{VERSION 6 0 "IBM INTEL NT" "6.0" } {USTYLETAB {CSTYLE "Maple Input" -1 0 "Courier" 0 1 255 0 0 1 0 1 0 0 1 0 0 0 0 1 }{CSTYLE "2D Math" -1 2 "Times" 0 1 0 0 0 0 0 0 2 0 0 0 0 0 0 1 }{CSTYLE "2D Comment" 2 18 "" 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 1 } {CSTYLE "2D Output" 2 20 "" 0 1 0 0 255 1 0 0 0 0 0 0 0 0 0 1 } {CSTYLE "Blue Emphasis" -1 256 "Times" 0 0 0 0 255 1 0 1 0 0 0 0 0 0 0 1 }{CSTYLE "Green Emphasis" -1 257 "Times" 1 12 0 128 0 1 0 1 0 0 0 0 0 0 0 1 }{CSTYLE "Maroon Emphasis" -1 258 "Times" 1 12 128 0 128 1 0 1 0 0 0 0 0 0 0 1 }{CSTYLE "Dark Red Emphasis" -1 259 "Times" 1 12 128 0 0 1 0 1 0 0 0 0 0 0 0 0 }{CSTYLE "Purple Emphasis" -1 260 "Times " 1 12 102 0 230 1 0 1 0 0 0 0 0 0 0 0 }{CSTYLE "Red Emphasis" -1 261 "Times" 1 12 255 0 0 1 0 1 0 0 0 0 0 0 0 0 }{CSTYLE "Grey Emphasis" -1 262 "Times" 1 12 96 52 84 1 0 1 0 0 0 0 0 0 0 0 }{CSTYLE "Magenta E mphasis" -1 263 "Times" 1 12 200 0 200 1 0 1 0 0 0 0 0 0 0 0 }{CSTYLE "Orange Emphasis" -1 264 "Times" 1 12 225 100 10 1 0 1 0 0 0 0 0 0 0 0 }{PSTYLE "Normal" -1 0 1 {CSTYLE "" -1 -1 "Times" 1 12 0 0 0 1 2 2 2 2 2 2 1 1 1 1 }1 1 0 0 0 0 1 0 1 0 2 2 0 1 }{PSTYLE "Heading 1" -1 3 1 {CSTYLE "" -1 -1 "Times" 1 18 0 0 128 1 2 1 2 2 2 2 1 1 1 1 }1 1 0 0 8 4 3 0 3 0 2 2 0 1 }{PSTYLE "Heading 2" -1 4 1 {CSTYLE "" -1 -1 " Times" 1 14 128 0 0 1 2 1 2 2 2 2 1 1 1 1 }1 1 0 0 8 2 1 0 1 0 2 2 0 1 }{PSTYLE "Maple Output" -1 11 1 {CSTYLE "" -1 -1 "Times" 1 12 0 0 0 1 2 2 2 2 2 2 1 1 1 1 }3 3 0 0 0 0 1 0 1 0 2 2 0 1 }{PSTYLE "Maple Out put" -1 12 1 {CSTYLE "" -1 -1 "Times" 1 12 0 0 0 1 2 2 2 2 2 2 1 1 1 1 }1 3 0 0 0 0 1 0 1 0 2 2 0 1 }{PSTYLE "Maple Plot" -1 13 1 {CSTYLE " " -1 -1 "Times" 1 12 0 0 0 1 2 2 2 2 2 2 1 1 1 1 }3 1 0 0 0 0 1 0 1 0 2 2 0 1 }{PSTYLE "Normal" -1 256 1 {CSTYLE "" -1 -1 "Times" 1 12 0 0 0 1 2 2 2 2 2 2 1 1 1 1 }3 1 0 0 0 0 1 0 1 0 2 2 0 1 }{PSTYLE "Normal " -1 257 1 {CSTYLE "" -1 -1 "Times" 1 12 0 0 0 1 2 2 2 2 2 2 1 1 1 1 } 1 1 0 0 0 0 1 0 1 0 2 2 0 1 }{PSTYLE "Bullet Item" -1 258 1 {CSTYLE " " -1 -1 "Times" 1 12 0 0 0 1 2 2 2 2 2 2 1 1 1 1 }3 1 0 0 3 3 1 0 1 0 2 2 15 2 }} {SECT 0 {PARA 3 "" 0 "" {TEXT -1 50 "Derivation of 5 stage, order 4 Ru nge-Kutta schemes" }}{PARA 0 "" 0 "" {TEXT -1 45 "by Peter Stone, Gabr iola Island, B.C., Canada" }}{PARA 0 "" 0 "" {TEXT -1 19 "Version: 13 .5.2009" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 8 "restart;" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 " ;" }}}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 53 "load procedures for constru cting Runge-Kutta schemes " }}{PARA 0 "" 0 "" {TEXT -1 17 "The Maple m -file " }{TEXT 262 9 "butcher.m" }{TEXT -1 32 " is required by this wo rksheet. " }}{PARA 0 "" 0 "" {TEXT -1 121 "It can be read into a Maple session by a command similar to the one that follows, where the file \+ path gives its location." }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 35 " read \"K:\\\\Maple/procdrs/butcher.m\";" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}} {SECT 1 {PARA 4 "" 0 "" {TEXT -1 55 "Equations for 5 stage, order 4 Ru nge-Kutta coefficients" }}{PARA 0 "" 0 "" {TEXT -1 1 " " }}{PARA 256 " " 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "matrix([[c[1] = 0, ``, ``, ``, ` `,``], [c[2], a[2,1], ``, ``, ``,``], [c[3], a[3,1], a[3,2], ``, ``,`` ], [c[4], a[4,1], a[4,2], a[4,3], ``,``], [c[5], a[5,1], a[5,2], a[5, 3], a[5,4],``],[``, b[1], b[2], b[3], b[4],b[5]]])" "6#-%'matrixG6#7(7 (/&%\"cG6#\"\"\"\"\"!%!GF.F.F.F.7(&F*6#\"\"#&%\"aG6$F2F,F.F.F.F.7(&F*6 #\"\"$&F46$F9F,&F46$F9F2F.F.F.7(&F*6#\"\"%&F46$FAF,&F46$FAF2&F46$FAF9F .F.7(&F*6#\"\"&&F46$FKF,&F46$FKF2&F46$FKF9&F46$FKFAF.7(F.&%\"bG6#F,&FV 6#F2&FV6#F9&FV6#FA&FV6#FK" }{TEXT -1 1 " " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT 262 30 "Simplifying row sum conditions" } {TEXT -1 2 ". " }}{PARA 258 "" 0 "" {TEXT -1 2 " " }{XPPEDIT 18 0 "PI ECEWISE([a[2,1] = c[2], ``],[a[3,1]+a[3,2] = c[3], ``],[a[4,1]+a[4,2]+ a[4,3] = c[4], ``],[a[5,1]+a[5,2]+a[5,3]+a[5,4] = c[5], ``]);" "6#-%*P IECEWISEG6&7$/&%\"aG6$\"\"#\"\"\"&%\"cG6#F+%!G7$/,&&F)6$\"\"$F,F,&F)6$ F6F+F,&F.6#F6F07$/,(&F)6$\"\"%F,F,&F)6$F@F+F,&F)6$F@F6F,&F.6#F@F07$/,* &F)6$\"\"&F,F,&F)6$FLF+F,&F)6$FLF6F,&F)6$FLF@F,&F.6#FLF0" }{TEXT -1 2 " " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT 262 29 "O rder 4 quadrature conditions" }{TEXT -1 2 ". " }}{PARA 258 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "PIECEWISE([b*e = 1, ``],[b*c = 1/2, ``] ,[b*c^2 = 1/3, ``],[b*c^3 = 1/4, ``]);" "6#-%*PIECEWISEG6&7$/*&%\"bG\" \"\"%\"eGF*F*%!G7$/*&F)F*%\"cGF**&F*F*\"\"#!\"\"F,7$/*&F)F**$F0F2F**&F *F*\"\"$F3F,7$/*&F)F**$F0F9F**&F*F*\"\"%F3F," }{TEXT -1 1 " " }}{PARA 0 "" 0 "" {TEXT -1 10 " that is, " }}{PARA 256 "" 0 "" {TEXT -1 2 " \+ " }{XPPEDIT 18 0 "PIECEWISE([b[1]+b[2]+b[3]+b[4]+b[5] = 1, ``],[b[2]*c [2]+b[3]*c[3]+b[4]*c[4]+b[5]*c[5] = 1/2, ``],[b[2]*c[2]^2+b[3]*c[3]^2+ b[4]*c[4]^2+b[5]*c[5]^2 = 1/3, ``],[b[2]*c[2]^3+b[3]*c[3]^3+b[4]*c[4]^ 3+b[5]*c[5]^3=1/4,``])" "6#-%*PIECEWISEG6&7$/,,&%\"bG6#\"\"\"F,&F*6#\" \"#F,&F*6#\"\"$F,&F*6#\"\"%F,&F*6#\"\"&F,F,%!G7$/,**&&F*6#F/F,&%\"cG6# F/F,F,*&&F*6#F2F,&FA6#F2F,F,*&&F*6#F5F,&FA6#F5F,F,*&&F*6#F8F,&FA6#F8F, F,*&F,F,F/!\"\"F97$/,**&&F*6#F/F,*$&FA6#F/F/F,F,*&&F*6#F2F,*$&FA6#F2F/ F,F,*&&F*6#F5F,*$&FA6#F5F/F,F,*&&F*6#F8F,*$&FA6#F8F/F,F,*&F,F,F2FSF97$ /,**&&F*6#F/F,*$&FA6#F/F2F,F,*&&F*6#F2F,*$&FA6#F2F2F,F,*&&F*6#F5F,*$&F A6#F5F2F,F,*&&F*6#F8F,*$&FA6#F8F2F,F,*&F,F,F5FSF9" }{TEXT -1 2 ". " }} {PARA 0 "" 0 "" {TEXT 262 33 "Order 4 sub-quadrature conditions" } {TEXT -1 2 ". " }}{PARA 258 "" 0 "" {XPPEDIT 18 0 "b*``(a*c) = 1/6" "6 #/*&%\"bG\"\"\"-%!G6#*&%\"aGF&%\"cGF&F&*&F&F&\"\"'!\"\"" }{TEXT -1 1 " ," }}{PARA 0 "" 0 "" {TEXT -1 3 "or " }}{PARA 256 "" 0 "" {TEXT -1 2 " " }{XPPEDIT 18 0 "Sum(Sum(b[i]*a[i,j]*c[j],j = 2 .. i-1),i = 3 .. 5) =1/6" "6#/-%$SumG6$-F%6$*(&%\"bG6#%\"iG\"\"\"&%\"aG6$F-%\"jGF.&%\"cG6# F2F./F2;\"\"#,&F-F.F.!\"\"/F-;\"\"$\"\"&*&F.F.\"\"'F:" }{TEXT -1 2 ", \+ " }}{PARA 0 "" 0 "" {TEXT -1 8 "that is," }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "b[3]*a[3,2]*c[2]+b[4]*(a[4,2]*c[2]+a[4,3]*c[3 ])+ b[5]*(a[5,2]*c[2]+a[5,3]*c[3]+a[5,4]*c[4])=1/6" "6#/,(*(&%\"bG6#\" \"$\"\"\"&%\"aG6$F)\"\"#F*&%\"cG6#F.F*F**&&F'6#\"\"%F*,&*&&F,6$F5F.F*& F06#F.F*F**&&F,6$F5F)F*&F06#F)F*F*F*F**&&F'6#\"\"&F*,(*&&F,6$FDF.F*&F0 6#F.F*F**&&F,6$FDF)F*&F06#F)F*F**&&F,6$FDF5F*&F06#F5F*F*F*F**&F*F*\"\" '!\"\"" }{TEXT -1 2 ". " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 258 " " 0 "" {XPPEDIT 18 0 "b*``(a*c^2) = 1/12" "6#/*&%\"bG\"\"\"-%!G6#*&%\" aGF&*$%\"cG\"\"#F&F&*&F&F&\"#7!\"\"" }{TEXT -1 2 ", " }}{PARA 0 "" 0 " " {TEXT -1 3 "or " }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 " Sum(b[i]*Sum(a[i,j]*c[j]^2,j = 2 .. i-1),i = 3 .. 5) = 1/12;" "6#/-%$S umG6$*&&%\"bG6#%\"iG\"\"\"-F%6$*&&%\"aG6$F+%\"jGF,*$&%\"cG6#F3\"\"#F,/ F3;F8,&F+F,F,!\"\"F,/F+;\"\"$\"\"&*&F,F,\"#7F<" }{TEXT -1 2 ", " }} {PARA 0 "" 0 "" {TEXT -1 9 "that is, " }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "b[3]*a[3,2]*c[2]^2+b[4]*(a[4,2]*c[2]^2+a[4,3]*c[3] ^2)+b[5]*(a[5,2]*c[2]^2+a[5,3]*c[3]^2+a[5,4]*c[4]^2)=1/12" "6#/,(*(&% \"bG6#\"\"$\"\"\"&%\"aG6$F)\"\"#F*&%\"cG6#F.F.F**&&F'6#\"\"%F*,&*&&F,6 $F5F.F**$&F06#F.F.F*F**&&F,6$F5F)F**$&F06#F)F.F*F*F*F**&&F'6#\"\"&F*,( *&&F,6$FFF.F**$&F06#F.F.F*F**&&F,6$FFF)F**$&F06#F)F.F*F**&&F,6$FFF5F** $&F06#F5F.F*F*F*F**&F*F*\"#7!\"\"" }{TEXT -1 1 " " }}{PARA 0 "" 0 "" {TEXT -1 1 " " }}{PARA 258 "" 0 "" {XPPEDIT 18 0 "b*``(a*``(a*c)) = 1/ 24" "6#/*&%\"bG\"\"\"-%!G6#*&%\"aGF&-F(6#*&F+F&%\"cGF&F&F&*&F&F&\"#C! \"\"" }{TEXT -1 2 ", " }}{PARA 0 "" 0 "" {TEXT -1 3 "or " }}{PARA 256 "" 0 "" {TEXT -1 2 " " }{XPPEDIT 18 0 "Sum(b[i]*Sum(a[i,j]*Sum(a[j,k] *c[k],k = 2 .. j-1),j = 3 .. i-1),i = 4 .. 5) = 1/24;" "6#/-%$SumG6$*& &%\"bG6#%\"iG\"\"\"-F%6$*&&%\"aG6$F+%\"jGF,-F%6$*&&F16$F3%\"kGF,&%\"cG 6#F9F,/F9;\"\"#,&F3F,F,!\"\"F,/F3;\"\"$,&F+F,F,FAF,/F+;\"\"%\"\"&*&F,F ,\"#CFA" }{TEXT -1 2 ", " }}{PARA 0 "" 0 "" {TEXT -1 9 "that is, " }} {PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "b[4]*a[4,3]*a[3,2]*c[ 2]+b[5]*(a[5,3]*a[3,2]*c[2]+a[5,4]*(a[4,2]*c[2]+a[4,3]*c[3]))=1/24" "6 #/,&**&%\"bG6#\"\"%\"\"\"&%\"aG6$F)\"\"$F*&F,6$F.\"\"#F*&%\"cG6#F1F*F* *&&F'6#\"\"&F*,&*(&F,6$F8F.F*&F,6$F.F1F*&F36#F1F*F**&&F,6$F8F)F*,&*&&F ,6$F)F1F*&F36#F1F*F**&&F,6$F)F.F*&F36#F.F*F*F*F*F*F**&F*F*\"#C!\"\"" } {TEXT -1 2 ". " }}{PARA 0 "" 0 "" {TEXT -1 2 " " }}{PARA 258 "" 0 "" {XPPEDIT 18 0 "b*c*``(a*c) = 1/8" "6#/*(%\"bG\"\"\"%\"cGF&-%!G6#*&%\"a GF&F'F&F&*&F&F&\"\")!\"\"" }{TEXT -1 2 ", " }}{PARA 0 "" 0 "" {TEXT -1 3 "or " }}{PARA 256 "" 0 "" {TEXT -1 2 " " }{XPPEDIT 18 0 "Sum(b[i ]*c[i]*Sum(a[i,j]*c[j],j = 2 .. i-1),i = 3 .. 5) = 1/8;" "6#/-%$SumG6$ *(&%\"bG6#%\"iG\"\"\"&%\"cG6#F+F,-F%6$*&&%\"aG6$F+%\"jGF,&F.6#F6F,/F6; \"\"#,&F+F,F,!\"\"F,/F+;\"\"$\"\"&*&F,F,\"\")F=" }{TEXT -1 2 ", " }} {PARA 0 "" 0 "" {TEXT -1 9 "that is, " }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "b[3]*c[3]*a[3,2]*c[2]+b[4]*c[4]*(a[4,2]*c[2]+a[4,3 ]*c[3])+b[5]*c[5]*(a[5,2]*c[2]+a[5,3]*c[3]+a[5,4]*c[4])=1/8" "6#/,(**& %\"bG6#\"\"$\"\"\"&%\"cG6#F)F*&%\"aG6$F)\"\"#F*&F,6#F1F*F**(&F'6#\"\"% F*&F,6#F7F*,&*&&F/6$F7F1F*&F,6#F1F*F**&&F/6$F7F)F*&F,6#F)F*F*F*F**(&F' 6#\"\"&F*&F,6#FHF*,(*&&F/6$FHF1F*&F,6#F1F*F**&&F/6$FHF)F*&F,6#F)F*F**& &F/6$FHF7F*&F,6#F7F*F*F*F**&F*F*\"\")!\"\"" }{TEXT -1 2 ". " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 85 "We can select s ome of the following order 5 conditions (principal error conditions). \+ " }}{PARA 0 "" 0 "" {TEXT 262 20 "Quadrature condition" }{TEXT -1 3 " \+ : " }{XPPEDIT 18 0 "b*c^4 = 1/5" "6#/*&%\"bG\"\"\"*$%\"cG\"\"%F&*&F&F& \"\"&!\"\"" }{TEXT -1 2 ". " }}{PARA 258 "" 0 "" {TEXT -1 1 " " } {XPPEDIT 18 0 "b[2]*c[2]^4+b[3]*c[3]^4+b[4]*c[4]^4+b[5]*c[5]^4=1/5" "6 #/,**&&%\"bG6#\"\"#\"\"\"*$&%\"cG6#F)\"\"%F*F**&&F'6#\"\"$F**$&F-6#F3F /F*F**&&F'6#F/F**$&F-6#F/F/F*F**&&F'6#\"\"&F**$&F-6#F@F/F*F**&F*F*F@! \"\"" }{TEXT -1 3 ". " }}{PARA 0 "" 0 "" {TEXT -1 1 "\n" }{TEXT 262 25 "Sub-quadrature conditions" }{TEXT -1 2 ": " }{XPPEDIT 18 0 "b*a*a* c^2" "6#**%\"bG\"\"\"%\"aGF%F&F%%\"cG\"\"#" }{TEXT -1 3 ", " } {XPPEDIT 18 0 "b*a*a*a*c" "6#*,%\"bG\"\"\"%\"aGF%F&F%F&F%%\"cGF%" } {TEXT -1 3 ", " }{XPPEDIT 18 0 "b*c*a*c^2" "6#**%\"bG\"\"\"%\"cGF%%\" aGF%F&\"\"#" }{TEXT -1 3 ", " }{XPPEDIT 18 0 "b*c^2*a*c" "6#**%\"bG\" \"\"*$%\"cG\"\"#F%%\"aGF%F'F%" }{TEXT -1 3 ", " }{XPPEDIT 18 0 "b*c*a *a*c" "6#*,%\"bG\"\"\"%\"cGF%%\"aGF%F'F%F&F%" }{TEXT -1 3 ", " } {XPPEDIT 18 0 "b*a*c*a*c" "6#*,%\"bG\"\"\"%\"aGF%%\"cGF%F&F%F'F%" } {TEXT -1 3 ", " }{XPPEDIT 18 0 "b*``(a*c^3) = 1/20" "6#/*&%\"bG\"\"\" -%!G6#*&%\"aGF&*$%\"cG\"\"$F&F&*&F&F&\"#?!\"\"" }{TEXT -1 2 ", " }} {PARA 258 "" 0 "" {XPPEDIT 18 0 "b*``(a*c^3) = 1/20" "6#/*&%\"bG\"\"\" -%!G6#*&%\"aGF&*$%\"cG\"\"$F&F&*&F&F&\"#?!\"\"" }{TEXT -1 4 ", \n " } {XPPEDIT 18 0 "Sum(b[i]*Sum(a[i,j]*c[j]^3,j = 2 .. i-1),i = 3 .. 5) = \+ 1/20;" "6#/-%$SumG6$*&&%\"bG6#%\"iG\"\"\"-F%6$*&&%\"aG6$F+%\"jGF,*$&% \"cG6#F3\"\"$F,/F3;\"\"#,&F+F,F,!\"\"F,/F+;F8\"\"&*&F,F,\"#?F=" } {TEXT -1 2 ", " }}{PARA 0 "" 0 "" {TEXT -1 9 "that is, " }}{PARA 256 " " 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "b[3]*a[3,2]*c[2]^3+b[4]*(a[4,2]* c[2]^3+a[4,3]*c[3]^3)+b[5]*(a[5,2]*c[2]^3+a[5,3]*c[3]^3+a[5,4]*c[4]^3) =1/20" "6#/,(*(&%\"bG6#\"\"$\"\"\"&%\"aG6$F)\"\"#F*&%\"cG6#F.F)F**&&F' 6#\"\"%F*,&*&&F,6$F5F.F**$&F06#F.F)F*F**&&F,6$F5F)F**$&F06#F)F)F*F*F*F **&&F'6#\"\"&F*,(*&&F,6$FFF.F**$&F06#F.F)F*F**&&F,6$FFF)F**$&F06#F)F)F *F**&&F,6$FFF5F**$&F06#F5F)F*F*F*F**&F*F*\"#?!\"\"" }{TEXT -1 2 ". " } }{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 258 "" 0 "" {TEXT -1 1 " " } {XPPEDIT 18 0 "b*``(a*``(c^2*a)) = 1/60" "6#/*&%\"bG\"\"\"-%!G6#*&%\"a GF&-F(6#*&%\"cG\"\"#F+F&F&F&*&F&F&\"#g!\"\"" }{TEXT -1 2 ", " }}{PARA 0 "" 0 "" {TEXT -1 3 "or " }}{PARA 256 "" 0 "" {TEXT -1 2 " " } {XPPEDIT 18 0 "Sum(b[i]*Sum(a[i,j]*Sum(a[j,k]*c[k]^2,k = 2 .. j-1),j = 3 .. i-1),i = 4 .. 5) = 1/60;" "6#/-%$SumG6$*&&%\"bG6#%\"iG\"\"\"-F%6 $*&&%\"aG6$F+%\"jGF,-F%6$*&&F16$F3%\"kGF,*$&%\"cG6#F9\"\"#F,/F9;F>,&F3 F,F,!\"\"F,/F3;\"\"$,&F+F,F,FBF,/F+;\"\"%\"\"&*&F,F,\"#gFB" }{TEXT -1 2 ", " }}{PARA 0 "" 0 "" {TEXT -1 9 "that is, " }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "b[4]*a[4,3]*a[3,2]*c[2]^2+b[5]*(a[5,3]* a[3,2]*c[2]^2+a[5,4]*(a[4,2]*c[2]^2+a[4,3]*c[3]^2))=1/60" "6#/,&**&%\" bG6#\"\"%\"\"\"&%\"aG6$F)\"\"$F*&F,6$F.\"\"#F*&%\"cG6#F1F1F**&&F'6#\" \"&F*,&*(&F,6$F8F.F*&F,6$F.F1F*&F36#F1F1F**&&F,6$F8F)F*,&*&&F,6$F)F1F* *$&F36#F1F1F*F**&&F,6$F)F.F**$&F36#F.F1F*F*F*F*F*F**&F*F*\"#g!\"\"" } {TEXT -1 2 ". " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 258 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "b*``(a*``(a*``(a*c))) = 1/120" "6#/*&% \"bG\"\"\"-%!G6#*&%\"aGF&-F(6#*&F+F&-F(6#*&F+F&%\"cGF&F&F&F&*&F&F&\"$? \"!\"\"" }{TEXT -1 3 ", " }}{PARA 0 "" 0 "" {TEXT -1 3 "or " }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "Sum(b[i]*Sum(a[i,j]*Sum(a[j ,k]*Sum(a[k,m]*c[m],m = 2 .. k-1),k = 3 .. j-1),j = 4 .. i-1),i = 5 .. 5) = 1/120;" "6#/-%$SumG6$*&&%\"bG6#%\"iG\"\"\"-F%6$*&&%\"aG6$F+%\"jG F,-F%6$*&&F16$F3%\"kGF,-F%6$*&&F16$F9%\"mGF,&%\"cG6#F?F,/F?;\"\"#,&F9F ,F,!\"\"F,/F9;\"\"$,&F3F,F,FGF,/F3;\"\"%,&F+F,F,FGF,/F+;\"\"&FR*&F,F, \"$?\"FG" }{TEXT -1 2 ", " }}{PARA 0 "" 0 "" {TEXT -1 9 "that is, " }} {PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "b[5]*a[5,4]*a[4,3]*a[ 3,2]*c[2]=1/120" "6#/*,&%\"bG6#\"\"&\"\"\"&%\"aG6$F(\"\"%F)&F+6$F-\"\" $F)&F+6$F0\"\"#F)&%\"cG6#F3F)*&F)F)\"$?\"!\"\"" }{TEXT -1 2 ". " }} {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 258 "" 0 "" {TEXT -1 1 " " } {XPPEDIT 18 0 "b*c*``(a*c^2) = 1/15" "6#/*(%\"bG\"\"\"%\"cGF&-%!G6#*&% \"aGF&*$F'\"\"#F&F&*&F&F&\"#:!