,numsteps] ]);``;\nmthds := [`Sharp-Verner scheme with `*``(c[7]=19/20),`scheme w ith `*``(c[7]=24/25),`scheme with `*``(c[7]=35/36),`scheme with `*``(c [7]=63/64),`scheme with `*``(c[7]=125/126),`Papakostas' scheme with `* ``(c[7]=199/200)]: errs := []:\nDigits := 20:\nfor ct to 6 do\n Kn_R K6_||ct := RK6_||ct(evalf(K(x,y)),x,y,x0,evalf(y0),hh,numsteps,false); \n sm := 0: numpts := nops(Kn_RK6_||ct):\n for ii to numpts do\n \+ sm := sm+(Kn_RK6_||ct[ii,2]-k(Kn_RK6_||ct[ii,1]))^2;\n end do:\n errs := [op(errs),sqrt(sm/numpts)];\nend do:\nDigits := 10:\nlinalg [transpose]([mthds,evalf(errs)]);" }}{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)pprint196\"" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%!G" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#K%'matrixG6#7(7$ *&%:Sharp-Verner~scheme~with~G\"\"\"-%!G6#/&%\"cG6#\"\"(#\"#>\"#?F*$\" +p)z`g%!#:7$*&%-scheme~with~GF*-F,6#/F/#\"#C\"#DF*$\"+!Q:y^%F87$*&F;F* -F,6#/F/#\"#N\"#OF*$\"+:k')eXF87$*&F;F*-F,6#/F/#\"#j\"#kF*$\"+L(z'QYF8 7$*&F;F*-F,6#/F/#\"$D\"\"$E\"F*$\"+%e!3MZF87$*&%9Papakostas'~scheme~wi th~GF*-F,6#/F/#\"$*>\"$+#F*$\"+$4Een%F8Q)pprint206\"" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 30 "The following code con structs " }{TEXT 260 20 "numerical procedures" }{TEXT -1 56 " for solu tions based on each of the Runge-Kutta schemes." }}{PARA 0 "" 0 "" {TEXT -1 75 "The error in the value obtained by each of the methods at the point where " }{XPPEDIT 18 0 "x = 0;" "6#/%\"xG\"\"!" }{TEXT -1 20 ".995 is also given." }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 711 "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]]);``;\n mthds := [`Sharp-Verner scheme with `*``(c[7]=19/20),`scheme with `*`` (c[7]=24/25),`scheme with `*``(c[7]=35/36),`scheme with `*``(c[7]=63/6 4),`scheme with `*``(c[7]=125/126),`Papakostas' scheme with `*``(c[7]= 199/200)]: errs := []:\nDigits := 20:\nfor ct to 6 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 6 do\n errs := [op(er rs),abs(kn_RK6_||ct(xx)-kxx)];\nend do:\nDigits := 10:\nlinalg[transpo se]([mthds,evalf(errs)]);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#K%'matrix G6#7&7$%0slope~field:~~~G,$**\"#K\"\"\"%\"xGF,-%\"yG6#F-F,-%#lnG6#\"\" #F,!\"\"7$%0initial~point:~G-%!G6$F5#F,\"\")7$%/step~width:~~~G$F,!\"# 7$%1no.~of~steps:~~~G\"$+#Q)pprint216\"" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%!G" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#K%'matrixG6#7(7$*&%:Sharp -Verner~scheme~with~G\"\"\"-%!G6#/&%\"cG6#\"\"(#\"#>\"#?F*$\"+P%4\")o% !#?7$*&%-scheme~with~GF*-F,6#/F/#\"#C\"#DF*$\"+Oh(*[nF87$*&F;F*-F,6#/F /#\"#N\"#OF*$\"+KaB1iF87$*&F;F*-F,6#/F/#\"#j\"#kF*$\"+x4f(o&F87$*&F;F* -F,6#/F/#\"$D\"\"$E\"F*$\"+%p&G@`F87$*&%9Papakostas'~scheme~with~GF*-F ,6#/F/#\"$*>\"$+#F*$\"+Zp\"y)\\F8Q)pprint226\"" }}}{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 met hod is estimated as follows using the special procedure " }{TEXT 0 5 "NCint" }{TEXT -1 98 " to perform numerical integration by the 7 poin t Newton-Cotes method over 100 equal subintervals." }}{EXCHG {PARA 0 " > " 0 "" {MPLTEXT 1 0 446 "mthds := [`Sharp-Verner scheme with `*``(c[ 7]=19/20),`scheme with `*``(c[7]=24/25),`scheme with `*``(c[7]=35/36), `scheme with `*``(c[7]=63/64),`scheme with `*``(c[7]=125/126),`Papakos tas' scheme with `*``(c[7]=199/200)]: errs := []:\nDigits := 20:\nfor \+ ct to 6 do\n sm := NCint((k(x)-'kn_RK6_||ct'(x))^2,x=-1..1,adaptive= false,numpoints=7,factor=100);\n errs := [op(errs),sqrt(sm/2)];\nend do:\nDigits := 10:\nlinalg[transpose]([mthds,evalf(errs)]);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#K%'matrixG6#7(7$*&%:Sharp-Verner~scheme~with ~G\"\"\"-%!G6#/&%\"cG6#\"\"(#\"#>\"#?F*$\"+o9)oh%!#:7$*&%-scheme~with~ GF*-F,6#/F/#\"#C\"#DF*$\"+[w4HXF87$*&F;F*-F,6#/F/#\"#N\"#OF*$\"+!Q^-d% F87$*&F;F*-F,6#/F/#\"#j\"#kF*$\"+!4k-l%F87$*&F;F*-F,6#/F/#\"$D\"\"$E\" F*$\"+iJ!fu%F87$*&%9Papakostas'~scheme~with~GF*-F,6#/F/#\"$*>\"$+#F*$ \"+8K](o%F8Q)pprint236\"" }}}{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 0 542 "evalf[20](plot([k(x)-'kn_RK6_1'(x),k(x)-'kn_RK6_2' (x),k(x)-'kn_RK6_3'(x),k(x)-'kn_RK6_4'(x),\nk(x)-'kn_RK6_5'(x),k(x)-'k n_RK6_6'(x)],x=-1..1,font=[HELVETICA,9],\ncolor=[COLOR(RGB,.95,0,.95), COLOR(RGB,.5,0,.95),COLOR(RGB,.95,.45,0),COLOR(RGB,.2,.85,0),\nCOLOR(R GB,0,.3,.95),COLOR(RGB,.95,0,.2)],legend=[`Sharp-Verner scheme with c[ 7]=19/20`,`scheme with c[7]=24/25`,`scheme with c[7]=35/36`,`scheme wi th c[7]=63/64`,`scheme with c[7]=125/126`,`Papakostas' scheme with c[7 ]=199/200`],title=`error curves for 8 stage order 6 Runge-Kutta method s`));" }}{PARA 13 "" 1 "" {GLPLOT2D 932 827 827 {PLOTDATA 2 "6,-%'CURV ESG6%7eo7$$!\"\"\"\"!$F*F*7$$!5nmmmm;p0k&*!#?$\",\"[w!z@\"F/7$$!5LLLL$ 37$$!5nmmmm\"4m(G $)F/$\",+m:Qx$F<7$$!5LLLL$3i.9!zF/$\",&=tY&4*F<7$$!5mmmm;/R=0vF/$\",el Dw)=!#=7$$!5++++]P8#\\4(F/$\",bwQ;\"QFL7$$!5mmmm;/siqmF/$\",pS6gW(FL7$ $!5++++](y$pZiF/$\",!R$>JQ\"!#<7$$!5LLLLL$yaE\"eF/$\",UEv`\\#Ffn7$$!5m mmmm\">s%HaF/$\",5uP'QSFfn7$$!5+++++]$*4)*\\F/$\",wgBgm'Ffn7$$!5+++++] _&\\c%F/$\"-=d[\"f0\"Ffn7$$!5+++++]1aZTF/$\",o!=c!e\"!#;7$$!5mmmm;/#)[ 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$\",(pvl())*F`p7$$\"5lmmm\"z\\1A-\"F/$\",)o\"\\&y%*F`p7$$\"4LLLL$e\"*[ H7F<$\",Y%e;+!*F`p7$$\"5lmmm;HCjV9F/$\",A/NjW)F`p7$$\"4+++++qvxl\"F<$ \",8M!QYyF`p7$$\"5++++]7JFn=F/$\",:x=#HsF`p7$$\"4++++]_qn2#F<$\",lH;hf 'F`p7$$\"5++++vV+ZzAF/$\",V6g2)fF`p7$$\"4++++Dcp@[#F<$\",6;uOP&F`p7$$ \"5++++v=GB2FF/$\",7J]4s%F`p7$$\"4++++]2'HKHF<$\",&)e+75%F`p7$$\"5NLLL $3UDX8$F/$\",rK@'zNF`p7$$\"4nmmmmwanL$F<$\",\"*Rxh4$F`p7$$\"4+++++v+'o PF<$\",iX$G.AF`p7$$\"4LLLLeR<*fTF<$\",s4Z5wF\"F<7$$\"4++++DOl5;*F< $\",FLUNh$F/7$$\"4++++v.Uac*F<$\"+xm'**G%F/7$$\"\"\"F*$!+([*3cfF/-%&CO LORG6&%$RGBG$\"#&*!\"#F+Fibl-%'LEGENDG6#%DSharp-Verner~scheme~with~c[7 ]=19/20G-F$6%7eoF'7$F-$\",35$Ge5F/7$F3$\",(y1(H6%F/7$F8$\",_FwxF\"F<7$ F>$\",'\\^IHMF<7$FC$\",B2cUR)F<7$FH$\",OP)Rh)H))F`p7$Fdz$\",Z,1nG)F`p7$Fiz$\",'\\#*H)p(F`p7 $F^[l$\",s0YH4(F`p7$Fc[l$\",G/m=Z'F`p7$Fh[l$\",MW\"4oeF`p7$F]\\l$\",SS rd`'Ffn7$Fj^l$\",S9\">!*RFfn7$F__l$\",BGDBO#Ffn7$Fd_ l$\",XisII\"Ffn7$Fi_l$\",@A=H*pFL7$F^`l$\",)*Q?z[$FL7$Fc`l$\",HJ-mj\"F 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\\l$\",'Qac$z%F`p7$Fg\\l$\",rmVR;%F`p7$F\\]l$\",)[\\%Rj$F`p7$Fa]l$\",J #HjUJF`p7$Ff]l$\",X)>:NAF`p7$F[^l$\",#oA-$e\"F`p7$F`^l$\"-\\&e+^;Y#Ffn7$Fd_l$\",D\\ qDO\"Ffn7$Fi_l$\",(4Q4[tFL7$F^`l$\",E))oNp$FL7$Fc`l$\",x)o\"Gv\"FL7$Fh `l$\",'e$*H\"G)F<7$F]al$\",_&3.HKF<7$Fbal$\",>[pA?\"F<7$Fgal$\",03KSG$ F/7$F\\bl$\"++0zsIF/7$Fabl$!+uT%o9'F/-Ffbl6&FhblFiblF+Fa]o-F]cl6#%EPap akostas'~scheme~with~c[7]=199/200G-%%FONTG6$%*HELVETICAG\"\"*-%+AXESLA BELSG6$Q\"x6\"Q!Fcjp-%&TITLEG6#%Uerror~curves~for~8~stage~order~6~Rung e-Kutta~methodsG-%%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 scheme with c[7]=19/20 " "scheme with c[7]=24/25" "scheme with c[7]=35/36" "scheme with c[7]= 63/64" "scheme with c[7]=125/126" "Papakostas' scheme with c[7]=199/20 0" }}}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 546 "evalf[20](plot([k(x)-'kn_RK6_1'(x),k(x)-'kn_RK6_2'(x ),k(x)-'kn_RK6_3'(x),k(x)-'kn_RK6_4'(x),\nk(x)-'kn_RK6_5'(x),k(x)-'kn_ RK6_6'(x)],x=-0.1..0.1,font=[HELVETICA,9],\ncolor=[COLOR(RGB,.95,0,.95 ),COLOR(RGB,.5,0,.95),COLOR(RGB,.95,.45,0),COLOR(RGB,.2,.85,0),\nCOLOR (RGB,0,.3,.95),COLOR(RGB,.95,0,.2)],legend=[`Sharp-Verner scheme with \+ c[7]=19/20`,`scheme with c[7]=24/25`,`scheme with c[7]=35/36`,`scheme \+ with c[7]=63/64`,`scheme with c[7]=125/126`,`Papakostas' scheme with c [7]=199/200`],title=`error curves for 8 stage order 6 Runge-Kutta meth ods`));" }}{PARA 13 "" 1 "" {GLPLOT2D 604 604 604 {PLOTDATA 2 "6,-%'CU RVESG6%7S7$$!\"\"F)$\",bgOf_*!#;7$$!5nmmmm;p0k&*!#@$\",r&zY;'*F,7$$!5L LLL$3**)))***F,7$$!5++++]P8#\\4(F0$\"-lq\"el+\"F,7$$!5mmmm;/siqmF0$\"- I*=/J,\"F,7$$!5++++](y$pZiF0$\"-)pxi#>5F,7$$!5LLLLL$yaE\"eF0$\"-,;'=_- \"F,7$$!5mmmmm\">s%HaF0$\"-0m%G,.\"F,7$$!5+++++]$*4)*\\F0$\"-!)))yGN5F ,7$$!5+++++]_&\\c%F0$\"-D#ya+/\"F,7$$!5+++++]1aZTF0$\"-nM-EW5F,7$$!5mm mm;/#)[oPF0$\"-zaauZ5F,7$$!5LLLLL$=exJ$F0$\"-$Gyj90\"F,7$$!5LLLLLL2$f$ HF0$\"-/inDa5F,7$$!5********\\PYx\"\\#F0$\"-CP#zq0\"F,7$$!5LLLLLL7i)4# 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\"F,7$Fcq$\"-HQ(p()3\"F,7$Fhq$\"-*R.L24\"F,7$F]r$\"-=Zn=#4\"F,7$Fbr$\" -_&[pK4\"F,7$Fgr$\"-%*>?(Q4\"F,7$F\\s$\"--j45%4\"F,7$Fas$\"-uLc(Q4\"F, 7$Fgs$\"-/.cH$4\"F,7$F\\t$\"-Dz*F,-Fjz6&F\\[lF`[l$\"\"$F)F][l-Fc[l6#%9scheme~with~c[7]= 125/126G-F$6%7S7$F($\",&=k#>n*F,7$F.$\",!e[%Qw*F,7$F4$\",Ipc6%)*F,7$F9 $\",#)*y/D**F,7$F>$\"-;MRh+5F,7$FC$\"-je1M35F,7$FH$\"-Ks->:5F,7$FM$\"- qrl&>-\"F,7$FR$\"--QxfG5F,7$FW$\"-iZ#\\[.\"F,7$Ffn$\"-Y*Q))3/\"F,7$F[o $\"-p=J(e/\"F,7$F`o$\"-7!o.60\"F,7$Feo$\"-huL%f0\"F,7$Fjo$\"-#>)3@g5F, 7$F_p$\"-ec\\uj5F,7$Fdp$\"-N$o>v1\"F,7$Fip$\"-bM0Nq5F,7$F^q$\"-1fg@t5F ,7$Fcq$\"-6=jOv5F,7$Fhq$\"-\"=Y0t2\"F,7$F]r$\"-&=GT(y5F,7$Fbr$\"-%or5) z5F,7$Fgr$\"-,MeS!3\"F,7$F\\s$\"-t\")>j!3\"F,7$Fas$\"-IG%4/3\"F,7$Fgs$ \"-DUl$)z5F,7$F\\t$\"-0q@#)y5F,7$Fat$\"-f'oVt2\"F,7$Fft$\"-z>fZv5F,7$F [u$\"-cmPFt5F,7$F`u$\"-kKoPq5F,7$Feu$\"-$>Art1\"F,7$Fju$\"-%H=XP1\"F,7 $F_v$\"-^5F,7$F^w$\"-#[*o5Y5F,7$ Fcw$\"-tL$f2/\"F,7$Fhw$\"-2)=uZ.\"F,7$F]x$\"-y&)=kG5F,7$Fbx$\"-Z=`+A5F ,7$Fgx$\"-'RMl],\"F,7$F\\y$\"-OI5Q35F,7$Fay$\"-&zNm.+\"F,7$Ffy$\",m?a) G**F,7$F[z$\",#zc$f%)*F,7$F`z$\",6^$fj(*F,7$Fez$\",y\")p>n*F,-Fjz6&F\\ [lF][lF`[lFhhm-Fc[l6#%EPapakostas'~scheme~with~c[7]=199/200G-%%FONTG6$ %*HELVETICAG\"\"*-%+AXESLABELSG6$Q\"x6\"Q!F`]o-%&TITLEG6#%Uerror~curve s~for~8~stage~order~6~Runge-Kutta~methodsG-%%VIEWG6$;F(Fez%(DEFAULTG" 1 2 0 1 10 0 2 9 1 4 2 1.000000 45.000000 45.000000 0 1 "Sharp-Verner \+ scheme with c[7]=19/20" "scheme with c[7]=24/25" "scheme with c[7]=35/ 36" "scheme with c[7]=63/64" "scheme with c[7]=125/126" "Papakostas' s cheme with c[7]=199/200" }}}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 47 "Test 5 of 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,font=[HELVETICA,9],labels=[`x`,`y(x)`]);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%#deG/-%%diffG6$-%\"yG6#%\"xGF,,$*( \"#;\"\"\",&*&F/F0F,F0F0F0F0!\"\"F)F3F0" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%#icG/-%\"yG6#\"\"!\"\"\"" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/- %\"yG6#%\"xG*$,&*&\"\"#\"\"\"-%#lnG6#,&*&\"#;F,F'F,F,F,F,F,F,F,F,#F,F+ " }}{PARA 13 "" 1 "" {GLPLOT2D 503 318 318 {PLOTDATA 2 "6&-%'CURVESG6$ 7U7$$\"\"!F)$\"\"\"F)7$$\"3WmmmT&)G\\a!#?$\"3EP,(Qyl.3\"!#<7$$\"3ILLL3 x&)*3\"!#>$\"3?25A!pa&\\6F27$$\"3-+]i!R(*Rc\"F6$\"3oz*p77wF?\"F27$$\"3 umm\"H2P\"Q?F6$\"3]_vibZz]7F27$$\"3MLL$eRwX5$F6$\"3!Qb+fY'3W8F27$$\"3C LL$3x%3yTF6$\"31#\\\\E7=EU\"F27$$\"3=mm\"z%4\\Y_F6$\"3s(e$4OUg*[\"F27$ $\"3)HL$eR-/PiF6$\"3.fPtw=4W:F27$$\"3A***\\il'pisF6$\"3/07@a`R%f\"F27$ $\"3`KLe*)>VB$)F6$\"3K!\\`od36k\"F27$$\"3!))**\\7`l2Q*F6$\"3#HUv\"fmC$ o\"F27$$\"3smm;/j$o/\"!#=$\"3:'H!f>cuAjU6Fco$\"3K$o8QC! za=F27$$\"3)*****\\P[6j9Fco$\"39iuo+OIZ=F27$$\"3KL$e*[z(yb\" Fco$\"3Q:]fA\\>F27$$\"3))**\\iSj0x=Fco$\"3-5Hbh&QF%>F27$$\"3Wmmm\"pW` (>Fco$\"3So#znsrC'>F27$$\"35+]i!f#=$3#Fco$\"3w)>Y)R!pI)>F27$$\"3/+](=x pe=#Fco$\"3?*eB@.[<+#F27$$\"3smm\"H28IH#Fco$\"3/Fyh^(\\.-#F27$$\"3km;z pSS\"R#Fco$\"3)4US+%ypO?F27$$\"3GLL3_?`(\\#Fco$\"3#4Cj+a0O0#F27$$\"3#H Le*)>pxg#Fco$\"3ab\\mG7Vq?F27$$\"3u**\\Pf4t.FFco$\"3Cx7m@=^%3#F27$$\"3 2LLe*Gst!GFco$\"3Q>IFco$\"3&ocGC'[]F@F27$$\"3h**\\i!RU07$Fco$\"3HCH$Q\")f .9#F27$$\"3b***\\(=S2LKFco$\"3C`wrWc9a@F27$$\"3Kmmm\"p)=MLFco$\"3;=S,I A7m@F27$$\"3!*****\\(=]@W$Fco$\"3w4%eC\"p]y@F27$$\"35L$e*[$z*RNFco$\"3 UyOr,.R*=#F27$$\"3#*****\\iC$pk$Fco$\"3wIdFs1%4?#F27$$\"39m;H2qcZPFco$ \"3Qbx\"QY%\\6AF27$$\"3q**\\7.\"fF&QFco$\"3f+!e(oz@AAF27$$\"3Ymm;/OgbR Fco$\"36qG(yA8CB#F27$$\"3y**\\ilAFjSFco$\"3v.zLgjzUAF27$$\"3YLLL$)*pp; %Fco$\"3IImU*yHDD#F27$$\"3?LL3xe,tUFco$\"3I%R!fhiAiAF27$$\"3em;HdO=yVF co$\"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\"\"*-%+AXESLABELSG6$%\"xG%%y(x)G-%%VIEWG6$;F($\"\"&Fj[l;F( $\"#EFj[l" 1 2 0 1 10 0 2 9 1 4 2 1.000000 45.000000 45.000000 0 0 "Cu rve 1" }}}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 32 "The following code constructs a " }{TEXT 260 17 "discrete solution " }{TEXT -1 44 " based on each of the methods and gives the " }{TEXT 260 22 "root mean square error" }{TEXT -1 18 " of each solution." }} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 763 "S := (x,y) -> 16/((16*x+1)* y): hh := 0.005: numsteps := 100: x0 := 0: y0 := 1:\nmatrix([[`slope f ield: `,S(x,y)],[`initial point: `,``(x0,y0)],[`step width: `,hh], \n[`no. of steps: `,numsteps]]);``;\nmthds := [`Sharp-Verner scheme \+ with `*``(c[7]=19/20),`scheme with `*``(c[7]=24/25),`scheme with `*``( c[7]=35/36),`scheme with `*``(c[7]=63/64),`scheme with `*``(c[7]=125/1 26),`Papakostas' scheme with `*``(c[7]=199/200)]: errs := []:\nDigits \+ := 20:\nfor ct to 6 do\n Sn_RK6_||ct := RK6_||ct(S(x,y),x,y,x0,y0,hh ,numsteps,false);\n sm := 0: numpts := nops(Sn_RK6_||ct):\n for ii to numpts do\n sm := sm+(Sn_RK6_||ct[ii,2]-s(Sn_RK6_||ct[ii,1])) ^2;\n end do:\n errs := [op(errs),sqrt(sm/numpts)];\nend do:\nDigi ts := 10:\nlinalg[transpose]([mthds,evalf(errs)]);" }}{PARA 11 "" 1 " " {XPPMATH 20 "6#K%'matrixG6#7&7$%0slope~field:~~~G,$*(\"#;\"\"\",&*&F +F,%\"xGF,F,F,F,!\"\"%\"yGF0F,7$%0initial~point:~G-%!G6$\"\"!F,7$%/ste p~width:~~~G$\"\"&!\"$7$%1no.~of~steps:~~~G\"$+\"Q)pprint246\"" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#%!G" }}{PARA 11 "" 1 "" {XPPMATH 20 "6 #K%'matrixG6#7(7$*&%:Sharp-Verner~scheme~with~G\"\"\"-%!G6#/&%\"cG6#\" \"(#\"#>\"#?F*$\"+/%*33V!#?7$*&%-scheme~with~GF*-F,6#/F/#\"#C\"#DF*$\" +%QrZr)!#@7$*&F;F*-F,6#/F/#\"#N\"#OF*$\"+H7&QF'FD7$*&F;F*-F,6#/F/#\"#j \"#kF*$\"+n%3Ab#FD7$*&F;F*-F,6#/F/#\"$D\"\"$E\"F*$\"+/T*e;$!#A7$*&%9Pa pakostas'~scheme~with~GF*-F,6#/F/#\"$*>\"$+#F*$\"+@S\\)\\\"!