{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 }{CSTYLE "" -1 265 "" 0 1 0 0 0 0 0 0 1 0 0 0 0 0 0 0 }{CSTYLE "" -1 266 "" 0 1 0 0 0 0 1 0 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 267 "" 0 1 0 0 0 0 1 0 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 268 "" 0 1 0 0 0 0 1 0 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 269 "" 0 1 0 0 0 0 1 0 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 "Time s" 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 Outpu t" -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 4 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 18 "Version: 5.1 2.2011" }}{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 58 "load procedures for constru cting Runge-Kutta schemes etc. " }}{PARA 0 "" 0 "" {TEXT -1 18 "The Ma ple m-files " }{TEXT 262 9 "butcher.m" }{TEXT -1 5 " and " }{TEXT 262 6 "intg.m" }{TEXT -1 33 " are required by this worksheet. " }}{PARA 0 "" 0 "" {TEXT -1 134 "They can be read into a Maple session by command s similar to those that follow, where each file path gives the locatio n of the m-file." }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 68 "read \"C :\\\\Maple/procdrs/butcher.m\";\nread \"C:\\\\Maple/procdrs/intg.m\"; " }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}{PARA 0 "" 0 "" {TEXT -1 42 "#========== ===============================" }}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 37 "4 stage, order 4 Runge-Kutta schemes " }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}{PARA 0 "" 0 "" {TEXT -1 144 "We consider coeffi cient systems for order 4 Runge-Kutta schemes which can be used to con struct numerical solutions for the differential equation" }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "dy/dx=f(x,y)" "6#/*&%#dyG\"\"\" %#dxG!\"\"-%\"fG6$%\"xG%\"yG" }{TEXT -1 2 ". " }}{PARA 0 "" 0 "" {TEXT -1 35 "The solutions have the general form" }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "y(x+h)" "6#-%\"yG6#,&%\"xG\"\"\"%\"hGF( " }{TEXT -1 1 " " }{TEXT 265 1 "~" }{TEXT -1 1 " " }{XPPEDIT 18 0 "y(x )+Sum(b[j]*u[j],j = 1 .. 4)*h;" "6#,&-%\"yG6#%\"xG\"\"\"*&-%$SumG6$*&& %\"bG6#%\"jGF(&%\"uG6#F1F(/F1;F(\"\"%F(%\"hGF(F(" }{TEXT -1 15 " ---- --- (i), " }}{PARA 0 "" 0 "" {TEXT -1 6 "where " }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "u[1] = f(x,y);" "6#/&%\"uG6#\"\"\"-%\"f G6$%\"xG%\"yG" }{TEXT -1 2 ", " }}{PARA 0 "" 0 "" {TEXT -1 4 "and " }} {PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "u[j] = f(x+c[j]*h,y+S um(a[j,k]*u[k],k = 1 .. j-1)*h);" "6#/&%\"uG6#%\"jG-%\"fG6$,&%\"xG\"\" \"*&&%\"cG6#F'F-%\"hGF-F-,&%\"yGF-*&-%$SumG6$*&&%\"aG6$F'%\"kGF-&F%6#F =F-/F=;F-,&F'F-F-!\"\"F-F2F-F-" }{TEXT -1 5 ", " }{XPPEDIT 18 0 "2 \+ <= j;" "6#1\"\"#%\"jG" }{XPPEDIT 18 0 "``<=4" "6#1%!G\"\"%" }{TEXT -1 2 ". " }}{PARA 256 "" 0 "" {TEXT -1 1 " " }}{PARA 0 "" 0 "" {TEXT -1 18 "Then coefficients " }{XPPEDIT 18 0 "b[j]" "6#&%\"bG6#%\"jG" } {TEXT -1 2 ", " }{XPPEDIT 18 0 "c[j]" "6#&%\"cG6#%\"jG" }{TEXT -1 6 ", and " }{XPPEDIT 18 0 "a[j,k]" "6#&%\"aG6$%\"jG%\"kG" }{TEXT -1 43 " a re chosen so that this best approximates " }{XPPEDIT 18 0 "y(x+h);" "6 #-%\"yG6#,&%\"xG\"\"\"%\"hGF(" }{TEXT -1 19 " to the 4th order. " }} {PARA 0 "" 0 "" {TEXT -1 73 "The coefficients can be arranged in the f ollowing scheme due to Butcher. " }}{PARA 256 "" 0 "" {TEXT -1 1 " " } {XPPEDIT 18 0 "matrix([[c[1], ``, ``, ``,``], [c[2], a[2,1], ``, ``,`` ], [c[3], a[3,1], a[3,2], ``,``], [c[4], a[4,1], a[4,2], a[4,3],``],[` `, b[1], b[2], b[3],b[4]]])" "6#-%'matrixG6#7'7'&%\"cG6#\"\"\"%!GF,F,F ,7'&F)6#\"\"#&%\"aG6$F0F+F,F,F,7'&F)6#\"\"$&F26$F7F+&F26$F7F0F,F,7'&F) 6#\"\"%&F26$F?F+&F26$F?F0&F26$F?F7F,7'F,&%\"bG6#F+&FH6#F0&FH6#F7&FH6#F ?" }{TEXT -1 1 " " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 76 "Since the coefficients above the diagonal are all zero th e scheme is called " }{TEXT 260 8 "explicit" }{TEXT -1 101 ". We shall not consider implicit schemes in which there are non-zero coefficient s above the diagonal." }}{PARA 0 "" 0 "" {TEXT -1 61 "It is convenient to make Kutta's simplifying assumption that " }{XPPEDIT 18 0 "Sum(a[i ,j],j = 1 .. i-1) = c[i];" "6#/-%$SumG6$&%\"aG6$%\"iG%\"jG/F+;\"\"\",& F*F.F.!\"\"&%\"cG6#F*" }{TEXT -1 6 ", for " }{XPPEDIT 18 0 "i = 2,3,4; " "6%/%\"iG\"\"#\"\"$\"\"%" }{TEXT -1 10 ", that is," }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "PIECEWISE([a[2,1] = c[2], ``],[a[3 ,1]+a[3,2] = c[3], ``],[a[4,1]+a[4,2] +a[4,3]= c[3], ``])" "6#-%*PIECE WISEG6%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#F6F0" }{TEXT -1 2 ". " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 37 "These three equations constitute the " }{TEXT 260 18 "row sum cond itions" }{TEXT -1 2 ". " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 28 "In the particular case that " }{XPPEDIT 18 0 "f(x,y) " "6#-%\"fG6$%\"xG%\"yG" }{TEXT -1 19 " is independent of " }{TEXT 266 1 "y" }{TEXT -1 46 ", the scheme amounts to numerical integration. " }}{PARA 0 "" 0 "" {TEXT -1 44 "Considering a single step over the in terval " }{XPPEDIT 18 0 "[0,1]" "6#7$\"\"!\"\"\"" }{TEXT -1 6 " with \+ " }{XPPEDIT 18 0 "h=1" "6#/%\"hG\"\"\"" }{TEXT -1 5 " and " }{XPPEDIT 18 0 "f(x,y) = x^q;" "6#/-%\"fG6$%\"xG%\"yG)F'%\"qG" }{TEXT -1 36 ", e xact values will be obtained for " }{XPPEDIT 18 0 "Int(x^q,x = 0 .. 1) ;" "6#-%$IntG6$)%\"xG%\"qG/F';\"\"!\"\"\"" }{TEXT -1 6 " when " } {XPPEDIT 18 0 "r = 0,1,2,3;" "6&/%\"rG\"\"!\"\"\"\"\"#\"\"$" }{TEXT -1 14 " provided that" }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "Sum(b[i]*c[i]^r,i = 1 .. 4) = Int(x^q,x = 0 .. 1);" "6#/-%$SumG6 $*&&%\"bG6#%\"iG\"\"\")&%\"cG6#F+%\"rGF,/F+;F,\"\"%-%$IntG6$)%\"xG%\"q G/F9;\"\"!F," }{XPPEDIT 18 0 "`` = 1/(q+1);" "6#/%!G*&\"\"\"F&,&%\"qGF &F&F&!\"\"" }{TEXT -1 2 ". " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "PIECEWISE([b[1]+b[2]+b[3]+b [4]=1,``],[b[2]*c[2]+b[3]*c[3]+b[4]*c[4]=1/2,``],[b[2]*c[2]^2+b[3]*c[3 ]^2+b[4]*c[4]^2=1/3,``],[b[2]*c[2]^3+b[3]*c[3]^3+b[4]*c[4]^3=1/4,``]) " "6#-%*PIECEWISEG6&7$/,*&%\"bG6#\"\"\"F,&F*6#\"\"#F,&F*6#\"\"$F,&F*6# \"\"%F,F,%!G7$/,(*&&F*6#F/F,&%\"cG6#F/F,F,*&&F*6#F2F,&F>6#F2F,F,*&&F*6 #F5F,&F>6#F5F,F,*&F,F,F/!\"\"F67$/,(*&&F*6#F/F,*$&F>6#F/F/F,F,*&&F*6#F 2F,*$&F>6#F2F/F,F,*&&F*6#F5F,*$&F>6#F5F/F,F,*&F,F,F2FKF67$/,(*&&F*6#F/ F,*$&F>6#F/F2F,F,*&&F*6#F2F,*$&F>6#F2F2F,F,*&&F*6#F5F,*$&F>6#F5F2F,F,* &F,F,F5FKF6" }{TEXT -1 2 ". " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 36 "These four equations constitute the " }{TEXT 260 21 "quadrature conditions" }{TEXT -1 18 ". The cefficients " } {XPPEDIT 18 0 "c[1],c[2],c[3],c[4]" "6&&%\"cG6#\"\"\"&F$6#\"\"#&F$6#\" \"$&F$6#\"\"%" }{TEXT -1 16 " are called the " }{TEXT 260 5 "nodes" } {TEXT -1 25 ", while the coefficients " }{XPPEDIT 18 0 "b[1],b[2],b[3] ,b[4]" "6&&%\"bG6#\"\"\"&F$6#\"\"#&F$6#\"\"$&F$6#\"\"%" }{TEXT -1 23 " are the corresponding " }{TEXT 260 7 "weights" }{TEXT -1 2 ". " }} {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 48 "The remai ning order 4 conditions, involving the " }{TEXT 260 20 "linking coeffi cients" }{TEXT -1 1 " " }{XPPEDIT 18 0 "a[i,j]" "6#&%\"aG6$%\"iG%\"jG " }{TEXT -1 17 ", are as follows." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }} {PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "PIECEWISE([b[3]*a[3,2 ]*c[2]+b[4]*(a[4,2]*c[2]+a[4,3]*c[3])=1/6,``],[b[3]*a[3,2]*c[2]^2+b[4] *(a[4,2]*c[2]^2+a[4,3]*c[3]^2)=1/12,``],\n[b[4]*a[4,3]*a[3,2]*c[2]=1/2 4,``],[b[3]*c[3]*a[3,2]*c[2]+b[4]*c[4]*(a[4,2]*c[2]+a[4,3]*c[3])=1/8,` `])" "6#-%*PIECEWISEG6&7$/,&*(&%\"bG6#\"\"$\"\"\"&%\"aG6$F-\"\"#F.&%\" cG6#F2F.F.*&&F+6#\"\"%F.,&*&&F06$F9F2F.&F46#F2F.F.*&&F06$F9F-F.&F46#F- F.F.F.F.*&F.F.\"\"'!\"\"%!G7$/,&*(&F+6#F-F.&F06$F-F2F.&F46#F2F2F.*&&F+ 6#F9F.,&*&&F06$F9F2F.*$&F46#F2F2F.F.*&&F06$F9F-F.*$&F46#F-F2F.F.F.F.*& F.F.\"#7FGFH7$/**&F+6#F9F.&F06$F9F-F.&F06$F-F2F.&F46#F2F.*&F.F.\"#CFGF H7$/,&**&F+6#F-F.&F46#F-F.&F06$F-F2F.&F46#F2F.F.*(&F+6#F9F.&F46#F9F.,& *&&F06$F9F2F.&F46#F2F.F.*&&F06$F9F-F.&F46#F-F.F.F.F.*&F.F.\"\")FGFH" } {TEXT -1 2 ". " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 31 "These conditions may be called " }{TEXT 260 25 "sub-quadr ature conditions" }{TEXT -1 1 "." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }} {PARA 0 "" 0 "" {TEXT -1 74 "There are a total of 11 equations in 13 u nknown Runge-Kutta coefficients. " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}}{SECT 1 {PARA 4 "" 0 " " {TEXT -1 67 "A system of equations for 4 stage, order 4 Runge-Kutta \+ coefficients" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 39 "Abreviated form of the order conditions " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 61 "Abrev iated order conditions can obtained using the procedure " }{TEXT 0 15 "OrderConditions" }{TEXT -1 17 " from the m-file " }{TEXT 0 9 "butcher .m" }{TEXT -1 1 "." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 " > " 0 "" {MPLTEXT 1 0 160 "convert(ListTools[Enumerate](OrderCondition s(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#*&%\"aGF(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 50 "Order conditions and their associated rooted trees" }} {PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 280 "TT := AllTrees(4):\nm := 4: ## number of trees per row\nnn := nop s(TT): q := iquo(nn,m):\nfor i to nn do \n p||i := DrawTree(TT[i],hei ght=`if`(i<=4,2,3),show_ordercondition=true,\n scale=`if`(i<=4,.8,1. 2));\nend do:\nplots[display](convert([seq([p||((k-1)*m+1..m*k)],k=1.. 2)],array));" }}{PARA 13 "" 1 "" {GLPLOT2D 696 471 471 {PLOTDATA 2 "6_ p-%'CURVESG6#-%'LEGENDG6#QB__never_display_this_legend_entry6\"-%%TEXT G6&7$$\"\"!F0$!+++++5!\")Q(b~e~=~1F*-%&COLORG6&%$RGBGF/F/$\"\"\"F0-%%F ONTG6$%(COURIERG\"#5-F$6&7#7$F/$!+nmmmm!\"*-%'SYMBOLG6#%'CIRCLEG-F66&F 8F/F/F/-%&STYLEG6#%&POINTG-F$6&FB-FH6#%(DIAMONDGFKFM-F$6&FB-FH6#%&CROS SGFKFM-F$6%7#7$F/$\"+++++5F3-%'COLOURG6&F8$F?!\"\"F/F/-FN6#%%LINEG-F,6 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lF^pQ)~~~~~~~1F*F5F;-F,6&7$Fd\\lFdpQ)~~~~~~~_F*F5F;-F,6&7$Fd\\lF1Q)~~~ ~~~~2F*F5F;-F$6&7$7$Fd\\lFeq7$Fd\\lFjqFGFKFM-F$6&FiblFSFKFM-F$6&FiblFX FKFM-F$6%FiblFfrFhr-F$6%7#7$Fd\\lFinF[oF`o-%*AXESSTYLEG6#%%NONEG" 1 2 0 1 10 0 2 9 1 1 2 1.000000 45.000000 45.000000 0 0 "Curve 1" "Curve 2 " "Curve 3" "Curve 4" "Curve 5" "Curve 6" "Curve 7" "Curve 8" "Curve 9 " "Curve 10" "Curve 11" "Curve 12" "Curve 13" "Curve 14" "Curve 15" "C urve 16" "Curve 17" "Curve 18" "Curve 19" "Curve 20" "Curve 21" "Curve 22" "Curve 23" "Curve 24" "Curve 25" "Curve 26" "Curve 27" "Curve 28 " "Curve 29" "Curve 30" "Curve 31" "Curve 32" "Curve 33" "Curve 34" "C urve 35" "Curve 36" "Curve 37" "Curve 38" "Curve 39" "Curve 40" "Curve 41" "Curve 42" "Curve 43" "Curve 44" "Curve 45" "Curve 46" "Curve 47 " "Curve 48" "Curve 49" "Curve 50" "Curve 51" "Curve 52" "Curve 53" "C urve 54" "Curve 55" "Curve 56" "Curve 57" "Curve 58" "Curve 59" "Curve 60" "Curve 61" "Curve 62" "Curve 63" "Curve 64" "Curve 65" "Curve 66 " "Curve 67" "Curve 68" "Curve 69" "Curve 70" "Curve 71" "Curve 72" "C urve 73" "Curve 74" "Curve 75" "Curve 76" "Curve 77" "Curve 78" "Curve 79" "Curve 80" "Curve 81" "Curve 82" }}}}{PARA 0 "" 0 "" {TEXT -1 0 " " }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}}{SECT 1 {PARA 4 " " 0 "" {TEXT -1 45 "Long (summation) form of the order conditions" }} {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 75 "The order conditions in summation form can be obtained using the procedure " } {TEXT 0 15 "OrderConditions" }{TEXT -1 17 " from the m-file " }{TEXT 0 9 "butcher.m" }{TEXT -1 1 "." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 146 "ListTools[Enumerate](OrderC onditions(4,4)):\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%F4F)/ -F,6$*&F.F()F;F6F(F2#F(F?7%\"\"&F)/-F,6$*&F.F(-F,6$*&FGF(-F,6$*&&FH6$F J%\"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(F4Q(pprint06\"" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 66 "Order conditions together with row sum co nditions in expanded form" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 " " 0 "" {TEXT -1 90 "The system of equations for the Runge-Kutta coeffi cients can be set with Maple as follows." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 530 "[a[2,1]=c[2],a[3,1]+ a[3,2]=c[3],a[4,1]+a[4,2]+a[4,3]=c[4],b[1]+b[2]+b[3]+b[4]=1,\nb[2]*c[2 ]+b[3]*c[3]+b[4]*c[4]=1/2,b[3]*a[3,2]*c[2]+b[4]*(a[4,2]*c[2]+a[4,3]*c[ 3])=1/6,\nb[2]*c[2]^2+b[3]*c[3]^2+b[4]*c[4]^2=1/3,b[4]*a[4,3]*a[3,2]*c [2]=1/24, b[3]*c[3]*a[3,2]*c[2]+b[4]*c[4]*(a[4,2]*c[2]+a[4,3]*c[3])=1/ 8, b[3]*a[3,2]*c[2]^2+b[4]*(a[4,2]*c[2]^2+a[4,3]*c[3]^2)=1/12, b[2]*c[ 2]^3+b[3]*c[3]^3+b[4]*c[4]^3=1/4]:\nListTools[Enumerate](%):\nlinalg[a ugment](linalg[delcols](%,2..2),matrix([[` `]$(linalg[rowdim](%))]),l inalg[delcols](%,1..1));" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#K%'matrixG 6#7-7%\"\"\"%#~~G/&%\"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%FA F)/,*&%\"bG6#F(F(&FLF1F(&FLF;F(&FLFGF(F(7%\"\"&F)/,(*&FNF(F/F(F(*&FOF( F:F(F(*&FPF(FFF(F(#F(F.7%\"\"'F)/,&*(FOF(F8F(F/F(F(*&FPF(,&*&FBF(F/F(F (*&FDF(F:F(F(F(F(#F(FZ7%\"\"(F)/,(*&FNF()F/F.F(F(*&FOF()F:F.F(F(*&FPF( )FFF.F(F(#F(F77%\"\")F)/**FPF(FDF(F8F(F/F(#F(\"#C7%\"\"*F)/,&**FOF(F:F (F8F(F/F(F(*(FPF(FFF(FinF(F(#F(Fio7%\"#5F)/,&*(FOF(F8F(FboF(F(*&FPF(,& *&FBF(FboF(F(*&FDF(FdoF(F(F(F(#F(\"#77%\"#6F)/,(*&FNF()F/F7F(F(*&FOF() F:F7F(F(*&FPF()FFF7F(F(#F(FAQ(pprint16\"" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 29 "Alternatively the procedures " } {TEXT 0 16 "RowSumConditions" }{TEXT -1 5 " and " }{TEXT 0 15 "OrderCo nditions" }{TEXT -1 13 " can be used." }}{PARA 0 "" 0 "" {TEXT -1 0 " " }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 201 "[op(RowSumConditions(4, 'expanded')),op(OrderConditions(4,4,'expanded'))]:\nListTools[Enumerat e](%):\nlinalg[augment](linalg[delcols](%,2..2),matrix([[` `]$(linalg [rowdim](%))]),linalg[delcols](%,1..1));" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#K%'matrixG6#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$F AF7F(&F06#FA7%FAF)/,*&%\"bG6#F(F(&FLF1F(&FLF;F(&FLFGF(F(7%\"\"&F)/,(*& FNF(F/F(F(*&FOF(F:F(F(*&FPF(FFF(F(#F(F.7%\"\"'F)/,&*(FOF(F8F(F/F(F(*&F PF(,&*&FBF(F/F(F(*&FDF(F:F(F(F(F(#F(FZ7%\"\"(F)/,(*&FNF()F/F.F(F(*&FOF ()F:F.F(F(*&FPF()FFF.F(F(#F(F77%\"\")F)/**FPF(FDF(F8F(F/F(#F(\"#C7%\" \"*F)/,&**FOF(F:F(F8F(F/F(F(*(FPF(FFF(FinF(F(#F(Fio7%\"#5F)/,&*(FOF(F8 F(FboF(F(*&FPF(,&*&FBF(FboF(F(*&FDF(FdoF(F(F(F(#F(\"#77%\"#6F)/,(*&FNF ()F/F7F(F(*&FOF()F:F7F(F(*&FPF()FFF7F(F(#F(FAQ(pprint26\"" }}}{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 47 "Principal error terms and associated conditions" }} {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 116 "The prin cipal error terms are those terms which correspond to the order 5 term s in the Taylor series of a solution. " }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 35 "Abreviated \+ form of the error terms " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 148 "ListTools[Enumerate](PrincipalErro rTerms(4)):\nlinalg[augment](linalg[delcols](%,2..2),matrix([[` `]$(l inalg[rowdim](%))]),linalg[delcols](%,1..1));" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#K%'matrixG6#7+7%\"\"\"%#~~G,&*&%\"bGF(-%!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(pprint36\"" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 41 "Long (summation) form of the error terms " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 150 "ListTools[E numerate](PrincipalErrorTerms(4,4)):\nlinalg[augment](linalg[delcols]( %,2..2),matrix([[` `]$(linalg[rowdim](%))]),linalg[delcols](%,1..1)); " }}{PARA 11 "" 1 "" {XPPMATH 20 "6#K%'matrixG6#7+7%\"\"\"%#~~G,&-%$Su mG6$*&&%\"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(/F 9;F(,&F2F(F(FLF(/F2;F(\"\"%F(#F(\"$?\"FL7%\"\"#F),&-F,6$*&F/F(-F,6$*(F 6F(&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%FUF),&*&#F(FYF(-F,6$*&F /F(-F,6$*&F6F(-F,6$*&F=F()F`oFYF(FMF(FPF(FSF(F(#F(FWFL7%\"\"&F),&*&Fcp F(-F,6$*&F/F()-F,6$*&F6F(F[oF(FPFYF(FSF(F(#F(FcoFL7%\"\"'F),&*&FcpF(-F ,6$*(F/F(FjoF(-F,6$*&F6F()F[oFYF(FPF(FSF(F(#F(F_pFL7%\"\"(F),&*&FcpF(- 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 ()FjoFUF(FSF(F(#F(FWFLQ(pprint46\"" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" } }{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 32 "The error terms in expanded form" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 161 "ListTools[E numerate](PrincipalErrorTerms(4,4,'expanded')):\nlinalg[augment](linal g[delcols](%,2..2),matrix([[` `]$(linalg[rowdim](%))]),linalg[delcols ](%,1..1));" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#K%'matrixG6#7+7%\"\"\"% #~~G#!\"\"\"$?\"7%\"\"#F),&#F(\"#SF+*,&%\"bG6#\"\"%F(&%\"aG6$F6\"\"$F( &%\"cG6#F:F(&F86$F:F.F(&F<6#F.F(F(7%F:F),&#F(\"#IF+*,F3F(&F F(F@F(F(7%F6F),&#F(F,F+*&#F(F.F(**F3F(F7F(F>F()F@F.F(F(F(7%\"\"&F),(#F (F1F+*&FLF(*(&F4F=F()F>F.F(FNF(F(F(*&FLF(*&F3F(),&*&&F86$F6F.F(F@F(F(* &F7F(F;F(F(F.F(F(F(7%\"\"'F),(#F(FEF+*&FLF(**FUF(F;F(F>F(FNF(F(F(*&FLF (*(F3F(FGF(,&*&FfnF(FNF(F(*&F7F()F;F.F(F(F(F(F(7%\"\"(F),(#F(\"#?F+*&F LF(**FUF(FdoF(F>F(F@F(F(F(*&FLF(*(F3F()FGF.F(FZF(F(F(7%\"\")F),(#F(F,F +*&#F(FjnF(*(FUF(F>F()F@F:F(F(F(*&FdpF(*&F3F(,&*&FfnF(FfpF(F(*&F7F()F; F:F(F(F(F(F(7%\"\"*F),*#F(F,F+*&#F(\"#CF(*&&F4FAF()F@F6F(F(F(*&FbqF(*& FUF()F;F6F(F(F(*&FbqF(*&F3F()FGF6F(F(F(Q(pprint56\"" }}}{PARA 0 "" 0 " " {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}} {SECT 1 {PARA 4 "" 0 "" {TEXT -1 37 "The error conditions in expanded \+ form" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 83 "N ote that the first error condition is impossible to satisfy with a 4 s tage scheme." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 166 "ListTools[Enumerate](PrincipalErrorConditions(4,4 ,'expanded')):\nlinalg[augment](linalg[delcols](%,2..2),matrix([[` `] $(linalg[rowdim](%))]),linalg[delcols](%,1..1));" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#K%'matrixG6#7+7%\"\"\"%#~~G/\"\"!#F(\"$?\"7%\"\"#F)/*,& %\"bG6#\"\"%F(&%\"aG6$F5\"\"$F(&%\"cG6#F9F(&F76$F9F/F(&F;6#F/F(#F(\"#S 7%F9F)/*,F2F(&F;F4F(F6F(F=F(F?F(#F(\"#I7%F5F)/**F2F(F6F(F=F()F?F/F(#F( \"#g7%\"\"&F)/,&*(&F3F " 0 "" {MPLTEXT 1 0 1 ";" }}}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}}{PARA 0 "" 0 "" {TEXT -1 42 "#============== ===========================" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 49 "#============================================== ==" }}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 35 "The classical order 4 metho d with " }{XPPEDIT 18 0 "c[2] = c[3];" "6#/&%\"cG6#\"\"#&F%6#\"\"$" } {XPPEDIT 18 0 "``=1/2" "6#/%!G*&\"\"\"F&\"\"#!\"\"" }{TEXT -1 1 " " }} {PARA 0 "" 0 "" {TEXT -1 1 " " }}{PARA 256 "" 0 "" {TEXT -1 1 " " } {XPPEDIT 18 0 "matrix([[1/2, 1/2, ``, ``, ``], [1/2, 0, 1/2, ``, ``], \+ [1, 0, 0, 1, ``], [``, 1/6, 1/3, 1/3, 1/6]]);" "6#-%'matrixG6#7&7'*&\" \"\"F)\"\"#!\"\"*&F)F)F*F+%!GF-F-7'*&F)F)F*F+\"\"!*&F)F)F*F+F-F-7'F)F0 F0F)F-7'F-*&F)F)\"\"'F+*&F)F)\"\"$F+*&F)F)F7F+*&F)F)F5F+" }{TEXT -1 2 " " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 85 "RK4_4eqs := [op(RowSumConditions(4,'expanded')),op(Or derConditions(4,4,'expanded'))]:" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }} {PARA 0 "" 0 "" {TEXT -1 21 "We specify the node " }{XPPEDIT 18 0 "c[ 2]=1/2" "6#/&%\"cG6#\"\"#*&\"\"\"F)F'!\"\"" }{TEXT -1 42 " and the li nking coefficientcoefficient " }{XPPEDIT 18 0 "a[3,1] = 0;" "6#/&%\"a G6$\"\"$\"\"\"\"\"!" }{TEXT -1 61 ", so as to obtain a system of 11 eq uations with 11 unknowns. " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 213 "e1 := \{c[2]=1/2,a[3,1]=0\}:\ne2 : = solve(subs(e1,\{op(RK4_4eqs)\})):\ne3 := `union`(e1,e2);\nsubs(e3,ma trix([[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],``],[``,b[1],b[2],b[3],b[4]]]));" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%#e3G " 0 "" {MPLTEXT 1 0 43 "subs(e3,RK4_4eqs);\nmap(u->lhs(u)-r hs(u),%);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#7-/#\"\"\"\"\"#F%F$/F&F&F (F$/#F&\"\"'F*/#F&\"\"$F-/#F&\"#CF0/#F&\"\")F3/#F&\"#7F6/#F&\"\"%F9" } }{PARA 11 "" 1 "" {XPPMATH 20 "6#7-\"\"!F$F$F$F$F$F$F$F$F$F$" }}} {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 94 "We can ca lculate the principal error norm, that is, the 2-norm of the principal error terms. " }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 150 "errterms 4_4 := PrincipalErrorTerms(4,4,'expanded'):\nsimplify(subs(e3,sqrt(add (errterms4_4[i]^2,i=1..nops(errterms4_4)))));\nevalf[14](%): evalf[10] (%);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#,$*&\"%!)G!\"\"\"%X<#\"\"\"\" \"#F)" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#$\"+M#e/X\"!#6" }}}{PARA 0 " " 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 54 "None of the princi pal error conditions are satisfied. " }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 105 "RK4_4err_eqs := PrincipalErrorConditions(4,4,'expand ed'):\nsubs(e3,RK4_4err_eqs):\nmap(u->lhs(u)-rhs(u),%);" }}{PARA 11 " " 1 "" {XPPMATH 20 "6#7+#!\"\"\"$?\"#F%\"$S##\"\"\"F&#F*F(#F*\"#!)F'F+ F$F)" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}}{PARA 0 "" 0 "" {TEXT -1 49 "#================= ===============================" }}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 29 "The 3/8 order 4 method with " }{XPPEDIT 18 0 "c[2] = 1/3;" "6#/&% \"cG6#\"\"#*&\"\"\"F)\"\"$!\"\"" }{TEXT -1 7 " and " }{XPPEDIT 18 0 "c[3] = 2/3;" "6#/&%\"cG6#\"\"$*&\"\"#\"\"\"F'!\"\"" }{TEXT -1 1 " " } }{PARA 0 "" 0 "" {TEXT -1 1 " " }}{PARA 256 "" 0 "" {TEXT -1 2 " " } {XPPEDIT 18 0 "matrix([[1/3, 1/3, ``, ``, ``], [2/3, -1/3, 1, ``, ``], [1, 1, -1, 1, ``], [``, 1/8, 3/8, 3/8, 1/8]]);" "6#-%'matrixG6#7&7'*& \"\"\"F)\"\"$!\"\"*&F)F)F*F+%!GF-F-7'*&\"\"#F)F*F+,$*&F)F)F*F+F+F)F-F- 7'F)F),$F)F+F)F-7'F-*&F)F)\"\")F+*&F*F)F7F+*&F*F)F7F+*&F)F)F7F+" } {TEXT -1 2 " " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 85 "RK4_4eqs := [op(RowSumConditions(4,'expanded')), op(OrderConditions(4,4,'expanded'))]:" }}}{PARA 0 "" 0 "" {TEXT -1 0 " " }}{PARA 0 "" 0 "" {TEXT -1 26 "We specify the two nodes " } {XPPEDIT 18 0 "c[2] = 1/3" "6#/&%\"cG6#\"\"#*&\"\"\"F)\"\"$!\"\"" } {TEXT -1 7 " and " }{XPPEDIT 18 0 "c[3] = 2/3;" "6#/&%\"cG6#\"\"$*& \"\"#\"\"\"F'!\"\"" }{TEXT -1 60 ", so as to obtain a system of 11 equ ations with 11 unknowns." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 213 "e1 := \{c[2]=1/3,c[3]=2/3\}:\ne2 : = solve(subs(e1,\{op(RK4_4eqs)\})):\ne3 := `union`(e1,e2):\nsubs(e3,ma trix([[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],``],[``,b[1],b[2],b[3],b[4]]]));" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#K%'matrixG6#7&7'#\"\"\"\"\"$F(%!GF+F+7'#\"\"#F*#!\"\"F* F)F+F+7'F)F)F0F)F+7'F+#F)\"\")#F*F4F5F3Q(pprint56\"" }}}{PARA 0 "" 0 " " {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 7 "Check: " }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 43 "subs(e3,RK4_4eqs);\nmap(u->lhs(u)-rhs(u), %);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#7-/#\"\"\"\"\"$F%/#\"\"#F'F)/F& F&F+/#F&F*F-/#F&\"\"'F/F$/#F&\"#CF2/#F&\"\")F5/#F&\"#7F8/#F&\"\"%F;" } }{PARA 11 "" 1 "" {XPPMATH 20 "6#7-\"\"!F$F$F$F$F$F$F$F$F$F$" }}} {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 93 "We can ca lculate the principal error norm, that is, the 2-norm of the principal error terms. " }}{PARA 0 "" 0 "" {TEXT -1 53 "This value is smaller t han for the previous example. " }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 150 "errterms4_4 := PrincipalErrorTerms(4,4,'expanded'):\nsimplify(s ubs(e3,sqrt(add(errterms4_4[i]^2,i=1..nops(errterms4_4)))));\nevalf[14 ](%): evalf[10](%);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#,$*&\"%SK!\"\" \"%&o\"#\"\"\"\"\"#F)" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#$\"+vn$pE\"!# 6" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 54 "Non e of the principal error conditions are satisfied. " }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 105 "RK4_4err_eqs := PrincipalErrorConditions(4 ,4,'expanded'):\nsubs(e3,RK4_4err_eqs):\nmap(u->lhs(u)-rhs(u),%);" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#7+#!\"\"\"$?\"#\"\"\"\"$g$#F(F&#F%F)#F (\"$!=F'F+#F%\"$q##F(F/" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}}{PARA 0 "" 0 "" {TEXT -1 49 " #================================================" }}{SECT 1 {PARA 4 " " 0 "" {TEXT -1 29 "Gill's order 4 method with " }{XPPEDIT 18 0 "c[2 ] = c[3];" "6#/&%\"cG6#\"\"#&F%6#\"\"$" }{XPPEDIT 18 0 "``=1/2" "6#/%! G*&\"\"\"F&\"\"#!\"\"" }{TEXT -1 2 " " }}{PARA 0 "" 0 "" {TEXT -1 1 " " }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "matrix([[1/2, 1/ 2, ``, ``, ``], [1/2, 1/sqrt(2)-1/2, 1-1/sqrt(2), ``, ``], [1, 0, -1/s qrt(2), 1+1/sqrt(2), ``], [``, 1/6, 1/3-1/(3*sqrt(2)), 1/3+1/(3*sqrt(2 )), 1/6]]);" "6#-%'matrixG6#7&7'*&\"\"\"F)\"\"#!