\"\"" }{TEXT -1 3 ", " }}{PARA 0 "" 0 " " {TEXT -1 3 "or " }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 " Sum(b[i]*c[i]*Sum(a[i,j]*c[j]^2,j = 2 .. i-1),i = 3 .. 5) = 1/15;" "6# /-%$SumG6$*(&%\"bG6#%\"iG\"\"\"&%\"cG6#F+F,-F%6$*&&%\"aG6$F+%\"jGF,*$& F.6#F6\"\"#F,/F6;F:,&F+F,F,!\"\"F,/F+;\"\"$\"\"&*&F,F,\"#:F>" }{TEXT -1 3 ", " }}{PARA 0 "" 0 "" {TEXT -1 9 "that is, " }}{PARA 256 "" 0 " " {TEXT -1 1 " " }{XPPEDIT 18 0 "b[3]*c[3]*a[3,2]*c[2]^2+b[4]*c[4]*(a[ 4,2]*c[2]^2+a[4,3]*c[3]^2)+b[5]*c[5]*(a[5,2]*c[2]^2+a[5,3]*c[3]^2+a[5, 4]*c[4]^2)=1/15" "6#/,(**&%\"bG6#\"\"$\"\"\"&%\"cG6#F)F*&%\"aG6$F)\"\" #F*&F,6#F1F1F**(&F'6#\"\"%F*&F,6#F7F*,&*&&F/6$F7F1F**$&F,6#F1F1F*F**&& F/6$F7F)F**$&F,6#F)F1F*F*F*F**(&F'6#\"\"&F*&F,6#FJF*,(*&&F/6$FJF1F**$& F,6#F1F1F*F**&&F/6$FJF)F**$&F,6#F)F1F*F**&&F/6$FJF7F**$&F,6#F7F1F*F*F* F**&F*F*\"#:!\"\"" }{TEXT -1 2 ". " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }} {PARA 258 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "b*c^2*``(a*c) = 1/10 " "6#/*(%\"bG\"\"\"*$%\"cG\"\"#F&-%!G6#*&%\"aGF&F(F&F&*&F&F&\"#5!\"\" " }{TEXT -1 2 ", " }}{PARA 0 "" 0 "" {TEXT -1 3 "or " }}{PARA 256 "" 0 "" {TEXT -1 2 " " }{XPPEDIT 18 0 "Sum(b[i]*c[i]^2*Sum(a[i,j]*c[j],j = 2 .. i-1),i = 3 .. 5) = 1/10;" "6#/-%$SumG6$*(&%\"bG6#%\"iG\"\"\"*$ &%\"cG6#F+\"\"#F,-F%6$*&&%\"aG6$F+%\"jGF,&F/6#F8F,/F8;F1,&F+F,F,!\"\"F ,/F+;\"\"$\"\"&*&F,F,\"#5F>" }{TEXT -1 2 ", " }}{PARA 0 "" 0 "" {TEXT -1 9 "that is, " }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "b[ 3]*c[3]^2*a[3,2]*c[2]+b[4]*c[4]^2*(a[4,2]*c[2]+a[4,3]*c[3])+b[5]*c[5]^ 2*(a[5,2]*c[2]+a[5,3]*c[3]+a[5,4]*c[4])=1/10" "6#/,(**&%\"bG6#\"\"$\" \"\"*$&%\"cG6#F)\"\"#F*&%\"aG6$F)F/F*&F-6#F/F*F**(&F'6#\"\"%F**$&F-6#F 8F/F*,&*&&F16$F8F/F*&F-6#F/F*F**&&F16$F8F)F*&F-6#F)F*F*F*F**(&F'6#\"\" &F**$&F-6#FJF/F*,(*&&F16$FJF/F*&F-6#F/F*F**&&F16$FJF)F*&F-6#F)F*F**&&F 16$FJF8F*&F-6#F8F*F*F*F**&F*F*\"#5!\"\"" }{TEXT -1 2 ". " }}{PARA 0 " " 0 "" {TEXT -1 0 "" }}{PARA 258 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "b*c*``(a*``(a*c)) = 1/30" "6#/*(%\"bG\"\"\"%\"cGF&-%!G6#*&%\"aGF&-F )6#*&F,F&F'F&F&F&*&F&F&\"#I!\"\"" }{TEXT -1 2 ", " }}{PARA 0 "" 0 "" {TEXT -1 3 "or " }}{PARA 256 "" 0 "" {TEXT -1 2 " " }{XPPEDIT 18 0 "S um(b[i]*c[i]*Sum(a[i,j]*Sum(a[j,k]*c[k],k = 2 .. j-1),j = 3 .. i-1),i \+ = 4 .. 5) = 1/30;" "6#/-%$SumG6$*(&%\"bG6#%\"iG\"\"\"&%\"cG6#F+F,-F%6$ *&&%\"aG6$F+%\"jGF,-F%6$*&&F46$F6%\"kGF,&F.6#F " 0 "" {MPLTEXT 1 0 1 ";" }}}}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 69 " A system of equations for 5 stage, order 4 Runge-Kutta coefficients: \+ " }{TEXT 0 8 "RK4_5eqs" }{TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 35 "Abreviated \+ form of order conditions" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 " " 0 "" {TEXT -1 14 "The procedure " }{TEXT 0 15 "OrderConditions" } {TEXT -1 97 " can be used to obtain the order conditions for an order \+ 4 Runge-Kutta scheme in abreviated form." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 160 "convert(ListTools[En umerate](OrderConditions(4)),matrix):\nlinalg[augment](linalg[delcols] (%,2..2),matrix([[` `]$(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%\"\"$F)/*&F,F(-%!G6#*&%\"aG F(F2F(F(#F(\"\"'7%\"\"%F)/*&F,F()F2F/F(#F(F57%\"\"&F)/*&F,F(-F96#*&FF)/*(F,F(F2F(F8F(#F(\"\")7%\"\"(F)/*&F,F(-F96#*&FCF( F " 0 "" {MPLTEXT 1 0 1 ";" }}}} {SECT 1 {PARA 4 "" 0 "" {TEXT -1 41 "Long (summation) form of order co nditions" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 14 "The procedure " }{TEXT 0 15 "OrderConditions" }{TEXT -1 96 " can b e used to obtain the order conditions for an order 4 Runge-Kutta schem e in summation form." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 162 "convert(ListTools[Enumerate](OrderCondit ions(4,5)),matrix):\nlinalg[augment](linalg[delcols](%,2..2),matrix([[ ` `]$(linalg[rowdim](%))]),linalg[delcols](%,1..1));" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#K%'matrixG6#7*7%\"\"\"%#~~G/-%$SumG6$&%\"bG6#%\"iG /F1;F(\"\"&F(7%\"\"#F)/-F,6$*&F.F(&%\"cGF0F(F2#F(F67%\"\"$F)/-F,6$*&F. F(-F,6$*&&%\"aG6$F1%\"jGF(&F<6#FJF(/FJ;F(,&F1F(F(!\"\"F(F2#F(\"\"'7%\" \"%F)/-F,6$*&F.F()F;F6F(F2#F(F?7%F4F)/-F,6$*&F.F(-F,6$*&FGF(-F,6$*&&FH 6$FJ%\"kGF(&F<6#FboF(/Fbo;F(,&FJF(F(FPF(FMF(F2#F(\"#C7%FRF)/-F,6$*(F.F (F;F(FDF(F2#F(\"\")7%\"\"(F)/-F,6$*&F.F(-F,6$*&FGF()FKF6F(FMF(F2#F(\"# 77%F`pF)/-F,6$*&F.F()F;F?F(F2#F(FTQ(pprint16\"" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 75 "Set up order conditions together with r ow sum conditions in expanded form: " }{TEXT 0 8 "RK4_5eqs" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 236 "RK4 _5eqs := [op(RowSumConditions(5,'expanded')),op(OrderConditions(4,5,'e xpanded'))]:\nconvert(ListTools[Enumerate](RK4_5eqs),matrix):\nlinalg[ augment](linalg[delcols](%,2..2),matrix([[` `]$(linalg[rowdim](%))]), linalg[delcols](%,1..1));" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#K%'matrix G6#7.7%\"\"\"%#~~G/&%\"aG6$\"\"#F(&%\"cG6#F.7%F.F)/,&&F,6$\"\"$F(F(&F, 6$F7F.F(&F06#F77%F7F)/,(&F,6$\"\"%F(F(&F,6$FAF.F(&F,6$FAF7F(&F06#FA7%F AF)/,*&F,6$\"\"&F(F(&F,6$FMF.F(&F,6$FMF7F(&F,6$FMFAF(&F06#FM7%FMF)/,,& %\"bG6#F(F(&FZF1F(&FZF;F(&FZFGF(&FZFUF(F(7%\"\"'F)/,**&FfnF(F/F(F(*&Fg nF(F:F(F(*&FhnF(FFF(F(*&FinF(FTF(F(#F(F.7%\"\"(F)/,(*(FgnF(F8F(F/F(F(* &FhnF(,&*&FBF(F/F(F(*&FDF(F:F(F(F(F(*&FinF(,(*&FNF(F/F(F(*&FPF(F:F(F(* &FRF(FFF(F(F(F(#F(F[o7%\"\")F)/,**&FfnF()F/F.F(F(*&FgnF()F:F.F(F(*&Fhn F()FFF.F(F(*&FinF()FTF.F(F(#F(F77%\"\"*F)/,&**FhnF(FDF(F8F(F/F(F(*&Fin F(,&*(FPF(F8F(F/F(F(*&FRF(FioF(F(F(F(#F(\"#C7%\"#5F)/,(**FgnF(F:F(F8F( F/F(F(*(FhnF(FFF(FioF(F(*(FinF(FTF(F]pF(F(#F(Fcp7%\"#6F)/,(*(FgnF(F8F( FgpF(F(*&FhnF(,&*&FBF(FgpF(F(*&FDF(FipF(F(F(F(*&FinF(,(*&FNF(FgpF(F(*& FPF(FipF(F(*&FRF(F[qF(F(F(F(#F(\"#77%FasF)/,**&FfnF()F/F7F(F(*&FgnF()F :F7F(F(*&FhnF()FFF7F(F(*&FinF()FTF7F(F(#F(FAQ(pprint86\"" }}}{PARA 0 " " 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }} }}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 49 "Principal error terms and assoc iated conditions: " }{TEXT 0 14 "RK4_5err_terms" }{TEXT -1 2 ", " } {TEXT 0 12 "RK4_5err_eqs" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 "; " }}}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 31 "Abreviated form of error ter ms " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 167 "PrincipalErrorTerms(4):\nconvert(ListTools[Enumerate ](%),matrix):\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(%\"cGF(F(F(F(F(#F(\"$?\"!\"\"7%\"\"#F),&*&F,F(-F.6 #*(F1F(F8F(F5F(F(F(#F(\"#SF;7%\"\"$F),&*(F,F(F8F(F2F(F(#F(\"#IF;7%\"\" %F),$*&#F(F=F(-F.6#,&*&F,F(-F.6#*&F1F(-F.6#*&)F8F=F(F1F(F(F(F(#F(\"#gF ;F(F(7%\"\"&F),$*&FOF(-F.6#,&*&F,F()F5F=F(F(#F(\"#?F;F(F(7%\"\"'F),$*& FOF(-F.6#,&*(F,F(F8F(FWF(F(#F(\"#:F;F(F(7%\"\"(F),$*&FOF(-F.6#,&*(F,F( FZF(F5F(F(#F(\"#5F;F(F(7%\"\")F),$*&#F(FcoF(-F.6#,&*&F,F(-F.6#*&)F8FFF (F1F(F(F(#F(FaoF;F(F(7%\"\"*F),$*&#F(\"#CF(-F.6#,&*&F,F()F8FLF(F(#F(Fh nF;F(F(Q)pprint486\"" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 38 "Long (summation) form of error terms " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 169 "PrincipalErrorTer ms(4,5):\nconvert(ListTools[Enumerate](%),matrix):\nlinalg[augment](li nalg[delcols](%,2..2),matrix([[` `]$(linalg[rowdim](%))]),linalg[delc ols](%,1..1));" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#K%'matrixG6#7+7%\"\" \"%#~~G,&-%$SumG6$*&&%\"bG6#%\"iGF(-F,6$*&&%\"aG6$F2%\"jGF(-F,6$*&&F76 $F9%\"kGF(-F,6$*&&F76$F?%\"lGF(&%\"cG6#FEF(/FE;F(,&F?F(F(!\"\"F(/F?;F( ,&F9F(F(FLF(/F9;F(,&F2F(F(FLF(/F2;F(\"\"&F(#F(\"$?\"FL7%\"\"#F),&-F,6$ *&F/F(-F,6$*(F6F(&FG6#F9F(-F,6$*&F=F(&FG6#F?F(FMF(FPF(FSF(#F(\"#SFL7% \"\"$F),&-F,6$*(F/F(&FGF1F(-F,6$*&F6F(F]oF(FPF(FSF(#F(\"#IFL7%\"\"%F), &*&#F(FYF(-F,6$*&F/F(-F,6$*&F6F(-F,6$*&F=F()F`oFYF(FMF(FPF(FSF(F(#F(FW FL7%FUF),&*&FdpF(-F,6$*&F/F()-F,6$*&F6F(F[oF(FPFYF(FSF(F(#F(FcoFL7%\" \"'F),&*&FdpF(-F,6$*(F/F(FjoF(-F,6$*&F6F()F[oFYF(FPF(FSF(F(#F(F_pFL7% \"\"(F),&*&FdpF(-F,6$*(F/F()FjoFYF(FgqF(FSF(F(#F(\"#?FL7%\"\")F),&*&#F (F\\rF(-F,6$*&F/F(-F,6$*&F6F()F[oFeoF(FPF(FSF(F(#F(FWFL7%\"\"*F),&*&#F (\"#CF(-F,6$*&F/F()FjoFapF(FSF(F(#F(FWFLQ(pprint66\"" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}} {SECT 1 {PARA 4 "" 0 "" {TEXT -1 37 "Set up error terms in expanded fo rm: " }{TEXT 0 14 "RK4_5err_terms" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 151 "RK4_5err_terms := Principal ErrorTerms(4,5,'expanded'):\nfor ct to nops(RK4_5err_terms) do\n err ||ct := RK4_5err_terms[ct]\nend do:\nnops(RK4_5err_terms);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#\"\"*" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }} {PARA 0 "" 0 "" {TEXT -1 12 "An example: " }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 5 "err9;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#,,#\"\"\"\"$? \"!\"\"*&#F%\"#CF%*&&%\"bG6#\"\"#F%)&%\"cGF.\"\"%F%F%F%*&F)F%*&&F-6#\" \"$F%)&F2F7F3F%F%F%*&F)F%*&&F-6#F3F%)&F2F>F3F%F%F%*&F)F%*&&F-6#\"\"&F% )&F2FDF3F%F%F%" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> \+ " 0 "" {MPLTEXT 1 0 1 ";" }}}}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 42 "Set up error conditions in expanded form: " }{TEXT 0 12 "RK4_5err_eqs" }} {PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 161 "RK4_5err_eqs := PrincipalErrorConditions(4,5,'expanded'):\nfor ct to nops(RK4_5err_eqs) do\n print(`equation `||ct);print(RK4_5err_eq s[ct]);print(``); \nend do:" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%+equ ation~1G" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/*,&%\"bG6#\"\"&\"\"\"&%\" aG6$F(\"\"%F)&F+6$F-\"\"$F)&F+6$F0\"\"#F)&%\"cG6#F3F)#F)\"$?\"" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#%!G" }}{PARA 11 "" 1 "" {XPPMATH 20 "6 #%+equation~2G" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/,&*,&%\"bG6#\"\"%\" \"\"&%\"aG6$F)\"\"$F*&%\"cG6#F.F*&F,6$F.\"\"#F*&F06#F4F*F**&&F'6#\"\"& F*,&**&F,6$F:F.F*F/F*F2F*F5F*F**(&F,6$F:F)F*&F0F(F*,&*&&F,6$F)F4F*F5F* F**&F+F*F/F*F*F*F*F*F*#F*\"#S" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%!G" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%+equation~3G" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/,&*,&%\"bG6#\"\"%\"\"\"&%\"cGF(F*&%\"aG6$F)\"\"$F*&F.6 $F0\"\"#F*&F,6#F3F*F**(&F'6#\"\"&F*&F,F8F*,&*(&F.6$F9F0F*F1F*F4F*F**&& F.6$F9F)F*,&*&&F.6$F)F3F*F4F*F**&F-F*&F,6#F0F*F*F*F*F*F*#F*\"#I" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#%!G" }}{PARA 11 "" 1 "" {XPPMATH 20 "6 #%+equation~4G" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/,&**&%\"bG6#\"\"%\" \"\"&%\"aG6$F)\"\"$F*&F,6$F.\"\"#F*)&%\"cG6#F1F1F*F**&&F'6#\"\"&F*,&*( &F,6$F9F.F*F/F*F2F*F**&&F,6$F9F)F*,&*&&F,6$F)F1F*F2F*F**&F+F*)&F46#F.F 1F*F*F*F*F*F*#F*\"#g" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%!G" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%+equation~5G" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/,(*(&%\"bG6#\"\"$\"\"\")&%\"aG6$F)\"\"#F/F*)&%\"cG6#F/F/F*F**&& F'6#\"\"%F*),&*&&F-6$F7F/F*F1F*F**&&F-6$F7F)F*&F2F(F*F*F/F*F**&&F'6#\" \"&F*),(*&&F-6$FDF/F*F1F*F**&&F-6$FDF)F*F@F*F**&&F-6$FDF7F*&F2F6F*F*F/ F*F*#F*\"#?" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%!G" }}{PARA 11 "" 1 " " {XPPMATH 20 "6#%+equation~6G" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/,(* *&%\"bG6#\"\"$\"\"\"&%\"cGF(F*&%\"aG6$F)\"\"#F*)&F,6#F0F0F*F**(&F'6#\" \"%F*&F,F6F*,&*&&F.6$F7F0F*F1F*F**&&F.6$F7F)F*)F+F0F*F*F*F**(&F'6#\"\" &F*&F,FCF*,(*&&F.6$FDF0F*F1F*F**&&F.6$FDF)F*F@F*F**&&F.6$FDF7F*)F8F0F* F*F*F*#F*\"#:" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%!G" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%+equation~7G" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/,( **&%\"bG6#\"\"$\"\"\")&%\"cGF(\"\"#F*&%\"aG6$F)F.F*&F-6#F.F*F**(&F'6# \"\"%F*)&F-F6F.F*,&*&&F06$F7F.F*F2F*F**&&F06$F7F)F*F,F*F*F*F**(&F'6#\" \"&F*)&F-FCF.F*,(*&&F06$FDF.F*F2F*F**&&F06$FDF)F*F,F*F**&&F06$FDF7F*F9 F*F*F*F*#F*\"#5" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%!G" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%+equation~8G" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/ ,(*(&%\"bG6#\"\"$\"\"\"&%\"aG6$F)\"\"#F*)&%\"cG6#F.F)F*F**&&F'6#\"\"%F *,&*&&F,6$F6F.F*F/F*F**&&F,6$F6F)F*)&F1F(F)F*F*F*F**&&F'6#\"\"&F*,(*&& F,6$FCF.F*F/F*F**&&F,6$FCF)F*F>F*F**&&F,6$FCF6F*)&F1F5F)F*F*F*F*#F*\"# ?" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%!G" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%+equation~9G" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/,**&&%\"bG6#\" \"#\"\"\")&%\"cGF(\"\"%F*F**&&F'6#\"\"$F*)&F-F1F.F*F**&&F'6#F.F*)&F-F7 F.F*F**&&F'6#\"\"&F*)&F-F " 0 "" {MPLTEXT 1 0 1 ";" }}}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}} }{SECT 1 {PARA 4 "" 0 "" {TEXT -1 39 "Stage-order expressions and cond itions " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 78 "StageOrderConditions(2,5,'expanded'):\nconvert(ListTo ols[Enumerate](%),matrix);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#K%'matri xG6#7%7$\"\"\"/*&&%\"aG6$\"\"$\"\"#F(&%\"cG6#F/F(,$*&#F(F/F(*$)&F16#F. F/F(F(F(7$F//,&*&&F,6$\"\"%F/F(F0F(F(*&&F,6$F@F.F(F8F(F(,$*&F5F(*$)&F1 6#F@F/F(F(F(7$F./,(*&&F,6$\"\"&F/F(F0F(F(*&&F,6$FPF.F(F8F(F(*&&F,6$FPF @F(FHF(F(,$*&F5F(*$)&F16#FPF/F(F(F(Q)pprint536\"" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 78 "StageOrderCo nditions(3,5,'expanded'):\nconvert(ListTools[Enumerate](%),matrix);" } }{PARA 11 "" 1 "" {XPPMATH 20 "6#K%'matrixG6#7%7$\"\"\"/*&&%\"aG6$\"\" $\"\"#F()&%\"cG6#F/F/F(,$*&#F(F.F(*$)&F26#F.F.F(F(F(7$F//,&*&&F,6$\"\" %F/F(F0F(F(*&&F,6$FAF.F()F9F/F(F(,$*&F6F(*$)&F26#FAF.F(F(F(7$F./,(*&&F ,6$\"\"&F/F(F0F(F(*&&F,6$FRF.F(FEF(F(*&&F,6$FRFAF()FJF/F(F(,$*&F6F(*$) &F26#FRF.F(F(F(Q)pprint556\"" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}}{SECT 1 {PARA 4 "" 0 " " {TEXT -1 63 "Examples of systems of 5 stage order 4 Runge-Kutta coef ficients" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 105 "It is possible with 5 stages to obtain schemes which satisfy up t o 7 of the 9 principal error conditions." }}{PARA 0 "" 0 "" {TEXT -1 114 "The \"type\" of each of the following schemes refers to the numbe r of principal error conditions that are satisfied." }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 80 "F ehlberg's 5 stage (type 0) order 4 Runge-Kutta method ..001839 stag e order 3" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "matrix([[1/4, 1/4, ``, ``, ``, ``], [3/8, 3/3 2, 9/32, ``, ``, ``], [12/13, 1932/2197, -7200/2197, 7296/2197, ``, `` ], [1, 439/216, -8, 3680/513, -845/4104, ``], [``, 25/216, 0, 1408/256 5, 2197/4104, -1/5]])" "6#-%'matrixG6#7'7(*&\"\"\"F)\"\"%!