#BQ)pprint 256\"" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 30 "The following code constructs " }{TEXT 260 20 "numerical procedures" }{TEXT -1 56 " for solutions based on each of the Runge-Kutta schemes. " }}{PARA 0 "" 0 "" {TEXT -1 75 "The error in the value obtained by ea ch of the methods at the point where " }{XPPEDIT 18 0 "x = 0;" "6#/% \"xG\"\"!" }{TEXT -1 21 ".4995 is also given." }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 695 "S := (x,y) -> 16/((16*x+1)*y): hh := 0.005: num steps := 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 with `*``(c[7]=19/20 ),`scheme with `*``(c[7]=24/25),`scheme with `*``(c[7]=35/36),`scheme \+ with `*``(c[7]=63/64),`scheme with `*``(c[7]=125/126),`Papakostas' sch eme with `*``(c[7]=199/200)]: errs := []:\nDigits := 25:\nfor ct to 6 \+ do\n sn_RK6_||ct := RK6_||ct(S(x,y),x,y,x0,y0,hh,numsteps,true);\nen d do:\nxx := 0.4995: sxx := evalf(s(xx)):\nfor ct to 6 do\n errs := \+ [op(errs),abs(sn_RK6_||ct(xx)-sxx)];\nend do:\nDigits := 10:\nlinalg[t ranspose]([mthds,evalf(errs)]);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#K%' matrixG6#7&7$%0slope~field:~~~G,$*(\"#;\"\"\",&*&F+F,%\"xGF,F,F,F,!\" \"%\"yGF0F,7$%0initial~point:~G-%!G6$\"\"!F,7$%/step~width:~~~G$\"\"&! \"$7$%1no.~of~steps:~~~G\"$+\"Q)pprint266\"" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%!G" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#K%'matrixG6#7(7$ *&%:Sharp-Verner~scheme~with~G\"\"\"-%!G6#/&%\"cG6#\"\"(#\"#>\"#?F*$\" +%4V#fN!#?7$*&%-scheme~with~GF*-F,6#/F/#\"#C\"#DF*$\"+,#e#)=(!#@7$*&F; F*-F,6#/F/#\"#N\"#OF*$\"+1b8u^FD7$*&F;F*-F,6#/F/#\"#j\"#kF*$\"+MY].@FD 7$*&F;F*-F,6#/F/#\"$D\"\"$E\"F*$\"+>+92E!#A7$*&%9Papakostas'~scheme~wi th~GF*-F,6#/F/#\"$*>\"$+#F*$\"+>HtU8!#BQ)pprint276\"" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 4 "The " }{TEXT 260 22 "ro ot mean square error" }{TEXT -1 110 " over the interval [0, 0.5] of \+ each Runge-Kutta method is estimated as follows using the special proc edure " }{TEXT 0 5 "NCint" }{TEXT -1 97 " to perform numerical integ ration by the 7 point Newton-Cotes method over 50 equal subintervals. " }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 448 "mthds := [`Sharp-Verner scheme with `*``(c[7]=19/20),`scheme with `*``(c[7]=24/25),`scheme wi th `*``(c[7]=35/36),`scheme with `*``(c[7]=63/64),`scheme with `*``(c[ 7]=125/126),`Papakostas' scheme with `*``(c[7]=199/200)]: errs := []: \nDigits := 20:\nfor ct to 6 do\n sm := NCint((s(x)-'sn_RK6_||ct'(x) )^2,x=0..0.5,adaptive=false,numpoints=7,factor=50);\n errs := [op(er rs),sqrt(sm/0.5)];\nend do:\nDigits := 10:\nlinalg[transpose]([mthds,e valf(errs)]);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#K%'matrixG6#7(7$*&%:S harp-Verner~scheme~with~G\"\"\"-%!G6#/&%\"cG6#\"\"(#\"#>\"#?F*$\"+T_h1 V!#?7$*&%-scheme~with~GF*-F,6#/F/#\"#C\"#DF*$\"+kx!>r)!#@7$*&F;F*-F,6# /F/#\"#N\"#OF*$\"+87\"=F'FD7$*&F;F*-F,6#/F/#\"#j\"#kF*$\"+!*3V^DFD7$*& F;F*-F,6#/F/#\"$D\"\"$E\"F*$\"+LynlJ!#A7$*&%9Papakostas'~scheme~with~G F*-F,6#/F/#\"$*>\"$+#F*$\"+K!='3:!#BQ)pprint286\"" }}}{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 0 543 "evalf[20](plot([s(x)-'sn_R K6_1'(x),s(x)-'sn_RK6_2'(x),s(x)-'sn_RK6_3'(x),s(x)-'sn_RK6_4'(x),\ns( x)-'sn_RK6_5'(x),s(x)-'sn_RK6_6'(x)],x=0..0.5,font=[HELVETICA,9],\ncol or=[COLOR(RGB,.95,0,.95),COLOR(RGB,.5,0,.95),COLOR(RGB,.95,.45,0),COLO R(RGB,.2,.85,0),\nCOLOR(RGB,0,.3,.95),COLOR(RGB,.95,0,.2)],legend=[`Sh arp-Verner scheme with c[7]=19/20`,`scheme with c[7]=24/25`,`scheme wi th c[7]=35/36`,`scheme with c[7]=63/64`,`scheme with c[7]=125/126`,`Pa pakostas' scheme with c[7]=199/200`],title=`error curves for 8 stage o rder 6 Runge-Kutta methods`));" }}{PARA 13 "" 1 "" {GLPLOT2D 1042 656 656 {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$$\"5NLLL$3FWYs#F 4$\"(_>_)F07$$\"5-++vo/[AlIF4$\")v'['=F07$$\"5omm;aQ`!eS$F4$\")W\"=u$F 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$\"*v+z;%F07$$\"5,++](=xpe=#F\\y$\"*ppxg#F\\y$\"*\"R1#*RF07$$\"5,++]Pf4t.FF\\y$\"*B+^'RF07 $$\"5MLLLe*Gst!GF\\y$\"*7;u$RF07$$\"5,++++DRW9HF\\y$\"*J`-\"RF07$$\"5, +++DJE>>IF\\y$\"*)o(\\)QF07$$\"5,++]i!RU07$F\\y$\"*pW;'QF07$$\"5,+++v= S2LKF\\y$\"*PF07$$\"5ommm;/OgbRF\\y$\"*N8Cq$F07$$\"5,++]ilAFjSF\\y$\"*,t_o$F07$$ \"5NLLLL$)*pp;%F\\y$\"*o[$pOF07$$\"5NLLL3xe,tUF\\y$\"*0@Ol$F07$$\"5omm ;HdO=yVF\\y$\"*=[&QOF07$$\"5,++++D>#[Z%F\\y$\"*)H8DOF07$$\"5ommmT&G!e& e%F\\y$\"*EU-h$F07$$\"5NLLLL$)Qk%o%F\\y$\"*PRtf$F07$$\"5-++]iSjE!z%F\\ y$\"*l\"*Re$F07$$\"5-++]P40O\"*[F\\y$\"*6#frNF07$$\"\"&!\"\"$\"*Jc'eNF 0-%&COLORG6&%$RGBG$\"#&*!\"#F(Fbel-%'LEGENDG6#%DSharp-Verner~scheme~wi th~c[7]=19/20G-F$6%7`pF'7$F+$\"$x\"F07$F2$\"&Z+#F07$F8$\"&$)**)F07$F=$ \"'4NIF07$FB$\"'W/%)F07$FG$\"(5V,#F07$FL$\"(\"RBVF07$FQ$\"(Ic])F07$FV$ \")rYf:F07$Fen$\")CV(p#F07$Fjn$\")H)GW%F07$F_o$\")2l=qF07$Fdo$\")^nq$* F07$Fio$\")lCI$*F07$F^p$\")Yp^#*F07$Fcp$\")xzz\"*F07$$\"50++v$4@\">_rF 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#&*!\"#-%'LEGENDG6#%9scheme~with~c[7]=125/126G-F$6%7irF'7$$\"5qmm;aQ`! eS$F-$\"\"\"F07$F+$\"#BF07$$\"5,++Dc,;u@5F4$\"$O$F07$F2$\"%R@F07$$\"5N LL3FpE!Hq\"F4$\"%%e)F07$F8$\"&%fDF07$$\"5omm\"zptjSQ#F4$\"&w<'F07$F=$ \"'+o7F07$FB$\"''zF#F07$FG$\"'*fj$F07$FL$\"'Mq^F07$FQ$\"'bdkF07$FV$\"' 5DnF07$Fen$\"'^gZF07$Fjn$\"'g&G\"F07$F_o$\"'1!G\"F07$$\"5SLL3_v!p)*y&F 4$\"'yu7F07$Fdo$\"'Cr7F07$$\"5qmm\"HK9I5Z'F4F][m7$Fio$\"'P)H\"F07$F]p$ \"'7l8F07$Fbp$\"'Z2:F07$Fgp$\"'`g$F07$F[r$\"'QlNF07$F`r$\"'2!R$F07$Fer$\"'+a?F07$Fjr$\"'&\\6\"F0 7$F`s$\"'(46\"F07$Fes$\"'?26F07$$\"5+]ilZQ9\\>6F\\s$\"'q/6F07$Fjs$\"'h /6F07$$\"5M$3-j()*)e(y6F\\s$\"'H56F07$F_t$\"'SF6F07$Fit$\"',G7F07$Fcu$ \"'`k9F07$$\"5M$eRA'z7cc8F\\s$\"'#zi\"F07$$\"5++Dcw4]>'Q\"F\\s$\"'d'y \"F07$$\"5n;a)3*R(GeT\"F\\s$\"'\"o(=F07$Fhu$\"'u)y\"F07$$\"5+]7`>+i4v9 F\\s$\"'U\\8F07$$\"5nmT&Q.$*HZ]\"F\\s$\"&MC&F07$$\"5M$3x\"[gOOM:F\\s$ \"&%H_F07$F]v$\"&d@&F07$$\"5MLe9\"4&[EB;F\\s$\"&N>&F07$Fbv$\"&BA&F07$$ \"5++vV[r(*zTF\\s$\"&kV'F07$$\"5ML3-j7'p)y>F\\s$ \"&`C$F07$$\"5+]PMxUL]3?F\\s$!&l+\"F07$Faw$!&U+\"F07$$\"5MLe*)fI&*y/@F \\s$!%$)**F07$$\"5++]7G))>Wr@F\\s$!%l(*F07$$\"5nmTN'fW%4QAF\\s$!%b')F0 7$$\"5MLLek.pu/BF\\s$!%haF07$$\"5n;zp[KJ2QBF\\s$!%+KF07$$\"5++D\"G8O*R rBF\\s$!%g9F07$$\"5M$3Fp,fDZS#F\\s$!%3AF07$$\"5nm;/,>=0QCF\\s$!%;\"*F0 7$$\"5+]i:&y/y8Z#F\\s$!&b$GF07$$\"5ML3FpwUq/DF\\s$!&R>'F07$$\"5n;aQ`00 .QDF\\s$!&)zhF07$Ffw$!&g;'F07$$\"5nm\"Hd?>4!QEF\\s$!&m8'F07$$\"5ML$eR( \\;m/FF\\s$!&()3'F07$$\"5++v=U2TJrFF\\s$!&$yfF07$$\"5nmmT5ll'z$GF\\s$! &9\"eF07$$\"5++](o/[r7(HF\\s$!&Q)yF07$F[x$!'h.5F07$$\"5MLLLL$eI8k$F\\s $!'!)z7F07$F`x$!'Sr9F07$$\"5-++]PfyG7ZF\\s$!'q,;F07$Fex$!'Q)o\"F07$$\" 5++++v$fl\"=F07$Fiy$!'*oz \"F07$F^z$!']wO\"F07$F^el$!'5d8F07$Fcel$!'2_8F07$F hel$!'RZ8F07$F]fl$!'^U8F0-Fcfl6&FeflFhflF($\"\"#F_fl-F\\gl6#%EPapakost as'~scheme~with~c[7]=199/200G-%%FONTG6$%*HELVETICAG\"\"*-%+AXESLABELSG 6$Q\"x6\"Q!Ffdn-%&TITLEG6#%Uerror~curves~for~8~stage~order~6~Runge-Kut ta~methodsG-%%VIEWG6$;F(F]fl%(DEFAULTG" 1 2 0 1 10 0 2 9 1 4 2 1.000000 45.000000 45.000000 0 1 "scheme with c[7]=125/126" "Papakosta s' scheme with c[7]=199/200" }}}}{PARA 0 "" 0 "" {TEXT -1 0 "" }} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 2 " ;" }}}}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 46 "Test 6 of 8 stage, order 6 Runge-Kutta methods" }} {PARA 256 "" 0 "" {TEXT -1 2 " " }{XPPEDIT 18 0 "dy/dx = (1+2*(x+1)*s in(3*x))*exp(-y);" "6#/*&%#dyG\"\"\"%#dxG!\"\"*&,&F&F&*(\"\"#F&,&%\"xG F&F&F&F&-%$sinG6#*&\"\"$F&F.F&F&F&F&-%$expG6#,$%\"yGF(F&" }{TEXT -1 6 ", " }{XPPEDIT 18 0 "y(0) = 0;" "6#/-%\"yG6#\"\"!F'" }{TEXT -1 1 " " }}{PARA 0 "" 0 "" {TEXT -1 10 "Solution: " }}{PARA 256 "" 0 "" {TEXT -1 2 " " }{XPPEDIT 18 0 "y=ln(x+2/9*sin(3*x)-2/3*x*cos(3*x)-2/3 *cos(3*x)+5/3)" "6#/%\"yG-%#lnG6#,,%\"xG\"\"\"*(\"\"#F*\"\"*!\"\"-%$si nG6#*&\"\"$F*F)F*F*F***F,F*F3F.F)F*-%$cosG6#*&F3F*F)F*F*F.*(F,F*F3F.-F 66#*&F3F*F)F*F*F.*&\"\"&F*F3F.F*" }{TEXT -1 2 ". " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 169 "de := diff( y(x),x)=(1+2*(x+1)*sin(3*x))*exp(-y(x));\nic := y(0)=0;\ndsolve(\{de,i c\},y(x));\nu := unapply(rhs(%),x):\nplot(u(x),x=0..5,font=[HELVETICA, 9],labels=[`x`,`y(x)`]);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%#deG/-%% diffG6$-%\"yG6#%\"xGF,*&,&\"\"\"F/*(\"\"#F/,&F,F/F/F/F/-%$sinG6#,$*&\" \"$F/F,F/F/F/F/F/-%$expG6#,$F)!\"\"F/" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%#icG/-%\"yG6#\"\"!F)" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/-%\"yG6# %\"xG-%#lnG6#,,F'\"\"\"*&#\"\"#\"\"*F,-%$sinG6#,$*&\"\"$F,F'F,F,F,F,*& #F/F6F,*&F'F,-%$cosGF3F,F,!\"\"*&#F/F6F,F:F,F<#\"\"&F6F," }}{PARA 13 " " 1 "" {GLPLOT2D 503 318 318 {PLOTDATA 2 "6&-%'CURVESG6$7bp7$$\"\"!F)F (7$$\"3GLLL3x&)*3\"!#=$\"3QWK+t!=.P\"F-7$$\"3umm\"H2P\"Q?F-$\"3pUCE&Gm M$HF-7$$\"3MLL$eRwX5$F-$\"3l!G\"yWq,6\\F-7$$\"33ML$3x%3yTF-$\"3dz%)zau hMpF-7$$\"3emm\"z%4\\Y_F-$\"3,G5kQO>C))F-7$$\"3`LLeR-/PiF-$\"36YrjIBvP 5!#<7$$\"3]***\\il'pisF-$\"3wGtPF*HL<\"FI7$$\"3>MLe*)>VB$)F-$\"3kc:o\" [JMG\"FI7$$\"3Y++DJbw!Q*F-$\"3/$y^LV#)3O\"FI7$$\"3+N$ekGkX#**F-$\"3'R4 -T\"[w(Q\"FI7$$\"3%ommTIOo/\"FI$\"3i$3slO!o09FI7$$\"3E+]7GTt%4\"FI$\"3 qC'f&H6%RT\"FI7$$\"3YLL3_>jU6FI$\"3#[%e$e^!3:9FI7$$\"3ym;HdNb'>\"FI$\" 3CNMY.Kv29FI7$$\"37++]i^Z]7FI$\"3=<^j^\"=7R\"FI7$$\"35+++v\"=YI\"FI$\" 3![4*)z8y_O\"FI7$$\"33++](=h(e8FI$\"3yj8C4s%*H8FI7$$\"3/++]P[6j9FI$\"3 ![\"=po,dN7FI7$$\"3UL$e*[z(yb\"FI$\"3%eEi/R377\"FI7$$\"3wmm;a/cq;FI$\" 3!frK?-E\\b*F-7$$\"3%ommmJFI$\"3yKBC%\\'\\`dF-7$$\"3gmmm\"pW`(>FI$\"3$ )>nU%[*3T`F-7$$\"3_ek.HW#)))>FI$\"3kKE$Q%*GSE&F-7$$\"3?]iSmTI-?FI$\"3N xPe(47T?&F-7$$\"3*=/wP!Ry:?FI$\"3MId=-(\\?;&F-7$$\"3dLe9TOEH?FI$\"3+V! )\\%**R%Q^F-7$$\"3EDc^yLuU?FI$\"3=k(\\#*\\cP8&F-7$$\"3'pT&)e6Bi0#FI$\" 3[P(R22Q$[^F-7$$\"3k3_D`Gqp?FI$\"3m1F&*)f\"Q#=&F-7$$\"3K+]i!f#=$3#FI$ \"3z7KO0x$fB&F-7$$\"3/++D\"=EX8#FI$\"31#)Gask8:cF-7$$\"3?+](=xpe=#FI$ \"3qCq#fO$Q]iF-7$$\"37nm\"H28IH#FI$\"3sUF;\"\\Mi=)F-7$$\"3$p;a8d3AM#FI $\"3NM9xIK\")Q#*F-7$$\"3um;zpSS\"R#FI$\"3r*)ek&)o0L5FI7$$\"3-+v$41oWW# FI$\"3if$>8HM6:\"FI7$$\"3GLL3_?`(\\#FI$\"31,>EjcGm7FI7$$\"3AL3_D1l_DFI $\"3+C:%e=o-Q\"FI7$$\"3fL$e*)>pxg#FI$\"3%R*>K#Rqn[\"FI7$$\"33+]Pf4t.FF I$\"3A89rRoB^;FI7$$\"3uLLe*Gst!GFI$\"3eR;lLLE'z\"FI7$$\"30+++DRW9HFI$ \"36)ejqPI#4>FI7$$\"3K+]7y#=o'HFI$\"3?c()=o%e2&>FI7$$\"3:++DJE>>IFI$\" 3s.\"e&4:F$)>FI7$$\"3A+v$4^n)pIFI$\"39;D$z^_h+#FI7$$\"3F+]i!RU07$FI$\" 3vuj8s:f??FI7$$\"3?]il(Hv'[JFI$\"3&\\3XiGc\\-#FI7$$\"39+vo/#3o<$FI$\"3 (>zS&>rqE?FI7$$\"32](=<6T\\?$FI$\"3X%zI>hOe-#FI7$$\"3+++v=S2LKFI$\"3'[ xpLqNB-#FI7$$\"3;L$3_NJOG$FI$\"3:>!e))R4?FI7$$\"3Jmmm\"p)=MLFI$\"3Or POD8'y)>FI7$$\"3GLLeR%p\")Q$FI$\"3S(z'\\,LGb>FI7$$\"3B++](=]@W$FI$\"3K :aBe2s7>FI7$$\"35L$e*[$z*RNFI$\"3!>**fc?u*4=FI7$$\"3e++]iC$pk$FI$\"3'e \\^`F#Qg;FI7$$\"3[m;H2qcZPFI$\"351&p,mkr[\"FI7$$\"3O+]7.\"fF&QFI$\"3o; 2PnR[#G\"FI7$$\"3Ymm;/OgbRFI$\"3/lp%G(QG#3\"FI7$$\"3*G$e*[$zV4SFI$\"3P 9Az61+7**F-7$$\"3w**\\ilAFjSFI$\"3!p>ERLl'*>*F-7$$\"3#G3_]p'>*3%FI$\"3 O9*z2=!e_*)F-7$$\"3ym\"zW7@^6%FI$\"3w4XE;/Rz()F-7$$\"3w3F>RL3GTFI$\"3J eP:9JjA()F-7$$\"3t]i!RbX59%FI$\"3mH1#H$\\k'o)F-7$$\"3#=z>'ox+aTFI$\"3o Lr_-o*=n)F-7$$\"3yLLL$)*pp;%FI$\"3A7j1wipy')F-7$$\"3!Q3_+sD-=%FI$\"32p cM,k23()F-7$$\"3#Q$3xc9[$>%FI$\"3Gri,**=4g()F-7$$\"3'Qe*[$>Pn?%FI$\"3s e,X+?^M))F-7$$\"3)QL3-$H**>UFI$\"3Z**e,OD#4$*)F-7$$\"3#R$ek.W]YUFI$\"3 i#fiyx0s=*F-7$$\"3)RL$3xe,tUFI$\"3[2R[)*eVA&*F-7$$\"3Cn;HdO=yVFI$\"3#) >Y<=$f\\9\"FI7$$\"3MMe9\"z-lU%FI$\"3)4DVDmlMD\"FI7$$\"3a+++D>#[Z%FI$\" 3qZKS'GmoO\"FI7$$\"3TM$3_5,-`%FI$\"3CFB-Gn\\(\\\"FI7$$\"3SnmT&G!e&e%FI $\"3t\\(p9r/Xi\"FI7$$\"3m+]P%37^j%FI$\"3_eaMDR_K " 0 "" {MPLTEXT 1 0 775 "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 with `*``(c[7]=19/20),`scheme with `*``(c[7]= 24/25),`scheme with `*``(c[7]=35/36),`scheme with `*``(c[7]=63/64),`sc heme with `*``(c[7]=125/126),`Papakostas' scheme with `*``(c[7]=199/20 0)]: errs := []:\nDigits := 25:\nfor ct to 6 do\n Un_RK6_||ct := RK6 _||ct(U(x,y),x,y,x0,y0,hh,numsteps,false);\n sm := 0: numpts := nops (Un_RK6_||ct):\n for ii to numpts do\n sm := sm+(Un_RK6_||ct[ii ,2]-u(Un_RK6_||ct[ii,1]))^2;\n end do:\n errs := [op(errs),sqrt(sm /numpts)];\nend do:\nDigits := 10:\nlinalg[transpose]([mthds,evalf(err s)]);" }}{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~wi dth:~~~G$F+!\"#7$%1no.~of~steps:~~~G\"$+&Q)pprint296\"" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%!G" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#K%'matrixG6 #7(7$*&%:Sharp-Verner~scheme~with~G\"\"\"-%!G6#/&%\"cG6#\"\"(#\"#>\"#? F*$\"+7Z2y`!#C7$*&%-scheme~with~GF*-F,6#/F/#\"#C\"#DF*$\"+Q8'[2$F87$*& F;F*-F,6#/F/#\"#N\"#OF*$\"+4UG8@F87$*&F;F*-F,6#/F/#\"#j\"#kF*$\"++[Y&= \"F87$*&F;F*-F,6#/F/#\"$D\"\"$E\"F*$\"+!f*)[f'!#D7$*&%9Papakostas'~sch eme~with~GF*-F,6#/F/#\"$*>\"$+#F*$\"+?#R8w%F\\oQ)pprint306\"" }}} {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 30 "The follo wing 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 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 0 706 "U := (x,y) -> (1+2*(x+1)*sin(3*x))*exp(-y) : hh := 0.01: numsteps := 500: x0 := 0: y0 := 0:\nmatrix([[`slope fiel d: `,U(x,y)],[`initial point: `,``(x0,y0)],[`step width: `,hh],\n[ `no. of steps: `,numsteps]]);``;\nmthds := [`Sharp-Verner scheme wit h `*``(c[7]=19/20),`scheme with `*``(c[7]=24/25),`scheme with `*``(c[7 ]=35/36),`scheme with `*``(c[7]=63/64),`scheme with `*``(c[7]=125/126) ,`Papakostas' scheme with `*``(c[7]=199/200)]: errs := []:\nDigits := \+ 30:\nfor ct to 6 do\n un_RK6_||ct := RK6_||ct(U(x,y),x,y,x0,y0,hh,nu msteps,true);\nend do:\nxx := 4.999: uxx := evalf(u(xx)):\nfor ct to 6 do\n errs := [op(errs),abs(un_RK6_||ct(xx)-uxx)];\nend do:\nDigits \+ := 10:\nlinalg[transpose]([mthds,evalf(errs)]);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#K%'matrixG6#7&7$%0slope~field:~~~G*&,&\"\"\"F+*(\"\"#F+ ,&%\"xGF+F+F+F+-%$sinG6#,$*&\"\"$F+F/F+F+F+F+F+-%$expG6#,$%\"yG!\"\"F+ 7$%0initial~point:~G-%!G6$\"\"!FA7$%/step~width:~~~G$F+!\"#7$%1no.~of~ steps:~~~G\"$+&Q)pprint336\"" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%!G" } }{PARA 11 "" 1 "" {XPPMATH 20 "6#K%'matrixG6#7(7$*&%:Sharp-Verner~sche me~with~G\"\"\"-%!G6#/&%\"cG6#\"\"(#\"#>\"#?F*$\"+v&)y(Q$!#C7$*&%-sche me~with~GF*-F,6#/F/#\"#C\"#DF*$\"+!HL+p'!#D7$*&F;F*-F,6#/F/#\"#N\"#OF* $\"+z')=SWFD7$*&F;F*-F,6#/F/#\"#j\"#kF*$\"+X+Eo&*!