\"\"*&F)F)F*F+%!GF-F-7 '*&F)F)F*F+,&*&F)F)-%%sqrtG6#F*F+F)*&F)F)F*F+F+,&F)F)*&F)F)-F36#F*F+F+ F-F-7'F)\"\"!,$*&F)F)-F36#F*F+F+,&F)F)*&F)F)-F36#F*F+F)F-7'F-*&F)F)\" \"'F+,&*&F)F)\"\"$F+F)*&F)F)*&FIF)-F36#F*F)F+F+,&*&F)F)FIF+F)*&F)F)*&F IF)-F36#F*F)F+F)*&F)F)FFF+" }{TEXT -1 1 " " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 85 "RK4_4eqs := [op(Ro wSumConditions(4,'expanded')),op(OrderConditions(4,4,'expanded'))]:" } }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 21 "We spec ify the node " }{XPPEDIT 18 0 "c[2] = 1/2" "6#/&%\"cG6#\"\"#*&\"\"\"F )F'!\"\"" }{TEXT -1 31 " and the linking coefficient " }{XPPEDIT 18 0 "a[3,2] = 1-1/sqrt(2);" "6#/&%\"aG6$\"\"$\"\"#,&\"\"\"F**&F*F*-%%sqr tG6#F(!\"\"F/" }{TEXT -1 61 ", so as to obtain a system of 11 equation s with 11 unknowns. " }}{PARA 0 "" 0 "" {TEXT 260 4 "Note" }{TEXT -1 27 ": Including the condition " }{XPPEDIT 18 0 "c[3]<>0" "6#0&%\"cG6# \"\"$\"\"!" }{TEXT -1 46 " ensures that we obtain the desired solutio n." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 250 "e1 := \{c[2]=1/2,a[3,2]=1-1/sqrt(2)\};\ne2 := solve( subs(e1,\{op(RK4_4eqs),c[3]<>0\}));\ne3 := `union`(e1,e2);\n``;\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],``],[``,b[1],b[2],b[3],b[4]]]));\n``;\nevalf(%%); " }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%#e1G<$/&%\"cG6#\"\"##\"\"\"F*/&% \"aG6$\"\"$F*,&F,F,*&F*!\"\"F*F+F4" }}{PARA 11 "" 1 "" {XPPMATH 20 "6# >%#e2G<-/&%\"aG6$\"\"#\"\"\"#F+F*/&%\"bG6#F*,&#F+\"\"$F+*&\"\"'!\"\"F* F,F6/&F(6$\"\"%F+\"\"!/&%\"cG6#F:F+/&F>6#F3F,/&F/F?#F+F5/&F(6$F:F*,$*& F*F6F*F,F6/&F/FB,&F2F+*&F5F6F*F,F+/&F(6$F:F3,&F+F+*&F*F6F*F,F+/&F/6#F+ FE/&F(6$F3F+,&#F+F*F6*&F*F6F*F,F+" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#> %#e3G/&F/6$FBF*,$*&F*F4F*F+F4/&F:FI,&FF4F*F+F,/ &F/6$FBF1,&F,F,*&F*F4F*F+F,/&F:6#F,FL/&F/6$F1F,,&#F,F*F4*&F*F4F*F+F," }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%!G" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#K%'matrixG6#7&7'#\"\"\"\"\"#F(%!GF+F+7'F(,&#F)F*!\"\"*&F*F/F*F(F),& F)F)*&F*F/F*F(F/F+F+7'F)\"\"!,$*&F*F/F*F(F/,&F)F)*&F*F/F*F(F)F+7'F+#F) \"\"',&#F)\"\"$F)*&F;F/F*F(F/,&F=F)*&F;F/F*F(F)F:Q(pprint86\"" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#%!G" }}{PARA 11 "" 1 "" {XPPMATH 20 "6 #K%'matrixG6#7&7'$\"+++++]!#5F(%!GF+F+7'F($\"+5y1r?F*$\"+!>K*GHF*F+F+7 '$\"\"\"\"\"!$F4F4$!+5y1rqF*$\"+\"y1rq\"!\"*F+7'F+$\"+nmmm;F*$\"*H2Jw* F*$\"+PfN!p&F*F " 0 "" {MPLTEXT 1 0 53 "simplify(subs(e3,RK4_4eqs));\nmap(u->lhs(u)-rhs(u),%) ;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#7-/#\"\"\"\"\"#F%F$/F&F&F(F$/#F& \"\"'F*/#F&\"\"$F-/#F&\"#CF0/#F&\"\")F3/#F&\"#7F6/#F&\"\"%F9" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#7-\"\"!F$F$F$F$F$F$F$F$F$F$" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 93 "We can calculate the principal error norm, that is, the 2-norm of the principal error term s. " }}{PARA 0 "" 0 "" {TEXT -1 59 "This value is between those for th e previous two examples. " }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 150 "errterms4_4 := PrincipalErrorTerms(4,4,'expanded'):\nsimplify(sub s(e3,sqrt(add(errterms4_4[i]^2,i=1..nops(errterms4_4)))));\nevalf[14]( %): evalf[10](%);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#,$*&\"%!)G!\"\",& *&F%\"\"\"\"\"##F)F*F&\"%DbF)F+F)" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#$ \"+aR7B8!#6" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 54 "None of the principal error conditions are satisfied. " }} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 115 "RK4_4err_eqs := PrincipalEr rorConditions(4,4,'expanded'):\nsimplify(subs(e3,RK4_4err_eqs)):\nmap( u->lhs(u)-rhs(u),%);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#7+#!\"\"\"$?\" #F%\"$S##\"\"\"F&#F*F(,&*&\"#[F%\"\"##F*F/F%#F*\"#IF*F'F+F$F)" }}} {PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}}{PARA 0 "" 0 "" {TEXT -1 49 "#============================== ==================" }}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 31 "Dormand's o rder 4 method with " }{XPPEDIT 18 0 "c[2] = 2/5;" "6#/&%\"cG6#\"\"#*& F'\"\"\"\"\"&!\"\"" }{TEXT -1 7 " and " }{XPPEDIT 18 0 "c[3] = 3/5; " "6#/&%\"cG6#\"\"$*&F'\"\"\"\"\"&!\"\"" }{TEXT -1 1 " " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 " matrix([[2/5, 2/5, ``, ``, ``], [3/5, -3/20, 3/4, ``, ``], [1, 19/44, \+ -15/44, 10/11, ``], [``, 11/72, 25/72, 25/72, 11/72]])" "6#-%'matrixG6 #7&7'*&\"\"#\"\"\"\"\"&!\"\"*&F)F*F+F,%!GF.F.7'*&\"\"$F*F+F,,$*&F1F*\" #?F,F,*&F1F*\"\"%F,F.F.7'F**&\"#>F*\"#WF,,$*&\"#:F*F:F,F,*&\"#5F*\"#6F ,F.7'F.*&F@F*\"#sF,*&\"#DF*FCF,*&FEF*FCF,*&F@F*FCF," }{TEXT -1 1 " " } }{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 85 "RK4_4eqs := [op(RowSumConditions(4,'expanded')),op(OrderConditio ns(4,4,'expanded'))]:" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 26 "We specify the two nodes " }{XPPEDIT 18 0 "c[2] = 2 /5;" "6#/&%\"cG6#\"\"#*&F'\"\"\"\"\"&!\"\"" }{TEXT -1 7 " and " } {XPPEDIT 18 0 "c[3] = 3/5;" "6#/&%\"cG6#\"\"$*&F'\"\"\"\"\"&!\"\"" } {TEXT -1 62 ", so as to obtain a system of 11 equations with 11 unknow ns. " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 228 "e1 := \{c[2]=2/5,c[3]=3/5\}:\ne2 := solve(subs(e1,\{ op(RK4_4eqs)\})):\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],``] ,[``,b[1],b[2],b[3],b[4]]]));\n``;\nevalf(%%);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#K%'matrixG6#7&7'#\"\"#\"\"&F(%!GF+F+7'#\"\"$F*#!\"$\"#? #F.\"\"%F+F+7'\"\"\"#\"#>\"#W#!#:F8#\"#5\"#6F+7'F+#F=\"#s#\"#DF@FAF?Q) pprint126\"" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%!G" }}{PARA 11 "" 1 " " {XPPMATH 20 "6#K%'matrixG6#7&7'$\"+++++S!#5F(%!GF+F+7'$\"+++++gF*$!+ ++++:F*$\"+++++vF*F+F+7'$\"\"\"\"\"!$\"+====VF*$!+4444MF*$\"+\"4444*F* F+7'F+$\"+yxxF:F*$\"+AAAsMF*F@F>Q)pprint136\"" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 7 "Check: " }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 61 "expand(simplify(subs(e3,RK4_4eqs)));\nmap(u ->lhs(u)-rhs(u),%);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#7-/#\"\"#\"\"&F %/#\"\"$F'F)/\"\"\"F,F+/#F,F&F./#F,\"\"'F0/#F,F*F3/#F,\"#CF5/#F,\"\")F 8/#F,\"#7F;/#F,\"\"%F>" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#7-\"\"!F$F$F $F$F$F$F$F$F$F$" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 92 "We can calculate the principal error norm, that is, the 2 -norm of the principal error terms." }}{PARA 0 "" 0 "" {TEXT -1 76 "Th e principal error norm is the lowest of all the methods considered so \+ far." }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 150 "errterms4_4 := Prin cipalErrorTerms(4,4,'expanded'):\nsimplify(subs(e3,sqrt(add(errterms4_ 4[i]^2,i=1..nops(errterms4_4)))));\nevalf[14](%): evalf[10](%);" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#,$*(\"&+'R!\"\"\"'T!>\"#\"\"\"\"\"#F*F (F)" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#$\"+1G;K7!#6" }}}{PARA 0 "" 0 " " {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 54 "Four of the principal e rror conditions are satisfied. " }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 123 "RK4_4err_eqs := PrincipalErrorConditions(4,4,'expanded'):\nex pand(simplify(subs(e3,RK4_4err_eqs))):\nmap(u->lhs(u)-rhs(u),%);" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#7+#!\"\"\"$?\"\"\"!#\"\"\"F&F'#\"\"$\" $S%F'F'#F%\"$]\"#F)F." }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 111 "4 of the 9 principal error conditions are satisfied . These conditions are given in abreviated form as follows." }} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 296 "ind := []:\nfor ct to nops( RK4_4err_eqs) do\n if simplify(subs(e3,RK4_4err_eqs[ct])) then ind : = [op(ind),ct] end if:\nend do:\nind;\nerr4 := PrincipalErrorCondition s(4):\n[seq([i,err4[i]],i=ind)]:\nlinalg[augment](linalg[delcols](%,2. .2),matrix([[` `]$(linalg[rowdim](%))]),linalg[delcols](%,1..1));" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#7&\"\"#\"\"%\"\"'\"\"(" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#K%'matrixG6#7&7%\"\"#%#~~G/*&%\"bG\"\"\"-%!G6#*(% \"aGF-%\"cGF--F/6#*&F2F-F3F-F-F-#F-\"#S7%\"\"%F)/*&F,F--F/6#*&F2F--F/6 #*&)F3F(F-F2F-F-F-#F-\"#g7%\"\"'F)/*(F,F-F3F-F@F-#F-\"#:7%\"\"(F)/*(F, F-FCF-F4F-#F-\"#5Q)pprint756\"" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }} {PARA 0 "" 0 "" {TEXT -1 51 "These error equations are given in full a s follows." }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 142 "[seq([i,RK4_4 err_eqs[i]],i=[2,4,6,7])]:\nlinalg[augment](linalg[delcols](%,2..2),ma trix([[` `]$(linalg[rowdim](%))]),linalg[delcols](%,1..1));" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#K%'matrixG6#7&7%\"\"#%#~~G/*,&%\"bG6#\"\"%\" \"\"&%\"aG6$F/\"\"$F0&%\"cG6#F4F0&F26$F4F(F0&F66#F(F0#F0\"#S7%F/F)/**F ,F0F1F0F8F0)F:F(F0#F0\"#g7%\"\"'F)/,&**&F-F7F0F5F0F8F0FAF0F0*(F,F0&F6F .F0,&*&&F26$F/F(F0FAF0F0*&F1F0)F5F(F0F0F0F0#F0\"#:7%\"\"(F)/,&**FIF0FQ F0F8F0F:F0F0*(F,F0)FKF(F0,&*&FNF0F:F0F0*&F1F0F5F0F0F0F0#F0\"#5Q)pprint 776\"" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 27 "The method can be obtained " }{TEXT 260 35 "without specifying any co efficients" }{TEXT -1 147 " by incorporating the previous error condit ions along with the order conditions and the row sum conditions in the system of equations to be solved." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 188 "e4 := solve(\{op(RK4_4eqs), seq(RK4_4err_eqs[i],i=[2,4,6,7])\}):\nsubs(e4,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],``],[`` ,b[1],b[2],b[3],b[4]]]));" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#K%'matrix G6#7&7'#\"\"#\"\"&F(%!GF+F+7'#\"\"$F*#!\"$\"#?#F.\"\"%F+F+7'\"\"\"#\"# >\"#W#!#:F8#\"#5\"#6F+7'F+#F=\"#s#\"#DF@FAF?Q)pprint746\"" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }} }}{PARA 0 "" 0 "" {TEXT -1 49 "#====================================== ==========" }}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 22 " order 4 method wit h " }{XPPEDIT 18 0 "c[2]=3/7" "6#/&%\"cG6#\"\"#*&\"\"$\"\"\"\"\"(!\" \"" }{TEXT -1 7 " and " }{XPPEDIT 18 0 "c[3]=6/11" "6#/&%\"cG6#\"\"$ *&\"\"'\"\"\"\"#6!\"\"" }{TEXT -1 1 " " }}{PARA 0 "" 0 "" {TEXT -1 0 " " }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "matrix([[3/7, 3/7 , ``, ``, ``], [6/11, 3/121, 63/121, ``, ``], [1, 83/351, -406/1053, 1 210/1053, ``], [``, 35/216, 343/1296, 1331/3240, 13/80]])" "6#-%'matri xG6#7&7'*&\"\"$\"\"\"\"\"(!\"\"*&F)F*F+F,%!GF.F.7'*&\"\"'F*\"#6F,*&F)F *\"$@\"F,*&\"#jF*F4F,F.F.7'F**&\"#$)F*\"$^$F,,$*&\"$1%F*\"%`5F,F,*&\"% 57F*F>F,F.7'F.*&\"#NF*\"$;#F,*&\"$V$F*\"%'H\"F,*&\"%J8F*\"%SKF,*&\"#8F *\"#!)F," }{TEXT -1 1 " " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 85 "RK4_4eqs := [op(RowSumConditions(4, 'expanded')),op(OrderConditions(4,4,'expanded'))]:" }}}{PARA 0 "" 0 " " {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 26 "We specify the two node s " }{XPPEDIT 18 0 "c[2] = 3/7" "6#/&%\"cG6#\"\"#*&\"\"$\"\"\"\"\"(! \"\"" }{TEXT -1 7 " and " }{XPPEDIT 18 0 "c[3] = 6/11;" "6#/&%\"cG6# \"\"$*&\"\"'\"\"\"\"#6!\"\"" }{TEXT -1 62 ", so as to obtain a system \+ of 11 equations with 11 unknowns. " }}{PARA 0 "" 0 "" {TEXT -1 0 "" } }{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 224 "e1 := \{c[2]=3/7,c[3]=6/11 \}:\ne2 := solve(subs(e1,\{op(RK4_4eqs)\})):\ne3 := `union`(e1,e2):\ns ubs(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],``],[``,b[1],b[2],b[3],b[4]]]));\nevalf(%);" } }{PARA 11 "" 1 "" {XPPMATH 20 "6#K%'matrixG6#7&7'#\"\"$\"\"(F(%!GF+F+7 '#\"\"'\"#6#F)\"$@\"#\"#jF1F+F+7'\"\"\"#\"#$)\"$^$#!$1%\"%`5#\"%57F;F+ 7'F+#\"#N\"$;##\"$V$\"%'H\"#\"%J8\"%SK#\"#8\"#!)Q)pprint146\"" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#K%'matrixG6#7&7'$\"+'G9dG%!#5F(%!GF+F+ 7'$\"+baaaaF*$\"+V)Q$zC!#6$\"+q:h1_F*F+F+7'$\"\"\"\"\"!$\"+lBnkBF*$!+A 0lbQF*$\"+;y4\\6!\"*F+7'F+$\"+q.P?;F*$\"+Q\\gYEF*$\"+\"pC!3TF*$\"++++D ;F*Q)pprint156\"" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 7 "Check: " }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 61 "expan d(simplify(subs(e3,RK4_4eqs)));\nmap(u->lhs(u)-rhs(u),%);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#7-/#\"\"$\"\"(F%/#\"\"'\"#6F)/\"\"\"F-F,/#F-\" \"#F//#F-F*F2/#F-F&F4/#F-\"#CF6/#F-\"\")F9/#F-\"#7F " 0 "" {MPLTEXT 1 0 150 "errterms4_4 := P rincipalErrorTerms(4,4,'expanded'):\nsimplify(subs(e3,sqrt(add(errterm s4_4[i]^2,i=1..nops(errterms4_4)))));\nevalf[14](%): evalf[10](%);" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#,$*(\"'?,7!\"\"\"'&*zJ#\"\"\"\"\"#\"\" (F(F)" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#$\"+AO1U7!#6" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 54 "None of the principa l error conditions are satisfied. " }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 123 "RK4_4err_eqs := PrincipalErrorConditions(4,4,'expand ed'):\nexpand(simplify(subs(e3,RK4_4err_eqs))):\nmap(u->lhs(u)-rhs(u), %);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#7+#!\"\"\"$?\"#F%\"$S%#\"\"\"F& #F*\"$S)#\"#H\"%?d#F%F,#F*F(#!\"$\"$&Q#\"\"$F4" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}}{PARA 0 "" 0 "" {TEXT -1 49 "#============================================== ==" }}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 21 "order 4 method with " } {XPPEDIT 18 0 "c[2]=2/5" "6#/&%\"cG6#\"\"#*&F'\"\"\"\"\"&!\"\"" } {TEXT -1 7 " and " }{XPPEDIT 18 0 "a[3,2]=2/3" "6#/&%\"aG6$\"\"$\"\" #*&F(\"\"\"F'!\"\"" }{TEXT -1 1 " " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }} {PARA 256 "" 0 "" {TEXT -1 2 " " }{XPPEDIT 18 0 "matrix([[2/5, 2/5, ` `, ``, ``], [1/5+sqrt(33)/15, -7/15+sqrt(33)/15, 2/3, ``, ``], [1, -16 757/23728+4515/23728*sqrt(33), 19245/23728-4965/23728*sqrt(33), 2655/2 966+225/11864*sqrt(33), ``], [``, 1/192+5/192*sqrt(33), 325/576-25/576 *sqrt(33), 75/296+25/1184*sqrt(33), 59/333-5/1332*sqrt(33)]]);" "6#-%' matrixG6#7&7'*&\"\"#\"\"\"\"\"&!\"\"*&F)F*F+F,%!GF.F.7',&*&F*F*F+F,F** &-%%sqrtG6#\"#LF*\"#:F,F*,&*&\"\"(F*F7F,F,*&-F46#F6F*F7F,F**&F)F*\"\"$ F,F.F.7'F*,&*&\"&dn\"F*\"&GP#F,F,*(\"%:XF*FDF,-F46#F6F*F*,&*&\"&X#>F*F DF,F**(\"%l\\F*FDF,-F46#F6F*F,,&*&\"%bEF*\"%mHF,F**(\"$D#F*\"&k=\"F,-F 46#F6F*F*F.7'F.,&*&F*F*\"$#>F,F**(F+F*FfnF,-F46#F6F*F*,&*&\"$D$F*\"$w& F,F**(\"#DF*F]oF,-F46#F6F*F,,&*&\"#vF*\"$'HF,F**(F_oF*\"%%=\"F,-F46#F6 F*F*,&*&\"#fF*\"$L$F,F**(F+F*\"%K8F,-F46#F6F*F," }{TEXT -1 1 " " }} {PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 85 "RK4_4eqs := [op(RowSumConditions(4,'expanded')),op(OrderConditions (4,4,'expanded'))]:" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 " " {TEXT -1 21 "We specify the node " }{XPPEDIT 18 0 "c[2] = 2/5;" "6# /&%\"cG6#\"\"#*&F'\"\"\"\"\"&!\"\"" }{TEXT -1 31 " and the linking co efficient " }{XPPEDIT 18 0 "a[3,2] = 2/3" "6#/&%\"aG6$\"\"$\"\"#*&F( \"\"\"F'!\"\"" }{TEXT -1 61 " so as to obtain a system of 11 equation s with 11 unknowns. " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 " " {TEXT 260 4 "Note" }{TEXT -1 28 ": We include the condition " } {XPPEDIT 18 0 "c[3]" "6#&%\"cG6#\"\"$" }{TEXT -1 48 " > 0 in order to avoid an undesirable solution." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 248 "e1 := \{c[2]=2/5,a[3,2]=2/3 \}:\ne2 := allvalues(solve(subs(e1,\{op(RK4_4eqs),c[3]>0\})));\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],``],[``,b[1],b[2],b[3],b[4]] ]));\n``;\nevalf(%%);" }}{PARA 12 "" 1 "" {XPPMATH 20 "6#>%#e2G<-/&%\" cG6#\"\"%\"\"\"/&%\"aG6$F*F+,&#\"&dn\"\"&GP#!\"\"*(\"%:XF+F3F4\"#L#F+ \"\"#F+/&%\"bGF),&#\"#f\"$L$F+*(\"\"&F+\"%K8F4F7F8F4/&F(6#\"\"$,&#F+FB F+*&\"#:F4F7F8F+/&F.6$F*F9,&#\"&X#>F3F+*(\"%l\\F+F3F4F7F8F4/&FF+*(FBF+FepF4F7F8F+/&F.6$F9F+#F9FB" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#K%'matrixG6#7&7'#\"\"#\"\"&F(%!GF+F+ 7',&#\"\"\"F*F/*&\"#:!\"\"\"#L#F/F)F/,&#\"\"(F1F2*&F1F2F3F4F/#F)\"\"$F +F+7'F/,&#\"&dn\"\"&GP#F2*(\"%:XF/F?F2F3F4F/,&#\"&X#>F?F/*(\"%l\\F/F?F 2F3F4F2,&#\"%bE\"%mHF/*(\"$D#F/\"&k=\"F2F3F4F/F+7'F+,&#F/\"$#>F/*(F*F/ FQF2F3F4F/,&#\"$D$\"$w&F/*(\"#DF/FVF2F3F4F2,&#\"#v\"$'HF/*(FXF/\"%%=\" F2F3F4F/,&#\"#f\"$L$F/*(F*F/\"%K8F2F3F4F2Q)pprint166\"" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%!G" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#K%'matrixG6 #7&7'$\"+++++S!#5F(%!GF+F+7'$\"+K%3(HeF*$!*N#ep$)F*$\"+nmmmmF*F+F+7'$ \"\"\"\"\"!$\"+\\1soQF*$!+VIi4RF*$\"+R-4/5!\"*F+7'F+$\"+*=j![:F*$\"+^8 1\\JF*$\"+C/uYPF*$\"+O]8c:F*Q)pprint176\"" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 7 "Check: " }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 61 "expand(simplify(subs(e3,RK4_4eqs)));\nmap(u->lhs (u)-rhs(u),%);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#7-/#\"\"#\"\"&F%/,&# \"\"\"F'F+*&\"#:!\"\"\"#L#F+F&F+F)/F+F+F1/F0F0/#F+\"\"'F4/#F+\"\"$F7/# F+\"#CF:/#F+\"\")F=/#F+\"#7F@/#F+\"\"%FC" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#7-\"\"!F$F$F$F$F$F$F$F$F$F$" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }} {PARA 0 "" 0 "" {TEXT -1 92 "We can calculate the principal error norm , that is, the 2-norm of the principal error terms." }}{PARA 0 "" 0 " " {TEXT -1 145 "The principal error norm is just marginally lower than that of the previous example and so it is the lowest of all the metho ds considered so far." }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 150 "er rterms4_4 := PrincipalErrorTerms(4,4,'expanded'):\nsimplify(subs(e3,sq rt(add(errterms4_4[i]^2,i=1..nops(errterms4_4)))));\nevalf[14](%): eva lf[10](%);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#,$*&\")+G.K!\"\",&\".E8I w36\"\"\"\"*&\"-7>\")ep;F)\"#L#F)\"\"#F&F-F)" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#$\"+wV=;7!#6" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 53 "Two of the principal error conditions are satis fied. " }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 123 "RK4_4err_eqs := P rincipalErrorConditions(4,4,'expanded'):\nexpand(simplify(subs(e3,RK4_ 4err_eqs))):\nmap(u->lhs(u)-rhs(u),%);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#7+#!\"\"\"$?\",&#\"\"\"\"#gF%*&\"$g$F%\"#L#F)\"\"#F)#F)F&\"\"!,& #\"$u&\"&XA#F)*(\"$j(F)\"'_N@F%F-F.F%F1,&#F)F*F)*&F,F%F-F.F%,&#F)\"#vF %*&\"$+*F%F-F.F),&#F)F>F)*&F@F%F-F.F%" }}}{PARA 0 "" 0 "" {TEXT -1 0 " " }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}}{PARA 0 "" 0 "" {TEXT -1 50 " #================================================" }} {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 41 "#======== ================================" }}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 36 "Test-bed procedures for the examples" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 7 "RK4step" }} {PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 2036 "rk4step := proc(x_rk4step::realcons)\n local c2,c3,c4,a21,a31, a32,a41,a42,a43,f1,f2,f3,f4,\n b1,b2,b3,b4,xk,yk,t,jF,jM,jS,n,h,data ,fn,xx,ys,saveDigits;\n options `Copyright 2004 by Peter Stone`;\n \+ \n data := SOLN_;\n\n saveDigits := Digits;\n Digits := max(tru nc(evalhf(Digits)),Digits+5);\n\n # procedure to evaluate the gradie nt field\n fn := proc(X_,Y_)\n local val; \n val := traper ror(evalf(FXY_));\n if val=lasterror or not type(val,numeric) the n\n error \"evaluation of gradient field failed at %1\",evalf( [X_,Y_],saveDigits);\n end if;\n val;\n end 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\",evalf(data[1,1]) ..evalf(data[n,1]);\n end if;\n\n c2 := c2_; c3 := c3_; c4 := c4_; \n a21 := c2; a31 := a31_; a32 := a32_; \n a41 := a41_; a42 := a4 2_; a43 := a43_;\n b1 := b1_; b2 := b2_; b3 := b3_; b4 := b4_; \n \+ # Perform a binary search for the interval containing x.\n n := nops (data);\n jF := 0;\n jS := n+1;\n\n if data[1,1]1 do\n jM := trunc((jF+jS)/2);\n if \+ xx>=data[jM,1] then jF := jM else jS := jM end if;\n end do;\n \+ if jM = n then jF := n-1; jS := n end if;\n else\n while jS- jF> 1 do\n jM := trunc((jF+jS)/2);\n if xx<=data[jM,1] t hen jF := jM else jS := jM end if;\n end do;\n if jM = n the n jF := n-1; jS := n end if;\n end if;\n \n # Get the data needed from the list.\n xk := data[jF,1];\n yk := data[jF,2];\n\n # Do one step with step-size ..\n h := xx-xk;\n f1 := fn(xk,yk);\n t := a21*f1;\n f2 := fn(xk + c2*h,yk + t*h);\n t := a31*f1 + a32*f2 ;\n f3 := fn(xk + c3*h,yk + t*h);\n t := a41*f1 + a42*f2 + a43*f3; \n f4 := fn(xk + c4*h,yk + t*h);\n \n t := b1*f1 + b2*f2 + b3*f3 + b4*f4; \n ys := yk + t*h;\n\n evalf[saveDigits](ys);\nend pr oc: # of rk4step" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 23 "RK 4_1 Classical method " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1354 "RK4_1 := proc(fxy,x,y,xx,yy,h,stps,bb) \n local c2,c3,c4,a21,a31,a32,a41,a42,a43,f1,f2,f3,f4,\n b1,b2,b3, b4,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(f xy,x,y);\n\n A := matrix([[1/2,1/2,0,0,0],[1/2,0,1/2,0,0],\n [1,0 ,0,1,0],[0,1/6,1/3,1/3,1/6]]);\n \n c2 := evalf(A[1,1]);\n c3 := e valf(A[2,1]);\n c4 := evalf(A[3,1]);\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 b1 := evalf(A[4,2]);\n b2 := evalf(A[4,3]);\n b3 := evalf(A[4,4]);\n b4 := evalf(A[4,5 ]);\n xk := evalf(xx);\n yk := evalf(yy);\n soln := [xk,yk]; \n \+ for k from 1 to stps do\n f1 := fn(xk,yk);\n t := a21*f1; \n f2 := fn(xk + c2*h,yk + t*h);\n t := a31*f1 + a32*f2;\n \+ f3 := fn(xk + c3*h,yk + t*h);\n t := a41*f1 + a42*f2 + a43*f3 ;\n f4 := fn(xk + c4*h,yk + t*h);\n\n yk := yk + (b1*f1 + b2 *f2 + b3*f3 + b4*f4)*h;\n xk := xk + h:\n soln := soln,[xk,y k];\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,a31_=a31,a32_=a32,a41_= a41,a42_=a42,a43_=a43,\n b1_=b1,b2_=b2,b3_=b3,b4_=b4\};\n \+ return subs(eqns,eval(rk4step)); \n else\n return evalf[saveDi gits]([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 24 "RK4_2 3/8 method with " }{XPPEDIT 18 0 "c[2] = 1 /3;" "6#/&%\"cG6#\"\"#*&\"\"\"F)\"\"$!\"\"" }{TEXT -1 7 " and " } {XPPEDIT 18 0 "c[3] = 2/3;" "6#/&%\"cG6#\"\"$*&\"\"#\"\"\"F'!\"\"" } {TEXT -1 1 " " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1356 "RK4_2 := proc(fxy,x,y,xx,yy,h,stps,bb)\n loc al c2,c3,c4,a21,a31,a32,a41,a42,a43,f1,f2,f3,f4,\n b1,b2,b3,b4,t,k,f n,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([[1/3,1/3,0,0,0],[2/3,-1/3,1,0,0],\n [1,1,-1,1,0 ],[0,1/8,3/8,3/8,1/8]]);\n \n c2 := evalf(A[1,1]);\n c3 := evalf(A [2,1]);\n c4 := evalf(A[3,1]);\n a21 := c2;\n a31 := evalf(A[2,2 ]);\n a32 := evalf(A[2,3]);\n a41 := evalf(A[3,2]);\n a42 := eva lf(A[3,3]);\n a43 := evalf(A[3,4]);\n b1 := evalf(A[4,2]);\n b2 \+ := evalf(A[4,3]);\n b3 := evalf(A[4,4]);\n b4 := evalf(A[4,5]);\n \+ xk := evalf(xx);\n yk := evalf(yy);\n soln := [xk,yk]; \n for \+ k from 1 to stps do\n f1 := fn(xk,yk);\n t := a21*f1;\n \+ f2 := fn(xk + c2*h,yk + t*h);\n t := a31*f1 + a32*f2;\n f3 \+ := fn(xk + c3*h,yk + t*h);\n t := a41*f1 + a42*f2 + a43*f3;\n \+ f4 := fn(xk + c4*h,yk + t*h);\n\n yk := yk + (b1*f1 + b2*f2 + b 3*f3 + b4*f4)*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,a31_=a31,a32_=a32,a41_=a41,a42 _=a42,a43_=a43,\n b1_=b1,b2_=b2,b3_=b3,b4_=b4\};\n return subs(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 26 "RK4_3 Gill's method with " }{XPPEDIT 18 0 "c[2] = c[3] ;" "6#/&%\"cG6#\"\"#&F%6#\"\"$" }{XPPEDIT 18 0 "``=1/2" "6#/%!G*&\"\" \"F&\"\"#!\"\"" }{TEXT -1 4 " " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1413 "RK4_3 := proc(fxy,x,y,xx,y y,h,stps,bb)\n local c2,c3,c4,a21,a31,a32,a41,a42,a43,f1,f2,f3,f4,\n b1,b2,b3,b4,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([[1/2,1/2,0,0,0],[1/2,-1/2+sqrt (2)/2,1-sqrt(2)/2,0,0],\n [1,0,-sqrt(2)/2,1+sqrt(2)/2,0],[0,1/6,1/3- sqrt(2)/6,sqrt(2)/6+1/3,1/6]]);\n \n c2 := evalf(A[1,1]);\n c3 := \+ evalf(A[2,1]);\n c4 := evalf(A[3,1]);\n a21 := c2;\n a31 := eval f(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 b1 := evalf(A[4,2]); \n b2 := evalf(A[4,3]);\n b3 := evalf(A[4,4]);\n b4 := evalf(A[4 ,5]);\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\n yk := yk + (b1*f1 + \+ b2*f2 + b3*f3 + b4*f4)*h;\n xk := xk + h:\n soln := soln,[xk ,yk];\n end do;\n if bb=true then\n eqns := \{SOLN_=[soln],FX Y_=fxy,X_=x,Y_=y,c2_=c2,c3_=c3,\n c4_=c4,a31_=a31,a32_=a32,a41 _=a41,a42_=a42,a43_=a43,\n b1_=b1,b2_=b2,b3_=b3,b4_=b4\};\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 29 "RK4_4 Dormand's method with " }{XPPEDIT 18 0 " c[2] = 2/5;" "6#/&%\"cG6#\"\"#*&F'\"\"\"\"\"&!\"\"" }{TEXT -1 7 " and " }{XPPEDIT 18 0 "c[3] = 3/5;" "6#/&%\"cG6#\"\"$*&F'\"\"\"\"\"&!\"\" " }{TEXT -1 2 " " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1379 "RK4_4 := proc(fxy,x,y,xx,yy,h,stps,bb)\n \+ local c2,c3,c4,a21,a31,a32,a41,a42,a43,f1,f2,f3,f4,\n b1,b2,b3,b4,t, k,fn,xk,yk,soln,eqns,A,saveDigits;\n\n saveDigits := Digits;\n Dig its := max(trunc(evalhf(Digits)),Digits+5);\n\n fn := unapply(fxy,x, y);\n\n A := matrix([[2/5,2/5,0,0,0],[3/5,-3/20,3/4,0,0],\n [1,19 /44,-15/44,10/11,0],[0,11/72,25/72,25/72,11/72]]);\n \n c2 := evalf( A[1,1]);\n c3 := evalf(A[2,1]);\n c4 := evalf(A[3,1]);\n a21 := \+ c2;\n a31 := evalf(A[2,2]);\n a32 := evalf(A[2,3]);\n a41 := eva lf(A[3,2]);\n a42 := evalf(A[3,3]);\n a43 := evalf(A[3,4]);\n b1 := evalf(A[4,2]);\n b2 := evalf(A[4,3]);\n b3 := evalf(A[4,4]);\n b4 := evalf(A[4,5]);\n xk := evalf(xx);\n yk := evalf(yy);\n \+ soln := [xk,yk]; \n for k from 1 to stps do\n f1 := fn(xk,yk); \n t := a21*f1;\n f2 := fn(xk + c2*h,yk + t*h);\n t := \+ a31*f1 + a32*f2;\n f3 := fn(xk + c3*h,yk + t*h);\n t := a41* f1 + a42*f2 + a43*f3;\n f4 := fn(xk + c4*h,yk + t*h);\n\n yk := yk + (b1*f1 + b2*f2 + b3*f3 + b4*f4)*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,a31 _=a31,a32_=a32,a41_=a41,a42_=a42,a43_=a43,\n b1_=b1,b2_=b2,b3_ =b3,b4_=b4\};\n return subs(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 19 "RK4_5 method with " } {XPPEDIT 18 0 "c[2]=3/7" "6#/&%\"cG6#\"\"#*&\"\"$\"\"\"\"\"(!