\"\"*&F)F)F* F+%!GF-F-F-7(*&\"\"$F)\"\")F+*&F0F)\"#KF+*&\"\"*F)F3F+F-F-F-7(*&\"#7F) \"#8F+*&\"%K>F)\"%(>#F+,$*&\"%+sF)F " 0 "" {MPLTEXT 1 0 456 "##Fehlberg\nee := \{c[2]=1/4,c[3]=3/8,c[4]=12/13,c[5]=1,\na[2,1]= 1/4,a[3,1]=3/32,a[3,2]=9/32,a[4,1]=1932/2197,a[4,2]=-7200/2197,a[4,3]= 7296/2197,\na[5,1]=439/216,a[5,2]=-8,a[5,3]=3680/513,a[5,4]=-845/4104, \nb[1]=25/216,b[2]=0,b[3]=1408/2565,b[4]=2197/4104,b[5]=-1/5\}:\nsubs( ee,matrix([[c[2],a[2,1],``,``,``,``],\n[c[3],a[3,1],a[3,2],``,``,``],[ c[4],a[4,1],a[4,2],a[4,3],``,``],\n[c[5],a[5,1],a[5,2],a[5,3],a[5,4],` `],\n[``,b[1],b[2],b[3],b[4],b[5]]]));\nevalf[8](%);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#K%'matrixG6#7'7(#\"\"\"\"\"%F(%!GF+F+F+7(#\"\"$\"\") #F.\"#K#\"\"*F1F+F+F+7(#\"#7\"#8#\"%K>\"%(>##!%+sF:#\"%'H(F:F+F+7(F)# \"$R%\"$;#!\")#\"%!o$\"$8&#!$X)\"%/TF+7(F+#\"#DFB\"\"!#\"%39\"%lD#F:FI #!\"\"\"\"&Q)pprint816\"" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#K%'matrixG 6#7'7($\")+++D!\")F(%!GF+F+F+7($\")++]PF*$\")++v$*!\"*$\")+]7GF*F+F+F+ 7($\")#p2B*F*$\")(4Qz)F*$!)i>xK!\"($\")@*3K$F;F+F+7($\"\"\"\"\"!$\")uS K?F;$F*FA$\")$*[trF;$!)p'*e?F*F+7(F+$\")uSd6F*$FAFA$\"))y#*[&F*$\")QJ` `F*$!)+++?F*Q)pprint826\"" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 112 "We can check that the appropriate order conditio ns are satisfied along with the simplifying row sum conditions. " }} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 129 "RK4_5eqs := [op(RowSumCondi tions(5,'expanded')),op(OrderConditions(4,5,'expanded'))]:\nsubs(ee,RK 4_5eqs);\nmap(u->lhs(u)-rhs(u),%);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6# 7./#\"\"\"\"\"%F%/#\"\"$\"\")F)/#\"#7\"#8F-/F&F&F0/#F&\"\"#F2/#F&\"\"' F5/#F&F*F8/#F&\"#CF:/#F&F+F=/#F&F.F?F$" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#7.\"\"!F$F$F$F$F$F$F$F$F$F$F$" }}}{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 3 inclusive. " }}{EXCHG {PARA 0 "> \+ " 0 "" {MPLTEXT 1 0 85 "for ct from 2 to 4 do\n so||ct||_5 := StageO rderConditions(ct,5,'expanded');\nend do:" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 58 "Stages 3 to 5 have the followin g respective stage-orders. " }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 173 "[seq([seq(expand(subs(ee,so||i||_5[j])),i=2..4)],j=1..3)];\nmap(p roc(L) local i; for i to nops(L) do if not evalb(L[i]) then break end \+ if end do; i end proc,%):\nsimplify(%); " }}{PARA 11 "" 1 "" {XPPMATH 20 "6#7%7%/#\"\"*\"$G\"F&/#F'\"$7&F*/#F'\"%[?#\"#\")\"&%Q;7%/#\"#s\"$p \"F4/#\"$w&\"%(>#F8/#\"%*3\"\"%)y)#\"%%=&\"&h&G7%/#\"\"\"\"\"#FD/#FE\" \"$FH/#\"#>\"$3##FE\"\"%" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#7%\"\"$F$F $" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 55 "Non e of the principal error conditions are satisfied. " }}{PARA 0 "" 0 " " {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 105 "RK4_5err_ eqs := PrincipalErrorConditions(4,5,'expanded'):\nsubs(ee,RK4_5err_eqs );\nmap(u->lhs(u)-rhs(u),%);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#7+/#\" \"\"\"$/\"#F&\"$?\"/#\"#@\"$K)#F&\"#S/#\"\"&\"$c\"#F&\"#I/#\"\"(\"$;%# F&\"#g/#\"#$)\"%k;#F&\"#?/#F>\"%[7#F&\"#:/#F>F-#F&\"#5/#F,F9F@/#F>F9#F &F2" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#7+#\"\"\"\"$!y#F%\"%gT#!\"\"F&# F%\"%Si#F*\"%?$)#F*F,#F*F(#F%\"%!3##F*F2" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 142 "Incorporating the order 2 and 3 stage-order conditions f or stages 3, 4 and 5 gives a total of 18 equations which are satisfied by this scheme." }}{PARA 0 "" 0 "" {TEXT -1 35 "Specifying the three \+ nodes, namely " }{XPPEDIT 18 0 "c[2] = 1/4;" "6#/&%\"cG6#\"\"#*&\"\"\" F)\"\"%!\"\"" }{TEXT -1 2 ", " }{XPPEDIT 18 0 "c[4]=12/13" "6#/&%\"cG6 #\"\"%*&\"#7\"\"\"\"#8!\"\"" }{TEXT -1 6 " and " }{XPPEDIT 18 0 "c[5] = 1;" "6#/&%\"cG6#\"\"&\"\"\"" }{TEXT -1 86 ", enables the remaining \+ coefficients to be calculated by means of these 18 equations. " }} {PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 376 "e1 := \{c[2]=1/4,c[4]=12/13,c[5]=1\}:\ne2 := solve(subs(e1,\{op(R K4_5eqs),op(StageOrderConditions(2,5,'expanded')),\n op(StageOrderC onditions(3,5,'expanded'))\})):\ne3 := `union`(e1,e2):\nsubs(e3,matrix ([[c[2],a[2,1],``,``,``,``],\n[c[3],a[3,1],a[3,2],``,``,``],[c[4],a[4, 1],a[4,2],a[4,3],``,``],\n[c[5],a[5,1],a[5,2],a[5,3],a[5,4],``],\n[``, b[1],b[2],b[3],b[4],b[5]]]));\nevalf[8](%);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#K%'matrixG6#7'7(#\"\"\"\"\"%F(%!GF+F+F+7(#\"\"$\"\")#F. \"#K#\"\"*F1F+F+F+7(#\"#7\"#8#\"%K>\"%(>##!%+sF:#\"%'H(F:F+F+7(F)#\"$R %\"$;#!\")#\"%!o$\"$8&#!$X)\"%/TF+7(F+#\"#DFB\"\"!#\"%39\"%lD#F:FI#!\" \"\"\"&Q)pprint836\"" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#K%'matrixG6#7' 7($\")+++D!\")F(%!GF+F+F+7($\")++]PF*$\")++v$*!\"*$\")+]7GF*F+F+F+7($ \")#p2B*F*$\")(4Qz)F*$!)i>xK!\"($\")@*3K$F;F+F+7($\"\"\"\"\"!$\")uSK?F ;$F*FA$\")$*[trF;$!)p'*e?F*F+7(F+$\")uSd6F*$FAFA$\"))y#*[&F*$\")QJ``F* $!)+++?F*Q)pprint846\"" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 7 "Check: " }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 70 " expand(simplify(subs(e3,RK4_5eqs)));\nmap(u->lhs(u)-rhs(u),%);\nnops(% );" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#7./#\"\"\"\"\"%F%/#\"\"$\"\")F)/ #\"#7\"#8F-/F&F&F0/#F&\"\"#F2/#F&\"\"'F5/#F&F*F8/#F&\"#CF:/#F&F+F=/#F& F.F?F$" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#7.\"\"!F$F$F$F$F$F$F$F$F$F$F $" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#\"#7" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 26 "Calculate the error norm. " }} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 147 "errterms := PrincipalErrorT erms(4,5,'expanded'):\nsm := 0:\nfor i to nops(errterms) do\n sm := \+ sm+(evalf(subs(ee,errterms[i])))^2;\nend do:\nsqrt(sm);" }}{PARA 11 " " 1 "" {XPPMATH 20 "6#$\"+=MCR=!#7" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" } }{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 76 "Prince's 5 stage (type 5) order 4 Runge-Kutta method . . 0.00199 (1,2,3,4,6) " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 256 " " 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "matrix([[1/5, 1/5, ``, ``, ``,`` ], [2/5, 0, 2/5, ``, ``,``], [4/5, 6/5, -12/5, 2, ``,``],[1, -17/8, 5, -5/2, 5/8,``], [``,13/96, 0,25/48, 25/96, 1/12]])" "6#-%'matrixG6#7'7 (*&\"\"\"F)\"\"&!\"\"*&F)F)F*F+%!GF-F-F-7(*&\"\"#F)F*F+\"\"!*&F0F)F*F+ F-F-F-7(*&\"\"%F)F*F+*&\"\"'F)F*F+,$*&\"#7F)F*F+F+F0F-F-7(F),$*&\"# " 0 "" {MPLTEXT 1 0 407 "##Prince\nee := \{c[2]= 1/5,c[3]=2/5,c[4]=4/5,c[5]=1,\na[2,1]=1/5,a[3,1]=0,a[3,2]=2/5,a[4,1]=6 /5,a[4,2]=-12/5,a[4,3]=2,\na[5,1]=-17/8,a[5,2]=5,a[5,3]=-5/2,a[5,4]=5/ 8,\nb[1]=13/96,b[2]=0,b[3]=25/48,b[4]=25/96,b[5]=1/12\}:\nsubs(ee,matr ix([[c[2],a[2,1],``,``,``,``],\n[c[3],a[3,1],a[3,2],``,``,``],[c[4],a[ 4,1],a[4,2],a[4,3],``,``],\n[c[5],a[5,1],a[5,2],a[5,3],a[5,4],``],\n[` `,b[1],b[2],b[3],b[4],b[5]]]));\nevalf[8](%);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#K%'matrixG6#7'7(#\"\"\"\"\"&F(%!GF+F+F+7(#\"\"#F*\"\"!F -F+F+F+7(#\"\"%F*#\"\"'F*#!#7F*F.F+F+7(F)#!#<\"\")F*#!\"&F.#F*F:F+7(F+ #\"#8\"#'*F/#\"#D\"#[#FCFA#F)\"#7Q(pprint06\"" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#K%'matrixG6#7'7($\")+++?!\")F(%!GF+F+F+7($\")+++SF*$\" \"!F0F-F+F+F+7($\")+++!)F*$\")+++7!\"($!)+++CF6$\"\"#F0F+F+7($\"\"\"F0 $!)++D@F6$\"\"&F0$!)+++DF6$\")++]iF*F+7(F+$\")n;a8F*F/$\")LL3_F*$\")n; /EF*$\")LLL$)!\"*Q(pprint06\"" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }} {PARA 0 "" 0 "" {TEXT -1 112 "We can check that the appropriate order \+ conditions are satisfied along with the simplifying row sum conditions . " }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 129 "RK4_5eqs := [op(RowSu mConditions(5,'expanded')),op(OrderConditions(4,5,'expanded'))]:\nsubs (ee,RK4_5eqs);\nmap(u->lhs(u)-rhs(u),%);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#7./#\"\"\"\"\"&F%/#\"\"#F'F)/#\"\"%F'F,/F&F&F./#F&F*F0/#F&\"\"'F 2/#F&\"\"$F5/#F&\"#CF8/#F&\"\")F;/#F&\"#7F>/#F&F-FA" }}{PARA 11 "" 1 " " {XPPMATH 20 "6#7.\"\"!F$F$F$F$F$F$F$F$F$F$F$" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 86 "Next we set-up stage-orde r condtions to check for stage-orders from 2 to 3 inclusive. " }} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 85 "for ct from 2 to 3 do\n so ||ct||_5 := StageOrderConditions(ct,5,'expanded');\nend do:" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 58 "Stages 3 to 5 h ave the following respective stage-orders. " }}{EXCHG {PARA 0 "> " 0 " " {MPLTEXT 1 0 173 "[seq([seq(expand(subs(ee,so||i||_5[j])),i=2..3)],j =1..3)];\nmap(proc(L) local i; for i to nops(L) do if not evalb(L[i]) \+ then break end if end do; i end proc,%):\nsimplify(%); " }}{PARA 11 " " 1 "" {XPPMATH 20 "6#7%7$/#\"\"#\"#DF&/#F'\"$D\"#\"\")\"$v$7$/#F-F(F1 /#\"#GF+#\"#kF.7$/#\"\"\"F'F9/#F:\"\"&#F:\"\"$" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#7%\"\"#F$F$" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 46 "The principal error conditions are as follows. " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 172 "PrincipalErrorConditions(4):\nconvert(ListTools[Enumerate](%) ,matrix):\nlinalg[augment](linalg[delcols](%,2..2),matrix([[` `]$(lin alg[rowdim](%))]),linalg[delcols](%,1..1));" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#K%'matrixG6#7+7%\"\"\"%#~~G/*&%\"bGF(-%!G6#*&%\"aGF(-F. 6#*&F1F(-F.6#*&F1F(%\"cGF(F(F(F(#F(\"$?\"7%\"\"#F)/*&F,F(-F.6#*(F1F(F8 F(F5F(F(#F(\"#S7%\"\"$F)/*(F,F(F8F(F2F(#F(\"#I7%\"\"%F)/*&F,F(-F.6#*&F 1F(-F.6#*&)F8F " 0 "" {MPLTEXT 1 0 325 "RK4_5err_eqs := P rincipalErrorConditions(4,5,'expanded'):\nPE := PrincipalErrorConditio ns(4):\nRK4_5err_eqsA := []:\nfor i to nops(RK4_5err_eqs) do\n t := \+ expand(subs(ee,RK4_5err_eqs[i]));\n if lhs(t)=rhs(t) then \n pr int(i,` .. `,PE[i]);\n RK4_5err_eqsA := [op(RK4_5err_eqsA),RK 4_5err_eqs[i]]; \n end if;\nend do:" }}{PARA 11 "" 1 "" {XPPMATH 20 "6%\"\"\"%)~..~~~~~G/*&%\"bGF#-%!G6#*&%\"aGF#-F)6#*&F,F#-F)6#*&F,F#%\" cGF#F#F#F##F#\"$?\"" }}{PARA 11 "" 1 "" {XPPMATH 20 "6%\"\"$%)~..~~~~~ G/*(%\"bG\"\"\"%\"cGF(-%!G6#*&%\"aGF(-F+6#*&F.F(F)F(F(F(#F(\"#I" }} {PARA 11 "" 1 "" {XPPMATH 20 "6%\"\"%%)~..~~~~~G/*&%\"bG\"\"\"-%!G6#*& %\"aGF(-F*6#*&)%\"cG\"\"#F(F-F(F(F(#F(\"#g" }}{PARA 11 "" 1 "" {XPPMATH 20 "6%\"\"'%)~..~~~~~G/*(%\"bG\"\"\"%\"cGF(-%!G6#*&)F)\"\"#F( %\"aGF(F(#F(\"#:" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 70 "This gives a total of 16 equations which are satisfied by this scheme." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 100 "RK4_5eqsA := [op(RK4_5eqs),op(RK4_5err_eqsA)]: \nnops(%);\nsubs(ee,RK4_5eqsA):\nmap(u->lhs(u)-rhs(u),%);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#\"#;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#72\"\"! F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }} {PARA 0 "" 0 "" {TEXT -1 24 "We can set up the list " }{TEXT 0 9 "RK4 _5eqsA" }{TEXT -1 100 " of the principal error conditions that are sa tisfied independently from the previous calculations." }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 345 "RK4_5err_eqs := PrincipalErrorConditions (4,5,'expanded'):\nPE := PrincipalErrorConditions(4):\nRK4_5err_eqsA : = []:\nfor ct to nops(RK4_5err_eqs) do\n if ct in [1,3,4,6] then \n \+ print(ct,` .. `,PE[ct]);\n RK4_5err_eqsA := [op(RK4_5err _eqsA),RK4_5err_eqs[ct]]; \n end if;\nend do:\nRK4_5eqsA := [op(RK4_ 5eqs),op(RK4_5err_eqsA)]:\nnops(%);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6 %\"\"\"%)~..~~~~~G/*&%\"bGF#-%!G6#*&%\"aGF#-F)6#*&F,F#-F)6#*&F,F#%\"cG F#F#F#F##F#\"$?\"" }}{PARA 11 "" 1 "" {XPPMATH 20 "6%\"\"$%)~..~~~~~G/ *(%\"bG\"\"\"%\"cGF(-%!G6#*&%\"aGF(-F+6#*&F.F(F)F(F(F(#F(\"#I" }} {PARA 11 "" 1 "" {XPPMATH 20 "6%\"\"%%)~..~~~~~G/*&%\"bG\"\"\"-%!G6#*& %\"aGF(-F*6#*&)%\"cG\"\"#F(F-F(F(F(#F(\"#g" }}{PARA 11 "" 1 "" {XPPMATH 20 "6%\"\"'%)~..~~~~~G/*(%\"bG\"\"\"%\"cGF(-%!G6#*&)F)\"\"#F( %\"aGF(F(#F(\"#:" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#\"#;" }}}{PARA 0 " " 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 136 "Incorporating the order 2 stage-order conditions for stages 3, 4 and 5 gives a total of 19 equations which are satisfied by this scheme." }}{PARA 0 "" 0 "" {TEXT -1 36 "Specifying two of the nodes, namely " }{XPPEDIT 18 0 "c[2 ] = 1/5;" "6#/&%\"cG6#\"\"#*&\"\"\"F)\"\"&!\"\"" }{TEXT -1 6 " and " }{XPPEDIT 18 0 "c[4] = 4/5;" "6#/&%\"cG6#\"\"%*&F'\"\"\"\"\"&!\"\"" } {TEXT -1 86 ", enables the remaining coefficients to be calculated by \+ means of these 19 equations. " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 322 "e1 := \{c[2]=1/5,c[4]=4/5\} :\ne2 := solve(subs(e1,\{op(RK4_5eqsA),op(StageOrderConditions(2,5,'ex panded'))\})):\ne3 := `union`(e1,e2):\nsubs(e3,matrix([[c[2],a[2,1],`` ,``,``,``],\n[c[3],a[3,1],a[3,2],``,``,``],[c[4],a[4,1],a[4,2],a[4,3], ``,``],\n[c[5],a[5,1],a[5,2],a[5,3],a[5,4],``],\n[``,b[1],b[2],b[3],b[ 4],b[5]]]));\nevalf[8](%);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#K%'matri xG6#7'7(#\"\"\"\"\"&F(%!GF+F+F+7(#\"\"#F*\"\"!F-F+F+F+7(#\"\"%F*#\"\"' F*#!