#E7$*&F;F*-F,6#/F/# \"$D\"\"$E\"F*$\"+p9,d6FD7$*&%9Papakostas'~scheme~with~GF*-F,6#/F/#\"$ *>\"$+#F*$\"+zzzD;FDQ)pprint346\"" }}}{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 0 445 "mthds := [`Sharp-Verner scheme with `*``(c[7]=19/20),`scheme \+ with `*``(c[7]=24/25),`scheme with `*``(c[7]=35/36),`scheme with `*``( c[7]=63/64),`scheme with `*``(c[7]=125/126),`Papakostas' scheme with ` *``(c[7]=199/200)]: errs := []:\nDigits := 20:\nfor ct to 6 do\n sm \+ := NCint((u(x)-'un_RK6_||ct'(x))^2,x=0..5,adaptive=false,numpoints=7,f actor=200);\n errs := [op(errs),sqrt(sm/5)];\nend do:\nDigits := 10: \nlinalg[transpose]([mthds,evalf(errs)]);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#K%'matrixG6#7(7$*&%:Sharp-Verner~scheme~with~G\"\"\"-%! G6#/&%\"cG6#\"\"(#\"#>\"#?F*$\"+@\"p9Q&!#C7$*&%-scheme~with~GF*-F,6#/F /#\"#C\"#DF*$\"+$y=O2$F87$*&F;F*-F,6#/F/#\"#N\"#OF*$\"+Fx^7@F87$*&F;F* -F,6#/F/#\"#j\"#kF*$\"+y<9&=\"F87$*&F;F*-F,6#/F/#\"$D\"\"$E\"F*$\"+![% *\\f'!#D7$*&%9Papakostas'~scheme~with~GF*-F,6#/F/#\"$*>\"$+#F*$\"+x.xi ZF\\oQ)pprint356\"" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 " " {TEXT -1 92 "The following error graphs are constructed using the nu merical procedures for the solutions." }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 541 "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),'un_RK6_6 '(x)-u(x)],x=0..5,font=[HELVETICA,9],\ncolor=[COLOR(RGB,.95,0,.95),COL OR(RGB,.5,0,.95),COLOR(RGB,.95,.45,0),COLOR(RGB,.2,.85,0),\nCOLOR(RGB, 0,.3,.95),COLOR(RGB,.95,0,.2)],legend=[`Sharp-Verner scheme with c[7]= 19/20`,`scheme with c[7]=24/25`,`scheme with c[7]=35/36`,`scheme with \+ c[7]=63/64`,`scheme with c[7]=125/126`,`Papakostas' scheme with c[7]=1 99/200`],title=`error curves for 8 stage order 6 Runge-Kutta methods`) );" }}{PARA 13 "" 1 "" {GLPLOT2D 967 587 587 {PLOTDATA 2 "6,-%'CURVESG 6%7as7$$\"\"!F)F(7$$\"5qmmmmT&)G\\a!#@$!'?G[F-7$$\"5MLLLL3x&)*3\"!#?$! 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Fjcw-%&TITLEG6#%Uerror~curves~for~8~stage~order~6~Runge-Kutta~methodsG -%%VIEWG6$;F(Fgdm%(DEFAULTG" 1 2 0 1 10 0 2 9 1 4 2 1.000000 45.000000 45.000000 0 1 "Sharp-Verner scheme with c[7]=19/20" "scheme \+ with c[7]=24/25" "scheme with c[7]=35/36" "scheme with c[7]=63/64" "sc heme with c[7]=125/126" "Papakostas' scheme with c[7]=199/200" }}}} {PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 47 "Test 7 of 8 stage, orde r 6 Runge-Kutta methods " }}{PARA 256 "" 0 "" {TEXT -1 2 " " } {XPPEDIT 18 0 "dy/dx=-(1+4*cos(3*x))*(y-1/3)" "6#/*&%#dyG\"\"\"%#dxG! \"\",$*&,&F&F&*&\"\"%F&-%$cosG6#*&\"\"$F&%\"xGF&F&F&F&,&%\"yGF&*&F&F&F 2F(F(F&F(" }{TEXT -1 5 ", " }{XPPEDIT 18 0 "y(0)=1" "6#/-%\"yG6#\" \"!\"\"\"" }{TEXT -1 2 ". " }}{PARA 0 "" 0 "" {TEXT -1 10 "Solution: \+ " }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "y = 1/3" "6#/%\"y G*&\"\"\"F&\"\"$!\"\"" }{TEXT -1 1 " " }{XPPEDIT 18 0 "exp(-4/3*sin(3* x)+8/3*sin(3/2*x)*cos(3/2*x))+2/3" "6#,&-%$expG6#,&*(\"\"%\"\"\"\"\"$! \"\"-%$sinG6#*&F+F*%\"xGF*F*F,**\"\")F*F+F,-F.6#*(F+F*\"\"#F,F1F*F*-%$ cosG6#*(F+F*F7F,F1F*F*F*F**&F7F*F+F,F*" }{TEXT -1 1 " " }{XPPEDIT 18 0 "exp(-4/3*sin(3*x)-x)" "6#-%$expG6#,&*(\"\"%\"\"\"\"\"$!\"\"-%$sinG6 #*&F*F)%\"xGF)F)F+F0F+" }{TEXT -1 2 ". " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 181 "de := diff(y(x),x)=-(1 +4*cos(3*x))*(y(x)-1/3);\nic := y(0)=1;\nsimplify(dsolve(\{de,ic\},y(x )));\nv := unapply(rhs(%),x):\nplot(v(x),x=0..5,0..1.1,font=[HELVETICA ,9],labels=[`x`,`y(x)`]);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%#deG/-% %diffG6$-%\"yG6#%\"xGF,,$*&,&\"\"\"F0*&\"\"%F0-%$cosG6#,$*&\"\"$F0F,F0 F0F0F0F0,&F)F0#F0F8!\"\"F0F;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%#icG /-%\"yG6#\"\"!\"\"\"" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/-%\"yG6#%\"xG ,&*&#\"\"\"\"\"$F+-%$expG6#,&*&#\"\"%F,F+-%$sinG6#,$*&F,F+F'F+F+F+!\" \"*&#\"\")F,F+*&-F56#,$*(F,F+\"\"#F9F'F+F+F+-%$cosGF?F+F+F+F+F+*&#FBF, F+-F.6#,&F'F9*&#F3F,F+F4F+F9F+F+" }}{PARA 13 "" 1 "" {GLPLOT2D 503 318 318 {PLOTDATA 2 "6&-%'CURVESG6$7ap7$$\"\"!F)$\"\"\"F)7$$\"3gmmTN@K i8!#>$\"3W+7cSy5h&*!#=7$$\"3ALL$3FWYs#F/$\"3KtP[t*Q;:*F27$$\"3%)***\\i Smp3%F/$\"3g.\"H>f!3q()F27$$\"3WmmmT&)G\\aF/$\"36p*p.:G\\T)F27$$\"3m** **\\7G$R<)F/$\"3a?glh]$zx(F27$$\"3GLLL3x&)*3\"F2$\"3IM[S(o-#HsF27$$\"3 em\"z%\\v#pK\"F2$\"3i=)H'*Q$=:oF27$$\"3))**\\i!R(*Rc\"F2$\"3w,'pRB0LX' F27$$\"3&edV F27$$\"3%QL$3_DG1qF2$\"3'fN^hMLe*)>VB$)F2$\"3DB(Rfp)*\\j %F27$$\"3Y++DJbw!Q*F2$\"3%GsCu$*)zK]F27$$\"3+N$ekGkX#**F2$\"3u>+\\,YW? `F27$$\"3%ommTIOo/\"!#<$\"3q]2x8ZEqcF27$$\"3E+]7GTt%4\"Fgt$\"39b$=$pWl HgF27$$\"3YLL3_>jU6Fgt$\"3nwYdkc=KkF27$$\"3ym;HdNb'>\"Fgt$\"3l[hQOW]Bp F27$$\"37++]i^Z]7Fgt$\"3IVnF)*yXIuF27$$\"35+++v\"=YI\"Fgt$\"3ahS!3L%e= zF27$$\"33++](=h(e8Fgt$\"3l&QV-<82M)F27$$\"3&*****\\7!Q4T\"Fgt$\"3^]H \"3wS2k)F27$$\"3/++]P[6j9Fgt$\"3ur)[IAj$)z)F27$$\"3'=HKkAg\\Z\"Fgt$\"3 3z^;ogY6))F27$$\"3W$ek`h0o[\"Fgt$\"3h=q?g>u:))F27$$\"3/voH/5l)\\\"Fgt$ \"3p\\\\U!)G36))F27$$\"3%o;HKR'\\5:Fgt$\"3%G4&GMdV(z)F27$$\"3-]P4rr=M: Fgt$\"3Erd.MaCV()F27$$\"3UL$e*[z(yb\"Fgt$\"3m)))[\\1qQl)F27$$\"34+Dc,# >Uh\"Fgt$\"3(fTb\\\\y3J)F27$$\"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 0 770 "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 with `*``(c[7]=19/20),`scheme with `*``(c [7]=24/25),`scheme with `*``(c[7]=35/36),`scheme with `*``(c[7]=63/64) ,`scheme with `*``(c[7]=125/126),`Papakostas' scheme with `*``(c[7]=19 9/200)]: errs := []:\nDigits := 30:\nfor ct to 6 do\n Vn_RK6_||ct := RK6_||ct(V(x,y),x,y,x0,y0,hh,numsteps,false);\n sm := 0: numpts := \+ nops(Vn_RK6_||ct):\n for ii to numpts do\n sm := sm+(Vn_RK6_||c t[ii,2]-v(Vn_RK6_||ct[ii,1]))^2;\n end do:\n errs := [op(errs),sqr t(sm/numpts)];\nend do:\nDigits := 10:\nlinalg[transpose]([mthds,evalf (errs)]);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#K%'matrixG6#7&7$%0slope~f ield:~~~G,$*&,&\"\"\"F,*&\"\"%F,-%$cosG6#,$*&\"\"$F,%\"xGF,F,F,F,F,,&% \"yGF,#F,F4!\"\"F,F97$%0initial~point:~G-%!G6$\"\"!F,7$%/step~width:~~ ~G$\"\"#!\"#7$%1no.~of~steps:~~~G\"$]#Q)pprint366\"" }}{PARA 11 "" 1 " " {XPPMATH 20 "6#%!G" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#K%'matrixG6#7( 7$*&%:Sharp-Verner~scheme~with~G\"\"\"-%!G6#/&%\"cG6#\"\"(#\"#>\"#?F*$ \"+ITT+6!#@7$*&%-scheme~with~GF*-F,6#/F/#\"#C\"#DF*$\"+0c#>%[!#A7$*&F; F*-F,6#/F/#\"#N\"#OF*$\"+u`a\\QFD7$*&F;F*-F,6#/F/#\"#j\"#kF*$\"+kte7HF D7$*&F;F*-F,6#/F/#\"$D\"\"$E\"F*$\"+!p\\GP#FD7$*&%9Papakostas'~scheme~ with~GF*-F,6#/F/#\"$*>\"$+#F*$\"+s,dt?FDQ)pprint376\"" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 30 "The following code c onstructs " }{TEXT 260 20 "numerical procedures" }{TEXT -1 56 " for so lutions based on each of the Runge-Kutta schemes." }}{PARA 0 "" 0 "" {TEXT -1 75 "The error in the value obtained by each of the methods at the point where " }{XPPEDIT 18 0 "x = 4.999;" "6#/%\"xG-%&FloatG6$\" %**\\!\"$" }{TEXT -1 16 " is also given." }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 701 "V := (x,y) -> -(1+4*cos(3*x))*(y-1/3): hh := 0.02: n umsteps := 250: x0 := 0: y0 := 1:\nmatrix([[`slope field: `,V(x,y)], [`initial point: `,``(x0,y0)],[`step width: `,hh],\n[`no. of steps: \+ `,numsteps]]);``;\nmthds := [`Sharp-Verner scheme with `*``(c[7]=19/ 20),`scheme with `*``(c[7]=24/25),`scheme with `*``(c[7]=35/36),`schem e with `*``(c[7]=63/64),`scheme with `*``(c[7]=125/126),`Papakostas' s cheme with `*``(c[7]=199/200)]: errs := []:\nDigits := 30:\nfor ct to \+ 6 do\n vn_RK6_||ct := RK6_||ct(V(x,y),x,y,x0,y0,hh,numsteps,true);\n end do:\nxx := 4.999: vxx := evalf(v(xx)):\nfor ct to 6 do\n errs := [op(errs),abs(vn_RK6_||ct(xx)-vxx)];\nend do:\nDigits := 10:\nlinalg[ transpose]([mthds,evalf(errs)]);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#K% 'matrixG6#7&7$%0slope~field:~~~G,$*&,&\"\"\"F,*&\"\"%F,-%$cosG6#,$*&\" \"$F,%\"xGF,F,F,F,F,,&%\"yGF,#F,F4!\"\"F,F97$%0initial~point:~G-%!G6$ \"\"!F,7$%/step~width:~~~G$\"\"#!\"#7$%1no.~of~steps:~~~G\"$]#Q)pprint 386\"" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%!G" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#K%'matrixG6#7(7$*&%:Sharp-Verner~scheme~with~G\"\"\"-%! G6#/&%\"cG6#\"\"(#\"#>\"#?F*$\"+!4jYC#!#B7$*&%-scheme~with~GF*-F,6#/F/ #\"#C\"#DF*$\"+9D![:%F87$*&F;F*-F,6#/F/#\"#N\"#OF*$\"+$F87$*&F;F*-F,6#/F/#\"$D\"\"$E\"F*$\"+a8hNE F87$*&%9Papakostas'~scheme~with~GF*-F,6#/F/#\"$*>\"$+#F*$\"+l#)evAF8Q) pprint396\"" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 4 "The " }{TEXT 260 22 "root mean square error" }{TEXT -1 20 " over the interval " }{XPPEDIT 18 0 "[0, 5];" "6#7$\"\"!\"\"&" }{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 100 equal subi ntervals." }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 445 "mthds := [`Sha rp-Verner scheme with `*``(c[7]=19/20),`scheme with `*``(c[7]=24/25),` scheme with `*``(c[7]=35/36),`scheme with `*``(c[7]=63/64),`scheme wit h `*``(c[7]=125/126),`Papakostas' scheme with `*``(c[7]=199/200)]: err s := []:\nDigits := 20:\nfor ct to 6 do\n sm := NCint((v(x)-'vn_RK6_ ||ct'(x))^2,x=0..5,adaptive=false,numpoints=7,factor=100);\n errs := [op(errs),sqrt(sm/5)];\nend do:\nDigits := 10:\nlinalg[transpose]([mt hds,evalf(errs)]);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#K%'matrixG6#7(7$ *&%:Sharp-Verner~scheme~with~G\"\"\"-%!G6#/&%\"cG6#\"\"(#\"#>\"#?F*$\" +(p**R5\"!#@7$*&%-scheme~with~GF*-F,6#/F/#\"#C\"#DF*$\"+3zL\"z%!#A7$*& F;F*-F,6#/F/#\"#N\"#OF*$\"+m#=&)z$FD7$*&F;F*-F,6#/F/#\"#j\"#kF*$\"+w\\ 6kGFD7$*&F;F*-F,6#/F/#\"$D\"\"$E\"F*$\"+fZ8KBFD7$*&%9Papakostas'~schem e~with~GF*-F,6#/F/#\"$*>\"$+#F*$\"+:=%z.#FDQ)pprint406\"" }}}{PARA 0 " " 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 92 "The following erro r graphs are constructed using the numerical procedures for the soluti ons." }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 541 "evalf[20](plot([v(x )-'vn_RK6_1'(x),v(x)-'vn_RK6_2'(x),v(x)-'vn_RK6_3'(x),v(x)-'vn_RK6_4'( x),\nv(x)-'vn_RK6_5'(x),v(x)-'vn_RK6_6'(x)],x=0..5,font=[HELVETICA,9], \ncolor=[COLOR(RGB,.95,0,.95),COLOR(RGB,.5,0,.95),COLOR(RGB,.95,.45,0) ,COLOR(RGB,.2,.85,0),\nCOLOR(RGB,0,.3,.95),COLOR(RGB,.95,0,.2)],legend =[`Sharp-Verner scheme with c[7]=19/20`,`scheme with c[7]=24/25`,`sche me with c[7]=35/36`,`scheme with c[7]=63/64`,`scheme with c[7]=125/126 `,`Papakostas' scheme with c[7]=199/200`],title=`error curves for 8 st age order 6 Runge-Kutta methods`));" }}{PARA 13 "" 1 "" {GLPLOT2D 1042 623 623 {PLOTDATA 2 "6,-%'CURVESG6%7]q7$$\"\"!F)F(7$$\"5qmmmmT&)G \\a!#@$!)(e6c$!#?7$$\"5MLLLL3x&)*3\"F0$!)`$eg&F07$$\"5+]ilZQ9\\>6F0$!) p\"z`&F07$$\"5nm\"z>'o^7\\6F0$!)zm2bF07$$\"5M$3-j()*)e(y6F0$!)XZ&e&F07 $$\"5++]i!*GER37F0$!)U5ldF07$$\"5ML3F>*3gwE\"F0$!)h!fg&F07$$\"5nmm\"z% \\v#pK\"F0$!)EJeaF07$$\"5ML$3_+ZiaW\"F0$!)l&oI&F07$$\"5+++]i!R(*Rc\"F0 $!)8:%*\\F07$$\"5MLL3xJs1,=F0$!)YcKSF07$$\"5nmmm\"H2P\"Q?F0$!)#[N4$F07 $$\"5+++]PMnNrDF0$!(q=v)F07$$\"5MLLL$eRwX5$F0$\")n'\\L\"F07$$\"5MLLLL$ eI8k$F0$\")n&yB$F07$$\"5NLLL$3x%3yTF0$\")!em;%F07$$\"5-++]PfyG7ZF0$\") 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" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 " > " 0 "" {MPLTEXT 1 0 178 "de := diff(y(x),x)=x*(9-x^2)/(1+y(x)^2);\ni c := y(0)=0;\ndsolve(\{de,ic\},y(x));\nw := unapply(rhs(%),x):\nplot(w (x),x=0..4,0..3.7,numpoints=75,font=[HELVETICA,9],labels=[`x`,`y(x)`]) ;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%#deG/-%%diffG6$-%\"yG6#%\"xGF,* (F,\"\"\",&\"\"*F.*$)F,\"\"#F.!\"\"F.,&F.F.*$)F)F3F.F.F4" }}{PARA 11 " " 1 "" {XPPMATH 20 "6#>%#icG/-%\"yG6#\"\"!F)" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/-%\"yG6#%\"xG,&*&\"\"#!\"\",(*&\"\"$\"\"\")F'\"\"%F/F+ *&\"#aF/)F'F*F/F/*$,*\"#kF/*&\"\"*F/)F'\"\")F/F/*&\"$C$F/)F'\"\"'F/F+* &\"%;HF/F0F/F/#F/F*F/#F/F.F/*&F*F/F,#F+F.F+" }}{PARA 13 "" 1 "" {GLPLOT2D 503 318 318 {PLOTDATA 2 "6&-%'CURVESG6$7io7$$\"\"!F)F(7$$\"3 ()=*=*=*Qx#G!#>$\"3_LLtbH2)f$!#?7$$\"3uPy$y$yZbcF-$\"3ZF^'eEW*Q9F-7$$ \"3;_8N^$ye6)F-$\"3C$\"3aT8Yqv-h6F>7$$\"3oKCVKs3o@F>$\"3c?q**e5wz?F>7$$\"3$4\" 3\"3T.Ds#F>$\"3+#H`Y\")*G6KF>7$$\"3jy$y$y\"=lB$F>$\"3L\\!fpl0?S%F>7$$ 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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 0 764 "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 with `*``(c[7]=19/20),`scheme wit h `*``(c[7]=24/25),`scheme with `*``(c[7]=35/36),`scheme with `*``(c[7 ]=63/64),`scheme with `*``(c[7]=125/126),`Papakostas' scheme with `*`` (c[7]=199/200)]: errs := []:\nDigits := 30:\nfor ct to 6 do\n Wn_RK6 _||ct := RK6_||ct(W(x,y),x,y,x0,y0,hh,numsteps,false);\n sm := 0: nu mpts := 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(e rrs),sqrt(sm/numpts)];\nend do:\nDigits := 10:\nlinalg[transpose]([mth ds,evalf(errs)]);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#K%'matrixG6#7&7$% 0slope~field:~~~G*(%\"xG\"\"\",&\"\"*F+*$)F*\"\"#F+!\"\"F+,&F+F+*$)%\" yGF0F+F+F17$%0initial~point:~G-%!G6$\"\"!F;7$%/step~width:~~~G$F+!\"#7 $%1no.~of~steps:~~~G\"$+%Q)pprint416\"" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%!G" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#K%'matrixG6#7(7$*&%:Sharp -Verner~scheme~with~G\"\"\"-%!G6#/&%\"cG6#\"\"(#\"#>\"#?F*$\"+iNJd$*!# C7$*&%-scheme~with~GF*-F,6#/F/#\"#C\"#DF*$\"+LzKvJF87$*&F;F*-F,6#/F/# \"#N\"#OF*$\"+mRjd@F87$*&F;F*-F,6#/F/#\"#j\"#kF*$\"+;vqT5F87$*&F;F*-F, 6#/F/#\"$D\"\"$E\"F*$\"+aR>zU!#D7$*&%9Papakostas'~scheme~with~GF*-F,6# /F/#\"$*>\"$+#F*$\"+*4q*oCF\\oQ)pprint426\"" }}}{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 ba sed 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 poin t where " }{XPPEDIT 18 0 "x = 3.499;" "6#/%\"xG-%&FloatG6$\"%*\\$!\"$ " }{TEXT -1 16 " is also given." }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 695 "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 with `*``(c[7]=19/20),`scheme wit h `*``(c[7]=24/25),`scheme with `*``(c[7]=35/36),`scheme with `*``(c[7 ]=63/64),`scheme with `*``(c[7]=125/126),`Papakostas' scheme with `*`` (c[7]=199/200)]: errs := []:\nDigits := 30:\nfor ct to 6 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 6 do\n errs := [op(errs),abs( wn_RK6_||ct(xx)-wxx)];\nend do:\nDigits := 10:\nlinalg[transpose]([mth ds,evalf(errs)]);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#K%'matrixG6#7&7$% 0slope~field:~~~G*(%\"xG\"\"\",&\"\"*F+*$)F*\"\"#F+!