\"\"" } {TEXT -1 7 " and " }{XPPEDIT 18 0 "c[3]=6/11" "6#/&%\"cG6#\"\"$*&\" \"'\"\"\"\"#6!\"\"" }{TEXT -1 1 " " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1399 "RK4_5 := proc(fxy,x,y,xx,y y,h,stps,bb)\n local c2,c3,c4,a21,a31,a32,a41,a42,a43,f1,f2,f3,f4,\n b1,b2,b3,b4,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([[3/7,3/7,0,0,0],[6/11,3/121,63 /121,0,0],\n [1,83/351,-406/1053,1210/1053,0],[0,35/216,343/1296,13 31/3240,13/80]]);\n \n c2 := evalf(A[1,1]);\n c3 := evalf(A[2,1]); \n c4 := evalf(A[3,1]);\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 b1 := evalf(A[4,2]);\n b2 := eval f(A[4,3]);\n b3 := evalf(A[4,4]);\n b4 := evalf(A[4,5]);\n xk := evalf(xx);\n yk := evalf(yy);\n soln := [xk,yk]; \n for k from \+ 1 to stps do\n f1 := fn(xk,yk);\n t := a21*f1;\n f2 := \+ fn(xk + c2*h,yk + t*h);\n t := a31*f1 + a32*f2;\n f3 := fn(x k + c3*h,yk + t*h);\n t := a41*f1 + a42*f2 + a43*f3;\n f4 := fn(xk + c4*h,yk + t*h);\n\n yk := yk + (b1*f1 + b2*f2 + b3*f3 + \+ b4*f4)*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,a31_=a31,a32_=a32,a41_=a41,a42_=a42,a 43_=a43,\n b1_=b1,b2_=b2,b3_=b3,b4_=b4\};\n return subs(e qns,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 19 "RK4_6 method with " }{XPPEDIT 18 0 "c[2]=2/5" "6#/&%\"cG 6#\"\"#*&F'\"\"\"\"\"&!\"\"" }{TEXT -1 7 " and " }{XPPEDIT 18 0 "a[3 ,2]=2/3" "6#/&%\"aG6$\"\"$\"\"#*&F(\"\"\"F'!\"\"" }{TEXT -1 1 " " }} {PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1616 "RK4_6 := proc(fxy,x,y,xx,yy,h,stps,bb)\n local c2,c3,c4,a21,a3 1,a32,a41,a42,a43,f1,f2,f3,f4,\n b1,b2,b3,b4,t,k,fn,xk,yk,soln,eqns, A,saveDigits,SQRT;\n\n saveDigits := Digits;\n Digits := max(trunc (evalhf(Digits)),Digits+5);\n SQRT := evalf(sqrt(33));\n\n fn := u napply(fxy,x,y);\n\n A := matrix([[2/5,2/5,0,0,0],\n [1/5+SQRT /15,-7/15+SQRT/15,2/3,0,0],\n [1,-16757/23728+4515/23728*SQRT,19 245/23728-4965/23728*SQRT,\n 2655/2966+225/11 864*SQRT,0],\n [0,1/192+5/192*SQRT,325/576-25/576*SQRT,75/296+25 /1184*SQRT,\n 59/333-5/1332*SQRT]]);\n \n c 2 := evalf(A[1,1]);\n c3 := evalf(A[2,1]);\n c4 := evalf(A[3,1]); \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 b1 := evalf(A[4,2]);\n b2 := evalf(A[4,3]);\n b3 := eval f(A[4,4]);\n b4 := evalf(A[4,5]);\n xk := evalf(xx);\n yk := eva lf(yy);\n soln := [xk,yk]; \n for k from 1 to stps do\n f1 := fn(xk,yk);\n t := a21*f1;\n f2 := fn(xk + c2*h,yk + t*h);\n t := a31*f1 + a32*f2;\n f3 := fn(xk + c3*h,yk + t*h);\n \+ t := a41*f1 + a42*f2 + a43*f3;\n f4 := fn(xk + c4*h,yk + t*h); \n\n yk := yk + (b1*f1 + b2*f2 + b3*f3 + b4*f4)*h;\n xk := x k + h:\n soln := soln,[xk,yk];\n end do;\n if bb=true then\n \+ eqns := \{SOLN_=[soln],FXY_=fxy,X_=x,Y_=y,c2_=c2,c3_=c3,\n \+ c4_=c4,a31_=a31,a32_=a32,a41_=a41,a42_=a42,a43_=a43,\n b1_=b 1,b2_=b2,b3_=b3,b4_=b4\};\n return subs(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 ";" }}}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}} }{SECT 1 {PARA 4 "" 0 "" {TEXT -1 20 "Testing the examples" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 39 "Test 1 of order 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 6 ", " } {XPPEDIT 18 0 "y(0)=1" "6#/-%\"yG6#\"\"!\"\"\"" }{TEXT -1 1 " " }} {PARA 0 "" 0 "" {TEXT -1 10 "Solution: " }}{PARA 256 "" 0 "" {TEXT -1 2 " " }{XPPEDIT 18 0 "y=exp(-12/17*x*cos(4*x)*exp(-x)+180/289*exp(-x) *cos(4*x)+48/17*exp(-x)*sin(4*x)*x+96/289*exp(-x)*sin(4*x)-180/289)" " 6#/%\"yG-%$expG6#,,*,\"#7\"\"\"\"# " 0 "" {MPLTEXT 1 0 229 "de := diff( y(x),x)=12*x*cos(4*x)*exp(-x)*y(x);\nic := y(0)=1;\ndsolve(\{de,ic\},y (x)):\ny(x)=simplify(numer(rhs(%))/convert(denom(rhs(%)),exp));\nf := \+ unapply(rhs(%),x):\nplot(f(x),x=0..5,0..1.45,font=[HELVETICA,9],labels =[`x`,`y(x)`]);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%#deG/-%%diffG6$-% \"yG6#%\"xGF,,$*,\"#7\"\"\"F,F0-%$cosG6#,$*&\"\"%F0F,F0F0F0-%$expG6#,$ F,!\"\"F0F)F0F0" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%#icG/-%\"yG6#\"\" !\"\"\"" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/-%\"yG6#%\"xG-%$expG6#,,*& #\"#7\"#<\"\"\"*(F'F0-%$cosG6#,$*&\"\"%F0F'F0F0F0-F)6#,$F'!\"\"F0F0F;* &#\"$!=\"$*GF0*&F8F0F2F0F0F0*&#\"#[F/F0*(F8F0-%$sinGF4F0F'F0F0F0*&#\"# '*F?F0*&F8F0FEF0F0F0#F>F?F;" }}{PARA 13 "" 1 "" {GLPLOT2D 577 286 286 {PLOTDATA 2 "6&-%'CURVESG6$7er7$$\"\"!F)$\"\"\"F)7$$\"3gmmTN@Ki8!#>$\" 3Fk>e\"G.6+\"!#<7$$\"3ALL$3FWYs#F/$\"3!H*fm:2P/5F27$$\"3%)***\\iSmp3%F /$\"3Qn()\\Dat45F27$$\"3WmmmT&)G\\aF/$\"34$Q7t`Dr,\"F27$$\"3m****\\7G$ 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lAFjSF2$\"3[i8#)*p//*\\FJ7$$\"3ym\"zW7@^6%F2$\"3>C%QCunR#[FJ7$$\"3yLLL $)*pp;%F2$\"3g*yCm#3E'p%FJ7$$\"3)QL3-$H**>UF2$\"3$*o:W?mr0YFJ7$$\"3)RL $3xe,tUF2$\"3!\\Bp&*))oXb%FJ7$$\"3h+v=n(*fDVF2$\"3kIpK$)H$3a%FJ7$$\"3C n;HdO=yVF2$\"3u&G6!oNOhXFJ7$$\"3MMe9\"z-lU%F2$\"3kC\">#=Lu2YFJ7$$\"3a+ ++D>#[Z%F2$\"3w_(eqj7vn%FJ7$$\"3SnmT&G!e&e%F2$\"3W>T$>g**p!\\FJ7$$\"3# RLLL)Qk%o%F2$\"3'yDBP_q:;&FJ7$$\"37+]iSjE!z%F2$\"3J;fP@m(pV&FJ7$$\"3a+ ]P40O\"*[F2$\"3!>+$=fU-gcFJ7$$\"\"&F)$\"3h(Q0fOqh\"eFJ-%'COLOURG6&%$RG BG$\"#5!\"\"F(F(-%%FONTG6$%*HELVETICAG\"\"*-%+AXESLABELSG6$%\"xG%%y(x) G-%%VIEWG6$;F(F]am;F($\"$X\"!\"#" 1 2 0 1 10 0 2 9 1 4 2 1.000000 45.000000 45.000000 0 0 "Curve 1" }}}}{PARA 0 "" 0 "" {TEXT -1 0 "" }} {PARA 0 "" 0 "" {TEXT -1 32 "The following code constructs a " }{TEXT 260 17 "discrete solution" }{TEXT -1 44 " based on each of the methods and gives the " }{TEXT 260 22 "root mean square error" }{TEXT -1 18 " of each solution." }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 821 "F := \+ (x,y) -> 12*x*cos(4*x)*exp(-x)*y: hh := 0.01: numsteps := 500: x0 := 0 : y0 := 1:\nmatrix([[`slope field: `,F(x,y)],[`initial point: `,``(x 0,y0)],[`step width: `,hh],\n[`no. of steps: `,numsteps]]);``;\nmt hds := [`classical method with `*(c[2]=1/2,c[3]=1/2),`3/8 method with \+ `*(c[2]=1/3,c[3]=2/3),`Gill's method with `*(c[2]=1/2,c[3]=1/2),`Dorma nd's method with `*(c[2]=2/5,c[3]=3/5),`method with `*(c[2]=3/7,c[3]=6 /11),`method with `*(c[2]=2/5,a[3,2]=2/3)]: errs := []:\nDigits := 25: \nfor ct to 6 do\n Fn_RK4_||ct := RK4_||ct(F(x,y),x,y,x0,y0,hh,numst eps,false);\n sm := 0: numpts := nops(Fn_RK4_||ct):\n for ii to nu mpts do\n sm := sm+(Fn_RK4_||ct[ii,2]-f(Fn_RK4_||ct[ii,1]))^2;\n \+ end do:\n errs := [op(errs),sqrt(sm/numpts)];\nend do:\nDigits := \+ 10:\nlinalg[transpose](convert([mthds,evalf(errs)],matrix));" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#K%'matrixG6#7&7$%0slope~field:~~~G,$*,\"#7\" \"\"%\"xGF,-%$cosG6#,$*&\"\"%F,F-F,F,F,-%$expG6#,$F-!\"\"F,%\"yGF,F,7$ %0initial~point:~G-%!G6$\"\"!F,7$%/step~width:~~~G$F,!\"#7$%1no.~of~st eps:~~~G\"$+&Q)pprint136\"" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%!G" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#K%'matrixG6#7(7$*&%7classical~method~w ith~G\"\"\"6$/&%\"cG6#\"\"##F*F0/&F.6#\"\"$F1F*$\"+\"f2&[#)!#=7$*&%13/ 8~method~with~GF*6$/F-#F*F5/F3#F0F5F*$\"+8+xpLF87$*&%4Gill's~method~wi th~GF*F+F*F67$*&%7Dormand's~method~with~GF*6$/F-#F0\"\"&/F3#F5FLF*$\"+ ;sJvAF87$*&%-method~with~GF*6$/F-#F5\"\"(/F3#\"\"'\"#6F*$\"+sZoCUF87$* &FSF*6$FJ/&%\"aG6$F5F0F@F*$\"+!>r>a#F8Q)pprint146\"" }}}{PARA 0 "" 0 " " {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 30 "The following code cons tructs " }{TEXT 260 20 "numerical procedures" }{TEXT -1 56 " for solut ions based on each of the Runge-Kutta schemes." }}{PARA 0 "" 0 "" {TEXT -1 75 "The error in the value obtained by each of the methods at the point where " }{XPPEDIT 18 0 "x = 4.999;" "6#/%\"xG-%&FloatG6$\" %**\\!\"$" }{TEXT -1 16 " is also given." }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 752 "F := (x,y) -> 12*x*cos(4*x)*exp(-x)*y: hh := 0.01: n umsteps := 500: x0 := 0: y0 := 1:\nmatrix([[`slope field: `,F(x,y)], [`initial point: `,``(x0,y0)],[`step width: `,hh],\n[`no. of steps: \+ `,numsteps]]);``;\nmthds := [`classical method with `*(c[2]=1/2,c[3] =1/2),`3/8 method with `*(c[2]=1/3,c[3]=2/3),`Gill's method with `*(c[ 2]=1/2,c[3]=1/2),`Dormand's method with `*(c[2]=2/5,c[3]=3/5),`method \+ with `*(c[2]=3/7,c[3]=6/11),`method with `*(c[2]=2/5,a[3,2]=2/3)]: err s := []:\nDigits := 25:\nfor ct to 6 do\n fn_RK4_||ct := RK4_||ct(F( x,y),x,y,x0,y0,hh,numsteps,true);\nend do:\nxx := 4.999: fxx := evalf( f(xx)):\nfor ct to 6 do\n errs := [op(errs),abs(fn_RK4_||ct(xx)-fxx) ];\nend do:\nDigits := 10:\nlinalg[transpose](convert([mthds,evalf(err s)],matrix));" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#K%'matrixG6#7&7$%0slo pe~field:~~~G,$*,\"#7\"\"\"%\"xGF,-%$cosG6#,$*&\"\"%F,F-F,F,F,-%$expG6 #,$F-!\"\"F,%\"yGF,F,7$%0initial~point:~G-%!G6$\"\"!F,7$%/step~width:~ ~~G$F,!\"#7$%1no.~of~steps:~~~G\"$+&Q(pprint86\"" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%!G" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#K%'matrixG6#7(7$ *&%7classical~method~with~G\"\"\"6$/&%\"cG6#\"\"##F*F0/&F.6#\"\"$F1F*$ \"+B\"G)y&)!#=7$*&%13/8~method~with~GF*6$/F-#F*F5/F3#F0F5F*$\"+cW$>x$F 87$*&%4Gill's~method~with~GF*F+F*F67$*&%7Dormand's~method~with~GF*6$/F -#F0\"\"&/F3#F5FLF*$\"+\"ejVw(!#>7$*&%-method~with~GF*6$/F-#F5\"\"(/F3 #\"\"'\"#6F*$\"+.ZXSOF87$*&FTF*6$FJ/&%\"aG6$F5F0F@F*$\"+%3bJC\"F8Q(ppr int96\"" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 4 "The " }{TEXT 260 22 "root mean square error" }{TEXT -1 20 " over th e interval " }{XPPEDIT 18 0 "[0, 5];" "6#7$\"\"!\"\"&" }{TEXT -1 82 " of each Runge-Kutta method is estimated as follows using the special procedure " }{TEXT 0 5 "NCint" }{TEXT -1 98 " to perform numerical \+ integration by the 7 point Newton-Cotes method over 200 equal subinter vals." }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 496 "mthds := [`classic al method with `*(c[2]=1/2,c[3]=1/2),`3/8 method with `*(c[2]=1/3,c[3] =2/3),`Gill's method with `*(c[2]=1/2,c[3]=1/2),`Dormand's method with `*(c[2]=2/5,c[3]=3/5),`method with `*(c[2]=3/7,c[3]=6/11),`method wit h `*(c[2]=2/5,a[3,2]=2/3)]: errs := []:\nDigits := 20:\nfor ct to 6 do \n sm := NCint((f(x)-'fn_RK4_||ct'(x))^2,x=0..5,adaptive=false,numpo ints=7,factor=200);\n errs := [op(errs),sqrt(sm/5)];\nend do:\nDigit s := 10:\nlinalg[transpose](convert([mthds,evalf(errs)],matrix));" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#K%'matrixG6#7(7$*&%7classical~method~w ith~G\"\"\"6$/&%\"cG6#\"\"##F*F0/&F.6#\"\"$F1F*$\"+A69^#)!#=7$*&%13/8~ method~with~GF*6$/F-#F*F5/F3#F0F5F*$\"+(y4!pLF87$*&%4Gill's~method~wit h~GF*F+F*F67$*&%7Dormand's~method~with~GF*6$/F-#F0\"\"&/F3#F5FLF*$\"+q LXwAF87$*&%-method~with~GF*6$/F-#F5\"\"(/F3#\"\"'\"#6F*$\"+q-tEUF87$*& FSF*6$FJ/&%\"aG6$F5F0F@F*$\"+QQ@VDF8Q)pprint106\"" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 92 "The following error graph s are constructed using the numerical procedures for the solutions." } }{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 456 "evalf[20](plot(['fn_RK4_1' (x)-f(x),'fn_RK4_2'(x)-f(x),'fn_RK4_3'(x)-f(x),'fn_RK4_4'(x)-f(x),\n'f n_RK4_5'(x)-f(x),'fn_RK4_6'(x)-f(x)],x=0..5,font=[HELVETICA,9],thickne ss=[1$2,2,1$3],\ncolor=[red,blue,COLOR(RGB,0,.65,0),COLOR(RGB,.4,0,1), magenta,brown],\nlegend=[`classical method`,`3/8 method`,`Gill's metho d`,`Dormand's method`,`method with c[2]=3/7 and c[3]=6/11`,`method wit h c[2]=2/5 and a[3,2]=2/3`],title=`error curves for order 4 Runge-Kutt a methods`));" }}{PARA 13 "" 1 "" {GLPLOT2D 768 458 458 {PLOTDATA 2 "6 ,-%'CURVESG6&7jr7$$\"\"!F)F(7$$\"5MLLLL3x&)*3\"!#?$\"+Z2YS**!#>7$$\"5n mmm\"H2P\"Q?F-$\",n[l(=@F07$$\"5MLLL$eRwX5$F-$\",ZKzKY$F07$$\"5MLLLL$e I8k$F-$\",f)4\"\\$RF07$$\"5NLLL$3x%3yTF-$\",HGtE=%F07$$\"5-+](oHaN;J%F -$\",a.qu?%F07$$\"5ommT5:j=XWF-$\",=7eU>%F07$$\"5NL$eRs3P(yXF-$\",X%RZ nTF07$$\"5-++]PfyG7ZF-$\",ziyH7%F07$$\"5NLLek.%*Qz\\F-$\",ls_l(RF07$$ 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]P%)*3#H)3[\"F0$\"-b)>5Rt#F-7$$\"5nT5:8oU%Q[\"F0$\"-wQ(4Rw#F-7$Fc]q$\" -\"ebH1y#F-7$Fh]q$\"-bH+r[FF-7$F]^q$\"-G0))4^FF-7$Fb^q$\"-AE*3fr#F-7$F g^q$\"-`5TVWFF-7$F\\_q$\"-3a\\\\WFF-7$Ff[l$\"-:kA5mEF-7$Fd_q$\"-J1+\"\\F07$Ff`l$\",+=?#F-7$Ff`o$\"-W)y&*)[>F-7$Feal$\"-fl98Z #F-7$Fgdl$ \",Q.ML3#F-7$Fael$\",d>,y4#F-7$Ffel$\",'=s)QC#F-7$F[fl$\",ajsIW#F-7$F` fl$\",KF-7$Fefl$\",,-2%[TF-7$Fjfl$\",mvx_W'F-7$F_gl$\",a_*GN*)F-7 $Fdgl$\"-hzLfD7F-7$Figl$\"-&3(zUf:F-7$F^hl$\"-qXI15([(**F-7$Fdam$\"-w@) )*H2\"F-7$Fiam$\"-8*4:3:\"F-7$F^bm$\"-IsS'*37F-7$Fcbm$\"-/>b=V7F--Fhbm 6&Fjbm$\")#)eqkF]cm$\"))eqk\"F]cmFajuF^cm-Fccm6#%Dmethod~with~c[2]=2/5 ~and~a[3,2]=2/3G-%%FONTG6$%*HELVETICAG\"\"*-%+AXESLABELSG6$Q\"x6\"Q!F_ [v-%&TITLEG6#%Merror~curves~for~order~4~Runge-Kutta~methodsG-%%VIEWG6$ ;F(Fcbm%(DEFAULTG" 1 2 0 1 10 0 2 9 1 4 2 1.000000 45.000000 45.000000 0 1 "classical method" "3/8 method" "Gill's method" "Dormand 's method" "method with c[2]=3/7 and c[3]=6/11" "method with c[2]=2/5 \+ and a[3,2]=2/3" }}}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> \+ " 0 "" {MPLTEXT 1 0 1 ";" }}}}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 38 "Tes t 2 of order 4 Runge-Kutta methods " }}{PARA 256 "" 0 "" {TEXT -1 1 " \+ " }{XPPEDIT 18 0 "dy/dx=x/y" "6#/*&%#dyG\"\"\"%#dxG!\"\"*&%\"xGF&%\"yG F(" }{TEXT -1 10 ", " }{XPPEDIT 18 0 "y(0)=1" "6#/-%\"yG6#\"\" !\"\"\"" }{TEXT -1 1 " " }}{PARA 0 "" 0 "" {TEXT -1 10 "Solution: " }} {PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "y=sqrt(1+x^2)" "6#/% \"yG-%%sqrtG6#,&\"\"\"F)*$%\"xG\"\"#F)" }{TEXT -1 2 ". " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 32 "The following code c onstructs a " }{TEXT 260 17 "discrete solution" }{TEXT -1 44 " based o n each of the methods and gives the " }{TEXT 260 22 "root mean square \+ error" }{TEXT -1 18 " of each solution." }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 824 "G := (x,y) -> x/y: hh := 0.05: numsteps := 200: x0 : = 0: y0 := 1:\nmatrix([[`slope field: `,G(x,y)],[`initial point: `,` `(x0,y0)],[`step width: `,hh],\n[`no. of steps: `,numsteps]]);``; \nmthds := [`classical method with `*(c[2]=1/2,c[3]=1/2),`3/8 method w ith `*(c[2]=1/3,c[3]=2/3),`Gill's method with `*(c[2]=1/2,c[3]=1/2),`D ormand's method with `*(c[2]=2/5,c[3]=3/5),`method with `*(c[2]=3/7,c[ 3]=6/11),`method with `*(c[2]=2/5,a[3,2]=2/3)]: errs := []:\nDigits := 20:\ng := x -> sqrt(1+x^2):\nfor ct to 6 do\n Gn_RK4_||ct := RK4_|| ct(G(x,y),x,y,x0,y0,hh,numsteps,false);\n sm := 0: numpts := nops(Gn _RK4_||ct):\n for ii to numpts do\n sm := sm+(Gn_RK4_||ct[ii,2] -g(Gn_RK4_||ct[ii,1]))^2;\n end do:\n errs := [op(errs),sqrt(sm/nu mpts)];\nend do:\nDigits := 10:\nlinalg[transpose](convert([mthds,eval f(errs)],matrix));" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#K%'matrixG6#7&7$ %0slope~field:~~~G*&%\"xG\"\"\"%\"yG!\"\"7$%0initial~point:~G-%!G6$\" \"!F+7$%/step~width:~~~G$\"\"&!\"#7$%1no.~of~steps:~~~G\"$+#Q)pprint15 6\"" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%!G" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#K%'matrixG6#7(7$*&%7classical~method~with~G\"\"\"6$/&% \"cG6#\"\"##F*F0/&F.6#\"\"$F1F*$\"+wcqhk!#=7$*&%13/8~method~with~GF*6$ /F-#F*F5/F3#F0F5F*$\"+oc$))Q%F87$*&%4Gill's~method~with~GF*F+F*$\"+qvz hPF87$*&%7Dormand's~method~with~GF*6$/F-#F0\"\"&/F3#F5FNF*$\"+-sF,ZF87 $*&%-method~with~GF*6$/F-#F5\"\"(/F3#\"\"'\"#6F*$\"+nzEMWF87$*&FUF*6$F L/&%\"aG6$F5F0F@F*$\"+[3N0WF8Q)pprint166\"" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 30 "The following code constructs \+ " }{TEXT 260 20 "numerical procedures" }{TEXT -1 56 " for solutions ba sed on each of the Runge-Kutta schemes." }}{PARA 0 "" 0 "" {TEXT -1 75 "The error in the value obtained by each of the methods at the poin t where " }{XPPEDIT 18 0 "9.99;" "6#-%&FloatG6$\"$***!\"#" }{TEXT -1 16 " is also given." }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 754 "G : = (x,y) -> x/y: hh := 0.05: numsteps := 200: x0 := 0: y0 := 1:\nmatrix ([[`slope field: `,G(x,y)],[`initial point: `,``(x0,y0)],[`step widt h: `,hh],\n[`no. of steps: `,numsteps]]);``;\nmthds := [`classical method with `*(c[2]=1/2,c[3]=1/2),`3/8 method with `*(c[2]=1/3,c[3]=2 /3),`Gill's method with `*(c[2]=1/2,c[3]=1/2),`Dormand's method with ` *(c[2]=2/5,c[3]=3/5),`method with `*(c[2]=3/7,c[3]=6/11),`method with \+ `*(c[2]=2/5,a[3,2]=2/3)]: errs := []:\nDigits := 25:\nfor ct to 6 do\n gn_RK4_||ct := RK4_||ct(G(x,y),x,y,x0,y0,hh,numsteps,true);\nend do :\ng := x -> sqrt(1+x^2):\nxx := 9.99: gxx := evalf(g(xx)):\nfor ct to 6 do\n errs := [op(errs),abs(gn_RK4_||ct(xx)-gxx)];\nend do:\nDigit s := 10:\nlinalg[transpose](convert([mthds,evalf(errs)],matrix));" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#K%'matrixG6#7&7$%0slope~field:~~~G*&% \"xG\"\"\"%\"yG!\"\"7$%0initial~point:~G-%!G6$\"\"!F+7$%/step~width:~~ ~G$\"\"&!\"#7$%1no.~of~steps:~~~G\"$+#Q)pprint176\"" }}{PARA 11 "" 1 " " {XPPMATH 20 "6#%!G" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#K%'matrixG6#7( 7$*&%7classical~method~with~G\"\"\"6$/&%\"cG6#\"\"##F*F0/&F.6#\"\"$F1F *$\"+YJ%HA#!#=7$*&%13/8~method~with~GF*6$/F-#F*F5/F3#F0F5F*$\"+3U)3^\" F87$*&%4Gill's~method~with~GF*F+F*$\"+I6z$H\"F87$*&%7Dormand's~method~ with~GF*6$/F-#F0\"\"&/F3#F5FNF*$\"+=.#*=;F87$*&%-method~with~GF*6$/F-# F5\"\"(/F3#\"\"'\"#6F*$\"+^'es_\"F87$*&FUF*6$FL/&%\"aG6$F5F0F@F*$\"+JD _<:F8Q)pprint186\"" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 " " {TEXT -1 4 "The " }{TEXT 260 22 "root mean square error" }{TEXT -1 20 " over the interval " }{XPPEDIT 18 0 "[0, 10]" "6#7$\"\"!\"#5" } {TEXT -1 82 " of each Runge-Kutta method is estimated as follows usin g the special procedure " }{TEXT 0 5 "NCint" }{TEXT -1 98 " to perfo rm numerical integration by the 7 point Newton-Cotes method over 100 e qual subintervals." }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 521 "mthds := [`classical method with `*(c[2]=1/2,c[3]=1/2),`3/8 method with `*( c[2]=1/3,c[3]=2/3),`Gill's method with `*(c[2]=1/2,c[3]=1/2),`Dormand' s method with `*(c[2]=2/5,c[3]=3/5),`method with `*(c[2]=3/7,c[3]=6/11 ),`method with `*(c[2]=2/5,a[3,2]=2/3)]: errs := []:\nDigits := 20:\ng := x -> sqrt(1+x^2):\nfor ct to 6 do\n sm := NCint((g(x)-'gn_RK4_|| ct'(x))^2,x=0..10,adaptive=false,numpoints=7,factor=100);\n errs := \+ [op(errs),sqrt(sm/10)];\nend do:\nDigits := 10:\nlinalg[transpose](con vert([mthds,evalf(errs)],matrix));" }}{PARA 11 "" 1 "" {XPPMATH 20 "6# K%'matrixG6#7(7$*&%7classical~method~with~G\"\"\"6$/&%\"cG6#\"\"##F*F0 /&F.6#\"\"$F1F*$\"+X88Yk!#=7$*&%13/8~method~with~GF*6$/F-#F*F5/F3#F0F5 F*$\"+deByVF87$*&%4Gill's~method~with~GF*F+F*$\"+nmt_PF87$*&%7Dormand' s~method~with~GF*6$/F-#F0\"\"&/F3#F5FNF*$\"+vI%**o%F87$*&%-method~with ~GF*6$/F-#F5\"\"(/F3#\"\"'\"#6F*$\"+(f\"fBWF87$*&FUF*6$FL/&%\"aG6$F5F0 F@F*$\"+&fQZR%F8Q)pprint196\"" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }} {PARA 0 "" 0 "" {TEXT -1 92 "The following error graphs are constructe d using the numerical procedures for the solutions." }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 453 "evalf[20](plot(['gn_RK4_1'(x)-g(x),'gn_RK4 _2'(x)-g(x),'gn_RK4_3'(x)-g(x),'gn_RK4_4'(x)-g(x),\n'gn_RK4_5'(x)-g(x) ,'gn_RK4_6'(x)-g(x)],x=0..10,font=[HELVETICA,9],\ncolor=[red,blue,COLO R(RGB,0,.65,0),COLOR(RGB,.4,0,1),magenta,brown],thickness=[1$5,2],\nle gend=[`classical method`,`3/8 method`,`Gill's method`,`Dormand's metho d`,`method with c[2]=3/7 and c[3]=6/11`,`method with c[2]=2/5 and a[3, 2]=2/3`],title=`error curves for order 4 Runge-Kutta methods`));" }} {PARA 13 "" 1 "" {GLPLOT2D 1061 690 690 {PLOTDATA 2 "6,-%'CURVESG6&7co 7$$\"\"!F)F(7$$\"5lmmmmT&)G\\a!#@$\"+tI=B;!#>7$$\"5LLLLL3x&)*3\"!#?$\" 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8**)F07$Fdt$\",U^&4+%)F07$Fit $\",t<)=TxF07$F^u$\",gj]&GrF07$Fcu$\",jr:Sc'F07$Fhu$\",:TX(>hF07$F]v$ \",(zy'yn&F07$Fbv$\",4)R,'G&F07$Fgv$\",<)eb_\\F07$F\\w$\",*>^%>o%F07$F aw$\",>U2TR%F07$Ffw$\",N>=`<%F07$F[x$\",#*)\\pXRF07$F`x$\",dBd;#p=MF07$F_y$\",,/nkE$F07$Fdy$\",qS#)y8$F07$Fi y$\",,YW)4IF07$F^z$\",wF07$F_^l$\",j;,H#>F07$Fd^l$ \",UK>F(=F07$Fi^l$\",n8'yE=F07$F^_l$\",;aq?y\"F07$Fc_l$\",E0A)RsGf:F07$Faal$\"+^luD:Feal-Fgal6&FialFjalF(FjalF]bl-F bbl6#%Cmethod~with~c[2]=3/7~and~c[3]=6/11G-F$6&7ioF'7$F_\\o$\"'D3QF07$ Fd\\o$\")-AOCF07$Fi\\o$\"*u')Hx#F07$F+$\"*#)3$*G*F07$Fa]o$\"*2T*R%*F07 $Ff]o$\"+<[\"Q@\"F07$F[^o$\"+kt(Hy#F07$F2$\"+?3+DSF07$F8$\"+[Syr!*F07$ F=$\",S_qOd\"F07$FB$\",z(zE`BF07$FG$\",Ac$41KF07$FL$\",Vq8v3%F07$FQ$\" ,1X*fd\\F07$FV$\",[B%=rdF07$Fen$\",f5]=^'F07$Fjn$\",d!=ykrF07$F_o$\",T 2RIs(F07$Fdo$\",%**e>e&)F07$Fio$\",dS7D1*F07$F^p$\",]AlF07$Fhu$\",gEb23'F07$F]v$\",fS*oTcF07$Fbv$\",$>!HBD&F07$Fgv$ \",s>$*4#\\F07$F\\w$\",]50@l%F07$Faw$\",3I+hO%F07$Ffw$\",$RYq[TF07$F[x $\",J;W0#RF07$F`x$\",jMyxt$F07$Fex$\",**e)GbNF07$Fjx$\",fW**oR$F07$F_y $\",KZWcC$F07$Fdy$\",\"o$zy6$F07$Fiy$\",F\\d1*HF07$F^z$\",.f\\)oGF07$F cz$\",dzy/x#F07$Fhz$\",l'3UrEF07$F][l$\",Tc\\hd#F07$Fb[l$\",z^5#*[#F07 $Fg[l$\",)4.V5CF07$F\\\\l$\",qGL&GBF07$Fa\\l$\",:O)\\fAF07$Ff\\l$\",;p @,>#F07$F[]l$\",9z*zI@F07$F`]l$\",#QF[p?F07$Fe]l$\",CJ(*[,#F07$Fj]l$\" ,!p)43'>F07$F_^l$\",[)Gk5>F07$Fd^l$\",H%4yg=F07$Fi^l$\",;dS^\"=F07$F^_ l$\",4.52x\"F07$Fc_l$\",&p3tG " 0 "" {MPLTEXT 1 0 1 ";" }}}}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 38 "Test 3 of order 4 Runge-Kutta methods " }}{PARA 256 " " 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "dy/dx = -x*y;" "6#/*&%#dyG\"\"\" %#dxG!\"\",$*&%\"xGF&%\"yGF&F(" }{TEXT -1 11 ", " }{XPPEDIT 18 0 "y(0)=1" "6#/-%\"yG6#\"\"!\"\"\"" }{TEXT -1 1 " " }}{PARA 0 "" 0 "" {TEXT -1 10 "Solution: " }}{PARA 256 "" 0 "" {TEXT -1 1 " " } {XPPEDIT 18 0 "y = exp(-x^2/2);" "6#/%\"yG-%$expG6#,$*&%\"xG\"\"#F+!\" \"F," }{TEXT -1 2 ". " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 32 "The following code constructs a " }{TEXT 260 17 "discr ete solution" }{TEXT -1 44 " based on each of the methods and gives th e " }{TEXT 260 22 "root mean square error" }{TEXT -1 18 " of each solu tion." }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 824 "H := (x,y) -> -x*y : hh := 0.1: numsteps := 100: x0 := 0: y0 := 1:\nmatrix([[`slope field : `,H(x,y)],[`initial point: `,``(x0,y0)],[`step width: `,hh],\n[` no. of steps: `,numsteps]]);``;\nmthds := [`classical method with `* (c[2]=1/2,c[3]=1/2),`3/8 method with `*(c[2]=1/3,c[3]=2/3),`Gill's met hod with `*(c[2]=1/2,c[3]=1/2),`Dormand's method with `*(c[2]=2/5,c[3] =3/5),`method with `*(c[2]=3/7,c[3]=6/11),`method with `*(c[2]=2/5,a[3 ,2]=2/3)]: errs := []:\nDigits := 20:\nh := x -> exp(-x^2/2):\nfor ct \+ to 6 do\n Hn_RK4_||ct := RK4_||ct(H(x,y),x,y,x0,y0,hh,numsteps,false );\n sm := 0: numpts := nops(Hn_RK4_||ct):\n for ii to numpts do\n sm := sm+(Hn_RK4_||ct[ii,2]-h(Hn_RK4_||ct[ii,1]))^2;\n end do: \n errs := [op(errs),sqrt(sm/numpts)];\nend do:\nDigits := 10:\nlina lg[transpose](convert([mthds,evalf(errs)],matrix));" }}{PARA 11 "" 1 " " {XPPMATH 20 "6#K%'matrixG6#7&7$%0slope~field:~~~G,$*&%\"xG\"\"\"%\"y GF,!\"\"7$%0initial~point:~G-%!G6$\"\"!F,7$%/step~width:~~~G$F,F.7$%1n o.~of~steps:~~~G\"$+\"Q)pprint206\"" }}{PARA 11 "" 1 "" {XPPMATH 20 "6 #%!G" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#K%'matrixG6#7(7$*&%7classical~ method~with~G\"\"\"6$/&%\"cG6#\"\"##F*F0/&F.6#\"\"$F1F*$\"+x>RUh!#;7$* &%13/8~method~with~GF*6$/F-#F*F5/F3#F0F5F*$\"+%pM%H_F87$*&%4Gill's~met hod~with~GF*F+F*F67$*&%7Dormand's~method~with~GF*6$/F-#F0\"\"&/F3#F5FL F*$\"+Ag'*HbF87$*&%-method~with~GF*6$/F-#F5\"\"(/F3#\"\"'\"#6F*$\"+)GS 0c&F87$*&FSF*6$FJ/&%\"aG6$F5F0F@F*$\"+#R[iW&F8Q)pprint216\"" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 30 "The following c ode constructs " }{TEXT 260 20 "numerical procedures" }{TEXT -1 56 " f or solutions based on each of the Runge-Kutta schemes." }}{PARA 0 "" 0 "" {TEXT -1 75 "The error in the value obtained by each of the metho ds at the point where " }{XPPEDIT 18 0 "9.99;" "6#-%&FloatG6$\"$***! \"#" }{TEXT -1 16 " is also given." }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 754 "H := (x,y) -> -x*y: hh := 0.1: numsteps := 100: x0 : = 0: y0 := 1:\nmatrix([[`slope field: `,H(x,y)],[`initial point: `,` `(x0,y0)],[`step width: `,hh],\n[`no. of steps: `,numsteps]]);``; \nmthds := [`classical method with `*(c[2]=1/2,c[3]=1/2),`3/8 method w ith `*(c[2]=1/3,c[3]=2/3),`Gill's method with `*(c[2]=1/2,c[3]=1/2),`D ormand's method with `*(c[2]=2/5,c[3]=3/5),`method with `*(c[2]=3/7,c[ 3]=6/11),`method with `*(c[2]=2/5,a[3,2]=2/3)]: errs := []:\nDigits := 20:\nfor ct to 6 do\n hn_RK4_||ct := RK4_||ct(H(x,y),x,y,x0,y0,hh,n umsteps,true);\nend do:\nh := x -> exp(-x^2/2):\nxx := 9.99: hxx := ev alf(h(xx)):\nfor ct to 6 do\n errs := [op(errs),abs(hn_RK4_||ct(xx)- hxx)];\nend do:\nDigits := 10:\nlinalg[transpose](convert([mthds,evalf (errs)],matrix));" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#K%'matrixG6#7&7$% 0slope~field:~~~G,$*&%\"xG\"\"\"%\"yGF,!\"\"7$%0initial~point:~G-%!G6$ \"\"!F,7$%/step~width:~~~G$F,F.7$%1no.