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" }}{PARA 0 "" 0 "" {TEXT -1 134 "They can be read \+ into a Maple session by commands similar to those that follow, where e ach 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/procdrs/intg.m\" ;" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}}{EXCHG {PARA 0 " > " 0 "" {MPLTEXT 1 0 1 ";" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 51 "#============================================== ====" }}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 55 "Step by step construction of the Prince-Dormand scheme " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }} {PARA 0 "" 0 "" {TEXT -1 5 "See: " }}{PARA 0 "" 0 "" {TEXT -1 83 "(1) \+ High order embedded Runge\226Kutta formulae, by P.J. Prince and J. \+ R. Dormand, " }}{PARA 0 "" 0 "" {TEXT -1 88 " Journal of Comput ational and Applied Mathematics, Vol.7, (1981), pages 67 to 75." }} {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 64 "(2) On Runge-Kutta Processes of High Order, by J. C. Butcher," }}{PARA 0 "" 0 "" {TEXT -1 90 " Journal of the Australian Mathematical Soci ety, Vol. 4, (1964), pages 179 to 194." }}{PARA 0 "" 0 "" {TEXT -1 0 " " }}{PARA 0 "" 0 "" {TEXT -1 155 "As suggested by Prince and Dormand i n (1) their scheme can be constructed in a step-by-step manner similar to that used by Butcher for his order 6 schemes." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 92 "------------------------- -------------------------------------------------------------------" } }{PARA 0 "" 0 "" {TEXT -1 17 "We specify that " }{XPPEDIT 18 0 "c[2] \+ = 1/10;" "6#/&%\"cG6#\"\"#*&\"\"\"F)\"#5!\"\"" }{TEXT -1 3 ", " } {XPPEDIT 18 0 "c[3] = 2/9;" "6#/&%\"cG6#\"\"$*&\"\"#\"\"\"\"\"*!\"\"" }{TEXT -1 3 ", " }{XPPEDIT 18 0 "c[5] = 3/5;" "6#/&%\"cG6#\"\"&*&\"\" $\"\"\"F'!\"\"" }{TEXT -1 3 ", " }{XPPEDIT 18 0 "c[6] = 4/5;" "6#/&% \"cG6#\"\"'*&\"\"%\"\"\"\"\"&!\"\"" }{TEXT -1 3 ", " }{XPPEDIT 18 0 " c[7]=1" "6#/&%\"cG6#\"\"(\"\"\"" }{TEXT -1 3 ", " }{XPPEDIT 18 0 "c[8 ]=1" "6#/&%\"cG6#\"\")\"\"\"" }{TEXT -1 3 ", " }{XPPEDIT 18 0 "b[2]=0 " "6#/&%\"bG6#\"\"#\"\"!" }{TEXT -1 3 ", " }{XPPEDIT 18 0 "b[8]=1/10 " "6#/&%\"bG6#\"\")*&\"\"\"F)\"#5!\"\"" }{TEXT -1 7 " and " } {XPPEDIT 18 0 "a[8,7] = 0;" "6#/&%\"aG6$\"\")\"\"(\"\"!" }{TEXT -1 1 " ." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 15 "" 0 "" {TEXT 281 6 "Ste p 1" }{TEXT -1 1 " " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 " " {TEXT -1 21 "We use the relation " }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "c[4]=c[3]/(15*c[3]^2-10*c[3]+2)" "6#/&%\"cG6#\"\"%* &&F%6#\"\"$\"\"\",(*&\"#:F,*$&F%6#F+\"\"#F,F,*&\"#5F,&F%6#F+F,!\"\"F3F ,F8" }{TEXT -1 1 " " }}{PARA 0 "" 0 "" {TEXT -1 14 "to determine " } {XPPEDIT 18 0 "c[4]" "6#&%\"cG6#\"\"%" }{TEXT -1 36 " and use the qua drature conditions:" }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "Sum(b[i],i = 1 .. 7) = 1;" "6#/-%$SumG6$&%\"bG6#%\"iG/F*;\"\"\"\"\" (F-" }{TEXT -1 15 ", " }{XPPEDIT 18 0 "Sum(b[i]*c[i]^(k-1 ),i = 2 .. 7) = 1/k;" "6#/-%$SumG6$*&&%\"bG6#%\"iG\"\"\")&%\"cG6#F+,&% \"kGF,F,!\"\"F,/F+;\"\"#\"\"(*&F,F,F2F3" }{TEXT -1 7 ", " } {XPPEDIT 18 0 "k = 2;" "6#/%\"kG\"\"#" }{TEXT -1 8 " . . 6, " }}{PARA 0 "" 0 "" {TEXT -1 21 "to find the weights " }{XPPEDIT 18 0 "b[1]" "6 #&%\"bG6#\"\"\"" }{TEXT -1 2 ", " }{XPPEDIT 18 0 "b[3]" "6#&%\"bG6#\" \"$" }{TEXT -1 3 ", " }{XPPEDIT 18 0 "b[4];" "6#&%\"bG6#\"\"%" } {TEXT -1 3 ", " }{XPPEDIT 18 0 "b[5];" "6#&%\"bG6#\"\"&" }{TEXT -1 3 ", " }{XPPEDIT 18 0 "b[6]" "6#&%\"bG6#\"\"'" }{TEXT -1 3 ", " } {XPPEDIT 18 0 "b[7]" "6#&%\"bG6#\"\"(" }{TEXT -1 1 "." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 356 "e0 := \{ c[2]=1/10,c[3]=2/9,c[5]=3/5,c[6]=4/5,c[7]=1,c[8]=1,b[2]=0,b[8]=1/10,a[ 8,7]=0\}:\ne1 := `union`(e0,\{subs(e0,c[4]=c[3]/(15*c[3]^2-10*c[3]+2)) \}):\nc[4]=subs(e1,c[4]);\n``;\nquad_cdns := [add(b[i],i=1..8)=1,seq(a dd(b[i]*c[i]^(k-1),i=2..8)=1/k,k=2..6)]:\nquad_eqns := subs(e1,quad_cd ns):\nmatrix(ListTools[Enumerate](quad_eqns));\n``;\nindets(quad_eqns) ;\nnops(%);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/&%\"cG6#\"\"%#\"\"$\" \"(" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%!G" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#K%'matrixG6#7(7$\"\"\"/,0&%\"bG6#F(F(#F(\"#5F(&F,6#\"\" $F(&F,6#\"\"%F(&F,6#\"\"&F(&F,6#\"\"'F(&F,6#\"\"(F(F(7$\"\"#/,.F.F(*&# F@\"\"*F(F0F(F(*&#F2F>F(F3F(F(*&#F2F8F(F6F(F(*&#F5F8F(F9F(F(F " 0 "" {MPLTEXT 1 0 83 "e2 := solve(\{op(quad_eqns)\},\{b[1 ],b[3],b[4],b[5],b[6],b[7]\}):\ne3 := `union`(e1,e2):" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}} {SECT 1 {PARA 4 "" 0 "" {TEXT -1 2 "e3" }}{PARA 0 "" 0 "" {TEXT -1 0 " " }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 233 "e3 := \{c[2] = 1/10, b[ 2] = 0, c[6] = 4/5, c[5] = 3/5, c[3] = 2/9, c[7] = 1, c[4] = 3/7, b[7] = -37/1120, b[1] = 61/864, b[6] = 1375/5408, b[5] = 1375/7344, b[4] = 16807/146016, b[3] = 98415/321776, c[8] = 1, b[8] = 1/10, a[8,7] = 0 \}:" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 " " {TEXT -1 41 "The weights of the scheme are as follows." }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 31 "seq(b[i]=subs(e3,b[i]),i=1..8);" }} {PARA 11 "" 1 "" {XPPMATH 20 "6*/&%\"bG6#\"\"\"#\"#h\"$k)/&F%6#\"\"#\" \"!/&F%6#\"\"$#\"&:%)*\"'w " 0 "" {MPLTEXT 1 0 385 "cdns1 := [b[6]*(1-c[6])*(a[ 6,5]*c[5]*(c[5]-c[3])+a[6,4]*c[4]*(c[4]-c[3]))+\n b[5]*(1-c[5])*a [5,4]*c[4]*(c[4]-c[3])=1/60-c[3]/24,\n b[6]*(1-c[6])*a[6,5]*c[5]*(c[ 5]-c[3])*(c[5]-c[4])=1/120-(c[3]+c[4])/60+c[3]*c[4]/24,\n b[6]*(1-c[ 6])*a[6,5]*a[5,4]*c[4]*(c[4]-c[3])=1/360-c[3]/120]:\neqns1 := simplify (subs(e3,cdns1)):\nmatrix(ListTools[Enumerate](eqns1));\n``;\nindets(e qns1);\nnops(%);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#K%'matrixG6#7%7$\" \"\"/,(*&#\"$(=\"&Ci\"F(&%\"aG6$\"\"'\"\"&F(F(*&#\"$v#\"&_6'F(&F06$F2 \"\"%F(F(*&#\"%vN\"'%yR&F(&F06$F3F:F(F(#F(\"$N\"7$\"\"#/,$*&#F-\"&SY*F (F/F(F(#\"#6\"%gv7$\"\"$/,$*&F5F(*&F/F(F?F(F(F(#F(\"%!3\"Q)pprint256\" " }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%!G" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#<%&%\"aG6$\"\"'\"\"&&F%6$F'\"\"%&F%6$F(F+" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#\"\"$" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 140 "e4 := solve(\{op(eqns1)\},\{a[5,4],a[6,4 ],a[6,5]\}):\ne5 := `union`(e3,e4):\na[5,4]=subs(e5,a[5,4]),a[6,4]=sub s(e5,a[6,4]),a[6,5]=subs(e5,a[6,5]);" }}{PARA 11 "" 1 "" {XPPMATH 20 " 6%/&%\"aG6$\"\"&\"\"%#\"%)*\\\"&vy\"/&F%6$\"\"'F(#!&1U#\"&Dr$/&F%6$F/F '#\"$Q$\"$f%" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 2 "e5" }} {PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 295 "e5 := \{c[2] = 1/10, b[2] = 0, c[6] = 4/5, c[5] = 3/5, c[3] = 2/9 , c[7] = 1, c[4] = 3/7, b[7] = -37/1120, b[1] = 61/864, a[6,4] = -2420 6/37125, a[5,4] = 4998/17875, b[6] = 1375/5408, b[5] = 1375/7344, b[4] = 16807/146016, b[3] = 98415/321776, c[8] = 1, a[6,5] = 338/459, b[8] = 1/10, a[8,7] = 0\}:" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}}{PARA 0 "" 0 "" {TEXT -1 0 " " }}{PARA 15 "" 0 "" {TEXT -1 1 " " }{TEXT 284 6 "Step 3" }{TEXT -1 1 " " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 18 "We \+ can determine " }{XPPEDIT 18 0 "a[3,2]" "6#&%\"aG6$\"\"$\"\"#" } {TEXT -1 32 " from the stage-order condition" }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "a[3,2]*c[2] = 1/2" "6#/*&&%\"aG6$\"\"$ \"\"#\"\"\"&%\"cG6#F)F**&F*F*F)!\"\"" }{TEXT -1 1 " " }{XPPEDIT 18 0 " c[3]^2" "6#*$&%\"cG6#\"\"$\"\"#" }{TEXT -1 1 "." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 113 "a[3,2]*c[2] =1/2*c[3]^2:\nsubs(e5,%);\ne6 := \{a[3,2]=solve(%,a[3,2])\}:\ne7 := `u nion`(e5,e6):\na[3,2]=subs(e7,a[3,2]);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/,$*&#\"\"\"\"#5F'&%\"aG6$\"\"$\"\"#F'F'#F-\"#\")" }}{PARA 11 " " 1 "" {XPPMATH 20 "6#/&%\"aG6$\"\"$\"\"##\"#?\"#\")" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}} {SECT 1 {PARA 4 "" 0 "" {TEXT -1 2 "e7" }}{PARA 0 "" 0 "" {TEXT -1 0 " " }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 311 "e7 := \{c[2] = 1/10, b[ 2] = 0, c[6] = 4/5, c[5] = 3/5, c[3] = 2/9, c[7] = 1, c[4] = 3/7, b[7] = -37/1120, b[1] = 61/864, a[6,4] = -24206/37125, a[5,4] = 4998/17875 , b[6] = 1375/5408, b[5] = 1375/7344, b[4] = 16807/146016, b[3] = 9841 5/321776, c[8] = 1, a[6,5] = 338/459, a[3,2] = 20/81, b[8] = 1/10, a[8 ,7] = 0\}:" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 " " {MPLTEXT 1 0 1 ";" }}}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 15 "" 0 "" {TEXT -1 1 " " }{TEXT 285 6 "Step 4" }{TEXT -1 1 " " }}{PARA 0 " " 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 41 "We can now determi ne the 6 coefficients " }{XPPEDIT 18 0 "a[4,2]" "6#&%\"aG6$\"\"%\"\"# " }{TEXT -1 3 ", " }{XPPEDIT 18 0 "a[5, 2];" "6#&%\"aG6$\"\"&\"\"#" } {TEXT -1 3 ", " }{XPPEDIT 18 0 "a[6, 2];" "6#&%\"aG6$\"\"'\"\"#" } {TEXT -1 3 ", " }{XPPEDIT 18 0 "a[4,3];" "6#&%\"aG6$\"\"%\"\"$" } {TEXT -1 3 ", " }{XPPEDIT 18 0 "a[5,3];" "6#&%\"aG6$\"\"&\"\"$" } {TEXT -1 3 ", " }{XPPEDIT 18 0 "a[6,3]" "6#&%\"aG6$\"\"'\"\"$" } {TEXT -1 56 " by using the three alternative simplifying conditions: " }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "Sum(b[i]*(1-c[i]) *a[i,2],i=3..6)=0" "6#/-%$SumG6$*(&%\"bG6#%\"iG\"\"\",&F,F,&%\"cG6#F+! \"\"F,&%\"aG6$F+\"\"#F,/F+;\"\"$\"\"'\"\"!" }{TEXT -1 2 ", " }}{PARA 0 "" 0 "" {TEXT -1 8 "that is," }}{PARA 256 "" 0 "" {TEXT -1 1 " " } {XPPEDIT 18 0 "b[3]*(1-c[3])*a[3,2]+b[4]*(1-c[4])*a[4,2]+b[5]*(1-c[5]) *a[5,2]+b[6]*(1-c[6])*a[6,2] = 0" "6#/,**(&%\"bG6#\"\"$\"\"\",&F*F*&% \"cG6#F)!\"\"F*&%\"aG6$F)\"\"#F*F**(&F'6#\"\"%F*,&F*F*&F-6#F7F/F*&F16$ F7F3F*F**(&F'6#\"\"&F*,&F*F*&F-6#F@F/F*&F16$F@F3F*F**(&F'6#\"\"'F*,&F* F*&F-6#FIF/F*&F16$FIF3F*F*\"\"!" }{TEXT -1 2 ", " }}{PARA 0 "" 0 "" {TEXT -1 3 "and" }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "Su m(b[i]*(1-c[i])*Sum(a[i,j]*a[j,2],j = 3 .. i-1),i = 3 .. 6) = 0" "6#/- %$SumG6$*(&%\"bG6#%\"iG\"\"\",&F,F,&%\"cG6#F+!\"\"F,-F%6$*&&%\"aG6$F+% \"jGF,&F66$F8\"\"#F,/F8;\"\"$,&F+F,F,F1F,/F+;F>\"\"'\"\"!" }{TEXT -1 2 ", " }}{PARA 0 "" 0 "" {TEXT -1 8 "that is," }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "b[4]*(1-c[4])*a[4,3]*a[3,2]+b[5]*(1-c[5 ])*(a[5,3]*a[3,2]+a[5,4]*a[4,2])+b[6]*(1-c[6])*(a[6,3]*a[3,2]+a[6,4]*a [4,2]+a[6,5]*a[5,2]) = 0" "6#/,(**&%\"bG6#\"\"%\"\"\",&F*F*&%\"cG6#F)! \"\"F*&%\"aG6$F)\"\"$F*&F16$F3\"\"#F*F**(&F'6#\"\"&F*,&F*F*&F-6#F:F/F* ,&*&&F16$F:F3F*&F16$F3F6F*F**&&F16$F:F)F*&F16$F)F6F*F*F*F**(&F'6#\"\"' F*,&F*F*&F-6#FLF/F*,(*&&F16$FLF3F*&F16$F3F6F*F**&&F16$FLF)F*&F16$F)F6F *F**&&F16$FLF:F*&F16$F:F6F*F*F*F*\"\"!" }{TEXT -1 2 ", " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 " Sum(b[i]*(c[i]-1)*(c[i]-c[6])*a[i,2],i = 3 .. 5) = 0;" "6#/-%$SumG6$** &%\"bG6#%\"iG\"\"\",&&%\"cG6#F+F,F,!\"\"F,,&&F/6#F+F,&F/6#\"\"'F1F,&% \"aG6$F+\"\"#F,/F+;\"\"$\"\"&\"\"!" }{TEXT -1 2 ", " }}{PARA 0 "" 0 " " {TEXT -1 9 "that is, " }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "b[3]*(c[3]-1)*(c[3]-c[6])*a[3,2]+b[4]*(c[4]-1)*(c[4]-c[6])*a[4,2 ]+b[5]*(c[5]-1)*(c[5]-c[6])*a[5,2]=0" "6#/,(**&%\"bG6#\"\"$\"\"\",&&% \"cG6#F)F*F*!\"\"F*,&&F-6#F)F*&F-6#\"\"'F/F*&%\"aG6$F)\"\"#F*F***&F'6# \"\"%F*,&&F-6#F=F*F*F/F*,&&F-6#F=F*&F-6#F5F/F*&F76$F=F9F*F***&F'6#\"\" &F*,&&F-6#FKF*F*F/F*,&&F-6#FKF*&F-6#F5F/F*&F76$FKF9F*F*\"\"!" }{TEXT -1 2 ", " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 48 "together with the three stage-order conditions: " }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "Sum(a[4,j]*c[j],j = 2 .. 3) = 1/2; " "6#/-%$SumG6$*&&%\"aG6$\"\"%%\"jG\"\"\"&%\"cG6#F,F-/F,;\"\"#\"\"$*&F -F-F3!\"\"" }{TEXT -1 1 " " }{XPPEDIT 18 0 "c[4]^2;" "6#*$&%\"cG6#\"\" %\"\"#" }{TEXT -1 8 " , " }{XPPEDIT 18 0 "Sum(a[5,j]*c[j],j=2..4) =1/2" "6#/-%$SumG6$*&&%\"aG6$\"\"&%\"jG\"\"\"&%\"cG6#F,F-/F,;\"\"#\"\" %*&F-F-F3!\"\"" }{TEXT -1 1 " " }{XPPEDIT 18 0 "c[5]^2" "6#*$&%\"cG6# \"\"&\"\"#" }{TEXT -1 2 " " }}{PARA 0 "" 0 "" {TEXT -1 4 "and " }} {PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "Sum(a[6,j]*c[j],j=2.. 5)=1/2" "6#/-%$SumG6$*&&%\"aG6$\"\"'%\"jG\"\"\"&%\"cG6#F,F-/F,;\"\"#\" \"&*&F-F-F3!\"\"" }{TEXT -1 1 " " }{XPPEDIT 18 0 "c[6]^2" "6#*$&%\"cG6 #\"\"'\"\"#" }{TEXT -1 1 "." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 291 "c dns2 := [add(b[i]*(1-c[i])*a[i,2],i=3..6)=0,add(b[i]*(1-c[i])*add(a[i, j]*a[j,2],j=3..i-1),i=3..6)=0,\n add(b[i]*(1-c[i])*(c[6]-c[i])*a[i,2] ,i=3..5)=0,seq(add(a[i,j]*c[j],j=2..i-1)=1/2*c[i]^2,i=4..6)]:\neqns2 : = subs(e7,cdns2):\nmatrix(ListTools[Enumerate](eqns2));\n``;\nindets(e qns2);\nnops(%);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#K%'matrixG6#7(7$\" \"\"/,*#\"$v'\"&#\\6F(*&#\"%,C\"&/l$F(&%\"aG6$\"\"%\"\"#F(F(*&#\"$v#\" %sOF(&F36$\"\"&F6F(F(*&#F9\"%3aF(&F36$\"\"'F6F(F(\"\"!7$F6/,,*&#\"&0? \"\"'1#R(F(&F36$F5\"\"$F(F(*&#\"%v8\"&eV(F(&F36$F=FNF(F(*&#\"$V$\"&!3G F(F2F(!\"\"*&#FQ\"'7&4\"F(&F36$FCFNF(F(*&#F9\"%WtF(F;F(F(FD7$FN/,(#\"# :\"$U%F(*&#FW\"&SS\"F(F2F(F(*&#\"#bF:F(F;F(F(FD7$F5/,&*&#F(\"#5F(F2F(F (*&#F6\"\"*F(FLF(F(#F`p\"#)*7$F=/,(*&F\\pF(F;F(F(*&F_pF(FSF(F(#\"%U@\" &vy\"F(#F`p\"#]7$FC/,(*&F\\pF(FAF(F(*&F_pF(FgnF(F(#\"%'z$\"&vL#F(#\"\" )\"#DQ)pprint266\"" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%!G" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#<(&%\"aG6$\"\"%\"\"#&F%6$\"\"&F(&F%6$\"\"'F(&F% 6$F'\"\"$&F%6$F+F1&F%6$F.F1" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#\"\"'" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 164 "e8 := solve(\{op(eqns2)\},\{a[4,2],a[5,2],a[6,2],a[4,3],a[5,3 ],a[6,3]\}):\ne9 := `union`(e7,e8):\nseq(a[i,2]=subs(e9,a[i,2]),i=4..6 );\nseq(a[i,3]=subs(e9,a[i,3]),i=4..6);\n" }}{PARA 11 "" 1 "" {XPPMATH 20 "6%/&%\"aG6$\"\"%\"\"##!$q#\"$V$/&F%6$\"\"&F(#!#a\"#b/&F%6 $\"\"'F(#\"#sF2" }}{PARA 11 "" 1 "" {XPPMATH 20 "6%/&%\"aG6$\"\"%\"\"$ #\"%`5\"%s8/&F%6$\"\"&F(#\"&\\4&\"&+:(/&F%6$\"\"'F(#\"%3G\"&vL#" }}} {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 0 426 "e9 := \{c[2 ] = 1/10, b[2] = 0, c[6] = 4/5, c[5] = 3/5, c[3] = 2/9, c[7] = 1, c[4] = 3/7, b[7] = -37/1120, b[1] = 61/864, a[6,4] = -24206/37125, a[5,4] \+ = 4998/17875, a[5,2] = -54/55, a[4,3] = 1053/1372, a[6,3] = 2808/23375 , a[5,3] = 50949/71500, b[6] = 1375/5408, b[5] = 1375/7344, b[4] = 168 07/146016, a[6,2] = 72/55, b[3] = 98415/321776, c[8] = 1, a[6,5] = 338 /459, a[4,2] = -270/343, a[3,2] = 20/81, b[8] = 1/10, a[8,7] = 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 0 "" }}{PARA 0 "" 0 "" {TEXT -1 39 "The four column simplifying con ditions " }}{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$F+%\"jGF,/F+;,&F0F,F,F,\"\")*&&F)6#F0F,,&F,F,&%\"cG6#F0!\" \"F," }{TEXT -1 5 ", " }{XPPEDIT 18 0 "j = 2;" "6#/%\"jG\"\"#" } {TEXT -1 10 ", 4, 5, 6 " }}{PARA 0 "" 0 "" {TEXT -1 19 "give the equat ions " }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "b[3]*a[3,2]+ b[4]*a[4,2]+b[5]*a[5,2]+b[6]*a[6,2]+b[7]*a[7,2]+b[8]*a[8,2] = b[2]*(1- c[2]);" "6#/,.*&&%\"bG6#\"\"$\"\"\"&%\"aG6$F)\"\"#F*F**&&F'6#\"\"%F*&F ,6$F2F.F*F**&&F'6#\"\"&F*&F,6$F8F.F*F**&&F'6#\"\"'F*&F,6$F>F.F*F**&&F' 6#\"\"(F*&F,6$FDF.F*F**&&F'6#\"\")F*&F,6$FJF.F*F**&&F'6#F.F*,&F*F*&%\" cG6#F.!\"\"F*" }{TEXT -1 2 ", " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }} {PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "b[5]*a[5,4]+b[6]*a[6, 4]+b[7]*a[7,4]+b[8]*a[8,4] = b[4]*(1-c[4]);" "6#/,**&&%\"bG6#\"\"&\"\" \"&%\"aG6$F)\"\"%F*F**&&F'6#\"\"'F*&F,6$F2F.F*F**&&F'6#\"\"(F*&F,6$F8F .F*F**&&F'6#\"\")F*&F,6$F>F.F*F**&&F'6#F.F*,&F*F*&%\"cG6#F.!\"\"F*" } {TEXT -1 2 ", " }}{PARA 257 "" 0 "" {TEXT -1 0 "" }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "b[6]*a[6,5]+b[7]*a[7,5]+b[8]*a[8,5] = b [5]*(1-c[5]);" "6#/,(*&&%\"bG6#\"\"'\"\"\"&%\"aG6$F)\"\"&F*F**&&F'6#\" \"(F*&F,6$F2F.F*F**&&F'6#\"\")F*&F,6$F8F.F*F**&&F'6#F.F*,&F*F*&%\"cG6# F.!\"\"F*" }{TEXT -1 2 ", " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "b[7]*a[7,6]+b[8]*a[8,6] = b [6]*(1-c[6]);" "6#/,&*&&%\"bG6#\"\"(\"\"\"&%\"aG6$F)\"\"'F*F**&&F'6#\" \")F*&F,6$F2F.F*F**&&F'6#F.F*,&F*F*&%\"cG6#F.!\"\"F*" }{TEXT -1 2 ", \+ " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 28 "from \+ which expressions for " }{XPPEDIT 18 0 "a[8,2];" "6#&%\"aG6$\"\")\"\" #" }{TEXT -1 3 ", " }{XPPEDIT 18 0 "a[8,4];" "6#&%\"aG6$\"\")\"\"%" } {TEXT -1 3 ", " }{XPPEDIT 18 0 "a[8,5];" "6#&%\"aG6$\"\")\"\"&" } {TEXT -1 7 " and " }{XPPEDIT 18 0 "a[8,6];" "6#&%\"aG6$\"\")\"\"'" } {TEXT -1 31 " can be obtained in terms of " }{XPPEDIT 18 0 "a[7,2]; " "6#&%\"aG6$\"\"(\"\"#" }{TEXT -1 3 ", " }{XPPEDIT 18 0 "a[7,4];" "6 #&%\"aG6$\"\"(\"\"%" }{TEXT -1 3 ", " }{XPPEDIT 18 0 "a[7,5];" "6#&% \"aG6$\"\"(\"\"&" }{TEXT -1 7 " and " }{XPPEDIT 18 0 "a[7,6];" "6#&% \"aG6$\"\"(\"\"'" }{TEXT -1 1 "." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 194 "cdns3 := [seq(add(b[i]*a[i, j],i=j+1..8)=b[j]*(1-c[j]),j=[2,4,5,6])]:\neqns3 := subs(e9,cdns3):\nm atrix(ListTools[Enumerate](eqns3));\n``;\nindets(eqns3) minus \{a[7,2] ,a[7,4],a[7,5],a[7,6]\};\nnops(%);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6# K%'matrixG6#7&7$\"\"\"/,(#\"#:\"$7\"F(*&#\"#P\"%?6F(&%\"aG6$\"\"(\"\"# F(!\"\"*&#F(\"#5F(&F36$\"\")F6F(F(\"\"!7$F6/,(#\"#\\\"$K%F7*&#F0F1F(&F 36$F5\"\"%F(F7*&F9F(&F36$F=FIF(F(#\"%,C\"&/l$7$\"\"$/,(#\"%v8\"%WtF(*& #F0F1F(&F36$F5\"\"&F(F7*&F9F(&F36$F=FenF(F(#\"$v#\"%sO7$FI/,&*&#F0F1F( &F36$F5\"\"'F(F7*&F9F(&F36$F=FcoF(F(#Fjn\"%3aQ)pprint286\"" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%!G" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#<&&% \"aG6$\"\")\"\"#&F%6$F'\"\"%&F%6$F'\"\"&&F%6$F'\"\"'" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#\"\"%" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 122 "e10 := solve(\{op(eqns3)\},\{a[8,2 ],a[8,4],a[8,5],a[8,6]\}):\ne11 := `union`(e9,e10):\nseq(a[7,j]=subs(e 11,a[8,j]),j=[2,4,5,6]);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6&/&%\"aG6$ \"\"(\"\"#,&#\"#v\"#c!\"\"*&#\"#P\"$7\"\"\"\"F$F2F2/&F%6$F'\"\"%,&#\"& 0=#\"&o@\"F2*&F/F2F4F2F2/&F%6$F'\"\"&,&#\"%v8\"%C7F-*&F/F2F=F2F2/&F%6$ F'\"\"',&*&F/F2FFF2F2#FB\"%/FF2" }}}{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 0 563 "e11 := \{c[2] = 1/10, b[2] = 0, c[6] = 4/5, \+ c[5] = 3/5, c[3] = 2/9, c[7] = 1, c[4] = 3/7, b[7] = -37/1120, b[1] = \+ 61/864, a[6,4] = -24206/37125, a[5,4] = 4998/17875, a[8,4] = 21805/121 68+37/112*a[7,4], a[8,5] = -1375/1224+37/112*a[7,5], a[8,6] = 37/112*a [7,6]+1375/2704, a[5,2] = -54/55, a[8,2] = -75/56+37/112*a[7,2], a[4,3 ] = 1053/1372, a[6,3] = 2808/23375, a[5,3] = 50949/71500, b[6] = 1375/ 5408, b[5] = 1375/7344, b[4] = 16807/146016, a[6,2] = 72/55, b[3] = 98 415/321776, c[8] = 1, a[6,5] = 338/459, a[4,2] = -270/343, a[3,2] = 20 /81, b[8] = 1/10, a[8,7] = 0\}:" }{TEXT -1 0 "" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 15 "" 0 "" {TEXT -1 1 " " }{TEXT 286 6 "Step 6" }{TEXT -1 1 " " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 " " 0 "" {TEXT -1 27 "The stage-order condition: " }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "Sum(a[7,j]*c[j],j = 2 .. 6) = 1/2;" "6# /-%$SumG6$*&&%\"aG6$\"\"(%\"jG\"\"\"&%\"cG6#F,F-/F,;\"\"#\"\"'*&F-F-F3 !\"\"" }{TEXT -1 1 " " }{XPPEDIT 18 0 "c[7]^2;" "6#*$&%\"cG6#\"\"(\"\" #" }{TEXT -1 8 " and " }{XPPEDIT 18 0 "Sum(a[8,j]*c[j],j = 2 .. 7) \+ = 1/2;" "6#/-%$SumG6$*&&%\"aG6$\"\")%\"jG\"\"\"&%\"cG6#F,F-/F,;\"\"#\" \"(*&F-F-F3!\"\"" }{TEXT -1 1 " " }{XPPEDIT 18 0 "c[8]^2;" "6#*$&%\"cG 6#\"\")\"\"#" }{TEXT -1 1 " " }}{PARA 0 "" 0 "" {TEXT -1 11 "determine " }{XPPEDIT 18 0 "a[7,3]" "6#&%\"aG6$\"\"(\"\"$" }{TEXT -1 7 " and \+ " }{XPPEDIT 18 0 "a[8,3]" "6#&%\"aG6$\"\")\"\"$" }{TEXT -1 14 " in te rms of " }{XPPEDIT 18 0 "a[8,2]" "6#&%\"aG6$\"\")\"\"#" }{TEXT -1 3 " , " }{XPPEDIT 18 0 "a[8,4]" "6#&%\"aG6$\"\")\"\"%" }{TEXT -1 3 ", " }{XPPEDIT 18 0 "a[8,5]" "6#&%\"aG6$\"\")\"\"&" }{TEXT -1 7 " and " } {XPPEDIT 18 0 "a[8,6]" "6#&%\"aG6$\"\")\"\"'" }{TEXT -1 1 "." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 208 "c dns4 := [add(a[7,j]*c[j],j=2..6)=1/2*c[7]^2,add(a[8,j]*c[j],j=2..7)=1/ 2*c[8]^2]:\neqns4 := subs(e11,cdns4):\nmatrix(ListTools[Enumerate](eqn s4));\n``;\nindets(eqns4) minus \{a[7,2],a[7,4],a[7,5],a[7,6]\};\nnops (%);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#K%'matrixG6#7$7$\"\"\"/,,*&#F( \"#5F(&%\"aG6$\"\"(\"\"#F(F(*&#F2\"\"*F(&F/6$F1\"\"$F(F(*&#F8F1F(&F/6$ F1\"\"%F(F(*&#F8\"\"&F(&F/6$F1F@F(F(*&#F=F@F(&F/6$F1\"\"'F(F(#F(F27$F2 /,.#\"'X!=\"\"'w " 0 "" {MPLTEXT 1 0 117 "e12 := solve(\{op(eqns4)\},\{a[7,3],a[8,3]\}):\ne13 \+ := `union`(e11,e12):\na[7,3]=subs(e13,a[7,3]);\na[8,3]=subs(e13,a[8,3] );" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/&%\"aG6$\"\"(\"\"$,,*&#\"\"*\"# ?\"\"\"&F%6$F'\"\"#F.!\"\"#F,\"\"%F.*&#\"#F\"#9F.&F%6$F'F4F.F2*&#F7\"# 5F.&F%6$F'\"\"&F.F2*&#\"#=F@F.&F%6$F'\"\"'F.F2" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/&%\"aG6$\"\")\"\"$,,#\"'(e&Q\"'_Nk\"\"\"*&#\"$L$\"%SAF -&F%6$\"\"(\"\"#F-!\"\"*&#\"$***\"%o:F-&F%6$F4\"\"%F-F6*&#F9\"%?6F-&F% 6$F4\"\"&F-F6*&#F0\"$!GF-&F%6$F4\"\"'F-F6" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 3 "e13" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 715 "e13 := \{a[7,3] = -9/20*a[7 ,2]+9/4-27/14*a[7,4]-27/10*a[7,5]-18/5*a[7,6], c[2] = 1/10, b[2] = 0, \+ c[6] = 4/5, c[5] = 3/5, c[3] = 2/9, c[7] = 1, c[4] = 3/7, b[7] = -37/1 120, b[1] = 61/864, a[6,4] = -24206/37125, a[5,4] = 4998/17875, a[8,3] = 385587/643552-333/2240*a[7,2]-999/1568*a[7,4]-999/1120*a[7,5]-333/2 80*a[7,6], a[8,4] = 21805/12168+37/112*a[7,4], a[8,5] = -1375/1224+37/ 112*a[7,5], a[8,6] = 37/112*a[7,6]+1375/2704, a[5,2] = -54/55, a[8,2] \+ = -75/56+37/112*a[7,2], a[4,3] = 1053/1372, a[6,3] = 2808/23375, a[5,3 ] = 50949/71500, b[6] = 1375/5408, b[5] = 1375/7344, b[4] = 16807/1460 16, a[6,2] = 72/55, b[3] = 98415/321776, c[8] = 1, a[6,5] = 338/459, a [4,2] = -270/343, a[3,2] = 20/81, b[8] = 1/10, a[8,7] = 0\}:" }}} {PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 15 "" 0 "" {TEXT -1 1 " " }{TEXT 287 6 "Step 7" }{TEXT -1 1 " " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 24 "The row-sum conditions: " }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "Sum(a[i,j],j=1..i-1)=c[i]" "6#/-%$SumG6$&%\"aG6$%\"iG%\"jG/F+;\"\"\",&F*F.F.!\"\"&%\"cG6#F*" } {TEXT -1 8 ", for " }{XPPEDIT 18 0 "i=2" "6#/%\"iG\"\"#" }{TEXT -1 8 " . . 8, " }}{PARA 0 "" 0 "" {TEXT -1 42 "can be used to find numeri cal values for " }{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 3 ", " }{XPPEDIT 18 0 "a[5,1]" "6#&%\"aG6$\"\"&\"\"\"" } {TEXT -1 7 " and " }{XPPEDIT 18 0 "a[6,1]" "6#&%\"aG6$\"\"'\"\"\"" } {TEXT -1 28 " give an expressions for " }{XPPEDIT 18 0 "a[7,1];" "6 #&%\"aG6$\"\"(\"\"\"" }{TEXT -1 7 " and " }{XPPEDIT 18 0 "a[8,1]" "6 #&%\"aG6$\"\")\"\"\"" }{TEXT -1 15 " in terms of " }{XPPEDIT 18 0 "a [8,2]" "6#&%\"aG6$\"\")\"\"#" }{TEXT -1 3 ", " }{XPPEDIT 18 0 "a[8,4] " "6#&%\"aG6$\"\")\"\"%" }{TEXT -1 3 ", " }{XPPEDIT 18 0 "a[8,5]" "6# &%\"aG6$\"\")\"\"&" }{TEXT -1 7 " and " }{XPPEDIT 18 0 "a[8,6]" "6#& %\"aG6$\"\")\"\"'" }{TEXT -1 1 "." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 176 "cdns5 := [seq(add(a[i,j],j= 1..i-1)=c[i],i=2..8)]:\neqns5 := subs(e13,cdns5):\nmatrix(ListTools[En umerate](eqns5));\n``;\nindets(eqns5) minus \{a[7,2],a[7,4],a[7,5],a[7 ,6]\};\nnops(%);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#K%'matrixG6#7)7$\" \"\"/&%\"aG6$\"\"#F(#F(\"#57$F-/,&&F+6$\"\"$F(F(#\"#?\"#\")F(#F-\"\"*7 $F5/,&&F+6$\"\"%F(F(#\"#F\"%s8!\"\"#F5\"\"(7$F@/,&&F+6$\"\"&F(F(#\"#d \"%+bF(#F5FL7$FL/,&&F+6$\"\"'F(F(#\"&#>c\"&Dr$F(#F@FL7$FV/,.&F+6$FFF(F (*&#\"#6F7F(&F+6$FFF-F(F(#F:F@F(*&#\"#8\"#9F(&F+6$FFF@F(FD*&#\"# " 0 "" {MPLTEXT 1 0 82 "e14 := solve(\{op(eqns5) \},\{seq(a[i,1],i=2..8)\}):\ne15 := `union`(subs(e14,e13),e14):" }}} {PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 86 "seq(a[i,1]=subs(e15,a[i,1]),i=2..6);\na[7,1]=subs(e15,a[7,1]);\na[ 8,1]=subs(e15,a[8,1]);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6'/&%\"aG6$\" \"#\"\"\"#F(\"#5/&F%6$\"\"$F(#!\"#\"#\")/&F%6$\"\"%F(#\"$:'\"%s8/&F%6$ \"\"&F(#\"%VK\"%+b/&F%6$\"\"'F(#!&#\\E\"&Dr$" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/&%\"aG6$\"\"(\"\"\",,*&#\"#<\"#5F(&F%6$F'\"\"&F(F(*&# \"#6\"#?F(&F%6$F'\"\"#F(!\"\"#F0\"\"%F8*&#\"#8\"#9F(&F%6$F'F:F(F(*&#F= F0F(&F%6$F'\"\"'F(F(" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/&%\"aG6$\"\") \"\"\",,*&#\"$H'\"%?6F(&F%6$\"\"(\"\"&F(F(#\"%N6\"%;?F(*&#\"$2%\"%SAF( &F%6$F0\"\"#F(!\"\"*&#\"$\"[\"%o:F(&F%6$F0\"\"%F(F(*&#F?\"$g&F(&F%6$F0 \"\"'F(F(" }}}{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 0 956 "e15 := \{a[6,1] = -26492/37125, a[5,1] = 3243/5500, a[4,1] = 615/ 1372, a[7,3] = -9/20*a[7,2]+9/4-27/14*a[7,4]-27/10*a[7,5]-18/5*a[7,6], a[3,1] = -2/81, c[2] = 1/10, b[2] = 0, c[6] = 4/5, c[5] = 3/5, c[3] = 2/9, c[7] = 1, c[4] = 3/7, b[7] = -37/1120, b[1] = 61/864, a[6,4] = - 24206/37125, a[5,4] = 4998/17875, a[8,3] = 385587/643552-333/2240*a[7, 2]-999/1568*a[7,4]-999/1120*a[7,5]-333/280*a[7,6], a[8,4] = 21805/1216 8+37/112*a[7,4], a[8,5] = -1375/1224+37/112*a[7,5], a[8,6] = 37/112*a[ 7,6]+1375/2704, a[5,2] = -54/55, a[8,2] = -75/56+37/112*a[7,2], a[4,3] = 1053/1372, a[6,3] = 2808/23375, a[5,3] = 50949/71500, b[6] = 1375/5 408, b[5] = 1375/7344, b[4] = 16807/146016, a[6,2] = 72/55, b[3] = 984 15/321776, c[8] = 1, a[6,5] = 338/459, a[4,2] = -270/343, a[3,2] = 20/ 81, a[2,1] = 1/10, b[8] = 1/10, a[8,7] = 0, a[8,1] = 629/1120*a[7,5]+1 135/2016-407/2240*a[7,2]+481/1568*a[7,4]+481/560*a[7,6], a[7,1] = 17/1 0*a[7,5]-11/20*a[7,2]-5/4+13/14*a[7,4]+13/5*a[7,6]\}:" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}} {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 200 "At this \+ point we have an 8 stage order 6 scheme in which the linking coefficie nts in rows 2 to 6 have numerical values while the coefficients in row s 7 and 8 are expressed in terms of the parameters " }{XPPEDIT 18 0 " a[7, 2];" "6#&%\"aG6$\"\"(\"\"#" }{TEXT -1 3 ", " }{XPPEDIT 18 0 "a[7 ,4];" "6#&%\"aG6$\"\"(\"\"%" }{TEXT -1 3 ", " }{XPPEDIT 18 0 "a[7,5]; " "6#&%\"aG6$\"\"(\"\"&" }{TEXT -1 7 " and " }{XPPEDIT 18 0 "a[7,6]; " "6#&%\"aG6$\"\"(\"\"'" }{TEXT -1 1 "." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 456 "subs(e15,matrix([seq([ c[i],seq(a[i,j],j=1..i-1),``$(6-i)],i=2..6)]));\nprint(``);\nfor ii fr om 7 to 8 do\n print(c[ii]=subs(e15,c[ii]));print(``); \n for jj t o ii-1 do\n coeff_val := subs(e15,a[ii,jj]);\n if coeff_val= a[ii,jj] then \n print(a[ii,jj]*` is a parameter.`)\n els e\n print(a[ii,jj]=coeff_val);\n end if;\n end do:\n \+ print(`__________________________________`):\nend do:\nprint(``);\nseq (b[i]=subs(e15,b[i]),i=1..8);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#K%'ma trixG6#7'7(#\"\"\"\"#5F(%!GF+F+F+7(#\"\"#\"\"*#!\"#\"#\")#\"#?F2F+F+F+ 7(#\"\"$\"\"(#\"$:'\"%s8#!$q#\"$V$#\"%`5F;F+F+7(#F7\"\"&#\"%VK\"%+b#!# a\"#b#\"&\\4&\"&+:(#\"%)*\\\"&vy\"F+7(#\"\"%FC#!&#\\E\"&Dr$#\"#sFI#\"% 3G\"&vL##!&1U#FU#\"$Q$\"$f%Q)pprint346\"" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%!G" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/&%\"cG6#\"\"(\"\"\"" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#%!G" }}{PARA 11 "" 1 "" {XPPMATH 20 "6 #/&%\"aG6$\"\"(\"\"\",,*&#\"#<\"#5F(&F%6$F'\"\"&F(F(*&#\"#6\"#?F(&F%6$ F'\"\"#F(!\"\"#F0\"\"%F8*&#\"#8\"#9F(&F%6$F'F:F(F(*&#F=F0F(&F%6$F'\"\" 'F(F(" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#*&&%\"aG6$\"\"(\"\"#\"\"\"%1~ is~a~parameter.GF)" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/&%\"aG6$\"\"(\" \"$,,*&#\"\"*\"#?\"\"\"&F%6$F'\"\"#F.!\"\"#F,\"\"%F.*&#\"#F\"#9F.&F%6$ F'F4F.F2*&#F7\"#5F.&F%6$F'\"\"&F.F2*&#\"#=F@F.&F%6$F'\"\"'F.F2" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#*&&%\"aG6$\"\"(\"\"%\"\"\"%1~is~a~para meter.GF)" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#*&&%\"aG6$\"\"(\"\"&\"\" \"%1~is~a~parameter.GF)" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#*&&%\"aG6$ \"\"(\"\"'\"\"\"%1~is~a~parameter.GF)" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%C__________________________________G" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/&%\"cG6#\"\")\"\"\"" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%!G" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#/&%\"aG6$\"\")\"\"\",,*&#\"$H'\"%?6F(& F%6$\"\"(\"\"&F(F(#\"%N6\"%;?F(*&#\"$2%\"%SAF(&F%6$F0\"\"#F(!\"\"*&#\" $\"[\"%o:F(&F%6$F0\"\"%F(F(*&#F?\"$g&F(&F%6$F0\"\"'F(F(" }}{PARA 11 " " 1 "" {XPPMATH 20 "6#/&%\"aG6$\"\")\"\"#,&#\"#v\"#c!\"\"*&#\"#P\"$7\" \"\"\"&F%6$\"\"(F(F2F2" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/&%\"aG6$\" \")\"\"$,,#\"'(e&Q\"'_Nk\"\"\"*&#\"$L$\"%SAF-&F%6$\"\"(\"\"#F-!\"\"*&# \"$***\"%o:F-&F%6$F4\"\"%F-F6*&#F9\"%?6F-&F%6$F4\"\"&F-F6*&#F0\"$!GF-& F%6$F4\"\"'F-F6" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/&%\"aG6$\"\")\"\"% ,&#\"&0=#\"&o@\"\"\"\"*&#\"#P\"$7\"F-&F%6$\"\"(F(F-F-" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/&%\"aG6$\"\")\"\"&,&#\"%v8\"%C7!\"\"*&#\"#P\"$7\" \"\"\"&F%6$\"\"(F(F2F2" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/&%\"aG6$\" \")\"\"',&*&#\"#P\"$7\"\"\"\"&F%6$\"\"(F(F.F.#\"%v8\"%/FF." }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/&%\"aG6$\"\")\"\"(\"\"!" }}{PARA 11 "" 1 " " {XPPMATH 20 "6#%C__________________________________G" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%!G" }}{PARA 11 "" 1 "" {XPPMATH 20 "6*/&%\"bG6#\" \"\"#\"#h\"$k)/&F%6#\"\"#\"\"!/&F%6#\"\"$#\"&:%)*\"'w " 0 "" {MPLTEXT 1 0 73 "OrderConditions(6,8,'expan ded'):\nsubs(e15,%):\nmap(u_->lhs(u_)-rhs(u_),%);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#7G\"\"!F$F$F$F$F$F$F$F$F$F$F$F$F$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 15 "" 0 "" {TEXT 289 6 "Step 8" }{TEXT -1 1 " " }}{PARA 0 "" 0 "" {TEXT -1 57 "Turning attention to the order 5 embedded scheme we set " } {XPPEDIT 18 0 "`b*`[2]=0" "6#/&%#b*G6#\"\"#\"\"!" }{TEXT -1 3 ", " } {XPPEDIT 18 0 "`b*`[7]=3/50" "6#/&%#b*G6#\"\"(*&\"\"$\"\"\"\"#]!\"\"" }{TEXT -1 3 ", " }{XPPEDIT 18 0 "`b*`[8]=0" "6#/&%#b*G6#\"\")\"\"!" } {TEXT -1 36 " and use the quadrature conditions:" }}{PARA 256 "" 0 " " {TEXT -1 1 " " }{XPPEDIT 18 0 "Sum(`b*`[i],i = 1 .. 8) = 1;" "6#/-%$ SumG6$&%#b*G6#%\"iG/F*;\"\"\"\"\")F-" }{TEXT -1 15 ", " } {XPPEDIT 18 0 "Sum(`b*`[i]*c[i]^(k-1),i = 2 .. 7) = 1/k;" "6#/-%$SumG6 $*&&%#b*G6#%\"iG\"\"\")&%\"cG6#F+,&%\"kGF,F,!\"\"F,/F+;\"\"#\"\"(*&F,F ,F2F3" }{TEXT -1 7 ", " }{XPPEDIT 18 0 "k = 2;" "6#/%\"kG\"\"#" } {TEXT -1 8 " . . 5, " }}{PARA 0 "" 0 "" {TEXT -1 21 "to find the weigh ts " }{XPPEDIT 18 0 "`b*`[1];" "6#&%#b*G6#\"\"\"" }{TEXT -1 2 ", " } {XPPEDIT 18 0 "`b*`[3];" "6#&%#b*G6#\"\"$" }{TEXT -1 3 ", " } {XPPEDIT 18 0 "`b*`[4];" "6#&%#b*G6#\"\"%" }{TEXT -1 3 ", " } {XPPEDIT 18 0 "`b*`[5];" "6#&%#b*G6#\"\"&" }{TEXT -1 7 " and " } {XPPEDIT 18 0 "`b*`[6];" "6#&%#b*G6#\"\"'" }{TEXT -1 1 "." }}{PARA 0 " " 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 284 "`qua d_cdns*` := [add(`b*`[i],i=1..8)=1,seq(add(`b*`[i]*c[i]^(k-1),i=2..8)= 1/k,k=2..5)]:\ne16 := \{`b*`[2]=0,`b*`[7]=3/50,`b*`[8]=0\}:\ne17 := `u nion`(e15,e16):\n`quad_eqns*` := simplify(subs(e17,`quad_cdns*`)):\nma trix(ListTools[Enumerate](`quad_eqns*`));\n``;\nindets(`quad_eqns*`); \nnops(%);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#K%'matrixG6#7'7$\"\"\"/, .&%#b*G6#F(F(#\"\"$\"#]F(&F,6#F/F(&F,6#\"\"%F(&F,6#\"\"&F(&F,6#\"\"'F( F(7$\"\"#/,,F.F(*&#F=\"\"*F(F1F(F(*&#F/\"\"(F(F3F(F(*&#F/F8F(F6F(F(*&# F5F8F(F9F(F(#F(F=7$F//,,F.F(*&#F5\"#\")F(F1F(F(*&#FB\"#\\F(F3F(F(*&#FB \"#DF(F6F(F(*&#\"#;FVF(F9F(F(#F(F/7$F5/,,F.F(*&#\"\")\"$H(F(F1F(F(*&# \"#F\"$V$F(F3F(F(*&#F^o\"$D\"F(F6F(F(*&#\"#kFboF(F9F(F(#F(F57$F8/,,F.F (*&#FY\"%hlF(F1F(F(*&#FP\"%,CF(F3F(F(*&#FP\"$D'F(F6F(F(*&#\"$c#FbpF(F9 F(F(#F(F8Q)pprint356\"" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%!G" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#<'&%#b*G6#\"\"$&F%6#\"\"&&F%6#\"\"'&F% 6#\"\"%&F%6#\"\"\"" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#\"\"&" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 110 "e 18 := solve(\{op(`quad_eqns*`)\},\{`b*`[1],`b*`[3],`b*`[4],`b*`[5],`b* `[6]\}):\ne19 := `union`(subs(e18,e17),e18):" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 50 "The weights of the embedd ed scheme are as follows." }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 38 "seq(`b*`[i]=subs(e19,`b*`[i]),i=1..8);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6*/&%#b*G6#\"\"\"#\"$@)\"&+3\"/&F%6#\"\"#\"\"!/&F%6#\"\"$#\"&$o>\" &D=(/&F%6#\"\"%#\"'t_<\"'+E\"*/&F%6#\"\"&#\"$&R\"%sO/&F%6#\"\"'#\"$&y \"%/F/&F%6#\"\"(#F3\"#]/&F%6#\"\")F/" }}}{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 1107 "e19 := \{a[6,1] = -26492/37125, a[5,1] \+ = 3243/5500, a[4,1] = 615/1372, a[7,3] = -9/20*a[7,2]+9/4-27/14*a[7,4] -27/10*a[7,5]-18/5*a[7,6], a[3,1] = -2/81, `b*`[5] = 395/3672, c[2] = \+ 1/10, b[2] = 0, c[6] = 4/5, c[5] = 3/5, c[3] = 2/9, c[7] = 1, c[4] = 3 /7, `b*`[8] = 0, b[7] = -37/1120, `b*`[4] = 175273/912600, a[6,4] = -2 4206/37125, a[5,4] = 4998/17875, `b*`[2] = 0, `b*`[7] = 3/50, a[8,3] = 385587/643552-333/2240*a[7,2]-999/1568*a[7,4]-999/1120*a[7,5]-333/280 *a[7,6], `b*`[3] = 19683/71825, a[8,4] = 21805/12168+37/112*a[7,4], a[ 8,5] = -1375/1224+37/112*a[7,5], a[8,6] = 37/112*a[7,6]+1375/2704, a[5 ,2] = -54/55, a[8,2] = -75/56+37/112*a[7,2], a[4,3] = 1053/1372, a[6,3 ] = 2808/23375, a[5,3] = 50949/71500, b[6] = 1375/5408, b[5] = 1375/73 44, b[4] = 16807/146016, a[6,2] = 72/55, b[3] = 98415/321776, c[8] = 1 , a[6,5] = 338/459, b[1] = 61/864, a[4,2] = -270/343, a[3,2] = 20/81, \+ a[2,1] = 1/10, `b*`[1] = 821/10800, b[8] = 1/10, `b*`[6] = 785/2704, a [8,7] = 0, a[8,1] = 629/1120*a[7,5]+1135/2016-407/2240*a[7,2]+481/1568 *a[7,4]+481/560*a[7,6], a[7,1] = 17/10*a[7,5]-11/20*a[7,2]-5/4+13/14*a [7,4]+13/5*a[7,6]\}:" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}}{PARA 0 "" 0 "" {TEXT -1 0 "" }} {PARA 15 "" 0 "" {TEXT -1 1 " " }{TEXT 288 6 "Step 9" }}{PARA 0 "" 0 " " {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 39 "The four column simplif ying conditions " }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "S um(`b*`[i]*a[i,j],i = j+1 .. 8) = `b*`[j]*(1-c[j]);" "6#/-%$SumG6$*&&% #b*G6#%\"iG\"\"\"&%\"aG6$F+%\"jGF,/F+;,&F0F,F,F,\"\")*&&F)6#F0F,,&F,F, &%\"cG6#F0!\"\"F," }{TEXT -1 5 ", " }{XPPEDIT 18 0 "j = 3;" "6#/%\" jG\"\"$" }{TEXT -1 10 ", 4, 5, 6 " }}{PARA 0 "" 0 "" {TEXT -1 19 "give the equations " }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "`b *`[4]*a[4,3]+`b*`[5]*a[5,3]+`b*`[6]*a[6,3]+`b*`[7]*a[7,3]+`b*`[8]*a[8, 3] = `b*`[3]*(1-c[3])" "6#/,,*&&%#b*G6#\"\"%\"\"\"&%\"aG6$F)\"\"$F*F** &&F'6#\"\"&F*&F,6$F2F.F*F**&&F'6#\"\"'F*&F,6$F8F.F*F**&&F'6#\"\"(F*&F, 6$F>F.F*F**&&F'6#\"\")F*&F,6$FDF.F*F**&&F'6#F.F*,&F*F*&%\"cG6#F.!\"\"F *" }{TEXT -1 1 " " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 256 "" 0 " " {TEXT -1 1 " " }{XPPEDIT 18 0 "`b*`[5]*a[5,4]+`b*`[6]*a[6,4]+`b*`[7] *a[7,4]+`b*`[8]*a[8,4] = `b*`[4]*(1-c[4])" "6#/,**&&%#b*G6#\"\"&\"\"\" &%\"aG6$F)\"\"%F*F**&&F'6#\"\"'F*&F,6$F2F.F*F**&&F'6#\"\"(F*&F,6$F8F.F *F**&&F'6#\"\")F*&F,6$F>F.F*F**&&F'6#F.F*,&F*F*&%\"cG6#F.!\"\"F*" } {TEXT -1 1 " " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "`b*`[6]*a[6,5]+`b*`[7]*a[7,5]+`b*`[8]*a [8,5] = `b*`[5]*(1-c[5])" "6#/,(*&&%#b*G6#\"\"'\"\"\"&%\"aG6$F)\"\"&F* F**&&F'6#\"\"(F*&F,6$F2F.F*F**&&F'6#\"\")F*&F,6$F8F.F*F**&&F'6#F.F*,&F *F*&%\"cG6#F.!\"\"F*" }{TEXT -1 1 " " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 26 "from which the values of " }{XPPEDIT 18 0 "a[7,2];" "6#&%\"aG6$\"\"(\"\"#" }{TEXT -1 3 ", " }{XPPEDIT 18 0 "a[7,4];" "6#&%\"aG6$\"\"(\"\"%" }{TEXT -1 3 ", " }{XPPEDIT 18 0 "a [7,5];" "6#&%\"aG6$\"\"(\"\"&" }{TEXT -1 7 " and " }{XPPEDIT 18 0 "a [7,6];" "6#&%\"aG6$\"\"(\"\"'" }{TEXT -1 108 " can be obtained. We ca n then obtain numerical values for the linking coefficients in the 7th and 8th rows." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 155 "`eqns*` := \+ subs(e19,[seq(add(`b*`[i]*a[i,j],i=j+1..8)=`b*`[j]*(1-c[j]),j=[3,4,5,6 ])]):\nmatrix(ListTools[Enumerate](`eqns*`));\n``;\nindets(`eqns*`);\n nops(%);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#K%'matrixG6#7&7$\"\"\"/,,# \"&LZ\"\"&+u$F(*&#\"#F\"%+5F(&%\"aG6$\"\"(\"\"#F(!\"\"*&#\"#\")\"$+(F( &F36$F5\"\"%F(F7*&#F:\"$+&F(&F36$F5\"\"&F(F7*&#F0\"$D\"F(&F36$F5\"\"'F (F7#\"&4`\"\"&D=(7$F6/,&#\"%d%*\"&+%fF7*&#\"\"$\"#]F(F/,$*&FUF(FHF(F(#\"$d \"\"%/FQ)pprint366\"" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%!G" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#<&&%\"aG6$\"\"(\"\"#&F%6$F'\"\"&&F%6$F'\"\"% &F%6$F'\"\"'" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#\"\"%" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 93 "e20 := \+ solve(\{op(`eqns*`)\},\{a[7,2],a[7,4],a[7,5],a[7,6]\}):\ne21 := `union `(subs(e20,e19),e20):" }}}{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 0 945 "e21 := \{a[6,1] = -26492/37125, a[5,1] = 3243/5500, \+ a[4,1] = 615/1372, a[3,1] = -2/81, `b*`[5] = 395/3672, c[2] = 1/10, b[ 2] = 0, c[6] = 4/5, c[5] = 3/5, c[3] = 2/9, a[7,1] = 5561/2376, a[8,1] = 465467/266112, a[7,5] = -5225/1836, c[7] = 1, c[4] = 3/7, `b*`[8] = 0, b[7] = -37/1120, `b*`[4] = 175273/912600, a[7,3] = -24117/31603, a [8,3] = -5610201/14158144, a[8,4] = 10513573/3212352, a[6,4] = -24206/ 37125, a[5,4] = 4998/17875, `b*`[2] = 0, `b*`[7] = 3/50, `b*`[3] = 196 83/71825, a[5,2] = -54/55, a[4,3] = 1053/1372, a[6,3] = 2808/23375, a[ 8,5] = -424325/205632, a[7,4] = 899983/200772, a[5,3] = 50949/71500, b [6] = 1375/5408, b[5] = 1375/7344, b[4] = 16807/146016, a[7,2] = -35/1 1, a[6,2] = 72/55, b[3] = 98415/321776, c[8] = 1, a[6,5] = 338/459, b[ 1] = 61/864, a[4,2] = -270/343, a[3,2] = 20/81, a[2,1] = 1/10, a[7,6] \+ = 3925/4056, `b*`[1] = 821/10800, b[8] = 1/10, `b*`[6] = 785/2704, a[8 ,7] = 0, a[8,6] = 376225/454272, a[8,2] = -2945/1232\}:" }}}{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 73 "seq(a[7,i]=subs(e21,a[7,i]),i=1..6);\nseq(a[8,i]=subs(e21,a[8,i] ),i=1..7);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6(/&%\"aG6$\"\"(\"\"\"#\"% hb\"%wB/&F%6$F'\"\"##!#N\"#6/&F%6$F'\"\"$#!& " 0 "" {MPLTEXT 1 0 120 "subs(e21,matrix([seq([c[i],seq(a[i,j],j=1..i-1),` `$(9-i)],i=2..8),\n[`b`,seq(b[i],i=1..8)],[`b*`,seq(`b*`[i],i=1..8)]]) );" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#K%'matrixG6#7+7+#\"\"\"\"#5F(%!G F+F+F+F+F+F+7+#\"\"#\"\"*#!\"#\"#\")#\"#?F2F+F+F+F+F+F+7+#\"\"$\"\"(# \"$:'\"%s8#!$q#\"$V$#\"%`5F;F+F+F+F+F+7+#F7\"\"&#\"%VK\"%+b#!#a\"#b#\" &\\4&\"&+:(#\"%)*\\\"&vy\"F+F+F+F+7+#\"\"%FC#!&#\\E\"&Dr$#\"#sFI#\"%3G \"&vL##!&1U#FU#\"$Q$\"$f%F+F+F+7+F)#\"%hb\"%wB#!#N\"#6#!&\"&D=(#\"'t_<\"'+E\"*#\"$&R\"%sO#\"$&y\"%/F #F7\"#]F`qQ)pprint376\"" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 " " 0 "" {TEXT -1 6 "Check:" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 82 "OrderConditions(6,8,'expanded'):\nsubs(e21,%):\nmap(u_->lhs(u_)-rhs(u _),%);\nnops(%);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#7G\"\"!F$F$F$F$F$F $F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#\"#P" }}}{PARA 0 "" 0 "" {TEXT -1 0 " " }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 95 "subs(b=`b*`,OrderConditi ons(5,8,'expanded')):\nsubs(e21,%):\nmap(u_->lhs(u_)-rhs(u_),%);\nnops (%);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#73\"\"!F$F$F$F$F$F$F$F$F$F$F$F $F$F$F$F$" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#\"#<" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}}{PARA 0 "" 0 "" {TEXT -1 51 "#============================================== ====" }}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 45 "Step by step construction of a general scheme" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 " " {TEXT -1 5 "See: " }}{PARA 0 "" 0 "" {TEXT -1 83 "(1) High order \+ embedded Runge\226Kutta formulae, by P.J. Prince and J. R. Dormand, " }}{PARA 0 "" 0 "" {TEXT -1 88 " Journal of Computational and Ap plied Mathematics, Vol.7, (1981), pages 67 to 75." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 64 "(2) On Runge-Kutta Pro cesses of High Order, by J. C. Butcher," }}{PARA 0 "" 0 "" {TEXT -1 90 " Journal of the Australian Mathematical Society, Vol. 4, ( 1964), pages 179 to 194." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 " " 0 "" {TEXT -1 155 "As suggested by Prince and Dormand in (1) their s cheme can be constructed in a step-by-step manner similar to that used by Butcher for his order 6 schemes." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 92 "--------------------------------------- -----------------------------------------------------" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 166 "We construct expressi ons for the weights and linking coefficients of a general 8 stage orde r 6 scheme with a embedded (7 stage) order 5 scheme in terms of the no des " }{XPPEDIT 18 0 "c[2]" "6#&%\"cG6#\"\"#" }{TEXT -1 3 ", " } {XPPEDIT 18 0 "c[3]" "6#&%\"cG6#\"\"$" }{TEXT -1 3 ", " }{XPPEDIT 18 0 "c[5]" "6#&%\"cG6#\"\"&" }{TEXT -1 3 ", " }{XPPEDIT 18 0 "c[6]" "6# &%\"cG6#\"\"'" }{TEXT -1 19 " and the weights " }{XPPEDIT 18 0 "b[8] " "6#&%\"bG6#\"\")" }{TEXT -1 7 " and " }{XPPEDIT 18 0 "`b*`[7]" "6# &%#b*G6#\"\"(" }{TEXT -1 1 "." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }} {PARA 0 "" 0 "" {TEXT -1 27 "Initially we specify that " }{XPPEDIT 18 0 "b[2]=0" "6#/&%\"bG6#\"\"#\"\"!" }{TEXT -1 7 " and " }{XPPEDIT 18 0 "c[7]=1" "6#/&%\"cG6#\"\"(\"\"\"" }{TEXT -1 3 ", " }{XPPEDIT 18 0 "c[8]=1" "6#/&%\"cG6#\"\")\"\"\"" }{TEXT -1 7 " and " }{XPPEDIT 18 0 "a[8,7]=0" "6#/&%\"aG6$\"\")\"\"(\"\"!" }{TEXT -1 1 "." }}{PARA 0 "" 0 "" {TEXT -1 34 "We also incorporate the relation " }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "c[4]=c[3]/(15*c[3]^2-10*c[3 ]+2)" "6#/&%\"cG6#\"\"%*&&F%6#\"\"$\"\"\",(*&\"#:F,*$&F%6#F+\"\"#F,F,* &\"#5F,&F%6#F+F,!\"\"F3F,F8" }{TEXT -1 1 "." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 15 "" 0 "" {TEXT 290 6 "Step 1" }{TEXT -1 1 " " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 " " 0 "" {TEXT -1 33 "We use the quadrature conditions:" }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "Sum(b[i],i = 1 .. 8) = 1;" "6#/-%$ SumG6$&%\"bG6#%\"iG/F*;\"\"\"\"\")F-" }{TEXT -1 15 ", " } {XPPEDIT 18 0 "Sum(b[i]*c[i]^(k-1),i = 2 .. 8) = 1/k;" "6#/-%$SumG6$*& &%\"bG6#%\"iG\"\"\")&%\"cG6#F+,&%\"kGF,F,!\"\"F,/F+;\"\"#\"\")*&F,F,F2 F3" }{TEXT -1 7 ", " }{XPPEDIT 18 0 "k = 2;" "6#/%\"kG\"\"#" } {TEXT -1 8 " . . 6, " }}{PARA 0 "" 0 "" {TEXT -1 21 "to find the weigh ts " }{XPPEDIT 18 0 "b[1]" "6#&%\"bG6#\"\"\"" }{TEXT -1 2 ", " } {XPPEDIT 18 0 "b[3]" "6#&%\"bG6#\"\"$" }{TEXT -1 3 ", " }{XPPEDIT 18 0 "b[4];" "6#&%\"bG6#\"\"%" }{TEXT -1 3 ", " }{XPPEDIT 18 0 "b[5];" " 6#&%\"bG6#\"\"&" }{TEXT -1 3 ", " }{XPPEDIT 18 0 "b[6]" "6#&%\"bG6#\" \"'" }{TEXT -1 3 ", " }{XPPEDIT 18 0 "b[7]" "6#&%\"bG6#\"\"(" }{TEXT -1 25 " in terms of the nodes " }{XPPEDIT 18 0 "c[3]" "6#&%\"cG6#\" \"$" }{TEXT -1 3 ", " }{XPPEDIT 18 0 "c[5]" "6#&%\"cG6#\"\"&" }{TEXT -1 7 " and " }{XPPEDIT 18 0 "c[6]" "6#&%\"cG6#\"\"'" }{TEXT -1 1 ". " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 289 "e1 := \{b[2]=0,c[4]=c[3]/(15*c[3]^2-10*c[3]+2),c[7]=1,c[8]=1, a[8,7]=0\}:\n``;\nquad_cdns := [add(b[i],i=1..8)=1,seq(add(b[i]*c[i]^( k-1),i=2..8)=1/k,k=2..6)]:\nquad_eqns := subs(e1,quad_cdns):\nmatrix(L istTools[Enumerate](quad_eqns));\n``;\nindets(quad_eqns) minus \{c[3], c[5],c[6],b[8]\};\n;\nnops(%);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%!G " }}{PARA 11 "" 1 "" {XPPMATH 20 "6#K%'matrixG6#7(7$\"\"\"/,0&%\"bG6#F (F(&F,6#\"\"$F(&F,6#\"\"%F(&F,6#\"\"&F(&F,6#\"\"'F(&F,6#\"\"(F(&F,6#\" \")F(F(7$\"\"#/,.*&F.F(&%\"cGF/F(F(*(F1F(FEF(,(*&\"#:F()FEFAF(F(*&\"#5 F(FEF(!\"\"FAF(FNF(*&F4F(&FFF5F(F(*&F7F(&FFF8F(F(F:F(F=F(#F(FA7$F0/,.* &F.F(FKF(F(*(F1F(FEFAFH!\"#F(*&F4F()FPFAF(F(*&F7F()FRFAF(F(F:F(F=F(#F( F07$F3/,.*&F.F()FEF0F(F(*(F1F(FEF0FH!\"$F(*&F4F()FPF0F(F(*&F7F()FRF0F( F(F:F(F=F(#F(F37$F6/,.*&F.F()FEF3F(F(*(F1F(FEF3FH!\"%F(*&F4F()FPF3F(F( *&F7F()FRF3F(F(F:F(F=F(#F(F67$F9/,.*&F.F()FEF6F(F(*(F1F(FEF6FH!\"&F(*& F4F()FPF6F(F(*&F7F()FRF6F(F(F:F(F=F(#F(F9Q)pprint386\"" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%!G" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#<(&%\"bG6# \"\"\"&F%6#\"\"%&F%6#\"\"$&F%6#\"\"(&F%6#\"\"'&F%6#\"\"&" }}{PARA 11 " " 1 "" {XPPMATH 20 "6#\"\"'" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 91 "e2 := factor(solve(\{op(quad _eqns)\},\{b[1],b[3],b[4],b[5],b[6],b[7]\})):\ne3 := `union`(e1,e2):" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 2 "e3" }}{PARA 0 "" 0 " " {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1672 "e3 := \{ c[7] = 1, c[8] = 1, a[8,7] = 0, b[4] = 1/60*(15*c[3]^2-10*c[3]+2)^5*(3 *c[3]-5*c[5]*c[3]-5*c[6]*c[3]+10*c[5]*c[6]*c[3]-5*c[5]*c[6]-2+3*c[6]+3 *c[5])/(15*c[3]^2-10*c[3]+1)/c[3]^2/(3*c[3]-1)/(5*c[3]-2)/(2*c[6]-10*c [6]*c[3]+15*c[6]*c[3]^2-c[3])/(15*c[5]*c[3]^2-10*c[5]*c[3]+2*c[5]-c[3] ), b[2] = 0, c[4] = c[3]/(15*c[3]^2-10*c[3]+2), b[6] = 1/60*(1-5*c[3]+ 5*c[3]^2)*(15*c[5]*c[3]-9*c[3]+4-6*c[5])/c[6]/(-1+c[6])/(2*c[6]-10*c[6 ]*c[3]+15*c[6]*c[3]^2-c[3])/(c[6]-c[3])/(-c[6]+c[5]), b[7] = -1/60*(20 -24*c[5]-24*c[6]-136*c[3]+225*c[3]^3*c[5]-120*b[8]+780*c[5]*c[6]*b[8]* c[3]-300*c[5]*c[3]^3*c[6]+455*c[5]*c[6]*c[3]^2+225*c[3]^3*c[6]-780*c[6 ]*b[8]*c[3]+900*c[5]*b[8]*c[3]^3*c[6]-900*b[8]*c[3]^3*c[6]+1560*b[8]*c [6]*c[3]^2+120*b[8]*c[6]-180*c[3]^3+285*c[3]^2-120*c[5]*b[8]*c[6]+30*c [5]*c[6]+120*b[8]*c[5]+1560*c[5]*b[8]*c[3]^2-780*c[5]*b[8]*c[3]-900*c[ 5]*b[8]*c[3]^3-1560*c[6]*c[5]*b[8]*c[3]^2-1560*b[8]*c[3]^2+780*b[8]*c[ 3]+900*b[8]*c[3]^3-210*c[5]*c[6]*c[3]-350*c[5]*c[3]^2+165*c[5]*c[3]-35 0*c[6]*c[3]^2+165*c[6]*c[3])/(c[3]-1)/(-1+c[6])/(c[5]-1)/(3*c[3]-1)/(5 *c[3]-2), b[5] = -1/60*(1-5*c[3]+5*c[3]^2)*(15*c[6]*c[3]-9*c[3]+4-6*c[ 6])/(-c[6]+c[5])/c[5]/(c[5]-1)/(15*c[5]*c[3]^2-10*c[5]*c[3]+2*c[5]-c[3 ])/(c[5]-c[3]), b[3] = -1/60*(75*c[5]*c[6]*c[3]^2-60*c[5]*c[6]*c[3]+10 *c[5]*c[6]-45*c[5]*c[3]^2+35*c[5]*c[3]-6*c[5]-45*c[6]*c[3]^2+35*c[6]*c [3]-6*c[6]+30*c[3]^2-23*c[3]+4)/(c[6]-c[3])/(c[5]-c[3])/c[3]^2/(c[3]-1 )/(15*c[3]^2-10*c[3]+1), b[1] = -1/60*1/c[3]^2*(150*c[5]*c[3]^3*c[6]-7 5*c[3]^3*c[6]+45*c[3]^3-75*c[3]^3*c[5]+105*c[5]*c[3]^2-205*c[5]*c[6]*c [3]^2+105*c[6]*c[3]^2-65*c[3]^2+29*c[3]-45*c[5]*c[3]+80*c[5]*c[6]*c[3] -45*c[6]*c[3]+6*c[5]-10*c[5]*c[6]-4+6*c[6])/c[5]/c[6]\}:" }}}{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 13 "The weig hts " }{XPPEDIT 18 0 "b[1]" "6#&%\"bG6#\"\"\"" }{TEXT -1 6 " to " } {XPPEDIT 18 0 "b[7]" "6#&%\"bG6#\"\"(" }{TEXT -1 31 " of the scheme a re as follows." }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 60 "for ii to \+ 7 do print(b[ii]=subs(e3,b[ii]));print(``) end do:" }}{PARA 11 "" 1 " " {XPPMATH 20 "6#/&%\"bG6#\"\"\",$*&#F'\"#gF'**&%\"cG6#\"\"$!\"#,B**\" $]\"F'&F.6#\"\"&F')F-F0F'&F.6#\"\"'F'F'*(\"#vF'F8F'F9F'!\"\"*&\"#XF'F8 F'F'*(F=F'F8F'F5F'F>*(\"$0\"F'F5F')F-\"\"#F'F'**\"$0#F'F5F'F9F'FDF'F>* (FCF'F9F'FDF'F'*&\"#lF'FDF'F>*&\"#HF'F-F'F'*(F@F'F5F'F-F'F>**\"#!)F'F5 F'F9F'F-F'F'*(F@F'F9F'F-F'F>*&F;F'F5F'F'*(\"#5F'F5F'F9F'F>\"\"%F>*&F;F 'F9F'F'F'F5F>F9F>F'F>" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%!G" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/&%\"bG6#\"\"#\"\"!" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%!G" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/&%\"bG6#\"\"$,$ *&#\"\"\"\"#gF+*.,:**\"#vF+&%\"cG6#\"\"&F+&F26#\"\"'F+)&F2F&\"\"#F+F+* *F,F+F1F+F5F+F9F+!\"\"*(\"#5F+F1F+F5F+F+*(\"#XF+F1F+F8F+F<*(\"#NF+F1F+ F9F+F+*&F7F+F1F+F<*(F@F+F5F+F8F+F<*(FBF+F5F+F9F+F+*&F7F+F5F+F<*&\"#IF+ F8F+F+*&\"#BF+F9F+F<\"\"%F+F+,&F5F+F9FF+F9F+FF+FAF+ F2F+F+*(F:F+F>F+FAF+F9F6F9*&F5F+FAF+F+*&F5F+F>F+F+F+,(*&F0F+F1F+F+*&F8 F+F2F+F9F+F+F9F2!\"#,&*&F5F+F2F+F+F+F9F9,&*&F:F+F2F+F+F6F9F9,**&F6F+FA F+F+*(F8F+FAF+F2F+F9*(F0F+FAF+F1F+F+F2F9F9,**(F0F+F>F+F1F+F+*(F8F+F>F+ F2F+F9*&F6F+F>F+F+F2F9F9F+F+" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%!G" } }{PARA 11 "" 1 "" {XPPMATH 20 "6#/&%\"bG6#\"\"&,$*&#\"\"\"\"#gF+*0,(F+ F+*&F'F+&%\"cG6#\"\"$F+!\"\"*&F'F+)F0\"\"#F+F+F+,**(\"#:F+&F16#\"\"'F+ F0F+F+*&\"\"*F+F0F+F4\"\"%F+*&F=F+F;F+F4F+,&F;F4&F1F&F+F4FCF4,&FCF+F+F 4F4,**(F:F+FCF+F6F+F+*(\"#5F+FCF+F0F+F4*&F7F+FCF+F+F0F4F4,&FCF+F0F4F4F +F4" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%!G" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/&%\"bG6#\"\"',$*&#\"\"\"\"#gF+*0,(F+F+*&\"\"&F+&%\"cG6 #\"\"$F+!\"\"*&F0F+)F1\"\"#F+F+F+,**(\"#:F+&F26#F0F+F1F+F+*&\"\"*F+F1F +F5\"\"%F+*&F'F+F " 0 "" {MPLTEXT 1 0 399 "cdns1 := \+ [b[6]*(1-c[6])*(a[6,5]*c[5]*(c[5]-c[3])+a[6,4]*c[4]*(c[4]-c[3]))+\n \+ b[5]*(1-c[5])*a[5,4]*c[4]*(c[4]-c[3])=1/60-c[3]/24,\n b[6]*(1-c[6 ])*a[6,5]*c[5]*(c[5]-c[3])*(c[5]-c[4])=1/120-(c[3]+c[4])/60+c[3]*c[4]/ 24,\n b[6]*(1-c[6])*a[6,5]*a[5,4]*c[4]*(c[4]-c[3])=1/360-c[3]/120]: \neqns1 := simplify(subs(e3,cdns1)):\ne4 := factor(solve(\{op(eqns1)\} ,\{a[5,4],a[6,4],a[6,5]\})):\ne5 := `union`(e3,e4):" }}}{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 2406 "e5 := \{a[6,4] = -1/6* c[6]*(15*c[3]^2-10*c[3]+2)^2*(2*c[6]-10*c[6]*c[3]+15*c[6]*c[3]^2-c[3]) *(c[6]-c[3])*(90*c[3]^3*c[6]-120*c[6]*c[3]^2+48*c[6]*c[3]-6*c[6]-225*c [3]^3*c[5]+225*c[5]^2*c[3]^3-240*c[5]^2*c[3]^2+9*c[3]^2+255*c[5]*c[3]^ 2-100*c[5]*c[3]-4*c[3]+90*c[5]^2*c[3]+14*c[5]-12*c[5]^2)/c[3]^2/(15*c[ 3]^2-10*c[3]+1)/(1-5*c[3]+5*c[3]^2)/(15*c[5]*c[3]^2-10*c[5]*c[3]+2*c[5 ]-c[3])/(15*c[5]*c[3]-9*c[3]+4-6*c[5]), c[7] = 1, c[8] = 1, b[4] = 1/6 0*(15*c[3]^2-10*c[3]+2)^5*(3*c[3]-5*c[5]*c[3]-5*c[6]*c[3]+10*c[5]*c[6] *c[3]-5*c[5]*c[6]-2+3*c[6]+3*c[5])/(15*c[3]^2-10*c[3]+1)/c[3]^2/(3*c[3 ]-1)/(5*c[3]-2)/(2*c[6]-10*c[6]*c[3]+15*c[6]*c[3]^2-c[3])/(15*c[5]*c[3 ]^2-10*c[5]*c[3]+2*c[5]-c[3]), b[2] = 0, a[8,7] = 0, a[6,5] = (3*c[3]- 1)*(2*c[6]-10*c[6]*c[3]+15*c[6]*c[3]^2-c[3])*(c[6]-c[3])*(-c[6]+c[5])* c[6]/(15*c[5]*c[3]^2-10*c[5]*c[3]+2*c[5]-c[3])/(c[5]-c[3])/(15*c[5]*c[ 3]-9*c[3]+4-6*c[5])/c[5], c[4] = c[3]/(15*c[3]^2-10*c[3]+2), b[6] = 1/ 60*(1-5*c[3]+5*c[3]^2)*(15*c[5]*c[3]-9*c[3]+4-6*c[5])/c[6]/(-1+c[6])/( 2*c[6]-10*c[6]*c[3]+15*c[6]*c[3]^2-c[3])/(c[6]-c[3])/(-c[6]+c[5]), b[7 ] = -1/60*(20-24*c[5]-24*c[6]-136*c[3]+225*c[3]^3*c[5]-120*b[8]+780*c[ 5]*c[6]*b[8]*c[3]-300*c[5]*c[3]^3*c[6]+455*c[5]*c[6]*c[3]^2+225*c[3]^3 *c[6]-780*c[6]*b[8]*c[3]+900*c[5]*b[8]*c[3]^3*c[6]-900*b[8]*c[3]^3*c[6 ]+1560*b[8]*c[6]*c[3]^2+120*b[8]*c[6]-180*c[3]^3+285*c[3]^2-120*c[5]*b [8]*c[6]+30*c[5]*c[6]+120*b[8]*c[5]+1560*c[5]*b[8]*c[3]^2-780*c[5]*b[8 ]*c[3]-900*c[5]*b[8]*c[3]^3-1560*c[6]*c[5]*b[8]*c[3]^2-1560*b[8]*c[3]^ 2+780*b[8]*c[3]+900*b[8]*c[3]^3-210*c[5]*c[6]*c[3]-350*c[5]*c[3]^2+165 *c[5]*c[3]-350*c[6]*c[3]^2+165*c[6]*c[3])/(c[3]-1)/(-1+c[6])/(c[5]-1)/ (3*c[3]-1)/(5*c[3]-2), b[5] = -1/60*(1-5*c[3]+5*c[3]^2)*(15*c[6]*c[3]- 9*c[3]+4-6*c[6])/(-c[6]+c[5])/c[5]/(c[5]-1)/(15*c[5]*c[3]^2-10*c[5]*c[ 3]+2*c[5]-c[3])/(c[5]-c[3]), b[3] = -1/60*(75*c[5]*c[6]*c[3]^2-60*c[5] *c[6]*c[3]+10*c[5]*c[6]-45*c[5]*c[3]^2+35*c[5]*c[3]-6*c[5]-45*c[6]*c[3 ]^2+35*c[6]*c[3]-6*c[6]+30*c[3]^2-23*c[3]+4)/(c[6]-c[3])/(c[5]-c[3])/c [3]^2/(c[3]-1)/(15*c[3]^2-10*c[3]+1), a[5,4] = -1/6*(15*c[3]^2-10*c[3] +2)^2*c[5]*(c[5]-c[3])*(15*c[5]*c[3]^2-10*c[5]*c[3]+2*c[5]-c[3])/c[3]^ 2/(1-5*c[3]+5*c[3]^2)/(15*c[3]^2-10*c[3]+1), b[1] = -1/60*1/c[3]^2*(15 0*c[5]*c[3]^3*c[6]-75*c[3]^3*c[6]+45*c[3]^3-75*c[3]^3*c[5]+105*c[5]*c[ 3]^2-205*c[5]*c[6]*c[3]^2+105*c[6]*c[3]^2-65*c[3]^2+29*c[3]-45*c[5]*c[ 3]+80*c[5]*c[6]*c[3]-45*c[6]*c[3]+6*c[5]-10*c[5]*c[6]-4+6*c[6])/c[5]/c [6]\}:" }}}{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 20 "The coefficients " }{XPPEDIT 18 0 "a[6,5]" "6#&%\"aG 6$\"\"'\"\"&" }{TEXT -1 3 ", " }{XPPEDIT 18 0 "a[5,4]" "6#&%\"aG6$\" \"&\"\"%" }{TEXT -1 7 " and " }{XPPEDIT 18 0 "a[6,4]" "6#&%\"aG6$\" \"'\"\"%" }{TEXT -1 17 " are as follows." }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 77 "a[5,4]=subs(e5,a[5,4]);``;\na[6,4]=subs(e5,a[6,4]);`` ;\na[6,5]=subs(e5,a[6,5]);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/&%\"aG6 $\"\"&\"\"%,$*&#\"\"\"\"\"'F,*0,(*&\"#:F,)&%\"cG6#\"\"$\"\"#F,F,*&\"#5 F,F3F,!\"\"F7F,F7&F46#F'F,,&F;F,F3F:F,,**(F1F,F;F,F2F,F,*(F9F,F;F,F3F, F:*&F7F,F;F,F,F3F:F,F3!\"#,(F,F,*&F'F,F3F,F:*&F'F,F2F,F,F:,(*&F1F,F2F, F,*&F9F,F3F,F:F,F,F:F,F:" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%!G" }} {PARA 12 "" 1 "" {XPPMATH 20 "6#/&%\"aG6$\"\"'\"\"%,$*&#\"\"\"F'F,*6&% \"cG6#F'F,,(*&\"#:F,)&F/6#\"\"$\"\"#F,F,*&\"#5F,F5F,!\"\"F8F,F8,**&F8F ,F.F,F,*(F:F,F.F,F5F,F;*(F3F,F.F,F4F,F,F5F;F,,&F.F,F5F;F,,>*(\"#!*F,)F 5F7F,F.F,F,*(\"$?\"F,F.F,F4F,F;*(\"#[F,F.F,F5F,F,*&F'F,F.F,F;*(\"$D#F, FDF,&F/6#\"\"&F,F;*(FKF,)FLF8F,FDF,F,*(\"$S#F,FPF,F4F,F;*&\"\"*F,F4F,F ,*(\"$b#F,FLF,F4F,F,*(\"$+\"F,FLF,F5F,F;*&F(F,F5F,F;*(FCF,FPF,F5F,F,*& \"#9F,FLF,F,*&\"#7F,FPF,F;F,F5!\"#,(*&F3F,F4F,F,*&F:F,F5F,F;F,F,F;,(F, F,*&FNF,F5F,F;*&FNF,F4F,F,F;,**(F3F,FLF,F4F,F,*(F:F,FLF,F5F,F;*&F8F,FL F,F,F5F;F;,**(F3F,FLF,F5F,F,*&FTF,F5F,F;F(F,*&F'F,FLF,F;F;F,F;" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#%!G" }}{PARA 11 "" 1 "" {XPPMATH 20 "6 #/&%\"aG6$\"\"'\"\"&*4,&*&\"\"$\"\"\"&%\"cG6#F,F-F-F-!\"\"F-,**&\"\"#F -&F/6#F'F-F-*(\"#5F-F5F-F.F-F1*(\"#:F-F5F-)F.F4F-F-F.F1F-,&F5F-F.F1F-, &F5F1&F/6#F(F-F-F5F-,**(F:F-F>F-F;F-F-*(F8F-F>F-F.F-F1*&F4F-F>F-F-F.F1 F1,&F>F-F.F1F1,**(F:F-F>F-F.F-F-*&\"\"*F-F.F-F1\"\"%F-*&F'F-F>F-F1F1F> F1" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 15 "" 0 "" {TEXT -1 1 " " }{TEXT 297 6 "Step 3" }{TEXT -1 1 " " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 18 "We can determine " }{XPPEDIT 18 0 "a[3,2]" "6#&%\"aG6$\" \"$\"\"#" }{TEXT -1 32 " from the stage-order condition" }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "a[3,2]*c[2] = 1/2" "6#/*&&%\"aG 6$\"\"$\"\"#\"\"\"&%\"cG6#F)F**&F*F*F)!\"\"" }{TEXT -1 1 " " } {XPPEDIT 18 0 "c[3]^2" "6#*$&%\"cG6#\"\"$\"\"#" }{TEXT -1 1 "." }} {PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 106 "a[3,2]*c[2]=1/2*c[3]^2:\nsubs(e5,%);\ne6 := solve(%,\{a[3,2]\}): \ne7 := `union`(e5,e6):\na[3,2]=subs(e7,a[3,2]);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/*&&%\"aG6$\"\"$\"\"#\"\"\"&%\"cG6#F)F*,$*&#F*F)F**$)&F ,6#F(F)F*F*F*" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/&%\"aG6$\"\"$\"\"#,$ *&#\"\"\"F(F,*&&%\"cG6#F'F(&F/6#F(!\"\"F,F," }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 2 "e7" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 2432 "e7 := \{a[6,4] = -1/6*c[6] *(15*c[3]^2-10*c[3]+2)^2*(2*c[6]-10*c[6]*c[3]+15*c[6]*c[3]^2-c[3])*(c[ 6]-c[3])*(90*c[3]^3*c[6]-120*c[6]*c[3]^2+48*c[6]*c[3]-6*c[6]-225*c[3]^ 3*c[5]+225*c[5]^2*c[3]^3-240*c[5]^2*c[3]^2+9*c[3]^2+255*c[5]*c[3]^2-10 0*c[5]*c[3]-4*c[3]+90*c[5]^2*c[3]+14*c[5]-12*c[5]^2)/c[3]^2/(15*c[3]^2 -10*c[3]+1)/(1-5*c[3]+5*c[3]^2)/(15*c[5]*c[3]^2-10*c[5]*c[3]+2*c[5]-c[ 3])/(15*c[5]*c[3]-9*c[3]+4-6*c[5]), c[7] = 1, c[8] = 1, b[4] = 1/60*(1 5*c[3]^2-10*c[3]+2)^5*(3*c[3]-5*c[5]*c[3]-5*c[6]*c[3]+10*c[5]*c[6]*c[3 ]-5*c[5]*c[6]-2+3*c[6]+3*c[5])/(15*c[3]^2-10*c[3]+1)/c[3]^2/(3*c[3]-1) /(5*c[3]-2)/(2*c[6]-10*c[6]*c[3]+15*c[6]*c[3]^2-c[3])/(15*c[5]*c[3]^2- 10*c[5]*c[3]+2*c[5]-c[3]), b[2] = 0, a[8,7] = 0, a[6,5] = (3*c[3]-1)*( 2*c[6]-10*c[6]*c[3]+15*c[6]*c[3]^2-c[3])*(c[6]-c[3])*(-c[6]+c[5])*c[6] /(15*c[5]*c[3]^2-10*c[5]*c[3]+2*c[5]-c[3])/(c[5]-c[3])/(15*c[5]*c[3]-9 *c[3]+4-6*c[5])/c[5], c[4] = c[3]/(15*c[3]^2-10*c[3]+2), b[6] = 1/60*( 1-5*c[3]+5*c[3]^2)*(15*c[5]*c[3]-9*c[3]+4-6*c[5])/c[6]/(-1+c[6])/(2*c[ 6]-10*c[6]*c[3]+15*c[6]*c[3]^2-c[3])/(c[6]-c[3])/(-c[6]+c[5]), b[7] = \+ -1/60*(20-24*c[5]-24*c[6]-136*c[3]+225*c[3]^3*c[5]-120*b[8]+780*c[5]*c [6]*b[8]*c[3]-300*c[5]*c[3]^3*c[6]+455*c[5]*c[6]*c[3]^2+225*c[3]^3*c[6 ]-780*c[6]*b[8]*c[3]+900*c[5]*b[8]*c[3]^3*c[6]-900*b[8]*c[3]^3*c[6]+15 60*b[8]*c[6]*c[3]^2+120*b[8]*c[6]-180*c[3]^3+285*c[3]^2-120*c[5]*b[8]* c[6]+30*c[5]*c[6]+120*b[8]*c[5]+1560*c[5]*b[8]*c[3]^2-780*c[5]*b[8]*c[ 3]-900*c[5]*b[8]*c[3]^3-1560*c[6]*c[5]*b[8]*c[3]^2-1560*b[8]*c[3]^2+78 0*b[8]*c[3]+900*b[8]*c[3]^3-210*c[5]*c[6]*c[3]-350*c[5]*c[3]^2+165*c[5 ]*c[3]-350*c[6]*c[3]^2+165*c[6]*c[3])/(c[3]-1)/(-1+c[6])/(c[5]-1)/(3*c [3]-1)/(5*c[3]-2), b[5] = -1/60*(1-5*c[3]+5*c[3]^2)*(15*c[6]*c[3]-9*c[ 3]+4-6*c[6])/(-c[6]+c[5])/c[5]/(c[5]-1)/(15*c[5]*c[3]^2-10*c[5]*c[3]+2 *c[5]-c[3])/(c[5]-c[3]), b[3] = -1/60*(75*c[5]*c[6]*c[3]^2-60*c[5]*c[6 ]*c[3]+10*c[5]*c[6]-45*c[5]*c[3]^2+35*c[5]*c[3]-6*c[5]-45*c[6]*c[3]^2+ 35*c[6]*c[3]-6*c[6]+30*c[3]^2-23*c[3]+4)/(c[6]-c[3])/(c[5]-c[3])/c[3]^ 2/(c[3]-1)/(15*c[3]^2-10*c[3]+1), a[5,4] = -1/6*(15*c[3]^2-10*c[3]+2)^ 2*c[5]*(c[5]-c[3])*(15*c[5]*c[3]^2-10*c[5]*c[3]+2*c[5]-c[3])/c[3]^2/(1 -5*c[3]+5*c[3]^2)/(15*c[3]^2-10*c[3]+1), b[1] = -1/60*1/c[3]^2*(150*c[ 5]*c[3]^3*c[6]-75*c[3]^3*c[6]+45*c[3]^3-75*c[3]^3*c[5]+105*c[5]*c[3]^2 -205*c[5]*c[6]*c[3]^2+105*c[6]*c[3]^2-65*c[3]^2+29*c[3]-45*c[5]*c[3]+8 0*c[5]*c[6]*c[3]-45*c[6]*c[3]+6*c[5]-10*c[5]*c[6]-4+6*c[6])/c[5]/c[6], a[3,2] = 1/2*c[3]^2/c[2]\}:" }}}{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 298 6 "Step 4" } {TEXT -1 1 " " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 41 "We can now determine the 6 coefficients " }{XPPEDIT 18 0 "a[4,2]" "6#&%\"aG6$\"\"%\"\"#" }{TEXT -1 3 ", " }{XPPEDIT 18 0 "a[ 5, 2];" "6#&%\"aG6$\"\"&\"\"#" }{TEXT -1 3 ", " }{XPPEDIT 18 0 "a[6, \+ 2];" "6#&%\"aG6$\"\"'\"\"#" }{TEXT -1 3 ", " }{XPPEDIT 18 0 "a[4,3]; " "6#&%\"aG6$\"\"%\"\"$" }{TEXT -1 3 ", " }{XPPEDIT 18 0 "a[5,3];" "6 #&%\"aG6$\"\"&\"\"$" }{TEXT -1 3 ", " }{XPPEDIT 18 0 "a[6,3]" "6#&%\" aG6$\"\"'\"\"$" }{TEXT -1 56 " by using the three alternative simplif ying conditions:" }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "S um(b[i]*(1-c[i])*a[i,2],i=3..6)=0" "6#/-%$SumG6$*(&%\"bG6#%\"iG\"\"\", &F,F,&%\"cG6#F+!\"\"F,&%\"aG6$F+\"\"#F,/F+;\"\"$\"\"'\"\"!" }{TEXT -1 2 ", " }}{PARA 0 "" 0 "" {TEXT -1 8 "that is," }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "b[3]*(1-c[3])*a[3,2]+b[4]*(1-c[4])*a[4, 2]+b[5]*(1-c[5])*a[5,2]+b[6]*(1-c[6])*a[6,2] = 0" "6#/,**(&%\"bG6#\"\" $\"\"\",&F*F*&%\"cG6#F)!\"\"F*&%\"aG6$F)\"\"#F*F**(&F'6#\"\"%F*,&F*F*& F-6#F7F/F*&F16$F7F3F*F**(&F'6#\"\"&F*,&F*F*&F-6#F@F/F*&F16$F@F3F*F**(& F'6#\"\"'F*,&F*F*&F-6#FIF/F*&F16$FIF3F*F*\"\"!" }{TEXT -1 2 ", " }} {PARA 0 "" 0 "" {TEXT -1 3 "and" }}{PARA 256 "" 0 "" {TEXT -1 1 " " } {XPPEDIT 18 0 "Sum(b[i]*(1-c[i])*Sum(a[i,j]*a[j,2],j = 3 .. i-1),i = 3 .. 6) = 0" "6#/-%$SumG6$*(&%\"bG6#%\"iG\"\"\",&F,F,&%\"cG6#F+!\"\"F,- F%6$*&&%\"aG6$F+%\"jGF,&F66$F8\"\"#F,/F8;\"\"$,&F+F,F,F1F,/F+;F>\"\"' \"\"!" }{TEXT -1 2 ", " }}{PARA 0 "" 0 "" {TEXT -1 8 "that is," }} {PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "b[4]*(1-c[4])*a[4,3]* a[3,2]+b[5]*(1-c[5])*(a[5,3]*a[3,2]+a[5,4]*a[4,2])+b[6]*(1-c[6])*(a[6, 3]*a[3,2]+a[6,4]*a[4,2]+a[6,5]*a[5,2]) = 0" "6#/,(**&%\"bG6#\"\"%\"\" \",&F*F*&%\"cG6#F)!\"\"F*&%\"aG6$F)\"\"$F*&F16$F3\"\"#F*F**(&F'6#\"\"& F*,&F*F*&F-6#F:F/F*,&*&&F16$F:F3F*&F16$F3F6F*F**&&F16$F:F)F*&F16$F)F6F *F*F*F**(&F'6#\"\"'F*,&F*F*&F-6#FLF/F*,(*&&F16$FLF3F*&F16$F3F6F*F**&&F 16$FLF)F*&F16$F)F6F*F**&&F16$FLF:F*&F16$F:F6F*F*F*F*\"\"!" }{TEXT -1 2 ", " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "Sum(b[i]*(c[i]-1)*(c[i]-c[6])*a[i,2],i = 3 .. 5) = 0;" "6#/-%$SumG6$**&%\"bG6#%\"iG\"\"\",&&%\"cG6#F+F,F,!\"\"F,,&&F/6#F +F,&F/6#\"\"'F1F,&%\"aG6$F+\"\"#F,/F+;\"\"$\"\"&\"\"!" }{TEXT -1 2 ", \+ " }}{PARA 0 "" 0 "" {TEXT -1 9 "that is, " }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "b[3]*(c[3]-1)*(c[3]-c[6])*a[3,2]+b[4]*(c[4]-1 )*(c[4]-c[6])*a[4,2]+b[5]*(c[5]-1)*(c[5]-c[6])*a[5,2]=0" "6#/,(**&%\"b G6#\"\"$\"\"\",&&%\"cG6#F)F*F*!\"\"F*,&&F-6#F)F*&F-6#\"\"'F/F*&%\"aG6$ F)\"\"#F*F***&F'6#\"\"%F*,&&F-6#F=F*F*F/F*,&&F-6#F=F*&F-6#F5F/F*&F76$F =F9F*F***&F'6#\"\"&F*,&&F-6#FKF*F*F/F*,&&F-6#FKF*&F-6#F5F/F*&F76$FKF9F *F*\"\"!" }{TEXT -1 2 ", " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 " " 0 "" {TEXT -1 48 "together with the three stage-order conditions: " }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "Sum(a[4,j]*c[j],j = 2 .. 3) = 1/2;" "6#/-%$SumG6$*&&%\"aG6$\"\"%%\"jG\"\"\"&%\"cG6#F,F-/F ,;\"\"#\"\"$*&F-F-F3!\"\"" }{TEXT -1 1 " " }{XPPEDIT 18 0 "c[4]^2;" "6 #*$&%\"cG6#\"\"%\"\"#" }{TEXT -1 8 " , " }{XPPEDIT 18 0 "Sum(a[5, j]*c[j],j=2..4)=1/2" "6#/-%$SumG6$*&&%\"aG6$\"\"&%\"jG\"\"\"&%\"cG6#F, F-/F,;\"\"#\"\"%*&F-F-F3!\"\"" }{TEXT -1 1 " " }{XPPEDIT 18 0 "c[5]^2 " "6#*$&%\"cG6#\"\"&\"\"#" }{TEXT -1 2 " " }}{PARA 0 "" 0 "" {TEXT -1 4 "and " }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "Sum(a[6 ,j]*c[j],j=2..5)=1/2" "6#/-%$SumG6$*&&%\"aG6$\"\"'%\"jG\"\"\"&%\"cG6#F ,F-/F,;\"\"#\"\"&*&F-F-F3!\"\"" }{TEXT -1 1 " " }{XPPEDIT 18 0 "c[6]^2 " "6#*$&%\"cG6#\"\"'\"\"#" }{TEXT -1 1 "." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 336 "cdns2 := [add(b[i]*( 1-c[i])*a[i,2],i=3..6)=0,add(b[i]*(1-c[i])*add(a[i,j]*a[j,2],j=3..i-1) ,i=3..6)=0,\n add(b[i]*(1-c[i])*(c[6]-c[i])*a[i,2],i=3..5)=0,seq(add( a[i,j]*c[j],j=2..i-1)=1/2*c[i]^2,i=4..6)]:\neqns2 := simplify(subs(e7, cdns2)):\ne8 := factor(solve(\{op(eqns2)\},\{a[4,2],a[5,2],a[6,2],a[4, 3],a[5,3],a[6,3]\})):\ne9 := `union`(e7,e8):" }}}{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 3581 "e9 := \{a[6,4] = -1/6*c[6] *(15*c[3]^2-10*c[3]+2)^2*(2*c[6]-10*c[6]*c[3]+15*c[6]*c[3]^2-c[3])*(c[ 6]-c[3])*(90*c[3]^3*c[6]-120*c[6]*c[3]^2+48*c[6]*c[3]-6*c[6]-225*c[3]^ 3*c[5]+225*c[5]^2*c[3]^3-240*c[5]^2*c[3]^2+9*c[3]^2+255*c[5]*c[3]^2-10 0*c[5]*c[3]-4*c[3]+90*c[5]^2*c[3]+14*c[5]-12*c[5]^2)/c[3]^2/(15*c[3]^2 -10*c[3]+1)/(1-5*c[3]+5*c[3]^2)/(15*c[5]*c[3]^2-10*c[5]*c[3]+2*c[5]-c[ 3])/(15*c[5]*c[3]-9*c[3]+4-6*c[5]), c[7] = 1, c[8] = 1, a[8,7] = 0, b[ 4] = 1/60*(15*c[3]^2-10*c[3]+2)^5*(3*c[3]-5*c[5]*c[3]-5*c[6]*c[3]+10*c [5]*c[6]*c[3]-5*c[5]*c[6]-2+3*c[6]+3*c[5])/(15*c[3]^2-10*c[3]+1)/c[3]^ 2/(3*c[3]-1)/(5*c[3]-2)/(2*c[6]-10*c[6]*c[3]+15*c[6]*c[3]^2-c[3])/(15* c[5]*c[3]^2-10*c[5]*c[3]+2*c[5]-c[3]), a[5,2] = 1/2*c[3]*c[5]/c[2]*(-5 *c[5]+1+5*c[5]^2)/(1-5*c[3]+5*c[3]^2), b[2] = 0, a[6,5] = (3*c[3]-1)*( 2*c[6]-10*c[6]*c[3]+15*c[6]*c[3]^2-c[3])*(c[6]-c[3])*(-c[6]+c[5])*c[6] /(15*c[5]*c[3]^2-10*c[5]*c[3]+2*c[5]-c[3])/(c[5]-c[3])/(15*c[5]*c[3]-9 *c[3]+4-6*c[5])/c[5], c[4] = c[3]/(15*c[3]^2-10*c[3]+2), b[6] = 1/60*( 1-5*c[3]+5*c[3]^2)*(15*c[5]*c[3]-9*c[3]+4-6*c[5])/c[6]/(-1+c[6])/(2*c[ 6]-10*c[6]*c[3]+15*c[6]*c[3]^2-c[3])/(c[6]-c[3])/(-c[6]+c[5]), b[7] = \+ -1/60*(20-24*c[5]-24*c[6]-136*c[3]+225*c[3]^3*c[5]-120*b[8]+780*c[5]*c [6]*b[8]*c[3]-300*c[5]*c[3]^3*c[6]+455*c[5]*c[6]*c[3]^2+225*c[3]^3*c[6 ]-780*c[6]*b[8]*c[3]+900*c[5]*b[8]*c[3]^3*c[6]-900*b[8]*c[3]^3*c[6]+15 60*b[8]*c[6]*c[3]^2+120*b[8]*c[6]-180*c[3]^3+285*c[3]^2-120*c[5]*b[8]* c[6]+30*c[5]*c[6]+120*b[8]*c[5]+1560*c[5]*b[8]*c[3]^2-780*c[5]*b[8]*c[ 3]-900*c[5]*b[8]*c[3]^3-1560*c[6]*c[5]*b[8]*c[3]^2-1560*b[8]*c[3]^2+78 0*b[8]*c[3]+900*b[8]*c[3]^3-210*c[5]*c[6]*c[3]-350*c[5]*c[3]^2+165*c[5 ]*c[3]-350*c[6]*c[3]^2+165*c[6]*c[3])/(c[3]-1)/(-1+c[6])/(c[5]-1)/(3*c [3]-1)/(5*c[3]-2), a[4,2] = 1/2*c[3]^2*(4-30*c[3]+45*c[3]^2)/c[2]/(15* c[3]^2-10*c[3]+2)^3, a[5,3] = -1/6*c[5]*(150*c[3]^3*c[5]-145*c[5]*c[3] ^2+40*c[5]*c[3]-4*c[5]-30*c[3]^3+20*c[3]^2-c[3])*(c[5]-c[3])/(1-5*c[3] +5*c[3]^2)/c[3]^2/(15*c[3]^2-10*c[3]+1), a[6,2] = 1/2*c[6]*c[3]/c[2]*( 5*c[6]^2-5*c[6]+1)/(1-5*c[3]+5*c[3]^2), a[6,3] = -1/6*(c[6]-c[3])*(-90 0*c[3]^5*c[6]*c[5]-45*c[3]^3*c[5]+24*c[5]^2*c[6]+450*c[3]^5*c[5]-300*c [3]^4*c[5]-975*c[5]*c[3]^4*c[6]+2535*c[5]*c[3]^3*c[6]-1450*c[5]*c[6]*c [3]^2-1815*c[3]^4*c[6]+820*c[3]^3*c[6]-1350*c[3]^5*c[6]^2-270*c[3]^5+3 00*c[3]^4+1350*c[3]^5*c[6]-89*c[3]^3+4*c[3]^2+12*c[6]^2+2250*c[5]^2*c[ 3]^4*c[6]-3075*c[5]^2*c[3]^3*c[6]-300*c[5]^2*c[6]*c[3]+1470*c[5]^2*c[3 ]^2*c[6]-28*c[5]*c[6]+328*c[5]*c[6]*c[3]+35*c[5]*c[3]^2+2*c[5]*c[3]-14 8*c[6]*c[3]^2+10*c[6]*c[3]-450*c[5]^2*c[3]^4+480*c[5]^2*c[3]^3-135*c[5 ]^2*c[3]^2+6*c[5]^2*c[3]+2700*c[6]^2*c[3]^4-2100*c[6]^2*c[3]^3+810*c[6 ]^2*c[3]^2-156*c[6]^2*c[3])*c[6]/(1-5*c[3]+5*c[3]^2)/(15*c[3]^2-10*c[3 ]+1)/(15*c[5]*c[3]-9*c[3]+4-6*c[5])/(c[5]-c[3])/c[3]^2, a[4,3] = -(15* c[3]^2-10*c[3]+1)*c[3]/(15*c[3]^2-10*c[3]+2)^3, b[5] = -1/60*(1-5*c[3] +5*c[3]^2)*(15*c[6]*c[3]-9*c[3]+4-6*c[6])/(-c[6]+c[5])/c[5]/(c[5]-1)/( 15*c[5]*c[3]^2-10*c[5]*c[3]+2*c[5]-c[3])/(c[5]-c[3]), b[3] = -1/60*(75 *c[5]*c[6]*c[3]^2-60*c[5]*c[6]*c[3]+10*c[5]*c[6]-45*c[5]*c[3]^2+35*c[5 ]*c[3]-6*c[5]-45*c[6]*c[3]^2+35*c[6]*c[3]-6*c[6]+30*c[3]^2-23*c[3]+4)/ (c[6]-c[3])/(c[5]-c[3])/c[3]^2/(c[3]-1)/(15*c[3]^2-10*c[3]+1), a[5,4] \+ = -1/6*(15*c[3]^2-10*c[3]+2)^2*c[5]*(c[5]-c[3])*(15*c[5]*c[3]^2-10*c[5 ]*c[3]+2*c[5]-c[3])/c[3]^2/(1-5*c[3]+5*c[3]^2)/(15*c[3]^2-10*c[3]+1), \+ b[1] = -1/60*1/c[3]^2*(150*c[5]*c[3]^3*c[6]-75*c[3]^3*c[6]+45*c[3]^3-7 5*c[3]^3*c[5]+105*c[5]*c[3]^2-205*c[5]*c[6]*c[3]^2+105*c[6]*c[3]^2-65* c[3]^2+29*c[3]-45*c[5]*c[3]+80*c[5]*c[6]*c[3]-45*c[6]*c[3]+6*c[5]-10*c [5]*c[6]-4+6*c[6])/c[5]/c[6], a[3,2] = 1/2*c[3]^2/c[2]\}:" }}}{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 0 143 "a[4,2]=subs(e9,a[4,2]);\na[4,3]=subs(e9,a[4,3]);\na[5,2]=subs(e 9,a[5,2]);\na[5,3]=subs(e9,a[5,3]);\na[6,2]=subs(e9,a[6,2]);\na[6,3]=s ubs(e9,a[6,3]);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/&%\"aG6$\"\"%\"\"# ,$*&#\"\"\"F(F,**&%\"cG6#\"\"$F(,(F'F,*&\"#IF,F.F,!\"\"*&\"#XF,)F.F(F, F,F,&F/6#F(F5,(*&\"#:F,F8F,F,*&\"#5F,F.F,F5F(F,!\"$F,F," }}{PARA 11 " " 1 "" {XPPMATH 20 "6#/&%\"aG6$\"\"%\"\"$,$*(,(*&\"#:\"\"\")&%\"cG6#F( \"\"#F.F.*&\"#5F.F0F.!\"\"F.F.F.F0F.,(*&F-F.F/F.F.*&F5F.F0F.F6F3F.!\"$ F6" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/&%\"aG6$\"\"&\"\"#,$*&#\"\"\"F( F,*,&%\"cG6#\"\"$F,&F/6#F'F,&F/6#F(!\"\",(*&F'F,F2F,F6F,F,*&F'F,)F2F(F ,F,F,,(F,F,*&F'F,F.F,F6*&F'F,)F.F(F,F,F6F,F," }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/&%\"aG6$\"\"&\"\"$,$*&#\"\"\"\"\"'F,*.&%\"cG6#F'F,,0*( \"$]\"F,)&F06#F(F(F,F/F,F,*(\"$X\"F,F/F,)F6\"\"#F,!\"\"*(\"#SF,F/F,F6F ,F,*&\"\"%F,F/F,F<*&\"#IF,F5F,F<*&\"#?F,F:F,F,F6F " 0 "" {MPLTEXT 1 0 1 ";" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 15 "" 0 "" {TEXT -1 1 " " }{TEXT 292 6 "Step 5" }{TEXT -1 1 " " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 " " 0 "" {TEXT -1 39 "The four column simplifying conditions " }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "Sum(b[i]*a[i,j],i = j+1 .. \+ 8) = b[j]*(1-c[j]);" "6#/-%$SumG6$*&&%\"bG6#%\"iG\"\"\"&%\"aG6$F+%\"jG F,/F+;,&F0F,F,F,\"\")*&&F)6#F0F,,&F,F,&%\"cG6#F0!\"\"F," }{TEXT -1 5 " , " }{XPPEDIT 18 0 "j = 2;" "6#/%\"jG\"\"#" }{TEXT -1 10 ", 4, 5, 6 " }}{PARA 0 "" 0 "" {TEXT -1 19 "give the equations " }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "b[3]*a[3,2]+b[4]*a[4,2]+b[5]*a[5,2 ]+b[6]*a[6,2]+b[7]*a[7,2]+b[8]*a[8,2] = b[2]*(1-c[2]);" "6#/,.*&&%\"bG 6#\"\"$\"\"\"&%\"aG6$F)\"\"#F*F**&&F'6#\"\"%F*&F,6$F2F.F*F**&&F'6#\"\" &F*&F,6$F8F.F*F**&&F'6#\"\"'F*&F,6$F>F.F*F**&&F'6#\"\"(F*&F,6$FDF.F*F* *&&F'6#\"\")F*&F,6$FJF.F*F**&&F'6#F.F*,&F*F*&%\"cG6#F.!\"\"F*" }{TEXT -1 2 ", " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "b[5]*a[5,4]+b[6]*a[6,4]+b[7]*a[7,4]+b[8]*a[8, 4] = b[4]*(1-c[4]);" "6#/,**&&%\"bG6#\"\"&\"\"\"&%\"aG6$F)\"\"%F*F**&& F'6#\"\"'F*&F,6$F2F.F*F**&&F'6#\"\"(F*&F,6$F8F.F*F**&&F'6#\"\")F*&F,6$ F>F.F*F**&&F'6#F.F*,&F*F*&%\"cG6#F.!\"\"F*" }{TEXT -1 3 ", " }}{PARA 256 "" 0 "" {TEXT -1 0 "" }}{PARA 256 "" 0 "" {TEXT -1 1 " " } {XPPEDIT 18 0 "b[6]*a[6,5]+b[7]*a[7,5]+b[8]*a[8,5] = b[5]*(1-c[5]);" " 6#/,(*&&%\"bG6#\"\"'\"\"\"&%\"aG6$F)\"\"&F*F**&&F'6#\"\"(F*&F,6$F2F.F* F**&&F'6#\"\")F*&F,6$F8F.F*F**&&F'6#F.F*,&F*F*&%\"cG6#F.!\"\"F*" } {TEXT -1 2 ", " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "b[7]*a[7,6]+b[8]*a[8,6] = b[6]*(1-c[6]) ;" "6#/,&*&&%\"bG6#\"\"(\"\"\"&%\"aG6$F)\"\"'F*F**&&F'6#\"\")F*&F,6$F2 F.F*F**&&F'6#F.F*,&F*F*&%\"cG6#F.!\"\"F*" }{TEXT -1 2 ", " }}{PARA 0 " " 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 28 "from which express ions for " }{XPPEDIT 18 0 "a[8,2]" "6#&%\"aG6$\"\")\"\"#" }{TEXT -1 3 ", " }{XPPEDIT 18 0 "a[8,4];" "6#&%\"aG6$\"\")\"\"%" }{TEXT -1 3 ", " }{XPPEDIT 18 0 "a[8,5];" "6#&%\"aG6$\"\")\"\"&" }{TEXT -1 7 " and " }{XPPEDIT 18 0 "a[8,6];" "6#&%\"aG6$\"\")\"\"'" }{TEXT -1 31 " ca n be obtained in terms of " }{XPPEDIT 18 0 "a[7,2]" "6#&%\"aG6$\"\"( \"\"#" }{TEXT -1 3 ", " }{XPPEDIT 18 0 "a[7,4];" "6#&%\"aG6$\"\"(\"\" %" }{TEXT -1 3 ", " }{XPPEDIT 18 0 "a[7,5];" "6#&%\"aG6$\"\"(\"\"&" } {TEXT -1 7 " and " }{XPPEDIT 18 0 "a[7,6];" "6#&%\"aG6$\"\"(\"\"'" } {TEXT -1 1 "." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 192 "cdns3 := [s eq(add(b[i]*a[i,j],i=j+1..8)=b[j]*(1-c[j]),j=[2,4,5,6])]:\neqns3 := si mplify(subs(e9,cdns3)):\ne10 := factor(solve(\{op(eqns3)\},\{a[8,2],a[ 8,4],a[8,5],a[8,6]\})):\ne11 := `union`(e9,e10):" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 21 "indets(map(r hs,e11));" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#<+&%\"cG6#\"\"#&F%6#\"\"' &F%6#\"\"&&F%6#\"\"$&%\"aG6$\"\"(F'&%\"bG6#\"\")&F26$F4F-&F26$F4\"\"%& F26$F4F*" }}}{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 23726 "e11 := \{a[6,2] = 1/2*c[6]*c[3]/c[2]*(5*c[6]^2-5*c[6]+1)/(1-5*c [3]+5*c[3]^2), a[6,3] = -1/6*(c[6]-c[3])*(450*c[3]^5*c[5]+300*c[3]^4-2 70*c[3]^5+12*c[6]^2+2700*c[6]^2*c[3]^4-2100*c[6]^2*c[3]^3+810*c[6]^2*c [3]^2-450*c[5]^2*c[3]^4+328*c[5]*c[6]*c[3]+2535*c[5]*c[3]^3*c[6]-1450* c[5]*c[6]*c[3]^2-900*c[3]^5*c[6]*c[5]+2250*c[5]^2*c[3]^4*c[6]-3075*c[5 ]^2*c[3]^3*c[6]-300*c[5]^2*c[6]*c[3]+1470*c[5]^2*c[3]^2*c[6]+24*c[5]^2 *c[6]+4*c[3]^2-156*c[6]^2*c[3]+10*c[6]*c[3]-148*c[6]*c[3]^2+820*c[3]^3 *c[6]-45*c[3]^3*c[5]+480*c[5]^2*c[3]^3-135*c[5]^2*c[3]^2+35*c[5]*c[3]^ 2+2*c[5]*c[3]+6*c[5]^2*c[3]-28*c[5]*c[6]-1350*c[3]^5*c[6]^2-89*c[3]^3- 300*c[3]^4*c[5]-1815*c[3]^4*c[6]+1350*c[3]^5*c[6]-975*c[5]*c[3]^4*c[6] )*c[6]/(1-5*c[3]+5*c[3]^2)/(15*c[3]^2-10*c[3]+1)/(15*c[5]*c[3]-9*c[3]+ 4-6*c[5])/(c[5]-c[3])/c[3]^2, b[2] = 0, b[4] = 1/60*(15*c[3]^2-10*c[3] +2)^5*(3*c[3]-5*c[5]*c[3]-5*c[6]*c[3]+10*c[5]*c[6]*c[3]-5*c[5]*c[6]-2+ 3*c[6]+3*c[5])/(15*c[3]^2-10*c[3]+1)/c[3]^2/(3*c[3]-1)/(5*c[3]-2)/(2*c [6]-10*c[6]*c[3]+15*c[6]*c[3]^2-c[3])/(15*c[5]*c[3]^2-10*c[5]*c[3]+2*c [5]-c[3]), c[7] = 1, c[4] = c[3]/(15*c[3]^2-10*c[3]+2), a[6,5] = (3*c[ 3]-1)*(2*c[6]-10*c[6]*c[3]+15*c[6]*c[3]^2-c[3])*(c[6]-c[3])*(-c[6]+c[5 ])*c[6]/(15*c[5]*c[3]^2-10*c[5]*c[3]+2*c[5]-c[3])/(c[5]-c[3])/(15*c[5] *c[3]-9*c[3]+4-6*c[5])/c[5], a[8,5] = 1/60*(8+2835*c[3]^5*c[5]-18*c[6] -10*c[5]-110*c[3]+2710*c[3]^4-2145*c[3]^5-780*a[7,5]*c[5]*b[8]*c[3]^3* c[6]^2+120*a[7,5]*c[5]*b[8]*c[6]^2*c[3]^2+240*a[7,5]*c[5]^5*b[8]*c[6]+ 2760*a[7,5]*c[5]^5*b[8]*c[3]-24*a[7,5]*c[5]*c[6]^2*c[3]^2+165*a[7,5]*c [5]*c[3]^3*c[6]^2-951*c[5]*c[6]^2*c[3]^2+168*c[5]*c[6]^2*c[3]+48*a[7,5 ]*c[5]^5+2*c[5]^2+12*c[6]^2+8300*a[7,5]*c[5]^5*c[3]^3*c[6]+240*a[7,5]* c[5]^3*b[8]*c[6]^2-13500*a[7,5]*c[5]^4*b[8]*c[3]^6-3375*a[7,5]*c[5]^2* c[3]^6*c[6]^2+72*a[7,5]*c[5]^2*c[6]^2*c[3]+2850*a[7,5]*c[5]^2*c[3]^3*c [6]^2-705*a[7,5]*c[5]^2*c[6]^2*c[3]^2+7200*a[7,5]*c[5]^2*c[3]^5*c[6]^2 +1100*a[7,5]*c[5]^3*c[3]^4*c[6]^2+2510*a[7,5]*c[5]^3*c[3]^3*c[6]^2+346 0*a[7,5]*c[5]^4*c[3]^2*c[6]^2-3460*a[7,5]*c[5]^5*c[3]^2*c[6]-4500*a[7, 5]*c[5]^4*c[3]^5*c[6]^2+4500*a[7,5]*c[5]^5*c[3]^5*c[6]-720*a[7,5]*c[5] ^4*c[6]^2*c[3]+720*a[7,5]*c[5]^5*c[6]*c[3]-12720*a[7,5]*c[5]^5*b[8]*c[ 3]^2+13500*a[7,5]*c[5]^4*b[8]*c[3]^5*c[6]^2-13500*a[7,5]*c[5]^5*b[8]*c [3]^5*c[6]+29100*a[7,5]*c[5]^4*c[6]^2*b[8]*c[3]^3-29100*a[7,5]*c[5]^5* c[6]*b[8]*c[3]^3-12720*a[7,5]*c[5]^4*c[6]^2*b[8]*c[3]^2+12720*a[7,5]*c [5]^5*c[6]*b[8]*c[3]^2-32400*a[7,5]*c[5]^4*b[8]*c[3]^4*c[6]^2+32400*a[ 7,5]*c[5]^5*b[8]*c[3]^4*c[6]-13500*a[7,5]*c[5]^3*b[8]*c[3]^6*c[6]^2+13 500*a[7,5]*c[5]^4*b[8]*c[3]^6*c[6]+18900*a[7,5]*c[5]^3*b[8]*c[3]^5*c[6 ]^2+13500*a[7,5]*c[5]^2*b[8]*c[3]^6*c[6]^2-900*a[7,5]*c[5]*b[8]*c[3]^5 *c[6]^2+4380*c[6]^2*c[3]^4-2775*c[6]^2*c[3]^3+951*c[6]^2*c[3]^2+835*c[ 5]^2*c[3]^4-12*c[5]*c[6]^2+1125*c[3]^6*c[6]^2+225*c[3]^6*c[5]^2-690*c[ 3]^5*c[5]^2-900*c[3]^6*c[5]+1575*c[3]^6*c[5]*c[6]-1125*c[5]*c[3]^6*c[6 ]^2-249*c[5]*c[6]*c[3]-4011*c[5]*c[3]^3*c[6]+1392*c[5]*c[6]*c[3]^2-498 0*c[3]^5*c[6]*c[5]+675*c[3]^6+2400*a[7,5]*c[5]^3*b[8]*c[3]^4*c[6]^2-31 500*a[7,5]*c[5]^2*b[8]*c[3]^5*c[6]^2-2400*a[7,5]*c[5]^3*c[6]^2*b[8]*c[ 3]-14820*a[7,5]*c[5]^3*b[8]*c[3]^3*c[6]^2+9180*a[7,5]*c[5]^3*c[6]^2*b[ 8]*c[3]^2-32400*a[7,5]*c[5]^5*b[8]*c[3]^4+13500*a[7,5]*c[5]^5*b[8]*c[3 ]^5+29100*a[7,5]*c[5]^5*b[8]*c[3]^3-360*a[7,5]*c[5]^2*c[6]^2*b[8]*c[3] -13500*a[7,5]*c[5]^2*b[8]*c[3]^3*c[6]^2+3420*a[7,5]*c[5]^2*c[6]^2*b[8] *c[3]^2+2760*a[7,5]*c[5]^4*c[6]^2*b[8]*c[3]-2760*a[7,5]*c[5]^5*c[6]*b[ 8]*c[3]+28440*a[7,5]*c[5]^2*b[8]*c[3]^4*c[6]^2+1560*a[7,5]*c[5]*b[8]*c [3]^4*c[6]^2+611*c[3]^2-168*c[6]^2*c[3]+9825*a[7,5]*c[5]^4*c[3]^4*c[6] ^2-9825*a[7,5]*c[5]^5*c[3]^4*c[6]-8300*a[7,5]*c[5]^4*c[3]^3*c[6]^2-350 *a[7,5]*c[5]*c[3]^4*c[6]^2+225*a[7,5]*c[5]*c[3]^5*c[6]^2-6195*a[7,5]*c [5]^2*c[3]^4*c[6]^2+249*c[6]*c[3]-1392*c[6]*c[3]^2+4011*c[3]^3*c[6]+22 62*c[3]^3*c[5]-513*c[5]^2*c[3]^3+170*c[5]^2*c[3]^2-781*c[5]*c[3]^2+139 *c[5]*c[3]-29*c[5]^2*c[3]+18*c[5]*c[6]+480*a[7,5]*c[5]^3*c[6]^2*c[3]-1 780*a[7,5]*c[5]^3*c[3]^2*c[6]^2+4500*a[7,5]*c[5]^3*c[3]^6*c[6]^2-4500* a[7,5]*c[5]^4*c[3]^6*c[6]-6450*a[7,5]*c[5]^3*c[3]^5*c[6]^2-240*a[7,5]* c[5]^4*b[8]*c[6]^2-3525*c[3]^5*c[6]^2-1575*c[3]^6*c[6]-6425*a[7,5]*c[5 ]^5*c[3]^3+2710*a[7,5]*c[5]^5*c[3]^2-570*a[7,5]*c[5]^5*c[3]+60*a[7,5]* c[5]^4*c[6]^2-60*a[7,5]*c[5]^5*c[6]+3375*a[7,5]*c[5]^4*c[3]^6-48*a[7,5 ]*c[5]^3*c[6]^2+2775*c[5]*c[6]^2*c[3]^3-4380*c[5]*c[3]^4*c[6]^2+3525*c [5]*c[3]^5*c[6]^2-1749*c[3]^3-40*a[7,5]*c[5]^4-8600*a[7,5]*c[5]^4*c[3] ^4*c[6]+3915*a[7,5]*c[5]^4*c[3]^3*c[6]-240*a[7,5]*c[5]^3*b[8]*c[6]+342 0*a[7,5]*c[5]^3*b[8]*c[3]^2-360*a[7,5]*c[5]^3*b[8]*c[3]-13500*a[7,5]*c [5]^3*b[8]*c[3]^3+28440*a[7,5]*c[5]^3*b[8]*c[3]^4-900*a[7,5]*c[5]^2*b[ 8]*c[3]^5+13500*a[7,5]*c[5]^3*b[8]*c[3]^6+2700*a[7,5]*c[5]^2*c[3]^6*c[ 6]-60*a[7,5]*c[5]^2*c[6]*c[3]-2515*a[7,5]*c[5]^2*c[3]^3*c[6]+608*a[7,5 ]*c[5]^2*c[6]*c[3]^2+120*a[7,5]*c[5]^2*b[8]*c[3]^2-780*a[7,5]*c[5]^2*b [8]*c[3]^3-6075*a[7,5]*c[5]^2*c[3]^5*c[6]+5620*a[7,5]*c[5]^3*c[3]^4*c[ 6]-5040*a[7,5]*c[5]^3*c[3]^3*c[6]-32400*a[7,5]*c[5]^4*b[8]*c[3]^5*c[6] -14280*a[7,5]*c[5]^4*c[6]*b[8]*c[3]^3+3540*a[7,5]*c[5]^4*c[6]*b[8]*c[3 ]^2+30000*a[7,5]*c[5]^4*b[8]*c[3]^4*c[6]+12600*a[7,5]*c[5]^3*b[8]*c[3] ^5*c[6]-13500*a[7,5]*c[5]^2*b[8]*c[3]^6*c[6]+5850*a[7,5]*c[5]^3*c[3]^5 +2190*a[7,5]*c[5]^4*c[3]^3-1495*a[7,5]*c[5]^4*c[3]^2+400*a[7,5]*c[5]^4 *c[3]+900*a[7,5]*c[5]*b[8]*c[3]^5*c[6]-30840*a[7,5]*c[5]^3*b[8]*c[3]^4 *c[6]+32400*a[7,5]*c[5]^2*b[8]*c[3]^5*c[6]+2760*a[7,5]*c[5]^3*c[6]*b[8 ]*c[3]+28320*a[7,5]*c[5]^3*b[8]*c[3]^3*c[6]-12600*a[7,5]*c[5]^3*c[6]*b [8]*c[3]^2-930*a[7,5]*c[5]^4*c[3]^2*c[6]+9825*a[7,5]*c[5]^4*c[3]^5*c[6 ]+90*a[7,5]*c[5]^4*c[6]*c[3]+9180*a[7,5]*c[5]^4*b[8]*c[3]^2-31500*a[7, 5]*c[5]^3*b[8]*c[3]^5+180*a[7,5]*c[5]^2*c[3]^5-5080*a[7,5]*c[5]^3*c[3] ^4-2700*a[7,5]*c[5]^3*c[3]^6+360*a[7,5]*c[5]^2*c[6]*b[8]*c[3]+14280*a[ 7,5]*c[5]^2*b[8]*c[3]^3*c[6]-3540*a[7,5]*c[5]^2*c[6]*b[8]*c[3]^2-360*a [7,5]*c[5]^4*c[6]*b[8]*c[3]-30000*a[7,5]*c[5]^2*b[8]*c[3]^4*c[6]-1560* a[7,5]*c[5]*b[8]*c[3]^4*c[6]+2400*a[7,5]*c[5]^4*b[8]*c[3]^4+18900*a[7, 5]*c[5]^4*b[8]*c[3]^5-14820*a[7,5]*c[5]^4*b[8]*c[3]^3+285*a[7,5]*c[5]* c[3]^4*c[6]-180*a[7,5]*c[5]*c[3]^5*c[6]+5430*a[7,5]*c[5]^2*c[3]^4*c[6] +40*a[7,5]*c[5]^3*c[6]+136*a[7,5]*c[5]^2*c[3]^3+2350*a[7,5]*c[5]^3*c[3 ]^3-584*a[7,5]*c[5]^3*c[3]^2-20*a[7,5]*c[5]^2*c[3]^2+60*a[7,5]*c[5]^3* c[3]+240*a[7,5]*c[5]^4*b[8]+575*a[7,5]*c[5]^4*c[3]^4-4800*a[7,5]*c[5]^ 4*c[3]^5-285*a[7,5]*c[5]^2*c[3]^4+780*a[7,5]*c[5]*b[8]*c[3]^3*c[6]-120 *a[7,5]*c[5]*b[8]*c[6]*c[3]^2-472*a[7,5]*c[5]^3*c[6]*c[3]+2200*a[7,5]* c[5]^3*c[3]^2*c[6]-2400*a[7,5]*c[5]^3*c[3]^5*c[6]+1560*a[7,5]*c[5]^2*b [8]*c[3]^4-2400*a[7,5]*c[5]^4*b[8]*c[3]+20*a[7,5]*c[5]*c[6]*c[3]^2-136 *a[7,5]*c[5]*c[3]^3*c[6]-240*a[7,5]*c[5]^5*b[8]+7500*a[7,5]*c[5]^5*c[3 ]^4-3375*a[7,5]*c[5]^5*c[3]^5-3545*c[3]^4*c[5]-6255*c[3]^4*c[6]+4980*c [3]^5*c[6]+6255*c[5]*c[3]^4*c[6])/b[8]/c[5]/(c[5]-1)/(c[3]-1)/(3*c[3]- 1)/(5*c[3]-2)/(15*c[5]*c[3]^2-10*c[5]*c[3]+2*c[5]-c[3])/(c[5]-c[3])/(- 1+c[6])/(c[6]-c[5]), b[1] = -1/60*1/c[3]^2*(150*c[5]*c[3]^3*c[6]-75*c[ 3]^3*c[6]+45*c[3]^3-75*c[3]^3*c[5]+105*c[5]*c[3]^2-205*c[5]*c[6]*c[3]^ 2+105*c[6]*c[3]^2-65*c[3]^2+29*c[3]-45*c[5]*c[3]+80*c[5]*c[6]*c[3]-45* c[6]*c[3]+6*c[5]-10*c[5]*c[6]-4+6*c[6])/c[5]/c[6], a[5,4] = -1/6*(15*c [3]^2-10*c[3]+2)^2*c[5]*(c[5]-c[3])*(15*c[5]*c[3]^2-10*c[5]*c[3]+2*c[5 ]-c[3])/c[3]^2/(1-5*c[3]+5*c[3]^2)/(15*c[3]^2-10*c[3]+1), a[6,4] = -1/ 6*c[6]*(15*c[3]^2-10*c[3]+2)^2*(2*c[6]-10*c[6]*c[3]+15*c[6]*c[3]^2-c[3 ])*(c[6]-c[3])*(90*c[3]^3*c[6]-120*c[6]*c[3]^2+48*c[6]*c[3]-6*c[6]-225 *c[3]^3*c[5]+225*c[5]^2*c[3]^3-240*c[5]^2*c[3]^2+9*c[3]^2+255*c[5]*c[3 ]^2-100*c[5]*c[3]-4*c[3]+90*c[5]^2*c[3]+14*c[5]-12*c[5]^2)/c[3]^2/(15* c[3]^2-10*c[3]+1)/(1-5*c[3]+5*c[3]^2)/(15*c[5]*c[3]^2-10*c[5]*c[3]+2*c [5]-c[3])/(15*c[5]*c[3]-9*c[3]+4-6*c[5]), a[5,3] = -1/6*c[5]*(150*c[3] ^3*c[5]-145*c[5]*c[3]^2+40*c[5]*c[3]-4*c[5]-30*c[3]^3+20*c[3]^2-c[3])* (c[5]-c[3])/(1-5*c[3]+5*c[3]^2)/c[3]^2/(15*c[3]^2-10*c[3]+1), a[8,6] = 1/60*(-8+40*a[7,6]*c[6]^4-5670*c[3]^5*c[5]+3460*a[7,6]*c[5]*c[6]^5*c[ 3]^2-8300*a[7,6]*c[5]*c[6]^5*c[3]^3+9825*a[7,6]*c[5]*c[6]^5*c[3]^4+830 0*a[7,6]*c[5]^2*c[6]^4*c[3]^3-9825*a[7,6]*c[5]^2*c[6]^4*c[3]^4-720*a[7 ,6]*c[5]*c[6]^5*c[3]-2700*a[7,6]*c[3]^6*c[5]*c[6]^2-4500*a[7,6]*c[5]*c [3]^5*c[6]^5+4500*a[7,6]*c[5]*c[3]^6*c[6]^4+4500*a[7,6]*c[5]^2*c[3]^5* c[6]^4-4500*a[7,6]*c[5]^2*c[3]^6*c[6]^3+720*a[7,6]*c[5]^2*c[6]^4*c[3]- 3460*a[7,6]*c[5]^2*c[6]^4*c[3]^2+8*c[6]+20*c[5]+110*c[3]-2710*c[3]^4+2 145*c[3]^5+12600*a[7,6]*c[6]^3*c[5]*b[8]*c[3]^2-3420*a[7,6]*c[6]^2*c[5 ]^2*b[8]*c[3]^2-2760*a[7,6]*c[5]*c[6]^3*b[8]*c[3]+360*a[7,6]*c[5]^2*c[ 6]^2*b[8]*c[3]-28320*a[7,6]*c[5]*b[8]*c[3]^3*c[6]^3+30000*a[7,6]*c[5]* b[8]*c[3]^4*c[6]^2+13500*a[7,6]*c[5]^2*b[8]*c[3]^3*c[6]^2+584*a[7,6]*c [6]^3*c[3]^2-1560*a[7,6]*c[5]^2*b[8]*c[3]^4*c[6]+285*a[7,6]*c[3]^4*c[6 ]^2-180*a[7,6]*c[3]^5*c[6]^2-575*a[7,6]*c[6]^4*c[3]^4+5080*a[7,6]*c[3] ^4*c[6]^3-2350*a[7,6]*c[3]^3*c[6]^3-28440*a[7,6]*b[8]*c[3]^4*c[6]^3+31 500*a[7,6]*b[8]*c[3]^5*c[6]^3-12*c[5]^2+13500*a[7,6]*c[5]*b[8]*c[3]^6* c[6]^2-2760*a[7,6]*b[8]*c[6]^5*c[3]+136*a[7,6]*c[5]*c[3]^3*c[6]-20*a[7 ,6]*c[5]*c[6]*c[3]^2+60*a[7,6]*c[5]*c[6]^2*c[3]+2515*a[7,6]*c[5]*c[3]^ 3*c[6]^2-165*a[7,6]*c[5]^2*c[3]^3*c[6]-608*a[7,6]*c[5]*c[6]^2*c[3]^2+2 4*a[7,6]*c[5]^2*c[6]*c[3]^2+780*a[7,6]*b[8]*c[3]^3*c[6]^2-120*a[7,6]*b [8]*c[6]^2*c[3]^2-12720*a[7,6]*c[6]^5*c[5]*b[8]*c[3]^2+29100*a[7,6]*c[ 6]^5*c[5]*b[8]*c[3]^3-32400*a[7,6]*c[6]^5*c[5]*b[8]*c[3]^4+12720*a[7,6 ]*c[6]^4*c[5]^2*b[8]*c[3]^2-29100*a[7,6]*c[6]^4*c[5]^2*b[8]*c[3]^3+324 00*a[7,6]*c[6]^4*c[5]^2*b[8]*c[3]^4+2760*a[7,6]*c[5]*c[6]^5*b[8]*c[3]- 2760*a[7,6]*c[5]^2*c[6]^4*b[8]*c[3]+13500*a[7,6]*c[5]*b[8]*c[3]^5*c[6] ^5-13500*a[7,6]*c[5]*b[8]*c[3]^6*c[6]^4-13500*a[7,6]*c[5]^2*b[8]*c[3]^ 5*c[6]^4+13500*a[7,6]*c[5]^2*b[8]*c[3]^6*c[6]^3+3375*a[7,6]*c[3]^5*c[6 ]^5+570*a[7,6]*c[6]^5*c[3]-60*a[7,6]*c[5]^2*c[6]^4+240*a[7,6]*b[8]*c[6 ]^5+60*a[7,6]*c[5]*c[6]^5-60*a[7,6]*c[6]^3*c[3]-29100*a[7,6]*c[6]^5*b[ 8]*c[3]^3+32400*a[7,6]*b[8]*c[3]^4*c[6]^5-13500*a[7,6]*b[8]*c[3]^5*c[6 ]^5+13500*a[7,6]*b[8]*c[3]^6*c[6]^4-240*a[7,6]*c[5]*b[8]*c[6]^5+240*a[ 7,6]*c[5]^2*b[8]*c[6]^4-48*a[7,6]*c[6]^5-900*a[7,6]*c[5]*b[8]*c[3]^5*c [6]-285*a[7,6]*c[3]^4*c[5]*c[6]+180*a[7,6]*c[3]^5*c[5]*c[6]-5850*a[7,6 ]*c[3]^5*c[6]^3+2700*a[7,6]*c[3]^6*c[6]^3-4380*c[5]^2*c[3]^4-1125*c[3] ^6*c[5]^2+3525*c[3]^5*c[5]^2+1800*c[3]^6*c[5]+930*a[7,6]*c[5]*c[6]^4*c [3]^2-3915*a[7,6]*c[5]*c[6]^4*c[3]^3+8600*a[7,6]*c[5]*c[6]^4*c[3]^4-25 10*a[7,6]*c[5]^2*c[6]^3*c[3]^3-1100*a[7,6]*c[5]^2*c[6]^3*c[3]^4-90*a[7 ,6]*c[5]*c[6]^4*c[3]-1800*c[3]^6*c[5]*c[6]-2200*a[7,6]*c[5]*c[6]^3*c[3 ]^2+705*a[7,6]*c[5]^2*c[6]^2*c[3]^2-40*a[7,6]*c[5]*c[6]^3+3540*a[7,6]* c[6]^2*c[5]*b[8]*c[3]^2-120*a[7,6]*c[6]*c[5]^2*b[8]*c[3]^2-360*a[7,6]* c[5]*c[6]^2*b[8]*c[3]-14280*a[7,6]*c[5]*b[8]*c[3]^3*c[6]^2+780*a[7,6]* c[5]^2*b[8]*c[3]^3*c[6]+278*c[5]*c[6]*c[3]+4524*c[5]*c[3]^3*c[6]-1562* c[5]*c[6]*c[3]^2+5670*c[3]^5*c[6]*c[5]+4380*c[5]^2*c[3]^4*c[6]-2775*c[ 5]^2*c[3]^3*c[6]-168*c[5]^2*c[6]*c[3]+951*c[5]^2*c[3]^2*c[6]-675*c[3]^ 6+12720*a[7,6]*b[8]*c[6]^5*c[3]^2-9825*a[7,6]*c[5]*c[3]^5*c[6]^4+6450* a[7,6]*c[5]^2*c[3]^5*c[6]^3+3375*a[7,6]*c[5]^2*c[3]^6*c[6]^2-480*a[7,6 ]*c[5]^2*c[6]^3*c[3]+1780*a[7,6]*c[5]^2*c[6]^3*c[3]^2+6075*a[7,6]*c[5] *c[3]^5*c[6]^2+900*a[7,6]*b[8]*c[3]^5*c[6]^2+2400*a[7,6]*b[8]*c[6]^4*c [3]-225*a[7,6]*c[3]^5*c[5]^2*c[6]+1560*a[7,6]*c[5]*b[8]*c[3]^4*c[6]+11 25*c[3]^6*c[5]^2*c[6]-3525*c[5]^2*c[3]^5*c[6]+12*c[5]^2*c[6]-611*c[3]^ 2-3540*a[7,6]*c[6]^4*c[5]*b[8]*c[3]^2+14280*a[7,6]*c[6]^4*c[5]*b[8]*c[ 3]^3-30000*a[7,6]*c[6]^4*c[5]*b[8]*c[3]^4-9180*a[7,6]*c[6]^3*c[5]^2*b[ 8]*c[3]^2+14820*a[7,6]*c[6]^3*c[5]^2*b[8]*c[3]^3-2400*a[7,6]*c[6]^3*c[ 5]^2*b[8]*c[3]^4-2190*a[7,6]*c[6]^4*c[3]^3+20*a[7,6]*c[6]^2*c[3]^2-136 *a[7,6]*c[3]^3*c[6]^2+360*a[7,6]*c[5]*c[6]^4*b[8]*c[3]+2400*a[7,6]*c[5 ]^2*c[6]^3*b[8]*c[3]+1495*a[7,6]*c[6]^4*c[3]^2+32400*a[7,6]*c[5]*b[8]* c[3]^5*c[6]^4+30840*a[7,6]*c[5]*b[8]*c[3]^4*c[6]^3-12600*a[7,6]*c[5]*b [8]*c[3]^5*c[6]^3-32400*a[7,6]*c[5]*b[8]*c[3]^5*c[6]^2-18900*a[7,6]*c[ 5]^2*b[8]*c[3]^5*c[6]^3-28440*a[7,6]*c[5]^2*b[8]*c[3]^4*c[6]^2+31500*a [7,6]*c[5]^2*b[8]*c[3]^5*c[6]^2-13500*a[7,6]*c[5]^2*b[8]*c[3]^6*c[6]^2 +900*a[7,6]*c[5]^2*b[8]*c[3]^5*c[6]+4800*a[7,6]*c[3]^5*c[6]^4-400*a[7, 6]*c[6]^4*c[3]+48*a[7,6]*c[5]^2*c[6]^3-240*a[7,6]*b[8]*c[6]^4-7500*a[7 ,6]*c[6]^5*c[3]^4+472*a[7,6]*c[5]*c[6]^3*c[3]-72*a[7,6]*c[5]^2*c[6]^2* c[3]+5040*a[7,6]*c[5]*c[3]^3*c[6]^3-5430*a[7,6]*c[5]*c[3]^4*c[6]^2-285 0*a[7,6]*c[5]^2*c[3]^3*c[6]^2+350*a[7,6]*c[5]^2*c[3]^4*c[6]-110*c[6]*c [3]+611*c[6]*c[3]^2-1749*c[3]^3*c[6]-4524*c[3]^3*c[5]+2775*c[5]^2*c[3] ^3-951*c[5]^2*c[3]^2+1562*c[5]*c[3]^2-278*c[5]*c[3]+168*c[5]^2*c[3]-20 *c[5]*c[6]+675*c[3]^6*c[6]+1749*c[3]^3+120*a[7,6]*c[6]*c[5]*b[8]*c[3]^ 2+360*a[7,6]*c[6]^3*b[8]*c[3]+13500*a[7,6]*b[8]*c[3]^3*c[6]^3-1560*a[7 ,6]*b[8]*c[3]^4*c[6]^2-3420*a[7,6]*b[8]*c[6]^3*c[3]^2+240*a[7,6]*c[5]* b[8]*c[6]^3+14820*a[7,6]*c[6]^4*b[8]*c[3]^3-2400*a[7,6]*b[8]*c[3]^4*c[ 6]^4-18900*a[7,6]*b[8]*c[3]^5*c[6]^4-13500*a[7,6]*b[8]*c[3]^6*c[6]^3-2 40*a[7,6]*c[5]^2*b[8]*c[6]^3-780*a[7,6]*c[5]*b[8]*c[3]^3*c[6]-9180*a[7 ,6]*b[8]*c[6]^4*c[3]^2-5620*a[7,6]*c[5]*c[3]^4*c[6]^3+2400*a[7,6]*c[5] *c[3]^5*c[6]^3+6195*a[7,6]*c[5]^2*c[3]^4*c[6]^2-7200*a[7,6]*c[5]^2*c[3 ]^5*c[6]^2-3375*a[7,6]*c[3]^6*c[6]^4+6425*a[7,6]*c[6]^5*c[3]^3-2710*a[ 7,6]*c[6]^5*c[3]^2+7090*c[3]^4*c[5]+2710*c[3]^4*c[6]-2145*c[3]^5*c[6]- 7090*c[5]*c[3]^4*c[6])/b[8]/c[6]/(2*c[6]-10*c[6]*c[3]+15*c[6]*c[3]^2-c [3])/(c[6]-c[3])/(c[6]-c[5])/(c[3]-1)/(-1+c[6])/(c[5]-1)/(3*c[3]-1)/(5 *c[3]-2), a[5,2] = 1/2*c[3]*c[5]/c[2]*(-5*c[5]+1+5*c[5]^2)/(1-5*c[3]+5 *c[3]^2), a[3,2] = 1/2*c[3]^2/c[2], a[8,2] = 1/120*(-240*a[7,2]*c[2]*b [8]-48*c[6]*a[7,2]*c[2]-10*c[3]-272*a[7,2]*c[2]*c[3]+240*a[7,2]*c[2]*c [5]*b[8]-1560*a[7,2]*c[2]*c[5]*b[8]*c[3]-700*a[7,2]*c[2]*c[5]*c[3]^2-1 800*a[7,2]*c[2]*c[5]*b[8]*c[3]^3+450*a[7,2]*c[2]*c[5]*c[3]^3+3120*a[7, 2]*c[2]*c[5]*b[8]*c[3]^2-360*a[7,2]*c[2]*c[3]^3+60*c[6]*a[7,2]*c[2]*c[ 5]-20*c[5]*c[6]*c[3]+45*c[5]*c[6]*c[3]^2-700*c[6]*a[7,2]*c[2]*c[3]^2+3 120*c[6]*a[7,2]*c[2]*b[8]*c[3]^2-1560*c[6]*a[7,2]*c[2]*b[8]*c[3]+910*c [6]*a[7,2]*c[2]*c[5]*c[3]^2+1800*c[6]*a[7,2]*c[2]*c[5]*b[8]*c[3]^3-600 *c[6]*a[7,2]*c[2]*c[5]*c[3]^3-3120*a[7,2]*c[2]*b[8]*c[3]^2+330*c[6]*a[ 7,2]*c[2]*c[3]+240*c[6]*a[7,2]*c[2]*b[8]+450*c[6]*a[7,2]*c[2]*c[3]^3-3 120*c[6]*a[7,2]*c[2]*c[5]*b[8]*c[3]^2+21*c[3]^2+570*a[7,2]*c[2]*c[3]^2 +1800*a[7,2]*c[2]*b[8]*c[3]^3-1800*c[6]*a[7,2]*c[2]*b[8]*c[3]^3+40*a[7 ,2]*c[2]+1560*a[7,2]*c[2]*b[8]*c[3]+330*a[7,2]*c[2]*c[5]*c[3]+14*c[6]* c[3]-30*c[6]*c[3]^2-30*c[5]*c[3]^2+14*c[5]*c[3]-240*c[6]*a[7,2]*c[2]*c [5]*b[8]+1560*c[6]*a[7,2]*c[2]*c[5]*b[8]*c[3]-48*a[7,2]*c[2]*c[5]-420* c[6]*a[7,2]*c[2]*c[5]*c[3])/b[8]/c[2]/(c[3]-1)/(3*c[3]-1)/(5*c[3]-2)/( c[5]-1)/(-1+c[6]), a[4,2] = 1/2*c[3]^2*(4-30*c[3]+45*c[3]^2)/c[2]/(15* c[3]^2-10*c[3]+2)^3, b[6] = 1/60*(1-5*c[3]+5*c[3]^2)*(15*c[5]*c[3]-9*c [3]+4-6*c[5])/c[6]/(-1+c[6])/(2*c[6]-10*c[6]*c[3]+15*c[6]*c[3]^2-c[3]) /(c[6]-c[3])/(-c[6]+c[5]), b[7] = -1/60*(20-24*c[6]-24*c[5]-136*c[3]-2 10*c[5]*c[6]*c[3]-300*c[5]*c[3]^3*c[6]+455*c[5]*c[6]*c[3]^2-780*c[6]*b [8]*c[3]-900*b[8]*c[3]^3*c[6]+1560*b[8]*c[6]*c[3]^2-120*c[5]*b[8]*c[6] +1560*c[5]*b[8]*c[3]^2-780*c[5]*b[8]*c[3]-900*c[5]*b[8]*c[3]^3-120*b[8 ]+285*c[3]^2+165*c[6]*c[3]-350*c[6]*c[3]^2+225*c[3]^3*c[6]+225*c[3]^3* c[5]-350*c[5]*c[3]^2+165*c[5]*c[3]+30*c[5]*c[6]+120*b[8]*c[6]+120*b[8] *c[5]-1560*b[8]*c[3]^2+780*b[8]*c[3]+900*b[8]*c[3]^3-180*c[3]^3+780*c[ 5]*c[6]*b[8]*c[3]+900*c[5]*b[8]*c[3]^3*c[6]-1560*c[6]*c[5]*b[8]*c[3]^2 )/(c[3]-1)/(-1+c[6])/(c[5]-1)/(3*c[3]-1)/(5*c[3]-2), b[3] = -1/60*(75* c[5]*c[6]*c[3]^2-60*c[5]*c[6]*c[3]+10*c[5]*c[6]-45*c[5]*c[3]^2+35*c[5] *c[3]-6*c[5]-45*c[6]*c[3]^2+35*c[6]*c[3]-6*c[6]+30*c[3]^2-23*c[3]+4)/( c[6]-c[3])/(c[5]-c[3])/c[3]^2/(c[3]-1)/(15*c[3]^2-10*c[3]+1), a[8,4] = 1/360*(384+59763580*c[3]^5*c[5]-688500*a[7,4]*c[3]^9*c[5]-470241675*c [3]^8*c[6]+508180500*c[3]^9*c[6]+932917500*c[3]^10*c[6]*c[5]+860490000 *c[3]^9*c[6]^2*c[5]+860490000*c[3]^9*c[5]^2*c[6]-864*c[6]+282487500*c[ 3]^11*c[6]*c[5]^2-960*c[5]-10800*c[3]+6008104*c[3]^4+3645000*a[7,4]*c[ 3]^9*b[8]*c[6]-688500*a[7,4]*c[3]^9*c[6]-23157444*c[3]^5+11880*a[7,4]* c[3]^5*b[8]-223256250*c[3]^11*c[6]^2*c[5]^2+160116750*c[3]^10-63519187 5*c[3]^10*c[6]*c[5]^2-664638750*c[3]^9*c[6]^2*c[5]^2-497712*c[5]*c[6]^ 2*c[3]^2+38016*c[5]*c[6]^2*c[3]+243000*a[7,4]*c[3]^10*c[5]-66960*a[7,4 ]*c[3]^6*b[8]+261225000*c[3]^10*c[6]^2+576*c[5]^2+576*c[6]^2-56953125* c[3]^12*c[6]^2*c[5]+11670*a[7,4]*c[3]^6+120*a[7,4]*c[3]^4+36450*a[7,4] *c[3]^8-16200*a[7,4]*c[3]^9-2016*a[7,4]*c[3]^5-1215000*a[7,4]*c[3]^10* b[8]*c[6]-1215000*a[7,4]*c[3]^10*c[5]*b[8]-69558750*c[3]^11-194400*a[7 ,4]*c[3]^8*b[8]+169200*a[7,4]*c[3]^7*b[8]+3645000*a[7,4]*c[3]^9*c[5]*b [8]-31893750*c[3]^12*c[6]+81000*a[7,4]*c[3]^9*b[8]-65610000*a[7,4]*c[3 ]^10*c[5]*b[8]*c[6]+66825000*a[7,4]*c[3]^10*c[5]*b[8]*c[6]^2-5208480*a [7,4]*c[3]^5*c[5]*b[8]*c[6]^2-21985200*a[7,4]*c[3]^6*c[6]^2*c[5]^2*b[8 ]+57933000*a[7,4]*c[3]^7*c[5]^2*b[8]*c[6]^2+674400*a[7,4]*c[3]^7*c[5]^ 2-114665625*c[3]^11*c[6]^2+95526000*a[7,4]*c[3]^8*c[5]*b[8]*c[6]^2-548 47800*a[7,4]*c[3]^7*c[5]*b[8]*c[6]^2+95526000*a[7,4]*c[3]^8*c[5]^2*b[8 ]*c[6]-54847800*a[7,4]*c[3]^7*c[5]^2*b[8]*c[6]+101979000*a[7,4]*c[3]^9 *c[5]*b[8]*c[6]-91076400*a[7,4]*c[3]^8*c[5]*b[8]*c[6]+109350000*a[7,4] *c[3]^9*c[5]^2*b[8]*c[6]^2+808200*a[7,4]*c[3]^8*c[5]+243000*a[7,4]*c[3 ]^10*c[6]-4449600*a[7,4]*c[3]^8*c[5]*b[8]+16193700*a[7,4]*c[3]^8*c[5]* c[6]-20992500*a[7,4]*c[3]^8*c[5]*c[6]^2+5999100*a[7,4]*c[3]^6*c[5]^2*c [6]^2-1456800*a[7,4]*c[3]^5*c[5]^2*c[6]^2+223320*a[7,4]*c[3]^4*c[5]^2* c[6]^2-19440*a[7,4]*c[3]^3*c[5]^2*c[6]^2-12288*a[7,4]*c[3]^3*c[5]*c[6] -840540*a[7,4]*c[3]^5*c[5]*c[6]+134100*a[7,4]*c[3]^4*c[5]*c[6]+1440*a[ 7,4]*c[3]^3*c[6]*b[8]-15795000*a[7,4]*c[3]^10*c[5]^2*c[6]-3726000*a[7, 4]*c[3]^9*b[8]*c[6]^2+354082500*c[3]^7*c[5]-15795000*a[7,4]*c[3]^10*c[ 5]*c[6]^2+22781250*c[3]^12*c[5]^2+2880*c[6]*a[7,4]*c[3]^2*b[8]*c[5]^2+ 9297240*c[6]^2*c[3]^4-1710408*c[6]^2*c[3]^3+213792*c[6]^2*c[3]^2+92972 40*c[5]^2*c[3]^4-1344*c[5]*c[6]^2+102802200*c[3]^6*c[6]^2-215452350*c[ 3]^7*c[5]^2+102802200*c[3]^6*c[5]^2-36130980*c[3]^5*c[5]^2-169540850*c [3]^6*c[5]-30420*a[7,4]*c[3]^7+329510250*c[3]^8*c[5]^2+582970500*c[3]^ 9*c[5]-369187875*c[3]^10*c[6]+329510250*c[3]^8*c[6]^2+251640*a[7,4]*c[ 3]^5*b[8]*c[6]-30240*a[7,4]*c[3]^4*b[8]*c[6]-30240*a[7,4]*c[3]^4*c[5]* b[8]+1440*a[7,4]*c[3]^3*c[5]*b[8]+251640*a[7,4]*c[3]^5*c[5]*b[8]-90438 00*a[7,4]*c[3]^7*c[6]*c[5]-4372800*a[7,4]*c[3]^6*c[5]^2*c[6]+1077840*a [7,4]*c[3]^5*c[5]^2*c[6]+15120*a[7,4]*c[3]^3*c[5]^2*c[6]-168960*a[7,4] *c[3]^4*c[5]^2*c[6]+13668750*c[3]^12-2880*a[7,4]*c[3]^2*c[5]*b[8]*c[6] -19679760*a[7,4]*c[3]^6*c[5]*b[8]*c[6]+23955750*a[7,4]*c[3]^9*c[6]^2*c [5]-20992500*a[7,4]*c[3]^8*c[5]^2*c[6]-18643500*a[7,4]*c[3]^9*c[5]*c[6 ]-32805000*a[7,4]*c[3]^9*c[5]^2*c[6]^2+28968750*a[7,4]*c[3]^8*c[5]^2*c [6]^2+11759400*a[7,4]*c[3]^7*c[5]^2*c[6]-1440*a[7,4]*c[3]^3*c[6]^2*b[8 ]-263520*a[7,4]*c[3]^5*b[8]*c[6]^2+365309800*c[3]^6*c[5]*c[6]-24165180 0*c[5]*c[3]^6*c[6]^2+15986400*c[5]^2*c[6]^2*c[3]^4-2912160*c[5]^2*c[6] ^2*c[3]^3+360960*c[5]^2*c[6]^2*c[3]^2-27360*c[5]^2*c[6]^2*c[3]+288*a[7 ,4]*c[3]^3*c[6]^2-240*a[7,4]*c[3]^3*c[6]+5088*a[7,4]*c[3]^4*c[6]-42870 *a[7,4]*c[3]^5*c[6]-42870*a[7,4]*c[3]^5*c[5]+54780*a[7,4]*c[3]^5*c[5]^ 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5]^2*b[8]-1440*a[7,4]*c[3]^3*c[5]^2*b[8]-263520*a[7,4]*c[3]^5*c[5]^2*b [8]+1186200*a[7,4]*c[3]^6*b[8]*c[6]^2+2916000*a[7,4]*c[3]^7*b[8]*c[6]- 514950*a[7,4]*c[3]^7*c[5]-252450*a[7,4]*c[3]^6*c[6]^2+54780*a[7,4]*c[3 ]^5*c[6]^2-6300*a[7,4]*c[3]^4*c[6]^2-252450*a[7,4]*c[3]^6*c[5]^2-10485 00*a[7,4]*c[3]^8*c[6]^2+877500*a[7,4]*c[3]^9*c[5]^2-357854625*c[3]^9*c [5]^2-423275625*c[3]^10*c[5]-357854625*c[3]^9*c[6]^2+827280*a[7,4]*c[3 ]^4*c[6]^2*c[5]*b[8]-2880*a[7,4]*c[3]^2*c[5]^2*b[8]*c[6]^2+1186200*a[7 ,4]*c[3]^6*c[5]^2*b[8]+2916000*a[7,4]*c[3]^7*c[5]*b[8]-168960*a[7,4]*c [3]^4*c[5]*c[6]^2+15120*a[7,4]*c[3]^3*c[5]*c[6]^2-858240*a[7,4]*c[3]^4 *c[6]^2*c[5]^2*b[8]+5472000*a[7,4]*c[3]^5*c[5]^2*b[8]*c[6]^2+20799000* a[7,4]*c[3]^6*c[5]*b[8]*c[6]^2+20799000*a[7,4]*c[3]^6*c[5]^2*b[8]*c[6] +51931800*a[7,4]*c[3]^7*c[5]*b[8]*c[6]+76320*a[7,4]*c[3]^3*c[5]^2*c[6] ^2*b[8]-74880*a[7,4]*c[3]^3*c[5]^2*c[6]*b[8]-5208480*a[7,4]*c[3]^5*c[5 ]^2*b[8]*c[6]+827280*a[7,4]*c[3]^4*c[6]*c[5]^2*b[8]-1048500*a[7,4]*c[3 ]^8*c[5]^2-410973750*c[3]^11*c[6]*c[5]-100170000*a[7,4]*c[3]^8*c[5]^2* b[8]*c[6]^2-105624000*a[7,4]*c[3]^9*c[5]*b[8]*c[6]^2-3726000*a[7,4]*c[ 3]^9*c[5]^2*b[8]-576*a[7,4]*c[3]^2*c[5]*c[6]^2-3085200*a[7,4]*c[3]^7*c [5]^2*b[8]+4644000*a[7,4]*c[3]^8*b[8]*c[6]^2-3085200*a[7,4]*c[3]^7*b[8 ]*c[6]^2-4449600*a[7,4]*c[3]^8*b[8]*c[6]+4644000*a[7,4]*c[3]^8*c[5]^2* b[8]-15416180*c[3]^4*c[5]-13569188*c[3]^4*c[6]+52412260*c[3]^5*c[6]+33 120200*c[5]*c[3]^4*c[6])/b[8]/c[3]^2/(c[5]-1)/(c[3]-1)/(3*c[3]-1)/(5*c [3]-2)/(15*c[3]^2-10*c[3]+1)/(15*c[5]*c[3]^2-10*c[5]*c[3]+2*c[5]-c[3]) /(-1+c[6])/(2*c[6]-10*c[6]*c[3]+15*c[6]*c[3]^2-c[3]), c[8] = 1, a[4,3] = -(15*c[3]^2-10*c[3]+1)*c[3]/(15*c[3]^2-10*c[3]+2)^3, b[5] = -1/60*( 1-5*c[3]+5*c[3]^2)*(15*c[6]*c[3]-9*c[3]+4-6*c[6])/(-c[6]+c[5])/c[5]/(c [5]-1)/(15*c[5]*c[3]^2-10*c[5]*c[3]+2*c[5]-c[3])/(c[5]-c[3]), a[8,7] = 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 293 6 "Step 6" }{TEXT -1 1 " " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 27 "The stage-order condit ion: " }}{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 8 " and " }{XPPEDIT 18 0 "S um(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 1 " " }} {PARA 0 "" 0 "" {TEXT -1 11 "determine " }{XPPEDIT 18 0 "a[7,3]" "6#& %\"aG6$\"\"(\"\"$" }{TEXT -1 7 " and " }{XPPEDIT 18 0 "a[8,3]" "6#&% \"aG6$\"\")\"\"$" }{TEXT -1 14 " in terms of " }{XPPEDIT 18 0 "a[7,2] ;" "6#&%\"aG6$\"\"(\"\"#" }{TEXT -1 3 ", " }{XPPEDIT 18 0 "a[7,4];" " 6#&%\"aG6$\"\"(\"\"%" }{TEXT -1 3 ", " }{XPPEDIT 18 0 "a[7,5];" "6#&% \"aG6$\"\"(\"\"&" }{TEXT -1 7 " and " }{XPPEDIT 18 0 "a[7,6];" "6#&% \"aG6$\"\"(\"\"'" }{TEXT -1 1 "." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 185 "cdns4 := [add(a[7,j]*c[j],j =2..6)=1/2*c[7]^2,add(a[8,j]*c[j],j=2..7)=1/2*c[8]^2]:\neqns4 := simpl ify(subs(e11,cdns4)):\ne12 := solve(\{op(eqns4)\},\{a[7,3],a[8,3]\}): \ne13 := `union`(e11,e12):" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 21 "indets(map(rhs,e13));" }}{PARA 11 " " 1 "" {XPPMATH 20 "6#<+&%\"cG6#\"\"#&F%6#\"\"'&F%6#\"\"&&F%6#\"\"$&% \"aG6$\"\"(F'&%\"bG6#\"\")&F26$F4F-&F26$F4\"\"%&F26$F4F*" }}}{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 49240 "e13 := \{a[6,2] = 1/ 2*c[6]*c[3]/c[2]*(5*c[6]^2-5*c[6]+1)/(1-5*c[3]+5*c[3]^2), a[6,3] = -1/ 6*(c[6]-c[3])*(450*c[3]^5*c[5]+300*c[3]^4-270*c[3]^5+12*c[6]^2+2700*c[ 6]^2*c[3]^4-2100*c[6]^2*c[3]^3+810*c[6]^2*c[3]^2-450*c[5]^2*c[3]^4+328 *c[5]*c[6]*c[3]+2535*c[5]*c[3]^3*c[6]-1450*c[5]*c[6]*c[3]^2-900*c[3]^5 *c[6]*c[5]+2250*c[5]^2*c[3]^4*c[6]-3075*c[5]^2*c[3]^3*c[6]-300*c[5]^2* c[6]*c[3]+1470*c[5]^2*c[3]^2*c[6]+24*c[5]^2*c[6]+4*c[3]^2-156*c[6]^2*c [3]+10*c[6]*c[3]-148*c[6]*c[3]^2+820*c[3]^3*c[6]-45*c[3]^3*c[5]+480*c[ 5]^2*c[3]^3-135*c[5]^2*c[3]^2+35*c[5]*c[3]^2+2*c[5]*c[3]+6*c[5]^2*c[3] -28*c[5]*c[6]-1350*c[3]^5*c[6]^2-89*c[3]^3-300*c[3]^4*c[5]-1815*c[3]^4 *c[6]+1350*c[3]^5*c[6]-975*c[5]*c[3]^4*c[6])*c[6]/(1-5*c[3]+5*c[3]^2)/ (15*c[3]^2-10*c[3]+1)/(15*c[5]*c[3]-9*c[3]+4-6*c[5])/(c[5]-c[3])/c[3]^ 2, b[2] = 0, b[4] = 1/60*(15*c[3]^2-10*c[3]+2)^5*(3*c[3]-5*c[5]*c[3]-5 *c[6]*c[3]+10*c[5]*c[6]*c[3]-5*c[5]*c[6]-2+3*c[6]+3*c[5])/(15*c[3]^2-1 0*c[3]+1)/c[3]^2/(3*c[3]-1)/(5*c[3]-2)/(2*c[6]-10*c[6]*c[3]+15*c[6]*c[ 3]^2-c[3])/(15*c[5]*c[3]^2-10*c[5]*c[3]+2*c[5]-c[3]), c[7] = 1, c[4] = c[3]/(15*c[3]^2-10*c[3]+2), a[6,5] = (3*c[3]-1)*(2*c[6]-10*c[6]*c[3]+ 15*c[6]*c[3]^2-c[3])*(c[6]-c[3])*(-c[6]+c[5])*c[6]/(15*c[5]*c[3]^2-10* c[5]*c[3]+2*c[5]-c[3])/(c[5]-c[3])/(15*c[5]*c[3]-9*c[3]+4-6*c[5])/c[5] , a[8,5] = 1/60*(8+2835*c[3]^5*c[5]-18*c[6]-10*c[5]-110*c[3]+2710*c[3] ^4-2145*c[3]^5-780*a[7,5]*c[5]*b[8]*c[3]^3*c[6]^2+120*a[7,5]*c[5]*b[8] *c[6]^2*c[3]^2+240*a[7,5]*c[5]^5*b[8]*c[6]+2760*a[7,5]*c[5]^5*b[8]*c[3 ]-24*a[7,5]*c[5]*c[6]^2*c[3]^2+165*a[7,5]*c[5]*c[3]^3*c[6]^2-951*c[5]* c[6]^2*c[3]^2+168*c[5]*c[6]^2*c[3]+48*a[7,5]*c[5]^5+2*c[5]^2+12*c[6]^2 +8300*a[7,5]*c[5]^5*c[3]^3*c[6]+240*a[7,5]*c[5]^3*b[8]*c[6]^2-13500*a[ 7,5]*c[5]^4*b[8]*c[3]^6-3375*a[7,5]*c[5]^2*c[3]^6*c[6]^2+72*a[7,5]*c[5 ]^2*c[6]^2*c[3]+2850*a[7,5]*c[5]^2*c[3]^3*c[6]^2-705*a[7,5]*c[5]^2*c[6 ]^2*c[3]^2+7200*a[7,5]*c[5]^2*c[3]^5*c[6]^2+1100*a[7,5]*c[5]^3*c[3]^4* c[6]^2+2510*a[7,5]*c[5]^3*c[3]^3*c[6]^2+3460*a[7,5]*c[5]^4*c[3]^2*c[6] ^2-3460*a[7,5]*c[5]^5*c[3]^2*c[6]-4500*a[7,5]*c[5]^4*c[3]^5*c[6]^2+450 0*a[7,5]*c[5]^5*c[3]^5*c[6]-720*a[7,5]*c[5]^4*c[6]^2*c[3]+720*a[7,5]*c [5]^5*c[6]*c[3]-12720*a[7,5]*c[5]^5*b[8]*c[3]^2+13500*a[7,5]*c[5]^4*b[ 8]*c[3]^5*c[6]^2-13500*a[7,5]*c[5]^5*b[8]*c[3]^5*c[6]+29100*a[7,5]*c[5 ]^4*c[6]^2*b[8]*c[3]^3-29100*a[7,5]*c[5]^5*c[6]*b[8]*c[3]^3-12720*a[7, 5]*c[5]^4*c[6]^2*b[8]*c[3]^2+12720*a[7,5]*c[5]^5*c[6]*b[8]*c[3]^2-3240 0*a[7,5]*c[5]^4*b[8]*c[3]^4*c[6]^2+32400*a[7,5]*c[5]^5*b[8]*c[3]^4*c[6 ]-13500*a[7,5]*c[5]^3*b[8]*c[3]^6*c[6]^2+13500*a[7,5]*c[5]^4*b[8]*c[3] ^6*c[6]+18900*a[7,5]*c[5]^3*b[8]*c[3]^5*c[6]^2+13500*a[7,5]*c[5]^2*b[8 ]*c[3]^6*c[6]^2-900*a[7,5]*c[5]*b[8]*c[3]^5*c[6]^2+4380*c[6]^2*c[3]^4- 2775*c[6]^2*c[3]^3+951*c[6]^2*c[3]^2+835*c[5]^2*c[3]^4-12*c[5]*c[6]^2+ 1125*c[3]^6*c[6]^2+225*c[3]^6*c[5]^2-690*c[3]^5*c[5]^2-900*c[3]^6*c[5] +1575*c[3]^6*c[5]*c[6]-1125*c[5]*c[3]^6*c[6]^2-249*c[5]*c[6]*c[3]-4011 *c[5]*c[3]^3*c[6]+1392*c[5]*c[6]*c[3]^2-4980*c[3]^5*c[6]*c[5]+675*c[3] ^6+2400*a[7,5]*c[5]^3*b[8]*c[3]^4*c[6]^2-31500*a[7,5]*c[5]^2*b[8]*c[3] ^5*c[6]^2-2400*a[7,5]*c[5]^3*c[6]^2*b[8]*c[3]-14820*a[7,5]*c[5]^3*b[8] *c[3]^3*c[6]^2+9180*a[7,5]*c[5]^3*c[6]^2*b[8]*c[3]^2-32400*a[7,5]*c[5] ^5*b[8]*c[3]^4+13500*a[7,5]*c[5]^5*b[8]*c[3]^5+29100*a[7,5]*c[5]^5*b[8 ]*c[3]^3-360*a[7,5]*c[5]^2*c[6]^2*b[8]*c[3]-13500*a[7,5]*c[5]^2*b[8]*c [3]^3*c[6]^2+3420*a[7,5]*c[5]^2*c[6]^2*b[8]*c[3]^2+2760*a[7,5]*c[5]^4* c[6]^2*b[8]*c[3]-2760*a[7,5]*c[5]^5*c[6]*b[8]*c[3]+28440*a[7,5]*c[5]^2 *b[8]*c[3]^4*c[6]^2+1560*a[7,5]*c[5]*b[8]*c[3]^4*c[6]^2+611*c[3]^2-168 *c[6]^2*c[3]+9825*a[7,5]*c[5]^4*c[3]^4*c[6]^2-9825*a[7,5]*c[5]^5*c[3]^ 4*c[6]-8300*a[7,5]*c[5]^4*c[3]^3*c[6]^2-350*a[7,5]*c[5]*c[3]^4*c[6]^2+ 225*a[7,5]*c[5]*c[3]^5*c[6]^2-6195*a[7,5]*c[5]^2*c[3]^4*c[6]^2+249*c[6 ]*c[3]-1392*c[6]*c[3]^2+4011*c[3]^3*c[6]+2262*c[3]^3*c[5]-513*c[5]^2*c [3]^3+170*c[5]^2*c[3]^2-781*c[5]*c[3]^2+139*c[5]*c[3]-29*c[5]^2*c[3]+1 8*c[5]*c[6]+480*a[7,5]*c[5]^3*c[6]^2*c[3]-1780*a[7,5]*c[5]^3*c[3]^2*c[ 6]^2+4500*a[7,5]*c[5]^3*c[3]^6*c[6]^2-4500*a[7,5]*c[5]^4*c[3]^6*c[6]-6 450*a[7,5]*c[5]^3*c[3]^5*c[6]^2-240*a[7,5]*c[5]^4*b[8]*c[6]^2-3525*c[3 ]^5*c[6]^2-1575*c[3]^6*c[6]-6425*a[7,5]*c[5]^5*c[3]^3+2710*a[7,5]*c[5] ^5*c[3]^2-570*a[7,5]*c[5]^5*c[3]+60*a[7,5]*c[5]^4*c[6]^2-60*a[7,5]*c[5 ]^5*c[6]+3375*a[7,5]*c[5]^4*c[3]^6-48*a[7,5]*c[5]^3*c[6]^2+2775*c[5]*c [6]^2*c[3]^3-4380*c[5]*c[3]^4*c[6]^2+3525*c[5]*c[3]^5*c[6]^2-1749*c[3] ^3-40*a[7,5]*c[5]^4-8600*a[7,5]*c[5]^4*c[3]^4*c[6]+3915*a[7,5]*c[5]^4* c[3]^3*c[6]-240*a[7,5]*c[5]^3*b[8]*c[6]+3420*a[7,5]*c[5]^3*b[8]*c[3]^2 -360*a[7,5]*c[5]^3*b[8]*c[3]-13500*a[7,5]*c[5]^3*b[8]*c[3]^3+28440*a[7 ,5]*c[5]^3*b[8]*c[3]^4-900*a[7,5]*c[5]^2*b[8]*c[3]^5+13500*a[7,5]*c[5] ^3*b[8]*c[3]^6+2700*a[7,5]*c[5]^2*c[3]^6*c[6]-60*a[7,5]*c[5]^2*c[6]*c[ 3]-2515*a[7,5]*c[5]^2*c[3]^3*c[6]+608*a[7,5]*c[5]^2*c[6]*c[3]^2+120*a[ 7,5]*c[5]^2*b[8]*c[3]^2-780*a[7,5]*c[5]^2*b[8]*c[3]^3-6075*a[7,5]*c[5] ^2*c[3]^5*c[6]+5620*a[7,5]*c[5]^3*c[3]^4*c[6]-5040*a[7,5]*c[5]^3*c[3]^ 3*c[6]-32400*a[7,5]*c[5]^4*b[8]*c[3]^5*c[6]-14280*a[7,5]*c[5]^4*c[6]*b [8]*c[3]^3+3540*a[7,5]*c[5]^4*c[6]*b[8]*c[3]^2+30000*a[7,5]*c[5]^4*b[8 ]*c[3]^4*c[6]+12600*a[7,5]*c[5]^3*b[8]*c[3]^5*c[6]-13500*a[7,5]*c[5]^2 *b[8]*c[3]^6*c[6]+5850*a[7,5]*c[5]^3*c[3]^5+2190*a[7,5]*c[5]^4*c[3]^3- 1495*a[7,5]*c[5]^4*c[3]^2+400*a[7,5]*c[5]^4*c[3]+900*a[7,5]*c[5]*b[8]* c[3]^5*c[6]-30840*a[7,5]*c[5]^3*b[8]*c[3]^4*c[6]+32400*a[7,5]*c[5]^2*b [8]*c[3]^5*c[6]+2760*a[7,5]*c[5]^3*c[6]*b[8]*c[3]+28320*a[7,5]*c[5]^3* b[8]*c[3]^3*c[6]-12600*a[7,5]*c[5]^3*c[6]*b[8]*c[3]^2-930*a[7,5]*c[5]^ 4*c[3]^2*c[6]+9825*a[7,5]*c[5]^4*c[3]^5*c[6]+90*a[7,5]*c[5]^4*c[6]*c[3 ]+9180*a[7,5]*c[5]^4*b[8]*c[3]^2-31500*a[7,5]*c[5]^3*b[8]*c[3]^5+180*a [7,5]*c[5]^2*c[3]^5-5080*a[7,5]*c[5]^3*c[3]^4-2700*a[7,5]*c[5]^3*c[3]^ 6+360*a[7,5]*c[5]^2*c[6]*b[8]*c[3]+14280*a[7,5]*c[5]^2*b[8]*c[3]^3*c[6 ]-3540*a[7,5]*c[5]^2*c[6]*b[8]*c[3]^2-360*a[7,5]*c[5]^4*c[6]*b[8]*c[3] -30000*a[7,5]*c[5]^2*b[8]*c[3]^4*c[6]-1560*a[7,5]*c[5]*b[8]*c[3]^4*c[6 ]+2400*a[7,5]*c[5]^4*b[8]*c[3]^4+18900*a[7,5]*c[5]^4*b[8]*c[3]^5-14820 *a[7,5]*c[5]^4*b[8]*c[3]^3+285*a[7,5]*c[5]*c[3]^4*c[6]-180*a[7,5]*c[5] *c[3]^5*c[6]+5430*a[7,5]*c[5]^2*c[3]^4*c[6]+40*a[7,5]*c[5]^3*c[6]+136* a[7,5]*c[5]^2*c[3]^3+2350*a[7,5]*c[5]^3*c[3]^3-584*a[7,5]*c[5]^3*c[3]^ 2-20*a[7,5]*c[5]^2*c[3]^2+60*a[7,5]*c[5]^3*c[3]+240*a[7,5]*c[5]^4*b[8] +575*a[7,5]*c[5]^4*c[3]^4-4800*a[7,5]*c[5]^4*c[3]^5-285*a[7,5]*c[5]^2* c[3]^4+780*a[7,5]*c[5]*b[8]*c[3]^3*c[6]-120*a[7,5]*c[5]*b[8]*c[6]*c[3] ^2-472*a[7,5]*c[5]^3*c[6]*c[3]+2200*a[7,5]*c[5]^3*c[3]^2*c[6]-2400*a[7 ,5]*c[5]^3*c[3]^5*c[6]+1560*a[7,5]*c[5]^2*b[8]*c[3]^4-2400*a[7,5]*c[5] ^4*b[8]*c[3]+20*a[7,5]*c[5]*c[6]*c[3]^2-136*a[7,5]*c[5]*c[3]^3*c[6]-24 0*a[7,5]*c[5]^5*b[8]+7500*a[7,5]*c[5]^5*c[3]^4-3375*a[7,5]*c[5]^5*c[3] ^5-3545*c[3]^4*c[5]-6255*c[3]^4*c[6]+4980*c[3]^5*c[6]+6255*c[5]*c[3]^4 *c[6])/b[8]/c[5]/(c[5]-1)/(c[3]-1)/(3*c[3]-1)/(5*c[3]-2)/(15*c[5]*c[3] ^2-10*c[5]*c[3]+2*c[5]-c[3])/(c[5]-c[3])/(-1+c[6])/(c[6]-c[5]), b[1] = -1/60*1/c[3]^2*(150*c[5]*c[3]^3*c[6]-75*c[3]^3*c[6]+45*c[3]^3-75*c[3] ^3*c[5]+105*c[5]*c[3]^2-205*c[5]*c[6]*c[3]^2+105*c[6]*c[3]^2-65*c[3]^2 +29*c[3]-45*c[5]*c[3]+80*c[5]*c[6]*c[3]-45*c[6]*c[3]+6*c[5]-10*c[5]*c[ 6]-4+6*c[6])/c[5]/c[6], a[7,3] = -1/2*(30*a[7,2]*c[2]*c[3]^2-20*a[7,2] *c[2]*c[3]+4*a[7,2]*c[2]+2*a[7,4]*c[3]+30*a[7,5]*c[5]*c[3]^2-20*a[7,5] *c[5]*c[3]+4*a[7,5]*c[5]+30*a[7,6]*c[6]*c[3]^2-20*a[7,6]*c[6]*c[3]+4*a [7,6]*c[6]-15*c[3]^2+10*c[3]-2)/c[3]/(15*c[3]^2-10*c[3]+2), a[5,4] = - 1/6*(15*c[3]^2-10*c[3]+2)^2*c[5]*(c[5]-c[3])*(15*c[5]*c[3]^2-10*c[5]*c [3]+2*c[5]-c[3])/c[3]^2/(1-5*c[3]+5*c[3]^2)/(15*c[3]^2-10*c[3]+1), a[6 ,4] = -1/6*c[6]*(15*c[3]^2-10*c[3]+2)^2*(2*c[6]-10*c[6]*c[3]+15*c[6]*c [3]^2-c[3])*(c[6]-c[3])*(90*c[3]^3*c[6]-120*c[6]*c[3]^2+48*c[6]*c[3]-6 *c[6]-225*c[3]^3*c[5]+225*c[5]^2*c[3]^3-240*c[5]^2*c[3]^2+9*c[3]^2+255 *c[5]*c[3]^2-100*c[5]*c[3]-4*c[3]+90*c[5]^2*c[3]+14*c[5]-12*c[5]^2)/c[ 3]^2/(15*c[3]^2-10*c[3]+1)/(1-5*c[3]+5*c[3]^2)/(15*c[5]*c[3]^2-10*c[5] *c[3]+2*c[5]-c[3])/(15*c[5]*c[3]-9*c[3]+4-6*c[5]), a[5,3] = -1/6*c[5]* (150*c[3]^3*c[5]-145*c[5]*c[3]^2+40*c[5]*c[3]-4*c[5]-30*c[3]^3+20*c[3] ^2-c[3])*(c[5]-c[3])/(1-5*c[3]+5*c[3]^2)/c[3]^2/(15*c[3]^2-10*c[3]+1), a[8,6] = 1/60*(-8+40*a[7,6]*c[6]^4-5670*c[3]^5*c[5]+3460*a[7,6]*c[5]* c[6]^5*c[3]^2-8300*a[7,6]*c[5]*c[6]^5*c[3]^3+9825*a[7,6]*c[5]*c[6]^5*c [3]^4+8300*a[7,6]*c[5]^2*c[6]^4*c[3]^3-9825*a[7,6]*c[5]^2*c[6]^4*c[3]^ 4-720*a[7,6]*c[5]*c[6]^5*c[3]-2700*a[7,6]*c[3]^6*c[5]*c[6]^2-4500*a[7, 6]*c[5]*c[3]^5*c[6]^5+4500*a[7,6]*c[5]*c[3]^6*c[6]^4+4500*a[7,6]*c[5]^ 2*c[3]^5*c[6]^4-4500*a[7,6]*c[5]^2*c[3]^6*c[6]^3+720*a[7,6]*c[5]^2*c[6 ]^4*c[3]-3460*a[7,6]*c[5]^2*c[6]^4*c[3]^2+8*c[6]+20*c[5]+110*c[3]-2710 *c[3]^4+2145*c[3]^5+12600*a[7,6]*c[6]^3*c[5]*b[8]*c[3]^2-3420*a[7,6]*c [6]^2*c[5]^2*b[8]*c[3]^2-2760*a[7,6]*c[5]*c[6]^3*b[8]*c[3]+360*a[7,6]* c[5]^2*c[6]^2*b[8]*c[3]-28320*a[7,6]*c[5]*b[8]*c[3]^3*c[6]^3+30000*a[7 ,6]*c[5]*b[8]*c[3]^4*c[6]^2+13500*a[7,6]*c[5]^2*b[8]*c[3]^3*c[6]^2+584 *a[7,6]*c[6]^3*c[3]^2-1560*a[7,6]*c[5]^2*b[8]*c[3]^4*c[6]+285*a[7,6]*c [3]^4*c[6]^2-180*a[7,6]*c[3]^5*c[6]^2-575*a[7,6]*c[6]^4*c[3]^4+5080*a[ 7,6]*c[3]^4*c[6]^3-2350*a[7,6]*c[3]^3*c[6]^3-28440*a[7,6]*b[8]*c[3]^4* c[6]^3+31500*a[7,6]*b[8]*c[3]^5*c[6]^3-12*c[5]^2+13500*a[7,6]*c[5]*b[8 ]*c[3]^6*c[6]^2-2760*a[7,6]*b[8]*c[6]^5*c[3]+136*a[7,6]*c[5]*c[3]^3*c[ 6]-20*a[7,6]*c[5]*c[6]*c[3]^2+60*a[7,6]*c[5]*c[6]^2*c[3]+2515*a[7,6]*c [5]*c[3]^3*c[6]^2-165*a[7,6]*c[5]^2*c[3]^3*c[6]-608*a[7,6]*c[5]*c[6]^2 *c[3]^2+24*a[7,6]*c[5]^2*c[6]*c[3]^2+780*a[7,6]*b[8]*c[3]^3*c[6]^2-120 *a[7,6]*b[8]*c[6]^2*c[3]^2-12720*a[7,6]*c[6]^5*c[5]*b[8]*c[3]^2+29100* a[7,6]*c[6]^5*c[5]*b[8]*c[3]^3-32400*a[7,6]*c[6]^5*c[5]*b[8]*c[3]^4+12 720*a[7,6]*c[6]^4*c[5]^2*b[8]*c[3]^2-29100*a[7,6]*c[6]^4*c[5]^2*b[8]*c [3]^3+32400*a[7,6]*c[6]^4*c[5]^2*b[8]*c[3]^4+2760*a[7,6]*c[5]*c[6]^5*b [8]*c[3]-2760*a[7,6]*c[5]^2*c[6]^4*b[8]*c[3]+13500*a[7,6]*c[5]*b[8]*c[ 3]^5*c[6]^5-13500*a[7,6]*c[5]*b[8]*c[3]^6*c[6]^4-13500*a[7,6]*c[5]^2*b [8]*c[3]^5*c[6]^4+13500*a[7,6]*c[5]^2*b[8]*c[3]^6*c[6]^3+3375*a[7,6]*c [3]^5*c[6]^5+570*a[7,6]*c[6]^5*c[3]-60*a[7,6]*c[5]^2*c[6]^4+240*a[7,6] *b[8]*c[6]^5+60*a[7,6]*c[5]*c[6]^5-60*a[7,6]*c[6]^3*c[3]-29100*a[7,6]* c[6]^5*b[8]*c[3]^3+32400*a[7,6]*b[8]*c[3]^4*c[6]^5-13500*a[7,6]*b[8]*c [3]^5*c[6]^5+13500*a[7,6]*b[8]*c[3]^6*c[6]^4-240*a[7,6]*c[5]*b[8]*c[6] ^5+240*a[7,6]*c[5]^2*b[8]*c[6]^4-48*a[7,6]*c[6]^5-900*a[7,6]*c[5]*b[8] *c[3]^5*c[6]-285*a[7,6]*c[3]^4*c[5]*c[6]+180*a[7,6]*c[3]^5*c[5]*c[6]-5 850*a[7,6]*c[3]^5*c[6]^3+2700*a[7,6]*c[3]^6*c[6]^3-4380*c[5]^2*c[3]^4- 1125*c[3]^6*c[5]^2+3525*c[3]^5*c[5]^2+1800*c[3]^6*c[5]+930*a[7,6]*c[5] *c[6]^4*c[3]^2-3915*a[7,6]*c[5]*c[6]^4*c[3]^3+8600*a[7,6]*c[5]*c[6]^4* c[3]^4-2510*a[7,6]*c[5]^2*c[6]^3*c[3]^3-1100*a[7,6]*c[5]^2*c[6]^3*c[3] ^4-90*a[7,6]*c[5]*c[6]^4*c[3]-1800*c[3]^6*c[5]*c[6]-2200*a[7,6]*c[5]*c [6]^3*c[3]^2+705*a[7,6]*c[5]^2*c[6]^2*c[3]^2-40*a[7,6]*c[5]*c[6]^3+354 0*a[7,6]*c[6]^2*c[5]*b[8]*c[3]^2-120*a[7,6]*c[6]*c[5]^2*b[8]*c[3]^2-36 0*a[7,6]*c[5]*c[6]^2*b[8]*c[3]-14280*a[7,6]*c[5]*b[8]*c[3]^3*c[6]^2+78 0*a[7,6]*c[5]^2*b[8]*c[3]^3*c[6]+278*c[5]*c[6]*c[3]+4524*c[5]*c[3]^3*c [6]-1562*c[5]*c[6]*c[3]^2+5670*c[3]^5*c[6]*c[5]+4380*c[5]^2*c[3]^4*c[6 ]-2775*c[5]^2*c[3]^3*c[6]-168*c[5]^2*c[6]*c[3]+951*c[5]^2*c[3]^2*c[6]- 675*c[3]^6+12720*a[7,6]*b[8]*c[6]^5*c[3]^2-9825*a[7,6]*c[5]*c[3]^5*c[6 ]^4+6450*a[7,6]*c[5]^2*c[3]^5*c[6]^3+3375*a[7,6]*c[5]^2*c[3]^6*c[6]^2- 480*a[7,6]*c[5]^2*c[6]^3*c[3]+1780*a[7,6]*c[5]^2*c[6]^3*c[3]^2+6075*a[ 7,6]*c[5]*c[3]^5*c[6]^2+900*a[7,6]*b[8]*c[3]^5*c[6]^2+2400*a[7,6]*b[8] *c[6]^4*c[3]-225*a[7,6]*c[3]^5*c[5]^2*c[6]+1560*a[7,6]*c[5]*b[8]*c[3]^ 4*c[6]+1125*c[3]^6*c[5]^2*c[6]-3525*c[5]^2*c[3]^5*c[6]+12*c[5]^2*c[6]- 611*c[3]^2-3540*a[7,6]*c[6]^4*c[5]*b[8]*c[3]^2+14280*a[7,6]*c[6]^4*c[5 ]*b[8]*c[3]^3-30000*a[7,6]*c[6]^4*c[5]*b[8]*c[3]^4-9180*a[7,6]*c[6]^3* c[5]^2*b[8]*c[3]^2+14820*a[7,6]*c[6]^3*c[5]^2*b[8]*c[3]^3-2400*a[7,6]* c[6]^3*c[5]^2*b[8]*c[3]^4-2190*a[7,6]*c[6]^4*c[3]^3+20*a[7,6]*c[6]^2*c [3]^2-136*a[7,6]*c[3]^3*c[6]^2+360*a[7,6]*c[5]*c[6]^4*b[8]*c[3]+2400*a [7,6]*c[5]^2*c[6]^3*b[8]*c[3]+1495*a[7,6]*c[6]^4*c[3]^2+32400*a[7,6]*c [5]*b[8]*c[3]^5*c[6]^4+30840*a[7,6]*c[5]*b[8]*c[3]^4*c[6]^3-12600*a[7, 6]*c[5]*b[8]*c[3]^5*c[6]^3-32400*a[7,6]*c[5]*b[8]*c[3]^5*c[6]^2-18900* a[7,6]*c[5]^2*b[8]*c[3]^5*c[6]^3-28440*a[7,6]*c[5]^2*b[8]*c[3]^4*c[6]^ 2+31500*a[7,6]*c[5]^2*b[8]*c[3]^5*c[6]^2-13500*a[7,6]*c[5]^2*b[8]*c[3] ^6*c[6]^2+900*a[7,6]*c[5]^2*b[8]*c[3]^5*c[6]+4800*a[7,6]*c[3]^5*c[6]^4 -400*a[7,6]*c[6]^4*c[3]+48*a[7,6]*c[5]^2*c[6]^3-240*a[7,6]*b[8]*c[6]^4 -7500*a[7,6]*c[6]^5*c[3]^4+472*a[7,6]*c[5]*c[6]^3*c[3]-72*a[7,6]*c[5]^ 2*c[6]^2*c[3]+5040*a[7,6]*c[5]*c[3]^3*c[6]^3-5430*a[7,6]*c[5]*c[3]^4*c [6]^2-2850*a[7,6]*c[5]^2*c[3]^3*c[6]^2+350*a[7,6]*c[5]^2*c[3]^4*c[6]-1 10*c[6]*c[3]+611*c[6]*c[3]^2-1749*c[3]^3*c[6]-4524*c[3]^3*c[5]+2775*c[ 5]^2*c[3]^3-951*c[5]^2*c[3]^2+1562*c[5]*c[3]^2-278*c[5]*c[3]+168*c[5]^ 2*c[3]-20*c[5]*c[6]+675*c[3]^6*c[6]+1749*c[3]^3+120*a[7,6]*c[6]*c[5]*b [8]*c[3]^2+360*a[7,6]*c[6]^3*b[8]*c[3]+13500*a[7,6]*b[8]*c[3]^3*c[6]^3 -1560*a[7,6]*b[8]*c[3]^4*c[6]^2-3420*a[7,6]*b[8]*c[6]^3*c[3]^2+240*a[7 ,6]*c[5]*b[8]*c[6]^3+14820*a[7,6]*c[6]^4*b[8]*c[3]^3-2400*a[7,6]*b[8]* c[3]^4*c[6]^4-18900*a[7,6]*b[8]*c[3]^5*c[6]^4-13500*a[7,6]*b[8]*c[3]^6 *c[6]^3-240*a[7,6]*c[5]^2*b[8]*c[6]^3-780*a[7,6]*c[5]*b[8]*c[3]^3*c[6] -9180*a[7,6]*b[8]*c[6]^4*c[3]^2-5620*a[7,6]*c[5]*c[3]^4*c[6]^3+2400*a[ 7,6]*c[5]*c[3]^5*c[6]^3+6195*a[7,6]*c[5]^2*c[3]^4*c[6]^2-7200*a[7,6]*c [5]^2*c[3]^5*c[6]^2-3375*a[7,6]*c[3]^6*c[6]^4+6425*a[7,6]*c[6]^5*c[3]^ 3-2710*a[7,6]*c[6]^5*c[3]^2+7090*c[3]^4*c[5]+2710*c[3]^4*c[6]-2145*c[3 ]^5*c[6]-7090*c[5]*c[3]^4*c[6])/b[8]/c[6]/(2*c[6]-10*c[6]*c[3]+15*c[6] *c[3]^2-c[3])/(c[6]-c[3])/(c[6]-c[5])/(c[3]-1)/(-1+c[6])/(c[5]-1)/(3*c [3]-1)/(5*c[3]-2), a[5,2] = 1/2*c[3]*c[5]/c[2]*(-5*c[5]+1+5*c[5]^2)/(1 -5*c[3]+5*c[3]^2), a[3,2] = 1/2*c[3]^2/c[2], a[8,2] = 1/120*(-240*a[7, 2]*c[2]*b[8]-48*c[6]*a[7,2]*c[2]-10*c[3]-272*a[7,2]*c[2]*c[3]+240*a[7, 2]*c[2]*c[5]*b[8]-1560*a[7,2]*c[2]*c[5]*b[8]*c[3]-700*a[7,2]*c[2]*c[5] *c[3]^2-1800*a[7,2]*c[2]*c[5]*b[8]*c[3]^3+450*a[7,2]*c[2]*c[5]*c[3]^3+ 3120*a[7,2]*c[2]*c[5]*b[8]*c[3]^2-360*a[7,2]*c[2]*c[3]^3+60*c[6]*a[7,2 ]*c[2]*c[5]-20*c[5]*c[6]*c[3]+45*c[5]*c[6]*c[3]^2-700*c[6]*a[7,2]*c[2] *c[3]^2+3120*c[6]*a[7,2]*c[2]*b[8]*c[3]^2-1560*c[6]*a[7,2]*c[2]*b[8]*c [3]+910*c[6]*a[7,2]*c[2]*c[5]*c[3]^2+1800*c[6]*a[7,2]*c[2]*c[5]*b[8]*c [3]^3-600*c[6]*a[7,2]*c[2]*c[5]*c[3]^3-3120*a[7,2]*c[2]*b[8]*c[3]^2+33 0*c[6]*a[7,2]*c[2]*c[3]+240*c[6]*a[7,2]*c[2]*b[8]+450*c[6]*a[7,2]*c[2] *c[3]^3-3120*c[6]*a[7,2]*c[2]*c[5]*b[8]*c[3]^2+21*c[3]^2+570*a[7,2]*c[ 2]*c[3]^2+1800*a[7,2]*c[2]*b[8]*c[3]^3-1800*c[6]*a[7,2]*c[2]*b[8]*c[3] ^3+40*a[7,2]*c[2]+1560*a[7,2]*c[2]*b[8]*c[3]+330*a[7,2]*c[2]*c[5]*c[3] +14*c[6]*c[3]-30*c[6]*c[3]^2-30*c[5]*c[3]^2+14*c[5]*c[3]-240*c[6]*a[7, 2]*c[2]*c[5]*b[8]+1560*c[6]*a[7,2]*c[2]*c[5]*b[8]*c[3]-48*a[7,2]*c[2]* c[5]-420*c[6]*a[7,2]*c[2]*c[5]*c[3])/b[8]/c[2]/(c[3]-1)/(3*c[3]-1)/(5* c[3]-2)/(c[5]-1)/(-1+c[6]), a[4,2] = 1/2*c[3]^2*(4-30*c[3]+45*c[3]^2)/ c[2]/(15*c[3]^2-10*c[3]+2)^3, b[6] = 1/60*(1-5*c[3]+5*c[3]^2)*(15*c[5] *c[3]-9*c[3]+4-6*c[5])/c[6]/(-1+c[6])/(2*c[6]-10*c[6]*c[3]+15*c[6]*c[3 ]^2-c[3])/(c[6]-c[3])/(-c[6]+c[5]), b[7] = -1/60*(20-24*c[6]-24*c[5]-1 36*c[3]-210*c[5]*c[6]*c[3]-300*c[5]*c[3]^3*c[6]+455*c[5]*c[6]*c[3]^2-7 80*c[6]*b[8]*c[3]-900*b[8]*c[3]^3*c[6]+1560*b[8]*c[6]*c[3]^2-120*c[5]* b[8]*c[6]+1560*c[5]*b[8]*c[3]^2-780*c[5]*b[8]*c[3]-900*c[5]*b[8]*c[3]^ 3-120*b[8]+285*c[3]^2+165*c[6]*c[3]-350*c[6]*c[3]^2+225*c[3]^3*c[6]+22 5*c[3]^3*c[5]-350*c[5]*c[3]^2+165*c[5]*c[3]+30*c[5]*c[6]+120*b[8]*c[6] +120*b[8]*c[5]-1560*b[8]*c[3]^2+780*b[8]*c[3]+900*b[8]*c[3]^3-180*c[3] ^3+780*c[5]*c[6]*b[8]*c[3]+900*c[5]*b[8]*c[3]^3*c[6]-1560*c[6]*c[5]*b[ 8]*c[3]^2)/(c[3]-1)/(-1+c[6])/(c[5]-1)/(3*c[3]-1)/(5*c[3]-2), b[3] = - 1/60*(75*c[5]*c[6]*c[3]^2-60*c[5]*c[6]*c[3]+10*c[5]*c[6]-45*c[5]*c[3]^ 2+35*c[5]*c[3]-6*c[5]-45*c[6]*c[3]^2+35*c[6]*c[3]-6*c[6]+30*c[3]^2-23* c[3]+4)/(c[6]-c[3])/(c[5]-c[3])/c[3]^2/(c[3]-1)/(15*c[3]^2-10*c[3]+1), a[8,4] = 1/360*(384+59763580*c[3]^5*c[5]-688500*a[7,4]*c[3]^9*c[5]-47 0241675*c[3]^8*c[6]+508180500*c[3]^9*c[6]+932917500*c[3]^10*c[6]*c[5]+ 860490000*c[3]^9*c[6]^2*c[5]+860490000*c[3]^9*c[5]^2*c[6]-864*c[6]+282 487500*c[3]^11*c[6]*c[5]^2-960*c[5]-10800*c[3]+6008104*c[3]^4+3645000* a[7,4]*c[3]^9*b[8]*c[6]-688500*a[7,4]*c[3]^9*c[6]-23157444*c[3]^5+1188 0*a[7,4]*c[3]^5*b[8]-223256250*c[3]^11*c[6]^2*c[5]^2+160116750*c[3]^10 -635191875*c[3]^10*c[6]*c[5]^2-664638750*c[3]^9*c[6]^2*c[5]^2-497712*c [5]*c[6]^2*c[3]^2+38016*c[5]*c[6]^2*c[3]+243000*a[7,4]*c[3]^10*c[5]-66 960*a[7,4]*c[3]^6*b[8]+261225000*c[3]^10*c[6]^2+576*c[5]^2+576*c[6]^2- 56953125*c[3]^12*c[6]^2*c[5]+11670*a[7,4]*c[3]^6+120*a[7,4]*c[3]^4+364 50*a[7,4]*c[3]^8-16200*a[7,4]*c[3]^9-2016*a[7,4]*c[3]^5-1215000*a[7,4] *c[3]^10*b[8]*c[6]-1215000*a[7,4]*c[3]^10*c[5]*b[8]-69558750*c[3]^11-1 94400*a[7,4]*c[3]^8*b[8]+169200*a[7,4]*c[3]^7*b[8]+3645000*a[7,4]*c[3] ^9*c[5]*b[8]-31893750*c[3]^12*c[6]+81000*a[7,4]*c[3]^9*b[8]-65610000*a [7,4]*c[3]^10*c[5]*b[8]*c[6]+66825000*a[7,4]*c[3]^10*c[5]*b[8]*c[6]^2- 5208480*a[7,4]*c[3]^5*c[5]*b[8]*c[6]^2-21985200*a[7,4]*c[3]^6*c[6]^2*c [5]^2*b[8]+57933000*a[7,4]*c[3]^7*c[5]^2*b[8]*c[6]^2+674400*a[7,4]*c[3 ]^7*c[5]^2-114665625*c[3]^11*c[6]^2+95526000*a[7,4]*c[3]^8*c[5]*b[8]*c [6]^2-54847800*a[7,4]*c[3]^7*c[5]*b[8]*c[6]^2+95526000*a[7,4]*c[3]^8*c [5]^2*b[8]*c[6]-54847800*a[7,4]*c[3]^7*c[5]^2*b[8]*c[6]+101979000*a[7, 4]*c[3]^9*c[5]*b[8]*c[6]-91076400*a[7,4]*c[3]^8*c[5]*b[8]*c[6]+1093500 00*a[7,4]*c[3]^9*c[5]^2*b[8]*c[6]^2+808200*a[7,4]*c[3]^8*c[5]+243000*a [7,4]*c[3]^10*c[6]-4449600*a[7,4]*c[3]^8*c[5]*b[8]+16193700*a[7,4]*c[3 ]^8*c[5]*c[6]-20992500*a[7,4]*c[3]^8*c[5]*c[6]^2+5999100*a[7,4]*c[3]^6 *c[5]^2*c[6]^2-1456800*a[7,4]*c[3]^5*c[5]^2*c[6]^2+223320*a[7,4]*c[3]^ 4*c[5]^2*c[6]^2-19440*a[7,4]*c[3]^3*c[5]^2*c[6]^2-12288*a[7,4]*c[3]^3* c[5]*c[6]-840540*a[7,4]*c[3]^5*c[5]*c[6]+134100*a[7,4]*c[3]^4*c[5]*c[6 ]+1440*a[7,4]*c[3]^3*c[6]*b[8]-15795000*a[7,4]*c[3]^10*c[5]^2*c[6]-372 6000*a[7,4]*c[3]^9*b[8]*c[6]^2+354082500*c[3]^7*c[5]-15795000*a[7,4]*c 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59400*a[7,4]*c[3]^7*c[5]*c[6]^2-16243500*a[7,4]*c[3]^7*c[5]^2*c[6]^2-5 8672*c[5]*c[6]*c[3]-6107416*c[5]*c[3]^3*c[6]+765728*c[5]*c[6]*c[3]^2-1 28503420*c[3]^5*c[6]*c[5]-21693360*c[5]^2*c[3]^4*c[6]+3984000*c[5]^2*c [3]^3*c[6]+38016*c[5]^2*c[6]*c[3]-497712*c[5]^2*c[3]^2*c[6]+2880*a[7,4 ]*c[3]^2*c[5]*b[8]*c[6]^2-74880*a[7,4]*c[3]^3*c[5]*c[6]^2*b[8]+193620* a[7,4]*c[3]^6*c[5]+193620*a[7,4]*c[3]^6*c[6]-514950*a[7,4]*c[3]^7*c[6] +73440*a[7,4]*c[3]^3*c[5]*c[6]*b[8]+4956840*a[7,4]*c[3]^5*c[5]*b[8]*c[ 6]-797040*a[7,4]*c[3]^4*c[6]*c[5]*b[8]+3374730*a[7,4]*c[3]^6*c[5]*c[6] +65340370*c[3]^6+66825000*a[7,4]*c[3]^10*c[5]^2*b[8]*c[6]-785011500*c[ 3]^8*c[6]^2*c[5]+509442000*c[3]^7*c[6]^2*c[5]-785011500*c[3]^8*c[5]^2* c[6]+509442000*c[3]^7*c[5]^2*c[6]-241651800*c[3]^6*c[5]^2*c[6]-7655605 50*c[3]^7*c[5]*c[6]+600007500*c[5]^2*c[3]^8*c[6]^2-385554000*c[5]^2*c[ 3]^7*c[6]^2+181188000*c[5]^2*c[3]^6*c[6]^2+84559200*c[5]^2*c[3]^5*c[6] -303750*a[7,4]*c[3]^10*c[6]^2-56953125*c[3]^12*c[5]^2*c[6]-114665625*c 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*a[7,4]*c[3]^4*c[5]*c[6]^2+15120*a[7,4]*c[3]^3*c[5]*c[6]^2-858240*a[7, 4]*c[3]^4*c[6]^2*c[5]^2*b[8]+5472000*a[7,4]*c[3]^5*c[5]^2*b[8]*c[6]^2+ 20799000*a[7,4]*c[3]^6*c[5]*b[8]*c[6]^2+20799000*a[7,4]*c[3]^6*c[5]^2* b[8]*c[6]+51931800*a[7,4]*c[3]^7*c[5]*b[8]*c[6]+76320*a[7,4]*c[3]^3*c[ 5]^2*c[6]^2*b[8]-74880*a[7,4]*c[3]^3*c[5]^2*c[6]*b[8]-5208480*a[7,4]*c [3]^5*c[5]^2*b[8]*c[6]+827280*a[7,4]*c[3]^4*c[6]*c[5]^2*b[8]-1048500*a [7,4]*c[3]^8*c[5]^2-410973750*c[3]^11*c[6]*c[5]-100170000*a[7,4]*c[3]^ 8*c[5]^2*b[8]*c[6]^2-105624000*a[7,4]*c[3]^9*c[5]*b[8]*c[6]^2-3726000* a[7,4]*c[3]^9*c[5]^2*b[8]-576*a[7,4]*c[3]^2*c[5]*c[6]^2-3085200*a[7,4] *c[3]^7*c[5]^2*b[8]+4644000*a[7,4]*c[3]^8*b[8]*c[6]^2-3085200*a[7,4]*c [3]^7*b[8]*c[6]^2-4449600*a[7,4]*c[3]^8*b[8]*c[6]+4644000*a[7,4]*c[3]^ 8*c[5]^2*b[8]-15416180*c[3]^4*c[5]-13569188*c[3]^4*c[6]+52412260*c[3]^ 5*c[6]+33120200*c[5]*c[3]^4*c[6])/b[8]/c[3]^2/(c[5]-1)/(c[3]-1)/(3*c[3 ]-1)/(5*c[3]-2)/(15*c[3]^2-10*c[3]+1)/(15*c[5]*c[3]^2-10*c[5]*c[3]+2*c [5]-c[3])/(-1+c[6])/(2*c[6]-10*c[6]*c[3]+15*c[6]*c[3]^2-c[3]), a[8,3] \+ = -1/360*(96+471770*c[3]^5*c[5]-1215000*c[6]^2*a[7,2]*c[2]*b[8]*c[3]^9 *c[5]-1593000*c[6]*a[7,2]*c[2]*b[8]*c[3]^8*c[5]-3085200*c[6]^2*a[7,2]* c[2]*b[8]*c[3]^6-15480*c[3]^3*b[8]*c[5]^2+173400*a[7,6]*c[3]^6*c[5]*c[ 6]^2-1363500*a[7,6]*c[5]^2*c[3]^6*c[6]^3+148500*a[7,2]*c[2]*c[5]*c[3]^ 8*c[6]+2476800*c[6]*a[7,2]*c[2]*b[8]*c[3]^7*c[5]+1899000*c[6]*a[7,2]*c [2]*b[8]*c[3]^6+5940*c[6]*a[7,2]*c[2]*c[5]^2*c[3]^2-288*c[6]*a[7,2]*c[ 2]*c[5]^2*c[3]-850500*c[6]^2*a[7,2]*c[2]*c[3]^8*c[5]+20250*a[7,4]*c[3] ^9*c[5]-116280*c[3]^4*b[8]*c[5]-133650*c[3]^8*c[6]+303750*c[3]^9*c[6]+ 202500*c[3]^10*c[6]*c[5]-202500*c[3]^9*c[6]^2*c[5]-202500*c[3]^9*c[5]^ 2*c[6]-216*c[6]-240*c[5]-1680*c[3]+129508*c[3]^4+1899000*a[7,6]*c[5]*b [8]*c[3]^6*c[6]-81000*a[7,4]*c[3]^9*b[8]*c[6]+240*a[7,6]*c[3]^3*c[6]+2 0250*a[7,4]*c[3]^9*c[6]+303750*c[3]^10*a[7,6]*c[6]^2-205758*c[3]^5-243 000*c[3]^10*a[7,2]*c[2]-46860*c[6]*a[7,2]*c[2]*c[5]^2*c[3]^3+232560*c[ 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[3]^2-288*a[7,5]*c[5]^3*c[6]*c[3]+5940*a[7,5]*c[5]^3*c[3]^2*c[6]+18015 0*a[7,5]*c[5]^3*c[3]^6*c[6]-352200*a[7,5]*c[5]^3*c[3]^5*c[6]+232560*a[ 7,5]*c[5]^2*b[8]*c[3]^4-240*a[7,5]*c[5]*c[6]*c[3]^2+4944*a[7,5]*c[5]*c [3]^3*c[6]-593100*c[3]^6*b[8]-3085200*a[7,6]*b[8]*c[3]^6*c[6]^3+154260 0*c[3]^7*b[8]-229500*c[3]^8*c[5]+2511000*c[3]^9*a[7,6]*c[5]*b[8]*c[6]- 519900*a[7,6]*c[3]^7*c[5]*c[6]^2+1154250*a[7,6]*c[5]^2*c[3]^7*c[6]^3+4 88160*c[5]*c[3]^5*b[8]*c[6]-116280*c[5]*c[3]^3*b[8]*c[6]^2-352200*c[6] *a[7,2]*c[2]*c[5]^2*c[3]^5+183090*c[6]*a[7,2]*c[2]*c[5]^2*c[3]^4-91800 0*c[6]*a[7,2]*c[2]*c[5]^2*b[8]*c[3]^7-1558800*c[6]*a[7,2]*c[2]*c[5]^2* b[8]*c[3]^6+1899000*c[6]*a[7,2]*c[2]*c[5]^2*b[8]*c[3]^5-15480*c[3]^4*b [8]+459000*c[3]^7*b[8]*c[5]^2*c[6]-593100*c[3]^5*b[8]*c[5]^2-1296000*a [7,2]*c[2]*b[8]*c[3]^9*c[5]*c[6]-918000*a[7,2]*c[2]*b[8]*c[3]^8*c[5]-6 34500*c[3]^9*a[7,2]*c[2]*c[6]-607500*c[3]^10*b[8]+11880*a[7,4]*c[3]^4* c[5]^2*b[8]-720*a[7,4]*c[3]^3*c[5]^2*b[8]-66960*a[7,4]*c[3]^5*c[5]^2*b [8]+169200*a[7,4]*c[3]^6*b[8]*c[6]^2+25200*a[7,4]*c[3]^7*b[8]*c[6]-295 20*a[7,6]*c[5]*b[8]*c[3]^3*c[6]+877500*a[7,2]*c[2]*c[5]^2*c[3]^8+77940 0*c[3]^6*b[8]*c[5]^2*c[6]-607500*c[3]^9*b[8]*c[6]^2+131760*c[3]^5*b[8] -949500*c[3]^5*b[8]*c[5]^2*c[6]+607500*c[3]^10*b[8]*c[5]+183090*a[7,6] *c[5]*c[3]^4*c[6]^3-352200*a[7,6]*c[5]*c[3]^5*c[6]^3+1863000*c[3]^9*b[ 8]+183090*a[7,6]*c[5]^2*c[3]^4*c[6]^2-352200*a[7,6]*c[5]^2*c[3]^5*c[6] ^2+750*a[7,4]*c[3]^7*c[5]-37200*a[7,4]*c[3]^6*c[6]^2+14160*a[7,4]*c[3] ^5*c[6]^2-2430*a[7,4]*c[3]^4*c[6]^2-37200*a[7,4]*c[3]^6*c[5]^2-20250*a [7,4]*c[3]^8*c[6]^2+151875*c[3]^9*c[5]^2-151875*c[3]^10*c[5]+151875*c[ 3]^9*c[6]^2-922680*c[6]*a[7,2]*c[2]*c[5]^2*b[8]*c[3]^4+486000*c[6]*a[7 ,2]*c[2]*c[5]^2*c[3]^7-1558800*a[7,2]*c[2]*b[8]*c[3]^7*c[5]+55080*a[7, 4]*c[3]^4*c[6]^2*c[5]*b[8]-720*a[7,4]*c[3]^2*c[5]^2*b[8]*c[6]^2+169200 *a[7,4]*c[3]^6*c[5]^2*b[8]+25200*a[7,4]*c[3]^7*c[5]*b[8]-11100*a[7,4]* c[3]^4*c[5]*c[6]^2+2250*a[7,4]*c[3]^3*c[5]*c[6]^2-2322000*c[3]^8*b[8]+ 14760*c[6]*c[3]^2*b[8]*c[5]^2-66960*a[7,4]*c[3]^4*c[6]^2*c[5]^2*b[8]+1 69200*a[7,4]*c[3]^5*c[5]^2*b[8]*c[6]^2+25200*a[7,4]*c[3]^6*c[5]*b[8]*c [6]^2+25200*a[7,4]*c[3]^6*c[5]^2*b[8]*c[6]-138600*a[7,4]*c[3]^7*c[5]*b [8]*c[6]+11880*a[7,4]*c[3]^3*c[5]^2*c[6]^2*b[8]-11160*a[7,4]*c[3]^3*c[ 5]^2*c[6]*b[8]-102240*a[7,4]*c[3]^5*c[5]^2*b[8]*c[6]+55080*a[7,4]*c[3] ^4*c[6]*c[5]^2*b[8]-20250*a[7,4]*c[3]^8*c[5]^2-29520*c[6]^2*a[7,2]*c[2 ]*c[5]*b[8]*c[3]^2+877500*c[6]^2*a[7,2]*c[2]*c[3]^8+2476800*a[7,6]*c[5 ]*b[8]*c[3]^7*c[6]^2+1215000*c[3]^10*a[7,6]*b[8]*c[6]-144*a[7,4]*c[3]^ 2*c[5]*c[6]^2-194400*a[7,4]*c[3]^7*c[5]^2*b[8]+81000*a[7,4]*c[3]^8*b[8 ]*c[6]^2-194400*a[7,4]*c[3]^7*b[8]*c[6]^2+113400*a[7,4]*c[3]^8*b[8]*c[ 6]+81000*a[7,4]*c[3]^8*c[5]^2*b[8]-299700*c[3]^6*a[7,6]*c[6]^2+178800* c[3]^7*a[7,6]*c[6]^2+877500*c[3]^8*a[7,6]*c[6]^3+461340*c[3]^4*b[8]*c[ 5]^2*c[6]-1255500*c[3]^9*b[8]*c[6]-850500*a[7,2]*c[2]*c[5]^2*c[3]^8*c[ 6]+405000*a[7,2]*c[2]*c[5]^2*c[3]^9*c[6]+240*c[6]*a[7,2]*c[2]*c[5]*c[3 ]-330540*c[3]^4*c[5]-255474*c[3]^4*c[6]+311120*c[3]^5*c[6]+724810*c[5] *c[3]^4*c[6]-3726000*a[7,6]*c[5]^2*b[8]*c[3]^7*c[6]^3+2511000*c[3]^9*a [7,5]*c[5]*b[8]*c[6]-634500*c[3]^9*a[7,5]*c[5]*c[6]-1558800*c[3]^7*a[7 ,6]*c[5]*b[8]*c[6]-299700*c[3]^6*a[7,6]*c[5]*c[6]+178800*c[3]^7*a[7,6] *c[5]*c[6]-1048500*c[3]^7*a[7,6]*c[6]^3)/b[8]/c[3]^2/(5275*c[3]^5*c[5] +6975*c[3]^8*c[6]-3375*c[3]^9*c[6]-732*c[3]^4+3295*c[3]^5+646*c[5]*c[6 ]^2*c[3]^2-82*c[5]*c[6]^2*c[3]-2550*c[3]^7*c[5]+3295*c[6]^2*c[3]^4-732 *c[6]^2*c[3]^3+86*c[6]^2*c[3]^2+3295*c[5]^2*c[3]^4+4*c[5]*c[6]^2+12900 *c[3]^6*c[6]^2-10350*c[3]^7*c[5]^2+12900*c[3]^6*c[5]^2-8570*c[3]^5*c[5 ]^2-4330*c[3]^6*c[5]+3375*c[3]^8*c[5]^2-3375*c[3]^9*c[5]+3375*c[3]^8*c [6]^2+6880*c[3]^6*c[5]*c[6]-2550*c[5]*c[3]^6*c[6]^2-8570*c[5]^2*c[6]^2 *c[3]^4+3295*c[5]^2*c[6]^2*c[3]^3-732*c[5]^2*c[6]^2*c[3]^2+86*c[5]^2*c [6]^2*c[3]+3375*c[3]^9*c[6]*c[5]+78*c[5]*c[6]*c[3]+1917*c[5]*c[3]^3*c[ 6]-564*c[5]*c[6]*c[3]^2-945*c[3]^5*c[6]*c[5]+5275*c[5]^2*c[3]^4*c[6]-2 563*c[5]^2*c[3]^3*c[6]-82*c[5]^2*c[6]*c[3]+646*c[5]^2*c[3]^2*c[6]-8570 *c[3]^6-3375*c[3]^8*c[6]^2*c[5]+6975*c[3]^7*c[6]^2*c[5]-3375*c[3]^8*c[ 5]^2*c[6]+6975*c[3]^7*c[5]^2*c[6]-2550*c[3]^6*c[5]^2*c[6]-4425*c[3]^7* c[5]*c[6]+3375*c[5]^2*c[3]^7*c[6]^2-10350*c[5]^2*c[3]^6*c[6]^2-4330*c[ 5]^2*c[3]^5*c[6]+4*c[5]^2*c[6]-4*c[3]^2+12900*c[3]^7-4*c[6]^2*c[3]-4*c [5]^2*c[6]^2-10350*c[3]^7*c[6]^2+4*c[6]*c[3]-82*c[6]*c[3]^2+646*c[3]^3 *c[6]+646*c[3]^3*c[5]-732*c[5]^2*c[3]^3+86*c[5]^2*c[3]^2-82*c[5]*c[3]^ 2+4*c[5]*c[3]-4*c[5]^2*c[3]-4*c[5]*c[6]-8570*c[3]^5*c[6]^2-4330*c[3]^6 *c[6]-2563*c[5]*c[6]^2*c[3]^3+5275*c[5]*c[3]^4*c[6]^2-4330*c[5]*c[3]^5 *c[6]^2+86*c[3]^3-3600*c[3]^8*c[6]*c[5]+12900*c[5]^2*c[6]^2*c[3]^5-255 0*c[3]^7*c[6]-10350*c[3]^8+3375*c[3]^9+6975*c[3]^8*c[5]-2563*c[3]^4*c[ 5]-2563*c[3]^4*c[6]+5275*c[3]^5*c[6]-2712*c[5]*c[3]^4*c[6]), c[8] = 1, a[4,3] = -(15*c[3]^2-10*c[3]+1)*c[3]/(15*c[3]^2-10*c[3]+2)^3, b[5] = \+ -1/60*(1-5*c[3]+5*c[3]^2)*(15*c[6]*c[3]-9*c[3]+4-6*c[6])/(-c[6]+c[5])/ c[5]/(c[5]-1)/(15*c[5]*c[3]^2-10*c[5]*c[3]+2*c[5]-c[3])/(c[5]-c[3]), a [8,7] = 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 294 6 "Step 7" }{TEXT -1 1 " " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 24 "The row-sum condi tions: " }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "Sum(a[i,j] ,j=1..i-1)=c[i]" "6#/-%$SumG6$&%\"aG6$%\"iG%\"jG/F+;\"\"\",&F*F.F.!\" \"&%\"cG6#F*" }{TEXT -1 8 ", for " }{XPPEDIT 18 0 "i=2" "6#/%\"iG\" \"#" }{TEXT -1 8 " . . 8, " }}{PARA 0 "" 0 "" {TEXT -1 42 "can be used to find numerical values for " }{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$\"\"%\"\" \"" }{TEXT -1 3 ", " }{XPPEDIT 18 0 "a[5,1]" "6#&%\"aG6$\"\"&\"\"\"" }{TEXT -1 7 " and " }{XPPEDIT 18 0 "a[6,1]" "6#&%\"aG6$\"\"'\"\"\"" }{TEXT -1 28 " give an expressions for " }{XPPEDIT 18 0 "a[7,1];" " 6#&%\"aG6$\"\"(\"\"\"" }{TEXT -1 7 " and " }{XPPEDIT 18 0 "a[8,1]" " 6#&%\"aG6$\"\")\"\"\"" }{TEXT -1 15 " in terms of " }{XPPEDIT 18 0 " a[8,2]" "6#&%\"aG6$\"\")\"\"#" }{TEXT -1 3 ", " }{XPPEDIT 18 0 "a[8,4 ]" "6#&%\"aG6$\"\")\"\"%" }{TEXT -1 3 ", " }{XPPEDIT 18 0 "a[8,5]" "6 #&%\"aG6$\"\")\"\"&" }{TEXT -1 7 " and " }{XPPEDIT 18 0 "a[8,6]" "6# &%\"aG6$\"\")\"\"'" }{TEXT -1 1 "." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 186 "cdns5 := [seq(add(a[i,j],j= 1..i-1)=c[i],i=2..8)]:\neqns5 := simplify(subs(e13,cdns5)):\ne14 := fa ctor(solve(\{op(eqns5)\},\{seq(a[i,1],i=2..8)\})):\ne15 := `union`(sim plify(subs(e14,e13)),e14):" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 3 "e15" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 66988 "e15 := \{a[6,2] = 1/2*c[6]*c[3]/c[2]*(5*c[6]^2-5*c [6]+1)/(1-5*c[3]+5*c[3]^2), b[2] = 0, b[4] = 1/60*(15*c[3]^2-10*c[3]+2 )^5*(3*c[3]-5*c[5]*c[3]-5*c[6]*c[3]+10*c[5]*c[6]*c[3]-5*c[5]*c[6]-2+3* c[6]+3*c[5])/(15*c[3]^2-10*c[3]+1)/c[3]^2/(3*c[3]-1)/(5*c[3]-2)/(2*c[6 ]-10*c[6]*c[3]+15*c[6]*c[3]^2-c[3])/(15*c[5]*c[3]^2-10*c[5]*c[3]+2*c[5 ]-c[3]), c[7] = 1, a[2,1] = c[2], c[4] = c[3]/(15*c[3]^2-10*c[3]+2), a [8,5] = 1/60*(8+2835*c[3]^5*c[5]-18*c[6]-10*c[5]-110*c[3]+2710*c[3]^4- 2145*c[3]^5-780*a[7,5]*c[5]*b[8]*c[3]^3*c[6]^2+120*a[7,5]*c[5]*b[8]*c[ 6]^2*c[3]^2+240*a[7,5]*c[5]^5*b[8]*c[6]+2760*a[7,5]*c[5]^5*b[8]*c[3]-2 4*a[7,5]*c[5]*c[6]^2*c[3]^2+165*a[7,5]*c[5]*c[3]^3*c[6]^2-951*c[5]*c[6 ]^2*c[3]^2+168*c[5]*c[6]^2*c[3]+48*a[7,5]*c[5]^5+2*c[5]^2+12*c[6]^2+83 00*a[7,5]*c[5]^5*c[3]^3*c[6]+240*a[7,5]*c[5]^3*b[8]*c[6]^2-13500*a[7,5 ]*c[5]^4*b[8]*c[3]^6-3375*a[7,5]*c[5]^2*c[3]^6*c[6]^2+72*a[7,5]*c[5]^2 *c[6]^2*c[3]+2850*a[7,5]*c[5]^2*c[3]^3*c[6]^2-705*a[7,5]*c[5]^2*c[6]^2 *c[3]^2+7200*a[7,5]*c[5]^2*c[3]^5*c[6]^2+1100*a[7,5]*c[5]^3*c[3]^4*c[6 ]^2+2510*a[7,5]*c[5]^3*c[3]^3*c[6]^2+3460*a[7,5]*c[5]^4*c[3]^2*c[6]^2- 3460*a[7,5]*c[5]^5*c[3]^2*c[6]-4500*a[7,5]*c[5]^4*c[3]^5*c[6]^2+4500*a [7,5]*c[5]^5*c[3]^5*c[6]-720*a[7,5]*c[5]^4*c[6]^2*c[3]+720*a[7,5]*c[5] ^5*c[6]*c[3]-12720*a[7,5]*c[5]^5*b[8]*c[3]^2+13500*a[7,5]*c[5]^4*b[8]* c[3]^5*c[6]^2-13500*a[7,5]*c[5]^5*b[8]*c[3]^5*c[6]+29100*a[7,5]*c[5]^4 *c[6]^2*b[8]*c[3]^3-29100*a[7,5]*c[5]^5*c[6]*b[8]*c[3]^3-12720*a[7,5]* c[5]^4*c[6]^2*b[8]*c[3]^2+12720*a[7,5]*c[5]^5*c[6]*b[8]*c[3]^2-32400*a [7,5]*c[5]^4*b[8]*c[3]^4*c[6]^2+32400*a[7,5]*c[5]^5*b[8]*c[3]^4*c[6]-1 3500*a[7,5]*c[5]^3*b[8]*c[3]^6*c[6]^2+13500*a[7,5]*c[5]^4*b[8]*c[3]^6* c[6]+18900*a[7,5]*c[5]^3*b[8]*c[3]^5*c[6]^2+13500*a[7,5]*c[5]^2*b[8]*c [3]^6*c[6]^2-900*a[7,5]*c[5]*b[8]*c[3]^5*c[6]^2+4380*c[6]^2*c[3]^4-277 5*c[6]^2*c[3]^3+951*c[6]^2*c[3]^2+835*c[5]^2*c[3]^4-12*c[5]*c[6]^2+112 5*c[3]^6*c[6]^2+225*c[3]^6*c[5]^2-690*c[3]^5*c[5]^2-900*c[3]^6*c[5]+15 75*c[3]^6*c[5]*c[6]-1125*c[5]*c[3]^6*c[6]^2-249*c[5]*c[6]*c[3]-4011*c[ 5]*c[3]^3*c[6]+1392*c[5]*c[6]*c[3]^2-4980*c[3]^5*c[6]*c[5]+675*c[3]^6+ 2400*a[7,5]*c[5]^3*b[8]*c[3]^4*c[6]^2-31500*a[7,5]*c[5]^2*b[8]*c[3]^5* c[6]^2-2400*a[7,5]*c[5]^3*c[6]^2*b[8]*c[3]-14820*a[7,5]*c[5]^3*b[8]*c[ 3]^3*c[6]^2+9180*a[7,5]*c[5]^3*c[6]^2*b[8]*c[3]^2-32400*a[7,5]*c[5]^5* b[8]*c[3]^4+13500*a[7,5]*c[5]^5*b[8]*c[3]^5+29100*a[7,5]*c[5]^5*b[8]*c [3]^3-360*a[7,5]*c[5]^2*c[6]^2*b[8]*c[3]-13500*a[7,5]*c[5]^2*b[8]*c[3] ^3*c[6]^2+3420*a[7,5]*c[5]^2*c[6]^2*b[8]*c[3]^2+2760*a[7,5]*c[5]^4*c[6 ]^2*b[8]*c[3]-2760*a[7,5]*c[5]^5*c[6]*b[8]*c[3]+28440*a[7,5]*c[5]^2*b[ 8]*c[3]^4*c[6]^2+1560*a[7,5]*c[5]*b[8]*c[3]^4*c[6]^2+611*c[3]^2-168*c[ 6]^2*c[3]+9825*a[7,5]*c[5]^4*c[3]^4*c[6]^2-9825*a[7,5]*c[5]^5*c[3]^4*c [6]-8300*a[7,5]*c[5]^4*c[3]^3*c[6]^2-350*a[7,5]*c[5]*c[3]^4*c[6]^2+225 *a[7,5]*c[5]*c[3]^5*c[6]^2-6195*a[7,5]*c[5]^2*c[3]^4*c[6]^2+249*c[6]*c [3]-1392*c[6]*c[3]^2+4011*c[3]^3*c[6]+2262*c[3]^3*c[5]-513*c[5]^2*c[3] ^3+170*c[5]^2*c[3]^2-781*c[5]*c[3]^2+139*c[5]*c[3]-29*c[5]^2*c[3]+18*c [5]*c[6]+480*a[7,5]*c[5]^3*c[6]^2*c[3]-1780*a[7,5]*c[5]^3*c[3]^2*c[6]^ 2+4500*a[7,5]*c[5]^3*c[3]^6*c[6]^2-4500*a[7,5]*c[5]^4*c[3]^6*c[6]-6450 *a[7,5]*c[5]^3*c[3]^5*c[6]^2-240*a[7,5]*c[5]^4*b[8]*c[6]^2-3525*c[3]^5 *c[6]^2-1575*c[3]^6*c[6]-6425*a[7,5]*c[5]^5*c[3]^3+2710*a[7,5]*c[5]^5* c[3]^2-570*a[7,5]*c[5]^5*c[3]+60*a[7,5]*c[5]^4*c[6]^2-60*a[7,5]*c[5]^5 *c[6]+3375*a[7,5]*c[5]^4*c[3]^6-48*a[7,5]*c[5]^3*c[6]^2+2775*c[5]*c[6] ^2*c[3]^3-4380*c[5]*c[3]^4*c[6]^2+3525*c[5]*c[3]^5*c[6]^2-1749*c[3]^3- 40*a[7,5]*c[5]^4-8600*a[7,5]*c[5]^4*c[3]^4*c[6]+3915*a[7,5]*c[5]^4*c[3 ]^3*c[6]-240*a[7,5]*c[5]^3*b[8]*c[6]+3420*a[7,5]*c[5]^3*b[8]*c[3]^2-36 0*a[7,5]*c[5]^3*b[8]*c[3]-13500*a[7,5]*c[5]^3*b[8]*c[3]^3+28440*a[7,5] *c[5]^3*b[8]*c[3]^4-900*a[7,5]*c[5]^2*b[8]*c[3]^5+13500*a[7,5]*c[5]^3* b[8]*c[3]^6+2700*a[7,5]*c[5]^2*c[3]^6*c[6]-60*a[7,5]*c[5]^2*c[6]*c[3]- 2515*a[7,5]*c[5]^2*c[3]^3*c[6]+608*a[7,5]*c[5]^2*c[6]*c[3]^2+120*a[7,5 ]*c[5]^2*b[8]*c[3]^2-780*a[7,5]*c[5]^2*b[8]*c[3]^3-6075*a[7,5]*c[5]^2* c[3]^5*c[6]+5620*a[7,5]*c[5]^3*c[3]^4*c[6]-5040*a[7,5]*c[5]^3*c[3]^3*c [6]-32400*a[7,5]*c[5]^4*b[8]*c[3]^5*c[6]-14280*a[7,5]*c[5]^4*c[6]*b[8] *c[3]^3+3540*a[7,5]*c[5]^4*c[6]*b[8]*c[3]^2+30000*a[7,5]*c[5]^4*b[8]*c [3]^4*c[6]+12600*a[7,5]*c[5]^3*b[8]*c[3]^5*c[6]-13500*a[7,5]*c[5]^2*b[ 8]*c[3]^6*c[6]+5850*a[7,5]*c[5]^3*c[3]^5+2190*a[7,5]*c[5]^4*c[3]^3-149 5*a[7,5]*c[5]^4*c[3]^2+400*a[7,5]*c[5]^4*c[3]+900*a[7,5]*c[5]*b[8]*c[3 ]^5*c[6]-30840*a[7,5]*c[5]^3*b[8]*c[3]^4*c[6]+32400*a[7,5]*c[5]^2*b[8] *c[3]^5*c[6]+2760*a[7,5]*c[5]^3*c[6]*b[8]*c[3]+28320*a[7,5]*c[5]^3*b[8 ]*c[3]^3*c[6]-12600*a[7,5]*c[5]^3*c[6]*b[8]*c[3]^2-930*a[7,5]*c[5]^4*c [3]^2*c[6]+9825*a[7,5]*c[5]^4*c[3]^5*c[6]+90*a[7,5]*c[5]^4*c[6]*c[3]+9 180*a[7,5]*c[5]^4*b[8]*c[3]^2-31500*a[7,5]*c[5]^3*b[8]*c[3]^5+180*a[7, 5]*c[5]^2*c[3]^5-5080*a[7,5]*c[5]^3*c[3]^4-2700*a[7,5]*c[5]^3*c[3]^6+3 60*a[7,5]*c[5]^2*c[6]*b[8]*c[3]+14280*a[7,5]*c[5]^2*b[8]*c[3]^3*c[6]-3 540*a[7,5]*c[5]^2*c[6]*b[8]*c[3]^2-360*a[7,5]*c[5]^4*c[6]*b[8]*c[3]-30 000*a[7,5]*c[5]^2*b[8]*c[3]^4*c[6]-1560*a[7,5]*c[5]*b[8]*c[3]^4*c[6]+2 400*a[7,5]*c[5]^4*b[8]*c[3]^4+18900*a[7,5]*c[5]^4*b[8]*c[3]^5-14820*a[ 7,5]*c[5]^4*b[8]*c[3]^3+285*a[7,5]*c[5]*c[3]^4*c[6]-180*a[7,5]*c[5]*c[ 3]^5*c[6]+5430*a[7,5]*c[5]^2*c[3]^4*c[6]+40*a[7,5]*c[5]^3*c[6]+136*a[7 ,5]*c[5]^2*c[3]^3+2350*a[7,5]*c[5]^3*c[3]^3-584*a[7,5]*c[5]^3*c[3]^2-2 0*a[7,5]*c[5]^2*c[3]^2+60*a[7,5]*c[5]^3*c[3]+240*a[7,5]*c[5]^4*b[8]+57 5*a[7,5]*c[5]^4*c[3]^4-4800*a[7,5]*c[5]^4*c[3]^5-285*a[7,5]*c[5]^2*c[3 ]^4+780*a[7,5]*c[5]*b[8]*c[3]^3*c[6]-120*a[7,5]*c[5]*b[8]*c[6]*c[3]^2- 472*a[7,5]*c[5]^3*c[6]*c[3]+2200*a[7,5]*c[5]^3*c[3]^2*c[6]-2400*a[7,5] *c[5]^3*c[3]^5*c[6]+1560*a[7,5]*c[5]^2*b[8]*c[3]^4-2400*a[7,5]*c[5]^4* b[8]*c[3]+20*a[7,5]*c[5]*c[6]*c[3]^2-136*a[7,5]*c[5]*c[3]^3*c[6]-240*a [7,5]*c[5]^5*b[8]+7500*a[7,5]*c[5]^5*c[3]^4-3375*a[7,5]*c[5]^5*c[3]^5- 3545*c[3]^4*c[5]-6255*c[3]^4*c[6]+4980*c[3]^5*c[6]+6255*c[5]*c[3]^4*c[ 6])/b[8]/c[5]/(c[5]-1)/(c[3]-1)/(3*c[3]-1)/(5*c[3]-2)/(15*c[5]*c[3]^2- 10*c[5]*c[3]+2*c[5]-c[3])/(c[5]-c[3])/(-1+c[6])/(c[6]-c[5]), a[7,1] = \+ 1/2*(-2-4*a[7,5]*c[3]+4*a[7,6]*c[6]-4*a[7,6]*c[3]-30*a[7,2]*c[3]^3+20* a[7,4]*c[3]^2+14*c[3]-20*a[7,6]*c[6]*c[3]+30*a[7,6]*c[6]*c[3]^2-20*a[7 ,2]*c[2]*c[3]+20*a[7,2]*c[3]^2+4*a[7,5]*c[5]+20*a[7,5]*c[3]^2-30*a[7,4 ]*c[3]^3-20*a[7,5]*c[5]*c[3]-35*c[3]^2+30*a[7,2]*c[2]*c[3]^2+20*a[7,6] *c[3]^2+4*a[7,2]*c[2]-30*a[7,6]*c[3]^3-4*a[7,2]*c[3]+30*c[3]^3-2*a[7,4 ]*c[3]+30*a[7,5]*c[5]*c[3]^2-30*a[7,5]*c[3]^3)/c[3]/(15*c[3]^2-10*c[3] +2), b[1] = -1/60*1/c[3]^2*(150*c[5]*c[3]^3*c[6]-75*c[3]^3*c[6]+45*c[3 ]^3-75*c[3]^3*c[5]+105*c[5]*c[3]^2-205*c[5]*c[6]*c[3]^2+105*c[6]*c[3]^ 2-65*c[3]^2+29*c[3]-45*c[5]*c[3]+80*c[5]*c[6]*c[3]-45*c[6]*c[3]+6*c[5] -10*c[5]*c[6]-4+6*c[6])/c[5]/c[6], a[7,3] = -1/2*(30*a[7,2]*c[2]*c[3]^ 2-20*a[7,2]*c[2]*c[3]+4*a[7,2]*c[2]+2*a[7,4]*c[3]+30*a[7,5]*c[5]*c[3]^ 2-20*a[7,5]*c[5]*c[3]+4*a[7,5]*c[5]+30*a[7,6]*c[6]*c[3]^2-20*a[7,6]*c[ 6]*c[3]+4*a[7,6]*c[6]-15*c[3]^2+10*c[3]-2)/c[3]/(15*c[3]^2-10*c[3]+2), a[5,4] = -1/6*(15*c[3]^2-10*c[3]+2)^2*c[5]*(c[5]-c[3])*(15*c[5]*c[3]^ 2-10*c[5]*c[3]+2*c[5]-c[3])/c[3]^2/(1-5*c[3]+5*c[3]^2)/(15*c[3]^2-10*c [3]+1), a[6,4] = -1/6*c[6]*(15*c[3]^2-10*c[3]+2)^2*(2*c[6]-10*c[6]*c[3 ]+15*c[6]*c[3]^2-c[3])*(c[6]-c[3])*(90*c[3]^3*c[6]-120*c[6]*c[3]^2+48* c[6]*c[3]-6*c[6]-225*c[3]^3*c[5]+225*c[5]^2*c[3]^3-240*c[5]^2*c[3]^2+9 *c[3]^2+255*c[5]*c[3]^2-100*c[5]*c[3]-4*c[3]+90*c[5]^2*c[3]+14*c[5]-12 *c[5]^2)/c[3]^2/(15*c[3]^2-10*c[3]+1)/(1-5*c[3]+5*c[3]^2)/(15*c[5]*c[3 ]^2-10*c[5]*c[3]+2*c[5]-c[3])/(15*c[5]*c[3]-9*c[3]+4-6*c[5]), a[5,3] = -1/6*c[5]*(150*c[3]^3*c[5]-145*c[5]*c[3]^2+40*c[5]*c[3]-4*c[5]-30*c[3 ]^3+20*c[3]^2-c[3])*(c[5]-c[3])/(1-5*c[3]+5*c[3]^2)/c[3]^2/(15*c[3]^2- 10*c[3]+1), a[8,6] = 1/60*(-8+40*a[7,6]*c[6]^4-5670*c[3]^5*c[5]+3460*a [7,6]*c[5]*c[6]^5*c[3]^2-8300*a[7,6]*c[5]*c[6]^5*c[3]^3+9825*a[7,6]*c[ 5]*c[6]^5*c[3]^4+8300*a[7,6]*c[5]^2*c[6]^4*c[3]^3-9825*a[7,6]*c[5]^2*c [6]^4*c[3]^4-720*a[7,6]*c[5]*c[6]^5*c[3]-2700*a[7,6]*c[3]^6*c[5]*c[6]^ 2-4500*a[7,6]*c[5]*c[3]^5*c[6]^5+4500*a[7,6]*c[5]*c[3]^6*c[6]^4+4500*a [7,6]*c[5]^2*c[3]^5*c[6]^4-4500*a[7,6]*c[5]^2*c[3]^6*c[6]^3+720*a[7,6] *c[5]^2*c[6]^4*c[3]-3460*a[7,6]*c[5]^2*c[6]^4*c[3]^2+8*c[6]+20*c[5]+11 0*c[3]-2710*c[3]^4+2145*c[3]^5+12600*a[7,6]*c[6]^3*c[5]*b[8]*c[3]^2-34 20*a[7,6]*c[6]^2*c[5]^2*b[8]*c[3]^2-2760*a[7,6]*c[5]*c[6]^3*b[8]*c[3]+ 360*a[7,6]*c[5]^2*c[6]^2*b[8]*c[3]-28320*a[7,6]*c[5]*b[8]*c[3]^3*c[6]^ 3+30000*a[7,6]*c[5]*b[8]*c[3]^4*c[6]^2+13500*a[7,6]*c[5]^2*b[8]*c[3]^3 *c[6]^2+584*a[7,6]*c[6]^3*c[3]^2-1560*a[7,6]*c[5]^2*b[8]*c[3]^4*c[6]+2 85*a[7,6]*c[3]^4*c[6]^2-180*a[7,6]*c[3]^5*c[6]^2-575*a[7,6]*c[6]^4*c[3 ]^4+5080*a[7,6]*c[3]^4*c[6]^3-2350*a[7,6]*c[3]^3*c[6]^3-28440*a[7,6]*b [8]*c[3]^4*c[6]^3+31500*a[7,6]*b[8]*c[3]^5*c[6]^3-12*c[5]^2+13500*a[7, 6]*c[5]*b[8]*c[3]^6*c[6]^2-2760*a[7,6]*b[8]*c[6]^5*c[3]+136*a[7,6]*c[5 ]*c[3]^3*c[6]-20*a[7,6]*c[5]*c[6]*c[3]^2+60*a[7,6]*c[5]*c[6]^2*c[3]+25 15*a[7,6]*c[5]*c[3]^3*c[6]^2-165*a[7,6]*c[5]^2*c[3]^3*c[6]-608*a[7,6]* c[5]*c[6]^2*c[3]^2+24*a[7,6]*c[5]^2*c[6]*c[3]^2+780*a[7,6]*b[8]*c[3]^3 *c[6]^2-120*a[7,6]*b[8]*c[6]^2*c[3]^2-12720*a[7,6]*c[6]^5*c[5]*b[8]*c[ 3]^2+29100*a[7,6]*c[6]^5*c[5]*b[8]*c[3]^3-32400*a[7,6]*c[6]^5*c[5]*b[8 ]*c[3]^4+12720*a[7,6]*c[6]^4*c[5]^2*b[8]*c[3]^2-29100*a[7,6]*c[6]^4*c[ 5]^2*b[8]*c[3]^3+32400*a[7,6]*c[6]^4*c[5]^2*b[8]*c[3]^4+2760*a[7,6]*c[ 5]*c[6]^5*b[8]*c[3]-2760*a[7,6]*c[5]^2*c[6]^4*b[8]*c[3]+13500*a[7,6]*c [5]*b[8]*c[3]^5*c[6]^5-13500*a[7,6]*c[5]*b[8]*c[3]^6*c[6]^4-13500*a[7, 6]*c[5]^2*b[8]*c[3]^5*c[6]^4+13500*a[7,6]*c[5]^2*b[8]*c[3]^6*c[6]^3+33 75*a[7,6]*c[3]^5*c[6]^5+570*a[7,6]*c[6]^5*c[3]-60*a[7,6]*c[5]^2*c[6]^4 +240*a[7,6]*b[8]*c[6]^5+60*a[7,6]*c[5]*c[6]^5-60*a[7,6]*c[6]^3*c[3]-29 100*a[7,6]*c[6]^5*b[8]*c[3]^3+32400*a[7,6]*b[8]*c[3]^4*c[6]^5-13500*a[ 7,6]*b[8]*c[3]^5*c[6]^5+13500*a[7,6]*b[8]*c[3]^6*c[6]^4-240*a[7,6]*c[5 ]*b[8]*c[6]^5+240*a[7,6]*c[5]^2*b[8]*c[6]^4-48*a[7,6]*c[6]^5-900*a[7,6 ]*c[5]*b[8]*c[3]^5*c[6]-285*a[7,6]*c[3]^4*c[5]*c[6]+180*a[7,6]*c[3]^5* c[5]*c[6]-5850*a[7,6]*c[3]^5*c[6]^3+2700*a[7,6]*c[3]^6*c[6]^3-4380*c[5 ]^2*c[3]^4-1125*c[3]^6*c[5]^2+3525*c[3]^5*c[5]^2+1800*c[3]^6*c[5]+930* a[7,6]*c[5]*c[6]^4*c[3]^2-3915*a[7,6]*c[5]*c[6]^4*c[3]^3+8600*a[7,6]*c [5]*c[6]^4*c[3]^4-2510*a[7,6]*c[5]^2*c[6]^3*c[3]^3-1100*a[7,6]*c[5]^2* c[6]^3*c[3]^4-90*a[7,6]*c[5]*c[6]^4*c[3]-1800*c[3]^6*c[5]*c[6]-2200*a[ 7,6]*c[5]*c[6]^3*c[3]^2+705*a[7,6]*c[5]^2*c[6]^2*c[3]^2-40*a[7,6]*c[5] *c[6]^3+3540*a[7,6]*c[6]^2*c[5]*b[8]*c[3]^2-120*a[7,6]*c[6]*c[5]^2*b[8 ]*c[3]^2-360*a[7,6]*c[5]*c[6]^2*b[8]*c[3]-14280*a[7,6]*c[5]*b[8]*c[3]^ 3*c[6]^2+780*a[7,6]*c[5]^2*b[8]*c[3]^3*c[6]+278*c[5]*c[6]*c[3]+4524*c[ 5]*c[3]^3*c[6]-1562*c[5]*c[6]*c[3]^2+5670*c[3]^5*c[6]*c[5]+4380*c[5]^2 *c[3]^4*c[6]-2775*c[5]^2*c[3]^3*c[6]-168*c[5]^2*c[6]*c[3]+951*c[5]^2*c [3]^2*c[6]-675*c[3]^6+12720*a[7,6]*b[8]*c[6]^5*c[3]^2-9825*a[7,6]*c[5] *c[3]^5*c[6]^4+6450*a[7,6]*c[5]^2*c[3]^5*c[6]^3+3375*a[7,6]*c[5]^2*c[3 ]^6*c[6]^2-480*a[7,6]*c[5]^2*c[6]^3*c[3]+1780*a[7,6]*c[5]^2*c[6]^3*c[3 ]^2+6075*a[7,6]*c[5]*c[3]^5*c[6]^2+900*a[7,6]*b[8]*c[3]^5*c[6]^2+2400* a[7,6]*b[8]*c[6]^4*c[3]-225*a[7,6]*c[3]^5*c[5]^2*c[6]+1560*a[7,6]*c[5] *b[8]*c[3]^4*c[6]+1125*c[3]^6*c[5]^2*c[6]-3525*c[5]^2*c[3]^5*c[6]+12*c [5]^2*c[6]-611*c[3]^2-3540*a[7,6]*c[6]^4*c[5]*b[8]*c[3]^2+14280*a[7,6] *c[6]^4*c[5]*b[8]*c[3]^3-30000*a[7,6]*c[6]^4*c[5]*b[8]*c[3]^4-9180*a[7 ,6]*c[6]^3*c[5]^2*b[8]*c[3]^2+14820*a[7,6]*c[6]^3*c[5]^2*b[8]*c[3]^3-2 400*a[7,6]*c[6]^3*c[5]^2*b[8]*c[3]^4-2190*a[7,6]*c[6]^4*c[3]^3+20*a[7, 6]*c[6]^2*c[3]^2-136*a[7,6]*c[3]^3*c[6]^2+360*a[7,6]*c[5]*c[6]^4*b[8]* c[3]+2400*a[7,6]*c[5]^2*c[6]^3*b[8]*c[3]+1495*a[7,6]*c[6]^4*c[3]^2+324 00*a[7,6]*c[5]*b[8]*c[3]^5*c[6]^4+30840*a[7,6]*c[5]*b[8]*c[3]^4*c[6]^3 -12600*a[7,6]*c[5]*b[8]*c[3]^5*c[6]^3-32400*a[7,6]*c[5]*b[8]*c[3]^5*c[ 6]^2-18900*a[7,6]*c[5]^2*b[8]*c[3]^5*c[6]^3-28440*a[7,6]*c[5]^2*b[8]*c [3]^4*c[6]^2+31500*a[7,6]*c[5]^2*b[8]*c[3]^5*c[6]^2-13500*a[7,6]*c[5]^ 2*b[8]*c[3]^6*c[6]^2+900*a[7,6]*c[5]^2*b[8]*c[3]^5*c[6]+4800*a[7,6]*c[ 3]^5*c[6]^4-400*a[7,6]*c[6]^4*c[3]+48*a[7,6]*c[5]^2*c[6]^3-240*a[7,6]* b[8]*c[6]^4-7500*a[7,6]*c[6]^5*c[3]^4+472*a[7,6]*c[5]*c[6]^3*c[3]-72*a [7,6]*c[5]^2*c[6]^2*c[3]+5040*a[7,6]*c[5]*c[3]^3*c[6]^3-5430*a[7,6]*c[ 5]*c[3]^4*c[6]^2-2850*a[7,6]*c[5]^2*c[3]^3*c[6]^2+350*a[7,6]*c[5]^2*c[ 3]^4*c[6]-110*c[6]*c[3]+611*c[6]*c[3]^2-1749*c[3]^3*c[6]-4524*c[3]^3*c [5]+2775*c[5]^2*c[3]^3-951*c[5]^2*c[3]^2+1562*c[5]*c[3]^2-278*c[5]*c[3 ]+168*c[5]^2*c[3]-20*c[5]*c[6]+675*c[3]^6*c[6]+1749*c[3]^3+120*a[7,6]* c[6]*c[5]*b[8]*c[3]^2+360*a[7,6]*c[6]^3*b[8]*c[3]+13500*a[7,6]*b[8]*c[ 3]^3*c[6]^3-1560*a[7,6]*b[8]*c[3]^4*c[6]^2-3420*a[7,6]*b[8]*c[6]^3*c[3 ]^2+240*a[7,6]*c[5]*b[8]*c[6]^3+14820*a[7,6]*c[6]^4*b[8]*c[3]^3-2400*a [7,6]*b[8]*c[3]^4*c[6]^4-18900*a[7,6]*b[8]*c[3]^5*c[6]^4-13500*a[7,6]* b[8]*c[3]^6*c[6]^3-240*a[7,6]*c[5]^2*b[8]*c[6]^3-780*a[7,6]*c[5]*b[8]* c[3]^3*c[6]-9180*a[7,6]*b[8]*c[6]^4*c[3]^2-5620*a[7,6]*c[5]*c[3]^4*c[6 ]^3+2400*a[7,6]*c[5]*c[3]^5*c[6]^3+6195*a[7,6]*c[5]^2*c[3]^4*c[6]^2-72 00*a[7,6]*c[5]^2*c[3]^5*c[6]^2-3375*a[7,6]*c[3]^6*c[6]^4+6425*a[7,6]*c [6]^5*c[3]^3-2710*a[7,6]*c[6]^5*c[3]^2+7090*c[3]^4*c[5]+2710*c[3]^4*c[ 6]-2145*c[3]^5*c[6]-7090*c[5]*c[3]^4*c[6])/b[8]/c[6]/(2*c[6]-10*c[6]*c [3]+15*c[6]*c[3]^2-c[3])/(c[6]-c[3])/(c[6]-c[5])/(c[3]-1)/(-1+c[6])/(c [5]-1)/(3*c[3]-1)/(5*c[3]-2), a[5,2] = 1/2*c[3]*c[5]/c[2]*(-5*c[5]+1+5 *c[5]^2)/(1-5*c[3]+5*c[3]^2), a[3,2] = 1/2*c[3]^2/c[2], a[8,2] = 1/120 *(-240*a[7,2]*c[2]*b[8]-48*c[6]*a[7,2]*c[2]-10*c[3]-272*a[7,2]*c[2]*c[ 3]+240*a[7,2]*c[2]*c[5]*b[8]-1560*a[7,2]*c[2]*c[5]*b[8]*c[3]-700*a[7,2 ]*c[2]*c[5]*c[3]^2-1800*a[7,2]*c[2]*c[5]*b[8]*c[3]^3+450*a[7,2]*c[2]*c [5]*c[3]^3+3120*a[7,2]*c[2]*c[5]*b[8]*c[3]^2-360*a[7,2]*c[2]*c[3]^3+60 *c[6]*a[7,2]*c[2]*c[5]-20*c[5]*c[6]*c[3]+45*c[5]*c[6]*c[3]^2-700*c[6]* a[7,2]*c[2]*c[3]^2+3120*c[6]*a[7,2]*c[2]*b[8]*c[3]^2-1560*c[6]*a[7,2]* c[2]*b[8]*c[3]+910*c[6]*a[7,2]*c[2]*c[5]*c[3]^2+1800*c[6]*a[7,2]*c[2]* c[5]*b[8]*c[3]^3-600*c[6]*a[7,2]*c[2]*c[5]*c[3]^3-3120*a[7,2]*c[2]*b[8 ]*c[3]^2+330*c[6]*a[7,2]*c[2]*c[3]+240*c[6]*a[7,2]*c[2]*b[8]+450*c[6]* a[7,2]*c[2]*c[3]^3-3120*c[6]*a[7,2]*c[2]*c[5]*b[8]*c[3]^2+21*c[3]^2+57 0*a[7,2]*c[2]*c[3]^2+1800*a[7,2]*c[2]*b[8]*c[3]^3-1800*c[6]*a[7,2]*c[2 ]*b[8]*c[3]^3+40*a[7,2]*c[2]+1560*a[7,2]*c[2]*b[8]*c[3]+330*a[7,2]*c[2 ]*c[5]*c[3]+14*c[6]*c[3]-30*c[6]*c[3]^2-30*c[5]*c[3]^2+14*c[5]*c[3]-24 0*c[6]*a[7,2]*c[2]*c[5]*b[8]+1560*c[6]*a[7,2]*c[2]*c[5]*b[8]*c[3]-48*a [7,2]*c[2]*c[5]-420*c[6]*a[7,2]*c[2]*c[5]*c[3])/b[8]/c[2]/(c[3]-1)/(3* c[3]-1)/(5*c[3]-2)/(c[5]-1)/(-1+c[6]), a[4,2] = 1/2*c[3]^2*(4-30*c[3]+ 45*c[3]^2)/c[2]/(15*c[3]^2-10*c[3]+2)^3, b[5] = 1/60*(1-5*c[3]+5*c[3]^ 2)*(15*c[6]*c[3]-9*c[3]+4-6*c[6])/(c[6]-c[5])/c[5]/(c[5]-1)/(15*c[5]*c [3]^2-10*c[5]*c[3]+2*c[5]-c[3])/(c[5]-c[3]), a[4,1] = 1/2*c[3]*(-4*c[3 ]+30*c[3]^2-45*c[3]^3+350*c[3]^2*c[2]-100*c[3]*c[2]+10*c[2]+450*c[2]*c [3]^4-600*c[3]^3*c[2])/c[2]/(15*c[3]^2-10*c[3]+2)^3, b[7] = -1/60*(20- 24*c[6]-24*c[5]-136*c[3]-210*c[5]*c[6]*c[3]-300*c[5]*c[3]^3*c[6]+455*c [5]*c[6]*c[3]^2-780*c[6]*b[8]*c[3]-900*b[8]*c[3]^3*c[6]+1560*b[8]*c[6] *c[3]^2-120*c[5]*b[8]*c[6]+1560*c[5]*b[8]*c[3]^2-780*c[5]*b[8]*c[3]-90 0*c[5]*b[8]*c[3]^3-120*b[8]+285*c[3]^2+165*c[6]*c[3]-350*c[6]*c[3]^2+2 25*c[3]^3*c[6]+225*c[3]^3*c[5]-350*c[5]*c[3]^2+165*c[5]*c[3]+30*c[5]*c [6]+120*b[8]*c[6]+120*b[8]*c[5]-1560*b[8]*c[3]^2+780*b[8]*c[3]+900*b[8 ]*c[3]^3-180*c[3]^3+780*c[5]*c[6]*b[8]*c[3]+900*c[5]*b[8]*c[3]^3*c[6]- 1560*c[6]*c[5]*b[8]*c[3]^2)/(c[3]-1)/(-1+c[6])/(c[5]-1)/(3*c[3]-1)/(5* c[3]-2), b[3] = -1/60*(75*c[5]*c[6]*c[3]^2-60*c[5]*c[6]*c[3]+10*c[5]*c [6]-45*c[5]*c[3]^2+35*c[5]*c[3]-6*c[5]-45*c[6]*c[3]^2+35*c[6]*c[3]-6*c [6]+30*c[3]^2-23*c[3]+4)/(c[6]-c[3])/(c[5]-c[3])/c[3]^2/(c[3]-1)/(15*c [3]^2-10*c[3]+1), a[5,1] = 1/6*c[5]*(-225*c[5]*c[2]*c[3]^5+300*c[5]*c[ 2]*c[3]^4+45*c[2]*c[3]^4+225*c[5]^2*c[2]*c[3]^4-9*c[5]*c[2]*c[3]+4*c[5 ]^2*c[2]-40*c[3]^3*c[2]-15*c[5]^2*c[3]^3-3*c[3]^3-190*c[5]*c[2]*c[3]^3 -300*c[5]^2*c[2]*c[3]^3+15*c[3]^3*c[5]+11*c[3]^2*c[2]+175*c[5]^2*c[2]* c[3]^2+50*c[5]*c[2]*c[3]^2-40*c[5]^2*c[2]*c[3])/c[3]^2/c[2]/(1-5*c[3]+ 5*c[3]^2), a[6,3] = 1/6*(c[6]-c[3])*(-450*c[3]^5*c[5]-300*c[3]^4+270*c [3]^5-12*c[6]^2-2700*c[6]^2*c[3]^4+2100*c[6]^2*c[3]^3-810*c[6]^2*c[3]^ 2+450*c[5]^2*c[3]^4-328*c[5]*c[6]*c[3]-2535*c[5]*c[3]^3*c[6]+1450*c[5] *c[6]*c[3]^2+900*c[3]^5*c[6]*c[5]-2250*c[5]^2*c[3]^4*c[6]+3075*c[5]^2* c[3]^3*c[6]+300*c[5]^2*c[6]*c[3]-1470*c[5]^2*c[3]^2*c[6]-24*c[5]^2*c[6 ]-4*c[3]^2+156*c[6]^2*c[3]-10*c[6]*c[3]+148*c[6]*c[3]^2-820*c[3]^3*c[6 ]+45*c[3]^3*c[5]-480*c[5]^2*c[3]^3+135*c[5]^2*c[3]^2-35*c[5]*c[3]^2-2* c[5]*c[3]-6*c[5]^2*c[3]+28*c[5]*c[6]+1350*c[3]^5*c[6]^2+89*c[3]^3+300* c[3]^4*c[5]+1815*c[3]^4*c[6]-1350*c[3]^5*c[6]+975*c[5]*c[3]^4*c[6])*c[ 6]/(1-5*c[3]+5*c[3]^2)/(15*c[3]^2-10*c[3]+1)/(15*c[5]*c[3]-9*c[3]+4-6* c[5])/(c[5]-c[3])/c[3]^2, a[8,4] = 1/360*(384+59763580*c[3]^5*c[5]-688 500*a[7,4]*c[3]^9*c[5]-470241675*c[3]^8*c[6]+508180500*c[3]^9*c[6]+932 917500*c[3]^10*c[6]*c[5]+860490000*c[3]^9*c[6]^2*c[5]+860490000*c[3]^9 *c[5]^2*c[6]-864*c[6]+282487500*c[3]^11*c[6]*c[5]^2-960*c[5]-10800*c[3 ]+6008104*c[3]^4+3645000*a[7,4]*c[3]^9*b[8]*c[6]-688500*a[7,4]*c[3]^9* c[6]-23157444*c[3]^5+11880*a[7,4]*c[3]^5*b[8]-223256250*c[3]^11*c[6]^2 *c[5]^2+160116750*c[3]^10-635191875*c[3]^10*c[6]*c[5]^2-664638750*c[3] ^9*c[6]^2*c[5]^2-497712*c[5]*c[6]^2*c[3]^2+38016*c[5]*c[6]^2*c[3]+2430 00*a[7,4]*c[3]^10*c[5]-66960*a[7,4]*c[3]^6*b[8]+261225000*c[3]^10*c[6] ^2+576*c[5]^2+576*c[6]^2-56953125*c[3]^12*c[6]^2*c[5]+11670*a[7,4]*c[3 ]^6+120*a[7,4]*c[3]^4+36450*a[7,4]*c[3]^8-16200*a[7,4]*c[3]^9-2016*a[7 ,4]*c[3]^5-1215000*a[7,4]*c[3]^10*b[8]*c[6]-1215000*a[7,4]*c[3]^10*c[5 ]*b[8]-69558750*c[3]^11-194400*a[7,4]*c[3]^8*b[8]+169200*a[7,4]*c[3]^7 *b[8]+3645000*a[7,4]*c[3]^9*c[5]*b[8]-31893750*c[3]^12*c[6]+81000*a[7, 4]*c[3]^9*b[8]-65610000*a[7,4]*c[3]^10*c[5]*b[8]*c[6]+66825000*a[7,4]* c[3]^10*c[5]*b[8]*c[6]^2-5208480*a[7,4]*c[3]^5*c[5]*b[8]*c[6]^2-219852 00*a[7,4]*c[3]^6*c[6]^2*c[5]^2*b[8]+57933000*a[7,4]*c[3]^7*c[5]^2*b[8] *c[6]^2+674400*a[7,4]*c[3]^7*c[5]^2-114665625*c[3]^11*c[6]^2+95526000* a[7,4]*c[3]^8*c[5]*b[8]*c[6]^2-54847800*a[7,4]*c[3]^7*c[5]*b[8]*c[6]^2 +95526000*a[7,4]*c[3]^8*c[5]^2*b[8]*c[6]-54847800*a[7,4]*c[3]^7*c[5]^2 *b[8]*c[6]+101979000*a[7,4]*c[3]^9*c[5]*b[8]*c[6]-91076400*a[7,4]*c[3] ^8*c[5]*b[8]*c[6]+109350000*a[7,4]*c[3]^9*c[5]^2*b[8]*c[6]^2+808200*a[ 7,4]*c[3]^8*c[5]+243000*a[7,4]*c[3]^10*c[6]-4449600*a[7,4]*c[3]^8*c[5] *b[8]+16193700*a[7,4]*c[3]^8*c[5]*c[6]-20992500*a[7,4]*c[3]^8*c[5]*c[6 ]^2+5999100*a[7,4]*c[3]^6*c[5]^2*c[6]^2-1456800*a[7,4]*c[3]^5*c[5]^2*c [6]^2+223320*a[7,4]*c[3]^4*c[5]^2*c[6]^2-19440*a[7,4]*c[3]^3*c[5]^2*c[ 6]^2-12288*a[7,4]*c[3]^3*c[5]*c[6]-840540*a[7,4]*c[3]^5*c[5]*c[6]+1341 00*a[7,4]*c[3]^4*c[5]*c[6]+1440*a[7,4]*c[3]^3*c[6]*b[8]-15795000*a[7,4 ]*c[3]^10*c[5]^2*c[6]-3726000*a[7,4]*c[3]^9*b[8]*c[6]^2+354082500*c[3] ^7*c[5]-15795000*a[7,4]*c[3]^10*c[5]*c[6]^2+22781250*c[3]^12*c[5]^2+28 80*c[6]*a[7,4]*c[3]^2*b[8]*c[5]^2+9297240*c[6]^2*c[3]^4-1710408*c[6]^2 *c[3]^3+213792*c[6]^2*c[3]^2+9297240*c[5]^2*c[3]^4-1344*c[5]*c[6]^2+10 2802200*c[3]^6*c[6]^2-215452350*c[3]^7*c[5]^2+102802200*c[3]^6*c[5]^2- 36130980*c[3]^5*c[5]^2-169540850*c[3]^6*c[5]-30420*a[7,4]*c[3]^7+32951 0250*c[3]^8*c[5]^2+582970500*c[3]^9*c[5]-369187875*c[3]^10*c[6]+329510 250*c[3]^8*c[6]^2+251640*a[7,4]*c[3]^5*b[8]*c[6]-30240*a[7,4]*c[3]^4*b [8]*c[6]-30240*a[7,4]*c[3]^4*c[5]*b[8]+1440*a[7,4]*c[3]^3*c[5]*b[8]+25 1640*a[7,4]*c[3]^5*c[5]*b[8]-9043800*a[7,4]*c[3]^7*c[6]*c[5]-4372800*a [7,4]*c[3]^6*c[5]^2*c[6]+1077840*a[7,4]*c[3]^5*c[5]^2*c[6]+15120*a[7,4 ]*c[3]^3*c[5]^2*c[6]-168960*a[7,4]*c[3]^4*c[5]^2*c[6]+13668750*c[3]^12 -2880*a[7,4]*c[3]^2*c[5]*b[8]*c[6]-19679760*a[7,4]*c[3]^6*c[5]*b[8]*c[ 6]+23955750*a[7,4]*c[3]^9*c[6]^2*c[5]-20992500*a[7,4]*c[3]^8*c[5]^2*c[ 6]-18643500*a[7,4]*c[3]^9*c[5]*c[6]-32805000*a[7,4]*c[3]^9*c[5]^2*c[6] ^2+28968750*a[7,4]*c[3]^8*c[5]^2*c[6]^2+11759400*a[7,4]*c[3]^7*c[5]^2* c[6]-1440*a[7,4]*c[3]^3*c[6]^2*b[8]-263520*a[7,4]*c[3]^5*b[8]*c[6]^2+3 65309800*c[3]^6*c[5]*c[6]-241651800*c[5]*c[3]^6*c[6]^2+15986400*c[5]^2 *c[6]^2*c[3]^4-2912160*c[5]^2*c[6]^2*c[3]^3+360960*c[5]^2*c[6]^2*c[3]^ 2-27360*c[5]^2*c[6]^2*c[3]+288*a[7,4]*c[3]^3*c[6]^2-240*a[7,4]*c[3]^3* c[6]+5088*a[7,4]*c[3]^4*c[6]-42870*a[7,4]*c[3]^5*c[6]-42870*a[7,4]*c[3 ]^5*c[5]+54780*a[7,4]*c[3]^5*c[5]^2-6300*a[7,4]*c[3]^4*c[5]^2+5088*a[7 ,4]*c[3]^4*c[5]-240*a[7,4]*c[3]^3*c[5]+288*a[7,4]*c[3]^3*c[5]^2+674400 *a[7,4]*c[3]^7*c[6]^2+808200*a[7,4]*c[3]^8*c[6]+261225000*c[3]^10*c[5] ^2+282487500*c[3]^11*c[6]^2*c[5]-1274670000*c[3]^9*c[6]*c[5]-36450000* c[3]^12*c[5]-576*c[6]*a[7,4]*c[3]^2*c[5]^2+23955750*c[6]*a[7,4]*c[3]^9 *c[5]^2-1119240*a[7,4]*c[3]^6*b[8]*c[6]+30960*a[7,4]*c[3]^4*b[8]*c[6]^ 2-1119240*a[7,4]*c[3]^6*c[5]*b[8]+720*a[7,4]*c[3]^2*c[5]^2*c[6]^2+480* a[7,4]*c[3]^2*c[5]*c[6]+1077840*a[7,4]*c[3]^5*c[5]*c[6]^2-4372800*a[7, 4]*c[3]^6*c[5]*c[6]^2+11759400*a[7,4]*c[3]^7*c[5]*c[6]^2-16243500*a[7, 4]*c[3]^7*c[5]^2*c[6]^2-58672*c[5]*c[6]*c[3]-6107416*c[5]*c[3]^3*c[6]+ 765728*c[5]*c[6]*c[3]^2-128503420*c[3]^5*c[6]*c[5]-21693360*c[5]^2*c[3 ]^4*c[6]+3984000*c[5]^2*c[3]^3*c[6]+38016*c[5]^2*c[6]*c[3]-497712*c[5] ^2*c[3]^2*c[6]+2880*a[7,4]*c[3]^2*c[5]*b[8]*c[6]^2-74880*a[7,4]*c[3]^3 *c[5]*c[6]^2*b[8]+193620*a[7,4]*c[3]^6*c[5]+193620*a[7,4]*c[3]^6*c[6]- 514950*a[7,4]*c[3]^7*c[6]+73440*a[7,4]*c[3]^3*c[5]*c[6]*b[8]+4956840*a [7,4]*c[3]^5*c[5]*b[8]*c[6]-797040*a[7,4]*c[3]^4*c[6]*c[5]*b[8]+337473 0*a[7,4]*c[3]^6*c[5]*c[6]+65340370*c[3]^6+66825000*a[7,4]*c[3]^10*c[5] ^2*b[8]*c[6]-785011500*c[3]^8*c[6]^2*c[5]+509442000*c[3]^7*c[6]^2*c[5] -785011500*c[3]^8*c[5]^2*c[6]+509442000*c[3]^7*c[5]^2*c[6]-241651800*c [3]^6*c[5]^2*c[6]-765560550*c[3]^7*c[5]*c[6]+600007500*c[5]^2*c[3]^8*c [6]^2-385554000*c[5]^2*c[3]^7*c[6]^2+181188000*c[5]^2*c[3]^6*c[6]^2+84 559200*c[5]^2*c[3]^5*c[6]-303750*a[7,4]*c[3]^10*c[6]^2-56953125*c[3]^1 2*c[5]^2*c[6]-114665625*c[3]^11*c[5]^2+184680000*c[3]^11*c[5]-1344*c[5 ]^2*c[6]+140392*c[3]^2+18225000*a[7,4]*c[3]^11*c[5]*b[8]*c[6]-68040000 *a[7,4]*c[3]^10*c[5]^2*b[8]*c[6]^2-18225000*a[7,4]*c[3]^11*c[5]*b[8]*c [6]^2-105624000*c[6]*a[7,4]*c[3]^9*c[5]^2*b[8]-135774810*c[3]^7-607500 0*c[6]^2*a[7,4]*c[3]^11*c[5]^2-16320*c[6]^2*c[3]+21363750*a[7,4]*c[3]^ 10*c[5]^2*c[6]^2+960*c[5]^2*c[6]^2-215452350*c[3]^7*c[6]^2-635191875*c [3]^10*c[6]^2*c[5]+24304*c[6]*c[3]-316160*c[6]*c[3]^2+2512408*c[3]^3*c [6]+2841712*c[3]^3*c[5]-1710408*c[5]^2*c[3]^3+213792*c[5]^2*c[3]^2-355 736*c[5]*c[3]^2+27184*c[5]*c[3]-16320*c[5]^2*c[3]+2080*c[5]*c[6]+45562 50*a[7,4]*c[3]^11*c[5]*c[6]^2-36130980*c[3]^5*c[6]^2-720*a[7,4]*c[3]^4 *b[8]-3645000*a[7,4]*c[3]^11*c[5]*c[6]-148270460*c[3]^6*c[6]+3984000*c [5]*c[6]^2*c[3]^3-21693360*c[5]*c[3]^4*c[6]^2+84559200*c[5]*c[3]^5*c[6 ]^2+22781250*c[3]^12*c[6]^2-1114296*c[3]^3+1171746000*c[3]^8*c[6]*c[5] -62842800*c[5]^2*c[6]^2*c[3]^5+309059250*c[3]^7*c[6]+205830450*c[3]^8- 221489100*c[3]^9+18225000*c[6]^2*a[7,4]*c[3]^11*c[5]^2*b[8]-303750*a[7 ,4]*c[3]^10*c[5]^2+82012500*c[3]^12*c[5]*c[6]+4556250*a[7,4]*c[3]^11*c [5]^2*c[6]+1215000*a[7,4]*c[3]^10*c[5]^2*b[8]+1215000*a[7,4]*c[3]^10*b [8]*c[6]^2-18225000*a[7,4]*c[3]^11*c[5]^2*b[8]*c[6]+496125000*c[3]^10* c[6]^2*c[5]^2+877500*a[7,4]*c[3]^9*c[6]^2+161291250*c[3]^11*c[6]-53932 6125*c[3]^8*c[5]+45562500*c[3]^12*c[6]^2*c[5]^2+12453750*a[7,4]*c[3]^1 0*c[5]*c[6]+30960*a[7,4]*c[3]^4*c[5]^2*b[8]-1440*a[7,4]*c[3]^3*c[5]^2* b[8]-263520*a[7,4]*c[3]^5*c[5]^2*b[8]+1186200*a[7,4]*c[3]^6*b[8]*c[6]^ 2+2916000*a[7,4]*c[3]^7*b[8]*c[6]-514950*a[7,4]*c[3]^7*c[5]-252450*a[7 ,4]*c[3]^6*c[6]^2+54780*a[7,4]*c[3]^5*c[6]^2-6300*a[7,4]*c[3]^4*c[6]^2 -252450*a[7,4]*c[3]^6*c[5]^2-1048500*a[7,4]*c[3]^8*c[6]^2+877500*a[7,4 ]*c[3]^9*c[5]^2-357854625*c[3]^9*c[5]^2-423275625*c[3]^10*c[5]-3578546 25*c[3]^9*c[6]^2+827280*a[7,4]*c[3]^4*c[6]^2*c[5]*b[8]-2880*a[7,4]*c[3 ]^2*c[5]^2*b[8]*c[6]^2+1186200*a[7,4]*c[3]^6*c[5]^2*b[8]+2916000*a[7,4 ]*c[3]^7*c[5]*b[8]-168960*a[7,4]*c[3]^4*c[5]*c[6]^2+15120*a[7,4]*c[3]^ 3*c[5]*c[6]^2-858240*a[7,4]*c[3]^4*c[6]^2*c[5]^2*b[8]+5472000*a[7,4]*c [3]^5*c[5]^2*b[8]*c[6]^2+20799000*a[7,4]*c[3]^6*c[5]*b[8]*c[6]^2+20799 000*a[7,4]*c[3]^6*c[5]^2*b[8]*c[6]+51931800*a[7,4]*c[3]^7*c[5]*b[8]*c[ 6]+76320*a[7,4]*c[3]^3*c[5]^2*c[6]^2*b[8]-74880*a[7,4]*c[3]^3*c[5]^2*c [6]*b[8]-5208480*a[7,4]*c[3]^5*c[5]^2*b[8]*c[6]+827280*a[7,4]*c[3]^4*c [6]*c[5]^2*b[8]-1048500*a[7,4]*c[3]^8*c[5]^2-410973750*c[3]^11*c[6]*c[ 5]-100170000*a[7,4]*c[3]^8*c[5]^2*b[8]*c[6]^2-105624000*a[7,4]*c[3]^9* c[5]*b[8]*c[6]^2-3726000*a[7,4]*c[3]^9*c[5]^2*b[8]-576*a[7,4]*c[3]^2*c [5]*c[6]^2-3085200*a[7,4]*c[3]^7*c[5]^2*b[8]+4644000*a[7,4]*c[3]^8*b[8 ]*c[6]^2-3085200*a[7,4]*c[3]^7*b[8]*c[6]^2-4449600*a[7,4]*c[3]^8*b[8]* c[6]+4644000*a[7,4]*c[3]^8*c[5]^2*b[8]-15416180*c[3]^4*c[5]-13569188*c [3]^4*c[6]+52412260*c[3]^5*c[6]+33120200*c[5]*c[3]^4*c[6])/b[8]/c[3]^2 /(c[5]-1)/(c[3]-1)/(3*c[3]-1)/(5*c[3]-2)/(15*c[3]^2-10*c[3]+1)/(15*c[5 ]*c[3]^2-10*c[5]*c[3]+2*c[5]-c[3])/(-1+c[6])/(2*c[6]-10*c[6]*c[3]+15*c [6]*c[3]^2-c[3]), a[6,1] = 1/6*c[6]*(485*c[5]*c[6]*c[2]*c[3]^2-6*c[2]* c[3]^2*c[6]-435*c[5]^2*c[6]*c[2]*c[3]^2+90*c[6]*c[2]*c[3]^5-120*c[6]*c [2]*c[3]^4+5850*c[6]*c[2]*c[5]^2*c[3]^5+4425*c[6]^2*c[2]*c[5]^2*c[3]^3 -1650*c[6]^2*c[2]*c[5]^2*c[3]^2+3375*c[6]^2*c[2]*c[5]^2*c[3]^5+5100*c[ 6]*c[2]*c[5]*c[3]^4-6300*c[6]*c[2]*c[5]*c[3]^5+50*c[5]*c[2]*c[3]^2-487 5*c[6]^2*c[2]*c[5]*c[3]^3+1890*c[6]*c[2]*c[5]^2*c[3]^3-4650*c[6]*c[2]* c[5]^2*c[3]^4+6300*c[6]^2*c[2]*c[5]*c[3]^4-3375*c[6]^2*c[2]*c[5]*c[3]^ 5-5850*c[6]^2*c[2]*c[5]^2*c[3]^4-352*c[5]*c[6]^2*c[2]*c[3]+300*c[5]^2* c[6]^2*c[2]*c[3]+3375*c[2]*c[3]^6*c[6]*c[5]-1350*c[2]*c[3]^6*c[6]^2-87 0*c[5]^2*c[2]*c[3]^4+405*c[5]^2*c[2]*c[3]^3-66*c[5]^2*c[2]*c[3]^2+660* c[5]*c[2]*c[3]^4-307*c[5]*c[2]*c[3]^3-24*c[6]^2*c[5]^2*c[2]-45*c[5]^2* c[3]^4+675*c[5]^2*c[2]*c[3]^5+1870*c[5]*c[6]^2*c[2]*c[3]^2-2140*c[5]*c [6]*c[2]*c[3]^3-225*c[5]^2*c[6]^2*c[3]^4+90*c[5]^2*c[6]^2*c[3]^3+60*c[ 5]*c[3]^3*c[6]+225*c[5]^2*c[3]^4*c[6]-90*c[5]^2*c[3]^3*c[6]+1350*c[6]^ 3*c[2]*c[3]^5-2700*c[6]^3*c[2]*c[3]^4+2100*c[6]^3*c[2]*c[3]^3-2190*c[6 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[3]^9*c[5]-116280*c[3]^4*b[8]*c[5]-133650*c[3]^8*c[6]+303750*c[3]^9*c[ 6]+202500*c[3]^10*c[6]*c[5]-202500*c[3]^9*c[6]^2*c[5]-202500*c[3]^9*c[ 5]^2*c[6]-216*c[6]-240*c[5]-1680*c[3]+129508*c[3]^4+1899000*a[7,6]*c[5 ]*b[8]*c[3]^6*c[6]-81000*a[7,4]*c[3]^9*b[8]*c[6]+240*a[7,6]*c[3]^3*c[6 ]+20250*a[7,4]*c[3]^9*c[6]+303750*c[3]^10*a[7,6]*c[6]^2-205758*c[3]^5- 243000*c[3]^10*a[7,2]*c[2]-46860*c[6]*a[7,2]*c[2]*c[5]^2*c[3]^3+232560 *c[6]*a[7,2]*c[2]*c[5]^2*b[8]*c[3]^3+720*c[5]^2*c[6]^2*b[8]*c[3]+11880 *a[7,4]*c[3]^5*b[8]-29520*c[6]*a[7,2]*c[2]*c[5]^2*b[8]*c[3]^2-1215000* c[3]^10*a[7,5]*c[5]^2*b[8]-207780*c[3]^6*a[7,5]*c[5]+552150*c[3]^7*a[7 ,5]*c[5]-29520*a[7,6]*c[6]^3*c[5]*b[8]*c[3]^2-29520*a[7,6]*c[6]^2*c[5] ^2*b[8]*c[3]^2+1440*a[7,6]*c[5]*c[6]^3*b[8]*c[3]+1440*a[7,6]*c[5]^2*c[ 6]^2*b[8]*c[3]+232560*a[7,6]*c[5]*b[8]*c[3]^3*c[6]^3+690120*a[7,6]*c[5 ]*b[8]*c[3]^4*c[6]^2+232560*a[7,6]*c[5]^2*b[8]*c[3]^3*c[6]^2+288*a[7,6 ]*c[6]^3*c[3]^2-3085200*c[3]^7*a[7,5]*c[5]*b[8]+1215000*c[3]^10*c[6]*a 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[2]*c[5]^2*b[8]*c[3]^8-194400*a[7,4]*c[3]^8*b[8]+169200*a[7,4]*c[3]^7* b[8]-3726000*c[3]^9*a[7,2]*c[2]*b[8]-1215000*c[3]^9*a[7,6]*b[8]*c[6]^3 *c[5]-81000*a[7,4]*c[3]^9*c[5]*b[8]-850500*c[3]^8*a[7,6]*c[5]^2*c[6]^2 -1558800*c[3]^7*a[7,6]*b[8]*c[6]^2+1215000*c[3]^10*a[7,2]*c[2]*b[8]-10 48500*c[3]^7*a[7,6]*c[5]^2*c[6]+674400*c[3]^6*a[7,6]*c[5]^2*c[6]+18990 00*c[3]^6*a[7,6]*b[8]*c[6]^2-340200*a[7,6]*c[5]*b[8]*c[3]^6*c[6]^2+810 00*a[7,4]*c[3]^9*b[8]-15480*c[3]^3*b[8]*c[6]^2+720*c[5]^2*b[8]*c[3]^2+ 303750*c[3]^10*a[7,5]*c[5]^2+4944*a[7,6]*c[5]*c[3]^3*c[6]-240*a[7,6]*c [5]*c[6]*c[3]^2-1440*a[7,6]*b[8]*c[3]^3*c[6]+1186200*a[7,6]*b[8]*c[3]^ 6*c[6]+240*a[7,6]*c[5]*c[6]^2*c[3]+33060*a[7,6]*c[5]*c[3]^3*c[6]^2-630 0*a[7,6]*c[5]^2*c[3]^3*c[6]-4656*a[7,6]*c[5]*c[6]^2*c[3]^2+288*a[7,6]* c[5]^2*c[6]*c[3]^2-29520*a[7,6]*b[8]*c[3]^3*c[6]^2+1440*a[7,6]*b[8]*c[ 6]^2*c[3]^2-116280*c[3]^4*b[8]*c[6]+461340*c[3]^5*b[8]*c[6]+303750*c[3 ]^10*a[7,2]*c[2]*c[6]-303750*c[3]^9*a[7,2]*c[2]*c[5]^2+4644000*a[7,6]* c[5]^2*b[8]*c[3]^6*c[6]^3+180150*a[7,5]*c[5]^2*c[3]^6*c[6]^2-288*a[7,5 ]*c[5]^2*c[6]^2*c[3]-46860*a[7,5]*c[5]^2*c[3]^3*c[6]^2+153000*a[7,2]*c [2]*c[5]*c[3]^5-39000*c[6]*a[7,2]*c[2]*c[3]^4+30960*a[7,2]*c[2]*b[8]*c [3]^4+153000*c[6]*a[7,2]*c[2]*c[3]^5+720*b[8]*c[6]^2*c[3]^2-102240*a[7 ,4]*c[3]^5*c[5]*b[8]*c[6]^2-194400*a[7,4]*c[3]^6*c[6]^2*c[5]^2*b[8]+81 000*a[7,4]*c[3]^7*c[5]^2*b[8]*c[6]^2+45000*a[7,4]*c[3]^7*c[5]^2-949500 *c[3]^5*b[8]*c[5]*c[6]^2+779400*c[3]^6*b[8]*c[5]*c[6]^2-634500*c[3]^9* a[7,5]*c[5]^2-922680*a[7,2]*c[2]*c[5]*b[8]*c[3]^5+232560*a[7,2]*c[2]*c [5]*b[8]*c[3]^4+779400*c[3]^7*b[8]*c[6]-81000*a[7,4]*c[3]^8*c[5]*b[8]* c[6]^2+113400*a[7,4]*c[3]^7*c[5]*b[8]*c[6]^2-81000*a[7,4]*c[3]^8*c[5]^ 2*b[8]*c[6]+113400*a[7,4]*c[3]^7*c[5]^2*b[8]*c[6]+81000*a[7,4]*c[3]^9* c[5]*b[8]*c[6]-32400*a[7,4]*c[3]^8*c[5]*b[8]*c[6]-28800*a[7,4]*c[3]^8* c[5]-720*c[5]*c[6]^2*b[8]*c[3]-240*a[7,2]*c[2]*c[5]*c[3]^2-29520*a[7,2 ]*c[2]*c[5]*b[8]*c[3]^3+5940*a[7,5]*c[5]^2*c[6]^2*c[3]^2-352200*a[7,5] *c[5]^2*c[3]^5*c[6]^2-322200*a[7,5]*c[5]^3*c[3]^4*c[6]^2+69360*a[7,5]* c[5]^3*c[3]^3*c[6]^2+45300*a[7,2]*c[2]*c[3]^5-5232*a[7,2]*c[2]*c[3]^4+ 459000*c[3]^7*b[8]*c[5]*c[6]^2+14760*c[6]^2*c[5]*c[3]^2*b[8]+4944*a[7, 2]*c[2]*c[5]*c[3]^3+113400*a[7,4]*c[3]^8*c[5]*b[8]+18450*a[7,4]*c[3]^8 *c[5]*c[6]+27000*a[7,4]*c[3]^8*c[5]*c[6]^2+58950*a[7,4]*c[3]^6*c[5]^2* c[6]^2-48000*a[7,4]*c[3]^5*c[5]^2*c[6]^2+18030*a[7,4]*c[3]^4*c[5]^2*c[ 6]^2-3060*a[7,4]*c[3]^3*c[5]^2*c[6]^2-1728*a[7,4]*c[3]^3*c[5]*c[6]-516 0*a[7,4]*c[3]^5*c[5]*c[6]+6990*a[7,4]*c[3]^4*c[5]*c[6]+720*a[7,4]*c[3] ^3*c[6]*b[8]+546750*a[7,2]*c[2]*c[3]^9*c[5]*c[6]-1558800*c[3]^6*a[7,6] *c[5]*b[8]*c[6]^3-918000*c[3]^7*a[7,6]*c[5]*b[8]*c[6]^3+1440*a[7,2]*c[ 2]*c[5]*b[8]*c[3]^2-243000*c[3]^10*a[7,6]*c[6]-2322000*c[3]^7*b[8]*c[5 ]^2+163800*c[3]^7*c[5]+4644000*a[7,5]*c[5]^3*b[8]*c[3]^6*c[6]^2-308520 0*a[7,5]*c[5]^3*b[8]*c[3]^5*c[6]^2-1558800*a[7,5]*c[5]^2*b[8]*c[3]^6*c [6]^2+1186200*a[7,5]*c[5]*b[8]*c[3]^5*c[6]^2+1863000*c[3]^8*b[8]*c[5]^ 2-918000*c[3]^7*a[7,6]*c[5]^2*b[8]*c[6]^2+2511000*c[3]^8*a[7,6]*c[5]^2 *b[8]*c[6]^2+4644000*c[3]^7*a[7,6]*c[5]^2*b[8]*c[6]+240*a[7,2]*c[2]*c[ 3]^3+720*c[6]*a[7,4]*c[3]^2*b[8]*c[5]^2-922680*a[7,6]*c[5]*b[8]*c[3]^5 *c[6]-39000*a[7,6]*c[3]^4*c[5]*c[6]+153000*a[7,6]*c[3]^5*c[5]*c[6]-252 450*a[7,6]*c[3]^5*c[6]^3+674400*a[7,6]*c[3]^6*c[6]^3-405000*c[3]^10*a[ 7,5]*c[5]^2*c[6]-607500*c[3]^9*b[8]*c[5]^2+131760*c[3]^4*b[8]*c[5]^2-9 18000*c[6]^2*a[7,2]*c[2]*b[8]*c[3]^7*c[5]+2511000*c[6]^2*a[7,2]*c[2]*b [8]*c[3]^8*c[5]+172890*c[6]^2*c[3]^4-76968*c[6]^2*c[3]^3+19044*c[6]^2* c[3]^2+232560*c[6]*a[7,2]*c[2]*b[8]*c[3]^4-106140*c[6]*a[7,2]*c[2]*c[5 ]*c[3]^4-976320*c[6]*a[7,2]*c[2]*c[5]*b[8]*c[3]^5-3726000*c[3]^8*a[7,6 ]*b[8]*c[6]^3+180150*c[3]^6*a[7,6]*c[5]*c[6]^3+486000*c[3]^7*a[7,6]*c[ 5]*c[6]^3+486000*c[3]^7*a[7,6]*c[5]^2*c[6]^2+172890*c[5]^2*c[3]^4-336* c[5]*c[6]^2-80100*c[3]^6*c[6]^2+417825*c[3]^7*c[5]^2-80100*c[3]^6*c[5] ^2-174900*c[3]^5*c[5]^2-357225*c[3]^6*c[5]+461340*c[3]^5*b[8]*c[5]-304 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8000*c[6]*a[7,2]*c[2]*c[5]^2*b[8]*c[3]^7-1558800*c[6]*a[7,2]*c[2]*c[5] ^2*b[8]*c[3]^6+1899000*c[6]*a[7,2]*c[2]*c[5]^2*b[8]*c[3]^5-15480*c[3]^ 4*b[8]+459000*c[3]^7*b[8]*c[5]^2*c[6]-593100*c[3]^5*b[8]*c[5]^2-129600 0*a[7,2]*c[2]*b[8]*c[3]^9*c[5]*c[6]-918000*a[7,2]*c[2]*b[8]*c[3]^8*c[5 ]-634500*c[3]^9*a[7,2]*c[2]*c[6]-607500*c[3]^10*b[8]+11880*a[7,4]*c[3] ^4*c[5]^2*b[8]-720*a[7,4]*c[3]^3*c[5]^2*b[8]-66960*a[7,4]*c[3]^5*c[5]^ 2*b[8]+169200*a[7,4]*c[3]^6*b[8]*c[6]^2+25200*a[7,4]*c[3]^7*b[8]*c[6]- 29520*a[7,6]*c[5]*b[8]*c[3]^3*c[6]+877500*a[7,2]*c[2]*c[5]^2*c[3]^8+77 9400*c[3]^6*b[8]*c[5]^2*c[6]-607500*c[3]^9*b[8]*c[6]^2+131760*c[3]^5*b [8]-949500*c[3]^5*b[8]*c[5]^2*c[6]+607500*c[3]^10*b[8]*c[5]+183090*a[7 ,6]*c[5]*c[3]^4*c[6]^3-352200*a[7,6]*c[5]*c[3]^5*c[6]^3+1863000*c[3]^9 *b[8]+183090*a[7,6]*c[5]^2*c[3]^4*c[6]^2-352200*a[7,6]*c[5]^2*c[3]^5*c [6]^2+750*a[7,4]*c[3]^7*c[5]-37200*a[7,4]*c[3]^6*c[6]^2+14160*a[7,4]*c [3]^5*c[6]^2-2430*a[7,4]*c[3]^4*c[6]^2-37200*a[7,4]*c[3]^6*c[5]^2-2025 0*a[7,4]*c[3]^8*c[6]^2+151875*c[3]^9*c[5]^2-151875*c[3]^10*c[5]+151875 *c[3]^9*c[6]^2-922680*c[6]*a[7,2]*c[2]*c[5]^2*b[8]*c[3]^4+486000*c[6]* a[7,2]*c[2]*c[5]^2*c[3]^7-1558800*a[7,2]*c[2]*b[8]*c[3]^7*c[5]+55080*a [7,4]*c[3]^4*c[6]^2*c[5]*b[8]-720*a[7,4]*c[3]^2*c[5]^2*b[8]*c[6]^2+169 200*a[7,4]*c[3]^6*c[5]^2*b[8]+25200*a[7,4]*c[3]^7*c[5]*b[8]-11100*a[7, 4]*c[3]^4*c[5]*c[6]^2+2250*a[7,4]*c[3]^3*c[5]*c[6]^2-2322000*c[3]^8*b[ 8]+14760*c[6]*c[3]^2*b[8]*c[5]^2-66960*a[7,4]*c[3]^4*c[6]^2*c[5]^2*b[8 ]+169200*a[7,4]*c[3]^5*c[5]^2*b[8]*c[6]^2+25200*a[7,4]*c[3]^6*c[5]*b[8 ]*c[6]^2+25200*a[7,4]*c[3]^6*c[5]^2*b[8]*c[6]-138600*a[7,4]*c[3]^7*c[5 ]*b[8]*c[6]+11880*a[7,4]*c[3]^3*c[5]^2*c[6]^2*b[8]-11160*a[7,4]*c[3]^3 *c[5]^2*c[6]*b[8]-102240*a[7,4]*c[3]^5*c[5]^2*b[8]*c[6]+55080*a[7,4]*c [3]^4*c[6]*c[5]^2*b[8]-20250*a[7,4]*c[3]^8*c[5]^2-29520*c[6]^2*a[7,2]* c[2]*c[5]*b[8]*c[3]^2+877500*c[6]^2*a[7,2]*c[2]*c[3]^8+2476800*a[7,6]* c[5]*b[8]*c[3]^7*c[6]^2+1215000*c[3]^10*a[7,6]*b[8]*c[6]-144*a[7,4]*c[ 3]^2*c[5]*c[6]^2-194400*a[7,4]*c[3]^7*c[5]^2*b[8]+81000*a[7,4]*c[3]^8* b[8]*c[6]^2-194400*a[7,4]*c[3]^7*b[8]*c[6]^2+113400*a[7,4]*c[3]^8*b[8] *c[6]+81000*a[7,4]*c[3]^8*c[5]^2*b[8]-299700*c[3]^6*a[7,6]*c[6]^2+1788 00*c[3]^7*a[7,6]*c[6]^2+877500*c[3]^8*a[7,6]*c[6]^3+461340*c[3]^4*b[8] *c[5]^2*c[6]-1255500*c[3]^9*b[8]*c[6]-850500*a[7,2]*c[2]*c[5]^2*c[3]^8 *c[6]+405000*a[7,2]*c[2]*c[5]^2*c[3]^9*c[6]+240*c[6]*a[7,2]*c[2]*c[5]* c[3]-330540*c[3]^4*c[5]-255474*c[3]^4*c[6]+311120*c[3]^5*c[6]+724810*c [5]*c[3]^4*c[6]-3726000*a[7,6]*c[5]^2*b[8]*c[3]^7*c[6]^3+2511000*c[3]^ 9*a[7,5]*c[5]*b[8]*c[6]-634500*c[3]^9*a[7,5]*c[5]*c[6]-1558800*c[3]^7* a[7,6]*c[5]*b[8]*c[6]-299700*c[3]^6*a[7,6]*c[5]*c[6]+178800*c[3]^7*a[7 ,6]*c[5]*c[6]-1048500*c[3]^7*a[7,6]*c[6]^3)/b[8]/c[3]^2/(5275*c[3]^5*c [5]+6975*c[3]^8*c[6]-3375*c[3]^9*c[6]-732*c[3]^4+3295*c[3]^5+646*c[5]* c[6]^2*c[3]^2-82*c[5]*c[6]^2*c[3]-2550*c[3]^7*c[5]+3295*c[6]^2*c[3]^4- 732*c[6]^2*c[3]^3+86*c[6]^2*c[3]^2+3295*c[5]^2*c[3]^4+4*c[5]*c[6]^2+12 900*c[3]^6*c[6]^2-10350*c[3]^7*c[5]^2+12900*c[3]^6*c[5]^2-8570*c[3]^5* c[5]^2-4330*c[3]^6*c[5]+3375*c[3]^8*c[5]^2-3375*c[3]^9*c[5]+3375*c[3]^ 8*c[6]^2+6880*c[3]^6*c[5]*c[6]-2550*c[5]*c[3]^6*c[6]^2-8570*c[5]^2*c[6 ]^2*c[3]^4+3295*c[5]^2*c[6]^2*c[3]^3-732*c[5]^2*c[6]^2*c[3]^2+86*c[5]^ 2*c[6]^2*c[3]+3375*c[3]^9*c[6]*c[5]+78*c[5]*c[6]*c[3]+1917*c[5]*c[3]^3 *c[6]-564*c[5]*c[6]*c[3]^2-945*c[3]^5*c[6]*c[5]+5275*c[5]^2*c[3]^4*c[6 ]-2563*c[5]^2*c[3]^3*c[6]-82*c[5]^2*c[6]*c[3]+646*c[5]^2*c[3]^2*c[6]-8 570*c[3]^6-3375*c[3]^8*c[6]^2*c[5]+6975*c[3]^7*c[6]^2*c[5]-3375*c[3]^8 *c[5]^2*c[6]+6975*c[3]^7*c[5]^2*c[6]-2550*c[3]^6*c[5]^2*c[6]-4425*c[3] ^7*c[5]*c[6]+3375*c[5]^2*c[3]^7*c[6]^2-10350*c[5]^2*c[3]^6*c[6]^2-4330 *c[5]^2*c[3]^5*c[6]+4*c[5]^2*c[6]-4*c[3]^2+12900*c[3]^7-4*c[6]^2*c[3]- 4*c[5]^2*c[6]^2-10350*c[3]^7*c[6]^2+4*c[6]*c[3]-82*c[6]*c[3]^2+646*c[3 ]^3*c[6]+646*c[3]^3*c[5]-732*c[5]^2*c[3]^3+86*c[5]^2*c[3]^2-82*c[5]*c[ 3]^2+4*c[5]*c[3]-4*c[5]^2*c[3]-4*c[5]*c[6]-8570*c[3]^5*c[6]^2-4330*c[3 ]^6*c[6]-2563*c[5]*c[6]^2*c[3]^3+5275*c[5]*c[3]^4*c[6]^2-4330*c[5]*c[3 ]^5*c[6]^2+86*c[3]^3-3600*c[3]^8*c[6]*c[5]+12900*c[5]^2*c[6]^2*c[3]^5- 2550*c[3]^7*c[6]-10350*c[3]^8+3375*c[3]^9+6975*c[3]^8*c[5]-2563*c[3]^4 *c[5]-2563*c[3]^4*c[6]+5275*c[3]^5*c[6]-2712*c[5]*c[3]^4*c[6]), c[8] = 1, a[4,3] = -(15*c[3]^2-10*c[3]+1)*c[3]/(15*c[3]^2-10*c[3]+2)^3, a[6, 5] = -(3*c[3]-1)*(2*c[6]-10*c[6]*c[3]+15*c[6]*c[3]^2-c[3])*(c[6]-c[3]) *(c[6]-c[5])*c[6]/(15*c[5]*c[3]^2-10*c[5]*c[3]+2*c[5]-c[3])/(c[5]-c[3] )/(15*c[5]*c[3]-9*c[3]+4-6*c[5])/c[5], a[8,1] = 1/360*(81000*c[2]*c[3] ^6*c[6]^2*a[7,5]*c[5]^3*b[8]-174600*b[8]*a[7,2]*c[6]*c[2]^2*c[5]^2*c[3 ]^4-101250*c[5]^2*c[2]*c[3]^8+162000*c[5]*c[2]*c[3]^8+174600*c[6]*b[8] *a[7,6]*c[2]*c[3]^5*c[5]^2-288*c[6]*a[7,5]*c[2]*c[5]^3*c[3]-174600*b[8 ]*a[7,2]*c[6]^2*c[2]^2*c[3]^4*c[5]+38550*a[7,6]*c[6]^3*c[2]*c[3]^4*c[5 ]-194400*c[6]^2*c[5]*c[2]*a[7,4]*c[3]^6*b[8]-2100*a[7,5]*c[6]*c[2]*c[3 ]^5*c[5]^2+38550*a[7,5]*c[6]*c[2]*c[5]^3*c[3]^4-4536*c[5]*c[2]*c[3]-19 4400*c[6]*c[5]^2*c[2]*a[7,4]*c[3]^6*b[8]-60750*c[2]*c[3]^8-76320*b[8]* a[7,5]*c[6]^2*c[2]*c[5]^3*c[3]^3-38700*c[5]^2*c[2]*c[6]^2*c[3]^6*a[7,6 ]+113400*c[5]^2*c[2]*c[6]^2*c[3]^6*a[7,6]*b[8]-81000*c[6]*a[7,2]*c[2]* b[8]*c[3]^7*c[5]+288*c[6]*a[7,2]*c[2]*c[5]^2*c[3]^2-79528*c[5]*c[6]*c[ 2]*c[3]^2+81000*c[5]*c[2]*c[6]^2*a[7,4]*c[3]^7*b[8]+174600*b[8]*a[7,5] *c[6]^2*c[2]*c[3]^5*c[5]-98280*b[8]*a[7,5]*c[6]^2*c[2]*c[5]^2*c[3]^4-1 440*c[5]^3*c[6]^2*b[8]*a[7,5]*c[2]*c[3]-96*c[2]+20250*c[5]^2*c[2]^2*c[ 6]*a[7,2]*c[3]^6-20250*c[5]^2*c[2]*c[3]^7*a[7,6]*c[6]+33264*c[2]*c[3]^ 2*c[6]+51252*c[5]^2*c[6]*c[2]*c[3]^2-2832*c[6]*a[7,2]*c[2]^2*c[5]*c[3] ^2+16560*c[6]*b[8]*a[7,2]*c[2]^2*c[5]*c[3]^2+76320*c[6]*b[8]*a[7,5]*c[ 2]*c[3]^4*c[5]-3420*c[6]*a[7,2]*c[2]*c[5]^2*c[3]^3+16560*c[6]*a[7,2]*c [2]*c[5]^2*b[8]*c[3]^3-796050*c[6]*c[2]*c[3]^5-38550*a[7,5]*c[6]^2*c[2 ]*c[3]^5*c[5]+17790*a[7,5]*c[6]^2*c[2]*c[5]^2*c[3]^4-1440*c[6]*a[7,2]* c[2]*c[5]^2*b[8]*c[3]^2+81000*c[2]*c[5]^2*c[6]^2*c[3]^7*b[8]+441000*c[ 6]*c[2]*c[3]^4-1281150*c[6]*c[2]*c[5]^2*c[3]^5-288*c[6]^2*c[5]^2*a[7,5 ]*c[2]*c[3]+1440*c[6]^2*c[5]*b[8]*a[7,2]*c[2]^2*c[3]-288*c[6]^2*c[5]*a [7,2]*c[2]^2*c[3]+1367550*c[2]*c[3]^7*c[6]*c[5]-1440*c[6]^2*c[5]*b[8]* a[7,5]*c[2]*c[3]^2-720*c[6]^2*c[5]*b[8]*a[7,4]*c[2]*c[3]^2+15120*c[6]^ 2*b[8]*a[7,6]*c[2]*c[5]*c[3]^2+194400*c[6]*c[5]^2*c[2]^2*a[7,2]*b[8]*c [3]^5+113400*c[6]^2*c[2]*c[5]^2*c[3]^6*a[7,5]*b[8]-1440*c[6]*c[5]^2*b[ 8]*a[7,5]*c[2]*c[3]-1098000*c[6]^2*c[2]*c[5]^2*c[3]^6+170910*c[6]^2*c[ 2]*c[5]^2*c[3]^3-36360*c[6]^2*c[2]*c[5]^2*c[3]^2+942375*c[6]^2*c[2]*c[ 5]^2*c[3]^5+253125*c[2]*c[5]*c[6]^2*c[3]^8-1065010*c[6]*c[2]*c[5]*c[3] ^4+1942305*c[6]*c[2]*c[5]*c[3]^5-1440*c[5]^2*c[6]^2*b[8]*a[7,2]*c[2]^2 *c[3]+54720*b[8]*c[6]^2*c[2]*c[5]^2*c[3]^3+163620*b[8]*c[6]^2*c[2]*c[3 ]^4*c[5]+81000*c[5]*c[2]*c[6]^2*c[3]^7*a[7,5]*b[8]+37224*c[5]*c[2]*c[3 ]^2-81000*c[2]*c[5]^2*c[6]^2*c[3]^7*a[7,6]*b[8]+27000*c[2]*c[5]^2*c[6] ^2*c[3]^7*a[7,6]-144*c[6]^2*c[2]+194400*c[6]*c[5]^3*c[2]*a[7,5]*b[8]*c [3]^5-1440*c[5]^2*c[6]^3*b[8]*a[7,6]*c[2]*c[3]-239280*c[6]^2*c[2]*c[5] *c[3]^3-239280*c[6]*c[2]*c[5]^2*c[3]^3+1458225*c[6]^2*c[2]*c[5]*c[3]^6 +27000*c[2]*c[5]^2*c[6]^2*c[3]^7*a[7,5]-234900*c[6]^2*c[2]*c[5]^2*c[3] ^6*b[8]+2016*c[6]*a[7,4]*c[2]*c[3]^3*c[5]-36450*c[6]*c[5]*c[2]*c[3]^6* a[7,5]+30420*c[6]*a[7,4]*c[2]*c[3]^5*c[5]+694920*c[6]*c[2]*c[5]^2*c[3] ^4+694920*c[6]^2*c[2]*c[5]*c[3]^4-1281150*c[6]^2*c[2]*c[5]*c[3]^5-5023 50*c[6]^2*c[2]*c[5]^2*c[3]^4+20250*c[5]*c[2]*c[3]^6*a[7,6]*c[6]^3-1656 0*c[5]^3*c[6]*b[8]*a[7,5]*c[2]*c[3]^2-6264*c[5]*c[6]^2*c[2]*c[3]-16200 *c[6]*c[5]*c[2]^2*a[7,2]*c[3]^6+31500*c[6]*a[7,6]*c[2]*c[3]^5*c[5]-597 60*c[5]^2*c[6]*b[8]*a[7,5]*c[2]*c[3]^3+16560*c[5]^2*c[6]*b[8]*a[7,6]*c [2]*c[3]^3+58950*c[5]^2*c[2]*c[6]^3*a[7,6]*c[3]^5+4440*c[5]^2*c[6]^2*c [2]*c[3]-38550*c[6]*a[7,6]*c[2]*c[3]^5*c[5]^2-4320*c[5]^2*a[7,2]*c[6]^ 2*c[2]^2*c[3]^2-9720*c[5]^2*b[8]*c[6]^2*c[2]*c[3]^2+45000*c[6]*c[5]^2* c[2]*c[3]^6*a[7,6]+194400*c[6]*c[5]*c[2]*c[3]^6*a[7,6]*b[8]-202500*c[2 ]*c[5]^2*c[6]^2*c[3]^8-931500*c[2]*c[5]*c[6]^2*c[3]^7+216*c[6]*c[2]+34 5750*c[2]*c[3]^5-27000*c[2]*c[3]^6*c[6]^2*a[7,5]*c[5]^3+384750*c[2]*c[ 3]^7*c[6]^2-11670*c[6]*a[7,4]*c[2]*c[3]^4*c[5]+81000*c[5]*c[2]*c[6]^2* c[3]^7*a[7,6]*b[8]-2179350*c[2]*c[3]^6*c[6]*c[5]-619200*c[2]*c[3]^6*c[ 6]^2-304650*c[5]^2*c[2]*c[3]^4+105480*c[5]^2*c[2]*c[3]^3-22572*c[5]^2* c[2]*c[3]^2+2736*c[5]^2*c[2]*c[3]+497100*c[5]*c[2]*c[3]^4-173028*c[5]* c[2]*c[3]^3+384750*c[5]^2*c[2]*c[3]^7+234900*c[5]*c[2]*c[6]^2*c[3]^6*b [8]-20250*c[5]*c[2]*c[6]^2*c[3]^7*a[7,5]-20250*c[5]*c[2]*c[6]^2*a[7,4] *c[3]^7-618300*c[5]*c[2]*c[3]^7-31500*c[6]*a[7,2]*c[2]^2*c[3]^4*c[5]-1 3380*c[6]*a[7,5]*c[2]*c[3]^4*c[5]+31500*c[6]*a[7,5]*c[2]*c[3]^5*c[5]-2 40*c[6]^2*c[5]^2*c[2]-81000*c[6]*c[5]*c[2]*c[3]^7*a[7,5]*b[8]+66960*c[ 6]*b[8]*a[7,4]*c[2]*c[3]^4*c[5]+174600*c[6]*b[8]*a[7,2]*c[2]^2*c[3]^4* c[5]+194400*c[6]*c[5]*c[2]*c[3]^6*a[7,5]*b[8]+194400*c[6]*c[5]*c[2]*a[ 7,4]*c[3]^6*b[8]-169200*c[6]*b[8]*a[7,4]*c[2]*c[3]^5*c[5]-16560*c[6]*b 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^2*c[3]^4-37200*a[7,4]*c[6]^2*c[2]*c[3]^5*c[5]-18030*a[7,4]*c[6]^2*c[2 ]*c[5]^2*c[3]^4+27000*a[7,4]*c[6]^2*c[2]*c[3]^7*c[5]^2-58950*a[7,4]*c[ 6]^2*c[2]*c[3]^6*c[5]^2+48000*a[7,4]*c[6]^2*c[2]*c[3]^5*c[5]^2-16260*c [6]^3*a[7,6]*c[2]*c[3]^3*c[5]-900600*c[5]*c[2]*c[3]^5+98280*c[6]*b[8]* a[7,5]*c[2]*c[5]^2*c[3]^4-16260*c[6]*a[7,5]*c[2]*c[5]^3*c[3]^3-4068*c[ 2]*c[6]*c[3]+1440*c[6]*b[8]*a[7,5]*c[2]*c[5]^3*c[3]-1440*c[6]*b[8]*a[7 ,6]*c[2]*c[5]^2*c[3]^2+144*c[6]*a[7,4]*c[2]*c[5]^2*c[3]^2-81000*c[2]*c [5]^2*c[6]^2*c[3]^7*a[7,5]*b[8]+20250*c[5]*c[2]^2*c[6]^2*a[7,2]*c[3]^6 -2100*c[6]^2*a[7,6]*c[2]*c[3]^5*c[5]+17790*c[6]^2*a[7,6]*c[2]*c[5]^2*c [3]^4-1440*c[6]*c[5]*b[8]*a[7,2]*c[2]^2*c[3]+1440*c[6]*c[5]*b[8]*a[7,5 ]*c[2]*c[3]^2+720*c[6]*c[5]*b[8]*a[7,4]*c[2]*c[3]^2+1440*c[6]*b[8]*a[7 ,6]*c[2]*c[5]*c[3]^2+81000*c[5]^2*c[2]*c[6]*a[7,4]*c[3]^7*b[8]-45000*c [6]^3*c[5]*c[2]*a[7,6]*c[3]^5+14160*c[6]*a[7,4]*c[2]*c[5]^2*c[3]^4-342 0*c[6]^2*a[7,2]*c[2]*c[5]*c[3]^3+16560*c[6]^2*a[7,2]*c[2]*c[5]*b[8]*c[ 3]^3+234900*c[5]^2*c[2]*c[6]*c[3]^6*b[8]-49800*c[6]^3*c[5]^2*c[2]*c[3] ^4*a[7,6]+67548*c[3]^3*c[2]-14652*c[3]^2*c[2]+11880*c[5]*b[8]*a[7,4]*c [2]*c[6]^2*c[3]^3+360*c[5]^2*c[6]^3*a[7,6]*c[2]*c[3]+9960*c[5]^2*c[6]* a[7,5]*c[2]*c[3]^3+16260*c[6]*a[7,6]*c[2]*c[5]^2*c[3]^4+194400*c[6]*a[ 7,2]*c[2]*b[8]*c[3]^6*c[5]+45000*c[6]*a[7,2]*c[2]*c[5]^2*c[3]^6-59760* c[5]*c[6]^2*b[8]*a[7,6]*c[2]*c[3]^3-19800*b[8]*a[7,5]*c[6]*c[2]*c[3]^5 *c[5]^2-20250*c[5]^2*c[2]*c[6]*a[7,4]*c[3]^7-364500*c[2]*c[3]^8*c[6]*c [5]+76320*c[6]*b[8]*a[7,2]*c[2]^2*c[5]^2*c[3]^3+174600*b[8]*a[7,2]*c[6 ]^2*c[2]^2*c[5]^2*c[3]^4+81000*b[8]*a[7,2]*c[6]^2*c[2]^2*c[3]^6*c[5]^2 -194400*b[8]*a[7,2]*c[6]^2*c[2]^2*c[3]^5*c[5]^2-76320*b[8]*a[7,2]*c[6] ^2*c[2]^2*c[5]^2*c[3]^3+3060*c[5]^2*a[7,4]*c[6]^2*c[2]*c[3]^3+38550*a[ 7,2]*c[6]*c[2]^2*c[5]^2*c[3]^4+20760*a[7,6]*c[6]^3*c[2]*c[5]^2*c[3]^3- 76320*c[6]^3*b[8]*a[7,6]*c[2]*c[5]^2*c[3]^3-13380*c[6]*a[7,6]*c[2]*c[3 ]^4*c[5]+54720*c[6]*b[8]*c[2]*c[3]^3*c[5]-144*c[5]^2*c[2]+16560*c[5]*c [6]^2*b[8]*a[7,5]*c[2]*c[3]^3-16560*c[5]*c[6]^3*b[8]*a[7,6]*c[2]*c[3]^ 2-931500*c[2]*c[5]^2*c[6]*c[3]^7+1458225*c[2]*c[5]^2*c[6]*c[3]^6+1440* c[6]^2*c[5]^2*b[8]*a[7,5]*c[2]*c[3]+3420*c[5]*c[6]^3*a[7,6]*c[2]*c[3]^ 2+1800*c[3]*c[2]-16560*c[5]*b[8]*a[7,2]*c[6]^2*c[2]^2*c[3]^2+288*c[6]* a[7,6]*c[2]*c[5]^2*c[3]^2+240*c[6]*c[5]^2*a[7,5]*c[2]*c[3]+240*c[6]*c[ 5]*a[7,2]*c[2]^2*c[3]-2430*c[5]^2*c[6]*a[7,4]*c[2]*c[3]^3-3420*c[5]^2* c[6]*a[7,6]*c[2]*c[3]^3-38550*c[6]*a[7,2]*c[2]*c[5]^2*c[3]^5+16260*c[6 ]*a[7,2]*c[2]*c[5]^2*c[3]^4+81000*c[6]*a[7,2]*c[2]*c[5]^2*b[8]*c[3]^7- 194400*c[6]*a[7,2]*c[2]*c[5]^2*b[8]*c[3]^6+174600*c[6]*a[7,2]*c[2]*c[5 ]^2*b[8]*c[3]^5-15240*c[6]*a[7,5]*c[2]*c[5]^2*c[3]^4+20760*a[7,2]*c[6] ^2*c[2]^2*c[5]^2*c[3]^3+38550*a[7,2]*c[6]^2*c[2]^2*c[3]^4*c[5]-163620* b[8]*c[6]^2*c[2]*c[5]^2*c[3]^4-194400*c[6]^2*c[5]*c[2]*c[3]^6*a[7,5]*b [8]+9960*c[5]*c[6]^2*a[7,6]*c[2]*c[3]^3-3420*c[5]*c[6]^2*a[7,5]*c[2]*c [3]^3-192450*c[2]*c[3]^4+9732*c[5]*c[2]*c[6]*c[3]-2430*c[5]*a[7,4]*c[6 ]^2*c[2]*c[3]^3-22572*c[6]^2*c[2]*c[3]^2+336*c[6]*c[5]^2*c[2]+3420*c[5 ]*a[7,2]*c[6]^2*c[2]^2*c[3]^2+9720*c[5]*b[8]*c[6]^2*c[2]*c[3]^2+81000* c[6]*c[2]*c[5]*c[3]^7*b[8]+2832*c[6]*a[7,6]*c[2]*c[3]^3*c[5]-36450*c[6 ]*c[5]*c[2]*a[7,4]*c[3]^6+271800*c[6]*c[5]*c[2]*c[3]^5*b[8]+16200*c[6] *c[5]*c[2]*c[3]^7*a[7,6]-288*c[6]^2*a[7,6]*c[2]*c[5]^2*c[3]+240*c[6]^2 *c[5]*a[7,6]*c[2]*c[3]-76320*c[6]*a[7,2]*c[2]*c[5]^2*b[8]*c[3]^4-20250 *c[6]*a[7,2]*c[2]*c[5]^2*c[3]^7+141750*c[2]*c[3]^8*c[6]-49800*c[6]^2*c [2]*c[5]^3*a[7,5]*c[3]^4+4800*c[6]^2*c[5]^2*c[2]*a[7,6]*c[3]^5+105480* c[6]^2*c[2]*c[3]^3-66960*c[6]*b[8]*a[7,4]*c[2]*c[5]^2*c[3]^4+4800*c[6] ^2*c[5]^2*c[2]*a[7,5]*c[3]^5+3420*c[5]^2*c[6]*a[7,2]*c[2]^2*c[3]^2+306 0*c[6]^2*a[7,5]*c[2]*c[5]^2*c[3]^2-1440*c[6]^2*a[7,2]*c[2]*c[5]*b[8]*c [3]^2+720*c[6]*c[5]*b[8]*c[2]*c[3]-426*c[5]*c[3]^4*c[6]-16260*c[6]*a[7 ,2]*c[2]^2*c[5]^2*c[3]^3-720*c[6]^2*c[5]*b[8]*c[2]*c[3]-174600*b[8]*a[ 7,5]*c[6]*c[2]*c[5]^3*c[3]^4+81000*c[5]^2*c[2]*c[6]*c[3]^7*a[7,5]*b[8] )/c[6]/c[5]/(c[3]-1)/(3*c[3]-1)/(5*c[3]-2)/(15*c[3]^2-10*c[3]+2)/(c[5] -1)/(-1+c[6])/c[3]^2/c[2]/b[8], a[8,7] = 0, b[6] = -1/60*(1-5*c[3]+5*c [3]^2)*(15*c[5]*c[3]-9*c[3]+4-6*c[5])/c[6]/(-1+c[6])/(2*c[6]-10*c[6]*c [3]+15*c[6]*c[3]^2-c[3])/(c[6]-c[3])/(c[6]-c[5]), a[3,1] = 1/2*c[3]*(- c[3]+2*c[2])/c[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 -1 140 "At this point we have an 8 stage order 6 scheme in which the linking coefficients in rows 2 to 6 are expressed in terms of the parameters " }{XPPEDIT 18 0 "c[2" "6#&%\"cG6#\"\"# " }{TEXT -1 3 ", " }{XPPEDIT 18 0 "c[3]" "6#&%\"cG6#\"\"$" }{TEXT -1 3 ", " }{XPPEDIT 18 0 "c[5]" "6#&%\"cG6#\"\"&" }{TEXT -1 7 " and " }{XPPEDIT 18 0 "c[6]" "6#&%\"cG6#\"\"'" }{TEXT -1 73 " while the coef ficients in rows 7 and 8 have the additional parameters " }{XPPEDIT 18 0 "a[7,2];" "6#&%\"aG6$\"\"(\"\"#" }{TEXT -1 3 ", " }{XPPEDIT 18 0 "a[7,4];" "6#&%\"aG6$\"\"(\"\"%" }{TEXT -1 3 ", " }{XPPEDIT 18 0 "a [7,5];" "6#&%\"aG6$\"\"(\"\"&" }{TEXT -1 7 " and " }{XPPEDIT 18 0 "a [7,6];" "6#&%\"aG6$\"\"(\"\"'" }{TEXT -1 1 "." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 21 "indets(map(r hs,e15));" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#<+&%\"cG6#\"\"'&F%6#\"\"$ &F%6#\"\"#&F%6#\"\"&&%\"aG6$\"\"(F0&%\"bG6#\"\")&F26$F4F-&F26$F4\"\"%& F26$F4F'" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 61 "We can check that the simple order conditions are satisfied. " }} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 56 "SO6_8 := SimpleOrderConditio ns(6,8,'expanded'):\nnops(%);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#\"#K " }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 142 "ind := []:\nfor ct to 32 do\n val := simplify(subs (e15,SO6_8[ct]));\n ind := [op(ind),lhs(val)-rhs(val)];\n print(ct );\nend do:\nind;\nnops(%);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#7B\"\"! F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#\"#K" }}}{PARA 0 "" 0 "" {TEXT -1 0 " " }}{PARA 15 "" 0 "" {TEXT 296 6 "Step 8" }{TEXT -1 1 " " }}{PARA 0 " " 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 57 "Turning attention \+ to the order 5 embedded scheme we set " }{XPPEDIT 18 0 "`b*`[2]=0" "6 #/&%#b*G6#\"\"#\"\"!" }{TEXT -1 7 " and " }{XPPEDIT 18 0 "`b*`[8]=0 " "6#/&%#b*G6#\"\")\"\"!" }{TEXT -1 36 " and use the quadrature condi tions:" }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "Sum(`b*`[i] ,i = 1 .. 8) = 1;" "6#/-%$SumG6$&%#b*G6#%\"iG/F*;\"\"\"\"\")F-" } {TEXT -1 15 ", " }{XPPEDIT 18 0 "Sum(`b*`[i]*c[i]^(k-1),i = 2 .. 7) = 1/k;" "6#/-%$SumG6$*&&%#b*G6#%\"iG\"\"\")&%\"cG6#F+,&%\"k GF,F,!\"\"F,/F+;\"\"#\"\"(*&F,F,F2F3" }{TEXT -1 7 ", " }{XPPEDIT 18 0 "k = 2;" "6#/%\"kG\"\"#" }{TEXT -1 8 " . . 5, " }}{PARA 0 "" 0 " " {TEXT -1 21 "to find the weights " }{XPPEDIT 18 0 "`b*`[1];" "6#&%# b*G6#\"\"\"" }{TEXT -1 2 ", " }{XPPEDIT 18 0 "`b*`[3];" "6#&%#b*G6#\" \"$" }{TEXT -1 3 ", " }{XPPEDIT 18 0 "`b*`[4];" "6#&%#b*G6#\"\"%" } {TEXT -1 3 ", " }{XPPEDIT 18 0 "`b*`[5];" "6#&%#b*G6#\"\"&" }{TEXT -1 7 " and " }{XPPEDIT 18 0 "`b*`[6];" "6#&%#b*G6#\"\"'" }{TEXT -1 15 " in terms of " }{XPPEDIT 18 0 "c[3]" "6#&%\"cG6#\"\"$" }{TEXT -1 3 ", " }{XPPEDIT 18 0 "c[5]" "6#&%\"cG6#\"\"&" }{TEXT -1 3 ", " } {XPPEDIT 18 0 "c[6]" "6#&%\"cG6#\"\"'" }{TEXT -1 7 " and " } {XPPEDIT 18 0 "`b*`[7]" "6#&%#b*G6#\"\"(" }{TEXT -1 1 "." }}{PARA 0 " " 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 264 "`qua d_cdns*` := [add(`b*`[i],i=1..8)=1,seq(add(`b*`[i]*c[i]^(k-1),i=2..8)= 1/k,k=2..5)]:\ne16 := \{`b*`[2]=0,`b*`[8]=0\}:\ne17 := `union`(e15,e16 ):\n`quad_eqns*` := simplify(subs(e19,`quad_cdns*`)):\nnops(%);\nindet s(`quad_eqns*`) minus \{c[3],c[5],c[6],`b*`[7]\};\nnops(%);\n" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#\"\"&" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#<'&%#b*G6#\"\"\"&F%6#\"\"$&F%6#\"\"%&F%6#\"\"&&F%6#\"\"'" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#\"\"&" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 22 "infolevel[solve] := 4:" }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 123 "e18 := solve(\{op(`quad_eqn s*`)\},\{`b*`[1],`b*`[3],`b*`[4],`b*`[5],`b*`[6]\}):\ne19 := `union`(e 17,e18):\ninfolevel[solve] := 0:" }}}{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 70886 "e19 := \{a[6,2] = 1/2*c[6]*c[3]/c[2]*(5*c[ 6]^2-5*c[6]+1)/(1-5*c[3]+5*c[3]^2), b[2] = 0, b[4] = 1/60*(15*c[3]^2-1 0*c[3]+2)^5*(3*c[3]-5*c[5]*c[3]-5*c[6]*c[3]+10*c[5]*c[6]*c[3]-5*c[5]*c [6]-2+3*c[6]+3*c[5])/(15*c[3]^2-10*c[3]+1)/c[3]^2/(3*c[3]-1)/(5*c[3]-2 )/(2*c[6]-10*c[6]*c[3]+15*c[6]*c[3]^2-c[3])/(15*c[5]*c[3]^2-10*c[5]*c[ 3]+2*c[5]-c[3]), b[7] = -1/60*(20-1560*c[6]*c[5]*b[8]*c[3]^2-24*c[6]-1 36*c[3]-900*c[5]*b[8]*c[3]^3+285*c[3]^2-780*c[6]*b[8]*c[3]+225*c[3]^3* c[5]+225*c[3]^3*c[6]-24*c[5]+780*c[5]*c[6]*b[8]*c[3]+900*c[5]*b[8]*c[3 ]^3*c[6]-900*b[8]*c[3]^3*c[6]+1560*b[8]*c[6]*c[3]^2-120*c[5]*b[8]*c[6] +1560*c[5]*b[8]*c[3]^2-780*c[5]*b[8]*c[3]+165*c[5]*c[3]+165*c[6]*c[3]+ 30*c[5]*c[6]-350*c[6]*c[3]^2-350*c[5]*c[3]^2-180*c[3]^3+120*b[8]*c[5]- 1560*b[8]*c[3]^2+780*b[8]*c[3]+900*b[8]*c[3]^3-120*b[8]+120*b[8]*c[6]- 210*c[5]*c[6]*c[3]-300*c[5]*c[3]^3*c[6]+455*c[5]*c[6]*c[3]^2)/(c[3]-1) /(-1+c[6])/(c[5]-1)/(3*c[3]-1)/(5*c[3]-2), a[6,3] = 1/6*(c[6]-c[3])*(4 50*c[5]^2*c[3]^4-810*c[6]^2*c[3]^2-12*c[6]^2-4*c[3]^2+156*c[6]^2*c[3]- 2700*c[6]^2*c[3]^4+45*c[3]^3*c[5]-820*c[3]^3*c[6]-1350*c[3]^5*c[6]+210 0*c[6]^2*c[3]^3+300*c[3]^4*c[5]-24*c[5]^2*c[6]+1815*c[3]^4*c[6]-2*c[5] *c[3]-10*c[6]*c[3]+28*c[5]*c[6]+148*c[6]*c[3]^2-35*c[5]*c[3]^2-450*c[3 ]^5*c[5]-480*c[5]^2*c[3]^3+135*c[5]^2*c[3]^2-6*c[5]^2*c[3]+1350*c[3]^5 *c[6]^2+89*c[3]^3-300*c[3]^4+270*c[3]^5-328*c[5]*c[6]*c[3]-2535*c[5]*c [3]^3*c[6]+1450*c[5]*c[6]*c[3]^2+900*c[3]^5*c[6]*c[5]+975*c[5]*c[3]^4* c[6]-2250*c[5]^2*c[3]^4*c[6]+3075*c[5]^2*c[3]^3*c[6]+300*c[5]^2*c[6]*c [3]-1470*c[5]^2*c[3]^2*c[6])*c[6]/(1-5*c[3]+5*c[3]^2)/(15*c[3]^2-10*c[ 3]+1)/(15*c[5]*c[3]-9*c[3]+4-6*c[5])/(c[5]-c[3])/c[3]^2, a[4,2] = 1/2* c[3]^2*(4-30*c[3]+45*c[3]^2)/c[2]/(15*c[3]^2-10*c[3]+2)^3, `b*`[5] = - 1/60*(-300*c[3]^3*c[6]+30*c[6]-210*c[6]*c[3]+455*c[6]*c[3]^2-1560*c[6] *c[3]^2*`b*`[7]+900*c[3]^3*c[6]*`b*`[7]+780*c[6]*c[3]*`b*`[7]-120*c[6] *`b*`[7]-24+165*c[3]-350*c[3]^2+225*c[3]^3+120*`b*`[7]+1560*`b*`[7]*c[ 3]^2-780*`b*`[7]*c[3]-900*`b*`[7]*c[3]^3)/(c[6]-c[5])/c[5]/(15*c[3]^3* c[5]-c[3]^2-10*c[5]*c[3]^2-15*c[5]^2*c[3]^2+3*c[5]*c[3]+10*c[5]^2*c[3] -2*c[5]^2), a[4,1] = 1/2*c[3]*(-4*c[3]+30*c[3]^2-45*c[3]^3+350*c[3]^2* c[2]-100*c[3]*c[2]+10*c[2]+450*c[2]*c[3]^4-600*c[3]^3*c[2])/c[2]/(15*c [3]^2-10*c[3]+2)^3, b[3] = -1/60*(75*c[5]*c[6]*c[3]^2-60*c[5]*c[6]*c[3 ]+10*c[5]*c[6]-45*c[5]*c[3]^2+35*c[5]*c[3]-6*c[5]-45*c[6]*c[3]^2+35*c[ 6]*c[3]-6*c[6]+30*c[3]^2-23*c[3]+4)/(c[6]-c[3])/(c[5]-c[3])/c[3]^2/(c[ 3]-1)/(15*c[3]^2-10*c[3]+1), b[5] = 1/60*(1-5*c[3]+5*c[3]^2)*(15*c[6]* c[3]-9*c[3]+4-6*c[6])/(c[6]-c[5])/c[5]/(c[5]-1)/(15*c[5]*c[3]^2-10*c[5 ]*c[3]+2*c[5]-c[3])/(c[5]-c[3]), `b*`[2] = 0, a[5,1] = 1/6*c[5]*(-225* c[5]*c[2]*c[3]^5+300*c[5]*c[2]*c[3]^4+45*c[2]*c[3]^4+225*c[5]^2*c[2]*c [3]^4-9*c[5]*c[2]*c[3]+4*c[5]^2*c[2]-40*c[3]^3*c[2]-15*c[5]^2*c[3]^3-3 *c[3]^3-190*c[5]*c[2]*c[3]^3-300*c[5]^2*c[2]*c[3]^3+15*c[3]^3*c[5]+11* c[3]^2*c[2]+175*c[5]^2*c[2]*c[3]^2+50*c[5]*c[2]*c[3]^2-40*c[5]^2*c[2]* c[3])/c[3]^2/c[2]/(1-5*c[3]+5*c[3]^2), a[8,2] = 1/120*(-48*c[6]*a[7,2] *c[2]+240*a[7,2]*c[2]*c[5]*b[8]-10*c[3]+21*c[3]^2-360*a[7,2]*c[2]*c[3] ^3-1800*c[6]*a[7,2]*c[2]*b[8]*c[3]^3-240*c[6]*a[7,2]*c[2]*c[5]*b[8]+31 20*c[6]*a[7,2]*c[2]*b[8]*c[3]^2-1560*c[6]*a[7,2]*c[2]*b[8]*c[3]+910*c[ 6]*a[7,2]*c[2]*c[5]*c[3]^2-240*a[7,2]*c[2]*b[8]-1560*a[7,2]*c[2]*c[5]* b[8]*c[3]-1800*a[7,2]*c[2]*c[5]*b[8]*c[3]^3+3120*a[7,2]*c[2]*c[5]*b[8] *c[3]^2-48*a[7,2]*c[2]*c[5]-420*c[6]*a[7,2]*c[2]*c[5]*c[3]+1800*c[6]*a [7,2]*c[2]*c[5]*b[8]*c[3]^3-600*c[6]*a[7,2]*c[2]*c[5]*c[3]^3-3120*c[6] *a[7,2]*c[2]*c[5]*b[8]*c[3]^2+14*c[5]*c[3]+14*c[6]*c[3]-30*c[6]*c[3]^2 -30*c[5]*c[3]^2+40*a[7,2]*c[2]-20*c[5]*c[6]*c[3]+45*c[5]*c[6]*c[3]^2-2 72*a[7,2]*c[2]*c[3]+570*a[7,2]*c[2]*c[3]^2-700*a[7,2]*c[2]*c[5]*c[3]^2 +450*a[7,2]*c[2]*c[5]*c[3]^3+60*c[6]*a[7,2]*c[2]*c[5]-700*c[6]*a[7,2]* c[2]*c[3]^2-3120*a[7,2]*c[2]*b[8]*c[3]^2+330*c[6]*a[7,2]*c[2]*c[3]+240 *c[6]*a[7,2]*c[2]*b[8]+450*c[6]*a[7,2]*c[2]*c[3]^3+1800*a[7,2]*c[2]*b[ 8]*c[3]^3+1560*a[7,2]*c[2]*b[8]*c[3]+330*a[7,2]*c[2]*c[5]*c[3]+1560*c[ 6]*a[7,2]*c[2]*c[5]*b[8]*c[3])/b[8]/c[2]/(c[3]-1)/(3*c[3]-1)/(5*c[3]-2 )/(c[5]-1)/(-1+c[6]), a[3,2] = 1/2*c[3]^2/c[2], a[5,2] = 1/2*c[3]*c[5] /c[2]*(-5*c[5]+1+5*c[5]^2)/(1-5*c[3]+5*c[3]^2), a[8,5] = 1/60*(8+13500 *a[7,5]*c[5]^4*b[8]*c[3]^5*c[6]^2-13500*a[7,5]*c[5]^5*b[8]*c[3]^5*c[6] +835*c[5]^2*c[3]^4+1125*c[3]^6*c[6]^2+9180*a[7,5]*c[5]^4*b[8]*c[3]^2-3 1500*a[7,5]*c[5]^3*b[8]*c[3]^5+2400*a[7,5]*c[5]^4*b[8]*c[3]^4+18900*a[ 7,5]*c[5]^4*b[8]*c[3]^5-12*c[5]*c[6]^2+29100*a[7,5]*c[5]^4*c[6]^2*b[8] *c[3]^3-29100*a[7,5]*c[5]^5*c[6]*b[8]*c[3]^3+951*c[6]^2*c[3]^2+2760*a[ 7,5]*c[5]^3*c[6]*b[8]*c[3]+28320*a[7,5]*c[5]^3*b[8]*c[3]^3*c[6]-18*c[6 ]-110*c[3]+12*c[6]^2+611*c[3]^2+3420*a[7,5]*c[5]^2*c[6]^2*b[8]*c[3]^2- 168*c[6]^2*c[3]-690*c[3]^5*c[5]^2+2400*a[7,5]*c[5]^3*b[8]*c[3]^4*c[6]^ 2-31500*a[7,5]*c[5]^2*b[8]*c[3]^5*c[6]^2-472*a[7,5]*c[5]^3*c[6]*c[3]-2 400*a[7,5]*c[5]^3*c[3]^5*c[6]+2200*a[7,5]*c[5]^3*c[3]^2*c[6]-900*c[3]^ 6*c[5]-40*a[7,5]*c[5]^4+12720*a[7,5]*c[5]^5*c[6]*b[8]*c[3]^2-32400*a[7 ,5]*c[5]^4*b[8]*c[3]^4*c[6]^2+4380*c[6]^2*c[3]^4-8600*a[7,5]*c[5]^4*c[ 3]^4*c[6]+3915*a[7,5]*c[5]^4*c[3]^3*c[6]-240*a[7,5]*c[5]^3*b[8]*c[6]+3 420*a[7,5]*c[5]^3*b[8]*c[3]^2-32400*a[7,5]*c[5]^4*b[8]*c[3]^5*c[6]-142 80*a[7,5]*c[5]^4*c[6]*b[8]*c[3]^3+360*a[7,5]*c[5]^2*c[6]*b[8]*c[3]+142 80*a[7,5]*c[5]^2*b[8]*c[3]^3*c[6]-1560*a[7,5]*c[5]*b[8]*c[3]^4*c[6]+28 440*a[7,5]*c[5]^2*b[8]*c[3]^4*c[6]^2-13500*a[7,5]*c[5]^3*b[8]*c[3]^6*c [6]^2+13500*a[7,5]*c[5]^4*b[8]*c[3]^6*c[6]+18900*a[7,5]*c[5]^3*b[8]*c[ 3]^5*c[6]^2+1575*c[3]^6*c[5]*c[6]+225*c[3]^6*c[5]^2-14820*a[7,5]*c[5]^ 3*b[8]*c[3]^3*c[6]^2+9180*a[7,5]*c[5]^3*c[6]^2*b[8]*c[3]^2-360*a[7,5]* c[5]^2*c[6]^2*b[8]*c[3]+2710*a[7,5]*c[5]^5*c[3]^2-780*a[7,5]*c[5]*b[8] *c[3]^3*c[6]^2+2262*c[3]^3*c[5]+120*a[7,5]*c[5]*b[8]*c[6]^2*c[3]^2+401 1*c[3]^3*c[6]+4980*c[3]^5*c[6]+13500*a[7,5]*c[5]^3*b[8]*c[3]^6+2700*a[ 7,5]*c[5]^2*c[3]^6*c[6]-60*a[7,5]*c[5]^2*c[6]*c[3]-2515*a[7,5]*c[5]^2* c[3]^3*c[6]-13500*a[7,5]*c[5]^2*b[8]*c[3]^3*c[6]^2+2760*a[7,5]*c[5]^4* c[6]^2*b[8]*c[3]-2760*a[7,5]*c[5]^5*c[6]*b[8]*c[3]-6425*a[7,5]*c[5]^5* c[3]^3+48*a[7,5]*c[5]^5+608*a[7,5]*c[5]^2*c[6]*c[3]^2-780*a[7,5]*c[5]^ 2*b[8]*c[3]^3+120*a[7,5]*c[5]^2*b[8]*c[3]^2-10*c[5]-12720*a[7,5]*c[5]^ 4*c[6]^2*b[8]*c[3]^2-2775*c[6]^2*c[3]^3-12600*a[7,5]*c[5]^3*c[6]*b[8]* c[3]^2+9825*a[7,5]*c[5]^4*c[3]^5*c[6]+90*a[7,5]*c[5]^4*c[6]*c[3]-3540* a[7,5]*c[5]^2*c[6]*b[8]*c[3]^2-360*a[7,5]*c[5]^4*c[6]*b[8]*c[3]-30000* a[7,5]*c[5]^2*b[8]*c[3]^4*c[6]+40*a[7,5]*c[5]^3*c[6]+780*a[7,5]*c[5]*b [8]*c[3]^3*c[6]-120*a[7,5]*c[5]*b[8]*c[6]*c[3]^2-3545*c[3]^4*c[5]+900* a[7,5]*c[5]*b[8]*c[3]^5*c[6]-30840*a[7,5]*c[5]^3*b[8]*c[3]^4*c[6]+3240 0*a[7,5]*c[5]^2*b[8]*c[3]^5*c[6]+180*a[7,5]*c[5]^2*c[3]^5-6075*a[7,5]* c[5]^2*c[3]^5*c[6]+5620*a[7,5]*c[5]^3*c[3]^4*c[6]-5040*a[7,5]*c[5]^3*c [3]^3*c[6]-930*a[7,5]*c[5]^4*c[3]^2*c[6]+1560*a[7,5]*c[5]*b[8]*c[3]^4* c[6]^2+13500*a[7,5]*c[5]^2*b[8]*c[3]^6*c[6]^2-900*a[7,5]*c[5]*b[8]*c[3 ]^5*c[6]^2-360*a[7,5]*c[5]^3*b[8]*c[3]-13500*a[7,5]*c[5]^3*b[8]*c[3]^3 +28440*a[7,5]*c[5]^3*b[8]*c[3]^4-900*a[7,5]*c[5]^2*b[8]*c[3]^5+3540*a[ 7,5]*c[5]^4*c[6]*b[8]*c[3]^2+30000*a[7,5]*c[5]^4*b[8]*c[3]^4*c[6]+1260 0*a[7,5]*c[5]^3*b[8]*c[3]^5*c[6]+5850*a[7,5]*c[5]^3*c[3]^5+1560*a[7,5] *c[5]^2*b[8]*c[3]^4-2400*a[7,5]*c[5]^4*b[8]*c[3]+20*a[7,5]*c[5]*c[6]*c [3]^2-136*a[7,5]*c[5]*c[3]^3*c[6]-6255*c[3]^4*c[6]-2400*a[7,5]*c[5]^3* c[6]^2*b[8]*c[3]+32400*a[7,5]*c[5]^5*b[8]*c[3]^4*c[6]-13500*a[7,5]*c[5 ]^2*b[8]*c[3]^6*c[6]+4500*a[7,5]*c[5]^3*c[3]^6*c[6]^2-4500*a[7,5]*c[5] ^4*c[3]^6*c[6]-6450*a[7,5]*c[5]^3*c[3]^5*c[6]^2-240*a[7,5]*c[5]^4*b[8] *c[6]^2-14820*a[7,5]*c[5]^4*b[8]*c[3]^3+285*a[7,5]*c[5]*c[3]^4*c[6]-18 0*a[7,5]*c[5]*c[3]^5*c[6]+5430*a[7,5]*c[5]^2*c[3]^4*c[6]+139*c[5]*c[3] +249*c[6]*c[3]+18*c[5]*c[6]-1392*c[6]*c[3]^2-781*c[5]*c[3]^2+2835*c[3] ^5*c[5]-513*c[5]^2*c[3]^3+170*c[5]^2*c[3]^2-29*c[5]^2*c[3]-3525*c[3]^5 *c[6]^2-1575*c[3]^6*c[6]-1749*c[3]^3+675*c[3]^6+2*c[5]^2+2710*c[3]^4-2 145*c[3]^5+240*a[7,5]*c[5]^5*b[8]*c[6]+2760*a[7,5]*c[5]^5*b[8]*c[3]-24 *a[7,5]*c[5]*c[6]^2*c[3]^2+165*a[7,5]*c[5]*c[3]^3*c[6]^2+8300*a[7,5]*c [5]^5*c[3]^3*c[6]+240*a[7,5]*c[5]^3*b[8]*c[6]^2-13500*a[7,5]*c[5]^4*b[ 8]*c[3]^6-3375*a[7,5]*c[5]^2*c[3]^6*c[6]^2+72*a[7,5]*c[5]^2*c[6]^2*c[3 ]+2850*a[7,5]*c[5]^2*c[3]^3*c[6]^2-705*a[7,5]*c[5]^2*c[6]^2*c[3]^2+720 0*a[7,5]*c[5]^2*c[3]^5*c[6]^2+1100*a[7,5]*c[5]^3*c[3]^4*c[6]^2+2510*a[ 7,5]*c[5]^3*c[3]^3*c[6]^2+3460*a[7,5]*c[5]^4*c[3]^2*c[6]^2-3460*a[7,5] *c[5]^5*c[3]^2*c[6]-4500*a[7,5]*c[5]^4*c[3]^5*c[6]^2+4500*a[7,5]*c[5]^ 5*c[3]^5*c[6]-720*a[7,5]*c[5]^4*c[6]^2*c[3]+720*a[7,5]*c[5]^5*c[6]*c[3 ]-12720*a[7,5]*c[5]^5*b[8]*c[3]^2-32400*a[7,5]*c[5]^5*b[8]*c[3]^4+1350 0*a[7,5]*c[5]^5*b[8]*c[3]^5+29100*a[7,5]*c[5]^5*b[8]*c[3]^3+9825*a[7,5 ]*c[5]^4*c[3]^4*c[6]^2-9825*a[7,5]*c[5]^5*c[3]^4*c[6]-8300*a[7,5]*c[5] ^4*c[3]^3*c[6]^2-350*a[7,5]*c[5]*c[3]^4*c[6]^2+225*a[7,5]*c[5]*c[3]^5* c[6]^2-6195*a[7,5]*c[5]^2*c[3]^4*c[6]^2+480*a[7,5]*c[5]^3*c[6]^2*c[3]- 1780*a[7,5]*c[5]^3*c[3]^2*c[6]^2-249*c[5]*c[6]*c[3]-951*c[5]*c[6]^2*c[ 3]^2+168*c[5]*c[6]^2*c[3]-1125*c[5]*c[3]^6*c[6]^2-4011*c[5]*c[3]^3*c[6 ]+1392*c[5]*c[6]*c[3]^2-4980*c[3]^5*c[6]*c[5]-570*a[7,5]*c[5]^5*c[3]+6 0*a[7,5]*c[5]^4*c[6]^2-60*a[7,5]*c[5]^5*c[6]+3375*a[7,5]*c[5]^4*c[3]^6 -48*a[7,5]*c[5]^3*c[6]^2+2775*c[5]*c[6]^2*c[3]^3-4380*c[5]*c[3]^4*c[6] ^2+3525*c[5]*c[3]^5*c[6]^2+2190*a[7,5]*c[5]^4*c[3]^3-1495*a[7,5]*c[5]^ 4*c[3]^2+400*a[7,5]*c[5]^4*c[3]-5080*a[7,5]*c[5]^3*c[3]^4-2700*a[7,5]* c[5]^3*c[3]^6+136*a[7,5]*c[5]^2*c[3]^3+2350*a[7,5]*c[5]^3*c[3]^3-584*a [7,5]*c[5]^3*c[3]^2-20*a[7,5]*c[5]^2*c[3]^2+60*a[7,5]*c[5]^3*c[3]+240* a[7,5]*c[5]^4*b[8]+575*a[7,5]*c[5]^4*c[3]^4-4800*a[7,5]*c[5]^4*c[3]^5- 285*a[7,5]*c[5]^2*c[3]^4-240*a[7,5]*c[5]^5*b[8]+7500*a[7,5]*c[5]^5*c[3 ]^4-3375*a[7,5]*c[5]^5*c[3]^5+6255*c[5]*c[3]^4*c[6])/b[8]/c[5]/(c[5]-1 )/(c[3]-1)/(3*c[3]-1)/(5*c[3]-2)/(15*c[5]*c[3]^2-10*c[5]*c[3]+2*c[5]-c [3])/(c[5]-c[3])/(-1+c[6])/(c[6]-c[5]), c[4] = c[3]/(15*c[3]^2-10*c[3] +2), a[2,1] = c[2], c[7] = 1, a[8,6] = 1/60*(-8-4380*c[5]^2*c[3]^4-982 5*a[7,6]*c[5]^2*c[6]^4*c[3]^4-2700*a[7,6]*c[3]^6*c[5]*c[6]^2-720*a[7,6 ]*c[5]*c[6]^5*c[3]-400*a[7,6]*c[6]^4*c[3]+48*a[7,6]*c[5]^2*c[6]^3-240* a[7,6]*b[8]*c[6]^4-12600*a[7,6]*c[5]*b[8]*c[3]^5*c[6]^3-32400*a[7,6]*c [5]*b[8]*c[3]^5*c[6]^2+8*c[6]+110*c[3]+30000*a[7,6]*c[5]*b[8]*c[3]^4*c [6]^2+360*a[7,6]*c[5]*c[6]^4*b[8]*c[3]+2400*a[7,6]*c[5]^2*c[6]^3*b[8]* c[3]-3540*a[7,6]*c[6]^4*c[5]*b[8]*c[3]^2+14280*a[7,6]*c[6]^4*c[5]*b[8] *c[3]^3-611*c[3]^2-90*a[7,6]*c[5]*c[6]^4*c[3]+705*a[7,6]*c[5]^2*c[6]^2 *c[3]^2-2200*a[7,6]*c[5]*c[6]^3*c[3]^2+3525*c[3]^5*c[5]^2+1560*a[7,6]* c[5]*b[8]*c[3]^4*c[6]-3525*c[5]^2*c[3]^5*c[6]-136*a[7,6]*c[3]^3*c[6]^2 +1495*a[7,6]*c[6]^4*c[3]^2+20*a[7,6]*c[6]^2*c[3]^2+1800*c[3]^6*c[5]+13 500*a[7,6]*c[5]^2*b[8]*c[3]^3*c[6]^2-2760*a[7,6]*b[8]*c[6]^5*c[3]-20*a [7,6]*c[5]*c[6]*c[3]^2+136*a[7,6]*c[5]*c[3]^3*c[6]+12720*a[7,6]*c[6]^4 *c[5]^2*b[8]*c[3]^2-29100*a[7,6]*c[6]^4*c[5]^2*b[8]*c[3]^3+32400*a[7,6 ]*c[6]^4*c[5]^2*b[8]*c[3]^4+570*a[7,6]*c[6]^5*c[3]-1800*c[3]^6*c[5]*c[ 6]+2760*a[7,6]*c[5]*c[6]^5*b[8]*c[3]-2760*a[7,6]*c[5]^2*c[6]^4*b[8]*c[ 3]+13500*a[7,6]*c[5]*b[8]*c[3]^5*c[6]^5+3375*a[7,6]*c[3]^5*c[6]^5-1125 *c[3]^6*c[5]^2-4524*c[3]^3*c[5]-1749*c[3]^3*c[6]-2145*c[3]^5*c[6]+1350 0*a[7,6]*b[8]*c[3]^6*c[6]^4+240*a[7,6]*c[5]^2*b[8]*c[6]^4-240*a[7,6]*c [5]*b[8]*c[6]^5-900*a[7,6]*c[5]*b[8]*c[3]^5*c[6]-285*a[7,6]*c[3]^4*c[5 ]*c[6]-5850*a[7,6]*c[3]^5*c[6]^3-30000*a[7,6]*c[6]^4*c[5]*b[8]*c[3]^4- 9180*a[7,6]*c[6]^3*c[5]^2*b[8]*c[3]^2+14820*a[7,6]*c[6]^3*c[5]^2*b[8]* c[3]^3-2190*a[7,6]*c[6]^4*c[3]^3-480*a[7,6]*c[5]^2*c[6]^3*c[3]+6075*a[ 7,6]*c[5]*c[3]^5*c[6]^2+1780*a[7,6]*c[5]^2*c[6]^3*c[3]^2+180*a[7,6]*c[ 3]^5*c[5]*c[6]-3915*a[7,6]*c[5]*c[6]^4*c[3]^3+930*a[7,6]*c[5]*c[6]^4*c [3]^2+20*c[5]-7500*a[7,6]*c[6]^5*c[3]^4-3375*a[7,6]*c[3]^6*c[6]^4+6425 *a[7,6]*c[6]^5*c[3]^3-2710*a[7,6]*c[6]^5*c[3]^2-780*a[7,6]*c[5]*b[8]*c [3]^3*c[6]-120*a[7,6]*b[8]*c[6]^2*c[3]^2-29100*a[7,6]*c[6]^5*b[8]*c[3] ^3+32400*a[7,6]*b[8]*c[3]^4*c[6]^5-13500*a[7,6]*b[8]*c[3]^5*c[6]^5-360 *a[7,6]*c[5]*c[6]^2*b[8]*c[3]-14280*a[7,6]*c[5]*b[8]*c[3]^3*c[6]^2+780 *a[7,6]*c[5]^2*b[8]*c[3]^3*c[6]+3460*a[7,6]*c[5]*c[6]^5*c[3]^2-8300*a[ 7,6]*c[5]*c[6]^5*c[3]^3+9825*a[7,6]*c[5]*c[6]^5*c[3]^4+8600*a[7,6]*c[5 ]*c[6]^4*c[3]^4-1100*a[7,6]*c[5]^2*c[6]^3*c[3]^4-2510*a[7,6]*c[5]^2*c[ 6]^3*c[3]^3+12720*a[7,6]*b[8]*c[6]^5*c[3]^2-9825*a[7,6]*c[5]*c[3]^5*c[ 6]^4+6450*a[7,6]*c[5]^2*c[3]^5*c[6]^3+3375*a[7,6]*c[5]^2*c[3]^6*c[6]^2 +7090*c[3]^4*c[5]+3540*a[7,6]*c[6]^2*c[5]*b[8]*c[3]^2-120*a[7,6]*c[6]* c[5]^2*b[8]*c[3]^2+12*c[5]^2*c[6]+13500*a[7,6]*c[5]*b[8]*c[3]^6*c[6]^2 +60*a[7,6]*c[5]*c[6]^2*c[3]+2515*a[7,6]*c[5]*c[3]^3*c[6]^2+8300*a[7,6] *c[5]^2*c[6]^4*c[3]^3+584*a[7,6]*c[6]^3*c[3]^2-4500*a[7,6]*c[5]*c[3]^5 *c[6]^5+4500*a[7,6]*c[5]*c[3]^6*c[6]^4+4500*a[7,6]*c[5]^2*c[3]^5*c[6]^ 4-4500*a[7,6]*c[5]^2*c[3]^6*c[6]^3+12600*a[7,6]*c[6]^3*c[5]*b[8]*c[3]^ 2-3420*a[7,6]*c[6]^2*c[5]^2*b[8]*c[3]^2-2400*a[7,6]*c[6]^3*c[5]^2*b[8] *c[3]^4+32400*a[7,6]*c[5]*b[8]*c[3]^5*c[6]^4+30840*a[7,6]*c[5]*b[8]*c[ 3]^4*c[6]^3+4800*a[7,6]*c[3]^5*c[6]^4-165*a[7,6]*c[5]^2*c[3]^3*c[6]-60 8*a[7,6]*c[5]*c[6]^2*c[3]^2+24*a[7,6]*c[5]^2*c[6]*c[3]^2+780*a[7,6]*b[ 8]*c[3]^3*c[6]^2+29100*a[7,6]*c[6]^5*c[5]*b[8]*c[3]^3-32400*a[7,6]*c[6 ]^5*c[5]*b[8]*c[3]^4-48*a[7,6]*c[6]^5+2710*c[3]^4*c[6]+720*a[7,6]*c[5] ^2*c[6]^4*c[3]-3460*a[7,6]*c[5]^2*c[6]^4*c[3]^2-28440*a[7,6]*b[8]*c[3] ^4*c[6]^3+31500*a[7,6]*b[8]*c[3]^5*c[6]^3+900*a[7,6]*b[8]*c[3]^5*c[6]^ 2+2400*a[7,6]*b[8]*c[6]^4*c[3]-225*a[7,6]*c[3]^5*c[5]^2*c[6]+472*a[7,6 ]*c[5]*c[6]^3*c[3]-12720*a[7,6]*c[6]^5*c[5]*b[8]*c[3]^2-278*c[5]*c[3]- 110*c[6]*c[3]-20*c[5]*c[6]+611*c[6]*c[3]^2+1562*c[5]*c[3]^2-5670*c[3]^ 5*c[5]+2775*c[5]^2*c[3]^3-951*c[5]^2*c[3]^2+168*c[5]^2*c[3]+675*c[3]^6 *c[6]+1749*c[3]^3-675*c[3]^6+40*a[7,6]*c[6]^4-12*c[5]^2-2710*c[3]^4+21 45*c[3]^5+278*c[5]*c[6]*c[3]+4524*c[5]*c[3]^3*c[6]-1562*c[5]*c[6]*c[3] ^2+5670*c[3]^5*c[6]*c[5]-7090*c[5]*c[3]^4*c[6]+285*a[7,6]*c[3]^4*c[6]^ 2-180*a[7,6]*c[3]^5*c[6]^2-575*a[7,6]*c[6]^4*c[3]^4+5080*a[7,6]*c[3]^4 *c[6]^3-2350*a[7,6]*c[3]^3*c[6]^3-60*a[7,6]*c[5]^2*c[6]^4+240*a[7,6]*b [8]*c[6]^5+60*a[7,6]*c[5]*c[6]^5-60*a[7,6]*c[6]^3*c[3]+2700*a[7,6]*c[3 ]^6*c[6]^3-40*a[7,6]*c[5]*c[6]^3+4380*c[5]^2*c[3]^4*c[6]-2775*c[5]^2*c [3]^3*c[6]-168*c[5]^2*c[6]*c[3]+951*c[5]^2*c[3]^2*c[6]+1125*c[3]^6*c[5 ]^2*c[6]-2760*a[7,6]*c[5]*c[6]^3*b[8]*c[3]+360*a[7,6]*c[5]^2*c[6]^2*b[ 8]*c[3]-28320*a[7,6]*c[5]*b[8]*c[3]^3*c[6]^3-1560*a[7,6]*c[5]^2*b[8]*c [3]^4*c[6]-13500*a[7,6]*c[5]*b[8]*c[3]^6*c[6]^4-13500*a[7,6]*c[5]^2*b[ 8]*c[3]^5*c[6]^4+13500*a[7,6]*c[5]^2*b[8]*c[3]^6*c[6]^3-18900*a[7,6]*c [5]^2*b[8]*c[3]^5*c[6]^3-28440*a[7,6]*c[5]^2*b[8]*c[3]^4*c[6]^2+31500* a[7,6]*c[5]^2*b[8]*c[3]^5*c[6]^2-13500*a[7,6]*c[5]^2*b[8]*c[3]^6*c[6]^ 2+900*a[7,6]*c[5]^2*b[8]*c[3]^5*c[6]-72*a[7,6]*c[5]^2*c[6]^2*c[3]+5040 *a[7,6]*c[5]*c[3]^3*c[6]^3-5430*a[7,6]*c[5]*c[3]^4*c[6]^2-2850*a[7,6]* c[5]^2*c[3]^3*c[6]^2+350*a[7,6]*c[5]^2*c[3]^4*c[6]+360*a[7,6]*c[6]^3*b [8]*c[3]+13500*a[7,6]*b[8]*c[3]^3*c[6]^3-1560*a[7,6]*b[8]*c[3]^4*c[6]^ 2-3420*a[7,6]*b[8]*c[6]^3*c[3]^2+240*a[7,6]*c[5]*b[8]*c[6]^3+14820*a[7 ,6]*c[6]^4*b[8]*c[3]^3-2400*a[7,6]*b[8]*c[3]^4*c[6]^4-18900*a[7,6]*b[8 ]*c[3]^5*c[6]^4-13500*a[7,6]*b[8]*c[3]^6*c[6]^3-240*a[7,6]*c[5]^2*b[8] *c[6]^3-9180*a[7,6]*b[8]*c[6]^4*c[3]^2-5620*a[7,6]*c[5]*c[3]^4*c[6]^3+ 2400*a[7,6]*c[5]*c[3]^5*c[6]^3+6195*a[7,6]*c[5]^2*c[3]^4*c[6]^2-7200*a [7,6]*c[5]^2*c[3]^5*c[6]^2+120*a[7,6]*c[6]*c[5]*b[8]*c[3]^2)/b[8]/c[6] /(2*c[6]-10*c[6]*c[3]+15*c[6]*c[3]^2-c[3])/(c[6]-c[3])/(c[6]-c[5])/(c[ 3]-1)/(-1+c[6])/(c[5]-1)/(3*c[3]-1)/(5*c[3]-2), a[5,3] = -1/6*c[5]*(15 0*c[3]^3*c[5]-145*c[5]*c[3]^2+40*c[5]*c[3]-4*c[5]-30*c[3]^3+20*c[3]^2- c[3])*(c[5]-c[3])/(1-5*c[3]+5*c[3]^2)/c[3]^2/(15*c[3]^2-10*c[3]+1), a[ 6,4] = -1/6*c[6]*(15*c[3]^2-10*c[3]+2)^2*(2*c[6]-10*c[6]*c[3]+15*c[6]* c[3]^2-c[3])*(c[6]-c[3])*(90*c[3]^3*c[6]-120*c[6]*c[3]^2+48*c[6]*c[3]- 6*c[6]-225*c[3]^3*c[5]+225*c[5]^2*c[3]^3-240*c[5]^2*c[3]^2+9*c[3]^2+25 5*c[5]*c[3]^2-100*c[5]*c[3]-4*c[3]+90*c[5]^2*c[3]+14*c[5]-12*c[5]^2)/c [3]^2/(15*c[3]^2-10*c[3]+1)/(1-5*c[3]+5*c[3]^2)/(15*c[5]*c[3]^2-10*c[5 ]*c[3]+2*c[5]-c[3])/(15*c[5]*c[3]-9*c[3]+4-6*c[5]), a[5,4] = -1/6*(15* c[3]^2-10*c[3]+2)^2*c[5]*(c[5]-c[3])*(15*c[5]*c[3]^2-10*c[5]*c[3]+2*c[ 5]-c[3])/c[3]^2/(1-5*c[3]+5*c[3]^2)/(15*c[3]^2-10*c[3]+1), a[7,3] = -1 /2*(30*a[7,2]*c[2]*c[3]^2-20*a[7,2]*c[2]*c[3]+4*a[7,2]*c[2]+2*a[7,4]*c [3]+30*a[7,5]*c[5]*c[3]^2-20*a[7,5]*c[5]*c[3]+4*a[7,5]*c[5]+30*a[7,6]* c[6]*c[3]^2-20*a[7,6]*c[6]*c[3]+4*a[7,6]*c[6]-15*c[3]^2+10*c[3]-2)/c[3 ]/(15*c[3]^2-10*c[3]+2), b[1] = -1/60*1/c[3]^2*(150*c[5]*c[3]^3*c[6]-7 5*c[3]^3*c[6]+45*c[3]^3-75*c[3]^3*c[5]+105*c[5]*c[3]^2-205*c[5]*c[6]*c [3]^2+105*c[6]*c[3]^2-65*c[3]^2+29*c[3]-45*c[5]*c[3]+80*c[5]*c[6]*c[3] -45*c[6]*c[3]+6*c[5]-10*c[5]*c[6]-4+6*c[6])/c[5]/c[6], a[7,1] = 1/2*(- 2-4*a[7,5]*c[3]-30*a[7,2]*c[3]^3+14*c[3]-35*c[3]^2-4*a[7,6]*c[3]+4*a[7 ,6]*c[6]+20*a[7,4]*c[3]^2-2*a[7,4]*c[3]+20*a[7,2]*c[3]^2+4*a[7,5]*c[5] +20*a[7,5]*c[3]^2-30*a[7,4]*c[3]^3+20*a[7,6]*c[3]^2+4*a[7,2]*c[2]-30*a [7,6]*c[3]^3-4*a[7,2]*c[3]-30*a[7,5]*c[3]^3+30*c[3]^3-20*a[7,6]*c[6]*c [3]+30*a[7,6]*c[6]*c[3]^2-20*a[7,2]*c[2]*c[3]-20*a[7,5]*c[5]*c[3]+30*a [7,2]*c[2]*c[3]^2+30*a[7,5]*c[5]*c[3]^2)/c[3]/(15*c[3]^2-10*c[3]+2), a [8,4] = 1/360*(384+9297240*c[5]^2*c[3]^4+960*c[5]^2*c[6]^2+30960*a[7,4 ]*c[3]^4*b[8]*c[6]^2+720*a[7,4]*c[3]^2*c[5]^2*c[6]^2-1119240*a[7,4]*c[ 3]^6*c[5]*b[8]+4556250*a[7,4]*c[3]^11*c[5]*c[6]^2-3645000*a[7,4]*c[3]^ 11*c[5]*c[6]+4556250*a[7,4]*c[3]^11*c[5]^2*c[6]+1215000*a[7,4]*c[3]^10 *c[5]^2*b[8]+102802200*c[3]^6*c[6]^2-1344*c[5]*c[6]^2+213792*c[6]^2*c[ 3]^2+36450*a[7,4]*c[3]^8-263520*a[7,4]*c[3]^5*c[5]^2*b[8]+2916000*a[7, 4]*c[3]^7*b[8]*c[6]+1186200*a[7,4]*c[3]^6*b[8]*c[6]^2-797040*a[7,4]*c[ 3]^4*c[6]*c[5]*b[8]+66825000*a[7,4]*c[3]^10*c[5]^2*b[8]*c[6]-1215000*a [7,4]*c[3]^10*b[8]*c[6]+3645000*a[7,4]*c[3]^9*b[8]*c[6]-864*c[6]-10800 *c[3]+22781250*c[3]^12*c[6]^2+184680000*c[3]^11*c[5]+11759400*a[7,4]*c [3]^7*c[5]*c[6]^2+3374730*a[7,4]*c[3]^6*c[5]*c[6]-16243500*a[7,4]*c[3] ^7*c[5]^2*c[6]^2+576*c[6]^2+140392*c[3]^2+261225000*c[3]^10*c[6]^2-656 10000*a[7,4]*c[3]^10*c[5]*b[8]*c[6]-114665625*c[3]^11*c[6]^2+120*a[7,4 ]*c[3]^4-16320*c[6]^2*c[3]+582970500*c[3]^9*c[5]-36130980*c[3]^5*c[5]^ 2+84559200*c[5]^2*c[3]^5*c[6]-169540850*c[3]^6*c[5]-114665625*c[3]^11* c[5]^2+261225000*c[3]^10*c[5]^2-252450*a[7,4]*c[3]^6*c[5]^2-1048500*a[ 7,4]*c[3]^8*c[6]^2+877500*a[7,4]*c[3]^9*c[5]^2-357854625*c[3]^9*c[5]^2 -16200*a[7,4]*c[3]^9+9297240*c[6]^2*c[3]^4-31893750*c[3]^12*c[6]-36918 7875*c[3]^10*c[6]+2880*a[7,4]*c[3]^2*c[5]*b[8]*c[6]^2-74880*a[7,4]*c[3 ]^3*c[5]*c[6]^2*b[8]-2016*a[7,4]*c[3]^5+365309800*c[3]^6*c[5]*c[6]+161 291250*c[3]^11*c[6]-423275625*c[3]^10*c[5]+102802200*c[3]^6*c[5]^2+955 26000*a[7,4]*c[3]^8*c[5]^2*b[8]*c[6]-54847800*a[7,4]*c[3]^7*c[5]^2*b[8 ]*c[6]+22781250*c[3]^12*c[5]^2+309059250*c[3]^7*c[6]+2841712*c[3]^3*c[ 5]+2512408*c[3]^3*c[6]+52412260*c[3]^5*c[6]+20799000*a[7,4]*c[3]^6*c[5 ]^2*b[8]*c[6]+51931800*a[7,4]*c[3]^7*c[5]*b[8]*c[6]+76320*a[7,4]*c[3]^ 3*c[5]^2*c[6]^2*b[8]-410973750*c[3]^11*c[6]*c[5]+11670*a[7,4]*c[3]^6+1 1759400*a[7,4]*c[3]^7*c[5]^2*c[6]-263520*a[7,4]*c[3]^5*b[8]*c[6]^2-144 0*a[7,4]*c[3]^3*c[6]^2*b[8]+57933000*a[7,4]*c[3]^7*c[5]^2*b[8]*c[6]^2+ 95526000*a[7,4]*c[3]^8*c[5]*b[8]*c[6]^2-54847800*a[7,4]*c[3]^7*c[5]*b[ 8]*c[6]^2+243000*a[7,4]*c[3]^10*c[6]-3726000*a[7,4]*c[3]^9*c[5]^2*b[8] -3085200*a[7,4]*c[3]^7*c[5]^2*b[8]-576*a[7,4]*c[3]^2*c[5]*c[6]^2+480*a [7,4]*c[3]^2*c[5]*c[6]-4372800*a[7,4]*c[3]^6*c[5]*c[6]^2+1077840*a[7,4 ]*c[3]^5*c[5]*c[6]^2+329510250*c[3]^8*c[6]^2-357854625*c[3]^9*c[6]^2-4 372800*a[7,4]*c[3]^6*c[5]^2*c[6]+15120*a[7,4]*c[3]^3*c[5]^2*c[6]+10778 40*a[7,4]*c[3]^5*c[5]^2*c[6]+354082500*c[3]^7*c[5]-470241675*c[3]^8*c[ 6]-100170000*a[7,4]*c[3]^8*c[5]^2*b[8]*c[6]^2-105624000*a[7,4]*c[3]^9* c[5]*b[8]*c[6]^2-960*c[5]+66825000*a[7,4]*c[3]^10*c[5]*b[8]*c[6]^2-520 8480*a[7,4]*c[3]^5*c[5]*b[8]*c[6]^2-21985200*a[7,4]*c[3]^6*c[6]^2*c[5] ^2*b[8]+674400*a[7,4]*c[3]^7*c[5]^2+827280*a[7,4]*c[3]^4*c[6]^2*c[5]*b [8]-2880*a[7,4]*c[3]^2*c[5]^2*b[8]*c[6]^2-1710408*c[6]^2*c[3]^3-182250 00*a[7,4]*c[3]^11*c[5]*b[8]*c[6]^2-105624000*c[6]*a[7,4]*c[3]^9*c[5]^2 *b[8]-6075000*c[6]^2*a[7,4]*c[3]^11*c[5]^2+21363750*a[7,4]*c[3]^10*c[5 ]^2*c[6]^2-15416180*c[3]^4*c[5]+223320*a[7,4]*c[3]^4*c[5]^2*c[6]^2-194 40*a[7,4]*c[3]^3*c[5]^2*c[6]^2-12288*a[7,4]*c[3]^3*c[5]*c[6]-840540*a[ 7,4]*c[3]^5*c[5]*c[6]+1440*a[7,4]*c[3]^3*c[5]*b[8]-9043800*a[7,4]*c[3] ^7*c[6]*c[5]+251640*a[7,4]*c[3]^5*c[5]*b[8]-36450000*c[3]^12*c[5]-576* c[6]*a[7,4]*c[3]^2*c[5]^2-1119240*a[7,4]*c[3]^6*b[8]*c[6]+23955750*c[6 ]*a[7,4]*c[3]^9*c[5]^2+73440*a[7,4]*c[3]^3*c[5]*c[6]*b[8]+18225000*a[7 ,4]*c[3]^11*c[5]*b[8]*c[6]-68040000*a[7,4]*c[3]^10*c[5]^2*b[8]*c[6]^2- 635191875*c[3]^10*c[6]^2*c[5]+4644000*a[7,4]*c[3]^8*b[8]*c[6]^2-308520 0*a[7,4]*c[3]^7*b[8]*c[6]^2-4449600*a[7,4]*c[3]^8*b[8]*c[6]+4644000*a[ 7,4]*c[3]^8*c[5]^2*b[8]+18225000*c[6]^2*a[7,4]*c[3]^11*c[5]^2*b[8]+121 5000*a[7,4]*c[3]^10*b[8]*c[6]^2+12453750*a[7,4]*c[3]^10*c[5]*c[6]-1344 *c[5]^2*c[6]+1186200*a[7,4]*c[3]^6*c[5]^2*b[8]+2916000*a[7,4]*c[3]^7*c [5]*b[8]-168960*a[7,4]*c[3]^4*c[5]*c[6]^2+15120*a[7,4]*c[3]^3*c[5]*c[6 ]^2-1215000*a[7,4]*c[3]^10*c[5]*b[8]-4449600*a[7,4]*c[3]^8*c[5]*b[8]+3 645000*a[7,4]*c[3]^9*c[5]*b[8]+134100*a[7,4]*c[3]^4*c[5]*c[6]+1440*a[7 ,4]*c[3]^3*c[6]*b[8]-15795000*a[7,4]*c[3]^10*c[5]^2*c[6]-3726000*a[7,4 ]*c[3]^9*b[8]*c[6]^2-539326125*c[3]^8*c[5]-2880*a[7,4]*c[3]^2*c[5]*b[8 ]*c[6]-19679760*a[7,4]*c[3]^6*c[5]*b[8]*c[6]-32805000*a[7,4]*c[3]^9*c[ 5]^2*c[6]^2+28968750*a[7,4]*c[3]^8*c[5]^2*c[6]^2-13569188*c[3]^4*c[6]- 18225000*a[7,4]*c[3]^11*c[5]^2*b[8]*c[6]+30960*a[7,4]*c[3]^4*c[5]^2*b[ 8]-1440*a[7,4]*c[3]^3*c[5]^2*b[8]+4956840*a[7,4]*c[3]^5*c[5]*b[8]*c[6] -215452350*c[3]^7*c[6]^2+16193700*a[7,4]*c[3]^8*c[5]*c[6]-20992500*a[7 ,4]*c[3]^8*c[5]*c[6]^2+5999100*a[7,4]*c[3]^6*c[5]^2*c[6]^2-1456800*a[7 ,4]*c[3]^5*c[5]^2*c[6]^2+101979000*a[7,4]*c[3]^9*c[5]*b[8]*c[6]-910764 00*a[7,4]*c[3]^8*c[5]*b[8]*c[6]+109350000*a[7,4]*c[3]^9*c[5]^2*b[8]*c[ 6]^2+808200*a[7,4]*c[3]^8*c[5]-168960*a[7,4]*c[3]^4*c[5]^2*c[6]+239557 50*a[7,4]*c[3]^9*c[6]^2*c[5]-20992500*a[7,4]*c[3]^8*c[5]^2*c[6]-186435 00*a[7,4]*c[3]^9*c[5]*c[6]-858240*a[7,4]*c[3]^4*c[6]^2*c[5]^2*b[8]+547 2000*a[7,4]*c[3]^5*c[5]^2*b[8]*c[6]^2+20799000*a[7,4]*c[3]^6*c[5]*b[8] *c[6]^2-74880*a[7,4]*c[3]^3*c[5]^2*c[6]*b[8]-5208480*a[7,4]*c[3]^5*c[5 ]^2*b[8]*c[6]+827280*a[7,4]*c[3]^4*c[6]*c[5]^2*b[8]-1048500*a[7,4]*c[3 ]^8*c[5]^2-15795000*a[7,4]*c[3]^10*c[5]*c[6]^2+251640*a[7,4]*c[3]^5*b[ 8]*c[6]-30240*a[7,4]*c[3]^4*b[8]*c[6]-30240*a[7,4]*c[3]^4*c[5]*b[8]+27 184*c[5]*c[3]+24304*c[6]*c[3]+2080*c[5]*c[6]-316160*c[6]*c[3]^2-355736 *c[5]*c[3]^2+59763580*c[3]^5*c[5]-1710408*c[5]^2*c[3]^3+213792*c[5]^2* c[3]^2-16320*c[5]^2*c[3]-36130980*c[3]^5*c[6]^2-148270460*c[3]^6*c[6]- 1114296*c[3]^3+65340370*c[3]^6+576*c[5]^2+6008104*c[3]^4-23157444*c[3] ^5+329510250*c[3]^8*c[5]^2-30420*a[7,4]*c[3]^7-215452350*c[3]^7*c[5]^2 +508180500*c[3]^9*c[6]+205830450*c[3]^8-221489100*c[3]^9+160116750*c[3 ]^10-69558750*c[3]^11+13668750*c[3]^12-135774810*c[3]^7-58672*c[5]*c[6 ]*c[3]-497712*c[5]*c[6]^2*c[3]^2+38016*c[5]*c[6]^2*c[3]-241651800*c[5] *c[3]^6*c[6]^2-6107416*c[5]*c[3]^3*c[6]+765728*c[5]*c[6]*c[3]^2-128503 420*c[3]^5*c[6]*c[5]+3984000*c[5]*c[6]^2*c[3]^3-21693360*c[5]*c[3]^4*c [6]^2+84559200*c[5]*c[3]^5*c[6]^2+33120200*c[5]*c[3]^4*c[6]-21693360*c [5]^2*c[3]^4*c[6]+3984000*c[5]^2*c[3]^3*c[6]+38016*c[5]^2*c[6]*c[3]-49 7712*c[5]^2*c[3]^2*c[6]-241651800*c[3]^6*c[5]^2*c[6]-688500*a[7,4]*c[3 ]^9*c[5]+932917500*c[3]^10*c[6]*c[5]+860490000*c[3]^9*c[6]^2*c[5]+8604 90000*c[3]^9*c[5]^2*c[6]+282487500*c[3]^11*c[6]*c[5]^2-688500*a[7,4]*c [3]^9*c[6]+11880*a[7,4]*c[3]^5*b[8]-223256250*c[3]^11*c[6]^2*c[5]^2-63 5191875*c[3]^10*c[6]*c[5]^2-664638750*c[3]^9*c[6]^2*c[5]^2+243000*a[7, 4]*c[3]^10*c[5]-66960*a[7,4]*c[3]^6*b[8]-56953125*c[3]^12*c[6]^2*c[5]- 194400*a[7,4]*c[3]^8*b[8]+169200*a[7,4]*c[3]^7*b[8]+81000*a[7,4]*c[3]^ 9*b[8]+15986400*c[5]^2*c[6]^2*c[3]^4-2912160*c[5]^2*c[6]^2*c[3]^3+3609 60*c[5]^2*c[6]^2*c[3]^2-27360*c[5]^2*c[6]^2*c[3]+288*a[7,4]*c[3]^3*c[6 ]^2-240*a[7,4]*c[3]^3*c[6]+5088*a[7,4]*c[3]^4*c[6]-42870*a[7,4]*c[3]^5 *c[6]-42870*a[7,4]*c[3]^5*c[5]+54780*a[7,4]*c[3]^5*c[5]^2-6300*a[7,4]* c[3]^4*c[5]^2+5088*a[7,4]*c[3]^4*c[5]-240*a[7,4]*c[3]^3*c[5]+288*a[7,4 ]*c[3]^3*c[5]^2+674400*a[7,4]*c[3]^7*c[6]^2+808200*a[7,4]*c[3]^8*c[6]+ 282487500*c[3]^11*c[6]^2*c[5]-1274670000*c[3]^9*c[6]*c[5]+193620*a[7,4 ]*c[3]^6*c[5]+193620*a[7,4]*c[3]^6*c[6]-514950*a[7,4]*c[3]^7*c[6]-7850 11500*c[3]^8*c[6]^2*c[5]+509442000*c[3]^7*c[6]^2*c[5]-785011500*c[3]^8 *c[5]^2*c[6]+509442000*c[3]^7*c[5]^2*c[6]-765560550*c[3]^7*c[5]*c[6]+6 00007500*c[5]^2*c[3]^8*c[6]^2-385554000*c[5]^2*c[3]^7*c[6]^2+181188000 *c[5]^2*c[3]^6*c[6]^2-303750*a[7,4]*c[3]^10*c[6]^2-56953125*c[3]^12*c[ 5]^2*c[6]-720*a[7,4]*c[3]^4*b[8]+1171746000*c[3]^8*c[6]*c[5]-62842800* c[5]^2*c[6]^2*c[3]^5-303750*a[7,4]*c[3]^10*c[5]^2+82012500*c[3]^12*c[5 ]*c[6]+496125000*c[3]^10*c[6]^2*c[5]^2+877500*a[7,4]*c[3]^9*c[6]^2+455 62500*c[3]^12*c[6]^2*c[5]^2-514950*a[7,4]*c[3]^7*c[5]-252450*a[7,4]*c[ 3]^6*c[6]^2+54780*a[7,4]*c[3]^5*c[6]^2-6300*a[7,4]*c[3]^4*c[6]^2+2880* c[6]*a[7,4]*c[3]^2*b[8]*c[5]^2)/b[8]/c[3]^2/(c[5]-1)/(c[3]-1)/(3*c[3]- 1)/(5*c[3]-2)/(15*c[3]^2-10*c[3]+1)/(15*c[5]*c[3]^2-10*c[5]*c[3]+2*c[5 ]-c[3])/(-1+c[6])/(2*c[6]-10*c[6]*c[3]+15*c[6]*c[3]^2-c[3]), a[6,1] = \+ 1/6*c[6]*(-45*c[5]^2*c[3]^4+405*c[5]^2*c[2]*c[3]^3-66*c[5]^2*c[2]*c[3] ^2+50*c[5]*c[2]*c[3]^2+1350*c[6]^3*c[2]*c[3]^5+6300*c[6]^2*c[2]*c[5]*c [3]^4-1350*c[2]*c[3]^6*c[6]^2-6300*c[6]*c[2]*c[5]*c[3]^5+1890*c[6]*c[2 ]*c[5]^2*c[3]^3-4875*c[6]^2*c[2]*c[5]*c[3]^3-3375*c[2]*c[5]^2*c[6]*c[3 ]^6-54*c[5]*c[2]*c[6]*c[3]-2700*c[6]^3*c[2]*c[3]^4+2100*c[6]^3*c[2]*c[ 3]^3-2190*c[6]^2*c[2]*c[3]^4-12*c[6]^3*c[2]-12*c[3]^3*c[5]-6*c[2]*c[3] ^2*c[6]+90*c[6]*c[2]*c[3]^5-120*c[6]*c[2]*c[3]^4-24*c[6]^2*c[5]^2*c[2] +675*c[5]^2*c[2]*c[3]^5+5100*c[6]*c[2]*c[5]*c[3]^4-495*c[5]*c[2]*c[3]^ 5+660*c[5]*c[2]*c[3]^4-870*c[5]^2*c[2]*c[3]^4-307*c[5]*c[2]*c[3]^3+27* c[3]^4*c[5]+18*c[6]^2*c[3]*c[2]+48*c[2]*c[6]*c[3]^3-204*c[6]^2*c[2]*c[ 3]^2+156*c[6]^3*c[2]*c[3]+930*c[6]^2*c[2]*c[3]^3+3375*c[2]*c[3]^6*c[6] *c[5]+1870*c[5]*c[6]^2*c[2]*c[3]^2-2140*c[5]*c[6]*c[2]*c[3]^3+54*c[2]* c[6]*c[5]^2*c[3]-3375*c[6]^2*c[2]*c[5]*c[3]^5-5850*c[6]^2*c[2]*c[5]^2* c[3]^4-352*c[5]*c[6]^2*c[2]*c[3]+300*c[5]^2*c[6]^2*c[2]*c[3]+2700*c[6] ^2*c[2]*c[3]^5-810*c[6]^3*c[2]*c[3]^2+28*c[6]^2*c[2]*c[5]+5850*c[6]*c[ 2]*c[5]^2*c[3]^5+4425*c[6]^2*c[2]*c[5]^2*c[3]^3-1650*c[6]^2*c[2]*c[5]^ 2*c[3]^2+3375*c[6]^2*c[2]*c[5]^2*c[3]^5-4650*c[6]*c[2]*c[5]^2*c[3]^4+4 85*c[5]*c[6]*c[2]*c[3]^2-435*c[5]^2*c[6]*c[2]*c[3]^2+18*c[5]^2*c[3]^3+ 60*c[5]*c[3]^3*c[6]-60*c[5]*c[6]^2*c[3]^3+135*c[5]*c[3]^4*c[6]^2-135*c [5]*c[3]^4*c[6]+225*c[5]^2*c[3]^4*c[6]-90*c[5]^2*c[3]^3*c[6]-225*c[5]^ 2*c[6]^2*c[3]^4+90*c[5]^2*c[6]^2*c[3]^3)/c[5]/c[3]^2/(15*c[5]*c[3]-9*c [3]+4-6*c[5])/c[2]/(1-5*c[3]+5*c[3]^2), a[8,3] = -1/360*(96+172890*c[5 ]^2*c[3]^4+2511000*c[3]^8*a[7,5]*c[5]^3*b[8]*c[6]+552150*c[3]^7*a[7,6] *c[6]+1863000*c[5]^2*c[3]^7*b[8]*c[6]^2+173400*a[7,6]*c[3]^6*c[5]*c[6] ^2+2511000*c[3]^8*a[7,6]*c[5]*b[8]*c[6]^3+240*c[5]^2*c[6]^2-3726000*c[ 3]^7*a[7,5]*c[5]^3*b[8]*c[6]^2+2511000*c[3]^8*a[7,5]*c[5]^2*b[8]*c[6]^ 2+4644000*c[3]^7*a[7,5]*c[5]*b[8]*c[6]^2+178800*c[3]^7*a[7,5]*c[5]^2+1 1880*a[7,4]*c[3]^4*b[8]*c[6]^2+180*a[7,4]*c[3]^2*c[5]^2*c[6]^2-102240* a[7,4]*c[3]^6*c[5]*b[8]-15480*c[3]^4*b[8]-80100*c[3]^6*c[6]^2+1186200* a[7,5]*c[5]^3*b[8]*c[3]^5-336*c[5]*c[6]^2+4644000*c[6]^2*a[7,2]*c[2]*b [8]*c[3]^7-3726000*c[6]^2*a[7,2]*c[2]*b[8]*c[3]^8-1558800*c[6]*a[7,2]* c[2]*b[8]*c[3]^7+19044*c[6]^2*c[3]^2+405000*a[7,2]*c[2]*c[5]^2*c[3]^9* c[6]-922680*c[6]^2*a[7,2]*c[2]*c[5]*b[8]*c[3]^4+486000*c[6]^2*a[7,2]*c [2]*c[5]*c[3]^7+1899000*a[7,6]*c[5]*b[8]*c[3]^5*c[6]^3-976320*a[7,6]*c [5]*b[8]*c[3]^5*c[6]^2+36450*a[7,4]*c[3]^8+1440*c[6]*a[7,2]*c[2]*c[5]^ 2*b[8]*c[3]+405000*c[6]^2*a[7,2]*c[2]*c[3]^9*c[5]-66960*a[7,4]*c[3]^5* c[5]^2*b[8]+25200*a[7,4]*c[3]^7*b[8]*c[6]+169200*a[7,4]*c[3]^6*b[8]*c[ 6]^2-43920*a[7,4]*c[3]^4*c[6]*c[5]*b[8]+1440*a[7,5]*c[5]^3*c[6]*b[8]*c [3]+232560*a[7,5]*c[5]^3*b[8]*c[3]^3*c[6]-14040*c[6]*c[5]*b[8]*c[3]^2- 81000*a[7,4]*c[3]^9*b[8]*c[6]-216*c[6]-1680*c[3]+690120*a[7,6]*c[5]*b[ 8]*c[3]^4*c[6]^2+1215000*c[3]^9*c[6]^2*a[7,2]*c[2]*b[8]+2511000*c[3]^9 *c[6]*a[7,2]*c[2]*b[8]+552150*a[7,2]*c[2]*c[3]^7-1440*a[7,6]*c[5]^2*c[ 6]^3*b[8]*c[3]+1899000*a[7,5]*c[5]^2*b[8]*c[3]^6+14760*c[5]*b[8]*c[3]^ 3-38700*a[7,4]*c[3]^7*c[5]*c[6]^2-19920*a[7,4]*c[3]^6*c[5]*c[6]-27000* a[7,4]*c[3]^7*c[5]^2*c[6]^2+144*c[6]^2+12544*c[3]^2-263520*a[7,2]*c[2] *b[8]*c[3]^5+5940*a[7,6]*c[5]^2*c[6]^2*c[3]^2+5940*a[7,6]*c[5]*c[6]^3* c[3]^2+120*a[7,4]*c[3]^4-1440*c[6]^2*a[7,2]*c[2]*c[5]^2*b[8]*c[3]-2952 0*a[7,5]*c[5]^2*c[6]^2*b[8]*c[3]^2-2544*c[6]^2*c[3]+240*a[7,2]*c[2]*c[ 3]^3+324000*c[3]^9*c[5]-174900*c[3]^5*c[5]^2-634500*a[7,2]*c[2]*c[3]^9 *c[5]+30960*a[7,5]*c[5]*b[8]*c[3]^4+1186200*a[7,5]*c[5]^3*b[8]*c[3]^4* c[6]^2+1899000*a[7,5]*c[5]^2*b[8]*c[3]^5*c[6]^2-288*a[7,5]*c[5]^3*c[6] *c[3]-352200*a[7,5]*c[5]^3*c[3]^5*c[6]+5940*a[7,5]*c[5]^3*c[3]^2*c[6]- 1363500*c[6]^2*a[7,2]*c[2]*c[5]^2*c[3]^6+1154250*c[6]^2*a[7,2]*c[2]*c[ 5]^2*c[3]^7+232560*a[7,6]*c[5]*b[8]*c[3]^4*c[6]+766800*c[5]^2*c[3]^5*c [6]+4944*a[7,6]*c[3]^3*c[6]^2-240*a[7,6]*c[6]^2*c[3]^2-357225*c[3]^6*c [5]-1558800*c[6]*a[7,2]*c[2]*c[5]^2*b[8]*c[3]^6-29520*c[6]*a[7,2]*c[2] *b[8]*c[3]^3+232560*a[7,6]*c[5]^2*b[8]*c[3]^3*c[6]^2-1593000*c[3]^8*a[ 7,6]*c[5]*b[8]*c[6]^2-1215000*c[3]^9*a[7,5]*c[5]^2*b[8]*c[6]^2-37200*a [7,4]*c[3]^6*c[5]^2-20250*a[7,4]*c[3]^8*c[6]^2+151875*c[3]^9*c[5]^2+25 11000*a[7,2]*c[2]*c[5]^2*b[8]*c[3]^8*c[6]-1215000*c[3]^9*a[7,6]*b[8]*c [6]^3*c[5]-16200*a[7,4]*c[3]^9+172890*c[6]^2*c[3]^4-519900*a[7,6]*c[3] ^7*c[5]*c[6]^2+1154250*a[7,6]*c[5]^2*c[3]^7*c[6]^3-1440*a[7,5]*c[5]^3* b[8]*c[3]^2-240*a[7,6]*c[5]*c[6]*c[3]^2+4944*a[7,6]*c[5]*c[3]^3*c[6]-1 440*a[7,5]*c[5]^2*c[6]*b[8]*c[3]-203040*a[7,5]*c[5]^2*b[8]*c[3]^3*c[6] -918000*c[3]^7*a[7,5]*c[5]^3*b[8]*c[6]-1593000*c[3]^8*a[7,5]*c[5]^2*b[ 8]*c[6]-1558800*c[3]^7*a[7,5]*c[5]*b[8]*c[6]-634500*c[3]^9*a[7,6]*c[6] ^2+232560*a[7,5]*c[5]*b[8]*c[3]^4*c[6]-720*c[5]^2*c[6]*b[8]*c[3]+12150 00*c[3]^10*a[7,5]*c[5]*b[8]-345060*c[5]*c[3]^4*b[8]*c[6]-922680*a[7,5] *c[5]^2*b[8]*c[3]^4*c[6]^2+148500*a[7,2]*c[2]*c[5]*c[3]^8*c[6]-922680* c[6]*a[7,2]*c[2]*b[8]*c[3]^5-243000*c[3]^10*a[7,5]*c[5]-3726000*c[3]^8 *a[7,5]*c[5]*b[8]*c[6]^2+339750*c[3]^8*a[7,5]*c[5]^2+607500*c[3]^9*b[8 ]*c[5]^2*c[6]+405000*c[3]^9*a[7,5]*c[5]^2*c[6]^2-151875*c[3]^10*c[6]+7 20*a[7,4]*c[3]^2*c[5]*b[8]*c[6]^2-11160*a[7,4]*c[3]^3*c[5]*c[6]^2*b[8] -303750*c[3]^9*a[7,6]*c[6]^3+779400*c[3]^7*b[8]*c[5]-5232*a[7,6]*c[3]^ 4*c[6]-2016*a[7,4]*c[3]^5+4644000*a[7,5]*c[5]^3*b[8]*c[3]^6*c[6]^2-308 5200*a[7,5]*c[5]^3*b[8]*c[3]^5*c[6]^2+687600*c[3]^6*c[5]*c[6]-151875*c [3]^10*c[5]+546750*a[7,2]*c[2]*c[3]^9*c[5]*c[6]-1558800*c[3]^6*a[7,6]* c[5]*b[8]*c[6]^3-80100*c[3]^6*c[5]^2-263520*a[7,5]*c[5]^3*b[8]*c[3]^3* c[6]^2+30960*a[7,5]*c[5]^3*c[6]^2*b[8]*c[3]^2+1440*a[7,5]*c[5]^2*c[6]^ 2*b[8]*c[3]-81000*a[7,4]*c[3]^8*c[5]^2*b[8]*c[6]+113400*a[7,4]*c[3]^7* c[5]^2*b[8]*c[6]+303750*c[3]^10*a[7,5]*c[5]*c[6]-3085200*a[7,2]*c[2]*b [8]*c[3]^7-31050*c[3]^7*c[6]-405000*c[6]^2*a[7,2]*c[2]*c[5]^2*c[3]^8-2 63520*c[6]^2*a[7,2]*c[2]*b[8]*c[3]^4+30960*a[7,5]*c[5]*b[8]*c[3]^3*c[6 ]^2+134372*c[3]^3*c[5]+1215000*c[3]^9*a[7,5]*c[5]*b[8]*c[6]^2-918000*c [3]^8*a[7,5]*c[5]*b[8]*c[6]-593100*c[3]^6*b[8]-1440*a[7,5]*c[5]*b[8]*c [6]^2*c[3]^2+111446*c[3]^3*c[6]+311120*c[3]^5*c[6]+1899000*c[6]*a[7,2] *c[2]*b[8]*c[3]^6+5940*c[6]*a[7,2]*c[2]*c[5]^2*c[3]^2+607500*c[3]^10*b [8]*c[6]-853200*c[3]^8*a[7,5]*c[5]-1440*a[7,2]*c[2]*c[5]^2*b[8]*c[3]^2 +25200*a[7,4]*c[3]^6*c[5]^2*b[8]*c[6]-138600*a[7,4]*c[3]^7*c[5]*b[8]*c [6]+11880*a[7,4]*c[3]^3*c[5]^2*c[6]^2*b[8]-288*c[6]^2*a[7,2]*c[2]*c[5] *c[3]+5940*c[6]^2*a[7,2]*c[2]*c[5]*c[3]^2-3085200*a[7,5]*c[5]^3*b[8]*c [3]^6+173400*a[7,5]*c[5]^2*c[3]^6*c[6]+240*a[7,5]*c[5]^2*c[6]*c[3]+330 60*a[7,5]*c[5]^2*c[3]^3*c[6]+11670*a[7,4]*c[3]^6-38700*a[7,4]*c[3]^7*c [5]^2*c[6]-66960*a[7,4]*c[3]^5*b[8]*c[6]^2-720*a[7,4]*c[3]^3*c[6]^2*b[ 8]+232560*a[7,5]*c[5]^2*b[8]*c[3]^3*c[6]^2+81000*a[7,4]*c[3]^7*c[5]^2* b[8]*c[6]^2-81000*a[7,4]*c[3]^8*c[5]*b[8]*c[6]^2+113400*a[7,4]*c[3]^7* c[5]*b[8]*c[6]^2-194400*a[7,4]*c[3]^7*c[5]^2*b[8]-144*a[7,4]*c[3]^2*c[ 5]*c[6]^2-46860*c[6]*a[7,2]*c[2]*c[5]^2*c[3]^3+232560*c[6]*a[7,2]*c[2] *c[5]^2*b[8]*c[3]^3-29520*c[6]*a[7,2]*c[2]*c[5]^2*b[8]*c[3]^2-1440*a[7 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*c[6]-3726000*a[7,6]*c[5]^2*b[8]*c[3]^7*c[6]^3+2511000*c[3]^9*a[7,5]*c [5]*b[8]*c[6]+163800*c[3]^7*c[5]-133650*c[3]^8*c[6]-4656*a[7,5]*c[5]^2 *c[6]*c[3]^2-29520*a[7,5]*c[5]^2*b[8]*c[3]^3+1440*a[7,5]*c[5]^2*b[8]*c [3]^2+153000*a[7,6]*c[3]^5*c[5]*c[6]-240*c[5]-102240*a[7,4]*c[3]^5*c[5 ]*b[8]*c[6]^2-194400*a[7,4]*c[3]^6*c[6]^2*c[5]^2*b[8]+45000*a[7,4]*c[3 ]^7*c[5]^2+55080*a[7,4]*c[3]^4*c[6]^2*c[5]*b[8]-720*a[7,4]*c[3]^2*c[5] ^2*b[8]*c[6]^2+131760*c[3]^5*b[8]-29520*a[7,6]*c[5]*b[8]*c[3]^3*c[6]+5 46750*c[3]^9*a[7,6]*c[6]^2*c[5]+405000*c[3]^9*a[7,6]*c[6]^3*c[5]-30375 0*c[3]^9*a[7,6]*c[5]^2*c[6]-593100*c[5]^2*c[3]^4*b[8]*c[6]^2-76968*c[6 ]^2*c[3]^3+1215000*c[3]^9*a[7,6]*c[5]^2*b[8]*c[6]-263520*c[6]^2*a[7,2] *c[2]*c[5]^2*b[8]*c[3]^3+1186200*c[6]^2*a[7,2]*c[2]*c[5]^2*b[8]*c[3]^4 -29520*a[7,5]*c[5]^3*c[6]*b[8]*c[3]^2+1440*a[7,6]*b[8]*c[6]^2*c[3]^2+7 20*c[5]*c[6]*b[8]*c[3]+101520*c[5]*b[8]*c[3]^3*c[6]+30960*a[7,2]*c[2]* c[5]^2*b[8]*c[3]^3+173400*c[6]*a[7,2]*c[2]*c[3]^6*c[5]+1899000*a[7,2]* 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[7,5]*c[5]*c[3]^3-607500*c[3]^9*b[8]*c[6]^2+607500*c[3]^10*b[8]*c[5]-2 99700*c[3]^6*a[7,6]*c[6]^2+178800*c[3]^7*a[7,6]*c[6]^2+877500*c[3]^8*a [7,6]*c[6]^3-1255500*c[3]^9*b[8]*c[6]-1048500*c[3]^7*a[7,6]*c[6]^3-121 5000*c[3]^10*a[7,5]*c[5]*b[8]*c[6]+1215000*a[7,2]*c[2]*c[5]^2*b[8]*c[3 ]^9+2511000*c[3]^9*a[7,6]*c[5]*b[8]*c[6]+877500*a[7,2]*c[2]*c[5]^2*c[3 ]^8+779400*c[3]^6*b[8]*c[5]^2*c[6]-949500*c[3]^5*b[8]*c[5]^2*c[6]+1476 0*c[6]*c[3]^2*b[8]*c[5]^2+877500*c[6]^2*a[7,2]*c[2]*c[3]^8+1215000*c[3 ]^10*a[7,6]*b[8]*c[6]+461340*c[3]^4*b[8]*c[5]^2*c[6]-634500*c[3]^9*a[7 ,5]*c[5]*c[6]-299700*c[3]^6*a[7,6]*c[5]*c[6]+178800*c[3]^7*a[7,6]*c[5] *c[6]-29520*c[6]^2*a[7,2]*c[2]*c[5]*b[8]*c[3]^2+2476800*a[7,6]*c[5]*b[ 8]*c[3]^7*c[6]^2)/b[8]/c[3]^2/(3295*c[5]^2*c[3]^4-4*c[5]^2*c[6]^2+1290 0*c[3]^6*c[6]^2+4*c[5]*c[6]^2+86*c[6]^2*c[3]^2-4*c[3]^2-4*c[6]^2*c[3]- 3375*c[3]^9*c[5]-8570*c[3]^5*c[5]^2-4330*c[5]^2*c[3]^5*c[6]-4330*c[3]^ 6*c[5]+3295*c[6]^2*c[3]^4+6880*c[3]^6*c[5]*c[6]+12900*c[3]^6*c[5]^2-25 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[3]+3375*c[3]^9*c[6]*c[5]-3375*c[3]^8*c[6]^2*c[5]+6975*c[3]^7*c[6]^2*c [5]-3375*c[3]^8*c[5]^2*c[6]+6975*c[3]^7*c[5]^2*c[6]-4425*c[3]^7*c[5]*c [6]+3375*c[5]^2*c[3]^7*c[6]^2-10350*c[5]^2*c[3]^6*c[6]^2-3600*c[3]^8*c [6]*c[5]+12900*c[5]^2*c[6]^2*c[3]^5), a[8,1] = 1/360*(105480*c[5]^2*c[ 2]*c[3]^3-22572*c[5]^2*c[2]*c[3]^2+37224*c[5]*c[2]*c[3]^2+58950*c[5]^3 *c[2]*c[6]^2*a[7,5]*c[3]^5+98280*c[6]^2*b[8]*a[7,6]*c[2]*c[3]^4*c[5]+6 94920*c[6]^2*c[2]*c[5]*c[3]^4-619200*c[2]*c[3]^6*c[6]^2+76320*c[6]*b[8 ]*a[7,2]*c[2]^2*c[5]^2*c[3]^3+3060*c[5]^2*a[7,4]*c[6]^2*c[2]*c[3]^3-61 9200*c[5]^2*c[2]*c[3]^6-194400*c[6]*c[5]^2*c[2]*a[7,4]*c[3]^6*b[8]+360 *c[5]^2*c[6]^3*a[7,6]*c[2]*c[3]+9960*c[5]^2*c[6]*a[7,5]*c[2]*c[3]^3-76 320*c[6]^2*a[7,2]*c[2]*c[5]*b[8]*c[3]^4-20250*c[6]^2*a[7,2]*c[2]*c[5]* c[3]^7+38550*a[7,2]*c[6]^2*c[2]^2*c[3]^4*c[5]-163620*b[8]*c[6]^2*c[2]* c[5]^2*c[3]^4+1942305*c[6]*c[2]*c[5]*c[3]^5-239280*c[6]*c[2]*c[5]^2*c[ 3]^3-239280*c[6]^2*c[2]*c[5]*c[3]^3-96*c[2]+169200*b[8]*a[7,4]*c[2]*c[ 6]*c[3]^5*c[5]^2-1440*c[5]^2*c[6]^2*b[8]*a[7,2]*c[2]^2*c[3]+54720*b[8] *c[6]^2*c[2]*c[5]^2*c[3]^3+271800*c[6]^2*c[5]^2*c[2]*b[8]*c[3]^5-54270 0*c[2]*c[3]^7*c[6]+27000*c[2]*c[5]^2*c[6]^2*c[3]^7*a[7,5]-234900*c[6]^ 2*c[2]*c[5]^2*c[3]^6*b[8]-58950*c[6]^2*a[7,2]*c[2]*c[5]^2*c[3]^6+27000 *c[6]^2*a[7,2]*c[2]*c[5]^2*c[3]^7-1530*c[5]^2*c[3]^5*c[6]-194400*c[6]* a[7,2]*c[2]*c[5]^2*b[8]*c[3]^6+2736*c[5]^2*c[2]*c[3]+720*c[6]*c[5]*b[8 ]*c[2]*c[3]-11940*c[6]^2*a[7,5]*c[2]*c[5]^2*c[3]^3+163620*b[8]*c[6]^2* c[2]*c[3]^4*c[5]+81000*c[5]*c[2]*c[6]^2*c[3]^7*a[7,5]*b[8]-81000*c[2]* c[5]^2*c[6]^2*c[3]^7*a[7,6]*b[8]+1458225*c[2]*c[5]^2*c[6]*c[3]^6+9732* c[5]*c[2]*c[6]*c[3]-38550*a[7,5]*c[6]^2*c[2]*c[3]^5*c[5]-11880*c[5]^2* b[8]*a[7,4]*c[2]*c[6]^2*c[3]^3-37200*a[7,4]*c[6]*c[2]*c[3]^5*c[5]^2+81 000*c[2]*c[3]^6*c[6]^2*a[7,5]*c[5]^3*b[8]-174600*b[8]*a[7,2]*c[6]*c[2] ^2*c[5]^2*c[3]^4+16200*c[6]*c[5]*c[2]*c[3]^7*a[7,6]-288*c[6]^2*a[7,6]* c[2]*c[5]^2*c[3]+240*c[6]^2*c[5]*a[7,6]*c[2]*c[3]+141750*c[2]*c[3]^8*c [6]-304650*c[6]^2*c[2]*c[3]^4+20250*c[5]*c[2]^2*c[6]^2*a[7,2]*c[3]^6-2 100*c[6]^2*a[7,6]*c[2]*c[3]^5*c[5]+17790*c[6]^2*a[7,6]*c[2]*c[5]^2*c[3 ]^4+336*c[6]*c[5]^2*c[2]-945*c[3]^6*c[5]*c[6]+194400*c[6]*c[5]^2*c[2]^ 2*a[7,2]*b[8]*c[3]^5+113400*c[6]^2*c[2]*c[5]^2*c[3]^6*a[7,5]*b[8]-1440 *c[6]*c[5]^2*b[8]*a[7,5]*c[2]*c[3]-20250*c[5]^2*c[2]*c[6]*c[3]^7*a[7,5 ]-20250*c[5]^2*c[2]*c[3]^7*a[7,6]*c[6]+33264*c[2]*c[3]^2*c[6]-796050*c [6]*c[2]*c[3]^5+441000*c[6]*c[2]*c[3]^4+58950*c[5]^2*c[2]*c[6]^3*a[7,6 ]*c[3]^5-38550*c[6]*a[7,6]*c[2]*c[3]^5*c[5]^2+216*c[6]*c[2]-2100*a[7,5 ]*c[6]*c[2]*c[3]^5*c[5]^2-60750*c[2]*c[3]^8+288*c[6]*a[7,2]*c[2]*c[5]^ 2*c[3]^2-1440*c[6]*b[8]*a[7,6]*c[2]*c[5]^2*c[3]^2+144*c[6]*a[7,4]*c[2] *c[5]^2*c[3]^2-81000*c[2]*c[5]^2*c[6]^2*c[3]^7*a[7,5]*b[8]+288*c[6]^2* a[7,2]*c[2]*c[5]*c[3]^2+3420*c[5]^3*c[6]*a[7,5]*c[2]*c[3]^2-15240*c[6] ^2*a[7,6]*c[2]*c[3]^4*c[5]-54720*c[6]^2*b[8]*c[2]*c[3]^3*c[5]+999900*c [5]*c[2]*c[3]^6-3420*c[6]*a[7,2]*c[2]*c[5]^2*c[3]^3+16560*c[6]*a[7,2]* c[2]*c[5]^2*b[8]*c[3]^3-1440*c[6]*a[7,2]*c[2]*c[5]^2*b[8]*c[3]^2+19440 0*c[6]^2*a[7,2]*c[2]*c[5]^2*b[8]*c[3]^6-174600*b[8]*a[7,2]*c[6]^2*c[2] ^2*c[3]^4*c[5]+38550*a[7,6]*c[6]^3*c[2]*c[3]^4*c[5]-194400*c[6]^2*c[5] *c[2]*a[7,4]*c[3]^6*b[8]+194400*c[6]*a[7,2]*c[2]*b[8]*c[3]^6*c[5]+4500 0*c[6]*a[7,2]*c[2]*c[5]^2*c[3]^6+59760*c[6]^2*b[8]*a[7,6]*c[2]*c[5]^2* c[3]^3+14160*c[6]^2*a[7,4]*c[2]*c[3]^4*c[5]-11940*c[6]^2*a[7,6]*c[2]*c [5]^2*c[3]^3+4800*c[6]^2*c[5]^2*c[2]*a[7,5]*c[3]^5+3420*c[5]^2*c[6]*a[ 7,2]*c[2]^2*c[3]^2+3060*c[6]^2*a[7,5]*c[2]*c[5]^2*c[3]^2-240*c[6]^2*c[ 5]^2*c[2]+554850*c[5]^2*c[2]*c[3]^5-240*c[6]*a[7,2]*c[2]*c[5]*c[3]^2-1 44*c[5]^2*c[2]-16260*c[6]^2*a[7,2]*c[2]^2*c[3]^3*c[5]-194400*c[6]*c[5] ^2*c[2]*c[3]^6*a[7,6]*b[8]+113400*c[5]^2*c[2]*c[6]^2*c[3]^6*a[7,6]*b[8 ]-2544*c[6]*a[7,5]*c[2]*c[5]^2*c[3]^2-81000*c[6]*c[5]*c[2]*a[7,4]*c[3] ^7*b[8]-16560*c[6]^2*a[7,2]*c[2]*c[5]^2*b[8]*c[3]^3+76320*c[6]^2*a[7,2 ]*c[2]*c[5]^2*b[8]*c[3]^4-38700*c[6]^2*c[2]*c[5]^2*c[3]^6*a[7,5]+76320 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8]*a[7,2]*c[2]^2*c[3]^2+28800*c[6]*c[2]*c[5]^2*c[3]^6*a[7,5]-27000*c[2 ]*c[3]^6*c[6]^2*a[7,5]*c[5]^3-11670*c[6]*a[7,4]*c[2]*c[3]^4*c[5]+81000 *c[5]*c[2]*c[6]^2*c[3]^7*a[7,6]*b[8]-4320*c[6]^3*a[7,6]*c[2]*c[5]^2*c[ 3]^2+234900*c[5]*c[2]*c[6]^2*c[3]^6*b[8]-20250*c[5]*c[2]*c[6]^2*c[3]^7 *a[7,5]-20250*c[5]*c[2]*c[6]^2*a[7,4]*c[3]^7-618300*c[5]*c[2]*c[3]^7+7 6320*c[6]*b[8]*a[7,5]*c[2]*c[5]^3*c[3]^3-54720*c[6]*b[8]*c[2]*c[5]^2*c [3]^3+16560*c[6]^3*b[8]*a[7,6]*c[2]*c[5]^2*c[3]^2-9720*c[6]*b[8]*c[2]* c[5]*c[3]^2+16200*c[6]*a[7,2]*c[2]*c[3]^7*c[5]-1440*c[5]^3*c[6]^2*b[8] *a[7,5]*c[2]*c[3]+20250*c[5]^2*c[2]^2*c[6]*a[7,2]*c[3]^6-81000*c[6]^2* a[7,2]*c[2]*c[5]^2*b[8]*c[3]^7-2544*c[6]^2*a[7,6]*c[2]*c[5]*c[3]^2+240 *c[2]*c[5]+2736*c[6]^2*c[3]*c[2]-154062*c[2]*c[6]*c[3]^3-22572*c[6]^2* c[2]*c[3]^2+45000*c[6]^2*a[7,2]*c[2]*c[5]*c[3]^6+105480*c[6]^2*c[2]*c[ 3]^3-163620*c[6]*b[8]*c[2]*c[3]^4*c[5]+233550*c[2]*c[3]^7-1440*c[6]^2* c[5]*b[8]*a[7,5]*c[2]*c[3]^2-720*c[6]^2*c[5]*b[8]*a[7,4]*c[2]*c[3]^2-4 5000*c[6]*c[5]^3*c[2]*a[7,5]*c[3]^5+720*c[5]^2*c[6]^2*b[8]*c[2]*c[3]+8 1000*c[5]^2*c[2]*c[3]^6*a[7,6]*b[8]*c[6]^3+253125*c[2]*c[5]^2*c[6]*c[3 ]^8+76320*c[6]^3*b[8]*a[7,6]*c[2]*c[3]^3*c[5]-194400*c[5]^2*c[2]*c[6]^ 3*a[7,6]*b[8]*c[3]^5+20250*c[5]^3*c[2]*c[6]*a[7,5]*c[3]^6-76320*c[6]*a [7,2]*c[2]*c[5]^2*b[8]*c[3]^4-20250*c[6]*a[7,2]*c[2]*c[5]^2*c[3]^7-810 00*c[6]*c[5]*c[2]*c[3]^7*a[7,5]*b[8]+66960*c[6]*b[8]*a[7,4]*c[2]*c[3]^ 4*c[5]+58950*a[7,2]*c[6]^2*c[2]^2*c[3]^5*c[5]^2-31500*c[6]*a[7,2]*c[2] ^2*c[3]^4*c[5]-13380*c[6]*a[7,5]*c[2]*c[3]^4*c[5]+31500*c[6]*a[7,5]*c[ 2]*c[3]^5*c[5]+16260*c[6]^2*a[7,5]*c[2]*c[3]^4*c[5]-288*c[6]*a[7,2]*c[ 2]^2*c[5]^2*c[3]-240*c[6]*c[5]*a[7,5]*c[2]*c[3]^2-120*c[6]*c[5]*a[7,4] *c[2]*c[3]^2-16560*c[6]*a[7,2]*c[2]*c[5]*b[8]*c[3]^3+2832*c[6]*a[7,2]* c[2]*c[5]*c[3]^3+1440*c[6]*a[7,2]*c[2]*c[5]*b[8]*c[3]^2+174600*c[6]*b[ 8]*a[7,6]*c[2]*c[3]^5*c[5]^2-288*c[6]*a[7,5]*c[2]*c[5]^3*c[3]+38550*a[ 7,5]*c[6]*c[2]*c[5]^3*c[3]^4-76320*b[8]*a[7,5]*c[6]^2*c[2]*c[5]^3*c[3] ^3-38700*c[5]^2*c[2]*c[6]^2*c[3]^6*a[7,6]-38550*c[6]*a[7,2]*c[2]*c[5]^ 2*c[3]^5+174600*c[6]*a[7,2]*c[2]*c[5]^2*b[8]*c[3]^5+31500*c[6]*a[7,6]* c[2]*c[3]^5*c[5]-59760*c[5]^2*c[6]*b[8]*a[7,5]*c[2]*c[3]^3+16560*c[5]^ 2*c[6]*b[8]*a[7,6]*c[2]*c[3]^3+98280*c[6]*b[8]*a[7,5]*c[2]*c[5]^2*c[3] ^4-16260*c[6]*a[7,5]*c[2]*c[5]^3*c[3]^3+1440*c[6]*b[8]*a[7,5]*c[2]*c[5 ]^3*c[3]-2179350*c[2]*c[3]^6*c[6]*c[5]+51252*c[5]*c[6]^2*c[2]*c[3]^2+3 69552*c[5]*c[6]*c[2]*c[3]^3-6264*c[2]*c[6]*c[5]^2*c[3]-3420*c[6]^2*a[7 ,2]*c[2]*c[5]*c[3]^3+16560*c[6]^2*a[7,2]*c[2]*c[5]*b[8]*c[3]^3-194400* c[6]^2*c[5]*c[2]*c[3]^6*a[7,5]*b[8]+9960*c[5]*c[6]^2*a[7,6]*c[2]*c[3]^ 3-3420*c[5]*c[6]^2*a[7,5]*c[2]*c[3]^3-1281150*c[6]^2*c[2]*c[5]*c[3]^5- 502350*c[6]^2*c[2]*c[5]^2*c[3]^4-6264*c[5]*c[6]^2*c[2]*c[3]+4440*c[5]^ 2*c[6]^2*c[2]*c[3]-81000*c[6]*c[5]*c[2]*c[3]^7*a[7,6]*b[8]-113400*c[6] *c[2]*c[5]^2*c[3]^6*a[7,5]*b[8]+288*c[6]^2*c[5]*a[7,5]*c[2]*c[3]^2+554 850*c[6]^2*c[2]*c[3]^5+336*c[6]^2*c[2]*c[5]+174600*c[6]*b[8]*a[7,2]*c[ 2]^2*c[3]^4*c[5]+194400*c[6]*c[5]*c[2]*c[3]^6*a[7,5]*b[8]-27000*a[7,2] *c[6]^2*c[2]^2*c[3]^6*c[5]^2-15240*c[6]*a[7,5]*c[2]*c[5]^2*c[3]^4-1944 00*c[6]^2*a[7,2]*c[2]*c[5]*b[8]*c[3]^6-1281150*c[6]*c[2]*c[5]^2*c[3]^5 +170910*c[6]^2*c[2]*c[5]^2*c[3]^3-36360*c[6]^2*c[2]*c[5]^2*c[3]^2+9423 75*c[6]^2*c[2]*c[5]^2*c[3]^5+81000*c[5]^2*c[2]*c[6]*a[7,4]*c[3]^7*b[8] -45000*c[6]^3*c[5]*c[2]*a[7,6]*c[3]^5+14160*c[6]*a[7,4]*c[2]*c[5]^2*c[ 3]^4+16260*c[6]^2*a[7,2]*c[2]*c[5]*c[3]^4-38550*c[6]^2*a[7,2]*c[2]*c[5 ]*c[3]^5+694920*c[6]*c[2]*c[5]^2*c[3]^4-79528*c[5]*c[6]*c[2]*c[3]^2+51 252*c[5]^2*c[6]*c[2]*c[3]^2+174600*b[8]*a[7,5]*c[6]^2*c[2]*c[3]^5*c[5] +194400*c[6]*c[5]*c[2]*a[7,4]*c[3]^6*b[8]-169200*c[6]*b[8]*a[7,4]*c[2] *c[3]^5*c[5]-49800*a[7,2]*c[6]^2*c[2]^2*c[5]^2*c[3]^4+16200*c[6]*c[5]* c[2]*a[7,4]*c[3]^7+13380*c[6]*a[7,2]*c[2]^2*c[3]^3*c[5]+49800*c[6]^2*a [7,2]*c[2]*c[5]^2*c[3]^5+54720*c[6]*b[8]*c[2]*c[3]^3*c[5]+16560*c[5]*c [6]^2*b[8]*a[7,5]*c[2]*c[3]^3-16560*c[5]*c[6]^3*b[8]*a[7,6]*c[2]*c[3]^ 2+1440*c[6]^2*c[5]^2*b[8]*a[7,5]*c[2]*c[3]+66960*b[8]*a[7,4]*c[2]*c[6] ^2*c[5]^2*c[3]^4-81000*b[8]*a[7,4]*c[2]*c[6]^2*c[3]^7*c[5]^2-288*c[6]^ 3*c[5]*a[7,6]*c[2]*c[3]+81000*c[2]*c[5]^2*c[6]^2*c[3]^7*b[8]+16560*c[5 ]^2*b[8]*a[7,2]*c[6]^2*c[2]^2*c[3]^2-11880*c[6]*b[8]*a[7,4]*c[2]*c[3]^ 3*c[5]+144*c[6]^2*c[5]*a[7,4]*c[2]*c[3]^2+174600*c[6]^2*c[2]*c[5]^3*a[ 7,5]*b[8]*c[3]^4-66960*c[6]^2*b[8]*a[7,4]*c[2]*c[3]^4*c[5]-271800*c[6] *c[5]^2*c[2]*c[3]^5*b[8]+16560*c[6]^2*b[8]*a[7,5]*c[2]*c[5]^3*c[3]^2+5 9760*c[6]^2*b[8]*a[7,5]*c[2]*c[5]^2*c[3]^3-45000*c[6]*c[5]^2*c[2]^2*a[ 7,2]*c[3]^5-98280*b[8]*a[7,5]*c[6]^2*c[2]*c[5]^2*c[3]^4-2832*c[6]*a[7, 2]*c[2]^2*c[5]*c[3]^2+16560*c[6]*b[8]*a[7,2]*c[2]^2*c[5]*c[3]^2-14652* c[3]^2*c[2]+1800*c[3]*c[2]-192450*c[2]*c[3]^4+67548*c[3]^3*c[2]+1350*c [5]*c[3]^6*c[6]^2+60*c[5]*c[3]^3*c[6]+1080*c[3]^5*c[6]*c[5]-84*c[5]*c[ 6]^2*c[3]^3+600*c[5]*c[3]^4*c[6]^2-1530*c[5]*c[3]^5*c[6]^2-426*c[5]*c[ 3]^4*c[6]+600*c[5]^2*c[3]^4*c[6]-84*c[5]^2*c[3]^3*c[6]+1350*c[3]^6*c[5 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0*c[6]*a[7,4]*c[2]*c[3]^5*c[5]-4320*c[5]^2*a[7,2]*c[6]^2*c[2]^2*c[3]^2 -9720*c[5]^2*b[8]*c[6]^2*c[2]*c[3]^2+45000*c[6]*c[5]^2*c[2]*c[3]^6*a[7 ,6]+194400*c[6]*c[5]*c[2]*c[3]^6*a[7,6]*b[8]-194400*c[6]*c[5]*c[2]^2*a [7,2]*b[8]*c[3]^5+720*c[5]^2*c[6]^2*b[8]*a[7,4]*c[2]*c[3]^2-15120*c[5] ^2*c[6]^2*b[8]*a[7,6]*c[2]*c[3]^2+880650*c[2]*c[3]^6*c[6]-101250*c[2]* c[3]^8*c[6]^2-520*c[2]*c[5]*c[6]-4068*c[2]*c[6]*c[3]-81000*c[5]*c[2]^2 *c[6]^2*a[7,2]*b[8]*c[3]^6-45000*c[6]^2*c[5]*c[2]^2*a[7,2]*c[3]^5+2880 0*c[6]^2*c[5]*c[2]*c[3]^6*a[7,6]+45000*c[6]*c[5]^2*c[2]*a[7,4]*c[3]^6- 16560*c[6]*b[8]*a[7,5]*c[2]*c[3]^3*c[5]+15120*c[6]*b[8]*a[7,5]*c[2]*c[ 5]^2*c[3]^2+11880*c[5]^2*c[6]*b[8]*a[7,4]*c[2]*c[3]^3-15120*c[6]^2*b[8 ]*a[7,5]*c[2]*c[5]^2*c[3]^2+16200*c[6]*c[5]*c[2]*c[3]^7*a[7,5]-234900* c[6]*c[5]*c[2]*c[3]^6*b[8]+169200*b[8]*a[7,4]*c[2]*c[6]^2*c[3]^5*c[5]+ 194400*b[8]*a[7,4]*c[2]*c[6]^2*c[3]^6*c[5]^2-169200*b[8]*a[7,4]*c[2]*c [6]^2*c[3]^5*c[5]^2+194400*c[6]^3*c[5]*c[2]*a[7,6]*b[8]*c[3]^5+81000*c [5]^2*c[2]*c[3]^7*a[7,6]*b[8]*c[6]+1440*c[6]^3*c[5]*b[8]*a[7,6]*c[2]*c [3]+360*c[5]^3*c[6]^2*a[7,5]*c[2]*c[3]+1440*c[6]*b[8]*a[7,2]*c[2]^2*c[ 5]^2*c[3]-81000*c[2]*c[5]*c[6]^2*c[3]^7*b[8]-720*c[6]*b[8]*a[7,4]*c[2] *c[5]^2*c[3]^2-240*c[6]*a[7,6]*c[2]*c[5]*c[3]^2-27000*c[5]^2*c[2]*c[3] ^6*a[7,6]*c[6]^3-180*c[5]^2*c[6]^2*a[7,4]*c[2]*c[3]^2-76320*c[6]^2*b[8 ]*a[7,5]*c[2]*c[3]^4*c[5]+1440*c[6]^2*b[8]*a[7,6]*c[2]*c[5]^2*c[3]-144 0*c[6]^2*c[5]*b[8]*a[7,6]*c[2]*c[3]+76320*c[6]*b[8]*a[7,6]*c[2]*c[3]^4 *c[5]-174600*c[6]*b[8]*a[7,6]*c[2]*c[3]^5*c[5]-81000*c[2]*c[5]^2*c[6]* c[3]^7*b[8]-4320*c[6]^2*a[7,5]*c[2]*c[5]^3*c[3]^2-20250*c[5]*c[2]*c[6] ^2*c[3]^7*a[7,6]-81000*c[5]^2*c[2]^2*c[6]*a[7,2]*b[8]*c[3]^6-113400*c[ 6]^2*c[5]*c[2]*c[3]^6*a[7,6]*b[8]-271800*c[6]^2*c[5]*c[2]*c[3]^5*b[8]+ 45000*c[6]^2*c[5]*c[2]*a[7,4]*c[3]^6+19800*c[6]^2*c[5]^2*c[2]*a[7,5]*b [8]*c[3]^5-36450*c[6]*c[5]*c[2]*c[3]^6*a[7,6]+36450*c[6]*c[5]*c[2]^2*a [7,2]*c[3]^5+81000*c[6]*c[5]*c[2]^2*a[7,2]*b[8]*c[3]^6+3060*c[5]^2*c[6 ]^2*a[7,6]*c[2]*c[3]^2+174600*c[6]^3*c[5]^2*c[2]*c[3]^4*a[7,6]*b[8]-17 4600*c[6]^3*b[8]*a[7,6]*c[2]*c[3]^4*c[5]-19800*c[6]^2*b[8]*a[7,6]*c[2] *c[3]^5*c[5]-98280*c[6]^2*b[8]*a[7,6]*c[2]*c[5]^2*c[3]^4-81000*c[5]^3* c[2]*c[6]*a[7,5]*b[8]*c[3]^6-1440*c[6]*c[5]*b[8]*a[7,2]*c[2]^2*c[3]+14 40*c[6]*c[5]*b[8]*a[7,5]*c[2]*c[3]^2+720*c[6]*c[5]*b[8]*a[7,4]*c[2]*c[ 3]^2+1440*c[6]*b[8]*a[7,6]*c[2]*c[5]*c[3]^2+234900*c[5]^2*c[2]*c[6]*c[ 3]^6*b[8]-49800*c[6]^3*c[5]^2*c[2]*c[3]^4*a[7,6]+11880*c[5]*b[8]*a[7,4 ]*c[2]*c[6]^2*c[3]^3+16260*c[6]*a[7,6]*c[2]*c[5]^2*c[3]^4-59760*c[5]*c [6]^2*b[8]*a[7,6]*c[2]*c[3]^3-19800*b[8]*a[7,5]*c[6]*c[2]*c[3]^5*c[5]^ 2-20250*c[5]^2*c[2]*c[6]*a[7,4]*c[3]^7+174600*b[8]*a[7,2]*c[6]^2*c[2]^ 2*c[5]^2*c[3]^4+81000*b[8]*a[7,2]*c[6]^2*c[2]^2*c[3]^6*c[5]^2-194400*b [8]*a[7,2]*c[6]^2*c[2]^2*c[3]^5*c[5]^2-76320*b[8]*a[7,2]*c[6]^2*c[2]^2 *c[5]^2*c[3]^3+38550*a[7,2]*c[6]*c[2]^2*c[5]^2*c[3]^4+20760*a[7,6]*c[6 ]^3*c[2]*c[5]^2*c[3]^3-76320*c[6]^3*b[8]*a[7,6]*c[2]*c[5]^2*c[3]^3-133 80*c[6]*a[7,6]*c[2]*c[3]^4*c[5]+3420*c[5]*c[6]^3*a[7,6]*c[2]*c[3]^2-16 560*c[5]*b[8]*a[7,2]*c[6]^2*c[2]^2*c[3]^2+288*c[6]*a[7,6]*c[2]*c[5]^2* c[3]^2+240*c[6]*c[5]^2*a[7,5]*c[2]*c[3]+240*c[6]*c[5]*a[7,2]*c[2]^2*c[ 3]-2430*c[5]^2*c[6]*a[7,4]*c[2]*c[3]^3-3420*c[5]^2*c[6]*a[7,6]*c[2]*c[ 3]^3+20760*a[7,2]*c[6]^2*c[2]^2*c[5]^2*c[3]^3-2430*c[5]*a[7,4]*c[6]^2* c[2]*c[3]^3+2832*c[6]*a[7,6]*c[2]*c[3]^3*c[5]-36450*c[6]*c[5]*c[2]*a[7 ,4]*c[3]^6+271800*c[6]*c[5]*c[2]*c[3]^5*b[8]+4800*c[6]^2*c[5]^2*c[2]*a [7,6]*c[3]^5-66960*c[6]*b[8]*a[7,4]*c[2]*c[5]^2*c[3]^4-16260*c[6]*a[7, 2]*c[2]^2*c[5]^2*c[3]^3-720*c[6]^2*c[5]*b[8]*c[2]*c[3]-174600*b[8]*a[7 ,5]*c[6]*c[2]*c[5]^3*c[3]^4+81000*c[5]^2*c[2]*c[6]*c[3]^7*a[7,5]*b[8]) /c[6]/c[5]/(c[3]-1)/(3*c[3]-1)/(5*c[3]-2)/(15*c[3]^2-10*c[3]+2)/(c[5]- 1)/(-1+c[6])/c[3]^2/c[2]/b[8], a[6,5] = -(3*c[3]-1)*(2*c[6]-10*c[6]*c[ 3]+15*c[6]*c[3]^2-c[3])*(c[6]-c[3])*(c[6]-c[5])*c[6]/(15*c[5]*c[3]^2-1 0*c[5]*c[3]+2*c[5]-c[3])/(c[5]-c[3])/(15*c[5]*c[3]-9*c[3]+4-6*c[5])/c[ 5], a[4,3] = -(15*c[3]^2-10*c[3]+1)*c[3]/(15*c[3]^2-10*c[3]+2)^3, c[8] = 1, a[8,7] = 0, b[6] = -1/60*(1-5*c[3]+5*c[3]^2)*(15*c[5]*c[3]-9*c[3 ]+4-6*c[5])/c[6]/(-1+c[6])/(2*c[6]-10*c[6]*c[3]+15*c[6]*c[3]^2-c[3])/( c[6]-c[3])/(c[6]-c[5]), a[3,1] = 1/2*c[3]*(-c[3]+2*c[2])/c[2], `b*`[8] = 0, `b*`[4] = -1/60*(192-240*c[6]-4080*c[3]+39360*c[3]^2+1012500*c[3 ]^9*c[5]-4710000*c[3]^6*c[5]+6660000*c[3]^6*c[5]*c[6]+5265000*c[3]^7*c [6]+285600*c[3]^3*c[5]+285600*c[3]^3*c[6]+2738000*c[3]^5*c[6]+5265000* c[3]^7*c[5]-3459375*c[3]^8*c[6]-240*c[5]-1075000*c[3]^4*c[5]-3459375*c [3]^8*c[5]-1075000*c[3]^4*c[6]+5120*c[5]*c[3]+5120*c[6]*c[3]+320*c[5]* c[6]-49600*c[6]*c[3]^2-49600*c[5]*c[3]^2+2738000*c[3]^5*c[5]-4710000*c [3]^6*c[6]-225600*c[3]^3+11137500*`b*`[7]*c[3]^8*c[5]+3037500*`b*`[7]* c[3]^9-17820000*c[5]*c[3]^7*`b*`[7]+3654000*c[3]^6-3037500*c[3]^9*c[6] *`b*`[7]+11137500*c[3]^8*c[6]*`b*`[7]-17820000*c[3]^7*c[6]*`b*`[7]+844 800*c[3]^4-2139000*c[3]^5+3037500*c[5]*c[3]^9*`b*`[7]*c[6]+1012500*c[3 ]^9*c[6]+2632500*c[3]^8-759375*c[3]^9-4050000*c[3]^7-6880*c[5]*c[6]*c[ 3]-390400*c[5]*c[3]^3*c[6]+67200*c[5]*c[6]*c[3]^2-3822000*c[3]^5*c[6]* c[5]+1484000*c[5]*c[3]^4*c[6]-1518750*c[3]^9*c[6]*c[5]-7560000*c[3]^7* c[5]*c[6]+5062500*c[3]^8*c[6]*c[5]-960*`b*`[7]-3037500*`b*`[7]*c[3]^9* c[5]-192000*`b*`[7]*c[3]^2+20160*`b*`[7]*c[3]-3996000*`b*`[7]*c[3]^4+1 084800*`b*`[7]*c[3]^3-16560000*`b*`[7]*c[3]^6+9924000*`b*`[7]*c[3]^5-1 1137500*`b*`[7]*c[3]^8+17820000*`b*`[7]*c[3]^7-11137500*c[3]^8*c[6]*`b *`[7]*c[5]-9924000*c[5]*c[3]^5*`b*`[7]+3996000*c[5]*c[3]^4*`b*`[7]-108 4800*c[5]*c[3]^3*`b*`[7]+192000*c[5]*c[3]^2*`b*`[7]-20160*c[5]*c[3]*`b *`[7]+960*c[5]*`b*`[7]-9924000*c[3]^5*c[6]*`b*`[7]+3996000*c[3]^4*c[6] *`b*`[7]+16560000*c[3]^6*c[6]*`b*`[7]+9924000*c[3]^5*c[6]*c[5]*`b*`[7] -16560000*c[3]^6*c[6]*c[5]*`b*`[7]-3996000*c[3]^4*c[6]*c[5]*`b*`[7]+10 84800*c[3]^3*c[6]*c[5]*`b*`[7]-192000*c[6]*c[3]^2*c[5]*`b*`[7]+20160*c [6]*c[3]*c[5]*`b*`[7]-1084800*c[3]^3*c[6]*`b*`[7]+192000*c[6]*c[3]^2*` b*`[7]-20160*c[6]*c[3]*`b*`[7]-960*c[5]*c[6]*`b*`[7]+960*c[6]*`b*`[7]+ 17820000*`b*`[7]*c[3]^7*c[5]*c[6]+16560000*`b*`[7]*c[3]^6*c[5])/c[3]^2 /(2*c[6]-10*c[6]*c[3]+15*c[6]*c[3]^2-c[3])/(15*c[5]*c[3]^2-10*c[5]*c[3 ]+2*c[5]-c[3])/(15*c[3]^2-10*c[3]+1), `b*`[1] = 1/60*(24-30*c[6]-165*c [3]+350*c[3]^2+300*c[3]^3*c[5]+300*c[3]^3*c[6]-30*c[5]+210*c[5]*c[3]+2 10*c[6]*c[3]+40*c[5]*c[6]-455*c[6]*c[3]^2-455*c[5]*c[3]^2-225*c[3]^3-2 90*c[5]*c[6]*c[3]-450*c[5]*c[3]^3*c[6]+660*c[5]*c[6]*c[3]^2-120*`b*`[7 ]-1560*`b*`[7]*c[3]^2+780*`b*`[7]*c[3]+900*`b*`[7]*c[3]^3-900*c[5]*c[3 ]^3*`b*`[7]+1560*c[5]*c[3]^2*`b*`[7]-780*c[5]*c[3]*`b*`[7]+120*c[5]*`b *`[7]+900*c[3]^3*c[6]*c[5]*`b*`[7]-1560*c[6]*c[3]^2*c[5]*`b*`[7]+780*c [6]*c[3]*c[5]*`b*`[7]-900*c[3]^3*c[6]*`b*`[7]+1560*c[6]*c[3]^2*`b*`[7] -780*c[6]*c[3]*`b*`[7]-120*c[5]*c[6]*`b*`[7]+120*c[6]*`b*`[7])/c[5]/c[ 6]/c[3]^2, `b*`[6] = -1/60*(-24+165*c[3]-350*c[3]^2-300*c[3]^3*c[5]+30 *c[5]-210*c[5]*c[3]+455*c[5]*c[3]^2+225*c[3]^3+120*`b*`[7]-1560*c[5]*c [3]^2*`b*`[7]+1560*`b*`[7]*c[3]^2-780*`b*`[7]*c[3]-900*`b*`[7]*c[3]^3+ 900*c[5]*c[3]^3*`b*`[7]+780*c[5]*c[3]*`b*`[7]-120*c[5]*`b*`[7])/c[6]/( 10*c[6]^2*c[3]^2+2*c[6]^3-10*c[6]^3*c[3]+15*c[5]*c[3]^3*c[6]-15*c[5]*c [6]^2*c[3]^2+15*c[6]^3*c[3]^2+10*c[5]*c[6]^2*c[3]+3*c[5]*c[6]*c[3]-c[5 ]*c[3]^2-10*c[5]*c[6]*c[3]^2+c[6]*c[3]^2-15*c[6]^2*c[3]^3-2*c[5]*c[6]^ 2-3*c[6]^2*c[3]), `b*`[3] = 1/60*(-300*c[5]*c[6]*c[3]^2+225*c[6]*c[3]^ 2-900*c[6]*c[3]^2*`b*`[7]+900*c[6]*c[3]^2*c[5]*`b*`[7]-660*c[6]*c[3]*c [5]*`b*`[7]-170*c[6]*c[3]+660*c[6]*c[3]*`b*`[7]+230*c[5]*c[6]*c[3]+30* c[6]-120*c[6]*`b*`[7]+120*c[5]*c[6]*`b*`[7]-40*c[5]*c[6]-180*c[3]^2+90 0*`b*`[7]*c[3]^2+225*c[5]*c[3]^2-900*c[5]*c[3]^2*`b*`[7]+135*c[3]-660* `b*`[7]*c[3]-170*c[5]*c[3]+660*c[5]*c[3]*`b*`[7]+30*c[5]-120*c[5]*`b*` [7]-24+120*`b*`[7])/(c[6]-c[3])/(-c[5]+c[3])/c[3]^2/(15*c[3]^2-10*c[3] +1)\}:" }}}{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 8 "Example:" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 26 "`b *`[6]=subs(e19,`b*`[6]);" }}{PARA 12 "" 1 "" {XPPMATH 20 "6#/&%#b*G6# \"\"',$*&#\"\"\"\"#gF+*(,B\"#C!\"\"*&\"$l\"F+&%\"cG6#\"\"$F+F+*&\"$]$F +)F3\"\"#F+F0*(\"$+$F+)F3F6F+&F46#\"\"&F+F0*&\"#IF+F>F+F+*(\"$5#F+F>F+ F3F+F0*(\"$b%F+F>F+F9F+F+*&\"$D#F+F=F+F+*&\"$?\"F+&F%6#\"\"(F+F+**\"%g :F+F>F+F9F+FKF+F0*(FOF+FKF+F9F+F+*(\"$!yF+FKF+F3F+F0*(\"$+*F+FKF+F=F+F 0**FTF+F>F+F=F+FKF+F+**FRF+F>F+F3F+FKF+F+*(FJF+F>F+FKF+F0F+&F4F&F0,>*( \"#5F+)FXF:F+F9F+F+*&F:F+)FXF6F+F+*(FenF+FhnF+F3F+F0**\"#:F+F>F+F=F+FX F+F+**F[oF+F>F+FfnF+F9F+F0*(F[oF+FhnF+F9F+F+**FenF+F>F+FfnF+F3F+F+**F6 F+F>F+FXF+F3F+F+*&F>F+F9F+F0**FenF+F>F+FXF+F9F+F0*&FXF+F9F+F+*(F[oF+Ff nF+F=F+F0*(F:F+F>F+FfnF+F0*(F6F+FfnF+F3F+F0F0F+F0" }}}{EXCHG {PARA 0 " > " 0 "" {MPLTEXT 1 0 1 ";" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 15 "" 0 "" {TEXT -1 1 " " }{TEXT 295 6 "Step 9" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 39 "The four column simplifyi ng 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#/-%$SumG6$*&&%#b *G6#%\"iG\"\"\"&%\"aG6$F+%\"jGF,/F+;,&F0F,F,F,\"\")*&&F)6#F0F,,&F,F,&% \"cG6#F0!\"\"F," }{TEXT -1 5 ", " }{XPPEDIT 18 0 "j = 3;" "6#/%\"jG \"\"$" }{TEXT -1 10 ", 4, 5, 6 " }}{PARA 0 "" 0 "" {TEXT -1 19 "give t he equations " }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "`b*` [4]*a[4,3]+`b*`[5]*a[5,3]+`b*`[6]*a[6,3]+`b*`[7]*a[7,3]+`b*`[8]*a[8,3] = `b*`[3]*(1-c[3])" "6#/,,*&&%#b*G6#\"\"%\"\"\"&%\"aG6$F)\"\"$F*F**&& F'6#\"\"&F*&F,6$F2F.F*F**&&F'6#\"\"'F*&F,6$F8F.F*F**&&F'6#\"\"(F*&F,6$ F>F.F*F**&&F'6#\"\")F*&F,6$FDF.F*F**&&F'6#F.F*,&F*F*&%\"cG6#F.!\"\"F* " }{TEXT -1 1 " " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "`b*`[5]*a[5,4]+`b*`[6]*a[6,4]+`b*`[7]*a [7,4]+`b*`[8]*a[8,4] = `b*`[4]*(1-c[4])" "6#/,**&&%#b*G6#\"\"&\"\"\"&% \"aG6$F)\"\"%F*F**&&F'6#\"\"'F*&F,6$F2F.F*F**&&F'6#\"\"(F*&F,6$F8F.F*F **&&F'6#\"\")F*&F,6$F>F.F*F**&&F'6#F.F*,&F*F*&%\"cG6#F.!\"\"F*" } {TEXT -1 1 " " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "`b*`[6]*a[6,5]+`b*`[7]*a[7,5]+`b*`[8]*a [8,5] = `b*`[5]*(1-c[5])" "6#/,(*&&%#b*G6#\"\"'\"\"\"&%\"aG6$F)\"\"&F* F**&&F'6#\"\"(F*&F,6$F2F.F*F**&&F'6#\"\")F*&F,6$F8F.F*F**&&F'6#F.F*,&F *F*&%\"cG6#F.!\"\"F*" }{TEXT -1 1 " " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 12 "from which " }{XPPEDIT 18 0 "a[8,2];" "6#&%\"aG6$\"\")\"\"#" }{TEXT -1 3 ", " }{XPPEDIT 18 0 "a[8,4];" "6#& %\"aG6$\"\")\"\"%" }{TEXT -1 3 ", " }{XPPEDIT 18 0 "a[8,5]" "6#&%\"aG 6$\"\")\"\"&" }{TEXT -1 7 " and " }{XPPEDIT 18 0 "a[8,6];" "6#&%\"aG 6$\"\")\"\"'" }{TEXT -1 52 " can be expressed in terms of the other p arameters." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 89 "We can then obtain numerical values for the linking coefficient s in the 7th and 8th rows." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 212 "`simp_cdns*` := [seq(add(`b*`[i]*a [i,j],i=j+1..8)=`b*`[j]*(1-c[j]),j=[3,4,5,6])]:\n`simp_eqns*` := simpl ify(subs(e19,`simp_cdns*`)):\nnops(%);\nindets(`simp_eqns*`) minus \{c [2],c[3],c[5],c[6],`b*`[7],b[8]\};\nnops(%);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#\"\"%" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#<&&%\"aG6$\"\" (\"\"&&F%6$F'\"\"#&F%6$F'\"\"%&F%6$F'\"\"'" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#\"\"%" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 22 "infolevel[solve] := 4:" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 164 "e20 := factor(solve(\{op(`simp_eqns*`)\} ,\{a[7,2],a[7,4],a[7,5],a[7,6]\})):\ne21 := `union`(map(u_->lhs(u_)=si mplify(subs(e20,rhs(u_))),e19),e20):\ninfolevel[solve] := 0:" }}} {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 48 "The coeff icients are now expressed in terms of " }{XPPEDIT 18 0 "c[2]" "6#&%\" cG6#\"\"#" }{TEXT -1 3 ", " }{XPPEDIT 18 0 "c[3]" "6#&%\"cG6#\"\"$" } {TEXT -1 3 ", " }{XPPEDIT 18 0 "c[5]" "6#&%\"cG6#\"\"&" }{TEXT -1 3 " , " }{XPPEDIT 18 0 "c[6]" "6#&%\"cG6#\"\"'" }{TEXT -1 3 ", " } {XPPEDIT 18 0 "b[8]" "6#&%\"bG6#\"\")" }{TEXT -1 7 " and " } {XPPEDIT 18 0 "`b*`[7]" "6#&%#b*G6#\"\"(" }{TEXT -1 1 "." }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 21 "indets(map(rhs,e21));" }}{PARA 11 " " 1 "" {XPPMATH 20 "6#<(&%\"cG6#\"\"'&F%6#\"\"$&F%6#\"\"#&F%6#\"\"&&% \"bG6#\"\")&%#b*G6#\"\"(" }}}{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 101449 "e21 := \{a[6,2] = 1/2*c[6]*c[3]/c[2]*(5*c[6]^2-5* c[6]+1)/(1-5*c[3]+5*c[3]^2), b[2] = 0, b[4] = 1/60*(15*c[3]^2-10*c[3]+ 2)^5*(3*c[3]-5*c[5]*c[3]-5*c[6]*c[3]+10*c[5]*c[6]*c[3]-5*c[5]*c[6]-2+3 *c[6]+3*c[5])/(15*c[3]^2-10*c[3]+1)/c[3]^2/(3*c[3]-1)/(5*c[3]-2)/(2*c[ 6]-10*c[6]*c[3]+15*c[6]*c[3]^2-c[3])/(15*c[5]*c[3]^2-10*c[5]*c[3]+2*c[ 5]-c[3]), a[5,3] = 1/6*c[5]*(150*c[3]^3*c[5]-145*c[5]*c[3]^2+40*c[5]*c [3]-4*c[5]-30*c[3]^3+20*c[3]^2-c[3])*(-c[5]+c[3])/(1-5*c[3]+5*c[3]^2)/ c[3]^2/(15*c[3]^2-10*c[3]+1), b[7] = -1/60*(20-1560*c[6]*c[5]*b[8]*c[3 ]^2-24*c[6]-136*c[3]-900*c[5]*b[8]*c[3]^3+285*c[3]^2-780*c[6]*b[8]*c[3 ]+225*c[3]^3*c[5]+225*c[3]^3*c[6]-24*c[5]+780*c[5]*c[6]*b[8]*c[3]+900* c[5]*b[8]*c[3]^3*c[6]-900*b[8]*c[3]^3*c[6]+1560*b[8]*c[6]*c[3]^2-120*c [5]*b[8]*c[6]+1560*c[5]*b[8]*c[3]^2-780*c[5]*b[8]*c[3]+165*c[5]*c[3]+1 65*c[6]*c[3]+30*c[5]*c[6]-350*c[6]*c[3]^2-350*c[5]*c[3]^2-180*c[3]^3+1 20*b[8]*c[5]-1560*b[8]*c[3]^2+780*b[8]*c[3]+900*b[8]*c[3]^3-120*b[8]+1 20*b[8]*c[6]-210*c[5]*c[6]*c[3]-300*c[5]*c[3]^3*c[6]+455*c[5]*c[6]*c[3 ]^2)/(c[3]-1)/(-1+c[6])/(c[5]-1)/(3*c[3]-1)/(5*c[3]-2), b[5] = -1/60*( 1-5*c[3]+5*c[3]^2)*(15*c[6]*c[3]-9*c[3]+4-6*c[6])/(c[6]-c[5])/c[5]/(c[ 5]-1)/(15*c[5]*c[3]^2-10*c[5]*c[3]+2*c[5]-c[3])/(-c[5]+c[3]), a[4,2] = 1/2*c[3]^2*(4-30*c[3]+45*c[3]^2)/c[2]/(15*c[3]^2-10*c[3]+2)^3, `b*`[5 ] = -1/60*(-300*c[3]^3*c[6]+30*c[6]-210*c[6]*c[3]+455*c[6]*c[3]^2-1560 *c[6]*c[3]^2*`b*`[7]+900*c[3]^3*c[6]*`b*`[7]+780*c[6]*c[3]*`b*`[7]-120 *c[6]*`b*`[7]-24+165*c[3]-350*c[3]^2+225*c[3]^3+120*`b*`[7]+1560*`b*`[ 7]*c[3]^2-780*`b*`[7]*c[3]-900*`b*`[7]*c[3]^3)/(c[6]-c[5])/c[5]/(15*c[ 3]^3*c[5]-c[3]^2-10*c[5]*c[3]^2-15*c[5]^2*c[3]^2+3*c[5]*c[3]+10*c[5]^2 *c[3]-2*c[5]^2), a[4,1] = 1/2*c[3]*(-4*c[3]+30*c[3]^2-45*c[3]^3+350*c[ 3]^2*c[2]-100*c[3]*c[2]+10*c[2]+450*c[2]*c[3]^4-600*c[3]^3*c[2])/c[2]/ (15*c[3]^2-10*c[3]+2)^3, a[5,4] = 1/6*(15*c[3]^2-10*c[3]+2)^2*c[5]*(-c [5]+c[3])*(15*c[5]*c[3]^2-10*c[5]*c[3]+2*c[5]-c[3])/(1-5*c[3]+5*c[3]^2 )/c[3]^2/(15*c[3]^2-10*c[3]+1), a[8,6] = 1/3600*(-480-256000*c[5]^2*c[ 3]^4+870000*c[3]^4*b[8]+473340*c[6]*c[5]*b[8]*c[3]^2+576*c[6]+6564*c[3 ]+1324500*c[5]*b[8]*c[3]^3-36280*c[3]^2+207375*c[3]^5*c[5]^2-273000*c[ 5]^2*c[3]^5*c[6]+104625*c[3]^6*c[5]+38520*c[6]*b[8]*c[3]+48600*c[5]^2* c[6]*b[8]*c[3]+2002800*c[5]*c[3]^4*b[8]*c[6]-135000*c[3]^6*c[5]*c[6]-6 7500*c[3]^6*c[5]^2-259430*c[3]^3*c[5]+202500*c[3]^6*b[8]-126300*c[3]^3 *c[6]-157500*c[3]^5*c[6]+1176*c[5]-666000*c[3]^5*b[8]-87120*c[5]*c[6]* b[8]*c[3]-1324500*c[5]*b[8]*c[3]^3*c[6]-877500*c[3]^5*b[8]*c[5]^2-2002 800*c[3]^4*b[8]*c[5]+405025*c[3]^4*c[5]-1965600*c[6]*c[5]*b[8]*c[3]^2* `b*`[7]+374400*c[5]*c[6]*b[8]*c[3]*`b*`[7]-1543500*c[5]*c[3]^5*b[8]*c[ 6]+900*c[5]^2*c[6]+745500*c[5]^2*c[3]^3*b[8]*c[6]+579000*b[8]*c[3]^3*c [6]-208140*b[8]*c[6]*c[3]^2+6480*c[5]*b[8]*c[6]-473340*c[5]*b[8]*c[3]^ 2+87120*c[5]*b[8]*c[3]-745500*c[3]^3*b[8]*c[5]^2+196750*c[3]^4*c[6]+47 2500*c[5]*c[3]^6*b[8]*c[6]-202500*c[3]^6*b[8]*c[6]-472500*c[3]^6*b[8]* c[5]-14400*b[8]*`b*`[7]-16200*c[5]*c[3]-7920*c[6]*c[3]-1440*c[5]*c[6]+ 44025*c[6]*c[3]^2+90235*c[5]*c[3]^2-324900*c[3]^5*c[5]+162525*c[5]^2*c [3]^3-56070*c[5]^2*c[3]^2+9990*c[5]^2*c[3]+50625*c[3]^6*c[6]+103445*c[ 3]^3-40500*c[3]^6-720*c[5]^2-6480*b[8]*c[5]+208140*b[8]*c[3]^2-38520*b [8]*c[3]-579000*b[8]*c[3]^3+2880*b[8]-160050*c[3]^4+127125*c[3]^5-2880 *b[8]*c[6]+982800*c[5]^2*b[8]*c[3]^2*c[6]*`b*`[7]+810000*c[3]^6*b[8]*c [6]*`b*`[7]-2808000*c[3]^5*b[8]*c[6]*`b*`[7]+3837600*c[3]^4*b[8]*c[6]* `b*`[7]+19980*c[5]*c[6]*c[3]+325050*c[5]*c[3]^3*c[6]-112140*c[5]*c[6]* c[3]^2+414750*c[3]^5*c[6]*c[5]-512000*c[5]*c[3]^4*c[6]+333025*c[5]^2*c [3]^4*c[6]-209100*c[5]^2*c[3]^3*c[6]-12600*c[5]^2*c[6]*c[3]+71400*c[5] ^2*c[3]^2*c[6]+90000*c[3]^6*c[5]^2*c[6]+14400*b[8]*c[6]*`b*`[7]+264960 0*b[8]*c[3]^3*`b*`[7]+28800*b[8]*c[5]*`b*`[7]+187200*b[8]*c[3]*`b*`[7] +265200*c[5]^2*b[8]*c[3]^2-870000*c[3]^4*b[8]*c[6]+666000*c[3]^5*b[8]* c[6]+1132800*c[3]^4*b[8]*c[5]^2+1543500*c[3]^5*b[8]*c[5]+270000*c[3]^6 *b[8]*c[5]^2-3600*c[5]^2*`b*`[7]*c[6]-270000*c[3]^6*b[8]*c[5]^2*c[6]+8 77500*c[3]^5*b[8]*c[5]^2*c[6]-265200*c[6]*c[3]^2*b[8]*c[5]^2-1132800*c [3]^4*b[8]*c[5]^2*c[6]+982800*b[8]*c[6]*c[3]^2*`b*`[7]+1965600*c[5]*b[ 8]*c[3]^2*`b*`[7]-5299200*c[5]*b[8]*c[3]^3*`b*`[7]-187200*c[6]*b[8]*c[ 3]*`b*`[7]-374400*c[5]*b[8]*c[3]*`b*`[7]+2880*`b*`[7]+208140*`b*`[7]*c [3]^2-38520*`b*`[7]*c[3]+870000*`b*`[7]*c[3]^4-579000*`b*`[7]*c[3]^3+2 02500*`b*`[7]*c[3]^6-666000*`b*`[7]*c[3]^5+5299200*c[5]*b[8]*c[3]^3*c[ 6]*`b*`[7]-982800*b[8]*c[3]^2*`b*`[7]-3600*c[5]^2*b[8]*c[6]+14400*c[5] ^2*b[8]*c[6]*`b*`[7]-187200*c[5]^2*b[8]*c[6]*`b*`[7]*c[3]+1543500*c[5] *c[3]^5*`b*`[7]-2002800*c[5]*c[3]^4*`b*`[7]+1324500*c[5]*c[3]^3*`b*`[7 ]-473340*c[5]*c[3]^2*`b*`[7]+87120*c[5]*c[3]*`b*`[7]-6480*c[5]*`b*`[7] -2649600*b[8]*c[3]^3*c[6]*`b*`[7]-28800*c[5]*b[8]*c[6]*`b*`[7]+666000* c[3]^5*c[6]*`b*`[7]-870000*c[3]^4*c[6]*`b*`[7]-202500*c[3]^6*c[6]*`b*` [7]-1543500*c[3]^5*c[6]*c[5]*`b*`[7]+472500*c[3]^6*c[6]*c[5]*`b*`[7]+2 002800*c[3]^4*c[6]*c[5]*`b*`[7]-1324500*c[3]^3*c[6]*c[5]*`b*`[7]+47334 0*c[6]*c[3]^2*c[5]*`b*`[7]-87120*c[6]*c[3]*c[5]*`b*`[7]+579000*c[3]^3* c[6]*`b*`[7]-208140*c[6]*c[3]^2*`b*`[7]+38520*c[6]*c[3]*`b*`[7]+6480*c [5]*c[6]*`b*`[7]-2880*c[6]*`b*`[7]-472500*`b*`[7]*c[3]^6*c[5]+270000*c [5]^2*c[3]^6*`b*`[7]-877500*c[5]^2*c[3]^5*`b*`[7]+1132800*c[5]^2*c[3]^ 4*`b*`[7]-745500*c[5]^2*c[3]^3*`b*`[7]+265200*c[5]^2*`b*`[7]*c[3]^2-27 0000*c[5]^2*c[3]^6*`b*`[7]*c[6]+877500*c[5]^2*c[3]^5*`b*`[7]*c[6]-1132 800*c[5]^2*c[3]^4*`b*`[7]*c[6]+745500*c[5]^2*c[3]^3*`b*`[7]*c[6]-26520 0*c[5]^2*c[3]^2*`b*`[7]*c[6]+48600*c[5]^2*`b*`[7]*c[6]*c[3]-48600*c[5] ^2*c[3]*`b*`[7]+3600*`b*`[7]*c[5]^2-14400*c[5]^2*b[8]*`b*`[7]-810000*c [3]^6*b[8]*`b*`[7]+2808000*c[3]^5*b[8]*`b*`[7]-3837600*c[3]^4*b[8]*`b* `[7]+3600*c[5]^2*b[8]+5616000*c[5]*b[8]*c[3]^5*c[6]*`b*`[7]-7675200*c[ 5]*b[8]*c[3]^4*c[6]*`b*`[7]-1620000*c[5]*b[8]*c[3]^6*c[6]*`b*`[7]+8100 00*c[5]^2*b[8]*c[3]^6*c[6]*`b*`[7]-2808000*c[5]^2*b[8]*c[3]^5*c[6]*`b* `[7]+3837600*c[5]^2*b[8]*c[3]^4*c[6]*`b*`[7]-2649600*c[5]^2*b[8]*c[3]^ 3*c[6]*`b*`[7]-48600*c[5]^2*b[8]*c[3]+187200*c[5]^2*b[8]*c[3]*`b*`[7]+ 7675200*c[5]*b[8]*c[3]^4*`b*`[7]+1620000*c[5]*b[8]*c[3]^6*`b*`[7]-5616 000*c[5]*b[8]*c[3]^5*`b*`[7]-810000*c[5]^2*b[8]*c[3]^6*`b*`[7]+2808000 *c[5]^2*b[8]*c[3]^5*`b*`[7]-3837600*c[5]^2*b[8]*c[3]^4*`b*`[7]+2649600 *c[5]^2*b[8]*c[3]^3*`b*`[7]-982800*c[5]^2*b[8]*c[3]^2*`b*`[7])/(5*c[3] -2)/(3*c[3]-1)/(c[5]-1)/(c[3]-1)/c[6]/b[8]/`b*`[7]/(2*c[6]-10*c[6]*c[3 ]+15*c[6]*c[3]^2-c[3])/(c[6]-c[3])/(c[6]-c[5]), `b*`[2] = 0, a[3,2] = \+ 1/2*c[3]^2/c[2], a[5,2] = 1/2*c[3]*c[5]/c[2]*(-5*c[5]+1+5*c[5]^2)/(1-5 *c[3]+5*c[3]^2), c[4] = c[3]/(15*c[3]^2-10*c[3]+2), a[2,1] = c[2], c[7 ] = 1, a[6,4] = -1/6*c[6]*(15*c[3]^2-10*c[3]+2)^2*(2*c[6]-10*c[6]*c[3] +15*c[6]*c[3]^2-c[3])*(c[6]-c[3])*(90*c[3]^3*c[6]-120*c[6]*c[3]^2+48*c [6]*c[3]-6*c[6]-225*c[3]^3*c[5]+225*c[5]^2*c[3]^3-240*c[5]^2*c[3]^2+9* c[3]^2+255*c[5]*c[3]^2-100*c[5]*c[3]-4*c[3]+90*c[5]^2*c[3]+14*c[5]-12* c[5]^2)/c[3]^2/(15*c[3]^2-10*c[3]+1)/(1-5*c[3]+5*c[3]^2)/(15*c[5]*c[3] ^2-10*c[5]*c[3]+2*c[5]-c[3])/(15*c[5]*c[3]-9*c[3]+4-6*c[5]), b[1] = -1 /60*1/c[3]^2*(150*c[5]*c[3]^3*c[6]-75*c[3]^3*c[6]+45*c[3]^3-75*c[3]^3* c[5]+105*c[5]*c[3]^2-205*c[5]*c[6]*c[3]^2+105*c[6]*c[3]^2-65*c[3]^2+29 *c[3]-45*c[5]*c[3]+80*c[5]*c[6]*c[3]-45*c[6]*c[3]+6*c[5]-10*c[5]*c[6]- 4+6*c[6])/c[5]/c[6], a[6,1] = 1/6*c[6]*(-45*c[5]^2*c[3]^4+405*c[5]^2*c [2]*c[3]^3-66*c[5]^2*c[2]*c[3]^2+50*c[5]*c[2]*c[3]^2+1350*c[6]^3*c[2]* c[3]^5+6300*c[6]^2*c[2]*c[5]*c[3]^4-1350*c[2]*c[3]^6*c[6]^2-6300*c[6]* c[2]*c[5]*c[3]^5+1890*c[6]*c[2]*c[5]^2*c[3]^3-4875*c[6]^2*c[2]*c[5]*c[ 3]^3-3375*c[2]*c[5]^2*c[6]*c[3]^6-54*c[5]*c[2]*c[6]*c[3]-2700*c[6]^3*c [2]*c[3]^4+2100*c[6]^3*c[2]*c[3]^3-2190*c[6]^2*c[2]*c[3]^4-12*c[6]^3*c [2]-12*c[3]^3*c[5]-6*c[2]*c[3]^2*c[6]+90*c[6]*c[2]*c[3]^5-120*c[6]*c[2 ]*c[3]^4-24*c[6]^2*c[5]^2*c[2]+675*c[5]^2*c[2]*c[3]^5+5100*c[6]*c[2]*c [5]*c[3]^4-495*c[5]*c[2]*c[3]^5+660*c[5]*c[2]*c[3]^4-870*c[5]^2*c[2]*c [3]^4-307*c[5]*c[2]*c[3]^3+27*c[3]^4*c[5]+18*c[6]^2*c[3]*c[2]+48*c[2]* c[6]*c[3]^3-204*c[6]^2*c[2]*c[3]^2+156*c[6]^3*c[2]*c[3]+930*c[6]^2*c[2 ]*c[3]^3+3375*c[2]*c[3]^6*c[6]*c[5]+1870*c[5]*c[6]^2*c[2]*c[3]^2-2140* c[5]*c[6]*c[2]*c[3]^3+54*c[2]*c[6]*c[5]^2*c[3]-3375*c[6]^2*c[2]*c[5]*c [3]^5-5850*c[6]^2*c[2]*c[5]^2*c[3]^4-352*c[5]*c[6]^2*c[2]*c[3]+300*c[5 ]^2*c[6]^2*c[2]*c[3]+2700*c[6]^2*c[2]*c[3]^5-810*c[6]^3*c[2]*c[3]^2+28 *c[6]^2*c[2]*c[5]+5850*c[6]*c[2]*c[5]^2*c[3]^5+4425*c[6]^2*c[2]*c[5]^2 *c[3]^3-1650*c[6]^2*c[2]*c[5]^2*c[3]^2+3375*c[6]^2*c[2]*c[5]^2*c[3]^5- 4650*c[6]*c[2]*c[5]^2*c[3]^4+485*c[5]*c[6]*c[2]*c[3]^2-435*c[5]^2*c[6] *c[2]*c[3]^2+18*c[5]^2*c[3]^3+60*c[5]*c[3]^3*c[6]-60*c[5]*c[6]^2*c[3]^ 3+135*c[5]*c[3]^4*c[6]^2-135*c[5]*c[3]^4*c[6]+225*c[5]^2*c[3]^4*c[6]-9 0*c[5]^2*c[3]^3*c[6]-225*c[5]^2*c[6]^2*c[3]^4+90*c[5]^2*c[6]^2*c[3]^3) /c[5]/c[3]^2/(15*c[5]*c[3]-9*c[3]+4-6*c[5])/c[2]/(1-5*c[3]+5*c[3]^2), \+ a[4,3] = -(15*c[3]^2-10*c[3]+1)*c[3]/(15*c[3]^2-10*c[3]+2)^3, c[8] = 1 , a[8,7] = 0, b[6] = -1/60*(1-5*c[3]+5*c[3]^2)*(15*c[5]*c[3]-9*c[3]+4- 6*c[5])/c[6]/(-1+c[6])/(2*c[6]-10*c[6]*c[3]+15*c[6]*c[3]^2-c[3])/(c[6] -c[3])/(c[6]-c[5]), a[7,3] = -1/360*(576+461475*c[5]^2*c[3]^4-576*c[6] -8136*c[3]+46488*c[3]^2-391350*c[3]^5*c[5]^2+396000*c[5]^2*c[3]^5*c[6] -165750*c[3]^6*c[5]+186750*c[3]^6*c[5]*c[6]+132750*c[3]^6*c[5]^2+39040 0*c[3]^3*c[5]+136240*c[3]^3*c[6]+179850*c[3]^5*c[6]-4500*c[3]^7*c[5]-1 584*c[5]-630875*c[3]^4*c[5]-1080*c[5]^2*c[6]-218255*c[3]^4*c[6]+22644* c[5]*c[3]+8118*c[6]*c[3]+1620*c[5]*c[6]-46260*c[6]*c[3]^2-130890*c[5]* c[3]^2+519525*c[3]^5*c[5]-279420*c[5]^2*c[3]^3+91980*c[5]^2*c[3]^2-156 60*c[5]^2*c[3]-59400*c[3]^6*c[6]-137340*c[3]^3+27000*c[5]*c[3]^7*`b*`[ 7]+63150*c[3]^6+1080*c[5]^2+221090*c[3]^4-184195*c[3]^5-1350*c[3]^7-23 220*c[5]*c[6]*c[3]-404695*c[5]*c[3]^3*c[6]+134750*c[5]*c[6]*c[3]^2-555 600*c[3]^5*c[6]*c[5]+661425*c[5]*c[3]^4*c[6]-464550*c[5]^2*c[3]^4*c[6] +280125*c[5]^2*c[3]^3*c[6]+15660*c[5]^2*c[6]*c[3]-92010*c[5]^2*c[3]^2* c[6]-135000*c[3]^6*c[5]^2*c[6]+2880*c[5]^2*`b*`[7]*c[6]-2880*`b*`[7]-2 29440*`b*`[7]*c[3]^2+40320*`b*`[7]*c[3]-1097700*`b*`[7]*c[3]^4+678360* `b*`[7]*c[3]^3-330300*`b*`[7]*c[3]^6+925440*`b*`[7]*c[3]^5+16200*`b*`[ 7]*c[3]^7-2267100*c[5]*c[3]^5*`b*`[7]+2786700*c[5]*c[3]^4*`b*`[7]-1733 340*c[5]*c[3]^3*`b*`[7]+583800*c[5]*c[3]^2*`b*`[7]-101760*c[5]*c[3]*`b *`[7]+7200*c[5]*`b*`[7]-684900*c[3]^5*c[6]*`b*`[7]+822360*c[3]^4*c[6]* `b*`[7]+226800*c[3]^6*c[6]*`b*`[7]+1991700*c[3]^5*c[6]*c[5]*`b*`[7]-67 5000*c[3]^6*c[6]*c[5]*`b*`[7]-2326500*c[3]^4*c[6]*c[5]*`b*`[7]+1394340 *c[3]^3*c[6]*c[5]*`b*`[7]-457620*c[6]*c[3]^2*c[5]*`b*`[7]+78600*c[6]*c [3]*c[5]*`b*`[7]-507540*c[3]^3*c[6]*`b*`[7]+171240*c[6]*c[3]^2*`b*`[7] -30120*c[6]*c[3]*`b*`[7]-5520*c[5]*c[6]*`b*`[7]+2160*c[6]*`b*`[7]+6975 00*`b*`[7]*c[3]^6*c[5]-513000*c[5]^2*c[3]^6*`b*`[7]+1532700*c[5]^2*c[3 ]^5*`b*`[7]-1812600*c[5]^2*c[3]^4*`b*`[7]+1096740*c[5]^2*c[3]^3*`b*`[7 ]-361440*c[5]^2*`b*`[7]*c[3]^2+405000*c[5]^2*c[3]^6*`b*`[7]*c[6]-11610 00*c[5]^2*c[3]^5*`b*`[7]*c[6]+1313100*c[5]^2*c[3]^4*`b*`[7]*c[6]-76320 0*c[5]^2*c[3]^3*`b*`[7]*c[6]+244620*c[5]^2*c[3]^2*`b*`[7]*c[6]-41400*c [5]^2*`b*`[7]*c[6]*c[3]+61920*c[5]^2*c[3]*`b*`[7]-4320*`b*`[7]*c[5]^2) /(c[6]-c[3])/(-c[5]+c[3])/(15*c[5]*c[3]-9*c[3]+4-6*c[5])/(15*c[3]^2-10 *c[3]+1)/c[3]^2/(1-5*c[3]+5*c[3]^2)/`b*`[7], a[7,6] = 1/60*(-1+c[6])*( -24+165*c[3]-350*c[3]^2-300*c[3]^3*c[5]+30*c[5]-210*c[5]*c[3]+455*c[5] *c[3]^2+225*c[3]^3+120*`b*`[7]-1560*c[5]*c[3]^2*`b*`[7]+1560*`b*`[7]*c [3]^2-780*`b*`[7]*c[3]-900*`b*`[7]*c[3]^3+900*c[5]*c[3]^3*`b*`[7]+780* c[5]*c[3]*`b*`[7]-120*c[5]*`b*`[7])/c[6]/`b*`[7]/(2*c[6]-10*c[6]*c[3]+ 15*c[6]*c[3]^2-c[3])/(c[6]-c[3])/(c[6]-c[5]), b[3] = 1/60*(75*c[5]*c[6 ]*c[3]^2-60*c[5]*c[6]*c[3]+10*c[5]*c[6]-45*c[5]*c[3]^2+35*c[5]*c[3]-6* c[5]-45*c[6]*c[3]^2+35*c[6]*c[3]-6*c[6]+30*c[3]^2-23*c[3]+4)/(c[6]-c[3 ])/(-c[5]+c[3])/c[3]^2/(c[3]-1)/(15*c[3]^2-10*c[3]+1), `b*`[8] = 0, `b *`[4] = -1/60*(192-240*c[6]-4080*c[3]+39360*c[3]^2+1012500*c[3]^9*c[5] -4710000*c[3]^6*c[5]+6660000*c[3]^6*c[5]*c[6]+5265000*c[3]^7*c[6]+2856 00*c[3]^3*c[5]+285600*c[3]^3*c[6]+2738000*c[3]^5*c[6]+5265000*c[3]^7*c [5]-3459375*c[3]^8*c[6]-240*c[5]-1075000*c[3]^4*c[5]-3459375*c[3]^8*c[ 5]-1075000*c[3]^4*c[6]+5120*c[5]*c[3]+5120*c[6]*c[3]+320*c[5]*c[6]-496 00*c[6]*c[3]^2-49600*c[5]*c[3]^2+2738000*c[3]^5*c[5]-4710000*c[3]^6*c[ 6]-225600*c[3]^3+11137500*`b*`[7]*c[3]^8*c[5]+3037500*`b*`[7]*c[3]^9-1 7820000*c[5]*c[3]^7*`b*`[7]+3654000*c[3]^6-3037500*c[3]^9*c[6]*`b*`[7] +11137500*c[3]^8*c[6]*`b*`[7]-17820000*c[3]^7*c[6]*`b*`[7]+844800*c[3] ^4-2139000*c[3]^5+3037500*c[5]*c[3]^9*`b*`[7]*c[6]+1012500*c[3]^9*c[6] +2632500*c[3]^8-759375*c[3]^9-4050000*c[3]^7-6880*c[5]*c[6]*c[3]-39040 0*c[5]*c[3]^3*c[6]+67200*c[5]*c[6]*c[3]^2-3822000*c[3]^5*c[6]*c[5]+148 4000*c[5]*c[3]^4*c[6]-1518750*c[3]^9*c[6]*c[5]-7560000*c[3]^7*c[5]*c[6 ]+5062500*c[3]^8*c[6]*c[5]-960*`b*`[7]-3037500*`b*`[7]*c[3]^9*c[5]-192 000*`b*`[7]*c[3]^2+20160*`b*`[7]*c[3]-3996000*`b*`[7]*c[3]^4+1084800*` b*`[7]*c[3]^3-16560000*`b*`[7]*c[3]^6+9924000*`b*`[7]*c[3]^5-11137500* `b*`[7]*c[3]^8+17820000*`b*`[7]*c[3]^7-11137500*c[3]^8*c[6]*`b*`[7]*c[ 5]-9924000*c[5]*c[3]^5*`b*`[7]+3996000*c[5]*c[3]^4*`b*`[7]-1084800*c[5 ]*c[3]^3*`b*`[7]+192000*c[5]*c[3]^2*`b*`[7]-20160*c[5]*c[3]*`b*`[7]+96 0*c[5]*`b*`[7]-9924000*c[3]^5*c[6]*`b*`[7]+3996000*c[3]^4*c[6]*`b*`[7] +16560000*c[3]^6*c[6]*`b*`[7]+9924000*c[3]^5*c[6]*c[5]*`b*`[7]-1656000 0*c[3]^6*c[6]*c[5]*`b*`[7]-3996000*c[3]^4*c[6]*c[5]*`b*`[7]+1084800*c[ 3]^3*c[6]*c[5]*`b*`[7]-192000*c[6]*c[3]^2*c[5]*`b*`[7]+20160*c[6]*c[3] *c[5]*`b*`[7]-1084800*c[3]^3*c[6]*`b*`[7]+192000*c[6]*c[3]^2*`b*`[7]-2 0160*c[6]*c[3]*`b*`[7]-960*c[5]*c[6]*`b*`[7]+960*c[6]*`b*`[7]+17820000 *`b*`[7]*c[3]^7*c[5]*c[6]+16560000*`b*`[7]*c[3]^6*c[5])/c[3]^2/(2*c[6] -10*c[6]*c[3]+15*c[6]*c[3]^2-c[3])/(15*c[5]*c[3]^2-10*c[5]*c[3]+2*c[5] -c[3])/(15*c[3]^2-10*c[3]+1), a[7,5] = 1/60*(96+4275*c[5]^2*c[3]^4-96* c[6]-876*c[3]+2885*c[3]^2+11925*c[3]^3*c[5]+4020*c[3]^3*c[6]-264*c[5]- 6075*c[3]^4*c[5]-180*c[5]^2*c[6]-2025*c[3]^4*c[6]+2463*c[5]*c[3]+873*c [6]*c[3]+270*c[5]*c[6]-2865*c[6]*c[3]^2-8305*c[5]*c[3]^2-8265*c[5]^2*c [3]^3+5660*c[5]^2*c[3]^2-1650*c[5]^2*c[3]-4050*c[3]^3+174*c[5]^2+2025* c[3]^4-2520*c[5]*c[6]*c[3]-12255*c[5]*c[3]^3*c[6]+8505*c[5]*c[6]*c[3]^ 2+6300*c[5]*c[3]^4*c[6]-4500*c[5]^2*c[3]^4*c[6]+8625*c[5]^2*c[3]^3*c[6 ]+1710*c[5]^2*c[6]*c[3]-5880*c[5]^2*c[3]^2*c[6]+720*c[5]^2*`b*`[7]*c[6 ]-480*`b*`[7]-13260*`b*`[7]*c[3]^2+4200*`b*`[7]*c[3]-8100*`b*`[7]*c[3] ^4+17640*`b*`[7]*c[3]^3+24300*c[5]*c[3]^4*`b*`[7]-52020*c[5]*c[3]^3*`b *`[7]+38220*c[5]*c[3]^2*`b*`[7]-11820*c[5]*c[3]*`b*`[7]+1320*c[5]*`b*` [7]+5400*c[3]^4*c[6]*`b*`[7]-18900*c[3]^4*c[6]*c[5]*`b*`[7]+40860*c[3] ^3*c[6]*c[5]*`b*`[7]-30420*c[6]*c[3]^2*c[5]*`b*`[7]+9540*c[6]*c[3]*c[5 ]*`b*`[7]-12060*c[3]^3*c[6]*`b*`[7]+9360*c[6]*c[3]^2*`b*`[7]-3060*c[6] *c[3]*`b*`[7]-1080*c[5]*c[6]*`b*`[7]+360*c[6]*`b*`[7]-16200*c[5]^2*c[3 ]^4*`b*`[7]+34380*c[5]^2*c[3]^3*`b*`[7]-24960*c[5]^2*`b*`[7]*c[3]^2+13 500*c[5]^2*c[3]^4*`b*`[7]*c[6]-28800*c[5]^2*c[3]^3*`b*`[7]*c[6]+21060* c[5]^2*c[3]^2*`b*`[7]*c[6]-6480*c[5]^2*`b*`[7]*c[6]*c[3]+7620*c[5]^2*c [3]*`b*`[7]-840*`b*`[7]*c[5]^2)/c[5]/`b*`[7]/(15*c[5]*c[3]-9*c[3]+4-6* c[5])/(-c[5]+c[3])/(15*c[5]*c[3]^2-10*c[5]*c[3]+2*c[5]-c[3])/(c[6]-c[5 ]), a[7,4] = -1/360*(15*c[3]^2-10*c[3]+2)^2*(1152+5098950*c[5]^2*c[3]^ 4-1152*c[6]-22032*c[3]+182958*c[3]^2-9601950*c[3]^5*c[5]^2+9579750*c[5 ]^2*c[3]^5*c[6]-15366375*c[3]^6*c[5]+15836625*c[3]^6*c[5]*c[6]+1114087 5*c[3]^6*c[5]^2+3327750*c[3]^7*c[6]+2438680*c[3]^3*c[5]+856930*c[3]^3* c[6]+4594350*c[3]^5*c[6]+9902250*c[3]^7*c[5]-911250*c[3]^8*c[6]-3168*c [5]-7182050*c[3]^4*c[5]-2160*c[5]^2*c[6]-2733750*c[3]^8*c[5]-2494190*c [3]^4*c[6]+61164*c[5]*c[3]+21996*c[6]*c[3]+3240*c[5]*c[6]-182340*c[6]* c[3]^2-512880*c[5]*c[3]^2+13384425*c[3]^5*c[5]-1713660*c[5]^2*c[3]^3+3 56760*c[5]^2*c[3]^2-42120*c[5]^2*c[3]-6075000*`b*`[7]*c[3]^8*c[5]^2-52 16175*c[3]^6*c[6]-861210*c[3]^3+9720000*`b*`[7]*c[3]^8*c[5]-37300500*c [5]*c[3]^7*`b*`[7]+5247900*c[3]^6+2430000*c[3]^8*c[6]*`b*`[7]-9598500* c[3]^7*c[6]*`b*`[7]+2160*c[5]^2+2509600*c[3]^4-4624905*c[3]^5+2025000* c[3]^8*c[5]^2-7256250*c[3]^7*c[5]^2+911250*c[3]^8-3341250*c[3]^7-62640 *c[5]*c[6]*c[3]-2503240*c[5]*c[3]^3*c[6]+525880*c[5]*c[6]*c[3]^2-13771 500*c[3]^5*c[6]*c[5]+7380400*c[5]*c[3]^4*c[6]-5090700*c[5]^2*c[3]^4*c[ 6]+1712100*c[5]^2*c[3]^3*c[6]+42120*c[5]^2*c[6]*c[3]-356640*c[5]^2*c[3 ]^2*c[6]-11110500*c[3]^6*c[5]^2*c[6]-2025000*c[3]^8*c[5]^2*c[6]+723937 5*c[3]^7*c[5]^2*c[6]-10229625*c[3]^7*c[5]*c[6]+2835000*c[3]^8*c[6]*c[5 ]+5760*c[5]^2*`b*`[7]*c[6]-5760*`b*`[7]-889080*`b*`[7]*c[3]^2+108720*` b*`[7]*c[3]-11718420*`b*`[7]*c[3]^4+4109880*`b*`[7]*c[3]^3-23082300*`b *`[7]*c[3]^6+21027960*`b*`[7]*c[3]^5-3645000*`b*`[7]*c[3]^8+14094000*` b*`[7]*c[3]^7+3037500*c[5]^2*c[3]^8*`b*`[7]*c[6]-12555000*c[5]^2*c[3]^ 7*`b*`[7]*c[6]-6682500*c[3]^8*c[6]*`b*`[7]*c[5]-54699300*c[5]*c[3]^5*` b*`[7]+30220500*c[5]*c[3]^4*`b*`[7]-10512120*c[5]*c[3]^3*`b*`[7]+22563 60*c[5]*c[3]^2*`b*`[7]-273840*c[5]*c[3]*`b*`[7]+14400*c[5]*`b*`[7]-148 63500*c[3]^5*c[6]*`b*`[7]+8411880*c[3]^4*c[6]*`b*`[7]+16032600*c[3]^6* c[6]*`b*`[7]+40324500*c[3]^5*c[6]*c[5]*`b*`[7]-43834500*c[3]^6*c[6]*c[ 5]*`b*`[7]-22597800*c[3]^4*c[6]*c[5]*`b*`[7]+7944480*c[3]^3*c[6]*c[5]* `b*`[7]-1718160*c[6]*c[3]^2*c[5]*`b*`[7]+209520*c[6]*c[3]*c[5]*`b*`[7] -2990640*c[3]^3*c[6]*`b*`[7]+654720*c[6]*c[3]^2*`b*`[7]-80880*c[6]*c[3 ]*`b*`[7]-11040*c[5]*c[6]*`b*`[7]+4320*c[6]*`b*`[7]+26365500*`b*`[7]*c [3]^7*c[5]*c[6]+60574500*`b*`[7]*c[3]^6*c[5]+23287500*c[5]^2*c[3]^7*`b *`[7]-37719000*c[5]^2*c[3]^6*`b*`[7]+33921900*c[5]^2*c[3]^5*`b*`[7]-18 644400*c[5]^2*c[3]^4*`b*`[7]+6446520*c[5]^2*c[3]^3*`b*`[7]-1374480*c[5 ]^2*`b*`[7]*c[3]^2+21613500*c[5]^2*c[3]^6*`b*`[7]*c[6]-20367000*c[5]^2 *c[3]^5*`b*`[7]*c[6]+11597400*c[5]^2*c[3]^4*`b*`[7]*c[6]-4118400*c[5]^ 2*c[3]^3*`b*`[7]*c[6]+895680*c[5]^2*c[3]^2*`b*`[7]*c[6]-109440*c[5]^2* `b*`[7]*c[6]*c[3]+165600*c[5]^2*c[3]*`b*`[7]-8640*`b*`[7]*c[5]^2)/`b*` [7]/c[3]^2/(1-5*c[3]+5*c[3]^2)/(15*c[3]^2-10*c[3]+1)/(15*c[5]*c[3]^2-1 0*c[5]*c[3]+2*c[5]-c[3])/(15*c[5]*c[3]-9*c[3]+4-6*c[5])/(2*c[6]-10*c[6 ]*c[3]+15*c[6]*c[3]^2-c[3]), `b*`[1] = 1/60*(24-30*c[6]-165*c[3]+350*c [3]^2+300*c[3]^3*c[5]+300*c[3]^3*c[6]-30*c[5]+210*c[5]*c[3]+210*c[6]*c [3]+40*c[5]*c[6]-455*c[6]*c[3]^2-455*c[5]*c[3]^2-225*c[3]^3-290*c[5]*c [6]*c[3]-450*c[5]*c[3]^3*c[6]+660*c[5]*c[6]*c[3]^2-120*`b*`[7]-1560*`b 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b*`[7]-16178040*c[5]*c[3]^3*`b*`[7]+4508820*c[5]*c[3]^2*`b*`[7]-681480 *c[5]*c[3]*`b*`[7]+43200*c[5]*`b*`[7]+33289200*b[8]*c[3]^3*c[6]*`b*`[7 ]+388800*c[5]*b[8]*c[6]*`b*`[7]-20923200*c[3]^5*c[6]*`b*`[7]+17324700* c[3]^4*c[6]*`b*`[7]+13581000*c[3]^6*c[6]*`b*`[7]+83425500*c[3]^5*c[6]* c[5]*`b*`[7]-54648000*c[3]^6*c[6]*c[5]*`b*`[7]-68378400*c[3]^4*c[6]*c[ 5]*`b*`[7]+32622120*c[3]^3*c[6]*c[5]*`b*`[7]-9090720*c[6]*c[3]^2*c[5]* `b*`[7]+1374120*c[6]*c[3]*c[5]*`b*`[7]-8354640*c[3]^3*c[6]*`b*`[7]+235 3740*c[6]*c[3]^2*`b*`[7]-359640*c[6]*c[3]*`b*`[7]-87120*c[5]*c[6]*`b*` [7]+23040*c[6]*`b*`[7]+14782500*`b*`[7]*c[3]^7*c[5]*c[6]+27013500*`b*` [7]*c[3]^6*c[5]+9315000*c[5]^2*c[3]^7*`b*`[7]-34317000*c[5]^2*c[3]^6*` b*`[7]+52137900*c[5]^2*c[3]^5*`b*`[7]-42477300*c[5]^2*c[3]^4*`b*`[7]+2 0125980*c[5]^2*c[3]^3*`b*`[7]-5567220*c[5]^2*`b*`[7]*c[3]^2+69957000*c [5]^2*c[3]^6*`b*`[7]*c[6]-106081200*c[5]^2*c[3]^5*`b*`[7]*c[6]+8630730 0*c[5]^2*c[3]^4*`b*`[7]*c[6]-40855680*c[5]^2*c[3]^3*`b*`[7]*c[6]+11294 820*c[5]^2*c[3]^2*`b*`[7]*c[6]-1693800*c[5]^2*`b*`[7]*c[6]*c[3]+835200 *c[5]^2*c[3]*`b*`[7]-52560*`b*`[7]*c[5]^2+1569600*c[5]^3*b[8]*c[3]*`b* `[7]+259200*c[5]^2*b[8]*`b*`[7]-7290000*c[3]^7*b[8]*`b*`[7]+28512000*c [3]^6*b[8]*`b*`[7]-45770400*c[3]^5*b[8]*`b*`[7]+39196800*c[3]^4*b[8]*` b*`[7]-52560*c[5]^2*b[8]+20729700*c[5]^2*c[3]^3*c[6]^2*`b*`[7]+8908075 *c[5]^3*c[3]^4*c[6]-11440575*c[5]^3*c[3]^5*c[6]-21321900*c[5]^3*c[3]^5 *`b*`[7]-6777000*c[6]^2*c[3]^6*`b*`[7]+10429200*c[6]^2*c[3]^5*`b*`[7]+ 1822500*c[6]^2*c[3]^7*`b*`[7]-8633700*c[6]^2*c[3]^4*`b*`[7]-5134650*c[ 5]^3*c[3]^4*c[6]^2+6633375*c[5]^3*c[3]^5*c[6]^2-4635000*c[5]^3*c[3]^6* c[6]^2-8137200*c[5]^3*c[3]^3*`b*`[7]+2237640*c[5]^3*`b*`[7]*c[3]^2-333 720*c[5]^3*c[3]*`b*`[7]-13845600*b[8]*c[3]^3*c[6]^2*`b*`[7]-29160000*c [5]^2*c[3]^7*b[8]*c[6]^2*`b*`[7]+12150000*c[5]^3*c[3]^7*b[8]*c[6]^2*`b *`[7]-46980000*c[5]^3*c[3]^6*b[8]*c[6]^2*`b*`[7]+113238000*c[5]^2*c[3] ^6*b[8]*c[6]^2*`b*`[7]+4871625*c[5]^3*c[3]^5-3355875*c[5]^3*c[3]^6+961 875*c[5]^3*c[3]^7+17274300*c[5]^3*c[3]^4*`b*`[7]+20880*c[5]^3*b[8]-100 800*c[5]^3*b[8]*`b*`[7]+89100*c[5]^3*c[6]^2*c[3]-617400*c[5]^3*c[6]^2* c[3]^2-54000*c[5]^2*b[8]*c[6]^2-42480*c[5]^3*b[8]*c[6]+21600*c[5]^3*b[ 8]*c[6]^2-3847500*c[5]^3*b[8]*c[3]^7-35253600*c[5]^3*`b*`[7]*c[3]^4*c[ 6]+16588200*c[5]^3*`b*`[7]*c[3]^3*c[6]+679320*c[5]^3*`b*`[7]*c[6]*c[3] -4557840*c[5]^3*`b*`[7]*c[3]^2*c[6]-3847500*c[5]^3*c[3]^7*`b*`[7]+1410 7500*c[5]^3*c[3]^6*`b*`[7]+30639600*c[5]^3*b[8]*c[3]^3*c[6]^2*`b*`[7]- 317584800*c[5]*b[8]*c[3]^5*c[6]*`b*`[7]+270540000*c[5]*b[8]*c[3]^4*c[6 ]*`b*`[7]-51030000*c[5]*b[8]*c[3]^7*c[6]*`b*`[7]+198774000*c[5]*b[8]*c [3]^6*c[6]*`b*`[7]+65610000*c[5]^2*b[8]*c[3]^7*c[6]*`b*`[7]-254178000* c[5]^2*b[8]*c[3]^6*c[6]*`b*`[7]+403509600*c[5]^2*b[8]*c[3]^5*c[6]*`b*` [7]-341258400*c[5]^2*b[8]*c[3]^4*c[6]*`b*`[7]+167043600*c[5]^2*b[8]*c[ 3]^3*c[6]*`b*`[7]+21870000*c[5]*b[8]*c[3]^7*c[6]^2*`b*`[7]-180273600*c [5]^2*b[8]*c[3]^5*c[6]^2*`b*`[7]+152949600*c[5]^2*b[8]*c[3]^4*c[6]^2*` b*`[7]-85536000*c[5]*b[8]*c[3]^6*c[6]^2*`b*`[7]+137311200*c[5]*b[8]*c[ 3]^5*c[6]^2*`b*`[7]-117590400*c[5]*b[8]*c[3]^4*c[6]^2*`b*`[7]-75124800 *c[5]^2*b[8]*c[3]^3*c[6]^2*`b*`[7]+58330800*c[5]*b[8]*c[3]^3*c[6]^2*`b *`[7]-26730000*c[5]^3*b[8]*c[3]^7*c[6]*`b*`[7]+103194000*c[5]^3*b[8]*c [3]^6*c[6]*`b*`[7]-163144800*c[5]^3*b[8]*c[3]^5*c[6]*`b*`[7]+74412000* c[5]^3*b[8]*c[3]^5*c[6]^2*`b*`[7]-62769600*c[5]^3*b[8]*c[3]^4*c[6]^2*` b*`[7]+137325600*c[5]^3*b[8]*c[3]^4*c[6]*`b*`[7]-66877200*c[5]^3*b[8]* c[3]^3*c[6]*`b*`[7]+1350000*c[5]^3*c[3]^7*c[6]^2-21321900*c[5]^3*b[8]* c[3]^5+17274300*c[5]^3*b[8]*c[3]^4-8137200*c[5]^3*b[8]*c[3]^3+2237640* c[5]^3*b[8]*c[3]^2-345600*c[5]^3*c[6]^2*c[3]*`b*`[7]+858600*c[5]^2*c[6 ]^2*c[3]*`b*`[7]+835200*c[5]^2*b[8]*c[3]-4017600*c[5]^2*b[8]*c[3]*`b*` [7]-16848000*c[5]*c[6]^2*b[8]*c[3]^2*`b*`[7]+2635200*c[5]*c[6]^2*b[8]* c[3]*`b*`[7]-5400*c[5]^3*c[6]^2+4165380*c[6]^2*`b*`[7]*c[3]^3-54000*c[ 6]^2*`b*`[7]*c[5]^2-1174500*c[6]^2*`b*`[7]*c[3]^2+21600*c[6]^2*`b*`[7] *c[5]^3+4071600*b[8]*c[6]^2*c[3]^2*`b*`[7]+679320*c[5]^3*c[6]*b[8]*c[3 ]+2320200*c[5]^3*c[6]^2*b[8]*c[3]^2-152949600*c[5]*b[8]*c[3]^4*`b*`[7] -113238000*c[5]*b[8]*c[3]^6*`b*`[7]+180273600*c[5]*b[8]*c[3]^5*`b*`[7] +140940000*c[5]^2*b[8]*c[3]^6*`b*`[7]-223236000*c[5]^2*b[8]*c[3]^5*`b* `[7]+188308800*c[5]^2*b[8]*c[3]^4*`b*`[7]-91918800*c[5]^2*b[8]*c[3]^3* `b*`[7]+26114400*c[5]^2*b[8]*c[3]^2*`b*`[7]-56214000*c[5]^3*b[8]*c[3]^ 6*`b*`[7]+88732800*c[5]^3*b[8]*c[3]^5*`b*`[7]-74556000*c[5]^3*b[8]*c[3 ]^4*`b*`[7]+36237600*c[5]^3*b[8]*c[3]^3*`b*`[7]-10249200*c[5]^3*b[8]*c [3]^2*`b*`[7]+7897500*c[5]^3*c[3]^7*c[6]*`b*`[7]-28890000*c[5]^3*c[3]^ 6*c[6]*`b*`[7]+14107500*c[5]^3*b[8]*c[3]^6+21481200*c[5]^2*b[8]*c[6]^2 *c[3]^2*`b*`[7]-3326400*c[5]^2*b[8]*c[6]^2*c[3]*`b*`[7]-172800*c[5]*c[ 6]^2*b[8]*`b*`[7]+43920*c[5]*c[6]^2*`b*`[7])/(-1+c[6])/(5*c[3]-2)/(3*c [3]-1)/(c[3]-1)/(c[5]-1)/c[5]/b[8]/`b*`[7]/(15*c[5]*c[3]-9*c[3]+4-6*c[ 5])/(-c[5]+c[3])/(15*c[5]*c[3]^2-10*c[5]*c[3]+2*c[5]-c[3])/(c[6]-c[5]) , `b*`[6] = -1/60*(-24+165*c[3]-350*c[3]^2-300*c[3]^3*c[5]+30*c[5]-210 *c[5]*c[3]+455*c[5]*c[3]^2+225*c[3]^3+120*`b*`[7]-1560*c[5]*c[3]^2*`b* `[7]+1560*`b*`[7]*c[3]^2-780*`b*`[7]*c[3]-900*`b*`[7]*c[3]^3+900*c[5]* c[3]^3*`b*`[7]+780*c[5]*c[3]*`b*`[7]-120*c[5]*`b*`[7])/c[6]/(10*c[6]^2 *c[3]^2+2*c[6]^3-10*c[6]^3*c[3]+15*c[5]*c[3]^3*c[6]-15*c[5]*c[6]^2*c[3 ]^2+15*c[6]^3*c[3]^2+10*c[5]*c[6]^2*c[3]+3*c[5]*c[6]*c[3]-c[5]*c[3]^2- 10*c[5]*c[6]*c[3]^2+c[6]*c[3]^2-15*c[6]^2*c[3]^3-2*c[5]*c[6]^2-3*c[6]^ 2*c[3]), a[6,3] = -1/6*(c[6]-c[3])*(450*c[5]^2*c[3]^4-810*c[6]^2*c[3]^ 2-12*c[6]^2-4*c[3]^2+156*c[6]^2*c[3]-2700*c[6]^2*c[3]^4+45*c[3]^3*c[5] -820*c[3]^3*c[6]-1350*c[3]^5*c[6]+2100*c[6]^2*c[3]^3+300*c[3]^4*c[5]-2 4*c[5]^2*c[6]+1815*c[3]^4*c[6]-2*c[5]*c[3]-10*c[6]*c[3]+28*c[5]*c[6]+1 48*c[6]*c[3]^2-35*c[5]*c[3]^2-450*c[3]^5*c[5]-480*c[5]^2*c[3]^3+135*c[ 5]^2*c[3]^2-6*c[5]^2*c[3]+1350*c[3]^5*c[6]^2+89*c[3]^3-300*c[3]^4+270* c[3]^5-328*c[5]*c[6]*c[3]-2535*c[5]*c[3]^3*c[6]+1450*c[5]*c[6]*c[3]^2+ 900*c[3]^5*c[6]*c[5]+975*c[5]*c[3]^4*c[6]-2250*c[5]^2*c[3]^4*c[6]+3075 *c[5]^2*c[3]^3*c[6]+300*c[5]^2*c[6]*c[3]-1470*c[5]^2*c[3]^2*c[6])*c[6] /(1-5*c[3]+5*c[3]^2)/(15*c[3]^2-10*c[3]+1)/(15*c[5]*c[3]-9*c[3]+4-6*c[ 5])/(-c[5]+c[3])/c[3]^2, a[7,2] = 1/24*(12*`b*`[7]-1)*c[3]/(1-5*c[3]+5 *c[3]^2)/`b*`[7]/c[2], `b*`[3] = 1/60*(-300*c[5]*c[6]*c[3]^2+225*c[6]* c[3]^2-900*c[6]*c[3]^2*`b*`[7]+900*c[6]*c[3]^2*c[5]*`b*`[7]-660*c[6]*c [3]*c[5]*`b*`[7]-170*c[6]*c[3]+660*c[6]*c[3]*`b*`[7]+230*c[5]*c[6]*c[3 ]+30*c[6]-120*c[6]*`b*`[7]+120*c[5]*c[6]*`b*`[7]-40*c[5]*c[6]-180*c[3] ^2+900*`b*`[7]*c[3]^2+225*c[5]*c[3]^2-900*c[5]*c[3]^2*`b*`[7]+135*c[3] -660*`b*`[7]*c[3]-170*c[5]*c[3]+660*c[5]*c[3]*`b*`[7]+30*c[5]-120*c[5] *`b*`[7]-24+120*`b*`[7])/(c[6]-c[3])/(-c[5]+c[3])/c[3]^2/(15*c[3]^2-10 *c[3]+1), a[5,1] = -1/6*c[5]*(225*c[5]*c[2]*c[3]^5-300*c[5]*c[2]*c[3]^ 4-45*c[2]*c[3]^4-225*c[5]^2*c[2]*c[3]^4+9*c[5]*c[2]*c[3]-4*c[5]^2*c[2] +40*c[3]^3*c[2]+15*c[5]^2*c[3]^3+3*c[3]^3+190*c[5]*c[2]*c[3]^3+300*c[5 ]^2*c[2]*c[3]^3-15*c[3]^3*c[5]-11*c[3]^2*c[2]-175*c[5]^2*c[2]*c[3]^2-5 0*c[5]*c[2]*c[3]^2+40*c[5]^2*c[2]*c[3])/c[3]^2/c[2]/(1-5*c[3]+5*c[3]^2 ), a[8,2] = 1/1440*c[3]*(-20+1560*c[6]*c[5]*b[8]*c[3]^2+24*c[6]+136*c[ 3]+900*c[5]*b[8]*c[3]^3-285*c[3]^2+780*c[6]*b[8]*c[3]-225*c[3]^3*c[5]- 225*c[3]^3*c[6]+24*c[5]-780*c[5]*c[6]*b[8]*c[3]-900*c[5]*b[8]*c[3]^3*c [6]-18720*c[6]*c[5]*b[8]*c[3]^2*`b*`[7]+9360*c[5]*c[6]*b[8]*c[3]*`b*`[ 7]+900*b[8]*c[3]^3*c[6]-1560*b[8]*c[6]*c[3]^2+120*c[5]*b[8]*c[6]-1560* c[5]*b[8]*c[3]^2+780*c[5]*b[8]*c[3]-1440*b[8]*`b*`[7]-165*c[5]*c[3]-16 5*c[6]*c[3]-30*c[5]*c[6]+350*c[6]*c[3]^2+350*c[5]*c[3]^2+180*c[3]^3-12 0*b[8]*c[5]+1560*b[8]*c[3]^2-780*b[8]*c[3]-900*b[8]*c[3]^3+120*b[8]-12 0*b[8]*c[6]+210*c[5]*c[6]*c[3]+300*c[5]*c[3]^3*c[6]-455*c[5]*c[6]*c[3] ^2+1440*b[8]*c[6]*`b*`[7]+10800*b[8]*c[3]^3*`b*`[7]+1440*b[8]*c[5]*`b* `[7]+9360*b[8]*c[3]*`b*`[7]+18720*b[8]*c[6]*c[3]^2*`b*`[7]+18720*c[5]* b[8]*c[3]^2*`b*`[7]-10800*c[5]*b[8]*c[3]^3*`b*`[7]-9360*c[6]*b[8]*c[3] *`b*`[7]-9360*c[5]*b[8]*c[3]*`b*`[7]+120*`b*`[7]+1560*`b*`[7]*c[3]^2-7 80*`b*`[7]*c[3]-900*`b*`[7]*c[3]^3+10800*c[5]*b[8]*c[3]^3*c[6]*`b*`[7] -18720*b[8]*c[3]^2*`b*`[7]+900*c[5]*c[3]^3*`b*`[7]-1560*c[5]*c[3]^2*`b *`[7]+780*c[5]*c[3]*`b*`[7]-120*c[5]*`b*`[7]-10800*b[8]*c[3]^3*c[6]*`b *`[7]-1440*c[5]*b[8]*c[6]*`b*`[7]-900*c[3]^3*c[6]*c[5]*`b*`[7]+1560*c[ 6]*c[3]^2*c[5]*`b*`[7]-780*c[6]*c[3]*c[5]*`b*`[7]+900*c[3]^3*c[6]*`b*` [7]-1560*c[6]*c[3]^2*`b*`[7]+780*c[6]*c[3]*`b*`[7]+120*c[5]*c[6]*`b*`[ 7]-120*c[6]*`b*`[7])/(1-5*c[3]+5*c[3]^2)/`b*`[7]/b[8]/c[2]/(c[3]-1)/(3 *c[3]-1)/(5*c[3]-2)/(c[5]-1)/(-1+c[6]), a[7,1] = 1/360*(-279450*c[5]^2 *c[2]*c[3]^3+91980*c[5]^2*c[2]*c[3]^2-131430*c[5]*c[2]*c[3]^2+135000*c [5]^2*c[2]*c[3]^6-555975*c[6]*c[2]*c[5]*c[3]^5+278775*c[6]*c[2]*c[5]^2 *c[3]^3+576*c[2]-15660*c[5]^2*c[2]*c[3]-135000*c[2]*c[5]^2*c[6]*c[3]^6 -23220*c[5]*c[2]*c[6]*c[3]-1080*c[6]*c[5]^2*c[2]-46260*c[2]*c[3]^2*c[6 ]+181350*c[6]*c[2]*c[3]^5-218700*c[6]*c[2]*c[3]^4-576*c[6]*c[2]-182250 *c[5]*c[2]*c[3]^6-393750*c[5]^2*c[2]*c[3]^5+1080*c[5]^2*c[2]+659850*c[ 6]*c[2]*c[5]*c[3]^4+539100*c[5]*c[2]*c[3]^5-641850*c[5]*c[2]*c[3]^4+46 2150*c[5]^2*c[2]*c[3]^4+22680*c[5]*c[2]*c[3]+393720*c[5]*c[2]*c[3]^3-1 82250*c[2]*c[3]^5+60480*c[2]*c[5]^2*c[3]*`b*`[7]+60750*c[2]*c[3]^6-432 0*c[2]*c[5]^2*`b*`[7]-1628640*`b*`[7]*c[3]^3*c[5]*c[2]-219960*c[2]*c[3 ]^2*`b*`[7]+629640*c[2]*c[3]^3*`b*`[7]-342360*`b*`[7]*c[3]^2*c[5]^2*c[ 2]-243000*c[2]*c[3]^6*`b*`[7]+772200*c[2]*c[3]^5*`b*`[7]-975600*c[2]*c [3]^4*`b*`[7]-1584*c[2]*c[5]+136260*c[2]*c[6]*c[3]^3-100080*c[2]*c[5]* c[3]*`b*`[7]-2880*c[2]*`b*`[7]+562320*c[2]*c[5]*`b*`[7]*c[3]^2+7200*c[ 5]*c[2]*`b*`[7]+189000*c[2]*c[3]^6*c[6]*c[5]-403855*c[5]*c[6]*c[2]*c[3 ]^3+15660*c[2]*c[6]*c[5]^2*c[3]+392625*c[6]*c[2]*c[5]^2*c[3]^5-460950* c[6]*c[2]*c[5]^2*c[3]^4+134630*c[5]*c[6]*c[2]*c[3]^2-91830*c[5]^2*c[6] *c[2]*c[3]^2-418020*`b*`[7]*c[3]^2*c[5]*c[6]*c[2]+39600*c[3]*c[2]*`b*` [7]+46470*c[3]^2*c[2]-8136*c[3]*c[2]+220200*c[2]*c[3]^4-137130*c[3]^3* c[2]-5520*`b*`[7]*c[5]*c[6]*c[2]-1828800*c[3]^4*`b*`[7]*c[5]*c[6]*c[2] +60*c[5]*c[3]^3*c[6]-135*c[5]*c[3]^4*c[6]+225*c[5]^2*c[3]^4*c[6]-90*c[ 5]^2*c[3]^3*c[6]-60750*c[2]*c[3]^6*c[6]+1620*c[2]*c[5]*c[6]+8118*c[2]* c[6]*c[3]+158760*c[3]^2*c[2]*c[6]*`b*`[7]+1189320*c[3]^3*`b*`[7]*c[5]* c[6]*c[2]-675000*`b*`[7]*c[3]^5*c[5]^2*c[6]*c[2]+202500*`b*`[7]*c[3]^6 *c[5]^2*c[6]*c[2]+675000*c[2]*c[3]^4*c[6]*`b*`[7]+162000*c[2]*c[3]^6*c [6]*`b*`[7]-523800*c[2]*c[3]^5*c[6]*`b*`[7]+2160*c[2]*c[6]*`b*`[7]+162 0*c[3]^4*c[6]*c[5]*`b*`[7]-720*c[3]^3*c[6]*c[5]*`b*`[7]-444960*c[3]^3* c[2]*c[6]*`b*`[7]-2700*c[5]^2*c[3]^4*`b*`[7]*c[6]+1080*c[5]^2*c[3]^3*` b*`[7]*c[6]+2880*c[2]*c[5]^2*c[6]*`b*`[7]+2552400*c[5]*c[2]*c[3]^4*`b* `[7]+648000*c[5]*c[2]*c[3]^6*`b*`[7]-2041200*c[5]*c[2]*c[3]^5*`b*`[7]- 405000*c[5]^2*c[2]*c[3]^6*`b*`[7]+1269000*c[5]^2*c[2]*c[3]^5*`b*`[7]-1 576800*c[5]^2*c[2]*c[3]^4*`b*`[7]+999000*c[5]^2*c[2]*c[3]^3*`b*`[7]-29 160*c[6]*c[3]*c[2]*`b*`[7]+1431000*c[5]*c[6]*c[2]*c[3]^5*`b*`[7]-44550 0*c[5]*c[6]*c[2]*c[3]^6*`b*`[7]+75720*`b*`[7]*c[3]*c[5]*c[6]*c[2]+8946 00*c[2]*c[5]^2*c[6]*c[3]^4*`b*`[7]-595800*c[2]*c[5]^2*c[6]*c[3]^3*`b*` [7]+213300*c[2]*c[5]^2*c[6]*c[3]^2*`b*`[7]-39240*c[3]*c[2]*c[5]^2*c[6] *`b*`[7])/c[2]/c[6]/c[5]/c[3]^2/(15*c[5]*c[3]-9*c[3]+4-6*c[5])/(1-5*c[ 3]+5*c[3]^2)/`b*`[7], a[8,4] = -1/21600*(92160+22150764240*c[5]^2*c[3] ^4+7438105434000*c[5]^2*c[3]^7*b[8]*c[6]^2+9545980005000*`b*`[7]*c[3]^ 11+596160*c[5]^2*c[6]^2+2073600*b[8]*c[6]^2*`b*`[7]-24014673120*c[3]^4 *b[8]+104984997900*c[3]^6*c[6]^2-449280*c[5]*c[6]^2+66777840*c[6]^2*c[ 3]^2-2496624960*c[6]*c[5]*b[8]*c[3]^2-202752*c[6]-3310848*c[3]+1697468 906250*c[3]^12*c[6]^2+7361963184375*c[3]^11*c[5]-12786981600*c[5]*b[8] *c[3]^3+110592*c[6]^2+55514208*c[3]^2+2024130420000*c[3]^10*c[6]^2-217 0516078125*c[3]^11*c[6]^2-3977856*c[6]^2*c[3]+4808183648250*c[3]^9*c[5 ]-117619625000*c[3]^5*c[5]^2+266611638000*c[5]^2*c[3]^5*c[6]-351771483 900*c[3]^6*c[5]-10125711365625*c[3]^11*c[5]^2+9378131619375*c[3]^10*c[ 5]^2-6549307053750*c[3]^9*c[5]^2+4991505240*c[6]^2*c[3]^4-307987666875 0*c[3]^12*c[6]-39381120*c[6]*b[8]*c[3]-184826880*c[5]^2*c[6]*b[8]*c[3] -184491592320*c[5]*c[3]^4*b[8]*c[6]-64532500605000*c[3]^9*b[8]*c[5]^2* c[6]-3684884630625*c[3]^10*c[6]-5791968424200*c[3]^7*b[8]*c[5]-552960* b[8]*c[6]^2+791897755200*c[3]^6*c[5]*c[6]+2108160*c[5]*c[6]^2*b[8]+394 5373515000*c[3]^11*c[6]-6852130565625*c[3]^10*c[5]+472683138600*c[3]^6 *c[5]^2+7971511725000*c[3]^12*c[5]^2+589234437850*c[3]^7*c[6]+23047278 80*c[3]^3*c[5]-494684846400*c[3]^6*b[8]+1270481000*c[3]^3*c[6]-7303680 0*c[3]*b[8]*c[6]^2*`b*`[7]-4599154687500*c[3]^13*c[5]^3*c[6]+976877724 3750*c[3]^10*c[5]^3*c[6]-10624281975000*c[3]^11*c[5]^3*c[6]+4812353544 0*c[3]^5*c[6]+18050859247500*c[3]^10*b[8]*c[6]-16733761936875*c[3]^11* c[6]*c[5]+19681920*c[3]*b[8]*c[6]^2-7324812787500*c[3]^12*`b*`[7]+7695 38731750*c[3]^8*c[6]^2+1503031030200*c[3]^7*b[8]-1423558406250*c[3]^9* c[6]^2-20145722625000*c[3]^11*c[5]^3*c[6]^2*`b*`[7]+18881663625000*c[3 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12500*`b*`[7]*c[3]^10*c[5]^2-71417053485000*`b*`[7]*c[3]^10*c[5]*c[6]+ 50737939263000*c[5]*c[3]^9*`b*`[7]*c[6]+466560*c[5]^3*c[6]-9791470800* c[5]^3*c[3]^4+1345923360*c[5]^3*c[3]^3-128057760*c[5]^3*c[3]^2+3515171 762500*c[3]^8*c[5]^2-3045571200*c[5]^3*c[3]^3*c[6]+7542720*c[5]^3*c[3] -207360*c[5]^3-1463179765500*c[3]^7*c[5]^2+2594837262000*c[3]^9*c[6]-5 8386973837500*`b*`[7]*c[3]^12*c[5]*c[6]-96992209687500*`b*`[7]*c[3]^11 *c[5]^2*c[6]+91172873362500*c[5]^2*c[3]^10*`b*`[7]*c[6]+289225440*c[5] ^3*c[3]^2*c[6]-17003520*c[5]^3*c[6]*c[3]+633814512150*c[3]^8-116967284 7750*c[3]^9+1658454304500*c[3]^10-1772493688125*c[3]^11+1380757590000* c[3]^12-266279397700*c[3]^7-369056250000*c[3]^15*c[6]*`b*`[7]+18452812 5000*c[3]^15*`b*`[7]+963646875000*c[5]^2*c[3]^15*`b*`[7]-2019432363750 0*c[5]^3*c[3]^11*`b*`[7]+18925015500000*c[5]^3*c[3]^10*`b*`[7]+2726915 625000*c[5]^3*c[3]^14*`b*`[7]-2030834115000*c[5]^3*c[3]^8*c[6]^2+38154 75937500*c[5]^3*c[3]^9*c[6]^2+11170456848000*c[5]^3*b[8]*c[3]^7*`b*`[7 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750*c[6]*c[2]*c[3]^9+4621860000*c[5]^2*c[2]*c[3]^8*b[8]*`b*`[7]-947700 000*c[5]^2*c[2]*c[3]^9*b[8]*`b*`[7]+364500000*c[5]^3*c[2]*c[3]^9*b[8]* `b*`[7]-1773900000*c[5]^3*c[2]*c[3]^8*b[8]*`b*`[7]+3714660000*c[5]^3*c [2]*c[3]^7*b[8]*`b*`[7]+7316416800*b[8]*c[2]*c[3]^5*c[5]*`b*`[7]+82590 84000*b[8]*c[2]*c[3]^7*c[5]*`b*`[7]-9837504000*b[8]*c[2]*c[3]^6*c[5]*` b*`[7]+801900000*b[8]*c[2]*c[3]^9*c[5]*`b*`[7]-3922020000*b[8]*c[2]*c[ 3]^8*c[5]*`b*`[7]+4100868000*b[8]*c[2]*c[3]^4*c[5]^2*`b*`[7]-970153200 0*b[8]*c[2]*c[3]^7*c[5]^2*`b*`[7]-4397328000*b[8]*c[2]*c[3]^6*c[5]^3*` b*`[7]+516780000*c[2]*c[3]^8*c[6]*`b*`[7]+2175350400*c[6]*b[8]*c[2]*c[ 5]^2*c[3]^3*`b*`[7])/b[8]/c[2]/c[3]^2/(-1+c[6])/(c[5]-1)/(5*c[3]-2)/(3 *c[3]-1)/(c[3]-1)/c[5]/c[6]/(15*c[5]*c[3]-9*c[3]+4-6*c[5])/(1-5*c[3]+5 *c[3]^2)/`b*`[7], a[6,5] = (3*c[3]-1)*(2*c[6]-10*c[6]*c[3]+15*c[6]*c[3 ]^2-c[3])*(c[6]-c[3])*(c[6]-c[5])*c[6]/(15*c[5]*c[3]^2-10*c[5]*c[3]+2* c[5]-c[3])/(-c[5]+c[3])/(15*c[5]*c[3]-9*c[3]+4-6*c[5])/c[5], a[3,1] = \+ -1/2*c[3]*(c[3]-2*c[2])/c[2], a[8,3] = -1/21600*(11520+206727120*c[5]^ 2*c[3]^4+2748838500*c[5]^2*c[3]^7*b[8]*c[6]^2+74520*c[5]^2*c[6]^2+2592 00*b[8]*c[6]^2*`b*`[7]-213499680*c[3]^4*b[8]+138144100*c[3]^6*c[6]^2-5 6160*c[5]*c[6]^2+2651310*c[6]^2*c[3]^2-98253960*c[6]*c[5]*b[8]*c[3]^2- 25344*c[6]-241056*c[3]-251644440*c[5]*b[8]*c[3]^3+13824*c[6]^2+2200416 *c[3]^2-289872*c[6]^2*c[3]+43233750*c[3]^9*c[5]-449431925*c[3]^5*c[5]^ 2+1030924525*c[5]^2*c[3]^5*c[6]-464232550*c[3]^6*c[5]-1012500*c[3]^10* c[5]^2-59613750*c[3]^9*c[5]^2+45735270*c[6]^2*c[3]^4-2849040*c[6]*b[8] *c[3]-13446000*c[5]^2*c[6]*b[8]*c[3]-1656355380*c[5]*c[3]^4*b[8]*c[6]- 551205000*c[3]^9*b[8]*c[5]^2*c[6]-303750*c[3]^10*c[6]-2042735700*c[3]^ 7*b[8]*c[5]-69120*b[8]*c[6]^2+1059511200*c[3]^6*c[5]*c[6]+263520*c[5]* c[6]^2*b[8]+506250*c[3]^10*c[5]+636040450*c[3]^6*c[5]^2+223287200*c[3] ^7*c[6]+46291580*c[3]^3*c[5]-619756500*c[3]^6*b[8]+25377350*c[3]^3*c[6 ]-5299200*c[3]*b[8]*c[6]^2*`b*`[7]+179864210*c[3]^5*c[6]+1215000*c[3]^ 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^7*c[5]*c[6]-225*c[5]^2*c[6]^2*c[3]^5)/c[3]^2/b[8]/(15*c[5]*c[3]-9*c[3 ]+4-6*c[5])/(1-5*c[3]+5*c[3]^2)/`b*`[7]\}:" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}} {SECT 1 {PARA 4 "" 0 "" {TEXT -1 20 "printed coefficients" }}{PARA 0 " " 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 605 "for \+ ii from 2 to 8 do\n if not member(ii,[4,7,8]) then print(c[ii]*` is \+ a parameter`)\n else print(c[ii]=subs(e21,c[ii]));\n end if;print( ``); \n for jj to ii-1 do\n print(a[ii,jj]=subs(e21,a[ii,jj])); \n end do:\n print(``);print(`************************************ `);\nend do:print(``);\nfor ii to 8 do\n if ii=8 then print(b[ii]*` \+ is a parameter`)\n else print(b[ii]=subs(e21,b[ii])) end if;\nend do :\nprint(``);print(`************************************`);\nfor ii to 8 do\n if ii=7 then print(`b*`[ii]*` is a parameter`)\n else prin t(`b*`[ii]=subs(e21,`b*`[ii])) end if;\nend do:print(``);\n" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#*&&%\"cG6#\"\"#\"\"\"%0~is~a~parameterGF(" } }{PARA 11 "" 1 "" {XPPMATH 20 "6#%!G" }}{PARA 11 "" 1 "" {XPPMATH 20 " 6#/&%\"aG6$\"\"#\"\"\"&%\"cG6#F'" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%! G" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%E******************************* *****G" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#*&&%\"cG6#\"\"$\"\"\"%0~is~a ~parameterGF(" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%!G" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/&%\"aG6$\"\"$\"\"\",$*&#F(\"\"#F(*(&%\"cG6#F'F(,&F. F(*&F,F(&F/6#F,F(!\"\"F(F3F5F(F5" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/& %\"aG6$\"\"$\"\"#,$*&#\"\"\"F(F,*&&%\"cG6#F'F(&F/6#F(!\"\"F,F," }} {PARA 11 "" 1 "" {XPPMATH 20 "6#%!G" }}{PARA 11 "" 1 "" {XPPMATH 20 "6 #%E************************************G" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/&%\"cG6#\"\"%*&&F%6#\"\"$\"\"\",(*&\"#:F,)F)\"\"#F,F,*&\"#5F,F) F,!\"\"F1F,F4" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%!G" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/&%\"aG6$\"\"%\"\"\",$*&#F(\"\"#F(**&%\"cG6#\"\"$F(, 2*&F'F(F.F(!\"\"*&\"#IF()F.F,F(F(*&\"#XF()F.F1F(F4*(\"$]$F(F7F(&F/6#F, F(F(*(\"$+\"F(F.F(F=F(F4*&\"#5F(F=F(F(*(\"$]%F(F=F()F.F'F(F(*(\"$+'F(F :F(F=F(F4F(F=F4,(*&\"#:F(F7F(F(*&FBF(F.F(F4F,F(!\"$F(F(" }}{PARA 11 " " 1 "" {XPPMATH 20 "6#/&%\"aG6$\"\"%\"\"#,$*&#\"\"\"F(F,**&%\"cG6#\"\" $F(,(F'F,*&\"#IF,F.F,!\"\"*&\"#XF,)F.F(F,F,F,&F/6#F(F5,(*&\"#:F,F8F,F, *&\"#5F,F.F,F5F(F,!\"$F,F," }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/&%\"aG6 $\"\"%\"\"$,$*(,(*&\"#:\"\"\")&%\"cG6#F(\"\"#F.F.*&\"#5F.F0F.!\"\"F.F. F.F0F.,(*&F-F.F/F.F.*&F5F.F0F.F6F3F.!\"$F6" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%!G" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%E************** **********************G" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#*&&%\"cG6# \"\"&\"\"\"%0~is~a~parameterGF(" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%!G " }}{PARA 12 "" 1 "" {XPPMATH 20 "6#/&%\"aG6$\"\"&\"\"\",$*&#F(\"\"'F( *,&%\"cG6#F'F(,B**\"$D#F(F.F(&F/6#\"\"#F()&F/6#\"\"$F'F(F(**\"$+$F(F.F (F4F()F8\"\"%F(!\"\"*(\"#XF(F4F(F=F(F?**F3F()F.F6F(F4F(F=F(F?**\"\"*F( F.F(F4F(F8F(F(*(F>F(FCF(F4F(F?*(\"#SF()F8F:F(F4F(F(*(\"#:F(FCF(FIF(F(* &F:F(FIF(F(**\"$!>F(F.F(F4F(FIF(F(**F#F(FNF(F@F(F8F(F=*(\"#7F(FJF(F@F(F=*(F[oF(FBF(F4F(F=**F'F(F@F(F EF(F-F(F=**\"#!*F(F-F(F@F(FKF(F(**\"$?\"F(F-F(F@F(F8F(F=**\"#CF(FNF(F3 F(F@F(F=**\"$v'F(F3F(F@F(FKF(F(*,\"%+^F(F-F(F@F(F4F(F8F(F(**\"$&\\F(F4 F(F@F(FKF(F=**\"$g'F(F4F(F@F(F8F(F(**\"$q)F(F3F(F@F(F8F(F=**\"$2$F(F4F (F@F(FBF(F=*(\"#FF(F8F(F4F(F(**\"#=F(FNF(F9F(F@F(F(**\"#[F(F@F(F-F(FBF 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TF,F1*.FjwF,F5F,FAF,FOF,FFF,FdpF,F,**FgtF,FAF,FHF,FdpF,F1**F-F,F4F,FAF ,FFF,F1*,F`rF,F4F,FAF,FFF,FdpF,F,*.FhvF,F4F,FAF,FFF,FdpF,F;F,F1**F[qF, F5F,FTF,FdpF,F,**FinF,F5F,F:F,FdpF,F1**FNF,F5F,FOF,FdpF,F,**FEF,F5F,FH F,FdpF,F1**F[pF,F5F,F;F,FdpF,F,*(FeqF,F5F,FdpF,F1*,FdvF,FAF,FOF,FFF,Fd pF,F1*,FfvF,F5F,FAF,FFF,FdpF,F1**FioF,FTF,FFF,FdpF,F,**F@F,F:F,FFF,Fdp F,F1**FaoF,FYF,FFF,FdpF,F1*,F[qF,FTF,FFF,F5F,FdpF,F1*,F\\rF,FYF,FFF,F5 F,FdpF,F,*,FinF,F:F,FFF,F5F,FdpF,F,*,FNF,FOF,FFF,F5F,FdpF,F1*,FEF,FFF, FHF,F5F,FdpF,F,*,F[pF,FFF,F;F,F5F,FdpF,F1**FaqF,FOF,FFF,FdpF,F,**FcqF, FFF,FHF,FdpF,F1**FenF,FFF,F;F,FdpF,F,**FeqF,F5F,FFF,FdpF,F,*(F`tF,FFF, FdpF,F1**F\\rF,FdpF,FYF,F5F,F1**FawF,F4F,FYF,FdpF,F,**F^pF,F4F,FTF,Fdp F,F1**F^wF,F4F,F:F,FdpF,F,**F_qF,F4F,FOF,FdpF,F1**FjvF,F4F,FdpF,FHF,F, *,FawF,F4F,FYF,FdpF,FFF,F1*,F^pF,F4F,FTF,FdpF,FFF,F,*,F^wF,F4F,F:F,Fdp F,FFF,F1*,F_qF,F4F,FOF,FdpF,FFF,F,*,FjvF,F4F,FHF,FdpF,FFF,F1*,FgnF,F4F ,FdpF,FFF,F;F,F,**FgnF,F4F,F;F,FdpF,F1*(F-F,FdpF,F4F,F,**F`rF,F4F,FAF, FdpF,F1**FitF,FYF,FAF,FdpF,F1**F[uF,FTF,FAF,FdpF,F,**F]uF,F:F,FAF,FdpF ,F1*(F-F,F4F,FAF,F,*.\"(+gh&F,F5F,FAF,FTF,FFF,FdpF,F,*.\"(+_n(F,F5F,FA F,F:F,FFF,FdpF,F1*.\"(++i\"F,F5F,FAF,FYF,FFF,FdpF,F1*.FitF,F4F,FAF,FYF ,FFF,FdpF,F,*.F[uF,F4F,FAF,FTF,FFF,FdpF,F1*.F]uF,F4F,FAF,F:F,FFF,FdpF, F,*.FdvF,F4F,FAF,FOF,FFF,FdpF,F1**FgnF,F4F,FAF,F;F,F1*,FhvF,F4F,FAF,F; F,FdpF,F,*,Fe[lF,F5F,FAF,F:F,FdpF,F,*,Fg[lF,F5F,FAF,FYF,FdpF,F,*,Fc[lF ,F5F,FAF,FTF,FdpF,F1*,FitF,F4F,FAF,FYF,FdpF,F1*,F[uF,F4F,FAF,FTF,FdpF, F,*,F]uF,F4F,FAF,F:F,FdpF,F1*,FdvF,F4F,FAF,FOF,FdpF,F,*,FgtF,F4F,FAF,F HF,FdpF,F1F,,&*&F8F,F;F,F,F9F1F1,&*&F=F,F;F,F,F,F1F1,&F5F,F,F1F1,&F;F, F,F1F1FFF1FAF1FdpF1,**&F9F,FFF,F,*(\"#5F,FFF,F;F,F1*(\"#:F,FFF,FHF,F,F ;F1F1,&FFF,F;F1F1,&FFF,F5F1F1F,F," }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/ &%\"aG6$\"\")\"\"(\"\"!" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%!G" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#%E************************************ G" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%!G" }}{PARA 12 "" 1 "" {XPPMATH 20 "6#/&%\"bG6#\"\"\",$*&#F'\"#gF'**&%\"cG6#\"\"$!\"#,B**\"$]\"F'&F.6# \"\"&F')F-F0F'&F.6#\"\"'F'F'*(\"#vF'F8F'F9F'!\"\"*&\"#XF'F8F'F'*(F=F'F 8F'F5F'F>*(\"$0\"F'F5F')F-\"\"#F'F'**\"$0#F'F5F'F9F'FDF'F>*(FCF'F9F'FD F'F'*&\"#lF'FDF'F>*&\"#HF'F-F'F'*(F@F'F5F'F-F'F>**\"#!)F'F5F'F9F'F-F'F '*(F@F'F9F'F-F'F>*&F;F'F5F'F'*(\"#5F'F5F'F9F'F>\"\"%F>*&F;F'F9F'F'F'F5 F>F9F>F'F>" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/&%\"bG6#\"\"#\"\"!" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#/&%\"bG6#\"\"$,$*&#\"\"\"\"#gF+*.,:** \"#vF+&%\"cG6#\"\"&F+&F26#\"\"'F+)&F2F&\"\"#F+F+**F,F+F1F+F5F+F9F+!\" \"*(\"#5F+F1F+F5F+F+*(\"#XF+F1F+F8F+F<*(\"#NF+F1F+F9F+F+*&F7F+F1F+F<*( F@F+F5F+F8F+F<*(FBF+F5F+F9F+F+*&F7F+F5F+F<*&\"#IF+F8F+F+*&\"#BF+F9F+F< \"\"%F+F+,&F5F+F9FF+F9F+FF+FAF+F2F+F+*(F:F+F>F+FAF+F9F6F9*&F5F+FAF+F+*&F5F+F>F+F+F+,(*&F0F+ F1F+F+*&F8F+F2F+F9F+F+F9F2!\"#,&*&F5F+F2F+F+F+F9F9,&*&F:F+F2F+F+F6F9F9 ,**&F6F+FAF+F+*(F8F+FAF+F2F+F9*(F0F+FAF+F1F+F+F2F9F9,**(F0F+F>F+F1F+F+ *(F8F+F>F+F2F+F9*&F6F+F>F+F+F2F9F9F+F+" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/&%\"bG6#\"\"&,$*&#\"\"\"\"#gF+*0,(F+F+*&F'F+&%\"cG6#\"\"$F+!\" \"*&F'F+)F0\"\"#F+F+F+,**(\"#:F+&F16#\"\"'F+F0F+F+*&\"\"*F+F0F+F4\"\"% F+*&F=F+F;F+F4F+,&F;F+&F1F&F4F4FCF4,&FCF+F+F4F4,**(F:F+FCF+F6F+F+*(\"# 5F+FCF+F0F+F4*&F7F+FCF+F+F0F4F4,&FCF4F0F+F4F+F4" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/&%\"bG6#\"\"',$*&#\"\"\"\"#gF+*0,(F+F+*&\"\"&F+&%\"cG6 #\"\"$F+!\"\"*&F0F+)F1\"\"#F+F+F+,**(\"#:F+&F26#F0F+F1F+F+*&\"\"*F+F1F +F5\"\"%F+*&F'F+FF+*&\"$S#F+&%\"cG6#\"\"'F+!\"\"*&\"%!3%F+&F36#\"\"$F+F6*&\"&g$ RF+)F9\"\"#F+F+*(\"(+D,\"F+)F9\"\"*F+&F36#\"\"&F+F+*(\"(++r%F+)F9F5F+F DF+F6**\"(++m'F+FIF+FDF+F2F+F+*(\"(+]E&F+)F9\"\"(F+F2F+F+*(\"'+cGF+)F9 F;F+FDF+F+*(FQF+FRF+F2F+F+*(\"(+!QFF+)F9FFF+F2F+F+*(FMF+FNF+FDF+F+*(\" (v$fMF+)F9\"\")F+F2F+F6*&F1F+FDF+F6*(\"(+]2\"F+)F9F'F+FDF+F6*(FYF+FZF+ FDF+F6*(FhnF+FinF+F2F+F6*(\"%?^F+FDF+F9F+F+*(F]oF+F2F+F9F+F+*(\"$?$F+F DF+F2F+F+*(\"&+'\\F+F2F+F>F+F6*(FboF+FDF+F>F+F6*(FUF+FVF+FDF+F+*(FHF+F 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+F2F+FDF+FjoF+F6*,FirF+FinF+F2F+FDF+FjoF+F6*,F[sF+FRF+F2F+FDF+FjoF+F+* ,FerF+F2F+F>F+FDF+FjoF+F6*,FgrF+F2F+F9F+FDF+FjoF+F+**F[sF+FRF+F2F+FjoF +F6**FerF+F2F+F>F+FjoF+F+**FgrF+F2F+F9F+FjoF+F6**FbrF+FDF+F2F+FjoF+F6* (FbrF+F2F+FjoF+F+*,F_pF+FjoF+FNF+FDF+F2F+F+**F]sF+FjoF+FIF+FDF+F+F+F9! \"#,**&F?F+F2F+F+*(\"#5F+F2F+F9F+F6*(\"#:F+F2F+F>F+F+F9F6F6,**(F_uF+FD F+F>F+F+*(F]uF+FDF+F9F+F6*&F?F+FDF+F+F9F6F6,(*&F_uF+F>F+F+*&F]uF+F9F+F 6F+F+F6F+F6" }}{PARA 12 "" 1 "" {XPPMATH 20 "6#/&%#b*G6#\"\"&,$*&#\"\" \"\"#gF+**,B*(\"$+$F+)&%\"cG6#\"\"$F5F+&F36#\"\"'F+!\"\"*&\"#IF+F6F+F+ *(\"$5#F+F6F+F2F+F9*(\"$b%F+F6F+)F2\"\"#F+F+**\"%g:F+F6F+F@F+&F%6#\"\" (F+F9**\"$+*F+F1F+F6F+FDF+F+**\"$!yF+F6F+F2F+FDF+F+*(\"$?\"F+F6F+FDF+F 9\"#CF9*&\"$l\"F+F2F+F+*&\"$]$F+F@F+F9*&\"$D#F+F1F+F+*&FLF+FDF+F+*(FCF +FDF+F@F+F+*(FJF+FDF+F2F+F9*(FHF+FDF+F1F+F9F+,&F6F+&F3F&F9F9FYF9,0*(\" #:F+F1F+FYF+F+*$F@F+F9*(\"#5F+FYF+F@F+F9*(FfnF+)FYFAF+F@F+F9*(F5F+FYF+ F2F+F+*(FinF+F[oF+F2F+F+*&FAF+F[oF+F9F9F+F9" }}{PARA 12 "" 1 "" {XPPMATH 20 "6#/&%#b*G6#\"\"',$*&#\"\"\"\"#gF+*(,B\"#C!\"\"*&\"$l\"F+& %\"cG6#\"\"$F+F+*&\"$]$F+)F3\"\"#F+F0*(\"$+$F+)F3F6F+&F46#\"\"&F+F0*& \"#IF+F>F+F+*(\"$5#F+F>F+F3F+F0*(\"$b%F+F>F+F9F+F+*&\"$D#F+F=F+F+*&\"$ ?\"F+&F%6#\"\"(F+F+**\"%g:F+F>F+F9F+FKF+F0*(FOF+FKF+F9F+F+*(\"$!yF+FKF +F3F+F0*(\"$+*F+FKF+F=F+F0**FTF+F>F+F=F+FKF+F+**FRF+F>F+F3F+FKF+F+*(FJ F+F>F+FKF+F0F+&F4F&F0,>*(\"#5F+)FXF:F+F9F+F+*&F:F+)FXF6F+F+*(FenF+FhnF +F3F+F0**\"#:F+F>F+F=F+FXF+F+**F[oF+F>F+FfnF+F9F+F0*(F[oF+FhnF+F9F+F+* *FenF+F>F+FfnF+F3F+F+**F6F+F>F+FXF+F3F+F+*&F>F+F9F+F0**FenF+F>F+FXF+F9 F+F0*&FXF+F9F+F+*(F[oF+FfnF+F=F+F0*(F:F+F>F+FfnF+F0*(F6F+FfnF+F3F+F0F0 F+F0" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#*&&%#b*G6#\"\"(\"\"\"%0~is~a~p arameterGF(" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/&%#b*G6#\"\")\"\"!" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#%!G" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 9 "Examples:" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 24 "a[7,2]=subs(e21,a[7,2]);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/&%\"aG6$\"\"(\"\"#,$*&#\"\"\"\"#CF,*,,&*&\"#7F,&% #b*G6#F'F,F,F,!\"\"F,&%\"cG6#\"\"$F,,(F,F,*&\"\"&F,F6F,F5*&F " 0 "" {MPLTEXT 1 0 24 "a[7,6]=subs(e21,a[7,6]);" }}{PARA 12 "" 1 "" {XPPMATH 20 "6#/&%\"aG6$\"\"(\"\"',$*&#\"\"\"\"#gF,*0,&F,! \"\"&%\"cG6#F(F,F,,B\"#CF0*&\"$l\"F,&F26#\"\"$F,F,*&\"$]$F,)F8\"\"#F,F 0*(\"$+$F,)F8F:F,&F26#\"\"&F,F0*&\"#IF,FBF,F,*(\"$5#F,FBF,F8F,F0*(\"$b %F,FBF,F=F,F,*&\"$D#F,FAF,F,*&\"$?\"F,&%#b*G6#F'F,F,**\"%g:F,FBF,F=F,F OF,F0*(FSF,FOF,F=F,F,*(\"$!yF,FOF,F8F,F0*(\"$+*F,FOF,FAF,F0**FXF,FBF,F AF,FOF,F,**FVF,FBF,F8F,FOF,F,*(FNF,FBF,FOF,F0F,F1F0FOF0,**&F>F,F1F,F,* (\"#5F,F1F,F8F,F0*(\"#:F,F1F,F=F,F,F8F0F0,&F1F,F8F0F0,&F1F,FBF0F0F,F, " }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 24 "a[8,2]=subs(e21,a[8,2]);" }}{PARA 12 "" 1 "" {XPPMATH 20 "6#/&%\"aG6$\"\")\"\"#,$*&#\"\"\"\"%S9F,*8&%\"cG6#\"\"$F,, \\s\"#?!\"\"*,\"%g:F,&F06#\"\"'F,&F06#\"\"&F,&%\"bG6#F'F,)F/F(F,F,*&\" #CF,F8F,F,*&\"$O\"F,F/F,F,**\"$+*F,F;F,F>F,)F/F2F,F,*&\"$&GF,FAF,F5** \"$!yF,F8F,F>F,F/F,F,*(\"$D#F,FHF,F;F,F5*(FNF,FHF,F8F,F5*&FCF,F;F,F,*, FLF,F;F,F8F,F>F,F/F,F5*,FGF,F;F,F>F,FHF,F8F,F5*.\"&?(=F,F8F,F;F,F>F,FA F,&%#b*G6#\"\"(F,F5*.\"%g$*F,F;F,F8F,F>F,F/F,FUF,F,**FGF,F>F,FHF,F8F,F ,**F7F,F>F,F8F,FAF,F5**\"$?\"F,F;F,F>F,F8F,F,**F7F,F;F,F>F,FAF,F5**FLF ,F;F,F>F,F/F,F,*(F-F,F>F,FUF,F5*(\"$l\"F,F;F,F/F,F5*(F]oF,F8F,F/F,F5*( \"#IF,F;F,F8F,F5*(\"$]$F,F8F,FAF,F,*(FboF,F;F,FAF,F,*&\"$!=F,FHF,F,*(F hnF,F>F,F;F,F5*(F7F,F>F,FAF,F,*(FLF,F>F,F/F,F5*(FGF,F>F,FHF,F5*&FhnF,F >F,F,*(FhnF,F>F,F8F,F5**\"$5#F,F;F,F8F,F/F,F,**\"$+$F,F;F,FHF,F8F,F,** \"$b%F,F;F,F8F,FAF,F5**F-F,F>F,F8F,FUF,F,**\"&+3\"F,F>F,FHF,FUF,F,**F- F,F>F,F;F,FUF,F,**FZF,F>F,F/F,FUF,F,*,FTF,F>F,F8F,FAF,FUF,F,*,FTF,F;F, F>F,FAF,FUF,F,*,FdpF,F;F,F>F,FHF,FUF,F5*,FZF,F8F,F>F,F/F,FUF,F5*,FZF,F ;F,F>F,F/F,FUF,F5*&FhnF,FUF,F,*(F7F,FUF,FAF,F,*(FLF,FUF,F/F,F5*(FGF,FU F,FHF,F5*.FdpF,F;F,F>F,FHF,F8F,FUF,F,**FTF,F>F,FAF,FUF,F5**FGF,F;F,FHF ,FUF,F,**F7F,F;F,FAF,FUF,F5**FLF,F;F,F/F,FUF,F,*(FhnF,F;F,FUF,F5*,FdpF ,F>F,FHF,F8F,FUF,F5*,F-F,F;F,F>F,F8F,FUF,F5*,FGF,FHF,F8F,F;F,FUF,F5*,F 7F,F8F,FAF,F;F,FUF,F,*,FLF,F8F,F/F,F;F,FUF,F5**FGF,FHF,F8F,FUF,F,**F7F ,F8F,FAF,FUF,F5**FLF,F8F,F/F,FUF,F,**FhnF,F;F,F8F,FUF,F,*(FhnF,F8F,FUF ,F5F,,(F,F,*&F=F,F/F,F5*&F=F,FAF,F,F5FUF5F>F5&F06#F(F5,&F/F,F,F5F5,&*& F2F,F/F,F,F,F5F5,&*&F=F,F/F,F,F(F5F5,&F;F,F,F5F5,&F,F5F8F,F5F,F," }}} {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 74 "We can ch eck that the main scheme satisfies the order 6 order conditions. " }} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 50 "RK6_8 := OrderConditions(6,8 ,'expanded'):\nnops(%);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#\"#P" }}} {PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 144 "ind := []:\nfor ct to 37 do\n val := simplify(subs(e21,RK6_8[ct ]));\n ind := [op(ind),lhs(val)-rhs(val)];\n ##print(ct);\nend do: \nind;\nnops(%);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#7G\"\"!F$F$F$F$F$F $F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#\"#P" }}}{PARA 0 "" 0 "" {TEXT -1 0 " " }}{PARA 0 "" 0 "" {TEXT -1 77 "We can check that the embedded scheme satisfies the order 5 order conditions." }}{PARA 0 "" 0 "" {TEXT 260 4 "Note" }{TEXT -1 11 ": Because " }{XPPEDIT 18 0 "a[8,7]=0" "6#/&%\" aG6$\"\")\"\"(\"\"!" }{TEXT -1 7 " and " }{XPPEDIT 18 0 "`b*`[8]=0" "6#/&%#b*G6#\"\")\"\"!" }{TEXT -1 37 " it is essentially a 7 stage sc heme." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 66 "`RK5_7*` := subs(b=`b*`,OrderConditions(5,7,'expanded ')):\nnops(%);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#\"#<" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 147 "ind := []:\nfor ct to 17 do\n val := simplify(subs(e21,`RK5_7*`[ct]));\n \+ ind := [op(ind),lhs(val)-rhs(val)];\n ##print(ct);\nend do:\nind;\n nops(%);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#73\"\"!F$F$F$F$F$F$F$F$F$F $F$F$F$F$F$F$" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#\"#<" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 14 "Substituting " } {XPPEDIT 18 0 "c[2] = 1/10;" "6#/&%\"cG6#\"\"#*&\"\"\"F)\"#5!\"\"" } {TEXT -1 3 ", " }{XPPEDIT 18 0 "c[3] = 2/9;" "6#/&%\"cG6#\"\"$*&\"\"# \"\"\"\"\"*!\"\"" }{TEXT -1 3 ", " }{XPPEDIT 18 0 "c[5] = 3/5;" "6#/& %\"cG6#\"\"&*&\"\"$\"\"\"F'!\"\"" }{TEXT -1 3 ", " }{XPPEDIT 18 0 "c[ 6] = 4/5;" "6#/&%\"cG6#\"\"'*&\"\"%\"\"\"\"\"&!\"\"" }{TEXT -1 3 ", \+ " }{XPPEDIT 18 0 "b[8]=1/10" "6#/&%\"bG6#\"\")*&\"\"\"F)\"#5!\"\"" } {TEXT -1 7 " and " }{XPPEDIT 18 0 "`b*`[7]=3/50" "6#/&%#b*G6#\"\"(*& \"\"$\"\"\"\"#]!\"\"" }{TEXT -1 34 " gives the Prince-Dormand scheme. " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 114 "e22 := \{c[2]=1/10,c[3]=2/9,c[5]=3/5,c[6]=4/5,b[8]=1/10,`b*`[ 7]=3/50\}:\ne23 := `union`(simplify(subs(e22,e21)),e22):" }}}{PARA 0 " " 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 48 "The Butcher tablea u of the scheme is as follows." }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 120 "subs(e23,matrix([seq([c[i],seq(a[i,j],j=1..i-1),``$(9-i)],i=2.. 8),\n[`b`,seq(b[i],i=1..8)],[`b*`,seq(`b*`[i],i=1..8)]]));" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#K%'matrixG6#7+7+#\"\"\"\"#5F(%!GF+F+F+F+F+F+ 7+#\"\"#\"\"*#!\"#\"#\")#\"#?F2F+F+F+F+F+F+7+#\"\"$\"\"(#\"$:'\"%s8#!$ q#\"$V$#\"%`5F;F+F+F+F+F+7+#F7\"\"&#\"%VK\"%+b#!#a\"#b#\"&\\4&\"&+:(# \"%)*\\\"&vy\"F+F+F+F+7+#\"\"%FC#!&#\\E\"&Dr$#\"#sFI#\"%3G\"&vL##!&1U# FU#\"$Q$\"$f%F+F+F+7+F)#\"%hb\"%wB#!#N\"#6#!&\"&D=(#\"'t_<\"'+E\"*#\"$&R\"%sO#\"$&y\"%/F#F7\"#]F`qQ( pprint06\"" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 " " {MPLTEXT 1 0 1 ";" }}}}{PARA 0 "" 0 "" {TEXT 299 31 "_______________ ________________" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 24 "#======================= " }}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 21 "Prince-Dormand scheme" }} {PARA 257 "" 0 "" {TEXT -1 1 " " }}{PARA 0 "" 0 "" {TEXT -1 77 "See: P .J. Prince and J. R. Dormand, High order embedded Runge-Kutta formulae ," }}{PARA 0 "" 0 "" {TEXT -1 78 " Journal of Computational and \+ Applied Mathematics, 7 (1981), pp. 67-75." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "matrix([[1/ 10, 1/10, ``, ``, ``, ``, ``, ``, ``], [2/9, -2/81, 20/81, ``, ``, ``, ``, ``, ``], [3/7, 615/1372, -270/343, 1053/1372, ``, ``, ``, ``, ``] , [3/5, 3243/5500, -54/55, 50949/71500, 4998/17875, ``, ``, ``, ``], [ 4/5, -26492/37125, 72/55, 2808/23375, -24206/37125, 338/459, ``, ``, ` `], [1, 5561/2376, -35/11, -24117/31603, 899983/200772, -5225/1836, 39 25/4056, ``, ``], [1, 465467/266112, -2945/1232, -5610201/14158144, 10 513573/3212352, -424325/205632, 376225/454272, 0, ``], [b, 61/864, 0, \+ 98415/321776, 16807/146016, 1375/7344, 1375/5408, -37/1120, 1/10], [`b *`, 821/10800, 0, 19683/71825, 175273/912600, 395/3672, 785/2704, 3/50 , 0]]);" "6#-%'matrixG6#7+7+*&\"\"\"F)\"#5!\"\"*&F)F)F*F+%!GF-F-F-F-F- F-7+*&\"\"#F)\"\"*F+,$*&F0F)\"#\")F+F+*&\"#?F)F4F+F-F-F-F-F-F-7+*&\"\" $F)\"\"(F+*&\"$:'F)\"%s8F+,$*&\"$q#F)\"$V$F+F+*&\"%`5F)F=F+F-F-F-F-F-7 +*&F9F)\"\"&F+*&\"%VKF)\"%+bF+,$*&\"#aF)\"#bF+F+*&\"&\\4&F)\"&+:(F+*& \"%)*\\F)\"&vy\"F+F-F-F-F-7+*&\"\"%F)FFF+,$*&\"&#\\EF)\"&Dr$F+F+*&\"#s F)FMF+*&\"%3GF)\"&vL#F+,$*&\"&1U#F)FZF+F+*&\"$Q$F)\"$f%F+F-F-F-7+F)*& \"%hbF)\"%wBF+,$*&\"#NF)\"#6F+F+,$*&\"&F)\"&D=( F+*&\"'t_ " 0 "" {MPLTEXT 1 0 1 ";" }}} {SECT 1 {PARA 4 "" 0 "" {TEXT -1 35 "coefficients of the combined sche me" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 872 "# coefficients by Prince-Dormand\nee := \{c[2]=1/10, c[3]=2/9,c[4]=3/7,c[5]=3/5,c[6]=4/5,c[7]=1,c[8]=1,\n\na[2,1]=1/10,\na[ 3,1]=-2/81,\na[3,2]=20/81,\na[4,1]=615/1372,\na[4,2]=-270/343,\na[4,3] =1053/1372,\na[5,1]=3243/5500,\na[5,2]=-54/55,\na[5,3]=50949/71500,\na [5,4]=4998/17875,\na[6,1]=-26492/37125,\na[6,2]=72/55,\na[6,3]=2808/23 375,\na[6,4]=-24206/37125,\na[6,5]=338/459,\na[7,1]=5561/2376,\na[7,2] =-35/11,\na[7,3]=-24117/31603,\na[7,4]=899983/200772,\na[7,5]=-5225/18 36,\na[7,6]=3925/4056,\na[8,1]=465467/266112,\na[8,2]=-2945/1232,\na[8 ,3]=-5610201/14158144,\na[8,4]=10513573/3212352,\na[8,5]=-424325/20563 2,\na[8,6]=376225/454272,\na[8,7]=0,\n\nb[1]=61/864,\nb[2]=0,\nb[3]=98 415/321776,\nb[4]=16807/146016,\nb[5]=1375/7344,\nb[6]=1375/5408,\nb[7 ]=-37/1120,\nb[8]=1/10,\n\n`b*`[1]=821/10800,\n`b*`[2]=0,\n`b*`[3]=196 83/71825,\n`b*`[4]=175273/912600,\n`b*`[5]=395/3672,\n`b*`[6]=785/2704 ,\n`b*`[7]=3/50,\n`b*`[8]=0\}:" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 146 "RK6_8eqs := [op(R owSumConditions(8,'expanded')),op(OrderConditions(6,8,'expanded'))]:\n `RK5_8eqs*` := subs(b=`b*`,OrderConditions(5,8,'expanded')):" }}} {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 7 "Check: " } }{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 128 "simplify(subs(ee,RK6_8eqs) ):\nmap(u->lhs(u)-rhs(u),%);\nnops(%);\nsimplify(subs(ee,`RK5_8eqs*`)) :\nmap(u->lhs(u)-rhs(u),%);\nnops(%);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#7N\"\"!F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$ F$F$F$F$F$F$F$F$F$F$F$F$F$" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#\"#W" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#73\"\"!F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F $" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#\"#<" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 86 "Next we set-up stage-order cond tions to check for stage-orders from 2 to 5 inclusive. " }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 85 "for ct from 2 to 5 do\n so||ct||_ 8 := StageOrderConditions(ct,8,'expanded');\nend do:" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 34 "Stages 3 to 8 have sta ge-order 2. " }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 173 "[seq([seq(e xpand(subs(ee,so||i||_8[j])),i=2..5)],j=1..6)]:\nmap(proc(L) local i; \+ for i to nops(L) do if not evalb(L[i]) then break end if end do; i end proc,%):\nsimplify(%); " }}{PARA 11 "" 1 "" {XPPMATH 20 "6#7(\"\"#F$F $F$F$F$" }}}{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#$\"+1qGEB!#8" }} }{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 79 "The 2 no rm of the principal error of the order 5 embedded scheme is as follows ." }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 153 "`errterms5_8*` := subs (b=`b*`,PrincipalErrorTerms(5,8,'expanded')):\nevalf(evalf[14](sqrt(ad d(subs(ee,`errterms5_8*`[i])^2,i=1.. nops(`errterms5_8*`)))));" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#$\"+3,ZX=!#8" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 52 "Two of the principal erro r conditions are satisfied." }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 291 "ind := []:\nfor ct to nops(errterms6_8) do\n if simplify(subs(e e,errterms6_8[ct]))=0 then ind := [op(ind),ct] end if:\nend do:\nerr6 \+ := PrincipalErrorConditions(6):\n[seq([i,err6[i]],i=ind)]:\nlinalg[aug ment](linalg[delcols](%,2..2),matrix([[` `]$(linalg[rowdim](%))]),lin alg[delcols](%,1..1));" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#K%'matrixG6# 7$7%\"\"\"%#~~G/*&%\"bGF(-%!G6#*&%\"aGF(-F.6#*&F1F(-F.6#*&F1F(-F.6#*&F 1F(-F.6#*&F1F(%\"cGF(F(F(F(F(F(#F(\"%S]7%\"\"&F)/*(F,F(F>F(F2F(#F(\"$S )Q*pprint1406\"" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> \+ " 0 "" {MPLTEXT 1 0 1 ";" }}}}{PARA 0 "" 0 "" {TEXT -1 52 "#---------- -----------------------------------------" }}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 35 "construction of the combined scheme" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 196 "First we construct the 8 stage order 6 scheme by making use of a selection of \"simple\" order conditions and incorporating the stage-order equations to ensure that stages 3 to 8 have stage-order 2." }}{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$F+%\"jGF,/F+;,&F0F, F,F,\"\")*&&F)6#F0F,,&F,F,&%\"cG6#F0!\"\"F," }{TEXT -1 1 " " }}{PARA 0 "" 0 "" {TEXT -1 5 "for " }{XPPEDIT 18 0 "j = 4;" "6#/%\"jG\"\"%" } {TEXT -1 18 " . . . 7. ( with " }{XPPEDIT 18 0 "c[1]=0" "6#/&%\"cG6# \"\"\"\"\"!" }{TEXT -1 55 " ), together with the further simplifying \+ conditions: " }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "Sum(b [i]*c[i]*a[i,2],i=3..8)=0" "6#/-%$SumG6$*(&%\"bG6#%\"iG\"\"\"&%\"cG6#F +F,&%\"aG6$F+\"\"#F,/F+;\"\"$\"\")\"\"!" }{TEXT -1 8 ", " } {XPPEDIT 18 0 "Sum(b[i]*c[i]*Sum(a[i,j]*a[j,2],j=3..i-1),i=3..8)=0" "6 #/-%$SumG6$*(&%\"bG6#%\"iG\"\"\"&%\"cG6#F+F,-F%6$*&&%\"aG6$F+%\"jGF,&F 46$F6\"\"#F,/F6;\"\"$,&F+F,F,!\"\"F,/F+;F<\"\")\"\"!" }{TEXT -1 8 ", \+ " }{XPPEDIT 18 0 "Sum(b[i]*c[i]^2*a[i,2],i=3..8)=0" "6#/-%$SumG6$ *(&%\"bG6#%\"iG\"\"\"*$&%\"cG6#F+\"\"#F,&%\"aG6$F+F1F,/F+;\"\"$\"\")\" \"!" }{TEXT -1 2 ". " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 " " {TEXT -1 32 "The scheme is constructed with " }{XPPEDIT 18 0 "a[7,3 ];" "6#&%\"aG6$\"\"(\"\"$" }{TEXT -1 3 ", " }{XPPEDIT 18 0 "a[7,4]" " 6#&%\"aG6$\"\"(\"\"%" }{TEXT -1 3 ", " }{XPPEDIT 18 0 "a[7,5]" "6#&% \"aG6$\"\"(\"\"&" }{TEXT -1 7 " and " }{XPPEDIT 18 0 "a[7,6]" "6#&% \"aG6$\"\"(\"\"'" }{TEXT -1 17 " as parameters. " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 76 "The simple order conditio ns used are given (in abreviated form) as follows. " }}{PARA 0 "" 0 " " {TEXT -1 94 "These include the 6 order 6 quadrature conditions toget her with 4 additional order conditions." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 184 "SO6 := SimpleOrderCond itions(6):\n[seq([i,SO6[i]],i=[1,2,4,8,13,16,24,28,29,32])]:\nlinalg[a ugment](linalg[delcols](%,2..2),matrix([[` `]$(linalg[rowdim](%))]),l inalg[delcols](%,1..1));" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#K%'matrixG 6#7,7%\"\"\"%#~~G/*&%\"bGF(%\"eGF(F(7%\"\"#F)/*&F,F(%\"cGF(#F(F/7%\"\" %F)/*&F,F()F2F/F(#F(\"\"$7%\"\")F)/*&F,F()F2F:F(#F(F57%\"#8F)/*(F,F(F2 F(-%!G6#*&F8F(%\"aGF(F(#F(\"#:7%\"#;F)/*&F,F()F2F5F(#F(\"\"&7%\"#CF)/* (F,F(F2F(-FF6#*&FIF(FEF(F(#F(\"#s7%\"#GF)/*(F,F(F8F(FEF(#F(\"#=7%\"#HF )/*(F,F(F2F(-FF6#*&F?F(FIF(F(#F(FT7%\"#KF)/*&F,F()F2FRF(#F(\"\"'Q)ppri nt816\"" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 513 "SO_eqs := [op(RowSumConditions(8,'expanded')),op(Sta geOrderConditions(2,8,'expanded'))]:\nSO6_8 := SimpleOrderConditions(6 ,8,'expanded'):\nord_cdns := [seq(SO6_8[i],i=[1,2,4,8,13,16,24,28,29,3 2])]:\nsimp_cdns := [add(b[i]*a[i,1],i=1+1..8)=b[1],seq(add(b[i]*a[i,j ],i=j+1..8)=b[j]*(1-c[j]),j=[4,5,6,7])]:\nsimp_cdns2 := [add(b[i]*c[i] *a[i,2],i=3..8)=0,add(b[i]*c[i]*add(a[i,j]*a[j,2],j=3..i-1),i=3..8)=0, \n add(b[i]*c[i]^2*a[i,2],i=3..8)=0]:\ncdns := [op(ord_ cdns),op(simp_cdns),op(simp_cdns2),op(SO_eqs)]:" }}}{EXCHG {PARA 0 "> \+ " 0 "" {MPLTEXT 1 0 1 ";" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 " " 0 "" {TEXT -1 21 "We specify the nodes:" }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "c[2]=1/10" "6#/&%\"cG6#\"\"#*&\"\"\"F)\"#5!\" \"" }{TEXT -1 3 ", " }{XPPEDIT 18 0 "c[3]=2/9" "6#/&%\"cG6#\"\"$*&\" \"#\"\"\"\"\"*!