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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 28 "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#\"\"$" }{TEXT -1 3 ", " }{XPPEDIT 18 0 "c[4]" "6# &%\"cG6#\"\"%" }{TEXT -1 3 ", " }{XPPEDIT 18 0 "c[5]" "6#&%\"cG6#\"\" &" }{TEXT -1 3 ", " }{XPPEDIT 18 0 "c[6]" "6#&%\"cG6#\"\"'" }{TEXT -1 3 ", " }{XPPEDIT 18 0 "c[7]" "6#&%\"cG6#\"\"(" }{TEXT -1 3 ", " } {XPPEDIT 18 0 "c[8]=1" "6#/&%\"cG6#\"\")\"\"\"" }{TEXT -1 3 ", " } {XPPEDIT 18 0 "c[9]=1" "6#/&%\"cG6#\"\"*\"\"\"" }{TEXT -1 11 " and le t " }{TEXT 278 2 "Id" }{TEXT -1 32 " be the 9 by 9 identity matrix. " }}{PARA 0 "" 0 "" {TEXT -1 5 "Let " }{XPPEDIT 18 0 "[c]" "6#7#%\"cG " }{TEXT -1 38 " be the row vector whose entries are " }{XPPEDIT 18 0 "c[1]=0" "6#/&%\"cG6#\"\"\"\"\"!" }{TEXT -1 3 ", " }{XPPEDIT 18 0 " c[2]" "6#&%\"cG6#\"\"#" }{TEXT -1 3 ", " }{XPPEDIT 18 0 "c[3]" "6#&% \"cG6#\"\"$" }{TEXT -1 3 ", " }{XPPEDIT 18 0 "c[4]" "6#&%\"cG6#\"\"% " }{TEXT -1 3 ", " }{XPPEDIT 18 0 "c[5]" "6#&%\"cG6#\"\"&" }{TEXT -1 3 ", " }{XPPEDIT 18 0 "c[6]" "6#&%\"cG6#\"\"'" }{TEXT -1 3 ", " } {XPPEDIT 18 0 "c[7]" "6#&%\"cG6#\"\"(" }{TEXT -1 3 ", " }{XPPEDIT 18 0 "c[8]=1" "6#/&%\"cG6#\"\")\"\"\"" }{TEXT -1 3 ", " }{XPPEDIT 18 0 " c[9]=1" "6#/&%\"cG6#\"\"*\"\"\"" }{TEXT -1 2 ". " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 238 "A := matrix ([seq([seq(a[i,j],j=1..i-1),seq(0,j=i..9)],i=1..9)]):\nB := matrix([[s eq(b[i],i=1..9)]]):\n`B*` := matrix([[seq(`b*`[i],i=1..9)]]):\nId := l inalg[diag](1$9):\nC := linalg[diag](seq(c[i],i=1..9)):\nc_ := matrix( [seq([c[i]],i=1..9)]):" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 49 "#================================================" } }{PARA 0 "" 0 "" {TEXT -1 34 "(1) The first order condition is: " }} {PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "[b]^T*([C]-Id)*[A]*(C -c[4]*Id)*(C-c[5]*Id)*[c] = Int((x-1)*Int((t-c[4])*(t-c[5])*t,t = 0 .. x),x = 0 .. 1);" "6#/*.)7#%\"bG%\"TG\"\"\",&7#%\"CGF)%#IdG!\"\"F)7#% \"AGF),&F,F)*&&%\"cG6#\"\"%F)F-F)F.F),&F,F)*&&F46#\"\"&F)F-F)F.F)7#F4F )-%$IntG6$*&,&%\"xGF)F)F.F)-F>6$*(,&%\"tGF)&F46#F6F.F),&FGF)&F46#F;F.F )FGF)/FG;\"\"!FBF)/FB;FOF)" }{TEXT -1 2 ". " }}{PARA 0 "" 0 "" {TEXT -1 37 "In summation form this condition is: " }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "Sum(Sum(b[i]*(c[i]-1)*a[i,j-1],i = j .. 7)*(c[j-1]-c[4])*(c[j-1]-c[5])*c[j-1],j = 2 .. 7) = -1/120+c[4]/60+c[ 5]/60-c[4]*c[5]/24;" "6#/-%$SumG6$**-F%6$*(&%\"bG6#%\"iG\"\"\",&&%\"cG 6#F.F/F/!\"\"F/&%\"aG6$F.,&%\"jGF/F/F4F//F.;F9\"\"(F/,&&F26#,&F9F/F/F4 F/&F26#\"\"%F4F/,&&F26#,&F9F/F/F4F/&F26#\"\"&F4F/&F26#,&F9F/F/F4F//F9; \"\"#F<,**&F/F/\"$?\"F4F4*&&F26#FCF/\"#gF4F/*&&F26#FJF/FWF4F/*(&F26#FC F/&F26#FJF/\"#CF4F4" }{TEXT -1 2 ". " }}{PARA 0 "" 0 "" {TEXT -1 1 " \+ " }}{PARA 0 "" 0 "" {TEXT -1 13 "We can omit " }{XPPEDIT 18 0 "j=2" " 6#/%\"jG\"\"#" }{TEXT -1 33 " in the outer summation because " } {XPPEDIT 18 0 "c[1]=0" "6#/&%\"cG6#\"\"\"\"\"!" }{TEXT -1 15 ", we can omit " }{XPPEDIT 18 0 "j=3" "6#/%\"jG\"\"$" }{TEXT -1 11 " because \+ " }{XPPEDIT 18 0 "a[i,2]=0" "6#/&%\"aG6$%\"iG\"\"#\"\"!" }{TEXT -1 8 ", for " }{XPPEDIT 18 0 "i=3" "6#/%\"iG\"\"$" }{TEXT -1 25 " . . 7, \+ and we can omit " }{XPPEDIT 18 0 "j=5" "6#/%\"jG\"\"&" }{TEXT -1 7 " \+ and " }{XPPEDIT 18 0 "j=6" "6#/%\"jG\"\"'" }{TEXT -1 34 " because o f obvious zero factors." }}{PARA 0 "" 0 "" {TEXT -1 13 "We can omit \+ " }{XPPEDIT 18 0 "j=4" "6#/%\"jG\"\"%" }{TEXT -1 30 " because (it tur ns out that) " }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "Sum( b[i]*(c[i]-1)*a[i,3],i=4..7)=0" "6#/-%$SumG6$*(&%\"bG6#%\"iG\"\"\",&&% \"cG6#F+F,F,!\"\"F,&%\"aG6$F+\"\"$F,/F+;\"\"%\"\"(\"\"!" }{TEXT -1 2 " . " }}{PARA 0 "" 0 "" {TEXT -1 24 "This gives the relation " }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "b[7]*(c[7]-1)*a[7,6]*(c[6]- c[4])*(c[6]-c[5])*c[6]=-1/120+c[4]/60+c[5]/60-c[4]*c[5]/24" "6#/*.&%\" bG6#\"\"(\"\"\",&&%\"cG6#F(F)F)!\"\"F)&%\"aG6$F(\"\"'F),&&F,6#F2F)&F,6 #\"\"%F.F),&&F,6#F2F)&F,6#\"\"&F.F)&F,6#F2F),**&F)F)\"$?\"F.F.*&&F,6#F 8F)\"#gF.F)*&&F,6#F>F)FGF.F)*(&F,6#F8F)&F,6#F>F)\"#CF.F." }{TEXT -1 1 "." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 61 "#-- ----------------------------------------------------------" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 62 "Int( (x-1)*Int((t-c[4])*(t-c[5])*t,t=0..x),x=0..1):\n%=value(%);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/-%$IntG6$*&,&%\"xG\"\"\"F*!\"\"F*-F%6$*(,&% \"tGF*&%\"cG6#\"\"%F+F*,&F0F*&F26#\"\"&F+F*F0F*/F0;\"\"!F)F*/F);F;F*,* #F*\"$?\"F+*&#F*\"#gF*F1F*F**&FBF*F6F*F**&#F*\"#CF**&F6F*F1F*F*F+" }}} {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 107 "We can u se the Sharp-Verner scheme to provide a numerical check for the order \+ condition in the matrix form." }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 159 "evalm(B &* (C-Id) &* A &* (C-c[4]*Id) &* (C-c[5]*Id) &* c_)[1,1 ]=int((x-1)*int((t-c[4])*(t-c[5])*t,t=0..x),x=0..1):\nsubs(\{c[1]=0,c[ 8]=1,c[9]=1\},%);\nsubs(ee,%);" }}{PARA 12 "" 1 "" {XPPMATH 20 "6#/,(* *,,*(&%\"bG6#\"\"$\"\"\",&&%\"cGF*F,F,!\"\"F,&%\"aG6$F+\"\"#F,F,*(&F)6 #\"\"%F,,&&F/F7F,F,F0F,&F26$F8F4F,F,*(&F)6#\"\"&F,,&F,F0&F/F?F,F,&F26$ F@F4F,F,*(&F)6#\"\"'F,,&F,F0&F/FGF,F,&F26$FHF4F,F,*(&F)6#\"\"(F,,&F,F0 &F/FOF,F,&F26$FPF4F,F,F,,&&F/6#F4F,F:F0F,,&FVF,FBF0F,FVF,F,**,**(F6F,F 9F,&F26$F8F+F,F,*(F>F,FAF,&F26$F@F+F,F,*(FFF,FIF,&F26$FHF+F,F,*(FNF,FQ F,&F26$FPF+F,F,F,,&F.F,F:F0F,,&F.F,FBF0F,F.F,F,*.FNF,FQF,&F26$FPFHF,,& F:F0FJF,F,,&FBF0FJF,F,FJF,F,,*#F,\"$?\"F0*&#F,\"#gF,F:F,F,*&F\\pF,FBF, F,*&#F,\"#CF,*&FBF,F:F,F,F0" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/#!\"\" \"%+=F$" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 80 "We can also make a numerical check of the order condition in the s ummation form." }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 151 "add(add(b [i]*(c[i]-1)*a[i,j-1],i=j..7)*(c[j-1]-c[4])*(c[j-1]-c[5])*c[j-1],j=2.. 7)=\n -1/120+c[4]/60+c[5]/60-c[4]*c[5]/24:\nsubs(\{c[1]=0\},%);\nsubs (ee,%);" }}{PARA 12 "" 1 "" {XPPMATH 20 "6#/,(**,,*(&%\"bG6#\"\"$\"\" \",&&%\"cGF*F,F,!\"\"F,&%\"aG6$F+\"\"#F,F,*(&F)6#\"\"%F,,&&F/F7F,F,F0F ,&F26$F8F4F,F,*(&F)6#\"\"&F,,&F,F0&F/F?F,F,&F26$F@F4F,F,*(&F)6#\"\"'F, ,&F,F0&F/FGF,F,&F26$FHF4F,F,*(&F)6#\"\"(F,,&F,F0&F/FOF,F,&F26$FPF4F,F, F,,&&F/6#F4F,F:F0F,,&FVF,FBF0F,FVF,F,**,**(F6F,F9F,&F26$F8F+F,F,*(F>F, FAF,&F26$F@F+F,F,*(FFF,FIF,&F26$FHF+F,F,*(FNF,FQF,&F26$FPF+F,F,F,,&F.F ,F:F0F,,&F.F,FBF0F,F.F,F,*.FNF,FQF,&F26$FPFHF,,&F:F0FJF,F,,&FBF0FJF,F, FJF,F,,*#F,\"$?\"F0*&#F,\"#gF,F:F,F,*&F\\pF,FBF,F,*&#F,\"#CF,*&FBF,F:F ,F,F0" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/#!\"\"\"%+=F$" }}}{PARA 0 " " 0 "" {TEXT -1 0 "" }{TEXT 277 0 "" }{TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 84 "After the omissions that we can make in the outer summati on we obtain the following." }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 132 "add(add(b[i]*(c[i]-1)*a[i,j-1],i=j..7)*(c[j-1]-c[4])*(c[j-1]-c[5] )*c[j-1],j=[7])=\n -1/120+c[4]/60+c[5]/60-c[4]*c[5]/24;\nsubs(ee,%); " }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/*.&%\"bG6#\"\"(\"\"\",&F)!\"\"&% \"cGF'F)F)&%\"aG6$F(\"\"'F),&&F-6#\"\"%F+&F-6#F1F)F),&&F-6#\"\"&F+F6F) F)F6F),*#F)\"$?\"F+*&#F)\"#gF)F3F)F)*&F@F)F9F)F)*&#F)\"#CF)*&F9F)F3F)F )F+" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/#!\"\"\"%+=F$" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 57 "#=================== =====================================" }}{PARA 0 "" 0 "" {TEXT -1 35 " (2) The second order condition is: " }}{PARA 256 "" 0 "" {TEXT -1 1 " \+ " }{XPPEDIT 18 0 "[`b*`]^T*[A]*(C-c[4]*Id)*(C-c[5]*Id)*[c] = Int(Int(( t-c[4])*(t-c[5])*t,t = 0 .. x),x = 0 .. 1);" "6#/*,)7#%#b*G%\"TG\"\"\" 7#%\"AGF),&%\"CGF)*&&%\"cG6#\"\"%F)%#IdGF)!\"\"F),&F-F)*&&F06#\"\"&F)F 3F)F4F)7#F0F)-%$IntG6$-F<6$*(,&%\"tGF)&F06#F2F4F),&FBF)&F06#F9F4F)FBF) /FB;\"\"!%\"xG/FK;FJF)" }{TEXT -1 2 ". " }}{PARA 0 "" 0 "" {TEXT -1 37 "In summation form this condition is: " }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "Sum(Sum(`b*`[i]*a[i,j-1],i = j .. 7)*(c[j-1]- c[4])*(c[j-1]-c[5])*c[j-1],j = 2 .. 9) = 1/20-c[4]/12-c[5]/12+c[4]*c[5 ]/6;" "6#/-%$SumG6$**-F%6$*&&%#b*G6#%\"iG\"\"\"&%\"aG6$F.,&%\"jGF/F/! \"\"F//F.;F4\"\"(F/,&&%\"cG6#,&F4F/F/F5F/&F;6#\"\"%F5F/,&&F;6#,&F4F/F/ F5F/&F;6#\"\"&F5F/&F;6#,&F4F/F/F5F//F4;\"\"#\"\"*,**&F/F/\"#?F5F/*&&F; 6#F@F/\"#7F5F5*&&F;6#FGF/FUF5F5*(&F;6#F@F/&F;6#FGF/\"\"'F5F/" }{TEXT -1 2 ". " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 13 "We can omit " }{XPPEDIT 18 0 "j=2" "6#/%\"jG\"\"#" }{TEXT -1 33 " in the outer summation because " }{XPPEDIT 18 0 "c[1]=0" "6#/&%\"cG6 #\"\"\"\"\"!" }{TEXT -1 15 ", we can omit " }{XPPEDIT 18 0 "j=3" "6#/ %\"jG\"\"$" }{TEXT -1 11 " because " }{XPPEDIT 18 0 "a[i,2]=0" "6#/& %\"aG6$%\"iG\"\"#\"\"!" }{TEXT -1 8 ", for " }{XPPEDIT 18 0 "i=3" "6 #/%\"iG\"\"$" }{TEXT -1 25 " . . 7, and we can omit " }{XPPEDIT 18 0 "j=5" "6#/%\"jG\"\"&" }{TEXT -1 7 " and " }{XPPEDIT 18 0 "j=6" "6#/% \"jG\"\"'" }{TEXT -1 34 " because of obvious zero factors." }}{PARA 0 "" 0 "" {TEXT -1 13 "We can omit " }{XPPEDIT 18 0 "j=4" "6#/%\"jG\" \"%" }{TEXT -1 11 " because " }}{PARA 256 "" 0 "" {TEXT -1 1 " " } {XPPEDIT 18 0 "Sum(`b*`[i]*a[i,3],i = 4 .. 9) = 0;" "6#/-%$SumG6$*&&%# b*G6#%\"iG\"\"\"&%\"aG6$F+\"\"$F,/F+;\"\"%\"\"*\"\"!" }{TEXT -1 2 ". \+ " }}{PARA 0 "" 0 "" {TEXT -1 24 "This gives the relation " }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "Sum(Sum(`b*`[i]*a[i,j-1],i \+ = j .. 7)*(c[j-1]-c[4])*(c[j-1]-c[5])*c[j-1],j = 7 .. 9) = 1/20-c[4]/1 2-c[5]/12+c[4]*c[5]/6;" "6#/-%$SumG6$**-F%6$*&&%#b*G6#%\"iG\"\"\"&%\"a G6$F.,&%\"jGF/F/!\"\"F//F.;F4\"\"(F/,&&%\"cG6#,&F4F/F/F5F/&F;6#\"\"%F5 F/,&&F;6#,&F4F/F/F5F/&F;6#\"\"&F5F/&F;6#,&F4F/F/F5F//F4;F8\"\"*,**&F/F /\"#?F5F/*&&F;6#F@F/\"#7F5F5*&&F;6#FGF/FTF5F5*(&F;6#F@F/&F;6#FGF/\"\"' F5F/" }{TEXT -1 2 ". " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 61 "#----------------------------------------------------- -------" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 56 "Int(Int((t-c[4])*(t-c[5])*t,t=0..x),x=0..1):\n%=value (%);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/-%$IntG6$-F%6$*(,&%\"tG\"\"\" &%\"cG6#\"\"%!\"\"F,,&F+F,&F.6#\"\"&F1F,F+F,/F+;\"\"!%\"xG/F9;F8F,,*#F ,\"#?F,*&#F,\"#7F,F-F,F1*&#F,FAF,F3F,F1*&#F,\"\"'F,*&F-F,F3F,F,F," }}} {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 107 "We can u se the Sharp-Verner scheme to provide a numerical check for the order \+ condition in the matrix form." }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 139 "evalm(`B*` &* A &* (C-c[4]*Id) &* (C-c[5]*Id) &* c_)[1,1]=int(i nt((t-c[4])*(t-c[5])*t,t=0..x),x=0..1):\nsubs(\{c[1]=0,c[8]=1\},%);\ns ubs(ee,%);" }}{PARA 12 "" 1 "" {XPPMATH 20 "6#/,,**,0*&&%#b*G6#\"\"$\" \"\"&%\"aG6$F+\"\"#F,F,*&&F)6#\"\"%F,&F.6$F4F0F,F,*&&F)6#\"\"&F,&F.6$F :F0F,F,*&&F)6#\"\"'F,&F.6$F@F0F,F,*&&F)6#\"\"(F,&F.6$FFF0F,F,*&&F)6#\" \")F,&F.6$FLF0F,F,*&&F)6#\"\"*F,&F.6$FRF0F,F,F,,&&%\"cG6#F0F,&FWF3!\" \"F,,&FVF,&FWF9FZF,FVF,F,**,.*&F2F,&F.6$F4F+F,F,*&F8F,&F.6$F:F+F,F,*&F >F,&F.6$F@F+F,F,*&FDF,&F.6$FFF+F,F,*&FJF,&F.6$FLF+F,F,*&FPF,&F.6$FRF+F ,F,F,,&&FWF*F,FYFZF,,&F\\pF,FfnFZF,F\\pF,F,**,(*&FDF,&F.6$FFF@F,F,*&FJ F,&F.6$FLF@F,F,*&FPF,&F.6$FRF@F,F,F,,&FYFZ&FWF?F,F,,&FfnFZFjpF,F,FjpF, F,**,&*&FJF,&F.6$FLFFF,F,*&FPF,&F.6$FRFFF,F,F,,&&FWFEF,FYFZF,,&FeqF,Ff nFZF,FeqF,F,**FPF,&F.6$FRFLF,,&F,F,FYFZF,,&F,F,FfnFZF,F,,*#F,\"#?F,*&# F,\"#7F,FYF,FZ*&#F,FarF,FfnF,FZ*&#F,F@F,*&FfnF,FYF,F,F," }}{PARA 11 " " 1 "" {XPPMATH 20 "6#/#\"\"\"\"$]\"F$" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 80 "We can also make a numerical check o f the order condition in the summation form." }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 152 "add(add(`b*`[i]*a[i,j-1],i=j..9)*(c[j-1]-c[4])*(c [j-1]-c[5])*c[j-1],j=[$2..9])=\n 1/20-c[4]/12-c[5]/12+c[4]*c[5]/6:\ns ubs(\{c[1]=0,c[8]=1\},%);\nsubs(ee,%);" }}{PARA 12 "" 1 "" {XPPMATH 20 "6#/,,**,0*&&%#b*G6#\"\"$\"\"\"&%\"aG6$F+\"\"#F,F,*&&F)6#\"\"%F,&F. 6$F4F0F,F,*&&F)6#\"\"&F,&F.6$F:F0F,F,*&&F)6#\"\"'F,&F.6$F@F0F,F,*&&F)6 #\"\"(F,&F.6$FFF0F,F,*&&F)6#\"\")F,&F.6$FLF0F,F,*&&F)6#\"\"*F,&F.6$FRF 0F,F,F,,&&%\"cG6#F0F,&FWF3!\"\"F,,&FVF,&FWF9FZF,FVF,F,**,.*&F2F,&F.6$F 4F+F,F,*&F8F,&F.6$F:F+F,F,*&F>F,&F.6$F@F+F,F,*&FDF,&F.6$FFF+F,F,*&FJF, &F.6$FLF+F,F,*&FPF,&F.6$FRF+F,F,F,,&&FWF*F,FYFZF,,&F\\pF,FfnFZF,F\\pF, F,**,(*&FDF,&F.6$FFF@F,F,*&FJF,&F.6$FLF@F,F,*&FPF,&F.6$FRF@F,F,F,,&FYF Z&FWF?F,F,,&FfnFZFjpF,F,FjpF,F,**,&*&FJF,&F.6$FLFFF,F,*&FPF,&F.6$FRFFF ,F,F,,&&FWFEF,FYFZF,,&FeqF,FfnFZF,FeqF,F,**FPF,&F.6$FRFLF,,&F,F,FYFZF, ,&F,F,FfnFZF,F,,*#F,\"#?F,*&#F,\"#7F,FYF,FZ*&#F,FarF,FfnF,FZ*&#F,F@F,* &FfnF,FYF,F,F," }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/#\"\"\"\"$]\"F$" }} }{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 84 "After th e omissions that we can make in the outer summation we obtain the foll owing." }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 124 "add(add(`b*`[i]*a [i,j-1],i=j..9)*(c[j-1]-c[4])*(c[j-1]-c[5])*c[j-1],j=7..9)=\n 1/20-c[ 4]/12-c[5]/12+c[4]*c[5]/6;\nsubs(ee,%);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/,(**,(*&&%#b*G6#\"\"(\"\"\"&%\"aG6$F+\"\"'F,F,*&&F)6#\"\")F,&F. 6$F4F0F,F,*&&F)6#\"\"*F,&F.6$F:F0F,F,F,,&&%\"cG6#\"\"%!\"\"&F?6#F0F,F, ,&&F?6#\"\"&FBFCF,F,FCF,F,**,&*&F2F,&F.6$F4F+F,F,*&F8F,&F.6$F:F+F,F,F, ,&&F?F*F,F>FBF,,&FRF,FFFBF,FRF,F,*,F8F,&F.6$F:F4F,,&&F?F3F,F>FBF,,&FXF ,FFFBF,FXF,F,,*#F,\"#?F,*&#F,\"#7F,F>F,FB*&#F,FinF,FFF,FB*&#F,F0F,*&FF F,F>F,F,F," }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/#\"\"\"\"$]\"F$" }}} {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 61 "#-------- ----------------------------------------------------" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}}{PARA 0 "" 0 "" {TEXT -1 60 "#----- ------------------------------------------------------" }}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 43 "Tsitouras' algorithm .. Sharp-Verner sche me" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 118 "In this subsection we illustrate the algorithm of Ch. Tsitouras by using it to construct a scheme of Sharp and Verner." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 161 "See: Completely Imbedded Runge-Kutta Pairs, by P. W. Sharp and J. H. Verner,\n SIAM Jou rnal on Numerical Analysis, Vol. 31, No. 4. (Aug., 1994), page 1185." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 22 "We spec ify the nodes: " }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "c[ 2] = 1/12;" "6#/&%\"cG6#\"\"#*&\"\"\"F)\"#7!\"\"" }{TEXT -1 3 ", " } {XPPEDIT 18 0 "c[4] = 1/5;" "6#/&%\"cG6#\"\"%*&\"\"\"F)\"\"&!\"\"" } {TEXT -1 3 ", " }{XPPEDIT 18 0 "c[5] = 8/15;" "6#/&%\"cG6#\"\"&*&\"\" )\"\"\"\"#:!\"\"" }{TEXT -1 3 ", " }{XPPEDIT 18 0 "c[6] = 2/3;" "6#/& %\"cG6#\"\"'*&\"\"#\"\"\"\"\"$!\"\"" }{TEXT -1 3 ", " }{XPPEDIT 18 0 "c[7] = 19/20;" "6#/&%\"cG6#\"\"(*&\"#>\"\"\"\"#?!\"\"" }{TEXT -1 3 ", " }{XPPEDIT 18 0 "c[8]=1" "6#/&%\"cG6#\"\")\"\"\"" }{TEXT -1 2 ", " }}{PARA 0 "" 0 "" {TEXT -1 31 "the zero linking coefficients: " }} {PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "a[4,2]=0" "6#/&%\"aG6 $\"\"%\"\"#\"\"!" }{TEXT -1 3 ", " }{XPPEDIT 18 0 "a[5,2]=0" "6#/&%\" aG6$\"\"&\"\"#\"\"!" }{TEXT -1 3 ", " }{XPPEDIT 18 0 "a[6,2]=0" "6#/& %\"aG6$\"\"'\"\"#\"\"!" }{TEXT -1 3 ", " }{XPPEDIT 18 0 "a[7,2]=0" "6 #/&%\"aG6$\"\"(\"\"#\"\"!" }{TEXT -1 3 ", " }{XPPEDIT 18 0 "a[8,2]=0 " "6#/&%\"aG6$\"\")\"\"#\"\"!" }{TEXT -1 2 ", " }}{PARA 0 "" 0 "" {TEXT -1 22 "and the zero weights: " }}{PARA 256 "" 0 "" {TEXT -1 1 " \+ " }{XPPEDIT 18 0 "b[2]=0" "6#/&%\"bG6#\"\"#\"\"!" }{TEXT -1 3 ", " } {XPPEDIT 18 0 "b[3]=0" "6#/&%\"bG6#\"\"$\"\"!" }{TEXT -1 3 ", " } {XPPEDIT 18 0 "`b*`[2]=0" "6#/&%#b*G6#\"\"#\"\"!" }{TEXT -1 3 ", " } {XPPEDIT 18 0 "`b*`[3]=0" "6#/&%#b*G6#\"\"$\"\"!" }{TEXT -1 3 ", " } {XPPEDIT 18 0 "`b*`[8]=0" "6#/&%#b*G6#\"\")\"\"!" }{TEXT -1 1 "." }} {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 90 "We shall \+ obtain expressions for all the coefficients of the scheme in terms of \+ the nodes " }{XPPEDIT 18 0 "c[2]" "6#&%\"cG6#\"\"#" }{TEXT -1 3 ", \+ " }{XPPEDIT 18 0 "c[4]" "6#&%\"cG6#\"\"%" }{TEXT -1 3 ", " }{XPPEDIT 18 0 "c[5]" "6#&%\"cG6#\"\"&" }{TEXT -1 3 ", " }{XPPEDIT 18 0 "c[6]" "6#&%\"cG6#\"\"'" }{TEXT -1 7 " and " }{XPPEDIT 18 0 "c[7]" "6#&%\"c G6#\"\"(" }{TEXT -1 1 "." }}{PARA 0 "" 0 "" {TEXT -1 10 "The node " } {XPPEDIT 18 0 "c[3]" "6#&%\"cG6#\"\"$" }{TEXT -1 27 " does not appear because " }{XPPEDIT 18 0 "c[3]=2/3" "6#/&%\"cG6#\"\"$*&\"\"#\"\"\"F' !\"\"" }{TEXT -1 1 " " }{XPPEDIT 18 0 "c[4]" "6#&%\"cG6#\"\"%" }{TEXT -1 2 ". " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 146 "e1 := \{c[2]=1/12,c[4]=1/5,c[5]=8/15,c[6]=2/3,c[7]=1 9/20,c[8]=1,c[9]=1,\n b[2]=0,b[3]=0,`b*`[2]=0,`b*`[3]=0,`b*`[8 ]=0,seq(a[i,2]=0,i=4..8)\}:" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 15 "" 0 "" {TEXT 279 6 "Step 1" }{TEXT -1 1 " " }}{PARA 0 "" 0 "" {TEXT -1 33 "We use the quadrature conditions:" }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "Sum(b[i],i=1..8)=1" "6#/-%$SumG6$&%\"bG 6#%\"iG/F*;\"\"\"\"\")F-" }{TEXT -1 15 ", " }{XPPEDIT 18 0 "Sum(b[i]*c[i]^(k-1),i = 2 .. 8) = 1/k;" "6#/-%$SumG6$*&&%\"bG6#%\"i G\"\"\")&%\"cG6#F+,&%\"kGF,F,!\"\"F,/F+;\"\"#\"\")*&F,F,F2F3" }{TEXT -1 7 ", " }{XPPEDIT 18 0 "k = 2;" "6#/%\"kG\"\"#" }{TEXT -1 8 " . . 6, " }}{PARA 0 "" 0 "" {TEXT -1 21 "to find the weights " } {XPPEDIT 18 0 "b[1]" "6#&%\"bG6#\"\"\"" }{TEXT -1 3 ", " }{XPPEDIT 18 0 "b[4];" "6#&%\"bG6#\"\"%" }{TEXT -1 3 ", " }{XPPEDIT 18 0 "b[5]; " "6#&%\"bG6#\"\"&" }{TEXT -1 3 ", " }{XPPEDIT 18 0 "b[6]" "6#&%\"bG6 #\"\"'" }{TEXT -1 3 ", " }{XPPEDIT 18 0 "b[7]" "6#&%\"bG6#\"\"(" } {TEXT -1 3 ", " }{XPPEDIT 18 0 "b[8]" "6#&%\"bG6#\"\")" }{TEXT -1 1 " ." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 142 "[add(b[i],i=1..8)=1,seq(add(b[i]*c[i]^(j-1),i=2..8)= 1/j,j=2..6)]:\ne2 := solve(\{op(subs(e1,%))\},\{seq(b[i],i=[1,$4..8]) \}):\ne3 := `union`(e1,e2):" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 49 "The weights of the order 6 scheme are as follow s." }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 31 "seq(b[i]=subs(e3,b[i]) ,i=1..8);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6*/&%\"bG6#\"\"\"#\"#`\"$7* /&F%6#\"\"#\"\"!/&F%6#\"\"$F//&F%6#\"\"%#\"\"&\"#;/&F%6#F9#\"#F\"$7\"/ &F%6#\"\"'#F?\"$O\"/&F%6#\"\"(#\"$c#\"$p*/&F%6#\"\")#!#D\"$O$" }}} {PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 2 "e3" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 300 "e3 := \{a[8 ,2] = 0, c[9] = 1, c[8] = 1, a[5,2] = 0, a[4,2] = 0, a[6,2] = 0, a[7,2 ] = 0, b[3] = 0, b[2] = 0, `b*`[3] = 0, `b*`[2] = 0, b[1] = 53/912, b[ 8] = -25/336, b[7] = 256/969, b[6] = 27/136, `b*`[8] = 0, c[4] = 1/5, \+ c[2] = 1/12, c[6] = 2/3, b[5] = 27/112, b[4] = 5/16, c[5] = 8/15, c[7] = 19/20\}:" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 15 "" 0 "" {TEXT -1 1 " " }{TEXT 280 6 "Step 2" }{TEXT -1 1 " " }}{PARA 0 " " 0 "" {TEXT -1 35 "We use the stage-order conditions: " }}{PARA 256 " " 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "a[3,2]*c[2] = 1/2" "6#/*&&%\"aG6 $\"\"$\"\"#\"\"\"&%\"cG6#F)F**&F*F*F)!\"\"" }{TEXT -1 1 " " }{XPPEDIT 18 0 "c[3]^2" "6#*$&%\"cG6#\"\"$\"\"#" }{TEXT -1 3 ", " }}{PARA 256 " " 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "Sum(a[4,j]*c[j],j = 2 .. 3) = 1/ 2;" "6#/-%$SumG6$*&&%\"aG6$\"\"%%\"jG\"\"\"&%\"cG6#F,F-/F,;\"\"#\"\"$* &F-F-F3!\"\"" }{TEXT -1 1 " " }{XPPEDIT 18 0 "c[4]^2;" "6#*$&%\"cG6#\" \"%\"\"#" }{TEXT -1 15 ", " }{XPPEDIT 18 0 "Sum(a[4,j]*c[ j]^2,j = 2 .. 3) = 1/3;" "6#/-%$SumG6$*&&%\"aG6$\"\"%%\"jG\"\"\"*$&%\" cG6#F,\"\"#F-/F,;F2\"\"$*&F-F-F5!\"\"" }{TEXT -1 1 " " }{XPPEDIT 18 0 "c[4]^3;" "6#*$&%\"cG6#\"\"%\"\"$" }{TEXT -1 1 " " }}{PARA 0 "" 0 "" {TEXT -1 9 "to find " }{XPPEDIT 18 0 "c[3]" "6#&%\"cG6#\"\"$" }{TEXT -1 3 ", " }{XPPEDIT 18 0 "a[3,2]" "6#&%\"aG6$\"\"$\"\"#" }{TEXT -1 7 " and " }{XPPEDIT 18 0 "a[4,3];" "6#&%\"aG6$\"\"%\"\"$" }{TEXT -1 2 ". " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" } }{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 269 "[a[3,2]*c[2] = c[3]^2/2,se q(add(a[4,j]*c[j]^(k-1),j=2..3)=c[4]^k/k,k=[2,3])]:\neqns1 := subs(e3, %):\neqns1[1];\neqns1[2];\neqns1[3];\ne4 := solve(\{op(eqns1)\},\{a[3, 2],c[3],a[4,3]\}):\ne5 := `union`(e3,e4):\n``;\nc[3]=subs(e5,c[3]);\na [3,2]=subs(e5,a[3,2]);\na[4,3]=subs(e5,a[4,3]);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/,$*&#\"\"\"\"#7F'&%\"aG6$\"\"$\"\"#F'F',$*&#F'F-F'*$)& %\"cG6#F,F-F'F'F'" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/*&&%\"aG6$\"\"% \"\"$\"\"\"&%\"cG6#F)F*#F*\"#]" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/*&& %\"aG6$\"\"%\"\"$\"\"\")&%\"cG6#F)\"\"#F*#F*\"$v$" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%!G" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/&%\"cG6#\"\"$# \"\"#\"#:" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/&%\"aG6$\"\"$\"\"##\"\") \"#v" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/&%\"aG6$\"\"%\"\"$#F(\"#?" }} }{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 2 "e5" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 343 "e5 := \{a[8 ,2] = 0, c[9] = 1, c[8] = 1, a[5,2] = 0, a[4,2] = 0, a[6,2] = 0, a[7,2 ] = 0, b[3] = 0, b[2] = 0, `b*`[3] = 0, `b*`[2] = 0, a[3,2] = 8/75, c[ 3] = 2/15, a[4,3] = 3/20, b[1] = 53/912, b[8] = -25/336, b[7] = 256/96 9, b[6] = 27/136, `b*`[8] = 0, c[4] = 1/5, c[2] = 1/12, c[6] = 2/3, b[ 5] = 27/112, b[4] = 5/16, c[5] = 8/15, c[7] = 19/20\}:" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}} {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 15 "" 0 "" {TEXT -1 1 " " } {TEXT 281 6 "Step 3" }{TEXT -1 1 " " }}{PARA 0 "" 0 "" {TEXT -1 35 "We use the stage-order conditions: " }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "Sum(a[5,j]*c[j],j = 2 .. 4) = 1/2;" "6#/-%$SumG6$*&&% \"aG6$\"\"&%\"jG\"\"\"&%\"cG6#F,F-/F,;\"\"#\"\"%*&F-F-F3!\"\"" }{TEXT -1 1 " " }{XPPEDIT 18 0 "c[5]^2" "6#*$&%\"cG6#\"\"&\"\"#" }{TEXT -1 15 ", " }{XPPEDIT 18 0 "Sum(a[5,j]*c[j]^(2),j=2..4)=1/3" "6#/-%$SumG6$*&&%\"aG6$\"\"&%\"jG\"\"\"*$&%\"cG6#F,\"\"#F-/F,;F2\"\"%* &F-F-\"\"$!\"\"" }{TEXT -1 1 " " }{XPPEDIT 18 0 "c[5]^3" "6#*$&%\"cG6# \"\"&\"\"$" }{TEXT -1 1 " " }}{PARA 0 "" 0 "" {TEXT -1 9 "to find " } {XPPEDIT 18 0 "a[5,3]" "6#&%\"aG6$\"\"&\"\"$" }{TEXT -1 7 " and " } {XPPEDIT 18 0 "a[5,4]" "6#&%\"aG6$\"\"&\"\"%" }{TEXT -1 1 "." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 209 "[ seq(add(a[5,j]*c[j]^(k-1),j=2..4)=c[5]^k/k,k=[2,3])]:\neqns2 := subs(e 5,%):\neqns2[1];\neqns2[2];\ne6 := solve(\{op(eqns2)\},\{a[5,3],a[5,4] \}):\ne7 := `union`(e5,subs(e4,e6)):\n``;\nseq(a[5,j]=subs(e7,a[5,j]), j=[3,4]);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/,&*&#\"\"#\"#:\"\"\"&%\" aG6$\"\"&\"\"$F)F)*&#F)F-F)&F+6$F-\"\"%F)F)#\"#K\"$D#" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/,&*&#\"\"%\"$D#\"\"\"&%\"aG6$\"\"&\"\"$F)F)*&#F) \"#DF)&F+6$F-F'F)F)#\"$7&\"&D,\"" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%! G" }}{PARA 11 "" 1 "" {XPPMATH 20 "6$/&%\"aG6$\"\"&\"\"$#!$7\"\"#X/&F% 6$F'\"\"%#\"#k\"#F" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 2 "e7 " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 377 "e7 := \{a[8,2] = 0, c[9] = 1, c[8] = 1, a[5,2] = 0, a[4,2] = \+ 0, a[6,2] = 0, a[7,2] = 0, b[3] = 0, b[2] = 0, `b*`[3] = 0, `b*`[2] = \+ 0, a[5,3] = -112/45, a[5,4] = 64/27, a[3,2] = 8/75, c[3] = 2/15, a[4,3 ] = 3/20, b[1] = 53/912, b[8] = -25/336, b[7] = 256/969, b[6] = 27/136 , `b*`[8] = 0, c[4] = 1/5, c[2] = 1/12, c[6] = 2/3, b[5] = 27/112, b[4 ] = 5/16, c[5] = 8/15, c[7] = 19/20\}:" }{TEXT -1 0 "" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}} {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 15 "" 0 "" {TEXT -1 1 " " } {TEXT 282 6 "Step 4" }{TEXT -1 1 " " }}{PARA 0 "" 0 "" {TEXT -1 41 "We use the (column) simplifying condition" }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "Sum(b[i]*a[i,j],i=j..8)=b[j]*(1-c[j])" "6#/-%$Su mG6$*&&%\"bG6#%\"iG\"\"\"&%\"aG6$F+%\"jGF,/F+;F0\"\")*&&F)6#F0F,,&F,F, &%\"cG6#F0!\"\"F," }{TEXT -1 7 " for " }{XPPEDIT 18 0 "j=7" "6#/%\"j G\"\"(" }{TEXT -1 1 "," }}{PARA 0 "" 0 "" {TEXT -1 8 "that is," }} {PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "b[8]*a[8,7] = b[7]*(1 -c[7])" "6#/*&&%\"bG6#\"\")\"\"\"&%\"aG6$F(\"\"(F)*&&F&6#F-F),&F)F)&% \"cG6#F-!\"\"F)" }{TEXT -1 2 ", " }}{PARA 0 "" 0 "" {TEXT -1 9 "to fin d " }{XPPEDIT 18 0 "a[8,7]" "6#&%\"aG6$\"\")\"\"(" }{TEXT -1 1 "." }} {PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 113 "b[8]*a[8,7] = b[7]*(1-c[7]);\nsubs(e7,%);\ne8 := solve(\{%\},\{a[ 8,7]\}):\ne9 := `union`(e7,e8):\na[8,7]=subs(e9,a[8,7]);" }}{PARA 11 " " 1 "" {XPPMATH 20 "6#/*&&%\"bG6#\"\")\"\"\"&%\"aG6$F(\"\"(F)*&&F&6#F- F),&F)F)&%\"cGF0!\"\"F)" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/,$*&#\"#D \"$O$\"\"\"&%\"aG6$\"\")\"\"(F)!\"\"#\"#k\"%X[" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/&%\"aG6$\"\")\"\"(#!%or\"&v.%" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 2 "e9" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 399 "e9 := \{a[8,2] = 0, c[9] = \+ 1, c[8] = 1, a[5,2] = 0, a[4,2] = 0, a[6,2] = 0, a[7,2] = 0, b[3] = 0, b[2] = 0, `b*`[3] = 0, `b*`[2] = 0, a[5,3] = -112/45, a[5,4] = 64/27, a[8,7] = -7168/40375, a[3,2] = 8/75, c[3] = 2/15, a[4,3] = 3/20, b[1] = 53/912, b[8] = -25/336, b[7] = 256/969, b[6] = 27/136, `b*`[8] = 0, c[4] = 1/5, c[2] = 1/12, c[6] = 2/3, b[5] = 27/112, b[4] = 5/16, c[5] = 8/15, c[7] = 19/20\}:" }{TEXT -1 0 "" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}}{PARA 0 "" 0 " " {TEXT -1 0 "" }}{PARA 15 "" 0 "" {TEXT -1 1 " " }{TEXT 283 6 "Step 5 " }{TEXT -1 1 " " }}{PARA 0 "" 0 "" {TEXT -1 41 "We use the \"alternat ive\" order condition " }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "[b]^T*([C]-Id)*[A]*(C-c[4]*Id)*(C-c[5]*Id)*[c] = Int((x-1)*Int(( t-c[4])*(t-c[5])*t,t = 0 .. x),x = 0 .. 1);" "6#/*.)7#%\"bG%\"TG\"\"\" ,&7#%\"CGF)%#IdG!\"\"F)7#%\"AGF),&F,F)*&&%\"cG6#\"\"%F)F-F)F.F),&F,F)* &&F46#\"\"&F)F-F)F.F)7#F4F)-%$IntG6$*&,&%\"xGF)F)F.F)-F>6$*(,&%\"tGF)& F46#F6F.F),&FGF)&F46#F;F.F)FGF)/FG;\"\"!FBF)/FB;FOF)" }{TEXT -1 2 ". \+ " }}{PARA 0 "" 0 "" {TEXT -1 39 "This condition amounts to the relatio n " }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "b[7]*(c[7]-1)*a [7,6]*(c[6]-c[4])*(c[6]-c[5])*c[6]=-1/120+c[4]/60+c[5]/60-c[4]*c[5]/24 " "6#/*.&%\"bG6#\"\"(\"\"\",&&%\"cG6#F(F)F)!\"\"F)&%\"aG6$F(\"\"'F),&& F,6#F2F)&F,6#\"\"%F.F),&&F,6#F2F)&F,6#\"\"&F.F)&F,6#F2F),**&F)F)\"$?\" F.F.*&&F,6#F8F)\"#gF.F)*&&F,6#F>F)FGF.F)*(&F,6#F8F)&F,6#F>F)\"#CF.F." }{TEXT -1 1 "," }}{PARA 0 "" 0 "" {TEXT -1 27 "which can be used to fi nd " }{XPPEDIT 18 0 "a[7,6]" "6#&%\"aG6$\"\"(\"\"'" }{TEXT -1 2 ". " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 179 "b[7]*(c[7]-1)*a[7,6]*(c[6]-c[4])*(c[6]-c[5])*c[6] = -1/120+c[4] /60+c[5]/60-c[4]*c[5]/24;\nsubs(e9,%);\ne10 := solve(\{%\},a[7,6]):\ne 11 := `union`(e9,e10):\n``;\na[7,6]=subs(e11,a[7,6]);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/*.&%\"bG6#\"\"(\"\"\",&&%\"cGF'F)F)!\"\"F)&%\"aG6 $F(\"\"'F),&&F,6#F1F)&F,6#\"\"%F-F),&F3F)&F,6#\"\"&F-F)F3F),*#F)\"$?\" F-*&#F)\"#gF)F5F)F)*&F@F)F9F)F)*&#F)\"#CF)*&F5F)F9F)F)F-" }}{PARA 11 " " 1 "" {XPPMATH 20 "6#/,$*&#\"%#z\"\"(v.F$\"\"\"&%\"aG6$\"\"(\"\"'F)! \"\"#F/\"%+=" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%!G" }}{PARA 11 "" 1 " " {XPPMATH 20 "6#/&%\"aG6$\"\"(\"\"'#\"&NX\"\"&OV\"" }}}{PARA 0 "" 0 " " {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}} {SECT 1 {PARA 4 "" 0 "" {TEXT -1 3 "e11" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 422 "e11 := \{a[8,2] = 0, c [9] = 1, c[8] = 1, a[5,2] = 0, a[4,2] = 0, a[6,2] = 0, a[7,2] = 0, b[3 ] = 0, b[2] = 0, `b*`[3] = 0, `b*`[2] = 0, a[5,3] = -112/45, a[5,4] = \+ 64/27, a[8,7] = -7168/40375, a[7,6] = 14535/14336, a[3,2] = 8/75, c[3] = 2/15, a[4,3] = 3/20, b[1] = 53/912, b[8] = -25/336, b[7] = 256/969, b[6] = 27/136, `b*`[8] = 0, c[4] = 1/5, c[2] = 1/12, c[6] = 2/3, b[5] = 27/112, b[4] = 5/16, c[5] = 8/15, c[7] = 19/20\}:" }{TEXT -1 0 "" } }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 15 "" 0 "" {TEXT -1 1 " " }{TEXT 284 6 "Step 6" }{TEXT -1 1 " " }}{PARA 0 "" 0 "" {TEXT -1 41 "We use the (column) simplifying condition" }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "Sum(b[i]*a[i,j],i=j..8)=b[j]*(1-c[j])" "6#/-%$SumG6$*&&%\"bG6#%\"iG\"\"\"&%\"aG6$F+%\"jGF,/F+;F0\"\")*&&F)6#F 0F,,&F,F,&%\"cG6#F0!\"\"F," }{TEXT -1 7 " for " }{XPPEDIT 18 0 "j = \+ 6;" "6#/%\"jG\"\"'" }{TEXT -1 1 "," }}{PARA 0 "" 0 "" {TEXT -1 8 "that is," }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "b[7]*a[7,6]+b [8]*a[8,6] = b[6]*(1-c[6])" "6#/,&*&&%\"bG6#\"\"(\"\"\"&%\"aG6$F)\"\"' F*F**&&F'6#\"\")F*&F,6$F2F.F*F**&&F'6#F.F*,&F*F*&%\"cG6#F.!\"\"F*" } {TEXT -1 2 ", " }}{PARA 0 "" 0 "" {TEXT -1 9 "to find " }{XPPEDIT 18 0 "a[8,6];" "6#&%\"aG6$\"\")\"\"'" }{TEXT -1 1 "." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 133 "add(b[i]*a[ i,6],i=7..8)=b[6]*(1-c[6]):\nsubs(e11,%);\ne12 := solve(\{%\},\{a[8,6] \}):\ne13 := `union`(e11,e12):\n``;\na[8,6]=subs(e13,a[8,6]);" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#/,&#\"#:\"#c\"\"\"*&#\"#D\"$O$F(&%\"aG 6$\"\")\"\"'F(!\"\"#\"\"*\"$O\"" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%!G " }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/&%\"aG6$\"\")\"\"'#\"%_6\"$D%" }} }{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 3 "e13" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 441 "e13 := \{a[ 8,2] = 0, c[9] = 1, c[8] = 1, a[5,2] = 0, a[4,2] = 0, a[6,2] = 0, a[7, 2] = 0, b[3] = 0, b[2] = 0, `b*`[3] = 0, `b*`[2] = 0, a[5,3] = -112/45 , a[5,4] = 64/27, a[8,7] = -7168/40375, a[7,6] = 14535/14336, a[3,2] = 8/75, c[3] = 2/15, a[4,3] = 3/20, b[1] = 53/912, b[8] = -25/336, b[7] = 256/969, b[6] = 27/136, `b*`[8] = 0, c[4] = 1/5, c[2] = 1/12, c[6] \+ = 2/3, b[5] = 27/112, b[4] = 5/16, a[8,6] = 1152/425, c[5] = 8/15, c[7 ] = 19/20\}:" }{TEXT -1 0 "" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 15 "" 0 "" {TEXT 285 6 "Step 7" }{TEXT -1 1 " " }} {PARA 0 "" 0 "" {TEXT -1 23 "We find the 6 weights " }{XPPEDIT 18 0 " `b*`[1];" "6#&%#b*G6#\"\"\"" }{TEXT -1 3 ", " }{XPPEDIT 18 0 "`b*`[4] ;" "6#&%#b*G6#\"\"%" }{TEXT -1 3 ", " }{XPPEDIT 18 0 "`b*`[5];" "6#&% #b*G6#\"\"&" }{TEXT -1 3 ", " }{XPPEDIT 18 0 "`b*`[6];" "6#&%#b*G6#\" \"'" }{TEXT -1 2 ", " }{XPPEDIT 18 0 "`b*`[7];" "6#&%#b*G6#\"\"(" } {TEXT -1 3 ", " }{XPPEDIT 18 0 "`b*`[9];" "6#&%#b*G6#\"\"*" }{TEXT -1 39 ", by using the 5 quadrature conditions:" }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "Sum(`b*`[i],i = 1 .. 9) = 1;" "6#/-%$Su mG6$&%#b*G6#%\"iG/F*;\"\"\"\"\"*F-" }{TEXT -1 15 ", " } {XPPEDIT 18 0 "Sum(`b*`[i]*c[i]^(k-1),i = 2 .. 9) = 1/k;" "6#/-%$SumG6 $*&&%#b*G6#%\"iG\"\"\")&%\"cG6#F+,&%\"kGF,F,!\"\"F,/F+;\"\"#\"\"**&F,F ,F2F3" }{TEXT -1 7 ", " }{XPPEDIT 18 0 "k = 2;" "6#/%\"kG\"\"#" } {TEXT -1 8 " . . 5, " }}{PARA 0 "" 0 "" {TEXT -1 48 "together with the \"alternative\" order condition " }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "[`b*`]^T*[A]*(C-c[4]*Id)*(C-c[5]*Id)*[c] = Int(Int((t- c[4])*(t-c[5])*t,t = 0 .. x),x = 0 .. 1);" "6#/*,)7#%#b*G%\"TG\"\"\"7# %\"AGF),&%\"CGF)*&&%\"cG6#\"\"%F)%#IdGF)!\"\"F),&F-F)*&&F06#\"\"&F)F3F )F4F)7#F0F)-%$IntG6$-F<6$*(,&%\"tGF)&F06#F2F4F),&FBF)&F06#F9F4F)FBF)/F B;\"\"!%\"xG/FK;FJF)" }{TEXT -1 2 ". " }}{PARA 0 "" 0 "" {TEXT -1 24 " This gives the relation " }}{PARA 256 "" 0 "" {TEXT -1 1 " " } {XPPEDIT 18 0 "Sum(Sum(`b*`[i]*a[i,j-1],i = j .. 7)*(c[j-1]-c[4])*(c[j -1]-c[5])*c[j-1],j = 7 .. 9) = -1/120+c[4]/60+c[5]/60-c[4]*c[5]/24;" " 6#/-%$SumG6$**-F%6$*&&%#b*G6#%\"iG\"\"\"&%\"aG6$F.,&%\"jGF/F/!\"\"F//F .;F4\"\"(F/,&&%\"cG6#,&F4F/F/F5F/&F;6#\"\"%F5F/,&&F;6#,&F4F/F/F5F/&F;6 #\"\"&F5F/&F;6#,&F4F/F/F5F//F4;F8\"\"*,**&F/F/\"$?\"F5F5*&&F;6#F@F/\"# gF5F/*&&F;6#FGF/FTF5F/*(&F;6#F@F/&F;6#FGF/\"#CF5F5" }{TEXT -1 2 ". " } }{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 41 "Expandin g the left-hand side gives . . . " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 76 "add(add(`b*`[i]*a[i,j-1],i=j ..9)*(c[j-1]-c[4])*(c[j-1]-c[5])*c[j-1],j=7..9);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#,(**,(*&&%#b*G6#\"\"(\"\"\"&%\"aG6$F*\"\"'F+F+*&&F(6#\" \")F+&F-6$F3F/F+F+*&&F(6#\"\"*F+&F-6$F9F/F+F+F+,&&%\"cG6#\"\"%!\"\"&F> 6#F/F+F+,&&F>6#\"\"&FAFBF+F+FBF+F+**,&*&F1F+&F-6$F3F*F+F+*&F7F+&F-6$F9 F*F+F+F+,&F=FA&F>F)F+F+,&FQF+FEFAF+FQF+F+*,F7F+&F-6$F9F3F+,&&F>F2F+F=F AF+,&FWF+FEFAF+FWF+F+" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT 260 4 "Note" }{TEXT -1 61 ": Since the last equation involv es the linking coefficients " }{XPPEDIT 18 0 "a[9,j]" "6#&%\"aG6$\"\" *%\"jG" }{TEXT -1 3 ", " }{XPPEDIT 18 0 "j=1" "6#/%\"jG\"\"\"" } {TEXT -1 43 " . . 8, we need to make the substitutions " }{XPPEDIT 18 0 "a[9,j]=b[j]" "6#/&%\"aG6$\"\"*%\"jG&%\"bG6#F(" }{TEXT -1 3 ", \+ " }{XPPEDIT 18 0 "j=1" "6#/%\"jG\"\"\"" }{TEXT -1 7 " . . 8." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 60 "First we set up the six equations for the six weights . . . " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 406 "`quad_eqs*` := [add(`b*`[i],i=1..9)=1,seq(add(`b*`[i]*c[i]^(j-1),i=2..9)=1/j,j=2. .5)]:\n`ord_eq*` := add(add(`b*`[i]*a[i,j-1],i=j..9)*(c[j-1]-c[4])*(c[ j-1]-c[5])*c[j-1],j=7..9)=\n 1/20-c[4]/12-c[5]/12+c[4] *c[5]/6:\nwt_eqs := [seq(a[9,i]=b[i],i=1..8)]:\n`eqns*` := simplify(su bs(e13,[op(`quad_eqs*`),subs(wt_eqs,`ord_eq*`)])):\nnops(`eqns*`);\nin dets(`eqns*`) minus \{c[4],c[5],c[6],c[7]\};\nnops(%);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#\"\"'" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#<(&%#b*G6 #\"\"'&F%6#\"\"*&F%6#\"\"(&F%6#\"\"&&F%6#\"\"%&F%6#\"\"\"" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#\"\"'" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }} {PARA 0 "" 0 "" {TEXT -1 30 " . . . and then we solve them." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 22 "in folevel[solve] := 4:" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 76 "e14 := solve(\{op(`eqns*`)\}):\ne15 := `union`(e13,e14):\ninfolevel[solve ] := 0:" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 49 "The weights of the order 5 scheme are as follows." }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 38 "seq(`b*`[i]=subs(e15,`b*`[i]),i=1..9);" } }{PARA 11 "" 1 "" {XPPMATH 20 "6+/&%#b*G6#\"\"\"#\"$<'\"&W4\"/&F%6#\" \"#\"\"!/&F%6#\"\"$F//&F%6#\"\"%#\"$T#\"$c(/&F%6#\"\"&#\"#p\"$?$/&F%6# \"\"'#\"$N%\"%/>/&F%6#\"\"(#\"&/.\"\"&0O%/&F%6#\"\")F//&F%6#\"\"*#!\" \"\"#=" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 3 "e15" }} {PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 559 "e15 := \{a[8,2] = 0, c[9] = 1, c[8] = 1, a[5,2] = 0, a[4,2] = 0, \+ a[6,2] = 0, a[7,2] = 0, b[3] = 0, b[2] = 0, `b*`[3] = 0, `b*`[2] = 0, \+ a[5,3] = -112/45, a[5,4] = 64/27, a[8,7] = -7168/40375, a[7,6] = 14535 /14336, a[3,2] = 8/75, c[3] = 2/15, a[4,3] = 3/20, `b*`[7] = 10304/436 05, b[1] = 53/912, b[8] = -25/336, b[7] = 256/969, b[6] = 27/136, `b*` [8] = 0, c[4] = 1/5, c[2] = 1/12, c[6] = 2/3, b[5] = 27/112, b[4] = 5/ 16, `b*`[5] = 69/320, `b*`[4] = 241/756, `b*`[6] = 435/1904, `b*`[1] = 617/10944, a[8,6] = 1152/425, c[5] = 8/15, c[7] = 19/20, `b*`[9] = -1 /18\}:" }{TEXT -1 0 "" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}}{PARA 0 "" 0 "" {TEXT -1 0 " " }}{PARA 15 "" 0 "" {TEXT 286 6 "Step 8" }{TEXT -1 1 " " }}{PARA 0 " " 0 "" {TEXT -1 22 "We use the relations: " }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "Sum(`b*`[i]*a[i,3],i=4..8)=0" "6#/-%$SumG6$*& &%#b*G6#%\"iG\"\"\"&%\"aG6$F+\"\"$F,/F+;\"\"%\"\")\"\"!" }{TEXT -1 6 " , " }{XPPEDIT 18 0 "Sum(b[i]*a[i,3],i=4..8)=b[3]*(1-c[3])" "6#/-%$ SumG6$*&&%\"bG6#%\"iG\"\"\"&%\"aG6$F+\"\"$F,/F+;\"\"%\"\")*&&F)6#F0F,, &F,F,&%\"cG6#F0!\"\"F," }{TEXT -1 6 ", " }{XPPEDIT 18 0 "Sum(b[i]* (c[i]-1)*a[i,3],i=4..7)=0" "6#/-%$SumG6$*(&%\"bG6#%\"iG\"\"\",&&%\"cG6 #F+F,F,!\"\"F,&%\"aG6$F+\"\"$F,/F+;\"\"%\"\"(\"\"!" }{TEXT -1 2 ", " } }{PARA 0 "" 0 "" {TEXT -1 9 "to find " }{XPPEDIT 18 0 "a[6,3]" "6#&% \"aG6$\"\"'\"\"$" }{TEXT -1 3 ", " }{XPPEDIT 18 0 "a[7,3]" "6#&%\"aG6 $\"\"(\"\"$" }{TEXT -1 7 " and " }{XPPEDIT 18 0 "a[8,3]" "6#&%\"aG6$ \"\")\"\"$" }{TEXT -1 1 "." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 272 "[add(`b*`[i]*a[i,3],i=4..8)=0,add( b[i]*a[i,3],i=4..8)=b[3]*(1-c[3]),add(b[i]*(c[i]-1)*a[i,3],i=4..7)=0]: \neqns3 := subs(e15,%):\neqns3[1];\neqns3[2];\neqns3[3];\ne16 := solve (\{op(eqns3)\},\{a[6,3],a[7,3],a[8,3]\}):\ne17 := `union`(e15,e16):\n` `;\nseq(a[i,3]=subs(e17,a[i,3]),i=6..8);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/,(#\"&>B\"\"&+_#!\"\"*&#\"$N%\"%/>\"\"\"&%\"aG6$\"\"'\"\"$F-F-* &#\"&/.\"\"&0O%F-&F/6$\"\"(F2F-F-\"\"!" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/,*#\"$x\"\"$?$!\"\"*&#\"#F\"$O\"\"\"\"&%\"aG6$\"\"'\"\"$F-F-*&# \"$c#\"$p*F-&F/6$\"\"(F2F-F-*&#\"#D\"$O$F-&F/6$\"\")F2F-F(\"\"!" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#/,(#\"#(*\"$+%\"\"\"*&#\"\"*\"$O\"F(&% \"aG6$\"\"'\"\"$F(!\"\"*&#\"#k\"%X[F(&F.6$\"\"(F1F(F2\"\"!" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%!G" }}{PARA 11 "" 1 "" {XPPMATH 20 "6%/&%\" aG6$\"\"'\"\"$#\"%!)Q\"$j*/&F%6$\"\"(F(#!(\\2+\"\"'Sya/&F%6$\"\")F(#!% $y'\"%S@" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 3 "e17" }} {PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 625 "e17 := \{a[8,2] = 0, c[9] = 1, c[8] = 1, a[5,2] = 0, a[4,2] = 0, \+ a[6,2] = 0, a[7,2] = 0, b[3] = 0, b[2] = 0, `b*`[3] = 0, `b*`[2] = 0, \+ a[5,3] = -112/45, a[5,4] = 64/27, a[8,7] = -7168/40375, a[7,6] = 14535 /14336, a[3,2] = 8/75, c[3] = 2/15, a[4,3] = 3/20, a[8,3] = -6783/2140 , a[6,3] = 3880/963, `b*`[7] = 10304/43605, b[1] = 53/912, a[7,3] = -1 000749/547840, b[8] = -25/336, b[7] = 256/969, b[6] = 27/136, `b*`[8] \+ = 0, c[4] = 1/5, c[2] = 1/12, c[6] = 2/3, b[5] = 27/112, b[4] = 5/16, \+ `b*`[5] = 69/320, `b*`[4] = 241/756, `b*`[6] = 435/1904, `b*`[1] = 617 /10944, a[8,6] = 1152/425, c[5] = 8/15, c[7] = 19/20, `b*`[9] = -1/18 \}:" }{TEXT -1 0 "" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}}{PARA 0 "" 0 "" {TEXT -1 0 "" }} {PARA 15 "" 0 "" {TEXT 287 6 "Step 9" }{TEXT -1 1 " " }}{PARA 0 "" 0 " " {TEXT -1 34 "We use the stage-order conditions:" }}{PARA 256 "" 0 " " {TEXT -1 1 " " }{XPPEDIT 18 0 "Sum(a[6,j]*c[j],j=2..5)=1/2" "6#/-%$S umG6$*&&%\"aG6$\"\"'%\"jG\"\"\"&%\"cG6#F,F-/F,;\"\"#\"\"&*&F-F-F3!\"\" " }{TEXT -1 1 " " }{XPPEDIT 18 0 "c[6]^2" "6#*$&%\"cG6#\"\"'\"\"#" } {TEXT -1 6 ", " }{XPPEDIT 18 0 "Sum(a[6,j]*c[j]^2,j=2..5)=1/3" "6# /-%$SumG6$*&&%\"aG6$\"\"'%\"jG\"\"\"*$&%\"cG6#F,\"\"#F-/F,;F2\"\"&*&F- F-\"\"$!\"\"" }{TEXT -1 1 " " }{XPPEDIT 18 0 "c[6]^3" "6#*$&%\"cG6#\" \"'\"\"$" }{TEXT -1 2 ", " }}{PARA 0 "" 0 "" {TEXT -1 9 "to find " } {XPPEDIT 18 0 "a[6,4]" "6#&%\"aG6$\"\"'\"\"%" }{TEXT -1 7 " and " } {XPPEDIT 18 0 "a[6,5]" "6#&%\"aG6$\"\"'\"\"&" }{TEXT -1 1 "." }}{PARA 0 "" 0 "" {TEXT -1 1 " " }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 217 " [seq(add(a[6,j]*c[j]^(k-1),j=2..5)=c[6]^k/k,k=[2,3])]:\neqns4 := subs( e17,%):\neqns4[1];\neqns4[2];\ne18 := solve(\{op(eqns4)\},\{a[6,4],a[6 ,5]\}):\ne19 := `union`(e17,e18):\n``;\na[6,4]=subs(e19,a[6,4]),a[6,5] =subs(e19,a[6,5]);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/,(#\"%_:\"%*)G \"\"\"*&#F(\"\"&F(&%\"aG6$\"\"'\"\"%F(F(*&#\"\")\"#:F(&F-6$F/F+F(F(#\" \"#\"\"*" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/,(#\"%/J\"&NL%\"\"\"*&#F( \"#DF(&%\"aG6$\"\"'\"\"%F(F(*&#\"#k\"$D#F(&F-6$F/\"\"&F(F(#\"\")\"#\") " }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%!G" }}{PARA 11 "" 1 "" {XPPMATH 20 "6$/&%\"aG6$\"\"'\"\"%#!%c%)\"%*)G/&F%6$F'\"\"&#\"$<#\"$G%" }}} {PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 3 "e19" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 664 "e19 := \{a[ 8,2] = 0, c[9] = 1, c[8] = 1, a[5,2] = 0, a[4,2] = 0, a[6,2] = 0, a[7, 2] = 0, b[3] = 0, b[2] = 0, `b*`[3] = 0, `b*`[2] = 0, a[5,3] = -112/45 , a[5,4] = 64/27, a[8,7] = -7168/40375, a[7,6] = 14535/14336, a[3,2] = 8/75, c[3] = 2/15, a[4,3] = 3/20, a[8,3] = -6783/2140, a[6,3] = 3880/ 963, `b*`[7] = 10304/43605, b[1] = 53/912, a[7,3] = -1000749/547840, b [8] = -25/336, b[7] = 256/969, b[6] = 27/136, `b*`[8] = 0, c[4] = 1/5, c[2] = 1/12, c[6] = 2/3, b[5] = 27/112, b[4] = 5/16, `b*`[5] = 69/320 , `b*`[4] = 241/756, `b*`[6] = 435/1904, `b*`[1] = 617/10944, a[8,6] = 1152/425, c[5] = 8/15, c[7] = 19/20, a[6,5] = 217/428, a[6,4] = -8456 /2889, `b*`[9] = -1/18\}:" }{TEXT -1 0 "" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}}{PARA 0 "" 0 " " {TEXT -1 0 "" }}{PARA 15 "" 0 "" {TEXT 288 7 "Step 10" }{TEXT -1 1 " " }}{PARA 0 "" 0 "" {TEXT -1 34 "We use the stage-order conditions:" }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "Sum(a[7,j]*c[j],j = 2 .. 6) = 1/2;" "6#/-%$SumG6$*&&%\"aG6$\"\"(%\"jG\"\"\"&%\"cG6#F,F-/F ,;\"\"#\"\"'*&F-F-F3!\"\"" }{TEXT -1 1 " " }{XPPEDIT 18 0 "c[7]^2;" "6 #*$&%\"cG6#\"\"(\"\"#" }{TEXT -1 6 ", " }{XPPEDIT 18 0 "Sum(a[7,j] *c[j]^2,j = 2 .. 6) = 1/3;" "6#/-%$SumG6$*&&%\"aG6$\"\"(%\"jG\"\"\"*$& %\"cG6#F,\"\"#F-/F,;F2\"\"'*&F-F-\"\"$!\"\"" }{TEXT -1 1 " " } {XPPEDIT 18 0 "c[7]^3;" "6#*$&%\"cG6#\"\"(\"\"$" }{TEXT -1 2 ", " }} {PARA 0 "" 0 "" {TEXT -1 9 "to find " }{XPPEDIT 18 0 "a[7,4];" "6#&% \"aG6$\"\"(\"\"%" }{TEXT -1 7 " and " }{XPPEDIT 18 0 "a[7, 5];" "6#& %\"aG6$\"\"(\"\"&" }{TEXT -1 1 "." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 217 "[seq(add(a[7,j]*c[j]^(k-1), j=2..6)=c[7]^k/k,k=[2,3])]:\neqns5 := subs(e19,%):\neqns5[1];\neqns5[2 ];\ne20 := solve(\{op(eqns5)\},\{a[7,4],a[7,5]\}):\ne21 := `union`(e19 ,e20):\n``;\na[7,4]=subs(e21,a[7,4]),a[7,5]=subs(e21,a[7,5]);" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#/,(#\"(8-H)\")+W<>\"\"\"*&#F(\"\"&F(&% \"aG6$\"\"(\"\"%F(F(*&#\"\")\"#:F(&F-6$F/F+F(F(#\"$h$\"$+)" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/,(#\")8<8g\"*+!3Q9\"\"\"*&#F(\"#DF(&%\"aG6$ \"\"(\"\"%F(F(*&#\"#k\"$D#F(&F-6$F/\"\"&F(F(#\"%fo\"&+S#" }}{PARA 11 " " 1 "" {XPPMATH 20 "6#%!G" }}{PARA 11 "" 1 "" {XPPMATH 20 "6$/&%\"aG6$ \"\"(\"\"%#\"'h#>)\"')[$Q/&F%6$F'\"\"&#!'v6n\"'Wl()" }}}{PARA 0 "" 0 " " {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}} {SECT 1 {PARA 4 "" 0 "" {TEXT -1 3 "e21" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 664 "e19 := \{a[8,2] = 0, c [9] = 1, c[8] = 1, a[5,2] = 0, a[4,2] = 0, a[6,2] = 0, a[7,2] = 0, b[3 ] = 0, b[2] = 0, `b*`[3] = 0, `b*`[2] = 0, a[5,3] = -112/45, a[5,4] = \+ 64/27, a[8,7] = -7168/40375, a[7,6] = 14535/14336, a[3,2] = 8/75, c[3] = 2/15, a[4,3] = 3/20, a[8,3] = -6783/2140, a[6,3] = 3880/963, `b*`[7 ] = 10304/43605, b[1] = 53/912, a[7,3] = -1000749/547840, b[8] = -25/3 36, b[7] = 256/969, b[6] = 27/136, `b*`[8] = 0, c[4] = 1/5, c[2] = 1/1 2, c[6] = 2/3, b[5] = 27/112, b[4] = 5/16, `b*`[5] = 69/320, `b*`[4] = 241/756, `b*`[6] = 435/1904, `b*`[1] = 617/10944, a[8,6] = 1152/425, \+ c[5] = 8/15, c[7] = 19/20, a[6,5] = 217/428, a[6,4] = -8456/2889, `b*` [9] = -1/18\}:" }{TEXT -1 0 "" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 15 "" 0 "" {TEXT 289 7 "Step 11" }{TEXT -1 1 " " }} {PARA 0 "" 0 "" {TEXT -1 34 "We use the stage-order conditions:" }} {PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "Sum(a[8,j]*c[j],j = 2 .. 7) = 1/2;" "6#/-%$SumG6$*&&%\"aG6$\"\")%\"jG\"\"\"&%\"cG6#F,F-/F,; \"\"#\"\"(*&F-F-F3!\"\"" }{TEXT -1 1 " " }{XPPEDIT 18 0 "c[8]^2;" "6#* $&%\"cG6#\"\")\"\"#" }{TEXT -1 6 ", " }{XPPEDIT 18 0 "Sum(a[8,j]*c [j]^2,j = 2 .. 7) = 1/3;" "6#/-%$SumG6$*&&%\"aG6$\"\")%\"jG\"\"\"*$&% \"cG6#F,\"\"#F-/F,;F2\"\"(*&F-F-\"\"$!\"\"" }{TEXT -1 1 " " }{XPPEDIT 18 0 "c[8]^3;" "6#*$&%\"cG6#\"\")\"\"$" }{TEXT -1 2 ", " }}{PARA 0 "" 0 "" {TEXT -1 9 "to find " }{XPPEDIT 18 0 "a[8,4];" "6#&%\"aG6$\"\") \"\"%" }{TEXT -1 7 " and " }{XPPEDIT 18 0 "a[8,5];" "6#&%\"aG6$\"\") \"\"&" }{TEXT -1 1 "." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 217 "[seq(add(a[8,j]*c[j]^(k-1),j=2..7)=c[8]^ k/k,k=[2,3])]:\neqns6 := subs(e21,%):\neqns6[1];\neqns6[2];\ne22 := so lve(\{op(eqns6)\},\{a[8,4],a[8,5]\}):\ne23 := `union`(e21,e22):\n``;\n a[8,4]=subs(e23,a[8,4]),a[8,5]=subs(e23,a[8,5]);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/,(#\"'6E;\"']P8\"\"\"*&#F(\"\"&F(&%\"aG6$\"\")\"\"%F(F (*&#F/\"#:F(&F-6$F/F+F(F(#F(\"\"#" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/ ,(#\"'>7**\"(DJ+\"\"\"\"*&#F(\"#DF(&%\"aG6$\"\")\"\"%F(F(*&#\"#k\"$D#F (&F-6$F/\"\"&F(F(#F(\"\"$" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%!G" }} {PARA 11 "" 1 "" {XPPMATH 20 "6$/&%\"aG6$\"\")\"\"%#\"&c4\"\"%vE/&F%6$ F'\"\"&#!&$\\Q\"&vL\"" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 3 "e23" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 757 "e23 := \{a[8,2] = 0, c[9] = 1, c[8] = 1, a[5,2] = 0, a[4,2] = 0, a[6,2] = 0, a[7,2] = 0, b[3] = 0, b[2] = 0, `b*`[3] = 0, \+ `b*`[2] = 0, a[5,3] = -112/45, a[5,4] = 64/27, a[8,7] = -7168/40375, a [7,6] = 14535/14336, a[3,2] = 8/75, c[3] = 2/15, a[4,3] = 3/20, a[8,3] = -6783/2140, a[6,3] = 3880/963, `b*`[7] = 10304/43605, a[7,5] = -671 175/876544, a[7,4] = 819261/383488, b[1] = 53/912, a[7,3] = -1000749/5 47840, b[8] = -25/336, b[7] = 256/969, b[6] = 27/136, `b*`[8] = 0, c[4 ] = 1/5, c[2] = 1/12, c[6] = 2/3, b[5] = 27/112, b[4] = 5/16, `b*`[5] \+ = 69/320, `b*`[4] = 241/756, `b*`[6] = 435/1904, `b*`[1] = 617/10944, \+ a[8,6] = 1152/425, c[5] = 8/15, c[7] = 19/20, a[6,5] = 217/428, a[6,4] = -8456/2889, a[8,5] = -38493/13375, a[8,4] = 10956/2675, `b*`[9] = - 1/18\}:" }{TEXT -1 0 "" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}}{PARA 0 "" 0 "" {TEXT -1 0 " " }}{PARA 0 "" 0 "" {TEXT -1 31 "We use the row-sum conditions: " }} {PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "Sum(a[i,j],j=1..i-1)= c[i]" "6#/-%$SumG6$&%\"aG6$%\"iG%\"jG/F+;\"\"\",&F*F.F.!\"\"&%\"cG6#F* " }{TEXT -1 7 ", for " }{XPPEDIT 18 0 "i=2" "6#/%\"iG\"\"#" }{TEXT -1 8 " . . 8, " }}{PARA 0 "" 0 "" {TEXT -1 9 "to find " }{XPPEDIT 18 0 "a[2,1]" "6#&%\"aG6$\"\"#\"\"\"" }{TEXT -1 3 ", " }{XPPEDIT 18 0 "a [3,1]" "6#&%\"aG6$\"\"$\"\"\"" }{TEXT -1 3 ", " }{XPPEDIT 18 0 "a[4,1 ]" "6#&%\"aG6$\"\"%\"\"\"" }{TEXT -1 11 ", . . . , " }{XPPEDIT 18 0 " a[8,1]" "6#&%\"aG6$\"\")\"\"\"" }{TEXT -1 2 ". " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 119 "[seq(add(a[ i,j],j=1..i-1)=c[i],i=2..8)]:\ne24 := solve(\{op(subs(e23,%))\},\{seq( a[i,1],i=2..8)\}):\ne25 := `union`(e23,e24):" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 60 "The linking coefficients \+ in the first column are as follows." }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 36 "seq(a[i,1]=subs(e25,a[i,1]),i=2..8);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6)/&%\"aG6$\"\"#\"\"\"#F(\"#7/&F%6$\"\"$F(#F'\"#v/&F %6$\"\"%F(#F(\"#?/&F%6$\"\"&F(#\"#))\"$N\"/&F%6$\"\"'F(#!&\"*3\"\"&c: \"/&F%6$\"\"(F(#\"(6*=<\"(?FQ%/&F%6$\"\")F(#\"&`^)\"'+L?" }}}{PARA 0 " " 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 22 "We use the equatio ns: " }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "a[9,i]=b[i]" "6#/&%\"aG6$\"\"*%\"iG&%\"bG6#F(" }{TEXT -1 3 ", " }{XPPEDIT 18 0 "i= 1" "6#/%\"iG\"\"\"" }{TEXT -1 8 " . . 8, " }}{PARA 0 "" 0 "" {TEXT -1 9 "to find " }{XPPEDIT 18 0 "a[9,i]" "6#&%\"aG6$\"\"*%\"iG" }{TEXT -1 2 ", " }{XPPEDIT 18 0 " i=1" "6#/%\"iG\"\"\"" }{TEXT -1 7 " . . 8. " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 116 "wt_eqs := [seq(a[9,i]=b[i],i=1..8)]:\ne26 := solve(\{op(subs( e25,%))\},\{seq(a[9,j],j=1..8)\}):\ne27 := `union`(e25,e26):\n" }}} {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 55 "The linki ng coefficients in the 9th row are as follows." }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 36 "seq(a[9,j]=subs(e27,a[9,j]),j=1..8);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6*/&%\"aG6$\"\"*\"\"\"#\"#`\"$7*/&F%6$F'\"\"# \"\"!/&F%6$F'\"\"$F0/&F%6$F'\"\"%#\"\"&\"#;/&F%6$F'F:#\"#F\"$7\"/&F%6$ F'\"\"'#F@\"$O\"/&F%6$F'\"\"(#\"$c#\"$p*/&F%6$F'\"\")#!#D\"$O$" }}} {PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 3 "e27" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1017 "e27 := \{a [8,2] = 0, c[9] = 1, c[8] = 1, a[5,2] = 0, a[4,2] = 0, a[6,2] = 0, a[7 ,2] = 0, a[9,3] = 0, a[9,2] = 0, b[3] = 0, b[2] = 0, `b*`[3] = 0, `b*` [2] = 0, a[7,1] = 1718911/4382720, a[8,1] = 85153/203300, a[5,3] = -11 2/45, a[5,4] = 64/27, a[6,1] = -10891/11556, a[5,1] = 88/135, a[8,7] = -7168/40375, a[7,6] = 14535/14336, a[4,1] = 1/20, a[3,1] = 2/75, a[3, 2] = 8/75, c[3] = 2/15, a[4,3] = 3/20, a[8,3] = -6783/2140, a[6,3] = 3 880/963, `b*`[7] = 10304/43605, a[7,5] = -671175/876544, a[7,4] = 8192 61/383488, b[1] = 53/912, a[7,3] = -1000749/547840, b[8] = -25/336, b[ 7] = 256/969, b[6] = 27/136, `b*`[8] = 0, c[4] = 1/5, c[2] = 1/12, c[6 ] = 2/3, b[5] = 27/112, a[9,1] = 53/912, a[9,4] = 5/16, a[9,5] = 27/11 2, a[9,6] = 27/136, a[9,7] = 256/969, a[9,8] = -25/336, b[4] = 5/16, a [2,1] = 1/12, `b*`[5] = 69/320, `b*`[4] = 241/756, `b*`[6] = 435/1904, `b*`[1] = 617/10944, a[8,6] = 1152/425, c[5] = 8/15, c[7] = 19/20, a[ 6,5] = 217/428, a[6,4] = -8456/2889, a[8,5] = -38493/13375, a[8,4] = 1 0956/2675, `b*`[9] = -1/18\}:" }{TEXT -1 0 "" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 49 "The Butcher tab leau for the scheme is as follows." }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 124 "subs(e27,matrix([seq([c[i],seq(a[i,j],j=1..i-1),``$( 10-i)],i=2..9),\n[`b`,seq(b[i],i=1..8),``],[`b*`,seq(`b*`[i],i=1..9)]] ));" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#K%'matrixG6#7,7,#\"\"\"\"#7F(%! GF+F+F+F+F+F+F+7,#\"\"#\"#:#F.\"#v#\"\")F1F+F+F+F+F+F+F+7,#F)\"\"&#F) \"#?\"\"!#\"\"$F8F+F+F+F+F+F+7,#F3F/#\"#))\"$N\"F9#!$7\"\"#X#\"#k\"#FF +F+F+F+F+7,#F.F;#!&\"*3\"\"&c:\"F9#\"%!)Q\"$j*#!%c%)\"%*)G#\"$<#\"$G%F +F+F+F+7,#\"#>F8#\"(6*=<\"(?FQ%F9#!(\\2+\"\"'Sya#\"'h#>)\"')[$Q#!'v6n \"'Wl()#\"&NX\"\"&OV\"F+F+F+7,F)#\"&`^)\"'+L?F9#!%$y'\"%S@#\"&c4\"\"%v E#!&$\\Q\"&vL\"#\"%_6\"$D%#!%or\"&v.%F+F+7,F)#\"#`\"$7*F9F9#F6\"#;#FF \"$7\"#FF\"$O\"#\"$c#\"$p*#!#D\"$O$F+7,%\"bGFepF9F9FhpFjpF\\qF^qFaqF+7 ,%#b*G#\"$<'\"&W4\"F9F9#\"$T#\"$c(#\"#p\"$?$#\"$N%\"%/>#\"&/.\"\"&0O%F 9#!\"\"\"#=Q(pprint06\"" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}}{PARA 0 "" 0 "" {TEXT -1 60 " #-----------------------------------------------------------" }}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 38 "Tsitouras' algorithm .. general scheme " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 129 "In t his subsection we use the algorithm of Ch. Tsitouras, as outlined in t he previous subsection, to construct a general scheme. " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 90 "We shall obtain expr essions for all the coefficients of the scheme in terms of the nodes \+ " }{XPPEDIT 18 0 "c[2]" "6#&%\"cG6#\"\"#" }{TEXT -1 3 ", " }{XPPEDIT 18 0 "c[4]" "6#&%\"cG6#\"\"%" }{TEXT -1 3 ", " }{XPPEDIT 18 0 "c[5]" "6#&%\"cG6#\"\"&" }{TEXT -1 3 ", " }{XPPEDIT 18 0 "c[6]" "6#&%\"cG6# \"\"'" }{TEXT -1 7 " and " }{XPPEDIT 18 0 "c[7]" "6#&%\"cG6#\"\"(" } {TEXT -1 1 "." }}{PARA 0 "" 0 "" {TEXT -1 10 "The node " }{XPPEDIT 18 0 "c[3]" "6#&%\"cG6#\"\"$" }{TEXT -1 27 " does not appear because \+ " }{XPPEDIT 18 0 "c[3]=2/3" "6#/&%\"cG6#\"\"$*&\"\"#\"\"\"F'!\"\"" } {TEXT -1 1 " " }{XPPEDIT 18 0 "c[4]" "6#&%\"cG6#\"\"%" }{TEXT -1 2 ". \+ " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT 260 4 "Note " }{TEXT -1 32 ": It is possible to also have " }{XPPEDIT 18 0 "`b*` [9]" "6#&%#b*G6#\"\"*" }{TEXT -1 123 " as a parameter but, since the \+ principal error norm and the stability polynomial for the order 6 sche me do not depend on " }{XPPEDIT 18 0 "`b*`[9]" "6#&%#b*G6#\"\"*" } {TEXT -1 116 ", it will be sufficient to obtain a solution that avoid s the occurrence of this extra parameter by requiring that " } {XPPEDIT 18 0 "`b*`[8]=0" "6#/&%#b*G6#\"\")\"\"!" }{TEXT -1 1 "." }} {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 52 "#-------- -------------------------------------------" }}{PARA 0 "" 0 "" {TEXT -1 22 "We specify the nodes: " }}{PARA 256 "" 0 "" {TEXT -1 1 " " } {XPPEDIT 18 0 "c[8]=1" "6#/&%\"cG6#\"\")\"\"\"" }{TEXT -1 3 ", " } {XPPEDIT 18 0 "c[9]=1" "6#/&%\"cG6#\"\"*\"\"\"" }{TEXT -1 1 "," }} {PARA 0 "" 0 "" {TEXT -1 31 "the zero linking coefficients: " }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "a[4,2]=0" "6#/&%\"aG6$\"\"% \"\"#\"\"!" }{TEXT -1 3 ", " }{XPPEDIT 18 0 "a[5,2]=0" "6#/&%\"aG6$\" \"&\"\"#\"\"!" }{TEXT -1 3 ", " }{XPPEDIT 18 0 "a[6,2]=0" "6#/&%\"aG6 $\"\"'\"\"#\"\"!" }{TEXT -1 3 ", " }{XPPEDIT 18 0 "a[7,2]=0" "6#/&%\" aG6$\"\"(\"\"#\"\"!" }{TEXT -1 3 ", " }{XPPEDIT 18 0 "a[8,2]=0" "6#/& %\"aG6$\"\")\"\"#\"\"!" }{TEXT -1 2 ", " }}{PARA 0 "" 0 "" {TEXT -1 22 "and the zero weights: " }}{PARA 256 "" 0 "" {TEXT -1 1 " " } {XPPEDIT 18 0 "b[2]=0" "6#/&%\"bG6#\"\"#\"\"!" }{TEXT -1 3 ", " } {XPPEDIT 18 0 "b[3]=0" "6#/&%\"bG6#\"\"$\"\"!" }{TEXT -1 3 ", " } {XPPEDIT 18 0 "`b*`[2]=0" "6#/&%#b*G6#\"\"#\"\"!" }{TEXT -1 3 ", " } {XPPEDIT 18 0 "`b*`[3]=0" "6#/&%#b*G6#\"\"$\"\"!" }{TEXT -1 3 ", " } {XPPEDIT 18 0 "`b*`[8]=0" "6#/&%#b*G6#\"\")\"\"!" }{TEXT -1 2 ". " }} {PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 87 "e1 := \{c[8]=1,c[9]=1,b[2]=0,b[3]=0,`b*`[2]=0,`b*`[3]=0,`b*`[8]=0, seq(a[i,2]=0,i=4..8)\}:" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 15 " " 0 "" {TEXT 266 6 "Step 1" }{TEXT -1 1 " " }}{PARA 0 "" 0 "" {TEXT -1 33 "We use the quadrature conditions:" }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "Sum(b[i],i=1..8)=1" "6#/-%$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 267 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 268 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 269 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 270 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 271 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 272 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 273 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 274 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 275 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 276 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 290 23 "______________ _________" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT 258 80 "Schemes that have linking coeffic ients with maximum magnitude no greater than 43" }{TEXT -1 1 " " }} {PARA 0 "" 0 "" {TEXT -1 34 "#=================================" }} {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 ord er 6(5), by Ch. Tsitouras and I. Th. Famelis,\n International Co nference of Numerical Analysis and Applied Mathematics, Crete, (2006) \+ Extended Abstract." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 257 "" 0 " " {TEXT -1 100 "The scheme constructed here is a minor modification of the scheme presented in the preceding paper. " }}{PARA 257 "" 0 "" {TEXT -1 237 "The order 6 scheme is exactly the same (except that all \+ the linking coefficents are given in exact rational form) but the embe dded order 5 scheme is altered so that it has similar stability charac tersitics 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 com bined 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,\n c[5]=71/125,\nc[6]=42/59,\nc[7]=199/200,\nc[8]=1,\nc[9]=1,\n\na[2,1]=1 7/183,\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]=-497008268 2619223281/2887511529739311186,\na[6,2]=0,\na[6,3]=97919278033879057/1 3556392158400522,\na[6,4]=-407131674007930877068/74078904949579652469, \na[6,5]=1237601855204268750000/1753200750473385108433,\na[7,1]=176597 685527535385020980411/42773485015591331328000000,\na[7,2]=0,\na[7,3]=- 6793162515552646891859/401628967282547712000,\na[7,4]=1270492601936128 7204873446554247/886659402653054716778496000000,\na[7,5]=-507288363345 09259632278125/32657591718008685915971584,\na[7,6]=5153622398279619070 3/51293749413888000000,\na[8,1]=299033520572337573523/6691872079381235 7519,\na[8,2]=0,\na[8,3]=-16550269823961899/902146153892364,\na[8,4]=4 9920346343238033627496282/3215735869387500624775563,\na[8,5]=-16864324 88955761721093750/978844996793357447730403,\na[8,6]=161901609084039/14 9698803705724,\na[8,7]=-305146137600000/54760341991955873,\na[9,1]=245 03/381483,\na[9,2]=0,\na[9,3]=0,\na[9,4]=1366847103121/4106349847584, \na[9,5]=20339599609375/75933913767768,\na[9,6]=35031290651/1947655461 44,\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]=20339599609375/75933913767768,\nb[6]=35031290651/194765546144, \nb[7]=16620160000000/11001207123543,\nb[8]=-14933/11016,\n\n`b*`[1]=6 1010485298317/979331468960880,\n`b*`[2]=0,\n`b*`[3]=0,\n`b*`[4]=320207 313882553286621/941222813406992395200,\n`b*`[5]=6845867841119140625/29 008216787127405534,\n`b*`[6]=124109197949158875473/5624956602501108163 20,\n`b*`[7]=19339714537078400000/16810691577722216811,\n`b*`[8]=-2110 29377951/210416202900,\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\"'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$>g#\\q7\"?+++'\\ynraIl-%f m))#!;D\"yAjf#4XLO)G2&\";%erf\"fo3!='z#)RAO:&\"5+++))QT\\P H^7&\"\"\"#\"6BNdPBd?N.*H\"5>vN7Qz?(=p'F;#!2**='R#)p-b;\"0kB*Q:Y@!*7&F +#\";#G'\\FO.QKMY.#*\\\":jbxC1](Qpet:K#!:]P4@;\"0Cdq.))p\\\"7&F+F+F+#!0++gPh90$\"2te&>*>MgZ&7&F\\p #\"&.X#\"'$[\"QF;F;7&F+#\".@J5ZoO\"\".%eZ)\\j5%#\"/v$4'*fR.#\"/oxw8R$f (#\",^1HJ]$\"-WhalZ>7&F+F+#\"/+++g,i;\"/VN727+6#!&L\\\"\"&;5\"7&F+%F__ ___________________________________GFgrFgr7&%\"bGFbqF;F;FeqF_r7&%#b*G# \"/<$)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,$\")JyX9!\")$\")*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(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(ee,RK6_8eqs) ):\nmap(u->`if`(lhs(u)=rhs(u),0,1),%);\nnops(%);\nsimplify(subs(ee,`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$" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#\"#<" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 " " {TEXT -1 86 "Next we set-up stage-order condtions to check for stage -orders from 2 to 5 inclusive. " }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 85 "for ct from 2 to 5 do\n so||ct||_8 := StageOrderConditions(c t,8,'expanded');\nend do:" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 34 "Stages 4 to 8 have stage-order 3. " }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 173 "[seq([seq(expand(subs(ee,so||i||_8 [j])),i=2..5)],j=1..6)]:\nmap(proc(L) local i; for i to nops(L) do if \+ not evalb(L[i]) then break end if end do; i end proc,%):\nsimplify(%); " }}{PARA 11 "" 1 "" {XPPMATH 20 "6#7(\"\"#\"\"$F%F%F%F%" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 53 "None of the princ ipal error conditions are satisfied." }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 123 "RK6_8err_eqs := PrincipalErrorConditions(6,8,'expand ed'):\nexpand(subs(ee,RK6_8err_eqs)):\nmap(u->`if`(lhs(u)=rhs(u),0,1), %);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#7R\"\"\"F$F$F$F$F$F$F$F$F$F$F$F $F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F $" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 114 "We can calculate the principal error norm of the order 6 scheme, that is , the 2-norm of the principal error terms." }}{EXCHG {PARA 0 "> " 0 " " {MPLTEXT 1 0 131 "errterms6_8 := PrincipalErrorTerms(6,8,'expanded') :\nevalf(evalf[14](sqrt(add(subs(ee,errterms6_8[i])^2,i=1.. nops(errte rms6_8)))));" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#$\"+O7OJ7!#9" }}} {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 79 "The 2 nor m of the principal error of the order 5 embedded scheme is as follows. " }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 153 "`errterms5_9*` := subs( b=`b*`,PrincipalErrorTerms(5,9,'expanded')):\nevalf(evalf[14](sqrt(add (subs(ee,`errterms5_9*`[i])^2,i=1.. nops(`errterms5_9*`)))));" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#$\"+U**zfd!#8" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}}{PARA 0 "" 0 "" {TEXT -1 52 "#---------------------------------------------- -----" }}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 44 "simultaneous constructio n of the two schemes" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 " " {TEXT -1 119 "We incorporate the stage-order equations to ensure tha t stage 2 has stage-order 2 and stages 3 to 8 have stage-order 3." }} {PARA 0 "" 0 "" {TEXT -1 48 "We also incorporate the simplifying condi tions: " }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "Sum(b[i]*a [i,j],i = j+1 .. 8) = b[j]*(1-c[j]);" "6#/-%$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] = 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] = 199/200;" "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 performed in the following subsection." }}{PARA 0 "" 0 " " {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 125 "The weights of the ord er 6 scheme provide the linking coefficients for the 9th stage of the \+ embedded order 5 scheme so that: " }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "a[9,i]=b[i]" "6#/&%\"aG6$\"\"*%\"iG&%\"bG6#F(" }{TEXT -1 3 ", " }{XPPEDIT 18 0 "i=1" "6#/%\"iG\"\"\"" }{TEXT -1 9 " . . . 8 ." }}{PARA 0 "" 0 "" {TEXT -1 22 "We also specify that " }{XPPEDIT 18 0 "c[9] = 1;" "6#/&%\"cG6#\"\"*\"\"\"" }{TEXT -1 3 ", " }{XPPEDIT 18 0 "`b*`[2] = 0;" "6#/&%#b*G6#\"\"#\"\"!" }{TEXT -1 3 ", " } {XPPEDIT 18 0 "`b*`[3] = 0;" "6#/&%#b*G6#\"\"$\"\"!" }{TEXT -1 7 " an d " }{XPPEDIT 18 0 "`b*`[9] = -1/150;" "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\}:\neqn s := subs(e1,cdns):\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[sol ve] := 0:\ne3 := `union`(e1,e2):" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}{SECT 1 {PARA 4 "" 0 " " {TEXT -1 2 "e3" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> \+ " 0 "" {MPLTEXT 1 0 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] \+ = 1366847103121/4106349847584, b[8] = -14933/11016, a[9,8] = -14933/11 016, 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 /75933913767768, a[8,5] = -1686432488955761721093750/97884499679335744 7730403, a[9,6] = 35031290651/194765546144, `b*`[1] = 61010485298317/9 79331468960880, a[8,3] = -16550269823961899/902146153892364, b[1] = 24 503/381483, b[3] = 0, a[7,3] = -6793162515552646891859/401628967282547 712000, a[4,1] = 9/166, b[6] = 35031290651/194765546144, b[7] = 166201 60000000/11001207123543, a[7,6] = 51536223982796190703/512937494138880 00000, a[9,4] = 1366847103121/4106349847584, a[6,3] = 9791927803387905 7/13556392158400522, a[5,1] = 207751751/316406250, a[2,1] = 17/183, a[ 5,3] = -526769377/210937500, a[9,1] = 24503/381483, a[8,7] = -30514613 7600000/54760341991955873, `b*`[4] = 320207313882553286621/94122281340 6992395200, a[8,1] = 299033520572337573523/66918720793812357519, b[5] \+ = 20339599609375/75933913767768, `b*`[7] = 19339714537078400000/168106 91577722216811, a[8,6] = 161901609084039/149698803705724, a[7,4] = 127 04926019361287204873446554247/886659402653054716778496000000, a[6,4] = -407131674007930877068/74078904949579652469, 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] = 3756/117113, `b*`[9] = -1/150, `b*`[5] = 6845867841119140625/2900 8216787127405534, a[4,3] = 27/166, a[7,1] = 17659768552753538502098041 1/42773485015591331328000000, `b*`[8] = -211029377951/210416202900, a[ 7,5] = -50728836334509259632278125/32657591718008685915971584, a[6,5] \+ = 1237601855204268750000/1753200750473385108433, a[8,4] = 499203463432 38033627496282/3215735869387500624775563, `b*`[6] = 124109197949158875 473/562495660250110816320\}:" }}{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)],[``,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(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?$F*$\"+Vs1D6F/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\"!\")$\"+K\")*G V\"FY$!+!4bLb\"FD$\"+wrs/5FDF+F+F+7,$\"\"\"F:$\"+VygoWFDF9$!+(=WX$=FY$ \"+sqP_:FY$!+@+)Gs\"FD$\"+N?S$F/$\"+9a(*fBF /$\"+kFS1AF/$\"+S8W]6FD$!+1T\"H+\"FD$!+nmmmmFioQ(pprint16\"" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 146 "R K6_8eqs := [op(RowSumConditions(8,'expanded')),op(OrderConditions(6,8, 'expanded'))]:\n`RK5_9eqs*` := subs(b=`b*`,OrderConditions(5,9,'expand ed')):" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 7 "Check: " }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 148 "simplify(subs(e 3,RK6_8eqs)):\nmap(u->`if`(lhs(u)=rhs(u),0,1),%);\nnops(%);\nsimplify( subs(e3,`RK5_9eqs*`)):\nmap(u->`if`(lhs(u)=rhs(u),0,1),%);\nnops(%);" }{TEXT -1 0 "" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#7N\"\"!F$F$F$F$F$F$F$ F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$ F$" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#\"#W" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#73\"\"!F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#\"#<" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 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 solution" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> \+ " 0 "" {MPLTEXT 1 0 71947 "eG := \{c[8] = 1, b[3] = 0, b[2] = 0, c[9] \+ = 1, b[5] = -1/60*(10*c[6]*c[7]*c[4]-5*c[6]*c[4]-5*c[7]*c[6]+3*c[6]-5* c[7]*c[4]+3*c[4]-2+3*c[7])/(-c[4]+c[5])/c[5]/(-c[5]^3+c[5]^2-c[7]*c[5] +c[7]*c[5]^2-c[6]*c[5]+c[6]*c[5]^2+c[7]*c[6]-c[6]*c[7]*c[5]), a[5,2] = 0, a[4,2] = 0, `b*`[3] = 0, `b*`[2] = 0, a[8,2] = 0, a[7,2] = 0, a[6, 2] = 0, a[6,5] = 1/2*c[6]*(30*c[6]^2*c[5]^2*c[4]^2-2*c[6]^2*c[4]^2+c[6 ]^3*c[4]-12*c[6]^2*c[4]*c[5]^2-3*c[5]^2*c[6]*c[4]+c[5]*c[6]*c[4]^2+c[6 ]*c[4]^3-12*c[6]^2*c[4]^2*c[5]-30*c[5]*c[6]^3*c[4]^2-20*c[5]*c[6]^2*c[ 4]^4+30*c[6]^2*c[5]*c[4]^3-c[5]*c[4]^3+20*c[4]^3*c[5]*c[6]^3-20*c[6]^2 *c[5]^2*c[4]^3+c[5]^2*c[4]^2-30*c[5]^2*c[6]*c[4]^3+12*c[5]^2*c[6]*c[4] ^2+2*c[6]^2*c[4]*c[5]+20*c[5]^2*c[6]*c[4]^4+2*c[6]^2*c[5]^2-2*c[6]^3*c [5]+12*c[6]^3*c[4]*c[5])/c[5]/(c[4]^3-2*c[5]*c[4]^2+30*c[5]^2*c[4]^3-6 *c[5]*c[4]^3-10*c[4]^4*c[5]^2-c[5]^3+10*c[5]^4*c[4]^2-30*c[5]^3*c[4]^2 +6*c[5]^3*c[4]+2*c[5]^2*c[4]), b[1] = 1/60*(30*c[5]*c[6]*c[7]*c[4]-10* c[5]*c[6]*c[4]-10*c[6]*c[7]*c[5]+5*c[6]*c[5]-10*c[6]*c[7]*c[4]+5*c[6]* c[4]+5*c[7]*c[6]-3*c[6]-10*c[5]*c[7]*c[4]+5*c[5]*c[4]+5*c[7]*c[5]-3*c[ 5]+5*c[7]*c[4]-3*c[4]+2-3*c[7])/c[5]/c[6]/c[7]/c[4], a[6,1] = -1/4*c[6 ]*(60*c[5]^2*c[6]*c[4]^4-90*c[5]^2*c[6]^3*c[4]^2-60*c[5]*c[6]^2*c[4]^4 -2*c[6]^3*c[4]^2+50*c[5]^4*c[4]^3*c[6]+40*c[6]^2*c[4]^5*c[5]-40*c[4]^5 *c[5]^3+36*c[4]*c[5]^2*c[6]^3-20*c[5]^4*c[6]^2*c[4]^2+8*c[6]^2*c[5]^3+ 4*c[6]^2*c[4]^3-24*c[5]*c[6]^3*c[4]^2-8*c[6]^2*c[4]^2*c[5]+24*c[6]^2*c [5]*c[4]^3+60*c[4]^3*c[5]*c[6]^3-4*c[6]*c[5]*c[4]^3-14*c[5]^3*c[6]*c[4 ]+7*c[5]^3*c[4]^2+6*c[5]^2*c[6]*c[4]^3+14*c[5]^2*c[6]*c[4]^2+66*c[5]^3 *c[6]*c[4]^2-80*c[5]^3*c[6]^2*c[4]^3+30*c[6]^2*c[5]^2*c[4]^3-24*c[6]^2 *c[5]^2*c[4]^2-240*c[6]*c[5]^3*c[4]^3-48*c[6]^2*c[5]^3*c[4]+150*c[6]^2 *c[5]^3*c[4]^2-24*c[4]^4*c[5]^2-9*c[5]^2*c[4]^3+6*c[4]^4*c[5]-40*c[5]^ 4*c[4]^4-20*c[6]^2*c[4]^4*c[5]^2-6*c[5]^2*c[6]^3-2*c[6]*c[4]^4-24*c[5] ^3*c[4]^3+60*c[6]^3*c[4]^3*c[5]^2+7*c[6]^3*c[4]*c[5]-40*c[5]^2*c[6]*c[ 4]^5-40*c[6]^3*c[4]^4*c[5]+110*c[4]^4*c[6]*c[5]^3+120*c[5]^3*c[4]^4)/c [5]/c[4]^2/(10*c[5]^3*c[4]^2+10*c[5]^2*c[4]^3-30*c[5]^2*c[4]^2+6*c[5]^ 2*c[4]-c[5]^2+6*c[5]*c[4]^2+c[5]*c[4]-c[4]^2), b[4] = 1/60*(-2+3*c[5]- 5*c[7]*c[6]+3*c[7]+3*c[6]+10*c[6]*c[7]*c[5]-5*c[6]*c[5]-5*c[7]*c[5])/c [4]/(-c[6]*c[4]^3-c[4]^3+c[4]^4+c[7]*c[4]^2+c[6]*c[4]^2-c[6]*c[7]*c[4] +c[6]*c[7]*c[4]^2-c[5]*c[4]^3+c[5]*c[4]^2-c[7]*c[4]^3-c[5]*c[7]*c[4]+c [5]*c[7]*c[4]^2-c[5]*c[6]*c[4]+c[5]*c[6]*c[4]^2+c[6]*c[7]*c[5]-c[5]*c[ 6]*c[7]*c[4]), b[6] = 1/60*(10*c[5]*c[7]*c[4]-5*c[5]*c[4]-5*c[7]*c[5]+ 3*c[5]-5*c[7]*c[4]+3*c[4]-2+3*c[7])/(-c[4]+c[6])/(-c[5]+c[6])/c[6]/(c[ 6]^2-c[6]+c[7]-c[7]*c[6]), `b*`[9] = 1/10*(50*c[5]^2*c[6]*c[4]^2-40*c[ 5]^2*c[6]*c[4]+10*c[6]*c[5]^2-40*c[5]*c[6]*c[4]^2+35*c[5]*c[6]*c[4]-10 *c[6]*c[5]+10*c[6]*c[4]^2-10*c[6]*c[4]+3*c[6]-5*c[5]^2*c[4]-5*c[5]*c[4 ]^2+10*c[5]*c[4]-c[5]-c[4])/(3*c[6]-28*c[5]*c[6]*c[4]^2-28*c[5]^2*c[6] *c[4]+8*c[6]*c[5]^2-9*c[6]*c[4]-9*c[6]*c[5]+28*c[5]*c[6]*c[4]+30*c[5]^ 2*c[6]*c[4]^2+8*c[6]*c[4]^2-4*c[5]^2*c[4]-4*c[5]*c[4]^2+9*c[5]*c[4]-c[ 5]-c[4]), a[7,5] = -1/2*c[7]*(-17*c[5]^2*c[6]*c[4]^4+5*c[6]*c[5]*c[4]^ 4+42*c[4]^2*c[5]^2*c[7]^3+17*c[7]^3*c[5]*c[4]^3-80*c[7]^3*c[5]^2*c[4]^ 3+c[7]*c[5]*c[4]^4+80*c[5]^2*c[4]^4*c[7]^2-60*c[4]^5*c[6]^2*c[5]^2+20* c[4]^3*c[7]^3*c[5]^3-20*c[7]^2*c[5]^2*c[4]^5+40*c[4]^5*c[6]*c[5]^3-5*c [5]*c[6]^2*c[4]^4+c[7]^3*c[4]^2-c[7]^2*c[4]^3-80*c[5]^3*c[4]^4*c[7]+20 *c[7]^3*c[5]^2*c[4]^4-200*c[6]^2*c[4]^5*c[7]*c[5]^3-290*c[5]^4*c[4]^2* c[7]^2*c[6]+20*c[6]^2*c[7]*c[5]^4*c[4]^2+40*c[6]*c[5]^2*c[4]^4*c[7]-14 0*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]^2-27*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]^2-10 0*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]), 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]-24 6*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]-2 5*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]-5 0*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]+1 04*c[4]^3*c[7]^2*c[5]*c[6]^2-12*c[7]^2*c[6]*c[5]*c[4]^3-36*c[7]^2*c[5] ^2*c[6]*c[4]^2-32*c[7]^2*c[5]^3*c[6]*c[4]-320*c[7]^2*c[5]^3*c[6]*c[4]^ 3-560*c[5]^3*c[6]^2*c[4]^4-4*c[6]^2*c[4]^3*c[7]^2+240*c[5]^3*c[6]^2*c[ 7]*c[4]^3-34*c[5]^3*c[7]^3*c[4]^2-2*c[7]^3*c[5]^2*c[4]+164*c[6]^2*c[7] *c[5]^3*c[4]^2-20*c[6]^2*c[7]^2*c[4]*c[5]^3+2*c[7]^2*c[5]^3*c[4]-130*c [5]^5*c[4]^3*c[6]*c[7]-90*c[7]^3*c[5]^2*c[6]^2*c[4]^3-16*c[7]^2*c[4]^4 *c[5]-60*c[7]^2*c[5]^3*c[6]^2*c[4]^2-300*c[6]^2*c[4]^5*c[5]*c[7]^2-150 *c[7]^2*c[4]^5*c[6]*c[5]^3+380*c[5]^2*c[6]*c[4]^5*c[7]^2+140*c[7]^3*c[ 5]^2*c[4]^3*c[6]-810*c[7]^3*c[5]^3*c[6]^2*c[4]^2-14*c[7]^3*c[6]*c[5]*c [4]^3-280*c[7]^3*c[6]^2*c[5]*c[4]^3+240*c[7]^3*c[5]^3*c[6]^2*c[4]-20*c [5]^4*c[7]^2*c[4]*c[6]-6*c[6]*c[7]^3*c[5]^3+22*c[5]^4*c[4]*c[7]*c[6]-2 80*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]^2-3 0*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]-14 0*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]^3-1 00*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+6 6*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]^2-2 8*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+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+3 20*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], 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]^2-3 60*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]-20 0*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]+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+12 0*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]^3-2 8*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]^2-6 0*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+30 0*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]-3 06*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+10 0*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+1 03*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+93 2*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+6 90*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]^2-6 00*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]+17 4*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 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2-6*c[4]^4*c[7]^2-400*c[5]^4*c[7]^2*c[4]^5*c[6]^2+415*c[7]^2*c[5]^4*c[ 4]^3-990*c[5]^3*c[7]*c[4]^6*c[6]+433*c[6]^2*c[4]^5*c[5]+4*c[4]^3-60*c[ 5]^5*c[4]^3+30*c[7]*c[6]^2*c[5]^3+4*c[7]*c[6]*c[5]^3-6*c[7]^2*c[6]*c[5 ]^3-350*c[5]^4*c[4]^6*c[6]^2-5*c[4]^4-250*c[5]^3*c[7]*c[4]^6-270*c[5]^ 4*c[4]^5*c[7]-330*c[7]*c[6]^2*c[5]*c[4]^5-1320*c[4]^6*c[6]^2*c[7]*c[5] ^2-55*c[5]^4*c[4]^2*c[7]^2-100*c[5]^5*c[7]^2*c[4]^3-96*c[6]*c[5]^2*c[4 ]^5*c[7]-562*c[4]^5*c[5]^3-600*c[5]^5*c[4]^4*c[7]^2*c[6]+380*c[5]^2*c[ 4]^7*c[7]+160*c[5]^5*c[4]^3*c[6]+62*c[7]*c[4]^4*c[6]*c[5]+80*c[5]^5*c[ 4]^4+30*c[5]*c[4]^6*c[7]^2+310*c[4]^4*c[6]^2*c[5]^5+450*c[5]^5*c[7]*c[ 4]^4*c[6]-356*c[7]^2*c[5]^3*c[4]^3+24*c[7]*c[4]^4*c[6]^2-200*c[5]^5*c[ 7]*c[4]^6*c[6]-900*c[5]^3*c[7]*c[4]^7*c[6]+60*c[5]^5*c[4]^2*c[6]^2*c[7 ]-450*c[5]^5*c[4]^4*c[6]^2*c[7]-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+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]^5-12 1*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 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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]-3280*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 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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 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60*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]+2 8*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]+21 0*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]-4 8*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+20 0*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+1 10*c[5]^3*c[6]*c[4]+110*c[5]^3*c[7]*c[4]+60*c[5]^2*c[4]-150*c[5]^3*c[6 ]*c[7]*c[4]+550*c[5]^3*c[4]^2-690*c[5]^2*c[6]*c[4]^3+520*c[5]^2*c[7]*c [4]^2+520*c[5]^2*c[6]*c[4]^2-690*c[5]^2*c[7]*c[4]^3-690*c[5]^3*c[7]*c[ 4]^2-690*c[5]^3*c[6]*c[4]^2+150*c[5]^2*c[4]^4*c[7]+900*c[6]*c[5]^3*c[4 ]^3+900*c[5]^3*c[7]*c[4]^3-200*c[5]^4*c[7]*c[4]^2*c[6]-429*c[5]^2*c[4] ^2-120*c[5]^4*c[4]^2-120*c[4]^4*c[5]^2+150*c[5]^4*c[4]^3-72*c[5]^2*c[6 ]*c[4]-72*c[5]^2*c[7]*c[4]+90*c[5]^2*c[6]*c[7]*c[4]+550*c[5]^2*c[4]^3+ 300*c[5]^3*c[4]^4*c[6]*c[7]+300*c[5]^4*c[4]^3*c[6]*c[7]-690*c[5]^3*c[4 ]^3-200*c[4]^4*c[6]*c[5]^3+150*c[5]^3*c[4]^4), a[8,3] = 3/4*(-1560*c[5 ]^2*c[6]*c[4]^4+585*c[6]*c[5]*c[4]^4-45*c[7]*c[5]*c[4]^4-12*c[5]^3*c[4 ]-450*c[4]^5*c[6]*c[5]^3+3*c[5]*c[7]*c[4]-150*c[5]^2*c[4]^5*c[7]-200*c [5]^3*c[4]^4*c[7]+300*c[6]*c[4]^5*c[5]*c[7]+2860*c[6]*c[5]^2*c[4]^4*c[ 7]+160*c[5]^4*c[4]^3*c[6]-150*c[5]^4*c[4]^4*c[6]-4*c[5]^2+2*c[4]^2-40* c[7]*c[6]*c[5]^3+132*c[5]*c[6]*c[7]*c[4]-900*c[6]*c[5]^2*c[4]^5*c[7]-3 0*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+31 40*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]-20 0*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+1 2*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+1 2*c[6]*c[4]^2-15*c[6]*c[4]^3-15*c[6]*c[7]*c[4]^2-87*c[5]*c[4]^3-72*c[5 ]*c[7]*c[4]^2-72*c[5]*c[6]*c[4]^2+60*c[5]*c[4]^2-12*c[5]*c[6]*c[4]+10* c[5]*c[4]-150*c[5]*c[7]*c[6]*c[4]^3-15*c[7]*c[5]^3-660*c[5]^2*c[6]*c[7 ]*c[4]^2+90*c[6]*c[5]*c[7]*c[4]^2-15*c[6]*c[7]*c[5]^2+150*c[6]*c[5]^4* c[4]^2+930*c[6]*c[5]^2*c[7]*c[4]^3+930*c[6]*c[5]^3*c[7]*c[4]^2-200*c[5 ]^4*c[4]^3*c[7]+110*c[7]*c[5]*c[4]^3+110*c[6]*c[5]*c[4]^3+110*c[5]^3*c [6]*c[4]+110*c[5]^3*c[7]*c[4]+60*c[5]^2*c[4]-150*c[5]^3*c[6]*c[7]*c[4] +550*c[5]^3*c[4]^2-690*c[5]^2*c[6]*c[4]^3+520*c[5]^2*c[7]*c[4]^2+520*c [5]^2*c[6]*c[4]^2-690*c[5]^2*c[7]*c[4]^3-690*c[5]^3*c[7]*c[4]^2-690*c[ 5]^3*c[6]*c[4]^2+150*c[5]^2*c[4]^4*c[7]+900*c[6]*c[5]^3*c[4]^3+900*c[5 ]^3*c[7]*c[4]^3-200*c[5]^4*c[7]*c[4]^2*c[6]-429*c[5]^2*c[4]^2-120*c[5] ^4*c[4]^2-120*c[4]^4*c[5]^2+150*c[5]^4*c[4]^3-72*c[5]^2*c[6]*c[4]-72*c [5]^2*c[7]*c[4]+90*c[5]^2*c[6]*c[7]*c[4]+550*c[5]^2*c[4]^3+300*c[5]^3* c[4]^4*c[6]*c[7]+300*c[5]^4*c[4]^3*c[6]*c[7]-690*c[5]^3*c[4]^3-200*c[4 ]^4*c[6]*c[5]^3+150*c[5]^3*c[4]^4), a[8,5] = -1/2*(893*c[5]^2*c[6]*c[4 ]^4-122*c[6]*c[5]*c[4]^4+850*c[5]^4*c[4]^4*c[6]^2*c[7]-101*c[7]*c[5]*c [4]^4-535*c[5]^2*c[4]^4*c[7]^2-44*c[5]^3*c[4]+160*c[4]^5*c[6]^2*c[5]^2 +100*c[7]^2*c[5]^2*c[4]^5+170*c[4]^5*c[6]*c[5]^3+75*c[5]*c[6]^2*c[4]^4 +16*c[7]^2*c[4]^3-9*c[5]*c[7]*c[4]-140*c[5]^2*c[4]^5*c[7]-780*c[5]^3*c [4]^4*c[7]-10*c[4]^4*c[6]^2-100*c[7]^2*c[4]^5*c[5]^3+450*c[6]^2*c[4]^5 *c[7]*c[5]^3+510*c[5]^4*c[4]^2*c[7]^2*c[6]+490*c[6]^2*c[7]*c[5]^4*c[4] ^2-2224*c[6]*c[5]^2*c[4]^4*c[7]+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 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[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+2 0*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]-7 50*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]-5 10*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+12 0*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]-1 80*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]+1 5*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]+1 5*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]-1 0*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 nod es " }{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 t he 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 for th e 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 260 "" 0 "" {TEXT -1 1 ":" }}{PARA 260 " " 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$6&Ff [lF\\]l-Fb\\l6$%(DIAMONDGFc]lFf\\l-F$6&Ff[lF\\]l-Fb\\l6$%&CROSSGFc]lFf \\l-%+AXESLABELSG6%Q%c[2]6\"Q!Fb^l-%%FONTG6#%(DEFAULTG-Fe^l6$%*HELVETI CAGF)-%%VIEWG6$;F(Fez;$\"&K]\"!#9$\"&W]\"Fb_l" 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 0 402 "c_4 := .216457345224: pp := .1503307088537e-9: \np1 := evalf[30](plot(prin_err_norm_sqrd(.926800182826e-1,c[4],.56721 0929396),c[4]=0.216454..0.2164608,\n color=COLOR(RGB,0,.7,.2))):\np 2 := plot([[[c_4,pp]]$4],style=point,symbol=[circle$2,diamond,cross],s ymbolsize=[12,10$3],\n color=[black,cyan$3]):\nplots[display ]([p1,p2],font=[HELVETICA,9],view=[0.216454..0.2164608,1.5032e-10..1.5 0442e-10]);" }}{PARA 13 "" 1 "" {GLPLOT2D 355 320 320 {PLOTDATA 2 "6*- %'CURVESG6$7S7$$\"'ak@!\"'$\"?bGc%=Hf8;Co1GsKHw7tJJ/:F-7$$\"?nmmmmm;>k'=xUX;#F1$\"?G!)*4l/eo!H+*oLU ]\"F-7$$\"?LLLLLLL)*QAAUak@F1$\"?%ew#em.*Hb4I?[T]\"F-7$$\"?LLLLLLL)G&> #oXX;#F1$\"?//$R,xCmG>LKmS]\"F-7$$\"?nmmmmm;*oF_8ZX;#F1$\"?\"f,1W*\\Hu [Z!**))R]\"F-7$$\"?LLLLLL$esuB[[X;#F1$\"?c*f$4\"R=ZJ%3**4#R]\"F-7$$\"? ++++++]_uEx)\\X;#F1$\"?>p&z*f5e/KJcV&Q]\"F-7$$\"?LLLLLL$e]n)>8bk@F1$\" ?oZt6w)=BkxrX*y.:F-7$$\"?++++++]A6%yv_X;#F1$\"?BD>2'fe#)*=yP)GP]\"F-7$ $\"?nmmmmmmOP(pBaX;#F1$\"?3aU;r$*y]OID2n.:F-7$$\"?LLLLLLL[XzRbbk@F1$\" ?D.vgXOnY4o.Ji.:F-7$$\"?+++++++5AY1qbk@F1$\"?oAwF&)H0B`\"y[tN]\"F-7$$ \"?+++++++]@:z%eX;#F1$\"?j7#op&\\(H'y>Ez_.:F-7$$\"?+++++++!zh$)*)fX;#F 1$\"?+ex^8&pYTap0)[.:F-7$$\"?LLLLLL$e5Sr=hX;#F1$\"?r;vl\"))z;S()fGbM] \"F-7$$\"?nmmmmmmw@i>Fck@F1$\"?rdL&\\(*zl$H>t0U.:F-7$$\"?nmmmmmm1b$y,k X;#F1$\"?hek\\ksd-q:%y%R.:F-7$$\"?++++++]Ki'z_lX;#F1$\"?$*)o$Ht5Zn8ig* oL]\"F-7$$\"?nmmmmmm1yokock@F1$\"?\"3](e^s.\")y_\\)\\L]\"F-7$$\"?+++++ +]KKGJ$oX;#F1$\"?!Hu[/H[4@\\1$HL.:F-7$$\"?++++++](*)GyspX;#F1$\"?uKHAp *3/,e%e2K.:F-7$$\"?nmmmmmm\"zx\\=rX;#F1$\"??J;9sZuelra@J.:F-7$$\"?nmmm 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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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ot3d(sqrt(prin_err_norm_sqrd(.926800182826e-1,c[4],c[5])),c[4]=0.21645 4..0.2164608,\n c[5]=0.5672045..0.5672175,axes=boxed,grid=[25,25],la bels=[`c[4]`,`c[5]`,``],orientation=[-75,40],\n color=COLOR(RGB,1 ,0,.2),font=[HELVETICA,9]);\nDigits := 10:" }}{PARA 13 "" 1 "" {GLPLOT3D 464 371 371 {PLOTDATA 3 "6'-%%GRIDG6&;$\"'ak@!\"'$\"(3Y;#!\" (;$\"(X?n&F,$\"(v@n&F,X,%)anythingG6\"6\"[gl'!%\"!!#\\bm\":\":3EE9B68B 9749249A3EE9B68DE683237A3EE9B69959C6AA683EE9B6ADF0DE36A23EE9B6CBAA8552 F63EE9B6F28C25673F3EE9B7228A8B9E6F3EE9B75BAE2E92C43EE9B79DF17944F13EE9 B7E95481E6E03EE9B83DD9F1FD523EE9B89B7C101E4E3EE9B9023B61F5533EE9B97219 E21AAA3EE9B9EB12D794993EE9BA6D2407D5CC3EE9BAF853FEA6B13EE9BB8C9728980C 3EE9BC29F60CCFAE3EE9BCD0681518F53EE9BD7FF09D7A273EE9BE388FBA64403EE9BE FA3E5B0C433EE9BFC4FE1CB8D83EE9C098CD78516F3EE9B692372BFA3F3EE9B68B83AA 9F793EE9B68DF7878FB33EE9B6998E75BA3F3EE9B6AE4CE8F7303EE9B6CC2E822F133E E9B6F32FD77AB23EE9B72357598C823EE9B75C9D6930023EE9B79F085CB0ED3EE9B7EA 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C0A73EE9B8AA45D1CF713EE9B84B153824793EE9B7F5147CA0923EE9B7A839686C113E E9B7648BF22F6D3EE9B72A0793A0AA3EE9B6F8AF781A443EE9B6D07F855EE63EE9B6B1 79E0197C3EE9B69B98B856153EE9B68EE45B8FD33EE9B68B5777103D3EE9B690F2EEB8 2D3EE9C0BE9E63CDCB3EE9BFE86675A8C83EE9BF1B5AEE98A73EE9BE5776CB87EB3EE9 BD9CBCCB19573EE9BCEB2E1110233EE9BC42C898E6893EE9BBA3900DE1593EE9BB0D80 CAB3373EE9BA80A07558913EE9B9FCE7BD6E433EE9B9825DB96FF43EE9B910F8F1F615 3EE9B8A8C59EA1493EE9B849B92D6EE53EE9B7F3D98699DD3EE9B7A727EDE7E13EE9B7 639C41F0673EE9B7293CD79D5F3EE9B6F80386E2713EE9B6CFF834BA1F3EE9B6B11730 73C03EE9B69B5E6F77F63EE9B68ECC49165A3EE9B68B633AA93B-%&COLORG6&%$RGBG$ \"\"\"\"\"!$F " 0 "" {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 43 " #==========================================" }}{SECT 1 {PARA 4 "" 0 " " {TEXT -1 38 "a modification of Papakostas' scheme " }{XPPEDIT 18 0 "c[7] = 199/200;" "6#/&%\"cG6#\"\"(*&\"$*>\"\"\"\"$+#!\"\"" }{TEXT -1 1 " " }}{PARA 257 "" 0 "" {TEXT -1 0 "" }}{PARA 257 "" 0 "" {TEXT -1 75 "The scheme constructed here is a modification of the previous sche me with " }{XPPEDIT 18 0 "c[7]=199/200" "6#/&%\"cG6#\"\"(*&\"$*>\"\" \"\"$+#!\"\"" }{TEXT -1 18 ", as before, but " }{XPPEDIT 18 0 "c[6]" "6#&%\"cG6#\"\"'" }{TEXT -1 19 " changed so that " }{XPPEDIT 18 0 "c [6]=73/99" "6#/&%\"cG6#\"\"'*&\"#t\"\"\"\"#**!\"\"" }{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. " }}{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 "check ing 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 2247 "ee := \{c[2]=1/1 1,\nc[3]=157/1104,\nc[4]=157/736,\nc[5]=123/218,\nc[6]=73/99,\nc[7]=19 9/200,\nc[8]=1,\nc[9]=1,\n\na[2,1]=1/11,\na[3,1]=75517/2437632,\na[3,2 ]=271139/2437632,\na[4,1]=157/2944,\na[4,2]=0,\na[4,3]=471/2944,\na[5, 1]=42928977471/63842339642,\na[5,2]=0,\na[5,3]=-81818872578/3192116982 1,\na[5,4]=78364952136/31921169821,\na[6,1]=-4529937293461155846201058 37/169351895877446242624608351,\na[6,2]=0,\na[6,3]=1697439018580941844 337032/152982742436717473012293,\na[6,4]=-3345789124301186824103356433 60/38759554641024302244921542187,\na[6,5]=183506358169025606635433230/ 193412520623391990592938849,\na[7,1]=237424966244801792240477643719598 303/34580776953279718763722454600000000,\na[7,2]=0,\na[7,3]=-201796740 02037028690247750335371/722117794714327571912012500000,\na[7,4]=128453 834152286986408251909051997320271/554455420013807792632312867531562500 0,\na[7,5]=-972708878592287689266824375699676121/505439891163443236912 399043150000000,\na[7,6]=199423777107739967674947/23991343305172460000 0000,\na[8,1]=5973266207518682347729280873413/802324115518564191458832 770081,\na[8,2]=0,\na[8,3]=-40870912569246870680146344/134706965418231 1812082183,\na[8,4]=112459036656658044255344022793238112/4474108065454 386923650087599365353,\na[8,5]=-4712736023245029582064053285795650/221 3621528406610002679128666238199,\na[8,6]=127657765502309225898/1426828 94776218689669,\na[8,7]=-541736066625000000/96886654438663742381,\na[9 ,1]=531842063/8415926910,\na[9,2]=0,\na[9,3]=0,\na[9,4]=69005738462239 78496/21081697499495907225,\na[9,5]=51126843079934806/1731604640341596 75,\na[9,6]=1147778420925807/6907762998459050,\na[9,7]=575779000000000 /411330558898641,\na[9,8]=-13426037/10725975,\n\nb[1]=531842063/841592 6910,\nb[2]=0,\nb[3]=0,\nb[4]=6900573846223978496/21081697499495907225 ,\nb[5]=51126843079934806/173160464034159675,\nb[6]=1147778420925807/6 907762998459050,\nb[7]=575779000000000/411330558898641,\nb[8]=-1342603 7/10725975,\n\n`b*`[1]=648854567388972967/10596897420732888642,\n`b*`[ 2]=0,\n`b*`[3]=0,\n`b*`[4]=8879698038617778833446469632/26544976952167 847097500780295,\n`b*`[5]=58500328551366236723248598/21803464957806968 2785496485,\n`b*`[6]=32934117770434118898396/161072145715862713160465, \n`b*`[7]=2584290081034720820000000/2589630224498721525333771,\n`b*`[8 ]=-4674424280471995/5402235928512618,\n`b*`[9]=-1/845\}:" }}}{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 Butc her 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],s eq(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)],[``,se q(`b*`[i],i=7..9)]]));" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#K%'matrixG6# 787&#\"\"\"\"#6F(%!GF+7&#\"$d\"\"%/6#\"&$7&F+F+F+#\",O@&\\OyFF7&#\"#t\"#**#!(=U:#\\C-V-TYbf(Q#\"RB1_7M>7&#\"$*>\" $+##\"E.$)f>Pkx/C#z,[Cm\\UP#\"D++++YXAPw=(zK&px!eMF:#!Ar`L]xC!pGq.-Snz ,#\"?++]7?\">dFVr%z<@s7&F+#\"Hr-K(*>04>D3k)pG_T$QXG\"\"F+]i:`nGJKEz2Q, ?aXa&#!E@hn*pvV#oE*o(G#fy)3F(*\"E+++]J/*R7pBVM;\"*)Ra]#\"9Z\\nn*Rx5xPU *>\"9++++Ys^IV8*R#7&F)#\"@8M(3GHxM#o=v?mK(f\"?\"3qF$)e9>k&=b6CB!)F:#!; Wj9!oqoCpD\"4(3%\":$=#37=J#=a'pqM\"7&F+#\"E7\"QKzASMbU/emlO!fC6\"C``O* f(3]O#pQaa13TZ%#!C]cz&G`S1#eH]CBgt7Z\"C*>QimG\"zE+5mSG:i8A#\"6)*eA4B]l xlF\"\"6p'*o=ix%*GoU\"7&F+F+F+#!3+++Dm1O&H!y+v4Zy;_p(\\aE#\";)f[KsOiO^&G.]e\"<&['\\&y#op!y&\\Y.=## \"8'R)*)=TVqx6MH$\"9l/;8F'erX@2h\"7&F+#\":+++?3sM53!H%e#\":rPLD:s)\\C- j*e##!1&*>Z!GCWn%\"1=E^GfB-a#!\"\"\"$X)Q(pprint26\"" }}}{PARA 0 "" 0 " " {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 136 "subs(ee,m atrix([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,$\")\"444*!\"*F(%!GF+F+F+F+F +F+F+7,$\")95A9!\")$\")c'z4$F*$\")\\I76F/F+F+F+F+F+F+F+7,$\")A:L@F/$\" )/)GL&F*$\"\"!F:$\")T')*f\"F/F+F+F+F+F+F+7,$\")=?UcF/$\")v@CnF/F9$!)S: jD!\"($\")C&\\X#FDF+F+F+F+F+7,$\")utttF/$!)m'[n#FDF9$\")Cc46!\"'$!)h;K ')FD$\")@#y[*F/F+F+F+F+7,$\")++]**F/$\")x!e'oFDF9$!)G^%z#FN$\")ov;BFN$ !))zW#>FD$\")BK7$)F/F+F+F+7,$\"\"\"F:$\")T&\\W(FDF9$!)31MIFN$\")@b8DFN $!)5(*G@FD$\")l&p%*)F/$!);W\"f&!#5F+F+7,F[o$\")5Z>jF*F9F9$\")MDtKF/$\" )-d_HF/$\")wdh;F/$\")jz*R\"FD$!)6t^7FDF+7,%\"bGF[pF9F9F]pF_pFapFcpFepF +7,%#b*G$\")<1BhF*F9F9$\")>:XLF/$\")\\2$o#F/$\")6oW?F/$\")(y$z**F/$!)& eFl)F/$!)?V$=\"FioQ(pprint36\"" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 146 "RK6_8eqs := [op(RowSumCondi tions(8,'expanded')),op(OrderConditions(6,8,'expanded'))]:\n`RK5_9eqs* ` := subs(b=`b*`,OrderConditions(5,9,'expanded')):" }}}{PARA 0 "" 0 " " {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 7 "Check: " }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 148 "simplify(subs(ee,RK6_8eqs)):\nmap(u->`if `(lhs(u)=rhs(u),0,1),%);\nnops(%);\nsimplify(subs(ee,`RK5_9eqs*`)):\nm ap(u->`if`(lhs(u)=rhs(u),0,1),%);\nnops(%);" }{TEXT -1 0 "" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#7N\"\"!F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F $F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#\"#W" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#73\"\"!F$F$F$F$ F$F$F$F$F$F$F$F$F$F$F$F$" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#\"#<" }}} {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 86 "Next we s et-up stage-order condtions to check for stage-orders from 2 to 5 incl usive. " }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 85 "for ct from 2 to \+ 5 do\n so||ct||_8 := StageOrderConditions(ct,8,'expanded');\nend do: " }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 34 "Stag es 4 to 8 have stage-order 3. " }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 173 "[seq([seq(expand(subs(ee,so||i||_8[j])),i=2..5)],j=1..6)]:\nmap (proc(L) local i; for i to nops(L) do if not evalb(L[i]) then break en d if end do; i end proc,%):\nsimplify(%); " }}{PARA 11 "" 1 "" {XPPMATH 20 "6#7(\"\"#\"\"$F%F%F%F%" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 53 "None of the principal error conditions \+ are satisfied." }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 123 "RK6_8err_ eqs := PrincipalErrorConditions(6,8,'expanded'):\nexpand(subs(ee,RK6_8 err_eqs)):\nmap(u->`if`(lhs(u)=rhs(u),0,1),%);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#7R\"\"\"F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F $F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 114 "We can calculate the pri ncipal error norm of the order 6 scheme, that is, the 2-norm of the pr incipal error terms." }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 131 "err terms6_8 := PrincipalErrorTerms(6,8,'expanded'):\nevalf(evalf[14](sqrt (add(subs(ee,errterms6_8[i])^2,i=1.. nops(errterms6_8)))));" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#$\"+.;%*G6!#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*`,PrincipalErrorTer ms(5,9,'expanded')):\nevalf(evalf[14](sqrt(add(subs(ee,`errterms5_9*`[ i])^2,i=1.. nops(`errterms5_9*`)))));" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#$\"+4)o&*>'!#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 stage-order equations to ensure that stage 2 has stage-order 2 an d 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 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] = 1/11;" "6# /&%\"cG6#\"\"#*&\"\"\"F)\"#6!\"\"" }{TEXT -1 3 ", " }{XPPEDIT 18 0 "c [4] = 157/736;" "6#/&%\"cG6#\"\"%*&\"$d\"\"\"\"\"$O(!\"\"" }{TEXT -1 3 ", " }{XPPEDIT 18 0 "c[5] = 123/218;" "6#/&%\"cG6#\"\"&*&\"$B\"\"\" \"\"$=#!\"\"" }{TEXT -1 3 ", " }{XPPEDIT 18 0 "c[6] = 73/99;" "6#/&% \"cG6#\"\"'*&\"#t\"\"\"\"#**!\"\"" }{TEXT -1 3 ", " }{XPPEDIT 18 0 "c [7] = 199/200;" "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 -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] = -1/845;" "6#/&%#b*G6#\" \"*,$*&\"\"\"F*\"$X)!\"\"F," }{TEXT -1 1 "." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 37 "We have 44 equations and 44 unk nowns." }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 215 "e1 := \{c[2]=1/11 ,c[4]=157/736,c[5]=123/218,c[6]=73/99,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/ 845\}:\neqns := subs(e1,cdns):\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)\}):\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 2366 "e3 := \{`b*`[2] = 0, a[5,1 ] = 42928977471/63842339642, a[7,1] = 23742496624480179224047764371959 8303/34580776953279718763722454600000000, c[9] = 1, c[8] = 1, b[3] = 0 , b[2] = 0, c[6] = 73/99, c[7] = 199/200, a[9,6] = 1147778420925807/69 07762998459050, a[6,3] = 1697439018580941844337032/1529827424367174730 12293, a[8,5] = -4712736023245029582064053285795650/221362152840661000 2679128666238199, c[4] = 157/736, c[5] = 123/218, c[2] = 1/11, a[7,2] \+ = 0, a[6,2] = 0, a[4,2] = 0, `b*`[3] = 0, a[8,2] = 0, a[5,2] = 0, a[9, 2] = 0, a[9,3] = 0, a[7,4] = 128453834152286986408251909051997320271/5 544554200138077926323128675315625000, b[4] = 6900573846223978496/21081 697499495907225, c[3] = 157/1104, a[8,7] = -541736066625000000/9688665 4438663742381, a[4,1] = 157/2944, a[2,1] = 1/11, a[9,7] = 575779000000 000/411330558898641, a[3,2] = 271139/2437632, a[8,1] = 597326620751868 2347729280873413/802324115518564191458832770081, a[6,5] = 183506358169 025606635433230/193412520623391990592938849, `b*`[8] = -46744242804719 95/5402235928512618, a[3,1] = 75517/2437632, b[1] = 531842063/84159269 10, `b*`[7] = 2584290081034720820000000/2589630224498721525333771, `b* `[1] = 648854567388972967/10596897420732888642, a[7,6] = 1994237771077 39967674947/239913433051724600000000, `b*`[9] = -1/845, `b*`[6] = 3293 4117770434118898396/161072145715862713160465, a[6,1] = -45299372934611 5584620105837/169351895877446242624608351, b[6] = 1147778420925807/690 7762998459050, a[7,5] = -972708878592287689266824375699676121/50543989 1163443236912399043150000000, a[5,4] = 78364952136/31921169821, a[6,4] = -334578912430118682410335643360/38759554641024302244921542187, a[9, 1] = 531842063/8415926910, b[8] = -13426037/10725975, `b*`[5] = 585003 28551366236723248598/218034649578069682785496485, b[5] = 5112684307993 4806/173160464034159675, a[8,3] = -40870912569246870680146344/13470696 54182311812082183, a[7,3] = -20179674002037028690247750335371/72211779 4714327571912012500000, b[7] = 575779000000000/411330558898641, a[9,5] = 51126843079934806/173160464034159675, a[9,4] = 6900573846223978496/ 21081697499495907225, a[9,8] = -13426037/10725975, a[5,3] = -818188725 78/31921169821, a[8,4] = 112459036656658044255344022793238112/44741080 65454386923650087599365353, a[4,3] = 471/2944, a[8,6] = 12765776550230 9225898/142682894776218689669, `b*`[4] = 8879698038617778833446469632/ 26544976952167847097500780295\}:" }{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 "s ubs(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%'matrix G6#787&#\"\"\"\"#6F(%!GF+7&#\"$d\"\"%/6#\"&$7&F+F+F+#\",O@&\\OyFF7&#\"#t\"#**#!(=U:#\\C-V-TYbf(Q#\"RB1_7M>7&#\"$* >\"$+##\"E.$)f>Pkx/C#z,[Cm\\UP#\"D++++YXAPw=(zK&px!eMF:#!Ar`L]xC!pGq.- Snz,#\"?++]7?\">dFVr%z<@s7&F+#\"Hr-K(*>04>D3k)pG_T$QXG\"\"F+]i:`nGJKEz 2Q,?aXa&#!E@hn*pvV#oE*o(G#fy)3F(*\"E+++]J/*R7pBVM;\"*)Ra]#\"9Z\\nn*Rx5 xPU*>\"9++++Ys^IV8*R#7&F)#\"@8M(3GHxM#o=v?mK(f\"?\"3qF$)e9>k&=b6CB!)F: #!;Wj9!oqoCpD\"4(3%\":$=#37=J#=a'pqM\"7&F+#\"E7\"QKzASMbU/emlO!fC6\"C` `O*f(3]O#pQaa13TZ%#!C]cz&G`S1#eH]CBgt7Z\"C*>QimG\"zE+5mSG:i8A#\"6)*eA4 B]lxlF\"\"6p'*o=ix%*GoU\"7&F+F+F+#!3+++Dm1O&H!y+v4Zy;_p(\\aE#\";)f[KsOiO^&G.]e\"<&['\\&y#op!y&\\Y. =##\"8'R)*)=TVqx6MH$\"9l/;8F'erX@2h\"7&F+#\":+++?3sM53!H%e#\":rPLD:s) \\C-j*e##!1&*>Z!GCWn%\"1=E^GfB-a#!\"\"\"$X)Q(pprint06\"" }}}{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`,s eq(b[i],i=1..8),``],[`b*`,seq(`b*`[i],i=1..9)]])):\nevalf(%);" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#K%'matrixG6#7,7,$\"+\"4444*!#6F(%!GF+F +F+F+F+F+F+7,$\"+\\95A9!#5$\"+nb'z4$F*$\"+$*[I76F/F+F+F+F+F+F+F+7,$\"+ u@:L@F/$\"+N/)GL&F*$\"\"!F:$\"+IT')*f\"F/F+F+F+F+F+F+7,$\"+N=?UcF/$\"+ du@CnF/F9$!+_R:jD!\"*$\"+!R_\\X#FDF+F+F+F+F+7,$\"+uttttF/$!+(fm[n#FDF9 $\"+?Cc46!\")$!+pg;K')FD$\"+q?#y[*F/F+F+F+F+7,$\"++++]**F/$\"+Xx!e'oFD F9$!+sF^%z#FN$\"+env;BFN$!+N)zW#>FD$\"+kAK7$)F/F+F+F+7,$\"\"\"F:$\"+$4 a\\W(FDF9$!+=31MIFN$\"+(3_N^#FN$!+$*4(*G@FD$\"+:l&p%*)F/$!+k:W\"f&!#7F +F+7,F[o$\"+I5Z>jF*F9F9$\"+@MDtKF/$\"+G-d_HF/$\"+#fx:m\"F/$\"+9jz*R\"F D$!+[6t^7FDF+7,%\"bGF[pF9F9F]pF_pFapFcpFepF+7,%#b*G$\"++<1BhF*F9F9$\"+ 6>:XLF/$\"+a[2$o#F/$\"+N6oW?F/$\"+T(y$z**F/$!+#\\eFl)F/$!+`>V$=\"FioQ( pprint16\"" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 " " {MPLTEXT 1 0 146 "RK6_8eqs := [op(RowSumConditions(8,'expanded')),op (OrderConditions(6,8,'expanded'))]:\n`RK5_9eqs*` := subs(b=`b*`,OrderC onditions(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#&%\"c G6#\"\"&" }{TEXT -1 1 " " }}{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 *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]^3-1 20*c[6]*c[5]*c[4]^8-10*c[6]*c[7]^2*c[4]^5-6*c[6]*c[4]^6+780*c[5]^4*c[4 ]^4*c[7]+32*c[7]^2*c[6]*c[4]^4+1200*c[5]^3*c[7]^2*c[4]^6*c[6]-750*c[6] *c[4]^5*c[5]*c[7]^2-6*c[4]^4*c[7]^2-400*c[5]^4*c[7]^2*c[4]^5*c[6]^2+41 5*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]^2-530*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+2 00*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]+28 5*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[ 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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] 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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* 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1100*c[5]^2*c[4]^4*c[7]*c[6]^2+2380*c[5]^3*c[4]^4*c[6]*c[7]-50*c[5]^4* c[4]^3*c[6]*c[7]-594*c[5]^3*c[4]^3-10*c[5]^5*c[6]*c[7]-3*c[5]^5*c[4]+3 28*c[7]^2*c[5]^3*c[4]^2+166*c[5]^5*c[6]*c[4]^2+150*c[5]^4*c[4]^5*c[6]^ 2-200*c[6]*c[7]^2*c[5]^4*c[4]^5-680*c[5]^4*c[4]^4*c[6]^2+850*c[5]^4*c[ 4]^3*c[6]^2+300*c[7]^2*c[5]^4*c[6]^2*c[4]^3-100*c[7]^2*c[5]^4*c[6]^2*c [4]^2-2570*c[5]^3*c[4]^4*c[7]^2*c[6]-1130*c[5]^4*c[4]^3*c[7]^2*c[6]+18 40*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]^4-1 067*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]+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] = 73/99;" "6#/&%\"cG6 #\"\"'*&\"#t\"\"\"\"#**!\"\"" }{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 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 73 "eA := \+ \{c[6]=73/99,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 16601 "eB := \{`b*`[2] = 0, a[9,6] = 3169966833/193633960*(990*c[5]*c[4]-395 *c[5]-395*c[4]+197)/(9801*c[5]*c[4]-7227*c[4]-7227*c[5]+5329), c[9] = \+ 1, `b*`[4] = -1/60*(10171176*c[4]+20384022*c[5]+27211480*c[5]^3-430870 36*c[5]^2+59923740*c[5]*c[4]^2+101900700*c[5]^3*c[4]^2-113634860*c[5]^ 3*c[4]+169487890*c[5]^2*c[4]-74680271*c[5]*c[4]-7843120*c[4]^2-1429042 10*c[5]^2*c[4]^2-2949711)/(194244*c[4]+150444*c[5]-116216*c[5]^2+10725 60*c[5]*c[4]^2+438000*c[5]^2*c[4]^3+602360*c[5]^2*c[4]-735265*c[5]*c[4 ]-267416*c[4]^2-923810*c[5]^2*c[4]^2-488000*c[5]*c[4]^3+116800*c[4]^3- 43581)/(99*c[4]-73)/(c[4]-c[5])/c[4], c[8] = 1, b[3] = 0, b[2] = 0, c[ 6] = 73/99, c[7] = 199/200, a[8,6] = -25227774/372373*(2452490*c[5]*c[ 4]+490597-981095*c[4]-981095*c[5])*(c[4]-1)*(-1+c[5])/(99*c[4]-73)/(99 *c[5]-73)/(46790*c[5]*c[4]-13625*c[5]-13625*c[4]+4293), a[9,1] = 1/871 620*(191800*c[5]*c[4]-33165*c[5]-33165*c[4]+9332)/c[5]/c[4], `b*`[5] = 1/60*(20384022*c[4]+10171176*c[5]-7843120*c[5]^2+169487890*c[5]*c[4]^ 2+101900700*c[5]^2*c[4]^3+59923740*c[5]^2*c[4]-74680271*c[5]*c[4]-4308 7036*c[4]^2-142904210*c[5]^2*c[4]^2-113634860*c[5]*c[4]^3+27211480*c[4 ]^3-2949711)/(150444*c[4]+194244*c[5]+116800*c[5]^3-267416*c[5]^2+6023 60*c[5]*c[4]^2+438000*c[5]^3*c[4]^2-488000*c[5]^3*c[4]+1072560*c[5]^2* c[4]-735265*c[5]*c[4]-116216*c[4]^2-923810*c[5]^2*c[4]^2-43581)/(99*c[ 5]-73)/(c[4]-c[5])/c[5], a[8,1] = 1/58108*(42995656350*c[5]^5*c[4]^3-4 06276386880*c[5]^4*c[4]^3+27508427400*c[5]^3*c[4]^6-286428071440*c[5]^ 3*c[4]^4-62171833800*c[4]^6*c[5]^2-8377720900*c[5]^5*c[4]^2+1675544180 0*c[4]^6*c[5]-255996188700*c[5]^4*c[4]^5+37179482800*c[5]^5*c[4]^5+146 246727060*c[4]^5*c[5]^2-4333433154*c[5]^3+1253476722*c[5]^2-6697953870 0*c[5]^5*c[4]^4-46865798590*c[4]^5*c[5]+8906560619*c[5]*c[4]^2+1208790 5279*c[5]^2*c[4]^3-105734589313*c[4]^4*c[5]^2+166144546665*c[5]^4*c[4] ^2-162550834555*c[5]^3*c[4]^2+39925986002*c[5]^3*c[4]-7080080909*c[5]^ 2*c[4]-1462389509*c[5]*c[4]+61810446450*c[4]^5*c[5]^3+49274456148*c[4] ^4*c[5]+481937823600*c[5]^4*c[4]^4+417825574*c[4]^2+329473484442*c[5]^ 3*c[4]^3-793982630*c[4]^5+3351088360*c[5]^4+2102094210*c[4]^4-34922844 810*c[5]^4*c[4]+13967645258*c[5]^2*c[4]^2-26472704504*c[5]*c[4]^3-1725 937154*c[4]^3+37179482800*c[5]^4*c[4]^6)/c[5]/c[4]^2/(467900*c[5]^4*c[ 4]^3+467900*c[5]^3*c[4]^4+13625*c[5]^3-4293*c[5]^2+25758*c[5]*c[4]^2+7 32420*c[5]^2*c[4]^3-136250*c[4]^4*c[5]^2-136250*c[5]^4*c[4]^2+732420*c [5]^3*c[4]^2-128540*c[5]^3*c[4]+25758*c[5]^2*c[4]+4293*c[5]*c[4]-4293* c[4]^2-1676200*c[5]^3*c[4]^3-245500*c[5]^2*c[4]^2-128540*c[5]*c[4]^3+1 3625*c[4]^3), b[1] = 1/871620*(191800*c[5]*c[4]-33165*c[5]-33165*c[4]+ 9332)/c[5]/c[4], a[9,5] = -1/60*(33165*c[4]-9332)/c[5]/(19800*c[5]^3*c [4]-54101*c[5]^2*c[4]+48828*c[5]*c[4]-14527*c[4]-19800*c[5]^4+54101*c[ 5]^3-48828*c[5]^2+14527*c[5]), b[6] = 3169966833/193633960*(990*c[5]*c [4]-395*c[5]-395*c[4]+197)/(99*c[4]-73)/(99*c[5]-73), a[8,7] = -416000 00000/1015099*(235*c[5]*c[4]-68*c[5]-68*c[4]+21)*(c[4]-1)*(-1+c[5])/(2 00*c[5]-199)/(200*c[4]-199)/(46790*c[5]*c[4]-13625*c[5]-13625*c[4]+429 3), b[4] = 1/60*(33165*c[5]-9332)/c[4]/(-54101*c[4]^3+19800*c[4]^4+488 28*c[4]^2-14527*c[4]-19800*c[5]*c[4]^3+54101*c[5]*c[4]^2-48828*c[5]*c[ 4]+14527*c[5]), a[6,4] = -73/192119202*(-14309460*c[5]^4*c[4]^3-429283 80*c[5]^3*c[4]^4+42928380*c[4]^5*c[5]^2-4220568*c[5]^3+2334102*c[5]^2- 21102840*c[4]^5*c[5]+12501834*c[5]*c[4]^2+14508957*c[5]^2*c[4]^3-85856 760*c[4]^4*c[5]^2+10551420*c[5]^4*c[4]^2-116389251*c[5]^3*c[4]^2+33193 611*c[5]^3*c[4]-12949470*c[5]^2*c[4]-2723119*c[5]*c[4]+53860018*c[4]^4 *c[5]+778034*c[4]^2+138975210*c[5]^3*c[4]^3+32041233*c[5]^2*c[4]^2-401 87157*c[5]*c[4]^3-1055142*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, `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], 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, b[8] = 1/1560*(46790*c[5]*c[4]-13625*c[5]-13625*c[4]+4293)/(c[4]-1)/(-1+c[ 5]), b[7] = -16000000000/3045297*(235*c[5]*c[4]-68*c[5]-68*c[4]+21)/(2 00*c[4]-199)/(200*c[5]-199), a[7,6] = -984949544601/233600000000*(5*c[ 5]*c[4]+1-2*c[4]-2*c[5])*(200*c[4]-199)*(200*c[5]-199)/(235*c[5]*c[4]- 68*c[5]-68*c[4]+21)/(99*c[4]-73)/(99*c[5]-73), `b*`[9] = 1/10*(3650*c[ 5]^2*c[4]^2-3415*c[5]^2*c[4]+730*c[5]^2-3415*c[5]*c[4]^2+3545*c[5]*c[4 ]-829*c[5]+730*c[4]^2-829*c[4]+219)/(219-2440*c[5]*c[4]^2-2440*c[5]^2* c[4]+584*c[5]^2-756*c[4]-756*c[5]+2935*c[5]*c[4]+2190*c[5]^2*c[4]^2+58 4*c[4]^2), a[6,5] = -73/192119202*(25383105*c[5]^2*c[4]^2-1055142*c[4] ^2+389017*c[4]-8477271*c[5]^2*c[4]-17285889*c[5]*c[4]^2+715473*c[4]^3- 10551420*c[4]^4*c[5]+22637171*c[5]*c[4]^3-32015610*c[5]^2*c[4]^3+57233 46*c[5]*c[4]+14309460*c[4]^4*c[5]^2+1055142*c[5]^2-778034*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] = -1/2*(92812580000*c[5]^5*c[4]^3-294987970660*c[5]^4*c[4]^3-3 09596948320*c[5]^3*c[4]^4-47216107000*c[5]^5*c[4]^2-91353045400*c[5]^4 *c[4]^5+208912787*c[4]+28618920000*c[5]^5*c[4]^5-50867303600*c[4]^5*c[ 5]^2-417825574*c[5]-4468994058*c[5]^3+2290313362*c[5]^2-84702760000*c[ 5]^5*c[4]^4+8377720900*c[4]^5*c[5]-10557561094*c[5]*c[4]^2-15827491475 3*c[5]^2*c[4]^3+147978728155*c[4]^4*c[5]^2+149958791825*c[5]^4*c[4]^2- 170551614506*c[5]^3*c[4]^2+42876253824*c[5]^3*c[4]-20356282516*c[5]^2* c[4]+2876639340*c[5]*c[4]+105223708100*c[4]^5*c[5]^3-23160636920*c[4]^ 4*c[5]+269878608400*c[5]^4*c[4]^4-862968577*c[4]^2+336517594960*c[5]^3 *c[4]^3+3738372270*c[5]^4+11629233000*c[5]^5*c[4]-396991315*c[4]^4-372 34756435*c[5]^4*c[4]+79229459352*c[5]^2*c[4]^2+22881663348*c[5]*c[4]^3 +1051047105*c[4]^3-1141866000*c[5]^5)/c[5]/(-59619008250*c[5]^5*c[4]^3 -9264420000*c[5]^7*c[4]^3+20055045650*c[5]^4*c[4]^3-50038684070*c[5]^3 *c[4]^4+29417850170*c[5]^5*c[4]^2-18747187900*c[5]^4*c[4]^5+9264420000 *c[5]^5*c[4]^5-19175427250*c[5]^6*c[4]^2-1979303750*c[4]^5*c[5]^2+6236 4411*c[5]^3-30491010000*c[5]^5*c[4]^4+124728822*c[5]*c[4]^2-2200253353 *c[5]^2*c[4]^3+15318690880*c[4]^4*c[5]^2-14651710300*c[5]^4*c[4]^2+186 9080720*c[5]^3*c[4]^2+118252295*c[5]^3*c[4]-124728822*c[5]^2*c[4]+1147 0694550*c[4]^5*c[5]^3-2334651705*c[4]^4*c[5]+67323740950*c[5]^4*c[4]^4 +4252078360*c[5]^3*c[4]^3-345184568*c[5]^4-4649283940*c[5]^5*c[4]+1979 30375*c[4]^4+46540447900*c[5]^6*c[4]^3+2113471813*c[5]^4*c[4]-29450838 6*c[5]^2*c[4]^2+323510284*c[5]*c[4]^3-62364411*c[4]^3-269775000*c[5]^6 +552352525*c[5]^5+2545092000*c[5]^6*c[4]+2697750000*c[5]^7*c[4]^2), 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[5] = -1/60*(33165*c[4]-9332)/(c[4]-c[5])/c[5]/(19800*c[5]^ 3-54101*c[5]^2+48828*c[5]-14527), a[7,1] = 199/467200000000*(861381303 8000*c[5]^5*c[4]^3-81413307765910*c[5]^4*c[4]^3+5524253580000*c[5]^3*c [4]^6-57277654290870*c[5]^3*c[4]^4-12444944452000*c[4]^6*c[5]^2-167554 4180000*c[5]^5*c[4]^2+3351088360000*c[4]^6*c[5]-51575676840000*c[5]^4* c[4]^5+7509596000000*c[5]^5*c[4]^5+29188793385740*c[4]^5*c[5]^2-863606 674368*c[5]^3+249441867678*c[5]^2-13468302420000*c[5]^5*c[4]^4-9349541 483400*c[4]^5*c[5]+1773625518950*c[5]*c[4]^2+2412700401128*c[5]^2*c[4] ^3-21064611142120*c[4]^4*c[5]^2+33262490787180*c[5]^4*c[4]^2-324103987 32576*c[5]^3*c[4]^2+7958274924054*c[5]^3*c[4]-1408902579938*c[5]^2*c[4 ]-291015512291*c[5]*c[4]+12423263913900*c[4]^5*c[5]^3+9815947899574*c[ 4]^4*c[5]+96790541403900*c[5]^4*c[4]^4+83147289226*c[4]^2+657441152647 10*c[5]^3*c[4]^3-159422720800*c[4]^5+670217672000*c[5]^4+420424196396* c[4]^4-6988563084800*c[5]^4*c[4]+2777786456469*c[5]^2*c[4]^2-527313867 7514*c[5]*c[4]^3-344054711054*c[4]^3+7509596000000*c[5]^4*c[4]^6)/(235 0*c[5]^4*c[4]^3+2350*c[5]^3*c[4]^4+68*c[5]^3-21*c[5]^2+126*c[5]*c[4]^2 +3660*c[5]^2*c[4]^3-680*c[4]^4*c[5]^2-680*c[5]^4*c[4]^2+3660*c[5]^3*c[ 4]^2-643*c[5]^3*c[4]+126*c[5]^2*c[4]+21*c[5]*c[4]-21*c[4]^2-8410*c[5]^ 3*c[4]^3-1211*c[5]^2*c[4]^2-643*c[5]*c[4]^3+68*c[4]^3)/c[5]/c[4]^2, a[ 6,3] = 73/42693156*(26930009*c[5]*c[4]^3-18680700*c[5]*c[4]^2+5371632* c[5]*c[4]-778034*c[5]+389017*c[4]-3517140*c[5]^3*c[4]^2-56996940*c[5]^ 2*c[4]^3+40227363*c[5]^2*c[4]^2-11303028*c[5]^2*c[4]+1406856*c[5]^2-10 551420*c[4]^4*c[5]-703428*c[4]^2+7154730*c[5]^3*c[4]^3+21464190*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), a[7,5] = -199/3200 000000*(18497990826000*c[5]^4*c[4]^3+36941023646600*c[5]^3*c[4]^4+5723 784000000*c[5]^4*c[4]^5-41573644613*c[4]+8494785566000*c[4]^5*c[5]^2+8 3147289226*c[5]+516519558330*c[5]^3-372537373895*c[5]^2-1675544180000* c[4]^5*c[5]+2275041434693*c[5]*c[4]^2+26767360263420*c[5]^2*c[4]^3-248 24372874170*c[4]^4*c[5]^2-9404640766000*c[5]^4*c[4]^2+20456644535770*c [5]^3*c[4]^2-5096140400035*c[5]^3*c[4]+3438007180584*c[5]^2*c[4]-61453 0394764*c[5]*c[4]-12540211880000*c[4]^5*c[5]^3+4700608021700*c[4]^4*c[ 5]-16905319880000*c[5]^4*c[4]^4+172027355527*c[4]^2-40277495907870*c[5 ]^3*c[4]^3-227231334000*c[5]^4+79711360400*c[4]^4+2315359233000*c[5]^4 *c[4]-13503766260132*c[5]^2*c[4]^2-4768433136187*c[5]*c[4]^3-210212098 198*c[4]^3)/c[5]/(936820*c[5]^5*c[4]^3-564290*c[5]^4*c[4]^3+926630*c[5 ]^3*c[4]^4-411980*c[5]^5*c[4]^2+232650*c[5]^4*c[4]^5+67320*c[5]^6*c[4] ^2+49640*c[4]^5*c[5]^2-1533*c[5]^3-3066*c[5]*c[4]^2+47206*c[5]^2*c[4]^ 3-330837*c[4]^4*c[5]^2+323412*c[5]^4*c[4]^2-41464*c[5]^3*c[4]^2+76*c[5 ]^3*c[4]+3066*c[5]^2*c[4]-238870*c[4]^5*c[5]^3+53671*c[4]^4*c[5]-76527 0*c[5]^4*c[4]^4-56232*c[5]^3*c[4]^3+7043*c[5]^4+63657*c[5]^5*c[4]-4964 *c[4]^4-232650*c[5]^6*c[4]^3-52681*c[5]^4*c[4]+4158*c[5]^2*c[4]^2-6313 *c[5]*c[4]^3+1533*c[4]^3-6732*c[5]^5), a[6,1] = 73/384238404*(-3577365 0*c[5]^4*c[4]^3-195137910*c[5]^3*c[4]^4+28618920*c[4]^5*c[5]^2-4220568 *c[5]^3+2334102*c[5]^2-21102840*c[4]^5*c[5]+13556976*c[5]*c[4]^2-34728 297*c[5]^2*c[4]^3-9089784*c[4]^4*c[5]^2+10551420*c[5]^4*c[4]^2-1331489 61*c[5]^3*c[4]^2+35340030*c[5]^3*c[4]-14004612*c[5]^2*c[4]-2723119*c[5 ]*c[4]+38811960*c[4]^5*c[5]^3+41393146*c[4]^4*c[5]+38811960*c[5]^4*c[4 ]^4+778034*c[4]^2+237206376*c[5]^3*c[4]^3+1430946*c[4]^4+37656612*c[5] ^2*c[4]^2-33140832*c[5]*c[4]^3-2110284*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[7,4] = -199/3200000000*(-8494785566000*c[5] ^5*c[4]^3+106822409459910*c[5]^4*c[4]^3-47004408000000*c[5]^3*c[4]^6+7 7455918156570*c[5]^3*c[4]^4+70365564289100*c[4]^6*c[5]^2+1675544180000 *c[5]^5*c[4]^2-18295779622600*c[4]^6*c[5]+82949152460000*c[5]^4*c[4]^5 -5723784000000*c[5]^5*c[4]^5-91476038445840*c[4]^5*c[5]^2+863606674368 *c[5]^3-249441867678*c[5]^2+12540211880000*c[5]^5*c[4]^4+2756120046582 0*c[4]^5*c[5]-2084011662386*c[5]*c[4]^2-4472164513378*c[5]^2*c[4]^3+47 320511740910*c[4]^4*c[5]^2-40044342948980*c[5]^4*c[4]^2+40229683516681 *c[5]^3*c[4]^2-8833940078644*c[5]^3*c[4]+1662908027255*c[5]^2*c[4]+291 015512291*c[5]*c[4]+9187125910200*c[4]^5*c[5]^3-20127244906949*c[4]^4* c[5]-139536331351900*c[5]^4*c[4]^4-83147289226*c[4]^2-89077341311640*c [5]^3*c[4]^3+227231334000*c[4]^5-670217672000*c[5]^4-516519558330*c[4] ^4+7658354339600*c[5]^4*c[4]-4107845654307*c[5]^2*c[4]^2+8109166887872 *c[5]*c[4]^3+372537373895*c[4]^3-17171352000000*c[5]^4*c[4]^6-19039424 580000*c[4]^7*c[5]^2+4544626680000*c[5]*c[4]^7+17171352000000*c[5]^3*c [4]^7)/(-2350*c[5]^5*c[4]^3+7730*c[5]^4*c[4]^3-7730*c[5]^3*c[4]^4+680* c[5]^5*c[4]^2-680*c[4]^5*c[5]^2+21*c[5]^3+42*c[5]*c[4]^2-568*c[5]^2*c[ 4]^3+3660*c[4]^4*c[5]^2-3660*c[5]^4*c[4]^2+568*c[5]^3*c[4]^2-58*c[5]^3 *c[4]-42*c[5]^2*c[4]+2350*c[4]^5*c[5]^3-643*c[4]^4*c[5]-68*c[5]^4+68*c [4]^4+643*c[5]^4*c[4]+58*c[5]*c[4]^3-21*c[4]^3)/(99*c[4]-73)/c[4]^2, ` b*`[6] = 32019867/7447460*(-663168*c[4]-663168*c[5]-576700*c[5]^3+1091 874*c[5]^2-6416454*c[5]*c[4]^2-6995390*c[5]^2*c[4]^3-6995390*c[5]^3*c[ 4]^2+3566550*c[5]^3*c[4]-6416454*c[5]^2*c[4]+3662697*c[5]*c[4]+1091874 *c[4]^2+4336200*c[5]^3*c[4]^3+11929650*c[5]^2*c[4]^2+3566550*c[5]*c[4] ^3-576700*c[4]^3+129429)/(219-2440*c[5]*c[4]^2-2440*c[5]^2*c[4]+584*c[ 5]^2-756*c[4]-756*c[5]+2935*c[5]*c[4]+2190*c[5]^2*c[4]^2+584*c[4]^2)/( 9801*c[5]*c[4]-7227*c[4]-7227*c[5]+5329), `b*`[7] = -80000000/3045297* (-87705*c[4]-87705*c[5]-99280*c[5]^3+169992*c[5]^2-1172829*c[5]*c[4]^2 -1559790*c[5]^2*c[4]^3-1559790*c[5]^3*c[4]^2+708260*c[5]^3*c[4]-117282 9*c[5]^2*c[4]+581067*c[5]*c[4]+169992*c[4]^2+1029300*c[5]^3*c[4]^3+247 8580*c[5]^2*c[4]^2+708260*c[5]*c[4]^3-99280*c[4]^3+13797)/(-38654556*c [4]-38654556*c[5]-23243200*c[5]^3+53215784*c[5]^2-266922640*c[5]*c[4]^ 2-184762000*c[5]^2*c[4]^3-184762000*c[5]^3*c[4]^2+120472000*c[5]^3*c[4 ]-266922640*c[5]^2*c[4]+185166535*c[5]*c[4]+53215784*c[4]^2+87600000*c [5]^3*c[4]^3+398350190*c[5]^2*c[4]^2+120472000*c[5]*c[4]^3-23243200*c[ 4]^3+8672619), a[7,3] = 597/6400000000*(-96158710000*c[5]^4*c[4]^3-904 978840000*c[5]^3*c[4]^4+1708505943*c[4]-288476130000*c[4]^5*c[5]^2-341 7011886*c[5]-9181064000*c[5]^3+11830228416*c[5]^2+68857980000*c[4]^5*c [5]-108393385461*c[5]*c[4]^2-1022465207910*c[5]^2*c[4]^3+945240831350* c[4]^4*c[5]^2+22952660000*c[5]^4*c[4]^2-432769992060*c[5]^3*c[4]^2+935 49868000*c[5]^3*c[4]-112439293146*c[5]^2*c[4]+26417185524*c[5]*c[4]+26 0172000000*c[4]^5*c[5]^3-226434716100*c[4]^4*c[5]+86724000000*c[5]^4*c [4]^4-5915114208*c[4]^2+969096475150*c[5]^3*c[4]^3+481520883170*c[5]^2 *c[4]^2+238933200070*c[5]*c[4]^3+4590532000*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)/(235*c[5]*c[4]-68*c[5]-68*c[4]+21), `b*`[1] = \+ 1/871620*(-20384022*c[4]+2949711-20384022*c[5]-27211480*c[5]^3+4308703 6*c[5]^2-376944086*c[5]*c[4]^2-683527560*c[5]^2*c[4]^3-683527560*c[5]^ 3*c[4]^2+252857540*c[5]^3*c[4]-376944086*c[5]^2*c[4]+162141233*c[5]*c[ 4]+43087036*c[4]^2+521942700*c[5]^3*c[4]^3+956247700*c[5]^2*c[4]^2+252 857540*c[5]*c[4]^3-27211480*c[4]^3)/(219-2440*c[5]*c[4]^2-2440*c[5]^2* c[4]+584*c[5]^2-756*c[4]-756*c[5]+2935*c[5]*c[4]+2190*c[5]^2*c[4]^2+58 4*c[4]^2)/c[5]/c[4], a[8,4] = -1/2*(50867303600*c[5]^5*c[4]^3-73459384 0635*c[5]^4*c[4]^3+278686325800*c[5]^3*c[4]^6-838023224030*c[5]^3*c[4] ^4-812015187820*c[4]^6*c[5]^2-8377720900*c[5]^5*c[4]^2+230422934110*c[ 4]^6*c[5]-1112876406500*c[5]^4*c[4]^5+91353045400*c[5]^5*c[4]^5-858567 60000*c[5]^4*c[4]^7+696109782615*c[4]^5*c[5]^2-4333433154*c[5]^3+12534 76722*c[5]^2-105223708100*c[5]^5*c[4]^4-239628040065*c[4]^5*c[5]+11418 66000*c[4]^6+11937350232*c[5]*c[4]^2+1544086110*c[5]^2*c[4]^3-25943553 2668*c[4]^4*c[5]^2+238556083460*c[5]^4*c[4]^2-246381050612*c[5]^3*c[4] ^2+48680445769*c[5]^3*c[4]-9617587841*c[5]^2*c[4]-1462389509*c[5]*c[4] +346234280540*c[4]^5*c[5]^3+141891004838*c[4]^4*c[5]+1232351244660*c[5 ]^4*c[4]^4+417825574*c[4]^2+649729411887*c[5]^3*c[4]^3-3738372270*c[4] ^5+3351088360*c[5]^4+4468994058*c[4]^4-41644425545*c[5]^4*c[4]+2906013 0582*c[5]^2*c[4]^2-51228868206*c[5]*c[4]^3-2290313362*c[4]^3-955607400 00*c[4]^8*c[5]^2+500713016200*c[5]^4*c[4]^6+448661572300*c[4]^7*c[5]^2 -114769311400*c[5]*c[4]^7-320449516200*c[5]^3*c[4]^7+85856760000*c[4]^ 8*c[5]^3+22837320000*c[5]*c[4]^8-28618920000*c[5]^5*c[4]^6)/(-467900*c [5]^5*c[4]^3+1539950*c[5]^4*c[4]^3-1539950*c[5]^3*c[4]^4+136250*c[5]^5 *c[4]^2-136250*c[4]^5*c[5]^2+4293*c[5]^3+8586*c[5]*c[4]^2-116960*c[5]^ 2*c[4]^3+732420*c[4]^4*c[5]^2-732420*c[5]^4*c[4]^2+116960*c[5]^3*c[4]^ 2-12133*c[5]^3*c[4]-8586*c[5]^2*c[4]+467900*c[4]^5*c[5]^3-128540*c[4]^ 4*c[5]-13625*c[5]^4+13625*c[4]^4+128540*c[5]^4*c[4]+12133*c[5]*c[4]^3- 4293*c[4]^3)/(-34301*c[4]+19800*c[4]^2+14527)/c[4]^2, a[8,3] = 3/4*(-2 413150*c[5]^4*c[4]^3-22661200*c[5]^3*c[4]^4+43143*c[4]-7239450*c[4]^5* c[5]^2-86286*c[5]-230680*c[5]^3+298302*c[5]^2+1730100*c[4]^5*c[5]-2734 830*c[5]*c[4]^2-25739520*c[5]^2*c[4]^3+23754880*c[4]^4*c[5]^2+576700*c [5]^4*c[4]^2-10864755*c[5]^3*c[4]^2+2349340*c[5]^3*c[4]-2834376*c[5]^2 *c[4]+666867*c[5]*c[4]+6504300*c[4]^5*c[5]^3-5697465*c[4]^4*c[5]+21681 00*c[5]^4*c[4]^4-149151*c[4]^2+24312200*c[5]^3*c[4]^3+12132299*c[5]^2* c[4]^2+6023125*c[5]*c[4]^3+115340*c[4]^3)/c[4]^2/(467900*c[5]^4*c[4]^3 +467900*c[5]^3*c[4]^4+13625*c[5]^3-4293*c[5]^2+25758*c[5]*c[4]^2+73242 0*c[5]^2*c[4]^3-136250*c[4]^4*c[5]^2-136250*c[5]^4*c[4]^2+732420*c[5]^ 3*c[4]^2-128540*c[5]^3*c[4]+25758*c[5]^2*c[4]+4293*c[5]*c[4]-4293*c[4] ^2-1676200*c[5]^3*c[4]^3-245500*c[5]^2*c[4]^2-128540*c[5]*c[4]^3+13625 *c[4]^3), a[9,7] = -16000000000/3045297*(235*c[5]*c[4]-68*c[5]-68*c[4] +21)/(40000*c[5]*c[4]-39800*c[4]-39800*c[5]+39601), a[9,8] = 1/1560*(4 6790*c[5]*c[4]-13625*c[5]-13625*c[4]+4293)/(-c[4]+c[5]*c[4]+1-c[5]), a [9,4] = 1/60*(33165*c[5]-9332)/c[4]/(-54101*c[4]^3+19800*c[4]^4+48828* c[4]^2-14527*c[4]-19800*c[5]*c[4]^3+54101*c[5]*c[4]^2-48828*c[5]*c[4]+ 14527*c[5])\}:" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> \+ " 0 "" {MPLTEXT 1 0 1 ";" }}}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 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 4572 "prin_ err_norm_sqrd := (u,v,w)->1/1280371276800000000*(-30539392089683400*u^ 2*w^2*v^3-207829906479846000*u^2*v^5*w^2+7604366664006000*u^2*w^3*v-78 87933537876000*w^8*v^5+20837153942973028*v^4*w^2-8546940072942172*w^4* v^2+2675272453913224*w^3*v^2-234702554273600000*v^9*w^2+11098654993104 00*w^8*v^4+531645366854000000*v^10*w^4-334073857768000000*v^10*w^3-341 3030775406776*w^2*v^3+253024520333216006*v^5*w^3+1043322809708844*v^4* w-343207085494549561*w^4*v^4+14015098260922500*w^8*v^6-882308245427585 0*w^3*v^3+1413852205936800000*v^9*w^3-4123159175173500*u^2*v^3*w-12295 865747214000*u^2*v^5*w+273398383115915400*u^2*w^3*v^3+5824384899214782 *w^5*v-245805201275922*w^3*v+2147055872733324000*v^8*w^5+4021677686336 80000*v^8*w^2-434522149855667980*w^4*v^6+28030196521845000*v^7*w^7-945 310689596763000*v^7*w^6+202831869186922500*v^8*w^6+1968229144918235680 *w^3*v^7-2332289579106960000*v^8*w^3-344918407980735780*w^6*v^3-194553 684511956*w^4*v+266389068202283*w^2*v^2-109500399019922*w*v^3-19682526 06695200000*v^9*w^4+88187765460568200*u^2*w^4*v^2-94530181372200*u^2*w *v-1134362176466400*u^2*w^2*v-221973099862080*w^7*v^2-4906943520360000 00*v^9*w^5-1713912679447140600*w^6*v^5+967509431604076020*w^6*v^4-1970 3963256575700*w^7*v^4+4402414620800000*w*v^8+2965894361105579400*w^5*v ^6-358693096525526080*v^7*w^2-11594975125280000*w*v^7-9506307664036008 *w^6*v+95174755583994450*u^2*w^2*v^4-12686099905231200*v^5*w*u+1355237 971337313000*v^4*u^2*w^4-1746811735959150000*u^2*v^7*w^4+1579201569978 669000*v^5*u^2*w^3+272962110136275000*u^2*v^6*w^2+52481143064000000*v^ 10*w^2+94530181372200*u^2*w^2-464300661686400*u^2*w^3+570122887219200* u^2*w^4+50833807529400*w^3*u-241606986421500000*u^2*v^8*w^3+3795511239 4500000*u^2*v^8*w^2-169740240144300000*u^2*v^7*w^2+1072905089926950000 *u^2*v^7*w^3-1826042916141045000*u^2*v^6*w^3+17451588317625000*u^2*v^6 *w^6+4651782171120000*u^2*v^6*w+339254941222890000*u^2*v^5*w^5-4156913 83164750000*u^2*v^6*w^5-12674467635679800*u*v^3*w^2-11612441402100000* u^2*v^5*w^6+5289493790734200*u*v^4*w-2642847359647635000*u^2*v^5*w^4+6 6691868693085000*u*v^3*w^3-50606816526000000*v^9*w^2*u-659077475336100 0*u*v^2*w^3+2086195018131000*u*v^2*w^2+14873848848576000*v^6*w*u-20988 93402144000*u^2*v^2*w^5-512658032323500000*v^9*w^4*u+38449352424262500 0*u^2*v^8*w^4-130616030943090000*u^2*v^4*w^5+123515801980200*u*v^2*w+2 5513857138924000*u^2*v^3*w^5+226320320192400000*v^8*w^2*u+193175769402 0000*u^2*v^4*w^6-1202950548729000*u*v^3*w-1410945184517400000*v^8*w^3* u-5631528940380000*v^7*w*u+163829394902250000*u^2*v^7*w^5-338686428856 078800*w^3*u*v^4+322142648562000000*v^9*w^3*u+2380552531590375000*v^7* w^3*u-146935945306656900*v^5*w^2*u+50916700189434600*u*v^4*w^2-1707808 92666300*u*v*w^2+297601125399468000*v^6*w^2*u+1050253805693925000*w^3* v^5*u-17025511474198800*w^4*u*v^2+2776051908575955000*w^4*v^6*u+892268 857933020000*w^5*v^6*u+614487775436577000*w^5*v^4*u-415201108546776*w^ 5+109556072541104*v^6-5685405927894000*w^5*u*v-3560266578144690000*w^4 *v^7*u-1024819889539455000*w^5*v^5*u-16719975342000000*w^5*v^8*u-20503 01568857193000*w^3*v^6*u-10433469769430400*u^2*v*w^4+17473852755125100 *w^4*u*v^3+19706621665500000*w^6*v^7*u-20146175134200*w^3*u*v-37090723 3390260000*v^7*w^2*u+2203337685150000000*w^4*v^8*u-1140170832556056000 *w^4*v^5*u+3393535156365600*w^4*v*u-570847287780000*w^6*u*v^2-21761796 8136384000*w^5*u*v^3+6957753208560000*w^6*u*v^3-76564921148100000*w^6* v^6*u-268695281843100000*w^5*v^7*u+76072324055310000*w^6*v^5*u+4659706 1624481000*w^5*u*v^2+915406584498000*u^2*w*v^2+2909425306747680*w^7*v^ 3+40655766143179206*w^5*v^3-18477804887449776*w^3*v^4+2040686409308652 49*v^6*w^2-251117707064400*w^4*u+310118811408000*w^5*u+831553324931246 24*w^4*v^3-956662593347935780*w^3*v^6-29835308102875092*w^5*v^2+100423 21767203992*v^6*w+697578060899255960*w^4*v^5-1299328172982872820*w^5*v ^5-1046620443553215700*w^4*v^7+76116018004662849*w^6*v^2+7134355761515 1600*w^7*v^5-3617533125429968400*w^5*v^7-79372146811489492*v^5*w^2+235 876411679751960*w^5*v^4-155059405704000*v^5*u-4370103400238018*v^5*w+2 517434577502750400*v^8*w^4+1808655714784224900*v^6*w^6-107754390179163 000*w^7*v^6-86446036890776*v^5+17082749097761*v^4+207908119143783000*w ^4*v^4*u-818606792539620000*u^2*w^3*v^4-33315857671635000*w^6*v^4*u+71 34567938719200*u^2*w^2*v^2-436366876929228000*u^2*w^4*v^3+102404947585 12200*u^2*v^4*w-58841796726090000*u^2*w^3*v^2+83365194781761*w^4+51716 6820253104*w^6+3029384961729945000*u^2*v^6*w^4+23632545343050*u^2*v^2- 116075165421600*u^2*v^3+142530721804800*u^2*v^4-25416903764700*v^3*u+1 25558853532200*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+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 46 "Preliminary investigations suggest that when \+ " }{XPPEDIT 18 0 "c[2] = 7/76;" "6#/&%\"cG6#\"\"#*&\"\"(\"\"\"\"#w!\" \"" }{TEXT -1 3 ", " }{XPPEDIT 18 0 "c[4] = 14/65;" "6#/&%\"cG6#\"\"% *&\"#9\"\"\"\"#l!\"\"" }{TEXT -1 7 " and " }{XPPEDIT 18 0 "c[5] = 21 /37;" "6#/&%\"cG6#\"\"&*&\"#@\"\"\"\"#P!\"\"" }{TEXT -1 56 " the prin cipal error norm is close to a minimum (with " }{XPPEDIT 18 0 "c[6]" "6#&%\"cG6#\"\"'" }{TEXT -1 7 " and " }{XPPEDIT 18 0 "c[7]" "6#&%\"c G6#\"\"(" }{TEXT -1 9 " fixed)." }}{PARA 0 "" 0 "" {TEXT -1 97 "These values can be used as starting values to minimize the (square of the) principal error norm." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 119 "Using a one dimensional minimization procedure and \+ cycling around the nodes gives slow convergence towards the minimum." }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 491 "Digits := 30:\nc_2 := 7/7 6: c_4 := 14/65: c_5 := 21/37:\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=locat ion)[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_sq rd(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 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#\"\"#$\"?OmAy(=DtUig6-+;*!#J/ &F%6#\"\"%$\"??,%ofU9f%=`OfeW@!#I/&F%6#\"\"&$\"?w)*44>A.L#=y<#)*))o7S@!#I/&F%6#\"\"&$\"?HY\\f$e/Iv3;tA\\l&F1" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#$\"?3(3RT$)Q6+s4\\6#)G\"!#R" }}{PARA 11 "" 1 "" {XPPMATH 20 "6%/&%\"cG6#\"\"#$\"?*HJ^^\\8ZtOXI5J7*!#J/&F%6# \"\"%$\"?V$\\gF0;7Q()[$fLP@!#I/&F%6#\"\"&$\"?2M,ogd)H6(Q#zE)\\cF1" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#$\"?$fPfVEgZww\"[u7!#R" }}{PARA 11 "" 1 "" {XPPMATH 20 "6%/&%\"cG6#\"\"#$\"?!=.fl&QsBTxm5n,\"*!#J/&F%6#\"\"%$\"?! 3*)H\"H$e'oH0=b;L@!#I/&F%6#\"\"&$\"?*Q:4\\`ThOE:xRAk&F1" }}{PARA 11 " " 1 "" {XPPMATH 20 "6#$\"?_df\"))R9V>Zww\"[u7!#R" }}{PARA 11 "" 1 "" {XPPMATH 20 "6%/&%\"cG6#\"\"#$\"?Zt.![yBP7un1r;5*!#J/&F%6#\"\"%$\"?q-- \"*G$e'oH0=b;L@!#I/&F%6#\"\"&$\"?7\"[.X`ThOE:xRAk&F1" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#$\"?5VtH`VJ%>Zww\"[u7!#R" }}{PARA 11 "" 1 "" {XPPMATH 20 "6%/&%\"cG6#\"\"#$\"?/%>%\\;OsBTxm5n,\"*!#J/&F%6#\"\"%$\"? w%RJ$H$e'oH0=b;L@!#I/&F%6#\"\"&$\"?=7qDN:9mj_r(RAk&F1" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#$\"?%H9\\9X9V>Zww\"[u7!#R" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 79 "The following graphs give a visual check that we have found a (local) minimum. " }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 391 "c_2 := .91016710668e-1: pp := .127 448176765e-9:\np1 := evalf[30](plot(prin_err_norm_sqrd(c[2],.213316551 805,.56422397715),c[2]=0.088..0.094,\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=[0.088..0.094,1.2742e-10..1. 27564e-10]);" }}{PARA 13 "" 1 "" {GLPLOT2D 367 347 347 {PLOTDATA 2 "6* -%'CURVESG6$7S7$$\"#))!\"$$\"?%)>`WXLbCF4@Akv7!#R7$$\"?+++++++++]#HyI \"))!#J$\"?\")et2&\\$e,ND%yVbF\"F-7$$\"?++++++++]([kdW#))F1$\"?+Dm&prd s\"px$ohaF\"F-7$$\"?+++++++++v;\\DP))F1$\"?L?l9uH.W_W)Ht`F\"F-7$$\"?++ +++++++Dq0]*)F1$\"?. .$Q/Eg)=\\W$)[xu7F-7$$\"?+++++++++]U80j*)F1$\"?$p/GvC4NaWEzEZF\"F-7$$ \"?+++++++++]!ytb(*)F1$\"?9i,t)Hg7\"GvAXou7F-7$$\"?++++++++](QNXp)*)F1 $\"?wt$G[Vj>7xBg\\YF\"F-7$$\"?+++++++++]asY+!*F1$\"?D!>eGZ$yd+\"4P7YF \"F-7$$\"?++++++++++y?#>,*F1$\"?[px&QA&QB)*\\!*F1$\"?zzA8Od%)\\ehOe^u7F-7$$\"?++++++++]is VIi!*F1$\"?wn#)3sI(>Eb!H:]u7F-7$$\"?+++++++++vo:;v!*F1$\"?X/g)>6*G6rxG 2\\u7F-7$$\"?++++++++]P)[op3*F1$\"?URB\"HC'Q)QsS_%[u7F-7$$\"?+++++++++ DYQq*4*F1$\"?JfGs%**y9y$*p\"=[u7F-7$$\"?++++++++](QIKH6*F1$\"?pIsN$)4q qwt%Q$[u7F-7$$\"?++++++++]7:xWC\"*F1$\"?HvQsQtI[d\"GQ)[u7F-7$$\"?+++++ ++++vuY)o8*F1$\"?!GcG)*>5qzt%zv\\u7F-7$$\"?++++++++++rKt\\\"*F1$\"?/U8 txR)QwqKA6XF\"F-7$$\"?+++++++++v:JIi\"*F1$\"?IM?M)\\q0iN_kGXF\"F-7$$\" ?++++++++](o3lW<*F1$\"?OYHYwGBi(otL\\XF\"F-7$$\"?+++++++++D#))oz=*F1$ \"?$[uA7w4jRG>twXF\"F-7$$\"?++++++++++VE5+#*F1$\"?(RuaO1rB_()QJ0YF\"F- 7$$\"?+++++++++]A!eI@*F1$\"?7:z\"4/w%>4'f(*RYF\"F-7$$\"?++++++++](=_(z C#*F1$\"?EZ%3H_F-]u;3vYF\"F-7$$\"?+++++++++]&*=jP#*F1$\"?=_[)))y2#yN=$ ['GgaA\\F\"F-7$$\"?+++ +++++++)RO+I*F1$\"?GTK6LJ;.C5BN)\\F\"F-7$$\"?+++++++++D0>w7$*F1$\"?%>x 'y2nytQ:l*\\]F\"F-7$$\"?++++++++]()Q?QD$*F1$\"?P@))*QS$3NtyN*>^F\"F-7$ $\"?++++++++++J'ypL*F1$\"?**4L!yci$[P*F1$\"?K(f!pT)=+N#4VKVv7F-7$$\"?++++++++]7hK'pQ *F1$\"?QcNb!G[!fnJJ'>bF\"F-7$$\"#%*F*$\"?1Yf9t&*R(*y8^mhv7F--%&COLORG6 &%$RGBG$\"\"&!\"\"$\"\"!Fa[l$\"\"*F_[l-F$6&7#7$$\"3K++!o1r;5*!#>$\"3!* ***\\ww\"[u7!#F-%'COLOURG6&F\\[lFa[lFa[lFa[l-%'SYMBOLG6$%'CIRCLEG\"#7- %&STYLEG6#%&POINTG-F$6&Ff[l-F_\\l6&F\\[l$\"*++++\"!\")F`[lF`[l-Fb\\l6$ Fd\\l\"#5Ff\\l-F$6&Ff[lF\\]l-Fb\\l6$%(DIAMONDGFc]lFf\\l-F$6&Ff[lF\\]l- Fb\\l6$%&CROSSGFc]lFf\\l-%+AXESLABELSG6%Q%c[2]6\"Q!Fb^l-%%FONTG6#%(DEF AULTG-Fe^l6$%*HELVETICAGFc[l-%%VIEWG6$;F(Fez;$\"&UF\"!#9$\"'kv7!#:" 1 2 0 1 10 0 2 9 1 4 2 1.000000 45.000000 45.000000 0 0 "Curve 1" "Curve 2" "Curve 3" "Curve 4" "Curve 5" }}}}{PARA 0 "" 0 "" {TEXT -1 0 "" }} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 401 "c_4 := .213316551805: pp := .127448176765e-9:\np1 := evalf[30](plot(prin_err_norm_sqrd(.910167106 68e-1,c[4],.56422397715),c[4]=0.2133118..0.2133212,\n color=COLOR(R GB,0,.7,.2))):\np2 := plot([[[c_4,pp]]$4],style=point,symbol=[circle$2 ,diamond,cross],symbolsize=[12,10$3],\n color=[black,cyan$3] ):\nplots[display]([p1,p2],font=[HELVETICA,9],view=[0.2133118..0.21332 12,1.2742e-10..1.27565e-10]);" }}{PARA 13 "" 1 "" {GLPLOT2D 355 320 320 {PLOTDATA 2 "6*-%'CURVESG6$7S7$$\"(=J8#!\"($\"?zK+'=n#ysDr8Xlv7!#R 7$$\"?nmmmmmm\"\\K*[+7L@!#I$\"?]^VtAa#*3WMlTyv$=_F\"F-7$$\"?nmmmmmT]cjD(HJ8#F1$\"?*Q$[e#4$\\^'>7NZ^F \"F-7$$\"?++++++v8(pQlJJ8#F1$\"?znj(y-U&RPa*fx]F\"F-7$$\"?nmmmmmTS@0[O 8L@F1$\"?&*[\")37#zlM/\">&4]F\"F-7$$\"?++++++v)>RejNJ8#F1$\"?6/YA$\\,V %y'GwX\\F\"F-7$$\"?LLLLLLL=D_!oPJ8#F1$\"?Mgp1Qa=&zs$fW)[F\"F-7$$\"?nmm mmm;*p![\"[RJ8#F1$\"?\\0x6ALQU%[p0M[F\"F-7$$\"?+++++++bI$*3:9L@F1$\"?5 4NN)*o6Hu$RM\"yu7F-7$$\"?+++++++D.rWN9L@F1$\"?1PR!>PuA#>&zqKZF\"F-7$$ \"?+++++++X*el]XJ8#F1$\"?Q@p\\[LFWNc4**ou7F-7$$\"?nmmmmmTSa5)GZJ8#F1$ \"?F8^a1)\\9PWr]aYF\"F-7$$\"?LLLLLLLQl`1%\\J8#F1$\"?2GZd*)pQ'*)=0q;YF \"F-7$$\"?LLLLLLL`b7,7:L@F1$\"?y5_+stifYZ?$)eu7F-7$$\"?++++++v.#f')G`J 8#F1$\"?cz@,\\(G+^xt^fXF\"F-7$$\"?LLLLLLL.-[O^:L@F1$\"??%otGAo/!em*zPX F\"F-7$$\"?++++++v.F$Q;dJ8#F1$\"?Qi-CL)Rmd8T0=XF\"F-7$$\"?++++++D6[u7F-7$$\"?nmmmmmTS41Eq;L@F1$\"?79j2*)eMiZJ] H[u7F-7$$\"?++++++DO?9I)oJ8#F1$\"?@tqm#R:6%y!HZ([u7F-7$$\"?nmmmmm;W!*f y2mm7ZF\"F-7$$\"?L LLLLL3P(fUX)=L@F1$\"?\"y1[OS5sU_g]bZF\"F-7$$\"?++++++vQ6(=V!>L@F1$\"?> :tws]TpgciZ![F\"F-7$$\"?LLLLLLLexMlB>L@F1$\"?T3Ff7E\\L@F1$\"?!*=]-RRT?F/wb\"\\F\"F-7$$\"?nmmmmmm'oN!Rj>L@ F1$\"?@H&zp(\\F4lHkh(\\F\"F-7$$\"?nmmmmm;*[)pK$)>L@F1$\"?]wi%GYK/VkEAU ]F\"F-7$$\"?LLLLLL3dF&)4.?L@F1$\"?8!pDjEeCy^t#=6v7F-7$$\"?+++++++!>_m7 -K8#F1$\"?wYwO*\\f\"Gx<\"Qz^F\"F-7$$\"?LLLLLL$eO4*3U?L@F1$\"?6z_)f\"en P-zR5Ev7F-7$$\"?nmmmmmm1]Irg?L@F1$\"?-YAYTc_'e3$3zLv7F-7$$\"?++++++v.s +d!3K8#F1$\"?&>6\\a)Qk4Ff_QUv7F-7$$\"?++++++Dwvdd*4K8#F1$\"?QC#eZX25Kx h'*4bF\"F-7$$\"(7K8#F*$\"?H><,Ojqj*p'4ngv7F--%&COLORG6&%$RGBG$\"\"!F^[ l$\"\"(!\"\"$\"\"#Fa[l-F$6&7#7$$\"3/++0=b;L@!#=$\"3!****\\ww\"[u7!#F-% 'COLOURG6&F\\[lF^[lF^[lF^[l-%'SYMBOLG6$%'CIRCLEG\"#7-%&STYLEG6#%&POINT G-F$6&Ff[l-F_\\l6&F\\[lF][l$\"*++++\"!\")F^]l-Fb\\l6$Fd\\l\"#5Ff\\l-F$ 6&Ff[lF\\]l-Fb\\l6$%(DIAMONDGFc]lFf\\l-F$6&Ff[lF\\]l-Fb\\l6$%&CROSSGFc ]lFf\\l-%+AXESLABELSG6%Q%c[4]6\"Q!Fb^l-%%FONTG6#%(DEFAULTG-Fe^l6$%*HEL VETICAG\"\"*-%%VIEWG6$;F(Fez;$\"&UF\"!#9$\"'lv7!#:" 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 0 393 "c_5 := .56422397715: pp := .127448176765e- 9:\np1 := evalf[30](plot(prin_err_norm_sqrd(.91016710668e-1,.213316551 805,c[5]),c[5]=0.5642153..0.5642327,\n color=COLOR(RGB,0.6,.2,.2)) ):\np2 := plot([[[c_5,pp]]$4],style=point,symbol=[circle$2,diamond,cro ss],symbolsize=[12,10$3],color=[black,green$3]):\nplots[display]([p1,p 2],font=[HELVETICA,9],view=[0.5642153..0.5642327,1.2742e-10..1.27565e- 10]);" }}{PARA 13 "" 1 "" {GLPLOT2D 368 366 366 {PLOTDATA 2 "6*-%'CURV 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95CA3EE7AF8FFB1BBE373EE7AF1F89F59A4E3EE7AEB9333425E03EE7AE5CFA411FEE3E E7AE0ADCE706653EE7ADC2DCF9C12B3EE7AD84F8B676DC3EE7AD5130307DE53EE7AD27 83AA74093EE7AD07F3E4E0863EE7ACF27EC21BA33EE7ACE723AA2FAE3EE7ACE5E12CCD F63EE7ACEEBBFFF6E83EE7B7E5396D87D03EE7B6FBB60EC23D3EE7B61C49D15CA23EE7 B546F712F16B3EE7B47BBF726E003EE7B3BA9FB657533EE7B3039C34D2353EE7B256B1 74B1D33EE7B1B3E47B3A5B3EE7B11B322BB1923EE7B08C9DCCC6B13EE7B00824295061 3EE7AF8DC6DAF2D73EE7AF1D85CAE3C73EE7AEB762C9752C3EE7AE5B5A94F5F63EE7AE 096E6CA53A3EE7ADC19FE5F6773EE7AD83EFC2482F3EE7AD50586DED493EE7AD26DC20 172A3EE7AD077ED154EA3EE7ACF23C0A9AAD3EE7ACE713BD8C933EE7ACE605A78BD7-% &COLORG6&%$RGBG$\"\"&!\"\"$\"\"!F=$\"\"\"F=-%*AXESSTYLEG6#%$BOXG-%+AXE SLABELSG6%%%c[4]G%%c[5]G%!G-%%FONTG6$%*HELVETICAG\"\"*-%+PROJECTIONG6% $!#vF=$\"#SF=F?" 1 2 0 1 10 0 2 1 1 2 2 1.000000 40.000000 -75.000000 1 0 "Curve 1" }}}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 36 "#-----------------------------------" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 152 "nds := [c[2]=.91016710668e-1,c[4]=.213316551 805,c[5]=.56422397715]:\nevalf[10](%);\nfor dgt from 6 by -1 to 3 do\n map(convert,nds,rational,dgt);\nend do;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#7%/&%\"cG6#\"\"#$\"+n5n,\"*!#6/&F&6#\"\"%$\"+=b;L@!#5/& F&6#\"\"&$\"+s(RAk&F2" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#7%/&%\"cG6#\" \"##\"#x\"$Y)/&F&6#\"\"%#\"$d\"\"$O(/&F&6#\"\"&#\"$B\"\"$=#" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#7%/&%\"cG6#\"\"##\"#w\"$N)/&F&6#\"\"%#\"#;\" #v/&F&6#\"\"&#\"$,\"\"$z\"" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#7%/&%\"c G6#\"\"##\"#w\"$N)/&F&6#\"\"%#\"#8\"#h/&F&6#\"\"&#\"#A\"#R" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#7%/&%\"cG6#\"\"##\"\"\"\"#6/&F&6#\"\"%#\"\"$ \"#9/&F&6#\"\"&#\"\"*\"#;" }}}{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_er r_norm_sqrd(.9101671067e-1,.2133165518,.5642239772)):\nevalf(sqrt(%)); " }}{PARA 11 "" 1 "" {XPPMATH 20 "6#$\"+z%H*G6!#9" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 6 "When " }{XPPEDIT 18 0 "c[ 2] = 1/11;" "6#/&%\"cG6#\"\"#*&\"\"\"F)\"#6!\"\"" }{TEXT -1 3 ", " } {XPPEDIT 18 0 "c[4] = 157/736;" "6#/&%\"cG6#\"\"%*&\"$d\"\"\"\"\"$O(! \"\"" }{TEXT -1 7 " and " }{XPPEDIT 18 0 "c[5] = 123/218;" "6#/&%\"c G6#\"\"&*&\"$B\"\"\"\"\"$=#!\"\"" }{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 68 "evalf[15](pr in_err_norm_sqrd(1/11,157/736,123/218)):\nevalf(sqrt(%));" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#$\"+.;%*G6!#9" }}}{PARA 0 "" 0 "" {TEXT -1 0 " " }}{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 35 "coefficients of the combined scheme" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 2247 "ee := \{c[ 2]=1/11,\nc[3]=157/1104,\nc[4]=157/736,\nc[5]=123/218,\nc[6]=73/99,\nc [7]=199/200,\nc[8]=1,\nc[9]=1,\n\na[2,1]=1/11,\na[3,1]=75517/2437632, \na[3,2]=271139/2437632,\na[4,1]=157/2944,\na[4,2]=0,\na[4,3]=471/2944 ,\na[5,1]=42928977471/63842339642,\na[5,2]=0,\na[5,3]=-81818872578/319 21169821,\na[5,4]=78364952136/31921169821,\na[6,1]=-452993729346115584 620105837/169351895877446242624608351,\na[6,2]=0,\na[6,3]=169743901858 0941844337032/152982742436717473012293,\na[6,4]=-334578912430118682410 335643360/38759554641024302244921542187,\na[6,5]=183506358169025606635 433230/193412520623391990592938849,\na[7,1]=23742496624480179224047764 3719598303/34580776953279718763722454600000000,\na[7,2]=0,\na[7,3]=-20 179674002037028690247750335371/722117794714327571912012500000,\na[7,4] =128453834152286986408251909051997320271/55445542001380779263231286753 15625000,\na[7,5]=-972708878592287689266824375699676121/50543989116344 3236912399043150000000,\na[7,6]=199423777107739967674947/2399134330517 24600000000,\na[8,1]=5973266207518682347729280873413/80232411551856419 1458832770081,\na[8,2]=0,\na[8,3]=-40870912569246870680146344/13470696 54182311812082183,\na[8,4]=112459036656658044255344022793238112/447410 8065454386923650087599365353,\na[8,5]=-4712736023245029582064053285795 650/2213621528406610002679128666238199,\na[8,6]=127657765502309225898/ 142682894776218689669,\na[8,7]=-541736066625000000/9688665443866374238 1,\na[9,1]=531842063/8415926910,\na[9,2]=0,\na[9,3]=0,\na[9,4]=6900573 846223978496/21081697499495907225,\na[9,5]=51126843079934806/173160464 034159675,\na[9,6]=1147778420925807/6907762998459050,\na[9,7]=57577900 0000000/411330558898641,\na[9,8]=-13426037/10725975,\n\nb[1]=531842063 /8415926910,\nb[2]=0,\nb[3]=0,\nb[4]=6900573846223978496/2108169749949 5907225,\nb[5]=51126843079934806/173160464034159675,\nb[6]=11477784209 25807/6907762998459050,\nb[7]=575779000000000/411330558898641,\nb[8]=- 13426037/10725975,\n\n`b*`[1]=648854567388972967/10596897420732888642, \n`b*`[2]=0,\n`b*`[3]=0,\n`b*`[4]=8879698038617778833446469632/2654497 6952167847097500780295,\n`b*`[5]=58500328551366236723248598/2180346495 78069682785496485,\n`b*`[6]=32934117770434118898396/161072145715862713 160465,\n`b*`[7]=2584290081034720820000000/2589630224498721525333771, \n`b*`[8]=-4674424280471995/5402235928512618,\n`b*`[9]=-1/845\}:" }}} {PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 5 " Let " }{XPPEDIT 18 0 "T[6, 8];" "6#&%\"TG6$\"\"'\"\")" }{TEXT -1 127 " denote the vector whose components are the principal error terms of the 8 stage, order 6 scheme (the error terms of order 7)." }}{PARA 0 "" 0 "" {TEXT -1 5 "Let " }{XPPEDIT 18 0 "`T*`[5,9];" "6#&%#T*G6$\"\" &\"\"*" }{TEXT -1 145 " denote the vector whose components are the pr incipal error terms of the embedded 9 stage, order 5 scheme (the error terms of order 6) and let " }{XPPEDIT 18 0 "`T*`[6,9];" "6#&%#T*G6$ \"\"'\"\"*" }{TEXT -1 99 " denote the vector whose components are the error terms of order 7 of the embedded order 5 scheme." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 39 "Denote the 2-norm of these vectors by " }{XPPEDIT 18 0 "abs(abs(T[6,8]));" "6#-%$absG6#-F $6#&%\"TG6$\"\"'\"\")" }{TEXT -1 3 ", " }{XPPEDIT 18 0 "abs(abs(`T*`[ 5,9]));" "6#-%$absG6#-F$6#&%#T*G6$\"\"&\"\"*" }{TEXT -1 7 " and " } {XPPEDIT 18 0 "abs(abs(`T*`[6,9]));" "6#-%$absG6#-F$6#&%#T*G6$\"\"'\" \"*" }{TEXT -1 15 " respectively." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }} {PARA 0 "" 0 "" {TEXT -1 8 "Define: " }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "A[7] = abs(abs(T[6, 8]));" "6#/&%\"AG6#\"\"(-%$absG 6#-F)6#&%\"TG6$\"\"'\"\")" }{TEXT -1 6 ", " }{XPPEDIT 18 0 "B[7] = abs(abs(`T*`[6,9]))/abs(abs(`T*`[5,9]));" "6#/&%\"BG6#\"\"(*&-%$absG6 #-F*6#&%#T*G6$\"\"'\"\"*\"\"\"-F*6#-F*6#&F/6$\"\"&F2!\"\"" }{TEXT -1 6 ", " }{XPPEDIT 18 0 "C[7] = abs(abs(`T*`[6,9]-T[6, 8]))/abs(abs( `T*`[5, 9]));" "6#/&%\"CG6#\"\"(*&-%$absG6#-F*6#,&&%#T*G6$\"\"'\"\"*\" \"\"&%\"TG6$F2\"\")!\"\"F4-F*6#-F*6#&F06$\"\"&F3F9" }{TEXT -1 1 "." }} {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 77 "Dormand a nd Prince have suggested that as well as attempting to ensure that " }{XPPEDIT 18 0 "A[7];" "6#&%\"AG6#\"\"(" }{TEXT -1 73 " is a minimum, if the embedded scheme is to be used for error control, " }{XPPEDIT 18 0 "B[7];" "6#&%\"BG6#\"\"(" }{TEXT -1 7 " and " }{XPPEDIT 18 0 "C [7];" "6#&%\"CG6#\"\"(" }{TEXT -1 27 " should be chosen so that " }} {PARA 0 "" 0 "" {TEXT -1 68 "they are \"similar in magnitude\" and als o not differ too much from 1." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 186 "errterms6_8 := PrincipalErr orTerms(6,8,'expanded'):\n`errterms6_9*` :=subs(b=`b*`,PrincipalErrorT erms(6,9,'expanded')):\n`errterms5_9*` := subs(b=`b*`,PrincipalErrorTe rms(5,9,'expanded')):" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 326 "snmB := sqrt(add(evalf(subs(ee,`er rterms6_9*`[i]))^2,i=1..nops(`errterms6_9*`))):\nsdnB := sqrt(add(eval f(subs(ee,`errterms5_9*`[i]))^2,i=1..nops(`errterms5_9*`))):\nsnmC := \+ sqrt(add((evalf(subs(ee,`errterms6_9*`[i])-subs(ee,errterms6_8[i])))^2 ,i=1..nops(errterms6_8))):\n'B[7]'= evalf[8](snmB/sdnB);\n'C[7]'= eval f[8](snmC/sdnB);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/&%\"BG6#\"\"($\") IJ&Q\"!\"(" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/&%\"CG6#\"\"($\")Z)pQ\" !\"(" }}}{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 2247 "ee := \{c[ 2]=1/11,\nc[3]=157/1104,\nc[4]=157/736,\nc[5]=123/218,\nc[6]=73/99,\nc [7]=199/200,\nc[8]=1,\nc[9]=1,\n\na[2,1]=1/11,\na[3,1]=75517/2437632, \na[3,2]=271139/2437632,\na[4,1]=157/2944,\na[4,2]=0,\na[4,3]=471/2944 ,\na[5,1]=42928977471/63842339642,\na[5,2]=0,\na[5,3]=-81818872578/319 21169821,\na[5,4]=78364952136/31921169821,\na[6,1]=-452993729346115584 620105837/169351895877446242624608351,\na[6,2]=0,\na[6,3]=169743901858 0941844337032/152982742436717473012293,\na[6,4]=-334578912430118682410 335643360/38759554641024302244921542187,\na[6,5]=183506358169025606635 433230/193412520623391990592938849,\na[7,1]=23742496624480179224047764 3719598303/34580776953279718763722454600000000,\na[7,2]=0,\na[7,3]=-20 179674002037028690247750335371/722117794714327571912012500000,\na[7,4] =128453834152286986408251909051997320271/55445542001380779263231286753 15625000,\na[7,5]=-972708878592287689266824375699676121/50543989116344 3236912399043150000000,\na[7,6]=199423777107739967674947/2399134330517 24600000000,\na[8,1]=5973266207518682347729280873413/80232411551856419 1458832770081,\na[8,2]=0,\na[8,3]=-40870912569246870680146344/13470696 54182311812082183,\na[8,4]=112459036656658044255344022793238112/447410 8065454386923650087599365353,\na[8,5]=-4712736023245029582064053285795 650/2213621528406610002679128666238199,\na[8,6]=127657765502309225898/ 142682894776218689669,\na[8,7]=-541736066625000000/9688665443866374238 1,\na[9,1]=531842063/8415926910,\na[9,2]=0,\na[9,3]=0,\na[9,4]=6900573 846223978496/21081697499495907225,\na[9,5]=51126843079934806/173160464 034159675,\na[9,6]=1147778420925807/6907762998459050,\na[9,7]=57577900 0000000/411330558898641,\na[9,8]=-13426037/10725975,\n\nb[1]=531842063 /8415926910,\nb[2]=0,\nb[3]=0,\nb[4]=6900573846223978496/2108169749949 5907225,\nb[5]=51126843079934806/173160464034159675,\nb[6]=11477784209 25807/6907762998459050,\nb[7]=575779000000000/411330558898641,\nb[8]=- 13426037/10725975,\n\n`b*`[1]=648854567388972967/10596897420732888642, \n`b*`[2]=0,\n`b*`[3]=0,\n`b*`[4]=8879698038617778833446469632/2654497 6952167847097500780295,\n`b*`[5]=58500328551366236723248598/2180346495 78069682785496485,\n`b*`[6]=32934117770434118898396/161072145715862713 160465,\n`b*`[7]=2584290081034720820000000/2589630224498721525333771, \n`b*`[8]=-4674424280471995/5402235928512618,\n`b*`[9]=-1/845\}:" }}} {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)*&#\"423@-t2(oHQ \"8gP8I-)R0&4Y'>F)*$)F'\"\"(F)F)F)*&#\"3\"4lV()zZRf%\"8W$4)ybCCZ$e&4#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$!+:UZfW!\"*" }}}{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],st yle=point,symbol=[circle,cross,diamond],color=black):\np3 := plot([[z0 ,0],[z0,1]],linestyle=3,color=COLOR(RGB,0,.5,0)):\nplots[display]([p1, p2,p3],view=[-5.19..0.49,-.07..1.47],font=[HELVETICA,9]);" }}{PARA 13 "" 1 "" {GLPLOT2D 360 253 253 {PLOTDATA 2 "6+-%'CURVESG6$7Z7$$!3Q+++++ +!>&!#<$\"3!4%)*o)429u$F*7$$!3QML3T![!f^F*$\"3A[f%*3J6^NF*7$$!3Ynm;#3' 4G^F*$\"37eEfVsWpLF*7$$!3a++DBT9(4&F*$\"33n4-:'pg>$F*7$$!3kLLLk@>m]F*$ \"3sQ8([s^1.$F*7$$!3E+]U'*)HB,&F*$\"3/f5j)[L3w#F*7$$!3!pm;&GwYe\\F*$\" 3FiINXzc7DF*7$$!3s+](\\(Q*y*[F*$\"3aNN#zu]sD#F*7$$!3nLLV@,KP[F*$\"3yTJ NhVHD?F*7$$!3'RLLd%[MwZF*$\"3'G&Q&*4f]8=F*7$$!3NLL.q&p`r%F*$\"3()ecgrm s@;F*7$$!3E+]<*4%oaYF*$\"3W[CXr;4\\9F*7$$!3;nmJG')*Rf%F*$\"3W(o4+WLJH \"F*7$$!3uLLyGAZ\"[%F*$\"3:uyn/^QV5F*7$$!3%3+])fw&\\O%F*$\"3D+sQPJV;$) !#=7$$!3$QL$)f7eWC%F*$\"3UuL9')39ZlF]p7$$!3A++lN]MCTF*$\"3K30\"ztMt8&F ]p7$$!3ummYeRz+SF*$\"3G_'[GrP4*RF]p7$$!3_LLV-,(>*QF*$\"3al+^7po\">$F]p 7$$!35++S:-YpPF*$\"3))[s!4%)p`[#F]p7$$!3K+++\"HZkk$F*$\"3DX+0F(=X%>F]p 7$$!3;++gW:!z_$F*$\"3;C6?,-H_:F]p7$$!3hLL)*\\1D?MF*$\"3sq@F\">H^G\"F]p 7$$!3'ommSKVAH$F*$\"3QhI7$ $!39++0(*RmdIF*$\"3C^Q)[AxHF)F`s7$$!39nmEI%3g%HF*$\"3e$pPeM>7$yF`s7$$! 3-++0xX]BGF*$\"3#)>gPDAn8xF`s7$$!3*)***\\\"R>&oq#F*$\"3?UqCffj**yF`s7$ $!3gmm;\\r8&e#F*$\"3wd,(o&R7d$)F`s7$$!3ymmrw\\OtCF*$\"3-/l7m@>')*)F`s7 $$!3SLL$))e.GN#F*$\"3%fd&HUU#)p)*F`s7$$!3nLL)**=uvA#F*$\"3c'e5FfX-5\"F ]p7$$!3K++:I;c=@F*$\"3yI^'[\\:l@\"F]p7$$!31LL.z]#3+#F*$\"3mQ*ek\"f2h8F ]p7$$!3M++?,<>z=F*$\"3E>U!=1*HK:F]p7$$!3;++!4<(>gF]p7$$!3H++q9zA<:F*$\"3!*RmOtJ/%>#F]p7$ $!3EnmEY;O-9F*$\"3Y--\"*f$*fgCF]p7$$!3#)*****pQ<(z7F*$\"3G(*y#z\">P\"y #F]p7$$!3)RL$efMeo6F*$\"3vz6fgD<3JF]p7$$!3I****fAZ3Z5F*$\"3j&F]p7$$!3aFLL*)4AjXF]p$\"3-*zFRI(4OjF]p7$$!33LLLO'R&eLF]p$\"3/TDad ZFZrF]p7$$!3Uim;`O$Q;#F]p$\"351/s,ZEa!)F]p7$$!3?*****>$H-m5F]p$\"3OwcI Q4$)))*)F]p7$$\"3v*QLLU?>#>F`s$\"3[-s.#30%>5F*7$$\"3%ymmY^avJ\"F]p$\"3 hAt,(HH39\"F*7$$\"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" "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 1387 "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 \+ 3829687077302210807/19646095053980230133760*z^7+459394779874365091/2 0955834724245578809344*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(RG B,.23,.12,.48)):\np2 := plots[polygonplot]([seq([pts[i-1],pts[i],[-2.2 ,0]],i=2..nops(pts))],\n style=patchnogrid,color=COLOR(RGB,.4 5,.23,.95)):\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:\np3 := plot(pts,color=COLOR(RGB,.23,.12 ,.48)):\np4 := plots[polygonplot]([seq([pts[i-1],pts[i],[1.90,4.72]],i =2..nops(pts))],\n style=patchnogrid,color=COLOR(RGB,.45,.23, .95)):\npts := []: z0 := 2-4.75*I:\nfor ct from 0 to 60 do\n zz := n ewton(R(z)=exp(ct*Pi/30*I),z=z0):\n z0 := zz:\n pts := [op(pts),[R e(zz),Im(zz)]]:\nend do:\np5 := plot(pts,color=COLOR(RGB,.23,.12,.48)) :\np6 := plots[polygonplot]([seq([pts[i-1],pts[i],[1.9,-4.72]],i=2..no ps(pts))],\n style=patchnogrid,color=COLOR(RGB,.45,.23,.95)): \np7 := plot([[[-5.19,0],[2.29,0]],[[0,-5.19],[0,5.19]]],color=black,l inestyle=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=box ed,scaling=constrained);" }}{PARA 13 "" 1 "" {GLPLOT2D 381 565 565 {PLOTDATA 2 "6/-%'CURVESG6$7]z7$$\"\"!F)F(7$F($\"3++++Fjzq:!#=7$$!3K++ +?nZ$>%!#G$\"3E+++WEfTJF-7$$!35+++?muFh!#F$\"3,+++<))Q7ZF-7$F($\"3]*** **>;%=$G'F-7$$\"36+++S7jLW!#D$\"3e*****z_wR&yF-7$$\"3E+++>6xJS!#C$\"3u *****HggZU*F-7$$\"3))*****4mp]:#!#B$\"31+++8Cb*4\"!#<7$$\"3n+++m_i@&)F L$\"3)******\\\"Qic7FO7$$\"31+++(eELt#!#A$\"3'******HF$o89FO7$$\"3w*** **\\ujPZ(FX$\"3'******HL92d\"FO7$$\"3-+++\"4p`z\"!#@$\"35+++U0oF#FO7$$\"3))*****R5%)49#F]p$\" 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" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }} {PARA 0 "" 0 "" {TEXT -1 148 "We can distort the boundary curve horizo ntally by taking the 11th root of the real part of points along the cu rve. In this way we see that there is " }{TEXT 260 53 "no largest inte rval on the nonnegative imaginary axis" }{TEXT -1 65 " that contains t he origin and lies inside the stability region. " }}{PARA 0 "" 0 "" {TEXT -1 119 "However the stability region intersects the nonnegative \+ imaginary axis in an interval that does not contain the origin." }} {PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 411 "R := z -> 1+z+1/2*z^2+1/6*z^3+1/24*z^4+1/120*z^5+1/720*z^6+\n \+ 3829687077302210807/19646095053980230133760*z^7+459394779874365091/20 955834724245578809344*z^8:\nDigits := 25:\npts := []: z0 := 0:\nfor ct from 0 to 107 do\n zz := newton(R(z)=exp(ct*Pi/100*I),z=z0):\n z0 := zz:\n pts := [op(pts),[surd(Re(zz),11),Im(zz)]]:\nend do:\nplot( pts,color=COLOR(RGB,.4,0,.9),thickness=2,font=[HELVETICA,9]);\nDigits \+ := 10:" }}{PARA 13 "" 1 "" {GLPLOT2D 261 276 276 {PLOTDATA 2 "6(-%'CUR VESG6#7hq7$$\"\"!F)F(7$$!:Y%*\\bo,Va2rS>#!#E$\":?o%et#y*e`EfTJF-7$$!:/ sLh$\\\\cT:kHOF-$\":w]-()RCyrI&=$G'F-7$$!:TDl3Ng2W&3]o[F-$\":\\vE]V\\Y 2'zxC%*F-7$$!:4o23rct;v45*fF-$\":\\%=%)>F)=91PmD\"!#D7$$!:LciXO/n>gf/. 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" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 " > " 0 "" {MPLTEXT 1 0 201 "Digits := 15:\nz0 := 0.60*I:\nfor ct from 1 8 to 21 do\n newton(R(z)=exp(ct*Pi/100*I),z=z0);\nend do;``;\nz0 := \+ 3.05*I:\nfor ct from 97 to 100 do\n newton(R(z)=exp(ct*Pi/100*I),z=z 0);\nend do;\nDigits := 10:" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#^$$!0&3 aTHA<6!#B$\"0#y39i'[l&!#:" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#^$$!0!oBX +Cp')!#C$\"0`v:BD!pf!#:" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#^$$\"0%*zql M)fM!#D$\"0k\"\\iT=$G'!#:" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#^$$\"0*Ga 16*p5#!#B$\"0t5O)HM(f'!#:" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%!G" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#^$$\"0a&owI@i6!#<$\"0lSLU@U+$!#9" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#^$$\"0$ox(\\+_D$!#=$\"0L-k!>'=.$!#9" } }{PARA 11 "" 1 "" {XPPMATH 20 "6#^$$!0h,#o\"*z*['!#=$\"0W7*=gEfI!#9" } }{PARA 11 "" 1 "" {XPPMATH 20 "6#^$$!0hYa.b4x\"!#<$\"0?0\"4pU'3$!#9" } }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 108 "Then w e apply the bisection method to calculate the parameter value associat ed with each intersection point." }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 331 "Digits := 15:\nreal_part := proc(u)\n Re(newton(R(z)=exp(u* Pi*I),z=0.60*I))\nend proc:\nu0 := bisect('real_part'(u),u=0.18..0.21) ;\nnewton(R(z)=exp(u0*Pi*I),z=0.60*I);``;\nreal_part := proc(u)\n Re (newton(R(z)=exp(u*Pi*I),z=3.05*I))\nend proc:\nu0 := bisect('real_par t'(u),u=0.97..1.0);\nnewton(R(z)=exp(u0*Pi*I),z=3.05*I);\nDigits := 10 :" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%#u0G$\"0-TWfnu*>!#:" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#^#$\"034bQG_F'!#:" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%!G" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%#u0G$\"00fC)[1 N)*!#:" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#^#$\"0.O@R)\\TI!#9" }}} {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 4 "The " } {TEXT 260 73 "stability region intersects the nonegative imaginary axi s in the interval" }{TEXT -1 39 " [ 0.6275, 3.0415 ] (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 sta ge, order 5 scheme is given as follows." }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 97 "subs(ee,subs(b=`b*`,StabilityFunction(5,9,'expanded') )):\n`R*` := unapply(%,z):\n'`R*`(z)'=`R*`(z);" }}{PARA 12 "" 1 "" {XPPMATH 20 "6#/-%#R*G6#%\"zG,6\"\"\"F)F'F)*&#F)\"\"#F)*$)F'F,F)F)F)*& #F)\"\"'F)*$)F'\"\"$F)F)F)*&#F)\"#CF)*$)F'\"\"%F)F)F)*&#F)\"$?\"F)*$)F '\"\"&F)F)F)*&#\"?(*o_E04$fn!\\!yQD8%\"B+s5D,P`y%)*zd9aF5HF)*$)F'F1F)F )F)*&#\">2F%e%Qm+9cMW&op@\"A+785L1_zDUFK>O9&*F)*$)F'\"\"(F)F)F)*&#\">L Nrd[p4)[wA\\GW)*\"C+KwNGs4!=KYVIFCmf'F)*$)F'\"\")F)F)F)*&#\"3\"4lV()zZ Rf%\";!o&*Q49v)>M!o2x\"F)*$)F'\"\"*F)F)!\"\"" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 124 "We can find the point wh ere the boundary of the stability region intersects the negative real \+ axis by solving the equation: " }}{PARA 256 "" 0 "" {TEXT -1 1 " " } {XPPEDIT 18 0 "`R*`(z) = -1;" "6#/-%#R*G6#%\"zG,$\"\"\"!\"\"" }{TEXT -1 2 ". " }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 33 "z_0 := newton(`R *`(z)=-1,z=-4.5);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%$z_0G$!+$*e%RY% !\"*" }}}{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],s tyle=point,symbol=[circle,cross,diamond],color=black):\np_3 := plot([[ z_0,0],[z_0,-1]],linestyle=3,color=COLOR(RGB,0,.5,0)):\nplots[display] ([p_1,p_2,p_3],view=[-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&!#<$!3'H>6wRUmK#F*7$$!3G++vz=Po\\F*$!3'G'p'HIs1+#F *7$$!3#**\\iE!Rai[F*$!3)Q]LUW]fu\"F*7$$!3r**\\Au#HNu%F*$!3]b,$48*f*[\" F*7$$!3x**\\dRdsBYF*$!3!3k#Qd7^h7F*7$$!3*)*\\7-h\"\\/XF*$!3gv3\"*\\y+i 5F*7$$!3s*\\i#4j%RR%F*$!3-vBFCi3&**)!#=7$$!35+D;`I[zUF*$!38.b$*3O'FK7$$!3!**\\7nc1J/%F*$!3kt3.IV=f]FK7 $$!3'****\\XoI<#RF*$!3#=X?%*H))*eSFK7$$!3++]ZTF#[\"QF*$!3Ui$oWwIRI$FK7 $$!3s***\\'=(pWp$F*$!3SMmR(oqxd#FK7$$!3#)***\\Z^AOd$F*$!3_fVM\"[;N'>FK 7$$!3))***\\8%Q;dMF*$!35NA%RWx]Y\"FK7$$!3#**\\i*3#39N$F*$!3]0Fjv*)R!3 \"FK7$$!3t***\\J`acA$F*$!3Mv?)yn***Qp!#>7$$!3l****fuY7>JF*$!3M(>gqd1!o TFhp7$$!3q*\\iQ70_*HF*$!3g`F%Q)3#eT\"Fhp7$$!3+++5C`^&)GF*$\"3+EEw^*z%> o!#?7$$!3q*\\i)G#o^w#F*$\"3>^F6QE..FFhp7$$!3W*\\(eM$p0l#F*$\"3ezUx9:sE WFhp7$$!3]**\\i5u*4`#F*$\"3oA07Q*o]3'Fhp7$$!3T*\\7\"eI>@CF*$\"3\"y(**) pQ[b`(Fhp7$$!3n**\\()HUv-BF*$\"3k5$*ym\")pu!*Fhp7$$!3y*\\iRdH(z@F*$\"3 g(f]#o8!)p5FK7$$!3o*\\P$\\ijs?F*$\"34A`^6v5<7FK7$$!3S**\\#[_sp&>F*$\"3 EJ#R9x&G'Q\"FK7$$!3y****pz0[P=F*$\"3sp09*Qjfd\"FK7$$!3')**\\_B5e?FK7$$!3i**\\2&R*) =[\"F*$\"3=^VAW-EpAFK7$$!3'*****4?a/p8F*$\"3gixR&*p1UDFK7$$!3O***\\2Rg &[7F*$\"3[-\\&)Q3boGFK7$$!36+DcYIQR6F*$\"3%GxO6&e!**>$FK7$$!3#*)**\\=P B+-\"F*$\"3lg\\(>L*y0OFK7$$!3c-]i)>_r2*FK$\"326\\1o\"QW.%FK7$$!3*)**\\ 74%3K!zFK$\"3x\\g$o$))*p`%FK7$$!3E****\\xPYbnFK$\"3YR/0b+z)3&FK7$$!3)4 +Dc^\")Qb&FK$\"3&QL'y2$)\\QdFK7$$!3)e****f)\\h'R%FK$\"3`#\\ga$eaUkFK7$ $!3F)**\\<\"G98KFK$\"33%zmlA`>D(FK7$$!3'G*\\i%Qq%R?FK$\"3'evmCib]:)FK7 $$!3xr****pJ()4'*Fhp$\"3%4s=KEVP3*FK7$$\"3k<+]_)f2v#Fhp$\"3(**3@bU*)y- \"F*7$$\"3)3++!Qdi!Q\"FK$\"3]lk&fQZ![6F*7$$\"3o4]PhBPfDFK$\"33a&odmr;H \"F*7$$\"3]/]i%G$e(o$FK$\"3#)=VF@#QfW\"F*7$$\"3!***************[FK$\"3 b;np8oJK;F*-%'COLOURG6&%$RGBG$\"*++++\"!\")$\"\"!Fb[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$FjoFg[l7$F_pFg[l7$FdpF g[l7$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[l7$F\\uFg[l7$FauFg[l7 $FfuFg[l7$F[vFg[l7$F`vFg[l7$FevFg[l7$FjvFg[l7$F_wFg[l7$FdwFg[l7$FiwFg[ l7$F^xFg[l7$FcxFg[l7$FhxFg[l7$F]yFg[l7$FbyFg[l7$FgyFg[l7$F\\zFg[l7$Faz Fg[l7$FfzFg[l-F[[l6&F][lFa[lFa[lF^[l-F$6&7#7$$!3++++$*e%RY%F*Fg[l-%'SY MBOLG6#%'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 $;$!$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 1552 "`R*` := z -> 1+z+1/2*z^2+1 /6*z^3+1/24*z^4+1/120*z^5+\n 413253878049067593090526526897/291027 541457799847853370125107200*z^6+\n 21696854434561400663845842707/9 5143619322742257952063310131200*z^7+\n 984428492276488096948577135 33/6596624273043463218009722835763200*z^8-\n 459394779874365091/17 707680341987514093895680*z^9:\npts := []: z0 := 0:\nfor ct from 0 to 2 00 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=CO LOR(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+4.5*I:\nfor ct from 0 to 50 do\n z z := newton(`R*`(z)=exp(ct*Pi/25*I),z=z0):\n z0 := zz:\n pts := [o p(pts),[Re(zz),Im(zz)]]:\nend do:\np_3 := plot(pts,color=COLOR(RGB,.2, 0,.4)):\np_4 := plots[polygonplot]([seq([pts[i-1],pts[i],[1.80,4.42]], i=2..nops(pts))],\n style=patchnogrid,color=COLOR(RGB,.4,0,.8 )):\npts := []: z0 := 1.9-4.5*I:\nfor ct from 0 to 50 do\n zz := new ton(`R*`(z)=exp(ct*Pi/25*I),z=z0):\n z0 := zz:\n pts := [op(pts),[ Re(zz),Im(zz)]]:\nend do:\np_5 := plot(pts,color=COLOR(RGB,.2,0,.4)): \np_6 := plots[polygonplot]([seq([pts[i-1],pts[i],[1.80,-4.42]],i=2..n ops(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,lin estyle=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$$\"3=+++E*z*eZ!#F$\"3++++F jzq:!#=7$$\"32+++7@'4@$!#D$\"3E+++WEfTJF07$$\"3%)*****H.w^&R!#C$\"31++ +<()Q7ZF07$$\"3?+++A#4zV#!#B$\"3++++wA=$G'F07$$\"3)******R$)*)Q-\"!#A$ \"35+++H4'R&yF07$$\"3m*****>^rtM$FF$\"3[*****>ixYU*F07$$\"3;+++G4#G7*F 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cG6#\"\"##\"\"\"\"#6/&F%6#\"\"$#\"$d\"\"%/6/&F%6#\"\"%#F0\"$O(/&F%6#\" \"&#\"$B\"\"$=#/&F%6#\"\"'#\"#t\"#**/&F%6#\"\"(#\"$*>\"$+#/&F%6#\"\")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(\"#6/&F%6$\"\"$F(#\"&$/&F%6$FJF:#\",O@&\\OyFV/&F%6$\"\"'F(#!(=U:#\\C-V-TYbf(Q/&F%6$FinFJ#\"RB1_7M>/&F%6$\"\"(F(#\"E.$)f>Pkx/C#z,[Cm\\UP#\"D++++Y XAPw=(zK&px!eM/&F%6$FepF'FA/&F%6$FepF.#!Ar`L]xC!pGq.-Snz,#\"?++]7?\">d FVr%z<@s/&F%6$FepF:#\"Hr-K(*>04>D3k)pG_T$QXG\"\"F+]i:`nGJKEz2Q,?aXa&/& F%6$FepFJ#!E@hn*pvV#oE*o(G#fy)3F(*\"E+++]J/*R7pBVM;\"*)Ra]/&F%6$FepFin #\"9Z\\nn*Rx5xPU*>\"9++++Ys^IV8*R#/&F%6$\"\")F(#\"@8M(3GHxM#o=v?mK(f\" ?\"3qF$)e9>k&=b6CB!)/&F%6$FgrF'FA/&F%6$FgrF.#!;Wj9!oqoCpD\"4(3%\":$=#3 7=J#=a'pqM\"/&F%6$FgrF:#\"E7\"QKzASMbU/emlO!fC6\"C``O*f(3]O#pQaa13TZ%/ &F%6$FgrFJ#!C]cz&G`S1#eH]CBgt7Z\"C*>QimG\"zE+5mSG:i8A/&F%6$FgrFin#\"6) *eA4B]lxlF\"\"6p'*o=ix%*GoU\"/&F%6$FgrFep#!3+++Dm1O " 0 "" {MPLTEXT 1 0 31 "seq(b[i]=s ubs(ee,b[i]),i=1..8);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6*/&%\"bG6#\"\" \"#\"*j?%=`\"+5p#fT)/&F%6#\"\"#\"\"!/&F%6#\"\"$F//&F%6#\"\"%#\"4'\\yRA YQd+p\"5Ds!f\\*\\(p\"3@/&F%6#\"\"&#\"21[$*zI%o7^\"3v'fT.k/;t\"/&F%6#\" \"'#\"12e#4Uyx9\"\"1]!f%)*Hw2p/&F%6#\"\"(#\"0++++!zdd\"0T')*)e0L6%/&F% 6#\"\")#!)PgU8\")vfs5" }}}{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#\"\"\"# \"3nH(*)QnX&)['\"5U'))Gt?u*of5/&F%6#\"\"#\"\"!/&F%6#\"\"$F//&F%6#\"\"% #\"=K'pkWL)yxhQ!)pz))\">&H!y+v4Zy;_p(\\aE/&F%6#\"\"&#\";)f[KsOiO^&G.]e \"<&['\\&y#op!y&\\Y.=#/&F%6#\"\"'#\"8'R)*)=TVqx6MH$\"9l/;8F'erX@2h\"/& F%6#\"\"(#\":+++?3sM53!H%e#\":rPLD:s)\\C-j*e#/&F%6#\"\")#!1&*>Z!GCWn% \"1=E^GfB-a/&F%6#\"\"*#!\"\"\"$X)" }}}{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 43 " #==========================================" }}{SECT 1 {PARA 4 "" 0 " " {TEXT -1 15 "a scheme with " }{XPPEDIT 18 0 "c[7] = 324/325;" "6#/& %\"cG6#\"\"(*&\"$C$\"\"\"\"$D$!\"\"" }{TEXT -1 1 " " }}{PARA 257 "" 0 "" {TEXT -1 33 "The scheme constructed here has " }{XPPEDIT 18 0 "c[6 ]=28/39" "6#/&%\"cG6#\"\"'*&\"#G\"\"\"\"#R!\"\"" }{TEXT -1 7 " and \+ " }{XPPEDIT 18 0 "c[7] = 324/325;" "6#/&%\"cG6#\"\"(*&\"$C$\"\"\"\"$D$ !\"\"" }{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 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 2132 "ee := \{c[ 2]=30/323,\nc[3]=96/667,\nc[4]=144/667,\nc[5]=161/284,\nc[6]=28/39,\nc [7]=324/325,\nc[8]=1,\nc[9]=1,\n\na[2,1]=30/323,\na[3,1]=72096/2224445 ,\na[3,2]=248064/2224445,\na[4,1]=36/667,\na[4,2]=0,\na[4,3]=108/667, \na[5,1]=627527529881/949970239488,\na[5,2]=0,\na[5,3]=-796051562201/3 16656746496,\na[5,4]=1149583311737/474985119744,\na[6,1]=-218087531946 93072629/11331655147076247477,\na[6,2]=0,\na[6,3]=23796070175842454861 /2956083951411194994,\na[6,4]=-1212069996160021962455905/1965529780132 81766346054,\na[6,5]=572100877243261794319360/753453082384246770993207 ,\na[7,1]=3900329744279593744638499197123/8121538124537291370807160156 25,\na[7,2]=0,\na[7,3]=-246193764192504580123435743/125327543297516166 36406250,\na[7,4]=224823365551024681432621834494295407/136038733848775 78159416276089843750,\na[7,5]=-21380486750050287941075671274033851392/ 12906219675382756068818499374186328125,\na[7,6]=4388242844049767568/45 30133947623046875,\na[8,1]=92443545592188947873141917/1833911860701423 5358999552,\na[8,2]=0,\na[8,3]=-750730245651799179235/3636895561547447 5776,\na[8,4]=2577779379770967239940697801161173/148529202838936128758 823432698880,\na[8,5]=-27329320985408825882348019928840/15495435293973 600997385336553683,\na[8,6]=9858553049752641/9733028110935040,\na[8,7] =-5581481368973046875/1697311052062486391808,\na[9,1]=806703661/126195 14880,\na[9,2]=0,\na[9,3]=0,\na[9,4]=220190408471162800979/66435307947 1007769600,\na[9,5]=120162483071349248/437171496550168695,\na[9,6]=587 917074771/3328097561600,\na[9,7]=16919101291015625/7106645968247808,\n a[9,8]=-716503/321645,\n\nb[1]=806703661/12619514880,\nb[2]=0,\nb[3]=0 ,\nb[4]=220190408471162800979/664353079471007769600,\nb[5]=12016248307 1349248/437171496550168695,\nb[6]=587917074771/3328097561600,\nb[7]=16 919101291015625/7106645968247808,\nb[8]=-716503/321645,\n\n`b*`[1]=390 94996190796419/631671402644956800,\n`b*`[2]=0,\n`b*`[3]=0,\n`b*`[4]=11 275925845175312052514787851/33254276852276902452679056000,\n`b*`[5]=12 7639567253281205865247168/525184180307710755436015425,\n`b*`[6]=807910 0125127495347/37019631452920928000,\n`b*`[7]=250555370757146308140625/ 142289622685217516228352,\n`b*`[8]=-1250061795577621/772799082319350, \n`b*`[9]=-1/180\}:" }}}{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 approxim ate 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],s eq(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&#\"#I\"$B$F(%!GF+7&#\" #'*\"$n'#\"&'4s\"(XWA##\"'k![#F2F+7&#\"$W\"F/#\"#OF/\"\"!#\"$3\"F/7&# \"$h\"\"$%G#\"-\"))Hv_F'\"-)[R-(*\\*F:#!-,Ac^gz\"-'\\Ynl;$7&F+F+F+#\". P^)\\Z7&#\"#G\"#R#!5HE2$p%>`(3=#\"5xuCwq9b;L6F:#\"5h[XUe< qgzB\"4%*\\>69&R3cH7&F+F+#!:0fXi>-gh**p?@\"\"9agMm#\"9g$>VzhK Cx35s&\"92K*4xYUQ#3`Mv7&#\"$C$\"$D$#\"@Br>*\\QYu$fzUuH.!R\"?Dc,;23P\"H PX7Q:7)F:#!kP>Y#\";]iSOmh^(HVvKD\"7&F+#\"E2aH%\\M=iK9oC5blL# [A\"D]P%)*3wiTf\"yv([QtQg8#!G#R^Q.u7nv5%zG]+v'[!Q@\"GD\"Gj=u$*\\=)ogv# Qv'>i!H\"#\"4ovw\\S%GC)Q%\"4vo/Bw%R8IX7&\"\"\"#\";<>9ty%*)=#fXNW#*\";_ &***e`B9qg=\"R$=F:#!6N#z\"*z^cCI2v\"5wdZuahb*oj$7&F+#\"Ct6;,ypS*Rs'4xz $zxd#\"B!)))pKM#)e(Gh$*QG?H&[\"#!AS)G*>![B)e#)3a)4KHt#\"A$o`lL&Q(*4gtR HNa\\:#\"1TEv\\Ibe)*\"1S]$46GIt*7&F+F+F+#!4vo/t*o8[\"e&\"73=R'[i?06tp \"7&F[p#\"*hOq1)\",!)[^>E\"F:F:7&F+#\"6z4!G;r%3/>?#\"6+'px+r%zINk'#\"3 [#\\82$[i,7\"3&po,b'\\rrV#\"-rZ2p\"\" 13yCofk1r#!'.lr\"'X;K7&F+%F_____________________________________GFfrFf r7&%\"bGFaqF:F:FdqF^r7&%#b*G#\"2>kz!>'*\\4R\"3+o&\\k-9nJ'F:F:7&F+#\">^ yy9D07`+g0zEX-pF_oFaK$#\"q$7&F+#\"9D193j9d2Pb0D\"9_$Gi^<_oA'*GU\"# !1@wd&zh+D\"\"0]$>B3*zs(#!\"\"\"$!=Q(pprint66\"" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 136 "subs(ee,mat rix([seq([c[i],seq(a[i,j],j=1..i-1),``$(10-i)],i=2..9),\n[`b`,seq(b[i] ,i=1..8),``],[`b*`,seq(`b*`[i],i=1..9)]])):\nevalf[8](%);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#K%'matrixG6#7,7,$\")d#zG*!\"*F(%!GF+F+F+F+F+F+F +7,$\")/GR9!\")$\")\"y5C$F*$\")D<:6F/F+F+F+F+F+F+F+7,$\")0#*e@F/$\")8I (R&F*$\"\"!F:$\")/>>;F/F+F+F+F+F+F+7,$\")T,pcF/$\")*fdg'F/F9$!)e#R^#! \"($\")7D?CFDF+F+F+F+F+7,$\")s[zrF/$!)neC>FDF9$\")G')\\!)FD$!)EjmhFD$ \")C0$f(F/F+F+F+F+7,$\")3Bp**F/$\")@X-[FDF9$!)FSk>!\"'$\")Bk_;FY$!)Lgc ;FD$\")Ry'o*F/F+F+F+7,$\"\"\"F:$\")XyS]FDF9$!)i?k?FY$\")r`N " 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%" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#\"#<" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 86 "Next we set-up \+ stage-order condtions to check for stage-orders from 2 to 5 inclusive. " }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 85 "for ct from 2 to 5 do\n so||ct||_8 := StageOrderConditions(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#$\"+G'pH;\"!#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#$ \"+r#fJ&f!#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] = 30/323;" " 6#/&%\"cG6#\"\"#*&\"#I\"\"\"\"$B$!\"\"" }{TEXT -1 3 ", " }{XPPEDIT 18 0 "c[4] = 144/667;" "6#/&%\"cG6#\"\"%*&\"$W\"\"\"\"\"$n'!\"\"" } {TEXT -1 3 ", " }{XPPEDIT 18 0 "c[5] = 161/284;" "6#/&%\"cG6#\"\"&*& \"$h\"\"\"\"\"$%G!\"\"" }{TEXT -1 3 ", " }{XPPEDIT 18 0 "c[6] = 28/39 ;" "6#/&%\"cG6#\"\"'*&\"#G\"\"\"\"#R!\"\"" }{TEXT -1 3 ", " } {XPPEDIT 18 0 "c[7] = 324/325;" "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 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 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] = -1/180;" "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 0 217 "e1 := \{c[2 ]=30/323,c[4]=144/667,c[5]=161/284,c[6]=28/39,c[7]=324/325,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]=-1/180\}:\neqns := subs(e1,cdns):\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(eq ns)\}):\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 2251 "e3 := \{`b*`[2] = 0, c [9] = 1, c[8] = 1, b[3] = 0, b[2] = 0, a[7,2] = 0, a[6,2] = 0, a[4,2] \+ = 0, `b*`[3] = 0, a[8,2] = 0, a[5,2] = 0, a[9,2] = 0, a[9,3] = 0, a[3, 2] = 248064/2224445, c[3] = 96/667, b[4] = 220190408471162800979/66435 3079471007769600, a[6,5] = 572100877243261794319360/753453082384246770 993207, `b*`[8] = -1250061795577621/772799082319350, a[2,1] = 30/323, \+ `b*`[4] = 11275925845175312052514787851/33254276852276902452679056000, a[8,1] = 92443545592188947873141917/18339118607014235358999552, a[7,4 ] = 224823365551024681432621834494295407/13603873384877578159416276089 843750, b[5] = 120162483071349248/437171496550168695, a[6,1] = -218087 53194693072629/11331655147076247477, a[7,5] = -21380486750050287941075 671274033851392/12906219675382756068818499374186328125, a[9,1] = 80670 3661/12619514880, a[8,6] = 9858553049752641/9733028110935040, a[6,4] = -1212069996160021962455905/196552978013281766346054, a[9,8] = -716503 /321645, a[8,5] = -27329320985408825882348019928840/154954352939736009 97385336553683, a[4,3] = 108/667, a[8,4] = 257777937977096723994069780 1161173/148529202838936128758823432698880, `b*`[6] = 80791001251274953 47/37019631452920928000, `b*`[7] = 250555370757146308140625/1422896226 85217516228352, b[6] = 587917074771/3328097561600, a[5,4] = 1149583311 737/474985119744, b[8] = -716503/321645, a[6,3] = 23796070175842454861 /2956083951411194994, b[1] = 806703661/12619514880, a[5,1] = 627527529 881/949970239488, a[9,4] = 220190408471162800979/664353079471007769600 , a[7,6] = 4388242844049767568/4530133947623046875, a[8,7] = -55814813 68973046875/1697311052062486391808, a[9,6] = 587917074771/332809756160 0, b[7] = 16919101291015625/7106645968247808, `b*`[1] = 39094996190796 419/631671402644956800, a[9,5] = 120162483071349248/437171496550168695 , c[2] = 30/323, c[4] = 144/667, c[5] = 161/284, c[6] = 28/39, c[7] = \+ 324/325, a[4,1] = 36/667, a[8,3] = -750730245651799179235/363689556154 74475776, a[7,3] = -246193764192504580123435743/1253275432975161663640 6250, a[3,1] = 72096/2224445, `b*`[9] = -1/180, a[9,7] = 1691910129101 5625/7106645968247808, a[7,1] = 3900329744279593744638499197123/812153 812453729137080716015625, a[5,3] = -796051562201/316656746496, `b*`[5] = 127639567253281205865247168/525184180307710755436015425\}:" }{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],se q(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&#\"#I\"$B$F(%!GF+7&#\"#'*\"$n'#\"& '4s\"(XWA##\"'k![#F2F+7&#\"$W\"F/#\"#OF/\"\"!#\"$3\"F/7&#\"$h\"\"$%G# \"-\"))Hv_F'\"-)[R-(*\\*F:#!-,Ac^gz\"-'\\Ynl;$7&F+F+F+#\".P^)\\Z7&#\"#G\"#R#!5HE2$p%>`(3=#\"5xuCwq9b;L6F:#\"5h[XUe 69&R3cH7&F+F+#!:0fXi>-gh**p?@\"\"9agMm#\"9g$>VzhKCx35s&\"92K* 4xYUQ#3`Mv7&#\"$C$\"$D$#\"@Br>*\\QYu$fzUuH.!R\"?Dc,;23P\"HPX7Q:7)F:#!< VdVB,e/D>kP>Y#\";]iSOmh^(HVvKD\"7&F+#\"E2aH%\\M=iK9oC5blL#[A\"D]P%)*3w iTf\"yv([QtQg8#!G#R^Q.u7nv5%zG]+v'[!Q@\"GD\"Gj=u$*\\=)ogv#Qv'>i!H\"#\" 4ovw\\S%GC)Q%\"4vo/Bw%R8IX7&\"\"\"#\";<>9ty%*)=#fXNW#*\";_&***e`B9qg= \"R$=F:#!6N#z\"*z^cCI2v\"5wdZuahb*oj$7&F+#\"Ct6;,ypS*Rs'4xz$zxd#\"B!)) )pKM#)e(Gh$*QG?H&[\"#!AS)G*>![B)e#)3a)4KHt#\"A$o`lL&Q(*4gtRHNa\\:#\"1T Ev\\Ibe)*\"1S]$46GIt*7&F+F+F+#!4vo/t*o8[\"e&\"73=R'[i?06tp\"7&F[p#\"*h Oq1)\",!)[^>E\"F:F:7&F+#\"6z4!G;r%3/>?#\"6+'px+r%zINk'#\"3[#\\82$[i,7 \"3&po,b'\\rrV#\"-rZ2p\"\"13yCofk1r#! '.lr\"'X;K7&F+%F_____________________________________GFfrFfr7&%\"bGFaq F:F:FdqF^r7&%#b*G#\"2>kz!>'*\\4R\"3+o&\\k-9nJ'F:F:7&F+#\">^yy9D07`+g0zEX-pF_oFaK$#\"q$7&F+#\"9D193j9d2Pb0D\"9_$Gi^<_oA'*GU\"#!1@wd&zh+D \"\"0]$>B3*zs(#!\"\"\"$!=Q(pprint46\"" }}}{PARA 0 "" 0 "" {TEXT -1 0 " " }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 136 "subs(e3,matrix([seq([c[ i],seq(a[i,j],j=1..i-1),``$(10-i)],i=2..9),\n[`b`,seq(b[i],i=1..8),``] ,[`b*`,seq(`b*`[i],i=1..9)]])):\nevalf[8](%);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#K%'matrixG6#7,7,$\")d#zG*!\"*F(%!GF+F+F+F+F+F+F+7,$\")/ GR9!\")$\")\"y5C$F*$\")D<:6F/F+F+F+F+F+F+F+7,$\")0#*e@F/$\")8I(R&F*$\" \"!F:$\")/>>;F/F+F+F+F+F+F+7,$\")T,pcF/$\")*fdg'F/F9$!)e#R^#!\"($\")7D ?CFDF+F+F+F+F+7,$\")s[zrF/$!)neC>FDF9$\")G')\\!)FD$!)EjmhFD$\")C0$f(F/ F+F+F+F+7,$\")3Bp**F/$\")@X-[FDF9$!)FSk>!\"'$\")Bk_;FY$!)Lgc;FD$\")Ry' o*F/F+F+F+7,$\"\"\"F:$\")XyS]FDF9$!)i?k?FY$\")r`N " 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`(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 "" }}{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 solution" }}{PARA 0 "" 0 "" {TEXT -1 0 " " }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 71947 "eG := \{c[8] = 1, b[3 ] = 0, b[2] = 0, c[9] = 1, b[5] = -1/60*(10*c[6]*c[7]*c[4]-5*c[6]*c[4] -5*c[7]*c[6]+3*c[6]-5*c[7]*c[4]+3*c[4]-2+3*c[7])/(-c[4]+c[5])/c[5]/(-c [5]^3+c[5]^2-c[7]*c[5]+c[7]*c[5]^2-c[6]*c[5]+c[6]*c[5]^2+c[7]*c[6]-c[6 ]*c[7]*c[5]), a[5,2] = 0, a[4,2] = 0, `b*`[3] = 0, `b*`[2] = 0, a[8,2] = 0, a[7,2] = 0, a[6,2] = 0, a[6,5] = 1/2*c[6]*(30*c[6]^2*c[5]^2*c[4] ^2-2*c[6]^2*c[4]^2+c[6]^3*c[4]-12*c[6]^2*c[4]*c[5]^2-3*c[5]^2*c[6]*c[4 ]+c[5]*c[6]*c[4]^2+c[6]*c[4]^3-12*c[6]^2*c[4]^2*c[5]-30*c[5]*c[6]^3*c[ 4]^2-20*c[5]*c[6]^2*c[4]^4+30*c[6]^2*c[5]*c[4]^3-c[5]*c[4]^3+20*c[4]^3 *c[5]*c[6]^3-20*c[6]^2*c[5]^2*c[4]^3+c[5]^2*c[4]^2-30*c[5]^2*c[6]*c[4] ^3+12*c[5]^2*c[6]*c[4]^2+2*c[6]^2*c[4]*c[5]+20*c[5]^2*c[6]*c[4]^4+2*c[ 6]^2*c[5]^2-2*c[6]^3*c[5]+12*c[6]^3*c[4]*c[5])/c[5]/(c[4]^3-2*c[5]*c[4 ]^2+30*c[5]^2*c[4]^3-6*c[5]*c[4]^3-10*c[4]^4*c[5]^2-c[5]^3+10*c[5]^4*c [4]^2-30*c[5]^3*c[4]^2+6*c[5]^3*c[4]+2*c[5]^2*c[4]), b[1] = 1/60*(30*c [5]*c[6]*c[7]*c[4]-10*c[5]*c[6]*c[4]-10*c[6]*c[7]*c[5]+5*c[6]*c[5]-10* c[6]*c[7]*c[4]+5*c[6]*c[4]+5*c[7]*c[6]-3*c[6]-10*c[5]*c[7]*c[4]+5*c[5] *c[4]+5*c[7]*c[5]-3*c[5]+5*c[7]*c[4]-3*c[4]+2-3*c[7])/c[5]/c[6]/c[7]/c [4], a[6,1] = -1/4*c[6]*(60*c[5]^2*c[6]*c[4]^4-90*c[5]^2*c[6]^3*c[4]^2 -60*c[5]*c[6]^2*c[4]^4-2*c[6]^3*c[4]^2+50*c[5]^4*c[4]^3*c[6]+40*c[6]^2 *c[4]^5*c[5]-40*c[4]^5*c[5]^3+36*c[4]*c[5]^2*c[6]^3-20*c[5]^4*c[6]^2*c [4]^2+8*c[6]^2*c[5]^3+4*c[6]^2*c[4]^3-24*c[5]*c[6]^3*c[4]^2-8*c[6]^2*c [4]^2*c[5]+24*c[6]^2*c[5]*c[4]^3+60*c[4]^3*c[5]*c[6]^3-4*c[6]*c[5]*c[4 ]^3-14*c[5]^3*c[6]*c[4]+7*c[5]^3*c[4]^2+6*c[5]^2*c[6]*c[4]^3+14*c[5]^2 *c[6]*c[4]^2+66*c[5]^3*c[6]*c[4]^2-80*c[5]^3*c[6]^2*c[4]^3+30*c[6]^2*c [5]^2*c[4]^3-24*c[6]^2*c[5]^2*c[4]^2-240*c[6]*c[5]^3*c[4]^3-48*c[6]^2* c[5]^3*c[4]+150*c[6]^2*c[5]^3*c[4]^2-24*c[4]^4*c[5]^2-9*c[5]^2*c[4]^3+ 6*c[4]^4*c[5]-40*c[5]^4*c[4]^4-20*c[6]^2*c[4]^4*c[5]^2-6*c[5]^2*c[6]^3 -2*c[6]*c[4]^4-24*c[5]^3*c[4]^3+60*c[6]^3*c[4]^3*c[5]^2+7*c[6]^3*c[4]* c[5]-40*c[5]^2*c[6]*c[4]^5-40*c[6]^3*c[4]^4*c[5]+110*c[4]^4*c[6]*c[5]^ 3+120*c[5]^3*c[4]^4)/c[5]/c[4]^2/(10*c[5]^3*c[4]^2+10*c[5]^2*c[4]^3-30 *c[5]^2*c[4]^2+6*c[5]^2*c[4]-c[5]^2+6*c[5]*c[4]^2+c[5]*c[4]-c[4]^2), b [4] = 1/60*(-2+3*c[5]-5*c[7]*c[6]+3*c[7]+3*c[6]+10*c[6]*c[7]*c[5]-5*c[ 6]*c[5]-5*c[7]*c[5])/c[4]/(-c[6]*c[4]^3-c[4]^3+c[4]^4+c[7]*c[4]^2+c[6] *c[4]^2-c[6]*c[7]*c[4]+c[6]*c[7]*c[4]^2-c[5]*c[4]^3+c[5]*c[4]^2-c[7]*c [4]^3-c[5]*c[7]*c[4]+c[5]*c[7]*c[4]^2-c[5]*c[6]*c[4]+c[5]*c[6]*c[4]^2+ c[6]*c[7]*c[5]-c[5]*c[6]*c[7]*c[4]), b[6] = 1/60*(10*c[5]*c[7]*c[4]-5* c[5]*c[4]-5*c[7]*c[5]+3*c[5]-5*c[7]*c[4]+3*c[4]-2+3*c[7])/(-c[4]+c[6]) /(-c[5]+c[6])/c[6]/(c[6]^2-c[6]+c[7]-c[7]*c[6]), `b*`[9] = 1/10*(50*c[ 5]^2*c[6]*c[4]^2-40*c[5]^2*c[6]*c[4]+10*c[6]*c[5]^2-40*c[5]*c[6]*c[4]^ 2+35*c[5]*c[6]*c[4]-10*c[6]*c[5]+10*c[6]*c[4]^2-10*c[6]*c[4]+3*c[6]-5* c[5]^2*c[4]-5*c[5]*c[4]^2+10*c[5]*c[4]-c[5]-c[4])/(3*c[6]-28*c[5]*c[6] *c[4]^2-28*c[5]^2*c[6]*c[4]+8*c[6]*c[5]^2-9*c[6]*c[4]-9*c[6]*c[5]+28*c [5]*c[6]*c[4]+30*c[5]^2*c[6]*c[4]^2+8*c[6]*c[4]^2-4*c[5]^2*c[4]-4*c[5] *c[4]^2+9*c[5]*c[4]-c[5]-c[4]), a[7,5] = -1/2*c[7]*(-17*c[5]^2*c[6]*c[ 4]^4+5*c[6]*c[5]*c[4]^4+42*c[4]^2*c[5]^2*c[7]^3+17*c[7]^3*c[5]*c[4]^3- 80*c[7]^3*c[5]^2*c[4]^3+c[7]*c[5]*c[4]^4+80*c[5]^2*c[4]^4*c[7]^2-60*c[ 4]^5*c[6]^2*c[5]^2+20*c[4]^3*c[7]^3*c[5]^3-20*c[7]^2*c[5]^2*c[4]^5+40* c[4]^5*c[6]*c[5]^3-5*c[5]*c[6]^2*c[4]^4+c[7]^3*c[4]^2-c[7]^2*c[4]^3-80 *c[5]^3*c[4]^4*c[7]+20*c[7]^3*c[5]^2*c[4]^4-200*c[6]^2*c[4]^5*c[7]*c[5 ]^3-290*c[5]^4*c[4]^2*c[7]^2*c[6]+20*c[6]^2*c[7]*c[5]^4*c[4]^2+40*c[6] *c[5]^2*c[4]^4*c[7]-140*c[5]^4*c[4]^3*c[6]-100*c[5]^4*c[4]^5*c[6]+190* c[5]^4*c[4]^4*c[6]-40*c[7]^2*c[4]^4*c[5]^3+20*c[5]^4*c[4]^4*c[7]+2*c[4 ]^4*c[7]^2-20*c[7]^2*c[5]^4*c[4]^3-4*c[6]^2*c[7]^2*c[5]-8*c[7]*c[6]^2* c[5]^3-3*c[7]*c[6]*c[5]^3+2*c[7]^2*c[6]*c[5]^3+60*c[7]*c[6]^2*c[5]*c[4 ]^5-40*c[6]*c[5]^2*c[4]^5*c[7]+17*c[7]*c[4]^4*c[6]*c[5]+80*c[7]^2*c[5] ^3*c[4]^3+5*c[7]*c[4]^4*c[6]^2+c[5]^2*c[7]^3-260*c[6]^2*c[7]*c[5]^2*c[ 4]^3-12*c[6]^2*c[7]^2*c[5]^2*c[4]+20*c[6]^2*c[7]^2*c[5]^2*c[4]^2-40*c[ 6]*c[7]*c[5]^3*c[4]^3-41*c[6]^2*c[7]^3*c[4]*c[5]+20*c[7]^2*c[6]*c[5]*c [4]^4+3*c[7]*c[6]*c[4]^3-2*c[6]^2*c[7]*c[4]^2-3*c[6]*c[7]^2*c[4]^2+4*c [7]^2*c[6]*c[4]^3+6*c[5]*c[7]^2*c[4]^3+4*c[6]*c[5]*c[7]^2*c[4]^2+c[7]^ 2*c[5]*c[6]*c[4]-11*c[5]*c[7]*c[6]*c[4]^3+2*c[6]^2*c[4]^2*c[5]-c[7]^3* c[4]*c[5]+17*c[5]^2*c[6]*c[7]*c[4]^2-200*c[5]^4*c[7]^2*c[4]^4*c[6]-50* c[4]^5*c[6]*c[7]*c[5]^3-150*c[5]^2*c[6]*c[7]^3*c[4]^4-c[6]^2*c[5]*c[4] ^3+2*c[7]^3*c[4]^2*c[6]*c[5]+2*c[6]^2*c[7]*c[4]*c[5]+108*c[7]*c[6]^2*c [5]*c[4]^3+4*c[6]^2*c[7]*c[5]^2+2*c[6]*c[5]*c[7]*c[4]^2-10*c[6]^2*c[7] ^3*c[5]^2+6*c[5]^4*c[7]*c[6]-8*c[6]*c[7]^3*c[5]^2+3*c[6]*c[7]^2*c[5]^2 +2*c[6]^2*c[7]^2*c[5]^2-27*c[6]^2*c[4]^2*c[7]*c[5]-62*c[6]^2*c[7]^2*c[ 4]^2*c[5]+46*c[6]*c[5]^4*c[4]^2+20*c[4]^5*c[7]*c[5]^3+2*c[7]^3*c[4]*c[ 6]*c[5]-27*c[6]^2*c[5]^2*c[7]*c[4]+120*c[6]^2*c[5]^2*c[7]*c[4]^2-35*c[ 6]*c[5]^2*c[7]*c[4]^3-37*c[6]*c[5]^3*c[7]*c[4]^2+53*c[6]^2*c[5]^3*c[7] *c[4]-54*c[7]^2*c[5]^2*c[4]^3+2*c[7]^2*c[5]*c[4]^2-c[7]*c[5]*c[4]^3-3* c[6]*c[5]*c[4]^3-2*c[4]^3*c[7]^3+63*c[7]^3*c[5]^2*c[6]*c[4]-16*c[7]^2* c[5]^2*c[6]*c[4]-5*c[6]*c[7]*c[4]^4+2*c[6]^2*c[7]^2*c[4]^2+2*c[6]^2*c[ 7]^2*c[4]-10*c[5]^4*c[7]^2*c[6]+3*c[5]^3*c[6]*c[4]+3*c[5]^3*c[7]*c[4]+ 20*c[5]^3*c[6]*c[7]*c[4]-2*c[5]^3*c[4]^2+22*c[6]^2*c[7]^2*c[4]*c[5]+7* c[5]^2*c[6]*c[4]^3-c[5]^2*c[7]*c[4]^2+c[5]^2*c[6]*c[4]^2-4*c[5]^2*c[7] *c[4]^3-8*c[5]^3*c[7]*c[4]^2-22*c[5]^3*c[6]*c[4]^2+240*c[5]^3*c[6]^2*c [4]^3+100*c[4]^3*c[7]^2*c[5]^2*c[6]^2+150*c[4]^5*c[5]^3*c[6]^2-3*c[7]* c[5]^4*c[4]+5*c[7]^2*c[5]^4*c[4]+12*c[5]^2*c[4]^4*c[7]-98*c[6]^2*c[5]^ 2*c[4]^3+25*c[6]^2*c[5]^2*c[4]^2+54*c[6]*c[5]^3*c[4]^3+8*c[6]^2*c[5]^3 *c[4]-63*c[6]^2*c[5]^3*c[4]^2+42*c[5]^3*c[7]*c[4]^3+60*c[5]^4*c[7]*c[4 ]^2*c[6]+200*c[5]^3*c[7]^3*c[4]^4*c[6]-4*c[6]^2*c[4]*c[5]^2+c[7]*c[6]^ 2*c[4]^3+200*c[7]^2*c[5]^2*c[6]*c[4]^3-240*c[7]^3*c[5]^2*c[6]*c[4]^2+1 80*c[7]^2*c[5]^3*c[6]*c[4]^2+3*c[5]^4*c[4]^2-80*c[5]^3*c[7]^3*c[6]*c[4 ]+c[7]^3*c[4]^2*c[6]-7*c[7]^3*c[4]^2*c[5]+20*c[4]^3*c[7]^2*c[5]*c[6]^2 -19*c[7]^2*c[6]*c[5]*c[4]^3-11*c[7]^2*c[5]^2*c[6]*c[4]^2-3*c[4]^4*c[5] ^2-27*c[7]^2*c[5]^3*c[6]*c[4]-480*c[7]^2*c[5]^3*c[6]*c[4]^3-380*c[5]^3 *c[6]^2*c[4]^4-5*c[5]^4*c[4]^3-10*c[6]^2*c[4]^3*c[7]^2+130*c[5]^3*c[6] ^2*c[7]*c[4]^3-5*c[7]^3*c[5]^2*c[4]-2*c[7]^2*c[5]^2*c[4]-4*c[5]^2*c[6] *c[7]*c[4]-160*c[6]^2*c[7]*c[5]^3*c[4]^2-80*c[6]^2*c[7]^2*c[4]*c[5]^3+ 2*c[5]^2*c[4]^3+12*c[7]^2*c[5]^2*c[4]^2+8*c[7]^2*c[5]^3*c[4]+400*c[7]^ 3*c[5]^2*c[6]^2*c[4]^3-17*c[7]^2*c[4]^4*c[5]+250*c[7]^2*c[5]^3*c[6]^2* c[4]^2-100*c[6]^2*c[4]^5*c[5]*c[7]^2-200*c[7]^2*c[4]^5*c[6]*c[5]^3+150 *c[5]^2*c[6]*c[4]^5*c[7]^2+380*c[7]^3*c[5]^2*c[4]^3*c[6]-20*c[7]^3*c[6 ]*c[5]*c[4]^3-210*c[7]^3*c[6]^2*c[5]*c[4]^3-6*c[5]^4*c[4]*c[6]+80*c[5] ^4*c[7]^2*c[4]*c[6]+10*c[6]*c[7]^3*c[5]^3-36*c[5]^4*c[4]*c[7]*c[6]-50* c[4]^5*c[6]^2*c[7]*c[5]^2+200*c[4]^5*c[6]^2*c[7]^2*c[5]^2+160*c[6]^2*c [4]^4*c[5]^2-450*c[7]^3*c[5]^3*c[4]^3*c[6]+5*c[6]^2*c[7]^3*c[4]^2-c[5] ^3*c[7]^2+6*c[6]^2*c[7]^3*c[5]-160*c[7]*c[6]^2*c[5]*c[4]^4+200*c[5]^3* c[4]^4*c[6]^2*c[7]-350*c[5]^3*c[4]^3*c[6]^2*c[7]^2-350*c[5]^4*c[4]^4*c [6]*c[7]+230*c[5]^2*c[4]^4*c[7]*c[6]^2+150*c[5]^3*c[4]^4*c[6]*c[7]+100 *c[5]^4*c[4]^3*c[6]*c[7]-3*c[6]^2*c[7]^3*c[4]-42*c[7]^2*c[5]^3*c[4]^2+ 50*c[5]^4*c[4]^4*c[6]^2-20*c[5]^4*c[4]^3*c[6]^2+500*c[5]^3*c[4]^4*c[7] ^2*c[6]+450*c[5]^4*c[4]^3*c[7]^2*c[6]-340*c[5]^2*c[4]^4*c[7]^2*c[6]-50 *c[5]^4*c[4]^3*c[6]^2*c[7]+150*c[7]^2*c[4]^4*c[6]^2*c[5]+200*c[7]^2*c[ 4]^4*c[5]^3*c[6]^2-350*c[7]^2*c[4]^4*c[6]^2*c[5]^2+10*c[7]^2*c[6]^2*c[ 5]^3-60*c[4]^4*c[6]*c[5]^3+100*c[7]^3*c[6]^2*c[5]*c[4]^4+200*c[5]^4*c[ 4]^5*c[6]*c[7]-200*c[7]^3*c[4]^4*c[6]^2*c[5]^2+5*c[5]^3*c[4]^4+80*c[6] ^2*c[7]^3*c[5]^2*c[4]-270*c[6]^2*c[7]^3*c[5]^2*c[4]^2+290*c[5]^3*c[7]^ 3*c[4]^2*c[6]+140*c[6]^2*c[5]*c[7]^3*c[4]^2-5*c[7]^3*c[4]*c[5]^3)/c[5] /(100*c[5]^2*c[6]*c[4]^4-18*c[6]*c[5]*c[4]^4+4*c[5]^3*c[4]+50*c[4]^5*c [6]^2*c[5]^2+40*c[5]*c[6]^2*c[4]^4+3*c[5]^5-5*c[4]^4*c[6]^2+180*c[5]^4 *c[4]^3*c[6]+100*c[5]^4*c[4]^5*c[6]-350*c[5]^4*c[4]^4*c[6]+140*c[5]^5* c[4]^2-180*c[5]^5*c[4]^3-2*c[5]^4+30*c[4]^5*c[5]^3+300*c[5]^5*c[4]^3*c [6]+2*c[6]*c[5]^3-50*c[5]^5*c[4]^2*c[6]^2-40*c[5]^4*c[6]^2*c[4]+240*c[ 5]^4*c[6]^2*c[4]^2-3*c[6]^2*c[5]^3+3*c[6]^2*c[4]^3+5*c[5]^4*c[6]^2-2*c [6]*c[4]^3+2*c[5]*c[4]^3+4*c[5]*c[6]*c[4]^2-6*c[6]^2*c[4]^2*c[5]-50*c[ 5]^4*c[4]^5-13*c[6]^2*c[5]*c[4]^3-40*c[6]*c[5]^4*c[4]^2+6*c[6]*c[5]*c[ 4]^3-15*c[5]^3*c[6]*c[4]-55*c[5]^2*c[6]*c[4]^3+6*c[5]^2*c[6]*c[4]^2+68 *c[5]^3*c[6]*c[4]^2-100*c[4]^5*c[5]^3*c[6]^2+100*c[6]^2*c[5]^2*c[4]^3- 100*c[6]*c[5]^3*c[4]^3+13*c[6]^2*c[5]^3*c[4]-100*c[6]^2*c[5]^3*c[4]^2- 4*c[5]^2*c[4]^2+6*c[6]^2*c[4]*c[5]^2-68*c[5]^4*c[4]^2+23*c[4]^4*c[5]^2 +350*c[5]^3*c[6]^2*c[4]^4-4*c[5]^2*c[6]*c[4]-9*c[5]^2*c[4]^3-3*c[4]^4* c[5]-5*c[5]^5*c[6]-30*c[5]^6*c[4]^2+50*c[5]^6*c[6]*c[4]^2+180*c[5]^4*c [4]^4+10*c[5]^4*c[4]*c[6]-240*c[6]^2*c[4]^4*c[5]^2+3*c[6]*c[4]^4-100*c [5]^6*c[4]^3*c[6]+68*c[5]^3*c[4]^3-23*c[5]^5*c[4]-210*c[5]^5*c[6]*c[4] ^2-350*c[5]^4*c[4]^3*c[6]^2-30*c[5]^2*c[6]*c[4]^5+50*c[5]^6*c[4]^3+60* c[4]^4*c[6]*c[5]^3+100*c[5]^5*c[6]^2*c[4]^3+9*c[5]^4*c[4]-140*c[5]^3*c [4]^4+40*c[5]^5*c[4]*c[6]), `b*`[6] = -1/60*(-30*c[5]^3*c[4]+103*c[5]* c[7]*c[4]-18*c[6]+6*c[5]+6*c[4]+27*c[7]*c[6]-9*c[5]^2-9*c[4]^2-100*c[7 ]*c[6]*c[5]^3+522*c[5]*c[6]*c[7]*c[4]-111*c[6]*c[5]^2+75*c[6]*c[5]+60* c[6]*c[5]^3-9*c[7]*c[5]+600*c[6]*c[7]*c[5]^3*c[4]^3+14*c[7]*c[5]^2-100 *c[7]*c[6]*c[4]^3+14*c[7]*c[4]^2-9*c[7]*c[4]-117*c[6]*c[7]*c[5]-111*c[ 6]*c[4]^2+60*c[6]*c[4]^3-117*c[6]*c[7]*c[4]+180*c[6]*c[7]*c[4]^2-30*c[ 5]*c[4]^3-174*c[5]*c[7]*c[4]^2+486*c[5]*c[6]*c[4]^2+107*c[5]*c[4]^2-31 8*c[5]*c[6]*c[4]-66*c[5]*c[4]+75*c[6]*c[4]+500*c[5]*c[7]*c[6]*c[4]^3+1 420*c[5]^2*c[6]*c[7]*c[4]^2-840*c[6]*c[5]*c[7]*c[4]^2+180*c[6]*c[7]*c[ 5]^2-900*c[6]*c[5]^2*c[7]*c[4]^3-900*c[6]*c[5]^3*c[7]*c[4]^2+50*c[7]*c [5]*c[4]^3-280*c[6]*c[5]*c[4]^3-280*c[5]^3*c[6]*c[4]+50*c[5]^3*c[7]*c[ 4]+107*c[5]^2*c[4]+500*c[5]^3*c[6]*c[7]*c[4]+50*c[5]^3*c[4]^2+470*c[5] ^2*c[6]*c[4]^3+310*c[5]^2*c[7]*c[4]^2-770*c[5]^2*c[6]*c[4]^2-90*c[5]^2 *c[7]*c[4]^3-90*c[5]^3*c[7]*c[4]^2+470*c[5]^3*c[6]*c[4]^2-300*c[6]*c[5 ]^3*c[4]^3-180*c[5]^2*c[4]^2+486*c[5]^2*c[6]*c[4]-174*c[5]^2*c[7]*c[4] -840*c[5]^2*c[6]*c[7]*c[4]+50*c[5]^2*c[4]^3)/(-9*c[5]*c[7]*c[4]-3*c[7] *c[6]+3*c[6]^2-28*c[5]*c[6]*c[7]*c[4]-c[6]*c[5]+c[7]*c[5]+8*c[6]^2*c[4 ]^2+c[7]*c[4]+9*c[6]*c[7]*c[5]+9*c[6]*c[7]*c[4]-8*c[6]*c[7]*c[4]^2+4*c [5]*c[7]*c[4]^2-4*c[5]*c[6]*c[4]^2+9*c[5]*c[6]*c[4]-c[6]*c[4]-28*c[6]^ 2*c[4]^2*c[5]+28*c[6]^2*c[4]*c[5]-30*c[5]^2*c[6]*c[7]*c[4]^2+28*c[6]*c [5]*c[7]*c[4]^2-8*c[6]*c[7]*c[5]^2+8*c[6]^2*c[5]^2-9*c[6]^2*c[5]+30*c[ 6]^2*c[5]^2*c[4]^2-28*c[6]^2*c[4]*c[5]^2-9*c[6]^2*c[4]-4*c[5]^2*c[6]*c [4]+4*c[5]^2*c[7]*c[4]+28*c[5]^2*c[6]*c[7]*c[4])/(c[5]*c[4]-c[6]*c[4]- c[6]*c[5]+c[6]^2)/c[6], a[8,7] = (10*c[5]*c[6]*c[4]-5*c[6]*c[5]-5*c[6] *c[4]+3*c[6]-5*c[5]*c[4]+3*c[5]+3*c[4]-2)/c[7]/(c[6]-c[7])/(c[5]-c[7]) /(c[4]-c[7])/(-20*c[5]*c[6]*c[4]+30*c[5]*c[6]*c[7]*c[4]+15*c[6]*c[5]-2 0*c[6]*c[7]*c[5]+15*c[6]*c[4]-20*c[6]*c[7]*c[4]-12*c[6]+15*c[7]*c[6]+1 5*c[5]*c[4]-20*c[5]*c[7]*c[4]-12*c[5]+15*c[7]*c[5]-12*c[4]+15*c[7]*c[4 ]+10-12*c[7])*(c[4]-1)*(-1+c[5])*(-1+c[6])*(-1+c[7]), a[6,3] = 3/4*1/c [4]^2*c[6]*(60*c[4]^3*c[5]*c[6]^3-90*c[5]*c[6]^3*c[4]^2+36*c[6]^3*c[4] *c[5]-6*c[6]^3*c[5]+3*c[6]^3*c[4]-20*c[6]^2*c[5]^3*c[4]^2-80*c[6]^2*c[ 5]^2*c[4]^3+150*c[6]^2*c[5]^2*c[4]^2-48*c[6]^2*c[4]*c[5]^2+8*c[6]^2*c[ 5]^2-60*c[5]*c[6]^2*c[4]^4+90*c[6]^2*c[5]*c[4]^3-48*c[6]^2*c[4]^2*c[5] +4*c[6]^2*c[4]*c[5]-4*c[6]^2*c[4]^2+30*c[6]*c[5]^3*c[4]^3+90*c[5]^2*c[ 6]*c[4]^4-180*c[5]^2*c[6]*c[4]^3+54*c[5]^2*c[6]*c[4]^2-12*c[5]^2*c[6]* c[4]+18*c[6]*c[5]*c[4]^3+6*c[5]*c[6]*c[4]^2+3*c[5]^2*c[4]^2-3*c[5]*c[4 ]^3)/(10*c[5]^3*c[4]^2+10*c[5]^2*c[4]^3-30*c[5]^2*c[4]^2+6*c[5]^2*c[4] -c[5]^2+6*c[5]*c[4]^2+c[5]*c[4]-c[4]^2), b[7] = -1/60*(10*c[5]*c[6]*c[ 4]-5*c[6]*c[5]-5*c[6]*c[4]+3*c[6]-5*c[5]*c[4]+3*c[5]+3*c[4]-2)/(c[4]-c [7])/(c[5]-c[7])/(c[6]-c[7])/c[7]/(-1+c[7]), a[7,1] = 1/4*c[7]*(-51*c[ 5]^2*c[6]*c[4]^4+14*c[6]*c[5]*c[4]^4+12*c[4]^2*c[5]^2*c[7]^3+2700*c[5] ^4*c[4]^4*c[6]^2*c[7]+12*c[7]^3*c[5]*c[4]^3-98*c[7]^3*c[5]^2*c[4]^3-2* c[7]*c[5]*c[4]^4+132*c[5]^2*c[4]^4*c[7]^2-480*c[4]^5*c[6]^2*c[5]^2+200 *c[4]^3*c[7]^3*c[5]^3-200*c[7]^2*c[5]^2*c[4]^5-760*c[4]^5*c[6]*c[5]^3- 10*c[5]*c[6]^2*c[4]^4+200*c[5]^5*c[4]^3*c[7]^2*c[6]^2-400*c[5]^2*c[4]^ 6*c[6]^2*c[7]^2+4*c[4]^5*c[5]*c[7]-34*c[5]^2*c[4]^5*c[7]+400*c[5]^3*c[ 4]^6*c[6]^2*c[7]-98*c[5]^3*c[4]^4*c[7]+420*c[7]^2*c[4]^5*c[5]^3+200*c[ 7]^3*c[5]^2*c[4]^4-300*c[6]^2*c[4]^5*c[7]*c[5]^3-40*c[7]*c[5]^2*c[4]^6 *c[6]+20*c[5]^4*c[4]^2*c[7]^2*c[6]-280*c[6]^2*c[7]*c[5]^4*c[4]^2-44*c[ 6]*c[4]^5*c[5]*c[7]-246*c[6]*c[5]^2*c[4]^4*c[7]+120*c[4]^6*c[6]^2*c[5] ^2-100*c[5]^2*c[7]^2*c[4]^6*c[6]+141*c[5]^4*c[4]^3*c[6]+1220*c[5]^4*c[ 4]^5*c[6]+400*c[5]^5*c[4]^5*c[6]^2-760*c[5]^4*c[4]^4*c[6]+100*c[5]^4*c [7]^3*c[4]^4-400*c[7]^2*c[4]^4*c[5]^3-200*c[5]^4*c[4]^5*c[7]^2+200*c[5 ]^4*c[4]^4*c[7]-2*c[4]^3*c[7]^3*c[6]-8*c[7]^2*c[6]*c[4]^4-380*c[7]^3*c [5]^3*c[4]^4-40*c[6]*c[4]^5*c[5]*c[7]^2+2*c[4]^4*c[7]^2-40*c[5]^2*c[7] ^3*c[4]^5-200*c[7]^2*c[5]^4*c[4]^3-40*c[7]^3*c[5]^4*c[4]^3+200*c[5]^3* c[7]*c[4]^6*c[6]-10*c[6]^2*c[7]^3*c[4]^3+30*c[6]^2*c[4]^5*c[5]+12*c[7] *c[6]^2*c[5]^3+4*c[7]^2*c[6]*c[5]^3+400*c[5]^4*c[4]^6*c[6]^2-40*c[5]^3 *c[7]*c[4]^6-380*c[5]^4*c[4]^5*c[7]+320*c[7]*c[6]^2*c[5]*c[4]^5+100*c[ 4]^6*c[6]^2*c[7]*c[5]^2+34*c[5]^4*c[4]^2*c[7]^2+40*c[5]^5*c[7]^2*c[4]^ 3+240*c[6]*c[5]^2*c[4]^5*c[7]+60*c[5]^4*c[7]^3*c[4]^2*c[6]+26*c[7]*c[4 ]^4*c[6]*c[5]-50*c[4]^4*c[6]^2*c[5]^5+200*c[5]^5*c[7]*c[4]^4*c[6]+132* c[7]^2*c[5]^3*c[4]^3-2*c[7]*c[4]^4*c[6]^2+60*c[5]^5*c[4]^2*c[6]^2*c[7] -500*c[5]^5*c[4]^4*c[6]^2*c[7]+160*c[5]^5*c[4]^4*c[6]+134*c[6]^2*c[5]^ 4*c[7]*c[4]+24*c[5]^4*c[6]^2*c[4]-169*c[5]^4*c[6]^2*c[4]^2-500*c[5]^3* c[4]^6*c[6]^2-40*c[5]^5*c[4]^4*c[7]+34*c[4]^5*c[5]*c[7]^2-100*c[5]^3*c [7]^2*c[4]^6-100*c[5]^5*c[4]^4*c[7]^2+160*c[4]^6*c[6]*c[5]^3-6*c[6]^2* c[7]*c[5]^2*c[4]^3+68*c[6]^2*c[7]^2*c[5]^2*c[4]-126*c[6]^2*c[7]^2*c[5] ^2*c[4]^2-600*c[6]*c[7]*c[5]^3*c[4]^3-21*c[6]^2*c[7]^3*c[4]*c[5]+40*c[ 7]^2*c[6]^2*c[5]^4+4*c[5]^4*c[7]*c[4]^2+58*c[7]^2*c[6]*c[5]*c[4]^4-100 *c[7]^2*c[5]^5*c[6]^2*c[4]^2-320*c[5]^4*c[4]*c[7]^2*c[6]^2+20*c[6]^2*c [4]^4*c[7]^2+40*c[5]^2*c[4]^6*c[7]^2+6*c[7]^2*c[6]*c[4]^3-8*c[6]*c[5]* c[7]^2*c[4]^2-4*c[5]*c[7]*c[6]*c[4]^3+14*c[5]^2*c[6]*c[7]*c[4]^2-910*c [4]^5*c[6]*c[7]*c[5]^3-340*c[5]^2*c[6]*c[7]^3*c[4]^4-4*c[6]^2*c[5]*c[4 ]^3+5*c[7]^3*c[4]^2*c[6]*c[5]+72*c[7]*c[6]^2*c[5]*c[4]^3+18*c[6]^2*c[7 ]^3*c[5]^2-12*c[6]^2*c[7]^2*c[5]^2-10*c[6]^2*c[4]^2*c[7]*c[5]-44*c[6]^ 2*c[7]^2*c[4]^2*c[5]-25*c[6]*c[5]^4*c[4]^2-24*c[7]*c[6]^2*c[5]^4+200*c [4]^5*c[7]*c[5]^3-2*c[6]^2*c[5]^2*c[7]*c[4]-42*c[6]^2*c[5]^2*c[7]*c[4] ^2+74*c[6]*c[5]^2*c[7]*c[4]^3+80*c[6]*c[5]^3*c[7]*c[4]^2-50*c[6]^2*c[5 ]^3*c[7]*c[4]-34*c[5]^4*c[4]^3*c[7]-24*c[7]^2*c[5]^2*c[4]^3+420*c[5]^4 *c[7]^2*c[4]^4-2*c[4]^3*c[7]^3+c[7]^3*c[5]^2*c[6]*c[4]+4*c[7]^2*c[5]^2 *c[6]*c[4]-6*c[6]*c[7]*c[4]^4-4*c[6]^2*c[7]^2*c[4]^2+50*c[5]^5*c[4]^3* c[7]^2*c[6]-18*c[5]^3*c[6]*c[7]*c[4]+14*c[6]^2*c[7]^2*c[4]*c[5]-20*c[5 ]^2*c[6]*c[4]^3+2*c[5]^2*c[7]*c[4]^3-2*c[5]^3*c[7]*c[4]^2+16*c[5]^3*c[ 6]*c[4]^2-96*c[5]^3*c[6]^2*c[4]^3+100*c[5]^5*c[4]^3*c[6]^2*c[7]-60*c[4 ]^3*c[7]^2*c[5]^2*c[6]^2+10*c[6]*c[7]*c[4]^5+1540*c[4]^5*c[5]^3*c[6]^2 -34*c[7]^3*c[4]^4*c[5]-4*c[7]^2*c[5]^4*c[4]+12*c[5]^2*c[4]^4*c[7]-41*c [6]^2*c[5]^2*c[4]^3+14*c[6]^2*c[5]^2*c[4]^2-54*c[6]*c[5]^3*c[4]^3-12*c [6]^2*c[5]^3*c[4]+47*c[6]^2*c[5]^3*c[4]^2+12*c[5]^3*c[7]*c[4]^3-98*c[5 ]^4*c[7]*c[4]^2*c[6]-50*c[5]^3*c[7]^3*c[4]^4*c[6]+4*c[7]*c[6]^2*c[4]^3 +156*c[7]^2*c[5]^2*c[6]*c[4]^3-23*c[7]^3*c[5]^2*c[6]*c[4]^2+188*c[7]^2 *c[5]^3*c[6]*c[4]^2+66*c[5]^3*c[7]^3*c[6]*c[4]+2*c[7]^3*c[4]^2*c[5]+10 0*c[5]^5*c[4]^5*c[7]+104*c[4]^3*c[7]^2*c[5]*c[6]^2-12*c[7]^2*c[6]*c[5] *c[4]^3-36*c[7]^2*c[5]^2*c[6]*c[4]^2-32*c[7]^2*c[5]^3*c[6]*c[4]-320*c[ 7]^2*c[5]^3*c[6]*c[4]^3-560*c[5]^3*c[6]^2*c[4]^4-4*c[6]^2*c[4]^3*c[7]^ 2+240*c[5]^3*c[6]^2*c[7]*c[4]^3-34*c[5]^3*c[7]^3*c[4]^2-2*c[7]^3*c[5]^ 2*c[4]+164*c[6]^2*c[7]*c[5]^3*c[4]^2-20*c[6]^2*c[7]^2*c[4]*c[5]^3+2*c[ 7]^2*c[5]^3*c[4]-130*c[5]^5*c[4]^3*c[6]*c[7]-90*c[7]^3*c[5]^2*c[6]^2*c [4]^3-16*c[7]^2*c[4]^4*c[5]-60*c[7]^2*c[5]^3*c[6]^2*c[4]^2-300*c[6]^2* c[4]^5*c[5]*c[7]^2-150*c[7]^2*c[4]^5*c[6]*c[5]^3+380*c[5]^2*c[6]*c[4]^ 5*c[7]^2+140*c[7]^3*c[5]^2*c[4]^3*c[6]-810*c[7]^3*c[5]^3*c[6]^2*c[4]^2 -14*c[7]^3*c[6]*c[5]*c[4]^3-280*c[7]^3*c[6]^2*c[5]*c[4]^3+240*c[7]^3*c [5]^3*c[6]^2*c[4]-20*c[5]^4*c[7]^2*c[4]*c[6]-6*c[6]*c[7]^3*c[5]^3+22*c [5]^4*c[4]*c[7]*c[6]-280*c[4]^5*c[6]^2*c[7]*c[5]^2+400*c[4]^5*c[6]^2*c [7]^2*c[5]^2+253*c[6]^2*c[4]^4*c[5]^2+470*c[7]^3*c[5]^3*c[4]^3*c[6]+10 0*c[5]^2*c[4]^5*c[7]^3*c[6]+6*c[6]^2*c[7]^3*c[4]^2-300*c[5]^5*c[4]^5*c [6]-120*c[5]*c[7]*c[4]^6*c[6]^2-10*c[6]^2*c[7]*c[4]^5-1100*c[5]^4*c[4] ^5*c[6]^2*c[7]-226*c[7]*c[6]^2*c[5]*c[4]^4-770*c[5]^3*c[4]^4*c[6]^2*c[ 7]+280*c[5]^3*c[4]^3*c[6]^2*c[7]^2-1000*c[5]^4*c[4]^4*c[6]*c[7]+240*c[ 5]^2*c[4]^4*c[7]*c[6]^2+1240*c[5]^3*c[4]^4*c[6]*c[7]+600*c[5]^4*c[4]^3 *c[6]*c[7]+1200*c[7]^3*c[5]^3*c[6]^2*c[4]^3+40*c[7]^3*c[4]^4*c[5]*c[6] -16*c[7]^2*c[5]^3*c[4]^2-1250*c[5]^4*c[4]^5*c[6]^2-600*c[6]^2*c[7]^3*c [5]^3*c[4]^4+200*c[6]^2*c[7]^2*c[5]^3*c[4]^5-200*c[6]^2*c[7]^3*c[4]^5* c[5]-140*c[5]^4*c[4]^4*c[6]^2+560*c[5]^4*c[4]^3*c[6]^2-1850*c[7]^2*c[5 ]^4*c[6]^2*c[4]^3+1230*c[7]^2*c[5]^4*c[6]^2*c[4]^2-150*c[5]^4*c[7]^3*c [4]^3*c[6]-300*c[5]^4*c[4]^6*c[6]+310*c[5]^3*c[4]^4*c[7]^2*c[6]-70*c[5 ]^4*c[4]^3*c[7]^2*c[6]-380*c[5]^2*c[4]^4*c[7]^2*c[6]-850*c[5]^4*c[4]^3 *c[6]^2*c[7]-40*c[7]^2*c[4]^4*c[6]^2*c[5]-450*c[7]^2*c[4]^4*c[5]^3*c[6 ]^2+250*c[7]^2*c[4]^4*c[6]^2*c[5]^2-4*c[7]^2*c[4]^5+136*c[5]^2*c[6]*c[ 4]^5+400*c[6]^2*c[7]^3*c[5]^2*c[4]^5+800*c[6]^2*c[7]^2*c[5]^4*c[4]^4+4 *c[7]^3*c[4]^4+100*c[5]^4*c[4]^6*c[7]+437*c[4]^4*c[6]*c[5]^3-22*c[6]*c [5]*c[4]^5-60*c[5]^5*c[6]^2*c[4]^3+200*c[6]^2*c[4]^6*c[5]*c[7]^2+420*c [7]^3*c[6]^2*c[5]*c[4]^4+400*c[5]^4*c[4]^5*c[6]*c[7]-500*c[7]^3*c[4]^4 *c[6]^2*c[5]^2+100*c[5]^3*c[7]^3*c[4]^5-103*c[6]^2*c[7]^3*c[5]^2*c[4]+ 260*c[6]^2*c[7]^3*c[5]^2*c[4]^2-300*c[5]^3*c[7]^3*c[4]^2*c[6]+97*c[6]^ 2*c[5]*c[7]^3*c[4]^2-30*c[6]^2*c[7]^3*c[5]^3+4*c[7]^3*c[4]*c[5]^3)/c[6 ]/(-50*c[5]^2*c[6]*c[4]^4+23*c[5]^3*c[4]+100*c[5]^4*c[4]^3*c[6]+2*c[5] ^2+2*c[4]^2-3*c[5]^3-3*c[4]^3-3*c[6]*c[5]^2+5*c[6]*c[5]^3-3*c[6]*c[4]^ 2+5*c[6]*c[4]^3+23*c[5]*c[4]^3+18*c[5]*c[6]*c[4]^2-12*c[5]*c[4]^2+3*c[ 5]*c[6]*c[4]-2*c[5]*c[4]-50*c[6]*c[5]^4*c[4]^2-40*c[6]*c[5]*c[4]^3-40* c[5]^3*c[6]*c[4]-12*c[5]^2*c[4]-140*c[5]^3*c[4]^2+240*c[5]^2*c[6]*c[4] ^3-140*c[5]^2*c[6]*c[4]^2+240*c[5]^3*c[6]*c[4]^2-400*c[6]*c[5]^3*c[4]^ 3+91*c[5]^2*c[4]^2+30*c[5]^4*c[4]^2+30*c[4]^4*c[5]^2-50*c[5]^4*c[4]^3+ 18*c[5]^2*c[6]*c[4]-140*c[5]^2*c[4]^3+210*c[5]^3*c[4]^3+100*c[4]^4*c[6 ]*c[5]^3-50*c[5]^3*c[4]^4)/c[5]/c[4]^2, `b*`[8] = 0, a[5,3] = -3/4*c[5 ]^2*(2*c[5]-3*c[4])/c[4]^2, `b*`[4] = -1/60*(-30*c[5]^3*c[4]-42*c[7]*c [6]^2+89*c[5]*c[7]*c[4]-18*c[6]+6*c[5]+6*c[4]+27*c[7]*c[6]-9*c[5]^2+18 0*c[7]*c[6]^2*c[5]^3-100*c[7]*c[6]*c[5]^3+27*c[6]^2+186*c[5]*c[6]*c[7] *c[4]-97*c[6]*c[5]^2+66*c[6]*c[5]+60*c[6]*c[5]^3-100*c[6]^2*c[5]^3-9*c [7]*c[5]+14*c[7]*c[5]^2-100*c[6]^2*c[7]*c[4]^2+120*c[6]^2*c[7]*c[4]+60 *c[6]^2*c[4]^2-9*c[7]*c[4]-103*c[6]*c[7]*c[5]-36*c[6]*c[4]^2-61*c[6]*c [7]*c[4]+60*c[6]*c[7]*c[4]^2-30*c[5]*c[7]*c[4]^2+126*c[5]*c[6]*c[4]^2+ 18*c[5]*c[4]^2-112*c[5]*c[6]*c[4]-57*c[5]*c[4]+39*c[6]*c[4]-280*c[6]^2 *c[4]^2*c[5]+330*c[6]^2*c[4]*c[5]+380*c[5]^2*c[6]*c[7]*c[4]^2-560*c[6] ^2*c[7]*c[4]*c[5]+192*c[6]^2*c[7]*c[5]-310*c[6]^2*c[7]*c[5]^2-230*c[6] *c[5]*c[7]*c[4]^2+156*c[6]*c[7]*c[5]^2+180*c[6]^2*c[5]^2+500*c[6]^2*c[ 4]^2*c[7]*c[5]+950*c[6]^2*c[5]^2*c[7]*c[4]-900*c[6]^2*c[5]^2*c[7]*c[4] ^2-300*c[6]*c[5]^3*c[7]*c[4]^2-600*c[6]^2*c[5]^3*c[7]*c[4]-117*c[6]^2* c[5]-130*c[5]^3*c[6]*c[4]+50*c[5]^3*c[7]*c[4]+93*c[5]^2*c[4]+230*c[5]^ 3*c[6]*c[7]*c[4]+50*c[5]^2*c[7]*c[4]^2-190*c[5]^2*c[6]*c[4]^2+150*c[5] ^3*c[6]*c[4]^2+470*c[6]^2*c[5]^2*c[4]^2+320*c[6]^2*c[5]^3*c[4]-300*c[6 ]^2*c[5]^3*c[4]^2-30*c[5]^2*c[4]^2-530*c[6]^2*c[4]*c[5]^2-75*c[6]^2*c[ 4]+156*c[5]^2*c[6]*c[4]-150*c[5]^2*c[7]*c[4]-270*c[5]^2*c[6]*c[7]*c[4] +600*c[6]^2*c[7]*c[5]^3*c[4]^2)/(-9*c[5]*c[7]*c[4]-3*c[7]*c[6]-c[4]^2- 28*c[5]*c[6]*c[7]*c[4]+c[7]*c[5]+c[7]*c[4]+9*c[6]*c[7]*c[5]-9*c[6]*c[4 ]^2+8*c[6]*c[4]^3+9*c[6]*c[7]*c[4]-8*c[6]*c[7]*c[4]^2-4*c[5]*c[4]^3+4* c[5]*c[7]*c[4]^2+28*c[5]*c[6]*c[4]^2+9*c[5]*c[4]^2-9*c[5]*c[6]*c[4]-c[ 5]*c[4]+3*c[6]*c[4]-30*c[5]^2*c[6]*c[7]*c[4]^2+28*c[6]*c[5]*c[7]*c[4]^ 2-8*c[6]*c[7]*c[5]^2-28*c[6]*c[5]*c[4]^3+30*c[5]^2*c[6]*c[4]^3-28*c[5] ^2*c[6]*c[4]^2-4*c[5]^2*c[4]^2+8*c[5]^2*c[6]*c[4]+4*c[5]^2*c[7]*c[4]+2 8*c[5]^2*c[6]*c[7]*c[4])/(-c[4]+c[6])/(-c[4]+c[5])/c[4], a[8,6] = 1/2* (5*c[5]*c[6]*c[4]+c[6]-2*c[6]*c[4]-2*c[6]*c[5]+25*c[5]*c[7]*c[4]+6*c[5 ]+9*c[7]+10*c[7]^2*c[4]-4+6*c[4]-6*c[7]^2-14*c[7]*c[4]-10*c[5]*c[4]-14 *c[7]*c[5]-20*c[7]^2*c[5]*c[4]+10*c[7]^2*c[5])/(c[6]-c[7])/(-c[4]+c[6] )/(-c[5]+c[6])/c[6]/(-20*c[5]*c[6]*c[4]+30*c[5]*c[6]*c[7]*c[4]+15*c[6] *c[5]-20*c[6]*c[7]*c[5]+15*c[6]*c[4]-20*c[6]*c[7]*c[4]-12*c[6]+15*c[7] *c[6]+15*c[5]*c[4]-20*c[5]*c[7]*c[4]-12*c[5]+15*c[7]*c[5]-12*c[4]+15*c [7]*c[4]+10-12*c[7])*(c[4]-1)*(-1+c[5])*(-1+c[6]), b[8] = 1/60*(-20*c[ 5]*c[6]*c[4]+30*c[5]*c[6]*c[7]*c[4]+15*c[6]*c[5]-20*c[6]*c[7]*c[5]+15* c[6]*c[4]-20*c[6]*c[7]*c[4]-12*c[6]+15*c[7]*c[6]+15*c[5]*c[4]-20*c[5]* c[7]*c[4]-12*c[5]+15*c[7]*c[5]-12*c[4]+15*c[7]*c[4]+10-12*c[7])/(c[4]- 1)/(-1+c[5])/(-1+c[6])/(-1+c[7]), a[5,1] = 1/4*c[5]*(2*c[5]^2-5*c[5]*c [4]+4*c[4]^2)/c[4]^2, `b*`[5] = 1/60*(-42*c[7]*c[6]^2+89*c[5]*c[7]*c[4 ]-18*c[6]+6*c[5]+6*c[4]+27*c[7]*c[6]-9*c[4]^2+27*c[6]^2+186*c[5]*c[6]* c[7]*c[4]-36*c[6]*c[5]^2+39*c[6]*c[5]-9*c[7]*c[5]-100*c[6]^2*c[4]^3+60 0*c[6]^2*c[7]*c[5]^2*c[4]^3-100*c[7]*c[6]*c[4]^3-310*c[6]^2*c[7]*c[4]^ 2+192*c[6]^2*c[7]*c[4]+180*c[6]^2*c[4]^2+14*c[7]*c[4]^2-9*c[7]*c[4]-61 *c[6]*c[7]*c[5]-97*c[6]*c[4]^2+60*c[6]*c[4]^3-103*c[6]*c[7]*c[4]+156*c [6]*c[7]*c[4]^2-30*c[5]*c[4]^3-150*c[5]*c[7]*c[4]^2+156*c[5]*c[6]*c[4] ^2+93*c[5]*c[4]^2-112*c[5]*c[6]*c[4]-57*c[5]*c[4]+66*c[6]*c[4]+230*c[5 ]*c[7]*c[6]*c[4]^3-530*c[6]^2*c[4]^2*c[5]+330*c[6]^2*c[4]*c[5]+380*c[5 ]^2*c[6]*c[7]*c[4]^2+320*c[6]^2*c[5]*c[4]^3-560*c[6]^2*c[7]*c[4]*c[5]- 600*c[7]*c[6]^2*c[5]*c[4]^3+120*c[6]^2*c[7]*c[5]-100*c[6]^2*c[7]*c[5]^ 2-270*c[6]*c[5]*c[7]*c[4]^2+60*c[6]*c[7]*c[5]^2+60*c[6]^2*c[5]^2+950*c [6]^2*c[4]^2*c[7]*c[5]+500*c[6]^2*c[5]^2*c[7]*c[4]-900*c[6]^2*c[5]^2*c [7]*c[4]^2-300*c[6]*c[5]^2*c[7]*c[4]^3-75*c[6]^2*c[5]+50*c[7]*c[5]*c[4 ]^3-130*c[6]*c[5]*c[4]^3+18*c[5]^2*c[4]+150*c[5]^2*c[6]*c[4]^3+50*c[5] ^2*c[7]*c[4]^2-190*c[5]^2*c[6]*c[4]^2-300*c[6]^2*c[5]^2*c[4]^3+470*c[6 ]^2*c[5]^2*c[4]^2-30*c[5]^2*c[4]^2-280*c[6]^2*c[4]*c[5]^2+180*c[7]*c[6 ]^2*c[4]^3-117*c[6]^2*c[4]+126*c[5]^2*c[6]*c[4]-30*c[5]^2*c[7]*c[4]-23 0*c[5]^2*c[6]*c[7]*c[4])/(-4*c[5]^3*c[4]-9*c[5]*c[7]*c[4]-3*c[7]*c[6]- c[5]^2-28*c[5]*c[6]*c[7]*c[4]-9*c[6]*c[5]^2+3*c[6]*c[5]+8*c[6]*c[5]^3+ c[7]*c[5]+c[7]*c[4]+9*c[6]*c[7]*c[5]+9*c[6]*c[7]*c[4]-8*c[6]*c[7]*c[4] ^2+4*c[5]*c[7]*c[4]^2+8*c[5]*c[6]*c[4]^2-9*c[5]*c[6]*c[4]-c[5]*c[4]-30 *c[5]^2*c[6]*c[7]*c[4]^2+28*c[6]*c[5]*c[7]*c[4]^2-8*c[6]*c[7]*c[5]^2-2 8*c[5]^3*c[6]*c[4]+9*c[5]^2*c[4]-28*c[5]^2*c[6]*c[4]^2+30*c[5]^3*c[6]* c[4]^2-4*c[5]^2*c[4]^2+28*c[5]^2*c[6]*c[4]+4*c[5]^2*c[7]*c[4]+28*c[5]^ 2*c[6]*c[7]*c[4])/(-c[5]+c[6])/(-c[4]+c[5])/c[5], a[5,4] = c[5]^2*(-c[ 4]+c[5])/c[4]^2, a[7,6] = 1/2*(5*c[5]*c[4]+1-2*c[4]-2*c[5])*(c[4]-c[7] )*(c[5]-c[7])*(c[6]-c[7])*c[7]/(10*c[5]*c[6]*c[4]-5*c[6]*c[5]-5*c[6]*c [4]+3*c[6]-5*c[5]*c[4]+3*c[5]+3*c[4]-2)/(-c[4]+c[6])/(-c[5]+c[6])/c[6] , c[3] = 2/3*c[4], a[4,3] = 3/4*c[4], a[3,2] = 2/9*c[4]^2/c[2], a[7,4] = -1/2*c[7]*(-68*c[5]^2*c[6]*c[4]^4-7*c[6]*c[5]*c[4]^4+7*c[4]^2*c[5]^ 2*c[7]^3-1400*c[5]^4*c[4]^4*c[6]^2*c[7]+11*c[7]^3*c[5]*c[4]^3-72*c[7]^ 3*c[5]^2*c[4]^3-3*c[7]*c[5]*c[4]^4-58*c[5]^2*c[4]^4*c[7]^2-120*c[6]*c[ 4]^7*c[7]*c[5]+320*c[4]^5*c[6]^2*c[5]^2-200*c[4]^3*c[7]^3*c[5]^3-100*c [4]^5*c[6]*c[5]^3-2*c[5]*c[6]^2*c[4]^4-600*c[4]^5*c[7]^3*c[6]*c[5]^3-2 00*c[5]^5*c[4]^3*c[7]^2*c[6]^2+400*c[5]^2*c[4]^6*c[6]^2*c[7]^2-21*c[4] ^5*c[5]*c[7]+120*c[5]^2*c[4]^5*c[7]-600*c[5]^3*c[4]^6*c[6]^2*c[7]-198* c[5]^3*c[4]^4*c[7]-280*c[7]^2*c[4]^5*c[5]^3+240*c[7]^3*c[5]^2*c[4]^4+1 250*c[6]^2*c[4]^5*c[7]*c[5]^3+120*c[7]*c[5]^2*c[4]^6*c[6]-340*c[5]^4*c [4]^2*c[7]^2*c[6]+400*c[6]^2*c[7]*c[5]^4*c[4]^2-240*c[5]^2*c[4]^6*c[7] -300*c[6]*c[5]*c[7]^2*c[4]^6-228*c[6]*c[4]^5*c[5]*c[7]-119*c[6]*c[5]^2 *c[4]^4*c[7]-200*c[5]*c[7]^3*c[4]^6*c[6]-600*c[5]^4*c[7]*c[4]^6*c[6]-1 20*c[4]^6*c[6]^2*c[5]^2+600*c[5]^2*c[7]^2*c[4]^6*c[6]+100*c[5]^2*c[7]^ 3*c[4]^6-187*c[5]^4*c[4]^3*c[6]-910*c[5]^4*c[4]^5*c[6]-320*c[5]^2*c[6] *c[4]^6+620*c[5]^4*c[4]^4*c[6]-100*c[5]^4*c[7]^3*c[4]^4-40*c[7]^2*c[4] ^4*c[5]^3+40*c[7]^3*c[4]^5*c[5]+10*c[6]*c[7]^2*c[4]^5-10*c[6]*c[7]^3*c [4]^4+100*c[5]^4*c[4]^5*c[7]^2+80*c[5]^4*c[4]^4*c[7]+8*c[4]^3*c[7]^3*c [6]-2*c[7]^2*c[6]*c[4]^4+280*c[7]^3*c[5]^3*c[4]^4+80*c[6]*c[4]^5*c[5]* c[7]^2+c[4]^4*c[7]^2-360*c[5]^2*c[7]^3*c[4]^5-80*c[7]^2*c[5]^4*c[4]^3+ 40*c[7]^3*c[5]^4*c[4]^3-1200*c[5]^3*c[7]*c[4]^6*c[6]+10*c[6]^2*c[7]^3* c[4]^3-10*c[6]^2*c[4]^5*c[5]-12*c[7]*c[6]^2*c[5]^3-4*c[7]^2*c[6]*c[5]^ 3-80*c[5]^4*c[4]^5*c[7]-240*c[7]*c[6]^2*c[5]*c[4]^5-3*c[5]^4*c[4]^2*c[ 7]^2+20*c[5]^5*c[7]^2*c[4]^3+180*c[6]*c[5]^2*c[4]^5*c[7]-60*c[5]^4*c[7 ]^3*c[4]^2*c[6]+400*c[5]^2*c[7]^3*c[4]^6*c[6]-25*c[4]^5*c[5]^3+200*c[5 ]^5*c[4]^4*c[7]^2*c[6]+60*c[5]^2*c[4]^7*c[7]+100*c[7]*c[4]^4*c[6]*c[5] -40*c[5]*c[4]^6*c[7]^2-150*c[4]^4*c[6]^2*c[5]^5+50*c[5]^5*c[7]*c[4]^4* c[6]+174*c[7]^2*c[5]^3*c[4]^3+8*c[7]*c[4]^4*c[6]^2+600*c[5]^3*c[7]*c[4 ]^7*c[6]-60*c[5]^5*c[4]^2*c[6]^2*c[7]+200*c[5]^5*c[4]^4*c[6]^2*c[7]-40 *c[5]^5*c[4]^4*c[6]-200*c[5]^5*c[7]*c[4]^5*c[6]-149*c[6]^2*c[5]^4*c[7] *c[4]-24*c[5]^4*c[6]^2*c[4]+189*c[5]^4*c[6]^2*c[4]^2+300*c[5]^3*c[4]^6 *c[6]^2-20*c[5]^5*c[4]^4*c[7]-5*c[4]^5*c[5]*c[7]^2+600*c[4]^6*c[6]*c[5 ]^3-84*c[6]^2*c[7]*c[5]^2*c[4]^3-74*c[6]^2*c[7]^2*c[5]^2*c[4]+162*c[6] ^2*c[7]^2*c[5]^2*c[4]^2+311*c[6]*c[7]*c[5]^3*c[4]^3+21*c[6]^2*c[7]^3*c [4]*c[5]-40*c[7]^2*c[6]^2*c[5]^4+11*c[5]^4*c[7]*c[4]^2+107*c[7]^2*c[6] *c[5]*c[4]^4+100*c[7]^2*c[5]^5*c[6]^2*c[4]^2+320*c[5]^4*c[4]*c[7]^2*c[ 6]^2-10*c[6]^2*c[4]^4*c[7]^2+300*c[5]^2*c[4]^6*c[7]^2-3*c[7]^2*c[6]*c[ 4]^3-2*c[5]*c[7]^2*c[4]^3+15*c[6]*c[5]*c[7]^2*c[4]^2-8*c[5]*c[7]*c[6]* c[4]^3-14*c[5]^2*c[6]*c[7]*c[4]^2-2000*c[5]^4*c[7]^2*c[4]^4*c[6]+200*c [4]^5*c[6]*c[7]*c[5]^3+250*c[5]^2*c[6]*c[7]^3*c[4]^4+4*c[6]^2*c[5]*c[4 ]^3-26*c[7]^3*c[4]^2*c[6]*c[5]-69*c[7]*c[6]^2*c[5]*c[4]^3-18*c[6]^2*c[ 7]^3*c[5]^2+12*c[6]^2*c[7]^2*c[5]^2+10*c[6]^2*c[4]^2*c[7]*c[5]+50*c[6] ^2*c[7]^2*c[4]^2*c[5]+31*c[6]*c[5]^4*c[4]^2+24*c[7]*c[6]^2*c[5]^4+240* c[4]^5*c[7]*c[5]^3+2*c[6]^2*c[5]^2*c[7]*c[4]+39*c[6]^2*c[5]^2*c[7]*c[4 ]^2-17*c[6]*c[5]^2*c[7]*c[4]^3-73*c[6]*c[5]^3*c[7]*c[4]^2+59*c[6]^2*c[ 5]^3*c[7]*c[4]-12*c[5]^4*c[4]^3*c[7]-12*c[7]^2*c[5]^2*c[4]^3+40*c[5]^4 *c[7]^2*c[4]^4-c[4]^3*c[7]^3+17*c[7]^3*c[5]^2*c[6]*c[4]-9*c[7]^2*c[5]^ 2*c[6]*c[4]+3*c[6]*c[7]*c[4]^4+4*c[4]^5*c[5]+4*c[6]^2*c[7]^2*c[4]^2-15 0*c[5]^5*c[4]^3*c[7]^2*c[6]+21*c[5]^3*c[6]*c[7]*c[4]-14*c[6]^2*c[7]^2* c[4]*c[5]+23*c[5]^2*c[6]*c[4]^3+13*c[5]^2*c[7]*c[4]^3-11*c[5]^3*c[7]*c [4]^2-17*c[5]^3*c[6]*c[4]^2+168*c[5]^3*c[6]^2*c[4]^3+50*c[5]^5*c[4]^3* c[6]^2*c[7]-220*c[4]^3*c[7]^2*c[5]^2*c[6]^2-6*c[6]*c[7]*c[4]^5-580*c[4 ]^5*c[5]^3*c[6]^2-31*c[7]^3*c[4]^4*c[5]-2*c[7]^2*c[5]^4*c[4]-4*c[5]^2* c[4]^4*c[7]+53*c[6]^2*c[5]^2*c[4]^3-14*c[6]^2*c[5]^2*c[4]^2+47*c[6]*c[ 5]^3*c[4]^3+12*c[6]^2*c[5]^3*c[4]-59*c[6]^2*c[5]^3*c[4]^2+28*c[5]^3*c[ 7]*c[4]^3+127*c[5]^4*c[7]*c[4]^2*c[6]+1250*c[5]^3*c[7]^3*c[4]^4*c[6]+1 0*c[6]*c[5]*c[4]^6+240*c[5]*c[4]^6*c[7]*c[6]-4*c[7]*c[6]^2*c[4]^3+11*c [7]^2*c[5]^2*c[6]*c[4]^3-80*c[7]^3*c[5]^2*c[6]*c[4]^2-121*c[7]^2*c[5]^ 3*c[6]*c[4]^2-96*c[5]^3*c[7]^3*c[6]*c[4]+c[7]^3*c[4]^2*c[5]-68*c[4]^3* c[7]^2*c[5]*c[6]^2-80*c[7]^2*c[6]*c[5]*c[4]^3+68*c[7]^2*c[5]^2*c[6]*c[ 4]^2-10*c[4]^4*c[5]^2+20*c[7]^2*c[5]^3*c[6]*c[4]-240*c[7]^2*c[5]^3*c[6 ]*c[4]^3-9*c[5]^4*c[4]^3-2*c[6]^2*c[4]^3*c[7]^2+260*c[5]^3*c[6]^2*c[7] *c[4]^3+49*c[5]^3*c[7]^3*c[4]^2-c[7]^3*c[5]^2*c[4]-228*c[6]^2*c[7]*c[5 ]^3*c[4]^2+30*c[6]^2*c[7]^2*c[4]*c[5]^3-2*c[7]^2*c[5]^2*c[4]^2+5*c[7]^ 2*c[5]^3*c[4]+40*c[5]^5*c[4]^3*c[6]*c[7]+36*c[4]^6*c[7]*c[5]+90*c[7]^3 *c[5]^2*c[6]^2*c[4]^3+16*c[7]^2*c[4]^4*c[5]-20*c[7]^2*c[5]^3*c[6]^2*c[ 4]^2+300*c[6]^2*c[4]^5*c[5]*c[7]^2+250*c[7]^2*c[4]^5*c[6]*c[5]^3-490*c [5]^2*c[6]*c[4]^5*c[7]^2+120*c[7]^3*c[5]^2*c[4]^3*c[6]+810*c[7]^3*c[5] ^3*c[6]^2*c[4]^2+111*c[7]^3*c[6]*c[5]*c[4]^3+280*c[7]^3*c[6]^2*c[5]*c[ 4]^3-240*c[7]^3*c[5]^3*c[6]^2*c[4]+15*c[5]^4*c[4]^4+60*c[5]^4*c[7]^2*c [4]*c[6]+6*c[6]*c[7]^3*c[5]^3-31*c[5]^4*c[4]*c[7]*c[6]-200*c[4]^5*c[6] ^2*c[7]*c[5]^2-500*c[4]^5*c[6]^2*c[7]^2*c[5]^2+9*c[4]^5*c[5]^2-181*c[6 ]^2*c[4]^4*c[5]^2-1280*c[7]^3*c[5]^3*c[4]^3*c[6]-600*c[5]^2*c[4]^5*c[7 ]^3*c[6]-6*c[4]^6*c[5]-6*c[6]^2*c[7]^3*c[4]^2+100*c[5]^5*c[4]^5*c[6]+1 20*c[5]*c[7]*c[4]^6*c[6]^2+420*c[6]*c[4]^5*c[7]^3*c[5]+600*c[5]^4*c[4] ^5*c[6]^2*c[7]+175*c[7]*c[6]^2*c[5]*c[4]^4-430*c[5]^3*c[4]^4*c[6]^2*c[ 7]+200*c[5]^3*c[4]^3*c[6]^2*c[7]^2-450*c[5]^4*c[4]^4*c[6]*c[7]+160*c[5 ]^2*c[4]^4*c[7]*c[6]^2-200*c[5]^3*c[4]^4*c[6]*c[7]-280*c[5]^4*c[4]^3*c [6]*c[7]+6*c[5]^3*c[4]^3-100*c[5]^2*c[4]^7*c[7]^2-1200*c[7]^3*c[5]^3*c [6]^2*c[4]^3-320*c[7]^3*c[4]^4*c[5]*c[6]-30*c[7]^2*c[5]^3*c[4]^2-300*c [6]*c[5]^3*c[4]^7-350*c[5]^4*c[4]^5*c[6]^2+600*c[6]^2*c[7]^3*c[5]^3*c[ 4]^4+600*c[6]*c[7]^2*c[5]^4*c[4]^5+200*c[6]^2*c[7]^3*c[4]^5*c[5]+200*c [7]^2*c[4]^7*c[6]*c[5]+1100*c[5]^4*c[4]^4*c[6]^2-720*c[5]^4*c[4]^3*c[6 ]^2+1750*c[7]^2*c[5]^4*c[6]^2*c[4]^3-1230*c[7]^2*c[5]^4*c[6]^2*c[4]^2+ 150*c[5]^4*c[7]^3*c[4]^3*c[6]+300*c[5]^4*c[4]^6*c[6]+480*c[5]^3*c[4]^4 *c[7]^2*c[6]+1420*c[5]^4*c[4]^3*c[7]^2*c[6]+20*c[5]^2*c[4]^4*c[7]^2*c[ 6]+130*c[5]^4*c[4]^3*c[6]^2*c[7]-40*c[7]^2*c[4]^4*c[6]^2*c[5]-350*c[7] ^2*c[4]^4*c[5]^3*c[6]^2+230*c[7]^2*c[4]^4*c[6]^2*c[5]^2+10*c[4]^6*c[5] ^2+188*c[5]^2*c[6]*c[4]^5-400*c[6]^2*c[7]^3*c[5]^2*c[4]^5-600*c[6]^2*c [7]^2*c[5]^4*c[4]^4+120*c[5]^2*c[4]^7*c[6]-94*c[4]^4*c[6]*c[5]^3+6*c[6 ]*c[5]*c[4]^5+60*c[5]^5*c[6]^2*c[4]^3-200*c[6]^2*c[4]^6*c[5]*c[7]^2-42 0*c[7]^3*c[6]^2*c[5]*c[4]^4+1550*c[5]^4*c[4]^5*c[6]*c[7]+500*c[7]^3*c[ 4]^4*c[6]^2*c[5]^2-400*c[5]^2*c[4]^7*c[6]*c[7]^2+6*c[5]^3*c[4]^4+103*c [6]^2*c[7]^3*c[5]^2*c[4]-260*c[6]^2*c[7]^3*c[5]^2*c[4]^2+540*c[5]^3*c[ 7]^3*c[4]^2*c[6]-97*c[6]^2*c[5]*c[7]^3*c[4]^2+30*c[6]^2*c[7]^3*c[5]^3- 4*c[7]^3*c[4]*c[5]^3)/(-240*c[5]^2*c[6]*c[4]^4+40*c[6]*c[5]*c[4]^4-9*c [5]^3*c[4]-100*c[4]^5*c[6]*c[5]^3-350*c[5]^4*c[4]^3*c[6]+30*c[5]^5*c[4 ]^2+2*c[5]^3-2*c[4]^3-50*c[5]^5*c[4]^3-3*c[5]^4+3*c[4]^4+50*c[4]^5*c[5 ]^3+100*c[5]^5*c[4]^3*c[6]-3*c[6]*c[5]^3+3*c[6]*c[4]^3+9*c[5]*c[4]^3-6 *c[5]*c[6]*c[4]^2+4*c[5]*c[4]^2+240*c[6]*c[5]^4*c[4]^2+5*c[6]*c[5]^4-1 3*c[6]*c[5]*c[4]^3+13*c[5]^3*c[6]*c[4]-4*c[5]^2*c[4]+68*c[5]^3*c[4]^2+ 100*c[5]^2*c[6]*c[4]^3-100*c[5]^3*c[6]*c[4]^2-140*c[5]^4*c[4]^2+140*c[ 4]^4*c[5]^2+180*c[5]^4*c[4]^3+6*c[5]^2*c[6]*c[4]-68*c[5]^2*c[4]^3-23*c [4]^4*c[5]-40*c[5]^4*c[4]*c[6]-30*c[4]^5*c[5]^2-5*c[6]*c[4]^4-50*c[5]^ 5*c[6]*c[4]^2+50*c[5]^2*c[6]*c[4]^5+350*c[4]^4*c[6]*c[5]^3+23*c[5]^4*c [4]-180*c[5]^3*c[4]^4)/(-c[4]+c[6])/c[4]^2, `b*`[7] = 1/60*(-30*c[5]^3 *c[4]-18*c[6]+6*c[5]+6*c[4]-9*c[5]^2-9*c[4]^2+27*c[6]^2-97*c[6]*c[5]^2 +66*c[6]*c[5]+60*c[6]*c[5]^3-100*c[6]^2*c[5]^3-100*c[6]^2*c[4]^3+180*c [6]^2*c[4]^2-97*c[6]*c[4]^2+60*c[6]*c[4]^3-30*c[5]*c[4]^3+312*c[5]*c[6 ]*c[4]^2+107*c[5]*c[4]^2-215*c[5]*c[6]*c[4]-66*c[5]*c[4]+66*c[6]*c[4]- 840*c[6]^2*c[4]^2*c[5]+522*c[6]^2*c[4]*c[5]+500*c[6]^2*c[5]*c[4]^3+180 *c[6]^2*c[5]^2-117*c[6]^2*c[5]-230*c[6]*c[5]*c[4]^3-230*c[5]^3*c[6]*c[ 4]+107*c[5]^2*c[4]+50*c[5]^3*c[4]^2+380*c[5]^2*c[6]*c[4]^3-460*c[5]^2* c[6]*c[4]^2+380*c[5]^3*c[6]*c[4]^2+600*c[5]^3*c[6]^2*c[4]^3-900*c[6]^2 *c[5]^2*c[4]^3+1420*c[6]^2*c[5]^2*c[4]^2-300*c[6]*c[5]^3*c[4]^3+500*c[ 6]^2*c[5]^3*c[4]-900*c[6]^2*c[5]^3*c[4]^2-180*c[5]^2*c[4]^2-840*c[6]^2 *c[4]*c[5]^2-117*c[6]^2*c[4]+312*c[5]^2*c[6]*c[4]+50*c[5]^2*c[4]^3)/c[ 7]/(-3*c[7]^3*c[6]+3*c[7]^2*c[6]^2-9*c[6]^2*c[7]^2*c[5]-8*c[7]*c[6]^2* c[5]^3+8*c[7]^2*c[6]*c[5]^3-c[5]*c[6]*c[7]*c[4]-c[7]^2*c[4]^2-30*c[6]^ 2*c[7]*c[5]^2*c[4]^3-28*c[6]^2*c[7]^2*c[5]^2*c[4]+30*c[6]^2*c[7]^2*c[5 ]^2*c[4]^2-30*c[6]*c[7]*c[5]^3*c[4]^3+9*c[6]^2*c[7]*c[4]^2-3*c[6]^2*c[ 7]*c[4]+2*c[6]*c[7]^2*c[4]-9*c[6]*c[7]^2*c[4]^2-2*c[7]^2*c[5]*c[4]-c[7 ]^2*c[5]^2+c[6]*c[7]*c[4]^2+8*c[7]^2*c[6]*c[4]^3-4*c[5]*c[7]^2*c[4]^3+ c[5]*c[7]*c[4]^2-c[5]*c[6]*c[4]^2+32*c[6]*c[5]*c[7]^2*c[4]^2-9*c[7]^2* c[5]*c[6]*c[4]-4*c[5]*c[7]*c[6]*c[4]^3-9*c[6]^2*c[4]^2*c[5]+3*c[6]^2*c [4]*c[5]+c[7]^3*c[4]+9*c[7]^3*c[4]*c[6]-9*c[7]^3*c[4]*c[5]-20*c[5]^2*c [6]*c[7]*c[4]^2+8*c[6]^2*c[5]*c[4]^3+28*c[7]^3*c[4]^2*c[6]*c[5]+18*c[6 ]^2*c[7]*c[4]*c[5]+28*c[7]*c[6]^2*c[5]*c[4]^3-3*c[6]^2*c[7]*c[5]+9*c[6 ]^2*c[7]*c[5]^2+c[6]*c[7]*c[5]^2-8*c[6]*c[7]^3*c[5]^2-9*c[6]*c[7]^2*c[ 5]^2+8*c[6]^2*c[7]^2*c[5]^2+2*c[6]*c[7]^2*c[5]-36*c[6]^2*c[4]^2*c[7]*c [5]-28*c[6]^2*c[7]^2*c[4]^2*c[5]-28*c[7]^3*c[4]*c[6]*c[5]-36*c[6]^2*c[ 5]^2*c[7]*c[4]+56*c[6]^2*c[5]^2*c[7]*c[4]^2+28*c[6]*c[5]^2*c[7]*c[4]^3 +28*c[6]*c[5]^3*c[7]*c[4]^2+28*c[6]^2*c[5]^3*c[7]*c[4]+9*c[6]*c[7]^3*c [5]+9*c[7]^2*c[5]*c[4]^2+28*c[7]^3*c[5]^2*c[6]*c[4]+32*c[7]^2*c[5]^2*c [6]*c[4]+8*c[6]^2*c[7]^2*c[4]^2-9*c[6]^2*c[7]^2*c[4]-4*c[5]^3*c[6]*c[7 ]*c[4]+28*c[6]^2*c[7]^2*c[4]*c[5]-4*c[5]^2*c[6]*c[4]^3-9*c[5]^2*c[7]*c [4]^2+9*c[5]^2*c[6]*c[4]^2+4*c[5]^2*c[7]*c[4]^3+4*c[5]^3*c[7]*c[4]^2-4 *c[5]^3*c[6]*c[4]^2+30*c[5]^3*c[6]^2*c[4]^3-28*c[6]^2*c[5]^2*c[4]^3+28 *c[6]^2*c[5]^2*c[4]^2+8*c[6]^2*c[5]^3*c[4]-28*c[6]^2*c[5]^3*c[4]^2-9*c [6]^2*c[4]*c[5]^2-8*c[7]*c[6]^2*c[4]^3+30*c[7]^2*c[5]^2*c[6]*c[4]^3-30 *c[7]^3*c[5]^2*c[6]*c[4]^2+30*c[7]^2*c[5]^3*c[6]*c[4]^2-8*c[7]^3*c[4]^ 2*c[6]+4*c[7]^3*c[4]^2*c[5]+c[7]^3*c[5]-28*c[7]^2*c[6]*c[5]*c[4]^3-56* c[7]^2*c[5]^2*c[6]*c[4]^2-28*c[7]^2*c[5]^3*c[6]*c[4]-c[5]^2*c[6]*c[4]+ c[5]^2*c[7]*c[4]+4*c[7]^3*c[5]^2*c[4]+9*c[7]^2*c[5]^2*c[4]-30*c[6]^2*c [7]*c[5]^3*c[4]^2-8*c[7]^2*c[5]^2*c[4]^2-4*c[7]^2*c[5]^3*c[4]), a[2,1] = c[2], a[6,4] = -1/2*(120*c[5]^2*c[6]*c[4]^4-12*c[6]*c[5]*c[4]^4-90* c[5]^2*c[6]^3*c[4]^2-60*c[5]*c[6]^2*c[4]^4-2*c[6]^3*c[4]^2+20*c[5]^4*c [4]^3*c[6]+40*c[6]^2*c[4]^5*c[5]+36*c[4]*c[5]^2*c[6]^3-20*c[5]^4*c[6]^ 2*c[4]^2+8*c[6]^2*c[5]^3+2*c[6]^2*c[4]^3-24*c[5]*c[6]^3*c[4]^2-6*c[6]^ 2*c[4]^2*c[5]+36*c[6]^2*c[5]*c[4]^3+60*c[4]^3*c[5]*c[6]^3-3*c[6]*c[5]* c[4]^3-11*c[5]^3*c[6]*c[4]+3*c[5]^3*c[4]^2-24*c[5]^2*c[6]*c[4]^3+13*c[ 5]^2*c[6]*c[4]^2+48*c[5]^3*c[6]*c[4]^2-60*c[5]^3*c[6]^2*c[4]^3-30*c[6] ^2*c[5]^2*c[4]^3-12*c[6]^2*c[5]^2*c[4]^2-150*c[6]*c[5]^3*c[4]^3-48*c[6 ]^2*c[5]^3*c[4]+150*c[6]^2*c[5]^3*c[4]^2-2*c[6]^2*c[4]*c[5]^2-5*c[5]^2 *c[4]^3+2*c[4]^4*c[5]-6*c[5]^2*c[6]^3+60*c[6]^3*c[4]^3*c[5]^2+7*c[6]^3 *c[4]*c[5]-60*c[5]^2*c[6]*c[4]^5-40*c[6]^3*c[4]^4*c[5]+60*c[4]^4*c[6]* c[5]^3)*c[6]/(c[4]^3-2*c[5]*c[4]^2+30*c[5]^2*c[4]^3-6*c[5]*c[4]^3-10*c [4]^4*c[5]^2-c[5]^3+10*c[5]^4*c[4]^2-30*c[5]^3*c[4]^2+6*c[5]^3*c[4]+2* c[5]^2*c[4])/c[4]^2, a[3,1] = -2/9*c[4]*(c[4]-3*c[2])/c[2], a[4,1] = 1 /4*c[4], a[7,3] = 3/4*1/c[4]^2*c[7]*(-480*c[5]^2*c[6]*c[4]^4+15*c[6]*c [5]*c[4]^4+300*c[4]^2*c[5]^2*c[7]^3+60*c[7]^3*c[5]*c[4]^3-510*c[7]^3*c [5]^2*c[4]^3+54*c[7]*c[5]*c[4]^4+450*c[5]^2*c[4]^4*c[7]^2+150*c[4]^3*c [7]^3*c[5]^3-150*c[7]^2*c[5]^2*c[4]^5-450*c[4]^5*c[6]*c[5]^3-3*c[7]^3* c[4]^2+90*c[5]^2*c[4]^5*c[7]+120*c[5]^3*c[4]^4*c[7]+150*c[7]^3*c[5]^2* c[4]^4+100*c[5]^4*c[4]^2*c[7]^2*c[6]-180*c[6]*c[4]^5*c[5]*c[7]+30*c[6] *c[5]^2*c[4]^4*c[7]+60*c[5]^4*c[4]^3*c[6]-150*c[5]^4*c[4]^4*c[6]-200*c [7]^2*c[4]^4*c[5]^3+300*c[6]*c[4]^5*c[5]*c[7]^2-50*c[7]^2*c[5]^4*c[4]^ 3+24*c[7]*c[6]*c[5]^3-40*c[7]^2*c[6]*c[5]^3-6*c[5]*c[6]*c[7]*c[4]+360* c[7]*c[4]^4*c[6]*c[5]+150*c[7]^2*c[5]^3*c[4]^3+2*c[7]^2*c[4]^2+6*c[5]^ 2*c[7]^3+330*c[6]*c[7]*c[5]^3*c[4]^3-450*c[7]^2*c[6]*c[5]*c[4]^4-12*c[ 7]*c[6]*c[4]^3-6*c[6]*c[7]^2*c[4]-2*c[7]^2*c[5]*c[4]-4*c[7]^2*c[5]^2+6 *c[6]*c[7]*c[4]^2+20*c[7]^2*c[6]*c[4]^3-10*c[5]*c[7]^2*c[4]^3+6*c[5]*c [4]^3-6*c[5]*c[7]*c[4]^2-6*c[5]*c[6]*c[4]^2+150*c[6]*c[5]*c[7]^2*c[4]^ 2-72*c[7]^2*c[5]*c[6]*c[4]-270*c[5]*c[7]*c[6]*c[4]^3+9*c[7]^3*c[4]*c[6 ]+3*c[7]^3*c[4]*c[5]-306*c[5]^2*c[6]*c[7]*c[4]^2+900*c[4]^5*c[6]*c[7]* c[5]^3+600*c[5]^2*c[6]*c[7]^3*c[4]^4-420*c[7]^3*c[4]^2*c[6]*c[5]+90*c[ 6]*c[5]*c[7]*c[4]^2-12*c[6]*c[7]*c[5]^2+30*c[6]*c[7]^3*c[5]^2+12*c[6]* c[7]^2*c[5]+123*c[7]^3*c[4]*c[6]*c[5]+360*c[6]*c[5]^2*c[7]*c[4]^3+360* c[6]*c[5]^3*c[7]*c[4]^2-18*c[6]*c[7]^3*c[5]+30*c[5]^4*c[4]^3*c[7]-80*c [7]^2*c[5]^2*c[4]^3+24*c[7]^2*c[5]*c[4]^2-30*c[7]*c[5]*c[4]^3+3*c[6]*c [5]*c[4]^3-240*c[7]^3*c[5]^2*c[6]*c[4]-24*c[5]^3*c[6]*c[4]-12*c[5]^3*c [7]*c[4]-144*c[5]^3*c[6]*c[7]*c[4]+9*c[5]^3*c[4]^2+294*c[5]^2*c[6]*c[4 ]^3-42*c[5]^2*c[7]*c[4]^2-75*c[5]^2*c[6]*c[4]^2+252*c[5]^2*c[7]*c[4]^3 +18*c[5]^3*c[7]*c[4]^2+189*c[5]^3*c[6]*c[4]^2-360*c[5]^2*c[4]^4*c[7]-7 20*c[6]*c[5]^3*c[4]^3-120*c[5]^3*c[7]*c[4]^3-60*c[5]^4*c[7]*c[4]^2*c[6 ]-6*c[5]^2*c[4]^2-780*c[7]^2*c[5]^2*c[6]*c[4]^3+810*c[7]^3*c[5]^2*c[6] *c[4]^2-1230*c[7]^2*c[5]^3*c[6]*c[4]^2-15*c[7]^3*c[4]^2*c[6]-21*c[7]^3 *c[4]^2*c[5]+20*c[7]^2*c[6]*c[5]*c[4]^3+220*c[7]^2*c[5]^2*c[6]*c[4]^2+ 15*c[4]^4*c[5]^2+320*c[7]^2*c[5]^3*c[6]*c[4]+1850*c[7]^2*c[5]^3*c[6]*c [4]^3-60*c[5]^3*c[7]^3*c[4]^2+12*c[5]^2*c[6]*c[4]+12*c[5]^2*c[7]*c[4]- 66*c[7]^3*c[5]^2*c[4]+24*c[7]^2*c[5]^2*c[4]+72*c[5]^2*c[6]*c[7]*c[4]-1 10*c[7]^2*c[5]^2*c[4]^2+20*c[7]^2*c[5]^3*c[4]-9*c[4]^4*c[5]-60*c[7]^2* c[4]^4*c[5]-600*c[5]^2*c[6]*c[4]^5*c[7]^2-1200*c[7]^3*c[5]^2*c[4]^3*c[ 6]+630*c[7]^3*c[6]*c[5]*c[4]^3+300*c[5]^4*c[4]^4*c[6]*c[7]-1800*c[5]^3 *c[4]^4*c[6]*c[7]-15*c[5]^3*c[4]^3-300*c[7]^3*c[4]^4*c[5]*c[6]-20*c[7] ^2*c[5]^3*c[4]^2-800*c[5]^3*c[4]^4*c[7]^2*c[6]-200*c[5]^4*c[4]^3*c[7]^ 2*c[6]+1150*c[5]^2*c[4]^4*c[7]^2*c[6]+180*c[5]^2*c[6]*c[4]^5+1140*c[4] ^4*c[6]*c[5]^3)/(10*c[5]^3*c[4]^2+10*c[5]^2*c[4]^3-30*c[5]^2*c[4]^2+6* c[5]^2*c[4]-c[5]^2+6*c[5]*c[4]^2+c[5]*c[4]-c[4]^2)/(10*c[5]*c[6]*c[4]- 5*c[6]*c[5]-5*c[6]*c[4]+3*c[6]-5*c[5]*c[4]+3*c[5]+3*c[4]-2), `b*`[1] = 1/60*(30*c[5]^3*c[4]+42*c[7]*c[6]^2-103*c[5]*c[7]*c[4]+18*c[6]-6*c[5] -6*c[4]-27*c[7]*c[6]+9*c[5]^2+9*c[4]^2-180*c[7]*c[6]^2*c[5]^3+100*c[7] *c[6]*c[5]^3-27*c[6]^2-354*c[5]*c[6]*c[7]*c[4]+97*c[6]*c[5]^2-66*c[6]* c[5]-60*c[6]*c[5]^3+100*c[6]^2*c[5]^3+9*c[7]*c[5]+100*c[6]^2*c[4]^3-20 40*c[6]^2*c[7]*c[5]^2*c[4]^3-600*c[6]*c[7]*c[5]^3*c[4]^3-14*c[7]*c[5]^ 2+100*c[7]*c[6]*c[4]^3+310*c[6]^2*c[7]*c[4]^2-192*c[6]^2*c[7]*c[4]-180 *c[6]^2*c[4]^2-14*c[7]*c[4]^2+9*c[7]*c[4]+103*c[6]*c[7]*c[5]+97*c[6]*c [4]^2-60*c[6]*c[4]^3+103*c[6]*c[7]*c[4]-156*c[6]*c[7]*c[4]^2+30*c[5]*c [4]^3+174*c[5]*c[7]*c[4]^2-312*c[5]*c[6]*c[4]^2-107*c[5]*c[4]^2+215*c[ 5]*c[6]*c[4]+66*c[5]*c[4]-66*c[6]*c[4]-410*c[5]*c[7]*c[6]*c[4]^3+840*c [6]^2*c[4]^2*c[5]-522*c[6]^2*c[4]*c[5]-760*c[5]^2*c[6]*c[7]*c[4]^2-500 *c[6]^2*c[5]*c[4]^3+932*c[6]^2*c[7]*c[4]*c[5]+1020*c[7]*c[6]^2*c[5]*c[ 4]^3-192*c[6]^2*c[7]*c[5]+310*c[6]^2*c[7]*c[5]^2+526*c[6]*c[5]*c[7]*c[ 4]^2-156*c[6]*c[7]*c[5]^2-180*c[6]^2*c[5]^2-1610*c[6]^2*c[4]^2*c[7]*c[ 5]-1610*c[6]^2*c[5]^2*c[7]*c[4]+2970*c[6]^2*c[5]^2*c[7]*c[4]^2+690*c[6 ]*c[5]^2*c[7]*c[4]^3+690*c[6]*c[5]^3*c[7]*c[4]^2+1020*c[6]^2*c[5]^3*c[ 7]*c[4]+117*c[6]^2*c[5]-50*c[7]*c[5]*c[4]^3+230*c[6]*c[5]*c[4]^3+230*c [5]^3*c[6]*c[4]-50*c[5]^3*c[7]*c[4]-107*c[5]^2*c[4]-410*c[5]^3*c[6]*c[ 7]*c[4]-50*c[5]^3*c[4]^2-380*c[5]^2*c[6]*c[4]^3-310*c[5]^2*c[7]*c[4]^2 +460*c[5]^2*c[6]*c[4]^2+90*c[5]^2*c[7]*c[4]^3+90*c[5]^3*c[7]*c[4]^2-38 0*c[5]^3*c[6]*c[4]^2-600*c[5]^3*c[6]^2*c[4]^3+900*c[6]^2*c[5]^2*c[4]^3 -1420*c[6]^2*c[5]^2*c[4]^2+300*c[6]*c[5]^3*c[4]^3-500*c[6]^2*c[5]^3*c[ 4]+900*c[6]^2*c[5]^3*c[4]^2+180*c[5]^2*c[4]^2+840*c[6]^2*c[4]*c[5]^2-1 80*c[7]*c[6]^2*c[4]^3+117*c[6]^2*c[4]+1500*c[5]^3*c[6]^2*c[7]*c[4]^3-3 12*c[5]^2*c[6]*c[4]+174*c[5]^2*c[7]*c[4]+526*c[5]^2*c[6]*c[7]*c[4]-204 0*c[6]^2*c[7]*c[5]^3*c[4]^2-50*c[5]^2*c[4]^3)/(3*c[6]-28*c[5]*c[6]*c[4 ]^2-28*c[5]^2*c[6]*c[4]+8*c[6]*c[5]^2-9*c[6]*c[4]-9*c[6]*c[5]+28*c[5]* c[6]*c[4]+30*c[5]^2*c[6]*c[4]^2+8*c[6]*c[4]^2-4*c[5]^2*c[4]-4*c[5]*c[4 ]^2+9*c[5]*c[4]-c[5]-c[4])/c[5]/c[6]/c[7]/c[4], a[8,4] = -1/2*(734*c[5 ]^2*c[6]*c[4]^4-203*c[6]*c[5]*c[4]^4+3190*c[5]^4*c[4]^4*c[6]^2*c[7]-93 *c[7]*c[5]*c[4]^4-636*c[5]^2*c[4]^4*c[7]^2-10*c[5]^3*c[4]-520*c[6]*c[4 ]^7*c[7]*c[5]-481*c[4]^5*c[6]^2*c[5]^2+645*c[7]^2*c[5]^2*c[4]^5-574*c[ 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6]*c[5]*c[7]*c[4]^2+18*c[6]^2*c[7]^2*c[5]^2-38*c[6]^2*c[4]^2*c[7]*c[5] +102*c[6]^2*c[7]^2*c[4]^2*c[5]+86*c[6]*c[5]^4*c[4]^2-24*c[7]*c[6]^2*c[ 5]^4+585*c[4]^5*c[7]*c[5]^3+c[4]^5+35*c[6]^2*c[5]^2*c[7]*c[4]-26*c[6]^ 2*c[5]^2*c[7]*c[4]^2+1128*c[6]*c[5]^2*c[7]*c[4]^3+706*c[6]*c[5]^3*c[7] *c[4]^2-138*c[6]^2*c[5]^3*c[7]*c[4]-685*c[5]^4*c[4]^3*c[7]+354*c[7]^2* c[5]^2*c[4]^3-370*c[5]^4*c[7]^2*c[4]^4-6*c[7]^2*c[5]*c[4]^2+70*c[7]*c[ 5]*c[4]^3+89*c[6]*c[5]*c[4]^3+200*c[4]^8*c[5]*c[6]*c[7]+390*c[6]*c[4]^ 7*c[5]-31*c[7]^2*c[5]^2*c[6]*c[4]-27*c[6]*c[7]*c[4]^4+10*c[4]^5*c[5]-8 0*c[5]^4*c[4]^6+200*c[5]^5*c[4]^5*c[7]^2*c[6]+6*c[6]^2*c[7]^2*c[4]^2+2 2*c[5]^3*c[6]*c[4]+19*c[5]^3*c[7]*c[4]+400*c[5]^5*c[4]^3*c[7]^2*c[6]+4 *c[5]^2*c[4]-135*c[5]^3*c[6]*c[7]*c[4]+71*c[5]^3*c[4]^2-21*c[6]^2*c[7] ^2*c[4]*c[5]-480*c[5]^2*c[6]*c[4]^3+42*c[5]^2*c[7]*c[4]^2+37*c[5]^2*c[ 6]*c[4]^2-480*c[5]^2*c[7]*c[4]^3-104*c[5]^3*c[7]*c[4]^2-97*c[5]^3*c[6] *c[4]^2-1536*c[5]^3*c[6]^2*c[4]^3+190*c[5]^5*c[4]^3*c[6]^2*c[7]+557*c[ 4]^3*c[7]^2*c[5]^2*c[6]^2-10*c[6]*c[7]*c[4]^5-730*c[4]^5*c[5]^3*c[6]^2 -14*c[7]*c[5]^4*c[4]+10*c[7]^2*c[5]^4*c[4]+646*c[5]^2*c[4]^4*c[7]+498* c[6]^2*c[5]^2*c[4]^3-95*c[6]^2*c[5]^2*c[4]^2+744*c[6]*c[5]^3*c[4]^3-46 *c[6]^2*c[5]^3*c[4]+334*c[6]^2*c[5]^3*c[4]^2+854*c[5]^3*c[7]*c[4]^3-49 0*c[5]^4*c[7]*c[4]^2*c[6]-30*c[7]*c[5]*c[4]^7-19*c[5]^2*c[4]^2+18*c[6] ^2*c[4]*c[5]^2-472*c[6]*c[5]*c[4]^6+360*c[5]*c[4]^6*c[7]*c[6]-10*c[7]* c[6]^2*c[4]^3-811*c[7]^2*c[5]^2*c[6]*c[4]^3-993*c[7]^2*c[5]^3*c[6]*c[4 ]^2-44*c[5]^4*c[4]^2-451*c[4]^3*c[7]^2*c[5]*c[6]^2+10*c[6]*c[7]*c[4]^6 -14*c[7]^2*c[6]*c[5]*c[4]^3-20*c[5]^5*c[4]^5+118*c[7]^2*c[5]^2*c[6]*c[ 4]^2-249*c[4]^4*c[5]^2+170*c[7]^2*c[5]^3*c[6]*c[4]+3829*c[7]^2*c[5]^3* c[6]*c[4]^3+2208*c[5]^3*c[6]^2*c[4]^4+339*c[5]^4*c[4]^3-6*c[6]^2*c[4]^ 3*c[7]^2+1871*c[5]^3*c[6]^2*c[7]*c[4]^3-10*c[5]^2*c[6]*c[4]-9*c[5]^2*c [7]*c[4]+6*c[7]^2*c[5]^2*c[4]+33*c[5]^2*c[6]*c[7]*c[4]+600*c[7]*c[4]^8 *c[6]*c[5]^3+187*c[6]^2*c[7]*c[5]^3*c[4]^2+341*c[6]^2*c[7]^2*c[4]*c[5] 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6]^2*c[4]^3+600*c[5]^2*c[7]*c[4]^7*c[6]^2+200*c[6]^2*c[4]^6*c[5]*c[7]^ 2-1510*c[5]^4*c[4]^5*c[6]*c[7]+6*c[5]^4*c[4]+600*c[5]^2*c[4]^7*c[6]*c[ 7]^2+814*c[5]^3*c[4]^4)/(690*c[5]^2*c[6]*c[4]^4-110*c[6]*c[5]*c[4]^4-1 10*c[7]*c[5]*c[4]^4+48*c[5]^3*c[4]+200*c[4]^5*c[6]*c[5]^3-150*c[5]^2*c [4]^5*c[7]-750*c[5]^3*c[4]^4*c[7]-930*c[6]*c[5]^2*c[4]^4*c[7]-200*c[5] ^5*c[4]^3*c[7]+750*c[5]^4*c[4]^3*c[6]-120*c[5]^5*c[4]^2-10*c[5]^3+10*c [4]^3+150*c[5]^5*c[4]^3-15*c[7]*c[6]*c[5]^3+12*c[5]^4-12*c[4]^4+200*c[ 6]*c[5]^2*c[4]^5*c[7]-150*c[4]^5*c[5]^3-200*c[5]^5*c[4]^3*c[6]+150*c[7 ]*c[4]^4*c[6]*c[5]+12*c[6]*c[5]^3+15*c[4]^4*c[7]-690*c[5]^4*c[7]*c[4]^ 2-15*c[5]^4*c[7]+150*c[5]^5*c[7]*c[4]^2+15*c[7]*c[6]*c[4]^3-12*c[7]*c[ 4]^3-12*c[6]*c[4]^3-48*c[5]*c[4]^3+24*c[5]*c[7]*c[4]^2+24*c[5]*c[6]*c[ 4]^2-20*c[5]*c[4]^2-70*c[5]*c[7]*c[6]*c[4]^3+12*c[7]*c[5]^3-300*c[4]^5 *c[6]*c[7]*c[5]^3-30*c[6]*c[5]*c[7]*c[4]^2+20*c[5]^4*c[7]*c[6]-690*c[6 ]*c[5]^4*c[4]^2-15*c[6]*c[5]^4+200*c[4]^5*c[7]*c[5]^3+510*c[6]*c[5]^2* c[7]*c[4]^3-510*c[6]*c[5]^3*c[7]*c[4]^2+750*c[5]^4*c[4]^3*c[7]+57*c[7] *c[5]*c[4]^3+57*c[6]*c[5]*c[4]^3-20*c[6]*c[7]*c[4]^4-57*c[5]^3*c[6]*c[ 4]-57*c[5]^3*c[7]*c[4]+20*c[5]^2*c[4]-200*c[5]^5*c[4]^2*c[6]*c[7]+70*c [5]^3*c[6]*c[7]*c[4]-342*c[5]^3*c[4]^2-410*c[5]^2*c[6]*c[4]^3-410*c[5] ^2*c[7]*c[4]^3+410*c[5]^3*c[7]*c[4]^2+410*c[5]^3*c[6]*c[4]^2+110*c[7]* c[5]^4*c[4]+690*c[5]^2*c[4]^4*c[7]+930*c[5]^4*c[7]*c[4]^2*c[6]+550*c[5 ]^4*c[4]^2-550*c[4]^4*c[5]^2-570*c[5]^4*c[4]^3-24*c[5]^2*c[6]*c[4]-24* c[5]^2*c[7]*c[4]+30*c[5]^2*c[6]*c[7]*c[4]+342*c[5]^2*c[4]^3+87*c[4]^4* c[5]+300*c[5]^5*c[4]^3*c[6]*c[7]+110*c[5]^4*c[4]*c[6]-150*c[5]^4*c[4]* c[7]*c[6]+120*c[4]^5*c[5]^2+15*c[6]*c[4]^4+1100*c[5]^3*c[4]^4*c[6]*c[7 ]-1100*c[5]^4*c[4]^3*c[6]*c[7]+150*c[5]^5*c[6]*c[4]^2-150*c[5]^2*c[6]* c[4]^5-750*c[4]^4*c[6]*c[5]^3-87*c[5]^4*c[4]+570*c[5]^3*c[4]^4)/(c[6]* c[4]-c[4]^2-c[7]*c[6]+c[7]*c[4])/c[4]^2, a[8,1] = 1/4*(-2816*c[5]^2*c[ 6]*c[4]^4+424*c[6]*c[5]*c[4]^4-9720*c[5]^4*c[4]^4*c[6]^2*c[7]+372*c[7] *c[5]*c[4]^4+1752*c[5]^2*c[4]^4*c[7]^2+20*c[5]^3*c[4]-1320*c[4]^5*c[6] ^2*c[5]^2-1320*c[7]^2*c[5]^2*c[4]^5-4880*c[4]^5*c[6]*c[5]^3-264*c[5]*c [6]^2*c[4]^4-12*c[7]^2*c[4]^3+700*c[5]^5*c[4]^3*c[7]^2*c[6]^2-200*c[5] ^2*c[4]^6*c[6]^2*c[7]^2-246*c[4]^5*c[5]*c[7]+1818*c[5]^2*c[4]^5*c[7]+1 300*c[5]^3*c[4]^6*c[6]^2*c[7]-800*c[5]^3*c[4]^6*c[6]^2*c[7]^2+5526*c[5 ]^3*c[4]^4*c[7]+32*c[4]^4*c[6]^2+3160*c[7]^2*c[4]^5*c[5]^3-7740*c[6]^2 *c[4]^5*c[7]*c[5]^3+780*c[7]*c[5]^2*c[4]^6*c[6]-325*c[5]^4*c[4]^2*c[7] ^2*c[6]-20*c[6]^2*c[4]^5-1339*c[6]^2*c[7]*c[5]^4*c[4]^2-280*c[5]^2*c[4 ]^6*c[7]+668*c[6]*c[4]^5*c[5]*c[7]+6574*c[6]*c[5]^2*c[4]^4*c[7]+1500*c [5]^4*c[7]*c[4]^6*c[6]+200*c[4]^6*c[6]^2*c[5]^2-280*c[5]^5*c[4]^3*c[7] -600*c[5]^2*c[7]^2*c[4]^6*c[6]+2092*c[5]^4*c[4]^3*c[6]+3640*c[5]^4*c[4 ]^5*c[6]-320*c[5]^2*c[6]*c[4]^6+400*c[5]^5*c[4]^5*c[6]^2-4880*c[5]^4*c [4]^4*c[6]-4080*c[7]^2*c[4]^4*c[5]^3+50*c[6]*c[7]^2*c[4]^5-2400*c[5]^4 *c[4]^5*c[7]^2-4160*c[5]^4*c[4]^4*c[7]-76*c[7]^2*c[6]*c[4]^4+1600*c[5] ^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]^2-710 0*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]-7 0*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]-97 2*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]^5-216 0*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+18*c [7]*c[4]^3+20*c[6]*c[4]^3+26*c[7]^2*c[6]*c[4]^3+72*c[5]*c[7]^2*c[4]^3+ 48*c[5]*c[4]^3-18*c[5]*c[7]*c[4]^2-20*c[5]*c[6]*c[4]^2+8*c[5]*c[4]^2-2 3*c[6]*c[5]*c[7]^2*c[4]^2+266*c[5]*c[7]*c[6]*c[4]^3+12*c[6]^2*c[4]^2*c [5]-1240*c[5]^4*c[4]^5+287*c[5]^2*c[6]*c[7]*c[4]^2-8120*c[5]^4*c[7]^2* c[4]^4*c[6]+11330*c[4]^5*c[6]*c[7]*c[5]^3+72*c[6]^2*c[5]*c[4]^3+14*c[6 ]^2*c[7]*c[4]*c[5]+76*c[7]*c[6]^2*c[5]*c[4]^3-12*c[6]^2*c[7]*c[5]^2+40 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2*c[4]^3+72*c[7]^2*c[5]^2*c[4]^2+32*c[7]^2*c[5]^3*c[4]-160*c[4]^4*c[5] +690*c[5]^5*c[4]^3*c[6]*c[7]-264*c[7]^2*c[4]^4*c[5]-1833*c[7]^2*c[5]^3 *c[6]^2*c[4]^2-260*c[6]^2*c[4]^5*c[5]*c[7]^2-8270*c[7]^2*c[4]^5*c[6]*c [5]^3+3780*c[5]^2*c[6]*c[4]^5*c[7]^2+1720*c[5]^4*c[4]^4+32*c[5]^4*c[4] *c[6]+30*c[5]^4*c[7]^2*c[4]*c[6]-60*c[5]^4*c[4]*c[7]*c[6]+2480*c[4]^5* c[6]^2*c[7]*c[5]^2-1120*c[4]^5*c[6]^2*c[7]^2*c[5]^2-772*c[4]^5*c[5]^2- 40*c[6]^2*c[7]^2*c[4]^5+1752*c[6]^2*c[4]^4*c[5]^2-1700*c[7]^2*c[6]^2*c [5]^5*c[4]^4-600*c[5]^5*c[4]^5*c[6]-120*c[5]*c[7]*c[4]^6*c[6]^2+50*c[6 ]^2*c[7]*c[4]^5-52*c[6]*c[4]^4-1200*c[5]^4*c[7]*c[4]^6*c[6]^2+7150*c[5 ]^4*c[4]^5*c[6]^2*c[7]+198*c[7]*c[6]^2*c[5]*c[4]^4+10560*c[5]^3*c[4]^4 *c[6]^2*c[7]+5240*c[5]^3*c[4]^3*c[6]^2*c[7]^2+11240*c[5]^4*c[4]^4*c[6] *c[7]-3649*c[5]^2*c[4]^4*c[7]*c[6]^2-14207*c[5]^3*c[4]^4*c[6]*c[7]-449 1*c[5]^4*c[4]^3*c[6]*c[7]+1064*c[5]^3*c[4]^3-264*c[7]^2*c[5]^3*c[4]^2- 2400*c[5]^4*c[4]^5*c[6]^2+6020*c[6]^2*c[7]^2*c[5]^3*c[4]^5+6800*c[6]*c [7]^2*c[5]^4*c[4]^5+3160*c[5]^4*c[4]^4*c[6]^2-1320*c[5]^4*c[4]^3*c[6]^ 2-5800*c[7]^2*c[5]^4*c[6]^2*c[4]^3+1940*c[7]^2*c[5]^4*c[6]^2*c[4]^2-60 0*c[5]^4*c[4]^6*c[6]+28*c[4]^5*c[7]+9740*c[5]^3*c[4]^4*c[7]^2*c[6]+292 0*c[5]^4*c[4]^3*c[7]^2*c[6]-4775*c[5]^2*c[4]^4*c[7]^2*c[6]+4980*c[5]^4 *c[4]^3*c[6]^2*c[7]+210*c[7]^2*c[4]^4*c[6]^2*c[5]-8860*c[7]^2*c[4]^4*c [5]^3*c[6]^2+2140*c[7]^2*c[4]^4*c[6]^2*c[5]^2+120*c[4]^6*c[5]^2-20*c[7 ]^2*c[4]^5+2092*c[5]^2*c[6]*c[4]^5+9920*c[6]^2*c[7]^2*c[5]^4*c[4]^4-50 0*c[5]^4*c[4]^6*c[7]-48*c[7]^2*c[6]^2*c[5]^3+6404*c[4]^4*c[6]*c[5]^3-2 84*c[6]*c[5]*c[4]^5+200*c[5]^5*c[6]^2*c[4]^3+200*c[6]^2*c[4]^6*c[5]*c[ 7]^2-8820*c[5]^4*c[4]^5*c[6]*c[7]-12*c[5]^4*c[4]-2324*c[5]^3*c[4]^4)/c [6]/c[5]/c[7]/c[4]^2/(150*c[5]^2*c[6]*c[4]^4-87*c[5]^3*c[4]-12*c[5]*c[ 7]*c[4]-200*c[5]^3*c[4]^4*c[7]-200*c[6]*c[5]^2*c[4]^4*c[7]-200*c[5]^4* c[4]^3*c[6]-10*c[5]^2-10*c[4]^2+12*c[5]^3+12*c[4]^3+20*c[7]*c[6]*c[5]^ 3+15*c[5]*c[6]*c[7]*c[4]+12*c[6]*c[5]^2-15*c[6]*c[5]^3-1300*c[6]*c[7]* c[5]^3*c[4]^3+150*c[5]^4*c[7]*c[4]^2+12*c[7]*c[5]^2+20*c[7]*c[6]*c[4]^ 3+12*c[7]*c[4]^2-15*c[7]*c[4]^3+12*c[6]*c[4]^2-15*c[6]*c[4]^3-15*c[6]* c[7]*c[4]^2-87*c[5]*c[4]^3-72*c[5]*c[7]*c[4]^2-72*c[5]*c[6]*c[4]^2+60* c[5]*c[4]^2-12*c[5]*c[6]*c[4]+10*c[5]*c[4]-150*c[5]*c[7]*c[6]*c[4]^3-1 5*c[7]*c[5]^3-660*c[5]^2*c[6]*c[7]*c[4]^2+90*c[6]*c[5]*c[7]*c[4]^2-15* c[6]*c[7]*c[5]^2+150*c[6]*c[5]^4*c[4]^2+930*c[6]*c[5]^2*c[7]*c[4]^3+93 0*c[6]*c[5]^3*c[7]*c[4]^2-200*c[5]^4*c[4]^3*c[7]+110*c[7]*c[5]*c[4]^3+ 110*c[6]*c[5]*c[4]^3+110*c[5]^3*c[6]*c[4]+110*c[5]^3*c[7]*c[4]+60*c[5] ^2*c[4]-150*c[5]^3*c[6]*c[7]*c[4]+550*c[5]^3*c[4]^2-690*c[5]^2*c[6]*c[ 4]^3+520*c[5]^2*c[7]*c[4]^2+520*c[5]^2*c[6]*c[4]^2-690*c[5]^2*c[7]*c[4 ]^3-690*c[5]^3*c[7]*c[4]^2-690*c[5]^3*c[6]*c[4]^2+150*c[5]^2*c[4]^4*c[ 7]+900*c[6]*c[5]^3*c[4]^3+900*c[5]^3*c[7]*c[4]^3-200*c[5]^4*c[7]*c[4]^ 2*c[6]-429*c[5]^2*c[4]^2-120*c[5]^4*c[4]^2-120*c[4]^4*c[5]^2+150*c[5]^ 4*c[4]^3-72*c[5]^2*c[6]*c[4]-72*c[5]^2*c[7]*c[4]+90*c[5]^2*c[6]*c[7]*c [4]+550*c[5]^2*c[4]^3+300*c[5]^3*c[4]^4*c[6]*c[7]+300*c[5]^4*c[4]^3*c[ 6]*c[7]-690*c[5]^3*c[4]^3-200*c[4]^4*c[6]*c[5]^3+150*c[5]^3*c[4]^4), a [8,3] = 3/4*(-1560*c[5]^2*c[6]*c[4]^4+585*c[6]*c[5]*c[4]^4-45*c[7]*c[5 ]*c[4]^4-12*c[5]^3*c[4]-450*c[4]^5*c[6]*c[5]^3+3*c[5]*c[7]*c[4]-150*c[ 5]^2*c[4]^5*c[7]-200*c[5]^3*c[4]^4*c[7]+300*c[6]*c[4]^5*c[5]*c[7]+2860 *c[6]*c[5]^2*c[4]^4*c[7]+160*c[5]^4*c[4]^3*c[6]-150*c[5]^4*c[4]^4*c[6] -4*c[5]^2+2*c[4]^2-40*c[7]*c[6]*c[5]^3+132*c[5]*c[6]*c[7]*c[4]-900*c[6 ]*c[5]^2*c[4]^5*c[7]-30*c[6]*c[5]^2+12*c[6]*c[5]-960*c[7]*c[4]^4*c[6]* c[5]+24*c[6]*c[5]^3+3140*c[6]*c[7]*c[5]^3*c[4]^3+6*c[7]*c[5]^2+20*c[7] *c[6]*c[4]^3-3*c[7]*c[4]^2-18*c[6]*c[7]*c[5]+15*c[6]*c[4]^2-12*c[6]*c[ 4]^3+9*c[6]*c[7]*c[4]-24*c[6]*c[7]*c[4]^2-54*c[5]*c[4]^3-12*c[5]*c[7]* c[4]^2+348*c[5]*c[6]*c[4]^2+9*c[5]*c[4]^2-87*c[5]*c[6]*c[4]-2*c[5]*c[4 ]-6*c[6]*c[4]+1035*c[5]*c[7]*c[6]*c[4]^3+1593*c[5]^2*c[6]*c[7]*c[4]^2+ 900*c[4]^5*c[6]*c[7]*c[5]^3-534*c[6]*c[5]*c[7]*c[4]^2+48*c[6]*c[7]*c[5 ]^2-60*c[6]*c[5]^4*c[4]^2-3060*c[6]*c[5]^2*c[7]*c[4]^3-1560*c[6]*c[5]^ 3*c[7]*c[4]^2-50*c[5]^4*c[4]^3*c[7]+80*c[7]*c[5]*c[4]^3-652*c[6]*c[5]* c[4]^3-208*c[5]^3*c[6]*c[4]+20*c[5]^3*c[7]*c[4]+54*c[5]^2*c[4]+360*c[5 ]^3*c[6]*c[7]*c[4]+82*c[5]^3*c[4]^2+1734*c[5]^2*c[6]*c[4]^3+388*c[5]^2 *c[7]*c[4]^2-944*c[5]^2*c[6]*c[4]^2-900*c[5]^2*c[7]*c[4]^3-135*c[5]^3* c[7]*c[4]^2+879*c[5]^3*c[6]*c[4]^2+660*c[5]^2*c[4]^4*c[7]-1680*c[6]*c[ 5]^3*c[4]^3+420*c[5]^3*c[7]*c[4]^3+100*c[5]^4*c[7]*c[4]^2*c[6]-246*c[5 ]^2*c[4]^2-405*c[4]^4*c[5]^2+30*c[5]^4*c[4]^3+240*c[5]^2*c[6]*c[4]-84* c[5]^2*c[7]*c[4]-396*c[5]^2*c[6]*c[7]*c[4]+562*c[5]^2*c[4]^3+30*c[4]^4 *c[5]+90*c[4]^5*c[5]^2+300*c[5]^4*c[4]^4*c[6]*c[7]-3000*c[5]^3*c[4]^4* c[6]*c[7]-300*c[5]^4*c[4]^3*c[6]*c[7]-255*c[5]^3*c[4]^3+480*c[5]^2*c[6 ]*c[4]^5+1540*c[4]^4*c[6]*c[5]^3-180*c[6]*c[5]*c[4]^5+120*c[5]^3*c[4]^ 4)/c[4]^2/(150*c[5]^2*c[6]*c[4]^4-87*c[5]^3*c[4]-12*c[5]*c[7]*c[4]-200 *c[5]^3*c[4]^4*c[7]-200*c[6]*c[5]^2*c[4]^4*c[7]-200*c[5]^4*c[4]^3*c[6] -10*c[5]^2-10*c[4]^2+12*c[5]^3+12*c[4]^3+20*c[7]*c[6]*c[5]^3+15*c[5]*c [6]*c[7]*c[4]+12*c[6]*c[5]^2-15*c[6]*c[5]^3-1300*c[6]*c[7]*c[5]^3*c[4] ^3+150*c[5]^4*c[7]*c[4]^2+12*c[7]*c[5]^2+20*c[7]*c[6]*c[4]^3+12*c[7]*c [4]^2-15*c[7]*c[4]^3+12*c[6]*c[4]^2-15*c[6]*c[4]^3-15*c[6]*c[7]*c[4]^2 -87*c[5]*c[4]^3-72*c[5]*c[7]*c[4]^2-72*c[5]*c[6]*c[4]^2+60*c[5]*c[4]^2 -12*c[5]*c[6]*c[4]+10*c[5]*c[4]-150*c[5]*c[7]*c[6]*c[4]^3-15*c[7]*c[5] ^3-660*c[5]^2*c[6]*c[7]*c[4]^2+90*c[6]*c[5]*c[7]*c[4]^2-15*c[6]*c[7]*c [5]^2+150*c[6]*c[5]^4*c[4]^2+930*c[6]*c[5]^2*c[7]*c[4]^3+930*c[6]*c[5] ^3*c[7]*c[4]^2-200*c[5]^4*c[4]^3*c[7]+110*c[7]*c[5]*c[4]^3+110*c[6]*c[ 5]*c[4]^3+110*c[5]^3*c[6]*c[4]+110*c[5]^3*c[7]*c[4]+60*c[5]^2*c[4]-150 *c[5]^3*c[6]*c[7]*c[4]+550*c[5]^3*c[4]^2-690*c[5]^2*c[6]*c[4]^3+520*c[ 5]^2*c[7]*c[4]^2+520*c[5]^2*c[6]*c[4]^2-690*c[5]^2*c[7]*c[4]^3-690*c[5 ]^3*c[7]*c[4]^2-690*c[5]^3*c[6]*c[4]^2+150*c[5]^2*c[4]^4*c[7]+900*c[6] *c[5]^3*c[4]^3+900*c[5]^3*c[7]*c[4]^3-200*c[5]^4*c[7]*c[4]^2*c[6]-429* c[5]^2*c[4]^2-120*c[5]^4*c[4]^2-120*c[4]^4*c[5]^2+150*c[5]^4*c[4]^3-72 *c[5]^2*c[6]*c[4]-72*c[5]^2*c[7]*c[4]+90*c[5]^2*c[6]*c[7]*c[4]+550*c[5 ]^2*c[4]^3+300*c[5]^3*c[4]^4*c[6]*c[7]+300*c[5]^4*c[4]^3*c[6]*c[7]-690 *c[5]^3*c[4]^3-200*c[4]^4*c[6]*c[5]^3+150*c[5]^3*c[4]^4), a[8,5] = -1/ 2*(893*c[5]^2*c[6]*c[4]^4-122*c[6]*c[5]*c[4]^4+850*c[5]^4*c[4]^4*c[6]^ 2*c[7]-101*c[7]*c[5]*c[4]^4-535*c[5]^2*c[4]^4*c[7]^2-44*c[5]^3*c[4]+16 0*c[4]^5*c[6]^2*c[5]^2+100*c[7]^2*c[5]^2*c[4]^5+170*c[4]^5*c[6]*c[5]^3 +75*c[5]*c[6]^2*c[4]^4+16*c[7]^2*c[4]^3-9*c[5]*c[7]*c[4]-140*c[5]^2*c[ 4]^5*c[7]-780*c[5]^3*c[4]^4*c[7]-10*c[4]^4*c[6]^2-100*c[7]^2*c[4]^5*c[ 5]^3+450*c[6]^2*c[4]^5*c[7]*c[5]^3+510*c[5]^4*c[4]^2*c[7]^2*c[6]+490*c [6]^2*c[7]*c[5]^4*c[4]^2-2224*c[6]*c[5]^2*c[4]^4*c[7]+10*c[5]^5*c[4]^3 *c[7]-146*c[5]^4*c[4]^3*c[6]+90*c[5]^4*c[4]^5*c[6]+40*c[5]^4*c[4]^4*c[ 6]+570*c[7]^2*c[4]^4*c[5]^3+130*c[5]^4*c[4]^4*c[7]+25*c[7]^2*c[6]*c[4] ^4+18*c[5]^5*c[4]^2-4*c[5]^2-4*c[4]^2-10*c[4]^4*c[7]^2+150*c[7]^2*c[5] ^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]^3-70*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+20 8*c[6]^2*c[5]^3*c[7]*c[4]-225*c[5]^4*c[4]^3*c[7]+704*c[7]^2*c[5]^2*c[4 ]^3-100*c[5]^4*c[7]^2*c[4]^4+29*c[7]^2*c[5]*c[4]^2+152*c[7]*c[5]*c[4]^ 3+181*c[6]*c[5]*c[4]^3-178*c[7]^2*c[5]^2*c[6]*c[4]-36*c[6]*c[7]*c[4]^4 -18*c[6]^2*c[7]^2*c[4]^2+3*c[6]^2*c[7]^2*c[4]+10*c[5]^4*c[7]^2*c[6]+77 *c[5]^3*c[6]*c[4]+75*c[5]^3*c[7]*c[4]+90*c[7]*c[6]*c[5]^5*c[4]+29*c[5] ^2*c[4]-350*c[5]^5*c[4]^2*c[6]*c[7]-234*c[5]^3*c[6]*c[7]*c[4]+279*c[5] ^3*c[4]^2+54*c[6]^2*c[7]^2*c[4]*c[5]-1192*c[5]^2*c[6]*c[4]^3+450*c[5]^ 2*c[7]*c[4]^2+522*c[5]^2*c[6]*c[4]^2-990*c[5]^2*c[7]*c[4]^3-502*c[5]^3 *c[7]*c[4]^2-614*c[5]^3*c[6]*c[4]^2-1638*c[5]^3*c[6]^2*c[4]^3+260*c[4] ^3*c[7]^2*c[5]^2*c[6]^2-310*c[4]^5*c[5]^3*c[6]^2-5*c[7]*c[5]^4*c[4]-5* c[7]^2*c[5]^4*c[4]+737*c[5]^2*c[4]^4*c[7]+948*c[6]^2*c[5]^2*c[4]^3-426 *c[6]^2*c[5]^2*c[4]^2+1443*c[6]*c[5]^3*c[4]^3-152*c[6]^2*c[5]^3*c[4]+7 72*c[6]^2*c[5]^3*c[4]^2+1076*c[5]^3*c[7]*c[4]^3+120*c[5]^4*c[7]*c[4]^2 *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]^3-23 11*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]^3-40 0*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]*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]^4-200* c[5]^4*c[4]^5*c[6]^2*c[7]+15*c[7]*c[6]^2*c[5]*c[4]^4-1990*c[5]^3*c[4]^ 4*c[6]^2*c[7]-750*c[5]^3*c[4]^3*c[6]^2*c[7]^2+210*c[5]^4*c[4]^4*c[6]*c [7]+1100*c[5]^2*c[4]^4*c[7]*c[6]^2+2380*c[5]^3*c[4]^4*c[6]*c[7]-50*c[5 ]^4*c[4]^3*c[6]*c[7]-594*c[5]^3*c[4]^3-10*c[5]^5*c[6]*c[7]-3*c[5]^5*c[ 4]+328*c[7]^2*c[5]^3*c[4]^2+166*c[5]^5*c[6]*c[4]^2+150*c[5]^4*c[4]^5*c [6]^2-200*c[6]*c[7]^2*c[5]^4*c[4]^5-680*c[5]^4*c[4]^4*c[6]^2+850*c[5]^ 4*c[4]^3*c[6]^2+300*c[7]^2*c[5]^4*c[6]^2*c[4]^3-100*c[7]^2*c[5]^4*c[6] ^2*c[4]^2-2570*c[5]^3*c[4]^4*c[7]^2*c[6]-1130*c[5]^4*c[4]^3*c[7]^2*c[6 ]+1840*c[5]^2*c[4]^4*c[7]^2*c[6]-1060*c[5]^4*c[4]^3*c[6]^2*c[7]-170*c[ 7]^2*c[4]^4*c[6]^2*c[5]+500*c[7]^2*c[4]^4*c[5]^3*c[6]^2-110*c[7]^2*c[4 ]^4*c[6]^2*c[5]^2-160*c[5]^2*c[6]*c[4]^5-200*c[6]^2*c[7]^2*c[5]^4*c[4] ^4-1067*c[4]^4*c[6]*c[5]^3-70*c[5]^5*c[6]^2*c[4]^3-200*c[5]^4*c[4]^5*c [6]*c[7]+14*c[5]^4*c[4]+434*c[5]^3*c[4]^4-52*c[5]^5*c[4]*c[6])/c[5]/(7 2*c[5]^2*c[6]*c[4]^4-12*c[6]*c[5]*c[4]^4+1100*c[5]^4*c[4]^4*c[6]^2*c[7 ]-12*c[7]*c[5]*c[4]^4-110*c[5]^2*c[4]^4*c[7]^2+120*c[4]^5*c[6]*c[5]^3+ 15*c[5]*c[6]^2*c[4]^4-300*c[5]^5*c[4]^3*c[7]^2*c[6]^2+10*c[5]^5-440*c[ 5]^3*c[4]^4*c[7]-150*c[7]^2*c[4]^5*c[5]^3+180*c[5]^4*c[4]^2*c[7]^2*c[6 ]+180*c[6]^2*c[7]*c[5]^4*c[4]^2+350*c[6]*c[5]^2*c[4]^4*c[7]-570*c[5]^5 *c[4]^3*c[7]-150*c[5]^5*c[4]*c[6]*c[7]^2-150*c[5]^5*c[4]*c[6]^2*c[7]+4 10*c[5]^4*c[4]^3*c[6]-120*c[5]^4*c[4]^4*c[6]+690*c[7]^2*c[4]^4*c[5]^3+ 200*c[5]^4*c[4]^5*c[7]^2-120*c[5]^4*c[4]^4*c[7]-15*c[7]^2*c[6]*c[4]^4+ 342*c[5]^5*c[4]^2-12*c[7]*c[6]^2*c[5]^3+10*c[7]*c[6]*c[5]^3-12*c[7]^2* c[6]*c[5]^3+15*c[5]^6*c[7]+410*c[5]^4*c[4]^2*c[7]^2+750*c[5]^5*c[7]^2* c[4]^3-120*c[6]*c[5]^2*c[4]^5*c[7]+570*c[5]^6*c[4]^2*c[7]-570*c[5]^5*c [4]^3*c[6]-57*c[7]*c[4]^4*c[6]*c[5]+87*c[5]^6*c[4]-570*c[5]^5*c[4]^4-1 100*c[5]^5*c[7]*c[4]^4*c[6]-410*c[7]^2*c[5]^3*c[4]^3-15*c[7]*c[4]^4*c[ 6]^2-15*c[5]^5*c[7]^2+780*c[5]^5*c[4]^2*c[6]^2*c[7]+750*c[5]^5*c[4]^4* c[6]+300*c[5]^5*c[7]*c[4]^5*c[6]-690*c[5]^5*c[4]^2*c[6]^2-40*c[6]^2*c[ 5]^4*c[7]*c[4]-690*c[5]^5*c[4]^2*c[7]^2-57*c[5]^4*c[6]^2*c[4]+410*c[5] ^4*c[6]^2*c[4]^2+750*c[5]^5*c[4]^4*c[7]+340*c[6]^2*c[7]*c[5]^2*c[4]^3- 30*c[6]^2*c[7]^2*c[5]^2*c[4]-750*c[6]*c[7]*c[5]^3*c[4]^3-20*c[7]^2*c[6 ]^2*c[5]^4-342*c[5]^4*c[7]*c[4]^2+90*c[7]^2*c[6]*c[5]*c[4]^4+200*c[7]^ 2*c[5]^5*c[6]^2*c[4]^2-10*c[5]^4*c[7]+140*c[5]^5*c[7]*c[4]^2+150*c[5]^ 4*c[4]*c[7]^2*c[6]^2+20*c[6]^2*c[4]^4*c[7]^2-10*c[7]*c[6]*c[4]^3-30*c[ 5]^5*c[7]*c[4]+12*c[5]^4*c[6]^2+12*c[7]^2*c[6]*c[4]^3-12*c[5]*c[7]^2*c [4]^3-24*c[6]*c[5]*c[7]^2*c[4]^2+24*c[5]*c[7]*c[6]*c[4]^3-12*c[5]^6+15 0*c[7]^2*c[5]^6*c[4]^2-120*c[5]^4*c[4]^5+48*c[5]^2*c[6]*c[7]*c[4]^2+11 00*c[5]^4*c[7]^2*c[4]^4*c[6]-150*c[4]^5*c[6]*c[7]*c[5]^3-12*c[6]^2*c[5 ]*c[4]^3-42*c[7]*c[6]^2*c[5]*c[4]^3+20*c[6]*c[5]*c[7]*c[4]^2+12*c[5]^4 *c[7]*c[6]-24*c[6]^2*c[4]^2*c[7]*c[5]+30*c[6]^2*c[7]^2*c[4]^2*c[5]-342 *c[6]*c[5]^4*c[4]^2-10*c[6]*c[5]^4+120*c[4]^5*c[7]*c[5]^3+24*c[6]^2*c[ 5]^2*c[7]*c[4]-30*c[6]^2*c[5]^2*c[7]*c[4]^2-243*c[6]*c[5]^2*c[7]*c[4]^ 3+372*c[6]*c[5]^3*c[7]*c[4]^2+87*c[6]^2*c[5]^3*c[7]*c[4]+410*c[5]^4*c[ 4]^3*c[7]+57*c[7]^2*c[5]^2*c[4]^3-750*c[5]^4*c[7]^2*c[4]^4+10*c[7]*c[5 ]*c[4]^3+10*c[6]*c[5]*c[4]^3+24*c[7]^2*c[5]^2*c[6]*c[4]+12*c[6]*c[7]*c [4]^4+200*c[7]*c[5]^7*c[4]^3-110*c[5]^6*c[7]*c[4]+15*c[6]*c[5]^6+20*c[ 5]^3*c[6]*c[4]+20*c[5]^3*c[7]*c[4]+150*c[7]*c[6]*c[5]^5*c[4]-900*c[5]^ 5*c[4]^3*c[7]^2*c[6]-750*c[5]^5*c[4]^2*c[6]*c[7]-96*c[5]^3*c[6]*c[7]*c [4]+200*c[6]*c[5]^7*c[4]^3+20*c[5]^3*c[4]^2-36*c[5]^2*c[6]*c[4]^3-20*c [5]^2*c[7]*c[4]^2-20*c[5]^2*c[6]*c[4]^2-36*c[5]^2*c[7]*c[4]^3-24*c[5]^ 3*c[7]*c[4]^2-24*c[5]^3*c[6]*c[4]^2-410*c[5]^3*c[6]^2*c[4]^3-900*c[5]^ 5*c[4]^3*c[6]^2*c[7]-510*c[4]^3*c[7]^2*c[5]^2*c[6]^2-630*c[7]*c[5]^6*c [6]*c[4]^2-150*c[4]^5*c[5]^3*c[6]^2+72*c[7]*c[5]^4*c[4]-57*c[7]^2*c[5] ^4*c[4]+72*c[5]^2*c[4]^4*c[7]+57*c[6]^2*c[5]^2*c[4]^3+24*c[6]^2*c[5]^2 *c[4]^2+285*c[6]*c[5]^3*c[4]^3-24*c[6]^2*c[5]^3*c[4]+285*c[5]^3*c[7]*c [4]^3+270*c[5]^4*c[7]*c[4]^2*c[6]+12*c[7]*c[6]^2*c[4]^3+340*c[7]^2*c[5 ]^2*c[6]*c[4]^3-410*c[7]^2*c[5]^3*c[6]*c[4]^2-200*c[5]^5*c[4]^5*c[7]+7 0*c[4]^3*c[7]^2*c[5]*c[6]^2-42*c[7]^2*c[6]*c[5]*c[4]^3+150*c[5]^5*c[4] ^5-30*c[7]^2*c[5]^2*c[6]*c[4]^2+12*c[4]^4*c[5]^2+87*c[7]^2*c[5]^3*c[6] *c[4]+510*c[7]^2*c[5]^3*c[6]*c[4]^3+690*c[5]^3*c[6]^2*c[4]^4-342*c[5]^ 4*c[4]^3-15*c[6]^2*c[4]^3*c[7]^2+510*c[5]^3*c[6]^2*c[7]*c[4]^3-20*c[5] ^2*c[6]*c[7]*c[4]-410*c[6]^2*c[7]*c[5]^3*c[4]^2-70*c[6]^2*c[7]^2*c[4]* c[5]^3-10*c[5]^2*c[4]^3+24*c[7]^2*c[5]^2*c[4]^2-24*c[7]^2*c[5]^3*c[4]- 200*c[7]^2*c[5]^6*c[4]^2*c[6]+110*c[7]^2*c[4]*c[5]^5+1350*c[5]^5*c[4]^ 3*c[6]*c[7]+15*c[7]^2*c[4]^4*c[5]+300*c[6]^2*c[7]*c[5]^6*c[4]^3+510*c[ 7]^2*c[5]^3*c[6]^2*c[4]^2-550*c[5]^6*c[4]^2-600*c[7]*c[5]^6*c[4]^3+20* c[7]*c[6]^2*c[5]^5+570*c[5]^6*c[6]*c[4]^2+150*c[5]^2*c[6]*c[4]^5*c[7]^ 2+550*c[5]^4*c[4]^4+120*c[5]^7*c[4]^2+72*c[5]^4*c[4]*c[6]-40*c[5]^4*c[ 7]^2*c[4]*c[6]-57*c[5]^4*c[4]*c[7]*c[6]+150*c[6]^2*c[5]^6*c[4]^2+150*c [4]^5*c[6]^2*c[7]*c[5]^2-200*c[4]^5*c[6]^2*c[7]^2*c[5]^2-150*c[5]^7*c[ 7]*c[4]^2-110*c[6]^2*c[4]^4*c[5]^2+780*c[5]^5*c[4]^2*c[7]^2*c[6]-200*c [5]^5*c[4]^5*c[6]-600*c[5]^6*c[4]^3*c[6]-200*c[6]^2*c[7]*c[5]^6*c[4]^2 -300*c[5]^4*c[4]^5*c[6]^2*c[7]+90*c[7]*c[6]^2*c[5]*c[4]^4+200*c[6]*c[5 ]^7*c[7]*c[4]^2-300*c[6]*c[5]^7*c[7]*c[4]^3-180*c[5]^3*c[4]^4*c[6]^2*c [7]-570*c[5]^4*c[4]^4*c[6]*c[7]-540*c[5]^2*c[4]^4*c[7]*c[6]^2+660*c[5] ^3*c[4]^4*c[6]*c[7]+60*c[5]^4*c[4]^3*c[6]*c[7]+48*c[5]^3*c[4]^3-15*c[6 ]^2*c[5]^5-200*c[5]^6*c[6]^2*c[4]^3-15*c[5]^5*c[6]*c[7]-48*c[5]^5*c[4] +140*c[5]^5*c[6]*c[4]^2+200*c[5]^4*c[4]^5*c[6]^2+300*c[6]^2*c[7]^2*c[5 ]^3*c[4]^5-300*c[6]*c[7]^2*c[5]^4*c[4]^5+700*c[7]*c[5]^6*c[4]^3*c[6]-7 50*c[5]^4*c[4]^4*c[6]^2+20*c[5]^5*c[7]^2*c[6]+1100*c[7]^2*c[5]^4*c[6]^ 2*c[4]^3-930*c[7]^2*c[5]^4*c[6]^2*c[4]^2+300*c[7]^2*c[5]^6*c[4]^3*c[6] -20*c[5]^6*c[7]*c[6]-180*c[5]^3*c[4]^4*c[7]^2*c[6]-750*c[5]^4*c[4]^3*c [7]^2*c[6]-540*c[5]^2*c[4]^4*c[7]^2*c[6]-750*c[5]^4*c[4]^3*c[6]^2*c[7] -150*c[7]^2*c[4]^4*c[6]^2*c[5]-1100*c[7]^2*c[4]^4*c[5]^3*c[6]^2+930*c[ 7]^2*c[4]^4*c[6]^2*c[5]^2+150*c[5]^6*c[4]*c[7]*c[6]-110*c[5]^6*c[4]*c[ 6]-150*c[5]^7*c[6]*c[4]^2+570*c[5]^6*c[4]^3-150*c[5]^7*c[4]^3+15*c[7]^ 2*c[6]^2*c[5]^3-440*c[4]^4*c[6]*c[5]^3+750*c[5]^5*c[6]^2*c[4]^3-200*c[ 7]^2*c[5]^6*c[4]^3+12*c[5]^4*c[7]^2+200*c[5]^4*c[4]^5*c[6]*c[7]-20*c[5 ]^4*c[4]-87*c[5]^3*c[4]^4-30*c[5]^5*c[4]*c[6]+110*c[5]^5*c[6]^2*c[4]), a[9,7] = -1/60*(10*c[5]*c[6]*c[4]-5*c[6]*c[5]-5*c[6]*c[4]+3*c[6]-5*c[ 5]*c[4]+3*c[5]+3*c[4]-2)/c[7]/(-c[5]*c[6]*c[4]+c[5]*c[6]*c[7]*c[4]+c[5 ]*c[7]*c[4]-c[7]^2*c[5]*c[4]+c[6]*c[7]*c[4]-c[6]*c[7]^2*c[4]-c[7]^2*c[ 4]+c[7]^3*c[4]+c[6]*c[7]*c[5]-c[6]*c[7]^2*c[5]-c[7]^2*c[5]+c[7]^3*c[5] -c[7]^2*c[6]+c[7]^3*c[6]+c[7]^3-c[7]^4), a[9,2] = 0, a[9,3] = 0, a[9,5 ] = -1/60*(10*c[6]*c[7]*c[4]-5*c[6]*c[4]-5*c[7]*c[6]+3*c[6]-5*c[7]*c[4 ]+3*c[4]-2+3*c[7])/c[5]/(c[5]^3*c[4]-c[5]^2*c[4]+c[5]*c[7]*c[4]-c[5]^2 *c[7]*c[4]+c[5]*c[6]*c[4]-c[5]^2*c[6]*c[4]-c[6]*c[7]*c[4]+c[5]*c[6]*c[ 7]*c[4]-c[5]^4+c[5]^3-c[7]*c[5]^2+c[7]*c[5]^3-c[6]*c[5]^2+c[6]*c[5]^3+ c[6]*c[7]*c[5]-c[6]*c[7]*c[5]^2), a[9,8] = 1/60*(-20*c[5]*c[6]*c[4]+30 *c[5]*c[6]*c[7]*c[4]+15*c[6]*c[5]-20*c[6]*c[7]*c[5]+15*c[6]*c[4]-20*c[ 6]*c[7]*c[4]-12*c[6]+15*c[7]*c[6]+15*c[5]*c[4]-20*c[5]*c[7]*c[4]-12*c[ 5]+15*c[7]*c[5]-12*c[4]+15*c[7]*c[4]+10-12*c[7])/(-c[4]+c[7]*c[4]+c[6] *c[4]-c[6]*c[7]*c[4]+c[5]*c[4]-c[5]*c[7]*c[4]-c[5]*c[6]*c[4]+c[5]*c[6] *c[7]*c[4]+1-c[7]-c[6]+c[7]*c[6]-c[5]+c[7]*c[5]+c[6]*c[5]-c[6]*c[7]*c[ 5]), a[9,1] = 1/60*(30*c[5]*c[6]*c[7]*c[4]-10*c[5]*c[6]*c[4]-10*c[6]*c [7]*c[5]+5*c[6]*c[5]-10*c[6]*c[7]*c[4]+5*c[6]*c[4]+5*c[7]*c[6]-3*c[6]- 10*c[5]*c[7]*c[4]+5*c[5]*c[4]+5*c[7]*c[5]-3*c[5]+5*c[7]*c[4]-3*c[4]+2- 3*c[7])/c[5]/c[6]/c[7]/c[4], a[9,4] = 1/60*(-2+3*c[5]-5*c[7]*c[6]+3*c[ 7]+3*c[6]+10*c[6]*c[7]*c[5]-5*c[6]*c[5]-5*c[7]*c[5])/c[4]/(-c[6]*c[4]^ 3-c[4]^3+c[4]^4+c[7]*c[4]^2+c[6]*c[4]^2-c[6]*c[7]*c[4]+c[6]*c[7]*c[4]^ 2-c[5]*c[4]^3+c[5]*c[4]^2-c[7]*c[4]^3-c[5]*c[7]*c[4]+c[5]*c[7]*c[4]^2- c[5]*c[6]*c[4]+c[5]*c[6]*c[4]^2+c[6]*c[7]*c[5]-c[5]*c[6]*c[7]*c[4]), a [9,6] = 1/60*(10*c[5]*c[7]*c[4]-5*c[5]*c[4]-5*c[7]*c[5]+3*c[5]-5*c[7]* c[4]+3*c[4]-2+3*c[7])/c[6]/(c[6]^2*c[4]*c[5]-c[5]*c[6]*c[4]+c[5]*c[7]* c[4]-c[5]*c[6]*c[7]*c[4]-c[6]^3*c[4]+c[6]^2*c[4]-c[6]*c[7]*c[4]+c[6]^2 *c[7]*c[4]-c[6]^3*c[5]+c[6]^2*c[5]-c[6]*c[7]*c[5]+c[6]^2*c[7]*c[5]+c[6 ]^4-c[6]^3+c[7]*c[6]^2-c[6]^3*c[7])\}:" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 33 "#======================== ========" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 22 "We specify the nodes " }{XPPEDIT 18 0 "c[6]=28/39" "6#/&%\"cG6#\" \"'*&\"#G\"\"\"\"#R!\"\"" }{TEXT -1 7 " and " }{XPPEDIT 18 0 "c[7]=3 24/325" "6#/&%\"cG6#\"\"(*&\"$C$\"\"\"\"$D$!\"\"" }{TEXT -1 27 " and d etermine 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 minimi ze 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 for th e 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]=28/39,c[7]=324/ 325\}:\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 16195 "eB := \{`b*`[2] = 0, c[9] = 1, c[7] = 324/325, c[6] = 28/39, a[8,6] = -652509/7616*(216155*c[5] *c[4]+43234-86465*c[4]-86465*c[5])*(c[4]-1)*(-1+c[5])/(39*c[4]-28)/(39 *c[5]-28)/(27565*c[5]*c[4]-7500*c[5]-7500*c[4]+1998), c[8] = 1, b[3] = 0, b[2] = 0, a[9,6] = 2313441/1675520*(1615*c[5]*c[4]-645*c[5]-645*c[ 4]+322)/(1521*c[5]*c[4]-1092*c[4]-1092*c[5]+784), a[8,3] = 3/4*(-15171 50*c[5]^4*c[4]^3-14208200*c[5]^3*c[4]^4+27048*c[4]-4551450*c[4]^5*c[5] ^2-54096*c[5]-144480*c[5]^3+187572*c[5]^2+1083600*c[4]^5*c[5]-1715205* c[5]*c[4]^2-16229970*c[5]^2*c[4]^3+14956305*c[4]^4*c[5]^2+361200*c[5]^ 4*c[4]^2-6819930*c[5]^3*c[4]^2+1473740*c[5]^3*c[4]-1785486*c[5]^2*c[4] +418362*c[5]*c[4]+4069800*c[4]^5*c[5]^3-3573990*c[4]^4*c[5]+1356600*c[ 5]^4*c[4]^4-93786*c[4]^2+15273075*c[5]^3*c[4]^3+7646014*c[5]^2*c[4]^2+ 3782750*c[5]*c[4]^3+72240*c[4]^3)/c[4]^2/(275650*c[5]^4*c[4]^3+275650* c[5]^3*c[4]^4+7500*c[5]^3-1998*c[5]^2+11988*c[5]*c[4]^2+410370*c[5]^2* c[4]^3-75000*c[4]^4*c[5]^2-75000*c[5]^4*c[4]^2+410370*c[5]^3*c[4]^2-72 565*c[5]^3*c[4]+11988*c[5]^2*c[4]+1998*c[5]*c[4]-1998*c[4]^2-976950*c[ 5]^3*c[4]^3-122375*c[5]^2*c[4]^2-72565*c[5]*c[4]^3+7500*c[4]^3), `b*`[ 8] = 0, a[9,5] = -1/60*(20065*c[4]-5502)/c[5]/(12675*c[5]^3*c[4]-34411 *c[5]^2*c[4]+30808*c[5]*c[4]-9072*c[4]-12675*c[5]^4+34411*c[5]^3-30808 *c[5]^2+9072*c[5]), 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*(20065*c[4]-5502)/(c[4]-c[5])/c[5]/(12675*c[5]^3- 34411*c[5]^2+30808*c[5]-9072), `b*`[9] = 1/10*(1400*c[5]^2*c[4]^2-1315 *c[5]^2*c[4]+280*c[5]^2-1315*c[5]*c[4]^2+1370*c[5]*c[4]-319*c[5]+280*c [4]^2-319*c[4]+84)/(-291*c[4]-291*c[5]+224*c[5]^2-940*c[5]*c[4]^2-940* c[5]^2*c[4]+1135*c[5]*c[4]+224*c[4]^2+840*c[5]^2*c[4]^2+84), a[7,5] = \+ -162/11156640625*(12075507715375*c[5]^4*c[4]^3+23922538774225*c[5]^3*c [4]^4+3725562750000*c[5]^4*c[4]^5-26500981248*c[4]+5448973296000*c[4]^ 5*c[5]^2+53001962496*c[5]+334755607680*c[5]^3-239468555520*c[5]^2-1064 962080000*c[4]^5*c[5]+1443922408928*c[5]*c[4]^2+17202365160070*c[5]^2* c[4]^3-15944397437195*c[4]^4*c[5]^2-6144358286625*c[5]^4*c[4]^2+132678 88706670*c[5]^3*c[4]^2-3307220146360*c[5]^3*c[4]+2212075069664*c[5]^2* c[4]-391368495744*c[5]*c[4]-8108344530000*c[4]^5*c[5]^3+2989046114200* c[4]^4*c[5]-11021925655000*c[5]^4*c[4]^4+110268801792*c[4]^2-261094443 35520*c[5]^3*c[4]^3-148334004000*c[5]^4+51081872400*c[4]^4+15135098355 00*c[5]^4*c[4]-8679788894672*c[5]^2*c[4]^2-3029515617952*c[5]*c[4]^3-1 34869015008*c[4]^3)/c[5]/(132220*c[5]^5*c[4]^3-77840*c[5]^4*c[4]^3+126 980*c[5]^3*c[4]^4-55580*c[5]^5*c[4]^2+33150*c[5]^4*c[4]^5+8970*c[5]^6* c[4]^2+6440*c[4]^5*c[5]^2-168*c[5]^3-336*c[5]*c[4]^2+4651*c[5]^2*c[4]^ 3-43977*c[4]^4*c[5]^2+41052*c[5]^4*c[4]^2-4144*c[5]^3*c[4]^2-104*c[5]^ 3*c[4]+336*c[5]^2*c[4]-32770*c[4]^5*c[5]^3+7141*c[4]^4*c[5]-108420*c[5 ]^4*c[4]^4-5772*c[5]^3*c[4]^3+878*c[5]^4+8697*c[5]^5*c[4]-644*c[4]^4-3 3150*c[5]^6*c[4]^3-6751*c[5]^4*c[4]+468*c[5]^2*c[4]^2-598*c[5]*c[4]^3+ 168*c[4]^3-897*c[5]^5), b[1] = 1/544320*(118175*c[5]*c[4]-20065*c[5]-2 0065*c[4]+5502)/c[5]/c[4], 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], a[7,4] = -162/11156640625*(-5448973296000*c[5]^5 *c[4]^3+68743476900735*c[5]^4*c[4]^3-30377665500000*c[5]^3*c[4]^6+5082 8485549470*c[5]^3*c[4]^4+45939076214850*c[4]^6*c[5]^2+1064962080000*c[ 5]^5*c[4]^2-11906174383350*c[4]^6*c[5]+53869707385000*c[5]^4*c[4]^5-37 25562750000*c[5]^5*c[4]^5-59467588007515*c[4]^5*c[5]^2+551826390528*c[ 5]^3-159005887488*c[5]^2+8108344530000*c[5]^5*c[4]^4+17858819627720*c[ 4]^5*c[5]-1335965705856*c[5]*c[4]^2-2889102820288*c[5]^2*c[4]^3+306163 50210110*c[4]^4*c[5]^2-25651544245080*c[5]^4*c[4]^2+25947501777376*c[5 ]^3*c[4]^2-5674477077024*c[5]^3*c[4]+1065239009280*c[5]^2*c[4]+1855068 68736*c[5]*c[4]+5340535496075*c[4]^5*c[5]^3-12981700098104*c[4]^4*c[5] -90244338280525*c[5]^4*c[4]^4-53001962496*c[4]^2-57796435409190*c[5]^3 *c[4]^3+148334004000*c[4]^5-425984832000*c[5]^4-334755607680*c[4]^4+48 88743927600*c[5]^4*c[4]-2633043781472*c[5]^2*c[4]^2+5212377226912*c[5] *c[4]^3+239468555520*c[4]^3-11176688250000*c[5]^4*c[4]^6-1247011623000 0*c[4]^7*c[5]^2+2966680080000*c[5]*c[4]^7+11176688250000*c[5]^3*c[4]^7 )/(-850*c[5]^5*c[4]^3+2780*c[5]^4*c[4]^3-2780*c[5]^3*c[4]^4+230*c[5]^5 *c[4]^2-230*c[4]^5*c[5]^2+6*c[5]^3+12*c[5]*c[4]^2-148*c[5]^2*c[4]^3+12 60*c[4]^4*c[5]^2-1260*c[5]^4*c[4]^2+148*c[5]^3*c[4]^2-13*c[5]^3*c[4]-1 2*c[5]^2*c[4]+850*c[4]^5*c[5]^3-223*c[4]^4*c[5]-23*c[5]^4+23*c[4]^4+22 3*c[5]^4*c[4]+13*c[5]*c[4]^3-6*c[4]^3)/(39*c[4]-28)/c[4]^2, a[7,6] = - 297432/2734375*(5*c[5]*c[4]+1-2*c[4]-2*c[5])*(325*c[4]-324)*(325*c[5]- 324)/(85*c[5]*c[4]-23*c[5]-23*c[4]+6)/(39*c[4]-28)/(39*c[5]-28), a[9,4 ] = 1/60*(20065*c[5]-5502)/c[4]/(-34411*c[4]^3+12675*c[4]^4+30808*c[4] ^2-9072*c[4]-12675*c[5]*c[4]^3+34411*c[5]*c[4]^2-30808*c[5]*c[4]+9072* c[5]), a[8,4] = -1/2*(20046298100*c[5]^5*c[4]^3-290508427085*c[5]^4*c[ 4]^3+109342832425*c[5]^3*c[4]^6-335541159630*c[5]^3*c[4]^4-32502178897 0*c[4]^6*c[5]^2-3276806400*c[5]^5*c[4]^2+91863985935*c[4]^6*c[5]-44353 7955875*c[5]^4*c[4]^5+36419880900*c[5]^5*c[4]^5+277706571290*c[4]^5*c[ 5]^2-1701653184*c[5]^3+490758912*c[5]^2-41726102600*c[5]^5*c[4]^4-9518 4112490*c[4]^5*c[5]+4696501672*c[5]*c[4]^2+721528410*c[5]^2*c[4]^3-103 211957553*c[4]^4*c[5]^2+93980308160*c[5]^4*c[4]^2-97557091877*c[5]^3*c [4]^2+19204603274*c[5]^3*c[4]-3780394536*c[5]^2*c[4]-572552064*c[5]*c[ 4]+141163808090*c[4]^5*c[5]^3+56154782648*c[4]^4*c[5]+489328280610*c[5 ]^4*c[4]^4+163586304*c[4]^2+258452586102*c[5]^3*c[4]^3-1491232795*c[4] ^5+1310722560*c[5]^4+1772606043*c[4]^4-16353353570*c[5]^4*c[4]+1142927 2897*c[5]^2*c[4]^2-20212723801*c[5]*c[4]^3-902780552*c[4]^3-3845975250 0*c[4]^8*c[5]^2+200170235200*c[5]^4*c[4]^6+180125762050*c[4]^7*c[5]^2- 45902301900*c[5]*c[4]^7+34389810000*c[4]^8*c[5]^3+9156420000*c[5]*c[4] ^8+457821000*c[4]^6-11463270000*c[5]^5*c[4]^6-34389810000*c[5]^4*c[4]^ 7-127753735200*c[5]^3*c[4]^7)/(-275650*c[5]^5*c[4]^3+901950*c[5]^4*c[4 ]^3-901950*c[5]^3*c[4]^4+75000*c[5]^5*c[4]^2-75000*c[4]^5*c[5]^2+1998* c[5]^3+3996*c[5]*c[4]^2-49810*c[5]^2*c[4]^3+410370*c[4]^4*c[5]^2-41037 0*c[5]^4*c[4]^2+49810*c[5]^3*c[4]^2-4488*c[5]^3*c[4]-3996*c[5]^2*c[4]+ 275650*c[4]^5*c[5]^3-72565*c[4]^4*c[5]-7500*c[5]^4+7500*c[4]^4+72565*c [5]^4*c[4]+4488*c[5]*c[4]^3-1998*c[4]^3)/(-21736*c[4]+12675*c[4]^2+907 2)/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[9,2] = 0, a[9,3] = 0, a[7,1] = 81/78096484375*(1374522 357000*c[5]^5*c[4]^3-12976637047465*c[5]^4*c[4]^3+869057995000*c[5]^3* c[4]^6-9217343941505*c[5]^3*c[4]^4-1982816628000*c[4]^6*c[5]^2-2662405 20000*c[5]^5*c[4]^2+532481040000*c[4]^6*c[5]-8157932510000*c[5]^4*c[4] ^5+1181162125000*c[5]^5*c[4]^5+4663886331510*c[4]^5*c[5]^2-13795659763 2*c[5]^3+39751471872*c[5]^2-2140050380000*c[5]^5*c[4]^4-1487764535850* c[4]^5*c[5]+283118587200*c[5]*c[4]^2+386785644322*c[5]^2*c[4]^3-336673 2266255*c[4]^4*c[5]^2+5296417529195*c[5]^4*c[4]^2-5195247843574*c[5]^3 *c[4]^2+1274369094096*c[5]^3*c[4]-224664933312*c[5]^2*c[4]-46376717184 *c[5]*c[4]+2009941468600*c[4]^5*c[5]^3+1561620782276*c[4]^4*c[5]+15397 641703600*c[5]^4*c[4]^4+13250490624*c[4]^2+10563086093790*c[5]^3*c[4]^ 3-25540936200*c[4]^5+106496208000*c[5]^4+67434507504*c[4]^4-1112163382 200*c[5]^4*c[4]+440310895856*c[5]^2*c[4]^2-839047037536*c[5]*c[4]^3-55 134400896*c[4]^3+1181162125000*c[5]^4*c[4]^6)/(850*c[5]^4*c[4]^3+850*c [5]^3*c[4]^4+23*c[5]^3-6*c[5]^2+36*c[5]*c[4]^2+1260*c[5]^2*c[4]^3-230* c[4]^4*c[5]^2-230*c[5]^4*c[4]^2+1260*c[5]^3*c[4]^2-223*c[5]^3*c[4]+36* c[5]^2*c[4]+6*c[5]*c[4]-6*c[4]^2-3010*c[5]^3*c[4]^3-371*c[5]^2*c[4]^2- 223*c[5]*c[4]^3+23*c[4]^3)/c[5]/c[4]^2, a[5,4] = -c[5]^2*(c[4]-c[5])/c [4]^2, b[7] = -55783203125/1057536*(85*c[5]*c[4]-23*c[5]-23*c[4]+6)/(3 25*c[4]-324)/(325*c[5]-324), a[7,3] = 729/11156640625*(-53291095000*c[ 5]^4*c[4]^3-499746130000*c[5]^3*c[4]^4+946463616*c[4]-159873285000*c[4 ]^5*c[5]^2-1892927232*c[5]-5071248000*c[5]^3+6569361792*c[5]^2+3803436 0000*c[4]^5*c[5]-60049438832*c[5]*c[4]^2-568996172920*c[5]^2*c[4]^3+52 4894175575*c[4]^4*c[5]^2+12678120000*c[5]^4*c[4]^2-239498594595*c[5]^3 *c[4]^2+51743926000*c[5]^3*c[4]-62544323552*c[5]^2*c[4]+14642244288*c[ 5]*c[4]+143290875000*c[4]^5*c[5]^3-125336541325*c[4]^4*c[5]+4776362500 0*c[5]^4*c[4]^4-3284680896*c[4]^2+536581912425*c[5]^3*c[4]^3+267914248 790*c[5]^2*c[4]^2+132506026090*c[5]*c[4]^3+2535624000*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)/(85*c[5]*c[4]-23*c[5]-23*c[4]+6), a[ 9,8] = 1/660*(27565*c[5]*c[4]-7500*c[5]-7500*c[4]+1998)/(-c[4]+c[5]*c[ 4]+1-c[5]), b[6] = 2313441/1675520*(1615*c[5]*c[4]-645*c[5]-645*c[4]+3 22)/(39*c[5]-28)/(39*c[4]-28), a[5,3] = 3/4*c[5]^2*(-2*c[5]+3*c[4])/c[ 4]^2, a[6,5] = -14/2313441*(1487655*c[5]^2*c[4]^2-61152*c[4]^2+21952*c [4]-494676*c[5]^2*c[4]-982884*c[5]*c[4]^2+42588*c[4]^3-611520*c[4]^4*c [5]+1297001*c[5]*c[4]^3-1889160*c[5]^2*c[4]^3+324576*c[5]*c[4]+851760* c[4]^4*c[5]^2+61152*c[5]^2-43904*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+3 0*c[5]^3*c[4]^2-6*c[5]^3*c[4]-2*c[5]^2*c[4]), a[6,3] = 7/257049*(15525 29*c[5]*c[4]^3-1062600*c[5]*c[4]^2+304192*c[5]*c[4]-43904*c[5]+21952*c [4]-203840*c[5]^3*c[4]^2-3370640*c[5]^2*c[4]^3+2354703*c[5]^2*c[4]^2-6 59568*c[5]^2*c[4]+81536*c[5]^2-611520*c[4]^4*c[5]-40768*c[4]^2+425880* c[5]^3*c[4]^3+1277640*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] = 59319/152320*(-416598*c[4]-416598*c[5]-361200*c[5] ^3+685689*c[5]^2-4040319*c[5]*c[4]^2-4391290*c[5]^2*c[4]^3-4391290*c[5 ]^3*c[4]^2+2239550*c[5]^3*c[4]-4040319*c[5]^2*c[4]+2306742*c[5]*c[4]+6 85689*c[4]^2+2713200*c[5]^3*c[4]^3+7510900*c[5]^2*c[4]^2+2239550*c[5]* c[4]^3-361200*c[4]^3+81144)/(-291*c[4]-291*c[5]+224*c[5]^2-940*c[5]*c[ 4]^2-940*c[5]^2*c[4]+1135*c[5]*c[4]+224*c[4]^2+840*c[5]^2*c[4]^2+84)/( 1521*c[5]*c[4]-1092*c[4]-1092*c[5]+784), b[4] = 1/60*(20065*c[5]-5502) /c[4]/(-34411*c[4]^3+12675*c[4]^4+30808*c[4]^2-9072*c[4]-12675*c[5]*c[ 4]^3+34411*c[5]*c[4]^2-30808*c[5]*c[4]+9072*c[5]), a[6,1] = 7/2313441* (-2129400*c[5]^4*c[4]^3-11802960*c[5]^3*c[4]^4+1703520*c[4]^5*c[5]^2-2 44608*c[5]^3+131712*c[5]^2-1223040*c[4]^5*c[5]+771456*c[5]*c[4]^2-1956 057*c[5]^2*c[4]^3-520104*c[4]^4*c[5]^2+611520*c[5]^4*c[4]^2-7812441*c[ 5]^3*c[4]^2+2063880*c[5]^3*c[4]-790272*c[5]^2*c[4]-153664*c[5]*c[4]+23 72760*c[4]^5*c[5]^3+2356726*c[4]^4*c[5]+2372760*c[5]^4*c[4]^4+43904*c[ 4]^2+14090856*c[5]^3*c[4]^3+85176*c[4]^4+2113272*c[5]^2*c[4]^2-1880592 *c[5]*c[4]^3-122304*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[8,5] = -1/2*(37234608125*c[5]^5*c[4]^3-117762583610*c[5]^4*c[4 ]^3-122852190970*c[5]^3*c[4]^4-18953223250*c[5]^5*c[4]^2-36419880900*c [5]^4*c[4]^5+81793152*c[4]+11463270000*c[5]^5*c[4]^5-20046298100*c[4]^ 5*c[5]^2-163586304*c[5]-1772606043*c[5]^3+902780552*c[5]^2-33956747500 *c[5]^5*c[4]^4+3276806400*c[4]^5*c[5]-4113681649*c[5]*c[4]^2-623942705 88*c[5]^2*c[4]^3+58340590505*c[4]^4*c[5]^2+59889386950*c[5]^4*c[4]^2-6 7712912801*c[5]^3*c[4]^2+17032468804*c[5]^3*c[4]-8033222711*c[5]^2*c[4 ]+1125535640*c[5]*c[4]+41726102600*c[4]^5*c[5]^3-9053152320*c[4]^4*c[5 ]+107678333275*c[5]^4*c[4]^4-339989592*c[4]^2+133579138410*c[5]^3*c[4] ^3+1491232795*c[5]^4+4669913625*c[5]^5*c[4]-156832990*c[4]^4-148764885 10*c[5]^4*c[4]+31230420342*c[5]^2*c[4]^2+8928078233*c[5]*c[4]^3+415029 430*c[4]^3-457821000*c[5]^5)/c[5]/(-22105482000*c[5]^5*c[4]^3-34938637 50*c[5]^7*c[4]^3+7551148650*c[5]^4*c[4]^3-18022054095*c[5]^3*c[4]^4+10 231544070*c[5]^5*c[4]^2-6942153400*c[5]^4*c[4]^5+3493863750*c[5]^5*c[4 ]^5-6831639750*c[5]^6*c[4]^2-680400000*c[4]^5*c[5]^2+18125856*c[5]^3-1 1432216250*c[5]^5*c[4]^4+36251712*c[5]*c[4]^2-574752138*c[5]^2*c[4]^3+ 5395211980*c[4]^4*c[5]^2-4805546800*c[5]^4*c[4]^2+502525620*c[5]^3*c[4 ]^2+46141920*c[5]^3*c[4]-36251712*c[5]^2*c[4]+4130896800*c[4]^5*c[5]^3 -821329680*c[4]^4*c[5]+24806224950*c[5]^4*c[4]^4+1139555560*c[5]^3*c[4 ]^3-111468528*c[5]^4-1634158240*c[5]^5*c[4]+68040000*c[4]^4+1742374465 0*c[5]^6*c[4]^3+705211548*c[5]^4*c[4]-86857056*c[5]^2*c[4]^2+84143664* c[5]*c[4]^3-18125856*c[4]^3-95062500*c[5]^6+188344650*c[5]^5+919761375 *c[5]^6*c[4]+950625000*c[5]^7*c[4]^2), `b*`[7] = -171640625/1057536*(- 10530*c[4]+1512-10530*c[5]-12880*c[5]^3+21507*c[5]^2-155109*c[5]*c[4]^ 2-214590*c[5]^2*c[4]^3-214590*c[5]^3*c[4]^2+95210*c[5]^3*c[4]-155109*c [5]^2*c[4]+74082*c[5]*c[4]+21507*c[4]^2+142800*c[5]^3*c[4]^3+337180*c[ 5]^2*c[4]^2+95210*c[5]*c[4]^3-12880*c[4]^3)/(-39393216*c[4]-39393216*c [5]-23587200*c[5]^3+54156924*c[5]^2+8817984-272517015*c[5]*c[4]^2-1877 39500*c[5]^2*c[4]^3-187739500*c[5]^3*c[4]^2+122642000*c[5]^3*c[4]-2725 17015*c[5]^2*c[4]+189304860*c[5]*c[4]+54156924*c[4]^2+88725000*c[5]^3* c[4]^3+406028215*c[5]^2*c[4]^2+122642000*c[5]*c[4]^3-23587200*c[4]^3), a[6,4] = -14/2313441*(-851760*c[5]^4*c[4]^3-2555280*c[5]^3*c[4]^4+255 5280*c[4]^5*c[5]^2-244608*c[5]^3+131712*c[5]^2-1223040*c[4]^5*c[5]+710 304*c[5]*c[4]^2+918867*c[5]^2*c[4]^3-5110560*c[4]^4*c[5]^2+611520*c[5] ^4*c[4]^2-6808581*c[5]^3*c[4]^2+1936116*c[5]^3*c[4]-729120*c[5]^2*c[4] -153664*c[5]*c[4]+3105058*c[4]^4*c[5]+43904*c[4]^2+8222760*c[5]^3*c[4] ^3+1788948*c[5]^2*c[4]^2-2290092*c[5]*c[4]^3-61152*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[ 8,7] = -9440234375/88128*(85*c[5]*c[4]-23*c[5]-23*c[4]+6)*(c[4]-1)*(-1 +c[5])/(325*c[5]-324)/(325*c[4]-324)/(27565*c[5]*c[4]-7500*c[5]-7500*c [4]+1998), `b*`[4] = -1/60*(2161446*c[4]+4471362*c[5]+6156080*c[5]^3-9 661881*c[5]^2+13119290*c[5]*c[4]^2+23062200*c[5]^3*c[4]^2-25824310*c[5 ]^3*c[4]+38159065*c[5]^2*c[4]-16422441*c[5]*c[4]-1661520*c[4]^2-320319 10*c[5]^2*c[4]^2-624456)/(121584*c[4]+94284*c[5]-72576*c[5]^2+673435*c [5]*c[4]^2+273000*c[5]^2*c[4]^3+377360*c[5]^2*c[4]-462315*c[5]*c[4]-16 7151*c[4]^2-577660*c[5]^2*c[4]^2-305500*c[5]*c[4]^3+72800*c[4]^3-27216 )/(39*c[4]-28)/(c[4]-c[5])/c[4], b[8] = 1/660*(27565*c[5]*c[4]-7500*c[ 5]-7500*c[4]+1998)/(c[4]-1)/(-1+c[5]), a[8,1] = 1/9072*(4225061150*c[5 ]^5*c[4]^3-39882153495*c[5]^4*c[4]^3+2666420100*c[5]^3*c[4]^6-28368124 935*c[5]^3*c[4]^4-6097627450*c[4]^6*c[5]^2-819201600*c[5]^5*c[4]^2+163 8403200*c[4]^6*c[5]-24992665050*c[5]^4*c[4]^5+3613100950*c[5]^5*c[4]^5 +14367374565*c[4]^5*c[5]^2-425413296*c[5]^3+122689728*c[5]^2-656388630 0*c[5]^5*c[4]^4-4584662160*c[4]^5*c[5]+873460356*c[5]*c[4]^2+119120247 1*c[5]^2*c[4]^3-10382642262*c[4]^4*c[5]^2+16286811960*c[5]^4*c[4]^2-16 015879620*c[5]^3*c[4]^2+3929316323*c[5]^3*c[4]-693426516*c[5]^2*c[4]-1 43138016*c[5]*c[4]+6170310550*c[4]^5*c[5]^3+4816503252*c[4]^4*c[5]+472 60687150*c[5]^4*c[4]^4+40896576*c[4]^2+32549100858*c[5]^3*c[4]^3-78416 495*c[4]^5+327680640*c[5]^4+207514715*c[4]^4-3420858815*c[5]^4*c[4]+13 59572292*c[5]^2*c[4]^2-2588088096*c[5]*c[4]^3-169994796*c[4]^3+3613100 950*c[5]^4*c[4]^6)/c[5]/c[4]^2/(275650*c[5]^4*c[4]^3+275650*c[5]^3*c[4 ]^4+7500*c[5]^3-1998*c[5]^2+11988*c[5]*c[4]^2+410370*c[5]^2*c[4]^3-750 00*c[4]^4*c[5]^2-75000*c[5]^4*c[4]^2+410370*c[5]^3*c[4]^2-72565*c[5]^3 *c[4]+11988*c[5]^2*c[4]+1998*c[5]*c[4]-1998*c[4]^2-976950*c[5]^3*c[4]^ 3-122375*c[5]^2*c[4]^2-72565*c[5]*c[4]^3+7500*c[4]^3), a[9,7] = -55783 203125/1057536*(85*c[5]*c[4]-23*c[5]-23*c[4]+6)/(105625*c[5]*c[4]-1053 00*c[4]-105300*c[5]+104976), `b*`[5] = 1/60*(4471362*c[4]+2161446*c[5] -1661520*c[5]^2+38159065*c[5]*c[4]^2+23062200*c[5]^2*c[4]^3+13119290*c [5]^2*c[4]-16422441*c[5]*c[4]-9661881*c[4]^2-32031910*c[5]^2*c[4]^2-25 824310*c[5]*c[4]^3+6156080*c[4]^3-624456)/(94284*c[4]+121584*c[5]+7280 0*c[5]^3-167151*c[5]^2+377360*c[5]*c[4]^2+273000*c[5]^3*c[4]^2-305500* c[5]^3*c[4]+673435*c[5]^2*c[4]-462315*c[5]*c[4]-72576*c[4]^2-577660*c[ 5]^2*c[4]^2-27216)/(39*c[5]-28)/(c[4]-c[5])/c[5], `b*`[1] = 1/544320*( -4471362*c[4]-4471362*c[5]-6156080*c[5]^3+9661881*c[5]^2-86704431*c[5] *c[4]^2-159971010*c[5]^2*c[4]^3-159971010*c[5]^3*c[4]^2+58451590*c[5]^ 3*c[4]-86704431*c[5]^2*c[4]+36659418*c[5]*c[4]+9661881*c[4]^2+12232920 0*c[5]^3*c[4]^3+223180700*c[5]^2*c[4]^2+58451590*c[5]*c[4]^3-6156080*c [4]^3+624456)/(-291*c[4]-291*c[5]+224*c[5]^2-940*c[5]*c[4]^2-940*c[5]^ 2*c[4]+1135*c[5]*c[4]+224*c[4]^2+840*c[5]^2*c[4]^2+84)/c[5]/c[4], a[9, 1] = 1/544320*(118175*c[5]*c[4]-20065*c[5]-20065*c[4]+5502)/c[5]/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 98 "A lengthy computation gives an expression for the squar e 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(e rrterms6_8) do\n print(ct);\n sm := sm+(simplify(subs(eB,errterms6 _8[ct])))^2;\nend do:\nsm := simplify(sm):\nprin_err_norm_sqrd := unap ply(%,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 4493 "prin_ err_norm_sqrd := (u,v,w)->1/524688418800000000*(99302196172200660*w^5* v^4+2075951388328200*u*v^4*w-4973061305733300*u*v^3*w^2-85817558855776 500*w^5*u*v^3-2209904484480000*v^7*w*u+88835619650400000*v^8*w^2*u+120 4492725217845000*u^2*v^6*w^4+423336106227825000*u^2*v^7*w^3-6931074177 12150000*u^2*v^7*w^4-57751147303299900*v^5*w^2*u-1620383634103500*v^3* u^2*w+359332232238000*v^2*u^2*w-23208433879050000*v^2*u^2*w^3-32340379 9269757500*v^4*u^2*w^3-81591626030953500*v^5*u^2*w^2-444539024294400*v *u^2*w^2+37304519825673450*v^4*u^2*w^2+2996330275071000*v*u^2*w^3-1340 7647819085000*w^6*v^4*u-173230154111763000*v^3*u^2*w^4-37044918691200* u^2*v*w+538598166826923000*v^4*u^2*w^4+2794486234531950*v^2*u^2*w^2+10 7928806193765900*v^3*u^2*w^3+30535175900385000*w^6*v^5*u-8484835053240 00*u^2*v^2*w^5+875027697075000000*w^4*v^8*u+152023854520125000*u^2*v^8 *w^4-52894788677640000*u^2*v^4*w^5-168408459969750000*u^2*v^6*w^5+3496 1747945947200*u^2*v^2*w^4+4030296495271200*u^2*v^4*w-4130187449018400* u^2*v*w^4+10322597716704000*u^2*v^3*w^5+624236903581599000*v^5*u^2*w^3 -11960430399716400*v^3*u^2*w^2-458090776318626000*w^4*v^5*u+6618505482 2250000*u^2*v^7*w^5+798544803420000*u^2*v^4*w^6-1051027019164710000*u^ 2*v^5*w^4+49130053351200*v^4*u+1713522316800000*w*v^8-9245801295841600 00*v^8*w^3+1183734108903812400*w^5*v^6+6465656321756*v^4-8119546232199 03000*w^3*v^6*u+416125318407975000*w^3*v^5*u+346181792704920000*w^5*v^ 6*u+42124479296109*v^6+126713267352000000*v^9*w^3*u+941701496801625000 *v^7*w^3*u+242016922335267000*w^5*v^4*u+32145617985756*w^4-14585413483 7460000*v^7*w^2*u+1327288846287600*w^4*v*u+84830809241493000*w^4*v^4*u +201877676648109*w^6-4845543086506500*u^2*v^5*w+117032266276240500*v^6 *w^2*u-66626714737800000*u^2*v^7*w^2-161076058293096*w^5-4983636846557 700*v^5*w*u+137472513129315000*u^2*v^5*w^5-4796822721600000*u^2*v^5*w^ 6-4519569213880000*w*v^7-92125827785600000*v^9*w^2+839870067028804000* v^8*w^5+20536578944000000*v^10*w^2-6956307852224700*w^7*v^4-1314063513 28000000*v^10*w^3-76927593762180*w^7*v^2-3717719938361268*w^6*v+558311 318322800000*v^9*w^3+10231584359745000*v^7*w^7+375804860798760420*w^6* v^4+783119689628700280*w^3*v^7-362743653568023000*v^7*w^6-280551450639 6000*w^8*v^5-39111296598423000*w^7*v^6-661756783376307600*w^6*v^5-7837 37767159200000*v^9*w^4+77766399425872500*v^8*w^6-435303575046664700*w^ 4*v^7-190396156356000000*v^9*w^5-30536411232600000*w^6*v^6*u-895918835 7000000*w^5*v^8*u+1110114790161930000*w^4*v^6*u+210205823534000000*v^1 0*w^4-141499536532906180*v^7*w^2-160690557234663580*w^4*v^6+1583121719 27280000*v^8*w^2+7654529200500000*w^6*v^7*u-99365742327600000*w^5*v^7* u-556842491937900000*v^8*w^3*u+7203577870125000*u^2*v^6*w^6-9507286004 1912*w^3*v+1042018363826904*w^3*v^2-3400295333960112*w^4*v^2+404900464 291224*v^4*w-135489156263045481*w^4*v^4-3410821743799450*w^3*v^3+10122 3401025601826*v^5*w^3-202698472693500000*v^9*w^4*u+1829621999520000*u^ 2*v^6*w+19995192384321600*u*v^4*w^2-41952217865912*w*v^3+1029105503892 68*w^2*v^2-71822302665576*w^4*v+2259666569390772*w^5*v-98260106702400* w^4*u-134213493564739800*w^3*u*v^4+694961188964327900*v^6*w^6-31239005 721690832*v^5*w^2+1022668463852280*w^7*v^3+225386375218200*u^2*w^4-329 67324117096*v^5-1706060944518028*v^5*w+5115792179872500*w^8*v^6+804917 20447652029*v^6*w^2-7633108299815796*w^3*v^4+121974799968000*w^5*u+101 3306034399050900*v^8*w^4+8180543204319088*v^4*w^2-1333489838733096*w^2 *v^3-381562258015502380*w^3*v^6-182750293454400*u^2*w^3-14273381851384 91400*w^5*v^7+14322754532569026*w^5*v^3+384637968810900*w^8*v^4+197881 86902400*w^3*u+29753722467421629*w^6*v^2+25454072003028600*w^7*v^5+370 44918691200*u^2*w^2+32923402940006604*w^4*v^3+3919826025878732*v^6*w+2 73250047342984660*w^4*v^5-526296134345928220*w^5*v^5-11457079065528432 *w^5*v^2-134543360426302380*w^6*v^3+56346593804550*u^2*v^4+92612296728 00*u^2*v^2-60987399984000*v^5*u-45687573363600*u^2*v^3-9894093451200*v ^3*u+6264506105072100*w^4*u*v^3-5577896023200*w^3*u*v-4020186677514300 00*w^5*v^5*u-1419174425465865000*w^4*v^7*u+2801659802760000*w^6*u*v^3- 721761178612320000*u^2*v^6*w^3+107289356276025000*u^2*v^6*w^2+58474750 89771000*v^6*w*u-471269326284000*u*v^3*w-95034950514000000*u^2*v^8*w^3 +14852347272000000*u^2*v^8*w^2+817548243876000*u*v^2*w^2+4820473969920 0*u*v^2*w-66727199044800*u*v*w^2-19803129696000000*v^9*w^2*u-661712597 0142300*w^4*u*v^2+18373145951376000*w^5*u*v^2-229591514880000*w^6*u*v^ 2+26436616815240000*u*v^3*w^3-2618627522526000*u*v^2*w^3-2239675007724 000*w^5*u*v)/(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 53 "Preliminary investigation suggested that the values " }{XPPEDIT 18 0 "c[2]=10/107" "6#/&%\"cG6#\"\"#*&\"#5\"\"\"\"$2\"!