\"\"F+,&F+F+*$)%\" yGF0F+F+F17$%0initial~point:~G-%!G6$\"\"!F;7$%/step~width:~~~G$F+!\"#7 $%1no.~of~steps:~~~G\"$+%Q)pprint436\"" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%!G" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#K%'matrixG6#7(7$*&%:Sharp -Verner~scheme~with~G\"\"\"-%!G6#/&%\"cG6#\"\"(#\"#>\"#?F*$\"+22TZM!#C 7$*&%-scheme~with~GF*-F,6#/F/#\"#C\"#DF*$\"+^fqG8F87$*&F;F*-F,6#/F/#\" #N\"#OF*$\"+^JKS!*!#D7$*&F;F*-F,6#/F/#\"#j\"#kF*$\"+?*3eO%FN7$*&F;F*-F ,6#/F/#\"$D\"\"$E\"F*$\"+H?NJ\"$+#F*$\"+!GhBB*!#EQ)pprint446\"" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 4 "The " }{TEXT 260 22 "root mean s quare error" }{TEXT -1 20 " over the interval " }{XPPEDIT 18 0 "[0,4] " "6#7$\"\"!\"\"%" }{TEXT -1 82 " of each Runge-Kutta method is estim ated as follows using the special procedure " }{TEXT 0 5 "NCint" } {TEXT -1 98 " to perform numerical integration by the 7 point Newton- Cotes method over 200 equal subintervals." }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 445 "mthds := [`Sharp-Verner scheme with `*``(c[7]=19/20) ,`scheme with `*``(c[7]=24/25),`scheme with `*``(c[7]=35/36),`scheme w ith `*``(c[7]=63/64),`scheme with `*``(c[7]=125/126),`Papakostas' sche me with `*``(c[7]=199/200)]: errs := []:\nDigits := 20:\nfor ct to 6 d o\n sm := NCint((w(x)-'wn_RK6_||ct'(x))^2,x=0..4,adaptive=false,nump oints=7,factor=200);\n errs := [op(errs),sqrt(sm/4)];\nend do:\nDigi ts := 10:\nlinalg[transpose]([mthds,evalf(errs)]);" }}{PARA 11 "" 1 " " {XPPMATH 20 "6#K%'matrixG6#7(7$*&%:Sharp-Verner~scheme~with~G\"\"\"- %!G6#/&%\"cG6#\"\"(#\"#>\"#?F*$\"+?;$QK*!#C7$*&%-scheme~with~GF*-F,6#/ F/#\"#C\"#DF*$\"+T]7dJF87$*&F;F*-F,6#/F/#\"#N\"#OF*$\"+f2^V@F87$*&F;F* -F,6#/F/#\"#j\"#kF*$\"+%4_J.\"F87$*&F;F*-F,6#/F/#\"$D\"\"$E\"F*$\"+E$* 4DU!#D7$*&%9Papakostas'~scheme~with~GF*-F,6#/F/#\"$*>\"$+#F*$\"+6n46CF \\oQ)pprint456\"" }}}{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 0 541 "evalf[20](plot([w(x)-'wn_RK6_1'(x),w(x)-'wn_RK6_2'(x ),w(x)-'wn_RK6_3'(x),w(x)-'wn_RK6_4'(x),\nw(x)-'wn_RK6_5'(x),w(x)-'wn_ RK6_6'(x)],x=0..4,font=[HELVETICA,9],\ncolor=[COLOR(RGB,.95,0,.95),COL OR(RGB,.5,0,.95),COLOR(RGB,.95,.45,0),COLOR(RGB,.2,.85,0),\nCOLOR(RGB, 0,.3,.95),COLOR(RGB,.95,0,.2)],legend=[`Sharp-Verner scheme with c[7]= 19/20`,`scheme with c[7]=24/25`,`scheme with c[7]=35/36`,`scheme with \+ c[7]=63/64`,`scheme with c[7]=125/126`,`Papakostas' scheme with c[7]=1 99/200`],title=`error curves for 8 stage order 6 Runge-Kutta methods`) );" }}{PARA 13 "" 1 "" {GLPLOT2D 1000 574 574 {PLOTDATA 2 "6,-%'CURVES G6%7eq7$$\"\"!F)F(7$$\"5mmmmm;arz@!#@$\"'<'4\"!#A7$$\"5LLLLLL3VfVF-$\" '0(['F07$$\"5lmmmmT&)G\\aF-$\"'4&H\"F-7$$\"5+++++]i9RlF-$\"'HzBF-7$$\" 5NLLLLeR+HwF-$\"'n1TF-7$$\"5mmmmmm;')=()F-$\"'X$y'F-7$$\"5MLLLeR?ah5!# ?$\"(#Hp9F-7$$\"5++++]7z>^7FM$\"(j:#GF-7$$\"5mmmmT&y`3W\"FM$\"(r['[F-7 $$\"5LLLLLe'40j\"FM$\"'cCwFM7$$\"5+++]i!f`rt\"FM$\"'tM#*FM7$$\"5mmmm\" H_(zV=FM$\"(&p&4\"FM7$$\"5LLL$3_XT/&>FM$\"(XYF\"FM7$$\"5++++](Q&3d?FM$ \"(v^X\"FM7$$\"5nmm;z>$HP;#FM$\"(p;j\"FM7$$\"5MLLL3_KPqAFM$\"(V%)z\"FM 7$$\"5+++]P%=FM7$$\"5nmmmm;6m$[#FM$\"(&\\\"3#FM7$$\"5nmm m;a>,\"f#FM$\"(C3>#FM7$$\"5nmmmm\"zi$)p#FM$\"(4LF#FM7$$\"5nmm;/,0?DFFM $\"(nLF#FM7$$\"5nmmmT5#Q?v#FM$\"(fgE#FM7$$\"5nmm;z>f()yFFM$\"(4!oAFM7$ $\"5nmmm;HOr0GFM$\"(TEI#FM7$$\"5nmm;aQ8bKGFM$\"(\\TH#FM7$$\"5nmmm\"z/* QfGFM$\"(]lG#FM7$$\"5nmm;HdnA')GFM$\"($[!H#FM7$$\"5nmmmmmW18HFM$\"(AaI 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"" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 48 "Test 9 of 8 stage, orde r 6 Runge-Kutta methods " }}{PARA 256 "" 0 "" {TEXT -1 3 " " } {XPPEDIT 18 0 "dy/dx=-(1+cos(2*x))*y^3" "6#/*&%#dyG\"\"\"%#dxG!\"\",$* &,&F&F&-%$cosG6#*&\"\"#F&%\"xGF&F&F&*$%\"yG\"\"$F&F(" }{TEXT -1 3 ", \+ " }{XPPEDIT 18 0 "y(0) = sqrt(2);" "6#/-%\"yG6#\"\"!-%%sqrtG6#\"\"#" } {TEXT -1 1 " " }}{PARA 0 "" 0 "" {TEXT -1 10 "Solution: " }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "y = 1/sqrt(sin(2*x)+2*x+1/2)" " 6#/%\"yG*&\"\"\"F&-%%sqrtG6#,(-%$sinG6#*&\"\"#F&%\"xGF&F&*&F/F&F0F&F&* &F&F&F/!\"\"F&F3" }{TEXT -1 2 ". " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 172 "de := diff(y(x),x)=-(1+cos( 2*x))*y(x)^3;\nic := y(0)=sqrt(2);\ndsolve(\{de,ic\},y(x));\nm := unap ply(rhs(%),x):\nplot(m(x),x=0..3,0..1.42,font=[HELVETICA,9],labels=[`x `,`y(x)`]);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%#deG/-%%diffG6$-%\"yG 6#%\"xGF,,$*&,&\"\"\"F0-%$cosG6#,$*&\"\"#F0F,F0F0F0F0)F)\"\"$F0!\"\"" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%#icG/-%\"yG6#\"\"!*$\"\"##\"\"\"F+ " }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/-%\"yG6#%\"xG*&\"\"\"F)*$,(*(\"\" #F)-%$cosGF&F)-%$sinGF&F)F)*&F-F)F'F)F)#F)F-F)#F)F-!\"\"" }}{PARA 13 " " 1 "" {GLPLOT2D 503 318 318 {PLOTDATA 2 "6&-%'CURVESG6$7Y7$$\"\"!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 0 777 "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 with `*` `(c[7]=19/20),`scheme with `*``(c[7]=24/25),`scheme with `*``(c[7]=35/ 36),`scheme with `*``(c[7]=63/64),`scheme with `*``(c[7]=125/126),`Pap akostas' scheme with `*``(c[7]=199/200)]: errs := []:\nDigits := 20:\n for ct to 6 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:\nDigit s := 10:\nlinalg[transpose]([mthds,evalf(errs)]);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#K%'matrixG6#7&7$%0slope~field:~~~G,$*&,&\"\"\"F,-%$cosG 6#,$*&\"\"#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)pp rint466\"" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%!G" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#K%'matrixG6#7(7$*&%:Sharp-Verner~scheme~with~G\"\"\"-%! G6#/&%\"cG6#\"\"(#\"#>\"#?F*$\"+P'[%*z&!#A7$*&%-scheme~with~GF*-F,6#/F /#\"#C\"#DF*$\"+-YvkNF87$*&F;F*-F,6#/F/#\"#N\"#OF*$\"+&zK.f#F87$*&F;F* -F,6#/F/#\"#j\"#kF*$\"+#RbYR\"F87$*&F;F*-F,6#/F/#\"$D\"\"$E\"F*$\"+e!H rV'!#B7$*&%9Papakostas'~scheme~with~GF*-F,6#/F/#\"$*>\"$+#F*$\"+s\\nsL F\\oQ)pprint476\"" }}}{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 \+ = 2.999;" "6#/%\"xG-%&FloatG6$\"%**H!\"$" }{TEXT -1 16 " is also give n." }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 708 "M := (x,y) -> -(1+cos (2*x))*y^3: hh := 0.01: numsteps := 300: x0 := 0: y0 := sqrt(2):\nmatr ix([[`slope field: `,M(x,y)],[`initial point: `,``(x0,y0)],[`step wi dth: `,hh],\n[`no. of steps: `,numsteps]]);``;\nmthds := [`Sharp-V erner scheme with `*``(c[7]=19/20),`scheme with `*``(c[7]=24/25),`sche me with `*``(c[7]=35/36),`scheme with `*``(c[7]=63/64),`scheme with `* ``(c[7]=125/126),`Papakostas' scheme with `*``(c[7]=199/200)]: errs := []:\nDigits := 25:\nfor ct to 6 do\n mn_RK6_||ct := RK6_||ct(M(x,y) ,x,y,x0,evalf(y0),hh,numsteps,true);\nend do:\nxx := 2.999: mxx := eva lf(m(xx)):\nfor ct to 6 do\n errs := [op(errs),abs(mn_RK6_||ct(xx)-m xx)];\nend do:\nDigits := 10:\nlinalg[transpose]([mthds,evalf(errs)]); " }}{PARA 11 "" 1 "" {XPPMATH 20 "6#K%'matrixG6#7&7$%0slope~field:~~~G ,$*&,&\"\"\"F,-%$cosG6#,$*&\"\"#F,%\"xGF,F,F,F,)%\"yG\"\"$F,!\"\"7$%0i nitial~point:~G-%!G6$\"\"!*$F2#F,F27$%/step~width:~~~G$F,!\"#7$%1no.~o f~steps:~~~G\"$+$Q)pprint486\"" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%!G " }}{PARA 11 "" 1 "" {XPPMATH 20 "6#K%'matrixG6#7(7$*&%:Sharp-Verner~s cheme~with~G\"\"\"-%!G6#/&%\"cG6#\"\"(#\"#>\"#?F*$\"+To(*H6!#A7$*&%-sc heme~with~GF*-F,6#/F/#\"#C\"#DF*$\"+lnXro!#B7$*&F;F*-F,6#/F/#\"#N\"#OF *$\"+-qp.]FD7$*&F;F*-F,6#/F/#\"#j\"#kF*$\"+$QC6r#FD7$*&F;F*-F,6#/F/#\" $D\"\"$E\"F*$\"+\"e/@F\"FD7$*&%9Papakostas'~scheme~with~GF*-F,6#/F/#\" $*>\"$+#F*$\"+LqHko!#CQ)pprint496\"" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 4 "The " }{TEXT 260 22 "root mean square er ror" }{TEXT -1 20 " over the interval " }{XPPEDIT 18 0 "[0, 3];" "6#7 $\"\"!\"\"$" }{TEXT -1 82 " of each Runge-Kutta method is estimated a s 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 150 equal subintervals." }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 445 "mthds := [`Sharp-Verner scheme with `*``(c[7]=19/20),`scheme \+ with `*``(c[7]=24/25),`scheme with `*``(c[7]=35/36),`scheme with `*``( c[7]=63/64),`scheme with `*``(c[7]=125/126),`Papakostas' scheme with ` *``(c[7]=199/200)]: errs := []:\nDigits := 20:\nfor ct to 6 do\n sm \+ := NCint((m(x)-'mn_RK6_||ct'(x))^2,x=0..3,adaptive=false,numpoints=7,f actor=150);\n errs := [op(errs),sqrt(sm/3)];\nend do:\nDigits := 10: \nlinalg[transpose]([mthds,evalf(errs)]);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#K%'matrixG6#7(7$*&%:Sharp-Verner~scheme~with~G\"\"\"-%! G6#/&%\"cG6#\"\"(#\"#>\"#?F*$\"+k,%Qs&!#A7$*&%-scheme~with~GF*-F,6#/F/ #\"#C\"#DF*$\"+/I!z^$F87$*&F;F*-F,6#/F/#\"#N\"#OF*$\"+4vRcDF87$*&F;F*- F,6#/F/#\"#j\"#kF*$\"+Eabw8F87$*&F;F*-F,6#/F/#\"$D\"\"$E\"F*$\"+@)=bN' !#B7$*&%9Papakostas'~scheme~with~GF*-F,6#/F/#\"$*>\"$+#F*$\"+5swJLF\\o Q)pprint506\"" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 92 "The following error graphs are constructed using the nume rical procedures for the solutions." }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 543 "evalf[20](plot([m(x)-'mn_RK6_1'(x),m(x)-'mn_RK6_2'(x ),m(x)-'mn_RK6_3'(x),m(x)-'mn_RK6_4'(x),\nm(x)-'mn_RK6_5'(x),m(x)-'mn_ RK6_6'(x)],x=0..0.5,font=[HELVETICA,9],\ncolor=[COLOR(RGB,.95,0,.95),C OLOR(RGB,.5,0,.95),COLOR(RGB,.95,.45,0),COLOR(RGB,.2,.85,0),\nCOLOR(RG B,0,.3,.95),COLOR(RGB,.95,0,.2)],legend=[`Sharp-Verner scheme with c[7 ]=19/20`,`scheme with c[7]=24/25`,`scheme with c[7]=35/36`,`scheme wit h c[7]=63/64`,`scheme with c[7]=125/126`,`Papakostas' scheme with c[7] =199/200`],title=`error curves for 8 stage order 6 Runge-Kutta methods `));" }}{PARA 13 "" 1 "" {GLPLOT2D 1009 490 490 {PLOTDATA 2 "6,-%'CURV ESG6%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$$\"50++]P4'\\/8'F- $\"'Vh\\F07$$\"5SLLL3x1h6oF-$\"(8#=5F07$$\"5qmm;zWF07$ $\"50+++]7G$R<)F-$\"(Mm^$F07$$\"5qmmTNYL^9&)F-$\"(?tj%F07$$\"5SLL$3-)Q 4b))F-$\"(tr/'F07$$\"50++D19Wn&>*F-$\"(+Y!yF07$$\"5qmmm\"z%\\DO&*F-$\" (\\n(**F07$$\"5NLL3x\"[No()*F-$\")$RSE\"F07$$\"5+++Dc,;u@5!#@$\")5\"3P \"F07$$\"5nm;z%\\l*zb5Fdp$\")([cO\"F07$$\"5MLLLL3x&)*3\"Fdp$\")(=0O\"F 07$$\"5nmm\"z%\\v#pK\"Fdp$\")21E8F07$$\"5+++]i!R(*Rc\"Fdp$\")\\V38F07$ $\"5nm;z>6B`#o\"Fdp$\")4hN8F07$$\"5MLL3xJs1,=Fdp$\")dOP9F07$$\"5nm\"Hd ?pM.'=Fdp$\")$H+a\"F07$$\"5++]PM_@g>>Fdp$\")*Hkp\"F07$$\"5ML3-j7'p)y>F dp$\")F07$$\"5nmmm\"H2P\"Q?Fdp$\")zaA?F07$$\"5MLLek.pu/BFdp$\")DKo 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^l7$Fecl$\"(_(f_Fa^l7$Fjcl$\"(8Q2&Fa^l7$F_dl$\"(YJ!\\Fa^l7$Fddl$\"(rxu %Fa^l7$Fidl$\"(@`e%Fa^l7$F^el$\"()4[WFa^l7$Fcel$\"(N(4VFa^l7$Fhel$\"(5 8>%Fa^l7$F]fl$\"(S'oSFa^l7$Fbfl$\"(3%fRFa^l7$Fgfl$\"(A6&QFa^l7$F\\gl$ \"(L2v$Fa^l7$Fagl$\"(;5l$Fa^l7$Ffgl$\"(w)fNFa^l7$F[hl$\"(.8Z$Fa^l7$F`h l$\"((z(Q$Fa^l7$Fehl$\"(>YJ$Fa^l7$Fjhl$\"(?ZB$Fa^l7$F_il$\"($fmJFa^l7$ Fdil$\"(fs4$Fa^l7$Fiil$\"(!)Q.$Fa^l7$F^jl$\"()))oHFa^l-Fdjl6&FfjlFgjlF (Fjdp-F[[m6#%EPapakostas'~scheme~with~c[7]=199/200G-%%FONTG6$%*HELVETI CAG\"\"*-%+AXESLABELSG6$Q\"x6\"Q!Fg^s-%&TITLEG6#%Uerror~curves~for~8~s tage~order~6~Runge-Kutta~methodsG-%%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 w ith c[7]=19/20" "scheme with c[7]=24/25" "scheme with c[7]=35/36" "sch eme with c[7]=63/64" "scheme with c[7]=125/126" "Papakostas' scheme wi th c[7]=199/200" }}}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 47 "Te st 10 of 8 stage, order 6 Runge-Kutta methods" }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "dy/dx = -(2*sin(5*x)+3*cos(7*x))*sinh(y );" "6#/*&%#dyG\"\"\"%#dxG!\"\",$*&,&*&\"\"#F&-%$sinG6#*&\"\"&F&%\"xGF &F&F&*&\"\"$F&-%$cosG6#*&\"\"(F&F3F&F&F&F&-%%sinhG6#%\"yGF&F(" }{TEXT -1 5 " , " }{XPPEDIT 18 0 "y(0)=sqrt(5)/2" "6#/-%\"yG6#\"\"!*&-%%sqr tG6#\"\"&\"\"\"\"\"#!\"\"" }{TEXT -1 1 " " }}{PARA 0 "" 0 "" {TEXT -1 10 "Solution: " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 207 "de := diff(y(x),x)=-(2*sin(5*x)+3*cos(7*x))*sin h(y(x));\nic := y(0)=sqrt(5)/2;\ndsolve(\{de,ic\},y(x));\nsimplify(con vert(%,exp));\np := unapply(rhs(%),x):\nplot(p(x),x=0..5,font=[HELVETI CA,9],labels=[`x`,`y(x)`]);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%#deG/ -%%diffG6$-%\"yG6#%\"xGF,,$*&,&*&\"\"#\"\"\"-%$sinG6#,$*&\"\"&F2F,F2F2 F2F2*&\"\"$F2-%$cosG6#,$*&\"\"(F2F,F2F2F2F2F2-%%sinhG6#F)F2!\"\"" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#>%#icG/-%\"yG6#\"\"!,$*&\"\"#!\"\"\"\" &#\"\"\"F,F0" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/-%\"yG6#%\"xG-%#lnG6# -%%tanhG6#,*#\"\"\"\"\"&F0*&#F0\"\"#F0-F)6#,$*&,&-%$expG6#,$*&F4!\"\"F 1F3F0F0F0F0F0,&F:F0F0F?F?F?F0F0*&#\"\"$\"#9F0-%$sinG6#,$*&\"\"(F0F'F0F 0F0F0*&#F0F1F0-%$cosG6#,$*&F1F0F'F0F0F0F?" }}{PARA 12 "" 1 "" {XPPMATH 20 "6#/-%\"yG6#%\"xG-%#lnG6#*&,*-%$expG6#,4#\"\"#\"\"&\"\"\"* &#\"\"$\"\"(F4-%$sinGF&F4!\"\"*&#\"$#>F8F4*&)-%$cosGF&\"\"'F4F9F4F4F4* &#\"$S#F8F4*&)FA\"\"%F4F9F4F4F;*&#\"#sF8F4*&)FAF2F4F9F4F4F4*&#\"#KF3F4 *$)FAF3F4F4F;*&\"\")F4)FAF7F4F4*&F2F4FAF4F;*&F2F;F3#F4F2F4F4-F.6#,2F1F 4*&#F7F8F4F9F4F;*&F=F4F?F4F4*&#FFF8F4FGF4F;*&FKF4FMF4F4*&#FQF3F4FRF4F; *&FUF4FVF4F4*&F2F4FAF4F;F4-F.6#,$*&F2F;F3FYF4F4F4F;F4,*F-F4FZF4FaoF;F4 F4F;" }}{PARA 13 "" 1 "" {GLPLOT2D 552 388 388 {PLOTDATA 2 "6&-%'CURVE SG6$7av7$$\"\"!F)$\"38&*)\\())R.=6!#<7$$\"3ALL$3FWYs#!#>$\"3`uw,WSt45F ,7$$\"3WmmmT&)G\\aF0$\"3cSB%H69F5*!#=7$$\"3m****\\7G$R<)F0$\"3OF6@7@G; #)F87$$\"3GLLL3x&)*3\"F8$\"3_]\"Hlv:eW(F87$$\"3))**\\i!R(*Rc\"F8$\"3WS 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\"386but%F,$\"3_@+X\\xqwZF87$$\"37+]iSjE!z%F,$\"3<\"z**[W(=JcF87$ $\"3y*\\7G))Rb\"[F,$\"3OXJ\"GK]h>'F87$$\"3L+++DM\"3%[F,$\"3!**fZR-9n(o F87$$\"3)3](=np3m[F,$\"3#eM]7.#Q?&=\\F,$\"3'4(*[g6ply*F87$$\"3s+voa-oX\\F,$\"3?(Q=b([U56 F,7$$\"3O]PMF,%G(\\F,$\"3_;`pzy]b7F,7$$\"\"&F)$\"3.ug')yo>49F,-%'COLOU RG6&%$RGBG$\"#5!