~of~steps:~~~G\"$+\"Q)pprint226 \"" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%!G" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#K%'matrixG6#7(7$*&%7classical~method~with~G\"\"\"6$/&% \"cG6#\"\"##F*F0/&F.6#\"\"$F1F*$\"+%)4&o!o!#K7$*&%13/8~method~with~GF* 6$/F-#F*F5/F3#F0F5F*$\"+-!*y-oF87$*&%4Gill's~method~with~GF*F+F*F67$*& %7Dormand's~method~with~GF*6$/F-#F0\"\"&/F3#F5FLF*$\"+Y$oW!oF87$*&%-me thod~with~GF*6$/F-#F5\"\"(/F3#\"\"'\"#6F*$\"+;v&))y'F87$*&FSF*6$FJ/&% \"aG6$F5F0F@F*$\"+s%zPz'F8Q)pprint236\"" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 4 "The " }{TEXT 260 22 "root mean squa re error" }{TEXT -1 110 " over the interval [0, 0.5] of each Runge-K utta method is estimated as follows using the special procedure " } {TEXT 0 5 "NCint" }{TEXT -1 97 " to perform numerical integration by \+ the 7 point Newton-Cotes method over 50 equal subintervals." }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 520 "mthds := [`classical method with ` *(c[2]=1/2,c[3]=1/2),`3/8 method with `*(c[2]=1/3,c[3]=2/3),`Gill's me thod with `*(c[2]=1/2,c[3]=1/2),`Dormand's method with `*(c[2]=2/5,c[3 ]=3/5),`method with `*(c[2]=3/7,c[3]=6/11),`method with `*(c[2]=2/5,a[ 3,2]=2/3)]: errs := []:\nDigits := 20:\nh := x -> exp(-x^2/2):\nfor ct to 6 do\n sm := NCint((h(x)-'hn_RK4_||ct'(x))^2,x=0..10,adaptive=fa lse,numpoints=7,factor=50);\n errs := [op(errs),sqrt(sm/10)];\nend d o:\nDigits := 10:\nlinalg[transpose](convert([mthds,evalf(errs)],matri x));" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#K%'matrixG6#7(7$*&%7classical~ method~with~G\"\"\"6$/&%\"cG6#\"\"##F*F0/&F.6#\"\"$F1F*$\"+xQT4e!#;7$* &%13/8~method~with~GF*6$/F-#F*F5/F3#F0F5F*$\"+EW*o$\\F87$*&%4Gill's~me thod~with~GF*F+F*F67$*&%7Dormand's~method~with~GF*6$/F-#F0\"\"&/F3#F5F LF*$\"+29B@_F87$*&%-method~with~GF*6$/F-#F5\"\"(/F3#\"\"'\"#6F*$\"+sII ^_F87$*&FSF*6$FJ/&%\"aG6$F5F0F@F*$\"+y%o;9&F8Q)pprint246\"" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 92 "The following e rror graphs are constructed using the numerical procedures for the sol utions." }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 456 "evalf[20](plot([ 'hn_RK4_1'(x)-h(x),'hn_RK4_2'(x)-h(x),'hn_RK4_3'(x)-h(x),'hn_RK4_4'(x) -h(x),\n'hn_RK4_5'(x)-h(x),'hn_RK4_6'(x)-h(x)],x=0..5,font=[HELVETICA, 9],thickness=[1$2,2,1$3],\ncolor=[red,blue,COLOR(RGB,0,.65,0),COLOR(RG B,.4,0,1),magenta,brown],\nlegend=[`classical method`,`3/8 method`,`Gi ll's method`,`Dormand's method`,`method with c[2]=3/7 and c[3]=6/11`,` method with c[2]=2/5 and a[3,2]=2/3`],title=`error curves for order 4 \+ Runge-Kutta methods`));" }}{PARA 13 "" 1 "" {GLPLOT2D 866 570 570 {PLOTDATA 2 "6,-%'CURVESG6&7et7$$\"\"!F)F(7$$\"5MLLLL3x&)*3\"!#?$!+I%G \"*f#F-7$$\"5nmmm\"H2P\"Q?F-$!,jA\"fBDF-7$$\"5MLLL$eRwX5$F-$!,)zdjC!)F 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0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 38 "Test 4 of order 4 Runge -Kutta methods " }}{PARA 0 "" 0 "" {TEXT -1 81 "F. G. Lether: Mathemat ics of Computation, Vol. 20, no. 95, (July 1966) page 381. " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "dy/dx = -32*x*y*ln(2);" "6#/*&%#dyG\"\"\"%#dxG!\"\",$**\"#KF&%\" xGF&%\"yGF&-%#lnG6#\"\"#F&F(" }{TEXT -1 6 ", " }{XPPEDIT 18 0 "y(- 1) = 1/8;" "6#/-%\"yG6#,$\"\"\"!\"\"*&F(F(\"\")F)" }{TEXT -1 1 " " }} {PARA 0 "" 0 "" {TEXT -1 10 "Solution: " }}{PARA 256 "" 0 "" {TEXT -1 2 " " }{XPPEDIT 18 0 "y = 2^(13-6*x^2);" "6#/%\"yG)\"\"#,&\"#8\"\"\"* &\"\"'F)*$%\"xGF&F)!\"\"" }{TEXT -1 2 ". " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 191 "de := diff(y(x),x)=- 32*x*y(x)*ln(2);\nic := y(-1)=1/8;\ndsolve(\{de,ic\},y(x)):\ny(x)=2^si mplify(log[2](rhs(%)));\nk := unapply(rhs(%),x):\nplot(k(x),x=-1..1,fo nt=[HELVETICA,9],labels=[`x`,`y(x)`]);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%#deG/-%%diffG6$-%\"yG6#%\"xGF,,$**\"#K\"\"\"F,F0F)F0-%#lnG6#\" \"#F0!\"\"" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%#icG/-%\"yG6#!\"\"#\" \"\"\"\")" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/-%\"yG6#%\"xG)\"\"#,&\"# 8\"\"\"*&\"#;F,)F'F)F,!\"\"" }}{PARA 13 "" 1 "" {GLPLOT2D 577 286 286 {PLOTDATA 2 "6&-%'CURVESG6$7eo7$$!\"\"\"\"!$\"3+++++++]7!#=7$$!3ommm;p 0k&*F-$\"3!Hg[[\"[$)=KF-7$$!3wKL$3$3(F-7$$!3mmmmT%p \"e()F-$\"3!=E-TWD`l\"!#<7$$!3:mmm\"4m(G$)F-$\"3M\"fONp()[t$F=7$$!3\"Q LL3i.9!zF-$\"3A!e'4A%*Rg!)F=7$$!3\"ommT!R=0vF-$\"3%z2Mbncie\"!#;7$$!3u ****\\P8#\\4(F-$\"3C>dT>$)H#3$FM7$$!3+nm;/siqmF-$\"3gp%*z`g)4*eFM7$$!3 [++](y$pZiF-$\"3%R6L-Y$zz5!#:7$$!33LLL$yaE\"eF-$\"3xvp\"p)==K>Fgn7$$!3 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***\\PQ#\\\")Fft$\"3!QM&=wHL5wFap7$$\"3ilm\"z\\1A-\"F-$\"3#*[#H(\\2i&H (Fap7$$\"3GKLLe\"*[H7F-$\"3))\\\\;@heFpFap7$$\"3ylm;HCjV9F-$\"3)e+$\\9 -Y,lFap7$$\"3I*******pvxl\"F-$\"3S%z:5s)zRgFap7$$\"3g)***\\7JFn=F-$\"3 1))p(30[[c&Fap7$$\"3#z****\\_qn2#F-$\"3ae5F\"zuv2&Fap7$$\"3=)**\\P/q%z AF-$\"3ZUhzOe!Rg%Fap7$$\"3U)***\\i&p@[#F-$\"3r&f%4uLbOTFap7$$\"3L)**\\ (=GB2FF-$\"3WV]5@%**Rj$Fap7$$\"3B)****\\2'HKHF-$\"3ul]=$GLo:$Fap7$$\"3 uJL$3UDX8$F-$\"3sKZjodBbFFap7$$\"3ElmmmZvOLF-$\"3!>\\-t_7IQ#Fap7$$\"3i ******\\2goPF-$\"3Q>G9F7l&p\"Fap7$$\"3UKL$eR<*fTF-$\"3?\"Fap 7$$\"3m******\\)Hxe%F-$\"3V-?C_;$p$zFgn7$$\"3ckm;H!o-*\\F-$\"31MiF2c]v ^Fgn7$$\"3y)***\\7k.6aF-$\"3#pB[/J``=$Fgn7$$\"3#emmmT9C#eF-$\"3&*=.D]9 +3>Fgn7$$\"33****\\i!*3`iF-$\"3%HX+j$our5Fgn7$$\"3%QLLL$*zym'F-$\"3!o4 *yfd(\\\"fFM7$$\"3wKLL3N1#4(F-$\"3!\\\\K5**)='4$FM7$$\"3Nmm;HYt7vF-$\" 3%o[)olFVm:FM7$$\"3Y*******p(G**yF-$\"3)3H-pcT.4)F=7$$\"3]mmmT6KU$)F-$ \"35omE\\#[Ck$F=7$$\"3fKLLLbdQ()F-$\"3TxwT%Qu%>ei< " 0 "" {MPLTEXT 1 0 831 "K := (x,y) -> -32*x*y(x)*ln(2): hh := 0.01: numsteps := 200: \+ x0 := -1: y0 := 1/8:\nmatrix([[`slope field: `,K(x,y)],[`initial poi nt: `,``(x0,y0)],[`step width: `,hh],\n[`no. of steps: `,numsteps] ]);``;\nmthds := [`classical method with `*(c[2]=1/2,c[3]=1/2),`3/8 me thod with `*(c[2]=1/3,c[3]=2/3),`Gill's method with `*(c[2]=1/2,c[3]=1 /2),`Dormand's method with `*(c[2]=2/5,c[3]=3/5),`method with `*(c[2]= 3/7,c[3]=6/11),`method with `*(c[2]=2/5,a[3,2]=2/3)]: errs := []:\nDig its := 20:\nfor ct to 6 do\n Kn_RK4_||ct := RK4_||ct(evalf(K(x,y)),x ,y,x0,evalf(y0),hh,numsteps,false);\n sm := 0: numpts := nops(Kn_RK4 _||ct):\n for ii to numpts do\n sm := sm+(Kn_RK4_||ct[ii,2]-k(K n_RK4_||ct[ii,1]))^2;\n end do:\n errs := [op(errs),sqrt(sm/numpts )];\nend do:\nDigits := 10:\nlinalg[transpose](convert([mthds,evalf(er rs)],matrix));" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#K%'matrixG6#7&7$%0sl ope~field:~~~G,$**\"#K\"\"\"%\"xGF,-%\"yG6#F-F,-%#lnG6#\"\"#F,!\"\"7$% 0initial~point:~G-%!G6$F5#F,\"\")7$%/step~width:~~~G$F,!\"#7$%1no.~of~ steps:~~~G\"$+#Q)pprint256\"" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%!G" } }{PARA 11 "" 1 "" {XPPMATH 20 "6#K%'matrixG6#7(7$*&%7classical~method~ with~G\"\"\"6$/&%\"cG6#\"\"##F*F0/&F.6#\"\"$F1F*$\"+_J3fA!#57$*&%13/8~ method~with~GF*6$/F-#F*F5/F3#F0F5F*$\"+n$onB#F87$*&%4Gill's~method~wit h~GF*F+F*F67$*&%7Dormand's~method~with~GF*6$/F-#F0\"\"&/F3#F5FLF*$\"+. piXAF87$*&%-method~with~GF*6$/F-#F5\"\"(/F3#\"\"'\"#6F*$\"+t'f " 0 "" {MPLTEXT 1 0 762 "K := (x,y) -> -32*x*y(x)*ln(2): hh := 0.01: numsteps := 200: x0 := -1 : y0 := 1/8:\nmatrix([[`slope field: `,K(x,y)],[`initial point: `,`` (x0,y0)],[`step width: `,hh],\n[`no. of steps: `,numsteps]]);``;\n mthds := [`classical method with `*(c[2]=1/2,c[3]=1/2),`3/8 method wit h `*(c[2]=1/3,c[3]=2/3),`Gill's method with `*(c[2]=1/2,c[3]=1/2),`Dor mand's method with `*(c[2]=2/5,c[3]=3/5),`method with `*(c[2]=3/7,c[3] =6/11),`method with `*(c[2]=2/5,a[3,2]=2/3)]: errs := []:\nDigits := 2 0:\nfor ct to 6 do\n kn_RK4_||ct := RK4_||ct(evalf(K(x,y)),x,y,x0,ev alf(y0),hh,numsteps,true);\nend do:\nxx := 0.995: kxx := evalf(k(xx)): \nfor ct to 6 do\n errs := [op(errs),abs(kn_RK4_||ct(xx)-kxx)];\nend do:\nDigits := 10:\nlinalg[transpose](convert([mthds,evalf(errs)],mat rix));" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#K%'matrixG6#7&7$%0slope~fiel d:~~~G,$**\"#K\"\"\"%\"xGF,-%\"yG6#F-F,-%#lnG6#\"\"#F,!\"\"7$%0initial ~point:~G-%!G6$F5#F,\"\")7$%/step~width:~~~G$F,!\"#7$%1no.~of~steps:~~ ~G\"$+#Q)pprint276\"" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%!G" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#K%'matrixG6#7(7$*&%7classical~method~with~G \"\"\"6$/&%\"cG6#\"\"##F*F0/&F.6#\"\"$F1F*$\"++L$)yD!#:7$*&%13/8~metho d~with~GF*6$/F-#F*F5/F3#F0F5F*$\"+F:MZDF87$*&%4Gill's~method~with~GF*F +F*F67$*&%7Dormand's~method~with~GF*6$/F-#F0\"\"&/F3#F5FLF*$\"+e=cgDF8 7$*&%-method~with~GF*6$/F-#F5\"\"(/F3#\"\"'\"#6F*$\"+BVHSDF87$*&FSF*6$ FJ/&%\"aG6$F5F0F@F*$\"+P " 0 "" {MPLTEXT 1 0 497 "mthds := [`classical method with `*(c[2]=1/2,c[3] =1/2),`3/8 method with `*(c[2]=1/3,c[3]=2/3),`Gill's method with `*(c[ 2]=1/2,c[3]=1/2),`Dormand's method with `*(c[2]=2/5,c[3]=3/5),`method \+ with `*(c[2]=3/7,c[3]=6/11),`method with `*(c[2]=2/5,a[3,2]=2/3)]: err s := []:\nDigits := 20:\nfor ct to 6 do\n sm := NCint((k(x)-'kn_RK4_ ||ct'(x))^2,x=-1..1,adaptive=false,numpoints=7,factor=100);\n errs : = [op(errs),sqrt(sm/2)];\nend do:\nDigits := 10:\nlinalg[transpose](co nvert([mthds,evalf(errs)],matrix));" }}{PARA 11 "" 1 "" {XPPMATH 20 "6 #K%'matrixG6#7(7$*&%7classical~method~with~G\"\"\"6$/&%\"cG6#\"\"##F*F 0/&F.6#\"\"$F1F*$\"+&yBZE#!#57$*&%13/8~method~with~GF*6$/F-#F*F5/F3#F0 F5F*$\"+!)GNUAF87$*&%4Gill's~method~with~GF*F+F*F67$*&%7Dormand's~meth od~with~GF*6$/F-#F0\"\"&/F3#F5FLF*$\"+5PB^AF87$*&%-method~with~GF*6$/F -#F5\"\"(/F3#\"\"'\"#6F*$\"+r>LPAF87$*&FSF*6$FJ/&%\"aG6$F5F0F@F*$\"+u5 gRAF8Q)pprint296\"" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 " " {TEXT -1 92 "The following error graphs are constructed using the nu merical procedures for the solutions." }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 457 "evalf[20](plot([k(x)-'kn_RK4_1'(x),k(x)-'kn_RK4_2'(x ),k(x)-'kn_RK4_3'(x),k(x)-'kn_RK4_4'(x),\nk(x)-'kn_RK4_5'(x),k(x)-'kn_ RK4_6'(x)],x=-1..1,font=[HELVETICA,9],thickness=[1$2,2,1$3],\ncolor=[r ed,blue,COLOR(RGB,0,.65,0),COLOR(RGB,.4,0,1),magenta,brown],\nlegend=[ `classical method`,`3/8 method`,`Gill's method`,`Dormand's method`,`me thod with c[2]=3/7 and c[3]=6/11`,`method with c[2]=2/5 and a[3,2]=2/3 `],title=`error curves for order 4 Runge-Kutta methods`));" }}{PARA 13 "" 1 "" {GLPLOT2D 804 583 583 {PLOTDATA 2 "6,-%'CURVESG6&7eo7$$!\" \"\"\"!$F*F*7$$!5nmmmm;p0k&*!#?$\"0+pXl:0O%F/7$$!5LLLL$3u\"F/7$$!5nmmmmT%p\"e()F/$\"0S&pClIgb!#>7$$!5nmmmm\"4m(G$)F/$\"1%R _K7DP`\"F<7$$!5LLLL$3i.9!zF/$\"1!H'Ry[9RQF<7$$!5mmmm;/R=0vF/$\"03MF1dC ?)!#=7$$!5++++]P8#\\4(F/$\"13\\'3xE3q\"FL7$$!5mmmm;/siqmF/$\"1y[@d$=lR $FL7$$!5++++](y$pZiF/$\"0_Pgc(3Fk!#<7$$!5LLLLL$yaE\"eF/$\"18)\\:'Qvy6F 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$\"18Ip&*=6a^F-7$Ffo$\"1*GSE_J+;&F-7$F[p$\"1BD\"))4G_;&F-7$F`p$\"1kn%H T@&p^F-7$Fep$\"1u1u,85u^F-7$Fjp$\"1=JX]u_x^F-7$F_q$\"1+\"Hf*o)4=&F-7$F dq$\"1c&[_t(>i,**=&F-7$Fbs$\"1Wx>,Vj*=&F -7$Fgs$\"19B&))=Y*)=&F-7$F\\t$\"16mDsrs(=&F-7$Fat$\"1dSGj)\\f=&F-7$Fft $\"1IfQ()Hq$=&F-7$F[u$\"1tr)\\SZ5=&F-7$F`u$\"1B(*=uxax^F-7$Feu$\"1?Th6 .\"R<&F-7$Fju$\"1Bv.&*\\^p^F-7$F_v$\"1,iV6a2l^F-7$Fdv$\"1_0QT*=(f^F-7$ Fiv$\"1$)z,Wn?a^F-7$F^w$\"1-*[;\")ez9&F-7$Fcw$\"1\"Q#3wbPT^F-7$Fhw$\"1 3@$ekuR8&F-7$F]x$\"16*[4[^j7&F-7$Fbx$\"1<2AaL1=^F-7$Fgx$\"14)[&evN4^F- 7$F\\y$\"1w7J>(Q45&F-7$Fay$\"1wVW#Ry24&F-7$Ffy$\"17=jNUB\"3&F-7$F[z$\" 1Ct!e?$fq]F-7$F`z$\"1V2c)zl*f]F-7$Fez$\"1T#[wcv![]F--Fhel6&F\\[l$\"\"% !\"\"F`[l$Fe[lFa[lFb[l-Fg[l6#%1Dormand's~methodG-F$6&7S7$F($\"1Zp,io7< ]F-7$F/$\"1/MOQ[+H]F-7$F5$\"1ccJ>/$*Q]F-7$F:$\"1:,]([L'\\]F-7$F?$\"1;< v#H5*f]F-7$FD$\"1lV2!=R'p]F-7$FI$\"1E)[0yE#y]F-7$FN$\"1IDeSrm'3&F-7$FS $\"1Jplb#4\\4&F-7$FX$\"1kAx%3FE5&F-7$Fgn$\"1O/G>![+6&F-7$F\\o$\"1#eK<% \\:;^F-7$Fao$\"12KZ=#RD7&F-7$Ffo$\"17'eSxB%G^F-7$F[p$\"1kM*)o>fL^F-7$F `p$\"1GC\\F6'y8&F-7$Fep$\"1'Qs=*HTU^F-7$Fjp$\"1H0kX$=e9&F-7$F_q$\"1j^0 -vD\\^F-7$Fdq$\"1yPue5$=:&F-7$Fiq$\"1yHa1/:a^F-7$F^r$\"1TM2bo'e:&F-7$F cr$\"1Bi#o&Q9d^F-7$Fhr$\"1(o9\"=O&y:&F-7$F]s$\"1M!4Zh@\"e^F-7$Fbs$\"1T NTYf&y:&F-7$Fgs$\"1SLoQ?a^F- 7$Fft$\"1wE?W*e>:&F-7$F[u$\"15\"H7i>$\\^F-7$F`u$\"1^D$oLTe9&F-7$Feu$\" 1\"RoAlDA9&F-7$Fju$\"16ABK_&y8&F-7$F_v$\"1YR?%RSM8&F-7$Fdv$\"1Ge)or;\" G^F-7$Fiv$\"1[]i&3QE7&F-7$F^w$\"1ugHEtU;^F-7$Fcw$\"1.@Pt2))4^F-7$Fhw$ \"1;`Mx?_-^F-7$F]x$\"1HZ*[dX\\4&F-7$Fbx$\"1quKvxq'3&F-7$Fgx$\"1[k]2K0y ]F-7$F\\y$\"1*)*p#\\,op]F-7$Fay$\"11'o4^z&f]F-7$Ffy$\"1b#46w$4]]F-7$F[ z$\"1Kp'f<<&R]F-7$F`z$\"1]56H=&*G]F-7$Fez$\"1i2J9Y7<]F--Fjz6&F\\[lF][l F`[lF][lFb[l-Fg[l6#%Cmethod~with~c[2]=3/7~and~c[3]=6/11G-F$6&7S7$F($\" 1#Q5v4,?-&F-7$F/$\"1#*y%yg!*Q.&F-7$F5$\"1,R05d#Q/&F-7$F:$\"15A*fGQX0&F -7$F?$\"1HSJ<<#[1&F-7$FD$\"1^qT2Rbu]F-7$FI$\"1@$*[C)\\J3&F-7$FN$\"1MO' [?)f\"4&F-7$FS$\"1J>R^t%)*4&F-7$FX$\"1*R.dWpv5&F-7$Fgn$\"1yWulO*\\6&F- 7$F\\o$\"17*)o&[167&F-7$Fao$\"1`GUYn\\F^F-7$Ffo$\"1me]3gQL^F-7$F[p$\"1 P&e$3jbQ^F-7$F`p$\"1-U#eyFG9&F-7$Fep$\"1GI5ISQZ^F-7$Fjp$\"1XrE>Dz]^F-7 $F_q$\"1[#*HSUBa^F-7$Fdq$\"1l^>]%3o:&F-7$Fiq$\"1jEp4$H\"f^F-7$F^r$\"1a 7&GSZ3;&F-7$Fcr$\"1aqwQb7i^F-7$Fhr$\"1BK<)pNG;&F-7$F]s$\"1!)R58L5j^F-7 $Fbs$\"1hN;pt$G;&F-7$Fgs$\"1Rr;2G:i^F-7$F\\t$\"1aUka+%4;&F-7$Fat$\"1Vj ic<%Ha^F-7$F`u$\"1Td-=E\"3:&F-7 $Feu$\"1GeqBQ>Z^F-7$Fju$\"1 \+ " 0 "" {MPLTEXT 1 0 1 ";" }}}}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 38 "Tes t 5 of order 4 Runge-Kutta methods " }}{PARA 256 "" 0 "" {TEXT -1 1 " \+ " }{XPPEDIT 18 0 "dy/dx=16/((16*x+1)*y)" "6#/*&%#dyG\"\"\"%#dxG!\"\"*& \"#;F&*&,&*&F*F&%\"xGF&F&F&F&F&%\"yGF&F(" }{TEXT -1 10 ", " } {XPPEDIT 18 0 "y(0)=1" "6#/-%\"yG6#\"\"!\"\"\"" }{TEXT -1 1 " " }} {PARA 0 "" 0 "" {TEXT -1 10 "Solution: " }}{PARA 256 "" 0 "" {TEXT -1 2 " " }{XPPEDIT 18 0 "y=sqrt (2*ln(16*x+1)+1)" "6#/%\"yG-%%sqrtG6#,&* &\"\"#\"\"\"-%#lnG6#,&*&\"#;F+%\"xGF+F+F+F+F+F+F+F+" }{TEXT -1 2 ". " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 165 "de := diff(y(x),x)=16/((16*x+1)*y(x));\nic := y(0)=1;\ndsolve( \{de,ic\},y(x));\ns := unapply(rhs(%),x):\nplot(s(x),x=0..0.5,0..2.6,f ont=[HELVETICA,9],labels=[`x`,`y(x)`]);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%#deG/-%%diffG6$-%\"yG6#%\"xGF,,$*(\"#;\"\"\",&*&F/F0F,F0F0F0F0 !\"\"F)F3F0" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%#icG/-%\"yG6#\"\"!\" \"\"" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/-%\"yG6#%\"xG*$,&*&\"\"#\"\" \"-%#lnG6#,&*&\"#;F,F'F,F,F,F,F,F,F,F,#F,F+" }}{PARA 13 "" 1 "" {GLPLOT2D 503 318 318 {PLOTDATA 2 "6&-%'CURVESG6$7U7$$\"\"!F)$\"\"\"F) 7$$\"3WmmmT&)G\\a!#?$\"3EP,(Qyl.3\"!#<7$$\"3ILLL3x&)*3\"!#>$\"3?25A!pa &\\6F27$$\"3-+]i!R(*Rc\"F6$\"3oz*p77wF?\"F27$$\"3umm\"H2P\"Q?F6$\"3]_v ibZz]7F27$$\"3MLL$eRwX5$F6$\"3!Qb+fY'3W8F27$$\"3CLL$3x%3yTF6$\"31#\\\\ E7=EU\"F27$$\"3=mm\"z%4\\Y_F6$\"3s(e$4OUg*[\"F27$$\"3)HL$eR-/PiF6$\"3. fPtw=4W:F27$$\"3A***\\il'pisF6$\"3/07@a`R%f\"F27$$\"3`KLe*)>VB$)F6$\"3 K!\\`od36k\"F27$$\"3!))**\\7`l2Q*F6$\"3#HUv\"fmC$o\"F27$$\"3smm;/j$o/ \"!#=$\"3:'H!f>cuAjU6Fco$\"3K$o8QC!za=F27$$\" 3)*****\\P[6j9Fco$\"39iuo+OIZ=F27$$\"3KL$e*[z(yb\"Fco$\"3Q:]fA\\>F27$ $\"3))**\\iSj0x=Fco$\"3-5Hbh&QF%>F27$$\"3Wmmm\"pW`(>Fco$\"3So#znsrC'>F 27$$\"35+]i!f#=$3#Fco$\"3w)>Y)R!pI)>F27$$\"3/+](=xpe=#Fco$\"3?*eB@.[<+ #F27$$\"3smm\"H28IH#Fco$\"3/Fyh^(\\.-#F27$$\"3km;zpSS\"R#Fco$\"3)4US+% ypO?F27$$\"3GLL3_?`(\\#Fco$\"3#4Cj+a0O0#F27$$\"3#HLe*)>pxg#Fco$\"3ab\\ mG7Vq?F27$$\"3u**\\Pf4t.FFco$\"3Cx7m@=^%3#F27$$\"32LLe*Gst!GFco$\"3Q

>IFco $\"3&ocGC'[]F@F27$$\"3h**\\i!RU07$Fco$\"3HCH$Q\")f.9#F27$$\"3b***\\(=S 2LKFco$\"3C`wrWc9a@F27$$\"3Kmmm\"p)=MLFco$\"3;=S,IA7m@F27$$\"3!*****\\ (=]@W$Fco$\"3w4%eC\"p]y@F27$$\"35L$e*[$z*RNFco$\"3UyOr,.R*=#F27$$\"3#* ****\\iC$pk$Fco$\"3wIdFs1%4?#F27$$\"39m;H2qcZPFco$\"3Qbx\"QY%\\6AF27$$ \"3q**\\7.\"fF&QFco$\"3f+!e(oz@AAF27$$\"3Ymm;/OgbRFco$\"36qG(yA8CB#F27 $$\"3y**\\ilAFjSFco$\"3v.zLgjzUAF27$$\"3YLLL$)*pp;%Fco$\"3IImU*yHDD#F2 7$$\"3?LL3xe,tUFco$\"3I%R!fhiAiAF27$$\"3em;HdO=yVFco$\"3?ogo1xfrAF27$$ \"3))*****\\#>#[Z%Fco$\"3EO%fx0/+G#F27$$\"3immT&G!e&e%Fco$\"3)zsS%e\"3 %*G#F27$$\"3;LLL$)Qk%o%Fco$\"35e8d5)>wH#F27$$\"37+]iSjE!z%Fco$\"3e%4h. zwhI#F27$$\"35+]P40O\"*[Fco$\"3Nwd2,K=9BF27$$\"3++++++++]Fco$\"3m'>()) [`fABF2-%'COLOURG6&%$RGBG$\"#5!\"\"F(F(-%%FONTG6$%*HELVETICAG\"\"*-%+A XESLABELSG6$%\"xG%%y(x)G-%%VIEWG6$;F($\"\"&Fj[l;F($\"#EFj[l" 1 2 0 1 10 0 2 9 1 4 2 1.000000 45.000000 45.000000 0 0 "Curve 1" }}}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 32 "The following cod e constructs a " }{TEXT 260 17 "discrete solution" }{TEXT -1 44 " base d on each of the methods and gives the " }{TEXT 260 22 "root mean squa re error" }{TEXT -1 18 " of each solution." }}{EXCHG {PARA 0 "> " 0 " " {MPLTEXT 1 0 814 "S := (x,y) -> 16/((16*x+1)*y): hh := 0.005: numste ps := 100: x0 := 0: y0 := 1:\nmatrix([[`slope field: `,S(x,y)],[`ini tial point: `,``(x0,y0)],[`step width: `,hh],\n[`no. of steps: `,n umsteps]]);``;\nmthds := [`classical method with `*(c[2]=1/2,c[3]=1/2) ,`3/8 method with `*(c[2]=1/3,c[3]=2/3),`Gill's method with `*(c[2]=1/ 2,c[3]=1/2),`Dormand's method with `*(c[2]=2/5,c[3]=3/5),`method with \+ `*(c[2]=3/7,c[3]=6/11),`method with `*(c[2]=2/5,a[3,2]=2/3)]: errs := \+ []:\nDigits := 20:\nfor ct to 6 do\n Sn_RK4_||ct := RK4_||ct(S(x,y), x,y,x0,y0,hh,numsteps,false);\n sm := 0: numpts := nops(Sn_RK4_||ct) :\n for ii to numpts do\n sm := sm+(Sn_RK4_||ct[ii,2]-s(Sn_RK4_ ||ct[ii,1]))^2;\n end do:\n errs := [op(errs),sqrt(sm/numpts)];\ne nd do:\nDigits := 10:\nlinalg[transpose](convert([mthds,evalf(errs)],m atrix));" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#K%'matrixG6#7&7$%0slope~fi eld:~~~G,$*(\"#;\"\"\",&*&F+F,%\"xGF,F,F,F,!\"\"%\"yGF0F,7$%0initial~p oint:~G-%!G6$\"\"!F,7$%/step~width:~~~G$\"\"&!\"$7$%1no.~of~steps:~~~G \"$+\"Q)pprint306\"" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%!G" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#K%'matrixG6#7(7$*&%7classical~method~with~G \"\"\"6$/&%\"cG6#\"\"##F*F0/&F.6#\"\"$F1F*$\"+KGT\\E!#;7$*&%13/8~metho d~with~GF*6$/F-#F*F5/F3#F0F5F*$\"+\"*p,i([*p\"F87$*&%7Dormand's~method~with~GF*6$/F-#F0\"\"&/F3#F5F NF*$\"+r#*=.?F87$*&%-method~with~GF*6$/F-#F5\"\"(/F3#\"\"'\"#6F*$\"+w9 Q=?F87$*&FUF*6$FL/&%\"aG6$F5F0F@F*$\"+2W%e&>F8Q)pprint316\"" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 30 "The following c ode constructs " }{TEXT 260 20 "numerical procedures" }{TEXT -1 56 " f or solutions based on each of the Runge-Kutta schemes." }}{PARA 0 "" 0 "" {TEXT -1 75 "The error in the value obtained by each of the metho ds at the point where " }{XPPEDIT 18 0 "x = 0;" "6#/%\"xG\"\"!" } {TEXT -1 21 ".4995 is also given." }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 746 "S := (x,y) -> 16/((16*x+1)*y): hh := 0.005: numsteps := 100: x0 := 0: y0 := 1:\nmatrix([[`slope field: `,S(x,y)],[`initi al point: `,``(x0,y0)],[`step width: `,hh],\n[`no. of steps: `,num steps]]);``;\nmthds := [`classical method with `*(c[2]=1/2,c[3]=1/2),` 3/8 method with `*(c[2]=1/3,c[3]=2/3),`Gill's method with `*(c[2]=1/2, c[3]=1/2),`Dormand's method with `*(c[2]=2/5,c[3]=3/5),`method with `* (c[2]=3/7,c[3]=6/11),`method with `*(c[2]=2/5,a[3,2]=2/3)]: errs := [] :\nDigits := 20:\nfor ct to 6 do\n sn_RK4_||ct := RK4_||ct(S(x,y),x, y,x0,y0,hh,numsteps,true);\nend do:\nxx := 0.4995: sxx := evalf(s(xx)) :\nfor ct to 6 do\n errs := [op(errs),abs(sn_RK4_||ct(xx)-sxx)];\nen d do:\nDigits := 10:\nlinalg[transpose](convert([mthds,evalf(errs)],ma trix));" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#K%'matrixG6#7&7$%0slope~fie ld:~~~G,$*(\"#;\"\"\",&*&F+F,%\"xGF,F,F,F,!\"\"%\"yGF0F,7$%0initial~po int:~G-%!G6$\"\"!F,7$%/step~width:~~~G$\"\"&!\"$7$%1no.~of~steps:~~~G \"$+\"Q)pprint326\"" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%!G" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#K%'matrixG6#7(7$*&%7classical~method~with~G \"\"\"6$/&%\"cG6#\"\"##F*F0/&F.6#\"\"$F1F*$\"+%oNkA#!#;7$*&%13/8~metho d~with~GF*6$/F-#F*F5/F3#F0F5F*$\"+kHhw9F87$*&%4Gill's~method~with~GF*F +F*$\"+2CjL9F87$*&%7Dormand's~method~with~GF*6$/F-#F0\"\"&/F3#F5FNF*$ \"+>Ej$o\"F87$*&%-method~with~GF*6$/F-#F5\"\"(/F3#\"\"'\"#6F*$\"+5Oc*p \"F87$*&FUF*6$FL/&%\"aG6$F5F0F@F*$\"+(zj`k\"F8Q)pprint336\"" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 4 "The " }{TEXT 260 22 "root mean square error" }{TEXT -1 110 " over the interval [0, 0.5] of each Runge-Kutta method is estimated as follows using the sp ecial procedure " }{TEXT 0 5 "NCint" }{TEXT -1 97 " to perform numer ical integration by the 7 point Newton-Cotes method over 50 equal subi ntervals." }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 499 "mthds := [`cla ssical method with `*(c[2]=1/2,c[3]=1/2),`3/8 method with `*(c[2]=1/3, c[3]=2/3),`Gill's method with `*(c[2]=1/2,c[3]=1/2),`Dormand's method \+ with `*(c[2]=2/5,c[3]=3/5),`method with `*(c[2]=3/7,c[3]=6/11),`method with `*(c[2]=2/5,a[3,2]=2/3)]: errs := []:\nDigits := 20:\nfor ct to \+ 6 do\n sm := NCint((s(x)-'sn_RK4_||ct'(x))^2,x=0..0.5,adaptive=false ,numpoints=7,factor=50);\n errs := [op(errs),sqrt(sm/0.5)];\nend do: \nDigits := 10:\nlinalg[transpose](convert([mthds,evalf(errs)],matrix) );" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#K%'matrixG6#7(7$*&%7classical~me thod~with~G\"\"\"6$/&%\"cG6#\"\"##F*F0/&F.6#\"\"$F1F*$\"+\\T>\\E!#;7$* &%13/8~method~with~GF*6$/F-#F*F5/F3#F0F5F*$\"+ZV%=w\"F87$*&%4Gill's~me thod~with~GF*F+F*$\"+jMP*p\"F87$*&%7Dormand's~method~with~GF*6$/F-#F0 \"\"&/F3#F5FNF*$\"+5.-.?F87$*&%-method~with~GF*6$/F-#F5\"\"(/F3#\"\"' \"#6F*$\"+()*H#=?F87$*&FUF*6$FL/&%\"aG6$F5F0F@F*$\"+]%)ob>F8Q)pprint34 6\"" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 92 "T he following error graphs are constructed using the numerical procedur es for the solutions." }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 436 "ev alf[20](plot(['sn_RK4_1'(x)-s(x),'sn_RK4_2'(x)-s(x),'sn_RK4_3'(x)-s(x) ,'sn_RK4_4'(x)-s(x),\n'sn_RK4_5'(x)-s(x),'sn_RK4_6'(x)-s(x)],x=0..0.5, font=[HELVETICA,9],\ncolor=[red,blue,COLOR(RGB,0,.65,0),COLOR(RGB,.4,0 ,1),magenta,brown],\nlegend=[`classical method`,`3/8 method`,`Gill's m ethod`,`Dormand's method`,`method with c[2]=3/7 and c[3]=6/11`,`method with c[2]=2/5 and a[3,2]=2/3`],title=`error curves for order 4 Runge- Kutta methods`));" }}{PARA 13 "" 1 "" {GLPLOT2D 1000 653 653 {PLOTDATA 2 "6,-%'CURVESG6%7]q7$$\"\"!F)F(7$$\"5SLLL3x1h6o!#B$\"**>SW8 !#>7$$\"5ommmTN@Ki8!#A$\"+vNP1TF07$$\"5-++]7.K[V?F4$\",(p[[yHF07$$\"5N LLL$3FWYs#F4$\"-\\f1n*>\"F07$$\"5-++vo/[AlIF4$\"-INM19@F07$$\"5omm;aQ` !eS$F4$\"-Ffsh,NF07$$\"5NLLeRseQYPF4$\"-`#>1l^&F07$$\"5-+++D1k'p3%F4$ 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/Rz()F-7$$\"3w3F>RL3GTFI$\"3JeP:9JjA()F-7$$\"3t]i!RbX59%FI$\"3mH1#H$\\ k'o)F-7$$\"3#=z>'ox+aTFI$\"3oLr_-o*=n)F-7$$\"3yLLL$)*pp;%FI$\"3A7j1wip y')F-7$$\"3!Q3_+sD-=%FI$\"32pcM,k23()F-7$$\"3#Q$3xc9[$>%FI$\"3Gri,**=4 g()F-7$$\"3'Qe*[$>Pn?%FI$\"3se,X+?^M))F-7$$\"3)QL3-$H**>UFI$\"3Z**e,OD #4$*)F-7$$\"3#R$ek.W]YUFI$\"3i#fiyx0s=*F-7$$\"3)RL$3xe,tUFI$\"3[2R[)*e VA&*F-7$$\"3Cn;HdO=yVFI$\"3#)>Y<=$f\\9\"FI7$$\"3MMe9\"z-lU%FI$\"3)4DVD mlMD\"FI7$$\"3a+++D>#[Z%FI$\"3qZKS'GmoO\"FI7$$\"3TM$3_5,-`%FI$\"3CFB-G n\\(\\\"FI7$$\"3SnmT&G!e&e%FI$\"3t\\(p9r/Xi\"FI7$$\"3m+]P%37^j%FI$\"3_ eaMDR_K " 0 "" {MPLTEXT 1 0 826 "U := (x,y) -> (1+2*(x+1)*sin(3*x))*exp(-y): hh := 0. 01: numsteps := 500: x0 := 0: y0 := 0:\nmatrix([[`slope field: `,U(x ,y)],[`initial point: `,``(x0,y0)],[`step width: `,hh],\n[`no. of st eps: `,numsteps]]);``;\nmthds := [`classical method with `*(c[2]=1/2 ,c[3]=1/2),`3/8 method with `*(c[2]=1/3,c[3]=2/3),`Gill's method with \+ `*(c[2]=1/2,c[3]=1/2),`Dormand's method with `*(c[2]=2/5,c[3]=3/5),`me thod with `*(c[2]=3/7,c[3]=6/11),`method with `*(c[2]=2/5,a[3,2]=2/3)] : errs := []:\nDigits := 25:\nfor ct to 6 do\n Un_RK4_||ct := RK4_|| ct(U(x,y),x,y,x0,y0,hh,numsteps,false);\n sm := 0: numpts := nops(Un _RK4_||ct):\n for ii to numpts do\n sm := sm+(Un_RK4_||ct[ii,2] -u(Un_RK4_||ct[ii,1]))^2;\n end do:\n errs := [op(errs),sqrt(sm/nu mpts)];\nend do:\nDigits := 10:\nlinalg[transpose](convert([mthds,eval f(errs)],matrix));" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#K%'matrixG6#7&7$ %0slope~field:~~~G*&,&\"\"\"F+*(\"\"#F+,&%\"xGF+F+F+F+-%$sinG6#,$*&\" \"$F+F/F+F+F+F+F+-%$expG6#,$%\"yG!\"\"F+7$%0initial~point:~G-%!G6$\"\" !FA7$%/step~width:~~~G$F+!\"#7$%1no.~of~steps:~~~G\"$+&Q)pprint356\"" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%!G" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#K%'matrixG6#7(7$*&%7classical~method~with~G\"\"\"6$/&%\"cG6#\"\"##F *F0/&F.6#\"\"$F1F*$\"+X2T\"e(!#>7$*&%13/8~method~with~GF*6$/F-#F*F5/F3 #F0F5F*$\"+h^')>8!#=7$*&%4Gill's~method~with~GF*F+F*$\"+3nC?XF87$*&%7D ormand's~method~with~GF*6$/F-#F0\"\"&/F3#F5FOF*$\"+Z:Mv5FC7$*&%-method ~with~GF*6$/F-#F5\"\"(/F3#\"\"'\"#6F*$\"+Sbl6\")F87$*&FVF*6$FM/&%\"aG6 $F5F0F@F*$\"+q'3'p(*F8Q)pprint366\"" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 30 "The following code constructs " }{TEXT 260 20 "numerical procedures" }{TEXT -1 56 " for solutions based on ea ch of the Runge-Kutta schemes." }}{PARA 0 "" 0 "" {TEXT -1 75 "The err or in the value obtained by each of the methods at the point where " }{XPPEDIT 18 0 "x = 4.999;" "6#/%\"xG-%&FloatG6$\"%**\\!