#7F*F.F+F+7(F)#!#<\"\")F*#!\"&F.#F*F:F+7(F+#\"#8\"#'*F/#\"#D\"#[#F CFA#F)\"#7Q(pprint06\"" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#K%'matrixG6# 7'7($\")+++?!\")F(%!GF+F+F+7($\")+++SF*$\"\"!F0F-F+F+F+7($\")+++!)F*$ \")+++7!\"($!)+++CF6$\"\"#F0F+F+7($\"\"\"F0$!)++D@F6$\"\"&F0$!)+++DF6$ \")++]iF*F+7(F+$\")n;a8F*F/$\")LL3_F*$\")n;/EF*$\")LLL$)!\"*Q(pprint06 \"" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 7 "Che ck: " }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 71 "expand(simplify(subs (e3,RK4_5eqsA)));\nmap(u->lhs(u)-rhs(u),%);\nnops(%);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#72/#\"\"\"\"\"&F%/#\"\"#F'F)/#\"\"%F'F,/F&F&F./#F& F*F0/#F&\"\"'F2/#F&\"\"$F5/#F&\"#CF8/#F&\"\")F;/#F&\"#7F>/#F&F-FA/#F& \"$?\"FC/#F&\"#IFF/#F&\"#gFI/#F&\"#:FL" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#72\"\"!F$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 -1 102 "Calculate the error norm, that is, the root mean squar e of the residues of the principal error terms. " }}{EXCHG {PARA 0 "> \+ " 0 "" {MPLTEXT 1 0 149 "errterms := PrincipalErrorTerms(4,5,'expanded '):\nsm := 0:\nfor ct to nops(errterms) do\n sm := sm+(evalf(subs(e3 ,errterms[ct])))^2;\nend do:\nsqrt(sm);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#$\"+p0V$*>!#7" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 " " {TEXT -1 79 "The error norm is slightly greater than that obtained f or the Fehlberg scheme. " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 " " 0 "" {TEXT -1 60 "4 of the 9 of the principal error conditions are s atisfied. " }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 123 "RK4_5err_eqs \+ := PrincipalErrorConditions(4,5,'expanded'):\nexpand(subs(e3,RK4_5err_ eqs)):\nmap(u->`if`(lhs(u)=rhs(u),0,1),%);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#7+\"\"!\"\"\"F$F$F%F$F%F%F%" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 78 "5 stage (type 4) order 4 Runge-Kutta meth od c[4]=10/11 .. 0.00212 (4,6,8,9) " }}{PARA 0 "" 0 "" {TEXT -1 0 " " }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "matrix([[3/10, 3/ 10, ``, ``, ``, ``], [7/12, -61/108, 31/27, ``, ``, ``], [10/11, 16049 35/1900668, -204685/271524, 129645/158389, ``, ``], [1, 3611/1326, -36 5519/88842, 27450/9503, -18634/37453, ``], [``, 1/10, 9250/23919, 864/ 3655, 14641/43215, -13/210]])" "6#-%'matrixG6#7'7(*&\"\"$\"\"\"\"#5!\" \"*&F)F*F+F,%!GF.F.F.7(*&\"\"(F*\"#7F,,$*&\"#hF*\"$3\"F,F,*&\"#JF*\"#F F,F.F.F.7(*&F+F*\"#6F,*&\"(N\\g\"F*\"(o1!>F,,$*&\"'&o/#F*\"'C:FF,F,*& \"'X'H\"F*\"'*Qe\"F,F.F.7(F**&\"%6OF*\"%E8F,,$*&\"'>bOF*\"&U)))F,F,*& \"&]u#F*\"%.&*F,,$*&\"&M'=F*\"&`u$F,F,F.7(F.*&F*F*F+F,*&\"%]#*F*\"&>R# F,*&\"$k)F*\"%bOF,*&\"&TY\"F*\"&:K%F,,$*&\"#8F*\"$5#F,F," }{TEXT -1 1 " " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 498 "ee := \{c[2]=3/10,c[3]=7/12,c[4]=10/11,c[5]=1,\na[2, 1]=3/10,a[3,1]=-61/108,\na[3,2]=31/27,\na[4,1]=1604935/1900668,\na[4,2 ]=-204685/271524,a[4,3]=129645/158389,\na[5,1]=3611/1326,a[5,2]=-36551 9/88842,a[5,3]=27450/9503,a[5,4]=-18634/37453,\nb[1]=1/10,b[2]=9250/23 919,b[3]=864/3655,b[4]=14641/43215,b[5]=-13/210\}:\nsubs(ee,matrix([[c [2],a[2,1],``,``,``,``],\n[c[3],a[3,1],a[3,2],``,``,``],[c[4],a[4,1],a [4,2],a[4,3],``,``],\n[c[5],a[5,1],a[5,2],a[5,3],a[5,4],``],\n[``,b[1] ,b[2],b[3],b[4],b[5]]]));\nevalf[8](%);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#K%'matrixG6#7'7(#\"\"$\"#5F(%!GF+F+F+7(#\"\"(\"#7#!#h\"$3\"#\"#J \"#FF+F+F+7(#F*\"#6#\"(N\\g\"\"(o1!>#!'&o/#\"'C:F#\"'X'H\"\"'*Qe\"F+F+ 7(\"\"\"#\"%6O\"%E8#!'>bO\"&U)))#\"&]u#\"%.&*#!&M'=\"&`u$F+7(F+#FCF*# \"%]#*\"&>R##\"$k)\"%bO#\"&TY\"\"&:K%#!#8\"$5#Q)pprint936\"" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#K%'matrixG6#7'7($\")+++I!\")F(%!GF+F+F+7($\" )LLLeF*$!)\"[\"[cF*$\")\"[\"[6!\"(F+F+F+7($\")\"444*F*$\")w0W%)F*$!)gP QvF*$\")vA&=)F*F+F+7($\"\"\"\"\"!$\")yABFF3$!)!fU6%F3$\"):c))GF3$!)CIv \\F*F+7(F+$\")+++5F*$\")&=s'QF*$\")^)QO#F*$\")S%zQ$F*$!)iZ!>'!\"*Q)ppr int946\"" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 112 "We can check that the appropriate order conditions are satisfied \+ along with the simplifying row sum conditions. " }}{EXCHG {PARA 0 "> \+ " 0 "" {MPLTEXT 1 0 129 "RK4_5eqs := [op(RowSumConditions(5,'expanded' )),op(OrderConditions(4,5,'expanded'))]:\nsubs(ee,RK4_5eqs);\nmap(u->l hs(u)-rhs(u),%);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#7./#\"\"$\"#5F%/# \"\"(\"#7F)/#F'\"#6F-/\"\"\"F0F//#F0\"\"#F2/#F0\"\"'F5/#F0F&F8/#F0\"#C F:/#F0\"\")F=/#F0F+F@/#F0\"\"%FB" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#7. \"\"!F$F$F$F$F$F$F$F$F$F$F$" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 86 "Next we set-up stage-order condtions to check f or stage-orders from 2 to 3 inclusive. " }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 85 "for ct from 2 to 3 do\n so||ct||_5 := StageOrderCon ditions(ct,5,'expanded');\nend do:" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" } }{PARA 0 "" 0 "" {TEXT -1 58 "Stages 3 to 5 have the following respect ive stage-orders. " }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 173 "[seq( [seq(expand(subs(ee,so||i||_5[j])),i=2..3)],j=1..3)]:\nmap(proc(L) loc al i; for i to nops(L) do if not evalb(L[i]) then break end if end do; i end proc,%):\nsimplify(%); " }}{PARA 11 "" 1 "" {XPPMATH 20 "6#7%\" \"\"F$F$" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 46 "The principal error conditions are as follows." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 172 "PrincipalEr rorConditions(4):\nconvert(ListTools[Enumerate](%),matrix):\nlinalg[au gment](linalg[delcols](%,2..2),matrix([[` `]$(linalg[rowdim](%))]),li nalg[delcols](%,1..1));" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#K%'matrixG6 #7+7%\"\"\"%#~~G/*&%\"bGF(-%!G6#*&%\"aGF(-F.6#*&F1F(-F.6#*&F1F(%\"cGF( F(F(F(#F(\"$?\"7%\"\"#F)/*&F,F(-F.6#*(F1F(F8F(F5F(F(#F(\"#S7%\"\"$F)/* (F,F(F8F(F2F(#F(\"#I7%\"\"%F)/*&F,F(-F.6#*&F1F(-F.6#*&)F8F \+ " 0 "" {MPLTEXT 1 0 329 "RK4_5err_eqs := PrincipalErrorConditions(4,5, 'expanded'):\nPE := PrincipalErrorConditions(4):\nRK4_5err_eqsA := []: \nfor ct to nops(RK4_5err_eqs) do\n t := expand(subs(ee,RK4_5err_eqs [ct]));\n if lhs(t)=rhs(t) then \n print(ct,` .. `,PE[ct]); \n RK4_5err_eqsA := [op(RK4_5err_eqsA),RK4_5err_eqs[i]]; \n end if;\nend do:" }}{PARA 11 "" 1 "" {XPPMATH 20 "6%\"\"%%)~..~~~~~G/*&% \"bG\"\"\"-%!G6#*&%\"aGF(-F*6#*&)%\"cG\"\"#F(F-F(F(F(#F(\"#g" }}{PARA 11 "" 1 "" {XPPMATH 20 "6%\"\"'%)~..~~~~~G/*(%\"bG\"\"\"%\"cGF(-%!G6#* &)F)\"\"#F(%\"aGF(F(#F(\"#:" }}{PARA 11 "" 1 "" {XPPMATH 20 "6%\"\")%) ~..~~~~~G/*&%\"bG\"\"\"-%!G6#*&)%\"cG\"\"$F(%\"aGF(F(#F(\"#?" }}{PARA 11 "" 1 "" {XPPMATH 20 "6%\"\"*%)~..~~~~~G/*&%\"bG\"\"\")%\"cG\"\"%F(# F(\"\"&" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 70 "This gives a total of 16 equations which are satisfied by this sch eme." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 172 "PrincipalErrorConditions(4):\nconvert(ListTools[Enum erate](%),matrix):\nlinalg[augment](linalg[delcols](%,2..2),matrix([[` `]$(linalg[rowdim](%))]),linalg[delcols](%,1..1));" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#K%'matrixG6#7+7%\"\"\"%#~~G/*&%\"bGF(-%!G6#*&%\"aGF( -F.6#*&F1F(-F.6#*&F1F(%\"cGF(F(F(F(#F(\"$?\"7%\"\"#F)/*&F,F(-F.6#*(F1F (F8F(F5F(F(#F(\"#S7%\"\"$F)/*(F,F(F8F(F2F(#F(\"#I7%\"\"%F)/*&F,F(-F.6# *&F1F(-F.6#*&)F8F " 0 "" {MPLTEXT 1 0 345 "RK4_5err_eq s := PrincipalErrorConditions(4,5,'expanded'):\nPE := PrincipalErrorCo nditions(4):\nRK4_5err_eqsB := []:\nfor ct to nops(RK4_5err_eqs) do\n \+ if ct in [4,6,8,9] then \n print(ct,` .. `,PE[ct]);\n \+ RK4_5err_eqsB := [op(RK4_5err_eqsB),RK4_5err_eqs[ct]]; \n end if;\ne nd do:\nRK4_5eqsB := [op(RK4_5eqs),op(RK4_5err_eqsB)]:\nnops(%);" }} {PARA 11 "" 1 "" {XPPMATH 20 "6%\"\"%%)~..~~~~~G/*&%\"bG\"\"\"-%!G6#*& %\"aGF(-F*6#*&)%\"cG\"\"#F(F-F(F(F(#F(\"#g" }}{PARA 11 "" 1 "" {XPPMATH 20 "6%\"\"'%)~..~~~~~G/*(%\"bG\"\"\"%\"cGF(-%!G6#*&)F)\"\"#F( %\"aGF(F(#F(\"#:" }}{PARA 11 "" 1 "" {XPPMATH 20 "6%\"\")%)~..~~~~~G/* &%\"bG\"\"\"-%!G6#*&)%\"cG\"\"$F(%\"aGF(F(#F(\"#?" }}{PARA 11 "" 1 "" {XPPMATH 20 "6%\"\"*%)~..~~~~~G/*&%\"bG\"\"\")%\"cG\"\"%F(#F(\"\"&" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#\"#;" }}}{PARA 0 "" 0 "" {TEXT -1 0 " " }}{PARA 0 "" 0 "" {TEXT -1 22 "Specifying the nodes " }{XPPEDIT 18 0 "c[2] = 3/10;" "6#/&%\"cG6#\"\"#*&\"\"$\"\"\"\"#5!\"\"" }{TEXT -1 2 ", " }{XPPEDIT 18 0 "c[3] = 7/12" "6#/&%\"cG6#\"\"$*&\"\"(\"\"\"\"#7! \"\"" }{TEXT -1 2 ", " }{XPPEDIT 18 0 "c[4] = 10/11" "6#/&%\"cG6#\"\"% *&\"#5\"\"\"\"#6!\"\"" }{TEXT -1 2 ", " }{XPPEDIT 18 0 "c[5]=1" "6#/&% \"cG6#\"\"&\"\"\"" }{TEXT -1 66 " enables the remaining coefficients t o be calculated, except for " }{XPPEDIT 18 0 "a[3,2]" "6#&%\"aG6$\"\" $\"\"#" }{TEXT -1 2 ". " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 127 "e1 := \{c[2]=3/10,c[3]=7/12,c[4]=1 0/11,c[5]=1\}:\n#infolevel[solve] := 4:\nsolve(subs(e1,\{op(RK4_5eqsB) \}));\n#infolevel[solve] := 1:" }}{PARA 12 "" 1 "" {XPPMATH 20 "6#<1/& %\"aG6$\"\"&\"\"\",&*&#\"%CI\"$@#F)&F&6$\"\"$\"\"#F)F)#\"%85\"#y!\"\"/ &F&6$\"\"%F2,&*&#\"&gB(\"&FE#F)F/F)F6#\"&D?(\"&%oCF)/&F&6$F:F),&*&#F>F ?F)F/F)F)#\"'0hJ\"'/=6F6/&F&6$F1F),&F/F6#\"\"(\"#7F)/&F&6$F2F)#F1\"#5/ &F&6$F:F1#\"'X'H\"\"'*Qe\"/&F&6$F(F1#\"&]u#\"%.&*/&F&6$F(F:#!&M'=\"&`u $/&%\"bG6#F)#F)FW/&Ffo6#F2#\"%]#*\"&>R#/&Ffo6#F1#\"$k)\"%bO/&Ffo6#F:# \"&TY\"\"&:K%/&Ffo6#F(#!#8\"$5#/&F&6$F(F2,&*&#F-F.F)F/F)F6#\"(D-.\"\"& U)))F)/F/F/" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 301 "e1 := \{c[2]=3/10 ,c[3]=7/12,c[4]=10/11,c[5]=1\}:\ne2 := solve(subs(e1,\{op(RK4_5eqsB)\} ));\ne3 := `union`(e1,e2):\nsubs(e3,matrix([[c[2],a[2,1],``,``,``,``], \n[c[3],a[3,1],a[3,2],``,``,``],[c[4],a[4,1],a[4,2],a[4,3],``,``],\n[c [5],a[5,1],a[5,2],a[5,3],a[5,4],``],\n[``,b[1],b[2],b[3],b[4],b[5]]])) ;\nevalf[8](%);" }}{PARA 12 "" 1 "" {XPPMATH 20 "6#>%#e2G<1/&%\"aG6$\" \"&\"\"\",&*&#\"%CI\"$@#F+&F(6$\"\"$\"\"#F+F+#\"%85\"#y!\"\"/&F(6$\"\" %F4,&*&#\"&gB(\"&FE#F+F1F+F8#\"&D?(\"&%oCF+/&F(6$FR#/&Fho6#F3#\"$k)\"%bO/&Fho6#F<#\"&TY\"\"& :K%/&Fho6#F*#!#8\"$5#/&F(6$F*F4,&*&#F/F0F+F1F+F8#\"(D-.\"\"&U)))F+/F1F 1" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#K%'matrixG6#7'7(#\"\"$\"#5F(%!GF+ F+F+7(#\"\"(\"#7,&&%\"aG6$F)\"\"#!\"\"F-\"\"\"F1F+F+F+7(#F*\"#6,&*&#\" &gB(\"&FE#F6F1F6F6#\"'0hJ\"'/=6F5,&*&#F=F>F6F1F6F5#\"&D?(\"&%oCF6#\"'X 'H\"\"'*Qe\"F+F+7(F6,&*&#\"%CI\"$@#F6F1F6F6#\"%85\"#yF5,&*&#FOFPF6F1F6 F5#\"(D-.\"\"&U)))F6#\"&]u#\"%.&*#!&M'=\"&`u$F+7(F+#F6F*#\"%]#*\"&>R## \"$k)\"%bO#\"&TY\"\"&:K%#!#8\"$5#Q)pprint916\"" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#K%'matrixG6#7'7($\")+++I!\")F(%!GF+F+F+7($\")LLLeF*,&*& $\"\"\"\"\"!F2&%\"aG6$\"\"$\"\"#F2!\"\"F-F2F4F+F+F+7($\")\"444*F*,&*&$ \")%\\z>$!\"(F2F4F2F2$\")RJFGFAF9,&*&$F@FAF2F4F2F9$\")?)y\"HFAF2$\")vA &=)F*F+F+7($F2F3,&*&$\")eKo8!\"'F2F4F2F2$\")zr)H\"FQF9,&*&$FPFQF2F4F2F 9$\")[hf6FQF2$\"):c))GFA$!)CIv\\F*F+7(F+$\")+++5F*$\")&=s'QF*$\")^)QO# F*$\")S%zQ$F*$!)iZ!>'!\"*Q)pprint926\"" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 70 "expand(simplify(subs(e3 ,RK4_5eqs)));\nmap(u->lhs(u)-rhs(u),%);\nnops(%);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#7./#\"\"$\"#5F%/#\"\"(\"#7F)/#F'\"#6F-/\"\"\"F0F//#F0\" \"#F2/#F0\"\"'F5/#F0F&F8/#F0\"#CF:/#F0\"\")F=/#F0F+F@/#F0\"\"%FB" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#7.\"\"!F$F$F$F$F$F$F$F$F$F$F$" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#\"#7" }}}{PARA 0 "" 0 "" {TEXT -1 0 " " }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 71 "expand(simplify(subs(e3, RK4_5eqsB)));\nmap(u->lhs(u)-rhs(u),%);\nnops(%);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#72/#\"\"$\"#5F%/#\"\"(\"#7F)/#F'\"#6F-/\"\"\"F0F//#F0\" \"#F2/#F0\"\"'F5/#F0F&F8/#F0\"#CF:/#F0\"\")F=/#F0F+F@/#F0\"\"%FB/#F0\" #gFE/#F0\"#:FH/#F0\"#?FK/#F0\"\"&FN" }}{PARA 11 "" 1 "" {XPPMATH 20 "6 #72\"\"!F$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 0 195 "errterms := PrincipalErrorTerms(4,5,'expanded'):\n sm := 0:\nfor ct to nops(errterms) do\n sm := sm+((subs(e3,errterms[ ct])))^2;\nend do:\nsimplify(sm);\nf := unapply(subs(a[3,2]=x,%),x):\n 'f(x)'=f(x);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#,,#\".4-!Hvx9\".+cUm87 *\"\"\"*&#\"-(f,f!\\6\"-Sm*[W)=F'&%\"aG6$\"\"$\"\"#F'!\"\"*&#\"/\"z45( 3Cg\"/+!Qf%z2qF'*$)F,F0F'F'F'*&#\"/%)>4&Qzi\"\"0DcD?:;H\"F'*$)F,\"\"%F 'F'F'*&#\"-Kl9ot\")\".DrHZ&>:F'*$)F,F/F'F'F1" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/-%\"fG6#%\"xG,,#\".4-!Hvx9\".+cUm87*\"\"\"*&#\"-(f,f! \\6\"-Sm*[W)=F,F'F,!\"\"*&#\"/\"z45(3Cg\"/+!Qf%z2qF,*$)F'\"\"#F,F,F,*& #\"/%)>4&Qzi\"\"0DcD?:;H\"F,*$)F'\"\"%F,F,F,*&#\"-Kl9ot\")\".DrHZ&>:F, *$)F'\"\"$F,F,F1" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 32 "plot(f(x),x=1..1.2,thickness=2);" }}{PARA 13 "" 1 "" {GLPLOT2D 476 476 476 {PLOTDATA 2 "6&-%'CURVESG6$7in7$$\"\"\" \"\"!$\"3=!QCFX#*QY\"!#A7$$\"3BLL$3VfV+\"!#<$\"3j!eB2RAIW\"F-7$$\"3nm; H[D:35F1$\"39PiP:OEI9F-7$$\"3MLLe0$=C,\"F1$\"3\"yN9qx61U\"F-7$$\"3DLL3 RBr;5F1$\"3&QBn;xBYT\"F-7$$\"3tm;zjf)4-\"F1$\"36%f>_A,SM29F-7$$\"3TL$e zs$HL5F1$\"34XI.%f!)\\S\"F-7$$\"3,+]7iI_P5F1$\"3f=M?*Gv7S\"F-7$$\"3gmm ;_M(=/\"F1$\"3Wj]!H6)H&R\"F-7$$\"3OLL3y_qX5F1$\"3e#HLm\"yn(Q\"F-7$$\"3 !*****\\1!>+0\"F1$\"39Wn)\\gYfP\"F-7$$\"31++]Z/Na5F1$\"3Ab_d+7Og8F-7$$ \"31++]$fC&e5F1$\"3=DkR\"*QRT8F-7$$\"3AL$ez6:B1\"F1$\"3a$oxmm81K\"F-7$ $\"3cmm;=C#o1\"F1$\"3A5w1hsM\"H\"F-7$$\"3smmm#pS12\"F1$\"3-:,j\"*eji7F -7$$\"3)***\\i`A3v5F1$\"3[^V=RrvC7F-7$$\"3cmmm(y8!z5F1$\"3iW*3\")*zK(= \"F-7$$\"3'***\\i.tK$3\"F1$\"3!>Fok1yA9\"F-7$$\"3%***\\(3zMu3\"F1$\"33 UE%*QD\"e4\"F-7$$\"3mmm\"H_?<4\"F1$\"3C&y?