\"\"F(F(-%%FONTG6$%*HELVETICAG\"\"*-%+AXESLABELSG6$%\" xG%%y(x)G-%%VIEWG6$;F(Ficn%(DEFAULTG" 1 2 0 1 10 0 2 9 1 4 2 1.000000 45.000000 45.000000 0 0 "Curve 1" }}}}{PARA 0 "" 0 "" {TEXT -1 0 "" }} {PARA 0 "" 0 "" {TEXT -1 93 "The following code constructs a discrete \+ solution based on each of the methods and gives the " }{TEXT 260 22 "r oot mean square error" }{TEXT -1 18 " of each solution." }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 801 "P := (x,y) -> -(2*sin(5*x)+3*cos(7 *x))*sinh(y): hh := 0.01: numsteps := 500: x0 := 0: y0 := sqrt(5)/2:\n matrix([[`slope field: `,P(x,y)],[`initial point: `,``(x0,y0)],[`ste p width: `,hh],\n[`no. of steps: `,numsteps]]);``;\nmthds := [`Sha rp-Verner scheme with `*``(c[7]=19/20),`scheme with `*``(c[7]=24/25),` scheme with `*``(c[7]=35/36),`scheme with `*``(c[7]=63/64),`scheme wit h `*``(c[7]=125/126),`Papakostas' scheme with `*``(c[7]=199/200)]: err s := []:\nDigits := 30:\nfor ct to 6 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),s qrt(sm/numpts)];\nend do:\nDigits := 10:\nlinalg[transpose]([mthds,eva lf(errs)]);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#K%'matrixG6#7&7$%0slope ~field:~~~G,$*&,&*&\"\"#\"\"\"-%$sinG6#,$*&\"\"&F.%\"xGF.F.F.F.*&\"\"$ F.-%$cosG6#,$*&\"\"(F.F5F.F.F.F.F.-%%sinhG6#%\"yGF.!\"\"7$%0initial~po int:~G-%!G6$\"\"!,$*&F-FBF4#F.F-F.7$%/step~width:~~~G$F.!\"#7$%1no.~of ~steps:~~~G\"$+&Q(pprint46\"" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%!G" } }{PARA 11 "" 1 "" {XPPMATH 20 "6#K%'matrixG6#7(7$*&%:Sharp-Verner~sche me~with~G\"\"\"-%!G6#/&%\"cG6#\"\"(#\"#>\"#?F*$\"+.'*z2?!#@7$*&%-schem e~with~GF*-F,6#/F/#\"#C\"#DF*$\"+t_9\"z\"F87$*&F;F*-F,6#/F/#\"#N\"#OF* $\"++;#)38F87$*&F;F*-F,6#/F/#\"#j\"#kF*$\"+-*=[v(!#A7$*&F;F*-F,6#/F/# \"$D\"\"$E\"F*$\"+X#\\y<%FX7$*&%9Papakostas'~scheme~with~GF*-F,6#/F/# \"$*>\"$+#F*$\"+8\"3_!GFXQ(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 each of the Runge-Kutta schemes." }}{PARA 0 "" 0 "" {TEXT -1 75 "T he error in the value obtained by each of the methods at the point whe re " }{XPPEDIT 18 0 "x = 4.999;" "6#/%\"xG-%&FloatG6$\"%**\\!\"$" } {TEXT -1 16 " is also given." }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 725 "P := (x,y) -> -(2*sin(5*x)+3*cos(7*x))*sinh(y): hh := 0.01: num steps := 500: x0 := 0: y0 := sqrt(5)/2:\nmatrix([[`slope field: `,P( x,y)],[`initial point: `,``(x0,y0)],[`step width: `,hh],\n[`no. of s teps: `,numsteps]]);``;\nmthds := [`Sharp-Verner scheme with `*``(c[ 7]=19/20),`scheme with `*``(c[7]=24/25),`scheme with `*``(c[7]=35/36), `scheme with `*``(c[7]=63/64),`scheme with `*``(c[7]=125/126),`Papakos tas' scheme with `*``(c[7]=199/200)]: errs := []:\nDigits := 30:\nfor \+ ct to 6 do\n pn_RK6_||ct := RK6_||ct(P(x,y),x,y,x0,evalf(y0),hh,nums teps,true);\nend do:\nxx := 4.999: pxx := evalf(p(xx)):\nfor ct to 6 d o\n errs := [op(errs),abs(pn_RK6_||ct(xx)-pxx)];\nend do:\nDigits := 10:\nlinalg[transpose]([mthds,evalf(errs)]);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#K%'matrixG6#7&7$%0slope~field:~~~G,$*&,&*&\"\"#\"\"\"-% $sinG6#,$*&\"\"&F.%\"xGF.F.F.F.*&\"\"$F.-%$cosG6#,$*&\"\"(F.F5F.F.F.F. F.-%%sinhG6#%\"yGF.!\"\"7$%0initial~point:~G-%!G6$\"\"!,$*&F-FBF4#F.F- F.7$%/step~width:~~~G$F.!\"#7$%1no.~of~steps:~~~G\"$+&Q)pprint536\"" } }{PARA 11 "" 1 "" {XPPMATH 20 "6#%!G" }}{PARA 11 "" 1 "" {XPPMATH 20 " 6#K%'matrixG6#7(7$*&%:Sharp-Verner~scheme~with~G\"\"\"-%!G6#/&%\"cG6# \"\"(#\"#>\"#?F*$\"+T^+W5!#@7$*&%-scheme~with~GF*-F,6#/F/#\"#C\"#DF*$ \"+P_Xt\"$+#F*$\"+I^f[TFXQ)pprint546 \"" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 4 "The " }{TEXT 260 22 "root mean square error" }{TEXT -1 20 " over the inte rval " }{XPPEDIT 18 0 "[0, 5];" "6#7$\"\"!\"\"&" }{TEXT -1 82 " of e ach Runge-Kutta method is estimated as follows using the special proce dure " }{TEXT 0 5 "NCint" }{TEXT -1 98 " to perform numerical integr ation by the 7 point Newton-Cotes method over 200 equal subintervals. " }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 445 "mthds := [`Sharp-Verner scheme with `*``(c[7]=19/20),`scheme with `*``(c[7]=24/25),`scheme wi th `*``(c[7]=35/36),`scheme with `*``(c[7]=63/64),`scheme with `*``(c[ 7]=125/126),`Papakostas' scheme with `*``(c[7]=199/200)]: errs := []: \nDigits := 20:\nfor ct to 6 do\n sm := NCint((p(x)-'pn_RK6_||ct'(x) )^2,x=0..5,adaptive=false,numpoints=7,factor=200);\n errs := [op(err s),sqrt(sm/5)];\nend do:\nDigits := 10:\nlinalg[transpose]([mthds,eval f(errs)]);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#K%'matrixG6#7(7$*&%:Shar p-Verner~scheme~with~G\"\"\"-%!G6#/&%\"cG6#\"\"(#\"#>\"#?F*$\"+yL`.?!# @7$*&%-scheme~with~GF*-F,6#/F/#\"#C\"#DF*$\"+'e3ky\"F87$*&F;F*-F,6#/F/ #\"#N\"#OF*$\"+Q1[08F87$*&F;F*-F,6#/F/#\"#j\"#kF*$\"+\\ZHRx!#A7$*&F;F* -F,6#/F/#\"$D\"\"$E\"F*$\"+(['*R<%FX7$*&%9Papakostas'~scheme~with~GF*- F,6#/F/#\"$*>\"$+#F*$\"+)o&>0GFXQ)pprint556\"" }}}{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 0 544 "evalf[20](plot([p(x)-'pn_R K6_1'(x),p(x)-'pn_RK6_2'(x),p(x)-'pn_RK6_3'(x),p(x)-'pn_RK6_4'(x),\np( x)-'pn_RK6_5'(x),p(x)-'pn_RK6_6'(x)],x=0..2.2,font=[HELVETICA,9],\ncol or=[COLOR(RGB,.95,0,.95),COLOR(RGB,.5,0,.95),COLOR(RGB,.95,.45,0),COLO R(RGB,.2,.85,0),\nCOLOR(RGB,0,.3,.95),COLOR(RGB,.95,0,.2)],legend=[`Sh arp-Verner scheme with c[7]=19/20`,`scheme with c[7]=24/25`,`scheme wi th c[7]=35/36`,`scheme with c[7]=63/64`,`scheme with c[7]=125/126`,`Pa pakostas' scheme with c[7]=199/200`],title=`error curves for 8 stage o rder 6 Runge-Kutta methods`));\n" }}{PARA 13 "" 1 "" {GLPLOT2D 936 546 546 {PLOTDATA 2 "6,-%'CURVESG6%7dr7$$\"\"!F)F(7$$\"5mmmm;zM%))>\"! 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,/1n(F-7$Fbfp$\"(&H2Z\"F-7$F`dl$\")u66;F-7$Fedl$\"(Zr!=F17$Fjdl$\"(_)=?F17$F_ el$\"(eAF#F17$Fdel$\"(!R#R#F17$Fiel$\"(\\Af#F17$F^fl$\"(Da'GF17$Fcfl$ \"(ja)HF17$F]gl$\"(nmJ$F17$Fggl$\"(Cb]$F17$Fahl$\"(B&*y$F17$F[il$\"(\\ V'QF17$F`il$\"(ri!RF17$Feil$\"(ne'RF17$Fjil$\"(MJ3%F17$F_jl$\"(uUN%F1- Fejl6&FgjlFhjlF[[mFjio-F][m6#%EPapakostas'~scheme~with~c[7]=199/200G-% %FONTG6$%*HELVETICAG\"\"*-%+AXESLABELSG6$Q\"x6\"Q!Fg^r-%&TITLEG6#%Uerr or~curves~for~8~stage~order~6~Runge-Kutta~methodsG-%%VIEWG6$;F(F_jl%(D EFAULTG" 1 2 0 1 10 0 2 9 1 4 2 1.000000 45.000000 45.000000 0 1 "Shar p-Verner scheme with c[7]=19/20" "scheme with c[7]=24/25" "scheme with c[7]=35/36" "scheme with c[7]=63/64" "scheme with c[7]=125/126" "Papa kostas' scheme with c[7]=199/200" }}}}{PARA 0 "" 0 "" {TEXT -1 0 "" }} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}}{SECT 1 {PARA 4 "" 0 " " {TEXT -1 48 "Test 11 of 8 stage, order 6 Runge-Kutta methods " }} {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 143 "This exa mple is similar to one that appears in an article by F. G. Lether: Ma thematics of Computation, Vol. 20, no. 95, (July 1966) page 382. " }} {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 256 "" 0 "" {TEXT -1 1 " " } {XPPEDIT 18 0 "dy/dx=exp(-x)/(x-1)^2" "6#/*&%#dyG\"\"\"%#dxG!\"\"*&-%$ expG6#,$%\"xGF(F&*$,&F.F&F&F(\"\"#F(" }{TEXT -1 2 " " }{XPPEDIT 18 0 "cos(1/(x-1))-y" "6#,&-%$cosG6#*&\"\"\"F(,&%\"xGF(F(!\"\"F+F(%\"yGF+" }{TEXT -1 5 ", " }{XPPEDIT 18 0 "y(0)=sin*1" "6#/-%\"yG6#\"\"!*&%$s inG\"\"\"F*F*" }{TEXT -1 1 " " }}{PARA 0 "" 0 "" {TEXT -1 10 "Solution : " }}{PARA 256 "" 0 "" {TEXT -1 2 " " }{XPPEDIT 18 0 "y = -exp(-x)*s in(1/(x-1))" "6#/%\"yG,$*&-%$expG6#,$%\"xG!\"\"\"\"\"-%$sinG6#*&F-F-,& F+F-F-F,F,F-F," }{TEXT -1 1 " " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 185 "de := diff(y(x),x)=exp(-x)/ (x-1)^2*cos(1/(x-1))-y(x);\nic := y(0)=sin(1);\ndsolve(\{de,ic\},y(x)) ;\nq := unapply(rhs(%),x):\nplot(q(x),x=0..1-1/(6*Pi),font=[HELVETICA, 9],labels=[`x`,`y(x)`]);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%#deG/-%% diffG6$-%\"yG6#%\"xGF,,&*(-%$expG6#,$F,!\"\"\"\"\",&F,F4F4F3!\"#-%$cos G6#*&F4F4F5F3F4F4F)F3" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%#icG/-%\"yG 6#\"\"!-%$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$$\"3#>=\"*)>z2k?!#>$\"3m%p=Qu6DN)F,7$$\"3 kMQU\"3@+'QF0$\"3m=O<,5$yO#)F,7$ $\"38UQ!)p4'G\"zF0$\"3/Z^$zTlU<)F,7$$\"3BY$*R0>JO**F0$\"36ty1)z*36\")F ,7$$\"3wbXC%*4B\"=\"F,$\"3A;o(=P!Q^!)F,7$$\"3M!)fxC(zaP\"F,$\"3E8^!ydw !))zF,7$$\"3kgswR?Pw:F,$\"3T8>lD8j?zF,7$$\"3Qnlb!4?mx\"F,$\"3%p4\\s^P4 &yF,7$$\"3OsvSC)*f#)>F,$\"3/$H(=wa6wxF,7$$\"3)Rk+h)o-k@F,$\"3LR8d$>`qq (F,7$$\"3Q^Vo'yq#oBF,$\"3YB)Qc;#3DwF,7$$\"3?0sMKLNtDF,$\"3,;%fG`C(F,7$$\"3S+dSsVlWLF,$\"3&36sy[X09(F,7$ $\"3EOur83&\\b$F,$\"37)QgTzpp+(F,7$$\"3c#**Qwz)4TPF,$\"3M]xfz#>l(oF,7$ $\"3wx#p)QELXRF,$\"3UR-VbS%zr'F,7$$\"3\"yy;Gb6)RTF,$\"3%40quzD%\\lF,7$ $\"3p2KM(*)HFM%F,$\"3W'4!o9@F_jF,7$$\"3`G+(=Gs!HXF,$\"3S.Rv)o&))[hF,7$ 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$\"3M$o6)eV:lLF,7$$\"3oE3CP5fw#*F,$\"3Ma)HpV]]I!H!G$*F,$\"3a%4t07BP*GF,7$$\"3J=s\")G&))3M*F,$ \"3f(3)H;[%o+#F,7$$\"3w&\\Kr-[PN*F,$\"33n1kl[]%*F,$!3m(=[SoWqQ#F,7$$\"3%>saO,CmX*F,$! 3GDS5eP#en\"F,7$$\"3AhB\"Gw`IY*F,$!3U$3!Gg0_(o)F07$$\"3]++(>^$[p%*F,$! 3V'=8$[D+C:!#C-%'COLOURG6&%$RGBG$\"#5!\"\"F(F(-%%FONTG6$%*HELVETICAG\" \"*-%+AXESLABELSG6$%\"xG%%y(x)G-%%VIEWG6$;F($\"+BN[p%*!#5%(DEFAULTG" 1 2 0 1 10 0 2 9 1 4 2 1.000000 45.000000 45.000000 0 0 "Curve 1" }}}} {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 93 "The 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 0 817 "Q := ( x,y) -> exp(-x)/(x-1)^2*cos(1/(x-1))-y: hh := 1/500-1/(3000*Pi): numst eps := 500: x0 := 0: y0 := sin(1):\nmatrix([[`slope field: `,Q(x,y)] ,[`initial point: `,``(x0,y0)],[`step width: `,hh],\n[`no. of steps: `,numsteps]]);``;\nmthds := [`Sharp-Verner scheme with `*``(c[7]=19 /20),`scheme with `*``(c[7]=24/25),`scheme with `*``(c[7]=35/36),`sche me with `*``(c[7]=63/64),`scheme with `*``(c[7]=125/126),`Papakostas' \+ scheme with `*``(c[7]=199/200)]: errs := []:\nDigits := 30:\nfor ct to 6 do\n Qn_RK6_||ct := RK6_||ct(Q(x,y),x,y,x0,evalf[33](y0),evalf[33 ](hh),numsteps,false);\n sm := 0: numpts := nops(Qn_RK6_||ct):\n f or ii to numpts do\n sm := sm+(Qn_RK6_||ct[ii,2]-q(Qn_RK6_||ct[ii ,1]))^2;\n end do:\n errs := [op(errs),sqrt(sm/numpts)];\nend do: \nDigits := 10:\nlinalg[transpose]([mthds,evalf(errs)]);" }}{PARA 11 " " 1 "" {XPPMATH 20 "6#K%'matrixG6#7&7$%0slope~field:~~~G,&*(-%$expG6#, $%\"xG!\"\"\"\"\",&F/F1F1F0!\"#-%$cosG6#*&F1F1F2F0F1F1%\"yGF07$%0initi al~point:~G-%!G6$\"\"!-%$sinG6#F17$%/step~width:~~~G,&#F1\"$+&F1*&F1F1 *&\"%+IF1%#PiGF1F0F07$%1no.~of~steps:~~~GFFQ)pprint566\"" }}{PARA 11 " " 1 "" {XPPMATH 20 "6#%!G" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#K%'matrix G6#7(7$*&%:Sharp-Verner~scheme~with~G\"\"\"-%!G6#/&%\"cG6#\"\"(#\"#>\" #?F*$\"+gvM4E!#>7$*&%-scheme~with~GF*-F,6#/F/#\"#C\"#DF*$\"+:ad%R#!#?7 $*&F;F*-F,6#/F/#\"#N\"#OF*$\"+yjpJEFD7$*&F;F*-F,6#/F/#\"#j\"#kF*$\"+Q` N6KFD7$*&F;F*-F,6#/F/#\"$D\"\"$E\"F*$\"+0RdQQFD7$*&%9Papakostas'~schem e~with~GF*-F,6#/F/#\"$*>\"$+#F*$\"+b)))>d$FDQ)pprint576\"" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 30 "The following cod e constructs " }{TEXT 260 20 "numerical procedures" }{TEXT -1 56 " for solutions based on each of the Runge-Kutta schemes." }}{PARA 0 "" 0 " " {TEXT -1 75 "The error in the value obtained by each of the methods \+ at the point where " }{XPPEDIT 18 0 "x = 0;" "6#/%\"xG\"\"!" }{TEXT -1 21 ".9469 is also given." }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 741 "Q := (x,y) -> exp(-x)/(x-1)^2*cos(1/(x-1))-y: hh := 1/500-1/(3000 *Pi): numsteps := 500: x0 := 0: y0 := sin(1):\nmatrix([[`slope field: \+ `,Q(x,y)],[`initial point: `,``(x0,y0)],[`step width: `,hh],\n[`no . of steps: `,numsteps]]);``;\nmthds := [`Sharp-Verner scheme with ` *``(c[7]=19/20),`scheme with `*``(c[7]=24/25),`scheme with `*``(c[7]=3 5/36),`scheme with `*``(c[7]=63/64),`scheme with `*``(c[7]=125/126),`P apakostas' scheme with `*``(c[7]=199/200)]: errs := []:\nDigits := 30: \nfor ct to 6 do\n qn_RK6_||ct := RK6_||ct(Q(x,y),x,y,x0,evalf(y0),e valf(hh),numsteps,true);\nend do:\nxx := 0.9469: qxx := evalf(q(xx)): \nfor ct to 6 do\n errs := [op(errs),abs(qn_RK6_||ct(xx)-qxx)];\nend do:\nDigits := 10:\nlinalg[transpose]([mthds,evalf(errs)]);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#K%'matrixG6#7&7$%0slope~field:~~~G,&*(-%$exp G6#,$%\"xG!\"\"\"\"\",&F/F1F1F0!\"#-%$cosG6#*&F1F1F2F0F1F1%\"yGF07$%0i nitial~point:~G-%!G6$\"\"!-%$sinG6#F17$%/step~width:~~~G,&#F1\"$+&F1*& F1F1*&\"%+IF1%#PiGF1F0F07$%1no.~of~steps:~~~GFFQ)pprint586\"" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%!G" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#K%'ma trixG6#7(7$*&%:Sharp-Verner~scheme~with~G\"\"\"-%!G6#/&%\"cG6#\"\"(#\" #>\"#?F*$\"+1MwyW!#=7$*&%-scheme~with~GF*-F,6#/F/#\"#C\"#DF*$\"+Dfyoq! #?7$*&F;F*-F,6#/F/#\"#N\"#OF*$\"+?,&[F)FD7$*&F;F*-F,6#/F/#\"#j\"#kF*$ \"+s`D_D!#>7$*&F;F*-F,6#/F/#\"$D\"\"$E\"F*$\"+?G8$)RFY7$*&%9Papakostas '~scheme~with~GF*-F,6#/F/#\"$*>\"$+#F*$\"+0SgMLFYQ)pprint596\"" }}} {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 0 465 "mthds := [`Sharp-Verner scheme with `*``(c[7]=19/20) ,`scheme with `*``(c[7]=24/25),`scheme with `*``(c[7]=35/36),`scheme w ith `*``(c[7]=63/64),`scheme with `*``(c[7]=125/126),`Papakostas' sche me with `*``(c[7]=199/200)]: errs := []:\nDigits := 20:\nfor ct to 6 d o\n sm := NCint((q(x)-'qn_RK6_||ct'(x))^2,x=0..1-1/(6*Pi),adaptive=f alse,numpoints=7,factor=200);\n errs := [op(errs),sqrt(sm/(1-1/(6*Pi )))];\nend do:\nDigits := 10:\nlinalg[transpose]([mthds,evalf(errs)]); " }}{PARA 11 "" 1 "" {XPPMATH 20 "6#K%'matrixG6#7(7$*&%:Sharp-Verner~s cheme~with~G\"\"\"-%!G6#/&%\"cG6#\"\"(#\"#>\"#?F*$\"+&*=#yc\"!#>7$*&%- scheme~with~GF*-F,6#/F/#\"#C\"#DF*$\"+?'>\"=B!#?7$*&F;F*-F,6#/F/#\"#N \"#OF*$\"+Y&Q!)f#FD7$*&F;F*-F,6#/F/#\"#j\"#kF*$\"+\\?*3-$FD7$*&F;F*-F, 6#/F/#\"$D\"\"$E\"F*$\"+T=gDMFD7$*&%9Papakostas'~scheme~with~GF*-F,6#/ F/#\"$*>\"$+#F*$\"+%G*3qKFDQ)pprint606\"" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 92 "The following error graphs are con structed using the numerical procedures for the solutions." }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 557 "evalf[25](plot(['qn_RK6_1'(x)-q(x) ,'qn_RK6_2'(x)-q(x),'qn_RK6_3'(x)-q(x),'qn_RK6_4'(x)-q(x),\n'qn_RK6_5' (x)-q(x),'qn_RK6_6'(x)-q(x)],x=0..0.7,-8e-18..8e-18,font=[HELVETICA,9] ,\ncolor=[COLOR(RGB,.95,0,.95),COLOR(RGB,.5,0,.95),COLOR(RGB,.95,.45,0 ),COLOR(RGB,.2,.85,0),\nCOLOR(RGB,0,.3,.95),COLOR(RGB,.95,0,.2)],legen d=[`Sharp-Verner scheme with c[7]=19/20`,`scheme with c[7]=24/25`,`sch eme with c[7]=35/36`,`scheme with c[7]=63/64`,`scheme with c[7]=125/12 6`,`Papakostas' scheme with c[7]=199/200`],title=`error curves for 8 s tage order 6 Runge-Kutta methods`));" }}{PARA 13 "" 1 "" {GLPLOT2D 873 613 613 {PLOTDATA 2 "6,-%'CURVESG6%7fn7$$\"\"!F)F(7$$\":mmmmmmm;z+ e_\"!