\"$" }{TEXT -1 16 " is also given." }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 757 " U := (x,y) -> (1+2*(x+1)*sin(3*x))*exp(-y): hh := 0.01: numsteps := 50 0: x0 := 0: y0 := 0:\nmatrix([[`slope field: `,U(x,y)],[`initial poi nt: `,``(x0,y0)],[`step width: `,hh],\n[`no. of steps: `,numsteps] ]);``;\nmthds := [`classical method with `*(c[2]=1/2,c[3]=1/2),`3/8 me thod with `*(c[2]=1/3,c[3]=2/3),`Gill's method with `*(c[2]=1/2,c[3]=1 /2),`Dormand's method with `*(c[2]=2/5,c[3]=3/5),`method with `*(c[2]= 3/7,c[3]=6/11),`method with `*(c[2]=2/5,a[3,2]=2/3)]: errs := []:\nDig its := 25:\nfor ct to 6 do\n un_RK4_||ct := RK4_||ct(U(x,y),x,y,x0,y 0,hh,numsteps,true);\nend do:\nxx := 4.999: uxx := evalf(u(xx)):\nfor \+ ct to 6 do\n errs := [op(errs),abs(un_RK4_||ct(xx)-uxx)];\nend do:\n Digits := 10:\nlinalg[transpose](convert([mthds,evalf(errs)],matrix)); " }}{PARA 11 "" 1 "" {XPPMATH 20 "6#K%'matrixG6#7&7$%0slope~field:~~~G *&,&\"\"\"F+*(\"\"#F+,&%\"xGF+F+F+F+-%$sinG6#,$*&\"\"$F+F/F+F+F+F+F+-% $expG6#,$%\"yG!\"\"F+7$%0initial~point:~G-%!G6$\"\"!FA7$%/step~width:~ ~~G$F+!\"#7$%1no.~of~steps:~~~G\"$+&Q)pprint376\"" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%!G" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#K%'matrixG6#7(7$ *&%7classical~method~with~G\"\"\"6$/&%\"cG6#\"\"##F*F0/&F.6#\"\"$F1F*$ \"+=R[25!#=7$*&%13/8~method~with~GF*6$/F-#F*F5/F3#F0F5F*$\"+d`bE=F87$* &%4Gill's~method~with~GF*F+F*$\"+'>NOL&!#>7$*&%7Dormand's~method~with~ GF*6$/F-#F0\"\"&/F3#F5FOF*$\"+Y`$p[\"F87$*&%-method~with~GF*6$/F-#F5\" \"(/F3#\"\"'\"#6F*$\"+M))\\Q5F87$*&FVF*6$FM/&%\"aG6$F5F0F@F*$\"+5!o\"4 8F8Q)pprint386\"" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 4 "The " }{TEXT 260 22 "root mean square error" }{TEXT -1 20 " over the interval " }{XPPEDIT 18 0 "[0, 5];" "6#7$\"\"!\"\"&" } {TEXT -1 82 " of each Runge-Kutta method is estimated as follows usin g the special procedure " }{TEXT 0 5 "NCint" }{TEXT -1 98 " to perfo rm numerical integration by the 7 point Newton-Cotes method over 200 e qual subintervals." }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 496 "mthds := [`classical method with `*(c[2]=1/2,c[3]=1/2),`3/8 method with `*( c[2]=1/3,c[3]=2/3),`Gill's method with `*(c[2]=1/2,c[3]=1/2),`Dormand' s method with `*(c[2]=2/5,c[3]=3/5),`method with `*(c[2]=3/7,c[3]=6/11 ),`method with `*(c[2]=2/5,a[3,2]=2/3)]: errs := []:\nDigits := 20:\nf or ct to 6 do\n sm := NCint((u(x)-'un_RK4_||ct'(x))^2,x=0..5,adaptiv e=false,numpoints=7,factor=200);\n errs := [op(errs),sqrt(sm/5)];\ne nd do:\nDigits := 10:\nlinalg[transpose](convert([mthds,evalf(errs)],m atrix));" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#K%'matrixG6#7(7$*&%7classi cal~method~with~G\"\"\"6$/&%\"cG6#\"\"##F*F0/&F.6#\"\"$F1F*$\"+:AS`v!# >7$*&%13/8~method~with~GF*6$/F-#F*F5/F3#F0F5F*$\"+@M![J\"!#=7$*&%4Gill 's~method~with~GF*F+F*$\"+:U.0XF87$*&%7Dormand's~method~with~GF*6$/F-# F0\"\"&/F3#F5FOF*$\"+.[Pr5FC7$*&%-method~with~GF*6$/F-#F5\"\"(/F3#\"\" '\"#6F*$\"+:=A&3)F87$*&FVF*6$FM/&%\"aG6$F5F0F@F*$\"+l%Qat*F8Q)pprint39 6\"" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 92 "T he following error graphs are constructed using the numerical procedur es for the solutions." }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 434 "ev alf[20](plot(['un_RK4_1'(x)-u(x),'un_RK4_2'(x)-u(x),'un_RK4_3'(x)-u(x) ,'un_RK4_4'(x)-u(x),\n'un_RK4_5'(x)-u(x),'un_RK4_6'(x)-u(x)],x=0..5,fo nt=[HELVETICA,9],\ncolor=[red,blue,COLOR(RGB,0,.65,0),COLOR(RGB,.4,0,1 ),magenta,brown],\nlegend=[`classical method`,`3/8 method`,`Gill's met hod`,`Dormand's method`,`method with c[2]=3/7 and c[3]=6/11`,`method w ith c[2]=2/5 and a[3,2]=2/3`],title=`error curves for order 4 Runge-Ku tta methods`));" }}{PARA 13 "" 1 "" {GLPLOT2D 963 744 744 {PLOTDATA 2 "6,-%'CURVESG6%7ev7$$\"\"!F)F(7$$\"5qmmmmT&)G\\a!#@$\"-4p&*>\\7F-7$$\" 5MLLLL3x&)*3\"!#?$\",ys&\\)o#F37$$\"5+++]i!R(*Rc\"F3$\",9#p:5OF37$$\"5 nmmm\"H2P\"Q?F3$\",m/CFB%F37$$\"5++]7G))>Wr@F3$\",(Qdm+VF37$$\"5MLLek. pu/BF3$\",\"[.W=0QCF3$\",9*f46WF37$$\"5+++]PMnNrDF3$\" ,g<&\\,WF37$$\"5ML$eR(\\;m/FF3$\",>&\\87WF37$$\"5nmmT5ll'z$GF3$\",J6)* >N%F37$$\"5++](o/[r7(HF3$\",C*f3(G%F37$$\"5MLLL$eRwX5$F3$\",E&G([A%F37 $$\"5MLLLL$eI8k$F3$\",,V%*Q$QF37$$\"5NLLL$3x%3yTF3$\",VI^HV$F37$$\"5-+ +]PfyG7ZF3$\",KQ-98$F37$$\"5ommm\"z%4\\Y_F3$\",`Jez'HF37$$\"5NLLL$3FGT 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@*R5Feq7$F`gl$\",%e?aI5Feq7$F_is$\",+q80.\"Feq7$F^eq$\",!4$RJ.\"Feq7$F gis$\",/Mu\"H5Feq7$Fegl$\",_By!G5Feq7$Fgdo$\",v])QP5Feq7$Fjgl$\",g$)** 4/\"Feq7$F_hl$\",b%Q@g5Feq7$Fihl$\",K)ppu5Feq7$Fcil$\",&)fH23\"Feq7$F] jl$\",u@Ay2\"Feq7$Fgjl$\",&4,mj5Feq7$F\\[m$\",6?^a0\"Feq7$Fa[m$\",QIq' [5Feq7$Ff[m$\",mGHA0\"Feq7$F[\\m$\",48wI1\"Feq7$F_]m$\",AKSF3\"Feq7$Fc ^m$\",HPAd6\"Feq7$Fh^m$\",34%*Feq7$Ff`m$\"+(o@z;(Feq7$F[am$\",X( 3W7VF37$F`am$\",fz:]y#F37$Feam$\",=6y3L\"F37$Fjam$!,hO^[@\"F37$F_bm$!, _Dy=\"GF37$Fdbm$!,av-\"4WF37$Fibm$!,WKflf'F37$F^cm$!,Lu_aM)F37$Fccm$!, &p0;b&*F37$Fhcm$!--+#*e%>\"F37$F]dm$!-z\"\\,WC\"F37$Fbdm$!-`\\%Hd>\"F3 7$Fgdm$!-6%=[#H6F37$F\\em$!-(Q)\\mB5F37$Faem$!,J#y.pzF37$Ffem$!,^)4,Dk F37$F[fm$!,+F-_b%F37$Fefm$!,bfZ=O\"F37$F_gm$\"+e[Y8vF37$Figm$\",#G\\82 GF37$Fchm$\",D)=`;eF37$Fhhm$\"+#o=d6)Feq7$F]im$\"+>Q'y!)*Feq7$Fbim$\", .hl7?\"Feq7$Fgim$\",K:rFeq7$Fi\\n$\",hW0I%>Feq7$Fc]n$\",_ Z-a(>Feq7$Fh]n$\",*))4Nn>Feq7$F]^n$\",3n:R&>Feq7$Fb^n$\",\")*ymh=Feq7$ Fg^n$\",R\"[P0MUo8Feq7$Fcbn$\",=S/#R8Feq7$Fgcn$\",qN5AK\"Feq7$F\\dn$\",!RSN88Feq7$ Fadn$\",rAf)38Feq7$Ffdn$\",9!z958Feq7$F[en$\",V'*yFJ\"Feq-F`en6&Fben$ \")#)eqkFeen$\"))eqk\"FeenF`ax-Fgen6#%Dmethod~with~c[2]=2/5~and~a[3,2] =2/3G-%%FONTG6$%*HELVETICAG\"\"*-%+AXESLABELSG6$Q\"x6\"Q!F^bx-%&TITLEG 6#%Merror~curves~for~order~4~Runge-Kutta~methodsG-%%VIEWG6$;F(F[en%(DE FAULTG" 1 2 0 1 10 0 2 9 1 4 2 1.000000 45.000000 45.000000 0 1 "class ical method" "3/8 method" "Gill's method" "Dormand's method" "method w ith c[2]=3/7 and c[3]=6/11" "method with c[2]=2/5 and a[3,2]=2/3" }}}} {PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 38 "Test 7 of order 4 Runge -Kutta methods " }}{PARA 256 "" 0 "" {TEXT -1 2 " " }{XPPEDIT 18 0 "d y/dx=-(1+4*cos(3*x))*(y-1/3)" "6#/*&%#dyG\"\"\"%#dxG!\"\",$*&,&F&F&*& \"\"%F&-%$cosG6#*&\"\"$F&%\"xGF&F&F&F&,&%\"yGF&*&F&F&F2F(F(F&F(" } {TEXT -1 5 ", " }{XPPEDIT 18 0 "y(0)=1" "6#/-%\"yG6#\"\"!\"\"\"" } {TEXT 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" }}{PARA 0 "" 0 "" {TEXT -1 10 "Solution: " }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "y = 1/3" "6#/%\"yG*&\"\"\"F &\"\"$!\"\"" }{TEXT -1 1 " " }{XPPEDIT 18 0 "exp(-4/3*sin(3*x)+8/3*sin (3/2*x)*cos(3/2*x))+2/3" "6#,&-%$expG6#,&*(\"\"%\"\"\"\"\"$!\"\"-%$sin G6#*&F+F*%\"xGF*F*F,**\"\")F*F+F,-F.6#*(F+F*\"\"#F,F1F*F*-%$cosG6#*(F+ F*F7F,F1F*F*F*F**&F7F*F+F,F*" }{TEXT -1 1 " " }{XPPEDIT 18 0 "exp(-4/3 *sin(3*x)-x)" "6#-%$expG6#,&*(\"\"%\"\"\"\"\"$!\"\"-%$sinG6#*&F*F)%\"x GF)F)F+F0F+" }{TEXT -1 2 ". " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 181 "de := diff(y(x),x)=-(1+4*co s(3*x))*(y(x)-1/3);\nic := y(0)=1;\nsimplify(dsolve(\{de,ic\},y(x))); \nv := unapply(rhs(%),x):\nplot(v(x),x=0..5,0..1.1,font=[HELVETICA,9], labels=[`x`,`y(x)`]);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%#deG/-%%dif fG6$-%\"yG6#%\"xGF,,$*&,&\"\"\"F0*&\"\"%F0-%$cosG6#,$*&\"\"$F0F,F0F0F0 F0F0,&F)F0#F0F8!\"\"F0F;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%#icG/-% \"yG6#\"\"!\"\"\"" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/-%\"yG6#%\"xG,&* &#\"\"\"\"\"$F+-%$expG6#,&*&#\"\"%F,F+-%$sinG6#,$*&F,F+F'F+F+F+!\"\"*& #\"\")F,F+*&-F56#,$*(F,F+\"\"#F9F'F+F+F+-%$cosGF?F+F+F+F+F+*&#FBF,F+-F .6#,&F'F9*&#F3F,F+F4F+F9F+F+" }}{PARA 13 "" 1 "" {GLPLOT2D 503 318 318 {PLOTDATA 2 "6&-%'CURVESG6$7ap7$$\"\"!F)$\"\"\"F)7$$\"3gmmTN@Ki8!# >$\"3W+7cSy5h&*!#=7$$\"3ALL$3FWYs#F/$\"3KtP[t*Q;:*F27$$\"3%)***\\iSmp3 %F/$\"3g.\"H>f!3q()F27$$\"3WmmmT&)G\\aF/$\"36p*p.:G\\T)F27$$\"3m****\\ 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1 10 0 2 9 1 4 2 1.000000 45.000000 45.000000 0 0 "Curve 1" }}}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 " " 0 "" {TEXT -1 32 "The following code constructs a " }{TEXT 260 17 "d iscrete solution" }{TEXT -1 44 " based on each of the methods and give s the " }{TEXT 260 22 "root mean square error" }{TEXT -1 18 " of each \+ solution." }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 821 "V := (x,y) -> \+ -(1+4*cos(3*x))*(y-1/3): hh := 0.02: numsteps := 250: x0 := 0: y0 := 1 :\nmatrix([[`slope field: `,V(x,y)],[`initial point: `,``(x0,y0)],[` step width: `,hh],\n[`no. of steps: `,numsteps]]);``;\nmthds := [` classical method with `*(c[2]=1/2,c[3]=1/2),`3/8 method with `*(c[2]=1 /3,c[3]=2/3),`Gill's method with `*(c[2]=1/2,c[3]=1/2),`Dormand's meth od with `*(c[2]=2/5,c[3]=3/5),`method with `*(c[2]=3/7,c[3]=6/11),`met hod with `*(c[2]=2/5,a[3,2]=2/3)]: errs := []:\nDigits := 30:\nfor ct \+ to 6 do\n Vn_RK4_||ct := RK4_||ct(V(x,y),x,y,x0,y0,hh,numsteps,false );\n sm := 0: numpts := nops(Vn_RK4_||ct):\n for ii to numpts do\n sm := sm+(Vn_RK4_||ct[ii,2]-v(Vn_RK4_||ct[ii,1]))^2;\n end do: \n errs := [op(errs),sqrt(sm/numpts)];\nend do:\nDigits := 10:\nlina lg[transpose](convert([mthds,evalf(errs)],matrix));" }}{PARA 11 "" 1 " " {XPPMATH 20 "6#K%'matrixG6#7&7$%0slope~field:~~~G,$*&,&\"\"\"F,*&\" \"%F,-%$cosG6#,$*&\"\"$F,%\"xGF,F,F,F,F,,&%\"yGF,#F,F4!\"\"F,F97$%0ini tial~point:~G-%!G6$\"\"!F,7$%/step~width:~~~G$\"\"#!\"#7$%1no.~of~step s:~~~G\"$]#Q)pprint406\"" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%!G" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#K%'matrixG6#7(7$*&%7classical~method~w ith~G\"\"\"6$/&%\"cG6#\"\"##F*F0/&F.6#\"\"$F1F*$\"+X3U[;!#;7$*&%13/8~m ethod~with~GF*6$/F-#F*F5/F3#F0F5F*$\"+P'\\!46F87$*&%4Gill's~method~wit h~GF*F+F*F67$*&%7Dormand's~method~with~GF*6$/F-#F0\"\"&/F3#F5FLF*$\"+/ tz,8F87$*&%-method~with~GF*6$/F-#F5\"\"(/F3#\"\"'\"#6F*$\"+LA\"G\\\"F8 7$*&FSF*6$FJ/&%\"aG6$F5F0F@F*$\"+V+5l8F8Q)pprint416\"" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 30 "The following code c onstructs " }{TEXT 260 20 "numerical procedures" }{TEXT -1 56 " for so lutions based on each of the Runge-Kutta schemes." }}{PARA 0 "" 0 "" {TEXT -1 75 "The error in the value obtained by each of the methods at the point where " }{XPPEDIT 18 0 "x = 4.999;" "6#/%\"xG-%&FloatG6$\" %**\\!\"$" }{TEXT -1 16 " is also given." }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 752 "V := (x,y) -> -(1+4*cos(3*x))*(y-1/3): hh := 0.02: n umsteps := 250: x0 := 0: y0 := 1:\nmatrix([[`slope field: `,V(x,y)], [`initial point: `,``(x0,y0)],[`step width: `,hh],\n[`no. of steps: \+ `,numsteps]]);``;\nmthds := [`classical method with `*(c[2]=1/2,c[3] =1/2),`3/8 method with `*(c[2]=1/3,c[3]=2/3),`Gill's method with `*(c[ 2]=1/2,c[3]=1/2),`Dormand's method with `*(c[2]=2/5,c[3]=3/5),`method \+ with `*(c[2]=3/7,c[3]=6/11),`method with `*(c[2]=2/5,a[3,2]=2/3)]: err s := []:\nDigits := 30:\nfor ct to 6 do\n vn_RK4_||ct := RK4_||ct(V( x,y),x,y,x0,y0,hh,numsteps,true);\nend do:\nxx := 4.999: vxx := evalf( v(xx)):\nfor ct to 6 do\n errs := [op(errs),abs(vn_RK4_||ct(xx)-vxx) ];\nend do:\nDigits := 10:\nlinalg[transpose](convert([mthds,evalf(err s)],matrix));" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#K%'matrixG6#7&7$%0slo pe~field:~~~G,$*&,&\"\"\"F,*&\"\"%F,-%$cosG6#,$*&\"\"$F,%\"xGF,F,F,F,F ,,&%\"yGF,#F,F4!\"\"F,F97$%0initial~point:~G-%!G6$\"\"!F,7$%/step~widt h:~~~G$\"\"#!\"#7$%1no.~of~steps:~~~G\"$]#Q)pprint426\"" }}{PARA 11 " " 1 "" {XPPMATH 20 "6#%!G" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#K%'matrix G6#7(7$*&%7classical~method~with~G\"\"\"6$/&%\"cG6#\"\"##F*F0/&F.6#\" \"$F1F*$\"+qP@Ny!#=7$*&%13/8~method~with~GF*6$/F-#F*F5/F3#F0F5F*$\"+J? 2dZF87$*&%4Gill's~method~with~GF*F+F*F67$*&%7Dormand's~method~with~GF* 6$/F-#F0\"\"&/F3#F5FLF*$\"+H,+YeF87$*&%-method~with~GF*6$/F-#F5\"\"(/F 3#\"\"'\"#6F*$\"+A#*GumF87$*&FSF*6$FJ/&%\"aG6$F5F0F@F*$\"+?:sMgF8Q)ppr int436\"" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 4 "The " }{TEXT 260 22 "root mean square error" }{TEXT -1 20 " over th e interval " }{XPPEDIT 18 0 "[0, 5];" "6#7$\"\"!\"\"&" }{TEXT -1 82 " of each Runge-Kutta method is estimated as follows using the special procedure " }{TEXT 0 5 "NCint" }{TEXT -1 98 " to perform numerical \+ integration by the 7 point Newton-Cotes method over 100 equal subinter vals." }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 496 "mthds := [`classic al method with `*(c[2]=1/2,c[3]=1/2),`3/8 method with `*(c[2]=1/3,c[3] =2/3),`Gill's method with `*(c[2]=1/2,c[3]=1/2),`Dormand's method with `*(c[2]=2/5,c[3]=3/5),`method with `*(c[2]=3/7,c[3]=6/11),`method wit h `*(c[2]=2/5,a[3,2]=2/3)]: errs := []:\nDigits := 20:\nfor ct to 6 do \n sm := NCint((v(x)-'vn_RK4_||ct'(x))^2,x=0..5,adaptive=false,numpo ints=7,factor=100);\n errs := [op(errs),sqrt(sm/5)];\nend do:\nDigit s := 10:\nlinalg[transpose](convert([mthds,evalf(errs)],matrix));" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#K%'matrixG6#7(7$*&%7classical~method~w ith~G\"\"\"6$/&%\"cG6#\"\"##F*F0/&F.6#\"\"$F1F*$\"+_NCY;!#;7$*&%13/8~m ethod~with~GF*6$/F-#F*F5/F3#F0F5F*$\"+&p%)z5\"F87$*&%4Gill's~method~wi th~GF*F+F*F67$*&%7Dormand's~method~with~GF*6$/F-#F0\"\"&/F3#F5FLF*$\"+ vnR+8F87$*&%-method~with~GF*6$/F-#F5\"\"(/F3#\"\"'\"#6F*$\"+z^G\"\\\"F 87$*&FSF*6$FJ/&%\"aG6$F5F0F@F*$\"+:4vj8F8Q)pprint446\"" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 92 "The following error \+ graphs are constructed using the numerical procedures for the solution s." }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 456 "evalf[18](plot(['vn_R K4_1'(x)-v(x),'vn_RK4_2'(x)-v(x),'vn_RK4_3'(x)-v(x),'vn_RK4_4'(x)-v(x) ,\n'vn_RK4_5'(x)-v(x),'vn_RK4_6'(x)-v(x)],x=0..5,font=[HELVETICA,9],th ickness=[1$2,2,1$3],\ncolor=[red,blue,COLOR(RGB,0,.65,0),COLOR(RGB,.4, 0,1),magenta,brown],\nlegend=[`classical method`,`3/8 method`,`Gill's \+ method`,`Dormand's method`,`method with c[2]=3/7 and c[3]=6/11`,`metho d with c[2]=2/5 and a[3,2]=2/3`],title=`error curves for order 4 Runge -Kutta methods`));" }}{PARA 13 "" 1 "" {GLPLOT2D 1000 682 682 {PLOTDATA 2 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6F97$F^^m$\"-1px%*o#)F97$Fc^m$\"-PWQ+ZdF97$Fh^m$\"-ajBABRF97$F]_m$\"-t &y/Au#F97$Fb_m$\".*e`X&>)=F-7$Fg_m$\".!>*4JuF\"F-7$F\\`m$\"-H%)**RX*)F -7$Fa`m$\"-U&4c+O'F-7$Ff`m$\"-kr4Z5_F-7$F[am$\"-x]q><[F-7$F`am$\"-#3*R Iu]F-7$Feam$\"-M+T@[gF--Fjam6&F\\bm$\")#)eqkF_bm$\"))eqk\"F_bmFhhsF`bm -Febm6#%Dmethod~with~c[2]=2/5~and~a[3,2]=2/3G-%%FONTG6$%*HELVETICAG\" \"*-%+AXESLABELSG6$Q\"x6\"Q!Ffis-%&TITLEG6#%Merror~curves~for~order~4~ Runge-Kutta~methodsG-%%VIEWG6$;F(Feam%(DEFAULTG" 1 2 0 1 10 0 2 9 1 4 2 1.000000 45.000000 45.000000 0 1 "classical method" "3/8 method" "Gi ll's method" "Dormand's method" "method with c[2]=3/7 and c[3]=6/11" " method with c[2]=2/5 and a[3,2]=2/3" }}}}{PARA 0 "" 0 "" {TEXT -1 0 " " }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}}{SECT 1 {PARA 4 " " 0 "" {TEXT -1 38 "Test 8 of order 4 Runge-Kutta methods " }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "dy/dx=x*(9-x^2)/(1+y^2)" "6 #/*&%#dyG\"\"\"%#dxG!\"\"*(%\"xGF&,&\"\"*F&*$F*\"\"#F(F&,&F&F&*$%\"yGF .F&F(" }{TEXT -1 3 ", " }{XPPEDIT 18 0 "y(0)=0" "6#/-%\"yG6#\"\"!F'" }{TEXT -1 1 " " }}{PARA 0 "" 0 "" {TEXT -1 10 "Solution: " }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "y = rho(x)/2-2/rho(x);" "6# /%\"yG,&*&-%$rhoG6#%\"xG\"\"\"\"\"#!\"\"F+*&F,F+-F(6#F*F-F-" }{TEXT -1 2 ", " }}{PARA 0 "" 0 "" {TEXT -1 7 "where " }{XPPEDIT 18 0 "rho(x ) = (54*x^2-3*x^4+sqrt(64+9*x^8-324*x^6+2916*x^4))^(1/3);" "6#/-%$rhoG 6#%\"xG),(*&\"#a\"\"\"*$F'\"\"#F,F,*&\"\"$F,*$F'\"\"%F,!\"\"-%%sqrtG6# ,*\"#kF,*&\"\"*F,*$F'\"\")F,F,*&\"$C$F,*$F'\"\"'F,F3*&\"%;HF,*$F'F2F,F ,F,*&F,F,F0F3" }{TEXT -1 2 ". " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 178 "de := diff(y(x),x)=x*(9-x^2 )/(1+y(x)^2);\nic := y(0)=0;\ndsolve(\{de,ic\},y(x));\nw := unapply(rh s(%),x):\nplot(w(x),x=0..4,0..3.7,numpoints=75,font=[HELVETICA,9],labe ls=[`x`,`y(x)`]);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%#deG/-%%diffG6$ -%\"yG6#%\"xGF,*(F,\"\"\",&\"\"*F.*$)F,\"\"#F.!\"\"F.,&F.F.*$)F)F3F.F. 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-%'COLOURG6&%$RGBG$\"#5!\"\"F(F(-%%FONTG6$%*HELVETICAG\"\"*-%+AXESLABE LSG6$%\"xG%%y(x)G-%%VIEWG6$;F(F`cl;F($\"#PFjcl" 1 2 0 1 10 0 2 9 1 4 2 1.000000 45.000000 45.000000 0 0 "Curve 1" }}}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 32 "The following code constr ucts a " }{TEXT 260 17 "discrete solution" }{TEXT -1 44 " based on eac h of the methods and gives the " }{TEXT 260 22 "root mean square error " }{TEXT -1 18 " of each solution." }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 815 "W := (x,y) -> x*(9-x^2)/(1+y^2): hh := 0.01: numstep s := 400: x0 := 0: y0 := 0:\nmatrix([[`slope field: `,W(x,y)],[`init ial point: `,``(x0,y0)],[`step width: `,hh],\n[`no. of steps: `,nu msteps]]);``;\nmthds := [`classical method with `*(c[2]=1/2,c[3]=1/2), `3/8 method with `*(c[2]=1/3,c[3]=2/3),`Gill's method with `*(c[2]=1/2 ,c[3]=1/2),`Dormand's method with `*(c[2]=2/5,c[3]=3/5),`method with ` *(c[2]=3/7,c[3]=6/11),`method with `*(c[2]=2/5,a[3,2]=2/3)]: errs := [ ]:\nDigits := 30:\nfor ct to 6 do\n Wn_RK4_||ct := RK4_||ct(W(x,y),x ,y,x0,y0,hh,numsteps,false);\n sm := 0: numpts := nops(Wn_RK4_||ct): \n for ii to numpts do\n sm := sm+(Wn_RK4_||ct[ii,2]-w(Wn_RK4_| |ct[ii,1]))^2;\n end do:\n errs := [op(errs),sqrt(sm/numpts)];\nen d do:\nDigits := 10:\nlinalg[transpose](convert([mthds,evalf(errs)],ma trix));" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#K%'matrixG6#7&7$%0slope~fie ld:~~~G*(%\"xG\"\"\",&\"\"*F+*$)F*\"\"#F+!\"\"F+,&F+F+*$)%\"yGF0F+F+F1 7$%0initial~point:~G-%!G6$\"\"!F;7$%/step~width:~~~G$F+!\"#7$%1no.~of~ steps:~~~G\"$+%Q)pprint456\"" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%!G" } }{PARA 11 "" 1 "" {XPPMATH 20 "6#K%'matrixG6#7(7$*&%7classical~method~ with~G\"\"\"6$/&%\"cG6#\"\"##F*F0/&F.6#\"\"$F1F*$\"+CP@Z$*!#>7$*&%13/8 ~method~with~GF*6$/F-#F*F5/F3#F0F5F*$\"+vShNrF87$*&%4Gill's~method~wit h~GF*F+F*$\"+sdzt\\F87$*&%7Dormand's~method~with~GF*6$/F-#F0\"\"&/F3#F 5FNF*$\"+zl^BOF87$*&%-method~with~GF*6$/F-#F5\"\"(/F3#\"\"'\"#6F*$\"+q (4x*HF87$*&FUF*6$FL/&%\"aG6$F5F0F@F*$\"+()yd!f$F8Q)pprint466\"" }}} {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 30 "The follo wing code constructs " }{TEXT 260 20 "numerical procedures" }{TEXT -1 56 " for solutions based on each of the Runge-Kutta schemes." }}{PARA 0 "" 0 "" {TEXT -1 75 "The error in the value obtained by each of the \+ methods at the point where " }{XPPEDIT 18 0 "x = 3.499;" "6#/%\"xG-%& FloatG6$\"%*\\$!\"$" }{TEXT -1 16 " is also given." }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 746 "W := (x,y) -> x*(9-x^2)/(1+y^2): hh := 0.0 1: numsteps := 400: x0 := 0: y0 := 0:\nmatrix([[`slope field: `,W(x, y)],[`initial point: `,``(x0,y0)],[`step width: `,hh],\n[`no. of ste ps: `,numsteps]]);``;\nmthds := [`classical method with `*(c[2]=1/2, c[3]=1/2),`3/8 method with `*(c[2]=1/3,c[3]=2/3),`Gill's method with ` *(c[2]=1/2,c[3]=1/2),`Dormand's method with `*(c[2]=2/5,c[3]=3/5),`met hod with `*(c[2]=3/7,c[3]=6/11),`method with `*(c[2]=2/5,a[3,2]=2/3)]: errs := []:\nDigits := 30:\nfor ct to 6 do\n wn_RK4_||ct := RK4_||c t(W(x,y),x,y,x0,y0,hh,numsteps,true);\nend do:\nxx := 3.499: wxx := ev alf(w(xx)):\nfor ct to 6 do\n errs := [op(errs),abs(wn_RK4_||ct(xx)- wxx)];\nend do:\nDigits := 10:\nlinalg[transpose](convert([mthds,evalf (errs)],matrix));" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#K%'matrixG6#7&7$% 0slope~field:~~~G*(%\"xG\"\"\",&\"\"*F+*$)F*\"\"#F+!\"\"F+,&F+F+*$)%\" yGF0F+F+F17$%0initial~point:~G-%!G6$\"\"!F;7$%/step~width:~~~G$F+!\"#7 $%1no.~of~steps:~~~G\"$+%Q)pprint476\"" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%!G" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#K%'matrixG6#7(7$*&%7class ical~method~with~G\"\"\"6$/&%\"cG6#\"\"##F*F0/&F.6#\"\"$F1F*$\"+JZG\"o #!#>7$*&%13/8~method~with~GF*6$/F-#F*F5/F3#F0F5F*$\"+=kW9FF87$*&%4Gill 's~method~with~GF*F+F*$\"+wGN&z\"F87$*&%7Dormand's~method~with~GF*6$/F -#F0\"\"&/F3#F5FNF*$\"+ " 0 "" {MPLTEXT 1 0 496 "mthds := [`classi cal method with `*(c[2]=1/2,c[3]=1/2),`3/8 method with `*(c[2]=1/3,c[3 ]=2/3),`Gill's method with `*(c[2]=1/2,c[3]=1/2),`Dormand's method wit h `*(c[2]=2/5,c[3]=3/5),`method with `*(c[2]=3/7,c[3]=6/11),`method wi th `*(c[2]=2/5,a[3,2]=2/3)]: errs := []:\nDigits := 20:\nfor ct to 6 d o\n sm := NCint((w(x)-'wn_RK4_||ct'(x))^2,x=0..4,adaptive=false,nump oints=7,factor=200);\n errs := [op(errs),sqrt(sm/4)];\nend do:\nDigi ts := 10:\nlinalg[transpose](convert([mthds,evalf(errs)],matrix));" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#K%'matrixG6#7(7$*&%7classical~method~w ith~G\"\"\"6$/&%\"cG6#\"\"##F*F0/&F.6#\"\"$F1F*$\"+%y&z-#*!#>7$*&%13/8 ~method~with~GF*6$/F-#F*F5/F3#F0F5F*$\"+VQNYqF87$*&%4Gill's~method~wit h~GF*F+F*$\"+FcE2\\F87$*&%7Dormand's~method~with~GF*6$/F-#F0\"\"&/F3#F 5FNF*$\"+A. " 0 "" {MPLTEXT 1 0 434 "evalf[20](plot([ 'wn_RK4_1'(x)-w(x),'wn_RK4_2'(x)-w(x),'wn_RK4_3'(x)-w(x),'wn_RK4_4'(x) -w(x),\n'wn_RK4_5'(x)-w(x),'wn_RK4_6'(x)-w(x)],x=0..4,font=[HELVETICA, 9],\ncolor=[red,blue,COLOR(RGB,0,.65,0),COLOR(RGB,.4,0,1),magenta,brow n],\nlegend=[`classical method`,`3/8 method`,`Gill's method`,`Dormand' s method`,`method with c[2]=3/7 and c[3]=6/11`,`method with c[2]=2/5 a nd a[3,2]=2/3`],title=`error curves for order 4 Runge-Kutta methods`)) ;" }}{PARA 13 "" 1 "" {GLPLOT2D 888 605 605 {PLOTDATA 2 "6,-%'CURVESG6 %7fp7$$\"\"!F)F(7$$\"5lmmmmT&)G\\a!#A$!)gN))>!#?7$$\"5LLLLL3x&)*3\"!#@ $!*ivPf(F07$$\"5mmmmTN@Ki8F4$!*V\"o0xF07$$\"5++++]ilyM;F4$!*Gqcl*F07$$ \"5LLLLe*)4D2>F4$!+>>;;@F07$$\"5mmmmm;arz@F4$!+Y*Rx.$F07$$\"5+++++DJdp KF4$!+hofQoF07$$\"5LLLLLL3VfVF4$!,i*ou<7F07$$\"5+++++]i9RlF4$!,\">![#o FF07$$\"5mmmmmm;')=()F4$!,AKqv0&F07$$\"5MLLLeR?ah5F0$!,\\K,we(F07$$\"5 ++++]7z>^7F0$!-[?&Q91\"F07$$\"5mmmmT&y`3W\"F0$!-;xPI+9F07$$\"5LLLLLe'4 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!+(Qdem$F07$FV$!+![`\"4&)F07$Fen$!,5hj]c\"F07$F_o$!,eLk8C$F07$Fio$!,d3 u!\\]F07$Feim$!,&z#*=2aF07$F^p$!,Y`7uo&F07$$\"5LL$e*)f]e/(=F0$!,l)[F:d F07$$\"5+++D1*[>r*=F0$!,\"=b5`eF07$$\"5mm;a8s/yB>F0$!,/(GxseF07$F]jm$! ,+*>>oeF07$$\"5mmmTN@Mw.?F0$!,.>wr%fF07$Fcp$!,*pLGFfF07$Fhp$!,.hU**e&F 07$F]q$!,!eo8*\\%F07$Far$!,+AAFX#F07$F[s$!+/!Ggd#F07$F`s$\",*yq[5?F07$ Fes$\",ER\")z=%F07$Fg[n$\",0:r&*3'F07$Fjs$\",#*\\\\Kg(F07$F_\\n$\",jjy Po)F07$F_t$\",V6ACL*F07$Fg\\n$\",[zvmS*F07$F\\]n$\",P(=,:%*F07$Fa]n$\" ,+cDoO*F07$Fdt$\",n?87F*F07$F^^n$\",g547(*)F07$Fit$\",Er!Qu&)F07$F`_n$ \",i))f#4wF07$F^u$\",kn4Io'F07$Fcu$\",%)\\Ko<&F07$Fhu$\",1T/8:%F07$F]v $\",Aaq%eMF07$Fbv$\",Q0S\"=IF07$Fgv$\",!)e$zZEF07$F]w$\",5!oTjBF07$Fbw $\",maou9#F07$Fgw$\",AI)>&)>F07$F\\x$\",sXah#=F07$Fax$\",))**4Pr\"F07$ Ffx$\",#p`O-;F07$F[y$\",g[%H=:F07$F`y$\",)QgtP9F07$Fey$\",9oSDP\"F07$F jy$\",'*H7LJ\"F07$F_z$\",%yvCm7F07$Fdz$\",79s;A\"F07$Fiz$\",?)e!>=\"F0 7$F^[l$\",y$oN_6F07$Fc[l$\",UDHU7\"F07$Fh[l$\",yaf-5\"F07$F]\\l$\",[+' ))z5F07$Fb\\l$\",?d))Q1\"F07$Fg\\l$\",axW(\\5F07$F\\]l$\",;#e(*R5F07$F a]l$\",S7\"RK5F07$Ff]l$\",!zCgF5F07$F[^l$\",Q4!*R-\"F07$F`^l$\",?s@=- \"F07$Fe^l$\",qmh$>5F07$Fj^l$\",e&ed:5F07$F__l$\",YZ0]+\"F07$Fd_l$\"+3 38j)*F07$Fi_l$\"+]84a%*F07$F^`l$\"++H.-')F07$Fc`l$\"+/Oh`qF07$Fh`l$\"+ kT$Gx&F07$F]al$\"+qThMPF07$Fbal$\"+eMQ:6F07$Fgal$!++FA'p#F07$F\\bl$!+e tz(p*F07$Fabl$!,iR#pF>F07$F[cl$!,_zdbW$F07$Fecl$!,[()3<(fF07$F_dl$!,+% =w5zF07$Fidl$!-/Bj'Q1\"F07$F^el$!-KHK3Z7F07$Fcel$!-u1\"e!y9F07$Fhel$!- M4qv:;F07$F]fl$!-YFJlx " 0 "" {MPLTEXT 1 0 1 ";" }}}}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 39 "Test 9 of order 4 R unge-Kutta methods " }}{PARA 256 "" 0 "" {TEXT -1 3 " " }{XPPEDIT 18 0 "dy/dx=-(1+cos(2*x))*y^3" "6#/*&%#dyG\"\"\"%#dxG!\"\",$*&,&F&F&-% $cosG6#*&\"\"#F&%\"xGF&F&F&*$%\"yG\"\"$F&F(" }{TEXT -1 3 ", " } {XPPEDIT 18 0 "y(0) = sqrt(2);" "6#/-%\"yG6#\"\"!-%%sqrtG6#\"\"#" } {TEXT -1 1 " " }}{PARA 0 "" 0 "" {TEXT -1 10 "Solution: " }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "y = 1/sqrt(sin(2*x)+2*x+1/2)" " 6#/%\"yG*&\"\"\"F&-%%sqrtG6#,(-%$sinG6#*&\"\"#F&%\"xGF&F&*&F/F&F0F&F&* &F&F&F/!\"\"F&F3" }{TEXT -1 2 ". " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 172 "de := diff(y(x),x)=-(1+cos( 2*x))*y(x)^3;\nic := y(0)=sqrt(2);\ndsolve(\{de,ic\},y(x));\nm := unap ply(rhs(%),x):\nplot(m(x),x=0..3,0..1.42,font=[HELVETICA,9],labels=[`x `,`y(x)`]);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%#deG/-%%diffG6$-%\"yG 6#%\"xGF,,$*&,&\"\"\"F0-%$cosG6#,$*&\"\"#F0F,F0F0F0F0)F)\"\"$F0!\"\"" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%#icG/-%\"yG6#\"\"!*$\"\"##\"\"\"F+ " }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/-%\"yG6#%\"xG*&\"\"\"F)*$,(*(\"\" #F)-%$cosGF&F)-%$sinGF&F)F)*&F-F)F'F)F)#F)F-F)#F)F-!\"\"" }}{PARA 13 " " 1 "" {GLPLOT2D 503 318 318 {PLOTDATA 2 "6&-%'CURVESG6$7Y7$$\"\"!F)$ \"3:&4tBc8UT\"!#<7$$\"3$*****\\ilyM;!#>$\"3ozW7@k#*H8F,7$$\"3')*****\\ 7t&pKF0$\"3!G<)\\ef9f7F,7$$\"3z****\\(ofV!\\F0$\"3oe\"F,7$$\"3s ******\\i9RlF0$\"3kESFh\"zh9\"F,7$$\"33++vVV)RQ*F0$\"3'f)*)e-w\\p5F,7$ $\"3/++vVA)GA\"!