&Qc/W5F-7$$\"3am;zihl&4\"F1$ \"3**)G?&*eq+%**!#B7$$\"3OLL3#G,**4\"F1$\"3wmE$=uO$z$*F\\s7$$\"3FL$ezw 5V5\"F1$\"3s@.)fz98y)F\\s7$$\"3,+]PQ#\\\"36F1$\"39O__%*[![D)F\\s7$$\"3 GLLe\"*[H76F1$\"31tHLiG2)o(F\\s7$$\"3&******pvxl6\"F1$\"3eB,_FyN9rF\\s 7$$\"3-++D0xw?6F1$\"3Mx+%)z#*RvlF\\s7$$\"3!***\\i&p@[7\"F1$\"3)*Q\\$*) fRe3'F\\s7$$\"3%****\\2'HKH6F1$\"3%p_\\+=xGf&F\\s7$$\"3qmmmZvOL6F1$\"3 5eS@bqb3_F\\s7$$\"3$*****\\2goP6F1$\"3/,@'HYMgF\\s7$$\"3ALL3N1#4<\"F1$\"37lE `%Qz#GpF\\s7$$\"3qm;HYt7v6F1$\"3ut:rGnUh!)F\\s7$$\"3YLek6,1x6F1$\"3!3k +R')z4n)F\\s7$$\"3)******p(G**y6F1$\"3iZA,]r#)R$*F\\s7$$\"3CL$3U/37=\" F1$\"3aCl/LbC=5F-7$$\"3smmT6KU$=\"F1$\"3&y?@:,))46\"F-7$$\"39+]P$[/a= \"F1$\"3`6&RD#=V,7F-7$$\"3MLLLbdQ(=\"F1$\"35qE?&*)Q#*H\"F-7$$\"3km\"zW ?)\\*=\"F1$\"3!)>%o\\y[>T\"F-7$$\"3%***\\i`1h\">\"F1$\"3QcA:a_pL:F-7$$ \"3/+++PDj$>\"F1$\"3=Nf%))G6!f;F-7$$\"3!***\\P?Wl&>\"F1$\"3c)4`*)zUKz \"F-7$$\"3'*\\7G:3u'>\"F1$\"3+)e\\wRh\"p=F-7$$\"3/+v=5s#y>\"F1$\"3#R5w gA'zZ>F-7$$\"3))\\P40O\"*)>\"F1$\"3@fz:Ty>H?F-7$$\"3%**************>\" F1$\"3i`VR@sT8@F--%'COLOURG6&%$RGBG$\"#5!\"\"$F*F*F]_l-%*THICKNESSG6# \"\"#-%+AXESLABELSG6$Q\"x6\"Q!Ff_l-%%VIEWG6$;F($\"#7F\\_l%(DEFAULTG" 1 2 0 1 10 2 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 57 "We can ob tain a rational approximation for the value of " }{XPPEDIT 18 0 "x=a[ 3,2]" "6#/%\"xG&%\"aG6$\"\"$\"\"#" }{TEXT -1 37 " which minimizes the \+ principal error." }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 47 "diff(f(x ),x);\nfsolve(%);\nconvert(%,rational,4);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#,*#\"-(f,f!\\6\"-Sm*[W)=!\"\"*&#\"/\"z45(3Cg\"/+!pH(*Q] $\"\"\"%\"xGF,F,*&#\"/OzOSv6l\"0DcD?:;H\"F,*$)F-\"\"$F,F,F,*&#\".'fRW5 _C\".DrHZ&>:F,*$)F-\"\"#F,F,F'" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#$\"+ \"RD$[6!\"*" }}{PARA 11 "" 1 "" {XPPMATH 20 "6##\"#J\"#F" }}}{PARA 0 " " 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 46 "Now we can recalcu late the coefficients with " }{XPPEDIT 18 0 "a[3,2]=31/27" "6#/&%\"aG 6$\"\"$\"\"#*&\"#J\"\"\"\"#F!\"\"" }{TEXT -1 1 "." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 314 "e1 := \{c[2 ]=3/10,c[3]=7/12,c[4]=10/11,c[5]=1,a[3,2]=31/27\}:\ne2 := solve(subs(e 1,\{op(RK4_5eqsB)\}));\ne3 := `union`(e1,e2):\nsubs(e3,matrix([[c[2],a [2,1],``,``,``,``],\n[c[3],a[3,1],a[3,2],``,``,``],[c[4],a[4,1],a[4,2] ,a[4,3],``,``],\n[c[5],a[5,1],a[5,2],a[5,3],a[5,4],``],\n[``,b[1],b[2] ,b[3],b[4],b[5]]]));\nevalf[8](%);" }}{PARA 12 "" 1 "" {XPPMATH 20 "6# >%#e2G<0/&%\"aG6$\"\"#\"\"\"#\"\"$\"#5/&F(6$\"\"%F-#\"'X'H\"\"'*Qe\"/& F(6$\"\"&F-#\"&]u#\"%.&*/&F(6$F9F2#!&M'=\"&`u$/&%\"bG6#F+#F+F./&FE6#F* #\"%]#*\"&>R#/&FE6#F-#\"$k)\"%bO/&FE6#F2#\"&TY\"\"&:K%/&FE6#F9#!#8\"$5 #/&F(6$F9F*#!'>bO\"&U)))/&F(6$F9F+#\"%6O\"%E8/&F(6$F2F*#!'&o/#\"'C:F/& F(6$F2F+#\"(N\\g\"\"(o1!>/&F(6$F-F+#!#h\"$3\"" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#K%'matrixG6#7'7(#\"\"$\"#5F(%!GF+F+F+7(#\"\"(\"#7#!#h\" $3\"#\"#J\"#FF+F+F+7(#F*\"#6#\"(N\\g\"\"(o1!>#!'&o/#\"'C:F#\"'X'H\"\"' *Qe\"F+F+7(\"\"\"#\"%6O\"%E8#!'>bO\"&U)))#\"&]u#\"%.&*#!&M'=\"&`u$F+7( F+#FCF*#\"%]#*\"&>R##\"$k)\"%bO#\"&TY\"\"&:K%#!#8\"$5#Q)pprint896\"" } }{PARA 11 "" 1 "" {XPPMATH 20 "6#K%'matrixG6#7'7($\")+++I!\")F(%!GF+F+ F+7($\")LLLeF*$!)\"[\"[cF*$\")\"[\"[6!\"(F+F+F+7($\")\"444*F*$\")w0W%) F*$!)gPQvF*$\")vA&=)F*F+F+7($\"\"\"\"\"!$\")yABFF3$!)!fU6%F3$\"):c))GF 3$!)CIv\\F*F+7(F+$\")+++5F*$\")&=s'QF*$\")^)QO#F*$\")S%zQ$F*$!)iZ!>'! \"*Q)pprint906\"" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 26 "Calculate the error norm. " }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 149 "errterms := PrincipalErrorTerms(4,5,'expanded'):\nsm := 0:\nfor ct to nops(errterms) do\n sm := sm+(evalf(subs(e3,errter ms[ct])))^2;\nend do:\nsqrt(sm);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#$ \"+!3&>?@!#7" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 60 "4 of the 9 of the principal error conditions are satisfie d. " }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 123 "RK4_5err_eqs := Prin cipalErrorConditions(4,5,'expanded'):\nexpand(subs(e3,RK4_5err_eqs)): \nmap(u->`if`(lhs(u)=rhs(u),0,1),%);" }}{PARA 11 "" 1 "" {XPPMATH 20 " 6#7+\"\"\"F$F$\"\"!F$F%F$F%F%" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}}{SECT 1 {PARA 4 "" 0 " " {TEXT -1 84 "5 stage (type 7) order 4 Runge-Kutta method c[4]=19/20 \+ .. 0.000347 (2,4,5,6,7,8,9) " }}{PARA 0 "" 0 "" {TEXT -1 1 " " }} {PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "matrix([[1/5, 1/5, `` , ``, ``, ``], [7/18, 7/648, 245/648, ``, ``, ``], [19/20, 2489/1400, \+ -5567/1360, 155439/47600, ``, ``], [1, 19659/6650, -1181/170, 1527174/ 300475, -176/1919, ``], [``, 97/798, 0, 4374/7777, 4000/5757, -25/66]] )" "6#-%'matrixG6#7'7(*&\"\"\"F)\"\"&!\"\"*&F)F)F*F+%!GF-F-F-7(*&\"\"( F)\"#=F+*&F0F)\"$['F+*&\"$X#F)F3F+F-F-F-7(*&\"#>F)\"#?F+*&\"%*[#F)\"%+ 9F+,$*&\"%nbF)\"%g8F+F+*&\"'Ra:F)\"&+w%F+F-F-7(F)*&\"&f'>F)\"%]mF+,$*& \"%\"=\"F)\"$q\"F+F+*&\"(ur_\"F)\"'v/IF+,$*&\"$w\"F)\"%>>F+F+F-7(F-*& \"#(*F)\"$)zF+\"\"!*&\"%uVF)\"%xxF+*&\"%+SF)\"%ddF+,$*&\"#DF)\"#mF+F+ " }{TEXT -1 1 " " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 54 "The order 4 principal error conditions are as follows." } }{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 172 "PrincipalErrorConditions(4):\nconvert(ListTools[Enumerate](%),m atrix):\nlinalg[augment](linalg[delcols](%,2..2),matrix([[` `]$(linal g[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(%\"cGF(F(F(F(#F(\"$?\"7%\"\"#F)/*&F,F(-F.6#*(F1F(F8 F(F5F(F(#F(\"#S7%\"\"$F)/*(F,F(F8F(F2F(#F(\"#I7%\"\"%F)/*&F,F(-F.6#*&F 1F(-F.6#*&)F8F " 0 "" {MPLTEXT 1 0 346 "RK4_5err_eq s := PrincipalErrorConditions(4,5,'expanded'):\nPE := PrincipalErrorCo nditions(4):\nRK4_5err_eqsC := []:\nfor i to nops(RK4_5err_eqs) do\n \+ if i in [2,4,5,6,7,8,9] then \n print(i,` .. `,PE[i]);\n \+ RK4_5err_eqsC := [op(RK4_5err_eqsC),RK4_5err_eqs[i]]; \n end if;\n end do:\nRK4_5eqsC := [op(RK4_5eqs),op(RK4_5err_eqsC)]:\nnops(%);" }} {PARA 11 "" 1 "" {XPPMATH 20 "6%\"\"#%)~..~~~~~G/*&%\"bG\"\"\"-%!G6#*( %\"aGF(%\"cGF(-F*6#*&F-F(F.F(F(F(#F(\"#S" }}{PARA 11 "" 1 "" {XPPMATH 20 "6%\"\"%%)~..~~~~~G/*&%\"bG\"\"\"-%!G6#*&%\"aGF(-F*6#*&)%\"cG\"\"#F (F-F(F(F(#F(\"#g" }}{PARA 11 "" 1 "" {XPPMATH 20 "6%\"\"&%)~..~~~~~G/* &%\"bG\"\"\")-%!G6#*&%\"aGF(%\"cGF(\"\"#F(#F(\"#?" }}{PARA 11 "" 1 "" {XPPMATH 20 "6%\"\"'%)~..~~~~~G/*(%\"bG\"\"\"%\"cGF(-%!G6#*&)F)\"\"#F( %\"aGF(F(#F(\"#:" }}{PARA 11 "" 1 "" {XPPMATH 20 "6%\"\"(%)~..~~~~~G/* (%\"bG\"\"\")%\"cG\"\"#F(-%!G6#*&%\"aGF(F*F(F(#F(\"#5" }}{PARA 11 "" 1 "" {XPPMATH 20 "6%\"\")%)~..~~~~~G/*&%\"bG\"\"\"-%!G6#*&)%\"cG\"\"$F (%\"aGF(F(#F(\"#?" }}{PARA 11 "" 1 "" {XPPMATH 20 "6%\"\"*%)~..~~~~~G/ *&%\"bG\"\"\")%\"cG\"\"%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 22 "Specifying the nodes \+ " }{XPPEDIT 18 0 "c[2] = 1/5;" "6#/&%\"cG6#\"\"#*&\"\"\"F)\"\"&!\"\"" }{TEXT -1 3 ", " }{XPPEDIT 18 0 "c[4] = 19/20;" "6#/&%\"cG6#\"\"%*&\" #>\"\"\"\"#?!\"\"" }{TEXT -1 5 " and " }{XPPEDIT 18 0 "c[5]=1" "6#/&% \"cG6#\"\"&\"\"\"" }{TEXT -1 54 " enables the remaining coefficients t o be calculated. " }}{PARA 0 "" 0 "" {TEXT -1 98 "Experimentation sugg ests that the residue of each of the remaining order conditions tends \+ to 0 as " }{XPPEDIT 18 0 "c[4]" "6#&%\"cG6#\"\"%" }{TEXT -1 91 " tends to 1. For small step size and accurate computation it would be advant ageous to take " }{XPPEDIT 18 0 "c[4]" "6#&%\"cG6#\"\"%" }{TEXT -1 82 " very close to 1. For more moderate step-sizes it seems desirable to \+ avoid taking " }{XPPEDIT 18 0 "c[4]" "6#&%\"cG6#\"\"%" }{TEXT -1 17 " \+ too close to 1. " }}{PARA 0 "" 0 "" {TEXT -1 11 "The choice " } {XPPEDIT 18 0 "c[4]=19/20" "6#/&%\"cG6#\"\"%*&\"#>\"\"\"\"#?!\"\"" } {TEXT -1 29 " is a reasonable compromise. " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 22 "infolevel[solve] : = 0:" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 309 "e1 := \{c[2]=1/5,c [4]=19/20,c[5]=1\}:\nsolve(subs(e1,\{op(RK4_5eqsC)\})):\ne2 := allvalu es([%])[1]:\ne3 := `union`(e1,e2):\nsubs(e3,matrix([[c[2],a[2,1],``,`` ,``,``],\n[c[3],a[3,1],a[3,2],``,``,``],[c[4],a[4,1],a[4,2],a[4,3],``, ``],\n[c[5],a[5,1],a[5,2],a[5,3],a[5,4],``],\n[``,b[1],b[2],b[3],b[4], b[5]]]));\nevalf[8](%);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#K%'matrixG6 #7'7(#\"\"\"\"\"&F(%!GF+F+F+7(#\"\"(\"#=#F.\"$['#\"$X#F1F+F+F+7(#\"#> \"#?#\"%*[#\"%+9#!%nb\"%g8#\"'Ra:\"&+w%F+F+7(F)#\"&f'>\"%]m#!%\"=\"\"$ q\"#\"(ur_\"\"'v/I#!$w\"\"%>>F+7(F+#\"#(*\"$)z\"\"!#\"%uV\"%xx#\"%+S\" %dd#!#D\"#mQ)pprint756\"" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#K%'matrixG 6#7'7($\")+++?!\")F(%!GF+F+F+7($\")*)))))QF*$\")pC!3\"!\"*$\")U'3y$F*F +F+F+7($\")+++&*F*$\")r&yx\"!\"($!)CQ$4%F9$\")__lKF9F+F+7($\"\"\"\"\"! $\")1CcHF9$!))eq%pF9$\")F`#3&F9$!)NWr\"*F1F+7(F+$\"))Qb@\"F*$FAFA$\")n FCcF*$\")K1[pF*$!))yyy$F*Q)pprint766\"" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 71 "expand(simplify(subs(e3 ,RK4_5eqsC))):\nmap(u->lhs(u)-rhs(u),%);\nnops(%);" }}{PARA 11 "" 1 " " {XPPMATH 20 "6#75\"\"!F$F$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 0 149 "errterms := PrincipalErrorT erms(4,5,'expanded'):\nsm := 0:\nfor ct to nops(errterms) do\n sm := sm+(evalf(subs(e3,errterms[ct])))^2;\nend do:\nsqrt(sm);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#$\"+6(4iY$!#8" }}}{PARA 0 "" 0 "" {TEXT -1 0 " " }}{PARA 0 "" 0 "" {TEXT -1 86 "Next we set-up stage-order condtions \+ to check for stage-orders from 2 to 4 inclusive. " }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 85 "for ct from 2 to 4 do\n so||ct||_5 := Stage OrderConditions(ct,5,'expanded');\nend do:" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 38 "Stages 3, 4 and 5 have stage-or der 2. " }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 173 "[seq([seq(expand (subs(e3,so||i||_5[j])),i=2..4)],j=1..3)]:\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$" }} }{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 60 "7 of the 9 of the principal error conditions are satisfied. " }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 123 "RK4_5err_eqs := PrincipalErrorConditions (4,5,'expanded'):\nexpand(subs(e3,RK4_5err_eqs)):\nmap(u->`if`(lhs(u)= rhs(u),0,1),%);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#7+\"\"\"\"\"!F$F%F% F%F%F%F%" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 80 "5 stage (t ype 7) order 4 Runge-Kutta method c[4]=2/3 .. 0.000341 (1,2,3,4,5,6,7) " }}{PARA 257 "" 0 "" {TEXT -1 1 " " }}{PARA 256 "" 0 "" {TEXT -1 1 " \+ " }{XPPEDIT 18 0 "matrix([[1/7, 1/7, ``, ``, ``, ``], [7/26+sqrt(394)/ 65, -7161/16900-3/169*sqrt(394), 11711/16900+28/845*sqrt(394), ``, ``, ``], [2/3, -47443/7398+2953/7398*sqrt(394), 1895075/1348902-202237/13 48902*sqrt(394), 11481950/2023353-504290/2023353*sqrt(394), ``, ``], [ 1, 34917/1370-1761/1370*sqrt(394), -263981921/41216450+14019733/412164 50*sqrt(394), -5397690166/283644115+274957918/283644115*sqrt(394), 197 289/208175-5022/208175*sqrt(394), ``], [``, -19/36+sqrt(394)/36, 11353 3/216612-2401/216612*sqrt(394), 44511750/79089089-7000000/711801801*sq rt(394), 10755/33308-243/33308*sqrt(394), 811/6876+sqrt(394)/2292]]); " "6#-%'matrixG6#7'7(*&\"\"\"F)\"\"(!\"\"*&F)F)F*F+%!GF-F-F-7(,&*&F*F) \"#EF+F)*&-%%sqrtG6#\"$%RF)\"#lF+F),&*&\"%hrF)\"&+p\"F+F+*(\"\"$F)\"$p \"F+-F46#F6F)F+,&*&\"&6<\"F)F;F+F)*(\"#GF)\"$X)F+-F46#F6F)F)F-F-F-7(*& \"\"#F)F=F+,&*&\"&Vu%F)\"%)R(F+F+*(\"%`HF)FOF+-F46#F6F)F),&*&\"(v]*=F) \"(-*[8F+F)*(\"'PA?F)FWF+-F46#F6F)F+,&*&\")]>[6F)\"(`L-#F+F)*(\"'!H/&F )FinF+-F46#F6F)F+F-F-7(F),&*&\"&<\\$F)\"%q8F+F)*(\"%h)REF)\")]k@TF+F+*(\")L(>S\"F)FjoF+-F46#F6F)F),&*&\"+m,p(R&F )\"*:Tk$GF+F+*(\"*=z&\\FF)FbpF+-F46#F6F)F),&*&\"'*G(>F)\"'v\"3#F+F)*( \"%A]F)FjpF+-F46#F6F)F+F-7(F-,&*&\"#>F)\"#OF+F+*&-F46#F6F)FcqF+F),&*& \"'LN6F)\"'7m@F+F)*(\"%,CF)FjqF+-F46#F6F)F+,&*&\")]<^WF)\")*3*3zF+F)*( \"(+++(F)\"*,=!=rF+-F46#F6F)F+,&*&\"&b2\"F)\"&3L$F+F)*(\"$V#F)F[sF+-F4 6#F6F)F+,&*&\"$6)F)\"%woF+F)*&-F46#F6F)\"%#H#F+F)" }{TEXT -1 2 " " }} {PARA 257 "" 0 "" {TEXT -1 1 " " }}{PARA 0 "" 0 "" {TEXT -1 48 "Consid er the order 4 principal error conditions." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 172 "PrincipalErrorCon ditions(4):\nconvert(ListTools[Enumerate](%),matrix):\nlinalg[augment] (linalg[delcols](%,2..2),matrix([[` `]$(linalg[rowdim](%))]),linalg[d elcols](%,1..1));" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#K%'matrixG6#7+7% \"\"\"%#~~G/*&%\"bGF(-%!G6#*&%\"aGF(-F.6#*&F1F(-F.6#*&F1F(%\"cGF(F(F(F (#F(\"$?\"7%\"\"#F)/*&F,F(-F.6#*(F1F(F8F(F5F(F(#F(\"#S7%\"\"$F)/*(F,F( F8F(F2F(#F(\"#I7%\"\"%F)/*&F,F(-F.6#*&F1F(-F.6#*&)F8F " 0 "" {MPLTEXT 1 0 351 "RK4_5err_eqs := PrincipalErrorConditions(4,5,'e xpanded'):\nPE := PrincipalErrorConditions(4):\nRK4_5err_eqsB := []:\n for ct to nops(RK4_5err_eqs) do\n if ct in [1,2,3,4,5,6,7] then \n \+ print(ct,` .. `,PE[ct]);\n RK4_5err_eqsB := [op(RK4_5err_ eqsB),RK4_5err_eqs[ct]]; \n end if;\nend do:\nRK4_5eqsB := [op(RK4_5 eqs),op(RK4_5err_eqsB)]:\nnops(%);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6% \"\"\"%)~..~~~~~G/*&%\"bGF#-%!G6#*&%\"aGF#-F)6#*&F,F#-F)6#*&F,F#%\"cGF #F#F#F##F#\"$?\"" }}{PARA 11 "" 1 "" {XPPMATH 20 "6%\"\"#%)~..~~~~~G/* &%\"bG\"\"\"-%!G6#*(%\"aGF(%\"cGF(-F*6#*&F-F(F.F(F(F(#F(\"#S" }}{PARA 11 "" 1 "" {XPPMATH 20 "6%\"\"$%)~..~~~~~G/*(%\"bG\"\"\"%\"cGF(-%!G6#* &%\"aGF(-F+6#*&F.F(F)F(F(F(#F(\"#I" }}{PARA 11 "" 1 "" {XPPMATH 20 "6% \"\"%%)~..~~~~~G/*&%\"bG\"\"\"-%!G6#*&%\"aGF(-F*6#*&)%\"cG\"\"#F(F-F(F (F(#F(\"#g" }}{PARA 11 "" 1 "" {XPPMATH 20 "6%\"\"&%)~..~~~~~G/*&%\"bG \"\"\")-%!G6#*&%\"aGF(%\"cGF(\"\"#F(#F(\"#?" }}{PARA 11 "" 1 "" {XPPMATH 20 "6%\"\"'%)~..~~~~~G/*(%\"bG\"\"\"%\"cGF(-%!G6#*&)F)\"\"#F( %\"aGF(F(#F(\"#:" }}{PARA 11 "" 1 "" {XPPMATH 20 "6%\"\"(%)~..