\"\" " }{TEXT -1 3 ", " }{XPPEDIT 18 0 "c[4]=163/755" "6#/&%\"cG6#\"\"%*& \"$j\"\"\"\"\"$b(!\"\"" }{TEXT -1 7 " and " }{XPPEDIT 18 0 "c[5]=233 /411" "6#/&%\"cG6#\"\"&*&\"$L#\"\"\"\"$6%!\"\"" }{TEXT -1 89 " give a value for the (square of the) principal error norm that is close to t he minimum." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 " " {MPLTEXT 1 0 59 "prin_err_norm_sqrd(10/107,163/755,233/411):\nevalf( sqrt(%));" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#$\"0nZ0'4.j6!#>" }}} {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 124 "Using a \+ one dimensional minimization procedure and cycling around the nodes gi ves very slow convergence towards the minimum." }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 494 "Digits := 30:\nc_2 := 10/107: c_4 := 163/755: c _5 := 233/411:\nfor ct to 20000 do\n c_2 := findmin(prin_err_norm_sq rd(c2,c_4,c_5),c2=\{0.05,c_2,0.15\},convergence=location)[1];\n c_4 \+ := findmin(prin_err_norm_sqrd(c_2,c4,c_5),c4=\{0.2,c_4,0.23\},converge nce=location)[1];\n mn := findmin(prin_err_norm_sqrd(c_2,c_4,c5),c5= \{0.5,c_5,0.65\},convergence=location); \n c_5 := mn[1]:\n if `mod `(ct,500)=0 then\n print(c[2]=c_2,c[4]=c_4,c[5]=c_5);\n prin t(mn[2]);\n end if;\nend do:\nDigits := 10:" }}{PARA 11 "" 1 "" {XPPMATH 20 "6%/&%\"cG6#\"\"#$\"?DeM80<`*Gpi^-!)G*!#J/&F%6#\"\"%$\"?[7 dB1aSy!*y&G_*e@!#I/&F%6#\"\"&$\"?Fe(>1p:*z&[,,\"4pcF1" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#$\"?;2)['4l\\&*[`)zACN\"!#R" }}{PARA 11 "" 1 "" {XPPMATH 20 "6%/&%\"cG6#\"\"#$\"?w(>;0g=9>QL!Q)zG*!#J/&F%6#\"\"%$\"?c% [<^/6_Y/_V[*e@!#I/&F%6#\"\"&$\"?NqijUn)3@+B#Q3pcF1" }}{PARA 11 "" 1 " " {XPPMATH 20 "6#$\"?Vlr1VmfIcG[FU_8!#R" }}{PARA 11 "" 1 "" {XPPMATH 20 "6%/&%\"cG6#\"\"#$\"?wh&e><#)Q^hvVozG*!#J/&F%6#\"\"%$\"?Xta))>'3L8) Qt_%*e@!#I/&F%6#\"\"&$\"?tXjj8qby,>?z2pcF1" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#$\"?2S!f=WR6!pS9FU_8!#R" }}{PARA 260 "" 0 "" {TEXT -1 1 ":" }}{PARA 260 "" 0 "" {TEXT -1 1 ":" }}{PARA 11 "" 1 "" {XPPMATH 20 "6%/&%\"cG6#\"\"#$\"?2J\"4)z8T?Hq*)z*yG*!#J/&F%6#\"\"%$\"?Q[*fVQ0Eo fvxI*e@!#I/&F%6#\"\"&$\"?'ziWZW$\\xYJh30pcF1" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#$\"?XsNc\\JNAhIWEU_8!#R" }}{PARA 11 "" 1 "" {XPPMATH 20 "6%/&%\"cG6#\"\"#$\"?Ov;U(=)*><%*p(z*yG*!#J/&F%6#\"\"%$\"?k8epB$p%y Y%\\xI*e@!#I/&F%6#\"\"&$\"?Pq&[T'GvtLVc30pcF1" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#$\"?cNg*3$R=*41VkACN\"!#R" }}{PARA 11 "" 1 "" {XPPMATH 20 "6%/&%\"cG6#\"\"#$\"?4(obO$36>'fl'z*yG*!#J/&F%6#\"\"%$\"?ajmc.,8Awz s2$*e@!#I/&F%6#\"\"&$\"?IOkV5\"**ecDC&30pcF1" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#$\"?-v$Q " 0 "" {MPLTEXT 1 0 530 "Digits := 30:\nc_2 : = .9287897966559619e-1: c_4 := .2158930772797622: c_5 := .566905085242 5566:\nfor ct to 100000 do\n c_2 := findmin(prin_err_norm_sqrd(c2,c_ 4,c_5),c2=\{0.05,c_2,0.15\},convergence=location)[1];\n c_4 := findm in(prin_err_norm_sqrd(c_2,c4,c_5),c4=\{0.2,c_4,0.23\},convergence=loca tion)[1];\n mn := findmin(prin_err_norm_sqrd(c_2,c_4,c5),c5=\{0.5,c_ 5,0.65\},convergence=location); \n c_5 := mn[1]:\n if `mod`(ct,100 0)=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#\"\"#$\"?D4.e07(33e4&z*yG*!#J/&F%6#\"\"%$\"?OT5q3yt&Q(e p2$*e@!#I/&F%6#\"\"&$\"?1xN'3br/n 0XLOI;k%HwI*e@!#I/&F%6#\"\"&$\"?#fz3M+#3*yOS$30pcF1" }}{PARA 11 "" 1 " " {XPPMATH 20 "6#$\"?p**eW7mD^gIWEU_8!#R" }}{PARA 11 "" 1 "" {XPPMATH 20 "6%/&%\"cG6#\"\"#$\"?i9IXWq&[S$o=z*yG*!#J/&F%6#\"\"%$\"?mgajLOI;k%H wI*e@!#I/&F%6#\"\"&$\"?4W-)Q+#3*yOS$30pcF1" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#$\"?=Yb!3bc701VkACN\"!#R" }}{PARA 11 "" 1 "" {XPPMATH 20 "6%/&%\"cG6#\"\"#$\"?.?V!4\"e&[S$o=z*yG*!#J/&F%6#\"\"%$\"?%\\$f'Rj. jTYHwI*e@!#I/&F%6#\"\"&$\"?#Hu0W+#3*yOS$30pcF1" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#$\"?e34PfkD^gIWEU_8!#R" }}}{PARA 0 "" 0 "" {TEXT -1 0 " " }}{PARA 0 "" 0 "" {TEXT -1 78 "The following graphs give a visual ch eck that we have found a (local) minimum." }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 379 "c_2 := .92878979187e-1: pp := .13524226443e-9:\np1 : = evalf[30](plot(prin_err_norm_sqrd(c[2],0.215893076295,.56690508340), c[2]=0.08..0.106,\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=[HE LVETICA,9],view=[0.08..0.106,1.35e-10..1.3767e-10]);" }}{PARA 13 "" 1 "" {GLPLOT2D 367 347 347 {PLOTDATA 2 "6*-%'CURVESG6$7S7$$\"\")!\"#$\"? Hn<0>^1$p=+=;q.jgObP=O\"F-7$$\"?nmmmmmmmm;y[NW&)F1$\"?&4p(GX OD5PC0Obg8F-7$$\"?LLLLLLLLL3:'oTf)F1$\"?b_[*Ge#Q_fqI1]f8F-7$$\"?++++++ +++]%3Z-l)F1$\"?Qd/)e!)z97\"3mDSe8F-7$$\"?+++++++++]3w)F1$\"?Y$4-K2aY!*y!p%3lN\"F-7$$\"?LL LLLLLL$eM`'45))F1$\"?I8ebHe7OqE4-yb8F-7$$\"?nmmmmmmmm;O9po))F1$\"?hc5[ ?=WYS8)>2]N\"F-7$$\"?nmmmmmmmmm/!H$=*)F1$\"?2E?O%y4&H&o\"p8Va8F-7$$\"? ++++++++]7(Hpg(*)F1$\"?0ikU<\\s;v>GF&QN\"F-7$$\"?nmmmmmmmmmR#zr-*F1$\" ?o#*zwo`0mF$QNAMN\"F-7$$\"?++++++++]7Z\\D$3*F1$\"?mULC?B36Zni&QIN\"F-7 $$\"?++++++++]P\"G_m8*F1$\"?3p(fl]5XxAY2fFN\"F-7$$\"?nmmmmmmmm\"zzmB>* F1$\"?$H<-A\"[n%*o%['ob_8F-7$$\"?nmmmmmmm;H;,`V#*F1$\"?l=].&*e%=`8F-7$$\"?+++++++++Do,)*p&*F1$\"?gO ,Z*=3**ow;!Hf`8F-7$$\"?++++++++]7V?oA'*F1$\"?[enk?Mt)*HsK52a8F-7$$\"?+ ++++++++v*[)>\"o*F1$\"?&HUNu$=#>p4;j(pa8F-7$$\"?nmmmmmmmmm>\"yPt*F1$\" ?(z%yqiHD[4-plMb8F-7$$\"?+++++++++](4=**y*F1$\"?)f7QTG&*o_'z8#HhN\"F-7 $$\"?LLLLLLLL$e9E*yS)*F1$\"?O5Pl\"Ghtm9!p%=pN\"F-7$$\"?+++++++++]!)[S' *)*F1$\"?$Qij(4N([HZUVoyN\"F-7$$\"?nmmmmmmm;zV[t[**F1$\"?fX;Wsz2:NIb`% )e8F-7$$\"?++++++++DOtMM+5!#I$\"?#p.D%QVkN2&Gi_*f8F-7$$\"?nmmmmmmmmT(Q \"p05F`w$\"?V6&)*)o]qCH4l-7h8F-7$$\"?++++++++D\"y:!H65F`w$\"?J'=n')*en ,kp:GVi8F-7$$\"?LLLLLLLLL8RCo;5F`w$\"?EK&=i8_k'4!G7%yj8F-7$$\"?LLLLLLL L$3c#o>A5F`w$\"?3JF\\&z^m2)z$[a_O\"F-7$$\"?nmmmmmmm\"z,blw-\"F`w$\"?0* ej6S[,O/ \"F`w$\"?WzSW1&)H()y^e#4=P\"F-7$$\"?++++++++Dr\\Q4\\5F`w$\"?=Nhw[ZTu^2 g%3PP\"F-7$$\"?++++++++v[Y2Na5F`w$\"?t<0GfU^=kNX$4cP\"F-7$$\"$1\"!\"$$ \"?,1k#Hye'4hb>Fux8F--%&COLORG6&%$RGBG$\"\"&!\"\"$\"\"!Fc[l$\"\"*Fa[l- F$6&7#7$$\"3%*****p=z*yG*!#>$\"3)*****HWEU_8!#F-%'COLOURG6&F^[lFc[lFc[ lFc[l-%'SYMBOLG6$%'CIRCLEG\"#7-%&STYLEG6#%&POINTG-F$6&Fh[l-Fa\\l6&F^[l $\"*++++\"!\")Fb[lFb[l-Fd\\l6$Ff\\l\"#5Fh\\l-F$6&Fh[lF^]l-Fd\\l6$%(DIA MONDGFe]lFh\\l-F$6&Fh[lF^]l-Fd\\l6$%&CROSSGFe]lFh\\l-%%FONTG6$%*HELVET ICAGFe[l-%+AXESLABELSG6%Q%c[2]6\"Q!Fh^l-Fa^l6#%(DEFAULTG-%%VIEWG6$;F(F fz;$\"$N\"!#7$\"&nP\"!#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 0 389 "c_4 := 0.215893076295: pp := .13524226443e-9:\np1 := evalf[30](pl ot(prin_err_norm_sqrd(.92878979187e-1,c[4],.56690508340),c[4]=0.21587. .0.21592,\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],font=[HELVETICA ,9],view=[0.21587..0.21592,1.34e-10..1.397e-10]);" }}{PARA 13 "" 1 "" {GLPLOT2D 400 369 369 {PLOTDATA 2 "6*-%'CURVESG6$7S7$$\"&(e@!\"&$\"?>T U)zR>w%R?#3/qR\"!#R7$$\"?LLLLLLL$3x&)*3re@!#I$\"?V![leOYjEohg3HR\"F-7$ $\"?nmmmmm;H2P\"Q?(e@F1$\"?'[7N/!z!)GXTCW]*Q\"F-7$$\"?LLLLLLLeRwX5te@F 1$\"?yj$)*Hg>Cgj_K`eQ\"F-7$$\"?LLLLLLL3x%3yT(e@F1$\"?&*4t_7i;N+.kxO#Q \"F-7$$\"?nmmmmm;z%4\\Y_(e@F1$\"?$)*[)G!zFhD90.v PTt8F-7$$\"?LLLLLL$e*)>VB$ye@F1$\"?!o4i$>ViPyQ\\#*pq8F-7$$\"?++++++]7` l2Qze@F1$\"?;V\\8n.q*RuJYz\"o8F-7$$\"?nmmmmmm;/j$o/)e@F1$\"?^teWrX!*f+ #oc\"yl8F-7$$\"?LLLLLLL3_>jU\")e@F1$\"?W-t:N$=Z$eo`E$QO\"F-7$$\"?+++++ ++]i^Z]#)e@F1$\"?9FP\"z9Wg]s)*p@=O\"F-7$$\"?+++++++](=h(e$)e@F1$\"?003 CBTn/s*))z(**f8F-7$$\"?+++++++]P[6j%)e@F1$\"?nkx**[\"olvz&GbUe8F-7$$\" ?LLLLLL$e*[z(yb)e@F1$\"?\\u?Iyg#=c,^ebrN\"F-7$$\"?nmmmmmm;a/cq')e@F1$ \"?\"RTrx9)R3>'oCTeN\"F-7$$\"?nmmmmmmm;t,m()e@F1$\"?_gS^lb@)G)>$R%*[N \"F-7$$\"?++++++]iSj0x))e@F1$\"?tTO?Ydo/hBDa)RN\"F-7$$\"?nmmmmmmm\"pW` (*)e@F1$\"?`0Hh.!3G3XEr`LN\"F-7$$\"?++++++]i!f#=$3*e@F1$\"?ds$eD:gwwS` hZGN\"F-7$$\"?++++++](=xpe=*e@F1$\"?5WBXlCTZaDdxa_8F-7$$\"?nmmmmmm\"H2 8IH*e@F1$\"?B%*o^$y)Hr#))yWCCN\"F-7$$\"?nmmmmm;zpSS\"R*e@F1$\"?a!R$>)4 q!*3+l\">[_8F-7$$\"?LLLLLLL3_?`(\\*e@F1$\"?%4CF5HrV],GLFFN\"F-7$$\"?LL LLLL$e*)>pxg*e@F1$\"?b$)Rwi\\#eX760%=`8F-7$$\"?++++++]Pf4t.(*e@F1$\"?! )yX]^\">W<$4J#\\PN\"F-7$$\"?LLLLLLLe*Gst!)*e@F1$\"?\"zXQbN\"F-7$$\"?+++++++DJE>>+f@F1 $\"?G/ab3>#)y'= !e8F-7$$\"?+++++++v=S2L-f@F1$\"?8P2Fqe5pcaf\"y'f8F-7$$\"?nmmmmmmm\"p)= M.f@F1$\"?Pj8(p$GN_JM!e`8O\"F-7$$\"?+++++++](=]@W!f@F1$\"?GABt31aF&4f> NLO\"F-7$$\"?LLLLLL$e*[$z*R0f@F1$\"?^CJ(\\YdI2)=SHIl8F-7$$\"?+++++++]i C$pk!f@F1$\"?y3nhOJAPz2b;kn8F-7$$\"?nmmmmm;H2qcZ2f@F1$\"?\"y$*z(*>kmLk p>@+P\"F-7$$\"?++++++]7.\"fF&3f@F1$\"?z#3@)G(y#=LqHUps8F-7$$\"?nmmmmmm ;/Ogb4f@F1$\"?C!pLkPJG^rmM\"\\v8F-7$$\"?++++++]ilAFj5f@F1$\"?)e8Dp%4iD ?QUXhy8F-7$$\"?LLLLLLLL$)*pp;\"f@F1$\"?i!>EZ(f[I+#e@6=Q\"F-7$$\"?LLLLL LL3xe,t7f@F1$\"?zM8hsM%Q\\D`&>F&Q\"F-7$$\"?nmmmmm;HdO=y8f@F1$\"?KLBNF3 SU\"HUo&*))Q\"F-7$$\"?++++++++D>#[Z\"f@F1$\"?cHgwR`4*)4OkRR#R\"F-7$$\" ?nmmmmmmT&G!e&e\"f@F1$\"?gd;3$pU!yz%p--mR\"F-7$$\"?LLLLLLLL$)Qk%o\"f@F 1$\"??1Br@6Dsk[8da+9F-7$$\"?++++++]iSjE!z\"f@F1$\"?QYYgankd:,')y$\\S\" F-7$$\"?++++++]P40O\"*=f@F1$\"?(yR]3Am&ROC&3B$49F-7$$\"&#f@F*$\"?vS34` KF&)yr " 0 "" {MPLTEXT 1 0 383 "c_5 := .56690508340: pp := .13524226443e-9:\np1 := evalf[30](plot (prin_err_norm_sqrd(.92878979187e-1,0.215893076295,c[5]),c[5]=0.56686. .0.566955,\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],c olor=[black,green$3]):\nplots[display]([p1,p2],font=[HELVETICA,9],view =[0.56686..0.566955,1.34e-10..1.402e-10]);" }}{PARA 13 "" 1 "" {GLPLOT2D 389 359 359 {PLOTDATA 2 "6*-%'CURVESG6$7S7$$\"&'oc!\"&$\"?h6 #Q-rNV!*p>$R%>S\"!#R7$$\"?LLLLLLLekH22ioc!#I$\"?m4B')*\\A%\\#f:!y[(R\" F-7$$\"?nmmmmmT&Q/YsQ'ocF1$\"?-1l\">3$\\=**4l9-<$)Qz pl8F-7$$\"?+++++++v3G!fP)ocF1$\"?;kSK'fLV7aSjpMO\"F-7$$\"?+++++++Dcik \"e)ocF1$\"?Rct,HvN,VJ0&Q9O\"F-7$$\"?+++++++D\">=*z()ocF1$\"??nEq6\"*[ )HyJhw'f8F-7$$\"?LLLLLL3-.\"o*f*)ocF1$\"?H&G&)H^1*)*=2]FCe8F-7$$\"?nmm mmmm\"H'[1u\"*ocF1$\"?g'>FF%Rc))Q+uLuc8F-7$$\"?nmmmmmmm,HVb$*ocF1$\"? \")Qa1D:5!=yr&ykb8F-7$$\"?++++++v=ZqSm&*ocF1$\"?BM%f#)zS07Ey\"\\da8F-7 $$\"?nmmmmmm;9\\:`(*ocF1$\"?kbh'*pCsv]_!y0QN\"F-7$$\"?++++++v=Ap/e**oc F1$\"?K^4y**\\Eax8$)o:`8F-7$$\"?++++++DcmD:`,pcF1$\"?h))p0^&)\\N\"F-7$$\"?+ +++++]P**fYOpcF1$\"?.ToP? ^1#f=_l/tN\"F-7$$\"?++++++]iN1%G9#pcF1$\"?%pZxG#*)[@$\\&)*H))e8F-7$$\" ?nmmmmmm;9&e\\L#pcF1$\"?Z9OHp;*e_jX/*[g8F-7$$\"?+++++++Dc`3SDpcF1$\"?D u%)*pjjhy>y&**Ri8F-7$$\"?LLLLLL3-j2'fs#pcF1$\"?8d$*[D<(H3zAK1VO\"F-7$$ \"?+++++++vy;cmr'QQ\"F-7$$\"?nmmmmmT&)[*[&=VpcF 1$\"?Jj!zS6rIZM$ebV(Q\"F-7$$\"?+++++++]d;;-XpcF1$\"?f&)Q!p\"\\+>oNeF)3 R\"F-7$$\"?nmmmmm;HUDg7ZpcF1$\"?uBbI(pUC%)eIjJ]R\"F-7$$\"?LLLLLLLLyL#3 !\\pcF1$\"?@*RPh1a2QmyY@*)R\"F-7$$\"?++++++v=Zg],^pcF1$\"?/!evn3cL`]$> \\D.9F-7$$\"?++++++D\"y'\\e$H&pcF1$\"?-g1fyGh[p*HK#e29F-7$$\"'bpc!\"'$ \"??uk2:6-rZ>P&GCT\"F--%&COLORG6&%$RGBG$\"\"'!\"\"$\"\"#F`[lFa[l-F$6&7 #7$$\"35+++M30pc!#=$\"3)*****HWEU_8!#F-%'COLOURG6&F][l\"\"!F`\\lF`\\l- %'SYMBOLG6$%'CIRCLEG\"#7-%&STYLEG6#%&POINTG-F$6&Fe[l-F^\\l6&F][l$F`\\l F`\\l$\"*++++\"!\")F^]l-Fb\\l6$Fd\\l\"#5Ff\\l-F$6&Fe[lF\\]l-Fb\\l6$%(D 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CB8853EFACF461CC3D9C03EF8CEC72CCCA74D3EF6DDD16FC454763EF4FFB1F16800B53 EF338D1497E6D4E3EF18F141C23D2DB3EF00A4A8CC4649B3EED6922C4BC352C3EEB34B 3669AD8B43EE9931C8238F7DD3EE89E89D25ECC253EE86845E87DF6C33EE8F23B22B4B 5ED3F0D10FADD6BE9483F0BDDD6E1A62ADD3F0AAD0D918DF3B63F097EAFBCFB47D53F0 852D1F1227E823F07298D51817D753F060300AD0676A33F04DF51E2A9786C3F03BEAFB 642883A3F02A15433F557203F018787E3F04FBF3F0071A61137937F3EFEC04532A3845 93EFCA723C52A23573EFA9967A4564EEC3EF899060E89072D3EF6A893F78C6BA33EF4C B803B1023B73EF30661FBE2CCA33EF15F5CAA6244B13EEFBD22D6E86AB23EED1CF7DAB 9CB0A3EEAF7CE1438F8A63EE969CF62BE86AC3EE88CA70C7A83723F0F6BC2FD423BF73 F0E33513CDAD8E03F0CFD2595D1D0D73F0BC94B74F459AC3F0A97D0E262DA463F0968C 70CE662573F083C42F9D561083F07125E643941593F05EB38D9E63B2F3F04C6F92FCCB D673F03A5CF633806543F0287F71945AA2B3F016DBAEE4CD4A63F005778ED225C063EF E8B513A5CA6373EFC71C6A5D63A373EFA63DF5427C4F33EF863A2EF2C6DE83EF673BFD A464D603EF497C82A1FFD893EF2D4836DBAF5713EF13056404083623EEF676A630F421 33EECD2F808F718C03EEABD993E276B72-%&COLORG6&%$RGBG$\"\"\"\"\"!$\"\"&! \"\"$\"\"#F>-%%FONTG6$%*HELVETICAG\"\"*-%+AXESLABELSG6%%%c[4]G%%c[5]G% !G-%+LIGHTMODELG6#%(LIGHT_4G-%*AXESSTYLEG6#%$BOXG-%%VIEWG6%F&F,;$F;F;$ \"#v!\"'-%+PROJECTIONG6%$!#vF;$\"#SF;F:" 1 2 0 1 10 0 2 1 6 2 2 1.000000 40.000000 -75.000000 1 0 "Curve 1" }}}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 36 "#------------------------------ -----" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 166 "nds := [c[2]=.9287 897918683405e-1,c[4]=.2158930762946416,c[5]=.5669050834036789]:\nevalf [10](%);\nfor dgt from 6 by -1 to 4 do\n map(convert,nds,rational,dg t);\nend do;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#7%/&%\"cG6#\"\"#$\"+>z *yG*!#6/&F&6#\"\"%$\"+j2$*e@!#5/&F&6#\"\"&$\"+M30pcF2" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#7%/&%\"cG6#\"\"##\"#I\"$B$/&F&6#\"\"%#\"$W\"\"$n'/ &F&6#\"\"&#\"$h\"\"$%G" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#7%/&%\"cG6# \"\"##\"#I\"$B$/&F&6#\"\"%#\"#>\"#))/&F&6#\"\"&#\"#s\"$F\"" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#7%/&%\"cG6#\"\"##\"#8\"$S\"/&F&6#\"\"%#\"#6 \"#^/&F&6#\"\"&#\"#<\"#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 0 104 "evalf[25](prin_er r_norm_sqrd(.9287897918683405e-1,.2158930762946416,.5669050834036789)) :\nevalf(sqrt(%));" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#$\"+xq$H;\"!#9" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 9 "Setting " }{XPPEDIT 18 0 "c[2] = 30/323" "6#/&%\"cG6#\"\"#*&\"#I\"\"\"\"$B$! \"\"" }{TEXT -1 3 ", " }{XPPEDIT 18 0 "c[4] = 144/667" "6#/&%\"cG6#\" \"%*&\"$W\"\"\"\"\"$n'!\"\"" }{TEXT -1 7 " and " }{XPPEDIT 18 0 "c[5 ] = 161/284" "6#/&%\"cG6#\"\"&*&\"$h\"\"\"\"\"$%G!\"\"" }{TEXT -1 57 " gives the following value for the principal error norm." }}{PARA 0 " " 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 59 "prin_ err_norm_sqrd(30/323,144/667,161/284):\nevalf(sqrt(%));" }}{PARA 11 " " 1 "" {XPPMATH 20 "6#$\"+G'pH;\"!#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 2132 "ee := \{c[2]=30/323,\nc[ 3]=96/667,\nc[4]=144/667,\nc[5]=161/284,\nc[6]=28/39,\nc[7]=324/325,\n c[8]=1,\nc[9]=1,\n\na[2,1]=30/323,\na[3,1]=72096/2224445,\na[3,2]=2480 64/2224445,\na[4,1]=36/667,\na[4,2]=0,\na[4,3]=108/667,\na[5,1]=627527 529881/949970239488,\na[5,2]=0,\na[5,3]=-796051562201/316656746496,\na [5,4]=1149583311737/474985119744,\na[6,1]=-21808753194693072629/113316 55147076247477,\na[6,2]=0,\na[6,3]=23796070175842454861/29560839514111 94994,\na[6,4]=-1212069996160021962455905/196552978013281766346054,\na [6,5]=572100877243261794319360/753453082384246770993207,\na[7,1]=39003 29744279593744638499197123/812153812453729137080716015625,\na[7,2]=0, \na[7,3]=-246193764192504580123435743/12532754329751616636406250,\na[7 ,4]=224823365551024681432621834494295407/13603873384877578159416276089 843750,\na[7,5]=-21380486750050287941075671274033851392/12906219675382 756068818499374186328125,\na[7,6]=4388242844049767568/4530133947623046 875,\na[8,1]=92443545592188947873141917/18339118607014235358999552,\na [8,2]=0,\na[8,3]=-750730245651799179235/36368955615474475776,\na[8,4]= 2577779379770967239940697801161173/148529202838936128758823432698880, \na[8,5]=-27329320985408825882348019928840/154954352939736009973853365 53683,\na[8,6]=9858553049752641/9733028110935040,\na[8,7]=-55814813689 73046875/1697311052062486391808,\na[9,1]=806703661/12619514880,\na[9,2 ]=0,\na[9,3]=0,\na[9,4]=220190408471162800979/664353079471007769600,\n a[9,5]=120162483071349248/437171496550168695,\na[9,6]=587917074771/332 8097561600,\na[9,7]=16919101291015625/7106645968247808,\na[9,8]=-71650 3/321645,\n\nb[1]=806703661/12619514880,\nb[2]=0,\nb[3]=0,\nb[4]=22019 0408471162800979/664353079471007769600,\nb[5]=120162483071349248/43717 1496550168695,\nb[6]=587917074771/3328097561600,\nb[7]=169191012910156 25/7106645968247808,\nb[8]=-716503/321645,\n\n`b*`[1]=3909499619079641 9/631671402644956800,\n`b*`[2]=0,\n`b*`[3]=0,\n`b*`[4]=112759258451753 12052514787851/33254276852276902452679056000,\n`b*`[5]=127639567253281 205865247168/525184180307710755436015425,\n`b*`[6]=8079100125127495347 /37019631452920928000,\n`b*`[7]=250555370757146308140625/1422896226852 17516228352,\n`b*`[8]=-1250061795577621/772799082319350,\n`b*`[9]=-1/1 80\}:" }}}{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#\"\"($\")e&pO\"!\"(" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/&%\"CG6#\"\" ($\")c$*o8!\"(" }}}{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 2132 "ee := \{c[2]=30/323,\nc[3]=96/667,\nc[4]=144/667,\nc[5]=161/284,\nc[6]=28/3 9,\nc[7]=324/325,\nc[8]=1,\nc[9]=1,\n\na[2,1]=30/323,\na[3,1]=72096/22 24445,\na[3,2]=248064/2224445,\na[4,1]=36/667,\na[4,2]=0,\na[4,3]=108/ 667,\na[5,1]=627527529881/949970239488,\na[5,2]=0,\na[5,3]=-7960515622 01/316656746496,\na[5,4]=1149583311737/474985119744,\na[6,1]=-21808753 194693072629/11331655147076247477,\na[6,2]=0,\na[6,3]=2379607017584245 4861/2956083951411194994,\na[6,4]=-1212069996160021962455905/196552978 013281766346054,\na[6,5]=572100877243261794319360/75345308238424677099 3207,\na[7,1]=3900329744279593744638499197123/812153812453729137080716 015625,\na[7,2]=0,\na[7,3]=-246193764192504580123435743/12532754329751 616636406250,\na[7,4]=224823365551024681432621834494295407/13603873384 877578159416276089843750,\na[7,5]=-21380486750050287941075671274033851 392/12906219675382756068818499374186328125,\na[7,6]=438824284404976756 8/4530133947623046875,\na[8,1]=92443545592188947873141917/183391186070 14235358999552,\na[8,2]=0,\na[8,3]=-750730245651799179235/363689556154 74475776,\na[8,4]=2577779379770967239940697801161173/14852920283893612 8758823432698880,\na[8,5]=-27329320985408825882348019928840/1549543529 3973600997385336553683,\na[8,6]=9858553049752641/9733028110935040,\na[ 8,7]=-5581481368973046875/1697311052062486391808,\na[9,1]=806703661/12 619514880,\na[9,2]=0,\na[9,3]=0,\na[9,4]=220190408471162800979/6643530 79471007769600,\na[9,5]=120162483071349248/437171496550168695,\na[9,6] =587917074771/3328097561600,\na[9,7]=16919101291015625/710664596824780 8,\na[9,8]=-716503/321645,\n\nb[1]=806703661/12619514880,\nb[2]=0,\nb[ 3]=0,\nb[4]=220190408471162800979/664353079471007769600,\nb[5]=1201624 83071349248/437171496550168695,\nb[6]=587917074771/3328097561600,\nb[7 ]=16919101291015625/7106645968247808,\nb[8]=-716503/321645,\n\n`b*`[1] =39094996190796419/631671402644956800,\n`b*`[2]=0,\n`b*`[3]=0,\n`b*`[4 ]=11275925845175312052514787851/33254276852276902452679056000,\n`b*`[5 ]=127639567253281205865247168/525184180307710755436015425,\n`b*`[6]=80 79100125127495347/37019631452920928000,\n`b*`[7]=250555370757146308140 625/142289622685217516228352,\n`b*`[8]=-1250061795577621/7727990823193 50,\n`b*`[9]=-1/180\}:" }}}{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 st age, order 6 scheme is given as follows." }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 75 "subs(ee,StabilityFunction(6,8,'expanded')):\nR := una pply(%,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)*&#\"381M,@sg(e#\"7S)[gjiIJF-L\"F)*$)F'\"\"(F)F)F)*&#\"/eQ z*\\nv\"\"3:!H0'HOxK!)F)*$)F'\"\")F)F)F)" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 124 "We can find the point where the b oundary of the stability region intersects the negative real axis by s olving the equation: " }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "R(z) = 1;" "6#/-%\"RG6#%\"zG\"\"\"" }{TEXT -1 2 ". " }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 28 "z0 := newton(R(z)=1,z=-4.4);" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#>%#z0G$!+G#piX%!\"*" }}}{PARA 0 "" 0 " " {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 303 "z0 := new ton(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 := 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],fo nt=[HELVETICA,9]);" }}{PARA 13 "" 1 "" {GLPLOT2D 360 253 253 {PLOTDATA 2 "6+-%'CURVESG6$7Z7$$!3Q++++++!>&!#<$\"3'[nz+l'[bPF*7$$!3QM L3T![!f^F*$\"3osq%4msZc$F*7$$!3Ynm;#3'4G^F*$\"3)ek^!GUp#Q$F*7$$!3a++DB T9(4&F*$\"3dPdI^:\"*3KF*7$$!3kLLLk@>m]F*$\"3!y%3x1i4VIF*7$$!3E+]U'*)HB ,&F*$\"3e\\^I2YgsFF*7$$!3!pm;&GwYe\\F*$\"3eQ'y!>\"*oBDF*7$$!3s+](\\(Q* y*[F*$\"3KqErP\"pwE#F*7$$!3nLLV@,KP[F*$\"3sy?G<)R].#F*7$$!3'RLLd%[MwZF *$\"3#y$4USSgA=F*7$$!3NLL.q&p`r%F*$\"3)RzW5W2-j\"F*7$$!3E+]<*4%oaYF*$ \"3u6B#4X()pX\"F*7$$!3;nmJG')*Rf%F*$\"3nj8q$=t/I\"F*7$$!3uLLyGAZ\"[%F* $\"3%[*4hz'o(\\5F*7$$!3%3+])fw&\\O%F*$\"37HU79vNr$)!#=7$$!3$QL$)f7eWC% F*$\"3iDL\">=aQf'F]p7$$!3A++lN]MCTF*$\"3AC\"yqW=o<&F]p7$$!3ummYeRz+SF* $\"3vww0W$4R-%F]p7$$!3_LLV-,(>*QF*$\"3oqWCc'R'>KF]p7$$!35++S:-YpPF*$\" 3LD<-@yS3DF]p7$$!3K+++\"HZkk$F*$\"37V2M%=LL'>F]p7$$!3;++gW:!z_$F*$\"3q -+Ci,kn:F]p7$$!3hLL)*\\1D?MF*$\"3)G96^\"Qz(H\"F]p7$$!3'ommSKVAH$F*$\"3 kjCTv,[m5F]p7$$!3/nmEGV!Q=$F*$\"37$$!39++0(*RmdIF*$\"3cC1 &oq#F*$\"30qZ/,6F]p7$$!3K++:I;c=@F *$\"3r$=-q)))3<7F]p7$$!31LL.z]#3+#F*$\"3_N&*=y!o9O\"F]p7$$!3M++?,<>z=F *$\"3!4Z?b8dD`\"F]p7$$!3;++!4<(>gyJ>F]p7$$!3H++q9zA<:F*$\"3_,!)[\"R/T>#F]p7$$!3EnmEY;O-9F*$\" 3eVFc\"GN1Y#F]p7$$!3#)*****pQ<(z7F*$\"3a!yGv?\"R\"y#F]p7$$!3)RL$efMeo6 F*$\"3%eA+i%H=3JF]p7$$!3I****fAZ3Z5F*$\"3yQvcYIk4NF]p7$$!3xqm;(zQwK*F] p$\"3pbewCxmMRF]p7$$!3&z***\\)ecE8)F]p$\"3EQ)\\_J%3MWF]p7$$!3'3nmm0VV' pF]p$\"3a4&R#=Qf$)\\F]p7$$!3P)***\\iqATdF]p$\"3RQD&)e!))>j&F]p7$$!3aFL L*)4AjXF]p$\"3Hy+%)>t4OjF]p7$$!33LLLO'R&eLF]p$\"3U&[M%fZFZrF]p7$$!3Uim ;`O$Q;#F]p$\"3Cv*3=qkU0)F]p7$$!3?*****>$H-m5F]p$\"3?4jIQ4$)))*)F]p7$$ \"3v*QLLU?>#>F`s$\"3[-s.#30%>5F*7$$\"3%ymmY^avJ\"F]p$\"31Nq,(HH39\"F*7 $$\"3E0+]HcU&!\"#$\"#\\Fid l;$!\"(Fidl$\"$Z\"Fidl" 1 2 0 1 10 0 2 9 1 4 2 1.000000 45.000000 45.000000 0 0 "Curve 1" "Curve 2" "Curve 3" "Curve 4" "Curve 5" "Curve 6" }}}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 50 "T he following picture shows the stability region. " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1377 "R := z -> \+ 1+z+1/2*z^2+1/6*z^3+1/24*z^4+1/120*z^5+1/720*z^6+\n 25876072210134 0613/1330227313062636048840*z^7+17567499793858/803277362960529015*z^8: \npts := []: z0 := 0:\nfor ct from 0 to 240 do\n zz := newton(R(z)=e xp(ct*Pi/20*I),z=z0):\n z0 := zz:\n pts := [op(pts),[Re(zz),Im(zz) ]]:\nend do:\np1 := plot(pts,color=COLOR(RGB,.48,.23,.08)):\np2 := plo ts[polygonplot]([seq([pts[i-1],pts[i],[-2.2,0]],i=2..nops(pts))],\n \+ style=patchnogrid,color=COLOR(RGB,.95,.45,.15)):\npts := []: z0 := 2+4.75*I:\nfor ct from 0 to 60 do\n zz := newton(R(z)=exp(ct*Pi/ 30*I),z=z0):\n z0 := zz:\n pts := [op(pts),[Re(zz),Im(zz)]]:\nend \+ do:\np3 := plot(pts,color=COLOR(RGB,.48,.23,.08)):\np4 := plots[polygo nplot]([seq([pts[i-1],pts[i],[1.89,4.73]],i=2..nops(pts))],\n \+ style=patchnogrid,color=COLOR(RGB,.95,.45,.15)):\npts := []: z0 := 2- 4.75*I:\nfor ct from 0 to 60 do\n zz := newton(R(z)=exp(ct*Pi/30*I), z=z0):\n z0 := zz:\n pts := [op(pts),[Re(zz),Im(zz)]]:\nend do:\np 5 := plot(pts,color=COLOR(RGB,.48,.23,.08)):\np6 := plots[polygonplot] ([seq([pts[i-1],pts[i],[1.89,-4.73]],i=2..nops(pts))],\n styl e=patchnogrid,color=COLOR(RGB,.95,.45,.15)):\np7 := plot([[[-5.19,0],[ 2.29,0]],[[0,-5.19],[0,5.19]]],color=black,linestyle=3):\nplots[displa y]([p||(1..7)],view=[-5.19..2.29,-5.19..5.19],font=[HELVETICA,9],\n \+ labels=[`Re(z)`,`Im(z)`],axes=boxed,scaling=constrained);" }}{PARA 13 "" 1 "" {GLPLOT2D 381 565 565 {PLOTDATA 2 "6/-%'CURVESG6$7] z7$$\"\"!F)F(7$F($\"3++++Fjzq:!#=7$$!3#)*****f,u!\\v!#G$\"35+++UEfTJF- 7$$!3/+++:@Bg9!#E$\"35+++)z)Q7ZF-7$$!37+++a0]c#)F7$\"3W*****p-%=$G'F-7 $$!3%)*****REN^w$F7$\"3y******Rf(R&yF-7$$\"36+++C#=<.#!#C$\"3Y*****zue ZU*F-7$$\"3)******H(Rz&\\\"!#B$\"35+++_>b*4\"!#<7$$\"3u*****zBkBp'FM$ \"3-+++zGic7FP7$$\"3%)*****Rw%Q*G#!#A$\"3.+++G#FP7$$\"3%******4DsU*>F^ p$\"3-+++E$=AN#FP7$$\"31+++-))G#)GF^p$\"3#)*****fgHe]#FP7$$\"3$)****** 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38Iz%F_\\oF[\\w7%F_iw7$Fb[u$!+j$3my%F_\\oF[\\w7%Fciw7$F\\[u$!+IzvzZF_ \\oF[\\w7%Fgiw7$Ffjt$!+#3.Dx%F_\\oF[\\w7%F[jw7$F`jt$!+no)[w%F_\\oF[\\w 7%F_jw7$Feit$!+^V&pv%F_\\oF[\\w7%FcjwFe[wF[\\wFaesFies-F$6%7$7$$!3Q+++ +++!>&FPF(7$$\"3/++++++!H#FPF(-%'COLOURG6&FfhnF)F)F)-%*LINESTYLEG6#\" \"$-F$6%7$7$F(F[[x7$F($\"3Q++++++!>&FPF`[xFc[x-%%FONTG6$%*HELVETICAG\" \"*-%+AXESLABELSG6%%&Re(z)G%&Im(z)G-F_\\x6#%(DEFAULTG-%*AXESSTYLEG6#%$ BOXG-%(SCALINGG6#%,CONSTRAINEDG-%%VIEWG6$;$!$>&Fihn$\"$H#Fihn;Fg]x$\"$ >&Fihn" 1 2 0 1 10 0 2 9 1 2 1 1.000000 45.000000 45.000000 0 0 "Curve 1" "Curve 2" "Curve 3" "Curve 4" "Curve 5" "Curve 6" "Curve 7" "Curve 8" }}}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 4 "Th e " }{TEXT 260 30 "interval of absolute stability" }{TEXT -1 89 " (or stability interval) is the intersection of the stability region with \+ the real line." }}{PARA 0 "" 0 "" {TEXT -1 59 "For this scheme the sta bility interval is (approximately) " }{XPPEDIT 18 0 "[-4.4563, 0];" " 6#7$,$-%&FloatG6$\"&jX%!\"%!\"\"\"\"!" }{TEXT -1 2 ". " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 148 "We can distort the \+ boundary curve horizontally by taking the 11th root of the real part o f points along the curve. In this way we see that there is " }{TEXT 260 53 "no largest interval on the nonnegative imaginary axis" }{TEXT -1 65 " that contains the origin and lies inside the stability region. " }}{PARA 0 "" 0 "" {TEXT -1 119 "However the stability region inter sects the nonnegative imaginary axis in an interval that does not cont ain the origin." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> \+ " 0 "" {MPLTEXT 1 0 402 "R := z -> 1+z+1/2*z^2+1/6*z^3+1/24*z^4+1/120* z^5+1/720*z^6+\n 258760722101340613/1330227313062636048840*z^7+175 67499793858/803277362960529015*z^8:\nDigits := 25:\npts := []: z0 := 0 :\nfor ct from 0 to 107 do\n zz := newton(R(z)=exp(ct*Pi/100*I),z=z0 ):\n z0 := zz:\n pts := [op(pts),[surd(Re(zz),11),Im(zz)]]:\nend d o:\nplot(pts,color=COLOR(RGB,.85,.25,0),thickness=2,font=[HELVETICA,9] );\nDigits := 10:" }}{PARA 13 "" 1 "" {GLPLOT2D 261 276 276 {PLOTDATA 2 "6(-%'CURVESG6#7hq7$$\"\"!F)F(7$$!:Q*)fVi%e@I$\\'*G#!#E$\":)Qyx\\\"y *e`EfTJF-7$$!:Z8.XEP+bh^()y$F-$\":1u3*o&3yrI&=$G'F-7$$!:!QX7V\\Ozj3=%3 &F-$\":cIwdYzV2'zxC%*F-7$$!:ld;Y%)He%prTgiF-$\":(G@%G*4oThqjc7!#D7$$!: Ey@EVK,m*[&GN(F-$\":vEis%\\TqEjzq:F?7$$!:y=n0+)o]s3R!Q)F-$\":.%*zr;HI= fb\\)=F?7$$!:.ci_/XyL<8WN*F-$\":YW&[!RLml&[6*>#F?7$$!:X\"Q.D>/^)3E#G5F ?$\":S(o%G*osZ?TF8DF?7$$!::W:Lj]#o()>)o6\"F?$\":QGS&[U'4GQLu#GF?7$$!:g !p\")R%GsEYMX5$3*HG\"F?$\":\"oW;) >iNs*=vbMF?7$$!:jRsZP:YJ6l2O\"F?$\":,77IUqmZ96*pPF?7$$!:xLT!4N9 F?$\":/sA![0f8\"QqS3%F?7$$!:Dol'Q#eGeBAf]\"F?$\":?n&)3w*)e6gH#)R%F?7$$ !:HSsa6avGnLJd\"F?$\":C(\\uRZe&zz)Q7ZF?7$$!:B-!4&pLH1K,lj\"F?$\":!4\"* )G;Y+E'zaE]F?7$$!:5#G7[l)3[Uycp\"F?$\":j\\xSEXLP322M&F?7$$!:8QyxE&R$yb Z,v\"F?$\":/j'37(\\Qr9m[l&F?7$$!:^r2qakAp!o8*z\"F?$\":!QV=Z#)pFN^-pfF? 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$\":V&*\\!Hpsf1*[e\">Ffu7$$\":2P>SFw>Gwy%y\\F?$\":.$R`OJ%*GWdFfu7$$ \":g$4?[2nr?1CS]F?$\":ffGTx_Fsl)[y>Ffu7$$\":>\"zl-l=OZ@#45&F?$\":=$ycU )[0lt&y4?Ffu7$$\":Nd(*o$=6MrO[g^F?$\":.qEn_^o.'[1T?Ffu7$$\":dq:+2.(y^9 ))=_F?$\":btG*GkIAyOKs?Ffu7$$\":$G\"R!G#pYTMngF&F?$\":t8?GM!ph%ffN5#Ff u7$$\":lDGP\"p%HKT))>L&F?$\":q)QU?ayS`(pZ8#Ffu7$$\":t.%4&[y/]['e'Q&F?$ \":sJq0NOtB-^f;#Ffu7$$\":S5![.Z\"pgi(zRaF?$\":=!ft,6kay**4(>#Ffu7$$\": y7+%GPiTO:b\"\\&F?$\":P4%zCF^f%*G@GAFfu7$$\":(pA`lG`'o$3xTbF?$\":Y\")) fjoAfEdGfAFfu7$$\":*R>OT2p(oAq.f&F?$\":i(y(*>uWa.TJ!H#Ffu7$$\":8\"pP3 \")o,\"[bsj&F?$\":wym!*Q'y1>LH@BFfu7$$\":2\"e&)oUhKUAK#o&F?$\":jk;,sp. cK=AN#Ffu7$$\":$[(*y`eOJdXXDdF?$\":SO.C]2j:t$3$Q#Ffu7$$\":zYF^Z>o(yI_m dF?$\":^WO=boq>s/ f'[d0'He]#Ffu7$$\":cPnSa%fHU7'p!fF?$\":;[a5S[%fNVIODFfu7$$\":s/yMF=W%p \"3^$fF?$\":$pV/\\7]m$=!omDFfu7$$\":@Y'H()[rP)ec*ffF?$\":[v4r_oL5e\\pf #Ffu7$$\":<_0r(4\\463<\")fF?$\":j%o\\C$[-*HZ5FEFfu7$$\":xa-8epy`o\\$)* fF?$\":S(*p5L-kbjPrl#Ffu7$$\":Jm;UsK?%)H15,'F?$\":WMU'*Rx3:9Sqo#Ffu7$$ \":gsE[Vm9J!*Q&=gF?$\":&QtS/&eXo'R!or#Ffu7$$\":]%>#o&eU;&4)=?gF?$\":zs ]*G?XEX2UYFFfu7$$\":m0z;Ip%yE;(\\,'F?$\":kib3^'Hok?)ex#Ffu7$$\":@#R/Yr tKt6e,gF?$\":=!)oBI6gm]z^!GFfu7$$\":?3x2V2a&R2@yfF?$\":\"f5+0Bpk%o/V$G Ffu7$$\":Et]qxQ55k`A%fF?$\":$p*)yR:@;!G\\K'GFfu7$$\":@`I7\"[@!*HSs*)eF ?$\":b,::X&30w]+#*GFfu7$$\":Csti4?[++JS\"eF?$\":3Nifz!)*3')Rc?HFfu7$$ \":&oGx,D$)*H:lHq&F?$\":%\\sgQfeM!3=*[HFfu7$$\":95d%eZwl`F?$\":r1*>8TM\"H>M ,1$Ffu7$$!:a*)f%=LN(3$Ffu7$$!:N'*Q$3`3Yf.$f(fF ?$\":gO51>l!pC8K9JFfu7$$!:?:%3p?TrX]TohF?$\":]2A)3(yR5$f.TJFfu7$$!:B\" Hom&4$)>_r.L'F?$\":\"y#*yz,*3.S*[nJFfu7$$!:nUrOQ[j2w$Ffu7$$!:OP'z'Q&Qdp3m*f'F?$\":xID&)pi!>k&)e>KFfu7$$!:RAcA%[a$zRth r'F?$\":9ew\"[+&o\\^A_C$Ffu7$$!:a#oTe1#R&p&GS#oF?$\":U-Y " 0 " " {MPLTEXT 1 0 198 "Digits := 15:\nz0 := 0.8*I:\nfor ct from 24 to 27 \+ do\n newton(R(z)=exp(ct*Pi/100*I),z=z0);\nend do;``;\nz0 := 3.0*I:\n for ct from 96 to 99 do\n newton(R(z)=exp(ct*Pi/100*I),z=z0);\nend d o;\nDigits := 10:" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#^$$!0YS\"4p.['*!# B$\"0Dl1.=)Rv!#:" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#^$$!0!)HfBN^w$!#B$ \"0(exRf(R&y!#:" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#^$$\"0)=]i)zTI)!#B$ \"0[`a]L\"o\")!#:" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#^$$\"0S.:KG5)H!#A $\"03Uzl!H#[)!#:" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%!G" }}{PARA 11 " " 1 "" {XPPMATH 20 "6#^$$\"0#zNIx8x9!#<$\"0)*oegfq(H!#9" }}{PARA 11 " " 1 "" {XPPMATH 20 "6#^$$\"0-\\./kQh(!#=$\"0u$\\-5)\\+$!#9" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#^$$!0;y7)G\\f\")!#>$\"0CGP\"\\nKI!#9" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#^$$!0`zS:o:1\"!#<$\"0W8H>M,1$!#9" }}} {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 108 "Then we \+ apply the bisection method to calculate the parameter value associated with each intersection point." }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 328 "Digits := 15:\nreal_part := proc(u)\n Re(newton(R(z)=exp(u*Pi *I),z=0.8*I))\nend proc:\nu0 := bisect('real_part'(u),u=0.24..0.27);\n newton(R(z)=exp(u0*Pi*I),z=0.8*I);``;\nreal_part := proc(u)\n Re(new ton(R(z)=exp(u*Pi*I),z=3.0*I))\nend proc:\nu0 := bisect('real_part'(u) ,u=0.96..0.99);\nnewton(R(z)=exp(u0*Pi*I),z=3.0*I);\nDigits := 10:" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#>%#u0G$\"0oD\\I]&QD!#:" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#^#$\"0WWNo%3vz!#:" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%!G" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%#u0G$\"0igi%3(4z*!#:" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#^$$\"0F=$HwwBS!#H$\"0$G'*=R=II!#9" }}} {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 4 "The " } {TEXT 260 73 "stability region intersects the nonegative imaginary axi s in the interval" }{TEXT -1 39 " [ 0.7975, 3.0302 ] (approximately )." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 37 "#-- ----------------------------------" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }} {PARA 0 "" 0 "" {TEXT -1 80 "The stability function R* for the 9 sta ge, order 5 scheme is given as follows." }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 97 "subs(ee,subs(b=`b*`,StabilityFunction(5,9,'expanded') )):\n`R*` := unapply(%,z):\n'`R*`(z)'=`R*`(z);" }}{PARA 12 "" 1 "" {XPPMATH 20 "6#/-%#R*G6#%\"zG,6\"\"\"F)F'F)*&#F)\"\"#F)*$)F'F,F)F)F)*& #F)\"\"'F)*$)F'\"\"$F)F)F)*&#F)\"#CF)*$)F'\"\"%F)F)F)*&#F)\"$?\"F)*$)F '\"\"&F)F)F)*&#\"=(o(\\6k*zAF7RgxE#\"@+E0c,/A[vjjoK!)f\"F)*$)F'F1F)F)F )*&#\"W`J\"@+otuoQHk+&[[-rI@F)*$)F'\"\")F)F)F)*&#\".Hp*)\\Py)\"5]8 hZkmi\\HsF)*$)F'\"\"*F)F)!\"\"" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }} {PARA 0 "" 0 "" {TEXT -1 124 "We can find the point where the boundary of the stability region intersects the negative real axis by solving \+ the equation: " }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "`R* `(z) = -1;" "6#/-%#R*G6#%\"zG,$\"\"\"!\"\"" }{TEXT -1 2 ". " }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 33 "z_0 := newton(`R*`(z)=-1,z=-4.5);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%$z_0G$!+p7jeW!\"*" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 324 "z_0 := newton(`R*`(z)=-1,z=-4.5):\np_1 := plot([`R*`(z),-1],z=-5.09..0.49,co lor=[red,blue]):\np_2 := plot([[[z_0,-1]]$3],style=point,symbol=[circl e,cross,diamond],color=black):\np_3 := plot([[z_0,0],[z_0,-1]],linesty le=3,color=COLOR(RGB,0,.5,0)):\nplots[display]([p_1,p_2,p_3],view=[-5. 09..0.49,-1.57..1.47],font=[HELVETICA,9]);" }}{PARA 13 "" 1 "" {GLPLOT2D 398 284 284 {PLOTDATA 2 "6+-%'CURVESG6$7S7$$!3')************ *3&!#<$!3\"G>0dpi.I#F*7$$!3G++vz=Po\\F*$!3'**y9$***>k)>F*7$$!3#**\\iE! 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" } }{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1536 "`R*` := z -> 1+z+1/2*z^2+1/6*z^3+1/24*z^4+1/120*z^5+\n 226 7760391227227996411497687/1598032686363754822040156052600*z^6+\n 5 86593890678654298639022337/2556852298182007715264249684160*z^7+\n \+ 31534419331876883995680667/2130710248485006429386874736800*z^8-\n \+ 8783749896929/72294962666447611350*z^9:\npts := []: z0 := 0:\nfor ct f rom 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(pt s,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,colo r=COLOR(RGB,.75,.2,0)):\npts := []: z0 := 1.9+4.45*I:\nfor ct from 0 t o 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 ],[1.81,4.40]],i=2..nops(pts))],\n 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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 2132 "ee := \{c[2]=30/323,\nc[3]=96/667 ,\nc[4]=144/667,\nc[5]=161/284,\nc[6]=28/39,\nc[7]=324/325,\nc[8]=1,\n c[9]=1,\n\na[2,1]=30/323,\na[3,1]=72096/2224445,\na[3,2]=248064/222444 5,\na[4,1]=36/667,\na[4,2]=0,\na[4,3]=108/667,\na[5,1]=627527529881/94 9970239488,\na[5,2]=0,\na[5,3]=-796051562201/316656746496,\na[5,4]=114 9583311737/474985119744,\na[6,1]=-21808753194693072629/113316551470762 47477,\na[6,2]=0,\na[6,3]=23796070175842454861/2956083951411194994,\na [6,4]=-1212069996160021962455905/196552978013281766346054,\na[6,5]=572 100877243261794319360/753453082384246770993207,\na[7,1]=39003297442795 93744638499197123/812153812453729137080716015625,\na[7,2]=0,\na[7,3]=- 246193764192504580123435743/12532754329751616636406250,\na[7,4]=224823 365551024681432621834494295407/13603873384877578159416276089843750,\na [7,5]=-21380486750050287941075671274033851392/129062196753827560688184 99374186328125,\na[7,6]=4388242844049767568/4530133947623046875,\na[8, 1]=92443545592188947873141917/18339118607014235358999552,\na[8,2]=0,\n a[8,3]=-750730245651799179235/36368955615474475776,\na[8,4]=2577779379 770967239940697801161173/148529202838936128758823432698880,\na[8,5]=-2 7329320985408825882348019928840/15495435293973600997385336553683,\na[8 ,6]=9858553049752641/9733028110935040,\na[8,7]=-5581481368973046875/16 97311052062486391808,\na[9,1]=806703661/12619514880,\na[9,2]=0,\na[9,3 ]=0,\na[9,4]=220190408471162800979/664353079471007769600,\na[9,5]=1201 62483071349248/437171496550168695,\na[9,6]=587917074771/3328097561600, \na[9,7]=16919101291015625/7106645968247808,\na[9,8]=-716503/321645,\n \nb[1]=806703661/12619514880,\nb[2]=0,\nb[3]=0,\nb[4]=2201904084711628 00979/664353079471007769600,\nb[5]=120162483071349248/4371714965501686 95,\nb[6]=587917074771/3328097561600,\nb[7]=16919101291015625/71066459 68247808,\nb[8]=-716503/321645,\n\n`b*`[1]=39094996190796419/631671402 644956800,\n`b*`[2]=0,\n`b*`[3]=0,\n`b*`[4]=11275925845175312052514787 851/33254276852276902452679056000,\n`b*`[5]=12763956725328120586524716 8/525184180307710755436015425,\n`b*`[6]=8079100125127495347/3701963145 2920928000,\n`b*`[7]=250555370757146308140625/142289622685217516228352 ,\n`b*`[8]=-1250061795577621/772799082319350,\n`b*`[9]=-1/180\}:" }}} {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#\"\"##\"#I\"$B$/&F%6#\"\"$#\"#'*\"$n'/&F%6#\"\"%#\"$W\"F1/&F%6#\" \"&#\"$h\"\"$%G/&F%6#\"\"'#\"#G\"#R/&F%6#\"\"(#\"$C$\"$D$/&F%6#\"\")\" \"\"/&F%6#\"\"*FQ" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT 259 51 "linking coefficients for the 9 stage order 5 scheme" } {TEXT -1 1 ":" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 49 "seq(seq(a[i ,j]=subs(ee,a[i,j]),j=1..i-1),i=1..9);" }}{PARA 12 "" 1 "" {XPPMATH 20 "6F/&%\"aG6$\"\"#\"\"\"#\"#I\"$B$/&F%6$\"\"$F(#\"&'4s\"(XWA#/&F%6$F /F'#\"'k![#F2/&F%6$\"\"%F(#\"#O\"$n'/&F%6$F;F'\"\"!/&F%6$F;F/#\"$3\"F> /&F%6$\"\"&F(#\"-\"))Hv_F'\"-)[R-(*\\*/&F%6$FKF'FB/&F%6$FKF/#!-,Ac^gz \"-'\\Ynl;$/&F%6$FKF;#\".P^)\\Z/&F%6$\"\"'F(#!5HE2$p%>`(3 =#\"5xuCwq9b;L6/&F%6$F[oF'FB/&F%6$F[oF/#\"5h[XUe69&R3cH/& F%6$F[oF;#!:0fXi>-gh**p?@\"\"9agMm/&F%6$F[oFK#\"9g$>VzhKCx35s &\"92K*4xYUQ#3`Mv/&F%6$\"\"(F(#\"@Br>*\\QYu$fzUuH.!R\"?Dc,;23P\"HPX7Q: 7)/&F%6$FgpF'FB/&F%6$FgpF/#!kP>Y#\";]iSOmh^(HVvKD\"/&F%6$Fgp F;#\"E2aH%\\M=iK9oC5blL#[A\"D]P%)*3wiTf\"yv([QtQg8/&F%6$FgpFK#!G#R^Q.u 7nv5%zG]+v'[!Q@\"GD\"Gj=u$*\\=)ogv#Qv'>i!H\"/&F%6$FgpF[o#\"4ovw\\S%GC) Q%\"4vo/Bw%R8IX/&F%6$\"\")F(#\";<>9ty%*)=#fXNW#*\";_&***e`B9qg=\"R$=/& F%6$FirF'FB/&F%6$FirF/#!6N#z\"*z^cCI2v\"5wdZuahb*oj$/&F%6$FirF;#\"Ct6; ,ypS*Rs'4xz$zxd#\"B!)))pKM#)e(Gh$*QG?H&[\"/&F%6$FirFK#!AS)G*>![B)e#)3a )4KHt#\"A$o`lL&Q(*4gtRHNa\\:/&F%6$FirF[o#\"1TEv\\Ibe)*\"1S]$46GIt*/&F% 6$FirFgp#!4vo/t*o8[\"e&\"73=R'[i?06tp\"/&F%6$\"\"*F(#\"*hOq1)\",!)[^>E \"/&F%6$FauF'FB/&F%6$FauF/FB/&F%6$FauF;#\"6z4!G;r%3/>?#\"6+'px+r%zINk' /&F%6$FauFK#\"3[#\\82$[i,7\"3&po,b'\\rrV/&F%6$FauF[o#\"-rZ2p\"\"13yCofk1r/&F%6$FauFir#!'.lr\"'X;K" }} }{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#\"\"\"#\"*hOq1)\",!)[^>E\"/&F%6#\"\"#\" \"!/&F%6#\"\"$F//&F%6#\"\"%#\"6z4!G;r%3/>?#\"6+'px+r%zINk'/&F%6#\"\"&# \"3[#\\82$[i,7\"3&po,b'\\rrV/&F%6#\"\"'#\"-rZ2p\"\"13yCofk1r/&F%6#\"\")#!'.lr\"'X;K" }}}{PARA 0 "" 0 " " {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT 259 38 "weights for the 9 stag e order 5 scheme" }{TEXT -1 1 ":" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 37 "seq(`b*`[i]=subs(ee,`b*`[i]),i=1..9);" }}{PARA 12 "" 1 "" {XPPMATH 20 "6+/&%#b*G6#\"\"\"#\"2>kz!>'*\\4R\"3+o&\\k-9nJ'/&F%6#\"\"# \"\"!/&F%6#\"\"$F//&F%6#\"\"%#\">^yy9D07`+g0zEX-pF_oFaK$/&F% 6#\"\"&#\"q$/&F%6#\"\"(#\"9D193j9d2Pb0D\"9_$Gi^<_oA'*GU\"/&F%6#\" \")#!1@wd&zh+D\"\"0]$>B3*zs(/&F%6#\"\"*#!\"\"\"$!=" }}}{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 15 "a scheme with " }{XPPEDIT 18 0 "c[7] = 5 75/576;" "6#/&%\"cG6#\"\"(*&\"$v&\"\"\"\"$w&!\"\"" }{TEXT -1 1 " " }} {PARA 257 "" 0 "" {TEXT -1 33 "The scheme constructed here has " } {XPPEDIT 18 0 "c[6] = 29/39;" "6#/&%\"cG6#\"\"'*&\"#H\"\"\"\"#R!\"\"" }{TEXT -1 7 " and " }{XPPEDIT 18 0 "c[7] = 575/576;" "6#/&%\"cG6#\" \"(*&\"$v&\"\"\"\"$w&!\"\"" }{TEXT -1 1 "." }}{PARA 0 "" 0 "" {TEXT -1 6 "With " }{XPPEDIT 18 0 "c[6]" "6#&%\"cG6#\"\"'" }{TEXT -1 7 " a nd " }{XPPEDIT 18 0 "c[7]" "6#&%\"cG6#\"\"(" }{TEXT -1 39 " having t hese 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 " ar e chosen to minimize the principal error norm." }}{PARA 257 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}{PARA 0 "" 0 "" {TEXT -1 52 "#---------------------------------------------- -----" }}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 28 "checking the combined 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 2472 "ee := \{c[2]=21/229,\nc[3]=398/28 05,\nc[4]=199/935,\nc[5]=541/959,\nc[6]=29/39,\nc[7]=575/576,\nc[8]=1, \nc[9]=1,\n\na[2,1]=21/229,\na[3,1]=5306932/165228525,\na[3,2]=1813725 8/165228525,\na[4,1]=199/3740,\na[4,2]=0,\na[4,3]=597/3740,\na[5,1]=94 539185952859/139708222009916,\na[5,2]=0,\na[5,3]=-360526602615135/1397 08222009916,\na[5,4]=86200229584590/34927055502479,\na[6,1]=-897896928 8480000243857587089/3094782216331416263956902924,\na[6,2]=0,\na[6,3]=9 9268298611547543700685925/8262922327440626705984748,\na[6,4]=-16240947 2645307771282700363150/17326138018112792098834790979,\na[6,5]=40197004 738132352835791693452/40001804824576007393862935391,\na[7,1]=202866047 31599462128958430510992102863475/2582125222615604140408598429414209880 064,\na[7,2]=0,\na[7,3]=-7596345291112508505437918843804759875/2377293 92100642727921981994337042432,\na[7,4]=6620788516589830027125628007103 689280061615375/250839001756308652298222060087067489645625344,\na[7,5] =-16378468176189294528981106978962032922592625/77246843805996938538267 20717982679148003328,\na[7,6]=97768503635786059874525/1190611171358059 61207808,\na[8,1]=13855115178421131108323299597371007/1715122015582524 727359996149290620,\na[8,2]=0,\na[8,3]=-239541724384503198568188548325 /7288002275830304575859927972,\na[8,4]=9565183091966721153066883446850 8100881883600/3523904935082188448634512810545504833479151,\na[8,5]=-74 1690472086763685807332786432434772072/33803569324846358753294490754564 7623719,\na[8,6]=257132510132640765480/305396035838269868597,\na[8,7]= -1187059899803728084992/657869227273850599426895,\na[9,1]=1697710672/2 6928207375,\na[9,2]=0,\na[9,3]=0,\na[9,4]=164490038363253411875/503598 049753427901952,\na[9,5]=688263291863311978681/2293101349345635227520, \na[9,6]=237419997287733/1436816713308800,\na[9,7]=2129892164719607808 /556154447061193625,\na[9,8]=-680161433/184588800,\n\nb[1]=1697710672/ 26928207375,\nb[2]=0,\nb[3]=0,\nb[4]=164490038363253411875/50359804975 3427901952,\nb[5]=688263291863311978681/2293101349345635227520,\nb[6]= 237419997287733/1436816713308800,\nb[7]=2129892164719607808/5561544470 61193625,\nb[8]=-680161433/184588800,\n\n`b*`[1]=109792437751912778236 9/17974423922545824426000,\n`b*`[2]=0,\n`b*`[3]=0,\n`b*`[4]=2294865662 6877224955778831923125/68757710895541194720834617432064,\n`b*`[5]=6422 693908420175795081848535941/23428040725205156020096373368320,\n`b*`[6] =5409978194277871906843931/26640747567082588647769600,\n`b*`[7]=687569 9405989996627114328064/2577985433915837392040353375,\n`b*`[8]=-1117082 0866846625947/4400427318713395200,\n`b*`[9]=-1/1764\}:" }}}{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 Butch er tableau in exact and approximate form is as follows." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 576 "subs(e e,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)],[``$2,b[7], b[8]],\n [`b*`,seq(`b*`[i],i=1..3)],[``,seq(`b*`[i],i=4..6)],[``,seq( `b*`[i],i=7..9)]]));" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#K%'matrixG6#78 7&#\"#@\"$H#F(%!GF+7&#\"$)R\"%0G#\"(KpI&\"*D&G_;#\")es8=F2F+7&#\"$*>\" $N*#F7\"%SP\"\"!#\"$(fF:7&#\"$T&\"$f*#\"/fG&f=RX*\"0;*4?A3(R\"F;#!0N^h -m_g$FD7&F+F+F+#\"/!f%eH-?')\"/zC]bq#\\$7&#\"#H\"#R#!=*3(edQC++[)Gp*y* )\"=CH!p&RE;9L;Ay%4$F;#\";Dfo+PaZ:h)Ho#**\":[Z)fqE1WFB#HE)7&F+F+#!?]JO +FGrxIXEZ4C;\">z4zM))4#z7\"=!QhK<#\">_Mp\"zNGNK\"QZ+(>S\">\"RNH'QR2gdC [!=+S7&#\"$v&\"$w&#\"JvM'G5#*4^I%e*G@Y*fJZg'G?\"Ik+))4UTH%)f3/9/chA_7# e#F;#!Fv)fZ!Q%)=zV0&3D6\"HXjf(\"EKC/PV*>)>#zsU15#RHxB7&F+#\"Ov`hh+G*o. r+Gc7F+$)*e;&)y?m\"NW`iX'*[nq3g?A)H_'3jv,!R3D#!MDEfAH.i*yp5\")*GXH*=w \"o%yj\"\"LGL+[\"zE)zr?n#Q&Qp*f!Q%oCx#\"8DX()fgyNO]ox*\"93y?hf!e8<61> \"7&\"\"\"#\"D25P(f*HB$368@%y^6bQ\"\"C?1H\\h**ftsCDe:?7:.XQC4$=l&*\"L^\" zM$[]X0\"G^M'[%)=#3N\\!R_$#!Hs?xMCV'yKt!eojn3s/pT(\"H>PiZca2\\%H`(ej%[ KpN!Q$#\"6![l2kK,^KrD\"6(fo)p#Qe.'R0$7&F+F+F+#!7#*\\3GP!)**)fq=\"\"9&* oU*f]QFF#pyl7&F[p#\"+s1r(p\"\",vt?Gp#F;F;7&F+#\"6v=T`KOQ+\\k\"\"6_>!zU `(\\!)f.&#\"6\"oy>Jj=Hj#)o\"7?vANcM\\85$H##\"0LxG(**>uB\"1+)3Lr;oV\"7& F+F+#\"43yg>Z;#*)H@\"3DO>hqWahb#!*L9;!o\"*+))e%=7&F+%F________________ _____________________GFfrFfr7&%\"bGFaqF;F;FdqF^r7&%#b*G#\"7pByF\">vPCz 4\"\"8+gUCeaARUuz\"F;F;7&F+#\"ADJ#>$)yd&\\Axoic'[H#\"Ak?VTb*3rd (o#\"@Tf`[=3&zv,U3RpAk\"A?$oLP'4?g:0_sS!GM##\":JR%o!>(yF%>y*4a\";+'pxk )e#3nvuSm#7&F+#\"=k!GV6Fm***)fS*pvo\"=vLNS?RPe\"RV&)zd##!5ZfiYo'3#3<6 \"4+_R8(=tU+W#!\"\"\"%k " 0 "" {MPLTEXT 1 0 136 "subs(ee,matrix([seq([c[ i],seq(a[i,j],j=1..i-1),``$(10-i)],i=2..9),\n[`b`,seq(b[i],i=1..8),``] ,[`b*`,seq(`b*`[i],i=1..9)]])):\nevalf[8](%);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#K%'matrixG6#7,7,$\")dIq\"*!\"*F(%!GF+F+F+F+F+F+F+7,$\") [*)=9!\")$\")S(=@$F*$\")uq(4\"F/F+F+F+F+F+F+F+7,$\")AMG@F/$\")c&3K&F*$ \"\"!F:$\")nD'f\"F/F+F+F+F+F+F+7,$\")IHTcF/$\")@!pw'F/F9$!)$o0e#!\"($ \")t+oCFDF+F+F+F+F+7,$\")u*eV(F/$!)^K,HFDF9$\").P,7!\"'$!)%oOP*FD$\")) z[+\"FDF+F+F+F+7,$\")*QE)**F/$\")MbcyFDF9$!)\\P&>$FN$\")uXREFN$!)nF?@F D$\")Ki6#)F/F+F+F+7,$\"\"\"F:$\")'4#y!)FDF9$!)az'G$FN$\")/P9FFN$!))=T> #FD$\")9k>%)F/$!)5S/=!#5F+F+7,F[o$\")6e/jF*F9F9$\")iHmKF/$\") \+ " 0 "" {MPLTEXT 1 0 146 "RK6_8eqs := [op(RowSumConditions(8,'expanded' )),op(OrderConditions(6,8,'expanded'))]:\n`RK5_9eqs*` := subs(b=`b*`,O rderConditions(5,9,'expanded')):" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }} {PARA 0 "" 0 "" {TEXT -1 7 "Check: " }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 148 "simplify(subs(ee,RK6_8eqs)):\nmap(u->`if`(lhs(u)=rhs (u),0,1),%);\nnops(%);\nsimplify(subs(ee,`RK5_9eqs*`)):\nmap(u->`if`(l hs(u)=rhs(u),0,1),%);\nnops(%);" }{TEXT -1 0 "" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#7N\"\"!F$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$" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 114 "We can calculate the principal erro r norm of the order 6 scheme, that is, the 2-norm of the principal err or terms." }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 131 "errterms6_8 := PrincipalErrorTerms(6,8,'expanded'):\nevalf(evalf[14](sqrt(add(subs(e e,errterms6_8[i])^2,i=1.. nops(errterms6_8)))));" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#$\"+XuaP5!#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#$\"+Am\"QI' !#8" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}}{PARA 0 "" 0 "" {TEXT -1 52 "#----------------- ----------------------------------" }}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 44 "simultaneous construction of the two schemes" }}{PARA 0 "" 0 " " {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 119 "We incorporate the sta ge-order equations to ensure that stage 2 has stage-order 2 and stages 3 to 8 have stage-order 3." }}{PARA 0 "" 0 "" {TEXT -1 48 "We also in corporate the simplifying conditions: " }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "Sum(b[i]*a[i,j],i = j+1 .. 8) = b[j]*(1-c[j]);" "6#/-%$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(\"\"$7%\"\")F)/*&F,F()F2F :F(#F(F57%\"#7F)/*&F,F(-%!G6#*&%\"aGF(-FF6#*&F8F(FIF(F(F(#F(\"#g7%\"#: F)/*&F,F(-FF6#*&F?F(FIF(F(#F(\"#?7%\"#;F)/*&F,F()F2F5F(#F(\"\"&Q)pprin t186\"" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 582 "SO6_8 := SimpleOrderConditions(6,8,'expanded'):\nSO_ eqs := [op(RowSumConditions(8,'expanded')),op(StageOrderConditions(2,8 ,'expanded')),\n op(StageOrderConditions(3,4..8,'expande d'))]:\n`SO5_9*` := subs(b=`b*`,SimpleOrderConditions(5,9,'expanded')) :\nord_cdns := [seq(SO6_8[i],i=[1,2,4,8,16,24,29,32])]:\n`ord_cdns*` : = [seq(`SO5_9*`[i],i=[1,2,4,8,12,15,16])]:\nsimp_eqs := [add(b[i]*a[i, 1],i=1+1..8)=b[1],seq(add(b[i]*a[i,j],i=j+1..8)=b[j]*(1-c[j]),j=[6,7]) ]:\nwt_eqns := [seq(b[i]=a[9,i],i=1..8)]:\ncdns := [op(ord_cdns),op(si mp_eqs),op(SO_eqs),op(wt_eqns),op(`ord_cdns*`)]:" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 22 "We specify the nodes: " } }{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "c[2] = 21/229;" "6#/ &%\"cG6#\"\"#*&\"#@\"\"\"\"$H#!\"\"" }{TEXT -1 3 ", " }{XPPEDIT 18 0 "c[4] = 199/935;" "6#/&%\"cG6#\"\"%*&\"$*>\"\"\"\"$N*!\"\"" }{TEXT -1 3 ", " }{XPPEDIT 18 0 "c[5] = 541/959;" "6#/&%\"cG6#\"\"&*&\"$T&\"\" \"\"$f*!\"\"" }{TEXT -1 3 ", " }{XPPEDIT 18 0 "c[6] = 29/39;" "6#/&% \"cG6#\"\"'*&\"#H\"\"\"\"#R!\"\"" }{TEXT -1 3 ", " }{XPPEDIT 18 0 "c[ 7] = 575/576;" "6#/&%\"cG6#\"\"(*&\"$v&\"\"\"\"$w&!\"\"" }{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] = -1/1764;" "6#/&%#b*G6# \"\"*,$*&\"\"\"F*\"%k " 0 "" {MPLTEXT 1 0 218 "e1 := \{c[2 ]=21/229,c[4]=199/935,c[5]=541/959,c[6]=29/39,c[7]=575/576,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]=-1/1764\}:\neqns := subs(e1,cdns):\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(eq ns)\}):\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 2591 "e3 := \{`b*`[2] = 0, c [9] = 1, c[8] = 1, b[3] = 0, b[2] = 0, b[4] = 164490038363253411875/50 3598049753427901952, `b*`[6] = 5409978194277871906843931/2664074756708 2588647769600, a[6,3] = 99268298611547543700685925/8262922327440626705 984748, a[7,2] = 0, a[6,2] = 0, a[4,2] = 0, `b*`[3] = 0, a[8,2] = 0, a [5,2] = 0, a[9,2] = 0, a[9,3] = 0, a[4,1] = 199/3740, b[5] = 688263291 863311978681/2293101349345635227520, a[5,3] = -360526602615135/1397082 22009916, c[2] = 21/229, a[9,5] = 688263291863311978681/22931013493456 35227520, c[3] = 398/2805, `b*`[7] = 6875699405989996627114328064/2577 985433915837392040353375, b[6] = 237419997287733/1436816713308800, a[8 ,5] = -741690472086763685807332786432434772072/33803569324846358753294 4907545647623719, a[8,4] = 9565183091966721153066883446850810088188360 0/3523904935082188448634512810545504833479151, a[9,6] = 23741999728773 3/1436816713308800, a[7,4] = 66207885165898300271256280071036892800616 15375/250839001756308652298222060087067489645625344, a[9,7] = 21298921 64719607808/556154447061193625, a[8,6] = 257132510132640765480/3053960 35838269868597, `b*`[4] = 22948656626877224955778831923125/68757710895 541194720834617432064, c[4] = 199/935, c[5] = 541/959, c[6] = 29/39, c [7] = 575/576, `b*`[9] = -1/1764, b[7] = 2129892164719607808/556154447 061193625, a[4,3] = 597/3740, b[8] = -680161433/184588800, a[7,6] = 97 768503635786059874525/119061117135805961207808, a[8,7] = -118705989980 3728084992/657869227273850599426895, a[6,4] = -16240947264530777128270 0363150/17326138018112792098834790979, a[2,1] = 21/229, `b*`[1] = 1097 924377519127782369/17974423922545824426000, b[1] = 1697710672/26928207 375, a[9,8] = -680161433/184588800, a[9,4] = 164490038363253411875/503 598049753427901952, a[7,1] = 20286604731599462128958430510992102863475 /2582125222615604140408598429414209880064, a[5,1] = 94539185952859/139 708222009916, `b*`[8] = -11170820866846625947/4400427318713395200, a[3 ,2] = 18137258/165228525, a[5,4] = 86200229584590/34927055502479, a[6, 1] = -8978969288480000243857587089/3094782216331416263956902924, a[8,1 ] = 13855115178421131108323299597371007/171512201558252472735999614929 0620, a[3,1] = 5306932/165228525, `b*`[5] = 64226939084201757950818485 35941/23428040725205156020096373368320, a[9,1] = 1697710672/2692820737 5, a[8,3] = -239541724384503198568188548325/72880022758303045758599279 72, a[7,3] = -7596345291112508505437918843804759875/237729392100642727 921981994337042432, a[7,5] = -1637846817618929452898110697896203292259 2625/7724684380599693853826720717982679148003328, a[6,5] = 40197004738 132352835791693452/40001804824576007393862935391\}:" }{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 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(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) ],[``$2,b[7],b[8]],\n [`b*`,seq(`b*`[i],i=1..3)],[``,seq(`b*`[i],i=4. .6)],[``,seq(`b*`[i],i=7..9)]]));" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#K %'matrixG6#787&#\"#@\"$H#F(%!GF+7&#\"$)R\"%0G#\"(KpI&\"*D&G_;#\")es8=F 2F+7&#\"$*>\"$N*#F7\"%SP\"\"!#\"$(fF:7&#\"$T&\"$f*#\"/fG&f=RX*\"0;*4?A 3(R\"F;#!0N^h-m_g$FD7&F+F+F+#\"/!f%eH-?')\"/zC]bq#\\$7&#\"#H\"#R#!=*3( edQC++[)Gp*y*)\"=CH!p&RE;9L;Ay%4$F;#\";Dfo+PaZ:h)Ho#**\":[Z)fqE1WFB#HE )7&F+F+#!?]JO+FGrxIXEZ4C;\">z4zM))4#z7\"=!QhK<#\">_Mp\"zNGNK\"QZ+(>S\" >\"RNH'QR2gdC[!=+S7&#\"$v&\"$w&#\"JvM'G5#*4^I%e*G@Y*fJZg'G?\"Ik+))4UTH %)f3/9/chA_7#e#F;#!Fv)fZ!Q%)=zV0&3D6\"HXjf(\"EKC/PV*>)>#zsU15#RHxB7&F+ #\"Ov`hh+G*o.r+Gc7F+$)*e;&)y?m\"NW`iX'*[nq3g?A)H_'3jv,!R3D#!MDEfAH.i*y p5\")*GXH*=w\"o%yj\"\"LGL+[\"zE)zr?n#Q&Qp*f!Q%oCx#\"8DX()fgyNO]ox*\"93 y?hf!e8<61>\"7&\"\"\"#\"D25P(f*HB$368@%y^6bQ\"\"C?1H\\h**ftsCDe:?7:.XQC4 $=l&*\"L^\"zM$[]X0\"G^M'[%)=#3N\\!R_$#!Hs?xMCV'yKt!eojn3s/pT(\"H>PiZca 2\\%H`(ej%[KpN!Q$#\"6![l2kK,^KrD\"6(fo)p#Qe.'R0$7&F+F+F+#!7#*\\3GP!)** )fq=\"\"9&*oU*f]QFF#pyl7&F[p#\"+s1r(p\"\",vt?Gp#F;F;7&F+#\"6v=T`KOQ+\\ k\"\"6_>!zU`(\\!)f.&#\"6\"oy>Jj=Hj#)o\"7?vANcM\\85$H##\"0LxG(**>uB\"1+ )3Lr;oV\"7&F+F+#\"43yg>Z;#*)H@\"3DO>hqWahb#!*L9;!o\"*+))e%=7&F+%F_____ ________________________________GFfrFfr7&%\"bGFaqF;F;FdqF^r7&%#b*G#\"7 pByF\">vPCz4\"\"8+gUCeaARUuz\"F;F;7&F+#\"ADJ#>$)yd&\\Axoic'[H#\"Ak?VTb*3rd(o#\"@Tf`[=3&zv,U3RpAk\"A?$oLP'4?g:0_sS!GM##\":JR%o!>(yF%>y *4a\";+'pxk)e#3nvuSm#7&F+#\"=k!GV6Fm***)fS*pvo\"=vLNS?RPe\"RV&)zd##!5Z fiYo'3#3<6\"4+_R8(=tU+W#!\"\"\"%k " 0 "" {MPLTEXT 1 0 136 "subs(e3,mat rix([seq([c[i],seq(a[i,j],j=1..i-1),``$(10-i)],i=2..9),\n[`b`,seq(b[i] ,i=1..8),``],[`b*`,seq(`b*`[i],i=1..9)]])):\nevalf[8](%);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#K%'matrixG6#7,7,$\")dIq\"*!\"*F(%!GF+F+F+F+F+F+ F+7,$\")[*)=9!\")$\")S(=@$F*$\")uq(4\"F/F+F+F+F+F+F+F+7,$\")AMG@F/$\") c&3K&F*$\"\"!F:$\")nD'f\"F/F+F+F+F+F+F+7,$\")IHTcF/$\")@!pw'F/F9$!)$o0 e#!\"($\")t+oCFDF+F+F+F+F+7,$\")u*eV(F/$!)^K,HFDF9$\").P,7!\"'$!)%oOP* FD$\"))z[+\"FDF+F+F+F+7,$\")*QE)**F/$\")MbcyFDF9$!)\\P&>$FN$\")uXREFN$ !)nF?@FD$\")Ki6#)F/F+F+F+7,$\"\"\"F:$\")'4#y!)FDF9$!)az'G$FN$\")/P9FFN $!))=T>#FD$\")9k>%)F/$!)5S/=!#5F+F+7,F[o$\")6e/jF*F9F9$\")iHmKF/$\") " 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 0 "" 0 "" {TEXT -1 0 "" }} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}{SECT 1 {PARA 4 "" 0 " " {TEXT -1 22 "eG .. general solution" }}{PARA 0 "" 0 "" {TEXT -1 0 " " }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 71947 "eG := \{c[8] = 1, b[3 ] = 0, b[2] = 0, c[9] = 1, b[5] = -1/60*(10*c[6]*c[7]*c[4]-5*c[6]*c[4] -5*c[7]*c[6]+3*c[6]-5*c[7]*c[4]+3*c[4]-2+3*c[7])/(-c[4]+c[5])/c[5]/(-c [5]^3+c[5]^2-c[7]*c[5]+c[7]*c[5]^2-c[6]*c[5]+c[6]*c[5]^2+c[7]*c[6]-c[6 ]*c[7]*c[5]), a[5,2] = 0, a[4,2] = 0, `b*`[3] = 0, `b*`[2] = 0, a[8,2] = 0, a[7,2] = 0, a[6,2] = 0, a[6,5] = 1/2*c[6]*(30*c[6]^2*c[5]^2*c[4] ^2-2*c[6]^2*c[4]^2+c[6]^3*c[4]-12*c[6]^2*c[4]*c[5]^2-3*c[5]^2*c[6]*c[4 ]+c[5]*c[6]*c[4]^2+c[6]*c[4]^3-12*c[6]^2*c[4]^2*c[5]-30*c[5]*c[6]^3*c[ 4]^2-20*c[5]*c[6]^2*c[4]^4+30*c[6]^2*c[5]*c[4]^3-c[5]*c[4]^3+20*c[4]^3 *c[5]*c[6]^3-20*c[6]^2*c[5]^2*c[4]^3+c[5]^2*c[4]^2-30*c[5]^2*c[6]*c[4] ^3+12*c[5]^2*c[6]*c[4]^2+2*c[6]^2*c[4]*c[5]+20*c[5]^2*c[6]*c[4]^4+2*c[ 6]^2*c[5]^2-2*c[6]^3*c[5]+12*c[6]^3*c[4]*c[5])/c[5]/(c[4]^3-2*c[5]*c[4 ]^2+30*c[5]^2*c[4]^3-6*c[5]*c[4]^3-10*c[4]^4*c[5]^2-c[5]^3+10*c[5]^4*c [4]^2-30*c[5]^3*c[4]^2+6*c[5]^3*c[4]+2*c[5]^2*c[4]), b[1] = 1/60*(30*c [5]*c[6]*c[7]*c[4]-10*c[5]*c[6]*c[4]-10*c[6]*c[7]*c[5]+5*c[6]*c[5]-10* c[6]*c[7]*c[4]+5*c[6]*c[4]+5*c[7]*c[6]-3*c[6]-10*c[5]*c[7]*c[4]+5*c[5] *c[4]+5*c[7]*c[5]-3*c[5]+5*c[7]*c[4]-3*c[4]+2-3*c[7])/c[5]/c[6]/c[7]/c [4], a[6,1] = -1/4*c[6]*(60*c[5]^2*c[6]*c[4]^4-90*c[5]^2*c[6]^3*c[4]^2 -60*c[5]*c[6]^2*c[4]^4-2*c[6]^3*c[4]^2+50*c[5]^4*c[4]^3*c[6]+40*c[6]^2 *c[4]^5*c[5]-40*c[4]^5*c[5]^3+36*c[4]*c[5]^2*c[6]^3-20*c[5]^4*c[6]^2*c [4]^2+8*c[6]^2*c[5]^3+4*c[6]^2*c[4]^3-24*c[5]*c[6]^3*c[4]^2-8*c[6]^2*c [4]^2*c[5]+24*c[6]^2*c[5]*c[4]^3+60*c[4]^3*c[5]*c[6]^3-4*c[6]*c[5]*c[4 ]^3-14*c[5]^3*c[6]*c[4]+7*c[5]^3*c[4]^2+6*c[5]^2*c[6]*c[4]^3+14*c[5]^2 *c[6]*c[4]^2+66*c[5]^3*c[6]*c[4]^2-80*c[5]^3*c[6]^2*c[4]^3+30*c[6]^2*c [5]^2*c[4]^3-24*c[6]^2*c[5]^2*c[4]^2-240*c[6]*c[5]^3*c[4]^3-48*c[6]^2* c[5]^3*c[4]+150*c[6]^2*c[5]^3*c[4]^2-24*c[4]^4*c[5]^2-9*c[5]^2*c[4]^3+ 6*c[4]^4*c[5]-40*c[5]^4*c[4]^4-20*c[6]^2*c[4]^4*c[5]^2-6*c[5]^2*c[6]^3 -2*c[6]*c[4]^4-24*c[5]^3*c[4]^3+60*c[6]^3*c[4]^3*c[5]^2+7*c[6]^3*c[4]* c[5]-40*c[5]^2*c[6]*c[4]^5-40*c[6]^3*c[4]^4*c[5]+110*c[4]^4*c[6]*c[5]^ 3+120*c[5]^3*c[4]^4)/c[5]/c[4]^2/(10*c[5]^3*c[4]^2+10*c[5]^2*c[4]^3-30 *c[5]^2*c[4]^2+6*c[5]^2*c[4]-c[5]^2+6*c[5]*c[4]^2+c[5]*c[4]-c[4]^2), b [4] = 1/60*(-2+3*c[5]-5*c[7]*c[6]+3*c[7]+3*c[6]+10*c[6]*c[7]*c[5]-5*c[ 6]*c[5]-5*c[7]*c[5])/c[4]/(-c[6]*c[4]^3-c[4]^3+c[4]^4+c[7]*c[4]^2+c[6] *c[4]^2-c[6]*c[7]*c[4]+c[6]*c[7]*c[4]^2-c[5]*c[4]^3+c[5]*c[4]^2-c[7]*c [4]^3-c[5]*c[7]*c[4]+c[5]*c[7]*c[4]^2-c[5]*c[6]*c[4]+c[5]*c[6]*c[4]^2+ c[6]*c[7]*c[5]-c[5]*c[6]*c[7]*c[4]), b[6] = 1/60*(10*c[5]*c[7]*c[4]-5* c[5]*c[4]-5*c[7]*c[5]+3*c[5]-5*c[7]*c[4]+3*c[4]-2+3*c[7])/(-c[4]+c[6]) /(-c[5]+c[6])/c[6]/(c[6]^2-c[6]+c[7]-c[7]*c[6]), `b*`[9] = 1/10*(50*c[ 5]^2*c[6]*c[4]^2-40*c[5]^2*c[6]*c[4]+10*c[6]*c[5]^2-40*c[5]*c[6]*c[4]^ 2+35*c[5]*c[6]*c[4]-10*c[6]*c[5]+10*c[6]*c[4]^2-10*c[6]*c[4]+3*c[6]-5* c[5]^2*c[4]-5*c[5]*c[4]^2+10*c[5]*c[4]-c[5]-c[4])/(3*c[6]-28*c[5]*c[6] *c[4]^2-28*c[5]^2*c[6]*c[4]+8*c[6]*c[5]^2-9*c[6]*c[4]-9*c[6]*c[5]+28*c [5]*c[6]*c[4]+30*c[5]^2*c[6]*c[4]^2+8*c[6]*c[4]^2-4*c[5]^2*c[4]-4*c[5] *c[4]^2+9*c[5]*c[4]-c[5]-c[4]), a[7,5] = -1/2*c[7]*(-17*c[5]^2*c[6]*c[ 4]^4+5*c[6]*c[5]*c[4]^4+42*c[4]^2*c[5]^2*c[7]^3+17*c[7]^3*c[5]*c[4]^3- 80*c[7]^3*c[5]^2*c[4]^3+c[7]*c[5]*c[4]^4+80*c[5]^2*c[4]^4*c[7]^2-60*c[ 4]^5*c[6]^2*c[5]^2+20*c[4]^3*c[7]^3*c[5]^3-20*c[7]^2*c[5]^2*c[4]^5+40* c[4]^5*c[6]*c[5]^3-5*c[5]*c[6]^2*c[4]^4+c[7]^3*c[4]^2-c[7]^2*c[4]^3-80 *c[5]^3*c[4]^4*c[7]+20*c[7]^3*c[5]^2*c[4]^4-200*c[6]^2*c[4]^5*c[7]*c[5 ]^3-290*c[5]^4*c[4]^2*c[7]^2*c[6]+20*c[6]^2*c[7]*c[5]^4*c[4]^2+40*c[6] *c[5]^2*c[4]^4*c[7]-140*c[5]^4*c[4]^3*c[6]-100*c[5]^4*c[4]^5*c[6]+190* c[5]^4*c[4]^4*c[6]-40*c[7]^2*c[4]^4*c[5]^3+20*c[5]^4*c[4]^4*c[7]+2*c[4 ]^4*c[7]^2-20*c[7]^2*c[5]^4*c[4]^3-4*c[6]^2*c[7]^2*c[5]-8*c[7]*c[6]^2* c[5]^3-3*c[7]*c[6]*c[5]^3+2*c[7]^2*c[6]*c[5]^3+60*c[7]*c[6]^2*c[5]*c[4 ]^5-40*c[6]*c[5]^2*c[4]^5*c[7]+17*c[7]*c[4]^4*c[6]*c[5]+80*c[7]^2*c[5] ^3*c[4]^3+5*c[7]*c[4]^4*c[6]^2+c[5]^2*c[7]^3-260*c[6]^2*c[7]*c[5]^2*c[ 4]^3-12*c[6]^2*c[7]^2*c[5]^2*c[4]+20*c[6]^2*c[7]^2*c[5]^2*c[4]^2-40*c[ 6]*c[7]*c[5]^3*c[4]^3-41*c[6]^2*c[7]^3*c[4]*c[5]+20*c[7]^2*c[6]*c[5]*c [4]^4+3*c[7]*c[6]*c[4]^3-2*c[6]^2*c[7]*c[4]^2-3*c[6]*c[7]^2*c[4]^2+4*c [7]^2*c[6]*c[4]^3+6*c[5]*c[7]^2*c[4]^3+4*c[6]*c[5]*c[7]^2*c[4]^2+c[7]^ 2*c[5]*c[6]*c[4]-11*c[5]*c[7]*c[6]*c[4]^3+2*c[6]^2*c[4]^2*c[5]-c[7]^3* c[4]*c[5]+17*c[5]^2*c[6]*c[7]*c[4]^2-200*c[5]^4*c[7]^2*c[4]^4*c[6]-50* c[4]^5*c[6]*c[7]*c[5]^3-150*c[5]^2*c[6]*c[7]^3*c[4]^4-c[6]^2*c[5]*c[4] ^3+2*c[7]^3*c[4]^2*c[6]*c[5]+2*c[6]^2*c[7]*c[4]*c[5]+108*c[7]*c[6]^2*c [5]*c[4]^3+4*c[6]^2*c[7]*c[5]^2+2*c[6]*c[5]*c[7]*c[4]^2-10*c[6]^2*c[7] ^3*c[5]^2+6*c[5]^4*c[7]*c[6]-8*c[6]*c[7]^3*c[5]^2+3*c[6]*c[7]^2*c[5]^2 +2*c[6]^2*c[7]^2*c[5]^2-27*c[6]^2*c[4]^2*c[7]*c[5]-62*c[6]^2*c[7]^2*c[ 4]^2*c[5]+46*c[6]*c[5]^4*c[4]^2+20*c[4]^5*c[7]*c[5]^3+2*c[7]^3*c[4]*c[ 6]*c[5]-27*c[6]^2*c[5]^2*c[7]*c[4]+120*c[6]^2*c[5]^2*c[7]*c[4]^2-35*c[ 6]*c[5]^2*c[7]*c[4]^3-37*c[6]*c[5]^3*c[7]*c[4]^2+53*c[6]^2*c[5]^3*c[7] *c[4]-54*c[7]^2*c[5]^2*c[4]^3+2*c[7]^2*c[5]*c[4]^2-c[7]*c[5]*c[4]^3-3* c[6]*c[5]*c[4]^3-2*c[4]^3*c[7]^3+63*c[7]^3*c[5]^2*c[6]*c[4]-16*c[7]^2* c[5]^2*c[6]*c[4]-5*c[6]*c[7]*c[4]^4+2*c[6]^2*c[7]^2*c[4]^2+2*c[6]^2*c[ 7]^2*c[4]-10*c[5]^4*c[7]^2*c[6]+3*c[5]^3*c[6]*c[4]+3*c[5]^3*c[7]*c[4]+ 20*c[5]^3*c[6]*c[7]*c[4]-2*c[5]^3*c[4]^2+22*c[6]^2*c[7]^2*c[4]*c[5]+7* c[5]^2*c[6]*c[4]^3-c[5]^2*c[7]*c[4]^2+c[5]^2*c[6]*c[4]^2-4*c[5]^2*c[7] *c[4]^3-8*c[5]^3*c[7]*c[4]^2-22*c[5]^3*c[6]*c[4]^2+240*c[5]^3*c[6]^2*c [4]^3+100*c[4]^3*c[7]^2*c[5]^2*c[6]^2+150*c[4]^5*c[5]^3*c[6]^2-3*c[7]* c[5]^4*c[4]+5*c[7]^2*c[5]^4*c[4]+12*c[5]^2*c[4]^4*c[7]-98*c[6]^2*c[5]^ 2*c[4]^3+25*c[6]^2*c[5]^2*c[4]^2+54*c[6]*c[5]^3*c[4]^3+8*c[6]^2*c[5]^3 *c[4]-63*c[6]^2*c[5]^3*c[4]^2+42*c[5]^3*c[7]*c[4]^3+60*c[5]^4*c[7]*c[4 ]^2*c[6]+200*c[5]^3*c[7]^3*c[4]^4*c[6]-4*c[6]^2*c[4]*c[5]^2+c[7]*c[6]^ 2*c[4]^3+200*c[7]^2*c[5]^2*c[6]*c[4]^3-240*c[7]^3*c[5]^2*c[6]*c[4]^2+1 80*c[7]^2*c[5]^3*c[6]*c[4]^2+3*c[5]^4*c[4]^2-80*c[5]^3*c[7]^3*c[6]*c[4 ]+c[7]^3*c[4]^2*c[6]-7*c[7]^3*c[4]^2*c[5]+20*c[4]^3*c[7]^2*c[5]*c[6]^2 -19*c[7]^2*c[6]*c[5]*c[4]^3-11*c[7]^2*c[5]^2*c[6]*c[4]^2-3*c[4]^4*c[5] ^2-27*c[7]^2*c[5]^3*c[6]*c[4]-480*c[7]^2*c[5]^3*c[6]*c[4]^3-380*c[5]^3 *c[6]^2*c[4]^4-5*c[5]^4*c[4]^3-10*c[6]^2*c[4]^3*c[7]^2+130*c[5]^3*c[6] ^2*c[7]*c[4]^3-5*c[7]^3*c[5]^2*c[4]-2*c[7]^2*c[5]^2*c[4]-4*c[5]^2*c[6] *c[7]*c[4]-160*c[6]^2*c[7]*c[5]^3*c[4]^2-80*c[6]^2*c[7]^2*c[4]*c[5]^3+ 2*c[5]^2*c[4]^3+12*c[7]^2*c[5]^2*c[4]^2+8*c[7]^2*c[5]^3*c[4]+400*c[7]^ 3*c[5]^2*c[6]^2*c[4]^3-17*c[7]^2*c[4]^4*c[5]+250*c[7]^2*c[5]^3*c[6]^2* c[4]^2-100*c[6]^2*c[4]^5*c[5]*c[7]^2-200*c[7]^2*c[4]^5*c[6]*c[5]^3+150 *c[5]^2*c[6]*c[4]^5*c[7]^2+380*c[7]^3*c[5]^2*c[4]^3*c[6]-20*c[7]^3*c[6 ]*c[5]*c[4]^3-210*c[7]^3*c[6]^2*c[5]*c[4]^3-6*c[5]^4*c[4]*c[6]+80*c[5] ^4*c[7]^2*c[4]*c[6]+10*c[6]*c[7]^3*c[5]^3-36*c[5]^4*c[4]*c[7]*c[6]-50* c[4]^5*c[6]^2*c[7]*c[5]^2+200*c[4]^5*c[6]^2*c[7]^2*c[5]^2+160*c[6]^2*c [4]^4*c[5]^2-450*c[7]^3*c[5]^3*c[4]^3*c[6]+5*c[6]^2*c[7]^3*c[4]^2-c[5] ^3*c[7]^2+6*c[6]^2*c[7]^3*c[5]-160*c[7]*c[6]^2*c[5]*c[4]^4+200*c[5]^3* c[4]^4*c[6]^2*c[7]-350*c[5]^3*c[4]^3*c[6]^2*c[7]^2-350*c[5]^4*c[4]^4*c [6]*c[7]+230*c[5]^2*c[4]^4*c[7]*c[6]^2+150*c[5]^3*c[4]^4*c[6]*c[7]+100 *c[5]^4*c[4]^3*c[6]*c[7]-3*c[6]^2*c[7]^3*c[4]-42*c[7]^2*c[5]^3*c[4]^2+ 50*c[5]^4*c[4]^4*c[6]^2-20*c[5]^4*c[4]^3*c[6]^2+500*c[5]^3*c[4]^4*c[7] ^2*c[6]+450*c[5]^4*c[4]^3*c[7]^2*c[6]-340*c[5]^2*c[4]^4*c[7]^2*c[6]-50 *c[5]^4*c[4]^3*c[6]^2*c[7]+150*c[7]^2*c[4]^4*c[6]^2*c[5]+200*c[7]^2*c[ 4]^4*c[5]^3*c[6]^2-350*c[7]^2*c[4]^4*c[6]^2*c[5]^2+10*c[7]^2*c[6]^2*c[ 5]^3-60*c[4]^4*c[6]*c[5]^3+100*c[7]^3*c[6]^2*c[5]*c[4]^4+200*c[5]^4*c[ 4]^5*c[6]*c[7]-200*c[7]^3*c[4]^4*c[6]^2*c[5]^2+5*c[5]^3*c[4]^4+80*c[6] ^2*c[7]^3*c[5]^2*c[4]-270*c[6]^2*c[7]^3*c[5]^2*c[4]^2+290*c[5]^3*c[7]^ 3*c[4]^2*c[6]+140*c[6]^2*c[5]*c[7]^3*c[4]^2-5*c[7]^3*c[4]*c[5]^3)/c[5] /(100*c[5]^2*c[6]*c[4]^4-18*c[6]*c[5]*c[4]^4+4*c[5]^3*c[4]+50*c[4]^5*c [6]^2*c[5]^2+40*c[5]*c[6]^2*c[4]^4+3*c[5]^5-5*c[4]^4*c[6]^2+180*c[5]^4 *c[4]^3*c[6]+100*c[5]^4*c[4]^5*c[6]-350*c[5]^4*c[4]^4*c[6]+140*c[5]^5* c[4]^2-180*c[5]^5*c[4]^3-2*c[5]^4+30*c[4]^5*c[5]^3+300*c[5]^5*c[4]^3*c [6]+2*c[6]*c[5]^3-50*c[5]^5*c[4]^2*c[6]^2-40*c[5]^4*c[6]^2*c[4]+240*c[ 5]^4*c[6]^2*c[4]^2-3*c[6]^2*c[5]^3+3*c[6]^2*c[4]^3+5*c[5]^4*c[6]^2-2*c [6]*c[4]^3+2*c[5]*c[4]^3+4*c[5]*c[6]*c[4]^2-6*c[6]^2*c[4]^2*c[5]-50*c[ 5]^4*c[4]^5-13*c[6]^2*c[5]*c[4]^3-40*c[6]*c[5]^4*c[4]^2+6*c[6]*c[5]*c[ 4]^3-15*c[5]^3*c[6]*c[4]-55*c[5]^2*c[6]*c[4]^3+6*c[5]^2*c[6]*c[4]^2+68 *c[5]^3*c[6]*c[4]^2-100*c[4]^5*c[5]^3*c[6]^2+100*c[6]^2*c[5]^2*c[4]^3- 100*c[6]*c[5]^3*c[4]^3+13*c[6]^2*c[5]^3*c[4]-100*c[6]^2*c[5]^3*c[4]^2- 4*c[5]^2*c[4]^2+6*c[6]^2*c[4]*c[5]^2-68*c[5]^4*c[4]^2+23*c[4]^4*c[5]^2 +350*c[5]^3*c[6]^2*c[4]^4-4*c[5]^2*c[6]*c[4]-9*c[5]^2*c[4]^3-3*c[4]^4* c[5]-5*c[5]^5*c[6]-30*c[5]^6*c[4]^2+50*c[5]^6*c[6]*c[4]^2+180*c[5]^4*c [4]^4+10*c[5]^4*c[4]*c[6]-240*c[6]^2*c[4]^4*c[5]^2+3*c[6]*c[4]^4-100*c [5]^6*c[4]^3*c[6]+68*c[5]^3*c[4]^3-23*c[5]^5*c[4]-210*c[5]^5*c[6]*c[4] ^2-350*c[5]^4*c[4]^3*c[6]^2-30*c[5]^2*c[6]*c[4]^5+50*c[5]^6*c[4]^3+60* c[4]^4*c[6]*c[5]^3+100*c[5]^5*c[6]^2*c[4]^3+9*c[5]^4*c[4]-140*c[5]^3*c [4]^4+40*c[5]^5*c[4]*c[6]), `b*`[6] = -1/60*(-30*c[5]^3*c[4]+103*c[5]* c[7]*c[4]-18*c[6]+6*c[5]+6*c[4]+27*c[7]*c[6]-9*c[5]^2-9*c[4]^2-100*c[7 ]*c[6]*c[5]^3+522*c[5]*c[6]*c[7]*c[4]-111*c[6]*c[5]^2+75*c[6]*c[5]+60* c[6]*c[5]^3-9*c[7]*c[5]+600*c[6]*c[7]*c[5]^3*c[4]^3+14*c[7]*c[5]^2-100 *c[7]*c[6]*c[4]^3+14*c[7]*c[4]^2-9*c[7]*c[4]-117*c[6]*c[7]*c[5]-111*c[ 6]*c[4]^2+60*c[6]*c[4]^3-117*c[6]*c[7]*c[4]+180*c[6]*c[7]*c[4]^2-30*c[ 5]*c[4]^3-174*c[5]*c[7]*c[4]^2+486*c[5]*c[6]*c[4]^2+107*c[5]*c[4]^2-31 8*c[5]*c[6]*c[4]-66*c[5]*c[4]+75*c[6]*c[4]+500*c[5]*c[7]*c[6]*c[4]^3+1 420*c[5]^2*c[6]*c[7]*c[4]^2-840*c[6]*c[5]*c[7]*c[4]^2+180*c[6]*c[7]*c[ 5]^2-900*c[6]*c[5]^2*c[7]*c[4]^3-900*c[6]*c[5]^3*c[7]*c[4]^2+50*c[7]*c [5]*c[4]^3-280*c[6]*c[5]*c[4]^3-280*c[5]^3*c[6]*c[4]+50*c[5]^3*c[7]*c[ 4]+107*c[5]^2*c[4]+500*c[5]^3*c[6]*c[7]*c[4]+50*c[5]^3*c[4]^2+470*c[5] ^2*c[6]*c[4]^3+310*c[5]^2*c[7]*c[4]^2-770*c[5]^2*c[6]*c[4]^2-90*c[5]^2 *c[7]*c[4]^3-90*c[5]^3*c[7]*c[4]^2+470*c[5]^3*c[6]*c[4]^2-300*c[6]*c[5 ]^3*c[4]^3-180*c[5]^2*c[4]^2+486*c[5]^2*c[6]*c[4]-174*c[5]^2*c[7]*c[4] -840*c[5]^2*c[6]*c[7]*c[4]+50*c[5]^2*c[4]^3)/(-9*c[5]*c[7]*c[4]-3*c[7] *c[6]+3*c[6]^2-28*c[5]*c[6]*c[7]*c[4]-c[6]*c[5]+c[7]*c[5]+8*c[6]^2*c[4 ]^2+c[7]*c[4]+9*c[6]*c[7]*c[5]+9*c[6]*c[7]*c[4]-8*c[6]*c[7]*c[4]^2+4*c [5]*c[7]*c[4]^2-4*c[5]*c[6]*c[4]^2+9*c[5]*c[6]*c[4]-c[6]*c[4]-28*c[6]^ 2*c[4]^2*c[5]+28*c[6]^2*c[4]*c[5]-30*c[5]^2*c[6]*c[7]*c[4]^2+28*c[6]*c [5]*c[7]*c[4]^2-8*c[6]*c[7]*c[5]^2+8*c[6]^2*c[5]^2-9*c[6]^2*c[5]+30*c[ 6]^2*c[5]^2*c[4]^2-28*c[6]^2*c[4]*c[5]^2-9*c[6]^2*c[4]-4*c[5]^2*c[6]*c [4]+4*c[5]^2*c[7]*c[4]+28*c[5]^2*c[6]*c[7]*c[4])/(c[5]*c[4]-c[6]*c[4]- c[6]*c[5]+c[6]^2)/c[6], a[8,7] = (10*c[5]*c[6]*c[4]-5*c[6]*c[5]-5*c[6] *c[4]+3*c[6]-5*c[5]*c[4]+3*c[5]+3*c[4]-2)/c[7]/(c[6]-c[7])/(c[5]-c[7]) /(c[4]-c[7])/(-20*c[5]*c[6]*c[4]+30*c[5]*c[6]*c[7]*c[4]+15*c[6]*c[5]-2 0*c[6]*c[7]*c[5]+15*c[6]*c[4]-20*c[6]*c[7]*c[4]-12*c[6]+15*c[7]*c[6]+1 5*c[5]*c[4]-20*c[5]*c[7]*c[4]-12*c[5]+15*c[7]*c[5]-12*c[4]+15*c[7]*c[4 ]+10-12*c[7])*(c[4]-1)*(-1+c[5])*(-1+c[6])*(-1+c[7]), a[6,3] = 3/4*1/c [4]^2*c[6]*(60*c[4]^3*c[5]*c[6]^3-90*c[5]*c[6]^3*c[4]^2+36*c[6]^3*c[4] *c[5]-6*c[6]^3*c[5]+3*c[6]^3*c[4]-20*c[6]^2*c[5]^3*c[4]^2-80*c[6]^2*c[ 5]^2*c[4]^3+150*c[6]^2*c[5]^2*c[4]^2-48*c[6]^2*c[4]*c[5]^2+8*c[6]^2*c[ 5]^2-60*c[5]*c[6]^2*c[4]^4+90*c[6]^2*c[5]*c[4]^3-48*c[6]^2*c[4]^2*c[5] +4*c[6]^2*c[4]*c[5]-4*c[6]^2*c[4]^2+30*c[6]*c[5]^3*c[4]^3+90*c[5]^2*c[ 6]*c[4]^4-180*c[5]^2*c[6]*c[4]^3+54*c[5]^2*c[6]*c[4]^2-12*c[5]^2*c[6]* c[4]+18*c[6]*c[5]*c[4]^3+6*c[5]*c[6]*c[4]^2+3*c[5]^2*c[4]^2-3*c[5]*c[4 ]^3)/(10*c[5]^3*c[4]^2+10*c[5]^2*c[4]^3-30*c[5]^2*c[4]^2+6*c[5]^2*c[4] -c[5]^2+6*c[5]*c[4]^2+c[5]*c[4]-c[4]^2), b[7] = -1/60*(10*c[5]*c[6]*c[ 4]-5*c[6]*c[5]-5*c[6]*c[4]+3*c[6]-5*c[5]*c[4]+3*c[5]+3*c[4]-2)/(c[4]-c [7])/(c[5]-c[7])/(c[6]-c[7])/c[7]/(-1+c[7]), a[7,1] = 1/4*c[7]*(-51*c[ 5]^2*c[6]*c[4]^4+14*c[6]*c[5]*c[4]^4+12*c[4]^2*c[5]^2*c[7]^3+2700*c[5] ^4*c[4]^4*c[6]^2*c[7]+12*c[7]^3*c[5]*c[4]^3-98*c[7]^3*c[5]^2*c[4]^3-2* c[7]*c[5]*c[4]^4+132*c[5]^2*c[4]^4*c[7]^2-480*c[4]^5*c[6]^2*c[5]^2+200 *c[4]^3*c[7]^3*c[5]^3-200*c[7]^2*c[5]^2*c[4]^5-760*c[4]^5*c[6]*c[5]^3- 10*c[5]*c[6]^2*c[4]^4+200*c[5]^5*c[4]^3*c[7]^2*c[6]^2-400*c[5]^2*c[4]^ 6*c[6]^2*c[7]^2+4*c[4]^5*c[5]*c[7]-34*c[5]^2*c[4]^5*c[7]+400*c[5]^3*c[ 4]^6*c[6]^2*c[7]-98*c[5]^3*c[4]^4*c[7]+420*c[7]^2*c[4]^5*c[5]^3+200*c[ 7]^3*c[5]^2*c[4]^4-300*c[6]^2*c[4]^5*c[7]*c[5]^3-40*c[7]*c[5]^2*c[4]^6 *c[6]+20*c[5]^4*c[4]^2*c[7]^2*c[6]-280*c[6]^2*c[7]*c[5]^4*c[4]^2-44*c[ 6]*c[4]^5*c[5]*c[7]-246*c[6]*c[5]^2*c[4]^4*c[7]+120*c[4]^6*c[6]^2*c[5] ^2-100*c[5]^2*c[7]^2*c[4]^6*c[6]+141*c[5]^4*c[4]^3*c[6]+1220*c[5]^4*c[ 4]^5*c[6]+400*c[5]^5*c[4]^5*c[6]^2-760*c[5]^4*c[4]^4*c[6]+100*c[5]^4*c [7]^3*c[4]^4-400*c[7]^2*c[4]^4*c[5]^3-200*c[5]^4*c[4]^5*c[7]^2+200*c[5 ]^4*c[4]^4*c[7]-2*c[4]^3*c[7]^3*c[6]-8*c[7]^2*c[6]*c[4]^4-380*c[7]^3*c [5]^3*c[4]^4-40*c[6]*c[4]^5*c[5]*c[7]^2+2*c[4]^4*c[7]^2-40*c[5]^2*c[7] ^3*c[4]^5-200*c[7]^2*c[5]^4*c[4]^3-40*c[7]^3*c[5]^4*c[4]^3+200*c[5]^3* c[7]*c[4]^6*c[6]-10*c[6]^2*c[7]^3*c[4]^3+30*c[6]^2*c[4]^5*c[5]+12*c[7] *c[6]^2*c[5]^3+4*c[7]^2*c[6]*c[5]^3+400*c[5]^4*c[4]^6*c[6]^2-40*c[5]^3 *c[7]*c[4]^6-380*c[5]^4*c[4]^5*c[7]+320*c[7]*c[6]^2*c[5]*c[4]^5+100*c[ 4]^6*c[6]^2*c[7]*c[5]^2+34*c[5]^4*c[4]^2*c[7]^2+40*c[5]^5*c[7]^2*c[4]^ 3+240*c[6]*c[5]^2*c[4]^5*c[7]+60*c[5]^4*c[7]^3*c[4]^2*c[6]+26*c[7]*c[4 ]^4*c[6]*c[5]-50*c[4]^4*c[6]^2*c[5]^5+200*c[5]^5*c[7]*c[4]^4*c[6]+132* c[7]^2*c[5]^3*c[4]^3-2*c[7]*c[4]^4*c[6]^2+60*c[5]^5*c[4]^2*c[6]^2*c[7] -500*c[5]^5*c[4]^4*c[6]^2*c[7]+160*c[5]^5*c[4]^4*c[6]+134*c[6]^2*c[5]^ 4*c[7]*c[4]+24*c[5]^4*c[6]^2*c[4]-169*c[5]^4*c[6]^2*c[4]^2-500*c[5]^3* c[4]^6*c[6]^2-40*c[5]^5*c[4]^4*c[7]+34*c[4]^5*c[5]*c[7]^2-100*c[5]^3*c [7]^2*c[4]^6-100*c[5]^5*c[4]^4*c[7]^2+160*c[4]^6*c[6]*c[5]^3-6*c[6]^2* c[7]*c[5]^2*c[4]^3+68*c[6]^2*c[7]^2*c[5]^2*c[4]-126*c[6]^2*c[7]^2*c[5] ^2*c[4]^2-600*c[6]*c[7]*c[5]^3*c[4]^3-21*c[6]^2*c[7]^3*c[4]*c[5]+40*c[ 7]^2*c[6]^2*c[5]^4+4*c[5]^4*c[7]*c[4]^2+58*c[7]^2*c[6]*c[5]*c[4]^4-100 *c[7]^2*c[5]^5*c[6]^2*c[4]^2-320*c[5]^4*c[4]*c[7]^2*c[6]^2+20*c[6]^2*c [4]^4*c[7]^2+40*c[5]^2*c[4]^6*c[7]^2+6*c[7]^2*c[6]*c[4]^3-8*c[6]*c[5]* c[7]^2*c[4]^2-4*c[5]*c[7]*c[6]*c[4]^3+14*c[5]^2*c[6]*c[7]*c[4]^2-910*c [4]^5*c[6]*c[7]*c[5]^3-340*c[5]^2*c[6]*c[7]^3*c[4]^4-4*c[6]^2*c[5]*c[4 ]^3+5*c[7]^3*c[4]^2*c[6]*c[5]+72*c[7]*c[6]^2*c[5]*c[4]^3+18*c[6]^2*c[7 ]^3*c[5]^2-12*c[6]^2*c[7]^2*c[5]^2-10*c[6]^2*c[4]^2*c[7]*c[5]-44*c[6]^ 2*c[7]^2*c[4]^2*c[5]-25*c[6]*c[5]^4*c[4]^2-24*c[7]*c[6]^2*c[5]^4+200*c [4]^5*c[7]*c[5]^3-2*c[6]^2*c[5]^2*c[7]*c[4]-42*c[6]^2*c[5]^2*c[7]*c[4] ^2+74*c[6]*c[5]^2*c[7]*c[4]^3+80*c[6]*c[5]^3*c[7]*c[4]^2-50*c[6]^2*c[5 ]^3*c[7]*c[4]-34*c[5]^4*c[4]^3*c[7]-24*c[7]^2*c[5]^2*c[4]^3+420*c[5]^4 *c[7]^2*c[4]^4-2*c[4]^3*c[7]^3+c[7]^3*c[5]^2*c[6]*c[4]+4*c[7]^2*c[5]^2 *c[6]*c[4]-6*c[6]*c[7]*c[4]^4-4*c[6]^2*c[7]^2*c[4]^2+50*c[5]^5*c[4]^3* c[7]^2*c[6]-18*c[5]^3*c[6]*c[7]*c[4]+14*c[6]^2*c[7]^2*c[4]*c[5]-20*c[5 ]^2*c[6]*c[4]^3+2*c[5]^2*c[7]*c[4]^3-2*c[5]^3*c[7]*c[4]^2+16*c[5]^3*c[ 6]*c[4]^2-96*c[5]^3*c[6]^2*c[4]^3+100*c[5]^5*c[4]^3*c[6]^2*c[7]-60*c[4 ]^3*c[7]^2*c[5]^2*c[6]^2+10*c[6]*c[7]*c[4]^5+1540*c[4]^5*c[5]^3*c[6]^2 -34*c[7]^3*c[4]^4*c[5]-4*c[7]^2*c[5]^4*c[4]+12*c[5]^2*c[4]^4*c[7]-41*c [6]^2*c[5]^2*c[4]^3+14*c[6]^2*c[5]^2*c[4]^2-54*c[6]*c[5]^3*c[4]^3-12*c [6]^2*c[5]^3*c[4]+47*c[6]^2*c[5]^3*c[4]^2+12*c[5]^3*c[7]*c[4]^3-98*c[5 ]^4*c[7]*c[4]^2*c[6]-50*c[5]^3*c[7]^3*c[4]^4*c[6]+4*c[7]*c[6]^2*c[4]^3 +156*c[7]^2*c[5]^2*c[6]*c[4]^3-23*c[7]^3*c[5]^2*c[6]*c[4]^2+188*c[7]^2 *c[5]^3*c[6]*c[4]^2+66*c[5]^3*c[7]^3*c[6]*c[4]+2*c[7]^3*c[4]^2*c[5]+10 0*c[5]^5*c[4]^5*c[7]+104*c[4]^3*c[7]^2*c[5]*c[6]^2-12*c[7]^2*c[6]*c[5] *c[4]^3-36*c[7]^2*c[5]^2*c[6]*c[4]^2-32*c[7]^2*c[5]^3*c[6]*c[4]-320*c[ 7]^2*c[5]^3*c[6]*c[4]^3-560*c[5]^3*c[6]^2*c[4]^4-4*c[6]^2*c[4]^3*c[7]^ 2+240*c[5]^3*c[6]^2*c[7]*c[4]^3-34*c[5]^3*c[7]^3*c[4]^2-2*c[7]^3*c[5]^ 2*c[4]+164*c[6]^2*c[7]*c[5]^3*c[4]^2-20*c[6]^2*c[7]^2*c[4]*c[5]^3+2*c[ 7]^2*c[5]^3*c[4]-130*c[5]^5*c[4]^3*c[6]*c[7]-90*c[7]^3*c[5]^2*c[6]^2*c [4]^3-16*c[7]^2*c[4]^4*c[5]-60*c[7]^2*c[5]^3*c[6]^2*c[4]^2-300*c[6]^2* c[4]^5*c[5]*c[7]^2-150*c[7]^2*c[4]^5*c[6]*c[5]^3+380*c[5]^2*c[6]*c[4]^ 5*c[7]^2+140*c[7]^3*c[5]^2*c[4]^3*c[6]-810*c[7]^3*c[5]^3*c[6]^2*c[4]^2 -14*c[7]^3*c[6]*c[5]*c[4]^3-280*c[7]^3*c[6]^2*c[5]*c[4]^3+240*c[7]^3*c [5]^3*c[6]^2*c[4]-20*c[5]^4*c[7]^2*c[4]*c[6]-6*c[6]*c[7]^3*c[5]^3+22*c [5]^4*c[4]*c[7]*c[6]-280*c[4]^5*c[6]^2*c[7]*c[5]^2+400*c[4]^5*c[6]^2*c [7]^2*c[5]^2+253*c[6]^2*c[4]^4*c[5]^2+470*c[7]^3*c[5]^3*c[4]^3*c[6]+10 0*c[5]^2*c[4]^5*c[7]^3*c[6]+6*c[6]^2*c[7]^3*c[4]^2-300*c[5]^5*c[4]^5*c [6]-120*c[5]*c[7]*c[4]^6*c[6]^2-10*c[6]^2*c[7]*c[4]^5-1100*c[5]^4*c[4] ^5*c[6]^2*c[7]-226*c[7]*c[6]^2*c[5]*c[4]^4-770*c[5]^3*c[4]^4*c[6]^2*c[ 7]+280*c[5]^3*c[4]^3*c[6]^2*c[7]^2-1000*c[5]^4*c[4]^4*c[6]*c[7]+240*c[ 5]^2*c[4]^4*c[7]*c[6]^2+1240*c[5]^3*c[4]^4*c[6]*c[7]+600*c[5]^4*c[4]^3 *c[6]*c[7]+1200*c[7]^3*c[5]^3*c[6]^2*c[4]^3+40*c[7]^3*c[4]^4*c[5]*c[6] -16*c[7]^2*c[5]^3*c[4]^2-1250*c[5]^4*c[4]^5*c[6]^2-600*c[6]^2*c[7]^3*c [5]^3*c[4]^4+200*c[6]^2*c[7]^2*c[5]^3*c[4]^5-200*c[6]^2*c[7]^3*c[4]^5* c[5]-140*c[5]^4*c[4]^4*c[6]^2+560*c[5]^4*c[4]^3*c[6]^2-1850*c[7]^2*c[5 ]^4*c[6]^2*c[4]^3+1230*c[7]^2*c[5]^4*c[6]^2*c[4]^2-150*c[5]^4*c[7]^3*c [4]^3*c[6]-300*c[5]^4*c[4]^6*c[6]+310*c[5]^3*c[4]^4*c[7]^2*c[6]-70*c[5 ]^4*c[4]^3*c[7]^2*c[6]-380*c[5]^2*c[4]^4*c[7]^2*c[6]-850*c[5]^4*c[4]^3 *c[6]^2*c[7]-40*c[7]^2*c[4]^4*c[6]^2*c[5]-450*c[7]^2*c[4]^4*c[5]^3*c[6 ]^2+250*c[7]^2*c[4]^4*c[6]^2*c[5]^2-4*c[7]^2*c[4]^5+136*c[5]^2*c[6]*c[ 4]^5+400*c[6]^2*c[7]^3*c[5]^2*c[4]^5+800*c[6]^2*c[7]^2*c[5]^4*c[4]^4+4 *c[7]^3*c[4]^4+100*c[5]^4*c[4]^6*c[7]+437*c[4]^4*c[6]*c[5]^3-22*c[6]*c [5]*c[4]^5-60*c[5]^5*c[6]^2*c[4]^3+200*c[6]^2*c[4]^6*c[5]*c[7]^2+420*c [7]^3*c[6]^2*c[5]*c[4]^4+400*c[5]^4*c[4]^5*c[6]*c[7]-500*c[7]^3*c[4]^4 *c[6]^2*c[5]^2+100*c[5]^3*c[7]^3*c[4]^5-103*c[6]^2*c[7]^3*c[5]^2*c[4]+ 260*c[6]^2*c[7]^3*c[5]^2*c[4]^2-300*c[5]^3*c[7]^3*c[4]^2*c[6]+97*c[6]^ 2*c[5]*c[7]^3*c[4]^2-30*c[6]^2*c[7]^3*c[5]^3+4*c[7]^3*c[4]*c[5]^3)/c[6 ]/(-50*c[5]^2*c[6]*c[4]^4+23*c[5]^3*c[4]+100*c[5]^4*c[4]^3*c[6]+2*c[5] ^2+2*c[4]^2-3*c[5]^3-3*c[4]^3-3*c[6]*c[5]^2+5*c[6]*c[5]^3-3*c[6]*c[4]^ 2+5*c[6]*c[4]^3+23*c[5]*c[4]^3+18*c[5]*c[6]*c[4]^2-12*c[5]*c[4]^2+3*c[ 5]*c[6]*c[4]-2*c[5]*c[4]-50*c[6]*c[5]^4*c[4]^2-40*c[6]*c[5]*c[4]^3-40* c[5]^3*c[6]*c[4]-12*c[5]^2*c[4]-140*c[5]^3*c[4]^2+240*c[5]^2*c[6]*c[4] ^3-140*c[5]^2*c[6]*c[4]^2+240*c[5]^3*c[6]*c[4]^2-400*c[6]*c[5]^3*c[4]^ 3+91*c[5]^2*c[4]^2+30*c[5]^4*c[4]^2+30*c[4]^4*c[5]^2-50*c[5]^4*c[4]^3+ 18*c[5]^2*c[6]*c[4]-140*c[5]^2*c[4]^3+210*c[5]^3*c[4]^3+100*c[4]^4*c[6 ]*c[5]^3-50*c[5]^3*c[4]^4)/c[5]/c[4]^2, `b*`[8] = 0, a[5,3] = -3/4*c[5 ]^2*(2*c[5]-3*c[4])/c[4]^2, `b*`[4] = -1/60*(-30*c[5]^3*c[4]-42*c[7]*c [6]^2+89*c[5]*c[7]*c[4]-18*c[6]+6*c[5]+6*c[4]+27*c[7]*c[6]-9*c[5]^2+18 0*c[7]*c[6]^2*c[5]^3-100*c[7]*c[6]*c[5]^3+27*c[6]^2+186*c[5]*c[6]*c[7] *c[4]-97*c[6]*c[5]^2+66*c[6]*c[5]+60*c[6]*c[5]^3-100*c[6]^2*c[5]^3-9*c [7]*c[5]+14*c[7]*c[5]^2-100*c[6]^2*c[7]*c[4]^2+120*c[6]^2*c[7]*c[4]+60 *c[6]^2*c[4]^2-9*c[7]*c[4]-103*c[6]*c[7]*c[5]-36*c[6]*c[4]^2-61*c[6]*c [7]*c[4]+60*c[6]*c[7]*c[4]^2-30*c[5]*c[7]*c[4]^2+126*c[5]*c[6]*c[4]^2+ 18*c[5]*c[4]^2-112*c[5]*c[6]*c[4]-57*c[5]*c[4]+39*c[6]*c[4]-280*c[6]^2 *c[4]^2*c[5]+330*c[6]^2*c[4]*c[5]+380*c[5]^2*c[6]*c[7]*c[4]^2-560*c[6] ^2*c[7]*c[4]*c[5]+192*c[6]^2*c[7]*c[5]-310*c[6]^2*c[7]*c[5]^2-230*c[6] *c[5]*c[7]*c[4]^2+156*c[6]*c[7]*c[5]^2+180*c[6]^2*c[5]^2+500*c[6]^2*c[ 4]^2*c[7]*c[5]+950*c[6]^2*c[5]^2*c[7]*c[4]-900*c[6]^2*c[5]^2*c[7]*c[4] ^2-300*c[6]*c[5]^3*c[7]*c[4]^2-600*c[6]^2*c[5]^3*c[7]*c[4]-117*c[6]^2* c[5]-130*c[5]^3*c[6]*c[4]+50*c[5]^3*c[7]*c[4]+93*c[5]^2*c[4]+230*c[5]^ 3*c[6]*c[7]*c[4]+50*c[5]^2*c[7]*c[4]^2-190*c[5]^2*c[6]*c[4]^2+150*c[5] ^3*c[6]*c[4]^2+470*c[6]^2*c[5]^2*c[4]^2+320*c[6]^2*c[5]^3*c[4]-300*c[6 ]^2*c[5]^3*c[4]^2-30*c[5]^2*c[4]^2-530*c[6]^2*c[4]*c[5]^2-75*c[6]^2*c[ 4]+156*c[5]^2*c[6]*c[4]-150*c[5]^2*c[7]*c[4]-270*c[5]^2*c[6]*c[7]*c[4] +600*c[6]^2*c[7]*c[5]^3*c[4]^2)/(-9*c[5]*c[7]*c[4]-3*c[7]*c[6]-c[4]^2- 28*c[5]*c[6]*c[7]*c[4]+c[7]*c[5]+c[7]*c[4]+9*c[6]*c[7]*c[5]-9*c[6]*c[4 ]^2+8*c[6]*c[4]^3+9*c[6]*c[7]*c[4]-8*c[6]*c[7]*c[4]^2-4*c[5]*c[4]^3+4* c[5]*c[7]*c[4]^2+28*c[5]*c[6]*c[4]^2+9*c[5]*c[4]^2-9*c[5]*c[6]*c[4]-c[ 5]*c[4]+3*c[6]*c[4]-30*c[5]^2*c[6]*c[7]*c[4]^2+28*c[6]*c[5]*c[7]*c[4]^ 2-8*c[6]*c[7]*c[5]^2-28*c[6]*c[5]*c[4]^3+30*c[5]^2*c[6]*c[4]^3-28*c[5] ^2*c[6]*c[4]^2-4*c[5]^2*c[4]^2+8*c[5]^2*c[6]*c[4]+4*c[5]^2*c[7]*c[4]+2 8*c[5]^2*c[6]*c[7]*c[4])/(-c[4]+c[6])/(-c[4]+c[5])/c[4], a[8,6] = 1/2* (5*c[5]*c[6]*c[4]+c[6]-2*c[6]*c[4]-2*c[6]*c[5]+25*c[5]*c[7]*c[4]+6*c[5 ]+9*c[7]+10*c[7]^2*c[4]-4+6*c[4]-6*c[7]^2-14*c[7]*c[4]-10*c[5]*c[4]-14 *c[7]*c[5]-20*c[7]^2*c[5]*c[4]+10*c[7]^2*c[5])/(c[6]-c[7])/(-c[4]+c[6] )/(-c[5]+c[6])/c[6]/(-20*c[5]*c[6]*c[4]+30*c[5]*c[6]*c[7]*c[4]+15*c[6] *c[5]-20*c[6]*c[7]*c[5]+15*c[6]*c[4]-20*c[6]*c[7]*c[4]-12*c[6]+15*c[7] *c[6]+15*c[5]*c[4]-20*c[5]*c[7]*c[4]-12*c[5]+15*c[7]*c[5]-12*c[4]+15*c [7]*c[4]+10-12*c[7])*(c[4]-1)*(-1+c[5])*(-1+c[6]), b[8] = 1/60*(-20*c[ 5]*c[6]*c[4]+30*c[5]*c[6]*c[7]*c[4]+15*c[6]*c[5]-20*c[6]*c[7]*c[5]+15* c[6]*c[4]-20*c[6]*c[7]*c[4]-12*c[6]+15*c[7]*c[6]+15*c[5]*c[4]-20*c[5]* c[7]*c[4]-12*c[5]+15*c[7]*c[5]-12*c[4]+15*c[7]*c[4]+10-12*c[7])/(c[4]- 1)/(-1+c[5])/(-1+c[6])/(-1+c[7]), a[5,1] = 1/4*c[5]*(2*c[5]^2-5*c[5]*c [4]+4*c[4]^2)/c[4]^2, `b*`[5] = 1/60*(-42*c[7]*c[6]^2+89*c[5]*c[7]*c[4 ]-18*c[6]+6*c[5]+6*c[4]+27*c[7]*c[6]-9*c[4]^2+27*c[6]^2+186*c[5]*c[6]* c[7]*c[4]-36*c[6]*c[5]^2+39*c[6]*c[5]-9*c[7]*c[5]-100*c[6]^2*c[4]^3+60 0*c[6]^2*c[7]*c[5]^2*c[4]^3-100*c[7]*c[6]*c[4]^3-310*c[6]^2*c[7]*c[4]^ 2+192*c[6]^2*c[7]*c[4]+180*c[6]^2*c[4]^2+14*c[7]*c[4]^2-9*c[7]*c[4]-61 *c[6]*c[7]*c[5]-97*c[6]*c[4]^2+60*c[6]*c[4]^3-103*c[6]*c[7]*c[4]+156*c [6]*c[7]*c[4]^2-30*c[5]*c[4]^3-150*c[5]*c[7]*c[4]^2+156*c[5]*c[6]*c[4] ^2+93*c[5]*c[4]^2-112*c[5]*c[6]*c[4]-57*c[5]*c[4]+66*c[6]*c[4]+230*c[5 ]*c[7]*c[6]*c[4]^3-530*c[6]^2*c[4]^2*c[5]+330*c[6]^2*c[4]*c[5]+380*c[5 ]^2*c[6]*c[7]*c[4]^2+320*c[6]^2*c[5]*c[4]^3-560*c[6]^2*c[7]*c[4]*c[5]- 600*c[7]*c[6]^2*c[5]*c[4]^3+120*c[6]^2*c[7]*c[5]-100*c[6]^2*c[7]*c[5]^ 2-270*c[6]*c[5]*c[7]*c[4]^2+60*c[6]*c[7]*c[5]^2+60*c[6]^2*c[5]^2+950*c [6]^2*c[4]^2*c[7]*c[5]+500*c[6]^2*c[5]^2*c[7]*c[4]-900*c[6]^2*c[5]^2*c [7]*c[4]^2-300*c[6]*c[5]^2*c[7]*c[4]^3-75*c[6]^2*c[5]+50*c[7]*c[5]*c[4 ]^3-130*c[6]*c[5]*c[4]^3+18*c[5]^2*c[4]+150*c[5]^2*c[6]*c[4]^3+50*c[5] ^2*c[7]*c[4]^2-190*c[5]^2*c[6]*c[4]^2-300*c[6]^2*c[5]^2*c[4]^3+470*c[6 ]^2*c[5]^2*c[4]^2-30*c[5]^2*c[4]^2-280*c[6]^2*c[4]*c[5]^2+180*c[7]*c[6 ]^2*c[4]^3-117*c[6]^2*c[4]+126*c[5]^2*c[6]*c[4]-30*c[5]^2*c[7]*c[4]-23 0*c[5]^2*c[6]*c[7]*c[4])/(-4*c[5]^3*c[4]-9*c[5]*c[7]*c[4]-3*c[7]*c[6]- c[5]^2-28*c[5]*c[6]*c[7]*c[4]-9*c[6]*c[5]^2+3*c[6]*c[5]+8*c[6]*c[5]^3+ c[7]*c[5]+c[7]*c[4]+9*c[6]*c[7]*c[5]+9*c[6]*c[7]*c[4]-8*c[6]*c[7]*c[4] ^2+4*c[5]*c[7]*c[4]^2+8*c[5]*c[6]*c[4]^2-9*c[5]*c[6]*c[4]-c[5]*c[4]-30 *c[5]^2*c[6]*c[7]*c[4]^2+28*c[6]*c[5]*c[7]*c[4]^2-8*c[6]*c[7]*c[5]^2-2 8*c[5]^3*c[6]*c[4]+9*c[5]^2*c[4]-28*c[5]^2*c[6]*c[4]^2+30*c[5]^3*c[6]* c[4]^2-4*c[5]^2*c[4]^2+28*c[5]^2*c[6]*c[4]+4*c[5]^2*c[7]*c[4]+28*c[5]^ 2*c[6]*c[7]*c[4])/(-c[5]+c[6])/(-c[4]+c[5])/c[5], a[5,4] = c[5]^2*(-c[ 4]+c[5])/c[4]^2, a[7,6] = 1/2*(5*c[5]*c[4]+1-2*c[4]-2*c[5])*(c[4]-c[7] )*(c[5]-c[7])*(c[6]-c[7])*c[7]/(10*c[5]*c[6]*c[4]-5*c[6]*c[5]-5*c[6]*c [4]+3*c[6]-5*c[5]*c[4]+3*c[5]+3*c[4]-2)/(-c[4]+c[6])/(-c[5]+c[6])/c[6] , c[3] = 2/3*c[4], a[4,3] = 3/4*c[4], a[3,2] = 2/9*c[4]^2/c[2], a[7,4] = -1/2*c[7]*(-68*c[5]^2*c[6]*c[4]^4-7*c[6]*c[5]*c[4]^4+7*c[4]^2*c[5]^ 2*c[7]^3-1400*c[5]^4*c[4]^4*c[6]^2*c[7]+11*c[7]^3*c[5]*c[4]^3-72*c[7]^ 3*c[5]^2*c[4]^3-3*c[7]*c[5]*c[4]^4-58*c[5]^2*c[4]^4*c[7]^2-120*c[6]*c[ 4]^7*c[7]*c[5]+320*c[4]^5*c[6]^2*c[5]^2-200*c[4]^3*c[7]^3*c[5]^3-100*c [4]^5*c[6]*c[5]^3-2*c[5]*c[6]^2*c[4]^4-600*c[4]^5*c[7]^3*c[6]*c[5]^3-2 00*c[5]^5*c[4]^3*c[7]^2*c[6]^2+400*c[5]^2*c[4]^6*c[6]^2*c[7]^2-21*c[4] ^5*c[5]*c[7]+120*c[5]^2*c[4]^5*c[7]-600*c[5]^3*c[4]^6*c[6]^2*c[7]-198* c[5]^3*c[4]^4*c[7]-280*c[7]^2*c[4]^5*c[5]^3+240*c[7]^3*c[5]^2*c[4]^4+1 250*c[6]^2*c[4]^5*c[7]*c[5]^3+120*c[7]*c[5]^2*c[4]^6*c[6]-340*c[5]^4*c [4]^2*c[7]^2*c[6]+400*c[6]^2*c[7]*c[5]^4*c[4]^2-240*c[5]^2*c[4]^6*c[7] -300*c[6]*c[5]*c[7]^2*c[4]^6-228*c[6]*c[4]^5*c[5]*c[7]-119*c[6]*c[5]^2 *c[4]^4*c[7]-200*c[5]*c[7]^3*c[4]^6*c[6]-600*c[5]^4*c[7]*c[4]^6*c[6]-1 20*c[4]^6*c[6]^2*c[5]^2+600*c[5]^2*c[7]^2*c[4]^6*c[6]+100*c[5]^2*c[7]^ 3*c[4]^6-187*c[5]^4*c[4]^3*c[6]-910*c[5]^4*c[4]^5*c[6]-320*c[5]^2*c[6] *c[4]^6+620*c[5]^4*c[4]^4*c[6]-100*c[5]^4*c[7]^3*c[4]^4-40*c[7]^2*c[4] ^4*c[5]^3+40*c[7]^3*c[4]^5*c[5]+10*c[6]*c[7]^2*c[4]^5-10*c[6]*c[7]^3*c [4]^4+100*c[5]^4*c[4]^5*c[7]^2+80*c[5]^4*c[4]^4*c[7]+8*c[4]^3*c[7]^3*c [6]-2*c[7]^2*c[6]*c[4]^4+280*c[7]^3*c[5]^3*c[4]^4+80*c[6]*c[4]^5*c[5]* c[7]^2+c[4]^4*c[7]^2-360*c[5]^2*c[7]^3*c[4]^5-80*c[7]^2*c[5]^4*c[4]^3+ 40*c[7]^3*c[5]^4*c[4]^3-1200*c[5]^3*c[7]*c[4]^6*c[6]+10*c[6]^2*c[7]^3* c[4]^3-10*c[6]^2*c[4]^5*c[5]-12*c[7]*c[6]^2*c[5]^3-4*c[7]^2*c[6]*c[5]^ 3-80*c[5]^4*c[4]^5*c[7]-240*c[7]*c[6]^2*c[5]*c[4]^5-3*c[5]^4*c[4]^2*c[ 7]^2+20*c[5]^5*c[7]^2*c[4]^3+180*c[6]*c[5]^2*c[4]^5*c[7]-60*c[5]^4*c[7 ]^3*c[4]^2*c[6]+400*c[5]^2*c[7]^3*c[4]^6*c[6]-25*c[4]^5*c[5]^3+200*c[5 ]^5*c[4]^4*c[7]^2*c[6]+60*c[5]^2*c[4]^7*c[7]+100*c[7]*c[4]^4*c[6]*c[5] -40*c[5]*c[4]^6*c[7]^2-150*c[4]^4*c[6]^2*c[5]^5+50*c[5]^5*c[7]*c[4]^4* c[6]+174*c[7]^2*c[5]^3*c[4]^3+8*c[7]*c[4]^4*c[6]^2+600*c[5]^3*c[7]*c[4 ]^7*c[6]-60*c[5]^5*c[4]^2*c[6]^2*c[7]+200*c[5]^5*c[4]^4*c[6]^2*c[7]-40 *c[5]^5*c[4]^4*c[6]-200*c[5]^5*c[7]*c[4]^5*c[6]-149*c[6]^2*c[5]^4*c[7] *c[4]-24*c[5]^4*c[6]^2*c[4]+189*c[5]^4*c[6]^2*c[4]^2+300*c[5]^3*c[4]^6 *c[6]^2-20*c[5]^5*c[4]^4*c[7]-5*c[4]^5*c[5]*c[7]^2+600*c[4]^6*c[6]*c[5 ]^3-84*c[6]^2*c[7]*c[5]^2*c[4]^3-74*c[6]^2*c[7]^2*c[5]^2*c[4]+162*c[6] ^2*c[7]^2*c[5]^2*c[4]^2+311*c[6]*c[7]*c[5]^3*c[4]^3+21*c[6]^2*c[7]^3*c [4]*c[5]-40*c[7]^2*c[6]^2*c[5]^4+11*c[5]^4*c[7]*c[4]^2+107*c[7]^2*c[6] *c[5]*c[4]^4+100*c[7]^2*c[5]^5*c[6]^2*c[4]^2+320*c[5]^4*c[4]*c[7]^2*c[ 6]^2-10*c[6]^2*c[4]^4*c[7]^2+300*c[5]^2*c[4]^6*c[7]^2-3*c[7]^2*c[6]*c[ 4]^3-2*c[5]*c[7]^2*c[4]^3+15*c[6]*c[5]*c[7]^2*c[4]^2-8*c[5]*c[7]*c[6]* c[4]^3-14*c[5]^2*c[6]*c[7]*c[4]^2-2000*c[5]^4*c[7]^2*c[4]^4*c[6]+200*c [4]^5*c[6]*c[7]*c[5]^3+250*c[5]^2*c[6]*c[7]^3*c[4]^4+4*c[6]^2*c[5]*c[4 ]^3-26*c[7]^3*c[4]^2*c[6]*c[5]-69*c[7]*c[6]^2*c[5]*c[4]^3-18*c[6]^2*c[ 7]^3*c[5]^2+12*c[6]^2*c[7]^2*c[5]^2+10*c[6]^2*c[4]^2*c[7]*c[5]+50*c[6] ^2*c[7]^2*c[4]^2*c[5]+31*c[6]*c[5]^4*c[4]^2+24*c[7]*c[6]^2*c[5]^4+240* c[4]^5*c[7]*c[5]^3+2*c[6]^2*c[5]^2*c[7]*c[4]+39*c[6]^2*c[5]^2*c[7]*c[4 ]^2-17*c[6]*c[5]^2*c[7]*c[4]^3-73*c[6]*c[5]^3*c[7]*c[4]^2+59*c[6]^2*c[ 5]^3*c[7]*c[4]-12*c[5]^4*c[4]^3*c[7]-12*c[7]^2*c[5]^2*c[4]^3+40*c[5]^4 *c[7]^2*c[4]^4-c[4]^3*c[7]^3+17*c[7]^3*c[5]^2*c[6]*c[4]-9*c[7]^2*c[5]^ 2*c[6]*c[4]+3*c[6]*c[7]*c[4]^4+4*c[4]^5*c[5]+4*c[6]^2*c[7]^2*c[4]^2-15 0*c[5]^5*c[4]^3*c[7]^2*c[6]+21*c[5]^3*c[6]*c[7]*c[4]-14*c[6]^2*c[7]^2* c[4]*c[5]+23*c[5]^2*c[6]*c[4]^3+13*c[5]^2*c[7]*c[4]^3-11*c[5]^3*c[7]*c [4]^2-17*c[5]^3*c[6]*c[4]^2+168*c[5]^3*c[6]^2*c[4]^3+50*c[5]^5*c[4]^3* c[6]^2*c[7]-220*c[4]^3*c[7]^2*c[5]^2*c[6]^2-6*c[6]*c[7]*c[4]^5-580*c[4 ]^5*c[5]^3*c[6]^2-31*c[7]^3*c[4]^4*c[5]-2*c[7]^2*c[5]^4*c[4]-4*c[5]^2* c[4]^4*c[7]+53*c[6]^2*c[5]^2*c[4]^3-14*c[6]^2*c[5]^2*c[4]^2+47*c[6]*c[ 5]^3*c[4]^3+12*c[6]^2*c[5]^3*c[4]-59*c[6]^2*c[5]^3*c[4]^2+28*c[5]^3*c[ 7]*c[4]^3+127*c[5]^4*c[7]*c[4]^2*c[6]+1250*c[5]^3*c[7]^3*c[4]^4*c[6]+1 0*c[6]*c[5]*c[4]^6+240*c[5]*c[4]^6*c[7]*c[6]-4*c[7]*c[6]^2*c[4]^3+11*c [7]^2*c[5]^2*c[6]*c[4]^3-80*c[7]^3*c[5]^2*c[6]*c[4]^2-121*c[7]^2*c[5]^ 3*c[6]*c[4]^2-96*c[5]^3*c[7]^3*c[6]*c[4]+c[7]^3*c[4]^2*c[5]-68*c[4]^3* c[7]^2*c[5]*c[6]^2-80*c[7]^2*c[6]*c[5]*c[4]^3+68*c[7]^2*c[5]^2*c[6]*c[ 4]^2-10*c[4]^4*c[5]^2+20*c[7]^2*c[5]^3*c[6]*c[4]-240*c[7]^2*c[5]^3*c[6 ]*c[4]^3-9*c[5]^4*c[4]^3-2*c[6]^2*c[4]^3*c[7]^2+260*c[5]^3*c[6]^2*c[7] *c[4]^3+49*c[5]^3*c[7]^3*c[4]^2-c[7]^3*c[5]^2*c[4]-228*c[6]^2*c[7]*c[5 ]^3*c[4]^2+30*c[6]^2*c[7]^2*c[4]*c[5]^3-2*c[7]^2*c[5]^2*c[4]^2+5*c[7]^ 2*c[5]^3*c[4]+40*c[5]^5*c[4]^3*c[6]*c[7]+36*c[4]^6*c[7]*c[5]+90*c[7]^3 *c[5]^2*c[6]^2*c[4]^3+16*c[7]^2*c[4]^4*c[5]-20*c[7]^2*c[5]^3*c[6]^2*c[ 4]^2+300*c[6]^2*c[4]^5*c[5]*c[7]^2+250*c[7]^2*c[4]^5*c[6]*c[5]^3-490*c [5]^2*c[6]*c[4]^5*c[7]^2+120*c[7]^3*c[5]^2*c[4]^3*c[6]+810*c[7]^3*c[5] ^3*c[6]^2*c[4]^2+111*c[7]^3*c[6]*c[5]*c[4]^3+280*c[7]^3*c[6]^2*c[5]*c[ 4]^3-240*c[7]^3*c[5]^3*c[6]^2*c[4]+15*c[5]^4*c[4]^4+60*c[5]^4*c[7]^2*c [4]*c[6]+6*c[6]*c[7]^3*c[5]^3-31*c[5]^4*c[4]*c[7]*c[6]-200*c[4]^5*c[6] ^2*c[7]*c[5]^2-500*c[4]^5*c[6]^2*c[7]^2*c[5]^2+9*c[4]^5*c[5]^2-181*c[6 ]^2*c[4]^4*c[5]^2-1280*c[7]^3*c[5]^3*c[4]^3*c[6]-600*c[5]^2*c[4]^5*c[7 ]^3*c[6]-6*c[4]^6*c[5]-6*c[6]^2*c[7]^3*c[4]^2+100*c[5]^5*c[4]^5*c[6]+1 20*c[5]*c[7]*c[4]^6*c[6]^2+420*c[6]*c[4]^5*c[7]^3*c[5]+600*c[5]^4*c[4] ^5*c[6]^2*c[7]+175*c[7]*c[6]^2*c[5]*c[4]^4-430*c[5]^3*c[4]^4*c[6]^2*c[ 7]+200*c[5]^3*c[4]^3*c[6]^2*c[7]^2-450*c[5]^4*c[4]^4*c[6]*c[7]+160*c[5 ]^2*c[4]^4*c[7]*c[6]^2-200*c[5]^3*c[4]^4*c[6]*c[7]-280*c[5]^4*c[4]^3*c [6]*c[7]+6*c[5]^3*c[4]^3-100*c[5]^2*c[4]^7*c[7]^2-1200*c[7]^3*c[5]^3*c [6]^2*c[4]^3-320*c[7]^3*c[4]^4*c[5]*c[6]-30*c[7]^2*c[5]^3*c[4]^2-300*c [6]*c[5]^3*c[4]^7-350*c[5]^4*c[4]^5*c[6]^2+600*c[6]^2*c[7]^3*c[5]^3*c[ 4]^4+600*c[6]*c[7]^2*c[5]^4*c[4]^5+200*c[6]^2*c[7]^3*c[4]^5*c[5]+200*c [7]^2*c[4]^7*c[6]*c[5]+1100*c[5]^4*c[4]^4*c[6]^2-720*c[5]^4*c[4]^3*c[6 ]^2+1750*c[7]^2*c[5]^4*c[6]^2*c[4]^3-1230*c[7]^2*c[5]^4*c[6]^2*c[4]^2+ 150*c[5]^4*c[7]^3*c[4]^3*c[6]+300*c[5]^4*c[4]^6*c[6]+480*c[5]^3*c[4]^4 *c[7]^2*c[6]+1420*c[5]^4*c[4]^3*c[7]^2*c[6]+20*c[5]^2*c[4]^4*c[7]^2*c[ 6]+130*c[5]^4*c[4]^3*c[6]^2*c[7]-40*c[7]^2*c[4]^4*c[6]^2*c[5]-350*c[7] ^2*c[4]^4*c[5]^3*c[6]^2+230*c[7]^2*c[4]^4*c[6]^2*c[5]^2+10*c[4]^6*c[5] ^2+188*c[5]^2*c[6]*c[4]^5-400*c[6]^2*c[7]^3*c[5]^2*c[4]^5-600*c[6]^2*c [7]^2*c[5]^4*c[4]^4+120*c[5]^2*c[4]^7*c[6]-94*c[4]^4*c[6]*c[5]^3+6*c[6 ]*c[5]*c[4]^5+60*c[5]^5*c[6]^2*c[4]^3-200*c[6]^2*c[4]^6*c[5]*c[7]^2-42 0*c[7]^3*c[6]^2*c[5]*c[4]^4+1550*c[5]^4*c[4]^5*c[6]*c[7]+500*c[7]^3*c[ 4]^4*c[6]^2*c[5]^2-400*c[5]^2*c[4]^7*c[6]*c[7]^2+6*c[5]^3*c[4]^4+103*c [6]^2*c[7]^3*c[5]^2*c[4]-260*c[6]^2*c[7]^3*c[5]^2*c[4]^2+540*c[5]^3*c[ 7]^3*c[4]^2*c[6]-97*c[6]^2*c[5]*c[7]^3*c[4]^2+30*c[6]^2*c[7]^3*c[5]^3- 4*c[7]^3*c[4]*c[5]^3)/(-240*c[5]^2*c[6]*c[4]^4+40*c[6]*c[5]*c[4]^4-9*c [5]^3*c[4]-100*c[4]^5*c[6]*c[5]^3-350*c[5]^4*c[4]^3*c[6]+30*c[5]^5*c[4 ]^2+2*c[5]^3-2*c[4]^3-50*c[5]^5*c[4]^3-3*c[5]^4+3*c[4]^4+50*c[4]^5*c[5 ]^3+100*c[5]^5*c[4]^3*c[6]-3*c[6]*c[5]^3+3*c[6]*c[4]^3+9*c[5]*c[4]^3-6 *c[5]*c[6]*c[4]^2+4*c[5]*c[4]^2+240*c[6]*c[5]^4*c[4]^2+5*c[6]*c[5]^4-1 3*c[6]*c[5]*c[4]^3+13*c[5]^3*c[6]*c[4]-4*c[5]^2*c[4]+68*c[5]^3*c[4]^2+ 100*c[5]^2*c[6]*c[4]^3-100*c[5]^3*c[6]*c[4]^2-140*c[5]^4*c[4]^2+140*c[ 4]^4*c[5]^2+180*c[5]^4*c[4]^3+6*c[5]^2*c[6]*c[4]-68*c[5]^2*c[4]^3-23*c [4]^4*c[5]-40*c[5]^4*c[4]*c[6]-30*c[4]^5*c[5]^2-5*c[6]*c[4]^4-50*c[5]^ 5*c[6]*c[4]^2+50*c[5]^2*c[6]*c[4]^5+350*c[4]^4*c[6]*c[5]^3+23*c[5]^4*c [4]-180*c[5]^3*c[4]^4)/(-c[4]+c[6])/c[4]^2, `b*`[7] = 1/60*(-30*c[5]^3 *c[4]-18*c[6]+6*c[5]+6*c[4]-9*c[5]^2-9*c[4]^2+27*c[6]^2-97*c[6]*c[5]^2 +66*c[6]*c[5]+60*c[6]*c[5]^3-100*c[6]^2*c[5]^3-100*c[6]^2*c[4]^3+180*c [6]^2*c[4]^2-97*c[6]*c[4]^2+60*c[6]*c[4]^3-30*c[5]*c[4]^3+312*c[5]*c[6 ]*c[4]^2+107*c[5]*c[4]^2-215*c[5]*c[6]*c[4]-66*c[5]*c[4]+66*c[6]*c[4]- 840*c[6]^2*c[4]^2*c[5]+522*c[6]^2*c[4]*c[5]+500*c[6]^2*c[5]*c[4]^3+180 *c[6]^2*c[5]^2-117*c[6]^2*c[5]-230*c[6]*c[5]*c[4]^3-230*c[5]^3*c[6]*c[ 4]+107*c[5]^2*c[4]+50*c[5]^3*c[4]^2+380*c[5]^2*c[6]*c[4]^3-460*c[5]^2* c[6]*c[4]^2+380*c[5]^3*c[6]*c[4]^2+600*c[5]^3*c[6]^2*c[4]^3-900*c[6]^2 *c[5]^2*c[4]^3+1420*c[6]^2*c[5]^2*c[4]^2-300*c[6]*c[5]^3*c[4]^3+500*c[ 6]^2*c[5]^3*c[4]-900*c[6]^2*c[5]^3*c[4]^2-180*c[5]^2*c[4]^2-840*c[6]^2 *c[4]*c[5]^2-117*c[6]^2*c[4]+312*c[5]^2*c[6]*c[4]+50*c[5]^2*c[4]^3)/c[ 7]/(-3*c[7]^3*c[6]+3*c[7]^2*c[6]^2-9*c[6]^2*c[7]^2*c[5]-8*c[7]*c[6]^2* c[5]^3+8*c[7]^2*c[6]*c[5]^3-c[5]*c[6]*c[7]*c[4]-c[7]^2*c[4]^2-30*c[6]^ 2*c[7]*c[5]^2*c[4]^3-28*c[6]^2*c[7]^2*c[5]^2*c[4]+30*c[6]^2*c[7]^2*c[5 ]^2*c[4]^2-30*c[6]*c[7]*c[5]^3*c[4]^3+9*c[6]^2*c[7]*c[4]^2-3*c[6]^2*c[ 7]*c[4]+2*c[6]*c[7]^2*c[4]-9*c[6]*c[7]^2*c[4]^2-2*c[7]^2*c[5]*c[4]-c[7 ]^2*c[5]^2+c[6]*c[7]*c[4]^2+8*c[7]^2*c[6]*c[4]^3-4*c[5]*c[7]^2*c[4]^3+ c[5]*c[7]*c[4]^2-c[5]*c[6]*c[4]^2+32*c[6]*c[5]*c[7]^2*c[4]^2-9*c[7]^2* c[5]*c[6]*c[4]-4*c[5]*c[7]*c[6]*c[4]^3-9*c[6]^2*c[4]^2*c[5]+3*c[6]^2*c [4]*c[5]+c[7]^3*c[4]+9*c[7]^3*c[4]*c[6]-9*c[7]^3*c[4]*c[5]-20*c[5]^2*c [6]*c[7]*c[4]^2+8*c[6]^2*c[5]*c[4]^3+28*c[7]^3*c[4]^2*c[6]*c[5]+18*c[6 ]^2*c[7]*c[4]*c[5]+28*c[7]*c[6]^2*c[5]*c[4]^3-3*c[6]^2*c[7]*c[5]+9*c[6 ]^2*c[7]*c[5]^2+c[6]*c[7]*c[5]^2-8*c[6]*c[7]^3*c[5]^2-9*c[6]*c[7]^2*c[ 5]^2+8*c[6]^2*c[7]^2*c[5]^2+2*c[6]*c[7]^2*c[5]-36*c[6]^2*c[4]^2*c[7]*c [5]-28*c[6]^2*c[7]^2*c[4]^2*c[5]-28*c[7]^3*c[4]*c[6]*c[5]-36*c[6]^2*c[ 5]^2*c[7]*c[4]+56*c[6]^2*c[5]^2*c[7]*c[4]^2+28*c[6]*c[5]^2*c[7]*c[4]^3 +28*c[6]*c[5]^3*c[7]*c[4]^2+28*c[6]^2*c[5]^3*c[7]*c[4]+9*c[6]*c[7]^3*c [5]+9*c[7]^2*c[5]*c[4]^2+28*c[7]^3*c[5]^2*c[6]*c[4]+32*c[7]^2*c[5]^2*c [6]*c[4]+8*c[6]^2*c[7]^2*c[4]^2-9*c[6]^2*c[7]^2*c[4]-4*c[5]^3*c[6]*c[7 ]*c[4]+28*c[6]^2*c[7]^2*c[4]*c[5]-4*c[5]^2*c[6]*c[4]^3-9*c[5]^2*c[7]*c [4]^2+9*c[5]^2*c[6]*c[4]^2+4*c[5]^2*c[7]*c[4]^3+4*c[5]^3*c[7]*c[4]^2-4 *c[5]^3*c[6]*c[4]^2+30*c[5]^3*c[6]^2*c[4]^3-28*c[6]^2*c[5]^2*c[4]^3+28 *c[6]^2*c[5]^2*c[4]^2+8*c[6]^2*c[5]^3*c[4]-28*c[6]^2*c[5]^3*c[4]^2-9*c [6]^2*c[4]*c[5]^2-8*c[7]*c[6]^2*c[4]^3+30*c[7]^2*c[5]^2*c[6]*c[4]^3-30 *c[7]^3*c[5]^2*c[6]*c[4]^2+30*c[7]^2*c[5]^3*c[6]*c[4]^2-8*c[7]^3*c[4]^ 2*c[6]+4*c[7]^3*c[4]^2*c[5]+c[7]^3*c[5]-28*c[7]^2*c[6]*c[5]*c[4]^3-56* c[7]^2*c[5]^2*c[6]*c[4]^2-28*c[7]^2*c[5]^3*c[6]*c[4]-c[5]^2*c[6]*c[4]+ c[5]^2*c[7]*c[4]+4*c[7]^3*c[5]^2*c[4]+9*c[7]^2*c[5]^2*c[4]-30*c[6]^2*c [7]*c[5]^3*c[4]^2-8*c[7]^2*c[5]^2*c[4]^2-4*c[7]^2*c[5]^3*c[4]), a[2,1] = c[2], a[6,4] = -1/2*(120*c[5]^2*c[6]*c[4]^4-12*c[6]*c[5]*c[4]^4-90* c[5]^2*c[6]^3*c[4]^2-60*c[5]*c[6]^2*c[4]^4-2*c[6]^3*c[4]^2+20*c[5]^4*c [4]^3*c[6]+40*c[6]^2*c[4]^5*c[5]+36*c[4]*c[5]^2*c[6]^3-20*c[5]^4*c[6]^ 2*c[4]^2+8*c[6]^2*c[5]^3+2*c[6]^2*c[4]^3-24*c[5]*c[6]^3*c[4]^2-6*c[6]^ 2*c[4]^2*c[5]+36*c[6]^2*c[5]*c[4]^3+60*c[4]^3*c[5]*c[6]^3-3*c[6]*c[5]* c[4]^3-11*c[5]^3*c[6]*c[4]+3*c[5]^3*c[4]^2-24*c[5]^2*c[6]*c[4]^3+13*c[ 5]^2*c[6]*c[4]^2+48*c[5]^3*c[6]*c[4]^2-60*c[5]^3*c[6]^2*c[4]^3-30*c[6] ^2*c[5]^2*c[4]^3-12*c[6]^2*c[5]^2*c[4]^2-150*c[6]*c[5]^3*c[4]^3-48*c[6 ]^2*c[5]^3*c[4]+150*c[6]^2*c[5]^3*c[4]^2-2*c[6]^2*c[4]*c[5]^2-5*c[5]^2 *c[4]^3+2*c[4]^4*c[5]-6*c[5]^2*c[6]^3+60*c[6]^3*c[4]^3*c[5]^2+7*c[6]^3 *c[4]*c[5]-60*c[5]^2*c[6]*c[4]^5-40*c[6]^3*c[4]^4*c[5]+60*c[4]^4*c[6]* c[5]^3)*c[6]/(c[4]^3-2*c[5]*c[4]^2+30*c[5]^2*c[4]^3-6*c[5]*c[4]^3-10*c [4]^4*c[5]^2-c[5]^3+10*c[5]^4*c[4]^2-30*c[5]^3*c[4]^2+6*c[5]^3*c[4]+2* c[5]^2*c[4])/c[4]^2, a[3,1] = -2/9*c[4]*(c[4]-3*c[2])/c[2], a[4,1] = 1 /4*c[4], a[7,3] = 3/4*1/c[4]^2*c[7]*(-480*c[5]^2*c[6]*c[4]^4+15*c[6]*c [5]*c[4]^4+300*c[4]^2*c[5]^2*c[7]^3+60*c[7]^3*c[5]*c[4]^3-510*c[7]^3*c [5]^2*c[4]^3+54*c[7]*c[5]*c[4]^4+450*c[5]^2*c[4]^4*c[7]^2+150*c[4]^3*c [7]^3*c[5]^3-150*c[7]^2*c[5]^2*c[4]^5-450*c[4]^5*c[6]*c[5]^3-3*c[7]^3* c[4]^2+90*c[5]^2*c[4]^5*c[7]+120*c[5]^3*c[4]^4*c[7]+150*c[7]^3*c[5]^2* c[4]^4+100*c[5]^4*c[4]^2*c[7]^2*c[6]-180*c[6]*c[4]^5*c[5]*c[7]+30*c[6] *c[5]^2*c[4]^4*c[7]+60*c[5]^4*c[4]^3*c[6]-150*c[5]^4*c[4]^4*c[6]-200*c [7]^2*c[4]^4*c[5]^3+300*c[6]*c[4]^5*c[5]*c[7]^2-50*c[7]^2*c[5]^4*c[4]^ 3+24*c[7]*c[6]*c[5]^3-40*c[7]^2*c[6]*c[5]^3-6*c[5]*c[6]*c[7]*c[4]+360* c[7]*c[4]^4*c[6]*c[5]+150*c[7]^2*c[5]^3*c[4]^3+2*c[7]^2*c[4]^2+6*c[5]^ 2*c[7]^3+330*c[6]*c[7]*c[5]^3*c[4]^3-450*c[7]^2*c[6]*c[5]*c[4]^4-12*c[ 7]*c[6]*c[4]^3-6*c[6]*c[7]^2*c[4]-2*c[7]^2*c[5]*c[4]-4*c[7]^2*c[5]^2+6 *c[6]*c[7]*c[4]^2+20*c[7]^2*c[6]*c[4]^3-10*c[5]*c[7]^2*c[4]^3+6*c[5]*c [4]^3-6*c[5]*c[7]*c[4]^2-6*c[5]*c[6]*c[4]^2+150*c[6]*c[5]*c[7]^2*c[4]^ 2-72*c[7]^2*c[5]*c[6]*c[4]-270*c[5]*c[7]*c[6]*c[4]^3+9*c[7]^3*c[4]*c[6 ]+3*c[7]^3*c[4]*c[5]-306*c[5]^2*c[6]*c[7]*c[4]^2+900*c[4]^5*c[6]*c[7]* c[5]^3+600*c[5]^2*c[6]*c[7]^3*c[4]^4-420*c[7]^3*c[4]^2*c[6]*c[5]+90*c[ 6]*c[5]*c[7]*c[4]^2-12*c[6]*c[7]*c[5]^2+30*c[6]*c[7]^3*c[5]^2+12*c[6]* c[7]^2*c[5]+123*c[7]^3*c[4]*c[6]*c[5]+360*c[6]*c[5]^2*c[7]*c[4]^3+360* c[6]*c[5]^3*c[7]*c[4]^2-18*c[6]*c[7]^3*c[5]+30*c[5]^4*c[4]^3*c[7]-80*c [7]^2*c[5]^2*c[4]^3+24*c[7]^2*c[5]*c[4]^2-30*c[7]*c[5]*c[4]^3+3*c[6]*c [5]*c[4]^3-240*c[7]^3*c[5]^2*c[6]*c[4]-24*c[5]^3*c[6]*c[4]-12*c[5]^3*c [7]*c[4]-144*c[5]^3*c[6]*c[7]*c[4]+9*c[5]^3*c[4]^2+294*c[5]^2*c[6]*c[4 ]^3-42*c[5]^2*c[7]*c[4]^2-75*c[5]^2*c[6]*c[4]^2+252*c[5]^2*c[7]*c[4]^3 +18*c[5]^3*c[7]*c[4]^2+189*c[5]^3*c[6]*c[4]^2-360*c[5]^2*c[4]^4*c[7]-7 20*c[6]*c[5]^3*c[4]^3-120*c[5]^3*c[7]*c[4]^3-60*c[5]^4*c[7]*c[4]^2*c[6 ]-6*c[5]^2*c[4]^2-780*c[7]^2*c[5]^2*c[6]*c[4]^3+810*c[7]^3*c[5]^2*c[6] *c[4]^2-1230*c[7]^2*c[5]^3*c[6]*c[4]^2-15*c[7]^3*c[4]^2*c[6]-21*c[7]^3 *c[4]^2*c[5]+20*c[7]^2*c[6]*c[5]*c[4]^3+220*c[7]^2*c[5]^2*c[6]*c[4]^2+ 15*c[4]^4*c[5]^2+320*c[7]^2*c[5]^3*c[6]*c[4]+1850*c[7]^2*c[5]^3*c[6]*c [4]^3-60*c[5]^3*c[7]^3*c[4]^2+12*c[5]^2*c[6]*c[4]+12*c[5]^2*c[7]*c[4]- 66*c[7]^3*c[5]^2*c[4]+24*c[7]^2*c[5]^2*c[4]+72*c[5]^2*c[6]*c[7]*c[4]-1 10*c[7]^2*c[5]^2*c[4]^2+20*c[7]^2*c[5]^3*c[4]-9*c[4]^4*c[5]-60*c[7]^2* c[4]^4*c[5]-600*c[5]^2*c[6]*c[4]^5*c[7]^2-1200*c[7]^3*c[5]^2*c[4]^3*c[ 6]+630*c[7]^3*c[6]*c[5]*c[4]^3+300*c[5]^4*c[4]^4*c[6]*c[7]-1800*c[5]^3 *c[4]^4*c[6]*c[7]-15*c[5]^3*c[4]^3-300*c[7]^3*c[4]^4*c[5]*c[6]-20*c[7] ^2*c[5]^3*c[4]^2-800*c[5]^3*c[4]^4*c[7]^2*c[6]-200*c[5]^4*c[4]^3*c[7]^ 2*c[6]+1150*c[5]^2*c[4]^4*c[7]^2*c[6]+180*c[5]^2*c[6]*c[4]^5+1140*c[4] ^4*c[6]*c[5]^3)/(10*c[5]^3*c[4]^2+10*c[5]^2*c[4]^3-30*c[5]^2*c[4]^2+6* c[5]^2*c[4]-c[5]^2+6*c[5]*c[4]^2+c[5]*c[4]-c[4]^2)/(10*c[5]*c[6]*c[4]- 5*c[6]*c[5]-5*c[6]*c[4]+3*c[6]-5*c[5]*c[4]+3*c[5]+3*c[4]-2), `b*`[1] = 1/60*(30*c[5]^3*c[4]+42*c[7]*c[6]^2-103*c[5]*c[7]*c[4]+18*c[6]-6*c[5] -6*c[4]-27*c[7]*c[6]+9*c[5]^2+9*c[4]^2-180*c[7]*c[6]^2*c[5]^3+100*c[7] *c[6]*c[5]^3-27*c[6]^2-354*c[5]*c[6]*c[7]*c[4]+97*c[6]*c[5]^2-66*c[6]* c[5]-60*c[6]*c[5]^3+100*c[6]^2*c[5]^3+9*c[7]*c[5]+100*c[6]^2*c[4]^3-20 40*c[6]^2*c[7]*c[5]^2*c[4]^3-600*c[6]*c[7]*c[5]^3*c[4]^3-14*c[7]*c[5]^ 2+100*c[7]*c[6]*c[4]^3+310*c[6]^2*c[7]*c[4]^2-192*c[6]^2*c[7]*c[4]-180 *c[6]^2*c[4]^2-14*c[7]*c[4]^2+9*c[7]*c[4]+103*c[6]*c[7]*c[5]+97*c[6]*c [4]^2-60*c[6]*c[4]^3+103*c[6]*c[7]*c[4]-156*c[6]*c[7]*c[4]^2+30*c[5]*c [4]^3+174*c[5]*c[7]*c[4]^2-312*c[5]*c[6]*c[4]^2-107*c[5]*c[4]^2+215*c[ 5]*c[6]*c[4]+66*c[5]*c[4]-66*c[6]*c[4]-410*c[5]*c[7]*c[6]*c[4]^3+840*c [6]^2*c[4]^2*c[5]-522*c[6]^2*c[4]*c[5]-760*c[5]^2*c[6]*c[7]*c[4]^2-500 *c[6]^2*c[5]*c[4]^3+932*c[6]^2*c[7]*c[4]*c[5]+1020*c[7]*c[6]^2*c[5]*c[ 4]^3-192*c[6]^2*c[7]*c[5]+310*c[6]^2*c[7]*c[5]^2+526*c[6]*c[5]*c[7]*c[ 4]^2-156*c[6]*c[7]*c[5]^2-180*c[6]^2*c[5]^2-1610*c[6]^2*c[4]^2*c[7]*c[ 5]-1610*c[6]^2*c[5]^2*c[7]*c[4]+2970*c[6]^2*c[5]^2*c[7]*c[4]^2+690*c[6 ]*c[5]^2*c[7]*c[4]^3+690*c[6]*c[5]^3*c[7]*c[4]^2+1020*c[6]^2*c[5]^3*c[ 7]*c[4]+117*c[6]^2*c[5]-50*c[7]*c[5]*c[4]^3+230*c[6]*c[5]*c[4]^3+230*c [5]^3*c[6]*c[4]-50*c[5]^3*c[7]*c[4]-107*c[5]^2*c[4]-410*c[5]^3*c[6]*c[ 7]*c[4]-50*c[5]^3*c[4]^2-380*c[5]^2*c[6]*c[4]^3-310*c[5]^2*c[7]*c[4]^2 +460*c[5]^2*c[6]*c[4]^2+90*c[5]^2*c[7]*c[4]^3+90*c[5]^3*c[7]*c[4]^2-38 0*c[5]^3*c[6]*c[4]^2-600*c[5]^3*c[6]^2*c[4]^3+900*c[6]^2*c[5]^2*c[4]^3 -1420*c[6]^2*c[5]^2*c[4]^2+300*c[6]*c[5]^3*c[4]^3-500*c[6]^2*c[5]^3*c[ 4]+900*c[6]^2*c[5]^3*c[4]^2+180*c[5]^2*c[4]^2+840*c[6]^2*c[4]*c[5]^2-1 80*c[7]*c[6]^2*c[4]^3+117*c[6]^2*c[4]+1500*c[5]^3*c[6]^2*c[7]*c[4]^3-3 12*c[5]^2*c[6]*c[4]+174*c[5]^2*c[7]*c[4]+526*c[5]^2*c[6]*c[7]*c[4]-204 0*c[6]^2*c[7]*c[5]^3*c[4]^2-50*c[5]^2*c[4]^3)/(3*c[6]-28*c[5]*c[6]*c[4 ]^2-28*c[5]^2*c[6]*c[4]+8*c[6]*c[5]^2-9*c[6]*c[4]-9*c[6]*c[5]+28*c[5]* c[6]*c[4]+30*c[5]^2*c[6]*c[4]^2+8*c[6]*c[4]^2-4*c[5]^2*c[4]-4*c[5]*c[4 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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]-3280* 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+130 0*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]^4-1 10*c[7]*c[5]*c[4]^4+48*c[5]^3*c[4]+200*c[4]^5*c[6]*c[5]^3-150*c[5]^2*c [4]^5*c[7]-750*c[5]^3*c[4]^4*c[7]-930*c[6]*c[5]^2*c[4]^4*c[7]-200*c[5] ^5*c[4]^3*c[7]+750*c[5]^4*c[4]^3*c[6]-120*c[5]^5*c[4]^2-10*c[5]^3+10*c [4]^3+150*c[5]^5*c[4]^3-15*c[7]*c[6]*c[5]^3+12*c[5]^4-12*c[4]^4+200*c[ 6]*c[5]^2*c[4]^5*c[7]-150*c[4]^5*c[5]^3-200*c[5]^5*c[4]^3*c[6]+150*c[7 ]*c[4]^4*c[6]*c[5]+12*c[6]*c[5]^3+15*c[4]^4*c[7]-690*c[5]^4*c[7]*c[4]^ 2-15*c[5]^4*c[7]+150*c[5]^5*c[7]*c[4]^2+15*c[7]*c[6]*c[4]^3-12*c[7]*c[ 4]^3-12*c[6]*c[4]^3-48*c[5]*c[4]^3+24*c[5]*c[7]*c[4]^2+24*c[5]*c[6]*c[ 4]^2-20*c[5]*c[4]^2-70*c[5]*c[7]*c[6]*c[4]^3+12*c[7]*c[5]^3-300*c[4]^5 *c[6]*c[7]*c[5]^3-30*c[6]*c[5]*c[7]*c[4]^2+20*c[5]^4*c[7]*c[6]-690*c[6 ]*c[5]^4*c[4]^2-15*c[6]*c[5]^4+200*c[4]^5*c[7]*c[5]^3+510*c[6]*c[5]^2* c[7]*c[4]^3-510*c[6]*c[5]^3*c[7]*c[4]^2+750*c[5]^4*c[4]^3*c[7]+57*c[7] *c[5]*c[4]^3+57*c[6]*c[5]*c[4]^3-20*c[6]*c[7]*c[4]^4-57*c[5]^3*c[6]*c[ 4]-57*c[5]^3*c[7]*c[4]+20*c[5]^2*c[4]-200*c[5]^5*c[4]^2*c[6]*c[7]+70*c [5]^3*c[6]*c[7]*c[4]-342*c[5]^3*c[4]^2-410*c[5]^2*c[6]*c[4]^3-410*c[5] ^2*c[7]*c[4]^3+410*c[5]^3*c[7]*c[4]^2+410*c[5]^3*c[6]*c[4]^2+110*c[7]* c[5]^4*c[4]+690*c[5]^2*c[4]^4*c[7]+930*c[5]^4*c[7]*c[4]^2*c[6]+550*c[5 ]^4*c[4]^2-550*c[4]^4*c[5]^2-570*c[5]^4*c[4]^3-24*c[5]^2*c[6]*c[4]-24* c[5]^2*c[7]*c[4]+30*c[5]^2*c[6]*c[7]*c[4]+342*c[5]^2*c[4]^3+87*c[4]^4* c[5]+300*c[5]^5*c[4]^3*c[6]*c[7]+110*c[5]^4*c[4]*c[6]-150*c[5]^4*c[4]* c[7]*c[6]+120*c[4]^5*c[5]^2+15*c[6]*c[4]^4+1100*c[5]^3*c[4]^4*c[6]*c[7 ]-1100*c[5]^4*c[4]^3*c[6]*c[7]+150*c[5]^5*c[6]*c[4]^2-150*c[5]^2*c[6]* c[4]^5-750*c[4]^4*c[6]*c[5]^3-87*c[5]^4*c[4]+570*c[5]^3*c[4]^4)/(c[6]* c[4]-c[4]^2-c[7]*c[6]+c[7]*c[4])/c[4]^2, a[8,1] = 1/4*(-2816*c[5]^2*c[ 6]*c[4]^4+424*c[6]*c[5]*c[4]^4-9720*c[5]^4*c[4]^4*c[6]^2*c[7]+372*c[7] *c[5]*c[4]^4+1752*c[5]^2*c[4]^4*c[7]^2+20*c[5]^3*c[4]-1320*c[4]^5*c[6] ^2*c[5]^2-1320*c[7]^2*c[5]^2*c[4]^5-4880*c[4]^5*c[6]*c[5]^3-264*c[5]*c [6]^2*c[4]^4-12*c[7]^2*c[4]^3+700*c[5]^5*c[4]^3*c[7]^2*c[6]^2-200*c[5] ^2*c[4]^6*c[6]^2*c[7]^2-246*c[4]^5*c[5]*c[7]+1818*c[5]^2*c[4]^5*c[7]+1 300*c[5]^3*c[4]^6*c[6]^2*c[7]-800*c[5]^3*c[4]^6*c[6]^2*c[7]^2+5526*c[5 ]^3*c[4]^4*c[7]+32*c[4]^4*c[6]^2+3160*c[7]^2*c[4]^5*c[5]^3-7740*c[6]^2 *c[4]^5*c[7]*c[5]^3+780*c[7]*c[5]^2*c[4]^6*c[6]-325*c[5]^4*c[4]^2*c[7] ^2*c[6]-20*c[6]^2*c[4]^5-1339*c[6]^2*c[7]*c[5]^4*c[4]^2-280*c[5]^2*c[4 ]^6*c[7]+668*c[6]*c[4]^5*c[5]*c[7]+6574*c[6]*c[5]^2*c[4]^4*c[7]+1500*c [5]^4*c[7]*c[4]^6*c[6]+200*c[4]^6*c[6]^2*c[5]^2-280*c[5]^5*c[4]^3*c[7] -600*c[5]^2*c[7]^2*c[4]^6*c[6]+2092*c[5]^4*c[4]^3*c[6]+3640*c[5]^4*c[4 ]^5*c[6]-320*c[5]^2*c[6]*c[4]^6+400*c[5]^5*c[4]^5*c[6]^2-4880*c[5]^4*c [4]^4*c[6]-4080*c[7]^2*c[4]^4*c[5]^3+50*c[6]*c[7]^2*c[4]^5-2400*c[5]^4 *c[4]^5*c[7]^2-4160*c[5]^4*c[4]^4*c[7]-76*c[7]^2*c[6]*c[4]^4+1600*c[5] ^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]^2-710 0*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]-7 0*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]-97 2*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]^5-216 0*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 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3*c[6]*c[5]*c[7]^2*c[4]^2+266*c[5]*c[7]*c[6]*c[4]^3+12*c[6]^2*c[4]^2*c [5]-1240*c[5]^4*c[4]^5+287*c[5]^2*c[6]*c[7]*c[4]^2-8120*c[5]^4*c[7]^2* c[4]^4*c[6]+11330*c[4]^5*c[6]*c[7]*c[5]^3+72*c[6]^2*c[5]*c[4]^3+14*c[6 ]^2*c[7]*c[4]*c[5]+76*c[7]*c[6]^2*c[5]*c[4]^3-12*c[6]^2*c[7]*c[5]^2+40 *c[6]*c[5]*c[7]*c[4]^2+18*c[6]^2*c[7]^2*c[5]^2-101*c[6]^2*c[4]^2*c[7]* c[5]+132*c[6]^2*c[7]^2*c[4]^2*c[5]-284*c[6]*c[5]^4*c[4]^2-24*c[7]*c[6] ^2*c[5]^4-4160*c[4]^5*c[7]*c[5]^3-12*c[4]^5+95*c[6]^2*c[5]^2*c[7]*c[4] -356*c[6]^2*c[5]^2*c[7]*c[4]^2-2598*c[6]*c[5]^2*c[7]*c[4]^3-698*c[6]*c [5]^3*c[7]*c[4]^2-300*c[6]^2*c[5]^3*c[7]*c[4]+1818*c[5]^4*c[4]^3*c[7]- 692*c[7]^2*c[5]^2*c[4]^3+3160*c[5]^4*c[7]^2*c[4]^4+12*c[7]^2*c[5]*c[4] ^2-108*c[7]*c[5]*c[4]^3-120*c[6]*c[5]*c[4]^3+29*c[7]^2*c[5]^2*c[6]*c[4 ]+114*c[6]*c[7]*c[4]^4+104*c[4]^5*c[5]+200*c[5]^4*c[4]^6-1200*c[5]^5*c [4]^5*c[7]^2*c[6]+6*c[6]^2*c[7]^2*c[4]^2-52*c[5]^3*c[6]*c[4]-46*c[5]^3 *c[7]*c[4]-450*c[5]^5*c[4]^3*c[7]^2*c[6]-8*c[5]^2*c[4]+60*c[5]^3*c[6]* c[7]*c[4]-160*c[5]^3*c[4]^2-21*c[6]^2*c[7]^2*c[4]*c[5]+1144*c[5]^2*c[6 ]*c[4]^3-108*c[5]^2*c[7]*c[4]^2-120*c[5]^2*c[6]*c[4]^2+1024*c[5]^2*c[7 ]*c[4]^3+372*c[5]^3*c[7]*c[4]^2+424*c[5]^3*c[6]*c[4]^2+1752*c[5]^3*c[6 ]^2*c[4]^3-660*c[5]^5*c[4]^3*c[6]^2*c[7]-1153*c[4]^3*c[7]^2*c[5]^2*c[6 ]^2-72*c[6]*c[7]*c[4]^5+3160*c[4]^5*c[5]^3*c[6]^2+28*c[7]*c[5]^4*c[4]- 20*c[7]^2*c[5]^4*c[4]-2472*c[5]^2*c[4]^4*c[7]-692*c[6]^2*c[5]^2*c[4]^3 +72*c[6]^2*c[5]^2*c[4]^2-2816*c[6]*c[5]^3*c[4]^3+32*c[6]^2*c[5]^3*c[4] -264*c[6]^2*c[5]^3*c[4]^2-2472*c[5]^3*c[7]*c[4]^3+566*c[5]^4*c[7]*c[4] ^2*c[6]+48*c[5]^2*c[4]^2+1200*c[6]^2*c[4]^5*c[5]^5*c[7]^2-12*c[6]^2*c[ 4]*c[5]^2+38*c[7]*c[6]^2*c[4]^3+1804*c[7]^2*c[5]^2*c[6]*c[4]^3+258*c[7 ]^2*c[5]^3*c[6]*c[4]^2+104*c[5]^4*c[4]^2-500*c[5]^5*c[4]^5*c[7]-256*c[ 4]^3*c[7]^2*c[5]*c[6]^2-176*c[7]^2*c[6]*c[5]*c[4]^3+200*c[5]^5*c[4]^5- 212*c[7]^2*c[5]^2*c[6]*c[4]^2+1064*c[4]^4*c[5]^2+8*c[7]^2*c[5]^3*c[6]* c[4]-3410*c[7]^2*c[5]^3*c[6]*c[4]^3-4080*c[5]^3*c[6]^2*c[4]^4-772*c[5] ^4*c[4]^3+1200*c[7]^2*c[6]^2*c[4]^6*c[5]^4-36*c[6]^2*c[4]^3*c[7]^2-536 8*c[5]^3*c[6]^2*c[7]*c[4]^3+20*c[5]^2*c[6]*c[4]+18*c[5]^2*c[7]*c[4]-12 *c[7]^2*c[5]^2*c[4]-44*c[5]^2*c[6]*c[7]*c[4]+400*c[5]^4*c[4]^6*c[7]^2+ 1438*c[6]^2*c[7]*c[5]^3*c[4]^2+426*c[6]^2*c[7]^2*c[4]*c[5]^3-452*c[5]^ 2*c[4]^3+72*c[7]^2*c[5]^2*c[4]^2+32*c[7]^2*c[5]^3*c[4]-160*c[4]^4*c[5] +690*c[5]^5*c[4]^3*c[6]*c[7]-264*c[7]^2*c[4]^4*c[5]-1833*c[7]^2*c[5]^3 *c[6]^2*c[4]^2-260*c[6]^2*c[4]^5*c[5]*c[7]^2-8270*c[7]^2*c[4]^5*c[6]*c [5]^3+3780*c[5]^2*c[6]*c[4]^5*c[7]^2+1720*c[5]^4*c[4]^4+32*c[5]^4*c[4] *c[6]+30*c[5]^4*c[7]^2*c[4]*c[6]-60*c[5]^4*c[4]*c[7]*c[6]+2480*c[4]^5* c[6]^2*c[7]*c[5]^2-1120*c[4]^5*c[6]^2*c[7]^2*c[5]^2-772*c[4]^5*c[5]^2- 40*c[6]^2*c[7]^2*c[4]^5+1752*c[6]^2*c[4]^4*c[5]^2-1700*c[7]^2*c[6]^2*c [5]^5*c[4]^4-600*c[5]^5*c[4]^5*c[6]-120*c[5]*c[7]*c[4]^6*c[6]^2+50*c[6 ]^2*c[7]*c[4]^5-52*c[6]*c[4]^4-1200*c[5]^4*c[7]*c[4]^6*c[6]^2+7150*c[5 ]^4*c[4]^5*c[6]^2*c[7]+198*c[7]*c[6]^2*c[5]*c[4]^4+10560*c[5]^3*c[4]^4 *c[6]^2*c[7]+5240*c[5]^3*c[4]^3*c[6]^2*c[7]^2+11240*c[5]^4*c[4]^4*c[6] *c[7]-3649*c[5]^2*c[4]^4*c[7]*c[6]^2-14207*c[5]^3*c[4]^4*c[6]*c[7]-449 1*c[5]^4*c[4]^3*c[6]*c[7]+1064*c[5]^3*c[4]^3-264*c[7]^2*c[5]^3*c[4]^2- 2400*c[5]^4*c[4]^5*c[6]^2+6020*c[6]^2*c[7]^2*c[5]^3*c[4]^5+6800*c[6]*c [7]^2*c[5]^4*c[4]^5+3160*c[5]^4*c[4]^4*c[6]^2-1320*c[5]^4*c[4]^3*c[6]^ 2-5800*c[7]^2*c[5]^4*c[6]^2*c[4]^3+1940*c[7]^2*c[5]^4*c[6]^2*c[4]^2-60 0*c[5]^4*c[4]^6*c[6]+28*c[4]^5*c[7]+9740*c[5]^3*c[4]^4*c[7]^2*c[6]+292 0*c[5]^4*c[4]^3*c[7]^2*c[6]-4775*c[5]^2*c[4]^4*c[7]^2*c[6]+4980*c[5]^4 *c[4]^3*c[6]^2*c[7]+210*c[7]^2*c[4]^4*c[6]^2*c[5]-8860*c[7]^2*c[4]^4*c [5]^3*c[6]^2+2140*c[7]^2*c[4]^4*c[6]^2*c[5]^2+120*c[4]^6*c[5]^2-20*c[7 ]^2*c[4]^5+2092*c[5]^2*c[6]*c[4]^5+9920*c[6]^2*c[7]^2*c[5]^4*c[4]^4-50 0*c[5]^4*c[4]^6*c[7]-48*c[7]^2*c[6]^2*c[5]^3+6404*c[4]^4*c[6]*c[5]^3-2 84*c[6]*c[5]*c[4]^5+200*c[5]^5*c[6]^2*c[4]^3+200*c[6]^2*c[4]^6*c[5]*c[ 7]^2-8820*c[5]^4*c[4]^5*c[6]*c[7]-12*c[5]^4*c[4]-2324*c[5]^3*c[4]^4)/c [6]/c[5]/c[7]/c[4]^2/(150*c[5]^2*c[6]*c[4]^4-87*c[5]^3*c[4]-12*c[5]*c[ 7]*c[4]-200*c[5]^3*c[4]^4*c[7]-200*c[6]*c[5]^2*c[4]^4*c[7]-200*c[5]^4* c[4]^3*c[6]-10*c[5]^2-10*c[4]^2+12*c[5]^3+12*c[4]^3+20*c[7]*c[6]*c[5]^ 3+15*c[5]*c[6]*c[7]*c[4]+12*c[6]*c[5]^2-15*c[6]*c[5]^3-1300*c[6]*c[7]* c[5]^3*c[4]^3+150*c[5]^4*c[7]*c[4]^2+12*c[7]*c[5]^2+20*c[7]*c[6]*c[4]^ 3+12*c[7]*c[4]^2-15*c[7]*c[4]^3+12*c[6]*c[4]^2-15*c[6]*c[4]^3-15*c[6]* c[7]*c[4]^2-87*c[5]*c[4]^3-72*c[5]*c[7]*c[4]^2-72*c[5]*c[6]*c[4]^2+60* c[5]*c[4]^2-12*c[5]*c[6]*c[4]+10*c[5]*c[4]-150*c[5]*c[7]*c[6]*c[4]^3-1 5*c[7]*c[5]^3-660*c[5]^2*c[6]*c[7]*c[4]^2+90*c[6]*c[5]*c[7]*c[4]^2-15* c[6]*c[7]*c[5]^2+150*c[6]*c[5]^4*c[4]^2+930*c[6]*c[5]^2*c[7]*c[4]^3+93 0*c[6]*c[5]^3*c[7]*c[4]^2-200*c[5]^4*c[4]^3*c[7]+110*c[7]*c[5]*c[4]^3+ 110*c[6]*c[5]*c[4]^3+110*c[5]^3*c[6]*c[4]+110*c[5]^3*c[7]*c[4]+60*c[5] ^2*c[4]-150*c[5]^3*c[6]*c[7]*c[4]+550*c[5]^3*c[4]^2-690*c[5]^2*c[6]*c[ 4]^3+520*c[5]^2*c[7]*c[4]^2+520*c[5]^2*c[6]*c[4]^2-690*c[5]^2*c[7]*c[4 ]^3-690*c[5]^3*c[7]*c[4]^2-690*c[5]^3*c[6]*c[4]^2+150*c[5]^2*c[4]^4*c[ 7]+900*c[6]*c[5]^3*c[4]^3+900*c[5]^3*c[7]*c[4]^3-200*c[5]^4*c[7]*c[4]^ 2*c[6]-429*c[5]^2*c[4]^2-120*c[5]^4*c[4]^2-120*c[4]^4*c[5]^2+150*c[5]^ 4*c[4]^3-72*c[5]^2*c[6]*c[4]-72*c[5]^2*c[7]*c[4]+90*c[5]^2*c[6]*c[7]*c [4]+550*c[5]^2*c[4]^3+300*c[5]^3*c[4]^4*c[6]*c[7]+300*c[5]^4*c[4]^3*c[ 6]*c[7]-690*c[5]^3*c[4]^3-200*c[4]^4*c[6]*c[5]^3+150*c[5]^3*c[4]^4), a [8,3] = 3/4*(-1560*c[5]^2*c[6]*c[4]^4+585*c[6]*c[5]*c[4]^4-45*c[7]*c[5 ]*c[4]^4-12*c[5]^3*c[4]-450*c[4]^5*c[6]*c[5]^3+3*c[5]*c[7]*c[4]-150*c[ 5]^2*c[4]^5*c[7]-200*c[5]^3*c[4]^4*c[7]+300*c[6]*c[4]^5*c[5]*c[7]+2860 *c[6]*c[5]^2*c[4]^4*c[7]+160*c[5]^4*c[4]^3*c[6]-150*c[5]^4*c[4]^4*c[6] -4*c[5]^2+2*c[4]^2-40*c[7]*c[6]*c[5]^3+132*c[5]*c[6]*c[7]*c[4]-900*c[6 ]*c[5]^2*c[4]^5*c[7]-30*c[6]*c[5]^2+12*c[6]*c[5]-960*c[7]*c[4]^4*c[6]* c[5]+24*c[6]*c[5]^3+3140*c[6]*c[7]*c[5]^3*c[4]^3+6*c[7]*c[5]^2+20*c[7] *c[6]*c[4]^3-3*c[7]*c[4]^2-18*c[6]*c[7]*c[5]+15*c[6]*c[4]^2-12*c[6]*c[ 4]^3+9*c[6]*c[7]*c[4]-24*c[6]*c[7]*c[4]^2-54*c[5]*c[4]^3-12*c[5]*c[7]* c[4]^2+348*c[5]*c[6]*c[4]^2+9*c[5]*c[4]^2-87*c[5]*c[6]*c[4]-2*c[5]*c[4 ]-6*c[6]*c[4]+1035*c[5]*c[7]*c[6]*c[4]^3+1593*c[5]^2*c[6]*c[7]*c[4]^2+ 900*c[4]^5*c[6]*c[7]*c[5]^3-534*c[6]*c[5]*c[7]*c[4]^2+48*c[6]*c[7]*c[5 ]^2-60*c[6]*c[5]^4*c[4]^2-3060*c[6]*c[5]^2*c[7]*c[4]^3-1560*c[6]*c[5]^ 3*c[7]*c[4]^2-50*c[5]^4*c[4]^3*c[7]+80*c[7]*c[5]*c[4]^3-652*c[6]*c[5]* c[4]^3-208*c[5]^3*c[6]*c[4]+20*c[5]^3*c[7]*c[4]+54*c[5]^2*c[4]+360*c[5 ]^3*c[6]*c[7]*c[4]+82*c[5]^3*c[4]^2+1734*c[5]^2*c[6]*c[4]^3+388*c[5]^2 *c[7]*c[4]^2-944*c[5]^2*c[6]*c[4]^2-900*c[5]^2*c[7]*c[4]^3-135*c[5]^3* c[7]*c[4]^2+879*c[5]^3*c[6]*c[4]^2+660*c[5]^2*c[4]^4*c[7]-1680*c[6]*c[ 5]^3*c[4]^3+420*c[5]^3*c[7]*c[4]^3+100*c[5]^4*c[7]*c[4]^2*c[6]-246*c[5 ]^2*c[4]^2-405*c[4]^4*c[5]^2+30*c[5]^4*c[4]^3+240*c[5]^2*c[6]*c[4]-84* c[5]^2*c[7]*c[4]-396*c[5]^2*c[6]*c[7]*c[4]+562*c[5]^2*c[4]^3+30*c[4]^4 *c[5]+90*c[4]^5*c[5]^2+300*c[5]^4*c[4]^4*c[6]*c[7]-3000*c[5]^3*c[4]^4* c[6]*c[7]-300*c[5]^4*c[4]^3*c[6]*c[7]-255*c[5]^3*c[4]^3+480*c[5]^2*c[6 ]*c[4]^5+1540*c[4]^4*c[6]*c[5]^3-180*c[6]*c[5]*c[4]^5+120*c[5]^3*c[4]^ 4)/c[4]^2/(150*c[5]^2*c[6]*c[4]^4-87*c[5]^3*c[4]-12*c[5]*c[7]*c[4]-200 *c[5]^3*c[4]^4*c[7]-200*c[6]*c[5]^2*c[4]^4*c[7]-200*c[5]^4*c[4]^3*c[6] -10*c[5]^2-10*c[4]^2+12*c[5]^3+12*c[4]^3+20*c[7]*c[6]*c[5]^3+15*c[5]*c [6]*c[7]*c[4]+12*c[6]*c[5]^2-15*c[6]*c[5]^3-1300*c[6]*c[7]*c[5]^3*c[4] ^3+150*c[5]^4*c[7]*c[4]^2+12*c[7]*c[5]^2+20*c[7]*c[6]*c[4]^3+12*c[7]*c [4]^2-15*c[7]*c[4]^3+12*c[6]*c[4]^2-15*c[6]*c[4]^3-15*c[6]*c[7]*c[4]^2 -87*c[5]*c[4]^3-72*c[5]*c[7]*c[4]^2-72*c[5]*c[6]*c[4]^2+60*c[5]*c[4]^2 -12*c[5]*c[6]*c[4]+10*c[5]*c[4]-150*c[5]*c[7]*c[6]*c[4]^3-15*c[7]*c[5] ^3-660*c[5]^2*c[6]*c[7]*c[4]^2+90*c[6]*c[5]*c[7]*c[4]^2-15*c[6]*c[7]*c [5]^2+150*c[6]*c[5]^4*c[4]^2+930*c[6]*c[5]^2*c[7]*c[4]^3+930*c[6]*c[5] ^3*c[7]*c[4]^2-200*c[5]^4*c[4]^3*c[7]+110*c[7]*c[5]*c[4]^3+110*c[6]*c[ 5]*c[4]^3+110*c[5]^3*c[6]*c[4]+110*c[5]^3*c[7]*c[4]+60*c[5]^2*c[4]-150 *c[5]^3*c[6]*c[7]*c[4]+550*c[5]^3*c[4]^2-690*c[5]^2*c[6]*c[4]^3+520*c[ 5]^2*c[7]*c[4]^2+520*c[5]^2*c[6]*c[4]^2-690*c[5]^2*c[7]*c[4]^3-690*c[5 ]^3*c[7]*c[4]^2-690*c[5]^3*c[6]*c[4]^2+150*c[5]^2*c[4]^4*c[7]+900*c[6] *c[5]^3*c[4]^3+900*c[5]^3*c[7]*c[4]^3-200*c[5]^4*c[7]*c[4]^2*c[6]-429* c[5]^2*c[4]^2-120*c[5]^4*c[4]^2-120*c[4]^4*c[5]^2+150*c[5]^4*c[4]^3-72 *c[5]^2*c[6]*c[4]-72*c[5]^2*c[7]*c[4]+90*c[5]^2*c[6]*c[7]*c[4]+550*c[5 ]^2*c[4]^3+300*c[5]^3*c[4]^4*c[6]*c[7]+300*c[5]^4*c[4]^3*c[6]*c[7]-690 *c[5]^3*c[4]^3-200*c[4]^4*c[6]*c[5]^3+150*c[5]^3*c[4]^4), a[8,5] = -1/ 2*(893*c[5]^2*c[6]*c[4]^4-122*c[6]*c[5]*c[4]^4+850*c[5]^4*c[4]^4*c[6]^ 2*c[7]-101*c[7]*c[5]*c[4]^4-535*c[5]^2*c[4]^4*c[7]^2-44*c[5]^3*c[4]+16 0*c[4]^5*c[6]^2*c[5]^2+100*c[7]^2*c[5]^2*c[4]^5+170*c[4]^5*c[6]*c[5]^3 +75*c[5]*c[6]^2*c[4]^4+16*c[7]^2*c[4]^3-9*c[5]*c[7]*c[4]-140*c[5]^2*c[ 4]^5*c[7]-780*c[5]^3*c[4]^4*c[7]-10*c[4]^4*c[6]^2-100*c[7]^2*c[4]^5*c[ 5]^3+450*c[6]^2*c[4]^5*c[7]*c[5]^3+510*c[5]^4*c[4]^2*c[7]^2*c[6]+490*c [6]^2*c[7]*c[5]^4*c[4]^2-2224*c[6]*c[5]^2*c[4]^4*c[7]+10*c[5]^5*c[4]^3 *c[7]-146*c[5]^4*c[4]^3*c[6]+90*c[5]^4*c[4]^5*c[6]+40*c[5]^4*c[4]^4*c[ 6]+570*c[7]^2*c[4]^4*c[5]^3+130*c[5]^4*c[4]^4*c[7]+25*c[7]^2*c[6]*c[4] ^4+18*c[5]^5*c[4]^2-4*c[5]^2-4*c[4]^2-10*c[4]^4*c[7]^2+150*c[7]^2*c[5] ^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]^3-70*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+20 8*c[6]^2*c[5]^3*c[7]*c[4]-225*c[5]^4*c[4]^3*c[7]+704*c[7]^2*c[5]^2*c[4 ]^3-100*c[5]^4*c[7]^2*c[4]^4+29*c[7]^2*c[5]*c[4]^2+152*c[7]*c[5]*c[4]^ 3+181*c[6]*c[5]*c[4]^3-178*c[7]^2*c[5]^2*c[6]*c[4]-36*c[6]*c[7]*c[4]^4 -18*c[6]^2*c[7]^2*c[4]^2+3*c[6]^2*c[7]^2*c[4]+10*c[5]^4*c[7]^2*c[6]+77 *c[5]^3*c[6]*c[4]+75*c[5]^3*c[7]*c[4]+90*c[7]*c[6]*c[5]^5*c[4]+29*c[5] ^2*c[4]-350*c[5]^5*c[4]^2*c[6]*c[7]-234*c[5]^3*c[6]*c[7]*c[4]+279*c[5] ^3*c[4]^2+54*c[6]^2*c[7]^2*c[4]*c[5]-1192*c[5]^2*c[6]*c[4]^3+450*c[5]^ 2*c[7]*c[4]^2+522*c[5]^2*c[6]*c[4]^2-990*c[5]^2*c[7]*c[4]^3-502*c[5]^3 *c[7]*c[4]^2-614*c[5]^3*c[6]*c[4]^2-1638*c[5]^3*c[6]^2*c[4]^3+260*c[4] 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7]^2*c[4]^4*c[6]^2*c[5]+500*c[7]^2*c[4]^4*c[5]^3*c[6]^2-110*c[7]^2*c[4 ]^4*c[6]^2*c[5]^2-160*c[5]^2*c[6]*c[4]^5-200*c[6]^2*c[7]^2*c[5]^4*c[4] ^4-1067*c[4]^4*c[6]*c[5]^3-70*c[5]^5*c[6]^2*c[4]^3-200*c[5]^4*c[4]^5*c [6]*c[7]+14*c[5]^4*c[4]+434*c[5]^3*c[4]^4-52*c[5]^5*c[4]*c[6])/c[5]/(7 2*c[5]^2*c[6]*c[4]^4-12*c[6]*c[5]*c[4]^4+1100*c[5]^4*c[4]^4*c[6]^2*c[7 ]-12*c[7]*c[5]*c[4]^4-110*c[5]^2*c[4]^4*c[7]^2+120*c[4]^5*c[6]*c[5]^3+ 15*c[5]*c[6]^2*c[4]^4-300*c[5]^5*c[4]^3*c[7]^2*c[6]^2+10*c[5]^5-440*c[ 5]^3*c[4]^4*c[7]-150*c[7]^2*c[4]^5*c[5]^3+180*c[5]^4*c[4]^2*c[7]^2*c[6 ]+180*c[6]^2*c[7]*c[5]^4*c[4]^2+350*c[6]*c[5]^2*c[4]^4*c[7]-570*c[5]^5 *c[4]^3*c[7]-150*c[5]^5*c[4]*c[6]*c[7]^2-150*c[5]^5*c[4]*c[6]^2*c[7]+4 10*c[5]^4*c[4]^3*c[6]-120*c[5]^4*c[4]^4*c[6]+690*c[7]^2*c[4]^4*c[5]^3+ 200*c[5]^4*c[4]^5*c[7]^2-120*c[5]^4*c[4]^4*c[7]-15*c[7]^2*c[6]*c[4]^4+ 342*c[5]^5*c[4]^2-12*c[7]*c[6]^2*c[5]^3+10*c[7]*c[6]*c[5]^3-12*c[7]^2* c[6]*c[5]^3+15*c[5]^6*c[7]+410*c[5]^4*c[4]^2*c[7]^2+750*c[5]^5*c[7]^2* c[4]^3-120*c[6]*c[5]^2*c[4]^5*c[7]+570*c[5]^6*c[4]^2*c[7]-570*c[5]^5*c [4]^3*c[6]-57*c[7]*c[4]^4*c[6]*c[5]+87*c[5]^6*c[4]-570*c[5]^5*c[4]^4-1 100*c[5]^5*c[7]*c[4]^4*c[6]-410*c[7]^2*c[5]^3*c[4]^3-15*c[7]*c[4]^4*c[ 6]^2-15*c[5]^5*c[7]^2+780*c[5]^5*c[4]^2*c[6]^2*c[7]+750*c[5]^5*c[4]^4* c[6]+300*c[5]^5*c[7]*c[4]^5*c[6]-690*c[5]^5*c[4]^2*c[6]^2-40*c[6]^2*c[ 5]^4*c[7]*c[4]-690*c[5]^5*c[4]^2*c[7]^2-57*c[5]^4*c[6]^2*c[4]+410*c[5] ^4*c[6]^2*c[4]^2+750*c[5]^5*c[4]^4*c[7]+340*c[6]^2*c[7]*c[5]^2*c[4]^3- 30*c[6]^2*c[7]^2*c[5]^2*c[4]-750*c[6]*c[7]*c[5]^3*c[4]^3-20*c[7]^2*c[6 ]^2*c[5]^4-342*c[5]^4*c[7]*c[4]^2+90*c[7]^2*c[6]*c[5]*c[4]^4+200*c[7]^ 2*c[5]^5*c[6]^2*c[4]^2-10*c[5]^4*c[7]+140*c[5]^5*c[7]*c[4]^2+150*c[5]^ 4*c[4]*c[7]^2*c[6]^2+20*c[6]^2*c[4]^4*c[7]^2-10*c[7]*c[6]*c[4]^3-30*c[ 5]^5*c[7]*c[4]+12*c[5]^4*c[6]^2+12*c[7]^2*c[6]*c[4]^3-12*c[5]*c[7]^2*c [4]^3-24*c[6]*c[5]*c[7]^2*c[4]^2+24*c[5]*c[7]*c[6]*c[4]^3-12*c[5]^6+15 0*c[7]^2*c[5]^6*c[4]^2-120*c[5]^4*c[4]^5+48*c[5]^2*c[6]*c[7]*c[4]^2+11 00*c[5]^4*c[7]^2*c[4]^4*c[6]-150*c[4]^5*c[6]*c[7]*c[5]^3-12*c[6]^2*c[5 ]*c[4]^3-42*c[7]*c[6]^2*c[5]*c[4]^3+20*c[6]*c[5]*c[7]*c[4]^2+12*c[5]^4 *c[7]*c[6]-24*c[6]^2*c[4]^2*c[7]*c[5]+30*c[6]^2*c[7]^2*c[4]^2*c[5]-342 *c[6]*c[5]^4*c[4]^2-10*c[6]*c[5]^4+120*c[4]^5*c[7]*c[5]^3+24*c[6]^2*c[ 5]^2*c[7]*c[4]-30*c[6]^2*c[5]^2*c[7]*c[4]^2-243*c[6]*c[5]^2*c[7]*c[4]^ 3+372*c[6]*c[5]^3*c[7]*c[4]^2+87*c[6]^2*c[5]^3*c[7]*c[4]+410*c[5]^4*c[ 4]^3*c[7]+57*c[7]^2*c[5]^2*c[4]^3-750*c[5]^4*c[7]^2*c[4]^4+10*c[7]*c[5 ]*c[4]^3+10*c[6]*c[5]*c[4]^3+24*c[7]^2*c[5]^2*c[6]*c[4]+12*c[6]*c[7]*c [4]^4+200*c[7]*c[5]^7*c[4]^3-110*c[5]^6*c[7]*c[4]+15*c[6]*c[5]^6+20*c[ 5]^3*c[6]*c[4]+20*c[5]^3*c[7]*c[4]+150*c[7]*c[6]*c[5]^5*c[4]-900*c[5]^ 5*c[4]^3*c[7]^2*c[6]-750*c[5]^5*c[4]^2*c[6]*c[7]-96*c[5]^3*c[6]*c[7]*c [4]+200*c[6]*c[5]^7*c[4]^3+20*c[5]^3*c[4]^2-36*c[5]^2*c[6]*c[4]^3-20*c [5]^2*c[7]*c[4]^2-20*c[5]^2*c[6]*c[4]^2-36*c[5]^2*c[7]*c[4]^3-24*c[5]^ 3*c[7]*c[4]^2-24*c[5]^3*c[6]*c[4]^2-410*c[5]^3*c[6]^2*c[4]^3-900*c[5]^ 5*c[4]^3*c[6]^2*c[7]-510*c[4]^3*c[7]^2*c[5]^2*c[6]^2-630*c[7]*c[5]^6*c [6]*c[4]^2-150*c[4]^5*c[5]^3*c[6]^2+72*c[7]*c[5]^4*c[4]-57*c[7]^2*c[5] ^4*c[4]+72*c[5]^2*c[4]^4*c[7]+57*c[6]^2*c[5]^2*c[4]^3+24*c[6]^2*c[5]^2 *c[4]^2+285*c[6]*c[5]^3*c[4]^3-24*c[6]^2*c[5]^3*c[4]+285*c[5]^3*c[7]*c [4]^3+270*c[5]^4*c[7]*c[4]^2*c[6]+12*c[7]*c[6]^2*c[4]^3+340*c[7]^2*c[5 ]^2*c[6]*c[4]^3-410*c[7]^2*c[5]^3*c[6]*c[4]^2-200*c[5]^5*c[4]^5*c[7]+7 0*c[4]^3*c[7]^2*c[5]*c[6]^2-42*c[7]^2*c[6]*c[5]*c[4]^3+150*c[5]^5*c[4] ^5-30*c[7]^2*c[5]^2*c[6]*c[4]^2+12*c[4]^4*c[5]^2+87*c[7]^2*c[5]^3*c[6] *c[4]+510*c[7]^2*c[5]^3*c[6]*c[4]^3+690*c[5]^3*c[6]^2*c[4]^4-342*c[5]^ 4*c[4]^3-15*c[6]^2*c[4]^3*c[7]^2+510*c[5]^3*c[6]^2*c[7]*c[4]^3-20*c[5] ^2*c[6]*c[7]*c[4]-410*c[6]^2*c[7]*c[5]^3*c[4]^2-70*c[6]^2*c[7]^2*c[4]* c[5]^3-10*c[5]^2*c[4]^3+24*c[7]^2*c[5]^2*c[4]^2-24*c[7]^2*c[5]^3*c[4]- 200*c[7]^2*c[5]^6*c[4]^2*c[6]+110*c[7]^2*c[4]*c[5]^5+1350*c[5]^5*c[4]^ 3*c[6]*c[7]+15*c[7]^2*c[4]^4*c[5]+300*c[6]^2*c[7]*c[5]^6*c[4]^3+510*c[ 7]^2*c[5]^3*c[6]^2*c[4]^2-550*c[5]^6*c[4]^2-600*c[7]*c[5]^6*c[4]^3+20* c[7]*c[6]^2*c[5]^5+570*c[5]^6*c[6]*c[4]^2+150*c[5]^2*c[6]*c[4]^5*c[7]^ 2+550*c[5]^4*c[4]^4+120*c[5]^7*c[4]^2+72*c[5]^4*c[4]*c[6]-40*c[5]^4*c[ 7]^2*c[4]*c[6]-57*c[5]^4*c[4]*c[7]*c[6]+150*c[6]^2*c[5]^6*c[4]^2+150*c [4]^5*c[6]^2*c[7]*c[5]^2-200*c[4]^5*c[6]^2*c[7]^2*c[5]^2-150*c[5]^7*c[ 7]*c[4]^2-110*c[6]^2*c[4]^4*c[5]^2+780*c[5]^5*c[4]^2*c[7]^2*c[6]-200*c [5]^5*c[4]^5*c[6]-600*c[5]^6*c[4]^3*c[6]-200*c[6]^2*c[7]*c[5]^6*c[4]^2 -300*c[5]^4*c[4]^5*c[6]^2*c[7]+90*c[7]*c[6]^2*c[5]*c[4]^4+200*c[6]*c[5 ]^7*c[7]*c[4]^2-300*c[6]*c[5]^7*c[7]*c[4]^3-180*c[5]^3*c[4]^4*c[6]^2*c [7]-570*c[5]^4*c[4]^4*c[6]*c[7]-540*c[5]^2*c[4]^4*c[7]*c[6]^2+660*c[5] ^3*c[4]^4*c[6]*c[7]+60*c[5]^4*c[4]^3*c[6]*c[7]+48*c[5]^3*c[4]^3-15*c[6 ]^2*c[5]^5-200*c[5]^6*c[6]^2*c[4]^3-15*c[5]^5*c[6]*c[7]-48*c[5]^5*c[4] +140*c[5]^5*c[6]*c[4]^2+200*c[5]^4*c[4]^5*c[6]^2+300*c[6]^2*c[7]^2*c[5 ]^3*c[4]^5-300*c[6]*c[7]^2*c[5]^4*c[4]^5+700*c[7]*c[5]^6*c[4]^3*c[6]-7 50*c[5]^4*c[4]^4*c[6]^2+20*c[5]^5*c[7]^2*c[6]+1100*c[7]^2*c[5]^4*c[6]^ 2*c[4]^3-930*c[7]^2*c[5]^4*c[6]^2*c[4]^2+300*c[7]^2*c[5]^6*c[4]^3*c[6] -20*c[5]^6*c[7]*c[6]-180*c[5]^3*c[4]^4*c[7]^2*c[6]-750*c[5]^4*c[4]^3*c [7]^2*c[6]-540*c[5]^2*c[4]^4*c[7]^2*c[6]-750*c[5]^4*c[4]^3*c[6]^2*c[7] -150*c[7]^2*c[4]^4*c[6]^2*c[5]-1100*c[7]^2*c[4]^4*c[5]^3*c[6]^2+930*c[ 7]^2*c[4]^4*c[6]^2*c[5]^2+150*c[5]^6*c[4]*c[7]*c[6]-110*c[5]^6*c[4]*c[ 6]-150*c[5]^7*c[6]*c[4]^2+570*c[5]^6*c[4]^3-150*c[5]^7*c[4]^3+15*c[7]^ 2*c[6]^2*c[5]^3-440*c[4]^4*c[6]*c[5]^3+750*c[5]^5*c[6]^2*c[4]^3-200*c[ 7]^2*c[5]^6*c[4]^3+12*c[5]^4*c[7]^2+200*c[5]^4*c[4]^5*c[6]*c[7]-20*c[5 ]^4*c[4]-87*c[5]^3*c[4]^4-30*c[5]^5*c[4]*c[6]+110*c[5]^5*c[6]^2*c[4]), a[9,7] = -1/60*(10*c[5]*c[6]*c[4]-5*c[6]*c[5]-5*c[6]*c[4]+3*c[6]-5*c[ 5]*c[4]+3*c[5]+3*c[4]-2)/c[7]/(-c[5]*c[6]*c[4]+c[5]*c[6]*c[7]*c[4]+c[5 ]*c[7]*c[4]-c[7]^2*c[5]*c[4]+c[6]*c[7]*c[4]-c[6]*c[7]^2*c[4]-c[7]^2*c[ 4]+c[7]^3*c[4]+c[6]*c[7]*c[5]-c[6]*c[7]^2*c[5]-c[7]^2*c[5]+c[7]^3*c[5] -c[7]^2*c[6]+c[7]^3*c[6]+c[7]^3-c[7]^4), a[9,2] = 0, a[9,3] = 0, a[9,5 ] = -1/60*(10*c[6]*c[7]*c[4]-5*c[6]*c[4]-5*c[7]*c[6]+3*c[6]-5*c[7]*c[4 ]+3*c[4]-2+3*c[7])/c[5]/(c[5]^3*c[4]-c[5]^2*c[4]+c[5]*c[7]*c[4]-c[5]^2 *c[7]*c[4]+c[5]*c[6]*c[4]-c[5]^2*c[6]*c[4]-c[6]*c[7]*c[4]+c[5]*c[6]*c[ 7]*c[4]-c[5]^4+c[5]^3-c[7]*c[5]^2+c[7]*c[5]^3-c[6]*c[5]^2+c[6]*c[5]^3+ c[6]*c[7]*c[5]-c[6]*c[7]*c[5]^2), a[9,8] = 1/60*(-20*c[5]*c[6]*c[4]+30 *c[5]*c[6]*c[7]*c[4]+15*c[6]*c[5]-20*c[6]*c[7]*c[5]+15*c[6]*c[4]-20*c[ 6]*c[7]*c[4]-12*c[6]+15*c[7]*c[6]+15*c[5]*c[4]-20*c[5]*c[7]*c[4]-12*c[ 5]+15*c[7]*c[5]-12*c[4]+15*c[7]*c[4]+10-12*c[7])/(-c[4]+c[7]*c[4]+c[6] *c[4]-c[6]*c[7]*c[4]+c[5]*c[4]-c[5]*c[7]*c[4]-c[5]*c[6]*c[4]+c[5]*c[6] *c[7]*c[4]+1-c[7]-c[6]+c[7]*c[6]-c[5]+c[7]*c[5]+c[6]*c[5]-c[6]*c[7]*c[ 5]), a[9,1] = 1/60*(30*c[5]*c[6]*c[7]*c[4]-10*c[5]*c[6]*c[4]-10*c[6]*c [7]*c[5]+5*c[6]*c[5]-10*c[6]*c[7]*c[4]+5*c[6]*c[4]+5*c[7]*c[6]-3*c[6]- 10*c[5]*c[7]*c[4]+5*c[5]*c[4]+5*c[7]*c[5]-3*c[5]+5*c[7]*c[4]-3*c[4]+2- 3*c[7])/c[5]/c[6]/c[7]/c[4], a[9,4] = 1/60*(-2+3*c[5]-5*c[7]*c[6]+3*c[ 7]+3*c[6]+10*c[6]*c[7]*c[5]-5*c[6]*c[5]-5*c[7]*c[5])/c[4]/(-c[6]*c[4]^ 3-c[4]^3+c[4]^4+c[7]*c[4]^2+c[6]*c[4]^2-c[6]*c[7]*c[4]+c[6]*c[7]*c[4]^ 2-c[5]*c[4]^3+c[5]*c[4]^2-c[7]*c[4]^3-c[5]*c[7]*c[4]+c[5]*c[7]*c[4]^2- c[5]*c[6]*c[4]+c[5]*c[6]*c[4]^2+c[6]*c[7]*c[5]-c[5]*c[6]*c[7]*c[4]), a [9,6] = 1/60*(10*c[5]*c[7]*c[4]-5*c[5]*c[4]-5*c[7]*c[5]+3*c[5]-5*c[7]* c[4]+3*c[4]-2+3*c[7])/c[6]/(c[6]^2*c[4]*c[5]-c[5]*c[6]*c[4]+c[5]*c[7]* c[4]-c[5]*c[6]*c[7]*c[4]-c[6]^3*c[4]+c[6]^2*c[4]-c[6]*c[7]*c[4]+c[6]^2 *c[7]*c[4]-c[6]^3*c[5]+c[6]^2*c[5]-c[6]*c[7]*c[5]+c[6]^2*c[7]*c[5]+c[6 ]^4-c[6]^3+c[7]*c[6]^2-c[6]^3*c[7])\}:" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 33 "#======================== ========" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 22 "We specify the nodes " }{XPPEDIT 18 0 "c[6] = 29/39;" "6#/&%\"cG6 #\"\"'*&\"#H\"\"\"\"#R!