#=$\"3V)o6<$fq15F,7$$\"3;+]iSS\"Ga\"FJ$\"3IyW%eHk>[*FJ 7$$\"3+++]Peui=FJ$\"3#4`!o2+#G**)FJ7$$\"37+++]$)z%=#FJ$\"3OGH4wwYu&)FJ 7$$\"3A++]i3&o]#FJ$\"3=1g%=M2W@)FJ7$$\"3%)***\\(oX*y9$FJ$\"31u2v$Q9&Gw FJ7$$\"3z***\\P9CAu$FJ$\"3=XIMTf7+sFJ7$$\"3!)***\\P*zhdVFJ$\"3P$G(zQ8# 4%oFJ7$$\"31++v$>fS*\\FJ$\"3X'3%RcqqPlFJ7$$\"3$)***\\(=$f%GcFJ$\"3mYY% G?7\"*G'FJ7$$\"3Q+++Dy,\"G'FJ$\"3?u@-35zxgFJ7$$\"33++]7[X$ocbFJ7$$\"3))***\\PpnsM*FJ $\"3!\\;$Q)fJR[&FJ7$$\"3,++]siL-5F,$\"3&3j/q(Qq8aFJ7$$\"3-+++!R5'f5F,$ \"3q`:6QhHm`FJ7$$\"3)***\\P/QBE6F,$\"3@Igj*yDKK&FJ7$$\"3!******\\\"o?& =\"F,$\"3i/K.-M\\%H&FJ7$$\"31+]Pa&4*\\7F,$\"3OjcS#)ygr_FJ7$$\"33+]7j=_ 68F,$\"3'e4m\")R`oD&FJ7$$\"33++vVy!eP\"F,$\"3a@U-1/NZ_FJ7$$\"34+](=WU[ V\"F,$\"3Nrr*HO\"oU_FJ7$$\"3)****\\7B>&)\\\"F,$\"3'HX%)zwR1C&FJ7$$\"3) ***\\P>:mk:F,$\"3<^\"Q\"4\"y-C&FJ7$$\"3'***\\iv&QAi\"F,$\"3:*4?^OZ,C&F J7$$\"31++vtLU%o\"F,$\"3\"3gSMou)Q_FJ7$$\"3!******\\Nm'[F,$\"3[h+0^h(R>&FJ7$$\"3z*****\\@80+#F, $\"3!zBIi>A%o^FJ7$$\"31++]7,Hl?F,$\"3<)30`]&>L^FJ7$$\"3()**\\P4w)R7#F, $\"3!Qwx>a)*Q4&FJ7$$\"3;++]x%f\")=#F,$\"3q$pQbJ#)G/&FJ7$$\"3!)**\\P/-a [AF,$\"3gJla\"HTu)\\FJ7$$\"3/+](=Yb;J#F,$\"3c:[>;?IA\\FJ7$$\"3')****\\ i@OtBF,$\"3m09))4iC_[FJ7$$\"3')**\\PfL'zV#F,$\"3%Gjf])o8tZFJ7$$\"3>+++ !*>=+DF,$\"3[G/4+_V#p%FJ7$$\"3-++DE&4Qc#F,$\"3!**R*=7x[1YFJ7$$\"3=+]P% >5pi#F,$\"3f7E:iH**=XFJ7$$\"39+++bJ*[o#F,$\"3cgVvc$ovV%FJ7$$\"33++Dr\" [8v#F,$\"3Ln\\jDQ5WVFJ7$$\"3++++Ijy5GF,$\"3OZ!)Q%zK7E%FJ7$$\"31+]P/)fT (GF,$\"3)*4_&egIW<%FJ7$$\"31+]i0j\"[$HF,$\"3qns]&)H\\$4%FJ7$$\"\"$F)$ \"3ntdq;jW4SFJ-%'COLOURG6&%$RGBG$\"#5!\"\"F(F(-%%FONTG6$%*HELVETICAG\" \"*-%+AXESLABELSG6$%\"xG%%y(x)G-%%VIEWG6$;F(Fc\\l;F($\"$U\"!\"#" 1 2 0 1 10 0 2 9 1 4 2 1.000000 45.000000 45.000000 0 0 "Curve 1" }}}} {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 32 "The follo wing code constructs a " }{TEXT 260 17 "discrete solution" }{TEXT -1 44 " based on each of the methods and gives the " }{TEXT 260 22 "root \+ mean square error" }{TEXT -1 18 " of each solution." }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 828 "M := (x,y) -> -(1+cos(2*x))*y^3: hh := 0.0 1: numsteps := 300: x0 := 0: y0 := sqrt(2):\nmatrix([[`slope field: \+ `,M(x,y)],[`initial point: `,``(x0,y0)],[`step width: `,hh],\n[`no. \+ of steps: `,numsteps]]);``;\nmthds := [`classical method with `*(c[2 ]=1/2,c[3]=1/2),`3/8 method with `*(c[2]=1/3,c[3]=2/3),`Gill's method \+ with `*(c[2]=1/2,c[3]=1/2),`Dormand's method with `*(c[2]=2/5,c[3]=3/5 ),`method with `*(c[2]=3/7,c[3]=6/11),`method with `*(c[2]=2/5,a[3,2]= 2/3)]: errs := []:\nDigits := 20:\nfor ct to 6 do\n Mn_RK4_||ct := R K4_||ct(M(x,y),x,y,x0,evalf(y0),hh,numsteps,false);\n sm := 0: numpt s := nops(Mn_RK4_||ct):\n for ii to numpts do\n sm := sm+(Mn_RK 4_||ct[ii,2]-m(Mn_RK4_||ct[ii,1]))^2;\n end do:\n errs := [op(errs ),sqrt(sm/numpts)];\nend do:\nDigits := 10:\nlinalg[transpose](convert ([mthds,evalf(errs)],matrix));" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#K%'m atrixG6#7&7$%0slope~field:~~~G,$*&,&\"\"\"F,-%$cosG6#,$*&\"\"#F,%\"xGF ,F,F,F,)%\"yG\"\"$F,!\"\"7$%0initial~point:~G-%!G6$\"\"!*$F2#F,F27$%/s tep~width:~~~G$F,!\"#7$%1no.~of~steps:~~~G\"$+$Q)pprint506\"" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%!G" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#K%'ma trixG6#7(7$*&%7classical~method~with~G\"\"\"6$/&%\"cG6#\"\"##F*F0/&F.6 #\"\"$F1F*$\"+9PM:?!#=7$*&%13/8~method~with~GF*6$/F-#F*F5/F3#F0F5F*$\" +i!)G!R'F87$*&%4Gill's~method~with~GF*F+F*$\"+lV.w=!#<7$*&%7Dormand's~ method~with~GF*6$/F-#F0\"\"&/F3#F5FOF*$\"+[FKXh!#>7$*&%-method~with~GF *6$/F-#F5\"\"(/F3#\"\"'\"#6F*$\"+%H+z:\"FH7$*&FWF*6$FM/&%\"aG6$F5F0F@F *$\"+`VKAiF8Q)pprint516\"" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 30 "The following code constructs " }{TEXT 260 20 "nu merical procedures" }{TEXT -1 56 " for solutions based on each of the \+ Runge-Kutta schemes." }}{PARA 0 "" 0 "" {TEXT -1 75 "The error in the \+ value obtained by each of the methods at the point where " }{XPPEDIT 18 0 "x = 2.999;" "6#/%\"xG-%&FloatG6$\"%**H!\"$" }{TEXT -1 16 " is a lso given." }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 759 "M := (x,y) -> -(1+cos(2*x))*y^3: hh := 0.01: numsteps := 300: x0 := 0: y0 := sqrt(2 ):\nmatrix([[`slope field: `,M(x,y)],[`initial point: `,``(x0,y0)],[ `step width: `,hh],\n[`no. of steps: `,numsteps]]);``;\nmthds := [ `classical method with `*(c[2]=1/2,c[3]=1/2),`3/8 method with `*(c[2]= 1/3,c[3]=2/3),`Gill's method with `*(c[2]=1/2,c[3]=1/2),`Dormand's met hod with `*(c[2]=2/5,c[3]=3/5),`method with `*(c[2]=3/7,c[3]=6/11),`me thod with `*(c[2]=2/5,a[3,2]=2/3)]: errs := []:\nDigits := 20:\nfor ct to 6 do\n mn_RK4_||ct := RK4_||ct(M(x,y),x,y,x0,evalf(y0),hh,numste ps,true);\nend do:\nxx := 2.999: mxx := evalf(m(xx)):\nfor ct to 6 do \n errs := [op(errs),abs(mn_RK4_||ct(xx)-mxx)];\nend do:\nDigits := \+ 10:\nlinalg[transpose](convert([mthds,evalf(errs)],matrix));" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#K%'matrixG6#7&7$%0slope~field:~~~G,$*&,&\"\" \"F,-%$cosG6#,$*&\"\"#F,%\"xGF,F,F,F,)%\"yG\"\"$F,!\"\"7$%0initial~poi nt:~G-%!G6$\"\"!*$F2#F,F27$%/step~width:~~~G$F,!\"#7$%1no.~of~steps:~~ ~G\"$+$Q)pprint526\"" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%!G" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#K%'matrixG6#7(7$*&%7classical~method~with~G \"\"\"6$/&%\"cG6#\"\"##F*F0/&F.6#\"\"$F1F*$\"+23sGN!#>7$*&%13/8~method ~with~GF*6$/F-#F*F5/F3#F0F5F*$\"+$*omw9!#=7$*&%4Gill's~method~with~GF* F+F*$\"+t2rrRFC7$*&%7Dormand's~method~with~GF*6$/F-#F0\"\"&/F3#F5FOF*$ \"+5fe&R#!#?7$*&%-method~with~GF*6$/F-#F5\"\"(/F3#\"\"'\"#6F*$\"+%f**G S#FC7$*&FWF*6$FM/&%\"aG6$F5F0F@F*$\"+QwXT7FCQ)pprint536\"" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 4 "The " }{TEXT 260 22 "root mean square error" }{TEXT -1 20 " over the interval " } {XPPEDIT 18 0 "[0, 3];" "6#7$\"\"!\"\"$" }{TEXT -1 82 " of each Runge -Kutta method is estimated as follows using the special procedure " } {TEXT 0 5 "NCint" }{TEXT -1 98 " to perform numerical integration by \+ the 7 point Newton-Cotes method over 150 equal subintervals." }} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 496 "mthds := [`classical method with `*(c[2]=1/2,c[3]=1/2),`3/8 method with `*(c[2]=1/3,c[3]=2/3),`Gi ll's method with `*(c[2]=1/2,c[3]=1/2),`Dormand's method with `*(c[2]= 2/5,c[3]=3/5),`method with `*(c[2]=3/7,c[3]=6/11),`method with `*(c[2] =2/5,a[3,2]=2/3)]: errs := []:\nDigits := 20:\nfor ct to 6 do\n sm : = NCint((m(x)-'mn_RK4_||ct'(x))^2,x=0..3,adaptive=false,numpoints=7,fa ctor=150);\n errs := [op(errs),sqrt(sm/3)];\nend do:\nDigits := 10: \nlinalg[transpose](convert([mthds,evalf(errs)],matrix));" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#K%'matrixG6#7(7$*&%7classical~method~with~G\"\" \"6$/&%\"cG6#\"\"##F*F0/&F.6#\"\"$F1F*$\"+I^)=*>!#=7$*&%13/8~method~wi th~GF*6$/F-#F*F5/F3#F0F5F*$\"+Ejn@jF87$*&%4Gill's~method~with~GF*F+F*$ \"+RSeb=!#<7$*&%7Dormand's~method~with~GF*6$/F-#F0\"\"&/F3#F5FOF*$\"+) pfJ2'!#>7$*&%-method~with~GF*6$/F-#F5\"\"(/F3#\"\"'\"#6F*$\"+$Rz^9\"FH 7$*&FWF*6$FM/&%\"aG6$F5F0F@F*$\"+da@`hF8Q)pprint546\"" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 92 "The following error \+ graphs are constructed using the numerical procedures for the solution s." }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 434 "evalf[20](plot(['mn_R K4_1'(x)-m(x),'mn_RK4_2'(x)-m(x),'mn_RK4_3'(x)-m(x),'mn_RK4_4'(x)-m(x) ,\n'mn_RK4_5'(x)-m(x),'mn_RK4_6'(x)-m(x)],x=0..3,font=[HELVETICA,9],\n color=[red,blue,COLOR(RGB,0,.65,0),COLOR(RGB,.4,0,1),magenta,brown],\n legend=[`classical method`,`3/8 method`,`Gill's method`,`Dormand's met hod`,`method with c[2]=3/7 and c[3]=6/11`,`method with c[2]=2/5 and a[ 3,2]=2/3`],title=`error curves for order 4 Runge-Kutta methods`));" }} {PARA 13 "" 1 "" {GLPLOT2D 943 671 671 {PLOTDATA 2 "6,-%'CURVESG6%7bp7 $$\"\"!F)F(7$$\"5+++]7.K[V?!#A$\")Gr)z#!#>7$$\"5++++D1k'p3%F-$\"*)p#\\ o(F07$$\"5+++]P4'\\/8'F-$\"+v#3]$\\F07$$\"5++++]7G$R<)F-$\",WK7^s\"F07 $$\"5+++Dc,;u@5!#@$\",L([8/RF07$$\"5+++](=#**3E7FC$\",zRZ-#QF07$$\"5++ +v=U#Q/V\"FC$\",uMiwz$F07$$\"5++++]ilyM;FC$\",+InT.%F07$$\"5+++D\"G)[8 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y(0)=sqrt(5)/2;\ndsolve(\{de,ic\}, y(x));\nsimplify(convert(%,exp));\np := unapply(rhs(%),x):\nplot(p(x), x=0..5,font=[HELVETICA,9],labels=[`x`,`y(x)`]);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%#deG/-%%diffG6$-%\"yG6#%\"xGF,,$*&,&*&\"\"#\"\"\"-%$s inG6#,$*&\"\"&F2F,F2F2F2F2*&\"\"$F2-%$cosG6#,$*&\"\"(F2F,F2F2F2F2F2-%% sinhG6#F)F2!\"\"" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%#icG/-%\"yG6#\" \"!,$*&\"\"#!\"\"\"\"&#\"\"\"F,F0" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/ -%\"yG6#%\"xG-%#lnG6#-%%tanhG6#,*#\"\"\"\"\"&F0*&#F0\"\"#F0-F)6#,$*&,& -%$expG6#,$*&F4!\"\"F1F3F0F0F0F0F0,&F:F0F0F?F?F?F0F0*&#\"\"$\"#9F0-%$s inG6#,$*&\"\"(F0F'F0F0F0F0*&#F0F1F0-%$cosG6#,$*&F1F0F'F0F0F0F?" }} {PARA 12 "" 1 "" {XPPMATH 20 "6#/-%\"yG6#%\"xG-%#lnG6#,$*&,*-%$expG6#, 4#\"\"#\"\"&\"\"\"*&#\"\"$\"\"(F5-%$sinGF&F5!\"\"*&#\"$#>F9F5*&)-%$cos GF&\"\"'F5F:F5F5F5*&#\"$S#F9F5*&)FB\"\"%F5F:F5F5F<*&#\"#sF9F5*&)FBF3F5 F:F5F5F5*&#\"#KF4F5*$)FBF4F5F5F<*&\"\")F5)FBF8F5F5*&F3F5FBF5F<*&F3FF5F@F5F5*&#FGF9F5FHF5F<*&FLF5FN 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g#R;G9D:%F87$$\"3/]i:NK'zf%F,$\"3*f`:i,h67%F87$$\"3fLe*[=Y.h%F,$\"33)> WD]()H5%F87$$\"386but%F,$\"3_@+X\\xqwZF87$$\"37+]iSjE!z%F,$\"3<\" z**[W(=JcF87$$\"3y*\\7G))Rb\"[F,$\"3OXJ\"GK]h>'F87$$\"3L+++DM\"3%[F,$ \"3!**fZR-9n(oF87$$\"3)3](=np3m[F,$\"3#eM]7.#Q?&=\\F,$\"3'4(*[g6ply*F87$$\"3s+voa-oX\\F, $\"3?(Q=b([U56F,7$$\"3O]PMF,%G(\\F,$\"3_;`pzy]b7F,7$$\"\"&F)$\"3.ug')y o>49F,-%'COLOURG6&%$RGBG$\"#5!\"\"F(F(-%%FONTG6$%*HELVETICAG\"\"*-%+AX ESLABELSG6$%\"xG%%y(x)G-%%VIEWG6$;F(Ficn%(DEFAULTG" 1 2 0 1 10 0 2 9 1 4 2 1.000000 45.000000 45.000000 0 0 "Curve 1" }}}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 93 "The following code constr ucts a discrete solution based on each of the methods and gives the " }{TEXT 260 22 "root mean square error" }{TEXT -1 18 " of each solution ." }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 852 "P := (x,y) -> -(2*sin( 5*x)+3*cos(7*x))*sinh(y): hh := 0.01: numsteps := 500: x0 := 0: y0 := \+ sqrt(5)/2:\nmatrix([[`slope field: `,P(x,y)],[`initial point: `,``(x 0,y0)],[`step width: `,hh],\n[`no. of steps: `,numsteps]]);``;\nmt hds := [`classical method with `*(c[2]=1/2,c[3]=1/2),`3/8 method with \+ `*(c[2]=1/3,c[3]=2/3),`Gill's method with `*(c[2]=1/2,c[3]=1/2),`Dorma nd's method with `*(c[2]=2/5,c[3]=3/5),`method with `*(c[2]=3/7,c[3]=6 /11),`method with `*(c[2]=2/5,a[3,2]=2/3)]: errs := []:\nDigits := 30: \nfor ct to 6 do\n Pn_RK4_||ct := RK4_||ct(P(x,y),x,y,x0,evalf(y0),h h,numsteps,false);\n sm := 0: numpts := nops(Pn_RK4_||ct):\n for i i to numpts do\n sm := sm+(Pn_RK4_||ct[ii,2]-evalf(p(Pn_RK4_||ct[ ii,1])))^2;\n end do:\n errs := [op(errs),sqrt(sm/numpts)];\nend d o:\nDigits := 10:\nlinalg[transpose](convert([mthds,evalf(errs)],matri x));" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#K%'matrixG6#7&7$%0slope~field: ~~~G,$*&,&*&\"\"#\"\"\"-%$sinG6#,$*&\"\"&F.%\"xGF.F.F.F.*&\"\"$F.-%$co sG6#,$*&\"\"(F.F5F.F.F.F.F.-%%sinhG6#%\"yGF.!\"\"7$%0initial~point:~G- %!G6$\"\"!,$*&F-FBF4#F.F-F.7$%/step~width:~~~G$F.!\"#7$%1no.~of~steps: ~~~G\"$+&Q)pprint936\"" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%!G" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#K%'matrixG6#7(7$*&%7classical~method~w ith~G\"\"\"6$/&%\"cG6#\"\"##F*F0/&F.6#\"\"$F1F*$\"+!*zLaO!#<7$*&%13/8~ method~with~GF*6$/F-#F*F5/F3#F0F5F*$\"+-eIJaF87$*&%4Gill's~method~with ~GF*F+F*$\"+PVbjWF87$*&%7Dormand's~method~with~GF*6$/F-#F0\"\"&/F3#F5F NF*$\"+;.xFRF87$*&%-method~with~GF*6$/F-#F5\"\"(/F3#\"\"'\"#6F*$\"+,'H e@$F87$*&FUF*6$FL/&%\"aG6$F5F0F@F*$\"+I;kWOF8Q)pprint946\"" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 30 "The following c ode constructs " }{TEXT 260 20 "numerical procedures" }{TEXT -1 56 " f or solutions based on each of the Runge-Kutta schemes." }}{PARA 0 "" 0 "" {TEXT -1 75 "The error in the value obtained by each of the metho ds at the point where " }{XPPEDIT 18 0 "x = 4.999;" "6#/%\"xG-%&Float G6$\"%**\\!\"$" }{TEXT -1 16 " is also given." }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 776 "P := (x,y) -> -(2*sin(5*x)+3*cos(7*x))*sinh(y): hh := 0.01: numsteps := 500: x0 := 0: y0 := sqrt(5)/2:\nmatrix([[`slo pe field: `,P(x,y)],[`initial point: `,``(x0,y0)],[`step width: `, hh],\n[`no. of steps: `,numsteps]]);``;\nmthds := [`classical method with `*(c[2]=1/2,c[3]=1/2),`3/8 method with `*(c[2]=1/3,c[3]=2/3),`Gi ll's method with `*(c[2]=1/2,c[3]=1/2),`Dormand's method with `*(c[2]= 2/5,c[3]=3/5),`method with `*(c[2]=3/7,c[3]=6/11),`method with `*(c[2] =2/5,a[3,2]=2/3)]: errs := []:\nDigits := 30:\nfor ct to 6 do\n pn_R K4_||ct := RK4_||ct(P(x,y),x,y,x0,evalf(y0),hh,numsteps,true);\nend do :\nxx := 4.999: pxx := evalf(p(xx)):\nfor ct to 6 do\n errs := [op(e rrs),abs(pn_RK4_||ct(xx)-pxx)];\nend do:\nDigits := 10:\nlinalg[transp ose](convert([mthds,evalf(errs)],matrix));" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#K%'matrixG6#7&7$%0slope~field:~~~G,$*&,&*&\"\"#\"\"\"-% $sinG6#,$*&\"\"&F.%\"xGF.F.F.F.*&\"\"$F.-%$cosG6#,$*&\"\"(F.F5F.F.F.F. F.-%%sinhG6#%\"yGF.!\"\"7$%0initial~point:~G-%!G6$\"\"!,$*&F-FBF4#F.F- F.7$%/step~width:~~~G$F.!\"#7$%1no.~of~steps:~~~G\"$+&Q*pprint1146\"" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%!G" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#K%'matrixG6#7(7$*&%7classical~method~with~G\"\"\"6$/&%\"cG6#\"\"##F *F0/&F.6#\"\"$F1F*$\"+%y-0-\"!#;7$*&%13/8~method~with~GF*6$/F-#F*F5/F3 #F0F5F*$\"+py*\\Y(!#<7$*&%4Gill's~method~with~GF*F+F*$\"+g\"o$Q5F87$*& %7Dormand's~method~with~GF*6$/F-#F0\"\"&/F3#F5FOF*$\"+Htu#)yFC7$*&%-me thod~with~GF*6$/F-#F5\"\"(/F3#\"\"'\"#6F*$\"+k\\Sz$)FC7$*&FVF*6$FM/&% \"aG6$F5F0F@F*$\"+8U8xxFCQ*pprint1156\"" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 4 "The " }{TEXT 260 22 "root mean squa re error" }{TEXT -1 20 " over the interval " }{XPPEDIT 18 0 "[0, 5]; " "6#7$\"\"!\"\"&" }{TEXT -1 82 " of each Runge-Kutta method is estim ated as follows using the special procedure " }{TEXT 0 5 "NCint" } {TEXT -1 98 " to perform numerical integration by the 7 point Newton- Cotes method over 200 equal subintervals." }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 496 "mthds := [`classical method with `*(c[2]=1/2,c[3]=1/ 2),`3/8 method with `*(c[2]=1/3,c[3]=2/3),`Gill's method with `*(c[2]= 1/2,c[3]=1/2),`Dormand's method with `*(c[2]=2/5,c[3]=3/5),`method wit h `*(c[2]=3/7,c[3]=6/11),`method with `*(c[2]=2/5,a[3,2]=2/3)]: errs : = []:\nDigits := 20:\nfor ct to 6 do\n sm := NCint((p(x)-'pn_RK4_||c t'(x))^2,x=0..5,adaptive=false,numpoints=7,factor=200);\n errs := [o p(errs),sqrt(sm/5)];\nend do:\nDigits := 10:\nlinalg[transpose](conver t([mthds,evalf(errs)],matrix));" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#K%' matrixG6#7(7$*&%7classical~method~with~G\"\"\"6$/&%\"cG6#\"\"##F*F0/&F .6#\"\"$F1F*$\"+#3iSi$!#<7$*&%13/8~method~with~GF*6$/F-#F*F5/F3#F0F5F* $\"+8W1CaF87$*&%4Gill's~method~with~GF*F+F*$\"+yD%3W%F87$*&%7Dormand's ~method~with~GF*6$/F-#F0\"\"&/F3#F5FNF*$\"+DS\"Q\"RF87$*&%-method~with ~GF*6$/F-#F5\"\"(/F3#\"\"'\"#6F*$\"+V(fj>$F87$*&FUF*6$FL/&%\"aG6$F5F0F @F*$\"+m-$*HOF8Q)pprint596\"" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }} {PARA 0 "" 0 "" {TEXT -1 92 "The following error graphs are constructe d using the numerical procedures for the solutions." }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 436 "evalf[30](plot(['pn_RK4_1'(x)-p(x),'pn_RK4 _2'(x)-p(x),'pn_RK4_3'(x)-p(x),'pn_RK4_4'(x)-p(x),\n'pn_RK4_5'(x)-p(x) ,'pn_RK4_6'(x)-p(x)],x=0..2.2,font=[HELVETICA,9],\ncolor=[red,blue,COL OR(RGB,0,.65,0),COLOR(RGB,.4,0,1),magenta,brown],\nlegend=[`classical \+ method`,`3/8 method`,`Gill's method`,`Dormand's method`,`method with c [2]=3/7 and c[3]=6/11`,`method with c[2]=2/5 and a[3,2]=2/3`],title=`e rror curves for order 4 Runge-Kutta methods`));" }}{PARA 13 "" 1 "" {GLPLOT2D 868 572 572 {PLOTDATA 2 "6,-%'CURVESG6%7gu7$$\"\"!F)F(7$$\"? 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=:(4([W:TnxEF-7$Fc`n$!8cCv8'R,`)H[d#F-7$F_[l$!8258#)3.$e_*R[#F-7$Fjbq$ !8?RWC$4)4@<'\\CF-7$F[an$!8%p\"pAqhhIeGU#F-7$Fbcq$!8f4P3')Rtm(=0CF-7$F d[l$!8Z.(*4d6dQ`>R#F-7$Fcan$!8Io5Hc)*)3\\8sBF-7$Fhan$!8(\\YAdHofFBdBF- 7$F\\cn$!8A\\^xUX^hlxM#F-7$Fi[l$!8g8['eM$yY.YM#F-7$Ficn$!8f-\\Xny#)>!* )[BF-7$F^\\l$!8?:42v#fzUIbBF-7$Ffdn$!8=Rjc4jvB;DP#F-7$Fc\\l$!8Dy+6*\\i >e7#Q#F-7$F]fn$!8R8mP253-dST#F-7$Fh\\l$!8%*e!fdF]\\![NX#F-7$Fefn$!8Emu **zel')H$*[#F-7$F]]l$!8TmMv;+))HJ-`#F-7$F^[o$!8J'p>Hg0jTI>EF-7$Fb]l$!8 af^'GIRm&3?x#F-7$Fg]l$!8h&4R2WuRQhcHF-7$F\\^l$!8yx^&QHmfUUZKF-7$Fa\\o$ !8t%QQ,0js#G/X$F-7$Fa^l$!8%y@DGG*3<$*oo$F-7$Fi\\o$!87,n#>$)o\">_='RF-7 $Ff^l$!8:1dAGb`hvNG%F-7$F[_l$!8,F%R9SW8.1UYF-7$F`_l$!7.1arE6(Q()f0&F17 $Fe_l$!7a2*[vw*z#pT[&F17$Fj_l$!7FZ4!pcu:sG%fF17$F_`l$!7Y_$4!eN[BvekF17 $Fd`l$!707BTEW*RTO)pF17$F^al$!7DUkN9q@ZwhuF17$Fhal$!7mUK-COpE$es(F1-F^ bl6&F`bl$\")#)eqkFcbl$\"))eqk\"FcblFe^u-Ffbl6#%Dmethod~with~c[2]=2/5~a nd~a[3,2]=2/3G-%%FONTG6$%*HELVETICAG\"\"*-%&TITLEG6#%Merror~curves~for ~order~4~Runge-Kutta~methodsG-%+AXESLABELSG6$Q\"x6\"Q!Fg_u-%%VIEWG6$;F (Fhal%(DEFAULTG" 1 2 0 1 10 0 2 9 1 4 2 1.000000 45.000000 45.000000 0 1 "classical method" "3/8 method" "Gill's method" "Dormand's method " "method with c[2]=3/7 and c[3]=6/11" "method with c[2]=2/5 and a[3,2 ]=2/3" }}}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 39 "Test 11 of order 4 Runge-Kutta methods " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }} {PARA 0 "" 0 "" {TEXT -1 143 "This example is similar to one that appe ars in an article by F. G. Lether: Mathematics of Computation, Vol. 2 0, no. 95, (July 1966) page 382. " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }} {PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "dy/dx=exp(-x)/(x-1)^2 " "6#/*&%#dyG\"\"\"%#dxG!\"\"*&-%$expG6#,$%\"xGF(F&*$,&F.F&F&F(\"\"#F( " }{TEXT -1 2 " " }{XPPEDIT 18 0 "cos(1/(x-1))-y" "6#,&-%$cosG6#*&\" \"\"F(,&%\"xGF(F(!\"\"F+F(%\"yGF+" }{TEXT -1 5 ", " }{XPPEDIT 18 0 "y(0)=sin*1" "6#/-%\"yG6#\"\"!*&%$sinG\"\"\"F*F*" }{TEXT -1 1 " " }} {PARA 0 "" 0 "" {TEXT -1 10 "Solution: " }}{PARA 256 "" 0 "" {TEXT -1 2 " " }{XPPEDIT 18 0 "y = -exp(-x)*sin(1/(x-1))" "6#/%\"yG,$*&-%$expG 6#,$%\"xG!\"\"\"\"\"-%$sinG6#*&F-F-,&F+F-F-F,F,F-F," }{TEXT -1 1 " " } }{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 185 "de := diff(y(x),x)=exp(-x)/(x-1)^2*cos(1/(x-1))-y(x);\nic := y( 0)=sin(1);\ndsolve(\{de,ic\},y(x));\nq := unapply(rhs(%),x):\nplot(q(x ),x=0..1-1/(6*Pi),font=[HELVETICA,9],labels=[`x`,`y(x)`]);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%#deG/-%%diffG6$-%\"yG6#%\"xGF,,&*(-%$expG6 #,$F,!\"\"\"\"\",&F,F4F4F3!\"#-%$cosG6#*&F4F4F5F3F4F4F)F3" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%#icG/-%\"yG6#\"\"!-%$sinG6#\"\"\"" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/-%\"yG6#%\"xG,$*&-%$expG6#,$F'!\"\"\"\"\"-% $sinG6#*&F/F/,&F'F/F/F.F.F/F." }}{PARA 13 "" 1 "" {GLPLOT2D 603 396 396 {PLOTDATA 2 "6&-%'CURVESG6$7[r7$$\"\"!F)$\"30l*y![)4ZT)!#=7$$\"3#> =\"*)>z2k?!#>$\"3m%p=Qu6DN)F,7$$\"3kMQU\"3@+'QF0$\"3m=O<,5$yO#)F,7$$\"38UQ!)p4'G\"zF0$\"3/Z^$zTlU<)F,7 $$\"3BY$*R0>JO**F0$\"36ty1)z*36\")F,7$$\"3wbXC%*4B\"=\"F,$\"3A;o(=P!Q^ !)F,7$$\"3M!)fxC(zaP\"F,$\"3E8^!ydw!))zF,7$$\"3kgswR?Pw:F,$\"3T8>lD8j? zF,7$$\"3Qnlb!4?mx\"F,$\"3%p4\\s^P4&yF,7$$\"3OsvSC)*f#)>F,$\"3/$H(=wa6 wxF,7$$\"3)Rk+h)o-k@F,$\"3LR8d$>`qq(F,7$$\"3Q^Vo'yq#oBF,$\"3YB)Qc;#3Dw F,7$$\"3?0sMKLNtDF,$\"3,;%fG`C(F, 7$$\"3S+dSsVlWLF,$\"3&36sy[X09(F,7$$\"3EOur83&\\b$F,$\"37)QgTzpp+(F,7$ $\"3c#**Qwz)4TPF,$\"3M]xfz#>l(oF,7$$\"3wx#p)QELXRF,$\"3UR-VbS%zr'F,7$$ \"3\"yy;Gb6)RTF,$\"3%40quzD%\\lF,7$$\"3p2KM(*)HFM%F,$\"3W'4!o9@F_jF,7$ $\"3`G+(=Gs!HXF,$\"3S.Rv)o&))[hF,7$$\"3&\\%Rcec>>`F,7$$\" 39pTj@J(oJ&F,$\"3-j_%RM!Rk\\F,7$$\"3QD(p)Qdl>bF,$\"34_#)R=svWXF,7$$\"3 #)Qm@o*Q!=dF,$\"3Iba)Q0\")Q2%F,7$$\"3#oP&GV\\)*4fF,$\"3;HPYk\"[Jb$F,7$ $\"3qUjqA#3J7'F,$\"3iI9Us9n*)GF,7$$\"3-koIo*3YJ'F,$\"3G0-$>\\f\"3AF,7$ $\"3fQ6D*)o2>lF,$\"3CG]Vmkt$Q\"F,7$$\"35Y]`:_N/nF,$\"33Dv;1s&pZ&F07$$ \"3.\"Q#ekL\"p!pF,$!3OST&zF\\gd%F07$$\"3%yB5rz/v4(F,$!3cn!**p^J2Z\"F,7 $$\"33-p_gxs'H(F,$!3A9e\"Q#e(4b#F,7$$\"3%324>i/:\\(F,$!3S!=W0GJd`$F,7$ $\"39FEN$GhMf(F,$!3U'\\`Wd^A(RF,7$$\"3c%='zWzT&p(F,$!3f\\'**z(4Q8VF,7$ $\"3,%e*>Oh^WxF,$!3G(>*pycjJWF,7$$\"3K#)HgFVh$z(F,$!3)Q%ReLd!G^%F,7$$ \"3h#o/LUj\"=yF,$!3p>V&oBmw`%F,7$$\"3y\"Q1!>DrUyF,$!3MH+5_!>5b%F,7$$\" 3%433Zhhs'yF,$!3KtI_'\\k?b%F,7$$\"3?\"y4/r5=*yF,$!3)*fMuc7)*RXF,7$$\"3 @o'GDq??%zF,$!3)R70$\\E%3Z%F,7$$\"3=bvk%pIA*zF,$!3(yFHocWfL%F,7$$\"3IV kw'oSC/)F,$!3&[P'\\m/)z7%F,7$$\"3II`))y1l#4)F,$!3E`XP\"[()*RQF,7$$\"3? gE&\\8RA>)F,$!3'*pxl2UD4IF,7$$\"37!**>5fF=H)F,$!3Fu3OFE=:=F,7$$\"3$)fW ::LeP$)F,$!374'4ii(p\\6F,7$$\"3kI*)GR!RLQ)F,$!3q\\t;%4rO@%F07$$\"3Z,MU jZ4H%)F,$\"3c]$R4W&G]NF07$$\"3=ryb([][Z)F,$\"37Ccp$Rpv:\"F,7$$\"3%QzY4 r\"HF&)F,$\"3CcG)))y\"yp?F,7$$\"3g()F,$\"3y_cgRCTxTF,7$$\"3Wx*okV>:t)F,$\"37VUul#=X<%F,7$$\"3 q%oYTYr\\v)F,$\"3Ri.A'QV25%F,7$$\"3'>RC=\\B%y()F,$\"3x'=W7Fx&HRF,7$$\" 3]1)zraF`#))F,$\"3QFG%plB4F$F,7$$\"39A_`-;Bs))F,$\"3700_RQTz@F,7$$\"31 ^1[bjB(*))F,$\"3OuNH?R:M9F,7$$\"34\"3E%36CA*)F,$\"3E<)GsFg<(fF07$$\"37 6:PheCZ*)F,$!3ocHHl.]EIF07$$\"39TpJ91Ds*)F,$!38U'y=D^^A\"F,7$$\"3=rBEn `D(**)F,$!3j&4Nt%[@=@F,7$$\"3@,y??,EA!*F,$!3!e@9jLE#=HF,7$$\"3EJK:t[EZ !*F,$!3I(>(y]J<_NF,7$$\"3Gh')4E'pA2*F,$!3U`$=&457URF,7$$\"3'3]u3\"GDy! *F,$!3U'ykJ>F2*RF,7$$\"3cT.l&*fB%3*F,$!3AH^N$RP+-%F,7$$\"3E#=E/=>-4*F, $!3&Rn/8H]\"HSF,7$$\"3'=--_O-i4*F,$!39=u$p*4B>*F,$!39yV6L@mxvF07$$\"3oYr;\"p**Q?*F,$!3 _9%**Gult/#!#?7$$\"3(p#)=21me@*F,$\"3=-0hZ^rztF07$$\"3E20FIC$yA*F,$\"3 MLbSaq^*[\"F,7$$\"3b(=A)*z)zR#*F,$\"3Qccz;X$>?#F,7$$\"3%*oQPp^w^#*F,$ \"3\\RTgHcmRGF,7$$\"3C\\b#*Q:tj#*F,$\"3M$o6)eV:lLF,7$$\"3oE3CP5fw#*F,$ \"3Ma)HpV]]I!H!G$*F,$\"3a %4t07BP*GF,7$$\"3J=s\")G&))3M*F,$\"3f(3)H;[%o+#F,7$$\"3w&\\Kr-[PN*F,$ \"33n1kl[]%*F,$ !3m(=[SoWqQ#F,7$$\"3%>saO,CmX*F,$!3GDS5eP#en\"F,7$$\"3AhB\"Gw`IY*F,$!3 U$3!Gg0_(o)F07$$\"3]++(>^$[p%*F,$!3V'=8$[D+C:!#C-%'COLOURG6&%$RGBG$\"# 5!\"\"F(F(-%%FONTG6$%*HELVETICAG\"\"*-%+AXESLABELSG6$%\"xG%%y(x)G-%%VI EWG6$;F($\"+BN[p%*!#5%(DEFAULTG" 1 2 0 1 10 0 2 9 1 4 2 1.000000 45.000000 45.000000 0 0 "Curve 1" }}}}{PARA 0 "" 0 "" {TEXT -1 0 "" }} {PARA 0 "" 0 "" {TEXT -1 93 "The following code constructs a discrete \+ solution based on each of the methods and gives the " }{TEXT 260 22 "r oot mean square error" }{TEXT -1 18 " of each solution." }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 868 "Q := (x,y) -> exp(-x)/(x-1)^2*cos( 1/(x-1))-y: hh := 1/500-1/(3000*Pi): numsteps := 500: x0 := 0: y0 := s in(1):\nmatrix([[`slope field: `,Q(x,y)],[`initial point: `,``(x0,y0 )],[`step width: `,hh],\n[`no. of steps: `,numsteps]]);``;\nmthds \+ := [`classical method with `*(c[2]=1/2,c[3]=1/2),`3/8 method with `*(c [2]=1/3,c[3]=2/3),`Gill's method with `*(c[2]=1/2,c[3]=1/2),`Dormand's method with `*(c[2]=2/5,c[3]=3/5),`method with `*(c[2]=3/7,c[3]=6/11) ,`method with `*(c[2]=2/5,a[3,2]=2/3)]: errs := []:\nDigits := 30:\nfo r ct to 6 do\n Qn_RK4_||ct := RK4_||ct(Q(x,y),x,y,x0,evalf[33](y0),e valf[33](hh),numsteps,false);\n sm := 0: numpts := nops(Qn_RK4_||ct) :\n for ii to numpts do\n sm := sm+(Qn_RK4_||ct[ii,2]-q(Qn_RK4_ ||ct[ii,1]))^2;\n end do:\n errs := [op(errs),sqrt(sm/numpts)];\ne nd do:\nDigits := 10:\nlinalg[transpose](convert([mthds,evalf(errs)],m atrix));" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#K%'matrixG6#7&7$%0slope~fi eld:~~~G,&*(-%$expG6#,$%\"xG!\"\"\"\"\",&F/F1F1F0!\"#-%$cosG6#*&F1F1F2 F0F1F1%\"yGF07$%0initial~point:~G-%!G6$\"\"!-%$sinG6#F17$%/step~width: ~~~G,&#F1\"$+&F1*&F1F1*&\"%+IF1%#PiGF1F0F07$%1no.~of~steps:~~~GFFQ)ppr int606\"" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%!G" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#K%'matrixG6#7(7$*&%7classical~method~with~G\"\"\"6$/&% \"cG6#\"\"##F*F0/&F.6#\"\"$F1F*$\"+7(4Dg\"!#:7$*&%13/8~method~with~GF* 6$/F-#F*F5/F3#F0F5F*$\"+mD)>6(!#;7$*&%4Gill's~method~with~GF*F+F*F67$* &%7Dormand's~method~with~GF*6$/F-#F0\"\"&/F3#F5FMF*$\"+rgd\"G\"F87$*&% -method~with~GF*6$/F-#F5\"\"(/F3#\"\"'\"#6F*$\"++^M)\\\"F87$*&FTF*6$FK /&%\"aG6$F5F0F@F*$\"+2NEO8F8Q)pprint616\"" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 30 "The following code constructs \+ " }{TEXT 260 20 "numerical procedures" }{TEXT -1 56 " for solutions ba sed on each of the Runge-Kutta schemes." }}{PARA 0 "" 0 "" {TEXT -1 75 "The error in the value obtained by each of the methods at the poin t where " }{XPPEDIT 18 0 "x = 0;" "6#/%\"xG\"\"!" }{TEXT -1 21 ".9469 is also given." }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 792 "Q := (x ,y) -> exp(-x)/(x-1)^2*cos(1/(x-1))-y: hh := 1/500-1/(3000*Pi): numste ps := 500: x0 := 0: y0 := sin(1):\nmatrix([[`slope field: `,Q(x,y)], [`initial point: `,``(x0,y0)],[`step width: `,hh],\n[`no. of steps: \+ `,numsteps]]);``;\nmthds := [`classical method with `*(c[2]=1/2,c[3] =1/2),`3/8 method with `*(c[2]=1/3,c[3]=2/3),`Gill's method with `*(c[ 2]=1/2,c[3]=1/2),`Dormand's method with `*(c[2]=2/5,c[3]=3/5),`method \+ with `*(c[2]=3/7,c[3]=6/11),`method with `*(c[2]=2/5,a[3,2]=2/3)]: err s := []:\nDigits := 30:\nfor ct to 6 do\n qn_RK4_||ct := RK4_||ct(Q( x,y),x,y,x0,evalf(y0),evalf(hh),numsteps,true);\nend do:\nxx := 0.9469 : qxx := evalf(q(xx)):\nfor ct to 6 do\n errs := [op(errs),abs(qn_RK 4_||ct(xx)-qxx)];\nend do:\nDigits := 10:\nlinalg[transpose](convert([ mthds,evalf(errs)],matrix));" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#K%'mat rixG6#7&7$%0slope~field:~~~G,&*(-%$expG6#,$%\"xG!