\"\"" }{TEXT -1 3 ", " }{XPPEDIT 18 0 "c[5]=3/5" "6#/& %\"cG6#\"\"&*&\"\"$\"\"\"F'!\"\"" }{TEXT -1 3 ", " }{XPPEDIT 18 0 "c[ 6]=4/5" "6#/&%\"cG6#\"\"'*&\"\"%\"\"\"\"\"&!\"\"" }{TEXT -1 3 ", " } {XPPEDIT 18 0 "c[7]=1" "6#/&%\"cG6#\"\"(\"\"\"" }{TEXT -1 3 ", " } {XPPEDIT 18 0 "c[8]=1" "6#/&%\"cG6#\"\")\"\"\"" }{TEXT -1 2 ", " }} {PARA 0 "" 0 "" {TEXT -1 17 "and the weights: " }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "b[2]=0" "6#/&%\"bG6#\"\"#\"\"!" }{TEXT -1 3 ", " }{XPPEDIT 18 0 "b[8]=1/10" "6#/&%\"bG6#\"\")*&\"\"\"F)\"#5! \"\"" }{TEXT -1 2 ". " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 170 "e1 := \{c[2]=1/10,c[3]=2/9,c[5]=3/5,c[6] =4/5,c[7]=1,c[8]=1,b[2]=0,b[8]=1/10\}:\neqns := subs(e1,cdns):\nnops(e qns);\nindets(eqns) minus \{a[7,3],a[7,4],a[7,5],a[7,6]\};\nnops(%);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#\"#J" }}{PARA 11 "" 1 "" {XPPMATH 20 "6# " 0 "" {MPLTEXT 1 0 22 "infolevel[solve] := 4:" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 118 "e2 := solve(\{op(eqns)\},indets(eqns) minus \{a[7,3] ,a[7,4],a[7,5],a[7,6]\}):\ninfolevel[solve] := 0:\ne3 := `union`(e1,e2 ):" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 3 "e3 " }} {PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 928 "e3 := \{c[6] = 4/5, c[7] = 1, b[8] = 1/10, c[8] = 1, c[5] = 3/5, \+ c[3] = 2/9, c[2] = 1/10, a[2,1] = 1/10, a[7,1] = 1232/333*a[8,3]-20063 /1332+368/37*a[8,4]+560/37*a[8,5]+784/37*a[8,6], c[4] = 3/7, a[3,1] = \+ -2/81, a[3,2] = 20/81, a[4,1] = 615/1372, a[4,2] = -270/343, a[4,3] = \+ 1053/1372, a[5,1] = 3243/5500, a[5,2] = -54/55, a[5,3] = 50949/71500, \+ a[5,4] = 4998/17875, a[6,1] = -26492/37125, a[6,2] = 72/55, a[6,3] = 2 808/23375, a[6,4] = -24206/37125, a[6,5] = 338/459, a[8,7] = 0, b[1] = 61/864, b[2] = 0, b[3] = 98415/321776, b[4] = 16807/146016, b[5] = 13 75/7344, b[6] = 1375/5408, b[7] = -37/1120, a[7,2] = -2240/333*a[8,3]+ 710/37-480/37*a[8,4]-672/37*a[8,5]-896/37*a[8,6], a[8,1] = 11/9*a[8,3] +23/7*a[8,4]+5*a[8,5]+7*a[8,6]-4, a[7,6] = 112/37*a[8,6]-9625/6253, a[ 8,2] = -20/9*a[8,3]-30/7*a[8,4]-6*a[8,5]-8*a[8,6]+5, a[7,5] = 19250/56 61+112/37*a[8,5], a[7,4] = -305270/56277+112/37*a[8,4], a[7,3] = 112/3 7*a[8,3]+185535/425204\}:" }{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 0 452 "subs(e3,m atrix([seq([c[i],seq(a[i,j],j=1..i-1),``$(6-i)],i=2..6)]));\nprint(``) ;\nfor ii from 7 to 8 do\n print(c[ii]=subs(e3,c[ii]));print(``); \n for jj to ii-1 do\n coeff_val := subs(e3,a[ii,jj]);\n if \+ coeff_val=a[ii,jj] then \n print(a[ii,jj]*` is a parameter.`) \n else\n print(a[ii,jj]=coeff_val);\n end if;\n e nd do:\n print(`__________________________________`):\nend do:\nprin t(``);\nseq(b[i]=subs(e3,b[i]),i=1..8);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#K%'matrixG6#7'7(#\"\"\"\"#5F(%!GF+F+F+7(#\"\"#\"\"*#!\"#\"#\")# \"#?F2F+F+F+7(#\"\"$\"\"(#\"$:'\"%s8#!$q#\"$V$#\"%`5F;F+F+7(#F7\"\"&# \"%VK\"%+b#!#a\"#b#\"&\\4&\"&+:(#\"%)*\\\"&vy\"F+7(#\"\"%FC#!&#\\E\"&D r$#\"#sFI#\"%3G\"&vL##!&1U#FU#\"$Q$\"$f%Q)pprint176\"" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%!G" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/&%\"cG6#\" \"(\"\"\"" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%!G" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/&%\"aG6$\"\"(\"\"\",,*&#\"%K7\"$L$F(&F%6$\"\")\"\"$F(F (#\"&j+#\"%K8!\"\"*&#\"$o$\"#PF(&F%6$F0\"\"%F(F(*&#\"$g&F9F(&F%6$F0\" \"&F(F(*&#\"$%yF9F(&F%6$F0\"\"'F(F(" }}{PARA 11 "" 1 "" {XPPMATH 20 "6 #/&%\"aG6$\"\"(\"\"#,,*&#\"%SA\"$L$\"\"\"&F%6$\"\")\"\"$F.!\"\"#\"$5( \"#PF.*&#\"$![F6F.&F%6$F1\"\"%F.F3*&#\"$s'F6F.&F%6$F1\"\"&F.F3*&#\"$'* )F6F.&F%6$F1\"\"'F.F3" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/&%\"aG6$\"\" (\"\"$,&*&#\"$7\"\"#P\"\"\"&F%6$\"\")F(F.F.#\"'Nb=\"'/_UF." }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/&%\"aG6$\"\"(\"\"%,&#\"'q_I\"&xi&!\"\"*&#\" $7\"\"#P\"\"\"&F%6$\"\")F(F2F2" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/&% \"aG6$\"\"(\"\"&,&#\"&]#>\"%hc\"\"\"*&#\"$7\"\"#PF-&F%6$\"\")F(F-F-" } }{PARA 11 "" 1 "" {XPPMATH 20 "6#/&%\"aG6$\"\"(\"\"',&#\"%D'*\"%`i!\" \"*&#\"$7\"\"#P\"\"\"&F%6$\"\")F(F2F2" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%C__________________________________G" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/&%\"cG6#\"\")\"\"\"" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%!G" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#/&%\"aG6$\"\")\"\"\",,*&#\"#6\"\"*F(&F %6$F'\"\"$F(F(*&#\"#B\"\"(F(&F%6$F'\"\"%F(F(*&\"\"&F(&F%6$F'F9F(F(*&F4 F(&F%6$F'\"\"'F(F(F7!\"\"" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/&%\"aG6$ \"\")\"\"#,,*&#\"#?\"\"*\"\"\"&F%6$F'\"\"$F.!\"\"*&#\"#I\"\"(F.&F%6$F' \"\"%F.F2*&\"\"'F.&F%6$F'\"\"&F.F2*&F'F.&F%6$F'F;F.F2F>F." }}{PARA 11 "" 1 "" {XPPMATH 20 "6#*&&%\"aG6$\"\")\"\"$\"\"\"%1~is~a~parameter.GF) " }}{PARA 11 "" 1 "" {XPPMATH 20 "6#*&&%\"aG6$\"\")\"\"%\"\"\"%1~is~a~ parameter.GF)" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#*&&%\"aG6$\"\")\"\"& \"\"\"%1~is~a~parameter.GF)" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#*&&%\"a G6$\"\")\"\"'\"\"\"%1~is~a~parameter.GF)" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/&%\"aG6$\"\")\"\"(\"\"!" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%C__ ________________________________G" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#% !G" }}{PARA 11 "" 1 "" {XPPMATH 20 "6*/&%\"bG6#\"\"\"#\"#h\"$k)/&F%6# \"\"#\"\"!/&F%6#\"\"$#\"&:%)*\"'w " 0 "" {MPLTEXT 1 0 1 ";" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 66 "We now turn our attention to the embedded 8 stage, order 5 scheme." }}{PARA 0 "" 0 "" {TEXT -1 35 "We use the simplifying conditions: " }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "Sum(`b*`[i]*a[i,2],i = 3 .. 8) = 0;" "6#/-%$SumG 6$*&&%#b*G6#%\"iG\"\"\"&%\"aG6$F+\"\"#F,/F+;\"\"$\"\")\"\"!" }{TEXT -1 8 ", " }{XPPEDIT 18 0 "Sum(`b*`[i]*Sum(a[i,j]*a[j,2],j = 3 .. i-1),i = 2 .. 8) = 0;" "6#/-%$SumG6$*&&%#b*G6#%\"iG\"\"\"-F%6$*&&%\"a G6$F+%\"jGF,&F16$F3\"\"#F,/F3;\"\"$,&F+F,F,!\"\"F,/F+;F6\"\")\"\"!" } {TEXT -1 1 "." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 101 "We use a selection of \"simple\" order conditions as giv en (in abreviated form) in the following table." }}{PARA 0 "" 0 "" {TEXT -1 95 "These include the 5 order 5 quadrature conditions togethe r with one additional order condition." }}{PARA 0 "" 0 "" {TEXT -1 0 " " }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 191 "`SO5*` := subs(b=`b*`,S impleOrderConditions(5)):\n[seq([i,`SO5*`[i]],i=[1,2,4,8,15,16])]:\nli nalg[augment](linalg[delcols](%,2..2),matrix([[` `]$(linalg[rowdim](% ))]),linalg[delcols](%,1..1));" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#K%'m atrixG6#7(7%\"\"\"%#~~G/*&%#b*GF(%\"eGF(F(7%\"\"#F)/*&F,F(%\"cGF(#F(F/ 7%\"\"%F)/*&F,F()F2F/F(#F(\"\"$7%\"\")F)/*&F,F()F2F:F(#F(F57%\"#:F)/*& F,F(-%!G6#*&F?F(%\"aGF(F(#F(\"#?7%\"#;F)/*&F,F()F2F5F(#F(\"\"&Q)pprint 836\"" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 262 "`SO5_7*` := subs(b=`b*`,SimpleOrderConditions(5,7,'e xpanded')):\n`ord_cdns*` := [seq(`SO5_7*`[i],i=[1,2,4,8,15,16])]:\n`si mp_cdns*` := [add(`b*`[i]*a[i,2],i=3..7)=0,add(`b*`[i]*add(a[i,j]*a[j, 2],j=3..i-1),i=2..7)=0]:\n`cdns*` := [op(`ord_cdns*`),op(`simp_cdns*`) ]: " }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 17 "W e specify that " }{XPPEDIT 18 0 "`b*`[2] = 0;" "6#/&%#b*G6#\"\"#\"\"! " }{TEXT -1 3 ", " }{XPPEDIT 18 0 "`b*`[7] = 3/50;" "6#/&%#b*G6#\"\"( *&\"\"$\"\"\"\"#]!\"\"" }{TEXT -1 7 " and " }{XPPEDIT 18 0 "`b*`[8] \+ = 0;" "6#/&%#b*G6#\"\")\"\"!" }{TEXT -1 2 ". " }}{PARA 0 "" 0 "" {TEXT 260 4 "Note" }{TEXT -1 71 ": Although we shall construct the sch eme as an 8 stage scheme, setting " }{XPPEDIT 18 0 "`b*`[8]=0" "6#/&%# b*G6#\"\")\"\"!" }{TEXT -1 101 " means that it may also be regarded a s an 7 stage scheme, where the 8th stage makes no contribution." }} {PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 63 "e4 := \{`b*`[2]=0,`b*`[7]=3/50,`b*`[8]=0\}:\ne5 := `union`(e3,e4): " }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 84 "`eqns*` := subs(e5,`cdns*`):\nnops(`eqns*`);\nindets( `eqns*`) minus \{a[7,6]\};\nnops(%);" }}{PARA 11 "" 1 "" {XPPMATH 20 " 6#\"\")" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#<*&%#b*G6#\"\"\"&F%6#\"\"$& F%6#\"\"%&F%6#\"\"&&F%6#\"\"'&%\"aG6$\"\"(F0&F56$F7F-&F56$F7F*" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#\"\")" }}}{PARA 0 "" 0 "" {TEXT -1 0 " " }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 22 "infolevel[solve] := 4:" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 112 "e6 := solve(\{op(`eqns*` )\},indets(`eqns*`) minus \{a[7,6]\}):\ne7 := `union`(subs(e6,e5),e6): \ninfolevel[solve] := 0:" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 " " 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 39 "The order 5 scheme is given as follows." }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 94 "sub s(e7,matrix([seq([c[i],seq(a[i,j],j=1..i-1),``$(9-i)],i=2..8),\n[``,se q(`b*`[i],i=1..8)]]));" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#K%'matrixG6# 7*7+#\"\"\"\"#5F(%!GF+F+F+F+F+F+7+#\"\"#\"\"*#!\"#\"#\")#\"#?F2F+F+F+F +F+F+7+#\"\"$\"\"(#\"$:'\"%s8#!$q#\"$V$#\"%`5F;F+F+F+F+F+7+#F7\"\"&#\" %VK\"%+b#!#a\"#b#\"&\\4&\"&+:(#\"%)*\\\"&vy\"F+F+F+F+7+#\"\"%FC#!&#\\E \"&Dr$#\"#sFI#\"%3G\"&vL##!&1U#FU#\"$Q$\"$f%F+F+F+7+F),&#\"#r\"#WF)*&# \"$p\"\"$D#F)&%\"aG6$F8\"\"'F)F)#!#N\"#6,&#\"&h[#\"%C(*F)*&#\"%e9\"$D% F)FcoF)!\"\",&#\"%,t\"%[^Fbp*&#F;FboF)FcoF)F),&#\"$v)\"$7'F)*&#\"$w'\" $`\"F)FcoF)FbpFcoF+F+7+F),&#\"%$=%\"%sFF)*&#\"%`i\"&+_#F)FcoF)F)#!%XH \"%K7,&#\"&@H(\"'/T5F)*&#\"&tp#\"&+Q#F)FcoF)Fbp,&#\"(drT\"\"(%yq5F)*&# \"%8=\"$+*F)FcoF)F),&#\"$D\"\"$#>Fbp*&#Fhq\"%%G%F)FcoF)Fbp,&*&#\"#P\"$ 7\"F)FcoF)F)#\"%v8\"%/FF)\"\"!F+7+F+#\"$@)\"&+3\"F\\t#\"&$o>\"&D=(#\"' t_<\"'+E\"*#\"$&R\"%sO#\"$&yF[t#F7\"#]F\\tQ)pprint766\"" }}}{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 1055 "e7 := \{a[5,2] = -54/5 5, b[4] = 16807/146016, `b*`[2] = 0, `b*`[8] = 0, b[6] = 1375/5408, b[ 1] = 61/864, a[6,2] = 72/55, a[6,4] = -24206/37125, a[5,4] = 4998/1787 5, b[3] = 98415/321776, c[4] = 3/7, c[2] = 1/10, c[3] = 2/9, c[5] = 3/ 5, c[6] = 4/5, c[7] = 1, b[8] = 1/10, a[6,1] = -26492/37125, a[8,7] = \+ 0, a[7,2] = -35/11, a[4,3] = 1053/1372, a[7,5] = 875/612-676/153*a[7,6 ], a[7,4] = -7301/5148+1372/225*a[7,6], a[7,3] = 24861/9724-1458/425*a [7,6], a[4,2] = -270/343, a[8,2] = -2945/1232, `b*`[1] = 821/10800, `b *`[3] = 19683/71825, `b*`[4] = 175273/912600, `b*`[5] = 395/3672, `b*` [6] = 785/2704, `b*`[7] = 3/50, a[7,6] = a[7,6], a[6,3] = 2808/23375, \+ b[7] = -37/1120, a[8,6] = 37/112*a[7,6]+1375/2704, a[4,1] = 615/1372, \+ b[2] = 0, a[2,1] = 1/10, a[3,1] = -2/81, a[5,3] = 50949/71500, a[8,4] \+ = 1417157/1070784+1813/900*a[7,6], a[5,1] = 3243/5500, a[8,1] = 4183/2 772+6253/25200*a[7,6], a[8,3] = 72921/104104-26973/23800*a[7,6], a[6,5 ] = 338/459, a[8,5] = -125/192-6253/4284*a[7,6], a[7,1] = 71/44+169/22 5*a[7,6], a[3,2] = 20/81, b[5] = 1375/7344, c[8] = 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 0 146 "RK6_8eqs := [op(RowSumConditions(8,'expand ed')),op(OrderConditions(6,8,'expanded'))]:\n`RK5_8eqs*` := subs(b=`b* `,OrderConditions(5,8,'expanded')):" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 7 "Check: " }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 128 "simplify(subs(e7,RK6_8eqs)):\nmap(u->lhs(u)-rhs(u),% );\nnops(%);\nsimplify(subs(e7,`RK5_8eqs*`)):\nmap(u->lhs(u)-rhs(u),%) ;\nnops(%);" }}{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#\"#<" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }} }{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 35 "The Prin ce and Dormand scheme has " }{XPPEDIT 18 0 "a[7,6]=3925/4056" "6#/&% \"aG6$\"\"(\"\"'*&\"%DR\"\"\"\"%cS!\"\"" }{TEXT -1 2 ". " }}{PARA 0 " " 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 55 "This value arises \+ by applying the simplifying condition" }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "Sum(`b*`[i]*a[i,j],i = j+1 .. s) = `b*`[j]*(1-c[j] );" "6#/-%$SumG6$*&&%#b*G6#%\"iG\"\"\"&%\"aG6$F+%\"jGF,/F+;,&F0F,F,F,% \"sG*&&F)6#F0F,,&F,F,&%\"cG6#F0!\"\"F," }{TEXT -1 5 ", " }{XPPEDIT 18 0 "j = 2" "6#/%\"jG\"\"#" }{TEXT -1 7 " . . . " }{TEXT 265 1 "s" } {TEXT -1 1 "," }}{PARA 0 "" 0 "" {TEXT -1 6 "with " }{XPPEDIT 18 0 "s =8" "6#/%\"sG\"\")" }{TEXT -1 23 " for the eighth stage." }}{PARA 0 " " 0 "" {TEXT -1 8 "Taking " }{XPPEDIT 18 0 "s = 8;" "6#/%\"sG\"\")" } {TEXT -1 7 " and " }{XPPEDIT 18 0 "j = 6;" "6#/%\"jG\"\"'" }{TEXT -1 9 ", we have" }}{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#/,&*&&%#b*G6#\"\"( \"\"\"&%\"aG6$F)\"\"'F*F**&&F'6#\"\")F*&F,6$F2F.F*F**&&F'6#F.F*,&F*F*& %\"cG6#F.!\"\"F*" }{TEXT -1 2 ". " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }} {PARA 0 "" 0 "" {TEXT -1 93 "Applying this equation to embedded order \+ 5 scheme gives an equation which can be solved for " }{XPPEDIT 18 0 " a[7,6];" "6#&%\"aG6$\"\"(\"\"'" }{TEXT -1 1 "." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 90 "subs(j=6,'ad d'(`b*`[i]*a[i,j],i=j+1..8)=`b*`[j]*(1-c[j]));\nvalue(%);\nsubs(e7,%); \nsolve(%);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/-%$addG6$*&&%#b*G6#%\" iG\"\"\"&%\"aG6$F+\"\"'F,/F+;\"\"(\"\")*&&F)6#F0F,,&F,F,&%\"cGF7!\"\"F ," }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/,&*&&%#b*G6#\"\"(\"\"\"&%\"aG6$F )\"\"'F*F**&&F'6#\"\")F*&F,6$F2F.F*F**&&F'6#F.F*,&F*F*&%\"cGF7!\"\"F* " }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/,$*&#\"\"$\"#]\"\"\"&%\"aG6$\"\"( \"\"'F)F)#\"$d\"\"%/F" }}{PARA 11 "" 1 "" {XPPMATH 20 "6##\"%DR\"%cS" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 39 "We now complete the construction with " }{XPPEDIT 18 0 "a[7,6] = 3925/4056; " "6#/&%\"aG6$\"\"(\"\"'*&\"%DR\"\"\"\"%cS!\"\"" }{TEXT -1 2 ". " }} {PARA 0 "" 0 "" {TEXT 260 4 "Note" }{TEXT -1 88 ": See the following s ubsection for further remarks concerning this choice of value for " } {XPPEDIT 18 0 "a[7,6];" "6#&%\"aG6$\"\"(\"\"'" }{TEXT -1 1 "." }} {PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 60 "e8 := `union`(subs(a[7,6]=3925/4056,e7),\{a[7,6]=3925/4056\}):" }} }{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 2 "e8" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 967 "e8 := \{a[5 ,2] = -54/55, b[4] = 16807/146016, 3925/4056 = 3925/4056, `b*`[2] = 0, `b*`[8] = 0, b[6] = 1375/5408, b[1] = 61/864, a[6,2] = 72/55, a[6,4] \+ = -24206/37125, a[5,4] = 4998/17875, b[3] = 98415/321776, c[4] = 3/7, \+ c[2] = 1/10, c[3] = 2/9, c[5] = 3/5, c[6] = 4/5, c[7] = 1, b[8] = 1/10 , a[6,1] = -26492/37125, a[8,7] = 0, a[7,1] = 5561/2376, a[7,2] = -35/ 11, a[7,3] = -24117/31603, a[7,4] = 899983/200772, a[7,5] = -5225/1836 , a[7,6] = 3925/4056, a[4,3] = 1053/1372, a[4,2] = -270/343, a[8,2] = \+ -2945/1232, a[8,3] = -5610201/14158144, a[8,4] = 10513573/3212352, a[8 ,5] = -424325/205632, a[8,6] = 376225/454272, `b*`[1] = 821/10800, `b* `[3] = 19683/71825, `b*`[4] = 175273/912600, `b*`[5] = 395/3672, `b*`[ 6] = 785/2704, `b*`[7] = 3/50, a[8,1] = 465467/266112, a[6,3] = 2808/2 3375, b[7] = -37/1120, a[4,1] = 615/1372, b[2] = 0, a[2,1] = 1/10, a[3 ,1] = -2/81, a[5,3] = 50949/71500, a[5,1] = 3243/5500, a[6,5] = 338/45 9, a[3,2] = 20/81, b[5] = 1375/7344, c[8] = 1\}:" }}}{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 0 119 "s ubs(e8,matrix([seq([c[i],seq(a[i,j],j=1..i-1),``$(9-i)],i=2..8),\n[`b` ,seq(b[i],i=1..8)],[`b*`,seq(`b*`[i],i=1..8)]]));" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#K%'matrixG6#7+7+#\"\"\"\"#5F(%!GF+F+F+F+F+F+7+#\"\"#\" \"*#!\"#\"#\")#\"#?F2F+F+F+F+F+F+7+#\"\"$\"\"(#\"$:'\"%s8#!$q#\"$V$#\" %`5F;F+F+F+F+F+7+#F7\"\"&#\"%VK\"%+b#!#a\"#b#\"&\\4&\"&+:(#\"%)*\\\"&v y\"F+F+F+F+7+#\"\"%FC#!&#\\E\"&Dr$#\"#sFI#\"%3G\"&vL##!&1U#FU#\"$Q$\"$ f%F+F+F+7+F)#\"%hb\"%wB#!#N\"#6#!&\"&D=(#\"'t_<\"'+E\"*#\"$&R\"%sO#\"$&y\"%/F#F7\"#]F`qQ)pprint636\" " }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT 260 4 "Note" }{TEXT -1 13 ": The value " }{XPPEDIT 18 0 "b[8]=1/10" "6#/&%\"bG6#\"\")*&\"\"\"F)\"#5! \"\"" }{TEXT -1 67 " ensures that two of the principal error conditio ns are satisfied." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 56 "errterms6_8 := PrincipalErrorConditions(6,8,' expanded'):" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 64 "e_8 := e8 minus \{b[8]=1/10\}:\nsubs(e_8,errterms6 _8[1]);\nsolve(%);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/,&#\"%4e\")+'pl %!\"\"*&#\"&\\]\"\"(gpl%\"\"\"&%\"bG6#\"\")F-F-#F-\"%S]" }}{PARA 11 " " 1 "" {XPPMATH 20 "6##\"\"\"\"#5" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 64 "e_8 := e8 minus \{b[8]=1/10 \}:\nsubs(e_8,errterms6_8[5]);\nsolve(%);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/,&#\"&db\"\")+[GB\"\"\"*&#\"&j@\"\"(![GBF(&%\"bG6#\"\" )F(F(#F(\"$S)" }}{PARA 11 "" 1 "" {XPPMATH 20 "6##\"\"\"\"#5" }}} {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 39 "The order 6 scheme is given as follows." }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 91 "subs(e8,matrix([seq([c[i],seq(a[i,j],j=1..i-1),``$(9-i)],i=2..8) ,\n[``,seq(b[i],i=1..8)]]));" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#K%'mat rixG6#7*7+#\"\"\"\"#5F(%!GF+F+F+F+F+F+7+#\"\"#\"\"*#!\"#\"#\")#\"#?F2F +F+F+F+F+F+7+#\"\"$\"\"(#\"$:'\"%s8#!$q#\"$V$#\"%`5F;F+F+F+F+F+7+#F7\" \"&#\"%VK\"%+b#!#a\"#b#\"&\\4&\"&+:(#\"%)*\\\"&vy\"F+F+F+F+7+#\"\"%FC# !&#\\E\"&Dr$#\"#sFI#\"%3G\"&vL##!&1U#FU#\"$Q$\"$f%F+F+F+7+F)#\"%hb\"%w B#!#N\"#6#!& " 0 "" {MPLTEXT 1 0 94 "subs(e8,matrix([seq([c[i],seq(a[i,j],j=1..i-1),``$(8- i)],i=2..7),\n[``,seq(`b*`[i],i=1..7)]]));" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#K%'matrixG6#7)7*#\"\"\"\"#5F(%!GF+F+F+F+F+7*#\"\"#\"\"* #!\"#\"#\")#\"#?F2F+F+F+F+F+7*#\"\"$\"\"(#\"$:'\"%s8#!$q#\"$V$#\"%`5F; F+F+F+F+7*#F7\"\"&#\"%VK\"%+b#!#a\"#b#\"&\\4&\"&+:(#\"%)*\\\"&vy\"F+F+ F+7*#\"\"%FC#!&#\\E\"&Dr$#\"#sFI#\"%3G\"&vL##!&1U#FU#\"$Q$\"$f%F+F+7*F )#\"%hb\"%wB#!#N\"#6#!&\"&D=(#\"'t_<\"'+E\"*#\"$&R\"%sO#\"$&y\" %/F#F7\"#]Q)pprint666\"" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}}{PARA 0 "" 0 "" {TEXT -1 52 " #---------------------------------------------------" }}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 31 "note concerning the value for " }{XPPEDIT 18 0 "a[7, 6];" "6#&%\"aG6$\"\"(\"\"'" }{TEXT -1 1 " " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 46 "e7: coefficients of the combined scheme with " }{XPPEDIT 18 0 "a[7,6]" "6#&%\"aG6$\"\"(\"\"'" }{TEXT -1 16 " as a parameter" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> \+ " 0 "" {MPLTEXT 1 0 1055 "e7 := \{a[5,2] = -54/55, b[4] = 16807/146016 , `b*`[2] = 0, `b*`[8] = 0, b[6] = 1375/5408, b[1] = 61/864, a[6,2] = \+ 72/55, a[6,4] = -24206/37125, a[5,4] = 4998/17875, b[3] = 98415/321776 , c[4] = 3/7, c[2] = 1/10, c[3] = 2/9, c[5] = 3/5, c[6] = 4/5, c[7] = \+ 1, b[8] = 1/10, a[6,1] = -26492/37125, a[8,7] = 0, a[7,2] = -35/11, a[ 4,3] = 1053/1372, a[7,5] = 875/612-676/153*a[7,6], a[7,4] = -7301/5148 +1372/225*a[7,6], a[7,3] = 24861/9724-1458/425*a[7,6], a[4,2] = -270/3 43, a[8,2] = -2945/1232, `b*`[1] = 821/10800, `b*`[3] = 19683/71825, ` b*`[4] = 175273/912600, `b*`[5] = 395/3672, `b*`[6] = 785/2704, `b*`[7 ] = 3/50, a[7,6] = a[7,6], a[6,3] = 2808/23375, b[7] = -37/1120, a[8,6 ] = 37/112*a[7,6]+1375/2704, a[4,1] = 615/1372, b[2] = 0, a[2,1] = 1/1 0, a[3,1] = -2/81, a[5,3] = 50949/71500, a[8,4] = 1417157/1070784+1813 /900*a[7,6], a[5,1] = 3243/5500, a[8,1] = 4183/2772+6253/25200*a[7,6], a[8,3] = 72921/104104-26973/23800*a[7,6], a[6,5] = 338/459, a[8,5] = \+ -125/192-6253/4284*a[7,6], a[7,1] = 71/44+169/225*a[7,6], a[3,2] = 20/ 81, b[5] = 1375/7344, c[8] = 1\}:" }{TEXT -1 0 "" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 5 "Let " } {XPPEDIT 18 0 "T[6, 8];" "6#&%\"TG6$\"\"'\"\")" }{TEXT -1 127 " denot e the vector whose components are the principal error terms of the 8 s tage, order 6 scheme (the error terms of order 7)." }}{PARA 0 "" 0 "" {TEXT -1 5 "Let " }{XPPEDIT 18 0 "`T*`[5,8];" "6#&%#T*G6$\"\"&\"\")" }{TEXT -1 145 " denote the vector whose components are the principal \+ error terms of the embedded 8 stage, order 5 scheme (the error terms o f order 6) and let " }{XPPEDIT 18 0 "`T*`[6,8];" "6#&%#T*G6$\"\"'\"\" )" }{TEXT -1 99 " denote the vector whose components are the error te rms of order 7 of the embedded order 5 scheme." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 39 "Denote the 2-norm of thes e vectors by " }{XPPEDIT 18 0 "abs(abs(T[6,8]));" "6#-%$absG6#-F$6#&% \"TG6$\"\"'\"\")" }{TEXT -1 3 ", " }{XPPEDIT 18 0 "abs(abs(`T*`[5,8]) );" "6#-%$absG6#-F$6#&%#T*G6$\"\"&\"\")" }{TEXT -1 7 " and " } {XPPEDIT 18 0 "abs(abs(`T*`[6,8]));" "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*`[6,8]))/abs(abs(`T*`[5,8]));" "6#/&%\"BG6#\"\"(*&-%$absG6 #-F*6#&%#T*G6$\"\"'\"\")\"\"\"-F*6#-F*6#&F/6$\"\"&F2!\"\"" }{TEXT -1 6 ", " }{XPPEDIT 18 0 "C[7] = abs(abs(`T*`[6,8]-T[6,8]))/abs(abs(` T*`[5,8]));" "6#/&%\"CG6#\"\"(*&-%$absG6#-F*6#,&&%#T*G6$\"\"'\"\")\"\" \"&%\"TG6$F2F3!\"\"F4-F*6#-F*6#&F06$\"\"&F3F8" }{TEXT -1 1 "." }} {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 77 "Dormand a nd Prince have suggested that as well as attempting to ensure that " }{XPPEDIT 18 0 "A[7]" "6#&%\"AG6#\"\"(" }{TEXT -1 73 " is a minimum, \+ if the embedded scheme is to be used for error control, " }{XPPEDIT 18 0 "B[7]" "6#&%\"BG6#\"\"(" }{TEXT -1 7 " and " }{XPPEDIT 18 0 "C[ 7]" "6#&%\"CG6#\"\"(" }{TEXT -1 27 " should be chosen so that " }} {PARA 0 "" 0 "" {TEXT -1 68 "they are \"similar in magnitude\" and als o not differ too much from 1." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 471 "errterms6_8 := PrincipalErr orTerms(6,8,'expanded'):\n`errterms5_8*` := subs(b=`b*`,PrincipalError Terms(5,8,'expanded')):\n`errterms6_8*` := subs(b=`b*`,PrincipalErrorT erms(6,8,'expanded')):\nsnmB := sqrt(add(subs(e7,`errterms6_8*`[i])^2, i=1..nops(`errterms6_8*`))):\nsdnB := sqrt(add(subs(e7,`errterms5_8*`[ i])^2,i=1..nops(`errterms5_8*`))):\nsnmC := sqrt(add((subs(e7,`errterm s6_8*`[i])-subs(e7,errterms6_8[i]))^2,i=1..nops(errterms6_8))):\nB_7 : = snmB/sdnB:\nC_7 := snmC/sdnB:" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 93 "'B[7]' = evalf[8](eval(B_7,a [7,6]=3925/4056));\n'C[7]' = evalf[8](eval(C_7,a[7,6]=3925/4056));" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#/&%\"BG6#\"\"($\")8f'>#!\"(" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/&%\"CG6#\"\"($\"):r0:!\"(" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 6 "When " }{XPPEDIT 18 0 "a[7,6] = 3925/4056" "6#/&%\"aG6$\"\"(\"\"'*&\"%DR\"\"\"\"%cS!\"\"" } {TEXT -1 44 ", as in the Prince-Dormand scheme, we have " }{XPPEDIT 18 0 "B[7]" "6#&%\"BG6#\"\"(" }{TEXT -1 1 " " }{TEXT 266 1 "~" }{TEXT -1 14 " 2.196 and " }{XPPEDIT 18 0 "C[7]" "6#&%\"CG6#\"\"(" }{TEXT -1 1 " " }{TEXT 267 1 "~" }{TEXT -1 8 " 1.506. " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 43 "The following picture sho ws the graphs of " }{XPPEDIT 18 0 "B[7]" "6#&%\"BG6#\"\"(" }{TEXT -1 5 " in " }{TEXT 259 5 "brown" }{TEXT -1 6 " and " }{XPPEDIT 18 0 "C[ 7]" "6#&%\"CG6#\"\"(" }{TEXT -1 5 " in " }{TEXT 261 3 "red" }{TEXT -1 3 ". \n" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 518 "aa := 3925/40 56:\nbb := evalf(eval(B_7,a[7,6]=aa)):\ncc := evalf(eval(C_7,a[7,6]=aa )):\np1 := plot([B_7,C_7,1],a[7,6]=-.12..1.65,0..2.8,color=[brown,red, blue]):\np2 := plot([[aa,0],[aa,2.9]],color=COLOR(RGB,0,.7,0)):\np3 := plot([[[aa,bb],[aa,cc]]$3],style=point,symbol=[circle,diamond,cross], color=[black,coral$2]):\np4 := plot([[[aa,0]]$3],style=point,symbol=[c ircle,diamond,cross],color=black):\np5 := plot([[[aa,bb],[0,bb]],[[aa, cc],[0,cc]]],linestyle=3,color=[brown,red]):\nplots[display]([p||(1..5 )],font=[HELVETICA,9]);" }}{PARA 13 "" 1 "" {GLPLOT2D 528 392 392 {PLOTDATA 2 "61-%'CURVESG6$7do7$$!3%**************>\"!#=$\"3Vh/\"[+]sg \"!#<7$$!3i****\\7P!>9)!#>$\"3[A\\!z)4d2;F-7$$!3'***\\(=w%*\\y%F1$\"3( *>?Vn:#zg\"F-7$$!3b++ve&*z45F1$\"3(*3QL\">7%3;F-7$$\"3!3+]()3?/z#F1$\" 3PKuQp=.4;F-7$$\"3[(*\\ibzdslF1$\"3F()G;FO!)4;F-7$$\"3p*\\7[C7z+\"F*$ \"3ok&eT?!p5;F-7$$\"3u*\\7jh%*4P\"F*$\"3tB)pvsB=h\"F-7$$\"3)**\\7V#\\ \\Y,i\"F-7$$\"3G++D v?oEKF*$\"30JVQJ'pQi\"F-7$$\"3s***\\Pg9+h$F*$\"3=xy4y#\\(G;F-7$$\"3[++ vClUzRF*$\"3dgdAd<)[j\"F-7$$\"3[*\\7$Rz)[J%F*$\"3Xa'>gz,@k\"F-7$$\"3G* **\\x+%y8ZF*$\"3K^xTsF``;F-7$$\"3/+++,8q^]F*$\"3!Hi6]i4mm\"F-7$$\"3%)* \\7eWzZW&F*$\"3'*oV'\\)y`(o\"F-7$$\"3e****\\3-s#z&F*$\"3\\ORmGxe8%>\"=F -7$$\"3g\\7GNEiFnF*$\"3un7pL$4@&=F-7$$\"3]+]7yiE0`,>F-7$$ \"3Y]PfU$=94(F*$\"3[t5;J-!o&>F-7$$\"3Y+D12/dlsF*$\"3\\h+1I8;B?F-7$$\"3 gD1RrO\\ftF*$\"3Y&zggi6S1#F-7$$\"3u](=d$pT`uF*$\"3Rzf8%QU&3@F-7$$\"3yu o/+-MZvF*$\"3S.ac1Bqc@F-7$$\"3#***\\PkMETwF*$\"3Ufdf87C3AF-7$$\"3Gv$f$ RL#)QxF*$\"39]W\"QP!zkAF-7$$\"3^\\PM9KQOyF*$\"3SQf%R8OMK#F-7$$\"3vB\"G $*3VR$zF*$\"3OUf.Nv2`#F-7$$\"3w)\\(='f27P)F*$\"31KY0'=W\\f#F-7$$\"3 'fo/[))oqT)F*$\"3pEF-7$$\"3\\ awM1@sJ&)F*$\"386_d8o?@EF-7$$\"3k[il]Fla&)F*$\"3H'H/'Hi$>i#F-7$$\"3%pV t#RS^+')F*$\"3JZ*pjKK6i#F-7$$\"39C1*yKvjk)F*$\"3n^f#3r$Q%z[*F*$\"3)fpi;#ywxAF-7$ $\"3i[(=())4Ln'*F*$\"3TG`!R'4^+AF-7$$\"3Y)\\7G1?n%)*F*$\"3&pjHFcdB8#F- 7$$\"3-v$fW5!f/5F-$\"3I&3&e'[St1#F-7$$\"34+vj-#3X-\"F-$\"3T.GFR')[7?F- 7$$\"3'*\\Pu*\\0C/\"F-$\"3'=OaOx'oq>F-7$$\"3%)***\\oz-.1\"F-$\"3*HTzP, %)[$>F-7$$\"3')**\\Pm6_)4\"F-$\"3o\\Mp$oAV(=F-7$$\"3\")\\7`*o_J8\"F-$ \"3#yEGF#*z\"F-7$$\"3m\\7e?(Qm? 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" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 27 "z0 := newton(R(z)=- 1,z=-4);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%#z0G$!+J(HT&R!\"*" }}} {PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 306 "z0 := newton(R(z)=-1,z=-4):\np1 := plot([R(z),-1],z=-4.49..0.49,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 ,color=COLOR(RGB,0,.5,0)):\nplots[display]([p1,p2,p3],view=[-4.49..0.4 9,-1.47..1.47],font=[HELVETICA,9]);" }}{PARA 13 "" 1 "" {GLPLOT2D 360 253 253 {PLOTDATA 2 "6+-%'CURVESG6$7U7$$!3A++++++!\\%!#<$!3iK'p5:D&)o# F*7$$!3;+]7'3DdV%F*$!3'yBEJ]7![CF*7$$!35++Ds,X\"Q%F*$!33#HMw%y!=%F*$!3[%)Q[\"\\_ya\"F*7$$!35+]#oviQ2%F*$!3?Vq%yrMWE\"F*7$$! 3A+ve,&\\u'RF*$!3=u.%e0+,T)!#=7$$!3;+ v.8ajmPF*$!33s5ua7+&y'FU7$$!3C+v$Q<')4m$F*$!3'oly[$*y)z`FU7$$!3G+v3`dn bNF*$!3rXMsSCn;UFU7$$!37++0T5NZMF*$!31(G3#p8$)GKFU7$$!3?+]sv&Q>N$F*$!3 lAf+n4Z1DFU7$$!3E++:Qn_WKF*$!3YB],/wpJ=FU7$$!3#****\\s&QnOJF*$!3?]#*f2 #*ew7FU7$$!3&****\\=iPF.$F*$!3udd/!fT_O)!#>7$$!3Q+v$Gc`$QHF*$!3[J')\\M FVE]Fhp7$$!3+++l(y@h#GF*$!3OVJc7l=4dQp\"F*$\"3 %>lkG9%)R#=FU7$$!3)*****p]Q@(e\"F*$\"3pL`rFNZO?FU7$$!3I+]FRT)G[\"F*$\" 3[txl0H&[E#FU7$$!3')*\\P*y(R>Q\"F*$\"3xZAkgb03DFU7$$!3++]Kx#e)p7F*$\"3 %fe2:*pC2GFU7$$!3=++5j![\"p6F*$\"3>kW?L%\\b5$FU7$$!3s***\\KT=;1\"F*$\" 3UL#R=.)feMFU7$$!3'***\\P%o0=k*FU$\"39v_l:!)y7QFU7$$!3](***\\tEbw&)FU$ \"3D$***QX#)[TUFU7$$!39-]P2EBuvFU$\"3sh<4*e0()o%FU7$$!3>)*\\(GL>l_'FU$ \"3#p9(z$>Cm?&FU7$$!3z++]-\")=-bFU$\"3C)eGR&QBodFU7$$!3$*)*\\PM#3)HWFU $\"3O%[Bd&[>@kFU7$$!3;(****f'*ypR$FU$\"3^sjV#z`)>rFU7$$!3L)**\\U'=wSBF U$\"3D'>n%p_,8zFU7$$!3?$*\\Pt2H$H\"FU$\"3+V,>Fw%oy)FU7$$!33R****pit2LF hp$\"3+r)*)36PYn*FU7$$\"3Z^+]FkzBxFhp$\"3QUWS=\"*H!3\"F*7$$\"35*****z2 `!f\"F*7$$\"3l+]i_F06GFU$\"35LwZUIfC8F*7$$\"3\\.]Pt1& z\"QFU$\"3]WLRM=\"\\Y\"F*7$$\"3!***************[FU$\"3Ko[\"G8;Bj\"F*-% 'COLOURG6&%$RGBG$\"*++++\"!\")$\"\"!F\\\\lF[\\l-F$6$7S7$F($!\"\"F\\\\l 7$F3Fa\\l7$F=Fa\\l7$FBFa\\l7$FGFa\\l7$FLFa\\l7$FQFa\\l7$FWFa\\l7$FfnFa \\l7$F[oFa\\l7$F`oFa\\l7$FeoFa\\l7$FjoFa\\l7$F_pFa\\l7$FdpFa\\l7$FjpFa \\l7$F_qFa\\l7$FdqFa\\l7$FjqFa\\l7$F_rFa\\l7$FdrFa\\l7$FirFa\\l7$F^sFa \\l7$FcsFa\\l7$FhsFa\\l7$F]tFa\\l7$FbtFa\\l7$FgtFa\\l7$F\\uFa\\l7$FauF a\\l7$FfuFa\\l7$F[vFa\\l7$F`vFa\\l7$FevFa\\l7$FjvFa\\l7$F_wFa\\l7$FdwF a\\l7$FiwFa\\l7$F^xFa\\l7$FcxFa\\l7$FhxFa\\l7$F]yFa\\l7$FbyFa\\l7$FgyF a\\l7$F\\zFa\\l7$FazFa\\l7$FfzFa\\l7$F[[lFa\\l7$F`[lFa\\l-Fe[l6&Fg[lF[ \\lF[\\lFh[l-F$6&7#7$$!35+++J(HT&RF*Fa\\l-%'SYMBOLG6#%'CIRCLEG-Fe[l6&F g[lF\\\\lF\\\\lF\\\\l-%&STYLEG6#%&POINTG-F$6&Fg_l-F\\`l6#%&CROSSGF_`lF a`l-F$6&Fg_l-F\\`l6#%(DIAMONDGF_`lFa`l-F$6%7$7$Fi_lF[\\lFh_l-%&COLORG6 &Fg[lF[\\l$\"\"&Fb\\lF[\\l-%*LINESTYLEG6#\"\"$-%+AXESLABELSG6%Q\"z6\"Q !F`bl-%%FONTG6#%(DEFAULTG-Fcbl6$%*HELVETICAG\"\"*-%%VIEWG6$;$!$\\%!\"# $\"#\\F`cl;$!$Z\"F`cl$\"$Z\"F`cl" 1 2 0 1 10 0 2 9 1 4 2 1.000000 45.000000 45.000000 0 0 "Curve 1" "Curve 2" "Curve 3" "Curve 4" "Curve 5" "Curve 6" }}}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 50 "The following picture shows the stability region. " }} {PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1284 "R := z -> 1+z+1/2*z^2+1/6*z^3+1/24*z^4+1/120*z^5+1/720*z^6+1/504 0*z^7:\npts := []:\nz0 := 0:\nfor ct from 0 to 200 do\n zz := newton (R(z)=exp(ct*Pi/20*I),z=z0):\n z0 := zz:\n pts := [op(pts),[Re(zz) ,Im(zz)]]:\nend do:\np1 := plot(pts,color=COLOR(RGB,.28,.15,.5)):\np2 \+ := plots[polygonplot]([seq([pts[i-1],pts[i],[-2,0]],i=2..nops(pts))], \n style=patchnogrid,color=COLOR(RGB,.55,.3,1)):\npts := []: \+ z0 := 1.5+4.3*I:\nfor ct from 0 to 60 do\n zz := newton(R(z)=exp(ct* Pi/30*I),z=z0):\n z0 := zz:\n pts := [op(pts),[Re(zz),Im(zz)]]:\ne nd do:\np3 := plot(pts,color=COLOR(RGB,.28,.15,.5)):\np4 := plots[poly gonplot]([seq([pts[i-1],pts[i],[1.39,4.2]],i=2..nops(pts))],\n \+ style=patchnogrid,color=COLOR(RGB,.55,.3,1)):\npts := []: z0 := 1.5- 4.3*I:\nfor ct from 0 to 60 do\n zz := newton(R(z)=exp(ct*Pi/30*I),z =z0):\n z0 := zz:\n pts := [op(pts),[Re(zz),Im(zz)]]:\nend do:\np5 := plot(pts,color=COLOR(RGB,.28,.15,.5)):\np6 := plots[polygonplot]([ seq([pts[i-1],pts[i],[1.39,-4.2]],i=2..nops(pts))],\n style=p atchnogrid,color=COLOR(RGB,.55,.3,1)):\np7 := plot([[[-4.49,0],[1.79,0 ]],[[0,-4.69],[0,4.69]]],color=black,linestyle=3):\nplots[display]([p| |(1..7)],view=[-4.49..1.79,-4.69..4.69],font=[HELVETICA,9],\n \+ labels=[`Re(z)`,`Im(z)`],axes=boxed,scaling=constrained);" }} {PARA 13 "" 1 "" {GLPLOT2D 381 565 565 {PLOTDATA 2 "6/-%'CURVESG6$7ew7 $$\"\"!F)F(7$F($\"3++++Fjzq:!#=7$$\"3!)*****\\QO5E#!#E$\"3'******pk#fT JF-7$$\"3!******4`NP]&!#D$\"3))*****\\t)Q7ZF-7$$\"3Y+++5i'))4&!#C$\"3] +++@@=$G'F-7$$\"3*)*****498?u#!#B$\"3#*******='eR&yF-7$$\"3(******HR0# G5!#A$\"3B+++.PmC%*F-7$$\"36+++IKeUHFI$\"3!******fv8&*4\"!#<7$$\"3M*** **>c%*4p'FI$\"3/+++&e*\\c7FQ7$$\"30+++YW#Q?\"!#@$\"3-+++)oWLT\"FQ7$$\" 3#******pUgyb\"FZ$\"3)******\\#e!*p:FQ7$$\"3_+++tN#Gd'FI$\"3++++dC&fs \"FQ7$$!39+++*>;_$RFZ$\"3++++KH:\")=FQ7$$!3-+++0t1!o\"!#?$\"3*)*****zJ 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59 "For this scheme the stability interval is (approxima tely) " }{XPPEDIT 18 0 "[-3.9541, 0];" "6#7$,$-%&FloatG6$\"&T&R!\"%! \"\"\"\"!" }{TEXT -1 2 ". " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 260 "We can distort the boundary curve horizontally b y taking the 11th root of the real part of points along the curve. In \+ this way we see that the largest interval on the nonnegative imaginary axis that contains the origin and lies inside the stability region is " }{XPPEDIT 18 0 "[0, 1.76];" "6#7$\"\"!-%&FloatG6$\"$w\"!\"#" } {TEXT -1 18 " approximately. " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 320 "R := z -> 1+z+1/2*z^2+1/6*z ^3+1/24*z^4+1/120*z^5+1/720*z^6+1/5040*z^7:\nDigits := 25:\npts := []: z0 := 0:\nfor ct from 0 to 60 do\n zz := newton(R(z)=exp(ct*Pi/100* I),z=z0):\n z0 := zz:\n pts := [op(pts),[surd(Re(zz),11),Im(zz)]]: \nend do:\nplot(pts,color=COLOR(RGB,.55,0,1),thickness=2,font=[HELVETI CA,9]);\nDigits := 10:" }}{PARA 13 "" 1 "" {GLPLOT2D 261 276 276 {PLOTDATA 2 "6(-%'CURVESG6#7in7$$\"\"!F)F(7$$\":(GMFj2VC9eNyI!#E$\":yZ vsJz*e`EfTJF-7$$\":=SG%)Gykc@*o&4&F-$\":w13[GbzrI&=$G'F-7$$\":!)3))4>t ?9qG@%oF-$\":!y*ezx3o2'zxC%*F-7$$\":A+QI!=3it^IK%)F-$\":d8Yy#)>M91PmD \"!#D7$$\":P)>V*eS67jI[\"**F-$\":CoYlW5#yEjzq:F?7$$\":obXc>7S_R:;8\"F? $\":'o/^.pz3#fb\\)=F?7$$\":-C)pR\"f*)Q1o_E\"F?$\":sb85*4%\\s&[6*>#F?7$ $\":BxXKDBKxyXNR\"F?$\":qcv5pP(*>7uK^#F?7$$\":\")ya<9'Q3^cB<:F?$\":/E/ 5]Z7dQLu#GF?7$$\":`#fswJ!y/!>#pj\"F?$\":Sx3Px>1rk#fTJF?7$$\":)3\"orXVL N#=0`vbMF?7$$\":7=K)y)pe\"3h(f'=F?$\":tc!G!=d_5:6*p PF?7$$\":V-d8&)zz2'[(f(>F?$\":'HrDDI8b\"QqS3%F?7$$\":3Z1;+9g!y]F$3#F?$ \":1=2]nS+GeH#)R%F?7$$\":.*GNchB@wF1)=#F?$\":#\\y2`DfKN()Q7ZF?7$$\":=S .17J!GX;\\!H#F?$\":Qo#Q+x[`4yaE]F?7$$\":92+c%))*>_O*o!R#F?$\":\"[Y*))e 3G'eR&yF?7 $$\":#e*[ypTY`WbZ?$F?$\":(=*p@)z#*4u$3\"o\")F?7$$\":*o)*)z_7DE&)=iG$F? $\":$4%3^;1X4#[D#[)F?7$$\":YFq!Q%=snbNfO$F?$\":Lh([eqW06qR'z)F?7$$\":H FMZ`e6An&*QW$F?$\":hpFExG*fyP`5\"*F?7$$\":(or$oWGI)y93?NF?$\":qr]p'=,E .PmC%*F?7$$\":p,g_r>zQ2nWf$F?$\":)Hu'R!=omo]yQ(*F?7$$\":!\\/.rl\"Ffu7$$\":*or%4d75p1\"H=TF?$\":crMI=1y(y.6D7Ffu7$$\":u,0qbVG)R y[tTF?$\":'[y&HZym_e*\\c7Ffu7$$\":.BK_a()=f%4\"fA%F?$\":!38Ffu7$$\":\"4M([@ZU(3%e9K%F ?$\":\"y;5zPltw4j]8Ffu7$$\":24d-,$4jtL&RO%F?$\":$ej\\b=!Q$yE*>Q\"Ffu7$ $\":w[$3ox#))\\L!Q-WF?$\":6qt.Htd\")oWLT\"Ffu7$$\":B'*Qv:g,*\\+>OWF?$ \":U[\\Okk`Di&oW9Ffu7$$\":JEMO$yF-*orYY%F?$\":PG)=^MV)e&R,w9Ffu7$$\":2 o-:pO$pp?)o[%F?$\":[&fP>#e.P,Gt]\"Ffu7$$\":(3;:2S))p#GI:]%F?$\":RJ%H6' )4omfiQ:Ffu7$$\":Y/@x)ezm#oyn]%F?$\":Czo'QWBTDe!*p:Ffu7$$\":^K#=r^T]'4 u)*\\%F?$\":X3cJ\">XGOa;,;Ffu7$$\":.mk\"3u_kVmRwWF?$\":Y\\#HbrabyCSK;F fu7$$\":Di#*Hae;=/:&GWF?$\":[!p,&*4/!GY9Om\"Ffu7$$\":DgMc=qsT-\\+M%F?$ \":Rs&yR5'e3t)z%p\"Ffu7$$\":lMP\"Qh'yWjKn;%F?$\":))4keas\"RdC&fs\"Ffu7 $$\":ye43b)4t(3B6j$F?$\":-_9>CrKJlsqv\"Ffu7$$!:![sV6%4$>0k\"34%F?$\":c RZc'\\X')ph:)y\"Ffu7$$!:y]OyGQ:9QF)oWF?$\":rq*>>=Ffu7$$!:[XNpEO \"33:R6ZF?$\":lFWI%*[M0#)*>]=Ffu7$$!:l!\\o*4H$e=[)G!\\F?$\":I/Ka`&4tJH :\")=Ffu-%*THICKNESSG6#\"\"#-%%FONTG6$%*HELVETICAG\"\"*-%&COLORG6&%$RG BG$\"#b!\"#F($\"\"\"F)-%+AXESLABELSG6$Q!6\"Fh_l-%%VIEWG6$%(DEFAULTGF]` l" 1 2 0 1 10 2 2 6 1 4 2 1.000000 46.000000 44.000000 0 0 "Curve 1" } }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 123 "The r elevant intersection point of the boundary curve with the imaginary ax is can be determined more accurately as follows." }}{PARA 0 "" 0 "" {TEXT -1 86 "First we look for points on the boundary curve either sid e of the intersection point. " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 112 "Digits := 15:\nz0 := 1.76*I :\nfor ct from 55 to 58 do\n newton(R(z)=exp(ct*Pi/100*I),z=z0);\nen d do;\nDigits := 10:" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#^$$\"02+GdBGd' !#>$\"0t\"RdC&fs\"!#9" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#^$$\"0o0E_SpW \"!#>$\"0rKJlsqv\"!#9" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#^$$!05]am.#p` !#>$\"0bk)ph:)y\"!#9" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#^$$!0-k9J![>9! #=$\"0j>rq*>>=!#9" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 107 "Then we apply the bisection method to calculate the para meter value associated with the intersection point." }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 178 "real_part := proc(u)\n Re(newton(R(z)=ex p(u*Pi*I),z=1.76*I))\nend proc:\nDigits := 15:\nu0 := bisect('real_par t'(u),u=0.55..0.58);\nnewton(R(z)=exp(u0*Pi*I),z=1.76*I);\nDigits := 1 0:" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%#u0G$\"0')fUmIOi&!#:" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#^#$\"0T`XK@Ww\"!#9" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 4 "The " }{TEXT 260 112 "larg est interval on the nonnegative imaginary axis that contains the origi n and lies inside the stability region" }{TEXT -1 5 " is " }{XPPEDIT 18 0 "[0, 1.7644];" "6#7$\"\"!-%&FloatG6$\"&Ww\"!\"%" }{TEXT -1 18 " \+ (approximately)." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 37 "#------------------------------------" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 80 "The stability function R * for the 8 stage, order 5 scheme is given as follows." }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 97 "subs(ee,subs(b=`b*`,StabilityFuncti on(5,8,'expanded'))):\n`R*` := unapply(%,z):\n'`R*`(z)'=`R*`(z);" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#/-%#R*G6#%\"zG,2\"\"\"F)F'F)*&#F)\"\"# F)*$)F'F,F)F)F)*&#F)\"\"'F)*$)F'\"\"$F)F)F)*&#F)\"#CF)*$)F'\"\"%F)F)F) *&#F)\"$?\"F)*$)F'\"\"&F)F)F)*&#\"$V*\"'+IpF)*$)F'F1F)F)F)*&#\"$d\"FDF )*$)F'\"\"(F)F)F)" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 124 "We can find the point where the boundary of the stabilit y region intersects the negative real axis by solving the equation: " }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "`R*`(z) = -1;" "6#/ -%#R*G6#%\"zG,$\"\"\"!\"\"" }{TEXT -1 2 ". " }}{EXCHG {PARA 0 "> " 0 " " {MPLTEXT 1 0 31 "z_0 := newton(`R*`(z)=-1,z=-4);" }}{PARA 11 "" 1 " " {XPPMATH 20 "6#>%$z_0G$!+XoDPP!\"*" }}}{PARA 0 "" 0 "" {TEXT -1 0 " " }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 322 "z_0 := newton(`R*`(z)=- 1,z=-4):\np_1 := plot([`R*`(z),-1],z=-4.29..0.49,color=[red,blue]):\np _2 := plot([[[z_0,-1]]$3],style=point,symbol=[circle,cross,diamond],co lor=black):\np_3 := plot([[z_0,0],[z_0,-1]],linestyle=3,color=COLOR(RG B,0,.5,0)):\nplots[display]([p_1,p_2,p_3],view=[-4.29..0.49,-1.47..1.4 7],font=[HELVETICA,9]);" }}{PARA 13 "" 1 "" {GLPLOT2D 398 284 284 {PLOTDATA 2 "6+-%'CURVESG6$7U7$$!3/++++++!H%!#<$!3&f-9z=2*=GF*7$$!3;n; a,[!zB%F*$!3#*H(o.hvQd#F*7$$!3QLL3.'4e=%F*$!37Ux?npDZBF*7$$!3`]iX\\=[S TF*$!3)>8pF?GT;#F*7$$!3ym\"He4a^4%F*$!3!o0MqV*G$*>F*7$$!3EL$el\\-K*RF* $!3%36n)owJ];F*7$$!3QL$3f4v0*QF*$!35Q/F<(4nN\"F*7$$!3am\"z`YN%)y$F*$!3 6xhxw!z\"46F*7$$!35Le4`*QPp$F*$!3%*3@KhT`U\"*!#=7$$!3-+Dm*>'o&f$F*$!3Q #e\\blyyU(FU7$$!3ELez,*zU\\$F*$!382R\"G(R7NfFU7$$!35+D@:))>$R$F*$!3CD7 y@hC!p%FU7$$!3amm@$\\C#*G$F*$!3QZ^A$==Zi$FU7$$!39L$3Q&Qk(>$F*$!37^K.>? 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" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1321 "`R*` := z -> 1+z+1/2*z^2+1/6*z^3+ 1/24*z^4+1/120*z^5+943/693000*z^6+157/693000*z^7:\npts := []:\nz0 := 0 :\nfor ct from 0 to 200 do\n zz := newton(`R*`(z)=exp(ct*Pi/20*I),z= z0):\n z0 := zz:\n pts := [op(pts),[Re(zz),Im(zz)]]:\nend do:\np_1 := plot(pts,color=COLOR(RGB,.18,.1,.43)):\np_2 := plots[polygonplot]( [seq([pts[i-1],pts[i],[-1.8,0]],i=2..nops(pts))],\n style=pat chnogrid,color=COLOR(RGB,.36,.2,.85)):\npts := []: z0 := 1.6+4.3*I:\nf or ct from 0 to 60 do\n zz := newton(`R*`(z)=exp(ct*Pi/30*I),z=z0): \n z0 := zz:\n pts := [op(pts),[Re(zz),Im(zz)]]:\nend do:\np_3 := \+ plot(pts,color=COLOR(RGB,.18,.1,.43)):\np_4 := plots[polygonplot]([seq ([pts[i-1],pts[i],[1.52,4.22]],i=2..nops(pts))],\n style=patc hnogrid,color=COLOR(RGB,.36,.2,.85)):\npts := []: z0 := 1.6-4.3*I:\nfo r 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 := pl ot(pts,color=COLOR(RGB,.18,.1,.43)):\np_6 := plots[polygonplot]([seq([ pts[i-1],pts[i],[1.52,-4.22]],i=2..nops(pts))],\n style=patch nogrid,color=COLOR(RGB,.36,.2,.85)):\np_7 := plot([[[-4.39,0],[2.19,0] ],[[0,-4.69],[0,4.69]]],color=black,linestyle=3):\nplots[display]([p_| |(1..7)],view=[-4.39..2.19,-4.69..4.69],font=[HELVETICA,9],\n \+ labels=[`Re(z)`,`Im(z)`],axes=boxed,scaling=constrained);" }}{PARA 13 "" 1 "" {GLPLOT2D 408 561 561 {PLOTDATA 2 "6/-%'CURVESG6$7ew7$$\"\" !F)F(7$$!39+++D_$*zR!#F$\"33+++Gjzq:!#=7$$!37+++u*zS3#!#D$\"3t*****>\" GfTJF07$$!3(******R:af`\"!#C$\"35+++G9R7ZF07$$!3w*****p6'z8DF:$\"3q*** **z4,KG'F07$$\"3&******R1z,u\"!#B$\"31+++,?/ayF07$$\"3-+++GoBt8!#A$\"3 p******)[K\\U*F07$$\"3?+++lb:gbFK$\"3)******z&Ge*4\"!#<7$$\"3%*******f K%=l\"!#@$\"3'******fsZmD\"FS7$$\"31+++-Y2()RFW$\"3!******fy9OT\"FS7$$ \"3s*****\\fyV;)FW$\"33+++?JKq:FS7$$\"3/+++2QhT9!#?$\"3*******Hxvks\"F S7$$\"3;+++m@8$=#Fao$\"3!******Ry)f\")=FS7$$\"3z*****\\%fAEFFao$\"3.++ +#\\K].#FS7$$\"3))*****HYJ>U#Fao$\"39+++`l(f=#FS7$$\"3&******pCT/I#FW$ \"34+++#zzNL#FS7$$!3(******H^!zO_Fao$\"31+++e5-xCFS7$$!31+++l_>c:!#>$ \"3*******Hg7bh#FS7$$!3;+++$fG'HKF`q$\"33+++y^?[FFS7$$!3K+++FB9ocF`q$ \"33+++B(eS(GFS7$$!3\"******42Bd$*)F`q$\"3%******p)e!=*HFS7$$!3/+++ZAV ,8F0$\"3++++3m6+JFS7$$!3%******4F*R!y\"F0$\"37+++!QKz>$FS7$$!3'******H F(4:BF0$\"3')*****H%zw%G$FS7$$!3')*****4'p$*))GF0$\"39+++!G$ygLFS7$$!3 C+++\")pS([$F0$\"3z*****p]DmU$FS7$$!3;+++j-Z*4%F0$\"3!)*****>Q)=$[$FS7 $$!3E+++%*[[jc'F0$\"32+++x:JMOFS 7$$!3!******phNU<(F0$\"37+++RH%ol$FS7$$!3/+++<2awxF0$\"3-+++))**HuOFS7 $$!3U+++JF%HP)F0$\"35+++Wd.(o$FS7$$!3w*****RLHL'*)F0$\"3=+++?^L&p$FS7$ $!3Q+++K8qZ&*F0$\"3')*****RkE%*p$FS7$$!3++++1Ph75FS$\"3!******4#>\\*p$ FS7$$!33+++Mx()p5FS$\"3'******f7tcp$FS7$$!3++++'[#eE6FS$\"3%******f*)y !)o$FS7$$!31+++Tav#=\"FS$\"3!*******y*)ywOFS7$$!3#******RFL%Q7FS$\"3=+ ++'*z&=m$FS7$$!3%*******[Zm$H\"FS$\"30+++_'=Lk$FS7$$!3/+++MY^[8FS$\"3: +++-[=@OFS7$$!32+++B(oIS\"FS$\"3!)*****R%[X&f$FS7$$!33+++P,Wd9FS$\"3?+ ++Dh6mNFS7$$!33+++;px6:FS$\"3%******zH^J`$FS7$$!3!******Hlqic\"FS$\"3- +++.![l\\$FS7$$!31+++S_;@;FS$\"3/+++'p8jX$FS7$$!3'*******zMww;FS$\"32+ ++Ey\\7MFS7$$!3(*******\\qULKAlLFS7$$!3-+++6Ac\"z\"FS$\"34 +++Djs9LFS7$$!33+++/6e^=FS$\"3%)*****\\=+9E$FS7$$!3-+++.0$Q\">FS$\"3@+ ++GQ#e?$FS7$$!33+++cF\\y>FS$\"3'******pP^([JFS7$$!3%******zf)[X?FS$\"3 ;+++cl.\"4$FS7$$!3;+++ZBV9@FS$\"3(******4*o\\LIFS7$$!3;+++Xun%=#FS$\"3 /+++XhwwHFS7$$!3y*****pURaD#FS$\"31+++3;?@HFS7$$!3>+++4x$fK#FS$\"3/+++ o%zo'GFS7$$!35+++5#*\\&R#FS$\"33+++<%eO\"GFS7$$!3#)*****pr.OY#FS$\"3-+ ++)pm7w#FS7$$!38+++mK))HDFS$\"3#)******>yP4FFS7$$!3y*****\\&y4%f#FS$\" 3++++S[5cEFS$\"3-+++WI\"eg#FS7$$!3'*******35$er#FS $\"3#******\\%Hc`DFS7$$!3'******f*3DtFFS$\"3()*****\\q\"o+DFS7$$!38+++ Z2PGGFS$\"3-+++,i(pW#FS7$$!3/+++7y@\")GFS$\"37+++)\\&G#R#FS7$$!3*)**** *HIL=$HFS$\"3*******z+wkL#FS7$$!35+++'4o-)HFS$\"3%******>*pVzAFS7$$!3y *****4N!eEIFS$\"36+++zn2@AFS7$$!3#******HVM32$FS$\"3!)*****p#)>8;#FS7$ $!3$)*****Rq+J6$FS$\"32+++cY5+@FS7$$!3)******p&eX`JFS$\"3z*****zm#QP?F S7$$!35+++6N)>>$FS$\"31+++ay6t>FS7$$!3&)******=[xGKFS$\"3,+++ztG2>FS7$ $!3-+++!zGRE$FS$\"33+++EI))R=FS7$$!3/+++F?b(H$FS$\"3)******\\V84x\"FS7 $$!3-+++GvvHLFS$\"36+++`pS+MFS$\"3'******zz4. 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" }} {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 79 "The stabi lity regions for the two schemes can be displayed in the same picture. 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al7%Fjial7$Fc[_l$!+TO`OVFccnFb\\al7%F^jal7$F][_l$!+![]+L%FccnFb\\al7%F bjal7$Fgj^l$!+B1!GK%FccnFb\\al7%Ffjal7$F\\j^l$!+&)\\#[J%FccnFb\\al7%Fj jalF\\\\alFb\\alF^f]lFj_r-%*AXESSTYLEG6#%$BOXG-%+AXESLABELSG6%%&Re(z)G %&Im(z)G-%%FONTG6#%(DEFAULTG-%(SCALINGG6#%,CONSTRAINEDG-Fh[bl6$%*HELVE TICAG\"\"*-%%VIEWG6$;$!$\\%Fj_n$\"$>#Fj_n;$!$p%Fj_n$\"$p%Fj_n" 1 2 0 1 10 0 2 9 1 2 1 1.000000 45.000000 45.000000 0 0 "Curve 1" "Curve 2" "Curve 3" "Curve 4" "Curve 5" "Curve 6" "Curve 7" "Curve 8" "Curve 9" "Curve 10" "Curve 11" "Curve 12" "Curve 13" "Curve 14" }}}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}} {PARA 0 "" 0 "" {TEXT -1 52 "#---------------------------------------- -----------" }}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 35 "coefficients of th e combined scheme" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 35 "coe fficients of the combined scheme" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 872 "# coefficients by Prince-Do rmand\nee := \{c[2]=1/10,c[3]=2/9,c[4]=3/7,c[5]=3/5,c[6]=4/5,c[7]=1,c[ 8]=1,\n\na[2,1]=1/10,\na[3,1]=-2/81,\na[3,2]=20/81,\na[4,1]=615/1372, \na[4,2]=-270/343,\na[4,3]=1053/1372,\na[5,1]=3243/5500,\na[5,2]=-54/5 5,\na[5,3]=50949/71500,\na[5,4]=4998/17875,\na[6,1]=-26492/37125,\na[6 ,2]=72/55,\na[6,3]=2808/23375,\na[6,4]=-24206/37125,\na[6,5]=338/459, \na[7,1]=5561/2376,\na[7,2]=-35/11,\na[7,3]=-24117/31603,\na[7,4]=8999 83/200772,\na[7,5]=-5225/1836,\na[7,6]=3925/4056,\na[8,1]=465467/26611 2,\na[8,2]=-2945/1232,\na[8,3]=-5610201/14158144,\na[8,4]=10513573/321 2352,\na[8,5]=-424325/205632,\na[8,6]=376225/454272,\na[8,7]=0,\n\nb[1 ]=61/864,\nb[2]=0,\nb[3]=98415/321776,\nb[4]=16807/146016,\nb[5]=1375/ 7344,\nb[6]=1375/5408,\nb[7]=-37/1120,\nb[8]=1/10,\n\n`b*`[1]=821/1080 0,\n`b*`[2]=0,\n`b*`[3]=19683/71825,\n`b*`[4]=175273/912600,\n`b*`[5]= 395/3672,\n`b*`[6]=785/2704,\n`b*`[7]=3/50,\n`b*`[8]=0\}:" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }} }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT 259 5 "nodes" }{TEXT -1 2 ": " }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 31 "seq(c[i]= subs(ee,c[i]),i=2..8);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6)/&%\"cG6#\" \"##\"\"\"\"#5/&F%6#\"\"$#F'\"\"*/&F%6#\"\"%#F.\"\"(/&F%6#\"\"&#F.F:/& F%6#\"\"'#F4F:/&F%6#F6F)/&F%6#\"\")F)" }}}{PARA 0 "" 0 "" {TEXT -1 0 " " }}{PARA 0 "" 0 "" {TEXT 259 20 "linking coefficients" }{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..8);" }}{PARA 12 "" 1 "" {XPPMATH 20 "6>/&%\"aG6 $\"\"#\"\"\"#F(\"#5/&F%6$\"\"$F(#!\"#\"#\")/&F%6$F.F'#\"#?F1/&F%6$\"\" %F(#\"$:'\"%s8/&F%6$F:F'#!$q#\"$V$/&F%6$F:F.#\"%`5F=/&F%6$\"\"&F(#\"%V K\"%+b/&F%6$FLF'#!#a\"#b/&F%6$FLF.#\"&\\4&\"&+:(/&F%6$FLF:#\"%)*\\\"&v y\"/&F%6$\"\"'F(#!&#\\E\"&Dr$/&F%6$F_oF'#\"#sFU/&F%6$F_oF.#\"%3G\"&vL# /&F%6$F_oF:#!&1U#Fbo/&F%6$F_oFL#\"$Q$\"$f%/&F%6$\"\"(F(#\"%hb\"%wB/&F% 6$F\\qF'#!#N\"#6/&F%6$F\\qF.#!& " 0 "" {MPLTEXT 1 0 31 "seq(b[i]=subs(ee,b[i]),i=1..8);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6*/&%\"bG6#\"\"\"#\"#h\"$k)/&F%6#\"\"#\"\"!/&F%6#\" \"$#\"&:%)*\"'w " 0 "" {MPLTEXT 1 0 37 "seq(`b*`[i]=subs(ee,`b*`[i]),i=1..8);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6*/&%#b*G6#\"\"\"#\"$@)\"&+3\"/&F%6#\"\"#\"\"!/&F%6#\"\"$#\"&$o>\" &D=(/&F%6#\"\"%#\"'t_<\"'+E\"*/&F%6#\"\"&#\"$&R\"%sO/&F%6#\"\"'#\"$&y \"%/F/&F%6#\"\"(#F3\"#]/&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 42 "#=========================================" }}{SECT 1 {PARA 4 " " 0 "" {TEXT -1 43 "a modification of the Prince-Dormand scheme" }} {PARA 257 "" 0 "" {TEXT -1 0 "" }}{PARA 256 "" 0 "" {TEXT -1 1 " " } {XPPEDIT 18 0 "matrix([[7/39, 7/39, ``, ``, ``, ``, ``, ``, ``], [2/9, 16/189, 26/189, ``, ``, ``, ``, ``, ``], [3/7, 957/9604, -1053/2401, \+ 1053/1372, ``, ``, ``, ``, ``], [23/33, 2563741/11068596, -18239/10248 7, 3243/761332, 3284078/5138991, ``, ``, ``, ``], [24/31, -11597952/14 8686881, 92664/208537, 98740944/564271331, -26004300/372178963, 936877 5900/30948112231, ``, ``, ``], [1, 38665819/91808640, -897/1232, 15639 9/1505504, 1592286101/1436292000, -2279466607/2965053280, 972169703/11 26224000, ``, ``], [1, 118627013/607606272, -1527/3136, 26560509/49818 496, 576719677/1357948800, -6116292391/16604298368, 5233891417/7453555 200, 0, ``], [b, 14459/198720, 0, 19683/68432, 8252237/43524000, 14349 6441/1058947600, 28629151/119448000, 11/1120, 13/200], [`b*`, 1236443/ 16593120, 0, 43680951/157136980, 379485253/1817127000, 1629060147/1768 4424920, 30137260793/109712988000, 12/167, 0]])" "6#-%'matrixG6#7+7+*& \"\"(\"\"\"\"#R!\"\"*&F)F*F+F,%!GF.F.F.F.F.F.7+*&\"\"#F*\"\"*F,*&\"#;F *\"$*=F,*&\"#EF*F5F,F.F.F.F.F.F.7+*&\"\"$F*F)F,*&\"$d*F*\"%/'*F,,$*&\" %`5F*\"%,CF,F,*&F@F*\"%s8F,F.F.F.F.F.7+*&\"#BF*\"#LF,*&\"(TPc#F*\")'fo 5\"F,,$*&\"&R#=F*\"'([-\"F,F,*&\"%VKF*\"'K8wF,*&\"(ySG$F*\"(\"**Q^F,F. F.F.F.7+*&\"#CF*\"#JF,,$*&\")_zf6F*\"*\")oo[\"F,F,*&\"&kE*F*\"'P&3#F,* &\")W4u)*F*\"*J8Fk&F,,$*&\")+V+EF*\"*j*y@PF,F,*&\"++fxo$*F*\",JA6[4$F, F.F.F.7+F**&\")>emQF*\")S'3=*F,,$*&\"$(*)F*\"%K7F,F,*&\"'*Rc\"F*\"(/b] \"F,*&\"+,hG#f\"F*\"++?HO9F,,$*&\"+2mYzAF*\"+!G`]'HF,F,*&\"*.(p@(*F*\" ++SAE6F,F.F.7+F**&\"*8qi=\"F*\"*sig2'F,,$*&\"%F:F*\"%OJF,F,*&\")40cEF* \")'\\=)\\F,*&\"*x'>ndF*\"++)[zN\"F,,$*&\"+\"R#H;hF*\",o$)H/m\"F,F,*& \"+<9*QB&F*\"++_b`uF,\"\"!F.7+%\"bG*&\"&fW\"F*\"'?()>F,F^r*&\"&$o>F*\" &K%oF,*&\"(PAD)F*\")+S_VF,*&\"*Tk\\V\"F*\"++w%*e5F,*&\")^\"H'GF*\"*+![ %>\"F,*&\"#6F*\"%?6F,*&\"#8F*\"$+#F,7+%#b*G*&\"(VkB\"F*\")?Jf;F,F^r*& \")^4oVF*\"*!)p8d\"F,*&\"*`_[z$F*\"++q7<=F,*&\"+Z,1H;F*\",?\\U%o " 0 "" {MPLTEXT 1 0 1 ";" }}}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 40 "ee: coefficients for the combined scheme" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1064 "ee := \{c[ 2]=7/39,c[3]=2/9,c[4]=3/7,c[5]=23/33,c[6]=24/31,c[7]=1,c[8]=1,\n\na[2, 1]=7/39,\na[3,1]=16/189,\na[3,2]=26/189,\na[4,1]=957/9604,\na[4,2]=-10 53/2401,\na[4,3]=1053/1372,\na[5,1]=2563741/11068596,\na[5,2]=-18239/1 02487,\na[5,3]=3243/761332,\na[5,4]=3284078/5138991,\na[6,1]=-11597952 /148686881,\na[6,2]=92664/208537,\na[6,3]=98740944/564271331,\na[6,4]= -26004300/372178963,\na[6,5]=9368775900/30948112231,\na[7,1]=38665819/ 91808640,\na[7,2]=-897/1232,\na[7,3]=156399/1505504,\na[7,4]=159228610 1/1436292000,\na[7,5]=-2279466607/2965053280,\na[7,6]=972169703/112622 4000,\na[8,1]=118627013/607606272,\na[8,2]=-1527/3136,\na[8,3]=2656050 9/49818496,\na[8,4]=576719677/1357948800,\na[8,5]=-6116292391/16604298 368,\na[8,6]=5233891417/7453555200,\na[8,7]=0,\n\nb[1]=14459/198720,\n b[2]=0,\nb[3]=19683/68432,\nb[4]=8252237/43524000,\nb[5]=143496441/105 8947600,\nb[6]=28629151/119448000,\nb[7]=11/1120,\nb[8]=13/200,\n\n`b* `[1]=1236443/16593120,\n`b*`[2]=0,\n`b*`[3]=43680951/157136980,\n`b*`[ 4]=379485253/1817127000,\n`b*`[5]=1629060147/17684424920,\n`b*`[6]=301 37260793/109712988000,\n`b*`[7]=12/167,\n`b*`[8]=0\}:" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}} {PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 146 "RK6_8eqs := [op(RowSumConditions(8,'expanded')),op(OrderCondition s(6,8,'expanded'))]:\n`RK5_8eqs*` := subs(b=`b*`,OrderConditions(5,8,' expanded')):" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 7 "Check: " }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 128 "simp lify(subs(ee,RK6_8eqs)):\nmap(u->lhs(u)-rhs(u),%);\nnops(%);\nsimplify (subs(ee,`RK5_8eqs*`)):\nmap(u->lhs(u)-rhs(u),%);\nnops(%);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#7N\"\"!F$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 3 to 8 have stage-order 2. " }}{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$F$F$" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }} {PARA 0 "" 0 "" {TEXT -1 114 "We can calculate the principal error nor m of the order 6 scheme, that is, the 2-norm of the principal error te rms." }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 131 "errterms6_8 := Prin cipalErrorTerms(6,8,'expanded'):\nevalf(evalf[14](sqrt(add(subs(ee,err terms6_8[i])^2,i=1.. nops(errterms6_8)))));" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#$\"+n(3j5#!#8" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }} {PARA 0 "" 0 "" {TEXT -1 79 "The 2 norm of the principal error of the \+ order 5 embedded scheme is as follows." }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 153 "`errterms5_8*` := subs(b=`b*`,PrincipalErrorTerms(5, 8,'expanded')):\nevalf(evalf[14](sqrt(add(subs(ee,`errterms5_8*`[i])^2 ,i=1.. nops(`errterms5_8*`)))));" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#$ \"+e-)[#=!#8" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 52 "Two of the principal error conditions are satisfied." }} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 291 "ind := []:\nfor ct to nops( errterms6_8) do\n if simplify(subs(ee,errterms6_8[ct]))=0 then ind : = [op(ind),ct] end if:\nend do:\nerr6 := PrincipalErrorConditions(6): \n[seq([i,err6[i]],i=ind)]:\nlinalg[augment](linalg[delcols](%,2..2),m atrix([[` `]$(linalg[rowdim](%))]),linalg[delcols](%,1..1));" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#K%'matrixG6#7$7%\"\"\"%#~~G/*&%\"bGF(- %!G6#*&%\"aGF(-F.6#*&F1F(-F.6#*&F1F(-F.6#*&F1F(-F.6#*&F1F(%\"cGF(F(F(F (F(F(#F(\"%S]7%\"\"&F)/*(F,F(F>F(F2F(#F(\"$S)Q(pprint86\"" }}}{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 "construction of \+ the combined scheme " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 " " {TEXT -1 196 "First we construct the 8 stage order 6 scheme by makin g use of a selection of \"simple\" order conditions and incorporating \+ the stage-order equations to ensure that stages 3 to 8 have stage-orde r 2." }}{PARA 0 "" 0 "" {TEXT -1 48 "We also incorporate the simplifyi ng conditions: " }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "Su m(b[i]*a[i,j],i = j+1 .. 8) = b[j]*(1-c[j]);" "6#/-%$SumG6$*&&%\"bG6#% \"iG\"\"\"&%\"aG6$F+%\"jGF,/F+;,&F0F,F,F,\"\")*&&F)6#F0F,,&F,F,&%\"cG6 #F0!\"\"F," }{TEXT -1 1 " " }}{PARA 0 "" 0 "" {TEXT -1 5 "for " } {XPPEDIT 18 0 "j = 4;" "6#/%\"jG\"\"%" }{TEXT -1 18 " . . . 7. ( with \+ " }{XPPEDIT 18 0 "c[1]=0" "6#/&%\"cG6#\"\"\"\"\"!" }{TEXT -1 55 " ), \+ together with the further simplifying conditions: " }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "Sum(b[i]*c[i]*a[i,2],i=3..8)=0" "6#/ -%$SumG6$*(&%\"bG6#%\"iG\"\"\"&%\"cG6#F+F,&%\"aG6$F+\"\"#F,/F+;\"\"$\" \")\"\"!" }{TEXT -1 8 ", " }{XPPEDIT 18 0 "Sum(b[i]*c[i]*Sum(a[i ,j]*a[j,2],j=3..i-1),i=3..8)=0" "6#/-%$SumG6$*(&%\"bG6#%\"iG\"\"\"&%\" cG6#F+F,-F%6$*&&%\"aG6$F+%\"jGF,&F46$F6\"\"#F,/F6;\"\"$,&F+F,F,!\"\"F, /F+;F<\"\")\"\"!" }{TEXT -1 8 ", " }{XPPEDIT 18 0 "Sum(b[i]*c[i] ^2*a[i,2],i=3..8)=0" "6#/-%$SumG6$*(&%\"bG6#%\"iG\"\"\"*$&%\"cG6#F+\" \"#F,&%\"aG6$F+F1F,/F+;\"\"$\"\")\"\"!" }{TEXT -1 2 ". " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 32 "The scheme is constr ucted with " }{XPPEDIT 18 0 "a[7,3];" "6#&%\"aG6$\"\"(\"\"$" }{TEXT -1 3 ", " }{XPPEDIT 18 0 "a[7,4]" "6#&%\"aG6$\"\"(\"\"%" }{TEXT -1 3 ", " }{XPPEDIT 18 0 "a[7,5]" "6#&%\"aG6$\"\"(\"\"&" }{TEXT -1 7 " an d " }{XPPEDIT 18 0 "a[7,6]" "6#&%\"aG6$\"\"(\"\"'" }{TEXT -1 17 " as parameters. " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 76 "The simple order conditions used are given (in abreviated form) as follows. " }}{PARA 0 "" 0 "" {TEXT -1 94 "These include the \+ 6 order 6 quadrature conditions together with 4 additional order condi tions." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 184 "SO6 := SimpleOrderConditions(6):\n[seq([i,SO6[i]],i= [1,2,4,8,13,16,24,28,29,32])]:\nlinalg[augment](linalg[delcols](%,2..2 ),matrix([[` `]$(linalg[rowdim](%))]),linalg[delcols](%,1..1));" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#K%'matrixG6#7,7%\"\"\"%#~~G/*&%\"bGF(% \"eGF(F(7%\"\"#F)/*&F,F(%\"cGF(#F(F/7%\"\"%F)/*&F,F()F2F/F(#F(\"\"$7% \"\")F)/*&F,F()F2F:F(#F(F57%\"#8F)/*(F,F(F2F(-%!G6#*&F8F(%\"aGF(F(#F( \"#:7%\"#;F)/*&F,F()F2F5F(#F(\"\"&7%\"#CF)/*(F,F(F2F(-FF6#*&FIF(FEF(F( #F(\"#s7%\"#GF)/*(F,F(F8F(FEF(#F(\"#=7%\"#HF)/*(F,F(F2F(-FF6#*&F?F(FIF (F(#F(FT7%\"#KF)/*&F,F()F2FRF(#F(\"\"'Q)pprint816\"" }}}{PARA 0 "" 0 " " {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 513 "SO_eqs := [op(RowSumConditions(8,'expanded')),op(StageOrderConditions(2,8,'expa nded'))]:\nSO6_8 := SimpleOrderConditions(6,8,'expanded'):\nord_cdns : = [seq(SO6_8[i],i=[1,2,4,8,13,16,24,28,29,32])]:\nsimp_cdns := [add(b[ i]*a[i,1],i=1+1..8)=b[1],seq(add(b[i]*a[i,j],i=j+1..8)=b[j]*(1-c[j]),j =[4,5,6,7])]:\nsimp_cdns2 := [add(b[i]*c[i]*a[i,2],i=3..8)=0,add(b[i]* c[i]*add(a[i,j]*a[j,2],j=3..i-1),i=3..8)=0,\n add(b[i]* c[i]^2*a[i,2],i=3..8)=0]:\ncdns := [op(ord_cdns),op(simp_cdns),op(simp _cdns2),op(SO_eqs)]:" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" } }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 21 "We spec ify the nodes:" }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "c[2 ] = 7/39;" "6#/&%\"cG6#\"\"#*&\"\"(\"\"\"\"#R!\"\"" }{TEXT -1 3 ", " }{XPPEDIT 18 0 "c[3] = 2/9;" "6#/&%\"cG6#\"\"$*&\"\"#\"\"\"\"\"*!\"\" " }{TEXT -1 3 ", " }{XPPEDIT 18 0 "c[5] = 23/33;" "6#/&%\"cG6#\"\"&*& \"#B\"\"\"\"#L!\"\"" }{TEXT -1 3 ", " }{XPPEDIT 18 0 "c[6] = 24/31;" "6#/&%\"cG6#\"\"'*&\"#C\"\"\"\"#J!\"\"" }{TEXT -1 3 ", " }{XPPEDIT 18 0 "c[7]=1" "6#/&%\"cG6#\"\"(\"\"\"" }{TEXT -1 3 ", " }{XPPEDIT 18 0 "c[8]=1" "6#/&%\"cG6#\"\")\"\"\"" }{TEXT -1 2 ", " }}{PARA 0 "" 0 " " {TEXT -1 17 "and the weights: " }}{PARA 256 "" 0 "" {TEXT -1 1 " " } {XPPEDIT 18 0 "b[2]=0" "6#/&%\"bG6#\"\"#\"\"!" }{TEXT -1 3 ", " } {XPPEDIT 18 0 "b[8] = 13/200;" "6#/&%\"bG6#\"\")*&\"#8\"\"\"\"$+#!\"\" " }{TEXT -1 2 ". " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 176 "e1 := \{c[2]=7/39,c[3]=2/9,c[5]=23/33,c[6]=2 4/31,c[7]=1,c[8]=1,b[2]=0,b[8]=13/200\}:\neqns := subs(e1,cdns):\nnops (eqns);\nindets(eqns) minus \{a[7,3],a[7,4],a[7,5],a[7,6]\};\nnops(%); " }}{PARA 11 "" 1 "" {XPPMATH 20 "6#\"#J" }}{PARA 11 "" 1 "" {XPPMATH 20 "6# " 0 "" {MPLTEXT 1 0 22 "infolevel[solve] := 4:" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 118 "e2 := solve(\{op(eqns)\},indets(eqns) minus \{a[7,3],a[7,4],a[7,5 ],a[7,6]\}):\ninfolevel[solve] := 0:\ne3 := `union`(e1,e2):" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 3 "e3 " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1094 "e3 := \{a[5,4] = 3284078/5138991, a[6,4] = -26004300/372178963, a[8,3] = 488259/889616 -55/364*a[7,3], c[8] = 1, a[5,2] = -18239/102487, b[4] = 8252237/43524 000, a[8,5] = -233450745/481821158-55/364*a[7,5], a[5,3] = 3243/761332 , c[3] = 2/9, b[2] = 0, a[6,5] = 9368775900/30948112231, a[4,2] = -105 3/2401, b[6] = 28629151/119448000, a[8,2] = -57/56+55/294*a[7,3]+495/1 372*a[7,4]+115/196*a[7,5]+990/1519*a[7,6], b[5] = 143496441/1058947600 , b[7] = 11/1120, a[8,4] = 837697/1414530-55/364*a[7,4], a[7,2] = -26/ 21*a[7,3]-117/49*a[7,4]-299/77*a[7,5]-936/217*a[7,6]+39/14, a[6,3] = 9 8740944/564271331, a[8,7] = 0, c[7] = 1, a[6,2] = 92664/208537, a[4,1] = 957/9604, c[2] = 7/39, c[5] = 23/33, c[6] = 24/31, b[8] = 13/200, a [8,6] = -55/364*a[7,6]+923521/1109160, a[6,1] = -11597952/148686881, a [3,1] = 16/189, c[4] = 3/7, a[4,3] = 1053/1372, b[3] = 19683/68432, a[ 8,1] = 478031/904176-275/7644*a[7,3]-935/4459*a[7,4]-555/1274*a[7,5]-3 9545/78988*a[7,6], b[1] = 14459/198720, a[7,1] = 5/21*a[7,3]+68/49*a[7 ,4]+222/77*a[7,5]+719/217*a[7,6]-25/14, a[2,1] = 7/39, a[5,1] = 256374 1/11068596, a[3,2] = 26/189\}:" }}}{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 0 452 "subs(e3,matrix([s eq([c[i],seq(a[i,j],j=1..i-1),``$(6-i)],i=2..6)]));\nprint(``);\nfor i i from 7 to 8 do\n print(c[ii]=subs(e3,c[ii]));print(``); \n for j j to ii-1 do\n coeff_val := subs(e3,a[ii,jj]);\n if coeff_va l=a[ii,jj] then \n print(a[ii,jj]*` is a parameter.`)\n e lse\n print(a[ii,jj]=coeff_val);\n end if;\n end do:\n \+ print(`__________________________________`):\nend do:\nprint(``);\ns eq(b[i]=subs(e3,b[i]),i=1..8);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#K%'m atrixG6#7'7(#\"\"(\"#RF(%!GF+F+F+7(#\"\"#\"\"*#\"#;\"$*=#\"#EF2F+F+F+7 (#\"\"$F)#\"$d*\"%/'*#!%`5\"%,C#\"%`5\"%s8F+F+7(#\"#B\"#L#\"(TPc#\")'f o5\"#!&R#=\"'([-\"#\"%VK\"'K8w#\"(ySG$\"(\"**Q^F+7(#\"#C\"#J#!)_zf6\"* \")oo[\"#\"&kE*\"'P&3##\")W4u)*\"*J8Fk&#!)+V+E\"*j*y@P#\"++fxo$*\",JA6 [4$Q(pprint46\"" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%!G" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/&%\"cG6#\"\"(\"\"\"" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%!G" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/&%\"aG6$\"\"(\"\"\",,*&# \"\"&\"#@F(&F%6$F'\"\"$F(F(*&#\"#o\"#\\F(&F%6$F'\"\"%F(F(*&#\"$A#\"#xF (&F%6$F'F,F(F(*&#\"$>(\"$<#F(&F%6$F'\"\"'F(F(#\"#D\"#9!\"\"" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/&%\"aG6$\"\"(\"\"#,,*&#\"#E\"#@\"\"\"&F%6$F '\"\"$F.!\"\"*&#\"$<\"\"#\\F.&F%6$F'\"\"%F.F2*&#\"$*H\"#xF.&F%6$F'\"\" &F.F2*&#\"$O*\"$<#F.&F%6$F'\"\"'F.F2#\"#R\"#9F." }}{PARA 11 "" 1 "" {XPPMATH 20 "6#*&&%\"aG6$\"\"(\"\"$\"\"\"%1~is~a~parameter.GF)" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#*&&%\"aG6$\"\"(\"\"%\"\"\"%1~is~a~para meter.GF)" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#*&&%\"aG6$\"\"(\"\"&\"\" \"%1~is~a~parameter.GF)" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#*&&%\"aG6$ \"\"(\"\"'\"\"\"%1~is~a~parameter.GF)" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%C__________________________________G" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/&%\"cG6#\"\")\"\"\"" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%!G" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#/&%\"aG6$\"\")\"\"\",,#\"'J!y%\"'wT!*F (*&#\"$v#\"%WwF(&F%6$\"\"(\"\"$F(!\"\"*&#\"$N*\"%fWF(&F%6$F3\"\"%F(F5* &#\"$b&\"%u7F(&F%6$F3\"\"&F(F5*&#\"&X&R\"&))*yF(&F%6$F3\"\"'F(F5" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#/&%\"aG6$\"\")\"\"#,,#\"#d\"#c!\"\"*&# \"#b\"$%H\"\"\"&F%6$\"\"(\"\"$F2F2*&#\"$&\\\"%s8F2&F%6$F5\"\"%F2F2*&# \"$:\"\"$'>F2&F%6$F5\"\"&F2F2*&#\"$!**\"%>:F2&F%6$F5\"\"'F2F2" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#/&%\"aG6$\"\")\"\"$,&#\"'f#)[\"';'*)) \"\"\"*&#\"#b\"$k$F-&F%6$\"\"(F(F-!\"\"" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/&%\"aG6$\"\")\"\"%,&#\"'(pP)\"(IXT\"\"\"\"*&#\"#b\"$k$F-&F%6$\" \"(F(F-!\"\"" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/&%\"aG6$\"\")\"\"&,&# \"*X2XL#\"*e6#=[!\"\"*&#\"#b\"$k$\"\"\"&F%6$\"\"(F(F2F-" }}{PARA 11 " " 1 "" {XPPMATH 20 "6#/&%\"aG6$\"\")\"\"',&*&#\"#b\"$k$\"\"\"&F%6$\"\" (F(F.!\"\"#\"'@N#*\"(g\"46F." }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/&%\"a G6$\"\")\"\"(\"\"!" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%C______________ ____________________G" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%!G" }}{PARA 11 "" 1 "" {XPPMATH 20 "6*/&%\"bG6#\"\"\"#\"&fW\"\"'?()>/&F%6#\"\"#\" \"!/&F%6#\"\"$#\"&$o>\"&K%o/&F%6#\"\"%#\"(PAD)\")+S_V/&F%6#\"\"&#\"*Tk \\V\"\"++w%*e5/&F%6#\"\"'#\")^\"H'G\"*+![%>\"/&F%6#\"\"(#\"#6\"%?6/&F% 6#\"\")#\"#8\"$+#" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}} {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 66 "We now tu rn our attention to the embedded 8 stage, order 5 scheme." }}{PARA 0 " " 0 "" {TEXT -1 35 "We use the simplifying conditions: " }}{PARA 256 " " 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "Sum(`b*`[i]*a[i,2],i = 3 .. 8) = 0;" "6#/-%$SumG6$*&&%#b*G6#%\"iG\"\"\"&%\"aG6$F+\"\"#F,/F+;\"\"$\"\") \"\"!" }{TEXT -1 8 ", " }{XPPEDIT 18 0 "Sum(`b*`[i]*Sum(a[i,j]*a [j,2],j = 3 .. i-1),i = 2 .. 8) = 0;" "6#/-%$SumG6$*&&%#b*G6#%\"iG\"\" \"-F%6$*&&%\"aG6$F+%\"jGF,&F16$F3\"\"#F,/F3;\"\"$,&F+F,F,!\"\"F,/F+;F6 \"\")\"\"!" }{TEXT -1 1 "." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 101 "We use a selection of \"simple\" order condition s as given (in abreviated form) in the following table." }}{PARA 0 "" 0 "" {TEXT -1 95 "These include the 5 order 5 quadrature conditions to gether with one additional order condition." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 191 "`SO5*` := subs(b= `b*`,SimpleOrderConditions(5)):\n[seq([i,`SO5*`[i]],i=[1,2,4,8,15,16]) ]:\nlinalg[augment](linalg[delcols](%,2..2),matrix([[` `]$(linalg[row dim](%))]),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(\"\"$7%\"\")F)/*&F,F()F2F:F(#F(F57%\"# :F)/*&F,F(-%!G6#*&F?F(%\"aGF(F(#F(\"#?7%\"#;F)/*&F,F()F2F5F(#F(\"\"&Q) pprint836\"" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 262 "`SO5_7*` := subs(b=`b*`,SimpleOrderConditions(5,7 ,'expanded')):\n`ord_cdns*` := [seq(`SO5_7*`[i],i=[1,2,4,8,15,16])]:\n `simp_cdns*` := [add(`b*`[i]*a[i,2],i=3..7)=0,add(`b*`[i]*add(a[i,j]*a [j,2],j=3..i-1),i=2..7)=0]:\n`cdns*` := [op(`ord_cdns*`),op(`simp_cdns *`)]: " }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 17 "We specify that " }{XPPEDIT 18 0 "`b*`[2] = 0;" "6#/&%#b*G6#\"\"# \"\"!" }{TEXT -1 3 ", " }{XPPEDIT 18 0 "`b*`[7] = 12/167;" "6#/&%#b*G 6#\"\"(*&\"#7\"\"\"\"$n\"!\"\"" }{TEXT -1 7 " and " }{XPPEDIT 18 0 " `b*`[8] = 0;" "6#/&%#b*G6#\"\")\"\"!" }{TEXT -1 2 ". " }}{PARA 0 "" 0 "" {TEXT 260 4 "Note" }{TEXT -1 71 ": Although we shall construct the \+ scheme as an 8 stage scheme, setting " }{XPPEDIT 18 0 "`b*`[8]=0" "6#/ &%#b*G6#\"\")\"\"!" }{TEXT -1 101 " means that it may also be regarde d as an 7 stage scheme, where the 8th stage makes no contribution." }} {PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 65 "e4 := \{`b*`[2]=0,`b*`[7]=12/167,`b*`[8]=0\}:\ne5 := `union`(e3,e4 ):" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 84 "`eqns*` := subs(e5,`cdns*`):\nnops(`eqns*`);\nindets( `eqns*`) minus \{a[7,6]\};\nnops(%);" }}{PARA 11 "" 1 "" {XPPMATH 20 " 6#\"\")" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#<*&%#b*G6#\"\"\"&F%6#\"\"$& F%6#\"\"%&F%6#\"\"&&F%6#\"\"'&%\"aG6$\"\"(F*&F56$F7F-&F56$F7F0" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#\"\")" }}}{PARA 0 "" 0 "" {TEXT -1 0 " " }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 22 "infolevel[solve] := 4:" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 112 "e6 := solve(\{op(`eqns*` )\},indets(`eqns*`) minus \{a[7,6]\}):\ne7 := `union`(subs(e6,e5),e6): \ninfolevel[solve] := 0:" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 2 "e7" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1331 "e7 := \{`b*`[3] = 43680951/157136980, a[8,1] = 5675 675/25316928-1194875/35630036*a[7,6], a[7,1] = 97577/425040+152075/685 193*a[7,6], a[5,4] = 3284078/5138991, a[6,4] = -26004300/372178963, c[ 8] = 1, a[5,2] = -18239/102487, `b*`[1] = 1236443/16593120, b[4] = 825 2237/43524000, `b*`[8] = 0, `b*`[7] = 12/167, a[5,3] = 3243/761332, c[ 3] = 2/9, b[2] = 0, `b*`[2] = 0, a[6,5] = 9368775900/30948112231, a[4, 2] = -1053/2401, b[6] = 28629151/119448000, b[5] = 143496441/105894760 0, b[7] = 11/1120, a[6,3] = 98740944/564271331, a[8,7] = 0, c[7] = 1, \+ a[8,5] = -126334216877/215855878784+3261282750/12978240613*a[7,6], a[6 ,2] = 92664/208537, a[4,1] = 957/9604, c[2] = 7/39, c[5] = 23/33, c[6] = 24/31, b[8] = 13/200, a[8,6] = -55/364*a[7,6]+923521/1109160, a[6,1 ] = -11597952/148686881, a[3,1] = 16/189, c[4] = 3/7, a[4,3] = 1053/13 72, a[7,4] = -194747/2127840+16691752/12005773*a[7,6], b[3] = 19683/68 432, a[8,3] = 2549109/6227312+237562875/1656409391*a[7,6], a[7,5] = 25 026793/37532320-1660289400/998326201*a[7,6], a[7,3] = 217176/235235-17 277300/18202301*a[7,6], `b*`[6] = 30137260793/109712988000, b[1] = 144 59/198720, `b*`[5] = 1629060147/17684424920, a[2,1] = 7/39, `b*`[4] = \+ 379485253/1817127000, a[5,1] = 2563741/11068596, a[7,2] = -897/1232, a [8,2] = -1527/3136, a[8,4] = 54864553/90529920-32787370/156075049*a[7, 6], a[3,2] = 26/189\}:" }}}{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 39 "The order 5 scheme is given as follow s." }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 405 "subs(e7,matrix([seq([ c[i],seq(a[i,j],j=1..i-1),``$(7-i)],i=2..6)]));\nfor ii from 7 to 8 do \n print(``);\n print(c[ii]=subs(e7,c[ii])); \n for jj to ii-1 d o\n coeff_val := subs(e7,a[ii,jj]);\n if coeff_val=a[ii,jj] \+ then \n print(a[ii,jj]*` is a parameter.`)\n else\n \+ print(a[ii,jj]=coeff_val);\n end if;\n end do:\nend do:\npri nt(``);\n[seq(`b*`[i]=subs(e7,`b*`[i]),i=1..8)];" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#K%'matrixG6#7'7)#\"\"(\"#RF(%!GF+F+F+F+7)#\"\"#\"\"*#\" #;\"$*=#\"#EF2F+F+F+F+7)#\"\"$F)#\"$d*\"%/'*#!%`5\"%,C#\"%`5\"%s8F+F+F +7)#\"#B\"#L#\"(TPc#\")'fo5\"#!&R#=\"'([-\"#\"%VK\"'K8w#\"(ySG$\"(\"** Q^F+F+7)#\"#C\"#J#!)_zf6\"*\")oo[\"#\"&kE*\"'P&3##\")W4u)*\"*J8Fk&#!)+ V+E\"*j*y@P#\"++fxo$*\",JA6[4$F+Q(pprint56\"" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%!G" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/&%\"cG6#\"\"(\" \"\"" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/&%\"aG6$\"\"(\"\"\",&#\"&xv* \"'S]UF(*&#\"'v?:\"'$>&oF(&F%6$F'\"\"'F(F(" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/&%\"aG6$\"\"(\"\"##!$(*)\"%K7" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/&%\"aG6$\"\"(\"\"$,&#\"'wr@\"'N_B\"\"\"*&#\")+tF<\"),B ?=F-&F%6$F'\"\"'F-!\"\"" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/&%\"aG6$\" \"(\"\"%,&#\"'ZZ>\"(Sy7#!\"\"*&#\")_\"\")O+jNF(&F%6$\"\"(\"\"'F(!\"\"" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/&%\"aG6$\"\")\"\"##!%F:\"%OJ" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#/&%\"aG6$\"\")\"\"$,&#\"(4\"\\D\"(7tA' \"\"\"*&#\"*vGcP#\"+\"R4kl\"F-&F%6$\"\"(\"\"'F-F-" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/&%\"aG6$\"\")\"\"%,&#\")`X'[&\")?*H0*\"\"\"*&#\")qtyK \"*\\]2c\"F-&F%6$\"\"(\"\"'F-!\"\"" }}{PARA 11 "" 1 "" {XPPMATH 20 "6# /&%\"aG6$\"\")\"\"&,&#\"-xo@Mj7\"-%yye&e@!\"\"*&#\"+]FGhK\",81CyH\"\" \"\"&F%6$\"\"(\"\"'F2F2" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/&%\"aG6$\" \")\"\"',&*&#\"#b\"$k$\"\"\"&F%6$\"\"(F(F.!\"\"#\"'@N#*\"(g\"46F." }} {PARA 11 "" 1 "" {XPPMATH 20 "6#/&%\"aG6$\"\")\"\"(\"\"!" }}{PARA 11 " " 1 "" {XPPMATH 20 "6#%!G" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#7*/&%#b*G 6#\"\"\"#\"(VkB\"\")?Jf;/&F&6#\"\"#\"\"!/&F&6#\"\"$#\")^4oV\"*!)p8d\"/ &F&6#\"\"%#\"*`_[z$\"++q7<=/&F&6#\"\"&#\"+Z,1H;\",?\\U%o " 0 "" {MPLTEXT 1 0 146 "RK6_ 8eqs := [op(RowSumConditions(8,'expanded')),op(OrderConditions(6,8,'ex panded'))]:\n`RK5_8eqs*` := subs(b=`b*`,OrderConditions(5,8,'expanded' )):" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 7 "Ch eck: " }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 128 "simplify(subs(e7,R K6_8eqs)):\nmap(u->lhs(u)-rhs(u),%);\nnops(%);\nsimplify(subs(e7,`RK5_ 8eqs*`)):\nmap(u->lhs(u)-rhs(u),%);\nnops(%);" }}{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#\"#<" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 21 "We find a value for " }{XPPEDIT 18 0 " a[7,6]" "6#&%\"aG6$\"\"(\"\"'" }{TEXT -1 39 " by applying the simplif ying condition" }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "Sum (`b*`[i]*a[i,j],i = j+1 .. s) = `b*`[j]*(1-c[j]);" "6#/-%$SumG6$*&&%#b *G6#%\"iG\"\"\"&%\"aG6$F+%\"jGF,/F+;,&F0F,F,F,%\"sG*&&F)6#F0F,,&F,F,&% \"cG6#F0!\"\"F," }{TEXT -1 5 ", " }{XPPEDIT 18 0 "j = 2" "6#/%\"jG \"\"#" }{TEXT -1 7 " . . . " }{TEXT 271 1 "s" }{TEXT -1 1 "," }}{PARA 0 "" 0 "" {TEXT -1 6 "with " }{XPPEDIT 18 0 "s=8" "6#/%\"sG\"\")" } {TEXT -1 23 " for the eighth stage." }}{PARA 0 "" 0 "" {TEXT -1 8 "Ta king " }{XPPEDIT 18 0 "s = 8;" "6#/%\"sG\"\")" }{TEXT -1 7 " and " }{XPPEDIT 18 0 "j = 6;" "6#/%\"jG\"\"'" }{TEXT -1 9 ", we have" }} {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#/,&*&&%#b*G6#\"\"(\"\"\"&%\"aG6$F)\" \"'F*F**&&F'6#\"\")F*&F,6$F2F.F*F**&&F'6#F.F*,&F*F*&%\"cG6#F.!\"\"F*" }{TEXT -1 2 ". " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 93 "Applying this equation to embedded order 5 scheme gives a n equation which can be solved for " }{XPPEDIT 18 0 "a[7,6];" "6#&%\" aG6$\"\"(\"\"'" }{TEXT -1 1 "." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 97 "subs(j=6,'add'(`b*`[i]*a[i,j ],i=j+1..8)=`b*`[j]*(1-c[j]));\nvalue(%);\nsubs(e7,%);\na[7,6]=solve(% );" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/-%$addG6$*&&%#b*G6#%\"iG\"\"\"& %\"aG6$F+\"\"'F,/F+;\"\"(\"\")*&&F)6#F0F,,&F,F,&%\"cGF7!\"\"F," }} {PARA 11 "" 1 "" {XPPMATH 20 "6#/,&*&&%#b*G6#\"\"(\"\"\"&%\"aG6$F)\"\" 'F*F**&&F'6#\"\")F*&F,6$F2F.F*F**&&F'6#F.F*,&F*F*&%\"cGF7!\"\"F*" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#/,$*&#\"#7\"$n\"\"\"\"&%\"aG6$\"\"(\" \"'F)F)#\"*.(p@(*\",+SGtc\"" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/&%\"aG 6$\"\"(\"\"'#\"*.(p@(*\"++SAE6" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }} {PARA 0 "" 0 "" {TEXT -1 39 "We now complete the construction with " }{XPPEDIT 18 0 "a[7,6]=972169703/1126224000" "6#/&%\"aG6$\"\"(\"\"'*& \"*.(p@(*\"\"\"\"++SAE6!\"\"" }{TEXT -1 2 ". " }}{PARA 0 "" 0 "" {TEXT 260 4 "Note" }{TEXT -1 88 ": See the following subsection for fu rther remarks concerning this choice of value for " }{XPPEDIT 18 0 "a [7,6];" "6#&%\"aG6$\"\"(\"\"'" }{TEXT -1 1 "." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 82 "e8 := `union `(subs(a[7,6]=972169703/1126224000,e7),\{a[7,6]=972169703/1126224000\} ):" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 2 "e8" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1169 " e8 := \{`b*`[3] = 43680951/157136980, a[8,5] = -6116292391/16604298368 , a[8,3] = 26560509/49818496, a[5,4] = 3284078/5138991, a[6,4] = -2600 4300/372178963, c[8] = 1, a[5,2] = -18239/102487, `b*`[1] = 1236443/16 593120, b[4] = 8252237/43524000, `b*`[8] = 0, `b*`[7] = 12/167, a[5,3] = 3243/761332, c[3] = 2/9, b[2] = 0, `b*`[2] = 0, a[6,5] = 9368775900 /30948112231, a[4,2] = -1053/2401, b[6] = 28629151/119448000, b[5] = 1 43496441/1058947600, b[7] = 11/1120, a[6,3] = 98740944/564271331, a[8, 7] = 0, c[7] = 1, a[6,2] = 92664/208537, a[4,1] = 957/9604, c[2] = 7/3 9, c[5] = 23/33, c[6] = 24/31, b[8] = 13/200, a[6,1] = -11597952/14868 6881, a[8,6] = 5233891417/7453555200, a[7,4] = 1592286101/1436292000, \+ a[7,5] = -2279466607/2965053280, a[3,1] = 16/189, c[4] = 3/7, a[4,3] = 1053/1372, a[7,6] = 972169703/1126224000, b[3] = 19683/68432, a[7,3] \+ = 156399/1505504, a[8,4] = 576719677/1357948800, `b*`[6] = 30137260793 /109712988000, b[1] = 14459/198720, `b*`[5] = 1629060147/17684424920, \+ a[2,1] = 7/39, a[8,1] = 118627013/607606272, a[7,1] = 38665819/9180864 0, `b*`[4] = 379485253/1817127000, a[5,1] = 2563741/11068596, a[7,2] = -897/1232, a[8,2] = -1527/3136, a[3,2] = 26/189\}:" }}}{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 0 119 "subs(e8,matrix([seq([c[i],seq(a[i,j],j=1..i-1),``$(9-i)],i=2..8), \n[`b`,seq(b[i],i=1..8)],[`b*`,seq(`b*`[i],i=1..8)]]));" }}{PARA 11 " " 1 "" {XPPMATH 20 "6#K%'matrixG6#7+7+#\"\"(\"#RF(%!GF+F+F+F+F+F+7+#\" \"#\"\"*#\"#;\"$*=#\"#EF2F+F+F+F+F+F+7+#\"\"$F)#\"$d*\"%/'*#!%`5\"%,C# \"%`5\"%s8F+F+F+F+F+7+#\"#B\"#L#\"(TPc#\")'fo5\"#!&R#=\"'([-\"#\"%VK\" 'K8w#\"(ySG$\"(\"**Q^F+F+F+F+7+#\"#C\"#J#!)_zf6\"*\")oo[\"#\"&kE*\"'P& 3##\")W4u)*\"*J8Fk&#!)+V+E\"*j*y@P#\"++fxo$*\",JA6[4$F+F+F+7+\"\"\"#\" )>emQ\")S'3=*#!$(*)\"%K7#\"'*Rc\"\"(/b]\"#\"+,hG#f\"\"++?HO9#!+2mYzA\" +!G`]'H#\"*.(p@(*\"++SAE6F+F+7+F_o#\"*8qi=\"\"*sig2'#!%F:\"%OJ#\")40cE \")'\\=)\\#\"*x'>nd\"++)[zN\"#!+\"R#H;h\",o$)H/m\"#\"+<9*QB&\"++_b`u\" \"!F+7+%\"bG#\"&fW\"\"'?()>Feq#\"&$o>\"&K%o#\"(PAD)\")+S_V#\"*Tk\\V\" \"++w%*e5#\")^\"H'G\"*+![%>\"#\"#6\"%?6#\"#8\"$+#7+%#b*G#\"(VkB\"\")?J f;Feq#\")^4oV\"*!)p8d\"#\"*`_[z$\"++q7<=#\"+Z,1H;\",?\\U%o<#\",$zgs8I \"-+!))Hr4\"#\"#7\"$n\"FeqQ(pprint76\"" }}}{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 31 "note concerning the value for " }{XPPEDIT 18 0 "a[7, 6];" "6#&%\"aG6$\"\"(\"\"'" }{TEXT -1 1 " " }} {PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 46 "e7: coefficients of the \+ combined scheme with " }{XPPEDIT 18 0 "a[7,6]" "6#&%\"aG6$\"\"(\"\"' " }{TEXT -1 16 " as a parameter" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1331 "e7 := \{`b*`[3] = 43680951 /157136980, a[8,1] = 5675675/25316928-1194875/35630036*a[7,6], a[7,1] \+ = 97577/425040+152075/685193*a[7,6], a[5,4] = 3284078/5138991, a[6,4] \+ = -26004300/372178963, c[8] = 1, a[5,2] = -18239/102487, `b*`[1] = 123 6443/16593120, b[4] = 8252237/43524000, `b*`[8] = 0, `b*`[7] = 12/167, a[5,3] = 3243/761332, c[3] = 2/9, b[2] = 0, `b*`[2] = 0, a[6,5] = 936 8775900/30948112231, a[4,2] = -1053/2401, b[6] = 28629151/119448000, b [5] = 143496441/1058947600, b[7] = 11/1120, a[6,3] = 98740944/56427133 1, a[8,7] = 0, c[7] = 1, a[8,5] = -126334216877/215855878784+326128275 0/12978240613*a[7,6], a[6,2] = 92664/208537, a[4,1] = 957/9604, c[2] = 7/39, c[5] = 23/33, c[6] = 24/31, b[8] = 13/200, a[8,6] = -55/364*a[7 ,6]+923521/1109160, a[6,1] = -11597952/148686881, a[3,1] = 16/189, c[4 ] = 3/7, a[4,3] = 1053/1372, a[7,4] = -194747/2127840+16691752/1200577 3*a[7,6], b[3] = 19683/68432, a[8,3] = 2549109/6227312+237562875/16564 09391*a[7,6], a[7,5] = 25026793/37532320-1660289400/998326201*a[7,6], \+ a[7,3] = 217176/235235-17277300/18202301*a[7,6], `b*`[6] = 30137260793 /109712988000, b[1] = 14459/198720, `b*`[5] = 1629060147/17684424920, \+ a[2,1] = 7/39, `b*`[4] = 379485253/1817127000, a[5,1] = 2563741/110685 96, a[7,2] = -897/1232, a[8,2] = -1527/3136, a[8,4] = 54864553/9052992 0-32787370/156075049*a[7,6], a[3,2] = 26/189\}:" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 5 "Let " } {XPPEDIT 18 0 "T[6, 8];" "6#&%\"TG6$\"\"'\"\")" }{TEXT -1 127 " denot e the vector whose components are the principal error terms of the 8 s tage, order 6 scheme (the error terms of order 7)." }}{PARA 0 "" 0 "" {TEXT -1 5 "Let " }{XPPEDIT 18 0 "`T*`[5,8];" "6#&%#T*G6$\"\"&\"\")" }{TEXT -1 145 " denote the vector whose components are the principal \+ error terms of the embedded 8 stage, order 5 scheme (the error terms o f order 6) and let " }{XPPEDIT 18 0 "`T*`[6,8];" "6#&%#T*G6$\"\"'\"\" )" }{TEXT -1 99 " denote the vector whose components are the error te rms of order 7 of the embedded order 5 scheme." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 39 "Denote the 2-norm of thes e vectors by " }{XPPEDIT 18 0 "abs(abs(T[6,8]));" "6#-%$absG6#-F$6#&% \"TG6$\"\"'\"\")" }{TEXT -1 3 ", " }{XPPEDIT 18 0 "abs(abs(`T*`[5,8]) );" "6#-%$absG6#-F$6#&%#T*G6$\"\"&\"\")" }{TEXT -1 7 " and " } {XPPEDIT 18 0 "abs(abs(`T*`[6,8]));" "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*`[6,8]))/abs(abs(`T*`[5,8]));" "6#/&%\"BG6#\"\"(*&-%$absG6 #-F*6#&%#T*G6$\"\"'\"\")\"\"\"-F*6#-F*6#&F/6$\"\"&F2!\"\"" }{TEXT -1 6 ", " }{XPPEDIT 18 0 "C[7] = abs(abs(`T*`[6,8]-T[6,8]))/abs(abs(` T*`[5,8]));" "6#/&%\"CG6#\"\"(*&-%$absG6#-F*6#,&&%#T*G6$\"\"'\"\")\"\" \"&%\"TG6$F2F3!\"\"F4-F*6#-F*6#&F06$\"\"&F3F8" }{TEXT -1 1 "." }} {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 77 "Dormand a nd Prince have suggested that as well as attempting to ensure that " }{XPPEDIT 18 0 "A[7]" "6#&%\"AG6#\"\"(" }{TEXT -1 73 " is a minimum, \+ if the embedded scheme is to be used for error control, " }{XPPEDIT 18 0 "B[7]" "6#&%\"BG6#\"\"(" }{TEXT -1 7 " and " }{XPPEDIT 18 0 "C[ 7]" "6#&%\"CG6#\"\"(" }{TEXT -1 27 " should be chosen so that " }} {PARA 0 "" 0 "" {TEXT -1 68 "they are \"similar in magnitude\" and als o not differ too much from 1." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 471 "errterms6_8 := PrincipalErr orTerms(6,8,'expanded'):\n`errterms5_8*` := subs(b=`b*`,PrincipalError Terms(5,8,'expanded')):\n`errterms6_8*` := subs(b=`b*`,PrincipalErrorT erms(6,8,'expanded')):\nsnmB := sqrt(add(subs(e7,`errterms6_8*`[i])^2, i=1..nops(`errterms6_8*`))):\nsdnB := sqrt(add(subs(e7,`errterms5_8*`[ i])^2,i=1..nops(`errterms5_8*`))):\nsnmC := sqrt(add((subs(e7,`errterm s6_8*`[i])-subs(e7,errterms6_8[i]))^2,i=1..nops(errterms6_8))):\nB_7 : = snmB/sdnB:\nC_7 := snmC/sdnB:" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 115 "'B[7]' = evalf[8](eval(B_7, a[7,6]=972169703/1126224000));\n'C[7]' = evalf[8](eval(C_7,a[7,6]=9721 69703/1126224000));" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/&%\"BG6#\"\"($ \")EKc?!\"(" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/&%\"CG6#\"\"($\")C#zX \"!\"(" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 6 "When " }{XPPEDIT 18 0 "a[7,6]=972169703/1126224000" "6#/&%\"aG6$\"\" (\"\"'*&\"*.(p@(*\"\"\"\"++SAE6!\"\"" }{TEXT -1 11 ", we have " } {XPPEDIT 18 0 "B[7]" "6#&%\"BG6#\"\"(" }{TEXT -1 1 " " }{TEXT 272 1 "~ " }{TEXT -1 14 " 2.056 and " }{XPPEDIT 18 0 "C[7]" "6#&%\"CG6#\"\"( " }{TEXT -1 1 " " }{TEXT 273 1 "~" }{TEXT -1 8 " 1.458. " }}{PARA 0 " " 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 43 "The following pict ure shows the graphs of " }{XPPEDIT 18 0 "B[7]" "6#&%\"BG6#\"\"(" } {TEXT -1 5 " in " }{TEXT 259 5 "brown" }{TEXT -1 6 " and " } {XPPEDIT 18 0 "C[7]" "6#&%\"CG6#\"\"(" }{TEXT -1 5 " in " }{TEXT 261 3 "red" }{TEXT -1 3 ". \n" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 529 "aa := 972169703/1126224000:\nbb := evalf(eval(B_7,a[7,6]=aa)):\nc c := evalf(eval(C_7,a[7,6]=aa)):\np1 := plot([B_7,C_7,1],a[7,6]=-.12.. 1.85,0..2.8,color=[brown,red,blue]):\np2 := plot([[aa,0],[aa,2.9]],col or=COLOR(RGB,0,.7,0)):\np3 := plot([[[aa,bb],[aa,cc]]$3],style=point,s ymbol=[circle,diamond,cross],color=[black,coral$2]):\np4 := plot([[[aa ,0]]$3],style=point,symbol=[circle,diamond,cross],color=black):\np5 := plot([[[aa,bb],[0,bb]],[[aa,cc],[0,cc]]],linestyle=3,color=[brown,red ]):\nplots[display]([p||(1..5)],font=[HELVETICA,9]);" }}{PARA 13 "" 1 "" {GLPLOT2D 528 392 392 {PLOTDATA 2 "61-%'CURVESG6$7`o7$$!3%********* 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K)F*$\"3QT/\")))=-j9F-7$Fiw$\"3)o%=)*QN!*f9F-7$$\"3]'Rs8>sM[)F*$\"3ik> +\"*4%*e9F-7$$\"3gU5lA-uN&)F*$\"3ag]trwJe9F-7$$\"3q)oHRD3!)e)F*$\"39pi bYE*zX\"F-7$F^x$\"3Ax4!4#z#zX\"F-7$$\"3]rU/!3oXp)F*$\"3iZ?`#)*)4e9F-7$ $\"3A3-)[()f)[()F*$\"3k1d9**zZe9F-7$$\"3/Yhrp;:.))F*$\"3OIr'eLL!f9F-7$ Fcx$\"3-^?!***otf9F-7$$\"3_fRAaq-m*)F*$\"3IJ$fRm#\\h9F-7$Fhx$\"3-P@j+@ ej9F-7$Fby$\"3Ct,Cf*o>Z\"F-7$F\\z$\"3kHtMoT(4[\"F-7$Faz$\"3;x&GVRg\"*[ \"F-7$Ffz$\"3#HZ]D4Pf\\\"F-7$F[[l$\"3uPo!3wj9]\"F-7$F`[l$\"3*\\-UhY=m] \"F-7$Fe[l$\"30!H\"o)3N0^\"F-7$Fj[l$\"32tKw[@p^ \"F-7$Fd\\l$\"3'o5>H/*f>:F-7$Fi\\l$\"3&\\G5[c4=_\"F-7$F^]l$\"3RTX$\\l` Q_\"F-7$Fc]l$\"3L*zK " 0 "" {MPLTEXT 1 0 27 "minimize(C_7):\nevalf[8](% );" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#$\")0#zX\"!\"(" }}}{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 40 "ee: coeffic ients for the combined scheme" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1064 "ee := \{c[2]=7/39,c[3]=2/9 ,c[4]=3/7,c[5]=23/33,c[6]=24/31,c[7]=1,c[8]=1,\n\na[2,1]=7/39,\na[3,1] =16/189,\na[3,2]=26/189,\na[4,1]=957/9604,\na[4,2]=-1053/2401,\na[4,3] =1053/1372,\na[5,1]=2563741/11068596,\na[5,2]=-18239/102487,\na[5,3]=3 243/761332,\na[5,4]=3284078/5138991,\na[6,1]=-11597952/148686881,\na[6 ,2]=92664/208537,\na[6,3]=98740944/564271331,\na[6,4]=-26004300/372178 963,\na[6,5]=9368775900/30948112231,\na[7,1]=38665819/91808640,\na[7,2 ]=-897/1232,\na[7,3]=156399/1505504,\na[7,4]=1592286101/1436292000,\na [7,5]=-2279466607/2965053280,\na[7,6]=972169703/1126224000,\na[8,1]=11 8627013/607606272,\na[8,2]=-1527/3136,\na[8,3]=26560509/49818496,\na[8 ,4]=576719677/1357948800,\na[8,5]=-6116292391/16604298368,\na[8,6]=523 3891417/7453555200,\na[8,7]=0,\n\nb[1]=14459/198720,\nb[2]=0,\nb[3]=19 683/68432,\nb[4]=8252237/43524000,\nb[5]=143496441/1058947600,\nb[6]=2 8629151/119448000,\nb[7]=11/1120,\nb[8]=13/200,\n\n`b*`[1]=1236443/165 93120,\n`b*`[2]=0,\n`b*`[3]=43680951/157136980,\n`b*`[4]=379485253/181 7127000,\n`b*`[5]=1629060147/17684424920,\n`b*`[6]=30137260793/1097129 88000,\n`b*`[7]=12/167,\n`b*`[8]=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 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,2\"\"\"F)F'F)*&#F)\"\"#F)*$)F'F,F)F)F)*&#F)\"\"'F)*$) F'\"\"$F)F)F)*&#F)\"#CF)*$)F'\"\"%F)F)F)*&#F)\"$?\"F)*$)F'\"\"&F)F)F)* &#F)\"$?(F)*$)F'F1F)F)F)*&#F)\"%S]F)*$)F'\"\"(F)F)F)" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 124 "We can find the point where the boundary of the stability region intersects the negative re al axis by solving the equation: " }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "R(z) = -1;" "6#/-%\"RG6#%\"zG,$\"\"\"!\"\"" }{TEXT -1 2 ". " }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 27 "z0 := newton(R(z)=- 1,z=-4);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%#z0G$!+J(HT&R!\"*" }}} {PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 306 "z0 := newton(R(z)=-1,z=-4):\np1 := plot([R(z),-1],z=-4.49..0.49,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 ,color=COLOR(RGB,0,.5,0)):\nplots[display]([p1,p2,p3],view=[-4.49..0.4 9,-1.47..1.47],font=[HELVETICA,9]);" }}{PARA 13 "" 1 "" {GLPLOT2D 360 253 253 {PLOTDATA 2 "6+-%'CURVESG6$7U7$$!3A++++++!\\%!#<$!3iK'p5:D&)o# F*7$$!3;+]7'3DdV%F*$!3'yBEJ]7![CF*7$$!35++Ds,X\"Q%F*$!33#HMw%y!=%F*$!3[%)Q[\"\\_ya\"F*7$$!35+]#oviQ2%F*$!3?Vq%yrMWE\"F*7$$! 3A+ve,&\\u'RF*$!3=u.%e0+,T)!#=7$$!3;+ v.8ajmPF*$!33s5ua7+&y'FU7$$!3C+v$Q<')4m$F*$!3'oly[$*y)z`FU7$$!3G+v3`dn bNF*$!3rXMsSCn;UFU7$$!37++0T5NZMF*$!31(G3#p8$)GKFU7$$!3?+]sv&Q>N$F*$!3 lAf+n4Z1DFU7$$!3E++:Qn_WKF*$!3YB],/wpJ=FU7$$!3#****\\s&QnOJF*$!3?]#*f2 #*ew7FU7$$!3&****\\=iPF.$F*$!3udd/!fT_O)!#>7$$!3Q+v$Gc`$QHF*$!3[J')\\M FVE]Fhp7$$!3+++l(y@h#GF*$!3OVJc7l=4dQp\"F*$\"3 %>lkG9%)R#=FU7$$!3)*****p]Q@(e\"F*$\"3pL`rFNZO?FU7$$!3I+]FRT)G[\"F*$\" 3[txl0H&[E#FU7$$!3')*\\P*y(R>Q\"F*$\"3xZAkgb03DFU7$$!3++]Kx#e)p7F*$\"3 %fe2:*pC2GFU7$$!3=++5j![\"p6F*$\"3>kW?L%\\b5$FU7$$!3s***\\KT=;1\"F*$\" 3UL#R=.)feMFU7$$!3'***\\P%o0=k*FU$\"39v_l:!)y7QFU7$$!3](***\\tEbw&)FU$ \"3D$***QX#)[TUFU7$$!39-]P2EBuvFU$\"3sh<4*e0()o%FU7$$!3>)*\\(GL>l_'FU$ \"3#p9(z$>Cm?&FU7$$!3z++]-\")=-bFU$\"3C)eGR&QBodFU7$$!3$*)*\\PM#3)HWFU $\"3O%[Bd&[>@kFU7$$!3;(****f'*ypR$FU$\"3^sjV#z`)>rFU7$$!3L)**\\U'=wSBF U$\"3D'>n%p_,8zFU7$$!3?$*\\Pt2H$H\"FU$\"3+V,>Fw%oy)FU7$$!33R****pit2LF hp$\"3+r)*)36PYn*FU7$$\"3Z^+]FkzBxFhp$\"3QUWS=\"*H!3\"F*7$$\"35*****z2 `!f\"F*7$$\"3l+]i_F06GFU$\"35LwZUIfC8F*7$$\"3\\.]Pt1& z\"QFU$\"3]WLRM=\"\\Y\"F*7$$\"3!***************[FU$\"3Ko[\"G8;Bj\"F*-% 'COLOURG6&%$RGBG$\"*++++\"!\")$\"\"!F\\\\lF[\\l-F$6$7S7$F($!\"\"F\\\\l 7$F3Fa\\l7$F=Fa\\l7$FBFa\\l7$FGFa\\l7$FLFa\\l7$FQFa\\l7$FWFa\\l7$FfnFa \\l7$F[oFa\\l7$F`oFa\\l7$FeoFa\\l7$FjoFa\\l7$F_pFa\\l7$FdpFa\\l7$FjpFa \\l7$F_qFa\\l7$FdqFa\\l7$FjqFa\\l7$F_rFa\\l7$FdrFa\\l7$FirFa\\l7$F^sFa \\l7$FcsFa\\l7$FhsFa\\l7$F]tFa\\l7$FbtFa\\l7$FgtFa\\l7$F\\uFa\\l7$FauF a\\l7$FfuFa\\l7$F[vFa\\l7$F`vFa\\l7$FevFa\\l7$FjvFa\\l7$F_wFa\\l7$FdwF a\\l7$FiwFa\\l7$F^xFa\\l7$FcxFa\\l7$FhxFa\\l7$F]yFa\\l7$FbyFa\\l7$FgyF a\\l7$F\\zFa\\l7$FazFa\\l7$FfzFa\\l7$F[[lFa\\l7$F`[lFa\\l-Fe[l6&Fg[lF[ \\lF[\\lFh[l-F$6&7#7$$!35+++J(HT&RF*Fa\\l-%'SYMBOLG6#%'CIRCLEG-Fe[l6&F g[lF\\\\lF\\\\lF\\\\l-%&STYLEG6#%&POINTG-F$6&Fg_l-F\\`l6#%&CROSSGF_`lF a`l-F$6&Fg_l-F\\`l6#%(DIAMONDGF_`lFa`l-F$6%7$7$Fi_lF[\\lFh_l-%&COLORG6 &Fg[lF[\\l$\"\"&Fb\\lF[\\l-%*LINESTYLEG6#\"\"$-%+AXESLABELSG6%Q\"z6\"Q !F`bl-%%FONTG6#%(DEFAULTG-Fcbl6$%*HELVETICAG\"\"*-%%VIEWG6$;$!$\\%!\"# $\"#\\F`cl;$!$Z\"F`cl$\"$Z\"F`cl" 1 2 0 1 10 0 2 9 1 4 2 1.000000 45.000000 45.000000 0 0 "Curve 1" "Curve 2" "Curve 3" "Curve 4" "Curve 5" "Curve 6" }}}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 50 "The following picture shows the stability region. " }} {PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1284 "R := z -> 1+z+1/2*z^2+1/6*z^3+1/24*z^4+1/120*z^5+1/720*z^6+1/504 0*z^7:\npts := []:\nz0 := 0:\nfor ct from 0 to 200 do\n zz := newton (R(z)=exp(ct*Pi/20*I),z=z0):\n z0 := zz:\n pts := [op(pts),[Re(zz) ,Im(zz)]]:\nend do:\np1 := plot(pts,color=COLOR(RGB,.1,.35,0)):\np2 := plots[polygonplot]([seq([pts[i-1],pts[i],[-2,0]],i=2..nops(pts))],\n \+ style=patchnogrid,color=COLOR(RGB,.2,.85,.15)):\npts := []: z 0 := 1.5+4.3*I:\nfor ct from 0 to 60 do\n zz := newton(R(z)=exp(ct*P i/30*I),z=z0):\n z0 := zz:\n pts := [op(pts),[Re(zz),Im(zz)]]:\nen d do:\np3 := plot(pts,color=COLOR(RGB,.1,.35,0)):\np4 := plots[polygon plot]([seq([pts[i-1],pts[i],[1.36,4.2]],i=2..nops(pts))],\n s tyle=patchnogrid,color=COLOR(RGB,.2,.85,.15)):\npts := []: z0 := 1.5-4 .3*I:\nfor ct from 0 to 60 do\n zz := newton(R(z)=exp(ct*Pi/30*I),z= z0):\n z0 := zz:\n pts := [op(pts),[Re(zz),Im(zz)]]:\nend do:\np5 \+ := 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scheme the stability interval is (approximately) " }{XPPEDIT 18 0 "[-3.9541, 0];" "6#7$,$-%&FloatG6$\"&T&R!\"%!\"\"\"\"!" }{TEXT -1 2 ". " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 260 "We can distort the boundary curve horizontally by taking the 11th root of the real part of points along the curve. In this way we see t hat the largest interval on the nonnegative imaginary axis that contai ns the origin and lies inside the stability region is " }{XPPEDIT 18 0 "[0, 1.76];" "6#7$\"\"!-%&FloatG6$\"$w\"!\"#" }{TEXT -1 18 " approx imately. " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 " " {MPLTEXT 1 0 320 "R := z -> 1+z+1/2*z^2+1/6*z^3+1/24*z^4+1/120*z^5+1 /720*z^6+1/5040*z^7:\nDigits := 25:\npts := []: z0 := 0:\nfor ct from \+ 0 to 60 do\n zz := newton(R(z)=exp(ct*Pi/100*I),z=z0):\n z0 := zz: \n pts := [op(pts),[surd(Re(zz),11),Im(zz)]]:\nend do:\nplot(pts,col or=COLOR(RGB,0,.75,0),thickness=2,font=[HELVETICA,9]);\nDigits := 10: " }}{PARA 13 "" 1 "" {GLPLOT2D 261 276 276 {PLOTDATA 2 "6(-%'CURVESG6# 7in7$$\"\"!F)F(7$$\":(GMFj2VC9eNyI!#E$\":yZvsJz*e`EfTJF-7$$\":=SG%)Gyk c@*o&4&F-$\":w13[GbzrI&=$G'F-7$$\":!)3))4>t?9qG@%oF-$\":!y*ezx3o2'zxC% *F-7$$\":A+QI!=3it^IK%)F-$\":d8Yy#)>M91PmD\"!#D7$$\":P)>V*eS67jI[\"**F -$\":CoYlW5#yEjzq:F?7$$\":obXc>7S_R:;8\"F?$\":'o/^.pz3#fb\\)=F?7$$\":- C)pR\"f*)Q1o_E\"F?$\":sb85*4%\\s&[6*>#F?7$$\":BxXKDBKxyXNR\"F?$\":qcv5 pP(*>7uK^#F?7$$\":\")ya<9'Q3^cB<:F?$\":/E/5]Z7dQLu#GF?7$$\":`#fswJ!y/! >#pj\"F?$\":Sx3Px>1rk#fTJF?7$$\":)3\"orXVLN#=0`vbMF ?7$$\":7=K)y)pe\"3h(f'=F?$\":tc!G!=d_5:6*pPF?7$$\":V-d8&)zz2'[(f(>F?$ \":'HrDDI8b\"QqS3%F?7$$\":3Z1;+9g!y]F$3#F?$\":1=2]nS+GeH#)R%F?7$$\":.* GNchB@wF1)=#F?$\":#\\y2`DfKN()Q7ZF?7$$\":=S.17J!GX;\\!H#F?$\":Qo#Q+x[` 4yaE]F?7$$\":92+c%))*>_O*o!R#F?$\":\"[Y*))e3G'eR&yF?7$$\":#e*[ypTY`WbZ?$F?$\":(= *p@)z#*4u$3\"o\")F?7$$\":*o)*)z_7DE&)=iG$F?$\":$4%3^;1X4#[D#[)F?7$$\": YFq!Q%=snbNfO$F?$\":Lh([eqW06qR'z)F?7$$\":HFMZ`e6An&*QW$F?$\":hpFExG*f yP`5\"*F?7$$\":(or$oWGI)y93?NF?$\":qr]p'=,E.PmC%*F?7$$\":p,g_r>zQ2nWf$ F?$\":)Hu'R!=omo]yQ(*F?7$$\":!\\/.rl\"Ffu7$$\":*or%4d75p 1\"H=TF?$\":crMI=1y(y.6D7Ffu7$$\":u,0qbVG)Ry[tTF?$\":'[y&HZym_e*\\c7Ff u7$$\":.BK_a()=f%4\"fA%F?$\":!38Ffu7$$\":\"4M([@ZU(3%e9K%F?$\":\"y;5zPltw4j]8Ffu7$$\" :24d-,$4jtL&RO%F?$\":$ej\\b=!Q$yE*>Q\"Ffu7$$\":w[$3ox#))\\L!Q-WF?$\":6 qt.Htd\")oWLT\"Ffu7$$\":B'*Qv:g,*\\+>OWF?$\":U[\\Okk`Di&oW9Ffu7$$\":JE MO$yF-*orYY%F?$\":PG)=^MV)e&R,w9Ffu7$$\":2o-:pO$pp?)o[%F?$\":[&fP>#e.P ,Gt]\"Ffu7$$\":(3;:2S))p#GI:]%F?$\":RJ%H6')4omfiQ:Ffu7$$\":Y/@x)ezm#oy n]%F?$\":Czo'QWBTDe!*p:Ffu7$$\":^K#=r^T]'4u)*\\%F?$\":X3cJ\">XGOa;,;Ff u7$$\":.mk\"3u_kVmRwWF?$\":Y\\#HbrabyCSK;Ffu7$$\":Di#*Hae;=/:&GWF?$\": [!p,&*4/!GY9Om\"Ffu7$$\":DgMc=qsT-\\+M%F?$\":Rs&yR5'e3t)z%p\"Ffu7$$\": lMP\"Qh'yWjKn;%F?$\":))4keas\"RdC&fs\"Ffu7$$\":ye43b)4t(3B6j$F?$\":-_9 >CrKJlsqv\"Ffu7$$!:![sV6%4$>0k\"34%F?$\":cRZc'\\X')ph:)y\"Ffu7$$!:y]Oy GQ:9QF)oWF?$\":rq*>>=Ffu7$$!:[XNpEO\"33:R6ZF?$\":lFWI%*[M0#)*>] =Ffu7$$!:l!\\o*4H$e=[)G!\\F?$\":I/Ka`&4tJH:\")=Ffu-%+AXESLABELSG6$Q!6 \"Ff^l-%&COLORG6&%$RGBGF($\"#v!\"#F(-%*THICKNESSG6#\"\"#-%%FONTG6$%*HE LVETICAG\"\"*-%%VIEWG6$%(DEFAULTGF[`l" 1 2 0 1 10 2 2 6 1 4 2 1.000000 45.000000 45.000000 0 0 "Curve 1" }}}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 123 "The relevant intersection poin t of the boundary curve with the imaginary axis can be determined more accurately as follows." }}{PARA 0 "" 0 "" {TEXT -1 86 "First we look \+ for points on the boundary curve either side of the intersection point . " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 112 "Digits := 15:\nz0 := 1.76*I:\nfor ct from 55 to 58 d o\n newton(R(z)=exp(ct*Pi/100*I),z=z0);\nend do;\nDigits := 10:" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#^$$\"02+GdBGd'!#>$\"0t\"RdC&fs\"!#9" } }{PARA 11 "" 1 "" {XPPMATH 20 "6#^$$\"0o0E_SpW\"!#>$\"0rKJlsqv\"!#9" } }{PARA 11 "" 1 "" {XPPMATH 20 "6#^$$!05]am.#p`!#>$\"0bk)ph:)y\"!#9" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#^$$!0-k9J![>9!#=$\"0j>rq*>>=!#9" }}} {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 107 "Then we \+ apply the bisection method to calculate the parameter value associated with the intersection point." }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 178 "real_part := proc(u)\n Re(newton(R(z)=exp(u*Pi*I),z=1.76*I)) \nend proc:\nDigits := 15:\nu0 := bisect('real_part'(u),u=0.55..0.58); \nnewton(R(z)=exp(u0*Pi*I),z=1.76*I);\nDigits := 10:" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%#u0G$\"0')fUmIOi&!#:" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#^#$\"0T`XK@Ww\"!#9" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 " " 0 "" {TEXT -1 4 "The " }{TEXT 260 112 "largest interval on the nonne gative imaginary axis that contains the origin and lies inside the sta bility region" }{TEXT -1 5 " is " }{XPPEDIT 18 0 "[0, 1.7644];" "6#7$ \"\"!-%&FloatG6$\"&Ww\"!\"%" }{TEXT -1 18 " (approximately)." }} {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 37 "#-------- ----------------------------" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 80 "The stability function R* for the 7 stage, or der 5 scheme is given as follows." }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 97 "subs(ee,subs(b=`b*`,StabilityFunction(5,8,'expanded') )):\n`R*` := unapply(%,z):\n'`R*`(z)'=`R*`(z);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/-%#R*G6#%\"zG,2\"\"\"F)F'F)*&#F)\"\"#F)*$)F'F,F)F)F)*& #F)\"\"'F)*$)F'\"\"$F)F)F)*&#F)\"#CF)*$)F'\"\"%F)F)F)*&#F)\"$?\"F)*$)F '\"\"&F)F)F)*&#\"&J!R\"))G,(GF)*$)F'F1F)F)F)*&#\"&LE$\"*Sk]V\"F)*$)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 regi on intersects the negative real axis by solving the equation: " }} {PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "`R*`(z) = -1;" "6#/-% #R*G6#%\"zG,$\"\"\"!\"\"" }{TEXT -1 2 ". " }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 33 "z_0 := newton(`R*`(z)=-1,z=-3.7);" }}{PARA 11 "" 1 " " {XPPMATH 20 "6#>%$z_0G$!+d\"R>t$!\"*" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 322 "z_0 := newton(`R*`(z)= -1,z=-4):\np_1 := plot([`R*`(z),-1],z=-4.29..0.49,color=[red,blue]):\n p_2 := plot([[[z_0,-1]]$3],style=point,symbol=[circle,cross,diamond],c olor=black):\np_3 := plot([[z_0,0],[z_0,-1]],linestyle=3,color=COLOR(R GB,0,.5,0)):\nplots[display]([p_1,p_2,p_3],view=[-4.29..0.49,-1.47..1. 47],font=[HELVETICA,9]);" }}{PARA 13 "" 1 "" {GLPLOT2D 398 284 284 {PLOTDATA 2 "6+-%'CURVESG6$7U7$$!3/++++++!H%!#<$!3'>82po7o%GF*7$$!3;n; a,[!zB%F*$!3I)=-HD`&*f#F*7$$!3QLL3.'4e=%F*$!3GUSim3'3P#F*7$$!3`]iX\\=[ STF*$!3SBMTJh/'=#F*7$$!3ym\"He4a^4%F*$!3(*)p^ygDO,#F*7$$!3EL$el\\-K*RF *$!3UdW1a#\\um\"F*7$$!3QL$3f4v0*QF*$!3%)GY)4Si5P\"F*7$$!3am\"z`YN%)y$F *$!3$*e9w0(e67\"F*7$$!35Le4`*QPp$F*$!3Y8-w\"\\9MC*!#=7$$!3-+Dm*>'o&f$F *$!33#>G%Ru#>^(FU7$$!3ELez,*zU\\$F*$!3:ZC`QVN/gFU7$$!35+D@:))>$R$F*$!3 5**p!>0\"*pu%FU7$$!3amm@$\\C#*G$F*$!3asK)z\\$oqOFU7$$!39L$3Q&Qk(>$F*$! 3Y['*\\e^OxGFU7$$!3a***\\YuXX4$F*$!3hrXO9WGI@FU7$$!3!)***\\ZIC5*HF*$!3 C02%**[#G5:FU7$$!3#)***\\`@i7*GF*$!3M$\\Y7u`X,\"FU7$$!3)H$ez!oo1!GF*$! 3Q'zKVG@cN'!#>7$$!3gmm\"e?WHp#F*$!3'yH<(fHyjDF]q7$$!3HmmEXuo,EF*$\"35: FoAt90;!#?7$$!3#)*\\i$QT`&\\#F*$\"3U%=2eDd?*GF]q7$$!3Tmmwu/d,CF*$\"3sm pQm1**4]F]q7$$!3q*\\iLWx%)H#F*$\"3mcx^^>L*4(F]q7$$!3%)*\\(3)\\3.?#F*$ \"3S<\"=%oW%R$*)F]q7$$!3Im;H-&zy4#F*$\"3;()*y)yTWv5FU7$$!3Dm\"z#4x\"Q+ #F*$\"3b;FCo1HR7FU7$$!3NL$3#e$fB!>F*$\"3j6G)>9$G<9FU7$$!3/LezXE(pz\"F* $\"3!ej\\UL#H3;FU7$$!3\"**\\PG?L_q\"F*$\"3M([6Q1aJy\"FU7$$!38L$e64_hg \"F*$\"3+$o@Jd:V)>FU7$$!3w****p2;z.:F*$\"32VckV$>*3AFU7$$!3p**\\_W=l.9 F*$\"3qaJ#eVD%[CFU7$$!3!)*\\iXZhnI\"F*$\"3^fKrP_y,FFU7$$!3)***\\2Q7=*> \"F*$\"3]5W+e!z:,$FU7$$!3$pmm2h:D5\"F*$\"3qp^E`8x=LFU7$$!3+(***\\2U/$* **FU$\"3'f@^XNb0o$FU7$$!3WJ$eRU(zd!*FU$\"3P:d&=je=/%FU7$$!3O%***\\ecKN !)FU$\"3_A(yJbksZ%FU7$$!3sl;H5%fK2(FU$\"3*=N(G^+`H\\FU7$$!3i(*\\7uHing FU$\"3/ij&)y<5^aFU7$$!3[mm;W&HW3&FU$\"3D6LDfWI9gFU7$$!3u**\\iSr6bSFU$ \"3;Jxg#)RNmmFU7$$!3)\\KL$fpwjIFU$\"3G)=HnW#4htFU7$$!3sGL3:#o*\\?FU$\" 3ILf(4.*\\Y\")FU7$$!3&Hm\"HOUcW5FU$\"3uR\"or]S\"3!*FU7$$!3e)*)***pR-27 F]q$\"3#[n&QMJ-!))*FU7$$\"33'RLeGv9Q*F]q$\"3%pV%pfiN)4\"F*7$$\"31ummCb >&)=FU$\"358X\\!pgu?\"F*7$$\"3[4]P;i%\\*GFU$\"34.t$F*Fa\\l-%'SYMBOLG6#%'CIRCLEG-Fe[l6&Fg[l F\\\\lF\\\\lF\\\\l-%&STYLEG6#%&POINTG-F$6&Fg_l-F\\`l6#%&CROSSGF_`lFa`l -F$6&Fg_l-F\\`l6#%(DIAMONDGF_`lFa`l-F$6%7$7$Fi_lF[\\lFh_l-%&COLORG6&Fg [lF[\\l$\"\"&Fb\\lF[\\l-%*LINESTYLEG6#\"\"$-%+AXESLABELSG6%Q\"z6\"Q!F` bl-%%FONTG6#%(DEFAULTG-Fcbl6$%*HELVETICAG\"\"*-%%VIEWG6$;$!