\"\"" }{TEXT -1 7 " and " }{XPPEDIT 18 0 "c[7 ] = 575/576;" "6#/&%\"cG6#\"\"(*&\"$v&\"\"\"\"$w&!\"\"" }{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 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 73 "eA := \{c[6]=29/39,c[7]= 575/576\}:\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 16525 "eB := \{c[7] = 575/57 6, c[6] = 29/39, a[7,3] = 575/48922361856*(-10987991250*c[5]+549399562 5*c[4]+37927910800*c[5]^2-29377881600*c[5]^3-18963955400*c[4]^2-348181 824083*c[5]*c[4]^2+220334112000*c[4]^5*c[5]-920188517760*c[4]^5*c[5]^2 +14688940800*c[4]^3+298959091968*c[5]^3*c[4]+84891902900*c[5]*c[4]+828 411494400*c[4]^5*c[5]^3-360045306134*c[5]^2*c[4]-3267704221258*c[5]^2* c[4]^3+3017541359930*c[4]^4*c[5]^2+73444704000*c[5]^4*c[4]^2-138210289 0356*c[5]^3*c[4]^2+766247024682*c[5]*c[4]^3+1540738194222*c[5]^2*c[4]^ 2-725118345772*c[4]^4*c[5]+276137164800*c[5]^4*c[4]^4+3091388543970*c[ 5]^3*c[4]^3-2883741012480*c[5]^3*c[4]^4-306729505920*c[5]^4*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)/(95*c[5]*c[4]-28*c[5]-28*c[4]+ 9), b[6] = 10024911/11060600*(2870*c[5]*c[4]-1147*c[5]-1147*c[4]+573)/ (39*c[5]-29)/(39*c[4]-29), a[8,7] = -73383542784/219305*(95*c[5]*c[4]- 28*c[5]-28*c[4]+9)*(c[4]-1)*(-1+c[5])/(576*c[5]-575)/(576*c[4]-575)/(5 4630*c[5]*c[4]-16133*c[5]-16133*c[4]+5217), a[7,2] = 0, a[8,2] = 0, a[ 6,2] = 0, a[4,2] = 0, a[5,2] = 0, a[6,5] = 29/4626882*(1572597*c[5]^2* c[4]^2-65598*c[4]^2+24389*c[4]-525915*c[5]^2*c[4]-1081149*c[5]*c[4]^2+ 44109*c[4]^3-655980*c[4]^4*c[5]+1412431*c[5]*c[4]^3-1979250*c[5]^2*c[4 ]^3+358266*c[5]*c[4]+882180*c[4]^4*c[5]^2+65598*c[5]^2-48778*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[9,2] = 0, a[8,5] = 1/2*(-554176950*c[5]+277088475*c[4]+3026620866 *c[5]^2-5885145786*c[5]^3-1141675545*c[4]^2-14020684702*c[5]*c[4]^2+11 093210500*c[4]^5*c[5]-67092714640*c[4]^5*c[5]^2+4907141934*c[5]^4-5247 88715*c[4]^4+1389375785*c[4]^3+56413723656*c[5]^3*c[4]+3816530652*c[5] *c[4]+138300937380*c[4]^5*c[5]^3-26883145836*c[5]^2*c[4]-208870106185* c[5]^2*c[4]^3+195196458571*c[4]^4*c[5]^2+196573967177*c[5]^4*c[4]^2-22 4326098698*c[5]^3*c[4]^2+30347431508*c[5]*c[4]^3+104622887224*c[5]^2*c [4]^2-119695007640*c[5]^4*c[4]^5+15205320000*c[5]^5*c[4]-48829516947*c [5]^4*c[4]-110651235840*c[5]^5*c[4]^4+37393574400*c[5]^5*c[4]^5-149444 0064*c[5]^5-30682311008*c[4]^4*c[5]+353563131520*c[5]^4*c[4]^4+4423978 37976*c[5]^3*c[4]^3-406901254528*c[5]^3*c[4]^4-386519716044*c[5]^4*c[4 ]^3-61708395456*c[5]^5*c[4]^2+121255176960*c[5]^5*c[4]^3)/c[5]/(-86993 475*c[5]^3-173986950*c[5]*c[4]^2+2690177750*c[4]^5*c[5]^2+362411712*c[ 5]^6+473153768*c[5]^4-269017775*c[4]^4+86993475*c[4]^3-61816539420*c[5 ]^6*c[4]^3-155328911*c[5]^3*c[4]-15422234070*c[4]^5*c[5]^3+173986950*c [5]^2*c[4]+3112742889*c[5]^2*c[4]^3-20693837424*c[4]^4*c[5]^2+20042653 692*c[5]^4*c[4]^2-2636389776*c[5]^3*c[4]^2-3401678592*c[5]^6*c[4]-3624 117120*c[5]^7*c[4]^2+12272083200*c[5]^7*c[4]^3-457079068*c[5]*c[4]^3+4 08271986*c[5]^2*c[4]^2+25000289820*c[5]^4*c[4]^5+25720229730*c[5]^6*c[ 4]^2+6265982628*c[5]^5*c[4]-2884220325*c[5]^4*c[4]+40440366720*c[5]^5* c[4]^4-12272083200*c[5]^5*c[4]^5-748462845*c[5]^5+3156330057*c[4]^4*c[ 5]-89848747830*c[5]^4*c[4]^4-5977210608*c[5]^3*c[4]^3+67225622102*c[5] ^3*c[4]^4-26782940978*c[5]^4*c[4]^3-39731180282*c[5]^5*c[4]^2+79550752 170*c[5]^5*c[4]^3), a[9,7] = -1761205026816/5482625*(95*c[5]*c[4]-28*c [5]-28*c[4]+9)/(331776*c[5]*c[4]-331200*c[4]-331200*c[5]+330625), a[6, 4] = -29/4626882*(-146334*c[5]^2+262392*c[5]^3-48778*c[4]^2-782130*c[5 ]*c[4]^2+1311960*c[4]^5*c[5]-2646540*c[4]^5*c[5]^2+65598*c[4]^3-205955 1*c[5]^3*c[4]+170723*c[5]*c[4]+812406*c[5]^2*c[4]-875841*c[5]^2*c[4]^3 +5293080*c[4]^4*c[5]^2-655980*c[5]^4*c[4]^2+7215039*c[5]^3*c[4]^2+2511 777*c[5]*c[4]^3-2015181*c[5]^2*c[4]^2-3354170*c[4]^4*c[5]-8584290*c[5] ^3*c[4]^3+2646540*c[5]^3*c[4]^4+882180*c[5]^4*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, `b*`[7] \+ = -3057647616/5482625*(-14625*c[5]-14625*c[4]+27984*c[5]^2-16240*c[5]^ 3+27984*c[4]^2-190821*c[5]*c[4]^2-16240*c[4]^3+114740*c[5]^3*c[4]+9545 1*c[5]*c[4]-190821*c[5]^2*c[4]-251070*c[5]^2*c[4]^3-251070*c[5]^3*c[4] ^2+114740*c[5]*c[4]^3+400180*c[5]^2*c[4]^2+2349+165300*c[5]^3*c[4]^3)/ (-128001900*c[5]-128001900*c[4]+176065000*c[5]^2-76838400*c[5]^3+17606 5000*c[4]^2-881601800*c[5]*c[4]^2-76838400*c[4]^3+28764375+397573632*c [5]^3*c[4]+612101387*c[5]*c[4]-881601800*c[5]^2*c[4]-609303168*c[5]^2* c[4]^3-609303168*c[5]^3*c[4]^2+397573632*c[5]*c[4]^3+1314702438*c[5]^2 *c[4]^2+288645120*c[5]^3*c[4]^3), a[7,6] = -2409065425/78819360768*(5* c[5]*c[4]+1-2*c[4]-2*c[5])*(576*c[4]-575)*(576*c[5]-575)/(95*c[5]*c[4] -28*c[5]-28*c[4]+9)/(39*c[4]-29)/(39*c[5]-29), c[8] = 1, a[6,1] = 29/9 253764*(146334*c[5]^2-262392*c[5]^3+48778*c[4]^2+847728*c[5]*c[4]^2-13 11960*c[4]^5*c[5]+1764360*c[4]^5*c[5]^2+88218*c[4]^4-131196*c[4]^3+219 1878*c[5]^3*c[4]-170723*c[5]*c[4]+2372760*c[4]^5*c[5]^3-878004*c[5]^2* c[4]-2178093*c[5]^2*c[4]^3-566904*c[4]^4*c[5]^2+655980*c[5]^4*c[4]^2-8 246277*c[5]^3*c[4]^2-2074080*c[5]*c[4]^3+2364660*c[5]^2*c[4]^2+2587586 *c[4]^4*c[5]+2372760*c[5]^4*c[4]^4+14633736*c[5]^3*c[4]^3-11970270*c[5 ]^3*c[4]^4-2205450*c[5]^4*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[9,8] = 1/600*(54630*c[5]*c[4]-16133*c[5]-16133*c[4]+5217 )/(-c[4]+c[5]*c[4]+1-c[5]), a[8,3] = 3/4*(-99702*c[5]+49851*c[4]+34397 4*c[5]^2-266104*c[5]^3-171987*c[4]^2-3158382*c[5]*c[4]^2+1995780*c[4]^ 5*c[5]-8331570*c[4]^5*c[5]^2+133052*c[4]^3+2707500*c[5]^3*c[4]+770199* c[5]*c[4]+7490700*c[4]^5*c[5]^3-3264984*c[5]^2*c[4]-29618496*c[5]^2*c[ 4]^3+27334840*c[4]^4*c[5]^2+665260*c[5]^4*c[4]^2-12515511*c[5]^3*c[4]^ 2+6948561*c[5]*c[4]^3+13969455*c[5]^2*c[4]^2-6571365*c[4]^4*c[5]+24969 00*c[5]^4*c[4]^4+27986960*c[5]^3*c[4]^3-26090160*c[5]^3*c[4]^4-2777190 *c[5]^4*c[4]^3)/c[4]^2/(-5217*c[5]^2+16133*c[5]^3-5217*c[4]^2+31302*c[ 5]*c[4]^2+16133*c[4]^3-151428*c[5]^3*c[4]+5217*c[5]*c[4]+31302*c[5]^2* c[4]+863940*c[5]^2*c[4]^3-161330*c[4]^4*c[5]^2-161330*c[5]^4*c[4]^2+86 3940*c[5]^3*c[4]^2-151428*c[5]*c[4]^3-295476*c[5]^2*c[4]^2-1961560*c[5 ]^3*c[4]^3+546300*c[5]^3*c[4]^4+546300*c[5]^4*c[4]^3), c[9] = 1, `b*`[ 3] = 0, a[9,6] = 10024911/11060600*(2870*c[5]*c[4]-1147*c[5]-1147*c[4] +573)/(1521*c[5]*c[4]-1131*c[4]-1131*c[5]+841), `b*`[9] = 1/10*(1450*c [5]^2*c[4]^2-1355*c[5]^2*c[4]+290*c[5]^2-1355*c[5]*c[4]^2+1405*c[5]*c[ 4]-329*c[5]+290*c[4]^2-329*c[4]+87)/(87-968*c[5]*c[4]^2-968*c[5]^2*c[4 ]+232*c[5]^2-300*c[4]-300*c[5]+1163*c[5]*c[4]+870*c[5]^2*c[4]^2+232*c[ 4]^2), a[9,4] = -1/60*(38497*c[5]-10916)/c[4]/(61593*c[4]^3-22464*c[4] ^4-55804*c[4]^2+16675*c[4]+22464*c[5]*c[4]^3-61593*c[5]*c[4]^2+55804*c [5]*c[4]-16675*c[5]), a[6,3] = 29/1028196*(1677085*c[5]*c[4]^3-1168236 *c[5]*c[4]^2+336400*c[5]*c[4]-48778*c[5]+24389*c[4]-218660*c[5]^3*c[4] ^2-3521180*c[5]^2*c[4]^3+2493231*c[5]^2*c[4]^2-701220*c[5]^2*c[4]+8746 4*c[5]^2-655980*c[4]^4*c[5]-43732*c[4]^2+441090*c[5]^3*c[4]^3+1323270* 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*`[2] = 0, a[9,5] = 1/60*(38497*c[4]-10916)/c[5]/(-22464*c[5]^3*c[4]+61593*c[5]^ 2*c[4]-55804*c[5]*c[4]+16675*c[4]+22464*c[5]^4-61593*c[5]^3+55804*c[5] ^2-16675*c[5]), a[9,3] = 0, b[3] = 0, `b*`[8] = 0, a[5,3] = -3/4*c[5]^ 2*(2*c[5]-3*c[4])/c[4]^2, a[5,1] = 1/4*c[5]*(2*c[5]^2-5*c[5]*c[4]+4*c[ 4]^2)/c[4]^2, b[2] = 0, b[8] = 1/600*(54630*c[5]*c[4]-16133*c[5]-16133 *c[4]+5217)/(c[4]-1)/(-1+c[5]), b[5] = 1/60*(38497*c[4]-10916)/(-c[4]+ c[5])/c[5]/(22464*c[5]^3-61593*c[5]^2+55804*c[5]-16675), a[7,1] = 575/ 12768736444416*(955955238750*c[5]^2-3299728239600*c[5]^3-605400480000* c[4]^5+318651746250*c[4]^2+6787405046750*c[5]*c[4]^2-35717537151352*c[ 4]^5*c[5]+111112217407436*c[4]^5*c[5]^2+12779378496000*c[4]^6*c[5]+255 5875699200*c[5]^4+1600593075900*c[4]^4-1313719971950*c[4]^3+3036865901 0358*c[5]^3*c[4]-1115281111875*c[5]*c[4]+47094411551580*c[4]^5*c[5]^3- 5399587284650*c[5]^2*c[4]+9086192600424*c[5]^2*c[4]^3-80238403082008*c [4]^4*c[5]^2+126582278364396*c[5]^4*c[4]^2-123586602459552*c[5]^3*c[4] ^2-20189620293914*c[5]*c[4]^3+10692054291949*c[5]^2*c[4]^2-19558062759 1680*c[5]^4*c[4]^5-26614841481216*c[5]^4*c[4]-51093173571840*c[5]^5*c[ 4]^4+20978626440960*c[5]^3*c[4]^6+28462100889600*c[5]^4*c[4]^6+2846210 0889600*c[5]^5*c[4]^5-47316929230080*c[4]^6*c[5]^2+37560320535174*c[4] ^4*c[5]+367390145972700*c[5]^4*c[4]^4+250331445277590*c[5]^3*c[4]^3-21 7703140727238*c[5]^3*c[4]^4-309427124217894*c[5]^4*c[4]^3-638968924800 0*c[5]^5*c[4]^2+32739471815040*c[5]^5*c[4]^3)/(-9*c[5]^2+28*c[5]^3-9*c [4]^2+54*c[5]*c[4]^2+28*c[4]^3-263*c[5]^3*c[4]+9*c[5]*c[4]+54*c[5]^2*c [4]+1500*c[5]^2*c[4]^3-280*c[4]^4*c[5]^2-280*c[5]^4*c[4]^2+1500*c[5]^3 *c[4]^2-263*c[5]*c[4]^3-511*c[5]^2*c[4]^2-3410*c[5]^3*c[4]^3+950*c[5]^ 3*c[4]^4+950*c[5]^4*c[4]^3)/c[5]/c[4]^2, a[9,1] = 1/1000500*(221280*c[ 5]*c[4]-38497*c[5]-38497*c[4]+10916)/c[5]/c[4], a[8,1] = 1/13340*(3325 06170*c[5]^2-1147153290*c[5]^3-209915486*c[4]^5+110835390*c[4]^2+23602 71543*c[5]*c[4]^2-12412832998*c[4]^5*c[5]+38608834324*c[4]^5*c[5]^2+44 37284200*c[4]^6*c[5]+887456840*c[5]^4+555750314*c[4]^4-456670218*c[4]^ 3+10557015114*c[5]^3*c[4]-387923865*c[5]*c[4]+16322686330*c[4]^5*c[5]^ 3-1878130593*c[5]^2*c[4]+3157261915*c[5]^2*c[4]^3-27898761829*c[4]^4*c [5]^2+43936731037*c[5]^4*c[4]^2-42957669063*c[5]^3*c[4]^2-7020149672*c [5]*c[4]^3+3719770938*c[5]^2*c[4]^2-67736275420*c[5]^4*c[4]^5-92394279 86*c[5]^4*c[4]-17707250860*c[5]^5*c[4]^4+7274233640*c[5]^3*c[4]^6+9848 684400*c[5]^4*c[4]^6+9848684400*c[5]^5*c[4]^5-16424702440*c[4]^6*c[5]^ 2+13059755652*c[4]^4*c[5]+127380087280*c[5]^4*c[4]^4+86989056706*c[5]^ 3*c[4]^3-75593384072*c[5]^3*c[4]^4-107367117176*c[5]^4*c[4]^3-22186421 00*c[5]^5*c[4]^2+11361083510*c[5]^5*c[4]^3)/c[5]/c[4]^2/(-5217*c[5]^2+ 16133*c[5]^3-5217*c[4]^2+31302*c[5]*c[4]^2+16133*c[4]^3-151428*c[5]^3* c[4]+5217*c[5]*c[4]+31302*c[5]^2*c[4]+863940*c[5]^2*c[4]^3-161330*c[4] ^4*c[5]^2-161330*c[5]^4*c[4]^2+863940*c[5]^3*c[4]^2-151428*c[5]*c[4]^3 -295476*c[5]^2*c[4]^2-1961560*c[5]^3*c[4]^3+546300*c[5]^3*c[4]^4+54630 0*c[5]^4*c[4]^3), a[5,4] = c[5]^2*(-c[4]+c[5])/c[4]^2, `b*`[6] = 25704 9/1106060*(-765216*c[5]-765216*c[4]+1259130*c[5]^2-665260*c[5]^3+14955 3+1259130*c[4]^2-7387158*c[5]*c[4]^2-665260*c[4]^3+4107710*c[5]^3*c[4] +4219629*c[5]*c[4]-7387158*c[5]^2*c[4]-8051670*c[5]^2*c[4]^3-8051670*c [5]^3*c[4]^2+4107710*c[5]*c[4]^3+13724650*c[5]^2*c[4]^2+4993800*c[5]^3 *c[4]^3)/(87-968*c[5]*c[4]^2-968*c[5]^2*c[4]+232*c[5]^2-300*c[4]-300*c [5]+1163*c[5]*c[4]+870*c[5]^2*c[4]^2+232*c[4]^2)/(1521*c[5]*c[4]-1131* c[4]-1131*c[5]+841), a[7,5] = 575/220150628352*(318651746250*c[5]-1593 25873125*c[4]-1421535792575*c[5]^2+1962058156050*c[5]^3+656859985975*c [4]^2+8721475284757*c[5]*c[4]^2-6389689248000*c[4]^5*c[5]+322514949830 40*c[4]^5*c[5]^2-859303036800*c[5]^4+302700240000*c[4]^4-800296537950* c[4]^3-19334742159579*c[5]^3*c[4]-2354504634100*c[5]*c[4]-473969562393 60*c[4]^5*c[5]^3+13105438091536*c[5]^2*c[4]+101895037140716*c[5]^2*c[4 ]^3-94378602658330*c[4]^4*c[5]^2-35493584075136*c[5]^4*c[4]^2+77561823 507018*c[5]^3*c[4]^2-18256352533059*c[5]*c[4]^3-51452507890044*c[5]^2* c[4]^2+21538698854400*c[5]^4*c[4]^5+8744553440064*c[5]^4*c[4]+17960797 658300*c[4]^4*c[5]-63689048962560*c[5]^4*c[4]^4-152585361380286*c[5]^3 *c[4]^3+139793610440040*c[5]^3*c[4]^4+69758611551360*c[5]^4*c[4]^3)/c[ 5]/(261*c[5]^3+522*c[5]*c[4]^2-8120*c[4]^5*c[5]^2-1163*c[5]^4+812*c[4] ^4-261*c[4]^3+37050*c[5]^6*c[4]^3-52*c[5]^3*c[4]+38470*c[4]^5*c[5]^3-5 22*c[5]^2*c[4]-8206*c[5]^2*c[4]^3+53757*c[4]^4*c[5]^2-53172*c[5]^4*c[4 ]^2+7192*c[5]^3*c[4]^2+1105*c[5]*c[4]^3-702*c[5]^2*c[4]^2-37050*c[5]^4 *c[4]^5-10920*c[5]^6*c[4]^2-10257*c[5]^5*c[4]+8641*c[5]^4*c[4]+1092*c[ 5]^5-8719*c[4]^4*c[5]+122070*c[5]^4*c[4]^4+9672*c[5]^3*c[4]^3-149270*c [5]^3*c[4]^4+90770*c[5]^4*c[4]^3+66620*c[5]^5*c[4]^2-149620*c[5]^5*c[4 ]^3), a[2,1] = c[2], a[3,1] = -2/9*c[4]*(c[4]-3*c[2])/c[2], a[7,4] = - 575/220150628352*(955955238750*c[5]^2-3299728239600*c[5]^3-85930303680 0*c[4]^5+318651746250*c[4]^2+7966550014250*c[5]*c[4]^2-104738983126660 *c[4]^5*c[5]+346428738999488*c[4]^5*c[5]^2+69338609466216*c[4]^6*c[5]+ 2555875699200*c[5]^4+1962058156050*c[4]^4-64616096563200*c[5]^3*c[4]^7 -1421535792575*c[4]^3+33683272809708*c[5]^3*c[4]-1115281111875*c[5]*c[ 4]-34977242331720*c[4]^5*c[5]^3-6364466422775*c[5]^2*c[4]+168535754599 14*c[5]^2*c[4]^3-179576175900286*c[4]^4*c[5]^2+152153530852140*c[5]^4* c[4]^2-153146479124529*c[5]^3*c[4]^2-30959494853864*c[5]*c[4]^3+157469 63183299*c[5]^2*c[4]^2-312743813149440*c[5]^4*c[4]^5-29143952908416*c[ 5]^4*c[4]-47396956239360*c[5]^5*c[4]^4+177280059801600*c[5]^3*c[4]^6+6 4616096563200*c[5]^4*c[4]^6+21538698854400*c[5]^5*c[4]^5+7177470438528 0*c[4]^7*c[5]^2-17186060736000*c[5]*c[4]^7-265824412456620*c[4]^6*c[5] ^2+76695954230349*c[4]^4*c[5]+527539937515740*c[5]^4*c[4]^4+3383238399 20904*c[5]^3*c[4]^3-293237560179162*c[5]^3*c[4]^4-404987336400294*c[5] ^4*c[4]^3-6389689248000*c[5]^5*c[4]^2+32251494983040*c[5]^5*c[4]^3)/(- 9*c[5]^3-18*c[5]*c[4]^2+280*c[4]^5*c[5]^2+28*c[5]^4-28*c[4]^4+9*c[4]^3 +26*c[5]^3*c[4]-950*c[4]^5*c[5]^3+18*c[5]^2*c[4]+248*c[5]^2*c[4]^3-150 0*c[4]^4*c[5]^2+1500*c[5]^4*c[4]^2-248*c[5]^3*c[4]^2-26*c[5]*c[4]^3-26 3*c[5]^4*c[4]+263*c[4]^4*c[5]+3130*c[5]^3*c[4]^4-3130*c[5]^4*c[4]^3-28 0*c[5]^5*c[4]^2+950*c[5]^5*c[4]^3)/(39*c[4]-29)/c[4]^2, a[3,2] = 2/9*c [4]^2/c[2], b[7] = -1761205026816/5482625*(95*c[5]*c[4]-28*c[5]-28*c[4 ]+9)/(576*c[4]-575)/(576*c[5]-575), b[4] = -1/60*(38497*c[5]-10916)/c[ 4]/(61593*c[4]^3-22464*c[4]^4-55804*c[4]^2+16675*c[4]+22464*c[5]*c[4]^ 3-61593*c[5]*c[4]^2+55804*c[5]*c[4]-16675*c[5]), a[8,4] = -1/2*(373935 74400*c[5]^5*c[4]^6-29888801280*c[5]*c[4]^8-1662530850*c[5]^2-11218072 3200*c[4]^8*c[5]^3+112180723200*c[5]^4*c[4]^7+5735766450*c[5]^3+490714 1934*c[4]^5-554176950*c[4]^2-15795660456*c[5]*c[4]^2+315533251201*c[4] ^5*c[5]-914170761183*c[4]^5*c[5]^2-302737137926*c[4]^6*c[5]-4437284200 *c[5]^4-1494440064*c[4]^6-5885145786*c[4]^4+419763846600*c[5]^3*c[4]^7 +3026620866*c[4]^3-64294948961*c[5]^3*c[4]+1939619325*c[5]*c[4]-450593 985924*c[4]^5*c[5]^3+12734749113*c[5]^2*c[4]-1793105630*c[5]^2*c[4]^3+ 341237751980*c[4]^4*c[5]^2-314772147564*c[5]^4*c[4]^2+324843699572*c[5 ]^3*c[4]^2+67700547286*c[5]*c[4]^3-38482768878*c[5]^2*c[4]^2+145902973 5700*c[5]^4*c[4]^5+55035157201*c[5]^4*c[4]+124773592320*c[4]^8*c[5]^2+ 138300937380*c[5]^5*c[4]^4-367470462120*c[5]^3*c[4]^6-655188674760*c[5 ]^4*c[4]^6-119695007640*c[5]^5*c[4]^5-586852163820*c[4]^7*c[5]^2+15047 4679320*c[5]*c[4]^7+1064215236948*c[4]^6*c[5]^2-187226497470*c[4]^4*c[ 5]-1619225296372*c[5]^4*c[4]^4-854706888455*c[5]^3*c[4]^3+109890369603 8*c[5]^3*c[4]^4+967377786795*c[5]^4*c[4]^3+11093210500*c[5]^5*c[4]^2-6 7092714640*c[5]^5*c[4]^3)/(-5217*c[5]^3-10434*c[5]*c[4]^2+161330*c[4]^ 5*c[5]^2+16133*c[5]^4-16133*c[4]^4+5217*c[4]^3+15169*c[5]^3*c[4]-54630 0*c[4]^5*c[5]^3+10434*c[5]^2*c[4]+144048*c[5]^2*c[4]^3-863940*c[4]^4*c [5]^2+863940*c[5]^4*c[4]^2-144048*c[5]^3*c[4]^2-15169*c[5]*c[4]^3-1514 28*c[5]^4*c[4]+151428*c[4]^4*c[5]+1800230*c[5]^3*c[4]^4-1800230*c[5]^4 *c[4]^3-161330*c[5]^5*c[4]^2+546300*c[5]^5*c[4]^3)/(-39129*c[4]+22464* c[4]^2+16675)/c[4]^2, a[4,1] = 1/4*c[4], `b*`[4] = 1/60*(9567750*c[5]+ 4820160*c[4]-20094716*c[5]^2+12656760*c[5]^3-3725456*c[4]^2+28192204*c [5]*c[4]^2-1398351-52800540*c[5]^3*c[4]-35047639*c[5]*c[4]+78985834*c[ 5]^2*c[4]+47441100*c[5]^3*c[4]^2-66749250*c[5]^2*c[4]^2)/(172500*c[5]+ 222612*c[4]-133400*c[5]^2-306200*c[4]^2+1226488*c[5]*c[4]^2-50025+1336 32*c[4]^3-841525*c[5]*c[4]+690232*c[5]^2*c[4]+501120*c[5]^2*c[4]^3-557 568*c[5]*c[4]^3-1057818*c[5]^2*c[4]^2)/(39*c[4]-29)/(-c[4]+c[5])/c[4], a[8,6] = -593190/55303*(2708770*c[5]*c[4]+541767-1083521*c[4]-1083521 *c[5])*(c[4]-1)*(-1+c[5])/(39*c[4]-29)/(39*c[5]-29)/(54630*c[5]*c[4]-1 6133*c[5]-16133*c[4]+5217), `b*`[5] = -1/60*(4820160*c[5]+9567750*c[4] -3725456*c[5]^2-20094716*c[4]^2+78985834*c[5]*c[4]^2-1398351+12656760* c[4]^3-35047639*c[5]*c[4]+28192204*c[5]^2*c[4]+47441100*c[5]^2*c[4]^3- 52800540*c[5]*c[4]^3-66749250*c[5]^2*c[4]^2)/(222612*c[5]+172500*c[4]- 306200*c[5]^2+133632*c[5]^3-133400*c[4]^2+690232*c[5]*c[4]^2-50025-557 568*c[5]^3*c[4]-841525*c[5]*c[4]+1226488*c[5]^2*c[4]+501120*c[5]^3*c[4 ]^2-1057818*c[5]^2*c[4]^2)/(39*c[5]-29)/(-c[4]+c[5])/c[5], `b*`[1] = 1 /1000500*(-9567750*c[5]-9567750*c[4]+20094716*c[5]^2-12656760*c[5]^3+1 398351+20094716*c[4]^2-174395854*c[5]*c[4]^2-12656760*c[4]^3+116794260 *c[5]^3*c[4]+75415849*c[5]*c[4]-174395854*c[5]^2*c[4]-314440680*c[5]^2 *c[4]^3-314440680*c[5]^3*c[4]^2+116794260*c[5]*c[4]^3+440348820*c[5]^2 *c[4]^2+239954700*c[5]^3*c[4]^3)/(87-968*c[5]*c[4]^2-968*c[5]^2*c[4]+2 32*c[5]^2-300*c[4]-300*c[5]+1163*c[5]*c[4]+870*c[5]^2*c[4]^2+232*c[4]^ 2)/c[5]/c[4], c[3] = 2/3*c[4], a[4,3] = 3/4*c[4], b[1] = 1/1000500*(22 1280*c[5]*c[4]-38497*c[5]-38497*c[4]+10916)/c[5]/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 98 "A length y computation gives an expression for the square of the principal erro r 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]):\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 4505 "prin_err_norm_sqrd := (u,v,w)->1/ 549361814077440000*(-14589079478649000*w^6*v^4*u-902507025941638200*w^ 3*v^6*u+595355484308974200*v^4*u^2*w^4-8897517248450620*w^7*v^4-684036 78033600*v^5*u-567297551804778204*w^5*v^5-13261062748847740*w^5*v^2+62 794150473600*u^2*v^4-95813066166890400*w^5*u*v^3+32114909542879760*w^7 *v^5+36630795973353584*w^4*v^3+4444360109996168*v^6*w-2485788305364000 *v^7*w*u+10464908104350*u^2*v^2-51269287154400*u^2*v^3-11274294460500* v^3*u-5610007928766600*u*v^3*w^2+55541939225400*v^4*u-3758369338526068 *w^4*v^2+462845201151492*v^4*w+2339108794355400*u*v^4*w+11101553568293 0994*v^5*w^3-48768396875766*w*v^3+118394139354849*w^2*v^2-100777992763 840*w^7*v^2-151316843378255707*w^4*v^4-3926841823192782*w^3*v^3+308303 599411176136*w^4*v^5+90117430995127891*v^6*w^2-7995435248190416*w^3*v^ 4+251176601894400*u^2*w^4-1935010684867798*v^5*w+6266447357876300*w^8* v^6+800491894053167820*v^6*w^6-35082516129206140*v^5*w^2+1315453429141 600*w^7*v^3-87953647424508*w^4*v+2581503207365802*w^5*v-11108387845080 0*w^4*u+136807356067200*w^5*u+1100565307949435200*v^8*w^4+921826583302 7532*v^4*w^2-1512025277934888*w^2*v^3-420232240888678124*w^3*v^6-20507 7148617600*u^2*w^3+33598276326069491*w^6*v^2+22548588921000*w^3*u+1017 16918762686536*w^5*v^4-1590454720112032240*w^5*v^7+18515112967124594*w ^5*v^3+503889963819200*w^8*v^4-4197592183685432*w^6*v+6223145336658880 00*v^9*w^3+12532894715752600*v^7*w^7+427400351895157468*w^6*v^4+865081 649590658400*w^3*v^7-418446344435956200*v^7*w^6-2505845321557200*w^5*u *v-863083651011680000*v^9*w^4+89795279250116300*v^8*w^6-45270325000941 0620*w^4*v^7+29368608859474200*u*v^3*w^3-3553927362792800*w^8*v^5-4826 0466235636200*w^7*v^6-757962254014531560*w^6*v^5-2899697305743000*u*v^ 2*w^3-216600479716320000*v^9*w^5+233550833487440000*v^10*w^4-158325976 178607040*v^7*w^2-250358815980000*w^6*u*v^2-195538190765764132*w^4*v^6 +177506908263126400*v^8*w^2-109212083963766*w^3*v+20524398295318200*w^ 5*u*v^2+1186572926133912*w^3*v^2+23182231741760000*v^10*w^2-1036203530 32896000*v^9*w^2-7546354458930000*w^4*u*v^2+1947321224576000*w*v^8+418 59632417400*u^2*w^2-1025804570530480000*v^8*w^3-22354294893840000*v^9* w^2*u-184068215505288*w^5-38487818265288*v^5+7631698634883*v^4+2285818 48576592*w^6+37068250598883*w^4-5129190785068800*w*v^7+945901791824567 200*v^8*w^5-75682515798900*u*v*w^2+48603329856592*v^6+1300269365606572 440*w^5*v^6-152266171017900524*w^6*v^3-147162890221760000*v^10*w^3-225 209732291460000*v^9*w^4*u-149143227355443600*w^3*u*v^4+999196261388640 00*v^8*w^2*u-766886514906330000*u^2*v^7*w^4+472370199121674000*u^2*v^7 *w^3+1329694514644923000*u^2*v^6*w^4+42128052350822550*v^4*u^2*w^2-502 315589008800*v*u^2*w^2-91885007482052400*v^5*u^2*w^2-36074672827653600 0*v^4*u^2*w^3-25958667820402800*v^2*u^2*w^3+404965655967600*v^2*u^2*w- 1822829247609300*v^3*u^2*w-64977022983667500*v^5*w^2*u-191812563226821 600*v^3*u^2*w^4+3356568110523600*v*u^2*w^3-57111323424246000*u^2*v^4*w ^5+168907299218595000*u^2*v^8*w^4+967023985369680000*w^4*v^8*u-9193113 55872000*u^2*v^2*w^5+33313406360130000*w^6*v^5*u+120547778682303000*v^ 3*u^2*w^3+3160350454615200*v^2*u^2*w^2-41859632417400*u^2*v*w+45227410 48035000*u^2*v^4*w+38791046868600600*u^2*v^2*w^4-181526029407450000*u^ 2*v^6*w^5+11164587017220000*u^2*v^3*w^5-4592790739958400*u^2*v*w^4+715 65204773790000*u^2*v^7*w^5-499074166547841600*w^4*v^5*u-13526850361361 400*v^3*u^2*w^2+695541887439197400*v^5*u^2*w^3+841174539660000*u^2*v^4 *w^6-1160404011374265000*u^2*v^5*w^4+462399392705866200*w^3*v^5*u+1047 742637602293000*v^7*w^3*u+141907072713840000*v^9*w^3*u+393994434365028 000*w^5*v^6*u+131499687411432000*v^6*w^2*u-5424018857888400*u^2*v^5*w+ 90516435310009800*w^4*v^4*u+1501079356893600*w^4*v*u-16376419215218160 0*v^7*w^2*u+270530088717339000*w^5*v^4*u+148231234230534000*u^2*v^5*w^ 5-5605247279808000*v^5*w*u-74939719604148000*u^2*v^7*w^2-6670475143920 000*w^5*v^8*u-33563629286460000*w^6*v^6*u-5050341632700000*u^2*v^5*w^6 +8694151551540000*w^6*v^7*u+1216722328723305000*w^4*v^6*u+758045835319 5000*u^2*v^6*w^6-621153372875112000*v^8*w^3*u-120003473755620000*w^5*v ^7*u+22529339942139000*u*v^4*w^2+2052110341008000*u^2*v^6*w+7952874079 644900*w^4*u*v^3-9963979209000*w^3*u*v-451456550735373000*w^5*v^5*u-15 61428821761614000*w^4*v^7*u+3048369747696000*w^6*u*v^3+120536866803364 200*u^2*v^6*w^2-803895990501807000*u^2*v^6*w^3+6565269385195200*v^6*w* u-532426014027000*u*v^3*w-106430304535380000*u^2*v^8*w^3+1676572117038 0000*u^2*v^8*w^2+923657724585000*u*v^2*w^2+54752699590200*u*v^2*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 investigations suggested that the values " }{XPPEDIT 18 0 "c[2]=11/118" "6#/&%\"cG6#\"\"#*&\"#6\"\"\"\"$=\"!\"\"" }{TEXT -1 3 ", " }{XPPEDIT 18 0 "c[4]=63/296" "6#/&%\"cG6#\"\"%*&\"#j\"\"\"\"$'H! \"\"" }{TEXT -1 7 " and " }{XPPEDIT 18 0 "c[5]=409/725" "6#/&%\"cG6# \"\"&*&\"$4%\"\"\"\"$D(!\"\"" }{TEXT -1 89 " give a value for the (sq uare of the) principal error norm that is close to the minimum." }} {PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 58 "prin_err_norm_sqrd(11/118,63/296,409/725):\nevalf(sqrt(%));" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#$\"+@9pP5!#9" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 124 "Using a one dimensional \+ minimization procedure and cycling around the nodes gives very slow co nvergence towards the minimum." }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 495 "Digits := 30:\nc_2 := 11/118: c_4 := 63/296: c_5 := 409/725:\nf or ct to 120000 do\n c_2 := findmin(prin_err_norm_sqrd(c2,c_4,c_5),c 2=\{0.05,c_2,0.15\},convergence=location)[1];\n c_4 := findmin(prin_ err_norm_sqrd(c_2,c4,c_5),c4=\{0.2,c_4,0.23\},convergence=location)[1] ;\n mn := findmin(prin_err_norm_sqrd(c_2,c_4,c5),c5=\{0.5,c_5,0.65\} ,convergence=location); \n c_5 := mn[1]:\n if `mod`(ct,1000)=0 the n\n print(c[2]=c_2,c[4]=c_4,c[5]=c_5);\n print(mn[2]);\n e nd if;\nend do:\nDigits := 10:" }}{PARA 11 "" 1 "" {XPPMATH 20 "6%/&% \"cG6#\"\"#$\"?SnKB'GT\"[=()f@Ot1=a7w$G@! #I/&F%6#\"\"&$\"?`S&R;6X5J;e2U8k&F1" }}{PARA 11 "" 1 "" {XPPMATH 20 "6 #$\"?8[0aT+$pq%GwS[w5!#R" }}{PARA 11 "" 1 "" {XPPMATH 20 "6%/&%\"cG6# \"\"#$\"?>&ynyi(=Vomg+Kq\"*!#J/&F%6#\"\"%$\"?@Zu-*3!)zwDaQk$G@!#I/&F%6 #\"\"&$\"?#*R'Q2XRr)o&R&4KTcF1" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#$\"? %H_M(*R!pd!37%R[w5!#R" }}{PARA 11 "" 1 "" {XPPMATH 20 "6%/&%\"cG6#\"\" #$\"?$=.VJ#ev9=Jt]Gq\"*!#J/&F%6#\"\"%$\"?G/j(H0W)R?AGxNG@!#I/&F%6#\"\" &$\"?Yb'f$>5:Cx%o(*38k&F1" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#$\"?!=rk> 0\\/Hyx*Q[w5!#R" }}{PARA 260 "" 0 "" {TEXT -1 1 ":" }}{PARA 260 "" 0 " " {TEXT -1 1 ":" }}{PARA 11 "" 1 "" {XPPMATH 20 "6%/&%\"cG6#\"\"#$\"?p )o8D-FE))4&e#R-<*!#J/&F%6#\"\"%$\"?v=adq-Z7l&4,\\$G@!#I/&F%6#\"\"&$\"? !Q%RVo))))[fc$H$HTcF1" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#$\"??b'o1gt6I ,s(Q[w5!#R" }}{PARA 11 "" 1 "" {XPPMATH 20 "6%/&%\"cG6#\"\"#$\"?tj.`Jr i#))4&e#R-<*!#J/&F%6#\"\"%$\"?hZ9,r-Z7l&4,\\$G@!#I/&F%6#\"\"&$\"?^%oP# p))))[fc$H$HTcF1" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#$\"?A/O1WR<,8?xQ[w 5!#R" }}{PARA 11 "" 1 "" {XPPMATH 20 "6%/&%\"cG6#\"\"#$\"?#[gmI9FE))4& e#R-<*!#J/&F%6#\"\"%$\"?bPMuq-Z7l&4,\\$G@!#I/&F%6#\"\"&$\"?\"Rlc(o)))) [fc$H$HTcF1" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#$\"?Y\\Ff_R<,8?xQ[w5!#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) mini mum." }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 381 "c_2 := .91702392585 1e-1: pp := .107648387720e-9:\np1 := evalf[30](plot(prin_err_norm_sqrd (c[2],.212834901096,.56412932936),c[2]=0.076..0.107,\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=[0.076..0.107,1.07e -10..1.107e-10]);" }}{PARA 13 "" 1 "" {GLPLOT2D 367 347 347 {PLOTDATA 2 "6*-%'CURVESG6$7S7$$\"#w!\"$$\"??%HFf\"RzYNc#f%*p5\"!#R7$$\"?nmmmmmm mm\"zDW5\"F-7$$\"?LLLLLLLL3_)\\kjs(F1$\"?@$3n *)QT:Be`]\"G-6F-7$$\"?nmmmmmmm;aOP[#z(F1$\"?Yh*Q[^@PQza!H(**4\"F-7$$\" ?nmmmmmmm;zb7/fyF1$\"?z40i#G!3wolv#ex4\"F-7$$\"?LLLLLLLL3xQCGDzF1$\"?y #)*=C]j`#)R^-jc4\"F-7$$\"?nmmmmmmmT&[\\'p')zF1$\"?(=heGkl\\xu?Z(G]!)F1$\"?-y_[$))y81GQ([+#4\"F-7$$\"?nmmmmmmmTN$y_g 6)F1$\"?#*3nL'[IWd+FbN-4\"F-7$$\"?++++++++v$HY2;=)F1$\"?]w6=`^wyb%>Vy& 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-7$$\"?++++++++v=#GOZ`*F1$\"?Ln'*F1$\"?Y#*o,$z1@RH4 #*R&z5F-7$$\"?+++++++++D;J8M(*F1$\"?l;!yV^:g9s8h=/3\"F-7$$\"?nmmmmmmmT N'>(y%z*F1$\"?3l(H)yruQfe)f583\"F-7$$\"?+++++++++vE\")4h)*F1$\"?!*3,Wl $>ZtUv&**Q#3\"F-7$$\"?LLLLLLLL3_W:\\B**F1$\"?srzl(4Xo&Qh>\\]$3\"F-7$$ \"?++++++++v$Rk5())**F1$\"?w`=^u462BvkLx%3\"F-7$$\"?LLLLLLLL$eMUZ_+\"! #I$\"?oj)Rl9H!z'GHF:h3\"F-7$$\"?+++++++](o/)G#>,\"Few$\"?-(fcayFtd2)Rz i(3\"F-7$$\"?nmmmmmmmm'*Q@N=5Few$\"?aO!*GS^7*[]d2*=*3\"F-7$$\"?nmmmmmm m\"zV)p#\\-\"Few$\"?)>%Q**)R\\+#4)HO\"*34\"F-7$$\"?LLLLLLL$3_nQZ9.\"Fe w$\"?O()))z3x@T8)R@&o#4\"F-7$$\"?+++++++++Nf*Qu.\"Few$\"?DN,Sd+f+TWUjU %4\"F-7$$\"?LLLLLLLLepxfIW5Few$\"?,sri4]3)Heu8Jl4\"F-7$$\"?nmmmmmmmmw? zW]5Few$\"?g4%y2YG5\\bAc7&)4\"F-7$$\"?+++++++](=J^'*p0\"Few$\"?$>')Gw1 8btt6,G25\"F-7$$\"?+++++++]7e^VEj5Few$\"?ie,2LZgIE(G)y%H5\"F-7$$\"$2\" F*$\"?8^+aiU\\1L(3#=W06F--%&COLORG6&%$RGBG$\"\"&!\"\"$\"\"!Fb[l$\"\"*F `[l-F$6&7#7$$\"3Q++^e#R-<*!#>$\"3%*****>xQ[w5!#F-%'COLOURG6&F][lFb[lFb [lFb[l-%'SYMBOLG6$%'CIRCLEG\"#7-%&STYLEG6#%&POINTG-F$6&Fg[l-F`\\l6&F][ l$\"*++++\"!\")Fa[lFa[l-Fc\\l6$Fe\\l\"#5Fg\\l-F$6&Fg[lF]]l-Fc\\l6$%(DI AMONDGFd]lFg\\l-F$6&Fg[lF]]l-Fc\\l6$%&CROSSGFd]lFg\\l-%%FONTG6$%*HELVE TICAGFd[l-%+AXESLABELSG6%Q%c[2]6\"Q!Fg^l-F`^l6#%(DEFAULTG-%%VIEWG6$;F( Ffz;$Fgz!#7$\"%26!#8" 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 0 390 "c_4 := .212834901096: pp := .107648387720e-9:\np1 := evalf[30](pl ot(prin_err_norm_sqrd(.917023925851e-1,c[4],.56412932936),c[4]=0.21281 ..0.21286,\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],font=[HELVETIC A,9],view=[0.21281..0.21286,1.07e-10..1.106e-10]);" }}{PARA 13 "" 1 " " {GLPLOT2D 400 369 369 {PLOTDATA 2 "6*-%'CURVESG6$7S7$$\"&\"G@!\"&$\" ?O=>D.:CFpv_R*e5\"!#R7$$\"?LLLLLLL$3x&)*36G@!#I$\"?@#>IpfsWpHV,$Q.6F-7 $$\"?nmmmmm;H2P\"Q?\"G@F1$\"?g,s]pNsd$3<*))G,6F-7$$\"?LLLLLLLeRwX58G@F 1$\"?%[IX[ta'zWbhY.*4\"F-7$$\"?LLLLLLL3x%3yT\"G@F1$\"?Efmxm&*\\h%eqPto 4\"F-7$$\"?nmmmmm;z%4\\Y_\"G@F1$\"?P)f>))f>c\\oH#*H[4\"F-7$$\"?LLLLLL$ eR-/Pi\"G@F1$\"?SBn_pmn1Xj58.$4\"F-7$$\"?++++++]il'pis\"G@F1$\"?e#[JTz 1ylUPHm74\"F-7$$\"?LLLLLL$e*)>VB$=G@F1$\"?q][)GZZ6ilY;X&*3\"F-7$$\"?++ ++++]7`l2Q>G@F1$\"?oDD\"3lLBhmN8Nz3\"F-7$$\"?nmmmmmm;/j$o/#G@F1$\"?.E$ QY$yZ`TH#3*Q'3\"F-7$$\"?LLLLLLL3_>jU@G@F1$\"?%yY(y*HbBfr?%)>^3\"F-7$$ \"?+++++++]i^Z]AG@F1$\"?(4=c^.;H>'3\\Zz$3\"F-7$$\"?+++++++](=h(eBG@F1$ \"?9hq&H[cB,^t)[d#3\"F-7$$\"?+++++++]P[6jCG@F1$\"?BA#eO%4a0o#yL/:3\"F- 7$$\"?LLLLLL$e*[z(yb#G@F1$\"?)p'))4O5Y4]&3a@13\"F-7$$\"?nmmmmmm;a/cqEG @F1$\"?aNf,[/=\\.9-Eoz5F-7$$\"?nmmmmmmm;t,mFG@F1$\"?jk.hX.z,!3qV\")*y5 F-7$$\"?++++++]iSj0xGG@F1$\"?Ct6)*G)3'*ojplu#y5F-7$$\"?nmmmmmmm\"pW`(H G@F1$\"?#H'eon7SKf(RxYx2\"F-7$$\"?++++++]i!f#=$3$G@F1$\"?DRh,'GgG^DxFt s2\"F-7$$\"?++++++](=xpe=$G@F1$\"?')p*o1\"o!pbN*\\_#p2\"F-7$$\"?nmmmmm m\"H28IH$G@F1$\"?fDpjefV(p'Hk\"pm2\"F-7$$\"?nmmmmm;zpSS\"R$G@F1$\"?!*y =Yz#\\z;x\"e.`w5F-7$$\"?LLLLLLL3_?`(\\$G@F1$\"?h%f,GAPg.&4ST[w5F-7$$\" ?LLLLLL$e*)>pxg$G@F1$\"?6R=bQ%>#G7\"y&*\\l2\"F-7$$\"?++++++]Pf4t.PG@F1 $\"?lx0xv53P@,afBgrj^\\sX'p2\"F- 7$$\"?++++++++DRW9RG@F1$\"?/Yg8AW>SG@F1$ \"?4*[^]'z()RQU59#y2\"F-7$$\"?++++++]i!RU07%G@F1$\"?m<40d>+E)Q(*Q$Qy5F -7$$\"?+++++++v=S2LUG@F1$\"?i/&))**H,F3S.wA\"z5F-7$$\"?nmmmmmmm\"p)=MV G@F1$\"?q))*R/!R^&e9^w!*)z5F-7$$\"?+++++++](=]@W%G@F1$\"?t'z^0n@T%Gxd \">33\"F-7$$\"?LLLLLL$e*[$z*RXG@F1$\"?tUy78RVP?Sgrv\"3\"F-7$$\"?++++++ +]iC$pk%G@F1$\"?#3!y)Rz-]mS1(z)G3\"F-7$$\"?nmmmmm;H2qcZZG@F1$\"?[V=Svt uo7$3^_S3\"F-7$$\"?++++++]7.\"fF&[G@F1$\"?6tk!o=>Rwn!oSP&3\"F-7$$\"?nm mmmmm;/Ogb\\G@F1$\"?OL1TyWN9#R$)3\"F-7$$\"?LLLLLLLL$)*pp;&G@F1$\"?h3EK))f[56Gxq&**3\"F-7$ $\"?LLLLLLL3xe,t_G@F1$\"?'=r4Q)4\"zhU'H!><4\"F-7$$\"?nmmmmm;HdO=y`G@F1 $\"?%HUhvc#y!pYcytN4\"F-7$$\"?++++++++D>#[Z&G@F1$\"?4Z:>c$R..bGZs`4\"F -7$$\"?nmmmmmmT&G!e&e&G@F1$\"?Y!>qXv4\"F-7$$\"?LLLLLLLL$)Qk %o&G@F1$\"?JoC^$f'>Fw\">B*e*4\"F-7$$\"?++++++]iSjE!z&G@F1$\"?1&4wQ#ocK 0]SM(=5\"F-7$$\"?++++++]P40O\"*eG@F1$\"?fkRxN5\\gNpQ6;/6F-7$$\"&'G@F*$ \"?Av+)o#4p)3)fZ-t16F--%&COLORG6&%$RGBG$\"\"!F^[l$\"\"(!\"\"$\"\"#Fa[l -F$6&7#7$$\"32++'4,\\$G@!#=$\"3%*****>xQ[w5!#F-%'COLOURG6&F\\[lF^[lF^[ lF^[l-%'SYMBOLG6$%'CIRCLEG\"#7-%&STYLEG6#%&POINTG-F$6&Ff[l-F_\\l6&F\\[ lF][l$\"*++++\"!\")F^]l-Fb\\l6$Fd\\l\"#5Ff\\l-F$6&Ff[lF\\]l-Fb\\l6$%(D IAMONDGFc]lFf\\l-F$6&Ff[lF\\]l-Fb\\l6$%&CROSSGFc]lFf\\l-%%FONTG6$%*HEL VETICAG\"\"*-%+AXESLABELSG6%Q%c[4]6\"Q!Fg^l-F_^l6#%(DEFAULTG-%%VIEWG6$ ;F(Fez;$\"$2\"!#7$\"%16!#8" 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 0 387 "c_5 := .56412932936: pp := .107648387720e-9:\np1 := evalf[30](plo t(prin_err_norm_sqrd(.917023925851e-1,.212834901096,c[5]),c[5]=0.56408 5..0.564174,\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],vi ew=[0.564085..0.564174,1.07e-10..1.1055e-10]);" }}{PARA 13 "" 1 "" {GLPLOT2D 389 359 359 {PLOTDATA 2 "6*-%'CURVESG6$7S7$$\"'&3k&!\"'$\"?/ *oeF6ZMqjP[eb5\"!#R7$$\"?LLLLLLL3sY*Rp3k&!#I$\"?zrR8^Fy?lI5\"F-7$$ \"?nmmmmm\"z*)R)yi)3k&F1$\"?rBo%>Y!*Qx7=/()45\"F-7$$\"?LLLLLL$e%)f9E04 k&F1$\"?Zm.2Cm-w^q)G^()4\"F-7$$\"?LLLLLL$3#*3*pV#4k&F1$\"?\"o))[9mOXT' zU!4m4\"F-7$$\"?nmmmmm\"H()QvQV4k&F1$\"?7(*yEzn'flm'R[e%4\"F-7$$\"?LLL LLLekiJ>5'4k&F1$\"?'*)>'oh#GU!G4\"F-7$$\"?++++++D\"[+gFz4k&F1$\"? sn\"R66<_<&H))y0\"4\"F-7$$\"?LLLLLLe9%*3d\")*4k&F1$\"??d,CM^gwNq!*fN*3 \"F-7$$\"?++++++Dckixp,TcF1$\"?SE$p!yaQjO-:\\w(3\"F-7$$\"?nmmmmmmT@'oL O5k&F1$\"?Q(=s)4-s3]:,!Qi3\"F-7$$\"?LLLLLL$3Z([)Q`5k&F1$\"?&oTpJy3GLcE ?&)\\3\"F-7$$\"?+++++++D*yXes5k&F1$\"?)p-qJ[#4$o.'G!yO3\"F-7$$\"?+++++ ++v8\\f=4TcF1$\"?cs=%Gr%p()R!fRvC3\"F-7$$\"?+++++++v5WM/6TcF1$\"?6sS\" H\\>!Qq<&f?93\"F-7$$\"?LLLLLLe9\\F-t7TcF1$\"?2:b!f(HC;%*Hm7b!3\"F-7$$ \"?nmmmmmmT3wft9TcF1$\"?]*)))pwK[(o_g!=gI-s&)\\>[l2\"F-7$$\"?++ ++++vo26k7LTcF1$\"?%>wdonkaQm`N'pw5F-7$$\"?LLLLLL$eanAr\\8k&F1$\"?]1S) pFT\"*HP!3H&p2\"F-7$$\"?+++++++]'=5xo8k&F1$\"?Q$o+NvlF\"ee'=Bt2\"F-7$$ \"?++++++]i$)G;uQTcF1$\"?GW\")43E`=:$p())yx5F-7$$\"?++++++DJbacaSTcF1$ \"?OKb)yqPIDMmzO$y5F-7$$\"?++++++]P`<([D9k&F1$\"?x?z#RE^<\\itFd!z5F-7$ $\"?nmmmmmm;re&[V9k&F1$\"?#\\(4!eRKK!ysR_!)z5F-7$$\"?+++++++vLt-FYTcF1 $\"?k%Re_xYy;w<*)323\"F-7$$\"?LLLLLLe9TK;,[TcF1$\"?`SE#Gf/:HX9Q@;3\"F- 7$$\"?+++++++D$yR:*\\TcF1$\"?jsVZ9%R[[Wh!3s#3\"F-7$$\"?nmmmmm\"zHFp1<: k&F1$\"?\\(**fD0A\")=4]O_Q3\"F-7$$\"?++++++Dc.7\"zN:k&F1$\"?pt9$[O`s\\ ShuN^3\"F-7$$\"?nmmmmmmT:W(4a:k&F1$\"?3'G!*Rc8#4sWG)*['3\"F-7$$\"?++++ ++D\"GjCEt:k&F1$\"?^4&)4PLB\"Q'[>E,)3\"F-7$$\"?LLLLLLLLql?Y?K**ev(yG\"4\"F-7 $$\"?nmmmmm\"z*4p;$H;k&F1$\"?oLeoojkKvkWO3$4\"F-7$$\"?+++++++]EI=lkTcF 1$\"?q%3ffS'**>3j8T#[4\"F-7$$\"?nmmmmm;/3HLimTcF1$\"?ScBWAIUpn#QDDp4\" F-7$$\"?LLLLLLLL7hmQoTcF1$\"?$*o%QJ5lUoDOu+*)4\"F-7$$\"?++++++DJ'3um-< k&F1$\"?xQ[w5!#F-%'COLOURG6&F\\[l\" \"!F_\\lF_\\l-%'SYMBOLG6$%'CIRCLEG\"#7-%&STYLEG6#%&POINTG-F$6&Fd[l-F] \\l6&F\\[l$F_\\lF_\\l$\"*++++\"!\")F]]l-Fa\\l6$Fc\\l\"#5Fe\\l-F$6&Fd[l F[]l-Fa\\l6$%(DIAMONDGFc]lFe\\l-F$6&Fd[lF[]l-Fa\\l6$%&CROSSGFc]lFe\\l- %%FONTG6$%*HELVETICAG\"\"*-%+AXESLABELSG6%Q%c[5]6\"Q!Fg^l-F_^l6#%(DEFA ULTG-%%VIEWG6$;F(Fez;$\"$2\"!#7$\"&b5\"!#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 0 313 "Digits := 20:\nplot3d(sqrt(prin_err_norm_sqrd(. 917023925851e-1,c[4],c[5])),c[4]=0.2125..0.213,c[5]=0.564..0.565,\n ax es=boxed,grid=[25,25],labels=[`c[4]`,`c[5]`,``],view=[0.2125..0.213,0. 564..0.565,0..0.000053],\n orientation=[-25,60],color=COLOR(RGB,0.5 ,1,.2),lightmodel=light4,font=[HELVETICA,9]);\nDigits := 10:" }}{PARA 13 "" 1 "" {GLPLOT3D 433 395 395 {PLOTDATA 3 "6)-%%GRIDG6&;$\"%D@!\"%$ \"$8#!\"$;$\"$k&F,$\"$l&F,X,%)anythingG6\"6\"[gl'!%\"!!#\\bm\":\":3EF5 5EB68FEC383A3EF6C0AF7B7554703EF82807B9A23D543EF9939FCC2A68F13EFB0292E0 5E95E93EFC7429635144623EFDE7CEE5D890333EFF5D0A770BB1B23F0069BC6A0A2F81 3F012563F977BAA23F01E159CB1EE1C63F029D8150771CA63F0359C280AF79643F0416 09061170493F04D24396CCFCC63F058E636F97A5C83F064A5BE7C7B3FE3F0706221A93 012D3F07C1AC9FCC94933F087CF351956BF53F0937EF1CAE52783F09F299D855E8223F 0AACEE2514D7093F0B66E7514BE69B3F0C208141688F513EF42A16A0F2802B3EF58670 3B1131263EF6E9698E695CDC3EF8519F8BA939053EF9BDF992B4FDD23EFB2D982224D8 3B3EFC9FC7CA6AA3803EFE13F75ABB6CAC3EFF89B07196DD983F008048EB08B6D73F01 3C259095F69F3F01F84CB379169A3F02B4A24F122B9A3F03710ECB52D1723F042D7E2F A9B8673F04E9DF7F9084983F05A62438750A923F06623FE7FA55643F071E27D76FA4C0 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641293293565949]:\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#\"\"#$\"+f#R-<*!#6/&F&6#\"\"%$\"+6!\\$G@!#5/&F&6#\"\"&$\" +%H$HTcF2" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#7%/&%\"cG6#\"\"##\"$*f\"% Kl/&F&6#\"\"%#\"$i#\"%J7/&F&6#\"\"&#\"$T&\"$f*" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#7%/&%\"cG6#\"\"##\"#@\"$H#/&F&6#\"\"%#\"$*>\"$N*/&F&6# \"\"&#\"$>&\"$?*" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#7%/&%\"cG6#\"\"## \"#@\"$H#/&F&6#\"\"%#\"#j\"$'H/&F&6#\"\"&#\"#A\"#R" }}{PARA 11 "" 1 " " {XPPMATH 20 "6#7%/&%\"cG6#\"\"##\"#5\"$4\"/&F&6#\"\"%#F*\"#Z/&F&6#\" \"&#\"#A\"#R" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 56 "The minimum value for the principal error norm is . . . \+ " }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 105 "evalf[25](prin_err_norm _sqrd(.9170239258509883e-1,.21283490109565125,.5641293293565949)):\nev alf(sqrt(%));" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#$\"+5u`P5!#9" }}} {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 9 "Setting \+ " }{XPPEDIT 18 0 "c[2] = 21/229;" "6#/&%\"cG6#\"\"#*&\"#@\"\"\"\"$H#! \"\"" }{TEXT -1 3 ", " }{XPPEDIT 18 0 "c[4] = 199/935;" "6#/&%\"cG6# \"\"%*&\"$*>\"\"\"\"$N*!\"\"" }{TEXT -1 7 " and " }{XPPEDIT 18 0 "c[ 5] = 541/959;" "6#/&%\"cG6#\"\"&*&\"$T&\"\"\"\"$f*!\"\"" }{TEXT -1 57 " gives the following value for the principal error norm." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 59 "prin _err_norm_sqrd(21/229,199/935,541/959):\nevalf(sqrt(%));" }}{PARA 11 " " 1 "" {XPPMATH 20 "6#$\"+XuaP5!#9" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" } }{PARA 0 "" 0 "" {TEXT -1 56 "#--------------------------------------- ----------------" }}{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 2472 "ee := \{c[2]=21/229,\nc[ 3]=398/2805,\nc[4]=199/935,\nc[5]=541/959,\nc[6]=29/39,\nc[7]=575/576, \nc[8]=1,\nc[9]=1,\n\na[2,1]=21/229,\na[3,1]=5306932/165228525,\na[3,2 ]=18137258/165228525,\na[4,1]=199/3740,\na[4,2]=0,\na[4,3]=597/3740,\n a[5,1]=94539185952859/139708222009916,\na[5,2]=0,\na[5,3]=-36052660261 5135/139708222009916,\na[5,4]=86200229584590/34927055502479,\na[6,1]=- 8978969288480000243857587089/3094782216331416263956902924,\na[6,2]=0, \na[6,3]=99268298611547543700685925/8262922327440626705984748,\na[6,4] =-162409472645307771282700363150/17326138018112792098834790979,\na[6,5 ]=40197004738132352835791693452/40001804824576007393862935391,\na[7,1] =20286604731599462128958430510992102863475/258212522261560414040859842 9414209880064,\na[7,2]=0,\na[7,3]=-75963452911125085054379188438047598 75/237729392100642727921981994337042432,\na[7,4]=662078851658983002712 5628007103689280061615375/25083900175630865229822206008706748964562534 4,\na[7,5]=-16378468176189294528981106978962032922592625/7724684380599 693853826720717982679148003328,\na[7,6]=97768503635786059874525/119061 117135805961207808,\na[8,1]=13855115178421131108323299597371007/171512 2015582524727359996149290620,\na[8,2]=0,\na[8,3]=-23954172438450319856 8188548325/7288002275830304575859927972,\na[8,4]=956518309196672115306 68834468508100881883600/3523904935082188448634512810545504833479151,\n a[8,5]=-741690472086763685807332786432434772072/3380356932484635875329 44907545647623719,\na[8,6]=257132510132640765480/305396035838269868597 ,\na[8,7]=-1187059899803728084992/657869227273850599426895,\na[9,1]=16 97710672/26928207375,\na[9,2]=0,\na[9,3]=0,\na[9,4]=164490038363253411 875/503598049753427901952,\na[9,5]=688263291863311978681/2293101349345 635227520,\na[9,6]=237419997287733/1436816713308800,\na[9,7]=212989216 4719607808/556154447061193625,\na[9,8]=-680161433/184588800,\n\nb[1]=1 697710672/26928207375,\nb[2]=0,\nb[3]=0,\nb[4]=164490038363253411875/5 03598049753427901952,\nb[5]=688263291863311978681/22931013493456352275 20,\nb[6]=237419997287733/1436816713308800,\nb[7]=2129892164719607808/ 556154447061193625,\nb[8]=-680161433/184588800,\n\n`b*`[1]=10979243775 19127782369/17974423922545824426000,\n`b*`[2]=0,\n`b*`[3]=0,\n`b*`[4]= 22948656626877224955778831923125/68757710895541194720834617432064,\n`b *`[5]=6422693908420175795081848535941/23428040725205156020096373368320 ,\n`b*`[6]=5409978194277871906843931/26640747567082588647769600,\n`b*` [7]=6875699405989996627114328064/2577985433915837392040353375,\n`b*`[8 ]=-11170820866846625947/4400427318713395200,\n`b*`[9]=-1/1764\}:" }}} {PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 5 " Let " }{XPPEDIT 18 0 "T[6, 8];" "6#&%\"TG6$\"\"'\"\")" }{TEXT -1 127 " denote the vector whose components are the principal error terms of the 8 stage, order 6 scheme (the error terms of order 7)." }}{PARA 0 "" 0 "" {TEXT -1 5 "Let " }{XPPEDIT 18 0 "`T*`[5,9];" "6#&%#T*G6$\"\" &\"\"*" }{TEXT -1 145 " denote the vector whose components are the pr incipal error terms of the embedded 9 stage, order 5 scheme (the error terms of order 6) and let " }{XPPEDIT 18 0 "`T*`[6,9];" "6#&%#T*G6$ \"\"'\"\"*" }{TEXT -1 99 " denote the vector whose components are the error terms of order 7 of the embedded order 5 scheme." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 39 "Denote the 2-norm of these vectors by " }{XPPEDIT 18 0 "abs(abs(T[6,8]));" "6#-%$absG6#-F $6#&%\"TG6$\"\"'\"\")" }{TEXT -1 3 ", " }{XPPEDIT 18 0 "abs(abs(`T*`[ 5,9]));" "6#-%$absG6#-F$6#&%#T*G6$\"\"&\"\"*" }{TEXT -1 7 " and " } {XPPEDIT 18 0 "abs(abs(`T*`[6,9]));" "6#-%$absG6#-F$6#&%#T*G6$\"\"'\" \"*" }{TEXT -1 15 " respectively." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }} {PARA 0 "" 0 "" {TEXT -1 8 "Define: " }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "A[7] = abs(abs(T[6, 8]));" "6#/&%\"AG6#\"\"(-%$absG 6#-F)6#&%\"TG6$\"\"'\"\")" }{TEXT -1 6 ", " }{XPPEDIT 18 0 "B[7] = abs(abs(`T*`[6,9]))/abs(abs(`T*`[5,9]));" "6#/&%\"BG6#\"\"(*&-%$absG6 #-F*6#&%#T*G6$\"\"'\"\"*\"\"\"-F*6#-F*6#&F/6$\"\"&F2!\"\"" }{TEXT -1 6 ", " }{XPPEDIT 18 0 "C[7] = abs(abs(`T*`[6,9]-T[6, 8]))/abs(abs( `T*`[5, 9]));" "6#/&%\"CG6#\"\"(*&-%$absG6#-F*6#,&&%#T*G6$\"\"'\"\"*\" \"\"&%\"TG6$F2\"\")!\"\"F4-F*6#-F*6#&F06$\"\"&F3F9" }{TEXT -1 1 "." }} {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 77 "Dormand a nd Prince have suggested that as well as attempting to ensure that " }{XPPEDIT 18 0 "A[7];" "6#&%\"AG6#\"\"(" }{TEXT -1 73 " is a minimum, if the embedded scheme is to be used for error control, " }{XPPEDIT 18 0 "B[7];" "6#&%\"BG6#\"\"(" }{TEXT -1 7 " and " }{XPPEDIT 18 0 "C [7];" "6#&%\"CG6#\"\"(" }{TEXT -1 27 " should be chosen so that " }} {PARA 0 "" 0 "" {TEXT -1 68 "they are \"similar in magnitude\" and als o not differ too much from 1." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 186 "errterms6_8 := PrincipalErr orTerms(6,8,'expanded'):\n`errterms6_9*` :=subs(b=`b*`,PrincipalErrorT erms(6,9,'expanded')):\n`errterms5_9*` := subs(b=`b*`,PrincipalErrorTe rms(5,9,'expanded')):" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 326 "snmB := sqrt(add(evalf(subs(ee,`er rterms6_9*`[i]))^2,i=1..nops(`errterms6_9*`))):\nsdnB := sqrt(add(eval f(subs(ee,`errterms5_9*`[i]))^2,i=1..nops(`errterms5_9*`))):\nsnmC := \+ sqrt(add((evalf(subs(ee,`errterms6_9*`[i])-subs(ee,errterms6_8[i])))^2 ,i=1..nops(errterms6_8))):\n'B[7]'= evalf[8](snmB/sdnB);\n'C[7]'= eval f[8](snmC/sdnB);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/&%\"BG6#\"\"($\") .U#R\"!\"(" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/&%\"CG6#\"\"($\")$[NR\" !\"(" }}}{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 2472 "ee := \{c[ 2]=21/229,\nc[3]=398/2805,\nc[4]=199/935,\nc[5]=541/959,\nc[6]=29/39, \nc[7]=575/576,\nc[8]=1,\nc[9]=1,\n\na[2,1]=21/229,\na[3,1]=5306932/16 5228525,\na[3,2]=18137258/165228525,\na[4,1]=199/3740,\na[4,2]=0,\na[4 ,3]=597/3740,\na[5,1]=94539185952859/139708222009916,\na[5,2]=0,\na[5, 3]=-360526602615135/139708222009916,\na[5,4]=86200229584590/3492705550 2479,\na[6,1]=-8978969288480000243857587089/30947822163314162639569029 24,\na[6,2]=0,\na[6,3]=99268298611547543700685925/82629223274406267059 84748,\na[6,4]=-162409472645307771282700363150/17326138018112792098834 790979,\na[6,5]=40197004738132352835791693452/400018048245760073938629 35391,\na[7,1]=20286604731599462128958430510992102863475/2582125222615 604140408598429414209880064,\na[7,2]=0,\na[7,3]=-759634529111250850543 7918843804759875/237729392100642727921981994337042432,\na[7,4]=6620788 516589830027125628007103689280061615375/250839001756308652298222060087 067489645625344,\na[7,5]=-16378468176189294528981106978962032922592625 /7724684380599693853826720717982679148003328,\na[7,6]=9776850363578605 9874525/119061117135805961207808,\na[8,1]=1385511517842113110832329959 7371007/1715122015582524727359996149290620,\na[8,2]=0,\na[8,3]=-239541 724384503198568188548325/7288002275830304575859927972,\na[8,4]=9565183 0919667211530668834468508100881883600/35239049350821884486345128105455 04833479151,\na[8,5]=-741690472086763685807332786432434772072/33803569 3248463587532944907545647623719,\na[8,6]=257132510132640765480/3053960 35838269868597,\na[8,7]=-1187059899803728084992/6578692272738505994268 95,\na[9,1]=1697710672/26928207375,\na[9,2]=0,\na[9,3]=0,\na[9,4]=1644 90038363253411875/503598049753427901952,\na[9,5]=688263291863311978681 /2293101349345635227520,\na[9,6]=237419997287733/1436816713308800,\na[ 9,7]=2129892164719607808/556154447061193625,\na[9,8]=-680161433/184588 800,\n\nb[1]=1697710672/26928207375,\nb[2]=0,\nb[3]=0,\nb[4]=164490038 363253411875/503598049753427901952,\nb[5]=688263291863311978681/229310 1349345635227520,\nb[6]=237419997287733/1436816713308800,\nb[7]=212989 2164719607808/556154447061193625,\nb[8]=-680161433/184588800,\n\n`b*`[ 1]=1097924377519127782369/17974423922545824426000,\n`b*`[2]=0,\n`b*`[3 ]=0,\n`b*`[4]=22948656626877224955778831923125/68757710895541194720834 617432064,\n`b*`[5]=6422693908420175795081848535941/234280407252051560 20096373368320,\n`b*`[6]=5409978194277871906843931/2664074756708258864 7769600,\n`b*`[7]=6875699405989996627114328064/25779854339158373920403 53375,\n`b*`[8]=-11170820866846625947/4400427318713395200,\n`b*`[9]=-1 /1764\}:" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 " " {TEXT -1 77 "The stability function R for the 8 stage, order 6 schem e is given as follows." }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 75 "su bs(ee,StabilityFunction(6,8,'expanded')):\nR := unapply(%,z):\n'R(z)'= R(z);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/-%\"RG6#%\"zG,4\"\"\"F)F'F)* &#F)\"\"#F)*$)F'F,F)F)F)*&#F)\"\"'F)*$)F'\"\"$F)F)F)*&#F)\"#CF)*$)F'\" \"%F)F)F)*&#F)\"$?\"F)*$)F'\"\"&F)F)F)*&#F)\"$?(F)*$)F'F1F)F)F)*&#\"6j '=F[ewxW2(*\":+Sz1n$ytH`j\")\\F)*$)F'\"\"(F)F)F)*&#\"5H@$>$p]tl4F\":+& )pw\"fWVK)3aC\"F)*$)F'\"\")F)F)F)" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }} {PARA 0 "" 0 "" {TEXT -1 124 "We can find the point where the boundary of the stability region intersects the negative real axis by solving \+ the equation: " }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "R(z ) = 1;" "6#/-%\"RG6#%\"zG\"\"\"" }{TEXT -1 2 ". " }}{EXCHG {PARA 0 "> \+ " 0 "" {MPLTEXT 1 0 28 "z0 := newton(R(z)=1,z=-4.5);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%#z0G$!+c=prW!\"*" }}}{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 := 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$$!3Q++++++!>&!#<$\"3!f2\"3]'=;m$F*7$$!3QML3T![!f^F*$\"3mo,$=$H 4vMF*7$$!3Ynm;#3'4G^F*$\"3S1c5/(QqH$F*7$$!3a++DBT9(4&F*$\"3\"*3qw#p@r7 $F*7$$!3kLLLk@>m]F*$\"3#))\\$GW)=]'HF*7$$!3E+]U'*)HB,&F*$\"3E5BSIzj+FF *7$$!3!pm;&GwYe\\F*$\"3#RD,PX6uX#F*7$$!3s+](\\(Q*y*[F*$\"30!f)f:oJ2AF* 7$$!3nLLV@,KP[F*$\"33^M#47X,)>F*7$$!3'RLLd%[MwZF*$\"3C[6))zIws5F*7$$!3%3+])f w&\\O%F*$\"3I'pW[+>+7)!#=7$$!3$QL$)f7eWC%F*$\"3QZ\">\"3-h!R'F]p7$$!3A+ +lN]MCTF*$\"3GF+r&z>L,&F]p7$$!3ummYeRz+SF*$\"3]Qp\"*ok/%*QF]p7$$!3_LLV -,(>*QF*$\"3Y\"HVC6KU6$F]p7$$!35++S:-YpPF*$\"3()=171hjDCF]p7$$!3K+++\" HZkk$F*$\"3q\"*fKyM!*)*=F]p7$$!3;++gW:!z_$F*$\"3%*=!4znDu^\"F]p7$$!3hL L)*\\1D?MF*$\"3QGed4a.e7F]p7$$!3'ommSKVAH$F*$\"3_?D\\:HkO5F]p7$$!3/nmE GV!Q=$F*$\"31d-,Ok>\"4*!#>7$$!39++0(*RmdIF*$\"3wE(*Q(ylU;)F`s7$$!39nmE I%3g%HF*$\"3i.FdHi(4v(F`s7$$!3-++0xX]BGF*$\"3'*f\\o;e&pl(F`s7$$!3*)*** \\\"R>&oq#F*$\"3QmEB^8ZfyF`s7$$!3gmm;\\r8&e#F*$\"3*>9xmrm&H$)F`s7$$!3y mmrw\\OtCF*$\"3K4wc^;,n*)F`s7$$!3SLL$))e.GN#F*$\"3#e\")o:l)4d)*F`s7$$! 3nLL)**=uvA#F*$\"3?'f9&=SV*4\"F]p7$$!3K++:I;c=@F*$\"3v2x6?(yf@\"F]p7$$ !31LL.z]#3+#F*$\"3u(3'R76ug8F]p7$$!3M++?,<>z=F*$\"3h,.d@)*4K:F]p7$$!3; ++!4<(>gF]p7$$!3 H++q9zA<:F*$\"3Kt]jf'4S>#F]p7$$!3EnmEY;O-9F*$\"3)[2=`1#egCF]p7$$!3#)** ***pQ<(z7F*$\"35&=O4%RO\"y#F]p7$$!3)RL$efMeo6F*$\"3PIT]#))o\"3JF]p7$$! 3I****fAZ3Z5F*$\"3A-ex8nj4NF]p7$$!3xqm;(zQwK*F]p$\"3w:8m%)\\mMRF]p7$$! 3&z***\\)ecE8)F]p$\"3-+-R(H$3MWF]p7$$!3'3nmm0VV'pF]p$\"3oL-'\\[$f$)\\F ]p7$$!3P)***\\iqATdF]p$\"3Ns+Xvz)>j&F]p7$$!3aFLL*)4AjXF]p$\"3wGRn.t4Oj F]p7$$!33LLLO'R&eLF]p$\"3Ay&3wvus9(F]p7$$!3Uim;`O$Q;#F]p$\"3gCys,ZEa!) F]p7$$!3?*****>$H-m5F]p$\"3IcdIQ4$)))*)F]p7$$\"3v*QLLU?>#>F`s$\"3[-s.# 30%>5F*7$$\"3%ymmY^avJ\"F]p$\"3xfs,(HH39\"F*7$$\"3E0+]HcU&!\"#$\"#\\Fidl;$!\"(Fidl$\"$Z\"Fid l" 1 2 0 1 10 0 2 9 1 4 2 1.000000 45.000000 45.000000 0 0 "Curve 1" " Curve 2" "Curve 3" "Curve 4" "Curve 5" "Curve 6" }}}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 50 "The following picture sho ws the stability region. " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1390 "R := z -> 1+z+1/2*z^2+1/6*z^3+1/2 4*z^4+1/120*z^5+1/720*z^6+\n 970744777658482718663/498163532973783 6706794000*z^7+27096573506931932129/1245408832434459176698500*z^8:\npt s := []: z0 := 0:\nfor ct from 0 to 240 do\n zz := newton(R(z)=exp(c t*Pi/20*I),z=z0):\n z0 := zz:\n pts := [op(pts),[Re(zz),Im(zz)]]: \nend do:\np1 := plot(pts,color=COLOR(RGB,.08,.4,.13)):\np2 := plots[p olygonplot]([seq([pts[i-1],pts[i],[-2.2,0]],i=2..nops(pts))],\n \+ style=patchnogrid,color=COLOR(RGB,.15,.8,.25)):\npts := []: z0 := 2 +4.75*I:\nfor ct from 0 to 60 do\n zz := newton(R(z)=exp(ct*Pi/30*I) ,z=z0):\n z0 := zz:\n pts := [op(pts),[Re(zz),Im(zz)]]:\nend do:\n p3 := plot(pts,color=COLOR(RGB,.08,.4,.13)):\np4 := plots[polygonplot] ([seq([pts[i-1],pts[i],[1.89,4.72]],i=2..nops(pts))],\n style =patchnogrid,color=COLOR(RGB,.15,.8,.25)):\npts := []: z0 := 2-4.75*I: \nfor ct from 0 to 60 do\n zz := newton(R(z)=exp(ct*Pi/30*I),z=z0): \n z0 := zz:\n pts := [op(pts),[Re(zz),Im(zz)]]:\nend do:\np5 := p lot(pts,color=COLOR(RGB,.08,.4,.13)):\np6 := plots[polygonplot]([seq([ pts[i-1],pts[i],[1.89,-4.72]],i=2..nops(pts))],\n style=patch nogrid,color=COLOR(RGB,.15,.8,.25)):\np7 := plot([[[-5.19,0],[2.29,0]] ,[[0,-5.19],[0,5.19]]],color=black,linestyle=3):\nplots[display]([p||( 1..7)],view=[-5.19..2.29,-5.19..5.19],font=[HELVETICA,9],\n \+ labels=[`Re(z)`,`Im(z)`],axes=boxed,scaling=constrained);" }}{PARA 13 "" 1 "" {GLPLOT2D 381 565 565 {PLOTDATA 2 "6/-%'CURVESG6$7]z7$$\"\" !F)F(7$F($\"3++++Fjzq:!#=7$$!3C+++3(4.N$!#G$\"3=+++VEfTJF-7$$!3))***** 4FbN<%!#F$\"3:+++7))Q7ZF-7$$\"3!*******39-0#!#B$\"3)*******Q?b*4\"!#<7$$\"3(*******)QLK^)FO$\"35+++ NFic7FR7$$\"35+++_;S(p#!#A$\"3#******Rb!o89FR7$$\"3b+++srT5tFen$\"31++ +H#32d\"FR7$$\"33+++94RV?(F`o$\"3/+++1>,T?FR7$$\"3++++*[v`E\"! #?$\"3')*****pm,q>#FR7$$\"36+++WfP8?F`p$\"3-+++z.0_BFR7$$\"3++++Y3zhGF `p$\"3&******4(Rc0DFR7$$\"30+++^!*R7NF`p$\"3/+++jtucEFR7$$\"3%******z[ 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(approximately) " }{XPPEDIT 18 0 "[-4.4717, 0];" "6#7$,$-%&FloatG6$ \"& " 0 "" {MPLTEXT 1 0 426 "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 \+ 970744777658482718663/4981635329737836706794000*z^7+\n 2709657350 6931932129/1245408832434459176698500*z^8:\nDigits := 25:\npts := []: z 0 := 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,0,.75,.2),thickness=2,font=[HELVET ICA,9]);\nDigits := 10:" }}{PARA 13 "" 1 "" {GLPLOT2D 261 276 276 {PLOTDATA 2 "6(-%'CURVESG6#7hq7$$\"\"!F)F(7$$!:Z@d$F-$\":#[GV3<#yrI&=$G'F-7$$!:u!R#=/&) Q)R:b!z%F-$\":.eBa/.Y2'zxC%*F-7$$!:!ys7nb^*pT3O*eF-$\":]@iL_Z=91PmD\"! #D7$$!:yf$zS05;#)[#Q\"pF-$\":7!*G37.7nK'zq:F?7$$!:`KS>')*3*o+I!oyF-$\" :.&R'3`Fe=fb\\)=F?7$$!:yl^Q-\"=IQ=yl()F-$\":QdX+UwZm&[6*>#F?7$$!:0eac$ RfaVv`7'*F-$\":7Xl\"z5?o?TF8DF?7$$!:!\\nd6>aOlD5T5F?$\":\"emKm[)pKQLu# 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\":4cJAZy*>BR[%)=Ffu7$$\":u+CU6\"zw7ptS\\F?$\":K[rvQqQ'Q(=e\">Ffu7$$\" :1-Xl@*Q0Cy2-]F?$\":z:Ffu7$$\":(p>z%=WHhQ(Ri]F?$\":Bxu8D[q< O[%y>Ffu7$$\":OP_QNt9$*3b;7&F?$\":LH#*f=?2GXR(4?Ffu7$$\":ZPMB\"p/rCn!) z^F?$\":*p+o$=z2g!>,T?Ffu7$$\":4O![z-d/SS!oB&F?$\":F$GF<#Qy,KjA2#Ffu7$ $\":nc7(HYR'zM%f#H&F?$\":n;!4#\\f0!\\5\\.@Ffu7$$\":rQU4.&fJy+7Z`F?$\": xM9i@X&F?$\":rATEM;Imm,q>#Ffu7$$\":NgBB\")3*3y%\\C]&F?$\":\"\\^ )Gm*Q\"ow-\"GAFfu7$$\":%og;PZm[@PA^bF?$\":sn#4*QGVHwi\"fAFfu7$$\":8\\J &y(>$)G2^$)f&F?$\":b'=,3)zgeCx,H#Ffu7$$\":l^F;#RL1=.tVcF?$\":hBnn9NFhZ T6K#Ffu7$$\":]XvKPsE2C\\so&F?$\":TC![/o4@z.0_BFfu7$$\":t+Mw:i*4EHyGdF? $\":3wOttJ5Oa)*GQ#Ffu7$$\":=vu9u)3'zm\">odF?$\":!QP0]\")zDA-o8CFfu7$$ \":*[AqEOA^c$=`!eF?$\":$R3\"Gk?4vL*QWCFfu7$$\":$p/*pq*yzy^)*ReF?$\":f$ y5)QOl**[>]Z#Ffu7$$\":QM=\"zAMT(R*)>(eF?$\":p/'HL=oIrRc0DFfu7$$\":8(=` #>ZEj(y4,fF?$\":trBGBt<7y:g`#Ffu7$$\":H!>&\\bP7t()Rq#fF?$\":$>(o$)p4x$ [wOmDFfu7$$\":jj%)[d3)fFu\\\\fF?$\":[yq9XZe&3A4 ofF?$\":K476#GJBn7uEEFfu7$$\":Yb;`1PMtMnB)fF?$\":RP\"*4v,[KOZnl#Ffu7$$ \":p(>b2p:3V:w\"*fF?$\":vvOIg%zRrAi'o#Ffu7$$\":-**)*))\\a\"p48d&*fF?$ \":9R\\]Zzn[!yN;FFfu7$$\":aS3$4I[jDT*G*fF?$\":`3)4^Euz*pXfu#Ffu7$$\":2 4qJZ%oww&RD)fF?$\":47%omPjX`wPvFFfu7$$\":o^-RMKM&>S)G'fF?$\":1f=\"\\*p `>OXY!GFfu7$$\":a6WIad2V3>;$fF?$\":9gxrN7#fc0uLGFfu7$$\":)y*yqO2]\\1%G &)eF?$\":P7%)f3^1!Q]liGFfu7$$\":<#H]9QnR:*4$=eF?$\":OzBEe#GI22Q\"*GFfu 7$$\":592\"y#e!y!z)y?dF?$\":oWt(**3M_$f4*>HFfu7$$\":Duv`HU_hjA@d&F?$\" :*erFcr2\"p(QB[HFfu7$$\":eD_Ivo&RqnO:`F?$\":r4N3!)p8w!fMwHFfu7$$\":2JJ ?U!=IbV0zXF?$\":i+*[DI\"p'>#QU+$Ffu7$$!:@=4+zN[!oK)p<&F?$\":'RNHQ*z!HS N!>.$Ffu7$$!:#yXAhMU&)**4K!3[*4i'Gl-T'F?$\":E 3Y!fz\\&\\`%emJFfu7$$!:dt#4w*GlFe$zTlF?$\":;TxpvA:Z9YF>$Ffu7$$!:(ePZ?1 h-Ye`hmF?$\":oiPi))))GwBN'=KFfu7$$!:o " 0 "" {MPLTEXT 1 0 198 "Dig its := 15:\nz0 := 0.6*I:\nfor ct from 17 to 20 do\n newton(R(z)=exp( ct*Pi/100*I),z=z0);\nend do;``;\nz0 := 3.0*I:\nfor ct from 96 to 99 do \n newton(R(z)=exp(ct*Pi/100*I),z=z0);\nend do;\nDigits := 10:" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#^$$!0:&z3&)e(R&!#C$\"0&[J;rqS`!#:" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#^$$!0]?uK\\*fM!#C$\"0m[l&!#:" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#^$$\"0VTU(=(Q#G!#C$\"0n<7?D!pf!#:" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#^$$\"0-]4S@]q\"!#B$\"0`Zt6%=$G'!#:" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#%!G" }}{PARA 11 "" 1 "" {XPPMATH 20 "6 #^$$\"0W^/8z*o&*!#=$\"0q8w!fMwH!#9" }}{PARA 11 "" 1 "" {XPPMATH 20 "6# ^$$\"0L.6=6e&=!#=$\"08p'>#QU+$!#9" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#^ $$!0QbBLo)er!#=$\"0!3HSN!>.$!#9" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#^$$ !00P(**=?d " 0 "" {MPLTEXT 1 0 328 "Digits := 15:\nreal_part \+ := proc(u)\n Re(newton(R(z)=exp(u*Pi*I),z=0.6*I))\nend proc:\nu0 := \+ bisect('real_part'(u),u=0.17..0.20);\nnewton(R(z)=exp(u0*Pi*I),z=0.6*I );``;\nreal_part := proc(u)\n Re(newton(R(z)=exp(u*Pi*I),z=3.0*I))\n end proc:\nu0 := bisect('real_part'(u),u=0.96..0.99);\nnewton(R(z)=exp (u0*Pi*I),z=3.0*I);\nDigits := 10:" }}{PARA 11 "" 1 "" {XPPMATH 20 "6# >%#u0G$\"0^IjaWe'=!#:" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#^#$\"0u/ixA<' e!#:" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%!G" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%#u0G$\"0(zc0p&=s*!#:" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#^#$\"0_Vej/.,$!#9" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 4 "The " }{TEXT 260 73 "stability region intersects the no negative imaginary axis in the interval" }{TEXT -1 39 " [ 0.5862, 3. 0103 ] (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 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 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)*&#\"B$=R&e22#ek;Y#o,o1O#\"E+kK&\\!> (=XF1aj-]1Em\"F)*$)F'F1F)F)F)*&#\"A(f!>6tPca)e-5 " 0 "" {MPLTEXT 1 0 33 "z_0 := newton(`R*`(z)=-1,z=-4.5);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%$z_0G$!+i**prW!\"*" }}}{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 := pl ot([[[z_0,-1]]$3],style=point,symbol=[circle,cross,diamond],color=blac k):\np_3 := plot([[z_0,0],[z_0,-1]],linestyle=3,color=COLOR(RGB,0,.5,0 )):\nplots[display]([p_1,p_2,p_3],view=[-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&!#<$!3eaLd*)f22BF*7$$!3G++vz=Po \\F*$!3_7S7K5b#)>F*7$$!3#**\\iE!Rai[F*$!3!p*fp**z=HKuxr(3!*\\FK7$$!3'****\\XoI<#RF*$!3=G!eRJe1+%FK7$$!3++]ZTF#[ 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INTG-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$%*HELVETICAG\"\"*-%+AXESLABELSG6%Q\"z6\"Q!F[bl-Fcal6#%( DEFAULTG-%%VIEWG6$;$!$4&!\"#$\"#\\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" "C urve 3" "Curve 4" "Curve 5" "Curve 6" }}}}{PARA 0 "" 0 "" {TEXT -1 0 " " }}{PARA 0 "" 0 "" {TEXT -1 82 "The following picture shows the stabi lity region for the 9 stage, order 5 scheme. " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1581 "`R*` := z \+ -> 1+z+1/2*z^2+1/6*z^3+1/24*z^4+1/120*z^5+\n 236066801682461664582 070758539183/166260650026354062745187190495326400*z^6+\n 944445417 10025885456377311190597/415651625065885156862967976238316000*z^7+\n \+ 317152706820901582856750854213/21315467952096674710921434678888000*z ^8-\n 3870939072418847447/313843025773483712528022000*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)]]:\n end do:\np_1 := plot(pts,color=COLOR(RGB,0,.33,.08)):\np_2 := plots[po lygonplot]([seq([pts[i-1],pts[i],[-2.2,0]],i=2..nops(pts))],\n \+ style=patchnogrid,color=COLOR(RGB,0,.65,.15)):\npts := []: z0 := 1.9 +4.5*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,.33,.08)):\np_4 := plots[polygonpl ot]([seq([pts[i-1],pts[i],[1.80,4.42]],i=2..nops(pts))],\n st yle=patchnogrid,color=COLOR(RGB,0,.65,.15)):\npts := []: z0 := 1.9-4.5 *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,0,.33,.08)):\np_6 := plots[polygonplot]( [seq([pts[i-1],pts[i],[1.80,-4.42]],i=2..nops(pts))],\n style =patchnogrid,color=COLOR(RGB,0,.65,.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 \+ labels=[`Re(z)`,`Im(z)`],axes=boxed,scaling=constrained);" }} {PARA 13 "" 1 "" {GLPLOT2D 407 552 552 {PLOTDATA 2 "6/-%'CURVESG6$7ew7 $$\"\"!F)F(7$$\"3f*****pyezt%!#F$\"3++++Fjzq:!#=7$$\"3%******Rm([#>$!# D$\"35+++UEfTJF07$$\"3u*****Rxa[#R!#C$\"3>+++&o)Q7ZF07$$\"3#******\\\\ rQT#!#B$\"3[*****\\0#=$G'F07$$\"3,+++evZ65!#A$\"33+++.+'R&yF07$$\"3?++ 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46.000000 44.000000 0 0 "Curve 1" "Curve 2" "C urve 3" "Curve 4" "Curve 5" "Curve 6" "Curve 7" "Curve 8" "Curve 9" "C urve 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 t he combined scheme" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 " > " 0 "" {MPLTEXT 1 0 1 ";" }}}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 35 "co efficients of the combined scheme" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 2472 "ee := \{c[2]=21/229,\nc[3] =398/2805,\nc[4]=199/935,\nc[5]=541/959,\nc[6]=29/39,\nc[7]=575/576,\n c[8]=1,\nc[9]=1,\n\na[2,1]=21/229,\na[3,1]=5306932/165228525,\na[3,2]= 18137258/165228525,\na[4,1]=199/3740,\na[4,2]=0,\na[4,3]=597/3740,\na[ 5,1]=94539185952859/139708222009916,\na[5,2]=0,\na[5,3]=-3605266026151 35/139708222009916,\na[5,4]=86200229584590/34927055502479,\na[6,1]=-89 78969288480000243857587089/3094782216331416263956902924,\na[6,2]=0,\na [6,3]=99268298611547543700685925/8262922327440626705984748,\na[6,4]=-1 62409472645307771282700363150/17326138018112792098834790979,\na[6,5]=4 0197004738132352835791693452/40001804824576007393862935391,\na[7,1]=20 286604731599462128958430510992102863475/258212522261560414040859842941 4209880064,\na[7,2]=0,\na[7,3]=-7596345291112508505437918843804759875/ 237729392100642727921981994337042432,\na[7,4]=662078851658983002712562 8007103689280061615375/250839001756308652298222060087067489645625344, \na[7,5]=-16378468176189294528981106978962032922592625/772468438059969 3853826720717982679148003328,\na[7,6]=97768503635786059874525/11906111 7135805961207808,\na[8,1]=13855115178421131108323299597371007/17151220 15582524727359996149290620,\na[8,2]=0,\na[8,3]=-2395417243845031985681 88548325/7288002275830304575859927972,\na[8,4]=95651830919667211530668 834468508100881883600/3523904935082188448634512810545504833479151,\na[ 8,5]=-741690472086763685807332786432434772072/338035693248463587532944 907545647623719,\na[8,6]=257132510132640765480/305396035838269868597, \na[8,7]=-1187059899803728084992/657869227273850599426895,\na[9,1]=169 7710672/26928207375,\na[9,2]=0,\na[9,3]=0,\na[9,4]=1644900383632534118 75/503598049753427901952,\na[9,5]=688263291863311978681/22931013493456 35227520,\na[9,6]=237419997287733/1436816713308800,\na[9,7]=2129892164 719607808/556154447061193625,\na[9,8]=-680161433/184588800,\n\nb[1]=16 97710672/26928207375,\nb[2]=0,\nb[3]=0,\nb[4]=164490038363253411875/50 3598049753427901952,\nb[5]=688263291863311978681/229310134934563522752 0,\nb[6]=237419997287733/1436816713308800,\nb[7]=2129892164719607808/5 56154447061193625,\nb[8]=-680161433/184588800,\n\n`b*`[1]=109792437751 9127782369/17974423922545824426000,\n`b*`[2]=0,\n`b*`[3]=0,\n`b*`[4]=2 2948656626877224955778831923125/68757710895541194720834617432064,\n`b* `[5]=6422693908420175795081848535941/23428040725205156020096373368320, \n`b*`[6]=5409978194277871906843931/26640747567082588647769600,\n`b*`[ 7]=6875699405989996627114328064/2577985433915837392040353375,\n`b*`[8] =-11170820866846625947/4400427318713395200,\n`b*`[9]=-1/1764\}:" }}} {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#\"\"##\"#@\"$H#/&F%6#\"\"$#\"$)R\"%0G/&F%6#\"\"%#\"$*>\"$N*/&F%6# \"\"&#\"$T&\"$f*/&F%6#\"\"'#\"#H\"#R/&F%6#\"\"(#\"$v&\"$w&/&F%6#\"\") \"\"\"/&F%6#\"\"*FR" }}}{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$\"\"#\"\"\"#\"#@\"$H#/&F%6$\"\"$F(#\"(KpI&\"*D&G_;/&F%6 $F/F'#\")es8=F2/&F%6$\"\"%F(#\"$*>\"%SP/&F%6$F;F'\"\"!/&F%6$F;F/#\"$(f F>/&F%6$\"\"&F(#\"/fG&f=RX*\"0;*4?A3(R\"/&F%6$FKF'FB/&F%6$FKF/#!0N^h-m _g$FN/&F%6$FKF;#\"/!f%eH-?')\"/zC]bq#\\$/&F%6$\"\"'F(#!=*3(edQC++[)Gp* y*)\"=CH!p&RE;9L;Ay%4$/&F%6$FjnF'FB/&F%6$FjnF/#\";Dfo+PaZ:h)Ho#**\":[Z )fqE1WFB#HE)/&F%6$FjnF;#!?]JO+FGrxIXEZ4C;\">z4zM))4#z7\"=!QhK_Mp\"zNGNK\"QZ+(>S\">\"RNH'QR2gdC[!=+S/&F%6$\"\"(F(#\"JvM'G5#*4 ^I%e*G@Y*fJZg'G?\"Ik+))4UTH%)f3/9/chA_7#e#/&F%6$FfpF'FB/&F%6$FfpF/#!Fv )fZ!Q%)=zV0&3D6\"HXjf(\"EKC/PV*>)>#zsU15#RHxB/&F%6$FfpF;#\"Ov`hh+G*o.r +Gc7F+$)*e;&)y?m\"NW`iX'*[nq3g?A)H_'3jv,!R3D/&F%6$FfpFK#!MDEfAH.i*yp5 \")*GXH*=w\"o%yj\"\"LGL+[\"zE)zr?n#Q&Qp*f!Q%oCx/&F%6$FfpFjn#\"8DX()fgy NO]ox*\"93y?hf!e8<61>\"/&F%6$\"\")F(#\"D25P(f*HB$368@%y^6bQ\"\"C?1H\\h **ftsCDe:?7:.XQC4$=l&*\"L^\"zM$[]X0\"G^M'[% )=#3N\\!R_$/&F%6$FhrFK#!Hs?xMCV'yKt!eojn3s/pT(\"H>PiZca2\\%H`(ej%[KpN! Q$/&F%6$FhrFjn#\"6![l2kK,^KrD\"6(fo)p#Qe.'R0$/&F%6$FhrFfp#!7#*\\3GP!)* *)fq=\"\"9&*oU*f]QFF#pyl/&F%6$\"\"*F(#\"+s1r(p\"\",vt?Gp#/&F%6$F`uF'FB /&F%6$F`uF/FB/&F%6$F`uF;#\"6v=T`KOQ+\\k\"\"6_>!zU`(\\!)f.&/&F%6$F`uFK# \"6\"oy>Jj=Hj#)o\"7?vANcM\\85$H#/&F%6$F`uFjn#\"0LxG(**>uB\"1+)3Lr;oV\" /&F%6$F`uFfp#\"43yg>Z;#*)H@\"3DO>hqWahb/&F%6$F`uFhr#!*L9;!o\"*+))e%=" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT 259 38 "weigh ts 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#\"\"\"#\"+s1r(p\"\",vt?Gp#/&F%6#\"\"#\" \"!/&F%6#\"\"$F//&F%6#\"\"%#\"6v=T`KOQ+\\k\"\"6_>!zU`(\\!)f.&/&F%6#\" \"&#\"6\"oy>Jj=Hj#)o\"7?vANcM\\85$H#/&F%6#\"\"'#\"0LxG(**>uB\"1+)3Lr;o V\"/&F%6#\"\"(#\"43yg>Z;#*)H@\"3DO>hqWahb/&F%6#\"\")#!*L9;!o\"*+))e%= " }}}{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#\"\"\"#\"7pByF\">vPCz4\"\"8+g UCeaARUuz\"/&F%6#\"\"#\"\"!/&F%6#\"\"$F//&F%6#\"\"%#\"ADJ#>$)yd&\\Axoi c'[H#\"Ak?VTb*3rd(o/&F%6#\"\"&#\"@Tf`[=3&zv,U3RpAk\"A?$oLP'4?g: 0_sS!GM#/&F%6#\"\"'#\":JR%o!>(yF%>y*4a\";+'pxk)e#3nvuSm#/&F%6#\"\"(#\" =k!GV6Fm***)fS*pvo\"=vLNS?RPe\"RV&)zd#/&F%6#\"\")#!5ZfiYo'3#3<6\"4+_R8 (=tU+W/&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 30 " #=============================" }}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 15 "a scheme with " }{XPPEDIT 18 0 "c[7] = 1728/1729;" "6#/&%\"cG6#\"\"( *&\"%G<\"\"\"\"%H " 0 "" {MPLTEXT 1 0 1 ";" }}}{PARA 0 "" 0 "" {TEXT -1 52 "#---------------------------------------------------" }}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 28 "checking the combined scheme" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 35 "coefficients of the combined s cheme" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 2419 "ee := \{c[2]=55/604,\nc[3]=86/611,\nc[4]=129/611,\n c[5]=562/999,\nc[6]=25/33,\nc[7]=1728/1729,\nc[8]=1,\nc[9]=1,\na[2,1]= 55/604,\na[3,1]=656438/20532655,\na[3,2]=2233592/20532655,\na[4,1]=129 /2444,\na[4,2]=0,\na[4,3]=387/2444,\na[5,1]=11379576325181/16591126906 359,\na[5,2]=0,\na[5,3]=-14480836320821/5530375635453,\na[5,4]=4139647 9505524/16591126906359,\na[6,1]=-3277828022520854505720678625/93094198 4189004121287101412,\na[6,2]=0, a[6,3]=16036828475594748052837775/1104 320265941879147434284,\na[6,4]=-2024967022933348076278835034350/177666 633425593828346456771343,\na[6,5]=128250153920078340306971850/11111405 7854856180218323669,\na[7,1]=1027311288352892762679823552816060640736/ 100017465428353477655102490880994099675,\na[7,2]=0,\na[7,3]=-319583399 218917235748122654831799520/7666271645206398506276676005113153,\na[7,4 ]=309853242826319046243113700571575488375374176/9059548279238747452639 357500349347045575147,\na[7,5]=-48209067061862081514667691555571147998 3232/188177190150438948277052975318150206923289,\na[7,6]=9793528151803 2740868758496/126352318943704270751022325,\na[8,1]=6176672051962930287 68837394929401/59572901637951905260312643716200,\na[8,2]=0,\na[8,3]=-7 34325479049899076018546413/17448823759340364442192828,\na[8,4]=1026118 80576141156159106897686663174423254/2972016175809293986626508592155810 158783,\na[8,5]=-297880443826969314952760262689300658642/1149791657445 32084288043320814139766233,\na[8,6]=59645563090975120908/7631491769126 4991025,\na[8,7]=-2372974894108974768977/4049500807421551820510712,\na [9,1]=2350382879/37582963200,\na[9,2]=0,\na[9,3]=0,\na[9,4]=1830473214 85658386133/564833555402036603760,\na[9,5]=27899748449743503627/896742 43702534905040,\na[9,6]=42494109833439/260652887700800,\na[9,7]=740190 076116618685831/69155160690290757120,\na[9,8]=-18738157/1773760,\n\nb[ 1]=2350382879/37582963200,\nb[2]=0,\nb[3]=0,\nb[4]=1830473214856583861 33/564833555402036603760,\nb[5]=27899748449743503627/89674243702534905 040,\nb[6]=42494109833439/260652887700800,\nb[7]=740190076116618685831 /69155160690290757120,\nb[8]=-18738157/1773760,\n\n`b*`[1]=25840735977 4721212871/4258903365184545792000,\n`b*`[2]=0,\n`b*`[3]=0,\n`b*`[4]=32 5668356622185113177778435233/984722587474007074909647594240,\n`b*`[5]= 117216732464037657080802634257/406475600385980177774696356096,\n`b*`[6 ]=48441628589025019393881/244109082100582095488000,\n`b*`[7]=870665371 500134942827763822063/120564099141830684284932218880,\n`b*`[8]=-199841 4293009470233/281403607000355840,\n`b*`[9]=1/560\}:" }}}{PARA 0 "" 0 " " {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}} {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 64 "The Butch er tableau in exact and approximate form is as follows." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 576 "subs(e e,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#78 7&#\"#b\"$/'F(%!GF+7&#\"#')\"$6'#\"'Qkl\")bE`?#\"(#fLAF2F+7&#\"$H\"F/# F7\"%WC\"\"!#\"$(QF97&#\"$i&\"$***#\"/\"=Djdz8\"\"/fj!p7\"f;F:#!/@3KO3 [9\".`ajv.`&7&F+F+F+#\"/Cb]zkRTFC7&#\"#D\"#L#!=D'y1s0X&3_A!GyF$\"<795( G@T+*=%)>%4$*F:#\";vx$G0[Zfv%Go.;\":%GMu9z=%fE?V5\"7&F+F+#!@]V.N)yi2[L $H-n\\-#\"?V8xckMGQfDMjmw<#\"<]=(pIS$y+#R:]#G\"\"$\"C`J60gnwi])R1_krim(7&F+#\"NwTPv$)[v:d+P6Vi/>j#GC `)4$\"LZ^dXqM\\.]d$REXZ(Q#z#[&f!*#!KKK)*z9rbb\"pnY^\"3i=1n!4#[\"K*GBp? ]\"=`(H0x#[*Q/:!>x\")=#\";'\\e(o3uK!=:GNz*\"0sm<'\"A+irVEJg_!>&zj,HdfF:#!<8ka=g2**)\\!zaKM(\";GG >UWOS$fP#)[u\"7&F+#\"KaKUuJm'o(*o5fh:Thd!)=h-\"\"I$ye,\"e:#f3li')RH4e< ;?(H#!HU'e1I*oi-w_\\Jpp#QW!)yH\"HLiwRT\"3KV!)G%3KXul\"z\\6#\"5347v44jb kf\"5D5*\\E\"p<\\Jw7&F+F+F+#!7x*oZ(*3T*[(HP#\":72^?=b@u!3]\\S7&Fjo#\"+ zGQ]B\",+K'HePF:F:7&F+#\"6LhQec[@t/$=\"6gPgO?SbN$[c#\"5FO]V(\\%[(**y# \"5S]!\\`-PCu'*)#\"/RM$)4T\\U\"0+3q()Glg#7&F+F+#\"6Jeo=m6w+>S(\"5?rv!H !pg^:p#!)d\"Q(=\"(gPx\"7&F+%F_____________________________________GFer Fer7&%\"bGF`qF:F:FcqF]r7&%#b*G#\"6rG@@ZxftSe#\"7+?zXX=lL!*eUF:F:7&F+# \"?L_Vyx<8^=AmNocK\"?SUfZ'4\\22SZ(eAZ)*#\"?dUj-33dw.kCt;s6\"?'4cjpux]-*eG;W[\"9+!)[&4#e+@34TC7&F+#\"?j?#QwFG%\\8+:Pl1() \"?!))=A$\\G%oI=9*4k07#!4L-Z4IH9%)*>\"3SeN+qg.9G#Fjo\"$g&Q(pprint26\" " }}}{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,$ \").'f5*!\"*F(%!GF+F+F+F+F+F+F+7,$\")'GvS\"!\")$\")R/(>$F*$\")U#y3\"F/ F+F+F+F+F+F+F+7,$\")IH6@F/$\")CBy_F*$\"\"!F:$\")(pMe\"F/F+F+F+F+F+F+7, $\")ciDcF/$\")F$)eoF/F9$!)#=%=E!\"($\")v4&\\#FDF+F+F+F+F+7,$\")wvvvF/$ !)*z4_$FDF9$\")+>_9!\"'$!)lvR6FN$\")3Aa6FDF+F+F+F+7,$\")j@%***F/$\")>8 F5FNF9$!)RpoTFN$\")U=?MFN$!)v*=c#FD$\")$o4v(F/F+F+F+7,$\"\"\"F:$\")e#o .\"FNF9$!)EX3UFN$\")*fe!#6F+F+7,F[o$\") ?&QD'F*F9F9$\").tSKF/$\")KB6JF/$\")]HI;F/$\")CLq5FN$!))3k0\"FNF+7,%\"b GF[pF9F9F]pF_pFapFcpFepF+7,%#b*G$\")CYngF*F9F9$\")#4sI$F/$\")Nt$)GF/$ \")bU%)>F/$\")tf@sFD$!)Wf,rFD$\")Vr&y\"!#5Q(pprint36\"" }}}{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(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 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 := StageOrderCon ditions(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(proc(L) local i; for i to nops(L) do if not evalb(L[i]) then break end if end do; i end proc,%):\nsimpl ify(%); " }}{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 t he principal error conditions are satisfied." }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 123 "RK6_8err_eqs := PrincipalErrorConditions(6,8,'exp anded'):\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,'expande d'):\nevalf(evalf[14](sqrt(add(subs(ee,errterms6_8[i])^2,i=1.. nops(er rterms6_8)))));" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#$\"+yZ?H(*!#:" }}} {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 79 "The 2 nor m of the principal error of the order 5 embedded scheme is as follows. " }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 153 "`errterms5_9*` := subs( b=`b*`,PrincipalErrorTerms(5,9,'expanded')):\nevalf(evalf[14](sqrt(add (subs(ee,`errterms5_9*`[i])^2,i=1.. nops(`errterms5_9*`)))));" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#$\"+D$*HEl!#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] = 55/604;" "6#/&%\"cG6#\"\"#*&\"#b\"\"\"\"$/'!\"\"" }{TEXT -1 3 ", " }{XPPEDIT 18 0 "c[4] = 129/611;" "6#/&%\"cG6#\"\"%*&\"$H\" \"\"\"\"$6'!\"\"" }{TEXT -1 3 ", " }{XPPEDIT 18 0 "c[5] = 562/999;" " 6#/&%\"cG6#\"\"&*&\"$i&\"\"\"\"$***!\"\"" }{TEXT -1 3 ", " }{XPPEDIT 18 0 "c[6] = 25/33;" "6#/&%\"cG6#\"\"'*&\"#D\"\"\"\"#L!\"\"" }{TEXT -1 3 ", " }{XPPEDIT 18 0 "c[7] = 1728/1729;" "6#/&%\"cG6#\"\"(*&\"%G< \"\"\"\"%H " 0 "" {MPLTEXT 1 0 218 "e1 := \{c[2]=55/604,c[4]=129/611,c[5]=562/999,c[6]=25/33,c[7]=172 8/1729,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]=1/560\}:\neqns := subs(e1,cdns):\nnops(%);\nind ets(eqns);\nnops(%);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#\"#W" }}{PARA 12 "" 1 "" {XPPMATH 20 "6# " 0 "" {MPLTEXT 1 0 22 "infolevel[so lve] := 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 2539 "e3 := \{a[ 7,3] = -319583399218917235748122654831799520/7666271645206398506276676 005113153, a[4,2] = 0, b[5] = 27899748449743503627/8967424370253490504 0, b[8] = -18738157/1773760, c[8] = 1, c[9] = 1, a[6,2] = 0, a[6,1] = \+ -3277828022520854505720678625/930941984189004121287101412, a[7,4] = 30 9853242826319046243113700571575488375374176/90595482792387474526393575 00349347045575147, a[5,2] = 0, a[8,4] = 102611880576141156159106897686 663174423254/2972016175809293986626508592155810158783, a[7,1] = 102731 1288352892762679823552816060640736/10001746542835347765510249088099409 9675, b[1] = 2350382879/37582963200, a[6,5] = 128250153920078340306971 850/111114057854856180218323669, b[6] = 42494109833439/260652887700800 , `b*`[8] = -1998414293009470233/281403607000355840, `b*`[4] = 3256683 56622185113177778435233/984722587474007074909647594240, a[8,5] = -2978 80443826969314952760262689300658642/1149791657445320842880433208141397 66233, `b*`[1] = 258407359774721212871/4258903365184545792000, a[6,4] \+ = -2024967022933348076278835034350/177666633425593828346456771343, a[5 ,4] = 41396479505524/16591126906359, a[7,6] = 979352815180327408687584 96/126352318943704270751022325, `b*`[3] = 0, `b*`[2] = 0, b[2] = 0, b[ 3] = 0, a[9,3] = 0, a[9,2] = 0, a[8,2] = 0, a[7,2] = 0, `b*`[5] = 1172 16732464037657080802634257/406475600385980177774696356096, b[4] = 1830 47321485658386133/564833555402036603760, b[7] = 740190076116618685831/ 69155160690290757120, a[8,3] = -734325479049899076018546413/1744882375 9340364442192828, a[6,3] = 16036828475594748052837775/1104320265941879 147434284, a[5,1] = 11379576325181/16591126906359, a[5,3] = -144808363 20821/5530375635453, a[8,7] = -2372974894108974768977/4049500807421551 820510712, a[9,1] = 2350382879/37582963200, a[8,6] = 59645563090975120 908/76314917691264991025, a[9,6] = 42494109833439/260652887700800, `b* `[6] = 48441628589025019393881/244109082100582095488000, a[2,1] = 55/6 04, a[9,7] = 740190076116618685831/69155160690290757120, c[2] = 55/604 , c[4] = 129/611, c[5] = 562/999, c[6] = 25/33, c[7] = 1728/1729, c[3] = 86/611, a[8,1] = 617667205196293028768837394929401/5957290163795190 5260312643716200, a[3,2] = 2233592/20532655, a[9,5] = 2789974844974350 3627/89674243702534905040, a[9,8] = -18738157/1773760, `b*`[7] = 87066 5371500134942827763822063/120564099141830684284932218880, a[4,1] = 129 /2444, a[9,4] = 183047321485658386133/564833555402036603760, `b*`[9] = 1/560, a[4,3] = 387/2444, a[7,5] = -482090670618620815146676915555711 479983232/188177190150438948277052975318150206923289, a[3,1] = 656438/ 20532655\}:" }{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 64 "The Butcher tableau in exact and appr oximate form is as follows." }}{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],s eq(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&#\"#b\"$/'F(%!GF+7&#\" #')\"$6'#\"'Qkl\")bE`?#\"(#fLAF2F+7&#\"$H\"F/#F7\"%WC\"\"!#\"$(QF97&# \"$i&\"$***#\"/\"=Djdz8\"\"/fj!p7\"f;F:#!/@3KO3[9\".`ajv.`&7&F+F+F+#\" /Cb]zkRTFC7&#\"#D\"#L#!=D'y1s0X&3_A!GyF$\"<795(G@T+*=%)>%4$*F:#\";vx$G 0[Zfv%Go.;\":%GMu9z=%fE?V5\"7&F+F+#!@]V.N)yi2[L$H-n\\-#\"?V8xckMGQfDMj mw<#\"<]=(pIS$y+#R:]#G\"\"$\"C `J60gnwi])R1_krim(7&F+#\"NwTPv$)[v:d+P6Vi/>j#GC`)4$\"LZ^dXqM\\.]d$REXZ (Q#z#[&f!*#!KKK)*z9rbb\"pnY^\"3i=1n!4#[\"K*GBp?]\"=`(H0x#[*Q/:!>x\")=# \";'\\e(o3uK!=:GNz*\"0sm<' \"A+irVEJg_!>&zj,HdfF:#!<8ka=g2**)\\!zaKM(\";GG>UWOS$fP#)[u\"7&F+#\"Ka KUuJm'o(*o5fh:Thd!)=h-\"\"I$ye,\"e:#f3li')RH4e<;?(H#!HU'e1I*oi-w_\\Jpp #QW!)yH\"HLiwRT\"3KV!)G%3KXul\"z\\6#\"5347v44jbkf\"5D5*\\E\"p<\\Jw7&F+ F+F+#!7x*oZ(*3T*[(HP#\":72^?=b@u!3]\\S7&Fjo#\"+zGQ]B\",+K'HePF:F:7&F+# \"6LhQec[@t/$=\"6gPgO?SbN$[c#\"5FO]V(\\%[(**y#\"5S]!\\`-PCu'*)#\"/RM$) 4T\\U\"0+3q()Glg#7&F+F+#\"6Jeo=m6w+>S(\"5?rv!H!pg^:p#!)d\"Q(=\"(gPx\"7 &F+%F_____________________________________GFerFer7&%\"bGF`qF:F:FcqF]r7 &%#b*G#\"6rG@@ZxftSe#\"7+?zXX=lL!*eUF:F:7&F+#\"?L_Vyx<8^=AmNocK\"?SUfZ '4\\22SZ(eAZ)*#\"?dUj-33dw.kCt;s6\"?'4cjpux]-*eG;W [\"9+!)[&4#e+@34TC7&F+#\"?j?#QwFG%\\8+:Pl1()\"?!))=A$\\G%oI=9*4k07#!4L -Z4IH9%)*>\"3SeN+qg.9G#Fjo\"$g&Q(pprint06\"" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 136 "subs(e3,matrix([s eq([c[i],seq(a[i,j],j=1..i-1),``$(10-i)],i=2..9),\n[`b`,seq(b[i],i=1.. 8),``],[`b*`,seq(`b*`[i],i=1..9)]])):\nevalf[8](%);" }}{PARA 11 "" 1 " " {XPPMATH 20 "6#K%'matrixG6#7,7,$\").'f5*!\"*F(%!GF+F+F+F+F+F+F+7,$\" )'GvS\"!\")$\")R/(>$F*$\")U#y3\"F/F+F+F+F+F+F+F+7,$\")IH6@F/$\")CBy_F* $\"\"!F:$\")(pMe\"F/F+F+F+F+F+F+7,$\")ciDcF/$\")F$)eoF/F9$!)#=%=E!\"($ \")v4&\\#FDF+F+F+F+F+7,$\")wvvvF/$!)*z4_$FDF9$\")+>_9!\"'$!)lvR6FN$\") 3Aa6FDF+F+F+F+7,$\")j@%***F/$\")>8F5FNF9$!)RpoTFN$\")U=?MFN$!)v*=c#FD$ \")$o4v(F/F+F+F+7,$\"\"\"F:$\")e#o.\"FNF9$!)EX3UFN$\")*fe!#6F+F+7,F[o$\")?&QD'F*F9F9$\").tSKF/$\")KB6JF/$\")] HI;F/$\")CLq5FN$!))3k0\"FNF+7,%\"bGF[pF9F9F]pF_pFapFcpFepF+7,%#b*G$\") CYngF*F9F9$\")#4sI$F/$\")Nt$)GF/$\")bU%)>F/$\")tf@sFD$!)Wf,rFD$\")Vr&y \"!#5Q(pprint16\"" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 " > " 0 "" {MPLTEXT 1 0 146 "RK6_8eqs := [op(RowSumConditions(8,'expande d')),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,`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 "" }}{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 0 "" 0 "" {TEXT -1 0 "" }} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}{SECT 1 {PARA 4 "" 0 " " {TEXT -1 22 "eG .. general solution" }}{PARA 0 "" 0 "" {TEXT -1 0 " " }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 71947 "eG := \{c[8] = 1, b[3 ] = 0, b[2] = 0, c[9] = 1, b[5] = -1/60*(10*c[6]*c[7]*c[4]-5*c[6]*c[4] -5*c[7]*c[6]+3*c[6]-5*c[7]*c[4]+3*c[4]-2+3*c[7])/(-c[4]+c[5])/c[5]/(-c [5]^3+c[5]^2-c[7]*c[5]+c[7]*c[5]^2-c[6]*c[5]+c[6]*c[5]^2+c[7]*c[6]-c[6 ]*c[7]*c[5]), a[5,2] = 0, a[4,2] = 0, `b*`[3] = 0, `b*`[2] = 0, a[8,2] = 0, a[7,2] = 0, a[6,2] = 0, a[6,5] = 1/2*c[6]*(30*c[6]^2*c[5]^2*c[4] ^2-2*c[6]^2*c[4]^2+c[6]^3*c[4]-12*c[6]^2*c[4]*c[5]^2-3*c[5]^2*c[6]*c[4 ]+c[5]*c[6]*c[4]^2+c[6]*c[4]^3-12*c[6]^2*c[4]^2*c[5]-30*c[5]*c[6]^3*c[ 4]^2-20*c[5]*c[6]^2*c[4]^4+30*c[6]^2*c[5]*c[4]^3-c[5]*c[4]^3+20*c[4]^3 *c[5]*c[6]^3-20*c[6]^2*c[5]^2*c[4]^3+c[5]^2*c[4]^2-30*c[5]^2*c[6]*c[4] ^3+12*c[5]^2*c[6]*c[4]^2+2*c[6]^2*c[4]*c[5]+20*c[5]^2*c[6]*c[4]^4+2*c[ 6]^2*c[5]^2-2*c[6]^3*c[5]+12*c[6]^3*c[4]*c[5])/c[5]/(c[4]^3-2*c[5]*c[4 ]^2+30*c[5]^2*c[4]^3-6*c[5]*c[4]^3-10*c[4]^4*c[5]^2-c[5]^3+10*c[5]^4*c [4]^2-30*c[5]^3*c[4]^2+6*c[5]^3*c[4]+2*c[5]^2*c[4]), b[1] = 1/60*(30*c [5]*c[6]*c[7]*c[4]-10*c[5]*c[6]*c[4]-10*c[6]*c[7]*c[5]+5*c[6]*c[5]-10* c[6]*c[7]*c[4]+5*c[6]*c[4]+5*c[7]*c[6]-3*c[6]-10*c[5]*c[7]*c[4]+5*c[5] *c[4]+5*c[7]*c[5]-3*c[5]+5*c[7]*c[4]-3*c[4]+2-3*c[7])/c[5]/c[6]/c[7]/c [4], a[6,1] = -1/4*c[6]*(60*c[5]^2*c[6]*c[4]^4-90*c[5]^2*c[6]^3*c[4]^2 -60*c[5]*c[6]^2*c[4]^4-2*c[6]^3*c[4]^2+50*c[5]^4*c[4]^3*c[6]+40*c[6]^2 *c[4]^5*c[5]-40*c[4]^5*c[5]^3+36*c[4]*c[5]^2*c[6]^3-20*c[5]^4*c[6]^2*c [4]^2+8*c[6]^2*c[5]^3+4*c[6]^2*c[4]^3-24*c[5]*c[6]^3*c[4]^2-8*c[6]^2*c [4]^2*c[5]+24*c[6]^2*c[5]*c[4]^3+60*c[4]^3*c[5]*c[6]^3-4*c[6]*c[5]*c[4 ]^3-14*c[5]^3*c[6]*c[4]+7*c[5]^3*c[4]^2+6*c[5]^2*c[6]*c[4]^3+14*c[5]^2 *c[6]*c[4]^2+66*c[5]^3*c[6]*c[4]^2-80*c[5]^3*c[6]^2*c[4]^3+30*c[6]^2*c [5]^2*c[4]^3-24*c[6]^2*c[5]^2*c[4]^2-240*c[6]*c[5]^3*c[4]^3-48*c[6]^2* c[5]^3*c[4]+150*c[6]^2*c[5]^3*c[4]^2-24*c[4]^4*c[5]^2-9*c[5]^2*c[4]^3+ 6*c[4]^4*c[5]-40*c[5]^4*c[4]^4-20*c[6]^2*c[4]^4*c[5]^2-6*c[5]^2*c[6]^3 -2*c[6]*c[4]^4-24*c[5]^3*c[4]^3+60*c[6]^3*c[4]^3*c[5]^2+7*c[6]^3*c[4]* c[5]-40*c[5]^2*c[6]*c[4]^5-40*c[6]^3*c[4]^4*c[5]+110*c[4]^4*c[6]*c[5]^ 3+120*c[5]^3*c[4]^4)/c[5]/c[4]^2/(10*c[5]^3*c[4]^2+10*c[5]^2*c[4]^3-30 *c[5]^2*c[4]^2+6*c[5]^2*c[4]-c[5]^2+6*c[5]*c[4]^2+c[5]*c[4]-c[4]^2), b [4] = 1/60*(-2+3*c[5]-5*c[7]*c[6]+3*c[7]+3*c[6]+10*c[6]*c[7]*c[5]-5*c[ 6]*c[5]-5*c[7]*c[5])/c[4]/(-c[6]*c[4]^3-c[4]^3+c[4]^4+c[7]*c[4]^2+c[6] *c[4]^2-c[6]*c[7]*c[4]+c[6]*c[7]*c[4]^2-c[5]*c[4]^3+c[5]*c[4]^2-c[7]*c [4]^3-c[5]*c[7]*c[4]+c[5]*c[7]*c[4]^2-c[5]*c[6]*c[4]+c[5]*c[6]*c[4]^2+ c[6]*c[7]*c[5]-c[5]*c[6]*c[7]*c[4]), b[6] = 1/60*(10*c[5]*c[7]*c[4]-5* c[5]*c[4]-5*c[7]*c[5]+3*c[5]-5*c[7]*c[4]+3*c[4]-2+3*c[7])/(-c[4]+c[6]) /(-c[5]+c[6])/c[6]/(c[6]^2-c[6]+c[7]-c[7]*c[6]), `b*`[9] = 1/10*(50*c[ 5]^2*c[6]*c[4]^2-40*c[5]^2*c[6]*c[4]+10*c[6]*c[5]^2-40*c[5]*c[6]*c[4]^ 2+35*c[5]*c[6]*c[4]-10*c[6]*c[5]+10*c[6]*c[4]^2-10*c[6]*c[4]+3*c[6]-5* c[5]^2*c[4]-5*c[5]*c[4]^2+10*c[5]*c[4]-c[5]-c[4])/(3*c[6]-28*c[5]*c[6] *c[4]^2-28*c[5]^2*c[6]*c[4]+8*c[6]*c[5]^2-9*c[6]*c[4]-9*c[6]*c[5]+28*c [5]*c[6]*c[4]+30*c[5]^2*c[6]*c[4]^2+8*c[6]*c[4]^2-4*c[5]^2*c[4]-4*c[5] *c[4]^2+9*c[5]*c[4]-c[5]-c[4]), a[7,5] = -1/2*c[7]*(-17*c[5]^2*c[6]*c[ 4]^4+5*c[6]*c[5]*c[4]^4+42*c[4]^2*c[5]^2*c[7]^3+17*c[7]^3*c[5]*c[4]^3- 80*c[7]^3*c[5]^2*c[4]^3+c[7]*c[5]*c[4]^4+80*c[5]^2*c[4]^4*c[7]^2-60*c[ 4]^5*c[6]^2*c[5]^2+20*c[4]^3*c[7]^3*c[5]^3-20*c[7]^2*c[5]^2*c[4]^5+40* c[4]^5*c[6]*c[5]^3-5*c[5]*c[6]^2*c[4]^4+c[7]^3*c[4]^2-c[7]^2*c[4]^3-80 *c[5]^3*c[4]^4*c[7]+20*c[7]^3*c[5]^2*c[4]^4-200*c[6]^2*c[4]^5*c[7]*c[5 ]^3-290*c[5]^4*c[4]^2*c[7]^2*c[6]+20*c[6]^2*c[7]*c[5]^4*c[4]^2+40*c[6] *c[5]^2*c[4]^4*c[7]-140*c[5]^4*c[4]^3*c[6]-100*c[5]^4*c[4]^5*c[6]+190* c[5]^4*c[4]^4*c[6]-40*c[7]^2*c[4]^4*c[5]^3+20*c[5]^4*c[4]^4*c[7]+2*c[4 ]^4*c[7]^2-20*c[7]^2*c[5]^4*c[4]^3-4*c[6]^2*c[7]^2*c[5]-8*c[7]*c[6]^2* c[5]^3-3*c[7]*c[6]*c[5]^3+2*c[7]^2*c[6]*c[5]^3+60*c[7]*c[6]^2*c[5]*c[4 ]^5-40*c[6]*c[5]^2*c[4]^5*c[7]+17*c[7]*c[4]^4*c[6]*c[5]+80*c[7]^2*c[5] ^3*c[4]^3+5*c[7]*c[4]^4*c[6]^2+c[5]^2*c[7]^3-260*c[6]^2*c[7]*c[5]^2*c[ 4]^3-12*c[6]^2*c[7]^2*c[5]^2*c[4]+20*c[6]^2*c[7]^2*c[5]^2*c[4]^2-40*c[ 6]*c[7]*c[5]^3*c[4]^3-41*c[6]^2*c[7]^3*c[4]*c[5]+20*c[7]^2*c[6]*c[5]*c [4]^4+3*c[7]*c[6]*c[4]^3-2*c[6]^2*c[7]*c[4]^2-3*c[6]*c[7]^2*c[4]^2+4*c [7]^2*c[6]*c[4]^3+6*c[5]*c[7]^2*c[4]^3+4*c[6]*c[5]*c[7]^2*c[4]^2+c[7]^ 2*c[5]*c[6]*c[4]-11*c[5]*c[7]*c[6]*c[4]^3+2*c[6]^2*c[4]^2*c[5]-c[7]^3* c[4]*c[5]+17*c[5]^2*c[6]*c[7]*c[4]^2-200*c[5]^4*c[7]^2*c[4]^4*c[6]-50* c[4]^5*c[6]*c[7]*c[5]^3-150*c[5]^2*c[6]*c[7]^3*c[4]^4-c[6]^2*c[5]*c[4] ^3+2*c[7]^3*c[4]^2*c[6]*c[5]+2*c[6]^2*c[7]*c[4]*c[5]+108*c[7]*c[6]^2*c [5]*c[4]^3+4*c[6]^2*c[7]*c[5]^2+2*c[6]*c[5]*c[7]*c[4]^2-10*c[6]^2*c[7] ^3*c[5]^2+6*c[5]^4*c[7]*c[6]-8*c[6]*c[7]^3*c[5]^2+3*c[6]*c[7]^2*c[5]^2 +2*c[6]^2*c[7]^2*c[5]^2-27*c[6]^2*c[4]^2*c[7]*c[5]-62*c[6]^2*c[7]^2*c[ 4]^2*c[5]+46*c[6]*c[5]^4*c[4]^2+20*c[4]^5*c[7]*c[5]^3+2*c[7]^3*c[4]*c[ 6]*c[5]-27*c[6]^2*c[5]^2*c[7]*c[4]+120*c[6]^2*c[5]^2*c[7]*c[4]^2-35*c[ 6]*c[5]^2*c[7]*c[4]^3-37*c[6]*c[5]^3*c[7]*c[4]^2+53*c[6]^2*c[5]^3*c[7] *c[4]-54*c[7]^2*c[5]^2*c[4]^3+2*c[7]^2*c[5]*c[4]^2-c[7]*c[5]*c[4]^3-3* c[6]*c[5]*c[4]^3-2*c[4]^3*c[7]^3+63*c[7]^3*c[5]^2*c[6]*c[4]-16*c[7]^2* c[5]^2*c[6]*c[4]-5*c[6]*c[7]*c[4]^4+2*c[6]^2*c[7]^2*c[4]^2+2*c[6]^2*c[ 7]^2*c[4]-10*c[5]^4*c[7]^2*c[6]+3*c[5]^3*c[6]*c[4]+3*c[5]^3*c[7]*c[4]+ 20*c[5]^3*c[6]*c[7]*c[4]-2*c[5]^3*c[4]^2+22*c[6]^2*c[7]^2*c[4]*c[5]+7* c[5]^2*c[6]*c[4]^3-c[5]^2*c[7]*c[4]^2+c[5]^2*c[6]*c[4]^2-4*c[5]^2*c[7] *c[4]^3-8*c[5]^3*c[7]*c[4]^2-22*c[5]^3*c[6]*c[4]^2+240*c[5]^3*c[6]^2*c [4]^3+100*c[4]^3*c[7]^2*c[5]^2*c[6]^2+150*c[4]^5*c[5]^3*c[6]^2-3*c[7]* c[5]^4*c[4]+5*c[7]^2*c[5]^4*c[4]+12*c[5]^2*c[4]^4*c[7]-98*c[6]^2*c[5]^ 2*c[4]^3+25*c[6]^2*c[5]^2*c[4]^2+54*c[6]*c[5]^3*c[4]^3+8*c[6]^2*c[5]^3 *c[4]-63*c[6]^2*c[5]^3*c[4]^2+42*c[5]^3*c[7]*c[4]^3+60*c[5]^4*c[7]*c[4 ]^2*c[6]+200*c[5]^3*c[7]^3*c[4]^4*c[6]-4*c[6]^2*c[4]*c[5]^2+c[7]*c[6]^ 2*c[4]^3+200*c[7]^2*c[5]^2*c[6]*c[4]^3-240*c[7]^3*c[5]^2*c[6]*c[4]^2+1 80*c[7]^2*c[5]^3*c[6]*c[4]^2+3*c[5]^4*c[4]^2-80*c[5]^3*c[7]^3*c[6]*c[4 ]+c[7]^3*c[4]^2*c[6]-7*c[7]^3*c[4]^2*c[5]+20*c[4]^3*c[7]^2*c[5]*c[6]^2 -19*c[7]^2*c[6]*c[5]*c[4]^3-11*c[7]^2*c[5]^2*c[6]*c[4]^2-3*c[4]^4*c[5] ^2-27*c[7]^2*c[5]^3*c[6]*c[4]-480*c[7]^2*c[5]^3*c[6]*c[4]^3-380*c[5]^3 *c[6]^2*c[4]^4-5*c[5]^4*c[4]^3-10*c[6]^2*c[4]^3*c[7]^2+130*c[5]^3*c[6] ^2*c[7]*c[4]^3-5*c[7]^3*c[5]^2*c[4]-2*c[7]^2*c[5]^2*c[4]-4*c[5]^2*c[6] *c[7]*c[4]-160*c[6]^2*c[7]*c[5]^3*c[4]^2-80*c[6]^2*c[7]^2*c[4]*c[5]^3+ 2*c[5]^2*c[4]^3+12*c[7]^2*c[5]^2*c[4]^2+8*c[7]^2*c[5]^3*c[4]+400*c[7]^ 3*c[5]^2*c[6]^2*c[4]^3-17*c[7]^2*c[4]^4*c[5]+250*c[7]^2*c[5]^3*c[6]^2* c[4]^2-100*c[6]^2*c[4]^5*c[5]*c[7]^2-200*c[7]^2*c[4]^5*c[6]*c[5]^3+150 *c[5]^2*c[6]*c[4]^5*c[7]^2+380*c[7]^3*c[5]^2*c[4]^3*c[6]-20*c[7]^3*c[6 ]*c[5]*c[4]^3-210*c[7]^3*c[6]^2*c[5]*c[4]^3-6*c[5]^4*c[4]*c[6]+80*c[5] ^4*c[7]^2*c[4]*c[6]+10*c[6]*c[7]^3*c[5]^3-36*c[5]^4*c[4]*c[7]*c[6]-50* c[4]^5*c[6]^2*c[7]*c[5]^2+200*c[4]^5*c[6]^2*c[7]^2*c[5]^2+160*c[6]^2*c [4]^4*c[5]^2-450*c[7]^3*c[5]^3*c[4]^3*c[6]+5*c[6]^2*c[7]^3*c[4]^2-c[5] ^3*c[7]^2+6*c[6]^2*c[7]^3*c[5]-160*c[7]*c[6]^2*c[5]*c[4]^4+200*c[5]^3* c[4]^4*c[6]^2*c[7]-350*c[5]^3*c[4]^3*c[6]^2*c[7]^2-350*c[5]^4*c[4]^4*c [6]*c[7]+230*c[5]^2*c[4]^4*c[7]*c[6]^2+150*c[5]^3*c[4]^4*c[6]*c[7]+100 *c[5]^4*c[4]^3*c[6]*c[7]-3*c[6]^2*c[7]^3*c[4]-42*c[7]^2*c[5]^3*c[4]^2+ 50*c[5]^4*c[4]^4*c[6]^2-20*c[5]^4*c[4]^3*c[6]^2+500*c[5]^3*c[4]^4*c[7] ^2*c[6]+450*c[5]^4*c[4]^3*c[7]^2*c[6]-340*c[5]^2*c[4]^4*c[7]^2*c[6]-50 *c[5]^4*c[4]^3*c[6]^2*c[7]+150*c[7]^2*c[4]^4*c[6]^2*c[5]+200*c[7]^2*c[ 4]^4*c[5]^3*c[6]^2-350*c[7]^2*c[4]^4*c[6]^2*c[5]^2+10*c[7]^2*c[6]^2*c[ 5]^3-60*c[4]^4*c[6]*c[5]^3+100*c[7]^3*c[6]^2*c[5]*c[4]^4+200*c[5]^4*c[ 4]^5*c[6]*c[7]-200*c[7]^3*c[4]^4*c[6]^2*c[5]^2+5*c[5]^3*c[4]^4+80*c[6] ^2*c[7]^3*c[5]^2*c[4]-270*c[6]^2*c[7]^3*c[5]^2*c[4]^2+290*c[5]^3*c[7]^ 3*c[4]^2*c[6]+140*c[6]^2*c[5]*c[7]^3*c[4]^2-5*c[7]^3*c[4]*c[5]^3)/c[5] /(100*c[5]^2*c[6]*c[4]^4-18*c[6]*c[5]*c[4]^4+4*c[5]^3*c[4]+50*c[4]^5*c [6]^2*c[5]^2+40*c[5]*c[6]^2*c[4]^4+3*c[5]^5-5*c[4]^4*c[6]^2+180*c[5]^4 *c[4]^3*c[6]+100*c[5]^4*c[4]^5*c[6]-350*c[5]^4*c[4]^4*c[6]+140*c[5]^5* c[4]^2-180*c[5]^5*c[4]^3-2*c[5]^4+30*c[4]^5*c[5]^3+300*c[5]^5*c[4]^3*c [6]+2*c[6]*c[5]^3-50*c[5]^5*c[4]^2*c[6]^2-40*c[5]^4*c[6]^2*c[4]+240*c[ 5]^4*c[6]^2*c[4]^2-3*c[6]^2*c[5]^3+3*c[6]^2*c[4]^3+5*c[5]^4*c[6]^2-2*c [6]*c[4]^3+2*c[5]*c[4]^3+4*c[5]*c[6]*c[4]^2-6*c[6]^2*c[4]^2*c[5]-50*c[ 5]^4*c[4]^5-13*c[6]^2*c[5]*c[4]^3-40*c[6]*c[5]^4*c[4]^2+6*c[6]*c[5]*c[ 4]^3-15*c[5]^3*c[6]*c[4]-55*c[5]^2*c[6]*c[4]^3+6*c[5]^2*c[6]*c[4]^2+68 *c[5]^3*c[6]*c[4]^2-100*c[4]^5*c[5]^3*c[6]^2+100*c[6]^2*c[5]^2*c[4]^3- 100*c[6]*c[5]^3*c[4]^3+13*c[6]^2*c[5]^3*c[4]-100*c[6]^2*c[5]^3*c[4]^2- 4*c[5]^2*c[4]^2+6*c[6]^2*c[4]*c[5]^2-68*c[5]^4*c[4]^2+23*c[4]^4*c[5]^2 +350*c[5]^3*c[6]^2*c[4]^4-4*c[5]^2*c[6]*c[4]-9*c[5]^2*c[4]^3-3*c[4]^4* c[5]-5*c[5]^5*c[6]-30*c[5]^6*c[4]^2+50*c[5]^6*c[6]*c[4]^2+180*c[5]^4*c [4]^4+10*c[5]^4*c[4]*c[6]-240*c[6]^2*c[4]^4*c[5]^2+3*c[6]*c[4]^4-100*c [5]^6*c[4]^3*c[6]+68*c[5]^3*c[4]^3-23*c[5]^5*c[4]-210*c[5]^5*c[6]*c[4] ^2-350*c[5]^4*c[4]^3*c[6]^2-30*c[5]^2*c[6]*c[4]^5+50*c[5]^6*c[4]^3+60* c[4]^4*c[6]*c[5]^3+100*c[5]^5*c[6]^2*c[4]^3+9*c[5]^4*c[4]-140*c[5]^3*c [4]^4+40*c[5]^5*c[4]*c[6]), `b*`[6] = -1/60*(-30*c[5]^3*c[4]+103*c[5]* c[7]*c[4]-18*c[6]+6*c[5]+6*c[4]+27*c[7]*c[6]-9*c[5]^2-9*c[4]^2-100*c[7 ]*c[6]*c[5]^3+522*c[5]*c[6]*c[7]*c[4]-111*c[6]*c[5]^2+75*c[6]*c[5]+60* c[6]*c[5]^3-9*c[7]*c[5]+600*c[6]*c[7]*c[5]^3*c[4]^3+14*c[7]*c[5]^2-100 *c[7]*c[6]*c[4]^3+14*c[7]*c[4]^2-9*c[7]*c[4]-117*c[6]*c[7]*c[5]-111*c[ 6]*c[4]^2+60*c[6]*c[4]^3-117*c[6]*c[7]*c[4]+180*c[6]*c[7]*c[4]^2-30*c[ 5]*c[4]^3-174*c[5]*c[7]*c[4]^2+486*c[5]*c[6]*c[4]^2+107*c[5]*c[4]^2-31 8*c[5]*c[6]*c[4]-66*c[5]*c[4]+75*c[6]*c[4]+500*c[5]*c[7]*c[6]*c[4]^3+1 420*c[5]^2*c[6]*c[7]*c[4]^2-840*c[6]*c[5]*c[7]*c[4]^2+180*c[6]*c[7]*c[ 5]^2-900*c[6]*c[5]^2*c[7]*c[4]^3-900*c[6]*c[5]^3*c[7]*c[4]^2+50*c[7]*c [5]*c[4]^3-280*c[6]*c[5]*c[4]^3-280*c[5]^3*c[6]*c[4]+50*c[5]^3*c[7]*c[ 4]+107*c[5]^2*c[4]+500*c[5]^3*c[6]*c[7]*c[4]+50*c[5]^3*c[4]^2+470*c[5] ^2*c[6]*c[4]^3+310*c[5]^2*c[7]*c[4]^2-770*c[5]^2*c[6]*c[4]^2-90*c[5]^2 *c[7]*c[4]^3-90*c[5]^3*c[7]*c[4]^2+470*c[5]^3*c[6]*c[4]^2-300*c[6]*c[5 ]^3*c[4]^3-180*c[5]^2*c[4]^2+486*c[5]^2*c[6]*c[4]-174*c[5]^2*c[7]*c[4] -840*c[5]^2*c[6]*c[7]*c[4]+50*c[5]^2*c[4]^3)/(-9*c[5]*c[7]*c[4]-3*c[7] *c[6]+3*c[6]^2-28*c[5]*c[6]*c[7]*c[4]-c[6]*c[5]+c[7]*c[5]+8*c[6]^2*c[4 ]^2+c[7]*c[4]+9*c[6]*c[7]*c[5]+9*c[6]*c[7]*c[4]-8*c[6]*c[7]*c[4]^2+4*c [5]*c[7]*c[4]^2-4*c[5]*c[6]*c[4]^2+9*c[5]*c[6]*c[4]-c[6]*c[4]-28*c[6]^ 2*c[4]^2*c[5]+28*c[6]^2*c[4]*c[5]-30*c[5]^2*c[6]*c[7]*c[4]^2+28*c[6]*c [5]*c[7]*c[4]^2-8*c[6]*c[7]*c[5]^2+8*c[6]^2*c[5]^2-9*c[6]^2*c[5]+30*c[ 6]^2*c[5]^2*c[4]^2-28*c[6]^2*c[4]*c[5]^2-9*c[6]^2*c[4]-4*c[5]^2*c[6]*c [4]+4*c[5]^2*c[7]*c[4]+28*c[5]^2*c[6]*c[7]*c[4])/(c[5]*c[4]-c[6]*c[4]- c[6]*c[5]+c[6]^2)/c[6], a[8,7] = (10*c[5]*c[6]*c[4]-5*c[6]*c[5]-5*c[6] *c[4]+3*c[6]-5*c[5]*c[4]+3*c[5]+3*c[4]-2)/c[7]/(c[6]-c[7])/(c[5]-c[7]) /(c[4]-c[7])/(-20*c[5]*c[6]*c[4]+30*c[5]*c[6]*c[7]*c[4]+15*c[6]*c[5]-2 0*c[6]*c[7]*c[5]+15*c[6]*c[4]-20*c[6]*c[7]*c[4]-12*c[6]+15*c[7]*c[6]+1 5*c[5]*c[4]-20*c[5]*c[7]*c[4]-12*c[5]+15*c[7]*c[5]-12*c[4]+15*c[7]*c[4 ]+10-12*c[7])*(c[4]-1)*(-1+c[5])*(-1+c[6])*(-1+c[7]), a[6,3] = 3/4*1/c [4]^2*c[6]*(60*c[4]^3*c[5]*c[6]^3-90*c[5]*c[6]^3*c[4]^2+36*c[6]^3*c[4] *c[5]-6*c[6]^3*c[5]+3*c[6]^3*c[4]-20*c[6]^2*c[5]^3*c[4]^2-80*c[6]^2*c[ 5]^2*c[4]^3+150*c[6]^2*c[5]^2*c[4]^2-48*c[6]^2*c[4]*c[5]^2+8*c[6]^2*c[ 5]^2-60*c[5]*c[6]^2*c[4]^4+90*c[6]^2*c[5]*c[4]^3-48*c[6]^2*c[4]^2*c[5] +4*c[6]^2*c[4]*c[5]-4*c[6]^2*c[4]^2+30*c[6]*c[5]^3*c[4]^3+90*c[5]^2*c[ 6]*c[4]^4-180*c[5]^2*c[6]*c[4]^3+54*c[5]^2*c[6]*c[4]^2-12*c[5]^2*c[6]* c[4]+18*c[6]*c[5]*c[4]^3+6*c[5]*c[6]*c[4]^2+3*c[5]^2*c[4]^2-3*c[5]*c[4 ]^3)/(10*c[5]^3*c[4]^2+10*c[5]^2*c[4]^3-30*c[5]^2*c[4]^2+6*c[5]^2*c[4] -c[5]^2+6*c[5]*c[4]^2+c[5]*c[4]-c[4]^2), b[7] = -1/60*(10*c[5]*c[6]*c[ 4]-5*c[6]*c[5]-5*c[6]*c[4]+3*c[6]-5*c[5]*c[4]+3*c[5]+3*c[4]-2)/(c[4]-c [7])/(c[5]-c[7])/(c[6]-c[7])/c[7]/(-1+c[7]), a[7,1] = 1/4*c[7]*(-51*c[ 5]^2*c[6]*c[4]^4+14*c[6]*c[5]*c[4]^4+12*c[4]^2*c[5]^2*c[7]^3+2700*c[5] ^4*c[4]^4*c[6]^2*c[7]+12*c[7]^3*c[5]*c[4]^3-98*c[7]^3*c[5]^2*c[4]^3-2* c[7]*c[5]*c[4]^4+132*c[5]^2*c[4]^4*c[7]^2-480*c[4]^5*c[6]^2*c[5]^2+200 *c[4]^3*c[7]^3*c[5]^3-200*c[7]^2*c[5]^2*c[4]^5-760*c[4]^5*c[6]*c[5]^3- 10*c[5]*c[6]^2*c[4]^4+200*c[5]^5*c[4]^3*c[7]^2*c[6]^2-400*c[5]^2*c[4]^ 6*c[6]^2*c[7]^2+4*c[4]^5*c[5]*c[7]-34*c[5]^2*c[4]^5*c[7]+400*c[5]^3*c[ 4]^6*c[6]^2*c[7]-98*c[5]^3*c[4]^4*c[7]+420*c[7]^2*c[4]^5*c[5]^3+200*c[ 7]^3*c[5]^2*c[4]^4-300*c[6]^2*c[4]^5*c[7]*c[5]^3-40*c[7]*c[5]^2*c[4]^6 *c[6]+20*c[5]^4*c[4]^2*c[7]^2*c[6]-280*c[6]^2*c[7]*c[5]^4*c[4]^2-44*c[ 6]*c[4]^5*c[5]*c[7]-246*c[6]*c[5]^2*c[4]^4*c[7]+120*c[4]^6*c[6]^2*c[5] ^2-100*c[5]^2*c[7]^2*c[4]^6*c[6]+141*c[5]^4*c[4]^3*c[6]+1220*c[5]^4*c[ 4]^5*c[6]+400*c[5]^5*c[4]^5*c[6]^2-760*c[5]^4*c[4]^4*c[6]+100*c[5]^4*c [7]^3*c[4]^4-400*c[7]^2*c[4]^4*c[5]^3-200*c[5]^4*c[4]^5*c[7]^2+200*c[5 ]^4*c[4]^4*c[7]-2*c[4]^3*c[7]^3*c[6]-8*c[7]^2*c[6]*c[4]^4-380*c[7]^3*c [5]^3*c[4]^4-40*c[6]*c[4]^5*c[5]*c[7]^2+2*c[4]^4*c[7]^2-40*c[5]^2*c[7] ^3*c[4]^5-200*c[7]^2*c[5]^4*c[4]^3-40*c[7]^3*c[5]^4*c[4]^3+200*c[5]^3* c[7]*c[4]^6*c[6]-10*c[6]^2*c[7]^3*c[4]^3+30*c[6]^2*c[4]^5*c[5]+12*c[7] *c[6]^2*c[5]^3+4*c[7]^2*c[6]*c[5]^3+400*c[5]^4*c[4]^6*c[6]^2-40*c[5]^3 *c[7]*c[4]^6-380*c[5]^4*c[4]^5*c[7]+320*c[7]*c[6]^2*c[5]*c[4]^5+100*c[ 4]^6*c[6]^2*c[7]*c[5]^2+34*c[5]^4*c[4]^2*c[7]^2+40*c[5]^5*c[7]^2*c[4]^ 3+240*c[6]*c[5]^2*c[4]^5*c[7]+60*c[5]^4*c[7]^3*c[4]^2*c[6]+26*c[7]*c[4 ]^4*c[6]*c[5]-50*c[4]^4*c[6]^2*c[5]^5+200*c[5]^5*c[7]*c[4]^4*c[6]+132* c[7]^2*c[5]^3*c[4]^3-2*c[7]*c[4]^4*c[6]^2+60*c[5]^5*c[4]^2*c[6]^2*c[7] -500*c[5]^5*c[4]^4*c[6]^2*c[7]+160*c[5]^5*c[4]^4*c[6]+134*c[6]^2*c[5]^ 4*c[7]*c[4]+24*c[5]^4*c[6]^2*c[4]-169*c[5]^4*c[6]^2*c[4]^2-500*c[5]^3* c[4]^6*c[6]^2-40*c[5]^5*c[4]^4*c[7]+34*c[4]^5*c[5]*c[7]^2-100*c[5]^3*c [7]^2*c[4]^6-100*c[5]^5*c[4]^4*c[7]^2+160*c[4]^6*c[6]*c[5]^3-6*c[6]^2* c[7]*c[5]^2*c[4]^3+68*c[6]^2*c[7]^2*c[5]^2*c[4]-126*c[6]^2*c[7]^2*c[5] ^2*c[4]^2-600*c[6]*c[7]*c[5]^3*c[4]^3-21*c[6]^2*c[7]^3*c[4]*c[5]+40*c[ 7]^2*c[6]^2*c[5]^4+4*c[5]^4*c[7]*c[4]^2+58*c[7]^2*c[6]*c[5]*c[4]^4-100 *c[7]^2*c[5]^5*c[6]^2*c[4]^2-320*c[5]^4*c[4]*c[7]^2*c[6]^2+20*c[6]^2*c [4]^4*c[7]^2+40*c[5]^2*c[4]^6*c[7]^2+6*c[7]^2*c[6]*c[4]^3-8*c[6]*c[5]* c[7]^2*c[4]^2-4*c[5]*c[7]*c[6]*c[4]^3+14*c[5]^2*c[6]*c[7]*c[4]^2-910*c [4]^5*c[6]*c[7]*c[5]^3-340*c[5]^2*c[6]*c[7]^3*c[4]^4-4*c[6]^2*c[5]*c[4 ]^3+5*c[7]^3*c[4]^2*c[6]*c[5]+72*c[7]*c[6]^2*c[5]*c[4]^3+18*c[6]^2*c[7 ]^3*c[5]^2-12*c[6]^2*c[7]^2*c[5]^2-10*c[6]^2*c[4]^2*c[7]*c[5]-44*c[6]^ 2*c[7]^2*c[4]^2*c[5]-25*c[6]*c[5]^4*c[4]^2-24*c[7]*c[6]^2*c[5]^4+200*c [4]^5*c[7]*c[5]^3-2*c[6]^2*c[5]^2*c[7]*c[4]-42*c[6]^2*c[5]^2*c[7]*c[4] ^2+74*c[6]*c[5]^2*c[7]*c[4]^3+80*c[6]*c[5]^3*c[7]*c[4]^2-50*c[6]^2*c[5 ]^3*c[7]*c[4]-34*c[5]^4*c[4]^3*c[7]-24*c[7]^2*c[5]^2*c[4]^3+420*c[5]^4 *c[7]^2*c[4]^4-2*c[4]^3*c[7]^3+c[7]^3*c[5]^2*c[6]*c[4]+4*c[7]^2*c[5]^2 *c[6]*c[4]-6*c[6]*c[7]*c[4]^4-4*c[6]^2*c[7]^2*c[4]^2+50*c[5]^5*c[4]^3* c[7]^2*c[6]-18*c[5]^3*c[6]*c[7]*c[4]+14*c[6]^2*c[7]^2*c[4]*c[5]-20*c[5 ]^2*c[6]*c[4]^3+2*c[5]^2*c[7]*c[4]^3-2*c[5]^3*c[7]*c[4]^2+16*c[5]^3*c[ 6]*c[4]^2-96*c[5]^3*c[6]^2*c[4]^3+100*c[5]^5*c[4]^3*c[6]^2*c[7]-60*c[4 ]^3*c[7]^2*c[5]^2*c[6]^2+10*c[6]*c[7]*c[4]^5+1540*c[4]^5*c[5]^3*c[6]^2 -34*c[7]^3*c[4]^4*c[5]-4*c[7]^2*c[5]^4*c[4]+12*c[5]^2*c[4]^4*c[7]-41*c [6]^2*c[5]^2*c[4]^3+14*c[6]^2*c[5]^2*c[4]^2-54*c[6]*c[5]^3*c[4]^3-12*c [6]^2*c[5]^3*c[4]+47*c[6]^2*c[5]^3*c[4]^2+12*c[5]^3*c[7]*c[4]^3-98*c[5 ]^4*c[7]*c[4]^2*c[6]-50*c[5]^3*c[7]^3*c[4]^4*c[6]+4*c[7]*c[6]^2*c[4]^3 +156*c[7]^2*c[5]^2*c[6]*c[4]^3-23*c[7]^3*c[5]^2*c[6]*c[4]^2+188*c[7]^2 *c[5]^3*c[6]*c[4]^2+66*c[5]^3*c[7]^3*c[6]*c[4]+2*c[7]^3*c[4]^2*c[5]+10 0*c[5]^5*c[4]^5*c[7]+104*c[4]^3*c[7]^2*c[5]*c[6]^2-12*c[7]^2*c[6]*c[5] *c[4]^3-36*c[7]^2*c[5]^2*c[6]*c[4]^2-32*c[7]^2*c[5]^3*c[6]*c[4]-320*c[ 7]^2*c[5]^3*c[6]*c[4]^3-560*c[5]^3*c[6]^2*c[4]^4-4*c[6]^2*c[4]^3*c[7]^ 2+240*c[5]^3*c[6]^2*c[7]*c[4]^3-34*c[5]^3*c[7]^3*c[4]^2-2*c[7]^3*c[5]^ 2*c[4]+164*c[6]^2*c[7]*c[5]^3*c[4]^2-20*c[6]^2*c[7]^2*c[4]*c[5]^3+2*c[ 7]^2*c[5]^3*c[4]-130*c[5]^5*c[4]^3*c[6]*c[7]-90*c[7]^3*c[5]^2*c[6]^2*c [4]^3-16*c[7]^2*c[4]^4*c[5]-60*c[7]^2*c[5]^3*c[6]^2*c[4]^2-300*c[6]^2* c[4]^5*c[5]*c[7]^2-150*c[7]^2*c[4]^5*c[6]*c[5]^3+380*c[5]^2*c[6]*c[4]^ 5*c[7]^2+140*c[7]^3*c[5]^2*c[4]^3*c[6]-810*c[7]^3*c[5]^3*c[6]^2*c[4]^2 -14*c[7]^3*c[6]*c[5]*c[4]^3-280*c[7]^3*c[6]^2*c[5]*c[4]^3+240*c[7]^3*c [5]^3*c[6]^2*c[4]-20*c[5]^4*c[7]^2*c[4]*c[6]-6*c[6]*c[7]^3*c[5]^3+22*c [5]^4*c[4]*c[7]*c[6]-280*c[4]^5*c[6]^2*c[7]*c[5]^2+400*c[4]^5*c[6]^2*c [7]^2*c[5]^2+253*c[6]^2*c[4]^4*c[5]^2+470*c[7]^3*c[5]^3*c[4]^3*c[6]+10 0*c[5]^2*c[4]^5*c[7]^3*c[6]+6*c[6]^2*c[7]^3*c[4]^2-300*c[5]^5*c[4]^5*c [6]-120*c[5]*c[7]*c[4]^6*c[6]^2-10*c[6]^2*c[7]*c[4]^5-1100*c[5]^4*c[4] ^5*c[6]^2*c[7]-226*c[7]*c[6]^2*c[5]*c[4]^4-770*c[5]^3*c[4]^4*c[6]^2*c[ 7]+280*c[5]^3*c[4]^3*c[6]^2*c[7]^2-1000*c[5]^4*c[4]^4*c[6]*c[7]+240*c[ 5]^2*c[4]^4*c[7]*c[6]^2+1240*c[5]^3*c[4]^4*c[6]*c[7]+600*c[5]^4*c[4]^3 *c[6]*c[7]+1200*c[7]^3*c[5]^3*c[6]^2*c[4]^3+40*c[7]^3*c[4]^4*c[5]*c[6] -16*c[7]^2*c[5]^3*c[4]^2-1250*c[5]^4*c[4]^5*c[6]^2-600*c[6]^2*c[7]^3*c [5]^3*c[4]^4+200*c[6]^2*c[7]^2*c[5]^3*c[4]^5-200*c[6]^2*c[7]^3*c[4]^5* c[5]-140*c[5]^4*c[4]^4*c[6]^2+560*c[5]^4*c[4]^3*c[6]^2-1850*c[7]^2*c[5 ]^4*c[6]^2*c[4]^3+1230*c[7]^2*c[5]^4*c[6]^2*c[4]^2-150*c[5]^4*c[7]^3*c [4]^3*c[6]-300*c[5]^4*c[4]^6*c[6]+310*c[5]^3*c[4]^4*c[7]^2*c[6]-70*c[5 ]^4*c[4]^3*c[7]^2*c[6]-380*c[5]^2*c[4]^4*c[7]^2*c[6]-850*c[5]^4*c[4]^3 *c[6]^2*c[7]-40*c[7]^2*c[4]^4*c[6]^2*c[5]-450*c[7]^2*c[4]^4*c[5]^3*c[6 ]^2+250*c[7]^2*c[4]^4*c[6]^2*c[5]^2-4*c[7]^2*c[4]^5+136*c[5]^2*c[6]*c[ 4]^5+400*c[6]^2*c[7]^3*c[5]^2*c[4]^5+800*c[6]^2*c[7]^2*c[5]^4*c[4]^4+4 *c[7]^3*c[4]^4+100*c[5]^4*c[4]^6*c[7]+437*c[4]^4*c[6]*c[5]^3-22*c[6]*c [5]*c[4]^5-60*c[5]^5*c[6]^2*c[4]^3+200*c[6]^2*c[4]^6*c[5]*c[7]^2+420*c [7]^3*c[6]^2*c[5]*c[4]^4+400*c[5]^4*c[4]^5*c[6]*c[7]-500*c[7]^3*c[4]^4 *c[6]^2*c[5]^2+100*c[5]^3*c[7]^3*c[4]^5-103*c[6]^2*c[7]^3*c[5]^2*c[4]+ 260*c[6]^2*c[7]^3*c[5]^2*c[4]^2-300*c[5]^3*c[7]^3*c[4]^2*c[6]+97*c[6]^ 2*c[5]*c[7]^3*c[4]^2-30*c[6]^2*c[7]^3*c[5]^3+4*c[7]^3*c[4]*c[5]^3)/c[6 ]/(-50*c[5]^2*c[6]*c[4]^4+23*c[5]^3*c[4]+100*c[5]^4*c[4]^3*c[6]+2*c[5] ^2+2*c[4]^2-3*c[5]^3-3*c[4]^3-3*c[6]*c[5]^2+5*c[6]*c[5]^3-3*c[6]*c[4]^ 2+5*c[6]*c[4]^3+23*c[5]*c[4]^3+18*c[5]*c[6]*c[4]^2-12*c[5]*c[4]^2+3*c[ 5]*c[6]*c[4]-2*c[5]*c[4]-50*c[6]*c[5]^4*c[4]^2-40*c[6]*c[5]*c[4]^3-40* c[5]^3*c[6]*c[4]-12*c[5]^2*c[4]-140*c[5]^3*c[4]^2+240*c[5]^2*c[6]*c[4] ^3-140*c[5]^2*c[6]*c[4]^2+240*c[5]^3*c[6]*c[4]^2-400*c[6]*c[5]^3*c[4]^ 3+91*c[5]^2*c[4]^2+30*c[5]^4*c[4]^2+30*c[4]^4*c[5]^2-50*c[5]^4*c[4]^3+ 18*c[5]^2*c[6]*c[4]-140*c[5]^2*c[4]^3+210*c[5]^3*c[4]^3+100*c[4]^4*c[6 ]*c[5]^3-50*c[5]^3*c[4]^4)/c[5]/c[4]^2, `b*`[8] = 0, a[5,3] = -3/4*c[5 ]^2*(2*c[5]-3*c[4])/c[4]^2, `b*`[4] = -1/60*(-30*c[5]^3*c[4]-42*c[7]*c [6]^2+89*c[5]*c[7]*c[4]-18*c[6]+6*c[5]+6*c[4]+27*c[7]*c[6]-9*c[5]^2+18 0*c[7]*c[6]^2*c[5]^3-100*c[7]*c[6]*c[5]^3+27*c[6]^2+186*c[5]*c[6]*c[7] *c[4]-97*c[6]*c[5]^2+66*c[6]*c[5]+60*c[6]*c[5]^3-100*c[6]^2*c[5]^3-9*c [7]*c[5]+14*c[7]*c[5]^2-100*c[6]^2*c[7]*c[4]^2+120*c[6]^2*c[7]*c[4]+60 *c[6]^2*c[4]^2-9*c[7]*c[4]-103*c[6]*c[7]*c[5]-36*c[6]*c[4]^2-61*c[6]*c [7]*c[4]+60*c[6]*c[7]*c[4]^2-30*c[5]*c[7]*c[4]^2+126*c[5]*c[6]*c[4]^2+ 18*c[5]*c[4]^2-112*c[5]*c[6]*c[4]-57*c[5]*c[4]+39*c[6]*c[4]-280*c[6]^2 *c[4]^2*c[5]+330*c[6]^2*c[4]*c[5]+380*c[5]^2*c[6]*c[7]*c[4]^2-560*c[6] ^2*c[7]*c[4]*c[5]+192*c[6]^2*c[7]*c[5]-310*c[6]^2*c[7]*c[5]^2-230*c[6] *c[5]*c[7]*c[4]^2+156*c[6]*c[7]*c[5]^2+180*c[6]^2*c[5]^2+500*c[6]^2*c[ 4]^2*c[7]*c[5]+950*c[6]^2*c[5]^2*c[7]*c[4]-900*c[6]^2*c[5]^2*c[7]*c[4] ^2-300*c[6]*c[5]^3*c[7]*c[4]^2-600*c[6]^2*c[5]^3*c[7]*c[4]-117*c[6]^2* c[5]-130*c[5]^3*c[6]*c[4]+50*c[5]^3*c[7]*c[4]+93*c[5]^2*c[4]+230*c[5]^ 3*c[6]*c[7]*c[4]+50*c[5]^2*c[7]*c[4]^2-190*c[5]^2*c[6]*c[4]^2+150*c[5] ^3*c[6]*c[4]^2+470*c[6]^2*c[5]^2*c[4]^2+320*c[6]^2*c[5]^3*c[4]-300*c[6 ]^2*c[5]^3*c[4]^2-30*c[5]^2*c[4]^2-530*c[6]^2*c[4]*c[5]^2-75*c[6]^2*c[ 4]+156*c[5]^2*c[6]*c[4]-150*c[5]^2*c[7]*c[4]-270*c[5]^2*c[6]*c[7]*c[4] +600*c[6]^2*c[7]*c[5]^3*c[4]^2)/(-9*c[5]*c[7]*c[4]-3*c[7]*c[6]-c[4]^2- 28*c[5]*c[6]*c[7]*c[4]+c[7]*c[5]+c[7]*c[4]+9*c[6]*c[7]*c[5]-9*c[6]*c[4 ]^2+8*c[6]*c[4]^3+9*c[6]*c[7]*c[4]-8*c[6]*c[7]*c[4]^2-4*c[5]*c[4]^3+4* c[5]*c[7]*c[4]^2+28*c[5]*c[6]*c[4]^2+9*c[5]*c[4]^2-9*c[5]*c[6]*c[4]-c[ 5]*c[4]+3*c[6]*c[4]-30*c[5]^2*c[6]*c[7]*c[4]^2+28*c[6]*c[5]*c[7]*c[4]^ 2-8*c[6]*c[7]*c[5]^2-28*c[6]*c[5]*c[4]^3+30*c[5]^2*c[6]*c[4]^3-28*c[5] ^2*c[6]*c[4]^2-4*c[5]^2*c[4]^2+8*c[5]^2*c[6]*c[4]+4*c[5]^2*c[7]*c[4]+2 8*c[5]^2*c[6]*c[7]*c[4])/(-c[4]+c[6])/(-c[4]+c[5])/c[4], a[8,6] = 1/2* (5*c[5]*c[6]*c[4]+c[6]-2*c[6]*c[4]-2*c[6]*c[5]+25*c[5]*c[7]*c[4]+6*c[5 ]+9*c[7]+10*c[7]^2*c[4]-4+6*c[4]-6*c[7]^2-14*c[7]*c[4]-10*c[5]*c[4]-14 *c[7]*c[5]-20*c[7]^2*c[5]*c[4]+10*c[7]^2*c[5])/(c[6]-c[7])/(-c[4]+c[6] )/(-c[5]+c[6])/c[6]/(-20*c[5]*c[6]*c[4]+30*c[5]*c[6]*c[7]*c[4]+15*c[6] *c[5]-20*c[6]*c[7]*c[5]+15*c[6]*c[4]-20*c[6]*c[7]*c[4]-12*c[6]+15*c[7] *c[6]+15*c[5]*c[4]-20*c[5]*c[7]*c[4]-12*c[5]+15*c[7]*c[5]-12*c[4]+15*c [7]*c[4]+10-12*c[7])*(c[4]-1)*(-1+c[5])*(-1+c[6]), b[8] = 1/60*(-20*c[ 5]*c[6]*c[4]+30*c[5]*c[6]*c[7]*c[4]+15*c[6]*c[5]-20*c[6]*c[7]*c[5]+15* c[6]*c[4]-20*c[6]*c[7]*c[4]-12*c[6]+15*c[7]*c[6]+15*c[5]*c[4]-20*c[5]* c[7]*c[4]-12*c[5]+15*c[7]*c[5]-12*c[4]+15*c[7]*c[4]+10-12*c[7])/(c[4]- 1)/(-1+c[5])/(-1+c[6])/(-1+c[7]), a[5,1] = 1/4*c[5]*(2*c[5]^2-5*c[5]*c [4]+4*c[4]^2)/c[4]^2, `b*`[5] = 1/60*(-42*c[7]*c[6]^2+89*c[5]*c[7]*c[4 ]-18*c[6]+6*c[5]+6*c[4]+27*c[7]*c[6]-9*c[4]^2+27*c[6]^2+186*c[5]*c[6]* c[7]*c[4]-36*c[6]*c[5]^2+39*c[6]*c[5]-9*c[7]*c[5]-100*c[6]^2*c[4]^3+60 0*c[6]^2*c[7]*c[5]^2*c[4]^3-100*c[7]*c[6]*c[4]^3-310*c[6]^2*c[7]*c[4]^ 2+192*c[6]^2*c[7]*c[4]+180*c[6]^2*c[4]^2+14*c[7]*c[4]^2-9*c[7]*c[4]-61 *c[6]*c[7]*c[5]-97*c[6]*c[4]^2+60*c[6]*c[4]^3-103*c[6]*c[7]*c[4]+156*c [6]*c[7]*c[4]^2-30*c[5]*c[4]^3-150*c[5]*c[7]*c[4]^2+156*c[5]*c[6]*c[4] ^2+93*c[5]*c[4]^2-112*c[5]*c[6]*c[4]-57*c[5]*c[4]+66*c[6]*c[4]+230*c[5 ]*c[7]*c[6]*c[4]^3-530*c[6]^2*c[4]^2*c[5]+330*c[6]^2*c[4]*c[5]+380*c[5 ]^2*c[6]*c[7]*c[4]^2+320*c[6]^2*c[5]*c[4]^3-560*c[6]^2*c[7]*c[4]*c[5]- 600*c[7]*c[6]^2*c[5]*c[4]^3+120*c[6]^2*c[7]*c[5]-100*c[6]^2*c[7]*c[5]^ 2-270*c[6]*c[5]*c[7]*c[4]^2+60*c[6]*c[7]*c[5]^2+60*c[6]^2*c[5]^2+950*c [6]^2*c[4]^2*c[7]*c[5]+500*c[6]^2*c[5]^2*c[7]*c[4]-900*c[6]^2*c[5]^2*c [7]*c[4]^2-300*c[6]*c[5]^2*c[7]*c[4]^3-75*c[6]^2*c[5]+50*c[7]*c[5]*c[4 ]^3-130*c[6]*c[5]*c[4]^3+18*c[5]^2*c[4]+150*c[5]^2*c[6]*c[4]^3+50*c[5] ^2*c[7]*c[4]^2-190*c[5]^2*c[6]*c[4]^2-300*c[6]^2*c[5]^2*c[4]^3+470*c[6 ]^2*c[5]^2*c[4]^2-30*c[5]^2*c[4]^2-280*c[6]^2*c[4]*c[5]^2+180*c[7]*c[6 ]^2*c[4]^3-117*c[6]^2*c[4]+126*c[5]^2*c[6]*c[4]-30*c[5]^2*c[7]*c[4]-23 0*c[5]^2*c[6]*c[7]*c[4])/(-4*c[5]^3*c[4]-9*c[5]*c[7]*c[4]-3*c[7]*c[6]- c[5]^2-28*c[5]*c[6]*c[7]*c[4]-9*c[6]*c[5]^2+3*c[6]*c[5]+8*c[6]*c[5]^3+ c[7]*c[5]+c[7]*c[4]+9*c[6]*c[7]*c[5]+9*c[6]*c[7]*c[4]-8*c[6]*c[7]*c[4] ^2+4*c[5]*c[7]*c[4]^2+8*c[5]*c[6]*c[4]^2-9*c[5]*c[6]*c[4]-c[5]*c[4]-30 *c[5]^2*c[6]*c[7]*c[4]^2+28*c[6]*c[5]*c[7]*c[4]^2-8*c[6]*c[7]*c[5]^2-2 8*c[5]^3*c[6]*c[4]+9*c[5]^2*c[4]-28*c[5]^2*c[6]*c[4]^2+30*c[5]^3*c[6]* c[4]^2-4*c[5]^2*c[4]^2+28*c[5]^2*c[6]*c[4]+4*c[5]^2*c[7]*c[4]+28*c[5]^ 2*c[6]*c[7]*c[4])/(-c[5]+c[6])/(-c[4]+c[5])/c[5], a[5,4] = c[5]^2*(-c[ 4]+c[5])/c[4]^2, a[7,6] = 1/2*(5*c[5]*c[4]+1-2*c[4]-2*c[5])*(c[4]-c[7] )*(c[5]-c[7])*(c[6]-c[7])*c[7]/(10*c[5]*c[6]*c[4]-5*c[6]*c[5]-5*c[6]*c [4]+3*c[6]-5*c[5]*c[4]+3*c[5]+3*c[4]-2)/(-c[4]+c[6])/(-c[5]+c[6])/c[6] , c[3] = 2/3*c[4], a[4,3] = 3/4*c[4], a[3,2] = 2/9*c[4]^2/c[2], a[7,4] = -1/2*c[7]*(-68*c[5]^2*c[6]*c[4]^4-7*c[6]*c[5]*c[4]^4+7*c[4]^2*c[5]^ 2*c[7]^3-1400*c[5]^4*c[4]^4*c[6]^2*c[7]+11*c[7]^3*c[5]*c[4]^3-72*c[7]^ 3*c[5]^2*c[4]^3-3*c[7]*c[5]*c[4]^4-58*c[5]^2*c[4]^4*c[7]^2-120*c[6]*c[ 4]^7*c[7]*c[5]+320*c[4]^5*c[6]^2*c[5]^2-200*c[4]^3*c[7]^3*c[5]^3-100*c [4]^5*c[6]*c[5]^3-2*c[5]*c[6]^2*c[4]^4-600*c[4]^5*c[7]^3*c[6]*c[5]^3-2 00*c[5]^5*c[4]^3*c[7]^2*c[6]^2+400*c[5]^2*c[4]^6*c[6]^2*c[7]^2-21*c[4] ^5*c[5]*c[7]+120*c[5]^2*c[4]^5*c[7]-600*c[5]^3*c[4]^6*c[6]^2*c[7]-198* c[5]^3*c[4]^4*c[7]-280*c[7]^2*c[4]^5*c[5]^3+240*c[7]^3*c[5]^2*c[4]^4+1 250*c[6]^2*c[4]^5*c[7]*c[5]^3+120*c[7]*c[5]^2*c[4]^6*c[6]-340*c[5]^4*c [4]^2*c[7]^2*c[6]+400*c[6]^2*c[7]*c[5]^4*c[4]^2-240*c[5]^2*c[4]^6*c[7] -300*c[6]*c[5]*c[7]^2*c[4]^6-228*c[6]*c[4]^5*c[5]*c[7]-119*c[6]*c[5]^2 *c[4]^4*c[7]-200*c[5]*c[7]^3*c[4]^6*c[6]-600*c[5]^4*c[7]*c[4]^6*c[6]-1 20*c[4]^6*c[6]^2*c[5]^2+600*c[5]^2*c[7]^2*c[4]^6*c[6]+100*c[5]^2*c[7]^ 3*c[4]^6-187*c[5]^4*c[4]^3*c[6]-910*c[5]^4*c[4]^5*c[6]-320*c[5]^2*c[6] *c[4]^6+620*c[5]^4*c[4]^4*c[6]-100*c[5]^4*c[7]^3*c[4]^4-40*c[7]^2*c[4] ^4*c[5]^3+40*c[7]^3*c[4]^5*c[5]+10*c[6]*c[7]^2*c[4]^5-10*c[6]*c[7]^3*c [4]^4+100*c[5]^4*c[4]^5*c[7]^2+80*c[5]^4*c[4]^4*c[7]+8*c[4]^3*c[7]^3*c [6]-2*c[7]^2*c[6]*c[4]^4+280*c[7]^3*c[5]^3*c[4]^4+80*c[6]*c[4]^5*c[5]* c[7]^2+c[4]^4*c[7]^2-360*c[5]^2*c[7]^3*c[4]^5-80*c[7]^2*c[5]^4*c[4]^3+ 40*c[7]^3*c[5]^4*c[4]^3-1200*c[5]^3*c[7]*c[4]^6*c[6]+10*c[6]^2*c[7]^3* c[4]^3-10*c[6]^2*c[4]^5*c[5]-12*c[7]*c[6]^2*c[5]^3-4*c[7]^2*c[6]*c[5]^ 3-80*c[5]^4*c[4]^5*c[7]-240*c[7]*c[6]^2*c[5]*c[4]^5-3*c[5]^4*c[4]^2*c[ 7]^2+20*c[5]^5*c[7]^2*c[4]^3+180*c[6]*c[5]^2*c[4]^5*c[7]-60*c[5]^4*c[7 ]^3*c[4]^2*c[6]+400*c[5]^2*c[7]^3*c[4]^6*c[6]-25*c[4]^5*c[5]^3+200*c[5 ]^5*c[4]^4*c[7]^2*c[6]+60*c[5]^2*c[4]^7*c[7]+100*c[7]*c[4]^4*c[6]*c[5] -40*c[5]*c[4]^6*c[7]^2-150*c[4]^4*c[6]^2*c[5]^5+50*c[5]^5*c[7]*c[4]^4* c[6]+174*c[7]^2*c[5]^3*c[4]^3+8*c[7]*c[4]^4*c[6]^2+600*c[5]^3*c[7]*c[4 ]^7*c[6]-60*c[5]^5*c[4]^2*c[6]^2*c[7]+200*c[5]^5*c[4]^4*c[6]^2*c[7]-40 *c[5]^5*c[4]^4*c[6]-200*c[5]^5*c[7]*c[4]^5*c[6]-149*c[6]^2*c[5]^4*c[7] *c[4]-24*c[5]^4*c[6]^2*c[4]+189*c[5]^4*c[6]^2*c[4]^2+300*c[5]^3*c[4]^6 *c[6]^2-20*c[5]^5*c[4]^4*c[7]-5*c[4]^5*c[5]*c[7]^2+600*c[4]^6*c[6]*c[5 ]^3-84*c[6]^2*c[7]*c[5]^2*c[4]^3-74*c[6]^2*c[7]^2*c[5]^2*c[4]+162*c[6] ^2*c[7]^2*c[5]^2*c[4]^2+311*c[6]*c[7]*c[5]^3*c[4]^3+21*c[6]^2*c[7]^3*c [4]*c[5]-40*c[7]^2*c[6]^2*c[5]^4+11*c[5]^4*c[7]*c[4]^2+107*c[7]^2*c[6] *c[5]*c[4]^4+100*c[7]^2*c[5]^5*c[6]^2*c[4]^2+320*c[5]^4*c[4]*c[7]^2*c[ 6]^2-10*c[6]^2*c[4]^4*c[7]^2+300*c[5]^2*c[4]^6*c[7]^2-3*c[7]^2*c[6]*c[ 4]^3-2*c[5]*c[7]^2*c[4]^3+15*c[6]*c[5]*c[7]^2*c[4]^2-8*c[5]*c[7]*c[6]* c[4]^3-14*c[5]^2*c[6]*c[7]*c[4]^2-2000*c[5]^4*c[7]^2*c[4]^4*c[6]+200*c [4]^5*c[6]*c[7]*c[5]^3+250*c[5]^2*c[6]*c[7]^3*c[4]^4+4*c[6]^2*c[5]*c[4 ]^3-26*c[7]^3*c[4]^2*c[6]*c[5]-69*c[7]*c[6]^2*c[5]*c[4]^3-18*c[6]^2*c[ 7]^3*c[5]^2+12*c[6]^2*c[7]^2*c[5]^2+10*c[6]^2*c[4]^2*c[7]*c[5]+50*c[6] ^2*c[7]^2*c[4]^2*c[5]+31*c[6]*c[5]^4*c[4]^2+24*c[7]*c[6]^2*c[5]^4+240* c[4]^5*c[7]*c[5]^3+2*c[6]^2*c[5]^2*c[7]*c[4]+39*c[6]^2*c[5]^2*c[7]*c[4 ]^2-17*c[6]*c[5]^2*c[7]*c[4]^3-73*c[6]*c[5]^3*c[7]*c[4]^2+59*c[6]^2*c[ 5]^3*c[7]*c[4]-12*c[5]^4*c[4]^3*c[7]-12*c[7]^2*c[5]^2*c[4]^3+40*c[5]^4 *c[7]^2*c[4]^4-c[4]^3*c[7]^3+17*c[7]^3*c[5]^2*c[6]*c[4]-9*c[7]^2*c[5]^ 2*c[6]*c[4]+3*c[6]*c[7]*c[4]^4+4*c[4]^5*c[5]+4*c[6]^2*c[7]^2*c[4]^2-15 0*c[5]^5*c[4]^3*c[7]^2*c[6]+21*c[5]^3*c[6]*c[7]*c[4]-14*c[6]^2*c[7]^2* c[4]*c[5]+23*c[5]^2*c[6]*c[4]^3+13*c[5]^2*c[7]*c[4]^3-11*c[5]^3*c[7]*c [4]^2-17*c[5]^3*c[6]*c[4]^2+168*c[5]^3*c[6]^2*c[4]^3+50*c[5]^5*c[4]^3* c[6]^2*c[7]-220*c[4]^3*c[7]^2*c[5]^2*c[6]^2-6*c[6]*c[7]*c[4]^5-580*c[4 ]^5*c[5]^3*c[6]^2-31*c[7]^3*c[4]^4*c[5]-2*c[7]^2*c[5]^4*c[4]-4*c[5]^2* c[4]^4*c[7]+53*c[6]^2*c[5]^2*c[4]^3-14*c[6]^2*c[5]^2*c[4]^2+47*c[6]*c[ 5]^3*c[4]^3+12*c[6]^2*c[5]^3*c[4]-59*c[6]^2*c[5]^3*c[4]^2+28*c[5]^3*c[ 7]*c[4]^3+127*c[5]^4*c[7]*c[4]^2*c[6]+1250*c[5]^3*c[7]^3*c[4]^4*c[6]+1 0*c[6]*c[5]*c[4]^6+240*c[5]*c[4]^6*c[7]*c[6]-4*c[7]*c[6]^2*c[4]^3+11*c [7]^2*c[5]^2*c[6]*c[4]^3-80*c[7]^3*c[5]^2*c[6]*c[4]^2-121*c[7]^2*c[5]^ 3*c[6]*c[4]^2-96*c[5]^3*c[7]^3*c[6]*c[4]+c[7]^3*c[4]^2*c[5]-68*c[4]^3* c[7]^2*c[5]*c[6]^2-80*c[7]^2*c[6]*c[5]*c[4]^3+68*c[7]^2*c[5]^2*c[6]*c[ 4]^2-10*c[4]^4*c[5]^2+20*c[7]^2*c[5]^3*c[6]*c[4]-240*c[7]^2*c[5]^3*c[6 ]*c[4]^3-9*c[5]^4*c[4]^3-2*c[6]^2*c[4]^3*c[7]^2+260*c[5]^3*c[6]^2*c[7] *c[4]^3+49*c[5]^3*c[7]^3*c[4]^2-c[7]^3*c[5]^2*c[4]-228*c[6]^2*c[7]*c[5 ]^3*c[4]^2+30*c[6]^2*c[7]^2*c[4]*c[5]^3-2*c[7]^2*c[5]^2*c[4]^2+5*c[7]^ 2*c[5]^3*c[4]+40*c[5]^5*c[4]^3*c[6]*c[7]+36*c[4]^6*c[7]*c[5]+90*c[7]^3 *c[5]^2*c[6]^2*c[4]^3+16*c[7]^2*c[4]^4*c[5]-20*c[7]^2*c[5]^3*c[6]^2*c[ 4]^2+300*c[6]^2*c[4]^5*c[5]*c[7]^2+250*c[7]^2*c[4]^5*c[6]*c[5]^3-490*c [5]^2*c[6]*c[4]^5*c[7]^2+120*c[7]^3*c[5]^2*c[4]^3*c[6]+810*c[7]^3*c[5] ^3*c[6]^2*c[4]^2+111*c[7]^3*c[6]*c[5]*c[4]^3+280*c[7]^3*c[6]^2*c[5]*c[ 4]^3-240*c[7]^3*c[5]^3*c[6]^2*c[4]+15*c[5]^4*c[4]^4+60*c[5]^4*c[7]^2*c [4]*c[6]+6*c[6]*c[7]^3*c[5]^3-31*c[5]^4*c[4]*c[7]*c[6]-200*c[4]^5*c[6] ^2*c[7]*c[5]^2-500*c[4]^5*c[6]^2*c[7]^2*c[5]^2+9*c[4]^5*c[5]^2-181*c[6 ]^2*c[4]^4*c[5]^2-1280*c[7]^3*c[5]^3*c[4]^3*c[6]-600*c[5]^2*c[4]^5*c[7 ]^3*c[6]-6*c[4]^6*c[5]-6*c[6]^2*c[7]^3*c[4]^2+100*c[5]^5*c[4]^5*c[6]+1 20*c[5]*c[7]*c[4]^6*c[6]^2+420*c[6]*c[4]^5*c[7]^3*c[5]+600*c[5]^4*c[4] ^5*c[6]^2*c[7]+175*c[7]*c[6]^2*c[5]*c[4]^4-430*c[5]^3*c[4]^4*c[6]^2*c[ 7]+200*c[5]^3*c[4]^3*c[6]^2*c[7]^2-450*c[5]^4*c[4]^4*c[6]*c[7]+160*c[5 ]^2*c[4]^4*c[7]*c[6]^2-200*c[5]^3*c[4]^4*c[6]*c[7]-280*c[5]^4*c[4]^3*c [6]*c[7]+6*c[5]^3*c[4]^3-100*c[5]^2*c[4]^7*c[7]^2-1200*c[7]^3*c[5]^3*c [6]^2*c[4]^3-320*c[7]^3*c[4]^4*c[5]*c[6]-30*c[7]^2*c[5]^3*c[4]^2-300*c [6]*c[5]^3*c[4]^7-350*c[5]^4*c[4]^5*c[6]^2+600*c[6]^2*c[7]^3*c[5]^3*c[ 4]^4+600*c[6]*c[7]^2*c[5]^4*c[4]^5+200*c[6]^2*c[7]^3*c[4]^5*c[5]+200*c [7]^2*c[4]^7*c[6]*c[5]+1100*c[5]^4*c[4]^4*c[6]^2-720*c[5]^4*c[4]^3*c[6 ]^2+1750*c[7]^2*c[5]^4*c[6]^2*c[4]^3-1230*c[7]^2*c[5]^4*c[6]^2*c[4]^2+ 150*c[5]^4*c[7]^3*c[4]^3*c[6]+300*c[5]^4*c[4]^6*c[6]+480*c[5]^3*c[4]^4 *c[7]^2*c[6]+1420*c[5]^4*c[4]^3*c[7]^2*c[6]+20*c[5]^2*c[4]^4*c[7]^2*c[ 6]+130*c[5]^4*c[4]^3*c[6]^2*c[7]-40*c[7]^2*c[4]^4*c[6]^2*c[5]-350*c[7] ^2*c[4]^4*c[5]^3*c[6]^2+230*c[7]^2*c[4]^4*c[6]^2*c[5]^2+10*c[4]^6*c[5] ^2+188*c[5]^2*c[6]*c[4]^5-400*c[6]^2*c[7]^3*c[5]^2*c[4]^5-600*c[6]^2*c [7]^2*c[5]^4*c[4]^4+120*c[5]^2*c[4]^7*c[6]-94*c[4]^4*c[6]*c[5]^3+6*c[6 ]*c[5]*c[4]^5+60*c[5]^5*c[6]^2*c[4]^3-200*c[6]^2*c[4]^6*c[5]*c[7]^2-42 0*c[7]^3*c[6]^2*c[5]*c[4]^4+1550*c[5]^4*c[4]^5*c[6]*c[7]+500*c[7]^3*c[ 4]^4*c[6]^2*c[5]^2-400*c[5]^2*c[4]^7*c[6]*c[7]^2+6*c[5]^3*c[4]^4+103*c [6]^2*c[7]^3*c[5]^2*c[4]-260*c[6]^2*c[7]^3*c[5]^2*c[4]^2+540*c[5]^3*c[ 7]^3*c[4]^2*c[6]-97*c[6]^2*c[5]*c[7]^3*c[4]^2+30*c[6]^2*c[7]^3*c[5]^3- 4*c[7]^3*c[4]*c[5]^3)/(-240*c[5]^2*c[6]*c[4]^4+40*c[6]*c[5]*c[4]^4-9*c [5]^3*c[4]-100*c[4]^5*c[6]*c[5]^3-350*c[5]^4*c[4]^3*c[6]+30*c[5]^5*c[4 ]^2+2*c[5]^3-2*c[4]^3-50*c[5]^5*c[4]^3-3*c[5]^4+3*c[4]^4+50*c[4]^5*c[5 ]^3+100*c[5]^5*c[4]^3*c[6]-3*c[6]*c[5]^3+3*c[6]*c[4]^3+9*c[5]*c[4]^3-6 *c[5]*c[6]*c[4]^2+4*c[5]*c[4]^2+240*c[6]*c[5]^4*c[4]^2+5*c[6]*c[5]^4-1 3*c[6]*c[5]*c[4]^3+13*c[5]^3*c[6]*c[4]-4*c[5]^2*c[4]+68*c[5]^3*c[4]^2+ 100*c[5]^2*c[6]*c[4]^3-100*c[5]^3*c[6]*c[4]^2-140*c[5]^4*c[4]^2+140*c[ 4]^4*c[5]^2+180*c[5]^4*c[4]^3+6*c[5]^2*c[6]*c[4]-68*c[5]^2*c[4]^3-23*c [4]^4*c[5]-40*c[5]^4*c[4]*c[6]-30*c[4]^5*c[5]^2-5*c[6]*c[4]^4-50*c[5]^ 5*c[6]*c[4]^2+50*c[5]^2*c[6]*c[4]^5+350*c[4]^4*c[6]*c[5]^3+23*c[5]^4*c [4]-180*c[5]^3*c[4]^4)/(-c[4]+c[6])/c[4]^2, `b*`[7] = 1/60*(-30*c[5]^3 *c[4]-18*c[6]+6*c[5]+6*c[4]-9*c[5]^2-9*c[4]^2+27*c[6]^2-97*c[6]*c[5]^2 +66*c[6]*c[5]+60*c[6]*c[5]^3-100*c[6]^2*c[5]^3-100*c[6]^2*c[4]^3+180*c [6]^2*c[4]^2-97*c[6]*c[4]^2+60*c[6]*c[4]^3-30*c[5]*c[4]^3+312*c[5]*c[6 ]*c[4]^2+107*c[5]*c[4]^2-215*c[5]*c[6]*c[4]-66*c[5]*c[4]+66*c[6]*c[4]- 840*c[6]^2*c[4]^2*c[5]+522*c[6]^2*c[4]*c[5]+500*c[6]^2*c[5]*c[4]^3+180 *c[6]^2*c[5]^2-117*c[6]^2*c[5]-230*c[6]*c[5]*c[4]^3-230*c[5]^3*c[6]*c[ 4]+107*c[5]^2*c[4]+50*c[5]^3*c[4]^2+380*c[5]^2*c[6]*c[4]^3-460*c[5]^2* c[6]*c[4]^2+380*c[5]^3*c[6]*c[4]^2+600*c[5]^3*c[6]^2*c[4]^3-900*c[6]^2 *c[5]^2*c[4]^3+1420*c[6]^2*c[5]^2*c[4]^2-300*c[6]*c[5]^3*c[4]^3+500*c[ 6]^2*c[5]^3*c[4]-900*c[6]^2*c[5]^3*c[4]^2-180*c[5]^2*c[4]^2-840*c[6]^2 *c[4]*c[5]^2-117*c[6]^2*c[4]+312*c[5]^2*c[6]*c[4]+50*c[5]^2*c[4]^3)/c[ 7]/(-3*c[7]^3*c[6]+3*c[7]^2*c[6]^2-9*c[6]^2*c[7]^2*c[5]-8*c[7]*c[6]^2* c[5]^3+8*c[7]^2*c[6]*c[5]^3-c[5]*c[6]*c[7]*c[4]-c[7]^2*c[4]^2-30*c[6]^ 2*c[7]*c[5]^2*c[4]^3-28*c[6]^2*c[7]^2*c[5]^2*c[4]+30*c[6]^2*c[7]^2*c[5 ]^2*c[4]^2-30*c[6]*c[7]*c[5]^3*c[4]^3+9*c[6]^2*c[7]*c[4]^2-3*c[6]^2*c[ 7]*c[4]+2*c[6]*c[7]^2*c[4]-9*c[6]*c[7]^2*c[4]^2-2*c[7]^2*c[5]*c[4]-c[7 ]^2*c[5]^2+c[6]*c[7]*c[4]^2+8*c[7]^2*c[6]*c[4]^3-4*c[5]*c[7]^2*c[4]^3+ c[5]*c[7]*c[4]^2-c[5]*c[6]*c[4]^2+32*c[6]*c[5]*c[7]^2*c[4]^2-9*c[7]^2* c[5]*c[6]*c[4]-4*c[5]*c[7]*c[6]*c[4]^3-9*c[6]^2*c[4]^2*c[5]+3*c[6]^2*c [4]*c[5]+c[7]^3*c[4]+9*c[7]^3*c[4]*c[6]-9*c[7]^3*c[4]*c[5]-20*c[5]^2*c [6]*c[7]*c[4]^2+8*c[6]^2*c[5]*c[4]^3+28*c[7]^3*c[4]^2*c[6]*c[5]+18*c[6 ]^2*c[7]*c[4]*c[5]+28*c[7]*c[6]^2*c[5]*c[4]^3-3*c[6]^2*c[7]*c[5]+9*c[6 ]^2*c[7]*c[5]^2+c[6]*c[7]*c[5]^2-8*c[6]*c[7]^3*c[5]^2-9*c[6]*c[7]^2*c[ 5]^2+8*c[6]^2*c[7]^2*c[5]^2+2*c[6]*c[7]^2*c[5]-36*c[6]^2*c[4]^2*c[7]*c [5]-28*c[6]^2*c[7]^2*c[4]^2*c[5]-28*c[7]^3*c[4]*c[6]*c[5]-36*c[6]^2*c[ 5]^2*c[7]*c[4]+56*c[6]^2*c[5]^2*c[7]*c[4]^2+28*c[6]*c[5]^2*c[7]*c[4]^3 +28*c[6]*c[5]^3*c[7]*c[4]^2+28*c[6]^2*c[5]^3*c[7]*c[4]+9*c[6]*c[7]^3*c [5]+9*c[7]^2*c[5]*c[4]^2+28*c[7]^3*c[5]^2*c[6]*c[4]+32*c[7]^2*c[5]^2*c [6]*c[4]+8*c[6]^2*c[7]^2*c[4]^2-9*c[6]^2*c[7]^2*c[4]-4*c[5]^3*c[6]*c[7 ]*c[4]+28*c[6]^2*c[7]^2*c[4]*c[5]-4*c[5]^2*c[6]*c[4]^3-9*c[5]^2*c[7]*c [4]^2+9*c[5]^2*c[6]*c[4]^2+4*c[5]^2*c[7]*c[4]^3+4*c[5]^3*c[7]*c[4]^2-4 *c[5]^3*c[6]*c[4]^2+30*c[5]^3*c[6]^2*c[4]^3-28*c[6]^2*c[5]^2*c[4]^3+28 *c[6]^2*c[5]^2*c[4]^2+8*c[6]^2*c[5]^3*c[4]-28*c[6]^2*c[5]^3*c[4]^2-9*c [6]^2*c[4]*c[5]^2-8*c[7]*c[6]^2*c[4]^3+30*c[7]^2*c[5]^2*c[6]*c[4]^3-30 *c[7]^3*c[5]^2*c[6]*c[4]^2+30*c[7]^2*c[5]^3*c[6]*c[4]^2-8*c[7]^3*c[4]^ 2*c[6]+4*c[7]^3*c[4]^2*c[5]+c[7]^3*c[5]-28*c[7]^2*c[6]*c[5]*c[4]^3-56* c[7]^2*c[5]^2*c[6]*c[4]^2-28*c[7]^2*c[5]^3*c[6]*c[4]-c[5]^2*c[6]*c[4]+ c[5]^2*c[7]*c[4]+4*c[7]^3*c[5]^2*c[4]+9*c[7]^2*c[5]^2*c[4]-30*c[6]^2*c [7]*c[5]^3*c[4]^2-8*c[7]^2*c[5]^2*c[4]^2-4*c[7]^2*c[5]^3*c[4]), a[2,1] = c[2], a[6,4] = -1/2*(120*c[5]^2*c[6]*c[4]^4-12*c[6]*c[5]*c[4]^4-90* c[5]^2*c[6]^3*c[4]^2-60*c[5]*c[6]^2*c[4]^4-2*c[6]^3*c[4]^2+20*c[5]^4*c [4]^3*c[6]+40*c[6]^2*c[4]^5*c[5]+36*c[4]*c[5]^2*c[6]^3-20*c[5]^4*c[6]^ 2*c[4]^2+8*c[6]^2*c[5]^3+2*c[6]^2*c[4]^3-24*c[5]*c[6]^3*c[4]^2-6*c[6]^ 2*c[4]^2*c[5]+36*c[6]^2*c[5]*c[4]^3+60*c[4]^3*c[5]*c[6]^3-3*c[6]*c[5]* c[4]^3-11*c[5]^3*c[6]*c[4]+3*c[5]^3*c[4]^2-24*c[5]^2*c[6]*c[4]^3+13*c[ 5]^2*c[6]*c[4]^2+48*c[5]^3*c[6]*c[4]^2-60*c[5]^3*c[6]^2*c[4]^3-30*c[6] ^2*c[5]^2*c[4]^3-12*c[6]^2*c[5]^2*c[4]^2-150*c[6]*c[5]^3*c[4]^3-48*c[6 ]^2*c[5]^3*c[4]+150*c[6]^2*c[5]^3*c[4]^2-2*c[6]^2*c[4]*c[5]^2-5*c[5]^2 *c[4]^3+2*c[4]^4*c[5]-6*c[5]^2*c[6]^3+60*c[6]^3*c[4]^3*c[5]^2+7*c[6]^3 *c[4]*c[5]-60*c[5]^2*c[6]*c[4]^5-40*c[6]^3*c[4]^4*c[5]+60*c[4]^4*c[6]* c[5]^3)*c[6]/(c[4]^3-2*c[5]*c[4]^2+30*c[5]^2*c[4]^3-6*c[5]*c[4]^3-10*c [4]^4*c[5]^2-c[5]^3+10*c[5]^4*c[4]^2-30*c[5]^3*c[4]^2+6*c[5]^3*c[4]+2* c[5]^2*c[4])/c[4]^2, a[3,1] = -2/9*c[4]*(c[4]-3*c[2])/c[2], a[4,1] = 1 /4*c[4], a[7,3] = 3/4*1/c[4]^2*c[7]*(-480*c[5]^2*c[6]*c[4]^4+15*c[6]*c [5]*c[4]^4+300*c[4]^2*c[5]^2*c[7]^3+60*c[7]^3*c[5]*c[4]^3-510*c[7]^3*c [5]^2*c[4]^3+54*c[7]*c[5]*c[4]^4+450*c[5]^2*c[4]^4*c[7]^2+150*c[4]^3*c [7]^3*c[5]^3-150*c[7]^2*c[5]^2*c[4]^5-450*c[4]^5*c[6]*c[5]^3-3*c[7]^3* c[4]^2+90*c[5]^2*c[4]^5*c[7]+120*c[5]^3*c[4]^4*c[7]+150*c[7]^3*c[5]^2* c[4]^4+100*c[5]^4*c[4]^2*c[7]^2*c[6]-180*c[6]*c[4]^5*c[5]*c[7]+30*c[6] *c[5]^2*c[4]^4*c[7]+60*c[5]^4*c[4]^3*c[6]-150*c[5]^4*c[4]^4*c[6]-200*c [7]^2*c[4]^4*c[5]^3+300*c[6]*c[4]^5*c[5]*c[7]^2-50*c[7]^2*c[5]^4*c[4]^ 3+24*c[7]*c[6]*c[5]^3-40*c[7]^2*c[6]*c[5]^3-6*c[5]*c[6]*c[7]*c[4]+360* c[7]*c[4]^4*c[6]*c[5]+150*c[7]^2*c[5]^3*c[4]^3+2*c[7]^2*c[4]^2+6*c[5]^ 2*c[7]^3+330*c[6]*c[7]*c[5]^3*c[4]^3-450*c[7]^2*c[6]*c[5]*c[4]^4-12*c[ 7]*c[6]*c[4]^3-6*c[6]*c[7]^2*c[4]-2*c[7]^2*c[5]*c[4]-4*c[7]^2*c[5]^2+6 *c[6]*c[7]*c[4]^2+20*c[7]^2*c[6]*c[4]^3-10*c[5]*c[7]^2*c[4]^3+6*c[5]*c [4]^3-6*c[5]*c[7]*c[4]^2-6*c[5]*c[6]*c[4]^2+150*c[6]*c[5]*c[7]^2*c[4]^ 2-72*c[7]^2*c[5]*c[6]*c[4]-270*c[5]*c[7]*c[6]*c[4]^3+9*c[7]^3*c[4]*c[6 ]+3*c[7]^3*c[4]*c[5]-306*c[5]^2*c[6]*c[7]*c[4]^2+900*c[4]^5*c[6]*c[7]* c[5]^3+600*c[5]^2*c[6]*c[7]^3*c[4]^4-420*c[7]^3*c[4]^2*c[6]*c[5]+90*c[ 6]*c[5]*c[7]*c[4]^2-12*c[6]*c[7]*c[5]^2+30*c[6]*c[7]^3*c[5]^2+12*c[6]* c[7]^2*c[5]+123*c[7]^3*c[4]*c[6]*c[5]+360*c[6]*c[5]^2*c[7]*c[4]^3+360* c[6]*c[5]^3*c[7]*c[4]^2-18*c[6]*c[7]^3*c[5]+30*c[5]^4*c[4]^3*c[7]-80*c [7]^2*c[5]^2*c[4]^3+24*c[7]^2*c[5]*c[4]^2-30*c[7]*c[5]*c[4]^3+3*c[6]*c [5]*c[4]^3-240*c[7]^3*c[5]^2*c[6]*c[4]-24*c[5]^3*c[6]*c[4]-12*c[5]^3*c [7]*c[4]-144*c[5]^3*c[6]*c[7]*c[4]+9*c[5]^3*c[4]^2+294*c[5]^2*c[6]*c[4 ]^3-42*c[5]^2*c[7]*c[4]^2-75*c[5]^2*c[6]*c[4]^2+252*c[5]^2*c[7]*c[4]^3 +18*c[5]^3*c[7]*c[4]^2+189*c[5]^3*c[6]*c[4]^2-360*c[5]^2*c[4]^4*c[7]-7 20*c[6]*c[5]^3*c[4]^3-120*c[5]^3*c[7]*c[4]^3-60*c[5]^4*c[7]*c[4]^2*c[6 ]-6*c[5]^2*c[4]^2-780*c[7]^2*c[5]^2*c[6]*c[4]^3+810*c[7]^3*c[5]^2*c[6] *c[4]^2-1230*c[7]^2*c[5]^3*c[6]*c[4]^2-15*c[7]^3*c[4]^2*c[6]-21*c[7]^3 *c[4]^2*c[5]+20*c[7]^2*c[6]*c[5]*c[4]^3+220*c[7]^2*c[5]^2*c[6]*c[4]^2+ 15*c[4]^4*c[5]^2+320*c[7]^2*c[5]^3*c[6]*c[4]+1850*c[7]^2*c[5]^3*c[6]*c [4]^3-60*c[5]^3*c[7]^3*c[4]^2+12*c[5]^2*c[6]*c[4]+12*c[5]^2*c[7]*c[4]- 66*c[7]^3*c[5]^2*c[4]+24*c[7]^2*c[5]^2*c[4]+72*c[5]^2*c[6]*c[7]*c[4]-1 10*c[7]^2*c[5]^2*c[4]^2+20*c[7]^2*c[5]^3*c[4]-9*c[4]^4*c[5]-60*c[7]^2* c[4]^4*c[5]-600*c[5]^2*c[6]*c[4]^5*c[7]^2-1200*c[7]^3*c[5]^2*c[4]^3*c[ 6]+630*c[7]^3*c[6]*c[5]*c[4]^3+300*c[5]^4*c[4]^4*c[6]*c[7]-1800*c[5]^3 *c[4]^4*c[6]*c[7]-15*c[5]^3*c[4]^3-300*c[7]^3*c[4]^4*c[5]*c[6]-20*c[7] ^2*c[5]^3*c[4]^2-800*c[5]^3*c[4]^4*c[7]^2*c[6]-200*c[5]^4*c[4]^3*c[7]^ 2*c[6]+1150*c[5]^2*c[4]^4*c[7]^2*c[6]+180*c[5]^2*c[6]*c[4]^5+1140*c[4] ^4*c[6]*c[5]^3)/(10*c[5]^3*c[4]^2+10*c[5]^2*c[4]^3-30*c[5]^2*c[4]^2+6* c[5]^2*c[4]-c[5]^2+6*c[5]*c[4]^2+c[5]*c[4]-c[4]^2)/(10*c[5]*c[6]*c[4]- 5*c[6]*c[5]-5*c[6]*c[4]+3*c[6]-5*c[5]*c[4]+3*c[5]+3*c[4]-2), `b*`[1] = 1/60*(30*c[5]^3*c[4]+42*c[7]*c[6]^2-103*c[5]*c[7]*c[4]+18*c[6]-6*c[5] -6*c[4]-27*c[7]*c[6]+9*c[5]^2+9*c[4]^2-180*c[7]*c[6]^2*c[5]^3+100*c[7] *c[6]*c[5]^3-27*c[6]^2-354*c[5]*c[6]*c[7]*c[4]+97*c[6]*c[5]^2-66*c[6]* c[5]-60*c[6]*c[5]^3+100*c[6]^2*c[5]^3+9*c[7]*c[5]+100*c[6]^2*c[4]^3-20 40*c[6]^2*c[7]*c[5]^2*c[4]^3-600*c[6]*c[7]*c[5]^3*c[4]^3-14*c[7]*c[5]^ 2+100*c[7]*c[6]*c[4]^3+310*c[6]^2*c[7]*c[4]^2-192*c[6]^2*c[7]*c[4]-180 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0*c[5]^3*c[6]*c[4]^2-600*c[5]^3*c[6]^2*c[4]^3+900*c[6]^2*c[5]^2*c[4]^3 -1420*c[6]^2*c[5]^2*c[4]^2+300*c[6]*c[5]^3*c[4]^3-500*c[6]^2*c[5]^3*c[ 4]+900*c[6]^2*c[5]^3*c[4]^2+180*c[5]^2*c[4]^2+840*c[6]^2*c[4]*c[5]^2-1 80*c[7]*c[6]^2*c[4]^3+117*c[6]^2*c[4]+1500*c[5]^3*c[6]^2*c[7]*c[4]^3-3 12*c[5]^2*c[6]*c[4]+174*c[5]^2*c[7]*c[4]+526*c[5]^2*c[6]*c[7]*c[4]-204 0*c[6]^2*c[7]*c[5]^3*c[4]^2-50*c[5]^2*c[4]^3)/(3*c[6]-28*c[5]*c[6]*c[4 ]^2-28*c[5]^2*c[6]*c[4]+8*c[6]*c[5]^2-9*c[6]*c[4]-9*c[6]*c[5]+28*c[5]* c[6]*c[4]+30*c[5]^2*c[6]*c[4]^2+8*c[6]*c[4]^2-4*c[5]^2*c[4]-4*c[5]*c[4 ]^2+9*c[5]*c[4]-c[5]-c[4])/c[5]/c[6]/c[7]/c[4], a[8,4] = -1/2*(734*c[5 ]^2*c[6]*c[4]^4-203*c[6]*c[5]*c[4]^4+3190*c[5]^4*c[4]^4*c[6]^2*c[7]-93 *c[7]*c[5]*c[4]^4-636*c[5]^2*c[4]^4*c[7]^2-10*c[5]^3*c[4]-520*c[6]*c[4 ]^7*c[7]*c[5]-481*c[4]^5*c[6]^2*c[5]^2+645*c[7]^2*c[5]^2*c[4]^5-574*c[ 4]^5*c[6]*c[5]^3-172*c[5]*c[6]^2*c[4]^4+6*c[7]^2*c[4]^3+100*c[5]^5*c[4 ]^3*c[7]^2*c[6]^2+60*c[4]^8*c[5]^2-600*c[5]^2*c[4]^6*c[6]^2*c[7]^2+9*c [4]^5*c[5]*c[7]-164*c[5]^2*c[4]^5*c[7]+900*c[5]^3*c[4]^6*c[6]^2*c[7]+6 00*c[5]^3*c[4]^6*c[6]^2*c[7]^2-1104*c[5]^3*c[4]^4*c[7]-18*c[4]^4*c[6]^ 2+90*c[7]^2*c[4]^5*c[5]^3+1380*c[6]^2*c[4]^5*c[7]*c[5]^3-750*c[7]*c[5] ^2*c[4]^6*c[6]+675*c[5]^4*c[4]^2*c[7]^2*c[6]+8*c[6]^2*c[4]^5-339*c[6]^ 2*c[7]*c[5]^4*c[4]^2-315*c[5]^2*c[4]^6*c[7]+640*c[6]*c[5]*c[7]^2*c[4]^ 6+46*c[6]*c[4]^5*c[5]*c[7]-1014*c[6]*c[5]^2*c[4]^4*c[7]+2100*c[5]^4*c[ 7]*c[4]^6*c[6]-390*c[6]^2*c[4]^6*c[5]+860*c[4]^6*c[6]^2*c[5]^2+140*c[5 ]^5*c[4]^3*c[7]-1740*c[5]^2*c[7]^2*c[4]^6*c[6]-613*c[5]^4*c[4]^3*c[6]- 100*c[4]^8*c[5]^2*c[7]+1140*c[5]^4*c[4]^5*c[6]+968*c[5]^2*c[6]*c[4]^6- 150*c[5]^5*c[4]^5*c[6]^2+263*c[5]^4*c[4]^4*c[6]+250*c[7]^2*c[4]^4*c[5] ^3-120*c[6]*c[5]*c[4]^8-10*c[6]*c[7]^2*c[4]^5-6*c[6]*c[4]^6+780*c[5]^4 *c[4]^4*c[7]+32*c[7]^2*c[6]*c[4]^4+1200*c[5]^3*c[7]^2*c[4]^6*c[6]-750* 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] 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5]^4+585*c[4]^5*c[7]*c[5]^3+c[4]^5+35*c[6]^2*c[5]^2*c[7]*c[4]-26*c[6]^ 2*c[5]^2*c[7]*c[4]^2+1128*c[6]*c[5]^2*c[7]*c[4]^3+706*c[6]*c[5]^3*c[7] *c[4]^2-138*c[6]^2*c[5]^3*c[7]*c[4]-685*c[5]^4*c[4]^3*c[7]+354*c[7]^2* c[5]^2*c[4]^3-370*c[5]^4*c[7]^2*c[4]^4-6*c[7]^2*c[5]*c[4]^2+70*c[7]*c[ 5]*c[4]^3+89*c[6]*c[5]*c[4]^3+200*c[4]^8*c[5]*c[6]*c[7]+390*c[6]*c[4]^ 7*c[5]-31*c[7]^2*c[5]^2*c[6]*c[4]-27*c[6]*c[7]*c[4]^4+10*c[4]^5*c[5]-8 0*c[5]^4*c[4]^6+200*c[5]^5*c[4]^5*c[7]^2*c[6]+6*c[6]^2*c[7]^2*c[4]^2+2 2*c[5]^3*c[6]*c[4]+19*c[5]^3*c[7]*c[4]+400*c[5]^5*c[4]^3*c[7]^2*c[6]+4 *c[5]^2*c[4]-135*c[5]^3*c[6]*c[7]*c[4]+71*c[5]^3*c[4]^2-21*c[6]^2*c[7] ^2*c[4]*c[5]-480*c[5]^2*c[6]*c[4]^3+42*c[5]^2*c[7]*c[4]^2+37*c[5]^2*c[ 6]*c[4]^2-480*c[5]^2*c[7]*c[4]^3-104*c[5]^3*c[7]*c[4]^2-97*c[5]^3*c[6] *c[4]^2-1536*c[5]^3*c[6]^2*c[4]^3+190*c[5]^5*c[4]^3*c[6]^2*c[7]+557*c[ 4]^3*c[7]^2*c[5]^2*c[6]^2-10*c[6]*c[7]*c[4]^5-730*c[4]^5*c[5]^3*c[6]^2 -14*c[7]*c[5]^4*c[4]+10*c[7]^2*c[5]^4*c[4]+646*c[5]^2*c[4]^4*c[7]+498* c[6]^2*c[5]^2*c[4]^3-95*c[6]^2*c[5]^2*c[4]^2+744*c[6]*c[5]^3*c[4]^3-46 *c[6]^2*c[5]^3*c[4]+334*c[6]^2*c[5]^3*c[4]^2+854*c[5]^3*c[7]*c[4]^3-49 0*c[5]^4*c[7]*c[4]^2*c[6]-30*c[7]*c[5]*c[4]^7-19*c[5]^2*c[4]^2+18*c[6] ^2*c[4]*c[5]^2-472*c[6]*c[5]*c[4]^6+360*c[5]*c[4]^6*c[7]*c[6]-10*c[7]* c[6]^2*c[4]^3-811*c[7]^2*c[5]^2*c[6]*c[4]^3-993*c[7]^2*c[5]^3*c[6]*c[4 ]^2-44*c[5]^4*c[4]^2-451*c[4]^3*c[7]^2*c[5]*c[6]^2+10*c[6]*c[7]*c[4]^6 -14*c[7]^2*c[6]*c[5]*c[4]^3-20*c[5]^5*c[4]^5+118*c[7]^2*c[5]^2*c[6]*c[ 4]^2-249*c[4]^4*c[5]^2+170*c[7]^2*c[5]^3*c[6]*c[4]+3829*c[7]^2*c[5]^3* c[6]*c[4]^3+2208*c[5]^3*c[6]^2*c[4]^4+339*c[5]^4*c[4]^3-6*c[6]^2*c[4]^ 3*c[7]^2+1871*c[5]^3*c[6]^2*c[7]*c[4]^3-10*c[5]^2*c[6]*c[4]-9*c[5]^2*c [7]*c[4]+6*c[7]^2*c[5]^2*c[4]+33*c[5]^2*c[6]*c[7]*c[4]+600*c[7]*c[4]^8 *c[6]*c[5]^3+187*c[6]^2*c[7]*c[5]^3*c[4]^2+341*c[6]^2*c[7]^2*c[4]*c[5] ^3+202*c[5]^2*c[4]^3-29*c[7]^2*c[5]^2*c[4]^2-10*c[7]^2*c[5]^3*c[4]+40* c[4]^4*c[5]-270*c[4]^7*c[5]^2-450*c[5]^5*c[4]^3*c[6]*c[7]+35*c[4]^6*c[ 7]*c[5]+80*c[7]^2*c[4]^4*c[5]-1163*c[7]^2*c[5]^3*c[6]^2*c[4]^2+20*c[5] *c[4]^7-640*c[6]^2*c[4]^5*c[5]*c[7]^2+850*c[7]^2*c[4]^5*c[6]*c[5]^3+15 30*c[5]^2*c[6]*c[4]^5*c[7]^2-506*c[5]^4*c[4]^4+1420*c[5]^2*c[4]^7*c[6] *c[7]-16*c[5]^4*c[4]*c[6]-85*c[5]^4*c[7]^2*c[4]*c[6]+72*c[5]^4*c[4]*c[ 7]*c[6]+230*c[4]^5*c[6]^2*c[7]*c[5]^2+1740*c[4]^5*c[6]^2*c[7]^2*c[5]^2 -600*c[6]*c[4]^8*c[5]^2*c[7]-82*c[4]^5*c[5]^2-480*c[6]^2*c[4]^4*c[5]^2 +320*c[4]^8*c[5]^2*c[6]-37*c[4]^6*c[5]+300*c[5]^3*c[4]^7*c[6]^2-90*c[5 ]^5*c[4]^5*c[6]+520*c[5]*c[7]*c[4]^6*c[6]^2-10*c[6]^2*c[7]*c[4]^5+9*c[ 6]*c[4]^4+600*c[5]^4*c[7]*c[4]^6*c[6]^2-2350*c[5]^4*c[4]^5*c[6]^2*c[7] -600*c[5]^3*c[7]^2*c[6]*c[4]^7-204*c[7]*c[6]^2*c[5]*c[4]^4-3630*c[5]^3 *c[4]^4*c[6]^2*c[7]+740*c[5]^3*c[4]^3*c[6]^2*c[7]^2-2120*c[5]^4*c[4]^4 *c[6]*c[7]+1460*c[5]^2*c[4]^4*c[7]*c[6]^2+2387*c[5]^3*c[4]^4*c[6]*c[7] +2548*c[5]^4*c[4]^3*c[6]*c[7]-498*c[5]^3*c[4]^3+100*c[5]^2*c[4]^7*c[7] ^2+26*c[7]^2*c[5]^3*c[4]^2+600*c[6]*c[5]^3*c[4]^7+100*c[6]*c[5]^5*c[4] 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7]^2+814*c[5]^3*c[4]^4)/(690*c[5]^2*c[6]*c[4]^4-110*c[6]*c[5]*c[4]^4-1 10*c[7]*c[5]*c[4]^4+48*c[5]^3*c[4]+200*c[4]^5*c[6]*c[5]^3-150*c[5]^2*c [4]^5*c[7]-750*c[5]^3*c[4]^4*c[7]-930*c[6]*c[5]^2*c[4]^4*c[7]-200*c[5] ^5*c[4]^3*c[7]+750*c[5]^4*c[4]^3*c[6]-120*c[5]^5*c[4]^2-10*c[5]^3+10*c [4]^3+150*c[5]^5*c[4]^3-15*c[7]*c[6]*c[5]^3+12*c[5]^4-12*c[4]^4+200*c[ 6]*c[5]^2*c[4]^5*c[7]-150*c[4]^5*c[5]^3-200*c[5]^5*c[4]^3*c[6]+150*c[7 ]*c[4]^4*c[6]*c[5]+12*c[6]*c[5]^3+15*c[4]^4*c[7]-690*c[5]^4*c[7]*c[4]^ 2-15*c[5]^4*c[7]+150*c[5]^5*c[7]*c[4]^2+15*c[7]*c[6]*c[4]^3-12*c[7]*c[ 4]^3-12*c[6]*c[4]^3-48*c[5]*c[4]^3+24*c[5]*c[7]*c[4]^2+24*c[5]*c[6]*c[ 4]^2-20*c[5]*c[4]^2-70*c[5]*c[7]*c[6]*c[4]^3+12*c[7]*c[5]^3-300*c[4]^5 *c[6]*c[7]*c[5]^3-30*c[6]*c[5]*c[7]*c[4]^2+20*c[5]^4*c[7]*c[6]-690*c[6 ]*c[5]^4*c[4]^2-15*c[6]*c[5]^4+200*c[4]^5*c[7]*c[5]^3+510*c[6]*c[5]^2* c[7]*c[4]^3-510*c[6]*c[5]^3*c[7]*c[4]^2+750*c[5]^4*c[4]^3*c[7]+57*c[7] *c[5]*c[4]^3+57*c[6]*c[5]*c[4]^3-20*c[6]*c[7]*c[4]^4-57*c[5]^3*c[6]*c[ 4]-57*c[5]^3*c[7]*c[4]+20*c[5]^2*c[4]-200*c[5]^5*c[4]^2*c[6]*c[7]+70*c [5]^3*c[6]*c[7]*c[4]-342*c[5]^3*c[4]^2-410*c[5]^2*c[6]*c[4]^3-410*c[5] ^2*c[7]*c[4]^3+410*c[5]^3*c[7]*c[4]^2+410*c[5]^3*c[6]*c[4]^2+110*c[7]* c[5]^4*c[4]+690*c[5]^2*c[4]^4*c[7]+930*c[5]^4*c[7]*c[4]^2*c[6]+550*c[5 ]^4*c[4]^2-550*c[4]^4*c[5]^2-570*c[5]^4*c[4]^3-24*c[5]^2*c[6]*c[4]-24* c[5]^2*c[7]*c[4]+30*c[5]^2*c[6]*c[7]*c[4]+342*c[5]^2*c[4]^3+87*c[4]^4* c[5]+300*c[5]^5*c[4]^3*c[6]*c[7]+110*c[5]^4*c[4]*c[6]-150*c[5]^4*c[4]* c[7]*c[6]+120*c[4]^5*c[5]^2+15*c[6]*c[4]^4+1100*c[5]^3*c[4]^4*c[6]*c[7 ]-1100*c[5]^4*c[4]^3*c[6]*c[7]+150*c[5]^5*c[6]*c[4]^2-150*c[5]^2*c[6]* c[4]^5-750*c[4]^4*c[6]*c[5]^3-87*c[5]^4*c[4]+570*c[5]^3*c[4]^4)/(c[6]* c[4]-c[4]^2-c[7]*c[6]+c[7]*c[4])/c[4]^2, a[8,1] = 1/4*(-2816*c[5]^2*c[ 6]*c[4]^4+424*c[6]*c[5]*c[4]^4-9720*c[5]^4*c[4]^4*c[6]^2*c[7]+372*c[7] *c[5]*c[4]^4+1752*c[5]^2*c[4]^4*c[7]^2+20*c[5]^3*c[4]-1320*c[4]^5*c[6] ^2*c[5]^2-1320*c[7]^2*c[5]^2*c[4]^5-4880*c[4]^5*c[6]*c[5]^3-264*c[5]*c 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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]-7 0*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]-97 2*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]^5-216 0*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] 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^2*c[5]^4-4160*c[4]^5*c[7]*c[5]^3-12*c[4]^5+95*c[6]^2*c[5]^2*c[7]*c[4] -356*c[6]^2*c[5]^2*c[7]*c[4]^2-2598*c[6]*c[5]^2*c[7]*c[4]^3-698*c[6]*c [5]^3*c[7]*c[4]^2-300*c[6]^2*c[5]^3*c[7]*c[4]+1818*c[5]^4*c[4]^3*c[7]- 692*c[7]^2*c[5]^2*c[4]^3+3160*c[5]^4*c[7]^2*c[4]^4+12*c[7]^2*c[5]*c[4] ^2-108*c[7]*c[5]*c[4]^3-120*c[6]*c[5]*c[4]^3+29*c[7]^2*c[5]^2*c[6]*c[4 ]+114*c[6]*c[7]*c[4]^4+104*c[4]^5*c[5]+200*c[5]^4*c[4]^6-1200*c[5]^5*c [4]^5*c[7]^2*c[6]+6*c[6]^2*c[7]^2*c[4]^2-52*c[5]^3*c[6]*c[4]-46*c[5]^3 *c[7]*c[4]-450*c[5]^5*c[4]^3*c[7]^2*c[6]-8*c[5]^2*c[4]+60*c[5]^3*c[6]* c[7]*c[4]-160*c[5]^3*c[4]^2-21*c[6]^2*c[7]^2*c[4]*c[5]+1144*c[5]^2*c[6 ]*c[4]^3-108*c[5]^2*c[7]*c[4]^2-120*c[5]^2*c[6]*c[4]^2+1024*c[5]^2*c[7 ]*c[4]^3+372*c[5]^3*c[7]*c[4]^2+424*c[5]^3*c[6]*c[4]^2+1752*c[5]^3*c[6 ]^2*c[4]^3-660*c[5]^5*c[4]^3*c[6]^2*c[7]-1153*c[4]^3*c[7]^2*c[5]^2*c[6 ]^2-72*c[6]*c[7]*c[4]^5+3160*c[4]^5*c[5]^3*c[6]^2+28*c[7]*c[5]^4*c[4]- 20*c[7]^2*c[5]^4*c[4]-2472*c[5]^2*c[4]^4*c[7]-692*c[6]^2*c[5]^2*c[4]^3 +72*c[6]^2*c[5]^2*c[4]^2-2816*c[6]*c[5]^3*c[4]^3+32*c[6]^2*c[5]^3*c[4] -264*c[6]^2*c[5]^3*c[4]^2-2472*c[5]^3*c[7]*c[4]^3+566*c[5]^4*c[7]*c[4] ^2*c[6]+48*c[5]^2*c[4]^2+1200*c[6]^2*c[4]^5*c[5]^5*c[7]^2-12*c[6]^2*c[ 4]*c[5]^2+38*c[7]*c[6]^2*c[4]^3+1804*c[7]^2*c[5]^2*c[6]*c[4]^3+258*c[7 ]^2*c[5]^3*c[6]*c[4]^2+104*c[5]^4*c[4]^2-500*c[5]^5*c[4]^5*c[7]-256*c[ 4]^3*c[7]^2*c[5]*c[6]^2-176*c[7]^2*c[6]*c[5]*c[4]^3+200*c[5]^5*c[4]^5- 212*c[7]^2*c[5]^2*c[6]*c[4]^2+1064*c[4]^4*c[5]^2+8*c[7]^2*c[5]^3*c[6]* c[4]-3410*c[7]^2*c[5]^3*c[6]*c[4]^3-4080*c[5]^3*c[6]^2*c[4]^4-772*c[5] ^4*c[4]^3+1200*c[7]^2*c[6]^2*c[4]^6*c[5]^4-36*c[6]^2*c[4]^3*c[7]^2-536 8*c[5]^3*c[6]^2*c[7]*c[4]^3+20*c[5]^2*c[6]*c[4]+18*c[5]^2*c[7]*c[4]-12 *c[7]^2*c[5]^2*c[4]-44*c[5]^2*c[6]*c[7]*c[4]+400*c[5]^4*c[4]^6*c[7]^2+ 1438*c[6]^2*c[7]*c[5]^3*c[4]^2+426*c[6]^2*c[7]^2*c[4]*c[5]^3-452*c[5]^ 2*c[4]^3+72*c[7]^2*c[5]^2*c[4]^2+32*c[7]^2*c[5]^3*c[4]-160*c[4]^4*c[5] +690*c[5]^5*c[4]^3*c[6]*c[7]-264*c[7]^2*c[4]^4*c[5]-1833*c[7]^2*c[5]^3 *c[6]^2*c[4]^2-260*c[6]^2*c[4]^5*c[5]*c[7]^2-8270*c[7]^2*c[4]^5*c[6]*c [5]^3+3780*c[5]^2*c[6]*c[4]^5*c[7]^2+1720*c[5]^4*c[4]^4+32*c[5]^4*c[4] *c[6]+30*c[5]^4*c[7]^2*c[4]*c[6]-60*c[5]^4*c[4]*c[7]*c[6]+2480*c[4]^5* c[6]^2*c[7]*c[5]^2-1120*c[4]^5*c[6]^2*c[7]^2*c[5]^2-772*c[4]^5*c[5]^2- 40*c[6]^2*c[7]^2*c[4]^5+1752*c[6]^2*c[4]^4*c[5]^2-1700*c[7]^2*c[6]^2*c [5]^5*c[4]^4-600*c[5]^5*c[4]^5*c[6]-120*c[5]*c[7]*c[4]^6*c[6]^2+50*c[6 ]^2*c[7]*c[4]^5-52*c[6]*c[4]^4-1200*c[5]^4*c[7]*c[4]^6*c[6]^2+7150*c[5 ]^4*c[4]^5*c[6]^2*c[7]+198*c[7]*c[6]^2*c[5]*c[4]^4+10560*c[5]^3*c[4]^4 *c[6]^2*c[7]+5240*c[5]^3*c[4]^3*c[6]^2*c[7]^2+11240*c[5]^4*c[4]^4*c[6] *c[7]-3649*c[5]^2*c[4]^4*c[7]*c[6]^2-14207*c[5]^3*c[4]^4*c[6]*c[7]-449 1*c[5]^4*c[4]^3*c[6]*c[7]+1064*c[5]^3*c[4]^3-264*c[7]^2*c[5]^3*c[4]^2- 2400*c[5]^4*c[4]^5*c[6]^2+6020*c[6]^2*c[7]^2*c[5]^3*c[4]^5+6800*c[6]*c [7]^2*c[5]^4*c[4]^5+3160*c[5]^4*c[4]^4*c[6]^2-1320*c[5]^4*c[4]^3*c[6]^ 2-5800*c[7]^2*c[5]^4*c[6]^2*c[4]^3+1940*c[7]^2*c[5]^4*c[6]^2*c[4]^2-60 0*c[5]^4*c[4]^6*c[6]+28*c[4]^5*c[7]+9740*c[5]^3*c[4]^4*c[7]^2*c[6]+292 0*c[5]^4*c[4]^3*c[7]^2*c[6]-4775*c[5]^2*c[4]^4*c[7]^2*c[6]+4980*c[5]^4 *c[4]^3*c[6]^2*c[7]+210*c[7]^2*c[4]^4*c[6]^2*c[5]-8860*c[7]^2*c[4]^4*c [5]^3*c[6]^2+2140*c[7]^2*c[4]^4*c[6]^2*c[5]^2+120*c[4]^6*c[5]^2-20*c[7 ]^2*c[4]^5+2092*c[5]^2*c[6]*c[4]^5+9920*c[6]^2*c[7]^2*c[5]^4*c[4]^4-50 0*c[5]^4*c[4]^6*c[7]-48*c[7]^2*c[6]^2*c[5]^3+6404*c[4]^4*c[6]*c[5]^3-2 84*c[6]*c[5]*c[4]^5+200*c[5]^5*c[6]^2*c[4]^3+200*c[6]^2*c[4]^6*c[5]*c[ 7]^2-8820*c[5]^4*c[4]^5*c[6]*c[7]-12*c[5]^4*c[4]-2324*c[5]^3*c[4]^4)/c [6]/c[5]/c[7]/c[4]^2/(150*c[5]^2*c[6]*c[4]^4-87*c[5]^3*c[4]-12*c[5]*c[ 7]*c[4]-200*c[5]^3*c[4]^4*c[7]-200*c[6]*c[5]^2*c[4]^4*c[7]-200*c[5]^4* c[4]^3*c[6]-10*c[5]^2-10*c[4]^2+12*c[5]^3+12*c[4]^3+20*c[7]*c[6]*c[5]^ 3+15*c[5]*c[6]*c[7]*c[4]+12*c[6]*c[5]^2-15*c[6]*c[5]^3-1300*c[6]*c[7]* c[5]^3*c[4]^3+150*c[5]^4*c[7]*c[4]^2+12*c[7]*c[5]^2+20*c[7]*c[6]*c[4]^ 3+12*c[7]*c[4]^2-15*c[7]*c[4]^3+12*c[6]*c[4]^2-15*c[6]*c[4]^3-15*c[6]* c[7]*c[4]^2-87*c[5]*c[4]^3-72*c[5]*c[7]*c[4]^2-72*c[5]*c[6]*c[4]^2+60* c[5]*c[4]^2-12*c[5]*c[6]*c[4]+10*c[5]*c[4]-150*c[5]*c[7]*c[6]*c[4]^3-1 5*c[7]*c[5]^3-660*c[5]^2*c[6]*c[7]*c[4]^2+90*c[6]*c[5]*c[7]*c[4]^2-15* c[6]*c[7]*c[5]^2+150*c[6]*c[5]^4*c[4]^2+930*c[6]*c[5]^2*c[7]*c[4]^3+93 0*c[6]*c[5]^3*c[7]*c[4]^2-200*c[5]^4*c[4]^3*c[7]+110*c[7]*c[5]*c[4]^3+ 110*c[6]*c[5]*c[4]^3+110*c[5]^3*c[6]*c[4]+110*c[5]^3*c[7]*c[4]+60*c[5] ^2*c[4]-150*c[5]^3*c[6]*c[7]*c[4]+550*c[5]^3*c[4]^2-690*c[5]^2*c[6]*c[ 4]^3+520*c[5]^2*c[7]*c[4]^2+520*c[5]^2*c[6]*c[4]^2-690*c[5]^2*c[7]*c[4 ]^3-690*c[5]^3*c[7]*c[4]^2-690*c[5]^3*c[6]*c[4]^2+150*c[5]^2*c[4]^4*c[ 7]+900*c[6]*c[5]^3*c[4]^3+900*c[5]^3*c[7]*c[4]^3-200*c[5]^4*c[7]*c[4]^ 2*c[6]-429*c[5]^2*c[4]^2-120*c[5]^4*c[4]^2-120*c[4]^4*c[5]^2+150*c[5]^ 4*c[4]^3-72*c[5]^2*c[6]*c[4]-72*c[5]^2*c[7]*c[4]+90*c[5]^2*c[6]*c[7]*c [4]+550*c[5]^2*c[4]^3+300*c[5]^3*c[4]^4*c[6]*c[7]+300*c[5]^4*c[4]^3*c[ 6]*c[7]-690*c[5]^3*c[4]^3-200*c[4]^4*c[6]*c[5]^3+150*c[5]^3*c[4]^4), a [8,3] = 3/4*(-1560*c[5]^2*c[6]*c[4]^4+585*c[6]*c[5]*c[4]^4-45*c[7]*c[5 ]*c[4]^4-12*c[5]^3*c[4]-450*c[4]^5*c[6]*c[5]^3+3*c[5]*c[7]*c[4]-150*c[ 5]^2*c[4]^5*c[7]-200*c[5]^3*c[4]^4*c[7]+300*c[6]*c[4]^5*c[5]*c[7]+2860 *c[6]*c[5]^2*c[4]^4*c[7]+160*c[5]^4*c[4]^3*c[6]-150*c[5]^4*c[4]^4*c[6] -4*c[5]^2+2*c[4]^2-40*c[7]*c[6]*c[5]^3+132*c[5]*c[6]*c[7]*c[4]-900*c[6 ]*c[5]^2*c[4]^5*c[7]-30*c[6]*c[5]^2+12*c[6]*c[5]-960*c[7]*c[4]^4*c[6]* c[5]+24*c[6]*c[5]^3+3140*c[6]*c[7]*c[5]^3*c[4]^3+6*c[7]*c[5]^2+20*c[7] *c[6]*c[4]^3-3*c[7]*c[4]^2-18*c[6]*c[7]*c[5]+15*c[6]*c[4]^2-12*c[6]*c[ 4]^3+9*c[6]*c[7]*c[4]-24*c[6]*c[7]*c[4]^2-54*c[5]*c[4]^3-12*c[5]*c[7]* c[4]^2+348*c[5]*c[6]*c[4]^2+9*c[5]*c[4]^2-87*c[5]*c[6]*c[4]-2*c[5]*c[4 ]-6*c[6]*c[4]+1035*c[5]*c[7]*c[6]*c[4]^3+1593*c[5]^2*c[6]*c[7]*c[4]^2+ 900*c[4]^5*c[6]*c[7]*c[5]^3-534*c[6]*c[5]*c[7]*c[4]^2+48*c[6]*c[7]*c[5 ]^2-60*c[6]*c[5]^4*c[4]^2-3060*c[6]*c[5]^2*c[7]*c[4]^3-1560*c[6]*c[5]^ 3*c[7]*c[4]^2-50*c[5]^4*c[4]^3*c[7]+80*c[7]*c[5]*c[4]^3-652*c[6]*c[5]* c[4]^3-208*c[5]^3*c[6]*c[4]+20*c[5]^3*c[7]*c[4]+54*c[5]^2*c[4]+360*c[5 ]^3*c[6]*c[7]*c[4]+82*c[5]^3*c[4]^2+1734*c[5]^2*c[6]*c[4]^3+388*c[5]^2 *c[7]*c[4]^2-944*c[5]^2*c[6]*c[4]^2-900*c[5]^2*c[7]*c[4]^3-135*c[5]^3* c[7]*c[4]^2+879*c[5]^3*c[6]*c[4]^2+660*c[5]^2*c[4]^4*c[7]-1680*c[6]*c[ 5]^3*c[4]^3+420*c[5]^3*c[7]*c[4]^3+100*c[5]^4*c[7]*c[4]^2*c[6]-246*c[5 ]^2*c[4]^2-405*c[4]^4*c[5]^2+30*c[5]^4*c[4]^3+240*c[5]^2*c[6]*c[4]-84* c[5]^2*c[7]*c[4]-396*c[5]^2*c[6]*c[7]*c[4]+562*c[5]^2*c[4]^3+30*c[4]^4 *c[5]+90*c[4]^5*c[5]^2+300*c[5]^4*c[4]^4*c[6]*c[7]-3000*c[5]^3*c[4]^4* c[6]*c[7]-300*c[5]^4*c[4]^3*c[6]*c[7]-255*c[5]^3*c[4]^3+480*c[5]^2*c[6 ]*c[4]^5+1540*c[4]^4*c[6]*c[5]^3-180*c[6]*c[5]*c[4]^5+120*c[5]^3*c[4]^ 4)/c[4]^2/(150*c[5]^2*c[6]*c[4]^4-87*c[5]^3*c[4]-12*c[5]*c[7]*c[4]-200 *c[5]^3*c[4]^4*c[7]-200*c[6]*c[5]^2*c[4]^4*c[7]-200*c[5]^4*c[4]^3*c[6] -10*c[5]^2-10*c[4]^2+12*c[5]^3+12*c[4]^3+20*c[7]*c[6]*c[5]^3+15*c[5]*c [6]*c[7]*c[4]+12*c[6]*c[5]^2-15*c[6]*c[5]^3-1300*c[6]*c[7]*c[5]^3*c[4] ^3+150*c[5]^4*c[7]*c[4]^2+12*c[7]*c[5]^2+20*c[7]*c[6]*c[4]^3+12*c[7]*c [4]^2-15*c[7]*c[4]^3+12*c[6]*c[4]^2-15*c[6]*c[4]^3-15*c[6]*c[7]*c[4]^2 -87*c[5]*c[4]^3-72*c[5]*c[7]*c[4]^2-72*c[5]*c[6]*c[4]^2+60*c[5]*c[4]^2 -12*c[5]*c[6]*c[4]+10*c[5]*c[4]-150*c[5]*c[7]*c[6]*c[4]^3-15*c[7]*c[5] ^3-660*c[5]^2*c[6]*c[7]*c[4]^2+90*c[6]*c[5]*c[7]*c[4]^2-15*c[6]*c[7]*c [5]^2+150*c[6]*c[5]^4*c[4]^2+930*c[6]*c[5]^2*c[7]*c[4]^3+930*c[6]*c[5] ^3*c[7]*c[4]^2-200*c[5]^4*c[4]^3*c[7]+110*c[7]*c[5]*c[4]^3+110*c[6]*c[ 5]*c[4]^3+110*c[5]^3*c[6]*c[4]+110*c[5]^3*c[7]*c[4]+60*c[5]^2*c[4]-150 *c[5]^3*c[6]*c[7]*c[4]+550*c[5]^3*c[4]^2-690*c[5]^2*c[6]*c[4]^3+520*c[ 5]^2*c[7]*c[4]^2+520*c[5]^2*c[6]*c[4]^2-690*c[5]^2*c[7]*c[4]^3-690*c[5 ]^3*c[7]*c[4]^2-690*c[5]^3*c[6]*c[4]^2+150*c[5]^2*c[4]^4*c[7]+900*c[6] *c[5]^3*c[4]^3+900*c[5]^3*c[7]*c[4]^3-200*c[5]^4*c[7]*c[4]^2*c[6]-429* c[5]^2*c[4]^2-120*c[5]^4*c[4]^2-120*c[4]^4*c[5]^2+150*c[5]^4*c[4]^3-72 *c[5]^2*c[6]*c[4]-72*c[5]^2*c[7]*c[4]+90*c[5]^2*c[6]*c[7]*c[4]+550*c[5 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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+20 8*c[6]^2*c[5]^3*c[7]*c[4]-225*c[5]^4*c[4]^3*c[7]+704*c[7]^2*c[5]^2*c[4 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4]^3*c[7]+57*c[7]^2*c[5]^2*c[4]^3-750*c[5]^4*c[7]^2*c[4]^4+10*c[7]*c[5 ]*c[4]^3+10*c[6]*c[5]*c[4]^3+24*c[7]^2*c[5]^2*c[6]*c[4]+12*c[6]*c[7]*c [4]^4+200*c[7]*c[5]^7*c[4]^3-110*c[5]^6*c[7]*c[4]+15*c[6]*c[5]^6+20*c[ 5]^3*c[6]*c[4]+20*c[5]^3*c[7]*c[4]+150*c[7]*c[6]*c[5]^5*c[4]-900*c[5]^ 5*c[4]^3*c[7]^2*c[6]-750*c[5]^5*c[4]^2*c[6]*c[7]-96*c[5]^3*c[6]*c[7]*c [4]+200*c[6]*c[5]^7*c[4]^3+20*c[5]^3*c[4]^2-36*c[5]^2*c[6]*c[4]^3-20*c [5]^2*c[7]*c[4]^2-20*c[5]^2*c[6]*c[4]^2-36*c[5]^2*c[7]*c[4]^3-24*c[5]^ 3*c[7]*c[4]^2-24*c[5]^3*c[6]*c[4]^2-410*c[5]^3*c[6]^2*c[4]^3-900*c[5]^ 5*c[4]^3*c[6]^2*c[7]-510*c[4]^3*c[7]^2*c[5]^2*c[6]^2-630*c[7]*c[5]^6*c [6]*c[4]^2-150*c[4]^5*c[5]^3*c[6]^2+72*c[7]*c[5]^4*c[4]-57*c[7]^2*c[5] ^4*c[4]+72*c[5]^2*c[4]^4*c[7]+57*c[6]^2*c[5]^2*c[4]^3+24*c[6]^2*c[5]^2 *c[4]^2+285*c[6]*c[5]^3*c[4]^3-24*c[6]^2*c[5]^3*c[4]+285*c[5]^3*c[7]*c [4]^3+270*c[5]^4*c[7]*c[4]^2*c[6]+12*c[7]*c[6]^2*c[4]^3+340*c[7]^2*c[5 ]^2*c[6]*c[4]^3-410*c[7]^2*c[5]^3*c[6]*c[4]^2-200*c[5]^5*c[4]^5*c[7]+7 0*c[4]^3*c[7]^2*c[5]*c[6]^2-42*c[7]^2*c[6]*c[5]*c[4]^3+150*c[5]^5*c[4] ^5-30*c[7]^2*c[5]^2*c[6]*c[4]^2+12*c[4]^4*c[5]^2+87*c[7]^2*c[5]^3*c[6] *c[4]+510*c[7]^2*c[5]^3*c[6]*c[4]^3+690*c[5]^3*c[6]^2*c[4]^4-342*c[5]^ 4*c[4]^3-15*c[6]^2*c[4]^3*c[7]^2+510*c[5]^3*c[6]^2*c[7]*c[4]^3-20*c[5] ^2*c[6]*c[7]*c[4]-410*c[6]^2*c[7]*c[5]^3*c[4]^2-70*c[6]^2*c[7]^2*c[4]* c[5]^3-10*c[5]^2*c[4]^3+24*c[7]^2*c[5]^2*c[4]^2-24*c[7]^2*c[5]^3*c[4]- 200*c[7]^2*c[5]^6*c[4]^2*c[6]+110*c[7]^2*c[4]*c[5]^5+1350*c[5]^5*c[4]^ 3*c[6]*c[7]+15*c[7]^2*c[4]^4*c[5]+300*c[6]^2*c[7]*c[5]^6*c[4]^3+510*c[ 7]^2*c[5]^3*c[6]^2*c[4]^2-550*c[5]^6*c[4]^2-600*c[7]*c[5]^6*c[4]^3+20* c[7]*c[6]^2*c[5]^5+570*c[5]^6*c[6]*c[4]^2+150*c[5]^2*c[6]*c[4]^5*c[7]^ 2+550*c[5]^4*c[4]^4+120*c[5]^7*c[4]^2+72*c[5]^4*c[4]*c[6]-40*c[5]^4*c[ 7]^2*c[4]*c[6]-57*c[5]^4*c[4]*c[7]*c[6]+150*c[6]^2*c[5]^6*c[4]^2+150*c [4]^5*c[6]^2*c[7]*c[5]^2-200*c[4]^5*c[6]^2*c[7]^2*c[5]^2-150*c[5]^7*c[ 7]*c[4]^2-110*c[6]^2*c[4]^4*c[5]^2+780*c[5]^5*c[4]^2*c[7]^2*c[6]-200*c [5]^5*c[4]^5*c[6]-600*c[5]^6*c[4]^3*c[6]-200*c[6]^2*c[7]*c[5]^6*c[4]^2 -300*c[5]^4*c[4]^5*c[6]^2*c[7]+90*c[7]*c[6]^2*c[5]*c[4]^4+200*c[6]*c[5 ]^7*c[7]*c[4]^2-300*c[6]*c[5]^7*c[7]*c[4]^3-180*c[5]^3*c[4]^4*c[6]^2*c [7]-570*c[5]^4*c[4]^4*c[6]*c[7]-540*c[5]^2*c[4]^4*c[7]*c[6]^2+660*c[5] ^3*c[4]^4*c[6]*c[7]+60*c[5]^4*c[4]^3*c[6]*c[7]+48*c[5]^3*c[4]^3-15*c[6 ]^2*c[5]^5-200*c[5]^6*c[6]^2*c[4]^3-15*c[5]^5*c[6]*c[7]-48*c[5]^5*c[4] +140*c[5]^5*c[6]*c[4]^2+200*c[5]^4*c[4]^5*c[6]^2+300*c[6]^2*c[7]^2*c[5 ]^3*c[4]^5-300*c[6]*c[7]^2*c[5]^4*c[4]^5+700*c[7]*c[5]^6*c[4]^3*c[6]-7 50*c[5]^4*c[4]^4*c[6]^2+20*c[5]^5*c[7]^2*c[6]+1100*c[7]^2*c[5]^4*c[6]^ 2*c[4]^3-930*c[7]^2*c[5]^4*c[6]^2*c[4]^2+300*c[7]^2*c[5]^6*c[4]^3*c[6] -20*c[5]^6*c[7]*c[6]-180*c[5]^3*c[4]^4*c[7]^2*c[6]-750*c[5]^4*c[4]^3*c [7]^2*c[6]-540*c[5]^2*c[4]^4*c[7]^2*c[6]-750*c[5]^4*c[4]^3*c[6]^2*c[7] -150*c[7]^2*c[4]^4*c[6]^2*c[5]-1100*c[7]^2*c[4]^4*c[5]^3*c[6]^2+930*c[ 7]^2*c[4]^4*c[6]^2*c[5]^2+150*c[5]^6*c[4]*c[7]*c[6]-110*c[5]^6*c[4]*c[ 6]-150*c[5]^7*c[6]*c[4]^2+570*c[5]^6*c[4]^3-150*c[5]^7*c[4]^3+15*c[7]^ 2*c[6]^2*c[5]^3-440*c[4]^4*c[6]*c[5]^3+750*c[5]^5*c[6]^2*c[4]^3-200*c[ 7]^2*c[5]^6*c[4]^3+12*c[5]^4*c[7]^2+200*c[5]^4*c[4]^5*c[6]*c[7]-20*c[5 ]^4*c[4]-87*c[5]^3*c[4]^4-30*c[5]^5*c[4]*c[6]+110*c[5]^5*c[6]^2*c[4]), a[9,7] = -1/60*(10*c[5]*c[6]*c[4]-5*c[6]*c[5]-5*c[6]*c[4]+3*c[6]-5*c[ 5]*c[4]+3*c[5]+3*c[4]-2)/c[7]/(-c[5]*c[6]*c[4]+c[5]*c[6]*c[7]*c[4]+c[5 ]*c[7]*c[4]-c[7]^2*c[5]*c[4]+c[6]*c[7]*c[4]-c[6]*c[7]^2*c[4]-c[7]^2*c[ 4]+c[7]^3*c[4]+c[6]*c[7]*c[5]-c[6]*c[7]^2*c[5]-c[7]^2*c[5]+c[7]^3*c[5] -c[7]^2*c[6]+c[7]^3*c[6]+c[7]^3-c[7]^4), a[9,2] = 0, a[9,3] = 0, a[9,5 ] = -1/60*(10*c[6]*c[7]*c[4]-5*c[6]*c[4]-5*c[7]*c[6]+3*c[6]-5*c[7]*c[4 ]+3*c[4]-2+3*c[7])/c[5]/(c[5]^3*c[4]-c[5]^2*c[4]+c[5]*c[7]*c[4]-c[5]^2 *c[7]*c[4]+c[5]*c[6]*c[4]-c[5]^2*c[6]*c[4]-c[6]*c[7]*c[4]+c[5]*c[6]*c[ 7]*c[4]-c[5]^4+c[5]^3-c[7]*c[5]^2+c[7]*c[5]^3-c[6]*c[5]^2+c[6]*c[5]^3+ c[6]*c[7]*c[5]-c[6]*c[7]*c[5]^2), a[9,8] = 1/60*(-20*c[5]*c[6]*c[4]+30 *c[5]*c[6]*c[7]*c[4]+15*c[6]*c[5]-20*c[6]*c[7]*c[5]+15*c[6]*c[4]-20*c[ 6]*c[7]*c[4]-12*c[6]+15*c[7]*c[6]+15*c[5]*c[4]-20*c[5]*c[7]*c[4]-12*c[ 5]+15*c[7]*c[5]-12*c[4]+15*c[7]*c[4]+10-12*c[7])/(-c[4]+c[7]*c[4]+c[6] *c[4]-c[6]*c[7]*c[4]+c[5]*c[4]-c[5]*c[7]*c[4]-c[5]*c[6]*c[4]+c[5]*c[6] *c[7]*c[4]+1-c[7]-c[6]+c[7]*c[6]-c[5]+c[7]*c[5]+c[6]*c[5]-c[6]*c[7]*c[ 5]), a[9,1] = 1/60*(30*c[5]*c[6]*c[7]*c[4]-10*c[5]*c[6]*c[4]-10*c[6]*c [7]*c[5]+5*c[6]*c[5]-10*c[6]*c[7]*c[4]+5*c[6]*c[4]+5*c[7]*c[6]-3*c[6]- 10*c[5]*c[7]*c[4]+5*c[5]*c[4]+5*c[7]*c[5]-3*c[5]+5*c[7]*c[4]-3*c[4]+2- 3*c[7])/c[5]/c[6]/c[7]/c[4], a[9,4] = 1/60*(-2+3*c[5]-5*c[7]*c[6]+3*c[ 7]+3*c[6]+10*c[6]*c[7]*c[5]-5*c[6]*c[5]-5*c[7]*c[5])/c[4]/(-c[6]*c[4]^ 3-c[4]^3+c[4]^4+c[7]*c[4]^2+c[6]*c[4]^2-c[6]*c[7]*c[4]+c[6]*c[7]*c[4]^ 2-c[5]*c[4]^3+c[5]*c[4]^2-c[7]*c[4]^3-c[5]*c[7]*c[4]+c[5]*c[7]*c[4]^2- c[5]*c[6]*c[4]+c[5]*c[6]*c[4]^2+c[6]*c[7]*c[5]-c[5]*c[6]*c[7]*c[4]), a [9,6] = 1/60*(10*c[5]*c[7]*c[4]-5*c[5]*c[4]-5*c[7]*c[5]+3*c[5]-5*c[7]* c[4]+3*c[4]-2+3*c[7])/c[6]/(c[6]^2*c[4]*c[5]-c[5]*c[6]*c[4]+c[5]*c[7]* c[4]-c[5]*c[6]*c[7]*c[4]-c[6]^3*c[4]+c[6]^2*c[4]-c[6]*c[7]*c[4]+c[6]^2 *c[7]*c[4]-c[6]^3*c[5]+c[6]^2*c[5]-c[6]*c[7]*c[5]+c[6]^2*c[7]*c[5]+c[6 ]^4-c[6]^3+c[7]*c[6]^2-c[6]^3*c[7])\}:" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 33 "#======================== ========" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 22 "We specify the nodes " }{XPPEDIT 18 0 "c[6] = 25/33;" "6#/&%\"cG6 #\"\"'*&\"#D\"\"\"\"#L!\"\"" }{TEXT -1 7 " and " }{XPPEDIT 18 0 "c[7 ] = 1728/1729;" "6#/&%\"cG6#\"\"(*&\"%G<\"\"\"\"%H " 0 "" {MPLTEXT 1 0 75 "eA := \+ \{c[6]=25/33,c[7]=1728/1729\}:\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 16924 "eB := \{a[8,7] = -8936757492481/2980584*(85*c[5]*c[4]-26*c[5]-26*c[4] +9)*(c[4]-1)*(-1+c[5])/(1729*c[5]-1728)/(1729*c[4]-1728)/(146875*c[5]* c[4]-44949*c[5]-44949*c[4]+15582), a[7,2] = 0, a[8,2] = 0, a[6,2] = 0, a[4,2] = 0, a[5,2] = 0, a[9,2] = 0, `b*`[9] = 1/10*(1250*c[5]^2*c[4]^ 2-1165*c[5]^2*c[4]+250*c[5]^2-1165*c[5]*c[4]^2+1205*c[5]*c[4]-283*c[5] +250*c[4]^2-283*c[4]+75)/(-258*c[5]-258*c[4]+200*c[5]^2+200*c[4]^2-832 *c[5]*c[4]^2+997*c[5]*c[4]-832*c[5]^2*c[4]+750*c[5]^2*c[4]^2+75), a[9, 1] = 1/2592000*(578795*c[5]*c[4]-101926*c[5]-101926*c[4]+29367)/c[5]/c [4], a[8,6] = -143748/344975*(118722445*c[5]*c[4]+23744555-47489044*c[ 4]-47489044*c[5])*(c[4]-1)*(-1+c[5])/(33*c[4]-25)/(33*c[5]-25)/(146875 *c[5]*c[4]-44949*c[5]-44949*c[4]+15582), `b*`[4] = 1/60*(22595283*c[5] +11583258*c[4]-46857966*c[5]^2+29363000*c[5]^3-8977800*c[4]^2+66704648 *c[5]*c[4]^2-122144080*c[5]^3*c[4]-82636479*c[5]*c[4]+183718183*c[5]^2 *c[4]+110096250*c[5]^3*c[4]^2-155795830*c[5]^2*c[4]^2-3367575)/(445824 *c[5]+575499*c[4]-345600*c[5]^2-791682*c[4]^2+3161509*c[5]*c[4]^2+3458 00*c[4]^3-2168898*c[5]*c[4]+1783496*c[5]^2*c[4]+1296750*c[5]^2*c[4]^3- 1438528*c[5]*c[4]^3-2734528*c[5]^2*c[4]^2-129600)/(33*c[4]-25)/(-c[4]+ c[5])/c[4], c[8] = 1, b[1] = 1/2592000*(578795*c[5]*c[4]-101926*c[5]-1 01926*c[4]+29367)/c[5]/c[4], c[9] = 1, `b*`[7] = -8936757492481/143068 0320*(-12141*c[5]-12141*c[4]+22674*c[5]^2-13000*c[5]^3+22674*c[4]^2-15 1077*c[5]*c[4]^2-13000*c[4]^3+90080*c[5]^3*c[4]+77001*c[5]*c[4]-151077 *c[5]^2*c[4]-194550*c[5]^2*c[4]^3-194550*c[5]^3*c[4]^2+90080*c[5]*c[4] ^3+311980*c[5]^2*c[4]^2+127500*c[5]^3*c[4]^3+2025)/(-994462272*c[5]-99 4462272*c[4]+1368026496*c[5]^2-597542400*c[5]^3+1368026496*c[4]^2-6831 905730*c[5]*c[4]^2-597542400*c[4]^3+3083664584*c[5]^3*c[4]+4742893515* c[5]*c[4]-6831905730*c[5]^2*c[4]-4727998912*c[5]^2*c[4]^3-4727998912*c [5]^3*c[4]^2+3083664584*c[5]*c[4]^3+10191513445*c[5]^2*c[4]^2+22420807 50*c[5]^3*c[4]^3+223948800), `b*`[3] = 0, c[6] = 25/33, c[7] = 1728/17 29, a[7,5] = 864/8936757492481*(6442260480000*c[5]-3221130240000*c[4]- 28538486998272*c[5]^2+39117777909696*c[5]^3+13218551872128*c[4]^2+1766 98175206562*c[5]*c[4]^2-128957119200000*c[4]^5*c[5]+646065496860420*c[ 4]^5*c[5]^2-17022339734400*c[5]^4+6078751031808*c[4]^4-16076577288960* c[4]^3-384938003176717*c[5]^3*c[4]-47620222835328*c[5]*c[4]-9430146955 23810*c[4]^5*c[5]^3+262818882502652*c[5]^2*c[4]+2042367141068623*c[5]^ 2*c[4]^3-1890796698177644*c[4]^4*c[5]^2-701874994525887*c[5]^4*c[4]^2+ 1543842294530889*c[5]^3*c[4]^2-369332231948572*c[5]*c[4]^3-10319203268 17598*c[5]^2*c[4]^2+425928080077500*c[5]^4*c[4]^5+172969146904554*c[5] ^4*c[4]+362771512736104*c[4]^4*c[5]-1259234027702560*c[5]^4*c[4]^4-303 6120904228746*c[5]^3*c[4]^3+2781115815816685*c[5]^3*c[4]^4+13792338715 12315*c[5]^4*c[4]^3)/c[5]/(225*c[5]^3+450*c[5]*c[4]^2-6500*c[4]^5*c[5] ^2-947*c[5]^4+650*c[4]^4-225*c[4]^3+28050*c[5]^6*c[4]^3-106*c[5]^3*c[4 ]+29830*c[4]^5*c[5]^3-450*c[5]^2*c[4]-7324*c[5]^2*c[4]^3+42453*c[4]^4* c[5]^2-42948*c[5]^4*c[4]^2+6400*c[5]^3*c[4]^2+997*c[5]*c[4]^3-594*c[5] ^2*c[4]^2-28050*c[5]^4*c[4]^5-8580*c[5]^6*c[4]^2-7953*c[5]^5*c[4]+6949 *c[5]^4*c[4]+858*c[5]^5-6883*c[4]^4*c[5]+92730*c[5]^4*c[4]^4+8448*c[5] ^3*c[4]^3-115790*c[5]^3*c[4]^4+70250*c[5]^4*c[4]^3+52040*c[5]^5*c[4]^2 -113980*c[5]^5*c[4]^3), `b*`[2] = 0, `b*`[6] = 395307/6899500*(-198339 9*c[5]-1983399*c[4]+3262848*c[5]^2-1726500*c[5]^3+3262848*c[4]^2-19097 727*c[5]*c[4]^2-1726500*c[4]^3+10636490*c[5]^3*c[4]+10912560*c[5]*c[4] -19097727*c[5]^2*c[4]-20843560*c[5]^2*c[4]^3-20843560*c[5]^3*c[4]^2+10 636490*c[5]*c[4]^3+35467930*c[5]^2*c[4]^2+12952500*c[5]^3*c[4]^3+38835 0)/(-258*c[5]-258*c[4]+200*c[5]^2+200*c[4]^2-832*c[5]*c[4]^2+997*c[5]* c[4]-832*c[5]^2*c[4]+750*c[5]^2*c[4]^2+75)/(1089*c[5]*c[4]-825*c[4]-82 5*c[5]+625), a[9,3] = 0, b[3] = 0, `b*`[8] = 0, a[5,3] = -3/4*c[5]^2*( 2*c[5]-3*c[4])/c[4]^2, a[5,1] = 1/4*c[5]*(2*c[5]^2-5*c[5]*c[4]+4*c[4]^ 2)/c[4]^2, b[2] = 0, a[7,4] = -864/8936757492481*(19326781440000*c[5]^ 2-66506510073600*c[5]^3-17022339734400*c[4]^5+6442260480000*c[4]^2+160 358327379072*c[5]*c[4]^2-2089365078727472*c[4]^5*c[5]+6884753795765851 *c[4]^5*c[5]^2+1377745679207244*c[4]^6*c[5]+51582847680000*c[5]^4+3911 7777909696*c[4]^4-1277784240232500*c[5]^3*c[4]^7-28538486998272*c[4]^3 +676249438279092*c[5]^3*c[4]-22547911680000*c[5]*c[4]-733904923199135* c[4]^5*c[5]^3-128253488268672*c[5]^2*c[4]+335236127580028*c[5]^2*c[4]^ 3-3582396151411565*c[4]^4*c[5]^2+3051999063602220*c[5]^4*c[4]^2-306537 9583778860*c[5]^3*c[4]^2-621927542320900*c[5]*c[4]^3+317535736782062*c [5]^2*c[4]^2-6199523041943140*c[5]^4*c[4]^5-586159537919592*c[5]^4*c[4 ]-943014695523810*c[5]^5*c[4]^4+3523586844277500*c[5]^3*c[4]^6+1277784 240232500*c[5]^4*c[4]^6+425928080077500*c[5]^5*c[4]^5+1416702169963080 *c[4]^7*c[5]^2-340446794688000*c[5]*c[4]^7-5262936025374930*c[4]^6*c[5 ]^2+1536191183207345*c[4]^4*c[5]+10497637430591065*c[5]^4*c[4]^4+67417 12261776987*c[5]^3*c[4]^3-5797912051119000*c[5]^3*c[4]^4-8093380020786 372*c[5]^4*c[4]^3-128957119200000*c[5]^5*c[4]^2+646065496860420*c[5]^5 *c[4]^3)/(-9*c[5]^3-18*c[5]*c[4]^2+260*c[4]^5*c[5]^2+26*c[5]^4-26*c[4] ^4+9*c[4]^3+28*c[5]^3*c[4]-850*c[4]^5*c[5]^3+18*c[5]^2*c[4]+256*c[5]^2 *c[4]^3-1380*c[4]^4*c[5]^2+1380*c[5]^4*c[4]^2-256*c[5]^3*c[4]^2-28*c[5 ]*c[4]^3-241*c[5]^4*c[4]+241*c[4]^4*c[5]+2810*c[5]^3*c[4]^4-2810*c[5]^ 4*c[4]^3-260*c[5]^5*c[4]^2+850*c[5]^5*c[4]^3)/(33*c[4]-25)/c[4]^2, a[7 ,3] = 3888/8936757492481*(-257690419200*c[5]+128845209600*c[4]+8867534 67648*c[5]^2-687771302400*c[5]^3-443376733824*c[4]^2-8159773666226*c[5 ]*c[4]^2+5158284768000*c[4]^5*c[5]-21465184393380*c[4]^5*c[5]^2+343885 651200*c[4]^3+6988652400184*c[5]^3*c[4]+1989519249024*c[5]*c[4]+193603 67276250*c[4]^5*c[5]^3-8404384614812*c[5]^2*c[4]-76165232941294*c[5]^2 *c[4]^3+70357664625605*c[4]^4*c[5]^2+1719428256000*c[5]^4*c[4]^2-32287 060524150*c[5]^3*c[4]^2+17929568853019*c[5]*c[4]^3+35943063943565*c[5] ^2*c[4]^2-16967143042534*c[4]^4*c[5]+6453455758750*c[5]^4*c[4]^4+72146 574394155*c[5]^3*c[4]^3-67340980410340*c[5]^3*c[4]^4-7155061464460*c[5 ]^4*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)/(85*c[5]*c[4]-26 *c[5]-26*c[4]+9), a[5,4] = c[5]^2*(-c[4]+c[5])/c[4]^2, a[2,1] = c[2], \+ a[3,1] = -2/9*c[4]*(c[4]-3*c[2])/c[2], a[3,2] = 2/9*c[4]^2/c[2], a[7,1 ] = 432/223418937312025*(19326781440000*c[5]^2-66506510073600*c[5]^3-1 2157502063616*c[4]^5+6442260480000*c[4]^2+136997484541056*c[5]*c[4]^2- 720968726347176*c[4]^5*c[5]+2232490363750632*c[4]^5*c[5]^2+25791423840 0000*c[4]^6*c[5]+51582847680000*c[5]^4+32153154577920*c[4]^4-264371037 44256*c[4]^3+610899153008820*c[5]^3*c[4]-22547911680000*c[5]*c[4]+9421 45541690380*c[4]^5*c[5]^3-109144537347456*c[5]^2*c[4]+180480535808179* c[5]^2*c[4]^3-1611845180182958*c[4]^4*c[5]^2+2548917965090774*c[5]^4*c [4]^2-2483807388342385*c[5]^3*c[4]^2-408170504832964*c[5]*c[4]^3+21737 3396544686*c[5]^2*c[4]^2-3944640449386910*c[5]^4*c[4]^5-53630643207741 6*c[5]^4*c[4]-1028443315884530*c[5]^5*c[4]^4+423584229834220*c[5]^3*c[ 4]^6+575472474549700*c[5]^4*c[4]^6+575472474549700*c[5]^5*c[4]^5-95168 4199032840*c[4]^6*c[5]^2+758968230066740*c[4]^4*c[5]+7392126190256380* c[5]^4*c[4]^4+5021920183081242*c[5]^3*c[4]^3-4361881757099171*c[5]^3*c [4]^4-6223543176691243*c[5]^4*c[4]^3-128957119200000*c[5]^5*c[4]^2+658 204630470660*c[5]^5*c[4]^3)/(-9*c[5]^2+26*c[5]^3-9*c[4]^2+54*c[5]*c[4] ^2+26*c[4]^3-241*c[5]^3*c[4]+9*c[5]*c[4]+54*c[5]^2*c[4]+1380*c[5]^2*c[ 4]^3-260*c[4]^4*c[5]^2-260*c[5]^4*c[4]^2+1380*c[5]^3*c[4]^2-241*c[5]*c [4]^3-497*c[5]^2*c[4]^2-3070*c[5]^3*c[4]^3+850*c[5]^3*c[4]^4+850*c[5]^ 4*c[4]^3)/c[5]/c[4]^2, b[6] = 13045131/55196000*(8635*c[5]*c[4]-3453*c [5]-3453*c[4]+1726)/(33*c[5]-25)/(33*c[4]-25), a[4,1] = 1/4*c[4], a[6, 5] = 25/2371842*(981387*c[5]^2*c[4]^2-41250*c[4]^2+15625*c[4]-329175*c [5]^2*c[4]-689025*c[5]*c[4]^2+27225*c[4]^3-412500*c[4]^4*c[5]+895313*c [5]*c[4]^3-1229250*c[5]^2*c[4]^3+228750*c[5]*c[4]+544500*c[4]^4*c[5]^2 +41250*c[5]^2-31250*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[6,4] = -25/2371842*(-93750*c[5]^2+16 5000*c[5]^3-31250*c[4]^2-498750*c[5]*c[4]^2+825000*c[4]^5*c[5]-1633500 *c[4]^5*c[5]^2+41250*c[4]^3-1289475*c[5]^3*c[4]+109375*c[5]*c[4]+52125 0*c[5]^2*c[4]-514335*c[5]^2*c[4]^3+3267000*c[4]^4*c[5]^2-412500*c[5]^4 *c[4]^2+4508361*c[5]^3*c[4]^2+1598325*c[5]*c[4]^3-1299825*c[5]^2*c[4]^ 2-2117326*c[4]^4*c[5]-5321250*c[5]^3*c[4]^3+1633500*c[5]^3*c[4]^4+5445 00*c[5]^4*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, c[3] = 2/3*c[4], a[4,3] = 3/4*c[4], b[5] = 1 /60*(101926*c[4]-29367)/(-c[4]+c[5])/c[5]/(57057*c[5]^3-157306*c[5]^2+ 143449*c[5]-43200), b[4] = -1/60*(101926*c[5]-29367)/c[4]/(157306*c[4] ^3-57057*c[4]^4-143449*c[4]^2+43200*c[4]+57057*c[5]*c[4]^3-157306*c[5] *c[4]^2+143449*c[5]*c[4]-43200*c[5]), a[9,4] = -1/60*(101926*c[5]-2936 7)/c[4]/(157306*c[4]^3-57057*c[4]^4-143449*c[4]^2+43200*c[4]+57057*c[5 ]*c[4]^3-157306*c[5]*c[4]^2+143449*c[5]*c[4]-43200*c[5]), a[9,8] = 1/4 80*(146875*c[5]*c[4]-44949*c[5]-44949*c[4]+15582)/(-c[4]+c[5]*c[4]+1-c [5]), a[9,7] = -15451653704499649/1430680320*(85*c[5]*c[4]-26*c[5]-26* c[4]+9)/(2989441*c[5]*c[4]-2987712*c[4]-2987712*c[5]+2985984), a[6,1] \+ = 25/4743684*(93750*c[5]^2-165000*c[5]^3+31250*c[4]^2+540000*c[5]*c[4] ^2-825000*c[4]^5*c[5]+1089000*c[4]^5*c[5]^2+54450*c[4]^4-82500*c[4]^3+ 1371150*c[5]^3*c[4]-109375*c[5]*c[4]+1437480*c[4]^5*c[5]^3-562500*c[5] ^2*c[4]-1396167*c[5]^2*c[4]^3-358512*c[4]^4*c[5]^2+412500*c[5]^4*c[4]^ 2-5142159*c[5]^3*c[4]^2-1323600*c[5]*c[4]^3+1520100*c[5]^2*c[4]^2+1646 878*c[4]^4*c[5]+1437480*c[5]^4*c[4]^4+9046488*c[5]^3*c[4]^3-7307190*c[ 5]^3*c[4]^4-1361250*c[5]^4*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), b[7] = -15451653704499649/1430680320*(85*c[5]*c[4]-26*c[5 ]-26*c[4]+9)/(1729*c[4]-1728)/(1729*c[5]-1728), a[6,3] = 25/527076*(10 58663*c[5]*c[4]^3-744300*c[5]*c[4]^2+215000*c[5]*c[4]-31250*c[5]+15625 *c[4]-137500*c[5]^3*c[4]^2-2183500*c[5]^2*c[4]^3+1557237*c[5]^2*c[4]^2 -438900*c[5]^2*c[4]+55000*c[5]^2-412500*c[4]^4*c[5]-27500*c[4]^2+27225 0*c[5]^3*c[4]^3+816750*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*`[1] = 1/2592000*(-22595283*c[5]-22595283*c[4]+46857966* c[5]^2-29363000*c[5]^3+46857966*c[4]^2-400290459*c[5]*c[4]^2+3367575-2 9363000*c[4]^3+267266080*c[5]^3*c[4]+174938295*c[5]*c[4]-400290459*c[5 ]^2*c[4]-713797770*c[5]^2*c[4]^3-713797770*c[5]^3*c[4]^2+267266080*c[5 ]*c[4]^3+1001375060*c[5]^2*c[4]^2+544192500*c[5]^3*c[4]^3)/(-258*c[5]- 258*c[4]+200*c[5]^2+200*c[4]^2-832*c[5]*c[4]^2+997*c[5]*c[4]-832*c[5]^ 2*c[4]+750*c[5]^2*c[4]^2+75)/c[5]/c[4], a[8,5] = 1/2*(-3728160000*c[5] +1864080000*c[4]+20243987660*c[5]^2-39154400586*c[5]^3-7648046640*c[4] ^2-94595786671*c[5]*c[4]^2+74584800000*c[4]^5*c[5]-448266390980*c[4]^5 *c[5]^2+32489463976*c[5]^4-3514027552*c[4]^4+9297994192*c[4]^3+3748510 67425*c[5]^3*c[4]+25691262290*c[5]*c[4]+919125720500*c[4]^5*c[5]^3-179 644305899*c[5]^2*c[4]-1395467936760*c[5]^2*c[4]^3+1303761550133*c[4]^4 *c[5]^2+1299510061240*c[5]^4*c[4]^2-1490504188187*c[5]^3*c[4]^2+204406 559168*c[5]*c[4]^3+699373095846*c[5]^2*c[4]^2-791787727020*c[5]^4*c[4] ^5+100092185193*c[5]^5*c[4]-322854289009*c[5]^4*c[4]-728478065770*c[5] ^5*c[4]^4+246343597500*c[5]^5*c[4]^5-9850891050*c[5]^5-206358674787*c[ 4]^4*c[5]+2337466376155*c[5]^4*c[4]^4+2938558959027*c[5]^3*c[4]^3-2702 877158179*c[5]^3*c[4]^4-2554823885342*c[5]^4*c[4]^3-406135135588*c[5]^ 5*c[4]^2+798028309715*c[5]^5*c[4]^3)/c[5]/(-673142400*c[5]^3-134628480 0*c[5]*c[4]^2+19417968000*c[4]^5*c[5]^2+2564655093*c[5]^6+3503876718*c [5]^4-1941796800*c[4]^4+673142400*c[4]^3-424294675930*c[5]^6*c[4]^3-10 27102236*c[5]^3*c[4]-108510923010*c[4]^5*c[5]^3+1346284800*c[5]^2*c[4] +24910502181*c[5]^2*c[4]^3-147380608774*c[4]^4*c[5]^2+147506135596*c[5 ]^4*c[4]^2-20933177148*c[5]^3*c[4]^2-23768177433*c[5]^6*c[4]-256465509 30*c[5]^7*c[4]^2+83802468750*c[5]^7*c[4]^3-3659137518*c[5]*c[4]^3+3124 159836*c[5]^2*c[4]^2+172887269680*c[5]^4*c[4]^5+181172678790*c[5]^6*c[ 4]^2+44530343632*c[5]^5*c[4]-21084043659*c[5]^4*c[4]+277053957180*c[5] ^5*c[4]^4-83802468750*c[5]^5*c[4]^5-5395154475*c[5]^5+22501873101*c[4] ^4*c[5]-622894835040*c[5]^4*c[4]^4-47220525547*c[5]^3*c[4]^3+472684144 893*c[5]^3*c[4]^4-184468665972*c[5]^4*c[4]^3-283865269488*c[5]^5*c[4]^ 2+550233079260*c[5]^5*c[4]^3), a[8,4] = -1/2*(246343597500*c[5]^5*c[4] ^6-197017821000*c[5]*c[4]^8-11184480000*c[5]^2-739030792500*c[4]^8*c[5 ]^3+739030792500*c[5]^4*c[4]^7+38481091200*c[5]^3+32489463976*c[4]^5-3 728160000*c[4]^2-105851175010*c[5]*c[4]^2+2098109212403*c[4]^5*c[5]-60 56001918800*c[4]^5*c[5]^2-2006415040797*c[4]^6*c[5]-29833920000*c[5]^4 -9850891050*c[4]^6-39154400586*c[4]^4+2776536757500*c[5]^3*c[4]^7+2024 3987660*c[4]^3-429783808142*c[5]^3*c[4]+13048560000*c[5]*c[4]-29337679 57382*c[4]^5*c[5]^3+85413299460*c[5]^2*c[4]-9948841212*c[5]^2*c[4]^3+2 266302492459*c[4]^4*c[5]^2-2104242114191*c[5]^4*c[4]^2+2165122945892*c [5]^3*c[4]^2+452718215935*c[5]*c[4]^3-258042586171*c[5]^2*c[4]^2+96575 68426895*c[5]^4*c[4]^5+368852159048*c[5]^4*c[4]+819736777860*c[4]^8*c[ 5]^2+919125720500*c[5]^5*c[4]^4-2460133627315*c[5]^3*c[4]^6-4324751601 130*c[5]^4*c[4]^6-791787727020*c[5]^5*c[4]^5-3864922292710*c[4]^7*c[5] ^2+994317617310*c[5]*c[4]^7+7028647549114*c[4]^6*c[5]^2-1248909568841* c[4]^4*c[5]-10753033388151*c[5]^4*c[4]^4-5675354679414*c[5]^3*c[4]^3+7 257930070161*c[5]^3*c[4]^4+6446409645029*c[5]^4*c[4]^3+74584800000*c[5 ]^5*c[4]^2-448266390980*c[5]^5*c[4]^3)/(-15582*c[5]^3-31164*c[5]*c[4]^ 2+449490*c[4]^5*c[5]^2+44949*c[5]^4-44949*c[4]^4+15582*c[4]^3+48543*c[ 5]^3*c[4]-1468750*c[4]^5*c[5]^3+31164*c[5]^2*c[4]+443404*c[5]^2*c[4]^3 -2385540*c[4]^4*c[5]^2+2385540*c[5]^4*c[4]^2-443404*c[5]^3*c[4]^2-4854 3*c[5]*c[4]^3-416569*c[5]^4*c[4]+416569*c[4]^4*c[5]+4855740*c[5]^3*c[4 ]^4-4855740*c[5]^4*c[4]^3-449490*c[5]^5*c[4]^2+1468750*c[5]^5*c[4]^3)/ (-100249*c[4]+57057*c[4]^2+43200)/c[4]^2, `b*`[5] = -1/60*(11583258*c[ 5]+22595283*c[4]-8977800*c[5]^2-46857966*c[4]^2+183718183*c[5]*c[4]^2+ 29363000*c[4]^3-82636479*c[5]*c[4]+66704648*c[5]^2*c[4]+110096250*c[5] ^2*c[4]^3-122144080*c[5]*c[4]^3-155795830*c[5]^2*c[4]^2-3367575)/(5754 99*c[5]+445824*c[4]-791682*c[5]^2+345800*c[5]^3-345600*c[4]^2+1783496* c[5]*c[4]^2-1438528*c[5]^3*c[4]-2168898*c[5]*c[4]+3161509*c[5]^2*c[4]+ 1296750*c[5]^3*c[4]^2-2734528*c[5]^2*c[4]^2-129600)/(33*c[5]-25)/(-c[4 ]+c[5])/c[5], a[9,6] = 13045131/55196000*(8635*c[5]*c[4]-3453*c[5]-345 3*c[4]+1726)/(1089*c[5]*c[4]-825*c[4]-825*c[5]+625), a[8,3] = 3/4*(-25 8900*c[5]+129450*c[4]+890766*c[5]^2-690600*c[5]^3-445383*c[4]^2-819727 5*c[5]*c[4]^2+5179500*c[4]^5*c[5]-21550470*c[4]^5*c[5]^2+345300*c[4]^3 +7016996*c[5]^3*c[4]+1998783*c[5]*c[4]+19428750*c[4]^5*c[5]^3-8442138* c[5]^2*c[4]-76495416*c[5]^2*c[4]^3+70648755*c[4]^4*c[5]^2+1726500*c[5] ^4*c[4]^2-32416791*c[5]^3*c[4]^2+18010142*c[5]*c[4]^3+36102490*c[5]^2* c[4]^2-17039745*c[4]^4*c[5]+6476250*c[5]^4*c[4]^4+72430545*c[5]^3*c[4] ^3-67591460*c[5]^3*c[4]^4-7183490*c[5]^4*c[4]^3)/c[4]^2/(-15582*c[5]^2 +44949*c[5]^3-15582*c[4]^2+93492*c[5]*c[4]^2+44949*c[4]^3-416569*c[5]^ 3*c[4]+15582*c[5]*c[4]+93492*c[5]^2*c[4]+2385540*c[5]^2*c[4]^3-449490* c[4]^4*c[5]^2-449490*c[5]^4*c[4]^2+2385540*c[5]^3*c[4]^2-416569*c[5]*c [4]^3-859973*c[5]^2*c[4]^2-5305230*c[5]^3*c[4]^3+1468750*c[5]^3*c[4]^4 +1468750*c[5]^4*c[4]^3), a[9,5] = 1/60*(101926*c[4]-29367)/c[5]/(-5705 7*c[5]^3*c[4]+157306*c[5]^2*c[4]-143449*c[5]*c[4]+43200*c[4]+57057*c[5 ]^4-157306*c[5]^3+143449*c[5]^2-43200*c[5]), a[7,6] = -428452988832/22 3418937312025*(5*c[5]*c[4]+1-2*c[4]-2*c[5])*(1729*c[4]-1728)*(1729*c[5 ]-1728)/(85*c[5]*c[4]-26*c[5]-26*c[4]+9)/(33*c[4]-25)/(33*c[5]-25), b[ 8] = 1/480*(146875*c[5]*c[4]-44949*c[5]-44949*c[4]+15582)/(c[4]-1)/(-1 +c[5]), a[8,1] = 1/5400*(349515000*c[5]^2-1202534100*c[5]^3-219626722* c[4]^5+116505000*c[4]^2+2477329965*c[5]*c[4]^2-13034689611*c[4]^5*c[5] +40360389093*c[4]^5*c[5]^2+4661550000*c[4]^6*c[5]+932310000*c[5]^4+581 124637*c[4]^4-478002915*c[4]^3+11045700937*c[5]^3*c[4]-407767500*c[5]* c[4]+17016577580*c[4]^5*c[5]^3-1973825865*c[5]^2*c[4]+3262944656*c[5]^ 2*c[4]^3-29146331694*c[4]^4*c[5]^2+46063748739*c[5]^4*c[4]^2-449082170 94*c[5]^3*c[4]^2-7380753210*c[5]*c[4]^3+3931346340*c[5]^2*c[4]^2-71233 419690*c[5]^4*c[4]^5-9692571322*c[5]^4*c[4]-18576143070*c[5]^5*c[4]^4+ 7652669430*c[5]^3*c[4]^6+10388940350*c[5]^4*c[4]^6+10388940350*c[5]^5* c[4]^5-17199155780*c[4]^6*c[5]^2+13723975806*c[4]^4*c[5]+133539087080* c[5]^4*c[4]^4+90789586656*c[5]^3*c[4]^3-78835900959*c[5]^3*c[4]^4-1124 58551157*c[5]^4*c[4]^3-2330775000*c[5]^5*c[4]^2+11893978720*c[5]^5*c[4 ]^3)/c[5]/c[4]^2/(-15582*c[5]^2+44949*c[5]^3-15582*c[4]^2+93492*c[5]*c [4]^2+44949*c[4]^3-416569*c[5]^3*c[4]+15582*c[5]*c[4]+93492*c[5]^2*c[4 ]+2385540*c[5]^2*c[4]^3-449490*c[4]^4*c[5]^2-449490*c[5]^4*c[4]^2+2385 540*c[5]^3*c[4]^2-416569*c[5]*c[4]^3-859973*c[5]^2*c[4]^2-5305230*c[5] ^3*c[4]^3+1468750*c[5]^3*c[4]^4+1468750*c[5]^4*c[4]^3)\}:" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }} }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 98 "A lengt hy computation gives an expression for the square of the principal err or 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]):\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 4589 "prin_err_norm_sqrd := (u,v,w)->1/ 1518886662733440000*(179799467002200*u^2*v^4-1594535049296523404*w^5*v ^5-38722233156856014*w^5*v^2+12839496177444594*v^6*w+97700021108830560 *w^7*v^5+104773970081469682*w^4*v^3-422364468568000000*v^10*w^3-439432 725928904748*w^6*v^3+3692938109020793400*w^5*v^6+1697174021340781200*v ^4*u^2*w^4-196509840885000*v^5*u-27247291594590900*w^7*v^4-10710802764 651484*w^4*v^2+1342748222275290*v^4*w-312084946194840*w^7*v^2-43363433 1833857595*w^4*v^4-11437164012617066*w^3*v^3+315232501819676858*v^5*w^ 3-142751455504814*w*v^3+344733826332221*w^2*v^2-264196439524710*w^4*v- 5590825984981190*v^5*w+18796281701700300*w^8*v^6+888082609600513024*w^ 4*v^5-100891847084921614*v^5*w^2+4038803574067080*w^7*v^3+719197868008 800*u^2*w^4+7488008941914810*w^5*v-320472117009000*w^4*u+2330658703470 935420*v^6*w^6+258724236163204339*v^6*w^2-22129503712590318*w^3*v^4+39 3019681770000*w^5*u+3107730891697390200*v^8*w^4+26556934634428516*v^4* w^2-4369527036660174*w^2*v^3-1195817690294333548*w^3*v^6-5894015697900 00*u^2*w^3-4551622846396721440*w^5*v^7-426407104022744700*w^3*u*v^4-41 239520259993000*w^6*v^4*u-2580502347523330200*w^3*v^6*u-64361294080146 0000*v^9*w^4*u-16169778781167600*u*v^3*w^2+6737285337330600*u*v^4*w-27 4556835209229300*w^5*u*v^3+3787656247742373000*u^2*v^6*w^4+28747688558 2200000*v^8*w^2*u-7151795710200000*v^7*w*u+1353214625616189000*u^2*v^7 *w^3-2186968826897280000*u^2*v^7*w^4-187019860587069600*v^5*w^2*u-5245 911796761900*v^3*u^2*w+1166596319092500*v^2*u^2*w-74474247517043400*v^ 2*u^2*w^3-1033430883068699100*v^4*u^2*w^3-264574102034918700*v^5*u^2*w ^2+56913095544932058*w^5*v^3+1560424730974200*w^8*v^4+9685535770172913 9*w^6*v^2-1449090283500000*v*u^2*w^2+121488442788022650*v^4*u^2*w^2+96 37854346918800*v*u^2*w^3+65327365350000*w^3*u+277499747526135424*w^5*v ^4-12100203237181806*w^6*v-547306432049345400*v^3*u^2*w^4-120757523625 000*u^2*v*w+9121169935553400*v^2*u^2*w^2+94288215537945000*w^6*v^5*u+3 45580015639429800*v^3*u^2*w^3-2593366907076000*u^2*v^2*w^5+48270970560 1095000*u^2*v^8*w^4+2755976516023860000*w^4*v^8*u+1781760435594928000* v^9*w^3+37592563403400600*v^7*w^7+1236546649552742748*w^6*v^4-71892172 15468200*w^5*u*v-160795476712530000*u^2*v^4*w^5+110805947958587400*u^2 *v^2*w^4-510559713057240000*u^2*v^6*w^5-13135019733514800*u^2*v*w^4+12 998559367008000*u^2*v^4*w-39040279063275600*v^3*u^2*w^2+19912726560541 37400*v^5*u^2*w^3+31463676763794000*u^2*v^3*w^5-1411720387687623600*w^ 4*v^5*u+201603611514690000*u^2*v^7*w^5+2465427910354263880*w^3*v^7-121 9116518320952400*v^7*w^6-10831469484490200*w^8*v^5-8274586901844600*u* v^2*w^3+2337865657380000*u^2*v^4*w^6-3305768620111785000*u^2*v^5*w^4+1 321970278390578000*w^3*v^5*u+407280023262000000*v^9*w^3*u+113980811703 1038000*w^5*v^6*u+2997135416601657000*v^7*w^3*u+775652064395463000*w^5 *v^4*u-470769209292063600*v^7*w^2*u-147350392447500*u^2*v^3+4331247224 635800*w^4*v*u-15567227508880800*u^2*v^5*w+253118379668223600*w^4*v^4* u+378198097481150100*v^6*w^2*u-145272098663672400*w^7*v^6-220076509785 0654600*w^6*v^5-2455132292577920000*v^9*w^4+83953040746618800*u*v^3*w^ 3-215607664186650000*u^2*v^7*w^2-95349492547650000*w^6*v^6*u-140302389 18960000*u^2*v^5*w^6+417020572152459000*u^2*v^5*w^5-16128470672982000* v^5*w*u-14199548790420000*w^5*v^8*u+25101349511040000*w^6*v^7*u+345277 7424239211000*w^4*v^6*u+667450457127440000*v^10*w^4-454359103359873240 *v^7*w^2-584071928388730852*w^4*v^6+58833276198535800*w^5*u*v^2+261735 558829140300*v^8*w^6-1247781770806518900*w^4*v^7-626502620745120000*v^ 9*w^5-708597925200000*w^6*u*v^2-357107913223290000*w^5*v^7*u-177915615 7815252000*v^8*w^3*u+21049927163595000*u^2*v^6*w^6+5895295226550000*u^ 2*v^6*w-1297362224867139000*w^5*v^5*u-34290755905500*w^3*u*v+242342547 45903000*w^4*u*v^3+64895856035077800*u*v^4*w^2-4439814877746549000*w^4 *v^7*u+509908179179398400*v^8*w^2-317740051004814*w^3*v+34364877665598 26*w^3*v^2-21867109633438200*w^4*u*v^2+8617396730382000*w^6*u*v^3+3466 13859421261200*u^2*v^6*w^2-2300892720129765000*u^2*v^6*w^3+26643183061 95000*u*v^2*w^2+48323983387500000*u^2*v^8*w^2-305460017446500000*u^2*v ^8*w^3-1535700139704000*u*v^3*w+18871904744692200*v^6*w*u+668183474000 00000*v^10*w^2-298124177640800000*v^9*w^2+5632060036000000*w*v^8-64431 977850000000*v^9*w^2*u+158369809837500*u*v^2*w+120757523625000*u^2*w^2 -2930026914888656000*v^8*w^3+2722003036848309800*v^8*w^5-2187485716500 00*u*v*w^2-534009607890174*w^5-112911851470174*v^5+22521274452407*v^4+ 659924025809742*w^6+108059852452407*w^4-14824971408096000*w*v^7+141706 396201742*v^6-32663682675000*v^3*u+30189380906250*u^2*v^2+160236058504 500*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+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 investigations suggested that the values \+ " }{XPPEDIT 18 0 "c[2] = 29/316;" "6#/&%\"cG6#\"\"#*&\"#H\"\"\"\"$;$! \"\"" }{TEXT -1 3 ", " }{XPPEDIT 18 0 "c[4] = 110/521;" "6#/&%\"cG6# \"\"%*&\"$5\"\"\"\"\"$@&!\"\"" }{TEXT -1 7 " and " }{XPPEDIT 18 0 "c [5] = 526/935;" "6#/&%\"cG6#\"\"&*&\"$E&\"\"\"\"$N*!\"\"" }{TEXT -1 89 " give a value for the (square of the) principal error norm that i s close to the minimum." }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 59 "p rin_err_norm_sqrd(29/316,110/521,526/935):\nevalf(sqrt(%));" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#$\"+LXbH(*!#:" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 124 "Using a one dimensional minimizat ion procedure and cycling around the nodes gives very slow convergence towards the minimum." }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 496 "Di gits := 30:\nc_2 := 29/316: c_4 := 110/521: c_5 := 526/935:\nfor ct to 120000 do\n c_2 := findmin(prin_err_norm_sqrd(c2,c_4,c_5),c2=\{0.05 ,c_2,0.15\},convergence=location)[1];\n c_4 := findmin(prin_err_norm _sqrd(c_2,c4,c_5),c4=\{0.2,c_4,0.22\},convergence=location)[1];\n mn := findmin(prin_err_norm_sqrd(c_2,c_4,c5),c5=\{0.5,c_5,0.63\},converg ence=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;\n end do:\nDigits := 10:" }}{PARA 11 "" 1 "" {XPPMATH 20 "6%/&%\"cG6#\" \"#$\"?$>GE.a\\E^;RUGf5*!#J/&F%6#\"\"%$\"?O)eQc-UEg0g/586#!#I/&F%6#\" \"&$\"?i5'R`/$e7Ph!o\\ci&F1" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#$\"?3XY oN(RokF')o)pl%*!#S" }}{PARA 11 "" 1 "" {XPPMATH 20 "6%/&%\"cG6#\"\"#$ \"?*oRt4uul&Gsyu*e5*!#J/&F%6#\"\"%$\"?z'[.u@lE?7rS/86#!#I/&F%6#\"\"&$ \"?\"=i(*HU`rD&*=vRci&F1" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#$\"?++,w*o e.K(zR%)pl%*!#S" }}{PARA 11 "" 1 "" {XPPMATH 20 "6%/&%\"cG6#\"\"#$\"?z 53X'*\\wb,A26)e5*!#J/&F%6#\"\"%$\"?@,tD`!*R0$*)QU,86#!#I/&F%6#\"\"&$\" ?Ar[G)\\]>x*=*\\Mci&F1" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#$\"?YrW8Ej!3 Zb,P)pl%*!#S" }}{PARA 260 "" 0 "" {TEXT -1 1 ":" }}{PARA 260 "" 0 "" {TEXT -1 1 ":" }}{PARA 11 "" 1 "" {XPPMATH 20 "6%/&%\"cG6#\"\"#$\"?(pH @^pCh$*Hzrie5*!#J/&F%6#\"\"%$\"?'e#4Sc-%Gn8NB2p34V$)pl%*!# S" }}{PARA 11 "" 1 "" {XPPMATH 20 "6%/&%\"cG6#\"\"#$\"?::*)[yZ7O*Hzrie 5*!#J/&F%6#\"\"%$\"?m6*ymDSG " 0 "" {MPLTEXT 1 0 378 "c_2 := .91058627179e-1: pp := .94656983431e-10:\np1 := evalf[30](plot(prin_err_norm_sqrd(c[2] ,.21112980730,.56256285991),c[2]=0.075..0.107,\n color=COLOR(RGB,.5 ,0,.9))):\np2 := plot([[[c_2,pp]]$4],style=point,symbol=[circle$2,diam ond,cross],symbolsize=[12,10$3],\n color=[black,red$3]):\npl ots[display]([p1,p2],font=[HELVETICA,9],view=[0.075..0.107,9.4e-11..9. 755e-11]);" }}{PARA 13 "" 1 "" {GLPLOT2D 367 347 347 {PLOTDATA 2 "6*-% 'CURVESG6$7S7$$\"#v!\"$$\"?:Q()4!yh*>(z<2\"4c(*!#S7$$\"?LLLLLLLLLL$*3v pv!#J$\"?\\x6r3 \"yDh[tk*F-7$$\"?LLLLLLLLLLv0<**yF1$\"?!)[/\\7))RW^7*Qn'H'*F-7$$\"?+++ +++++++'e7['zF1$\"?A6f(4Q\"H[rfCLJ7'*F-7$$\"?LLLLLLLLLLZ'*pK!)F1$\"?L# R#eVLo#*yK0lQ&f*F-7$$\"?++++++++++%**o.5)F1$\"?]_X,#f2CeLX#paz&*F-7$$ \"?nmmmmmmmmmM_(*p\")F1$\"?K.VFtP(3YaT'*HVc*F-7$$\"?LLLLLLLLLL\\WGJ#)F 1$\"?,xOKv;oL%pZoI=b*F-7$$\"?++++++++++/TI+$)F1$\"?xzm[E,DBWR$Rs(Q&*F- 7$$\"?++++++++++grgp$)F1$\"?g.HO,_MK>#R'*Rn_*F-7$$\"?++++++++++'\\$RO% )F1$\"?Qoy/P*R,?.?(z;;&*F-7$$\"?LLLLLLLLLL()=/(\\)F1$\"?d[I#RjNrM(f/zV 2&*F-7$$\"?nmmmmmmmmm!pe\"p&)F1$\"?i`lzxK3Rd)f=N\")\\*F-7$$\"?nmmmmmmm mm#3^-j)F1$\"?C/1L)yy-C6S(4<\"\\*F-7$$\"?++++++++++egJ,()F1$\"?,bXeYo8 qgUEv7%[*F-7$$\"?nmmmmmmmmm-1Ak()F1$\"?T%H9id\"e%p1;'=%)y%*F-7$$\"?+++ +++++++eoBL))F1$\"??Lm!>0Z*F-7$$\"?nmmmmmmmmmm$Gv'*)F1$\"?1w'yX?QS'eL^K&yY*F-7$ $\"?nmmmmmmmmm/')\\I!*F1$\"?mj=6**y4X*ps#zLm%*F-7$$\"?LLLLLLLLLL80U)4* F1$\"?s#z._-))R'3)*zXql%*F-7$$\"?LLLLLLLLLL(Gs*o\"*F1$\"?(Gy?FTQ-@&RSo 9m%*F-7$$\"?++++++++++9yQI#*F1$\"?E&pzen3(Q*z')\\WuY*F-7$$\"?LLLLLLLLL Ll#=nH*F1$\"?DIc#[Nhp.]D=+)p%*F-7$$\"?++++++++++7TCl$*F1$\"?9_bI^U&Q[V -YuKZ*F-7$$\"?++++++++++%G$GK%*F1$\">%\\;\"\\oY#)*yminpx%*!#R7$$\"?+++ +++++++Ir9(\\*F1$\">wz\\QOS2&=Q6!RH[*Fjt7$$\"?++++++++++st;p&*F1$\">X' z&31O.v?:xp)*[*Fjt7$$\"?nmmmmmmmmmi2)Qj*F1$\">Gns1zArby&yP4(\\*Fjt7$$ \"?++++++++++?h(Hq*F1$\">#332Bj^Bnmr![e]*Fjt7$$\"?LLLLLLLLLL$y'el(*F1$ \">G+K>_(QHE!*\\$4Z^*Fjt7$$\"?++++++++++wn.M)*F1$\">HdzKd4'ecd%G2a_*Fj t7$$\"?nmmmmmmmmm%)GW)*)*F1$\">*e$>N4S#)RoXwOk`*Fjt7$$\"?++++++++++Eew l**F1$\">$>C$Qp\"Rd1w*Rk*[&*Fjt7$$\"?nmmmmmmmmmI'eJ+\"!#I$\">&)=Q/k[\" Rip*H*>i&*Fjt7$$\"?++++++++++D%\\+,\"Ffw$\">7K(y2Vx8@O\\/5x&*Fjt7$$\"? LLLLLLLLL$*ygo;5Ffw$\">L%)R74]bP\\v)HY#f*Fjt7$$\"?LLLLLLLLL8;IZB5Ffw$ \">&z\"z+E4\"HL#pJ*>4'*Fjt7$$\"?nmmmmmmmm1aP?I5Ffw$\">pT&)G)*)[T367;#o i*Fjt7$$\"?+++++++++?.')QO5Ffw$\">(op55vT`3.%>9Rk*Fjt7$$\"?nmmmmmmmmEQ rZV5Ffw$\">JxI5AhqdUntjXm*Fjt7$$\"?LLLLLLLLL`3s\")\\5Ffw$\">o*))*3'3?= see;*Ro*Fjt7$$\"?+++++++++!e/xl0\"Ffw$\">UxeJu#4wQ-bIq0(*Fjt7$$\"?++++ +++++gsq/j5Ffw$\">2Xc0Ij5+*=GvWF(*Fjt7$$\"$2\"F*$\">Ptjz>c%R0p!*e'=v*F jt-%&COLORG6&%$RGBG$\"\"&!\"\"$\"\"!Fc[l$\"\"*Fa[l-F$6&7#7$$\"3&)***** yrie5*!#>$\"3')****4V$)pl%*!#G-%'COLOURG6&F^[lFc[lFc[lFc[l-%'SYMBOLG6$ %'CIRCLEG\"#7-%&STYLEG6#%&POINTG-F$6&Fh[l-Fa\\l6&F^[l$\"*++++\"!\")Fb[ lFb[l-Fd\\l6$Ff\\l\"#5Fh\\l-F$6&Fh[lF^]l-Fd\\l6$%(DIAMONDGFe]lFh\\l-F$ 6&Fh[lF^]l-Fd\\l6$%&CROSSGFe]lFh\\l-%%FONTG6$%*HELVETICAGFe[l-%+AXESLA BELSG6%Q%c[2]6\"Q!Fh^l-Fa^l6#%(DEFAULTG-%%VIEWG6$;F(Fgz;$\"#%*!#7$\"%b (*!#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 0 383 "c_4 := .211129807 30: pp := .94656983431e-10:\np1 := evalf[30](plot(prin_err_norm_sqrd(. 91058627179e-1,c[4],.56256285991),c[4]=0.2111..0.21116,\n color=COL OR(RGB,0,.7,.2))):\np2 := plot([[[c_4,pp]]$4],style=point,symbol=[circ le$2,diamond,cross],symbolsize=[12,10$3],\n color=[black,cya n$3]):\nplots[display]([p1,p2],font=[HELVETICA,9],view=[0.2111..0.2111 6,9.4e-11..9.8e-11]);" }}{PARA 13 "" 1 "" {GLPLOT2D 400 369 369 {PLOTDATA 2 "6*-%'CURVESG6$7S7$$\"%6@!\"%$\"?]D!=k=UZNY9M****z*!#S7$$ \"?++++++++DHyI,6@!#I$\"?Uh8*QYHbXgib)Qr(*F-7$$\"?+++++++v[kdW-6@F1$\" ?o9aOk*HHZn\"y(Hvu*F-7$$\"?+++++++]n\"\\DP56#F1$\"?T=nUsgo@?*4!*\\=s*F -7$$\"?+++++++]s,P,06@F1$\"?)f8#H`1Q7#y'[AB(p*F-7$$\"?+++++++v8*y&H16@ F1$\"?Y^SN!=F(R%Q8GgRn*F-7$$\"?+++++++vG[W[26@F1$\"?g:Z!e\\yn&>t_+[`'* F-7$$\"?+++++++v)fB:(36@F1$\"?%QC`36TE.Ua%fQL'*F-7$$\"?+++++++vQ=\"))* 46@F1$\"?Wu)>j!zY)=o]`&z8'*F-7$$\"?+++++++vj=pD66@F1$\"?XuZ7SmLHlbFSZ& f*F-7$$\"?++++++++lN?c76@F1$\"?%R5!G`\")yBedAa)yd*F-7$$\"?+++++++]U$e6 P66#F1$\"?V[/G7vWk-'*3.Xj&*F-7$$\"?++++++++&>q0]66#F1$\"?K?Yc2.^j;8G_Q [&*F-7$$\"?++++++++DM^I;6@F1$\"?C`x&*)ei[]+!oD_M&*F-7$$\"?++++++++0ytb <6@F1$\"?,*335$3atDMjHOA&*F-7$$\"?+++++++vQNXp=6@F1$\"?lAk/'Gy;Ua5vTB^ *F-7$$\"?++++++++XDn/?6@F1$\"?Y?]=rGy:#y4A\"p,&*F-7$$\"?++++++++!y?#>@ 6@F1$\"?%*)z&eMm&\\7!f*QXP\\*F-7$$\"?+++++++v3wY_A6@F1$\"?o)Hbc4H,cNG) eu&[*F-7$$\"?++++++++IOTqB6@F1$\"?&>*zlNR3n;Z%\\\"yz%*F-7$$\"?+++++++v 3\">)*\\76#F1$\"?F0`SK1n-fYI[Wu%*F-7$$\"?+++++++DEP/BE6@F1$\"?6Qkf^y?S 8eR!Q0Z*F-7$$\"?+++++++](o:;v76#F1$\"?vg=$4&e&H4^Ke%on%*F-7$$\"?++++++ +v$)[opG6@F1$\"?!>7W'y_(=%e*F-7$$\"?+++++++v=n#f([6@F1$\"?\\W'pyec? r`7]&>-'*F-7$$\"?++++++++!)RO+]6@F1$\"?v:IYd7+!4dm3[2i*F-7$$\"?+++++++ ]_!>w7:6#F1$\"?ND9y8*fmR,#*Q[4k*F-7$$\"?+++++++v)Q?QD:6#F1$\"?JCy**)QT :6pLM3Am*F-7$$\"?++++++++5jyp`6@F1$\"?.-d?vW0e3RW?#Go*F-7$$\"?+++++++] Ujp-b6@F1$\"?OIlw^eI-2z'R=xq*F-7$$\"?++++++++gEd@c6@F1$\"?bt%y+egqCe$* pO6t*F-7$$\"?+++++++v3'>$[d6@F1$\"?Q%yj*HH_LIq&G.tv*F-7$$\"?+++++++D6E jpe6@F1$\"?!eRO.LcG;&**Hn]$y*F-7$$\"&;6#!\"&$\"?&e.ZEmPKHNUAIH\")*F--% &COLORG6&%$RGBG$\"\"!F_[l$\"\"(!\"\"$\"\"#Fb[l-F$6&7#7$$\"3++++t!)H6@! #=$\"3')****4V$)pl%*!#G-%'COLOURG6&F][lF_[lF_[lF_[l-%'SYMBOLG6$%'CIRCL EG\"#7-%&STYLEG6#%&POINTG-F$6&Fg[l-F`\\l6&F][lF^[l$\"*++++\"!\")F_]l-F c\\l6$Fe\\l\"#5Fg\\l-F$6&Fg[lF]]l-Fc\\l6$%(DIAMONDGFd]lFg\\l-F$6&Fg[lF ]]l-Fc\\l6$%&CROSSGFd]lFg\\l-%%FONTG6$%*HELVETICAG\"\"*-%+AXESLABELSG6 %Q%c[4]6\"Q!Fh^l-F`^l6#%(DEFAULTG-%%VIEWG6$;F(Fez;$\"#%*!#7$\"#)*Fd_l " 1 2 0 1 10 0 2 9 1 4 2 1.000000 45.000000 45.000000 0 0 "Curve 1" "C urve 2" "Curve 3" "Curve 4" "Curve 5" }}}}{PARA 0 "" 0 "" {TEXT -1 0 " " }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 380 "c_5 := .562 56285991: pp := .94656983431e-10:\np1 := evalf[30](plot(prin_err_norm_ sqrd(.91058627179e-1,.21112980730,c[5]),c[5]=0.56251..0.562616,\n \+ color=COLOR(RGB,0.6,.2,.2))):\np2 := plot([[[c_5,pp]]$4],style=point,s ymbol=[circle$2,diamond,cross],symbolsize=[12,10$3],color=[black,green $3]):\nplots[display]([p1,p2],font=[HELVETICA,9],view=[0.56251..0.5626 16,9.4e-11..9.81e-11]);" }}{PARA 13 "" 1 "" {GLPLOT2D 389 359 359 {PLOTDATA 2 "6*-%'CURVESG6$7S7$$\"&^i&!\"&$\"?zLw;na::p;A[43)*!#S7$$\" ?nmmmmmm;M)\\5B^i&!#I$\"?<]W!*pJ$f(=mkswy(*F-7$$\"?LLLLLL$e%f]3K9DcF1$ \"?>D(Rc/RRv4j<@Vv*F-7$$\"?nmmmmmm\"f>q\"e;DcF1$\"?Cp6Iir-5801)>!G(*F- 7$$\"?nmmmmmmTrRv&)=DcF1$\"?\\MDGRprs;yPn\"Gq*F-7$$\"?LLLLLL$e43cA6_i& F1$\"?i^RM'>(G(p$)ei+!z'*F-7$$\"?nmmmmm;zIDDABDcF1$\"?_008$)zv.,ab)[!e '*F-7$$\"?++++++]7\"p\"pRDDcF1$\"?`'[;\\lUvDWGb)\\P'*F-7$$\"?nmmmmm;z \"enXw_i&F1$\"?v2D>H%zo$)o:@puh*F-7$$\"?++++++]i#HA())HDcF1$\"?<>E$pW[ VYH\"[>u)f*F-7$$\"?LLLLLLL$['HH>KDcF1$\"?R4oHV\"f7fV#zuw!e*F-7$$\"?nmm mmmmTQ(zBU`i&F1$\"?o#fLFql3bA?nzg/NI='*G]*F-7$$\"?LLL LLLLL6n&Ruai&F1$\"?2U`^*=p^b)fE@w%\\*F-7$$\"?++++++]7U%f$z\\DcF1$\"?k* 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1.000000 50.000000 113.000000 1 0 "Curve 1" }}}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 36 "#------------------------------ -----" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 166 "nds := [c[2]=.9105 862717929936e-1,c[4]=.2111298073004173,c[5]=.5625628599143128]:\nevalf [10](%);\nfor dgt from 7 by -1 to 4 do\n map(convert,nds,rational,dg t);\nend do;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#7%/&%\"cG6#\"\"#$\"+=F 'e5*!#6/&F&6#\"\"%$\"+t!)H6@!#5/&F&6#\"\"&$\"+*fGci&F2" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#7%/&%\"cG6#\"\"##\"$m\"\"%B=/&F&6#\"\"%#\"$E'\"%lH /&F&6#\"\"&#\"$i&\"$***" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#7%/&%\"cG6# \"\"##\"$6\"\"%>7/&F&6#\"\"%#\"$H\"\"$6'/&F&6#\"\"&#\"$`&\"$$)*" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#7%/&%\"cG6#\"\"##\"#b\"$/'/&F&6#\"\"%# \"#>\"#!*/&F&6#\"\"&#\"$`&\"$$)*" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#7% /&%\"cG6#\"\"##\"#b\"$/'/&F&6#\"\"%#\"#:\"#r/&F&6#\"\"&#\"\"*\"#;" }}} {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 56 "The minim um value for the principal error norm is . . . " }}{EXCHG {PARA 0 "> \+ " 0 "" {MPLTEXT 1 0 104 "evalf[25](prin_err_norm_sqrd(.910586271792993 6e-1,.2111298073004173,.5625628599143128)):\nevalf(sqrt(%));" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#$\"+`?=H(*!#:" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 9 "Setting " }{XPPEDIT 18 0 "c[2] = 5 5/604;" "6#/&%\"cG6#\"\"#*&\"#b\"\"\"\"$/'!\"\"" }{TEXT -1 3 ", " } {XPPEDIT 18 0 "c[4] = 129/611;" "6#/&%\"cG6#\"\"%*&\"$H\"\"\"\"\"$6'! \"\"" }{TEXT -1 7 " and " }{XPPEDIT 18 0 "c[5] = 562/999;" "6#/&%\"c G6#\"\"&*&\"$i&\"\"\"\"$***!\"\"" }{TEXT -1 57 " gives the following \+ value for the principal error norm." }}{PARA 0 "" 0 "" {TEXT -1 0 "" } }{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 59 "prin_err_norm_sqrd(55/604,1 29/611,562/999):\nevalf(sqrt(%));" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#$ \"+!y/#H(*!#:" }}}{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 s cheme" }}{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 2419 "ee := \{c[2]=55/604,\nc[3]=86/611 ,\nc[4]=129/611,\nc[5]=562/999,\nc[6]=25/33,\nc[7]=1728/1729,\nc[8]=1, \nc[9]=1,\na[2,1]=55/604,\na[3,1]=656438/20532655,\na[3,2]=2233592/205 32655,\na[4,1]=129/2444,\na[4,2]=0,\na[4,3]=387/2444,\na[5,1]=11379576 325181/16591126906359,\na[5,2]=0,\na[5,3]=-14480836320821/553037563545 3,\na[5,4]=41396479505524/16591126906359,\na[6,1]=-3277828022520854505 720678625/930941984189004121287101412,\na[6,2]=0, a[6,3]=1603682847559 4748052837775/1104320265941879147434284,\na[6,4]=-20249670229333480762 78835034350/177666633425593828346456771343,\na[6,5]=128250153920078340 306971850/111114057854856180218323669,\na[7,1]=10273112883528927626798 23552816060640736/100017465428353477655102490880994099675,\na[7,2]=0, \na[7,3]=-319583399218917235748122654831799520/76662716452063985062766 76005113153,\na[7,4]=309853242826319046243113700571575488375374176/905 9548279238747452639357500349347045575147,\na[7,5]=-4820906706186208151 46676915555711479983232/188177190150438948277052975318150206923289,\na [7,6]=97935281518032740868758496/126352318943704270751022325,\na[8,1]= 617667205196293028768837394929401/59572901637951905260312643716200,\na [8,2]=0,\na[8,3]=-734325479049899076018546413/174488237593403644421928 28,\na[8,4]=102611880576141156159106897686663174423254/297201617580929 3986626508592155810158783,\na[8,5]=-2978804438269693149527602626893006 58642/114979165744532084288043320814139766233,\na[8,6]=596455630909751 20908/76314917691264991025,\na[8,7]=-2372974894108974768977/4049500807 421551820510712,\na[9,1]=2350382879/37582963200,\na[9,2]=0,\na[9,3]=0, \na[9,4]=183047321485658386133/564833555402036603760,\na[9,5]=27899748 449743503627/89674243702534905040,\na[9,6]=42494109833439/260652887700 800,\na[9,7]=740190076116618685831/69155160690290757120,\na[9,8]=-1873 8157/1773760,\n\nb[1]=2350382879/37582963200,\nb[2]=0,\nb[3]=0,\nb[4]= 183047321485658386133/564833555402036603760,\nb[5]=2789974844974350362 7/89674243702534905040,\nb[6]=42494109833439/260652887700800,\nb[7]=74 0190076116618685831/69155160690290757120,\nb[8]=-18738157/1773760,\n\n `b*`[1]=258407359774721212871/4258903365184545792000,\n`b*`[2]=0,\n`b* `[3]=0,\n`b*`[4]=325668356622185113177778435233/9847225874740070749096 47594240,\n`b*`[5]=117216732464037657080802634257/40647560038598017777 4696356096,\n`b*`[6]=48441628589025019393881/244109082100582095488000, \n`b*`[7]=870665371500134942827763822063/12056409914183068428493221888 0,\n`b*`[8]=-1998414293009470233/281403607000355840,\n`b*`[9]=1/560\}: " }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 " " {TEXT -1 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#\"\"($\")0/29!\"(" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/&%\"CG6#\"\"($ \")(RzS\"!\"(" }}}{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 2419 "ee := \{c[2]=55/604,\nc[3]=86/611,\nc[4]=129/611,\nc[5]=562/999,\nc[6]=25/3 3,\nc[7]=1728/1729,\nc[8]=1,\nc[9]=1,\na[2,1]=55/604,\na[3,1]=656438/2 0532655,\na[3,2]=2233592/20532655,\na[4,1]=129/2444,\na[4,2]=0,\na[4,3 ]=387/2444,\na[5,1]=11379576325181/16591126906359,\na[5,2]=0,\na[5,3]= -14480836320821/5530375635453,\na[5,4]=41396479505524/16591126906359, \na[6,1]=-3277828022520854505720678625/930941984189004121287101412,\na [6,2]=0, a[6,3]=16036828475594748052837775/1104320265941879147434284, \na[6,4]=-2024967022933348076278835034350/1776666334255938283464567713 43,\na[6,5]=128250153920078340306971850/111114057854856180218323669,\n a[7,1]=1027311288352892762679823552816060640736/1000174654283534776551 02490880994099675,\na[7,2]=0,\na[7,3]=-3195833992189172357481226548317 99520/7666271645206398506276676005113153,\na[7,4]=30985324282631904624 3113700571575488375374176/9059548279238747452639357500349347045575147, \na[7,5]=-482090670618620815146676915555711479983232/18817719015043894 8277052975318150206923289,\na[7,6]=97935281518032740868758496/12635231 8943704270751022325,\na[8,1]=617667205196293028768837394929401/5957290 1637951905260312643716200,\na[8,2]=0,\na[8,3]=-73432547904989907601854 6413/17448823759340364442192828,\na[8,4]=10261188057614115615910689768 6663174423254/2972016175809293986626508592155810158783,\na[8,5]=-29788 0443826969314952760262689300658642/11497916574453208428804332081413976 6233,\na[8,6]=59645563090975120908/76314917691264991025,\na[8,7]=-2372 974894108974768977/4049500807421551820510712,\na[9,1]=2350382879/37582 963200,\na[9,2]=0,\na[9,3]=0,\na[9,4]=183047321485658386133/5648335554 02036603760,\na[9,5]=27899748449743503627/89674243702534905040,\na[9,6 ]=42494109833439/260652887700800,\na[9,7]=740190076116618685831/691551 60690290757120,\na[9,8]=-18738157/1773760,\n\nb[1]=2350382879/37582963 200,\nb[2]=0,\nb[3]=0,\nb[4]=183047321485658386133/5648335554020366037 60,\nb[5]=27899748449743503627/89674243702534905040,\nb[6]=42494109833 439/260652887700800,\nb[7]=740190076116618685831/69155160690290757120, \nb[8]=-18738157/1773760,\n\n`b*`[1]=258407359774721212871/42589033651 84545792000,\n`b*`[2]=0,\n`b*`[3]=0,\n`b*`[4]=325668356622185113177778 435233/984722587474007074909647594240,\n`b*`[5]=1172167324640376570808 02634257/406475600385980177774696356096,\n`b*`[6]=48441628589025019393 881/244109082100582095488000,\n`b*`[7]=870665371500134942827763822063/ 120564099141830684284932218880,\n`b*`[8]=-1998414293009470233/28140360 7000355840,\n`b*`[9]=1/560\}:" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 77 "The stability function R for th e 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)*&#\"7\"4]_])R9G9g>\";gtF)[@JtIN5^+\"F)*$)F'\"\"(F)F) F)*&#\"58[z[o=PW:=\"9!y*o!zEW*3'>fP)F)*$)F'\"\")F)F)F)" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 124 "We can find the poi nt where the boundary of the stability region intersects the negative \+ real axis by solving the equation: " }}{PARA 256 "" 0 "" {TEXT -1 1 " \+ " }{XPPEDIT 18 0 "R(z) = 1;" "6#/-%\"RG6#%\"zG\"\"\"" }{TEXT -1 2 ". \+ " }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 28 "z0 := newton(R(z)=1,z=-4 .4);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%#z0G$!+NOV\"[%!\"*" }}} {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,c olor=[red,blue]):\np2 := plot([[[z0,1]]$3],style=point,symbol=[circle, cross,diamond],color=black):\np3 := plot([[z0,0],[z0,1]],linestyle=3,c olor=COLOR(RGB,0,.5,0)):\nplots[display]([p1,p2,p3],view=[-5.19..0.49, -.07..1.47],font=[HELVETICA,9]);" }}{PARA 13 "" 1 "" {GLPLOT2D 387 248 248 {PLOTDATA 2 "6+-%'CURVESG6$7Z7$$!3Q++++++!>&!#<$\"3&4>av`6Eg$F *7$$!3QML3T![!f^F*$\"3e#Gh'QUv=MF*7$$!3Ynm;#3'4G^F*$\"3xjv**eBEVKF*7$$ !3a++DBT9(4&F*$\"3Y)R+[D0e2$F*7$$!3kLLLk@>m]F*$\"3bNL\"*)Rjg\"HF*7$$!3 E+]U'*)HB,&F*$\"3k0b0]`cbEF*7$$!3!pm;&GwYe\\F*$\"3Mc\"op&)\\fT#F*7$$!3 s+](\\(Q*y*[F*$\"3/LOj\"z6'p@F*7$$!3nLLV@,KP[F*$\"3OK:Zjd*e%>F*7$$!3'R LLd%[MwZF*$\"3s$HLu^4Z\"QF]p7$$!3_LLV-,(>*QF*$\"31us<[p/]IF]p7$$!35++ S:-YpPF*$\"3c1N/U/UvBF]p7$$!3K+++\"HZkk$F*$\"31e136i'*f=F]p7$$!3;++gW: !z_$F*$\"39,=Ewl>([\"F]p7$$!3hLL)*\\1D?MF*$\"33**ow>#)=M7F]p7$$!3'ommS KVAH$F*$\"3*[/'G9t#)=5F]p7$$!3/nmEGV!Q=$F*$\"3%=US`7$$!39++0( *RmdIF*$\"3cl6(zu&oq#F*$\"3JG_O$er$>yF`s7$$!3gmm; \\r8&e#F*$\"3UyOV)*QG,$)F`s7$$!3ymmrw\\OtCF*$\"3c87nwzxY*)F`s7$$!3SLL$ ))e.GN#F*$\"3E0#R,$\\BV)*F`s7$$!3nLL)**=uvA#F*$\"39m\")*oz;&)4\"F]p7$$ !3K++:I;c=@F*$\"3h9M_5/N:7F]p7$$!31LL.z]#3+#F*$\"3a%p2D`K.O\"F]p7$$!3M ++?,<>z=F*$\"3s#eu,uW=`\"F]p7$$!3;++!4<(>g;:$>F]p7$$!3H++q9zA<:F*$\"3q7OPY\"eR>#F]p7$$!3EnmEY; O-9F*$\"3eMbJ*Q`0Y#F]p7$$!3#)*****pQ<(z7F*$\"3_joH*R\\8y#F]p7$$!3)RL$e fMeo6F*$\"3a[Euc9;3JF]p7$$!3I****fAZ3Z5F*$\"3=6D#)3Mj4NF]p7$$!3xqm;(zQ wK*F]p$\"3\"eY-9djY$RF]p7$$!3&z***\\)ecE8)F]p$\"3`\"H3$zF3MWF]p7$$!3'3 nmm0VV'pF]p$\"3Ig@nj&F]p7$$! 3aFLL*)4AjXF]p$\"3$*zu\"eH(4OjF]p7$$!33LLLO'R&eLF]p$\"3w6utcZFZrF]p7$$ 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*-%+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" "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 1403 "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 \+ 1960142814398505250091/10051103530733121488277360*z^7+\n 1815443 7186848794813/837591960894426790689780*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 z 0 := zz:\n pts := [op(pts),[Re(zz),Im(zz)]]:\nend do:\np1 := plot(pt s,color=COLOR(RGB,.48,.12,.48)):\np2 := plots[polygonplot]([seq([pts[i -1],pts[i],[-2.2,0]],i=2..nops(pts))],\n style=patchnogrid,co lor=COLOR(RGB,.95,.23,.95)):\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,.12,.48)):\np4 := plots[polygonplot]([seq([pts[i-1],pts[ i],[1.89,4.72]],i=2..nops(pts))],\n style=patchnogrid,color=C OLOR(RGB,.95,.23,.95)):\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 p ts := [op(pts),[Re(zz),Im(zz)]]:\nend do:\np5 := plot(pts,color=COLOR( RGB,.48,.12,.48)):\np6 := plots[polygonplot]([seq([pts[i-1],pts[i],[1. 89,-4.72]],i=2..nops(pts))],\n style=patchnogrid,color=COLOR( RGB,.95,.23,.95)):\np7 := plot([[[-5.19,0],[2.29,0]],[[0,-5.19],[0,5.1 9]]],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$$\"\"!F)F(7$F($ \"3++++Fjzq:!#=7$F($\"3E+++WEfTJF-7$$\"3,+++%=*o+7!#F$\"3<+++>))Q7ZF-7 $$\"3L+++#3x\\!o!#E$\"3))*****H:%=$G'F-7$$\"3]+++,m;h!)!#D$\"3')*****R QwR&yF-7$$\"39+++zRx&Q&!#C$\"39+++#ofZU*F-7$$\"3!)******GmRSD!#B$\"31+ ++;?b*4\"!#<7$$\"3y*****f*G\"=R*FL$\"35+++![AmD\"FO7$$\"3!******R+[%)) G!#A$\"3-+++A&zOT\"FO7$$\"3m+++A4%3n(FX$\"31+++n^qq:FO7$$\"3,+++#QfB!= !#@$\"3'******zbgws\"FO7$$\"3!******f.Df!QF]o$\"31+++`uY%)=FO7$$\"3'** *****f**)yG(F]o$\"3>+++8'z4/#FO7$$\"3)******4Hi!p7!#?$\"3#)*****frVp># FO7$$\"3;+++D%=3+#F]p$\"3A+++jW&>N#FO7$$\"3.+++q?88GF]p$\"3?+++cqT0DFO 7$$\"33+++,Oo(R$F]p$\"35+++%*)Qll#FO7$$\"3#******R;-&oJF]p$\"38+++!3rV !GFO7$$\"3/+++3ICP7F]p$\"3'******>J)*y%HFO7$$!3%)*****zMlU^$F]p$\"39++ +N39'3$FO7$$!3++++o,fE7!#>$\"33+++G(=#=KFO7$$!3))******4.n/EFar$\"34++ +y2OVLFO7$$!3?+++EwmaXFar$\"3A+++(***)3Y$FO7$$!3g*****p6&o+rFar$\"3;++ +_RBqNFO7$$!30+++O39A5F-$\"3%)*****>?w4n$FO7$$!3))*****pIUfQ\"F-$\"31+ ++L*>Hw$FO7$$!3#*******ycj$z\"F-$\"33+++D$>h%QFO7$$!3#******>'H#oB#F-$ \"3*)*****>Qd3#RFO7$$!3#)*****Hqvwq#F-$\"3++++PHe()RFO7$$!3=+++.*H&*>$ F-$\"3#)******z_$o/%FO7$$!3:+++?c/2PF-$\"3K+++O#z\"*4%FO7$$!3@+++v!4hA %F-$\"3-+++V,;XTFO7$$!3!)*****4`FOv%F-$\"3w******o!y_=%FO7$$!3I+++>$3t G&F-$\"3O+++qj(*>UFO7$$!3%******paka#eF-$\"3%)*****zJR'\\UFO7$$!3++++! 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%%&Re(z)G%&Im(z)G-Fhjw6#%(DEFAULTG-%*AXESSTYLEG6#%$BOXG-%(SCALINGG6#%, CONSTRAINEDG-%%VIEWG6$;$!$>&F`hn$\"$H#F`hn;F`\\x$\"$>&F`hn" 1 2 0 1 10 0 2 9 1 2 1 1.000000 45.000000 45.000000 0 0 "Curve 1" "Curve 2" "C urve 3" "Curve 4" "Curve 5" "Curve 6" "Curve 7" "Curve 8" }}}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 4 "The " }{TEXT 260 30 "interval of absolute stability" }{TEXT -1 89 " (or stability inte rval) is the intersection of the stability region with the real line. " }}{PARA 0 "" 0 "" {TEXT -1 59 "For this scheme the stability interva l is (approximately) " }{XPPEDIT 18 0 "[-4.4814, 0];" "6#7$,$-%&Float G6$\"&9[%!\"%!\"\"\"\"!" }{TEXT -1 2 ". " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 148 "We can distort the boundary curve horizontally by taking the 11th root of the real part of points along the curve. In this way we see that there is " }{TEXT 260 53 "no large st interval on the nonnegative imaginary axis" }{TEXT -1 65 " that con tains the origin and lies inside the stability region. " }}{PARA 0 " " 0 "" {TEXT -1 119 "However the stability region intersects the nonne gative imaginary axis in an interval that does not contain the origin. " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 428 "R := z -> 1+z+1/2*z^2+1/6*z^3+1/24*z^4+1/120*z^5+1/720*z^6+\n 1960142814398505250091/10051103530733121488277360*z^7+\n 1815 4437186848794813/837591960894426790689780*z^8:\nDigits := 25:\npts := \+ []: z0 := 0:\nfor ct from 0 to 107 do\n zz := newton(R(z)=exp(ct*Pi/ 100*I),z=z0):\n z0 := zz:\n pts := [op(pts),[surd(Re(zz),11),Im(zz )]]:\nend do:\nplot(pts,color=COLOR(RGB,.85,0,.85),thickness=2,font=[H ELVETICA,9]);\nDigits := 10:" }}{PARA 13 "" 1 "" {GLPLOT2D 261 276 276 {PLOTDATA 2 "6(-%'CURVESG6#7hq7$$\"\"!F)F(7$$!:i/.Q)4#y22H(Q?!#E$ \":S$R%))Hy*e`EfTJF-7$$!:+4NB$*G`NAU+P$F-$\":\"Q,y)fFyrI&=$G'F-7$$!:$y :E;:=B81E9XF-$\":#Q6upJqugzxC%*F-7$$!:)HE.Kg/z&G$>WbF-$\":@ra'*)>#>91P mD\"!#D7$$!:d#*4Tw7D=um&)['F-$\":(4\"Hg)[brEjzq:F?7$$!:Azg;&\\?>F]rgtF -$\":Z.QcZtq=fb\\)=F?7$$!:+'G<:#F?7$$!:ro v!e]L-3Df/*)F-$\":%==L#4gs27uK^#F?7$$!:R\"3m)[zi>xF.d*F-$\":G2qP/nsMQL u#GF?7$$!:t***)*\\w!=(R%=\\,\"F?$\":'R=)H_K4Pk#fTJF?7$$!:*fC?N$*GX?\\4 h5F?$\":!oVLOym$)**=vbMF?7$$!:.KNm.QTR&**H)3\"F?$\":%o_=K)**f#[]\"F?$\":rgy a\"Q=&)HrqS`F?7$$\":47F?$\":yz#H.^LClHM(f'F?7$$\":F?$\": Y)HI'*ysz?q<#F?$\":k/RLC86**Q=)RvF?7$$\":v%R2.C8*e22`E#F?$\":.4r&*z#G#QQ wR&yF?7$$\":^ox,l2p[t5DN#F?$\":.7GN8twP/M\"o\")F?7$$\":TzIc%)H7c%\\tQC F?$\":Mu.ucT&e(H\"H#[)F?7$$\":esc9D8B\"p(eS_#F?$\":IS\\84Gy&f![kz)F?7$ $\":iz*f-e4R*pV&3EF?$\":xzCzaD(4GUg5\"*F?7$$\":X,wr=cf')4QAp#F?$\":b/' )og3vCofZU*F?7$$\":]W'QNIZe(G!=vFF?$\":lIYy+v$RzU\"*Q(*F?7$$\":T.G%4c1 n\\2SdGF?$\":zGb8@e][y1`+\"!#C7$$\":#3l`)fN*>OP#*QHF?$\":\\sk\\Df=(=?s O5Ffu7$$\":0M.Bq0h`Xo(>IF?$\":apb[')z%zJF?$\":))4yV5g4qrm48\"Ffu7$$ \":h\\&*QW!Q!)*ym$eKF?$\":2*[ZR3K'p;\"Qi6Ffu7$$\":slAq^oi?'ohOLF?$\":s N]sf^_wJ&z$>\"Ffu7$$\":'))*4)[fMWqRB9MF?$\":f?GUJJY964_A\"Ffu7$$\":0(R e\"\\=eT.?7\\$F?$\":x.MP(HH*)zCic7Ffu7$$\":hC$)HcQvE>vvc$F?$\":.#4*pH# z!=MN!)G\"Ffu7$$\"::$z<(z8I[&zHVOF?$\":%\\eo/;O2,wW>8Ffu7$$\":PH7h\"e/ Q7`Q=PF?$\"::#*fD&H)=T9f3N\"Ffu7$$\":p*>s`0M6CG$Gz$F?$\":vi$)p%Hb/P)pA Q\"Ffu7$$\":8,dkR\\!4vYjmQF?$\":D75M=onB_zOT\"Ffu7$$\":Tro@&3C[9PyRRF? $\":g(REr/kK:!)3X9Ffu7$$\":4JA![YDdu9F7SF?$\":0qlG=hW(*4&\\w9Ffu7$$\": wpcwEbLG<)3%3%F?$\":pF=**Q\"QQB0!z]\"Ffu7$$\":r]lM*GzW)pA_:%F?$\":WS(p prcw#*RIR:Ffu7$$\":-T?')oc9UeicA%F?$\":b,!3rp8Un^qq:Ffu7$$\":[R\\([o#H W'RR&H%F?$\":J/5r)3I``O5-;Ffu7$$\":IF55*eu->:SkVF?$\":Rj)[(47Lp**)\\L; Ffu7$$\":A6!HllX,/%oEV%F?$\":Ptx]/AJan!*[m\"Ffu7$$\":qYB')4Hqz;w,]%F?$ \":uE=Y%z,W!4yip\"Ffu7$$\":))3>psc&=aY!pc%F?$\":G%GM4-.)ybgws\"Ffu7$$ \":EV/t>rkx)=$Gj%F?$\":6&\\NjdaM)HP!fG1ex*yYR$zp%F?$\":u;% zs$pBlU2/z\"Ffu7$$\":K%=1ap$)Q;[=iZF?$\":u9Si(zX()R*p<#=Ffu7$$\":MyZ%o \\\"eUAcb#[F?$\":UaRwHJxlqBJ&=Ffu7$$\":cBGfMImeXMVr9N&\\\\F?$\":8aJpN,x\"\\(*z:>Ffu7$$\":Cf()o)pa+_I 25]F?$\":Ffu7$$\":v`z(4*)4W\"y\"fp]F?$\":/COq5rAOTB%y>F fu7$$\":=#[.s7y95&[!G^F?$\":?LKKi`lt,6(4?Ffu7$$\":/*Hum4YwXmR&=&F?$\": V)=#GH=%f7'z4/#Ffu7$$\":h,P-([h`aaeT_F?$\":!\\2_5Z!yUxEA2#Ffu7$$\":$* \\V/g18djflH&F?$\":*>5O2`W!G$)\\M5#Ffu7$$\":he;c5^**\\$)e-N&F?$\":X$zR ,rx1fekM@Ffu7$$\":BME>do$y%3b:iNX&F?$\":LKvxM\")HcrVp>#Ffu7$$\":[^u\"4<\"*>)z9I]&F?$\":y\"eM#Rl nbEQ!GAFfu7$$\":reQ))[\"y+_y)3b&F?$\":H*3O4jKF*>\"4fAFfu7$$\":\"Ht7P\\ '>M=&3(f&F?$\":]S$HF$=Pk4)4!H#Ffu7$$\":rwTJy\"Q]7/]TcF?$\":4/-!=HKD5U0 @BFfu7$$\":wWhN`I6zO:So&F?$\":tWhN#Ffu7$$\":/(R;H7!3C>)\\CdF?$ \":a\"p(>A\"zRXMz#Q#Ffu7$$\":hwNt#=+b$*4!Gw&F?$\":(>,nx[s2!HYSlLMJkUW#Ffu7$$\":,w_s#\\;D%Hw@$eF?$\":= G_gQK7Tv$)[Z#Ffu7$$\":-')>^2=()Q.UG'eF?$\":*HkZ_q(oi0mDFfu7$$\":WvY2#G-@@]eNfF?$\":M@#R!e$[P_\"Hkf#Ffu7$$\":)f?:t[*4lfVA&f F?$\":>\"fm7M*QAqXli#Ffu7$$\":j3nZ34pPxNV'fF?$\":;#)=!HQI9%*)Qll#Ffu7$ $\":e$*Q&\\D=G(GL7(fF?$\":We$e4cr\"=g'fF?$\":VCVt(=-9;xpXFFfu7$$\":?[A8 1&p#=$)z9&fF?$\":hL?^M$ywmk6vFFfu7$$\":!GtwE:W\"QN!fEfF?$\":QyVR.!)=.3 rV!GFfu7$$\":7I(=(zCzeR-Hr!)4>bB'GFfu7$$\":4QP_&zyD!\\sBv&F?$\":ISE8qIMe`o5*GFfu7$ $\":=[A,\"R#p!)z&yKcF?$\":$3YHFfu7$$\":v(oP]\"y%*z\\()4W&F ?$\":Tp+db*f*=J)*y%HFfu7$$\":mFU=CR)G/M,b]F?$\":aH[Iqf@LR**f(HFfu7$$!: 6_jI_uh!zHG=ZF?$\":$y5J&pBxHK\")Q+$Ffu7$$!:dY$G+&[)z)GT`U&F?$\":$Qx_%4 Ey&po`JIFfu7$$!:iKqVeI0t?>@v&F?$\":eD!*[J.A5**e*eIFfu7$$!:BbC=pM!f]il# )fF?$\":VRo]d_GY$39'3$Ffu7$$!:J8+R8'4d#4$*y(e'F?$\":)oh]ijg3qZL#>$Ffu7$$!:XzqV^A&H( [KEq'F?$\":P/S^G]Q%G(=#=KFfu7$$!:?C&>N6O%oCt#4oF?$\":AsD;;oUY'\\#QC$Ff u7$$!:i\"eB\"*o7ch'f\"4pF?$\":vov=ggG$zx9pKFfu-%%FONTG6$%*HELVETICAG\" \"*-%+AXESLABELSG6$Q!6\"Ff]m-%&COLORG6&%$RGBG$\"#&)!\"#F(F\\^m-%*THICK NESSG6#\"\"#-%%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 points of the \+ boundary curve with the imaginary axis can be determined as follows." }}{PARA 0 "" 0 "" {TEXT -1 87 "First we look for points on the boundar y curve either side of each intersection point. " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 198 "Digits := 1 5:\nz0 := 0.4*I:\nfor ct from 12 to 15 do\n newton(R(z)=exp(ct*Pi/10 0*I),z=z0);\nend do;``;\nz0 := 3.0*I:\nfor ct from 95 to 98 do\n new ton(R(z)=exp(ct*Pi/100*I),z=z0);\nend do;\nDigits := 10:" }}{PARA 11 " " 1 "" {XPPMATH 20 "6#^$$!0;df8*\\OD!#D$\"06W%\\6\"*pP!#:" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#^$$!0X4*>7PA@!#D$\"0xF\"*QqS3%!#:" }}{PARA 11 " " 1 "" {XPPMATH 20 "6#^$$\"0qB<5\"Hq8!#D$\"0fDUhH#)R%!#:" }}{PARA 11 " " 1 "" {XPPMATH 20 "6#^$$\"0%*QTE*o+7!#C$\"0)R^=))Q7Z!#:" }}{PARA 11 " " 1 "" {XPPMATH 20 "6#%!G" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#^$$\"0!Qy 2ICP7!#<$\"0+'*=J)*y%H!#9" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#^$$\"0(oQ n[Q2b!#=$\"0g@LR**f(H!#9" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#^$$!0oZNZC +e#!#=$\"0CxHK\")Q+$!#9" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#^$$!0QQfkk' )>\"!#<$\"0Ey&po`JI!#9" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 108 "Then we apply the bisection method to calculate the parameter value associated with each intersection point." }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 328 "Digits := 15:\nreal_part := proc(u )\n Re(newton(R(z)=exp(u*Pi*I),z=0.4*I))\nend proc:\nu0 := bisect('r eal_part'(u),u=0.12..0.15);\nnewton(R(z)=exp(u0*Pi*I),z=0.4*I);``;\nre al_part := proc(u)\n Re(newton(R(z)=exp(u*Pi*I),z=3.0*I))\nend proc: \nu0 := bisect('real_part'(u),u=0.95..0.98);\nnewton(R(z)=exp(u0*Pi*I) ,z=3.0*I);\nDigits := 10:" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%#u0G$\" 07'H%3=VP\"!#:" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#^#$\"0qm&3va%#u0G$\"0;OK$Qwp'*!#:" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#^#$\"023`W ua*H!#9" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 4 "The " }{TEXT 260 73 "stability region intersects the nonegative ima ginary axis in the interval" }{TEXT -1 39 " [ 0.4318, 2.9955 ] (app roximately)." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 37 "#------------------------------------" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 80 "The stability function R* for the 9 stage, order 5 scheme is given as follows." }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 97 "subs(ee,subs(b=`b*`,StabilityFunction(5,9,'ex panded'))):\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)*&#\"D^X,\"4]wX\"=\"e3)4(\\KK6\"F+7H1M,\">\"p;Bv?F1TH (zF)*$)F'F1F)F)F)*&#\"@.HioF7b`y;rvYAX$\"D+'\\#e:5H!e7.)[`M$QF:F)*$)F' \"\"(F)F)F)*&#\"B2;L!3\"*yi\\+x`q>Y*=\"FDF)*$)F'\"\")F)F)F)*&#\"58[z[o =PW:=\"<+oF'y-!z35)\\^!p%F)*$)F'\"\"*F)F)F)" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 124 "We can find the point where th e boundary of the stability region intersects the negative real axis b y 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$!+j].#[%! \"*" }}}{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],s tyle=point,symbol=[circle,cross,diamond],color=black):\np_3 := plot([[ z_0,0],[z_0,-1]],linestyle=3,color=COLOR(RGB,0,.5,0)):\nplots[display] ([p_1,p_2,p_3],view=[-5.09..0.49,-1.57..1.47],font=[HELVETICA,9]);" }} {PARA 13 "" 1 "" {GLPLOT2D 398 284 284 {PLOTDATA 2 "6+-%'CURVESG6$7S7$ $!3')*************3&!#<$!3#HZm.v(=%H#F*7$$!3G++vz=Po\\F*$!3aot(>GRs'>F *7$$!3#**\\iE!Rai[F*$!3S31\"*zrx7'*4)Q*fFK7$$!3!**\\7nc1J/%F*$!3%o@RS]\"R# )[FK7$$!3'****\\XoI<#RF*$!3!y,y#)e?p!RFK7$$!3++]ZTF#[\"QF*$!3!3iy6Cy?< $FK7$$!3s***\\'=(pWp$F*$!3#*)>&H<*)pmCFK7$$!3#)***\\Z^AOd$F*$!3I7@N)3 \"4r=FK7$$!3))***\\8%Q;dMF*$!3oWMYD6])Q\"FK7$$!3#**\\i*3#39N$F*$!33Al0 kyX;5FK7$$!3t***\\J`acA$F*$!3]WW)Q$G7$$!3l****fuY7>JF*$!3YF,Wjag ^PFhp7$$!3q*\\iQ70_*HF*$!31TjpB9$44\"Fhp7$$!3+++5C`^&)GF*$\"3?]eY&)=s' R*!#?7$$!3q*\\i)G#o^w#F*$\"3qw!*4EkA+HFhp7$$!3W*\\(eM$p0l#F*$\"3i$pWX# =TxXFhp7$$!3]**\\i5u*4`#F*$\"3Ufmk=<7(>'Fhp7$$!3T*\\7\"eI>@CF*$\"3M%H! *GBK'>wFhp7$$!3n**\\()HUv-BF*$\"3#\\0!**4LJN\"*Fhp7$$!3y*\\iRdH(z@F*$ \"3'4HC>OES2\"FK7$$!3o*\\P$\\ijs?F*$\"3F;_1`\\8?7FK7$$!3S**\\#[_sp&>F* $\"3ul6C))>N)Q\"FK7$$!3y****pz0[P=F*$\"3m9td\"FK7$$!3')**\\_B5e?< F*$\"3)zAB\\!4)3y\"FK7$$!3g*\\i?puug\"F*$\"39j\\t&=T*)*>FK7$$!3i**\\2& R*)=[\"F*$\"3sT5c\"*zdpAFK7$$!3'*****4?a/p8F*$\"3?v3EsDDUDFK7$$!3O*** \\2Rg&[7F*$\"39-Vdn*\\'oGFK7$$!36+DcYIQR6F*$\"3)*[&G:'*e**>$FK7$$!3#*) **\\=PB+-\"F*$\"3m@]ivU\"eg$FK7$$!3c-]i)>_r2*FK$\"3q03&fS\\W.%FK7$$!3* )**\\74%3K!zFK$\"3?Ds%p>.q`%FK7$$!3E****\\xPYbnFK$\"3!e0lua\"z)3&FK7$$ !3)4+Dc^\")Qb&FK$\"3u!y(=+()\\QdFK7$$!3)e****f)\\h'R%FK$\"3WhPf:faUkFK 7$$!3F)**\\<\"G98KFK$\"3/\")o=OK&>D(FK7$$!3'G*\\i%Qq%R?FK$\"3o/m#Hib]: )FK7$$!3xr****pJ()4'*Fhp$\"3%Q/AKEVP3*FK7$$\"3k<+]_)f2v#Fhp$\"3(**3@bU *)y-\"F*7$$\"3)3++!Qdi!Q\"FK$\"378h&fQZ![6F*7$$\"3o4]PhBPfDFK$\"3)3a(o l;n\"H\"F*7$$\"3]/]i%G$e(o$FK$\"3]O7)*>#QfW\"F*7$$\"3!***************[ FK$\"32.OC.oJK;F*-%'COLOURG6&%$RGBG$\"*++++\"!\")$\"\"!Fb[lFa[l-F$6$7S 7$F($!\"\"Fb[l7$F.Fg[l7$F3Fg[l7$F8Fg[l7$F=Fg[l7$FBFg[l7$FGFg[l7$FMFg[l 7$FRFg[l7$FWFg[l7$FfnFg[l7$F[oFg[l7$F`oFg[l7$FeoFg[l7$FjoFg[l7$F_pFg[l 7$FdpFg[l7$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[l7$F\\uFg[l7$Fa uFg[l7$FfuFg[l7$F[vFg[l7$F`vFg[l7$FevFg[l7$FjvFg[l7$F_wFg[l7$FdwFg[l7$ FiwFg[l7$F^xFg[l7$FcxFg[l7$FhxFg[l7$F]yFg[l7$FbyFg[l7$FgyFg[l7$F\\zFg[ l7$FazFg[l7$FfzFg[l-F[[l6&F][lFa[lFa[lF^[l-F$6&7#7$$!3m*****H1N?[%F*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#\"\"$-%%FO NTG6$%*HELVETICAG\"\"*-%+AXESLABELSG6%Q\"z6\"Q!F[bl-Fcal6#%(DEFAULTG-% %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 1588 "`R*` := z -> 1+z+1/ 2*z^2+1/6*z^3+1/24*z^4+1/120*z^5+\n 113232497098085811814576500910 14551/7972941062720752316691191013406291200*z^6+\n 345224675711678 5355122768622903/15273833453488031258029101558249600*z^7+\n 118946 197053770049627891080331607/7972941062720752316691191013406291200*z^8+ \n 18154437186848794813/469051498100879002786276800*z^9:\npts := [ ]: z0 := 0:\nfor ct from 0 to 200 do\n zz := newton(`R*`(z)=exp(ct*P i/20*I),z=z0):\n z0 := zz:\n pts := [op(pts),[Re(zz),Im(zz)]]:\nen d do:\np_1 := plot(pts,color=COLOR(RGB,.42,0,.42)):\np_2 := plots[poly gonplot]([seq([pts[i-1],pts[i],[-2.2,0]],i=2..nops(pts))],\n \+ style=patchnogrid,color=COLOR(RGB,.83,0,.83)):\npts := []: z0 := 1.9+4 .5*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:\n p_3 := plot(pts,color=COLOR(RGB,.42,0,.42)):\np_4 := plots[polygonplot ]([seq([pts[i-1],pts[i],[1.79,4.43]],i=2..nops(pts))],\n styl e=patchnogrid,color=COLOR(RGB,.83,0,.83)):\npts := []: z0 := 1.9-4.5*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,.42,0,.42)):\np_6 := plots[polygonplot]([s eq([pts[i-1],pts[i],[1.79,-4.43]],i=2..nops(pts))],\n style=p atchnogrid,color=COLOR(RGB,.83,0,.83)):\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$$\"3w******)\\U`y%!#F$\"3++++Fjzq:!#=7$$\"39+++=v#H@$!#D$ \"3x*****zj#fTJF07$$\"3$)*****zV`)HR!#C$\"3D+++7')Q7ZF07$$\"34+++WP'GS #!#B$\"3A+++a:=$G'F07$$\"3-+++dx'3+\"!#A$\"3Y*****>'y&R&yF07$$\"3#)*** **>%G!fC$FF$\"3++++\\#oYU*F07$$\"3M******[iFy()FF$\"3#******46<&*4\"!# <7$$\"39+++^1Ne?!#@$\"3%*******3C^c7FS7$$\"3++++91!>G%FW$\"3%******HV! 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/281403607000355840,\n`b*`[9]=1/560\}:" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT 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#\"\"##\"#b\"$/'/&F%6# \"\"$#\"#')\"$6'/&F%6#\"\"%#\"$H\"F1/&F%6#\"\"&#\"$i&\"$***/&F%6#\"\"' #\"#D\"#L/&F%6#\"\"(#\"%G<\"%H " 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$\"\"#\"\"\" #\"#b\"$/'/&F%6$\"\"$F(#\"'Qkl\")bE`?/&F%6$F/F'#\"(#fLAF2/&F%6$\"\"%F( #\"$H\"\"%WC/&F%6$F;F'\"\"!/&F%6$F;F/#\"$(QF>/&F%6$\"\"&F(#\"/\"=Djdz8 \"\"/fj!p7\"f;/&F%6$FKF'FB/&F%6$FKF/#!/@3KO3[9\".`ajv.`&/&F%6$FKF;#\"/ Cb]zkRTFN/&F%6$\"\"'F(#!=D'y1s0X&3_A!GyF$\"<795(G@T+*=%)>%4$*/&F%6$Fjn F'FB/&F%6$FjnF/#\";vx$G0[Zfv%Go.;\":%GMu9z=%fE?V5\"/&F%6$FjnF;#!@]V.N) yi2[L$H-n\\-#\"?V8xckMGQfDMjmw$\"C`J60gnwi] )R1_krim(/&F%6$FfpF;#\"NwTPv$)[v:d+P6Vi/>j#GC`)4$\"LZ^dXqM\\.]d$REXZ(Q #z#[&f!*/&F%6$FfpFK#!KKK)*z9rbb\"pnY^\"3i=1n!4#[\"K*GBp?]\"=`(H0x#[*Q/ :!>x\")=/&F%6$FfpFjn#\";'\\e(o3uK!=:GNz*\"0sm<'\"A+irVEJg_!>&zj,Hdf/&F%6$FhrF'FB/&F%6$FhrF/ #!<8ka=g2**)\\!zaKM(\";GG>UWOS$fP#)[u\"/&F%6$FhrF;#\"KaKUuJm'o(*o5fh:T hd!)=h-\"\"I$ye,\"e:#f3li')RH4e<;?(H/&F%6$FhrFK#!HU'e1I*oi-w_\\Jpp#QW! )yH\"HLiwRT\"3KV!)G%3KXul\"z\\6/&F%6$FhrFjn#\"5347v44jbkf\"5D5*\\E\"p< \\Jw/&F%6$FhrFfp#!7x*oZ(*3T*[(HP#\":72^?=b@u!3]\\S/&F%6$\"\"*F(#\"+zGQ ]B\",+K'HeP/&F%6$F`uF'FB/&F%6$F`uF/FB/&F%6$F`uF;#\"6LhQec[@t/$=\"6gPgO ?SbN$[c/&F%6$F`uFK#\"5FO]V(\\%[(**y#\"5S]!\\`-PCu'*)/&F%6$F`uFjn#\"/RM $)4T\\U\"0+3q()Glg#/&F%6$F`uFfp#\"6Jeo=m6w+>S(\"5?rv!H!pg^:p/&F%6$F`uF hr#!)d\"Q(=\"(gPx\"" }}}{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#\"\"\"#\"+zGQ]B\", +K'HeP/&F%6#\"\"#\"\"!/&F%6#\"\"$F//&F%6#\"\"%#\"6LhQec[@t/$=\"6gPgO?S bN$[c/&F%6#\"\"&#\"5FO]V(\\%[(**y#\"5S]!\\`-PCu'*)/&F%6#\"\"'#\"/RM$)4 T\\U\"0+3q()Glg#/&F%6#\"\"(#\"6Jeo=m6w+>S(\"5?rv!H!pg^:p/&F%6#\"\")#!) d\"Q(=\"(gPx\"" }}}{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#\"\"\"#\"6rG@@ ZxftSe#\"7+?zXX=lL!*eU/&F%6#\"\"#\"\"!/&F%6#\"\"$F//&F%6#\"\"%#\"?L_Vy x<8^=AmNocK\"?SUfZ'4\\22SZ(eAZ)*/&F%6#\"\"&#\"?dUj-33dw.kCt;s6\"?'4cjp ux]-*eG;W[\"9+!)[&4#e+@34TC/&F%6#\"\"(# \"?j?#QwFG%\\8+:Pl1()\"?!))=A$\\G%oI=9*4k07/&F%6#\"\")#!4L-Z4IH9%)*>\" 3SeN+qg.9G/&F%6#\"\"*#F'\"$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 30 " #=============================" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }} {PARA 0 "" 0 "" {TEXT 258 78 "Schemes with maximum magnitude of the li nking coefficients of the order of 200" }}{PARA 0 "" 0 "" {TEXT -1 30 "#=============================" }}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 24 "Tsitouras' scheme with " }{XPPEDIT 18 0 "c[7] = 544/545;" "6#/&% \"cG6#\"\"(*&\"$W&\"\"\"\"$X&!\"\"" }{TEXT -1 1 " " }}{PARA 257 "" 0 " " {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 86 "See: A parameter study \+ of explicit Runge-Kutta pairs of orders 6(5), by Ch. Tsitouras," }} {PARA 0 "" 0 "" {TEXT -1 68 " Applied Mathematics Letters, Vol. \+ 11, pages 65 to 69, (1998)." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 257 "" 0 "" {TEXT -1 83 "The scheme constructed is a minor modificatio n of a scheme of Tsitouras which has " }{XPPEDIT 18 0 "c[6] = 35/36; " "6#/&%\"cG6#\"\"'*&\"#N\"\"\"\"#O!\"\"" }{TEXT -1 7 " and " } {XPPEDIT 18 0 "c[7] = 544/545;" "6#/&%\"cG6#\"\"(*&\"$W&\"\"\"\"$X&!\" \"" }{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 no des " }{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 m inimize the principal error norm." }}{PARA 257 "" 0 "" {TEXT -1 0 "" } }{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}{PARA 0 "" 0 "" {TEXT -1 52 "#---------------------------------------------------" }}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 28 "checking the combined scheme" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 35 "coefficients of the combined s cheme" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1968 "ee := \{c[2]=3/59,\nc[3]=19/198,\nc[4]=19/132,\nc[5 ]=88/177,\nc[6]=35/36,\nc[7]=544/545,\nc[8]=1,\nc[9]=1,\n\na[2,1]=3/59 ,\na[3,1]=1273/235224,\na[3,2]=21299/235224,\na[4,1]=19/528,\na[4,2]=0 ,\na[4,3]=19/176,\na[5,1]=292792984/222425457,\na[5,2]=0,\na[5,3]=-373 191104/74141819,\na[5,4]=312454912/74141819,\na[6,1]=-9328224680414006 5/2230303430271708,\na[6,2]=0,\na[6,3]=14352497130136165/9011326990996 8,\na[6,4]=-2430460528059531110/19920664729472301,\na[6,5]=53730709552 104745/9712013829327216,\na[7,1]=-2815838771905694350037121443888/5228 7718910334988576674096875,\na[7,2]=0, a[7,3]=3895971979602604572891204 0192/190137159673945413006087625,\na[7,4]=-801496182066760414629068567 466496/5106133423043804066278483169375,\na[7,5]=2654592673856028993946 9575124336/3828996342298708285360778643125,\na[7,6]=-16804913133796526 592/942707650019020409375,\na[8,1]=-76006376179258679683396673/1389345 361299174809573120,\na[8,2]=0,\na[8,3]=109830477025712790000/527674313 813796947,\na[8,4]=-194391745260058069863027995198/1219164295706831920 486709627,\na[8,5]=444873349778574018937254390317/63269358937392669883 372236224,\na[8,6]=-49046624888634/2929654772721215,\na[8,7]=-45835339 680636651875/22882961544684083693312,\na[9,1]=65760917/1910092800,\na[ 9,2]=0,\na[9,3]=0,\na[9,4]=6404556346056/24802664014135,\na[9,5]=29464 48255213557/7004255810773760,\na[9,6]=16630875288/3684943675,\na[9,7]= -496773593038619375/9868243105287168,\na[9,8]=27826183/603420,\n\nb[1] =65760917/1910092800,\nb[2]=0,\nb[3]=0,\nb[4]=6404556346056/2480266401 4135,\nb[5]=2946448255213557/7004255810773760,\nb[6]=16630875288/36849 43675,\nb[7]=-496773593038619375/9868243105287168,\nb[8]=27826183/6034 20,\n\n`b*`[1]=46875680038130591/1245460832642611200,\n`b*`[2]=0,\n`b* `[3]=0,\n`b*`[4]=2029502937748023643719/8086189994224235216645,\n`b*`[ 5]=1969638800119487132740431/4567069345598357799703040,\n`b*`[6]=44507 93456682602037/1201369121360654225,\n`b*`[7]=-253394715948809936175102 125/6434509503742783282563072,\n`b*`[8]=28284253367140001/786910432449 360,\n`b*`[9]=1/80\}:" }}}{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 appr oximate 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],s eq(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&#\"\"$\"#fF(%!GF+7&#\" #>\"$)>#\"%t7\"'C_B#\"&*H@F2F+7&#F.\"$K\"#F.\"$G&\"\"!#F.\"$w\"7&#\"#) )\"$x\"#\"*%)Hz#H\"*daUA#F:#!*/6>t$\")>=9u7&F+F+F+#\"*7\\X7$FF7&#\"#N \"#O#!2l+9/oC#G$*\"13#\"2XZ5_&42t`\"1;sKHQ,7(*7&#\"$W&\"$X&#!@))QW@r.]Vp0 >xQe\"G\">vo4umd))\\L5*=xG_F:#\">#>S?\"*Gd/Egz>(f*Q\"7& F+#!B'\\muco!HYTgn1#='\\,)\"@v$pJ[yi1/Q/BM81^#\"AOV7v&p%R**Gg&Qn#faE\" @DJky2O&G3()HUj**GQ#!5#fElzLJ\"\\!o\"\"6v$4/->+l2F%*7&\"\"\"#!;tmR$oz' e#zhP1g(\":?Jd4[<*Hh`M*Q\"F:#\"6++z7d-x/$)4\"\"3Zpz8QJuw_7&F+#!?)>&*z- j)p!e+EX\"=F'4n[?>$oq&Hk\">7#\"?<.Ras$*=Sdy(\\L([W\">CiBsL))pERP*e $pK'#!/M'))[iY!\\\"1:7ssZlHH7&F+F+F+#!5v=lO1oR`$e%\"87Lp$3%oW:'H)G#7&F jo#\")<4wl\"++G45>F:F:7&F+#\".cgMcXS'\"/NT,kE![##\"1dN@b#[k%H\"1gPx5eD /q#\",)Gv3j;\"+vO%\\o$7&F+F+#!3v$>'QIftn\\\"1orG0JCo)*#\")$=Ey#\"'?Mg7 &F+%F_____________________________________GFerFer7&%\"bGF`qF:F:FcqF]r7 &%#b*G#\"2\"fI\"Q+ovo%\"4+7hUE$3YX7F:F:7&F+#\"7>PkB![x$H]H?\"7Xm@NUA%* **='3)#\":J/uKr[>,!)Q'p>\":SIq*zd$)fX$pqc%#\"4P?g#ocMz]W\"4DUlg87p8?\" 7&F+#! " 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%'matri xG6#7,7,$\")eu%3&!\"*F(%!GF+F+F+F+F+F+F+7,$\")'fff*F*$\")F'=T&!#5$\")L xa!*F*F+F+F+F+F+F+F+7,$\")RRR9!\")$\")[[)f$F*$\"\"!F;$\")baz5F7F+F+F+F +F+F+7,$\")9vr\\F7$\")XO;8!\"(F:$!)lZL]FC$\")rG9UFCF+F+F+F+F+7,$\")AAA (*F7$!)8\\#=%!\"'F:$\")ur#f\"!\"&$!)+2?7FP$\")jRKbFCF+F+F+F+7,$\")9l\" )**F7$!)vF&Q&FMF:$\")C.\\?FP$!)Mnp:FP$\")\"oG$pFC$!)>i#y\"F*F+F+F+7,$ \"\"\"F;$!)6mqaFMF:$\")lS\"3#FP$!)sY%f\"FP$\")'=9.(FC$!)N9u;F*$!)N..?F 1F+F+7,F]o$\")F\"GW$F*F:F:$\")]?#e#F7$\")Vl1UF7$\")m>8XFC$!)K1M]FM$\") @T6YFMF+7,%\"bGF\\pF:F:F^pF`pFbpFdpFfpF+7,%#b*G$\") " 0 "" {MPLTEXT 1 0 146 "RK6_8eqs := [op(RowSumConditions(8,'expanded')),op(OrderConditi ons(6,8,'expanded'))]:\n`RK5_9eqs*` := subs(b=`b*`,OrderConditions(5,9 ,'expanded')):" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 7 "Check: " }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 148 "simp lify(subs(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#$\"+h5vmE!#:" }} }{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#$\"+s[q()[!#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] = 3/59;" "6#/&%\"cG6#\"\"#*&\"\"$\"\"\"\"#f!\"\"" }{TEXT -1 3 ", " }{XPPEDIT 18 0 "c[4] = 19/132;" "6#/&%\"cG6#\"\"%*&\"#>\"\" \"\"$K\"!\"\"" }{TEXT -1 3 ", " }{XPPEDIT 18 0 "c[5] = 88/177;" "6#/& %\"cG6#\"\"&*&\"#))\"\"\"\"$x\"!\"\"" }{TEXT -1 3 ", " }{XPPEDIT 18 0 "c[6] = 35/36;" "6#/&%\"cG6#\"\"'*&\"#N\"\"\"\"#O!\"\"" }{TEXT -1 3 ", " }{XPPEDIT 18 0 "c[7] = 544/545;" "6#/&%\"cG6#\"\"(*&\"$W&\"\"\" \"$X&!\"\"" }{TEXT -1 3 ", " }{XPPEDIT 18 0 "c[8]=1" "6#/&%\"cG6#\"\" )\"\"\"" }{TEXT -1 2 ", " }}{PARA 0 "" 0 "" {TEXT -1 31 "the zero link ing 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 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] = 1/80;" "6# /&%#b*G6#\"\"**&\"\"\"F)\"#!)!\"\"" }{TEXT -1 2 ". " }}{PARA 0 "" 0 " " {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 37 "We have 44 equations an d 44 unknowns." }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 211 "e1 := \{c [2]=3/59,c[4]=19/132,c[5]=88/177,c[6]=35/36,c[7]=544/545,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]=1/80\}:\neqns := subs(e1,cdns):\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)\}):\ninfolev el[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 2087 "e3 := \{c[7] = 544/545, c[3] = 19/198, \+ c[4] = 19/132, c[5] = 88/177, c[6] = 35/36, a[2,1] = 3/59, a[3,1] = 12 73/235224, a[6,5] = 53730709552104745/9712013829327216, a[3,2] = 21299 /235224, c[2] = 3/59, a[4,1] = 19/528, a[4,3] = 19/176, a[5,1] = 29279 2984/222425457, a[5,3] = -373191104/74141819, a[5,4] = 312454912/74141 819, a[6,3] = 14352497130136165/90113269909968, a[6,4] = -243046052805 9531110/19920664729472301, a[7,1] = -2815838771905694350037121443888/5 2287718910334988576674096875, a[6,1] = -93282246804140065/223030343027 1708, a[7,3] = 38959719796026045728912040192/1901371596739454130060876 25, a[7,4] = -801496182066760414629068567466496/5106133423043804066278 483169375, a[7,5] = 26545926738560289939469575124336/38289963422987082 85360778643125, a[7,6] = -16804913133796526592/942707650019020409375, \+ a[8,1] = -76006376179258679683396673/1389345361299174809573120, a[9,1] = 65760917/1910092800, a[9,4] = 6404556346056/24802664014135, a[9,5] \+ = 2946448255213557/7004255810773760, a[9,6] = 16630875288/3684943675, \+ a[9,7] = -496773593038619375/9868243105287168, a[8,3] = 10983047702571 2790000/527674313813796947, a[8,4] = -194391745260058069863027995198/1 219164295706831920486709627, a[8,5] = 444873349778574018937254390317/6 3269358937392669883372236224, a[8,6] = -49046624888634/292965477272121 5, a[8,7] = -45835339680636651875/22882961544684083693312, a[9,8] = 27 826183/603420, b[1] = 65760917/1910092800, b[4] = 6404556346056/248026 64014135, b[5] = 2946448255213557/7004255810773760, b[6] = 16630875288 /3684943675, b[7] = -496773593038619375/9868243105287168, b[8] = 27826 183/603420, b[2] = 0, a[5,2] = 0, a[6,2] = 0, a[7,2] = 0, a[8,2] = 0, \+ a[4,2] = 0, `b*`[1] = 46875680038130591/1245460832642611200, `b*`[8] = 28284253367140001/786910432449360, a[9,2] = 0, a[9,3] = 0, `b*`[9] = \+ 1/80, c[9] = 1, c[8] = 1, b[3] = 0, `b*`[2] = 0, `b*`[3] = 0, `b*`[7] \+ = -253394715948809936175102125/6434509503742783282563072, `b*`[6] = 44 50793456682602037/1201369121360654225, `b*`[4] = 202950293774802364371 9/8086189994224235216645, `b*`[5] = 1969638800119487132740431/45670693 45598357799703040\}:" }}}{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 approxim ate form is as follows." }}{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],s eq(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&#\"\"$\"#fF(%!GF+7&#\" #>\"$)>#\"%t7\"'C_B#\"&*H@F2F+7&#F.\"$K\"#F.\"$G&\"\"!#F.\"$w\"7&#\"#) )\"$x\"#\"*%)Hz#H\"*daUA#F:#!*/6>t$\")>=9u7&F+F+F+#\"*7\\X7$FF7&#\"#N \"#O#!2l+9/oC#G$*\"13#\"2XZ5_&42t`\"1;sKHQ,7(*7&#\"$W&\"$X&#!@))QW@r.]Vp0 >xQe\"G\">vo4umd))\\L5*=xG_F:#\">#>S?\"*Gd/Egz>(f*Q\"7& F+#!B'\\muco!HYTgn1#='\\,)\"@v$pJ[yi1/Q/BM81^#\"AOV7v&p%R**Gg&Qn#faE\" @DJky2O&G3()HUj**GQ#!5#fElzLJ\"\\!o\"\"6v$4/->+l2F%*7&\"\"\"#!;tmR$oz' e#zhP1g(\":?Jd4[<*Hh`M*Q\"F:#\"6++z7d-x/$)4\"\"3Zpz8QJuw_7&F+#!?)>&*z- j)p!e+EX\"=F'4n[?>$oq&Hk\">7#\"?<.Ras$*=Sdy(\\L([W\">CiBsL))pERP*e $pK'#!/M'))[iY!\\\"1:7ssZlHH7&F+F+F+#!5v=lO1oR`$e%\"87Lp$3%oW:'H)G#7&F jo#\")<4wl\"++G45>F:F:7&F+#\".cgMcXS'\"/NT,kE![##\"1dN@b#[k%H\"1gPx5eD /q#\",)Gv3j;\"+vO%\\o$7&F+F+#!3v$>'QIftn\\\"1orG0JCo)*#\")$=Ey#\"'?Mg7 &F+%F_____________________________________GFerFer7&%\"bGF`qF:F:FcqF]r7 &%#b*G#\"2\"fI\"Q+ovo%\"4+7hUE$3YX7F:F:7&F+#\"7>PkB![x$H]H?\"7Xm@NUA%* **='3)#\":J/uKr[>,!)Q'p>\":SIq*zd$)fX$pqc%#\"4P?g#ocMz]W\"4DUlg87p8?\" 7&F+#! " 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%'matrix G6#7,7,$\")eu%3&!\"*F(%!GF+F+F+F+F+F+F+7,$\")'fff*F*$\")F'=T&!#5$\")Lx a!*F*F+F+F+F+F+F+F+7,$\")RRR9!\")$\")[[)f$F*$\"\"!F;$\")baz5F7F+F+F+F+ F+F+7,$\")9vr\\F7$\")XO;8!\"(F:$!)lZL]FC$\")rG9UFCF+F+F+F+F+7,$\")AAA( *F7$!)8\\#=%!\"'F:$\")ur#f\"!\"&$!)+2?7FP$\")jRKbFCF+F+F+F+7,$\")9l\") **F7$!)vF&Q&FMF:$\")C.\\?FP$!)Mnp:FP$\")\"oG$pFC$!)>i#y\"F*F+F+F+7,$\" \"\"F;$!)6mqaFMF:$\")lS\"3#FP$!)sY%f\"FP$\")'=9.(FC$!)N9u;F*$!)N..?F1F +F+7,F]o$\")F\"GW$F*F:F:$\")]?#e#F7$\")Vl1UF7$\")m>8XFC$!)K1M]FM$\")@T 6YFMF+7,%\"bGF\\pF:F:F^pF`pFbpFdpFfpF+7,%#b*G$\") " 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(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 4 " .. " }{XPPEDIT 18 0 "c[7] = 544/545;" "6#/&%\"cG6#\"\"(*& \"$W&\"\"\"\"$X&!\"\"" }{TEXT -1 1 " " }}{PARA 4 "" 0 "" {TEXT -1 0 " " }}{PARA 0 "" 0 "" {TEXT -1 82 "In this subsection we obtain the node s 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*(30*c[5]*c[6]*c[7]*c[4]-10*c[5]*c[6]*c[4]-10*c[6]*c[7]*c[5]+ 5*c[6]*c[5]-10*c[6]*c[7]*c[4]+5*c[6]*c[4]+5*c[7]*c[6]-3*c[6]-10*c[5]*c [7]*c[4]+5*c[5]*c[4]+5*c[7]*c[5]-3*c[5]+5*c[7]*c[4]-3*c[4]+2-3*c[7])/c [5]/c[6]/c[7]/c[4], a[6,1] = -1/4*c[6]*(60*c[5]^2*c[6]*c[4]^4-90*c[5]^ 2*c[6]^3*c[4]^2-60*c[5]*c[6]^2*c[4]^4-2*c[6]^3*c[4]^2+50*c[5]^4*c[4]^3 *c[6]+40*c[6]^2*c[4]^5*c[5]-40*c[4]^5*c[5]^3+36*c[4]*c[5]^2*c[6]^3-20* c[5]^4*c[6]^2*c[4]^2+8*c[6]^2*c[5]^3+4*c[6]^2*c[4]^3-24*c[5]*c[6]^3*c[ 4]^2-8*c[6]^2*c[4]^2*c[5]+24*c[6]^2*c[5]*c[4]^3+60*c[4]^3*c[5]*c[6]^3- 4*c[6]*c[5]*c[4]^3-14*c[5]^3*c[6]*c[4]+7*c[5]^3*c[4]^2+6*c[5]^2*c[6]*c [4]^3+14*c[5]^2*c[6]*c[4]^2+66*c[5]^3*c[6]*c[4]^2-80*c[5]^3*c[6]^2*c[4 ]^3+30*c[6]^2*c[5]^2*c[4]^3-24*c[6]^2*c[5]^2*c[4]^2-240*c[6]*c[5]^3*c[ 4]^3-48*c[6]^2*c[5]^3*c[4]+150*c[6]^2*c[5]^3*c[4]^2-24*c[4]^4*c[5]^2-9 *c[5]^2*c[4]^3+6*c[4]^4*c[5]-40*c[5]^4*c[4]^4-20*c[6]^2*c[4]^4*c[5]^2- 6*c[5]^2*c[6]^3-2*c[6]*c[4]^4-24*c[5]^3*c[4]^3+60*c[6]^3*c[4]^3*c[5]^2 +7*c[6]^3*c[4]*c[5]-40*c[5]^2*c[6]*c[4]^5-40*c[6]^3*c[4]^4*c[5]+110*c[ 4]^4*c[6]*c[5]^3+120*c[5]^3*c[4]^4)/c[5]/c[4]^2/(10*c[5]^3*c[4]^2+10*c [5]^2*c[4]^3-30*c[5]^2*c[4]^2+6*c[5]^2*c[4]-c[5]^2+6*c[5]*c[4]^2+c[5]* c[4]-c[4]^2), b[4] = 1/60*(-2+3*c[5]-5*c[7]*c[6]+3*c[7]+3*c[6]+10*c[6] *c[7]*c[5]-5*c[6]*c[5]-5*c[7]*c[5])/c[4]/(-c[6]*c[4]^3-c[4]^3+c[4]^4+c [7]*c[4]^2+c[6]*c[4]^2-c[6]*c[7]*c[4]+c[6]*c[7]*c[4]^2-c[5]*c[4]^3+c[5 ]*c[4]^2-c[7]*c[4]^3-c[5]*c[7]*c[4]+c[5]*c[7]*c[4]^2-c[5]*c[6]*c[4]+c[ 5]*c[6]*c[4]^2+c[6]*c[7]*c[5]-c[5]*c[6]*c[7]*c[4]), b[6] = 1/60*(10*c[ 5]*c[7]*c[4]-5*c[5]*c[4]-5*c[7]*c[5]+3*c[5]-5*c[7]*c[4]+3*c[4]-2+3*c[7 ])/(-c[4]+c[6])/(-c[5]+c[6])/c[6]/(c[6]^2-c[6]+c[7]-c[7]*c[6]), `b*`[9 ] = 1/10*(50*c[5]^2*c[6]*c[4]^2-40*c[5]^2*c[6]*c[4]+10*c[6]*c[5]^2-40* c[5]*c[6]*c[4]^2+35*c[5]*c[6]*c[4]-10*c[6]*c[5]+10*c[6]*c[4]^2-10*c[6] *c[4]+3*c[6]-5*c[5]^2*c[4]-5*c[5]*c[4]^2+10*c[5]*c[4]-c[5]-c[4])/(3*c[ 6]-28*c[5]*c[6]*c[4]^2-28*c[5]^2*c[6]*c[4]+8*c[6]*c[5]^2-9*c[6]*c[4]-9 *c[6]*c[5]+28*c[5]*c[6]*c[4]+30*c[5]^2*c[6]*c[4]^2+8*c[6]*c[4]^2-4*c[5 ]^2*c[4]-4*c[5]*c[4]^2+9*c[5]*c[4]-c[5]-c[4]), a[7,5] = -1/2*c[7]*(-17 *c[5]^2*c[6]*c[4]^4+5*c[6]*c[5]*c[4]^4+42*c[4]^2*c[5]^2*c[7]^3+17*c[7] ^3*c[5]*c[4]^3-80*c[7]^3*c[5]^2*c[4]^3+c[7]*c[5]*c[4]^4+80*c[5]^2*c[4] ^4*c[7]^2-60*c[4]^5*c[6]^2*c[5]^2+20*c[4]^3*c[7]^3*c[5]^3-20*c[7]^2*c[ 5]^2*c[4]^5+40*c[4]^5*c[6]*c[5]^3-5*c[5]*c[6]^2*c[4]^4+c[7]^3*c[4]^2-c [7]^2*c[4]^3-80*c[5]^3*c[4]^4*c[7]+20*c[7]^3*c[5]^2*c[4]^4-200*c[6]^2* c[4]^5*c[7]*c[5]^3-290*c[5]^4*c[4]^2*c[7]^2*c[6]+20*c[6]^2*c[7]*c[5]^4 *c[4]^2+40*c[6]*c[5]^2*c[4]^4*c[7]-140*c[5]^4*c[4]^3*c[6]-100*c[5]^4*c [4]^5*c[6]+190*c[5]^4*c[4]^4*c[6]-40*c[7]^2*c[4]^4*c[5]^3+20*c[5]^4*c[ 4]^4*c[7]+2*c[4]^4*c[7]^2-20*c[7]^2*c[5]^4*c[4]^3-4*c[6]^2*c[7]^2*c[5] -8*c[7]*c[6]^2*c[5]^3-3*c[7]*c[6]*c[5]^3+2*c[7]^2*c[6]*c[5]^3+60*c[7]* c[6]^2*c[5]*c[4]^5-40*c[6]*c[5]^2*c[4]^5*c[7]+17*c[7]*c[4]^4*c[6]*c[5] +80*c[7]^2*c[5]^3*c[4]^3+5*c[7]*c[4]^4*c[6]^2+c[5]^2*c[7]^3-260*c[6]^2 *c[7]*c[5]^2*c[4]^3-12*c[6]^2*c[7]^2*c[5]^2*c[4]+20*c[6]^2*c[7]^2*c[5] ^2*c[4]^2-40*c[6]*c[7]*c[5]^3*c[4]^3-41*c[6]^2*c[7]^3*c[4]*c[5]+20*c[7 ]^2*c[6]*c[5]*c[4]^4+3*c[7]*c[6]*c[4]^3-2*c[6]^2*c[7]*c[4]^2-3*c[6]*c[ 7]^2*c[4]^2+4*c[7]^2*c[6]*c[4]^3+6*c[5]*c[7]^2*c[4]^3+4*c[6]*c[5]*c[7] ^2*c[4]^2+c[7]^2*c[5]*c[6]*c[4]-11*c[5]*c[7]*c[6]*c[4]^3+2*c[6]^2*c[4] ^2*c[5]-c[7]^3*c[4]*c[5]+17*c[5]^2*c[6]*c[7]*c[4]^2-200*c[5]^4*c[7]^2* c[4]^4*c[6]-50*c[4]^5*c[6]*c[7]*c[5]^3-150*c[5]^2*c[6]*c[7]^3*c[4]^4-c [6]^2*c[5]*c[4]^3+2*c[7]^3*c[4]^2*c[6]*c[5]+2*c[6]^2*c[7]*c[4]*c[5]+10 8*c[7]*c[6]^2*c[5]*c[4]^3+4*c[6]^2*c[7]*c[5]^2+2*c[6]*c[5]*c[7]*c[4]^2 -10*c[6]^2*c[7]^3*c[5]^2+6*c[5]^4*c[7]*c[6]-8*c[6]*c[7]^3*c[5]^2+3*c[6 ]*c[7]^2*c[5]^2+2*c[6]^2*c[7]^2*c[5]^2-27*c[6]^2*c[4]^2*c[7]*c[5]-62*c [6]^2*c[7]^2*c[4]^2*c[5]+46*c[6]*c[5]^4*c[4]^2+20*c[4]^5*c[7]*c[5]^3+2 *c[7]^3*c[4]*c[6]*c[5]-27*c[6]^2*c[5]^2*c[7]*c[4]+120*c[6]^2*c[5]^2*c[ 7]*c[4]^2-35*c[6]*c[5]^2*c[7]*c[4]^3-37*c[6]*c[5]^3*c[7]*c[4]^2+53*c[6 ]^2*c[5]^3*c[7]*c[4]-54*c[7]^2*c[5]^2*c[4]^3+2*c[7]^2*c[5]*c[4]^2-c[7] *c[5]*c[4]^3-3*c[6]*c[5]*c[4]^3-2*c[4]^3*c[7]^3+63*c[7]^3*c[5]^2*c[6]* c[4]-16*c[7]^2*c[5]^2*c[6]*c[4]-5*c[6]*c[7]*c[4]^4+2*c[6]^2*c[7]^2*c[4 ]^2+2*c[6]^2*c[7]^2*c[4]-10*c[5]^4*c[7]^2*c[6]+3*c[5]^3*c[6]*c[4]+3*c[ 5]^3*c[7]*c[4]+20*c[5]^3*c[6]*c[7]*c[4]-2*c[5]^3*c[4]^2+22*c[6]^2*c[7] ^2*c[4]*c[5]+7*c[5]^2*c[6]*c[4]^3-c[5]^2*c[7]*c[4]^2+c[5]^2*c[6]*c[4]^ 2-4*c[5]^2*c[7]*c[4]^3-8*c[5]^3*c[7]*c[4]^2-22*c[5]^3*c[6]*c[4]^2+240* c[5]^3*c[6]^2*c[4]^3+100*c[4]^3*c[7]^2*c[5]^2*c[6]^2+150*c[4]^5*c[5]^3 *c[6]^2-3*c[7]*c[5]^4*c[4]+5*c[7]^2*c[5]^4*c[4]+12*c[5]^2*c[4]^4*c[7]- 98*c[6]^2*c[5]^2*c[4]^3+25*c[6]^2*c[5]^2*c[4]^2+54*c[6]*c[5]^3*c[4]^3+ 8*c[6]^2*c[5]^3*c[4]-63*c[6]^2*c[5]^3*c[4]^2+42*c[5]^3*c[7]*c[4]^3+60* c[5]^4*c[7]*c[4]^2*c[6]+200*c[5]^3*c[7]^3*c[4]^4*c[6]-4*c[6]^2*c[4]*c[ 5]^2+c[7]*c[6]^2*c[4]^3+200*c[7]^2*c[5]^2*c[6]*c[4]^3-240*c[7]^3*c[5]^ 2*c[6]*c[4]^2+180*c[7]^2*c[5]^3*c[6]*c[4]^2+3*c[5]^4*c[4]^2-80*c[5]^3* c[7]^3*c[6]*c[4]+c[7]^3*c[4]^2*c[6]-7*c[7]^3*c[4]^2*c[5]+20*c[4]^3*c[7 ]^2*c[5]*c[6]^2-19*c[7]^2*c[6]*c[5]*c[4]^3-11*c[7]^2*c[5]^2*c[6]*c[4]^ 2-3*c[4]^4*c[5]^2-27*c[7]^2*c[5]^3*c[6]*c[4]-480*c[7]^2*c[5]^3*c[6]*c[ 4]^3-380*c[5]^3*c[6]^2*c[4]^4-5*c[5]^4*c[4]^3-10*c[6]^2*c[4]^3*c[7]^2+ 130*c[5]^3*c[6]^2*c[7]*c[4]^3-5*c[7]^3*c[5]^2*c[4]-2*c[7]^2*c[5]^2*c[4 ]-4*c[5]^2*c[6]*c[7]*c[4]-160*c[6]^2*c[7]*c[5]^3*c[4]^2-80*c[6]^2*c[7] ^2*c[4]*c[5]^3+2*c[5]^2*c[4]^3+12*c[7]^2*c[5]^2*c[4]^2+8*c[7]^2*c[5]^3 *c[4]+400*c[7]^3*c[5]^2*c[6]^2*c[4]^3-17*c[7]^2*c[4]^4*c[5]+250*c[7]^2 *c[5]^3*c[6]^2*c[4]^2-100*c[6]^2*c[4]^5*c[5]*c[7]^2-200*c[7]^2*c[4]^5* c[6]*c[5]^3+150*c[5]^2*c[6]*c[4]^5*c[7]^2+380*c[7]^3*c[5]^2*c[4]^3*c[6 ]-20*c[7]^3*c[6]*c[5]*c[4]^3-210*c[7]^3*c[6]^2*c[5]*c[4]^3-6*c[5]^4*c[ 4]*c[6]+80*c[5]^4*c[7]^2*c[4]*c[6]+10*c[6]*c[7]^3*c[5]^3-36*c[5]^4*c[4 ]*c[7]*c[6]-50*c[4]^5*c[6]^2*c[7]*c[5]^2+200*c[4]^5*c[6]^2*c[7]^2*c[5] ^2+160*c[6]^2*c[4]^4*c[5]^2-450*c[7]^3*c[5]^3*c[4]^3*c[6]+5*c[6]^2*c[7 ]^3*c[4]^2-c[5]^3*c[7]^2+6*c[6]^2*c[7]^3*c[5]-160*c[7]*c[6]^2*c[5]*c[4 ]^4+200*c[5]^3*c[4]^4*c[6]^2*c[7]-350*c[5]^3*c[4]^3*c[6]^2*c[7]^2-350* c[5]^4*c[4]^4*c[6]*c[7]+230*c[5]^2*c[4]^4*c[7]*c[6]^2+150*c[5]^3*c[4]^ 4*c[6]*c[7]+100*c[5]^4*c[4]^3*c[6]*c[7]-3*c[6]^2*c[7]^3*c[4]-42*c[7]^2 *c[5]^3*c[4]^2+50*c[5]^4*c[4]^4*c[6]^2-20*c[5]^4*c[4]^3*c[6]^2+500*c[5 ]^3*c[4]^4*c[7]^2*c[6]+450*c[5]^4*c[4]^3*c[7]^2*c[6]-340*c[5]^2*c[4]^4 *c[7]^2*c[6]-50*c[5]^4*c[4]^3*c[6]^2*c[7]+150*c[7]^2*c[4]^4*c[6]^2*c[5 ]+200*c[7]^2*c[4]^4*c[5]^3*c[6]^2-350*c[7]^2*c[4]^4*c[6]^2*c[5]^2+10*c [7]^2*c[6]^2*c[5]^3-60*c[4]^4*c[6]*c[5]^3+100*c[7]^3*c[6]^2*c[5]*c[4]^ 4+200*c[5]^4*c[4]^5*c[6]*c[7]-200*c[7]^3*c[4]^4*c[6]^2*c[5]^2+5*c[5]^3 *c[4]^4+80*c[6]^2*c[7]^3*c[5]^2*c[4]-270*c[6]^2*c[7]^3*c[5]^2*c[4]^2+2 90*c[5]^3*c[7]^3*c[4]^2*c[6]+140*c[6]^2*c[5]*c[7]^3*c[4]^2-5*c[7]^3*c[ 4]*c[5]^3)/c[5]/(100*c[5]^2*c[6]*c[4]^4-18*c[6]*c[5]*c[4]^4+4*c[5]^3*c [4]+50*c[4]^5*c[6]^2*c[5]^2+40*c[5]*c[6]^2*c[4]^4+3*c[5]^5-5*c[4]^4*c[ 6]^2+180*c[5]^4*c[4]^3*c[6]+100*c[5]^4*c[4]^5*c[6]-350*c[5]^4*c[4]^4*c [6]+140*c[5]^5*c[4]^2-180*c[5]^5*c[4]^3-2*c[5]^4+30*c[4]^5*c[5]^3+300* c[5]^5*c[4]^3*c[6]+2*c[6]*c[5]^3-50*c[5]^5*c[4]^2*c[6]^2-40*c[5]^4*c[6 ]^2*c[4]+240*c[5]^4*c[6]^2*c[4]^2-3*c[6]^2*c[5]^3+3*c[6]^2*c[4]^3+5*c[ 5]^4*c[6]^2-2*c[6]*c[4]^3+2*c[5]*c[4]^3+4*c[5]*c[6]*c[4]^2-6*c[6]^2*c[ 4]^2*c[5]-50*c[5]^4*c[4]^5-13*c[6]^2*c[5]*c[4]^3-40*c[6]*c[5]^4*c[4]^2 +6*c[6]*c[5]*c[4]^3-15*c[5]^3*c[6]*c[4]-55*c[5]^2*c[6]*c[4]^3+6*c[5]^2 *c[6]*c[4]^2+68*c[5]^3*c[6]*c[4]^2-100*c[4]^5*c[5]^3*c[6]^2+100*c[6]^2 *c[5]^2*c[4]^3-100*c[6]*c[5]^3*c[4]^3+13*c[6]^2*c[5]^3*c[4]-100*c[6]^2 *c[5]^3*c[4]^2-4*c[5]^2*c[4]^2+6*c[6]^2*c[4]*c[5]^2-68*c[5]^4*c[4]^2+2 3*c[4]^4*c[5]^2+350*c[5]^3*c[6]^2*c[4]^4-4*c[5]^2*c[6]*c[4]-9*c[5]^2*c [4]^3-3*c[4]^4*c[5]-5*c[5]^5*c[6]-30*c[5]^6*c[4]^2+50*c[5]^6*c[6]*c[4] ^2+180*c[5]^4*c[4]^4+10*c[5]^4*c[4]*c[6]-240*c[6]^2*c[4]^4*c[5]^2+3*c[ 6]*c[4]^4-100*c[5]^6*c[4]^3*c[6]+68*c[5]^3*c[4]^3-23*c[5]^5*c[4]-210*c [5]^5*c[6]*c[4]^2-350*c[5]^4*c[4]^3*c[6]^2-30*c[5]^2*c[6]*c[4]^5+50*c[ 5]^6*c[4]^3+60*c[4]^4*c[6]*c[5]^3+100*c[5]^5*c[6]^2*c[4]^3+9*c[5]^4*c[ 4]-140*c[5]^3*c[4]^4+40*c[5]^5*c[4]*c[6]), `b*`[6] = -1/60*(-30*c[5]^3 *c[4]+103*c[5]*c[7]*c[4]-18*c[6]+6*c[5]+6*c[4]+27*c[7]*c[6]-9*c[5]^2-9 *c[4]^2-100*c[7]*c[6]*c[5]^3+522*c[5]*c[6]*c[7]*c[4]-111*c[6]*c[5]^2+7 5*c[6]*c[5]+60*c[6]*c[5]^3-9*c[7]*c[5]+600*c[6]*c[7]*c[5]^3*c[4]^3+14* c[7]*c[5]^2-100*c[7]*c[6]*c[4]^3+14*c[7]*c[4]^2-9*c[7]*c[4]-117*c[6]*c [7]*c[5]-111*c[6]*c[4]^2+60*c[6]*c[4]^3-117*c[6]*c[7]*c[4]+180*c[6]*c[ 7]*c[4]^2-30*c[5]*c[4]^3-174*c[5]*c[7]*c[4]^2+486*c[5]*c[6]*c[4]^2+107 *c[5]*c[4]^2-318*c[5]*c[6]*c[4]-66*c[5]*c[4]+75*c[6]*c[4]+500*c[5]*c[7 ]*c[6]*c[4]^3+1420*c[5]^2*c[6]*c[7]*c[4]^2-840*c[6]*c[5]*c[7]*c[4]^2+1 80*c[6]*c[7]*c[5]^2-900*c[6]*c[5]^2*c[7]*c[4]^3-900*c[6]*c[5]^3*c[7]*c [4]^2+50*c[7]*c[5]*c[4]^3-280*c[6]*c[5]*c[4]^3-280*c[5]^3*c[6]*c[4]+50 *c[5]^3*c[7]*c[4]+107*c[5]^2*c[4]+500*c[5]^3*c[6]*c[7]*c[4]+50*c[5]^3* c[4]^2+470*c[5]^2*c[6]*c[4]^3+310*c[5]^2*c[7]*c[4]^2-770*c[5]^2*c[6]*c [4]^2-90*c[5]^2*c[7]*c[4]^3-90*c[5]^3*c[7]*c[4]^2+470*c[5]^3*c[6]*c[4] ^2-300*c[6]*c[5]^3*c[4]^3-180*c[5]^2*c[4]^2+486*c[5]^2*c[6]*c[4]-174*c [5]^2*c[7]*c[4]-840*c[5]^2*c[6]*c[7]*c[4]+50*c[5]^2*c[4]^3)/(-9*c[5]*c [7]*c[4]-3*c[7]*c[6]+3*c[6]^2-28*c[5]*c[6]*c[7]*c[4]-c[6]*c[5]+c[7]*c[ 5]+8*c[6]^2*c[4]^2+c[7]*c[4]+9*c[6]*c[7]*c[5]+9*c[6]*c[7]*c[4]-8*c[6]* c[7]*c[4]^2+4*c[5]*c[7]*c[4]^2-4*c[5]*c[6]*c[4]^2+9*c[5]*c[6]*c[4]-c[6 ]*c[4]-28*c[6]^2*c[4]^2*c[5]+28*c[6]^2*c[4]*c[5]-30*c[5]^2*c[6]*c[7]*c [4]^2+28*c[6]*c[5]*c[7]*c[4]^2-8*c[6]*c[7]*c[5]^2+8*c[6]^2*c[5]^2-9*c[ 6]^2*c[5]+30*c[6]^2*c[5]^2*c[4]^2-28*c[6]^2*c[4]*c[5]^2-9*c[6]^2*c[4]- 4*c[5]^2*c[6]*c[4]+4*c[5]^2*c[7]*c[4]+28*c[5]^2*c[6]*c[7]*c[4])/(c[5]* c[4]-c[6]*c[4]-c[6]*c[5]+c[6]^2)/c[6], a[8,7] = (10*c[5]*c[6]*c[4]-5*c [6]*c[5]-5*c[6]*c[4]+3*c[6]-5*c[5]*c[4]+3*c[5]+3*c[4]-2)/c[7]/(c[6]-c[ 7])/(c[5]-c[7])/(c[4]-c[7])/(-20*c[5]*c[6]*c[4]+30*c[5]*c[6]*c[7]*c[4] +15*c[6]*c[5]-20*c[6]*c[7]*c[5]+15*c[6]*c[4]-20*c[6]*c[7]*c[4]-12*c[6] +15*c[7]*c[6]+15*c[5]*c[4]-20*c[5]*c[7]*c[4]-12*c[5]+15*c[7]*c[5]-12*c [4]+15*c[7]*c[4]+10-12*c[7])*(c[4]-1)*(-1+c[5])*(-1+c[6])*(-1+c[7]), a [6,3] = 3/4*1/c[4]^2*c[6]*(60*c[4]^3*c[5]*c[6]^3-90*c[5]*c[6]^3*c[4]^2 +36*c[6]^3*c[4]*c[5]-6*c[6]^3*c[5]+3*c[6]^3*c[4]-20*c[6]^2*c[5]^3*c[4] ^2-80*c[6]^2*c[5]^2*c[4]^3+150*c[6]^2*c[5]^2*c[4]^2-48*c[6]^2*c[4]*c[5 ]^2+8*c[6]^2*c[5]^2-60*c[5]*c[6]^2*c[4]^4+90*c[6]^2*c[5]*c[4]^3-48*c[6 ]^2*c[4]^2*c[5]+4*c[6]^2*c[4]*c[5]-4*c[6]^2*c[4]^2+30*c[6]*c[5]^3*c[4] ^3+90*c[5]^2*c[6]*c[4]^4-180*c[5]^2*c[6]*c[4]^3+54*c[5]^2*c[6]*c[4]^2- 12*c[5]^2*c[6]*c[4]+18*c[6]*c[5]*c[4]^3+6*c[5]*c[6]*c[4]^2+3*c[5]^2*c[ 4]^2-3*c[5]*c[4]^3)/(10*c[5]^3*c[4]^2+10*c[5]^2*c[4]^3-30*c[5]^2*c[4]^ 2+6*c[5]^2*c[4]-c[5]^2+6*c[5]*c[4]^2+c[5]*c[4]-c[4]^2), b[7] = -1/60*( 10*c[5]*c[6]*c[4]-5*c[6]*c[5]-5*c[6]*c[4]+3*c[6]-5*c[5]*c[4]+3*c[5]+3* c[4]-2)/(c[4]-c[7])/(c[5]-c[7])/(c[6]-c[7])/c[7]/(-1+c[7]), a[7,1] = 1 /4*c[7]*(-51*c[5]^2*c[6]*c[4]^4+14*c[6]*c[5]*c[4]^4+12*c[4]^2*c[5]^2*c [7]^3+2700*c[5]^4*c[4]^4*c[6]^2*c[7]+12*c[7]^3*c[5]*c[4]^3-98*c[7]^3*c [5]^2*c[4]^3-2*c[7]*c[5]*c[4]^4+132*c[5]^2*c[4]^4*c[7]^2-480*c[4]^5*c[ 6]^2*c[5]^2+200*c[4]^3*c[7]^3*c[5]^3-200*c[7]^2*c[5]^2*c[4]^5-760*c[4] ^5*c[6]*c[5]^3-10*c[5]*c[6]^2*c[4]^4+200*c[5]^5*c[4]^3*c[7]^2*c[6]^2-4 00*c[5]^2*c[4]^6*c[6]^2*c[7]^2+4*c[4]^5*c[5]*c[7]-34*c[5]^2*c[4]^5*c[7 ]+400*c[5]^3*c[4]^6*c[6]^2*c[7]-98*c[5]^3*c[4]^4*c[7]+420*c[7]^2*c[4]^ 5*c[5]^3+200*c[7]^3*c[5]^2*c[4]^4-300*c[6]^2*c[4]^5*c[7]*c[5]^3-40*c[7 ]*c[5]^2*c[4]^6*c[6]+20*c[5]^4*c[4]^2*c[7]^2*c[6]-280*c[6]^2*c[7]*c[5] ^4*c[4]^2-44*c[6]*c[4]^5*c[5]*c[7]-246*c[6]*c[5]^2*c[4]^4*c[7]+120*c[4 ]^6*c[6]^2*c[5]^2-100*c[5]^2*c[7]^2*c[4]^6*c[6]+141*c[5]^4*c[4]^3*c[6] +1220*c[5]^4*c[4]^5*c[6]+400*c[5]^5*c[4]^5*c[6]^2-760*c[5]^4*c[4]^4*c[ 6]+100*c[5]^4*c[7]^3*c[4]^4-400*c[7]^2*c[4]^4*c[5]^3-200*c[5]^4*c[4]^5 *c[7]^2+200*c[5]^4*c[4]^4*c[7]-2*c[4]^3*c[7]^3*c[6]-8*c[7]^2*c[6]*c[4] ^4-380*c[7]^3*c[5]^3*c[4]^4-40*c[6]*c[4]^5*c[5]*c[7]^2+2*c[4]^4*c[7]^2 -40*c[5]^2*c[7]^3*c[4]^5-200*c[7]^2*c[5]^4*c[4]^3-40*c[7]^3*c[5]^4*c[4 ]^3+200*c[5]^3*c[7]*c[4]^6*c[6]-10*c[6]^2*c[7]^3*c[4]^3+30*c[6]^2*c[4] ^5*c[5]+12*c[7]*c[6]^2*c[5]^3+4*c[7]^2*c[6]*c[5]^3+400*c[5]^4*c[4]^6*c [6]^2-40*c[5]^3*c[7]*c[4]^6-380*c[5]^4*c[4]^5*c[7]+320*c[7]*c[6]^2*c[5 ]*c[4]^5+100*c[4]^6*c[6]^2*c[7]*c[5]^2+34*c[5]^4*c[4]^2*c[7]^2+40*c[5] ^5*c[7]^2*c[4]^3+240*c[6]*c[5]^2*c[4]^5*c[7]+60*c[5]^4*c[7]^3*c[4]^2*c [6]+26*c[7]*c[4]^4*c[6]*c[5]-50*c[4]^4*c[6]^2*c[5]^5+200*c[5]^5*c[7]*c [4]^4*c[6]+132*c[7]^2*c[5]^3*c[4]^3-2*c[7]*c[4]^4*c[6]^2+60*c[5]^5*c[4 ]^2*c[6]^2*c[7]-500*c[5]^5*c[4]^4*c[6]^2*c[7]+160*c[5]^5*c[4]^4*c[6]+1 34*c[6]^2*c[5]^4*c[7]*c[4]+24*c[5]^4*c[6]^2*c[4]-169*c[5]^4*c[6]^2*c[4 ]^2-500*c[5]^3*c[4]^6*c[6]^2-40*c[5]^5*c[4]^4*c[7]+34*c[4]^5*c[5]*c[7] ^2-100*c[5]^3*c[7]^2*c[4]^6-100*c[5]^5*c[4]^4*c[7]^2+160*c[4]^6*c[6]*c [5]^3-6*c[6]^2*c[7]*c[5]^2*c[4]^3+68*c[6]^2*c[7]^2*c[5]^2*c[4]-126*c[6 ]^2*c[7]^2*c[5]^2*c[4]^2-600*c[6]*c[7]*c[5]^3*c[4]^3-21*c[6]^2*c[7]^3* c[4]*c[5]+40*c[7]^2*c[6]^2*c[5]^4+4*c[5]^4*c[7]*c[4]^2+58*c[7]^2*c[6]* c[5]*c[4]^4-100*c[7]^2*c[5]^5*c[6]^2*c[4]^2-320*c[5]^4*c[4]*c[7]^2*c[6 ]^2+20*c[6]^2*c[4]^4*c[7]^2+40*c[5]^2*c[4]^6*c[7]^2+6*c[7]^2*c[6]*c[4] ^3-8*c[6]*c[5]*c[7]^2*c[4]^2-4*c[5]*c[7]*c[6]*c[4]^3+14*c[5]^2*c[6]*c[ 7]*c[4]^2-910*c[4]^5*c[6]*c[7]*c[5]^3-340*c[5]^2*c[6]*c[7]^3*c[4]^4-4* c[6]^2*c[5]*c[4]^3+5*c[7]^3*c[4]^2*c[6]*c[5]+72*c[7]*c[6]^2*c[5]*c[4]^ 3+18*c[6]^2*c[7]^3*c[5]^2-12*c[6]^2*c[7]^2*c[5]^2-10*c[6]^2*c[4]^2*c[7 ]*c[5]-44*c[6]^2*c[7]^2*c[4]^2*c[5]-25*c[6]*c[5]^4*c[4]^2-24*c[7]*c[6] ^2*c[5]^4+200*c[4]^5*c[7]*c[5]^3-2*c[6]^2*c[5]^2*c[7]*c[4]-42*c[6]^2*c [5]^2*c[7]*c[4]^2+74*c[6]*c[5]^2*c[7]*c[4]^3+80*c[6]*c[5]^3*c[7]*c[4]^ 2-50*c[6]^2*c[5]^3*c[7]*c[4]-34*c[5]^4*c[4]^3*c[7]-24*c[7]^2*c[5]^2*c[ 4]^3+420*c[5]^4*c[7]^2*c[4]^4-2*c[4]^3*c[7]^3+c[7]^3*c[5]^2*c[6]*c[4]+ 4*c[7]^2*c[5]^2*c[6]*c[4]-6*c[6]*c[7]*c[4]^4-4*c[6]^2*c[7]^2*c[4]^2+50 *c[5]^5*c[4]^3*c[7]^2*c[6]-18*c[5]^3*c[6]*c[7]*c[4]+14*c[6]^2*c[7]^2*c [4]*c[5]-20*c[5]^2*c[6]*c[4]^3+2*c[5]^2*c[7]*c[4]^3-2*c[5]^3*c[7]*c[4] ^2+16*c[5]^3*c[6]*c[4]^2-96*c[5]^3*c[6]^2*c[4]^3+100*c[5]^5*c[4]^3*c[6 ]^2*c[7]-60*c[4]^3*c[7]^2*c[5]^2*c[6]^2+10*c[6]*c[7]*c[4]^5+1540*c[4]^ 5*c[5]^3*c[6]^2-34*c[7]^3*c[4]^4*c[5]-4*c[7]^2*c[5]^4*c[4]+12*c[5]^2*c [4]^4*c[7]-41*c[6]^2*c[5]^2*c[4]^3+14*c[6]^2*c[5]^2*c[4]^2-54*c[6]*c[5 ]^3*c[4]^3-12*c[6]^2*c[5]^3*c[4]+47*c[6]^2*c[5]^3*c[4]^2+12*c[5]^3*c[7 ]*c[4]^3-98*c[5]^4*c[7]*c[4]^2*c[6]-50*c[5]^3*c[7]^3*c[4]^4*c[6]+4*c[7 ]*c[6]^2*c[4]^3+156*c[7]^2*c[5]^2*c[6]*c[4]^3-23*c[7]^3*c[5]^2*c[6]*c[ 4]^2+188*c[7]^2*c[5]^3*c[6]*c[4]^2+66*c[5]^3*c[7]^3*c[6]*c[4]+2*c[7]^3 *c[4]^2*c[5]+100*c[5]^5*c[4]^5*c[7]+104*c[4]^3*c[7]^2*c[5]*c[6]^2-12*c [7]^2*c[6]*c[5]*c[4]^3-36*c[7]^2*c[5]^2*c[6]*c[4]^2-32*c[7]^2*c[5]^3*c [6]*c[4]-320*c[7]^2*c[5]^3*c[6]*c[4]^3-560*c[5]^3*c[6]^2*c[4]^4-4*c[6] ^2*c[4]^3*c[7]^2+240*c[5]^3*c[6]^2*c[7]*c[4]^3-34*c[5]^3*c[7]^3*c[4]^2 -2*c[7]^3*c[5]^2*c[4]+164*c[6]^2*c[7]*c[5]^3*c[4]^2-20*c[6]^2*c[7]^2*c [4]*c[5]^3+2*c[7]^2*c[5]^3*c[4]-130*c[5]^5*c[4]^3*c[6]*c[7]-90*c[7]^3* c[5]^2*c[6]^2*c[4]^3-16*c[7]^2*c[4]^4*c[5]-60*c[7]^2*c[5]^3*c[6]^2*c[4 ]^2-300*c[6]^2*c[4]^5*c[5]*c[7]^2-150*c[7]^2*c[4]^5*c[6]*c[5]^3+380*c[ 5]^2*c[6]*c[4]^5*c[7]^2+140*c[7]^3*c[5]^2*c[4]^3*c[6]-810*c[7]^3*c[5]^ 3*c[6]^2*c[4]^2-14*c[7]^3*c[6]*c[5]*c[4]^3-280*c[7]^3*c[6]^2*c[5]*c[4] ^3+240*c[7]^3*c[5]^3*c[6]^2*c[4]-20*c[5]^4*c[7]^2*c[4]*c[6]-6*c[6]*c[7 ]^3*c[5]^3+22*c[5]^4*c[4]*c[7]*c[6]-280*c[4]^5*c[6]^2*c[7]*c[5]^2+400* c[4]^5*c[6]^2*c[7]^2*c[5]^2+253*c[6]^2*c[4]^4*c[5]^2+470*c[7]^3*c[5]^3 *c[4]^3*c[6]+100*c[5]^2*c[4]^5*c[7]^3*c[6]+6*c[6]^2*c[7]^3*c[4]^2-300* c[5]^5*c[4]^5*c[6]-120*c[5]*c[7]*c[4]^6*c[6]^2-10*c[6]^2*c[7]*c[4]^5-1 100*c[5]^4*c[4]^5*c[6]^2*c[7]-226*c[7]*c[6]^2*c[5]*c[4]^4-770*c[5]^3*c [4]^4*c[6]^2*c[7]+280*c[5]^3*c[4]^3*c[6]^2*c[7]^2-1000*c[5]^4*c[4]^4*c [6]*c[7]+240*c[5]^2*c[4]^4*c[7]*c[6]^2+1240*c[5]^3*c[4]^4*c[6]*c[7]+60 0*c[5]^4*c[4]^3*c[6]*c[7]+1200*c[7]^3*c[5]^3*c[6]^2*c[4]^3+40*c[7]^3*c [4]^4*c[5]*c[6]-16*c[7]^2*c[5]^3*c[4]^2-1250*c[5]^4*c[4]^5*c[6]^2-600* c[6]^2*c[7]^3*c[5]^3*c[4]^4+200*c[6]^2*c[7]^2*c[5]^3*c[4]^5-200*c[6]^2 *c[7]^3*c[4]^5*c[5]-140*c[5]^4*c[4]^4*c[6]^2+560*c[5]^4*c[4]^3*c[6]^2- 1850*c[7]^2*c[5]^4*c[6]^2*c[4]^3+1230*c[7]^2*c[5]^4*c[6]^2*c[4]^2-150* c[5]^4*c[7]^3*c[4]^3*c[6]-300*c[5]^4*c[4]^6*c[6]+310*c[5]^3*c[4]^4*c[7 ]^2*c[6]-70*c[5]^4*c[4]^3*c[7]^2*c[6]-380*c[5]^2*c[4]^4*c[7]^2*c[6]-85 0*c[5]^4*c[4]^3*c[6]^2*c[7]-40*c[7]^2*c[4]^4*c[6]^2*c[5]-450*c[7]^2*c[ 4]^4*c[5]^3*c[6]^2+250*c[7]^2*c[4]^4*c[6]^2*c[5]^2-4*c[7]^2*c[4]^5+136 *c[5]^2*c[6]*c[4]^5+400*c[6]^2*c[7]^3*c[5]^2*c[4]^5+800*c[6]^2*c[7]^2* c[5]^4*c[4]^4+4*c[7]^3*c[4]^4+100*c[5]^4*c[4]^6*c[7]+437*c[4]^4*c[6]*c [5]^3-22*c[6]*c[5]*c[4]^5-60*c[5]^5*c[6]^2*c[4]^3+200*c[6]^2*c[4]^6*c[ 5]*c[7]^2+420*c[7]^3*c[6]^2*c[5]*c[4]^4+400*c[5]^4*c[4]^5*c[6]*c[7]-50 0*c[7]^3*c[4]^4*c[6]^2*c[5]^2+100*c[5]^3*c[7]^3*c[4]^5-103*c[6]^2*c[7] ^3*c[5]^2*c[4]+260*c[6]^2*c[7]^3*c[5]^2*c[4]^2-300*c[5]^3*c[7]^3*c[4]^ 2*c[6]+97*c[6]^2*c[5]*c[7]^3*c[4]^2-30*c[6]^2*c[7]^3*c[5]^3+4*c[7]^3*c [4]*c[5]^3)/c[6]/(-50*c[5]^2*c[6]*c[4]^4+23*c[5]^3*c[4]+100*c[5]^4*c[4 ]^3*c[6]+2*c[5]^2+2*c[4]^2-3*c[5]^3-3*c[4]^3-3*c[6]*c[5]^2+5*c[6]*c[5] ^3-3*c[6]*c[4]^2+5*c[6]*c[4]^3+23*c[5]*c[4]^3+18*c[5]*c[6]*c[4]^2-12*c [5]*c[4]^2+3*c[5]*c[6]*c[4]-2*c[5]*c[4]-50*c[6]*c[5]^4*c[4]^2-40*c[6]* c[5]*c[4]^3-40*c[5]^3*c[6]*c[4]-12*c[5]^2*c[4]-140*c[5]^3*c[4]^2+240*c [5]^2*c[6]*c[4]^3-140*c[5]^2*c[6]*c[4]^2+240*c[5]^3*c[6]*c[4]^2-400*c[ 6]*c[5]^3*c[4]^3+91*c[5]^2*c[4]^2+30*c[5]^4*c[4]^2+30*c[4]^4*c[5]^2-50 *c[5]^4*c[4]^3+18*c[5]^2*c[6]*c[4]-140*c[5]^2*c[4]^3+210*c[5]^3*c[4]^3 +100*c[4]^4*c[6]*c[5]^3-50*c[5]^3*c[4]^4)/c[5]/c[4]^2, `b*`[8] = 0, a[ 5,3] = -3/4*c[5]^2*(2*c[5]-3*c[4])/c[4]^2, `b*`[4] = -1/60*(-30*c[5]^3 *c[4]-42*c[7]*c[6]^2+89*c[5]*c[7]*c[4]-18*c[6]+6*c[5]+6*c[4]+27*c[7]*c [6]-9*c[5]^2+180*c[7]*c[6]^2*c[5]^3-100*c[7]*c[6]*c[5]^3+27*c[6]^2+186 *c[5]*c[6]*c[7]*c[4]-97*c[6]*c[5]^2+66*c[6]*c[5]+60*c[6]*c[5]^3-100*c[ 6]^2*c[5]^3-9*c[7]*c[5]+14*c[7]*c[5]^2-100*c[6]^2*c[7]*c[4]^2+120*c[6] ^2*c[7]*c[4]+60*c[6]^2*c[4]^2-9*c[7]*c[4]-103*c[6]*c[7]*c[5]-36*c[6]*c [4]^2-61*c[6]*c[7]*c[4]+60*c[6]*c[7]*c[4]^2-30*c[5]*c[7]*c[4]^2+126*c[ 5]*c[6]*c[4]^2+18*c[5]*c[4]^2-112*c[5]*c[6]*c[4]-57*c[5]*c[4]+39*c[6]* c[4]-280*c[6]^2*c[4]^2*c[5]+330*c[6]^2*c[4]*c[5]+380*c[5]^2*c[6]*c[7]* c[4]^2-560*c[6]^2*c[7]*c[4]*c[5]+192*c[6]^2*c[7]*c[5]-310*c[6]^2*c[7]* c[5]^2-230*c[6]*c[5]*c[7]*c[4]^2+156*c[6]*c[7]*c[5]^2+180*c[6]^2*c[5]^ 2+500*c[6]^2*c[4]^2*c[7]*c[5]+950*c[6]^2*c[5]^2*c[7]*c[4]-900*c[6]^2*c [5]^2*c[7]*c[4]^2-300*c[6]*c[5]^3*c[7]*c[4]^2-600*c[6]^2*c[5]^3*c[7]*c [4]-117*c[6]^2*c[5]-130*c[5]^3*c[6]*c[4]+50*c[5]^3*c[7]*c[4]+93*c[5]^2 *c[4]+230*c[5]^3*c[6]*c[7]*c[4]+50*c[5]^2*c[7]*c[4]^2-190*c[5]^2*c[6]* c[4]^2+150*c[5]^3*c[6]*c[4]^2+470*c[6]^2*c[5]^2*c[4]^2+320*c[6]^2*c[5] ^3*c[4]-300*c[6]^2*c[5]^3*c[4]^2-30*c[5]^2*c[4]^2-530*c[6]^2*c[4]*c[5] ^2-75*c[6]^2*c[4]+156*c[5]^2*c[6]*c[4]-150*c[5]^2*c[7]*c[4]-270*c[5]^2 *c[6]*c[7]*c[4]+600*c[6]^2*c[7]*c[5]^3*c[4]^2)/(-9*c[5]*c[7]*c[4]-3*c[ 7]*c[6]-c[4]^2-28*c[5]*c[6]*c[7]*c[4]+c[7]*c[5]+c[7]*c[4]+9*c[6]*c[7]* c[5]-9*c[6]*c[4]^2+8*c[6]*c[4]^3+9*c[6]*c[7]*c[4]-8*c[6]*c[7]*c[4]^2-4 *c[5]*c[4]^3+4*c[5]*c[7]*c[4]^2+28*c[5]*c[6]*c[4]^2+9*c[5]*c[4]^2-9*c[ 5]*c[6]*c[4]-c[5]*c[4]+3*c[6]*c[4]-30*c[5]^2*c[6]*c[7]*c[4]^2+28*c[6]* c[5]*c[7]*c[4]^2-8*c[6]*c[7]*c[5]^2-28*c[6]*c[5]*c[4]^3+30*c[5]^2*c[6] *c[4]^3-28*c[5]^2*c[6]*c[4]^2-4*c[5]^2*c[4]^2+8*c[5]^2*c[6]*c[4]+4*c[5 ]^2*c[7]*c[4]+28*c[5]^2*c[6]*c[7]*c[4])/(-c[4]+c[6])/(-c[4]+c[5])/c[4] , a[8,6] = 1/2*(5*c[5]*c[6]*c[4]+c[6]-2*c[6]*c[4]-2*c[6]*c[5]+25*c[5]* c[7]*c[4]+6*c[5]+9*c[7]+10*c[7]^2*c[4]-4+6*c[4]-6*c[7]^2-14*c[7]*c[4]- 10*c[5]*c[4]-14*c[7]*c[5]-20*c[7]^2*c[5]*c[4]+10*c[7]^2*c[5])/(c[6]-c[ 7])/(-c[4]+c[6])/(-c[5]+c[6])/c[6]/(-20*c[5]*c[6]*c[4]+30*c[5]*c[6]*c[ 7]*c[4]+15*c[6]*c[5]-20*c[6]*c[7]*c[5]+15*c[6]*c[4]-20*c[6]*c[7]*c[4]- 12*c[6]+15*c[7]*c[6]+15*c[5]*c[4]-20*c[5]*c[7]*c[4]-12*c[5]+15*c[7]*c[ 5]-12*c[4]+15*c[7]*c[4]+10-12*c[7])*(c[4]-1)*(-1+c[5])*(-1+c[6]), b[8] = 1/60*(-20*c[5]*c[6]*c[4]+30*c[5]*c[6]*c[7]*c[4]+15*c[6]*c[5]-20*c[6 ]*c[7]*c[5]+15*c[6]*c[4]-20*c[6]*c[7]*c[4]-12*c[6]+15*c[7]*c[6]+15*c[5 ]*c[4]-20*c[5]*c[7]*c[4]-12*c[5]+15*c[7]*c[5]-12*c[4]+15*c[7]*c[4]+10- 12*c[7])/(c[4]-1)/(-1+c[5])/(-1+c[6])/(-1+c[7]), a[5,1] = 1/4*c[5]*(2* c[5]^2-5*c[5]*c[4]+4*c[4]^2)/c[4]^2, `b*`[5] = 1/60*(-42*c[7]*c[6]^2+8 9*c[5]*c[7]*c[4]-18*c[6]+6*c[5]+6*c[4]+27*c[7]*c[6]-9*c[4]^2+27*c[6]^2 +186*c[5]*c[6]*c[7]*c[4]-36*c[6]*c[5]^2+39*c[6]*c[5]-9*c[7]*c[5]-100*c [6]^2*c[4]^3+600*c[6]^2*c[7]*c[5]^2*c[4]^3-100*c[7]*c[6]*c[4]^3-310*c[ 6]^2*c[7]*c[4]^2+192*c[6]^2*c[7]*c[4]+180*c[6]^2*c[4]^2+14*c[7]*c[4]^2 -9*c[7]*c[4]-61*c[6]*c[7]*c[5]-97*c[6]*c[4]^2+60*c[6]*c[4]^3-103*c[6]* c[7]*c[4]+156*c[6]*c[7]*c[4]^2-30*c[5]*c[4]^3-150*c[5]*c[7]*c[4]^2+156 *c[5]*c[6]*c[4]^2+93*c[5]*c[4]^2-112*c[5]*c[6]*c[4]-57*c[5]*c[4]+66*c[ 6]*c[4]+230*c[5]*c[7]*c[6]*c[4]^3-530*c[6]^2*c[4]^2*c[5]+330*c[6]^2*c[ 4]*c[5]+380*c[5]^2*c[6]*c[7]*c[4]^2+320*c[6]^2*c[5]*c[4]^3-560*c[6]^2* c[7]*c[4]*c[5]-600*c[7]*c[6]^2*c[5]*c[4]^3+120*c[6]^2*c[7]*c[5]-100*c[ 6]^2*c[7]*c[5]^2-270*c[6]*c[5]*c[7]*c[4]^2+60*c[6]*c[7]*c[5]^2+60*c[6] ^2*c[5]^2+950*c[6]^2*c[4]^2*c[7]*c[5]+500*c[6]^2*c[5]^2*c[7]*c[4]-900* c[6]^2*c[5]^2*c[7]*c[4]^2-300*c[6]*c[5]^2*c[7]*c[4]^3-75*c[6]^2*c[5]+5 0*c[7]*c[5]*c[4]^3-130*c[6]*c[5]*c[4]^3+18*c[5]^2*c[4]+150*c[5]^2*c[6] *c[4]^3+50*c[5]^2*c[7]*c[4]^2-190*c[5]^2*c[6]*c[4]^2-300*c[6]^2*c[5]^2 *c[4]^3+470*c[6]^2*c[5]^2*c[4]^2-30*c[5]^2*c[4]^2-280*c[6]^2*c[4]*c[5] ^2+180*c[7]*c[6]^2*c[4]^3-117*c[6]^2*c[4]+126*c[5]^2*c[6]*c[4]-30*c[5] ^2*c[7]*c[4]-230*c[5]^2*c[6]*c[7]*c[4])/(-4*c[5]^3*c[4]-9*c[5]*c[7]*c[ 4]-3*c[7]*c[6]-c[5]^2-28*c[5]*c[6]*c[7]*c[4]-9*c[6]*c[5]^2+3*c[6]*c[5] +8*c[6]*c[5]^3+c[7]*c[5]+c[7]*c[4]+9*c[6]*c[7]*c[5]+9*c[6]*c[7]*c[4]-8 *c[6]*c[7]*c[4]^2+4*c[5]*c[7]*c[4]^2+8*c[5]*c[6]*c[4]^2-9*c[5]*c[6]*c[ 4]-c[5]*c[4]-30*c[5]^2*c[6]*c[7]*c[4]^2+28*c[6]*c[5]*c[7]*c[4]^2-8*c[6 ]*c[7]*c[5]^2-28*c[5]^3*c[6]*c[4]+9*c[5]^2*c[4]-28*c[5]^2*c[6]*c[4]^2+ 30*c[5]^3*c[6]*c[4]^2-4*c[5]^2*c[4]^2+28*c[5]^2*c[6]*c[4]+4*c[5]^2*c[7 ]*c[4]+28*c[5]^2*c[6]*c[7]*c[4])/(-c[5]+c[6])/(-c[4]+c[5])/c[5], a[5,4 ] = c[5]^2*(-c[4]+c[5])/c[4]^2, a[7,6] = 1/2*(5*c[5]*c[4]+1-2*c[4]-2*c [5])*(c[4]-c[7])*(c[5]-c[7])*(c[6]-c[7])*c[7]/(10*c[5]*c[6]*c[4]-5*c[6 ]*c[5]-5*c[6]*c[4]+3*c[6]-5*c[5]*c[4]+3*c[5]+3*c[4]-2)/(-c[4]+c[6])/(- c[5]+c[6])/c[6], c[3] = 2/3*c[4], a[4,3] = 3/4*c[4], a[3,2] = 2/9*c[4] ^2/c[2], a[7,4] = -1/2*c[7]*(-68*c[5]^2*c[6]*c[4]^4-7*c[6]*c[5]*c[4]^4 +7*c[4]^2*c[5]^2*c[7]^3-1400*c[5]^4*c[4]^4*c[6]^2*c[7]+11*c[7]^3*c[5]* c[4]^3-72*c[7]^3*c[5]^2*c[4]^3-3*c[7]*c[5]*c[4]^4-58*c[5]^2*c[4]^4*c[7 ]^2-120*c[6]*c[4]^7*c[7]*c[5]+320*c[4]^5*c[6]^2*c[5]^2-200*c[4]^3*c[7] ^3*c[5]^3-100*c[4]^5*c[6]*c[5]^3-2*c[5]*c[6]^2*c[4]^4-600*c[4]^5*c[7]^ 3*c[6]*c[5]^3-200*c[5]^5*c[4]^3*c[7]^2*c[6]^2+400*c[5]^2*c[4]^6*c[6]^2 *c[7]^2-21*c[4]^5*c[5]*c[7]+120*c[5]^2*c[4]^5*c[7]-600*c[5]^3*c[4]^6*c [6]^2*c[7]-198*c[5]^3*c[4]^4*c[7]-280*c[7]^2*c[4]^5*c[5]^3+240*c[7]^3* c[5]^2*c[4]^4+1250*c[6]^2*c[4]^5*c[7]*c[5]^3+120*c[7]*c[5]^2*c[4]^6*c[ 6]-340*c[5]^4*c[4]^2*c[7]^2*c[6]+400*c[6]^2*c[7]*c[5]^4*c[4]^2-240*c[5 ]^2*c[4]^6*c[7]-300*c[6]*c[5]*c[7]^2*c[4]^6-228*c[6]*c[4]^5*c[5]*c[7]- 119*c[6]*c[5]^2*c[4]^4*c[7]-200*c[5]*c[7]^3*c[4]^6*c[6]-600*c[5]^4*c[7 ]*c[4]^6*c[6]-120*c[4]^6*c[6]^2*c[5]^2+600*c[5]^2*c[7]^2*c[4]^6*c[6]+1 00*c[5]^2*c[7]^3*c[4]^6-187*c[5]^4*c[4]^3*c[6]-910*c[5]^4*c[4]^5*c[6]- 320*c[5]^2*c[6]*c[4]^6+620*c[5]^4*c[4]^4*c[6]-100*c[5]^4*c[7]^3*c[4]^4 -40*c[7]^2*c[4]^4*c[5]^3+40*c[7]^3*c[4]^5*c[5]+10*c[6]*c[7]^2*c[4]^5-1 0*c[6]*c[7]^3*c[4]^4+100*c[5]^4*c[4]^5*c[7]^2+80*c[5]^4*c[4]^4*c[7]+8* c[4]^3*c[7]^3*c[6]-2*c[7]^2*c[6]*c[4]^4+280*c[7]^3*c[5]^3*c[4]^4+80*c[ 6]*c[4]^5*c[5]*c[7]^2+c[4]^4*c[7]^2-360*c[5]^2*c[7]^3*c[4]^5-80*c[7]^2 *c[5]^4*c[4]^3+40*c[7]^3*c[5]^4*c[4]^3-1200*c[5]^3*c[7]*c[4]^6*c[6]+10 *c[6]^2*c[7]^3*c[4]^3-10*c[6]^2*c[4]^5*c[5]-12*c[7]*c[6]^2*c[5]^3-4*c[ 7]^2*c[6]*c[5]^3-80*c[5]^4*c[4]^5*c[7]-240*c[7]*c[6]^2*c[5]*c[4]^5-3*c [5]^4*c[4]^2*c[7]^2+20*c[5]^5*c[7]^2*c[4]^3+180*c[6]*c[5]^2*c[4]^5*c[7 ]-60*c[5]^4*c[7]^3*c[4]^2*c[6]+400*c[5]^2*c[7]^3*c[4]^6*c[6]-25*c[4]^5 *c[5]^3+200*c[5]^5*c[4]^4*c[7]^2*c[6]+60*c[5]^2*c[4]^7*c[7]+100*c[7]*c [4]^4*c[6]*c[5]-40*c[5]*c[4]^6*c[7]^2-150*c[4]^4*c[6]^2*c[5]^5+50*c[5] ^5*c[7]*c[4]^4*c[6]+174*c[7]^2*c[5]^3*c[4]^3+8*c[7]*c[4]^4*c[6]^2+600* c[5]^3*c[7]*c[4]^7*c[6]-60*c[5]^5*c[4]^2*c[6]^2*c[7]+200*c[5]^5*c[4]^4 *c[6]^2*c[7]-40*c[5]^5*c[4]^4*c[6]-200*c[5]^5*c[7]*c[4]^5*c[6]-149*c[6 ]^2*c[5]^4*c[7]*c[4]-24*c[5]^4*c[6]^2*c[4]+189*c[5]^4*c[6]^2*c[4]^2+30 0*c[5]^3*c[4]^6*c[6]^2-20*c[5]^5*c[4]^4*c[7]-5*c[4]^5*c[5]*c[7]^2+600* c[4]^6*c[6]*c[5]^3-84*c[6]^2*c[7]*c[5]^2*c[4]^3-74*c[6]^2*c[7]^2*c[5]^ 2*c[4]+162*c[6]^2*c[7]^2*c[5]^2*c[4]^2+311*c[6]*c[7]*c[5]^3*c[4]^3+21* c[6]^2*c[7]^3*c[4]*c[5]-40*c[7]^2*c[6]^2*c[5]^4+11*c[5]^4*c[7]*c[4]^2+ 107*c[7]^2*c[6]*c[5]*c[4]^4+100*c[7]^2*c[5]^5*c[6]^2*c[4]^2+320*c[5]^4 *c[4]*c[7]^2*c[6]^2-10*c[6]^2*c[4]^4*c[7]^2+300*c[5]^2*c[4]^6*c[7]^2-3 *c[7]^2*c[6]*c[4]^3-2*c[5]*c[7]^2*c[4]^3+15*c[6]*c[5]*c[7]^2*c[4]^2-8* c[5]*c[7]*c[6]*c[4]^3-14*c[5]^2*c[6]*c[7]*c[4]^2-2000*c[5]^4*c[7]^2*c[ 4]^4*c[6]+200*c[4]^5*c[6]*c[7]*c[5]^3+250*c[5]^2*c[6]*c[7]^3*c[4]^4+4* c[6]^2*c[5]*c[4]^3-26*c[7]^3*c[4]^2*c[6]*c[5]-69*c[7]*c[6]^2*c[5]*c[4] ^3-18*c[6]^2*c[7]^3*c[5]^2+12*c[6]^2*c[7]^2*c[5]^2+10*c[6]^2*c[4]^2*c[ 7]*c[5]+50*c[6]^2*c[7]^2*c[4]^2*c[5]+31*c[6]*c[5]^4*c[4]^2+24*c[7]*c[6 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]^2*c[4]^3+11*c[7]^2*c[5]^2*c[6]*c[4]^3-80*c[7]^3*c[5]^2*c[6]*c[4]^2-1 21*c[7]^2*c[5]^3*c[6]*c[4]^2-96*c[5]^3*c[7]^3*c[6]*c[4]+c[7]^3*c[4]^2* c[5]-68*c[4]^3*c[7]^2*c[5]*c[6]^2-80*c[7]^2*c[6]*c[5]*c[4]^3+68*c[7]^2 *c[5]^2*c[6]*c[4]^2-10*c[4]^4*c[5]^2+20*c[7]^2*c[5]^3*c[6]*c[4]-240*c[ 7]^2*c[5]^3*c[6]*c[4]^3-9*c[5]^4*c[4]^3-2*c[6]^2*c[4]^3*c[7]^2+260*c[5 ]^3*c[6]^2*c[7]*c[4]^3+49*c[5]^3*c[7]^3*c[4]^2-c[7]^3*c[5]^2*c[4]-228* c[6]^2*c[7]*c[5]^3*c[4]^2+30*c[6]^2*c[7]^2*c[4]*c[5]^3-2*c[7]^2*c[5]^2 *c[4]^2+5*c[7]^2*c[5]^3*c[4]+40*c[5]^5*c[4]^3*c[6]*c[7]+36*c[4]^6*c[7] *c[5]+90*c[7]^3*c[5]^2*c[6]^2*c[4]^3+16*c[7]^2*c[4]^4*c[5]-20*c[7]^2*c [5]^3*c[6]^2*c[4]^2+300*c[6]^2*c[4]^5*c[5]*c[7]^2+250*c[7]^2*c[4]^5*c[ 6]*c[5]^3-490*c[5]^2*c[6]*c[4]^5*c[7]^2+120*c[7]^3*c[5]^2*c[4]^3*c[6]+ 810*c[7]^3*c[5]^3*c[6]^2*c[4]^2+111*c[7]^3*c[6]*c[5]*c[4]^3+280*c[7]^3 *c[6]^2*c[5]*c[4]^3-240*c[7]^3*c[5]^3*c[6]^2*c[4]+15*c[5]^4*c[4]^4+60* c[5]^4*c[7]^2*c[4]*c[6]+6*c[6]*c[7]^3*c[5]^3-31*c[5]^4*c[4]*c[7]*c[6]- 200*c[4]^5*c[6]^2*c[7]*c[5]^2-500*c[4]^5*c[6]^2*c[7]^2*c[5]^2+9*c[4]^5 *c[5]^2-181*c[6]^2*c[4]^4*c[5]^2-1280*c[7]^3*c[5]^3*c[4]^3*c[6]-600*c[ 5]^2*c[4]^5*c[7]^3*c[6]-6*c[4]^6*c[5]-6*c[6]^2*c[7]^3*c[4]^2+100*c[5]^ 5*c[4]^5*c[6]+120*c[5]*c[7]*c[4]^6*c[6]^2+420*c[6]*c[4]^5*c[7]^3*c[5]+ 600*c[5]^4*c[4]^5*c[6]^2*c[7]+175*c[7]*c[6]^2*c[5]*c[4]^4-430*c[5]^3*c [4]^4*c[6]^2*c[7]+200*c[5]^3*c[4]^3*c[6]^2*c[7]^2-450*c[5]^4*c[4]^4*c[ 6]*c[7]+160*c[5]^2*c[4]^4*c[7]*c[6]^2-200*c[5]^3*c[4]^4*c[6]*c[7]-280* c[5]^4*c[4]^3*c[6]*c[7]+6*c[5]^3*c[4]^3-100*c[5]^2*c[4]^7*c[7]^2-1200* c[7]^3*c[5]^3*c[6]^2*c[4]^3-320*c[7]^3*c[4]^4*c[5]*c[6]-30*c[7]^2*c[5] ^3*c[4]^2-300*c[6]*c[5]^3*c[4]^7-350*c[5]^4*c[4]^5*c[6]^2+600*c[6]^2*c [7]^3*c[5]^3*c[4]^4+600*c[6]*c[7]^2*c[5]^4*c[4]^5+200*c[6]^2*c[7]^3*c[ 4]^5*c[5]+200*c[7]^2*c[4]^7*c[6]*c[5]+1100*c[5]^4*c[4]^4*c[6]^2-720*c[ 5]^4*c[4]^3*c[6]^2+1750*c[7]^2*c[5]^4*c[6]^2*c[4]^3-1230*c[7]^2*c[5]^4 *c[6]^2*c[4]^2+150*c[5]^4*c[7]^3*c[4]^3*c[6]+300*c[5]^4*c[4]^6*c[6]+48 0*c[5]^3*c[4]^4*c[7]^2*c[6]+1420*c[5]^4*c[4]^3*c[7]^2*c[6]+20*c[5]^2*c [4]^4*c[7]^2*c[6]+130*c[5]^4*c[4]^3*c[6]^2*c[7]-40*c[7]^2*c[4]^4*c[6]^ 2*c[5]-350*c[7]^2*c[4]^4*c[5]^3*c[6]^2+230*c[7]^2*c[4]^4*c[6]^2*c[5]^2 +10*c[4]^6*c[5]^2+188*c[5]^2*c[6]*c[4]^5-400*c[6]^2*c[7]^3*c[5]^2*c[4] ^5-600*c[6]^2*c[7]^2*c[5]^4*c[4]^4+120*c[5]^2*c[4]^7*c[6]-94*c[4]^4*c[ 6]*c[5]^3+6*c[6]*c[5]*c[4]^5+60*c[5]^5*c[6]^2*c[4]^3-200*c[6]^2*c[4]^6 *c[5]*c[7]^2-420*c[7]^3*c[6]^2*c[5]*c[4]^4+1550*c[5]^4*c[4]^5*c[6]*c[7 ]+500*c[7]^3*c[4]^4*c[6]^2*c[5]^2-400*c[5]^2*c[4]^7*c[6]*c[7]^2+6*c[5] ^3*c[4]^4+103*c[6]^2*c[7]^3*c[5]^2*c[4]-260*c[6]^2*c[7]^3*c[5]^2*c[4]^ 2+540*c[5]^3*c[7]^3*c[4]^2*c[6]-97*c[6]^2*c[5]*c[7]^3*c[4]^2+30*c[6]^2 *c[7]^3*c[5]^3-4*c[7]^3*c[4]*c[5]^3)/(-240*c[5]^2*c[6]*c[4]^4+40*c[6]* c[5]*c[4]^4-9*c[5]^3*c[4]-100*c[4]^5*c[6]*c[5]^3-350*c[5]^4*c[4]^3*c[6 ]+30*c[5]^5*c[4]^2+2*c[5]^3-2*c[4]^3-50*c[5]^5*c[4]^3-3*c[5]^4+3*c[4]^ 4+50*c[4]^5*c[5]^3+100*c[5]^5*c[4]^3*c[6]-3*c[6]*c[5]^3+3*c[6]*c[4]^3+ 9*c[5]*c[4]^3-6*c[5]*c[6]*c[4]^2+4*c[5]*c[4]^2+240*c[6]*c[5]^4*c[4]^2+ 5*c[6]*c[5]^4-13*c[6]*c[5]*c[4]^3+13*c[5]^3*c[6]*c[4]-4*c[5]^2*c[4]+68 *c[5]^3*c[4]^2+100*c[5]^2*c[6]*c[4]^3-100*c[5]^3*c[6]*c[4]^2-140*c[5]^ 4*c[4]^2+140*c[4]^4*c[5]^2+180*c[5]^4*c[4]^3+6*c[5]^2*c[6]*c[4]-68*c[5 ]^2*c[4]^3-23*c[4]^4*c[5]-40*c[5]^4*c[4]*c[6]-30*c[4]^5*c[5]^2-5*c[6]* c[4]^4-50*c[5]^5*c[6]*c[4]^2+50*c[5]^2*c[6]*c[4]^5+350*c[4]^4*c[6]*c[5 ]^3+23*c[5]^4*c[4]-180*c[5]^3*c[4]^4)/(-c[4]+c[6])/c[4]^2, `b*`[7] = 1 /60*(-30*c[5]^3*c[4]-18*c[6]+6*c[5]+6*c[4]-9*c[5]^2-9*c[4]^2+27*c[6]^2 -97*c[6]*c[5]^2+66*c[6]*c[5]+60*c[6]*c[5]^3-100*c[6]^2*c[5]^3-100*c[6] ^2*c[4]^3+180*c[6]^2*c[4]^2-97*c[6]*c[4]^2+60*c[6]*c[4]^3-30*c[5]*c[4] ^3+312*c[5]*c[6]*c[4]^2+107*c[5]*c[4]^2-215*c[5]*c[6]*c[4]-66*c[5]*c[4 ]+66*c[6]*c[4]-840*c[6]^2*c[4]^2*c[5]+522*c[6]^2*c[4]*c[5]+500*c[6]^2* c[5]*c[4]^3+180*c[6]^2*c[5]^2-117*c[6]^2*c[5]-230*c[6]*c[5]*c[4]^3-230 *c[5]^3*c[6]*c[4]+107*c[5]^2*c[4]+50*c[5]^3*c[4]^2+380*c[5]^2*c[6]*c[4 ]^3-460*c[5]^2*c[6]*c[4]^2+380*c[5]^3*c[6]*c[4]^2+600*c[5]^3*c[6]^2*c[ 4]^3-900*c[6]^2*c[5]^2*c[4]^3+1420*c[6]^2*c[5]^2*c[4]^2-300*c[6]*c[5]^ 3*c[4]^3+500*c[6]^2*c[5]^3*c[4]-900*c[6]^2*c[5]^3*c[4]^2-180*c[5]^2*c[ 4]^2-840*c[6]^2*c[4]*c[5]^2-117*c[6]^2*c[4]+312*c[5]^2*c[6]*c[4]+50*c[ 5]^2*c[4]^3)/c[7]/(-3*c[7]^3*c[6]+3*c[7]^2*c[6]^2-9*c[6]^2*c[7]^2*c[5] -8*c[7]*c[6]^2*c[5]^3+8*c[7]^2*c[6]*c[5]^3-c[5]*c[6]*c[7]*c[4]-c[7]^2* c[4]^2-30*c[6]^2*c[7]*c[5]^2*c[4]^3-28*c[6]^2*c[7]^2*c[5]^2*c[4]+30*c[ 6]^2*c[7]^2*c[5]^2*c[4]^2-30*c[6]*c[7]*c[5]^3*c[4]^3+9*c[6]^2*c[7]*c[4 ]^2-3*c[6]^2*c[7]*c[4]+2*c[6]*c[7]^2*c[4]-9*c[6]*c[7]^2*c[4]^2-2*c[7]^ 2*c[5]*c[4]-c[7]^2*c[5]^2+c[6]*c[7]*c[4]^2+8*c[7]^2*c[6]*c[4]^3-4*c[5] *c[7]^2*c[4]^3+c[5]*c[7]*c[4]^2-c[5]*c[6]*c[4]^2+32*c[6]*c[5]*c[7]^2*c [4]^2-9*c[7]^2*c[5]*c[6]*c[4]-4*c[5]*c[7]*c[6]*c[4]^3-9*c[6]^2*c[4]^2* c[5]+3*c[6]^2*c[4]*c[5]+c[7]^3*c[4]+9*c[7]^3*c[4]*c[6]-9*c[7]^3*c[4]*c [5]-20*c[5]^2*c[6]*c[7]*c[4]^2+8*c[6]^2*c[5]*c[4]^3+28*c[7]^3*c[4]^2*c [6]*c[5]+18*c[6]^2*c[7]*c[4]*c[5]+28*c[7]*c[6]^2*c[5]*c[4]^3-3*c[6]^2* c[7]*c[5]+9*c[6]^2*c[7]*c[5]^2+c[6]*c[7]*c[5]^2-8*c[6]*c[7]^3*c[5]^2-9 *c[6]*c[7]^2*c[5]^2+8*c[6]^2*c[7]^2*c[5]^2+2*c[6]*c[7]^2*c[5]-36*c[6]^ 2*c[4]^2*c[7]*c[5]-28*c[6]^2*c[7]^2*c[4]^2*c[5]-28*c[7]^3*c[4]*c[6]*c[ 5]-36*c[6]^2*c[5]^2*c[7]*c[4]+56*c[6]^2*c[5]^2*c[7]*c[4]^2+28*c[6]*c[5 ]^2*c[7]*c[4]^3+28*c[6]*c[5]^3*c[7]*c[4]^2+28*c[6]^2*c[5]^3*c[7]*c[4]+ 9*c[6]*c[7]^3*c[5]+9*c[7]^2*c[5]*c[4]^2+28*c[7]^3*c[5]^2*c[6]*c[4]+32* c[7]^2*c[5]^2*c[6]*c[4]+8*c[6]^2*c[7]^2*c[4]^2-9*c[6]^2*c[7]^2*c[4]-4* c[5]^3*c[6]*c[7]*c[4]+28*c[6]^2*c[7]^2*c[4]*c[5]-4*c[5]^2*c[6]*c[4]^3- 9*c[5]^2*c[7]*c[4]^2+9*c[5]^2*c[6]*c[4]^2+4*c[5]^2*c[7]*c[4]^3+4*c[5]^ 3*c[7]*c[4]^2-4*c[5]^3*c[6]*c[4]^2+30*c[5]^3*c[6]^2*c[4]^3-28*c[6]^2*c [5]^2*c[4]^3+28*c[6]^2*c[5]^2*c[4]^2+8*c[6]^2*c[5]^3*c[4]-28*c[6]^2*c[ 5]^3*c[4]^2-9*c[6]^2*c[4]*c[5]^2-8*c[7]*c[6]^2*c[4]^3+30*c[7]^2*c[5]^2 *c[6]*c[4]^3-30*c[7]^3*c[5]^2*c[6]*c[4]^2+30*c[7]^2*c[5]^3*c[6]*c[4]^2 -8*c[7]^3*c[4]^2*c[6]+4*c[7]^3*c[4]^2*c[5]+c[7]^3*c[5]-28*c[7]^2*c[6]* c[5]*c[4]^3-56*c[7]^2*c[5]^2*c[6]*c[4]^2-28*c[7]^2*c[5]^3*c[6]*c[4]-c[ 5]^2*c[6]*c[4]+c[5]^2*c[7]*c[4]+4*c[7]^3*c[5]^2*c[4]+9*c[7]^2*c[5]^2*c [4]-30*c[6]^2*c[7]*c[5]^3*c[4]^2-8*c[7]^2*c[5]^2*c[4]^2-4*c[7]^2*c[5]^ 3*c[4]), a[2,1] = c[2], a[6,4] = -1/2*(120*c[5]^2*c[6]*c[4]^4-12*c[6]* c[5]*c[4]^4-90*c[5]^2*c[6]^3*c[4]^2-60*c[5]*c[6]^2*c[4]^4-2*c[6]^3*c[4 ]^2+20*c[5]^4*c[4]^3*c[6]+40*c[6]^2*c[4]^5*c[5]+36*c[4]*c[5]^2*c[6]^3- 20*c[5]^4*c[6]^2*c[4]^2+8*c[6]^2*c[5]^3+2*c[6]^2*c[4]^3-24*c[5]*c[6]^3 *c[4]^2-6*c[6]^2*c[4]^2*c[5]+36*c[6]^2*c[5]*c[4]^3+60*c[4]^3*c[5]*c[6] ^3-3*c[6]*c[5]*c[4]^3-11*c[5]^3*c[6]*c[4]+3*c[5]^3*c[4]^2-24*c[5]^2*c[ 6]*c[4]^3+13*c[5]^2*c[6]*c[4]^2+48*c[5]^3*c[6]*c[4]^2-60*c[5]^3*c[6]^2 *c[4]^3-30*c[6]^2*c[5]^2*c[4]^3-12*c[6]^2*c[5]^2*c[4]^2-150*c[6]*c[5]^ 3*c[4]^3-48*c[6]^2*c[5]^3*c[4]+150*c[6]^2*c[5]^3*c[4]^2-2*c[6]^2*c[4]* c[5]^2-5*c[5]^2*c[4]^3+2*c[4]^4*c[5]-6*c[5]^2*c[6]^3+60*c[6]^3*c[4]^3* c[5]^2+7*c[6]^3*c[4]*c[5]-60*c[5]^2*c[6]*c[4]^5-40*c[6]^3*c[4]^4*c[5]+ 60*c[4]^4*c[6]*c[5]^3)*c[6]/(c[4]^3-2*c[5]*c[4]^2+30*c[5]^2*c[4]^3-6*c [5]*c[4]^3-10*c[4]^4*c[5]^2-c[5]^3+10*c[5]^4*c[4]^2-30*c[5]^3*c[4]^2+6 *c[5]^3*c[4]+2*c[5]^2*c[4])/c[4]^2, a[3,1] = -2/9*c[4]*(c[4]-3*c[2])/c [2], a[4,1] = 1/4*c[4], a[7,3] = 3/4*1/c[4]^2*c[7]*(-480*c[5]^2*c[6]*c [4]^4+15*c[6]*c[5]*c[4]^4+300*c[4]^2*c[5]^2*c[7]^3+60*c[7]^3*c[5]*c[4] ^3-510*c[7]^3*c[5]^2*c[4]^3+54*c[7]*c[5]*c[4]^4+450*c[5]^2*c[4]^4*c[7] ^2+150*c[4]^3*c[7]^3*c[5]^3-150*c[7]^2*c[5]^2*c[4]^5-450*c[4]^5*c[6]*c [5]^3-3*c[7]^3*c[4]^2+90*c[5]^2*c[4]^5*c[7]+120*c[5]^3*c[4]^4*c[7]+150 *c[7]^3*c[5]^2*c[4]^4+100*c[5]^4*c[4]^2*c[7]^2*c[6]-180*c[6]*c[4]^5*c[ 5]*c[7]+30*c[6]*c[5]^2*c[4]^4*c[7]+60*c[5]^4*c[4]^3*c[6]-150*c[5]^4*c[ 4]^4*c[6]-200*c[7]^2*c[4]^4*c[5]^3+300*c[6]*c[4]^5*c[5]*c[7]^2-50*c[7] ^2*c[5]^4*c[4]^3+24*c[7]*c[6]*c[5]^3-40*c[7]^2*c[6]*c[5]^3-6*c[5]*c[6] *c[7]*c[4]+360*c[7]*c[4]^4*c[6]*c[5]+150*c[7]^2*c[5]^3*c[4]^3+2*c[7]^2 *c[4]^2+6*c[5]^2*c[7]^3+330*c[6]*c[7]*c[5]^3*c[4]^3-450*c[7]^2*c[6]*c[ 5]*c[4]^4-12*c[7]*c[6]*c[4]^3-6*c[6]*c[7]^2*c[4]-2*c[7]^2*c[5]*c[4]-4* c[7]^2*c[5]^2+6*c[6]*c[7]*c[4]^2+20*c[7]^2*c[6]*c[4]^3-10*c[5]*c[7]^2* c[4]^3+6*c[5]*c[4]^3-6*c[5]*c[7]*c[4]^2-6*c[5]*c[6]*c[4]^2+150*c[6]*c[ 5]*c[7]^2*c[4]^2-72*c[7]^2*c[5]*c[6]*c[4]-270*c[5]*c[7]*c[6]*c[4]^3+9* c[7]^3*c[4]*c[6]+3*c[7]^3*c[4]*c[5]-306*c[5]^2*c[6]*c[7]*c[4]^2+900*c[ 4]^5*c[6]*c[7]*c[5]^3+600*c[5]^2*c[6]*c[7]^3*c[4]^4-420*c[7]^3*c[4]^2* c[6]*c[5]+90*c[6]*c[5]*c[7]*c[4]^2-12*c[6]*c[7]*c[5]^2+30*c[6]*c[7]^3* c[5]^2+12*c[6]*c[7]^2*c[5]+123*c[7]^3*c[4]*c[6]*c[5]+360*c[6]*c[5]^2*c [7]*c[4]^3+360*c[6]*c[5]^3*c[7]*c[4]^2-18*c[6]*c[7]^3*c[5]+30*c[5]^4*c [4]^3*c[7]-80*c[7]^2*c[5]^2*c[4]^3+24*c[7]^2*c[5]*c[4]^2-30*c[7]*c[5]* c[4]^3+3*c[6]*c[5]*c[4]^3-240*c[7]^3*c[5]^2*c[6]*c[4]-24*c[5]^3*c[6]*c [4]-12*c[5]^3*c[7]*c[4]-144*c[5]^3*c[6]*c[7]*c[4]+9*c[5]^3*c[4]^2+294* c[5]^2*c[6]*c[4]^3-42*c[5]^2*c[7]*c[4]^2-75*c[5]^2*c[6]*c[4]^2+252*c[5 ]^2*c[7]*c[4]^3+18*c[5]^3*c[7]*c[4]^2+189*c[5]^3*c[6]*c[4]^2-360*c[5]^ 2*c[4]^4*c[7]-720*c[6]*c[5]^3*c[4]^3-120*c[5]^3*c[7]*c[4]^3-60*c[5]^4* c[7]*c[4]^2*c[6]-6*c[5]^2*c[4]^2-780*c[7]^2*c[5]^2*c[6]*c[4]^3+810*c[7 ]^3*c[5]^2*c[6]*c[4]^2-1230*c[7]^2*c[5]^3*c[6]*c[4]^2-15*c[7]^3*c[4]^2 *c[6]-21*c[7]^3*c[4]^2*c[5]+20*c[7]^2*c[6]*c[5]*c[4]^3+220*c[7]^2*c[5] ^2*c[6]*c[4]^2+15*c[4]^4*c[5]^2+320*c[7]^2*c[5]^3*c[6]*c[4]+1850*c[7]^ 2*c[5]^3*c[6]*c[4]^3-60*c[5]^3*c[7]^3*c[4]^2+12*c[5]^2*c[6]*c[4]+12*c[ 5]^2*c[7]*c[4]-66*c[7]^3*c[5]^2*c[4]+24*c[7]^2*c[5]^2*c[4]+72*c[5]^2*c [6]*c[7]*c[4]-110*c[7]^2*c[5]^2*c[4]^2+20*c[7]^2*c[5]^3*c[4]-9*c[4]^4* c[5]-60*c[7]^2*c[4]^4*c[5]-600*c[5]^2*c[6]*c[4]^5*c[7]^2-1200*c[7]^3*c [5]^2*c[4]^3*c[6]+630*c[7]^3*c[6]*c[5]*c[4]^3+300*c[5]^4*c[4]^4*c[6]*c [7]-1800*c[5]^3*c[4]^4*c[6]*c[7]-15*c[5]^3*c[4]^3-300*c[7]^3*c[4]^4*c[ 5]*c[6]-20*c[7]^2*c[5]^3*c[4]^2-800*c[5]^3*c[4]^4*c[7]^2*c[6]-200*c[5] ^4*c[4]^3*c[7]^2*c[6]+1150*c[5]^2*c[4]^4*c[7]^2*c[6]+180*c[5]^2*c[6]*c [4]^5+1140*c[4]^4*c[6]*c[5]^3)/(10*c[5]^3*c[4]^2+10*c[5]^2*c[4]^3-30*c [5]^2*c[4]^2+6*c[5]^2*c[4]-c[5]^2+6*c[5]*c[4]^2+c[5]*c[4]-c[4]^2)/(10* c[5]*c[6]*c[4]-5*c[6]*c[5]-5*c[6]*c[4]+3*c[6]-5*c[5]*c[4]+3*c[5]+3*c[4 ]-2), `b*`[1] = 1/60*(30*c[5]^3*c[4]+42*c[7]*c[6]^2-103*c[5]*c[7]*c[4] +18*c[6]-6*c[5]-6*c[4]-27*c[7]*c[6]+9*c[5]^2+9*c[4]^2-180*c[7]*c[6]^2* c[5]^3+100*c[7]*c[6]*c[5]^3-27*c[6]^2-354*c[5]*c[6]*c[7]*c[4]+97*c[6]* c[5]^2-66*c[6]*c[5]-60*c[6]*c[5]^3+100*c[6]^2*c[5]^3+9*c[7]*c[5]+100*c [6]^2*c[4]^3-2040*c[6]^2*c[7]*c[5]^2*c[4]^3-600*c[6]*c[7]*c[5]^3*c[4]^ 3-14*c[7]*c[5]^2+100*c[7]*c[6]*c[4]^3+310*c[6]^2*c[7]*c[4]^2-192*c[6]^ 2*c[7]*c[4]-180*c[6]^2*c[4]^2-14*c[7]*c[4]^2+9*c[7]*c[4]+103*c[6]*c[7] *c[5]+97*c[6]*c[4]^2-60*c[6]*c[4]^3+103*c[6]*c[7]*c[4]-156*c[6]*c[7]*c [4]^2+30*c[5]*c[4]^3+174*c[5]*c[7]*c[4]^2-312*c[5]*c[6]*c[4]^2-107*c[5 ]*c[4]^2+215*c[5]*c[6]*c[4]+66*c[5]*c[4]-66*c[6]*c[4]-410*c[5]*c[7]*c[ 6]*c[4]^3+840*c[6]^2*c[4]^2*c[5]-522*c[6]^2*c[4]*c[5]-760*c[5]^2*c[6]* c[7]*c[4]^2-500*c[6]^2*c[5]*c[4]^3+932*c[6]^2*c[7]*c[4]*c[5]+1020*c[7] *c[6]^2*c[5]*c[4]^3-192*c[6]^2*c[7]*c[5]+310*c[6]^2*c[7]*c[5]^2+526*c[ 6]*c[5]*c[7]*c[4]^2-156*c[6]*c[7]*c[5]^2-180*c[6]^2*c[5]^2-1610*c[6]^2 *c[4]^2*c[7]*c[5]-1610*c[6]^2*c[5]^2*c[7]*c[4]+2970*c[6]^2*c[5]^2*c[7] *c[4]^2+690*c[6]*c[5]^2*c[7]*c[4]^3+690*c[6]*c[5]^3*c[7]*c[4]^2+1020*c [6]^2*c[5]^3*c[7]*c[4]+117*c[6]^2*c[5]-50*c[7]*c[5]*c[4]^3+230*c[6]*c[ 5]*c[4]^3+230*c[5]^3*c[6]*c[4]-50*c[5]^3*c[7]*c[4]-107*c[5]^2*c[4]-410 *c[5]^3*c[6]*c[7]*c[4]-50*c[5]^3*c[4]^2-380*c[5]^2*c[6]*c[4]^3-310*c[5 ]^2*c[7]*c[4]^2+460*c[5]^2*c[6]*c[4]^2+90*c[5]^2*c[7]*c[4]^3+90*c[5]^3 *c[7]*c[4]^2-380*c[5]^3*c[6]*c[4]^2-600*c[5]^3*c[6]^2*c[4]^3+900*c[6]^ 2*c[5]^2*c[4]^3-1420*c[6]^2*c[5]^2*c[4]^2+300*c[6]*c[5]^3*c[4]^3-500*c [6]^2*c[5]^3*c[4]+900*c[6]^2*c[5]^3*c[4]^2+180*c[5]^2*c[4]^2+840*c[6]^ 2*c[4]*c[5]^2-180*c[7]*c[6]^2*c[4]^3+117*c[6]^2*c[4]+1500*c[5]^3*c[6]^ 2*c[7]*c[4]^3-312*c[5]^2*c[6]*c[4]+174*c[5]^2*c[7]*c[4]+526*c[5]^2*c[6 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5]*c[7]^2*c[4]^6+46*c[6]*c[4]^5*c[5]*c[7]-1014*c[6]*c[5]^2*c[4]^4*c[7] +2100*c[5]^4*c[7]*c[4]^6*c[6]-390*c[6]^2*c[4]^6*c[5]+860*c[4]^6*c[6]^2 *c[5]^2+140*c[5]^5*c[4]^3*c[7]-1740*c[5]^2*c[7]^2*c[4]^6*c[6]-613*c[5] ^4*c[4]^3*c[6]-100*c[4]^8*c[5]^2*c[7]+1140*c[5]^4*c[4]^5*c[6]+968*c[5] ^2*c[6]*c[4]^6-150*c[5]^5*c[4]^5*c[6]^2+263*c[5]^4*c[4]^4*c[6]+250*c[7 ]^2*c[4]^4*c[5]^3-120*c[6]*c[5]*c[4]^8-10*c[6]*c[7]^2*c[4]^5-6*c[6]*c[ 4]^6+780*c[5]^4*c[4]^4*c[7]+32*c[7]^2*c[6]*c[4]^4+1200*c[5]^3*c[7]^2*c [4]^6*c[6]-750*c[6]*c[4]^5*c[5]*c[7]^2-6*c[4]^4*c[7]^2-400*c[5]^4*c[7] ^2*c[4]^5*c[6]^2+415*c[7]^2*c[5]^4*c[4]^3-990*c[5]^3*c[7]*c[4]^6*c[6]+ 433*c[6]^2*c[4]^5*c[5]+4*c[4]^3-60*c[5]^5*c[4]^3+30*c[7]*c[6]^2*c[5]^3 +4*c[7]*c[6]*c[5]^3-6*c[7]^2*c[6]*c[5]^3-350*c[5]^4*c[4]^6*c[6]^2-5*c[ 4]^4-250*c[5]^3*c[7]*c[4]^6-270*c[5]^4*c[4]^5*c[7]-330*c[7]*c[6]^2*c[5 ]*c[4]^5-1320*c[4]^6*c[6]^2*c[7]*c[5]^2-55*c[5]^4*c[4]^2*c[7]^2-100*c[ 5]^5*c[7]^2*c[4]^3-96*c[6]*c[5]^2*c[4]^5*c[7]-562*c[4]^5*c[5]^3-600*c[ 5]^5*c[4]^4*c[7]^2*c[6]+380*c[5]^2*c[4]^7*c[7]+160*c[5]^5*c[4]^3*c[6]+ 62*c[7]*c[4]^4*c[6]*c[5]+80*c[5]^5*c[4]^4+30*c[5]*c[4]^6*c[7]^2+310*c[ 4]^4*c[6]^2*c[5]^5+450*c[5]^5*c[7]*c[4]^4*c[6]-356*c[7]^2*c[5]^3*c[4]^ 3+24*c[7]*c[4]^4*c[6]^2-200*c[5]^5*c[7]*c[4]^6*c[6]-900*c[5]^3*c[7]*c[ 4]^7*c[6]+60*c[5]^5*c[4]^2*c[6]^2*c[7]-450*c[5]^5*c[4]^4*c[6]^2*c[7]-1 70*c[5]^5*c[4]^4*c[6]+200*c[5]^5*c[7]*c[4]^5*c[6]+143*c[6]^2*c[5]^4*c[ 7]*c[4]+34*c[5]^4*c[6]^2*c[4]-283*c[5]^4*c[6]^2*c[4]^2-530*c[5]^3*c[4] ^6*c[6]^2+9*c[4]^4*c[7]+600*c[5]^4*c[7]^2*c[4]^6*c[6]-140*c[5]^5*c[4]^ 4*c[7]-55*c[4]^5*c[5]*c[7]^2+10*c[6]^2*c[4]^3+100*c[5]^5*c[4]^4*c[7]^2 +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+28 4*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+1 0*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]+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]^3-10*c[5]^2*c[6]* c[4]-9*c[5]^2*c[7]*c[4]+6*c[7]^2*c[5]^2*c[4]+33*c[5]^2*c[6]*c[7]*c[4]+ 600*c[7]*c[4]^8*c[6]*c[5]^3+187*c[6]^2*c[7]*c[5]^3*c[4]^2+341*c[6]^2*c [7]^2*c[4]*c[5]^3+202*c[5]^2*c[4]^3-29*c[7]^2*c[5]^2*c[4]^2-10*c[7]^2* c[5]^3*c[4]+40*c[4]^4*c[5]-270*c[4]^7*c[5]^2-450*c[5]^5*c[4]^3*c[6]*c[ 7]+35*c[4]^6*c[7]*c[5]+80*c[7]^2*c[4]^4*c[5]-1163*c[7]^2*c[5]^3*c[6]^2 *c[4]^2+20*c[5]*c[4]^7-640*c[6]^2*c[4]^5*c[5]*c[7]^2+850*c[7]^2*c[4]^5 *c[6]*c[5]^3+1530*c[5]^2*c[6]*c[4]^5*c[7]^2-506*c[5]^4*c[4]^4+1420*c[5 ]^2*c[4]^7*c[6]*c[7]-16*c[5]^4*c[4]*c[6]-85*c[5]^4*c[7]^2*c[4]*c[6]+72 *c[5]^4*c[4]*c[7]*c[6]+230*c[4]^5*c[6]^2*c[7]*c[5]^2+1740*c[4]^5*c[6]^ 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7]^2*c[6]-3280*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]^4-1 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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]^2-400*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+ 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c[7]^2*c[6]^2*c[5]^5*c[4]^4-600*c[5]^5*c[4]^5*c[6]-120*c[5]*c[7]*c[4]^ 6*c[6]^2+50*c[6]^2*c[7]*c[4]^5-52*c[6]*c[4]^4-1200*c[5]^4*c[7]*c[4]^6* c[6]^2+7150*c[5]^4*c[4]^5*c[6]^2*c[7]+198*c[7]*c[6]^2*c[5]*c[4]^4+1056 0*c[5]^3*c[4]^4*c[6]^2*c[7]+5240*c[5]^3*c[4]^3*c[6]^2*c[7]^2+11240*c[5 ]^4*c[4]^4*c[6]*c[7]-3649*c[5]^2*c[4]^4*c[7]*c[6]^2-14207*c[5]^3*c[4]^ 4*c[6]*c[7]-4491*c[5]^4*c[4]^3*c[6]*c[7]+1064*c[5]^3*c[4]^3-264*c[7]^2 *c[5]^3*c[4]^2-2400*c[5]^4*c[4]^5*c[6]^2+6020*c[6]^2*c[7]^2*c[5]^3*c[4 ]^5+6800*c[6]*c[7]^2*c[5]^4*c[4]^5+3160*c[5]^4*c[4]^4*c[6]^2-1320*c[5] ^4*c[4]^3*c[6]^2-5800*c[7]^2*c[5]^4*c[6]^2*c[4]^3+1940*c[7]^2*c[5]^4*c [6]^2*c[4]^2-600*c[5]^4*c[4]^6*c[6]+28*c[4]^5*c[7]+9740*c[5]^3*c[4]^4* c[7]^2*c[6]+2920*c[5]^4*c[4]^3*c[7]^2*c[6]-4775*c[5]^2*c[4]^4*c[7]^2*c [6]+4980*c[5]^4*c[4]^3*c[6]^2*c[7]+210*c[7]^2*c[4]^4*c[6]^2*c[5]-8860* c[7]^2*c[4]^4*c[5]^3*c[6]^2+2140*c[7]^2*c[4]^4*c[6]^2*c[5]^2+120*c[4]^ 6*c[5]^2-20*c[7]^2*c[4]^5+2092*c[5]^2*c[6]*c[4]^5+9920*c[6]^2*c[7]^2*c [5]^4*c[4]^4-500*c[5]^4*c[4]^6*c[7]-48*c[7]^2*c[6]^2*c[5]^3+6404*c[4]^ 4*c[6]*c[5]^3-284*c[6]*c[5]*c[4]^5+200*c[5]^5*c[6]^2*c[4]^3+200*c[6]^2 *c[4]^6*c[5]*c[7]^2-8820*c[5]^4*c[4]^5*c[6]*c[7]-12*c[5]^4*c[4]-2324*c [5]^3*c[4]^4)/c[6]/c[5]/c[7]/c[4]^2/(150*c[5]^2*c[6]*c[4]^4-87*c[5]^3* c[4]-12*c[5]*c[7]*c[4]-200*c[5]^3*c[4]^4*c[7]-200*c[6]*c[5]^2*c[4]^4*c [7]-200*c[5]^4*c[4]^3*c[6]-10*c[5]^2-10*c[4]^2+12*c[5]^3+12*c[4]^3+20* c[7]*c[6]*c[5]^3+15*c[5]*c[6]*c[7]*c[4]+12*c[6]*c[5]^2-15*c[6]*c[5]^3- 1300*c[6]*c[7]*c[5]^3*c[4]^3+150*c[5]^4*c[7]*c[4]^2+12*c[7]*c[5]^2+20* c[7]*c[6]*c[4]^3+12*c[7]*c[4]^2-15*c[7]*c[4]^3+12*c[6]*c[4]^2-15*c[6]* c[4]^3-15*c[6]*c[7]*c[4]^2-87*c[5]*c[4]^3-72*c[5]*c[7]*c[4]^2-72*c[5]* c[6]*c[4]^2+60*c[5]*c[4]^2-12*c[5]*c[6]*c[4]+10*c[5]*c[4]-150*c[5]*c[7 ]*c[6]*c[4]^3-15*c[7]*c[5]^3-660*c[5]^2*c[6]*c[7]*c[4]^2+90*c[6]*c[5]* c[7]*c[4]^2-15*c[6]*c[7]*c[5]^2+150*c[6]*c[5]^4*c[4]^2+930*c[6]*c[5]^2 *c[7]*c[4]^3+930*c[6]*c[5]^3*c[7]*c[4]^2-200*c[5]^4*c[4]^3*c[7]+110*c[ 7]*c[5]*c[4]^3+110*c[6]*c[5]*c[4]^3+110*c[5]^3*c[6]*c[4]+110*c[5]^3*c[ 7]*c[4]+60*c[5]^2*c[4]-150*c[5]^3*c[6]*c[7]*c[4]+550*c[5]^3*c[4]^2-690 *c[5]^2*c[6]*c[4]^3+520*c[5]^2*c[7]*c[4]^2+520*c[5]^2*c[6]*c[4]^2-690* c[5]^2*c[7]*c[4]^3-690*c[5]^3*c[7]*c[4]^2-690*c[5]^3*c[6]*c[4]^2+150*c [5]^2*c[4]^4*c[7]+900*c[6]*c[5]^3*c[4]^3+900*c[5]^3*c[7]*c[4]^3-200*c[ 5]^4*c[7]*c[4]^2*c[6]-429*c[5]^2*c[4]^2-120*c[5]^4*c[4]^2-120*c[4]^4*c [5]^2+150*c[5]^4*c[4]^3-72*c[5]^2*c[6]*c[4]-72*c[5]^2*c[7]*c[4]+90*c[5 ]^2*c[6]*c[7]*c[4]+550*c[5]^2*c[4]^3+300*c[5]^3*c[4]^4*c[6]*c[7]+300*c [5]^4*c[4]^3*c[6]*c[7]-690*c[5]^3*c[4]^3-200*c[4]^4*c[6]*c[5]^3+150*c[ 5]^3*c[4]^4), a[8,3] = 3/4*(-1560*c[5]^2*c[6]*c[4]^4+585*c[6]*c[5]*c[4 ]^4-45*c[7]*c[5]*c[4]^4-12*c[5]^3*c[4]-450*c[4]^5*c[6]*c[5]^3+3*c[5]*c [7]*c[4]-150*c[5]^2*c[4]^5*c[7]-200*c[5]^3*c[4]^4*c[7]+300*c[6]*c[4]^5 *c[5]*c[7]+2860*c[6]*c[5]^2*c[4]^4*c[7]+160*c[5]^4*c[4]^3*c[6]-150*c[5 ]^4*c[4]^4*c[6]-4*c[5]^2+2*c[4]^2-40*c[7]*c[6]*c[5]^3+132*c[5]*c[6]*c[ 7]*c[4]-900*c[6]*c[5]^2*c[4]^5*c[7]-30*c[6]*c[5]^2+12*c[6]*c[5]-960*c[ 7]*c[4]^4*c[6]*c[5]+24*c[6]*c[5]^3+3140*c[6]*c[7]*c[5]^3*c[4]^3+6*c[7] *c[5]^2+20*c[7]*c[6]*c[4]^3-3*c[7]*c[4]^2-18*c[6]*c[7]*c[5]+15*c[6]*c[ 4]^2-12*c[6]*c[4]^3+9*c[6]*c[7]*c[4]-24*c[6]*c[7]*c[4]^2-54*c[5]*c[4]^ 3-12*c[5]*c[7]*c[4]^2+348*c[5]*c[6]*c[4]^2+9*c[5]*c[4]^2-87*c[5]*c[6]* c[4]-2*c[5]*c[4]-6*c[6]*c[4]+1035*c[5]*c[7]*c[6]*c[4]^3+1593*c[5]^2*c[ 6]*c[7]*c[4]^2+900*c[4]^5*c[6]*c[7]*c[5]^3-534*c[6]*c[5]*c[7]*c[4]^2+4 8*c[6]*c[7]*c[5]^2-60*c[6]*c[5]^4*c[4]^2-3060*c[6]*c[5]^2*c[7]*c[4]^3- 1560*c[6]*c[5]^3*c[7]*c[4]^2-50*c[5]^4*c[4]^3*c[7]+80*c[7]*c[5]*c[4]^3 -652*c[6]*c[5]*c[4]^3-208*c[5]^3*c[6]*c[4]+20*c[5]^3*c[7]*c[4]+54*c[5] ^2*c[4]+360*c[5]^3*c[6]*c[7]*c[4]+82*c[5]^3*c[4]^2+1734*c[5]^2*c[6]*c[ 4]^3+388*c[5]^2*c[7]*c[4]^2-944*c[5]^2*c[6]*c[4]^2-900*c[5]^2*c[7]*c[4 ]^3-135*c[5]^3*c[7]*c[4]^2+879*c[5]^3*c[6]*c[4]^2+660*c[5]^2*c[4]^4*c[ 7]-1680*c[6]*c[5]^3*c[4]^3+420*c[5]^3*c[7]*c[4]^3+100*c[5]^4*c[7]*c[4] ^2*c[6]-246*c[5]^2*c[4]^2-405*c[4]^4*c[5]^2+30*c[5]^4*c[4]^3+240*c[5]^ 2*c[6]*c[4]-84*c[5]^2*c[7]*c[4]-396*c[5]^2*c[6]*c[7]*c[4]+562*c[5]^2*c [4]^3+30*c[4]^4*c[5]+90*c[4]^5*c[5]^2+300*c[5]^4*c[4]^4*c[6]*c[7]-3000 *c[5]^3*c[4]^4*c[6]*c[7]-300*c[5]^4*c[4]^3*c[6]*c[7]-255*c[5]^3*c[4]^3 +480*c[5]^2*c[6]*c[4]^5+1540*c[4]^4*c[6]*c[5]^3-180*c[6]*c[5]*c[4]^5+1 20*c[5]^3*c[4]^4)/c[4]^2/(150*c[5]^2*c[6]*c[4]^4-87*c[5]^3*c[4]-12*c[5 ]*c[7]*c[4]-200*c[5]^3*c[4]^4*c[7]-200*c[6]*c[5]^2*c[4]^4*c[7]-200*c[5 ]^4*c[4]^3*c[6]-10*c[5]^2-10*c[4]^2+12*c[5]^3+12*c[4]^3+20*c[7]*c[6]*c [5]^3+15*c[5]*c[6]*c[7]*c[4]+12*c[6]*c[5]^2-15*c[6]*c[5]^3-1300*c[6]*c [7]*c[5]^3*c[4]^3+150*c[5]^4*c[7]*c[4]^2+12*c[7]*c[5]^2+20*c[7]*c[6]*c [4]^3+12*c[7]*c[4]^2-15*c[7]*c[4]^3+12*c[6]*c[4]^2-15*c[6]*c[4]^3-15*c [6]*c[7]*c[4]^2-87*c[5]*c[4]^3-72*c[5]*c[7]*c[4]^2-72*c[5]*c[6]*c[4]^2 +60*c[5]*c[4]^2-12*c[5]*c[6]*c[4]+10*c[5]*c[4]-150*c[5]*c[7]*c[6]*c[4] ^3-15*c[7]*c[5]^3-660*c[5]^2*c[6]*c[7]*c[4]^2+90*c[6]*c[5]*c[7]*c[4]^2 -15*c[6]*c[7]*c[5]^2+150*c[6]*c[5]^4*c[4]^2+930*c[6]*c[5]^2*c[7]*c[4]^ 3+930*c[6]*c[5]^3*c[7]*c[4]^2-200*c[5]^4*c[4]^3*c[7]+110*c[7]*c[5]*c[4 ]^3+110*c[6]*c[5]*c[4]^3+110*c[5]^3*c[6]*c[4]+110*c[5]^3*c[7]*c[4]+60* c[5]^2*c[4]-150*c[5]^3*c[6]*c[7]*c[4]+550*c[5]^3*c[4]^2-690*c[5]^2*c[6 ]*c[4]^3+520*c[5]^2*c[7]*c[4]^2+520*c[5]^2*c[6]*c[4]^2-690*c[5]^2*c[7] *c[4]^3-690*c[5]^3*c[7]*c[4]^2-690*c[5]^3*c[6]*c[4]^2+150*c[5]^2*c[4]^ 4*c[7]+900*c[6]*c[5]^3*c[4]^3+900*c[5]^3*c[7]*c[4]^3-200*c[5]^4*c[7]*c [4]^2*c[6]-429*c[5]^2*c[4]^2-120*c[5]^4*c[4]^2-120*c[4]^4*c[5]^2+150*c [5]^4*c[4]^3-72*c[5]^2*c[6]*c[4]-72*c[5]^2*c[7]*c[4]+90*c[5]^2*c[6]*c[ 7]*c[4]+550*c[5]^2*c[4]^3+300*c[5]^3*c[4]^4*c[6]*c[7]+300*c[5]^4*c[4]^ 3*c[6]*c[7]-690*c[5]^3*c[4]^3-200*c[4]^4*c[6]*c[5]^3+150*c[5]^3*c[4]^4 ), a[8,5] = -1/2*(893*c[5]^2*c[6]*c[4]^4-122*c[6]*c[5]*c[4]^4+850*c[5] ^4*c[4]^4*c[6]^2*c[7]-101*c[7]*c[5]*c[4]^4-535*c[5]^2*c[4]^4*c[7]^2-44 *c[5]^3*c[4]+160*c[4]^5*c[6]^2*c[5]^2+100*c[7]^2*c[5]^2*c[4]^5+170*c[4 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^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]^3-70*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 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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]*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]^4-200*c[5]^4*c[4]^5*c[6]^2*c[7]+15*c[7]*c[6]^2*c[5]*c[4]^4-19 90*c[5]^3*c[4]^4*c[6]^2*c[7]-750*c[5]^3*c[4]^3*c[6]^2*c[7]^2+210*c[5]^ 4*c[4]^4*c[6]*c[7]+1100*c[5]^2*c[4]^4*c[7]*c[6]^2+2380*c[5]^3*c[4]^4*c [6]*c[7]-50*c[5]^4*c[4]^3*c[6]*c[7]-594*c[5]^3*c[4]^3-10*c[5]^5*c[6]*c [7]-3*c[5]^5*c[4]+328*c[7]^2*c[5]^3*c[4]^2+166*c[5]^5*c[6]*c[4]^2+150* c[5]^4*c[4]^5*c[6]^2-200*c[6]*c[7]^2*c[5]^4*c[4]^5-680*c[5]^4*c[4]^4*c 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*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]^3-1 2*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 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^2*c[7]^2*c[4]*c[5]^3-10*c[5]^2*c[4]^3+24*c[7]^2*c[5]^2*c[4]^2-24*c[7] ^2*c[5]^3*c[4]-200*c[7]^2*c[5]^6*c[4]^2*c[6]+110*c[7]^2*c[4]*c[5]^5+13 50*c[5]^5*c[4]^3*c[6]*c[7]+15*c[7]^2*c[4]^4*c[5]+300*c[6]^2*c[7]*c[5]^ 6*c[4]^3+510*c[7]^2*c[5]^3*c[6]^2*c[4]^2-550*c[5]^6*c[4]^2-600*c[7]*c[ 5]^6*c[4]^3+20*c[7]*c[6]^2*c[5]^5+570*c[5]^6*c[6]*c[4]^2+150*c[5]^2*c[ 6]*c[4]^5*c[7]^2+550*c[5]^4*c[4]^4+120*c[5]^7*c[4]^2+72*c[5]^4*c[4]*c[ 6]-40*c[5]^4*c[7]^2*c[4]*c[6]-57*c[5]^4*c[4]*c[7]*c[6]+150*c[6]^2*c[5] ^6*c[4]^2+150*c[4]^5*c[6]^2*c[7]*c[5]^2-200*c[4]^5*c[6]^2*c[7]^2*c[5]^ 2-150*c[5]^7*c[7]*c[4]^2-110*c[6]^2*c[4]^4*c[5]^2+780*c[5]^5*c[4]^2*c[ 7]^2*c[6]-200*c[5]^5*c[4]^5*c[6]-600*c[5]^6*c[4]^3*c[6]-200*c[6]^2*c[7 ]*c[5]^6*c[4]^2-300*c[5]^4*c[4]^5*c[6]^2*c[7]+90*c[7]*c[6]^2*c[5]*c[4] ^4+200*c[6]*c[5]^7*c[7]*c[4]^2-300*c[6]*c[5]^7*c[7]*c[4]^3-180*c[5]^3* c[4]^4*c[6]^2*c[7]-570*c[5]^4*c[4]^4*c[6]*c[7]-540*c[5]^2*c[4]^4*c[7]* c[6]^2+660*c[5]^3*c[4]^4*c[6]*c[7]+60*c[5]^4*c[4]^3*c[6]*c[7]+48*c[5]^ 3*c[4]^3-15*c[6]^2*c[5]^5-200*c[5]^6*c[6]^2*c[4]^3-15*c[5]^5*c[6]*c[7] -48*c[5]^5*c[4]+140*c[5]^5*c[6]*c[4]^2+200*c[5]^4*c[4]^5*c[6]^2+300*c[ 6]^2*c[7]^2*c[5]^3*c[4]^5-300*c[6]*c[7]^2*c[5]^4*c[4]^5+700*c[7]*c[5]^ 6*c[4]^3*c[6]-750*c[5]^4*c[4]^4*c[6]^2+20*c[5]^5*c[7]^2*c[6]+1100*c[7] ^2*c[5]^4*c[6]^2*c[4]^3-930*c[7]^2*c[5]^4*c[6]^2*c[4]^2+300*c[7]^2*c[5 ]^6*c[4]^3*c[6]-20*c[5]^6*c[7]*c[6]-180*c[5]^3*c[4]^4*c[7]^2*c[6]-750* c[5]^4*c[4]^3*c[7]^2*c[6]-540*c[5]^2*c[4]^4*c[7]^2*c[6]-750*c[5]^4*c[4 ]^3*c[6]^2*c[7]-150*c[7]^2*c[4]^4*c[6]^2*c[5]-1100*c[7]^2*c[4]^4*c[5]^ 3*c[6]^2+930*c[7]^2*c[4]^4*c[6]^2*c[5]^2+150*c[5]^6*c[4]*c[7]*c[6]-110 *c[5]^6*c[4]*c[6]-150*c[5]^7*c[6]*c[4]^2+570*c[5]^6*c[4]^3-150*c[5]^7* c[4]^3+15*c[7]^2*c[6]^2*c[5]^3-440*c[4]^4*c[6]*c[5]^3+750*c[5]^5*c[6]^ 2*c[4]^3-200*c[7]^2*c[5]^6*c[4]^3+12*c[5]^4*c[7]^2+200*c[5]^4*c[4]^5*c [6]*c[7]-20*c[5]^4*c[4]-87*c[5]^3*c[4]^4-30*c[5]^5*c[4]*c[6]+110*c[5]^ 5*c[6]^2*c[4]), a[9,7] = -1/60*(10*c[5]*c[6]*c[4]-5*c[6]*c[5]-5*c[6]*c [4]+3*c[6]-5*c[5]*c[4]+3*c[5]+3*c[4]-2)/c[7]/(-c[5]*c[6]*c[4]+c[5]*c[6 ]*c[7]*c[4]+c[5]*c[7]*c[4]-c[7]^2*c[5]*c[4]+c[6]*c[7]*c[4]-c[6]*c[7]^2 *c[4]-c[7]^2*c[4]+c[7]^3*c[4]+c[6]*c[7]*c[5]-c[6]*c[7]^2*c[5]-c[7]^2*c [5]+c[7]^3*c[5]-c[7]^2*c[6]+c[7]^3*c[6]+c[7]^3-c[7]^4), a[9,2] = 0, a[ 9,3] = 0, a[9,5] = -1/60*(10*c[6]*c[7]*c[4]-5*c[6]*c[4]-5*c[7]*c[6]+3* c[6]-5*c[7]*c[4]+3*c[4]-2+3*c[7])/c[5]/(c[5]^3*c[4]-c[5]^2*c[4]+c[5]*c [7]*c[4]-c[5]^2*c[7]*c[4]+c[5]*c[6]*c[4]-c[5]^2*c[6]*c[4]-c[6]*c[7]*c[ 4]+c[5]*c[6]*c[7]*c[4]-c[5]^4+c[5]^3-c[7]*c[5]^2+c[7]*c[5]^3-c[6]*c[5] ^2+c[6]*c[5]^3+c[6]*c[7]*c[5]-c[6]*c[7]*c[5]^2), a[9,8] = 1/60*(-20*c[ 5]*c[6]*c[4]+30*c[5]*c[6]*c[7]*c[4]+15*c[6]*c[5]-20*c[6]*c[7]*c[5]+15* c[6]*c[4]-20*c[6]*c[7]*c[4]-12*c[6]+15*c[7]*c[6]+15*c[5]*c[4]-20*c[5]* c[7]*c[4]-12*c[5]+15*c[7]*c[5]-12*c[4]+15*c[7]*c[4]+10-12*c[7])/(-c[4] +c[7]*c[4]+c[6]*c[4]-c[6]*c[7]*c[4]+c[5]*c[4]-c[5]*c[7]*c[4]-c[5]*c[6] *c[4]+c[5]*c[6]*c[7]*c[4]+1-c[7]-c[6]+c[7]*c[6]-c[5]+c[7]*c[5]+c[6]*c[ 5]-c[6]*c[7]*c[5]), a[9,1] = 1/60*(30*c[5]*c[6]*c[7]*c[4]-10*c[5]*c[6] *c[4]-10*c[6]*c[7]*c[5]+5*c[6]*c[5]-10*c[6]*c[7]*c[4]+5*c[6]*c[4]+5*c[ 7]*c[6]-3*c[6]-10*c[5]*c[7]*c[4]+5*c[5]*c[4]+5*c[7]*c[5]-3*c[5]+5*c[7] *c[4]-3*c[4]+2-3*c[7])/c[5]/c[6]/c[7]/c[4], a[9,4] = 1/60*(-2+3*c[5]-5 *c[7]*c[6]+3*c[7]+3*c[6]+10*c[6]*c[7]*c[5]-5*c[6]*c[5]-5*c[7]*c[5])/c[ 4]/(-c[6]*c[4]^3-c[4]^3+c[4]^4+c[7]*c[4]^2+c[6]*c[4]^2-c[6]*c[7]*c[4]+ c[6]*c[7]*c[4]^2-c[5]*c[4]^3+c[5]*c[4]^2-c[7]*c[4]^3-c[5]*c[7]*c[4]+c[ 5]*c[7]*c[4]^2-c[5]*c[6]*c[4]+c[5]*c[6]*c[4]^2+c[6]*c[7]*c[5]-c[5]*c[6 ]*c[7]*c[4]), a[9,6] = 1/60*(10*c[5]*c[7]*c[4]-5*c[5]*c[4]-5*c[7]*c[5] +3*c[5]-5*c[7]*c[4]+3*c[4]-2+3*c[7])/c[6]/(c[6]^2*c[4]*c[5]-c[5]*c[6]* c[4]+c[5]*c[7]*c[4]-c[5]*c[6]*c[7]*c[4]-c[6]^3*c[4]+c[6]^2*c[4]-c[6]*c [7]*c[4]+c[6]^2*c[7]*c[4]-c[6]^3*c[5]+c[6]^2*c[5]-c[6]*c[7]*c[5]+c[6]^ 2*c[7]*c[5]+c[6]^4-c[6]^3+c[7]*c[6]^2-c[6]^3*c[7])\}:" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}} {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 33 "#======== ========================" }}{PARA 0 "" 0 "" {TEXT -1 22 "We specify th e nodes " }{XPPEDIT 18 0 "c[6] = 35/36;" "6#/&%\"cG6#\"\"'*&\"#N\"\" \"\"#O!\"\"" }{TEXT -1 7 " and " }{XPPEDIT 18 0 "c[7] = 544/545;" "6 #/&%\"cG6#\"\"(*&\"$W&\"\"\"\"$X&!\"\"" }{TEXT -1 27 " and determine v alues 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 t he 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 for th e 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]=35/36,c[7]=544/ 545\}:\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 16646 "eB := \{a[4,1] = 1/4*c[4] , a[3,1] = -2/9*c[4]*(c[4]-3*c[2])/c[2], a[5,4] = c[5]^2*(-c[4]+c[5])/ c[4]^2, a[2,1] = c[2], `b*`[6] = 139968/89075*(170730+4595262*c[5]*c[4 ]+4569600*c[5]*c[4]^3-8952310*c[5]^2*c[4]^3+1407471*c[5]^2-855591*c[5] -759500*c[5]^3-8031426*c[5]^2*c[4]+1407471*c[4]^2-855591*c[4]+14888490 *c[5]^2*c[4]^2-8952310*c[5]^3*c[4]^2-8031426*c[5]*c[4]^2+4569600*c[5]^ 3*c[4]+5701500*c[5]^3*c[4]^3-759500*c[4]^3)/(105-1124*c[5]*c[4]^2-1124 *c[5]^2*c[4]+280*c[5]^2-351*c[4]-351*c[5]+1304*c[5]*c[4]+1050*c[5]^2*c [4]^2+280*c[4]^2)/(1296*c[5]*c[4]-1260*c[4]-1260*c[5]+1225), a[7,1] = \+ 136/3087834771875*(65287186407300*c[4]^5*c[5]^3-373718478717460*c[5]^4 *c[4]^3-1375406950400*c[5]*c[4]-24802421661092*c[5]*c[4]^3+71210225754 23*c[5]^2*c[4]^3+1178920243200*c[5]^2-3945304218240*c[5]^3+12414632429 5040*c[4]^5*c[5]^2-86381637422335*c[4]^4*c[5]^2+24098678491500*c[5]^3* c[4]^6-6675195303488*c[5]^2*c[4]+392973414400*c[4]^2+39465438563300*c[ 5]^5*c[4]^3+14216185548798*c[5]^2*c[4]^2+15762359200000*c[4]^6*c[5]+45 6657453597300*c[5]^4*c[4]^4-259668019369620*c[5]^3*c[4]^4+153209344640 355*c[5]^4*c[4]^2-7881179600000*c[5]^5*c[4]^2+3152471840000*c[5]^4+390 71381360000*c[5]^5*c[4]^5-143768917995321*c[5]^3*c[4]^2+1960918575296* c[4]^4+39071381360000*c[5]^4*c[4]^6+45401553616654*c[4]^4*c[5]-4305522 7906170*c[4]^5*c[5]-64809443477250*c[5]^5*c[4]^4+8262659035328*c[5]*c[ 4]^2-32362360959080*c[5]^4*c[4]+35575638644436*c[5]^3*c[4]+28941336546 0440*c[5]^3*c[4]^3-779548502080*c[4]^5-55544422064200*c[4]^6*c[5]^2-15 74325953408*c[4]^3-256711711253250*c[5]^4*c[4]^5)/(1700*c[5]^4*c[4]^3+ 33*c[5]*c[4]-572*c[5]*c[4]^3+3360*c[5]^2*c[4]^3-33*c[5]^2+67*c[5]^3-67 0*c[4]^4*c[5]^2+198*c[5]^2*c[4]-33*c[4]^2-1624*c[5]^2*c[4]^2+1700*c[5] ^3*c[4]^4-670*c[5]^4*c[4]^2+3360*c[5]^3*c[4]^2+198*c[5]*c[4]^2-572*c[5 ]^3*c[4]-6440*c[5]^3*c[4]^3+67*c[4]^3)/c[5]/c[4]^2, b[6] = 5038848/890 75*(2715*c[5]*c[4]-1085*c[5]-1085*c[4]+542)/(36*c[4]-35)/(36*c[5]-35), a[7,5] = 272/88223850625*(-48318907924250*c[4]^5*c[5]^3+6416017207710 0*c[5]^4*c[4]^3-2942902849984*c[5]*c[4]-22461896126317*c[5]*c[4]^3+111 375565226850*c[5]^2*c[4]^3-1589690358656*c[5]^2+392973414400*c[5]+2007 341297280*c[5]^3+35878567192100*c[4]^5*c[5]^2-103406333831420*c[4]^4*c [5]^2+14520713284860*c[5]^2*c[4]+787162976704*c[4]^2-196486707200*c[4] -56777295446547*c[5]^2*c[4]^2-59044332161750*c[5]^4*c[4]^4+14016281736 1700*c[5]^3*c[4]^4-32747395489000*c[5]^4*c[4]^2-810635616000*c[5]^4+77 741355811585*c[5]^3*c[4]^2+389774251040*c[4]^4+21897089885135*c[4]^4*c [5]-7881179600000*c[4]^5*c[5]+10995643937522*c[5]*c[4]^2+8121294631800 *c[5]^4*c[4]-19502494389460*c[5]^3*c[4]-152092318006920*c[5]^3*c[4]^3- 980459287648*c[4]^3+20321856450000*c[5]^4*c[4]^5)/c[5]/(2412*c[5]^5+83 620*c[4]^5*c[5]^3+201950*c[5]^4*c[4]^3+5773*c[5]*c[4]^3-41536*c[5]^2*c [4]^3+1155*c[5]^3-23450*c[4]^5*c[5]^2+138192*c[4]^4*c[5]^2-2310*c[5]^2 *c[4]-267220*c[5]^5*c[4]^3-2376*c[5]^2*c[4]^2+61200*c[5]^6*c[4]^3+2077 20*c[5]^4*c[4]^4-24120*c[5]^6*c[4]^2-322910*c[5]^3*c[4]^4-155472*c[5]^ 4*c[4]^2+144410*c[5]^5*c[4]^2-3533*c[5]^4-20592*c[5]^5*c[4]+36820*c[5] ^3*c[4]^2+2345*c[4]^4-22432*c[4]^4*c[5]+2310*c[5]*c[4]^2+24736*c[5]^4* c[4]-2209*c[5]^3*c[4]+37872*c[5]^3*c[4]^3-1155*c[4]^3-61200*c[5]^4*c[4 ]^5), a[6,4] = -35/3359232*(907200*c[5]^4*c[4]^3+300125*c[5]*c[4]+4024 020*c[5]*c[4]^3-72420*c[5]^2*c[4]^3-257250*c[5]^2+352800*c[5]^3-272160 0*c[4]^5*c[5]^2+5443200*c[4]^4*c[5]^2+1455300*c[5]^2*c[4]-85750*c[4]^2 -3798270*c[5]^2*c[4]^2+2721600*c[5]^3*c[4]^4-882000*c[5]^4*c[4]^2+8932 248*c[5]^3*c[4]^2-4812008*c[4]^4*c[5]+1764000*c[4]^5*c[5]-1293600*c[5] *c[4]^2-2615760*c[5]^3*c[4]-9450000*c[5]^3*c[4]^3+88200*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[8,4] = -1/2*(-363614395790*c[4]^5*c[5]^3-584301010000*c[4]^7*c[5]^ 2+1125966162735*c[5]^4*c[4]^3+2528321600*c[5]*c[4]+82197016959*c[5]*c[ 4]^3-352242114*c[5]^2*c[4]^3-29802780000*c[5]*c[4]^8+111863430000*c[5] ^4*c[4]^7+444141148500*c[5]^3*c[4]^7-2167132800*c[5]^2+7248413760*c[5] ^3-991343610075*c[4]^5*c[5]^2+386173959496*c[4]^4*c[5]^2-456715162750* c[5]^3*c[4]^6-1490139000*c[4]^6+15957266294*c[5]^2*c[4]-722377600*c[4] ^2-80312165600*c[5]^5*c[4]^3-47905513941*c[5]^2*c[4]^2+37287810000*c[5 ]^5*c[4]^6+119705544000*c[4]^8*c[5]^2-326649178330*c[4]^6*c[5]-1801258 139430*c[5]^4*c[4]^4+1124391333620*c[5]^3*c[4]^4-382061344565*c[5]^4*c [4]^2+14460880000*c[5]^5*c[4]^2-111863430000*c[4]^8*c[5]^3-5784352000* c[5]^4-125966565250*c[5]^5*c[4]^5+377661385694*c[5]^3*c[4]^2-661348655 4*c[4]^4-672273543250*c[5]^4*c[4]^6-220400956880*c[4]^4*c[5]+356057689 935*c[4]^5*c[5]+154530040850*c[5]^5*c[4]^4-19604775684*c[5]*c[4]^2+689 70808910*c[5]^4*c[4]+155674662400*c[5]*c[4]^7-77570088010*c[5]^3*c[4]- 943679205024*c[5]^3*c[4]^3+5180771700*c[4]^5+1104232739140*c[4]^6*c[5] ^2+3645231454*c[4]^3+1554576977600*c[5]^4*c[4]^5)/(-923200*c[4]^5*c[5] ^3-3133150*c[5]^4*c[4]^3-70997*c[5]*c[4]^3+570250*c[5]^2*c[4]^3-17892* c[5]^3+363550*c[4]^5*c[5]^2-1823490*c[4]^4*c[5]^2+35784*c[5]^2*c[4]+92 3200*c[5]^5*c[4]^3+3133150*c[5]^3*c[4]^4+1823490*c[5]^4*c[4]^2-363550* c[5]^5*c[4]^2+36355*c[5]^4-570250*c[5]^3*c[4]^2-36355*c[4]^4+310450*c[ 4]^4*c[5]-35784*c[5]*c[4]^2-310450*c[5]^4*c[4]+70997*c[5]^3*c[4]+17892 *c[4]^3)/(-38659*c[4]+19620*c[4]^2+19040)/c[4]^2, `b*`[4] = 1/60*(-797 99414*c[5]*c[4]-42579304*c[5]^2+22451259*c[5]+25847500*c[5]^3+16122931 0*c[5]^2*c[4]-10177300*c[4]^2+12758289*c[4]-141920450*c[5]^2*c[4]^2-38 16645+96925500*c[5]^3*c[4]^2+66701040*c[5]*c[4]^2-103758200*c[5]^3*c[4 ])/(-900671*c[5]*c[4]-612580*c[5]*c[4]^3+572250*c[5]^2*c[4]^3-152320*c [5]^2+190944*c[5]+764056*c[5]^2*c[4]-343615*c[4]^2+248169*c[4]-1183780 *c[5]^2*c[4]^2+1322136*c[5]*c[4]^2-57120+152600*c[4]^3)/(36*c[4]-35)/( -c[4]+c[5])/c[4], `b*`[1] = 1/1142400*(151578938*c[5]*c[4]+208764500*c [5]*c[4]^3-518449740*c[5]^2*c[4]^3+42579304*c[5]^2-22451259*c[5]-25847 500*c[5]^3-318710024*c[5]^2*c[4]+42579304*c[4]^2-22451259*c[4]+7347082 60*c[5]^2*c[4]^2+3816645-518449740*c[5]^3*c[4]^2-318710024*c[5]*c[4]^2 +208764500*c[5]^3*c[4]+393771000*c[5]^3*c[4]^3-25847500*c[4]^3)/(105-1 124*c[5]*c[4]^2-1124*c[5]^2*c[4]+280*c[5]^2-351*c[4]-351*c[5]+1304*c[5 ]*c[4]+1050*c[5]^2*c[4]^2+280*c[4]^2)/c[5]/c[4], 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*(-79799414*c[5]*c [4]-103758200*c[5]*c[4]^3+96925500*c[5]^2*c[4]^3-10177300*c[5]^2+12758 289*c[5]+66701040*c[5]^2*c[4]-42579304*c[4]^2+22451259*c[4]-141920450* c[5]^2*c[4]^2-3816645+161229310*c[5]*c[4]^2+25847500*c[4]^3)/(-900671* c[5]*c[4]-343615*c[5]^2+248169*c[5]+152600*c[5]^3+1322136*c[5]^2*c[4]- 152320*c[4]^2+190944*c[4]-1183780*c[5]^2*c[4]^2+572250*c[5]^3*c[4]^2+7 64056*c[5]*c[4]^2-57120-612580*c[5]^3*c[4])/(36*c[5]-35)/(-c[4]+c[5])/ c[5], `b*`[2] = 0, a[5,3] = -3/4*c[5]^2*(2*c[5]-3*c[4])/c[4]^2, a[8,5] = 1/2*(-1490139000*c[5]^5+154530040850*c[4]^5*c[5]^3-397385219350*c[5 ]^4*c[4]^3+5046602256*c[5]*c[4]+39479926290*c[5]*c[4]^3-245971968236*c [5]^2*c[4]^3+3645231454*c[5]^2-722377600*c[5]-6613486554*c[5]^3-803121 65600*c[4]^5*c[5]^2+230169941275*c[4]^4*c[5]^2-32100547627*c[5]^2*c[4] -1445697886*c[4]^2+117882459050*c[5]^5*c[4]^3+361188800*c[4]+124569508 734*c[5]^2*c[4]^2+365875565800*c[5]^4*c[4]^4-447400523830*c[5]^3*c[4]^ 4+203071639205*c[5]^4*c[4]^2-60179390500*c[5]^5*c[4]^2+5180771700*c[5] ^4+37287810000*c[5]^5*c[4]^5+14926111200*c[5]^5*c[4]-247255884212*c[5] ^3*c[4]^2-713276890*c[4]^4-39504855605*c[4]^4*c[5]+14460880000*c[4]^5* c[5]-108426850750*c[5]^5*c[4]^4-18760175341*c[5]*c[4]^2-50776192105*c[ 5]^4*c[4]+62542837476*c[5]^3*c[4]+484197016270*c[5]^3*c[4]^3+179778597 6*c[4]^3-125966565250*c[5]^4*c[4]^5)/c[5]/(-1756488985*c[5]^5-31632207 450*c[4]^5*c[5]^3-48466871000*c[5]^4*c[4]^3-2043469708*c[5]*c[4]^3+139 53274063*c[5]^2*c[4]^3+713285100*c[5]^6-340663680*c[5]^3+6921992000*c[ 4]^5*c[5]^2-47434221250*c[4]^4*c[5]^2+681327360*c[5]^2*c[4]-6091029000 *c[5]^6*c[4]+138702173850*c[5]^5*c[4]^3-7132851000*c[5]^7*c[4]^2+13833 73656*c[5]^2*c[4]^2-97162391800*c[5]^6*c[4]^3-156901319650*c[5]^4*c[4] ^4+49831353250*c[5]^6*c[4]^2+136240504910*c[5]^3*c[4]^4+56764544350*c[ 5]^4*c[4]^2-88604596910*c[5]^5*c[4]^2+1383886028*c[5]^4-18113184000*c[ 5]^5*c[4]^5+13394647690*c[5]^5*c[4]-11559642080*c[5]^3*c[4]^2-69219920 0*c[4]^4+18113184000*c[5]^7*c[4]^3+7316415945*c[4]^4*c[5]+61472403000* c[5]^5*c[4]^4-681327360*c[5]*c[4]^2-7953558943*c[5]^4*c[4]-31590776*c[ 5]^3*c[4]-23438255890*c[5]^3*c[4]^3+340663680*c[4]^3+42822839800*c[5]^ 4*c[4]^5), a[7,4] = -272/88223850625*(-68312237151900*c[4]^5*c[5]^3+65 149618716000*c[4]^7*c[5]^2-439782929659260*c[5]^4*c[4]^3-1375406950400 *c[5]*c[4]-35426435564804*c[5]*c[4]^3+18835120278223*c[5]^2*c[4]^3-609 65569350000*c[5]^3*c[4]^7+1178920243200*c[5]^2-3945304218240*c[5]^3+34 8189438755040*c[4]^5*c[5]^2-191519095418675*c[4]^4*c[5]^2+181203220012 500*c[5]^3*c[4]^6-7498962161216*c[5]^2*c[4]+392973414400*c[4]^2+358785 67192100*c[5]^5*c[4]^3+18541337762142*c[5]^2*c[4]^2+68476500092200*c[4 ]^6*c[5]+541724724630100*c[5]^4*c[4]^4-265129056515320*c[5]^3*c[4]^4+1 73772052157735*c[5]^4*c[4]^2-7881179600000*c[5]^5*c[4]^2+3152471840000 *c[5]^4+20321856450000*c[5]^5*c[4]^5-167227493950561*c[5]^3*c[4]^2+200 7341297280*c[4]^4+60965569350000*c[5]^4*c[4]^6+84476042631014*c[4]^4*c [5]-109226410436910*c[4]^5*c[5]-48318907924250*c[5]^5*c[4]^4+928852650 6176*c[5]*c[4]^2-34435580669960*c[5]^4*c[4]-16212712320000*c[5]*c[4]^7 +38268523897396*c[5]^3*c[4]+346108759253220*c[5]^3*c[4]^3-810635616000 *c[4]^5-252876785719400*c[4]^6*c[5]^2-1589690358656*c[4]^3-30539717089 0250*c[5]^4*c[4]^5)/(-1700*c[4]^5*c[5]^3-5770*c[5]^4*c[4]^3-131*c[5]*c [4]^3+1052*c[5]^2*c[4]^3-33*c[5]^3+670*c[4]^5*c[5]^2-3360*c[4]^4*c[5]^ 2+66*c[5]^2*c[4]+1700*c[5]^5*c[4]^3+5770*c[5]^3*c[4]^4+3360*c[5]^4*c[4 ]^2-670*c[5]^5*c[4]^2+67*c[5]^4-1052*c[5]^3*c[4]^2-67*c[4]^4+572*c[4]^ 4*c[5]-66*c[5]*c[4]^2-572*c[5]^4*c[4]+131*c[5]^3*c[4]+33*c[4]^3)/(36*c [4]-35)/c[4]^2, `b*`[8] = 0, b[1] = 1/1142400*(282710*c[5]*c[4]-55965* c[5]-55965*c[4]+18463)/c[5]/c[4], b[4] = -1/60*(-18463+55965*c[5])/c[4 ]/(58279*c[4]^3-19620*c[4]^4-57699*c[4]^2+19040*c[4]+19620*c[5]*c[4]^3 -58279*c[5]*c[4]^2+57699*c[5]*c[4]-19040*c[5]), b[2] = 0, c[9] = 1, a[ 8,1] = 1/19040*(29789163990*c[4]^5*c[5]^3-171267998470*c[5]^4*c[4]^3-6 32080400*c[5]*c[4]-11395087142*c[5]*c[4]^3+3278777027*c[5]^2*c[4]^3+54 1783200*c[5]^2-1812103440*c[5]^3+57028455930*c[4]^5*c[5]^2-39720920962 *c[4]^4*c[5]^2+11058948450*c[5]^3*c[4]^6-3067496423*c[5]^2*c[4]+180594 400*c[4]^2+18087240450*c[5]^5*c[4]^3+6533148218*c[5]^2*c[4]^2+72304400 00*c[4]^6*c[5]+209047431990*c[5]^4*c[4]^4-119004152800*c[5]^3*c[4]^4+7 0247310825*c[5]^4*c[4]^2-3615220000*c[5]^5*c[4]^2+1446088000*c[5]^4+17 841675100*c[5]^5*c[4]^5-66016404586*c[5]^3*c[4]^2+898892988*c[4]^4+178 41675100*c[5]^4*c[4]^6+20862125334*c[4]^4*c[5]-19773547375*c[4]^5*c[5] -29649761550*c[5]^5*c[4]^4+3796033463*c[5]*c[4]^2-14841508845*c[5]^4*c [4]+16338505148*c[5]^3*c[4]+132829055238*c[5]^3*c[4]^3-356638445*c[4]^ 5-25475327550*c[4]^6*c[5]^2-722848943*c[4]^3-117347738400*c[5]^4*c[4]^ 5)/c[5]/c[4]^2/(923200*c[5]^4*c[4]^3+17892*c[5]*c[4]-310450*c[5]*c[4]^ 3+1823490*c[5]^2*c[4]^3-17892*c[5]^2+36355*c[5]^3-363550*c[4]^4*c[5]^2 +107352*c[5]^2*c[4]-17892*c[4]^2-880700*c[5]^2*c[4]^2+923200*c[5]^3*c[ 4]^4-363550*c[5]^4*c[4]^2+1823490*c[5]^3*c[4]^2+107352*c[5]*c[4]^2-310 450*c[5]^3*c[4]-3496700*c[5]^3*c[4]^3+36355*c[4]^3), a[8,6] = -23328/1 7815*(1191545*c[5]*c[4]+238381-476690*c[4]-476690*c[5])*(c[4]-1)*(-1+c [5])/(36*c[4]-35)/(36*c[5]-35)/(92320*c[5]*c[4]-36355*c[5]-36355*c[4]+ 17892), a[9,7] = -9616399718125/3322752*(170*c[5]*c[4]-67*c[5]-67*c[4] +33)/(297025*c[5]*c[4]-296480*c[4]-296480*c[5]+295936), a[7,6] = -6459 429888/3087834771875*(5*c[5]*c[4]+1-2*c[4]-2*c[5])*(545*c[4]-544)*(545 *c[5]-544)/(170*c[5]*c[4]-67*c[5]-67*c[4]+33)/(36*c[4]-35)/(36*c[5]-35 ), a[9,1] = 1/1142400*(282710*c[5]*c[4]-55965*c[5]-55965*c[4]+18463)/c [5]/c[4], `b*`[9] = 1/10*(1750*c[5]^2*c[4]^2-1580*c[5]^2*c[4]+350*c[5] ^2-1580*c[5]*c[4]^2+1585*c[5]*c[4]-386*c[5]+350*c[4]^2-386*c[4]+105)/( 105-1124*c[5]*c[4]^2-1124*c[5]^2*c[4]+280*c[5]^2-351*c[4]-351*c[5]+130 4*c[5]*c[4]+1050*c[5]^2*c[4]^2+280*c[4]^2), a[6,2] = 0, a[7,2] = 0, a[ 6,5] = 35/3359232*(1913976*c[5]^2*c[4]^2-88200*c[4]^2+42875*c[4]-66528 0*c[5]^2*c[4]-1770090*c[5]*c[4]^2+45360*c[4]^3-882000*c[4]^4*c[5]+2133 844*c[5]*c[4]^3-2242800*c[5]^2*c[4]^3+602700*c[5]*c[4]+907200*c[4]^4*c [5]^2+88200*c[5]^2-85750*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[9,2] = 0, a[9,3] = 0, a[9,6] = \+ 5038848/89075*(2715*c[5]*c[4]-1085*c[5]-1085*c[4]+542)/(1296*c[5]*c[4] -1260*c[4]-1260*c[5]+1225), a[8,2] = 0, `b*`[3] = 0, a[9,8] = 1/60*(92 320*c[5]*c[4]-36355*c[5]-36355*c[4]+17892)/(-c[4]+c[5]*c[4]+1-c[5]), a [6,3] = 35/746496*(2406004*c[5]*c[4]^3-1901130*c[5]*c[4]^2+573300*c[5] *c[4]-85750*c[5]+42875*c[4]-294000*c[5]^3*c[4]^2-3897600*c[5]^2*c[4]^3 +3068136*c[5]^2*c[4]^2-887040*c[5]^2*c[4]+117600*c[5]^2-882000*c[4]^4* c[5]-58800*c[4]^2+453600*c[5]^3*c[4]^3+1360800*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), a[4,2] = 0, a[5,2] = 0, a[9,5] = 1/ 60*(55965*c[4]-18463)/c[5]/(-19620*c[5]^3*c[4]+58279*c[5]^2*c[4]-57699 *c[5]*c[4]+19040*c[4]+19620*c[5]^4-58279*c[5]^3+57699*c[5]^2-19040*c[5 ]), `b*`[7] = -17644770125/3322752*(283014*c[5]*c[4]+283820*c[5]*c[4]^ 3-558900*c[5]^2*c[4]^3+86616*c[5]^2-52389*c[5]-46900*c[5]^3-497208*c[5 ]^2*c[4]+86616*c[4]^2-52389*c[4]+926620*c[5]^2*c[4]^2+10395-558900*c[5 ]^3*c[4]^2-497208*c[5]*c[4]^2+283820*c[5]^3*c[4]+357000*c[5]^3*c[4]^3- 46900*c[4]^3)/(625217129*c[5]*c[4]+416410520*c[5]*c[4]^3-645160100*c[5 ]^2*c[4]^3+186926560*c[5]^2-135003936*c[5]-83014400*c[5]^3-906512159*c [5]^2*c[4]+186926560*c[4]^2-135003936*c[4]+1364540440*c[5]^2*c[4]^2+31 073280-645160100*c[5]^3*c[4]^2-906512159*c[5]*c[4]^2+416410520*c[5]^3* c[4]+311876250*c[5]^3*c[4]^3-83014400*c[4]^3), a[7,3] = 408/8822385062 5*(2540232056250*c[4]^5*c[5]^3-904855815500*c[5]^4*c[4]^3+258462101952 *c[5]*c[4]+2302893876065*c[5]*c[4]^3-9417364581390*c[5]^2*c[4]^3+11272 2977664*c[5]^2-33683435520*c[5]-90070624000*c[5]^3-2714567446500*c[4]^ 5*c[5]^2+8798076183100*c[4]^4*c[5]^2-1050491197404*c[5]^2*c[4]-5636148 8832*c[4]^2+16841717760*c[4]+4471185413575*c[5]^2*c[4]^2+846744018750* c[5]^4*c[4]^4-8699887374500*c[5]^3*c[4]^4+225176560000*c[5]^4*c[4]^2-4 143277524165*c[5]^3*c[4]^2-2196422537175*c[4]^4*c[5]+675529680000*c[4] ^5*c[5]-1062065701074*c[5]*c[4]^2+902366070200*c[5]^3*c[4]+91849719221 00*c[5]^3*c[4]^3+45035312000*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)/(170*c[5]*c[4]-67*c[5]-67*c[4]+33), a[6,1] = 35/6718464*(1866 240*c[4]^5*c[5]^3-2268000*c[5]^4*c[4]^3-300125*c[5]*c[4]-3449460*c[5]* c[4]^3-3747756*c[5]^2*c[4]^3+257250*c[5]^2-352800*c[5]^3+1814400*c[4]^ 5*c[5]^2-719856*c[4]^4*c[5]^2-1543500*c[5]^2*c[4]+85750*c[4]^2+4282110 *c[5]^2*c[4]^2+1866240*c[5]^4*c[4]^4-10588320*c[5]^3*c[4]^4+882000*c[5 ]^4*c[4]^2-9935352*c[5]^3*c[4]^2+90720*c[4]^4+4081064*c[4]^4*c[5]-1764 000*c[4]^5*c[5]+1381800*c[5]*c[4]^2+2751840*c[5]^3*c[4]+15534144*c[5]^ 3*c[4]^3-176400*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) , b[8] = 1/60*(92320*c[5]*c[4]-36355*c[5]-36355*c[4]+17892)/(c[4]-1)/( -1+c[5]), b[5] = 1/60*(55965*c[4]-18463)/(-c[4]+c[5])/c[5]/(19620*c[5] ^3-58279*c[5]^2+57699*c[5]-19040), c[6] = 35/36, c[7] = 544/545, a[9,4 ] = -1/60*(-18463+55965*c[5])/c[4]/(58279*c[4]^3-19620*c[4]^4-57699*c[ 4]^2+19040*c[4]+19620*c[5]*c[4]^3-58279*c[5]*c[4]^2+57699*c[5]*c[4]-19 040*c[5]), a[8,3] = 3/4*(8552250*c[4]^5*c[5]^3-3050600*c[5]^4*c[4]^3+8 73267*c[5]*c[4]+7776740*c[5]*c[4]^3-31785510*c[5]^2*c[4]^3+380694*c[5] ^2-113820*c[5]-303800*c[5]^3-9151800*c[4]^5*c[5]^2+29676740*c[4]^4*c[5 ]^2-3547416*c[5]^2*c[4]-190347*c[4]^2+56910*c[4]+15095992*c[5]^2*c[4]^ 2+2850750*c[5]^4*c[4]^4-29306900*c[5]^3*c[4]^4+759500*c[5]^4*c[4]^2-13 970475*c[5]^3*c[4]^2-7412205*c[4]^4*c[5]+2278500*c[4]^5*c[5]-3587688*c [5]*c[4]^2+3043040*c[5]^3*c[4]+30961780*c[5]^3*c[4]^3+151900*c[4]^3)/c [4]^2/(923200*c[5]^4*c[4]^3+17892*c[5]*c[4]-310450*c[5]*c[4]^3+1823490 *c[5]^2*c[4]^3-17892*c[5]^2+36355*c[5]^3-363550*c[4]^4*c[5]^2+107352*c [5]^2*c[4]-17892*c[4]^2-880700*c[5]^2*c[4]^2+923200*c[5]^3*c[4]^4-3635 50*c[5]^4*c[4]^2+1823490*c[5]^3*c[4]^2+107352*c[5]*c[4]^2-310450*c[5]^ 3*c[4]-3496700*c[5]^3*c[4]^3+36355*c[4]^3), a[8,7] = -88223850625/2768 96*(170*c[5]*c[4]-67*c[5]-67*c[4]+33)*(c[4]-1)*(-1+c[5])/(545*c[5]-544 )/(545*c[4]-544)/(92320*c[5]*c[4]-36355*c[5]-36355*c[4]+17892), b[7] = -9616399718125/3322752*(170*c[5]*c[4]-67*c[5]-67*c[4]+33)/(545*c[4]-5 44)/(545*c[5]-544), a[4,3] = 3/4*c[4], a[3,2] = 2/9*c[4]^2/c[2], c[3] \+ = 2/3*c[4], c[8] = 1, b[3] = 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 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 4601 "prin_err_norm_sqrd := (u,v,w)->1/125719761484800000 0*(223968812622149*v^6+40593637666109*v^4-190700545749992*v^5+91748626 9534149*w^6+166651009106109*w^4-782048177317992*w^5+163685906055000*u^ 2*w^2+899527403968200*u^2*w^4-436648020861600*w^4*u+511977834270000*w^ 5*u-395187831416000000*v^9*w^2+93100462938000*w^3*u+359042955157885976 *v^5*w^3-3903951106252890000*u^2*v^5*w^4-9490761970200000*u*v^7*w-3520 66053744594000*u^2*v^5*w^2-1298918499317208000*u^2*v^4*w^3-28710285641 43390000*u^2*v^6*w^3-767437139952000*u^2*w^3+127153548321156746*w^4*v^ 3-230818769820218*w*v^3-6059695859701992*w^2*v^3-317222653442877000*v^ 6*w^7+7956418316000000*w*v^8-581570380635530980*v^7*w^2+22149364743450 0*u*v^2*w-303336600462000*u*v*w^2+381073980294000000*u*v^8*w^2+8648380 5611088600*u*v^4*w^2+8995440717423900*u*v^4*w-853474909304352400*w^4*v ^7+79504585384770000*v^7*w^7-1904275361405277000*v^7*w^6-2852347134918 400000*v^9*w^4+35724169341642748*v^4*w^2-26236299096189000*w^8*v^5-138 7193121520432480*w^3*v^6+2908226885716103680*w^3*v^7+4328954403414900* w^8*v^4-133150848448174018*v^5*w^2+2235620049196800000*v^9*w^3-8913884 36064000000*v^9*w^5-16545370083171678*w^6*v-3318415372874419200*w^6*v^ 5+137297429387329176*w^5*v^3-865790880682980*w^7*v^2+524781039191327*w ^2*v^2+1784286793083516720*w^6*v^4-69080800190192400*w^7*v^4+410892541 2710140800*w^5*v^6+661952606000560000*v^8*w^2-657920883770388000*u^2*v ^3*w^4-159983662091400000*u^2*v^4*w^5+31616763963624000*u^2*v^3*w^5+24 87601931219244000*u^2*v^5*w^3-43153008373800000*u*v^4*w^6+437630958121 505400*u^2*v^3*w^3+3609366267780000*u*v^2*w^2-2081783415981600*u*v^3*w +102508295040685800*u*v^3*w^3-9872805405006900*u*v^2*w^3-2170337652823 5000*u*v^3*w^2+33115506858000000*u*v^7*w^6-2624960985166800000*u^2*v^7 *w^4-503856306840600000*u^2*v^6*w^5-2631213051264000*u^2*v^2*w^5+20185 74661759173000*u^2*v^4*w^4+65564754412500000*u^2*v^8*w^2+1349136443035 33200*u^2*v^2*w^4+100443037140360000*u*v^5*w^6+596397749094000000*u^2* v^8*w^4-395487889335000000*u^2*v^8*w^3+1709974357610400000*u^2*v^7*w^3 +4474341894672720000*u^2*v^6*w^4+203393771658000000*u^2*v^7*w^5+328135 2451020000000*v^8*w^4*u+58608879966000000*v^8*w^5*u+411796511423640000 *u^2*v^5*w^5-106313075505000000*u*v^6*w^6-2243743921024200000*v^8*w^3* u-657905699115000000*v^7*w^5*u-795196998792000000*v^9*w^4*u+1924144528 320000*u^2*v^4*w^6-163685906055000*u^2*w*v-131444404873200*w^3*v*u+527 317185780000000*v^9*w^3*u-748794715200000*w^6*u*v^2+17341208514000000* u^2*v^6*w^6-11552832748800000*u^2*v^5*w^6-285805485220500000*u^2*v^7*w ^2-1964230872660000*v*u^2*w^2+991903088789982000*w^5*u*v^4+89990746599 60000*w^6*u*v^3-21323965516135200*w*v^5*u-348324433222860000*w^5*u*v^3 -31237752629646000*w^4*u*v^2+1557693291294000*v^2*u^2*w+59127273807300 00*w^4*u*v+74850369937632000*w^5*u*v^2-5123740115084040000*w^4*v^7*u-1 513399767364116000*w^4*v^5*u-3172795453100178000*w^3*v^6*u-92309505770 16000*w^5*u*v-53174269213222500*u^2*v^3*w^2+1616032736660520000*w^5*v^ 6*u+3713309952496200000*w^3*v^7*u+3624263243681648400*v^6*w^6+17698000 520700322*v^6*w-613128104953752480*w^6*v^3-536898843580935646*w^4*v^4- 5669891838338933400*w^5*v^7+233566926006186600*w^7*v^5+104420051033356 80*w^7*v^3+52192554339045600*w^4*v^3*u-522218113541074800*w^3*v^4*u-17 06195284478280000*w^5*v^5*u+226783259040420000*w^4*v^4*u+1242180614552 2200*u^2*v*w^3+16937577996122700*u^2*v^4*w+3874336945308720000*w^4*v^6 *u-95103596863119600*u^2*v^2*w^3-616456545897780000*w^2*v^7*u+49652807 7620088000*w^2*v^6*u-6931416866988600*u^2*v^3*w+1620419958587088000*w^ 3*v^5*u+24733025591652000*u*v^6*w-247229841510446400*w^2*v^5*u+1242021 4024235850*u^2*v^2*w^2+164146169397949200*u^2*v^4*w^2-1621662797908440 0*u^2*v*w^4-87419672550000000*u*v^9*w^2+455593103428365000*u^2*v^6*w^2 -20014323073353000*u^2*v^5*w+90657438200000000*v^10*w^2-10573942996796 67280*w^4*v^6+1929911806894284*v^4*w-7650762409557986*v^5*w+1113382296 39393060*w^5*v^4+824648739488000000*v^10*w^4-546847451920000000*v^10*w ^3-12295910189744452*w^4*v^2-61065008802154818*w^5*v^2-356189502310992 0000*v^8*w^3-1481471131658983920*w^5*v^5+10947358802849214*w^5*v+48703 67830418008*w^3*v^2+132934982350624674*w^6*v^2-17440905032438454*w^3*v ^4-20731055486960000*w*v^7+3642477175663011000*w^5*v^8+410902976540385 000*w^6*v^8-478124960560218*w^3*v-17120216381454464*w^3*v^3+3975229269 2385000*v^6*w^8+7679667514050000*u^2*v^6*w-538533677281716*w^4*v+32571 54435904454900*w^4*v^8+1160412694810121060*w^4*v^5+333439954298303074* v^6*w^2+40921476513750*u^2*v^2-191859284988000*u^2*v^3+224881850992050 *u^2*v^4-255988917135000*u*v^5+218324010430800*v^4*u-46550231469000*u* v^3)/(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] = 1/11;" "6#/&%\"cG6 #\"\"#*&\"\"\"F)\"#6!\"\"" }{TEXT -1 3 ", " }{XPPEDIT 18 0 "c[4] = 20 /139;" "6#/&%\"cG6#\"\"%*&\"#?\"\"\"\"$R\"!\"\"" }{TEXT -1 7 " and \+ " }{XPPEDIT 18 0 "c[5] = 88/177;" "6#/&%\"cG6#\"\"&*&\"#))\"\"\"\"$x\" !\"\"" }{TEXT -1 112 " used in Tsitouras' scheme can be used as start ing values to minimize the (square of the) principal error norm." }} {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 119 "Using a \+ one dimensional minimization procedure and cycling around the nodes gi ves slow convergence towards the minimum." }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 489 "Digits := 35:\nc_2 := 1/11: c_4 := 20/139: c_5 := 88 /177:\nfor ct to 1000 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=locat ion)[1];\n mn := findmin(prin_err_norm_sqrd(c_2,c_4,c5),c5=\{0.5,c_5 ,0.63\},convergence=location); \n c_5 := mn[1]:\n if `mod`(ct,50)= 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#\"\"#$\"D(**4Jmz$pd&4.Q[e9a$3&!#O/&F%6#\"\"%$\"DtTsJ9 G&3fXR9!#N/&F%6#\"\"&$\"D1BRnDEO!\\^Vx!eX^<(\\F1" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#$\"DG&3fXR9!#N/&F%6#\"\"&$\"DcBRnDEO!\\^Vx!eX^<(\\F1" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#$\"Di&*)))e\"y?0(Rh+Qu?+-r!#Y" }} {PARA 11 "" 1 "" {XPPMATH 20 "6%/&%\"cG6#\"\"#$\"DTumA\\Ppd&4.Q[e9a$3& !#O/&F%6#\"\"%$\"DS@vJ9G&3fXR9!#N/&F%6#\"\"&$\"D1CRnDEO!\\^Vx!eX ^<(\\F1" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#$\"D37`jwy?0(Rh+Qu?+-r!#Y" }}{PARA 260 "" 0 "" {TEXT -1 1 ":" }}{PARA 260 "" 0 "" {TEXT -1 1 ":" }}{PARA 11 "" 1 "" {XPPMATH 20 "6%/&%\"cG6#\"\"#$\"DC^A.LPpd&4.Q[e9a$3 &!#O/&F%6#\"\"%$\"DYP$=Vr@i#GA>G&3fXR9!#N/&F%6#\"\"&$\"D1JRnDEO!\\^Vx! eX^<(\\F1" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#$\"DU`'pu*y?0(Rh+Qu?+-r!# Y" }}{PARA 11 "" 1 "" {XPPMATH 20 "6%/&%\"cG6#\"\"#$\"D1&RbLv$pd&4.Q[e 9a$3&!#O/&F%6#\"\"%$\"DYP$=Vr@i#GA>G&3fXR9!#N/&F%6#\"\"&$\"DcJRnDEO!\\ ^Vx!eX^<(\\F1" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#$\"Dtlo,>z?0(Rh+Qu?+- r!#Y" }}{PARA 11 "" 1 "" {XPPMATH 20 "6%/&%\"cG6#\"\"#$\"Dj2*RSx$pd&4. Q[e9a$3&!#O/&F%6#\"\"%$\"DYP$=Vr@i#GA>G&3fXR9!#N/&F%6#\"\"&$\"D1KRnDEO !\\^Vx!eX^<(\\F1" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#$\"DB<%*fCz?0(Rh+Q u?+-r!#Y" }}{PARA 11 "" 1 "" {XPPMATH 20 "6%/&%\"cG6#\"\"#$\"D(RYQiu$p d&4.Q[e9a$3&!#O/&F%6#\"\"%$\"DYP$=Vr@i#GA>G&3fXR9!#N/&F%6#\"\"&$\"DcKR nDEO!\\^Vx!eX^<(\\F1" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#$\"D'*)GaZo2_q Rh+Qu?+-r!#Y" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 79 "The following graphs give a visual check that we have fou nd a (local) minimum. " }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 390 "c _2 := .50835414585e-1: pp := .71020020744e-11:\np1 := evalf[30](plot(p rin_err_norm_sqrd(c[2],.14394559085,.49717514558),c[2]=0.045..0.0566, \n color=COLOR(RGB,.5,0,.9))):\np2 := plot([[[c_2,pp]]$4],st yle=point,symbol=[circle$2,diamond,cross],symbolsize=[12,10$3],\n \+ color=[black,red$3]):\nplots[display]([p1,p2],font=[HELVETICA,9] ,view=[0.045..0.0566,7.1006e-12..7.117e-12]);" }}{PARA 13 "" 1 "" {GLPLOT2D 367 347 347 {PLOTDATA 2 "6*-%'CURVESG6$7S7$$\"#X!\"$$\"?mG%H [K%e?;qNJ*p6(!#T7$$\"?LLLLLLLLL$))p%GDX!#J$\"?df>$**>0;\\YM;Ad6(F-7$$ \"?nmmmmmmm;4!y%GZXF1$\"?uwZ$=epT[AS4iY6(F-7$$\"?LLLLLLLLLQsh-sXF1$\"? _sAr=15[A`Y3_8rF-7$$\"?LLLLLLLLLom:$pf%F1$\"?$*[)[^*p=Rf^okU7rF-7$$\"? nmmmmmmm;***e=J![5rF-7$$\"?++++++++]UiX\\oYF1$\"?%\\7OdxV?g`@\"ee4rF-7$ $\"?LLLLLLLL$e@i.Jp%F1$\"?!3\"\\r+0e+HzfJr3rF-7$$\"?++++++++]KgPj>Ez0rF-7$$\"?+++++++++]&fK_\"[F1$\"?Tb&>1;P_mMHMr^5(F -7$$\"?+++++++++IUEWR[F1$\"?RVxKM(G'z!zw?DY5(F-7$$\"?LLLLLLLL$eToF9'[F 1$\"?*4%*yg@Mk1,`(RC$pqAL.rF-7$$\"?++++++++]-rqZN\\F 1$\"?vDuF#zy`*=Q9s'H5(F-7$$\"?nmmmmmmmmYo*z#e\\F1$\"?j`a9xsKZquHGp-rF- 7$$\"?++++++++]-h$)H$)\\F1$\"?(Ge$*)ph#[^vqXWC5(F-7$$\"?++++++++]2(y@r +&F1$\"?\")R^C_%*)eC3H.&F1$\"?A2c/CrtE@D? \">@5(F-7$$\"?nmmmmmmm;>Wd![0&F1$\"?tG+-rF-7$$\"?LLLLLLLL$eTX-]5&F1$\"?qwKOFCY_ &31NA?5(F-7$$\"?++++++++]d#els7&F1$\"?@bn&fi%*RCs\"Ri3-rF-7$$\"?LLLLLL LLL=r.J^^F1$\"?f!QP!*>qY,Y'fU?-rF-7$$\"?+++++++++g!*4:w^F1$\"?Cb;.'Gd` '>aY'zB5(F-7$$\"?+++++++++X!p_/?&F1$\"?bVRCugsm^j.Qg-rF-7$$\"?++++++++ ]iMe'RA&F1$\"?HvrlE*)[V^E#=qG5(F-7$$\"?+++++++++NsJ2]_F1$\"?MU3V@*\\mX 6:)HA.rF-7$$\"?nmmmmmmmmYw<`t_F1$\"?2LCddy*zMkg<\"f.rF-7$$\"?+++++++++ ]V)y&)H&F1$\"?D3%[QKC*>W6)yPS5(F-7$$\"?LLLLLLLL$e*3_F@`F1$\"?c_yUZ9dTr X*=!\\/rF-7$$\"?+++++++++IJ$)3Y`F1$\"?SD\"**e)H-2Zi\"oO]5(F-7$$\"?nmmm mmmm;pXbVp`F1$\"?$f]$zt&pOv4\"*Q+c5(F-7$$\"?++++++++]#>6SQR&F1$\"?/JO \\:VZ%**f'H4C1rF-7$$\"?nmmmmmmmm;O+qVGE'H!y_sy\"p5(F-7$$\"? ++++++++]il\"zEW&F1$\"?g!>#)44zRH*Gw-o2rF-7$$\"?MLLLLLLLL8OqtmaF1$\"?B VR!*48Ta?!z\\m%3rF-7$$\"?MLLLLLLLL[$oR8\\&F1$\"?w`nRWpWs'HvAB$4rF-7$$ \"?nmmmmmmm;\\3'Qd^&F1$\"?ydXS+3CM]R(\\D-6(F-7$$\"?+++++++++g'oe\"QbF1 $\"?A&zbpT`h%[*4!356rF-7$$\"?nmmmmmmmm@EY&Qc&F1$\"?&31P(H*)*\\z#HG%e@6 (F-7$$\"?LLLLLLLLL$4QPoe&F1$\"?\"*GWmB[%G=Saj`J6(F-7$$\"?++++++++]-\"z T8h&F1$\"?0jci]'p#fn1ifE9rF-7$$\"?++++++++]$\"3$)****Ru?+-r!#H-%'COLOU RG6&F][lFb[lFb[lFb[l-%'SYMBOLG6$%'CIRCLEG\"#7-%&STYLEG6#%&POINTG-F$6&F g[l-F`\\l6&F][l$\"*++++\"!\")Fa[lFa[l-Fc\\l6$Fe\\l\"#5Fg\\l-F$6&Fg[lF] ]l-Fc\\l6$%(DIAMONDGFd]lFg\\l-F$6&Fg[lF]]l-Fc\\l6$%&CROSSGFd]lFg\\l-%% FONTG6$%*HELVETICAGFd[l-%+AXESLABELSG6%Q%c[2]6\"Q!Fg^l-F`^l6#%(DEFAULT G-%%VIEWG6$;F(Fez;$\"&15(!#;$\"% " 0 "" {MPLTEXT 1 0 454 "c_4 := .14394559085: pp := .71020020744e-8:\np1 := evalf[30](plot(10^3*prin_err_norm_sqrd(.50835414585e-1,c[4],.49717 514558),c[4]=0.143938..0.143953,\n color=COLOR(RGB,0,.7,.2))):\np2 \+ := plot([[[c_4,pp]]$4],style=point,symbol=[circle$2,diamond,cross],sym bolsize=[12,10$3],\n color=[black,cyan$3]):\nplots[display]( [p1,p2],font=[HELVETICA,9],labels=[`c[4]`,`principal error x 1000`],\n view=[0.143938..0.143953,7.1006e-9..7.1165e-9]);" }} {PARA 13 "" 1 "" {GLPLOT2D 473 386 386 {PLOTDATA 2 "6*-%'CURVESG6$7S7$ $\"'QR9!\"'$\"?;6'y\">2t,t%zt&Q;r!#Q7$$\"?+++++++DJdpKQR9!#I$\"?$>%)Gb %G'[Y,;9rF-7$$\"?+++++ +](=HPJ*QR9F1$\"?yf8Tz4xB[#RwsI6(F-7$$\"?++++++]7VDMDRR9F1$\"?4(oB(3NU @AzUz-7rF-7$$\"?++++++vVGZRdRR9F1$\"?A%*RA$)G(Rj\\=\\R56(F-7$$\"?+++++ +v=276()RR9F1$\"?'[$*o3a6C.JW))o,6(F-7$$\"?++++++vo**3)y,%R9F1$\"?5h-> z;&*\\dStQJ4rF-7$$\"?++++++vofHq\\SR9F1$\"?b\"4r:F-!3+1M$z%3rF-7$$\"?+ +++++v$f'HU\"3%R9F1$\"?07*ztSW&zG)*yxp2rF-7$$\"?+++++++D\"*309TR9F1$\" ?K@15$\\!4(ek$ei%p5(F-7$$\"?++++++]i&e*yUTR9F1$\"?e1bH%GeG0)HVR9F1$\"?Ost*RZ(R^fy87X-rF-7$$ \"?++++++DcJ4wbWR9F1$\"??s#*GsyjowS'foA5(F-7$$\"?++++++](=#R!z[%R9F1$ \"?PD%*[(>MLSWgcG@5(F-7$$\"?++++++v$4A@u^%R9F1$\"?!)3UTv.%p9Z3TX?5(F-7 $$\"?++++++]i:'f#\\XR9F1$\"?HQZW:;r%4dX[/?5(F-7$$\"?++++++vof2L#e%R9F1 $\"?p.eV0f9-(>Xc:?5(F-7$$\"?++++++D\"yG>6h%R9F1$\"?W)H\\/.]9e'Hp'p?5(F -7$$\"?++++++](oo6Ak%R9F1$\"??_$Goirxu+ieu@5(F-7$$\"?+++++++]xJLuYR9F1 $\"?NQq*4\"3Y]j$)pOL-rF-7$$\"?++++++]P*yddq%R9F1$\"?2&*[xPV)=GIy:RD5(F -7$$\"?++++++v=aM5(F- 7$$\"?+++++++Dc]kK[R9F1$\"?@T!4w\"yeJ65;/(Q5(F-7$$\"?++++++vo/Q*>'[R9F 1$\"?bxi-JKn%QX&=GH/rF-7$$\"?+++++++vQ(zS*[R9F1$\"?_sI%ee-0<0M$Q![5(F- 7$$\"?++++++v=-,FC\\R9F1$\"?%>MCu<%H8iL.;L0rF-7$$\"?++++++v$4tFe&\\R9F 1$\"?&oSVJWp2.g,$>$f5(F-7$$\"?+++++++D\"3\"o')\\R9F1$\"?;tsR'GF],l+%pc 1rF-7$$\"?++++++voz;)*=]R9F1$\"?!)[JVgwpe1`sEG2rF-7$$\"?++++++++&*44]] R9F1$\"?3=jDT]or[\"[E@!3rF-7$$\"?++++++]7jZ!>3&R9F1$\"?#ew\"fFibZ`Gol# )3rF-7$$\"?++++++v=(4bM6&R9F1$\"?(>Nr@Z?&3m\"Q7v'4rF-7$$\"?+++++++]xlW U^R9F1$\"?N:f8eS/fnF(o)\\5rF-7$$\"?++++++]i&3uc<&R9F1$\"?GOo70L?@wD4U \\6rF-7$$\"?++++++++lJR0_R9F1$\"?&*)z+lM&4Bv&3LJC6(F-7$$\"?++++++v=-*z qB&R9F1$\"?$)3)p(3#)[**y@(3zM6(F-7$$\"?++++++D\"G:3uE&R9F1$\"?RSOBB@p* >]=*)GX6(F-7$$\"'`R9F*$\"?@l'fjR4C\\'Gh#3d6(F--%&COLORG6&%$RGBG$\"\"!F ^[l$\"\"(!\"\"$\"\"#Fa[l-F$6&7#7$$\"3******\\3fXR9!#=$\"3K++Su?+-r!#E- %'COLOURG6&F\\[lF^[lF^[lF^[l-%'SYMBOLG6$%'CIRCLEG\"#7-%&STYLEG6#%&POIN TG-F$6&Ff[l-F_\\l6&F\\[lF][l$\"*++++\"!\")F^]l-Fb\\l6$Fd\\l\"#5Ff\\l-F $6&Ff[lF\\]l-Fb\\l6$%(DIAMONDGFc]lFf\\l-F$6&Ff[lF\\]l-Fb\\l6$%&CROSSGF c]lFf\\l-%%FONTG6$%*HELVETICAG\"\"*-%+AXESLABELSG6%%%c[4]G%7principal~ error~x~1000G-F_^l6#%(DEFAULTG-%%VIEWG6$;F(Fez;$\"&15(!#8$\"&l6(Fb_l" 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" }}}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 445 "c_5 := .49717514558: pp : = .71020020744e-8:\np1 := evalf[30](plot(10^3*prin_err_norm_sqrd(.5083 5414585e-1,.14394559085,c[5]),c[5]=0.4971685..0.4971818,\n color=C OLOR(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]):\n plots[display]([p1,p2],font=[HELVETICA,9],labels=[`c[5]`,`principal er ror x 1000`],\n view=[0.4971685..0.4971818,7.1006e-9..7.1175e -9]);" }}{PARA 13 "" 1 "" {GLPLOT2D 527 487 487 {PLOTDATA 2 "6*-%'CURV ESG6$7S7$$\"(&or\\!\"($\"?8aoc\"o$zy7H!>gv6(!#Q7$$\"?nmmmmm;/:-**yor\\ !#I$\"?\\LUk037?eHm+B;rF-7$$\"?LLLLL$eRhW9U!pr\\F1$\"?tA(4-&=![K.C3@^6 (F-7$$\"?nmmmmm\"H@t\"eKpr\\F1$\"?*ew(fHMtGGgjw#R6(F-7$$\"?nmmmmmT!\\0 P6'pr\\F1$\"?[VR*ysC!4lz-Qy7rF-7$$\"?LLLLL$e9#fmb*)pr\\F1$\"?H>@W+c$4E ili-<6(F-7$$\"?nmmmm;HPq_!f,<(\\F1$\"?7&)=DfWj1R:e7v5rF-7$$\"?+++++]i0 Jx=Vqr\\F1$\"?Pu&zSxp4UuG!z\")4rF-7$$\"?nmmmm;H#4H.92<(\\F1$\"?E['Qjt> $z$e'))z!*3rF-7$$\"?+++++]78j$G&*4<(\\F1$\"?y;%HVN'[,a%e,d!3rF-7$$\"?L LLLLL$3pXe%Grr\\F1$\"?TX^gvz#GQ@K/Ss5(F-7$$\"?nmmmmmTD*4SR:<(\\F1$\"?Z DEr,!=_X*)HZpl5(F-7$$\"?++++++]A$REE=<(\\F1$\"?up;HO_T[d(o]pe5(F-7$$\" ?++++++](eZI9@<(\\F1$\"?&H%4*4DqO7ME@D_5(F-7$$\"?++++++]xY&)=Rsr\\F1$ \"?P&z[\\$)e,/rF -7$$\"?LLLLLL$333pVH<(\\F1$\"?y@#os8=\\0gvo(p.rF-7$$\"?LLLLLLLB11w>tr \\F1$\"?sWa#y\\)H&GajGFL5(F-7$$\"?+++++]ig')pH\\tr\\F1$\"?'HG\"\\U!)o. RSzP&H5(F-7$$\"?LLLLLLL)zoTaP<(\\F1$\"?G0`>oBjADyHYn-rF-7$$\"?+++++]i5 pl7/ur\\F1$\"?UO6y#RqnJ4D'RU-rF-7$$\"?+++++](=$f8WJur\\F1$\"?i=2C2q&y. BgMRA5(F-7$$\"?LLLLLLeRx9%*fur\\F1$\"?ERvIs^1hc]cH5-rF-7$$\"?LLLLL$ekD [8h[<(\\F1$\"?>XQs;]E\"RH_HG?5(F-7$$\"?nmmmmmT&e_VV^<(\\F1$\"?vjqJGx#y 8H!e?+-rF-7$$\"?nmmmm;H#pgmOa<(\\F1$\"?*[AD!G+HeutoW.-rF-7$$\"?+++++]P >NC>pvr\\F1$\"?i()eR&RL'[&[$z@6-rF-7$$\"?nmmmmm\"H!H5w'f<(\\F1$\"?&p&Q ,(>MsN$*o&yC-rF-7$$\"?+++++++0%3U_i<(\\F1$\"?I.B)y%*zfh@7YWC5(F-7$$\"? ++++++D\"*R_5`wr\\F1$\"?R^>4BE!zr]>I#p-rF-7$$\"?+++++]i!fFk+o<(\\F1$\" ?NH-=(R5v#fi]V)H5(F-7$$\"?++++++v)*)o(**4xr\\F1$\"?Sz$pa$pz$)[EE)oL5(F -7$$\"?LLLLLLL)>>%*ot<(\\F1$\"?'*Hhl xr\\F1$\"?0?)e(Q#H&GMd_8D/rF-7$$\"?nmmmm;H#o]M;z<(\\F1$\"?MD])H-luP$z* RRZ5(F-7$$\"?++++++]-NS3?yr\\F1$\"?.y!4#\\/jH29]xK0rF-7$$\"?LLLLL$eRR# G&o%yr\\F1$\"?y'))\\,SF*)y%>(fLf5(F-7$$\"?+++++]7V@R$[(yr\\F1$\"?R,a\" *Q'G2mp,/@m5(F-7$$\"?LLLLLL$3(e0>-zr\\F1$\"?R7ek\"HMn$[EfmM2rF-7$$\"?+ ++++]ilA/$3$zr\\F1$\"?t&yJD*>S*4rF-7$$ \"?LLLLL$eRG&of9!=(\\F1$\"?tmfb@&3.#>1eN)36(F-7$$\"?+++++++0KEIS!=(\\F 1$\"?n[(*)\\[fP\"3k1z\"=6(F-7$$\"?LLLLLL3#fNk(p!=(\\F1$\"?5&4/'3RH49LL i%H6(F-7$$\"?nmmmmmm'HF:h4=(\\F1$\"?f!G$)z\\\"[cd\"RT2S6(F-7$$\"?+++++ ]igY3@C\"=(\\F1$\"?n%48v3;syV2$H>:rF-7$$\"?+++++]P\\&*=5^\"=(\\F1$\"?C 8pHZQa-1^$*)zj6(F-7$$\"(==(\\F*$\"?OU0*)yZI)oq+W7x6(F--%&COLORG6&%$RGB G$\"\"'!\"\"$\"\"#F_[lF`[l-F$6&7#7$$\"3y****zb9vr\\!#=$\"3K++Su?+-r!#E -%'COLOURG6&F\\[l\"\"!F_\\lF_\\l-%'SYMBOLG6$%'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$%*HELVETICAG\"\"*-%+AXESLABELSG6%%%c[5]G%7p rincipal~error~x~1000G-F_^l6#%(DEFAULTG-%%VIEWG6$;F(Fez;$\"&15(!#8$\"& v6(Fb_l" 1 2 0 1 10 0 2 9 1 4 2 1.000000 45.000000 45.000000 0 0 "Curv e 1" "Curve 2" "Curve 3" "Curve 4" "Curve 5" }}}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 36 "#------------------------ -----------" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 155 "nds := [c[2] =.5083541458484e-1,c[4]=.143945590853,c[5]=.497175145581]:\nevalf[10]( %);\nfor dgt from 6 by -1 to 4 do\n map(convert,nds,rational,dgt);\n end do;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#7%/&%\"cG6#\"\"#$\"+e9a$3&! #6/&F&6#\"\"%$\"+4fXR9!#5/&F&6#\"\"&$\"+c9vr\\F2" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#7%/&%\"cG6#\"\"##\"#q\"%x8/&F&6#\"\"%#\"$q\"\"%\"=\"/&F &6#\"\"&#\"#))\"$x\"" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#7%/&%\"cG6#\" \"##\"#q\"%x8/&F&6#\"\"%#\"#>\"$K\"/&F&6#\"\"&#\"#))\"$x\"" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#7%/&%\"cG6#\"\"##\"\"$\"#f/&F&6#\"\"%#\"#=\" $D\"/&F&6#\"\"&#\"#))\"$x\"" }}}{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 93 "evalf[25](prin_ err_norm_sqrd(.5083541458484e-1,.143945590853,.497175145581)):\nevalf( sqrt(%));" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#$\"+v\"e\\m#!#:" }}} {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 6 "When " } {XPPEDIT 18 0 "c[2] = 3/59;" "6#/&%\"cG6#\"\"#*&\"\"$\"\"\"\"#f!\"\"" }{TEXT -1 3 ", " }{XPPEDIT 18 0 "c[4] = 19/132;" "6#/&%\"cG6#\"\"%*& \"#>\"\"\"\"$K\"!\"\"" }{TEXT -1 7 " and " }{XPPEDIT 18 0 "c[5] = 88 /177;" "6#/&%\"cG6#\"\"&*&\"#))\"\"\"\"$x\"!\"\"" }{TEXT -1 67 ", the principal error norm is given (approximately) as follows. " }} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 66 "evalf[15](prin_err_norm_sqrd (3/59,19/132,88/177)):\nevalf(sqrt(%));" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#$\"+h5vmE!#:" }}}{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 embedd ed scheme" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 35 "coefficient s of the combined scheme" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1968 "ee := \{c[2]=3/59,\nc[3]=19/198, \nc[4]=19/132,\nc[5]=88/177,\nc[6]=35/36,\nc[7]=544/545,\nc[8]=1,\nc[9 ]=1,\n\na[2,1]=3/59,\na[3,1]=1273/235224,\na[3,2]=21299/235224,\na[4,1 ]=19/528,\na[4,2]=0,\na[4,3]=19/176,\na[5,1]=292792984/222425457,\na[5 ,2]=0,\na[5,3]=-373191104/74141819,\na[5,4]=312454912/74141819,\na[6,1 ]=-93282246804140065/2230303430271708,\na[6,2]=0,\na[6,3]=143524971301 36165/90113269909968,\na[6,4]=-2430460528059531110/19920664729472301, \na[6,5]=53730709552104745/9712013829327216,\na[7,1]=-2815838771905694 350037121443888/52287718910334988576674096875,\na[7,2]=0, a[7,3]=38959 719796026045728912040192/190137159673945413006087625,\na[7,4]=-8014961 82066760414629068567466496/5106133423043804066278483169375,\na[7,5]=26 545926738560289939469575124336/3828996342298708285360778643125,\na[7,6 ]=-16804913133796526592/942707650019020409375,\na[8,1]=-76006376179258 679683396673/1389345361299174809573120,\na[8,2]=0,\na[8,3]=10983047702 5712790000/527674313813796947,\na[8,4]=-194391745260058069863027995198 /1219164295706831920486709627,\na[8,5]=444873349778574018937254390317/ 63269358937392669883372236224,\na[8,6]=-49046624888634/292965477272121 5,\na[8,7]=-45835339680636651875/22882961544684083693312,\na[9,1]=6576 0917/1910092800,\na[9,2]=0,\na[9,3]=0,\na[9,4]=6404556346056/248026640 14135,\na[9,5]=2946448255213557/7004255810773760,\na[9,6]=16630875288/ 3684943675,\na[9,7]=-496773593038619375/9868243105287168,\na[9,8]=2782 6183/603420,\n\nb[1]=65760917/1910092800,\nb[2]=0,\nb[3]=0,\nb[4]=6404 556346056/24802664014135,\nb[5]=2946448255213557/7004255810773760,\nb[ 6]=16630875288/3684943675,\nb[7]=-496773593038619375/9868243105287168, \nb[8]=27826183/603420,\n\n`b*`[1]=46875680038130591/12454608326426112 00,\n`b*`[2]=0,\n`b*`[3]=0,\n`b*`[4]=2029502937748023643719/8086189994 224235216645,\n`b*`[5]=1969638800119487132740431/456706934559835779970 3040,\n`b*`[6]=4450793456682602037/1201369121360654225,\n`b*`[7]=-2533 94715948809936175102125/6434509503742783282563072,\n`b*`[8]=2828425336 7140001/786910432449360,\n`b*`[9]=1/80\}:" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 5 "Let " }{XPPEDIT 18 0 "T[6, 8];" "6#&%\"TG6$\"\"'\"\")" }{TEXT -1 127 " denote the vector whose components are the principal error terms of the 8 stage, order \+ 6 scheme (the error terms of order 7)." }}{PARA 0 "" 0 "" {TEXT -1 5 " Let " }{XPPEDIT 18 0 "`T*`[5,9];" "6#&%#T*G6$\"\"&\"\"*" }{TEXT -1 145 " denote the vector whose components are the principal error term s of the embedded 9 stage, order 5 scheme (the error terms of order 6) and let " }{XPPEDIT 18 0 "`T*`[6,9];" "6#&%#T*G6$\"\"'\"\"*" }{TEXT -1 99 " denote the vector whose components are the error terms of ord er 7 of the embedded order 5 scheme." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 39 "Denote the 2-norm of these vectors by \+ " }{XPPEDIT 18 0 "abs(abs(T[6,8]));" "6#-%$absG6#-F$6#&%\"TG6$\"\"'\" \")" }{TEXT -1 3 ", " }{XPPEDIT 18 0 "abs(abs(`T*`[5,9]));" "6#-%$abs G6#-F$6#&%#T*G6$\"\"&\"\"*" }{TEXT -1 7 " and " }{XPPEDIT 18 0 "abs( abs(`T*`[6,9]));" "6#-%$absG6#-F$6#&%#T*G6$\"\"'\"\"*" }{TEXT -1 15 " \+ respectively." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 8 "Define: " }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "A[7] = abs(abs(T[6, 8]));" "6#/&%\"AG6#\"\"(-%$absG6#-F)6#&%\"TG6$ \"\"'\"\")" }{TEXT -1 6 ", " }{XPPEDIT 18 0 "B[7] = abs(abs(`T*`[6 ,9]))/abs(abs(`T*`[5,9]));" "6#/&%\"BG6#\"\"(*&-%$absG6#-F*6#&%#T*G6$ \"\"'\"\"*\"\"\"-F*6#-F*6#&F/6$\"\"&F2!\"\"" }{TEXT -1 6 ", " } {XPPEDIT 18 0 "C[7] = abs(abs(`T*`[6,9]-T[6, 8]))/abs(abs(`T*`[5, 9])) ;" "6#/&%\"CG6#\"\"(*&-%$absG6#-F*6#,&&%#T*G6$\"\"'\"\"*\"\"\"&%\"TG6$ F2\"\")!\"\"F4-F*6#-F*6#&F06$\"\"&F3F9" }{TEXT -1 1 "." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 77 "Dormand and Prince h ave suggested that as well as attempting to ensure that " }{XPPEDIT 18 0 "A[7];" "6#&%\"AG6#\"\"(" }{TEXT -1 73 " is a minimum, if the em bedded scheme is to be used for error control, " }{XPPEDIT 18 0 "B[7] ;" "6#&%\"BG6#\"\"(" }{TEXT -1 7 " and " }{XPPEDIT 18 0 "C[7];" "6#& %\"CG6#\"\"(" }{TEXT -1 27 " should be chosen so that " }}{PARA 0 "" 0 "" {TEXT -1 68 "they are \"similar in magnitude\" and also not diffe r too much from 1." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 " > " 0 "" {MPLTEXT 1 0 186 "errterms6_8 := PrincipalErrorTerms(6,8,'exp anded'):\n`errterms6_9*` :=subs(b=`b*`,PrincipalErrorTerms(6,9,'expand ed')):\n`errterms5_9*` := subs(b=`b*`,PrincipalErrorTerms(5,9,'expande d')):" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 326 "snmB := sqrt(add(evalf(subs(ee,`errterms6_9*`[i]))^2 ,i=1..nops(`errterms6_9*`))):\nsdnB := sqrt(add(evalf(subs(ee,`errterm s5_9*`[i]))^2,i=1..nops(`errterms5_9*`))):\nsnmC := sqrt(add((evalf(su bs(ee,`errterms6_9*`[i])-subs(ee,errterms6_8[i])))^2,i=1..nops(errterm s6_8))):\n'B[7]'= evalf[8](snmB/sdnB);\n'C[7]'= evalf[8](snmC/sdnB);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/&%\"BG6#\"\"($\")\\;B " 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 "co efficients of the combined scheme" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1968 "ee := \{c[2]=3/59,\nc[3]=1 9/198,\nc[4]=19/132,\nc[5]=88/177,\nc[6]=35/36,\nc[7]=544/545,\nc[8]=1 ,\nc[9]=1,\n\na[2,1]=3/59,\na[3,1]=1273/235224,\na[3,2]=21299/235224, \na[4,1]=19/528,\na[4,2]=0,\na[4,3]=19/176,\na[5,1]=292792984/22242545 7,\na[5,2]=0,\na[5,3]=-373191104/74141819,\na[5,4]=312454912/74141819, \na[6,1]=-93282246804140065/2230303430271708,\na[6,2]=0,\na[6,3]=14352 497130136165/90113269909968,\na[6,4]=-2430460528059531110/199206647294 72301,\na[6,5]=53730709552104745/9712013829327216,\na[7,1]=-2815838771 905694350037121443888/52287718910334988576674096875,\na[7,2]=0, a[7,3] =38959719796026045728912040192/190137159673945413006087625,\na[7,4]=-8 01496182066760414629068567466496/5106133423043804066278483169375,\na[7 ,5]=26545926738560289939469575124336/3828996342298708285360778643125, \na[7,6]=-16804913133796526592/942707650019020409375,\na[8,1]=-7600637 6179258679683396673/1389345361299174809573120,\na[8,2]=0,\na[8,3]=1098 30477025712790000/527674313813796947,\na[8,4]=-19439174526005806986302 7995198/1219164295706831920486709627,\na[8,5]=444873349778574018937254 390317/63269358937392669883372236224,\na[8,6]=-49046624888634/29296547 72721215,\na[8,7]=-45835339680636651875/22882961544684083693312,\na[9, 1]=65760917/1910092800,\na[9,2]=0,\na[9,3]=0,\na[9,4]=6404556346056/24 802664014135,\na[9,5]=2946448255213557/7004255810773760,\na[9,6]=16630 875288/3684943675,\na[9,7]=-496773593038619375/9868243105287168,\na[9, 8]=27826183/603420,\n\nb[1]=65760917/1910092800,\nb[2]=0,\nb[3]=0,\nb[ 4]=6404556346056/24802664014135,\nb[5]=2946448255213557/70042558107737 60,\nb[6]=16630875288/3684943675,\nb[7]=-496773593038619375/9868243105 287168,\nb[8]=27826183/603420,\n\n`b*`[1]=46875680038130591/1245460832 642611200,\n`b*`[2]=0,\n`b*`[3]=0,\n`b*`[4]=2029502937748023643719/808 6189994224235216645,\n`b*`[5]=1969638800119487132740431/45670693455983 57799703040,\n`b*`[6]=4450793456682602037/1201369121360654225,\n`b*`[7 ]=-253394715948809936175102125/6434509503742783282563072,\n`b*`[8]=282 84253367140001/786910432449360,\n`b*`[9]=1/80\}:" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 77 "The stability f unction R for the 8 stage, order 6 scheme is given as follows." }} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 75 "subs(ee,StabilityFunction(6, 8,'expanded')):\nR := unapply(%,z):\n'R(z)'=R(z);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/-%\"RG6#%\"zG,4\"\"\"F)F'F)*&#F)\"\"#F)*$)F'F,F)F)F)*& #F)\"\"'F)*$)F'\"\"$F)F)F)*&#F)\"#CF)*$)F'\"\"%F)F)F)*&#F)\"$?\"F)*$)F '\"\"&F)F)F)*&#F)\"$?(F)*$)F'F1F)F)F)*&#\".$>AySkk\"2Se\"\\Y/0iKF)*$)F '\"\"(F)F)F)*&#\",BG`>7#\"1SI@_E176F)*$)F'\"\")F)F)F)" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 124 "We can find the poi nt where the boundary of the stability region intersects the negative \+ real axis by solving the equation: " }}{PARA 256 "" 0 "" {TEXT -1 1 " \+ " }{XPPEDIT 18 0 "R(z) = 1;" "6#/-%\"RG6#%\"zG\"\"\"" }{TEXT -1 2 ". \+ " }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 29 "z0 := newton(R(z)=1,z=-4 .85);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%#z0G$!++\")*4&[!\"*" }}} {PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 304 "z0 := newton(R(z)=1,z=-4.85):\np1 := plot([R(z),1],z=-5.49..0.49, color=[red,blue]):\np2 := plot([[[z0,1]]$3],style=point,symbol=[circle ,cross,diamond],color=black):\np3 := plot([[z0,0],[z0,1]],linestyle=3, color=COLOR(RGB,0,.5,0)):\nplots[display]([p1,p2,p3],view=[-5.49..0.49 ,-.07..1.47],font=[HELVETICA,9]);" }}{PARA 13 "" 1 "" {GLPLOT2D 385 263 263 {PLOTDATA 2 "6+-%'CURVESG6$7Z7$$!3A++++++!\\&!#<$\"3!z)3/EshyK F*7$$!3eL3_aKTdaF*$\"3wP\"4'\\vG)4$F*7$$!3(pmT!4l#[U&F*$\"3#35K\"H=rEH F*7$$!3M+Dcj(RAR&F*$\"3!Q;bToBNw#F*7$$!3sLL3=Ilf`F*$\"39QwJx'p$3EF*7$$ !3x]737f%HI&F*$\"3]V\">-.@lN#F*7$$!3%p;zg!)QiC&F*$\"3(pk7Uzmi7#F*7$$!3 O](oXtlC=&F*$\"3U48Zs(H6*=F*7$$!3#QLeIm#p=^F*$\"3G?&eRQ_\"z;F*7$$!3%RL LA'o\\a]F*$\"3y_dT`7;([\"F*7$$!3=L$391,.*\\F*$\"3P2Rkl%oZJ\"F*7$$!3l]( =?P5k#\\F*$\"3[18wR\"R4;\"F*7$$!3An\"HEo>D'[F*$\"3k'G9A'=BB5F*7$$!3%R$ eM()*\\Su%F*$\"3W63O?oxc!)!#=7$$!3P+D\"*z9Q@YF*$\"3;M\\S*RbzC'Fho7$$!3 ULe/Mv^%\\%F*$\"3=!)zMLJfpZFho7$$!3C+DYU/1oVF*$\"3Dg!Gy9A0i$Fho7$$!3Am m@!y$)zB%F*$\"3J?M8Br-7FFho7$$!3?L$3`=7M7%F*$\"3k)pk$)e+w4#Fho7$$!3G++ l0;#\\'QF*$\"3#p,J8F0L#*!#>7$$!3$Q$e/tzxEOF*$\"3G0I&[CL#RvFar7$$!3'p m;op4?\\$F*$\"3(>)o9HI<;iFar7$$!3)pmm#*GVyP$F*$\"3q&[^z'o&zb&Far7$$!35 +Dh;1/XKF*$\"35A#pg&o!z?&Far7$$!3!omm([x[FJF*$\"3#)eV$zo^Q>&Far7$$!37+ Dh@O^)*HF*$\"3Ks:Xg[\\LaFar7$$!3!**\\PGv*pvGF*$\"3+kQBb!Hj'eFar7$$!3am ;zkjbZFF*$\"3Uvqo'[='*\\'Far7$$!3mm\"HDt!))HEF*$\"3(f-[)pWQGsFar7$$!3s L$3dm^H]#F*$\"3?rV%4:vX;)Far7$$!3eLe/Q!36P#F*$\"3zvar9tD-$*Far7$$!3-+v esxLcAF*$\"3%zNW=5(fV5Fho7$$!3>L$e;u#QK@F*$\"3i)>OV>@A=\"Fho7$$!3C++ql ]K/?F*$\"3xda9z.)[M\"Fho7$$!3;+]-8c/z=F*$\"3x5&zz'oUD:Fho7$$!3;+D\"3IJ yv\"F*$\"3eR>7VE%Gs\"Fho7$$!35+]dtMCB;F*$\"3izv))Q\\ur>Fho7$$!3PnmwC.J -:F*$\"3YI'y]cUcA#Fho7$$!3')***\\dP)=t8F*$\"3bVHB=)*pKDFho7$$!3#Q$ek)p %=c7F*$\"3EIm*G%4BZGFho7$$!3b***\\[xo#G6F*$\"3Zj$)Qp&eeB$Fho7$$!3an\"z #f)4z+\"F*$\"3)Qv\"fP*y(\\OFho7$$!3-,]iE6+@))Fho$\"3]A?%=d:\"RTFho7$$! 3)ymmT*3)4f(Fho$\"31hT+%H!)3o%Fho7$$!3S**\\7.PE.jFho$\"3]$G!p\\\"zTK&F ho7$$!3MJLL***QI1&Fho$\"3m[h_/;>FgFho7$$!3[PL35,t%z$Fho$\"3k&e?()*)=A% oFho7$$!3Ei;zeM#p`#Fho$\"375ipM[IfxFho7$$!3S'****pxH6Q\"Fho$\"3#R#yxDG +5()Fho7$$!3s:nmT'yfk&!#?$\"3^?#=%*H*pV**Fho7$$\"3tommW3MG6Fho$\"3d?)G S=Y%>6F*7$$\"3)3+vQV&e\"R#Fho$\"3]5d;**)z,F\"F*7$$\"3F**\\7#pr1g$Fho$ \"3;\"[/8pDMV\"F*7$$\"3!***************[Fho$\"3P*)=e%>;Bj\"F*-%'COLOUR G6&%$RGBG$\"*++++\"!\")$\"\"!Fe]lFd]l-F$6$7S7$F($\"\"\"Fe]l7$F=Fj]l7$F GFj]l7$FQFj]l7$FenFj]l7$F_oFj]l7$FdoFj]l7$FjoFj]l7$F_pFj]l7$FdpFj]l7$F ipFj]l7$F^qFj]l7$FcqFj]l7$FhqFj]l7$F]rFj]l7$FcrFj]l7$FhrFj]l7$F]sFj]l7 $FbsFj]l7$FgsFj]l7$F\\tFj]l7$FatFj]l7$FftFj]l7$F[uFj]l7$F`uFj]l7$FeuFj ]l7$FjuFj]l7$F_vFj]l7$FdvFj]l7$FivFj]l7$F^wFj]l7$FcwFj]l7$FhwFj]l7$F]x Fj]l7$FbxFj]l7$FgxFj]l7$F\\yFj]l7$FayFj]l7$FfyFj]l7$F[zFj]l7$F`zFj]l7$ FezFj]l7$FjzFj]l7$F_[lFj]l7$Fd[lFj]l7$Fj[lFj]l7$F_\\lFj]l7$Fd\\lFj]l7$ Fi\\lFj]l-F^]l6&F`]lFd]lFd]lFa]l-F$6&7#7$$!3)********4)*4&[F*Fj]l-%'SY MBOLG6#%'CIRCLEG-F^]l6&F`]lFe]lFe]lFe]l-%&STYLEG6#%&POINTG-F$6&F`al-Fe al6#%&CROSSGFhalFjal-F$6&F`al-Feal6#%(DIAMONDGFhalFjal-F$6%7$7$FbalFd] lFaal-%&COLORG6&F`]lFd]l$\"\"&!\"\"Fd]l-%*LINESTYLEG6#\"\"$-%%FONTG6$% *HELVETICAG\"\"*-%+AXESLABELSG6%Q\"z6\"Q!F_dl-Fgcl6#%(DEFAULTG-%%VIEWG 6$;$!$\\&!\"#$\"#\\Fjdl;$!\"(Fjdl$\"$Z\"Fjdl" 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 1358 "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 \+ 6464407822193/32620504464915840*z^7+21219532823/1112062652213040*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,.38,.15,.15)):\np2 := \+ plots[polygonplot]([seq([pts[i-1],pts[i],[-2.4,0]],i=2..nops(pts))],\n style=patchnogrid,color=COLOR(RGB,.75,.3,.3)):\npts := []: z 0 := 1.9+4.7*I:\nfor ct from 0 to 50 do\n zz := newton(R(z)=exp(ct*P i/25*I),z=z0):\n z0 := zz:\n pts := [op(pts),[Re(zz),Im(zz)]]:\nen d do:\np3 := plot(pts,color=COLOR(RGB,.38,.15,.15)):\np4 := plots[poly gonplot]([seq([pts[i-1],pts[i],[1.80,4.67]],i=2..nops(pts))],\n \+ style=patchnogrid,color=COLOR(RGB,.75,.3,.3)):\npts := []: z0 := 1. 9-4.7*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:\n p5 := plot(pts,color=COLOR(RGB,.38,.15,.15)):\np6 := plots[polygonplot ]([seq([pts[i-1],pts[i],[1.80,-4.67]],i=2..nops(pts))],\n sty le=patchnogrid,color=COLOR(RGB,.75,.3,.3)):\np7 := plot([[[-5.49,0],[2 .29,0]],[[0,-5.19],[0,5.19]]],color=black,linestyle=3):\nplots[display ]([p||(1..7)],view=[-5.49..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]z 7$$\"\"!F)F(7$F($\"3++++Fjzq:!#=7$$\"3[++++Ewk^!#F$\"3\")*****>l#fTJF- 7$$\"3.+++M.Y78!#D$\"3y*****\\$*)Q7ZF-7$$\"3-+++Tj,%H\"!#C$\"3y******o Z=$G'F-7$$\"3)******p(*H-d(F=$\"3G+++Qy(R&yF-7$$\"35+++de/sJ!#B$\"3&)* *****f)eZU*F-7$$\"3'******R?D80\"!#A$\"3)*******f)\\&*4\"!#<7$$\"3$)** ****oEN?HFN$\"3/+++i>hc7FQ7$$\"3i******o*H^/(FN$\"31+++AUk89FQ7$$\"3.+ ++\\*z(3:!#@$\"33+++7\"41d\"FQ7$$\"31+++Qro.HFin$\"3#******4*[VF#FQ7$$\"35+++6*eM<\"F^p$\"3))*****\\ 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G-%*AXESSTYLEG6#%$BOXG-%%VIEWG6$;$!$\\&Fghn$\"$H#Fghn;$!$>&Fghn$\"$>&F ghn" 1 2 0 1 10 0 2 9 1 2 1 1.000000 45.000000 45.000000 0 0 "Curve 1 " "Curve 2" "Curve 3" "Curve 4" "Curve 5" "Curve 6" "Curve 7" "Curve 8 " }}}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 4 "The \+ " }{TEXT 260 30 "interval of absolute stability" }{TEXT -1 89 " (or s tability interval) is the intersection of the stability region with th e real line." }}{PARA 0 "" 0 "" {TEXT -1 59 "For this scheme the stabi lity interval is (approximately) " }{XPPEDIT 18 0 "[-4.8510, 0];" "6# 7$,$-%&FloatG6$\"&5&[!\"%!\"\"\"\"!" }{TEXT -1 2 ". " }}{PARA 0 "" 0 " " {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 260 "We can distort the bou ndary curve horizontally by taking the 11th root of the real part of p oints along the curve. In this way we see that the largest interval on the nonnegative imaginary axis that contains the origin and lies insi de the stability region is " }{XPPEDIT 18 0 "[0, 2.6];" "6#7$\"\"!-%& FloatG6$\"#E!\"\"" }{TEXT -1 18 " approximately. " }}{PARA 0 "" 0 " " {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 388 "R := z -> 1+z+1/2*z^2+1/6*z^3+1/24*z^4+1/120*z^5+1/720*z^6+\n 6464407822193 /32620504464915840*z^7+21219532823/1112062652213040*z^8:\nDigits := 25 :\npts := []: z0 := 0:\nfor ct from 0 to 90 do\n zz := newton(R(z)=e xp(ct*Pi/100*I),z=z0):\n z0 := zz:\n pts := [op(pts),[surd(Re(zz), 11),Im(zz)]]:\nend do:\nplot(pts,color=COLOR(RGB,.65,.17,.17),thicknes s=2,font=[HELVETICA,9]);\nDigits := 10:" }}{PARA 13 "" 1 "" {GLPLOT2D 261 276 276 {PLOTDATA 2 "6(-%'CURVESG6#7gp7$$\"\"!F)F(7$$\":2>dTw'p3K4 `$o#!#E$\":cJ>(\\#z*e`EfTJF-7$$\":DD'\\Rj9yGgTFv)*F-$\":0m-29![7#fb\\)=F?7$$\":(HP&z>IO#[4g/6F?$\":$GEH;)> =u&[6*>#F?7$$\":-MqNso(QU*[r@\"F?$\":&pH[eQRgATF8DF?7$$\":zwiWUJVxcueK \"F?$\":8:EG>G]vQLu#GF?7$$\":#>d-vf[c*)QIJ9F?$\":T*4%fS6/?l#fTJF?7$$\" :cy%*[l?T-TVQ`\"F?$\":gk2ua72b\">vbMF?7$$\":=NA&)Rap6N:Qj\"F?$\":#3`xJ O&ps<6*pPF?7$$\":1iP:f))=fU![JvHb72#>F?$\":5#*QI`jI[$*)Q7ZF ?7$$\":iZ$pP.&*p9(4E,#F?$\":?9)3P>x\"p;[l-&F?7$$\":>E!o\">?_mYtG5#F?$ \":%4svx6!e(ytqS`F?7$$\":^\"z]0*))y'QXh\">#F?$\":9*=yI')z=gl'[l&F?7$$ \":g'3r6W$Q'p(G*yAF?$\":3gHFTa)*opD!pfF?7$$\":NP\"=&=ZCue**[O#F?$\":(e 7,**H94pZ=$G'F?7$$\":7hApeY:t&**f\\CF?$\":vrFAZ6,)\\PM(f'F?7$$\":C-\") )R&4i8$R4LDF?$\":-KJMua@Gg-:\"pF?7$$\":bgZM;YqP*yV:EF?$\":r#fil!**p,Gh cA(F?7$$\":Hyf[lxt8d\"o'p#F?$\":#yh,(Gc7!>(>)RvF?7$$\":>70)oqEH#[)F?7$$\":*=D(G\"oo_)Rd9,$F?$\":8;pD 7*H4&3\\kz)F?7$$\":5&)G,:ov-,jx3$F?$\":crz5rQgvc/16*F?7$$\":(o$[Xs$*zD VkJ;$F?$\":_P\"op'4r)f)eZU*F?7$$\":Qh8-$>>]r9oPKF?$\":s6z2KM\"GW;\"*Q( *F?7$$\":-M#o:&z%eU5L6LF?$\":t*\\Q@c8C`iI05!#C7$$\":#3(>2'\\'p=OFTQ$F? 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$\":B1^c%o1#[i972#Ffu7$$\":$e&=Mp!3Bqu-(G&F?$\":YwdN+skNe9B5#Ffu7$$\": 0\")4[3#*pgEunJ&F?$\":mYt*[#fgRtwL8#Ffu7$$\":cy\"\\)=5nG/FNM&F?$\":b67 Mjj=XZ(Rk@Ffu7$$\":`C?t!pG]x3%pO&F?$\"::F$4&QUca(HP&>#Ffu7$$\":T%)))3C a'R6gf'Q&F?$\":\\4I$okTxx\"*HEAFfu7$$\":n))p)>XcBeL(>S&F?$\":kGInZm`Sz rrD#Ffu7$$\":qai#>N#zwz=CT&F?$\":-1e]Zn!46j)zG#Ffu7$$\":cd/%fBUs>%*3y%F?$\":\"Q!3E0z<*fwh@EFfu7$$!:HMXg()y/&f2?z^F?$\":^ Y_naa0SV78l#Ffu7$$!:DRVnqrdD*y0KaF?$\":1(pY_drD?V(3o#Ffu7$$!:?2W\\#eMz C+(zi&F?$\":YL?/2GZM$oH5FFfu7$$!:8[%*3Gz70t$y#z&F?$\":Gg&**=+\"=(eMdRF Ffu7$$!:uJdxA;p)[GyPfF?$\":G!fT0;kthwpoFFfu7$$!:Hwr]oF'=Bg#*ogF?$\":dG MaF4?k*Gm(z#Ffu-%+AXESLABELSG6$Q!6\"F\\hl-%*THICKNESSG6#\"\"#-%%FONTG6 $%*HELVETICAG\"\"*-%&COLORG6&%$RGBG$\"#l!\"#$\"# " 0 "" {MPLTEXT 1 0 111 "Digits := 15:\nz0 := 2. 6*I:\nfor ct from 82 to 85 do\n newton(R(z)=exp(ct*Pi/100*I),z=z0); \nend do;\nDigits := 10:" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#^$$\"0;:!* [EEW$!#=$\"0D'Gl^&=c#!#9" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#^$$\"0*pNQ .`?`!#>$\"0+u7W'z\"f#!#9" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#^$$!0z(Qf \\n*)H!#=$\"0z<*fwh@E!#9" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#^$$!0yPW7r E>(!#=$\"0b0SV78l#!#9" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 107 "Then we apply the bisection method to calculate the parameter value associated with the intersection point." }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 176 "real_part := proc(u)\n Re(newton (R(z)=exp(u*Pi*I),z=2.6*I))\nend proc:\nDigits := 15:\nu0 := bisect('r eal_part'(u),u=0.82..0.85);\nnewton(R(z)=exp(u0*Pi*I),z=2.6*I);\nDigit s := 10:" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%#u0G$\"0\"4(fhXjJ)!#:" } }{PARA 11 "" 1 "" {XPPMATH 20 "6#^#$\"0UN*z#zmf#!#9" }}}{PARA 0 "" 0 " " {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 4 "The " }{TEXT 260 112 "la rgest interval on the nonnegative imaginary axis that contains the ori gin and lies inside the stability region" }{TEXT -1 5 " is " } {XPPEDIT 18 0 "[0, 2.5967];" "6#7$\"\"!-%&FloatG6$\"&nf#!\"%" }{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 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 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)*&#\"9xM!*G#o*z)=)=-5\";+7X-vJmgW-)* *3(F)*$)F'F1F)F)F)*&#\"8FB.%)z(oL\\TF9FDF)*$)F'\"\"(F)F)F)*&#\"8$>R\\] BnvmPGW\"=+KC)3IuQe!))GR_DF)*$)F'\"\")F)F)F)*&#\",BG`>7#\"2+K/x@,l*))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#/- %#R*G6#%\"zG\"\"\"" }{TEXT -1 2 ". " }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 32 "z_0 := newton(`R*`(z)=1,z=-5.5);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%$z_0G$!+/Ij0b!\"*" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" } }{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 320 "z_0 := newton(`R*`(z)=1,z= -5.5):\np_1 := plot([`R*`(z),1],z=-6.09..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=[-6.09..0.49,-0.07..1.47],fo nt=[HELVETICA,9]);" }}{PARA 13 "" 1 "" {GLPLOT2D 398 284 284 {PLOTDATA 2 "6+-%'CURVESG6$7Z7$$!3')*************3'!#<$\"3off257XcHF*7 $$!3#H$eR\"oVT0'F*$\"3O8d$[+'yyFF*7$$!3)fm\"zitG=gF*$\"3$>E)[GpQ5EF*7$ $!3%**\\(=W5V#)fF*$\"3AP4R.Q%3X#F*7$$!3+LLeDZdYfF*$\"3i-(\\H[h(*H#F*7$ $!3+]PRVz<%)eF*$\"33LLZ@()*e0#F*7$$!36mT?h6y@eF*$\"3h'*)\\n33Y$=F*7$$! 3m\\iv`%4;v&F*$\"3I(3L59F*7$$!3. 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" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1506 "`R*` := z -> 1+z+1/2*z^2+1 /6*z^3+1/24*z^4+1/120*z^5+\n 100218818879968228903477/708998024460 66317502451200*z^6+\n 14274149336877984032327/70899802446066317502 451200*z^7+\n 44283766756723504939193/2552392888058387430088243200 *z^8+\n 21219532823/88965012177043200*z^9:\npts := []: z0 := 0:\nf or ct from 0 to 300 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_1 := \+ plot(pts,color=COLOR(RGB,.3,.1,.1)):\np_2 := plots[polygonplot]([seq([ pts[i-1],pts[i],[-2.75,0]],i=2..nops(pts))],\n style=patchnog rid,color=COLOR(RGB,.6,.2,.2)):\npts := []: z0 := 1.8+4.7*I:\nfor ct f rom 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,.3,.1,.1)):\np_4 := plots[polygonplot]([seq([pts[i-1] ,pts[i],[1.73,4.63]],i=2..nops(pts))],\n style=patchnogrid,co lor=COLOR(RGB,.6,.2,.2)):\npts := []: z0 := 1.8-4.7*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,.1,.1)):\np_6 := plots[polygonplot]([seq([pts[i-1],pts[i ],[1.73,-4.63]],i=2..nops(pts))],\n style=patchnogrid,color=C OLOR(RGB,.6,.2,.2)):\np_7 := plot([[[-6.19,0],[2.19,0]],[[0,-5.09],[0, 5.09]]],color=black,linestyle=3):\nplots[display]([p_||(1..7)],view=[- 6.19..2.19,-5.09..5.09],font=[HELVETICA,9],\n labels=[`Re(z )`,`Im(z)`],axes=boxed,scaling=constrained);" }}{PARA 13 "" 1 "" {GLPLOT2D 448 481 481 {PLOTDATA 2 "6/-%'CURVESG6$7i]l7$$\"\"!F)F(7$$\" 3!)******3p2!p*!#G$\"3%******41PmD\"!#=7$$\"3))******G;Xxh!#E$\"3#**** ***3TF8DF07$$\"3%)*****zFpy)p!#D$\"3?+++Y4\"*pPF07$$\"37+++Y)zZ)Q!#C$ \"3J+++RkaE]F07$$\"31+++O#)[f9!#B$\"3k******ol<$G'F07$$\"3M+++%R,lE%FF $\"3J+++Y0zRvF07$$\"3-+++E38X5!#A$\"3Y*****f2kjz)F07$$\"3++++VvQRAFQ$ \"3-+++x[G05!#<7$$\"39+++$G@tI%FQ$\"3%*******\\c\"48\"FY7$$\"3i******R km^vFQ$\"3-+++dP^c7FY7$$\"3'*******y$Qc@\"!#@$\"3/+++)=b?Q\"FY7$$\"39+ ++ki\"ez\"Fao$\"3'******RD.v]\"FY7$$\"3&******>;soS#Fao$\"3#******RJ/G j\"FY7$$\"3')******R#G+$GFao$\"3!*******)3$)yv\"FY7$$\"3()*****RCs`j#F ao$\"3'******\\3QE)=FY7$$\"3)******4f*Rv5Fao$\"3.+++Iy$p+#FY7$$!3&)*** **fd'GVIFao$\"3$)*****H#)=18#FY7$$!3'******Rb=*\\6!#?$\"31+++>f[`AFY7$ $!3%******\\:C'yEFeq$\"3y*****RK:`P#FY7$$!3!)*****Hz*z?_Feq$\"3#)***** *y#fe\\#FY7$$!3=*****zB;0>*Feq$\"3++++\"=][h#FY7$$!3'******>^!*y]\"!#> $\"34+++64+KFFY7$$!36+++fAgVBFjr$\"3%******\\K)*p%GFY7$$!3=+++VY)R[$Fj r$\"3)******R**)[fHFY7$$!3=+++no;&)\\Fjr$\"3!******HEg!pIFY7$$!3%)**** *>ope*oFjr$\"3!)*****p%\\AvJFY7$$!3e+++D1)3D*Fjr$\"39+++.@UxKFY7$$!3$* *****zD'\\17F0$\"3=+++Sc/vLFY7$$!3#******4^5H`\"F0$\"3))*****pf.vY$FY7 $$!3()*****44%4,>F0$\"3/++++^HaNFY7$$!3*******H\"*zfI#F0$\"3$*******R! 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350037121443888/52287718910334988576674096875,\na[7,2]=0, a[7,3]=38959 719796026045728912040192/190137159673945413006087625,\na[7,4]=-8014961 82066760414629068567466496/5106133423043804066278483169375,\na[7,5]=26 545926738560289939469575124336/3828996342298708285360778643125,\na[7,6 ]=-16804913133796526592/942707650019020409375,\na[8,1]=-76006376179258 679683396673/1389345361299174809573120,\na[8,2]=0,\na[8,3]=10983047702 5712790000/527674313813796947,\na[8,4]=-194391745260058069863027995198 /1219164295706831920486709627,\na[8,5]=444873349778574018937254390317/ 63269358937392669883372236224,\na[8,6]=-49046624888634/292965477272121 5,\na[8,7]=-45835339680636651875/22882961544684083693312,\na[9,1]=6576 0917/1910092800,\na[9,2]=0,\na[9,3]=0,\na[9,4]=6404556346056/248026640 14135,\na[9,5]=2946448255213557/7004255810773760,\na[9,6]=16630875288/ 3684943675,\na[9,7]=-496773593038619375/9868243105287168,\na[9,8]=2782 6183/603420,\n\nb[1]=65760917/1910092800,\nb[2]=0,\nb[3]=0,\nb[4]=6404 556346056/24802664014135,\nb[5]=2946448255213557/7004255810773760,\nb[ 6]=16630875288/3684943675,\nb[7]=-496773593038619375/9868243105287168, \nb[8]=27826183/603420,\n\n`b*`[1]=46875680038130591/12454608326426112 00,\n`b*`[2]=0,\n`b*`[3]=0,\n`b*`[4]=2029502937748023643719/8086189994 224235216645,\n`b*`[5]=1969638800119487132740431/456706934559835779970 3040,\n`b*`[6]=4450793456682602037/1201369121360654225,\n`b*`[7]=-2533 94715948809936175102125/6434509503742783282563072,\n`b*`[8]=2828425336 7140001/786910432449360,\n`b*`[9]=1/80\}:" }}}{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/& F%6#F)#\"#>\"$)>/&F%6#\"\"%#F/\"$K\"/&F%6#\"\"&#\"#))\"$x\"/&F%6#\"\"' #\"#N\"#O/&F%6#\"\"(#\"$W&\"$X&/&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$\"\"#\"\"\" #\"\"$\"#f/&F%6$F*F(#\"%t7\"'C_B/&F%6$F*F'#\"&*H@F1/&F%6$\"\"%F(#\"#> \"$G&/&F%6$F:F'\"\"!/&F%6$F:F*#F<\"$w\"/&F%6$\"\"&F(#\"*%)Hz#H\"*daUA# /&F%6$FJF'FA/&F%6$FJF*#!*/6>t$\")>=9u/&F%6$FJF:#\"*7\\X7$FV/&F%6$\"\"' F(#!2l+9/oC#G$*\"13/&F%6$FinFJ#\"2XZ5_&42t`\"1 ;sKHQ,7(*/&F%6$\"\"(F(#!@))QW@r.]Vp0>xQe\"G\">vo4umd))\\L5*=xG_/&F%6$F epF'FA/&F%6$FepF*#\">#>S?\"*Gd/Egz>(f*Q\"/&F%6$FepF:#!B '\\muco!HYTgn1#='\\,)\"@v$pJ[yi1/Q/BM81^/&F%6$FepFJ#\"AOV7v&p%R**Gg&Qn #faE\"@DJky2O&G3()HUj**GQ/&F%6$FepFin#!5#fElzLJ\"\\!o\"\"6v$4/->+l2F%* /&F%6$\"\")F(#!;tmR$oz'e#zhP1g(\":?Jd4[<*Hh`M*Q\"/&F%6$FgrF'FA/&F%6$Fg rF*#\"6++z7d-x/$)4\"\"3Zpz8QJuw_/&F%6$FgrF:#!?)>&*z-j)p!e+EX\"=F'4 n[?>$oq&Hk\">7/&F%6$FgrFJ#\"?<.Ras$*=Sdy(\\L([W\">CiBsL))pERP*e$pK'/&F %6$FgrFin#!/M'))[iY!\\\"1:7ssZlHH/&F%6$FgrFep#!5v=lO1oR`$e%\"87Lp$3%oW :'H)G#/&F%6$\"\"*F(#\")<4wl\"++G45>/&F%6$F_uF'FA/&F%6$F_uF*FA/&F%6$F_u F:#\".cgMcXS'\"/NT,kE![#/&F%6$F_uFJ#\"1dN@b#[k%H\"1gPx5eD/q/&F%6$F_uFi n#\",)Gv3j;\"+vO%\\o$/&F%6$F_uFep#!3v$>'QIftn\\\"1orG0JCo)*/&F%6$F_uFg r#\")$=Ey#\"'?Mg" }}}{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#\"\"\"#\")<4wl\"++G4 5>/&F%6#\"\"#\"\"!/&F%6#\"\"$F//&F%6#\"\"%#\".cgMcXS'\"/NT,kE![#/&F%6# \"\"&#\"1dN@b#[k%H\"1gPx5eD/q/&F%6#\"\"'#\",)Gv3j;\"+vO%\\o$/&F%6#\"\" (#!3v$>'QIftn\\\"1orG0JCo)*/&F%6#\"\")#\")$=Ey#\"'?Mg" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT 259 38 "weights for the 9 s tage order 5 scheme" }{TEXT -1 1 ":" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 37 "seq(`b*`[i]=subs(ee,`b*`[i]),i=1..9);" }}{PARA 12 "" 1 "" {XPPMATH 20 "6+/&%#b*G6#\"\"\"#\"2\"fI\"Q+ovo%\"4+7hUE$3YX7/&F%6# \"\"#\"\"!/&F%6#\"\"$F//&F%6#\"\"%#\"7>PkB![x$H]H?\"7Xm@NUA%***='3)/&F %6#\"\"&#\":J/uKr[>,!)Q'p>\":SIq*zd$)fX$pqc%/&F%6#\"\"'#\"4P?g#ocMz]W \"4DUlg87p8?\"/&F%6#\"\"(#! " 0 "" {MPLTEXT 1 0 1 ";" }}}}{PARA 0 "" 0 "" {TEXT -1 52 "#---------------------------------------------- -----" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}}{PARA 0 "" 0 "" {TEXT -1 45 "#============================================" }} {SECT 1 {PARA 4 "" 0 "" {TEXT -1 39 "Verner's \"most efficient\" schem e with " }{XPPEDIT 18 0 "c[7] = 1999/2000;" "6#/&%\"cG6#\"\"(*&\"%**> \"\"\"\"%+?!\"\"" }{TEXT -1 1 " " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }} {PARA 0 "" 0 "" {TEXT -1 120 "The scheme given here is a minor modific ation of Jim Verner's \"most efficient\" scheme which is available on \+ his website." }}{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 2487 "ee := \{c[2]=7/116,\nc[3]=208/215 7,\nc[4]=104/719,\nc[5]=121/243,\nc[6]=389/400,\nc[7]=1999/2000,\nc[8] =1,\nc[9]=1,\n\na[2,1]=7/116,\na[3,1]=631280/32568543,\na[3,2]=2509312 /32568543,\na[4,1]=26/719,\na[4,2]=0,\na[4,3]=78/719,\na[5,1]=40529883 5029/310395556224,\na[5,2]=0,\na[5,3]=-516775016989/103465185408,\na[5 ,4]=649792660033/155197778112,\na[6,1]=-613196378874450690363694258375 297/14810689674481071256371200000000,\na[6,2]=0,\na[6,3]=1931459030429 7630584070330403179/122402394003975795507200000000,\na[6,4]=-456903226 810742698201959542362145743/3777766287341706964636467200000000,\na[6,5 ]=233468684012501097630099419892501/42262363236718430355123200000000, \na[7,1]=-690994313677727892638163353214849278889337/12834810518108703 660371044352000000000000,\na[7,2]=0,\na[7,3]=5590270196811785636610822 8295253679031/272680756296260886366208000000000000,\na[7,4]=-314869670 545645544422296903990152470264703005657/200374588553351361015034269961 9328000000000000,\na[7,5]=39432881384982237198854165804094689535145653 9/56456885797940553200650756061648000000000000,\na[7,6]=-1841588041032 613773/98979602990157900500,\na[8,1]=-41346042665166483923424449659815 720863/764692984974231010310450749101004672,\na[8,2]=0,\na[8,3]=167329 7023243723294266377481831/8127169928568684375229008512,\na[8,4]=-50372 5499129961091532189480038365151123989843/31919113953833232441893004967 93324868133824,\na[8,5]=221222045246087052146775273050936702042442/315 51173215903875161610805733586403142833,\na[8,6]=-119908984351974400000 0/65355131597330942592057,\na[8,7]=-203964353120000000000/398405317752 764522861589,\na[9,1]=314492473/9032729376,\na[9,2]=0,\na[9,3]=0,\na[9 ,4]=11996228882183665169/46375417381391048160,\na[9,5]=172318679431540 59861/40981954688105953544,\na[9,6]=16704872960000000/3806497022432643 ,\na[9,7]=-370696000000000000/2109493746887301,\na[9,8]=1699767001/990 3960,\n\nb[1]=314492473/9032729376,\nb[2]=0,\nb[3]=0,\nb[4]=1199622888 2183665169/46375417381391048160,\nb[5]=17231867943154059861/4098195468 8105953544,\nb[6]=16704872960000000/3806497022432643,\nb[7]=-370696000 000000000/2109493746887301,\nb[8]=1699767001/9903960,\n\n`b*`[1]=35699 68140173481166883/93912409021561276605888,\n`b*`[2]=0,\n`b*`[3]=0,\n`b *`[4]=40411737031027877369640158198393/1607202350612752490229628676553 60,\n`b*`[5]=14130323574784057347070883452389/327757998459260025705754 45690544,\n`b*`[6]=1826359104228575029348000000/5073817873861059116774 01003,\n`b*`[7]=-38533860125250630597500000000/28118206883332247658864 1821,\n`b*`[8]=5883828582995007071/44004491070454772,\n`b*`[9]=1/80\}: " }}}{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&#\"$3#\"%d@#\"'!GJ '\")V&oD$#\"(7$4DF2F+7&#\"$/\"\"$>(#\"#EF8\"\"!#\"#yF87&#\"$@\"\"$V## \"-H]$))H0%\"-Cib&R5$F;#!-*)p,vn^\"-3a=lM57&F+F+F+#\"-L+m#z\\'\"-7\"yx >b\"7&#\"$*Q\"$+%#!B(Hv$eUpj.p]W()yj>8'\"A++++7Pc72\"[u'*o5[\"F;#\"AzJ SI.2%eIwH/.f9$>\"?++++s]&zvR+%R-C77&F+F+#!EVd9iBaf>?)pU2\"oA.pX\"C++++ sYOY'pqTtGmxx$#\"B,D*)>%*4Iw4,D,%ooMB\"A++++K7b.V=nBjBEU7&#\"%**>\"%+? #!KP$*))y#\\[@`L;QE*ysxOJ%*4p\"J++++++_V/r.m.(3\"=0\"[$G\"F;#\"GJ!zOD& HG#3hOcy6o>q-f&\"E++++++3iO')3E'Hc2os#7&F+#!Qdc+.ZEqC:!*R!pHAWaXcaq'p[ J\"O++++++G$>'*pU.:5O^Lb)eu.?#\"NRlX^`*o%4/e;a))>PA)\\Q\")GVR\"M++++++ [;1c2l+KbSzz&)oXc#!4tPhK5/)eT=\"5+0!z:!*Hgz*)*7&\"\"\"#!Gj3s:)f'\\WUBR [m^mUgMT\"EsY+,\"\\2X5.,JU(\\)Hpk(F;#\"@J=[xjE%HBPCBqHt;\"=7&3!H_P%oo& G*pr7)7&F+#!NV)*)R7^^OQ+[*=K:4h*H\"*\\DP]\"LCQ8o[K$z'\\+$*=WKK$Q&R6>>$ #\"KUC/-n$40t_xY@0(3Y_/A7A\"JLG9.keLd!3hh^(Q!f@t6bJ#!7+++W(>N%)*3*>\" \"8d?fU4L(fJ^Nl7&F+F+F+#!6+++++7`V'R?\"9*ehG_kFv<`S)R7&F\\p#\"*tC\\9$ \"+w$HF.*F;F;7&F+#\"5p^m$=#))Gi*>\"\"5g\"[5R\"Q)4%#\"2+++gH([q;\"1VEVAq\\1Q7&F+F+#!3++++++'pq$\"1,t)ou$\\4@# \"+,qw*p\"\"(gR!**7&F+%F_____________________________________GFgrFgr7& %\"bGFbqF;F;FeqF_r7&%#b*G#\"7$)o;\"[t,9o*pN\"8))egw7c@!4C\"R*F;F;7&F+# \"A$R)>e,kpt(y-Jqt6/%\"Bg`lnG'H-\\_Fh]B?2;#\"A*Q_M)32Zt0%yuNKIT\"\"AW0 pXadqD+Ef%)*zvF$#\"=+++[$H]dGU5fj#=\"<.5Sx;\"f5'Q(y\"Q2&7&F+#!>++++vfI 1DD,'Q`Q\"<@=k)ewCKL)o?=\"G#\"4rq+&*HeGQ)e\"2sZXq5\\/S%#F\\p\"#!)Q)ppr int166\"" }}}{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,$ \")G[Mg!\"*F(%!GF+F+F+F+F+F+F+7,$\")F-V'*F*$\")@JQ>F*$\")1r/xF*F+F+F+F +F+F+F+7,$\")MXY9!\")$\")N8;OF*$\"\"!F:$\"),%[3\"F6F+F+F+F+F+F+7,$\")R Uz\\F6$\")&\\dI\"!\"(F9$!)an%*\\FB$\")$oo=%FBF+F+F+F+F+7,$\")++D(*F6$! )%G-9%!\"'F9$\")&ezd\"!\"&$!)MX47FO$\")&pU_&FBF+F+F+F+7,$\")++&***F6$! );v$Q&FLF9$\")a6]?FO$!)_Sr:FO$\")4g%)pFB$!)Ldg=F*F+F+F+7,$\"\"\"F:$!)9 )oS&FLF9$\")E*)e?FO$!)78y:FO$\"):`6qFB$!)%HZ$=F*$!))=&>^!#6F+F+7,F\\o$ \")$*p\"[$F*F9F9$\")Yw'e#F6$\")`u/UF6$\")g^)Q%FB$!)ZFd " 0 "" {MPLTEXT 1 0 146 "R K6_8eqs := [op(RowSumConditions(8,'expanded')),op(OrderConditions(6,8, 'expanded'))]:\n`RK5_9eqs*` := subs(b=`b*`,OrderConditions(5,9,'expand ed')):" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 7 "Check: " }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 148 "simplify(subs(e e,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 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 := StageOrderCon ditions(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(proc(L) local i; for i to nops(L) do if not evalb(L[i]) then break end if end do; i end proc,%):\nsimpl ify(%); " }}{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 t he principal error conditions are satisfied." }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 123 "RK6_8err_eqs := PrincipalErrorConditions(6,8,'exp anded'):\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,'expande d'):\nevalf(evalf[14](sqrt(add(subs(ee,errterms6_8[i])^2,i=1.. nops(er rterms6_8)))));" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#$\"+rB)=T\"!#:" }}} {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 79 "The 2 nor m of the principal error of the order 5 embedded scheme is as follows. " }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 153 "`errterms5_9*` := subs( b=`b*`,PrincipalErrorTerms(5,9,'expanded')):\nevalf(evalf[14](sqrt(add (subs(ee,`errterms5_9*`[i])^2,i=1.. nops(`errterms5_9*`)))));" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#$\"+#*fP[\\!#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)pprint616\"" }}}{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*` )]:" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}{PARA 0 "" 0 " " {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 22 "We specify the nodes: \+ " }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "c[2] = 7/116;" "6 #/&%\"cG6#\"\"#*&\"\"(\"\"\"\"$;\"!\"\"" }{TEXT -1 3 ", " }{XPPEDIT 18 0 "c[4] = 104/719;" "6#/&%\"cG6#\"\"%*&\"$/\"\"\"\"\"$>(!\"\"" } {TEXT -1 3 ", " }{XPPEDIT 18 0 "c[5] = 121/243;" "6#/&%\"cG6#\"\"&*& \"$@\"\"\"\"\"$V#!\"\"" }{TEXT -1 3 ", " }{XPPEDIT 18 0 "c[6] = 389/4 00;" "6#/&%\"cG6#\"\"'*&\"$*Q\"\"\"\"$+%!\"\"" }{TEXT -1 3 ", " } {XPPEDIT 18 0 "c[7] = 1999/2000;" "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 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] = 1/80;" "6# /&%#b*G6#\"\"**&\"\"\"F)\"#!)!\"\"" }{TEXT -1 2 ". " }}{PARA 0 "" 0 " " {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 37 "We have 44 equations an d 44 unknowns." }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 220 "e1 := \{c [2]=7/116,c[4]=104/719,c[5]=121/243,c[6]=389/400,c[7]=1999/2000,\n \+ c[8]=1,c[9]=1,seq(a[i,2]=0,i=4..8),b[2]=0,b[3]=0,`b*`[2]=0,`b*`[3 ]=0,`b*`[9]=1/80\}:\neqns := subs(e1,cdns):\nnops(%);\nindets(eqns);\n nops(%);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#\"#W" }}{PARA 12 "" 1 "" {XPPMATH 20 "6# " 0 "" {MPLTEXT 1 0 22 "infolevel[solve] := 4:" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 69 "e2 := solve(\{op(eqns)\}) :\ninfolevel[solve] := 0:\ne3 := `union`(e1,e2):" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 2 "e3" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 2606 "e3 := \{`b*`[7] = -3853386 0125250630597500000000/281182068833322476588641821, `b*`[1] = 35699681 40173481166883/93912409021561276605888, b[2] = 0, a[5,2] = 0, a[6,2] = 0, a[7,2] = 0, a[8,2] = 0, a[4,2] = 0, a[9,2] = 0, a[9,3] = 0, `b*`[9 ] = 1/80, `b*`[5] = 14130323574784057347070883452389/32775799845926002 570575445690544, c[9] = 1, c[8] = 1, a[2,1] = 7/116, b[3] = 0, `b*`[2] = 0, `b*`[3] = 0, `b*`[6] = 1826359104228575029348000000/507381787386 105911677401003, c[7] = 1999/2000, c[4] = 104/719, c[2] = 7/116, `b*`[ 4] = 40411737031027877369640158198393/16072023506127524902296286765536 0, c[6] = 389/400, c[3] = 208/2157, a[5,3] = -516775016989/10346518540 8, c[5] = 121/243, `b*`[8] = 5883828582995007071/44004491070454772, b[ 8] = 1699767001/9903960, a[5,4] = 649792660033/155197778112, b[5] = 17 231867943154059861/40981954688105953544, a[7,6] = -1841588041032613773 /98979602990157900500, a[6,5] = 233468684012501097630099419892501/4226 2363236718430355123200000000, a[6,4] = -456903226810742698201959542362 145743/3777766287341706964636467200000000, a[6,3] = 193145903042976305 84070330403179/122402394003975795507200000000, a[7,3] = 55902701968117 856366108228295253679031/272680756296260886366208000000000000, a[8,7] \+ = -203964353120000000000/398405317752764522861589, a[8,6] = -119908984 3519744000000/65355131597330942592057, a[8,3] = 1673297023243723294266 377481831/8127169928568684375229008512, a[4,3] = 78/719, a[7,5] = 3943 28813849822371988541658040946895351456539/5645688579794055320065075606 1648000000000000, a[8,5] = 221222045246087052146775273050936702042442/ 31551173215903875161610805733586403142833, b[7] = -370696000000000000/ 2109493746887301, a[8,4] = -503725499129961091532189480038365151123989 843/3191911395383323244189300496793324868133824, a[8,1] = -41346042665 166483923424449659815720863/764692984974231010310450749101004672, a[5, 1] = 405298835029/310395556224, a[3,2] = 2509312/32568543, a[7,1] = -6 90994313677727892638163353214849278889337/1283481051810870366037104435 2000000000000, a[9,5] = 17231867943154059861/40981954688105953544, a[9 ,8] = 1699767001/9903960, a[9,7] = -370696000000000000/210949374688730 1, a[9,6] = 16704872960000000/3806497022432643, b[6] = 167048729600000 00/3806497022432643, a[7,4] = -314869670545645544422296903990152470264 703005657/2003745885533513610150342699619328000000000000, b[4] = 11996 228882183665169/46375417381391048160, b[1] = 314492473/9032729376, a[4 ,1] = 26/719, a[6,1] = -613196378874450690363694258375297/148106896744 81071256371200000000, a[9,1] = 314492473/9032729376, a[9,4] = 11996228 882183665169/46375417381391048160, a[3,1] = 631280/32568543\}:" }}} {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(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&#\"$3#\"%d@#\"'!GJ'\")V&oD$#\"(7$4DF 2F+7&#\"$/\"\"$>(#\"#EF8\"\"!#\"#yF87&#\"$@\"\"$V##\"-H]$))H0%\"-Cib&R 5$F;#!-*)p,vn^\"-3a=lM57&F+F+F+#\"-L+m#z\\'\"-7\"yx>b\"7&#\"$*Q\"$+%#! B(Hv$eUpj.p]W()yj>8'\"A++++7Pc72\"[u'*o5[\"F;#\"AzJSI.2%eIwH/.f9$>\"?+ +++s]&zvR+%R-C77&F+F+#!EVd9iBaf>?)pU2\"oA.pX\"C++++sYOY'pqTtGmxx$#\"B, D*)>%*4Iw4,D,%ooMB\"A++++K7b.V=nBjBEU7&#\"%**>\"%+?#!KP$*))y#\\[@`L;QE *ysxOJ%*4p\"J++++++_V/r.m.(3\"=0\"[$G\"F;#\"GJ!zOD&HG#3hOcy6o>q-f&\"E+ +++++3iO')3E'Hc2os#7&F+#!Qdc+.ZEqC:!*R!pHAWaXcaq'p[J\"O++++++G$>'*pU.: 5O^Lb)eu.?#\"NRlX^`*o%4/e;a))>PA)\\Q\")GVR\"M++++++[;1c2l+KbSzz&)oXc#! 4tPhK5/)eT=\"5+0!z:!*Hgz*)*7&\"\"\"#!Gj3s:)f'\\WUBR[m^mUgMT\"EsY+,\"\\ 2X5.,JU(\\)Hpk(F;#\"@J=[xjE%HBPCBqHt;\"=7&3!H_P%oo&G*pr7)7&F+#!NV)*)R7 ^^OQ+[*=K:4h*H\"*\\DP]\"LCQ8o[K$z'\\+$*=WKK$Q&R6>>$#\"KUC/-n$40t_xY@0( 3Y_/A7A\"JLG9.keLd!3hh^(Q!f@t6bJ#!7+++W(>N%)*3*>\"\"8d?fU4L(fJ^Nl7&F+F +F+#!6+++++7`V'R?\"9*ehG_kFv<`S)R7&F\\p#\"*tC\\9$\"+w$HF.*F;F;7&F+#\"5 p^m$=#))Gi*>\"\"5g\"[5R\"Q)4%#\"2+++gH( [q;\"1VEVAq\\1Q7&F+F+#!3++++++'pq$\"1,t)ou$\\4@#\"+,qw*p\"\"(gR!**7&F+ %F_____________________________________GFgrFgr7&%\"bGFbqF;F;FeqF_r7&%# b*G#\"7$)o;\"[t,9o*pN\"8))egw7c@!4C\"R*F;F;7&F+#\"A$R)>e,kpt(y-Jqt6/% \"Bg`lnG'H-\\_Fh]B?2;#\"A*Q_M)32Zt0%yuNKIT\"\"AW0pXadqD+Ef%)*zvF$#\"=+ ++[$H]dGU5fj#=\"<.5Sx;\"f5'Q(y\"Q2&7&F+#!>++++vfI1DD,'Q`Q\"<@=k)ewCKL) o?=\"G#\"4rq+&*HeGQ)e\"2sZXq5\\/S%#F\\p\"#!)Q)pprint146\"" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 136 "sub s(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,$\")G[Mg!\"*F(%!GF+F+F +F+F+F+F+7,$\")F-V'*F*$\")@JQ>F*$\")1r/xF*F+F+F+F+F+F+F+7,$\")MXY9!\") $\")N8;OF*$\"\"!F:$\"),%[3\"F6F+F+F+F+F+F+7,$\")RUz\\F6$\")&\\dI\"!\"( F9$!)an%*\\FB$\")$oo=%FBF+F+F+F+F+7,$\")++D(*F6$!)%G-9%!\"'F9$\")&ezd \"!\"&$!)MX47FO$\")&pU_&FBF+F+F+F+7,$\")++&***F6$!);v$Q&FLF9$\")a6]?FO $!)_Sr:FO$\")4g%)pFB$!)Ldg=F*F+F+F+7,$\"\"\"F:$!)9)oS&FLF9$\")E*)e?FO$ !)78y:FO$\"):`6qFB$!)%HZ$=F*$!))=&>^!#6F+F+7,F\\o$\")$*p\"[$F*F9F9$\") Yw'e#F6$\")`u/UF6$\")g^)Q%FB$!)ZFd " 0 "" {MPLTEXT 1 0 146 "RK6_8eqs := [op(RowSum Conditions(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,`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 "determinat ion 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 solution" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 71947 "eG := \{c[8] = 1, b[3] = 0, b[2] = 0, c[9] = 1, b[5] = -1/60*(10*c[6] *c[7]*c[4]-5*c[6]*c[4]-5*c[7]*c[6]+3*c[6]-5*c[7]*c[4]+3*c[4]-2+3*c[7]) /(-c[4]+c[5])/c[5]/(-c[5]^3+c[5]^2-c[7]*c[5]+c[7]*c[5]^2-c[6]*c[5]+c[6 ]*c[5]^2+c[7]*c[6]-c[6]*c[7]*c[5]), a[5,2] = 0, a[4,2] = 0, `b*`[3] = \+ 0, `b*`[2] = 0, a[8,2] = 0, a[7,2] = 0, a[6,2] = 0, a[6,5] = 1/2*c[6]* (30*c[6]^2*c[5]^2*c[4]^2-2*c[6]^2*c[4]^2+c[6]^3*c[4]-12*c[6]^2*c[4]*c[ 5]^2-3*c[5]^2*c[6]*c[4]+c[5]*c[6]*c[4]^2+c[6]*c[4]^3-12*c[6]^2*c[4]^2* c[5]-30*c[5]*c[6]^3*c[4]^2-20*c[5]*c[6]^2*c[4]^4+30*c[6]^2*c[5]*c[4]^3 -c[5]*c[4]^3+20*c[4]^3*c[5]*c[6]^3-20*c[6]^2*c[5]^2*c[4]^3+c[5]^2*c[4] ^2-30*c[5]^2*c[6]*c[4]^3+12*c[5]^2*c[6]*c[4]^2+2*c[6]^2*c[4]*c[5]+20*c [5]^2*c[6]*c[4]^4+2*c[6]^2*c[5]^2-2*c[6]^3*c[5]+12*c[6]^3*c[4]*c[5])/c [5]/(c[4]^3-2*c[5]*c[4]^2+30*c[5]^2*c[4]^3-6*c[5]*c[4]^3-10*c[4]^4*c[5 ]^2-c[5]^3+10*c[5]^4*c[4]^2-30*c[5]^3*c[4]^2+6*c[5]^3*c[4]+2*c[5]^2*c[ 4]), b[1] = 1/60*(30*c[5]*c[6]*c[7]*c[4]-10*c[5]*c[6]*c[4]-10*c[6]*c[7 ]*c[5]+5*c[6]*c[5]-10*c[6]*c[7]*c[4]+5*c[6]*c[4]+5*c[7]*c[6]-3*c[6]-10 *c[5]*c[7]*c[4]+5*c[5]*c[4]+5*c[7]*c[5]-3*c[5]+5*c[7]*c[4]-3*c[4]+2-3* c[7])/c[5]/c[6]/c[7]/c[4], a[6,1] = -1/4*c[6]*(60*c[5]^2*c[6]*c[4]^4-9 0*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+6 0*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]^4-100*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]), 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]^5-7 60*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+154 0*c[4]^5*c[5]^3*c[6]^2-34*c[7]^3*c[4]^4*c[5]-4*c[7]^2*c[5]^4*c[4]+12*c [5]^2*c[4]^4*c[7]-41*c[6]^2*c[5]^2*c[4]^3+14*c[6]^2*c[5]^2*c[4]^2-54*c [6]*c[5]^3*c[4]^3-12*c[6]^2*c[5]^3*c[4]+47*c[6]^2*c[5]^3*c[4]^2+12*c[5 ]^3*c[7]*c[4]^3-98*c[5]^4*c[7]*c[4]^2*c[6]-50*c[5]^3*c[7]^3*c[4]^4*c[6 ]+4*c[7]*c[6]^2*c[4]^3+156*c[7]^2*c[5]^2*c[6]*c[4]^3-23*c[7]^3*c[5]^2* c[6]*c[4]^2+188*c[7]^2*c[5]^3*c[6]*c[4]^2+66*c[5]^3*c[7]^3*c[6]*c[4]+2 *c[7]^3*c[4]^2*c[5]+100*c[5]^5*c[4]^5*c[7]+104*c[4]^3*c[7]^2*c[5]*c[6] ^2-12*c[7]^2*c[6]*c[5]*c[4]^3-36*c[7]^2*c[5]^2*c[6]*c[4]^2-32*c[7]^2*c [5]^3*c[6]*c[4]-320*c[7]^2*c[5]^3*c[6]*c[4]^3-560*c[5]^3*c[6]^2*c[4]^4 -4*c[6]^2*c[4]^3*c[7]^2+240*c[5]^3*c[6]^2*c[7]*c[4]^3-34*c[5]^3*c[7]^3 *c[4]^2-2*c[7]^3*c[5]^2*c[4]+164*c[6]^2*c[7]*c[5]^3*c[4]^2-20*c[6]^2*c [7]^2*c[4]*c[5]^3+2*c[7]^2*c[5]^3*c[4]-130*c[5]^5*c[4]^3*c[6]*c[7]-90* c[7]^3*c[5]^2*c[6]^2*c[4]^3-16*c[7]^2*c[4]^4*c[5]-60*c[7]^2*c[5]^3*c[6 ]^2*c[4]^2-300*c[6]^2*c[4]^5*c[5]*c[7]^2-150*c[7]^2*c[4]^5*c[6]*c[5]^3 +380*c[5]^2*c[6]*c[4]^5*c[7]^2+140*c[7]^3*c[5]^2*c[4]^3*c[6]-810*c[7]^ 3*c[5]^3*c[6]^2*c[4]^2-14*c[7]^3*c[6]*c[5]*c[4]^3-280*c[7]^3*c[6]^2*c[ 5]*c[4]^3+240*c[7]^3*c[5]^3*c[6]^2*c[4]-20*c[5]^4*c[7]^2*c[4]*c[6]-6*c [6]*c[7]^3*c[5]^3+22*c[5]^4*c[4]*c[7]*c[6]-280*c[4]^5*c[6]^2*c[7]*c[5] ^2+400*c[4]^5*c[6]^2*c[7]^2*c[5]^2+253*c[6]^2*c[4]^4*c[5]^2+470*c[7]^3 *c[5]^3*c[4]^3*c[6]+100*c[5]^2*c[4]^5*c[7]^3*c[6]+6*c[6]^2*c[7]^3*c[4] ^2-300*c[5]^5*c[4]^5*c[6]-120*c[5]*c[7]*c[4]^6*c[6]^2-10*c[6]^2*c[7]*c [4]^5-1100*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]^2-4 0*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+1 20*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]+3 9*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+2 8*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]+2 5*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+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]^3-10 3*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], 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+10 0*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+4 0*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]^2-14 0*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]^3-1 00*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]-3 6*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]^4-1 2*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]^2-36 0*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]+185 0*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]-2 00*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]+9 7*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]^2-19 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]+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]+10 20*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+90 0*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+84 0*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 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9*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]+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]^5-73 0*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]^3-10*c[5]^ 2*c[6]*c[4]-9*c[5]^2*c[7]*c[4]+6*c[7]^2*c[5]^2*c[4]+33*c[5]^2*c[6]*c[7 ]*c[4]+600*c[7]*c[4]^8*c[6]*c[5]^3+187*c[6]^2*c[7]*c[5]^3*c[4]^2+341*c [6]^2*c[7]^2*c[4]*c[5]^3+202*c[5]^2*c[4]^3-29*c[7]^2*c[5]^2*c[4]^2-10* c[7]^2*c[5]^3*c[4]+40*c[4]^4*c[5]-270*c[4]^7*c[5]^2-450*c[5]^5*c[4]^3* c[6]*c[7]+35*c[4]^6*c[7]*c[5]+80*c[7]^2*c[4]^4*c[5]-1163*c[7]^2*c[5]^3 *c[6]^2*c[4]^2+20*c[5]*c[4]^7-640*c[6]^2*c[4]^5*c[5]*c[7]^2+850*c[7]^2 *c[4]^5*c[6]*c[5]^3+1530*c[5]^2*c[6]*c[4]^5*c[7]^2-506*c[5]^4*c[4]^4+1 420*c[5]^2*c[4]^7*c[6]*c[7]-16*c[5]^4*c[4]*c[6]-85*c[5]^4*c[7]^2*c[4]* c[6]+72*c[5]^4*c[4]*c[7]*c[6]+230*c[4]^5*c[6]^2*c[7]*c[5]^2+1740*c[4]^ 5*c[6]^2*c[7]^2*c[5]^2-600*c[6]*c[4]^8*c[5]^2*c[7]-82*c[4]^5*c[5]^2-48 0*c[6]^2*c[4]^4*c[5]^2+320*c[4]^8*c[5]^2*c[6]-37*c[4]^6*c[5]+300*c[5]^ 3*c[4]^7*c[6]^2-90*c[5]^5*c[4]^5*c[6]+520*c[5]*c[7]*c[4]^6*c[6]^2-10*c [6]^2*c[7]*c[4]^5+9*c[6]*c[4]^4+600*c[5]^4*c[7]*c[4]^6*c[6]^2-2350*c[5 ]^4*c[4]^5*c[6]^2*c[7]-600*c[5]^3*c[7]^2*c[6]*c[4]^7-204*c[7]*c[6]^2*c [5]*c[4]^4-3630*c[5]^3*c[4]^4*c[6]^2*c[7]+740*c[5]^3*c[4]^3*c[6]^2*c[7 ]^2-2120*c[5]^4*c[4]^4*c[6]*c[7]+1460*c[5]^2*c[4]^4*c[7]*c[6]^2+2387*c [5]^3*c[4]^4*c[6]*c[7]+2548*c[5]^4*c[4]^3*c[6]*c[7]-498*c[5]^3*c[4]^3+ 100*c[5]^2*c[4]^7*c[7]^2+26*c[7]^2*c[5]^3*c[4]^2+600*c[6]*c[5]^3*c[4]^ 7+100*c[6]*c[5]^5*c[4]^6+300*c[6]*c[5]^4*c[4]^7-600*c[5]^4*c[7]*c[4]^7 *c[6]+1600*c[5]^4*c[4]^5*c[6]^2-1500*c[6]^2*c[7]^2*c[5]^3*c[4]^5-2700* c[6]*c[7]^2*c[5]^4*c[4]^5-200*c[7]^2*c[4]^7*c[6]*c[5]-2430*c[5]^4*c[4] ^4*c[6]^2+1429*c[5]^4*c[4]^3*c[6]^2-2010*c[7]^2*c[5]^4*c[6]^2*c[4]^3+1 410*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]-3280*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]+1 030*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+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] 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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]+25 8*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]-12 00*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+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]^2-1 01*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]+18 18*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] 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+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]^2-827 0*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+11 240*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]^2-13 20*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+12 0*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+640 4*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]^2-1 5*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]^2-7 2*c[5]*c[6]*c[4]^2+60*c[5]*c[4]^2-12*c[5]*c[6]*c[4]+10*c[5]*c[4]-150*c [5]*c[7]*c[6]*c[4]^3-15*c[7]*c[5]^3-660*c[5]^2*c[6]*c[7]*c[4]^2+90*c[6 ]*c[5]*c[7]*c[4]^2-15*c[6]*c[7]*c[5]^2+150*c[6]*c[5]^4*c[4]^2+930*c[6] *c[5]^2*c[7]*c[4]^3+930*c[6]*c[5]^3*c[7]*c[4]^2-200*c[5]^4*c[4]^3*c[7] +110*c[7]*c[5]*c[4]^3+110*c[6]*c[5]*c[4]^3+110*c[5]^3*c[6]*c[4]+110*c[ 5]^3*c[7]*c[4]+60*c[5]^2*c[4]-150*c[5]^3*c[6]*c[7]*c[4]+550*c[5]^3*c[4 ]^2-690*c[5]^2*c[6]*c[4]^3+520*c[5]^2*c[7]*c[4]^2+520*c[5]^2*c[6]*c[4] ^2-690*c[5]^2*c[7]*c[4]^3-690*c[5]^3*c[7]*c[4]^2-690*c[5]^3*c[6]*c[4]^ 2+150*c[5]^2*c[4]^4*c[7]+900*c[6]*c[5]^3*c[4]^3+900*c[5]^3*c[7]*c[4]^3 -200*c[5]^4*c[7]*c[4]^2*c[6]-429*c[5]^2*c[4]^2-120*c[5]^4*c[4]^2-120*c [4]^4*c[5]^2+150*c[5]^4*c[4]^3-72*c[5]^2*c[6]*c[4]-72*c[5]^2*c[7]*c[4] +90*c[5]^2*c[6]*c[7]*c[4]+550*c[5]^2*c[4]^3+300*c[5]^3*c[4]^4*c[6]*c[7 ]+300*c[5]^4*c[4]^3*c[6]*c[7]-690*c[5]^3*c[4]^3-200*c[4]^4*c[6]*c[5]^3 +150*c[5]^3*c[4]^4), a[8,3] = 3/4*(-1560*c[5]^2*c[6]*c[4]^4+585*c[6]*c [5]*c[4]^4-45*c[7]*c[5]*c[4]^4-12*c[5]^3*c[4]-450*c[4]^5*c[6]*c[5]^3+3 *c[5]*c[7]*c[4]-150*c[5]^2*c[4]^5*c[7]-200*c[5]^3*c[4]^4*c[7]+300*c[6] *c[4]^5*c[5]*c[7]+2860*c[6]*c[5]^2*c[4]^4*c[7]+160*c[5]^4*c[4]^3*c[6]- 150*c[5]^4*c[4]^4*c[6]-4*c[5]^2+2*c[4]^2-40*c[7]*c[6]*c[5]^3+132*c[5]* c[6]*c[7]*c[4]-900*c[6]*c[5]^2*c[4]^5*c[7]-30*c[6]*c[5]^2+12*c[6]*c[5] -960*c[7]*c[4]^4*c[6]*c[5]+24*c[6]*c[5]^3+3140*c[6]*c[7]*c[5]^3*c[4]^3 +6*c[7]*c[5]^2+20*c[7]*c[6]*c[4]^3-3*c[7]*c[4]^2-18*c[6]*c[7]*c[5]+15* c[6]*c[4]^2-12*c[6]*c[4]^3+9*c[6]*c[7]*c[4]-24*c[6]*c[7]*c[4]^2-54*c[5 ]*c[4]^3-12*c[5]*c[7]*c[4]^2+348*c[5]*c[6]*c[4]^2+9*c[5]*c[4]^2-87*c[5 ]*c[6]*c[4]-2*c[5]*c[4]-6*c[6]*c[4]+1035*c[5]*c[7]*c[6]*c[4]^3+1593*c[ 5]^2*c[6]*c[7]*c[4]^2+900*c[4]^5*c[6]*c[7]*c[5]^3-534*c[6]*c[5]*c[7]*c [4]^2+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+24 0*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+8 50*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]+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]^5-130 8*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* 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5]^4*c[4]^3*c[7]^2*c[6]+1840*c[5]^2*c[4]^4*c[7]^2*c[6]-1060*c[5]^4*c[4 ]^3*c[6]^2*c[7]-170*c[7]^2*c[4]^4*c[6]^2*c[5]+500*c[7]^2*c[4]^4*c[5]^3 *c[6]^2-110*c[7]^2*c[4]^4*c[6]^2*c[5]^2-160*c[5]^2*c[6]*c[4]^5-200*c[6 ]^2*c[7]^2*c[5]^4*c[4]^4-1067*c[4]^4*c[6]*c[5]^3-70*c[5]^5*c[6]^2*c[4] ^3-200*c[5]^4*c[4]^5*c[6]*c[7]+14*c[5]^4*c[4]+434*c[5]^3*c[4]^4-52*c[5 ]^5*c[4]*c[6])/c[5]/(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+1 20*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]^3-20*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]^2-20 0*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]+4 8*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]+11 00*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]+11 0*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]-2 0*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] = 389/400;" "6#/&%\"cG6#\"\"'*&\"$*Q\"\"\"\"$+%!