~~~~~G/* (%\"bG\"\"\")%\"cG\"\"#F(-%!G6#*&%\"aGF(F*F(F(#F(\"#5" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#\"#>" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 " " 0 "" {TEXT -1 22 "Specifying the nodes " }{XPPEDIT 18 0 "c[2] = 1/7 ;" "6#/&%\"cG6#\"\"#*&\"\"\"F)\"\"(!\"\"" }{TEXT -1 3 ", " }{XPPEDIT 18 0 "c[4] = 2/3;" "6#/&%\"cG6#\"\"%*&\"\"#\"\"\"\"\"$!\"\"" }{TEXT -1 2 ", " }{XPPEDIT 18 0 "c[5]=1" "6#/&%\"cG6#\"\"&\"\"\"" }{TEXT -1 54 " enables the remaining coefficients to be calculated. " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 305 "e1 \+ := \{c[2]=1/7,c[4]=2/3,c[5]=1\}:\nsolve(subs(e1,\{op(RK4_5eqsB)\})):\n e2 := allvalues(%)[1]:\ne3 := `union`(e1,e2):\nsubs(e3,matrix([[c[2],a [2,1],``,``,``,``],\n[c[3],a[3,1],a[3,2],``,``,``],[c[4],a[4,1],a[4,2] ,a[4,3],``,``],\n[c[5],a[5,1],a[5,2],a[5,3],a[5,4],``],\n[``,b[1],b[2] ,b[3],b[4],b[5]]]));\nevalf[8](%);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6# K%'matrixG6#7'7(#\"\"\"\"\"(F(%!GF+F+F+7(,&#F*\"#EF)*&\"#l!\"\"\"$%R#F )\"\"#F),&#\"%hr\"&+p\"F2*(\"\"$F)\"$p\"F2F3F4F2,&#\"&6<\"F9F)*(\"#GF) \"$X)F2F3F4F)F+F+F+7(#F5F;,&#\"&Vu%\"%)R(F2*(\"%`HF)FHF2F3F4F),&#\"(v] *=\"(-*[8F)*(\"'PA?F)FNF2F3F4F2,&#\")]>[6\"(`L-#F)*(\"'!H/&F)FTF2F3F4F 2F+F+7(F),&#\"&<\\$\"%q8F)*(\"%h)RE\")]k@TF2*(\" )L(>S\"F)F[oF2F3F4F),&#\"+m,p(R&\"*:Tk$GF2*(\"*=z&\\FF)FaoF2F3F4F),&# \"'*G(>\"'v\"3#F)*(\"%A]F)FgoF2F3F4F2F+7(F+,&#\"#>\"#OF2*&F^pF2F3F4F), &#\"'LN6\"'7m@F)*(\"%,CF)FcpF2F3F4F2,&#\")]<^W\")*3*3zF)*(\"(+++(F)\"* ,=!=rF2F3F4F2,&#\"&b2\"\"&3L$F)*(\"$V#F)F`qF2F3F4F2,&#\"$6)\"%woF)*&\" %#H#F2F3F4F)Q)pprint846\"" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#K%'matrix G6#7'7($\")9dG9!\")F(%!GF+F+F+7($\")l1YdF*$!)g%3w(F*$\")8p]8!\"(F+F+F+ 7($\")nmmmF*$\")))=5:F3$!)m1r:F3$\"(XaF(F3F+F+7($\"\"\"\"\"!$!&Iw#!\"' $\"(L*pMF3$\"'u<@FC$\")0h)o%F*F+7(F+$\"(O&fBF*$\")%H6/$F*$\")$>gn$F*$ \"))H3y\"F*$\")z1m7F*Q)pprint856\"" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" } }{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 71 "expand(simplify(subs(e3,RK4 _5eqsB))):\nmap(u->lhs(u)-rhs(u),%);\nnops(%);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#75\"\"!F$F$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 0 149 "errterms := PrincipalErrorT erms(4,5,'expanded'):\nsm := 0:\nfor ct to nops(errterms) do\n sm := sm+(evalf(subs(e3,errterms[ct])))^2;\nend do:\nsqrt(sm);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#$\"+UAl3M!#8" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 86 "Next we set-up stage-order condtions to check for stage-orders from 2 to 4 inclusive. " }}{EXCHG {PARA 0 "> \+ " 0 "" {MPLTEXT 1 0 85 "for ct from 2 to 4 do\n so||ct||_5 := StageO rderConditions(ct,5,'expanded');\nend do:" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 38 "Stages 3, 4 and 5 have stage-or der 1. " }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 173 "[seq([seq(expand (subs(e3,so||i||_5[j])),i=2..4)],j=1..3)]:\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$" } }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 60 "7 of th e 9 of the principal error conditions are satisfied. " }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 123 "RK4_5err_eqs := PrincipalErrorConditions (4,5,'expanded'):\nexpand(subs(e3,RK4_5err_eqs)):\nmap(u->`if`(lhs(u)= rhs(u),0,1),%);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#7+\"\"!F$F$F$F$F$F$ \"\"\"F%" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 81 "5 stage (t ype 6) order 4 Runge-Kutta method c[4]=14/15 .. 0.000118 (1,2,3,4,6,7) " }}{PARA 257 "" 0 "" {TEXT -1 0 "" }}{PARA 257 "" 0 "" {TEXT 258 7 " Comment" }{TEXT -1 51 ": I cannot remember how I came up with this sch eme." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 48 "C onsider the order 4 principal error conditions." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 172 "PrincipalEr rorConditions(4):\nconvert(ListTools[Enumerate](%),matrix):\nlinalg[au gment](linalg[delcols](%,2..2),matrix([[` `]$(linalg[rowdim](%))]),li nalg[delcols](%,1..1));" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#K%'matrixG6 #7+7%\"\"\"%#~~G/*&%\"bGF(-%!G6#*&%\"aGF(-F.6#*&F1F(-F.6#*&F1F(%\"cGF( F(F(F(#F(\"$?\"7%\"\"#F)/*&F,F(-F.6#*(F1F(F8F(F5F(F(#F(\"#S7%\"\"$F)/* (F,F(F8F(F2F(#F(\"#I7%\"\"%F)/*&F,F(-F.6#*&F1F(-F.6#*&)F8F " 0 "" {MPLTEXT 1 0 85 "RK4_5eqs := [op(RowSumConditions(5,'expanded' )),op(OrderConditions(4,5,'expanded'))]:" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 349 "RK4_5err_eqs := PrincipalErrorConditions(4,5,'expand ed'):\nPE := PrincipalErrorConditions(4):\nRK4_5err_eqsD := []:\nfor c t to nops(RK4_5err_eqs) do\n if ct in [1,2,3,4,6,7] then \n pri nt(ct,` .. `,PE[ct]);\n RK4_5err_eqsD := [op(RK4_5err_eqsD),R K4_5err_eqs[ct]]; \n end if;\nend do:\nRK4_5eqsD := [op(RK4_5eqs),op (RK4_5err_eqsD)]:\nnops(%);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6%\"\"\"% )~..~~~~~G/*&%\"bGF#-%!G6#*&%\"aGF#-F)6#*&F,F#-F)6#*&F,F#%\"cGF#F#F#F# #F#\"$?\"" }}{PARA 11 "" 1 "" {XPPMATH 20 "6%\"\"#%)~..~~~~~G/*&%\"bG \"\"\"-%!G6#*(%\"aGF(%\"cGF(-F*6#*&F-F(F.F(F(F(#F(\"#S" }}{PARA 11 "" 1 "" {XPPMATH 20 "6%\"\"$%)~..~~~~~G/*(%\"bG\"\"\"%\"cGF(-%!G6#*&%\"aG F(-F+6#*&F.F(F)F(F(F(#F(\"#I" }}{PARA 11 "" 1 "" {XPPMATH 20 "6%\"\"%% )~..~~~~~G/*&%\"bG\"\"\"-%!G6#*&%\"aGF(-F*6#*&)%\"cG\"\"#F(F-F(F(F(#F( \"#g" }}{PARA 11 "" 1 "" {XPPMATH 20 "6%\"\"'%)~..~~~~~G/*(%\"bG\"\"\" %\"cGF(-%!G6#*&)F)\"\"#F(%\"aGF(F(#F(\"#:" }}{PARA 11 "" 1 "" {XPPMATH 20 "6%\"\"(%)~..~~~~~G/*(%\"bG\"\"\")%\"cG\"\"#F(-%!G6#*&%\"a GF(F*F(F(#F(\"#5" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#\"#=" }}}{PARA 0 " " 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 22 "Specifying the nod es " }{XPPEDIT 18 0 "c[2] = 1/5;" "6#/&%\"cG6#\"\"#*&\"\"\"F)\"\"&!\" \"" }{TEXT -1 2 ", " }{XPPEDIT 18 0 "c[3]=41423/95610+2/47805*sqrt(286 834)" "6#/&%\"cG6#\"\"$,&*&\"&B9%\"\"\"\"&5c*!\"\"F+*(\"\"#F+\"&0y%F-- %%sqrtG6#\"'MoGF+F+" }{TEXT -1 3 ", " }{XPPEDIT 18 0 "c[4] = 14/15;" "6#/&%\"cG6#\"\"%*&\"#9\"\"\"\"#:!\"\"" }{TEXT -1 2 ", " }{XPPEDIT 18 0 "c[5]=1" "6#/&%\"cG6#\"\"&\"\"\"" }{TEXT -1 54 " enables the remaini ng coefficients to be calculated. " }}{PARA 0 "" 0 "" {TEXT -1 14 "The value for " }{XPPEDIT 18 0 "c[3]" "6#&%\"cG6#\"\"$" }{TEXT -1 112 " m ay have come from a previous analysis using a faulty error condition, \+ or from looking at the principal error. " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 328 "e1 := \{c[2]=1/5,c[3 ]=41423/95610+2/47805*sqrt(286834),c[4]=14/15,c[5]=1\}:\ne2 := solve(s ubs(e1,\{op(RK4_5eqsD)\})):\ne3 := `union`(e1,e2):\nsubs(e3,matrix([[c [2],a[2,1],``,``,``,``],\n[c[3],a[3,1],a[3,2],``,``,``],[c[4],a[4,1],a [4,2],a[4,3],``,``],\n[c[5],a[5,1],a[5,2],a[5,3],a[5,4],``],\n[``,b[1] ,b[2],b[3],b[4],b[5]]]));\nevalf[8](%);" }}{PARA 12 "" 1 "" {XPPMATH 20 "6#K%'matrixG6#7'7(#\"\"\"\"\"&F(%!GF+F+F+7(,&#\"&B9%\"&5c*F)*(\"\" #F)\"&0y%!\"\"\"'MoG#F)F2F),&#\"*hIFO\"\"+?WDG=F4*(\"&-Y%F)\"*0O1d%F4F 5F6F4,&#\"*nOOG*F:F)*(\"&CP'F)F=F4F5F6F)F+F+F+7(#\"#9\"#:,&#\"*@L-q%\" *g==k#F)*(\"&`T\"F)\")?11))F4F5F6F),&#\"+\\#>9I\"\"*gIM!QF4*(\"'tj5F)F QF4F5F6F),&#\".x:O%pn()\".0(H^y.MF)*(\"+T4,*\\\"F)FWF4F5F6F4F+F+7(F),& #\"+$)fgE?\"*6KT.&F)*(\"(hVD\"F)\"+WGl8?F4F5F6F),&#\"+=1(\\^#\"**\\81J F4*(\"%rUF)\"+'*RXU7F4F5F6F),&#\"33)=SOE)e86\"2F!\\y)4BE7#F)*(\"/,!G'H O+8F)FfoF4F5F6F4,&#\"'QZ6\"'(ya'F4*(\"\"*F)F\\pF4F5F6F4F+7(F+,&*(\"#DF )\"(uM]\"F4F5F6F4#\"&Un#\"'z0DF),&*(\"$D'F)\"(%)G!oF4F5F6F)#\"'D9N\"(U 9S$F),&*(\".%y!4H#3:F)\"2f$QuDWt\"f#F4F5F6F4#\"2Qwl>/g\"p7FaqF),&*(FFF )\"'')RHF4F5F6F4#\"'XG9FfqF),&*&\"&O%HF4F5F6F)#\"%m8\"%ftF4Q(pprint36 \"" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#K%'matrixG6#7'7($\")+++?!\")F(%! GF+F+F+7($\")1ccXF*$!)4+o7F*$\")9cCeF*F+F+F+7($\")LLL$*F*$\")=Cl=!\"($ !)C\">F$F8$\")S+SBF8F+F+7($\"\"\"\"\"!$\"):NfVF8$!)P%\\4)F8$\")H==\\F8 $!)'3f#=F*F+7(F+$\")2`\"y*!\"*$\")p?D:F*$\")YG&e%F*$\")Ck&e%F*$!)qGu;F *Q(pprint46\"" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 71 "expand(simplify(subs(e3,RK4_5eqsD))):\nmap(u->lh s(u)-rhs(u),%);\nnops(%);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#74\"\"!F$ 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 0 149 "errterms := PrincipalErrorTerms(4,5,'expanded'):\nsm := 0:\nfor ct to nops(errterms) do\n sm := sm+(evalf(subs(e3,errter ms[ct])))^2;\nend do:\nsqrt(sm);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#$ \"+1&G.=\"!#8" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 86 "Next we set-up stage-order condtions to check for stage-o rders from 2 to 4 inclusive. " }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 85 "for ct from 2 to 4 do\n so||ct||_5 := StageOrderConditions(ct, 5,'expanded');\nend do:" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 " " 0 "" {TEXT -1 38 "Stages 3, 4 and 5 have stage-order 1. " }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 173 "[seq([seq(expand(subs(e3,so||i||_5 [j])),i=2..4)],j=1..3)]:\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$" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 60 "6 of the 9 of the princip al error conditions are satisfied. " }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 123 "RK4_5err_eqs := PrincipalErrorConditions(4,5,'expand ed'):\nexpand(subs(e3,RK4_5err_eqs)):\nmap(u->`if`(lhs(u)=rhs(u),0,1), %);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#7+\"\"!F$F$F$\"\"\"F$F$F%F%" }} }{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 80 "5 stage (type 6) orde r 4 Runge-Kutta method c[4]=14/15 .. 0.000102 (1,2,3,4,6,7)" }}{PARA 257 "" 0 "" {TEXT -1 1 " " }}{PARA 0 "" 0 "" {TEXT -1 48 "Consider the order 4 principal error conditions." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 172 "PrincipalErrorConditions( 4):\nconvert(ListTools[Enumerate](%),matrix):\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(%\"cGF(F(F(F(#F(\"$? \"7%\"\"#F)/*&F,F(-F.6#*(F1F(F8F(F5F(F(#F(\"#S7%\"\"$F)/*(F,F(F8F(F2F( #F(\"#I7%\"\"%F)/*&F,F(-F.6#*&F1F(-F.6#*&)F8F " 0 "" {MPLTEXT 1 0 349 "RK4_5err_eqs := PrincipalErrorConditions(4,5,'expand ed'):\nPE := PrincipalErrorConditions(4):\nRK4_5err_eqsD := []:\nfor c t to nops(RK4_5err_eqs) do\n if ct in [1,2,3,4,6,7] then \n pri nt(ct,` .. `,PE[ct]);\n RK4_5err_eqsD := [op(RK4_5err_eqsD),R K4_5err_eqs[ct]]; \n end if;\nend do:\nRK4_5eqsD := [op(RK4_5eqs),op (RK4_5err_eqsD)]:\nnops(%);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6%\"\"\"% )~..~~~~~G/*&%\"bGF#-%!G6#*&%\"aGF#-F)6#*&F,F#-F)6#*&F,F#%\"cGF#F#F#F# #F#\"$?\"" }}{PARA 11 "" 1 "" {XPPMATH 20 "6%\"\"#%)~..~~~~~G/*&%\"bG \"\"\"-%!G6#*(%\"aGF(%\"cGF(-F*6#*&F-F(F.F(F(F(#F(\"#S" }}{PARA 11 "" 1 "" {XPPMATH 20 "6%\"\"$%)~..~~~~~G/*(%\"bG\"\"\"%\"cGF(-%!G6#*&%\"aG F(-F+6#*&F.F(F)F(F(F(#F(\"#I" }}{PARA 11 "" 1 "" {XPPMATH 20 "6%\"\"%% )~..~~~~~G/*&%\"bG\"\"\"-%!G6#*&%\"aGF(-F*6#*&)%\"cG\"\"#F(F-F(F(F(#F( \"#g" }}{PARA 11 "" 1 "" {XPPMATH 20 "6%\"\"'%)~..~~~~~G/*(%\"bG\"\"\" %\"cGF(-%!G6#*&)F)\"\"#F(%\"aGF(F(#F(\"#:" }}{PARA 11 "" 1 "" {XPPMATH 20 "6%\"\"(%)~..~~~~~G/*(%\"bG\"\"\")%\"cG\"\"#F(-%!G6#*&%\"a GF(F*F(F(#F(\"#5" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#\"#=" }}}{PARA 0 " " 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 21 "specifying the nod es " }{XPPEDIT 18 0 "c[2]=1/5" "6#/&%\"cG6#\"\"#*&\"\"\"F)\"\"&!\"\"" }{TEXT -1 3 ", " }{XPPEDIT 18 0 "c[4]=14/15" "6#/&%\"cG6#\"\"%*&\"#9 \"\"\"\"#:!\"\"" }{TEXT -1 3 ", " }{XPPEDIT 18 0 "c[5]=1" "6#/&%\"cG6 #\"\"&\"\"\"" }{TEXT -1 28 ", we obtain a solution with " }{XPPEDIT 18 0 "c[3]" "6#&%\"cG6#\"\"$" }{TEXT -1 17 " as a parameter. " }} {PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 99 "e1 := \{c[2]=1/5,c[4]=14/15,c[5]=1\}:\ne2 := solve(subs(e1,\{op(RK 4_5eqsD)\}))[1];\ne3 := `union`(e1,e2):" }}{PARA 12 "" 1 "" {XPPMATH 20 "6#>%#e2G<2/&%\"aG6$\"\"&\"\"#,$*&#\"#?\"#6\"\"\"*&,(*&\"$&pF1)&%\" cG6#\"\"$F+F1F1*&\"$*pF1F7F1!\"\"\"$C#F1F1,(*&\"$5$F1F6F1F1*&\"$Z#F1F7 F1F=\"#PF1F=F1F1/&F(6$F*F:,$*,F/F1,&F7F1F1F=F1,&*&\"#XF1F7F1F1FDF=F1,* *&\"%X!)F1F6F1F=*&\"%]YF1)F7F:F1F1*&\"%8SF1F7F1F1\"$=&F=F=F7F=F1/&F(6$ \"\"%F:,$*&#\"$a\"FMF1**,&*&\"#:F1F7F1F1\"#9F=F1,&F1F=*&F*F1F7F1F1F=F7 F=,&*&\"#!*F1F7F1F1\"#fF=F=F1F1/&F(6$FZF+,$*&#Fhn\"\"*F1*(,(*&\"#vF1F6 F1F1*&F^pF1F7F1F=\"#BF1F1F^oF=F`oF=F1F1/&F(6$FZF1,$*&#F]oFMF1*(,(*&\"$ b&F1F6F1F1*&\"$$[F1F7F1F=FhnF1F1F7F=F`oF=F1F=/&F(6$F:F+*&F^oF1F7F1/&F( 6$F*F1,$*&#F1\"\"(F1*(,(*&\"%'G\"F1F6F1F1*&\"%J6F1F7F1F=\"$q$F1F1,&*& \"#iF1F7F1F1FDF=F=F7F=F1F=/&F(6$F*FZ,$*&#Fbo\"#xF1*(,(*&FboF1F6F1F1*& \"$\\\"F1F7F1F=FcoF1F1F_rF=FjnF=F1F=/&F(6$F:F1,&*&F+F1F7F1F1*&F*F1F6F1 F=/&%\"bG6#F*,$*&#F1\"#'*F1*&F_rF1FJF=F1F=/&Ffs6#FZ,$*&#\"$D#\"%K7F1*& F`oF1FjnF=F1F1/&FfsF9,$*&#\"#D\"#CF1*&F1F1**F7F1FJF1F^oF1FjnF1F=F1F1/& F(6$F+F1#F1F*/&Ffs6#F+,$*&#F[u\"%c5F1*&,&*&\"$I\"F1F7F1F1\"#^F=F1F^oF= F1F1/&Ffs6#F1,$*&#F1\"$O$F1*&,&*&\"#AF1F7F1F1F[uF=F1F7F=F1F=/F7F7" }}} {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 37 "The princ ipal error can be minimised." }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 144 "errterms := PrincipalErrorTerms(4,5,'expanded'):\nsm := 0:\nfor c t to nops(errterms) do\n sm := sm+subs(e3,errterms[ct])^2;\nend do: \nsimplify(sm);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#,$*&#\"\"\"\"+++Cm= F&*(,,*&\"-+)y95\\$F&)&%\"cG6#\"\"$\"\"%F&F&*&\"-g`RFIhF&)F-F0F&!\"\"* &\"-3))4]TSF&)F-\"\"#F&F&*&\"-O&='p&=\"F&F-F&F5\",8pLjI\"F&F&,&*&\"#!* F&F-F&F&\"#fF5!\"#,&*&\"#iF&F-F&F&\"#PF5FAF&F&" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 32 "plot(sm,c[3] =0.35..0.5,0..1e-7);" }}{PARA 13 "" 1 "" {GLPLOT2D 357 199 199 {PLOTDATA 2 "6%-%'CURVESG6$7Z7$$\"3w*************\\$!#=$\"3(\\4HxF[tK) !#D7$$\"3e***\\7t&pKNF*$\"3g*Q@=g6An(F-7$$\"3#)*\\(=7T9hNF*$\"3t))Rq+z UGrF-7$$\"3')**\\(=HPJf$F*$\"3'eV9W.