$H%!\"#$\"# \\F`cl;$!$Z\"F`cl$\"$Z\"F`cl" 1 2 0 1 10 0 2 9 1 4 2 1.000000 45.000000 45.000000 0 0 "Curve 1" "Curve 2" "Curve 3" "Curve 4" "Curve 5" "Curve 6" }}}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 82 "The following picture shows the stability region for the \+ 7 stage, order 5 scheme. " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1315 "`R*` := z -> 1+z+1/2*z^2+1/6*z^3+ 1/24*z^4+1/120*z^5+39031/28701288*z^6+32633/143506440*z^7:\npts := []: \nz0 := 0:\nfor ct from 0 to 200 do\n zz := newton(`R*`(z)=exp(ct*Pi /20*I),z=z0):\n z0 := zz:\n pts := [op(pts),[Re(zz),Im(zz)]]:\nend do:\np_1 := plot(pts,color=COLOR(RGB,0,.2,0)):\np_2 := plots[polygonp lot]([seq([pts[i-1],pts[i],[-1.8,0]],i=2..nops(pts))],\n styl e=patchnogrid,color=COLOR(RGB,.2,.65,.1)):\npts := []: z0 := 1.6+4.3*I :\nfor ct from 0 to 60 do\n zz := newton(`R*`(z)=exp(ct*Pi/30*I),z=z 0):\n z0 := zz:\n pts := [op(pts),[Re(zz),Im(zz)]]:\nend do:\np_3 \+ := plot(pts,color=COLOR(RGB,0,.2,0)):\np_4 := plots[polygonplot]([seq( [pts[i-1],pts[i],[1.53,4.23]],i=2..nops(pts))],\n style=patch nogrid,color=COLOR(RGB,.2,.65,.1)):\npts := []: z0 := 1.6-4.3*I:\nfor \+ ct from 0 to 60 do\n zz := newton(`R*`(z)=exp(ct*Pi/30*I),z=z0):\n \+ z0 := zz:\n pts := [op(pts),[Re(zz),Im(zz)]]:\nend do:\np_5 := plot (pts,color=COLOR(RGB,0,.2,0)):\np_6 := plots[polygonplot]([seq([pts[i- 1],pts[i],[1.53,-4.23]],i=2..nops(pts))],\n style=patchnogrid ,color=COLOR(RGB,.2,.65,.1)):\np_7 := plot([[[-4.39,0],[2.19,0]],[[0,- 4.69],[0,4.69]]],color=black,linestyle=3):\nplots[display]([p_||(1..7) ],view=[-4.39..2.19,-4.69..4.69],font=[HELVETICA,9],\n labe ls=[`Re(z)`,`Im(z)`],axes=boxed,scaling=constrained);" }}{PARA 13 "" 1 "" {GLPLOT2D 408 561 561 {PLOTDATA 2 "6/-%'CURVESG6$7ew7$$\"\"!F)F(7 $$!3&******R*3R-T!#F$\"33+++Gjzq:!#=7$$!3#*******Hhb`@!#D$\"31+++;GfTJ F07$$!3+++++&)p)f\"!#C$\"39+++4:R7ZF07$$!3#******H(fuUFF:$\"3K+++p;?$G 'F07$$\"3(******zA[+r\"!#B$\"3'*******3X/ayF07$$\"31+++k%>OQ\"!#A$\"3H +++u0%\\U*F07$$\"3z*****R([&*QcFK$\"3#******p$\\e*4\"!#<7$$\"3))****** 4!\\9o\"!#@$\"31+++#=_mD\"FS7$$\"3=+++Tg*42%FW$\"3!******R!Hi89FS7$$\" 3s*****RnDPO)FW$\"3!******4iN.d\"FS7$$\"33+++ap7$[\"!#?$\"3-+++#G\"\\E $\"3)*******G>9:EFS7$$!3s*****\\I@)fJF`q$\"37+++Y*\\wu#FS7$$!3!)*****p 3l-e&F`q$\"3!*******f_HtGFS7$$!3&******H49l#))F`q$\"3=+++d*G3*HFS7$$!3 3+++FE2)G\"F0$\"3!)*****\\[U*)4$FS7$$!35+++scSk$FS7$$! 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cnF\\\\al7%F`ial7$Fc[_l$!+%H4AM%F`cnF\\\\al7%Fdial7$F][_l$!+TO`OVF`cnF \\\\al7%Fhial7$Fgj^l$!+![]+L%F`cnF\\\\al7%F\\jal7$Faj^l$!+B1!GK%F`cnF \\\\al7%F`jal7$Ffi^l$!+&)\\#[J%F`cnF\\\\al7%FdjalFf[alF\\\\alFhe]lFh_r -%*AXESSTYLEG6#%$BOXG-%+AXESLABELSG6%%&Re(z)G%&Im(z)G-%%FONTG6#%(DEFAU LTG-%(SCALINGG6#%,CONSTRAINEDG-Fb[bl6$%*HELVETICAG\"\"*-%%VIEWG6$;$!$ \\%Fe_r$\"$>#Fe_r;$!$p%Fe_r$\"$p%Fe_r" 1 2 0 1 10 0 2 9 1 2 1 1.000000 45.000000 45.000000 0 0 "Curve 1" "Curve 2" "Curve 3" "Curve \+ 4" "Curve 5" "Curve 6" "Curve 7" "Curve 8" "Curve 9" "Curve 10" "Curve 11" "Curve 12" "Curve 13" "Curve 14" }}}}{PARA 0 "" 0 "" {TEXT -1 0 " " }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}}{PARA 0 "" 0 "" {TEXT -1 52 "#---------------------------------------------------" }} {SECT 1 {PARA 4 "" 0 "" {TEXT -1 36 "coefficients for the combined sch eme" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 40 "ee: coeffic ients for the combined scheme" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1064 "ee := \{c[2]=7/39,c[3]=2/9 ,c[4]=3/7,c[5]=23/33,c[6]=24/31,c[7]=1,c[8]=1,\n\na[2,1]=7/39,\na[3,1] =16/189,\na[3,2]=26/189,\na[4,1]=957/9604,\na[4,2]=-1053/2401,\na[4,3] =1053/1372,\na[5,1]=2563741/11068596,\na[5,2]=-18239/102487,\na[5,3]=3 243/761332,\na[5,4]=3284078/5138991,\na[6,1]=-11597952/148686881,\na[6 ,2]=92664/208537,\na[6,3]=98740944/564271331,\na[6,4]=-26004300/372178 963,\na[6,5]=9368775900/30948112231,\na[7,1]=38665819/91808640,\na[7,2 ]=-897/1232,\na[7,3]=156399/1505504,\na[7,4]=1592286101/1436292000,\na [7,5]=-2279466607/2965053280,\na[7,6]=972169703/1126224000,\na[8,1]=11 8627013/607606272,\na[8,2]=-1527/3136,\na[8,3]=26560509/49818496,\na[8 ,4]=576719677/1357948800,\na[8,5]=-6116292391/16604298368,\na[8,6]=523 3891417/7453555200,\na[8,7]=0,\n\nb[1]=14459/198720,\nb[2]=0,\nb[3]=19 683/68432,\nb[4]=8252237/43524000,\nb[5]=143496441/1058947600,\nb[6]=2 8629151/119448000,\nb[7]=11/1120,\nb[8]=13/200,\n\n`b*`[1]=1236443/165 93120,\n`b*`[2]=0,\n`b*`[3]=43680951/157136980,\n`b*`[4]=379485253/181 7127000,\n`b*`[5]=1629060147/17684424920,\n`b*`[6]=30137260793/1097129 88000,\n`b*`[7]=12/167,\n`b*`[8]=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 259 5 "nodes" }{TEXT -1 2 ": " } }{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 31 "seq(c[i]=subs(ee,c[i]),i=2. .8);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6)/&%\"cG6#\"\"##\"\"(\"#R/&F%6# \"\"$#F'\"\"*/&F%6#\"\"%#F.F)/&F%6#\"\"&#\"#B\"#L/&F%6#\"\"'#\"#C\"#J/ &F%6#F)\"\"\"/&F%6#\"\")FG" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT 259 20 "linking coefficients" }{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..8);" }}{PARA 12 "" 1 "" {XPPMATH 20 "6>/&%\"aG6$\"\" #\"\"\"#\"\"(\"#R/&F%6$\"\"$F(#\"#;\"$*=/&F%6$F/F'#\"#EF2/&F%6$\"\"%F( #\"$d*\"%/'*/&F%6$F;F'#!%`5\"%,C/&F%6$F;F/#\"%`5\"%s8/&F%6$\"\"&F(#\"( TPc#\")'fo5\"/&F%6$FNF'#!&R#=\"'([-\"/&F%6$FNF/#\"%VK\"'K8w/&F%6$FNF;# \"(ySG$\"(\"**Q^/&F%6$\"\"'F(#!)_zf6\"*\")oo[\"/&F%6$FaoF'#\"&kE*\"'P& 3#/&F%6$FaoF/#\")W4u)*\"*J8Fk&/&F%6$FaoF;#!)+V+E\"*j*y@P/&F%6$FaoFN#\" ++fxo$*\",JA6[4$/&F%6$F*F(#\")>emQ\")S'3=*/&F%6$F*F'#!$(*)\"%K7/&F%6$F *F/#\"'*Rc\"\"(/b]\"/&F%6$F*F;#\"+,hG#f\"\"++?HO9/&F%6$F*FN#!+2mYzA\"+ !G`]'H/&F%6$F*Fao#\"*.(p@(*\"++SAE6/&F%6$\"\")F(#\"*8qi=\"\"*sig2'/&F% 6$FdsF'#!%F:\"%OJ/&F%6$FdsF/#\")40cE\")'\\=)\\/&F%6$FdsF;#\"*x'>nd\"++ )[zN\"/&F%6$FdsFN#!+\"R#H;h\",o$)H/m\"/&F%6$FdsFao#\"+<9*QB&\"++_b`u/& F%6$FdsF*\"\"!" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT 259 30 "weights for the 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#\"\"\"#\"&fW\"\"'?()>/ &F%6#\"\"#\"\"!/&F%6#\"\"$#\"&$o>\"&K%o/&F%6#\"\"%#\"(PAD)\")+S_V/&F%6 #\"\"&#\"*Tk\\V\"\"++w%*e5/&F%6#\"\"'#\")^\"H'G\"*+![%>\"/&F%6#\"\"(# \"#6\"%?6/&F%6#\"\")#\"#8\"$+#" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }} {PARA 0 "" 0 "" {TEXT 259 30 "weights for the order 5 scheme" }{TEXT -1 2 ": " }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 37 "seq(`b*`[i]=subs (ee,`b*`[i]),i=1..8);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6*/&%#b*G6#\"\" \"#\"(VkB\"\")?Jf;/&F%6#\"\"#\"\"!/&F%6#\"\"$#\")^4oV\"*!)p8d\"/&F%6# \"\"%#\"*`_[z$\"++q7<=/&F%6#\"\"&#\"+Z,1H;\",?\\U%o " 0 "" {MPLTEXT 1 0 1 ";" }}}} {PARA 0 "" 0 "" {TEXT -1 52 "#---------------------------------------- -----------" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 " " {MPLTEXT 1 0 1 ";" }}}}{PARA 0 "" 0 "" {TEXT -1 42 "#=============== ==========================" }}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 25 "Ver ner's (2004) scheme " }}{PARA 257 "" 0 "" {TEXT -1 0 "" }}{PARA 0 " " 0 "" {TEXT -1 60 "See: A Contrast of a New RK56 pair with DP56, by J im Verner," }}{PARA 0 "" 0 "" {TEXT -1 83 " Department of Mathe matics. Simon Fraser University, Burnaby, Canada, 2006." }}{PARA 0 " " 0 "" {TEXT -1 0 "" }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "matrix([[1/7, 1/7, ``, ``, ``, ``, ``, ``, ``], [2/9, 4/81, 14/81, \+ ``, ``, ``, ``, ``, ``], [3/7, 291/1372, -27/49, 1053/1372, ``, ``, `` , ``, ``], [2/3, 86/297, -14/33, 42/143, 1960/3861, ``, ``, ``, ``], [ 3/4, -267/22528, 189/704, 63099/585728, 58653/366080, 4617/20480, ``, \+ ``, ``], [1, 10949/6912, -69/32, -90891/68096, 112931/25920, -69861/17 920, 26378/10773, ``, ``], [1, 1501/19008, -21/88, 219519/347776, 1638 07/926640, -417/640, 1544/1539, 0, ``], [b, 79/1080, 0, 19683/69160, 1 6807/84240, 0, 2816/7695, 1/100, 187/2800], [`b*`, 763/10800, 0, 59049 /197600, 88837/526500, 243/4000, 12352/38475, 0, 2/25]]);" "6#-%'matri xG6#7+7+*&\"\"\"F)\"\"(!\"\"*&F)F)F*F+%!GF-F-F-F-F-F-7+*&\"\"#F)\"\"*F +*&\"\"%F)\"#\")F+*&\"#9F)F4F+F-F-F-F-F-F-7+*&\"\"$F)F*F+*&\"$\"HF)\"% s8F+,$*&\"#FF)\"#\\F+F+*&\"%`5F)FF)\"%hQF+F-F-F-F-7+*&F9F)F3 F+,$*&\"$n#F)\"&GD#F+F+*&\"$*=F)\"$/(F+*&\"&*4jF)\"'GdeF+*&\"&`'eF)\"' !3m$F+*&\"%F+,$*&\"#@F)\"#))F+F+*&\"'>&>#F) \"'wxMF+*&\"'2Q;F)\"'Sm#*F+,$*&\"$<%F)\"$S'F+F+*&\"%W:F)\"%R:F+\"\"!F- 7+%\"bG*&\"#zF)\"%!3\"F+Fhq*&\"&$o>F)\"&g\"pF+*&\"&2o\"F)\"&SU)F+Fhq*& \"%;GF)\"%&p(F+*&F)F)\"$+\"F+*&\"$(=F)\"%+GF+7+%#b*G*&\"$j(F)\"&+3\"F+ Fhq*&\"&\\!fF)\"'+w>F+*&\"&P)))F)\"'+l_F+*&\"$V#F)\"%+SF+*&\"&_B\"F)\" &v%QF+Fhq*&F0F)\"#DF+" }{TEXT -1 1 " " }}{PARA 0 "" 0 "" {TEXT -1 0 " " }}{PARA 0 "" 0 "" {TEXT -1 132 " The last-but-one row gives the weig hts for the order 6 scheme while the last row gives the weights for th e embedded order 5 scheme." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 52 "#------------------------------------------------ ---" }}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 28 "checking the combined sche me" }}{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 0 839 "# coefficients by Jim Verner 2004 \nee := \{c[2]=1/7,c[3]=2/9,c[4]=3/7,c[5]=2/3,c[6]=3/4,c[7]=1,c[8]=1, \n\na[2,1]=1/7,\na[3,1]=4/81,\na[3,2]=14/81,\na[4,1]=291/1372,\na[4,2] =-27/49,\na[4,3]=1053/1372,\na[5,1]=86/297,\na[5,2]=-14/33,\na[5,3]=42 /143,\na[5,4]=1960/3861,\na[6,1]=-267/22528,\na[6,2]=189/704,\na[6,3]= 63099/585728,\na[6,4]=58653/366080,\na[6,5]=4617/20480,\na[7,1]=10949/ 6912,\na[7,2]=-69/32,\na[7,3]=-90891/68096,\na[7,4]=112931/25920,\na[7 ,5]=-69861/17920,\na[7,6]=26378/10773,\na[8,1]=1501/19008,\na[8,2]=-21 /88,\na[8,3]=219519/347776,\na[8,4]=163807/926640,\na[8,5]=-417/640,\n a[8,6]=1544/1539,\na[8,7]=0,\n\nb[1]=79/1080,\nb[2]=0,\nb[3]=19683/691 60,\nb[4]=16807/84240,\nb[5]=0,\nb[6]=2816/7695,\nb[7]=1/100,\nb[8]=18 7/2800,\n\n`b*`[1]=763/10800,\n`b*`[2]=0,\n`b*`[3]=59049/197600,\n`b*` [4]=88837/526500,\n`b*`[5]=243/4000,\n`b*`[6]=12352/38475,\n`b*`[7]=0, \n`b*`[8]=2/25\}:" }}}{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 0 155 "RK6_8eqs := [op(RowSumCondi tions(8,'expanded')),op(OrderConditions(6,8,'expanded'))]:\n`RK5_8eqs* ` := subs(b=`b*`,OrderConditions(5,8,'expanded')):\nnops(%);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#\"#<" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }} {PARA 0 "" 0 "" {TEXT -1 7 "Check: " }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 128 "simplify(subs(ee,RK6_8eqs)):\nmap(u->lhs(u)-rhs(u),% );\nnops(%);\nsimplify(subs(ee,`RK5_8eqs*`)):\nmap(u->lhs(u)-rhs(u),%) ;\nnops(%);" }}{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 3 to 8 have stage-order 2. " }}{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$F$" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 114 "We can calculate th e principal error norm of the order 6 scheme, that is, the 2-norm of t he 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(errterms6_8)))));" } }{PARA 11 "" 1 "" {XPPMATH 20 "6#$\"+-!)))Q@!#8" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 79 "The 2 norm of the princip al error of the order 5 embedded scheme is as follows." }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 153 "`errterms5_8*` := subs(b=`b*`,Prin cipalErrorTerms(5,8,'expanded')):\nevalf(evalf[14](sqrt(add(subs(ee,`e rrterms5_8*`[i])^2,i=1.. nops(`errterms5_8*`)))));" }}{PARA 11 "" 1 " " {XPPMATH 20 "6#$\"+6'[/^\"!#8" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }} {PARA 0 "" 0 "" {TEXT -1 53 "Five of the principal error conditions ar e satisfied." }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 291 "ind := []: \nfor ct to nops(errterms6_8) do\n if simplify(subs(ee,errterms6_8[c t]))=0 then ind := [op(ind),ct] end if:\nend do:\nerr6 := PrincipalErr orConditions(6):\n[seq([i,err6[i]],i=ind)]:\nlinalg[augment](linalg[de lcols](%,2..2),matrix([[` `]$(linalg[rowdim](%))]),linalg[delcols](%, 1..1));" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#K%'matrixG6#7'7%\"\"\"%#~~G /*&%\"bGF(-%!G6#*&%\"aGF(-F.6#*&F1F(-F.6#*&F1F(-F.6#*&F1F(-F.6#*&F1F(% \"cGF(F(F(F(F(F(#F(\"%S]7%\"\"&F)/*(F,F(F>F(F2F(#F(\"$S)7%\"#F(-F.6#*&F1F(-F.6#*&)F>\"\"#F(F1F(F(F(F(#F(\"$/&7%\"#AF) /*(F,F(F;F(FNF(#F(\"$o\"7%\"#JF)/*(F,F(FTF(FNF(#F(\"#%)Q*pprint1416\" " }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}}{PARA 0 "" 0 "" {TEXT -1 52 "#----------------- ----------------------------------" }}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 35 "construction of the combined scheme" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 196 "First we construct the 8 stage order 6 scheme by making use of a selection of \"simple\" order condi tions and incorporating the stage-order equations to ensure that stage s 3 to 8 have stage-order 2." }}{PARA 0 "" 0 "" {TEXT -1 48 "We also i ncorporate 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 = 4;" "6#/%\"jG\"\"%" }{TEXT -1 18 " . . . 7. ( with " }{XPPEDIT 18 0 "c[1]=0" "6#/&%\"cG6#\"\"\"\"\" !" }{TEXT -1 55 " ), together with the further simplifying conditions : " }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "Sum(b[i]*c[i]*a [i,2],i=3..8)=0" "6#/-%$SumG6$*(&%\"bG6#%\"iG\"\"\"&%\"cG6#F+F,&%\"aG6 $F+\"\"#F,/F+;\"\"$\"\")\"\"!" }{TEXT -1 8 ", " }{XPPEDIT 18 0 " Sum(b[i]*c[i]*Sum(a[i,j]*a[j,2],j=3..i-1),i=3..8)=0" "6#/-%$SumG6$*(&% \"bG6#%\"iG\"\"\"&%\"cG6#F+F,-F%6$*&&%\"aG6$F+%\"jGF,&F46$F6\"\"#F,/F6 ;\"\"$,&F+F,F,!\"\"F,/F+;F<\"\")\"\"!" }{TEXT -1 8 ", " } {XPPEDIT 18 0 "Sum(b[i]*c[i]^2*a[i,2],i=3..8)=0" "6#/-%$SumG6$*(&%\"bG 6#%\"iG\"\"\"*$&%\"cG6#F+\"\"#F,&%\"aG6$F+F1F,/F+;\"\"$\"\")\"\"!" } {TEXT -1 2 ". " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 32 "The scheme is constructed with " }{XPPEDIT 18 0 "a[7,3]; " "6#&%\"aG6$\"\"(\"\"$" }{TEXT -1 3 ", " }{XPPEDIT 18 0 "a[7,4]" "6# &%\"aG6$\"\"(\"\"%" }{TEXT -1 3 ", " }{XPPEDIT 18 0 "a[7,5]" "6#&%\"a G6$\"\"(\"\"&" }{TEXT -1 7 " and " }{XPPEDIT 18 0 "a[7,6]" "6#&%\"aG 6$\"\"(\"\"'" }{TEXT -1 17 " as parameters. " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 76 "The simple order conditio ns used are given (in abreviated form) as follows. " }}{PARA 0 "" 0 " " {TEXT -1 94 "These include the 6 order 6 quadrature conditions toget her with 4 additional order conditions." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 184 "SO6 := SimpleOrderCond itions(6):\n[seq([i,SO6[i]],i=[1,2,4,8,13,16,24,28,29,32])]:\nlinalg[a ugment](linalg[delcols](%,2..2),matrix([[` `]$(linalg[rowdim](%))]),l inalg[delcols](%,1..1));" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#K%'matrixG 6#7,7%\"\"\"%#~~G/*&%\"bGF(%\"eGF(F(7%\"\"#F)/*&F,F(%\"cGF(#F(F/7%\"\" %F)/*&F,F()F2F/F(#F(\"\"$7%\"\")F)/*&F,F()F2F:F(#F(F57%\"#8F)/*(F,F(F2 F(-%!G6#*&F8F(%\"aGF(F(#F(\"#:7%\"#;F)/*&F,F()F2F5F(#F(\"\"&7%\"#CF)/* (F,F(F2F(-FF6#*&FIF(FEF(F(#F(\"#s7%\"#GF)/*(F,F(F8F(FEF(#F(\"#=7%\"#HF )/*(F,F(F2F(-FF6#*&F?F(FIF(F(#F(FT7%\"#KF)/*&F,F()F2FRF(#F(\"\"'Q)ppri nt816\"" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 513 "SO_eqs := [op(RowSumConditions(8,'expanded')),op(Sta geOrderConditions(2,8,'expanded'))]:\nSO6_8 := SimpleOrderConditions(6 ,8,'expanded'):\nord_cdns := [seq(SO6_8[i],i=[1,2,4,8,13,16,24,28,29,3 2])]:\nsimp_cdns := [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=[4,5,6,7])]:\nsimp_cdns2 := [add(b[i]*c[i] *a[i,2],i=3..8)=0,add(b[i]*c[i]*add(a[i,j]*a[j,2],j=3..i-1),i=3..8)=0, \n add(b[i]*c[i]^2*a[i,2],i=3..8)=0]:\ncdns := [op(ord_ cdns),op(simp_cdns),op(simp_cdns2),op(SO_eqs)]:" }}}{EXCHG {PARA 0 "> \+ " 0 "" {MPLTEXT 1 0 1 ";" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 " " 0 "" {TEXT -1 21 "We specify the nodes:" }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "c[2] = 1/7;" "6#/&%\"cG6#\"\"#*&\"\"\"F)\"\"( !\"\"" }{TEXT -1 3 ", " }{XPPEDIT 18 0 "c[3]=2/9" "6#/&%\"cG6#\"\"$*& \"\"#\"\"\"\"\"*!\"\"" }{TEXT -1 3 ", " }{XPPEDIT 18 0 "c[5] = 2/3;" "6#/&%\"cG6#\"\"&*&\"\"#\"\"\"\"\"$!\"\"" }{TEXT -1 3 ", " }{XPPEDIT 18 0 "c[6] = 3/4;" "6#/&%\"cG6#\"\"'*&\"\"$\"\"\"\"\"%!\"\"" }{TEXT -1 3 ", " }{XPPEDIT 18 0 "c[7]=1" "6#/&%\"cG6#\"\"(\"\"\"" }{TEXT -1 3 ", " }{XPPEDIT 18 0 "c[8]=1" "6#/&%\"cG6#\"\")\"\"\"" }{TEXT -1 2 " , " }}{PARA 0 "" 0 "" {TEXT -1 17 "and the weights: " }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "b[2]=0" "6#/&%\"bG6#\"\"#\"\"!" } {TEXT -1 3 ", " }{XPPEDIT 18 0 "b[7] = 1/100;" "6#/&%\"bG6#\"\"(*&\" \"\"F)\"$+\"!\"\"" }{TEXT -1 2 ". " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 170 "e1 := \{c[2]=1/7,c[3]=2/9,c [5]=2/3,c[6]=3/4,c[7]=1,c[8]=1,b[2]=0,b[7]=1/100\}:\neqns := subs(e1,c dns):\nnops(eqns);\nindets(eqns) minus \{a[7,3],a[7,4],a[7,5],a[7,6]\} ;\nnops(%);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#\"#J" }}{PARA 11 "" 1 " " {XPPMATH 20 "6# " 0 "" {MPLTEXT 1 0 22 "infolevel[solve] := 4:" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 118 "e2 := solve(\{op(eqns)\},indets(eqns) mi nus \{a[7,3],a[7,4],a[7,5],a[7,6]\}):\ninfolevel[solve] := 0:\ne3 := ` union`(e1,e2):" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> \+ " 0 "" {MPLTEXT 1 0 1 ";" }}}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 3 "e3 " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 913 "e3 := \{a[8,4] = 9604/11583-28/187*a[7,4], a[5,1] = 86/297, a[8 ,3] = 7245/16796-28/187*a[7,3], a[3,1] = 4/81, b[5] = 0, a[5,4] = 1960 /3861, b[1] = 79/1080, a[6,5] = 4617/20480, a[6,3] = 63099/585728, a[6 ,2] = 189/704, a[4,1] = 291/1372, a[6,4] = 58653/366080, a[8,2] = -203 /187+392/1683*a[7,3]+84/187*a[7,4]+392/561*a[7,5]+147/187*a[7,6], a[3, 2] = 14/81, c[6] = 3/4, c[5] = 2/3, b[4] = 16807/84240, c[4] = 3/7, c[ 2] = 1/7, b[7] = 1/100, c[3] = 2/9, c[7] = 1, a[8,7] = 0, a[4,3] = 105 3/1372, a[7,2] = -14/9*a[7,3]-3*a[7,4]-14/3*a[7,5]-21/4*a[7,6]+7/2, a[ 2,1] = 1/7, a[5,2] = -14/33, a[4,2] = -27/49, b[8] = 187/2800, b[3] = \+ 19683/69160, b[6] = 2816/7695, a[5,3] = 42/143, a[8,6] = 35840/26163-2 8/187*a[7,6], a[8,1] = 13945/20196-140/1683*a[7,3]-56/187*a[7,4]-28/51 *a[7,5]-7/11*a[7,6], a[6,1] = -267/22528, b[2] = 0, a[8,5] = -21/17-28 /187*a[7,5], c[8] = 1, a[7,1] = 5/9*a[7,3]+2*a[7,4]+11/3*a[7,5]+17/4*a [7,6]-5/2\}:" }{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 0 452 "subs(e3,matrix([s eq([c[i],seq(a[i,j],j=1..i-1),``$(6-i)],i=2..6)]));\nprint(``);\nfor i i from 7 to 8 do\n print(c[ii]=subs(e3,c[ii]));print(``); \n for j j to ii-1 do\n coeff_val := subs(e3,a[ii,jj]);\n if coeff_va l=a[ii,jj] then \n print(a[ii,jj]*` is a parameter.`)\n e lse\n print(a[ii,jj]=coeff_val);\n end if;\n end do:\n \+ print(`__________________________________`):\nend do:\nprint(``);\ns eq(b[i]=subs(e3,b[i]),i=1..8);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#K%'m atrixG6#7'7(#\"\"\"\"\"(F(%!GF+F+F+7(#\"\"#\"\"*#\"\"%\"#\")#\"#9F2F+F +F+7(#\"\"$F*#\"$\"H\"%s8#!#F\"#\\#\"%`5F:F+F+7(#F.F7#\"#')\"$(H#!#9\" #L#\"#U\"$V\"#\"%g>\"%hQF+7(#F7F1#!$n#\"&GD##\"$*=\"$/(#\"&*4j\"'Gde# \"&`'e\"'!3m$#\"%?F(*&#\"$S\"\"%$o \"F(&F%6$\"\"(\"\"$F(!\"\"*&#\"#c\"$(=F(&F%6$F3\"\"%F(F5*&#\"#G\"#^F(& F%6$F3\"\"&F(F5*&#F3\"#6F(&F%6$F3\"\"'F(F5" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/&%\"aG6$\"\")\"\"#,,#\"$.#\"$(=!\"\"*&#\"$#R\"%$o\"\" \"\"&F%6$\"\"(\"\"$F2F2*&#\"#%)F,F2&F%6$F5\"\"%F2F2*&#F0\"$h&F2&F%6$F5 \"\"&F2F2*&#\"$Z\"F,F2&F%6$F5\"\"'F2F2" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/&%\"aG6$\"\")\"\"$,&#\"%Xs\"&'z;\"\"\"*&#\"#G\"$(=F-&F%6$\"\"(F (F-!\"\"" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/&%\"aG6$\"\")\"\"%,&#\"%/ '*\"&$e6\"\"\"*&#\"#G\"$(=F-&F%6$\"\"(F(F-!\"\"" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/&%\"aG6$\"\")\"\"&,&#\"#@\"#\"&g\"p/&F%6#\"\"%#\"& 2o\"\"&SU)/&F%6#\"\"&F//&F%6#\"\"'#\"%;G\"%&p(/&F%6#\"\"(#F'\"$+\"/&F% 6#\"\")#\"$(=\"%+G" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}} {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 66 "We now tu rn our attention to the embedded 8 stage, order 5 scheme." }}{PARA 0 " " 0 "" {TEXT -1 35 "We use the simplifying conditions: " }}{PARA 256 " " 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "Sum(`b*`[i]*a[i,2],i = 3 .. 8) = 0;" "6#/-%$SumG6$*&&%#b*G6#%\"iG\"\"\"&%\"aG6$F+\"\"#F,/F+;\"\"$\"\") \"\"!" }{TEXT -1 8 ", " }{XPPEDIT 18 0 "Sum(`b*`[i]*Sum(a[i,j]*a [j,2],j = 3 .. i-1),i = 2 .. 8) = 0;" "6#/-%$SumG6$*&&%#b*G6#%\"iG\"\" \"-F%6$*&&%\"aG6$F+%\"jGF,&F16$F3\"\"#F,/F3;\"\"$,&F+F,F,!\"\"F,/F+;F6 \"\")\"\"!" }{TEXT -1 1 "." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 101 "We use a selection of \"simple\" order condition s as given (in abreviated form) in the following table." }}{PARA 0 "" 0 "" {TEXT -1 95 "These include the 5 order 5 quadrature conditions to gether with one additional order condition." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 191 "`SO5*` := subs(b= `b*`,SimpleOrderConditions(5)):\n[seq([i,`SO5*`[i]],i=[1,2,4,8,15,16]) ]:\nlinalg[augment](linalg[delcols](%,2..2),matrix([[` `]$(linalg[row dim](%))]),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%\"# :F)/*&F,F(-%!G6#*&F?F(%\"aGF(F(#F(\"#?7%\"#;F)/*&F,F()F2F5F(#F(\"\"&Q) pprint866\"" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 262 "`SO5_8*` := subs(b=`b*`,SimpleOrderConditions(5,8 ,'expanded')):\n`ord_cdns*` := [seq(`SO5_8*`[i],i=[1,2,4,8,15,16])]:\n `simp_cdns*` := [add(`b*`[i]*a[i,2],i=3..8)=0,add(`b*`[i]*add(a[i,j]*a [j,2],j=3..i-1),i=2..8)=0]:\n`cdns*` := [op(`ord_cdns*`),op(`simp_cdns *`)]: " }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 17 "We specify that " }{XPPEDIT 18 0 "`b*`[2] = 0;" "6#/&%#b*G6#\"\"# \"\"!" }{TEXT -1 3 ", " }{XPPEDIT 18 0 "`b*`[7] = 0;" "6#/&%#b*G6#\" \"(\"\"!" }{TEXT -1 7 " and " }{XPPEDIT 18 0 "`b*`[8] = 2/25;" "6#/& %#b*G6#\"\")*&\"\"#\"\"\"\"#D!\"\"" }{TEXT -1 2 ". " }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 63 "e4 := \{`b*`[2]=0,`b*`[7]=0,`b*`[8]=2/25\}: \ne5 := `union`(e3,e4):" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 84 "`eqns*` := subs(e5,`cdns*`):\nnops( `eqns*`);\nindets(`eqns*`) minus \{a[7,6]\};\nnops(%);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#\"\")" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#<*&%#b*G6 #\"\"\"&F%6#\"\"$&F%6#\"\"%&F%6#\"\"&&F%6#\"\"'&%\"aG6$\"\"(F0&F56$F7F -&F56$F7F*" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#\"\")" }}}{PARA 0 "" 0 " " {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 22 "infolevel[ solve] := 4:" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 112 "e6 := solv e(\{op(`eqns*`)\},indets(`eqns*`) minus \{a[7,6]\}):\ne7 := `union`(su bs(e6,e5),e6):\ninfolevel[solve] := 0:" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 39 "The \+ order 5 scheme is given as follows." }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 94 "subs(e7,matrix([seq([c[i],seq(a[i,j],j=1..i-1),``$(9- i)],i=2..8),\n[``,seq(`b*`[i],i=1..8)]]));" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#K%'matrixG6#7*7+#\"\"\"\"\"(F(%!GF+F+F+F+F+F+7+#\"\"#\" \"*#\"\"%\"#\")#\"#9F2F+F+F+F+F+F+7+#\"\"$F*#\"$\"H\"%s8#!#F\"#\\#\"%` 5F:F+F+F+F+F+7+#F.F7#\"#')\"$(H#!#9\"#L#\"#U\"$V\"#\"%g>\"%hQF+F+F+F+7 +#F7F1#!$n#\"&GD##\"$*=\"$/(#\"&*4j\"'Gde#\"&`'e\"'!3m$#\"%\"F)FboF)Fap#!#@\"#)),&#\"(0_f\"\"(#>7fF)*&#\"&Ff%\"'o6 JF)FboF)F),&#\")tB$=\"\")!)Gv:F)*&#\"&>c%\"'![%>F)FboF)Fap,&#\"&$G9\"& !)3\"Fap*&#\"&>B$\"'!o>\"F)FboF)F),&#\"&Se$\"&jh#F)*&#\"#G\"$(=F)FboF) Fap\"\"!F+7+F+#\"$j(\"&+3\"F]t#\"&\\!f\"'+w>#\"&P)))\"'+l_#\"$V#\"%+S# \"&_B\"\"&v%QF]t#F.\"#DQ)pprint876\"" }}}{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 1052 "e7 := \{a[7,5] = 9273/17920-4617/2560*a[7 ,6], a[7,4] = 6503/12480+6517/4160*a[7,6], a[5,1] = 86/297, a[7,3] = 5 0265/46592-6561/6656*a[7,6], `b*`[2] = 0, a[3,1] = 4/81, b[5] = 0, a[5 ,4] = 1960/3861, b[1] = 79/1080, a[6,5] = 4617/20480, a[7,2] = -69/32, a[8,2] = -21/88, `b*`[3] = 59049/197600, `b*`[4] = 88837/526500, `b*` [5] = 243/4000, `b*`[6] = 12352/38475, `b*`[7] = 0, `b*`[8] = 2/25, `b *`[1] = 763/10800, a[6,3] = 63099/585728, a[6,2] = 189/704, a[4,1] = 2 91/1372, a[6,4] = 58653/366080, a[3,2] = 14/81, a[8,5] = -14283/10880+ 32319/119680*a[7,6], a[8,1] = 51895/323136-399/11968*a[7,6], a[7,1] = \+ 5585/5376+57/256*a[7,6], c[6] = 3/4, c[5] = 2/3, b[4] = 16807/84240, c [4] = 3/7, c[2] = 1/7, b[7] = 1/100, c[3] = 2/9, c[7] = 1, a[8,7] = 0, a[4,3] = 1053/1372, a[2,1] = 1/7, a[5,2] = -14/33, a[4,2] = -27/49, b [8] = 187/2800, b[3] = 19683/69160, b[6] = 2816/7695, a[5,3] = 42/143, a[8,6] = 35840/26163-28/187*a[7,6], a[8,3] = 1595205/5912192+45927/31 1168*a[7,6], a[6,1] = -267/22528, b[2] = 0, a[8,4] = 11832373/15752880 -45619/194480*a[7,6], c[8] = 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 0 146 "R K6_8eqs := [op(RowSumConditions(8,'expanded')),op(OrderConditions(6,8, 'expanded'))]:\n`RK5_8eqs*` := subs(b=`b*`,OrderConditions(5,8,'expand ed')):" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 7 "Check: " }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 129 "simplify(subs(e 7,RK6_8eqs)):\nmap(u->lhs(u)-rhs(u),%);\n;nops(%);\nsimplify(subs(e7,` RK5_8eqs*`)):\nmap(u->lhs(u)-rhs(u),%);\nnops(%);" }}{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#\"#<" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 23 "The Verner scheme has " }{XPPEDIT 18 0 "a[7,6] = 26378/10773;" "6#/&%\"aG6$\"\"(\"\"'*&\"&yj#\"\"\"\"&t2\"! \"\"" }{TEXT -1 2 ". " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 55 "This value arises by applying the simplifying conditio n" }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "Sum(`b*`[i]*a[i, j],i = j+1 .. s) = `b*`[j]*(1-c[j]);" "6#/-%$SumG6$*&&%#b*G6#%\"iG\"\" \"&%\"aG6$F+%\"jGF,/F+;,&F0F,F,F,%\"sG*&&F)6#F0F,,&F,F,&%\"cG6#F0!\"\" F," }{TEXT -1 5 ", " }{XPPEDIT 18 0 "j = 2" "6#/%\"jG\"\"#" }{TEXT -1 7 " . . . " }{TEXT 268 1 "s" }{TEXT -1 1 "," }}{PARA 0 "" 0 "" {TEXT -1 6 "with " }{XPPEDIT 18 0 "s=8" "6#/%\"sG\"\")" }{TEXT -1 23 " for the eighth stage." }}{PARA 0 "" 0 "" {TEXT -1 8 "Taking " } {XPPEDIT 18 0 "s = 8;" "6#/%\"sG\"\")" }{TEXT -1 7 " and " } {XPPEDIT 18 0 "j = 6;" "6#/%\"jG\"\"'" }{TEXT -1 9 ", we have" }} {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#/,&*&&%#b*G6#\"\"(\"\"\"&%\"aG6$F)\" \"'F*F**&&F'6#\"\")F*&F,6$F2F.F*F**&&F'6#F.F*,&F*F*&%\"cG6#F.!\"\"F*" }{TEXT -1 2 ". " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 93 "Applying this equation to embedded order 5 scheme gives a n equation which can be solved for " }{XPPEDIT 18 0 "a[7,6];" "6#&%\" aG6$\"\"(\"\"'" }{TEXT -1 1 "." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 90 "subs(j=6,'add'(`b*`[i]*a[i,j ],i=j+1..8)=`b*`[j]*(1-c[j]));\nvalue(%);\nsubs(e7,%);\nsolve(%);" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#/-%$addG6$*&&%#b*G6#%\"iG\"\"\"&%\"aG6 $F+\"\"'F,/F+;\"\"(\"\")*&&F)6#F0F,,&F,F,&%\"cGF7!\"\"F," }}{PARA 11 " " 1 "" {XPPMATH 20 "6#/,&*&&%#b*G6#\"\"(\"\"\"&%\"aG6$F)\"\"'F*F**&&F' 6#\"\")F*&F,6$F2F.F*F**&&F'6#F.F*,&F*F*&%\"cGF7!\"\"F*" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/,&#\"&OV\"\"':38\"\"\"*&#\"#c\"%vYF(&%\"aG6$\"\"( \"\"'F(!\"\"#\"%)3$\"&v%Q" }}{PARA 11 "" 1 "" {XPPMATH 20 "6##\"&yj#\" &t2\"" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 39 "We now complete the construction with " }{XPPEDIT 18 0 "a[7,6] = 263 78/10773;" "6#/&%\"aG6$\"\"(\"\"'*&\"&yj#\"\"\"\"&t2\"!\"\"" }{TEXT -1 2 ". " }}{PARA 0 "" 0 "" {TEXT 260 4 "Note" }{TEXT -1 88 ": See the following subsection for further remarks concerning this choice of va lue for " }{XPPEDIT 18 0 "a[7,6];" "6#&%\"aG6$\"\"(\"\"'" }{TEXT -1 1 "." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 64 "e8 := `union`(subs(a[7,6]=26378/10773,e7),\{a[7,6]=26 378/10773\}):" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 2 "e8" }} {PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 910 "e8 := \{a[5,1] = 86/297, `b*`[2] = 0, a[3,1] = 4/81, b[5] = 0, a[ 5,4] = 1960/3861, b[1] = 79/1080, a[7,1] = 10949/6912, a[6,5] = 4617/2 0480, a[7,2] = -69/32, a[7,3] = -90891/68096, a[7,4] = 112931/25920, a [7,5] = -69861/17920, a[7,6] = 26378/10773, a[8,2] = -21/88, a[8,3] = \+ 219519/347776, a[8,4] = 163807/926640, a[8,5] = -417/640, a[8,6] = 154 4/1539, a[8,1] = 1501/19008, `b*`[3] = 59049/197600, `b*`[4] = 88837/5 26500, `b*`[5] = 243/4000, `b*`[6] = 12352/38475, `b*`[7] = 0, `b*`[8] = 2/25, `b*`[1] = 763/10800, a[6,3] = 63099/585728, a[6,2] = 189/704, a[4,1] = 291/1372, a[6,4] = 58653/366080, a[3,2] = 14/81, c[6] = 3/4, c[5] = 2/3, b[4] = 16807/84240, c[4] = 3/7, c[2] = 1/7, b[7] = 1/100, c[3] = 2/9, c[7] = 1, a[8,7] = 0, a[4,3] = 1053/1372, a[2,1] = 1/7, a [5,2] = -14/33, a[4,2] = -27/49, b[8] = 187/2800, b[3] = 19683/69160, \+ b[6] = 2816/7695, a[5,3] = 42/143, a[6,1] = -267/22528, b[2] = 0, c[8] = 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 0 119 "subs(e8,matrix([seq([c[ i],seq(a[i,j],j=1..i-1),``$(9-i)],i=2..8),\n[`b`,seq(b[i],i=1..8)],[`b *`,seq(`b*`[i],i=1..8)]]));" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#K%'matr ixG6#7+7+#\"\"\"\"\"(F(%!GF+F+F+F+F+F+7+#\"\"#\"\"*#\"\"%\"#\")#\"#9F2 F+F+F+F+F+F+7+#\"\"$F*#\"$\"H\"%s8#!#F\"#\\#\"%`5F:F+F+F+F+F+7+#F.F7# \"#')\"$(H#!#9\"#L#\"#U\"$V\"#\"%g>\"%hQF+F+F+F+7+#F7F1#!$n#\"&GD##\"$ *=\"$/(#\"&*4j\"'Gde#\"&`'e\"'!3m$#\"%#!#@\"#))#\"'>&>#\"'wxM#\"'2Q;\"'Sm#*#!$<%\"$S'#\"%W:\"%R: \"\"!F+7+%\"bG#\"#z\"%!3\"F_q#\"&$o>\"&g\"p#\"&2o\"\"&SU)F_q#\"%;G\"%& p(#F)\"$+\"#\"$(=\"%+G7+%#b*G#\"$j(\"&+3\"F_q#\"&\\!f\"'+w>#\"&P)))\"' +l_#\"$V#\"%+S#\"&_B\"\"&v%QF_q#F.\"#DQ)pprint886\"" }}}{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 31 "note concerning th e value for " }{XPPEDIT 18 0 "a[7, 6];" "6#&%\"aG6$\"\"(\"\"'" } {TEXT -1 1 " " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 46 "e7: co efficients of the combined scheme with " }{XPPEDIT 18 0 "a[7,6]" "6#& %\"aG6$\"\"(\"\"'" }{TEXT -1 16 " as a parameter" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1052 "e7 := \{a[ 7,5] = 9273/17920-4617/2560*a[7,6], a[7,4] = 6503/12480+6517/4160*a[7, 6], a[5,1] = 86/297, a[7,3] = 50265/46592-6561/6656*a[7,6], `b*`[2] = \+ 0, a[3,1] = 4/81, b[5] = 0, a[5,4] = 1960/3861, b[1] = 79/1080, a[6,5] = 4617/20480, a[7,2] = -69/32, a[8,2] = -21/88, `b*`[3] = 59049/19760 0, `b*`[4] = 88837/526500, `b*`[5] = 243/4000, `b*`[6] = 12352/38475, \+ `b*`[7] = 0, `b*`[8] = 2/25, `b*`[1] = 763/10800, a[6,3] = 63099/58572 8, a[6,2] = 189/704, a[4,1] = 291/1372, a[6,4] = 58653/366080, a[3,2] \+ = 14/81, a[8,5] = -14283/10880+32319/119680*a[7,6], a[8,1] = 51895/323 136-399/11968*a[7,6], a[7,1] = 5585/5376+57/256*a[7,6], c[6] = 3/4, c[ 5] = 2/3, b[4] = 16807/84240, c[4] = 3/7, c[2] = 1/7, b[7] = 1/100, c[ 3] = 2/9, c[7] = 1, a[8,7] = 0, a[4,3] = 1053/1372, a[2,1] = 1/7, a[5, 2] = -14/33, a[4,2] = -27/49, b[8] = 187/2800, b[3] = 19683/69160, b[6 ] = 2816/7695, a[5,3] = 42/143, a[8,6] = 35840/26163-28/187*a[7,6], a[ 8,3] = 1595205/5912192+45927/311168*a[7,6], a[6,1] = -267/22528, b[2] \+ = 0, a[8,4] = 11832373/15752880-45619/194480*a[7,6], c[8] = 1\}:" } {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 5 "Let " }{XPPEDIT 18 0 "T[6, 8];" "6#&%\"TG6$\"\"'\" \")" }{TEXT -1 127 " denote the vector whose components are the princ ipal error terms of the 8 stage, order 6 scheme (the error terms of or der 7)." }}{PARA 0 "" 0 "" {TEXT -1 5 "Let " }{XPPEDIT 18 0 "`T*`[5,8 ];" "6#&%#T*G6$\"\"&\"\")" }{TEXT -1 145 " denote the vector whose co mponents are the principal error terms of the embedded 8 stage, order \+ 5 scheme (the error terms of order 6) and let " }{XPPEDIT 18 0 "`T*`[ 6,8];" "6#&%#T*G6$\"\"'\"\")" }{TEXT -1 99 " denote the vector whose \+ components are the error terms of order 7 of the embedded order 5 sche me." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 39 "De note 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*`[5,8]));" "6#-%$absG6#-F$6#&%#T*G6$\"\"&\"\")" } {TEXT -1 7 " and " }{XPPEDIT 18 0 "abs(abs(`T*`[6,8]));" "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*`[6,8]))/abs(abs(`T*`[5,8]));" "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*` [6,8]-T[6,8]))/abs(abs(`T*`[5,8]));" "6#/&%\"CG6#\"\"(*&-%$absG6#-F*6# ,&&%#T*G6$\"\"'\"\")\"\"\"&%\"TG6$F2F3!\"\"F4-F*6#-F*6#&F06$\"\"&F3F8 " }{TEXT -1 1 "." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 77 "Dormand and Prince have suggested that as well as 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 " 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 0 471 "errterms6_8 := PrincipalErrorTerms(6,8,'expanded'):\n`errterms5_8*` := subs(b=`b* `,PrincipalErrorTerms(5,8,'expanded')):\n`errterms6_8*` := subs(b=`b*` ,PrincipalErrorTerms(6,8,'expanded')):\nsnmB := sqrt(add(subs(e7,`errt erms6_8*`[i])^2,i=1..nops(`errterms6_8*`))):\nsdnB := sqrt(add(subs(e7 ,`errterms5_8*`[i])^2,i=1..nops(`errterms5_8*`))):\nsnmC := sqrt(add(( subs(e7,`errterms6_8*`[i])-subs(e7,errterms6_8[i]))^2,i=1..nops(errter ms6_8))):\nB_7 := snmB/sdnB:\nC_7 := snmC/sdnB:" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 97 "'B[7]' = eva lf[8](eval(B_7,a[7,6]=26378/10773));\n'C[7]' = evalf[8](eval(C_7,a[7,6 ]=26378/10773));" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/&%\"BG6#\"\"($\") /#)3>!\"(" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/&%\"CG6#\"\"($\")Y)3X\"! \"(" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 6 "Wh en " }{XPPEDIT 18 0 "a[7,6] = 26738/10773;" "6#/&%\"aG6$\"\"(\"\"'*& \"&Qn#\"\"\"\"&t2\"!\"\"" }{TEXT -1 36 ", as in the Verner scheme, we \+ have " }{XPPEDIT 18 0 "B[7]" "6#&%\"BG6#\"\"(" }{TEXT -1 1 " " } {TEXT 269 1 "~" }{TEXT -1 15 " 1.9088 and " }{XPPEDIT 18 0 "C[7]" " 6#&%\"CG6#\"\"(" }{TEXT -1 1 " " }{TEXT 270 1 "~" }{TEXT -1 10 " 1.450 88. " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 43 "T he following picture shows the graphs of " }{XPPEDIT 18 0 "B[7]" "6#& %\"BG6#\"\"(" }{TEXT -1 5 " in " }{TEXT 259 5 "brown" }{TEXT -1 6 " a nd " }{XPPEDIT 18 0 "C[7]" "6#&%\"CG6#\"\"(" }{TEXT -1 5 " in " } {TEXT 261 3 "red" }{TEXT -1 3 ". \n" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 520 "aa := 26378/10773:\nbb := evalf(eval(B_7,a[7,6]=aa)) :\ncc := evalf(eval(C_7,a[7,6]=aa)):\np1 := plot([B_7,C_7,1],a[7,6]=-. 12..4.35,0..2.8,color=[brown,red,blue]):\np2 := plot([[aa,0],[aa,2.9]] ,color=COLOR(RGB,0,.7,0)):\np3 := plot([[[aa,bb],[aa,cc]]$3],style=poi nt,symbol=[circle,diamond,cross],color=[black,coral$2]):\np4 := plot([ [[aa,0]]$3],style=point,symbol=[circle,diamond,cross],color=black):\np 5 := plot([[[aa,bb],[0,bb]],[[aa,cc],[0,cc]]],linestyle=3,color=[brown ,red]):\nplots[display]([p||(1..5)],font=[HELVETICA,9]);" }}{PARA 13 " " 1 "" {GLPLOT2D 528 392 392 {PLOTDATA 2 "61-%'CURVESG6$7Y7$$!3%****** ********>\"!#=$\"3Yf06d@mw?!#<7$$!39++](3smD#!#>$\"3Yc!oSElt5#F-7$$\"3 g**\\(=VX4A'F1$\"3$G'p0EKJO@F-7$$\"3?+](yH\"\\v:F*$\"3EJSMA3`r@F-7$$\" 3X+]7&y2_`#F*$\"3H([WU\\#**4AF-7$$\"3w*\\Pu!HO!\\$F*$\"3OzrA$)RQ^AF-7$ $\"3s*\\(=uR\"fP%F*$\"3!o)Qsbva#H#F-7$$\"3S*\\(o53&GH&F*$\"3HH9\\^x(yL #F-7$$\"3f+vo)>[6C'F*$\"3=#HFwI7tQ#F-7$$\"3C*\\P\\Q/k=(F*$\"3'zK6yR&\\ QCF-7$$\"31,+Dflre\")F*$\"3AmrYV&H>\\#F-7$$\"3-**\\i^'H^,*F*$\"3GedfhD =QDF-7$$\"3)****\\F&zCz**F*$\"3o3\">pQ9qe#F-7$$\"3!)**\\i,Dt%4\"F-$\"3 HK\"4![ZPHEF-7$$\"3!***\\skY-)=\"F-$\"3])o%zw)\\,m#F-7$$\"3&\\(oavPQI7 F-$\"3o:\\t@g#)pEF-7$$\"3+](oj)Gus7F-$\"3yk\\EVkLwEF-7$$\"3m7GGEu#zH\" F-$\"3)ewchiy&yEF-7$$\"35vo>m>6B8F-$\"3=<,;)oM&zEF-7$$\"3aP461lH[8F-$ \"3Kes]pU:zEF-7$$\"3?+]-Y5[t8F-$\"3W)eUK5&RxEF-7$$\"32+Dcj-:;9F-$\"3A/ cyd$z6n#F-7$$\"3#*****4\"[>)e9F-$\"3!\\htPf?3m#F-7$$\"3!\\P4[#RX3:F-$ \"3Kr&3^4eNk#F-7$$\"3))\\(=&o$)3e:F-$\"3aE'RR!G%3i#F-7$$\"3q***\\V:efk \"F-$\"3-N\"*GHeMoDF-7$$\"3F](=gBlBu\"F-$\"3)prN3'yz&\\#F-7$$\"38]i0wv ;M=F-$\"3N;43aDm;CF-7$$\"3/+v=(o`*H>F-$\"3*>JqtPB(GBF-7$$\"3)*\\PRQ_\" z,#F-$\"37CK;qq9ZAF-7$$\"3o*\\iXl$z7@F-$\"37[Y<5*=<;#F-7$$\"3()\\(oQmX 8@#F-$\"35?s)y6?F-7$$\"3))*\\()o# 3z*Q#F-$\"3o'\\(oX6_Y>F-7$$\"3!****\\*oG^&[#F-$\"3ak$G#e>_')=F-7$$\"3k *\\PB@e\"zDF-$\"3wE<'G5`[$=F-7$$\"3w\\(=s*[wpEF-$\"3==SMV(45z\"F-7$$\" 3!)*\\iF+]Z@wNSJF-$\"3]KG-j:5R;F-7$$\"3l\\(=X!\\KIKF-$\"3P%o,)e%G2i\"F- 7$$\"3y\\P>QmOCLF-$\"3'o9Vj'35/;F-7$$\"3c**\\7A'4jT$F-$\"3+/y\\*fX+f\" F-7$$\"3i\\(oaSlD^$F-$\"3M1)3$3'Qtd\"F-7$$\"3,++5l6F0OF-$\"3U46()GHxm: F-7$$\"3***\\7T>w+q$F-$\"3=jbQW!Hub\"F-7$$\"3;](='*='4%z$F-$\"3K#p!)=? @%\\:F-7$$\"3k***\\4!3\\!)QF-$\"3Kig\\u,-V:F-7$$\"3e*\\i^x3&zRF-$\"3Mg Rhx,mO:F-7$$\"3k****pJ;2oSF-$\"3%f.X^!)R<`\"F-7$$\"3!)\\(=&3\")\\iTF-$ \"34&4<]P'=F:F-7$$\"3y\\7Q&HwGD%F-$\"3\"*f!*RmATB:F-7$$\"3k*********** *\\VF-$\"3()3t;O0\"*>:F--%'COLOURG6&%$RGBG$\")#)eqk!\")$\"))eqk\"F]]lF ^]l-F$6$7Z7$F($\"3shME@8%Rm\"F-7$F/$\"3MRUo&\\/rm\"F-7$F5$\"37_4\\K-KHn\"F-7$F?$\"3EgBCr9*en\"F-7$FD$\"39Gky\"[&fy;F-7$F I$\"3#=YRt))[2o\"F-7$FN$\"3=I-V!\\MCo\"F-7$FS$\"3wi!>XmMLo\"F-7$$\"3\" **\\7=HwPr'F*$\"3I,eM7'eLo\"F-7$FX$\"3%>\\2X=LIo\"F-7$$\"39]P4s/cswF*$ \"31LM<[SF#o\"F-7$Fgn$\"3A8))\\mw,\"o\"F-7$F\\o$\"3EL8X;?Px;F-7$Fao$\" 3k%R^U8J1n\"F-7$Ffo$\"3ueFY/u\\g;F-7$F[p$\"3a4]%)>H8Z;F-7$Fep$\"3hTuS0 @$=j\"F-7$Fiq$\"3%fI8Y)z.5;F-7$F^r$\"3U`4>T%Q)*f\"F-7$Fcr$\"3#3')Qp\"e @*e\"F-7$Fhr$\"33%[Zb9+ld\"F-7$F]s$\"3?>Tu*)Qhj:F-7$$\"3zuVVhK--;F-$\" 3_eB0xSD_:F-7$Fbs$\"3!\\)[z]l5T:F-7$$\"3)\\P%=&phTp\"F-$\"3ctvsk&*HH:F -7$Fgs$\"3$eSTI!p5=:F-7$$\"3?+v.1kE)y\"F-$\"3)pIh*QJ:3:F-7$F\\t$\"3%)o t\"RS$)*)\\\"F-7$Fat$\"3)f5;f2>E[\"F-7$Fft$\"3=rS'R8#)4Z\"F-7$F[u$\"3! 4S!yDp'=Y\"F-7$F`u$\"3=bEk^Mpb9F-7$Feu$\"3#yX4.[rDX\"F-7$Fju$\"3Ap)Rw$ e0^9F-7$F_v$\"3;rxXXF.^9F-7$Fdv$\"31VWe\"p=@X\"F-7$Fiv$\"3c!)p%Gh2RX\" F-7$F^w$\"3G.Drx*okX\"F-7$Fcw$\"3-Y,<3C5f9F-7$Fhw$\"3!=Ew%eg6i9F-7$F]x $\"3YQrv>\\\"F-7$Fd[l$\"3[I#p\"Q`,%\\\"F-7$Fi[l$\" 3%>7\\i<#4'\\\"F-7$F^\\l$\"3m&=f\"4%*)z\\\"F-7$Fc\\l$\"3_3^'yF-7$Fg\\m$\"3%******Hh%)3X\"F--Fh\\l6&Fj\\lFjhlFjhlFjhl -%'SYMBOLG6#%'CIRCLEG-%&STYLEG6#%&POINTG-F$6&Fd]m-Fh\\l6&Fj\\lFghl$\") AR!)\\F]]lFihl-F^^m6#%(DIAMONDGFa^m-F$6&Fd]mFg^m-F^^m6#%&CROSSGFa^m-F$ 6&7#Ff\\mF]^mF[^mFa^m-F$6&Fe_mF[_mF[^mFa^m-F$6&Fe_mF`_mF[^mFa^m-F$6%7$ Fe]m7$FihlFf]mFg\\l-%*LINESTYLEG6#\"\"$-F$6%7$Fh]m7$FihlFi]mFehlF^`m-% %FONTG6$%*HELVETICAG\"\"*-%+AXESLABELSG6%Q'a[7,6]6\"Q!F_am-Fg`m6#%(DEF AULTG-%%VIEWG6$;$!#7!\"#$\"$N%Fjam;Fihl$\"#GFa]m" 1 2 0 1 10 0 2 9 1 4 2 1.000000 45.000000 45.000000 0 0 "Curve 1" "Curve 2" "Curve 3" "Cu rve 4" "Curve 5" "Curve 6" "Curve 7" "Curve 8" "Curve 9" "Curve 10" "C urve 11" "Curve 12" }}}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 " " {TEXT 260 4 "Note" }{TEXT -1 23 ": The value taken for " }{XPPEDIT 18 0 "a[7,6]" "6#&%\"aG6$\"\"(\"\"'" }{TEXT -1 9 " makes " } {XPPEDIT 18 0 "C[7];" "6#&%\"CG6#\"\"(" }{TEXT -1 23 " close to its m inimum." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 27 "minimize(C_7):\nevalf[8](%);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#$\")m(3X\"!\"(" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}}{PARA 0 "" 0 "" {TEXT -1 52 "#---------------------------------------------------" }}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 27 "absolute stability regions " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }} }{SECT 1 {PARA 4 "" 0 "" {TEXT -1 35 "coefficients of the combined sch eme" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 839 "# coefficients by Jim Verner 2004\nee := \{c[2]=1/7, c[3]=2/9,c[4]=3/7,c[5]=2/3,c[6]=3/4,c[7]=1,c[8]=1,\n\na[2,1]=1/7,\na[3 ,1]=4/81,\na[3,2]=14/81,\na[4,1]=291/1372,\na[4,2]=-27/49,\na[4,3]=105 3/1372,\na[5,1]=86/297,\na[5,2]=-14/33,\na[5,3]=42/143,\na[5,4]=1960/3 861,\na[6,1]=-267/22528,\na[6,2]=189/704,\na[6,3]=63099/585728,\na[6,4 ]=58653/366080,\na[6,5]=4617/20480,\na[7,1]=10949/6912,\na[7,2]=-69/32 ,\na[7,3]=-90891/68096,\na[7,4]=112931/25920,\na[7,5]=-69861/17920,\na [7,6]=26378/10773,\na[8,1]=1501/19008,\na[8,2]=-21/88,\na[8,3]=219519/ 347776,\na[8,4]=163807/926640,\na[8,5]=-417/640,\na[8,6]=1544/1539,\na [8,7]=0,\n\nb[1]=79/1080,\nb[2]=0,\nb[3]=19683/69160,\nb[4]=16807/8424 0,\nb[5]=0,\nb[6]=2816/7695,\nb[7]=1/100,\nb[8]=187/2800,\n\n`b*`[1]=7 63/10800,\n`b*`[2]=0,\n`b*`[3]=59049/197600,\n`b*`[4]=88837/526500,\n` b*`[5]=243/4000,\n`b*`[6]=12352/38475,\n`b*`[7]=0,\n`b*`[8]=2/25\}:" } }}{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,Stabi lityFunction(6,8,'expanded')):\nR := unapply(%,z):\n'R(z)'=R(z);" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#/-%\"RG6#%\"zG,2\"\"\"F)F'F)*&#F)\"\"# F)*$)F'F,F)F)F)*&#F)\"\"'F)*$)F'\"\"$F)F)F)*&#F)\"#CF)*$)F'\"\"%F)F)F) *&#F)\"$?\"F)*$)F'\"\"&F)F)F)*&#F)\"$?(F)*$)F'F1F)F)F)*&#F)\"%S]F)*$)F '\"\"(F)F)F)" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 124 "We can find the point where the boundary of the 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 27 "z0 := newton(R(z)=-1,z=-4);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%#z0G$!+J(HT&R!\"*" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" } }{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 306 "z0 := newton(R(z)=-1,z=-4) :\np1 := plot([R(z),-1],z=-4.49..0.49,color=[red,blue]):\np2 := plot([ [[z0,-1]]$3],style=point,symbol=[circle,cross,diamond],color=black):\n p3 := plot([[z0,0],[z0,-1]],linestyle=3,color=COLOR(RGB,0,.5,0)):\nplo ts[display]([p1,p2,p3],view=[-4.49..0.49,-1.47..1.47],font=[HELVETICA, 9]);" }}{PARA 13 "" 1 "" {GLPLOT2D 360 253 253 {PLOTDATA 2 "6+-%'CURVE SG6$7U7$$!3A++++++!\\%!#<$!3iK'p5:D&)o#F*7$$!3;+]7'3DdV%F*$!3'yBEJ]7![ CF*7$$!35++Ds,X\"Q%F*$!33#HMw%y!=%F*$!3[%)Q[\"\\_ya\" F*7$$!35+]#oviQ2%F*$!3?Vq%yrMWE\"F*7$$!3A+ve,&\\u'RF*$!3=u.%e0+,T)!#=7$$!3;+v.8ajmPF*$!33s5ua7+&y'FU7$$!3C+ v$Q<')4m$F*$!3'oly[$*y)z`FU7$$!3G+v3`dnbNF*$!3rXMsSCn;UFU7$$!37++0T5NZ MF*$!31(G3#p8$)GKFU7$$!3?+]sv&Q>N$F*$!3lAf+n4Z1DFU7$$!3E++:Qn_WKF*$!3Y B],/wpJ=FU7$$!3#****\\s&QnOJF*$!3?]#*f2#*ew7FU7$$!3&****\\=iPF.$F*$!3u dd/!fT_O)!#>7$$!3Q+v$Gc`$QHF*$!3[J')\\MFVE]Fhp7$$!3+++l(y@h#GF*$!3OVJc 7l=4dQp\"F*$\"3%>lkG9%)R#=FU7$$!3)*****p]Q@(e \"F*$\"3pL`rFNZO?FU7$$!3I+]FRT)G[\"F*$\"3[txl0H&[E#FU7$$!3')*\\P*y(R>Q \"F*$\"3xZAkgb03DFU7$$!3++]Kx#e)p7F*$\"3%fe2:*pC2GFU7$$!3=++5j![\"p6F* $\"3>kW?L%\\b5$FU7$$!3s***\\KT=;1\"F*$\"3UL#R=.)feMFU7$$!3'***\\P%o0=k *FU$\"39v_l:!)y7QFU7$$!3](***\\tEbw&)FU$\"3D$***QX#)[TUFU7$$!39-]P2EBu vFU$\"3sh<4*e0()o%FU7$$!3>)*\\(GL>l_'FU$\"3#p9(z$>Cm?&FU7$$!3z++]-\")= -bFU$\"3C)eGR&QBodFU7$$!3$*)*\\PM#3)HWFU$\"3O%[Bd&[>@kFU7$$!3;(****f'* ypR$FU$\"3^sjV#z`)>rFU7$$!3L)**\\U'=wSBFU$\"3D'>n%p_,8zFU7$$!3?$*\\Pt2 H$H\"FU$\"3+V,>Fw%oy)FU7$$!33R****pit2LFhp$\"3+r)*)36PYn*FU7$$\"3Z^+]F kzBxFhp$\"3QUWS=\"*H!3\"F*7$$\"35*****z2`!f\"F*7$$\"3 l+]i_F06GFU$\"35LwZUIfC8F*7$$\"3\\.]Pt1&z\"QFU$\"3]WLRM=\"\\Y\"F*7$$\" 3!***************[FU$\"3Ko[\"G8;Bj\"F*-%'COLOURG6&%$RGBG$\"*++++\"!\") $\"\"!F\\\\lF[\\l-F$6$7S7$F($!\"\"F\\\\l7$F3Fa\\l7$F=Fa\\l7$FBFa\\l7$F GFa\\l7$FLFa\\l7$FQFa\\l7$FWFa\\l7$FfnFa\\l7$F[oFa\\l7$F`oFa\\l7$FeoFa \\l7$FjoFa\\l7$F_pFa\\l7$FdpFa\\l7$FjpFa\\l7$F_qFa\\l7$FdqFa\\l7$FjqFa \\l7$F_rFa\\l7$FdrFa\\l7$FirFa\\l7$F^sFa\\l7$FcsFa\\l7$FhsFa\\l7$F]tFa \\l7$FbtFa\\l7$FgtFa\\l7$F\\uFa\\l7$FauFa\\l7$FfuFa\\l7$F[vFa\\l7$F`vF a\\l7$FevFa\\l7$FjvFa\\l7$F_wFa\\l7$FdwFa\\l7$FiwFa\\l7$F^xFa\\l7$FcxF a\\l7$FhxFa\\l7$F]yFa\\l7$FbyFa\\l7$FgyFa\\l7$F\\zFa\\l7$FazFa\\l7$Ffz Fa\\l7$F[[lFa\\l7$F`[lFa\\l-Fe[l6&Fg[lF[\\lF[\\lFh[l-F$6&7#7$$!35+++J( HT&RF*Fa\\l-%'SYMBOLG6#%'CIRCLEG-Fe[l6&Fg[lF\\\\lF\\\\lF\\\\l-%&STYLEG 6#%&POINTG-F$6&Fg_l-F\\`l6#%&CROSSGF_`lFa`l-F$6&Fg_l-F\\`l6#%(DIAMONDG F_`lFa`l-F$6%7$7$Fi_lF[\\lFh_l-%&COLORG6&Fg[lF[\\l$\"\"&Fb\\lF[\\l-%*L INESTYLEG6#\"\"$-%+AXESLABELSG6%Q\"z6\"Q!F`bl-%%FONTG6#%(DEFAULTG-Fcbl 6$%*HELVETICAG\"\"*-%%VIEWG6$;$!$\\%!\"#$\"#\\F`cl;$!$Z\"F`cl$\"$Z\"F` cl" 1 2 0 1 10 0 2 9 1 4 2 1.000000 45.000000 45.000000 0 0 "Curve 1" "Curve 2" "Curve 3" "Curve 4" "Curve 5" "Curve 6" }}}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 50 "The following picture sho ws the stability region. " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1284 "R := z -> 1+z+1/2*z^2+1/6*z^3+1/2 4*z^4+1/120*z^5+1/720*z^6+1/5040*z^7:\npts := []:\nz0 := 0:\nfor ct fr om 0 to 200 do\n zz := newton(R(z)=exp(ct*Pi/20*I),z=z0):\n z0 := \+ zz:\n pts := [op(pts),[Re(zz),Im(zz)]]:\nend do:\np1 := plot(pts,col or=COLOR(RGB,.55,.15,0)):\np2 := plots[polygonplot]([seq([pts[i-1],pts [i],[-2,0]],i=2..nops(pts))],\n style=patchnogrid,color=COLOR (RGB,.9,.6,.15)):\npts := []: z0 := 1.5+4.3*I:\nfor ct from 0 to 60 do \n zz := newton(R(z)=exp(ct*Pi/30*I),z=z0):\n z0 := zz:\n pts := [op(pts),[Re(zz),Im(zz)]]:\nend do:\np3 := plot(pts,color=COLOR(RGB,. 55,.15,0)):\np4 := plots[polygonplot]([seq([pts[i-1],pts[i],[1.39,4.2] ],i=2..nops(pts))],\n style=patchnogrid,color=COLOR(RGB,.9,.6 ,.15)):\npts := []: z0 := 1.5-4.3*I:\nfor ct from 0 to 60 do\n zz := newton(R(z)=exp(ct*Pi/30*I),z=z0):\n z0 := zz:\n pts := [op(pts), [Re(zz),Im(zz)]]:\nend do:\np5 := plot(pts,color=COLOR(RGB,.55,.15,0)) 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Im(z)G-%%FONTG6#%(DEFAULTG-F^ev6$%*HELVETICAGF]^r-%*AXESSTYLEG6#%$BOXG -%(SCALINGG6#%,CONSTRAINEDG-%%VIEWG6$;$!$\\%Fc^n$\"$z\"Fc^n;$!$p%Fc^n$ \"$p%Fc^n" 1 2 0 1 10 0 2 9 1 2 1 1.000000 45.000000 45.000000 0 0 "Cu rve 1" "Curve 2" "Curve 3" "Curve 4" "Curve 5" "Curve 6" "Curve 7" "Cu rve 8" }}}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 4 "The " }{TEXT 260 30 "interval of absolute stability" }{TEXT -1 89 " \+ (or stability interval) is the intersection of the stability region wi th the real line." }}{PARA 0 "" 0 "" {TEXT -1 59 "For this scheme the \+ stability interval is (approximately) " }{XPPEDIT 18 0 "[-3.9541, 0]; " "6#7$,$-%&FloatG6$\"&T&R!\"%!\"\"\"\"!" }{TEXT -1 2 ". " }}{PARA 0 " " 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 260 "We can distort 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 the largest interv al on the nonnegative imaginary axis that contains the origin and lies inside the stability region is " }{XPPEDIT 18 0 "[0, 1.76];" "6#7$\" \"!-%&FloatG6$\"$w\"!\"#" }{TEXT -1 18 " approximately. " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 322 "R : = z -> 1+z+1/2*z^2+1/6*z^3+1/24*z^4+1/120*z^5+1/720*z^6+1/5040*z^7:\nD igits := 25:\npts := []: z0 := 0:\nfor ct from 0 to 60 do\n zz := ne wton(R(z)=exp(ct*Pi/100*I),z=z0):\n z0 := zz:\n pts := [op(pts),[s urd(Re(zz),11),Im(zz)]]:\nend do:\nplot(pts,color=COLOR(RGB,.85,.45,0) ,thickness=2,font=[HELVETICA,9]);\nDigits := 10:" }}{PARA 13 "" 1 "" {GLPLOT2D 261 276 276 {PLOTDATA 2 "6(-%'CURVESG6#7in7$$\"\"!F)F(7$$\": (GMFj2VC9eNyI!#E$\":yZvsJz*e`EfTJF-7$$\":=SG%)Gykc@*o&4&F-$\":w13[Gbzr I&=$G'F-7$$\":!)3))4>t?9qG@%oF-$\":!y*ezx3o2'zxC%*F-7$$\":A+QI!=3it^IK %)F-$\":d8Yy#)>M91PmD\"!#D7$$\":P)>V*eS67jI[\"**F-$\":CoYlW5#yEjzq:F?7 $$\":obXc>7S_R:;8\"F?$\":'o/^.pz3#fb\\)=F?7$$\":-C)pR\"f*)Q1o_E\"F?$\" :sb85*4%\\s&[6*>#F?7$$\":BxXKDBKxyXNR\"F?$\":qcv5pP(*>7uK^#F?7$$\":\") ya<9'Q3^cB<:F?$\":/E/5]Z7dQLu#GF?7$$\":`#fswJ!y/!>#pj\"F?$\":Sx3Px>1rk #fTJF?7$$\":)3\"orXVLN#=0`vbMF?7$$\":7=K)y)pe\"3h(f '=F?$\":tc!G!=d_5:6*pPF?7$$\":V-d8&)zz2'[(f(>F?$\":'HrDDI8b\"QqS3%F?7$ $\":3Z1;+9g!y]F$3#F?$\":1=2]nS+GeH#)R%F?7$$\":.*GNchB@wF1)=#F?$\":#\\y 2`DfKN()Q7ZF?7$$\":=S.17J!GX;\\!H#F?$\":Qo#Q+x[`4yaE]F?7$$\":92+c%))*> _O*o!R#F?$\":\"[Y*))e3G'eR&yF?7$$\":#e*[ypTY`WbZ?$F?$\":(=*p@)z#*4u$3\"o\")F?7$$ \":*o)*)z_7DE&)=iG$F?$\":$4%3^;1X4#[D#[)F?7$$\":YFq!Q%=snbNfO$F?$\":Lh ([eqW06qR'z)F?7$$\":HFMZ`e6An&*QW$F?$\":hpFExG*fyP`5\"*F?7$$\":(or$oWG I)y93?NF?$\":qr]p'=,E.PmC%*F?7$$\":p,g_r>zQ2nWf$F?$\":)Hu'R!=omo]yQ(*F ?7$$\":!\\/.rl\"Ffu7$$\":*or%4d75p1\"H=TF?$\":crMI=1y(y.6 D7Ffu7$$\":u,0qbVG)Ry[tTF?$\":'[y&HZym_e*\\c7Ffu7$$\":.BK_a()=f%4\"fA% F?$\":!38Ffu7$$ \":\"4M([@ZU(3%e9K%F?$\":\"y;5zPltw4j]8Ffu7$$\":24d-,$4jtL&RO%F?$\":$e j\\b=!Q$yE*>Q\"Ffu7$$\":w[$3ox#))\\L!Q-WF?$\":6qt.Htd\")oWLT\"Ffu7$$\" :B'*Qv:g,*\\+>OWF?$\":U[\\Okk`Di&oW9Ffu7$$\":JEMO$yF-*orYY%F?$\":PG)=^ MV)e&R,w9Ffu7$$\":2o-:pO$pp?)o[%F?$\":[&fP>#e.P,Gt]\"Ffu7$$\":(3;:2S)) p#GI:]%F?$\":RJ%H6')4omfiQ:Ffu7$$\":Y/@x)ezm#oyn]%F?$\":Czo'QWBTDe!*p: Ffu7$$\":^K#=r^T]'4u)*\\%F?$\":X3cJ\">XGOa;,;Ffu7$$\":.mk\"3u_kVmRwWF? $\":Y\\#HbrabyCSK;Ffu7$$\":Di#*Hae;=/:&GWF?$\":[!p,&*4/!GY9Om\"Ffu7$$ \":DgMc=qsT-\\+M%F?$\":Rs&yR5'e3t)z%p\"Ffu7$$\":lMP\"Qh'yWjKn;%F?$\":) )4keas\"RdC&fs\"Ffu7$$\":ye43b)4t(3B6j$F?$\":-_9>CrKJlsqv\"Ffu7$$!:![s V6%4$>0k\"34%F?$\":cRZc'\\X')ph:)y\"Ffu7$$!:y]OyGQ:9QF)oWF?$\": rq*>>=Ffu7$$!:[XNpEO\"33:R6ZF?$\":lFWI%*[M0#)*>]=Ffu7$$!:l!\\o*4H$e=[) G!\\F?$\":I/Ka`&4tJH:\")=Ffu-%*THICKNESSG6#\"\"#-%%FONTG6$%*HELVETICAG \"\"*-%&COLORG6&%$RGBG$\"#&)!\"#$\"#XFb_lF(-%+AXESLABELSG6$Q!6\"Fh_l-% %VIEWG6$%(DEFAULTGF]`l" 1 2 0 1 10 2 2 6 1 4 2 1.000000 45.000000 44.000000 0 0 "Curve 1" }}}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 " " 0 "" {TEXT -1 123 "The relevant intersection point of the boundary c urve with the imaginary axis can be determined more accurately as foll ows." }}{PARA 0 "" 0 "" {TEXT -1 86 "First we look for points on the b oundary curve either side of the intersection point. " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 112 "Digits : = 15:\nz0 := 1.76*I:\nfor ct from 55 to 58 do\n newton(R(z)=exp(ct*P i/100*I),z=z0);\nend do;\nDigits := 10:" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#^$$\"02+GdBGd'!#>$\"0t\"RdC&fs\"!#9" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#^$$\"0o0E_SpW\"!#>$\"0rKJlsqv\"!#9" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#^$$!05]am.#p`!#>$\"0bk)ph:)y\"!#9" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#^$$!0-k9J![>9!#=$\"0j>rq*>>=!#9" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 107 "Then we apply the bisect ion method to calculate the parameter value associated with the inters ection point." }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 178 "real_part \+ := proc(u)\n Re(newton(R(z)=exp(u*Pi*I),z=1.76*I))\nend proc:\nDigit s := 15:\nu0 := bisect('real_part'(u),u=0.55..0.58);\nnewton(R(z)=exp( u0*Pi*I),z=1.76*I);\nDigits := 10:" }}{PARA 11 "" 1 "" {XPPMATH 20 "6# >%#u0G$\"0')fUmIOi&!#:" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#^#$\"0T`XK@W w\"!#9" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 4 "The " }{TEXT 260 112 "largest interval on the nonnegative imaginary a xis that contains the origin and lies inside the stability region" } {TEXT -1 5 " is " }{XPPEDIT 18 0 "[0, 1.7644];" "6#7$\"\"!-%&FloatG6$ \"&Ww\"!\"%" }{TEXT -1 18 " (approximately)." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 37 "#------------------------ ------------" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 80 "The stability function R* for the 8 stage, order 5 scheme is \+ given as follows." }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 97 "subs(ee ,subs(b=`b*`,StabilityFunction(5,8,'expanded'))):\n`R*` := unapply(%,z ):\n'`R*`(z)'=`R*`(z);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/-%#R*G6#%\" zG,2\"\"\"F)F'F)*&#F)\"\"#F)*$)F'F,F)F)F)*&#F)\"\"'F)*$)F'\"\"$F)F)F)* &#F)\"#CF)*$)F'\"\"%F)F)F)*&#F)\"$?\"F)*$)F'\"\"&F)F)F)*&#\"%n:\"(+)36 F)*$)F'F1F)F)F)*&#\"$$>FDF)*$)F'\"\"(F)F)F)" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 124 "We can find the point where th e boundary of the stability region intersects the negative real axis b y solving the equation: " }}{PARA 256 "" 0 "" {TEXT -1 1 " " } {XPPEDIT 18 0 "`R*`(z) = -1;" "6#/-%#R*G6#%\"zG,$\"\"\"!\"\"" }{TEXT -1 2 ". " }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 31 "z_0 := newton(`R *`(z)=-1,z=-4);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%$z_0G$!+3dTZU!\"* " }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 322 "z_0 := newton(`R*`(z)=-1,z=-4):\np_1 := plot([`R*`(z ),-1],z=-4.79..0.49,color=[red,blue]):\np_2 := plot([[[z_0,-1]]$3],sty le=point,symbol=[circle,cross,diamond],color=black):\np_3 := plot([[z_ 0,0],[z_0,-1]],linestyle=3,color=COLOR(RGB,0,.5,0)):\nplots[display]([ p_1,p_2,p_3],view=[-4.79..0.49,-1.47..1.47],font=[HELVETICA,9]);" }} {PARA 13 "" 1 "" {GLPLOT2D 398 284 284 {PLOTDATA 2 "6+-%'CURVESG6$7U7$ $!3/++++++!z%!#<$!3'p*pFNeX9FF*7$$!3n*****H^bCt%F*$!3%Ry^+gq'eCF*7$$!3 =+++E5\"\\n%F*$!3p&zodI5PA#F*7$$!3[++bv=%[i%F*$!38x.R,`3N?F*7$$!3))*** *4DFxuXF*$!3uwRNLRAg=F*7$$!3Y++gKn:iWF*$!3U\"pM*HV%G^\"F*7$$!3%*****>[ Uz[VF*$!3+`e&=d#3?7F*7$$!3$)*****ebqfB%F*$!3of4)f))4Qx*!#=7$$!3U++q]&o 88%F*$!3OGHqE]^&*yFP7$$!3p****4L#fI-%F*$!3QL%p#4E6tiFP7$$!3&)*****=eX5 \"RF*$!3o&HCC'G,*)[FP7$$!3')*****)f6R*z$F*$!3#4OH)pq3gPFP7$$!3/++!G'3a %o$F*$!3td!*yV4$p\"GFP7$$!3K++ge1Q$e$F*$!3Z\")Rw(*3\"*Q@FP7$$!3+++SG#) \\pMF*$!3-pY/cs+<:FP7$$!33+++'=[^N$F*$!3j[w\"z4gf,\"FP7$$!3#*****fJ2& \\C$F*$!3g36U\"RcJF'!#>7$$!3:++!f)3)[9$F*$!3QpNHI'GgQ$Fhp7$$!33++Sg\") )e-$F*$!3%Q\"esBh!yy&!#?7$$!3p****f8d3DHF*$\"3?kJ=q,_'Q\"Fhp7$$!3v**** H/&Gy!GF*$\"33md$fOg'=LFhp7$$!3N++g0g./FF*$\"3s.'RYa`&*z%Fhp7$$!3-++I% =f,f#F*$\"3GAbm@Y.giFhp7$$!3))*****3_@<[#F*$\"3oW^sooOdvFhp7$$!3++++&> y&oBF*$\"3#Hq_Zi4Q())Fhp7$$!3)*****H-tnkAF*$\"3mHPJT0P45FP7$$!3C+++`hg _@F*$\"3a<]m\"*QGY6FP7$$!3')*****es&>O?F*$\"3woAV3lQ)H\"FP7$$!3******* o5g[$>F*$\"3$Q(pdz%y>W\"FP7$$!3r****>i[TD=F*$\"3*3]PS%*=;h\"FP7$$!3L++ ?:sM7&e0#HYC#FP7$$!3E++?OQ(eP\"F*$\"3#QarMkat_#FP7$$! 3I++gTn4p7F*$\"33\"\\$4n@q6GFP7$$!3w*****>S*3b6F*$\"3C8vc=X%3:$FP7$$!3 G++]2=y^5F*$\"3E#))[,-#\\$\\$FP7$$!3c&****f>$R)Q*FP$\"3wO]-#z-5\"RFP7$ $!3N-++.CpD$)FP$\"3w[JRU.S\\VFP7$$!3#o****4(Q'[@(FP$\"37.X.*=N.'[FP7$$ !3D-++Sf#)GhFP$\"3&RCC$Q[)yT&FP7$$!3'******\\([%=*\\FP$\"3w*3p1pf-2'FP 7$$!3'4++g(pz'*QFP$\"3UW^C!yQFx'FP7$$!3u-++QB&px#FP$\"3%=]!z&ow_d(FP7$ $!3/$*****y0Qm;FP$\"3-b'*GE%f]Y)FP7$$!3%)******>Z!)ekFhp$\"3cG4o/d`u$* FP7$$\"3:!*****R\"ysB&Fhp$\"3UZ')f%\\oP0\"F*7$$\"3++++3%R)p:FP$\"3w\\h :Do(*p6F*7$$\"3;0++dD@&o#FP$\"33f&)y&yG!38F*7$$\"3+/++zpw_PFP$\"3o#f.d 4%Rb9F*7$$\"3!***************[FP$\"38xuq/jJK;F*-%'COLOURG6&%$RGBG$\"*+ +++\"!\")$\"\"!F\\\\lF[\\l-F$6$7S7$F($!\"\"F\\\\l7$F3Fa\\l7$F=Fa\\l7$F BFa\\l7$FGFa\\l7$FLFa\\l7$FRFa\\l7$FWFa\\l7$FfnFa\\l7$F[oFa\\l7$F`oFa \\l7$FeoFa\\l7$FjoFa\\l7$F_pFa\\l7$FdpFa\\l7$FjpFa\\l7$F_qFa\\l7$FeqFa \\l7$FjqFa\\l7$F_rFa\\l7$FdrFa\\l7$FirFa\\l7$F^sFa\\l7$FcsFa\\l7$FhsFa \\l7$F]tFa\\l7$FbtFa\\l7$FgtFa\\l7$F\\uFa\\l7$FauFa\\l7$FfuFa\\l7$F[vF a\\l7$F`vFa\\l7$FevFa\\l7$FjvFa\\l7$F_wFa\\l7$FdwFa\\l7$FiwFa\\l7$F^xF a\\l7$FcxFa\\l7$FhxFa\\l7$F]yFa\\l7$FbyFa\\l7$FgyFa\\l7$F\\zFa\\l7$Faz Fa\\l7$FfzFa\\l7$F[[lFa\\l7$F`[lFa\\l-Fe[l6&Fg[lF[\\lF[\\lFh[l-F$6&7#7 $$!37+++3dTZUF*Fa\\l-%'SYMBOLG6#%'CIRCLEG-Fe[l6&Fg[lF\\\\lF\\\\lF\\\\l -%&STYLEG6#%&POINTG-F$6&Fg_l-F\\`l6#%&CROSSGF_`lFa`l-F$6&Fg_l-F\\`l6#% (DIAMONDGF_`lFa`l-F$6%7$7$Fi_lF[\\lFh_l-%&COLORG6&Fg[lF[\\l$\"\"&Fb\\l F[\\l-%*LINESTYLEG6#\"\"$-%+AXESLABELSG6%Q\"z6\"Q!F`bl-%%FONTG6#%(DEFA ULTG-Fcbl6$%*HELVETICAG\"\"*-%%VIEWG6$;$!$z%!\"#$\"#\\F`cl;$!$Z\"F`cl$ \"$Z\"F`cl" 1 2 0 1 10 0 2 9 1 4 2 1.000000 45.000000 45.000000 0 0 "C urve 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 pic ture shows the stability region for the 7 stage, order 5 scheme. " }} {PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1318 "`R*` := z -> 1+z+1/2*z^2+1/6*z^3+1/24*z^4+1/120*z^5+1567/1108800 *z^6+193/1108800*z^7:\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, .35,.1,0)):\np_2 := plots[polygonplot]([seq([pts[i-1],pts[i],[-1.8,0]] ,i=2..nops(pts))],\n style=patchnogrid,color=COLOR(RGB,.75,.3 5,.1)):\npts := []: z0 := 1.4+4.3*I:\nfor ct from 0 to 60 do\n zz := newton(`R*`(z)=exp(ct*Pi/30*I),z=z0):\n z0 := zz:\n pts := [op(pt s),[Re(zz),Im(zz)]]:\nend do:\np_3 := plot(pts,color=COLOR(RGB,.35,.1, 0)):\np_4 := plots[polygonplot]([seq([pts[i-1],pts[i],[1.27,4.16]],i=2 ..nops(pts))],\n style=patchnogrid,color=COLOR(RGB,.75,.35,.1 )):\npts := []: z0 := 1.4-4.3*I:\nfor ct from 0 to 60 do\n zz := new ton(`R*`(z)=exp(ct*Pi/30*I),z=z0):\n z0 := zz:\n pts := [op(pts),[ Re(zz),Im(zz)]]:\nend do:\np_5 := 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" }} {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 79 "The stabi lity regions for the two schemes can be displayed in the same picture. 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"Curve 12" "Curve 13" "Curve 14" }}}}{PARA 0 "" 0 "" {TEXT -1 0 " " }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}}{PARA 0 "" 0 "" {TEXT -1 52 "#---------------------------------------------------" }} {SECT 1 {PARA 4 "" 0 "" {TEXT -1 36 "coefficients for the combined sch eme" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 35 "coefficient s of the combined scheme" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 839 "# coefficients by Jim Verner 2004 \nee := \{c[2]=1/7,c[3]=2/9,c[4]=3/7,c[5]=2/3,c[6]=3/4,c[7]=1,c[8]=1, \n\na[2,1]=1/7,\na[3,1]=4/81,\na[3,2]=14/81,\na[4,1]=291/1372,\na[4,2] =-27/49,\na[4,3]=1053/1372,\na[5,1]=86/297,\na[5,2]=-14/33,\na[5,3]=42 /143,\na[5,4]=1960/3861,\na[6,1]=-267/22528,\na[6,2]=189/704,\na[6,3]= 63099/585728,\na[6,4]=58653/366080,\na[6,5]=4617/20480,\na[7,1]=10949/ 6912,\na[7,2]=-69/32,\na[7,3]=-90891/68096,\na[7,4]=112931/25920,\na[7 ,5]=-69861/17920,\na[7,6]=26378/10773,\na[8,1]=1501/19008,\na[8,2]=-21 /88,\na[8,3]=219519/347776,\na[8,4]=163807/926640,\na[8,5]=-417/640,\n a[8,6]=1544/1539,\na[8,7]=0,\n\nb[1]=79/1080,\nb[2]=0,\nb[3]=19683/691 60,\nb[4]=16807/84240,\nb[5]=0,\nb[6]=2816/7695,\nb[7]=1/100,\nb[8]=18 7/2800,\n\n`b*`[1]=763/10800,\n`b*`[2]=0,\n`b*`[3]=59049/197600,\n`b*` [4]=88837/526500,\n`b*`[5]=243/4000,\n`b*`[6]=12352/38475,\n`b*`[7]=0, \n`b*`[8]=2/25\}:" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 " > " 0 "" {MPLTEXT 1 0 1 ";" }}}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT 259 5 "nodes" }{TEXT -1 2 ": " }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 31 "seq(c[i]=subs(ee,c[i]),i=2..8);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6)/&%\"cG6#\"\"##\"\"\"\"\"(/&F%6#\"\"$#F'\"\"*/&F%6 #\"\"%#F.F*/&F%6#\"\"&#F'F./&F%6#\"\"'#F.F4/&F%6#F*F)/&F%6#\"\")F)" }} }{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT 259 20 "linking coefficients" }{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..8);" }}{PARA 12 " " 1 "" {XPPMATH 20 "6>/&%\"aG6$\"\"#\"\"\"#F(\"\"(/&F%6$\"\"$F(#\"\"% \"#\")/&F%6$F.F'#\"#9F1/&F%6$F0F(#\"$\"H\"%s8/&F%6$F0F'#!#F\"#\\/&F%6$ F0F.#\"%`5F\"%hQ/&F%6$\"\"'F(#!$n#\"&GD#/&F%6$F^oF'#\"$*=\" $/(/&F%6$F^oF.#\"&*4j\"'Gde/&F%6$F^oF0#\"&`'e\"'!3m$/&F%6$F^oFK#\"%/&F%6$FasF'#!#@\"#))/&F%6$FasF.#\"'>&>#\"' wxM/&F%6$FasF0#\"'2Q;\"'Sm#*/&F%6$FasFK#!$<%\"$S'/&F%6$FasF^o#\"%W:\"% R:/&F%6$FasF*\"\"!" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 " " {TEXT 259 30 "weights for the 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#\"\"\"#\"#z\"%!3\"/&F% 6#\"\"#\"\"!/&F%6#\"\"$#\"&$o>\"&g\"p/&F%6#\"\"%#\"&2o\"\"&SU)/&F%6#\" \"&F//&F%6#\"\"'#\"%;G\"%&p(/&F%6#\"\"(#F'\"$+\"/&F%6#\"\")#\"$(=\"%+G " }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT 259 30 "wei ghts for the order 5 scheme" }{TEXT -1 2 ": " }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 37 "seq(`b*`[i]=subs(ee,`b*`[i]),i=1..8);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6*/&%#b*G6#\"\"\"#\"$j(\"&+3\"/&F%6#\"\"#\"\"!/&F %6#\"\"$#\"&\\!f\"'+w>/&F%6#\"\"%#\"&P)))\"'+l_/&F%6#\"\"&#\"$V#\"%+S/ &F%6#\"\"'#\"&_B\"\"&v%Q/&F%6#\"\"(F//&F%6#\"\")#F.\"#D" }}}{PARA 0 " " 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}} }{PARA 0 "" 0 "" {TEXT -1 52 "#--------------------------------------- ------------" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}}{PARA 0 "" 0 "" {TEXT -1 32 "#============== =================" }}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 34 "a modificati on of Verner's scheme " }}{PARA 257 "" 0 "" {TEXT -1 0 "" }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "matrix([[1/6, 1/6, ``, ``, ``, \+ ``, ``, ``, ``], [2/9, 2/27, 4/27, ``, ``, ``, ``, ``, ``], [3/7, 183/ 1372, -162/343, 1053/1372, ``, ``, ``, ``, ``], [11/16, 1475/6144, -51 3/2048, 5427/53248, 95207/159744, ``, ``, ``, ``], [3/4, 3/88, 81/352, 44631/613184, 58653/265408, 4104/21373, ``, ``, ``], [1, 28909/24948, -120/77, -29605749/25485460, 17354309/5038605, -21971456/6732495, 384 688/161595, ``, ``], [1, 817/39204, -30/121, 31908963/40048580, -10850 56/55424655, -5961728/10579635, 257344/253935, 0, ``], [b, 79/1080, 0, 19683/69160, 16807/84240, 0, 2816/7695, 1/100, 187/2800], [`b*`, 2203 9/308880, 0, 5071653/17210960, 11339923/63516960, 75776/1389245, 32168 /100035, 0, 33/416]])" "6#-%'matrixG6#7+7+*&\"\"\"F)\"\"'!\"\"*&F)F)F* F+%!GF-F-F-F-F-F-7+*&\"\"#F)\"\"*F+*&F0F)\"#FF+*&\"\"%F)F3F+F-F-F-F-F- F-7+*&\"\"$F)\"\"(F+*&\"$$=F)\"%s8F+,$*&\"$i\"F)\"$V$F+F+*&\"%`5F)F#F)\"(&\\KnF+F+*&\"')o%QF)\"'&fh\"F+F-F-7 +F)*&\"$<)F)\"&/#RF+,$*&\"#IF)\"$@\"F+F+*&\")j*3>$F)\")!e[+%F+,$*&\"(c ]3\"F)\")bYUbF+F+,$*&\"(G<'fF)\")N'z0\"F+F+*&\"'WtDF)\"'NRDF+\"\"!F-7+ %\"bG*&\"#zF)\"%!3\"F+Fjq*&\"&$o>F)\"&g\"pF+*&\"&2o\"F)\"&SU)F+Fjq*&\" %;GF)\"%&p(F+*&F)F)\"$+\"F+*&\"$(=F)\"%+GF+7+%#b*G*&\"&R?#F)\"'!))3$F+ Fjq*&\"(`;2&F)\")g4@ " 0 "" {MPLTEXT 1 0 1 ";" }}}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 40 "ee: co efficients for the combined scheme" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 874 "ee := \{c[2]=1/6,c[3]=2/9,c [4]=3/7,c[5]=11/16,c[6]=3/4,c[7]=1,c[8]=1,\n\na[2,1]=1/6,\na[3,1]=2/27 ,\na[3,2]=4/27,\na[4,1]=183/1372,\na[4,2]=-162/343,\na[4,3]=1053/1372, \na[5,1]=1475/6144,\na[5,2]=-513/2048,\na[5,3]=5427/53248,\na[5,4]=952 07/159744,\na[6,1]=3/88,\na[6,2]=81/352,\na[6,3]=44631/613184,\na[6,4] =58653/265408,\na[6,5]=4104/21373,\na[7,1]=28909/24948,\na[7,2]=-120/7 7,\na[7,3]=-29605749/25485460,\na[7,4]=17354309/5038605,\na[7,5]=-2197 1456/6732495,\na[7,6]=384688/161595,\na[8,1]=817/39204,\na[8,2]=-30/12 1,\na[8,3]=31908963/40048580,\na[8,4]=-1085056/55424655,\na[8,5]=-5961 728/10579635,\na[8,6]=257344/253935,\na[8,7]=0,\n\nb[1]=79/1080,\nb[2] =0,\nb[3]=19683/69160,\nb[4]=16807/84240,\nb[5]=0,\nb[6]=2816/7695,\nb [7]=1/100,\nb[8]=187/2800,\n\n`b*`[1]=22039/308880,\n`b*`[2]=0,\n`b*`[ 3]=5071653/17210960,\n`b*`[4]=11339923/63516960,\n`b*`[5]=75776/138924 5,\n`b*`[6]=32168/100035,\n`b*`[7]=0,\n`b*`[8]=33/416\}:" }}}{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 0 155 "RK6_8eqs := [op(RowSumConditions(8,'expanded')),op(OrderConditi ons(6,8,'expanded'))]:\n`RK5_8eqs*` := subs(b=`b*`,OrderConditions(5,8 ,'expanded')):\nnops(%);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#\"#<" }}} {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 7 "Check: " } }{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 128 "simplify(subs(ee,RK6_8eqs) ):\nmap(u->lhs(u)-rhs(u),%);\nnops(%);\nsimplify(subs(ee,`RK5_8eqs*`)) :\nmap(u->lhs(u)-rhs(u),%);\nnops(%);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#7N\"\"!F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$ F$F$F$F$F$F$F$F$F$F$F$F$F$" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#\"#W" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#73\"\"!F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F $" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#\"#<" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 86 "Next we set-up stage-order cond tions to check for stage-orders from 2 to 5 inclusive. " }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 85 "for ct from 2 to 5 do\n so||ct||_ 8 := StageOrderConditions(ct,8,'expanded');\nend do:" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 34 "Stages 3 to 8 have sta ge-order 2. " }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 173 "[seq([seq(e xpand(subs(ee,so||i||_8[j])),i=2..5)],j=1..6)]:\nmap(proc(L) local i; \+ for i to nops(L) do if not evalb(L[i]) then break end if end do; i end proc,%):\nsimplify(%); " }}{PARA 11 "" 1 "" {XPPMATH 20 "6#7(\"\"#F$F $F$F$F$" }}}{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#$\"+AM!R7#!#8" } }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 79 "The 2 n orm of the principal error of the order 5 embedded scheme is as follow s." }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 153 "`errterms5_8*` := sub s(b=`b*`,PrincipalErrorTerms(5,8,'expanded')):\nevalf(evalf[14](sqrt(a dd(subs(ee,`errterms5_8*`[i])^2,i=1.. nops(`errterms5_8*`)))));" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#$\"+UDJ-:!#8" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 52 "Two of the principal erro r conditions are satisfied." }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 291 "ind := []:\nfor ct to nops(errterms6_8) do\n if simplify(subs(e e,errterms6_8[ct]))=0 then ind := [op(ind),ct] end if:\nend do:\nerr6 \+ := PrincipalErrorConditions(6):\n[seq([i,err6[i]],i=ind)]:\nlinalg[aug ment](linalg[delcols](%,2..2),matrix([[` `]$(linalg[rowdim](%))]),lin alg[delcols](%,1..1));" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#K%'matrixG6# 7$7%\"\"\"%#~~G/*&%\"bGF(-%!G6#*&%\"aGF(-F.6#*&F1F(-F.6#*&F1F(-F.6#*&F 1F(-F.6#*&F1F(%\"cGF(F(F(F(F(F(#F(\"%S]7%\"\"&F)/*(F,F(F>F(F2F(#F(\"$S )Q(pprint46\"" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}}{PARA 0 "" 0 "" {TEXT -1 52 "#------------ ---------------------------------------" }}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 35 "construction of the combined scheme" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 196 "First we construct the 8 stage order 6 scheme by making use of a selection of \"simple\" order conditions and incorporating the stage-order equations to ensure that stages 3 to 8 have stage-order 2." }}{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$F+%\"jGF,/F+;,&F0F, F,F,\"\")*&&F)6#F0F,,&F,F,&%\"cG6#F0!\"\"F," }{TEXT -1 1 " " }}{PARA 0 "" 0 "" {TEXT -1 5 "for " }{XPPEDIT 18 0 "j = 4;" "6#/%\"jG\"\"%" } {TEXT -1 18 " . . . 7. ( with " }{XPPEDIT 18 0 "c[1]=0" "6#/&%\"cG6# \"\"\"\"\"!" }{TEXT -1 55 " ), together with the further simplifying \+ conditions: " }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "Sum(b [i]*c[i]*a[i,2],i=3..8)=0" "6#/-%$SumG6$*(&%\"bG6#%\"iG\"\"\"&%\"cG6#F +F,&%\"aG6$F+\"\"#F,/F+;\"\"$\"\")\"\"!" }{TEXT -1 8 ", " } {XPPEDIT 18 0 "Sum(b[i]*c[i]*Sum(a[i,j]*a[j,2],j=3..i-1),i=3..8)=0" "6 #/-%$SumG6$*(&%\"bG6#%\"iG\"\"\"&%\"cG6#F+F,-F%6$*&&%\"aG6$F+%\"jGF,&F 46$F6\"\"#F,/F6;\"\"$,&F+F,F,!\"\"F,/F+;F<\"\")\"\"!" }{TEXT -1 8 ", \+ " }{XPPEDIT 18 0 "Sum(b[i]*c[i]^2*a[i,2],i=3..8)=0" "6#/-%$SumG6$ *(&%\"bG6#%\"iG\"\"\"*$&%\"cG6#F+\"\"#F,&%\"aG6$F+F1F,/F+;\"\"$\"\")\" \"!" }{TEXT -1 2 ". " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 " " {TEXT -1 32 "The scheme is constructed with " }{XPPEDIT 18 0 "a[7,3 ];" "6#&%\"aG6$\"\"(\"\"$" }{TEXT -1 3 ", " }{XPPEDIT 18 0 "a[7,4]" " 6#&%\"aG6$\"\"(\"\"%" }{TEXT -1 3 ", " }{XPPEDIT 18 0 "a[7,5]" "6#&% \"aG6$\"\"(\"\"&" }{TEXT -1 7 " and " }{XPPEDIT 18 0 "a[7,6]" "6#&% \"aG6$\"\"(\"\"'" }{TEXT -1 17 " as parameters. " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 76 "The simple order conditio ns used are given (in abreviated form) as follows. " }}{PARA 0 "" 0 " " {TEXT -1 94 "These include the 6 order 6 quadrature conditions toget her with 4 additional order conditions." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 184 "SO6 := SimpleOrderCond itions(6):\n[seq([i,SO6[i]],i=[1,2,4,8,13,16,24,28,29,32])]:\nlinalg[a ugment](linalg[delcols](%,2..2),matrix([[` `]$(linalg[rowdim](%))]),l inalg[delcols](%,1..1));" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#K%'matrixG 6#7,7%\"\"\"%#~~G/*&%\"bGF(%\"eGF(F(7%\"\"#F)/*&F,F(%\"cGF(#F(F/7%\"\" %F)/*&F,F()F2F/F(#F(\"\"$7%\"\")F)/*&F,F()F2F:F(#F(F57%\"#8F)/*(F,F(F2 F(-%!G6#*&F8F(%\"aGF(F(#F(\"#:7%\"#;F)/*&F,F()F2F5F(#F(\"\"&7%\"#CF)/* (F,F(F2F(-FF6#*&FIF(FEF(F(#F(\"#s7%\"#GF)/*(F,F(F8F(FEF(#F(\"#=7%\"#HF )/*(F,F(F2F(-FF6#*&F?F(FIF(F(#F(FT7%\"#KF)/*&F,F()F2FRF(#F(\"\"'Q)ppri nt816\"" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 513 "SO_eqs := [op(RowSumConditions(8,'expanded')),op(Sta geOrderConditions(2,8,'expanded'))]:\nSO6_8 := SimpleOrderConditions(6 ,8,'expanded'):\nord_cdns := [seq(SO6_8[i],i=[1,2,4,8,13,16,24,28,29,3 2])]:\nsimp_cdns := [add(b[i]*a[i,1],i=1+1..8)=b[1],seq(add(b[i]*a[i,j ],i=j+1..8)=b[j]*(1-c[j]),j=[4,5,6,7])]:\nsimp_cdns2 := [add(b[i]*c[i] *a[i,2],i=3..8)=0,add(b[i]*c[i]*add(a[i,j]*a[j,2],j=3..i-1),i=3..8)=0, \n add(b[i]*c[i]^2*a[i,2],i=3..8)=0]:\ncdns := [op(ord_ cdns),op(simp_cdns),op(simp_cdns2),op(SO_eqs)]:" }}}{EXCHG {PARA 0 "> \+ " 0 "" {MPLTEXT 1 0 1 ";" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 " " 0 "" {TEXT -1 21 "We specify the nodes:" }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "c[2] = 1/6;" "6#/&%\"cG6#\"\"#*&\"\"\"F)\"\"' !\"\"" }{TEXT -1 3 ", " }{XPPEDIT 18 0 "c[3]=2/9" "6#/&%\"cG6#\"\"$*& \"\"#\"\"\"\"\"*!\"\"" }{TEXT -1 3 ", " }{XPPEDIT 18 0 "c[5] = 11/16; " "6#/&%\"cG6#\"\"&*&\"#6\"\"\"\"#;!\"\"" }{TEXT -1 3 ", " }{XPPEDIT 18 0 "c[6] = 3/4;" "6#/&%\"cG6#\"\"'*&\"\"$\"\"\"\"\"%!\"\"" }{TEXT -1 3 ", " }{XPPEDIT 18 0 "c[7]=1" "6#/&%\"cG6#\"\"(\"\"\"" }{TEXT -1 3 ", " }{XPPEDIT 18 0 "c[8]=1" "6#/&%\"cG6#\"\")\"\"\"" }{TEXT -1 2 " , " }}{PARA 0 "" 0 "" {TEXT -1 17 "and the weights: " }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "b[2]=0" "6#/&%\"bG6#\"\"#\"\"!" } {TEXT -1 3 ", " }{XPPEDIT 18 0 "b[7] = 1/100;" "6#/&%\"bG6#\"\"(*&\" \"\"F)\"$+\"!\"\"" }{TEXT -1 2 ". " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 172 "e1 := \{c[2]=1/6,c[3]=2/9,c [5]=11/16,c[6]=3/4,c[7]=1,c[8]=1,b[2]=0,b[7]=1/100\}:\neqns := subs(e1 ,cdns):\nnops(eqns);\nindets(eqns) minus \{a[7,3],a[7,4],a[7,5],a[7,6] \};\nnops(%);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#\"#J" }}{PARA 11 "" 1 "" {XPPMATH 20 "6# " 0 "" {MPLTEXT 1 0 22 "infolevel[solve] := 4:" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 118 "e2 := solve(\{op(eqns)\},indets(eqns) mi nus \{a[7,3],a[7,4],a[7,5],a[7,6]\}):\ninfolevel[solve] := 0:\ne3 := ` union`(e1,e2):" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> \+ " 0 "" {MPLTEXT 1 0 1 ";" }}}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 3 "e3 " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 937 "e3 := \{a[6,5] = 4104/21373, b[7] = 1/100, b[4] = 16807/84240, \+ a[8,3] = -28/187*a[7,3]+700875/1125332, c[7] = 1, c[8] = 1, b[2] = 0, \+ a[8,7] = 0, a[6,2] = 81/352, b[5] = 0, b[6] = 2816/7695, c[3] = 2/9, a [2,1] = 1/6, a[8,1] = -28/561*a[7,3]+9973/20196-4/17*a[7,4]-175/374*a[ 7,5]-98/187*a[7,6], c[4] = 3/7, b[1] = 79/1080, a[4,1] = 183/1372, a[6 ,1] = 3/88, a[5,4] = 95207/159744, b[3] = 19683/69160, a[6,4] = 58653/ 265408, a[6,3] = 44631/613184, a[5,1] = 1475/6144, a[4,2] = -162/343, \+ a[5,3] = 5427/53248, a[8,2] = 112/561*a[7,3]-174/187+72/187*a[7,4]+21/ 34*a[7,5]+126/187*a[7,6], a[8,6] = 35840/26163-28/187*a[7,6], b[8] = 1 87/2800, a[3,2] = 4/27, a[7,1] = 1/3*a[7,3]+11/7*a[7,4]+25/8*a[7,5]+7/ 2*a[7,6]-2, a[7,2] = -4/3*a[7,3]-18/7*a[7,4]-33/8*a[7,5]-9/2*a[7,6]+3, a[8,5] = -1146880/1090023-28/187*a[7,5], a[3,1] = 2/27, a[8,4] = 2833 180/5710419-28/187*a[7,4], a[5,2] = -513/2048, c[2] = 1/6, c[6] = 3/4, c[5] = 11/16, a[4,3] = 1053/1372\}:" }{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 0 452 "subs(e3,matrix([seq([c[i],seq(a[i,j],j=1..i-1),``$(7-i)],i=2..6)] ));\nprint(``);\nfor ii from 7 to 8 do\n print(c[ii]=subs(e3,c[ii])) ;print(``); \n for jj to ii-1 do\n coeff_val := subs(e3,a[ii,jj ]);\n if coeff_val=a[ii,jj] then \n print(a[ii,jj]*` is a parameter.`)\n else\n print(a[ii,jj]=coeff_val);\n \+ end if;\n end do:\n print(`__________________________________`):\n end do:\nprint(``);\nseq(b[i]=subs(e3,b[i]),i=1..8);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#K%'matrixG6#7'7)#\"\"\"\"\"'F(%!GF+F+F+F+7)#\"\"#\" \"*#F.\"#F#\"\"%F1F+F+F+F+7)#\"\"$\"\"(#\"$$=\"%s8#!$i\"\"$V$#\"%`5F:F +F+F+7)#\"#6\"#;#\"%v9\"%Wh#!$8&\"%[?#\"%Fa\"&[K&#\"&2_*\"'W(f\"F+F+7) #F6F3#F6\"#))#\"#\")\"$_$#\"&JY%\"'%=8'#\"&`'e\"'3aE#\"%/T\"&t8#F+Q(pp rint76\"" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%!G" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/&%\"cG6#\"\"(\"\"\"" }}{PARA 11 "" 1 "" {XPPMATH 20 "6 #%!G" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/&%\"aG6$\"\"(\"\"\",,*&#F(\" \"$F(&F%6$F'F,F(F(*&#\"#6F'F(&F%6$F'\"\"%F(F(*&#\"#D\"\")F(&F%6$F'\"\" &F(F(*&#F'\"\"#F(&F%6$F'\"\"'F(F(F>!\"\"" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/&%\"aG6$\"\"(\"\"#,,*&#\"\"%\"\"$\"\"\"&F%6$F'F-F.!\"\"*&#\"#=F 'F.&F%6$F'F,F.F1*&#\"#L\"\")F.&F%6$F'\"\"&F.F1*&#\"\"*F(F.&F%6$F'\"\"' F.F1F-F." }}{PARA 11 "" 1 "" {XPPMATH 20 "6#*&&%\"aG6$\"\"(\"\"$\"\"\" %1~is~a~parameter.GF)" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#*&&%\"aG6$\" \"(\"\"%\"\"\"%1~is~a~parameter.GF)" }}{PARA 11 "" 1 "" {XPPMATH 20 "6 #*&&%\"aG6$\"\"(\"\"&\"\"\"%1~is~a~parameter.GF)" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#*&&%\"aG6$\"\"(\"\"'\"\"\"%1~is~a~parameter.GF)" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#%C__________________________________G " }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/&%\"cG6#\"\")\"\"\"" }}{PARA 11 " " 1 "" {XPPMATH 20 "6#%!G" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/&%\"aG6$ \"\")\"\"\",,*&#\"#G\"$h&F(&F%6$\"\"(\"\"$F(!\"\"#\"%t**\"&'>?F(*&#\" \"%\"#/r&\"\"\"*&#\"#G\"$(=F -&F%6$\"\"(F(F-!\"\"" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/&%\"aG6$\"\") \"\"&,&#\"(!)o9\"\"(B+4\"!\"\"*&#\"#G\"$(=\"\"\"&F%6$\"\"(F(F2F-" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#/&%\"aG6$\"\")\"\"',&#\"&Se$\"&jh#\"\" \"*&#\"#G\"$(=F-&F%6$\"\"(F(F-!\"\"" }}{PARA 11 "" 1 "" {XPPMATH 20 "6 #/&%\"aG6$\"\")\"\"(\"\"!" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%C_______ ___________________________G" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%!G" } }{PARA 11 "" 1 "" {XPPMATH 20 "6*/&%\"bG6#\"\"\"#\"#z\"%!3\"/&F%6#\"\" #\"\"!/&F%6#\"\"$#\"&$o>\"&g\"p/&F%6#\"\"%#\"&2o\"\"&SU)/&F%6#\"\"&F// &F%6#\"\"'#\"%;G\"%&p(/&F%6#\"\"(#F'\"$+\"/&F%6#\"\")#\"$(=\"%+G" }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 66 "We now turn our attention to th e embedded 8 stage, order 5 scheme." }}{PARA 0 "" 0 "" {TEXT -1 35 "We use the simplifying conditions: " }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "Sum(`b*`[i]*a[i,2],i = 3 .. 8) = 0;" "6#/-%$SumG6$*&&% #b*G6#%\"iG\"\"\"&%\"aG6$F+\"\"#F,/F+;\"\"$\"\")\"\"!" }{TEXT -1 8 ", \+ " }{XPPEDIT 18 0 "Sum(`b*`[i]*Sum(a[i,j]*a[j,2],j = 3 .. i-1),i \+ = 2 .. 8) = 0;" "6#/-%$SumG6$*&&%#b*G6#%\"iG\"\"\"-F%6$*&&%\"aG6$F+%\" jGF,&F16$F3\"\"#F,/F3;\"\"$,&F+F,F,!\"\"F,/F+;F6\"\")\"\"!" }{TEXT -1 1 "." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 101 " We use a selection of \"simple\" order conditions as given (in abrevia ted form) in the following table." }}{PARA 0 "" 0 "" {TEXT -1 95 "Thes e include the 5 order 5 quadrature conditions together with one additi onal order condition." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 191 "`SO5*` := subs(b=`b*`,SimpleOrderConditi ons(5)):\n[seq([i,`SO5*`[i]],i=[1,2,4,8,15,16])]:\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/*&%#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%\"#:F)/*&F,F(-%!G6#*&F?F( %\"aGF(F(#F(\"#?7%\"#;F)/*&F,F()F2F5F(#F(\"\"&Q)pprint866\"" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 262 "` SO5_8*` := subs(b=`b*`,SimpleOrderConditions(5,8,'expanded')):\n`ord_c dns*` := [seq(`SO5_8*`[i],i=[1,2,4,8,15,16])]:\n`simp_cdns*` := [add(` b*`[i]*a[i,2],i=3..8)=0,add(`b*`[i]*add(a[i,j]*a[j,2],j=3..i-1),i=2..8 )=0]:\n`cdns*` := [op(`ord_cdns*`),op(`simp_cdns*`)]: " }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 17 "We specify that " } {XPPEDIT 18 0 "`b*`[2] = 0;" "6#/&%#b*G6#\"\"#\"\"!" }{TEXT -1 3 ", \+ " }{XPPEDIT 18 0 "`b*`[7] = 0;" "6#/&%#b*G6#\"\"(\"\"!" }{TEXT -1 7 " \+ and " }{XPPEDIT 18 0 "`b*`[8] = 33/416;" "6#/&%#b*G6#\"\")*&\"#L\"\" \"\"$;%!\"\"" }{TEXT -1 2 ". " }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 65 "e4 := \{`b*`[2]=0,`b*`[7]=0,`b*`[8]=33/416\}:\ne5 := `union`(e3, e4):" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 84 "`eqns*` := subs(e5,`cdns*`):\nnops(`eqns*`);\nindets( `eqns*`) minus \{a[7,6]\};\nnops(%);" }}{PARA 11 "" 1 "" {XPPMATH 20 " 6#\"\")" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#<*&%\"aG6$\"\"(\"\"$&F%6$F' \"\"%&F%6$F'\"\"&&%#b*G6#\"\"\"&F06#F(&F06#F+&F06#F.&F06#\"\"'" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#\"\")" }}}{PARA 0 "" 0 "" {TEXT -1 0 " " }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 22 "infolevel[solve] := 4:" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 112 "e6 := solve(\{op(`eqns*` )\},indets(`eqns*`) minus \{a[7,6]\}):\ne7 := `union`(subs(e6,e5),e6): \ninfolevel[solve] := 0:" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 " " 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 39 "The order 5 scheme is given as follows." }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 94 "sub s(e7,matrix([seq([c[i],seq(a[i,j],j=1..i-1),``$(9-i)],i=2..8),\n[``,se q(`b*`[i],i=1..8)]]));" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#K%'matrixG6# 7*7+#\"\"\"\"\"'F(%!GF+F+F+F+F+F+7+#\"\"#\"\"*#F.\"#F#\"\"%F1F+F+F+F+F +F+7+#\"\"$\"\"(#\"$$=\"%s8#!$i\"\"$V$#\"%`5F:F+F+F+F+F+7+#\"#6\"#;#\" %v9\"%Wh#!$8&\"%[?#\"%Fa\"&[K&#\"&2_*\"'W(f\"F+F+F+F+7+#F6F3#F6\"#))# \"#\")\"$_$#\"&JY%\"'%=8'#\"&`'e\"'3aE#\"%/T\"&t8#F+F+F+7+F),&#\"%BH\" %!y$F)*&#\"#dFVF)&%\"aG6$F7F*F)F)#!$?\"\"#x,&#\"'$y'p\"(S8M\"F)*&#\"&$ o>\"&sy#F)FboF)!\"\",&#\"'[$)[\"'X)f&F)*&#\"%c%\"''*>GF)FboF)F`p,&#\"*/&H)*>\"*&z` )z\"F`p*&#\"''H>*\"(^n*RF)FboF)F),&#\"&Se$\"&jh#F)*&#\"#G\"$(=F)FboF)F `p\"\"!F+7+F+#\"&R?#\"'!))3$F\\t#\"(`;2&\")g4@<#\")B*R8\"\")gp^j#\"&wd (\"(X#*Q\"#\"&o@$\"'N+5F\\t#\"#L\"$;%Q(pprint16\"" }}}{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 1113 "e7 := \{c[8] = 1, c[7] = 1 , b[2] = 0, a[5,1] = 1475/6144, a[4,3] = 1053/1372, `b*`[8] = 33/416, \+ c[4] = 3/7, b[3] = 19683/69160, a[8,7] = 0, b[8] = 187/2800, a[8,6] = \+ 35840/26163-28/187*a[7,6], a[5,2] = -513/2048, a[7,3] = 696783/1341340 -19683/27872*a[7,6], a[6,3] = 44631/613184, a[5,4] = 95207/159744, b[1 ] = 79/1080, a[6,1] = 3/88, b[5] = 0, a[3,1] = 2/27, `b*`[7] = 0, a[3, 2] = 4/27, a[6,5] = 4104/21373, b[4] = 16807/84240, `b*`[2] = 0, c[6] \+ = 3/4, c[3] = 2/9, c[5] = 11/16, c[2] = 1/6, b[7] = 1/100, a[2,1] = 1/ 6, a[8,1] = 7933/100980-399/16456*a[7,6], a[8,3] = 371073867/680825860 +137781/1303016*a[7,6], a[8,5] = -199829504/179853795+919296/3996751*a [7,6], a[7,1] = 2923/3780+57/352*a[7,6], a[8,4] = 114803668/314073045- 45619/281996*a[7,6], `b*`[1] = 22039/308880, `b*`[6] = 32168/100035, ` b*`[5] = 75776/1389245, `b*`[4] = 11339923/63516960, a[7,5] = 378368/9 61785-32832/21373*a[7,6], a[7,4] = 488348/559845+6517/6032*a[7,6], a[4 ,2] = -162/343, b[6] = 2816/7695, `b*`[3] = 5071653/17210960, a[6,4] = 58653/265408, a[4,1] = 183/1372, a[5,3] = 5427/53248, a[6,2] = 81/352 , a[7,2] = -120/77, a[8,2] = -30/121\}:" }}}{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 0 146 "RK6_8eqs \+ := [op(RowSumConditions(8,'expanded')),op(OrderConditions(6,8,'expande d'))]:\n`RK5_8eqs*` := subs(b=`b*`,OrderConditions(5,8,'expanded')):" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 7 "Check: \+ " }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 129 "simplify(subs(e7,RK6_8e qs)):\nmap(u->lhs(u)-rhs(u),%);\n;nops(%);\nsimplify(subs(e7,`RK5_8eqs *`)):\nmap(u->lhs(u)-rhs(u),%);\nnops(%);" }}{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#\"#<" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 21 "We find a value for " }{XPPEDIT 18 0 " a[7,6]" "6#&%\"aG6$\"\"(\"\"'" }{TEXT -1 39 " by applying the simplif ying condition" }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "Sum (`b*`[i]*a[i,j],i = j+1 .. s) = `b*`[j]*(1-c[j]);" "6#/-%$SumG6$*&&%#b *G6#%\"iG\"\"\"&%\"aG6$F+%\"jGF,/F+;,&F0F,F,F,%\"sG*&&F)6#F0F,,&F,F,&% \"cG6#F0!\"\"F," }{TEXT -1 5 ", " }{XPPEDIT 18 0 "j = 2" "6#/%\"jG \"\"#" }{TEXT -1 7 " . . . " }{TEXT 300 1 "s" }{TEXT -1 1 "," }}{PARA 0 "" 0 "" {TEXT -1 6 "with " }{XPPEDIT 18 0 "s=8" "6#/%\"sG\"\")" } {TEXT -1 23 " for the eighth stage." }}{PARA 0 "" 0 "" {TEXT -1 8 "Ta king " }{XPPEDIT 18 0 "s = 8;" "6#/%\"sG\"\")" }{TEXT -1 7 " and " }{XPPEDIT 18 0 "j = 6;" "6#/%\"jG\"\"'" }{TEXT -1 9 ", we have" }} {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#/,&*&&%#b*G6#\"\"(\"\"\"&%\"aG6$F)\" \"'F*F**&&F'6#\"\")F*&F,6$F2F.F*F**&&F'6#F.F*,&F*F*&%\"cG6#F.!\"\"F*" }{TEXT -1 2 ". " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 93 "Applying this equation to embedded order 5 scheme gives a n equation which can be solved for " }{XPPEDIT 18 0 "a[7,6];" "6#&%\" aG6$\"\"(\"\"'" }{TEXT -1 1 "." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 90 "subs(j=6,'add'(`b*`[i]*a[i,j ],i=j+1..8)=`b*`[j]*(1-c[j]));\nvalue(%);\nsubs(e7,%);\nsolve(%);" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#/-%$addG6$*&&%#b*G6#%\"iG\"\"\"&%\"aG6 $F+\"\"'F,/F+;\"\"(\"\")*&&F)6#F0F,,&F,F,&%\"cGF7!\"\"F," }}{PARA 11 " " 1 "" {XPPMATH 20 "6#/,&*&&%#b*G6#\"\"(\"\"\"&%\"aG6$F)\"\"'F*F**&&F' 6#\"\")F*&F,6$F2F.F*F**&&F'6#F.F*,&F*F*&%\"cGF7!\"\"F*" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/,&#\"&?B\"\"'tL6\"\"\"*&#\"#@\"%o " 0 "" {MPLTEXT 1 0 68 "e8 := `union`(subs(a[7,6]=384688/161595,e7),\{a[ 7,6]=384688/161595\}):" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 2 "e8" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 979 "e8 := \{c[8] = 1, c[7] = 1, b[2] = 0, a[5,1] = 1475/ 6144, a[4,3] = 1053/1372, `b*`[8] = 33/416, c[4] = 3/7, b[3] = 19683/6 9160, a[8,7] = 0, b[8] = 187/2800, a[5,2] = -513/2048, a[6,3] = 44631/ 613184, a[7,4] = 17354309/5038605, a[7,5] = -21971456/6732495, a[5,4] \+ = 95207/159744, b[1] = 79/1080, a[6,1] = 3/88, b[5] = 0, a[3,1] = 2/27 , `b*`[7] = 0, a[3,2] = 4/27, a[6,5] = 4104/21373, b[4] = 16807/84240, `b*`[2] = 0, c[6] = 3/4, c[3] = 2/9, c[5] = 11/16, c[2] = 1/6, b[7] = 1/100, a[2,1] = 1/6, a[7,3] = -29605749/25485460, `b*`[1] = 22039/308 880, `b*`[6] = 32168/100035, `b*`[5] = 75776/1389245, `b*`[4] = 113399 23/63516960, a[4,2] = -162/343, b[6] = 2816/7695, `b*`[3] = 5071653/17 210960, a[7,6] = 384688/161595, a[6,4] = 58653/265408, a[4,1] = 183/13 72, a[5,3] = 5427/53248, a[6,2] = 81/352, a[8,1] = 817/39204, a[8,6] = 257344/253935, a[8,3] = 31908963/40048580, a[8,5] = -5961728/10579635 , a[7,1] = 28909/24948, a[7,2] = -120/77, a[8,4] = -1085056/55424655, \+ a[8,2] = -30/121\}:" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}}{PARA 0 "" 0 "" {TEXT -1 0 "" }} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 119 "subs(e8,matrix([seq([c[i],s eq(a[i,j],j=1..i-1),``$(9-i)],i=2..8),\n[`b`,seq(b[i],i=1..8)],[`b*`,s eq(`b*`[i],i=1..8)]]));" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#K%'matrixG6 #7+7+#\"\"\"\"\"'F(%!GF+F+F+F+F+F+7+#\"\"#\"\"*#F.\"#F#\"\"%F1F+F+F+F+ F+F+7+#\"\"$\"\"(#\"$$=\"%s8#!$i\"\"$V$#\"%`5F:F+F+F+F+F+7+#\"#6\"#;# \"%v9\"%Wh#!$8&\"%[?#\"%Fa\"&[K&#\"&2_*\"'W(f\"F+F+F+F+7+#F6F3#F6\"#)) #\"#\")\"$_$#\"&JY%\"'%=8'#\"&`'e\"'3aE#\"%/T\"&t8#F+F+F+7+F)#\"&4*G\" &[\\##!$?\"\"#x#!)\\dgH\")ga[D#\")4VN<\"(0'Q]#!)c9(>#\"(&\\Kn#\"')o%Q \"'&fh\"F+F+7+F)#\"$<)\"&/#R#!#I\"$@\"#\")j*3>$\")!e[+%#!(c]3\"\")bYUb #!(G<'f\")N'z0\"#\"'WtD\"'NRD\"\"!F+7+%\"bG#\"#z\"%!3\"F`q#\"&$o>\"&g \"p#\"&2o\"\"&SU)F`q#\"%;G\"%&p(#F)\"$+\"#\"$(=\"%+G7+%#b*G#\"&R?#\"'! ))3$F`q#\"(`;2&\")g4@<#\")B*R8\"\")gp^j#\"&wd(\"(X#*Q\"#\"&o@$\"'N+5F` q#\"#L\"$;%Q(pprint36\"" }}}{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 31 "note concerning the value for " }{XPPEDIT 18 0 "a[7, 6];" "6#&%\"aG6$\"\"(\"\"'" }{TEXT -1 1 " " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 46 "e7: coefficients of the combined scheme with " }{XPPEDIT 18 0 "a[7,6]" "6#&%\"aG6$\"\"(\"\"'" }{TEXT -1 16 " as a parameter" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> \+ " 0 "" {MPLTEXT 1 0 1113 "e7 := \{c[8] = 1, c[7] = 1, b[2] = 0, a[5,1] = 1475/6144, a[4,3] = 1053/1372, `b*`[8] = 33/416, c[4] = 3/7, b[3] = 19683/69160, a[8,7] = 0, b[8] = 187/2800, a[8,6] = 35840/26163-28/187 *a[7,6], a[5,2] = -513/2048, a[7,3] = 696783/1341340-19683/27872*a[7,6 ], a[6,3] = 44631/613184, a[5,4] = 95207/159744, b[1] = 79/1080, a[6,1 ] = 3/88, b[5] = 0, a[3,1] = 2/27, `b*`[7] = 0, a[3,2] = 4/27, a[6,5] \+ = 4104/21373, b[4] = 16807/84240, `b*`[2] = 0, c[6] = 3/4, c[3] = 2/9, c[5] = 11/16, c[2] = 1/6, b[7] = 1/100, a[2,1] = 1/6, a[8,1] = 7933/1 00980-399/16456*a[7,6], a[8,3] = 371073867/680825860+137781/1303016*a[ 7,6], a[8,5] = -199829504/179853795+919296/3996751*a[7,6], a[7,1] = 29 23/3780+57/352*a[7,6], a[8,4] = 114803668/314073045-45619/281996*a[7,6 ], `b*`[1] = 22039/308880, `b*`[6] = 32168/100035, `b*`[5] = 75776/138 9245, `b*`[4] = 11339923/63516960, a[7,5] = 378368/961785-32832/21373* a[7,6], a[7,4] = 488348/559845+6517/6032*a[7,6], a[4,2] = -162/343, b[ 6] = 2816/7695, `b*`[3] = 5071653/17210960, a[6,4] = 58653/265408, a[4 ,1] = 183/1372, a[5,3] = 5427/53248, a[6,2] = 81/352, a[7,2] = -120/77 , a[8,2] = -30/121\}:" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}}{PARA 0 "" 0 "" {TEXT -1 0 " " }}{PARA 0 "" 0 "" {TEXT -1 5 "Let " }{XPPEDIT 18 0 "T[6, 8];" "6#&% \"TG6$\"\"'\"\")" }{TEXT -1 127 " denote the vector whose components \+ are the principal error terms of the 8 stage, order 6 scheme (the erro r terms of order 7)." }}{PARA 0 "" 0 "" {TEXT -1 5 "Let " }{XPPEDIT 18 0 "`T*`[5,8];" "6#&%#T*G6$\"\"&\"\")" }{TEXT -1 145 " denote the v ector whose components are the principal error terms of the embedded 8 stage, order 5 scheme (the error terms of order 6) and let " } {XPPEDIT 18 0 "`T*`[6,8];" "6#&%#T*G6$\"\"'\"\")" }{TEXT -1 99 " deno te the vector whose components are the error terms of order 7 of the e mbedded order 5 scheme." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 39 "Denote the 2-norm of these vectors by " }{XPPEDIT 18 0 "abs(abs(T[6,8]));" "6#-%$absG6#-F$6#&%\"TG6$\"\"'\"\")" }{TEXT -1 3 ", " }{XPPEDIT 18 0 "abs(abs(`T*`[5,8]));" "6#-%$absG6#-F$6#&%#T *G6$\"\"&\"\")" }{TEXT -1 7 " and " }{XPPEDIT 18 0 "abs(abs(`T*`[6,8 ]));" "6#-%$absG6#-F$6#&%#T*G6$\"\"'\"\")" }{TEXT -1 15 " respectivel y." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 8 "Defi ne: " }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "A[7] = abs(ab s(T[6, 8]));" "6#/&%\"AG6#\"\"(-%$absG6#-F)6#&%\"TG6$\"\"'\"\")" } {TEXT -1 6 ", " }{XPPEDIT 18 0 "B[7] = abs(abs(`T*`[6,8]))/abs(abs (`T*`[5,8]));" "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*`[6,8]-T[6,8]))/abs(abs(`T*`[5,8]));" "6#/&%\"CG6#\"\" (*&-%$absG6#-F*6#,&&%#T*G6$\"\"'\"\")\"\"\"&%\"TG6$F2F3!\"\"F4-F*6#-F* 6#&F06$\"\"&F3F8" }{TEXT -1 1 "." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }} {PARA 0 "" 0 "" {TEXT -1 77 "Dormand and Prince have suggested that as well as attempting to ensure that " }{XPPEDIT 18 0 "A[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 "the y are \"similar in magnitude\" and also not differ too much from 1." } }{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 471 "errterms6_8 := PrincipalErrorTerms(6,8,'expanded'):\n`errterms5 _8*` := subs(b=`b*`,PrincipalErrorTerms(5,8,'expanded')):\n`errterms6_ 8*` := subs(b=`b*`,PrincipalErrorTerms(6,8,'expanded')):\nsnmB := sqrt (add(subs(e7,`errterms6_8*`[i])^2,i=1..nops(`errterms6_8*`))):\nsdnB : = sqrt(add(subs(e7,`errterms5_8*`[i])^2,i=1..nops(`errterms5_8*`))):\n snmC := sqrt(add((subs(e7,`errterms6_8*`[i])-subs(e7,errterms6_8[i]))^ 2,i=1..nops(errterms6_8))):\nB_7 := snmB/sdnB:\nC_7 := snmC/sdnB:" }}} {PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 101 "'B[7]' = evalf[8](eval(B_7,a[7,6]=384688/161595));\n'C[7]' = eval f[8](eval(C_7,a[7,6]=384688/161595));" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/&%\"BG6#\"\"($\")ac!)=!\"(" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/&% \"CG6#\"\"($\");/X9!\"(" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 " " 0 "" {TEXT -1 6 "When " }{XPPEDIT 18 0 "a[7,6]=384688/161595" "6#/& %\"aG6$\"\"(\"\"'*&\"')o%Q\"\"\"\"'&fh\"!\"\"" }{TEXT -1 11 ", we have " }{XPPEDIT 18 0 "B[7]" "6#&%\"BG6#\"\"(" }{TEXT -1 1 " " }{TEXT 301 1 "~" }{TEXT -1 15 " 1.8806 and " }{XPPEDIT 18 0 "C[7]" "6#&%\" CG6#\"\"(" }{TEXT -1 1 " " }{TEXT 302 1 "~" }{TEXT -1 9 " 1.4450. " }} {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 43 "The follo wing picture shows the graphs of " }{XPPEDIT 18 0 "B[7]" "6#&%\"BG6# \"\"(" }{TEXT -1 5 " in " }{TEXT 259 5 "brown" }{TEXT -1 6 " and " } {XPPEDIT 18 0 "C[7]" "6#&%\"CG6#\"\"(" }{TEXT -1 5 " in " }{TEXT 261 3 "red" }{TEXT -1 3 ". \n" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 523 "aa := 384688/161595:\nbb := evalf(eval(B_7,a[7,6]=aa)):\ncc := ev alf(eval(C_7,a[7,6]=aa)):\np1 := plot([B_7,C_7,1],a[7,6]=-2.52..4.35,0 ..2.8,color=[brown,red,blue]):\np2 := plot([[aa,0],[aa,2.9]],color=COL OR(RGB,0,.7,0)):\np3 := plot([[[aa,bb],[aa,cc]]$3],style=point,symbol= [circle,diamond,cross],color=[black,coral$2]):\np4 := plot([[[aa,0]]$3 ],style=point,symbol=[circle,diamond,cross],color=black):\np5 := plot( [[[aa,bb],[0,bb]],[[aa,cc],[0,cc]]],linestyle=3,color=[brown,red]):\np lots[display]([p||(1..5)],font=[HELVETICA,9]);" }}{PARA 13 "" 1 "" {GLPLOT2D 500 385 385 {PLOTDATA 2 "61-%'CURVESG6$7\\o7$$!3-++++++?D!#< 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makes " }{XPPEDIT 18 0 "C[7];" "6#&%\"CG6#\"\"(" } {TEXT -1 23 " close to its minimum." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 27 "minimize(C_7):\nevalf[8](% );" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#$\")i.X9!\"(" }}}{PARA 0 "" 0 " " {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}} {PARA 0 "" 0 "" {TEXT -1 52 "#---------------------------------------- -----------" }}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 27 "absolute stability regions " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 40 "ee: coeffic ients for the combined scheme" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 874 "ee := \{c[2]=1/6,c[3]=2/9,c [4]=3/7,c[5]=11/16,c[6]=3/4,c[7]=1,c[8]=1,\n\na[2,1]=1/6,\na[3,1]=2/27 ,\na[3,2]=4/27,\na[4,1]=183/1372,\na[4,2]=-162/343,\na[4,3]=1053/1372, \na[5,1]=1475/6144,\na[5,2]=-513/2048,\na[5,3]=5427/53248,\na[5,4]=952 07/159744,\na[6,1]=3/88,\na[6,2]=81/352,\na[6,3]=44631/613184,\na[6,4] =58653/265408,\na[6,5]=4104/21373,\na[7,1]=28909/24948,\na[7,2]=-120/7 7,\na[7,3]=-29605749/25485460,\na[7,4]=17354309/5038605,\na[7,5]=-2197 1456/6732495,\na[7,6]=384688/161595,\na[8,1]=817/39204,\na[8,2]=-30/12 1,\na[8,3]=31908963/40048580,\na[8,4]=-1085056/55424655,\na[8,5]=-5961 728/10579635,\na[8,6]=257344/253935,\na[8,7]=0,\n\nb[1]=79/1080,\nb[2] =0,\nb[3]=19683/69160,\nb[4]=16807/84240,\nb[5]=0,\nb[6]=2816/7695,\nb [7]=1/100,\nb[8]=187/2800,\n\n`b*`[1]=22039/308880,\n`b*`[2]=0,\n`b*`[ 3]=5071653/17210960,\n`b*`[4]=11339923/63516960,\n`b*`[5]=75776/138924 5,\n`b*`[6]=32168/100035,\n`b*`[7]=0,\n`b*`[8]=33/416\}:" }}}{PARA 0 " " 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}} }{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 77 "The stab ility function R for the 8 stage, order 6 scheme is given as follows. " }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 75 "subs(ee,StabilityFunctio n(6,8,'expanded')):\nR := unapply(%,z):\n'R(z)'=R(z);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/-%\"RG6#%\"zG,2\"\"\"F)F'F)*&#F)\"\"#F)*$)F'F,F)F )F)*&#F)\"\"'F)*$)F'\"\"$F)F)F)*&#F)\"#CF)*$)F'\"\"%F)F)F)*&#F)\"$?\"F )*$)F'\"\"&F)F)F)*&#F)\"$?(F)*$)F'F1F)F)F)*&#F)\"%S]F)*$)F'\"\"(F)F)F) " }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 124 "We \+ can find the point where the boundary of the stability region intersec ts the negative real axis by solving the equation: " }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "R(z) = -1;" "6#/-%\"RG6#%\"zG,$\"\" \"!\"\"" }{TEXT -1 2 ". " }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 27 " z0 := newton(R(z)=-1,z=-4);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%#z0G$ !+J(HT&R!\"*" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 306 "z0 := newton(R(z)=-1,z=-4):\np1 := plot([R(z),- 1],z=-4.49..0.49,color=[red,blue]):\np2 := plot([[[z0,-1]]$3],style=po int,symbol=[circle,cross,diamond],color=black):\np3 := plot([[z0,0],[z 0,-1]],linestyle=3,color=COLOR(RGB,0,.5,0)):\nplots[display]([p1,p2,p3 ],view=[-4.49..0.49,-1.47..1.47],font=[HELVETICA,9]);" }}{PARA 13 "" 1 "" {GLPLOT2D 360 253 253 {PLOTDATA 2 "6+-%'CURVESG6$7U7$$!3A++++++! \\%!#<$!3iK'p5:D&)o#F*7$$!3;+]7'3DdV%F*$!3'yBEJ]7![CF*7$$!35++Ds,X\"Q% F*$!33#HMw%y!=%F*$!3[%)Q[\"\\_ya\"F*7$$!35+]#oviQ2%F* $!3?Vq%yrMWE\"F*7$$!3A+ve,&\\u'RF*$!3=u.%e0+,T)!#=7$$!3;+v.8ajmPF*$!33s5ua7+&y'FU7$$!3C+v$Q<')4m$F*$!3'oly[ $*y)z`FU7$$!3G+v3`dnbNF*$!3rXMsSCn;UFU7$$!37++0T5NZMF*$!31(G3#p8$)GKFU 7$$!3?+]sv&Q>N$F*$!3lAf+n4Z1DFU7$$!3E++:Qn_WKF*$!3YB],/wpJ=FU7$$!3#*** *\\s&QnOJF*$!3?]#*f2#*ew7FU7$$!3&****\\=iPF.$F*$!3udd/!fT_O)!#>7$$!3Q+ v$Gc`$QHF*$!3[J')\\MFVE]Fhp7$$!3+++l(y@h#GF*$!3OVJc7l=4dQp\"F*$\"3%>lkG9%)R#=FU7$$!3)*****p]Q@(e\"F*$\"3pL`rFNZO?FU 7$$!3I+]FRT)G[\"F*$\"3[txl0H&[E#FU7$$!3')*\\P*y(R>Q\"F*$\"3xZAkgb03DFU 7$$!3++]Kx#e)p7F*$\"3%fe2:*pC2GFU7$$!3=++5j![\"p6F*$\"3>kW?L%\\b5$FU7$ $!3s***\\KT=;1\"F*$\"3UL#R=.)feMFU7$$!3'***\\P%o0=k*FU$\"39v_l:!)y7QFU 7$$!3](***\\tEbw&)FU$\"3D$***QX#)[TUFU7$$!39-]P2EBuvFU$\"3sh<4*e0()o%F U7$$!3>)*\\(GL>l_'FU$\"3#p9(z$>Cm?&FU7$$!3z++]-\")=-bFU$\"3C)eGR&QBodF U7$$!3$*)*\\PM#3)HWFU$\"3O%[Bd&[>@kFU7$$!3;(****f'*ypR$FU$\"3^sjV#z`)> rFU7$$!3L)**\\U'=wSBFU$\"3D'>n%p_,8zFU7$$!3?$*\\Pt2H$H\"FU$\"3+V,>Fw%o y)FU7$$!33R****pit2LFhp$\"3+r)*)36PYn*FU7$$\"3Z^+]FkzBxFhp$\"3QUWS=\"* H!3\"F*7$$\"35*****z2`!f\"F*7$$\"3l+]i_F06GFU$\"35LwZ UIfC8F*7$$\"3\\.]Pt1&z\"QFU$\"3]WLRM=\"\\Y\"F*7$$\"3!***************[F U$\"3Ko[\"G8;Bj\"F*-%'COLOURG6&%$RGBG$\"*++++\"!\")$\"\"!F\\\\lF[\\l-F $6$7S7$F($!\"\"F\\\\l7$F3Fa\\l7$F=Fa\\l7$FBFa\\l7$FGFa\\l7$FLFa\\l7$FQ Fa\\l7$FWFa\\l7$FfnFa\\l7$F[oFa\\l7$F`oFa\\l7$FeoFa\\l7$FjoFa\\l7$F_pF a\\l7$FdpFa\\l7$FjpFa\\l7$F_qFa\\l7$FdqFa\\l7$FjqFa\\l7$F_rFa\\l7$FdrF a\\l7$FirFa\\l7$F^sFa\\l7$FcsFa\\l7$FhsFa\\l7$F]tFa\\l7$FbtFa\\l7$FgtF a\\l7$F\\uFa\\l7$FauFa\\l7$FfuFa\\l7$F[vFa\\l7$F`vFa\\l7$FevFa\\l7$Fjv Fa\\l7$F_wFa\\l7$FdwFa\\l7$FiwFa\\l7$F^xFa\\l7$FcxFa\\l7$FhxFa\\l7$F]y Fa\\l7$FbyFa\\l7$FgyFa\\l7$F\\zFa\\l7$FazFa\\l7$FfzFa\\l7$F[[lFa\\l7$F `[lFa\\l-Fe[l6&Fg[lF[\\lF[\\lFh[l-F$6&7#7$$!35+++J(HT&RF*Fa\\l-%'SYMBO LG6#%'CIRCLEG-Fe[l6&Fg[lF\\\\lF\\\\lF\\\\l-%&STYLEG6#%&POINTG-F$6&Fg_l -F\\`l6#%&CROSSGF_`lFa`l-F$6&Fg_l-F\\`l6#%(DIAMONDGF_`lFa`l-F$6%7$7$Fi _lF[\\lFh_l-%&COLORG6&Fg[lF[\\l$\"\"&Fb\\lF[\\l-%*LINESTYLEG6#\"\"$-%+ AXESLABELSG6%Q\"z6\"Q!F`bl-%%FONTG6#%(DEFAULTG-Fcbl6$%*HELVETICAG\"\"* -%%VIEWG6$;$!$\\%!\"#$\"#\\F`cl;$!