\"\"" }{TEXT -1 7 " and " }{XPPEDIT 18 0 "c[7] = 1999/2000;" "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 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 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 77 "eA := \{c[6]=389/400,c[7]=1999/2000 \}:\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 18450 "eB := \{`b*`[2] = 0, a[9, 6] = 640000000/346599*(9990*c[5]*c[4]-3995*c[5]-3995*c[4]+1997)/(16000 0*c[5]*c[4]-155600*c[4]-155600*c[5]+151321), c[9] = 1, c[8] = 1, b[3] \+ = 0, b[2] = 0, c[6] = 389/400, c[7] = 1999/2000, a[6,3] = 1167/1024000 00000*(9907708140*c[5]*c[4]^3-7829671410*c[5]*c[4]^2+2361212884*c[5]*c [4]-353183214*c[5]+176591607*c[4]-1210568000*c[5]^3*c[4]^2-16045472000 *c[5]^2*c[4]^3+12632220000*c[5]^2*c[4]^2-3652243200*c[5]^2*c[4]+484227 200*c[5]^2-3631704000*c[4]^4*c[5]-242113600*c[4]^2+1867200000*c[5]^3*c [4]^3+5601600000*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[4] = 1/12*(457622*c[5]-151051)/c[4]/(-2377600*c[4]^3+800000*c[4 ]^4+2355211*c[4]^2-777611*c[4]-800000*c[5]*c[4]^3+2377600*c[5]*c[4]^2- 2355211*c[5]*c[4]+777611*c[5]), a[7,5] = -1999/32000000000000*(3930618 37840000000*c[5]^4*c[4]^3+859053779617280000*c[5]^3*c[4]^4+12435552000 0000000*c[5]^4*c[4]^5-1207543698040037*c[4]+219931147408400000*c[4]^5* c[5]^2+2415087396080074*c[5]+12319070855264000*c[5]^3-9763686598577990 *c[5]^2-48338010504200000*c[4]^5*c[5]+67556709561459860*c[5]*c[4]^2+68 3549516495743600*c[5]^2*c[4]^3-634315378627495800*c[4]^4*c[5]^2-200680 387928000000*c[5]^4*c[4]^2+476849085175376000*c[5]^3*c[4]^2-1196513585 40512000*c[5]^3*c[4]+89160067502867520*c[5]^2*c[4]-18081737674389039*c [5]*c[4]-295948125440000000*c[4]^5*c[5]^3+134419335197767900*c[4]^4*c[ 5]-361545662240000000*c[5]^4*c[4]^4+4832903475841795*c[4]^2-9326260986 73880000*c[5]^3*c[4]^3-4970489512000000*c[5]^4+2386366220000000*c[4]^4 +49780768896000000*c[5]^4*c[4]-348559147742899330*c[5]^2*c[4]^2-137971 826436250590*c[5]*c[4]^3-6011742025300000*c[4]^3)/c[5]/(33012100*c[5]^ 5*c[4]^3-24954350*c[5]^4*c[4]^3+39898350*c[5]^3*c[4]^4-17842050*c[5]^5 *c[4]^2+7560000*c[5]^4*c[4]^5+2980000*c[5]^6*c[4]^2+2898050*c[4]^5*c[5 ]^2-142763*c[5]^3-285526*c[5]*c[4]^2+5134100*c[5]^2*c[4]^3-17077040*c[ 4]^4*c[5]^2+19213040*c[5]^4*c[4]^2-4551300*c[5]^3*c[4]^2+273173*c[5]^3 *c[4]+285526*c[5]^2*c[4]-10332100*c[4]^5*c[5]^3+2772040*c[4]^4*c[5]-25 660000*c[5]^4*c[4]^4-4680000*c[5]^3*c[4]^3+436605*c[5]^4+2544000*c[5]^ 5*c[4]-289805*c[4]^4-7560000*c[5]^6*c[4]^3-3056840*c[5]^4*c[4]+293600* c[5]^2*c[4]^2-713573*c[5]*c[4]^3+142763*c[4]^3-298000*c[5]^5), b[8] = \+ 1/132*(755266*c[5]*c[4]-297644*c[5]-297644*c[4]+146593)/(c[4]-1)/(-1+c [5]), a[8,6] = -70400000/10503*(260010*c[5]*c[4]+52003-104005*c[4]-104 005*c[5])*(c[4]-1)*(-1+c[5])/(400*c[4]-389)/(400*c[5]-389)/(755266*c[5 ]*c[4]-297644*c[5]-297644*c[4]+146593), a[6,5] = -389/51200000000*(262 6732000*c[5]^2*c[4]^2-121056800*c[4]^2+58863869*c[4]-913060800*c[5]^2* c[4]-2430016870*c[5]*c[4]^2+62240000*c[4]^3-1210568000*c[4]^4*c[5]+292 9129380*c[5]*c[4]^3-3077768000*c[5]^2*c[4]^3+827423228*c[5]*c[4]+12448 00000*c[4]^4*c[5]^2+121056800*c[5]^2-117727738*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[5,4] = - c[5]^2*(c[4]-c[5])/c[4]^2, `b*`[5] = 1/30*(5106081584*c[4]+2902631540* c[5]-2315541950*c[5]^2+36657548525*c[5]*c[4]^2+22034243700*c[5]^2*c[4] ^3+15170683750*c[5]^2*c[4]-18148650920*c[5]*c[4]-9681132045*c[4]^2-322 69499550*c[5]^2*c[4]^2-23586377700*c[5]*c[4]^3+5875841110*c[4]^3-86833 7859)/(7798099*c[4]+10132099*c[5]+6224000*c[5]^3-14022888*c[5]^2+31195 508*c[5]*c[4]^2+23340000*c[5]^3*c[4]^2-24984000*c[5]^3*c[4]+53955508*c [5]^2*c[4]-36771508*c[5]*c[4]-6220888*c[4]^2-48312330*c[5]^2*c[4]^2-23 32833)/(400*c[5]-389)/(c[4]-c[5])/c[5], `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], 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, a[7,1] \+ = 1999/24896000000000000*(241817238888400000*c[5]^5*c[4]^3-22896931658 81960000*c[5]^4*c[4]^3+147658174400000000*c[5]^3*c[4]^6-15914750123131 27400*c[5]^3*c[4]^4-340452504576800000*c[4]^6*c[5]^2-48338010504200000 *c[5]^5*c[4]^2+96676021008400000*c[4]^6*c[5]-1569742811600000000*c[5]^ 4*c[4]^5+238723200000000000*c[5]^5*c[4]^5+761756464893511600*c[4]^5*c[ 5]^2-24226419437106770*c[5]^3+7245262188240222*c[5]^2-3965526700000000 00*c[5]^5*c[4]^4-264330526395535800*c[4]^5*c[5]+50760065423197863*c[5] *c[4]^2+43553719690854890*c[5]^2*c[4]^3-530313797405935500*c[4]^4*c[5] ^2+939126121440324000*c[5]^4*c[4]^2-882434531514289990*c[5]^3*c[4]^2+2 18403044621373360*c[5]^3*c[4]-41024170362571537*c[5]^2*c[4]-8452805886 280259*c[5]*c[4]+399117139832400000*c[4]^5*c[5]^3+278891109056501180*c [4]^4*c[5]+2795423031297600000*c[5]^4*c[4]^4+2415087396080074*c[4]^2+1 775575281812242800*c[5]^3*c[4]^3-4772732440000000*c[4]^5+1933520420168 0000*c[5]^4+12023484050600000*c[4]^4-198419923445440000*c[5]^4*c[4]+87 426221510183340*c[5]^2*c[4]^2-152366084946683720*c[5]*c[4]^3-966580695 1683590*c[4]^3+238723200000000000*c[5]^4*c[4]^6)/(18900*c[5]^4*c[4]^3+ 18900*c[5]^3*c[4]^4+745*c[5]^3-367*c[5]^2+2202*c[5]*c[4]^2+37360*c[5]^ 2*c[4]^3-7450*c[4]^4*c[5]^2-7450*c[5]^4*c[4]^2+37360*c[5]^3*c[4]^2-636 0*c[5]^3*c[4]+2202*c[5]^2*c[4]+367*c[5]*c[4]-367*c[4]^2-71600*c[5]^3*c [4]^3-18060*c[5]^2*c[4]^2-6360*c[5]*c[4]^3+745*c[4]^3)/c[5]/c[4]^2, b[ 7] = -2000000000000/161919*(1890*c[5]*c[4]-745*c[5]-745*c[4]+367)/(200 0*c[4]-1999)/(2000*c[5]-1999), a[8,3] = 3/20*(-124783300*c[5]^4*c[4]^3 -1198633000*c[5]^3*c[4]^4+2330499*c[4]-374349900*c[4]^5*c[5]^2-4660998 *c[5]-12432440*c[5]^3+15582928*c[5]^2+93243300*c[4]^5*c[5]-146895474*c [5]*c[4]^2-1300477660*c[5]^2*c[4]^3+1214023460*c[4]^4*c[5]^2+31081100* c[5]^4*c[4]^2-571557160*c[5]^3*c[4]^2+124507960*c[5]^3*c[4]-145180356* c[5]^2*c[4]+35757452*c[5]*c[4]+349749900*c[4]^5*c[5]^3-303358560*c[4]^ 4*c[5]+116583300*c[5]^4*c[4]^4-7791464*c[4]^2+1266490540*c[5]^3*c[4]^3 +617747123*c[5]^2*c[4]^2+318339385*c[5]*c[4]^3+6216220*c[4]^3)/c[4]^2/ (7552660*c[5]^4*c[4]^3+7552660*c[5]^3*c[4]^4+297644*c[5]^3-146593*c[5] ^2+879558*c[5]*c[4]^2+14926846*c[5]^2*c[4]^3-2976440*c[4]^4*c[5]^2-297 6440*c[5]^4*c[4]^2+14926846*c[5]^3*c[4]^2-2541130*c[5]^3*c[4]+879558*c [5]^2*c[4]+146593*c[5]*c[4]-146593*c[4]^2-28610860*c[5]^3*c[4]^3-72142 52*c[5]^2*c[4]^2-2541130*c[5]*c[4]^3+297644*c[4]^3), `b*`[7] = -100000 0000/161919*(-6474957*c[4]-6474957*c[5]-5796100*c[5]^3+10704580*c[5]^2 -61442440*c[5]*c[4]^2-69060900*c[5]^2*c[4]^3-69060900*c[5]^3*c[4]^2+35 072500*c[5]^3*c[4]-61442440*c[5]^2*c[4]+34975562*c[5]*c[4]+10704580*c[ 4]^2+44112600*c[5]^3*c[4]^3+114499820*c[5]^2*c[4]^2+35072500*c[5]*c[4] ^3-5796100*c[4]^3+1284867)/(-20254065901*c[4]-20254065901*c[5]-1244177 6000*c[5]^3+28031753112*c[5]^2-135902836492*c[5]*c[4]^2-96624660000*c[ 5]^2*c[4]^3-96624660000*c[5]^3*c[4]^2+62391016000*c[5]^3*c[4]-13590283 6492*c[5]^2*c[4]+93770442492*c[5]*c[4]+28031753112*c[4]^2+46680000000* c[5]^3*c[4]^3+204487363670*c[5]^2*c[4]^2+62391016000*c[5]*c[4]^3-12441 776000*c[4]^3+4663333167), a[8,4] = -1/10*(134143230992100*c[5]^5*c[4] ^3-1880297253643210*c[5]^4*c[4]^3+762371417312600*c[5]^3*c[4]^6-187717 1751019370*c[5]^3*c[4]^4-1843072838096600*c[4]^6*c[5]^2-24169005252100 *c[5]^5*c[4]^2+545471856008200*c[4]^6*c[5]-2594054966348400*c[5]^4*c[4 ]^5+210160982720000*c[5]^5*c[4]^5+186533280000000*c[4]^8*c[5]^3+165525 4961946540*c[4]^5*c[5]^2-12117456225008*c[5]^3+3624443315778*c[5]^2-25 7957448460000*c[5]^5*c[4]^4-594820871925440*c[4]^5*c[5]+2486488000000* c[4]^6+32776878926742*c[5]*c[4]^2+490091160197*c[5]^2*c[4]^3-644961896 453450*c[4]^4*c[5]^2+638222734790610*c[5]^4*c[4]^2-631011927099123*c[5 ]^3*c[4]^2+129638351204461*c[5]^3*c[4]-26682777592490*c[5]^2*c[4]-4228 517201741*c[5]*c[4]+606472567718500*c[4]^5*c[5]^3+368330598488410*c[4] ^4*c[5]+3006858473661300*c[5]^4*c[4]^4+1208147771926*c[4]^2+1576195706 267940*c[5]^3*c[4]^3-8649407136000*c[4]^5+9667602100840*c[5]^4+1104749 0211200*c[4]^4-115239858721140*c[5]^4*c[4]+80112195560025*c[5]^2*c[4]^ 2-137401385576171*c[5]*c[4]^3-6092718847126*c[4]^3-199653280000000*c[4 ]^8*c[5]^2+1121376548160000*c[5]^4*c[4]^6+974889100160000*c[4]^7*c[5]^ 2-259858318720000*c[5]*c[4]^7-740910188160000*c[5]^3*c[4]^7+4972976000 0000*c[5]*c[4]^8-62177760000000*c[5]^5*c[4]^6-186533280000000*c[5]^4*c [4]^7)/(-7552660*c[5]^5*c[4]^3+25634420*c[5]^4*c[4]^3-25634420*c[5]^3* c[4]^4+2976440*c[5]^5*c[4]^2-2976440*c[4]^5*c[5]^2+146593*c[5]^3+29318 6*c[5]*c[4]^2-4673122*c[5]^2*c[4]^3+14926846*c[4]^4*c[5]^2-14926846*c[ 5]^4*c[4]^2+4673122*c[5]^3*c[4]^2-581914*c[5]^3*c[4]-293186*c[5]^2*c[4 ]+7552660*c[4]^5*c[5]^3-2541130*c[4]^4*c[5]-297644*c[5]^4+297644*c[4]^ 4+2541130*c[5]^4*c[4]+581914*c[5]*c[4]^3-146593*c[4]^3)/(-1577600*c[4] +800000*c[4]^2+777611)/c[4]^2, b[5] = -1/12*(457622*c[4]-151051)/(c[4] -c[5])/c[5]/(800000*c[5]^3-2377600*c[5]^2+2355211*c[5]-777611), a[6,1] = 389/102400000000*(-3112000000*c[5]^4*c[4]^3-14526400000*c[5]^3*c[4] ^4+2489600000*c[4]^5*c[5]^2-484227200*c[5]^3+353183214*c[5]^2-24211360 00*c[4]^5*c[5]+1896960056*c[5]*c[4]^2-5145124140*c[5]^2*c[4]^3-9878320 00*c[4]^4*c[5]^2+1210568000*c[5]^4*c[4]^2-13635100000*c[5]^3*c[4]^2+37 76723200*c[5]^3*c[4]-2119099284*c[5]^2*c[4]-412047083*c[5]*c[4]+256000 0000*c[4]^5*c[5]^3+5602258760*c[4]^4*c[5]+2560000000*c[5]^4*c[4]^4+117 727738*c[4]^2+21315872000*c[5]^3*c[4]^3+124480000*c[4]^4+5879069810*c[ 5]^2*c[4]^2-4735553740*c[5]*c[4]^3-242113600*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[5,3] = 3/4*c[5]^2*(-2*c[5]+3*c[4])/c[ 4]^2, a[8,7] = -44000000000/53973*(1890*c[5]*c[4]-745*c[5]-745*c[4]+36 7)*(c[4]-1)*(-1+c[5])/(2000*c[5]-1999)/(2000*c[4]-1999)/(755266*c[5]*c [4]-297644*c[5]-297644*c[4]+146593), `b*`[4] = -1/30*(2902631540*c[4]+ 5106081584*c[5]+5875841110*c[5]^3-9681132045*c[5]^2+15170683750*c[5]*c [4]^2+22034243700*c[5]^3*c[4]^2-23586377700*c[5]^3*c[4]+36657548525*c[ 5]^2*c[4]-18148650920*c[5]*c[4]-2315541950*c[4]^2-32269499550*c[5]^2*c [4]^2-868337859)/(10132099*c[4]+7798099*c[5]-6220888*c[5]^2+53955508*c [5]*c[4]^2+23340000*c[5]^2*c[4]^3+31195508*c[5]^2*c[4]-36771508*c[5]*c [4]-14022888*c[4]^2-48312330*c[5]^2*c[4]^2-24984000*c[5]*c[4]^3+622400 0*c[4]^3-2332833)/(400*c[4]-389)/(c[4]-c[5])/c[4], `b*`[9] = 1/10*(194 50*c[5]^2*c[4]^2-17560*c[5]^2*c[4]+3890*c[5]^2-17560*c[5]*c[4]^2+17615 *c[5]*c[4]-4290*c[5]+3890*c[4]^2-4290*c[4]+1167)/(1167-12492*c[5]*c[4] ^2-12492*c[5]^2*c[4]+3112*c[5]^2-3901*c[4]-3901*c[5]+14492*c[5]*c[4]+1 1670*c[5]^2*c[4]^2+3112*c[4]^2), b[6] = 640000000/346599*(9990*c[5]*c[ 4]-3995*c[5]-3995*c[4]+1997)/(400*c[5]-389)/(400*c[4]-389), b[1] = 1/9 331332*(2310466*c[5]*c[4]-457622*c[5]-457622*c[4]+151051)/c[5]/c[4], a [7,6] = -53973/243125*(5*c[5]*c[4]+1-2*c[4]-2*c[5])*(2000*c[4]-1999)*( 2000*c[5]-1999)/(1890*c[5]*c[4]-745*c[5]-745*c[4]+367)/(400*c[4]-389)/ (400*c[5]-389), a[7,4] = -1999/32000000000000*(-219931147408400000*c[5 ]^5*c[4]^3+2695157079417760000*c[5]^4*c[4]^3-1108940349600000000*c[5]^ 3*c[4]^6+1627500326596807400*c[5]^3*c[4]^4+1549924837938560000*c[4]^6* c[5]^2+48338010504200000*c[5]^5*c[4]^2-420050649857280000*c[4]^6*c[5]+ 1869642648320000000*c[5]^4*c[4]^5-124355520000000000*c[5]^5*c[4]^5-213 4984890927911600*c[4]^5*c[5]^2+24226419437106770*c[5]^3-72452621882402 22*c[5]^2+295948125440000000*c[5]^5*c[4]^4+670357192643783800*c[4]^5*c [5]-57066395959758263*c[5]*c[4]^2-115171944257470890*c[5]^2*c[4]^3+117 4567254401699500*c[4]^4*c[5]^2-1065276318784324000*c[5]^4*c[4]^2+10265 26641073585990*c[5]^3*c[4]^2-234946905928685360*c[5]^3*c[4]+4608881072 1198737*c[5]^2*c[4]+8452805886280259*c[5]*c[4]+416812823133360000*c[4] ^5*c[5]^3-518701513149893180*c[4]^4*c[5]-3318263068769200000*c[5]^4*c[ 4]^4-2415087396080074*c[4]^2-2124246834026046800*c[5]^3*c[4]^3+4970489 512000000*c[4]^5-19335204201680000*c[5]^4-12319070855264000*c[4]^4+211 142782833440000*c[5]^4*c[4]-114021531800484540*c[5]^2*c[4]^2+217598606 891822520*c[5]*c[4]^3+9763686598577990*c[4]^3-373066560000000000*c[5]^ 4*c[4]^6-399156636480000000*c[4]^7*c[5]^2+99409790240000000*c[5]*c[4]^ 7+373066560000000000*c[5]^3*c[4]^7)/(-18900*c[5]^5*c[4]^3+64150*c[5]^4 *c[4]^3-64150*c[5]^3*c[4]^4+7450*c[5]^5*c[4]^2-7450*c[4]^5*c[5]^2+367* c[5]^3+734*c[5]*c[4]^2-11700*c[5]^2*c[4]^3+37360*c[4]^4*c[5]^2-37360*c [5]^4*c[4]^2+11700*c[5]^3*c[4]^2-1457*c[5]^3*c[4]-734*c[5]^2*c[4]+1890 0*c[4]^5*c[5]^3-6360*c[4]^4*c[5]-745*c[5]^4+745*c[4]^4+6360*c[5]^4*c[4 ]+1457*c[5]*c[4]^3-367*c[4]^3)/(400*c[4]-389)/c[4]^2, a[6,4] = -389/51 200000000*(-1244800000*c[5]^4*c[4]^3-3734400000*c[5]^3*c[4]^4+37344000 00*c[4]^5*c[5]^2-484227200*c[5]^3+353183214*c[5]^2-2421136000*c[4]^5*c [5]+1775903256*c[5]*c[4]^2+97779860*c[5]^2*c[4]^3-7468800000*c[4]^4*c[ 5]^2+1210568000*c[5]^4*c[4]^2-12258780000*c[5]^3*c[4]^2+3590003200*c[5 ]^3*c[4]-1998042484*c[5]^2*c[4]-412047083*c[5]*c[4]+6605138760*c[4]^4* c[5]+117727738*c[4]^2+12967704000*c[5]^3*c[4]^3+5214969010*c[5]^2*c[4] ^2-5524134540*c[5]*c[4]^3-121056800*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+3 0*c[5]^3*c[4]^2-6*c[5]^3*c[4]-2*c[5]^2*c[4])/c[4]^2, `b*`[6] = 1600000 /31509*(-35026887*c[4]-35026887*c[5]-31081100*c[5]^3+57606380*c[5]^2-3 28615640*c[5]*c[4]^2-366153900*c[5]^2*c[4]^3-366153900*c[5]^3*c[4]^2+1 86945500*c[5]^3*c[4]-328615640*c[5]^2*c[4]+188067742*c[5]*c[4]+5760638 0*c[4]^2+233166600*c[5]^3*c[4]^3+609023620*c[5]^2*c[4]^2+186945500*c[5 ]*c[4]^3-31081100*c[4]^3+6991497)/(1167-12492*c[5]*c[4]^2-12492*c[5]^2 *c[4]+3112*c[5]^2-3901*c[4]-3901*c[5]+14492*c[5]*c[4]+11670*c[5]^2*c[4 ]^2+3112*c[4]^2)/(160000*c[5]*c[4]-155600*c[4]-155600*c[5]+151321), a[ 7,3] = 5997/64000000000000*(-498945795600000*c[5]^4*c[4]^3-47937823824 00000*c[5]^3*c[4]^4+9312676334499*c[4]-1496837386800000*c[4]^5*c[5]^2- 18625352668998*c[5]-49704895120000*c[5]^3+62278713205930*c[5]^2+372786 713400000*c[4]^5*c[5]-587045574608220*c[5]*c[4]^2-5198347094969200*c[5 ]^2*c[4]^3+4853592734806600*c[4]^4*c[5]^2+124262237800000*c[5]^4*c[4]^ 2-2285282316916000*c[5]^3*c[4]^2+497807688960000*c[5]^3*c[4]-580243611 240240*c[5]^2*c[4]+142891472616953*c[5]*c[4]+1398999600000000*c[4]^5*c [5]^3-1212654858183300*c[4]^4*c[5]+466333200000000*c[5]^4*c[4]^4-31139 356602965*c[4]^2+5064243239240000*c[5]^3*c[4]^3+2469080309264910*c[5]^ 2*c[4]^2+1272306661710930*c[5]*c[4]^3+24852447560000*c[4]^3)/c[4]^2/(1 0*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)/(1890*c[5]*c[4]-745*c[5]-745*c[4]+367 ), `b*`[1] = 1/23328330*(-5106081584*c[4]-5106081584*c[5]-5875841110*c [5]^3+9681132045*c[5]^2-72431799395*c[5]*c[4]^2-117776474580*c[5]^2*c[ 4]^3-117776474580*c[5]^3*c[4]^2+47437644290*c[5]^3*c[4]-72431799395*c[ 5]^2*c[4]+34458475614*c[5]*c[4]+9681132045*c[4]^2+89442089250*c[5]^3*c [4]^3+166926766315*c[5]^2*c[4]^2+47437644290*c[5]*c[4]^3-5875841110*c[ 4]^3+868337859)/(1167-12492*c[5]*c[4]^2-12492*c[5]^2*c[4]+3112*c[5]^2- 3901*c[4]-3901*c[5]+14492*c[5]*c[4]+11670*c[5]^2*c[4]^2+3112*c[4]^2)/c [5]/c[4], a[8,1] = 1/15552220*(120879224824700*c[5]^5*c[4]^3-114455127 3269800*c[5]^4*c[4]^3+73837176503200*c[5]^3*c[4]^6-795538758840060*c[5 ]^3*c[4]^4-170219375264200*c[4]^6*c[5]^2-24169005252100*c[5]^5*c[4]^2+ 48338010504200*c[4]^6*c[5]-784126910199100*c[5]^4*c[4]^5+1192145684652 00*c[5]^5*c[4]^5+381024680608480*c[4]^5*c[5]^2-12117456225008*c[5]^3+3 624443315778*c[5]^2-198132192985700*c[5]^5*c[4]^4-132207913607460*c[4] ^5*c[5]+25390646869572*c[5]*c[4]^2+21799073945287*c[5]^2*c[4]^3-265332 002745060*c[4]^4*c[5]^2+469504523270740*c[5]^4*c[4]^2-441339990690593* c[5]^3*c[4]^2+109236873243546*c[5]^3*c[4]-20522077276120*c[5]^2*c[4]-4 228517201741*c[5]*c[4]+199269881140420*c[4]^5*c[5]^3+139510599523010*c [4]^4*c[5]+1396933046192320*c[5]^4*c[4]^4+1208147771926*c[4]^2+8879165 10342040*c[5]^3*c[4]^3-2384635812840*c[4]^5+9667602100840*c[5]^4+60106 33534870*c[4]^4-99203395096400*c[5]^4*c[4]+43734935739990*c[5]^2*c[4]^ 2-76215531491776*c[5]*c[4]^3-4834145493956*c[4]^3+119214568465200*c[5] ^4*c[4]^6)/c[5]/c[4]^2/(7552660*c[5]^4*c[4]^3+7552660*c[5]^3*c[4]^4+29 7644*c[5]^3-146593*c[5]^2+879558*c[5]*c[4]^2+14926846*c[5]^2*c[4]^3-29 76440*c[4]^4*c[5]^2-2976440*c[5]^4*c[4]^2+14926846*c[5]^3*c[4]^2-25411 30*c[5]^3*c[4]+879558*c[5]^2*c[4]+146593*c[5]*c[4]-146593*c[4]^2-28610 860*c[5]^3*c[4]^3-7214252*c[5]^2*c[4]^2-2541130*c[5]*c[4]^3+297644*c[4 ]^3), a[8,5] = -1/10*(196602416000000*c[5]^5*c[4]^3-663112859111700*c[ 5]^4*c[4]^3-746904825451500*c[5]^3*c[4]^4-100382200000000*c[5]^5*c[4]^ 2-210160982720000*c[5]^4*c[4]^5+604073885963*c[4]+62177760000000*c[5]^ 5*c[4]^5-134143230992100*c[4]^5*c[5]^2-1208147771926*c[5]-110474902112 00*c[5]^3+6092718847126*c[5]^2-180813080000000*c[5]^5*c[4]^4+241690052 52100*c[4]^5*c[5]-31375124742493*c[5]*c[4]^2-410942046662140*c[5]^2*c[ 4]^3+384483491898690*c[4]^4*c[5]^2+338919310843900*c[5]^4*c[4]^2-41290 0935576950*c[5]^3*c[4]^2+104458010814400*c[5]^3*c[4]-53646955314097*c[ 5]^2*c[4]+8440427637734*c[5]*c[4]+257957448460000*c[4]^5*c[5]^3-660355 41416570*c[4]^4*c[5]+610462272875800*c[5]^4*c[4]^4-2417072746978*c[4]^ 2+808437791965250*c[5]^3*c[4]^3+8649407136000*c[5]^4+24901592000000*c[ 5]^5*c[4]-1192317906420*c[4]^4-84757149024000*c[5]^4*c[4]+208156022222 521*c[5]^2*c[4]^2+66009381041155*c[5]*c[4]^3+3005316767435*c[4]^3-2486 488000000*c[5]^5)/c[5]/(-46313892487260*c[5]^5*c[4]^3-6042128000000*c[ 5]^7*c[4]^3+16195109370620*c[5]^4*c[4]^3-45515103220220*c[5]^3*c[4]^4+ 29601602334440*c[5]^5*c[4]^2-14296228416000*c[5]^4*c[4]^5+604212800000 0*c[5]^5*c[4]^5-16637108544000*c[5]^6*c[4]^2-2314512484840*c[4]^5*c[5] ^2+113992329323*c[5]^3-20507536000000*c[5]^5*c[4]^4+227984658646*c[5]* c[4]^2-4669172997942*c[5]^2*c[4]^3+15854281532906*c[4]^4*c[5]^2-189795 96912106*c[5]^4*c[4]^2+3868419871542*c[5]^3*c[4]^2+10027506146*c[5]^3* c[4]-227984658646*c[5]^2*c[4]+10568663239260*c[4]^5*c[5]^3-24455738148 30*c[4]^4*c[5]+52382337792000*c[5]^4*c[4]^4+7837848467200*c[5]^3*c[4]^ 3-462716365284*c[5]^4-4474417888000*c[5]^5*c[4]+231451248484*c[4]^4+32 422612416000*c[5]^6*c[4]^3+2659489366830*c[5]^4*c[4]-462530233600*c[5] ^2*c[4]^2+683767844254*c[5]*c[4]^3-113992329323*c[4]^3-238115200000*c[ 5]^6+586837574400*c[5]^5+2032904000000*c[5]^6*c[4]+2381152000000*c[5]^ 7*c[4]^2), a[9,7] = -2000000000000/161919*(1890*c[5]*c[4]-745*c[5]-745 *c[4]+367)/(4000000*c[5]*c[4]-3998000*c[4]-3998000*c[5]+3996001), a[9, 5] = -1/12*(457622*c[4]-151051)/c[5]/(800000*c[5]^3*c[4]-2377600*c[5]^ 2*c[4]+2355211*c[5]*c[4]-777611*c[4]-800000*c[5]^4+2377600*c[5]^3-2355 211*c[5]^2+777611*c[5]), a[9,8] = 1/132*(755266*c[5]*c[4]-297644*c[5]- 297644*c[4]+146593)/(-c[4]+c[5]*c[4]+1-c[5]), a[9,1] = 1/9331332*(2310 466*c[5]*c[4]-457622*c[5]-457622*c[4]+151051)/c[5]/c[4], a[9,4] = 1/12 *(457622*c[5]-151051)/c[4]/(-2377600*c[4]^3+800000*c[4]^4+2355211*c[4] ^2-777611*c[4]-800000*c[5]*c[4]^3+2377600*c[5]*c[4]^2-2355211*c[5]*c[4 ]+777611*c[5])\}:" }{TEXT -1 0 "" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{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 5293 "prin_err_norm_sqrd := (u,v,w)->1/188116992000000000 00000*(33627047181367785600000*v^9*w^3-85246269012869075528000*w^5*v^7 -919074302415147120120*w^5*v^2+1402237449249346800*w^3*u+7703971405276 230000*w^5*u+54757021535546158710000*w^5*v^8-11553789031632414000*u^2* w^3+13536040917962745000*u^2*w^4-6573522160853316000*w^4*u-15927222850 646648014200*w^4*v^6+7903086010568090163*w^2*v^2+164775057988126856580 *w^5*v+1999469410488610562700*w^6*v^2+1672510482551423740100*w^5*v^4+5 4470810160780727109000*v^6*w^6+17468629230748350500100*w^4*v^5-2228034 1594478849474400*w^5*v^5+3507282311462701272000*w^7*v^5-39397939481148 1290000*w^8*v^5-8224585328415456000000*v^10*w^3+6175355597930775690000 *w^6*v^8-261286946911451589120*w^3*v^4+65027048310488722500*w^8*v^4+99 63463374969422640000*v^8*w^2-9220416940365922598400*w^6*v^3-8118948505 429303182*w^4*v+29062015222980362418*v^4*w-8081093104087848346680*w^4* v^4-257984563123365379678*w^3*v^3+73358053672700272530*w^3*v^2-7200650 763828366642*w^3*v+5400019192881583961460*v^5*w^3-91259044153397849070 *w^2*v^3+537972801055913462376*v^4*w^2-185017984170321222424*w^4*v^2-1 15207352161710811420*v^5*w-2004934173328418136120*v^5*w^2+268275815823 86679025800*w^6*v^4-53582433341257070400000*v^8*w^3+266436434004185564 400*v^6*w+48964719640993136322500*v^8*w^4+12396949890274656000000*v^10 *w^4+1193501539473343380000*w^7*v^7+5020487932249919450700*v^6*w^2-498 83455165126499388000*w^6*v^5+61790040186275845812000*w^5*v^6-128141526 95996998608000*w^4*v^7-1037480025975045408000*w^7*v^4-8755538107107286 064500*v^7*w^2-311985707546003280000*w*v^7+43753560501848430405000*w^3 *v^7-3475813025798178642*w*v^3-248899934786585619600*w^6*v-13005409662 097744500*w^7*v^2+3286761080426658000*v^4*u-2888447257908103500*u^2*v^ 3+3384010229490686250*u^2*v^4+119716562381186400000*w*v^8+136412192051 9064000000*v^10*w^2-5947109578666468800000*v^9*w^2-2086917907672852791 8400*w^3*v^6-4762442802854474010000*w^7*v^6-28619206986291050010000*w^ 6*v^7+3368831482786547225*v^6-2869675440648003870*v^5+6111220691183109 21*v^4-3851985702638115000*v^5*u+616363948944535050*u^2*v^2-7011187246 24673400*v^3*u+2066647861891261817460*w^5*v^3+2465455795778140200*u^2* w^2+2509696254720246921*w^4+13804303288110147225*w^6-11771927634330723 870*w^5+1913698403411185274880*w^4*v^3+156828336934882725000*w^7*v^3-3 3749667550538843400000*v^8*w^3*u-5948137603586178000000*u^2*v^8*w^3-98 88208901301528000000*w^5*v^7*u-77041280266894083600000*w^4*v^7*u+55862 080567447678200000*v^7*w^3*u-39465992077655472000000*u^2*v^7*w^4+78647 7680982509022000*w^4*u*v^3+30360548500450442505000*u^2*v^4*w^4-9896573 736746202240000*u^2*v^3*w^4+986552460375394500000*u^2*v^8*w^2-24398858 1425818320000*u^2*v*w^4-2406841045241976000000*u^2*v^4*w^5+18698791467 3612381000*u^2*v*w^3-1431480244885239150000*u^2*v^2*w^3+25498633245675 6537000*u^2*v^4*w+475739540324244000000*u^2*v^3*w^5+187084380646619010 450*u^2*v^2*w^2-5300784599279328810000*u^2*v^5*w^2+2029617501231686310 000*u^2*v^2*w^4+6857533300938485445000*u^2*v^6*w^2+6194011632506460000 000*u^2*v^5*w^5-301218664783440915000*u^2*v^5*w-800922048062654511000* u^2*v^3*w^2+6586455421157738553000*u^2*v^3*w^3-58710950456553183600000 *u^2*v^5*w^4+67280059465413492600000*u^2*v^6*w^4-431915518484673111000 00*u^2*v^6*w^3+25721005267883809800000*u^2*v^7*w^3+3742940622730835421 0000*u^2*v^5*w^3+2472031439604048294000*u^2*v^4*w^2+260784018180000000 000*u^2*v^6*w^6-7577076142560960000000*u^2*v^6*w^5+3058168433720400000 000*u^2*v^7*w^5-39601676200968000000*u^2*v^2*w^5+23458076548393151700* v^2*u^2*w+3407806103990288400000*w^4*v^4*u-29585469549337682400*v*u^2* w^2+8965651259930778000000*u^2*v^8*w^4-19546756293072604680000*u^2*v^4 *w^3+24383646850801565700000*w^3*v^5*u+58259561672172065400000*w^4*v^6 *u-173823393744000000000*u^2*v^5*w^6+28965130588800000000*u^2*v^4*w^6- 4568811969652160400*u*v*w^2-4301034605999856900000*u^2*v^7*w^2-3268865 99261936955600*u*v^3*w^2-321013670693147316000*v^5*w*u+135446506031088 967500*u*v^4*w+54361811187570771000*u*v^2*w^2-31349863799086404600*u*v ^3*w+1302526704098203866000*u*v^4*w^2-649160929797192000000*w^6*v^4*u- 1598536092424320000000*w^6*v^6*u-22756579740839347200000*w^4*v^5*u+333 6084071763090300*u*v^2*w+497821441546800000000*w^6*v^7*u+1542667188841 755246000*u*v^3*w^3+1510642547548020000000*w^6*v^5*u-14855704164415048 7700*u*v^2*w^3+7930850138114904000000*v^9*w^3*u+880153450419852000000* w^5*v^8*u-1315403280500526000000*v^9*w^2*u-104370990241432792500*u^2*v ^3*w-3723051357617482602000*v^5*w^2*u+372230274765007620000*v^6*w*u-11 269550027736000000*w^6*u*v^2-142809878077788600000*v^7*w*u-78587901400 01021208000*w^3*u*v^4-1986299278933089600*w^3*u*v-92787833028742752600 00*v^7*w^2*u+135405568289628000000*w^6*u*v^3-470280952517430780000*w^4 *u*v^2+89008051713728274000*w^4*v*u-5239205620205985360000*w^5*u*v^3+1 125983256571931940000*w^5*u*v^2-138880734366210480000*w^5*u*v+74756397 46129534980000*v^6*w^2*u-11954201679907704000000*v^9*w^4*u+14917322707 064519670000*w^5*v^4*u+115559571079143450000*u^2*v^6*w-477378358926476 74650000*w^3*v^6*u-25655593203456422400000*w^5*v^5*u+24295142157546488 400000*w^5*v^6*u-2465455795778140200*u^2*v*w+49333780382381964000000*w ^4*v^8*u+5734712807999809200000*v^8*w^2*u+596750769736671690000*w^8*v^ 6-13398839601146496000000*w^5*v^9-42881221049428608000000*v^9*w^4)/(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] = 3/50;" "6#/&%\"cG6#\"\"#*&\"\"$ \"\"\"\"#]!\"\"" }{TEXT -1 3 ", " }{XPPEDIT 18 0 "c[4] = 1439/10000; " "6#/&%\"cG6#\"\"%*&\"%R9\"\"\"\"&++\"!\"\"" }{TEXT -1 7 " and " } {XPPEDIT 18 0 "c[5] = 4973/10000;" "6#/&%\"cG6#\"\"&*&\"%t\\\"\"\"\"&+ +\"!\"\"" }{TEXT -1 125 " in Verner's \"most efficient scheme\" give \+ a value for the (square of the) principal error norm that is close to \+ the minimum." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 124 "Using a one dimensional minimization procedure and cycling aro und the nodes gives very slow convergence towards the minimum." }} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 501 "Digits := 30:\nc_2 := 3/50: c_4 := 1439/10000: c_5 := 4973/10000:\nfor ct to 200000 do\n c_2 := findmin(prin_err_norm_sqrd(c2,c_4,c_5),c2=\{0.05,c_2,0.13\},convergen ce=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 mn := findmin(prin_err _norm_sqrd(c_2,c_4,c5),c5=\{0.5,c_5,0.63\},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 := 1 0:" }}{PARA 11 "" 1 "" {XPPMATH 20 "6%/&%\"cG6#\"\"#$\"?yT,Y!eT,<(4?!#T" } }{PARA 11 "" 1 "" {XPPMATH 20 "6%/&%\"cG6#\"\"#$\"?Y5)f5SOilRybO:)f!#J /&F%6#\"\"%$\"?B\"zy&)fBe*f!yO'*4W\"!#I/&F%6#\"\"&$\"?d[e\\O'oP>$pv=#[ (\\F1" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#$\"?>6m)e_0#HOi]jP0?!#T" }} {PARA 11 "" 1 "" {XPPMATH 20 "6%/&%\"cG6#\"\"#$\"?GHr^nUn,8$=F8%*)f!#J /&F%6#\"\"%$\"?iI1v)=*[PH/7kyT9!#I/&F%6#\"\"&$\"?<\"f$33!#T" }}{PARA 260 "" 0 "" {TEXT -1 1 ":" }}{PARA 260 "" 0 "" {TEXT -1 1 ":" }}{PARA 11 "" 1 "" {XPPMATH 20 "6%/&%\"cG6#\"\"#$\"?%yQ%\\O0k\\\"R(3?!#T" }}{PARA 11 " " 1 "" {XPPMATH 20 "6%/&%\"cG6#\"\"#$\"?lm2MiU5]\"R(3?I+__aiQu8JWz\\F1" }}{PARA 11 " " 1 "" {XPPMATH 20 "6#$\"?!G5\\h'e]#)HZc&4M*>!#T" }}{PARA 11 "" 1 "" {XPPMATH 20 "6%/&%\"cG6#\"\"#$\"?NrpU>3]]\"R(3?)H3n4?L_ZY9!#I/&F%6#\"\"&$\"?V$Gj6))yD'Qu8JWz\\F1" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#$\"?28yI(31D)HZc&4M*>!#T" }}{PARA 11 "" 1 "" {XPPMATH 20 "6%/&%\"cG6#\"\"#$\"?=FGt#>S3:R(3?!#T" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 11 "" 1 "" {TEXT -1 0 "" }}}{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 0 383 "c_2 := .60361720087e-1: pp := .19934095565e-11:\n p1 := evalf[30](plot(prin_err_norm_sqrd(c[2],.14464752332,.49794431137 ),c[2]=0.055..0.0657,\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=[0.055..0.0657,1.992e-12..2.0052e-12]);" }}{PARA 13 "" 1 "" {GLPLOT2D 367 347 347 {PLOTDATA 2 "6*-%'CURVESG6$7S7$$\"#b! 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++++]7=JhxnfF1$\"?y#[uZ'Hg$*>F-7$$\"?nmmmmmmmTg(z/2*fF1$\"?eX>\" R-ov--'oX\\$*>F-7$$\"?nmmmmmm;a$4Zg<,'F1$\"?95.pziM.h[iSV$*>F-7$$\"?LL LLLLLLe9f=ZMgF1$\"?\"fJ6n\")4u!RKv'4M*>F-7$$\"?LLLLLLL$3x&3E1egF1$\"?Q z\\$e.)y>o#GEHM*>F-7$$\"?+++++++]iID%)fygF1$\"?3:%Rb23-\\%eyN[$*>F-7$$ \"?LLLLLLLL3(*pwx+hF1$\"?$yK_$)o$y#zm1?\"e$*>F-7$$\"?+++++++++&***4pBh F1$\"?ZtK.\")y*3.SJaCP*>F-7$$\"?++++++++v3Bs5YhF1$\"?J6'oLung3QF -7$$\"?+++++++]PfrgznhF1$\"?;&fCA/F(4s,7?7%*>F-7$$\"?++++++++D,Sy(=>'F 1$\"?;'[ig&fy4\"*HglS%*>F-7$$\"?nmmmmmmmm,!Q;N@'F1$\"?Bzui:<]-kdBHq%*> F-7$$\"?++++++++]7S,iOiF1$\"?S/Zq_!4u_qU'=1&*>F-7$$\"?LLLLLLL$3x1ebvD' F1$\"?'y3K#pw!oU=E.Da*>F-7$$\"?++++++++](paV/G'F1$\"?../*zgi%==D>L'e*> F-7$$\"?nmmmmmm;/cR$z>I'F1$\"?wW[+J,,ddPb]J'*>F-7$$\"?+++++++](o![/\\C jF1$\"?S,qIFY@V*G_-Go*>F-7$$\"?nmmmmmmm;Hr\"*\\YjF1$\"?BZlZq\\Q)p/0%)p t*>F-7$$\"?+++++++]P%[ES&pjF1$\"?EtTj6s!zo*fc(zz*>F-7$$\"?MLLLLLLLLVw: t\"R'F1$\"?sZ!yT='*oIJjX3')*>F-7$$\"?MLLLLLLLep(RDWT'F1$\"?imV3baxJGG 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0087e-1,c[4],.49794431137),c[4]=0.144641..0.144654,\n color=COLOR(R GB,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],font=[HELVETICA,9],labels=[`c[4]`,`principa l error x 1000`],\n view=[0.144641..0.144654,1.992e-9..2.0043 e-9]);" }}{PARA 13 "" 1 "" {GLPLOT2D 520 423 423 {PLOTDATA 2 "6*-%'CUR VESG6$7S7$$\"'TY9!\"'$\"?8lk\"epiY$yl(QwV+#!#Q7$$\"?nmmmmmmT+jLGTY9!#I $\"?ys0QS?l\\)oiNWM+#F-7$$\"?LLLLLLe*Qc\"*H:kW\"F1$\"?n%H()**\\v98Ki1n E+#F-7$$\"?nmmmmm;H')*=2=kW\"F1$\"?s&\\oj;Bc*)p![.$=+#F-7$$\"?nmmmmm;/ /-j3UY9F1$\"?@wm!3zLP\"Hh*4G5+#F-7$$\"?LLLLLLekk(3kBkW\"F1$\"?gI)RDeSe X4q_p-+#F-7$$\"?nmmmmm\"Hi/j@EkW\"F1$\"?V]5P`#\\NT,,w,'**>F-7$$\"?++++ ++D18,$))GkW\"F1$\"?S#paV!eHsX#*[j%*)*>F-7$$\"?nmmmmm\"H]A(Q22$)*>F-7$$\"?++++++D\"Q!***QMkW\"F1$\"?#>k`*z\"ps@v8')3x* >F-7$$\"?LLLLLLL3RuF-7$$\"?nmmmmm;a2V3(Rk W\"F1$\"?)ysWszq32A!3@m'*>F-7$$\"?+++++++DUN7DWY9F1$\"?%3='GP*3k^\"oT* 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0 "> " 0 "" {MPLTEXT 1 0 386 "c_5 := .49794431137: pp := \+ .19934095565e-11:\np1 := evalf[30](plot(prin_err_norm_sqrd(.6036172008 7e-1,.14464752332,c[5]),c[5]=0.4979387..0.49795,\n color=COLOR(RGB ,0.6,.2,.2))):\np2 := plot([[[c_5,pp]]$4],style=point,symbol=[circle$2 ,diamond,cross],symbolsize=[12,10$3],color=[black,green$3]):\nplots[di splay]([p1,p2],font=[HELVETICA,9],view=[0.4979387..0.49795,1.992e-12.. 2.0047e-12]);" }}{PARA 13 "" 1 "" {GLPLOT2D 389 359 359 {PLOTDATA 2 "6 *-%'CURVESG6$7S7$$\"((Qz\\!\"($\"?!HGkxdA#**y.?F-7$$\"?nmmmm;z%y*=1;Rz\\F1$\"?puw:Y\"> +%3lwr(H+#F-7$$\"?LLLLLLeaEM;SRz\\F1$\"?bsY<@$R+t7ot-@+#F-7$$\"?LLLLLL 3#erCW'Rz\\F1$\"?%4_%Q\"y)yp`Z;[E,?F-7$$\"?nmmmm;HU&pq&))Rz\\F1$\"?\\2 \\U!47llc>.t/+#F-7$$\"?LLLLL$e9%4r&4,%z\\F1$\"?Hle#oiw1n2T^w(**>F-7$$ \"?+++++]7VWp8MSz\\F1$\"?c\"))zK#4]Zcp,M4**>F-7$$\"?LLLLL$ekHc4\"eSz\\ F1$\"?]Hm3Lhy\\sO\"pF%)*>F-7$$\"?+++++]i+,`+#3%z\\F1$\"?\">!fh<8vKsen` !y*>F-7$$\"?nmmmmm;u/]e1Tz\\F1$\"?rK\"\\noS)[9Y,#3s*>F-7$$\"?LLLLLL3<@ 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E98D8DD7533ED1F7D3D6C97F933ED1F7C2F5D875ED3ED1F7B77BF42F893ED1F7B0F7F7 B2913ED1F7AF8A2CF9B7-%&COLORG6&%$RGBG$\"\"!F<$\"#()!\"#F=-%%FONTG6$%*H ELVETICAG\"\"*-%+AXESLABELSG6%%%c[4]G%%c[5]G%!G-%*AXESSTYLEG6#%$BOXG-% +PROJECTIONG6%$!#jF<$\"#bF<\"\"\"" 1 2 0 1 10 0 2 1 1 2 2 1.000000 55.000000 -63.000000 1 0 "Curve 1" }}}}{PARA 0 "" 0 "" {TEXT -1 0 "" } }{PARA 0 "" 0 "" {TEXT -1 36 "#-----------------------------------" }} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 151 "nds := [c[2]=.60361720087e- 1,c[4]=.14464752332,c[5]=.49794431137]:\nevalf[10](%);\nfor dgt from 6 by -1 to 4 do\n map(convert,nds,rational,dgt);\nend do;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#7%/&%\"cG6#\"\"#$\"+4? " 0 "" {MPLTEXT 1 0 89 "evalf[25](prin_err_norm_sqrd (.60361720087e-1,.14464752332,.49794431137)):\nevalf(sqrt(%));" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#$\"+m:)=T\"!#:" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 9 "Setting " }{XPPEDIT 18 0 "c[2] = 7/116;" "6#/&%\"cG6#\"\"#*&\"\"(\"\"\"\"$;\"!\"\"" }{TEXT -1 7 " and " }{XPPEDIT 18 0 "c[7] = 121/243;" "6#/&%\"cG6#\"\"(*&\"$@\" \"\"\"\"$V#!\"\"" }{TEXT -1 46 " the principal error norm is a minimu m when " }{XPPEDIT 18 0 "c[4] = 104/719;" "6#/&%\"cG6#\"\"%*&\"$/\"\" \"\"\"$>(!\"\"" }{TEXT -1 19 " (approximately). " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 114 "mn := evalf [20](findmin(prin_err_norm_sqrd(7/116,c4,121/243),c4=0.144..0.145)):\n c[4]=mn[1];\nconvert(%,rational,6);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6 #/&%\"cG6#\"\"%$\"5.wg$[+`_kW\"!#?" }}{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] = 7/116;" "6#/&%\"c G6#\"\"#*&\"\"(\"\"\"\"$;\"!\"\"" }{TEXT -1 3 ", " }{XPPEDIT 18 0 "c[ 4] = 104/719;" "6#/&%\"cG6#\"\"%*&\"$/\"\"\"\"\"$>(!\"\"" }{TEXT -1 7 " and " }{XPPEDIT 18 0 "c[5] = 121/243;" "6#/&%\"cG6#\"\"&*&\"$@\"\" \"\"\"$V#!\"\"" }{TEXT -1 67 ", the principal error norm is given (ap proximately) as follows. " }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 69 "evalf[15](prin_err_norm_sqrd(7/116,104/719,121/243)):\nevalf(sqrt( %));" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#$\"+rB)=T\"!#:" }}}{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 2487 "ee := \{c[2]=7/116,\nc[3]=208/2157,\nc[4]=104/719,\nc[5]=121/243 ,\nc[6]=389/400,\nc[7]=1999/2000,\nc[8]=1,\nc[9]=1,\n\na[2,1]=7/116,\n a[3,1]=631280/32568543,\na[3,2]=2509312/32568543,\na[4,1]=26/719,\na[4 ,2]=0,\na[4,3]=78/719,\na[5,1]=405298835029/310395556224,\na[5,2]=0,\n a[5,3]=-516775016989/103465185408,\na[5,4]=649792660033/155197778112, \na[6,1]=-613196378874450690363694258375297/14810689674481071256371200 000000,\na[6,2]=0,\na[6,3]=19314590304297630584070330403179/1224023940 03975795507200000000,\na[6,4]=-456903226810742698201959542362145743/37 77766287341706964636467200000000,\na[6,5]=2334686840125010976300994198 92501/42262363236718430355123200000000,\na[7,1]=-690994313677727892638 163353214849278889337/12834810518108703660371044352000000000000,\na[7, 2]=0,\na[7,3]=55902701968117856366108228295253679031/27268075629626088 6366208000000000000,\na[7,4]=-3148696705456455444222969039901524702647 03005657/2003745885533513610150342699619328000000000000,\na[7,5]=39432 8813849822371988541658040946895351456539/56456885797940553200650756061 648000000000000,\na[7,6]=-1841588041032613773/98979602990157900500,\na [8,1]=-41346042665166483923424449659815720863/764692984974231010310450 749101004672,\na[8,2]=0,\na[8,3]=1673297023243723294266377481831/81271 69928568684375229008512,\na[8,4]=-503725499129961091532189480038365151 123989843/3191911395383323244189300496793324868133824,\na[8,5]=2212220 45246087052146775273050936702042442/3155117321590387516161080573358640 3142833,\na[8,6]=-1199089843519744000000/65355131597330942592057,\na[8 ,7]=-203964353120000000000/398405317752764522861589,\na[9,1]=314492473 /9032729376,\na[9,2]=0,\na[9,3]=0,\na[9,4]=11996228882183665169/463754 17381391048160,\na[9,5]=17231867943154059861/40981954688105953544,\na[ 9,6]=16704872960000000/3806497022432643,\na[9,7]=-370696000000000000/2 109493746887301,\na[9,8]=1699767001/9903960,\n\nb[1]=314492473/9032729 376,\nb[2]=0,\nb[3]=0,\nb[4]=11996228882183665169/46375417381391048160 ,\nb[5]=17231867943154059861/40981954688105953544,\nb[6]=1670487296000 0000/3806497022432643,\nb[7]=-370696000000000000/2109493746887301,\nb[ 8]=1699767001/9903960,\n\n`b*`[1]=3569968140173481166883/9391240902156 1276605888,\n`b*`[2]=0,\n`b*`[3]=0,\n`b*`[4]=4041173703102787736964015 8198393/160720235061275249022962867655360,\n`b*`[5]=141303235747840573 47070883452389/32775799845926002570575445690544,\n`b*`[6]=182635910422 8575029348000000/507381787386105911677401003,\n`b*`[7]=-38533860125250 630597500000000/281182068833322476588641821,\n`b*`[8]=5883828582995007 071/44004491070454772,\n`b*`[9]=1/80\}:" }}}{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$\"\"'\"\")" }{TEXT -1 127 " denote the vector who se components are the principal error terms of the 8 stage, order 6 sc heme (the error terms of order 7)." }}{PARA 0 "" 0 "" {TEXT -1 5 "Let \+ " }{XPPEDIT 18 0 "`T*`[5,9];" "6#&%#T*G6$\"\"&\"\"*" }{TEXT -1 145 " \+ denote the vector whose components are the principal error terms of t he embedded 9 stage, order 5 scheme (the error terms of order 6) and l et " }{XPPEDIT 18 0 "`T*`[6,9];" "6#&%#T*G6$\"\"'\"\"*" }{TEXT -1 99 " denote the vector whose components are the error terms of order 7 o f the embedded order 5 scheme." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }} {PARA 0 "" 0 "" {TEXT -1 39 "Denote the 2-norm of these vectors by " }{XPPEDIT 18 0 "abs(abs(T[6,8]));" "6#-%$absG6#-F$6#&%\"TG6$\"\"'\"\") " }{TEXT -1 3 ", " }{XPPEDIT 18 0 "abs(abs(`T*`[5,9]));" "6#-%$absG6# -F$6#&%#T*G6$\"\"&\"\"*" }{TEXT -1 7 " and " }{XPPEDIT 18 0 "abs(abs (`T*`[6,9]));" "6#-%$absG6#-F$6#&%#T*G6$\"\"'\"\"*" }{TEXT -1 15 " re spectively." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 8 "Define: " }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "A[7 ] = abs(abs(T[6, 8]));" "6#/&%\"AG6#\"\"(-%$absG6#-F)6#&%\"TG6$\"\"'\" \")" }{TEXT -1 6 ", " }{XPPEDIT 18 0 "B[7] = abs(abs(`T*`[6,9]))/a bs(abs(`T*`[5,9]));" "6#/&%\"BG6#\"\"(*&-%$absG6#-F*6#&%#T*G6$\"\"'\" \"*\"\"\"-F*6#-F*6#&F/6$\"\"&F2!\"\"" }{TEXT -1 6 ", " }{XPPEDIT 18 0 "C[7] = abs(abs(`T*`[6,9]-T[6, 8]))/abs(abs(`T*`[5, 9]));" "6#/&% \"CG6#\"\"(*&-%$absG6#-F*6#,&&%#T*G6$\"\"'\"\"*\"\"\"&%\"TG6$F2\"\")! \"\"F4-F*6#-F*6#&F06$\"\"&F3F9" }{TEXT -1 1 "." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 77 "Dormand and Prince have s uggested that as well as attempting to ensure that " }{XPPEDIT 18 0 " A[7];" "6#&%\"AG6#\"\"(" }{TEXT -1 73 " is a minimum, if the embedded scheme is to be used for error control, " }{XPPEDIT 18 0 "B[7];" "6# &%\"BG6#\"\"(" }{TEXT -1 7 " and " }{XPPEDIT 18 0 "C[7];" "6#&%\"CG6 #\"\"(" }{TEXT -1 27 " should be chosen so that " }}{PARA 0 "" 0 "" {TEXT -1 68 "they are \"similar in magnitude\" and also not differ too much from 1." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 186 "errterms6_8 := PrincipalErrorTerms(6,8,'expande d'):\n`errterms6_9*` :=subs(b=`b*`,PrincipalErrorTerms(6,9,'expanded') ):\n`errterms5_9*` := subs(b=`b*`,PrincipalErrorTerms(5,9,'expanded')) :" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 326 "snmB := sqrt(add(evalf(subs(ee,`errterms6_9*`[i]))^2 ,i=1..nops(`errterms6_9*`))):\nsdnB := sqrt(add(evalf(subs(ee,`errterm s5_9*`[i]))^2,i=1..nops(`errterms5_9*`))):\nsnmC := sqrt(add((evalf(su bs(ee,`errterms6_9*`[i])-subs(ee,errterms6_8[i])))^2,i=1..nops(errterm s6_8))):\n'B[7]'= evalf[8](snmB/sdnB);\n'C[7]'= evalf[8](snmC/sdnB);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/&%\"BG6#\"\"($\")q3A " 0 "" {MPLTEXT 1 0 1 ";" }}} }{PARA 0 "" 0 "" {TEXT -1 52 "#--------------------------------------- ------------" }}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 26 "absolute stabilit y regions" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 35 "coefficient s of the combined scheme" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 2487 "ee := \{c[2]=7/116,\nc[3]=208/215 7,\nc[4]=104/719,\nc[5]=121/243,\nc[6]=389/400,\nc[7]=1999/2000,\nc[8] =1,\nc[9]=1,\n\na[2,1]=7/116,\na[3,1]=631280/32568543,\na[3,2]=2509312 /32568543,\na[4,1]=26/719,\na[4,2]=0,\na[4,3]=78/719,\na[5,1]=40529883 5029/310395556224,\na[5,2]=0,\na[5,3]=-516775016989/103465185408,\na[5 ,4]=649792660033/155197778112,\na[6,1]=-613196378874450690363694258375 297/14810689674481071256371200000000,\na[6,2]=0,\na[6,3]=1931459030429 7630584070330403179/122402394003975795507200000000,\na[6,4]=-456903226 810742698201959542362145743/3777766287341706964636467200000000,\na[6,5 ]=233468684012501097630099419892501/42262363236718430355123200000000, \na[7,1]=-690994313677727892638163353214849278889337/12834810518108703 660371044352000000000000,\na[7,2]=0,\na[7,3]=5590270196811785636610822 8295253679031/272680756296260886366208000000000000,\na[7,4]=-314869670 545645544422296903990152470264703005657/200374588553351361015034269961 9328000000000000,\na[7,5]=39432881384982237198854165804094689535145653 9/56456885797940553200650756061648000000000000,\na[7,6]=-1841588041032 613773/98979602990157900500,\na[8,1]=-41346042665166483923424449659815 720863/764692984974231010310450749101004672,\na[8,2]=0,\na[8,3]=167329 7023243723294266377481831/8127169928568684375229008512,\na[8,4]=-50372 5499129961091532189480038365151123989843/31919113953833232441893004967 93324868133824,\na[8,5]=221222045246087052146775273050936702042442/315 51173215903875161610805733586403142833,\na[8,6]=-119908984351974400000 0/65355131597330942592057,\na[8,7]=-203964353120000000000/398405317752 764522861589,\na[9,1]=314492473/9032729376,\na[9,2]=0,\na[9,3]=0,\na[9 ,4]=11996228882183665169/46375417381391048160,\na[9,5]=172318679431540 59861/40981954688105953544,\na[9,6]=16704872960000000/3806497022432643 ,\na[9,7]=-370696000000000000/2109493746887301,\na[9,8]=1699767001/990 3960,\n\nb[1]=314492473/9032729376,\nb[2]=0,\nb[3]=0,\nb[4]=1199622888 2183665169/46375417381391048160,\nb[5]=17231867943154059861/4098195468 8105953544,\nb[6]=16704872960000000/3806497022432643,\nb[7]=-370696000 000000000/2109493746887301,\nb[8]=1699767001/9903960,\n\n`b*`[1]=35699 68140173481166883/93912409021561276605888,\n`b*`[2]=0,\n`b*`[3]=0,\n`b *`[4]=40411737031027877369640158198393/1607202350612752490229628676553 60,\n`b*`[5]=14130323574784057347070883452389/327757998459260025705754 45690544,\n`b*`[6]=1826359104228575029348000000/5073817873861059116774 01003,\n`b*`[7]=-38533860125250630597500000000/28118206883332247658864 1821,\n`b*`[8]=5883828582995007071/44004491070454772,\n`b*`[9]=1/80\}: " }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 " " {TEXT -1 77 "The stability function R for the 8 stage, order 6 schem e is given as follows." }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 75 "su bs(ee,StabilityFunction(6,8,'expanded')):\nR := unapply(%,z):\n'R(z)'= R(z);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/-%\"RG6#%\"zG,4\"\"\"F)F'F)* &#F)\"\"#F)*$)F'F,F)F)F)*&#F)\"\"'F)*$)F'\"\"$F)F)F)*&#F)\"#CF)*$)F'\" \"%F)F)F)*&#F)\"$?\"F)*$)F'\"\"&F)F)F)*&#F)\"$?(F)*$)F'F1F)F)F)*&#\"6. .*oCC*G(o1E\":?b6&4cLb?iK;8F)*$)F'\"\"(F)F)F)*&#\"48fk9neYLu\"\"8+3;iI )R4V:T\"*F)*$)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 th e stability region intersects the negative real axis by solving the eq uation: " }}{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=-4.8);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%#z0G$!+5f*z%[!\"*" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" } }{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 303 "z0 := newton(R(z)=1,z=-4.8 ):\np1 := plot([R(z),1],z=-5.39..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=[-5.39..0.49,-.07..1.47],font=[HELVETICA,9]) ;" }}{PARA 13 "" 1 "" {GLPLOT2D 360 253 253 {PLOTDATA 2 "6+-%'CURVESG6 $7Z7$$!3o*************Q&!#<$\"3Grb19kg_FF*7$$!3i**\\P$=ezN&F*$\"3P1X<( [80g#F*7$$!3e***\\nO;fK&F*$\"3ETynP=%eX#F*7$$!3_**\\7]X(QH&F*$\"3wh\"[ V_z#=BF*7$$!3[****\\LF$=E&F*$\"3?zkPwc_(=#F*7$$!3r*\\io!R21_F*$\"3=jUw 8bOv>F*7$$!3'***\\A!3:.:&F*$\"3KqJr`c\\\"y\"F*7$$!3!**\\P![%3w3&F*$\"3 ?-c3)eEOe\"F*7$$!3%)***\\e\"=!\\-&F*$\"3>#f@c&*e`S\"F*7$$!3s****RB&z<' \\F*$\"3--<(H72SC\"F*7$$!3g***\\4Bd')*[F*$\"3E63p:*G#*4\"F*7$$!3L*\\Px %\\$e$[F*$\"3#p`aP4!Q,(*!#=7$$!3'***\\_kE,tZF*$\"39[a?+HpY&)F^o7$$!3%* **\\#y1Cll%F*$\"3b-Vh&yOZs'F^o7$$!3k**\\Ato!f`%F*$\"3XP(3*Gfu7_F^o7$$! 3$***\\-)Rk6T%F*$\"3A9L'ptP)zRF^o7$$!3%***\\_t>#oG%F*$\"31`CJ+-ICIF^o7 $$!3W****H10#*eTF*$\"3c)owjN/;F#F^o7$$!3o***\\V#[EYSF*$\"3a6#f]()**[w \"F^o7$$!3Z*****)37W>RF*$\"3y)3e;+C\\L\"F^o7$$!3a****\\Vo4#z$F*$\"3m7/ +tMbA5F^o7$$!3Q****4^pPpOF*$\"3GnpMC0<1\")!#>7$$!3o**\\-Kb$zb$F*$\"3Q9 Qn#G'[!y'Far7$$!3P*****e!4UDMF*$\"3g,f4mXG(y&Far7$$!3g****fNO;8LF*$\"3 X!QJ*o6!zL&Far7$$!3m**\\UVGF*$\"3 z)p)4o$))H7'Far7$$!3W***\\ii;Mp#F*$\"3*zT#o\"36!=oFar7$$!3u**\\#R\")3x d#F*$\"3q8xf\"4!\\%f(Far7$$!3)****\\nI-HX#F*$\"3Mlxz_vMw&)Far7$$!3p** \\-AMEBBF*$\"3%=>'e)fH\"e(*Far7$$!3#)**\\x\"R7/@#F*$\"3EJ)=a:?H4\"F^o7 $$!3_***\\u=I&)3#F*$\"3\\\\#>9:*eN7F^o7$$!39++?WRhi>F*$\"3XE`[,P^-9F^o 7$$!3()***\\cYH%R=F*$\"3C7X%evate\"F^o7$$!3p**\\i[@C?hG)*F ^o$\"3jEHtJgDT#F^o$\"3K?y'oh/k&yF^o7$$!3Q%****>;%4w7F^o$\"30TtUF5(>!)) 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" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1389 "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 \+ 260668728924246890303/1316326220553356095115520*z^7+\n 1743346 586714645913/91411543093983062160800*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,.1,.4,.48)):\np2 := plots[polygonplot]([seq([pts[i-1], pts[i],[-2.4,0]],i=2..nops(pts))],\n style=patchnogrid,color= COLOR(RGB,.2,.8,.95)):\npts := []: z0 := 1.9+4.7*I:\nfor ct from 0 to \+ 50 do\n zz := newton(R(z)=exp(ct*Pi/25*I),z=z0):\n z0 := zz:\n p ts := [op(pts),[Re(zz),Im(zz)]]:\nend do:\np3 := plot(pts,color=COLOR( RGB,.1,.4,.48)):\np4 := plots[polygonplot]([seq([pts[i-1],pts[i],[1.79 ,4.66]],i=2..nops(pts))],\n style=patchnogrid,color=COLOR(RGB ,.2,.8,.95)):\npts := []: z0 := 1.9-4.7*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,.1,.4 ,.48)):\np6 := plots[polygonplot]([seq([pts[i-1],pts[i],[1.79,-4.66]], i=2..nops(pts))],\n style=patchnogrid,color=COLOR(RGB,.2,.8,. 95)):\np7 := plot([[[-5.39,0],[2.29,0]],[[0,-5.19],[0,5.19]]],color=bl ack,linestyle=3):\nplots[display]([p||(1..7)],view=[-5.39..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^******f:^S]!#F$\"3\")*****>l#fTJF-7$$\"3%******4Ix6G\"!#D$\"3v**** *z#*)Q7ZF-7$$\"39+++G(RNE\"!#C$\"3S+++BZ=$G'F-7$$\"3!)*****HY'Q%R(F=$ \"3g*****>kxR&yF-7$$\"3))*****R7G%*4$!#B$\"3#******pDeZU*F-7$$\"3#**** ***ybeF5!#A$\"3)******\\r\\&*4\"!#<7$$\"3')*****R;K^&GFN$\"31+++'o6mD \"FQ7$$\"3e+++dHq))oFN$\"3++++;Qk89FQ7$$\"3++++]d=v9!#@$\"3#******Hv31 d\"FQ7$$\"3)******fDlz$GFin$\"36+++f^VF6>L5 !#?$\"3)******z>*Q&>#FQ7$$\"3-+++sXPG6F^p$\"3#*******3YX\\BFQ7$$\"31++ +$>*owqFin$\"3&)*****H$pp,DFQ7$$!33+++'ezW(zFin$\"3=+++77T^EFQ7$$!3<++ +1v+2UF^p$\"33+++Tf\"yz#FQ7$$!31+++C([c0\"!#>$\"3A+++Fy4SHFQ7$$!3')*** **pR;*)4#Fhq$\"3++++0LVxIFQ7$$!3M+++Yq/fOFhq$\"3++++9&y*3KFQ7$$!3()*** ***HYx>eFhq$\"3#*******[O&QL$FQ7$$!3G+++AoEB')Fhq$\"31+++&zo6X$FQ7$$!3 1+++`]717F-$\"3'******>$*=,c$FQ7$$!3'*******4M$yg\"F-$\"3;+++$o?,m$FQ7 $$!31+++B]oe?F-$\"3;+++k&>4v$FQ7$$!3?+++/On[DF-$\"3()*****\\V5E$QFQ7$$ !3!******H3a#oIF-$\"34+++^7%FQ7$$!3Y+++dVW%)eF-$\"3O+++Q83gTF Q7$$!3a*****>l(*fY'F-$\"3S+++,%*)G>%FQ7$$!3%******z1Y&\\qF-$\"3.+++_`r ?UFQ7$$!3n*****4`kVj(F-$\"3$)*****p)4)QC%FQ7$$!3v*****>c$**>#)F-$\"3?+ ++FWliUFQ7$$!3C+++^)oh!))F-$\"3S+++*HdsF%FQ7$$!3I+++dox#R*F-$\"3i***** REryG%FQ7$$!3C+++=%H)z**F-$\"3G+++/Pk%H%FQ7$$!3'******RDXn0\"FQ$\"3%)* ****fj#p(H%FQ7$$!3%******pi)e:6FQ$\"3q*****>&46(H%FQ7$$!3++++v$\\X<\"F Q$\"3g*****\\?qHH%FQ7$$!3++++TynL7FQ$\"35+++#=C`G%FQ7$$!3!******H*3/$H \"FQ$\"31+++WD@uUFQ7$$!3++++K^s_8FQ$\"3=+++[\\mfUFQ7$$!3,+++`2%GT\"FQ$ \"31+++6hqTUFQ7$$!31+++Le_t9FQ$\"3(******f[i.A%FQ7$$!3$******\\c]\\`\" FQ$\"3W+++R8n&>%FQ7$$!3/+++c/K(f\"FQ$\"3%******>;$pnTFQ7$$!3/+++9n(3m \"FQ$\"3<+++D#Gl8%FQ7$$!3,+++V4*es\"FQ$\"3k*****f\\PB5%FQ7$$!3-+++)3]E z\"FQ$\"3m*****pXm`1%FQ7$$!3'******\\@F9'=FQ$\"3u******es'f-%FQ7$$!3#* *****z!eVK>FQ$\"3)******H>5Y)RFQ7$$!3)******4yrd+#FQ$\"3\")*****f!>(=% RFQ7$$!3A+++%*zN\"3#FQ$\"3!)*****z&)*Q)*QFQ7$$!3-+++lZ\"*e@FQ$\"3/+++v XyaQFQ7$$!38+++Mk(zB#FQ$\"3:+++a+d6QFQ7$$!31+++$fazJ#FQ$\"3=+++#\\%3pP FQ7$$!37+++y^A)R#FQ$\"3&)******RbYFPFQ7$$!3%)*****\\(y?yCFQ$\"3?+++'\\ rmo$FQ7$$!3++++LoTdDFQ$\"3'******pWIlk$FQ7$$!3$)*****\\Tzaj#FQ$\"3'*** ***p<&z1OFQ7$$!3++++W:87FFQ$\"3;+++l*)=nNFQ7$$!3#******p(4?(y#FQ$\"3\" )******yqVFNFQ7$$!3'******z:(egGFQ$\"3))*****z0&G([$FQ7$$!3!)*****z^UA $HFQ$\"3%******RG2lW$FQ7$$!3)******>U4\\S$FQ7$$!3?+ ++$HZ.2$FQ$\"33+++^uKiLFQ7$$!3&******4sYo8$FQ$\"3>+++0^i=LFQ7$$!3#)*** **p6,!GIF Q7$$!3!)*****>>q&fNFQ$\"3#******fS4W(HFQ7$$!3')******)R7Yh$FQ$\"3\"*** ***>D#G>HFQ7$$!3-+++/\">&oOFQ$\"3%)*****>8aE'GFQ7$$!3*)*****H$fN@PFQ$ \"3\")*****praX!GFQ7$$!3A+++]^=tPFQ$\"3&******HDF]u#FQ7$$!3&)*****pujS #QFQ$\"35+++Q%GTo#FQ7$$!31+++/7/uQFQ$\"3()*****f*y#>i#FQ7$$!3*******H3 dJ#RFQ$\"3/+++!32&eDFQ7$$!3#******zkQ9(RFQ$\"3)******pYdR\\#FQ7$$!3&** ****>y)*)=SFQ$\"31+++2!y$GCFQ7$$!3#)*****z]Mb1%FQ$\"31+++**=(=O#FQ7$$! 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" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 260 "We can distort the bound ary curve horizontally by taking the 11th root of the real part of poi nts along the curve. In this way we see that the largest interval on t he nonnegative imaginary axis that contains the origin and lies inside the stability region is " }{XPPEDIT 18 0 "[0, 2.6];" "6#7$\"\"!-%&Fl oatG6$\"#E!\"\"" }{TEXT -1 18 " approximately. " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 423 "R := z -> 1 +z+1/2*z^2+1/6*z^3+1/24*z^4+1/120*z^5+1/720*z^6+\n 260668728924246 890303/1316326220553356095115520*z^7+\n 1743346586714645913/914115 43093983062160800*z^8:\nDigits := 25:\npts := []: z0 := 0:\nfor ct fro m 0 to 90 do\n zz := newton(R(z)=exp(ct*Pi/100*I),z=z0):\n z0 := z z:\n pts := [op(pts),[surd(Re(zz),11),Im(zz)]]:\nend do:\nplot(pts,c olor=COLOR(RGB,0,.75,.95),thickness=2,font=[HELVETICA,9]);\nDigits := \+ 10:" }}{PARA 13 "" 1 "" {GLPLOT2D 261 276 276 {PLOTDATA 2 "6(-%'CURVES G6#7gp7$$\"\"!F)F(7$$\":c@QWJO>&38bxE!#E$\":[fhm?z*e`EfTJF-7$$\":7%=ts =ZnH?jKWF-$\":p@@&HL%zrI&=$G'F-7$$\":*>*\\yz]H$Q;t_fF-$\":[]U16gKD=[.?CI')F-$ \":$R#F?7$$\":01u6<<\\eA]W@\"F?$\": y7\")z.0;D7uK^#F?7$$\":w%=$=sB#\\U'QHK\"F?$\":r#)>S2J^tQLu#GF?7$$\":K5 RH9v;a7R\"G9F?$\":(\\\"*=G`:f^EfTJF?7$$\":`H\"p(>%))3JtXI:F?$\":bQ8zna 5Z\">vbMF?7$$\":#QQAntz=:[@I;F?$\":3*Qy_k'>e<6*pPF?7$$\":o/)*)[**>)\\v rws\"F?$\":!eN!GNXvMVqS3%F?7$$\":&3hnhG+#[BUI#=F?$\":&oI!yAK5aoH#)R%F? 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" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 32 "z_0 := new ton(`R*`(z)=1,z=-5.5);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%$z_0G$!+@! \\,^&!\"*" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 " " {MPLTEXT 1 0 320 "z_0 := newton(`R*`(z)=1,z=-5.5):\np_1 := plot([`R* `(z),1],z=-6.09..0.49,color=[red,blue]):\np_2 := plot([[[z_0,1]]$3],st yle=point,symbol=[circle,cross,diamond],color=black):\np_3 := plot([[z _0,0],[z_0,1]],linestyle=3,color=COLOR(RGB,0,.5,0)):\nplots[display]([ p_1,p_2,p_3],view=[-6.09..0.49,-0.07..1.47],font=[HELVETICA,9]);" }} {PARA 13 "" 1 "" {GLPLOT2D 398 284 284 {PLOTDATA 2 "6+-%'CURVESG6$7Z7$ $!3')*************3'!#<$\"3By#)f=v\\OHF*7$$!3#H$eR\"oVT0'F*$\"3I#**G_? V(fFF*7$$!3)fm\"zitG=gF*$\"3'\\j&owy@#f#F*7$$!3%**\\(=W5V#)fF*$\"3m]=G zE^LCF*7$$!3+LLeDZdYfF*$\"3gpoP]PB$G#F*7$$!3+]PRVz<%)eF*$\"3t:4*)z.pS? 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Ficl-%%FONTG6#%(DEFAULTG-F\\dl6$%*HELVETICAG\"\"*-%%VIEWG6$;$!$4'!\"#$ \"#\\Fidl;$!\"(Fidl$\"$Z\"Fidl" 1 2 0 1 10 0 2 9 1 4 2 1.000000 45.000000 45.000000 0 0 "Curve 1" "Curve 2" "Curve 3" "Curve 4" "Curve 5" "Curve 6" }}}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 81 "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 1582 "`R*` := z -> 1+z+1/2*z^2+1/6*z^3+ 1/24*z^4+1/120*z^5+\n 2289050183374887275340375255789143/161961132 8004340558993590293015500800*z^6+\n 264852150155595143186701687622 749/1315934204003526704182292113075094400*z^7+\n 56147093659987971 637029344062633/3239222656008681117987180586031001600*z^8+\n 17433 46586714645913/7312923447518644972864000*z^9:\npts := []: z0 := 0:\nfo r ct from 0 to 300 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_1 := \+ plot(pts,color=COLOR(RGB,0,.33,.4)):\np_2 := plots[polygonplot]([seq([ pts[i-1],pts[i],[-2.75,0]],i=2..nops(pts))],\n style=patchnog rid,color=COLOR(RGB,0,.65,.8)):\npts := []: z0 := 1.8+4.7*I:\nfor ct f rom 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,.33,.4)):\np_4 := plots[polygonplot]([seq([pts[i-1] ,pts[i],[1.72,4.63]],i=2..nops(pts))],\n style=patchnogrid,co lor=COLOR(RGB,0,.65,.8)):\npts := []: z0 := 1.8-4.7*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,0,.33,.4)):\np_6 := plots[polygonplot]([seq([pts[i-1],pts[i ],[1.72,-4.63]],i=2..nops(pts))],\n style=patchnogrid,color=C OLOR(RGB,0,.65,.8)):\np_7 := plot([[[-6.19,0],[2.29,0]],[[0,-5.19],[0, 5.19]]],color=black,linestyle=3):\nplots[display]([p_||(1..7)],view=[- 6.19..2.29,-5.19..5.19],font=[HELVETICA,9],\n labels=[`Re(z )`,`Im(z)`],axes=boxed,scaling=constrained);" }}{PARA 13 "" 1 "" {GLPLOT2D 407 552 552 {PLOTDATA 2 "6/-%'CURVESG6$7i]l7$$\"\"!F)F(7$$\" 3W*****zH?Qh*!#G$\"3%******41PmD\"!#=7$$\"3K+++<0DHh!#E$\"3#*******3TF 8DF07$$\"3j*****fV#3Mp!#D$\"3u*****p%4\"*pPF07$$\"3#*******\\FUbQ!#C$ \"3.+++\\kaE]F07$$\"3'******zQ,([9!#B$\"3Q+++=m<$G'F07$$\"3q******3UuN UFF$\"3()*****\\r!zRvF07$$\"3'******4@!zP5!#A$\"3i*****4ckjz)F07$$\"35 +++E)[SA#FQ$\"3'******p*\\G05!#<7$$\"3-+++f*R%yUFQ$\"36+++:f\"48\"FY7$ $\"3#)*****pE1<](FQ$\"35+++#H9lD\"FY7$$\"31+++87f27!#@$\"3)******4>c?Q \"FY7$$\"3-+++\"pKOy\"Fao$\"3++++B]]2:FY7$$\"3'*******R!>%*Q#Fao$\"3.+ ++ys!Gj\"FY7$$\"33+++J*oi!GFao$\"31+++Ty)yv\"FY7$$\"31+++Kqb/EFao$\"3/ +++:ak#)=FY7$$\"3(*******G[GP5Fao$\"3')*****4w[p+#FY7$$!33+++&GF#)3$Fa o$\"3*)*****4kM18#FY7$$!33+++\\?'\\:\"!#?$\"35+++*>3ND#FY7$$!3)******* f,+%o#Feq$\"3#)******\\fMvBFY7$$!35+++$\\PiA&Feq$\"37+++b/!f\\#FY7$$!3 C*****\\8hd>*Feq$\"3(******HV/\\h#FY7$$!3'******z.%Q3:!#>$\"3A+++o52KF 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"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 "coefficients of the combined sche me" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 2487 "ee := \{c[2]=7/116,\nc[3]=208/2157,\nc[4]=104/719, \nc[5]=121/243,\nc[6]=389/400,\nc[7]=1999/2000,\nc[8]=1,\nc[9]=1,\n\na [2,1]=7/116,\na[3,1]=631280/32568543,\na[3,2]=2509312/32568543,\na[4,1 ]=26/719,\na[4,2]=0,\na[4,3]=78/719,\na[5,1]=405298835029/310395556224 ,\na[5,2]=0,\na[5,3]=-516775016989/103465185408,\na[5,4]=649792660033/ 155197778112,\na[6,1]=-613196378874450690363694258375297/1481068967448 1071256371200000000,\na[6,2]=0,\na[6,3]=193145903042976305840703304031 79/122402394003975795507200000000,\na[6,4]=-45690322681074269820195954 2362145743/3777766287341706964636467200000000,\na[6,5]=233468684012501 097630099419892501/42262363236718430355123200000000,\na[7,1]=-69099431 3677727892638163353214849278889337/12834810518108703660371044352000000 000000,\na[7,2]=0,\na[7,3]=55902701968117856366108228295253679031/2726 80756296260886366208000000000000,\na[7,4]=-314869670545645544422296903 990152470264703005657/2003745885533513610150342699619328000000000000, \na[7,5]=394328813849822371988541658040946895351456539/564568857979405 53200650756061648000000000000,\na[7,6]=-1841588041032613773/9897960299 0157900500,\na[8,1]=-41346042665166483923424449659815720863/7646929849 74231010310450749101004672,\na[8,2]=0,\na[8,3]=16732970232437232942663 77481831/8127169928568684375229008512,\na[8,4]=-5037254991299610915321 89480038365151123989843/3191911395383323244189300496793324868133824,\n a[8,5]=221222045246087052146775273050936702042442/31551173215903875161 610805733586403142833,\na[8,6]=-1199089843519744000000/653551315973309 42592057,\na[8,7]=-203964353120000000000/398405317752764522861589,\na[ 9,1]=314492473/9032729376,\na[9,2]=0,\na[9,3]=0,\na[9,4]=1199622888218 3665169/46375417381391048160,\na[9,5]=17231867943154059861/40981954688 105953544,\na[9,6]=16704872960000000/3806497022432643,\na[9,7]=-370696 000000000000/2109493746887301,\na[9,8]=1699767001/9903960,\n\nb[1]=314 492473/9032729376,\nb[2]=0,\nb[3]=0,\nb[4]=11996228882183665169/463754 17381391048160,\nb[5]=17231867943154059861/40981954688105953544,\nb[6] =16704872960000000/3806497022432643,\nb[7]=-370696000000000000/2109493 746887301,\nb[8]=1699767001/9903960,\n\n`b*`[1]=3569968140173481166883 /93912409021561276605888,\n`b*`[2]=0,\n`b*`[3]=0,\n`b*`[4]=40411737031 027877369640158198393/160720235061275249022962867655360,\n`b*`[5]=1413 0323574784057347070883452389/32775799845926002570575445690544,\n`b*`[6 ]=1826359104228575029348000000/507381787386105911677401003,\n`b*`[7]=- 38533860125250630597500000000/281182068833322476588641821,\n`b*`[8]=58 83828582995007071/44004491070454772,\n`b*`[9]=1/80\}:" }}}{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]=s ubs(ee,c[i]),i=2..9);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6*/&%\"cG6#\"\" ##\"\"(\"$;\"/&F%6#\"\"$#\"$3#\"%d@/&F%6#\"\"%#\"$/\"\"$>(/&F%6#\"\"&# \"$@\"\"$V#/&F%6#\"\"'#\"$*Q\"$+%/&F%6#F)#\"%**>\"%+?/&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 2 ": " }}{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(#\"'!GJ'\")V&oD$/&F %6$F/F'#\"(7$4DF2/&F%6$\"\"%F(#\"#E\"$>(/&F%6$F;F'\"\"!/&F%6$F;F/#\"#y F>/&F%6$\"\"&F(#\"-H]$))H0%\"-Cib&R5$/&F%6$FKF'FB/&F%6$FKF/#!-*)p,vn^ \"-3a=lM5/&F%6$FKF;#\"-L+m#z\\'\"-7\"yx>b\"/&F%6$\"\"'F(#!B(Hv$eUpj.p] W()yj>8'\"A++++7Pc72\"[u'*o5[\"/&F%6$F[oF'FB/&F%6$F[oF/#\"AzJSI.2%eIwH /.f9$>\"?++++s]&zvR+%R-C7/&F%6$F[oF;#!EVd9iBaf>?)pU2\"oA.pX\"C++++sYOY 'pqTtGmxx$/&F%6$F[oFK#\"B,D*)>%*4Iw4,D,%ooMB\"A++++K7b.V=nBjBEU/&F%6$F *F(#!KP$*))y#\\[@`L;QE*ysxOJ%*4p\"J++++++_V/r.m.(3\"=0\"[$G\"/&F%6$F*F 'FB/&F%6$F*F/#\"GJ!zOD&HG#3hOcy6o>q-f&\"E++++++3iO')3E'Hc2os#/&F%6$F*F ;#!Qdc+.ZEqC:!*R!pHAWaXcaq'p[J\"O++++++G$>'*pU.:5O^Lb)eu.?/&F%6$F*FK# \"NRlX^`*o%4/e;a))>PA)\\Q\")GVR\"M++++++[;1c2l+KbSzz&)oXc/&F%6$F*F[o#! 4tPhK5/)eT=\"5+0!z:!*Hgz*)*/&F%6$\"\")F(#!Gj3s:)f'\\WUBR[m^mUgMT\"EsY+ ,\"\\2X5.,JU(\\)Hpk(/&F%6$FhrF'FB/&F%6$FhrF/#\"@J=[xjE%HBPCBqHt;\"=7&3 !H_P%oo&G*pr7)/&F%6$FhrF;#!NV)*)R7^^OQ+[*=K:4h*H\"*\\DP]\"LCQ8o[K$z'\\ +$*=WKK$Q&R6>>$/&F%6$FhrFK#\"KUC/-n$40t_xY@0(3Y_/A7A\"JLG9.keLd!3hh^(Q !f@t6bJ/&F%6$FhrF[o#!7+++W(>N%)*3*>\"\"8d?fU4L(fJ^Nl/&F%6$FhrF*#!6++++ +7`V'R?\"9*ehG_kFv<`S)R/&F%6$\"\"*F(#\"*tC\\9$\"+w$HF.*/&F%6$F`uF'FB/& F%6$F`uF/FB/&F%6$F`uF;#\"5p^m$=#))Gi*>\"\"5g\"[5R\"Q)4%/&F%6$F`uF[o#\"2+++gH([q;\"1VEVAq\\1Q/&F%6$ F`uF*#!3++++++'pq$\"1,t)ou$\\4@/&F%6$F`uFhr#\"+,qw*p\"\"(gR!**" }}} {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT 259 38 "weights \+ for the 8 stage order 6 scheme" }{TEXT -1 2 ": " }}{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#\"\"\"#\"*tC\\9$\"+w$HF.*/&F%6#\"\"#\" \"!/&F%6#\"\"$F//&F%6#\"\"%#\"5p^m$=#))Gi*>\"\"5g\"[5R\"Q)4%/&F%6#\"\"'#\"2+++gH([q;\"1VEVAq\\1Q/ &F%6#\"\"(#!3++++++'pq$\"1,t)ou$\\4@/&F%6#\"\")#\"+,qw*p\"\"(gR!**" }} }{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT 259 38 "weights for the 9 stage order 5 scheme" }{TEXT -1 2 ": " }}{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#\"\"\"#\"7$)o;\"[t,9o*pN\"8)) egw7c@!4C\"R*/&F%6#\"\"#\"\"!/&F%6#\"\"$F//&F%6#\"\"%#\"A$R)>e,kpt(y-J qt6/%\"Bg`lnG'H-\\_Fh]B?2;/&F%6#\"\"&#\"A*Q_M)32Zt0%yuNKIT\"\"AW0pXadq D+Ef%)*zvF$/&F%6#\"\"'#\"=+++[$H]dGU5fj#=\"<.5Sx;\"f5'Q(y\"Q2&/&F%6#\" \"(#!>++++vfI1DD,'Q`Q\"<@=k)ewCKL)o?=\"G/&F%6#\"\")#\"4rq+&*HeGQ)e\"2s ZXq5\\/S%/&F%6#\"\"*#F'\"#!)" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}}{PARA 0 "" 0 "" {TEXT -1 52 "#---------------------------------------------------" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}}{PARA 0 "" 0 "" {TEXT -1 45 " #============================================" }}{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 condition s etc." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 770 "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*`)]:\n\nerrterms6_8 := Pri ncipalErrorTerms(6,8,'expanded'):\n`errterms6_9*` :=subs(b=`b*`,Princi palErrorTerms(6,9,'expanded')):\n`errterms5_9*` := subs(b=`b*`,Princip alErrorTerms(5,9,'expanded')):" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}{SECT 1 {PARA 4 "" 0 "" {TEXT 0 13 "calc_RKcoeffs" }{TEXT -1 1 " " }}{PARA 0 " " 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1985 "cal c_RKcoeffs := proc()\n local eqns,sm,ct,Rz,stb6,stb5,nmB,snmB,dnB,sd nB,nmC,snmC,B_7,C_7,nrm;\n global e1,e2,e3;\n\n e1 := \{c[2]=c_2,c [4]=c_4,c[5]=c_5,c[6]=c_6,c[7]=c_7,c[8]=1,c[9]=1,seq(a[i,2]=0,i=4..8), \n b[2]=0,b[3]=0,`b*`[2]=0,`b*`[3]=0,`b*`[bs[1]]=bs[2]\};\n \+ eqns := subs(e1,cdns):\n e2 := solve(\{op(eqns)\});\n e3 := `unio n`(e1,e2);\n Digits := 14;\n sm := 0;\n for ct to nops(errterms6 _8) do\n sm := sm+(evalf(subs(e3,errterms6_8[ct])))^2;\n end do ;\n Rz := subs(e3,StabilityFunction(6,8,'expanded'));\n stb6 := ma x(fsolve(Rz=1,z=-9..-1e-7),fsolve(Rz=-1,z=-9..-1e-7));\n stb6 := eva lf[8](stb6);\n Rz := subs(e3,subs(b=`b*`,StabilityFunction(5,9,'expa nded')));\n stb5 := evalf[8](max(fsolve(Rz=1,z=-9..-1e-7),fsolve(Rz= -1,z=-9..-1e-7)));\n stb5 := evalf[8](stb5);\n nmB := 0;\n for c t to nops(`errterms6_9*`) do\n nmB := nmB+evalf(subs(e3,`errterms 6_9*`[ct]))^2;\n end do:\n snmB := sqrt(nmB);\n dnB := 0;\n fo r ct to nops(`errterms5_9*`) do\n dnB := dnB+evalf(subs(e3,`errte rms5_9*`[ct]))^2;\n end do;\n sdnB := sqrt(dnB);\n nmC := 0;\n \+ for ct to nops(errterms6_8) do\n nmC := nmC+(evalf(subs(e3,`errt erms6_9*`[ct]))-evalf(subs(e3,errterms6_8[ct])))^2;\n end do;\n sn mC := 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 nrm := eva lf(max(seq(seq(subs(e3,abs(a[i,j])),j=1..i-1),i=2..9)));\n print(inf inity*`-norm of linking coeffs`=evalf[10](nrm));\n print(`2-norm of \+ principal error of order 6 scheme` = evalf[10](sqrt(sm)));\n print(` 2-norm of principal error of order 5 scheme` = evalf[10](sdnB));\n p rint(`order 6 stability interval` = [stb6,0]);\n print(`order 5 stab ility interval` = [stb5,0]);\n print('B[7]'=B_7,'C[7]'=C_7);\nend pr oc:" }}}{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 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 := 2 7/40: 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/&%\"bG 6#\"\"\"$\"'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~linking~coeffsGF&$\"+ -0(3`\"!\")" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/%L2-norm~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~intervalG7$$!)4y= W!\"(\"\"!" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/%;order~5~stability~int ervalG7$$!)1#[T%!\"(\"\"!" }}{PARA 11 "" 1 "" {XPPMATH 20 "6$/&%\"BG6# \"\"($\"))o8I\"!\"(/&%\"CGF&$\")1-?8F*" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 41 "#=================================== =====" }}{PARA 0 "" 0 "" {TEXT -1 18 "Papakostas' scheme" }}{PARA 0 " " 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 108 "c_2 \+ := 17/183: c_4 := 18/83: c_5 := 71/125: c_6 := 42/59: c_7 := 199/200: \+ bs := [9,-1/150]:\ncalc_RKcoeffs();" }}{PARA 11 "" 1 "" {XPPMATH 20 "6 )%'nodes:G/&%\"cG6#\"\"##\"#<\"$$=/&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#\"\")$!'db8FE" }}{PARA 11 "" 1 "" {XPPMATH 20 "6*%1order~5~weights:G/&%#b*G6#\"\"\"$\"'\")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 54 "#---------------------------------------- -------------" }}{PARA 0 "" 0 "" {TEXT -1 42 "a minor modification of \+ Papakostas' scheme" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 110 "c_2 : = 19/205: c_4 := 50/231: c_5 := 173/305: c_6 := 42/59: c_7 := 199/200: bs := [9,-1/150]:\ncalc_RKcoeffs();" }}{PARA 11 "" 1 "" {XPPMATH 20 " 6)%'nodes:G/&%\"cG6#\"\"##\"#>\"$0#/&F&6#\"\"$#\"$+\"\"$$p/&F&6#\"\"%# \"#]\"$J#/&F&6#\"\"&#\"$t\"\"$0$/&F&6#\"\"'#\"#U\"#f/&F&6#\"\"(#\"$*> \"$+#" }}{PARA 11 "" 1 "" {XPPMATH 20 "6)%1order~6~weights:G/&%\"bG6# \"\"\"$\"'Z3k!\"(/&F&6#\"\"%$\"'KBL!\"'/&F&6#\"\"&$\"')4o#F2/&F&6#\"\" '$\"'%=!=F2/&F&6#\"\"($\"'!R^\"!\"&/&F&6#\"\")$!'ge8FE" }}{PARA 11 "" 1 "" {XPPMATH 20 "6*%1order~5~weights:G/&%#b*G6#\"\"\"$\"'')3i!\"(/&F& 6#\"\"%$\"')*)R$!\"'/&F&6#\"\"&$\"'>bBF2/&F&6#\"\"'$\"'&*=AF2/&F&6#\" \"($\"')Q9\"!\"&/&F&6#\"\")$!'=m**F2/&F&6#\"\"*$!'nmm!\")" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/*&%)infinityG\"\"\"%8-norm~of~linking~coeffsGF &$\"+ee.a=!\")" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/%L2-norm~of~princip al~error~of~order~6~schemeG$\"+SP%)G7!#9" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/%L2-norm~of~principal~error~of~order~5~schemeG$\"+&pY?'e!#8" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#/%;order~6~stability~intervalG7$$!),B] W!\"(\"\"!" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/%;order~5~stability~int ervalG7$$!)%[XX%!\"(\"\"!" }}{PARA 11 "" 1 "" {XPPMATH 20 "6$/&%\"BG6# \"\"($\")gGh8!\"(/&%\"CGF&$\")lqi8F*" }}}{PARA 0 "" 0 "" {TEXT -1 0 " " }}{PARA 0 "" 0 "" {TEXT -1 54 "#------------------------------------ -----------------" }}{PARA 0 "" 0 "" {TEXT -1 46 "a 2nd minor modifica tion of Papakostas' scheme" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 111 "c_2 := 19/205: c_4 := 121/559: c_5 := 173/305: c_6 := 42/59: c_7 \+ := 199/200: bs := [9,-1/152]:\ncalc_RKcoeffs();" }}{PARA 11 "" 1 "" {XPPMATH 20 "6)%'nodes:G/&%\"cG6#\"\"##\"#>\"$0#/&F&6#\"\"$#\"$U#\"%x; /&F&6#\"\"%#\"$@\"\"$f&/&F&6#\"\"&#\"$t\"\"$0$/&F&6#\"\"'#\"#U\"#f/&F& 6#\"\"(#\"$*>\"$+#" }}{PARA 11 "" 1 "" {XPPMATH 20 "6)%1order~6~weight s:G/&%\"bG6#\"\"\"$\"'z3k!\"(/&F&6#\"\"%$\"'RBL!\"'/&F&6#\"\"&$\"'n!o# F2/&F&6#\"\"'$\"'8-=F2/&F&6#\"\"($\"'u8:!\"&/&F&6#\"\")$!'Xe8FE" }} {PARA 11 "" 1 "" {XPPMATH 20 "6*%1order~5~weights:G/&%#b*G6#\"\"\"$\"' :3i!\"(/&F&6#\"\"%$\"'W*R$!\"'/&F&6#\"\"&$\"'?`BF2/&F&6#\"\"'$\"'Q@AF2 /&F&6#\"\"($\"'#=9\"!\"&/&F&6#\"\")$!'BZ**F2/&F&6#\"\"*$!'&*yl!\")" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#/*&%)infinityG\"\"\"%8-norm~of~linking ~coeffsGF&$\"+2eHl=!\")" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/%L2-norm~o f~principal~error~of~order~6~schemeG$\"+th5E7!#9" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/%L2-norm~of~principal~error~of~order~5~schemeG$\"+w8p& )e!#8" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/%;order~6~stability~interval G7$$!)Ws[W!\"(\"\"!" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/%;order~5~stab ility~intervalG7$$!)g=_W!\"(\"\"!" }}{PARA 11 "" 1 "" {XPPMATH 20 "6$/ &%\"BG6#\"\"($\")ofg8!\"(/&%\"CGF&$\")L#HO\"F*" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 54 "#------------------------ -----------------------------" }}{PARA 0 "" 0 "" {TEXT -1 46 "a 3rd mi nor modification of Papakostas' scheme" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 111 "c_2 := 17/183: c_4 := 121/559: c_5 := 173/305: c_6 : = 42/59: c_7 := 199/200: bs := [9,-1/152]:\ncalc_RKcoeffs();" }}{PARA 11 "" 1 "" {XPPMATH 20 "6)%'nodes:G/&%\"cG6#\"\"##\"#<\"$$=/&F&6#\"\"$ #\"$U#\"%x;/&F&6#\"\"%#\"$@\"\"$f&/&F&6#\"\"&#\"$t\"\"$0$/&F&6#\"\"'# \"#U\"#f/&F&6#\"\"(#\"$*>\"$+#" }}{PARA 11 "" 1 "" {XPPMATH 20 "6)%1or der~6~weights:G/&%\"bG6#\"\"\"$\"'z3k!\"(/&F&6#\"\"%$\"'RBL!\"'/&F&6# \"\"&$\"'n!o#F2/&F&6#\"\"'$\"'8-=F2/&F&6#\"\"($\"'u8:!\"&/&F&6#\"\")$! 'Xe8FE" }}{PARA 11 "" 1 "" {XPPMATH 20 "6*%1order~5~weights:G/&%#b*G6# \"\"\"$\"':3i!\"(/&F&6#\"\"%$\"'W*R$!\"'/&F&6#\"\"&$\"'?`BF2/&F&6#\"\" '$\"'Q@AF2/&F&6#\"\"($\"'#=9\"!\"&/&F&6#\"\")$!'BZ**F2/&F&6#\"\"*$!'&* yl!\")" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/*&%)infinityG\"\"\"%8-norm~ of~linking~coeffsGF&$\"+2eHl=!\")" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/ %L2-norm~of~principal~error~of~order~6~schemeG$\"+yc5E7!#9" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/%L2-norm~of~principal~error~of~order~5~sche meG$\"+a!yl)e!#8" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/%;order~6~stabili ty~intervalG7$$!)Ws[W!\"(\"\"!" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/%;o rder~5~stability~intervalG7$$!)g=_W!\"(\"\"!" }}{PARA 11 "" 1 "" {XPPMATH 20 "6$/&%\"BG6#\"\"($\")&y0O\"!\"(/&%\"CGF&$\")T\"HO\"F*" }}} {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 41 "#======== ================================" }}{PARA 0 "" 0 "" {TEXT -1 43 "a mod ification of Papakostas' scheme with " }{XPPEDIT 18 0 "c[6]=73/99" "6 #/&%\"cG6#\"\"'*&\"#t\"\"\"\"#**!\"\"" }{TEXT -1 1 " " }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 109 "c_2 := 1/11: c_4 := 157/736: c_5 := 123/ 218: c_6 := 73/99: c_7 := 199/200: bs := [9,-1/845]:\ncalc_RKcoeffs(); " }}{PARA 11 "" 1 "" {XPPMATH 20 "6)%'nodes:G/&%\"cG6#\"\"##\"\"\"\"#6 /&F&6#\"\"$#\"$d\"\"%/6/&F&6#\"\"%#F1\"$O(/&F&6#\"\"&#\"$B\"\"$=#/&F&6 #\"\"'#\"#t\"#**/&F&6#\"\"(#\"$*>\"$+#" }}{PARA 11 "" 1 "" {XPPMATH 20 "6)%1order~6~weights:G/&%\"bG6#\"\"\"$\"'Z>j!\"(/&F&6#\"\"%$\"'DtK! \"'/&F&6#\"\"&$\"'d_HF2/&F&6#\"\"'$\"'eh;F2/&F&6#\"\"($\"'!)*R\"!\"&/& F&6#\"\")$!'t^7FE" }}{PARA 11 "" 1 "" {XPPMATH 20 "6*%1order~5~weights :G/&%#b*G6#\"\"\"$\"'1Bh!\"(/&F&6#\"\"%$\"':XL!\"'/&F&6#\"\"&$\"'2$o#F 2/&F&6#\"\"'$\"'oW?F2/&F&6#\"\"($\"'Qz**F2/&F&6#\"\")$!'w_')F2/&F&6#\" \"*$!'V$=\"!\")" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/*&%)infinityG\"\" \"%8-norm~of~linking~coeffsGF&$\"+=31MI!\")" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/%L2-norm~of~principal~error~of~order~6~schemeG$\"+.;%* G6!#9" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/%L2-norm~of~principal~error~ of~order~5~schemeG$\"+3)o&*>'!#8" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/% ;order~6~stability~intervalG7$$!)UZfW!\"(\"\"!" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/%;order~5~stability~intervalG7$$!)f%RY%!\"(\"\"!" }} {PARA 11 "" 1 "" {XPPMATH 20 "6$/&%\"BG6#\"\"($\")IJ&Q\"!\"(/&%\"CGF&$ \")Z)pQ\"F*" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 40 "#=======================================" }}{PARA 0 "" 0 "" {TEXT -1 15 "a scheme with " }{XPPEDIT 18 0 "c[6]=28/39" "6#/&%\"cG6# \"\"'*&\"#G\"\"\"\"#R!\"\"" }{TEXT -1 7 " and " }{XPPEDIT 18 0 "c[7] =324/325" "6#/&%\"cG6#\"\"(*&\"$C$\"\"\"\"$D$!\"\"" }{TEXT -1 1 " " }} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 111 "c_2 := 30/323: c_4 := 144/6 67: c_5 := 161/284: c_6 := 28/39: c_7 := 324/325: bs := [9,-1/180]:\nc alc_RKcoeffs();" }}{PARA 11 "" 1 "" {XPPMATH 20 "6)%'nodes:G/&%\"cG6# \"\"##\"#I\"$B$/&F&6#\"\"$#\"#'*\"$n'/&F&6#\"\"%#\"$W\"F2/&F&6#\"\"&# \"$h\"\"$%G/&F&6#\"\"'#\"#G\"#R/&F&6#\"\"(#\"$C$\"$D$" }}{PARA 11 "" 1 "" {XPPMATH 20 "6)%1order~6~weights:G/&%\"bG6#\"\"\"$\"'^#R'!\"(/&F& 6#\"\"%$\"'O9L!\"'/&F&6#\"\"&$\"'j[FF2/&F&6#\"\"'$\"'`m " 0 "" {MPLTEXT 1 0 111 "c_2 : = 10/107: c_4 := 163/755: c_5 := 233/411: c_6 := 28/39: c_7 := 324/325 :\nbs := [9,-1/180]:\ncalc_RKcoeffs();" }}{PARA 11 "" 1 "" {XPPMATH 20 "6)%'nodes:G/&%\"cG6#\"\"##\"#5\"$2\"/&F&6#\"\"$#\"$E$\"%lA/&F&6#\" \"%#\"$j\"\"$b(/&F&6#\"\"&#\"$L#\"$6%/&F&6#\"\"'#\"#G\"#R/&F&6#\"\"(# \"$C$\"$D$" }}{PARA 11 "" 1 "" {XPPMATH 20 "6)%1order~6~weights:G/&%\" bG6#\"\"\"$\"'c#R'!\"(/&F&6#\"\"%$\"'R9L!\"'/&F&6#\"\"&$\"'r[FF2/&F&6# \"\"'$\"'Rm'!\"(/& F&6#\"\"%$\"'O!R$!\"'/&F&6#\"\"&$\"'\\KCF2/&F&6#\"\"'$\"'fz@F2/&F&6#\" \"($\"'#\\w\"!\"&/&F&6#\"\")$!'^@;FE/&F&6#\"\"*$!'cbb!\")" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/*&%)infinityG\"\"\"%8-norm~of~linking~coeffsGF &$\"+(=-/1#!\")" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/%L2-norm~of~princi pal~error~of~order~6~schemeG$\"+h4.j6!#9" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/%L2-norm~of~principal~error~of~order~5~schemeG$\"+@K!*=f!#8" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#/%;order~6~stability~intervalG7$$!)etc W!\"(\"\"!" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/%;order~5~stability~int ervalG7$$!)w2mW!\"(\"\"!" }}{PARA 11 "" 1 "" {XPPMATH 20 "6$/&%\"BG6# \"\"($\")z7n8!\"(/&%\"CGF&$\")0&)o8F*" }}}{PARA 0 "" 0 "" {TEXT -1 0 " " }}{PARA 0 "" 0 "" {TEXT -1 39 "#==================================== ==" }}{PARA 0 "" 0 "" {TEXT -1 13 "scheme with " }{XPPEDIT 18 0 "c[6] =29/39" "6#/&%\"cG6#\"\"'*&\"#H\"\"\"\"#R!\"\"" }{TEXT -1 7 " and " }{XPPEDIT 18 0 "c[7]=575/576" "6#/&%\"cG6#\"\"(*&\"$v&\"\"\"\"$w&!\"\" " }{TEXT -1 1 " " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> \+ " 0 "" {MPLTEXT 1 0 112 "c_2 := 21/229: c_4 := 199/935: c_5 := 541/959 : c_6 := 29/39: c_7 := 575/576: bs := [9,-1/1764]:\ncalc_RKcoeffs();" }}{PARA 11 "" 1 "" {XPPMATH 20 "6)%'nodes:G/&%\"cG6#\"\"##\"#@\"$H#/&F &6#\"\"$#\"$)R\"%0G/&F&6#\"\"%#\"$*>\"$N*/&F&6#\"\"&#\"$T&\"$f*/&F&6# \"\"'#\"#H\"#R/&F&6#\"\"(#\"$v&\"$w&" }}{PARA 11 "" 1 "" {XPPMATH 20 " 6)%1order~6~weights:G/&%\"bG6#\"\"\"$\"'e/j!\"(/&F&6#\"\"%$\"'ImK!\"'/ &F&6#\"\"&$\"'X,IF2/&F&6#\"\"'$\"'S_;F2/&F&6#\"\"($\"'oHQ!\"&/&F&6#\" \")$!'u%o$FE" }}{PARA 11 "" 1 "" {XPPMATH 20 "6*%1order~5~weights:G/&% #b*G6#\"\"\"$\"'E3h!\"(/&F&6#\"\"%$\"'hPL!\"'/&F&6#\"\"&$\"'YTFF2/&F&6 #\"\"'$\"'sI?F2/&F&6#\"\"($\"'3nE!\"&/&F&6#\"\")$!'eQDFE/&F&6#\"\"*$!' $*oc!\"*" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/*&%)infinityG\"\"\"%8-nor m~of~linking~coeffsGF&$\"+6az'G$!\")" }}{PARA 11 "" 1 "" {XPPMATH 20 " 6#/%L2-norm~of~principal~error~of~order~6~schemeG$\"+YuaP5!#9" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#/%L2-norm~of~principal~error~of~order~ 5~schemeG$\"+>m\"QI'!#8" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/%;order~6~ stability~intervalG7$$!)>prW!\"(\"\"!" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/%;order~5~stability~intervalG7$$!)+qrW!\"(\"\"!" }}{PARA 11 "" 1 " " {XPPMATH 20 "6$/&%\"BG6#\"\"($\").U#R\"!\"(/&%\"CGF&$\")$[NR\"F*" }} }{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 54 "#------- ----------------------------------------------" }}{PARA 0 "" 0 "" {TEXT -1 21 "another scheme with " }{XPPEDIT 18 0 "c[6]=29/39" "6#/&% \"cG6#\"\"'*&\"#H\"\"\"\"#R!\"\"" }{TEXT -1 7 " and " }{XPPEDIT 18 0 "c[7]=575/576" "6#/&%\"cG6#\"\"(*&\"$v&\"\"\"\"$w&!\"\"" }{TEXT -1 1 " " }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 111 "c_2 := 11/118: c_4 \+ := 63/296: c_5 := 409/725: c_6 := 29/39: c_7 := 575/576: bs := [9,-1/1 750]:\ncalc_RKcoeffs();" }}{PARA 11 "" 1 "" {XPPMATH 20 "6)%'nodes:G/& %\"cG6#\"\"##\"#6\"$=\"/&F&6#\"\"$#\"#@\"$[\"/&F&6#\"\"%#\"#j\"$'H/&F& 6#\"\"&#\"$4%\"$D(/&F&6#\"\"'#\"#H\"#R/&F&6#\"\"(#\"$v&\"$w&" }}{PARA 11 "" 1 "" {XPPMATH 20 "6)%1order~6~weights:G/&%\"bG6#\"\"\"$\"'q/j!\" (/&F&6#\"\"%$\"'NmK!\"'/&F&6#\"\"&$\"'Z,IF2/&F&6#\"\"'$\"'J_;F2/&F&6# \"\"($\"'qHQ!\"&/&F&6#\"\")$!'w%o$FE" }}{PARA 11 "" 1 "" {XPPMATH 20 " 6*%1order~5~weights:G/&%#b*G6#\"\"\"$\"'#*4h!\"(/&F&6#\"\"%$\"'5PL!\"' /&F&6#\"\"&$\"'\\VFF2/&F&6#\"\"'$\"'pF?F2/&F&6#\"\"($\"';wE!\"&/&F&6# \"\")$!'_ZDFE/&F&6#\"\"*$!'H9d!\"*" }}{PARA 11 "" 1 "" {XPPMATH 20 "6# /*&%)infinityG\"\"\"%8-norm~of~linking~coeffsGF&$\"+U@(\\G$!\")" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#/%L2-norm~of~principal~error~of~order~ 6~schemeG$\"+A9pP5!#9" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/%L2-norm~of~ principal~error~of~order~5~schemeG$\"+w%=:E'!#8" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/%;order~6~stability~intervalG7$$!)([=Z%!\"(\"\"!" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#/%;order~5~stability~intervalG7$$!)j(= [%!\"(\"\"!" }}{PARA 11 "" 1 "" {XPPMATH 20 "6$/&%\"BG6#\"\"($\")PS#R \"!\"(/&%\"CGF&$\")q\\$R\"F*" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }} {PARA 0 "" 0 "" {TEXT -1 39 "#======================================" }}{PARA 0 "" 0 "" {TEXT -1 13 "scheme with " }{XPPEDIT 18 0 "c[6]=25/ 33" "6#/&%\"cG6#\"\"'*&\"#D\"\"\"\"#L!\"\"" }{TEXT -1 7 " and " } {XPPEDIT 18 0 "c[7]=1728/1729" "6#/&%\"cG6#\"\"(*&\"%G<\"\"\"\"%H " 0 "" {MPLTEXT 1 0 112 "c_2 : = 55/604: c_4 := 129/611: c_5 := 562/999: c_6 := 25/33: c_7 := 1728/17 29: bs := [9,1/560]:\ncalc_RKcoeffs();" }}{PARA 11 "" 1 "" {XPPMATH 20 "6)%'nodes:G/&%\"cG6#\"\"##\"#b\"$/'/&F&6#\"\"$#\"#')\"$6'/&F&6#\" \"%#\"$H\"F2/&F&6#\"\"&#\"$i&\"$***/&F&6#\"\"'#\"#D\"#L/&F&6#\"\"(#\"% G<\"%H<" }}{PARA 11 "" 1 "" {XPPMATH 20 "6)%1order~6~weights:G/&%\"bG6 #\"\"\"$\"'&QD'!\"(/&F&6#\"\"%$\"'tSK!\"'/&F&6#\"\"&$\"'B6JF2/&F&6#\" \"'$\"'HI;F2/&F&6#\"\"($\"'Lq5!\"%/&F&6#\"\")$!'Tc5FE" }}{PARA 11 "" 1 "" {XPPMATH 20 "6*%1order~5~weights:G/&%#b*G6#\"\"\"$\"'Yng!\"(/&F&6 #\"\"%$\"'@2L!\"'/&F&6#\"\"&$\"'t$)GF2/&F&6#\"\"'$\"'V%)>F2/&F&6#\"\"( $\"'g@s!\"&/&F&6#\"\")$!'f,rFE/&F&6#\"\"*$\"'r&y\"!\")" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/*&%)infinityG\"\"\"%8-norm~of~linking~coeffsGF&$ \"+6EX3U!\")" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/%L2-norm~of~principal ~error~of~order~6~schemeG$\"+yZ?H(*!#:" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/%L2-norm~of~principal~error~of~order~5~schemeG$\"+F$*HEl!#8" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#/%;order~6~stability~intervalG7$$!)OV \"[%!\"(\"\"!" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/%;order~5~stability~ intervalG7$$!)^.#[%!\"(\"\"!" }}{PARA 11 "" 1 "" {XPPMATH 20 "6$/&%\"B G6#\"\"($\")0/29!\"(/&%\"CGF&$\")(RzS\"F*" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 49 "#------------------------------ ------------------" }}{PARA 0 "" 0 "" {TEXT -1 21 "another scheme with " }{XPPEDIT 18 0 "c[6]=25/33" "6#/&%\"cG6#\"\"'*&\"#D\"\"\"\"#L!\"\" " }{TEXT -1 7 " and " }{XPPEDIT 18 0 "c[7]=1728/1729" "6#/&%\"cG6#\" \"(*&\"%G<\"\"\"\"%H " 0 " " {MPLTEXT 1 0 112 "c_2 := 29/316: c_4 := 110/521: c_5 := 526/935: c_6 := 25/33: c_7 := 1728/1729: bs := [9,1/560]:\ncalc_RKcoeffs();" }} {PARA 11 "" 1 "" {XPPMATH 20 "6)%'nodes:G/&%\"cG6#\"\"##\"#H\"$;$/&F&6 #\"\"$#\"$?#\"%j:/&F&6#\"\"%#\"$5\"\"$@&/&F&6#\"\"&#\"$E&\"$N*/&F&6#\" \"'#\"#D\"#L/&F&6#\"\"(#\"%G<\"%H<" }}{PARA 11 "" 1 "" {XPPMATH 20 "6) %1order~6~weights:G/&%\"bG6#\"\"\"$\"'(RD'!\"(/&F&6#\"\"%$\"'xSK!\"'/& F&6#\"\"&$\"'?6JF2/&F&6#\"\"'$\"'HI;F2/&F&6#\"\"($\"'Jq5!\"%/&F&6#\"\" )$!'Rc5FE" }}{PARA 11 "" 1 "" {XPPMATH 20 "6*%1order~5~weights:G/&%#b* G6#\"\"\"$\"'*y1'!\"(/&F&6#\"\"%$\"'82L!\"'/&F&6#\"\"&$\"'2%)GF2/&F&6# \"\"'$\"'%Q)>F2/&F&6#\"\"($\"'7Fs!\"&/&F&6#\"\")$!'42rFE/&F&6#\"\"*$\" 'r&y\"!\")" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/*&%)infinityG\"\"\"%8-n orm~of~linking~coeffsGF&$\"+\"3`\"4U!\")" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/%L2-norm~of~principal~error~of~order~6~schemeG$\"+LXbH(*!#:" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#/%L2-norm~of~principal~error~of~order~ 5~schemeG$\"+abT>l!#8" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/%;order~6~st ability~intervalG7$$!)aM\"[%!\"(\"\"!" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/%;order~5~stability~intervalG7$$!)\")G%[%!\"(\"\"!" }}{PARA 11 "" 1 "" {XPPMATH 20 "6$/&%\"BG6#\"\"($\")L'pS\"!\"(/&%\"CGF&$\")n$zS\"F* " }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 39 "#=== ===================================" }}{PARA 0 "" 0 "" {TEXT -1 49 "Ts itouras' scheme with small principal error norm" }}{EXCHG {PARA 0 "> \+ " 0 "" {MPLTEXT 1 0 105 "c_2 := 1/11: c_4 := 20/139: c_5 := 88/177: c_ 6 := 35/36: c_7 := 544/545: bs := [9,1/20]:\ncalc_RKcoeffs();" }} {PARA 11 "" 1 "" {XPPMATH 20 "6)%'nodes:G/&%\"cG6#\"\"##\"\"\"\"#6/&F& 6#\"\"$#\"#S\"$<%/&F&6#\"\"%#\"#?\"$R\"/&F&6#\"\"&#\"#))\"$x\"/&F&6#\" \"'#\"#N\"#O/&F&6#\"\"(#\"$W&\"$X&" }}{PARA 11 "" 1 "" {XPPMATH 20 "6) %1order~6~weights:G/&%\"bG6#\"\"\"$\"'[QM!\"(/&F&6#\"\"%$\"'H#e#!\"'/& F&6#\"\"&$\"'C2UF2/&F&6#\"\"'$\"'I6X!\"&/&F&6#\"\"($!'dJ]!\"%/&F&6#\" \")$\"'54YFF" }}{PARA 11 "" 1 "" {XPPMATH 20 "6*%1order~5~weights:G/&% #b*G6#\"\"\"$\"'?AZ!\"(/&F&6#\"\"%$\"'!HH#!\"'/&F&6#\"\"&$\"'@JYF2/&F& 6#\"\"'$\"'\")y7!\"&/&F&6#\"\"($!'*>\\'F?/&F&6#\"\")$\"'bBaF?/&F&6#\" \"*$\"'++]F+" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/*&%)infinityG\"\"\"%8 -norm~of~linking~coeffsGF&$\"+m_>#3#!\"(" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/%L2-norm~of~principal~error~of~order~6~schemeG$\"+,LKuG!#:" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#/%L2-norm~of~principal~error~of~order~ 5~schemeG$\"+@Nn;?!#7" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/%;order~6~st ability~intervalG7$$!)N!G&[!\"(\"\"!" }}{PARA 11 "" 1 "" {XPPMATH 20 " 6#/%;order~5~stability~intervalG7$$!)1QtW!\"(\"\"!" }}{PARA 11 "" 1 " " {XPPMATH 20 "6$/&%\"BG6#\"\"($\")Zb? " 0 "" {MPLTEXT 1 0 105 "c_2 := 3/59: c_4 := 19/132: c_5 := 88/177: c_6 := 35/36: c_7 \+ := 544/545: bs := [9,1/80]:\ncalc_RKcoeffs();" }}{PARA 11 "" 1 "" {XPPMATH 20 "6)%'nodes:G/&%\"cG6#\"\"##\"\"$\"#f/&F&6#F*#\"#>\"$)>/&F& 6#\"\"%#F0\"$K\"/&F&6#\"\"&#\"#))\"$x\"/&F&6#\"\"'#\"#N\"#O/&F&6#\"\"( #\"$W&\"$X&" }}{PARA 11 "" 1 "" {XPPMATH 20 "6)%1order~6~weights:G/&% \"bG6#\"\"\"$\"'\"GW$!\"(/&F&6#\"\"%$\"'@#e#!\"'/&F&6#\"\"&$\"'l1UF2/& F&6#\"\"'$\"'?8X!\"&/&F&6#\"\"($!'1M]!\"%/&F&6#\"\")$\"'T6YFF" }} {PARA 11 "" 1 "" {XPPMATH 20 "6*%1order~5~weights:G/&%#b*G6#\"\"\"$\"' sjP!\"(/&F&6#\"\"%$\"'%)4D!\"'/&F&6#\"\"&$\"'q7VF2/&F&6#\"\"'$\"'x/P! \"&/&F&6#\"\"($!'1QR!\"%/&F&6#\"\")$\"'M%f$FF/&F&6#\"\"*$\"'+]7F+" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#/*&%)infinityG\"\"\"%8-norm~of~linking ~coeffsGF&$\"+XlS\"3#!\"(" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/%L2-norm ~of~principal~error~of~order~6~schemeG$\"+h5vmE!#:" }}{PARA 11 "" 1 " " {XPPMATH 20 "6#/%L2-norm~of~principal~error~of~order~5~schemeG$\"+s[ q()[!#8" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/%;order~6~stability~interv alG7$$!)\")*4&[!\"(\"\"!" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/%;order~5 ~stability~intervalG7$$!)Ij0b!\"(\"\"!" }}{PARA 11 "" 1 "" {XPPMATH 20 "6$/&%\"BG6#\"\"($\")\\;B " 0 "" {MPLTEXT 1 0 112 "c_2 := 7/116: c_4 := 104/719: c_5 := 121/243: c_6 := 389/400: c_7 := 1999/2000: bs := [9,1/80]:\ncalc_RKcoeffs();" }} {PARA 11 "" 1 "" {XPPMATH 20 "6)%'nodes:G/&%\"cG6#\"\"##\"\"(\"$;\"/&F &6#\"\"$#\"$3#\"%d@/&F&6#\"\"%#\"$/\"\"$>(/&F&6#\"\"&#\"$@\"\"$V#/&F&6 #\"\"'#\"$*Q\"$+%/&F&6#F*#\"%**>\"%+?" }}{PARA 11 "" 1 "" {XPPMATH 20 "6)%1order~6~weights:G/&%\"bG6#\"\"\"$\"'q\"[$!\"(/&F&6#\"\"%$\"'w'e#! \"'/&F&6#\"\"&$\"'v/UF2/&F&6#\"\"'$\"'_)Q%!\"&/&F&6#\"\"($!'Fd " 0 "" {MPLTEXT 1 0 117 "c_2 := 3/50: c_4 := 1439/10 000: c_5 := 4973/10000: c_6 := 389/400: c_7 := 1999/2000: bs := [9,1/3 0]:\ncalc_RKcoeffs();" }}{PARA 11 "" 1 "" {XPPMATH 20 "6)%'nodes:G/&% \"cG6#\"\"##\"\"$\"#]/&F&6#F*#\"%R9\"&+]\"/&F&6#\"\"%#F0\"&++\"/&F&6# \"\"&#\"%t\\F7/&F&6#\"\"'#\"$*Q\"$+%/&F&6#\"\"(#\"%**>\"%+?" }}{PARA 11 "" 1 "" {XPPMATH 20 "6)%1order~6~weights:G/&%\"bG6#\"\"\"$\"''*QM! \"(/&F&6#\"\"%$\"'i#e#!\"'/&F&6#\"\"&$\"'P4UF2/&F&6#\"\"'$\"'S0W!\"&/& F&6#\"\"($!'$[w\"!\"$/&F&6#\"\")$\"'kB8!#7" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/%;order~6~stability~intervalG7$$!)VFb[!\"( \"\"!" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/%;order~5~stability~interval G7$$!).!4$[!\"(\"\"!" }}{PARA 11 "" 1 "" {XPPMATH 20 "6$/&%\"BG6#\"\"( $\")aGA " 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 procedures for the examp les " }}{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,a75,a76,a81,a82,a 83,a84,a85,a86,a87,\n f1,f2,f3,f4,f5,f6,f7,f8,b1,b2,b3,b4,b5,b6,b7,b 8,\n xk,yk,t,jF,jM,jS,n,h,data,fn,xx,ys,saveDigits;\n options `Cop yright 2004 by Peter Stone`;\n \n data := SOLN_;\n\n saveDigits \+ := Digits;\n Digits := max(trunc(evalhf(Digits)),Digits+5);\n\n # \+ procedure to evaluate the slope field\n fn := proc(X_,Y_)\n loc al val; \n val := traperror(evalf(FXY_));\n if val=lasterror or not type(val,numeric) then\n error \"evaluation of slope f ield failed at %1\",evalf([X_,Y_],saveDigits);\n end if;\n v al;\n end proc;\n\n xx := evalf(x_rk6step);\n n := nops(data) ;\n\n if (data[1,1]data[n,1] or xxdata[1,1])) then \n error \"independent variable is outside the interpolation inte rval: %1\",evalf(data[1,1])..evalf(data[n,1]);\n end if;\n\n c2 := c2_; c3 := c3_; c4 := c4_; c5 := c5_; c6 := c6_; c7 := c7_; c8 := c8_ ;\n a21 := c2; a31 := a31_; a32 := a32_; a41 := a41_; a42 := a42_; a 43 := 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_; 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 # 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 while jS-jF> 1 do\n \+ jM := trunc((jF+jS)/2);\n if xx<=data[jM,1] then jF := j M else jS := jM end if;\n end do;\n if jM = n then jF := n-1 ; jS := n end if;\n end if;\n \n # Get the data needed from the l ist.\n xk := data[jF,1];\n yk := data[jF,2];\n\n # Do one step w ith step-size ..\n h := xx-xk;\n f1 := fn(xk,yk);\n t := a21*f1; \n f2 := fn(xk + c2*h,yk + t*h);\n t := a31*f1 + a32*f2;\n f3 := fn(xk + c3*h,yk + t*h);\n t := a41*f1 + a42*f2 + a43*f3;\n f4 := \+ fn(xk + c4*h,yk + t*h);\n t := a51*f1 + a52*f2 + a53*f3 + a54*f4;\n \+ f5 := fn(xk + c5*h,yk + t*h);\n t := a61*f1 + a62*f2 + a63*f3 + a6 4*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 + a8 7*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 eva lf[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 25 "RK6_1 Papakostas' scheme" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 4164 "RK6_1 := proc(fxy,x,y,xx,yy,h,stps,bb)\n local c2,c3,c4,c5,c6,c7,c8,a21,a 31,a32,a41,a42,a43,a51,\n a52,a53,a54,a61,a62,a63,a64,a65,a71,a72,a7 3,a74,a75,a76,\n a81,a82,a83,a84,a85,a86,a87,b1,b2,b3,b4,b5,b6,b7,b8 ,\n f1,f2,f3,f4,f5,f6,f7,f8,t,k,fn,xk,yk,soln,eqns,A,saveDigits;\n \+ saveDigits := Digits;\n Digits := max(trunc(evalhf(Digits)),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 [71/125,20775 1751/316406250,0,-526769377/210937500,1524242129/632812500,0,0,0,0],\n [42/59,-4970082682619223281/2887511529739311186,0,9791927803 3879057/13556392158400522,\n -407131674007930877068/7407 8904949579652469,\n 1237601855204268750000/175320075047 3385108433,0,0,0],\n [199/200,176597685527535385020980411/427 73485015591331328000000,0,\n -6793162515552646891859/401 628967282547712000,\n 12704926019361287204873446554247/ 886659402653054716778496000000,\n -507288363345092596322 78125/32657591718008685915971584,\n 5153622398279619070 3/51293749413888000000,0,0],\n [1,299033520572337573523/66918 720793812357519,0,-16550269823961899/902146153892364,\n \+ 49920346343238033627496282/3215735869387500624775563,\n \+ -1686432488955761721093750/978844996793357447730403,\n \+ 161901609084039/149698803705724,-305146137600000/54760341991955873,0] ,\n [0,24503/381483,0,0,1366847103121/4106349847584,203395996 09375/75933913767768,\n 35031290651/194765546144,166201 60000000/11001207123543,-14933/11016]]);\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 := ev alf(A[3,4]);\n a51 := evalf(A[4,2]);\n a52 := evalf(A[4,3]);\n a 53 := 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 := eval f(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 43 "RK6_2 a m odification of Papakostas' scheme" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 4517 "RK6_2 := proc(fxy,x,y,xx,y y,h,stps,bb)\n local c2,c3,c4,c5,c6,c7,c8,a21,a31,a32,a41,a42,a43,a5 1,\n a52,a53,a54,a61,a62,a63,a64,a65,a71,a72,a73,a74,a75,a76,\n a8 1,a82,a83,a84,a85,a86,a87,b1,b2,b3,b4,b5,b6,b7,b8,\n f1,f2,f3,f4,f5, f6,f7,f8,t,k,fn,xk,yk,soln,eqns,A,saveDigits;\n saveDigits := Digits ;\n Digits := max(trunc(evalhf(Digits)),Digits+5);\n\n fn := unapp ly(fxy,x,y);\n\n A := matrix([[1/11,1/11,0,0,0,0,0,0,0],\n \+ [157/1104,75517/2437632,271139/2437632,0,0,0,0,0,0],\n [157/7 36,157/2944,0,471/2944,0,0,0,0,0],\n [123/218,42928977471/638 42339642,0,-81818872578/31921169821,\n 78364952136/319211 69821,0,0,0,0],\n [73/99,-452993729346115584620105837/1693518 95877446242624608351,0,\n 1697439018580941844337032/15298 2742436717473012293,\n -334578912430118682410335643360/387 59554641024302244921542187,\n 183506358169025606635433230 /193412520623391990592938849,0,0,0],\n [199/200,2374249662448 01792240477643719598303/34580776953279718763722454600000000,0,\n \+ -20179674002037028690247750335371/72211779471432757191201250000 0,\n 128453834152286986408251909051997320271/554455420013 8077926323128675315625000,\n -9727088785922876892668243756 99676121/505439891163443236912399043150000000,\n 19942377 7107739967674947/239913433051724600000000,0,0],\n [1,59732662 07518682347729280873413/802324115518564191458832770081,0,\n \+ -40870912569246870680146344/1347069654182311812082183,\n \+ 112459036656658044255344022793238112/4474108065454386923650087599365 353,\n -4712736023245029582064053285795650/221362152840661 0002679128666238199,\n 127657765502309225898/142682894776 218689669,\n -541736066625000000/96886654438663742381,0], \n [0,531842063/8415926910,0,0,6900573846223978496/2108169749 9495907225,\n 51126843079934806/173160464034159675,114777 8420925807/6907762998459050,\n 575779000000000/4113305588 98641,-13426037/10725975]]);\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_3 scheme with " } {XPPEDIT 18 0 "c[7] = 324/325;" "6#/&%\"cG6#\"\"(*&\"$C$\"\"\"\"$D$!\" \"" }{TEXT -1 2 " " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 4415 "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, a54,a61,a62,a63,a64,a65,a71,a72,a73,a74,a75,a76,\n a81,a82,a83,a84,a 85,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([[30/323,30/323,0,0,0,0,0,0,0],\n [96/667,72 096/2224445,248064/2224445,0,0,0,0,0,0],\n [144/667,36/667,0, 108/667,0,0,0,0,0],\n [161/284,627527529881/949970239488,0,-7 96051562201/316656746496,\n 1149583311737/474985119744,0 ,0,0,0],\n [28/39,-21808753194693072629/11331655147076247477, 0,\n 23796070175842454861/2956083951411194994,\n \+ -1212069996160021962455905/196552978013281766346054,\n \+ 572100877243261794319360/753453082384246770993207,0,0,0],\n \+ [324/325,3900329744279593744638499197123/8121538124537291370807160 15625,0,\n -246193764192504580123435743/12532754329751616 636406250,\n 224823365551024681432621834494295407/136038 73384877578159416276089843750,\n -21380486750050287941075 671274033851392/12906219675382756068818499374186328125,\n \+ 4388242844049767568/4530133947623046875,0,0],\n [1,924435455 92188947873141917/18339118607014235358999552,0,\n -750730 245651799179235/36368955615474475776,\n 2577779379770967 239940697801161173/148529202838936128758823432698880,\n - 27329320985408825882348019928840/15495435293973600997385336553683,\n \+ 9858553049752641/9733028110935040,-5581481368973046875/16 97311052062486391808,0],\n [0,806703661/12619514880,0,0,220190 408471162800979/664353079471007769600,\n 120162483071349 248/437171496550168695,587917074771/3328097561600,\n 169 19101291015625/7106645968247808,-716503/321645]]);\n\n c2 := evalf(A [1,1]);\n c3 := evalf(A[2,1]);\n c4 := evalf(A[3,1]);\n c5 := ev alf(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 := ev alf(A[5,2]);\n a62 := evalf(A[5,3]);\n a63 := evalf(A[5,4]);\n a 64 := 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 := eval f(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 b 5 := 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 + a 74*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 e qns := \{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 a 42_=a42,a43_=a43,a51_=a51,a52_=a52,a53_=a53,a54_=a54,\n a61_=a 61,a62_=a62,a63_=a63,a64_=a64,a65_=a65,\n a71_=a71,a72_=a72,a7 3_=a73,a74_=a74,a75_=a75,a76_=a76,\n a81_=a81,a82_=a82,a83_=a8 3,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(rk6 step)); \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 scheme with " }{XPPEDIT 18 0 "c[7] = 575/576;" "6#/&%\"cG6#\"\"(*& \"$v&\"\"\"\"$w&!\"\"" }{TEXT -1 1 " " }}{PARA 0 "" 0 "" {TEXT -1 0 " " }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 4693 "RK6_4 := proc(fxy,x,y, xx,yy,h,stps,bb)\n local c2,c3,c4,c5,c6,c7,c8,a21,a31,a32,a41,a42,a4 3,a51,\n a52,a53,a54,a61,a62,a63,a64,a65,a71,a72,a73,a74,a75,a76,\n \+ a81,a82,a83,a84,a85,a86,a87,b1,b2,b3,b4,b5,b6,b7,b8,\n f1,f2,f3,f4 ,f5,f6,f7,f8,t,k,fn,xk,yk,soln,eqns,A,saveDigits;\n saveDigits := Di gits;\n Digits := max(trunc(evalhf(Digits)),Digits+5);\n\n fn := u napply(fxy,x,y);\n\n A := matrix([[21/229,21/229,0,0,0,0,0,0,0],\n \+ [398/2805,5306932/165228525,18137258/165228525,0,0,0,0,0,0],\n [199/935,199/3740,0,597/3740,0,0,0,0,0],\n [541/959 ,94539185952859/139708222009916,0,\n -360526602615135/139 708222009916,86200229584590/34927055502479,0,0,0,0],\n [29/39 ,-8978969288480000243857587089/3094782216331416263956902924,0,\n \+ 99268298611547543700685925/8262922327440626705984748,\n \+ -162409472645307771282700363150/17326138018112792098834790979, \n 40197004738132352835791693452/40001804824576007393862 935391,0,0,0],\n [575/576,20286604731599462128958430510992102 863475/2582125222615604140408598429414209880064,0,\n -759 6345291112508505437918843804759875/23772939210064272792198199433704243 2,\n 6620788516589830027125628007103689280061615375/2508 39001756308652298222060087067489645625344,\n -16378468176 189294528981106978962032922592625/772468438059969385382672071798267914 8003328,\n 97768503635786059874525/119061117135805961207 808,0,0],\n [1,13855115178421131108323299597371007/1715122015 582524727359996149290620,0,\n -23954172438450319856818854 8325/7288002275830304575859927972,\n 9565183091966721153 0668834468508100881883600/3523904935082188448634512810545504833479151, \n -741690472086763685807332786432434772072/3380356932484 63587532944907545647623719,\n 257132510132640765480/3053 96035838269868597,\n -1187059899803728084992/657869227273 850599426895,0],\n [0,1697710672/26928207375,0,0,164490038363 253411875/503598049753427901952,\n 688263291863311978681 /2293101349345635227520,237419997287733/1436816713308800,\n \+ 2129892164719607808/556154447061193625,-680161433/184588800]]);\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 := eval f(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 := ev alf(A[5,4]);\n a64 := evalf(A[5,5]);\n a65 := evalf(A[5,6]);\n a 71 := evalf(A[6,2]);\n a72 := evalf(A[6,3]);\n a73 := evalf(A[6,4] );\n a74 := evalf(A[6,5]);\n a75 := evalf(A[6,6]);\n a76 := eval f(A[6,7]);\n 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 := ev alf(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 := f n(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*f 6 + 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=tr ue 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 s ubs(eqns,eval(rk6step)); \n else\n return evalf[saveDigits]([so ln]);\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] = 1728/1729; " "6#/&%\"cG6#\"\"(*&\"%G<\"\"\"\"%H " 0 "" {MPLTEXT 1 0 4658 "RK 6_5 := proc(fxy,x,y,xx,yy,h,stps,bb)\n local c2,c3,c4,c5,c6,c7,c8,a2 1,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,saveDigits; \n saveDigits := Digits;\n Digits := max(trunc(evalhf(Digits)),Dig its+5);\n\n fn := unapply(fxy,x,y);\n\n A := matrix([[55/604,55/60 4,0,0,0,0,0,0,0],\n [86/611,656438/20532655,2233592/20532655, 0,0,0,0,0,0],\n [129/611,129/2444,0,387/2444,0,0,0,0,0],\n \+ [562/999,11379576325181/16591126906359,0,\n -14480 836320821/5530375635453,41396479505524/16591126906359,0,0,0,0],\n \+ [25/33,-3277828022520854505720678625/930941984189004121287101412, 0,\n 16036828475594748052837775/110432026594187914743428 4,\n -2024967022933348076278835034350/1776666334255938283 46456771343,\n 128250153920078340306971850/1111140578548 56180218323669,0,0,0],\n [1728/1729,1027311288352892762679823 552816060640736/100017465428353477655102490880994099675,0,\n \+ -319583399218917235748122654831799520/7666271645206398506276676005 113153,\n 309853242826319046243113700571575488375374176/ 9059548279238747452639357500349347045575147,\n -482090670 618620815146676915555711479983232/188177190150438948277052975318150206 923289,\n 97935281518032740868758496/1263523189437042707 51022325,0,0],\n [1,617667205196293028768837394929401/5957290 1637951905260312643716200,0,\n -7343254790498990760185464 13/17448823759340364442192828,\n 10261188057614115615910 6897686663174423254/2972016175809293986626508592155810158783,\n \+ -297880443826969314952760262689300658642/1149791657445320842880 43320814139766233,\n 59645563090975120908/76314917691264 991025,\n -2372974894108974768977/40495008074215518205107 12,0],\n [0,2350382879/37582963200,0,0,183047321485658386133/ 564833555402036603760,\n 27899748449743503627/8967424370 2534905040,42494109833439/260652887700800,\n 74019007611 6618685831/69155160690290757120,-18738157/1773760]]);\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 ";" }}}}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 31 "RK6 _6 Tsitouras' scheme with " }{XPPEDIT 18 0 "c[7] = 544/545;" "6#/&% \"cG6#\"\"(*&\"$W&\"\"\"\"$X&!\"\"" }{TEXT -1 1 " " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 4263 "RK6_6 := p roc(fxy,x,y,xx,yy,h,stps,bb)\n local c2,c3,c4,c5,c6,c7,c8,a21,a31,a3 2,a41,a42,a43,a51,\n a52,a53,a54,a61,a62,a63,a64,a65,a71,a72,a73,a74 ,a75,a76,\n a81,a82,a83,a84,a85,a86,a87,b1,b2,b3,b4,b5,b6,b7,b8,\n \+ f1,f2,f3,f4,f5,f6,f7,f8,t,k,fn,xk,yk,soln,eqns,A,saveDigits;\n save Digits := Digits;\n Digits := max(trunc(evalhf(Digits)),Digits+5);\n \n fn := unapply(fxy,x,y);\n\n A := matrix([[3/59,3/59,0,0,0,0,0,0 ,0],\n [19/198,1273/235224,21299/235224,0,0,0,0,0,0],\n \+ [19/132,19/528,0,19/176,0,0,0,0,0],\n [88/177,292792984/2 22425457,0,-373191104/74141819,312454912/74141819,0,0,0,0],\n \+ [35/36,-93282246804140065/2230303430271708,0,14352497130136165/901132 69909968,\n -2430460528059531110/19920664729472301,5373 0709552104745/9712013829327216,0,0,0],\n [544/545,-2815838771 905694350037121443888/52287718910334988576674096875,0,\n \+ 38959719796026045728912040192/190137159673945413006087625,\n \+ -801496182066760414629068567466496/51061334230438040662784831 69375,\n 26545926738560289939469575124336/382899634229 8708285360778643125,\n -16804913133796526592/9427076500 19020409375,0,0],\n [1,-76006376179258679683396673/1389345361 299174809573120,0,\n 109830477025712790000/52767431381 3796947,\n -194391745260058069863027995198/121916429570 6831920486709627,\n 444873349778574018937254390317/632 69358937392669883372236224,\n -49046624888634/292965477 2721215,-45835339680636651875/22882961544684083693312,0],\n [ 0,65760917/1910092800,0,0,6404556346056/24802664014135,\n \+ 2946448255213557/7004255810773760,16630875288/3684943675,\n \+ -496773593038619375/9868243105287168,27826183/603420]]);\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 46 "RK6_7 Verner's \"most efficient\" scheme with " } {XPPEDIT 18 0 "c[7] = 1999/2000;" "6#/&%\"cG6#\"\"(*&\"%**>\"\"\"\"%+? !\"\"" }{TEXT -1 1 " " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 4701 "RK6_7 := 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([[7/116,7/116,0,0,0,0,0,0,0],\n [208/2157, 631280/32568543,2509312/32568543,0,0,0,0,0,0],\n [104/719,26/ 719,0,78/719,0,0,0,0,0],\n [121/243,405298835029/310395556224 ,0,-516775016989/103465185408,649792660033/155197778112,0,0,0,0],\n \+ [389/400,-613196378874450690363694258375297/1481068967448107125 6371200000000,0,\n 19314590304297630584070330403179/1224 02394003975795507200000000,\n -45690322681074269820195954 2362145743/3777766287341706964636467200000000,\n 2334686 84012501097630099419892501/42262363236718430355123200000000,0,0,0],\n \+ [1999/2000,-690994313677727892638163353214849278889337/128348 10518108703660371044352000000000000,0,\n 559027019681178 56366108228295253679031/272680756296260886366208000000000000,\n \+ -314869670545645544422296903990152470264703005657/2003745885533 513610150342699619328000000000000,\n 3943288138498223719 88541658040946895351456539/5645688579794055320065075606164800000000000 0,\n -1841588041032613773/98979602990157900500,0,0],\n \+ [1,-41346042665166483923424449659815720863/76469298497423101031 0450749101004672,0,\n 1673297023243723294266377481831/81 27169928568684375229008512,\n -50372549912996109153218948 0038365151123989843/3191911395383323244189300496793324868133824,\n \+ 221222045246087052146775273050936702042442/3155117321590387 5161610805733586403142833,\n -1199089843519744000000/6535 5131597330942592057,\n -203964353120000000000/39840531775 2764522861589,0],\n [0,314492473/9032729376,0,0,1199622888218 3665169/46375417381391048160,\n 17231867943154059861/409 81954688105953544,16704872960000000/3806497022432643,\n - 370696000000000000/2109493746887301,1699767001/9903960]]);\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 := ev alf(A[4,3]);\n a53 := evalf(A[4,4]);\n a54 := evalf(A[4,5]);\n a 61 := evalf(A[5,2]);\n a62 := evalf(A[5,3]);\n a63 := evalf(A[5,4] );\n a64 := evalf(A[5,5]);\n a65 := evalf(A[5,6]);\n a71 := eval f(A[6,2]);\n a72 := evalf(A[6,3]);\n a73 := evalf(A[6,4]);\n a74 := evalf(A[6,5]);\n a75 := evalf(A[6,6]);\n a76 := evalf(A[6,7]); \n 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 := eval f(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*f4 + b5*f5 + b6*f6 + b7*f7 + b8*f8)*h;\n xk := x k + 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,a 83_=a83,a84_=a84,a85_=a85,a86_=a86,a87_=a87,\n b1_=b1,b2_=b2,b 3_=b3,b4_=b4,b5_=b5,b6_=b6,b7_=b7,b8_=b8\};\n return subs(eqns,ev al(rk6step)); \n else\n return evalf[saveDigits]([soln]);\n e nd if;\nend proc: " }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 20 "Testing the examples " }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}{PARA 0 "" 0 "" {TEXT -1 80 "These tests do not make use of the embedded order 4 metho d 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$$\"3emm\"z%4\\Y_FJ$\"3Yii#4W6uD \"F27$$\"3'QLL3FGT\\&FJ$\"3c!QStI8]>\"F27$$\"32++v$flW v*FJ7$$\"3I++vVVX$\\'FJ$\"3w/21T*\\F&*)FJ7$$\"31nm\"zWo)\\nFJ$\"3E>3;k 'H:;)FJ7$$\"3%QL$3_DG1qFJ$\"31le1yn9(R(FJ7$$\"3]***\\il'pisFJ$\"3E!)4G 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\\PMFwrmNF2$\"3R[i&\\xl(GsFJ7$$\"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$\"#5!\"\"F(F(-%+AXESLABELSG6$%\"xG%%y(x)G-%%FONTG6$%*HELVETICAG\"\" *-%%VIEWG6$;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 0 874 "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 := [`Papakostas' scheme with `*``(c[7]=199/200),`a modification of Papakostas' scheme with `*``(c[7]=199/200),`a scheme with `*``(c[7]=3 24/325),`a scheme with `*``(c[7]=575/576),`a scheme with `*``(c[7]=172 8/1729),`Tsitouras' scheme with `*``(c[7]=544/545),`Verner's \"most ef ficient\" scheme with `*``(c[7]=1999/2000)]: errs := []:\nDigits := 25 :\nfor ct to 7 do\n Fn_RK6_||ct := RK6_||ct(F(x,y),x,y,x0,y0,hh,nums teps,false);\n sm := 0: numpts := nops(Fn_RK6_||ct):\n for ii to n umpts do\n sm := sm+(Fn_RK6_||ct[ii,2]-f(Fn_RK6_||ct[ii,1]))^2;\n end do:\n errs := [op(errs),sqrt(sm/numpts)];\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$%0initial~p oint:~G-%!G6$\"\"!F,7$%/step~width:~~~G$F,!\"#7$%1no.~of~steps:~~~G\"$ +&Q)pprint346\"" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%!G" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#K%'matrixG6#7)7$*&%9Papakostas'~scheme~with~G\"\" \"-%!G6#/&%\"cG6#\"\"(#\"$*>\"$+#F*$\"+pP/M\\!#C7$*&%Ka~modification~o f~Papakostas'~scheme~with~GF*F+F*$\"+r2LjfF87$*&%/a~scheme~with~GF*-F, 6#/F/#\"$C$\"$D$F*$\"+'p&4gqF87$*&F@F*-F,6#/F/#\"$v&\"$w&F*$\"+Eu&3&)* F87$*&F@F*-F,6#/F/#\"%G<\"%H?\"!#B7$*&%8Tsitouras'~scheme~w ith~GF*-F,6#/F/#\"$W&\"$X&F*$\"+<&=?-$Fgn7$*&%GVerner's~\"most~efficie nt\"~scheme~with~GF*-F,6#/F/#\"%**>\"%+?F*$\"+S#4lA$FgnQ)pprint356\"" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 30 "The fo llowing code constructs " }{TEXT 260 20 "numerical procedures" }{TEXT -1 56 " for solutions based on each of the Runge-Kutta schemes." }} {PARA 0 "" 0 "" {TEXT -1 75 "The error in the value obtained by each o f the methods at the point where " }{XPPEDIT 18 0 "x = 4.999;" "6#/% \"xG-%&FloatG6$\"%**\\!\"$" }{TEXT -1 16 " is also given." }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 805 "F := (x,y) -> 12*x*cos(4*x)*exp(-x )*y: hh := 0.01: numsteps := 500: x0 := 0: y0 := 1:\nmatrix([[`slope f ield: `,F(x,y)],[`initial point: `,``(x0,y0)],[`step width: `,hh], \n[`no. of steps: `,numsteps]]);``;\nmthds := [`Papakostas' scheme w ith `*``(c[7]=199/200),`a modification of Papakostas' scheme with `*`` (c[7]=199/200),`a scheme with `*``(c[7]=324/325),`a scheme with `*``(c [7]=575/576),`a scheme with `*``(c[7]=1728/1729),`Tsitouras' scheme wi th `*``(c[7]=544/545),`Verner's \"most efficient\" scheme with `*``(c[ 7]=1999/2000)]: errs := []:\nDigits := 25:\nfor ct to 7 do\n fn_RK6_ ||ct := RK6_||ct(F(x,y),x,y,x0,y0,hh,numsteps,true);\nend do:\nxx := 4 .999: fxx := evalf(f(xx)):\nfor ct to 7 do\n errs := [op(errs),abs(f n_RK6_||ct(xx)-fxx)];\nend do:\nDigits := 10:\nlinalg[transpose]([mthd s,evalf(errs)]);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#K%'matrixG6#7&7$%0 slope~field:~~~G,$*,\"#7\"\"\"%\"xGF,-%$cosG6#,$*&\"\"%F,F-F,F,F,-%$ex pG6#,$F-!\"\"F,%\"yGF,F,7$%0initial~point:~G-%!G6$\"\"!F,7$%/step~widt h:~~~G$F,!\"#7$%1no.~of~steps:~~~G\"$+&Q)pprint316\"" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%!G" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#K%'matrixG6#7 )7$*&%9Papakostas'~scheme~with~G\"\"\"-%!G6#/&%\"cG6#\"\"(#\"$*>\"$+#F *$\"+)=V,R&!#C7$*&%Ka~modification~of~Papakostas'~scheme~with~GF*F+F*$ \"+?V#og'F87$*&%/a~scheme~with~GF*-F,6#/F/#\"$C$\"$D$F*$\"+v,TL#)F87$* &F@F*-F,6#/F/#\"$v&\"$w&F*$\"+:Tq^6!#B7$*&F@F*-F,6#/F/#\"%G<\"%H\" %+?F*$\"+ZZzqOFSQ)pprint326\"" }}}{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 0 549 "mthds := [`Papakostas' scheme with `*``(c[7]=199/200),`a modi fication of Papakostas' scheme with `*``(c[7]=199/200),`a scheme with \+ `*``(c[7]=324/325),`a scheme with `*``(c[7]=575/576),`a scheme with `* ``(c[7]=1728/1729),`Tsitouras' scheme with `*``(c[7]=544/545),`Verner' s \"most efficient\" scheme with `*``(c[7]=1999/2000)]: errs := []:\nD igits := 20:\nfor ct to 7 do\n sm := NCint((f(x)-'fn_RK6_||ct'(x))^2 ,x=0..5,adaptive=false,numpoints=7,factor=200);\n errs := [op(errs), sqrt(sm/5)];\nend do:\nDigits := 10:\nlinalg[transpose]([mthds,evalf(e rrs)]);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#K%'matrixG6#7)7$*&%9Papakos tas'~scheme~with~G\"\"\"-%!G6#/&%\"cG6#\"\"(#\"$*>\"$+#F*$\"+5$zv#\\!# C7$*&%Ka~modification~of~Papakostas'~scheme~with~GF*F+F*$\"+*yop&fF87$ *&%/a~scheme~with~GF*-F,6#/F/#\"$C$\"$D$F*$\"+d>)p0(F87$*&F@F*-F,6#/F/ #\"$v&\"$w&F*$\"+\\)G!\\)*F87$*&F@F*-F,6#/F/#\"%G<\"%H\"%+?F*$\" +W3tDKFgnQ)pprint336\"" }}}{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 0 679 "evalf[20](plot([f(x)-'fn_RK6_1'(x),f(x)-'fn_RK6_2'(x ),f(x)-'fn_RK6_3'(x),f(x)-'fn_RK6_4'(x),\nf(x)-'fn_RK6_5'(x),f(x)-'fn_ RK6_6'(x),f(x)-'fn_RK6_7'(x)],x=0..5,font=[HELVETICA,9],\ncolor=[COLOR (RGB,.95,0,.2),COLOR(RGB,.5,0,.95),COLOR(RGB,.95,.45,0),COLOR(RGB,0,.8 ,.25),\nCOLOR(RGB,.95,0,.95),COLOR(RGB,.75,.2,.2),COLOR(RGB,0,.7,.9)], legend=[`Papakostas' scheme with c[7]=199/200`,`a modification of Papa kostas' scheme with c[7]=199/200`,`a scheme with c[7]=324/325`,`a sche me with c[7]=575/576`,`a scheme with c[7]=1728/1729`,`Tsitouras' schem e with c[7]=544/545`,`Verner's \"most efficient\" scheme with c[7]=199 9/2000`],title=`error curves for 8 stage order 6 Runge-Kutta methods`) );" }}{PARA 13 "" 1 "" {GLPLOT2D 995 628 628 {PLOTDATA 2 "6--%'CURVESG 6%7it7$$\"\"!F)F(7$$\"5qmmmmT&)G\\a!#@$!$@#!#>7$$\"5MLLLL3x&)*3\"!#?$! %e5F07$$\"5++]i!*GER37F4$!$5*F07$$\"5nmm\"z%\\v#pK\"F4$!$.'F07$$\"5ML$ 3_+ZiaW\"F4$!$,\"F07$$\"5+++]i!R(*Rc\"F4$\"$^'F07$$\"5MLL3xJs1,=F4$\"% PTF07$$\"5nmmm\"H2P\"Q?F4$\"%MwF07$$\"5+++]PMnNrDF4$\"&h'=F07$$\"5MLLL $eRwX5$F4$\"&B$GF07$$\"5MLL$3F%\\wQKF4$\"&(>HF07$$\"5MLLLe*[`HP$F4$\"& l(HF07$$\"5MLL$ek.Ur]$F4$\"&K)HF07$$\"5MLLLL$eI8k$F4$\"&4'HF07$$\"5MLL L3xwq4RF4$\"&sy#F07$$\"5NLLL$3x%3yTF4$\"&7k#F07$$\"5-+](oHaN;J%F4$\"&# pDF07$$\"5ommT5:j=XWF4$\"&jb#F07$$\"5-+v=<,<'>^%F4$\"&me#F07$$\"5NL$eR s3P(yXF4$\"&de#F07$$\"5om\"H2LZ7bk%F4$\"&-k#F07$$\"5-++]PfyG7ZF4$\"&4v #F07$$\"5NLLek.%*Qz\\F4$\"&U5$F07$$\"5ommm\"z%4\\Y_F4$\"&4%QF07$$\"5NL 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EF$F47$Fb_m$\"(*H)*GF47$Fg_m$\"(xre#F47$Fa\\s$\"(g_L#F47$F\\`m$\"(?j:# F47$Fa`m$\"(un0#F47$Ff`m$\"(kQ+#F47$F[am$\"(sE*>F47$F`am$\"(0A*>F47$Fe am$\"(hI+#F47$Fjam$\"(BO-#F47$F_bm$\"(Y))4#F47$Fdbm$\"(lo@#F47$F^cm$\" (*=sDF47$Fccm$\"(7,0$F47$Fhcm$\"(\"RXOF47$F]dm$\"(u&HTF47$Fbdm$\"('>KV F47$Fgdm$\"(d!zWF47$F\\em$\"(cY_%F47$Faem$\"(llb%F47$Fddq$\"(.sc%F47$F fem$\"(MVd%F47$F\\eq$\"(1!yXF47$F[fm$\"(%GyXF47$Fe`s$\"(AYd%F47$F`fm$ \"(;qc%F47$F]as$\"(Ybb%F47$Fefm$\"(#RSXF47$Fjfm$\"(6&*\\%F47$F_gm$\"(! *fW%F47$Fdgm$\"(*H;VF47$Figm$\"(0!fTF47$F^hm$\"(d5z$F47$Fchm$\"(0bW$F4 7$Fhhm$\"(YT:$F47$F_io$\"(W%[IF47$F]im$\"(Wr'HF47$Fbim$\"(!H4HF47$Fgim $\"(\\l(GF47$F\\jm$\"([v'GF47$Fajm$\"('G!)GF47$Fahq$\"(b%4HF47$Ffjm$\" (sM&HF47$F[[n$\"(;&)4$F47$F`[n$\"(i$fKF47$Fe[n$\"(4IV$F47$Fj[n$\"(RLd$ F47$F_\\n$\"(!QrOF4-Fd\\n6&Ff\\nF($\"\"(F\\]n$\"\"*F\\]n-F^]n6#%UVerne r's~\"most~efficient\"~scheme~with~c[7]=1999/2000G-%+AXESLABELSG6$Q\"x 6\"Q!F\\ix-%&TITLEG6#%Uerror~curves~for~8~stage~order~6~Runge-Kutta~me thodsG-%%FONTG6$%*HELVETICAGFdhx-%%VIEWG6$;F(F_\\n%(DEFAULTG" 1 2 0 1 10 0 2 9 1 4 2 1.000000 45.000000 45.000000 0 1 "Papakostas' scheme wi th c[7]=199/200" "a modification of Papakostas' scheme with c[7]=199/2 00" "a scheme with c[7]=324/325" "a scheme with c[7]=575/576" "a schem e with c[7]=1728/1729" "Tsitouras' scheme with c[7]=544/545" "Verner's \"most efficient\" scheme with c[7]=1999/2000" }}}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 47 "Test 2 of 8 stage, order 6 Runge-Kutta \+ methods " }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "dy/dx=x/y " "6#/*&%#dyG\"\"\"%#dxG!\"\"*&%\"xGF&%\"yGF(" }{TEXT -1 10 ", \+ " }{XPPEDIT 18 0 "y(0)=1" "6#/-%\"yG6#\"\"!\"\"\"" }{TEXT -1 1 " " }} {PARA 0 "" 0 "" {TEXT -1 10 "Solution: " }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "y=sqrt(1+x^2)" "6#/%\"yG-%%sqrtG6#,&\"\"\"F)*$% \"xG\"\"#F)" }{TEXT -1 2 ". " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 32 "The following code constructs a " }{TEXT 260 17 "discrete solution" }{TEXT -1 44 " based on each of the methods and gives the " }{TEXT 260 22 "root mean square error" }{TEXT -1 18 " of \+ each solution." }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 877 "G := (x,y ) -> x/y: hh := 0.05: numsteps := 200: x0 := 0: y0 := 1:\nmatrix([[`sl ope field: `,G(x,y)],[`initial point: `,``(x0,y0)],[`step width: ` ,hh],\n[`no. of steps: `,numsteps]]);``;\nmthds := [`Papakostas' sch eme with `*``(c[7]=199/200),`a modification of Papakostas' scheme with `*``(c[7]=199/200),`a scheme with `*``(c[7]=324/325),`a scheme with ` *``(c[7]=575/576),`a scheme with `*``(c[7]=1728/1729),`Tsitouras' sche me with `*``(c[7]=544/545),`Verner's \"most efficient\" scheme with `* ``(c[7]=1999/2000)]: errs := []:\nDigits := 20:\ng := x -> sqrt(1+x^2) :\nfor ct to 7 do\n Gn_RK6_||ct := RK6_||ct(G(x,y),x,y,x0,y0,hh,nums teps,false);\n sm := 0: numpts := nops(Gn_RK6_||ct):\n for ii to n umpts do\n sm := sm+(Gn_RK6_||ct[ii,2]-g(Gn_RK6_||ct[ii,1]))^2;\n end do:\n errs := [op(errs),sqrt(sm/numpts)];\nend do:\nDigits := 10:\nlinalg[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\"$+#Q)pprint366\"" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#% !G" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#K%'matrixG6#7)7$*&%9Papakostas'~ scheme~with~G\"\"\"-%!G6#/&%\"cG6#\"\"(#\"$*>\"$+#F*$\"+UD*))*H!#C7$*& %Ka~modification~of~Papakostas'~scheme~with~GF*F+F*$\"+XDtfJF87$*&%/a~ scheme~with~GF*-F,6#/F/#\"$C$\"$D$F*$\"+.zWfHF87$*&F@F*-F,6#/F/#\"$v& \"$w&F*$\"+iY:sIF87$*&F@F*-F,6#/F/#\"%G<\"%H\"%+?F*$\"+EHY&>(F8Q) pprint376\"" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 30 "The following code constructs " }{TEXT 260 20 "numerical proced ures" }{TEXT -1 56 " for solutions based on each of the Runge-Kutta sc hemes." }}{PARA 0 "" 0 "" {TEXT -1 75 "The error in the value obtained by each of the methods at the point where " }{XPPEDIT 18 0 "9.99;" " 6#-%&FloatG6$\"$***!\"#" }{TEXT -1 16 " is also given." }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 807 "G := (x,y) -> x/y: hh := 0.05: num steps := 200: x0 := 0: y0 := 1:\nmatrix([[`slope field: `,G(x,y)],[` initial point: `,``(x0,y0)],[`step width: `,hh],\n[`no. of steps: \+ `,numsteps]]);``;\nmthds := [`Papakostas' scheme with `*``(c[7]=199/20 0),`a modification of Papakostas' scheme with `*``(c[7]=199/200),`a sc heme with `*``(c[7]=324/325),`a scheme with `*``(c[7]=575/576),`a sche me with `*``(c[7]=1728/1729),`Tsitouras' scheme with `*``(c[7]=544/545 ),`Verner's \"most efficient\" scheme with `*``(c[7]=1999/2000)]: errs := []:\nDigits := 25:\nfor ct to 7 do\n gn_RK6_||ct := RK6_||ct(G(x ,y),x,y,x0,y0,hh,numsteps,true);\nend do:\ng := x -> sqrt(1+x^2):\nxx \+ := 9.99: gxx := evalf(g(xx)):\nfor ct to 7 do\n errs := [op(errs),ab s(gn_RK6_||ct(xx)-gxx)];\nend do:\nDigits := 10:\nlinalg[transpose]([m thds,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\"$+#Q)pprint50 6\"" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%!G" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#K%'matrixG6#7)7$*&%9Papakostas'~scheme~with~G\"\"\"-%!G 6#/&%\"cG6#\"\"(#\"$*>\"$+#F*$\"*v.\"e5!#B7$*&%Ka~modification~of~Papa kostas'~scheme~with~GF*F+F*$\"*TZ16\"F87$*&%/a~scheme~with~GF*-F,6#/F/ #\"$C$\"$D$F*$\"*!)f<,\"F87$*&F@F*-F,6#/F/#\"$v&\"$w&F*$\"*N-v-\"F87$* &F@F*-F,6#/F/#\"%G<\"%H\"%+?F*$\"*N-aA#F8Q)pprint516\"" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 4 "The " }{TEXT 260 22 "root mean square error" }{TEXT -1 20 " over the interval " } {XPPEDIT 18 0 "[0, 10]" "6#7$\"\"!\"#5" }{TEXT -1 82 " of each Runge- Kutta method is estimated as follows using the special procedure " } {TEXT 0 5 "NCint" }{TEXT -1 98 " to perform numerical integration by \+ the 7 point Newton-Cotes method over 100 equal subintervals." }} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 574 "mthds := [`Papakostas' sche me with `*``(c[7]=199/200),`a modification of Papakostas' scheme with \+ `*``(c[7]=199/200),`a scheme with `*``(c[7]=324/325),`a scheme with `* ``(c[7]=575/576),`a scheme with `*``(c[7]=1728/1729),`Tsitouras' schem e with `*``(c[7]=544/545),`Verner's \"most efficient\" scheme with `*` `(c[7]=1999/2000)]: errs := []:\nDigits := 20:\ng := x -> sqrt(1+x^2): \nfor ct to 7 do\n sm := NCint((g(x)-'gn_RK6_||ct'(x))^2,x=0..10,ada ptive=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$*&%9Papakostas'~schem e~with~G\"\"\"-%!G6#/&%\"cG6#\"\"(#\"$*>\"$+#F*$\"+]S`!*H!#C7$*&%Ka~mo dification~of~Papakostas'~scheme~with~GF*F+F*$\"+B8%4:$F87$*&%/a~schem e~with~GF*-F,6#/F/#\"$C$\"$D$F*$\"+c,2^HF87$*&F@F*-F,6#/F/#\"$v&\"$w&F *$\"+I\">M1$F87$*&F@F*-F,6#/F/#\"%G<\"%H\"%+?F*$\"+!zea<(F8Q)pprin t406\"" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 92 "The following error graphs are constructed using the numerical pro cedures for the solutions." }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 680 "evalf[20](plot([g(x)-'gn_RK6_1'(x),g(x)-'gn_RK6_2'(x),g(x)-'gn_RK 6_3'(x),g(x)-'gn_RK6_4'(x),\ng(x)-'gn_RK6_5'(x),g(x)-'gn_RK6_6'(x),g(x )-'gn_RK6_7'(x)],x=0..10,font=[HELVETICA,9],\ncolor=[COLOR(RGB,.95,0,. 2),COLOR(RGB,.5,0,.95),COLOR(RGB,.95,.45,0),COLOR(RGB,0,.8,.25),\nCOLO R(RGB,.95,0,.95),COLOR(RGB,.75,.2,.2),COLOR(RGB,0,.7,.9)],legend=[`Pap akostas' scheme with c[7]=199/200`,`a modification of Papakostas' sche me with c[7]=199/200`,`a scheme with c[7]=324/325`,`a scheme with c[7] =575/576`,`a scheme with c[7]=1728/1729`,`Tsitouras' scheme with c[7]= 544/545`,`Verner's \"most efficient\" scheme with c[7]=1999/2000`],tit le=`error curves 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bl$\"&!\\eF-7$Fdcl$\"&)ReF-7$F^dl$\"&]$eF-7$FcdlFhjo7$Fhdl$\"&u$eF-7$F ]el$\"&$QeF-7$Fbel$\"&;$eF-7$Fgel$\"&4!eF-7$F\\fl$\"&0x&F-7$Fafl$\"&Ct &F-7$Fffl$\"&.p&F-7$F`gl$\"&\"RcF-7$Fbjl$\"&`g&F-7$F\\[m$\"&Zb&F-7$Fa[ m$\"&(>bF-7$Ff[m$\"&aY&F-7$F[\\m$\"&>U&F-7$F`\\m$\"&RQ&F-7$Fe\\m$\"&GL &F-7$Fj\\m$\"&hH&F-7$F_]m$\"&$[_F-7$Fd]m$\"&K?&F-7$Fi]m$\"&8;&F-7$F^^m $\"&T6&F-7$Fc^m$\"&&y]F-7$Fh^m$\"&;.&F-7$F]_m$\"&_*\\F-7$Fb_m$\"&4&\\F -7$Fg_m$\"&$4\\F-7$F\\`m$\"&%p[F-7$Fa`m$\"&t#[F-7$Ff`m$\"&(*y%F-7$F[am $\"&zu%F-7$F`am$\"&Ar%F-7$Feam$\"&$\"&7W'F-7$FC$\"&lV'F-7$FH$\"&LV'F-7$FM$\"&AV'F-7$FR$\"&TV'F-7$FW $\"&ZX'F-7$Ffn$\"&u['F-7$F[oF__n7$F`o$\"&IX'F-7$Feo$\"&*=kF-7$Fjo$\"&. R'F-7$F_p$\"&gQ'F-7$Fdp$\"&SQ'F-7$Fip$\"&aQ'F-7$F^q$\"&8R'F-7$Fcq$\"&M S'F-7$Fhq$\"&RU'F-7$F]r$\"&]U'F-7$Fbr$\"&2U'F-7$Fgr$\"&lT'F-7$F\\s$\"& AT'F-7$Fas$\"&2Q'F-7$Ffs$\"&+N'F-7$F[t$\"&aL'F-7$F^t$\"&QK'F-7$Fct$\"& *>jF-7$Fht$\"&#=jF-7$F[u$\"&%>jF-7$F`u$\"&ZK'F-7$Feu$\"&cL'F-7$Fju$\"& yM'F-7$F_v$\"&&RjF-7$Fdv$\"&6L'F-7$Fiv$\"&YJ'F-7$F^w$\"&\")H'F-7$Fcw$ \"&hE'F-7$Fhw$\"& &F-7$Fc^m$\"&H;&F-7$Fh^m$\"&_6&F-7$F]_m$\"&w2&F-7$Fb_m$\"&D.&F-7$Fg_m$ \"&+*\\F-7$F\\`m$\"&\"\\\\F-7$Fa`m$\"&j!\\F-7$Ff`m$\"&v'[F-7$F[am$\"&^ #[F-7$F`am$\"&&)y%F-7$Feam$\"&su%F-7$Fjam$\"&$4ZF-7$F_bm$\"&?n%F-7$Fdb m$\"&Sj%F-7$Fibm$\"&mf%F--F^cm6&F`cmFdcm$\"\")Ffcm$\"#DFccm-Fhcm6#%;a~ scheme~with~c[7]=575/576G-F$6%7ip7$F($\"&P(oF-7$F4$\"&%*z'F-7$F9$\"&Uy 'F-7$F>$\"&0x'F-7$FC$\"&\\w'F-7$FH$\"&0w'F-7$FM$\"&zv'F-7$FR$\"&yv'F-7 $FW$\"&:x'F-7$Ffn$\"&iz'F-7$F[o$\"&#ynF-7$F`o$\"&,w'F-7$Feo$\"&Vs'F-7$ Fjo$\"&Ip'F-7$F_p$\"&vo'F-7$Fdp$\"&So'F-7$Fip$\"&Ko'F-7$F^q$\"&lo'F-7$ Fcq$\"&^p'F-7$Fhq$\"&6r'F-7$Fbr$\"&pq'F-7$F\\s$\"&zp'F-7$Fas$\"&]m'F-7 $Ffs$\"&Gj'F-7$F[t$\"&sh'F-7$F^t$\"&Sg'F-7$Fct$\"&!*f'F-7$Fht$\"&ff'F- 7$F[u$\"&_f'F-7$F`u$\"&!)f'F-7$F^w$\"&Ic'F-7$Fcw$\"&&HlF-7$Fhw$\"&A]'F -7$F]x$\"&\")\\'F-7$Fbx$\"&l\\'F-7$FgxF]]r7$F\\y$\"&X]'F-7$F`z$\"&m]'F -7$Fjz$\"&z\\'F-7$F_[l$\"&$*['F-7$Fd[l$\"&1['F-7$Fi[l$\"&.X'F-7$F^\\l$ \"&4U'F-7$Fh\\l$\"&fS'F-7$Fb]l$\"&WR'F-7$Fg]l$\"&5R'F-7$F\\^l$\"&)*Q'F -7$Fa^l$\"&?R'F-7$Ff^l$\"&!)R'F-7$F[_l$\"&WO'F-7$F`_l$\"&5L'F-7$F^al$ \"&\"ziF-7$F`bl$\"&qC'F-7$Febl$\"&L@'F-7$Fjbl$\"&<='F-7$Fdcl$\"&1<'F-7 $F^dl$\"&I;'F-7$Fcdl$\"&7;'F-7$Fhdl$\"&:;'F-7$F]el$\"&3;'F-7$Fbel$\"&P :'F-7$Fgel$\"&77'F-7$F\\fl$\"&\"*3'F-7$Fafl$\"&U/'F-7$Fffl$\"&f*fF-7$F `gl$\"&3%fF-7$Fbjl$\"&%**eF-7$F\\[m$\"&f%eF-7$Fa[m$\"&Q!eF-7$Ff[m$\"&k u&F-7$F[\\m$\"&#*p&F-7$F`\\m$\"&gl&F-7$Fe\\m$\"&?g&F-7$Fj\\m$\"&*fbF-7 $F_]m$\"&(4bF-7$Fd]m$\"&-Y&F-7$Fi]mF_[o7$F^^m$\"&^O&F-7$Fc^m$\"&dK&F-7 $Fh^m$\"&lF&F-7$F]_m$\"&jB&F-7$Fb_m$\"&(*=&F-7$Fg_m$\"&c9&F-7$F\\`m$\" &D5&F-7$Fa`m$\"&$e]F-7$Ff`m$\"&u,&F-7$F[am$\"&P(\\F-7$F`am$\"&c$\\F-7$ Feam$\"&E*[F-7$Fjam$\"&M&[F-7$F_bm$\"&W\"[F-7$Fdbm$\"&_x%F-7$Fibm$\"&h t%F--F^cm6&F`cmFacmFdcmFacm-Fhcm6#%=a~scheme~with~c[7]=1728/1729G-%&TI TLEG6#%Uerror~curves~for~8~stage~order~6~Runge-Kutta~methodsG-%%FONTG6 $%*HELVETICAG\"\"*-%+AXESLABELSG6$Q\"x6\"Q!F^ir-%%VIEWG6$;F(Fibm;$\"#T !#;$\"#rFgir" 1 2 0 1 10 0 2 9 1 4 2 1.000000 45.000000 45.000000 0 1 "Papakostas' scheme with c[7]=199/200" "a modification of Papakostas' \+ scheme with c[7]=199/200" "a scheme with c[7]=324/325" "a scheme with \+ c[7]=575/576" "a scheme with c[7]=1728/1729" }}}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 47 "Test 3 of 8 stage, order 6 Runge-Kutta \+ methods " }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "dy/dx = - x*y;" "6#/*&%#dyG\"\"\"%#dxG!\"\",$*&%\"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-%$expG6#, $*&%\"xG\"\"#F+!\"\"F," }{TEXT -1 2 ". " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 32 "The following code constructs a " } {TEXT 260 17 "discrete solution" }{TEXT -1 44 " based on each of the m ethods and gives the " }{TEXT 260 22 "root mean square error" }{TEXT -1 18 " of each solution." }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 877 "H := (x,y) -> -x*y: hh := 0.1: numsteps := 100: x0 := 0: y0 := 1: \nmatrix([[`slope field: `,H(x,y)],[`initial point: `,``(x0,y0)],[`s tep width: `,hh],\n[`no. of steps: `,numsteps]]);``;\nmthds := [`P apakostas' scheme with `*``(c[7]=199/200),`a modification of Papakosta s' scheme with `*``(c[7]=199/200),`a scheme with `*``(c[7]=324/325),`a scheme with `*``(c[7]=575/576),`a scheme with `*``(c[7]=1728/1729),`T sitouras' scheme with `*``(c[7]=544/545),`Verner's \"most efficient\" \+ scheme with `*``(c[7]=1999/2000)]: errs := []:\nDigits := 20:\nh := x \+ -> exp(-x^2/2):\nfor ct to 7 do\n Hn_RK6_||ct := RK6_||ct(H(x,y),x,y ,x0,y0,hh,numsteps,false);\n sm := 0: numpts := nops(Hn_RK6_||ct):\n for ii to numpts do\n sm := sm+(Hn_RK6_||ct[ii,2]-h(Hn_RK6_||c t[ii,1]))^2;\n end do:\n errs := [op(errs),sqrt(sm/numpts)];\nend \+ do:\nDigits := 10:\nlinalg[transpose]([mthds,evalf(errs)]);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#K%'matrixG6#7&7$%0slope~field:~~~G,$*&%\"xG \"\"\"%\"yGF,!\"\"7$%0initial~point:~G-%!G6$\"\"!F,7$%/step~width:~~~G $F,F.7$%1no.~of~steps:~~~G\"$+\"Q)pprint416\"" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%!G" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#K%'matrixG6#7)7$ *&%9Papakostas'~scheme~with~G\"\"\"-%!G6#/&%\"cG6#\"\"(#\"$*>\"$+#F*$ \"+N))*\\#G!#?7$*&%Ka~modification~of~Papakostas'~scheme~with~GF*F+F*$ \"+>dW+FF87$*&%/a~scheme~with~GF*-F,6#/F/#\"$C$\"$D$F*$\"+s)\\/w#F87$* &F@F*-F,6#/F/#\"$v&\"$w&F*$\"+x)pFb#F87$*&F@F*-F,6#/F/#\"%G<\"%H \"%+?F*$\"+?9$\\U\"F8Q)pprint426\"" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" } }{PARA 0 "" 0 "" {TEXT -1 30 "The following code constructs " }{TEXT 260 20 "numerical procedures" }{TEXT -1 56 " for solutions based on ea ch of the Runge-Kutta schemes." }}{PARA 0 "" 0 "" {TEXT -1 75 "The err or in the value obtained by each of the methods at the point where " }{XPPEDIT 18 0 "9.99;" "6#-%&FloatG6$\"$***!\"#" }{TEXT -1 16 " is al so given." }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 807 "H := (x,y) -> \+ -x*y: hh := 0.1: numsteps := 100: x0 := 0: y0 := 1:\nmatrix([[`slope f ield: `,H(x,y)],[`initial point: `,``(x0,y0)],[`step width: `,hh], \n[`no. of steps: `,numsteps]]);``;\nmthds := [`Papakostas' scheme w ith `*``(c[7]=199/200),`a modification of Papakostas' scheme with `*`` (c[7]=199/200),`a scheme with `*``(c[7]=324/325),`a scheme with `*``(c [7]=575/576),`a scheme with `*``(c[7]=1728/1729),`Tsitouras' scheme wi th `*``(c[7]=544/545),`Verner's \"most efficient\" scheme with `*``(c[ 7]=1999/2000)]: errs := []:\nDigits := 20:\nfor ct to 7 do\n hn_RK6_ ||ct := RK6_||ct(H(x,y),x,y,x0,y0,hh,numsteps,true);\nend do:\nh := x \+ -> exp(-x^2/2):\nxx := 9.99: hxx := evalf(h(xx)):\nfor ct to 7 do\n \+ errs := [op(errs),abs(hn_RK6_||ct(xx)-hxx)];\nend do:\nDigits := 10:\n linalg[transpose]([mthds,evalf(errs)]);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#K%'matrixG6#7&7$%0slope~field:~~~G,$*&%\"xG\"\"\"%\"yGF,!