\"\"\"\"\",&F/F1F1F0! \"#-%$cosG6#*&F1F1F2F0F1F1%\"yGF07$%0initial~point:~G-%!G6$\"\"!-%$sin G6#F17$%/step~width:~~~G,&#F1\"$+&F1*&F1F1*&\"%+IF1%#PiGF1F0F07$%1no.~ of~steps:~~~GFFQ*pprint1126\"" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%!G" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#K%'matrixG6#7(7$*&%7classical~method ~with~G\"\"\"6$/&%\"cG6#\"\"##F*F0/&F.6#\"\"$F1F*$\"+70Ir=!#97$*&%13/8 ~method~with~GF*6$/F-#F*F5/F3#F0F5F*$\"+,<&GH)!#:7$*&%4Gill's~method~w ith~GF*F+F*F67$*&%7Dormand's~method~with~GF*6$/F-#F0\"\"&/F3#F5FMF*$\" +RA*f\\\"F87$*&%-method~with~GF*6$/F-#F5\"\"(/F3#\"\"'\"#6F*$\"+0E5\\< F87$*&FTF*6$FK/&%\"aG6$F5F0F@F*$\"+;ELf:F8Q*pprint1136\"" }}}{PARA 0 " " 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 4 "The " }{TEXT 260 22 "root mean square error" }{TEXT -1 19 " over the interval " } {XPPEDIT 18 0 " [0, 1-1/(6*Pi)] " "6#7$\"\"!,&\"\"\"F&*&F&F&*&\"\"'F&% #PiGF&!\"\"F+" }{TEXT -1 82 " of each Runge-Kutta method is estimated as follows using the special procedure " }{TEXT 0 5 "NCint" }{TEXT -1 98 " to perform numerical integration by the 7 point Newton-Cotes \+ method over 200 equal subintervals." }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 516 "mthds := [`classical method with `*(c[2]=1/2,c[3]=1/ 2),`3/8 method with `*(c[2]=1/3,c[3]=2/3),`Gill's method with `*(c[2]= 1/2,c[3]=1/2),`Dormand's method with `*(c[2]=2/5,c[3]=3/5),`method wit h `*(c[2]=3/7,c[3]=6/11),`method with `*(c[2]=2/5,a[3,2]=2/3)]: errs : = []:\nDigits := 20:\nfor ct to 6 do\n sm := NCint((q(x)-'qn_RK4_||c t'(x))^2,x=0..1-1/(6*Pi),adaptive=false,numpoints=7,factor=200);\n e rrs := [op(errs),sqrt(sm/(1-1/(6*Pi)))];\nend do:\nDigits := 10:\nlina lg[transpose](convert([mthds,evalf(errs)],matrix));" }}{PARA 11 "" 1 " " {XPPMATH 20 "6#K%'matrixG6#7(7$*&%7classical~method~with~G\"\"\"6$/& %\"cG6#\"\"##F*F0/&F.6#\"\"$F1F*$\"+EAE/9!#:7$*&%13/8~method~with~GF*6 $/F-#F*F5/F3#F0F5F*$\"+Sd;Oi!#;7$*&%4Gill's~method~with~GF*F+F*F67$*&% 7Dormand's~method~with~GF*6$/F-#F0\"\"&/F3#F5FMF*$\"+BC@B6F87$*&%-meth od~with~GF*6$/F-#F5\"\"(/F3#\"\"'\"#6F*$\"+;A;88F87$*&FTF*6$FK/&%\"aG6 $F5F0F@F*$\"+CRHr6F8Q)pprint646\"" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }} {PARA 0 "" 0 "" {TEXT -1 92 "The following error graphs are constructe d using the numerical procedures for the solutions." }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 474 "evalf[20](plot(['qn_RK4_1'(x)-q(x),'qn_RK4 _2'(x)-q(x),'qn_RK4_3'(x)-q(x),'qn_RK4_4'(x)-q(x),\n'qn_RK4_5'(x)-q(x) ,'qn_RK4_6'(x)-q(x)],x=0..0.7,-6e-12..1.9e-11,font=[HELVETICA,9],\ncol or=[red,blue,COLOR(RGB,0,.65,0),COLOR(RGB,.4,0,1),magenta,brown],thick ness=[1$2,2,1$3],\nlegend=[`classical method`,`3/8 method`,`Gill's met hod`,`Dormand's method`,`method with c[2]=3/7 and c[3]=6/11`,`method w ith c[2]=2/5 and a[3,2]=2/3`],title=`error curves for order 4 Runge-Ku tta methods`));" }}{PARA 13 "" 1 "" {GLPLOT2D 786 534 534 {PLOTDATA 2 "6,-%'CURVESG6&7fn7$$\"\"!F)F(7$$\"5mmmmm\"z+e_\"!#@$!(XwI\"!#?7$$\"5L LLL3->R`GF-$!(wLc#F07$$\"5mmmm;apSYVF-$!(DO4%F07$$\"5lmmm;z'=$\\eF-$!( &H$y&F07$$\"5KLLL3Ft3XtF-$!(dxk(F07$$\"5lmmmTNj&=t)F-$!(d/r*F07$$\"5++ +](=`xn,\"F0$!))=7=\"F07$$\"5mmm;ay/Gl6F0$!)sF0$!)B:NKF07$$\"5++++]s2O[?F0 $!)dwZPF07$$\"5mmm;aG\"H5=#F0$!)QU[UF07$$\"5LLLL$ej%yQBF0$!)nQ))[F07$$ \"5LLLLLVUUsCF0$!)j=HsF07$$\"5******\\(oibk\"HF0$!)[:]#)F07$$\"5******\\i!o<-1$ F0$!)&4TE*F07$$\"5LLLL3-$=-@$F0$!*e9]0\"F07$$\"5LLL$3xplzM$F0$!*#oL'= \"F07$$\"5mmmm\"H([a'\\$F0$!*%[yW8F07$$\"5mmm;ayo(3l$F0$!*j?>`\"F07$$ \"5******\\7VLA&y$F0$!*))znr\"F07$$\"5mmmmT07KIRF0$!*p%4J>F07$$\"5**** *****\\\\@-3%F0$!*1YJ=#F07$$\"5*******\\PopoA%F0$!*y\"GjCF07$$\"5***** *\\(oMf(oVF0$!*L(*Rt#F07$$\"5*******\\ii.j_%F0$!*K)\\,JF07$$\"5KLLLLoT 'ym%F0$!*C*43MF07$$\"5********\\i-,>[F0$!*+e#pPF07$$\"5mmm;a)3rf&\\F0$ !*NQr3%F07$$\"5********\\Zaq0^F0$!*.QUT%F07$$\"5KLL$3-\"QfY_F0$!*,Rwn% F07$$\"5******\\PWF'QR&F0$!*@_vz%F07$$\"5KLLL$e/Xy`&F0$!*2\"\\CZF07$$ \"5******\\(=<\"e)o&F0$!*G?cG%F07$$\"5lmmmmwzvLeF0$!*VT\"eKF07$$\"5lmm m\"zAAA)fF0$!*tnaX\"F07$$\"5KLL$3-7d%HhF0$\"*Ruci\"F07$$\"5*********\\ p]ZE'F0$\"*\\8mD'F07$$\"5lmm;HZ:GUjF0$\"+'ymc+\"F07$$\"5KLLLe*R7)>kF0$ \"+\\X\\/:F07$$\"5+++]7=p:*['F0$\"+9\\&*e>F07$$\"5lmmmmO9]elF0$\"+3Q%y i#F07$$\"5ILL3xcrVKmF0$\",XmAIZ$F-7$$\"5)*****\\(o(GP1nF0$\",:4qG^%F-7 $$\"5+++v$f$evTnF0$\",edmo2&F-7$$\"5+++++&zQrx'F0$\",G+eBe&F-7$$\"5+++ D1a<_7oF0$\"-w3;Y0i!#A7$$\"5)*****\\78Z!z%oF0$\",:Ron$pF-7$$\"5++]P%[` Gf)oF0$\",[(>z!z(F-7$$\"5+++DccB&R#pF0$\",EK!HH()F-7$$\"5++]7Gyh(>'pF0 $\",m%zte(*F-7$$\"\"(!\"\"$\"-x?#o&)3\"F--%'COLOURG6&%$RGBG$\"*++++\"! 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" 0 "" {MPLTEXT 1 0 1 ";" }}}}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 40 "Test 12 of order 4 Ru nge-Kutta methods " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 " " {TEXT -1 0 "" }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "dy/ dx=exp(-x)/(x-1)^2" "6#/*&%#dyG\"\"\"%#dxG!\"\"*&-%$expG6#,$%\"xGF(F&* $,&F.F&F&F(\"\"#F(" }{TEXT -1 2 " " }{XPPEDIT 18 0 "5*y*sin^7*7*x;" " 6#*,\"\"&\"\"\"%\"yGF%%$sinG\"\"(F(F%%\"xGF%" }{TEXT -1 5 ", " } {XPPEDIT 18 0 "y(0) = 1;" "6#/-%\"yG6#\"\"!\"\"\"" }{TEXT -1 1 " " }} {PARA 0 "" 0 "" {TEXT -1 10 "Solution: " }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "y = exp(16/49+5/3136*cos*49*x-cos*35*x/64+5/64*c os*21*x-25/64*cos*7*x);" "6#/%\"yG-%$expG6#,,*&\"#;\"\"\"\"#\\!\"\"F+* ,\"\"&F+\"%OJF-%$cosGF+F,F+%\"xGF+F+**F1F+\"#NF+F2F+\"#kF-F-*,F/F+F5F- F1F+\"#@F+F2F+F+*,\"#DF+F5F-F1F+\"\"(F+F2F+F-" }{TEXT -1 2 " " }} {PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 213 "de := diff(y(x),x)=5*y(x)*sin(7*x)^7;\nic := y(0)=1;\ndsolve(\{de ,ic\},y(x)):\ny(x)=combine((numer(rhs(%))/convert(denom(rhs(%)),exp))) ;\nr := unapply(rhs(%),x):\nplot(r(x),x=0..5,font=[HELVETICA,9],labels =[`x`,`y(x)`]);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%#deG/-%%diffG6$-% \"yG6#%\"xGF,,$*(\"\"&\"\"\"F)F0)-%$sinG6#,$*&\"\"(F0F,F0F0F7F0F0" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#>%#icG/-%\"yG6#\"\"!\"\"\"" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/-%\"yG6#%\"xG-%$expG6#,,*&#\"#D\"#k\"\"\"-% $cosG6#,$*&\"\"(F0F'F0F0F0!\"\"*&#F0F/F0-F26#,$*&\"#NF0F'F0F0F0F7*&#\" \"&\"%OJF0-F26#,$*&\"#\\F0F'F0F0F0F0*&#FAF/F0-F26#,$*&\"#@F0F'F0F0F0F0 #\"#;FGF0" }}{PARA 13 "" 1 "" {GLPLOT2D 806 286 286 {PLOTDATA 2 "6&-%' CURVESG6$7_^l7$$\"\"!F)$\"\"\"F)7$$\"3ALL$3FWYs#!#>$\"3RX36^,++5!#<7$$ \"3WmmmT&)G\\aF/$\"3/h:lL\\.+5F27$$\"3MKL3x1h6oF/$\"3N>!>$zG>+5F27$$\" 3m****\\7G$R<)F/$\"3^$*H^L`v+5F27$$\"3±z%\\DO&*F/$\"3J?*f5+AB+\"F27 $$\"3GLLL3x&)*3\"!#=$\"3x]lWM_&f+\"F27$$\"3em\"z%\\v#pK\"FJ$\"3$\\ClDT `A-\"F27$$\"3))**\\i!R(*Rc\"FJ$\"3%z>L#y^ah5F27$$\"3_;z>6B`#o\"FJ$\"3c ]f\"H$G$Q4\"F27$$\"3!**HO6F27$$\"34]PM_@g>>FJ$\"33 RW!o::'*=\"F27$$\"3umm\"H2P\"Q?FJ$\"3?q+*)GVj`7F27$$\"3C]7G))>Wr@FJ$\" 3L$)HfnG\"pL\"F27$$\"3YLek.pu/BFJ$\"3Gv`o46\")G9F27$$\"33D\"G8O*RrBFJ$ \"3+PI&HSekZ\"F27$$\"3o;/,>=0QCFJ$\"3mjV0*4cU_\"F27$$\"3c3FpwUq/DFJ$\" 3#pc5I%eTr:F27$$\"3!***\\PMnNrDFJ$\"3U+AysC:<;F27$$\"37$eR(\\;m/FFJ$\" 3)f:eA`w9q\"F27$$\"3MmT5ll'z$GFJ$\"35TSzD!3Dx\"F27$$\"37](o/[r7(HFJ$\" 3e:[6>4vF=F27$$\"3MLL$eRwX5$FJ$\"3ICf`%)[>n=F27$$\"3_LLe*[`HP$FJ$\"3!e Z8'p.63>F27$$\"3rLLL$eI8k$FJ$\"3%[F&)3\"\\`>>F27$$\"3_L$3-8>bx$FJ$\"3I prJEM*3#>F27$$\"3*QL$3xwq4RFJ$\"3rrv?(f28#>F27$$\"3EM$eRA'*Q/%FJ$\"3X^ 'fP/+9#>F27$$\"33ML$3x%3yTFJ$\"3cOdy3HT@>F27$$\"3h+]PfyG7ZFJ$\"3VH&=@l 89#>F27$$\"3emm\"z%4\\Y_FJ$\"3ml4uC:e?>F27$$\"3C+]P4'4.P&FJ$\"3`*yZBqt )=>F27$$\"3'QLL3FGT\\&FJ$\"3t^RtM))*[\">F27$$\"3Um;HKp%zh&FJ$\"3!*H'3: A\\o!>F27$$\"32++v$flV;F27$$\"3I++vVVX$\\'FJ$\"3s/S7/D%eb\"F27$$\"3pLL$eRh;i'FJ$\"3)z%[YI ??k9F27$$\"31nm\"zWo)\\nFJ$\"31\")Q9\"48QP\"F27$$\"3W++++b2yoFJ$\"3k#R e\"fbk*G\"F27$$\"3%QL$3_DG1qFJ$\"3jY.gupS:7F27$$\"3Anm;/'*[MrFJ$\"3y@x $3A]K:\"F27$$\"3]***\\il'pisFJ$\"3_zleRG$Q5\"F27$$\"3+MLe*[!)y_(FJ$\"3 :2l<5ljQ5F27$$\"3Qnm\"HKkIz(FJ$\"3z[=lFuU55F27$$\"35MLeRilDzFJ$\"3tR>* >I/Y+\"F27$$\"3!3+]i:[#e!)FJ$\"3ZybuD9w,5F27$$\"3[nm\"H2S3>)FJ$\"36f+O Z/c+5F27$$\"3>MLe*)>VB$)FJ$\"3q[OmV#R,+\"F27$$\"3wmmTg()4_))FJ$\"3!z[$ G++++5F27$$\"3Y++DJbw!Q*FJ$\"3(f!Q*>W.++\"F27$$\"3E$3_+A:n^*FJ$\"3$o[9 !4H.+5F27$$\"3=nT&)3\\m_'*FJ$\"3qlL!>[$=+5F27$$\"33^il(f9')y*FJ$\"39t! )=KMs+5F27$$\"3+N$ekGkX#**FJ$\"3)[%4Vj`B-5F27$$\"31]iSmjk>5F2$\"3?UoC! *)fG,\"F27$$\"3%ommTIOo/\"F2$\"3%pJ)[*fok/\"F27$$\"3?]i:g2\")e5F2$\"3' ob!*>J(\\t5F27$$\"3cLe9;_yq5F2$\"3!f)3IJ$>-6\"F27$$\"3!pTN@nfF3\"F2$\" 3O\\W6vrpd6F27$$\"3E+]7GTt%4\"F2$\"3&p\"fsjqH;7F27$$\"3i$e9Te3n5\"F2$ \"3e&f$*[!\\V&G\"F27$$\"3(p;/,/$o=6F2$\"3K1G\">I$Qj8F27$$\"3L]P4'\\d18 \"F2$\"3mAQZ9DAZ9F27$$\"3YLL3_>jU6F2$\"3A//J![AI`\"F27$$\"3)\\PMF:s$\\ 6F2$\"3%**=P`q)\\!e\"F27$$\"3u;aQ`B6c6F2$\"3vn0oImOE;F27$$\"3[ek.aD&G; 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$\"30z2N6y#HT\"F27$$\"3SfRs3!)QmUF2$\"3E%zqA:-pO\"F27$$\"3)RL$3xe,tUF2 $\"3Hzu(eD`EK\"F27$$\"3kv$4'\\=;'G%F2$\"32(*p'*eu>U7F27$$\"3G#[Z %F2$\"3S)\\h/++++\"F27$$\"3TM$3_5,-`%F2$\"3y/++\"F27$$\"3SnmT&G!e &e%F2$\"3$eA?a3RF+\"F27$$\"3fLe*[=Y.h%F2$\"3M:+/vy285F27$$\"3m+]P%37^j %F2$\"356M^f6but%F2$\"3%\\.QNQyVc\"F27$$\"3ID19>zl]ZF2$\"3S[]>-K'z\"F27$$\"37+]i SjE!z%F2$\"3&)4L&Q*oyW=F27$$\"3y*\\7G))Rb\"[F2$\"3)fCVDB`!)*=F27$$\"3L +++DM\"3%[F2$\"3$Gxb\\1an\">F27$$\"3i]P4'>]M&[F2$\"3p-qe54u>>F27$$\"3) 3](=np3m[F2$\"3zS9J5F#4#>F27$$\"3G]7GQPsy[F2$\"3j:7^7UI@>F27$$\"3a+]P4 0O\"*[F2$\"3L:$**HR(R@>F27$$\"3s+voa-oX\\F2$\"3%e\"e*Rr89#>F27$$\"\"&F )$\"3n\\kX'z.7#>F2-%'COLOURG6&%$RGBG$\"#5!\"\"F(F(-%%FONTG6$%*HELVETIC AG\"\"*-%+AXESLABELSG6$%\"xG%%y(x)G-%%VIEWG6$;F(F_[q%(DEFAULTG" 1 2 0 1 10 0 2 9 1 4 2 1.000000 45.000000 45.000000 0 0 "Curve 1" }}}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 32 "The following c ode constructs a " }{TEXT 260 17 "discrete solution" }{TEXT -1 44 " ba sed on each of the methods and gives the " }{TEXT 260 22 "root mean sq uare error" }{TEXT -1 18 " of each solution." }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 812 "R := (x,y) -> 5*y*sin(7*x)^7: hh := 0.01: numstep s := 500: x0 := 0: y0 := 1:\nmatrix([[`slope field: `,R(x,y)],[`init ial point: `,``(x0,y0)],[`step width: `,hh],\n[`no. of steps: `,nu msteps]]);``;\nmthds := [`classical method with `*(c[2]=1/2,c[3]=1/2), `3/8 method with `*(c[2]=1/3,c[3]=2/3),`Gill's method with `*(c[2]=1/2 ,c[3]=1/2),`Dormand's method with `*(c[2]=2/5,c[3]=3/5),`method with ` *(c[2]=3/7,c[3]=6/11),`method with `*(c[2]=2/5,a[3,2]=2/3)]: errs := [ ]:\nDigits := 30:\nfor ct to 6 do\n Rn_RK4_||ct := RK4_||ct(R(x,y),x ,y,x0,y0,hh,numsteps,false);\n sm := 0: numpts := nops(Rn_RK4_||ct): \n for ii to numpts do\n sm := sm+(Rn_RK4_||ct[ii,2]-r(Rn_RK4_| |ct[ii,1]))^2;\n end do:\n errs := [op(errs),sqrt(sm/numpts)];\nen d do:\nDigits := 10:\nlinalg[transpose](convert([mthds,evalf(errs)],ma trix));" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#K%'matrixG6#7&7$%0slope~fie ld:~~~G,$*(\"\"&\"\"\"%\"yGF,)-%$sinG6#,$*&\"\"(F,%\"xGF,F,F4F,F,7$%0i nitial~point:~G-%!G6$\"\"!F,7$%/step~width:~~~G$F,!\"#7$%1no.~of~steps :~~~G\"$+&Q)pprint656\"" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%!G" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#K%'matrixG6#7(7$*&%7classical~method~w ith~G\"\"\"6$/&%\"cG6#\"\"##F*F0/&F.6#\"\"$F1F*$\"+Ve[6H!#;7$*&%13/8~m ethod~with~GF*6$/F-#F*F5/F3#F0F5F*$\"+Y$*f0`F87$*&%4Gill's~method~with ~GF*F+F*F67$*&%7Dormand's~method~with~GF*6$/F-#F0\"\"&/F3#F5FLF*$\"+hC W`DF87$*&%-method~with~GF*6$/F-#F5\"\"(/F3#\"\"'\"#6F*$\"+%)QZ18F87$*& FSF*6$FJ/&%\"aG6$F5F0F@F*$\"+f#QqC#F8Q)pprint666\"" }}}{PARA 0 "" 0 " " {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 30 "The following code cons tructs " }{TEXT 260 20 "numerical procedures" }{TEXT -1 56 " for solut ions based on each of the Runge-Kutta schemes." }}{PARA 0 "" 0 "" {TEXT -1 75 "The error in the value obtained by each of the methods at the point where " }{XPPEDIT 18 0 "x = 4.999;" "6#/%\"xG-%&FloatG6$\" %**\\!\"$" }{TEXT -1 16 " is also given." }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 743 "R := (x,y) -> 5*y*sin(7*x)^7: hh := 0.01: numsteps : = 500: x0 := 0: y0 := 1:\nmatrix([[`slope field: `,R(x,y)],[`initial point: `,``(x0,y0)],[`step width: `,hh],\n[`no. of steps: `,numst eps]]);``;\nmthds := [`classical method with `*(c[2]=1/2,c[3]=1/2),`3/ 8 method with `*(c[2]=1/3,c[3]=2/3),`Gill's method with `*(c[2]=1/2,c[ 3]=1/2),`Dormand's method with `*(c[2]=2/5,c[3]=3/5),`method with `*(c [2]=3/7,c[3]=6/11),`method with `*(c[2]=2/5,a[3,2]=2/3)]: errs := []: \nDigits := 25:\nfor ct to 6 do\n rn_RK4_||ct := RK4_||ct(R(x,y),x,y ,x0,y0,hh,numsteps,true);\nend do:\nxx := 4.999: rxx := evalf(r(xx)): \nfor ct to 6 do\n errs := [op(errs),abs(rn_RK4_||ct(xx)-rxx)];\nend do:\nDigits := 10:\nlinalg[transpose](convert([mthds,evalf(errs)],mat rix));" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#K%'matrixG6#7&7$%0slope~fiel d:~~~G,$*(\"\"&\"\"\"%\"yGF,)-%$sinG6#,$*&\"\"(F,%\"xGF,F,F4F,F,7$%0in itial~point:~G-%!G6$\"\"!F,7$%/step~width:~~~G$F,!\"#7$%1no.~of~steps: ~~~G\"$+&Q*pprint1106\"" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%!G" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#K%'matrixG6#7(7$*&%7classical~method~w ith~G\"\"\"6$/&%\"cG6#\"\"##F*F0/&F.6#\"\"$F1F*$\"+E*H:V'!#;7$*&%13/8~ method~with~GF*6$/F-#F*F5/F3#F0F5F*$\"+JRz,**F87$*&%4Gill's~method~wit h~GF*F+F*F67$*&%7Dormand's~method~with~GF*6$/F-#F0\"\"&/F3#F5FLF*$\"+' GT*HOF87$*&%-method~with~GF*6$/F-#F5\"\"(/F3#\"\"'\"#6F*$\"+bOv4V!#<7$ *&FSF*6$FJ/&%\"aG6$F5F0F@F*$\"++-n9GF8Q*pprint1116\"" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 4 "The " }{TEXT 260 22 "ro ot mean square error" }{TEXT -1 20 " over the interval " }{XPPEDIT 18 0 "[0, 5];" "6#7$\"\"!\"\"&" }{TEXT -1 82 " of each Runge-Kutta me thod is estimated as follows using the special procedure " }{TEXT 0 5 "NCint" }{TEXT -1 98 " to perform numerical integration by the 7 po int Newton-Cotes method over 200 equal subintervals." }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 496 "mthds := [`classical method with `*(c[2] =1/2,c[3]=1/2),`3/8 method with `*(c[2]=1/3,c[3]=2/3),`Gill's method w ith `*(c[2]=1/2,c[3]=1/2),`Dormand's method with `*(c[2]=2/5,c[3]=3/5) ,`method with `*(c[2]=3/7,c[3]=6/11),`method with `*(c[2]=2/5,a[3,2]=2 /3)]: errs := []:\nDigits := 20:\nfor ct to 6 do\n sm := NCint((r(x) -'rn_RK4_||ct'(x))^2,x=0..5,adaptive=false,numpoints=7,factor=200);\n \+ errs := [op(errs),sqrt(sm/5)];\nend do:\nDigits := 10:\nlinalg[trans pose](convert([mthds,evalf(errs)],matrix));" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#K%'matrixG6#7(7$*&%7classical~method~with~G\"\"\"6$/&% \"cG6#\"\"##F*F0/&F.6#\"\"$F1F*$\"+:qk-H!#;7$*&%13/8~method~with~GF*6$ /F-#F*F5/F3#F0F5F*$\"+LSD(H&F87$*&%4Gill's~method~with~GF*F+F*F67$*&%7 Dormand's~method~with~GF*6$/F-#F0\"\"&/F3#F5FLF*$\"+e))z^DF87$*&%-meth od~with~GF*6$/F-#F5\"\"(/F3#\"\"'\"#6F*$\"+B(pfI\"F87$*&FSF*6$FJ/&%\"a G6$F5F0F@F*$\"+\"Q;hC#F8Q)pprint696\"" }}}{PARA 0 "" 0 "" {TEXT -1 0 " " }}{PARA 0 "" 0 "" {TEXT -1 92 "The following error graphs are constr ucted using the numerical procedures for the solutions." }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 456 "evalf[20](plot([r(x)-'rn_RK4_1'(x) ,r(x)-'rn_RK4_2'(x),r(x)-'rn_RK4_3'(x),r(x)-'rn_RK4_4'(x),\nr(x)-'rn_R K4_5'(x),r(x)-'rn_RK4_6'(x)],x=0..5,font=[HELVETICA,9],\ncolor=[red,bl ue,COLOR(RGB,0,.65,0),COLOR(RGB,.4,0,1),magenta,brown],thickness=[1$2, 2,1$3],\nlegend=[`classical method`,`3/8 method`,`Gill's method`,`Dorm and's method`,`method with c[2]=3/7 and c[3]=6/11`,`method with c[2]=2 /5 and a[3,2]=2/3`],title=`error curves for order 4 Runge-Kutta method s`));" }}{PARA 13 "" 1 "" {GLPLOT2D 977 637 637 {PLOTDATA 2 "6,-%'CURV ESG6&7cjl7$$\"\"!F)F(7$$\"5ommmTN@Ki8!#@$!*S]].#!#>7$$\"5NLLL$3FWYs#F- $!+N=N(*pF07$$\"5-+++D1k'p3%F-$!,4]AI/'F07$$\"5qmmmmT&)G\\aF-$!-L&>\"* )>8F07$$\"5SLLL3x1h6oF-$!-#\\m90q#F07$$\"50+++]7G$R<)F-$!-YJZibXF07$$ 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UI$eE]ZF07$F]bt$!.K2\"R`yZF07$Fbbt$!.f*eE0kXF07$Fgbt$!.,*\\jB4TF07$F\\ ct$!.q6WJ:f$F07$Fact$!.oj4/T0$F07$Ffct$!.s*f;weDF07$F[dt$!.:XzFCI#F07$ F`dt$!.PuRjw]#F07$Fedt$!.'*)o`!G+$F07$Fjdt$!.![F(G/5$F07$F_et$!.H8%Hhd JF07$Fdet$!.eW-1,<$F07$Fiet$!.\">8e6qJF07$F^ft$!.\"48/(*oJF07$Fcft$!.' \\MD)=;$F07$Fhft$!.6T0uy:$F07$F]gt$!.eFbha:$F07$Fbgt$!.>yT/58$F07$Fggt $!.RR%>.@JF07$F\\ht$!.*3Q>pHIF07$Faht$!.f[o8Qu#F0-Ffht6&Fhht$\")#)eqkF [it$\"))eqk\"F[itFif`mF\\it-Fait6#%Dmethod~with~c[2]=2/5~and~a[3,2]=2/ 3G-%%FONTG6$%*HELVETICAG\"\"*-%&TITLEG6#%Merror~curves~for~order~4~Run ge-Kutta~methodsG-%+AXESLABELSG6$Q\"x6\"Q!F[h`m-%%VIEWG6$;F(Faht%(DEFA ULTG" 1 2 0 1 10 0 2 9 1 4 2 1.000000 45.000000 45.000000 0 1 "classic al method" "3/8 method" "Gill's method" "Dormand's method" "method wit h c[2]=3/7 and c[3]=6/11" "method with c[2]=2/5 and a[3,2]=2/3" }}}} {PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 39 "Test 13 of order 4 Rung e-Kutta methods " }}{PARA 256 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 72 "See: \"Mathematica in Action\" by Stan Wagon, Springer-V erlag, page 302. " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 256 "" 0 " " {TEXT -1 3 " " }{XPPEDIT 18 0 "dy/dx = cos*x+2*y;" "6#/*&%#dyG\"\" \"%#dxG!\"\",&*&%$cosGF&%\"xGF&F&*&\"\"#F&%\"yGF&F&" }{TEXT -1 8 ", \+ " }{XPPEDIT 18 0 "y(0) = -2/5;" "6#/-%\"yG6#\"\"!,$*&\"\"#\"\"\"\" \"&!\"\"F-" }{TEXT -1 1 " " }}{PARA 0 "" 0 "" {TEXT -1 10 "Solution: \+ " }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "y = 1/5;" "6#/%\" yG*&\"\"\"F&\"\"&!\"\"" }{TEXT -1 1 " " }{XPPEDIT 18 0 "sin*x-2/5" "6# ,&*&%$sinG\"\"\"%\"xGF&F&*&\"\"#F&\"\"&!\"\"F+" }{TEXT -1 1 " " } {XPPEDIT 18 0 "cos*x" "6#*&%$cosG\"\"\"%\"xGF%" }{TEXT -1 3 " . " }} {PARA 0 "" 0 "" {TEXT 260 4 "Note" }{TEXT -1 2 ": " }}{PARA 0 "" 0 "" {TEXT -1 53 "The general solution of the differential equation " } {XPPEDIT 18 0 "dy/dx = cos*x+2*y;" "6#/*&%#dyG\"\"\"%#dxG!\"\",&*&%$co sGF&%\"xGF&F&*&\"\"#F&%\"yGF&F&" }{TEXT -1 64 " contains an exponenti al term, but with the initial condition " }{XPPEDIT 18 0 "y(0) = -2/5 " "6#/-%\"yG6#\"\"!,$*&\"\"#\"\"\"\"\"&!\"\"F-" }{TEXT -1 23 " this t erm disappears." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> \+ " 0 "" {MPLTEXT 1 0 50 "de := diff(y(x),x)=cos(x)+2*y(x);\ndsolve(de,y (x));" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%#deG/-%%diffG6$-%\"yG6#%\"x GF,,&-%$cosGF+\"\"\"*&\"\"#F0F)F0F0" }}{PARA 11 "" 1 "" {XPPMATH 20 "6 #/-%\"yG6#%\"xG,(*&#\"\"#\"\"&\"\"\"-%$cosGF&F-!\"\"*&#F-F,F--%$sinGF& F-F-*&-%$expG6#,$*&F+F-F'F-F-F-%$_C1GF-F-" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 105 "Any slight deviation of a nume rical solution from the correct solution tends to become rapidly magni fied." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 154 "de := diff(y(x),x)=cos(x)+2*y(x);\nic := y(0)=-2/5; \ndsolve(\{de,ic\},y(x));\ne := unapply(rhs(%),x):\nplot(e(x),x=0..8,f ont=[HELVETICA,9],labels=[`x`,`y(x)`]);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%#deG/-%%diffG6$-%\"yG6#%\"xGF,,&-%$cosGF+\"\"\"*&\"\"#F0F)F0F0 " }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%#icG/-%\"yG6#\"\"!#!\"#\"\"&" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#/-%\"yG6#%\"xG,&*&#\"\"#\"\"&\"\"\"-%$ cosGF&F-!\"\"*&#F-F,F--%$sinGF&F-F-" }}{PARA 13 "" 1 "" {GLPLOT2D 703 312 312 {PLOTDATA 2 "6&-%'CURVESG6$7gn7$$\"\"!F)$!3A+++++++S!#=7$$\"3E LLLLBxV5E$F,$!35;%fC]([[JF,7$$\"3MLL LLAKn\\F,$!3C&4%=OwYjDF,7$$\"3=LLLLc$\\o'F,$!31c1[)*fT**=F,7$$\"3)emmm ^&Q%R)F,$!39J7$$\"3))** ***\\YJ?;\"!#<$\"3m!=?Y3*>`CFK7$$\"3?LLL=\"\\g**FK7 $$\"3\")*****\\[A4]\"FO$\"3Xgu?U;&er\"F,7$$\"3wmmm'3Q\\n\"FO$\"3S\\4.g (y\\S#F,7$$\"3OLLLB6@G=FO$\"3e*[f2BGC&HF,7$$\"3&)******f-w+?FO$\"375@E VOJ&[$F,7$$\"3%*********y,u@FO$\"3VG2]n#=i\"RF,7$$\"3)*******RP)4M#FO$ \"3ym!)\\t%R1A%F,7$$\"3Umm;HUz;CFO$\"3:@(\\YT,0K%F,7$$\"3ILLL=Zg#\\#FO $\"3++xVHVa&R%F,7$$\"3;++]A2v#e#FO$\"3+<'Hh4))=X%F,7$$\"3cmmmEn*Gn#FO$ \"3a5#zx'*y?Z%F,7$$\"3qmmm;AE\\FFO$\"35^%H>#ywgWF,7$$\"3Tmmm1xiDGFO$\" 3(3\\(>4bXBWF,7$$\"3LLL$e#*eW\"HFO$\"3![MOl&\\jZVF,7$$\"3!)*****\\9!H. 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37+2hu:afHF,7$$\"35LLL.a#o$oFO$!3;\"e/Z#4*3N#F,7$$\"3ammm^Q40qFO$!3!4` 1I$pa!o\"F,7$$\"3y******z]rfrFO$!3pfL'*)RTA-\"F,7$$\"3gmmmc%GpL(FO$!3? j;%3XMsQ#FK7$$\"3/LLL8-V&\\(FO$\"3qi(R>/(R\"p%FK7$$\"3=+++XhUkwFO$\"3Z X^U-))=F,7$$\"\")F)$\"3s<7[GmrgDF,- %'COLOURG6&%$RGBG$\"#5!\"\"F(F(-%%FONTG6$%*HELVETICAG\"\"*-%+AXESLABEL SG6$%\"xG%%y(x)G-%%VIEWG6$;F(Fg]l%(DEFAULTG" 1 2 0 1 10 0 2 9 1 4 2 1.000000 45.000000 45.000000 0 0 "Curve 1" }}}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 32 "The following code constructs a " }{TEXT 260 17 "discrete solution" }{TEXT -1 44 " based on each of t he methods and gives the " }{TEXT 260 22 "root mean square error" } {TEXT -1 18 " of each solution." }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 818 "E := (x,y) -> cos(x)+2*y: hh := 0.02: numsteps := 400: x0 := \+ 0: y0 := -2/5:\nmatrix([[`slope field: `,E(x,y)],[`initial point: `, ``(x0,y0)],[`step width: `,hh],\n[`no. of steps: `,numsteps]]);``; \nmthds := [`classical method with `*(c[2]=1/2,c[3]=1/2),`3/8 method w ith `*(c[2]=1/3,c[3]=2/3),`Gill's method with `*(c[2]=1/2,c[3]=1/2),`D ormand's method with `*(c[2]=2/5,c[3]=3/5),`method with `*(c[2]=3/7,c[ 3]=6/11),`method with `*(c[2]=2/5,a[3,2]=2/3)]: errs := []:\nDigits := 20:\nfor ct to 6 do\n En_RK4_||ct := RK4_||ct(E(x,y),x,y,x0,evalf(y 0),hh,numsteps,false);\n sm := 0: numpts := nops(En_RK4_||ct):\n f or ii to numpts do\n sm := sm+(En_RK4_||ct[ii,2]-e(En_RK4_||ct[ii ,1]))^2;\n end do:\n errs := [op(errs),sqrt(sm/numpts)];\nend do: \nDigits := 10:\nlinalg[transpose](convert([mthds,evalf(errs)],matrix) );" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#K%'matrixG6#7&7$%0slope~field:~~ ~G,&-%$cosG6#%\"xG\"\"\"*&\"\"#F.%\"yGF.F.7$%0initial~point:~G-%!G6$\" \"!#!\"#\"\"&7$%/step~width:~~~G$F0F97$%1no.~of~steps:~~~G\"$+%Q)pprin t706\"" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%!G" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#K%'matrixG6#7(7$*&%7classical~method~with~G\"\"\"6$/&% \"cG6#\"\"##F*F0/&F.6#\"\"$F1F*$\"+t'o'oH!#77$*&%13/8~method~with~GF*6 $/F-#F*F5/F3#F0F5F*$\"+APf8:F87$*&%4Gill's~method~with~GF*F+F*F67$*&%7 Dormand's~method~with~GF*6$/F-#F0\"\"&/F3#F5FLF*$\"+$)4m4@F87$*&%-meth od~with~GF*6$/F-#F5\"\"(/F3#\"\"'\"#6F*$\"+VlCjBF87$*&FSF*6$FJ/&%\"aG6 $F5F0F@F*$\"+VO?8@F8Q)pprint716\"" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }} {PARA 0 "" 0 "" {TEXT -1 30 "The following code constructs " }{TEXT 260 20 "numerical procedures" }{TEXT -1 56 " for solutions based on ea ch of the Runge-Kutta schemes." }}{PARA 0 "" 0 "" {TEXT -1 75 "The err or in the value obtained by each of the methods at the point where " }{XPPEDIT 18 0 "x = 7.999;" "6#/%\"xG-%&FloatG6$\"%**z!\"$" }{TEXT -1 16 " is also given." }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 749 "E : = (x,y) -> cos(x)+2*y: hh := 0.02: numsteps := 400: x0 := 0: y0 := -2/ 5:\nmatrix([[`slope field: `,E(x,y)],[`initial point: `,``(x0,y0)],[ `step width: `,hh],\n[`no. of steps: `,numsteps]]);``;\nmthds := [ `classical method with `*(c[2]=1/2,c[3]=1/2),`3/8 method with `*(c[2]= 1/3,c[3]=2/3),`Gill's method with `*(c[2]=1/2,c[3]=1/2),`Dormand's met hod with `*(c[2]=2/5,c[3]=3/5),`method with `*(c[2]=3/7,c[3]=6/11),`me thod with `*(c[2]=2/5,a[3,2]=2/3)]: errs := []:\nDigits := 20:\nfor ct to 6 do\n en_RK4_||ct := RK4_||ct(E(x,y),x,y,x0,evalf(y0),hh,numste ps,true);\nend do:\nxx := 7.999: exx := evalf(e(xx)):\nfor ct to 6 do \n errs := [op(errs),abs(en_RK4_||ct(xx)-exx)];\nend do:\nDigits := \+ 10:\nlinalg[transpose](convert([mthds,evalf(errs)],matrix));" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#K%'matrixG6#7&7$%0slope~field:~~~G,&-%$cosG6 #%\"xG\"\"\"*&\"\"#F.%\"yGF.F.7$%0initial~point:~G-%!G6$\"\"!#!\"#\"\" &7$%/step~width:~~~G$F0F97$%1no.~of~steps:~~~G\"$+%Q*pprint1086\"" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#%!G" }}{PARA 11 "" 1 "" {XPPMATH 20 "6 #K%'matrixG6#7(7$*&%7classical~method~with~G\"\"\"6$/&%\"cG6#\"\"##F*F 0/&F.6#\"\"$F1F*$\"+JA1X;!#67$*&%13/8~method~with~GF*6$/F-#F*F5/F3#F0F 5F*$\"+v\"\\uQ)!