$Z\"F`cl$\"$Z\"F`cl" 1 2 0 1 10 0 2 9 1 4 2 1.000000 45.000000 45.000000 0 0 "Curve 1" "Curve 2" "Curve 3 " "Curve 4" "Curve 5" "Curve 6" }}}}{PARA 0 "" 0 "" {TEXT -1 0 "" }} {PARA 0 "" 0 "" {TEXT -1 50 "The following picture shows the stability region. " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1287 "R := z -> 1+z+1/2*z^2+1/6*z^3+1/24*z^4+1/120*z^5+1/ 720*z^6+1/5040*z^7:\npts := []:\nz0 := 0:\nfor ct from 0 to 200 do\n \+ zz := newton(R(z)=exp(ct*Pi/20*I),z=z0):\n z0 := zz:\n pts := [op (pts),[Re(zz),Im(zz)]]:\nend do:\np1 := plot(pts,color=COLOR(RGB,.45,0 ,.45)):\np2 := plots[polygonplot]([seq([pts[i-1],pts[i],[-2,0]],i=2..n ops(pts))],\n style=patchnogrid,color=COLOR(RGB,.85,.2,.85)): \npts := []: z0 := 1.5+4.3*I:\nfor ct from 0 to 60 do\n zz := newton (R(z)=exp(ct*Pi/30*I),z=z0):\n z0 := zz:\n pts := [op(pts),[Re(zz) ,Im(zz)]]:\nend do:\np3 := plot(pts,color=COLOR(RGB,.45,0,.45)):\np4 : = plots[polygonplot]([seq([pts[i-1],pts[i],[1.39,4.2]],i=2..nops(pts)) ],\n style=patchnogrid,color=COLOR(RGB,.85,.2,.85)):\npts := \+ []: z0 := 1.5-4.3*I:\nfor ct from 0 to 60 do\n zz := newton(R(z)=exp (ct*Pi/30*I),z=z0):\n z0 := zz:\n pts := [op(pts),[Re(zz),Im(zz)]] :\nend do:\np5 := plot(pts,color=COLOR(RGB,.45,0,.45)):\np6 := plots[p olygonplot]([seq([pts[i-1],pts[i],[1.39,-4.2]],i=2..nops(pts))],\n \+ style=patchnogrid,color=COLOR(RGB,.85,.2,.85)):\np7 := plot([[[- 4.49,0],[1.79,0]],[[0,-4.69],[0,4.69]]],color=black,linestyle=3):\nplo ts[display]([p||(1..7)],view=[-4.49..1.79,-4.69..4.69],font=[HELVETICA ,9],\n labels=[`Re(z)`,`Im(z)`],axes=boxed,scaling=constr ained);" }}{PARA 13 "" 1 "" {GLPLOT2D 381 565 565 {PLOTDATA 2 "6/-%'CU RVESG6$7ew7$$\"\"!F)F(7$F($\"3++++Fjzq:!#=7$$\"3!)*****\\QO5E#!#E$\"3' ******pk#fTJF-7$$\"3!******4`NP]&!#D$\"3))*****\\t)Q7ZF-7$$\"3Y+++5i') )4&!#C$\"3]+++@@=$G'F-7$$\"3*)*****498?u#!#B$\"3#*******='eR&yF-7$$\"3 (******HR0#G5!#A$\"3B+++.PmC%*F-7$$\"36+++IKeUHFI$\"3!******fv8&*4\"!# <7$$\"3M*****>c%*4p'FI$\"3/+++&e*\\c7FQ7$$\"30+++YW#Q?\"!#@$\"3-+++)oW LT\"FQ7$$\"3#******pUgyb\"FZ$\"3)******\\#e!*p:FQ7$$\"3_+++tN#Gd'FI$\" 3++++dC&fs\"FQ7$$!39+++*>;_$RFZ$\"3++++KH:\")=FQ7$$!3-+++0t1!o\"!#?$\" 3*)*****zJ!3N?FQ7$$!3:+++okGTXFio$\"3=+++b+C(=#FQ7$$!3%******\\9yf+\"! 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" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 260 "We can distort the boundary curve horizontally by taking the 11th root of the real part of points along the curve. In this way we see that the largest interval on the nonnegative imaginary axis th at contains the origin and lies inside the stability region is " } {XPPEDIT 18 0 "[0, 1.76];" "6#7$\"\"!-%&FloatG6$\"$w\"!\"#" }{TEXT -1 18 " approximately. " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 320 "R := z -> 1+z+1/2*z^2+1/6*z^3+1/24 *z^4+1/120*z^5+1/720*z^6+1/5040*z^7:\nDigits := 25:\npts := []: z0 := \+ 0:\nfor ct from 0 to 60 do\n zz := newton(R(z)=exp(ct*Pi/100*I),z=z0 ):\n z0 := zz:\n pts := [op(pts),[surd(Re(zz),11),Im(zz)]]:\nend d o:\nplot(pts,color=COLOR(RGB,.8,0,.8),thickness=2,font=[HELVETICA,9]); \nDigits := 10:" }}{PARA 13 "" 1 "" {GLPLOT2D 261 276 276 {PLOTDATA 2 "6(-%'CURVESG6#7in7$$\"\"!F)F(7$$\":(GMFj2VC9eNyI!#E$\":yZvsJz*e`EfTJF -7$$\":=SG%)Gykc@*o&4&F-$\":w13[GbzrI&=$G'F-7$$\":!)3))4>t?9qG@%oF-$\" :!y*ezx3o2'zxC%*F-7$$\":A+QI!=3it^IK%)F-$\":d8Yy#)>M91PmD\"!#D7$$\":P) >V*eS67jI[\"**F-$\":CoYlW5#yEjzq:F?7$$\":obXc>7S_R:;8\"F?$\":'o/^.pz3# fb\\)=F?7$$\":-C)pR\"f*)Q1o_E\"F?$\":sb85*4%\\s&[6*>#F?7$$\":BxXKDBKxy XNR\"F?$\":qcv5pP(*>7uK^#F?7$$\":\")ya<9'Q3^cB<:F?$\":/E/5]Z7dQLu#GF?7 $$\":`#fswJ!y/!>#pj\"F?$\":Sx3Px>1rk#fTJF?7$$\":)3\"orXVLN#=0`vbMF?7$$\":7=K)y)pe\"3h(f'=F?$\":tc!G!=d_5:6*pPF?7$$\":V-d8 &)zz2'[(f(>F?$\":'HrDDI8b\"QqS3%F?7$$\":3Z1;+9g!y]F$3#F?$\":1=2]nS+GeH #)R%F?7$$\":.*GNchB@wF1)=#F?$\":#\\y2`DfKN()Q7ZF?7$$\":=S.17J!GX;\\!H# F?$\":Qo#Q+x[`4yaE]F?7$$\":92+c%))*>_O*o!R#F?$\":\"[Y*))e3G'eR&yF?7$$\":#e*[ypTY `WbZ?$F?$\":(=*p@)z#*4u$3\"o\")F?7$$\":*o)*)z_7DE&)=iG$F?$\":$4%3^;1X4 #[D#[)F?7$$\":YFq!Q%=snbNfO$F?$\":Lh([eqW06qR'z)F?7$$\":HFMZ`e6An&*QW$ F?$\":hpFExG*fyP`5\"*F?7$$\":(or$oWGI)y93?NF?$\":qr]p'=,E.PmC%*F?7$$\" :p,g_r>zQ2nWf$F?$\":)Hu'R!=omo]yQ(*F?7$$\":!\\/.rl\"Ffu7$ $\":*or%4d75p1\"H=TF?$\":crMI=1y(y.6D7Ffu7$$\":u,0qbVG)Ry[tTF?$\":'[y& HZym_e*\\c7Ffu7$$\":.BK_a()=f%4\"fA%F?$\":!38Ffu7$$\":\"4M([@ZU(3%e9K%F?$\":\"y;5zPltw 4j]8Ffu7$$\":24d-,$4jtL&RO%F?$\":$ej\\b=!Q$yE*>Q\"Ffu7$$\":w[$3ox#))\\ L!Q-WF?$\":6qt.Htd\")oWLT\"Ffu7$$\":B'*Qv:g,*\\+>OWF?$\":U[\\Okk`Di&oW 9Ffu7$$\":JEMO$yF-*orYY%F?$\":PG)=^MV)e&R,w9Ffu7$$\":2o-:pO$pp?)o[%F?$ \":[&fP>#e.P,Gt]\"Ffu7$$\":(3;:2S))p#GI:]%F?$\":RJ%H6')4omfiQ:Ffu7$$\" :Y/@x)ezm#oyn]%F?$\":Czo'QWBTDe!*p:Ffu7$$\":^K#=r^T]'4u)*\\%F?$\":X3cJ \">XGOa;,;Ffu7$$\":.mk\"3u_kVmRwWF?$\":Y\\#HbrabyCSK;Ffu7$$\":Di#*Hae; =/:&GWF?$\":[!p,&*4/!GY9Om\"Ffu7$$\":DgMc=qsT-\\+M%F?$\":Rs&yR5'e3t)z% p\"Ffu7$$\":lMP\"Qh'yWjKn;%F?$\":))4keas\"RdC&fs\"Ffu7$$\":ye43b)4t(3B 6j$F?$\":-_9>CrKJlsqv\"Ffu7$$!:![sV6%4$>0k\"34%F?$\":cRZc'\\X')ph:)y\" Ffu7$$!:y]OyGQ:9QF)oWF?$\":rq*>>=Ffu7$$!:[XNpEO\"33:R6ZF?$\":lF WI%*[M0#)*>]=Ffu7$$!:l!\\o*4H$e=[)G!\\F?$\":I/Ka`&4tJH:\")=Ffu-%*THICK NESSG6#\"\"#-%%FONTG6$%*HELVETICAG\"\"*-%&COLORG6&%$RGBG$\"\")!\"\"F(F `_l-%+AXESLABELSG6$Q!6\"Ff_l-%%VIEWG6$%(DEFAULTGF[`l" 1 2 0 1 10 2 2 6 1 4 2 1.000000 45.000000 45.000000 0 0 "Curve 1" }}}}{PARA 0 "" 0 " " {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 123 "The relevant intersect ion point of the boundary curve with the imaginary axis can be determi ned more accurately as follows." }}{PARA 0 "" 0 "" {TEXT -1 86 "First \+ we look for points on the boundary curve either side of the intersecti on point. " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 " " {MPLTEXT 1 0 112 "Digits := 15:\nz0 := 1.76*I:\nfor ct from 55 to 58 do\n newton(R(z)=exp(ct*Pi/100*I),z=z0);\nend do;\nDigits := 10:" } }{PARA 11 "" 1 "" {XPPMATH 20 "6#^$$\"02+GdBGd'!#>$\"0t\"RdC&fs\"!#9" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#^$$\"0o0E_SpW\"!#>$\"0rKJlsqv\"!#9" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#^$$!05]am.#p`!#>$\"0bk)ph:)y\"!#9" } }{PARA 11 "" 1 "" {XPPMATH 20 "6#^$$!0-k9J![>9!#=$\"0j>rq*>>=!#9" }}} {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 107 "Then we \+ apply the bisection method to calculate the parameter value associated with the intersection point." }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 178 "real_part := proc(u)\n Re(newton(R(z)=exp(u*Pi*I),z=1.76*I)) \nend proc:\nDigits := 15:\nu0 := bisect('real_part'(u),u=0.55..0.58); \nnewton(R(z)=exp(u0*Pi*I),z=1.76*I);\nDigits := 10:" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%#u0G$\"0')fUmIOi&!#:" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#^#$\"0T`XK@Ww\"!#9" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 " " 0 "" {TEXT -1 4 "The " }{TEXT 260 112 "largest interval on the nonne gative imaginary axis that contains the origin and lies inside the sta bility region" }{TEXT -1 5 " is " }{XPPEDIT 18 0 "[0, 1.7644];" "6#7$ \"\"!-%&FloatG6$\"&Ww\"!\"%" }{TEXT -1 18 " (approximately)." }} {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 37 "#-------- ----------------------------" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 80 "The stability function R* for the 8 stage, or der 5 scheme is given as follows." }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 97 "subs(ee,subs(b=`b*`,StabilityFunction(5,8,'expanded') )):\n`R*` := unapply(%,z):\n'`R*`(z)'=`R*`(z);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/-%#R*G6#%\"zG,2\"\"\"F)F'F)*&#F)\"\"#F)*$)F'F,F)F)F)*& #F)\"\"'F)*$)F'\"\"$F)F)F)*&#F)\"#CF)*$)F'\"\"%F)F)F)*&#F)\"$?\"F)*$)F '\"\"&F)F)F)*&#\"&(eK\")SI1BF)*$)F'F1F)F)F)*&#\"%@SFDF)*$)F'\"\"(F)F)F )" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 124 "We can find the point where the boundary of the stability region interse cts the negative real axis by solving the equation: " }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "`R*`(z) = -1;" "6#/-%#R*G6#%\"zG,$ \"\"\"!\"\"" }{TEXT -1 2 ". " }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 33 "z_0 := newton(`R*`(z)=-1,z=-4.2);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%$z_0G$!+r9,VU!\"*" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 324 "z_0 := newton(`R*`(z)=-1,z=-4.2): \np_1 := plot([`R*`(z),-1],z=-4.79..0.49,color=[red,blue]):\np_2 := pl ot([[[z_0,-1]]$3],style=point,symbol=[circle,cross,diamond],color=blac k):\np_3 := plot([[z_0,0],[z_0,-1]],linestyle=3,color=COLOR(RGB,0,.5,0 )):\nplots[display]([p_1,p_2,p_3],view=[-4.79..0.49,-1.47..1.47],font= [HELVETICA,9]);" }}{PARA 13 "" 1 "" {GLPLOT2D 398 284 284 {PLOTDATA 2 "6+-%'CURVESG6$7U7$$!3/++++++!z%!#<$!3x)H;,:pWt#F*7$$!3n*****H^bCt%F*$ !3o!*zv')*)4xCF*7$$!3=+++E5\"\\n%F*$!3JQo9CDmSAF*7$$!3[++bv=%[i%F*$!3g @Wz'pO30#F*7$$!3))****4DFxuXF*$!3]00\"*=\"[[(=F*7$$!3Y++gKn:iWF*$!3Omz &fx#=D:F*7$$!3%*****>[Uz[VF*$!3RGVT#HO/B\"F*7$$!3$)*****ebqfB%F*$!3Y&3 CW&[Pg)*!#=7$$!3U++q]&o88%F*$!3miQEn?`ozFP7$$!3p****4L#fI-%F*$!3&z$Q!3 2dSL'FP7$$!3&)*****=eX5\"RF*$!3Z;:.x \\,6S,QFP7$$!3/++!G'3a%o$F*$!3u!G6fQt/&GFP7$$!3K++ge1Q$e$F*$!3!z\\AYe! om@FP7$$!3+++SG#)\\pMF*$!3X#*p$yc;$R:FP7$$!33+++'=[^N$F*$!3C<^o?)QP.\" FP7$$!3#*****fJ2&\\C$F*$!3QOsz[+)\\T'!#>7$$!3:++!f)3)[9$F*$!3pL:f(y\"y +NFhp7$$!33++Sg\"))e-$F*$!3I*)zOf(>@n'!#?7$$!3p****f8d3DHF*$\"3?bQHTI; ;8Fhp7$$!3v****H/&Gy!GF*$\"3WWy/n>ElKFhp7$$!3N++g0g./FF*$\"3iD%eZe<\"e ZFhp7$$!3-++I%=f,f#F*$\"3xZ?nw/,HiFhp7$$!3))*****3_@<[#F*$\"3qN/f)=*3M vFhp7$$!3++++&>y&oBF*$\"3wxt'[-)yc))Fhp7$$!3)*****H-tnkAF*$\"3%*)*>S\\ ,635FP7$$!3C+++`hg_@F*$\"3gkb5)G'QX6FP7$$!3')*****es&>O?F*$\"3]4@lYsw( H\"FP7$$!3*******o5g[$>F*$\"3i+1(3\"y`T9FP7$$!3r****>i[TD=F*$\"3R,1'3! )>8h\"FP7$$!3L++?:sM7&3t_#FP7$$!3I++gTn4p7F*$\"3P`u:P]n6GFP7$$!3w*****>S*3b6F*$\"3qh(*H q)H3:$FP7$$!3G++]2=y^5F*$\"3w(3\\62%[$\\$FP7$$!3c&****f>$R)Q*FP$\"3)pD 4E**)*4\"RFP7$$!3N-++.CpD$)FP$\"3_Ios&f)R\\VFP7$$!3#o****4(Q'[@(FP$\"3 -%[qT\\M.'[FP7$$!3D-++Sf#)GhFP$\"3mrJq$f%)yT&FP7$$!3'******\\([%=*\\FP $\"3#3I(GC'f-2'FP7$$!3'4++g(pz'*QFP$\"3w\\.Km(QFx'FP7$$!3u-++QB&px#FP$ \"3?kP6%ow_d(FP7$$!3/$*****y0Qm;FP$\"33r\"=iUf]Y)FP7$$!3%)******>Z!)ek Fhp$\"3i22o/d`u$*FP7$$\"3:!*****R\"ysB&Fhp$\"3'=k)f%\\oP0\"F*7$$\"3+++ +3%R)p:FP$\"33RD:Do(*p6F*7$$\"3;0++dD@&o#FP$\"3S*35dyG!38F*7$$\"3+/++z pw_PFP$\"39?U?&4%Rb9F*7$$\"3!***************[FP$\"3\\rto-jJK;F*-%'COLO URG6&%$RGBG$\"*++++\"!\")$\"\"!F\\\\lF[\\l-F$6$7S7$F($!\"\"F\\\\l7$F3F a\\l7$F=Fa\\l7$FBFa\\l7$FGFa\\l7$FLFa\\l7$FRFa\\l7$FWFa\\l7$FfnFa\\l7$ F[oFa\\l7$F`oFa\\l7$FeoFa\\l7$FjoFa\\l7$F_pFa\\l7$FdpFa\\l7$FjpFa\\l7$ F_qFa\\l7$FeqFa\\l7$FjqFa\\l7$F_rFa\\l7$FdrFa\\l7$FirFa\\l7$F^sFa\\l7$ FcsFa\\l7$FhsFa\\l7$F]tFa\\l7$FbtFa\\l7$FgtFa\\l7$F\\uFa\\l7$FauFa\\l7 $FfuFa\\l7$F[vFa\\l7$F`vFa\\l7$FevFa\\l7$FjvFa\\l7$F_wFa\\l7$FdwFa\\l7 $FiwFa\\l7$F^xFa\\l7$FcxFa\\l7$FhxFa\\l7$F]yFa\\l7$FbyFa\\l7$FgyFa\\l7 $F\\zFa\\l7$FazFa\\l7$FfzFa\\l7$F[[lFa\\l7$F`[lFa\\l-Fe[l6&Fg[lF[\\lF[ \\lFh[l-F$6&7#7$$!3)******4Z6IC%F*Fa\\l-%'SYMBOLG6#%'CIRCLEG-Fe[l6&Fg[ lF\\\\lF\\\\lF\\\\l-%&STYLEG6#%&POINTG-F$6&Fg_l-F\\`l6#%&CROSSGF_`lFa` l-F$6&Fg_l-F\\`l6#%(DIAMONDGF_`lFa`l-F$6%7$7$Fi_lF[\\lFh_l-%&COLORG6&F g[lF[\\l$\"\"&Fb\\lF[\\l-%*LINESTYLEG6#\"\"$-%%FONTG6$%*HELVETICAG\"\" *-%+AXESLABELSG6%Q\"z6\"Q!Febl-F]bl6#%(DEFAULTG-%%VIEWG6$;$!$z%!\"#$\" #\\F`cl;$!$Z\"F`cl$\"$Z\"F`cl" 1 2 0 1 10 0 2 9 1 4 2 1.000000 45.000000 45.000000 0 0 "Curve 1" "Curve 2" "Curve 3" "Curve 4" "Curve 5" "Curve 6" }}}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 82 "The following picture shows the stability region for the \+ 7 stage, order 5 scheme. " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1325 "`R*` := z -> 1+z+1/2*z^2+1/6*z^3+ 1/24*z^4+1/120*z^5+32587/23063040*z^6+4021/23063040*z^7:\npts := []:\n z0 := 0:\nfor ct from 0 to 200 do\n zz := newton(`R*`(z)=exp(ct*Pi/2 0*I),z=z0):\n z0 := zz:\n pts := [op(pts),[Re(zz),Im(zz)]]:\nend d o:\np_1 := plot(pts,color=COLOR(RGB,.25,0,.25)):\np_2 := plots[polygon plot]([seq([pts[i-1],pts[i],[-1.8,0]],i=2..nops(pts))],\n sty le=patchnogrid,color=COLOR(RGB,.65,.1,.65)):\npts := []: z0 := 1.4+4.3 *I:\nfor ct from 0 to 60 do\n zz := newton(`R*`(z)=exp(ct*Pi/30*I),z =z0):\n z0 := zz:\n pts := [op(pts),[Re(zz),Im(zz)]]:\nend do:\np_ 3 := plot(pts,color=COLOR(RGB,.25,0,.25)):\np_4 := plots[polygonplot]( [seq([pts[i-1],pts[i],[1.27,4.17]],i=2..nops(pts))],\n style= patchnogrid,color=COLOR(RGB,.65,.1,.65)):\npts := []: z0 := 1.4-4.3*I: \nfor ct from 0 to 60 do\n zz := newton(`R*`(z)=exp(ct*Pi/30*I),z=z0 ):\n z0 := zz:\n pts := [op(pts),[Re(zz),Im(zz)]]:\nend do:\np_5 : = plot(pts,color=COLOR(RGB,.25,0,.25)):\np_6 := plots[polygonplot]([se q([pts[i-1],pts[i],[1.27,-4.17]],i=2..nops(pts))],\n style=pa tchnogrid,color=COLOR(RGB,.65,.1,.65)):\np_7 := plot([[[-4.89,0],[1.89 ,0]],[[0,-4.69],[0,4.69]]],color=black,linestyle=3):\nplots[display]([ p_||(1..7)],view=[-4.89..1.89,-4.69..4.69],font=[HELVETICA,9],\n \+ labels=[`Re(z)`,`Im(z)`],axes=boxed,scaling=constrained);" }} {PARA 13 "" 1 "" {GLPLOT2D 390 543 543 {PLOTDATA 2 "6/-%'CURVESG6$7ew7 $$\"\"!F)F(7$$\"3%)*****pQ#esN!#F$\"3#******fK'zq:!#=7$$\"3'*******eP! =?#!#D$\"3&******f]#fTJF07$$\"30+++v%3YL#!#C$\"3u*****>V'Q7ZF07$$\"3++ ++MD*3;\"!#B$\"3))*****4*e;$G'F07$$\"3#******4T-()f$F@$\"3E+++>t)Q&yF0 7$$\"38+++.YZMtF@$\"3Z*****z!RVC%*F07$$\"3%)*****>Q!H!3(F@$\"3/+++\"pa %*4\"!#<7$$!35+++7:e(o\"!#A$\"3-+++bHPc7FR7$$!3++++8qGl6!#@$\"3$****** zY8JT\"FR7$$!3!******R8U!eSFfn$\"3!*******4lap:FR7$$!3)******>IsQ5\"!# ?$\"3,+++'H$\\DgK2)=FR7$$!3y*****>zp) 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>%FeanF[j`l7%Fgbal7$F^`_l$!+d9\"*=UFeanF[j`l7%F[cal7$Fh__l$!+'*[\"yB%F eanF[j`l7%F_cal7$Fb__l$!+Q!*)[D%FeanF[j`l7%Fccal7$F\\__l$!+_z8qUFeanF[ j`l7%Fgcal7$Ff^_l$!+&R*f$G%FeanF[j`l7%F[dal7$F`^_l$!+R*H`H%FeanF[j`l7% F_dal7$Fj]_l$!+8cR0VFeanF[j`l7%Fcdal7$Fd]_l$!+wh'QJ%FeanF[j`l7%Fgdal7$ F^]_l$!+M<\"3K%FeanF[j`l7%F[eal7$Fh\\_l$!+:3IEVFeanF[j`l7%F_eal7$Fb\\_ l$!+k$*RIVFeanF[j`l7%Fceal7$F\\\\_l$!+,. " 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 40 "e e: coefficients for the combined scheme" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 874 "ee := \{c[2]=1/6,c[3]= 2/9,c[4]=3/7,c[5]=11/16,c[6]=3/4,c[7]=1,c[8]=1,\n\na[2,1]=1/6,\na[3,1] =2/27,\na[3,2]=4/27,\na[4,1]=183/1372,\na[4,2]=-162/343,\na[4,3]=1053/ 1372,\na[5,1]=1475/6144,\na[5,2]=-513/2048,\na[5,3]=5427/53248,\na[5,4 ]=95207/159744,\na[6,1]=3/88,\na[6,2]=81/352,\na[6,3]=44631/613184,\na [6,4]=58653/265408,\na[6,5]=4104/21373,\na[7,1]=28909/24948,\na[7,2]=- 120/77,\na[7,3]=-29605749/25485460,\na[7,4]=17354309/5038605,\na[7,5]= -21971456/6732495,\na[7,6]=384688/161595,\na[8,1]=817/39204,\na[8,2]=- 30/121,\na[8,3]=31908963/40048580,\na[8,4]=-1085056/55424655,\na[8,5]= -5961728/10579635,\na[8,6]=257344/253935,\na[8,7]=0,\n\nb[1]=79/1080, \nb[2]=0,\nb[3]=19683/69160,\nb[4]=16807/84240,\nb[5]=0,\nb[6]=2816/76 95,\nb[7]=1/100,\nb[8]=187/2800,\n\n`b*`[1]=22039/308880,\n`b*`[2]=0, \n`b*`[3]=5071653/17210960,\n`b*`[4]=11339923/63516960,\n`b*`[5]=75776 /1389245,\n`b*`[6]=32168/100035,\n`b*`[7]=0,\n`b*`[8]=33/416\}:" }}} {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 "s eq(c[i]=subs(ee,c[i]),i=2..8);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6)/&% \"cG6#\"\"##\"\"\"\"\"'/&F%6#\"\"$#F'\"\"*/&F%6#\"\"%#F.\"\"(/&F%6#\" \"&#\"#6\"#;/&F%6#F*#F.F4/&F%6#F6F)/&F%6#\"\")F)" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT 259 20 "linking coefficients" } {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..8);" }}{PARA 12 "" 1 "" {XPPMATH 20 "6>/&%\"aG6$\"\"#\"\"\"#F(\"\"'/&F%6$\"\"$F(#F'\"#F/&F%6$F.F'#\"\"% F0/&F%6$F5F(#\"$$=\"%s8/&F%6$F5F'#!$i\"\"$V$/&F%6$F5F.#\"%`5F;/&F%6$\" \"&F(#\"%v9\"%Wh/&F%6$FJF'#!$8&\"%[?/&F%6$FJF.#\"%Fa\"&[K&/&F%6$FJF5# \"&2_*\"'W(f\"/&F%6$F*F(#F.\"#))/&F%6$F*F'#\"#\")\"$_$/&F%6$F*F.#\"&JY %\"'%=8'/&F%6$F*F5#\"&`'e\"'3aE/&F%6$F*FJ#\"%/T\"&t8#/&F%6$\"\"(F(#\"& 4*G\"&[\\#/&F%6$FjpF'#!$?\"\"#x/&F%6$FjpF.#!)\\dgH\")ga[D/&F%6$FjpF5# \")4VN<\"(0'Q]/&F%6$FjpFJ#!)c9(>#\"(&\\Kn/&F%6$FjpF*#\"')o%Q\"'&fh\"/& F%6$\"\")F(#\"$<)\"&/#R/&F%6$F_sF'#!#I\"$@\"/&F%6$F_sF.#\")j*3>$\")!e[ +%/&F%6$F_sF5#!(c]3\"\")bYUb/&F%6$F_sFJ#!(G<'f\")N'z0\"/&F%6$F_sF*#\"' WtD\"'NRD/&F%6$F_sFjp\"\"!" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT 259 30 "weights for the order 6 scheme" }{TEXT -1 2 ": " }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 31 "seq(b[i]=subs(ee,b[i]), i=1..8);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6*/&%\"bG6#\"\"\"#\"#z\"%!3 \"/&F%6#\"\"#\"\"!/&F%6#\"\"$#\"&$o>\"&g\"p/&F%6#\"\"%#\"&2o\"\"&SU)/& F%6#\"\"&F//&F%6#\"\"'#\"%;G\"%&p(/&F%6#\"\"(#F'\"$+\"/&F%6#\"\")#\"$( =\"%+G" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT 259 30 "weights for the order 5 scheme" }{TEXT -1 2 ": " }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 37 "seq(`b*`[i]=subs(ee,`b*`[i]),i=1..8);" }} {PARA 11 "" 1 "" {XPPMATH 20 "6*/&%#b*G6#\"\"\"#\"&R?#\"'!))3$/&F%6#\" \"#\"\"!/&F%6#\"\"$#\"(`;2&\")g4@ " 0 "" {MPLTEXT 1 0 1 ";" }}}}{PARA 0 "" 0 "" {TEXT -1 52 "#----------------- ----------------------------------" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}}{PARA 0 "" 0 "" {TEXT -1 35 "#==================================" }}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 38 "a 2nd modification of Verner's scheme " }}{PARA 257 "" 0 "" {TEXT -1 0 "" }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "ma trix([[5/28, 5/28, ``, ``, ``, ``, ``, ``, ``], [2/9, 34/405, 56/405, \+ ``, ``, ``, ``, ``, ``], [3/7, 699/6860, -108/245, 1053/1372, ``, ``, \+ ``, ``, ``], [37/53, 22960313/98258820, -1407924/8188235, -680319/8515 7644, 41155100/63868233, ``, ``, ``, ``], [27/35, -22691367/349002500, 28188/67375, 31655367/194652500, -37368/1203125, 141862511268/4951472 96875, ``, ``, ``], [1, 108451463/92067840, -4263/2816, -77302059309/5 7505310720, 3338370539/988416000, -154367434883/70954752000, 63635075/ 43047936, ``, ``], [1, 31846/494505, -42/121, 241942482/308866415, 625 387/21235500, -4513801763/25915114500, 10127075/15722586, 0, ``], [b, \+ 13093/179820, 0, 2932767/10210460, 2144093/11232000, 4600150423/365524 48000, 181575625/731814912, 1/100, 3333/51200], [`b*`, 364613/5106888, 0, 6580683/22305928, 16807/95850, 1317710327/8110074400, 354447625/16 23714336, 0, 11/142]])" "6#-%'matrixG6#7+7+*&\"\"&\"\"\"\"#G!\"\"*&F)F *F+F,%!GF.F.F.F.F.F.7+*&\"\"#F*\"\"*F,*&\"#MF*\"$0%F,*&\"#cF*F5F,F.F.F .F.F.F.7+*&\"\"$F*\"\"(F,*&\"$*pF*\"%goF,,$*&\"$3\"F*\"$X#F,F,*&\"%`5F *\"%s8F,F.F.F.F.F.7+*&\"#PF*\"#`F,*&\")8.'H#F*\")?)e#)*F,,$*&\"(CzS\"F *\"(N#)=)F,F,,$*&\"'>.oF*\")Ww:&)F,F,*&\")+^:TF*\")L#oQ'F,F.F.F.F.7+*& \"#FF*\"#NF,,$*&\")n8pAF*\"*+D+\\$F,F,*&\"&)=GF*\"&vt'F,*&\")n`lJF*\"* +Dl%>F,,$*&\"&ot$F*\"(DJ?\"F,F,*&\"-o7^i=9F*\"-voHZ^\\F,F.F.F.7+F**&\" *j9X3\"F*\")Sy1#*F,,$*&\"%jUF*\"%;GF,F,,$*&\",4$f?IxF*\",?2J0v&F,F,*& \"+R0PQLF*\"*+gT))*F,,$*&\"-$)[VnV:F*\",+?va4(F,F,*&\")v]jjF*\")Oz/VF, F.F.7+F**&\"&Y=$F*\"'0X\\F,,$*&\"#UF*\"$@\"F,F,*&\"*#[U>CF*\"*:k')3$F, *&\"'(QD'F*\")+bB@F,,$*&\"+j " 0 "" {MPLTEXT 1 0 1 ";" }}}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 40 "ee: coefficients for the combined scheme" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> \+ " 0 "" {MPLTEXT 1 0 1056 "ee := \{c[2]=5/28,c[3]=2/9,c[4]=3/7,c[5]=37/ 53,c[6]=27/35,c[7]=1,c[8]=1,\n\na[2,1]=5/28,\na[3,1]=34/405,\na[3,2]=5 6/405,\na[4,1]=699/6860,\na[4,2]=-108/245,\na[4,3]=1053/1372,\na[5,1]= 22960313/98258820,\na[5,2]=-1407924/8188235,\na[5,3]=-680319/85157644, \na[5,4]=41155100/63868233,\na[6,1]=-22691367/349002500,\na[6,2]=28188 /67375,\na[6,3]=31655367/194652500,\na[6,4]=-37368/1203125,\na[6,5]=14 1862511268/495147296875,\na[7,1]=108451463/92067840,\na[7,2]=-4263/281 6,\na[7,3]=-77302059309/57505310720,\na[7,4]=3338370539/988416000,\na[ 7,5]=-154367434883/70954752000,\na[7,6]=63635075/43047936,\na[8,1]=318 46/494505,\na[8,2]=-42/121,\na[8,3]=241942482/308866415,\na[8,4]=62538 7/21235500,\na[8,5]=-4513801763/25915114500,\na[8,6]=10127075/15722586 ,\na[8,7]=0,\n\nb[1]=13093/179820,\nb[2]=0,\nb[3]=2932767/10210460,\nb [4]=2144093/11232000,\nb[5]=4600150423/36552448000,\nb[6]=181575625/73 1814912,\nb[7]=1/100,\nb[8]=3333/51200,\n\n`b*`[1]=364613/5106888,\n`b *`[2]=0,\n`b*`[3]=6580683/22305928,\n`b*`[4]=16807/95850,\n`b*`[5]=131 7710327/8110074400,\n`b*`[6]=354447625/1623714336,\n`b*`[7]=0,\n`b*`[8 ]=11/142\}:" }}}{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 0 155 "RK6_8eqs := [op(RowSumConditions(8 ,'expanded')),op(OrderConditions(6,8,'expanded'))]:\n`RK5_8eqs*` := su bs(b=`b*`,OrderConditions(5,8,'expanded')):\nnops(%);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#\"#<" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 " " 0 "" {TEXT -1 7 "Check: " }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 128 "simplify(subs(ee,RK6_8eqs)):\nmap(u->lhs(u)-rhs(u),%);\nnops(%); \nsimplify(subs(ee,`RK5_8eqs*`)):\nmap(u->lhs(u)-rhs(u),%);\nnops(%); " }}{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(ct,8,'expanded') ;\nend do:" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 34 "Stages 3 to 8 have stage-order 2. " }}{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]) th en break end if end do; i end proc,%):\nsimplify(%); " }}{PARA 11 "" 1 "" {XPPMATH 20 "6#7(\"\"#F$F$F$F$F$" }}}{PARA 0 "" 0 "" {TEXT -1 0 " " }}{PARA 0 "" 0 "" {TEXT -1 114 "We can calculate the principal error norm of the order 6 scheme, that is, the 2-norm of the principal erro r terms." }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 131 "errterms6_8 := \+ PrincipalErrorTerms(6,8,'expanded'):\nevalf(evalf[14](sqrt(add(subs(ee ,errterms6_8[i])^2,i=1.. nops(errterms6_8)))));" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#$\"+gV]2@!#8" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 79 "The 2 norm of the principal error of the order \+ 5 embedded scheme is as follows." }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 153 "`errterms5_8*` := subs(b=`b*`,PrincipalErrorTerms(5,8,'expand ed')):\nevalf(evalf[14](sqrt(add(subs(ee,`errterms5_8*`[i])^2,i=1.. no ps(`errterms5_8*`)))));" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#$\"+83'y\\ \"!#8" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 52 "Two of the principal error conditions are satisfied." }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 291 "ind := []:\nfor ct to nops(errterms6_8) \+ do\n if simplify(subs(ee,errterms6_8[ct]))=0 then ind := [op(ind),ct ] end if:\nend do:\nerr6 := PrincipalErrorConditions(6):\n[seq([i,err6 [i]],i=ind)]:\nlinalg[augment](linalg[delcols](%,2..2),matrix([[` `]$ (linalg[rowdim](%))]),linalg[delcols](%,1..1));" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#K%'matrixG6#7$7%\"\"\"%#~~G/*&%\"bGF(-%!G6#*&%\"aGF(-F. 6#*&F1F(-F.6#*&F1F(-F.6#*&F1F(-F.6#*&F1F(%\"cGF(F(F(F(F(F(#F(\"%S]7%\" \"&F)/*(F,F(F>F(F2F(#F(\"$S)Q)pprint136\"" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}}{PARA 0 "" 0 "" {TEXT -1 52 "#--------------------------------------------------- " }}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 35 "construction of the combined \+ scheme" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 196 "First we construct the 8 stage order 6 scheme by making use of a \+ selection of \"simple\" order conditions and incorporating the stage-o rder equations to ensure that stages 3 to 8 have stage-order 2." }} {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$F+%\"jGF,/F+;,&F0F,F,F,\"\")*&&F)6#F0F,,&F,F,&%\"cG6#F0!\"\" F," }{TEXT -1 1 " " }}{PARA 0 "" 0 "" {TEXT -1 5 "for " }{XPPEDIT 18 0 "j = 4;" "6#/%\"jG\"\"%" }{TEXT -1 18 " . . . 7. ( with " } {XPPEDIT 18 0 "c[1]=0" "6#/&%\"cG6#\"\"\"\"\"!" }{TEXT -1 55 " ), tog ether with the further simplifying conditions: " }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "Sum(b[i]*c[i]*a[i,2],i=3..8)=0" "6#/-%$ SumG6$*(&%\"bG6#%\"iG\"\"\"&%\"cG6#F+F,&%\"aG6$F+\"\"#F,/F+;\"\"$\"\") \"\"!" }{TEXT -1 8 ", " }{XPPEDIT 18 0 "Sum(b[i]*c[i]*Sum(a[i,j] *a[j,2],j=3..i-1),i=3..8)=0" "6#/-%$SumG6$*(&%\"bG6#%\"iG\"\"\"&%\"cG6 #F+F,-F%6$*&&%\"aG6$F+%\"jGF,&F46$F6\"\"#F,/F6;\"\"$,&F+F,F,!\"\"F,/F+ ;F<\"\")\"\"!" }{TEXT -1 8 ", " }{XPPEDIT 18 0 "Sum(b[i]*c[i]^2* a[i,2],i=3..8)=0" "6#/-%$SumG6$*(&%\"bG6#%\"iG\"\"\"*$&%\"cG6#F+\"\"#F ,&%\"aG6$F+F1F,/F+;\"\"$\"\")\"\"!" }{TEXT -1 2 ". " }}{PARA 0 "" 0 " " {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 32 "The scheme is construct ed with " }{XPPEDIT 18 0 "a[7,3];" "6#&%\"aG6$\"\"(\"\"$" }{TEXT -1 3 ", " }{XPPEDIT 18 0 "a[7,4]" "6#&%\"aG6$\"\"(\"\"%" }{TEXT -1 3 ", \+ " }{XPPEDIT 18 0 "a[7,5]" "6#&%\"aG6$\"\"(\"\"&" }{TEXT -1 7 " and \+ " }{XPPEDIT 18 0 "a[7,6]" "6#&%\"aG6$\"\"(\"\"'" }{TEXT -1 17 " as pa rameters. " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 76 "The simple order conditions used are given (in abreviated form) as follows. " }}{PARA 0 "" 0 "" {TEXT -1 94 "These include the 6 orde r 6 quadrature conditions together with 4 additional order conditions. " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 184 "SO6 := SimpleOrderConditions(6):\n[seq([i,SO6[i]],i=[1,2,4,8, 13,16,24,28,29,32])]:\nlinalg[augment](linalg[delcols](%,2..2),matrix( [[` `]$(linalg[rowdim](%))]),linalg[delcols](%,1..1));" }}{PARA 11 " " 1 "" {XPPMATH 20 "6#K%'matrixG6#7,7%\"\"\"%#~~G/*&%\"bGF(%\"eGF(F(7% \"\"#F)/*&F,F(%\"cGF(#F(F/7%\"\"%F)/*&F,F()F2F/F(#F(\"\"$7%\"\")F)/*&F ,F()F2F:F(#F(F57%\"#8F)/*(F,F(F2F(-%!G6#*&F8F(%\"aGF(F(#F(\"#:7%\"#;F) /*&F,F()F2F5F(#F(\"\"&7%\"#CF)/*(F,F(F2F(-FF6#*&FIF(FEF(F(#F(\"#s7%\"# GF)/*(F,F(F8F(FEF(#F(\"#=7%\"#HF)/*(F,F(F2F(-FF6#*&F?F(FIF(F(#F(FT7%\" #KF)/*&F,F()F2FRF(#F(\"\"'Q)pprint816\"" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 513 "SO_eqs := [op(RowSum Conditions(8,'expanded')),op(StageOrderConditions(2,8,'expanded'))]:\n SO6_8 := SimpleOrderConditions(6,8,'expanded'):\nord_cdns := [seq(SO6_ 8[i],i=[1,2,4,8,13,16,24,28,29,32])]:\nsimp_cdns := [add(b[i]*a[i,1],i =1+1..8)=b[1],seq(add(b[i]*a[i,j],i=j+1..8)=b[j]*(1-c[j]),j=[4,5,6,7]) ]:\nsimp_cdns2 := [add(b[i]*c[i]*a[i,2],i=3..8)=0,add(b[i]*c[i]*add(a[ i,j]*a[j,2],j=3..i-1),i=3..8)=0,\n add(b[i]*c[i]^2*a[i, 2],i=3..8)=0]:\ncdns := [op(ord_cdns),op(simp_cdns),op(simp_cdns2),op( SO_eqs)]:" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}{PARA 0 " " 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 21 "We specify the nod es:" }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "c[2] = 5/28;" "6#/&%\"cG6#\"\"#*&\"\"&\"\"\"\"#G!\"\"" }{TEXT -1 3 ", " }{XPPEDIT 18 0 "c[3]=2/9" "6#/&%\"cG6#\"\"$*&\"\"#\"\"\"\"\"*!\"\"" }{TEXT -1 3 ", " }{XPPEDIT 18 0 "c[5] = 37/53;" "6#/&%\"cG6#\"\"&*&\"#P\"\"\"\"#` !\"\"" }{TEXT -1 3 ", " }{XPPEDIT 18 0 "c[6] = 27/35;" "6#/&%\"cG6#\" \"'*&\"#F\"\"\"\"#N!\"\"" }{TEXT -1 3 ", " }{XPPEDIT 18 0 "c[7]=1" "6 #/&%\"cG6#\"\"(\"\"\"" }{TEXT -1 3 ", " }{XPPEDIT 18 0 "c[8]=1" "6#/& %\"cG6#\"\")\"\"\"" }{TEXT -1 2 ", " }}{PARA 0 "" 0 "" {TEXT -1 17 "an d the weights: " }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "b[ 2]=0" "6#/&%\"bG6#\"\"#\"\"!" }{TEXT -1 3 ", " }{XPPEDIT 18 0 "b[7] = 1/100;" "6#/&%\"bG6#\"\"(*&\"\"\"F)\"$+\"!\"\"" }{TEXT -1 2 ". " }} {PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 175 "e1 := \{c[2]=5/28,c[3]=2/9,c[5]=37/53,c[6]=27/35,c[7]=1,c[8]=1,b[ 2]=0,b[7]=1/100\}:\neqns := subs(e1,cdns):\nnops(eqns);\nindets(eqns) \+ minus \{a[7,3],a[7,4],a[7,5],a[7,6]\};\nnops(%);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#\"#J" }}{PARA 11 "" 1 "" {XPPMATH 20 "6# " 0 "" {MPLTEXT 1 0 22 "infolevel[so lve] := 4:" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 118 "e2 := solve( \{op(eqns)\},indets(eqns) minus \{a[7,3],a[7,4],a[7,5],a[7,6]\}):\ninf olevel[solve] := 0:\ne3 := `union`(e1,e2):" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 3 "e3 " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1159 "e3 := \{c[8] = 1, a[7,1] = 11/45*a[7,3]+7/5*a[7,4]+771/265*a[7,5]+83/25*a[7,6]-9/5, a[5,3] = -68 0319/85157644, c[3] = 2/9, b[2] = 0, c[2] = 5/28, c[5] = 37/53, c[6] = 27/35, a[6,1] = -22691367/349002500, a[6,3] = 31655367/194652500, a[7 ,2] = -56/45*a[7,3]-12/5*a[7,4]-1036/265*a[7,5]-108/25*a[7,6]+14/5, a[ 6,4] = -37368/1203125, a[5,4] = 41155100/63868233, a[5,2] = -1407924/8 188235, b[4] = 2144093/11232000, a[6,5] = 141862511268/495147296875, a [8,1] = 312770927/599340060-512/13635*a[7,3]-3584/16665*a[7,4]-131584/ 294415*a[7,5]-42496/83325*a[7,6], a[4,2] = -108/245, b[8] = 3333/51200 , a[8,6] = 188650000/216542889-512/3333*a[7,6], b[1] = 13093/179820, a [8,7] = 0, a[8,5] = -329911432/648948735-512/3333*a[7,5], c[7] = 1, a[ 8,4] = 3207148/5849415-512/3333*a[7,4], a[3,2] = 56/405, b[6] = 181575 625/731814912, a[2,1] = 5/28, a[8,3] = 118970925/206251292-512/3333*a[ 7,3], a[3,1] = 34/405, a[6,2] = 28188/67375, b[5] = 4600150423/3655244 8000, c[4] = 3/7, a[4,3] = 1053/1372, a[5,1] = 22960313/98258820, b[7] = 1/100, b[3] = 2932767/10210460, a[4,1] = 699/6860, a[8,2] = -16828/ 16665+28672/149985*a[7,3]+2048/5555*a[7,4]+530432/883245*a[7,5]+18432/ 27775*a[7,6]\}:" }}}{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 0 452 "subs(e3,matrix([seq([c[i],seq(a[i, j],j=1..i-1),``$(6-i)],i=2..6)]));\nprint(``);\nfor ii from 7 to 8 do \n print(c[ii]=subs(e3,c[ii]));print(``); \n for jj to ii-1 do\n \+ coeff_val := subs(e3,a[ii,jj]);\n if coeff_val=a[ii,jj] then \+ \n print(a[ii,jj]*` is a parameter.`)\n else\n pr int(a[ii,jj]=coeff_val);\n end if;\n end do:\n print(`_______ ___________________________`):\nend do:\nprint(``);\nseq(b[i]=subs(e3, b[i]),i=1..8);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#K%'matrixG6#7'7(#\" \"&\"#GF(%!GF+F+F+7(#\"\"#\"\"*#\"#M\"$0%#\"#cF2F+F+F+7(#\"\"$\"\"(#\" $*p\"%go#!$3\"\"$X##\"%`5\"%s8F+F+7(#\"#P\"#`#\")8.'H#\")?)e#)*#!(CzS \"\"(N#)=)#!'>.o\")Ww:&)#\")+^:T\")L#oQ'F+7(#\"#F\"#N#!)n8pA\"*+D+\\$# \"&)=G\"&vt'#\")n`lJ\"*+Dl%>#!&ot$\"(DJ?\"#\"-o7^i=9\"-voHZ^\\Q(pprint 96\"" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%!G" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/&%\"cG6#\"\"(\"\"\"" }}{PARA 11 "" 1 "" {XPPMATH 20 "6 #%!G" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/&%\"aG6$\"\"(\"\"\",,*&#\"#6 \"#XF(&F%6$F'\"\"$F(F(*&#F'\"\"&F(&F%6$F'\"\"%F(F(*&#\"$r(\"$l#F(&F%6$ F'F3F(F(*&#\"#$)\"#DF(&F%6$F'\"\"'F(F(#\"\"*F3!\"\"" }}{PARA 11 "" 1 " " {XPPMATH 20 "6#/&%\"aG6$\"\"(\"\"#,,*&#\"#c\"#X\"\"\"&F%6$F'\"\"$F.! \"\"*&#\"#7\"\"&F.&F%6$F'\"\"%F.F2*&#\"%O5\"$l#F.&F%6$F'F6F.F2*&#\"$3 \"\"#DF.&F%6$F'\"\"'F.F2#\"#9F6F." }}{PARA 11 "" 1 "" {XPPMATH 20 "6#* &&%\"aG6$\"\"(\"\"$\"\"\"%1~is~a~parameter.GF)" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#*&&%\"aG6$\"\"(\"\"%\"\"\"%1~is~a~parameter.GF)" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#*&&%\"aG6$\"\"(\"\"&\"\"\"%1~is~a~para meter.GF)" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#*&&%\"aG6$\"\"(\"\"'\"\" \"%1~is~a~parameter.GF)" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%C_________ _________________________G" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/&%\"cG6 #\"\")\"\"\"" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%!G" }}{PARA 11 "" 1 " " {XPPMATH 20 "6#/&%\"aG6$\"\")\"\"\",,#\"*F4x7$\"*g+M*fF(*&#\"$7&\"&N O\"F(&F%6$\"\"(\"\"$F(!\"\"*&#\"%%e$\"&lm\"F(&F%6$F3\"\"%F(F5*&#\"'%eJ \"\"':WHF(&F%6$F3\"\"&F(F5*&#\"&'\\U\"&DL)F(&F%6$F3\"\"'F(F5" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/&%\"aG6$\"\")\"\"#,,#\"&Go\"\"&lm\"!\"\"*&# \"&s'G\"'&)*\\\"\"\"\"&F%6$\"\"(\"\"$F2F2*&#\"%[?\"%bbF2&F%6$F5\"\"%F2 F2*&#\"'K/`\"'XK))F2&F%6$F5\"\"&F2F2*&#\"&K%=\"&vx#F2&F%6$F5\"\"'F2F2 " }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/&%\"aG6$\"\")\"\"$,&#\"*D4(*=\"\" *#H^i?\"\"\"*&#\"$7&\"%LLF-&F%6$\"\"(F(F-!\"\"" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/&%\"aG6$\"\")\"\"%,&#\"([r?$\"(:%\\e\"\"\"*&#\"$7&\"%L LF-&F%6$\"\"(F(F-!\"\"" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/&%\"aG6$\" \")\"\"&,&#\"*K9\"*H$\"*N([*['!\"\"*&#\"$7&\"%LL\"\"\"&F%6$\"\"(F(F2F- " }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/&%\"aG6$\"\")\"\"',&#\"*++l)=\"** )Ga;#\"\"\"*&#\"$7&\"%LLF-&F%6$\"\"(F(F-!\"\"" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/&%\"aG6$\"\")\"\"(\"\"!" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%C__________________________________G" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%!G" }}{PARA 11 "" 1 "" {XPPMATH 20 "6*/&%\"bG6#\"\"\"# \"&$48\"'?)z\"/&F%6#\"\"#\"\"!/&F%6#\"\"$#\"(nF$H\")g/@5/&F%6#\"\"%#\" ($4W@\")+?B6/&F%6#\"\"&#\"+B/:+Y\",+![CbO/&F%6#\"\"'#\"*Dcd\"=\"*7\\\" =t/&F%6#\"\"(#F'\"$+\"/&F%6#\"\")#\"%LL\"&+7&" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 66 "We now turn our attention to the embedded 8 stage, o rder 5 scheme." }}{PARA 0 "" 0 "" {TEXT -1 35 "We use the simplifying \+ conditions: " }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "Sum(` b*`[i]*a[i,2],i = 3 .. 8) = 0;" "6#/-%$SumG6$*&&%#b*G6#%\"iG\"\"\"&%\" aG6$F+\"\"#F,/F+;\"\"$\"\")\"\"!" }{TEXT -1 8 ", " }{XPPEDIT 18 0 "Sum(`b*`[i]*Sum(a[i,j]*a[j,2],j = 3 .. i-1),i = 2 .. 8) = 0;" "6#/- %$SumG6$*&&%#b*G6#%\"iG\"\"\"-F%6$*&&%\"aG6$F+%\"jGF,&F16$F3\"\"#F,/F3 ;\"\"$,&F+F,F,!\"\"F,/F+;F6\"\")\"\"!" }{TEXT -1 1 "." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 101 "We use a selection of \"simple\" order conditions as given (in abreviated form) in the foll owing table." }}{PARA 0 "" 0 "" {TEXT -1 95 "These include the 5 order 5 quadrature conditions together with one additional order condition. " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 191 "`SO5*` := subs(b=`b*`,SimpleOrderConditions(5)):\n[seq([i,`SO 5*`[i]],i=[1,2,4,8,15,16])]:\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(\"\"$7% \"\")F)/*&F,F()F2F:F(#F(F57%\"#:F)/*&F,F(-%!G6#*&F?F(%\"aGF(F(#F(\"#?7 %\"#;F)/*&F,F()F2F5F(#F(\"\"&Q)pprint866\"" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 262 "`SO5_8*` := subs( b=`b*`,SimpleOrderConditions(5,8,'expanded')):\n`ord_cdns*` := [seq(`S O5_8*`[i],i=[1,2,4,8,15,16])]:\n`simp_cdns*` := [add(`b*`[i]*a[i,2],i= 3..8)=0,add(`b*`[i]*add(a[i,j]*a[j,2],j=3..i-1),i=2..8)=0]:\n`cdns*` : = [op(`ord_cdns*`),op(`simp_cdns*`)]: " }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 17 "We specify that " }{XPPEDIT 18 0 "` b*`[2] = 0;" "6#/&%#b*G6#\"\"#\"\"!" }{TEXT -1 3 ", " }{XPPEDIT 18 0 "`b*`[7] = 0;" "6#/&%#b*G6#\"\"(\"\"!" }{TEXT -1 7 " and " } {XPPEDIT 18 0 "`b*`[8] = 11/142;" "6#/&%#b*G6#\"\")*&\"#6\"\"\"\"$U\"! \"\"" }{TEXT -1 2 ". " }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 65 "e4 \+ := \{`b*`[2]=0,`b*`[7]=0,`b*`[8]=11/142\}:\ne5 := `union`(e3,e4):" }}} {PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 84 "`eqns*` := subs(e5,`cdns*`):\nnops(`eqns*`);\nindets(`eqns*`) minu s \{a[7,6]\};\nnops(%);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#\"\")" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#<*&%#b*G6#\"\"\"&F%6#\"\"$&F%6#\"\"%&F %6#\"\"&&F%6#\"\"'&%\"aG6$\"\"(F*&F56$F7F-&F56$F7F0" }}{PARA 11 "" 1 " " {XPPMATH 20 "6#\"\")" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 22 "infolevel[solve] := 4:" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 112 "e6 := solve(\{op(`eqns*`)\},indets (`eqns*`) minus \{a[7,6]\}):\ne7 := `union`(subs(e6,e5),e6):\ninfoleve l[solve] := 0:" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 39 "The order 5 scheme is giv en as follows." }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 458 "subs(e3,m atrix([seq([c[i],seq(a[i,j],j=1..i-1),``$(6-i)],i=2..6)]));\nprint(``) ;\nfor ii from 7 to 8 do\n print(c[ii]=subs(e7,c[ii]));print(``); \n for jj to ii-1 do\n coeff_val := subs(e7,a[ii,jj]);\n if \+ coeff_val=a[ii,jj] then \n print(a[ii,jj]*` is a parameter.`) \n else\n print(a[ii,jj]=coeff_val);\n end if;\n e nd do:\n print(`__________________________________`):\nend do:\nprin t(``);\nseq(`b*`[i]=subs(e7,`b*`[i]),i=1..8);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#K%'matrixG6#7'7(#\"\"&\"#GF(%!GF+F+F+7(#\"\"#\"\"*#\"#M \"$0%#\"#cF2F+F+F+7(#\"\"$\"\"(#\"$*p\"%go#!$3\"\"$X##\"%`5\"%s8F+F+7( #\"#P\"#`#\")8.'H#\")?)e#)*#!(CzS\"\"(N#)=)#!'>.o\")Ww:&)#\")+^:T\")L# oQ'F+7(#\"#F\"#N#!)n8pA\"*+D+\\$#\"&)=G\"&vt'#\")n`lJ\"*+Dl%>#!&ot$\"( DJ?\"#\"-o7^i=9\"-voHZ^\\Q)pprint116\"" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%!G" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/&%\"cG6#\"\"(\"\"\"" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#%!G" }}{PARA 11 "" 1 "" {XPPMATH 20 "6 #/&%\"aG6$\"\"(\"\"\",&#\"'oF(*&#\"&cq%\"'DmAF(&F%6$F'\"\"'F( F(" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/&%\"aG6$\"\"(\"\"##!%jU\"%;G" } }{PARA 11 "" 1 "" {XPPMATH 20 "6#/&%\"aG6$\"\"(\"\"$,&#\"(83/#\")G,[m! \"\"*&#\")G81;\")v[2=\"\"\"&F%6$F'\"\"'F2F-" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/&%\"aG6$\"\"(\"\"%,&#\")tsi&*\")+W*e'\"\"\"*&#\"&QH&\" &D1%F-&F%6$F'\"\"'F-F-" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/&%\"aG6$\" \"(\"\"&,&#\"+2D&[:$\",+/&4>9\"\"\"*&#\"+,M@'3#\"+vo4'G\"F-&F%6$F'\"\" 'F-!\"\"" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#*&&%\"aG6$\"\"(\"\"'\"\"\" %1~is~a~parameter.GF)" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%C___________ _______________________G" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/&%\"cG6# \"\")\"\"\"" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%!G" }}{PARA 11 "" 1 " " {XPPMATH 20 "6#/&%\"aG6$\"\")\"\"\",&#\"(EGM$\").q'*HF(*&#\")sE4C\"* D6Mb(F(&F%6$\"\"(\"\"'F(!\"\"" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/&%\" aG6$\"\")\"\"##!#U\"$@\"" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/&%\"aG6$ \"\")\"\"$,&#\"+-AHGO\"+$e,\"Ri\"\"\"*&#\"+7L8TF\",Dh=\"3?F-&F%6$\"\"( \"\"'F-F-" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/&%\"aG6$\"\")\"\"%,&#\"* T2p=%\"++8(oG\"\"\"\"*&#\"(_Z.*\")vV8XF-&F%6$\"\"(\"\"'F-!\"\"" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#/&%\"aG6$\"\")\"\"&,&#\"-f!e\"=?&)\".+ (QfXq:!\"\"*&#\"-/r3ZgN\".D\"GO&)G9\"\"\"&F%6$\"\"(\"\"'F2F2" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/&%\"aG6$\"\")\"\"',&#\"*++l)=\"**)Ga;#\"\" \"*&#\"$7&\"%LLF-&F%6$\"\"(F(F-!\"\"" }}{PARA 11 "" 1 "" {XPPMATH 20 " 6#/&%\"aG6$\"\")\"\"(\"\"!" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%C______ ____________________________G" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%!G" }}{PARA 11 "" 1 "" {XPPMATH 20 "6*/&%#b*G6#\"\"\"#\"'8YO\"())o5&/&F%6# \"\"#\"\"!/&F%6#\"\"$#\"($o!e'\")GfIA/&F%6#\"\"%#\"&2o\"\"&]e*/&F%6#\" \"&#\"+F.r<8\"++W25\")/&F%6#\"\"'#\"*DwWa$\"+OVrB;/&F%6#\"\"(F//&F%6# \"\")#\"#6\"$U\"" }}}{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 1355 "e7 := \{c[8] = 1, a[5,3] = -680319/85157644, a[7,4] = 95627273 /65894400+52938/40625*a[7,6], a[7,5] = 3154852507/14190950400-20862134 01/1286096875*a[7,6], c[3] = 2/9, b[2] = 0, `b*`[2] = 0, c[2] = 5/28, \+ c[5] = 37/53, c[6] = 27/35, a[6,1] = -22691367/349002500, a[6,3] = 316 55367/194652500, a[6,4] = -37368/1203125, a[5,4] = 41155100/63868233, \+ a[5,2] = -1407924/8188235, b[4] = 2144093/11232000, a[6,5] = 141862511 268/495147296875, a[7,1] = 594017/681984+47056/226625*a[7,6], a[7,3] = -2040813/66480128-16061328/18074875*a[7,6], `b*`[6] = 354447625/16237 14336, `b*`[5] = 1317710327/8110074400, `b*`[4] = 16807/95850, `b*`[3] = 6580683/22305928, a[4,2] = -108/245, b[8] = 3333/51200, a[8,6] = 18 8650000/216542889-512/3333*a[7,6], b[1] = 13093/179820, a[8,7] = 0, a[ 7,2] = -4263/2816, c[7] = 1, a[3,2] = 56/405, b[6] = 181575625/7318149 12, a[2,1] = 5/28, a[8,1] = 3342826/29967003-24092672/755341125*a[7,6] , a[8,5] = -852018158059/1570455938700+356047087104/1428853628125*a[7, 6], a[8,4] = 418690741/1286871300-9034752/45134375*a[7,6], a[8,3] = 36 28292202/6239101583+2741133312/20081186125*a[7,6], a[8,2] = -42/121, a [3,1] = 34/405, a[6,2] = 28188/67375, b[5] = 4600150423/36552448000, c [4] = 3/7, a[4,3] = 1053/1372, `b*`[1] = 364613/5106888, a[5,1] = 2296 0313/98258820, b[7] = 1/100, b[3] = 2932767/10210460, a[4,1] = 699/686 0, `b*`[7] = 0, `b*`[8] = 11/142\}:" }}}{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 0 146 "RK6_8eqs := [op(RowSumConditions(8,'expanded')),op(OrderConditions(6,8,'expanded' ))]:\n`RK5_8eqs*` := subs(b=`b*`,OrderConditions(5,8,'expanded')):" }} }{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 7 "Check: " }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 129 "simplify(subs(e7,RK6_8eqs )):\nmap(u->lhs(u)-rhs(u),%);\n;nops(%);\nsimplify(subs(e7,`RK5_8eqs*` )):\nmap(u->lhs(u)-rhs(u),%);\nnops(%);" }}{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$" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#\"#<" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 21 "We find a value for " }{XPPEDIT 18 0 "a[7,6]" "6#&% \"aG6$\"\"(\"\"'" }{TEXT -1 39 " by applying the simplifying conditio n" }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "Sum(`b*`[i]*a[i, j],i = j+1 .. s) = `b*`[j]*(1-c[j]);" "6#/-%$SumG6$*&&%#b*G6#%\"iG\"\" \"&%\"aG6$F+%\"jGF,/F+;,&F0F,F,F,%\"sG*&&F)6#F0F,,&F,F,&%\"cG6#F0!\"\" F," }{TEXT -1 5 ", " }{XPPEDIT 18 0 "j = 2" "6#/%\"jG\"\"#" }{TEXT -1 7 " . . . " }{TEXT 274 1 "s" }{TEXT -1 1 "," }}{PARA 0 "" 0 "" {TEXT -1 6 "with " }{XPPEDIT 18 0 "s=8" "6#/%\"sG\"\")" }{TEXT -1 23 " for the eighth stage." }}{PARA 0 "" 0 "" {TEXT -1 8 "Taking " } {XPPEDIT 18 0 "s = 8;" "6#/%\"sG\"\")" }{TEXT -1 7 " and " } {XPPEDIT 18 0 "j = 6;" "6#/%\"jG\"\"'" }{TEXT -1 9 ", we have" }} {PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "`b*`[7]*a[7,6]+`b*`[8 ]*a[8,6] = `b*`[6]*(1-c[6]);" "6#/,&*&&%#b*G6#\"\"(\"\"\"&%\"aG6$F)\" \"'F*F**&&F'6#\"\")F*&F,6$F2F.F*F**&&F'6#F.F*,&F*F*&%\"cG6#F.!\"\"F*" }{TEXT -1 2 ". " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 93 "Applying this equation to embedded order 5 scheme gives a n equation which can be solved for " }{XPPEDIT 18 0 "a[7,6];" "6#&%\" aG6$\"\"(\"\"'" }{TEXT -1 1 "." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 90 "subs(j=6,'add'(`b*`[i]*a[i,j ],i=j+1..8)=`b*`[j]*(1-c[j]));\nvalue(%);\nsubs(e7,%);\nsolve(%);" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#/-%$addG6$*&&%#b*G6#%\"iG\"\"\"&%\"aG6 $F+\"\"'F,/F+;\"\"(\"\")*&&F)6#F0F,,&F,F,&%\"cGF7!\"\"F," }}{PARA 11 " " 1 "" {XPPMATH 20 "6#/,&*&&%#b*G6#\"\"(\"\"\"&%\"aG6$F)\"\"'F*F**&&F' 6#\"\")F*&F,6$F2F.F*F**&&F'6#F.F*,&F*F*&%\"cGF7!\"\"F*" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/,&#\"++]dP5\",>^au`\"\"\"\"*&#\"$c#\"&8:#F(&%\"aG 6$\"\"(\"\"'F(!\"\"#\")vq75\"*#HkH?" }}{PARA 11 "" 1 "" {XPPMATH 20 "6 ##\")v]jj\")Oz/V" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 39 "We now complete the construction with " }{XPPEDIT 18 0 " a[7,6]=63635075/43047936" "6#/&%\"aG6$\"\"(\"\"'*&\")v]jj\"\"\"\")Oz/V !\"\"" }{TEXT -1 2 ". " }}{PARA 0 "" 0 "" {TEXT 260 4 "Note" }{TEXT -1 88 ": See the following subsection for further remarks concerning t his choice of value for " }{XPPEDIT 18 0 "a[7,6];" "6#&%\"aG6$\"\"(\" \"'" }{TEXT -1 1 "." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 76 "e8 := `union`(subs(a[7,6]=63635075/43047936 ,e7),\{a[7,6]=63635075/43047936\}):" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 2 "e8" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1161 "e8 := \{c[8] = 1, a[5,3] = -680319/85157644 , a[7,1] = 108451463/92067840, a[8,6] = 10127075/15722586, c[3] = 2/9, a[8,1] = 31846/494505, a[7,4] = 3338370539/988416000, b[2] = 0, `b*`[ 2] = 0, c[2] = 5/28, c[5] = 37/53, c[6] = 27/35, a[6,1] = -22691367/34 9002500, a[6,3] = 31655367/194652500, a[6,4] = -37368/1203125, a[5,4] \+ = 41155100/63868233, a[5,2] = -1407924/8188235, b[4] = 2144093/1123200 0, a[6,5] = 141862511268/495147296875, a[7,5] = -154367434883/70954752 000, a[7,3] = -77302059309/57505310720, `b*`[6] = 354447625/1623714336 , `b*`[5] = 1317710327/8110074400, `b*`[4] = 16807/95850, `b*`[3] = 65 80683/22305928, a[4,2] = -108/245, b[8] = 3333/51200, b[1] = 13093/179 820, a[8,7] = 0, a[7,2] = -4263/2816, c[7] = 1, a[3,2] = 56/405, b[6] \+ = 181575625/731814912, a[2,1] = 5/28, a[8,2] = -42/121, a[3,1] = 34/40 5, a[6,2] = 28188/67375, b[5] = 4600150423/36552448000, c[4] = 3/7, a[ 4,3] = 1053/1372, `b*`[1] = 364613/5106888, a[8,5] = -4513801763/25915 114500, a[5,1] = 22960313/98258820, b[7] = 1/100, b[3] = 2932767/10210 460, a[4,1] = 699/6860, a[8,3] = 241942482/308866415, a[8,4] = 625387/ 21235500, `b*`[7] = 0, `b*`[8] = 11/142, a[7,6] = 63635075/43047936\}: " }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 119 "subs(e8,matrix([seq([c[i],seq(a[i,j],j=1.. i-1),``$(9-i)],i=2..8),\n[`b`,seq(b[i],i=1..8)],[`b*`,seq(`b*`[i],i=1. .8)]]));" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#K%'matrixG6#7+7+#\"\"&\"#G F(%!GF+F+F+F+F+F+7+#\"\"#\"\"*#\"#M\"$0%#\"#cF2F+F+F+F+F+F+7+#\"\"$\" \"(#\"$*p\"%go#!$3\"\"$X##\"%`5\"%s8F+F+F+F+F+7+#\"#P\"#`#\")8.'H#\")? )e#)*#!(CzS\"\"(N#)=)#!'>.o\")Ww:&)#\")+^:T\")L#oQ'F+F+F+F+7+#\"#F\"#N #!)n8pA\"*+D+\\$#\"&)=G\"&vt'#\")n`lJ\"*+Dl%>#!&ot$\"(DJ?\"#\"-o7^i=9 \"-voHZ^\\F+F+F+7+\"\"\"#\"*j9X3\"\")Sy1#*#!%jU\"%;G#!,4$f?Ix\",?2J0v& #\"+R0PQL\"*+gT))*#!-$)[VnV:\",+?va4(#\")v]jj\")Oz/VF+F+7+F`o#\"&Y=$\" '0X\\#!#U\"$@\"#\"*#[U>C\"*:k')3$#\"'(QD'\")+bB@#!+j " 0 "" {MPLTEXT 1 0 1 ";" }}}}{PARA 0 "" 0 "" {TEXT -1 52 "#--------------------------------------------------- " }}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 31 "note concerning the value for " }{XPPEDIT 18 0 "a[7, 6];" "6#&%\"aG6$\"\"(\"\"'" }{TEXT -1 1 " " } }{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 46 "e7: coefficients of th e combined scheme with " }{XPPEDIT 18 0 "a[7,6]" "6#&%\"aG6$\"\"(\"\" '" }{TEXT -1 16 " as a parameter" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1355 "e7 := \{c[8] = 1, a[5,3] = -680319/85157644, a[7,4] = 95627273/65894400+52938/40625*a[7,6], a[7, 5] = 3154852507/14190950400-2086213401/1286096875*a[7,6], c[3] = 2/9, \+ b[2] = 0, `b*`[2] = 0, c[2] = 5/28, c[5] = 37/53, c[6] = 27/35, a[6,1] = -22691367/349002500, a[6,3] = 31655367/194652500, a[6,4] = -37368/1 203125, a[5,4] = 41155100/63868233, a[5,2] = -1407924/8188235, b[4] = \+ 2144093/11232000, a[6,5] = 141862511268/495147296875, a[7,1] = 594017/ 681984+47056/226625*a[7,6], a[7,3] = -2040813/66480128-16061328/180748 75*a[7,6], `b*`[6] = 354447625/1623714336, `b*`[5] = 1317710327/811007 4400, `b*`[4] = 16807/95850, `b*`[3] = 6580683/22305928, a[4,2] = -108 /245, b[8] = 3333/51200, a[8,6] = 188650000/216542889-512/3333*a[7,6], b[1] = 13093/179820, a[8,7] = 0, a[7,2] = -4263/2816, c[7] = 1, a[3,2 ] = 56/405, b[6] = 181575625/731814912, a[2,1] = 5/28, a[8,1] = 334282 6/29967003-24092672/755341125*a[7,6], a[8,5] = -852018158059/157045593 8700+356047087104/1428853628125*a[7,6], a[8,4] = 418690741/1286871300- 9034752/45134375*a[7,6], a[8,3] = 3628292202/6239101583+2741133312/200 81186125*a[7,6], a[8,2] = -42/121, a[3,1] = 34/405, a[6,2] = 28188/673 75, b[5] = 4600150423/36552448000, c[4] = 3/7, a[4,3] = 1053/1372, `b* `[1] = 364613/5106888, a[5,1] = 22960313/98258820, b[7] = 1/100, b[3] \+ = 2932767/10210460, a[4,1] = 699/6860, `b*`[7] = 0, `b*`[8] = 11/142\} :" }}}{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$\"\"'\"\")" }{TEXT -1 127 " denote the vector whose components are the principal \+ error terms of the 8 stage, order 6 scheme (the error terms of order 7 )." }}{PARA 0 "" 0 "" {TEXT -1 5 "Let " }{XPPEDIT 18 0 "`T*`[5,8];" " 6#&%#T*G6$\"\"&\"\")" }{TEXT -1 145 " denote the vector whose compone nts are the principal error terms of the embedded 8 stage, order 5 sch eme (the error terms of order 6) and let " }{XPPEDIT 18 0 "`T*`[6,8]; " "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*`[5,8]));" "6#-%$absG6#-F$6#&%#T*G6$\"\"&\"\")" }{TEXT -1 7 " and " }{XPPEDIT 18 0 "abs(abs(`T*`[6,8]));" "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*`[6,8]))/abs(abs(`T*`[5,8]));" "6#/&% \"BG6#\"\"(*&-%$absG6#-F*6#&%#T*G6$\"\"'\"\")\"\"\"-F*6#-F*6#&F/6$\"\" &F2!\"\"" }{TEXT -1 6 ", " }{XPPEDIT 18 0 "C[7] = abs(abs(`T*`[6,8 ]-T[6,8]))/abs(abs(`T*`[5,8]));" "6#/&%\"CG6#\"\"(*&-%$absG6#-F*6#,&&% #T*G6$\"\"'\"\")\"\"\"&%\"TG6$F2F3!\"\"F4-F*6#-F*6#&F06$\"\"&F3F8" } {TEXT -1 1 "." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 77 "Dormand and Prince have suggested that as well as attempt ing to ensure that " }{XPPEDIT 18 0 "A[7]" "6#&%\"AG6#\"\"(" }{TEXT -1 73 " is a minimum, if the embedded scheme is to be used for error \+ control, " }{XPPEDIT 18 0 "B[7]" "6#&%\"BG6#\"\"(" }{TEXT -1 7 " and " }{XPPEDIT 18 0 "C[7]" "6#&%\"CG6#\"\"(" }{TEXT -1 27 " should be \+ chosen so that " }}{PARA 0 "" 0 "" {TEXT -1 68 "they are \"similar in \+ magnitude\" and also not differ too much from 1." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 471 "errterms6_8 := PrincipalErrorTerms(6,8,'expanded'):\n`errterms5_8*` := subs(b=`b* `,PrincipalErrorTerms(5,8,'expanded')):\n`errterms6_8*` := subs(b=`b*` ,PrincipalErrorTerms(6,8,'expanded')):\nsnmB := sqrt(add(subs(e7,`errt erms6_8*`[i])^2,i=1..nops(`errterms6_8*`))):\nsdnB := sqrt(add(subs(e7 ,`errterms5_8*`[i])^2,i=1..nops(`errterms5_8*`))):\nsnmC := sqrt(add(( subs(e7,`errterms6_8*`[i])-subs(e7,errterms6_8[i]))^2,i=1..nops(errter ms6_8))):\nB_7 := snmB/sdnB:\nC_7 := snmC/sdnB:" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 109 "'B[7]' = ev alf[8](eval(B_7,a[7,6]=63635075/43047936));\n'C[7]' = evalf[8](eval(C_ 7,a[7,6]=63635075/43047936));" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/&%\" BG6#\"\"($\")a6'y\"!\"(" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/&%\"CG6#\" \"($\");6d9!\"(" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 6 "When " }{XPPEDIT 18 0 "a[7,6]=63635075/43047936" "6#/&%\" aG6$\"\"(\"\"'*&\")v]jj\"\"\"\")Oz/V!\"\"" }{TEXT -1 11 ", we have " }{XPPEDIT 18 0 "B[7]" "6#&%\"BG6#\"\"(" }{TEXT -1 1 " " }{TEXT 275 1 " ~" }{TEXT -1 15 " 1.7861 and " }{XPPEDIT 18 0 "C[7]" "6#&%\"CG6#\" \"(" }{TEXT -1 1 " " }{TEXT 276 1 "~" }{TEXT -1 9 " 1.4571. " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 43 "The following p icture shows the graphs of " }{XPPEDIT 18 0 "B[7]" "6#&%\"BG6#\"\"(" }{TEXT -1 5 " in " }{TEXT 259 5 "brown" }{TEXT -1 6 " and " } {XPPEDIT 18 0 "C[7]" "6#&%\"CG6#\"\"(" }{TEXT -1 5 " in " }{TEXT 261 3 "red" }{TEXT -1 3 ". \n" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 527 "aa := 63635075/43047936:\nbb := evalf(eval(B_7,a[7,6]=aa)):\ncc : = evalf(eval(C_7,a[7,6]=aa)):\np1 := plot([B_7,C_7,1],a[7,6]=-2.52..4. 35,0..2.8,color=[brown,red,blue]):\np2 := plot([[aa,0],[aa,2.9]],color =COLOR(RGB,0,.7,0)):\np3 := plot([[[aa,bb],[aa,cc]]$3],style=point,sym bol=[circle,diamond,cross],color=[black,coral$2]):\np4 := plot([[[aa,0 ]]$3],style=point,symbol=[circle,diamond,cross],color=black):\np5 := p lot([[[aa,bb],[0,bb]],[[aa,cc],[0,cc]]],linestyle=3,color=[brown,red]) :\nplots[display]([p||(1..5)],font=[HELVETICA,9]);" }}{PARA 13 "" 1 " " {GLPLOT2D 500 385 385 {PLOTDATA 2 "61-%'CURVESG6$7^o7$$!3-++++++?D!# <$\"3US>!ze$[)p\"F*7$$!3A+](3b`-P#F*$\"3RX-h%G')eq\"F*7$$!3$*\\7=m*f*R AF*$\"3tX$*p>$QHr\"F*7$$!3)**\\7K?JM4#F*$\"3Ql()z2nk@F*$\"3dAX;^%p8t\"F*7$$!3=]ivV@8*z\"F*$\"3&G$*R5$[:Up/\"32Lv\"F*7$$!32]7Vza5A:F*$\"3_FSo3SAm\"ROn==F*7$$!3#H+v$ypB+&*!#=$\"3\\>@_'Rg\"R=F*7$$!3[-+Dn7Z=!)F\\o $\"3O3@#)3['e'=F*7$$!3R/+v$G@1`'F\\o$\"3%ejq'3UX(*=F*7$$!3Q-+vK@!o4&F \\o$\"3\\0=%y%y`L>F*7$$!3q.DJ\")pv%z$F\\o$\"3mLphagDs>F*7$$!3#\\+](f$* \\YAF\\o$\"3.=s[HG[F?F*7$$!3if++!p?#\\$*!#>$\"3;&Qv75hQ3#F*7$$\"3#H$\\ (=?Tv!fF[q$\"3IUhy(4LG;#F*7$$\"3n$***\\jgBT>F\\o$\"3)=EB'pOBYAF*7$$\"3 F%\\(=&zGHU$F\\o$\"3cm,^?IV^BF*7$$\"3w&\\ibm]Q$[F\\o$\"3Q\"y@Ip%HdCF*7 $$\"3!eu=P9D*pbF\\o$\"3(fJ-pPI$3DF*7$$\"3&e*\\(=i**fI'F\\o$\"3'y=In(GI ]DF*7$$\"3#4iShpsRk'F\\o$\"3gJ_SBq![c#F*7$$\"3)fC1/xX>)pF\\o$\"37e^>\" >v`d#F*7$$\"3]e!RvIK4:(F\\o$\"3S3TdtF\\o$\"3S+F -G#Q8e#F*7$$\"3e$o/=Q0*)[(F\\o$\"3?_u(>N'R#e#F*7$$\"35'\\P*=>*yl(F\\o$ \"3])*pu1y2#e#F*7$$\"3LLM_Gn;SyF\\o$\"3%=A>*f,7!e#F*7$$\"3bq$4\"Q:WA!) 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6%7$Fafm7$FbamFbfmF^amFghm-%+AXESLABELSG6%Q'a[7,6]6\"Q!Fcim-%%FONTG6#% (DEFAULTG-Ffim6$%*HELVETICAG\"\"*-%%VIEWG6$;$!$_#!\"#$\"$N%Fcjm;Fbam$ \"#GFjem" 1 2 0 1 10 0 2 9 1 4 2 1.000000 45.000000 45.000000 0 0 "Cur ve 1" "Curve 2" "Curve 3" "Curve 4" "Curve 5" "Curve 6" "Curve 7" "Cur ve 8" "Curve 9" "Curve 10" "Curve 11" "Curve 12" }}}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT 260 4 "Note" }{TEXT -1 23 ": The value taken for " }{XPPEDIT 18 0 "a[7,6]" "6#&%\"aG6$\"\"(\"\"'" } {TEXT -1 9 " makes " }{XPPEDIT 18 0 "C[7];" "6#&%\"CG6#\"\"(" } {TEXT -1 23 " close to its minimum." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 27 "minimize(C_7):\nevalf[8](% );" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#$\")/6d9!\"(" }}}{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 40 "ee: coeffic ients for the combined scheme" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1056 "ee := \{c[2]=5/28,c[3]=2/9 ,c[4]=3/7,c[5]=37/53,c[6]=27/35,c[7]=1,c[8]=1,\n\na[2,1]=5/28,\na[3,1] =34/405,\na[3,2]=56/405,\na[4,1]=699/6860,\na[4,2]=-108/245,\na[4,3]=1 053/1372,\na[5,1]=22960313/98258820,\na[5,2]=-1407924/8188235,\na[5,3] =-680319/85157644,\na[5,4]=41155100/63868233,\na[6,1]=-22691367/349002 500,\na[6,2]=28188/67375,\na[6,3]=31655367/194652500,\na[6,4]=-37368/1 203125,\na[6,5]=141862511268/495147296875,\na[7,1]=108451463/92067840, \na[7,2]=-4263/2816,\na[7,3]=-77302059309/57505310720,\na[7,4]=3338370 539/988416000,\na[7,5]=-154367434883/70954752000,\na[7,6]=63635075/430 47936,\na[8,1]=31846/494505,\na[8,2]=-42/121,\na[8,3]=241942482/308866 415,\na[8,4]=625387/21235500,\na[8,5]=-4513801763/25915114500,\na[8,6] =10127075/15722586,\na[8,7]=0,\n\nb[1]=13093/179820,\nb[2]=0,\nb[3]=29 32767/10210460,\nb[4]=2144093/11232000,\nb[5]=4600150423/36552448000, \nb[6]=181575625/731814912,\nb[7]=1/100,\nb[8]=3333/51200,\n\n`b*`[1]= 364613/5106888,\n`b*`[2]=0,\n`b*`[3]=6580683/22305928,\n`b*`[4]=16807/ 95850,\n`b*`[5]=1317710327/8110074400,\n`b*`[6]=354447625/1623714336, \n`b*`[7]=0,\n`b*`[8]=11/142\}:" }}}{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,2\"\"\"F)F'F)*&#F)\"\"#F)*$)F'F,F)F)F)*&#F)\"\"'F)*$)F'\"\"$ F)F)F)*&#F)\"#CF)*$)F'\"\"%F)F)F)*&#F)\"$?\"F)*$)F'\"\"&F)F)F)*&#F)\"$ ?(F)*$)F'F1F)F)F)*&#F)\"%S]F)*$)F'\"\"(F)F)F)" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 124 "We can find the point wh ere the boundary of the stability region intersects the negative real \+ axis by solving the equation: " }}{PARA 256 "" 0 "" {TEXT -1 1 " " } {XPPEDIT 18 0 "R(z) = -1;" "6#/-%\"RG6#%\"zG,$\"\"\"!\"\"" }{TEXT -1 2 ". " }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 27 "z0 := newton(R(z)=- 1,z=-4);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%#z0G$!+J(HT&R!\"*" }}} {PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 306 "z0 := newton(R(z)=-1,z=-4):\np1 := plot([R(z),-1],z=-4.49..0.49,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 ,color=COLOR(RGB,0,.5,0)):\nplots[display]([p1,p2,p3],view=[-4.49..0.4 9,-1.47..1.47],font=[HELVETICA,9]);" }}{PARA 13 "" 1 "" {GLPLOT2D 360 253 253 {PLOTDATA 2 "6+-%'CURVESG6$7U7$$!3A++++++!\\%!#<$!3iK'p5:D&)o# F*7$$!3;+]7'3DdV%F*$!3'yBEJ]7![CF*7$$!35++Ds,X\"Q%F*$!33#HMw%y!=%F*$!3[%)Q[\"\\_ya\"F*7$$!35+]#oviQ2%F*$!3?Vq%yrMWE\"F*7$$! 3A+ve,&\\u'RF*$!3=u.%e0+,T)!#=7$$!3;+ v.8ajmPF*$!33s5ua7+&y'FU7$$!3C+v$Q<')4m$F*$!3'oly[$*y)z`FU7$$!3G+v3`dn bNF*$!3rXMsSCn;UFU7$$!37++0T5NZMF*$!31(G3#p8$)GKFU7$$!3?+]sv&Q>N$F*$!3 lAf+n4Z1DFU7$$!3E++:Qn_WKF*$!3YB],/wpJ=FU7$$!3#****\\s&QnOJF*$!3?]#*f2 #*ew7FU7$$!3&****\\=iPF.$F*$!3udd/!fT_O)!#>7$$!3Q+v$Gc`$QHF*$!3[J')\\M FVE]Fhp7$$!3+++l(y@h#GF*$!3OVJc7l=4dQp\"F*$\"3 %>lkG9%)R#=FU7$$!3)*****p]Q@(e\"F*$\"3pL`rFNZO?FU7$$!3I+]FRT)G[\"F*$\" 3[txl0H&[E#FU7$$!3')*\\P*y(R>Q\"F*$\"3xZAkgb03DFU7$$!3++]Kx#e)p7F*$\"3 %fe2:*pC2GFU7$$!3=++5j![\"p6F*$\"3>kW?L%\\b5$FU7$$!3s***\\KT=;1\"F*$\" 3UL#R=.)feMFU7$$!3'***\\P%o0=k*FU$\"39v_l:!)y7QFU7$$!3](***\\tEbw&)FU$ \"3D$***QX#)[TUFU7$$!39-]P2EBuvFU$\"3sh<4*e0()o%FU7$$!3>)*\\(GL>l_'FU$ \"3#p9(z$>Cm?&FU7$$!3z++]-\")=-bFU$\"3C)eGR&QBodFU7$$!3$*)*\\PM#3)HWFU $\"3O%[Bd&[>@kFU7$$!3;(****f'*ypR$FU$\"3^sjV#z`)>rFU7$$!3L)**\\U'=wSBF U$\"3D'>n%p_,8zFU7$$!3?$*\\Pt2H$H\"FU$\"3+V,>Fw%oy)FU7$$!33R****pit2LF hp$\"3+r)*)36PYn*FU7$$\"3Z^+]FkzBxFhp$\"3QUWS=\"*H!3\"F*7$$\"35*****z2 `!f\"F*7$$\"3l+]i_F06GFU$\"35LwZUIfC8F*7$$\"3\\.]Pt1& z\"QFU$\"3]WLRM=\"\\Y\"F*7$$\"3!***************[FU$\"3Ko[\"G8;Bj\"F*-% 'COLOURG6&%$RGBG$\"*++++\"!\")$\"\"!F\\\\lF[\\l-F$6$7S7$F($!\"\"F\\\\l 7$F3Fa\\l7$F=Fa\\l7$FBFa\\l7$FGFa\\l7$FLFa\\l7$FQFa\\l7$FWFa\\l7$FfnFa \\l7$F[oFa\\l7$F`oFa\\l7$FeoFa\\l7$FjoFa\\l7$F_pFa\\l7$FdpFa\\l7$FjpFa \\l7$F_qFa\\l7$FdqFa\\l7$FjqFa\\l7$F_rFa\\l7$FdrFa\\l7$FirFa\\l7$F^sFa \\l7$FcsFa\\l7$FhsFa\\l7$F]tFa\\l7$FbtFa\\l7$FgtFa\\l7$F\\uFa\\l7$FauF a\\l7$FfuFa\\l7$F[vFa\\l7$F`vFa\\l7$FevFa\\l7$FjvFa\\l7$F_wFa\\l7$FdwF a\\l7$FiwFa\\l7$F^xFa\\l7$FcxFa\\l7$FhxFa\\l7$F]yFa\\l7$FbyFa\\l7$FgyF a\\l7$F\\zFa\\l7$FazFa\\l7$FfzFa\\l7$F[[lFa\\l7$F`[lFa\\l-Fe[l6&Fg[lF[ \\lF[\\lFh[l-F$6&7#7$$!35+++J(HT&RF*Fa\\l-%'SYMBOLG6#%'CIRCLEG-Fe[l6&F g[lF\\\\lF\\\\lF\\\\l-%&STYLEG6#%&POINTG-F$6&Fg_l-F\\`l6#%&CROSSGF_`lF a`l-F$6&Fg_l-F\\`l6#%(DIAMONDGF_`lFa`l-F$6%7$7$Fi_lF[\\lFh_l-%&COLORG6 &Fg[lF[\\l$\"\"&Fb\\lF[\\l-%*LINESTYLEG6#\"\"$-%+AXESLABELSG6%Q\"z6\"Q !F`bl-%%FONTG6#%(DEFAULTG-Fcbl6$%*HELVETICAG\"\"*-%%VIEWG6$;$!$\\%!\"# $\"#\\F`cl;$!$Z\"F`cl$\"$Z\"F`cl" 1 2 0 1 10 0 2 9 1 4 2 1.000000 45.000000 45.000000 0 0 "Curve 1" "Curve 2" "Curve 3" "Curve 4" "Curve 5" "Curve 6" }}}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 50 "The following picture shows the stability region. " }} {PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1281 "R := z -> 1+z+1/2*z^2+1/6*z^3+1/24*z^4+1/120*z^5+1/720*z^6+1/504 0*z^7:\npts := []:\nz0 := 0:\nfor ct from 0 to 200 do\n zz := newton (R(z)=exp(ct*Pi/20*I),z=z0):\n z0 := zz:\n pts := [op(pts),[Re(zz) ,Im(zz)]]:\nend do:\np1 := plot(pts,color=COLOR(RGB,.1,.2,.45)):\np2 : = plots[polygonplot]([seq([pts[i-1],pts[i],[-2,0]],i=2..nops(pts))],\n style=patchnogrid,color=COLOR(RGB,.2,.4,.9)):\npts := []: z0 := 1.5+4.3*I:\nfor ct from 0 to 60 do\n zz := newton(R(z)=exp(ct*Pi /30*I),z=z0):\n z0 := zz:\n pts := [op(pts),[Re(zz),Im(zz)]]:\nend do:\np3 := plot(pts,color=COLOR(RGB,.1,.2,.45)):\np4 := plots[polygon plot]([seq([pts[i-1],pts[i],[1.39,4.2]],i=2..nops(pts))],\n s tyle=patchnogrid,color=COLOR(RGB,.2,.4,.9)):\npts := []: z0 := 1.5-4.3 *I:\nfor ct from 0 to 60 do\n zz := newton(R(z)=exp(ct*Pi/30*I),z=z0 ):\n z0 := zz:\n pts := [op(pts),[Re(zz),Im(zz)]]:\nend do:\np5 := plot(pts,color=COLOR(RGB,.1,.2,.45)):\np6 := plots[polygonplot]([seq( [pts[i-1],pts[i],[1.39,-4.2]],i=2..nops(pts))],\n style=patch nogrid,color=COLOR(RGB,.2,.4,.9)):\np7 := plot([[[-4.49,0],[1.79,0]],[ [0,-4.69],[0,4.69]]],color=black,linestyle=3):\nplots[display]([p||(1. .7)],view=[-4.49..1.79,-4.69..4.69],font=[HELVETICA,9],\n \+ labels=[`Re(z)`,`Im(z)`],axes=boxed,scaling=constrained);" }}{PARA 13 "" 1 "" {GLPLOT2D 381 565 565 {PLOTDATA 2 "6/-%'CURVESG6$7ew7$$\"\" !F)F(7$F($\"3++++Fjzq:!#=7$$\"3!)*****\\QO5E#!#E$\"3'******pk#fTJF-7$$ \"3!******4`NP]&!#D$\"3))*****\\t)Q7ZF-7$$\"3Y+++5i'))4&!#C$\"3]+++@@= $G'F-7$$\"3*)*****498?u#!#B$\"3#*******='eR&yF-7$$\"3(******HR0#G5!#A$ \"3B+++.PmC%*F-7$$\"36+++IKeUHFI$\"3!******fv8&*4\"!#<7$$\"3M*****>c%* 4p'FI$\"3/+++&e*\\c7FQ7$$\"30+++YW#Q?\"!#@$\"3-+++)oWLT\"FQ7$$\"3#**** **pUgyb\"FZ$\"3)******\\#e!*p:FQ7$$\"3_+++tN#Gd'FI$\"3++++dC&fs\"FQ7$$ !39+++*>;_$RFZ$\"3++++KH:\")=FQ7$$!3-+++0t1!o\"!#?$\"3*)*****zJ!3N?FQ7 $$!3:+++okGTXFio$\"3=+++b+C(=#FQ7$$!3%******\\9yf+\"!#>$\"3-+++'>-rL#F Q7$$!3!******zLXZ'>Fdp$\"3A++++*)4%[#FQ7$$!3/+++zx3(\\$Fdp$\"31+++q?aF EFQ7$$!3#)******=O1ydFdp$\"3%******H3Ykw#FQ7$$!3Q+++sKCf*)Fdp$\"3))*** **z]@$**GFQ7$$!38+++=g-78F-$\"3%)*****\\!)HT-$FQ7$$!3*******fg'=A=F-$ \"3%******zRK'QJFQ7$$!3.+++!Qf+T#F-$\"38+++JI4TKFQ7$$!3%)*****fF\\B0$F -$\"34+++#oX3L$FQ7$$!3z*****Rbb[T%F -$\"3')*****p'\\`uMFQ7$$!35+++-+u$FQ7$$!3'*******4\\tg6FQ$\"36+++R^TRPFQ7$$!3.+ ++5fR@7FQ$\"37+++Cc)\\t$FQ7$$!3#*******z-V\"G\"FQ$\"3?+++%>Cos$FQ7$$!3 3+++E&))3M\"FQ$\"34+++a[,:PFQ7$$!3/+++-t#)*R\"FQ$\"3#******fU>'*p$FQ7$ $!3%******pf7$e9FQ$\"3/+++P9o!o$FQ7$$!33+++ENU;:FQ$\"3%)******p*H#eOFQ 7$$!3-+++@nDu:FQ$\"3'******H&RGKOFQ7$$!36+++*HJ>j\"FQ$\"3,+++8v&Gg$FQ7 $$!3!******R:%f*o\"FQ$\"3/+++Wr'*pNFQ7$$!3-+++>]UZFQ$\"3(******\\'e%[S$FQ7$$!3'******p[Y_)>FQ$ \"3-+++?c)eN$FQ7$$!3?+++lBrZ?FQ$\"3!)*****z.QWI$FQ7$$!36+++Js&=6#FQ$\" 3%)*****R)=*4D$FQ7$$!3'******H4Xx<#FQ$\"3;+++FF8'>$FQ7$$!3%******R!*e_ C#FQ$\"30+++&f'[SJFQ7$$!3(******4'f39BFQ$\"3'******z`PY3$FQ7$$!37+++s1 v$Q#FQ$\"36+++T^/HIFQ7$$!33+++!f&o`CFQ$\"37+++_@*R(HFQ7$$!3'******fW8L _#FQ$\"3(******>Bx&>HFQ7$$!3/+++TG6#f#FQ$\"3=+++RbulGFQ7$$!33+++vYlfEF Q$\"3++++6sL7GFQ7$$!3#******H`6cs#FQ$\"3;+++E_8fFFQ7$$!3/+++`&\\(*y#FQ $\"3%)*****\\h-fq#FQ7$$!3=+++5C\">&GFQ$\"3-+++@oS_EFQ7$$!3&)*****z#R+7 HFQ$\"3#)*****4EL%)f#FQ7$$!3;+++[E(*pHFQ$\"33+++;8zVDFQ7$$!3>+++;&*zDI FQ$\"39+++/dJ)[#FQ7$$!3/+++4()[zIFQ$\"32+++P`'=V#FQ7$$!30+++Z61JJFQ$\" 3!******p1@VP#FQ7$$!39+++G,b!=$FQ$\"37+++RJe:BFQ7$$!3;+++X&)*zA$FQ$\"3 5+++9)obD#FQ7$$!35+++esXtKFQ$\"3/+++9/@%>#FQ7$$!3*)******[U)pJ$FQ$\"3+ +++GRXJ@FQ7$$!33+++'GW'eLFQ$\"3=+++>#es1#FQ7$$!33+++;)3&)R$FQ$\"3#)*** **\\)[f,?FQ7$$!3********zelOMFQ$\"36+++[([W$>FQ7$$!3)******RypJZ$FQ$\" 3!*******[*=e'=FQ7$$!3))*****zGS\"3NFQ$\"3#******Ra?dz\"FQ7$$!3&)***** R2h;a$FQ$\"35+++Ok=C(4YMOFQ$\"30+++Jbg,:FQ7$$!3)) *****ps)3jOFQ$\"34+++KN2D9FQ7$$!31+++S6n!p$FQ$\"3!******z8&eZ8FQ7$$!3! 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\\%FQF(7$$\"3/++++++!z\"FQF(-%'COLOURG6&F`^nF)F)F)-%*LINESTYLEG6#\"\"$ -F$6%7$7$F($!3Q++++++!p%FQ7$F($\"3Q++++++!p%FQFjcvF]dv-%%FONTG6$%*HELV ETICAGFb^r-%*AXESSTYLEG6#%$BOXG-%(SCALINGG6#%,CONSTRAINEDG-%+AXESLABEL SG6%%&Re(z)G%&Im(z)G-F[ev6#%(DEFAULTG-%%VIEWG6$;$!$\\%Fh^n$\"$z\"Fh^n; $!$p%Fh^n$\"$p%Fh^n" 1 2 0 1 10 0 2 9 1 2 1 1.000000 45.000000 45.000000 0 0 "Curve 1" "Curve 2" "Curve 3" "Curve 4" "Curve 5" "Curve 6" "Curve 7" "Curve 8" }}}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 " " 0 "" {TEXT -1 4 "The " }{TEXT 260 30 "interval of absolute stability " }{TEXT -1 89 " (or stability interval) is the intersection of the s tability region with the real line." }}{PARA 0 "" 0 "" {TEXT -1 59 "Fo r this scheme the stability interval is (approximately) " }{XPPEDIT 18 0 "[-3.9541, 0];" "6#7$,$-%&FloatG6$\"&T&R!\"%!\"\"\"\"!" }{TEXT -1 2 ". " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 260 "We can distort the boundary curve horizontally by taking the 11th root of the real part of points along the curve. In this way we see t hat the largest interval on the nonnegative imaginary axis that contai ns the origin and lies inside the stability region is " }{XPPEDIT 18 0 "[0, 1.76];" "6#7$\"\"!-%&FloatG6$\"$w\"!\"#" }{TEXT -1 18 " approx imately. " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 " " {MPLTEXT 1 0 319 "R := z -> 1+z+1/2*z^2+1/6*z^3+1/24*z^4+1/120*z^5+1 /720*z^6+1/5040*z^7:\nDigits := 25:\npts := []: z0 := 0:\nfor ct from \+ 0 to 60 do\n zz := newton(R(z)=exp(ct*Pi/100*I),z=z0):\n z0 := zz: \n pts := [op(pts),[surd(Re(zz),11),Im(zz)]]:\nend do:\nplot(pts,col or=COLOR(RGB,0,.2,1),thickness=2,font=[HELVETICA,9]);\nDigits := 10:" }}{PARA 13 "" 1 "" {GLPLOT2D 261 276 276 {PLOTDATA 2 "6(-%'CURVESG6#7i n7$$\"\"!F)F(7$$\":(GMFj2VC9eNyI!#E$\":yZvsJz*e`EfTJF-7$$\":=SG%)Gykc@ *o&4&F-$\":w13[GbzrI&=$G'F-7$$\":!)3))4>t?9qG@%oF-$\":!y*ezx3o2'zxC%*F -7$$\":A+QI!=3it^IK%)F-$\":d8Yy#)>M91PmD\"!#D7$$\":P)>V*eS67jI[\"**F-$ \":CoYlW5#yEjzq:F?7$$\":obXc>7S_R:;8\"F?$\":'o/^.pz3#fb\\)=F?7$$\":-C) pR\"f*)Q1o_E\"F?$\":sb85*4%\\s&[6*>#F?7$$\":BxXKDBKxyXNR\"F?$\":qcv5pP (*>7uK^#F?7$$\":\")ya<9'Q3^cB<:F?$\":/E/5]Z7dQLu#GF?7$$\":`#fswJ!y/!># pj\"F?$\":Sx3Px>1rk#fTJF?7$$\":)3\"orXVLN#=0`vbMF?7 $$\":7=K)y)pe\"3h(f'=F?$\":tc!G!=d_5:6*pPF?7$$\":V-d8&)zz2'[(f(>F?$\": 'HrDDI8b\"QqS3%F?7$$\":3Z1;+9g!y]F$3#F?$\":1=2]nS+GeH#)R%F?7$$\":.*GNc hB@wF1)=#F?$\":#\\y2`DfKN()Q7ZF?7$$\":=S.17J!GX;\\!H#F?$\":Qo#Q+x[`4ya E]F?7$$\":92+c%))*>_O*o!R#F?$\":\"[Y*))e3G'eR&yF?7$$\":#e*[ypTY`WbZ?$F?$\":(=*p @)z#*4u$3\"o\")F?7$$\":*o)*)z_7DE&)=iG$F?$\":$4%3^;1X4#[D#[)F?7$$\":YF q!Q%=snbNfO$F?$\":Lh([eqW06qR'z)F?7$$\":HFMZ`e6An&*QW$F?$\":hpFExG*fyP `5\"*F?7$$\":(or$oWGI)y93?NF?$\":qr]p'=,E.PmC%*F?7$$\":p,g_r>zQ2nWf$F? $\":)Hu'R!=omo]yQ(*F?7$$\":!\\/.rl\"Ffu7$$\":*or%4d75p1\" H=TF?$\":crMI=1y(y.6D7Ffu7$$\":u,0qbVG)Ry[tTF?$\":'[y&HZym_e*\\c7Ffu7$ $\":.BK_a()=f%4\"fA%F?$\":!38Ffu7$$\":\"4M([@ZU(3%e9K%F?$\":\"y;5zPltw4j]8Ffu7$$\":24d -,$4jtL&RO%F?$\":$ej\\b=!Q$yE*>Q\"Ffu7$$\":w[$3ox#))\\L!Q-WF?$\":6qt.H td\")oWLT\"Ffu7$$\":B'*Qv:g,*\\+>OWF?$\":U[\\Okk`Di&oW9Ffu7$$\":JEMO$y F-*orYY%F?$\":PG)=^MV)e&R,w9Ffu7$$\":2o-:pO$pp?)o[%F?$\":[&fP>#e.P,Gt] \"Ffu7$$\":(3;:2S))p#GI:]%F?$\":RJ%H6')4omfiQ:Ffu7$$\":Y/@x)ezm#oyn]%F ?$\":Czo'QWBTDe!*p:Ffu7$$\":^K#=r^T]'4u)*\\%F?$\":X3cJ\">XGOa;,;Ffu7$$ \":.mk\"3u_kVmRwWF?$\":Y\\#HbrabyCSK;Ffu7$$\":Di#*Hae;=/:&GWF?$\":[!p, &*4/!GY9Om\"Ffu7$$\":DgMc=qsT-\\+M%F?$\":Rs&yR5'e3t)z%p\"Ffu7$$\":lMP \"Qh'yWjKn;%F?$\":))4keas\"RdC&fs\"Ffu7$$\":ye43b)4t(3B6j$F?$\":-_9>Cr KJlsqv\"Ffu7$$!:![sV6%4$>0k\"34%F?$\":cRZc'\\X')ph:)y\"Ffu7$$!:y]OyGQ: 9QF)oWF?$\":rq*>>=Ffu7$$!:[XNpEO\"33:R6ZF?$\":lFWI%*[M0#)*>]=Ff u7$$!:l!\\o*4H$e=[)G!\\F?$\":I/Ka`&4tJH:\")=Ffu-%%FONTG6$%*HELVETICAG \"\"*-%&COLORG6&%$RGBGF($\"\"#!\"\"$\"\"\"F)-%+AXESLABELSG6$Q!6\"Fd_l- %*THICKNESSG6#F]_l-%%VIEWG6$%(DEFAULTGF\\`l" 1 2 0 1 10 2 2 6 1 4 2 1.000000 45.000000 45.000000 0 0 "Curve 1" }}}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 123 "The relevant intersection poin t of the boundary curve with the imaginary axis can be determined more accurately as follows." }}{PARA 0 "" 0 "" {TEXT -1 86 "First we look \+ for points on the boundary curve either side of the intersection point . " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 112 "Digits := 15:\nz0 := 1.76*I:\nfor ct from 55 to 58 d o\n newton(R(z)=exp(ct*Pi/100*I),z=z0);\nend do;\nDigits := 10:" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#^$$\"02+GdBGd'!#>$\"0t\"RdC&fs\"!#9" } }{PARA 11 "" 1 "" {XPPMATH 20 "6#^$$\"0o0E_SpW\"!#>$\"0rKJlsqv\"!#9" } }{PARA 11 "" 1 "" {XPPMATH 20 "6#^$$!05]am.#p`!#>$\"0bk)ph:)y\"!#9" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#^$$!0-k9J![>9!#=$\"0j>rq*>>=!#9" }}} {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 107 "Then we \+ apply the bisection method to calculate the parameter value associated with the intersection point." }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 178 "real_part := proc(u)\n Re(newton(R(z)=exp(u*Pi*I),z=1.76*I)) \nend proc:\nDigits := 15:\nu0 := bisect('real_part'(u),u=0.55..0.58); \nnewton(R(z)=exp(u0*Pi*I),z=1.76*I);\nDigits := 10:" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%#u0G$\"0')fUmIOi&!#:" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#^#$\"0T`XK@Ww\"!#9" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 " " 0 "" {TEXT -1 4 "The " }{TEXT 260 112 "largest interval on the nonne gative imaginary axis that contains the origin and lies inside the sta bility region" }{TEXT -1 5 " is " }{XPPEDIT 18 0 "[0, 1.7644];" "6#7$ \"\"!-%&FloatG6$\"&Ww\"!\"%" }{TEXT -1 18 " (approximately)." }} {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 37 "#-------- ----------------------------" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 80 "The stability function R* for the 8 stage, or der 5 scheme is given as follows." }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 97 "subs(ee,subs(b=`b*`,StabilityFunction(5,8,'expanded') )):\n`R*` := unapply(%,z):\n'`R*`(z)'=`R*`(z);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/-%#R*G6#%\"zG,2\"\"\"F)F'F)*&#F)\"\"#F)*$)F'F,F)F)F)*& #F)\"\"'F)*$)F'\"\"$F)F)F)*&#F)\"#CF)*$)F'\"\"%F)F)F)*&#F)\"$?\"F)*$)F '\"\"&F)F)F)*&#\"&J#Q\")];1FF)*$)F'F1F)F)F)*&#\"%iB\")D3`8F)*$)F'\"\"( F)F)F)" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 124 "We can find the point where the boundary of the stability region \+ intersects the negative real axis by solving the equation: " }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "`R*`(z) = -1;" "6#/-%#R*G6# %\"zG,$\"\"\"!\"\"" }{TEXT -1 2 ". " }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 33 "z_0 := newton(`R*`(z)=-1,z=-4.2);" }}{PARA 11 "" 1 " " {XPPMATH 20 "6#>%$z_0G$!+%p/(RU!\"*" }}}{PARA 0 "" 0 "" {TEXT -1 0 " " }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 324 "z_0 := newton(`R*`(z)=- 1,z=-4.2):\np_1 := plot([`R*`(z),-1],z=-4.69..0.49,color=[red,blue]): \np_2 := plot([[[z_0,-1]]$3],style=point,symbol=[circle,cross,diamond] ,color=black):\np_3 := plot([[z_0,0],[z_0,-1]],linestyle=3,color=COLOR (RGB,0,.5,0)):\nplots[display]([p_1,p_2,p_3],view=[-4.69..0.49,-1.47.. 1.47],font=[HELVETICA,9]);" }}{PARA 13 "" 1 "" {GLPLOT2D 398 284 284 {PLOTDATA 2 "6+-%'CURVESG6$7U7$$!3Q++++++!p%!#<$!3[ab([w1QJ#F*7$$!3Pouqy$*=F*7$$!3J]7Lq)pz _%F*$!3qm5)[Q(QK'y#e\"F*7$$!3On;R&)eOoV F*$!3_lP^O:l&G\"F*7$$!3En;u4L6smFP7$$!3H+DTEYePRF*$!3!RUV]za \\G&FP7$$!3Cn\"zeW#pFQF*$!3-FMK\\'y65%FP7$$!3X+D'4p_\"=PF*$!3euW6.:-NJ FP7$$!3pLL)))exag$F*$!3\">qU\\&*GpK#FP7$$!3!omTwHLi]$F*$!3bEP$R!e*\\u \"FP7$$!33++lJx]%R$F*$!3k\\r_uS$)47FP7$$!3/++v4MK#G$F*$!3qL$\\^JS!px!# >7$$!3_++NGI@uJF*$!3;Bbc*ex7R%Fcp7$$!3)o;z[WQg2$F*$!3ib&R'*[SG'=Fcp7$$ !3QLL[p$*HfHF*$\"3vVJK.&G)>i!#?7$$!3kLL$*fgSgGF*$\"3Yq%z\"fM-'Q#Fcp7$$ !3A+D6J'p`u#F*$\"3-Eg->i6]TFcp7$$!3lLLV**GaVEF*$\"3^a9z])f&HbFcp7$$!39 +D6OG#=`#F*$\"3ur*zfwY(=pFcp7$$!3J+vL;*QaU#F*$\"3[J4;iicy\")Fcp7$$!3NL $ekXQWJ#F*$\"3C@SRGt3![*Fcp7$$!3\\Lep$Q0D@#F*$\"3?%*[(>v#Hq5FP7$$!3&pm TSzcD5#F*$\"32k'RL*)o)37FP7$$!3'o;z)46N))>F*$\"3!)4fXpRzj8FP7$$!3C+v3E Z$*))=F*$\"3k8v8K`]5:FP7$$!3#pm\"*zIi:y\"F*$\"39!)y'z04So\"FP7$$!3?++q $4O1n\"F*$\"3]S$o$H+Y#)=FP7$$!3W+]-Mk6i:F*$\"3EX'QSK?$)4#FP7$$!3O+DJ$3 =rX\"F*$\"3jm2az[MIBFP7$$!3-+]d;``S8F*$\"3\\xlt'RZ!=EFP7$$!3!RLLa^!yN7 F*$\"3Dr'Qu@$z1HFP7$$!3?++v0C$R7\"F*$\"3qY/:>GX]KFP7$$!3&o;zWR\"eA5F*$ \"3mzYD]&Qpf$FP7$$!3?'***\\)ozx6*FP$\"3fFP$ \"3W?C=A(RA`&FP7$$!3e-]7G$*\\/[FP$\"3WVq**3T0&='FP7$$!3Wgmms4>IPFP$\"3 !)fGJ1;_')oFP7$$!3$zm;M^b:j#FP$\"3AJqFsLA'o(FP7$$!3NK$e/JLzt<\"F*7$$\"3q+]())Gfrs #FP$\"3Uu>3jq_88F*7$$\"3'3+Dr([\\uPFP$\"3_5G,*zf&e9F*7$$\"3!********** *****[FP$\"3#\\`h6I;Bj\"F*-%'COLOURG6&%$RGBG$\"*++++\"!\")$\"\"!F\\\\l F[\\l-F$6$7S7$F($!\"\"F\\\\l7$F3Fa\\l7$F=Fa\\l7$FBFa\\l7$FGFa\\l7$FLFa \\l7$FRFa\\l7$FWFa\\l7$FfnFa\\l7$F[oFa\\l7$F`oFa\\l7$FeoFa\\l7$FjoFa\\ l7$F_pFa\\l7$FepFa\\l7$FjpFa\\l7$F_qFa\\l7$FeqFa\\l7$FjqFa\\l7$F_rFa\\ l7$FdrFa\\l7$FirFa\\l7$F^sFa\\l7$FcsFa\\l7$FhsFa\\l7$F]tFa\\l7$FbtFa\\ l7$FgtFa\\l7$F\\uFa\\l7$FauFa\\l7$FfuFa\\l7$F[vFa\\l7$F`vFa\\l7$FevFa \\l7$FjvFa\\l7$F_wFa\\l7$FdwFa\\l7$FiwFa\\l7$F^xFa\\l7$FcxFa\\l7$FhxFa \\l7$F]yFa\\l7$FbyFa\\l7$FgyFa\\l7$F\\zFa\\l7$FazFa\\l7$FfzFa\\l7$F[[l Fa\\l7$F`[lFa\\l-Fe[l6&Fg[lF[\\lF[\\lFh[l-F$6&7#7$$!3G+++%p/(RUF*Fa\\l -%'SYMBOLG6#%'CIRCLEG-Fe[l6&Fg[lF\\\\lF\\\\lF\\\\l-%&STYLEG6#%&POINTG- F$6&Fg_l-F\\`l6#%&CROSSGF_`lFa`l-F$6&Fg_l-F\\`l6#%(DIAMONDGF_`lFa`l-F$ 6%7$7$Fi_lF[\\lFh_l-%&COLORG6&Fg[lF[\\l$\"\"&Fb\\lF[\\l-%*LINESTYLEG6# \"\"$-%+AXESLABELSG6%Q\"z6\"Q!F`bl-%%FONTG6#%(DEFAULTG-Fcbl6$%*HELVETI CAG\"\"*-%%VIEWG6$;$!$p%!\"#$\"#\\F`cl;$!$Z\"F`cl$\"$Z\"F`cl" 1 2 0 1 10 0 2 9 1 4 2 1.000000 45.000000 45.000000 0 0 "Curve 1" "Curve 2" "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 7 stage, order 5 scheme. " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1310 "`R*` := z \+ -> 1+z+1/2*z^2+1/6*z^3+1/24*z^4+1/120*z^5+38231/27061650*z^6+2362/1353 0825*z^7:\npts := []:\nz0 := 0:\nfor ct from 0 to 200 do\n zz := new ton(`R*`(z)=exp(ct*Pi/20*I),z=z0):\n z0 := zz:\n pts := [op(pts),[ Re(zz),Im(zz)]]:\nend do:\np_1 := plot(pts,color=COLOR(RGB,0,.1,.4)): \np_2 := plots[polygonplot]([seq([pts[i-1],pts[i],[-1.8,0]],i=2..nops( pts))],\n style=patchnogrid,color=COLOR(RGB,0,.2,.8)):\npts : = []: z0 := 1.4+4.3*I:\nfor ct from 0 to 60 do\n zz := newton(`R*`(z )=exp(ct*Pi/30*I),z=z0):\n z0 := zz:\n pts := [op(pts),[Re(zz),Im( zz)]]:\nend do:\np_3 := plot(pts,color=COLOR(RGB,0,.1,.4)):\np_4 := pl ots[polygonplot]([seq([pts[i-1],pts[i],[1.27,4.17]],i=2..nops(pts))], \n style=patchnogrid,color=COLOR(RGB,0,.2,.8)):\npts := []: z 0 := 1.4-4.3*I:\nfor ct from 0 to 60 do\n zz := newton(`R*`(z)=exp(c t*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,0,.1,.4)):\np_6 := plots[po lygonplot]([seq([pts[i-1],pts[i],[1.27,-4.17]],i=2..nops(pts))],\n \+ style=patchnogrid,color=COLOR(RGB,0,.2,.8)):\np_7 := plot([[[-4. 79,0],[1.89,0]],[[0,-4.69],[0,4.69]]],color=black,linestyle=3):\nplots [display]([p_||(1..7)],view=[-4.79..1.89,-4.69..4.69],font=[HELVETICA, 9],\n labels=[`Re(z)`,`Im(z)`],axes=boxed,scaling=constrain ed);" }}{PARA 13 "" 1 "" {GLPLOT2D 390 543 543 {PLOTDATA 2 "6/-%'CURVE SG6$7ew7$$\"\"!F)F(7$$\"3))*****>:58a$!#F$\"3#******fK'zq:!#=7$$\"3-++ +Uv0%=#!#D$\"36+++3DfTJF07$$\"3%*******G@e=B!#C$\"3\")*****HX'Q7ZF07$$ \"3-+++-_/b6!#B$\"3w*****p.mJG'F07$$\"3/+++*z25f$F@$\"3#*******ez)Q&yF 07$$\"3O+++`u$4O(F@$\"3Q+++sfVC%*F07$$\"3q******z$Q3G(F@$\"3++++A_X*4 \"!#<7$$!3&******f[HCh\"!#A$\"3%******>4ujD\"FR7$$!3&******\\z_S9\"!#@ $\"3#******\\a:JT\"FR7$$!38+++f*Qx+%Ffn$\"3%******pu\\&p:FR7$$!35+++Yj R$4\"!#?$\"31+++nu\\D].#FR7$$!3-+++lszc5!#>$\"3')*****z5'4)=#F 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840,\na[7,2]=-4263/2816,\na[7,3]=-77302059309/57505310720,\na[7,4]=333 8370539/988416000,\na[7,5]=-154367434883/70954752000,\na[7,6]=63635075 /43047936,\na[8,1]=31846/494505,\na[8,2]=-42/121,\na[8,3]=241942482/30 8866415,\na[8,4]=625387/21235500,\na[8,5]=-4513801763/25915114500,\na[ 8,6]=10127075/15722586,\na[8,7]=0,\n\nb[1]=13093/179820,\nb[2]=0,\nb[3 ]=2932767/10210460,\nb[4]=2144093/11232000,\nb[5]=4600150423/365524480 00,\nb[6]=181575625/731814912,\nb[7]=1/100,\nb[8]=3333/51200,\n\n`b*`[ 1]=364613/5106888,\n`b*`[2]=0,\n`b*`[3]=6580683/22305928,\n`b*`[4]=168 07/95850,\n`b*`[5]=1317710327/8110074400,\n`b*`[6]=354447625/162371433 6,\n`b*`[7]=0,\n`b*`[8]=11/142\}:" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT 259 5 "nodes" }{TEXT -1 2 ": " }} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 31 "seq(c[i]=subs(ee,c[i]),i=2.. 8);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6)/&%\"cG6#\"\"##\"\"&\"#G/&F%6# 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1134395721190865/1377175673936268], [23/28, 2334363 61745853683/373836194308813824, 158645421/143412416, -2222985817851376 045/1063980184927185792, 85304345532627205619/66923624494207595520], [ ``, ``, ``, ``, -872666029598803/9228174667747840], [1, -1178762513361 663864779/6649079911196633209152, -8522902869921/10868389582568, 30871 63579546988168202379/1791201064146135353344584, -376736459007705361142 6162077/10220229771451535362575529152], [``, ``, ``, 17701204856780353 690623/13557389052385415225632, -7678963110907452/10984484330835583], \+ [1, 1416449588677763/6015481297132284, -48943449/199463911, 8388448562 8625980/463004404677327933, 2690685696370602054377/6604522931171002998 060], [``, ``, 855085588386243786/608893946374021595, -107976726347212 80/10984484330835583, 0], [``, __________________________, ___________ _______________, __________________________, _________________________ _], [b, 2503727/33509160, 0, 116376653000/351539506521, 38494907685743 537/196546626900894600], [``, 117203823775061/154952625580200, -849245 60896/194121869325, 0, 4243913/53400600], [`b*`, 2644521142/3509665645 5, 0, 8764703092174/26729124547545, 737727050374863463/362213530821685 3050], [``, 2721770523832513/4007247067087950, -89022652768/2458877011 45, 2/57, 2/49]])" "6#-%'matrixG6#717'*&\"#Z\"\"\"\"$$>!\"\"*&F)F*F+F, %!GF.F.7'*&\"\"*F*\"#QF,*&\"#&)F*\"%WrF,*&\"&Lc\"F*\"'Od8F,F.F.7'*&\"$ U$F*\"$$oF,,$*&\"*;CI&pF*\",*QjZ(\\\"F,F,,$*&\"+9[\")p%*F*\",*QjZ(\\\" F,F,*&\"*G$=ePF*\"*()>h=$F,F.7'*&\"#rF*\"#*)F,*&\"/,AHp)4N$F*\"/O_#y_b ])F,*&\"+.\"fue$F*\"+xCc0YF,,$*&\"/Gc@2Xs@F*\"/$z^Dt?\"=F,F,*&\"1l3>@d RM6F*\"1oi$RnvrP\"F,7'*&\"#BF*\"#GF,*&\"3$o`euhjVL#F*\"3CQ\")3V>OQPF,* &\"*@ake\"F*\"*;CTV\"F,,$*&\"4XgP^y\"e)HA#F*\"4#z&=F\\=!)R1\"F,F,*&\"5 >c?FE`XVI&)F*\"5?bf2U\\CO#p'F,7'F.F.F.F.,$*&\"0.))fHgms)F*\"1SyunY6*z!\\mF,F,,$*&\".@*pG!H_)F*\" /oDe*Qo3\"F,F,*&\":zB?o\"))pazN;(3$F*\":%eWLNNh9k5?\"z\"F,,$*&\"=x?;E9 h`q2!fktw$F*\">_\"Hbdi``^9xH-A5F,F,7'F.F.F.*&\"8B1p`.yc[?,x\"F*\"8KcA: aQ_!*QdN\"F,,$*&\"1_u!46j*ywF*\"2$eN3L%[%)4\"F,F,7'F**&\"1jxn)e\\kT\"F *\"1%GKrH\"[:gF,,$*&\")\\M%*[F*\"*6RY*>F,F,*&\"2!)fiGc[%)Q)F*\"3LzKxYS /IYF,*&\"7xV0-1P'p&o!p#F*\"7g!)*H+r6$H_/mF,7'F.F.*&\"3'yViQ)e&3b)F*\"3 &f@SPYR*)3'F,,$*&\"2!G@Zjswz5F*\"2$eN3L%[%)4\"F,F,\"\"!7'F.%;_________ _________________GFirFirFir7'%\"bG*&\"(FP]#F*\")g\"4N$F,Fgr*&\"-+Ilwj6 F*\"-@l]R:NF,*&\"2PNu&o2\\\\QF*\"3+Y*3!piYl>F,7'F.*&\"0h]xBQ?<\"F*\"0+ -eDE&\\:F,,$*&\",'*3cC\\)F*\"-D$p=7%>F,F,Fgr*&\"(8RC%F*\")+1S`F,7'%#b* G*&\"+U6_WEF*\",bkl'4NF,Fgr*&\".u@4.Zw)F*\"/XvaC\"Hn#F,*&\"3jM'[P]qsP( F*\"4]I&o@3`8AOF,7'F.*&\"18D$Q_q " 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 co mbined scheme" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1633 "ee := \{c[2]=47/193,\nc[3]=9/38,\nc[4]=342/683 ,\nc[5]=71/89,\nc[6]=23/28,\nc[7]=1,\nc[8]=1,\n\na[2,1]=47/193,\na[3,1 ]=16515/135736,\na[3,2]=15633/135736,\na[4,1]=-695302416/14974763389, \na[4,2]=-9469814814/14974763389,\na[4,3]=375818328/318611987,\na[5,1] =33509869292201/85055527825236,\na[5,2]=3587459103/4605562477,\na[5,3] =-21724507215628/18120732551793,\na[5,4]=1134395721190865/137717567393 6268,\na[6,1]=233436361745853683/373836194308813824,\na[6,2]=158645421 /143412416,\na[6,3]=-2222985817851376045/1063980184927185792,\na[6,4]= 85304345532627205619/66923624494207595520,\na[6,5]=-872666029598803/92 28174667747840,\na[7,1]=-1178762513361663864779/6649079911196633209152 ,\na[7,2]=-8522902869921/10868389582568,\na[7,3]=308716357954698816820 2379/1791201064146135353344584,\na[7,4]=-3767364590077053611426162077/ 10220229771451535362575529152,\na[7,5]=17701204856780353690623/1355738 9052385415225632,\na[7,6]=-7678963110907452/10984484330835583,\na[8,1] =1416449588677763/6015481297132284,\na[8,2]=-48943449/199463911,\na[8, 3]=83884485628625980/463004404677327933,\na[8,4]=269068569637060205437 7/6604522931171002998060,\na[8,5]=855085588386243786/60889394637402159 5,\na[8,6]=-10797672634721280/10984484330835583,\na[8,7]=0,\n\nb[1]=25 03727/33509160,\nb[2]=0,\nb[3]=116376653000/351539506521,\nb[4]=384949 07685743537/196546626900894600,\nb[5]=117203823775061/154952625580200, \nb[6]=-84924560896/194121869325,\nb[7]=0,\nb[8]=4243913/53400600,\n\n `b*`[1]=2644521142/35096656455,\n`b*`[2]=0,\n`b*`[3]=8764703092174/267 29124547545,\n`b*`[4]=737727050374863463/3622135308216853050,\n`b*`[5] =2721770523832513/4007247067087950,\n`b*`[6]=-89022652768/245887701145 ,\n`b*`[7]=2/57,\n`b*`[8]=2/49\}:" }}}{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 48 "The Butcher tableau of the sche me is as follows." }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 418 "subs(e e,matrix([[c[2],a[2,1],``$3],\n [c[3],a[3,1],a[3,2],``$2],\n [c[4],s eq(a[4,i],i=1..3),``],\n [c[5],seq(a[5,i],i=1..4)],\n [c[6],seq(a[6, i],i=1..4)],[``$4,a[6,5]],\n [c[7],seq(a[7,i],i=1..4)],[``$3,a[7,5],a [7,6]],\n [c[8],seq(a[8,i],i=1..4)],[``$2,seq(a[8,i],i=5..7)],\n [`` ,`__________________________`$4],\n [`b`,seq(b[i],i=1..4)],[``,seq(b[ i],i=5..8)],\n [`b*`,seq(`b*`[i],i=1..4)],[``,seq(`b*`[i],i=5..8)]])) ;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#K%'matrixG6#717'#\"#Z\"$$>F(%!GF+ F+7'#\"\"*\"#Q#\"&:l\"\"'Od8#\"&Lc\"F2F+F+7'#\"$U$\"$$o#!*;CI&p\",*QjZ (\\\"#!+9[\")p%*F;#\"*G$=eP\"*()>h=$F+7'#\"#r\"#*)#\"/,AHp)4N$\"/O_#y_ b])#\"+.\"fue$\"+xCc0Y#!/Gc@2Xs@\"/$z^Dt?\"=#\"1l3>@dRM6\"1oi$RnvrP\"7 '#\"#B\"#G#\"3$o`euhjVL#\"3CQ\")3V>OQP#\"*@ake\"\"*;CTV\"#!4XgP^y\"e)H A#\"4#z&=F\\=!)R1\"#\"5>c?FE`XVI&)\"5?bf2U\\CO#p'7'F+F+F+F+#!0.))fHgms )\"1SyunY6*z!\\m#!.@*pG!H_)\"/oD e*Qo3\"#\":zB?o\"))pazN;(3$\":%eWLNNh9k5?\"z\"#!=x?;E9h`q2!fktw$\">_\" Hbdi``^9xH-A57'F+F+F+#\"8B1p`.yc[?,x\"\"8KcA:aQ_!*QdN\"#!1_u!46j*yw\"2 $eN3L%[%)4\"7'F`o#\"1jxn)e\\kT\"\"1%GKrH\"[:g#!)\\M%*[\"*6RY*>#\"2!)fi Gc[%)Q)\"3LzKxYS/IY#\"7xV0-1P'p&o!p#\"7g!)*H+r6$H_/m7'F+F+#\"3'yViQ)e& 3b)\"3&f@SPYR*)3'#!2!G@Zjswz5Fcp\"\"!7'F+%;__________________________G FiqFiqFiq7'%\"bG#\"(FP]#\")g\"4N$Fgq#\"-+Ilwj6\"-@l]R:N#\"2PNu&o2\\\\Q \"3+Y*3!piYl>7'F+#\"0h]xBQ?<\"\"0+-eDE&\\:#!,'*3cC\\)\"-D$p=7%>Fgq#\"( 8RC%\")+1S`7'%#b*G#\"+U6_WE\",bkl'4NFgq#\".u@4.Zw)\"/XvaC\"Hn##\"3jM'[ P]qsP(\"4]I&o@3`8AO7'F+#\"18D$Q_q " 0 "" {MPLTEXT 1 0 129 "subs(ee,matrix([seq([c[i],s eq(a[i,j],j=1..i-1),``$(9-i)],i=2..8),\n[`b`,seq(b[i],i=1..8)],[`b*`,s eq(`b*`[i],i=1..8)]])):\nevalf(%);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6# K%'matrixG6#7+7+$\"+hJBNC!#5F(%!GF+F+F+F+F+F+7+$\"+`5UoBF*$\"+l+q;7F*$ \"+))4s^6F*F+F+F+F+F+F+7+$\"+W1K2]F*$!+%GhJk%!#6$!+y$\\QK'F*$\"+:'[&z6 !\"*F+F+F+F+F+7+$\"+!4Gv(zF*$\"+SQwRRF*$\"+B\\S*y(F*$!+rd())>\"F=$\"+T q6P#)F*F+F+F+F+7+$\"+9dG9#)F*$\"+q)[VC'F*$\"+u\"=i5\"F=$!+&H6$*3#F=$\" +0Alu7F=$!+4+ac%*F8F+F+F+7+$\"\"\"\"\"!$!+%\\?Gx\"F*$!+5)=>%yF*$\"+Pf^ BF*$\"+V)[Qc(F*$!+lk![P%F*F]p$\"+ELJZzF87+ %#b*G$\"+Jg'\\`(F8F]p$\"+([$3zKF*$\"+T'=n.#F*$\"+r07#z'F*$!+h(f/i$F*$ \"+I>x3NF8$\"+`Ej\"3%F8Q)pprint196\"" }}}{PARA 0 "" 0 "" {TEXT -1 0 " " }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 155 "RK6_8eqs := [op(RowSumC onditions(8,'expanded')),op(OrderConditions(6,8,'expanded'))]:\n`RK5_8 eqs*` := subs(b=`b*`,OrderConditions(5,8,'expanded')):\nnops(%);" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#\"#<" }}}{PARA 0 "" 0 "" {TEXT -1 0 " " }}{PARA 0 "" 0 "" {TEXT -1 7 "Check: " }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 128 "simplify(subs(ee,RK6_8eqs)):\nmap(u->lhs(u)-rhs(u),% );\nnops(%);\nsimplify(subs(ee,`RK5_8eqs*`)):\nmap(u->lhs(u)-rhs(u),%) ;\nnops(%);" }}{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 3 to 8 have stage-order 2. " }}{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$F$" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 114 "We can calculate th e principal error norm of the order 6 scheme, that is, the 2-norm of t he 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(errterms6_8)))));" } }{PARA 11 "" 1 "" {XPPMATH 20 "6#$\"+\")\\Gs@!#8" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 79 "The 2 norm of the princip al error of the order 5 embedded scheme is as follows." }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 153 "`errterms5_8*` := subs(b=`b*`,Prin cipalErrorTerms(5,8,'expanded')):\nevalf(evalf[14](sqrt(add(subs(ee,`e rrterms5_8*`[i])^2,i=1.. nops(`errterms5_8*`)))));" }}{PARA 11 "" 1 " " {XPPMATH 20 "6#$\"+!4XT$=!#8" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }} {PARA 0 "" 0 "" {TEXT -1 53 "None of the principal error conditions ar e satisfied." }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 130 "ind := []: \nfor ct to nops(errterms6_8) do\n if simplify(subs(ee,errterms6_8[c t]))=0 then ind := [op(ind),ct] end if:\nend do:\nind;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#7\"" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}}{PARA 0 "" 0 "" {TEXT -1 52 " #---------------------------------------------------" }}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 35 "construction of the combined scheme" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 196 "First we constru ct the 8 stage order 6 scheme by making use of a selection of \"simple \" order conditions and incorporating the stage-order equations to ens ure that stages 3 to 8 have stage-order 2." }}{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 = 4;" "6#/%\"jG\" \"%" }{TEXT -1 18 " . . . 7. ( with " }{XPPEDIT 18 0 "c[1]=0" "6#/&% \"cG6#\"\"\"\"\"!" }{TEXT -1 55 " ), together with the further simpli fying conditions: " }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "Sum(b[i]*c[i]*a[i,2],i=3..8)=0" "6#/-%$SumG6$*(&%\"bG6#%\"iG\"\"\"&% \"cG6#F+F,&%\"aG6$F+\"\"#F,/F+;\"\"$\"\")\"\"!" }{TEXT -1 8 ", \+ " }{XPPEDIT 18 0 "Sum(b[i]*c[i]*Sum(a[i,j]*a[j,2],j=3..i-1),i=3..8)=0 " "6#/-%$SumG6$*(&%\"bG6#%\"iG\"\"\"&%\"cG6#F+F,-F%6$*&&%\"aG6$F+%\"jG F,&F46$F6\"\"#F,/F6;\"\"$,&F+F,F,!\"\"F,/F+;F<\"\")\"\"!" }{TEXT -1 8 ", " }{XPPEDIT 18 0 "Sum(b[i]*c[i]^2*a[i,2],i=3..8)=0" "6#/-%$Su mG6$*(&%\"bG6#%\"iG\"\"\"*$&%\"cG6#F+\"\"#F,&%\"aG6$F+F1F,/F+;\"\"$\" \")\"\"!" }{TEXT -1 2 ". " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 " " 0 "" {TEXT -1 32 "The scheme is constructed with " }{XPPEDIT 18 0 " a[7,3];" "6#&%\"aG6$\"\"(\"\"$" }{TEXT -1 3 ", " }{XPPEDIT 18 0 "a[7, 4]" "6#&%\"aG6$\"\"(\"\"%" }{TEXT -1 3 ", " }{XPPEDIT 18 0 "a[7,5]" " 6#&%\"aG6$\"\"(\"\"&" }{TEXT -1 7 " and " }{XPPEDIT 18 0 "a[7,6]" "6 #&%\"aG6$\"\"(\"\"'" }{TEXT -1 17 " as parameters. " }}{PARA 0 "" 0 " " {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 76 "The simple order condit ions used are given (in abreviated form) as follows. " }}{PARA 0 "" 0 "" {TEXT -1 94 "These include the 6 order 6 quadrature conditions toge ther with 4 additional order conditions." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 184 "SO6 := SimpleOrderCo nditions(6):\n[seq([i,SO6[i]],i=[1,2,4,8,13,16,24,28,29,32])]:\nlinalg [augment](linalg[delcols](%,2..2),matrix([[` `]$(linalg[rowdim](%))]) ,linalg[delcols](%,1..1));" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#K%'matri xG6#7,7%\"\"\"%#~~G/*&%\"bGF(%\"eGF(F(7%\"\"#F)/*&F,F(%\"cGF(#F(F/7%\" \"%F)/*&F,F()F2F/F(#F(\"\"$7%\"\")F)/*&F,F()F2F:F(#F(F57%\"#8F)/*(F,F( F2F(-%!G6#*&F8F(%\"aGF(F(#F(\"#:7%\"#;F)/*&F,F()F2F5F(#F(\"\"&7%\"#CF) /*(F,F(F2F(-FF6#*&FIF(FEF(F(#F(\"#s7%\"#GF)/*(F,F(F8F(FEF(#F(\"#=7%\"# HF)/*(F,F(F2F(-FF6#*&F?F(FIF(F(#F(FT7%\"#KF)/*&F,F()F2FRF(#F(\"\"'Q)pp rint816\"" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 " " {MPLTEXT 1 0 513 "SO_eqs := [op(RowSumConditions(8,'expanded')),op(S tageOrderConditions(2,8,'expanded'))]:\nSO6_8 := SimpleOrderConditions (6,8,'expanded'):\nord_cdns := [seq(SO6_8[i],i=[1,2,4,8,13,16,24,28,29 ,32])]:\nsimp_cdns := [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=[4,5,6,7])]:\nsimp_cdns2 := [add(b[i]*c[ i]*a[i,2],i=3..8)=0,add(b[i]*c[i]*add(a[i,j]*a[j,2],j=3..i-1),i=3..8)= 0,\n add(b[i]*c[i]^2*a[i,2],i=3..8)=0]:\ncdns := [op(or d_cdns),op(simp_cdns),op(simp_cdns2),op(SO_eqs)]:" }}}{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] = 47/193;" "6#/&%\"cG6#\"\"#*&\"#Z \"\"\"\"$$>!\"\"" }{TEXT -1 3 ", " }{XPPEDIT 18 0 "c[3] = 9/38;" "6#/ &%\"cG6#\"\"$*&\"\"*\"\"\"\"#Q!\"\"" }{TEXT -1 3 ", " }{XPPEDIT 18 0 "c[5] = 11/16;" "6#/&%\"cG6#\"\"&*&\"#6\"\"\"\"#;!\"\"" }{TEXT -1 3 ", " }{XPPEDIT 18 0 "c[6] = 3/4;" "6#/&%\"cG6#\"\"'*&\"\"$\"\"\"\"\"%! \"\"" }{TEXT -1 3 ", " }{XPPEDIT 18 0 "c[7]=1" "6#/&%\"cG6#\"\"(\"\" \"" }{TEXT -1 3 ", " }{XPPEDIT 18 0 "c[8]=1" "6#/&%\"cG6#\"\")\"\"\" " }{TEXT -1 2 ", " }}{PARA 0 "" 0 "" {TEXT -1 17 "and the weights: " } }{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "b[2]=0" "6#/&%\"bG6# \"\"#\"\"!" }{TEXT -1 3 ", " }{XPPEDIT 18 0 "b[7] = 0;" "6#/&%\"bG6# \"\"(\"\"!" }{TEXT -1 2 ". " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 174 "e1 := \{c[2]=47/193,c[3]=9/38,c[5] =71/89,c[6]=23/28,c[7]=1,c[8]=1,b[2]=0,b[7]=0\}:\neqns := subs(e1,cdns ):\nnops(eqns);\nindets(eqns) minus \{a[7,3],a[7,4],a[7,5],a[7,6]\};\n nops(%);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#\"#J" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#F8&F>6 #F1&F>6#\"\"'&F>6#F,&F>6#F'&F%6$F@FF&F%6$F@\"\"(&F%6$F@F+&F%6$F@F1&F%6 $F@F(&F%6$F@F'&F%6$F@F,&F%6$FOF,&F%6$FOF(&F%6$FFF+&F%6$FFF1&F%6$FFF(&F %6$FFF'&F%6$F1F+&F%6$FFF,&F%6$F1F'" }}{PARA 11 "" 1 "" {XPPMATH 20 "6# \"#J" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 22 "infolevel[solve] := 4:" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 118 "e2 := solve(\{op(eqns)\},indets(eqns) minus \{a[7,3] ,a[7,4],a[7,5],a[7,6]\}):\ninfolevel[solve] := 0:\ne3 := `union`(e1,e2 ):" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 3 "e3 " }} {PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1322 "e3 := \{c[7] = 1, c[8] = 1, a[5,3] = -21724507215628/18120732551 793, a[8,4] = 2690685696370602054377/6604522931171002998060, b[2] = 0, a[8,7] = 0, a[6,1] = 233436361745853683/373836194308813824, a[4,3] = \+ 375818328/318611987, c[2] = 47/193, a[2,1] = 47/193, b[5] = 1172038237 75061/154952625580200, b[7] = 0, b[8] = 4243913/53400600, a[4,1] = -69 5302416/14974763389, b[6] = -84924560896/194121869325, b[1] = 2503727/ 33509160, a[6,4] = 85304345532627205619/66923624494207595520, a[5,4] = 1134395721190865/1377175673936268, a[6,2] = 158645421/143412416, c[3] = 9/38, c[4] = 342/683, a[8,1] = 1416449588677763/6015481297132284, b [4] = 38494907685743537/196546626900894600, a[7,1] = -49/1786*a[7,3]+3 3905/32101*a[7,4]+9520/4183*a[7,5]+3123/1316*a[7,6]-99/94, c[6] = 23/2 8, c[5] = 71/89, a[5,2] = 3587459103/4605562477, a[6,5] = -87266602959 8803/9228174667747840, a[4,2] = -9469814814/14974763389, b[3] = 116376 653000/351539506521, a[8,6] = -10797672634721280/10984484330835583, a[ 7,2] = -1737/1786*a[7,3]-66006/32101*a[7,4]-13703/4183*a[7,5]-4439/131 6*a[7,6]+193/94, a[5,1] = 33509869292201/85055527825236, a[3,2] = 1563 3/135736, a[8,5] = 855085588386243786/608893946374021595, a[8,2] = -48 943449/199463911, a[6,3] = -2222985817851376045/1063980184927185792, a [8,3] = 83884485628625980/463004404677327933, a[3,1] = 16515/135736\}: " }{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 0 452 "subs(e3,matrix([seq([c[i],seq(a[i, j],j=1..i-1),``$(6-i)],i=2..6)]));\nprint(``);\nfor ii from 7 to 8 do \n print(c[ii]=subs(e3,c[ii]));print(``); \n for jj to ii-1 do\n \+ coeff_val := subs(e3,a[ii,jj]);\n if coeff_val=a[ii,jj] then \+ \n print(a[ii,jj]*` is a parameter.`)\n else\n pr int(a[ii,jj]=coeff_val);\n end if;\n end do:\n print(`_______ ___________________________`):\nend do:\nprint(``);\nseq(b[i]=subs(e3, b[i]),i=1..8);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#K%'matrixG6#7'7(#\"# Z\"$$>F(%!GF+F+F+7(#\"\"*\"#Q#\"&:l\"\"'Od8#\"&Lc\"F2F+F+F+7(#\"$U$\"$ $o#!*;CI&p\",*QjZ(\\\"#!+9[\")p%*F;#\"*G$=eP\"*()>h=$F+F+7(#\"#r\"#*)# \"/,AHp)4N$\"/O_#y_b])#\"+.\"fue$\"+xCc0Y#!/Gc@2Xs@\"/$z^Dt?\"=#\"1l3> @dRM6\"1oi$RnvrP\"F+7(#\"#B\"#G#\"3$o`euhjVL#\"3CQ\")3V>OQP#\"*@ake\" \"*;CTV\"#!4XgP^y\"e)HA#\"4#z&=F\\=!)R1\"#\"5>c?FE`XVI&)\"5?bf2U\\CO#p '#!0.))fHgms)\"1SyunY\"#%*F." }}{PARA 11 "" 1 "" {XPPMATH 20 "6#*&&%\"a G6$\"\"(\"\"$\"\"\"%1~is~a~parameter.GF)" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#*&&%\"aG6$\"\"(\"\"%\"\"\"%1~is~a~parameter.GF)" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#*&&%\"aG6$\"\"(\"\"&\"\"\"%1~is~a~parameter.GF)" } }{PARA 11 "" 1 "" {XPPMATH 20 "6#*&&%\"aG6$\"\"(\"\"'\"\"\"%1~is~a~par ameter.GF)" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%C______________________ ____________G" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/&%\"cG6#\"\")\"\"\" " }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%!G" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/&%\"aG6$\"\")\"\"\"#\"1jxn)e\\kT\"\"1%GKrH\"[:g" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/&%\"aG6$\"\")\"\"##!)\\M%*[\"*6RY*>" }}{PARA 11 " " 1 "" {XPPMATH 20 "6#/&%\"aG6$\"\")\"\"$#\"2!)fiGc[%)Q)\"3LzKxYS/IY" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/&%\"aG6$\"\")\"\"%#\"7xV0-1P'p&o!p# \"7g!)*H+r6$H_/m" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/&%\"aG6$\"\")\"\" &#\"3'yViQ)e&3b)\"3&f@SPYR*)3'" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/&% \"aG6$\"\")\"\"'#!2!G@Zjswz5\"2$eN3L%[%)4\"" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/&%\"aG6$\"\")\"\"(\"\"!" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%C__________________________________G" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%!G" }}{PARA 11 "" 1 "" {XPPMATH 20 "6*/&%\"bG6#\"\"\"# \"(FP]#\")g\"4N$/&F%6#\"\"#\"\"!/&F%6#\"\"$#\"-+Ilwj6\"-@l]R:N/&F%6#\" \"%#\"2PNu&o2\\\\Q\"3+Y*3!piYl>/&F%6#\"\"&#\"0h]xBQ?<\"\"0+-eDE&\\:/&F %6#\"\"'#!,'*3cC\\)\"-D$p=7%>/&F%6#\"\"(F//&F%6#\"\")#\"(8RC%\")+1S`" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 66 "We now turn our attention to the embedded 8 stage, order 5 scheme." }}{PARA 0 "" 0 "" {TEXT -1 35 "We use the simplifying conditions: " }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "Sum(`b*`[i]*a[i,2],i = 3 .. 8) = 0;" "6#/-%$SumG 6$*&&%#b*G6#%\"iG\"\"\"&%\"aG6$F+\"\"#F,/F+;\"\"$\"\")\"\"!" }{TEXT -1 8 ", " }{XPPEDIT 18 0 "Sum(`b*`[i]*Sum(a[i,j]*a[j,2],j = 3 .. i-1),i = 2 .. 8) = 0;" "6#/-%$SumG6$*&&%#b*G6#%\"iG\"\"\"-F%6$*&&%\"a G6$F+%\"jGF,&F16$F3\"\"#F,/F3;\"\"$,&F+F,F,!\"\"F,/F+;F6\"\")\"\"!" } {TEXT -1 1 "," }}{PARA 0 "" 0 "" {TEXT -1 30 "and the simplifying cond ition " }}{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$*&&%#b*G6#%\"i G\"\"\"&%\"aG6$F+%\"jGF,/F+;,&F0F,F,F,\"\")*&&F)6#F0F,,&F,F,&%\"cG6#F0 !\"\"F," }{TEXT -1 10 " for " }{XPPEDIT 18 0 "j = 6;" "6#/%\"jG\" \"'" }{TEXT -1 3 ", " }}{PARA 257 "" 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#/,&*&&%#b*G6#\"\"(\"\"\"&%\"aG6$F)\" \"'F*F**&&F'6#\"\")F*&F,6$F2F.F*F**&&F'6#F.F*,&F*F*&%\"cG6#F.!\"\"F*" }{TEXT -1 2 ". " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 101 "We use a selection of \"simple\" order conditions as giv en (in abreviated form) in the following table." }}{PARA 0 "" 0 "" {TEXT -1 95 "These include the 5 order 5 quadrature conditions togethe r with one additional order condition." }}{PARA 0 "" 0 "" {TEXT -1 0 " " }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 191 "`SO5*` := subs(b=`b*`,S impleOrderConditions(5)):\n[seq([i,`SO5*`[i]],i=[1,2,4,8,15,16])]:\nli nalg[augment](linalg[delcols](%,2..2),matrix([[` `]$(linalg[rowdim](% ))]),linalg[delcols](%,1..1));" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#K%'m atrixG6#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%\"#:F)/*& F,F(-%!G6#*&F?F(%\"aGF(F(#F(\"#?7%\"#;F)/*&F,F()F2F5F(#F(\"\"&Q)pprint 866\"" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 330 "`SO5_8*` := subs(b=`b*`,SimpleOrderConditions(5,8,'e xpanded')):\n`ord_cdns*` := [seq(`SO5_8*`[i],i=[1,2,4,8,15,16])]:\n`si mp_cdns*` := [add(`b*`[i]*a[i,2],i=3..8)=0,add(`b*`[i]*add(a[i,j]*a[j, 2],j=3..i-1),i=2..8)=0,\n `b*`[7]*a[7,6]+`b*`[8]*a[8, 6] = `b*`[6]*(1-c[6])]:\n`cdns*` := [op(`ord_cdns*`),op(`simp_cdns*`)] : " }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 17 "We specify that " }{XPPEDIT 18 0 "`b*`[2] = 0;" "6#/&%#b*G6#\"\"#\"\"! " }{TEXT -1 3 ", " }{XPPEDIT 18 0 "`b*`[7] = 2/57;" "6#/&%#b*G6#\"\"( *&\"\"#\"\"\"\"#d!\"\"" }{TEXT -1 7 " and " }{XPPEDIT 18 0 "`b*`[8] \+ = 2/49;" "6#/&%#b*G6#\"\")*&\"\"#\"\"\"\"#\\!