\"\"7$% 0initial~point:~G-%!G6$\"\"!F,7$%/step~width:~~~G$F,F.7$%1no.~of~steps :~~~G\"$+\"Q)pprint526\"" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%!G" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#K%'matrixG6#7)7$*&%9Papakostas'~scheme ~with~G\"\"\"-%!G6#/&%\"cG6#\"\"(#\"$*>\"$+#F*$\"+/HBtB!#N7$*&%Ka~modi fication~of~Papakostas'~scheme~with~GF*F+F*$\"+&**pS3#F87$*&%/a~scheme ~with~GF*-F,6#/F/#\"$C$\"$D$F*$\"+052+BF87$*&F@F*-F,6#/F/#\"$v&\"$w&F* $\"+tsy@?F87$*&F@F*-F,6#/F/#\"%G<\"%H\"%+?F*$\"+x-*QR\"F8Q)pprin t536\"" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 4 "The " }{TEXT 260 22 "root mean square error" }{TEXT -1 110 " over the interval [0, 0.5] of each Runge-Kutta method is estimated as follow s using the special procedure " }{TEXT 0 5 "NCint" }{TEXT -1 97 " to perform numerical integration by the 7 point Newton-Cotes method over 50 equal subintervals." }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 573 " mthds := [`Papakostas' scheme with `*``(c[7]=199/200),`a modification \+ of Papakostas' scheme with `*``(c[7]=199/200),`a scheme with `*``(c[7] =324/325),`a scheme with `*``(c[7]=575/576),`a scheme with `*``(c[7]=1 728/1729),`Tsitouras' scheme with `*``(c[7]=544/545),`Verner's \"most \+ efficient\" scheme with `*``(c[7]=1999/2000)]: errs := []:\nDigits := \+ 20:\nh := x -> exp(-x^2/2):\nfor ct to 7 do\n sm := NCint((h(x)-'hn_ RK6_||ct'(x))^2,x=0..10,adaptive=false,numpoints=7,factor=50);\n err s := [op(errs),sqrt(sm/10)];\nend do:\nDigits := 10:\nlinalg[transpose ]([mthds,evalf(errs)]);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#K%'matrixG6 #7)7$*&%9Papakostas'~scheme~with~G\"\"\"-%!G6#/&%\"cG6#\"\"(#\"$*>\"$+ #F*$\"+hCrVE!#?7$*&%Ka~modification~of~Papakostas'~scheme~with~GF*F+F* $\"+FoWGDF87$*&%/a~scheme~with~GF*-F,6#/F/#\"$C$\"$D$F*$\"+]\\\\$e#F87 $*&F@F*-F,6#/F/#\"$v&\"$w&F*$\"+pYe*Q#F87$*&F@F*-F,6#/F/#\"%G<\"%H \"%+?F*$\"+,\"GSM\"F8Q)pprint456\"" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" } }{PARA 0 "" 0 "" {TEXT -1 92 "The following error graphs are construct ed using the numerical procedures for the solutions." }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 693 "evalf[20](plot(['hn_RK6_1'(x)-h(x),'hn_R K6_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),'hn_RK6_7'(x)-h(x)],x=0..6,numpoints=100,font=[H ELVETICA,9],\ncolor=[COLOR(RGB,.95,0,.2),COLOR(RGB,.5,0,.95),COLOR(RGB ,.95,.45,0),COLOR(RGB,0,.8,.25),\nCOLOR(RGB,.95,0,.95),COLOR(RGB,.75,. 2,.2),COLOR(RGB,0,.7,.9)],legend=[`Papakostas' scheme with c[7]=199/20 0`,`a modification of Papakostas' scheme with c[7]=199/200`,`a scheme \+ with c[7]=324/325`,`a scheme with c[7]=575/576`,`a scheme with c[7]=17 28/1729`,`Tsitouras' scheme with c[7]=544/545`,`Verner's \"most effici ent\" scheme with c[7]=1999/2000`],title=`error curves for 8 stage ord er 6 Runge-Kutta methods`));" }}{PARA 13 "" 1 "" {GLPLOT2D 1034 752 752 {PLOTDATA 2 "6--%'CURVESG6%7g_l7$$\"\"!F)F(7$$\"5.......*4M'!#@$!' f_C!#?7$$\"5111111_#e=\"F0$!(/*f'*F07$$\"5%RRRRR**)H1=F0$!)J*p<\"F07$$ \"5&[[[[[o&)3V#F0$!)\"RY=#F07$$\"5CCCCCCQ]_IF0$!)83_QF07$$\"5777777M#) GOF0$!)c2SQF07$$\"5kjjjjj*obA%F0$!)c1#*fF07$$\"5nmmmmmSsU[F0$!)T4>mF07 $$\"5YXXXXX***yX&F0$!)i_t&)F07$$\"5+(pppp%f4;cF0$!)@>z&)F07$$\"5][[[[[ >HudF0$!)>T;*)F07$$\"5+++++]z[KfF0$!*p5-/\"F07$$\"5_^^^^^Ro!4'F0$!*#=M w6F07$$\"5++++++AOpjF0$!*tOj:\"F07$$\"5\\[[[[[//[mF0$!*%G%*\\6F07$$\"5 ++++++L!\\!oF0$!*:ig>\"F07$$\"5]^^^^^hwhpF0$!*tl^Q\"F07$$\"5+.....!H'= rF0$!*pnEY\"F07$$\"5baaaaa=\\vsF0$!*-yiW\"F07$$\"5POOOOO'>b!zF0$!*n2(G :F07$$\"5YXXXXX\"oE^)F0$!*>wug\"F07$$\"5edddddr,k!*F0$!*EcKm\"F07$$\"5 hgggggWi>(*F0$!*yT_b\"F07$$\"5CCCCC%)4]F5!#>$!*3kEH\"F07$$\"5FFFFF2b5# 4\"F\\s$!)X`7%)F07$$\"5[[[[)4v`#*4\"F\\s$!)S?*G&F07$$\"5qpppp%*>S16F\\ s$!)trF[F07$$\"5\"4444%Q-b86F\\s$!)#f&*y%F07$$\"577777#[)p?6F\\s$!)8P^ 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Q`_F-7$Fa[l$!1/i7:q[#o&F-7$Ff[l$!10%yxz!GgfF-7$F[\\l$!1&Q+CadGF'F-7$F` \\l$!1rf2P&=]\"pF--Fe\\l6&Fg\\lF($\"\"(F]]l$\"\"*F]]l-F_]l6#%UVerner's ~\"most~efficient\"~scheme~with~c[7]=1999/2000G-%+AXESLABELSG6$Q\"x6\" Q!F`^p-%&TITLEG6#%hnrelative~error~curves~for~8~stage~order~6~Runge-Ku tta~methodsG-%%FONTG6$%*HELVETICAGFh]p-%%VIEWG6$;F(F`\\l%(DEFAULTG" 1 2 0 1 10 0 2 9 1 4 2 1.000000 45.000000 45.000000 0 1 "Papakostas' sch eme with c[7]=199/200" "a modification of Papakostas' scheme with c[7] =199/200" "a scheme with c[7]=324/325" "a scheme with c[7]=575/576" "a scheme with c[7]=1728/1729" "Tsitouras' scheme with c[7]=544/545" "Ve rner's \"most efficient\" scheme with c[7]=1999/2000" }}}}{PARA 0 "" 0 "" {TEXT -1 1 " " }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}} {SECT 1 {PARA 4 "" 0 "" {TEXT -1 47 "Test 4 of 8 stage, order 6 Runge- Kutta methods " }}{PARA 0 "" 0 "" {TEXT -1 81 "F. G. Lether: Mathemati cs of Computation, Vol. 20, no. 95, (July 1966) page 381. " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "dy/dx = -32*x*y*ln(2);" "6#/*&%#dyG\"\"\"%#dxG!\"\",$**\"#KF&%\"xGF &%\"yGF&-%#lnG6#\"\"#F&F(" }{TEXT -1 6 ", " }{XPPEDIT 18 0 "y(-1) \+ = 1/8;" "6#/-%\"yG6#,$\"\"\"!\"\"*&F(F(\"\")F)" }{TEXT -1 1 " " }} {PARA 0 "" 0 "" {TEXT -1 10 "Solution: " }}{PARA 256 "" 0 "" {TEXT -1 2 " " }{XPPEDIT 18 0 "y = 2^(13-6*x^2);" "6#/%\"yG)\"\"#,&\"#8\"\"\"* &\"\"'F)*$%\"xGF&F)!\"\"" }{TEXT -1 2 ". " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 191 "de := diff(y(x),x)=- 32*x*y(x)*ln(2);\nic := y(-1)=1/8;\ndsolve(\{de,ic\},y(x)):\ny(x)=2^si mplify(log[2](rhs(%)));\nk := unapply(rhs(%),x):\nplot(k(x),x=-1..1,fo nt=[HELVETICA,9],labels=[`x`,`y(x)`]);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%#deG/-%%diffG6$-%\"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;p 0k&*F-$\"3!Hg[[\"[$)=KF-7$$!3wKL$3$3(F-7$$!3mmmmT%p \"e()F-$\"3!=E-TWD`l\"!#<7$$!3:mmm\"4m(G$)F-$\"3M\"fONp()[t$F=7$$!3\"Q LL3i.9!zF-$\"3A!e'4A%*Rg!)F=7$$!3\"ommT!R=0vF-$\"3%z2Mbncie\"!#;7$$!3u ****\\P8#\\4(F-$\"3C>dT>$)H#3$FM7$$!3+nm;/siqmF-$\"3gp%*z`g)4*eFM7$$!3 [++](y$pZiF-$\"3%R6L-Y$zz5!#:7$$!33LLL$yaE\"eF-$\"3xvp\"p)==K>Fgn7$$!3 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-> -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 := [`Papakostas' scheme with `*``(c[7]=199/200),`a modif ication of Papakostas' scheme with `*``(c[7]=199/200),`a scheme with ` *``(c[7]=324/325),`a scheme with `*``(c[7]=575/576),`a scheme with `*` `(c[7]=1728/1729),`Tsitouras' scheme with `*``(c[7]=544/545),`Verner's \"most efficient\" scheme with `*``(c[7]=1999/2000)]: errs := []:\nDi gits := 20:\nfor ct to 7 do\n Kn_RK6_||ct := RK6_||ct(evalf(K(x,y)), x,y,x0,evalf(y0),hh,numsteps,false);\n sm := 0: numpts := nops(Kn_RK 6_||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/numpt s)];\nend do:\nDigits := 10:\nlinalg[transpose]([mthds,evalf(errs)]); " }}{PARA 11 "" 1 "" {XPPMATH 20 "6#K%'matrixG6#7&7$%0slope~field:~~~G ,$**\"#K\"\"\"%\"xGF,-%\"yG6#F-F,-%#lnG6#\"\"#F,!\"\"7$%0initial~point :~G-%!G6$F5#F,\"\")7$%/step~width:~~~G$F,!\"#7$%1no.~of~steps:~~~G\"$+ #Q)pprint466\"" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%!G" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#K%'matrixG6#7)7$*&%9Papakostas'~scheme~with~G\"\" \"-%!G6#/&%\"cG6#\"\"(#\"$*>\"$+#F*$\"+$4Een%!#:7$*&%Ka~modification~o f~Papakostas'~scheme~with~GF*F+F*$\"+gss3UF87$*&%/a~scheme~with~GF*-F, 6#/F/#\"$C$\"$D$F*$\"+(3!enXF87$*&F@F*-F,6#/F/#\"$v&\"$w&F*$\"+!\\xs9% F87$*&F@F*-F,6#/F/#\"%G<\"%H\"%+?F*$\"+tsD.5F8Q)pprint476\"" } }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 30 "The fol lowing code constructs " }{TEXT 260 20 "numerical procedures" }{TEXT -1 56 " for solutions based on each of the Runge-Kutta schemes." }} {PARA 0 "" 0 "" {TEXT -1 75 "The error in the value obtained by each o f the methods at the point where " }{XPPEDIT 18 0 "x = 0;" "6#/%\"xG \"\"!" }{TEXT -1 20 ".995 is also given." }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 815 "K := (x,y) -> -32*x*y(x)*ln(2): hh := 0.01: numsteps := 200: x0 := -1: y0 := 1/8:\nmatrix([[`slope field: `,K(x,y)],[`in itial point: `,``(x0,y0)],[`step width: `,hh],\n[`no. of steps: `, numsteps]]);``;\nmthds := [`Papakostas' scheme with `*``(c[7]=199/200) ,`a modification of Papakostas' scheme with `*``(c[7]=199/200),`a sche me with `*``(c[7]=324/325),`a scheme with `*``(c[7]=575/576),`a scheme with `*``(c[7]=1728/1729),`Tsitouras' scheme with `*``(c[7]=544/545), `Verner's \"most efficient\" scheme with `*``(c[7]=1999/2000)]: errs : = []:\nDigits := 20:\nfor ct to 7 do\n kn_RK6_||ct := RK6_||ct(evalf (K(x,y)),x,y,x0,evalf(y0),hh,numsteps,true);\nend do:\nxx := 0.995: kx x := evalf(k(xx)):\nfor ct to 7 do\n errs := [op(errs),abs(kn_RK6_|| ct(xx)-kxx)];\nend do:\nDigits := 10:\nlinalg[transpose]([mthds,evalf( errs)]);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#K%'matrixG6#7&7$%0slope~fi eld:~~~G,$**\"#K\"\"\"%\"xGF,-%\"yG6#F-F,-%#lnG6#\"\"#F,!\"\"7$%0initi al~point:~G-%!G6$F5#F,\"\")7$%/step~width:~~~G$F,!\"#7$%1no.~of~steps: ~~~G\"$+#Q)pprint486\"" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%!G" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#K%'matrixG6#7)7$*&%9Papakostas'~scheme ~with~G\"\"\"-%!G6#/&%\"cG6#\"\"(#\"$*>\"$+#F*$\"+Zp\"y)\\!#?7$*&%Ka~m odification~of~Papakostas'~scheme~with~GF*F+F*$\"+\"yY&eWF87$*&%/a~sch eme~with~GF*-F,6#/F/#\"$C$\"$D$F*$\"+X:g(y%F87$*&F@F*-F,6#/F/#\"$v&\"$ w&F*$\"+!3o)RTF87$*&F@F*-F,6#/F/#\"%G<\"%H\"%+?F*$\"+s38:MF8Q)pprint 496\"" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 4 " The " }{TEXT 260 22 "root mean square error" }{TEXT -1 20 " over the i nterval " }{XPPEDIT 18 0 "[-1,1]" "6#7$,$\"\"\"!\"\"F%" }{TEXT -1 82 " of each Runge-Kutta method is estimated as follows using the specia l procedure " }{TEXT 0 5 "NCint" }{TEXT -1 98 " to perform numerical integration by the 7 point Newton-Cotes method over 100 equal subinte rvals." }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 550 "mthds := [`Papako stas' scheme with `*``(c[7]=199/200),`a modification of Papakostas' sc heme with `*``(c[7]=199/200),`a scheme with `*``(c[7]=324/325),`a sche me with `*``(c[7]=575/576),`a scheme with `*``(c[7]=1728/1729),`Tsitou ras' scheme with `*``(c[7]=544/545),`Verner's \"most efficient\" schem e with `*``(c[7]=1999/2000)]: errs := []:\nDigits := 20:\nfor ct to 7 \+ do\n sm := NCint((k(x)-'kn_RK6_||ct'(x))^2,x=-1..1,adaptive=false,nu mpoints=7,factor=100);\n errs := [op(errs),sqrt(sm/2)];\nend do:\nDi gits := 10:\nlinalg[transpose]([mthds,evalf(errs)]);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#K%'matrixG6#7)7$*&%9Papakostas'~scheme~with~G\"\"\"- %!G6#/&%\"cG6#\"\"(#\"$*>\"$+#F*$\"+8K](o%!#:7$*&%Ka~modification~of~P apakostas'~scheme~with~GF*F+F*$\"+'=Q#>UF87$*&%/a~scheme~with~GF*-F,6# /F/#\"$C$\"$D$F*$\"+\\p)*yXF87$*&F@F*-F,6#/F/#\"$v&\"$w&F*$\"+*)\\jdTF 87$*&F@F*-F,6#/F/#\"%G<\"%H\"%+?F*$\"+*fld+\"F8Q)pprint506\"" }} }{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 92 "The foll owing error graphs are constructed using the numerical procedures for \+ the solutions." }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 680 "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),k(x)-'kn_RK6_7'(x )],x=-1..1,font=[HELVETICA,9],\ncolor=[COLOR(RGB,.95,0,.2),COLOR(RGB,. 5,0,.95),COLOR(RGB,.95,.45,0),COLOR(RGB,0,.8,.25),\nCOLOR(RGB,.95,0,.9 5),COLOR(RGB,.75,.2,.2),COLOR(RGB,0,.7,.9)],legend=[`Papakostas' schem e with c[7]=199/200`,`a modification of Papakostas' scheme with c[7]=1 99/200`,`a scheme with c[7]=324/325`,`a scheme with c[7]=575/576`,`a s cheme with c[7]=1728/1729`,`Tsitouras' scheme with c[7]=544/545`,`Vern er's \"most efficient\" scheme with c[7]=1999/2000`],title=`error curv es for 8 stage order 6 Runge-Kutta methods`));" }}{PARA 13 "" 1 "" {GLPLOT2D 932 827 827 {PLOTDATA 2 "6--%'CURVESG6%7eo7$$!\"\"\"\"!$F*F* 7$$!5nmmmm;p0k&*!#?$\",m>:5;\"F/7$$!5LLLL$37$$!5nmmmm\"4m(G$)F/$\",`Cn_n$F<7$$!5LLLL$3i. 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" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 165 "de := diff(y(x),x)=16/((16*x+1)*y( x));\nic := y(0)=1;\ndsolve(\{de,ic\},y(x));\ns := unapply(rhs(%),x): \nplot(s(x),x=0..0.5,0..2.6,font=[HELVETICA,9],labels=[`x`,`y(x)`]);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%#deG/-%%diffG6$-%\"yG6#%\"xGF,,$*( \"#;\"\"\",&*&F/F0F,F0F0F0F0!\"\"F)F3F0" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%#icG/-%\"yG6#\"\"!\"\"\"" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/- %\"yG6#%\"xG*$,&*&\"\"#\"\"\"-%#lnG6#,&*&\"#;F,F'F,F,F,F,F,F,F,F,#F,F+ " }}{PARA 13 "" 1 "" {GLPLOT2D 503 318 318 {PLOTDATA 2 "6&-%'CURVESG6$ 7U7$$\"\"!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(-%+AXESLABE LSG6$%\"xG%%y(x)G-%%FONTG6$%*HELVETICAG\"\"*-%%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 0 867 "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 := [`Papakostas' scheme w ith `*``(c[7]=199/200),`a modification of Papakostas' scheme with `*`` (c[7]=199/200),`a scheme with `*``(c[7]=324/325),`a scheme with `*``(c [7]=575/576),`a scheme with `*``(c[7]=1728/1729),`Tsitouras' scheme wi th `*``(c[7]=544/545),`Verner's \"most efficient\" scheme with `*``(c[ 7]=1999/2000)]: errs := []:\nDigits := 20:\nfor ct to 7 do\n Sn_RK6_ ||ct := RK6_||ct(S(x,y),x,y,x0,y0,hh,numsteps,false);\n sm := 0: num pts := 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(er rs),sqrt(sm/numpts)];\nend do:\nDigits := 10:\nlinalg[transpose]([mthd s,evalf(errs)]);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#K%'matrixG6#7&7$%0 slope~field:~~~G,$*(\"#;\"\"\",&*&F+F,%\"xGF,F,F,F,!\"\"%\"yGF0F,7$%0i nitial~point:~G-%!G6$\"\"!F,7$%/step~width:~~~G$\"\"&!\"$7$%1no.~of~st eps:~~~G\"$+\"Q)pprint516\"" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%!G" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#K%'matrixG6#7)7$*&%9Papakostas'~scheme ~with~G\"\"\"-%!G6#/&%\"cG6#\"\"(#\"$*>\"$+#F*$\"+@S\\)\\\"!#B7$*&%Ka~ modification~of~Papakostas'~scheme~with~GF*F+F*$\"+HNNVG!#@7$*&%/a~sch eme~with~GF*-F,6#/F/#\"$C$\"$D$F*$\"+)[.n_#F87$*&FAF*-F,6#/F/#\"$v&\"$ w&F*$\"+OcP?EF>7$*&FAF*-F,6#/F/#\"%G<\"%H7$*&%8Tsitour as'~scheme~with~GF*-F,6#/F/#\"$W&\"$X&F*$\"+%[fRA$!#?7$*&%GVerner's~\" most~efficient\"~scheme~with~GF*-F,6#/F/#\"%**>\"%+?F*$\"+=%H09$FcoQ)p print526\"" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 30 "The following code constructs " }{TEXT 260 20 "numerical proced ures" }{TEXT -1 56 " for solutions based on each of the Runge-Kutta sc hemes." }}{PARA 0 "" 0 "" {TEXT -1 75 "The error in the value obtained by each of the methods at the point where " }{XPPEDIT 18 0 "x = 0;" "6#/%\"xG\"\"!" }{TEXT -1 21 ".4995 is also given." }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 799 "S := (x,y) -> 16/((16*x+1)*y): hh := 0.005 : numsteps := 100: x0 := 0: y0 := 1:\nmatrix([[`slope field: `,S(x,y )],[`initial point: `,``(x0,y0)],[`step width: `,hh],\n[`no. of step s: `,numsteps]]);``;\nmthds := [`Papakostas' scheme with `*``(c[7]=1 99/200),`a modification of Papakostas' scheme with `*``(c[7]=199/200), `a scheme with `*``(c[7]=324/325),`a scheme with `*``(c[7]=575/576),`a scheme with `*``(c[7]=1728/1729),`Tsitouras' scheme with `*``(c[7]=54 4/545),`Verner's \"most efficient\" scheme with `*``(c[7]=1999/2000)]: errs := []:\nDigits := 25:\nfor ct to 7 do\n sn_RK6_||ct := RK6_||c t(S(x,y),x,y,x0,y0,hh,numsteps,true);\nend do:\nxx := 0.4995: sxx := e valf(s(xx)):\nfor ct to 7 do\n errs := [op(errs),abs(sn_RK6_||ct(xx) -sxx)];\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,!\"\"%\"yGF0F,7$%0initial~point:~ G-%!G6$\"\"!F,7$%/step~width:~~~G$\"\"&!\"$7$%1no.~of~steps:~~~G\"$+\" Q)pprint556\"" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%!G" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#K%'matrixG6#7)7$*&%9Papakostas'~scheme~with~G\"\"\"- %!G6#/&%\"cG6#\"\"(#\"$*>\"$+#F*$\"+>HtU8!#B7$*&%Ka~modification~of~Pa pakostas'~scheme~with~GF*F+F*$\"+TXEKB!#@7$*&%/a~scheme~with~GF*-F,6#/ F/#\"$C$\"$D$F*$\"+eN'\\:#F87$*&FAF*-F,6#/F/#\"$v&\"$w&F*$\"+B:DX@F>7$ *&FAF*-F,6#/F/#\"%G<\"%H7$*&%8Tsitouras'~scheme~with~GF* -F,6#/F/#\"$W&\"$X&F*$\"+r6tYE!#?7$*&%GVerner's~\"most~efficient\"~sch eme~with~GF*-F,6#/F/#\"%**>\"%+?F*$\"+1)[wd#FcoQ)pprint566\"" }}} {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 4 "The " } {TEXT 260 22 "root mean square error" }{TEXT -1 110 " over the interva l [0, 0.5] of each Runge-Kutta method is estimated as follows using \+ the special procedure " }{TEXT 0 5 "NCint" }{TEXT -1 97 " to perform numerical integration by the 7 point Newton-Cotes method over 50 equa l subintervals." }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 552 "mthds := [`Papakostas' scheme with `*``(c[7]=199/200),`a modification of Papak ostas' scheme with `*``(c[7]=199/200),`a scheme with `*``(c[7]=324/325 ),`a scheme with `*``(c[7]=575/576),`a scheme with `*``(c[7]=1728/1729 ),`Tsitouras' scheme with `*``(c[7]=544/545),`Verner's \"most efficien t\" scheme with `*``(c[7]=1999/2000)]: errs := []:\nDigits := 20:\nfor ct to 7 do\n sm := NCint((s(x)-'sn_RK6_||ct'(x))^2,x=0..0.5,adaptiv e=false,numpoints=7,factor=50);\n errs := [op(errs),sqrt(sm/0.5)];\n end do:\nDigits := 10:\nlinalg[transpose]([mthds,evalf(errs)]);" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#K%'matrixG6#7)7$*&%9Papakostas'~scheme ~with~G\"\"\"-%!G6#/&%\"cG6#\"\"(#\"$*>\"$+#F*$\"+K!='3:!#B7$*&%Ka~mod ification~of~Papakostas'~scheme~with~GF*F+F*$\"+k-[UG!#@7$*&%/a~scheme ~with~GF*-F,6#/F/#\"$C$\"$D$F*$\"+-rZTDF87$*&FAF*-F,6#/F/#\"$v&\"$w&F* $\"+k[e>EF>7$*&FAF*-F,6#/F/#\"%G<\"%H7$*&%8Tsitouras'~sc heme~with~GF*-F,6#/F/#\"$W&\"$X&F*$\"+'ofGA$!#?7$*&%GVerner's~\"most~e fficient\"~scheme~with~GF*-F,6#/F/#\"%**>\"%+?F*$\"+t(e%RJFcoQ)pprint5 76\"" }}}{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 0 681 "e valf[20](plot([s(x)-'sn_RK6_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),s(x)-'sn_ RK6_7'(x)],x=0..0.5,font=[HELVETICA,9],\ncolor=[COLOR(RGB,.95,0,.2),CO LOR(RGB,.5,0,.95),COLOR(RGB,.95,.45,0),COLOR(RGB,0,.8,.25),\nCOLOR(RGB ,.95,0,.95),COLOR(RGB,.75,.2,.2),COLOR(RGB,0,.7,.9)],legend=[`Papakost as' scheme with c[7]=199/200`,`a modification of Papakostas' scheme wi th c[7]=199/200`,`a scheme with c[7]=324/325`,`a scheme with c[7]=575/ 576`,`a scheme with c[7]=1728/1729`,`Tsitouras' scheme with c[7]=544/5 45`,`Verner's \"most efficient\" scheme with c[7]=1999/2000`],title=`e rror curves for 8 stage order 6 Runge-Kutta methods`));" }}{PARA 13 " " 1 "" {GLPLOT2D 1042 656 656 {PLOTDATA 2 "6--%'CURVESG6%7ir7$$\"\"!F) 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scheme with c[7]=199/200`,`a scheme with c[7]=324/325`],title= `error curves for 8 stage order 6 Runge-Kutta methods`));" }}{PARA 13 "" 1 "" {GLPLOT2D 923 462 462 {PLOTDATA 2 "6(-%'CURVESG6%7]x7$$\"\"!F) F(7$$\"5IKKKKK#)>0m!#B$\"#>!#>7$$\"5YYYYYY'R5K\"!#A$\"%fF4$\"&c8#F07$$\"5]]]]]D%*oY@F4$\"&(3M F07$$\"5IJJJJ\"Q>=J#F4$\"&5?&F07$$\"567777P$\\pZ#F4$\"&ji(F07$$\"5#HHH HHHz?k#F4$\"'F07$$\"5MNNNNg\"pu8$F4$\"'vRDF07$$\"5:;;;;;\"*f-LF4$\"'%R>$F07$$\"5' pppp>2HxY$F4$\"'M5RF07$$\"5wxxxxF!fGj$F4$\"'>fYF07$$\"5deeee$)*))zz$F4 $\"'(eR&F07$$\"5QRRRRR*=J'RF4$\"'BegF07$$\"5>????&*)[#GTF4$\"')Rc'F07$ $\"5+,,,,^)yLH%F4$\"'<3oF07$$\"5\"====o!)3&eWF4$\"'!3m'F07$$\"5iiiiii( QOi%F4$\"'#['fF07$$\"5VVVVV=(o()y%F4$\"'ZMXF07$$\"5CCCCCu')*Q&\\F4$\"' ^F4$\"'V&G\"F07$$\"5&eeeeeeeTG&F4$\"'s#G\"F07$$\" 5IFFFF_L()eeF4$\"'&QF\"F07$$\"5qoooo=\")eLkF4$\"'tt7F07$$\"555555&)GI3 qF4$\"'uH8F07$$\"5]^^^^^w,$e(F4$\"'Ui:F07$$\"5?AAAsM]PqyF4$\"'$fz\"F07 $$\"5!HHHHzTKx:)F4$\"'zD@F07$$\"5DGGG`46T,$)F4$\"'ODBF07$$\"5gjjj8,)*3 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'`ACF07$Fj]n$!'B8CF07$F_^n$!'R0CF07$Fd^n$!'q'R#F07$Fi^n$!'^)Q#F07$F^_n $!'a!Q#F07$Fc_n$!'zsBF07$Fh_n$!']lBF07$F]`n$!'#yN#F07$Fb`n$!'n]BF07$Fg `n$!'JVBF07$F\\an$!'!oL#F07$Faan$!'wHBF07$Ffan$!';BBF07$F[bn$!'s;BF07$ F`bn$!'85BF07$Febn$!'?/BF07$Fjbn$!'B)H#F07$F_cn$!'z\"H#F07$Fdcn$!'1'G# F07$Ficn$!'L!G#F07$F^dn$!'guAF07$Fcdn$!'WpAF07$Fhdn$!'/kAF07$F]en$!'ye AF07$Fben$!'E`AF07$Fgen$!'Z[AF07$F\\fn$!'0VAF07$Fafn$!'CQAF07$Fffn$!'d LAF07$F[gn$!'jGAF07$F`gn$!'uBAF07$Fegn$!'R>AF07$Fjgn$!'$[@#F07$F_hn$!' X5AF07$Fdhn$!'z0AF07$Fihn$!'#=?#F07$F^in$!'K(>#F07$Fcin$!';$>#F07$Fhin $!'6*=#F07$F]jn$!'8&=#F07$Fbjn$!'6\"=#F07$Fgjn$!'#p<#F07$F\\[o$!'9t@F0 7$Fa[o$!'Yp@F07$Ff[o$!'dl@F07$F[\\o$!'xh@F07$F`\\o$!'Qe@F07$Fe\\o$!'ha @F0-F[]o6&F]]oF^]o$\"#XF`]oF(-Fd]o6#%;a~scheme~with~c[7]=324/325G-%+AX ESLABELSG6$Q\"x6\"Q!Ffaq-%&TITLEG6#%Uerror~curves~for~8~stage~order~6~ Runge-Kutta~methodsG-%%FONTG6$%*HELVETICAG\"\"*-%%VIEWG6$;F(Fe\\o%(DEF AULTG" 1 2 0 1 10 0 2 9 1 4 2 1.000000 45.000000 45.000000 0 1 "Papako stas' scheme with c[7]=199/200" "a scheme with c[7]=324/325" }}}} {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, ord er 6 Runge-Kutta methods" }}{PARA 256 "" 0 "" {TEXT -1 2 " " } {XPPEDIT 18 0 "dy/dx = (1+2*(x+1)*sin(3*x))*exp(-y);" "6#/*&%#dyG\"\" \"%#dxG!\"\"*&,&F&F&*(\"\"#F&,&%\"xGF&F&F&F&-%$sinG6#*&\"\"$F&F.F&F&F& F&-%$expG6#,$%\"yGF(F&" }{TEXT -1 6 ", " }{XPPEDIT 18 0 "y(0) = 0; " "6#/-%\"yG6#\"\"!F'" }{TEXT -1 1 " " }}{PARA 0 "" 0 "" {TEXT -1 10 " Solution: " }}{PARA 256 "" 0 "" {TEXT -1 2 " " }{XPPEDIT 18 0 "y=ln(x +2/9*sin(3*x)-2/3*x*cos(3*x)-2/3*cos(3*x)+5/3)" "6#/%\"yG-%#lnG6#,,%\" xG\"\"\"*(\"\"#F*\"\"*!\"\"-%$sinG6#*&\"\"$F*F)F*F*F***F,F*F3F.F)F*-%$ cosG6#*&F3F*F)F*F*F.*(F,F*F3F.-F66#*&F3F*F)F*F*F.*&\"\"&F*F3F.F*" } {TEXT -1 2 ". " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 169 "de := diff(y(x),x)=(1+2*(x+1)*sin(3*x))*exp(-y( x));\nic := y(0)=0;\ndsolve(\{de,ic\},y(x));\nu := unapply(rhs(%),x): \nplot(u(x),x=0..5,font=[HELVETICA,9],labels=[`x`,`y(x)`]);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%#deG/-%%diffG6$-%\"yG6#%\"xGF,*&,&\"\"\"F/ *(\"\"#F/,&F,F/F/F/F/-%$sinG6#,$*&\"\"$F/F,F/F/F/F/F/-%$expG6#,$F)!\" \"F/" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%#icG/-%\"yG6#\"\"!F)" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#/-%\"yG6#%\"xG-%#lnG6#,,F'\"\"\"*&#\" \"#\"\"*F,-%$sinG6#,$*&\"\"$F,F'F,F,F,F,*&#F/F6F,*&F'F,-%$cosGF3F,F,! \"\"*&#F/F6F,F:F,F<#\"\"&F6F," }}{PARA 13 "" 1 "" {GLPLOT2D 503 318 318 {PLOTDATA 2 "6&-%'CURVESG6$7bp7$$\"\"!F)F(7$$\"3GLLL3x&)*3\"!#=$\" 3QWK+t!=.P\"F-7$$\"3umm\"H2P\"Q?F-$\"3pUCE&GmM$HF-7$$\"3MLL$eRwX5$F-$ \"3l!G\"yWq,6\\F-7$$\"33ML$3x%3yTF-$\"3dz%)zauhMpF-7$$\"3emm\"z%4\\Y_F -$\"3,G5kQO>C))F-7$$\"3`LLeR-/PiF-$\"36YrjIBvP5!#<7$$\"3]***\\il'pisF- 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^6%FI$\"3w4XE;/Rz()F-7$$\"3w3F>RL3GTFI$\"3JeP:9JjA()F-7$$\"3t]i!RbX59% FI$\"3mH1#H$\\k'o)F-7$$\"3#=z>'ox+aTFI$\"3oLr_-o*=n)F-7$$\"3yLLL$)*pp; %FI$\"3A7j1wipy')F-7$$\"3!Q3_+sD-=%FI$\"32pcM,k23()F-7$$\"3#Q$3xc9[$>% FI$\"3Gri,**=4g()F-7$$\"3'Qe*[$>Pn?%FI$\"3se,X+?^M))F-7$$\"3)QL3-$H**> UFI$\"3Z**e,OD#4$*)F-7$$\"3#R$ek.W]YUFI$\"3i#fiyx0s=*F-7$$\"3)RL$3xe,t UFI$\"3[2R[)*eVA&*F-7$$\"3Cn;HdO=yVFI$\"3#)>Y<=$f\\9\"FI7$$\"3MMe9\"z- lU%FI$\"3)4DVDmlMD\"FI7$$\"3a+++D>#[Z%FI$\"3qZKS'GmoO\"FI7$$\"3TM$3_5, -`%FI$\"3CFB-Gn\\(\\\"FI7$$\"3SnmT&G!e&e%FI$\"3t\\(p9r/Xi\"FI7$$\"3m+] P%37^j%FI$\"3_eaMDR_K " 0 " " {MPLTEXT 1 0 879 "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 := [`Papakostas' scheme with `*``(c[ 7]=199/200),`a modification of Papakostas' scheme with `*``(c[7]=199/2 00),`a scheme with `*``(c[7]=324/325),`a scheme with `*``(c[7]=575/576 ),`a scheme with `*``(c[7]=1728/1729),`Tsitouras' scheme with `*``(c[7 ]=544/545),`Verner's \"most efficient\" scheme with `*``(c[7]=1999/200 0)]: errs := []:\nDigits := 25:\nfor ct to 7 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(pprint06\"" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%!G" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#K%'matrixG6 #7)7$*&%9Papakostas'~scheme~with~G\"\"\"-%!G6#/&%\"cG6#\"\"(#\"$*>\"$+ #F*$\"+?#R8w%!#D7$*&%Ka~modification~of~Papakostas'~scheme~with~GF*F+F *$\"+jR\"oQ%F87$*&%/a~scheme~with~GF*-F,6#/F/#\"$C$\"$D$F*$\"+VqJKNF87 $*&F@F*-F,6#/F/#\"$v&\"$w&F*$\"+'o8OZ#F87$*&F@F*-F,6#/F/#\"%G<\"%H>c \"%+?F*$\"+S2*3p$F8Q(pprint16\"" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }} {PARA 0 "" 0 "" {TEXT -1 30 "The following code constructs " }{TEXT 260 20 "numerical procedures" }{TEXT -1 56 " for solutions based on ea ch of the Runge-Kutta schemes." }}{PARA 0 "" 0 "" {TEXT -1 75 "The err or in the value obtained by each of the methods at the point where " }{XPPEDIT 18 0 "x = 4.999;" "6#/%\"xG-%&FloatG6$\"%**\\!\"$" }{TEXT -1 16 " is also given." }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 810 " U := (x,y) -> (1+2*(x+1)*sin(3*x))*exp(-y): hh := 0.01: numsteps := 50 0: x0 := 0: y0 := 0:\nmatrix([[`slope field: `,U(x,y)],[`initial poi nt: `,``(x0,y0)],[`step width: `,hh],\n[`no. of steps: `,numsteps] ]);``;\nmthds := [`Papakostas' scheme with `*``(c[7]=199/200),`a modif ication of Papakostas' scheme with `*``(c[7]=199/200),`a scheme with ` *``(c[7]=324/325),`a scheme with `*``(c[7]=575/576),`a scheme with `*` `(c[7]=1728/1729),`Tsitouras' scheme with `*``(c[7]=544/545),`Verner's \"most efficient\" scheme with `*``(c[7]=1999/2000)]: errs := []:\nDi gits := 25:\nfor ct to 7 do\n un_RK6_||ct := RK6_||ct(U(x,y),x,y,x0, y0,hh,numsteps,true);\nend do:\nxx := 4.999: uxx := evalf(u(xx)):\nfor ct to 7 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$%1 no.~of~steps:~~~G\"$+&Q(pprint26\"" }}{PARA 11 "" 1 "" {XPPMATH 20 "6# %!G" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#K%'matrixG6#7)7$*&%9Papakostas' ~scheme~with~G\"\"\"-%!G6#/&%\"cG6#\"\"(#\"$*>\"$+#F*$\"*(zzD;!#C7$*&% Ka~modification~of~Papakostas'~scheme~with~GF*F+F*$\"*yCXT\"F87$*&%/a~ scheme~with~GF*-F,6#/F/#\"$C$\"$D$F*$\"*MHmO#F87$*&F@F*-F,6#/F/#\"$v& \"$w&F*$\"*kD&=@F87$*&F@F*-F,6#/F/#\"%G<\"%H\"%+?F*$\"*j%[aGF8Q(ppr int36\"" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 4 "The " }{TEXT 260 22 "root mean square error" }{TEXT -1 20 " over th e interval " }{XPPEDIT 18 0 "[0, 5];" "6#7$\"\"!\"\"&" }{TEXT -1 82 " of each Runge-Kutta method is estimated as follows using the special procedure " }{TEXT 0 5 "NCint" }{TEXT -1 98 " to perform numerical \+ integration by the 7 point Newton-Cotes method over 200 equal subinter vals." }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 549 "mthds := [`Papakos tas' scheme with `*``(c[7]=199/200),`a modification of Papakostas' sch eme with `*``(c[7]=199/200),`a scheme with `*``(c[7]=324/325),`a schem e with `*``(c[7]=575/576),`a scheme with `*``(c[7]=1728/1729),`Tsitour as' scheme with `*``(c[7]=544/545),`Verner's \"most efficient\" scheme with `*``(c[7]=1999/2000)]: errs := []:\nDigits := 20:\nfor ct to 7 d o\n sm := NCint((u(x)-'un_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$*&%9Papakostas'~scheme~with~G\"\"\"-% !G6#/&%\"cG6#\"\"(#\"$*>\"$+#F*$\"+x.xiZ!#D7$*&%Ka~modification~of~Pap akostas'~scheme~with~GF*F+F*$\"+x@q)Q%F87$*&%/a~scheme~with~GF*-F,6#/F /#\"$C$\"$D$F*$\"+)zbF`$F87$*&F@F*-F,6#/F/#\"$v&\"$w&F*$\"+\\<]vCF87$* &F@F*-F,6#/F/#\"%G<\"%H$F87$*&%GVerner's~\"most~efficient\"~sch eme~with~GF*-F,6#/F/#\"%**>\"%+?F*$\"+RM,qOF8Q(pprint46\"" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 92 "The following err or graphs are constructed using the numerical procedures for the solut ions." }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 697 "evalf[20](plot(['u n_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),'un_RK6_7'(x)-u(x)],x=0..5 ,-1.3e-15..1.7e-15,font=[HELVETICA,9],\ncolor=[COLOR(RGB,.95,0,.2),COL OR(RGB,.5,0,.95),COLOR(RGB,.95,.45,0),COLOR(RGB,0,.8,.25),\nCOLOR(RGB, .95,0,.95),COLOR(RGB,.75,.2,.2),COLOR(RGB,0,.7,.9)],legend=[`Papakosta s' scheme with c[7]=199/200`,`a modification of Papakostas' scheme wit h c[7]=199/200`,`a scheme with c[7]=324/325`,`a scheme with c[7]=575/5 76`,`a scheme with c[7]=1728/1729`,`Tsitouras' scheme with c[7]=544/54 5`,`Verner's \"most efficient\" scheme with c[7]=1999/2000`],title=`er ror curves for 8 stage order 6 Runge-Kutta methods`));" }}{PARA 13 "" 1 "" {GLPLOT2D 967 587 587 {PLOTDATA 2 "6--%'CURVESG6%7eu7$$\"\"!F)F(7 $$\"5qmmmmT&)G\\a!#@$\"&i.%F-7$$\"5MLLLL3x&)*3\"!#?$\"%9xF37$$\"5++]i! *GER37F3$\"%(R)F37$$\"5nmm\"z%\\v#pK\"F3$\"%m')F37$$\"5ML$3_+ZiaW\"F3$ \"%O))F37$$\"5+++]i!R(*Rc\"F3$\"%4*)F37$$\"5nm;z>6B`#o\"F3$\"%))))F37$ $\"5MLL3xJs1,=F3$\"%3()F37$$\"5++]PM_@g>>F3$\"%2%)F37$$\"5nmmm\"H2P\"Q ?F3$\"%I!)F37$$\"5+++]PMnNrDF3$\"%)R&F37$$\"5MLLL$eRwX5$F3$\"%N>F37$$ 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&F)[F37$F]am$!&[*eF37$F]aw$!&IY'F37$Fbam$!&>0(F37$Fgam$!&58)F37$F\\bm$ !&Wl)F37$Ffbm$!&:<*F37$Fjcm$!'Q:5F37$Fddm$!'B#4\"F37$Fbfm$!'A#=\"F37$$ \"5-+v$4'\\=;'G%Fdr$!'8&>\"F37$Fgfm$!'z87F37$$\"5N3FpE:Vf-VFdr$!'()=7F 37$$\"5-]P%)R33)eI%Fdr$!&H@\"Fdr7$$\"5o\"z%*H:In\"4VFdr$!&d@\"Fdr7$$\" 5NLe9m%z`CJ%Fdr$!&*>7Fdr7$$\"5o;zW#4yE!>VFdr$!&K@\"Fdr7$F\\gm$!/Fdr 7$Ffgm$!&!y6Fdr7$F`hm$!&17\"Fdr7$Fjhm$!&u0\"Fdr7$Fdim$!%:**Fdr7$F^jm$! %&>*Fdr7$Fhjm$!%e&)Fdr7$F][n$!%EyFdr7$Fb[n$!%hrFdr7$Fg[n$!%1mFdr7$F\\ \\n$!%ygFdr7$F`]n$!%(H&Fdr7$Fj]n$!%tYFdr7$F_^n$!%hTFdr7$Fd^n$!%hPFdr7$ F^_n$!%>KFdr7$Fh_n$!%_GFdr-F[`n6&F]`nF($FaavFc`n$\"\"*Fc`n-Fe`n6#%UVer ner's~\"most~efficient\"~scheme~with~c[7]=1999/2000G-%&TITLEG6#%Uerror ~curves~for~8~stage~order~6~Runge-Kutta~methodsG-%%FONTG6$%*HELVETICAG Fd^y-%+AXESLABELSG6$Q\"x6\"Q!Fd_y-%%VIEWG6$;F(Fh_n;$!#8F]^o$\"# " 0 "" {MPLTEXT 1 0 1 ";" }}} }{SECT 1 {PARA 4 "" 0 "" {TEXT -1 47 "Test 7 of 8 stage, order 6 Runge -Kutta methods " }}{PARA 256 "" 0 "" {TEXT -1 2 " " }{XPPEDIT 18 0 "d y/dx=-(1+4*cos(3*x))*(y-1/3)" "6#/*&%#dyG\"\"\"%#dxG!\"\",$*&,&F&F&*& \"\"%F&-%$cosG6#*&\"\"$F&%\"xGF&F&F&F&,&%\"yGF&*&F&F&F2F(F(F&F(" } {TEXT -1 5 ", " }{XPPEDIT 18 0 "y(0)=1" "6#/-%\"yG6#\"\"!\"\"\"" } {TEXT -1 2 ". " }}{PARA 0 "" 0 "" {TEXT -1 10 "Solution: " }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "y = 1/3" "6#/%\"yG*&\"\"\"F &\"\"$!\"\"" }{TEXT -1 1 " " }{XPPEDIT 18 0 "exp(-4/3*sin(3*x)+8/3*sin (3/2*x)*cos(3/2*x))+2/3" "6#,&-%$expG6#,&*(\"\"%\"\"\"\"\"$!\"\"-%$sin G6#*&F+F*%\"xGF*F*F,**\"\")F*F+F,-F.6#*(F+F*\"\"#F,F1F*F*-%$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#-%$expG6#,&*(\"\"%\"\"\"\"\"$!\"\"-%$sinG6#*&F*F)%\"x GF)F)F+F0F+" }{TEXT -1 2 ". " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 181 "de := diff(y(x),x)=-(1+4*co s(3*x))*(y(x)-1/3);\nic := y(0)=1;\nsimplify(dsolve(\{de,ic\},y(x))); \nv := unapply(rhs(%),x):\nplot(v(x),x=0..5,0..1.1,font=[HELVETICA,9], labels=[`x`,`y(x)`]);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%#deG/-%%dif fG6$-%\"yG6#%\"xGF,,$*&,&\"\"\"F0*&\"\"%F0-%$cosG6#,$*&\"\"$F0F,F0F0F0 F0F0,&F)F0#F0F8!\"\"F0F;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%#icG/-% \"yG6#\"\"!\"\"\"" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/-%\"yG6#%\"xG,&* &#\"\"\"\"\"$F+-%$expG6#,&*&#\"\"%F,F+-%$sinG6#,$*&F,F+F'F+F+F+!\"\"*& #\"\")F,F+*&-F56#,$*(F,F+\"\"#F9F'F+F+F+-%$cosGF?F+F+F+F+F+*&#FBF,F+-F .6#,&F'F9*&#F3F,F+F4F+F9F+F+" }}{PARA 13 "" 1 "" {GLPLOT2D 503 318 318 {PLOTDATA 2 "6&-%'CURVESG6$7ap7$$\"\"!F)$\"\"\"F)7$$\"3gmmTN@Ki8!# >$\"3W+7cSy5h&*!#=7$$\"3ALL$3FWYs#F/$\"3KtP[t*Q;:*F27$$\"3%)***\\iSmp3 %F/$\"3g.\"H>f!3q()F27$$\"3WmmmT&)G\\aF/$\"36p*p.:G\\T)F27$$\"3m****\\ 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1 10 0 2 9 1 4 2 1.000000 45.000000 45.000000 0 0 "Curve 1" }}}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 " " 0 "" {TEXT -1 32 "The following code constructs a " }{TEXT 260 17 "d iscrete solution" }{TEXT -1 44 " based on each of the methods and give s the " }{TEXT 260 22 "root mean square error" }{TEXT -1 18 " of each \+ solution." }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 874 "V := (x,y) -> \+ -(1+4*cos(3*x))*(y-1/3): hh := 0.02: numsteps := 250: x0 := 0: y0 := 1 :\nmatrix([[`slope field: `,V(x,y)],[`initial point: `,``(x0,y0)],[` step width: `,hh],\n[`no. of steps: `,numsteps]]);``;\nmthds := [` Papakostas' scheme with `*``(c[7]=199/200),`a modification of Papakost as' scheme with `*``(c[7]=199/200),`a scheme with `*``(c[7]=324/325),` a scheme with `*``(c[7]=575/576),`a scheme with `*``(c[7]=1728/1729),` Tsitouras' scheme with `*``(c[7]=544/545),`Verner's \"most efficient\" scheme with `*``(c[7]=1999/2000)]: errs := []:\nDigits := 30:\nfor ct to 7 do\n Vn_RK6_||ct := RK6_||ct(V(x,y),x,y,x0,y0,hh,numsteps,fals e);\n sm := 0: numpts := nops(Vn_RK6_||ct):\n for ii to numpts do \n sm := sm+(Vn_RK6_||ct[ii,2]-v(Vn_RK6_||ct[ii,1]))^2;\n end 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$%0slope~field:~~~G,$*&,&\"\"\"F,*&\"\"%F,-%$cosG6#, $*&\"\"$F,%\"xGF,F,F,F,F,,&%\"yGF,#F,F4!\"\"F,F97$%0initial~point:~G-% !G6$\"\"!F,7$%/step~width:~~~G$\"\"#!\"#7$%1no.~of~steps:~~~G\"$]#Q(pp rint56\"" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%!G" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#K%'matrixG6#7)7$*&%9Papakostas'~scheme~with~G\"\"\"-%!G 6#/&%\"cG6#\"\"(#\"$*>\"$+#F*$\"+s,dt?!#A7$*&%Ka~modification~of~Papak ostas'~scheme~with~GF*F+F*$\"+k*fy.#F87$*&%/a~scheme~with~GF*-F,6#/F/# \"$C$\"$D$F*$\"+Fe!z.#F87$*&F@F*-F,6#/F/#\"$v&\"$w&F*$\"+a1b#y\"F87$*& F@F*-F,6#/F/#\"%G<\"%H\"%+?F*$\"+kk@/)*!#BQ(pprint66\"" }}} {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 0 805 "V := (x,y) -> -(1+4*cos(3*x))*(y-1/3): hh \+ := 0.02: numsteps := 250: x0 := 0: y0 := 1:\nmatrix([[`slope field: \+ `,V(x,y)],[`initial point: `,``(x0,y0)],[`step width: `,hh],\n[`no. \+ of steps: `,numsteps]]);``;\nmthds := [`Papakostas' scheme with `*`` (c[7]=199/200),`a modification of Papakostas' scheme with `*``(c[7]=19 9/200),`a scheme with `*``(c[7]=324/325),`a scheme with `*``(c[7]=575/ 576),`a scheme with `*``(c[7]=1728/1729),`Tsitouras' scheme with `*``( c[7]=544/545),`Verner's \"most efficient\" scheme with `*``(c[7]=1999/ 2000)]: errs := []:\nDigits := 30:\nfor ct to 7 do\n vn_RK6_||ct := \+ RK6_||ct(V(x,y),x,y,x0,y0,hh,numsteps,true);\nend do:\nxx := 4.999: vx x := evalf(v(xx)):\nfor ct to 7 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$%0slope~fi eld:~~~G,$*&,&\"\"\"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\"$]#Q(pprint76\"" }}{PARA 11 "" 1 " " {XPPMATH 20 "6#%!G" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#K%'matrixG6#7) 7$*&%9Papakostas'~scheme~with~G\"\"\"-%!G6#/&%\"cG6#\"\"(#\"$*>\"$+#F* $\"+l#)evA!#B7$*&%Ka~modification~of~Papakostas'~scheme~with~GF*F+F*$ \"+*oD!fAF87$*&%/a~scheme~with~GF*-F,6#/F/#\"$C$\"$D$F*$\"+>>6'>#F87$* &F@F*-F,6#/F/#\"$v&\"$w&F*$\"+JC2<>F87$*&F@F*-F,6#/F/#\"%G<\"%H \"%+?F*$\"+jLPWmFboQ(pprint86\"" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }} {PARA 0 "" 0 "" {TEXT -1 4 "The " }{TEXT 260 22 "root mean square erro r" }{TEXT -1 20 " over the interval " }{XPPEDIT 18 0 "[0, 5];" "6#7$ \"\"!\"\"&" }{TEXT -1 82 " of each Runge-Kutta method is estimated as follows using the special procedure " }{TEXT 0 5 "NCint" }{TEXT -1 98 " to perform numerical integration by the 7 point Newton-Cotes met hod over 100 equal subintervals." }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 549 "mthds := [`Papakostas' scheme with `*``(c[7]=199/200),`a modi fication of Papakostas' scheme with `*``(c[7]=199/200),`a scheme with \+ `*``(c[7]=324/325),`a scheme with `*``(c[7]=575/576),`a scheme with `* ``(c[7]=1728/1729),`Tsitouras' scheme with `*``(c[7]=544/545),`Verner' s \"most efficient\" scheme with `*``(c[7]=1999/2000)]: errs := []:\nD igits := 20:\nfor ct to 7 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]([mthds,evalf(e rrs)]);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#K%'matrixG6#7)7$*&%9Papakos tas'~scheme~with~G\"\"\"-%!G6#/&%\"cG6#\"\"(#\"$*>\"$+#F*$\"+:=%z.#!#A 7$*&%Ka~modification~of~Papakostas'~scheme~with~GF*F+F*$\"+k8x.?F87$*& %/a~scheme~with~GF*-F,6#/F/#\"$C$\"$D$F*$\"+s+T0?F87$*&F@F*-F,6#/F/#\" $v&\"$w&F*$\"+6Fea\"%+?F*$\"+jB'Gn*! #BQ(pprint96\"" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 92 "The following error graphs are constructed using the nume rical procedures for the solutions." }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 679 "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),v(x)-'vn_RK6_7'(x)],x=0..5,font=[HELVETICA,9],\ncolor=[COLOR (RGB,.95,0,.2),COLOR(RGB,.5,0,.95),COLOR(RGB,.95,.45,0),COLOR(RGB,0,.8 ,.25),\nCOLOR(RGB,.95,0,.95),COLOR(RGB,.75,.2,.2),COLOR(RGB,0,.7,.9)], legend=[`Papakostas' scheme with c[7]=199/200`,`a modification of Papa kostas' scheme with c[7]=199/200`,`a scheme with c[7]=324/325`,`a sche me with c[7]=575/576`,`a scheme with c[7]=1728/1729`,`Tsitouras' schem e with c[7]=544/545`,`Verner's \"most efficient\" scheme with c[7]=199 9/2000`],title=`error curves for 8 stage order 6 Runge-Kutta methods`) );" }}{PARA 13 "" 1 "" {GLPLOT2D 1025 579 579 {PLOTDATA 2 "6--%'CURVES G6%7\\w7$$\"\"!F)F(7$$\"5qmm;aQ`!eS$!#A$!$,\"!#?7$$\"5SLLL3x1h6oF-$!&y H\"F07$$\"5,++Dc,;u@5!#@$!'pGAF07$$\"5ommmTN@Ki8F9$!(**en\"F07$$\"5NLL 3FpE!Hq\"F9$!(\"36!)F07$$\"5-++]7.K[V?F9$!)*4nY#F07$$\"5omm\"zptjSQ#F9 $!)8:DCF07$$\"5NLLL$3FWYs#F9$!)=&eQ#F07$$\"5-++vo/[AlIF9$!),UnBF07$$\" 5omm;aQ`!eS$F9$!)!)emCF07$$\"5NLLeRseQYPF9$!)w<-IF07$$\"5-+++D1k'p3%F9 $!)Gs9TF07$$\"5pmmT5SpaFWF9$!)\\sXSF07$$\"5OLL$eRZF\"oZF9$!)OmzRF07$$ \"50++D\"y+3(3^F9$!))\\T$RF07$$\"5qmmmmT&)G\\aF9$!)DQ$*RF07$$\"5SLL3_v !p)*y&F9$!)hk?WF07$$\"50++]P4'\\/8'F9$!)1'*)3&F07$$\"5qmm\"HK9I5Z'F9$! )%*G/]F07$$F3F9$!).*G#\\F07$$\"50++v$4@\">_rF9$!)!>,'[F07$$\"5qmm;zW&F07$$\"50+++]7G$R<)F9$!)=RIb F07$$\"5SLL$3-)Q4b))F9$!)ie^`F07$$\"5qmmm\"z%\\DO&*F9$!)v!)p_F07$$\"5N 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" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 178 "de := \+ diff(y(x),x)=x*(9-x^2)/(1+y(x)^2);\nic := y(0)=0;\ndsolve(\{de,ic\},y( x));\nw := unapply(rhs(%),x):\nplot(w(x),x=0..4,0..3.7,numpoints=75,fo nt=[HELVETICA,9],labels=[`x`,`y(x)`]);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%#deG/-%%diffG6$-%\"yG6#%\"xGF,*(F,\"\"\",&\"\"*F.*$)F,\"\"#F.! \"\"F.,&F.F.*$)F)F3F.F.F4" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%#icG/-% \"yG6#\"\"!F)" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/-%\"yG6#%\"xG,&*&\" \"#!\"\",(*&\"\"$\"\"\")F'\"\"%F/F+*&\"#aF/)F'F*F/F/*$,*\"#kF/*&\"\"*F /)F'\"\")F/F/*&\"$C$F/)F'\"\"'F/F+*&\"%;HF/F0F/F/#F/F*F/#F/F.F/*&F*F/F ,#F+F.F+" }}{PARA 13 "" 1 "" {GLPLOT2D 503 318 318 {PLOTDATA 2 "6&-%'C URVESG6$7io7$$\"\"!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 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3a59(yGT$>NFho7$$\"3['['[1m/8NFho$\"3uI!*3pTpuMFho7$$\"3E#*=*=p]\"pNFh o$\"3s<,a='3,U$Fho7$$\"37.Fq-Y#3i$Fho$\"3w<#3Q&zxhLFho7$$\"31wcnbdutOF ho$\"3?^'pWBqHH$Fho7$$\"32dnv'*p'o+i*\\/FFFho7$$ \"\"%F)$\"3CxC=rRoRDFho-%'COLOURG6&%$RGBG$\"#5!\"\"F(F(-%%FONTG6$%*HEL VETICAG\"\"*-%+AXESLABELSG6$%\"xG%%y(x)G-%%VIEWG6$;F(F`cl;F($\"#PFjcl " 1 2 0 1 10 0 2 9 1 4 2 1.000000 45.000000 45.000000 0 0 "Curve 1" }} }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 32 "The fol lowing code constructs a " }{TEXT 260 17 "discrete solution" }{TEXT -1 44 " based on each of the methods and gives the " }{TEXT 260 22 "ro ot mean square error" }{TEXT -1 18 " of each solution." }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 868 "W := (x,y) -> x*(9-x^2)/(1+y^2): h h := 0.01: numsteps := 400: x0 := 0: y0 := 0:\nmatrix([[`slope field: \+ `,W(x,y)],[`initial point: `,``(x0,y0)],[`step width: `,hh],\n[`no . of steps: `,numsteps]]);``;\nmthds := [`Papakostas' scheme with `* ``(c[7]=199/200),`a modification of Papakostas' scheme with `*``(c[7]= 199/200),`a scheme with `*``(c[7]=324/325),`a scheme with `*``(c[7]=57 5/576),`a scheme with `*``(c[7]=1728/1729),`Tsitouras' scheme with `*` `(c[7]=544/545),`Verner's \"most efficient\" scheme with `*``(c[7]=199 9/2000)]: errs := []:\nDigits := 30:\nfor ct to 7 do\n Wn_RK6_||ct : = RK6_||ct(W(x,y),x,y,x0,y0,hh,numsteps,false);\n sm := 0: numpts := nops(Wn_RK6_||ct):\n for ii to numpts do\n sm := sm+(Wn_RK6_|| ct[ii,2]-w(Wn_RK6_||ct[ii,1]))^2;\n end do:\n errs := [op(errs),sq rt(sm/numpts)];\nend do:\nDigits := 10:\nlinalg[transpose]([mthds,eval f(errs)]);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#K%'matrixG6#7&7$%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)pprint106\"" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%!G " }}{PARA 11 "" 1 "" {XPPMATH 20 "6#K%'matrixG6#7)7$*&%9Papakostas'~sc heme~with~G\"\"\"-%!G6#/&%\"cG6#\"\"(#\"$*>\"$+#F*$\"+*4q*oC!#D7$*&%Ka ~modification~of~Papakostas'~scheme~with~GF*F+F*$\"+%)))[7@F87$*&%/a~s cheme~with~GF*-F,6#/F/#\"$C$\"$D$F*$\"+RNVPy!#E7$*&F@F*-F,6#/F/#\"$v& \"$w&F*$\"+&zB=)zFI7$*&F@F*-F,6#/F/#\"%G<\"%H\"%+?F*$\"+#pQv?\"F8 Q)pprint116\"" }}}{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 \+ = 3.499;" "6#/%\"xG-%&FloatG6$\"%*\\$!\"$" }{TEXT -1 16 " is also giv en." }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 799 "W := (x,y) -> x*(9-x ^2)/(1+y^2): hh := 0.01: numsteps := 400: x0 := 0: y0 := 0:\nmatrix([[ `slope field: `,W(x,y)],[`initial point: `,``(x0,y0)],[`step width: \+ `,hh],\n[`no. of steps: `,numsteps]]);``;\nmthds := [`Papakostas' \+ scheme with `*``(c[7]=199/200),`a modification of Papakostas' scheme w ith `*``(c[7]=199/200),`a scheme with `*``(c[7]=324/325),`a scheme wit h `*``(c[7]=575/576),`a scheme with `*``(c[7]=1728/1729),`Tsitouras' s cheme with `*``(c[7]=544/545),`Verner's \"most efficient\" scheme with `*``(c[7]=1999/2000)]: errs := []:\nDigits := 30:\nfor ct to 7 do\n \+ wn_RK6_||ct := RK6_||ct(W(x,y),x,y,x0,y0,hh,numsteps,true);\nend do: \nxx := 3.499: wxx := evalf(w(xx)):\nfor ct to 7 do\n errs := [op(er rs),abs(wn_RK6_||ct(xx)-wxx)];\nend do:\nDigits := 10:\nlinalg[transpo se]([mthds,evalf(errs)]);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#K%'matrix G6#7&7$%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\"$+%Q)pprint126\"" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%!G" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#K%'matrixG6#7)7$ *&%9Papakostas'~scheme~with~G\"\"\"-%!G6#/&%\"cG6#\"\"(#\"$*>\"$+#F*$ \"+!GhBB*!#E7$*&%Ka~modification~of~Papakostas'~scheme~with~GF*F+F*$\" +@*G(4\")F87$*&%/a~scheme~with~GF*-F,6#/F/#\"$C$\"$D$F*$\"+F=uqy!#F7$* &F@F*-F,6#/F/#\"$v&\"$w&F*$\"+kVm\"[#F87$*&F@F*-F,6#/F/#\"%G<\"%H \"%+?F*$\"+()**[6DF8Q)pprint136\"" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }} {PARA 0 "" 0 "" {TEXT -1 4 "The " }{TEXT 260 22 "root mean square erro r" }{TEXT -1 20 " over the interval " }{XPPEDIT 18 0 "[0,4]" "6#7$\" \"!\"\"%" }{TEXT -1 82 " of each Runge-Kutta method is estimated as f ollows using the special procedure " }{TEXT 0 5 "NCint" }{TEXT -1 98 " to perform numerical integration by the 7 point Newton-Cotes method over 200 equal subintervals." }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 549 "mthds := [`Papakostas' scheme with `*``(c[7]=199/200),`a modifi cation of Papakostas' scheme with `*``(c[7]=199/200),`a scheme with `* ``(c[7]=324/325),`a scheme with `*``(c[7]=575/576),`a scheme with `*`` (c[7]=1728/1729),`Tsitouras' scheme with `*``(c[7]=544/545),`Verner's \+ \"most efficient\" scheme with `*``(c[7]=1999/2000)]: errs := []:\nDig its := 20:\nfor ct to 7 do\n sm := NCint((w(x)-'wn_RK6_||ct'(x))^2,x =0..4,adaptive=false,numpoints=7,factor=200);\n errs := [op(errs),sq rt(sm/4)];\nend do:\nDigits := 10:\nlinalg[transpose]([mthds,evalf(err s)]);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#K%'matrixG6#7)7$*&%9Papakosta s'~scheme~with~G\"\"\"-%!G6#/&%\"cG6#\"\"(#\"$*>\"$+#F*$\"+6n46C!#D7$* &%Ka~modification~of~Papakostas'~scheme~with~GF*F+F*$\"+**>3g?F87$*&%/ a~scheme~with~GF*-F,6#/F/#\"$C$\"$D$F*$\"+[r=Rs!#E7$*&F@F*-F,6#/F/#\"$ v&\"$w&F*$\"+**yEUwFI7$*&F@F*-F,6#/F/#\"%G<\"%H\"%+?F*$\"+vXu/7F 8Q)pprint146\"" }}}{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 0 697 "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),w(x)-'wn_RK6_7'(x)],x=0..4,-9.5e-16..7.9e-16,font=[HELVETICA ,9],\ncolor=[COLOR(RGB,.95,0,.2),COLOR(RGB,.5,0,.95),COLOR(RGB,.95,.45 ,0),COLOR(RGB,0,.8,.25),\nCOLOR(RGB,.95,0,.95),COLOR(RGB,.75,.2,.2),CO LOR(RGB,0,.7,.9)],legend=[`Papakostas' scheme with c[7]=199/200`,`a mo dification of Papakostas' scheme with c[7]=199/200`,`a scheme with c[7 ]=324/325`,`a scheme with c[7]=575/576`,`a scheme with c[7]=1728/1729` ,`Tsitouras' scheme with c[7]=544/545`,`Verner's \"most efficient\" sc heme with c[7]=1999/2000`],title=`error curves for 8 stage order 6 Run ge-Kutta methods`));" }}{PARA 13 "" 1 "" {GLPLOT2D 989 655 655 {PLOTDATA 2 "6--%'CURVESG6%7fp7$$\"\"!F)F(7$$\"5mmmmm;arz@!#@$\"%XG!#A 7$$\"5LLLLLL3VfVF-$\"&'*=#F07$$\"5lmmmmT&)G\\aF-$\"%vTF-7$$\"5+++++]i9 RlF-$\"%/rF-7$$\"5NLLLLeR+HwF-$\"&V7\"F-7$$\"5mmmmmm;')=()F-$\"&Sr\"F- 7$$\"5++++]7z>^7!#?$\"&H=&F-7$$\"5LLLLLe'40j\"FM$\"%\\$*FM7$$\"5+++]i! f`rt\"FM$\"%@**FM7$$\"5mmmm\"H_(zV=FM$\"&0,\"FM7$$\"5+++D1*[>r*=FM$\"% 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$]#F[w7$Fi_l$\"$\\#F[w7$F^`l$\"$[#F[w7$Fc`lFcdr7$Fh`lFcdr7$F]alF]dr7$F bal$\"$g#F[w7$Fgal$\"$m#F[w7$F\\blFeer7$FablFber7$Ffbl$\"$Z#F[w7$F[cl$ \"$A#F[w7$F`clFbdn7$Fecl$\"#qF[w7$Fjcl$!$F\"F[w7$F_dlFj[q7$Fddl$!$$)*F [w7$FidlFfdl7$F^el$!%z@F[w7$Fcel$!%8HF[w7$Fhel$!%BNF[w7$F]fl$!%xSF[w7$ Fbfl$!%&f%F[w7$Fgfl$!%@YF[w7$F\\gl$!%)z%F[w7$Fagl$!%^gF[w-Ffgl6&Fhgl$F fgnF[hlF\\hlF\\hl-F`hl6#%DTsitouras'~scheme~with~c[7]=544/545G-F$6%7ds F'7$F+$!&D!zF07$F2$!'oxBF07$F<$!&[r&F-7$FF$!'Fw6F-7$F_bp$!'xj:F-7$Fe]o $!'zC?F-7$Fgbp$!'GaDF-7$FK$!'%[8$F-7$F]^o$!'PYPF-7$Fb^o$!'2eVF-7$Fg^o$ !&'H\\FM7$FQ$!&qT&FM7$FV$!&!odFM7$Fen$!&[$fFM7$F_o$!&%zeFM7$Fio$!&zc&F M7$F_eq$!&p-&FM7$Fe_o$!&u)\\FM7$$\"5ML$ekGI!R!>#FM$!&%eYFM7$Fgeq$!&j@% FM7$$\"5nm;/,pArVAFM$!&p?%FM7$F^p$!&E9%FM7$F_fq$!&QU$FM7$Fdfq$!&J?$FM7 $Fifq$!&E>$FM7$F]`o$!&D0$FM7$Fagq$!&#f?FM7$Ffgq$!&W0#FM7$F[hq$!&m.#FM7 $Fcp$!&Rw\"FM7$$\"5nmm;/E))\\5DFM$!%i&)FM7$Fgdp$!%C&)FM7$$\"5nmm;zWU)FM7$Fhp$!%)4$FM7$$\"5nmm;aj'\\yh#FM$\"%\\IFM7$F_ep$\"%]IFM7$$ \"5nmm;H#3D:n#FM$\"%6PFM7$F]q$\"&Z?\"FM7$$\"5nmm;/,0?DFFM$\"&#H8FM7$Fb q$\"&%G8FM7$$\"5nmm;z>f()yFFM$\"&1W\"FM7$Fgq$\"&89#FM7$$\"5nmm;aQ8bKGF M$\"&R8#FM7$F\\r$\"&H8#FM7$$\"5nmm;HdnA')GFM$\"&tG#FM7$Far$\"&Pn#FM7$F j[n$\"&Sm#FM7$F_\\n$\"&;m#FM7$Fd\\n$\"&(*z#FM7$Fi\\n$\"&8!HFM7$F^]n$\" &B(GFM7$Ffr$\"&x#GFM7$Ff]n$\"&N[#FM7$F[s$\"&A#>FM7$$\"5MLL3_D\"))eR$FM $\"&'>9FM7$F^^n$\"&;@\"FM7$$\"5ML$3-))e=gZ$FM$\"&:7\"FM7$$\"5nmm\"HKuG F]$FM$\"%MVFM7$$\"5++]il(*)Q%HNFM$\"%6VFM7$F`s$\"%3UFM7$$\"5nm;/^1#fGe $FM$\"%\"[#FM7$$\"5+++v$4Op&4OFM$!%BPFM7$$\"5ML$ek`^zij$FM$!%0PFM7$Ff^ n$!%vQFM7$$\"5++](=U#)*p*o$FM$!%8pFM7$$\"5MLLeky*4kr$FM$!&?8\"FM7$$\"5 nm;H2L,7VPFM$!&v7\"FM7$Fes$!&+;\"FM7$F^_n$!&r&=FM7$Fjs$!&qX#FM7$Ff_n$! &!QHFM7$F_t$!&MH$FM7$$\"5+++]i!\\c>A%FM$!&$HLFM7$$\"5mmmm\"HA?nC%FM$!& /J$FM7$$\"5LLL$3_&R[rUFM$!&GI$FM7$$\"5++++](oZiH%FM$!&RS$FM7$$\"5LLLL3 _^xXVFM$!&1T$FM7$Fdt$!&iV$FM7$$\"5++++D\"3I[W%FM$!&nU$FM7$$\"5LLLL$ead V\\%FM$!&jS$FM7$$\"5mmmmT5])Qa%FM$!&*yLFM7$Fit$!&wK$FM7$Fcu$!&$yIFM7$F ]v$!& " 0 "" {MPLTEXT 1 0 1 ";" }}}}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 48 "Test 9 of 8 stage, order 6 Runge-Kutta methods " }} {PARA 256 "" 0 "" {TEXT -1 3 " " }{XPPEDIT 18 0 "dy/dx=-(1+cos(2*x)) *y^3" "6#/*&%#dyG\"\"\"%#dxG!\"\",$*&,&F&F&-%$cosG6#*&\"\"#F&%\"xGF&F& F&*$%\"yG\"\"$F&F(" }{TEXT -1 3 ", " }{XPPEDIT 18 0 "y(0) = sqrt(2); " "6#/-%\"yG6#\"\"!-%%sqrtG6#\"\"#" }{TEXT -1 1 " " }}{PARA 0 "" 0 "" {TEXT -1 10 "Solution: " }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "y = 1/sqrt(sin(2*x)+2*x+1/2)" "6#/%\"yG*&\"\"\"F&-%%sqrtG6#,(-%$ sinG6#*&\"\"#F&%\"xGF&F&*&F/F&F0F&F&*&F&F&F/!\"\"F&F3" }{TEXT -1 2 ". \+ " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 172 "de := diff(y(x),x)=-(1+cos(2*x))*y(x)^3;\nic := y(0)=sqrt(2); \ndsolve(\{de,ic\},y(x));\nm := unapply(rhs(%),x):\nplot(m(x),x=0..3,0 ..1.42,font=[HELVETICA,9],labels=[`x`,`y(x)`]);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%#deG/-%%diffG6$-%\"yG6#%\"xGF,,$*&,&\"\"\"F0-%$cosG6# ,$*&\"\"#F0F,F0F0F0F0)F)\"\"$F0!\"\"" }}{PARA 11 "" 1 "" {XPPMATH 20 " 6#>%#icG/-%\"yG6#\"\"!*$\"\"##\"\"\"F+" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/-%\"yG6#%\"xG*&\"\"\"F)*$,(*(\"\"#F)-%$cosGF&F)-%$sinGF&F)F)*&F -F)F'F)F)#F)F-F)#F)F-!\"\"" }}{PARA 13 "" 1 "" {GLPLOT2D 503 318 318 {PLOTDATA 2 "6&-%'CURVESG6$7Y7$$\"\"!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$$\"3;+]iSS\"Ga\"FJ$\"3IyW%eHk>[*FJ7$$\"3+++]Peui=FJ$\"3#4`!o2+ #G**)FJ7$$\"37+++]$)z%=#FJ$\"3OGH4wwYu&)FJ7$$\"3A++]i3&o]#FJ$\"3=1g%=M 2W@)FJ7$$\"3%)***\\(oX*y9$FJ$\"31u2v$Q9&GwFJ7$$\"3z***\\P9CAu$FJ$\"3=X IMTf7+sFJ7$$\"3!)***\\P*zhdVFJ$\"3P$G(zQ8#4%oFJ7$$\"31++v$>fS*\\FJ$\"3 X'3%RcqqPlFJ7$$\"3$)***\\(=$f%GcFJ$\"3mYY%G?7\"*G'FJ7$$\"3Q+++Dy,\"G'F J$\"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$$\"3 1+]Pa&4*\\7F,$\"3OjcS#)ygr_FJ7$$\"33+]7j=_68F,$\"3'e4m\")R`oD&FJ7$$\"3 3++vVy!eP\"F,$\"3a@U-1/NZ_FJ7$$\"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$$\"3'***\\iv&QAi\"F,$\"3:*4?^OZ,C&FJ7$$\"31++vtLU%o\"F,$\"3\"3g SMou)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$;F(Fc\\l;F($\"$U\"!\"#" 1 2 0 1 10 0 2 9 1 4 2 1.000000 45.000000 45.000000 0 0 "Curve 1" }}}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 32 "The following code constructs a " }{TEXT 260 17 "discrete solution" }{TEXT -1 44 " based on each of t he methods and gives the " }{TEXT 260 22 "root mean square error" } {TEXT -1 18 " of each solution." }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 881 "M := (x,y) -> -(1+cos(2*x))*y^3: hh := 0.01: numsteps := 300: x0 := 0: y0 := sqrt(2):\nmatrix([[`slope field: `,M(x,y)],[`initial point: `,``(x0,y0)],[`step width: `,hh],\n[`no. of steps: `,numst eps]]);``;\nmthds := [`Papakostas' scheme with `*``(c[7]=199/200),`a m odification of Papakostas' scheme with `*``(c[7]=199/200),`a scheme wi th `*``(c[7]=324/325),`a scheme with `*``(c[7]=575/576),`a scheme with `*``(c[7]=1728/1729),`Tsitouras' scheme with `*``(c[7]=544/545),`Vern er's \"most efficient\" scheme with `*``(c[7]=1999/2000)]: errs := []: \nDigits := 20:\nfor ct to 7 do\n Mn_RK6_||ct := RK6_||ct(M(x,y),x,y ,x0,evalf(y0),hh,numsteps,false);\n sm := 0: numpts := nops(Mn_RK6_| |ct):\n for ii to numpts do\n sm := sm+(Mn_RK6_||ct[ii,2]-m(Mn_ RK6_||ct[ii,1]))^2;\n end do:\n errs := [op(errs),sqrt(sm/numpts)] ;\nend do:\nDigits := 10:\nlinalg[transpose]([mthds,evalf(errs)]);" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#K%'matrixG6#7&7$%0slope~field:~~~G,$*& ,&\"\"\"F,-%$cosG6#,$*&\"\"#F,%\"xGF,F,F,F,)%\"yG\"\"$F,!\"\"7$%0initi al~point:~G-%!G6$\"\"!*$F2#F,F27$%/step~width:~~~G$F,!\"#7$%1no.~of~st eps:~~~G\"$+$Q)pprint156\"" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%!G" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#K%'matrixG6#7)7$*&%9Papakostas'~scheme ~with~G\"\"\"-%!G6#/&%\"cG6#\"\"(#\"$*>\"$+#F*$\"+s\\nsL!#B7$*&%Ka~mod ification~of~Papakostas'~scheme~with~GF*F+F*$\"+&4$y,=F87$*&%/a~scheme ~with~GF*-F,6#/F/#\"$C$\"$D$F*$\"+o2\"3N\"F87$*&F@F*-F,6#/F/#\"$v&\"$w &F*$\"+**pzzBF87$*&F@F*-F,6#/F/#\"%G<\"%H\"%+?F*$\"+p_j&H\"!#AQ)ppr int166\"" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 30 "The following code constructs " }{TEXT 260 20 "numerical procedure s" }{TEXT -1 56 " for solutions based on each of the Runge-Kutta schem es." }}{PARA 0 "" 0 "" {TEXT -1 75 "The error in the value obtained by each of the methods at the point where " }{XPPEDIT 18 0 "x = 2.999; " "6#/%\"xG-%&FloatG6$\"%**H!\"$" }{TEXT -1 16 " is also given." }} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 812 "M := (x,y) -> -(1+cos(2*x)) *y^3: hh := 0.01: numsteps := 300: x0 := 0: y0 := sqrt(2):\nmatrix([[` slope field: `,M(x,y)],[`initial point: `,``(x0,y0)],[`step width: \+ `,hh],\n[`no. of steps: `,numsteps]]);``;\nmthds := [`Papakostas' s cheme with `*``(c[7]=199/200),`a modification of Papakostas' scheme wi th `*``(c[7]=199/200),`a scheme with `*``(c[7]=324/325),`a scheme with `*``(c[7]=575/576),`a scheme with `*``(c[7]=1728/1729),`Tsitouras' sc heme with `*``(c[7]=544/545),`Verner's \"most efficient\" scheme with \+ `*``(c[7]=1999/2000)]: errs := []:\nDigits := 25:\nfor ct to 7 do\n \+ mn_RK6_||ct := RK6_||ct(M(x,y),x,y,x0,evalf(y0),hh,numsteps,true);\nen d do:\nxx := 2.999: mxx := evalf(m(xx)):\nfor ct to 7 do\n errs := [ op(errs),abs(mn_RK6_||ct(xx)-mxx)];\nend do:\nDigits := 10:\nlinalg[tr anspose]([mthds,evalf(errs)]);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#K%'m atrixG6#7&7$%0slope~field:~~~G,$*&,&\"\"\"F,-%$cosG6#,$*&\"\"#F,%\"xGF ,F,F,F,)%\"yG\"\"$F,!\"\"7$%0initial~point:~G-%!G6$\"\"!*$F2#F,F27$%/s tep~width:~~~G$F,!\"#7$%1no.~of~steps:~~~G\"$+$Q)pprint176\"" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%!G" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#K%'ma trixG6#7)7$*&%9Papakostas'~scheme~with~G\"\"\"-%!G6#/&%\"cG6#\"\"(#\"$ *>\"$+#F*$\"+LqHko!#C7$*&%Ka~modification~of~Papakostas'~scheme~with~G F*F+F*$\"+z?*4*RF87$*&%/a~scheme~with~GF*-F,6#/F/#\"$C$\"$D$F*$\"+Tr1* )HF87$*&F@F*-F,6#/F/#\"$v&\"$w&F*$\"+MFA)z$F87$*&F@F*-F,6#/F/#\"%G<\"% H\"%+?F*$\"+L9$zK#FgnQ)pprint186\"" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 4 "The " }{TEXT 260 22 "root mean s quare error" }{TEXT -1 20 " over the interval " }{XPPEDIT 18 0 "[0, 3 ];" "6#7$\"\"!\"\"$" }{TEXT -1 82 " of each Runge-Kutta method is est imated as follows using the special procedure " }{TEXT 0 5 "NCint" } {TEXT -1 98 " to perform numerical integration by the 7 point Newton- Cotes method over 150 equal subintervals." }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 549 "mthds := [`Papakostas' scheme with `*``(c[7]=199/200 ),`a modification of Papakostas' scheme with `*``(c[7]=199/200),`a sch eme with `*``(c[7]=324/325),`a scheme with `*``(c[7]=575/576),`a schem e with `*``(c[7]=1728/1729),`Tsitouras' scheme with `*``(c[7]=544/545) ,`Verner's \"most efficient\" scheme with `*``(c[7]=1999/2000)]: errs \+ := []:\nDigits := 20:\nfor ct to 7 do\n sm := NCint((m(x)-'mn_RK6_|| ct'(x))^2,x=0..3,adaptive=false,numpoints=7,factor=150);\n errs := [ op(errs),sqrt(sm/3)];\nend do:\nDigits := 10:\nlinalg[transpose]([mthd s,evalf(errs)]);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#K%'matrixG6#7)7$*& %9Papakostas'~scheme~with~G\"\"\"-%!G6#/&%\"cG6#\"\"(#\"$*>\"$+#F*$\"+ 5swJL!#B7$*&%Ka~modification~of~Papakostas'~scheme~with~GF*F+F*$\"+1M' Hy\"F87$*&%/a~scheme~with~GF*-F,6#/F/#\"$C$\"$D$F*$\"+S4eO8F87$*&F@F*- F,6#/F/#\"$v&\"$w&F*$\"+IGlSBF87$*&F@F*-F,6#/F/#\"%G<\"%H@g-' F87$*&%8Tsitouras'~scheme~with~GF*-F,6#/F/#\"$W&\"$X&F*$\"+Zc)QV*F87$* &%GVerner's~\"most~efficient\"~scheme~with~GF*-F,6#/F/#\"%**>\"%+?F*$ \"+f4#oF\"!#AQ)pprint196\"" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 92 "The following error graphs are constructed usin g the numerical procedures for the solutions." }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 681 "evalf[20](plot(['mn_RK6_1'(x)-m(x),'mn_RK6_2'(x )-m(x),'mn_RK6_3'(x)-m(x),'mn_RK6_4'(x)-m(x),\n'mn_RK6_5'(x)-m(x),'mn_ RK6_6'(x)-m(x),'mn_RK6_7'(x)-m(x)],x=0..0.5,font=[HELVETICA,9],\ncolor =[COLOR(RGB,.95,0,.2),COLOR(RGB,.5,0,.95),COLOR(RGB,.95,.45,0),COLOR(R GB,0,.8,.25),\nCOLOR(RGB,.95,0,.95),COLOR(RGB,.75,.2,.2),COLOR(RGB,0,. 7,.9)],legend=[`Papakostas' scheme with c[7]=199/200`,`a modification \+ of Papakostas' scheme with c[7]=199/200`,`a scheme with c[7]=324/325`, `a scheme with c[7]=575/576`,`a scheme with c[7]=1728/1729`,`Tsitouras ' scheme with c[7]=544/545`,`Verner's \"most efficient\" scheme with c [7]=1999/2000`],title=`error curves for 8 stage order 6 Runge-Kutta me thods`));" }}{PARA 13 "" 1 "" {GLPLOT2D 967 619 619 {PLOTDATA 2 "6--%' CURVESG6%7hq7$$\"\"!F)F(7$$\"5ommmTN@Ki8!#A$!\"#!#>7$$\"5NLLL$3FWYs#F- $!$2#F07$$\"5omm;aQ`!eS$F-$!$G*F07$$\"5-+++D1k'p3%F-$!%*F-$!'WkVF07$$\"5qmmm\"z%\\DO&*F-$!'`*>&F07$$\"5NLL3 x\"[No()*F-$!':$4'F07$$\"5+++Dc,;u@5!#@$!'>5kF07$$\"5nm;z%\\l*zb5Fdp$! 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diff(y(x),x)=-(2*sin( 5*x)+3*cos(7*x))*sinh(y(x));\nic := y(0)=sqrt(5)/2;\ndsolve(\{de,ic\}, y(x));\nsimplify(convert(%,exp));\np := unapply(rhs(%),x):\nplot(p(x), x=0..5,font=[HELVETICA,9],labels=[`x`,`y(x)`]);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%#deG/-%%diffG6$-%\"yG6#%\"xGF,,$*&,&*&\"\"#\"\"\"-%$s inG6#,$*&\"\"&F2F,F2F2F2F2*&\"\"$F2-%$cosG6#,$*&\"\"(F2F,F2F2F2F2F2-%% sinhG6#F)F2!\"\"" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%#icG/-%\"yG6#\" \"!,$*&\"\"#!\"\"\"\"&#\"\"\"F,F0" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/ -%\"yG6#%\"xG-%#lnG6#-%%tanhG6#,*#\"\"\"\"\"&F0*&#F0\"\"#F0-F)6#,$*&,& -%$expG6#,$*&F4!\"\"F1F3F0F0F0F0F0,&F:F0F0F?F?F?F0F0*&#\"\"$\"#9F0-%$s inG6#,$*&\"\"(F0F'F0F0F0F0*&#F0F1F0-%$cosG6#,$*&F1F0F'F0F0F0F?" }} {PARA 12 "" 1 "" {XPPMATH 20 "6#/-%\"yG6#%\"xG-%#lnG6#,$*&,*-%$expG6#, 4#\"\"#\"\"&\"\"\"*&#\"$#>\"\"(F5*&-%$sinGF&F5)-%$cosGF&\"\"'F5F5F5*&# \"$S#F9F5*&F;F5)F>\"\"%F5F5!\"\"*&#\"#sF9F5*&F;F5)F>F3F5F5F5*&#\"\"$F9 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3E>(p'zRPv&)F87$$\"3!Q3_+sD-=%F,$\"3Ga;6C(p2d)F87$$\"3#Q$3xc9[$>%F,$\" 3MZZFn2MW&)F87$$\"3'Qe*[$>Pn?%F,$\"31`G!HhJc\\)F87$$\"3)QL3-$H**>UF,$ \"3[:#3'R6kC%)F87$$\"3#R$ek.W]YUF,$\"3+Xc4Am3=#)F87$$\"3)RL$3xe,tUF,$ \"3s/#zi!3VLzF87$$\"3Cn;HdO=yVF,$\"3E$>X)HduejF87$$\"3a+++D>#[Z%F,$\"3 gxAO;Hx\")\\F87$$\"3TM$3_5,-`%F,$\"3%GSzCT^zW%F87$$\"3SnmT&G!e&e%F,$\" 3=g#R;G9D:%F87$$\"3/]i:NK'zf%F,$\"3*f`:i,h67%F87$$\"3fLe*[=Y.h%F,$\"33 )>WD]()H5%F87$$\"386but%F,$\"3_@+X\\xqwZF87$$\"37+]iSjE!z%F,$ \"3<\"z**[W(=JcF87$$\"3y*\\7G))Rb\"[F,$\"3OXJ\"GK]h>'F87$$\"3L+++DM\"3 %[F,$\"3!**fZR-9n(oF87$$\"3)3](=np3m[F,$\"3QTt^2I*Ho(F87$$\"3a+]P40O\" *[F,$\"3k:x`tb,B')F87$$\"3>]7.#Q?&=\\F,$\"3'4(*[g6ply*F87$$\"3s+voa-oX \\F,$\"3?(Q=b([U56F,7$$\"3O]PMF,%G(\\F,$\"3_;`pzy]b7F,7$$\"\"&F)$\"3.u g')yo>49F,-%'COLOURG6&%$RGBG$\"#5!\"\"F(F(-%+AXESLABELSG6$%\"xG%%y(x)G -%%FONTG6$%*HELVETICAG\"\"*-%%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 con structs a discrete solution based on each of the methods and gives the " }{TEXT 260 22 "root mean square error" }{TEXT -1 18 " of each solut ion." }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 905 "P := (x,y) -> -(2*s in(5*x)+3*cos(7*x))*sinh(y): hh := 0.01: numsteps := 500: x0 := 0: y0 \+ := sqrt(5)/2:\nmatrix([[`slope field: `,P(x,y)],[`initial point: `,` `(x0,y0)],[`step width: `,hh],\n[`no. of steps: `,numsteps]]);``; \nmthds := [`Papakostas' scheme with `*``(c[7]=199/200),`a modificatio n of Papakostas' scheme with `*``(c[7]=199/200),`a scheme with `*``(c[ 7]=324/325),`a scheme with `*``(c[7]=575/576),`a scheme with `*``(c[7] =1728/1729),`Tsitouras' scheme with `*``(c[7]=544/545),`Verner's \"mos t efficient\" scheme with `*``(c[7]=1999/2000)]: errs := []:\nDigits : = 30:\nfor ct to 7 do\n Pn_RK6_||ct := RK6_||ct(P(x,y),x,y,x0,evalf( y0),hh,numsteps,false);\n sm := 0: numpts := nops(Pn_RK6_||ct):\n \+ for ii to numpts do\n sm := sm+(Pn_RK6_||ct[ii,2]-evalf(p(Pn_RK6_ ||ct[ii,1])))^2;\n end do:\n errs := [op(errs),sqrt(sm/numpts)];\n end 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(pprint06\"" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%!G" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#K%'matrixG6#7)7$*&%9Papakostas'~scheme~with~G\"\" \"-%!G6#/&%\"cG6#\"\"(#\"$*>\"$+#F*$\"+8\"3_!G!#A7$*&%Ka~modification~ of~Papakostas'~scheme~with~GF*F+F*$\"+9(f%yIF87$*&%/a~scheme~with~GF*- F,6#/F/#\"$C$\"$D$F*$\"+&\\4\")3#F87$*&F@F*-F,6#/F/#\"$v&\"$w&F*$\"+ME _F;F87$*&F@F*-F,6#/F/#\"%G<\"%H\"%+?F*$\"+?W!H$GF8Q(pprint16\"" }} }{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 30 "The foll owing 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 o f 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 0 829 "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 := [`Pap akostas' scheme with `*``(c[7]=199/200),`a modification of Papakostas' scheme with `*``(c[7]=199/200),`a scheme with `*``(c[7]=324/325),`a s cheme with `*``(c[7]=575/576),`a scheme with `*``(c[7]=1728/1729),`Tsi touras' scheme with `*``(c[7]=544/545),`Verner's \"most efficient\" sc heme with `*``(c[7]=1999/2000)]: errs := []:\nDigits := 30:\nfor ct to 7 do\n pn_RK6_||ct := RK6_||ct(P(x,y),x,y,x0,evalf(y0),hh,numsteps, true);\nend do:\nxx := 4.999: pxx := evalf(p(xx)):\nfor ct to 7 do\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)pprint226\"" } }{PARA 11 "" 1 "" {XPPMATH 20 "6#%!G" }}{PARA 11 "" 1 "" {XPPMATH 20 " 6#K%'matrixG6#7)7$*&%9Papakostas'~scheme~with~G\"\"\"-%!G6#/&%\"cG6#\" \"(#\"$*>\"$+#F*$\"+I^f[T!#A7$*&%Ka~modification~of~Papakostas'~scheme ~with~GF*F+F*$\"+<4#=A%F87$*&%/a~scheme~with~GF*-F,6#/F/#\"$C$\"$D$F*$ \"+\"%+?F*$\"+7TD0cF8Q)pprint236\"" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 4 "The " }{TEXT 260 22 "root mean s quare error" }{TEXT -1 20 " over the interval " }{XPPEDIT 18 0 "[0, 5 ];" "6#7$\"\"!\"\"&" }{TEXT -1 82 " of each Runge-Kutta method is est imated as follows using the special procedure " }{TEXT 0 5 "NCint" } {TEXT -1 98 " to perform numerical integration by the 7 point Newton- Cotes method over 200 equal subintervals." }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 549 "mthds := [`Papakostas' scheme with `*``(c[7]=199/200 ),`a modification of Papakostas' scheme with `*``(c[7]=199/200),`a sch eme with `*``(c[7]=324/325),`a scheme with `*``(c[7]=575/576),`a schem e with `*``(c[7]=1728/1729),`Tsitouras' scheme with `*``(c[7]=544/545) ,`Verner's \"most efficient\" scheme with `*``(c[7]=1999/2000)]: errs \+ := []:\nDigits := 20:\nfor ct to 7 do\n sm := NCint((p(x)-'pn_RK6_|| ct'(x))^2,x=0..5,adaptive=false,numpoints=7,factor=200);\n errs := [ op(errs),sqrt(sm/5)];\nend do:\nDigits := 10:\nlinalg[transpose]([mthd s,evalf(errs)]);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#K%'matrixG6#7)7$*& %9Papakostas'~scheme~with~G\"\"\"-%!G6#/&%\"cG6#\"\"(#\"$*>\"$+#F*$\"+ )o&>0G!#A7$*&%Ka~modification~of~Papakostas'~scheme~with~GF*F+F*$\"+%Q J43$F87$*&%/a~scheme~with~GF*-F,6#/F/#\"$C$\"$D$F*$\"+Uwp\"4#F87$*&F@F *-F,6#/F/#\"$v&\"$w&F*$\"+f(yej\"F87$*&F@F*-F,6#/F/#\"%G<\"%H\"% +?F*$\"+rc4UGF8Q)pprint246\"" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }} {PARA 0 "" 0 "" {TEXT -1 92 "The following error graphs are constructe d using the numerical procedures for the solutions." }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 682 "evalf[20](plot([p(x)-'pn_RK6_1'(x),p(x)-'p n_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),p(x)-'pn_RK6_7'(x)],x=0..2.2,font=[HELVETICA,9],\n color=[COLOR(RGB,.95,0,.2),COLOR(RGB,.5,0,.95),COLOR(RGB,.95,.45,0),CO LOR(RGB,0,.8,.25),\nCOLOR(RGB,.95,0,.95),COLOR(RGB,.75,.2,.2),COLOR(RG B,0,.7,.9)],legend=[`Papakostas' scheme with c[7]=199/200`,`a modifica tion of Papakostas' scheme with c[7]=199/200`,`a scheme with c[7]=324/ 325`,`a scheme with c[7]=575/576`,`a scheme with c[7]=1728/1729`,`Tsit ouras' scheme with c[7]=544/545`,`Verner's \"most efficient\" scheme w ith c[7]=1999/2000`],title=`error curves for 8 stage order 6 Runge-Kut ta methods`));\n" }}{PARA 13 "" 1 "" {GLPLOT2D 936 546 546 {PLOTDATA 2 "6--%'CURVESG6%7at7$$\"\"!F)F(7$$\"5mmmm;zM%))>\"!#@$!&#yO!#>7$$\"5L LLLLepo(R#F-$!&B\"eF07$$\"5mmm\"HKRUva#F-$!&!)z&F07$$\"5+++]7GyR(p#F-$ !&](eF07$$\"5LLL3-jKDZGF-$!&L<'F07$$\"5mmmm\"zp3r*HF-$!'/go!#?7$$\"5LL L$3xc>oH$F-$!'NxnFJ7$$\"5++++]P/`'f$F-$!'G2nFJ7$$\"5LLLL3x@&f>%F-$!'.c nFJ7$$\"5mmmmm;RP&z%F-$!'?QlFJ7$$\"5++++v=&)e\")oF-$!'N>QFJ7$$\"5LLLL$ 37.y'*)F-$\"&2=*FJ7$$\"5nmmm;9O,m8FJ$\"'\\v\"*FJ7$$\"5nmmmTN&*4%[\"FJ$ \"(?T5\"FJ7$$\"5nmmmmca=-;FJ$\"(s3L\"FJ7$$\"5nmmm\"zPr-s\"FJ$\"((G89FJ 7$$\"5nmmm;*Hd$Q=FJ$\"(SjZ\"FJ7$$\"5+++]7emSt?FJ$\"(f;c\"FJ7$$\"5LLLL3 $FJ$\"(avg\"FJ7$$\"5mmmmTv+JiOFJ$\"(MBj\"FJ7$$ \"5++++vLo`FTFJ$\"(<%*o\"FJ7$$\"5LLLLLQ(zgg%FJ$\"(sK#=FJ7$$\"5mmmm;*e! 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-7$F`z$\")T!HZ\"F-7$Fjz$\")dys9F-7$Fd[l$\")lkj9F-7$Fi[l$\")Z*)Q9F-7$F^ \\l$\")?r$R\"F-7$Fh\\l$\")A-(G\"F-7$F]]l$\")R7V6F-7$Fb]l$\")%G:-\"F-7$ Fg]l$\"(EYB*F-7$F\\^l$\"(7hd)F-7$Fa^l$\"(qM'zF-7$Ff^l$\"(B[R(F-7$F[_l$ \"(9!frF-7$F`_l$\"(Cu,(F-7$Fe_l$\"(Z$=qF-7$Fj_l$\"(-47(F-7$F_`l$\"(.rM (F-7$Fd`l$\"(K%\\xF-7$Fi`l$\"(Fbp)F-7$F^al$\"(oK*)*F-7$Fcal$\")S$[?\"F -7$Fhal$\")M2j8F-7$F]bl$\")*o:e\"F-7$Fbbl$\")cYy=F-7$Fgbl$\")cfxAF-7$F \\cl$\")a%y_#F-7$Facl$\"(6_%GF17$Ffcl$\"(v\\:$F17$F[dl$\"(\"R-NF17$Fed l$\"(Kx\"RF17$F_el$\"(\")[R%F17$Fdel$\"(1\\u%F17$Fiel$\"(kd'\\F17$Fcfl $\"(#zP`F17$Fggl$\"()RxdF1-F]hl6&F_hlFchl$\"\"(F*$\"\"*F*-Fghl6#%UVern er's~\"most~efficient\"~scheme~with~c[7]=1999/2000G-%&TITLEG6#%Uerror~ curves~for~8~stage~order~6~Runge-Kutta~methodsG-%%FONTG6$%*HELVETICAGF ^fs-%+AXESLABELSG6$Q\"x6\"Q!F^gs-%%VIEWG6$;F(Fggl%(DEFAULTG" 1 2 0 1 10 0 2 9 1 4 2 1.000000 45.000000 45.000000 0 1 "Papakostas' scheme wi th c[7]=199/200" "a modification of Papakostas' scheme with c[7]=199/2 00" "a scheme with c[7]=324/325" "a scheme with c[7]=575/576" "a schem e with c[7]=1728/1729" "Tsitouras' scheme with c[7]=544/545" "Verner's \"most efficient\" scheme with c[7]=1999/2000" }}}}{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 example is similar to one that appears in an article by F. \+ G. Lether: Mathematics of Computation, Vol. 20, no. 95, (July 1966) pa ge 382. " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "dy/dx=exp(-x)/(x-1)^2" "6#/*&%#dyG\"\"\"%#dxG !\"\"*&-%$expG6#,$%\"xGF(F&*$,&F.F&F&F(\"\"#F(" }{TEXT -1 2 " " } {XPPEDIT 18 0 "cos(1/(x-1))-y" "6#,&-%$cosG6#*&\"\"\"F(,&%\"xGF(F(!\" \"F+F(%\"yGF+" }{TEXT -1 5 ", " }{XPPEDIT 18 0 "y(0)=sin*1" "6#/-% \"yG6#\"\"!*&%$sinG\"\"\"F*F*" }{TEXT -1 1 " " }}{PARA 0 "" 0 "" {TEXT -1 10 "Solution: " }}{PARA 256 "" 0 "" {TEXT -1 2 " " } {XPPEDIT 18 0 "y = -exp(-x)*sin(1/(x-1))" "6#/%\"yG,$*&-%$expG6#,$%\"x G!\"\"\"\"\"-%$sinG6#*&F-F-,&F+F-F-F,F,F-F," }{TEXT -1 1 " " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 185 "d e := diff(y(x),x)=exp(-x)/(x-1)^2*cos(1/(x-1))-y(x);\nic := y(0)=sin(1 );\ndsolve(\{de,ic\},y(x));\nq := unapply(rhs(%),x):\nplot(q(x),x=0..1 -1/(6*Pi),font=[HELVETICA,9],labels=[`x`,`y(x)`]);" }}{PARA 11 "" 1 " " {XPPMATH 20 "6#>%#deG/-%%diffG6$-%\"yG6#%\"xGF,,&*(-%$expG6#,$F,!\" \"\"\"\",&F,F4F4F3!\"#-%$cosG6#*&F4F4F5F3F4F4F)F3" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%#icG/-%\"yG6#\"\"!-%$sinG6#\"\"\"" }}{PARA 11 "" 1 " " {XPPMATH 20 "6#/-%\"yG6#%\"xG,$*&-%$expG6#,$F'!\"\"\"\"\"-%$sinG6#*& F/F/,&F'F/F/F.F.F/F." }}{PARA 13 "" 1 "" {GLPLOT2D 603 396 396 {PLOTDATA 2 "6&-%'CURVESG6$7[r7$$\"\"!F)$\"30l*y![)4ZT)!#=7$$\"3#>=\"* )>z2k?!#>$\"3m%p=Qu6DN)F,7$$\"3kMQU\"3@+'QF0$\"3m=O<,5$yO#)F,7$$\"38UQ!)p4'G\"zF0$\"3/Z^$zTlU<)F,7$$\" 3BY$*R0>JO**F0$\"36ty1)z*36\")F,7$$\"3wbXC%*4B\"=\"F,$\"3A;o(=P!Q^!)F, 7$$\"3M!)fxC(zaP\"F,$\"3E8^!ydw!))zF,7$$\"3kgswR?Pw:F,$\"3T8>lD8j?zF,7 $$\"3Qnlb!4?mx\"F,$\"3%p4\\s^P4&yF,7$$\"3OsvSC)*f#)>F,$\"3/$H(=wa6wxF, 7$$\"3)Rk+h)o-k@F,$\"3LR8d$>`qq(F,7$$\"3Q^Vo'yq#oBF,$\"3YB)Qc;#3DwF,7$ $\"3?0sMKLNtDF,$\"3,;%fG`C(F,7$$ \"3S+dSsVlWLF,$\"3&36sy[X09(F,7$$\"3EOur83&\\b$F,$\"37)QgTzpp+(F,7$$\" 3c#**Qwz)4TPF,$\"3M]xfz#>l(oF,7$$\"3wx#p)QELXRF,$\"3UR-VbS%zr'F,7$$\"3 \"yy;Gb6)RTF,$\"3%40quzD%\\lF,7$$\"3p2KM(*)HFM%F,$\"3W'4!o9@F_jF,7$$\" 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LeP$)F,$!374'4ii(p\\6F,7$$\"3kI*)GR!RLQ)F,$!3q\\t;%4rO@%F07$$\"3Z,MUjZ 4H%)F,$\"3c]$R4W&G]NF07$$\"3=ryb([][Z)F,$\"37Ccp$Rpv:\"F,7$$\"3%QzY4r \"HF&)F,$\"3CcG)))y\"yp?F,7$$\"3g()F,$\"3y_cgRCTxTF,7$$\"3Wx*okV>:t)F,$\"37VUul#=X<%F,7$$\"3q% oYTYr\\v)F,$\"3Ri.A'QV25%F,7$$\"3'>RC=\\B%y()F,$\"3x'=W7Fx&HRF,7$$\"3] 1)zraF`#))F,$\"3QFG%plB4F$F,7$$\"39A_`-;Bs))F,$\"3700_RQTz@F,7$$\"31^1 [bjB(*))F,$\"3OuNH?R:M9F,7$$\"34\"3E%36CA*)F,$\"3E<)GsFg<(fF07$$\"376: PheCZ*)F,$!3ocHHl.]EIF07$$\"39TpJ91Ds*)F,$!38U'y=D^^A\"F,7$$\"3=rBEn`D (**)F,$!3j&4Nt%[@=@F,7$$\"3@,y??,EA!*F,$!3!e@9jLE#=HF,7$$\"3EJK:t[EZ!* F,$!3I(>(y]J<_NF,7$$\"3Gh')4E'pA2*F,$!3U`$=&457URF,7$$\"3'3]u3\"GDy!*F ,$!3U'ykJ>F2*RF,7$$\"3cT.l&*fB%3*F,$!3AH^N$RP+-%F,7$$\"3E#=E/=>-4*F,$! 3&Rn/8H]\"HSF,7$$\"3'=--_O-i4*F,$!39=u$p*4B>*F,$!39yV6L@mxvF07$$\"3oYr;\"p**Q?*F,$!3_9 %**Gult/#!#?7$$\"3(p#)=21me@*F,$\"3=-0hZ^rztF07$$\"3E20FIC$yA*F,$\"3ML bSaq^*[\"F,7$$\"3b(=A)*z)zR#*F,$\"3Qccz;X$>?#F,7$$\"3%*oQPp^w^#*F,$\"3 \\RTgHcmRGF,7$$\"3C\\b#*Q:tj#*F,$\"3M$o6)eV:lLF,7$$\"3oE3CP5fw#*F,$\"3 Ma)HpV]]I!H!G$*F,$\"3a%4t 07BP*GF,7$$\"3J=s\")G&))3M*F,$\"3f(3)H;[%o+#F,7$$\"3w&\\Kr-[PN*F,$\"33 n1kl[]%*F,$!3m (=[SoWqQ#F,7$$\"3%>saO,CmX*F,$!3GDS5eP#en\"F,7$$\"3AhB\"Gw`IY*F,$!3U$3 !Gg0_(o)F07$$\"3]++(>^$[p%*F,$!3V'=8$[D+C:!#C-%'COLOURG6&%$RGBG$\"#5! \"\"F(F(-%%FONTG6$%*HELVETICAG\"\"*-%+AXESLABELSG6$%\"xG%%y(x)G-%%VIEW G6$;F($\"+BN[p%*!#5%(DEFAULTG" 1 2 0 1 10 0 2 9 1 4 2 1.000000 45.000000 45.000000 0 0 "Curve 1" }}}}{PARA 0 "" 0 "" {TEXT -1 0 "" }} {PARA 0 "" 0 "" {TEXT -1 93 "The following code constructs a discrete \+ solution based on each of the methods and gives the " }{TEXT 260 22 "r oot mean square error" }{TEXT -1 18 " of each solution." }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 921 "Q := (x,y) -> exp(-x)/(x-1)^2*cos( 1/(x-1))-y: hh := 1/500-1/(3000*Pi): numsteps := 500: x0 := 0: y0 := s in(1):\nmatrix([[`slope field: `,Q(x,y)],[`initial point: `,``(x0,y0 )],[`step width: `,hh],\n[`no. of steps: `,numsteps]]);``;\nmthds \+ := [`Papakostas' scheme with `*``(c[7]=199/200),`a modification of Pap akostas' scheme with `*``(c[7]=199/200),`a scheme with `*``(c[7]=324/3 25),`a scheme with `*``(c[7]=575/576),`a scheme with `*``(c[7]=1728/17 29),`Tsitouras' scheme with `*``(c[7]=544/545),`Verner's \"most effici ent\" scheme with `*``(c[7]=1999/2000)]: errs := []:\nDigits := 30:\nf or ct to 7 do\n Qn_RK6_||ct := RK6_||ct(Q(x,y),x,y,x0,evalf[33](y0), evalf[33](hh),numsteps,false);\n sm := 0: numpts := nops(Qn_RK6_||ct ):\n for ii to numpts do\n sm := sm+(Qn_RK6_||ct[ii,2]-q(Qn_RK6 _||ct[ii,1]))^2;\n end do:\n errs := [op(errs),sqrt(sm/numpts)];\n end do:\nDigits := 10:\nlinalg[transpose]([mthds,evalf(errs)]);" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#K%'matrixG6#7&7$%0slope~field:~~~G,&*( -%$expG6#,$%\"xG!\"\"\"\"\",&F/F1F1F0!\"#-%$cosG6#*&F1F1F2F0F1F1%\"yGF 07$%0initial~point:~G-%!G6$\"\"!-%$sinG6#F17$%/step~width:~~~G,&#F1\"$ +&F1*&F1F1*&\"%+IF1%#PiGF1F0F07$%1no.~of~steps:~~~GFFQ)pprint256\"" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#%!G" }}{PARA 11 "" 1 "" {XPPMATH 20 "6 #K%'matrixG6#7)7$*&%9Papakostas'~scheme~with~G\"\"\"-%!G6#/&%\"cG6#\" \"(#\"$*>\"$+#F*$\"+b)))>d$!#?7$*&%Ka~modification~of~Papakostas'~sche me~with~GF*F+F*$\"+mukOQF87$*&%/a~scheme~with~GF*-F,6#/F/#\"$C$\"$D$F* $\"+&)e]pUF87$*&F@F*-F,6#/F/#\"$v&\"$w&F*$\"+!\\-(oSF87$*&F@F*-F,6#/F/ #\"%G<\"%H\"%+?F*$\"+rt)>\\%F8Q)pprint266\"" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 30 "The following code constr ucts " }{TEXT 260 20 "numerical procedures" }{TEXT -1 56 " for solutio ns based on each of the Runge-Kutta schemes." }}{PARA 0 "" 0 "" {TEXT -1 75 "The error in the value obtained by each of the methods at the p oint where " }{XPPEDIT 18 0 "x = 0;" "6#/%\"xG\"\"!" }{TEXT -1 21 ".9 469 is also given." }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 845 "Q := (x,y) -> exp(-x)/(x-1)^2*cos(1/(x-1))-y: hh := 1/500-1/(3000*Pi): num steps := 500: x0 := 0: y0 := sin(1):\nmatrix([[`slope field: `,Q(x,y )],[`initial point: `,``(x0,y0)],[`step width: `,hh],\n[`no. of step s: `,numsteps]]);``;\nmthds := [`Papakostas' scheme with `*``(c[7]=1 99/200),`a modification of Papakostas' scheme with `*``(c[7]=199/200), `a scheme with `*``(c[7]=324/325),`a scheme with `*``(c[7]=575/576),`a scheme with `*``(c[7]=1728/1729),`Tsitouras' scheme with `*``(c[7]=54 4/545),`Verner's \"most efficient\" scheme with `*``(c[7]=1999/2000)]: errs := []:\nDigits := 30:\nfor ct to 7 do\n qn_RK6_||ct := RK6_||c t(Q(x,y),x,y,x0,evalf(y0),evalf(hh),numsteps,true);\nend do:\nxx := 0. 9469: qxx := evalf(q(xx)):\nfor ct to 7 do\n errs := [op(errs),abs(q n_RK6_||ct(xx)-qxx)];\nend do:\nDigits := 10:\nlinalg[transpose]([mthd s,evalf(errs)]);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#K%'matrixG6#7&7$%0 slope~field:~~~G,&*(-%$expG6#,$%\"xG!\"\"\"\"\",&F/F1F1F0!\"#-%$cosG6# *&F1F1F2F0F1F1%\"yGF07$%0initial~point:~G-%!G6$\"\"!-%$sinG6#F17$%/ste p~width:~~~G,&#F1\"$+&F1*&F1F1*&\"%+IF1%#PiGF1F0F07$%1no.~of~steps:~~~ GFFQ)pprint276\"" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%!G" }}{PARA 11 " " 1 "" {XPPMATH 20 "6#K%'matrixG6#7)7$*&%9Papakostas'~scheme~with~G\" \"\"-%!G6#/&%\"cG6#\"\"(#\"$*>\"$+#F*$\"+0SgML!#>7$*&%Ka~modification~ of~Papakostas'~scheme~with~GF*F+F*$\"+*>@&GWF87$*&%/a~scheme~with~GF*- F,6#/F/#\"$C$\"$D$F*$\"+'G4m/&F87$*&F@F*-F,6#/F/#\"$v&\"$w&F*$\"+;kUp \\F87$*&F@F*-F,6#/F/#\"%G<\"%H\"%+?F*$\"+E#QXt%F8Q)pprint286\"" } }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 4 "The " } {TEXT 260 22 "root mean square error" }{TEXT -1 19 " over the interval " }{XPPEDIT 18 0 " [0, 1-1/(6*Pi)] " "6#7$\"\"!,&\"\"\"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 0 569 "mthds := [`Papakostas' scheme with `*``(c[7]=199/200 ),`a modification of Papakostas' scheme with `*``(c[7]=199/200),`a sch eme with `*``(c[7]=324/325),`a scheme with `*``(c[7]=575/576),`a schem e with `*``(c[7]=1728/1729),`Tsitouras' scheme with `*``(c[7]=544/545) ,`Verner's \"most efficient\" scheme with `*``(c[7]=1999/2000)]: errs \+ := []:\nDigits := 20:\nfor ct to 7 do\n sm := NCint((q(x)-'qn_RK6_|| ct'(x))^2,x=0..1-1/(6*Pi),adaptive=false,numpoints=7,factor=200);\n \+ errs := [op(errs),sqrt(sm/(1-1/(6*Pi)))];\nend do:\nDigits := 10:\nlin alg[transpose]([mthds,evalf(errs)]);" }}{PARA 11 "" 1 "" {XPPMATH 20 " 6#K%'matrixG6#7)7$*&%9Papakostas'~scheme~with~G\"\"\"-%!G6#/&%\"cG6#\" \"(#\"$*>\"$+#F*$\"+%G*3qK!#?7$*&%Ka~modification~of~Papakostas'~schem e~with~GF*F+F*$\"+jv^1LF87$*&%/a~scheme~with~GF*-F,6#/F/#\"$C$\"$D$F*$ \"+r`SXOF87$*&F@F*-F,6#/F/#\"$v&\"$w&F*$\"+DovCMF87$*&F@F*-F,6#/F/#\"% G<\"%H\"%+?F*$\"+%4,))*RF8Q)pprint296\"" }}}{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 0 699 "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'q n_RK6_5'(x)-q(x),'qn_RK6_6'(x)-q(x),'qn_RK6_7'(x)-q(x)],x=0..0.7,-1.6e -18..1.6e-18,font=[HELVETICA,9],\ncolor=[COLOR(RGB,.95,0,.2),COLOR(RGB ,.5,0,.95),COLOR(RGB,.95,.45,0),COLOR(RGB,0,.8,.25),\nCOLOR(RGB,.95,0, .95),COLOR(RGB,.75,.2,.2),COLOR(RGB,0,.7,.9)],legend=[`Papakostas' sch eme with c[7]=199/200`,`a modification of Papakostas' scheme with c[7] =199/200`,`a scheme with c[7]=324/325`,`a scheme with c[7]=575/576`,`a scheme with c[7]=1728/1729`,`Tsitouras' scheme with c[7]=544/545`,`Ve rner's \"most efficient\" scheme with c[7]=1999/2000`],title=`error cu rves for 8 stage order 6 Runge-Kutta methods`));" }}{PARA 13 "" 1 "" {GLPLOT2D 878 575 575 {PLOTDATA 2 "6--%'CURVESG6%7fn7$$\"\"!F)F(7$$\": mmmmmmm;z+e_\"!#E$!$^\"!#D7$$\":LLLLLL$3->R`GF-$!$1$F07$$\":mmmmmmmT&p 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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 0 865 "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 := [` Papakostas' scheme with `*``(c[7]=199/200),`a modification of Papakost as' scheme with `*``(c[7]=199/200),`a scheme with `*``(c[7]=324/325),` a scheme with `*``(c[7]=575/576),`a scheme with `*``(c[7]=1728/1729),` Tsitouras' scheme with `*``(c[7]=544/545),`Verner's \"most efficient\" scheme with `*``(c[7]=1999/2000)]: errs := []:\nDigits := 30:\nfor ct to 7 do\n Rn_RK6_||ct := RK6_||ct(R(x,y),x,y,x0,y0,hh,numsteps,fals e);\n sm := 0: numpts := nops(Rn_RK6_||ct):\n for ii to numpts do \n sm := sm+(Rn_RK6_||ct[ii,2]-r(Rn_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$%0slope~field:~~~G,$*(\"\"&\"\"\"%\"yGF,)-%$sinG6#, $*&\"\"(F,%\"xGF,F,F4F,F,7$%0initial~point:~G-%!G6$\"\"!F,7$%/step~wid th:~~~G$F,!\"#7$%1no.~of~steps:~~~G\"$+&Q)pprint306\"" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%!G" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#K%'matrixG6 #7)7$*&%9Papakostas'~scheme~with~G\"\"\"-%!G6#/&%\"cG6#\"\"(#\"$*>\"$+ #F*$\"+c^wBN!#@7$*&%Ka~modification~of~Papakostas'~scheme~with~GF*F+F* $\"+y[vY6!#?7$*&%/a~scheme~with~GF*-F,6#/F/#\"$C$\"$D$F*$\"+xAyi\")F87 $*&FAF*-F,6#/F/#\"$v&\"$w&F*$\"+'p'eU;F>7$*&FAF*-F,6#/F/#\"%G<\"%H7$*&%8Tsitouras'~scheme~with~GF*-F,6#/F/#\"$W&\"$X&F*$\"+^B h:6!#>7$*&%GVerner's~\"most~efficient\"~scheme~with~GF*-F,6#/F/#\"%**> \"%+?F*$\"+xoy>6FcoQ)pprint316\"" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }} {PARA 0 "" 0 "" {TEXT -1 30 "The following code constructs " }{TEXT 260 20 "numerical procedures" }{TEXT -1 56 " for solutions based on ea ch of the Runge-Kutta schemes." }}{PARA 0 "" 0 "" {TEXT -1 75 "The err or in the value obtained by each of the methods at the point where " }{XPPEDIT 18 0 "x = 4.999;" "6#/%\"xG-%&FloatG6$\"%**\\!\"$" }{TEXT -1 16 " is also given." }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 796 " R := (x,y) -> 5*y*sin(7*x)^7: hh := 0.01: numsteps := 500: x0 := 0: y0 := 1:\nmatrix([[`slope field: `,R(x,y)],[`initial point: `,``(x0,y0 )],[`step width: `,hh],\n[`no. of steps: `,numsteps]]);``;\nmthds \+ := [`Papakostas' scheme with `*``(c[7]=199/200),`a modification of Pap akostas' scheme with `*``(c[7]=199/200),`a scheme with `*``(c[7]=324/3 25),`a scheme with `*``(c[7]=575/576),`a scheme with `*``(c[7]=1728/17 29),`Tsitouras' scheme with `*``(c[7]=544/545),`Verner's \"most effici ent\" scheme with `*``(c[7]=1999/2000)]: errs := []:\nDigits := 25:\nf or ct to 7 do\n rn_RK6_||ct := RK6_||ct(R(x,y),x,y,x0,y0,hh,numsteps ,true);\nend do:\nxx := 4.999: rxx := evalf(r(xx)):\nfor ct to 7 do\n \+ errs := [op(errs),abs(rn_RK6_||ct(xx)-rxx)];\nend do:\nDigits := 10: \nlinalg[transpose]([mthds,evalf(errs)]);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#K%'matrixG6#7&7$%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\"$+&Q)pprint326\"" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#%!G" }}{PARA 11 "" 1 "" {XPPMATH 20 "6 #K%'matrixG6#7)7$*&%9Papakostas'~scheme~with~G\"\"\"-%!G6#/&%\"cG6#\" \"(#\"$*>\"$+#F*$\"+o(eVL(!#@7$*&%Ka~modification~of~Papakostas'~schem e~with~GF*F+F*$\"+)G:F\\#!#?7$*&%/a~scheme~with~GF*-F,6#/F/#\"$C$\"$D$ F*$\"+'G(ef7$*&FAF*-F,6#/F/#\"$v&\"$w&F*$\"+3)[fa$F>7$*&FAF*-F,6#/F /#\"%G<\"%Hon%F>7$*&%8Tsitouras'~scheme~with~GF*-F,6#/F/#\"$ W&\"$X&F*$\"+/=#=U#!#>7$*&%GVerner's~\"most~efficient\"~scheme~with~GF *-F,6#/F/#\"%**>\"%+?F*$\"+)Hb,V#FcoQ)pprint336\"" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 4 "The " }{TEXT 260 22 "root \+ mean square error" }{TEXT -1 20 " over the interval " }{XPPEDIT 18 0 "[0, 5];" "6#7$\"\"!\"\"&" }{TEXT -1 82 " of each Runge-Kutta method \+ is estimated as follows using the special procedure " }{TEXT 0 5 "NCi nt" }{TEXT -1 98 " to perform numerical integration by the 7 point Ne wton-Cotes method over 200 equal subintervals." }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 549 "mthds := [`Papakostas' scheme with `*``(c[7]=19 9/200),`a modification of Papakostas' scheme with `*``(c[7]=199/200),` a scheme with `*``(c[7]=324/325),`a scheme with `*``(c[7]=575/576),`a \+ scheme with `*``(c[7]=1728/1729),`Tsitouras' scheme with `*``(c[7]=544 /545),`Verner's \"most efficient\" scheme with `*``(c[7]=1999/2000)]: \+ errs := []:\nDigits := 20:\nfor ct to 7 do\n sm := NCint((r(x)-'rn_R K6_||ct'(x))^2,x=0..5,adaptive=false,numpoints=7,factor=200);\n errs := [op(errs),sqrt(sm/5)];\nend do:\nDigits := 10:\nlinalg[transpose]( [mthds,evalf(errs)]);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#K%'matrixG6#7 )7$*&%9Papakostas'~scheme~with~G\"\"\"-%!G6#/&%\"cG6#\"\"(#\"$*>\"$+#F *$\"+YZ1;N!#@7$*&%Ka~modification~of~Papakostas'~scheme~with~GF*F+F*$ \"+,>(Q9\"!#?7$*&%/a~scheme~with~GF*-F,6#/F/#\"$C$\"$D$F*$\"+))e=U\")F 87$*&FAF*-F,6#/F/#\"$v&\"$w&F*$\"+96_Q;F>7$*&FAF*-F,6#/F/#\"%G<\"%H7$*&%8Tsitouras'~scheme~with~GF*-F,6#/F/#\"$W&\"$X&F*$\"+9 Y&G6\"!#>7$*&%GVerner's~\"most~efficient\"~scheme~with~GF*-F,6#/F/#\"% **>\"%+?F*$\"+x'=q6\"FcoQ)pprint346\"" }}}{PARA 0 "" 0 "" {TEXT -1 0 " " }}{PARA 0 "" 0 "" {TEXT -1 92 "The following error graphs are constr ucted using the numerical procedures for the solutions." }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 679 "evalf[20](plot(['rn_RK6_1'(x)-r(x) ,'rn_RK6_2'(x)-r(x),'rn_RK6_3'(x)-r(x),'rn_RK6_4'(x)-r(x),\n'rn_RK6_5' (x)-r(x),'rn_RK6_6'(x)-r(x),'rn_RK6_7'(x)-r(x)],x=0..5,font=[HELVETICA ,9],\ncolor=[COLOR(RGB,.95,0,.2),COLOR(RGB,.5,0,.95),COLOR(RGB,.95,.45 ,0),COLOR(RGB,0,.8,.25),\nCOLOR(RGB,.95,0,.95),COLOR(RGB,.75,.2,.2),CO LOR(RGB,0,.7,.9)],legend=[`Papakostas' scheme with c[7]=199/200`,`a mo dification of Papakostas' scheme with c[7]=199/200`,`a scheme with c[7 ]=324/325`,`a scheme with c[7]=575/576`,`a scheme with c[7]=1728/1729` ,`Tsitouras' scheme with c[7]=544/545`,`Verner's \"most efficient\" sc heme with c[7]=1999/2000`],title=`error curves for 8 stage order 6 Run ge-Kutta methods`));" }}{PARA 13 "" 1 "" {GLPLOT2D 1131 636 636 {PLOTDATA 2 "6--%'CURVESG6%7cil7$$\"\"!F)F(7$$\"5SLLL3x1h6o!#A$!&.O#!# >7$$\"5ommmTN@Ki8!#@$!'VB\\F07$$\"5-++]7.K[V?F4$!']hqF07$$\"5NLLL$3FWY s#F4$!'gQpF07$$\"5-+++D1k'p3%F4$\"&5Z#F07$$\"5qmmmmT&)G\\aF4$\"'QF#)F0 7$$F,F4$\"('*yt\"F07$$\"50+++]7G$R<)F4$\"(^cJ#F07$$\"5SLL$3-)Q4b))F4$ \"(C[<#F07$$\"5qmmm\"z%\\DO&*F4$\"(@k$=F07$$\"5+++Dc,;u@5!#?$\"''f;)F0 7$$\"5MLLLL3x&)*3\"F[o$\"'>V@F07$$\"5++]i!*GER37F[o$!(8E0#F07$$\"5nmm \"z%\\v#pK\"F[o$!(X(eKF07$$\"5++Dcw4]>'Q\"F[o$!(/(yMF07$$\"5ML$3_+ZiaW \"F[o$!(kK%RF07$$\"5nmT&Q.$*HZ]\"F[o$!(Ho$RF07$$\"5+++]i!R(*Rc\"F[o$!( 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s$\"*Q*=^AF07$Fi\\s$\"*'36cAF07$F^]s$\"*?3bE#F07$Fc]s$\"*davF#F07$Fh]s $\"*Kg6G#F07$F]^s$\"*)oD\"G#F07$Fb^s$\"*$z#oF#F07$Fg^s$\"*#\\7pAF07$Fa _s$\"*4vmE#F07$F[`s$\"*o\">BF07$Fgcs$\"*eP\\L#F07$Fads$\"*i8(oBF07$Ffd s$\"*8>la#F07$F[es$\"*j<7)GF07$Fifs$\"*@U/J$F07$Fcgs$\"*)H[oOF07$Fais$ \"*,XD-%F07$F[js$\"*%eHfVF07$Fejs$\"*)oVVXF07$F_[t$\"*60)*o%F07$Fi[t$ \"*J_dp%F07$Fc\\t$\"*q0wr%F07$Fh\\t$\"*K%*es%F07$F]]t$\"*P*4GZF07$Fhda l$\"*gkYs%F07$Fb]t$\"*e\\7s%F07$F`eal$\"*k+Mr%F07$Fg]t$\"*=?%*p%F07$F[ _t$\"*h%p(o%F07$Fe_t$\"*Y%f\"o%F07$Fj_t$\"*qi%)o%F07$F_`t$\"*=.xp%F07$ Fcat$\"*^u#3ZF07$Fgbt$\"*`Luq%F07$Fhix$\"*EZ0q%F07$F\\ct$\"*y[vp%F07$F `jx$\"*AgNo%F07$Fact$\"*fo$oYF0-Ffct6&FhctFictF(Fict-F`dt6#%=a~scheme~ with~c[7]=1728/1729G-%&TITLEG6#%Uerror~curves~for~8~stage~order~6~Rung e-Kutta~methodsG-%%FONTG6$%*HELVETICAG\"\"*-%+AXESLABELSG6$Q\"x6\"Q!F^ _el-%%VIEWG6$;F(Fact%(DEFAULTG" 1 2 0 1 10 0 2 9 1 4 2 1.000000 45.000000 45.000000 0 1 "Papakostas' scheme with c[7]=199/200" "a modi fication of Papakostas' scheme with c[7]=199/200" "a scheme with c[7]= 324/325" "a scheme with c[7]=575/576" "a scheme with c[7]=1728/1729" } }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 48 "Test 13 of 8 stage, order 6 Runge-Kutta methods " }}{PARA 256 "" 0 "" {TEXT -1 0 "" }} {PARA 0 "" 0 "" {TEXT -1 72 "See: \"Mathematica in Action\" by Stan W agon, Springer-Verlag, page 302. " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }} {PARA 256 "" 0 "" {TEXT -1 3 " " }{XPPEDIT 18 0 "dy/dx = cos*x+2*y; " "6#/*&%#dyG\"\"\"%#dxG!\"\",&*&%$cosGF&%\"xGF&F&*&\"\"#F&%\"yGF&F&" }{TEXT -1 8 ", " }{XPPEDIT 18 0 "y(0) = -2/5;" "6#/-%\"yG6#\"\"! ,$*&\"\"#\"\"\"\"\"&!\"\"F-" }{TEXT -1 1 " " }}{PARA 0 "" 0 "" {TEXT -1 10 "Solution: " }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 " y = 1/5;" "6#/%\"yG*&\"\"\"F&\"\"&!\"\"" }{TEXT -1 1 " " }{XPPEDIT 18 0 "sin*x-2/5" "6#,&*&%$sinG\"\"\"%\"xGF&F&*&\"\"#F&\"\"&!\"\"F+" } {TEXT -1 1 " " }{XPPEDIT 18 0 "cos*x" "6#*&%$cosG\"\"\"%\"xGF%" } {TEXT -1 3 " . " }}{PARA 0 "" 0 "" {TEXT 260 4 "Note" }{TEXT -1 2 ": \+ " }}{PARA 0 "" 0 "" {TEXT -1 53 "The general solution of the different ial equation " }{XPPEDIT 18 0 "dy/dx = cos*x+2*y;" "6#/*&%#dyG\"\" \"%#dxG!\"\",&*&%$cosGF&%\"xGF&F&*&\"\"#F&%\"yGF&F&" }{TEXT -1 64 " c ontains an exponential term, but with the initial condition " } {XPPEDIT 18 0 "y(0) = -2/5" "6#/-%\"yG6#\"\"!,$*&\"\"#\"\"\"\"\"&!\"\" F-" }{TEXT -1 23 " this term disappears." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 50 "de := diff(y(x),x)=co s(x)+2*y(x);\ndsolve(de,y(x));" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%#d eG/-%%diffG6$-%\"yG6#%\"xGF,,&-%$cosGF+\"\"\"*&\"\"#F0F)F0F0" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/-%\"yG6#%\"xG,(*&#\"\"#\"\"&\"\"\"-%$cosGF& F-!\"\"*&#F-F,F--%$sinGF&F-F-*&-%$expG6#,$*&F+F-F'F-F-F-%$_C1GF-F-" }} }{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 105 "Any sli ght deviation of a numerical solution from the correct solution tends \+ to become rapidly magnified." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 154 "de := diff(y(x),x)=cos(x)+2 *y(x);\nic := y(0)=-2/5;\ndsolve(\{de,ic\},y(x));\ne := unapply(rhs(%) ,x):\nplot(e(x),x=0..8,font=[HELVETICA,9],labels=[`x`,`y(x)`]);" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#>%#deG/-%%diffG6$-%\"yG6#%\"xGF,,&-%$c osGF+\"\"\"*&\"\"#F0F)F0F0" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%#icG/- %\"yG6#\"\"!#!\"#\"\"&" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/-%\"yG6#%\" xG,&*&#\"\"#\"\"&\"\"\"-%$cosGF&F-!\"\"*&#F-F,F--%$sinGF&F-F-" }} {PARA 13 "" 1 "" {GLPLOT2D 703 312 312 {PLOTDATA 2 "6&-%'CURVESG6$7gn7 $$\"\"!F)$!3A+++++++S!#=7$$\"3ELLLLBxV5E$F,$!35;%fC]([[JF,7$$\"3MLLLLAKn\\F,$!3C&4%=OwYjDF,7$$\"3=LLLLc$\\ o'F,$!31c1[)*fT**=F,7$$\"3)emmm^&Q%R)F,$!39J7$$\"3))*****\\YJ?;\"!#<$\"3m!=?Y3*>`CFK7$$\"3?LL L=\"\\g**FK7$$\"3\")*****\\[A4]\"FO$\"3Xgu?U;&er\"F ,7$$\"3wmmm'3Q\\n\"FO$\"3S\\4.g(y\\S#F,7$$\"3OLLLB6@G=FO$\"3e*[f2BGC&H F,7$$\"3&)******f-w+?FO$\"375@EVOJ&[$F,7$$\"3%*********y,u@FO$\"3VG2]n #=i\"RF,7$$\"3)*******RP)4M#FO$\"3ym!)\\t%R1A%F,7$$\"3Umm;HUz;CFO$\"3: @(\\YT,0K%F,7$$\"3ILLL=Zg#\\#FO$\"3++xVHVa&R%F,7$$\"3;++]A2v#e#FO$\"3+ <'Hh4))=X%F,7$$\"3cmmmEn*Gn#FO$\"3a5#zx'*y?Z%F,7$$\"3qmmm;AE\\FFO$\"35 ^%H>#ywgWF,7$$\"3Tmmm1xiDGFO$\"3(3\\(>4bXBWF,7$$\"3LLL$e#*eW\"HFO$\"3! 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$!3;B6I^7jsMF,7$$\"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$\"3ZX^U-))=F,7$$\"\")F)$\"3s<7[GmrgDF,-%'COLOURG6&%$RGBG$\"#5!\"\"F(F(-%%FON TG6$%*HELVETICAG\"\"*-%+AXESLABELSG6$%\"xG%%y(x)G-%%VIEWG6$;F(Fg]l%(DE FAULTG" 1 2 0 1 10 0 2 9 1 4 2 1.000000 45.000000 45.000000 0 0 "Curve 1" }}}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 32 "T he following code constructs a " }{TEXT 260 17 "discrete solution" } {TEXT -1 44 " based on each of the methods and gives the " }{TEXT 260 22 "root mean square error" }{TEXT -1 18 " of each solution." }} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 871 "E := (x,y) -> cos(x)+2*y: h h := 0.02: numsteps := 400: x0 := 0: y0 := -2/5:\nmatrix([[`slope fiel d: `,E(x,y)],[`initial point: `,``(x0,y0)],[`step width: `,hh],\n[ `no. of steps: `,numsteps]]);``;\nmthds := [`Papakostas' scheme with `*``(c[7]=199/200),`a modification of Papakostas' scheme with `*``(c[ 7]=199/200),`a scheme with `*``(c[7]=324/325),`a scheme with `*``(c[7] =575/576),`a scheme with `*``(c[7]=1728/1729),`Tsitouras' scheme with \+ `*``(c[7]=544/545),`Verner's \"most efficient\" scheme with `*``(c[7]= 1999/2000)]: errs := []:\nDigits := 20:\nfor ct to 7 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)pprint356\"" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%!G" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#K%'matrixG6#7)7$*&%9Papakostas'~scheme ~with~G\"\"\"-%!G6#/&%\"cG6#\"\"(#\"$*>\"$+#F*$\"+jj>,7$*&%GVerner's~\"m ost~efficient\"~scheme~with~GF*-F,6#/F/#\"%**>\"%+?F*$\"+&GUp3(!#?Q)pp rint366\"" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 30 "The following code constructs " }{TEXT 260 20 "numerical proced ures" }{TEXT -1 56 " for solutions based on each of the Runge-Kutta sc hemes." }}{PARA 0 "" 0 "" {TEXT -1 75 "The error in the value obtained by each of the methods at the point where " }{XPPEDIT 18 0 "x = 7.99 9;" "6#/%\"xG-%&FloatG6$\"%**z!\"$" }{TEXT -1 16 " is also given." }} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 802 "E := (x,y) -> cos(x)+2*y: h h := 0.02: numsteps := 400: x0 := 0: y0 := -2/5:\nmatrix([[`slope fiel d: `,E(x,y)],[`initial point: `,``(x0,y0)],[`step width: `,hh],\n[ `no. of steps: `,numsteps]]);``;\nmthds := [`Papakostas' scheme with `*``(c[7]=199/200),`a modification of Papakostas' scheme with `*``(c[ 7]=199/200),`a scheme with `*``(c[7]=324/325),`a scheme with `*``(c[7] =575/576),`a scheme with `*``(c[7]=1728/1729),`Tsitouras' scheme with \+ `*``(c[7]=544/545),`Verner's \"most efficient\" scheme with `*``(c[7]= 1999/2000)]: errs := []:\nDigits := 20:\nfor ct to 7 do\n en_RK6_||c t := RK6_||ct(E(x,y),x,y,x0,evalf(y0),hh,numsteps,true);\nend do:\nxx \+ := 7.999: exx := evalf(e(xx)):\nfor ct to 7 do\n errs := [op(errs),a bs(en_RK6_||ct(xx)-exx)];\nend do:\nDigits := 10:\nlinalg[transpose]([ mthds,evalf(errs)]);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#K%'matrixG6#7& 7$%0slope~field:~~~G,&-%$cosG6#%\"xG\"\"\"*&\"\"#F.%\"yGF.F.7$%0initia l~point:~G-%!G6$\"\"!#!\"#\"\"&7$%/step~width:~~~G$F0F97$%1no.~of~step s:~~~G\"$+%Q)pprint376\"" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%!G" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#K%'matrixG6#7)7$*&%9Papakostas'~scheme ~with~G\"\"\"-%!G6#/&%\"cG6#\"\"(#\"$*>\"$+#F*$\"+JK.F%*!#=7$*&%Ka~mod ification~of~Papakostas'~scheme~with~GF*F+F*$\"+N1z&y)F87$*&%/a~scheme ~with~GF*-F,6#/F/#\"$C$\"$D$F*$\"+qT>N*)F87$*&F@F*-F,6#/F/#\"$v&\"$w&F *$\"+`]q\"p(F87$*&F@F*-F,6#/F/#\"%G<\"%H*RYj!#>7$*&%GVerner's~\"m ost~efficient\"~scheme~with~GF*-F,6#/F/#\"%**>\"%+?F*$\"+%4or#RFboQ)pp rint386\"" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 4 "The " }{TEXT 260 22 "root mean square error" }{TEXT -1 20 " over the interval " }{XPPEDIT 18 0 "[0, 8];" "6#7$\"\"!\"\")" }{TEXT -1 82 " of each Runge-Kutta method is estimated as follows using the spe cial procedure " }{TEXT 0 5 "NCint" }{TEXT -1 98 " to perform numeri cal integration by the 7 point Newton-Cotes method over 200 equal subi ntervals." }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 549 "mthds := [`Pap akostas' scheme with `*``(c[7]=199/200),`a modification of Papakostas' scheme with `*``(c[7]=199/200),`a scheme with `*``(c[7]=324/325),`a s cheme with `*``(c[7]=575/576),`a scheme with `*``(c[7]=1728/1729),`Tsi touras' scheme with `*``(c[7]=544/545),`Verner's \"most efficient\" sc heme with `*``(c[7]=1999/2000)]: errs := []:\nDigits := 20:\nfor ct to 7 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:\n Digits := 10:\nlinalg[transpose]([mthds,evalf(errs)]);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#K%'matrixG6#7)7$*&%9Papakostas'~scheme~with~G\"\" \"-%!G6#/&%\"cG6#\"\"(#\"$*>\"$+#F*$\"+Ch\")p;!#=7$*&%Ka~modification~ of~Papakostas'~scheme~with~GF*F+F*$\"+LCBc:F87$*&%/a~scheme~with~GF*-F ,6#/F/#\"$C$\"$D$F*$\"+rjp#e\"F87$*&F@F*-F,6#/F/#\"$v&\"$w&F*$\"+)QOCO \"F87$*&F@F*-F,6#/F/#\"%G<\"%HA7F87$*&%8Tsitouras'~scheme~w ith~GF*-F,6#/F/#\"$W&\"$X&F*$\"+r:9C6!#>7$*&%GVerner's~\"most~efficien t\"~scheme~with~GF*-F,6#/F/#\"%**>\"%+?F*$\"+km@cp!#?Q)pprint396\"" }} }{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 92 "The foll owing error graphs are constructed using the numerical procedures for \+ the solutions." }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 679 "evalf[20] (plot([e(x)-'en_RK6_1'(x),e(x)-'en_RK6_2'(x),e(x)-'en_RK6_3'(x),e(x)-' en_RK6_4'(x),\ne(x)-'en_RK6_5'(x),e(x)-'en_RK6_6'(x),e(x)-'en_RK6_7'(x )],x=0..2,font=[HELVETICA,9],\ncolor=[COLOR(RGB,.95,0,.2),COLOR(RGB,.5 ,0,.95),COLOR(RGB,.95,.45,0),COLOR(RGB,0,.8,.25),\nCOLOR(RGB,.95,0,.95 ),COLOR(RGB,.75,.2,.2),COLOR(RGB,0,.7,.9)],legend=[`Papakostas' scheme with c[7]=199/200`,`a modification of Papakostas' scheme with c[7]=19 9/200`,`a scheme with c[7]=324/325`,`a scheme with c[7]=575/576`,`a sc heme with c[7]=1728/1729`,`Tsitouras' scheme with c[7]=544/545`,`Verne r's \"most efficient\" scheme with c[7]=1999/2000`],title=`error curve s for 8 stage order 6 Runge-Kutta methods`));" }}{PARA 13 "" 1 "" {GLPLOT2D 745 641 641 {PLOTDATA 2 "6--%'CURVESG6%7S7$$\"\"!F)F(7$$\"5M LLLLL3VfV!#@$\"&H[\"!#?7$$\"5nmmmm\"H[D:)F-$\"&P/$F07$$\"5LLLLLe0$=C\" F0$\"&6t%F07$$\"5LLLLL3RBr;F0$\"&Ca'F07$$\"5nmmm;zjf)4#F0$\"&U[)F07$$ \"5MLLL$e4;[\\#F0$\"'F\\5F07$$\"5++++]i'y]!HF0$\"'9m7F07$$\"5MLLL$ezs$ HLF0$\"'1.:F07$$\"5++++]7iI_PF0$\"'Qf+&F0$\"'C^EF07$$\"5+++++]Z/Na 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H%F07$FK$\"%\"=&F07$FP$\"%ehF07$FU$\"%@sF07$FZ$\"%\\%)F07$Fin$\"%$\\*F 07$F^o$\"&$)3\"F07$Fco$\"&^A\"F07$Fho$\"&#o8F07$F]p$\"&*4:F07$Fbp$\"&T o\"F07$Fgp$\"&r%=F07$F\\q$\"&f/#F07$Faq$\"&wB#F07$Ffq$\"&>Y#F07$F[r$\" &Cp#F07$F`r$\"''>&HF-7$Fer$\"'`1KF-7$Fjr$\"'M.NF-7$F_s$\"'qMQF-7$Fes$ \"'4XTF-7$Fjs$\"(H^]%F^t7$F`t$\"'92\\F-7$Fet$\"'IK`F-7$Fjt$\"'3wdF-7$F _u$\"'_5jF-7$Fdu$\"&+$oF07$Fiu$\"&*GuF07$F^v$\"&q,)F07$Fcv$\"&$3()F07$ Fhv$\"&YT*F07$F]w$\"'F@5F07$Fbw$\"',16F07$Fgw$\"'R-7F07$F\\x$\"'0.8F07 $Fax$\"'![T\"F07$Ffx$\"'3N:F07$F[y$\"'Ya;F07$F`y$\"':.=F07$Fey$\"'TZ>F 07$Fjy$\"''R6#F07$F_z$\"')pG#F07$Fdz$\"'\"z[#F0-Fiz6&F[[lF($\"\"(F`[l$ \"\"*F`[l-Fb[l6#%UVerner's~\"most~efficient\"~scheme~with~c[7]=1999/20 00G-%&TITLEG6#%Uerror~curves~for~8~stage~order~6~Runge-Kutta~methodsG- %%FONTG6$%*HELVETICAGF_eo-%+AXESLABELSG6$Q\"x6\"Q!F_fo-%%VIEWG6$;F(Fdz %(DEFAULTG" 1 2 0 1 10 0 2 9 1 4 2 1.000000 45.000000 45.000000 0 1 "P apakostas' scheme with c[7]=199/200" "a modification of Papakostas' sc heme with c[7]=199/200" "a scheme with c[7]=324/325" "a scheme with c[ 7]=575/576" "a scheme with c[7]=1728/1729" "Tsitouras' scheme with c[7 ]=544/545" "Verner's \"most efficient\" scheme with c[7]=1999/2000" }} }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 679 "evalf[20](plot([e(x)-'en_RK6_1'(x),e(x)-'en_RK6_2'(x),e(x)-'en_ RK6_3'(x),e(x)-'en_RK6_4'(x),\ne(x)-'en_RK6_5'(x),e(x)-'en_RK6_6'(x),e (x)-'en_RK6_7'(x)],x=2..8,font=[HELVETICA,9],\ncolor=[COLOR(RGB,.95,0, .2),COLOR(RGB,.5,0,.95),COLOR(RGB,.95,.45,0),COLOR(RGB,0,.8,.25),\nCOL OR(RGB,.95,0,.95),COLOR(RGB,.75,.2,.2),COLOR(RGB,0,.7,.9)],legend=[`Pa pakostas' scheme with c[7]=199/200`,`a modification of Papakostas' sch eme with c[7]=199/200`,`a scheme with c[7]=324/325`,`a scheme with c[7 ]=575/576`,`a scheme with c[7]=1728/1729`,`Tsitouras' scheme with c[7] =544/545`,`Verner's \"most efficient\" scheme with c[7]=1999/2000`],ti tle=`error curves for 8 stage order 6 Runge-Kutta methods`));" }} {PARA 13 "" 1 "" 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cRem&F-7$Fgu$\"*R5?A(F-7$F\\v$\"*T4\"e$*F-7$Fav$\"+#)*oM=\"F-7$Ffv$\"+ LF!)H:F-7$F[w$\"+cisZ>F-7$F`w$\"+2V42DF-7$Few$\"+4a(*3KF-7$Fjw$\"+1W=b TF-7$F_x$\"+#3i$H`F-7$Fdx$\"+@*pR(oF-7$Fix$\",8%\\eZ))F^q7$F^y$\"..&p. r:6Fdq7$Fcy$\"-7=bIu7F^q7$Fhy$\"-n?XWb9F^q7$F]z$\"-ckh;R;F^q7$Fbz$\",' )4zg%=F-7$Fgz$\",RDRb4#F-7$F\\[l$\",z))3(yBF-7$Fa[l$\",K^5bo#F-7$Ff[l$ \",J'G)=.$F-7$F[\\l$\",e_&4OKF-7$F`\\l$\",$eH1aMF-7$Fe\\l$\",;f6no$F-7 $Fj\\l$\",cGI]$RF--F_]l6&Fa]lFe]l$\"\"(Fg]l$\"\"*Fg]l-Fi]l6#%UVerner's ~\"most~efficient\"~scheme~with~c[7]=1999/2000G-%&TITLEG6#%Uerror~curv es~for~8~stage~order~6~Runge-Kutta~methodsG-%%FONTG6$%*HELVETICAGFe`p- %+AXESLABELSG6$Q\"x6\"Q!Feap-%%VIEWG6$;F(Fj\\l%(DEFAULTG" 1 2 0 1 10 0 2 9 1 4 2 1.000000 45.000000 45.000000 0 1 "Papakostas' scheme with \+ c[7]=199/200" "a modification of Papakostas' scheme with c[7]=199/200 " "a scheme with c[7]=324/325" "a scheme with c[7]=575/576" "a scheme \+ with c[7]=1728/1729" "Tsitouras' scheme with c[7]=544/545" "Verner's \+ \"most efficient\" scheme with c[7]=1999/2000" }}}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 2 " ;" }}}} {SECT 1 {PARA 4 "" 0 "" {TEXT -1 48 "Test 14 of 8 stage, order 6 Runge -Kutta methods " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 256 "" 0 "" {TEXT -1 3 " " }{XPPEDIT 18 0 "dy/dx = 10*x*cos*x-10*y;" "6#/*&%#dyG \"\"\"%#dxG!\"\",&**\"#5F&%\"xGF&%$cosGF&F,F&F&*&F+F&%\"yGF&F(" } {TEXT -1 8 ", " }{XPPEDIT 18 0 "y(0) = sqrt(5);" "6#/-%\"yG6#\" \"!-%%sqrtG6#\"\"&" }{TEXT -1 1 " " }}{PARA 0 "" 0 "" {TEXT -1 10 "Sol ution: " }}{PARA 256 "" 0 "" {TEXT -1 2 " " }{XPPEDIT 18 0 "y=100/101 " "6#/%\"yG*&\"$+\"\"\"\"\"$,\"!\"\"" }{TEXT -1 1 " " }{XPPEDIT 18 0 " x*cos*x-990/10201" "6#,&*(%\"xG\"\"\"%$cosGF&F%F&F&*&\"$!**F&\"&,-\"! \"\"F+" }{TEXT -1 1 " " }{XPPEDIT 18 0 "cos*x+10/101" "6#,&*&%$cosG\" \"\"%\"xGF&F&*&\"#5F&\"$,\"!\"\"F&" }{TEXT -1 1 " " }{XPPEDIT 18 0 "x* sin*x-200/10201" "6#,&*(%\"xG\"\"\"%$sinGF&F%F&F&*&\"$+#F&\"&,-\"!\"\" F+" }{TEXT -1 1 " " }{XPPEDIT 18 0 "sin*x+(990/10201+sqrt(5))*exp(-10* x)" "6#,&*&%$sinG\"\"\"%\"xGF&F&*&,&*&\"$!**F&\"&,-\"!\"\"F&-%%sqrtG6# \"\"&F&F&-%$expG6#,$*&\"#5F&F'F&F-F&F&" }{TEXT -1 1 "." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 163 "de := \+ diff(y(x),x)=10*x*cos(x)-10*y(x);\nic := y(0)=sqrt(5);\ndsolve(\{de,ic \},y(x));\nb := unapply(rhs(%),x):\nplot(b(x),x=0..5,font=[HELVETICA,9 ],labels=[`x`,`y(x)`]);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%#deG/-%%d iffG6$-%\"yG6#%\"xGF,,&*(\"#5\"\"\"F,F0-%$cosGF+F0F0*&F/F0F)F0!\"\"" } }{PARA 11 "" 1 "" {XPPMATH 20 "6#>%#icG/-%\"yG6#\"\"!*$\"\"&#\"\"\"\" \"#" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/-%\"yG6#%\"xG,,*&#\"$+\"\"$,\" \"\"\"*&F'F--%$cosGF&F-F-F-*&#\"$!**\"&,-\"F-F/F-!\"\"*&#\"#5F,F-*&-%$ sinGF&F-F'F-F-F-*&#\"$+#F4F-F:F-F5*&-%$expG6#,$*&F8F-F'F-F5F-,&#F3F4F- *$\"\"&#F-\"\"#F-F-F-" }}{PARA 13 "" 1 "" {GLPLOT2D 703 312 312 {PLOTDATA 2 "6&-%'CURVESG6$7hn7$$\"\"!F)$\"3\")*y*\\xz1OA!#<7$$\"3ALL$ 3FWYs#!#>$\"3*pc)\\jF;12h\"H\"48F,7$$ 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7$$\"3um;zpSS\"R#F,$!3/xdj'GyI^\"F,7$$\"3GLL3_?`(\\#F,$!3MGc,i$*4jpxg#F,$!3XXbRF+x>IF,$!3r\"fP(*)y]OGF,7$$\"3F+]i!RU07$F,$!3[Tb]!*H%e)HF,7$$\"3+++v =S2LKF,$!3Tyi#Q\\'[=JF,7$$\"3Jmmm\"p)=MLF,$!3N`)=cl>V?$F,7$$\"3B++](=] @W$F,$!3Y_BiN[odKF,7$$\"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$F,$!3D1>x`HA5KF,7$$\"3[m;H2qcZPF ,$!3/[q%\\V.-<$F,7$$\"3O+]7.\"fF&QF,$!3KL?tX&>E0$F,7$$\"3Ymm;/OgbRF,$! 3KQEMNc$G*GF,7$$\"3w**\\ilAFjSF,$!3/QR)44g!yEF,7$$\"3yLLL$)*pp;%F,$!30 ,GW_`#fU#F,7$$\"3)RL$3xe,tUF,$!3*G#*H@1([B@F,7$$\"3Cn;HdO=yVF,$!35Q!)* 4x]5y\"F,7$$\"3a+++D>#[Z%F,$!3(y*pyl_QJ9F,7$$\"3SnmT&G!e&e%F,$!3]X/0\" RC%G**F@7$$\"3#RLLL)Qk%o%F,$!3u!*)Q\"4WH+dF@7$$\"37+]iSjE!z%F,$!3+r[gf MO'=*F07$$\"3a+]P40O\"*[F,$\"3+2*eSHde(QF@7$$\"\"&F)$\"3&Q8`\">jC3#*F@ -%'COLOURG6&%$RGBG$\"#5!\"\"F(F(-%+AXESLABELSG6$%\"xG%%y(x)G-%%FONTG6$ %*HELVETICAG\"\"*-%%VIEWG6$;F(F\\^l%(DEFAULTG" 1 2 0 1 10 0 2 9 1 4 2 1.000000 45.000000 45.000000 0 0 "Curve 1" }}}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 32 "The following code constructs a " }{TEXT 260 17 "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 0 887 "B := (x,y) -> 10*x*cos(x)-10*y: hh := 0.01: numsteps := 500: \+ x0 := 0: y0 := sqrt(5):\nmatrix([[`slope field: `,B(x,y)],[`initial \+ point: `,``(x0,y0)],[`step width: `,hh],\n[`no. of steps: `,numste ps]]);``;\nmthds := [`Papakostas' scheme with `*``(c[7]=199/200),`a mo dification of Papakostas' scheme with `*``(c[7]=199/200),`a scheme wit h `*``(c[7]=324/325),`a scheme with `*``(c[7]=575/576),`a scheme with \+ `*``(c[7]=1728/1729),`Tsitouras' scheme with `*``(c[7]=544/545),`Verne r's \"most efficient\" scheme with `*``(c[7]=1999/2000)]: errs := []: \nDigits := 20:\nfor ct to 7 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/n umpts)];\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~p oint:~G-%!G6$\"\"!*$\"\"&#F,\"\"#7$%/step~width:~~~G$F,!\"#7$%1no.~of~ steps:~~~G\"$+&Q(pprint26\"" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%!G" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#K%'matrixG6#7)7$*&%9Papakostas'~scheme ~with~G\"\"\"-%!G6#/&%\"cG6#\"\"(#\"$*>\"$+#F*$\"+WU#=\"o!#A7$*&%Ka~mo dification~of~Papakostas'~scheme~with~GF*F+F*$\"+'z_0(fF87$*&%/a~schem e~with~GF*-F,6#/F/#\"$C$\"$D$F*$\"+xdv)p'F87$*&F@F*-F,6#/F/#\"$v&\"$w& F*$\"++f0XgF87$*&F@F*-F,6#/F/#\"%G<\"%H\"%+?F*$\"+0%)z6TFboQ(pprin t36\"" }}}{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 = 4.999;" "6 #/%\"xG-%&FloatG6$\"%**\\!\"$" }{TEXT -1 16 " is also given." }} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 811 "B := (x,y) -> 10*x*cos(x)-1 0*y: hh := 0.01: numsteps := 500: x0 := 0: y0 := sqrt(5):\nmatrix([[`s lope field: `,B(x,y)],[`initial point: `,``(x0,y0)],[`step width: \+ `,hh],\n[`no. of steps: `,numsteps]]);``;\nmthds := [`Papakostas' sc heme with `*``(c[7]=199/200),`a modification of Papakostas' scheme wit h `*``(c[7]=199/200),`a scheme with `*``(c[7]=324/325),`a scheme with \+ `*``(c[7]=575/576),`a scheme with `*``(c[7]=1728/1729),`Tsitouras' sch eme with `*``(c[7]=544/545),`Verner's \"most efficient\" scheme with ` *``(c[7]=1999/2000)]: errs := []:\nDigits := 25:\nfor ct to 7 do\n b n_RK6_||ct := RK6_||ct(B(x,y),x,y,x0,evalf(y0),hh,numsteps,true);\nend do:\nxx := 4.999: bxx := evalf(b(xx)):\nfor ct to 7 do\n errs := [o p(errs),abs(bn_RK6_||ct(xx)-bxx)];\nend do:\nDigits := 10:\nlinalg[tra nspose]([mthds,evalf(errs)]);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#K%'ma trixG6#7&7$%0slope~field:~~~G,&*(\"#5\"\"\"%\"xGF,-%$cosG6#F-F,F,*&F+F ,%\"yGF,!\"\"7$%0initial~point:~G-%!G6$\"\"!*$\"\"&#F,\"\"#7$%/step~wi dth:~~~G$F,!\"#7$%1no.~of~steps:~~~G\"$+&Q)pprint426\"" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%!G" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#K%'matrixG6 #7)7$*&%9Papakostas'~scheme~with~G\"\"\"-%!G6#/&%\"cG6#\"\"(#\"$*>\"$+ #F*$\"+&*RRM#)!#B7$*&%Ka~modification~of~Papakostas'~scheme~with~GF*F+ F*$\"+Z&QTs(F87$*&%/a~scheme~with~GF*-F,6#/F/#\"$C$\"$D$F*$\"+f,4\")yF 87$*&F@F*-F,6#/F/#\"$v&\"$w&F*$\"+.t$H-(F87$*&F@F*-F,6#/F/#\"%G<\"%H\"%+?F*$\"+%GKS#=F8Q)pprint436\"" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 4 "The " }{TEXT 260 22 "root mean square error" }{TEXT -1 20 " over the interval " }{XPPEDIT 18 0 "[0, 5];" " 6#7$\"\"!\"\"&" }{TEXT -1 82 " of each Runge-Kutta method is estimate d 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 0 549 "mthds := [`Papakostas' scheme with `*``(c[7]=199/200 ),`a modification of Papakostas' scheme with `*``(c[7]=199/200),`a sch eme with `*``(c[7]=324/325),`a scheme with `*``(c[7]=575/576),`a schem e with `*``(c[7]=1728/1729),`Tsitouras' scheme with `*``(c[7]=544/545) ,`Verner's \"most efficient\" scheme with `*``(c[7]=1999/2000)]: errs \+ := []:\nDigits := 20:\nfor ct to 7 do\n sm := NCint((b(x)-'bn_RK6_|| ct'(x))^2,x=0..5,adaptive=false,numpoints=7,factor=200);\n errs := [ op(errs),sqrt(sm/5)];\nend do:\nDigits := 10:\nlinalg[transpose]([mthd s,evalf(errs)]);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#K%'matrixG6#7)7$*& %9Papakostas'~scheme~with~G\"\"\"-%!G6#/&%\"cG6#\"\"(#\"$*>\"$+#F*$\"+ jE2ql!#A7$*&%Ka~modification~of~Papakostas'~scheme~with~GF*F+F*$\"+;lr edF87$*&%/a~scheme~with~GF*-F,6#/F/#\"$C$\"$D$F*$\"+:^/hkF87$*&F@F*-F, 6#/F/#\"$v&\"$w&F*$\"+DJiIeF87$*&F@F*-F,6#/F/#\"%G<\"%H\"%+?F*$\"+? L2bRFboQ)pprint446\"" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 92 "The following error graphs are constructed using the n umerical procedures for the solutions." }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 696 "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),'bn_RK6_7'(x)-b(x)],x=0..0.65,numpoints=100,font=[HELVETICA, 9],\ncolor=[COLOR(RGB,.95,0,.2),COLOR(RGB,.5,0,.95),COLOR(RGB,.95,.45, 0),COLOR(RGB,0,.8,.25),\nCOLOR(RGB,.95,0,.95),COLOR(RGB,.75,.2,.2),COL OR(RGB,0,.7,.9)],legend=[`Papakostas' scheme with c[7]=199/200`,`a mod ification of Papakostas' scheme with c[7]=199/200`,`a scheme with c[7] =324/325`,`a scheme with c[7]=575/576`,`a scheme with c[7]=1728/1729`, `Tsitouras' scheme with c[7]=544/545`,`Verner's \"most efficient\" sch eme with c[7]=1999/2000`],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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#pm2.Vi=XF[hn7$F_em$\":(zeDZa8V'*Rn)f%F[hn7$Fdem$\":e&=,i69B(f\"fpWF[h n7$Fiem$\":[gNaYg4X1w)>WF[hn7$F^fm$\":z9u4q'R:&4\">QXF[hn7$Fcfm$\":J;B `Q>]]ZMEh%F[hn7$Fhfm$\":$RzN%f.^!=CU4YF[hn7$F]gm$\":\")*Qp;&>C&\\[\"=[ %F[hn7$Fbgm$\":3Q(Gl6cnq`FZF[hn7$F[im$\":?H()**3nPbZFyh%F[hn7$F`im$\":eVz3&p,)ym_^b%F[hn7 $Feim$\":lENJ6DYK)HX3XF[hn7$Fjim$\":;B&e2F:IiBUXYF[hn7$F_jm$\":!=$Q'y& HyI\\m[q%F[hn7$Fdjm$\":IGv/z.!yk:()zXF[hn7$Fijm$\"::WdBn\"R\\&)Hl5XF[h n7$F^[n$\":O\")[QqlKf)**z)z%F[hn7$Fc[n$\":#3uu&)p*e.Z]$pYF[hn7$Fh[n$\" :MmH8Ac9vmZ2b%F[hn7$F]\\n$\":!R/fEq(oN%3-dXF[hn7$Fb\\n$\":d>-sgs\\\"oP M)o%F[hn7$Fg\\n$\":%\\GWn&4ocR*=<[F[hn7$F\\]n$\":\"y!)*z`az4(3b#p%F[hn 7$Fa]n$\":\\!)[-5%4gCIl!e%F[hn7$Ff]n$\":gs%R'*>^&)3Z#)fYF[hn7$F[^n$\": Cj+\"*R9`$G.F!y%F[hn7$F`^n$\":9Ban]`H%*)eOqYF[hn7$Fe^n$\":*G(=UC%\\1H& [.e%F[hn7$Fj^n$\":Ld7;+#QO^E%yl%F[hn7$F__n$\":v&f&yU9/,+f7z%F[hn7$Fi_n $\":6&GwCb[-eMTAYF[hn7$Fabn$\":N:#Q'p1NZ.Lgb%F[hn7$Ffbn$\":r(pKx])H'>' *\\NYF[hn7$F[cn$\":a?Bjj2l?jchz%F[hn7$F`cn$\":)46/**>FdW0-%o%F[hn7$Fec n$\":Df.G-h?qbeqUmq#F[hn7$Fagn$\":;*G h,pk>**)[GI#F[hn7$Ffgn$\":fA\"f)Q+SD>ye(=F[hn-F]hn6&F_hnFbhn$\"\"(Fehn $\"\"*Fehn-Fghn6#%UVerner's~\"most~efficient\"~scheme~with~c[7]=1999/2 000G-%&TITLEG6#%Uerror~curves~for~8~stage~order~6~Runge-Kutta~methodsG -%%FONTG6$%*HELVETICAGFfe]l-%+AXESLABELSG6$Q\"x6\"Q!Fff]l-%%VIEWG6$;F( Ffgn%(DEFAULTG" 1 2 0 1 10 0 2 9 1 4 2 1.000000 45.000000 45.000000 0 1 "Papakostas' scheme with c[7]=199/200" "a modification of Papakostas ' scheme with c[7]=199/200" "a scheme with c[7]=324/325" "a scheme wit h c[7]=575/576" "a scheme with c[7]=1728/1729" "Tsitouras' scheme with c[7]=544/545" "Verner's \"most efficient\" scheme with c[7]=1999/2000 " }}}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 48 "Test 15 of 8 stage, order 6 Runge-Kutta methods " }}{PARA 0 "" 0 "" {TEXT -1 0 " " }}{PARA 0 "" 0 "" {TEXT -1 198 "See: Comparing Numerical Methods for Ordinary Differential Equations, Hull, Enright, Fellen and Sedgwick, \n Siam Journal on Numerical 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*arctan(y/x)+4*l n*x-2*ln*2-Pi = 0;" "6#/,,*&\"\"#\"\"\"-%#lnG6#*&,&*$%\"xGF&F'*$%\"yGF &F'F'*$F.F&!\"\"F'F'*&\"\"%F'-%'arctanG6#*&F0F'F.F2F'F'*(F4F'F)F'F.F'F '*(F&F'F)F'F&F'F2%#PiGF2\"\"!" }{TEXT -1 2 ". " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 73 "de := diff(y (x),x)=(y(x)-x)/(y(x)+x);\nic := y(1)=1;\ndsolve(\{de,ic\},y(x));" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#>%#deG/-%%diffG6$-%\"yG6#%\"xGF,*&,&F) \"\"\"F,!\"\"F/,&F)F/F,F/F0" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%#icG/ -%\"yG6#\"\"\"F)" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/-%\"yG6#%\"xG-%'R ootOfG6#,,*&\"\"#\"\"\"-%#lnG6#*&,&*$)F'F-F.F.*$)%#_ZGF-F.F.F.F'!\"#F. !\"\"*&\"\"%F.-%'arctanG6#*&F8F.F'F:F.F:*&F " 0 "" {MPLTEXT 1 0 1 ";" }}}{PARA 0 " " 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 41 "The solution can b e 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 293 1 "x" } {TEXT -1 47 " increasing) is the section of the polar curve " }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "r=sqrt(2)*exp(Pi/4-theta)" "6#/%\"rG*&-%%sqrtG6#\"\"#\"\"\"-%$expG6#,&*&%#PiGF*\"\"%!\"\"F*%&thet aGF2F*" }{TEXT -1 5 ", " }{XPPEDIT 18 0 "-Pi/4<=theta" "6#1,$*&%#Pi G\"\"\"\"\"%!\"\"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 291 1 "y" }{TEXT -1 25 " numerically in terms of " }{TEXT 292 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-$!+ " 0 "" {MPLTEXT 1 0 932 "C := (x,y) -> (y-x)/(y+x): hh := 0.01: numsteps := 3 75: x0 := 1: y0 := 1:\nmatrix([[`slope field: `,C(x,y)],[`initial po int: `,``(x0,y0)],[`step width: `,hh],\n[`no. of steps: `,numsteps ]]);``;\nmthds := [`Papakostas' scheme with `*``(c[7]=199/200),`a modi fication of Papakostas' scheme with `*``(c[7]=199/200),`a scheme with \+ `*``(c[7]=324/325),`a scheme with `*``(c[7]=575/576),`a scheme with `* ``(c[7]=1728/1729),`Tsitouras' scheme with `*``(c[7]=544/545),`Verner' s \"most efficient\" scheme with `*``(c[7]=1999/2000)]: errs := []: va ls := []:\nDigits := 25:\nfor ct to 7 do\n Cn_RK6_||ct := RK6_||ct( C(x,y),x,y,x0,y0,hh,numsteps,false);\n sm := 0: numpts := nops(Cn_RK 6_||ct):\n for ii to numpts do\n if ct=1 then vals := [op(vals) ,phi(Cn_RK6_||ct[ii,1])] end if;\n sm := sm+(Cn_RK6_||ct[ii,2]-va ls[ii])^2;\n end do:\n errs := [op(errs),sqrt(sm/numpts)];\nend do :\nDigits := 10:\nlinalg[transpose]([mthds,evalf(errs)]);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#K%'matrixG6#7&7$%0slope~field:~~~G*&,&%\"yG\"\" \"%\"xG!\"\"F,,&F-F,F+F,F.7$%0initial~point:~G-%!G6$F,F,7$%/step~width :~~~G$F,!\"#7$%1no.~of~steps:~~~G\"$v$Q)pprint456\"" }}{PARA 11 "" 1 " " {XPPMATH 20 "6#%!G" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#K%'matrixG6#7) 7$*&%9Papakostas'~scheme~with~G\"\"\"-%!G6#/&%\"cG6#\"\"(#\"$*>\"$+#F* $\"+)Q'\\d!)!#B7$*&%Ka~modification~of~Papakostas'~scheme~with~GF*F+F* $\"++%f\\w(F87$*&%/a~scheme~with~GF*-F,6#/F/#\"$C$\"$D$F*$\"+&*=P%G'F8 7$*&F@F*-F,6#/F/#\"$v&\"$w&F*$\"+Hg\"4u&F87$*&F@F*-F,6#/F/#\"%G<\"%H \"%+?F*$\"+I*yCt)F8Q)pprint466\"" }}}{PARA 0 "" 0 "" {TEXT -1 2 " " } }{PARA 0 "" 0 "" {TEXT -1 30 "The following code constructs " }{TEXT 260 20 "numerical procedures" }{TEXT -1 56 " for solutions based on ea ch of the Runge-Kutta schemes." }}{PARA 0 "" 0 "" {TEXT -1 75 "The err or in the value obtained by each of the methods at the point where " }{XPPEDIT 18 0 "x = 4.749;" "6#/%\"xG-%&FloatG6$\"%\\Z!\"$" }{TEXT -1 16 " is also given." }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 795 "C : = (x,y) -> (y-x)/(y+x): hh := 0.01: numsteps := 375: x0 := 1: y0 := 1: \nmatrix([[`slope field: `,C(x,y)],[`initial point: `,``(x0,y0)],[`s tep width: `,hh],\n[`no. of steps: `,numsteps]]);``;\nmthds := [`P apakostas' scheme with `*``(c[7]=199/200),`a modification of Papakosta s' scheme with `*``(c[7]=199/200),`a scheme with `*``(c[7]=324/325),`a scheme with `*``(c[7]=575/576),`a scheme with `*``(c[7]=1728/1729),`T sitouras' scheme with `*``(c[7]=544/545),`Verner's \"most efficient\" \+ scheme with `*``(c[7]=1999/2000)]: errs := []:\nDigits := 30:\nfor ct \+ to 7 do\n cn_RK6_||ct := RK6_||ct(C(x,y),x,y,x0,y0,hh,numsteps,true) ;\nend do:\nxx := 4.749: cxx := evalf(phi(xx)):\nfor ct to 7 do\n er rs := [op(errs),abs(cn_RK6_||ct(xx)-cxx)];\nend do:\nDigits := 10:\nli nalg[transpose]([mthds,evalf(errs)]);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#K%'matrixG6#7&7$%0slope~field:~~~G*&,&%\"yG\"\"\"%\"xG!\"\"F,,&F-F, F+F,F.7$%0initial~point:~G-%!G6$F,F,7$%/step~width:~~~G$F,!\"#7$%1no.~ of~steps:~~~G\"$v$Q)pprint476\"" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%!G " }}{PARA 11 "" 1 "" {XPPMATH 20 "6#K%'matrixG6#7)7$*&%9Papakostas'~sc heme~with~G\"\"\"-%!G6#/&%\"cG6#\"\"(#\"$*>\"$+#F*$\"+&[;;o*!#A7$*&%Ka ~modification~of~Papakostas'~scheme~with~GF*F+F*$\"+$>F6B*F87$*&%/a~sc heme~with~GF*-F,6#/F/#\"$C$\"$D$F*$\"+=Mm\"\\(F87$*&F@F*-F,6#/F/#\"$v& \"$w&F*$\"+&)H2)p'F87$*&F@F*-F,6#/F/#\"%G<\"%H\"%+?F*$\"+d,`c5!#@Q )pprint486\"" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 4 "The " }{TEXT 260 22 "root mean square error" }{TEXT -1 20 " over the interval " }{XPPEDIT 18 0 "[1, 4.75];" "6#7$\"\"\"-%&Float G6$\"$v%!\"#" }{TEXT -1 82 " of each Runge-Kutta method is estimated \+ as follows using the special procedure " }{TEXT 0 5 "NCint" }{TEXT -1 98 " to perform numerical integration by the 7 point Newton-Cotes \+ method over 200 equal subintervals." }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 556 "mthds := [`Papakostas' scheme with `*``(c[7]=199/200 ),`a modification of Papakostas' scheme with `*``(c[7]=199/200),`a sch eme with `*``(c[7]=324/325),`a scheme with `*``(c[7]=575/576),`a schem e with `*``(c[7]=1728/1729),`Tsitouras' scheme with `*``(c[7]=544/545) ,`Verner's \"most efficient\" scheme with `*``(c[7]=1999/2000)]: errs \+ := []:\nDigits := 20:\nfor ct to 7 do\n sm := NCint(('phi'(x)-'cn_RK 6_||ct'(x))^2,x=1..4.75,adaptive=false,numpoints=7,factor=200);\n er rs := [op(errs),sqrt(sm/5)];\nend do:\nDigits := 10:\nlinalg[transpose ]([mthds,evalf(errs)]);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#K%'matrixG6 #7)7$*&%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 constructe d using the numerical procedures for the solutions." }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 728 "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'c n_RK6_5'(x)-'phi'(x),'cn_RK6_6'(x)-'phi'(x),'cn_RK6_7'(x)-'phi'(x)],x= 1..3.75,-2.3e-19..8.1e-19,\nfont=[HELVETICA,9],color=[COLOR(RGB,.95,0, .2),COLOR(RGB,.5,0,.95),COLOR(RGB,.95,.45,0),COLOR(RGB,0,.8,.25),\nCOL OR(RGB,.95,0,.95),COLOR(RGB,.75,.2,.2),COLOR(RGB,0,.7,.9)],legend=[`Pa pakostas' scheme with c[7]=199/200`,`a modification of Papakostas' sch eme with c[7]=199/200`,`a scheme with c[7]=324/325`,`a scheme with c[7 ]=575/576`,`a scheme with c[7]=1728/1729`,`Tsitouras' scheme with c[7] =544/545`,`Verner's \"most efficient\" scheme with c[7]=1999/2000`],ti tle=`error curves for 8 stage order 6 Runge-Kutta methods`));" }} {PARA 13 "" 1 "" {GLPLOT2D 1041 597 597 {PLOTDATA 2 "6--%'CURVESG6%7S7 $$\"\"\"\"\"!$F*F*7$$\"?LLLLLLLL$eRR *F27$$\"?++++++++]PR8w(o\"F/$\"-N$y'p65F27$$\"?++++++++]7`'=tu\"F/$\"- Fy-n&3\"F27$$\"?++++++++]igJr/=F/$\"-*Grq\"f6F27$$\"?LLLLLLL$3F>(G$o&= F/$\"-uMh7D7F27$$\"?nmmmmmmm;z\\#3)=>F/$\"-W9)puI\"F27$$\"?nmmmmmmmm;C 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1.000000 45.000000 45.000000 0 1 "Papakostas' scheme with c[7]=1 99/200" "a modification of Papakostas' scheme with c[7]=199/200" "a sc heme with c[7]=324/325" "a scheme with c[7]=575/576" "a scheme with c[ 7]=1728/1729" "Tsitouras' scheme with c[7]=544/545" "Verner's \"most e fficient\" scheme with c[7]=1999/2000" }}}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 731 "evalf[30](plot(['cn_RK 6_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),'cn_RK 6_7'(x)-'phi'(x)],x=3.75..4.6,-4.9e-16..1.09e-15,\nfont=[HELVETICA,9], color=[COLOR(RGB,.95,0,.2),COLOR(RGB,.5,0,.95),COLOR(RGB,.95,.45,0),CO LOR(RGB,0,.8,.25),\nCOLOR(RGB,.95,0,.95),COLOR(RGB,.75,.2,.2),COLOR(RG B,0,.7,.9)],legend=[`Papakostas' scheme with c[7]=199/200`,`a modifica tion of Papakostas' scheme with c[7]=199/200`,`a scheme with c[7]=324/ 325`,`a scheme with c[7]=575/576`,`a scheme with c[7]=1728/1729`,`Tsit ouras' scheme with c[7]=544/545`,`Verner's \"most efficient\" scheme w ith c[7]=1999/2000`],title=`error curves for 8 stage order 6 Runge-Kut ta methods`));" }}{PARA 13 "" 1 "" {GLPLOT2D 953 698 698 {PLOTDATA 2 " 6--%'CURVESG6%7co7$$\"$v$!\"#$\"-^_s!=M)!#I7$$\"?nmmmmmmm;/\"eF&oP!#H$ \"-)f^6,e)F-7$$\"?LLLLLLL$eR-L[Yy$F1$\"-C=*)QU))F-7$$\"?nmmmmmmm\"H()z xF!QF1$\"-%pRl(p\"*F-7$$\"?nmmmmmmmT5Tu-@QF1$\"-U,a$o_*F-7$$\"?LLLLLLL $e9hM!>RQF1$\"-(*zu$H')*F-7$$\"?nmmmmmm;H2%oHg&QF1$\".+_yY+-\"F-7$$\"? +++++++]i:VeYtQF1$\"-sgZ+k5F17$$\"?nmmmmmm;H#QM)\\\"*QF1$\"-[2#*\\86F1 7$$\"?+++++++]7.9IZ4RF1$\"-!e,GD;\"F17$$\"?LLLLLLLL$3<,:F17$$\"?nmmmmmm;HK^#R[,%F1$\"-&)3CH'e\"F17$$\"?LLLLLLLL$3sF &*R.%F1$\"-I*>hRq\"F17$$\"?LLLLLLLLL$Q%HA]SF1$\"-o7$fQ$=F17$$\"?++++++ +]i!zd*4pSF1$\"->Xtog>F17$$\"?LLLLLLLLLe(f3e3%F1$\"-33t)o4#F17$$\"?+++ ++++]iSS59/TF1$\"-*4X2TH#F17$$\"?+++++++](=7'yf@TF1$\"-sp[pDDF17$$\"?L LLLLLLLeRAA\")RTF1$\"-4>a,*y#F17$$\"?LLLLLLL$ek=pQl:%F1$\"-Y[@)p*HF17$ 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L]\"3>5F17$FX$!,Be'RZ5F17$Fgn$!,v(f$z2\"F17$F\\o$!,'yeT76F17$Fao$!,s[' 3c6F17$Ffo$!,dtlo?\"F17$F[p$!,zPOFD\"F17$F`p$!,'e;p38F17$Fep$!,&HKH(Q \"F17$Fjp$!,^&z\"pZ\"F17$F_q$!,!=(oqc\"F17$Fdq$!,+B]0n\"F17$Fiq$!,Y'[$ \\#=F17$F^r$!,?T>a,#F17$Fcr$!,L!)G9*F17$F`z$!/etA>4\"=\"F17$Fez$!/$R%=B6Y:F17 $Fjz$!/FW5]\\2,^FF17$Fg]l$!/\"oT=Zk^GF17 $Fa^l$!/o^Z[gCIF17$Ff^l$!/yEE'4,X$F17$F[_l$!/EA.)e5q$F17$F`_l$!/V$*3#y ps$F17$Fe_l$!/?x#fx=w$F17$Fj_l$!/2(p:`!4QF17$F_`l$!/)*)Hr+z$RF17$Fd`l$ !/=qOEXoSF17$Fi`l$!/#*QzMWvUF17$F^al$!/Es4:?&f%F17$Fcal$!/bxxS\"o2&F1- Fial6&F[blF^bl$\"\"(Feal$\"\"*Feal-Fcbl6#%UVerner's~\"most~efficient\" ~scheme~with~c[7]=1999/2000G-%&TITLEG6#%Uerror~curves~for~8~stage~orde r~6~Runge-Kutta~methodsG-%%FONTG6$%*HELVETICAGF^[r-%+AXESLABELSG6$Q\"x 6\"Q!F^\\r-%%VIEWG6$;F(Fcal;$!#\\!#<$\"$4\"Fg\\r" 1 2 0 1 10 0 2 9 1 4 2 1.000000 45.000000 45.000000 0 1 "Papakostas' scheme with c[7]=199 /200" "a modification of Papakostas' scheme with c[7]=199/200" "a sche me with c[7]=324/325" "a scheme with c[7]=575/576" "a scheme with c[7] =1728/1729" "Tsitouras' scheme with c[7]=544/545" "Verner's \"most eff icient\" scheme with c[7]=1999/2000" }}}}{PARA 0 "" 0 "" {TEXT -1 0 " " }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 712 "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),'cn_RK6 _7'(x)-'phi'(x)],x=4.6..4.75,\nfont=[HELVETICA,9],color=[COLOR(RGB,.95 ,0,.2),COLOR(RGB,.5,0,.95),COLOR(RGB,.95,.45,0),COLOR(RGB,0,.8,.25),\n COLOR(RGB,.95,0,.95),COLOR(RGB,.75,.2,.2),COLOR(RGB,0,.7,.9)],legend=[ `Papakostas' scheme with c[7]=199/200`,`a modification of Papakostas' \+ scheme with c[7]=199/200`,`a scheme with c[7]=324/325`,`a scheme with \+ c[7]=575/576`,`a scheme with c[7]=1728/1729`,`Tsitouras' scheme with c [7]=544/545`,`Verner's \"most efficient\" scheme with c[7]=1999/2000`] ,title=`error curves for 8 stage order 6 Runge-Kutta methods`));" }} {PARA 13 "" 1 "" {GLPLOT2D 978 723 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