$RYlF-7$$\"3r**\\7VDMDOF*$\"3y9.\" )p7L#*fF-7$$\"3q*\\P%GZRdOF*$\"3O;N5#*z#GZ&F-7$$\"31+v=276(o$F*$\"3hi6 (>&yu>]F-7$$\"3e*\\(o**3)yr$F*$\"3?aBimggzXF-7$$\"3y*\\(ofHq\\PF*$\"3I u\"*ohDQbTF-7$$\"3#)*\\Pf'HU\"y$F*$\"33@dQL=pjPF-7$$\"3&)***\\7*309QF* $\"3%zKO9=cHR$F-7$$\"3&***\\i&e*yUQF*$\"39'H(\\$)e2$4$F-7$$\"3k***\\([ D9vQF*$\"3S)y:?jK[y#F-7$$\"3r***\\iNGw!RF*$\"3HC3iC#*)e]#F-7$$\"3>++D^ W$*QRF*$\"31F0da8@lAF-7$$\"3/+vo%Qjt'RF*$\"3e(4lxBk(p?F-7$$\"37++DO\"o 6+%F*$\"33eUgze!['=F-7$$\"3n*****\\>0)HSF*$\"3w`R$\\(=S8M\"F-7$$\"3#**\\i:$4wbTF*$\"3C)HpJ`m(f7F-7$$\"3E +](=#R!z=%F*$\"3Wq/J'4c2>\"F-7$$\"30+v$4A@u@%F*$\"3))3d7\"4^39\"F-7$$ \"35+]i:'f#\\UF*$\"3o@M_2Iz*4\"F-7$$\"3))*\\(of2L#G%F*$\"3em!or<9&p5F- 7$$\"3()*\\7yG>6J%F*$\"3Y>iC^*GB0\"F-7$$\"3#***\\(oo6AM%F*$\"3\\N?h'o' oU5F-7$$\"3%*****\\xJLuVF*$\"3+5G%*zBXU5F-7$$\"3'***\\P*yddS%F*$\"3i.W 479\"G0\"F-7$$\"35+v=xYr\"=\"F-7$$\"3\")***\\i0XE` %F*$\"3'\\#z_^[e%G\"F-7$$\"3s*\\(o/Q*>c%F*$\"3@z#yJ*znA9F-7$$\"3')*** \\(Q(zSf%F*$\"3%zi;M'[fU;F-7$$\"3!**\\(=-,FCYF*$\"3:ZwMh(H2%>F-7$$\"39 +v$4tFel%F*$\"3ADyk@\">IQ#F-7$$\"3%)***\\73\"o'o%F*$\"3+d;c.tH\"*HF-7$ $\"3/+voz;)*=ZF*$\"3#Q0*3,v:\")QF-7$$\"3K+++&*44]ZF*$\"3C-9vY:y%F*$\"3\"=q>9%QtEnF-7$$\"34+v=(4bM\"[F*$\"3#ReWym.W'*)F-7$ $\"3I++]xlWU[F*$\"3mB;*))Ht,<\"!#C7$$\"3G+DcJ.1f[F*$\"3[\\'fX[;PO\"F_y 7$$\"3E+]i&3uc([F*$\"3)zaJRPb\"*e\"F_y7$$\"3O+DJDO`!*[F*$\"3(*[W6AVx@= F_y7$$\"3W+++lJR0\\F*$\"3)=7^*=:v(3#F_y7$$\"3C]PfLlB@\\F*$\"3:=x'3$F_y7$$\"3E+D\"G:3u'\\F*$\"3S#H'\\(4x\"oOF_y7$$\"3?v$4Y6cb(\\F*$\"3 54Zu*Q4t%RF_y7$$\"37]iSwSq$)\\F*$\"3M()HFK[%pC%F_y7$$\"31DJ?Q?&=*\\F*$ \"3#*G,FgB`oXF_y7$$\"3++++++++]F*$\"3T&faQ6BO\"\\F_y-%'COLOURG6&%$RGBG $\"#5!\"\"$\"\"!Fd]lFc]l-%+AXESLABELSG6$Q%c[3]6\"Q!Fi]l-%%VIEWG6$;$\"# N!\"#$\"\"&Fb]l;Fc]l$\"\"\"!\"(" 1 2 0 1 10 0 2 9 1 4 2 1.000000 45.000000 45.000000 0 0 "Curve 1" }}}}{PARA 0 "" 0 "" {TEXT -1 0 "" }} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 25 "diff(sm,c[3]):\nsolve(%);\n " }}{PARA 11 "" 1 "" {XPPMATH 20 "6&,&#\"%R9\"%!G$\"\"\"*&^##F'F&F'\"% >b#F'\"\"#F',&F$F'*&^##!\"\"F&F'F+F,F',&#\"%$=&\"%-)*F'*(\"##)F'\"&0X# F2\"$q(F,F',&F4F'*(F8F'F9F2F:F,F2" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }} {PARA 0 "" 0 "" {TEXT -1 11 "The value " }{XPPEDIT 18 0 "c[3]=5183/98 02-82/24505*sqrt(770)" "6#/&%\"cG6#\"\"$,&*&\"%$=&\"\"\"\"%-)*!\"\"F+* (\"##)F+\"&0X#F--%%sqrtG6#\"$q(F+F-" }{TEXT -1 32 ", minimizes the pri ncipal error." }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 39 "5183/9802-8 2/24505*sqrt(770);\nevalf(%);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#,&#\" %$=&\"%-)*\"\"\"*(\"##)F'\"&0X#!\"\"\"$q(#F'\"\"#F+" }}{PARA 11 "" 1 " " {XPPMATH 20 "6#$\"+p![\"fV!#5" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 324 "e1 := \{c[2]=1/5,c[3]=5183/ 9802-82/24505*sqrt(770),c[4]=14/15,c[5]=1\}:\ne2 := solve(subs(e1,\{op (RK4_5eqsD)\})):\ne3 := `union`(e1,e2):\nsubs(e3,matrix([[c[2],a[2,1], ``,``,``,``],\n[c[3],a[3,1],a[3,2],``,``,``],[c[4],a[4,1],a[4,2],a[4,3 ],``,``],\n[c[5],a[5,1],a[5,2],a[5,3],a[5,4],``],\n[``,b[1],b[2],b[3], b[4],b[5]]]));\nevalf[8](%);" }}{PARA 12 "" 1 "" {XPPMATH 20 "6#K%'mat rixG6#7'7(#\"\"\"\"\"&F(%!GF+F+F+7(,&#\"%$=&\"%-)*F)*(\"##)F)\"&0X#!\" \"\"$q(#F)\"\"#F4,&#\")(*=&o$\")/#zg*F4*(\"(m7K\"F)\"*0!*4?\"F4F5F6F), &#\")jcl()F;F)*(\"([Js\"F)F>F4F5F6F4F+F+F+7(#\"#9\"#:,&#\"+V]$)o7\"*!= ?g>F)*(\")Z,DLF)FKF4F5F6F4,&#\"(4\\9&\"'))\\eF4*(\"%2%)F)\"&/G%F4F5F6F ),&*(\")V$f!QF)\"+0j9@9F4F5F6F4#\",V!4Z$z$\",,(*R`;\"F)F+F+7(F),&#\"*J m$f:\")9gC:F4*(\"*:aM4\"F)\"*'>WM@F4F5F6F),&#\"&&>H\"%yBF)*(\")v\"Q4\" F)\")#p9]\"F4F5F6F4,&#\"/]g_k@zA\"/hB'*\\&)f@F4*(\"1&\\,T-G2Q\"F)\"12w >RQy)>'F4F5F6F),&*(\"(D'3CF)\"*^pio$F4F5F6F4#\"&?&H\"(jty%F)F+7(F+,&# \"&@[\"\"'Ff=F)*(\"%D5F)\"(ib6\"F4F5F6F),&#\"$+)\"%nNF)*(\"$D'F)\"'[p: F4F5F6F4,&#\"3]#ymA$>g-@\"3\"[aDwjK'=XF)*(\"0+!GoepAyF)\"4VMm(G\"z*eb8 F4F5F6F),&#\"(Du)G\"(EZd*F)*(\"'v=pF)\"*')>K0\"F4F5F6F),&#\"%(*))\"'Eb 7F4*(FipF)\"'_5DF4F5F6F4Q)pprint486\"" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#K%'matrixG6#7'7($\")+++?!\")F(%!GF+F+F+7($\")\"[\"fVF*$!(**y#yF*$\" )!Q>9&F*F+F+F+7($\")LLL$*F*$\")t/m " 0 "" {MPLTEXT 1 0 325 "e1 := \{c[2 ]=1/20,c[2]=3806/7489-194/7489*sqrt(10^),c[4]=19/20,c[5]=1\}:\ne2 := s olve(subs(e1,\{op(RK4_5eqsD)\})):\ne3 := `union`(e1,e2):\nsubs(e3,matr ix([[c[2],a[2,1],``,``,``,``],\n[c[3],a[3,1],a[3,2],``,``,``],[c[4],a[ 4,1],a[4,2],a[4,3],``,``],\n[c[5],a[5,1],a[5,2],a[5,3],a[5,4],``],\n[` `,b[1],b[2],b[3],b[4],b[5]]]));\nevalf[8](%);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#74\"\"!F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$" }}{PARA 11 " " 1 "" {XPPMATH 20 "6#\"#=" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 149 "errterms := PrincipalErrorTerms(4, 5,'expanded'):\nsm := 0:\nfor ct to nops(errterms) do\n sm := sm+(ev alf(subs(e3,errterms[ct])))^2;\nend do:\nsqrt(sm);" }}{PARA 11 "" 1 " " {XPPMATH 20 "6#$\"+RxV?5!#8" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }} {PARA 0 "" 0 "" {TEXT -1 86 "Next we set-up stage-order condtions to c heck for stage-orders from 2 to 4 inclusive. " }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 85 "for ct from 2 to 4 do\n so||ct||_5 := StageOrd erConditions(ct,5,'expanded');\nend do:" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 38 "Stages 3, 4 and 5 have stage-order 1. " }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 173 "[seq([seq(expand(su bs(e3,so||i||_5[j])),i=2..4)],j=1..3)]:\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$F$" }}} {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 78 "We can ch eck that 6 of the 9 of the principal error conditions are satisfied. \+ " }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 123 "RK4_5err_eqs := Princip alErrorConditions(4,5,'expanded'):\nexpand(subs(e3,RK4_5err_eqs)):\nma p(u->`if`(lhs(u)=rhs(u),0,1),%);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#7+ \"\"!F$F$F$\"\"\"F$F$F%F%" }}}{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 36 "Test-bed p rocedures for the examples" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 7 "RK4step" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 2224 "rk4step := proc(x_rk4step::realcons)\n local c2,c3,c4,c5,a21,a31,a32,a41,a42,a 43,a51,a52,a53,a54,\n f1,f2,f3,f4,f5,b1,b2,b3,b4,b5,xk,yk,t,jF,jM,jS ,n,h,data,\n fn,xx,ys,saveDigits;\n options `Copyright 2004 by Pet er Stone`;\n \n data := SOLN_;\n\n saveDigits := Digits;\n Dig its := max(trunc(evalhf(Digits)),Digits+5);\n\n # procedure to evalu ate the gradient field\n fn := proc(X_,Y_)\n local val; \n \+ val := traperror(evalf(FXY_));\n if val=lasterror or not type(va l,numeric) then\n error \"evaluation of gradient field failed \+ at %1\",evalf([X_,Y_],saveDigits);\n end if;\n val;\n e nd proc;\n\n xx := evalf(x_rk4step);\n n := nops(data);\n\n if ( data[1,1]data[n,1] or xxdata[1,1])) then\n error \"independent variable is outside the interpolation interval: %1\",ev alf(data[1,1])..evalf(data[n,1]);\n end if;\n\n c2 := c2_; c3 := c 3_; c4 := c4_; c5 := c5_;\n a21 := c2; a31 := a31_; a32 := a32_; \n \+ a41 := a41_; a42 := a42_; a43 := a43_;\n a51 := a51_; a52 := a52_; a53 := a53_; a54 := a54_;\n b1 := b1_; b2 := b2_; b3 := b3_; b4 := \+ b4_; b5 := b5_;\n # Perform a binary search for the interval contain ing x.\n n := nops(data);\n jF := 0;\n jS := n+1;\n\n if data[ 1,1]1 do\n jM := trunc((jF+j S)/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 e lse\n while jS-jF> 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 end if;\n \n \+ # Get the data needed from the list.\n xk := data[jF,1];\n yk := d ata[jF,2];\n\n # Do one step with step-size ..\n h := xx-xk;\n f 1 := 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*f 1 + a52*f2 + a53*f3 + a54*f4;\n f5 := fn(xk + c5*h,yk + t*h);\n \n t := b1*f1 + b2*f2 + b3*f3 + b4*f4 + b5*f5; \n ys := yk + t*h; \n\n evalf[saveDigits](ys);\nend proc: # of rk4step" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}} {SECT 1 {PARA 4 "" 0 "" {TEXT -1 23 "RK4_1 Fehlberg (type 0)" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1783 " RK4_1 := proc(fxy,x,y,xx,yy,h,stps,bb)\n local c2,c3,c4,c5,a21,a31,a 32,a41,a42,a43,a51,a52,a53,a54,\n f1,f2,f3,f4,f5,b1,b2,b3,b4,b5,t,k, fn,xk,yk,soln,eqns,A,saveDigits;\n\n saveDigits := Digits;\n Digit s := max(trunc(evalhf(Digits)),Digits+5);\n\n fn := unapply(fxy,x,y) ;\n\n A := matrix([[1/4,1/4,0,0,0,0],\n [3/8,3/32,9/32,0,0,0],[1 2/13,1932/2197,-7200/2197,7296/2197,0,0],\n [1,439/216,-8,3680/513 ,-845/4104,0],\n [0,25/216,0,1408/2565,2197/4104,-1/5]]);\n \n c 2 := evalf(A[1,1]);\n c3 := evalf(A[2,1]);\n c4 := evalf(A[3,1]); \n c5 := evalf(A[4,1]);\n a21 := c2;\n a31 := evalf(A[2,2]);\n \+ a32 := evalf(A[2,3]);\n a41 := evalf(A[3,2]);\n a42 := evalf(A[3, 3]);\n a43 := evalf(A[3,4]);\n a51 := evalf(A[4,2]);\n a52 := ev alf(A[4,3]);\n a53 := evalf(A[4,4]);\n a54 := evalf(A[4,5]);\n b 1 := evalf(A[5,2]);\n b2 := evalf(A[5,3]);\n b3 := evalf(A[5,4]); \n b4 := evalf(A[5,5]);\n b5 := evalf(A[5,6]);\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,y k + t*h);\n t := a41*f1 + a42*f2 + a43*f3;\n f4 := fn(xk + c 4*h,yk + t*h);\n t := a51*f1 + a52*f2 + a53*f3 + a54*f4;\n f 5 := fn(xk + c5*h,yk + t*h);\n\n yk := yk + (b1*f1 + b2*f2 + b3*f 3 + b4*f4 + b5*f5)*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,a31_=a31,a32_=a32, a41_=a41,a42_=a42,\n a43_=a43,a51_=a51,a52_=a52,a53_=a53,a54_= a54,\n b1_=b1,b2_=b2,b3_=b3,b4_=b4,b5_=b5\};\n return sub s(eqns,eval(rk4step)); \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 "RK4_2 Prince (type 5)" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1722 "RK4_2 := proc(fxy,x,y,xx ,yy,h,stps,bb)\n local c2,c3,c4,c5,a21,a31,a32,a41,a42,a43,a51,a52,a 53,a54,\n f1,f2,f3,f4,f5,b1,b2,b3,b4,b5,t,k,fn,xk,yk,soln,eqns,A,sav eDigits;\n\n saveDigits := Digits;\n Digits := max(trunc(evalhf(Di gits)),Digits+5);\n\n fn := unapply(fxy,x,y);\n\n A := matrix([[1/ 5,1/5,0,0,0,0],[2/5,0,2/5,0,0,0],[4/5,6/5,-12/5,2,0,0],\n [1,-17/8,5 ,-5/2,5/8,0],[0,13/96,0,25/48,25/96,1/12]]);\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 a21 := c2;\n a31 := evalf(A[2,2]);\n a32 := evalf(A[2, 3]);\n a41 := evalf(A[3,2]);\n a42 := evalf(A[3,3]);\n a43 := ev alf(A[3,4]);\n a51 := evalf(A[4,2]);\n a52 := evalf(A[4,3]);\n a 53 := evalf(A[4,4]);\n a54 := evalf(A[4,5]);\n b1 := evalf(A[5,2]) ;\n b2 := evalf(A[5,3]);\n b3 := evalf(A[5,4]);\n b4 := evalf(A[ 5,5]);\n b5 := evalf(A[5,6]);\n xk := evalf(xx);\n yk := evalf(y y);\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,y k + t*h);\n\n yk := yk + (b1*f1 + b2*f2 + b3*f3 + b4*f4 + b5*f5)* h;\n xk := xk + h:\n soln := soln,[xk,yk];\n end do;\n i f bb=true then\n eqns := \{SOLN_=[soln],FXY_=fxy,X_=x,Y_=y,c2_=c2 ,c3_=c3,\n c4_=c4,c5_=c5,a31_=a31,a32_=a32,a41_=a41,a42_=a42, \n a43_=a43,a51_=a51,a52_=a52,a53_=a53,a54_=a54,\n b1_ =b1,b2_=b2,b3_=b3,b4_=b4,b5_=b5\};\n return subs(eqns,eval(rk4ste p)); \n else\n return evalf[saveDigits]([soln]);\n end if;\ne nd proc: " }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 " " {MPLTEXT 1 0 1 ";" }}}}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 14 "RK4_3 (t ype 4)" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1847 "RK4_3 := proc(fxy,x,y,xx,yy,h,stps,bb)\n local c2 ,c3,c4,c5,a21,a31,a32,a41,a42,a43,a51,a52,a53,a54,\n f1,f2,f3,f4,f5, b1,b2,b3,b4,b5,t,k,fn,xk,yk,soln,eqns,A,saveDigits;\n\n saveDigits : = Digits;\n Digits := max(trunc(evalhf(Digits)),Digits+5);\n\n fn \+ := unapply(fxy,x,y);\n\n A := matrix([[3/10,3/10,0,0,0,0],\n [ 7/12,-61/108,31/27,0,0,0],\n [10/11,1604935/1900668,-204685/2715 24,129645/158389,0,0],\n [1,3611/1326,-365519/88842,27450/9503,- 18634/37453,0],\n [0,1/10,9250/23919,864/3655,14641/43215,-13/21 0]]);\n \n c2 := evalf(A[1,1]);\n c3 := evalf(A[2,1]);\n c4 := e valf(A[3,1]);\n c5 := evalf(A[4,1]);\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 b1 := evalf(A[5,2]);\n b2 := evalf(A[5,3]);\n b3 := e valf(A[5,4]);\n b4 := evalf(A[5,5]);\n b5 := evalf(A[5,6]);\n xk := evalf(xx);\n yk := evalf(yy);\n soln := [xk,yk]; \n for k fr om 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 := f n(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\n yk := yk + (b1*f1 + b2*f2 + b3*f3 + b4*f4 + b5*f5)*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,a31_= a31,a32_=a32,a41_=a41,a42_=a42,\n a43_=a43,a51_=a51,a52_=a52,a 53_=a53,a54_=a54,\n b1_=b1,b2_=b2,b3_=b3,b4_=b4,b5_=b5\};\n \+ return subs(eqns,eval(rk4step)); \n else\n return evalf[save Digits]([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 15 "RK4_4 (type 7a)" }}{PARA 0 "" 0 "" {TEXT -1 0 " " }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1820 "RK4_4 := proc(fxy,x,y, xx,yy,h,stps,bb)\n local c2,c3,c4,c5,a21,a31,a32,a41,a42,a43,a51,a52 ,a53,a54,\n f1,f2,f3,f4,f5,b1,b2,b3,b4,b5,t,k,fn,xk,yk,soln,eqns,A,s aveDigits;\n\n saveDigits := Digits;\n Digits := max(trunc(evalhf( Digits)),Digits+5);\n\n fn := unapply(fxy,x,y);\n\n A := matrix([[ 1/5,1/5,0,0,0,0],\n [7/18,7/648,245/648,0,0,0],\n [19/20,24 89/1400,-5567/1360,155439/47600,0,0],\n [1,19659/6650,-1181/170,1 527174/300475,-176/1919,0],\n [0,97/798,0,4374/7777,4000/5757,-25 /66]]);\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 a21 := c2;\n a31 := eva lf(A[2,2]);\n a32 := evalf(A[2,3]);\n a41 := evalf(A[3,2]);\n a4 2 := 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 b1 := evalf(A[5,2]);\n b2 := evalf(A[5,3]);\n b3 := \+ evalf(A[5,4]);\n b4 := evalf(A[5,5]);\n b5 := evalf(A[5,6]);\n x k := evalf(xx);\n yk := evalf(yy);\n soln := [xk,yk]; \n for k f rom 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 f 4 := fn(xk + c4*h,yk + t*h);\n t := a51*f1 + a52*f2 + a53*f3 + a5 4*f4;\n f5 := fn(xk + c5*h,yk + t*h);\n\n