\"\"" }{TEXT -1 2 ". " }} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 66 "e4 := \{`b*`[2]=0,`b*`[7]=2/ 57,`b*`[8]=2/49\}:\ne5 := `union`(e3,e4):" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 69 "`eqns*` := subs(e5 ,`cdns*`):\nnops(`eqns*`);\nindets(`eqns*`);\nnops(%);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#\"\"*" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#<+&%#b*G6 #\"\"'&F%6#\"\"%&F%6#\"\"$&F%6#\"\"\"&F%6#\"\"&&%\"aG6$\"\"(F3&F56$F7F '&F56$F7F-&F56$F7F*" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#\"\"*" }}} {PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 22 "infolevel[solve] := 4:" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 81 "e6 := solve(\{op(`eqns*`)\}):\ne7 := `union`(subs(e6,e5),e6):\ninf olevel[solve] := 0:" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 146 "RK6_8eqs := [op(RowSumConditions(8,'expa nded')),op(OrderConditions(6,8,'expanded'))]:\n`RK5_8eqs*` := subs(b=` b*`,OrderConditions(5,8,'expanded')):" }}}{PARA 0 "" 0 "" {TEXT -1 0 " " }}{PARA 0 "" 0 "" {TEXT -1 7 "Check: " }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 129 "simplify(subs(e7,RK6_8eqs)):\nmap(u->lhs(u)-rhs(u),% );\n;nops(%);\nsimplify(subs(e7,`RK5_8eqs*`)):\nmap(u->lhs(u)-rhs(u),% );\nnops(%);" }}{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 2 "e7" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1732 "e7 := \{`b*`[7] = 2/57, b[5] = 117203823775061/1549 52625580200, b[2] = 0, a[8,7] = 0, b[1] = 2503727/33509160, a[6,4] = 8 5304345532627205619/66923624494207595520, c[2] = 47/193, a[2,1] = 47/1 93, c[8] = 1, a[5,3] = -21724507215628/18120732551793, c[7] = 1, a[6,1 ] = 233436361745853683/373836194308813824, a[4,3] = 375818328/31861198 7, `b*`[5] = 2721770523832513/4007247067087950, a[4,1] = -695302416/14 974763389, b[6] = -84924560896/194121869325, `b*`[6] = -89022652768/24 5887701145, `b*`[2] = 0, a[3,1] = 16515/135736, a[6,3] = -222298581785 1376045/1063980184927185792, a[5,1] = 33509869292201/85055527825236, a [3,2] = 15633/135736, b[3] = 116376653000/351539506521, a[6,5] = -8726 66029598803/9228174667747840, a[4,2] = -9469814814/14974763389, c[6] = 23/28, c[5] = 71/89, a[5,2] = 3587459103/4605562477, c[3] = 9/38, c[4 ] = 342/683, b[4] = 38494907685743537/196546626900894600, a[5,4] = 113 4395721190865/1377175673936268, a[6,2] = 158645421/143412416, b[7] = 0 , a[7,2] = -8522902869921/10868389582568, a[7,1] = -117876251336166386 4779/6649079911196633209152, a[8,3] = 83884485628625980/46300440467732 7933, b[8] = 4243913/53400600, `b*`[1] = 2644521142/35096656455, a[7,4 ] = -3767364590077053611426162077/10220229771451535362575529152, a[8,6 ] = -10797672634721280/10984484330835583, `b*`[8] = 2/49, a[8,5] = 855 085588386243786/608893946374021595, a[7,6] = -7678963110907452/1098448 4330835583, a[8,2] = -48943449/199463911, a[7,3] = 3087163579546988168 202379/1791201064146135353344584, a[8,1] = 1416449588677763/6015481297 132284, `b*`[3] = 8764703092174/26729124547545, a[7,5] = 1770120485678 0353690623/13557389052385415225632, `b*`[4] = 737727050374863463/36221 35308216853050, a[8,4] = 2690685696370602054377/6604522931171002998060 \}:" }{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 0 418 "subs(e7,matrix([[c[2],a[2,1 ],``$3],\n [c[3],a[3,1],a[3,2],``$2],\n [c[4],seq(a[4,i],i=1..3),``] ,\n [c[5],seq(a[5,i],i=1..4)],\n [c[6],seq(a[6,i],i=1..4)],[``$4,a[6 ,5]],\n [c[7],seq(a[7,i],i=1..4)],[``$3,a[7,5],a[7,6]],\n [c[8],seq( a[8,i],i=1..4)],[``$2,seq(a[8,i],i=5..7)],\n [``,`___________________ _______`$4],\n [`b`,seq(b[i],i=1..4)],[``,seq(b[i],i=5..8)],\n [`b*` ,seq(`b*`[i],i=1..4)],[``,seq(`b*`[i],i=5..8)]]));" }}{PARA 11 "" 1 " " {XPPMATH 20 "6#K%'matrixG6#717'#\"#Z\"$$>F(%!GF+F+7'#\"\"*\"#Q#\"&:l \"\"'Od8#\"&Lc\"F2F+F+7'#\"$U$\"$$o#!*;CI&p\",*QjZ(\\\"#!+9[\")p%*F;# \"*G$=eP\"*()>h=$F+7'#\"#r\"#*)#\"/,AHp)4N$\"/O_#y_b])#\"+.\"fue$\"+xC c0Y#!/Gc@2Xs@\"/$z^Dt?\"=#\"1l3>@dRM6\"1oi$RnvrP\"7'#\"#B\"#G#\"3$o`eu hjVL#\"3CQ\")3V>OQP#\"*@ake\"\"*;CTV\"#!4XgP^y\"e)HA#\"4#z&=F\\=!)R1\" #\"5>c?FE`XVI&)\"5?bf2U\\CO#p'7'F+F+F+F+#!0.))fHgms)\"1SyunY6*z!\\m#!.@*pG!H_)\"/oDe*Qo3\"#\":zB?o\")) pazN;(3$\":%eWLNNh9k5?\"z\"#!=x?;E9h`q2!fktw$\">_\"Hbdi``^9xH-A57'F+F+ F+#\"8B1p`.yc[?,x\"\"8KcA:aQ_!*QdN\"#!1_u!46j*yw\"2$eN3L%[%)4\"7'F`o# \"1jxn)e\\kT\"\"1%GKrH\"[:g#!)\\M%*[\"*6RY*>#\"2!)fiGc[%)Q)\"3LzKxYS/I Y#\"7xV0-1P'p&o!p#\"7g!)*H+r6$H_/m7'F+F+#\"3'yViQ)e&3b)\"3&f@SPYR*)3'# !2!G@Zjswz5Fcp\"\"!7'F+%;__________________________GFiqFiqFiq7'%\"bG# \"(FP]#\")g\"4N$Fgq#\"-+Ilwj6\"-@l]R:N#\"2PNu&o2\\\\Q\"3+Y*3!piYl>7'F+ #\"0h]xBQ?<\"\"0+-eDE&\\:#!,'*3cC\\)\"-D$p=7%>Fgq#\"(8RC%\")+1S`7'%#b* G#\"+U6_WE\",bkl'4NFgq#\".u@4.Zw)\"/XvaC\"Hn##\"3jM'[P]qsP(\"4]I&o@3`8 AO7'F+#\"18D$Q_q " 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 sche me" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1633 "ee := \{c[2]=47/193,\nc[3]=9/38,\nc[4]=342/683,\nc[ 5]=71/89,\nc[6]=23/28,\nc[7]=1,\nc[8]=1,\n\na[2,1]=47/193,\na[3,1]=165 15/135736,\na[3,2]=15633/135736,\na[4,1]=-695302416/14974763389,\na[4, 2]=-9469814814/14974763389,\na[4,3]=375818328/318611987,\na[5,1]=33509 869292201/85055527825236,\na[5,2]=3587459103/4605562477,\na[5,3]=-2172 4507215628/18120732551793,\na[5,4]=1134395721190865/1377175673936268, \na[6,1]=233436361745853683/373836194308813824,\na[6,2]=158645421/1434 12416,\na[6,3]=-2222985817851376045/1063980184927185792,\na[6,4]=85304 345532627205619/66923624494207595520,\na[6,5]=-872666029598803/9228174 667747840,\na[7,1]=-1178762513361663864779/6649079911196633209152,\na[ 7,2]=-8522902869921/10868389582568,\na[7,3]=3087163579546988168202379/ 1791201064146135353344584,\na[7,4]=-3767364590077053611426162077/10220 229771451535362575529152,\na[7,5]=17701204856780353690623/135573890523 85415225632,\na[7,6]=-7678963110907452/10984484330835583,\na[8,1]=1416 449588677763/6015481297132284,\na[8,2]=-48943449/199463911,\na[8,3]=83 884485628625980/463004404677327933,\na[8,4]=2690685696370602054377/660 4522931171002998060,\na[8,5]=855085588386243786/608893946374021595,\na [8,6]=-10797672634721280/10984484330835583,\na[8,7]=0,\n\nb[1]=2503727 /33509160,\nb[2]=0,\nb[3]=116376653000/351539506521,\nb[4]=38494907685 743537/196546626900894600,\nb[5]=117203823775061/154952625580200,\nb[6 ]=-84924560896/194121869325,\nb[7]=0,\nb[8]=4243913/53400600,\n\n`b*`[ 1]=2644521142/35096656455,\n`b*`[2]=0,\n`b*`[3]=8764703092174/26729124 547545,\n`b*`[4]=737727050374863463/3622135308216853050,\n`b*`[5]=2721 770523832513/4007247067087950,\n`b*`[6]=-89022652768/245887701145,\n`b *`[7]=2/57,\n`b*`[8]=2/49\}:" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 5 "Let " }{XPPEDIT 18 0 "T[6, 8]; " "6#&%\"TG6$\"\"'\"\")" }{TEXT -1 127 " denote the vector whose comp onents are the principal error terms of the 8 stage, order 6 scheme (t he error terms of order 7)." }}{PARA 0 "" 0 "" {TEXT -1 5 "Let " } {XPPEDIT 18 0 "`T*`[5,8];" "6#&%#T*G6$\"\"&\"\")" }{TEXT -1 145 " den ote the vector whose components are the principal error terms of the e mbedded 8 stage, order 5 scheme (the error terms of order 6) and let \+ " }{XPPEDIT 18 0 "`T*`[6,8];" "6#&%#T*G6$\"\"'\"\")" }{TEXT -1 99 " d enote the vector whose components are the error terms of order 7 of th e embedded order 5 scheme." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 39 "Denote the 2-norm of these vectors by " } {XPPEDIT 18 0 "abs(abs(T[6,8]));" "6#-%$absG6#-F$6#&%\"TG6$\"\"'\"\") " }{TEXT -1 3 ", " }{XPPEDIT 18 0 "abs(abs(`T*`[5,8]));" "6#-%$absG6# -F$6#&%#T*G6$\"\"&\"\")" }{TEXT -1 7 " and " }{XPPEDIT 18 0 "abs(abs (`T*`[6,8]));" "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*`[6,8]))/a bs(abs(`T*`[5,8]));" "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*`[6,8]-T[6,8]))/abs(abs(`T*`[5,8]));" "6#/&%\" CG6#\"\"(*&-%$absG6#-F*6#,&&%#T*G6$\"\"'\"\")\"\"\"&%\"TG6$F2F3!\"\"F4 -F*6#-F*6#&F06$\"\"&F3F8" }{TEXT -1 1 "." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 77 "Dormand and Prince have suggested \+ that as well as attempting to ensure that " }{XPPEDIT 18 0 "A[7];" "6 #&%\"AG6#\"\"(" }{TEXT -1 73 " is a minimum, if the embedded scheme i s 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 fro m 1." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 186 "errterms6_8 := PrincipalErrorTerms(6,8,'expanded'): \n`errterms6_8*` :=subs(b=`b*`,PrincipalErrorTerms(6,8,'expanded')):\n `errterms5_8*` := subs(b=`b*`,PrincipalErrorTerms(5,8,'expanded')):" } }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 326 "snmB := sqrt(add(evalf(subs(ee,`errterms6_8*`[i]))^2,i=1..nops( `errterms6_8*`))):\nsdnB := sqrt(add(evalf(subs(ee,`errterms5_8*`[i])) ^2,i=1..nops(`errterms5_8*`))):\nsnmC := sqrt(add((evalf(subs(ee,`errt erms6_8*`[i])-subs(ee,errterms6_8[i])))^2,i=1..nops(errterms6_8))):\n' B[7]'= evalf[8](snmB/sdnB);\n'C[7]'= evalf[8](snmC/sdnB);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/&%\"BG6#\"\"($\")k&>t\"!\"(" }}{PARA 11 "" 1 " " {XPPMATH 20 "6#/&%\"CG6#\"\"($\")Bw3;!\"(" }}}{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 co mbined scheme" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1633 "ee := \{c[2]=47/193,\nc[3]=9/38,\nc[4]=342/683 ,\nc[5]=71/89,\nc[6]=23/28,\nc[7]=1,\nc[8]=1,\n\na[2,1]=47/193,\na[3,1 ]=16515/135736,\na[3,2]=15633/135736,\na[4,1]=-695302416/14974763389, \na[4,2]=-9469814814/14974763389,\na[4,3]=375818328/318611987,\na[5,1] =33509869292201/85055527825236,\na[5,2]=3587459103/4605562477,\na[5,3] =-21724507215628/18120732551793,\na[5,4]=1134395721190865/137717567393 6268,\na[6,1]=233436361745853683/373836194308813824,\na[6,2]=158645421 /143412416,\na[6,3]=-2222985817851376045/1063980184927185792,\na[6,4]= 85304345532627205619/66923624494207595520,\na[6,5]=-872666029598803/92 28174667747840,\na[7,1]=-1178762513361663864779/6649079911196633209152 ,\na[7,2]=-8522902869921/10868389582568,\na[7,3]=308716357954698816820 2379/1791201064146135353344584,\na[7,4]=-3767364590077053611426162077/ 10220229771451535362575529152,\na[7,5]=17701204856780353690623/1355738 9052385415225632,\na[7,6]=-7678963110907452/10984484330835583,\na[8,1] =1416449588677763/6015481297132284,\na[8,2]=-48943449/199463911,\na[8, 3]=83884485628625980/463004404677327933,\na[8,4]=269068569637060205437 7/6604522931171002998060,\na[8,5]=855085588386243786/60889394637402159 5,\na[8,6]=-10797672634721280/10984484330835583,\na[8,7]=0,\n\nb[1]=25 03727/33509160,\nb[2]=0,\nb[3]=116376653000/351539506521,\nb[4]=384949 07685743537/196546626900894600,\nb[5]=117203823775061/154952625580200, \nb[6]=-84924560896/194121869325,\nb[7]=0,\nb[8]=4243913/53400600,\n\n `b*`[1]=2644521142/35096656455,\n`b*`[2]=0,\n`b*`[3]=8764703092174/267 29124547545,\n`b*`[4]=737727050374863463/3622135308216853050,\n`b*`[5] =2721770523832513/4007247067087950,\n`b*`[6]=-89022652768/245887701145 ,\n`b*`[7]=2/57,\n`b*`[8]=2/49\}:" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 68 "The stability function R for th e order 6 scheme is given as follows." }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 83 "expand(subs(ee,StabilityFunction(6,8,'expanded'))):\n R := unapply(%,z):\n'R(z)'=R(z);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/- %\"RG6#%\"zG,2\"\"\"F)F'F)*&#F)\"\"#F)*$)F'F,F)F)F)*&#F)\"\"'F)*$)F'\" \"$F)F)F)*&#F)\"#CF)*$)F'\"\"%F)F)F)*&#F)\"$?\"F)*$)F'\"\"&F)F)F)*&#F) \"$?(F)*$)F'F1F)F)F)*&#\"#6\"&SY&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 rea l axis by solving the equation: " }}{PARA 256 "" 0 "" {TEXT -1 1 " " } {XPPEDIT 18 0 "R(z) = -1;" "6#/-%\"RG6#%\"zG,$\"\"\"!\"\"" }{TEXT -1 2 ". " }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 27 "z0 := newton(R(z)=- 1,z=-4);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%#z0G$!+XmgKR!\"*" }}} {PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 306 "z0 := newton(R(z)=-1,z=-4):\np1 := plot([R(z),-1],z=-4.39..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 ,color=COLOR(RGB,0,.5,0)):\nplots[display]([p1,p2,p3],view=[-4.39..0.4 9,-1.47..1.47],font=[HELVETICA,9]);" }}{PARA 13 "" 1 "" {GLPLOT2D 360 253 253 {PLOTDATA 2 "6+-%'CURVESG6$7U7$$!3o*************Q%!#<$!3SLFPFc =^BF*7$$!3lLL$Q%\\\"oL%F*$!3-Sm8ygcS@F*7$$!3ummm())HOG%F*$!37#))pFGVi% >F*7$$!3\")**\\naQNPUF*$!3YF:V\\]d*y\"F*7$$!3'GL$o@y2\">%F*$!3ikHv$)4u V;F*7$$!3MmmwVL*p3%F*$!39BJ/\"G@?N\"F*7$$!3_mmOE*=A)RF*$!3(QrTCG)f.6F* 7$$!3#HL$[$[Uz(QF*$!3Q=vJq@1`*)!#=7$$!3Ammhs[E\"y$F*$!3k@*[L2_'HF*$!3%)RAt))\\8#R'!#>7$$!3Amm \"=76&pGF*$!3?M.8I3#GQ$Fhp7$$!3'GLLn*H`fFF*$!3%=\"*R*[$[!oO!#?7$$!3yKL $*)4njm#F*$\"3>,.HtbU6=Fhp7$$!3!)***\\:,$*zb#F*$\"37#4-AL9i,%Fhp7$$!35 LL$4ej?Y#F*$\"3e\"Rr\"oQ@]dFhp7$$!3g***\\:z8oN#F*$\"3ItI(p9wC\\(Fhp7$$ !3S***\\E5\"fcAF*$\"3M&fC'pk^c!*Fhp7$$!3'HLL3C>?:#F*$\"3%)3_:JOdk5FP7$ $!3sKL)yi*)f0#F*$\"3P]-EF*$\"3+P-k/3Ht8FP7$$!3 \\mm\"=E<[%=F*$\"3;j@,&=+4b\"FP7$$!3m***\\Oee6v\"F*$\"3*eG<`&=&fr\"FP7 $$!3OmmOXY+];F*$\"3Mcg46t.3>FP7$$!3))****>HF]X:F*$\"3cX1`>/cC@FP7$$!3b *****=*zEV9F*$\"3))QmDDD:dBFP7$$!3$)***\\ni]VM\"F*$\"3NST!3'4c/EFP7$$! 3y****pd(>XB\"F*$\"3'\\(pnxGW3HFP7$$!3MLL$p$=$e8\"F*$\"3;2+>*\\o3@$FP7 $$!3r*****pTh/.\"F*$\"37a.x\\U5oNFP7$$!3qlm;a:!)\\$*FP$\"3ka!*oDtyDRFP 7$$!3r$****f;RfI)FP$\"37W\"zRaWyN%FP7$$!3$RLL)3guBtFP$\"3:1.b1yj2[FP7$ $!3o&***\\`62(H'FP$\"35s$p#\\7ZF`FP7$$!3=HLLB)3LH&FP$\"3IKX1$p()**)eFP 7$$!36(***\\(oiCC%FP$\"3zcpkbqiUlFP7$$!3jcmmiHPIKFP$\"3#>BVW\\o%RsFP7$ $!3_jmmR]O&>#FP$\"3,QdAFhp$\"3YOQT13zw(*FP7$$\"3Ozmmcej_&)Fhp$\"3Qsf4t-H*3\"F*7$ $\"3-TLL,V7A=FP$\"3A'yS^0p)*>\"F*7$$\"330+]%[**H&GFP$\"3gc\"Gt%4;I8F*7 $$\"3C4+]r&y'RQFP$\"3Ez()RV#)4o9F*7$$\"3!***************[FP$\"3]nt^_hJ K;F*-%'COLOURG6&%$RGBG$\"*++++\"!\")$\"\"!F\\\\lF[\\l-F$6$7S7$F($!\"\" F\\\\l7$F3Fa\\l7$F=Fa\\l7$FBFa\\l7$FGFa\\l7$FLFa\\l7$FRFa\\l7$FWFa\\l7 $FfnFa\\l7$F[oFa\\l7$F`oFa\\l7$FeoFa\\l7$FjoFa\\l7$F_pFa\\l7$FdpFa\\l7 $FjpFa\\l7$F_qFa\\l7$FeqFa\\l7$FjqFa\\l7$F_rFa\\l7$FdrFa\\l7$FirFa\\l7 $F^sFa\\l7$FcsFa\\l7$FhsFa\\l7$F]tFa\\l7$FbtFa\\l7$FgtFa\\l7$F\\uFa\\l 7$FauFa\\l7$FfuFa\\l7$F[vFa\\l7$F`vFa\\l7$FevFa\\l7$FjvFa\\l7$F_wFa\\l 7$FdwFa\\l7$FiwFa\\l7$F^xFa\\l7$FcxFa\\l7$FhxFa\\l7$F]yFa\\l7$FbyFa\\l 7$FgyFa\\l7$F\\zFa\\l7$FazFa\\l7$FfzFa\\l7$F[[lFa\\l7$F`[lFa\\l-Fe[l6& Fg[lF[\\lF[\\lFh[l-F$6&7#7$$!3!******\\k1E$RF*Fa\\l-%'SYMBOLG6#%'CIRCL EG-Fe[l6&Fg[lF\\\\lF\\\\lF\\\\l-%&STYLEG6#%&POINTG-F$6&Fg_l-F\\`l6#%&C ROSSGF_`lFa`l-F$6&Fg_l-F\\`l6#%(DIAMONDGF_`lFa`l-F$6%7$7$Fi_lF[\\lFh_l -%&COLORG6&Fg[lF[\\l$\"\"&Fb\\lF[\\l-%*LINESTYLEG6#\"\"$-%%FONTG6$%*HE LVETICAG\"\"*-%+AXESLABELSG6%Q\"z6\"Q!Febl-F]bl6#%(DEFAULTG-%%VIEWG6$; $!$R%!\"#$\"#\\F`cl;$!$Z\"F`cl$\"$Z\"F`cl" 1 2 0 1 10 0 2 9 1 4 2 1.000000 45.000000 45.000000 0 0 "Curve 1" "Curve 2" "Curve 3" "Curve \+ 4" "Curve 5" "Curve 6" }}}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 50 "The following picture shows the stability region. " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1296 "R := z -> 1+z+1/2*z^2+1/6*z^3+1/24*z^4+1/120*z^5+1/720*z^6+11/ 54640*z^7:\npts := []:\nz0 := 0:\nfor ct from 0 to 200 do\n zz := ne wton(R(z)=exp(ct*Pi/20*I),z=z0):\n z0 := zz:\n pts := [op(pts),[Re (zz),Im(zz)]]:\nend do:\np1 := plot(pts,color=COLOR(RGB,.48,.08,.13)): \np2 := plots[polygonplot]([seq([pts[i-1],pts[i],[-2,0]],i=2..nops(pts ))],\n style=patchnogrid,color=COLOR(RGB,.95,.15,.25)):\npts \+ := []: z0 := 1.5+4.3*I:\nfor ct from 0 to 60 do\n zz := newton(R(z)= exp(ct*Pi/30*I),z=z0):\n z0 := zz:\n pts := [op(pts),[Re(zz),Im(zz )]]:\nend do:\np3 := plot(pts,color=COLOR(RGB,.48,.08,.13)):\np4 := pl ots[polygonplot]([seq([pts[i-1],pts[i],[1.4,4.2]],i=2..nops(pts))],\n \+ style=patchnogrid,color=COLOR(RGB,.95,.15,.25)):\npts := []: \+ z0 := 1.5-4.3*I:\nfor ct from 0 to 60 do\n zz := newton(R(z)=exp(ct* Pi/30*I),z=z0):\n z0 := zz:\n pts := [op(pts),[Re(zz),Im(zz)]]:\ne nd do:\np5 := 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"Curve 5" "Curve 6" "Curve 7" "Curve 8" }}}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 4 "The " }{TEXT 260 30 "inter val of absolute stability" }{TEXT -1 89 " (or stability interval) is \+ the intersection of the stability region with the real line." }}{PARA 0 "" 0 "" {TEXT -1 59 "For this scheme the stability interval is (appr oximately) " }{XPPEDIT 18 0 "[-3.9326, 0];" "6#7$,$-%&FloatG6$\"&E$R! \"%!\"\"\"\"!" }{TEXT -1 2 ". " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }} {PARA 0 "" 0 "" {TEXT -1 260 "We can distort the boundary curve horizo ntally by taking the 11th root of the real part of points along the cu rve. In this way we see that the largest interval on the nonnegative i maginary axis that contains the origin and lies inside the stability r egion is " }{XPPEDIT 18 0 "[0, 1.84];" "6#7$\"\"!-%&FloatG6$\"$%=!\"# " }{TEXT -1 18 " approximately. " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 324 "R := z -> 1+z+1/2*z^2+1/6*z ^3+1/24*z^4+1/120*z^5+1/720*z^6+11/54640*z^7:\nDigits := 25:\npts := [ ]: z0 := 0:\nfor ct from 0 to 60 do\n zz := newton(R(z)=exp(ct*Pi/10 0*I),z=z0):\n z0 := zz:\n pts := [op(pts),[surd(Re(zz),11),Im(zz)] ]:\nend do:\nplot(pts,color=COLOR(RGB,.95,0,.25),thickness=2,font=[HEL VETICA,9]);\nDigits := 10:" }}{PARA 13 "" 1 "" {GLPLOT2D 261 276 276 {PLOTDATA 2 "6(-%'CURVESG6#7in7$$\"\"!F)F(7$$\":5?!**fs=8gj^4J!#E$\":e ,VU>!)*e`EfTJF-7$$\":a5]%z)*yH$4/t9&F-$\":%eqXqt1=2`=$G'F-7$$\":WRZ\\F 63xj4:\"pF-$\":7(*yn8>(ygzxC%*F-7$$\":9,vmJN-qxUz^)F-$\":h[>a-Y[91PmD \"!#D7$$\":KWUTQuuL\\s:+\"F?$\":!)e*=X2)\\oK'zq:F?7$$\":N6![F<+YT*eJ9 \"F?$\":.c6#[M#HBfb\\)=F?7$$\":C?wLbzB&=C@y7F?$\":q&>'R\\eaz&[6*>#F?7$ $\":(=D\")[U$*Gh/&yS\"F?$\":5;(fw5'zP7uK^#F?7$$\":8q\")Q28KLxpG`\"F?$ \":JW8QNUU(*QLu#GF?7$$\":!os7Q#*H%>?iQl\"F?$\":691n>vbMF?7$$\":\"Q(=J\"H!zY6$[&)=F?$\":R6\"f -kOG!=6*pPF?7$$\":Lgf*>^;=xCv'*>F?$\":)em/N=)y?VqS3%F?7$$\":eg&*poF(ex 1K0@F?$\":+Pzz!fw[m'H#)R%F?7$$\":)Q)pF3rU8tx8@#F?$\":5*H)yJP)*)o))Q7ZF ?7$$\":r3qMNx-C9\"3:BF?$\":yZp\")p!o#f,[l-&F?7$$\":g=+!\\v>t0Bc;CF?$\" :Bd\\H_%3irqqS`F?7$$\":JjMo)\\Yuv7$f^#F?$\":%3&\\p>)4_#)f'[l&F?7$$\":k [_]WB#pt.G8EF?$\":&y?iEa.OrY-pfF?7$$\":2P)eL\\#f1&po3FF?$\":Ca$QJY+V>R*GF ?$\":^DBk+wqn4)\\6pF?7$$\":vVwc'Ge!R8UQ)HF?$\":z8gL'HkkJVlDsF?7$$\":$* *=5)y<:&*>=?2$F?$\":t)z8,7+aa#4)RvF?7$$\":$>Z_7twIwSZeJF?$\":p$=$)R0Kd 0C'R&yF?7$$\":qrHIgnQ\\lGKC$F?$\":L%Qr>B:a(>8\"o\")F?7$$\":%3([gz#*pwm $HELF?$\":6k>(p6oK*)3E#[)F?7$$\":h16@qY,%QRn2MF?$\":1fgNaIGbb/kz)F?7$$ \":b+!yWv2_'\\nt[$F?$\":0iCw$3.1_Ia5\"*F?7$$\":(f'fuml[1cl`c$F?$\":@)* p:K%z[x\\nC%*F?7$$\":N:8X4'zt#R_;k$F?$\":N)4x'o,.*H')zQ(*F?7$$\":#4oT5 PZW<]?;PF?$\":\"**o!Qa0=f>\"H05!#C7$$\":99]d^?4ez#**)y$F?$\":(p[_Fg.X^ 7qO5Ffu7$$\":P:$y1')peNo(*fQF?$\":&GiZ'H,5(Q(4\"o5Ffu7$$\":)R'4[Q*[\\7 #4\"HRF?$\":bPP%z#=?[J;&*4\"Ffu7$$\":1/(y0EeWm>L'*RF?$\":8/bS**3J#y0#4 8\"Ffu7$$\":;]Kp#*)Q`BedhSF?$\":K?R*o#y%fh?Ki6Ffu7$$\":yrL\\*[+lw&eZ7% F?$\":At%y1)4:UA?P>\"Ffu7$$\":%Hfr#H>NI.$y&=%F?$\":h4**RHw5TW9^A\"Ffu7 $$\":m`'p'QV1ESMXC%F?$\":j0\\(=@:u4S]c7Ffu7$$\":m]DV'=#z:&p(3I%F?$\":D p%z_$3g96))yG\"Ffu7$$\":,D6E?v\\Rn\\YN%F?$\":ktTe\\YmF$eE>8Ffu7$$\":&f btM&)**=E1m0WF?$\":L&43e8dXThj]8Ffu7$$\":y%)H,]gJrSzOX%F?$\":^ajPF)[w! )y*>Q\"Ffu7$$\":[#Q-![q/)4qU)\\%F?$\":T#p?45+kg(\\LT\"Ffu7$$\":^()p=J< P@'=cRXF?$\":\"*yYygmd%F?$\":yB\\WQ8b\\1 =gZ\"Ffu7$$\":#R#RpBllSq(=4YF?$\":_+05vs9(o6L2:Ffu7$$\":kQ#z1G)GZSdkj% F?$\":FD4(Hv')RdxiQ:Ffu7$$\":y!GhpihT=YcdYF?$\":R(G-+Ug)yv0*p:Ffu7$$\" :X[ZcBtW$yAGrYF?$\":9OJc'4oIAH;,;Ffu7$$\":xT*yW'GiS1$)en%F?$\":^=T%o+] aCoRK;Ffu7$$\":Ch&G*RK7@?;)oYF?$\":..UX)3*z\"e[gj;Ffu7$$\":hu72lhV6Fmg k%F?$\":f_'yjW!=gB%y%p\"Ffu7$$\":&R=70RL9l(=2g%F?$\":7x:)>9[M\"*>$fs\" Ffu7$$\":g1jroxzXF?$\":/(y8oFG*4)\\/d\")y\"Ffu7$$\":^=j(4<%f7\"H)\\*RF?$\":vWH2GuT)RK:>=Ff u7$$!:$QBURz')pd(y41%F?$\":>^i!\\I$[URT,&=Ffu7$$!:\"fVlq_M&>Q&)R`%F?$ \":TTt^0_!Qs03\")=Ffu-%+AXESLABELSG6$Q!6\"Ff^l-%%FONTG6$%*HELVETICAG\" \"*-%&COLORG6&%$RGBG$\"#&*!\"#F($\"#DFc_l-%*THICKNESSG6#\"\"#-%%VIEWG6 $%(DEFAULTGF]`l" 1 2 0 1 10 2 2 6 1 4 2 1.000000 45.000000 45.000000 0 0 "Curve 1" }}}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 123 "The relevant intersection point of the boundary curve wi th the imaginary axis can be determined more accurately as follows." } }{PARA 0 "" 0 "" {TEXT -1 86 "First we look for points on the boundary curve either side of the intersection point. " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 112 "Digits := 1 5:\nz0 := 1.84*I:\nfor ct from 57 to 60 do\n newton(R(z)=exp(ct*Pi/1 00*I),z=z0);\nend do;\nDigits := 10:" }}{PARA 11 "" 1 "" {XPPMATH 20 " 6#^$$\"0zf%Ge@-6!#=$\"0m#G!*)>\")y\"!#9" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#^$$\"0u3mw'zOT!#>$\"0uT)RK:>=!#9" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#^$$!0Vl1Y$z`\\!#>$\"0L[URT,&=!#9" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#^$$!0!pr5.\\k;!#=$\"0_!Qs03\")=!#9" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 107 "Then we apply the bisection metho d to calculate the parameter value associated with the intersection po int." }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 178 "real_part := proc(u )\n Re(newton(R(z)=exp(u*Pi*I),z=1.84*I))\nend proc:\nDigits := 15: \nu0 := bisect('real_part'(u),u=0.57..0.60);\nnewton(R(z)=exp(u0*Pi*I) ,z=1.84*I);\nDigits := 10:" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%#u0G$ \"0[RO(\\z[e!#:" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#^$$\"0K)f-')4oA!#H$ \"0wK(y(zU$=!#9" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 4 "The " }{TEXT 260 112 "largest interval on the nonnegative \+ imaginary axis that contains the origin and lies inside the stability \+ region" }{TEXT -1 5 " is " }{XPPEDIT 18 0 "[0, 1.8343];" "6#7$\"\"!-% &FloatG6$\"&V$=!\"%" }{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 71 "The stability function R* for the order 5 scheme is giv en as follows." }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 105 "expand(su bs(ee,subs(b=`b*`,StabilityFunction(5,8,'expanded')))):\n`R*` := unapp ly(%,z):\n'`R*`(z)'=`R*`(z);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/-%#R* G6#%\"zG,2\"\"\"F)F'F)*&#F)\"\"#F)*$)F'F,F)F)F)*&#F)\"\"'F)*$)F'\"\"$F )F)F)*&#F)\"#CF)*$)F'\"\"%F)F)F)*&#F)\"$?\"F)*$)F'\"\"&F)F)F)*&#\",Vs4 &Q<\"/SI#o#)[?\"F)*$)F'F1F)F)F)*&#\"+$[jN(=\"//:qrbC6F)*$)F'\"\"(F)F)F )" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 124 "We can find the point where the boundary of the stability region interse cts the negative real axis by solving the equation: " }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "`R*`(z) = -1;" "6#/-%#R*G6#%\"zG,$ \"\"\"!\"\"" }{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#>%$z_0G$!+OeRaW!\"*" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 324 "z_0 := newton(`R*`(z)=-1,z=-4.5): \np_1 := plot([`R*`(z),-1],z=-4.99..0.49,color=[red,blue]):\np_2 := pl ot([[[z_0,-1]]$3],style=point,symbol=[circle,cross,diamond],color=blac k):\np_3 := plot([[z_0,0],[z_0,-1]],linestyle=3,color=COLOR(RGB,0,.5,0 )):\nplots[display]([p_1,p_2,p_3],view=[-4.99..0.49,-1.47..1.47],font= [HELVETICA,9]);" }}{PARA 13 "" 1 "" {GLPLOT2D 398 284 284 {PLOTDATA 2 "6+-%'CURVESG6$7U7$$!3A++++++!*\\!#<$!3#[(*4A*GzbFF*7$$!3cLLe(zv-$\\F* $!3-$*[X6K2\"[#F*7$$!3#pmm^f^0([F*$!3<+&[?#[vHAF*7$$!3#3]()f)ee=[F*$!3 uM.w&y3)G?F*7$$!3%QL3o3KXF*$!3')Qk4Aodr6F*7$$!3\\L$3@f%)\\T%F*$!3.CbJo$z ;@*!#=7$$!3jm;u*Q?kI%F*$!3Q)[0fOBqH(FP7$$!3E+]ZY%3S>%F*$!3#G\\EH'4![m& FP7$$!3%omTR&=vxSF*$!3)p_nLmALH%FP7$$!3/+]x(4o='RF*$!3Ya=$FP7$$ !3OF*$!3)o()>'32e#4\"FP7$$!3?++]Qxz+NF*$!3of\")*)[Rm#['!#>7$$ !3/++5QhU'Q$F*$!3zP?r.YuTJFcp7$$!3mm;%zwlDG$F*$!3'>N;nNnfO(!#?7$$!3[LL BUd1fJF*$\"3I#pMM^M__\"Fcp7$$!3eLL$4-XW0$F*$\"3QF$yK@7#eIFcp7$$!3;+]n] iuKHF*$\"3EfYlYz@EXFcp7$$!3ILL$z@A]#GF*$\"3;BD1DzvKcFcp7$$!3!)**\\n!)= $oq#F*$\"3oO]mQt*>s'Fcp7$$!3')**\\-InG%f#F*$\"3O9vR&f>Hq(Fcp7$$!3ILL3s w&oZ#F*$\"3l;)\\]q*Fcp7$$!3#om; 4([q_AF*$\"3Q\"\\*4Z^O%3\"FP7$$!3xm;%z&\\)=8#F*$\"3\\M$[2Fo_@\"FP7$$!3 $***\\_o3rE?F*$\"3`^-R-KOU8FP7$$!3hmmhq*>J\">F*$\"3)GLU'>s\"f\\\"FP7$$ !35++?e%pdz\"F*$\"3evG1&4m\\n\"FP7$$!3+++:w['4o\"F*$\"3'*ye;&****G(=FP 7$$!3-+]()Rb))p:F*$\"3'*G%oULv%)3#FP7$$!3#)***\\a(3bY9F*$\"3-B$)y`B))e BFP7$$!3cLL$R>HdL\"F*$\"3R_xpA)eHj#FP7$$!3z****\\%R.u@\"F*$\"39)e(**)e V>'HFP7$$!3pm;aLE=56F*$\"33piY'3^hH$FP7$$!3E%****4@?'H**FP$\"3-[9UXkQ0 PFP7$$!3MML3+cmE))FP$\"3xsy\\.%)3PTFP7$$!3%H**\\(H-wtwFP$\"3F.i8O'GCk% FP7$$!3cOLL)\\%eYlFP$\"3KUhx(=Yi>&FP7$$!3g.+vof`m`FP$\"31psmh;.ZeFP7$$ !3Dkmm#)*3+B%FP$\"3o4@7oSy]lFP7$$!3Yjm;()funIFP$\"37rtYu[;etFP7$$!3HBL 3;r5:>FP$\"3#=_#*G$y5d#)FP7$$!3_^****>q^f&)Fcp$\"3E/gKfxlz\"*FP7$$\"3O Mnm\"G*fzNFcp$\"32%\\AGQWk.\"F*7$$\"3\"RLL8'ppV9FP$\"3e=')3W6Jb6F*7$$ \"3@0+D$4>8g#FP$\"3KXVv)=,rH\"F*7$$\"3Q,+v#=6$4PFP$\"3+'eV_N$3\\9F*7$$ \"3!***************[FP$\"3/u**[kmJK;F*-%'COLOURG6&%$RGBG$\"*++++\"!\") $\"\"!F\\\\lF[\\l-F$6$7S7$F($!\"\"F\\\\l7$F3Fa\\l7$F=Fa\\l7$FBFa\\l7$F GFa\\l7$FLFa\\l7$FRFa\\l7$FWFa\\l7$FfnFa\\l7$F[oFa\\l7$F`oFa\\l7$FeoFa \\l7$FjoFa\\l7$F_pFa\\l7$FepFa\\l7$FjpFa\\l7$F`qFa\\l7$FeqFa\\l7$FjqFa \\l7$F_rFa\\l7$FdrFa\\l7$FirFa\\l7$F^sFa\\l7$FcsFa\\l7$FhsFa\\l7$F]tFa \\l7$FbtFa\\l7$FgtFa\\l7$F\\uFa\\l7$FauFa\\l7$FfuFa\\l7$F[vFa\\l7$F`vF a\\l7$FevFa\\l7$FjvFa\\l7$F_wFa\\l7$FdwFa\\l7$FiwFa\\l7$F^xFa\\l7$FcxF a\\l7$FhxFa\\l7$F]yFa\\l7$FbyFa\\l7$FgyFa\\l7$F\\zFa\\l7$FazFa\\l7$Ffz Fa\\l7$F[[lFa\\l7$F`[lFa\\l-Fe[l6&Fg[lF[\\lF[\\lFh[l-F$6&7#7$$!3y***** f$eRaWF*Fa\\l-%'SYMBOLG6#%'CIRCLEG-Fe[l6&Fg[lF\\\\lF\\\\lF\\\\l-%&STYL EG6#%&POINTG-F$6&Fg_l-F\\`l6#%&CROSSGF_`lFa`l-F$6&Fg_l-F\\`l6#%(DIAMON DGF_`lFa`l-F$6%7$7$Fi_lF[\\lFh_l-%&COLORG6&Fg[lF[\\l$\"\"&Fb\\lF[\\l-% *LINESTYLEG6#\"\"$-%+AXESLABELSG6%Q\"z6\"Q!F`bl-%%FONTG6#%(DEFAULTG-Fc bl6$%*HELVETICAG\"\"*-%%VIEWG6$;$!$*\\!\"#$\"#\\F`cl;$!$Z\"F`cl$\"$Z\" F`cl" 1 2 0 1 10 0 2 9 1 4 2 1.000000 45.000000 45.000000 0 0 "Curve 1 " "Curve 2" "Curve 3" "Curve 4" "Curve 5" "Curve 6" }}}}{PARA 0 "" 0 " " {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 73 "The following picture s hows the stability region for the order 5 scheme. " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1554 "`R*` := z \+ -> 1+z+1/2*z^2+1/6*z^3+1/24*z^4+1/120*z^5+\n 17385097243/12048826 823040*z^6+1873563483/11245571701504*z^7:\npts := []: z0 := 0:\nfor ct to 301 do\n zz := newton(`R*`(z)=exp(ct*Pi/30*I),z=z0):\n z0 := z z:\n pts := [op(pts),[Re(zz),Im(zz)]]:\nend do:\np_1 := plot(pts,col or=COLOR(RGB,.4,0,.08)):\np_2 := plots[polygonplot]([seq([pts[i-1],pts [i],[-2.2,0]],i=2..nops(pts))],\n style=patchnogrid,color=COL OR(RGB,.8,0,.15)):\npts := []: z0 := 1.4+4.25*I: tt := 0:\nwhile tt<=6 1/30 do\n zz := newton(`R*`(z)=exp(tt*Pi*I),z=z0):\n z0 := zz:\n \+ if (1/2<=tt and tt<=7/5) then\n hh := 1/60\n else \n hh : = 1/30\n end if;\n tt := tt+hh;\n pts := [op(pts),[Re(zz),Im(zz) ]]:\nend do:\np_3 := plot(pts,color=COLOR(RGB,.4,0,.08)):\np_4 := plot s[polygonplot]([seq([pts[i-1],pts[i],[1.25,4.12]],i=2..nops(pts))],\n \+ style=patchnogrid,color=COLOR(RGB,.8,0,.15)):\npts := []: z0 \+ := 1.4-4.25*I: tt := 0:\nwhile tt<=61/30 do\n zz := newton(`R*`(z)=e xp(tt*Pi*I),z=z0):\n z0 := zz:\n if (3/5<=tt and tt<=3/2) then\n \+ hh := 1/60\n else \n hh := 1/30\n end if;\n tt := tt+hh ;\n pts := [op(pts),[Re(zz),Im(zz)]]:\nend do:\np_5 := plot(pts,colo r=COLOR(RGB,.4,0,.08)):\np_6 := plots[polygonplot]([seq([pts[i-1],pts[ i],[1.25,-4.12]],i=2..nops(pts))],\n style=patchnogrid,color= COLOR(RGB,.8,0,.15)):\np_7 := plot([[[-5.09,0],[1.89,0]],[[0,-4.79],[0 ,4.69]]],color=black,linestyle=3):\nplots[display]([p_||(1..7)],view=[ -5.09..1.89,-4.69..4.69],font=[HELVETICA,9],\n labels=[`R e(z)`,`Im(z)`],axes=boxed,scaling=constrained);" }}{PARA 13 "" 1 "" {GLPLOT2D 390 543 543 {PLOTDATA 2 "6/-%'CURVESG6$7i]l7$$\"3/+++%)R&>2( !#G$\"31+++^v>Z5!#=7$$\"3*)*****f$zEJW!#E$\"3)******p3&R%4#F-7$$\"3G++ +PvZm[!#D$\"3?+++&R#fTJF-7$$\"3=+++htn!f#!#C$\"3%******\\F)y)=%F-7$$\" 3i******)RT.<*F=$\"3C+++$fyfB&F-7$$\"3?+++#Rb[Z#!#B$\"3i*****Rk_JG'F-7 $$\"3k*****fiRTW&FH$\"3`*****HT(GItF-7$$\"3#*******e,105!#A$\"3W+++J%R tP)F-7$$\"35+++m*f?b\"FS$\"3_*****HBMUU*F-7$$\"3'******fO*)Q)=FS$\"3)* *****\\P&3Z5!#<7$$\"3-+++TA(*z7FS$\"31+++`Aq^6Fjn7$$!31+++rrk#o\"FS$\" 3'******RU\\iD\"Fjn7$$!3M*****zq6f^*FS$\"3#******f2%pg8Fjn7$$!3u***** \\2d=j#!#@$\"3#******\\l%*\\Y\"Fjn7$$!3Z+++?\"y\"QeF]p$\"3/+++J65p:Fjn 7$$!3'******>\\U)[6!#?$\"3/+++Cj&Hn\"Fjn7$$!3))*****p;Ok3#Fhp$\"3%**** ***f$)\\w'Rmz=Fjn7$$!3N+++!*ya>eFhp $\"3)******4'=R#)>Fjn7$$!3n*****Hh&Q?\"*Fhp$\"3)******4,DY3#Fjn7$$!3%* *****p!os\"Q\"!#>$\"3))*****4q4j=#Fjn7$$!34+++r-dK?Fbr$\"3++++z!*Q(G#F jn7$$!3!*******G_[8HFbr$\"3)******p.)y(Q#Fjn7$$!3-+++tL$33%Fbr$\"3?+++ AhQ([#Fjn7$$!3-+++Bnk(f&Fbr$\"39+++?c(fe#Fjn7$$!3?+++LI'4`(Fbr$\"33+++ _\\?$o#Fjn7$$!3e+++sirX**Fbr$\"3=+++4i^yFFjn7$$!33+++3oP*G\"F-$\"3y*** **f\\-6(GFjn7$$!3-+++#zs)R;F-$\"3=+++!=I*fHFjn7$$!3$********etT/#F-$\" 31+++K\"oQ/$Fjn7$$!3/+++#f@d\\#F-$\"3,+++g2\">7$Fjn7$$!3#******pRRW)HF -$\"3w*****zf(R$>$Fjn7$$!31+++!\\')*)\\$F-$\"39+++fG6eKFjn7$$!3\"***** *\\wE*GSF-$\"3*******poLiJ$Fjn7$$!3%*******>f&fc%F-$\"3))*****pU#>oLFj n7$$!3#)*****H!y9/^F-$\"3()*******QOXT$Fjn7$$!3j*****HuD'RcF-$\"3!)*** **4aGeX$Fjn7$$!3C+++g@0qhF-$\"3?+++**Hf#\\$Fjn7$$!3;++++D:%p'F-$\"39++ +SDHDNFjn7$$!3X+++>jM6sF-$\"3%******\\MCVb$Fjn7$$!3n*****H\"p[@xF-$\"3 /+++EW-!e$Fjn7$$!3Z*****z5(oC#)F-$\"37+++&GvEg$Fjn7$$!3m******pl?@()F- $\"3))*****p39Di$Fjn7$$!3O+++92Q6#*F-$\"3&)******)=S(ROFjn7$$!3w*****R $pd&p*F-$\"33+++9;_aOFjn7$$!3$******f%oT<5Fjn$\"3)******HX+qm$Fjn7$$!3 ++++]>vk5Fjn$\"3%)*****RF(HxOFjn7$$!3-+++Zxf66Fjn$\"37+++;Z^&o$Fjn7$$! 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cn$!+ofVrUFdcnFhbfl7%F`[hl7$$\"+$pNVO\"Fdcn$!+#z)ejUFdcnFhbfl7%Ff[hl7$ F]]el$!+X/%\\D%FdcnFhbfl7%F\\\\hl7$Fg\\el$!+dK`XUFdcnFhbflFb_alFa`r-%* AXESSTYLEG6#%$BOXG-%+AXESLABELSG6%%&Re(z)G%&Im(z)G-%%FONTG6#%(DEFAULTG -F]]hl6$%*HELVETICAG\"\"*-%(SCALINGG6#%,CONSTRAINEDG-%%VIEWG6$;$!$4&F^ `n$\"$*>F^`n;$!$*[F^`n$\"$*[F^`n" 1 2 0 1 10 0 2 9 1 2 1 1.000000 45.000000 45.000000 0 0 "Curve 1" "Curve 2" "Curve 3" "Curve 4" "Curve 5" "Curve 6" "Curve 7" "Curve 8" "Curve 9" "Curve 10" "Curve 11" "Cur ve 12" "Curve 13" "Curve 14" }}}}{PARA 0 "" 0 "" {TEXT -1 0 "" }} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}}{PARA 0 "" 0 "" {TEXT -1 52 "#---------------------------------------------------" }}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 36 "coefficients for the combined scheme" } }{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 35 "coefficients of the co mbined scheme" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1633 "ee := \{c[2]=47/193,\nc[3]=9/38,\nc[4]=342/683 ,\nc[5]=71/89,\nc[6]=23/28,\nc[7]=1,\nc[8]=1,\n\na[2,1]=47/193,\na[3,1 ]=16515/135736,\na[3,2]=15633/135736,\na[4,1]=-695302416/14974763389, \na[4,2]=-9469814814/14974763389,\na[4,3]=375818328/318611987,\na[5,1] =33509869292201/85055527825236,\na[5,2]=3587459103/4605562477,\na[5,3] =-21724507215628/18120732551793,\na[5,4]=1134395721190865/137717567393 6268,\na[6,1]=233436361745853683/373836194308813824,\na[6,2]=158645421 /143412416,\na[6,3]=-2222985817851376045/1063980184927185792,\na[6,4]= 85304345532627205619/66923624494207595520,\na[6,5]=-872666029598803/92 28174667747840,\na[7,1]=-1178762513361663864779/6649079911196633209152 ,\na[7,2]=-8522902869921/10868389582568,\na[7,3]=308716357954698816820 2379/1791201064146135353344584,\na[7,4]=-3767364590077053611426162077/ 10220229771451535362575529152,\na[7,5]=17701204856780353690623/1355738 9052385415225632,\na[7,6]=-7678963110907452/10984484330835583,\na[8,1] =1416449588677763/6015481297132284,\na[8,2]=-48943449/199463911,\na[8, 3]=83884485628625980/463004404677327933,\na[8,4]=269068569637060205437 7/6604522931171002998060,\na[8,5]=855085588386243786/60889394637402159 5,\na[8,6]=-10797672634721280/10984484330835583,\na[8,7]=0,\n\nb[1]=25 03727/33509160,\nb[2]=0,\nb[3]=116376653000/351539506521,\nb[4]=384949 07685743537/196546626900894600,\nb[5]=117203823775061/154952625580200, \nb[6]=-84924560896/194121869325,\nb[7]=0,\nb[8]=4243913/53400600,\n\n `b*`[1]=2644521142/35096656455,\n`b*`[2]=0,\n`b*`[3]=8764703092174/267 29124547545,\n`b*`[4]=737727050374863463/3622135308216853050,\n`b*`[5] =2721770523832513/4007247067087950,\n`b*`[6]=-89022652768/245887701145 ,\n`b*`[7]=2/57,\n`b*`[8]=2/49\}:" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT 259 5 "nodes" }{TEXT -1 2 ": " }} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 31 "seq(c[i]=subs(ee,c[i]),i=2.. 8);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6)/&%\"cG6#\"\"##\"#Z\"$$>/&F%6# \"\"$#\"\"*\"#Q/&F%6#\"\"%#\"$U$\"$$o/&F%6#\"\"&#\"#r\"#*)/&F%6#\"\"'# \"#B\"#G/&F%6#\"\"(\"\"\"/&F%6#\"\")FK" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT 259 20 "linking coefficients" }{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..8);" }}{PARA 12 "" 1 "" {XPPMATH 20 "6>/&%\"aG 6$\"\"#\"\"\"#\"#Z\"$$>/&F%6$\"\"$F(#\"&:l\"\"'Od8/&F%6$F/F'#\"&Lc\"F2 /&F%6$\"\"%F(#!*;CI&p\",*QjZ(\\\"/&F%6$F;F'#!+9[\")p%*F>/&F%6$F;F/#\"* G$=eP\"*()>h=$/&F%6$\"\"&F(#\"/,AHp)4N$\"/O_#y_b])/&F%6$FMF'#\"+.\"fue $\"+xCc0Y/&F%6$FMF/#!/Gc@2Xs@\"/$z^Dt?\"=/&F%6$FMF;#\"1l3>@dRM6\"1oi$R nvrP\"/&F%6$\"\"'F(#\"3$o`euhjVL#\"3CQ\")3V>OQP/&F%6$F`oF'#\"*@ake\"\" *;CTV\"/&F%6$F`oF/#!4XgP^y\"e)HA#\"4#z&=F\\=!)R1\"/&F%6$F`oF;#\"5>c?FE `XVI&)\"5?bf2U\\CO#p'/&F%6$F`oFM#!0.))fHgms)\"1SyunY6*z!\\m/&F%6$F_qF'#!.@*pG!H_)\"/oDe*Qo3\"/&F% 6$F_qF/#\":zB?o\"))pazN;(3$\":%eWLNNh9k5?\"z\"/&F%6$F_qF;#!=x?;E9h`q2! fktw$\">_\"Hbdi``^9xH-A5/&F%6$F_qFM#\"8B1p`.yc[?,x\"\"8KcA:aQ_!*QdN\"/ &F%6$F_qF`o#!1_u!46j*yw\"2$eN3L%[%)4\"/&F%6$\"\")F(#\"1jxn)e\\kT\"\"1% GKrH\"[:g/&F%6$FdsF'#!)\\M%*[\"*6RY*>/&F%6$FdsF/#\"2!)fiGc[%)Q)\"3LzKx YS/IY/&F%6$FdsF;#\"7xV0-1P'p&o!p#\"7g!)*H+r6$H_/m/&F%6$FdsFM#\"3'yViQ) e&3b)\"3&f@SPYR*)3'/&F%6$FdsF`o#!2!G@Zjswz5F`s/&F%6$FdsF_q\"\"!" }}} {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT 259 30 "weights \+ for the 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#\"\"\"#\"(FP]#\")g\"4N$/&F%6#\"\"#\"\"!/&F%6# \"\"$#\"-+Ilwj6\"-@l]R:N/&F%6#\"\"%#\"2PNu&o2\\\\Q\"3+Y*3!piYl>/&F%6# \"\"&#\"0h]xBQ?<\"\"0+-eDE&\\:/&F%6#\"\"'#!,'*3cC\\)\"-D$p=7%>/&F%6#\" \"(F//&F%6#\"\")#\"(8RC%\")+1S`" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }} {PARA 0 "" 0 "" {TEXT 259 30 "weights for the order 5 scheme" }{TEXT -1 2 ": " }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 37 "seq(`b*`[i]=subs (ee,`b*`[i]),i=1..8);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6*/&%#b*G6#\"\" \"#\"+U6_WE\",bkl'4N/&F%6#\"\"#\"\"!/&F%6#\"\"$#\".u@4.Zw)\"/XvaC\"Hn# /&F%6#\"\"%#\"3jM'[P]qsP(\"4]I&o@3`8AO/&F%6#\"\"&#\"18D$Q_q " 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 "#================= =========================" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 " " 0 "" {TEXT -1 23 "#======================" }}{SECT 1 {PARA 4 "" 0 " " {TEXT -1 23 "Abreviated calculations" }}{PARA 0 "" 0 "" {TEXT -1 0 " " }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 28 "Set up order conditions etc." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1030 "SO_eqs := \+ [op(RowSumConditions(8,'expanded')),op(StageOrderConditions(2,8,'expan ded'))]:\nSO6_8 := SimpleOrderConditions(6,8,'expanded'):\nord_cdns := [seq(SO6_8[i],i=[1,2,4,8,13,16,24,28,29,32])]:\nsimp_cdns := [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= [4,5,6,7])]:\nsimp_cdns2 := [add(b[i]*c[i]*a[i,2],i=3..8)=0,add(b[i]*c [i]*add(a[i,j]*a[j,2],j=3..i-1),i=3..8)=0,\n add(b[i]*c [i]^2*a[i,2],i=3..8)=0]:\ncdns := [op(ord_cdns),op(simp_cdns),op(simp_ cdns2),op(SO_eqs)]:\n`SO5_8*` := subs(b=`b*`,SimpleOrderConditions(5,8 ,'expanded')):\n`ord_cdns*` := [seq(`SO5_8*`[i],i=[1,2,4,8,15,16])]:\n `simp_cdns*` := [add(`b*`[i]*a[i,2],i=3..8)=0,add(`b*`[i]*add(a[i,j]*a [j,2],j=3..i-1),i=2..8)=0,\n `b*`[7]*a[7,6]+`b*`[8]*a[ 8,6]=`b*`[6]*(1-c[6])]:\n`cdns*` := [op(`ord_cdns*`),op(`simp_cdns*`)] : \n\nerrterms6_8 := PrincipalErrorTerms(6,8,'expanded'):\n`errterms5_ 8*` := subs(b=`b*`,PrincipalErrorTerms(5,8,'expanded')):\n`errterms6_8 *` := subs(b=`b*`,PrincipalErrorTerms(6,8,'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" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 2183 " calc_RKcoeffs := proc()\n local eqns,`eqns*`,val,sm,ct,Rz,stb6,stb5, nmB,snmB,dnB,sdnB,nmC,snmC,B_7,C_7,nrm;\n global cdns,e1,e2,e3,e4,e5 ,e6,e7;\n\n e1 := \{c[2]=c_2,c[3]=c_3,c[5]=c_5,c[6]=c_6,c[7]=1,c[8]= 1,b[2]=0,b[bb[1]]=bb[2]\};\n eqns := subs(e1,cdns);\n e2 := solve( \{op(eqns)\},indets(eqns) minus \{a[7,3],a[7,4],a[7,5],a[7,6]\});\n \+ e3 := `union`(e1,e2);\n e4 := \{`b*`[2]=0,`b*`[bsA[1]]=bsA[2],`b*`[b sB[1]]=bsB[2]\}:\n e5 := `union`(e3,e4):\n `eqns*` := subs(e5,`cdn s*`):\n e6 := solve(\{op(`eqns*`)\}):\n e7 := `union`(subs(e6,e5), e6):\n Digits := 14;\n sm := 0;\n for ct to nops(errterms6_8) do \n sm := sm+(evalf(subs(e7,errterms6_8[ct])))^2;\n end do;\n \+ Rz := subs(e7,StabilityFunction(6,8,'expanded'));\n stb6 := max(fsol ve(Rz=1,z=-8..-1e-7),fsolve(Rz=-1,z=-8..-1e-7));\n stb6 := evalf[8]( stb6);\n Rz := subs(e7,subs(b=`b*`,StabilityFunction(5,8,'expanded') ));\n stb5 := evalf[8](max(fsolve(Rz=1,z=-8..-1e-7),fsolve(Rz=-1,z=- 8..-1e-7)));\n stb5 := evalf[8](stb5);\n nmB := 0;\n for ct to n ops(`errterms6_8*`) do\n nmB := nmB+evalf(subs(e7,`errterms6_8*`[ ct]))^2;\n end do:\n snmB := sqrt(simplify(nmB));\n dnB := 0;\n \+ for ct to nops(`errterms5_8*`) do\n dnB := dnB+evalf(subs(e7,`e rrterms5_8*`[ct]))^2;\n end do;\n sdnB := sqrt(simplify(dnB));\n \+ nmC := 0;\n for ct to nops(errterms6_8) do\n nmC := nmC+(evalf (subs(e7,`errterms6_8*`[ct]))-evalf(subs(e7,errterms6_8[ct])))^2;\n \+ end do;\n snmC := sqrt(simplify(nmC));\n B_7 := evalf[8](snmB/sdnB );\n C_7 := evalf[8](snmC/sdnB);\n print(`nodes:`,c[2]=c_2,c[3]=c_ 3,c[4]=subs(e7,c[4]),c[5]=c_5,c[6]=c_6); \n print(`order 6 weights :`,seq(b[i]=evalf[6](subs(e7,b[i])),i=[1,$3..8]));\n print(`order 5 \+ weights:`,seq(`b*`[i]=evalf[6](subs(e7,`b*`[i])),i=[1,$3..8]));\n nr m := max(seq(seq(subs(e7,abs(a[i,j])),j=1..i-1),i=2..8));\n print(in finity*`-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 \+ print(`order 6 stability interval` = [stb6,0]);\n print(`order 5 sta bility interval` = [stb5,0]);\n print('B[7]'=B_7,'C[7]'=C_7);\nend p roc:" }}}{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 32 "#-------------------------------" }}{PARA 0 "" 0 "" {TEXT -1 21 "Prince-Dormand scheme" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 112 "c_2 := 1/10: c_3 := 2/9: c_5 := 3/5: c_6 := 4/5: bb \+ := [8,1/10]: bsA := [7,3/50]: bsB := [8,0]:\ncalc_RKcoeffs();" }} {PARA 11 "" 1 "" {XPPMATH 20 "6(%'nodes:G/&%\"cG6#\"\"##\"\"\"\"#5/&F& 6#\"\"$#F(\"\"*/&F&6#\"\"%#F/\"\"(/&F&6#\"\"&#F/F;/&F&6#\"\"'#F5F;" }} {PARA 11 "" 1 "" {XPPMATH 20 "6*%1order~6~weights:G/&%\"bG6#\"\"\"$\"' >gq!\"(/&F&6#\"\"$$\"'\\eI!\"'/&F&6#\"\"%$\"'/^6F2/&F&6#\"\"&$\"'Gs=F2 /&F&6#\"\"'$\"'`UDF2/&F&6#\"\"($!'d.LF+/&F&6#\"\")$\"'++5F2" }}{PARA 11 "" 1 "" {XPPMATH 20 "6*%1order~5~weights:G/&%#b*G6#\"\"\"$\"'&=g(! \"(/&F&6#\"\"$$\"'TSF!\"'/&F&6#\"\"%$\"'f?>F2/&F&6#\"\"&$\"'rv5F2/&F&6 #\"\"'$\"'6.HF2/&F&6#\"\"($\"'++gF+/&F&6#\"\")$\"\"!FP" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/*&%)infinityG\"\"\"%8-norm~of~linking~coeffsGF&$ \"+<@h#[%!\"*" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/%L2-norm~of~principa l~error~of~order~6~schemeG$\"+1qGEB!#8" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/%L2-norm~of~principal~error~of~order~5~schemeG$\"+3,ZX=!#8" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#/%;order~6~stability~intervalG7$$!)(HT &R!\"(\"\"!" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/%;order~5~stability~in tervalG7$$!)oDPP!\"(\"\"!" }}{PARA 11 "" 1 "" {XPPMATH 20 "6$/&%\"BG6# \"\"($\")8f'>#!\"(/&%\"CGF&$\"):r0:F*" }}}{PARA 0 "" 0 "" {TEXT -1 0 " " }}{PARA 0 "" 0 "" {TEXT -1 32 "#-------------------------------" }} {PARA 0 "" 0 "" {TEXT -1 43 "a modification of the Prince-Dormand sche me" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 120 "c_2 := 7/39: c_3 := 2 /9: c_5 := 23/33: c_6 := 24/31: bb := [8,13/200]: bsA := [7,12/167]: b sB := [8,0]:\ncalc_RKcoeffs();" }}{PARA 11 "" 1 "" {XPPMATH 20 "6(%'no des:G/&%\"cG6#\"\"##\"\"(\"#R/&F&6#\"\"$#F(\"\"*/&F&6#\"\"%#F/F*/&F&6# \"\"&#\"#B\"#L/&F&6#\"\"'#\"#C\"#J" }}{PARA 11 "" 1 "" {XPPMATH 20 "6* %1order~6~weights:G/&%\"bG6#\"\"\"$\"'2ws!\"(/&F&6#\"\"$$\"'HwG!\"'/&F &6#\"\"%$\"'-'*=F2/&F&6#\"\"&$\"'4b8F2/&F&6#\"\"'$\"'z'R#F2/&F&6#\"\"( $\"'V@)*!\")/&F&6#\"\")$\"'++lF+" }}{PARA 11 "" 1 "" {XPPMATH 20 "6*%1 order~5~weights:G/&%#b*G6#\"\"\"$\"'a^u!\"(/&F&6#\"\"$$\"'!)zF!\"'/&F& 6#\"\"%$\"'Q)3#F2/&F&6#\"\"&$\"'%=@*F+/&F&6#\"\"'$\"'#pu#F2/&F&6#\"\"( $\"'j&=(F+/&F&6#\"\")$\"\"!FP" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/*&%) infinityG\"\"\"%8-norm~of~linking~coeffsGF&$\"+0*3'36!\"*" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/%L2-norm~of~principal~error~of~order~6~schemeG $\"+n(3j5#!#8" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/%L2-norm~of~principa l~error~of~order~5~schemeG$\"+e-)[#=!#8" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/%;order~6~stability~intervalG7$$!)(HT&R!\"(\"\"!" }}{PARA 11 " " 1 "" {XPPMATH 20 "6#/%;order~5~stability~intervalG7$$!)#R>t$!\"(\"\" !" }}{PARA 11 "" 1 "" {XPPMATH 20 "6$/&%\"BG6#\"\"($\")DKc?!\"(/&%\"CG F&$\")B#zX\"F*" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> \+ " 0 "" {MPLTEXT 1 0 119 "subs(e7,matrix([seq([c[i],seq(a[i,j],j=1..i-1 ),``$(9-i)],i=2..8),\n[`b`,seq(b[i],i=1..8)],[`b*`,seq(`b*`[i],i=1..8) ]]));" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#K%'matrixG6#7+7+#\"\"(\"#RF(% !GF+F+F+F+F+F+7+#\"\"#\"\"*#\"#;\"$*=#\"#EF2F+F+F+F+F+F+7+#\"\"$F)#\"$ d*\"%/'*#!%`5\"%,C#\"%`5\"%s8F+F+F+F+F+7+#\"#B\"#L#\"(TPc#\")'fo5\"#!& R#=\"'([-\"#\"%VK\"'K8w#\"(ySG$\"(\"**Q^F+F+F+F+7+#\"#C\"#J#!)_zf6\"* \")oo[\"#\"&kE*\"'P&3##\")W4u)*\"*J8Fk&#!)+V+E\"*j*y@P#\"++fxo$*\",JA6 [4$F+F+F+7+\"\"\"#\")>emQ\")S'3=*#!$(*)\"%K7#\"'*Rc\"\"(/b]\"#\"+,hG#f \"\"++?HO9#!+2mYzA\"+!G`]'H#\"*.(p@(*\"++SAE6F+F+7+F_o#\"*8qi=\"\"*sig 2'#!%F:\"%OJ#\")40cE\")'\\=)\\#\"*x'>nd\"++)[zN\"#!+\"R#H;h\",o$)H/m\" #\"+<9*QB&\"++_b`u\"\"!F+7+%\"bG#\"&fW\"\"'?()>Feq#\"&$o>\"&K%o#\"(PAD )\")+S_V#\"*Tk\\V\"\"++w%*e5#\")^\"H'G\"*+![%>\"#\"#6\"%?6#\"#8\"$+#7+ %#b*G#\"(VkB\"\")?Jf;Feq#\")^4oV\"*!)p8d\"#\"*`_[z$\"++q7<=#\"+Z,1H;\" ,?\\U%o<#\",$zgs8I\"-+!))Hr4\"#\"#7\"$n\"FeqQ(pprint26\"" }}}{PARA 0 " " 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 40 "#================= ======================" }}{PARA 0 "" 0 "" {TEXT -1 20 "Verner's 2004 s cheme" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 112 "c_2 := 1/7: c_3 := 2/9: c_5 := 2/3: c_6 := 3/4: bb := [7,1/100]: bsA := [7,0]: bsB := [8 ,2/25]:\ncalc_RKcoeffs();" }}{PARA 11 "" 1 "" {XPPMATH 20 "6(%'nodes:G /&%\"cG6#\"\"##\"\"\"\"\"(/&F&6#\"\"$#F(\"\"*/&F&6#\"\"%#F/F+/&F&6#\" \"&#F(F//&F&6#\"\"'#F/F5" }}{PARA 11 "" 1 "" {XPPMATH 20 "6*%1order~6~ weights:G/&%\"bG6#\"\"\"$\"'\"[J(!\"(/&F&6#\"\"$$\"',YG!\"'/&F&6#\"\"% $\"'8&*>F2/&F&6#\"\"&$\"\"!F>/&F&6#\"\"'$\"'_fOF2/&F&6#\"\"($\"'++5F+/ &F&6#\"\")$\"'dymF+" }}{PARA 11 "" 1 "" {XPPMATH 20 "6*%1order~5~weigh ts:G/&%#b*G6#\"\"\"$\"'\"[1(!\"(/&F&6#\"\"$$\"'J))H!\"'/&F&6#\"\"%$\"' J(o\"F2/&F&6#\"\"&$\"'+vgF+/&F&6#\"\"'$\"'S5KF2/&F&6#\"\"($\"\"!FJ/&F& 6#\"\")$\"'++!)F+" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/*&%)infinityG\" \"\"%8-norm~of~linking~coeffsGF&$\"+ke!pN%!\"*" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/%L2-norm~of~principal~error~of~order~6~schemeG$\"+-!)) )Q@!#8" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/%L2-norm~of~principal~error ~of~order~5~schemeG$\"+7'[/^\"!#8" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/ %;order~6~stability~intervalG7$$!)(HT&R!\"(\"\"!" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/%;order~5~stability~intervalG7$$!)dTZU!\"(\"\"!" }} {PARA 11 "" 1 "" {XPPMATH 20 "6$/&%\"BG6#\"\"($\")0#)3>!\"(/&%\"CGF&$ \")Z)3X\"F*" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 34 "#---------------------------------" }}{PARA 0 "" 0 "" {TEXT -1 33 "a modification of Verner's scheme" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 116 "c_2 := 1/6: c_3 := 2/9: c_5 := 11/16: c_6 := 3/4: bb := [7,1/100]: bsA := [7,0]: bsB := [8,33/416]:\ncalc_RKcoeffs();" }} {PARA 11 "" 1 "" {XPPMATH 20 "6(%'nodes:G/&%\"cG6#\"\"##\"\"\"\"\"'/&F &6#\"\"$#F(\"\"*/&F&6#\"\"%#F/\"\"(/&F&6#\"\"&#\"#6\"#;/&F&6#F+#F/F5" }}{PARA 11 "" 1 "" {XPPMATH 20 "6*%1order~6~weights:G/&%\"bG6#\"\"\"$ \"'\"[J(!\"(/&F&6#\"\"$$\"',YG!\"'/&F&6#\"\"%$\"'8&*>F2/&F&6#\"\"&$\" \"!F>/&F&6#\"\"'$\"'_fOF2/&F&6#\"\"($\"'++5F+/&F&6#\"\")$\"'dymF+" }} {PARA 11 "" 1 "" {XPPMATH 20 "6*%1order~5~weights:G/&%#b*G6#\"\"\"$\"' 8Nr!\"(/&F&6#\"\"$$\"'wYH!\"'/&F&6#\"\"%$\"'M&y\"F2/&F&6#\"\"&$\"'ZaaF +/&F&6#\"\"'$\"'n:KF2/&F&6#\"\"($\"\"!FJ/&F&6#\"\")$\"'pKzF+" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/*&%)infinityG\"\"\"%8-norm~of~linking~coeff sGF&$\"+-'oUW$!\"*" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/%L2-norm~of~pri ncipal~error~of~order~6~schemeG$\"+AM!R7#!#8" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/%L2-norm~of~principal~error~of~order~5~schemeG$\"+UDJ- :!#8" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/%;order~6~stability~intervalG 7$$!)(HT&R!\"(\"\"!" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/%;order~5~stab ility~intervalG7$$!):,VU!\"(\"\"!" }}{PARA 11 "" 1 "" {XPPMATH 20 "6$/ &%\"BG6#\"\"($\")bc!)=!\"(/&%\"CGF&$\") " 0 "" {MPLTEXT 1 0 119 "subs(e7,mat rix([seq([c[i],seq(a[i,j],j=1..i-1),``$(9-i)],i=2..8),\n[`b`,seq(b[i], i=1..8)],[`b*`,seq(`b*`[i],i=1..8)]]));" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#K%'matrixG6#7+7+#\"\"\"\"\"'F(%!GF+F+F+F+F+F+7+#\"\"#\"\"*#F.\"# F#\"\"%F1F+F+F+F+F+F+7+#\"\"$\"\"(#\"$$=\"%s8#!$i\"\"$V$#\"%`5F:F+F+F+ F+F+7+#\"#6\"#;#\"%v9\"%Wh#!$8&\"%[?#\"%Fa\"&[K&#\"&2_*\"'W(f\"F+F+F+F +7+#F6F3#F6\"#))#\"#\")\"$_$#\"&JY%\"'%=8'#\"&`'e\"'3aE#\"%/T\"&t8#F+F +F+7+F)#\"&4*G\"&[\\##!$?\"\"#x#!)\\dgH\")ga[D#\")4VN<\"(0'Q]#!)c9(># \"(&\\Kn#\"')o%Q\"'&fh\"F+F+7+F)#\"$<)\"&/#R#!#I\"$@\"#\")j*3>$\")!e[+ %#!(c]3\"\")bYUb#!(G<'f\")N'z0\"#\"'WtD\"'NRD\"\"!F+7+%\"bG#\"#z\"%!3 \"F`q#\"&$o>\"&g\"p#\"&2o\"\"&SU)F`q#\"%;G\"%&p(#F)\"$+\"#\"$(=\"%+G7+ %#b*G#\"&R?#\"'!))3$F`q#\"(`;2&\")g4@<#\")B*R8\"\")gp^j#\"&wd(\"(X#*Q \"#\"&o@$\"'N+5F`q#\"#L\"$;%Q(pprint06\"" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 38 "#--------------------------------- ----" }}{PARA 0 "" 0 "" {TEXT -1 37 "a 2nd modification of Verner's sc heme" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 119 "c_2 := 5/28: c_3 := 2/9: c_5 := 37/53: c_6 := 27/35: bb := [7,1/100]: bsA := [7,0]: bsB : = [8,11/142]:\ncalc_RKcoeffs();" }}{PARA 11 "" 1 "" {XPPMATH 20 "6(%'n odes:G/&%\"cG6#\"\"##\"\"&\"#G/&F&6#\"\"$#F(\"\"*/&F&6#\"\"%#F/\"\"(/& F&6#F*#\"#P\"#`/&F&6#\"\"'#\"#F\"#N" }}{PARA 11 "" 1 "" {XPPMATH 20 "6 *%1order~6~weights:G/&%\"bG6#\"\"\"$\"'<\"G(!\"(/&F&6#\"\"$$\"'KsG!\"' /&F&6#\"\"%$\"'\"*3>F2/&F&6#\"\"&$\"'^e7F2/&F&6#\"\"'$\"'<\"[#F2/&F&6# \"\"($\"'++5F+/&F&6#\"\")$\"'x4lF+" }}{PARA 11 "" 1 "" {XPPMATH 20 "6* %1order~5~weights:G/&%#b*G6#\"\"\"$\"'jRr!\"(/&F&6#\"\"$$\"'>]H!\"'/&F &6#\"\"%$\"'Z` " 0 "" {MPLTEXT 1 0 119 "subs(e7,matrix([seq([c[i],seq(a[i,j],j=1..i-1), ``$(9-i)],i=2..8),\n[`b`,seq(b[i],i=1..8)],[`b*`,seq(`b*`[i],i=1..8)]] ));" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#K%'matrixG6#7+7+#\"\"&\"#GF(%!G F+F+F+F+F+F+7+#\"\"#\"\"*#\"#M\"$0%#\"#cF2F+F+F+F+F+F+7+#\"\"$\"\"(#\" $*p\"%go#!$3\"\"$X##\"%`5\"%s8F+F+F+F+F+7+#\"#P\"#`#\")8.'H#\")?)e#)*# !(CzS\"\"(N#)=)#!'>.o\")Ww:&)#\")+^:T\")L#oQ'F+F+F+F+7+#\"#F\"#N#!)n8p A\"*+D+\\$#\"&)=G\"&vt'#\")n`lJ\"*+Dl%>#!&ot$\"(DJ?\"#\"-o7^i=9\"-voHZ ^\\F+F+F+7+\"\"\"#\"*j9X3\"\")Sy1#*#!%jU\"%;G#!,4$f?Ix\",?2J0v&#\"+R0P QL\"*+gT))*#!-$)[VnV:\",+?va4(#\")v]jj\")Oz/VF+F+7+F`o#\"&Y=$\"'0X\\#! #U\"$@\"#\"*#[U>C\"*:k')3$#\"'(QD'\")+bB@#!+j " 0 "" {MPLTEXT 1 0 125 "c_2 := 47/193: c_3 := 9/38: c_5 := 217/272: c_6 := 2 774/3377: bb := [7,0]: bsA := [7,2/57]: bsB := [8,2/49]:\ncalc_RKcoeff s();" }}{PARA 11 "" 1 "" {XPPMATH 20 "6(%'nodes:G/&%\"cG6#\"\"##\"#Z\" $$>/&F&6#\"\"$#\"\"*\"#Q/&F&6#\"\"%#\"$U$\"$$o/&F&6#\"\"&#\"$<#\"$s#/& F&6#\"\"'#\"%uF\"%xL" }}{PARA 11 "" 1 "" {XPPMATH 20 "6*%1order~6~weig hts:G/&%\"bG6#\"\"\"$\"'\"=Z(!\"(/&F&6#\"\"$$\"'Y5L!\"'/&F&6#\"\"%$\"' pe>F2/&F&6#\"\"&$\"'huvF2/&F&6#\"\"'$!'v&Q%F2/&F&6#\"\"($\"\"!FJ/&F&6# \"\")$\"':[zF+" }}{PARA 11 "" 1 "" {XPPMATH 20 "6*%1order~5~weights:G/ &%#b*G6#\"\"\"$\"'8Nv!\"(/&F&6#\"\"$$\"'+zK!\"'/&F&6#\"\"%$\"'*p.#F2/& F&6#\"\"&$\"'J+oF2/&F&6#\"\"'$!'&)GOF2/&F&6#\"\"($\"'x3NF+/&F&6#\"\")$ \"'j\"3%F+" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/*&%)infinityG\"\"\"%8-n orm~of~linking~coeffsGF&$\"+XRm)3#!\"*" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/%L2-norm~of~principal~error~of~order~6~schemeG$\"+[[Gs@!#8" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#/%L2-norm~of~principal~error~of~order~ 5~schemeG$\"+W%Qe$=!#8" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/%;order~6~s tability~intervalG7$$!)mgKR!\"(\"\"!" }}{PARA 11 "" 1 "" {XPPMATH 20 " 6#/%;order~5~stability~intervalG7$$!)v3bW!\"(\"\"!" }}{PARA 11 "" 1 " " {XPPMATH 20 "6$/&%\"BG6#\"\"($\")%e9t\"!\"(/&%\"CGF&$\")(f)3;F*" }}} {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 79 "This sche me is obtained by simplifying the coefficients of the previous scheme. " }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 119 "c_2 := 47/193: c_3 := 9 /38: c_5 := 71/89: c_6 := 23/28: bb := [7,0]: bsA := [7,2/57]: bsB := \+ [8,2/49]:\ncalc_RKcoeffs();" }}{PARA 11 "" 1 "" {XPPMATH 20 "6(%'nodes :G/&%\"cG6#\"\"##\"#Z\"$$>/&F&6#\"\"$#\"\"*\"#Q/&F&6#\"\"%#\"$U$\"$$o/ &F&6#\"\"&#\"#r\"#*)/&F&6#\"\"'#\"#B\"#G" }}{PARA 11 "" 1 "" {XPPMATH 20 "6*%1order~6~weights:G/&%\"bG6#\"\"\"$\"'xru!\"(/&F&6#\"\"$$\"'\\5L !\"'/&F&6#\"\"%$\"'ce>F2/&F&6#\"\"&$\"'&Qc(F2/&F&6#\"\"'$!'\"[P%F2/&F& 6#\"\"($\"\"!FJ/&F&6#\"\")$\"'JZzF+" }}{PARA 11 "" 1 "" {XPPMATH 20 "6 *%1order~5~weights:G/&%#b*G6#\"\"\"$\"'(\\`(!\"(/&F&6#\"\"$$\"'3zK!\"' /&F&6#\"\"%$\"'sO?F2/&F&6#\"\"&$\"'7#z'F2/&F&6#\"\"'$!'Y?OF2/&F&6#\"\" ($\"'x3NF+/&F&6#\"\")$\"'j\"3%F+" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/* &%)infinityG\"\"\"%8-norm~of~linking~coeffsGF&$\"+&H6$*3#!\"*" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#/%L2-norm~of~principal~error~of~order~ 6~schemeG$\"+\")\\Gs@!#8" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/%L2-norm~ of~principal~error~of~order~5~schemeG$\"+!4XT$=!#8" }}{PARA 11 "" 1 " " {XPPMATH 20 "6#/%;order~6~stability~intervalG7$$!)mgKR!\"(\"\"!" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#/%;order~5~stability~intervalG7$$!)eRa W!\"(\"\"!" }}{PARA 11 "" 1 "" {XPPMATH 20 "6$/&%\"BG6#\"\"($\")k&>t\" !\"(/&%\"CGF&$\")Bw3;F*" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 " " 0 "" {TEXT -1 72 "Scheme with near maximal imaginary axis inclusion \+ from another worksheet" }}{EXCHG {PARA 11 "" 1 "" {XPPMATH 20 "6(%'nod es:G/&%\"cG6#\"\"##\"#Z\"$$>/&F&6#\"\"$#\"\"*\"#Q/&F&6#\"\"%#\"$U$\"$$ o/&F&6#\"\"&#\"$<#\"$s#/&F&6#\"\"'#\"%uF\"%xL" }}{PARA 11 "" 1 "" {XPPMATH 20 "6*%1order~6~weights:G/&%\"bG6#\"\"\"$\"'\"=Z(!\"(/&F&6#\" \"$$\"'Y5L!\"'/&F&6#\"\"%$\"'pe>F2/&F&6#\"\"&$\"'huvF2/&F&6#\"\"'$!'v& Q%F2/&F&6#\"\"($\"':[zF+/&F&6#\"\")$\"\"!FP" }}{PARA 11 "" 1 "" {XPPMATH 20 "6*%1order~5~weights:G/&%#b*G6#\"\"\"$\"'crt!\"(/&F&6#\"\" $$\"'FgL!\"'/&F&6#\"\"%$\"'sM=F2/&F&6#\"\"&$\"'Q+))F2/&F&6#\"\"'$!')Re &F2/&F&6#\"\"($\"'$yM%F+/&F&6#\"\")$\"'nmTF+" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/*&%)infinityG\"\"\"%8-norm~of~linking~coeffsGF&$\"+wU` @@!\"*" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/%L2-norm~of~principal~error ~of~order~6~schemeG$\"+[[Gs@!#8" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/%L 2-norm~of~principal~error~of~order~5~schemeG$\"+A:O#*R!#8" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/%;order~6~stability~intervalG7$$!)mgKR!\"(\"\" !" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/%;order~5~stability~intervalG7$$ !)Gv_S!\"(\"\"!" }}{PARA 11 "" 1 "" {XPPMATH 20 "6$/&%\"BG6#\"\"($\")# 3:\\\"!\"(/&%\"CGF&$\")z*Q^\"F*" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }} {PARA 0 "" 0 "" {TEXT -1 32 "#-------------------------------" }} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}}{PARA 0 "" 0 "" {TEXT -1 23 "#======================" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }} {PARA 0 "" 0 "" {TEXT -1 32 "#===============================" }} {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 28 "RK6_1 Prince-Dormand scheme" }}{PARA 0 " " 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 3385 "RK6 _1 := proc(fxy,x,y,xx,yy,h,stps,bb)\n local c2,c3,c4,c5,c6,c7,c8,a21 ,a31,a32,a41,a42,a43,a51,\n a52,a53,a54,a61,a62,a63,a64,a65,a71,a72, a73,a74,a75,a76,\n a81,a82,a83,a84,a85,a86,a87,b1,b2,b3,b4,b5,b6,b7, b8,\n f1,f2,f3,f4,f5,f6,f7,f8,t,k,fn,xk,yk,soln,eqns,A,saveDigits;\n saveDigits := Digits;\n Digits := max(trunc(evalhf(Digits)),Digit s+5);\n\n fn := unapply(fxy,x,y);\n\n A := matrix([[1/10,1/10,0,0, 0,0,0,0,0],\n [2/9,-2/81,20/81,0,0,0,0,0,0],\n [3/7,615/1372,- 270/343,1053/1372,0,0,0,0,0],\n [3/5,3243/5500,-54/55,50949/71500, 4998/17875,0,0,0,0],\n [4/5,-26492/37125,72/55, 2808/23375, -24206 /37125, 338/459,0,0,0],\n [1,5561/2376,-35/11,-24117/31603,899983/ 200772,-5225/1836,3925/4056,0,0],\n [1,465467/266112,-2945/1232,-5 610201/14158144,10513573/3212352,-424325/205632,\n 376225/ 454272,0,0],\n [0,61/864,0,98415/321776,16807/146016,1375/7344,137 5/5408,-37/1120,1/10]]);\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 := eva lf(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 := ev alf(A[5,3]);\n a63 := evalf(A[5,4]);\n a64 := evalf(A[5,5]);\n a 65 := 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 := eval f(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 := e valf(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 f or k from 1 to stps do\n f1 := fn(xk,yk);\n t := a21*f1;\n \+ f2 := fn(xk + c2*h,yk + t*h);\n t := a31*f1 + a32*f2;\n \+ f3 := fn(xk + c3*h,yk + t*h);\n t := a41*f1 + a42*f2 + a43*f3;\n \+ f4 := fn(xk + c4*h,yk + t*h);\n t := a51*f1 + a52*f2 + a53*f 3 + a54*f4;\n f5 := fn(xk + c5*h,yk + t*h);\n t := a61*f1 + \+ a62*f2 + a63*f3 + a64*f4 + a65*f5;\n f6 := fn(xk + c6*h,yk + t*h) ;\n t := a71*f1 + a72*f2 + a73*f3 + a74*f4 + a75*f5 + a76*f6;\n \+ f7 := fn(xk + c7*h,yk + t*h);\n t := a81*f1 + a82*f2 + a83*f3 + a84*f4 + a85*f5 + a86*f6 + a87*f7;\n f8 := fn(xk + c8*h,yk + t *h);\n\n yk := yk + (b1*f1 + b2*f2 + b3*f3 + b4*f4 + b5*f5 + b6*f 6 + 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_=f xy,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,a5 2_=a52,a53_=a53,a54_=a54,\n a61_=a61,a62_=a62,a63_=a63,a64_=a6 4,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 ret urn 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 37 "RK6_2 modified Prince-Dormand sc heme" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 3665 "RK6_2 := proc(fxy,x,y,xx,yy,h,stps,bb)\n local c2 ,c3,c4,c5,c6,c7,c8,a21,a31,a32,a41,a42,a43,a51,\n a52,a53,a54,a61,a6 2,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,sol n,eqns,A,saveDigits;\n saveDigits := Digits;\n Digits := max(trunc (evalhf(Digits)),Digits+5);\n\n fn := unapply(fxy,x,y);\n\n A := m atrix([[7/39,7/39,0,0,0,0,0,0,0],\n [2/9,16/189,26/189,0,0,0, 0,0,0],\n [3/7,957/9604,-1053/2401,1053/1372,0,0,0,0,0],\n \+ [23/33,2563741/11068596,-18239/102487,3243/761332,3284078/51389 91,0,0,0,0],\n [24/31,-11597952/148686881,92664/208537,987409 44/564271331,-26004300/372178963,\n 9368775900/30 948112231,0,0,0],\n [1,38665819/91808640,-897/1232,156399/150 5504,1592286101/1436292000,\n -2279466607/29650532 80,972169703/1126224000,0,0],\n [1,118627013/607606272,-1527/ 3136,26560509/49818496,576719677/1357948800,\n -61 16292391/16604298368,5233891417/7453555200,0,0],\n [0,14459/1 98720,0,19683/68432,8252237/43524000,143496441/1058947600,\n \+ 28629151/119448000,11/1120,13/200]]);\n\n c2 := evalf(A[ 1,1]);\n c3 := evalf(A[2,1]);\n c4 := evalf(A[3,1]);\n c5 := eva lf(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 := e valf(A[2,3]);\n a41 := evalf(A[3,2]);\n a42 := evalf(A[3,3]);\n \+ a43 := evalf(A[3,4]);\n a51 := evalf(A[4,2]);\n a52 := evalf(A[4,3 ]);\n a53 := evalf(A[4,4]);\n a54 := evalf(A[4,5]);\n a61 := eva lf(A[5,2]);\n a62 := evalf(A[5,3]);\n a63 := evalf(A[5,4]);\n a6 4 := evalf(A[5,5]);\n a65 := evalf(A[5,6]);\n a71 := evalf(A[6,2]) ;\n a72 := evalf(A[6,3]);\n a73 := evalf(A[6,4]);\n a74 := evalf (A[6,5]);\n a75 := evalf(A[6,6]);\n a76 := evalf(A[6,7]);\n 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 21 "RK6_3 Verner (2004) " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> \+ " 0 "" {MPLTEXT 1 0 3345 "RK6_3 := proc(fxy,x,y,xx,yy,h,stps,bb)\n l ocal c2,c3,c4,c5,c6,c7,c8,a21,a31,a32,a41,a42,a43,a51,\n a52,a53,a54 ,a61,a62,a63,a64,a65,a71,a72,a73,a74,a75,a76,\n a81,a82,a83,a84,a85, a86,a87,b1,b2,b3,b4,b5,b6,b7,b8,\n f1,f2,f3,f4,f5,f6,f7,f8,t,k,fn,xk ,yk,soln,eqns,A,saveDigits;\n saveDigits := Digits;\n Digits := ma x(trunc(evalhf(Digits)),Digits+5);\n\n fn := unapply(fxy,x,y);\n\n \+ A := matrix([[1/7,1/7,0,0,0,0,0,0,0],\n [2/9,4/81,14/81,0,0,0,0 ,0,0],\n [3/7,291/1372,-27/49,1053/1372,0,0,0,0,0],\n [2/3 ,86/297,-14/33,42/143,1960/3861,0,0,0,0],\n [3/4,-267/22528,189/ 704,63099/585728,58653/366080,4617/20480,0,0,0],\n [1,10949/6912 ,-69/32,-90891/68096,112931/25920,-69861/17920,26378/10773,0,0],\n \+ [1,1501/19008,-21/88,219519/347776,163807/926640,-417/640,1544/1539 ,0,0],\n [0,79/1080,0,19683/69160,16807/84240,0,2816/7695,1/100, 187/2800]]);\n\n c2 := evalf(A[1,1]);\n c3 := evalf(A[2,1]);\n c 4 := 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 := eva lf(A[4,2]);\n a52 := evalf(A[4,3]);\n a53 := evalf(A[4,4]);\n a5 4 := evalf(A[4,5]);\n a61 := evalf(A[5,2]);\n a62 := evalf(A[5,3]) ;\n a63 := evalf(A[5,4]);\n a64 := evalf(A[5,5]);\n a65 := evalf (A[5,6]);\n a71 := evalf(A[6,2]);\n a72 := evalf(A[6,3]);\n a73 \+ := evalf(A[6,4]);\n a74 := evalf(A[6,5]);\n a75 := evalf(A[6,6]); \n a76 := evalf(A[6,7]);\n a81 := evalf(A[7,2]);\n a82 := evalf( A[7,3]);\n a83 := evalf(A[7,4]);\n a84 := evalf(A[7,5]);\n a85 : = evalf(A[7,6]);\n a86 := evalf(A[7,7]);\n a87 := evalf(A[7,8]);\n 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(x k + 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 + a 63*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_=a3 1,a32_=a32,a41_=a41,\n a42_=a42,a43_=a43,a51_=a51,a52_=a52,a53 _=a53,a54_=a54,\n a61_=a61,a62_=a62,a63_=a63,a64_=a64,a65_=a65 ,\n a71_=a71,a72_=a72,a73_=a73,a74_=a74,a75_=a75,a76_=a76,\n \+ a81_=a81,a82_=a82,a83_=a83,a84_=a84,a85_=a85,a86_=a86,a87_=a87, \n b1_=b1,b2_=b2,b3_=b3,b4_=b4,b5_=b5,b6_=b6,b7_=b7,b8_=b8\}; \n return subs(eqns,eval(rk6step)); \n 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 37 "RK6_4 modified Verner (2004) scheme" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 3394 "RK6_4 := proc(fxy,x,y,xx,yy,h,stps,bb)\n local c2,c3,c4,c5,c 6,c7,c8,a21,a31,a32,a41,a42,a43,a51,\n a52,a53,a54,a61,a62,a63,a64,a 65,a71,a72,a73,a74,a75,a76,\n a81,a82,a83,a84,a85,a86,a87,b1,b2,b3,b 4,b5,b6,b7,b8,\n f1,f2,f3,f4,f5,f6,f7,f8,t,k,fn,xk,yk,soln,eqns,A,sa veDigits;\n saveDigits := Digits;\n Digits := max(trunc(evalhf(Dig its)),Digits+5);\n\n fn := unapply(fxy,x,y);\n\n A := matrix([[1/6 ,1/6,0,0,0,0,0,0,0],\n [2/9,2/27,4/27,0,0,0,0,0,0],\n [3/7 ,183/1372,-162/343,1053/1372,0,0,0,0,0],\n [11/16,1475/6144,-513 /2048,5427/53248,95207/159744,0,0,0,0],\n [3/4,3/88,81/352,44631 /613184,58653/265408,4104/21373,0,0,0],\n [1,28909/24948,-120/77 ,-29605749/25485460,17354309/5038605,-21971456/6732495,384688/161595,0 ,0],\n [1,817/39204,-30/121,31908963/40048580,-1085056/55424655, -5961728/10579635,257344/253935,0,0],\n [0,79/1080,0,19683/69160 ,16807/84240,0,2816/7695,1/100,187/2800]]);\n\n c2 := evalf(A[1,1]); \n c3 := evalf(A[2,1]);\n c4 := evalf(A[3,1]);\n c5 := evalf(A[4 ,1]);\n c6 := evalf(A[5,1]);\n c7 := evalf(A[6,1]);\n c8 := eval f(A[7,1]);\n 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 := e valf(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 := eva lf(A[7,2]);\n a82 := evalf(A[7,3]);\n a83 := evalf(A[7,4]);\n a8 4 := 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 := ev alf(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*f 1 + 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 + a 75*f5 + a76*f6;\n f7 := fn(xk + c7*h,yk + t*h);\n t := a81*f 1 + 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 + b 4*f4 + b5*f5 + b6*f6 + b7*f7 + b8*f8)*h;\n xk := xk + h:\n s oln := 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,a 43_=a43,a51_=a51,a52_=a52,a53_=a53,a54_=a54,\n a61_=a61,a62_=a 62,a63_=a63,a64_=a64,a65_=a65,\n a71_=a71,a72_=a72,a73_=a73,a7 4_=a74,a75_=a75,a76_=a76,\n a81_=a81,a82_=a82,a83_=a83,a84_=a8 4,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 pr oc: " }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 41 "RK6_5 2nd modified Verner (2004) scheme" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 3661 "RK6_5 := 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([[5/28,5/28,0,0,0,0,0,0,0],\n \+ [2/9,34/405,56/405,0,0,0,0,0,0],\n [3/7,699/6860,-108/245,105 3/1372,0,0,0,0,0],\n [37/53,22960313/98258820,-1407924/818823 5,-680319/85157644,41155100/63868233,0,0,0,0],\n [27/35,-2269 1367/349002500,28188/67375,31655367/194652500,-37368/1203125,\n \+ 141862511268/495147296875,0,0,0],\n [1,108451463/92067840, -4263/2816,-77302059309/57505310720,3338370539/988416000,\n \+ -154367434883/70954752000,63635075/43047936,0,0],\n \+ [1,31846/494505,-42/121,241942482/308866415,625387/21235500,-451380176 3/25915114500,\n 10127075/15722586,0,0],\n \+ [0,13093/179820,0,2932767/10210460,2144093/11232000,4600150423/36552 448000,\n 181575625/731814912,1/100,3333/51200]]); \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 := ev alf(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 := ev alf(A[6,7]);\n a81 := evalf(A[7,2]);\n a82 := evalf(A[7,3]);\n a 83 := 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 b 7 := 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,y k + 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 + a 72*f2 + a73*f3 + a74*f4 + a75*f5 + a76*f6;\n f7 := fn(xk + c7*h,y k + 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 a7 1_=a71,a72_=a72,a73_=a73,a74_=a74,a75_=a75,a76_=a76,\n a81_=a8 1,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]([s oln]);\n end if;\nend proc: " }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}}{SECT 1 {PARA 4 "" 0 " " {TEXT -1 51 "RK6_6 scheme with maximal imaginary axis inclusion" }} {PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 4267 "RK6_6 := proc(fxy,x,y,xx,yy,h,stps,bb)\n local c2,c3,c4,c5,c6, c7,c8,a21,a31,a32,a41,a42,a43,a51,\n a52,a53,a54,a61,a62,a63,a64,a65 ,a71,a72,a73,a74,a75,a76,\n a81,a82,a83,a84,a85,a86,a87,b1,b2,b3,b4, b5,b6,b7,b8,\n f1,f2,f3,f4,f5,f6,f7,f8,t,k,fn,xk,yk,soln,eqns,A,save Digits;\n saveDigits := Digits;\n Digits := max(trunc(evalhf(Digit s)),Digits+5);\n\n fn := unapply(fxy,x,y);\n\n A := matrix([[47/19 3,47/193,0,0,0,0,0,0,0],\n [9/38,16515/135736,15633/135736,0, 0,0,0,0,0],\n [342/683,-695302416/14974763389,-9469814814/149 74763389,375818328/318611987,0,0,0,0,0],\n [71/89,33509869292 201/85055527825236,3587459103/4605562477,\n -21724507 215628/18120732551793,1134395721190865/1377175673936268,0,0,0,0],\n \+ [23/28,233436361745853683/373836194308813824,158645421/14341241 6,\n -2222985817851376045/1063980184927185792,8530434 5532627205619/66923624494207595520,\n -87266602959880 3/9228174667747840,0,0,0],\n [1,-1178762513361663864779/66490 79911196633209152,-8522902869921/10868389582568,\n 3 087163579546988168202379/1791201064146135353344584,\n \+ -3767364590077053611426162077/10220229771451535362575529152,\n \+ 17701204856780353690623/13557389052385415225632,-767896311 0907452/10984484330835583,0,0],\n [1,1416449588677763/6015481 297132284,-48943449/199463911,\n 83884485628625980/4 63004404677327933,2690685696370602054377/6604522931171002998060,\n \+ 855085588386243786/608893946374021595,-1079767263472128 0/10984484330835583,0,0],\n [0,2503727/33509160,0,11637665300 0/351539506521,38494907685743537/196546626900894600,\n \+ 117203823775061/154952625580200,-84924560896/194121869325,0,4243913 /53400600]]);\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 := eva lf(A[4,2]);\n a52 := evalf(A[4,3]);\n a53 := evalf(A[4,4]);\n a5 4 := evalf(A[4,5]);\n a61 := evalf(A[5,2]);\n a62 := evalf(A[5,3]) ;\n a63 := evalf(A[5,4]);\n a64 := evalf(A[5,5]);\n a65 := evalf (A[5,6]);\n a71 := evalf(A[6,2]);\n a72 := evalf(A[6,3]);\n a73 \+ := evalf(A[6,4]);\n a74 := evalf(A[6,5]);\n a75 := evalf(A[6,6]); \n a76 := evalf(A[6,7]);\n a81 := evalf(A[7,2]);\n a82 := evalf( A[7,3]);\n a83 := evalf(A[7,4]);\n a84 := evalf(A[7,5]);\n a85 : = evalf(A[7,6]);\n a86 := evalf(A[7,7]);\n a87 := evalf(A[7,8]);\n 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(x k + 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 + a 63*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_=a3 1,a32_=a32,a41_=a41,\n a42_=a42,a43_=a43,a51_=a51,a52_=a52,a53 _=a53,a54_=a54,\n a61_=a61,a62_=a62,a63_=a63,a64_=a64,a65_=a65 ,\n a71_=a71,a72_=a72,a73_=a73,a74_=a74,a75_=a75,a76_=a76,\n \+ a81_=a81,a82_=a82,a83_=a83,a84_=a84,a85_=a85,a86_=a86,a87_=a87, \n b1_=b1,b2_=b2,b3_=b3,b4_=b4,b5_=b5,b6_=b6,b7_=b7,b8_=b8\}; \n return subs(eqns,eval(rk6step)); \n 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 ";" }}}} {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 5 method for error correction." }} {SECT 1 {PARA 4 "" 0 "" {TEXT -1 46 "Test 1 of 8 stage, order 6 Runge- Kutta methods" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "dy/dx=12*x*cos(4*x)*exp(-x)*y" "6#/*&%# dyG\"\"\"%#dxG!\"\"*,\"#7F&%\"xGF&-%$cosG6#*&\"\"%F&F+F&F&-%$expG6#,$F +F(F&%\"yGF&" }{TEXT -1 6 ", " }{XPPEDIT 18 0 "y(0)=1" "6#/-%\"yG6 #\"\"!\"\"\"" }{TEXT -1 1 " " }}{PARA 0 "" 0 "" {TEXT -1 10 "Solution: " }}{PARA 256 "" 0 "" {TEXT -1 2 " " }{XPPEDIT 18 0 "y=exp(-12/17*x* cos(4*x)*exp(-x)+180/289*exp(-x)*cos(4*x)+48/17*exp(-x)*sin(4*x)*x+96/ 289*exp(-x)*sin(4*x)-180/289)" "6#/%\"yG-%$expG6#,,*,\"#7\"\"\"\"# " 0 "" {MPLTEXT 1 0 229 "de := diff(y(x),x)=12*x*cos(4*x)*exp(-x)*y(x); \nic := y(0)=1;\ndsolve(\{de,ic\},y(x)):\ny(x)=simplify(numer(rhs(%))/ convert(denom(rhs(%)),exp));\nf := unapply(rhs(%),x):\nplot(f(x),x=0.. 5,0..1.45,font=[HELVETICA,9],labels=[`x`,`y(x)`]);" }}{PARA 11 "" 1 " " {XPPMATH 20 "6#>%#deG/-%%diffG6$-%\"yG6#%\"xGF,,$*,\"#7\"\"\"F,F0-%$ cosG6#,$*&\"\"%F0F,F0F0F0-%$expG6#,$F,!\"\"F0F)F0F0" }}{PARA 11 "" 1 " " {XPPMATH 20 "6#>%#icG/-%\"yG6#\"\"!\"\"\"" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/-%\"yG6#%\"xG-%$expG6#,,*&#\"#7\"#<\"\"\"*(F'F0-%$cosG 6#,$*&\"\"%F0F'F0F0F0-F)6#,$F'!\"\"F0F0F;*&#\"$!=\"$*GF0*&F2F0F8F0F0F0 *&#\"#[F/F0*(F8F0-%$sinGF4F0F'F0F0F0*&#\"#'*F?F0*&F8F0FEF0F0F0#F>F?F; " }}{PARA 13 "" 1 "" {GLPLOT2D 577 286 286 {PLOTDATA 2 "6&-%'CURVESG6$ 7er7$$\"\"!F)$\"\"\"F)7$$\"3gmmTN@Ki8!#>$\"3Fk>e\"G.6+\"!#<7$$\"3ALL$3 FWYs#F/$\"3!H*fm:2P/5F27$$\"3%)***\\iSmp3%F/$\"3Qn()\\Dat45F27$$\"3Wmm mT&)G\\aF/$\"34$Q7t`Dr,\"F27$$\"3m****\\7G$R<)F/$\"3S2-*\\9jw.\"F27$$ \"3GLLL3x&)*3\"!#=$\"3U([#>C\\El5F27$$\"3))**\\i!R(*Rc\"FJ$\"3>&=^@[0u 7\"F27$$\"3umm\"H2P\"Q?FJ$\"3k\\#o#G?)=?\"F27$$\"3!***\\PMnNrDFJ$\"3s_ j<)f!R*G\"F27$$\"3MLL$eRwX5$FJ$\"37'\\4u:c`O\"F27$$\"3_LLe*[`HP$FJ$\"3 [!\\'y0#yNR\"F27$$\"3rLLL$eI8k$FJ$\"3N\"Ha_9o@T\"F27$$\"3_L$3-8>bx$FJ$ \"3@))>@pAD<9F27$$\"3*QL$3xwq4RFJ$\"3a@g!fsi#>9F27$$\"3EM$eRA'*Q/%FJ$ \"3^DvP/8/=9F27$$\"33ML$3x%3yTFJ$\"3bF0p:\"oMT\"F27$$\"3h+]PfyG7ZFJ$\" 3e=U+Y19h8F27$$\"3emm\"z%4\\Y_FJ$\"3Yii#4W6uD\"F27$$\"3'QLL3FGT\\&FJ$ \"3c!QStI8]>\"F27$$\"32++v$flWv*FJ7$$\"3I++vVVX$\\'FJ$ \"3w/21T*\\F&*)FJ7$$\"31nm\"zWo)\\nFJ$\"3E>3;k'H:;)FJ7$$\"3%QL$3_DG1qF J$\"31le1yn9(R(FJ7$$\"3]***\\il'pisFJ$\"3E!)4GzFfsmFJ7$$\"3+MLe*[!)y_( FJ$\"3CJpN=**=vfFJ7$$\"3Qnm\"HKkIz(FJ$\"3'oU:>LtrL&FJ7$$\"3!3+]i:[#e!) 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ArFJ7$$\"3e++]iC$pk$F2$\"3ma\\oRiHQqFJ7$$\"3ILe*[t\\sp$F2$\"3'e9/wG(3M oFJ7$$\"3[m;H2qcZPF2$\"3CYQ8S*3be'FJ7$$\"3O+]7.\"fF&QF2$\"3**Q8E[N&3+' FJ7$$\"3Ymm;/OgbRF2$\"3kN#z0%oN^aFJ7$$\"3w**\\ilAFjSF2$\"3[i8#)*p//*\\ FJ7$$\"3ym\"zW7@^6%F2$\"3>C%QCunR#[FJ7$$\"3yLLL$)*pp;%F2$\"3g*yCm#3E'p %FJ7$$\"3)QL3-$H**>UF2$\"3$*o:W?mr0YFJ7$$\"3)RL$3xe,tUF2$\"3!\\Bp&*))o Xb%FJ7$$\"3h+v=n(*fDVF2$\"3kIpK$)H$3a%FJ7$$\"3Cn;HdO=yVF2$\"3u&G6!oNOh XFJ7$$\"3MMe9\"z-lU%F2$\"3kC\">#=Lu2YFJ7$$\"3a+++D>#[Z%F2$\"3w_(eqj7vn %FJ7$$\"3SnmT&G!e&e%F2$\"3W>T$>g**p!\\FJ7$$\"3#RLLL)Qk%o%F2$\"3'yDBP_q :;&FJ7$$\"37+]iSjE!z%F2$\"3J;fP@m(pV&FJ7$$\"3a+]P40O\"*[F2$\"3!>+$=fU- gcFJ7$$\"\"&F)$\"3h(Q0fOqh\"eFJ-%'COLOURG6&%$RGBG$\"#5!\"\"F(F(-%%FONT G6$%*HELVETICAG\"\"*-%+AXESLABELSG6$%\"xG%%y(x)G-%%VIEWG6$;F(F]am;F($ \"$X\"!\"#" 1 2 0 1 10 0 2 9 1 4 2 1.000000 45.000000 45.000000 0 0 "C urve 1" }}}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 32 "The following code constructs a " }{TEXT 260 17 "discrete solution " }{TEXT -1 44 " based on each of the methods and gives the " }{TEXT 260 22 "root mean square error" }{TEXT -1 18 " of each solution." }} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 786 "F := (x,y) -> 12*x*cos(4*x) *exp(-x)*y: hh := 0.01: numsteps := 500: x0 := 0: y0 := 1:\nmatrix([[` slope field: `,F(x,y)],[`initial point: `,``(x0,y0)],[`step width: \+ `,hh],\n[`no. of steps: `,numsteps]]);``;\nmthds := [`Prince-Dorman d scheme`,`a modification of the Prince-Dormand scheme`,`Verner's (200 4) scheme`,`a modification of Verner's scheme`,`a 2nd modification of \+ Verner's scheme`,`scheme with maximal imaginary axis inclusion`]: errs := []:\nDigits := 25:\nfor ct to 6 do\n Fn_RK6_||ct := RK6_||ct(F(x ,y),x,y,x0,y0,hh,numsteps,false);\n sm := 0: numpts := nops(Fn_RK6_| |ct):\n for ii to numpts do\n sm := sm+(Fn_RK6_||ct[ii,2]-evalf (f(Fn_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,$*,\"#7\"\"\"%\"xGF,-%$cosG6#,$*&\"\"%F,F-F,F,F,-%$expG6#,$F-!\"\" F,%\"yGF,F,7$%0initial~point:~G-%!G6$\"\"!F,7$%/step~width:~~~G$F,!\"# 7$%1no.~of~steps:~~~G\"$+&Q)pprint606\"" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%!G" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#K%'matrixG6#7(7$%6Prince- Dormand~schemeG$\"+_zi2?!#A7$%La~modification~of~the~Prince-Dormand~sc hemeG$\"+s&)3_EF+7$%7Verner's~(2004)~schemeG$\"+h$Qaa#F+7$%Ba~modifica tion~of~Verner's~schemeG$\"+Z&odq#F+7$%Fa~2nd~modification~of~Verner's ~schemeG$\"+![nXm#F+7$%Mscheme~with~maximal~imaginary~axis~inclusionG$ \"+W&QKq#F+Q)pprint616\"" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 " " 0 "" {TEXT -1 30 "The following code constructs " }{TEXT 260 20 "num erical procedures" }{TEXT -1 56 " for solutions based on each of the R unge-Kutta schemes." }}{PARA 0 "" 0 "" {TEXT -1 75 "The error in the v alue obtained by each of the methods at the point where " }{XPPEDIT 18 0 "x = 4.999;" "6#/%\"xG-%&FloatG6$\"%**\\!\"$" }{TEXT -1 16 " is \+ also given." }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 710 "F := (x,y) - > 12*x*cos(4*x)*exp(-x)*y: hh := 0.01: numsteps := 500: x0 := 0: y0 := 1:\nmatrix([[`slope field: `,F(x,y)],[`initial point: `,``(x0,y0)], [`step width: `,hh],\n[`no. of steps: `,numsteps]]);``;\nmthds := \+ [`Prince-Dormand scheme`,`a modification of the Prince-Dormand scheme` ,`Verner's (2004) scheme`,`a modification of Verner's scheme`,`a 2nd m odification of Verner's scheme`,`scheme with maximal imaginary axis in clusion`]: errs := []:\nDigits := 25:\nfor ct to 6 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 6 do\n errs := [op(errs),abs(fn_RK6 _||ct(xx)-fxx)];\nend do:\nDigits := 10:\nlinalg[transpose]([mthds,eva lf(errs)]);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#K%'matrixG6#7&7$%0slope ~field:~~~G,$*,\"#7\"\"\"%\"xGF,-%$cosG6#,$*&\"\"%F,F-F,F,F,-%$expG6#, $F-!\"\"F,%\"yGF,F,7$%0initial~point:~G-%!G6$\"\"!F,7$%/step~width:~~~ G$F,!\"#7$%1no.~of~steps:~~~G\"$+&Q)pprint106\"" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%!G" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#K%'matrixG6#7(7$ %6Prince-Dormand~schemeG$\"+kD&z+#!#A7$%La~modification~of~the~Prince- Dormand~schemeG$\"+gwAZGF+7$%7Verner's~(2004)~schemeG$\"+WE5;FF+7$%Ba~ modification~of~Verner's~schemeG$\"+ChB;HF+7$%Fa~2nd~modification~of~V erner's~schemeG$\"+.i3jGF+7$%Mscheme~with~maximal~imaginary~axis~inclu sionG$\"+5!eE)GF+Q)pprint116\"" }}}{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 0 454 "mthds := [`Prince-Dormand scheme`,`a modification of the Prin ce-Dormand scheme`,`Verner's (2004) scheme`,`a modification of Verner' s scheme`,`a 2nd modification of Verner's scheme`,`scheme with maximal imaginary axis inclusion`]: errs := []:\nDigits := 20:\nfor ct to 6 d o\n sm := NCint((f(x)-'fn_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$%6Prince-Dormand~schemeG$\"+S;\\3?!#A 7$%La~modification~of~the~Prince-Dormand~schemeG$\"+/gw_EF+7$%7Verner' s~(2004)~schemeG$\"+(pCja#F+7$%Ba~modification~of~Verner's~schemeG$\"+ *eklq#F+7$%Fa~2nd~modification~of~Verner's~schemeG$\"+RQDlEF+7$%Mschem e~with~maximal~imaginary~axis~inclusionG$\"+k8n.FF+Q)pprint126\"" }}} {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 92 "The follo wing error graphs are constructed using the numerical procedures for t he solutions." }}{EXCHG {PARA 0 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AXGF-7$F\\y$\")^@_GF-7$Fay$\")mtfGF-7$Ffy$\")3-mGF-7$F[z$ \")5CsGF-7$F`z$\")btxGF-7$Fez$\")y6$)GF--F[[l6&F][l$\"#&*FaelF`[l$\"#: Fael-Fd[l6#%Mscheme~with~maximal~imaginary~axis~inclusionG-%%FONTG6$%* HELVETICAG\"\"*-%&TITLEG6#%Uerror~curves~for~8~stage~order~6~Runge-Kut ta~methodsG-%+AXESLABELSG6$Q\"x6\"Q!Fg]o-%%VIEWG6$;F(Fez%(DEFAULTG" 1 2 0 1 10 0 2 9 1 4 2 1.000000 45.000000 45.000000 0 1 "Prince-Dormand \+ scheme" "a modification of the Prince-Dormand scheme" "Verner's (2004) scheme" "a modification of Verner's scheme" "a 2nd modification of Ve rner's scheme" "scheme with maximal imaginary axis inclusion" }}}} {PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 46 "Test 2 of 8 stage, orde r 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-%%sqrt G6#,&\"\"\"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 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 0 789 "G := (x,y) -> x/y: hh := 0.05: numsteps := 200: x0 := 0: y0 : = 1:\nmatrix([[`slope field: `,G(x,y)],[`initial point: `,``(x0,y0)] ,[`step width: `,hh],\n[`no. of steps: `,numsteps]]);``;\nmthds := [`Prince-Dormand scheme`,`a modification of the Prince-Dormand scheme `,`Verner's (2004) scheme`,`a modification of Verner's scheme`,`a 2nd \+ modification of Verner's scheme`,`scheme with maximal imaginary axis i nclusion`]: errs := []:\nDigits := 20:\ng := x -> sqrt(1+x^2):\nfor ct to 6 do\n Gn_RK6_||ct := RK6_||ct(G(x,y),x,y,x0,y0,hh,numsteps,fals e);\n sm := 0: numpts := nops(Gn_RK6_||ct):\n for ii to numpts do \n sm := sm+(Gn_RK6_||ct[ii,2]-evalf(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)pprint626\"" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#% !G" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#K%'matrixG6#7(7$%6Prince-Dormand ~schemeG$\"+Py(R[)!#B7$%La~modification~of~the~Prince-Dormand~schemeG$ \"+)yK@$fF+7$%7Verner's~(2004)~schemeG$\"++&QcA*F+7$%Ba~modification~o f~Verner's~schemeG$\"+%3=ks(F+7$%Fa~2nd~modification~of~Verner's~schem eG$\"+/RUifF+7$%Mscheme~with~maximal~imaginary~axis~inclusionG$\"+0LHo 8F+Q)pprint636\"" }}}{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 "9. 99;" "6#-%&FloatG6$\"$***!\"#" }{TEXT -1 16 " is also given." }} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 712 "G := (x,y) -> x/y: hh := 0. 05: numsteps := 200: x0 := 0: y0 := 1:\nmatrix([[`slope field: `,G(x ,y)],[`initial point: `,``(x0,y0)],[`step width: `,hh],\n[`no. of st eps: `,numsteps]]);``;\nmthds := [`Prince-Dormand scheme`,`a modific ation of the Prince-Dormand scheme`,`Verner's (2004) scheme`,`a modifi cation of Verner's scheme`,`a 2nd modification of Verner's scheme`,`sc heme with maximal imaginary axis inclusion`]: errs := []:\nDigits := 3 0:\nfor ct to 6 do\n gn_RK6_||ct := RK6_||ct(G(x,y),x,y,x0,y0,hh,num steps,true);\nend do:\ng := x -> sqrt(1+x^2):\nxx := 9.99: gxx := eval f(g(xx)):\nfor ct to 6 do\n errs := [op(errs),abs(gn_RK6_||ct(xx)-gx x)];\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)pprint156\"" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%!G" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#K%'matrixG6 #7(7$%6Prince-Dormand~schemeG$\"+6\"3]4#!#B7$%La~modification~of~the~P rince-Dormand~schemeG$\"+;'>?S\"F+7$%7Verner's~(2004)~schemeG$\"+qF^EB F+7$%Ba~modification~of~Verner's~schemeG$\"+@`5k=F+7$%Fa~2nd~modificat ion~of~Verner's~schemeG$\"+'4q\"*R\"F+7$%Mscheme~with~maximal~imaginar y~axis~inclusionG$\"+T#e6>&!#CQ)pprint166\"" }}}{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, 1 0]" "6#7$\"\"!\"#5" }{TEXT -1 82 " of each Runge-Kutta method is esti mated 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 0 479 "mthds := [`Prince-Dormand scheme`,`a modification of the Prince-Dormand scheme`,`Verner's (2004) scheme`,`a modification o f Verner's scheme`,`a 2nd modification of Verner's scheme`,`scheme wit h maximal imaginary axis inclusion`]: errs := []:\nDigits := 20:\ng := x -> sqrt(1+x^2):\nfor ct to 6 do\n sm := NCint((g(x)-'gn_RK6_||ct' (x))^2,x=0..10,adaptive=false,numpoints=7,factor=100);\n errs := [op (errs),sqrt(sm/10)];\nend do:\nDigits := 10:\nlinalg[transpose]([mthds ,evalf(errs)]);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#K%'matrixG6#7(7$%6P rince-Dormand~schemeG$\"+k0/e%)!#B7$%La~modification~of~the~Prince-Dor mand~schemeG$\"+6&)*Q\"fF+7$%7Verner's~(2004)~schemeG$\"+hz[(>*F+7$%Ba ~modification~of~Verner's~schemeG$\"+%*Qs-xF+7$%Fa~2nd~modification~of ~Verner's~schemeG$\"+t`3WfF+7$%Mscheme~with~maximal~imaginary~axis~inc lusionG$\"+'=&ok8F+Q)pprint176\"" }}}{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 0 574 "evalf[20](plot([g(x)-'gn_RK6_1'(x),g(x)-'g n_RK6_2'(x),g(x)-'gn_RK6_3'(x),\ng(x)-'gn_RK6_4'(x),g(x)-'gn_RK6_5'(x) ,g(x)-'gn_RK6_6'(x)],x=0..10,color=[COLOR(RGB,.4,0,1),\nCOLOR(RGB,0,.6 5,0),COLOR(RGB,1,.5,0),COLOR(RGB,.85,0,.85),COLOR(RGB,0,.3,1),COLOR(RG B,.95,0,.15)],\nlegend=[`Prince-Dormand scheme`,`a modification of the Prince-Dormand scheme`,`Verner's (2004) scheme`,`a modification of Ve rner's scheme`,`a 2nd modification of Verner's scheme`,`scheme with ma ximal imaginary axis inclusion`],font=[HELVETICA,9],title=`error curve s for 8 stage order 6 Runge-Kutta methods`));" }}{PARA 13 "" 1 "" {GLPLOT2D 943 574 574 {PLOTDATA 2 "6,-%'CURVESG6%7dq7$$\"\"!F)F(7$$\"5 lmmmmT&)G\\a!#@$\"&'Rf!#>7$$\"5LLLLL3x&)*3\"!#?$\"'**3DF07$$\"5mmmT5:Q (z:\"F4$\"'%z]#F07$$\"5+++](=#**3E7F4$\"'%e^#F07$$\"5LLLekGg?%H\"F4$\" 'JsDF07$$\"5mmmmTN@Ki8F4$\"'F4$\"'%HK'F07$$\"5mmmTN' 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(=f\"F07$Ffal$!'\"R\\\"F07$F[bl$!'[>9F07$F`bl$!'NT8F07$Febl$!'yy7F07$F jbl$!'K;7F07$F_cl$!'6i6F07$Fdcl$!'M56F07$Ficl$!'im5F07$F^dl$!'5B5F07$F cdl$!&T\")*F07$Fhdl$!&wZ*F07$F]el$!&'Q\"*F07$Fbel$!&F\"))F07$Fgel$!&_^ )F07$F\\fl$!&eC)F07$Fafl$!&b'zF07$Fffl$!&%HxF07$F[gl$!&?\\(F07$F`gl$!& \"*G(F07$Fegl$!&$zqF07$Fjgl$!&E*oF07$F_hl$!&wq'F07$Fdhl$!&g`'F07$Fihl$ !&aO'F07$F^il$!&$4iF07$Fcil$!&t0'F07$Fhil$!&P\"fF07$F]jl$!&wy&F07$Fbjl $!&&\\cF07$Fgjl$!&8`&F07$F\\[m$!&1T&F07$Fa[m$!&+I&F07$Ff[m$!%'=&Fj[m-F \\\\m6&F^\\m$\"#&*F^anF($\"#:F^an-Fe\\m6#%Mscheme~with~maximal~imagina ry~axis~inclusionG-%%FONTG6$%*HELVETICAG\"\"*-%&TITLEG6#%Uerror~curves ~for~8~stage~order~6~Runge-Kutta~methodsG-%+AXESLABELSG6$Q\"x6\"Q!Fhis -%%VIEWG6$;F(Ff[m%(DEFAULTG" 1 2 0 1 10 0 2 9 1 4 2 1.000000 45.000000 45.000000 0 1 "Prince-Dormand scheme" "a modification of the Prince-Dormand scheme" "Verner's (2004) scheme" "a modification of Ve rner's scheme" "a 2nd modification of Verner's scheme" "scheme with ma ximal imaginary axis inclusion" }}}}{PARA 0 "" 0 "" {TEXT -1 0 "" }} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}}{SECT 1 {PARA 4 "" 0 " " {TEXT -1 46 "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 methods and gives the " }{TEXT 260 22 "root mean square error" }{TEXT -1 18 " of each solution." }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 789 "H := \+ (x,y) -> -x*y: hh := 0.1: numsteps := 100: x0 := 0: y0 := 1:\nmatrix([ [`slope field: `,H(x,y)],[`initial point: `,``(x0,y0)],[`step width: `,hh],\n[`no. of steps: `,numsteps]]);``;\nmthds := [`Prince-Dorm and scheme`,`a modification of the Prince-Dormand scheme`,`Verner's (2 004) scheme`,`a modification of Verner's scheme`,`a 2nd modification o f Verner's scheme`,`scheme with maximal imaginary axis inclusion`]: er rs := []:\nDigits := 20:\nh := x -> exp(-x^2/2):\nfor ct to 6 do\n H n_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 := s m+(Hn_RK6_||ct[ii,2]-evalf(h(Hn_RK6_||ct[ii,1])))^2;\n end do:\n e rrs := [op(errs),sqrt(sm/numpts)];\nend do:\nDigits := 10:\nlinalg[tra nspose]([mthds,evalf(errs)]);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#K%'ma trixG6#7&7$%0slope~field:~~~G,$*&%\"xG\"\"\"%\"yGF,!\"\"7$%0initial~po int:~G-%!G6$\"\"!F,7$%/step~width:~~~G$F,F.7$%1no.~of~steps:~~~G\"$+\" Q)pprint646\"" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%!G" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#K%'matrixG6#7(7$%6Prince-Dormand~schemeG$\"+ZqH.g!#? 7$%La~modification~of~the~Prince-Dormand~schemeG$\"+%H$)za)F+7$%7Verne r's~(2004)~schemeG$\"+;fsAkF+7$%Ba~modification~of~Verner's~schemeG$\" +`>,jsF+7$%Fa~2nd~modification~of~Verner's~schemeG$\"+43dz%)F+7$%Msche me~with~maximal~imaginary~axis~inclusionG$\"+hNW85!#>Q)pprint656\"" }} }{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 "9.99;" "6#-%&Float G6$\"$***!\"#" }{TEXT -1 16 " is also given." }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 712 "H := (x,y) -> -x*y: hh := 0.1: numsteps := 100: x0 := 0: y0 := 1:\nmatrix([[`slope field: `,H(x,y)],[`initial point : `,``(x0,y0)],[`step width: `,hh],\n[`no. of steps: `,numsteps]]) ;``;\nmthds := [`Prince-Dormand scheme`,`a modification of the Prince- Dormand scheme`,`Verner's (2004) scheme`,`a modification of Verner's s cheme`,`a 2nd modification of Verner's scheme`,`scheme with maximal im aginary axis inclusion`]: errs := []:\nDigits := 20:\nfor ct to 6 do\n hn_RK6_||ct := RK6_||ct(H(x,y),x,y,x0,y0,hh,numsteps,true);\nend do :\nh := x -> exp(-x^2/2):\nxx := 9.99: hxx := evalf(h(xx)):\nfor ct to 6 do\n errs := [op(errs),abs(hn_RK6_||ct(xx)-hxx)];\nend do:\nDigit s := 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)pprint206\"" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#% !G" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#K%'matrixG6#7(7$%6Prince-Dormand ~schemeG$\"+,y`,7!#M7$%La~modification~of~the~Prince-Dormand~schemeG$ \"+LO'*e7F+7$%7Verner's~(2004)~schemeG$\"+z9#eA\"F+7$%Ba~modification~ of~Verner's~schemeG$\"+'e$*GC\"F+7$%Fa~2nd~modification~of~Verner's~sc hemeG$\"+X)\\!e7F+7$%Mscheme~with~maximal~imaginary~axis~inclusionG$\" +6FAh9F+Q)pprint216\"" }}}{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 es timated as follows using the special procedure " }{TEXT 0 5 "NCint" } {TEXT -1 97 " to perform numerical integration by the 7 point Newton- Cotes method over 50 equal subintervals." }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 478 "mthds := [`Prince-Dormand scheme`,`a modification of the Prince-Dormand scheme`,`Verner's (2004) scheme`,`a modification o f Verner's scheme`,`a 2nd modification of Verner's scheme`,`scheme wit h maximal imaginary axis inclusion`]: errs := []:\nDigits := 20:\nh := x -> exp(-x^2/2):\nfor ct to 6 do\n sm := NCint((h(x)-'hn_RK6_||ct' (x))^2,x=0..10,adaptive=false,numpoints=7,factor=50);\n errs := [op( errs),sqrt(sm/10)];\nend do:\nDigits := 10:\nlinalg[transpose]([mthds, evalf(errs)]);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#K%'matrixG6#7(7$%6Pr ince-Dormand~schemeG$\"+#Gobm&!#?7$%La~modification~of~the~Prince-Dorm and~schemeG$\"+:n#o.)F+7$%7Verner's~(2004)~schemeG$\"+W*oG/'F+7$%Ba~mo dification~of~Verner's~schemeG$\"+M'\\w#oF+7$%Fa~2nd~modification~of~V erner's~schemeG$\"+&=#oszF+7$%Mscheme~with~maximal~imaginary~axis~incl usionG$\"+SxKH&*F+Q)pprint226\"" }}}{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 0 587 "evalf[20](plot([h(x)-'hn_RK6_1'(x),h(x)-'h n_RK6_2'(x),h(x)-'hn_RK6_3'(x),\nh(x)-'hn_RK6_4'(x),h(x)-'hn_RK6_5'(x) ,h(x)-'hn_RK6_6'(x)],x=0..5,numpoints=150,color=[COLOR(RGB,.4,0,1),\nC OLOR(RGB,0,.65,0),COLOR(RGB,1,.5,0),COLOR(RGB,.85,0,.85),COLOR(RGB,0,. 3,1),COLOR(RGB,.95,0,.15)],\nlegend=[`Prince-Dormand scheme`,`a modifi cation of the Prince-Dormand scheme`,`Verner's (2004) scheme`,`a modif ication of Verner's scheme`,`a 2nd modification of Verner's scheme`,`s cheme with maximal imaginary axis inclusion`],font=[HELVETICA,9],title =`error curves for 8 stage order 6 Runge-Kutta methods`));" }}{PARA 13 "" 1 "" {GLPLOT2D 1028 549 549 {PLOTDATA 2 "6,-%'CURVESG6%7]\\m7$$ \"\"!F)F(7$$\"5gR49r15&4^$!#@$!&^2#!#?7$$\"5l)zp/d&3QQ]F-$!'nLPF07$$\" 5sd')zp/2\"ec'F-$!(Yr5$F07$$\"5IWm\\u6!)oCuF-$!(#)3J)F07$$\"5&3j%>z=`c $G)F-$!)o$f*>F07$$\"5S8+5F0$!)-U7 !*F07$$\"5OaJ(fR4YI<\"F0$!)\"4b**)F07$$\"5K[s3j%>gfM\"F0$!)dex*)F07$$ 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[[lH\"=MabF-7$Fi[l$\"2$z(ofC:Dd&F-7$F^\\l$\"27R&*G>Jyf&F-7$Fc\\l$\"27P M\"4RdTcF-7$Fh\\l$\"2\")**3(GUe*p&F-7$F]]l$\"20N0?f%=SdF-7$Fb]l$\"2c:- $\\u7*y&F-7$Fg]l$\"2!\\YUU+;XeF-7$F\\^l$\"2hNQ'fNq6fF-7$Fa^l$\"2lp8D!= !p*fF-7$Ff^l$\"2$pUZJ0o)4'F--F\\_l6&F^_lFb_l$\"\"&F*Fa_l-Fe_l6#%7Verne r's~(2004)~schemeG-F$6%7jn7$F($\"2!QkVvw[Sanx^2&F-7$F5$ \"25stfnx^2&F-7$F:$\"27yp0px^2&F-7$F?$\"2C\"o'*Qyv]F-7$FS$\"2o5Q6ZA_2&F-7$FX$\"2 Hl[f9)Hv]F-7$Fgn$\"2i_=Eoxa2&F-7$F\\o$\"2jIE\"\\9!e2&F-7$Fao$\"2pNZ![$ 3l2&F-7$Ffo$\"2A!>:02'y2&F-7$F[p$\"2hkmz'[?!3&F-7$F`p$\"2an)[%[hP3&F-7 $Fep$\"2/F(e#***y!4&F-7$Fjp$\"26$*)3IQO+^F-7$F_q$\"2T/z5>\\y6&F-7$Fdq$ \"2n@MrZd;9&F-7$Fiq$\"2<\\DG;%*3=&F-7$F^r$\"2WzR%fTPO_F-7$Fcr$\"2M.$3P Gku_F-7$Fhr$\"2#p&4%>n(3K&F-7$F]s$\"2yj)yj2er`F-7$Fbs$\"2)>j!*))p\\JaF -7$Fgs$\"20xo%*fP\"3bF-7$F\\t$\"2GAuwE'H*f&F-7$Fat$\"20[3+'y#Rg&F-7$Ff tFg]n7$F[u$\"2qZ3+'y#Rg&F-7$F`u$\"2%z%3+'y#Rg&F-7$Feu$\"2\\Z3+'y#Rg&F- 7$Fju$\"2))[3+'y#Rg&F-7$F_v$\"2]))3+'y#Rg&F-7$Fdv$\"2SY7+'y#Rg&F-7$Fiv $\"2?`'egy#Rg&F-7$F^w$\"2XfA^(y#Rg&F-7$Fcw$\"2C\\bf-GRg&F-7$Fhw$\"2`/ \\]))GRg&F-7$F]x$\"2h\"o$>BKRg&F-7$Fbx$\"2hO]Mrj`?dF-7$Fh\\l$\"2=;- i5_\"zdF-7$F]]l$\"23:dp-p,#eF-7$Fb]l$\"24#Qij>9&F-7$F[p$\"2%4kA*=NV9&F-7$F`p$\"2=3t&)zJz9&F-7$Fep$\"2b<'yJk. b^F-7$Fjp$\"2P2$ec0rk^F-7$F_q$\"2.;;S]tB=&F-7$Fdq$\"2&o%Q%ooT1_F-7$Fiq $\"2(GmZy5.Y_F-7$F^r$\"2$GwTh@.-`F-7$Fcr$\"2M22)[WlS`F-7$Fhr$\"2C%pm)) 3J(Q&F-7$F]s$\"2:bbypt%QaF-7$Fbs$\"2idot+F*)\\&F-7$Fgs$\"2IqYg@[id&F-7 $F\\t$\"2)ySay)3#ocF-7$Fat$\"2eT:*H5)Gn&F-7$Fft$\"2dT:*H5)Gn&F-7$F[u$ \"2?T:*H5)Gn&F-7$F`u$\"2XT:*H5)Gn&F-7$Feu$\"2.T:*H5)Gn&F-7$Fju$\"2hU:* H5)Gn&F-7$F_v$\"2+'e\"*H5)Gn&F-7$Fdv$\"2&>(>*H5)Gn&F-7$Fiv$\"2.+B0.\") Gn&F-7$F^w$\"2km$fX5)Gn&F-7$Fcw$\"2J(ox+7)Gn&F-7$Fhw$\"2r&=[!3#)Gn&F-7 $F]x$\"2,\">'p\\&)Gn&F-7$Fbx$\"2aletcF-7$Ffy$\"2,!p1b6ftcF-7$F[z$\"2'\\, fG+KucF-7$F`z$\"2XMy@]Idn&F-7$Fez$\"2)G1:ge=ycF-7$Fjz$\"2f*Q]qsi#o&F-7 $F_[l$\"2P)*>\\Vw(*o&F-7$Fd[l$\"2]gX7**4;q&F-7$Fi[l$\"2BtCn>r,s&F-7$F^ \\l$\"2Jk]dE7gu&F-7$Fc\\l$\"2JlGx2Q1z&F-7$Fh\\l$\"2iO\"[O2z\\eF-7$F]]l $\"2al$)\\[x6*eF-7$Fb]l$\"2vyW(3x0TfF-7$Fg]l$\"2-\\Kl:_\")*fF-7$F\\^l$ \"25>Uv=Wf1'F-7$Fa^l$\"2k?OG/DF:'F-7$Ff^l$\"2`S\"3TcPciF--F\\_l6&F^_lF a_l$\"\"$F*Fb_l-Fe_l6#%Fa~2nd~modification~of~Verner's~schemeG-F$6%7hn 7$F($\"27A(z&ozu(fF-7$F/$\"22!*3eozu(fF-7$F5$\"2lHvpozu(fF-7$F:$\"2d)o z8(zu(fF-7$F?$\"2xm1z&*zu(fF-7$FD$\"2q$f.z7[xfF-7$FI$\"2__Ba(e[xfF-7$F N$\"20^w!e.]xfF-7$FS$\"2%f;$f7Tv(fF-7$FX$\"2.mtOoSw(fF-7$Fgn$\"2uy4M!> (y(fF-7$F\\o$\"2.y>&y>GyfF-7$Fao$\"2'>T.xW;zfF-7$Ffo$\"2z4lK,J3)fF-7$F [p$\"2WhZ(Q#)o$)fF-7$F`p$\"2%fzzfX)z)fF-7$Fep$\"2^aC3h.k*fF-7$Fjp$\"2 \\iR'Gyy2gF-7$F_q$\"2c(fB9?WGgF-7$Fdq$\"2h-LbO)RcgF-7$Fiq$\"2rU@H-DA5' F-7$F^r$\"2@%)ew(\\qmhF-7$Fcr$\"2#eY_ga.6iF-7$Fhr$\"2S3(Q)ovWE'F-7$F]s $\"2Q*eqsa'HK'F-7$Fbs$\"2y\"z@JY&>R'F-7$Fgs$\"2[R&[)>Q+['F-7$F\\t$\"2# zkW=eh%e'F-7$Fft$\"2.oZ')QC**e'F-7$F`u$\"2$zwk)QC**e'F-7$FeuF]fo7$Fju$ \"2vtZ')QC**e'F-7$F_v$\"21.\\')QC**e'F-7$Fdv$\"2fCf')QC**e'F-7$Fiv$\"2 7k7**QC**e'F-7$F^w$\"2feVmTC**e'F-7$Fcw$\"2:mzZmC**e'F-7$Fhw$\"2`aDi(f #**e'F-7$F]x$\"2\"HCRB3$**e'F-7$Fbx$\"2:p.;][**e'F-7$Fgx$\"2E`kp;#***e 'F-7$F\\y$\"2eOL*3_5!f'F-7$Fay$\"2LfZ)zxK!f'F-7$Ffy$\"2\"=`pkE#3f'F-7$ F[z$\"2OqDEj>!e_!=&HFnF-7$Fh\\l$\"2%pMQ 3@K&z'F-7$F]]l$\"23o&f/LxUoF-7$Fb]l$\"2$Gv5#HX)**oF-7$Fg]l$\"2lW^ITY]' pF-7$F\\^l$\"2ei=GtCB/(F-7$Fa^l$\"2RA(pe;2TrF-7$Ff^l$\"2S57c%\\!)esF-- F\\_l6&F^_l$\"#&*Fi[mFa_l$\"#:Fi[m-Fe_l6#%Mscheme~with~maximal~imagina ry~axis~inclusionG-%%FONTG6$%*HELVETICAG\"\"*-%&TITLEG6#%hnrelative~er ror~curves~for~8~stage~order~6~Runge-Kutta~methodsG-%+AXESLABELSG6$Q\" x6\"Q!F`]p-%%VIEWG6$;F(Ff^l%(DEFAULTG" 1 2 0 1 10 0 2 9 1 4 2 1.000000 45.000000 45.000000 0 1 "Prince-Dormand scheme" "a modificati on of the Prince-Dormand scheme" "Verner's (2004) scheme" "a modificat ion of Verner's scheme" "a 2nd modification of Verner's scheme" "schem e with maximal imaginary axis inclusion" }}}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 46 "Test 4 of 8 stage, order 6 Runge-Kutta methods " }}{PARA 0 "" 0 "" {TEXT -1 81 "F. G. Lether: Mathematics of Computat ion, 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&-%#lnG 6#\"\"#F&F(" }{TEXT -1 6 ", " }{XPPEDIT 18 0 "y(-1) = 1/8;" "6#/-% \"yG6#,$\"\"\"!\"\"*&F(F(\"\")F)" }{TEXT -1 1 " " }}{PARA 0 "" 0 "" {TEXT -1 10 "Solution: " }}{PARA 256 "" 0 "" {TEXT -1 2 " " } {XPPEDIT 18 0 "y = 2^(13-6*x^2);" "6#/%\"yG)\"\"#,&\"#8\"\"\"*&\"\"'F) *$%\"xGF&F)!\"\"" }{TEXT -1 2 ". " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 191 "de := diff(y(x),x)=-32*x*y( x)*ln(2);\nic := y(-1)=1/8;\ndsolve(\{de,ic\},y(x)):\ny(x)=2^simplify( log[2](rhs(%)));\nk := unapply(rhs(%),x):\nplot(k(x),x=-1..1,font=[HEL VETICA,9],labels=[`x`,`y(x)`]);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%# deG/-%%diffG6$-%\"yG6#%\"xGF,,$**\"#K\"\"\"F,F0F)F0-%#lnG6#\"\"#F0!\" \"" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%#icG/-%\"yG6#!\"\"#\"\"\"\"\") " }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/-%\"yG6#%\"xG)\"\"#,&\"#8\"\"\"*& \"#;F,)F'F)F,!\"\"" }}{PARA 13 "" 1 "" {GLPLOT2D 577 286 286 {PLOTDATA 2 "6&-%'CURVESG6$7eo7$$!\"\"\"\"!$\"3+++++++]7!#=7$$!3ommm;p 0k&*F-$\"3!Hg[[\"[$)=KF-7$$!3wKL$3$3(F-7$$!3mmmmT%p \"e()F-$\"3!=E-TWD`l\"!#<7$$!3:mmm\"4m(G$)F-$\"3M\"fONp()[t$F=7$$!3\"Q LL3i.9!zF-$\"3A!e'4A%*Rg!)F=7$$!3\"ommT!R=0vF-$\"3%z2Mbncie\"!#;7$$!3u ****\\P8#\\4(F-$\"3C>dT>$)H#3$FM7$$!3+nm;/siqmF-$\"3gp%*z`g)4*eFM7$$!3 [++](y$pZiF-$\"3%R6L-Y$zz5!#:7$$!33LLL$yaE\"eF-$\"3xvp\"p)==K>Fgn7$$!3 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***\\PQ#\\\")Fft$\"3!QM&=wHL5wFap7$$\"3ilm\"z\\1A-\"F-$\"3#*[#H(\\2i&H (Fap7$$\"3GKLLe\"*[H7F-$\"3))\\\\;@heFpFap7$$\"3ylm;HCjV9F-$\"3)e+$\\9 -Y,lFap7$$\"3I*******pvxl\"F-$\"3S%z:5s)zRgFap7$$\"3g)***\\7JFn=F-$\"3 1))p(30[[c&Fap7$$\"3#z****\\_qn2#F-$\"3ae5F\"zuv2&Fap7$$\"3=)**\\P/q%z AF-$\"3ZUhzOe!Rg%Fap7$$\"3U)***\\i&p@[#F-$\"3r&f%4uLbOTFap7$$\"3L)**\\ (=GB2FF-$\"3WV]5@%**Rj$Fap7$$\"3B)****\\2'HKHF-$\"3ul]=$GLo:$Fap7$$\"3 uJL$3UDX8$F-$\"3sKZjodBbFFap7$$\"3ElmmmZvOLF-$\"3!>\\-t_7IQ#Fap7$$\"3i ******\\2goPF-$\"3Q>G9F7l&p\"Fap7$$\"3UKL$eR<*fTF-$\"3?\"Fap 7$$\"3m******\\)Hxe%F-$\"3V-?C_;$p$zFgn7$$\"3ckm;H!o-*\\F-$\"31MiF2c]v ^Fgn7$$\"3y)***\\7k.6aF-$\"3#pB[/J``=$Fgn7$$\"3#emmmT9C#eF-$\"3&*=.D]9 +3>Fgn7$$\"33****\\i!*3`iF-$\"3%HX+j$our5Fgn7$$\"3%QLLL$*zym'F-$\"3!o4 *yfd(\\\"fFM7$$\"3wKLL3N1#4(F-$\"3!\\\\K5**)='4$FM7$$\"3Nmm;HYt7vF-$\" 3%o[)olFVm:FM7$$\"3Y*******p(G**yF-$\"3)3H-pcT.4)F=7$$\"3]mmmT6KU$)F-$ \"35omE\\#[Ck$F=7$$\"3fKLLLbdQ()F-$\"3TxwT%Qu%>ei< " 0 "" {MPLTEXT 1 0 796 "K := (x,y) -> -32*x*y(x)*ln(2): hh := 0.01: numsteps := 200: \+ x0 := -1: y0 := 1/8:\nmatrix([[`slope field: `,K(x,y)],[`initial poi nt: `,``(x0,y0)],[`step width: `,hh],\n[`no. of steps: `,numsteps] ]);``;\nmthds := [`Prince-Dormand scheme`,`a modification of the Princ e-Dormand scheme`,`Verner's (2004) scheme`,`a modification of Verner's scheme`,`a 2nd modification of Verner's scheme`,`scheme with maximal \+ imaginary axis inclusion`]: errs := []:\nDigits := 20:\nfor ct to 6 do \n Kn_RK6_||ct := RK6_||ct(evalf(K(x,y)),x,y,x0,evalf(y0),hh,numstep s,false);\n sm := 0: numpts := nops(Kn_RK6_||ct):\n for ii to nump ts do\n sm := sm+(Kn_RK6_||ct[ii,2]-evalf(k(Kn_RK6_||ct[ii,1])))^ 2;\n end do:\n errs := [op(errs),sqrt(sm/numpts)];\nend do:\nDigit s := 10:\nlinalg[transpose]([mthds,evalf(errs)]);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#K%'matrixG6#7&7$%0slope~field:~~~G,$**\"#K\"\"\"%\"xGF, -%\"yG6#F-F,-%#lnG6#\"\"#F,!\"\"7$%0initial~point:~G-%!G6$F5#F,\"\")7$ %/step~width:~~~G$F,!\"#7$%1no.~of~steps:~~~G\"$+#Q)pprint666\"" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#%!G" }}{PARA 11 "" 1 "" {XPPMATH 20 "6 #K%'matrixG6#7(7$%6Prince-Dormand~schemeG$\"+fy>CG!#:7$%La~modificatio n~of~the~Prince-Dormand~schemeG$\"+K%HPk(!#;7$%7Verner's~(2004)~scheme G$\"+J[Zu?F+7$%Ba~modification~of~Verner's~schemeG$\"+b9UU9F+7$%Fa~2nd ~modification~of~Verner's~schemeG$\"+B\\-e!)F07$%Mscheme~with~maximal~ imaginary~axis~inclusionG$\"+pNUq?F0Q)pprint676\"" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 30 "The following code constr ucts " }{TEXT 260 20 "numerical procedures" }{TEXT -1 56 " for solutio ns based on each of the Runge-Kutta schemes." }}{PARA 0 "" 0 "" {TEXT -1 75 "The error in the value obtained by each of the methods at the p oint where " }{XPPEDIT 18 0 "x = 0;" "6#/%\"xG\"\"!" }{TEXT -1 20 ".9 95 is also given." }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 720 "K := \+ (x,y) -> -32*x*y(x)*ln(2): hh := 0.01: numsteps := 200: x0 := -1: y0 : = 1/8:\nmatrix([[`slope field: `,K(x,y)],[`initial point: `,``(x0,y0 )],[`step width: `,hh],\n[`no. of steps: `,numsteps]]);``;\nmthds \+ := [`Prince-Dormand scheme`,`a modification of the Prince-Dormand sche me`,`Verner's (2004) scheme`,`a modification of Verner's scheme`,`a 2n d modification of Verner's scheme`,`scheme with maximal imaginary axis inclusion`]: errs := []:\nDigits := 20:\nfor ct to 6 do\n kn_RK6_|| ct := RK6_||ct(evalf(K(x,y)),x,y,x0,evalf(y0),hh,numsteps,true);\nend \+ do:\nxx := 0.995: kxx := evalf(k(xx)):\nfor ct to 6 do\n errs := [op (errs),abs(kn_RK6_||ct(xx)-kxx)];\nend do:\nDigits := 10:\nlinalg[tran spose]([mthds,evalf(errs)]);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#K%'mat rixG6#7&7$%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\"$+#Q)pprint256\"" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%!G" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#K%'matrixG6#7(7$ %6Prince-Dormand~schemeG$\"+&>GY'Q!