#E$\"%r()!#D7$$\":LLLLLL$3->R`GF-$\"&ss\"F07$$\":mmmmmmmT&pSYVF-$ \"&Lw#F07$$\":lmmmmmm\"z'=$\\eF-$\"&[#RF07$$\":KLLLLL$3Ft3XtF-$\"&#H_F 07$$\":lmmmmm;aLc=t)F-$\"&Pr'F07$$\":++++++v=`xn,\"F0$\"&K?)F07$$\":mm mmmmT&y/Gl6F0$\"'()35F07$$\":++++++vV<2LJ\"F0$\"'oB7F07$$\":LLLLLLLe#3 dl9F0$\"'#pY\"F07$$\":mmmmmm;Ht%o*f\"F0$\"'J1F0$\"'QkBF07$$\":+++++++Dxg$[?F0$\"'viFF07 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"Papakostas' scheme with c[7]=199/200" }}}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 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+F2F+\"#kF-F-*,F/F+F5F-F1F+\"#@F+F2F+F+*,\"#DF+F 5F-F1F+\"\"(F+F2F+F-" }{TEXT -1 2 " " }}{PARA 0 "" 0 "" {TEXT -1 0 " " }}{EXCHG {PARA 0 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"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 0 761 "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 scheme with `*``(c[7]=19/20),`scheme with `*``(c[7]=24/25 ),`scheme with `*``(c[7]=35/36),`scheme with `*``(c[7]=63/64),`scheme \+ with `*``(c[7]=125/126),`Papakostas' scheme with `*``(c[7]=199/200)]: \+ errs := []:\nDigits := 30:\nfor ct to 6 do\n Rn_RK6_||ct := RK6_||ct (R(x,y),x,y,x0,y0,hh,numsteps,false);\n sm := 0: numpts := nops(Rn_R K6_||ct):\n for ii to numpts do\n sm := sm+(Rn_RK6_||ct[ii,2]-r (Rn_RK6_||ct[ii,1]))^2;\n end do:\n errs := [op(errs),sqrt(sm/nump ts)];\nend do:\nDigits := 10:\nlinalg[transpose]([mthds,evalf(errs)]); " }}{PARA 11 "" 1 "" {XPPMATH 20 "6#K%'matrixG6#7&7$%0slope~field:~~~G ,$*(\"\"&\"\"\"%\"yGF,)-%$sinG6#,$*&\"\"(F,%\"xGF,F,F4F,F,7$%0initial~ point:~G-%!G6$\"\"!F,7$%/step~width:~~~G$F,!\"#7$%1no.~of~steps:~~~G\" $+&Q)pprint616\"" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%!G" }}{PARA 11 " " 1 "" {XPPMATH 20 "6#K%'matrixG6#7(7$*&%:Sharp-Verner~scheme~with~G\" \"\"-%!G6#/&%\"cG6#\"\"(#\"#>\"#?F*$\"+*G%HU7!#>7$*&%-scheme~with~GF*- F,6#/F/#\"#C\"#DF*$\"+5jSLV!#?7$*&F;F*-F,6#/F/#\"#N\"#OF*$\"+Yw2;EFD7$ *&F;F*-F,6#/F/#\"#j\"#kF*$\"+'>TB5*!#@7$*&F;F*-F,6#/F/#\"$D\"\"$E\"F*$ \"+JO:z:FY7$*&%9Papakostas'~scheme~with~GF*-F,6#/F/#\"$*>\"$+#F*$\"+c^ wBNFYQ)pprint626\"" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 " " {TEXT -1 30 "The following code constructs " }{TEXT 260 20 "numerica l procedures" }{TEXT -1 56 " for solutions based on each of the Runge- Kutta schemes." }}{PARA 0 "" 0 "" {TEXT -1 75 "The error in the value \+ obtained by each of the methods at the point where " }{XPPEDIT 18 0 " x = 4.999;" "6#/%\"xG-%&FloatG6$\"%**\\!\"$" }{TEXT -1 16 " is also g iven." }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 691 "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 \+ scheme with `*``(c[7]=19/20),`scheme with `*``(c[7]=24/25),`scheme wit h `*``(c[7]=35/36),`scheme with `*``(c[7]=63/64),`scheme with `*``(c[7 ]=125/126),`Papakostas' scheme with `*``(c[7]=199/200)]: errs := []:Di gits := 25:\nfor ct to 6 do\n rn_RK6_||ct := RK6_||ct(R(x,y),x,y,x0, y0,hh,numsteps,true);\nend do:\nxx := 4.999: rxx := evalf(r(xx)):\nfor ct to 6 do\n errs := [op(errs),abs(rn_RK6_||ct(xx)-rxx)];\nend do: \nDigits := 10:\nlinalg[transpose]([mthds,evalf(errs)]);" }}{PARA 11 " " 1 "" {XPPMATH 20 "6#K%'matrixG6#7&7$%0slope~field:~~~G,$*(\"\"&\"\" \"%\"yGF,)-%$sinG6#,$*&\"\"(F,%\"xGF,F,F4F,F,7$%0initial~point:~G-%!G6 $\"\"!F,7$%/step~width:~~~G$F,!\"#7$%1no.~of~steps:~~~G\"$+&Q)pprint63 6\"" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%!G" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#K%'matrixG6#7(7$*&%:Sharp-Verner~scheme~with~G\"\"\"-%! G6#/&%\"cG6#\"\"(#\"#>\"#?F*$\"+$y=sT#!#>7$*&%-scheme~with~GF*-F,6#/F/ #\"#C\"#DF*$\"+#[JXq)!#?7$*&F;F*-F,6#/F/#\"#N\"#OF*$\"+^L!o>&FD7$*&F;F *-F,6#/F/#\"#j\"#kF*$\"+A%o5s\"FD7$*&F;F*-F,6#/F/#\"$D\"\"$E\"F*$\"+q* )o9N!#A7$*&%9Papakostas'~scheme~with~GF*-F,6#/F/#\"$*>\"$+#F*$\"+o(eVL (!#@Q)pprint646\"" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 4 "The " }{TEXT 260 22 "root mean square error" }{TEXT -1 20 " over the interval " }{XPPEDIT 18 0 "[0, 5];" "6#7$\"\"!\"\"&" } {TEXT -1 82 " of each Runge-Kutta method is estimated as follows usin g the special procedure " }{TEXT 0 5 "NCint" }{TEXT -1 98 " to perfo rm numerical integration by the 7 point Newton-Cotes method over 200 e qual subintervals." }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 445 "mthds := [`Sharp-Verner scheme with `*``(c[7]=19/20),`scheme with `*``(c[7] =24/25),`scheme with `*``(c[7]=35/36),`scheme with `*``(c[7]=63/64),`s cheme with `*``(c[7]=125/126),`Papakostas' scheme with `*``(c[7]=199/2 00)]: errs := []:\nDigits := 20:\nfor ct to 6 do\n sm := NCint((r(x) -'rn_RK6_||ct'(x))^2,x=0..5,adaptive=false,numpoints=7,factor=200);\n \+ errs := [op(errs),sqrt(sm/5)];\nend do:\nDigits := 10:\nlinalg[trans pose]([mthds,evalf(errs)]);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#K%'matr ixG6#7(7$*&%:Sharp-Verner~scheme~with~G\"\"\"-%!G6#/&%\"cG6#\"\"(#\"#> \"#?F*$\"+Tv))R7!#>7$*&%-scheme~with~GF*-F,6#/F/#\"#C\"#DF*$\"+k\\dCV! #?7$*&F;F*-F,6#/F/#\"#N\"#OF*$\"+1`$4h#FD7$*&F;F*-F,6#/F/#\"#j\"#kF*$ \"+*z*y(3*!#@7$*&F;F*-F,6#/F/#\"$D\"\"$E\"F*$\"+/Mj#e\"FY7$*&%9Papakos tas'~scheme~with~GF*-F,6#/F/#\"$*>\"$+#F*$\"+YZ1;NFYQ)pprint656\"" }}} {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 0 541 "evalf[20]( plot([r(x)-'rn_RK6_1'(x),r(x)-'rn_RK6_2'(x),r(x)-'rn_RK6_3'(x),r(x)-'r n_RK6_4'(x),\nr(x)-'rn_RK6_5'(x),r(x)-'rn_RK6_6'(x)],x=0..5,font=[HELV ETICA,9],\ncolor=[COLOR(RGB,.95,0,.95),COLOR(RGB,.5,0,.95),COLOR(RGB,. 95,.45,0),COLOR(RGB,.2,.85,0),\nCOLOR(RGB,0,.3,.95),COLOR(RGB,.95,0,.2 )],legend=[`Sharp-Verner scheme with c[7]=19/20`,`scheme with c[7]=24/ 25`,`scheme with c[7]=35/36`,`scheme with c[7]=63/64`,`scheme with c[7 ]=125/126`,`Papakostas' scheme with c[7]=199/200`],title=`error curves for 8 stage order 6 Runge-Kutta methods`));" }}{PARA 13 "" 1 "" {GLPLOT2D 1131 636 636 {PLOTDATA 2 "6,-%'CURVESG6%7a_l7$$\"\"!F)F(7$$ \"5NLLL$3FWYs#!#@$\"(>f3%!#>7$$\"5qmmmmT&)G\\aF-$\"(Eg1(F07$$\"5SLLL3x 1h6oF-$\"($p$\\#F07$$\"50+++]7G$R<)F-$!((GXpF07$$\"5qmmm\"z%\\DO&*F-$! 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" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 256 "" 0 " " {TEXT -1 3 " " }{XPPEDIT 18 0 "dy/dx = cos*x+2*y;" "6#/*&%#dyG\"\" \"%#dxG!\"\",&*&%$cosGF&%\"xGF&F&*&\"\"#F&%\"yGF&F&" }{TEXT -1 8 ", \+ " }{XPPEDIT 18 0 "y(0) = -2/5;" "6#/-%\"yG6#\"\"!,$*&\"\"#\"\"\"\" \"&!\"\"F-" }{TEXT -1 1 " " }}{PARA 0 "" 0 "" {TEXT -1 10 "Solution: \+ " }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "y = 1/5;" "6#/%\" yG*&\"\"\"F&\"\"&!\"\"" }{TEXT -1 1 " " }{XPPEDIT 18 0 "sin*x-2/5" "6# ,&*&%$sinG\"\"\"%\"xGF&F&*&\"\"#F&\"\"&!\"\"F+" }{TEXT -1 1 " " } {XPPEDIT 18 0 "cos*x" "6#*&%$cosG\"\"\"%\"xGF%" }{TEXT -1 3 " . " }} {PARA 0 "" 0 "" {TEXT 260 4 "Note" }{TEXT -1 2 ": " }}{PARA 0 "" 0 "" {TEXT -1 53 "The general solution of the differential equation " } {XPPEDIT 18 0 "dy/dx = cos*x+2*y;" "6#/*&%#dyG\"\"\"%#dxG!\"\",&*&%$co sGF&%\"xGF&F&*&\"\"#F&%\"yGF&F&" }{TEXT -1 64 " contains an exponenti al term, but with the initial condition " }{XPPEDIT 18 0 "y(0) = -2/5 " "6#/-%\"yG6#\"\"!,$*&\"\"#\"\"\"\"\"&!\"\"F-" }{TEXT -1 23 " this t erm disappears." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> \+ " 0 "" {MPLTEXT 1 0 50 "de := diff(y(x),x)=cos(x)+2*y(x);\ndsolve(de,y (x));" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%#deG/-%%diffG6$-%\"yG6#%\"x GF,,&-%$cosGF+\"\"\"*&\"\"#F0F)F0F0" }}{PARA 11 "" 1 "" {XPPMATH 20 "6 #/-%\"yG6#%\"xG,(*&#\"\"#\"\"&\"\"\"-%$cosGF&F-!\"\"*&#F-F,F--%$sinGF& F-F-*&-%$expG6#,$*&F+F-F'F-F-F-%$_C1GF-F-" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 105 "Any slight deviation of a nume rical solution from the correct solution tends to become rapidly magni fied." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 154 "de := diff(y(x),x)=cos(x)+2*y(x);\nic := y(0)=-2/5; \ndsolve(\{de,ic\},y(x));\ne := unapply(rhs(%),x):\nplot(e(x),x=0..8,f ont=[HELVETICA,9],labels=[`x`,`y(x)`]);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%#deG/-%%diffG6$-%\"yG6#%\"xGF,,&-%$cosGF+\"\"\"*&\"\"#F0F)F0F0 " }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%#icG/-%\"yG6#\"\"!#!\"#\"\"&" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#/-%\"yG6#%\"xG,&*&#\"\"#\"\"&\"\"\"-%$ cosGF&F-!\"\"*&#F-F,F--%$sinGF&F-F-" }}{PARA 13 "" 1 "" {GLPLOT2D 703 312 312 {PLOTDATA 2 "6&-%'CURVESG6$7gn7$$\"\"!F)$!3A+++++++S!#=7$$\"3E LLLLBxV5E$F,$!35;%fC]([[JF,7$$\"3MLL LLAKn\\F,$!3C&4%=OwYjDF,7$$\"3=LLLLc$\\o'F,$!31c1[)*fT**=F,7$$\"3)emmm ^&Q%R)F,$!39J7$$\"3))** ***\\YJ?;\"!#<$\"3m!=?Y3*>`CFK7$$\"3?LLL=\"\\g**FK7 $$\"3\")*****\\[A4]\"FO$\"3Xgu?U;&er\"F,7$$\"3wmmm'3Q\\n\"FO$\"3S\\4.g (y\\S#F,7$$\"3OLLLB6@G=FO$\"3e*[f2BGC&HF,7$$\"3&)******f-w+?FO$\"375@E VOJ&[$F,7$$\"3%*********y,u@FO$\"3VG2]n#=i\"RF,7$$\"3)*******RP)4M#FO$ \"3ym!)\\t%R1A%F,7$$\"3Umm;HUz;CFO$\"3:@(\\YT,0K%F,7$$\"3ILLL=Zg#\\#FO $\"3++xVHVa&R%F,7$$\"3;++]A2v#e#FO$\"3+<'Hh4))=X%F,7$$\"3cmmmEn*Gn#FO$ \"3a5#zx'*y?Z%F,7$$\"3qmmm;AE\\FFO$\"35^%H>#ywgWF,7$$\"3Tmmm1xiDGFO$\" 3(3\\(>4bXBWF,7$$\"3LLL$e#*eW\"HFO$\"3![MOl&\\jZVF,7$$\"3!)*****\\9!H. IFO$\"37X%)HL)HvB%F,7$$\"3Immm1:bgJFO$\"37d#H1rl8'RF,7$$\"3<+++X@4LLFO $\"3G,Fnxt@YNF,7$$\"31+++N;R(\\$FO$\"3+2Ml_]z_IF,7$$\"3wmmm;4#)oOFO$\" 3?6K>$)*R0X#F,7$$\"3jmmm6lCEQFO$\"3xp`%>:UP$=F,7$$\"3ELLL$G^g*RFO$\"38 $\\Qkcw!=6F,7$$\"3oKLL=2VsTFO$\"39U#4i*[S4MFK7$$\"3f*****\\`pfK%FO$!3E &Q)=hjJ]MFK7$$\"3!HLLLm&z\"\\%FO$!3u$z\"\\\">,j2\"F,7$$\"3s******z-6jY FO$!3_=%f%oq`+=F,7$$\"3<******4#32$[FO$!3Gvm#oI!>eCF,7$$\"3O*****\\#y' G*\\FO$!3Ak5yX#4\"HIF,7$$\"3G******H%=H<&FO$!3EIq1&[C$pNF,7$$\"35mmm1> qM`FO$!3%z'[2h*Gn&RF,7$$\"3%)*******HSu]&FO$!3%*oc=HW4cUF,7$$\"3'fmm\" HOq&e&FO$!3oqc`'[F/N%F,7$$\"3'HLL$ep'Rm&FO$!3$e%**GFr7=WF,7$$\"3D***\\ P?[nq&FO$!3TlAsE+sVWF,7$$\"3Umm;\\%H&\\dFO$!3y[ey96=hWF,7$$\"3eLLe%p5B z&FO$!3=)zg%Q%y/Z%F,7$$\"3')******R>4NeFO$!3waa0%)\\frWF,7$$\"3HLL$ed* f:fFO$!3]_J$4k<:X%F,7$$\"3#emm;@2h*fFO$!3V5vHeMg-WF,7$$\"37LLL))3E!3'F O$!3=l`a'y%*4K%F,7$$\"3]*****\\c9W;'FO$!3>=$e-d.)3UF,7$$\"3Lmmmmd'*GjF O$!3Gy*y<4!G/RF,7$$\"3j*****\\iN7]'FO$!3;B6I^7jsMF,7$$\"3aLLLt>:nmFO$! 37+2hu:afHF,7$$\"35LLL.a#o$oFO$!3;\"e/Z#4*3N#F,7$$\"3ammm^Q40qFO$!3!4` 1I$pa!o\"F,7$$\"3y******z]rfrFO$!3pfL'*)RTA-\"F,7$$\"3gmmmc%GpL(FO$!3? j;%3XMsQ#FK7$$\"3/LLL8-V&\\(FO$\"3qi(R>/(R\"p%FK7$$\"3=+++XhUkwFO$\"3Z X^U-))=F,7$$\"\")F)$\"3s<7[GmrgDF,- %'COLOURG6&%$RGBG$\"#5!\"\"F(F(-%%FONTG6$%*HELVETICAG\"\"*-%+AXESLABEL SG6$%\"xG%%y(x)G-%%VIEWG6$;F(Fg]l%(DEFAULTG" 1 2 0 1 10 0 2 9 1 4 2 1.000000 45.000000 45.000000 0 0 "Curve 1" }}}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 32 "The following code constructs a " }{TEXT 260 17 "discrete solution" }{TEXT -1 44 " based on each of t he methods and gives the " }{TEXT 260 22 "root mean square error" } {TEXT -1 18 " of each solution." }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 767 "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 with `*``(c[7]=19/20),`scheme with `* ``(c[7]=24/25),`scheme with `*``(c[7]=35/36),`scheme with `*``(c[7]=63 /64),`scheme with `*``(c[7]=125/126),`Papakostas' scheme with `*``(c[7 ]=199/200)]: errs := []:\nDigits := 20:\nfor ct to 6 do\n En_RK6_||c t := RK6_||ct(E(x,y),x,y,x0,evalf(y0),hh,numsteps,false);\n sm := 0: numpts := nops(En_RK6_||ct):\n for ii to numpts do\n sm := sm+ (En_RK6_||ct[ii,2]-e(En_RK6_||ct[ii,1]))^2;\n end do:\n errs := [o p(errs),sqrt(sm/numpts)];\nend do:\nDigits := 10:\nlinalg[transpose]([ mthds,evalf(errs)]);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#K%'matrixG6#7& 7$%0slope~field:~~~G,&-%$cosG6#%\"xG\"\"\"*&\"\"#F.%\"yGF.F.7$%0initia l~point:~G-%!G6$\"\"!#!\"#\"\"&7$%/step~width:~~~G$F0F97$%1no.~of~step s:~~~G\"$+%Q)pprint666\"" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%!G" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#K%'matrixG6#7(7$*&%:Sharp-Verner~schem e~with~G\"\"\"-%!G6#/&%\"cG6#\"\"(#\"#>\"#?F*$\"+3=5F@F87$*&F;F*-F,6#/F/#\"$D\" \"$E\"F*$\"+@f$G!>F87$*&%9Papakostas'~scheme~with~GF*-F,6#/F/#\"$*>\"$ +#F*$\"+jj>, " 0 "" {MPLTEXT 1 0 698 "E : = (x,y) -> cos(x)+2*y: hh := 0.02: numsteps := 400: x0 := 0: y0 := -2/ 5:\nmatrix([[`slope field: `,E(x,y)],[`initial point: `,``(x0,y0)],[ `step width: `,hh],\n[`no. of steps: `,numsteps]]);``;\nmthds := [ `Sharp-Verner scheme with `*``(c[7]=19/20),`scheme with `*``(c[7]=24/2 5),`scheme with `*``(c[7]=35/36),`scheme with `*``(c[7]=63/64),`scheme with `*``(c[7]=125/126),`Papakostas' scheme with `*``(c[7]=199/200)]: errs := []:\nDigits := 20:\nfor ct to 6 do\n en_RK6_||ct := RK6_||c t(E(x,y),x,y,x0,evalf(y0),hh,numsteps,true);\nend do:\nxx := 7.999: ex x := evalf(e(xx)):\nfor ct to 6 do\n errs := [op(errs),abs(en_RK6_|| ct(xx)-exx)];\nend do:\nDigits := 10:\nlinalg[transpose]([mthds,evalf( errs)]);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#K%'matrixG6#7&7$%0slope~fi eld:~~~G,&-%$cosG6#%\"xG\"\"\"*&\"\"#F.%\"yGF.F.7$%0initial~point:~G-% !G6$\"\"!#!\"#\"\"&7$%/step~width:~~~G$F0F97$%1no.~of~steps:~~~G\"$+%Q )pprint686\"" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%!G" }}{PARA 11 "" 1 " " {XPPMATH 20 "6#K%'matrixG6#7(7$*&%:Sharp-Verner~scheme~with~G\"\"\"- %!G6#/&%\"cG6#\"\"(#\"#>\"#?F*$\"+mheq&*!#=7$*&%-scheme~with~GF*-F,6#/ F/#\"#C\"#DF*$\"+gNy@)*F87$*&F;F*-F,6#/F/#\"#N\"#OF*$\"+/ko#>\"!#<7$*& F;F*-F,6#/F/#\"#j\"#kF*$\"+&)oUu6FN7$*&F;F*-F,6#/F/#\"$D\"\"$E\"F*$\"+ \\,Wa5FN7$*&%9Papakostas'~scheme~with~GF*-F,6#/F/#\"$*>\"$+#F*$\"+JK.F %*F8Q)pprint696\"" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 4 "The " }{TEXT 260 22 "root mean square error" }{TEXT -1 20 " over the interval " }{XPPEDIT 18 0 "[0, 8];" "6#7$\"\"!