yk := yk + (b1*f1 \+ + b2*f2 + b3*f3 + b4*f4 + b5*f5)*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,a31_ =a31,a32_=a32,a41_=a41,a42_=a42,\n a43_=a43,a51_=a51,a52_=a52, a53_=a53,a54_=a54,\n b1_=b1,b2_=b2,b3_=b3,b4_=b4,b5_=b5\};\n \+ return subs(eqns,eval(rk4step)); \n else\n return evalf[sav eDigits]([soln]);\n end if;\nend proc: " }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 16 "RK4_5 (type 7b) " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 2166 "RK4_5 := p roc(fxy,x,y,xx,yy,h,stps,bb)\n local c2,c3,c4,c5,a21,a31,a32,a41,a42 ,a43,a51,a52,a53,a54,\n f1,f2,f3,f4,f5,b1,b2,b3,b4,b5,t,k,fn,xk,yk,s oln,eqns,A,saveDigits,SQRT;\n\n saveDigits := Digits;\n Digits := \+ max(trunc(evalhf(Digits)),Digits+5);\n\n fn := unapply(fxy,x,y);\n\n SQRT := sqrt(394);\n\n A := matrix([[1/7,1/7,0,0,0,0],\n [7/ 26+SQRT/65,-7161/16900-3/169*SQRT,11711/16900+28/845*SQRT,0,0,0],\n \+ [2/3,-47443/7398+2953/7398*SQRT,1895075/1348902-202237/1348902*SQRT ,\n 11481950/2023353-504290/2023353*SQRT,0,0],\n [1,34917/13 70-1761/1370*SQRT,-263981921/41216450+14019733/41216450*SQRT,\n \+ -5397690166/283644115+274957918/283644115*SQRT,197289/208175-5022/2081 75*SQRT,0],\n [0,-19/36+SQRT/36,113533/216612-2401/216612*SQRT,\n \+ 44511750/79089089-7000000/711801801*SQRT,10755/33308-243/33308*SQRT,81 1/6876+SQRT/2292]]);\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 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 b1 := evalf(A[5,2]);\n b2 := evalf(A[5,3] );\n b3 := evalf(A[5,4]);\n b4 := evalf(A[5,5]);\n b5 := evalf(A [5,6]);\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 + a4 3*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\n yk : = yk + (b1*f1 + b2*f2 + b3*f3 + b4*f4 + b5*f5)*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,a31_=a31,a32_=a32,a41_=a41,a42_=a42,\n a43_=a43,a51_ =a51,a52_=a52,a53_=a53,a54_=a54,\n b1_=b1,b2_=b2,b3_=b3,b4_=b4 ,b5_=b5\};\n return subs(eqns,eval(rk4step)); \n else\n re turn 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 16 "RK4_6 (type 6a) " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 2488 "RK4_6 := proc(fxy,x,y,xx,yy,h,stps,bb)\n local c2,c3,c4,c5,a21,a31,a32,a4 1,a42,a43,a51,a52,a53,a54,\n f1,f2,f3,f4,f5,b1,b2,b3,b4,b5,t,k,fn,xk ,yk,soln,eqns,A,saveDigits,SQRT;\n\n saveDigits := Digits;\n Digit s := max(trunc(evalhf(Digits)),Digits+5);\n\n fn := unapply(fxy,x,y) ;\n\n SQRT := sqrt(286834);\n\n A := matrix([[1/5, 1/5,0,0,0,0],\n [41423/95610+2/47805*SQRT,-136273061/1828254420-44602/457063605*SQ RT,\n 928363667/1828254420+63724/45706360 5*SQRT,0,0,0],\n [14/15, 470023321/264181860+14153/88060620*SQRT,\n -1301419249/380343060+106373/380343060*SQRT,\n \+ 8767694361577/3403785129705-1499010941/3403785129705*SQRT,0,0], \n [1, 2026605983/503413211+1254361/2013652844*SQRT,\n - 2514970618/310613499+4271/1242453996*SQRT,\n 111358826364018808/ 21226230987849027-13003629628001/21226230987849027*SQRT,\n \+ -114738/654787-9/654787*SQRT,0],\n \+ [0,26742/250579-25/1503474*SQRT,351425/3401442+625/6802884*SQRT,\n \+ 12691600419657638/25917344257438359-1508229090784/25917344257438 359*SQRT,\n 142845/293986-15/293986*SQRT,-1366/7359+SQRT/29436]] );\n \n c2 := evalf(A[1,1]);\n c3 := evalf(A[2,1]);\n c4 := eval f(A[3,1]);\n c5 := evalf(A[4,1]);\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 b1 := evalf(A[5,2]);\n b2 := evalf(A[5,3]);\n b3 := evalf (A[5,4]);\n b4 := evalf(A[5,5]);\n b5 := evalf(A[5,6]);\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 := f n(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\n yk := yk + (b1*f1 + b2* f2 + b3*f3 + b4*f4 + b5*f5)*h;\n xk := xk + h:\n soln := sol n,[xk,yk];\n end do;\n if bb=true then\n eqns := \{SOLN_=[sol n],FXY_=fxy,X_=x,Y_=y,c2_=c2,c3_=c3,\n c4_=c4,c5_=c5,a31_=a31, a32_=a32,a41_=a41,a42_=a42,\n a43_=a43,a51_=a51,a52_=a52,a53_= a53,a54_=a54,\n b1_=b1,b2_=b2,b3_=b3,b4_=b4,b5_=b5\};\n r eturn subs(eqns,eval(rk4step)); \n else\n return evalf[saveDigi ts]([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 16 "RK4_7 (type 6b) " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 2385 "RK4_7 := proc(fxy,x,y,xx ,yy,h,stps,bb)\n local c2,c3,c4,c5,a21,a31,a32,a41,a42,a43,a51,a52,a 53,a54,\n f1,f2,f3,f4,f5,b1,b2,b3,b4,b5,t,k,fn,xk,yk,soln,eqns,A,sav eDigits,SQRT;\n\n saveDigits := Digits;\n Digits := max(trunc(eval hf(Digits)),Digits+5);\n\n fn := unapply(fxy,x,y);\n\n SQRT := sqr t(770);\n\n A := matrix([[1/5,1/5,0,0,0,0],\n [5183/9802-82/245 05*SQRT,-36851897/96079204+1321266/120099005*SQRT,\n 87655663/96 079204-1723148/120099005*SQRT,0,0,0],\n [14/15, 1268835043/1960201 80-33250147/196020180*SQRT,-5144909/584988+8407/42804*SQRT,\n -38 059343/1421146305*SQRT+37934709043/11653399701,0,0],\n [1,-1559366 31/15246014+109345415/213444196*SQRT, 29195/2378-10938175/15014692*SQR T,\n -22792164526050/21598549962361+1380728024101495/61987838391 97607*SQRT,\n -2408625/368626951*SQRT+29520/4787363,0],\n [0 ,14821/185927+1025/1115562*SQRT,800/3567-625/156948*SQRT,\n 21 0260193226678250/451863263762554481+782269586828000/135558979128766344 3*SQRT,\n 2887425/9574726+691875/105321986*SQRT,-8897/125526-10 25/251052*SQRT]]);\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 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 := ev alf(A[4,2]);\n a52 := evalf(A[4,3]);\n a53 := evalf(A[4,4]);\n a 54 := evalf(A[4,5]);\n b1 := evalf(A[5,2]);\n b2 := evalf(A[5,3]); \n b3 := evalf(A[5,4]);\n b4 := evalf(A[5,5]);\n b5 := evalf(A[5 ,6]);\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*f 1;\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\n yk := \+ yk + (b1*f1 + b2*f2 + b3*f3 + b4*f4 + b5*f5)*h;\n xk := xk + h:\n soln := soln,[xk,yk];\n end do;\n if bb=true then\n eqn s := \{SOLN_=[soln],FXY_=fxy,X_=x,Y_=y,c2_=c2,c3_=c3,\n c4_=c4 ,c5_=c5,a31_=a31,a32_=a32,a41_=a41,a42_=a42,\n a43_=a43,a51_=a 51,a52_=a52,a53_=a53,a54_=a54,\n b1_=b1,b2_=b2,b3_=b3,b4_=b4,b 5_=b5\};\n return subs(eqns,eval(rk4step)); \n else\n retu rn evalf[saveDigits]([soln]);\n end if;\nend proc: " }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}}{SECT 1 {PARA 4 "" 0 " " {TEXT -1 21 "Testing the examples " }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 38 "Test 1 of o rder 4 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 3 ", " }{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-%$expG6#,,*,\"#7\"\" \"\"# " 0 "" {MPLTEXT 1 0 190 "de := diff(y(x),x)=12*x*cos(4*x)*exp(-x) *y(x);\nic := y(0)=1;\ndsolve(\{de,ic\},y(x)):\ny(x)=simplify(numer(rh s(%))/convert(denom(rhs(%)),exp));\nf := unapply(rhs(%),x):\nplot(f(x) ,x=0..5,0..1.45);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%#deG/-%%diffG6$ -%\"yG6#%\"xGF,,$*,\"#7\"\"\"F,F0-%$cosG6#,$*&\"\"%F0F,F0F0F0-%$expG6# ,$F,!\"\"F0F)F0F0" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%#icG/-%\"yG6#\" \"!\"\"\"" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/-%\"yG6#%\"xG-%$expG6#,, *&#\"#7\"#<\"\"\"*(F'F0-%$cosG6#,$*&\"\"%F0F'F0F0F0-F)6#,$F'!\"\"F0F0F ;*&#\"$!=\"$*GF0*&F2F0F8F0F0F0*&#\"#[F/F0*(F'F0-%$sinGF4F0F8F0F0F0*&# \"#'*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$$\"3ALL$3FWYs#F/$\"3!H*fm:2P/5F27$$\"3%)***\\iSm p3%F/$\"3Qn()\\Dat45F27$$\"3WmmmT&)G\\aF/$\"34$Q7t`Dr,\"F27$$\"3m**** 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G6#,$*&\"\"%F0F1F0F0F0-%$expG6#,$F1!\"\"F09%F0F0F)F)F)" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 128 "Digits := 20:\nxx := 5.9;\nfxx := evalf(f(xx));\nfor ct to 7 do\n fnxx := \+ fn_RK4_||ct(xx);\n abs(fnxx-fxx);\nend do;\nDigits := 10:" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%#xxG$\"#f!\"\"" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%$fxxG$\"5NAx]XE__=^!#?" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%%fnxxG$\"5M.Y`5xg_=^!#?" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#$\" .*4)o-l])!#?" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%%fnxxG$\"5WGE1^ze_=^ !#?" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#$\".41\\b0`'!#?" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%%fnxxG$\"5.qYIOGY_=^!#?" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#$\".K_I?4)f!#?" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%%fn xxG$\"5U-h%pm/E&=^!#?" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#$\".2!QQ9-#)! #?" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%%fnxxG$\"5=++W\"4+D&=^!#?" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#$\".%%fnxxG$\"51K?pnce_=^!#?" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#$\".r4V=AI'!#?" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%%fnxxG$\"5E( H%o7^e_=^!#?" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#$\".\"\\dwrYi!#?" }}} {PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 403 "evalf[30](plot(['fn_RK4_1'(x)-f(x),'fn_RK4_2'(x)-f(x),'fn_RK4_3'( x)-f(x),\n'fn_RK4_4'(x)-f(x),'fn_RK4_5'(x)-f(x),'fn_RK4_6'(x)-f(x),'fn _RK4_7'(x)-f(x)],x=0..6,\n color=[blue,coral,aquamarine,COLOR(RGB,.7, .1,.7),brown,COLOR(RGB,.5,0,1),green],\nthickness=[1$6,2],\nlegend=[`F ehlberg`,`Prince`,`type 4`,`type 7a`,`type 7b`,`type 6a`,`type 6b`],\n title=`error curves for 5 stage order 4 Runge-Kutta methods`));" }} {PARA 13 "" 1 "" {GLPLOT2D 699 431 431 {PLOTDATA 2 "6,-%'CURVESG6&7_v7 $$\"\"!F)F(7$$\"?+++++++++]ilyM;!#J$\"4aq+Q3_X$QB!#H7$$\"?++++++++++DJ dpKF-$\"51mT_q64&Hp&F07$$\"?+++++++++](ofV!\\F-$\"5ik'\\J)=$GnT&F07$$ \"?++++++++++]i9RlF-$\"5#Qb>LhMXmj#F07$$\"?++++++++++v$>(3)*F-$!5(*>!= P%Q.o5iF07$$\"?++++++++++]#HyI\"!#I$!66Dmzr#*ekC'=F07$$\"?+++++++++voo 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EG6#%Uerror~curves~for~5~stage~order~4~Runge-Kutta~methodsG-%%VIEWG6$; F(F]cn%(DEFAULTG" 1 2 0 1 10 0 2 9 1 4 2 1.000000 45.000000 45.000000 0 1 "Fehlberg" "Prince" "type 4" "type 7a" "type 7b" "type 6a" "type 6 b" }}}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 38 "Test 2 of \+ order 4 Runge-Kutta methods " }}{PARA 256 "" 0 "" {TEXT -1 1 " " } {XPPEDIT 18 0 "dy/dx=x/y" "6#/*&%#dyG\"\"\"%#dxG!\"\"*&%\"xGF&%\"yGF( " }{TEXT -1 2 ", " }{XPPEDIT 18 0 "y(0)=1" "6#/-%\"yG6#\"\"!\"\"\"" } {TEXT -1 1 " " }}{PARA 0 "" 0 "" {TEXT -1 10 "Solution: " }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "y=sqrt(1+x^2)" "6#/%\"yG-%%sqrt G6#,&\"\"\"F)*$%\"xG\"\"#F)" }{TEXT -1 2 ". " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 126 "g := x -> sqrt(1+ x^2);\nDigits := 30:\nfor ct to 7 do\n gn_RK4_||ct := RK4_||ct(x/y,x ,y,0,1,.1,150,true);\nend do;\nDigits := 10:" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"gGf*6#%\"xG6\"6$%)operatorG%&arrowGF(-%%sqrtG6#,&\" \"\"F0*$)9$\"\"#F0F0F(F(F(" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 122 "Digits := 20:\nxx := 14.9;\ngxx := g(xx);\nfor ct to 7 do\n gnxx := gn_RK4_||ct(xx);\n abs(gnxx-gxx) ;\nend do;\nDigits := 10:" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%#xxG$\" $\\\"!\"\"" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%$gxxG$\"5rJhzSM>N$\\\" !#=" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%%gnxxG$\"5&fC&fBM>N$\\\"!#=" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#$\"+w&)3?%%gnxxG$\"5x\"oy1Z$>N$\\\"!#=" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#$\"+1]D))H!#=" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%%gnx xG$\"52VIG*Q$>N$\\\"!#=" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#$\"+k)38:&! #=" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%%gnxxG$\"5!QI%zSM>N$\\\"!#=" } }{PARA 11 "" 1 "" {XPPMATH 20 "6#$\"'\"z#=!#=" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%%gnxxG$\"5jtb!4W$>N$\\\"!#=" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#$\")#>W4\"!#=" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%%gnx xG$\"5$>eQ\"RM>N$\\\"!#=" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#$\"*y\\vl \"!#=" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%%gnxxG$\"5@?jfTM>N$\\\"!#= " }}{PARA 11 "" 1 "" {XPPMATH 20 "6#$\")])=+)!#=" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 404 "evalf[20](plot(['gn_RK4_1'(x)-g(x),'gn_RK4_2'(x)- g(x),'gn_RK4_3'(x)-g(x),\n'gn_RK4_4'(x)-g(x),'gn_RK4_5'(x)-g(x),'gn_RK 4_6'(x)-g(x),'gn_RK4_7'(x)-g(x)],x=0..15,\n color=[blue,coral,aquamar ine,COLOR(RGB,.7,.1,.7),brown,COLOR(RGB,.5,0,1),green],\nthickness=[1$ 6,2],\nlegend=[`Fehlberg`,`Prince`,`type 4`,`type 7a`,`type 7b`,`type \+ 6a`,`type 6b`],\ntitle=`error curves for 5 stage order 4 Runge-Kutta m ethods`));" }}{PARA 13 "" 1 "" {GLPLOT2D 782 520 520 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20:\nrt2 := evalf(sqrt(2)):\nf or ct to 7 do\n mn_RK4_||ct := RK4_||ct(M(x,y),x,y,0,rt2,.05,100,tru e);\nend do;\nDigits := 10:" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"MGf *6$%\"xG%\"yG6\"6$%)operatorG%&arrowGF),$*&,&\"\"\"F0-%$cosG6#,$*&\"\" #F09$F0F0F0F0)9%\"\"$F0!\"\"F)F)F)" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" } }{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 128 "xx := 4.9;\nDigits := 16: \nmxx := evalf(m(xx));\nfor ct to 7 do\n mnxx := mn_RK4_||ct(xx);\n \+ abs(mnxx-mxx);\nend do;\nDigits := 10:" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%#xxG$\"#\\!\"\"" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%$mxxG$\"1 FR*RjTG<$!#;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%%mnxxG$\"1%3)eFo%G<$ !#;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#$\"+dTf$>&!#;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%%mnxxG$\"1FVe#[RG<$!#;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#$\"++'49:#!#;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%%mnx xG$\"1u0!p*>)G<$!#;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#$\",Zm!HOS!#;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%%mnxxG$\"1q/3Ix%G<$!#;" 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