#>7$%La~modification~of~the~Prince- Dormand~schemeG$\"+/&o?a%F+7$%7Verner's~(2004)~schemeG$\"+h#eJ9%F+7$%B a~modification~of~Verner's~schemeG$\"+5U%[M%F+7$%Fa~2nd~modification~o f~Verner's~schemeG$\"+^W&4`%F+7$%Mscheme~with~maximal~imaginary~axis~i nclusionG$\"+A_st`F+Q)pprint266\"" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }} {PARA 0 "" 0 "" {TEXT -1 4 "The " }{TEXT 260 22 "root mean square erro r" }{TEXT -1 20 " over the interval " }{XPPEDIT 18 0 "[-1,1]" "6#7$,$ \"\"\"!\"\"F%" }{TEXT -1 82 " of each Runge-Kutta method is estimated as follows using the special procedure " }{TEXT 0 5 "NCint" }{TEXT -1 98 " to perform numerical integration by the 7 point Newton-Cotes \+ method over 100 equal subintervals." }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 455 "mthds := [`Prince-Dormand scheme`,`a modification of the Prince-Dormand scheme`,`Verner's (2004) scheme`,`a modification o f Verner's scheme`,`a 2nd modification of Verner's scheme`,`scheme wit h maximal imaginary axis inclusion`]: errs := []:\nDigits := 20:\nfor \+ ct to 6 do\n sm := NCint((k(x)-'kn_RK6_||ct'(x))^2,x=-1..1,adaptive= false,numpoints=7,factor=100);\n errs := [op(errs),sqrt(sm/2)];\nend do:\nDigits := 10:\nlinalg[transpose]([mthds,evalf(errs)]);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#K%'matrixG6#7(7$%6Prince-Dormand~schemeG$\"+ IsCJG!#:7$%La~modification~of~the~Prince-Dormand~schemeG$\"+\"RgFm(!#; 7$%7Verner's~(2004)~schemeG$\"+R3lz?F+7$%Ba~modification~of~Verner's~s chemeG$\"+v%=gW\"F+7$%Fa~2nd~modification~of~Verner's~schemeG$\"+f54y! )F07$%Mscheme~with~maximal~imaginary~axis~inclusionG$\"+T7iv?F0Q)pprin t276\"" }}}{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 0 574 "evalf[20](plot([k(x)-'kn_RK6_1'(x),k(x)-'kn_RK6_2'(x),k(x)-'kn_RK 6_3'(x),\nk(x)-'kn_RK6_4'(x),k(x)-'kn_RK6_5'(x),k(x)-'kn_RK6_6'(x)],x= -1..1,color=[COLOR(RGB,.4,0,1),\nCOLOR(RGB,0,.65,0),COLOR(RGB,1,.5,0), COLOR(RGB,.85,0,.85),COLOR(RGB,0,.3,1),COLOR(RGB,.95,0,.15)],\nlegend= [`Prince-Dormand scheme`,`a modification of the Prince-Dormand scheme` ,`Verner's (2004) scheme`,`a modification of Verner's scheme`,`a 2nd m odification of Verner's scheme`,`scheme with maximal imaginary axis in clusion`],font=[HELVETICA,9],title=`error curves for 8 stage order 6 R unge-Kutta methods`));" }}{PARA 13 "" 1 "" {GLPLOT2D 923 653 653 {PLOTDATA 2 "6,-%'CURVESG6%7eo7$$!\"\"\"\"!$F*F*7$$!5nmmmm;p0k&*!#?$\" +,g97'*F/7$$!5LLLL$37$$!5nmmmm\"4m(G$)F/$\",P\\cmr#F<7$$!5LLLL$3i.9!zF/$\",Ti$oZjF<7$$!5 mmmm;/R=0vF/$\",sSGyG\"!#=7$$!5++++]P8#\\4(F/$\",eCs%HaF/$\",\"f\"[ta#Ffn7$$!5+++++]$*4)*\\F/$\", -R`^;%Ffn7$$!5+++++]_&\\c%F/$\",u#)=.c'Ffn7$$!5+++++]1aZTF/$\"+hDPy(*! #;7$$!5mmmm;/#)[oPF/$\",cg4&f8F`p7$$!5LLLLL$=exJ$F/$\",o$G]K>F`p7$$!5L LLLLeW%o7$F/$\",__cJ@#F`p7$$!5LLLLLL2$f$HF/$\",ksRW^#F`p7$$!5mmmmT&o_Q 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!+o::\"3#F`p7$Fg\\l$!+Go)yy\"F`p7$F\\]l$!+0mkP:F`p7$Fa]l$!+y!RWI\"F`p7 $$\"5NLLLLexn_NF/$!+'[G#y5F`p7$Ff]l$!*U0Hu)F`p7$F[^l$!*C:Sg&F`p7$F`^l$ !+C\\q3IFfn7$Fe^l$!+(=NkH\"Ffn7$$\"5NLLL$3A_1?&F/$!*1\\?n&Ffn7$Fj^l$!) l1`')Ffn7$$\"5NLLLe9as;cF/$\"*k?Os#Ffn7$F__l$\"*a#G(H&Ffn7$Fd_l$\"*Eiq %yFfn7$Fi_l$\"+N([hL)FL7$F^`l$\"+!H')p$yFL7$Fc`l$\"+:T!=e'FL7$Fh`l$\", $=!RB8&F<7$F]al$\",([-wPMF<7$Fbal$\",xvXcN#F<7$Fgal$\"-$3Z*=e9F/7$F\\b l$\",+4`)*)))F/7$Fabl$\",QeRg@&F/-Ffbl6&Fhbl$\"#&*F^amF+$\"#:F^am-F\\c l6#%Mscheme~with~maximal~imaginary~axis~inclusionG-%%FONTG6$%*HELVETIC AG\"\"*-%&TITLEG6#%Uerror~curves~for~8~stage~order~6~Runge-Kutta~metho dsG-%+AXESLABELSG6$Q\"x6\"Q!Fh[q-%%VIEWG6$;F(Fabl%(DEFAULTG" 1 2 0 1 10 0 2 9 1 4 2 1.000000 45.000000 45.000000 0 1 "Prince-Dormand scheme " "a modification of the Prince-Dormand scheme" "Verner's (2004) schem e" "a modification of Verner's scheme" "a 2nd modification of Verner's scheme" "scheme with maximal imaginary axis inclusion" }}}}{PARA 0 " " 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}} }{SECT 1 {PARA 4 "" 0 "" {TEXT -1 46 "Test 5 of 8 stage, order 6 Runge -Kutta methods" }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "dy/ dx=16/((16*x+1)*y)" "6#/*&%#dyG\"\"\"%#dxG!\"\"*&\"#;F&*&,&*&F*F&%\"xG F&F&F&F&F&%\"yGF&F(" }{TEXT -1 10 ", " }{XPPEDIT 18 0 "y(0)=1 " "6#/-%\"yG6#\"\"!\"\"\"" }{TEXT -1 1 " " }}{PARA 0 "" 0 "" {TEXT -1 10 "Solution: " }}{PARA 256 "" 0 "" {TEXT -1 2 " " }{XPPEDIT 18 0 "y= sqrt (2*ln(16*x+1)+1)" "6#/%\"yG-%%sqrtG6#,&*&\"\"#\"\"\"-%#lnG6#,&*& \"#;F+%\"xGF+F+F+F+F+F+F+F+" }{TEXT -1 2 ". " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 165 "de := diff(y(x),x )=16/((16*x+1)*y(x));\nic := y(0)=1;\ndsolve(\{de,ic\},y(x));\ns := un apply(rhs(%),x):\nplot(s(x),x=0..0.5,0..2.6,font=[HELVETICA,9],labels= [`x`,`y(x)`]);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%#deG/-%%diffG6$-% \"yG6#%\"xGF,,$*(\"#;\"\"\",&*&F/F0F,F0F0F0F0!\"\"F)F3F0" }}{PARA 11 " " 1 "" {XPPMATH 20 "6#>%#icG/-%\"yG6#\"\"!\"\"\"" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/-%\"yG6#%\"xG*$,&*&\"\"#\"\"\"-%#lnG6#,&*&\"#;F,F'F,F, F,F,F,F,F,F,#F,F+" }}{PARA 13 "" 1 "" {GLPLOT2D 503 318 318 {PLOTDATA 2 "6&-%'CURVESG6$7U7$$\"\"!F)$\"\"\"F)7$$\"3WmmmT&)G\\a!#?$\"3EP,(Qyl. 3\"!#<7$$\"3ILLL3x&)*3\"!#>$\"3?25A!pa&\\6F27$$\"3-+]i!R(*Rc\"F6$\"3oz *p77wF?\"F27$$\"3umm\"H2P\"Q?F6$\"3]_vibZz]7F27$$\"3MLL$eRwX5$F6$\"3!Q b+fY'3W8F27$$\"3CLL$3x%3yTF6$\"31#\\\\E7=EU\"F27$$\"3=mm\"z%4\\Y_F6$\" 3s(e$4OUg*[\"F27$$\"3)HL$eR-/PiF6$\"3.fPtw=4W:F27$$\"3A***\\il'pisF6$ \"3/07@a`R%f\"F27$$\"3`KLe*)>VB$)F6$\"3K!\\`od36k\"F27$$\"3!))**\\7`l2 Q*F6$\"3#HUv\"fmC$o\"F27$$\"3smm;/j$o/\"!#=$\"3:'H!f>cuAjU6Fco$\"3K$o8QC!za=F27$$\"3)*****\\P[6j9Fco$\"39iuo+OIZ=F2 7$$\"3KL$e*[z(yb\"Fco$\"3Q:]fA\\>F27$$\"3))**\\iSj0x=Fco$\"3-5Hbh&QF% >F27$$\"3Wmmm\"pW`(>Fco$\"3So#znsrC'>F27$$\"35+]i!f#=$3#Fco$\"3w)>Y)R! pI)>F27$$\"3/+](=xpe=#Fco$\"3?*eB@.[<+#F27$$\"3smm\"H28IH#Fco$\"3/Fyh^ (\\.-#F27$$\"3km;zpSS\"R#Fco$\"3)4US+%ypO?F27$$\"3GLL3_?`(\\#Fco$\"3#4 Cj+a0O0#F27$$\"3#HLe*)>pxg#Fco$\"3ab\\mG7Vq?F27$$\"3u**\\Pf4t.FFco$\"3 Cx7m@=^%3#F27$$\"32LLe*Gst!GFco$\"3Q>IFco$\"3&ocGC'[]F@F27$$\"3h**\\i!RU0 7$Fco$\"3HCH$Q\")f.9#F27$$\"3b***\\(=S2LKFco$\"3C`wrWc9a@F27$$\"3Kmmm \"p)=MLFco$\"3;=S,IA7m@F27$$\"3!*****\\(=]@W$Fco$\"3w4%eC\"p]y@F27$$\" 35L$e*[$z*RNFco$\"3UyOr,.R*=#F27$$\"3#*****\\iC$pk$Fco$\"3wIdFs1%4?#F2 7$$\"39m;H2qcZPFco$\"3Qbx\"QY%\\6AF27$$\"3q**\\7.\"fF&QFco$\"3f+!e(oz@ AAF27$$\"3Ymm;/OgbRFco$\"36qG(yA8CB#F27$$\"3y**\\ilAFjSFco$\"3v.zLgjzU AF27$$\"3YLLL$)*pp;%Fco$\"3IImU*yHDD#F27$$\"3?LL3xe,tUFco$\"3I%R!fhiAi AF27$$\"3em;HdO=yVFco$\"3?ogo1xfrAF27$$\"3))*****\\#>#[Z%Fco$\"3EO%fx0 /+G#F27$$\"3immT&G!e&e%Fco$\"3)zsS%e\"3%*G#F27$$\"3;LLL$)Qk%o%Fco$\"35 e8d5)>wH#F27$$\"37+]iSjE!z%Fco$\"3e%4h.zwhI#F27$$\"35+]P40O\"*[Fco$\"3 Nwd2,K=9BF27$$\"3++++++++]Fco$\"3m'>())[`fABF2-%'COLOURG6&%$RGBG$\"#5! \"\"F(F(-%%FONTG6$%*HELVETICAG\"\"*-%+AXESLABELSG6$%\"xG%%y(x)G-%%VIEW G6$;F($\"\"&Fj[l;F($\"#EFj[l" 1 2 0 1 10 0 2 9 1 4 2 1.000000 45.000000 45.000000 0 0 "Curve 1" }}}}{PARA 0 "" 0 "" {TEXT -1 0 "" }} {PARA 0 "" 0 "" {TEXT -1 32 "The following code 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 0 779 "S := \+ (x,y) -> 16/((16*x+1)*y): hh := 0.005: numsteps := 100: x0 := 0: y0 := 1:\nmatrix([[`slope field: `,S(x,y)],[`initial point: `,``(x0,y0)], [`step width: `,hh],\n[`no. of steps: `,numsteps]]);``;\nmthds := \+ [`Prince-Dormand scheme`,`a modification of the Prince-Dormand scheme` ,`Verner's (2004) scheme`,`a modification of Verner's scheme`,`a 2nd m odification of Verner's scheme`,`scheme with maximal imaginary axis in clusion`]: errs := []:\nDigits := 20:\nfor ct to 6 do\n Sn_RK6_||ct \+ := RK6_||ct(S(x,y),x,y,x0,y0,hh,numsteps,false);\n sm := 0: numpts : = nops(Sn_RK6_||ct):\n for ii to numpts do\n sm := sm+(Sn_RK6_| |ct[ii,2]-evalf(s(Sn_RK6_||ct[ii,1])))^2;\n end do:\n errs := [op( errs),sqrt(sm/numpts)];\nend do:\nDigits := 10:\nlinalg[transpose]([mt hds,evalf(errs)]);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#K%'matrixG6#7&7$ %0slope~field:~~~G,$*(\"#;\"\"\",&*&F+F,%\"xGF,F,F,F,!\"\"%\"yGF0F,7$% 0initial~point:~G-%!G6$\"\"!F,7$%/step~width:~~~G$\"\"&!\"$7$%1no.~of~ steps:~~~G\"$+\"Q)pprint686\"" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%!G" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#K%'matrixG6#7(7$%6Prince-Dormand~sch emeG$\"+FN?'=\"!#>7$%La~modification~of~the~Prince-Dormand~schemeG$\"+ v4m9!*!#?7$%7Verner's~(2004)~schemeG$\"+X(plA\"F+7$%Ba~modification~of ~Verner's~schemeG$\"+bRsJ6F+7$%Fa~2nd~modification~of~Verner's~schemeG $\"+J\">o?*F07$%Mscheme~with~maximal~imaginary~axis~inclusionG$\"+s))y [ " 0 "" {MPLTEXT 1 0 704 "S := (x,y) -> 16/((16*x+1)*y): hh \+ := 0.005: numsteps := 100: x0 := 0: y0 := 1:\nmatrix([[`slope field: \+ `,S(x,y)],[`initial point: `,``(x0,y0)],[`step width: `,hh],\n[`no. of steps: `,numsteps]]);``;\nmthds := [`Prince-Dormand scheme`,`a m odification of the Prince-Dormand scheme`,`Verner's (2004) scheme`,`a \+ modification of Verner's scheme`,`a 2nd modification of Verner's schem e`,`scheme with maximal imaginary axis inclusion`]: errs := []:\nDigit s := 20:\nfor ct to 6 do\n sn_RK6_||ct := RK6_||ct(S(x,y),x,y,x0,y0, hh,numsteps,true);\nend do:\nxx := 0.4995: sxx := evalf(s(xx)):\nfor c t to 6 do\n errs := [op(errs),abs(sn_RK6_||ct(xx)-sxx)];\nend do:\nD igits := 10:\nlinalg[transpose]([mthds,evalf(errs)]);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#K%'matrixG6#7&7$%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\"$+\"Q)pprint306\"" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#%!G" }}{PARA 11 "" 1 "" {XPPMATH 20 "6 #K%'matrixG6#7(7$%6Prince-Dormand~schemeG$\"*_!R(z*!#>7$%La~modificati on~of~the~Prince-Dormand~schemeG$\"*P-=X(F+7$%7Verner's~(2004)~schemeG $\"+FD!Q,\"F+7$%Ba~modification~of~Verner's~schemeG$\"*qmVN*F+7$%Fa~2n d~modification~of~Verner's~schemeG$\"*I%[5wF+7$%Mscheme~with~maximal~i maginary~axis~inclusionG$\"*K/LX\"F+Q)pprint316\"" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 4 "The " }{TEXT 260 22 "root \+ mean square error" }{TEXT -1 110 " over the interval [0, 0.5] of eac h Runge-Kutta method is estimated as follows using the special procedu re " }{TEXT 0 5 "NCint" }{TEXT -1 97 " to perform numerical integrat ion by the 7 point Newton-Cotes method over 50 equal subintervals." }} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 457 "mthds := [`Prince-Dormand s cheme`,`a modification of the Prince-Dormand scheme`,`Verner's (2004) \+ scheme`,`a modification of Verner's scheme`,`a 2nd modification of Ver ner's scheme`,`scheme with maximal imaginary axis inclusion`]: errs := []:\nDigits := 20:\nfor ct to 6 do\n sm := NCint((s(x)-'sn_RK6_||ct '(x))^2,x=0..0.5,adaptive=false,numpoints=7,factor=50);\n errs := [o p(errs),sqrt(sm/0.5)];\nend do:\nDigits := 10:\nlinalg[transpose]([mth ds,evalf(errs)]);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#K%'matrixG6#7(7$% 6Prince-Dormand~schemeG$\"+u-!e=\"!#>7$%La~modification~of~the~Prince- Dormand~schemeG$\"+Rmf6!*!#?7$%7Verner's~(2004)~schemeG$\"+W::E7F+7$%B a~modification~of~Verner's~schemeG$\"+&[Q88\"F+7$%Fa~2nd~modification~ of~Verner's~schemeG$\"+<$*o.#*F07$%Mscheme~with~maximal~imaginary~axis ~inclusionG$\"+(o7#[ " 0 "" {MPLTEXT 1 0 575 "evalf[20](plot(['sn_RK6_1'(x)-s(x) ,'sn_RK6_2'(x)-s(x),'sn_RK6_3'(x)-s(x),\n'sn_RK6_4'(x)-s(x),'sn_RK6_5' (x)-s(x),'sn_RK6_6'(x)-s(x)],x=0..0.5,color=[COLOR(RGB,.4,0,1),\nCOLOR (RGB,0,.65,0),COLOR(RGB,1,.5,0),COLOR(RGB,.85,0,.85),COLOR(RGB,0,.3,1) ,COLOR(RGB,.95,0,.15)],\nlegend=[`Prince-Dormand scheme`,`a modificati on of the Prince-Dormand scheme`,`Verner's (2004) scheme`,`a modificat ion of Verner's scheme`,`a 2nd modification of Verner's scheme`,`schem e with maximal imaginary axis inclusion`],font=[HELVETICA,9],title=`er ror curves for 8 stage order 6 Runge-Kutta methods`));" }}{PARA 13 "" 1 "" {GLPLOT2D 1030 622 622 {PLOTDATA 2 "6,-%'CURVESG6%7^p7$$\"\"!F)F( 7$$\"5SLLL3x1h6o!#B$\"%W7$$\"5ommmTN@Ki8!#A$\"'ul?F07$$\"5NLL3FpE! 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ification of Verner's scheme" "a 2nd modification of Verner's scheme" "scheme with maximal imaginary axis inclusion" }}}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 46 "Test 6 of 8 stage, order 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*-%$cosG6#*&F3F*F)F*F*F.*(F ,F*F3F.-F66#*&F3F*F)F*F*F.*&\"\"&F*F3F.F*" }{TEXT -1 2 ". " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 169 "de \+ := diff(y(x),x)=(1+2*(x+1)*sin(3*x))*exp(-y(x));\nic := y(0)=0;\ndsolv e(\{de,ic\},y(x));\nu := unapply(rhs(%),x):\nplot(u(x),x=0..5,font=[HE LVETICA,9],labels=[`x`,`y(x)`]);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>% #deG/-%%diffG6$-%\"yG6#%\"xGF,*&,&\"\"\"F/*(\"\"#F/,&F,F/F/F/F/-%$sinG 6#,$*&\"\"$F/F,F/F/F/F/F/-%$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$7b p7$$\"\"!F)F(7$$\"3GLLL3x&)*3\"!#=$\"3QWK+t!=.P\"F-7$$\"3umm\"H2P\"Q?F -$\"3pUCE&GmM$HF-7$$\"3MLL$eRwX5$F-$\"3l!G\"yWq,6\\F-7$$\"33ML$3x%3yTF -$\"3dz%)zauhMpF-7$$\"3emm\"z%4\\Y_F-$\"3,G5kQO>C))F-7$$\"3`LLeR-/PiF- $\"36YrjIBvP5!#<7$$\"3]***\\il'pisF-$\"3wGtPF*HL<\"FI7$$\"3>MLe*)>VB$) 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GTFI$\"3JeP:9JjA()F-7$$\"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$3xe,tUFI$\"3[2R[)*eVA&*F-7$$\"3Cn;HdO=y VFI$\"3#)>Y<=$f\\9\"FI7$$\"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$\"3_eaMDR_K " 0 "" {MPLTEXT 1 0 791 "U := (x,y) -> \+ (1+2*(x+1)*sin(3*x))*exp(-y): hh := 0.01: numsteps := 500: x0 := 0: y0 := 0:\nmatrix([[`slope field: `,U(x,y)],[`initial point: `,``(x0,y0 )],[`step width: `,hh],\n[`no. of steps: `,numsteps]]);``;\nmthds \+ := [`Prince-Dormand scheme`,`a modification of the Prince-Dormand sche me`,`Verner's (2004) scheme`,`a modification of Verner's scheme`,`a 2n d modification of Verner's scheme`,`scheme with maximal imaginary axis inclusion`]: errs := []:\nDigits := 25:\nfor ct to 6 do\n Un_RK6_|| ct := RK6_||ct(U(x,y),x,y,x0,y0,hh,numsteps,false);\n sm := 0: numpt s := nops(Un_RK6_||ct):\n for ii to numpts do\n sm := sm+(Un_RK 6_||ct[ii,2]-evalf(u(Un_RK6_||ct[ii,1])))^2;\n end do:\n errs := [ op(errs),sqrt(sm/numpts)];\nend do:\nDigits := 10:\nlinalg[transpose]( [mthds,evalf(errs)]);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#K%'matrixG6#7 &7$%0slope~field:~~~G*&,&\"\"\"F+*(\"\"#F+,&%\"xGF+F+F+F+-%$sinG6#,$*& \"\"$F+F/F+F+F+F+F+-%$expG6#,$%\"yG!\"\"F+7$%0initial~point:~G-%!G6$\" \"!FA7$%/step~width:~~~G$F+!\"#7$%1no.~of~steps:~~~G\"$+&Q)pprint706\" " }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%!G" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#K%'matrixG6#7(7$%6Prince-Dormand~schemeG$\"+:LR:\")!#C7$%La~modi fication~of~the~Prince-Dormand~schemeG$\"+s/e-NF+7$%7Verner's~(2004)~s chemeG$\"+xQpScF+7$%Ba~modification~of~Verner's~schemeG$\"+(frt_%F+7$% Fa~2nd~modification~of~Verner's~schemeG$\"+vl(*zNF+7$%Mscheme~with~max imal~imaginary~axis~inclusionG$\"+JyrNoF+Q)pprint716\"" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 30 "The following code c onstructs " }{TEXT 260 20 "numerical procedures" }{TEXT -1 56 " for so lutions based on each of the Runge-Kutta schemes." }}{PARA 0 "" 0 "" {TEXT -1 75 "The error in the value obtained by each of the methods at the point where " }{XPPEDIT 18 0 "x = 4.999;" "6#/%\"xG-%&FloatG6$\" %**\\!\"$" }{TEXT -1 16 " is also given." }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 715 "U := (x,y) -> (1+2*(x+1)*sin(3*x))*exp(-y): hh := 0. 01: numsteps := 500: x0 := 0: y0 := 0:\nmatrix([[`slope field: `,U(x ,y)],[`initial point: `,``(x0,y0)],[`step width: `,hh],\n[`no. of st eps: `,numsteps]]);``;\nmthds := [`Prince-Dormand scheme`,`a modific ation of the Prince-Dormand scheme`,`Verner's (2004) scheme`,`a modifi cation of Verner's scheme`,`a 2nd modification of Verner's scheme`,`sc heme with maximal imaginary axis inclusion`]: errs := []:\nDigits := 3 0:\nfor ct to 6 do\n un_RK6_||ct := RK6_||ct(U(x,y),x,y,x0,y0,hh,num steps,true);\nend do:\nxx := 4.999: uxx := evalf(u(xx)):\nfor ct to 6 \+ do\n errs := [op(errs),abs(un_RK6_||ct(xx)-uxx)];\nend do:\nDigits : = 10:\nlinalg[transpose]([mthds,evalf(errs)]);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#K%'matrixG6#7&7$%0slope~field:~~~G*&,&\"\"\"F+*(\"\"#F+ ,&%\"xGF+F+F+F+-%$sinG6#,$*&\"\"$F+F/F+F+F+F+F+-%$expG6#,$%\"yG!\"\"F+ 7$%0initial~point:~G-%!G6$\"\"!FA7$%/step~width:~~~G$F+!\"#7$%1no.~of~ steps:~~~G\"$+&Q)pprint356\"" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%!G" } }{PARA 11 "" 1 "" {XPPMATH 20 "6#K%'matrixG6#7(7$%6Prince-Dormand~sche meG$\"+h*>%eY!#C7$%La~modification~of~the~Prince-Dormand~schemeG$\"+Oe gc9F+7$%7Verner's~(2004)~schemeG$\"+dUntR!#E7$%Ba~modification~of~Vern er's~schemeG$\"+Bb " 0 "" {MPLTEXT 1 0 454 "mthds := [`Pri nce-Dormand scheme`,`a modification of the Prince-Dormand scheme`,`Ver ner's (2004) scheme`,`a modification of Verner's scheme`,`a 2nd modifi cation of Verner's scheme`,`scheme with maximal imaginary axis inclusi on`]: errs := []:\nDigits := 20:\nfor ct to 6 do\n sm := NCint((u(x) -'un_RK6_||ct'(x))^2,x=0..5,adaptive=false,numpoints=7,factor=200);\n \+ errs := [op(errs),sqrt(sm/5)];\nend do:\nDigits := 10:\nlinalg[trans pose]([mthds,evalf(errs)]);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#K%'matr ixG6#7(7$%6Prince-Dormand~schemeG$\"+<_t(4)!#C7$%La~modification~of~th e~Prince-Dormand~schemeG$\"+1_)))[$F+7$%7Verner's~(2004)~schemeG$\"+08 kGcF+7$%Ba~modification~of~Verner's~schemeG$\"+XdQ:XF+7$%Fa~2nd~modifi cation~of~Verner's~schemeG$\"+WQDmNF+7$%Mscheme~with~maximal~imaginary ~axis~inclusionG$\"+c<&*HoF+Q)pprint376\"" }}}{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 0 573 "evalf[20](plot([u(x)-'un_RK 6_1'(x),u(x)-'un_RK6_2'(x),u(x)-'un_RK6_3'(x),\nu(x)-'un_RK6_4'(x),u(x )-'un_RK6_5'(x),u(x)-'un_RK6_6'(x)],x=0..5,color=[COLOR(RGB,.4,0,1),\n COLOR(RGB,0,.65,0),COLOR(RGB,1,.5,0),COLOR(RGB,.85,0,.85),COLOR(RGB,0, .3,1),COLOR(RGB,.95,0,.15)],\nlegend=[`Prince-Dormand scheme`,`a modif ication of the Prince-Dormand scheme`,`Verner's (2004) scheme`,`a modi fication of Verner's scheme`,`a 2nd modification of Verner's scheme`,` scheme with maximal imaginary axis inclusion`],font=[HELVETICA,9],titl e=`error curves for 8 stage order 6 Runge-Kutta methods`));" }}{PARA 13 "" 1 "" {GLPLOT2D 1006 619 619 {PLOTDATA 2 "6,-%'CURVESG6%7_t7$$\" \"!F)F(7$$\"5MLLLL3x&)*3\"!#?$\"'P?MF-7$$\"5+++]i!R(*Rc\"F-$\"'-j\\F-7 $$\"5nmmm\"H2P\"Q?F-$\"'p8iF-7$$\"5++]7G))>Wr@F-$\"'&zP'F-7$$\"5MLLek. pu/BF-$\"'M7nF-7$$\"5nm;/,>=0QCF-$\"'7xnF-7$$\"5+++]PMnNrDF-$\"'iNoF-7 $$\"5ML$eR(\\;m/FF-$\"'[2qF-7$$\"5nmmT5ll'z$GF-$\"'P()pF-7$$\"5++](o/[ r7(HF-$\"'GhpF-7$$\"5MLLL$eRwX5$F-$\"'O0qF-7$$\"5MLLLL$eI8k$F-$\"'WzmF -7$$\"5NLLL$3x%3yTF-$\"'(*3iF-7$$\"5ommm\"z%4\\Y_F-$\"'+V_F-7$$\"5NLLL 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" }}{PARA 0 "" 0 "" {TEXT -1 10 "Solution: " }}{PARA 256 "" 0 "" {TEXT -1 1 " " } {XPPEDIT 18 0 "y = 1/3" "6#/%\"yG*&\"\"\"F&\"\"$!\"\"" }{TEXT -1 1 " \+ " }{XPPEDIT 18 0 "exp(-4/3*sin(3*x)+8/3*sin(3/2*x)*cos(3/2*x))+2/3" "6 #,&-%$expG6#,&*(\"\"%\"\"\"\"\"$!\"\"-%$sinG6#*&F+F*%\"xGF*F*F,**\"\") F*F+F,-F.6#*(F+F*\"\"#F,F1F*F*-%$cosG6#*(F+F*F7F,F1F*F*F*F**&F7F*F+F,F *" }{TEXT -1 1 " " }{XPPEDIT 18 0 "exp(-4/3*sin(3*x)-x)" "6#-%$expG6#, &*(\"\"%\"\"\"\"\"$!\"\"-%$sinG6#*&F*F)%\"xGF)F)F+F0F+" }{TEXT -1 2 ". " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 181 "de := diff(y(x),x)=-(1+4*cos(3*x))*(y(x)-1/3);\nic : = y(0)=1;\nsimplify(dsolve(\{de,ic\},y(x)));\nv := unapply(rhs(%),x): \nplot(v(x),x=0..5,0..1.1,font=[HELVETICA,9],labels=[`x`,`y(x)`]);" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#>%#deG/-%%diffG6$-%\"yG6#%\"xGF,,$*&,& \"\"\"F0*&\"\"%F0-%$cosG6#,$*&\"\"$F0F,F0F0F0F0F0,&F)F0#F0F8!\"\"F0F; " }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%#icG/-%\"yG6#\"\"!\"\"\"" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#/-%\"yG6#%\"xG,&*&#\"\"\"\"\"$F+-%$exp G6#,&*&#\"\"%F,F+-%$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&-%'CU RVESG6$7ap7$$\"\"!F)$\"\"\"F)7$$\"3gmmTN@Ki8!#>$\"3W+7cSy5h&*!#=7$$\"3 ALL$3FWYs#F/$\"3KtP[t*Q;:*F27$$\"3%)***\\iSmp3%F/$\"3g.\"H>f!3q()F27$$ \"3WmmmT&)G\\aF/$\"36p*p.:G\\T)F27$$\"3m****\\7G$R<)F/$\"3a?glh]$zx(F2 7$$\"3GLLL3x&)*3\"F2$\"3IM[S(o-#HsF27$$\"3em\"z%\\v#pK\"F2$\"3i=)H'*Q$ =:oF27$$\"3))**\\i!R(*Rc\"F2$\"3w,'pRB0LX'F27$$\"3&edVF27$$\"3%QL$3_DG1qF2$\"3'fN^ hMLe*)>VB$)F2$\"3DB(Rfp)*\\j%F27$$\"3Y++DJbw!Q*F2$\"3%Gs Cu$*)zK]F27$$\"3+N$ekGkX#**F2$\"3u>+\\,YW?`F27$$\"3%ommTIOo/\"!#<$\"3q ]2x8ZEqcF27$$\"3E+]7GTt%4\"Fgt$\"39b$=$pWlHgF27$$\"3YLL3_>jU6Fgt$\"3nw Ydkc=KkF27$$\"3ym;HdNb'>\"Fgt$\"3l[hQOW]BpF27$$\"37++]i^Z]7Fgt$\"3IVnF )*yXIuF27$$\"35+++v\"=YI\"Fgt$\"3ahS!3L%e=zF27$$\"33++](=h(e8Fgt$\"3l& QV-<82M)F27$$\"3&*****\\7!Q4T\"Fgt$\"3^]H\"3wS2k)F27$$\"3/++]P[6j9Fgt$ \"3ur)[IAj$)z)F27$$\"3'=HKkAg\\Z\"Fgt$\"33z^;ogY6))F27$$\"3W$ek`h0o[\" Fgt$\"3h=q?g>u:))F27$$\"3/voH/5l)\\\"Fgt$\"3p\\\\U!)G36))F27$$\"3%o;HK R'\\5:Fgt$\"3%G4&GMdV(z)F27$$\"3-]P4rr=M:Fgt$\"3Erd.MaCV()F27$$\"3UL$e *[z(yb\"Fgt$\"3m)))[\\1qQl)F27$$\"34+Dc,#>Uh\"Fgt$\"3(fTb\\\\y3J)F27$$ \"3wmm;a/cq;Fgt$\"3-!y\"yF27$$\"3\"pm;a)))G=BtF 27$$\"3%ommmJFgt$\"3%RlX>.MR=&F2 7$$\"3gmmm\"pW`(>Fgt$\"3+6YS9:C2[F27$$\"3dLe9TOEH?Fgt$\"3!eWte3T%oWF27 $$\"3K+]i!f#=$3#Fgt$\"3:XZ<;2j,UF27$$\"3?+](=xpe=#Fgt$\"3E#Q(H44MbQF27 $$\"37nm\"H28IH#Fgt$\"3MH4)f2==l$F27$$\"3um;zpSS\"R#Fgt$\"3wpxg#Fgt$\"37l*=e[ EHY$F27$$\"33+]Pf4t.FFgt$\"35!4Ne]qiX$F27$$\"3uLLe*Gst!GFgt$\"3U+pq))z 7kMF27$$\"30+++DRW9HFgt$\"37'z:1TS%*[$F27$$\"3:++DJE>>IFgt$\"3N!o4Joz] `$F27$$\"3F+]i!RU07$Fgt$\"3=,?;D0\"Qg$F27$$\"3+++v=S2LKFgt$\"3wRH=fZn5 PF27$$\"3Jmmm\"p)=MLFgt$\"3RsXuk([b#QF27$$\"3B++](=]@W$Fgt$\"3%4[=*QOM SRF27$$\"3mm\"H#oZ1\"\\$Fgt$\"3QK??D+QyRF27$$\"35L$e*[$z*RNFgt$\"3UAxt ;S)>+%F27$$\"3%o;Hd!fX$f$Fgt$\"3+h91z&\\y+%F27$$\"3e++]iC$pk$Fgt$\"3eI Rs#H!Q\"*RF27$$\"3ILe*[t\\sp$Fgt$\"3m\"Rx)H&*[cRF27$$\"3[m;H2qcZPFgt$ \"3w)))[$RF!f!RF27$$\"3O+]7.\"fF&QFgt$\"3+Efp,iIqPF27$$\"3Ymm;/OgbRFgt $\"3W-Tml[`MOF27$$\"3w**\\ilAFjSFgt$\"3&zNMj#[Z%Fgt$\"3ADU\\K%G5O$F27$$\"3SnmT&G!e&e% Fgt$\"35gRzc#\\LF27$$\"\"&F) $\"3Ii#4)y!3AN$F2-%'COLOURG6&%$RGBG$\"#5!\"\"F(F(-%%FONTG6$%*HELVETICA G\"\"*-%+AXESLABELSG6$%\"xG%%y(x)G-%%VIEWG6$;F(Fiel;F($\"#6Fcfl" 1 2 0 1 10 0 2 9 1 4 2 1.000000 45.000000 45.000000 0 0 "Curve 1" }}}} {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 32 "The follo wing code constructs a " }{TEXT 260 17 "discrete solution" }{TEXT -1 44 " based on each of the methods and gives the " }{TEXT 260 22 "root \+ mean square error" }{TEXT -1 18 " of each solution." }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 786 "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 := [`Prince-Dormand scheme`,`a mo dification of the Prince-Dormand scheme`,`Verner's (2004) scheme`,`a m odification of Verner's scheme`,`a 2nd modification of Verner's scheme `,`scheme with maximal imaginary axis inclusion`]: errs := []:\nDigits := 30:\nfor ct to 6 do\n Vn_RK6_||ct := RK6_||ct(V(x,y),x,y,x0,y0,h h,numsteps,false);\n sm := 0: numpts := nops(Vn_RK6_||ct):\n for i i to numpts do\n sm := sm+(Vn_RK6_||ct[ii,2]-evalf(v(Vn_RK6_||ct[ ii,1])))^2;\n end do:\n errs := [op(errs),sqrt(sm/numpts)];\nend d o:\nDigits := 10:\nlinalg[transpose]([mthds,evalf(errs)]);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#K%'matrixG6#7&7$%0slope~field:~~~G,$*&,&\"\" \"F,*&\"\"%F,-%$cosG6#,$*&\"\"$F,%\"xGF,F,F,F,F,,&%\"yGF,#F,F4!\"\"F,F 97$%0initial~point:~G-%!G6$\"\"!F,7$%/step~width:~~~G$\"\"#!\"#7$%1no. ~of~steps:~~~G\"$]#Q)pprint726\"" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%! G" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#K%'matrixG6#7(7$%6Prince-Dormand~ schemeG$\"+NOWB7!#?7$%La~modification~of~the~Prince-Dormand~schemeG$\" +=J;8(*!#@7$%7Verner's~(2004)~schemeG$\"+hI_w5F+7$%Ba~modification~of~ Verner's~schemeG$\"+(oPA,\"F+7$%Fa~2nd~modification~of~Verner's~scheme G$\"+b " 0 "" {MPLTEXT 1 0 710 "V := (x,y) -> -(1+4* cos(3*x))*(y-1/3): hh := 0.02: numsteps := 250: x0 := 0: y0 := 1:\nmat rix([[`slope field: `,V(x,y)],[`initial point: `,``(x0,y0)],[`step w idth: `,hh],\n[`no. of steps: `,numsteps]]);``;\nmthds := [`Prince -Dormand scheme`,`a modification of the Prince-Dormand scheme`,`Verner 's (2004) scheme`,`a modification of Verner's scheme`,`a 2nd modificat ion of Verner's scheme`,`scheme with maximal imaginary axis inclusion` ]: errs := []:\nDigits := 30:\nfor ct to 6 do\n vn_RK6_||ct := RK6_| |ct(V(x,y),x,y,x0,y0,hh,numsteps,true);\nend do:\nxx := 4.999: vxx := \+ evalf(v(xx)):\nfor ct to 6 do\n errs := [op(errs),abs(vn_RK6_||ct(xx )-vxx)];\nend do:\nDigits := 10:\nlinalg[transpose]([mthds,evalf(errs) ]);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#K%'matrixG6#7&7$%0slope~field:~ ~~G,$*&,&\"\"\"F,*&\"\"%F,-%$cosG6#,$*&\"\"$F,%\"xGF,F,F,F,F,,&%\"yGF, #F,F4!\"\"F,F97$%0initial~point:~G-%!G6$\"\"!F,7$%/step~width:~~~G$\" \"#!\"#7$%1no.~of~steps:~~~G\"$]#Q)pprint406\"" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%!G" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#K%'matrixG6#7(7$ %6Prince-Dormand~schemeG$\"+pI&yo&!#A7$%La~modification~of~the~Prince- Dormand~schemeG$\"+?\\Ov^F+7$%7Verner's~(2004)~schemeG$\"+Fc'\\F&F+7$% Ba~modification~of~Verner's~schemeG$\"+?i[!>&F+7$%Fa~2nd~modification~ of~Verner's~schemeG$\"+)>4a;&F+7$%Mscheme~with~maximal~imaginary~axis~ inclusionG$\"+wo_<]F+Q)pprint416\"" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" } }{PARA 0 "" 0 "" {TEXT -1 4 "The " }{TEXT 260 22 "root mean square err or" }{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 0 454 "mthds := [`Prince-Dormand scheme`,`a modification of the Prin ce-Dormand scheme`,`Verner's (2004) scheme`,`a modification of Verner' s scheme`,`a 2nd modification of Verner's scheme`,`scheme with maximal imaginary axis inclusion`]: errs := []:\nDigits := 20:\nfor ct to 6 d o\n sm := NCint((v(x)-'vn_RK6_||ct'(x))^2,x=0..5,adaptive=false,nump oints=7,factor=100);\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$%6Prince-Dormand~schemeG$\"+)f&p@7!#? 7$%La~modification~of~the~Prince-Dormand~schemeG$\"+Qiy\"p*!#@7$%7Vern er's~(2004)~schemeG$\"+\"3kX2\"F+7$%Ba~modification~of~Verner's~scheme G$\"+CS855F+7$%Fa~2nd~modification~of~Verner's~schemeG$\"+*=M_n*F07$%M scheme~with~maximal~imaginary~axis~inclusionG$\"+N(4)\\!*F0Q)pprint426 \"" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 92 "Th e following error graphs are constructed using the numerical procedure s for the solutions." }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 573 "eva lf[20](plot([v(x)-'vn_RK6_1'(x),v(x)-'vn_RK6_2'(x),v(x)-'vn_RK6_3'(x), \nv(x)-'vn_RK6_4'(x),v(x)-'vn_RK6_5'(x),v(x)-'vn_RK6_6'(x)],x=0..5,col or=[COLOR(RGB,.4,0,1),\nCOLOR(RGB,0,.65,0),COLOR(RGB,1,.5,0),COLOR(RGB ,.85,0,.85),COLOR(RGB,0,.3,1),COLOR(RGB,.95,0,.15)],\nlegend=[`Prince- Dormand scheme`,`a modification of the Prince-Dormand scheme`,`Verner' s (2004) scheme`,`a modification of Verner's scheme`,`a 2nd modificati on of Verner's scheme`,`scheme with maximal imaginary axis inclusion`] ,font=[HELVETICA,9],title=`error curves for 8 stage order 6 Runge-Kutt a methods`));" }}{PARA 13 "" 1 "" {GLPLOT2D 1028 578 578 {PLOTDATA 2 " 6,-%'CURVESG6%7\\r7$$\"\"!F)F(7$$\"5qmm;aQ`!eS$!#A$\"%;9!#?7$$\"5SLLL3 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x6\"Q!Fg]o-%%VIEWG6$;F(Fez%(DEFAULTG" 1 2 0 1 10 0 2 9 1 4 2 1.000000 45.000000 45.000000 0 1 "Prince-Dormand scheme" "a modification of the Prince-Dormand scheme" "Verner's (2004) scheme" "a modification of Ve rner's scheme" "a 2nd modification of Verner's scheme" "scheme with ma ximal imaginary axis inclusion" }}}}{PARA 0 "" 0 "" {TEXT -1 0 "" }} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}}{SECT 1 {PARA 4 "" 0 " " {TEXT -1 46 "Test 8 of 8 stage, order 6 Runge-Kutta methods" }} {PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "dy/dx=x*(9-x^2)/(1+y^ 2)" "6#/*&%#dyG\"\"\"%#dxG!\"\"*(%\"xGF&,&\"\"*F&*$F*\"\"#F(F&,&F&F&*$ %\"yGF.F&F(" }{TEXT -1 3 ", " }{XPPEDIT 18 0 "y(0)=0" "6#/-%\"yG6#\" \"!F'" }{TEXT -1 1 " " }}{PARA 0 "" 0 "" {TEXT -1 10 "Solution: " }} {PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "y = rho(x)/2-2/rho(x) ;" "6#/%\"yG,&*&-%$rhoG6#%\"xG\"\"\"\"\"#!\"\"F+*&F,F+-F(6#F*F-F-" } {TEXT -1 2 ", " }}{PARA 0 "" 0 "" {TEXT -1 7 "where " }{XPPEDIT 18 0 "rho(x) = (54*x^2-3*x^4+sqrt(64+9*x^8-324*x^6+2916*x^4))^(1/3);" "6#/- %$rhoG6#%\"xG),(*&\"#a\"\"\"*$F'\"\"#F,F,*&\"\"$F,*$F'\"\"%F,!\"\"-%%s qrtG6#,*\"#kF,*&\"\"*F,*$F'\"\")F,F,*&\"$C$F,*$F'\"\"'F,F3*&\"%;HF,*$F 'F2F,F,F,*&F,F,F0F3" }{TEXT -1 2 ". " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 178 "de := diff(y(x),x)=x*(9-x ^2)/(1+y(x)^2);\nic := y(0)=0;\ndsolve(\{de,ic\},y(x));\nw := unapply( rhs(%),x):\nplot(w(x),x=0..4,0..3.7,numpoints=75,font=[HELVETICA,9],la bels=[`x`,`y(x)`]);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%#deG/-%%diffG 6$-%\"yG6#%\"xGF,*(F,\"\"\",&\"\"*F.*$)F,\"\"#F.!\"\"F.,&F.F.*$)F)F3F. F.F4" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%#icG/-%\"yG6#\"\"!F)" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#/-%\"yG6#%\"xG,&*&\"\"#!\"\",(*&\"\"$ \"\"\")F'\"\"%F/F+*&\"#aF/)F'F*F/F/*$,*\"#kF/*&\"\"*F/)F'\"\")F/F/*&\" $C$F/)F'\"\"'F/F+*&\"%;HF/F0F/F/#F/F*F/#F/F.F/*&F*F/F,#F+F.F+" }} {PARA 13 "" 1 "" {GLPLOT2D 503 318 318 {PLOTDATA 2 "6&-%'CURVESG6$7io7 $$\"\"!F)F(7$$\"3()=*=*=*Qx#G!#>$\"3_LLtbH2)f$!#?7$$\"3uPy$y$yZbcF-$\" 3ZF^'eEW*Q9F-7$$\"3;_8N^$ye6)F-$\"3C$\"3aT8Yqv-h6F>7$$\"3oKCVKs3o@F>$\"3c?q**e 5wz?F>7$$\"3$4\"3\"3T.Ds#F>$\"3+#H`Y\")*G6KF>7$$\"3jy$y$y\"=lB$F>$\"3L \\!fpl0?S%F>7$$\"3G(H(H(p](oPF>$\"3XzNO%Rmzr&F>7$$\"3/I(H(Hj=>VF>$\"3n )f#*4g%))4rF>7$$\"373\"3\"3n&y'[F>$\"3[oK(4I9f[)F>7$$\"3oKCVK;BKaF>$\" 3y!44&)e3J')*F>7$$\"3etH(HPL$HfF>$\"3xCnB_*)['F> $\"3g[#>`Yr+B\"Fho7$$\"3Ul['[')o30(F>$\"3?yA^q$e=N\"Fho7$$\"3)yH(H(pzB f(F>$\"3)GF.C0:VY\"Fho7$$\"3m53\"3TBT3)F>$\"3ipKusL^i:Fho7$$\"3u\\'['[ 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7$$\"3AdnvO#pkf#Fho$\"3[Y'[5$oF(e$Fho7$$\"3'>;i@A4pk#Fho$\"3e\"*)HR3Wu g$Fho7$$\"3$ovcn0fTq#Fho$\"3E1Ex&>gui$Fho7$$\"3;Yf%f1Ojv#Fho$\"3)*3G:e W%Hk$Fho7$$\"3[aS09&4M\"GFho$\"3#G5xO+Znl$Fho7$$\"3OnvcZUliGFho$\"3?xe _?\\\"fm$Fho7$$\"3163\"3DQ(=HFho$\"3I;`e!RAJn$Fho7$$\"3k*=*=HE\"H(HFho $\"3IP\"\\xLemn$Fho7$$\"3%pvcnh^q-$Fho$\"3e^O4S8lwOFho7$$\"3-dnvO9*43$ Fho$\"3y,I^j&GHn$Fho7$$\"3m(H(H<2\"G8$Fho$\"3]\\w?2VllOFho7$$\"3'>*=*= tG))=$Fho$\"3#R\\Gi]bMl$Fho7$$\"3-A;ihz@UKFho$\"3q^o+4DNPOFho7$$\"3!>* =*=h2%)H$Fho$\"3%>QOHhJ`h$Fho7$$\"3A;i@wFF\\LFho$\"3YxQ[8&f0f$Fho7$$\" 3)\\'['[I)[0MFho$\"3q>yn%)HPdNFho7$$\"3c0aS&)HLfMFho$\"3a59(yGT$>NFho7 $$\"3['['[1m/8NFho$\"3uI!*3pTpuMFho7$$\"3E#*=*=p]\"pNFho$\"3s<,a='3,U$ Fho7$$\"37.Fq-Y#3i$Fho$\"3w<#3Q&zxhLFho7$$\"31wcnbdutOFho$\"3?^'pWBqHH $Fho7$$\"32dnv'*p'o+i*\\/FFFho7$$\"\"%F)$\"3CxC=r RoRDFho-%'COLOURG6&%$RGBG$\"#5!\"\"F(F(-%%FONTG6$%*HELVETICAG\"\"*-%+A XESLABELSG6$%\"xG%%y(x)G-%%VIEWG6$;F(F`cl;F($\"#PFjcl" 1 2 0 1 10 0 2 9 1 4 2 1.000000 45.000000 45.000000 0 0 "Curve 1" }}}}{PARA 0 "" 0 " " {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 32 "The following code cons tructs a " }{TEXT 260 17 "discrete solution" }{TEXT -1 44 " based on e ach of the methods and gives the " }{TEXT 260 22 "root mean square err or" }{TEXT -1 18 " of each solution." }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 780 "W := (x,y) -> x*(9-x^2)/(1+y^2): hh := 0.01: numstep s := 400: x0 := 0: y0 := 0:\nmatrix([[`slope field: `,W(x,y)],[`init ial point: `,``(x0,y0)],[`step width: `,hh],\n[`no. of steps: `,nu msteps]]);``;\nmthds := [`Prince-Dormand scheme`,`a modification of th e Prince-Dormand scheme`,`Verner's (2004) scheme`,`a modification of V erner's scheme`,`a 2nd modification of Verner's scheme`,`scheme with m aximal imaginary axis inclusion`]: errs := []:\nDigits := 30:\nfor ct \+ to 6 do\n Wn_RK6_||ct := RK6_||ct(W(x,y),x,y,x0,y0,hh,numsteps,false );\n sm := 0: numpts := nops(Wn_RK6_||ct):\n for ii to numpts do\n sm := sm+(Wn_RK6_||ct[ii,2]-evalf(w(Wn_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\"\"\",&\"\"*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)pprint746\" " }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%!G" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#K%'matrixG6#7(7$%6Prince-Dormand~schemeG$\"+>[cO'*!#C7$%La~modif ication~of~the~Prince-Dormand~schemeG$\"+:hk>XF+7$%7Verner's~(2004)~sc hemeG$\"+)fsf*fF+7$%Ba~modification~of~Verner's~schemeG$\"+XoWsiF+7$%F a~2nd~modification~of~Verner's~schemeG$\"+c&oqu%F+7$%Mscheme~with~maxi mal~imaginary~axis~inclusionG$\"+:ooweF+Q)pprint756\"" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 30 "The following code c onstructs " }{TEXT 260 20 "numerical procedures" }{TEXT -1 56 " for so lutions based on each of the Runge-Kutta schemes." }}{PARA 0 "" 0 "" {TEXT -1 75 "The error in the value obtained by each of the methods at the point where " }{XPPEDIT 18 0 "x = 3.499;" "6#/%\"xG-%&FloatG6$\" %*\\$!\"$" }{TEXT -1 16 " is also given." }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 704 "W := (x,y) -> x*(9-x^2)/(1+y^2): hh := 0.01: numstep s := 400: x0 := 0: y0 := 0:\nmatrix([[`slope field: `,W(x,y)],[`init ial point: `,``(x0,y0)],[`step width: `,hh],\n[`no. of steps: `,nu msteps]]);``;\nmthds := [`Prince-Dormand scheme`,`a modification of th e Prince-Dormand scheme`,`Verner's (2004) scheme`,`a modification of V erner's scheme`,`a 2nd modification of Verner's scheme`,`scheme with m aximal imaginary axis inclusion`]: errs := []:\nDigits := 30:\nfor ct \+ to 6 do\n wn_RK6_||ct := RK6_||ct(W(x,y),x,y,x0,y0,hh,numsteps,true) ;\nend do:\nxx := 3.499: wxx := evalf(w(xx)):\nfor ct to 6 do\n errs := [op(errs),abs(wn_RK6_||ct(xx)-wxx)];\nend do:\nDigits := 10:\nlina lg[transpose]([mthds,evalf(errs)]);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6 #K%'matrixG6#7&7$%0slope~field:~~~G*(%\"xG\"\"\",&\"\"*F+*$)F*\"\"#F+! \"\"F+,&F+F+*$)%\"yGF0F+F+F17$%0initial~point:~G-%!G6$\"\"!F;7$%/step~ width:~~~G$F+!\"#7$%1no.~of~steps:~~~G\"$+%Q)pprint456\"" }}{PARA 11 " " 1 "" {XPPMATH 20 "6#%!G" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#K%'matrix G6#7(7$%6Prince-Dormand~schemeG$\"+#H;?E#!#C7$%La~modification~of~the~ Prince-Dormand~schemeG$\"+%zhi[\"F+7$%7Verner's~(2004)~schemeG$\"++r'* G9F+7$%Ba~modification~of~Verner's~schemeG$\"+P\"GL5#F+7$%Fa~2nd~modif ication~of~Verner's~schemeG$\"+u/!Rh\"F+7$%Mscheme~with~maximal~imagin ary~axis~inclusionG$\"+H*4BB#F+Q)pprint466\"" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 4 "The " }{TEXT 260 22 "root \+ mean square error" }{TEXT -1 20 " over the interval " }{XPPEDIT 18 0 "[0,4]" "6#7$\"\"!\"\"%" }{TEXT -1 82 " of each Runge-Kutta method is estimated as follows using the special procedure " }{TEXT 0 5 "NCint " }{TEXT -1 98 " to perform numerical integration by the 7 point Newt on-Cotes method over 200 equal subintervals." }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 454 "mthds := [`Prince-Dormand scheme`,`a modification of the Prince-Dormand scheme`,`Verner's (2004) scheme`,`a modificatio n of Verner's scheme`,`a 2nd modification of Verner's scheme`,`scheme \+ with maximal imaginary axis inclusion`]: errs := []:\nDigits := 20:\nf or ct to 6 do\n sm := NCint((w(x)-'wn_RK6_||ct'(x))^2,x=0..4,adaptiv e=false,numpoints=7,factor=200);\n errs := [op(errs),sqrt(sm/4)];\ne nd do:\nDigits := 10:\nlinalg[transpose]([mthds,evalf(errs)]);" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#K%'matrixG6#7(7$%6Prince-Dormand~schem eG$\"+)Rv\"R%*!#C7$%La~modification~of~the~Prince-Dormand~schemeG$\"+4 MiaWF+7$%7Verner's~(2004)~schemeG$\"+?c3geF+7$%Ba~modification~of~Vern er's~schemeG$\"+,w\\vhF+7$%Fa~2nd~modification~of~Verner's~schemeG$\"+ ]2b#o%F+7$%Mscheme~with~maximal~imaginary~axis~inclusionG$\"+bzbzdF+Q) pprint476\"" }}}{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 0 573 "evalf[20](plot([w(x)-'wn_RK6_1'(x),w(x)-'wn_RK6_2'(x),w(x)-'wn_ RK6_3'(x),\nw(x)-'wn_RK6_4'(x),w(x)-'wn_RK6_5'(x),w(x)-'wn_RK6_6'(x)], x=0..4,color=[COLOR(RGB,.4,0,1),\nCOLOR(RGB,0,.65,0),COLOR(RGB,1,.5,0) ,COLOR(RGB,.85,0,.85),COLOR(RGB,0,.3,1),COLOR(RGB,.95,0,.15)],\nlegend =[`Prince-Dormand scheme`,`a modification of the Prince-Dormand scheme `,`Verner's (2004) scheme`,`a modification of Verner's scheme`,`a 2nd \+ modification of Verner's scheme`,`scheme with maximal imaginary axis i nclusion`],font=[HELVETICA,9],title=`error curves for 8 stage order 6 \+ Runge-Kutta methods`));" }}{PARA 13 "" 1 "" {GLPLOT2D 906 673 673 {PLOTDATA 2 "6,-%'CURVESG6%7[r7$$\"\"!F)F(7$$\"5mmmmm;arz@!#@$!'jS5!#A 7$$\"5LLLLLL3VfVF-$!'&fS)F07$$\"5lmmmmT&)G\\aF-$!'wm^7FM$!(:k(RF-7$$\"5mmmmT&y`3W \"FM$!(*)el'F-7$$\"5LLLLLe'40j\"FM$!'nB**FM7$$\"5mmmm\"H_(zV=FM$!(GkJ 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" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 172 "de := diff(y(x),x )=-(1+cos(2*x))*y(x)^3;\nic := y(0)=sqrt(2);\ndsolve(\{de,ic\},y(x)); \nm := unapply(rhs(%),x):\nplot(m(x),x=0..3,0..1.42,font=[HELVETICA,9] ,labels=[`x`,`y(x)`]);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%#deG/-%%di ffG6$-%\"yG6#%\"xGF,,$*&,&\"\"\"F0-%$cosG6#,$*&\"\"#F0F,F0F0F0F0)F)\" \"$F0!\"\"" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%#icG/-%\"yG6#\"\"!*$\" \"##\"\"\"F+" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/-%\"yG6#%\"xG*&\"\"\" F)*$,(*(\"\"#F)-%$cosGF&F)-%$sinGF&F)F)*&F-F)F'F)F)#F)F-F)#F)F-!\"\"" }}{PARA 13 "" 1 "" {GLPLOT2D 503 318 318 {PLOTDATA 2 "6&-%'CURVESG6$7Y 7$$\"\"!F)$\"3:&4tBc8UT\"!#<7$$\"3$*****\\ilyM;!#>$\"3ozW7@k#*H8F,7$$ \"3')*****\\7t&pKF0$\"3!G<)\\ef9f7F,7$$\"3z****\\(ofV!\\F0$\"3oe\"F,7$$\"3s******\\i9RlF0$\"3kESFh\"zh9\"F,7$$\"33++vVV)RQ*F0$\"3'f)* )e-w\\p5F,7$$\"3/++vVA)GA\"!#=$\"3V)o6<$fq15F,7$$\"3;+]iSS\"Ga\"FJ$\"3 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)**\\P4w)R7#F,$\"3!Qwx>a)*Q4&FJ7$$\"3;++]x%f\")=#F,$\"3q$pQbJ#)G/&FJ7$ $\"3!)**\\P/-a[AF,$\"3gJla\"HTu)\\FJ7$$\"3/+](=Yb;J#F,$\"3c:[>;?IA\\FJ 7$$\"3')****\\i@OtBF,$\"3m09))4iC_[FJ7$$\"3')**\\PfL'zV#F,$\"3%Gjf])o8 tZFJ7$$\"3>+++!*>=+DF,$\"3[G/4+_V#p%FJ7$$\"3-++DE&4Qc#F,$\"3!**R*=7x[1 YFJ7$$\"3=+]P%>5pi#F,$\"3f7E:iH**=XFJ7$$\"39+++bJ*[o#F,$\"3cgVvc$ovV%F J7$$\"33++Dr\"[8v#F,$\"3Ln\\jDQ5WVFJ7$$\"3++++Ijy5GF,$\"3OZ!)Q%zK7E%FJ 7$$\"31+]P/)fT(GF,$\"3)*4_&egIW<%FJ7$$\"31+]i0j\"[$HF,$\"3qns]&)H\\$4% FJ7$$\"\"$F)$\"3ntdq;jW4SFJ-%'COLOURG6&%$RGBG$\"#5!\"\"F(F(-%%FONTG6$% *HELVETICAG\"\"*-%+AXESLABELSG6$%\"xG%%y(x)G-%%VIEWG6$;F(Fc\\l;F($\"$U \"!\"#" 1 2 0 1 10 0 2 9 1 4 2 1.000000 45.000000 45.000000 0 0 "Curve 1" }}}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 32 "T he following code constructs a " }{TEXT 260 17 "discrete solution" } {TEXT -1 44 " based on each of the methods and gives the " }{TEXT 260 22 "root mean square error" }{TEXT -1 18 " of each solution." }} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 793 "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 := [`Prince-Dorman d scheme`,`a modification of the Prince-Dormand scheme`,`Verner's (200 4) scheme`,`a modification of Verner's scheme`,`a 2nd modification of \+ Verner's scheme`,`scheme with maximal imaginary axis inclusion`]: errs := []:\nDigits := 20:\nfor ct to 6 do\n Mn_RK6_||ct := RK6_||ct(M(x ,y),x,y,x0,evalf(y0),hh,numsteps,false);\n sm := 0: numpts := nops(M n_RK6_||ct):\n for ii to numpts do\n sm := sm+(Mn_RK6_||ct[ii,2 ]-evalf(m(Mn_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,$*&,&\"\"\"F,-%$cosG6#,$*&\"\"#F,%\"xGF,F,F,F,)%\"yG\"\"$F, !\"\"7$%0initial~point:~G-%!G6$\"\"!*$F2#F,F27$%/step~width:~~~G$F,!\" #7$%1no.~of~steps:~~~G\"$+$Q)pprint766\"" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%!G" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#K%'matrixG6#7(7$%6Prince- Dormand~schemeG$\"+()4dpu!#A7$%La~modification~of~the~Prince-Dormand~s chemeG$\"+M>>$>\"!#@7$%7Verner's~(2004)~schemeG$\"+PN(*z6F07$%Ba~modif ication~of~Verner's~schemeG$\"+()=r!R\"F07$%Fa~2nd~modification~of~Ver ner's~schemeG$\"+E%*oz6F07$%Mscheme~with~maximal~imaginary~axis~inclus ionG$\"+F(pgT'F+Q)pprint776\"" }}}{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 = 2.999;" "6#/%\"xG-%&FloatG6$\"%**H!\"$" }{TEXT -1 16 " is also given." }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 717 "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]]);``;\n mthds := [`Prince-Dormand scheme`,`a modification of the Prince-Dorman d scheme`,`Verner's (2004) scheme`,`a modification of Verner's scheme` ,`a 2nd modification of Verner's scheme`,`scheme with maximal imaginar y axis inclusion`]: errs := []:\nDigits := 25:\nfor ct to 6 do\n mn_ RK6_||ct := RK6_||ct(M(x,y),x,y,x0,evalf(y0),hh,numsteps,true);\nend d o:\nxx := 2.999: mxx := evalf(m(xx)):\nfor ct to 6 do\n errs := [op( errs),abs(mn_RK6_||ct(xx)-mxx)];\nend do:\nDigits := 10:\nlinalg[trans pose]([mthds,evalf(errs)]);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#K%'matr ixG6#7&7$%0slope~field:~~~G,$*&,&\"\"\"F,-%$cosG6#,$*&\"\"#F,%\"xGF,F, F,F,)%\"yG\"\"$F,!\"\"7$%0initial~point:~G-%!G6$\"\"!*$F2#F,F27$%/step ~width:~~~G$F,!\"#7$%1no.~of~steps:~~~G\"$+$Q)pprint506\"" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%!G" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#K%'matri xG6#7(7$%6Prince-Dormand~schemeG$\"++cW99!#A7$%La~modification~of~the~ Prince-Dormand~schemeG$\"+O$4+I#F+7$%7Verner's~(2004)~schemeG$\"+CqUzA F+7$%Ba~modification~of~Verner's~schemeG$\"+'=dAo#F+7$%Fa~2nd~modifica tion~of~Verner's~schemeG$\"+G&f_F#F+7$%Mscheme~with~maximal~imaginary~ axis~inclusionG$\"+?$z(R7F+Q)pprint516\"" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 4 "The " }{TEXT 260 22 "root mean squa re error" }{TEXT -1 20 " over the interval " }{XPPEDIT 18 0 "[0, 3]; " "6#7$\"\"!\"\"$" }{TEXT -1 82 " of each Runge-Kutta method is estim ated as follows using the special procedure " }{TEXT 0 5 "NCint" } {TEXT -1 98 " to perform numerical integration by the 7 point Newton- Cotes method over 150 equal subintervals." }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 454 "mthds := [`Prince-Dormand scheme`,`a modification of the Prince-Dormand scheme`,`Verner's (2004) scheme`,`a modification o f Verner's scheme`,`a 2nd modification of Verner's scheme`,`scheme wit h maximal imaginary axis inclusion`]: errs := []:\nDigits := 20:\nfor \+ ct to 6 do\n sm := NCint((m(x)-'mn_RK6_||ct'(x))^2,x=0..3,adaptive=f alse,numpoints=7,factor=150);\n errs := [op(errs),sqrt(sm/3)];\nend \+ do:\nDigits := 10:\nlinalg[transpose]([mthds,evalf(errs)]);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#K%'matrixG6#7(7$%6Prince-Dormand~schemeG$\"+ +;unt!#A7$%La~modification~of~the~Prince-Dormand~schemeG$\"+A%ov<\"!#@ 7$%7Verner's~(2004)~schemeG$\"+Pgck6F07$%Ba~modification~of~Verner's~s chemeG$\"+nw\\s8F07$%Fa~2nd~modification~of~Verner's~schemeG$\"+*[bU; \"F07$%Mscheme~with~maximal~imaginary~axis~inclusionG$\"+\"GhBL'F+Q)pp rint526\"" }}}{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 0 575 "evalf[20](plot(['mn_RK6_1'(x)-m(x),'mn_RK6_2'(x)-m(x),'mn_RK6_3 '(x)-m(x),\n'mn_RK6_4'(x)-m(x),'mn_RK6_5'(x)-m(x),'mn_RK6_6'(x)-m(x)], x=0..0.5,color=[COLOR(RGB,.4,0,1),\nCOLOR(RGB,0,.65,0),COLOR(RGB,1,.5, 0),COLOR(RGB,.85,0,.85),COLOR(RGB,0,.3,1),COLOR(RGB,.95,0,.15)],\nlege nd=[`Prince-Dormand scheme`,`a modification of the Prince-Dormand sche me`,`Verner's (2004) scheme`,`a modification of Verner's scheme`,`a 2n d modification of Verner's scheme`,`scheme with maximal imaginary axis inclusion`],font=[HELVETICA,9],title=`error curves for 8 stage order \+ 6 Runge-Kutta methods`));" }}{PARA 13 "" 1 "" {GLPLOT2D 921 477 477 {PLOTDATA 2 "6,-%'CURVESG6%7jp7$$\"\"!F)F(7$$\"5ommmTN@Ki8!#A$!\"&!#>7 $$\"5NLLL$3FWYs#F-$!$T)F07$$\"5omm;aQ`!eS$F-$!%.ZF07$$\"5-+++D1k'p3%F- $!&\\#>F07$$\"5OLL$eRZF\"oZF-$!&#[jF07$$\"5qmmmmT&)G\\aF-$!')ey\"F07$$ \"5SLLL3x1h6oF-$!(>f+\"F07$$\"50+++]7G$R<)F-$!(J\\7%F07$$\"5qmmTNYL^9& )F-$!(;gl&F07$$\"5SLL$3-)Q4b))F-$!(()*ewF07$$\"50++D19Wn&>*F-$!)H>D5F0 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F.-%%sinhG6#%\"yGF.!\"\"7$%0initial~point:~G-%!G6$\"\"!,$*&F-FBF4#F.F- F.7$%/step~width:~~~G$F.!\"#7$%1no.~of~steps:~~~G\"$+&Q)pprint586\"" } }{PARA 11 "" 1 "" {XPPMATH 20 "6#%!G" }}{PARA 11 "" 1 "" {XPPMATH 20 " 6#K%'matrixG6#7(7$%6Prince-Dormand~schemeG$\"+K!H&zd!#@7$%La~modificat ion~of~the~Prince-Dormand~schemeG$\"+*3$)eI(F+7$%7Verner's~(2004)~sche meG$\"+;u?jmF+7$%Ba~modification~of~Verner's~schemeG$\"+.:I(H(F+7$%Fa~ 2nd~modification~of~Verner's~schemeG$\"+(**o')Q(F+7$%Mscheme~with~maxi mal~imaginary~axis~inclusionG$\"+pYih%)F+Q)pprint596\"" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 30 "The following code c onstructs " }{TEXT 260 20 "numerical procedures" }{TEXT -1 56 " for so lutions based on each of the Runge-Kutta schemes." }}{PARA 0 "" 0 "" {TEXT -1 75 "The error in the value obtained by each of the methods at the point where " }{XPPEDIT 18 0 "x = 4.999;" "6#/%\"xG-%&FloatG6$\" %**\\!\"$" }{TEXT -1 16 " is also given." }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 734 "P := (x,y) -> -(2*sin(5*x)+3*cos(7*x))*sinh(y): hh : = 0.01: numsteps := 500: x0 := 0: y0 := sqrt(5)/2:\nmatrix([[`slope fi eld: `,P(x,y)],[`initial point: `,``(x0,y0)],[`step width: `,hh], \n[`no. of steps: `,numsteps]]);``;\nmthds := [`Prince-Dormand schem e`,`a modification of the Prince-Dormand scheme`,`Verner's (2004) sche me`,`a modification of Verner's scheme`,`a 2nd modification of Verner' s scheme`,`scheme with maximal imaginary axis inclusion`]: errs := []: \nDigits := 30:\nfor ct to 6 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 6 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)pprint556\"" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%!G" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#K%'matrixG6#7(7$%6Prince-Dormand~schemeG$\"+)G172 \"!#?7$%La~modification~of~the~Prince-Dormand~schemeG$\"+$>z4e*!#@7$%7 Verner's~(2004)~schemeG$\"+yO2o5F+7$%Ba~modification~of~Verner's~schem eG$\"+Y6q85F+7$%Fa~2nd~modification~of~Verner's~schemeG$\"+BaH$p*F07$% Mscheme~with~maximal~imaginary~axis~inclusionG$\"+3#Grp*F0Q)pprint566 \"" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 4 "The " }{TEXT 260 22 "root mean square error" }{TEXT -1 20 " over the inte rval " }{XPPEDIT 18 0 "[0, 5];" "6#7$\"\"!\"\"&" }{TEXT -1 82 " of e ach Runge-Kutta method is estimated as follows using the special proce dure " }{TEXT 0 5 "NCint" }{TEXT -1 98 " to perform numerical integr ation by the 7 point Newton-Cotes method over 200 equal subintervals. " }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 454 "mthds := [`Prince-Dorma nd scheme`,`a modification of the Prince-Dormand scheme`,`Verner's (20 04) scheme`,`a modification of Verner's scheme`,`a 2nd modification of Verner's scheme`,`scheme with maximal imaginary axis inclusion`]: err s := []:\nDigits := 20:\nfor ct to 6 do\n sm := NCint((p(x)-'pn_RK6_ ||ct'(x))^2,x=0..5,adaptive=false,numpoints=7,factor=200);\n errs := [op(errs),sqrt(sm/5)];\nend do:\nDigits := 10:\nlinalg[transpose]([mt hds,evalf(errs)]);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#K%'matrixG6#7(7$ %6Prince-Dormand~schemeG$\"+ds\\]d!#@7$%La~modification~of~the~Prince- Dormand~schemeG$\"+ECR(G(F+7$%7Verner's~(2004)~schemeG$\"+hUgOmF+7$%Ba ~modification~of~Verner's~schemeG$\"+\\(\\bF(F+7$%Fa~2nd~modification~ of~Verner's~schemeG$\"+v.#)ptF+7$%Mscheme~with~maximal~imaginary~axis~ inclusionG$\"+(yLqW)F+Q)pprint576\"" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 92 "The following error graphs are construc ted using the numerical procedures for the solutions." }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 575 "evalf[20](plot(['pn_RK6_1'(x)-p(x),'pn_R K6_2'(x)-p(x),'pn_RK6_3'(x)-p(x),\n'pn_RK6_4'(x)-p(x),'pn_RK6_5'(x)-p( x),'pn_RK6_6'(x)-p(x)],x=0..2.2,color=[COLOR(RGB,.4,0,1),\nCOLOR(RGB,0 ,.65,0),COLOR(RGB,1,.5,0),COLOR(RGB,.85,0,.85),COLOR(RGB,0,.3,1),COLOR (RGB,.95,0,.15)],\nlegend=[`Prince-Dormand scheme`,`a modification of \+ the Prince-Dormand scheme`,`Verner's (2004) scheme`,`a modification of Verner's scheme`,`a 2nd modification of Verner's scheme`,`scheme with maximal imaginary axis inclusion`],font=[HELVETICA,9],title=`error cu rves for 8 stage order 6 Runge-Kutta methods`));" }}{PARA 13 "" 1 "" {GLPLOT2D 1023 502 502 {PLOTDATA 2 "6,-%'CURVESG6%7dr7$$\"\"!F)F(7$$\" 5LLLLLepo(R#!#@$!(o!4:!#>7$$\"5mmmmm;RP&z%F-$!)FY7D!#?7$$\"5NLL$3x@\"[ QeF-$!)L*f#GF67$$\"5++++v=&)e\")oF-$!)&f&\\IF67$$\"5lmm;z>epCzF-$!))y' *>$F67$$\"5LLLL$37.y'*)F-$!)>W!H$F67$$\"5&****\\P%[ZMa&*F-$!)(4^C$F67$ 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%F1$\"*0l_M#F-7$F_[l$\"*g9pP#F-7$$\"5++](o48%F1$\"*2uDQ#F-7$$\"5++]i:*3u,@%F1$\"*GGbP#F-7$Fd[l$\"*^OMN#F-7$ $\"5mmm;H'z(zWUF1$\"*lg2H#F-7$Fi[l$\"*e)o+AF-7$F^\\l$\"*#[*p%>F-7$Fc\\ l$\"*Y/1r\"F-7$Fh\\l$\"*9*e)[\"F-7$F]]l$\"*=oJK\"F-7$Fb]l$\"*;z3=\"F-7 $Fg]l$\"*W'ef5F-7$F\\^l$\")%4/o*F-7$$\"5MLL3x%H`1h%F1$\")#zRM*F-7$Fa^l $\")aqA#*F-7$$\"5nmm;H*)ozcYF1$\")\"z9F*F-7$Ff^l$\")\")fZ&*F-7$F[_l$\" *S]d+\"F-7$F`_l$\"*3SD3\"F-7$Fe_l$\"*[#e#=\"F-7$Fj_l$\"*&RG;8F-7$Fd`l$ \"*7[uu\"F-7$Fi`l$\"*zs#R?F-7$F^al$\"*n)3-CF-7$Fcal$\"*\"4arGF-7$Fhal$ \"*)p!p]$F-7$F]bl$\"*hfv\"RF-7$Fbbl$\")zlaWF17$Fgbl$\");'G*\\F17$F\\cl $\")SZ9cF17$Ffcl$\")#yTP'F17$F`dl$\")xOtsF17$Fedl$\")xKxzF17$Fjdl$\")w k)R)F17$F_el$\")AII\"*F17$Fdel$\"*b9h+\"F1-Fjel6&F\\fl$\"#&*F`dmF_fl$ \"#:F`dm-Fcfl6#%Mscheme~with~maximal~imaginary~axis~inclusionG-%%FONTG 6$%*HELVETICAG\"\"*-%&TITLEG6#%Uerror~curves~for~8~stage~order~6~Runge -Kutta~methodsG-%+AXESLABELSG6$Q\"x6\"Q!Fcbq-%%VIEWG6$;F(Fdel%(DEFAULT G" 1 2 0 1 10 0 2 9 1 4 2 1.000000 45.000000 45.000000 0 1 "Prince-Dor mand scheme" "a modification of the Prince-Dormand scheme" "Verner's ( 2004) scheme" "a modification of Verner's scheme" "a 2nd modification \+ of Verner's scheme" "scheme with maximal imaginary axis inclusion" }}} }{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 47 "Test 11 of 8 stage, o rder 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 a n article by F. G. Lether: Mathematics of Computation, Vol. 20, no. 9 5, (July 1966) page 382. " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "dy/dx=exp(-x)/(x-1)^2" "6#/ *&%#dyG\"\"\"%#dxG!\"\"*&-%$expG6#,$%\"xGF(F&*$,&F.F&F&F(\"\"#F(" } {TEXT -1 2 " " }{XPPEDIT 18 0 "cos(1/(x-1))-y" "6#,&-%$cosG6#*&\"\"\" F(,&%\"xGF(F(!\"\"F+F(%\"yGF+" }{TEXT -1 5 ", " }{XPPEDIT 18 0 "y(0 )=sin*1" "6#/-%\"yG6#\"\"!*&%$sinG\"\"\"F*F*" }{TEXT -1 1 " " }}{PARA 0 "" 0 "" {TEXT -1 10 "Solution: " }}{PARA 256 "" 0 "" {TEXT -1 2 " \+ " }{XPPEDIT 18 0 "y = -exp(-x)*sin(1/(x-1))" "6#/%\"yG,$*&-%$expG6#,$% \"xG!\"\"\"\"\"-%$sinG6#*&F-F-,&F+F-F-F,F,F-F," }{TEXT -1 1 " " }} {PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 185 "de := diff(y(x),x)=exp(-x)/(x-1)^2*cos(1/(x-1))-y(x);\nic := y(0) =sin(1);\ndsolve(\{de,ic\},y(x));\nq := unapply(rhs(%),x):\nplot(q(x), x=0..1-1/(6*Pi),font=[HELVETICA,9],labels=[`x`,`y(x)`]);" }}{PARA 11 " " 1 "" {XPPMATH 20 "6#>%#deG/-%%diffG6$-%\"yG6#%\"xGF,,&*(-%$expG6#,$F ,!\"\"\"\"\",&F,F4F4F3!\"#-%$cosG6#*&F4F4F5F3F4F4F)F3" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%#icG/-%\"yG6#\"\"!-%$sinG6#\"\"\"" }}{PARA 11 " " 1 "" {XPPMATH 20 "6#/-%\"yG6#%\"xG,$*&-%$expG6#,$F'!\"\"\"\"\"-%$sin G6#*&F/F/,&F'F/F/F.F.F/F." }}{PARA 13 "" 1 "" {GLPLOT2D 603 396 396 {PLOTDATA 2 "6&-%'CURVESG6$7[r7$$\"\"!F)$\"30l*y![)4ZT)!#=7$$\"3#>=\"* )>z2k?!#>$\"3m%p=Qu6DN)F,7$$\"3kMQU\"3@+'QF0$\"3m=O<,5$yO#)F,7$$\"38UQ!)p4'G\"zF0$\"3/Z^$zTlU<)F,7$$\" 3BY$*R0>JO**F0$\"36ty1)z*36\")F,7$$\"3wbXC%*4B\"=\"F,$\"3A;o(=P!Q^!)F, 7$$\"3M!)fxC(zaP\"F,$\"3E8^!ydw!))zF,7$$\"3kgswR?Pw:F,$\"3T8>lD8j?zF,7 $$\"3Qnlb!4?mx\"F,$\"3%p4\\s^P4&yF,7$$\"3OsvSC)*f#)>F,$\"3/$H(=wa6wxF, 7$$\"3)Rk+h)o-k@F,$\"3LR8d$>`qq(F,7$$\"3Q^Vo'yq#oBF,$\"3YB)Qc;#3DwF,7$ $\"3?0sMKLNtDF,$\"3,;%fG`C(F,7$$ \"3S+dSsVlWLF,$\"3&36sy[X09(F,7$$\"3EOur83&\\b$F,$\"37)QgTzpp+(F,7$$\" 3c#**Qwz)4TPF,$\"3M]xfz#>l(oF,7$$\"3wx#p)QELXRF,$\"3UR-VbS%zr'F,7$$\"3 \"yy;Gb6)RTF,$\"3%40quzD%\\lF,7$$\"3p2KM(*)HFM%F,$\"3W'4!o9@F_jF,7$$\" 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\\RTgHcmRGF,7$$\"3C\\b#*Q:tj#*F,$\"3M$o6)eV:lLF,7$$\"3oE3CP5fw#*F,$\"3 Ma)HpV]]I!H!G$*F,$\"3a%4t 07BP*GF,7$$\"3J=s\")G&))3M*F,$\"3f(3)H;[%o+#F,7$$\"3w&\\Kr-[PN*F,$\"33 n1kl[]%*F,$!3m (=[SoWqQ#F,7$$\"3%>saO,CmX*F,$!3GDS5eP#en\"F,7$$\"3AhB\"Gw`IY*F,$!3U$3 !Gg0_(o)F07$$\"3]++(>^$[p%*F,$!3V'=8$[D+C:!#C-%'COLOURG6&%$RGBG$\"#5! \"\"F(F(-%+AXESLABELSG6$%\"xG%%y(x)G-%%FONTG6$%*HELVETICAG\"\"*-%%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 0 826 "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 \+ := [`Prince-Dormand scheme`,`a modification of the Prince-Dormand sche me`,`Verner's (2004) scheme`,`a modification of Verner's scheme`,`a 2n d modification of Verner's scheme`,`scheme with maximal imaginary axis inclusion`]: errs := []:\nDigits := 30:\nfor ct to 6 do\n Qn_RK6_|| ct := RK6_||ct(Q(x,y),x,y,x0,evalf[33](y0),evalf[33](hh),numsteps,fals e);\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 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,&*(-%$expG6#,$%\"xG!\"\"\"\"\",& F/F1F1F0!\"#-%$cosG6#*&F1F1F2F0F1F1%\"yGF07$%0initial~point:~G-%!G6$\" \"!-%$sinG6#F17$%/step~width:~~~G,&#F1\"$+&F1*&F1F1*&\"%+IF1%#PiGF1F0F 07$%1no.~of~steps:~~~GFFQ(pprint06\"" }}{PARA 11 "" 1 "" {XPPMATH 20 " 6#%!G" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#K%'matrixG6#7(7$%6Prince-Dorm and~schemeG$\"+2]=9#*!#>7$%La~modification~of~the~Prince-Dormand~schem eG$\"+Z!H:?\"!#=7$%7Verner's~(2004)~schemeG$\"+b$*of7F07$%Ba~modificat ion~of~Verner's~schemeG$\"+fBnf7F07$%Fa~2nd~modification~of~Verner's~s chemeG$\"+iuG47F07$%Mscheme~with~maximal~imaginary~axis~inclusionG$\"+ `Z4i7F0Q(pprint16\"" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 " " {TEXT -1 30 "The following code constructs " }{TEXT 260 20 "numerica l procedures" }{TEXT -1 56 " for solutions based on each of the Runge- Kutta schemes." }}{PARA 0 "" 0 "" {TEXT -1 75 "The error in the value \+ obtained by each of the methods at the point where " }{XPPEDIT 18 0 " x = 0;" "6#/%\"xG\"\"!" }{TEXT -1 21 ".9469 is also given." }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 750 "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 \+ := [`Prince-Dormand scheme`,`a modification of the Prince-Dormand sche me`,`Verner's (2004) scheme`,`a modification of Verner's scheme`,`a 2n d modification of Verner's scheme`,`scheme with maximal imaginary axis inclusion`]: errs := []:\nDigits := 30:\nfor ct to 6 do\n qn_RK6_|| ct := RK6_||ct(Q(x,y),x,y,x0,evalf(y0),evalf(hh),numsteps,true);\nend \+ do:\nxx := 0.9469: qxx := evalf(q(xx)):\nfor ct to 6 do\n errs := [o p(errs),abs(qn_RK6_||ct(xx)-qxx)];\nend do:\nDigits := 10:\nlinalg[tra nspose]([mthds,evalf(errs)]);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#K%'ma trixG6#7&7$%0slope~field:~~~G,&*(-%$expG6#,$%\"xG!\"\"\"\"\",&F/F1F1F0 !\"#-%$cosG6#*&F1F1F2F0F1F1%\"yGF07$%0initial~point:~G-%!G6$\"\"!-%$si nG6#F17$%/step~width:~~~G,&#F1\"$+&F1*&F1F1*&\"%+IF1%#PiGF1F0F07$%1no. ~of~steps:~~~GFFQ(pprint26\"" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%!G" } }{PARA 11 "" 1 "" {XPPMATH 20 "6#K%'matrixG6#7(7$%6Prince-Dormand~sche meG$\"+=O)Hh\"!#<7$%La~modification~of~the~Prince-Dormand~schemeG$\"+# fQ>5#F+7$%7Verner's~(2004)~schemeG$\"+$)zO.AF+7$%Ba~modification~of~Ve rner's~schemeG$\"+-CM.AF+7$%Fa~2nd~modification~of~Verner's~schemeG$\" +)poa6#F+7$%Mscheme~with~maximal~imaginary~axis~inclusionG$\"+%3Iy?#F+ Q(pprint36\"" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 4 "The " }{TEXT 260 22 "root mean square error" }{TEXT -1 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-K utta method is estimated as follows using the special procedure " } {TEXT 0 5 "NCint" }{TEXT -1 98 " to perform numerical integration by \+ the 7 point Newton-Cotes method over 200 equal subintervals." }} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 474 "mthds := [`Prince-Dormand s cheme`,`a modification of the Prince-Dormand scheme`,`Verner's (2004) \+ scheme`,`a modification of Verner's scheme`,`a 2nd modification of Ver ner's scheme`,`scheme with maximal imaginary axis inclusion`]: errs := []:\nDigits := 20:\nfor ct to 6 do\n sm := NCint((q(x)-'qn_RK6_||ct '(x))^2,x=0..1-1/(6*Pi),adaptive=false,numpoints=7,factor=200);\n er rs := [op(errs),sqrt(sm/(1-1/(6*Pi)))];\nend do:\nDigits := 10:\nlinal g[transpose]([mthds,evalf(errs)]);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6# K%'matrixG6#7(7$%6Prince-Dormand~schemeG$\"+ww0g_!#>7$%La~modification ~of~the~Prince-Dormand~schemeG$\"+pP]soF+7$%7Verner's~(2004)~schemeG$ \"+V!)>3sF+7$%Ba~modification~of~Verner's~schemeG$\"+%ek!3sF+7$%Fa~2nd ~modification~of~Verner's~schemeG$\"+b')GsF+Q(pprint46\"" }}}{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 0 593 "evalf[20](plot(['qn_RK6_1' (x)-q(x),'qn_RK6_2'(x)-q(x),'qn_RK6_3'(x)-q(x),'qn_RK6_4'(x)-q(x),\n'q n_RK6_5'(x)-q(x),'qn_RK6_6'(x)-q(x)],x=0..0.7,-0.9e-18..4.9e-18,color= [COLOR(RGB,.4,0,1),\nCOLOR(RGB,0,.65,0),COLOR(RGB,1,.5,0),COLOR(RGB,.8 5,0,.85),COLOR(RGB,0,.3,1),COLOR(RGB,.95,0,.15)],\nlegend=[`Prince-Dor mand scheme`,`a modification of the Prince-Dormand scheme`,`Verner's ( 2004) scheme`,`a modification of Verner's scheme`,`a 2nd modification \+ of Verner's scheme`,`scheme with maximal imaginary axis inclusion`],fo nt=[HELVETICA,9],title=`error curves for 8 stage order 6 Runge-Kutta m ethods`));" }}{PARA 13 "" 1 "" {GLPLOT2D 685 540 540 {PLOTDATA 2 "6,-% 'CURVESG6%7fn7$$\"\"!F)F(7$$\"5mmmmm\"z+e_\"!#@$!\"#!#?7$$\"5LLLL3->R` GF-$!\"\"F07$$\"5mmmm;apSYVF-$!\"$F07$$\"5lmmm;z'=$\\eF-F.7$$\"5KLLL3F t3XtF-F.7$$\"5lmmmTNj&=t)F-F.7$$\"5+++](=`xn,\"F0$!\"%F07$$\"5mmm;ay/G l6F0$!\"&F07$$\"5+++]PurI88F0FL7$$\"5LLLL$e#3dl9F0$!\"'F07$$\"5mmmm\"H t%o*f\"F0FL7$$\"5++++]F_m]F0$!\"(F07$$\"5++++]s2O [?F0$!#5F07$$\"5mmm;aG\"H5=#F0F^o7$$\"5LLLL$ej%yQBF0$!#8F07$$\"5LLLLLV UUsCF0$!#9F07$$\"5******\\(o()yyi#F0$!#:F07$$\"5LLLLLoD[lFF0$!#=F07$$ 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27$$\"3ID19>zl]ZF2$\"3S[]>-K'z\"F27$$\"37+]iSjE!z%F2$\"3&)4L&Q*oyW=F27$ $\"3y*\\7G))Rb\"[F2$\"3)fCVDB`!)*=F27$$\"3L+++DM\"3%[F2$\"3$Gxb\\1an\" >F27$$\"3i]P4'>]M&[F2$\"3p-qe54u>>F27$$\"3)3](=np3m[F2$\"3zS9J5F#4#>F2 7$$\"3G]7GQPsy[F2$\"3j:7^7UI@>F27$$\"3a+]P40O\"*[F2$\"3L:$**HR(R@>F27$ $\"3s+voa-oX\\F2$\"3%e\"e*Rr89#>F27$$\"\"&F)$\"3n\\kX'z.7#>F2-%'COLOUR G6&%$RGBG$\"#5!\"\"F(F(-%+AXESLABELSG6$%\"xG%%y(x)G-%%FONTG6$%*HELVETI CAG\"\"*-%%VIEWG6$;F(F_[q%(DEFAULTG" 1 2 0 1 10 0 2 9 1 4 2 1.000000 45.000000 45.000000 0 0 "Curve 1" }}}}{PARA 0 "" 0 "" {TEXT -1 0 "" }} {PARA 0 "" 0 "" {TEXT -1 32 "The following code constructs a " }{TEXT 260 17 "discrete solution" }{TEXT -1 44 " based on each of the methods and gives the " }{TEXT 260 22 "root mean square error" }{TEXT -1 18 " of each solution." }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 770 "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 := [` Prince-Dormand scheme`,`a modification of the Prince-Dormand scheme`,` Verner's (2004) scheme`,`a modification of Verner's scheme`,`a 2nd mod ification of Verner's scheme`,`scheme with maximal imaginary axis incl usion`]: errs := []:\nDigits := 30:\nfor ct to 6 do\n Rn_RK6_||ct := RK6_||ct(R(x,y),x,y,x0,y0,hh,numsteps,false);\n sm := 0: numpts := \+ nops(Rn_RK6_||ct):\n for ii to numpts do\n sm := sm+(Rn_RK6_||c t[ii,2]-r(Rn_RK6_||ct[ii,1]))^2;\n end do:\n errs := [op(errs),sqr t(sm/numpts)];\nend do:\nDigits := 10:\nlinalg[transpose]([mthds,evalf (errs)]);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#K%'matrixG6#7&7$%0slope~f ield:~~~G,$*(\"\"&\"\"\"%\"yGF,)-%$sinG6#,$*&\"\"(F,%\"xGF,F,F4F,F,7$% 0initial~point:~G-%!G6$\"\"!F,7$%/step~width:~~~G$F,!\"#7$%1no.~of~ste ps:~~~G\"$+&Q(pprint56\"" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%!G" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#K%'matrixG6#7(7$%6Prince-Dormand~schem eG$\"+V2<^;!#>7$%La~modification~of~the~Prince-Dormand~schemeG$\"+dBr4 J!#?7$%7Verner's~(2004)~schemeG$\"+))GqfSF07$%Ba~modification~of~Verne r's~schemeG$\"+C\"=w3$F07$%Fa~2nd~modification~of~Verner's~schemeG$\"+ Vn)G,$F07$%Mscheme~with~maximal~imaginary~axis~inclusionG$\"+.Pi$[$F0Q (pprint66\"" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 30 "The following code constructs " }{TEXT 260 20 "numerical proced ures" }{TEXT -1 56 " for solutions based on each of the Runge-Kutta sc hemes." }}{PARA 0 "" 0 "" {TEXT -1 75 "The error in the value obtained by each of the methods at the point where " }{XPPEDIT 18 0 "x = 4.99 9;" "6#/%\"xG-%&FloatG6$\"%**\\!\"$" }{TEXT -1 16 " is also given." } }{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 701 "R := (x,y) -> 5*y*sin(7*x) ^7: hh := 0.01: numsteps := 500: x0 := 0: y0 := 1:\nmatrix([[`slope fi eld: `,R(x,y)],[`initial point: `,``(x0,y0)],[`step width: `,hh], \n[`no. of steps: `,numsteps]]);``;\nmthds := [`Prince-Dormand schem e`,`a modification of the Prince-Dormand scheme`,`Verner's (2004) sche me`,`a modification of Verner's scheme`,`a 2nd modification of Verner' s scheme`,`scheme with maximal imaginary axis inclusion`]: errs := []: \nDigits := 25:\nfor ct to 6 do\n rn_RK6_||ct := RK6_||ct(R(x,y),x,y ,x0,y0,hh,numsteps,true);\nend do:\nxx := 4.999: rxx := evalf(r(xx)): \nfor ct to 6 do\n errs := [op(errs),abs(rn_RK6_||ct(xx)-rxx)];\nend do:\nDigits := 10:\nlinalg[transpose]([mthds,evalf(errs)]);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#K%'matrixG6#7&7$%0slope~field:~~~G,$*(\"\"& \"\"\"%\"yGF,)-%$sinG6#,$*&\"\"(F,%\"xGF,F,F4F,F,7$%0initial~point:~G- %!G6$\"\"!F,7$%/step~width:~~~G$F,!\"#7$%1no.~of~steps:~~~G\"$+&Q(ppri nt76\"" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%!G" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#K%'matrixG6#7(7$%6Prince-Dormand~schemeG$\"+%eKV!H!#>7$ %La~modification~of~the~Prince-Dormand~schemeG$\"+So\"=#H!#@7$%7Verner 's~(2004)~schemeG$\"+1%*3gD!#?7$%Ba~modification~of~Verner's~schemeG$ \"+BL&R_#F57$%Fa~2nd~modification~of~Verner's~schemeG$\"+&ofpQ$F07$%Ms cheme~with~maximal~imaginary~axis~inclusionG$\"+L4?]hF5Q(pprint86\"" } }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 4 "The " } {TEXT 260 22 "root mean square error" }{TEXT -1 20 " over the interval " }{XPPEDIT 18 0 "[0, 5];" "6#7$\"\"!\"\"&" }{TEXT -1 82 " of each \+ Runge-Kutta method is estimated as follows using the special procedure " }{TEXT 0 5 "NCint" }{TEXT -1 98 " to perform numerical integratio n by the 7 point Newton-Cotes method over 200 equal subintervals." }} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 454 "mthds := [`Prince-Dormand s cheme`,`a modification of the Prince-Dormand scheme`,`Verner's (2004) \+ scheme`,`a modification of Verner's scheme`,`a 2nd modification of Ver ner's scheme`,`scheme with maximal imaginary axis inclusion`]: errs := []:\nDigits := 20:\nfor ct to 6 do\n sm := NCint((r(x)-'rn_RK6_||ct '(x))^2,x=0..5,adaptive=false,numpoints=7,factor=200);\n errs := [op (errs),sqrt(sm/5)];\nend do:\nDigits := 10:\nlinalg[transpose]([mthds, evalf(errs)]);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#K%'matrixG6#7(7$%6Pr ince-Dormand~schemeG$\"+YS#)[;!#>7$%La~modification~of~the~Prince-Dorm and~schemeG$\"+7)p;6$!#?7$%7Verner's~(2004)~schemeG$\"+9]*[1%F07$%Ba~m odification~of~Verner's~schemeG$\"+%p&R\"4$F07$%Fa~2nd~modification~of ~Verner's~schemeG$\"+![!*\\,$F07$%Mscheme~with~maximal~imaginary~axis~ inclusionG$\"+(efGZ$F0Q(pprint96\"" }}}{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 0 573 "evalf[20](plot([r(x)-'rn_RK6_1'(x),r(x)- 'rn_RK6_2'(x),r(x)-'rn_RK6_3'(x),\nr(x)-'rn_RK6_4'(x),r(x)-'rn_RK6_5'( x),r(x)-'rn_RK6_6'(x)],x=0..5,color=[COLOR(RGB,.4,0,1),\nCOLOR(RGB,0,. 65,0),COLOR(RGB,1,.5,0),COLOR(RGB,.85,0,.85),COLOR(RGB,0,.3,1),COLOR(R GB,.95,0,.15)],\nlegend=[`Prince-Dormand scheme`,`a modification of th e Prince-Dormand scheme`,`Verner's (2004) scheme`,`a modification of V erner's scheme`,`a 2nd modification of Verner's scheme`,`scheme with m aximal imaginary axis inclusion`],font=[HELVETICA,9],title=`error curv es for 8 stage order 6 Runge-Kutta methods`));" }}{PARA 13 "" 1 "" {GLPLOT2D 990 636 636 {PLOTDATA 2 "6,-%'CURVESG6%7\\[m7$$\"\"!F)F(7$$ \"5MLLLL3x&)*3\"!#?$\")$*f&**)!#>7$$\"5nmm\"z%\\v#pK\"F-$\")gC:%*F07$$ \"5+++]i!R(*Rc\"F-$\")yDt**F07$$\"5MLL3xJs1,=F-$\"*AtYl\"F07$$\"5nmmm \"H2P\"Q?F-$\"*5MBk#F07$$\"5MLe*)fI&*y/@F-$\"*D'R[OF07$$\"5++]7G))>Wr@ F-$\"*L>8\"RF07$$\"5nmTN'fW%4QAF-$\"*KPe^&F07$$\"5MLLek.pu/BF-$\"*\"*4 ,0)F07$$\"5n;zp[KJ2QBF-$\"*X$=(=)F07$$\"5++D\"G8O*RrBF-$\"*2p$*e)F07$$ \"5M$3Fp,fDZS#F-$\"+SH:Q6F07$$\"5nm;/,>=0QCF-$\"+h&4m:\"F07$$\"5+]i:&y /y8Z#F-$\"+b1$H?\"F07$$\"5ML3FpwUq/DF-$\"+]-_&\\\"F07$$\"5n;aQ`00.QDF- $\"+t*3y^\"F07$$\"5+++]PMnNrDF-$\"+F\\ij:F07$$\"5nm\"Hd?>4!QEF-$\"+(3% fO=F07$$\"5ML$eR(\\;m/FF-$\"+Wf9J?F07$$\"5++v=U2TJrFF-$\"+T=cz?F07$$\" 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0$!*8RXw&F07$F_bw$!*\\,\"QlF07$Fdbw$!*3VXI(F07$Fibw$!*%>%pf(F07$F^cw$! *x^Ne(F07$Fccw$!*?9(=rF07$Fhcw$!*8J\\A(F07$F]dw$!*$p];rF07$$\"5-Dc^G#R Z6q%F0$!*rVV;'F07$Fbdw$!*))\\(fiF07$$\"5oT&)e'ez[xq%F0$!*%4Q'4'F07$Fgd w$!*@#GR[F07$F\\ew$!*tm[\"\\F07$Faew$!*$3caZF07$Ffew$!*YD6_$F07$F[fw$! *%oPvNF07$F`fw$!*&4pqMF07$Fefw$!*JD]g#F07$Fjfw$!*<]Sk#F07$F_gw$!*9?5i# F07$Fdgw$!*fF\"GBF07$Figw$!*tX8O#F07$F^hw$!*jwRT#F07$Fchw$!*>))oo#F07$ Fhhw$!*51O#GF07$F]iw$!*q*\\PNF07$Fbiw$!*!GwWXF07$Fgiw$!*$>X5ZF07$F\\jw $!*HXKn&F07$Fajw$!*%*QOr&F07$Ffjw$!*f-'HeF07$F[[x$!*P50!oF07$F_\\x$!*W 9,(yF07$Fi\\x$!*i*fE()F07$F^]x$!*Y+d!*)F07$Fc]x$!*5:#z\"*F07$Fh]x$!*(* eO>*F07$F]^x$!*g*z9#*F07$Fb^x$!*L4y@*F07$Fg^x$!*'*R]A*F07$F\\_x$!*9!ew \"*F07$Fa_x$!*h/g#))F07$Ff_x$!*=/!G\")F07$F[`x$!*i&\\)G(F07$F``x$!*D)e xjF07$Fe`x$!*ozm'eF07$Fj`x$!*#*RZD'F07$F_ax$!*2>fm'F07$Fdax$!*c/GP'F07 $Fiax$!*oBr*fF07$F^bx$!*Rs3*fF07$Fcbx$!*,+)*)eF07$Fhbx$!*0?5'eF07$F]cx $!*%G[jeF07$Fbcx$!*)GemfF07$Fgcx$!*p#RljF0-F\\dx6&F^dx$\"#&*FadxF($\"# :Fadx-Fcdx6#%Mscheme~with~maximal~imaginary~axis~inclusionG-%&TITLEG6# %Uerror~curves~for~8~stage~order~6~Runge-Kutta~methodsG-%+AXESLABELSG6 $Q\"x6\"Q!Ffcjm-%%FONTG6$%*HELVETICAG\"\"*-%%VIEWG6$;F(Fgcx%(DEFAULTG " 1 2 0 1 10 0 2 9 1 4 2 1.000000 45.000000 45.000000 0 1 "a modificat ion of the Prince-Dormand scheme" "Verner's (2004) scheme" "a modifica tion of Verner's scheme" "a 2nd modification of Verner's scheme" "sche me with maximal imaginary axis inclusion" }}}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 47 "Test 13 of 8 stage, order 6 Runge-Kutta m ethods" }}{PARA 256 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 72 "See: \"Mathematica in Action\" by Stan Wagon, Springer-Verlag, pa ge 302. " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 256 "" 0 "" {TEXT -1 3 " " }{XPPEDIT 18 0 "dy/dx = cos*x+2*y;" "6#/*&%#dyG\"\"\"%#dxG! \"\",&*&%$cosGF&%\"xGF&F&*&\"\"#F&%\"yGF&F&" }{TEXT -1 8 ", " } {XPPEDIT 18 0 "y(0) = -2/5;" "6#/-%\"yG6#\"\"!,$*&\"\"#\"\"\"\"\"&!\" \"F-" }{TEXT -1 1 " " }}{PARA 0 "" 0 "" {TEXT -1 10 "Solution: " }} {PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "y = 1/5;" "6#/%\"yG*& \"\"\"F&\"\"&!\"\"" }{TEXT -1 1 " " }{XPPEDIT 18 0 "sin*x-2/5" "6#,&*& %$sinG\"\"\"%\"xGF&F&*&\"\"#F&\"\"&!\"\"F+" }{TEXT -1 1 " " }{XPPEDIT 18 0 "cos*x" "6#*&%$cosG\"\"\"%\"xGF%" }{TEXT -1 3 " . " }}{PARA 0 "" 0 "" {TEXT 260 4 "Note" }{TEXT -1 2 ": " }}{PARA 0 "" 0 "" {TEXT -1 53 "The general solution of the differential equation " }{XPPEDIT 18 0 "dy/dx = cos*x+2*y;" "6#/*&%#dyG\"\"\"%#dxG!\"\",&*&%$cosGF&%\"xG F&F&*&\"\"#F&%\"yGF&F&" }{TEXT -1 64 " contains an exponential term, \+ but with the initial condition " }{XPPEDIT 18 0 "y(0) = -2/5" "6#/-% \"yG6#\"\"!,$*&\"\"#\"\"\"\"\"&!\"\"F-" }{TEXT -1 23 " this term disa ppears." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 50 "de := diff(y(x),x)=cos(x)+2*y(x);\ndsolve(de,y(x));" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%#deG/-%%diffG6$-%\"yG6#%\"xGF,,&-% $cosGF+\"\"\"*&\"\"#F0F)F0F0" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/-%\"y G6#%\"xG,(*&#\"\"#\"\"&\"\"\"-%$cosGF&F-!\"\"*&#F-F,F--%$sinGF&F-F-*&- %$expG6#,$*&F+F-F'F-F-F-%$_C1GF-F-" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" } }{PARA 0 "" 0 "" {TEXT -1 105 "Any slight deviation of a numerical sol ution from the correct solution tends to become rapidly magnified." }} {PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 154 "de := diff(y(x),x)=cos(x)+2*y(x);\nic := y(0)=-2/5;\ndsolve(\{de, ic\},y(x));\ne := unapply(rhs(%),x):\nplot(e(x),x=0..8,font=[HELVETICA ,9],labels=[`x`,`y(x)`]);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%#deG/-% %diffG6$-%\"yG6#%\"xGF,,&-%$cosGF+\"\"\"*&\"\"#F0F)F0F0" }}{PARA 11 " " 1 "" {XPPMATH 20 "6#>%#icG/-%\"yG6#\"\"!#!\"#\"\"&" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/-%\"yG6#%\"xG,&*&#\"\"#\"\"&\"\"\"-%$cosGF&F-!\"\"* &#F-F,F--%$sinGF&F-F-" }}{PARA 13 "" 1 "" {GLPLOT2D 703 312 312 {PLOTDATA 2 "6&-%'CURVESG6$7gn7$$\"\"!F)$!3A+++++++S!#=7$$\"3ELLLLBxV< F,$!3'*f!G(3%\\Bf$F,7$$\"3ymmm;$>5E$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$$\"3?LLL=\"\\g**FK7$$\"3\") *****\\[A4]\"FO$\"3Xgu?U;&er\"F,7$$\"3wmmm'3Q\\n\"FO$\"3S\\4.g(y\\S#F, 7$$\"3OLLLB6@G=FO$\"3e*[f2BGC&HF,7$$\"3&)******f-w+?FO$\"375@EVOJ&[$F, 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square error" }{TEXT -1 18 " of each solution." }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 783 "E := \+ (x,y) -> cos(x)+2*y: hh := 0.02: numsteps := 400: x0 := 0: y0 := -2/5: \nmatrix([[`slope field: `,E(x,y)],[`initial point: `,``(x0,y0)],[`s tep width: `,hh],\n[`no. of steps: `,numsteps]]);``;\nmthds := [`P rince-Dormand scheme`,`a modification of the Prince-Dormand scheme`,`V erner's (2004) scheme`,`a modification of Verner's scheme`,`a 2nd modi fication of Verner's scheme`,`scheme with maximal imaginary axis inclu sion`]: errs := []:\nDigits := 20:\nfor ct to 6 do\n En_RK6_||ct := \+ RK6_||ct(E(x,y),x,y,x0,evalf(y0),hh,numsteps,false);\n sm := 0: nump ts := nops(En_RK6_||ct):\n for ii to numpts do\n sm := sm+(En_R K6_||ct[ii,2]-evalf(e(En_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,&-%$cosG6#%\"xG\"\"\"*&\"\"#F.%\"yGF.F.7$%0init ial~point:~G-%!G6$\"\"!#!\"#\"\"&7$%/step~width:~~~G$F0F97$%1no.~of~st eps:~~~G\"$+%Q)pprint106\"" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%!G" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#K%'matrixG6#7(7$%6Prince-Dormand~schem eG$\"+#3-Gx(!#=7$%La~modification~of~the~Prince-Dormand~schemeG$\"+AA% 3(=!#<7$%7Verner's~(2004)~schemeG$\"+_8L#Q\"F07$%Ba~modification~of~Ve rner's~schemeG$\"+bWD,@F0 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 \+ = 7.999;" "6#/%\"xG-%&FloatG6$\"%**z!\"$" }{TEXT -1 16 " is also give n." }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 707 "E := (x,y) -> cos(x)+ 2*y: hh := 0.02: numsteps := 400: x0 := 0: y0 := -2/5:\nmatrix([[`slop e field: `,E(x,y)],[`initial point: `,``(x0,y0)],[`step width: `,h h],\n[`no. of steps: `,numsteps]]);``;\nmthds := [`Prince-Dormand sc heme`,`a modification of the Prince-Dormand scheme`,`Verner's (2004) s cheme`,`a modification of Verner's scheme`,`a 2nd modification of Vern er's scheme`,`scheme with maximal imaginary axis inclusion`]: errs := \+ []:\nDigits := 20:\nfor ct to 6 do\n en_RK6_||ct := RK6_||ct(E(x,y), x,y,x0,evalf(y0),hh,numsteps,true);\nend do:\nxx := 7.999: exx := eval f(e(xx)):\nfor ct to 6 do\n errs := [op(errs),abs(en_RK6_||ct(xx)-ex x)];\nend do:\nDigits := 10:\nlinalg[transpose]([mthds,evalf(errs)]); " }}{PARA 11 "" 1 "" {XPPMATH 20 "6#K%'matrixG6#7&7$%0slope~field:~~~G ,&-%$cosG6#%\"xG\"\"\"*&\"\"#F.%\"yGF.F.7$%0initial~point:~G-%!G6$\"\" !#!\"#\"\"&7$%/step~width:~~~G$F0F97$%1no.~of~steps:~~~G\"$+%Q)pprint1 26\"" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%!G" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#K%'matrixG6#7(7$%6Prince-Dormand~schemeG$\"+%=JsI%!#<7$ %La~modification~of~the~Prince-Dormand~schemeG$\"+O6rO5!#;7$%7Verner's ~(2004)~schemeG$\"+V-2gwF+7$%Ba~modification~of~Verner's~schemeG$\"+)G btU*F+7$%Fa~2nd~modification~of~Verner's~schemeG$\"+ug*4.\"F07$%Mschem e~with~maximal~imaginary~axis~inclusionG$\"+zqCu6F0Q)pprint136\"" }}} {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 4 "The " } {TEXT 260 22 "root mean square error" }{TEXT -1 20 " over the interval " }{XPPEDIT 18 0 "[0, 8];" "6#7$\"\"!\"\")" }{TEXT -1 82 " of each \+ Runge-Kutta method is estimated as follows using the special procedure " }{TEXT 0 5 "NCint" }{TEXT -1 98 " to perform numerical integratio n by the 7 point Newton-Cotes method over 200 equal subintervals." }} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 454 "mthds := [`Prince-Dormand s cheme`,`a modification of the Prince-Dormand scheme`,`Verner's (2004) \+ scheme`,`a modification of Verner's scheme`,`a 2nd modification of Ver ner's scheme`,`scheme with maximal imaginary axis inclusion`]: errs := []:\nDigits := 20:\nfor ct to 6 do\n sm := NCint((e(x)-'en_RK6_||ct '(x))^2,x=0..8,adaptive=false,numpoints=7,factor=200);\n errs := [op (errs),sqrt(sm/8)];\nend do:\nDigits := 10:\nlinalg[transpose]([mthds, evalf(errs)]);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#K%'matrixG6#7(7$%6Pr ince-Dormand~schemeG$\"+F^UHw!#=7$%La~modification~of~the~Prince-Dorma nd~schemeG$\"+&pKj$=!#<7$%7Verner's~(2004)~schemeG$\"+!*G$oN\"F07$%Ba~ modification~of~Verner's~schemeG$\"+UJ()p;F07$%Fa~2nd~modification~of~ Verner's~schemeG$\"+r&4i#=F07$%Mscheme~with~maximal~imaginary~axis~inc lusionG$\"+64&*z?F0Q)pprint146\"" }}}{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 0 573 "evalf[20](plot(['en_RK6_1'(x)-e(x),'en_RK6 _2'(x)-e(x),'en_RK6_3'(x)-e(x),\n'en_RK6_4'(x)-e(x),'en_RK6_5'(x)-e(x) ,'en_RK6_6'(x)-e(x)],x=0..2,color=[COLOR(RGB,.4,0,1),\nCOLOR(RGB,0,.65 ,0),COLOR(RGB,1,.5,0),COLOR(RGB,.85,0,.85),COLOR(RGB,0,.3,1),COLOR(RGB ,.95,0,.15)],\nlegend=[`Prince-Dormand scheme`,`a modification of the \+ Prince-Dormand scheme`,`Verner's (2004) scheme`,`a modification of Ver ner's scheme`,`a 2nd modification of Verner's scheme`,`scheme with max imal imaginary axis inclusion`],font=[HELVETICA,9],title=`error curves for 8 stage order 6 Runge-Kutta methods`));" }}{PARA 13 "" 1 "" {GLPLOT2D 604 604 604 {PLOTDATA 2 "6,-%'CURVESG6%7S7$$\"\"!F)F(7$$\"5M LLLLL3VfV!#@$\"()eMz!#A7$$\"5nmmmm\"H[D:)F-$\"(E8i\"F-7$$\"5LLLLLe0$=C \"!#?$\"(K*3DF-7$$\"5LLLLL3RBr;F9$\"(=UX$F-7$$\"5nmmm;zjf)4#F9$\"(+'fW F-7$$\"5MLLL$e4;[\\#F9$\"(37\\&F-7$$\"5++++]i'y]!HF9$\"(orf'F-7$$\"5ML LL$ezs$HLF9$\"(')oz(F-7$$\"5++++]7iI_PF9$\"(A[3*F-7$$\"5nmmmm;_M(=%F9$ \")Ayc5F-7$$\"5MLLLL3y_qXF9$\")7Y#=\"F-7$$\"5+++++]1!>+&F9$\")CL\\8F-7 $$\"5+++++]Z/NaF9$\"(?>^\"F97$$\"5+++++]$fC&eF9$\"(%[!o\"F97$$\"5MLLL$ ez6:B'F9$\"(ie%=F97$$\"5nmmmm;=C#o'F9$\"(Z&\\?F97$$\"5nmmmmm#pS1(F9$\" 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\",,j>aa#F-7$Fds$\",\"3()Q;LF-7$Fis$\",EmK`<%F-7$F^t$\",(4:_a`F-7$Fct$ \",h!Q\\BpF-7$Fht$\",T8CC!*)F-7$F]u$\"-15rRN6F-7$Fbu$\".mv')[u[\"FH7$F gu$\".u0%)))f*=FH7$F\\v$\".y`%oxcCFH7$Fav$\"/iRL6&p5$Fgn7$Ffv$\".Sl%y< ;SFH7$F[w$\".7]k[L6&FH7$F`w$\".+Oh_=e'FH7$Few$\".Sk%[\\C%)FH7$Fjw$\".% p)*o&34\"F-7$F_x$\".Hmu7\"*R\"F-7$Fdx$\".WgI\"F-7$F:$\"*^t8a\"F-7$F?$\"*'*o%))>F- 7$FD$\"+;%pVc#FH7$FJ$\"+SP6ZKFH7$FO$\"+Q))zZTFH7$FT$\"+'>VTM&FH7$FY$\" ,')H!4#)oFgn7$Fin$\"+u%)>H*)FH7$F^o$\",o[UK7\"FH7$Fco$\",sy1XX\"FH7$Fh o$\",%H9v&)=FH7$F]p$\"+v_/ACF-7$Fbp$\"+^*[-/$F-7$Fgp$\"+(*e/%)RF-7$F\\ q$\"+))))f4]F-7$Faq$\"+\"\\x#RlF-7$Ffq$\"+wv*)y#)F-7$F[r$\",f')QC2\"F- 7$F`r$\",5$[;s8F-7$Fer$\",`3SXx\"F-7$Fjr$\",M!z?ZAF-7$F_s$\",1A9\"**GF -7$Fds$\",0$R?xPF-7$Fis$\",vw\"\\bZF-7$F^t$\",Vx?&)4'F-7$Fct$\",W*3\\& )yF-7$Fht$\"-Auz$R,\"F-7$F]u$\"-#)*GbJH\"F-7$Fbu$\".wtLATp\"FH7$Fgu$\" .9*RjUf@FH7$F\\v$\".3QPK\")z#FH7$Fav$\"/i0)QW'QNFgn7$Ffv$\".IIn-Ud%FH7 $F[w$\".iF\")=Q#eFH7$F`w$\".q9uij\\(FH7$Few$\".+CMH]f*FH7$Fjw$\".(y:^U U7F-7$F_x$\".xWL6Nf\"F-7$Fdx$\".c'=%e`0#F-7$Fix$\".MFE\"[XEF-7$F^y$\". 6xPTgL$F-7$Fcy$\".Y]\\]-\"QF-7$Fhy$\".fU6n=N%F-7$F]z$\".hi)f?,\\F-7$Fb z$\".viA)))>bF-7$Fgz$\".&=U#*yliF-7$F\\[l$\".1$\\O[7rF-7$Fa[l$\".t3:R) H!)F-7$Ff[l$\".'*f_8b1*F-7$F[\\l$\".\">0=7w'*F-7$F`\\l$\"/)3pw&yK5F-7$ Fe\\l$\"/OUA\"\\B5\"F-7$Fj\\l$\"/3'>#zfw6F--F_]l6&Fa]l$\"#&*F[ilFe]l$ \"#:F[il-Fi]l6#%Mscheme~with~maximal~imaginary~axis~inclusionG-%&TITLE G6#%Uerror~curves~for~8~stage~order~6~Runge-Kutta~methodsG-%+AXESLABEL SG6$Q\"x6\"Q!Fafo-%%FONTG6$%*HELVETICAG\"\"*-%%VIEWG6$;F(Fj\\l%(DEFAUL TG" 1 2 0 1 10 0 2 9 1 4 2 1.000000 45.000000 45.000000 0 1 "Prince-Do rmand scheme" "a modification of the Prince-Dormand scheme" "Verner's \+ (2004) scheme" "a modification of Verner's scheme" "a 2nd modification of Verner's scheme" "scheme with maximal imaginary axis inclusion" }} }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 2 " ;" }}}}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 47 "Test 14 of 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&%\"y GF&F(" }{TEXT -1 8 ", " }{XPPEDIT 18 0 "y(0) = sqrt(5);" "6#/-% \"yG6#\"\"!-%%sqrtG6#\"\"&" }{TEXT -1 1 " " }}{PARA 0 "" 0 "" {TEXT -1 10 "Solution: " }}{PARA 256 "" 0 "" {TEXT -1 2 " " }{XPPEDIT 18 0 "y=100/101" "6#/%\"yG*&\"$+\"\"\"\"\"$,\"!\"\"" }{TEXT -1 1 " " } {XPPEDIT 18 0 "x*cos*x-990/10201" "6#,&*(%\"xG\"\"\"%$cosGF&F%F&F&*&\" $!**F&\"&,-\"!\"\"F+" }{TEXT -1 1 " " }{XPPEDIT 18 0 "cos*x+10/101" "6 #,&*&%$cosG\"\"\"%\"xGF&F&*&\"#5F&\"$,\"!\"\"F&" }{TEXT -1 1 " " } {XPPEDIT 18 0 "x*sin*x-200/10201" "6#,&*(%\"xG\"\"\"%$sinGF&F%F&F&*&\" $+#F&\"&,-\"!\"\"F+" }{TEXT -1 1 " " }{XPPEDIT 18 0 "sin*x+(990/10201+ sqrt(5))*exp(-10*x)" "6#,&*&%$sinG\"\"\"%\"xGF&F&*&,&*&\"$!**F&\"&,-\" !\"\"F&-%%sqrtG6#\"\"&F&F&-%$expG6#,$*&\"#5F&F'F&F-F&F&" }{TEXT -1 1 " ." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 163 "de := diff(y(x),x)=10*x*cos(x)-10*y(x);\nic := y(0)= sqrt(5);\ndsolve(\{de,ic\},y(x));\nb := unapply(rhs(%),x):\nplot(b(x), x=0..5,font=[HELVETICA,9],labels=[`x`,`y(x)`]);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%#deG/-%%diffG6$-%\"yG6#%\"xGF,,&*(\"#5\"\"\"F,F0-%$co sGF+F0F0*&F/F0F)F0!\"\"" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%#icG/-%\" yG6#\"\"!*$\"\"&#\"\"\"\"\"#" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/-%\"y G6#%\"xG,,*&#\"$+\"\"$,\"\"\"\"*&F'F--%$cosGF&F-F-F-*&#\"$!**\"&,-\"F- F/F-!\"\"*&#\"#5F,F-*&-%$sinGF&F-F'F-F-F-*&#\"$+#F4F-F:F-F5*&-%$expG6# ,$*&F8F-F'F-F5F-,&#F3F4F-*$\"\"&#F-\"\"#F-F-F-" }}{PARA 13 "" 1 "" {GLPLOT2D 703 312 312 {PLOTDATA 2 "6&-%'CURVESG6$7hn7$$\"\"!F)$\"3\")* y*\\xz1OA!#<7$$\"3ALL$3FWYs#!#>$\"3*pc)\\jF;12h\"H\"48F,7$$\"3m****\\7G$R<)F0$\"3<_u(oLbK,\"F,7$$\"3GLLL3x&) 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!*H%e)HF,7$$\"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$$\"3%o;Hd!fX$f$F,$!3#*y!45U t-E$F,7$$\"3e++]iC$pk$F,$!3LO[nw')*)RKF,7$$\"3ILe*[t\\sp$F,$!3D1>x`HA5 KF,7$$\"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$$\"3a+++D>#[Z%F,$!3(y*pyl_QJ9F,7$$\"3Sn mT&G!e&e%F,$!3]X/0\"RC%G**F@7$$\"3#RLLL)Qk%o%F,$!3u!*)Q\"4WH+dF@7$$\"3 7+]iSjE!z%F,$!3+r[gfMO'=*F07$$\"3a+]P40O\"*[F,$\"3+2*eSHde(QF@7$$\"\"& F)$\"3&Q8`\">jC3#*F@-%'COLOURG6&%$RGBG$\"#5!\"\"F(F(-%+AXESLABELSG6$% \"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 follo wing code constructs a " }{TEXT 260 17 "discrete solution" }{TEXT -1 44 " based on each of the methods and gives the " }{TEXT 260 22 "root \+ mean square error" }{TEXT -1 18 " of each solution." }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 792 "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. o f steps: `,numsteps]]);``;\nmthds := [`Prince-Dormand scheme`,`a mod ification of the Prince-Dormand scheme`,`Verner's (2004) scheme`,`a mo dification of Verner's scheme`,`a 2nd modification of Verner's scheme` ,`scheme with maximal imaginary axis inclusion`]: errs := []:\nDigits \+ := 20:\nfor ct to 6 do\n Bn_RK6_||ct := RK6_||ct(B(x,y),x,y,x0,evalf (y0),hh,numsteps,false);\n sm := 0: numpts := nops(Bn_RK6_||ct):\n \+ for ii to numpts do\n sm := sm+(Bn_RK6_||ct[ii,2]-evalf(b(Bn_RK6 _||ct[ii,1])))^2;\n end do:\n errs := [op(errs),sqrt(sm/numpts)]; \nend do:\nDigits := 10:\nlinalg[transpose]([mthds,evalf(errs)]);" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#K%'matrixG6#7&7$%0slope~field:~~~G,&*( \"#5\"\"\"%\"xGF,-%$cosG6#F-F,F,*&F+F,%\"yGF,!\"\"7$%0initial~point:~G -%!G6$\"\"!*$\"\"&#F,\"\"#7$%/step~width:~~~G$F,!\"#7$%1no.~of~steps:~ ~~G\"$+&Q)pprint156\"" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%!G" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#K%'matrixG6#7(7$%6Prince-Dormand~schemeG$\"+ Sm_Eq!#A7$%La~modification~of~the~Prince-Dormand~schemeG$\"+arah5!#@7$ %7Verner's~(2004)~schemeG$\"+A#4\\)))F+7$%Ba~modification~of~Verner's~ schemeG$\"+UdK****F+7$%Fa~2nd~modification~of~Verner's~schemeG$\"+q0:d 5F07$%Mscheme~with~maximal~imaginary~axis~inclusionG$\"+5)*)RT\"F0Q)pp rint166\"" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 30 "The following code constructs " }{TEXT 260 20 "numerical proced ures" }{TEXT -1 56 " for solutions based on each of the Runge-Kutta sc hemes." }}{PARA 0 "" 0 "" {TEXT -1 75 "The error in the value obtained by each of the methods at the point where " }{XPPEDIT 18 0 "x = 4.99 9;" "6#/%\"xG-%&FloatG6$\"%**\\!\"$" }{TEXT -1 16 " is also given." } }{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 716 "B := (x,y) -> 10*x*cos(x)- 10*y: hh := 0.01: numsteps := 500: x0 := 0: y0 := sqrt(5):\nmatrix([[` slope field: `,B(x,y)],[`initial point: `,``(x0,y0)],[`step width: \+ `,hh],\n[`no. of steps: `,numsteps]]);``;\nmthds := [`Prince-Dorman d scheme`,`a modification of the Prince-Dormand scheme`,`Verner's (200 4) scheme`,`a modification of Verner's scheme`,`a 2nd modification of \+ Verner's scheme`,`scheme with maximal imaginary axis inclusion`]: errs := []:\nDigits := 25:\nfor ct to 6 do\n bn_RK6_||ct := RK6_||ct(B(x ,y),x,y,x0,evalf(y0),hh,numsteps,true);\nend do:\nxx := 4.999: bxx := \+ evalf(b(xx)):\nfor ct to 6 do\n errs := [op(errs),abs(bn_RK6_||ct(xx )-bxx)];\nend do:\nDigits := 10:\nlinalg[transpose]([mthds,evalf(errs) ]);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#K%'matrixG6#7&7$%0slope~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)pprint176\"" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%!G" } }{PARA 11 "" 1 "" {XPPMATH 20 "6#K%'matrixG6#7(7$%6Prince-Dormand~sche meG$\"+g75@R!#A7$%La~modification~of~the~Prince-Dormand~schemeG$\"+G'* yYdF+7$%7Verner's~(2004)~schemeG$\"+)em%3\\F+7$%Ba~modification~of~Ver ner's~schemeG$\"+WFl`aF+7$%Fa~2nd~modification~of~Verner's~schemeG$\"+ !=(=EdF+7$%Mscheme~with~maximal~imaginary~axis~inclusionG$\"++(=I\"eF+ Q)pprint186\"" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 4 "The " }{TEXT 260 22 "root mean square error" }{TEXT -1 20 " over the interval " }{XPPEDIT 18 0 "[0, 5];" "6#7$\"\"!\"\"&" } {TEXT -1 82 " of each Runge-Kutta method is estimated as follows usin g the special procedure " }{TEXT 0 5 "NCint" }{TEXT -1 98 " to perfo rm numerical integration by the 7 point Newton-Cotes method over 200 e qual subintervals." }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 454 "mthds := [`Prince-Dormand scheme`,`a modification of the Prince-Dormand sch eme`,`Verner's (2004) scheme`,`a modification of Verner's scheme`,`a 2 nd modification of Verner's scheme`,`scheme with maximal imaginary axi s inclusion`]: errs := []:\nDigits := 20:\nfor ct to 6 do\n sm := NC int((b(x)-'bn_RK6_||ct'(x))^2,x=0..5,adaptive=false,numpoints=7,factor =200);\n errs := [op(errs),sqrt(sm/5)];\nend do:\nDigits := 10:\nlin alg[transpose]([mthds,evalf(errs)]);" }}{PARA 11 "" 1 "" {XPPMATH 20 " 6#K%'matrixG6#7(7$%6Prince-Dormand~schemeG$\"+e\\irn!#A7$%La~modificat ion~of~the~Prince-Dormand~schemeG$\"+jdTB5!#@7$%7Verner's~(2004)~schem eG$\"+#)plk&)F+7$%Ba~modification~of~Verner's~schemeG$\"+5#p(R'*F+7$%F a~2nd~modification~of~Verner's~schemeG$\"+tW<>5F07$%Mscheme~with~maxim al~imaginary~axis~inclusionG$\"+!4+KO\"F0Q)pprint196\"" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 92 "The following error \+ graphs are constructed using the numerical procedures for the solution s." }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 590 "evalf[20](plot([b(x)- 'bn_RK6_1'(x),b(x)-'bn_RK6_2'(x),b(x)-'bn_RK6_3'(x),b(x)-'bn_RK6_4'(x) ,\nb(x)-'bn_RK6_5'(x),b(x)-'bn_RK6_6'(x)],x=0..0.65,numpoints=100,colo r=[COLOR(RGB,.4,0,1),\nCOLOR(RGB,0,.65,0),COLOR(RGB,1,.5,0),COLOR(RGB, .85,0,.85),COLOR(RGB,0,.3,1),COLOR(RGB,.95,0,.15)],\nlegend=[`Prince-D ormand scheme`,`a modification of the Prince-Dormand scheme`,`Verner's (2004) scheme`,`a modification of Verner's scheme`,`a 2nd modificatio n of Verner's scheme`,`scheme with maximal imaginary axis inclusion`], font=[HELVETICA,9],title=`error curves for 8 stage order 6 Runge-Kutta methods`));" }}{PARA 13 "" 1 "" {GLPLOT2D 938 575 575 {PLOTDATA 2 "6, -%'CURVESG6%7i^l7$$\"\"!F)F(7$$\"5SSSSSS:N<7$$\"5\"333333 .ZV$F-$\"%o6F07$$\"5+,,,,^)yLH%F-$\"%!*oF07$$\"5?@@@@@Y0_^F-$\"&J%HF07 $$\"5STTTT\"RI2,'F-$\"'-05F07$$\"5ihhhhhhSpoF-$\"'78HF07$$\"5XXXX&zNNl 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{TEXT -1 198 "See: Comparing Numerical Methods for Ordinary Differenti al Equations, Hull, Enright, Fellen and Sedgwick,\n Siam Journa l 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*ln*x-2*ln*2-Pi = 0;" "6#/, ,*&\"\"#\"\"\"-%#lnG6#*&,&*$%\"xGF&F'*$%\"yGF&F'F'*$F.F&!\"\"F'F'*&\" \"%F'-%'arctanG6#*&F0F'F.F2F'F'*(F4F'F)F'F.F'F'*(F&F'F)F'F&F'F2%#PiGF2 \"\"!" }{TEXT -1 2 ". " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 73 "de := diff(y(x),x)=(y(x)-x)/(y(x)+x );\nic := y(1)=1;\ndsolve(\{de,ic\},y(x));" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%#deG/-%%diffG6$-%\"yG6#%\"xGF,*&,&F)\"\"\"F,!\"\"F/,& F)F/F,F/F0" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%#icG/-%\"yG6#\"\"\"F) " }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/-%\"yG6#%\"xG-%'RootOfG6#,,*&\"\" #\"\"\"-%#lnG6#*&,&*$)F'F-F.F.*$)%#_ZGF-F.F.F.F'!\"#F.!\"\"*&\"\"%F.-% 'arctanG6#*&F8F.F'F:F.F:*&F " 0 "" {MPLTEXT 1 0 1 ";" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 41 "The solution can be given more \+ simply as " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 " ln(x^2+y^2)+2*arctan(y/x)=ln*2+Pi/2" "6#/,&- %#lnG6#,&*$%\"xG\"\"#\"\"\"*$%\"yGF+F,F,*&F+F,-%'arctanG6#*&F.F,F*!\" \"F,F,,&*&F&F,F+F,F,*&%#PiGF,F+F4F," }{TEXT -1 2 ". " }}{PARA 0 "" 0 " " {TEXT -1 18 "The solution (for " }{TEXT 279 1 "x" }{TEXT -1 47 " inc reasing) is the section of the polar curve " }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "r=sqrt(2)*exp(Pi/4-theta)" "6#/%\"rG*&- %%sqrtG6#\"\"#\"\"\"-%$expG6#,&*&%#PiGF*\"\"%!\"\"F*%&thetaGF2F*" } {TEXT -1 5 ", " }{XPPEDIT 18 0 "-Pi/4<=theta" "6#1,$*&%#PiG\"\"\"\" \"%!\"\"F)%&thetaG" }{XPPEDIT 18 0 "``<=Pi/4" "6#1%!G*&%#PiG\"\"\"\"\" %!\"\"" }{TEXT -1 2 ". " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 7 "Check: " }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 60 " ln((x^2+y^2))+2*arctan(y/x)=ln(2)+Pi/2;\nimplicitdiff(%,y,x);" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#/,&-%#lnG6#,&*$)%\"xG\"\"#\"\"\"F-*$)% \"yGF,F-F-F-*&F,F--%'arctanG6#*&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,linesty le=2):\np4 := plot([[[1,1],[uu,-uu]]$4],style=point,symbol=[circle$2,d iamond,cross],\n symbolsize=[12,10$3],color=[black,gr een$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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GF+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 277 1 "y " }{TEXT -1 25 " numerically in terms of " }{TEXT 278 1 "x" }{TEXT -1 1 "." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 209 "phi := proc(x) local y;\n fsolve(ln(x^2+y^2 )+2*arctan(y/x)=ln(2)+Pi/2,y=-x..7/2-x);\n end proc:\nuu := eval f(exp(Pi/2)):\nplot('phi'(x),x=1..uu,numpoints=100,font=[HELVETICA,9], labels=[`x`,`y(x)`]);" }}{PARA 13 "" 1 "" {GLPLOT2D 515 404 404 {PLOTDATA 2 "6&-%'CURVESG6$7jq7$$\"\"\"\"\"!$\"+++++5!\"*7$$\"+M.FS5F- $\"+!Hsf***!#57$$\"+N$4`2\"F-$\"+]6*f)**F37$$\"+eVr96F-$\"+Zyon**F37$$ \"+#e!Qa6F-$\"+z@\"=%**F37$$\"+!GeQ>\"F-$\"+^qu3**F37$$\"+f\"f/B\"F-$ \"+$)4mr)*F37$$\"+\\sNo7F-$\"+m$=o#)*F37$$\"+4:b28F-$\"+-Eit(*F37$$\"+ s+iY8F-$\"+@hw8(*F37$$\"+\"*o!oQ\"F-$\"+ig9X'*F37$$\"+jM?A9F-$\"+N0$)y &*F37$$\"+>;0i9F-$\"+tvj(\\*F37$$\"+'Rj?]\"F-$\"+2v=4%*F37$$\"+d@iS:F- 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F37$$\"+t#3\\U$F-$!+e$R\\f$F37$$\"+d)[KY$F-$!+\")RRuSF37$$\"+>h\\,NF-$ !+)*o;lXF37$$\"+LbWTNF-$!+6&)e\"4&F37$$\"+,/CyNF-$!+4mR*e&F37$$\"+zL#f h$F-$!+w:g7hF37$$\"+l)Hvl$F-$!+Zam1nF37$$\"+I6?&p$F-$!+q5-gsF37$$\"+aq sLPF-$!+g\\%=%yF37$$\"+Fp!Hx$F-$!+)QD3X)F37$$\"+a3#*3QF-$!+'*3rE!*F37$ $\"+M0JZQF-$!+?;We'*F37$$\"+*)zS&)QF-$!+6MXI5F-7$$\"+C/;ERF-$!+B0!=5\" F-7$$\"+$oA@'RF-$!+qNzm6F-7$$\"+kch.SF-$!+'>,VC\"F-7$$\"+!))f5/%F-$!+- [r;8F-7$$\"+v*3\"ySF-$!+Z4$3R\"F-7$$\"+b'[z6%F-$!+zwYt9F-7$$\"+$\\\\z: %F-$!+'fX(f:F-7$$\"+MVM%>%F-$!+DqOT;F-7$$\"+QS*HB%F-$!+A:fJF-7$$\"+G\"ypM%F-$!+AUgA?F-7$$ \"+M=h(Q%F-$!+:96P@F-7$$\"+`'4eU%F-$!+)[X5D#F-7$$\"+C&QOY%F-$!+d+zqBF- 7$$\"+([(\\,XF-$!+HVk)\\#F-7$$\"+L76SXF-$!+'GZ)QEF-7$$\"+d6/\"e%F-$!+$ pm1!GF-7$$\"+!*)p&=YF-$!+?m5kHF-7$$\"+obhbYF-$!+9%4S9$F-7$$\"+(y;_p%F- $!+/&R[O$F-7$$\"+*zJZt%F-$!+`i>JOF-7$$\"+M`Y_ZF-$!+El5vPF-7$$\"+o))>qZ F-$!+iFfWRF-7$$\"+'\\o-y%F-$!+G<\")eSF-7$$\"+D\"Q.z%F-$!+!p\\]>%F-7$$ \"+SHP&z%F-$!+M@kwUF-7$$\"+axS+[F-$!+ " 0 "" {MPLTEXT 1 0 837 "C := \+ (x,y) -> (y-x)/(y+x): hh := 0.01: numsteps := 375: x0 := 1: y0 := 1:\n matrix([[`slope field: `,C(x,y)],[`initial point: `,``(x0,y0)],[`ste p width: `,hh],\n[`no. of steps: `,numsteps]]);``;\nmthds := [`Pri nce-Dormand scheme`,`a modification of the Prince-Dormand scheme`,`Ver ner's (2004) scheme`,`a modification of Verner's scheme`,`a 2nd modifi cation of Verner's scheme`,`scheme with maximal imaginary axis inclusi on`]: errs := []: vals := []:\nDigits := 25:\nfor ct to 6 do\n Cn_R K6_||ct := RK6_||ct(C(x,y),x,y,x0,y0,hh,numsteps,false);\n sm := 0: \+ numpts := nops(Cn_RK6_||ct):\n for ii to numpts do\n if ct=1 th en vals := [op(vals),phi(Cn_RK6_||ct[ii,1])] end if;\n sm := sm+( Cn_RK6_||ct[ii,2]-vals[ii])^2;\n end do:\n errs := [op(errs),sqrt( sm/numpts)];\nend do:\nDigits := 10:\nlinalg[transpose]([mthds,evalf(e rrs)]);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#K%'matrixG6#7&7$%0slope~fie ld:~~~G*&,&%\"yG\"\"\"%\"xG!\"\"F,,&F-F,F+F,F.7$%0initial~point:~G-%!G 6$F,F,7$%/step~width:~~~G$F,!\"#7$%1no.~of~steps:~~~G\"$v$Q)pprint206 \"" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%!G" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#K%'matrixG6#7(7$%6Prince-Dormand~schemeG$\"+)oh&*p\"!#@ 7$%La~modification~of~the~Prince-Dormand~schemeG$\"+/?[!=\"F+7$%7Verne r's~(2004)~schemeG$\"+cZAT;F+7$%Ba~modification~of~Verner's~schemeG$\" +aL;E:F+7$%Fa~2nd~modification~of~Verner's~schemeG$\"+#H$\\17F+7$%Msch eme~with~maximal~imaginary~axis~inclusionG$\"+Z'eG@%!#BQ)pprint216\"" }}}{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 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 = 4.749;" "6 #/%\"xG-%&FloatG6$\"%\\Z!\"$" }{TEXT -1 16 " is also given." }} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 700 "C := (x,y) -> (y-x)/(y+x): \+ hh := 0.01: numsteps := 375: x0 := 1: y0 := 1:\nmatrix([[`slope field: `,C(x,y)],[`initial point: `,``(x0,y0)],[`step width: `,hh],\n[`n o. of steps: `,numsteps]]);``;\nmthds := [`Prince-Dormand scheme`,`a modification of the Prince-Dormand scheme`,`Verner's (2004) scheme`,` a modification of Verner's scheme`,`a 2nd modification of Verner's sch eme`,`scheme with maximal imaginary axis inclusion`]: errs := []:\nDig its := 30:\nfor ct to 6 do\n cn_RK6_||ct := RK6_||ct(C(x,y),x,y,x0,y 0,hh,numsteps,true);\nend do:\nxx := 4.749: cxx := evalf(phi(xx)):\nfo r ct to 6 do\n errs := [op(errs),abs(cn_RK6_||ct(xx)-cxx)];\nend do: \nDigits := 10:\nlinalg[transpose]([mthds,evalf(errs)]);" }}{PARA 11 " " 1 "" {XPPMATH 20 "6#K%'matrixG6#7&7$%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)pprint226\"" }}{PARA 11 "" 1 " " {XPPMATH 20 "6#%!G" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#K%'matrixG6#7( 7$%6Prince-Dormand~schemeG$\"+^8n)3#!#?7$%La~modification~of~the~Princ e-Dormand~schemeG$\"+%=B[W\"F+7$%7Verner's~(2004)~schemeG$\"+d&ez+#F+7 $%Ba~modification~of~Verner's~schemeG$\"+/y,n=F+7$%Fa~2nd~modification ~of~Verner's~schemeG$\"+P>jw9F+7$%Mscheme~with~maximal~imaginary~axis~ inclusionG$\"+PyK[Y!#AQ)pprint236\"" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 4 "The " }{TEXT 260 22 "root mean square er ror" }{TEXT -1 20 " over the interval " }{XPPEDIT 18 0 "[1, 4.75];" " 6#7$\"\"\"-%&FloatG6$\"$v%!\"#" }{TEXT -1 82 " of each Runge-Kutta me thod is estimated as follows using the special procedure " }{TEXT 0 5 "NCint" }{TEXT -1 98 " to perform numerical integration by the 7 po int Newton-Cotes method over 200 equal subintervals." }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 461 "mthds := [`Prince-Dormand scheme`,`a mod ification of the Prince-Dormand scheme`,`Verner's (2004) scheme`,`a mo dification of Verner's scheme`,`a 2nd modification of Verner's scheme` ,`scheme with maximal imaginary axis inclusion`]: errs := []:\nDigits \+ := 20:\nfor ct to 6 do\n sm := NCint(('phi'(x)-'cn_RK6_||ct'(x))^2,x =1..4.75,adaptive=false,numpoints=7,factor=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$%6Prince-D ormand~schemeG$\"+\"Gw%*>)!#A7$%La~modification~of~the~Prince-Dormand~ schemeG$\"+l;g0cF+7$%7Verner's~(2004)~schemeG$\"+DPP/yF+7$%Ba~modifica tion~of~Verner's~schemeG$\"+ZH9`sF+7$%Fa~2nd~modification~of~Verner's~ schemeG$\"+'G4.t&F+7$%Mscheme~with~maximal~imaginary~axis~inclusionG$ \"+#38=V#!#BQ)pprint246\"" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 92 "The following error graphs are constructed using \+ the numerical procedures for the solutions." }}{EXCHG {PARA 0 "> " 0 " " {MPLTEXT 1 0 611 "evalf[30](plot(['cn_RK6_1'(x)-'phi'(x),'cn_RK6_2'( x)-'phi'(x),'cn_RK6_3'(x)-'phi'(x),'cn_RK6_4'(x)-'phi'(x),\n'cn_RK6_5' (x)-'phi'(x),'cn_RK6_6'(x)-'phi'(x)],x=1..3.75,0..3.3e-17,color=[COLOR (RGB,.4,0,1),\nCOLOR(RGB,0,.65,0),COLOR(RGB,1,.5,0),COLOR(RGB,.85,0,.8 5),COLOR(RGB,0,.3,1),COLOR(RGB,.95,0,.15)],\nlegend=[`Prince-Dormand s cheme`,`a modification of the Prince-Dormand scheme`,`Verner's (2004) \+ scheme`,`a modification of Verner's scheme`,`a 2nd modification of Ver ner's scheme`,`scheme with maximal imaginary axis inclusion`],font=[HE LVETICA,9],title=`error curves for 8 stage order 6 Runge-Kutta methods `));" }}{PARA 13 "" 1 "" {GLPLOT2D 946 615 615 {PLOTDATA 2 "6,-%'CURVE SG6%7S7$$\"\"\"\"\"!$F*F*7$$\"?LLLLLLLL$eR(G$o&=F/$\".nBI$ohdF27$$\"?nmmmmmmm;z\\#3)=>F/$\".j&y UW3gF27$$\"?nmmmmmmmm;C&48(>F/$\".*f&\\g%>iF27$$\"?+++++++]PM()4QK?F/$ 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