\"\")" } {TEXT -1 82 " of each Runge-Kutta method is estimated as follows 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 0 445 "mthds := [`Sharp-Verner scheme with `*``(c[7]=19/20),`scheme with `*``(c[7] =24/25),`scheme with `*``(c[7]=35/36),`scheme with `*``(c[7]=63/64),`s cheme with `*``(c[7]=125/126),`Papakostas' scheme with `*``(c[7]=199/2 00)]: errs := []:\nDigits := 20:\nfor ct to 6 do\n sm := NCint((e(x) -'en_RK6_||ct'(x))^2,x=0..8,adaptive=false,numpoints=7,factor=200);\n \+ errs := [op(errs),sqrt(sm/8)];\nend do:\nDigits := 10:\nlinalg[trans pose]([mthds,evalf(errs)]);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#K%'matr ixG6#7(7$*&%:Sharp-Verner~scheme~with~G\"\"\"-%!G6#/&%\"cG6#\"\"(#\"#> \"#?F*$\"+:PC&p\"!#=7$*&%-scheme~with~GF*-F,6#/F/#\"#C\"#DF*$\"+D'Q(R< F87$*&F;F*-F,6#/F/#\"#N\"#OF*$\"+KFh7@F87$*&F;F*-F,6#/F/#\"#j\"#kF*$\" +\"\\p-3#F87$*&F;F*-F,6#/F/#\"$D\"\"$E\"F*$\"+Uitn=F87$*&%9Papakostas' ~scheme~with~GF*-F,6#/F/#\"$*>\"$+#F*$\"+Ch\")p;F8Q)pprint706\"" }}} {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 92 "The follo wing error graphs are constructed using the numerical procedures for t he solutions." }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 541 "evalf[20]( plot([e(x)-'en_RK6_1'(x),e(x)-'en_RK6_2'(x),e(x)-'en_RK6_3'(x),e(x)-'e n_RK6_4'(x),\ne(x)-'en_RK6_5'(x),e(x)-'en_RK6_6'(x)],x=0..2,font=[HELV ETICA,9],\ncolor=[COLOR(RGB,.95,0,.95),COLOR(RGB,.5,0,.95),COLOR(RGB,. 95,.45,0),COLOR(RGB,.2,.85,0),\nCOLOR(RGB,0,.3,.95),COLOR(RGB,.95,0,.2 )],legend=[`Sharp-Verner scheme with c[7]=19/20`,`scheme with c[7]=24/ 25`,`scheme with c[7]=35/36`,`scheme with c[7]=63/64`,`scheme with c[7 ]=125/126`,`Papakostas' scheme with c[7]=199/200`],title=`error curves for 8 stage order 6 Runge-Kutta methods`));" }}{PARA 13 "" 1 "" {GLPLOT2D 745 641 641 {PLOTDATA 2 "6,-%'CURVESG6%7S7$$\"\"!F)F(7$$\"5M LLLLL3VfV!#@$\"&)>9!#?7$$\"5nmmmm\"H[D:)F-$\"&&>HF07$$\"5LLLLLe0$=C\"F 0$\"&ja%F07$$\"5LLLLL3RBr;F0$\"&\")H'F07$$\"5nmmm;zjf)4#F0$\"&?=)F07$$ \"5MLLL$e4;[\\#F0$\"'o85F07$$\"5++++]i'y]!HF0$\"'ND7F07$$\"5MLLL$ezs$H LF0$\"'>d9F07$$\"5++++]7iI_PF0$\"'v3+&F0$\"'b*e#F07$$\"5+++++]Z/Na 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c[7]=24/25`,`scheme with c[7]=35/36`,`scheme with c[7]=63/64`,` scheme with c[7]=125/126`,`Papakostas' scheme with c[7]=199/200`],titl e=`error curves for 8 stage order 6 Runge-Kutta methods`));" }}{PARA 13 "" 1 "" {GLPLOT2D 813 648 648 {PLOTDATA 2 "6,-%'CURVESG6%7Z7$$\"\"# \"\"!$\"(]70'!#?7$$\"5+++++DHyI@!#>$\"(11#yF-7$$\"5++++v[kdWAF1$\"(nmx *F-7$$\"5++++]n\"\\DP#F1$\")f;e7F-7$$\"5++++]s,P,DF1$\")>DA;F-7$$\"5++ ++v8*y&HEF1$\")?R\"4#F-7$$\"5++++vG[W[FF1$\")zcZEF-7$$\"5++++v)fB:(GF1 $\")ZO\"Q$F-7$$\"5++++vQ=\"))*HF1$\")B3cVF-7$$\"5++++vj=pDJF1$\")vB4cF -7$$\"5+++++lN?cKF1$\")&ztF(F-7$$\"5++++]U$e6P$F1$\")xHa\"*F-7$$\"5+++ ++&>q0]$F1$\"*n*Q&=\"F-7$$\"5+++++DM^IOF1$\"*ESo`\"F-7$$\"5+++++0ytbPF 1$\"*\\7R(>F-7$$\"5++++vQNXpQF1$\"*s\\xZ#F-7$$\"5+++++XDn/SF1$\"*\"y&p C$F-7$$\"5+++++!y?#>TF1$\"+#\\1G3%!#@7$$\"5++++v3wY_UF1$\",yeN&H`!#A7$ $\"5+++++IOTqVF1$\"+\")QQZnF^q7$$\"5++++v3\">)*\\%F1$\"*(=bS()F-7$$\"5 ++++DEP/BYF1$\"+8LM=6F-7$$\"5++++](o:;v%F1$\"+'z%HY9F-7$$\"5++++v$)[op 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\",)y;rCeF-7$F`w$\",!*\\7v\\(F-7$Few$\",B8,lf*F-7$Fjw$\"-h!o:EC\"F-7$F _x$\"-70dv$f\"F-7$Fdx$\"-P\\Nnb?F-7$Fix$\".^)zn)ek#F^q7$F^y$\"/hrOEbOL Fdq7$Fcy$\".N=OM3\"QF^q7$Fhy$\".!Q#*Q`_VF^q7$F]z$\".*fho&>!\\F^q7$Fbz$ \"-X5Qt?bF-7$Fgz$\"-)3,\\nE'F-7$F\\[l$\"-0GId8rF-7$Fa[l$\"-6\\*o5.)F-7 $Ff[l$\"-h^=!p1*F-7$F[\\l$\"-_#e.wn*F-7$F`\\l$\".w<#R%H.\"F-7$Fe\\l$\" .rS#z^-6F-7$Fj\\l$\".9r3yn<\"F--F_]l6&Fa]l$F)Fihl$\"#&)Fd]lFe]l-Fg]l6# %7scheme~with~c[7]=63/64G-F$6%7Z7$F($\"(:')o'F-7$F/$\"(\"4P')F-7$F5$\" )$H!z5F-7$F:$\")4!zQ\"F-7$F?$\")eu)y\"F-7$FD$\")9S0BF-7$FI$\")j!z\"HF- 7$FN$\")n4EPF-7$FS$\")Wn*z%F-7$FX$\")S.!='F-7$Fgn$\")*3w,)F-7$F\\o$\"* tA&35F-7$Fao$\"*X?fI\"F-7$Ffo$\"*7.Jp\"F-7$F[p$\"*l@Y<#F-7$F`p$\"*;.(H FF-7$Fep$\"*icrd$F-7$Fjp$\"+)*p.)\\%F^q7$F`q$\",!pJgreFdq7$Ffq$\"+\\xq LuF^q7$F[r$\"*:x'H'*F-7$F`r$\"+E=6K7F-7$Fer$\"+e@V$f\"F-7$Fjr$\"+(*\\( y,#F-7$F_s$\"+U%pKg#F-7$Fds$\"+f7x\"R$F-7$Fis$\"+D*[-F%F-7$F^t$\"+/NDw aF-7$Fct$\"+Fc!43(F-7$Fht$\"+D(Q[5*F-7$F]u$\",M]87;\"F-7$Fbu$\",0/r7_ \"F-7$Fgu$\",vD,\"R>F-7$F\\v$\",N]TE^#F-7$Fav$\",+j+w<$F-7$Ffv$\",%\\G ]2TF-7$F[w$\",0%GiH_F-7$F`w$\",r$)>:t'F-7$Few$\",@nigh)F-7$Fjw$\"-h2Bm :6F-7$F_x$\"-=#oF4V\"F-7$Fdx$\"-6GJlX=F-7$Fix$\".4+^mbP#F^q7$F^y$\"/&y!RF^q7$F]z$\".:N[U6S%F^q7 $Fbz$\"-]W8qc\\F-7$Fgz$\"-i3!*\\EcF-7$F\\[l$\"-gOi!oQ'F-7$Fa[l$\"-))*3 l0@(F-7$Ff[l$\"-'e)3dS\")F-7$F[\\l$\"-HY&z))o)F-7$F`\\l$\"-*Hh>TF*F-7$ Fe\\l$\"-^E&y()*)*F-7$Fj\\l$\".,w8^l0\"F--F_]l6&Fa]lFe]l$\"\"$FihlFb]l -Fg]l6#%9scheme~with~c[7]=125/126G-F$6%7Z7$F($\"(X&zfF-7$F/$\"(d:s(F-7 $F5$\"(9mk*F-7$F:$\")[!3C\"F-7$F?$\")p<*f\"F-7$FD$\")l3h?F-7$FI$\")\"* o3EF-7$FN$\")4CJLF-7$FS$\")z0\"H%F-7$FX$\")![^_&F-7$Fgn$\")G+orF-7$F\\ o$\")R_;!*F-7$Fao$\"*\"f`n6F-7$Ffo$\"*-!p8:F-7$F[p$\"*P$=W>F-7$F`p$\"* NW/W#F-7$Fep$\"*B&4)>$F-7$Fjp$\"+g?R@SF^q7$F`q$\",)QXS\\_Fdq7$Ffq$\"++ j(fk'F^q7$F[r$\"*ZV#4')F-7$F`r$\"+Cya,6F-7$Fer$\"+,)zXU\"F-7$Fjr$\"+6` //=F-7$F_s$\"+?nSFBF-7$Fds$\"+FENKIF-7$Fis$\"+D)Rx\"QF-7$F^t$\"+$fYf*[ F-7$Fct$\"+]qbIjF-7$Fht$\"+P\"=+9)F-7$F]u$\",]9i\"Q5F-7$Fbu$\",88l+O\" F-7$Fgu$\",su=Ot\"F-7$F\\v$\",\\7#QYAF-7$Fav$\",Y\"p(3%GF-7$Ffv$\",)H$ RAn$F-7$F[w$\",**f]an%F-7$F`w$\",tp%>=gF-7$Few$\", " 0 "" {MPLTEXT 1 0 2 " ;" }}}}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 48 "Test 14 o f 8 stage, order 6 Runge-Kutta methods " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 256 "" 0 "" {TEXT -1 3 " " }{XPPEDIT 18 0 "dy/dx = 10*x*c os*x-10*y;" "6#/*&%#dyG\"\"\"%#dxG!\"\",&**\"#5F&%\"xGF&%$cosGF&F,F&F& *&F+F&%\"yGF&F(" }{TEXT -1 8 ", " }{XPPEDIT 18 0 "y(0) = sqrt(5) ;" "6#/-%\"yG6#\"\"!-%%sqrtG6#\"\"&" }{TEXT -1 1 " " }}{PARA 0 "" 0 " " {TEXT -1 10 "Solution: " }}{PARA 256 "" 0 "" {TEXT -1 2 " " } {XPPEDIT 18 0 "y=100/101" "6#/%\"yG*&\"$+\"\"\"\"\"$,\"!\"\"" }{TEXT -1 1 " " }{XPPEDIT 18 0 "x*cos*x-990/10201" "6#,&*(%\"xG\"\"\"%$cosGF& F%F&F&*&\"$!**F&\"&,-\"!\"\"F+" }{TEXT -1 1 " " }{XPPEDIT 18 0 "cos*x+ 10/101" "6#,&*&%$cosG\"\"\"%\"xGF&F&*&\"#5F&\"$,\"!\"\"F&" }{TEXT -1 1 " " }{XPPEDIT 18 0 "x*sin*x-200/10201" "6#,&*(%\"xG\"\"\"%$sinGF&F%F &F&*&\"$+#F&\"&,-\"!\"\"F+" }{TEXT -1 1 " " }{XPPEDIT 18 0 "sin*x+(990 /10201+sqrt(5))*exp(-10*x)" "6#,&*&%$sinG\"\"\"%\"xGF&F&*&,&*&\"$!**F& \"&,-\"!\"\"F&-%%sqrtG6#\"\"&F&F&-%$expG6#,$*&\"#5F&F'F&F-F&F&" } {TEXT -1 1 "." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 163 "de := diff(y(x),x)=10*x*cos(x)-10*y(x);\nic := \+ y(0)=sqrt(5);\ndsolve(\{de,ic\},y(x));\nb := unapply(rhs(%),x):\nplot( b(x),x=0..5,font=[HELVETICA,9],labels=[`x`,`y(x)`]);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%#deG/-%%diffG6$-%\"yG6#%\"xGF,,&*(\"#5\"\"\"F,F0-% $cosGF+F0F0*&F/F0F)F0!\"\"" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%#icG/- %\"yG6#\"\"!*$\"\"&#\"\"\"\"\"#" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/-% \"yG6#%\"xG,,*&#\"$+\"\"$,\"\"\"\"*&F'F--%$cosGF&F-F-F-*&#\"$!**\"&,- \"F-F/F-!\"\"*&#\"#5F,F-*&-%$sinGF&F-F'F-F-F-*&#\"$+#F4F-F:F-F5*&-%$ex pG6#,$*&F8F-F'F-F5F-,&#F3F4F-*$\"\"&#F-\"\"#F-F-F-" }}{PARA 13 "" 1 " " {GLPLOT2D 703 312 312 {PLOTDATA 2 "6&-%'CURVESG6$7hn7$$\"\"!F)$\"3\" )*y*\\xz1OA!#<7$$\"3ALL$3FWYs#!#>$\"3*pc)\\jF;12h\"H\"48F,7$$\"3m****\\7G$R<)F0$\"3<_u(oLbK,\"F,7$$\"3GLLL3x &)*3\"!#=$\"3(**[ro!GyVzF@7$$\"3))**\\i!R(*Rc\"F@$\"3A'ysO]2xW&F@7$$\" 3umm\"H2P\"Q?F@$\"3/$)oqvSKmSF@7$$\"3YLek.pu/BF@$\"3$Qjx*Gs<7OF@7$$\"3 !***\\PMnNrDF@$\"3M:4%*3rt@LF@7$$\"3MmT5ll'z$GF@$\"3/?Np5C\\bJF@7$$\"3 MLL$eRwX5$F@$\"3)GTJ!oG0$3$F@7$$\"3rLLL$eI8k$F@$\"3FyHM$p'GKJF@7$$\"33 ML$3x%3yTF@$\"3n$**Q]`\"yRLF@7$$\"3emm\"z%4\\Y_F@$\"3%)*)G8T#p2%RF@7$$ \"3`LLeR-/PiF@$\"3PZ.%R2Cm^%F@7$$\"3]***\\il'pisF@$\"31e'*fKlL9]F@7$$ \"3>MLe*)>VB$)F@$\"3%3)yy-pAk`F@7$$\"3Y++DJbw!Q*F@$\"3Kg$RQBm7^&F@7$$ \"3%ommTIOo/\"F,$\"3xrTB'zz$GaF@7$$\"3YLL3_>jU6F,$\"3!p\\3Dp!RX^F@7$$ \"37++]i^Z]7F,$\"3Kvxv\"*=%>e%F@7$$\"33++](=h(e8F,$\"3C^UQ-(*\\]PF@7$$ \"3/++]P[6j9F,$\"3icMTpV\"zp#F@7$$\"3UL$e*[z(yb\"F,$\"3C-jVJM+L:F@7$$ \"3wmm;a/cq;F,$!33XIKjA5+5F07$$\"3%ommmJF,$!3!\\F1VDv$)p&F@7$$\"3K+] i!f#=$3#F,$!3V$*[LN4F4!)F@7$$\"3?+](=xpe=#F,$!3$[6ral`?.\"F,7$$\"37nm \"H28IH#F,$!3s6ToLL*4G\"F,7$$\"3um;zpSS\"R#F,$!3/xdj'GyI^\"F,7$$\"3GLL 3_?`(\\#F,$!3MGc,i$*4jpxg#F,$!3XXbRF+x>IF,$!3r\"fP(*)y]OGF,7$$\"3F+]i!RU07$F, $!3[Tb]!*H%e)HF,7$$\"3+++v=S2LKF,$!3Tyi#Q\\'[=JF,7$$\"3Jmmm\"p)=MLF,$! 3N`)=cl>V?$F,7$$\"3B++](=]@W$F,$!3Y_BiN[odKF,7$$\"3mm\"H#oZ1\"\\$F,$!3 )o4&)z%=-oKF,7$$\"35L$e*[$z*RNF,$!3%)Q61)ek$pKF,7$$\"3%o;Hd!fX$f$F,$!3 #*y!45Ut-E$F,7$$\"3e++]iC$pk$F,$!3LO[nw')*)RKF,7$$\"3ILe*[t\\sp$F,$!3D 1>x`HA5KF,7$$\"3[m;H2qcZPF,$!3/[q%\\V.-<$F,7$$\"3O+]7.\"fF&QF,$!3KL?tX &>E0$F,7$$\"3Ymm;/OgbRF,$!3KQEMNc$G*GF,7$$\"3w**\\ilAFjSF,$!3/QR)44g!y EF,7$$\"3yLLL$)*pp;%F,$!30,GW_`#fU#F,7$$\"3)RL$3xe,tUF,$!3*G#*H@1([B@F ,7$$\"3Cn;HdO=yVF,$!35Q!)*4x]5y\"F,7$$\"3a+++D>#[Z%F,$!3(y*pyl_QJ9F,7$ $\"3SnmT&G!e&e%F,$!3]X/0\"RC%G**F@7$$\"3#RLLL)Qk%o%F,$!3u!*)Q\"4WH+dF@ 7$$\"37+]iSjE!z%F,$!3+r[gfMO'=*F07$$\"3a+]P40O\"*[F,$\"3+2*eSHde(QF@7$ $\"\"&F)$\"3&Q8`\">jC3#*F@-%'COLOURG6&%$RGBG$\"#5!\"\"F(F(-%%FONTG6$%* HELVETICAG\"\"*-%+AXESLABELSG6$%\"xG%%y(x)G-%%VIEWG6$;F(F\\^l%(DEFAULT G" 1 2 0 1 10 0 2 9 1 4 2 1.000000 45.000000 45.000000 0 0 "Curve 1" } }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 32 "The fo llowing code constructs a " }{TEXT 260 17 "discrete solution" }{TEXT -1 44 " based on each of the methods and gives the " }{TEXT 260 22 "ro ot mean square error" }{TEXT -1 18 " of each solution." }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 783 "B := (x,y) -> 10*x*cos(x)-10*y: hh := 0.01: numsteps := 500: x0 := 0: y0 := sqrt(5):\nmatrix([[`slope fi eld: `,B(x,y)],[`initial point: `,``(x0,y0)],[`step width: `,hh], \n[`no. of steps: `,numsteps]]);``;\nmthds := [`Sharp-Verner scheme \+ with `*``(c[7]=19/20),`scheme with `*``(c[7]=24/25),`scheme with `*``( c[7]=35/36),`scheme with `*``(c[7]=63/64),`scheme with `*``(c[7]=125/1 26),`Papakostas' scheme with `*``(c[7]=199/200)]: errs := []:\nDigits \+ := 20:\nfor ct to 6 do\n Bn_RK6_||ct := RK6_||ct(B(x,y),x,y,x0,evalf (y0),hh,numsteps,false);\n sm := 0: numpts := nops(Bn_RK6_||ct):\n \+ for ii to numpts do\n sm := sm+(Bn_RK6_||ct[ii,2]-evalf(b(Bn_RK6 _||ct[ii,1])))^2;\n end do:\n errs := [op(errs),sqrt(sm/numpts)]; \nend do:\nDigits := 10:\nlinalg[transpose]([mthds,evalf(errs)]);" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#K%'matrixG6#7&7$%0slope~field:~~~G,&*( \"#5\"\"\"%\"xGF,-%$cosG6#F-F,F,*&F+F,%\"yGF,!\"\"7$%0initial~point:~G -%!G6$\"\"!*$\"\"&#F,\"\"#7$%/step~width:~~~G$F,!\"#7$%1no.~of~steps:~ ~~G\"$+&Q(pprint66\"" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%!G" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#K%'matrixG6#7(7$*&%:Sharp-Verner~scheme~with ~G\"\"\"-%!G6#/&%\"cG6#\"\"(#\"#>\"#?F*$\"+#QX#o%)!#A7$*&%-scheme~with ~GF*-F,6#/F/#\"#C\"#DF*$\"+2wx4aF87$*&F;F*-F,6#/F/#\"#N\"#OF*$\"+Bx?Xd F87$*&F;F*-F,6#/F/#\"#j\"#kF*$\"+%G(\\giF87$*&F;F*-F,6#/F/#\"$D\"\"$E \"F*$\"+S9(Rw'F87$*&%9Papakostas'~scheme~with~GF*-F,6#/F/#\"$*>\"$+#F* $\"+WU#=\"oF8Q(pprint76\"" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 30 "The following code constructs " }{TEXT 260 20 "nu merical procedures" }{TEXT -1 56 " for solutions based on each of the \+ Runge-Kutta schemes." }}{PARA 0 "" 0 "" {TEXT -1 75 "The error in the \+ value obtained by each of the methods at the point where " }{XPPEDIT 18 0 "x = 4.999;" "6#/%\"xG-%&FloatG6$\"%**\\!\"$" }{TEXT -1 16 " is \+ also given." }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 707 "B := (x,y) - > 10*x*cos(x)-10*y: hh := 0.01: numsteps := 500: x0 := 0: y0 := sqrt(5 ):\nmatrix([[`slope field: `,B(x,y)],[`initial point: `,``(x0,y0)],[ `step width: `,hh],\n[`no. of steps: `,numsteps]]);``;\nmthds := [ `Sharp-Verner scheme with `*``(c[7]=19/20),`scheme with `*``(c[7]=24/2 5),`scheme with `*``(c[7]=35/36),`scheme with `*``(c[7]=63/64),`scheme with `*``(c[7]=125/126),`Papakostas' scheme with `*``(c[7]=199/200)]: errs := []:\nDigits := 25:\nfor ct to 6 do\n bn_RK6_||ct := RK6_||c t(B(x,y),x,y,x0,evalf(y0),hh,numsteps,true);\nend do:\nxx := 4.999: bx x := evalf(b(xx)):\nfor ct to 6 do\n errs := [op(errs),abs(bn_RK6_|| ct(xx)-bxx)];\nend do:\nDigits := 10:\nlinalg[transpose]([mthds,evalf( errs)]);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#K%'matrixG6#7&7$%0slope~fi eld:~~~G,&*(\"#5\"\"\"%\"xGF,-%$cosG6#F-F,F,*&F+F,%\"yGF,!\"\"7$%0init ial~point:~G-%!G6$\"\"!*$\"\"&#F,\"\"#7$%/step~width:~~~G$F,!\"#7$%1no .~of~steps:~~~G\"$+&Q)pprint736\"" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#% !G" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#K%'matrixG6#7(7$*&%:Sharp-Verner ~scheme~with~G\"\"\"-%!G6#/&%\"cG6#\"\"(#\"#>\"#?F*$\"+$\\s,[(!#B7$*&% -scheme~with~GF*-F,6#/F/#\"#C\"#DF*$\"+vy+P6!#A7$*&F;F*-F,6#/F/#\"#N\" #OF*$\"+g5p$4\"FD7$*&F;F*-F,6#/F/#\"#j\"#kF*$\"+VNk>**F87$*&F;F*-F,6#/ F/#\"$D\"\"$E\"F*$\"+GShD*)F87$*&%9Papakostas'~scheme~with~GF*-F,6#/F/ #\"$*>\"$+#F*$\"+&*RRM#)F8Q)pprint746\"" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 4 "The " }{TEXT 260 22 "root mean squa re error" }{TEXT -1 20 " over the interval " }{XPPEDIT 18 0 "[0, 5]; " "6#7$\"\"!\"\"&" }{TEXT -1 82 " of each Runge-Kutta method is estim ated as follows using the special procedure " }{TEXT 0 5 "NCint" } {TEXT -1 98 " to perform numerical integration by the 7 point Newton- Cotes method over 200 equal subintervals." }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 445 "mthds := [`Sharp-Verner scheme with `*``(c[7]=19/20) ,`scheme with `*``(c[7]=24/25),`scheme with `*``(c[7]=35/36),`scheme w ith `*``(c[7]=63/64),`scheme with `*``(c[7]=125/126),`Papakostas' sche me with `*``(c[7]=199/200)]: errs := []:\nDigits := 20:\nfor ct to 6 d o\n sm := NCint((b(x)-'bn_RK6_||ct'(x))^2,x=0..5,adaptive=false,nump oints=7,factor=200);\n errs := [op(errs),sqrt(sm/5)];\nend do:\nDigi ts := 10:\nlinalg[transpose]([mthds,evalf(errs)]);" }}{PARA 11 "" 1 " " {XPPMATH 20 "6#K%'matrixG6#7(7$*&%:Sharp-Verner~scheme~with~G\"\"\"- %!G6#/&%\"cG6#\"\"(#\"#>\"#?F*$\"+s-tn\")!#A7$*&%-scheme~with~GF*-F,6# /F/#\"#C\"#DF*$\"+\"G5u@&F87$*&F;F*-F,6#/F/#\"#N\"#OF*$\"+\"Qn5a&F87$* &F;F*-F,6#/F/#\"#j\"#kF*$\"+8+>QgF87$*&F;F*-F,6#/F/#\"$D\"\"$E\"F*$\"+ c8(Q_'F87$*&%9Papakostas'~scheme~with~GF*-F,6#/F/#\"$*>\"$+#F*$\"+jE2q lF8Q)pprint756\"" }}}{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 0 558 "evalf[20](plot(['bn_RK6_1'(x)-b(x),'bn_RK6_2'(x)-b(x ),'bn_RK6_3'(x)-b(x),'bn_RK6_4'(x)-b(x),\n'bn_RK6_5'(x)-b(x),'bn_RK6_6 '(x)-b(x)],x=0..0.65,numpoints=100,font=[HELVETICA,9],\ncolor=[COLOR(R GB,.95,0,.95),COLOR(RGB,.5,0,.95),COLOR(RGB,.95,.45,0),COLOR(RGB,.2,.8 5,0),\nCOLOR(RGB,0,.3,.95),COLOR(RGB,.95,0,.2)],legend=[`Sharp-Verner \+ scheme with c[7]=19/20`,`scheme with c[7]=24/25`,`scheme with c[7]=35/ 36`,`scheme with c[7]=63/64`,`scheme with c[7]=125/126`,`Papakostas' s cheme with c[7]=199/200`],title=`error curves for 8 stage order 6 Rung e-Kutta methods`));" }}{PARA 13 "" 1 "" {GLPLOT2D 938 575 575 {PLOTDATA 2 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Runge-Kutta methods " }}{PARA 0 "" 0 " " {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 198 "See: Comparing Numeric al Methods for Ordinary Differential Equations, Hull, Enright, Fellen \+ and Sedgwick,\n Siam Journal on Numerical Analysis, Vol. 9, No. 4 (Dec. 1972), page 617, Example A5." }}{PARA 0 "" 0 "" {TEXT -1 0 " " }}{PARA 256 "" 0 "" {TEXT -1 3 " " }{XPPEDIT 18 0 "dy/dx = (y-x)/( y+x);" "6#/*&%#dyG\"\"\"%#dxG!\"\"*&,&%\"yGF&%\"xGF(F&,&F+F&F,F&F(" } {TEXT -1 8 ", " }{XPPEDIT 18 0 "y(1) = 1;" "6#/-%\"yG6#\"\"\"F' " }{TEXT -1 1 " " }}{PARA 0 "" 0 "" {TEXT -1 10 "Solution: " }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "2*ln((x^2+y^2)/(x^2))+4*arc tan(y/x)+4*ln*x-2*ln*2-Pi = 0;" "6#/,,*&\"\"#\"\"\"-%#lnG6#*&,&*$%\"xG F&F'*$%\"yGF&F'F'*$F.F&!\"\"F'F'*&\"\"%F'-%'arctanG6#*&F0F'F.F2F'F'*(F 4F'F)F'F.F'F'*(F&F'F)F'F&F'F2%#PiGF2\"\"!" }{TEXT -1 2 ". " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 73 "de : = diff(y(x),x)=(y(x)-x)/(y(x)+x);\nic := y(1)=1;\ndsolve(\{de,ic\},y(x ));" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%#deG/-%%diffG6$-%\"yG6#%\"xGF ,*&,&F)\"\"\"F,!\"\"F/,&F)F/F,F/F0" }}{PARA 11 "" 1 "" {XPPMATH 20 "6# >%#icG/-%\"yG6#\"\"\"F)" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/-%\"yG6#% \"xG-%'RootOfG6#,,*&\"\"#\"\"\"-%#lnG6#*&,&*$)F'F-F.F.*$)%#_ZGF-F.F.F. F'!\"#F.!\"\"*&\"\"%F.-%'arctanG6#*&F8F.F'F:F.F:*&F " 0 "" {MPLTEXT 1 0 1 ";" }}} {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 41 "The solut ion can be given more simply as " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }} {PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 " ln(x^2+y^2)+2*arctan (y/x)=ln*2+Pi/2" "6#/,&-%#lnG6#,&*$%\"xG\"\"#\"\"\"*$%\"yGF+F,F,*&F+F, -%'arctanG6#*&F.F,F*!\"\"F,F,,&*&F&F,F+F,F,*&%#PiGF,F+F4F," }{TEXT -1 2 ". " }}{PARA 0 "" 0 "" {TEXT -1 18 "The solution (for " }{TEXT 268 1 "x" }{TEXT -1 47 " increasing) is the section of the polar curve " } }{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "r=sqrt(2)*exp(Pi/4-t heta)" "6#/%\"rG*&-%%sqrtG6#\"\"#\"\"\"-%$expG6#,&*&%#PiGF*\"\"%!\"\"F *%&thetaGF2F*" }{TEXT -1 5 ", " }{XPPEDIT 18 0 "-Pi/4<=theta" "6#1, $*&%#PiG\"\"\"\"\"%!\"\"F)%&thetaG" }{XPPEDIT 18 0 "``<=Pi/4" "6#1%!G* &%#PiG\"\"\"\"\"%!\"\"" }{TEXT -1 2 ". " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 7 "Check: " }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 60 "ln((x^2+y^2))+2*arctan(y/x)=ln(2)+Pi/2;\nimplicitdiff (%,y,x);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/,&-%#lnG6#,&*$)%\"xG\"\"# \"\"\"F-*$)%\"yGF,F-F-F-*&F,F--%'arctanG6#*&F0F-F+!\"\"F-F-,&-F&6#F,F- *&F,F6%#PiGF-F-" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#,$*&,&%\"xG\"\"\"% \"yG!\"\"F',&F(F'F&F'F)F)" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 471 "p1 := plot([sqrt(2)*exp(Pi/4-t),t, t=-Pi/4..Pi/4],coords=polar,thickness=2,color=red):\np2 := plot([sqrt( 2)*exp(Pi/4-t),t,t=Pi/4..2*Pi],coords=polar,color=black,linestyle=2): \np3 := plot([sqrt(2)*exp(Pi/4-t),t,t=-Pi/3..-Pi/4],coords=polar,color =black,linestyle=2):\np4 := plot([[[1,1],[uu,-uu]]$4],style=point,symb ol=[circle$2,diamond,cross],\n symbolsize=[12,10$3],c olor=[black,green$3]):\nplots[display]([p1,p2,p3,p4],font=[HELVETICA,9 ],labels=[`x`,`y(x)`]);" }}{PARA 13 "" 1 "" {GLPLOT2D 567 520 520 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G6#%&POINTG-F$6&Fh\\n-Fjz6&F\\[lF`[lF][lF`[l-Fb]n6$Fd]n\"#5Ff]n-F$6&Fh \\nF\\^n-Fb]n6$%(DIAMONDGF`^nFf]n-F$6&Fh\\nF\\^n-Fb]n6$%&CROSSGF`^nFf] n-%+AXESLABELSG6%%\"xG%%y(x)G-%%FONTG6#%(DEFAULTG-Fa_n6$%*HELVETICAG\" \"*-%%VIEWG6$Fc_nFc_n" 1 2 0 1 10 0 2 9 1 4 2 1.000000 45.000000 45.000000 0 0 "Curve 1" "Curve 2" "Curve 3" "Curve 4" "Curve 5" "Curve 6" "Curve 7" }}}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 29 "The following procedure uses " }{TEXT 0 6 "fsolve" } {TEXT -1 23 " to solve the equation " }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 " ln(x^2+y^2)+2*arctan(y/x)=ln*2+Pi/2" "6#/,&-%#lnG6 #,&*$%\"xG\"\"#\"\"\"*$%\"yGF+F,F,*&F+F,-%'arctanG6#*&F.F,F*!\"\"F,F,, &*&F&F,F+F,F,*&%#PiGF,F+F4F," }{TEXT -1 1 " " }}{PARA 0 "" 0 "" {TEXT -1 4 "for " }{TEXT 266 1 "y" }{TEXT -1 25 " numerically in terms of " }{TEXT 267 1 "x" }{TEXT -1 1 "." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 209 "phi := proc(x) local y;\n \+ fsolve(ln(x^2+y^2)+2*arctan(y/x)=ln(2)+Pi/2,y=-x..7/2-x);\n \+ end proc:\nuu := evalf(exp(Pi/2)):\nplot('phi'(x),x=1..uu,numpoint s=100,font=[HELVETICA,9],labels=[`x`,`y(x)`]);" }}{PARA 13 "" 1 "" {GLPLOT2D 515 404 404 {PLOTDATA 2 "6&-%'CURVESG6$7jq7$$\"\"\"\"\"!$\"+ ++++5!\"*7$$\"+M.FS5F-$\"+!Hsf***!#57$$\"+N$4`2\"F-$\"+]6*f)**F37$$\"+ eVr96F-$\"+Zyon**F37$$\"+#e!Qa6F-$\"+z@\"=%**F37$$\"+!GeQ>\"F-$\"+^qu3 **F37$$\"+f\"f/B\"F-$\"+$)4mr)*F37$$\"+\\sNo7F-$\"+m$=o#)*F37$$\"+4:b2 8F-$\"+-Eit(*F37$$\"+s+iY8F-$\"+@hw8(*F37$$\"+\"*o!oQ\"F-$\"+ig9X'*F37 $$\"+jM?A9F-$\"+N0$)y&*F37$$\"+>;0i9F-$\"+tvj(\\*F37$$\"+'Rj?]\"F-$\"+ 2v=4%*F37$$\"+d@iS:F-$\"+c3U<$*F37$$\"+1sjv:F-$\"+R4cG#*F37$$\"+'[tsh \"F-$\"+gf2;\"*F37$$\"+x[a_;F-$\"+=V+:!*F37$$\"+*)Qd$p\"F-$\"+F-$\"+rA&\\1)F37$$\"+-Hdj>F-$\"+oiz%*yF37$$\"+`3.**>F-$\"+4o(3u(F37$$ \"+6kKP?F-$\"+KCbovF37$$\"+q%*)o2#F-$\"+LT&QQ(F37$$\"+0Uf:@F-$\"+Okc'> (F37$$\"+EI/`@F-$\"+7w54qF37$$\"+cNi%>#F-$\"+Snr$z'F37$$\"+8b)>B#F-$\" +;Bi$f'F37$$\"+wt(=F#F-$\"+v62tjF37$$\"+L`-3BF-$\"+'*\\+nhF37$$\"+)fWv M#F-$\"+7i*[$fF37$$\"+Y\"HZQ#F-$\"+Yr#*4dF37$$\"+9xfBCF-$\"+\\X)yY&F37 $$\"+V))fhCF-$\"+8POC_F37$$\"+.DQ,DF-$\"+iJ/i\\F37$$\"+(*))pRDF-$\"+qG <-ZF37$$\"+VH))yDF-$\"+%H()*GWF37$$\"+EDuLBF 37$$\"+#GbZ)GF-$\"+E=zC?F37$$\"+O0_DHF-$\"+Z[&em\"F37$$\"+SOniHF-$\"+< &>.L\"F37$$\"+(H8L+$F-$\"+L)p\"R&*!#67$$\"+[%y$QIF-$\"+H`(4@'F]\\l7$$ \"+iNJyIF-$\"+j(QwK#F]\\l7$$\"+6*))o6$F-$!+[(['>:F]\\l7$$\"+f!Ra:$F-$! +RiBhaF]\\l7$$\"+)QZQ>$F-$!+d^2([*F]\\l7$$\"+oduIKF-$!+O[*\\M\"F37$$\" +%*QjqKF-$!+%Q!>%y\"F37$$\"+r,l3LF-$!+B!>N@#F37$$\"+70m[LF-$!+?8/xEF37 $$\"+0&z[Q$F-$!+_aG2JF37$$\"+t#3\\U$F-$!+e$R\\f$F37$$\"+d)[KY$F-$!+\") RRuSF37$$\"+>h\\,NF-$!+)*o;lXF37$$\"+LbWTNF-$!+6&)e\"4&F37$$\"+,/CyNF- $!+4mR*e&F37$$\"+zL#fh$F-$!+w:g7hF37$$\"+l)Hvl$F-$!+Zam1nF37$$\"+I6?&p $F-$!+q5-gsF37$$\"+aqsLPF-$!+g\\%=%yF37$$\"+Fp!Hx$F-$!+)QD3X)F37$$\"+a 3#*3QF-$!+'*3rE!*F37$$\"+M0JZQF-$!+?;We'*F37$$\"+*)zS&)QF-$!+6MXI5F-7$ $\"+C/;ERF-$!+B0!=5\"F-7$$\"+$oA@'RF-$!+qNzm6F-7$$\"+kch.SF-$!+'>,VC\" F-7$$\"+!))f5/%F-$!+-[r;8F-7$$\"+v*3\"ySF-$!+Z4$3R\"F-7$$\"+b'[z6%F-$! +zwYt9F-7$$\"+$\\\\z:%F-$!+'fX(f:F-7$$\"+MVM%>%F-$!+DqOT;F-7$$\"+QS*HB %F-$!+A:fJF-7$$\"+G \"ypM%F-$!+AUgA?F-7$$\"+M=h(Q%F-$!+:96P@F-7$$\"+`'4eU%F-$!+)[X5D#F-7$$ \"+C&QOY%F-$!+d+zqBF-7$$\"+([(\\,XF-$!+HVk)\\#F-7$$\"+L76SXF-$!+'GZ)QE F-7$$\"+d6/\"e%F-$!+$pm1!GF-7$$\"+!*)p&=YF-$!+?m5kHF-7$$\"+obhbYF-$!+9 %4S9$F-7$$\"+(y;_p%F-$!+/&R[O$F-7$$\"+*zJZt%F-$!+`i>JOF-7$$\"+M`Y_ZF-$ !+El5vPF-7$$\"+o))>qZF-$!+iFfWRF-7$$\"+'\\o-y%F-$!+G<\")eSF-7$$\"+D\"Q .z%F-$!+!p\\]>%F-7$$\"+SHP&z%F-$!+M@kwUF-7$$\"+axS+[F-$!+ " 0 "" {MPLTEXT 1 0 828 "C := (x,y) -> (y-x)/(y+x): hh := 0.01: numsteps := 3 75: x0 := 1: y0 := 1:\nmatrix([[`slope field: `,C(x,y)],[`initial po int: `,``(x0,y0)],[`step width: `,hh],\n[`no. of steps: `,numsteps ]]);``;\nmthds := [`Sharp-Verner scheme with `*``(c[7]=19/20),`scheme \+ with `*``(c[7]=24/25),`scheme with `*``(c[7]=35/36),`scheme with `*``( c[7]=63/64),`scheme with `*``(c[7]=125/126),`Papakostas' scheme with ` *``(c[7]=199/200)]: errs := []: vals := []:\nDigits := 25:\nfor ct to \+ 6 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),sqrt(sm/numpts)];\nend do:\nDigits := 10:\nlinalg[transpose]( [mthds,evalf(errs)]);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#K%'matrixG6#7 &7$%0slope~field:~~~G*&,&%\"yG\"\"\"%\"xG!\"\"F,,&F-F,F+F,F.7$%0initia l~point:~G-%!G6$F,F,7$%/step~width:~~~G$F,!\"#7$%1no.~of~steps:~~~G\"$ v$Q)pprint766\"" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%!G" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#K%'matrixG6#7(7$*&%:Sharp-Verner~scheme~with~G\"\" \"-%!G6#/&%\"cG6#\"\"(#\"#>\"#?F*$\"+ggU_m!#A7$*&%-scheme~with~GF*-F,6 #/F/#\"#C\"#DF*$\"+LkL7HF87$*&F;F*-F,6#/F/#\"#N\"#OF*$\"+!eTZE#F87$*&F ;F*-F,6#/F/#\"#j\"#kF*$\"+1&*[D9F87$*&F;F*-F,6#/F/#\"$D\"\"$E\"F*$\"+9 La@&*!#B7$*&%9Papakostas'~scheme~with~GF*-F,6#/F/#\"$*>\"$+#F*$\"+)Q' \\d!)F\\oQ)pprint776\"" }}}{PARA 0 "" 0 "" {TEXT -1 2 " " }}{PARA 0 " " 0 "" {TEXT -1 30 "The following code constructs " }{TEXT 260 20 "num erical procedures" }{TEXT -1 56 " for solutions based on each of the R unge-Kutta schemes." }}{PARA 0 "" 0 "" {TEXT -1 75 "The error in the v alue obtained by each of the methods at the point where " }{XPPEDIT 18 0 "x = 4.749;" "6#/%\"xG-%&FloatG6$\"%\\Z!\"$" }{TEXT -1 16 " is a lso given." }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 691 "C := (x,y) -> (y-x)/(y+x): hh := 0.01: numsteps := 375: x0 := 1: y0 := 1:\nmatrix([ [`slope field: `,C(x,y)],[`initial point: `,``(x0,y0)],[`step width: `,hh],\n[`no. of steps: `,numsteps]]);``;\nmthds := [`Sharp-Verne r scheme with `*``(c[7]=19/20),`scheme with `*``(c[7]=24/25),`scheme w ith `*``(c[7]=35/36),`scheme with `*``(c[7]=63/64),`scheme with `*``(c [7]=125/126),`Papakostas' scheme with `*``(c[7]=199/200)]: errs := []: \nDigits := 30:\nfor ct to 6 do\n cn_RK6_||ct := RK6_||ct(C(x,y),x,y ,x0,y0,hh,numsteps,true);\nend do:\nxx := 4.749: cxx := evalf(phi(xx)) :\nfor ct to 6 do\n errs := [op(errs),abs(cn_RK6_||ct(xx)-cxx)];\nen d do:\nDigits := 10:\nlinalg[transpose]([mthds,evalf(errs)]);" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#K%'matrixG6#7&7$%0slope~field:~~~G*&,& %\"yG\"\"\"%\"xG!\"\"F,,&F-F,F+F,F.7$%0initial~point:~G-%!G6$F,F,7$%/s tep~width:~~~G$F,!\"#7$%1no.~of~steps:~~~G\"$v$Q)pprint786\"" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%!G" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#K%'ma trixG6#7(7$*&%:Sharp-Verner~scheme~with~G\"\"\"-%!G6#/&%\"cG6#\"\"(#\" #>\"#?F*$\"+=,Is\")!#@7$*&%-scheme~with~GF*-F,6#/F/#\"#C\"#DF*$\"+tjjl NF87$*&F;F*-F,6#/F/#\"#N\"#OF*$\"+WXomFF87$*&F;F*-F,6#/F/#\"#j\"#kF*$ \"+#oz?t\"F87$*&F;F*-F,6#/F/#\"$D\"\"$E\"F*$\"+,))p[6F87$*&%9Papakosta s'~scheme~with~GF*-F,6#/F/#\"$*>\"$+#F*$\"+&[;;o*!#AQ)pprint796\"" }}} {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 4 "The " } {TEXT 260 22 "root mean square error" }{TEXT -1 20 " over the interval " }{XPPEDIT 18 0 "[1, 4.75];" "6#7$\"\"\"-%&FloatG6$\"$v%!\"#" } {TEXT -1 82 " of each Runge-Kutta method is estimated as follows usin g the special procedure " }{TEXT 0 5 "NCint" }{TEXT -1 98 " to perfo rm numerical integration by the 7 point Newton-Cotes method over 200 e qual subintervals." }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 452 "mthds := [`Sharp-Verner scheme with `*``(c[7]=19/20),`scheme with `*``(c[7] =24/25),`scheme with `*``(c[7]=35/36),`scheme with `*``(c[7]=63/64),`s cheme with `*``(c[7]=125/126),`Papakostas' scheme with `*``(c[7]=199/2 00)]: errs := []:\nDigits := 20:\nfor ct to 6 do\n sm := NCint(('phi '(x)-'cn_RK6_||ct'(x))^2,x=1..4.75,adaptive=false,numpoints=7,factor=2 00);\n errs := [op(errs),sqrt(sm/5)];\nend do:\nDigits := 10:\nlinal g[transpose]([mthds,evalf(errs)]);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6# K%'matrixG6#7)7$*&%9Papakostas'~scheme~with~G\"\"\"-%!G6#/&%\"cG6#\"\" (#\"$*>\"$+#F*$\"+cH$)zP!#B7$*&%Ka~modification~of~Papakostas'~scheme~ with~GF*F+F*$\"+s3l\"f$F87$*&%/a~scheme~with~GF*-F,6#/F/#\"$C$\"$D$F*$ \"+:5h5HF87$*&F@F*-F,6#/F/#\"$v&\"$w&F*$\"+yt^'e#F87$*&F@F*-F,6#/F/#\" %G<\"%HGF87$*&%GVerner's~\"most~efficient\"~scheme~with~GF*-F,6#/ F/#\"%**>\"%+?F*$\"+Jf:#*RF8Q)pprint496\"" }}}{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 0 586 "evalf[30](plot(['cn_RK6_1'( x)-'phi'(x),'cn_RK6_2'(x)-'phi'(x),'cn_RK6_3'(x)-'phi'(x),'cn_RK6_4'(x )-'phi'(x),\n'cn_RK6_5'(x)-'phi'(x),'cn_RK6_6'(x)-'phi'(x)],x=1..3.75, -3.9e-18..8.1e-19,\nfont=[HELVETICA,9],color=[COLOR(RGB,.95,0,.95),COL OR(RGB,.5,0,.95),COLOR(RGB,.95,.45,0),COLOR(RGB,.2,.85,0),\nCOLOR(RGB, 0,.3,.95),COLOR(RGB,.95,0,.2)],legend=[`Sharp-Verner scheme with c[7]= 19/20`,`scheme with c[7]=24/25`,`scheme with c[7]=35/36`,`scheme with \+ c[7]=63/64`,`scheme with c[7]=125/126`,`Papakostas' scheme with c[7]=1 99/200`],title=`error curves for 8 stage order 6 Runge-Kutta methods`) );" }}{PARA 13 "" 1 "" {GLPLOT2D 1041 597 597 {PLOTDATA 2 "6,-%'CURVES G6%7jn7$$\"\"\"\"\"!$F*F*7$$\"?LLLLLLLL$eR'>'F27$$\" ?LLLLLLLLeRiYzH7F/$!-(=YkSh(F27$$\"?nmmmmmm;a8-qb)G\"F/$!-q\"*)*4b()F2 7$$\"?LLLLLLL$3xJ@PIM\"F/$!-9#)z'yv*F27$$\"?+++++++]P4;$[%*R\"F/$!.^3l t]1\"F27$$\"?LLLLLLL$3F%f()yd9F/$!.JSMb^9\"F27$$\"?+++++++](=U5Uf^\"F/ $!.iep(3@7F27$$\"?nmmmmmmm;Hn*fdd\"F/$!.0%p#GKH\"F27$$\"?LLLLLLLLektvW G;F/$!.aNXNJN\"F27$$\"?++++++++]PR8w(o\"F/$!.6MxYsT\"F27$$\"?++++++++] 7`'=tu\"F/$!.N[)Q`z9F27$$\"?++++++++]igJr/=F/$!.4nD8%Q:F27$$\"?LLLLLLL $3F>(G$o&=F/$!.iT8%R!f\"F27$$\"?nmmmmmmm;z\\#3)=>F/$!.9V6CAl\"F27$$\"? nmmmmmmmm;C&48(>F/$!.:o_.Yq\"F27$$\"?+++++++]PM()4QK?F/$!.:!>,4lF27$$\"?nmmmmmmmT5!>d6E#F/$!.;VI6()*> F27$$\"?nmmmmmm;aQQAF:BF/$!.'f$*Qbc?F27$$\"?LLLLLLLLekGEktBF/$!.[._)>? 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