#77$*&%4Gill's~method~with~GF*F+F*F67$*&%7Dormand's~me thod~with~GF*6$/F-#F0\"\"&/F3#F5FMF*$\"+%[]!p6F87$*&%-method~with~GF*6 $/F-#F5\"\"(/F3#\"\"'\"#6F*$\"+\\Fd48F87$*&FTF*6$FK/&%\"aG6$F5F0F@F*$ \"+>O,r6F8Q*pprint1096\"" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 " " 0 "" {TEXT -1 4 "The " }{TEXT 260 22 "root mean square error" } {TEXT -1 20 " over the interval " }{XPPEDIT 18 0 "[0, 8];" "6#7$\"\"! \"\")" }{TEXT -1 82 " of each Runge-Kutta method is estimated as foll ows using the special procedure " }{TEXT 0 5 "NCint" }{TEXT -1 98 " \+ to perform numerical integration by the 7 point Newton-Cotes method ov er 200 equal subintervals." }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 496 "mthds := [`classical method with `*(c[2]=1/2,c[3]=1/2),`3/8 metho d with `*(c[2]=1/3,c[3]=2/3),`Gill's method with `*(c[2]=1/2,c[3]=1/2) ,`Dormand's method with `*(c[2]=2/5,c[3]=3/5),`method with `*(c[2]=3/7 ,c[3]=6/11),`method with `*(c[2]=2/5,a[3,2]=2/3)]: errs := []:\nDigits := 20:\nfor ct to 6 do\n sm := NCint((e(x)-'en_RK4_||ct'(x))^2,x=0. .8,adaptive=false,numpoints=7,factor=200);\n errs := [op(errs),sqrt( sm/8)];\nend do:\nDigits := 10:\nlinalg[transpose](convert([mthds,eval f(errs)],matrix));" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#K%'matrixG6#7(7$ *&%7classical~method~with~G\"\"\"6$/&%\"cG6#\"\"##F*F0/&F.6#\"\"$F1F*$ \"+c'3R\"H!#77$*&%13/8~method~with~GF*6$/F-#F*F5/F3#F0F5F*$\"+**Rn&[\" F87$*&%4Gill's~method~with~GF*F+F*F67$*&%7Dormand's~method~with~GF*6$/ F-#F0\"\"&/F3#F5FLF*$\"+6iuq?F87$*&%-method~with~GF*6$/F-#F5\"\"(/F3# \"\"'\"#6F*$\"+NTl>BF87$*&FSF*6$FJ/&%\"aG6$F5F0F@F*$\"+BNAu?F8Q)pprint 746\"" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 92 "The following error graphs are constructed using the numerical proced ures for the solutions." }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 458 " evalf[20](plot([e(x)-'en_RK4_1'(x),e(x)-'en_RK4_2'(x),e(x)-'en_RK4_3'( x),e(x)-'en_RK4_4'(x),\ne(x)-'en_RK4_5'(x),e(x)-'en_RK4_6'(x)],x=0..2, font=[HELVETICA,9],\ncolor=[red,blue,COLOR(RGB,0,.65,0),COLOR(RGB,.4,0 ,1),magenta,brown],thickness=[1$2,2,1$2,2],\nlegend=[`classical method `,`3/8 method`,`Gill's method`,`Dormand's method`,`method with c[2]=3/ 7 and c[3]=6/11`,`method with c[2]=2/5 and a[3,2]=2/3`],title=`error c urves for order 4 Runge-Kutta methods`));" }}{PARA 13 "" 1 "" {GLPLOT2D 604 604 604 {PLOTDATA 2 "6,-%'CURVESG6&7S7$$\"\"!F)F(7$$\"5M LLLLL3VfV!#@$\",Rv$*3F#!#?7$$\"5nmmmm\"H[D:)F-$\",ePC.o%F07$$\"5LLLLLe 0$=C\"F0$\",HJUdI(F07$$\"5LLLLL3RBr;F0$\"-F07$$\"5MLLL$ezs$HLF0$\"-zT)=YP#F07$$\"5++++]7iI_PF0$\"-M`)pLz# F07$$\"5nmmmm;_M(=%F0$\"->];YyKF07$$\"5MLLLL3y_qXF0$\"-Snh)\\p$F07$$\" 5+++++]1!>+&F0$\"--OI&)[UF07$$\"5+++++]Z/NaF0$\"-9UO)yz%F07$$\"5+++++] $fC&eF0$\"-d\"o`PP&F07$$\"5MLLL$ez6:B'F0$\"-$e*>,ZfF07$$\"5nmmmm;=C#o' F0$\"-=q@'Gl'F07$$\"5nmmmmm#pS1(F0$\"-WGA$fJ(F07$$\"5,+++]i`A3vF0$\"-k -q$p7)F07$$\"5nmmmmm(y8!zF0$\"-e&yn5\"*)F07$$\"5,+++]i.tK$)F0$\"-'G%4; 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7$FN$\"/8h&ynr8%F-7$FS$\"/]=95zH`F-7$FX$\"/H6`H?joF-7$Fgn$\"/K%R^4U!*) F-7$F\\o$\"0-&Qp@3?6F-7$Fao$\"0D)H@$*R]9F-7$Ffo$\"0E'Q6,U!)=F-7$F[p$\" 0dx)*=,_T#F-7$F`p$\"0!\\IJ6oJIF-7$Fep$\"0wbq\"G&G(RF-7$Fjp$\"1v=&)y.c& *\\F^q7$F`q$\"2e)fa7\\+@lFdq7$Ffq$\"1!f^E'>#eD)F^q7$F[r$\"1VTQ=tXp5F-7 $F`r$\"1tcx\"Hd$o8F-7$Fer$\"16!)>aTipQEF^q7$F^y$\"4Ot)\\\\q`&oK$F^ q7$Fcy$\"4K:LJUoe(*z$F^q7$Fhy$\"4WM@jH(Q))RVF^q7$F]z$\"4P!)Gf905x)[F^q 7$Fbz$\"4c6Hklk)o/bF^q7$Fgz$\"3`)Rp))pN&[iF-7$F\\[l$\"3C=`t'e)*G4(F-7$ Fa[l$\"3A.#eb)zs2!)F-7$Ff[l$\"3qvcwT/bS!*F-7$F[\\l$\"3e+77\"Hx%\\'*F-7 $F`\\l$\"4/:jG$[=%*H5F-7$Fe\\l$\"4r&4X!zk8$*4\"F-7$Fj\\l$\"4$o7gQ))zNt 6F--F_]l6&Fa]l$\")#)eqkFd]l$\"))eqk\"Fd]lFe^oF]]m-F[^l6#%Dmethod~with~ c[2]=2/5~and~a[3,2]=2/3G-%%FONTG6$%*HELVETICAG\"\"*-%&TITLEG6#%Merror~ curves~for~order~4~Runge-Kutta~methodsG-%+AXESLABELSG6$Q\"x6\"Q!Fg_o-% %VIEWG6$;F(Fj\\l%(DEFAULTG" 1 2 0 1 10 0 2 9 1 4 2 1.000000 45.000000 45.000000 0 1 "classical method" "3/8 method" "Gill's method" "Dormand 's method" "method with c[2]=3/7 and c[3]=6/11" "method with c[2]=2/5 \+ and a[3,2]=2/3" }}}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> \+ " 0 "" {MPLTEXT 1 0 2 " ;" }}}}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 39 "Te st 14 of order 4 Runge-Kutta methods " }}{PARA 0 "" 0 "" {TEXT -1 0 " " }}{PARA 256 "" 0 "" {TEXT -1 3 " " }{XPPEDIT 18 0 "dy/dx = 10*x*co s*x-10*y;" "6#/*&%#dyG\"\"\"%#dxG!\"\",&**\"#5F&%\"xGF&%$cosGF&F,F&F&* &F+F&%\"yGF&F(" }{TEXT -1 8 ", " }{XPPEDIT 18 0 "y(0) = sqrt(5); " "6#/-%\"yG6#\"\"!-%%sqrtG6#\"\"&" }{TEXT -1 1 " " }}{PARA 0 "" 0 "" {TEXT -1 10 "Solution: " }}{PARA 256 "" 0 "" {TEXT -1 2 " " } {XPPEDIT 18 0 "y=100/101" "6#/%\"yG*&\"$+\"\"\"\"\"$,\"!\"\"" }{TEXT -1 1 " " }{XPPEDIT 18 0 "x*cos*x-990/10201" "6#,&*(%\"xG\"\"\"%$cosGF& F%F&F&*&\"$!**F&\"&,-\"!\"\"F+" }{TEXT -1 1 " " }{XPPEDIT 18 0 "cos*x+ 10/101" "6#,&*&%$cosG\"\"\"%\"xGF&F&*&\"#5F&\"$,\"!\"\"F&" }{TEXT -1 1 " " }{XPPEDIT 18 0 "x*sin*x-200/10201" "6#,&*(%\"xG\"\"\"%$sinGF&F%F &F&*&\"$+#F&\"&,-\"!\"\"F+" }{TEXT -1 1 " " }{XPPEDIT 18 0 "sin*x+(990 /10201+sqrt(5))*exp(-10*x)" "6#,&*&%$sinG\"\"\"%\"xGF&F&*&,&*&\"$!**F& \"&,-\"!\"\"F&-%%sqrtG6#\"\"&F&F&-%$expG6#,$*&\"#5F&F'F&F-F&F&" } {TEXT -1 1 "." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 163 "de := diff(y(x),x)=10*x*cos(x)-10*y(x);\nic := \+ y(0)=sqrt(5);\ndsolve(\{de,ic\},y(x));\nb := unapply(rhs(%),x):\nplot( b(x),x=0..5,font=[HELVETICA,9],labels=[`x`,`y(x)`]);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%#deG/-%%diffG6$-%\"yG6#%\"xGF,,&*(\"#5\"\"\"F,F0-% $cosGF+F0F0*&F/F0F)F0!\"\"" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%#icG/- %\"yG6#\"\"!*$\"\"&#\"\"\"\"\"#" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/-% \"yG6#%\"xG,,*&#\"$+\"\"$,\"\"\"\"*&F'F--%$cosGF&F-F-F-*&#\"$!**\"&,- \"F-F/F-!\"\"*&#\"#5F,F-*&-%$sinGF&F-F'F-F-F-*&#\"$+#F4F-F:F-F5*&-%$ex pG6#,$*&F8F-F'F-F5F-,&#F3F4F-*$\"\"&#F-\"\"#F-F-F-" }}{PARA 13 "" 1 " " {GLPLOT2D 703 312 312 {PLOTDATA 2 "6&-%'CURVESG6$7hn7$$\"\"!F)$\"3\" )*y*\\xz1OA!#<7$$\"3ALL$3FWYs#!#>$\"3*pc)\\jF;12h\"H\"48F,7$$\"3m****\\7G$R<)F0$\"3<_u(oLbK,\"F,7$$\"3GLLL3x &)*3\"!#=$\"3(**[ro!GyVzF@7$$\"3))**\\i!R(*Rc\"F@$\"3A'ysO]2xW&F@7$$\" 3umm\"H2P\"Q?F@$\"3/$)oqvSKmSF@7$$\"3YLek.pu/BF@$\"3$Qjx*Gs<7OF@7$$\"3 !***\\PMnNrDF@$\"3M:4%*3rt@LF@7$$\"3MmT5ll'z$GF@$\"3/?Np5C\\bJF@7$$\"3 MLL$eRwX5$F@$\"3)GTJ!oG0$3$F@7$$\"3rLLL$eI8k$F@$\"3FyHM$p'GKJF@7$$\"33 ML$3x%3yTF@$\"3n$**Q]`\"yRLF@7$$\"3emm\"z%4\\Y_F@$\"3%)*)G8T#p2%RF@7$$ \"3`LLeR-/PiF@$\"3PZ.%R2Cm^%F@7$$\"3]***\\il'pisF@$\"31e'*fKlL9]F@7$$ \"3>MLe*)>VB$)F@$\"3%3)yy-pAk`F@7$$\"3Y++DJbw!Q*F@$\"3Kg$RQBm7^&F@7$$ \"3%ommTIOo/\"F,$\"3xrTB'zz$GaF@7$$\"3YLL3_>jU6F,$\"3!p\\3Dp!RX^F@7$$ \"37++]i^Z]7F,$\"3Kvxv\"*=%>e%F@7$$\"33++](=h(e8F,$\"3C^UQ-(*\\]PF@7$$ \"3/++]P[6j9F,$\"3icMTpV\"zp#F@7$$\"3UL$e*[z(yb\"F,$\"3C-jVJM+L:F@7$$ \"3wmm;a/cq;F,$!33XIKjA5+5F07$$\"3%ommmJF,$!3!\\F1VDv$)p&F@7$$\"3K+] i!f#=$3#F,$!3V$*[LN4F4!)F@7$$\"3?+](=xpe=#F,$!3$[6ral`?.\"F,7$$\"37nm \"H28IH#F,$!3s6ToLL*4G\"F,7$$\"3um;zpSS\"R#F,$!3/xdj'GyI^\"F,7$$\"3GLL 3_?`(\\#F,$!3MGc,i$*4jpxg#F,$!3XXbRF+x>IF,$!3r\"fP(*)y]OGF,7$$\"3F+]i!RU07$F, $!3[Tb]!*H%e)HF,7$$\"3+++v=S2LKF,$!3Tyi#Q\\'[=JF,7$$\"3Jmmm\"p)=MLF,$! 3N`)=cl>V?$F,7$$\"3B++](=]@W$F,$!3Y_BiN[odKF,7$$\"3mm\"H#oZ1\"\\$F,$!3 )o4&)z%=-oKF,7$$\"35L$e*[$z*RNF,$!3%)Q61)ek$pKF,7$$\"3%o;Hd!fX$f$F,$!3 #*y!45Ut-E$F,7$$\"3e++]iC$pk$F,$!3LO[nw')*)RKF,7$$\"3ILe*[t\\sp$F,$!3D 1>x`HA5KF,7$$\"3[m;H2qcZPF,$!3/[q%\\V.-<$F,7$$\"3O+]7.\"fF&QF,$!3KL?tX &>E0$F,7$$\"3Ymm;/OgbRF,$!3KQEMNc$G*GF,7$$\"3w**\\ilAFjSF,$!3/QR)44g!y EF,7$$\"3yLLL$)*pp;%F,$!30,GW_`#fU#F,7$$\"3)RL$3xe,tUF,$!3*G#*H@1([B@F ,7$$\"3Cn;HdO=yVF,$!35Q!)*4x]5y\"F,7$$\"3a+++D>#[Z%F,$!3(y*pyl_QJ9F,7$ $\"3SnmT&G!e&e%F,$!3]X/0\"RC%G**F@7$$\"3#RLLL)Qk%o%F,$!3u!*)Q\"4WH+dF@ 7$$\"37+]iSjE!z%F,$!3+r[gfMO'=*F07$$\"3a+]P40O\"*[F,$\"3+2*eSHde(QF@7$ $\"\"&F)$\"3&Q8`\">jC3#*F@-%'COLOURG6&%$RGBG$\"#5!\"\"F(F(-%%FONTG6$%* HELVETICAG\"\"*-%+AXESLABELSG6$%\"xG%%y(x)G-%%VIEWG6$;F(F\\^l%(DEFAULT G" 1 2 0 1 10 0 2 9 1 4 2 1.000000 45.000000 45.000000 0 0 "Curve 1" } }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 32 "The fo llowing code constructs a " }{TEXT 260 17 "discrete solution" }{TEXT -1 44 " based on each of the methods and gives the " }{TEXT 260 22 "ro ot mean square error" }{TEXT -1 18 " of each solution." }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 834 "B := (x,y) -> 10*x*cos(x)-10*y: hh := 0.01: numsteps := 500: x0 := 0: y0 := sqrt(5):\nmatrix([[`slope fi eld: `,B(x,y)],[`initial point: `,``(x0,y0)],[`step width: `,hh], \n[`no. of steps: `,numsteps]]);``;\nmthds := [`classical method wit h `*(c[2]=1/2,c[3]=1/2),`3/8 method with `*(c[2]=1/3,c[3]=2/3),`Gill's method with `*(c[2]=1/2,c[3]=1/2),`Dormand's method with `*(c[2]=2/5, c[3]=3/5),`method with `*(c[2]=3/7,c[3]=6/11),`method with `*(c[2]=2/5 ,a[3,2]=2/3)]: errs := []:\nDigits := 20:\nfor ct to 6 do\n Bn_RK4_| |ct := RK4_||ct(B(x,y),x,y,x0,evalf(y0),hh,numsteps,false);\n sm := \+ 0: numpts := nops(Bn_RK4_||ct):\n for ii to numpts do\n sm := s m+(Bn_RK4_||ct[ii,2]-evalf(b(Bn_RK4_||ct[ii,1])))^2;\n end do:\n e rrs := [op(errs),sqrt(sm/numpts)];\nend do:\nDigits := 10:\nlinalg[tra nspose](convert([mthds,evalf(errs)],matrix));" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#K%'matrixG6#7&7$%0slope~field:~~~G,&*(\"#5\"\"\"%\"xGF, -%$cosG6#F-F,F,*&F+F,%\"yGF,!\"\"7$%0initial~point:~G-%!G6$\"\"!*$\"\" &#F,\"\"#7$%/step~width:~~~G$F,!\"#7$%1no.~of~steps:~~~G\"$+&Q)pprint9 56\"" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%!G" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#K%'matrixG6#7(7$*&%7classical~method~with~G\"\"\"6$/&% \"cG6#\"\"##F*F0/&F.6#\"\"$F1F*$\"+ErB5:!#;7$*&%13/8~method~with~GF*6$ /F-#F*F5/F3#F0F5F*$\"+\"H'>,:F87$*&%4Gill's~method~with~GF*F+F*F67$*&% 7Dormand's~method~with~GF*6$/F-#F0\"\"&/F3#F5FLF*$\"+u70/:F87$*&%-meth od~with~GF*6$/F-#F5\"\"(/F3#\"\"'\"#6F*$\"+0:b0:F87$*&FSF*6$FJ/&%\"aG6 $F5F0F@F*$\"+w\">S]\"F8Q)pprint966\"" }}}{PARA 0 "" 0 "" {TEXT -1 0 " " }}{PARA 0 "" 0 "" {TEXT -1 30 "The following code constructs " } {TEXT 260 20 "numerical procedures" }{TEXT -1 56 " for solutions based on each of the Runge-Kutta schemes." }}{PARA 0 "" 0 "" {TEXT -1 75 "T he error in the value obtained by each of the methods at the point whe re " }{XPPEDIT 18 0 "x = 4.999;" "6#/%\"xG-%&FloatG6$\"%**\\!\"$" } {TEXT -1 16 " is also given." }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 758 "B := (x,y) -> 10*x*cos(x)-10*y: hh := 0.01: numsteps := 500: x0 := 0: y0 := sqrt(5):\nmatrix([[`slope field: `,B(x,y)],[`initial po int: `,``(x0,y0)],[`step width: `,hh],\n[`no. of steps: `,numsteps ]]);``;\nmthds := [`classical method with `*(c[2]=1/2,c[3]=1/2),`3/8 m ethod with `*(c[2]=1/3,c[3]=2/3),`Gill's method with `*(c[2]=1/2,c[3]= 1/2),`Dormand's method with `*(c[2]=2/5,c[3]=3/5),`method with `*(c[2] =3/7,c[3]=6/11),`method with `*(c[2]=2/5,a[3,2]=2/3)]: errs := []:\nDi gits := 20:\nfor ct to 6 do\n bn_RK4_||ct := RK4_||ct(B(x,y),x,y,x0, evalf(y0),hh,numsteps,true);\nend do:\nxx := 4.999: bxx := evalf(b(xx) ):\nfor ct to 6 do\n errs := [op(errs),abs(bn_RK4_||ct(xx)-bxx)];\ne nd do:\nDigits := 10:\nlinalg[transpose](convert([mthds,evalf(errs)],m atrix));" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#K%'matrixG6#7&7$%0slope~fi eld:~~~G,&*(\"#5\"\"\"%\"xGF,-%$cosG6#F-F,F,*&F+F,%\"yGF,!\"\"7$%0init ial~point:~G-%!G6$\"\"!*$\"\"&#F,\"\"#7$%/step~width:~~~G$F,!\"#7$%1no .~of~steps:~~~G\"$+&Q*pprint1046\"" }}{PARA 11 "" 1 "" {XPPMATH 20 "6# %!G" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#K%'matrixG6#7(7$*&%7classical~m ethod~with~G\"\"\"6$/&%\"cG6#\"\"##F*F0/&F.6#\"\"$F1F*$\"+[r[&[\"!#<7$ *&%13/8~method~with~GF*6$/F-#F*F5/F3#F0F5F*$\"+\\(f,:\"F87$*&%4Gill's~ method~with~GF*F+F*F67$*&%7Dormand's~method~with~GF*6$/F-#F0\"\"&/F3#F 5FLF*$\"+;ZLu7F87$*&%-method~with~GF*6$/F-#F5\"\"(/F3#\"\"'\"#6F*$\"+> exG8F87$*&FSF*6$FJ/&%\"aG6$F5F0F@F*$\"+Wusr7F8Q*pprint1056\"" }}} {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 4 "The " } {TEXT 260 22 "root mean square error" }{TEXT -1 20 " over the interval " }{XPPEDIT 18 0 "[0, 5];" "6#7$\"\"!\"\"&" }{TEXT -1 82 " of each \+ Runge-Kutta method is estimated as follows using the special procedure " }{TEXT 0 5 "NCint" }{TEXT -1 98 " to perform numerical integratio n by the 7 point Newton-Cotes method over 200 equal subintervals." }} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 496 "mthds := [`classical method with `*(c[2]=1/2,c[3]=1/2),`3/8 method with `*(c[2]=1/3,c[3]=2/3),`Gi ll's method with `*(c[2]=1/2,c[3]=1/2),`Dormand's method with `*(c[2]= 2/5,c[3]=3/5),`method with `*(c[2]=3/7,c[3]=6/11),`method with `*(c[2] =2/5,a[3,2]=2/3)]: errs := []:\nDigits := 20:\nfor ct to 6 do\n sm : = NCint((b(x)-'bn_RK4_||ct'(x))^2,x=0..5,adaptive=false,numpoints=7,fa ctor=200);\n errs := [op(errs),sqrt(sm/5)];\nend do:\nDigits := 10: \nlinalg[transpose](convert([mthds,evalf(errs)],matrix));" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#K%'matrixG6#7(7$*&%7classical~method~with~G\"\" \"6$/&%\"cG6#\"\"##F*F0/&F.6#\"\"$F1F*$\"+&QkCY\"!#;7$*&%13/8~method~w ith~GF*6$/F-#F*F5/F3#F0F5F*$\"+&y5PX\"F87$*&%4Gill's~method~with~GF*F+ F*F67$*&%7Dormand's~method~with~GF*6$/F-#F0\"\"&/F3#F5FLF*$\"+E\\Zc9F8 7$*&%-method~with~GF*6$/F-#F5\"\"(/F3#\"\"'\"#6F*$\"+Ev#zX\"F87$*&FSF* 6$FJ/&%\"aG6$F5F0F@F*$\"+sQWc9F8Q)pprint796\"" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 92 "The following error graph s are constructed using the numerical procedures for the solutions." } }{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 459 "evalf[20](plot(['bn_RK4_1' (x)-b(x),'bn_RK4_2'(x)-b(x),'bn_RK4_3'(x)-b(x),'bn_RK4_4'(x)-b(x),\n'b n_RK4_5'(x)-b(x),'bn_RK4_6'(x)-b(x)],x=0..0.65,font=[HELVETICA,9],\nco lor=[red,blue,COLOR(RGB,0,.65,0),COLOR(RGB,.4,0,1),magenta,brown],thic kness=[1$2,2,1$3],\nlegend=[`classical method`,`3/8 method`,`Gill's me thod`,`Dormand's method`,`method with c[2]=3/7 and c[3]=6/11`,`method \+ with c[2]=2/5 and a[3,2]=2/3`],title=`error curves for order 4 Runge-K utta methods`));" }}{PARA 13 "" 1 "" {GLPLOT2D 811 476 476 {PLOTDATA 2 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" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 73 "de := diff(y(x),x)=(y(x)-x)/(y(x)+x );\nic := y(1)=1;\ndsolve(\{de,ic\},y(x));" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%#deG/-%%diffG6$-%\"yG6#%\"xGF,*&,&F)\"\"\"F,!\"\"F/,& F)F/F,F/F0" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%#icG/-%\"yG6#\"\"\"F) " }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/-%\"yG6#%\"xG-%'RootOfG6#,,*&\"\" #\"\"\"-%#lnG6#*&,&*$)F'F-F.F.*$)%#_ZGF-F.F.F.F'!\"#F.!\"\"*&\"\"%F.-% 'arctanG6#*&F8F.F'F:F.F:*&F " 0 "" {MPLTEXT 1 0 1 ";" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 41 "The solution can be given more \+ simply as " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 " ln(x^2+y^2)+2*arctan(y/x)=ln*2+Pi/2" "6#/,&- %#lnG6#,&*$%\"xG\"\"#\"\"\"*$%\"yGF+F,F,*&F+F,-%'arctanG6#*&F.F,F*!\" \"F,F,,&*&F&F,F+F,F,*&%#PiGF,F+F4F," }{TEXT -1 2 ". " }}{PARA 0 "" 0 " " {TEXT -1 18 "The solution (for " }{TEXT 269 1 "x" }{TEXT -1 47 " inc reasing) is the section of the polar curve " }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "r=sqrt(2)*exp(Pi/4-theta)" "6#/%\"rG*&- %%sqrtG6#\"\"#\"\"\"-%$expG6#,&*&%#PiGF*\"\"%!\"\"F*%&thetaGF2F*" } {TEXT -1 5 ", " }{XPPEDIT 18 0 "-Pi/4<=theta" "6#1,$*&%#PiG\"\"\"\" \"%!\"\"F)%&thetaG" }{XPPEDIT 18 0 "``<=Pi/4" "6#1%!G*&%#PiG\"\"\"\"\" %!\"\"" }{TEXT -1 2 ". " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 7 "Check: " }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 60 " ln((x^2+y^2))+2*arctan(y/x)=ln(2)+Pi/2;\nimplicitdiff(%,y,x);" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#/,&-%#lnG6#,&*$)%\"xG\"\"#\"\"\"F-*$)% \"yGF,F-F-F-*&F,F--%'arctanG6#*&F0F-F+!\"\"F-F-,&-F&6#F,F-*&F,F6%#PiGF -F-" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#,$*&,&%\"xG\"\"\"%\"yG!\"\"F',& F(F'F&F'F)F)" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 471 "p1 := plot([sqrt(2)*exp(Pi/4-t),t,t=-Pi/4..Pi/4 ],coords=polar,thickness=2,color=red):\np2 := plot([sqrt(2)*exp(Pi/4-t ),t,t=Pi/4..2*Pi],coords=polar,color=black,linestyle=2):\np3 := plot([ sqrt(2)*exp(Pi/4-t),t,t=-Pi/3..-Pi/4],coords=polar,color=black,linesty le=2):\np4 := plot([[[1,1],[uu,-uu]]$4],style=point,symbol=[circle$2,d iamond,cross],\n symbolsize=[12,10$3],color=[black,gr een$3]):\nplots[display]([p1,p2,p3,p4],font=[HELVETICA,9],labels=[`x`, `y(x)`]);" }}{PARA 13 "" 1 "" {GLPLOT2D 567 520 520 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GL![F*$!3AMPAGam$=&F*7$$\"3B_c%y]m_![F*$!3C%yzJ*p\")G^F*7$$\"3'*o#foxs n![F*$!3?qs#R*Qny]F*7$$\"37Ot>[%z\"3[F*$!3@8Kvdp]@]F*7$$\"3$**\\xVK^\" 4[F*$!3[*Q1)f,lq\\F*7$$\"3s=V+(*H*)4[F*$!3ES2X[7r;\\F*7$$\"3@WmjN6K5[F *$!3i,7R(Rg`'[F*F'F_]mFa]m-F$6&7$7$$\"\"\"Fa[lFj\\n7$$\"3>+++#Qx/\"[F* $!3>+++#Qx/\"[F*F_]m-%'SYMBOLG6$%'CIRCLEG\"#7-%&STYLEG6#%&POINTG-F$6&F h\\n-Fjz6&F\\[lF`[lF][lF`[l-Fb]n6$Fd]n\"#5Ff]n-F$6&Fh\\nF\\^n-Fb]n6$%( DIAMONDGF`^nFf]n-F$6&Fh\\nF\\^n-Fb]n6$%&CROSSGF`^nFf]n-%+AXESLABELSG6% %\"xG%%y(x)G-%%FONTG6#%(DEFAULTG-Fa_n6$%*HELVETICAG\"\"*-%%VIEWG6$Fc_n Fc_n" 1 2 0 1 10 0 2 9 1 4 2 1.000000 45.000000 45.000000 0 0 "Curve 1 " "Curve 2" "Curve 3" "Curve 4" "Curve 5" "Curve 6" "Curve 7" }}}} {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 29 "The follo wing procedure uses " }{TEXT 0 6 "fsolve" }{TEXT -1 23 " to solve the \+ equation " }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 " ln(x^2+ y^2)+2*arctan(y/x)=ln*2+Pi/2" "6#/,&-%#lnG6#,&*$%\"xG\"\"#\"\"\"*$%\"y GF+F,F,*&F+F,-%'arctanG6#*&F.F,F*!\"\"F,F,,&*&F&F,F+F,F,*&%#PiGF,F+F4F ," }{TEXT -1 1 " " }}{PARA 0 "" 0 "" {TEXT -1 4 "for " }{TEXT 267 1 "y " }{TEXT -1 25 " numerically in terms of " }{TEXT 268 1 "x" }{TEXT -1 1 "." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 209 "phi := proc(x) local y;\n fsolve(ln(x^2+y^2 )+2*arctan(y/x)=ln(2)+Pi/2,y=-x..7/2-x);\n end proc:\nuu := eval f(exp(Pi/2)):\nplot('phi'(x),x=1..uu,numpoints=100,font=[HELVETICA,9], labels=[`x`,`y(x)`]);" }}{PARA 13 "" 1 "" {GLPLOT2D 515 404 404 {PLOTDATA 2 "6&-%'CURVESG6$7jq7$$\"\"\"\"\"!$\"+++++5!\"*7$$\"+M.FS5F- $\"+!Hsf***!#57$$\"+N$4`2\"F-$\"+]6*f)**F37$$\"+eVr96F-$\"+Zyon**F37$$ \"+#e!Qa6F-$\"+z@\"=%**F37$$\"+!GeQ>\"F-$\"+^qu3**F37$$\"+f\"f/B\"F-$ \"+$)4mr)*F37$$\"+\\sNo7F-$\"+m$=o#)*F37$$\"+4:b28F-$\"+-Eit(*F37$$\"+ s+iY8F-$\"+@hw8(*F37$$\"+\"*o!oQ\"F-$\"+ig9X'*F37$$\"+jM?A9F-$\"+N0$)y &*F37$$\"+>;0i9F-$\"+tvj(\\*F37$$\"+'Rj?]\"F-$\"+2v=4%*F37$$\"+d@iS:F- 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F37$$\"+t#3\\U$F-$!+e$R\\f$F37$$\"+d)[KY$F-$!+\")RRuSF37$$\"+>h\\,NF-$ !+)*o;lXF37$$\"+LbWTNF-$!+6&)e\"4&F37$$\"+,/CyNF-$!+4mR*e&F37$$\"+zL#f h$F-$!+w:g7hF37$$\"+l)Hvl$F-$!+Zam1nF37$$\"+I6?&p$F-$!+q5-gsF37$$\"+aq sLPF-$!+g\\%=%yF37$$\"+Fp!Hx$F-$!+)QD3X)F37$$\"+a3#*3QF-$!+'*3rE!*F37$ $\"+M0JZQF-$!+?;We'*F37$$\"+*)zS&)QF-$!+6MXI5F-7$$\"+C/;ERF-$!+B0!=5\" F-7$$\"+$oA@'RF-$!+qNzm6F-7$$\"+kch.SF-$!+'>,VC\"F-7$$\"+!))f5/%F-$!+- [r;8F-7$$\"+v*3\"ySF-$!+Z4$3R\"F-7$$\"+b'[z6%F-$!+zwYt9F-7$$\"+$\\\\z: %F-$!+'fX(f:F-7$$\"+MVM%>%F-$!+DqOT;F-7$$\"+QS*HB%F-$!+A:fJF-7$$\"+G\"ypM%F-$!+AUgA?F-7$$ \"+M=h(Q%F-$!+:96P@F-7$$\"+`'4eU%F-$!+)[X5D#F-7$$\"+C&QOY%F-$!+d+zqBF- 7$$\"+([(\\,XF-$!+HVk)\\#F-7$$\"+L76SXF-$!+'GZ)QEF-7$$\"+d6/\"e%F-$!+$ pm1!GF-7$$\"+!*)p&=YF-$!+?m5kHF-7$$\"+obhbYF-$!+9%4S9$F-7$$\"+(y;_p%F- $!+/&R[O$F-7$$\"+*zJZt%F-$!+`i>JOF-7$$\"+M`Y_ZF-$!+El5vPF-7$$\"+o))>qZ F-$!+iFfWRF-7$$\"+'\\o-y%F-$!+G<\")eSF-7$$\"+D\"Q.z%F-$!+!p\\]>%F-7$$ \"+SHP&z%F-$!+M@kwUF-7$$\"+axS+[F-$!+ " 0 "" {MPLTEXT 1 0 863 "C := \+ (x,y) -> (y-x)/(y+x): hh := 0.01: numsteps := 375: x0 := 1: y0 := 1:\n matrix([[`slope field: `,C(x,y)],[`initial point: `,``(x0,y0)],[`ste p width: `,hh],\n[`no. of steps: `,numsteps]]);``;\nmthds := [`cla ssical method with `*(c[2]=1/2,c[3]=1/2),`3/8 method with `*(c[2]=1/3, c[3]=2/3),`Gill's method with `*(c[2]=1/2,c[3]=1/2),`Dormand's method \+ with `*(c[2]=2/5,c[3]=3/5),`method with `*(c[2]=3/7,c[3]=6/11),`method with `*(c[2]=2/5,a[3,2]=2/3)]: errs := []: vals := []:\nDigits := 25: \nfor ct to 6 do\n Cn_RK4_||ct := RK4_||ct(C(x,y),x,y,x0,y0,hh,nums teps,false);\n sm := 0: numpts := nops(Cn_RK4_||ct):\n for ii to n umpts do\n if ct=1 then vals := [op(vals),phi(Cn_RK4_||ct[ii,1])] end if;\n sm := sm+(Cn_RK4_||ct[ii,2]-vals[ii])^2;\n end do:\n errs := [op(errs),sqrt(sm/numpts)];\nend do:\nDigits := 10:\nlinalg [transpose]([mthds,evalf(errs)]);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#K %'matrixG6#7&7$%0slope~field:~~~G*&,&%\"yG\"\"\"%\"xG!\"\"F,,&F-F,F+F, F.7$%0initial~point:~G-%!G6$F,F,7$%/step~width:~~~G$F,!\"#7$%1no.~of~s teps:~~~G\"$v$Q)pprint976\"" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%!G" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#K%'matrixG6#7(7$*&%7classical~method~w ith~G\"\"\"6$/&%\"cG6#\"\"##F*F0/&F.6#\"\"$F1F*$\"+!\\G/1\"!#<7$*&%13/ 8~method~with~GF*6$/F-#F*F5/F3#F0F5F*$\"+\"\\dv4(!#=7$*&%4Gill's~metho d~with~GF*F+F*$\"+j\"\\xj%FC7$*&%7Dormand's~method~with~GF*6$/F-#F0\" \"&/F3#F5FOF*$\"+V>!>M(FC7$*&%-method~with~GF*6$/F-#F5\"\"(/F3#\"\"'\" #6F*$\"+qAs9hFC7$*&FVF*6$FM/&%\"aG6$F5F0F@F*$\"+`Qa@kFCQ)pprint986\"" }}}{PARA 0 "" 0 "" {TEXT -1 2 " " }}{PARA 0 "" 0 "" {TEXT -1 30 "The \+ following code constructs " }{TEXT 260 20 "numerical procedures" } {TEXT -1 56 " for solutions based on each of the Runge-Kutta schemes. " }}{PARA 0 "" 0 "" {TEXT -1 75 "The error in the value obtained by ea ch of the methods at the point where " }{XPPEDIT 18 0 "x = 4.749;" "6 #/%\"xG-%&FloatG6$\"%\\Z!\"$" }{TEXT -1 16 " is also given." }} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 726 "C := (x,y) -> (y-x)/(y+x): \+ hh := 0.01: numsteps := 375: x0 := 1: y0 := 1:\nmatrix([[`slope field: `,C(x,y)],[`initial point: `,``(x0,y0)],[`step width: `,hh],\n[`n o. of steps: `,numsteps]]);``;\nmthds := [`classical method with `*( c[2]=1/2,c[3]=1/2),`3/8 method with `*(c[2]=1/3,c[3]=2/3),`Gill's meth od with `*(c[2]=1/2,c[3]=1/2),`Dormand's method with `*(c[2]=2/5,c[3]= 3/5),`method with `*(c[2]=3/7,c[3]=6/11),`method with `*(c[2]=2/5,a[3, 2]=2/3)]: errs := []:\nDigits := 25:\nfor ct to 6 do\n cn_RK4_||ct : = RK4_||ct(C(x,y),x,y,x0,y0,hh,numsteps,true);\nend do:\nxx := 4.749: \+ cxx := evalf(phi(xx)):\nfor ct to 6 do\n errs := [op(errs),abs(cn_RK 4_||ct(xx)-cxx)];\nend do:\nDigits := 10:\nlinalg[transpose]([mthds,ev alf(errs)]);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#K%'matrixG6#7&7$%0slop e~field:~~~G*&,&%\"yG\"\"\"%\"xG!\"\"F,,&F-F,F+F,F.7$%0initial~point:~ G-%!G6$F,F,7$%/step~width:~~~G$F,!\"#7$%1no.~of~steps:~~~G\"$v$Q*pprin t1016\"" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%!G" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#K%'matrixG6#7(7$*&%7classical~method~with~G\"\"\"6$/&% \"cG6#\"\"##F*F0/&F.6#\"\"$F1F*$\"+=+zf8!#;7$*&%13/8~method~with~GF*6$ /F-#F*F5/F3#F0F5F*$\"+Okoz!*!#<7$*&%4Gill's~method~with~GF*F+F*$\"+3C! R.'FC7$*&%7Dormand's~method~with~GF*6$/F-#F0\"\"&/F3#F5FOF*$\"+u\"[OU* FC7$*&%-method~with~GF*6$/F-#F5\"\"(/F3#\"\"'\"#6F*$\"+]'\\[!zFC7$*&FV F*6$FM/&%\"aG6$F5F0F@F*$\"+Y=Cu#)FCQ*pprint1026\"" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 4 "The " }{TEXT 260 22 "root \+ mean square error" }{TEXT -1 20 " over the interval " }{XPPEDIT 18 0 "[1, 4.75];" "6#7$\"\"\"-%&FloatG6$\"$v%!\"#" }{TEXT -1 82 " of each \+ Runge-Kutta method is estimated as follows using the special procedure " }{TEXT 0 5 "NCint" }{TEXT -1 98 " to perform numerical integratio n by the 7 point Newton-Cotes method over 200 equal subintervals." }} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 487 "mthds := [`classical method with `*(c[2]=1/2,c[3]=1/2),`3/8 method with `*(c[2]=1/3,c[3]=2/3),`Gi ll's method with `*(c[2]=1/2,c[3]=1/2),`Dormand's method with `*(c[2]= 2/5,c[3]=3/5),`method with `*(c[2]=3/7,c[3]=6/11),`method with `*(c[2] =2/5,a[3,2]=2/3)]: errs := []:\nDigits := 20:\nfor ct to 6 do\n sm : = NCint(('phi'(x)-'cn_RK4_||ct'(x))^2,x=1..4.75,adaptive=false,numpoin ts=7,factor=200);\n errs := [op(errs),sqrt(sm/5)];\nend do:\nDigits \+ := 10:\nlinalg[transpose]([mthds,evalf(errs)]);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#K%'matrixG6#7(7$*&%7classical~method~with~G\"\"\"6$/&% \"cG6#\"\"##F*F0/&F.6#\"\"$F1F*$\"+#p_Ye'!#=7$*&%13/8~method~with~GF*6 $/F-#F*F5/F3#F0F5F*$\"+n$yKX%F87$*&%4Gill's~method~with~GF*F+F*$\"+ZQu )y#F87$*&%7Dormand's~method~with~GF*6$/F-#F0\"\"&/F3#F5FNF*$\"+s\"4^c% F87$*&%-method~with~GF*6$/F-#F5\"\"(/F3#\"\"'\"#6F*$\"+)*f**QPF87$*&FU F*6$FL/&%\"aG6$F5F0F@F*$\"+gi[fRF8Q*pprint1036\"" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 92 "The following error graph s are constructed using the numerical procedures for the solutions." } }{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 460 "evalf[25](plot(['cn_RK4_1' (x)-'phi'(x),'cn_RK4_2'(x)-'phi'(x),'cn_RK4_3'(x)-'phi'(x),'cn_RK4_4'( x)-'phi'(x),\n'cn_RK4_5'(x)-'phi'(x),'cn_RK4_6'(x)-'phi'(x)],x=1..3.75 ,font=[HELVETICA,9],\ncolor=[red,blue,COLOR(RGB,0,.65,0),COLOR(RGB,.4, 0,1),magenta,brown],legend=[`classical method`,`3/8 method`,`Gill's me thod`,`Dormand's method`,`method with c[2]=3/7 and c[3]=6/11`,`method \+ with c[2]=2/5 and a[3,2]=2/3`],title=`error curves for order 4 Runge-K utta methods`));" }}{PARA 13 "" 1 "" {GLPLOT2D 918 655 655 {PLOTDATA 2 "6,-%'CURVESG6%7S7$$\"\"\"\"\"!$F*F*7$$\":LLLLLLeRF27$$\":LLLLL$eRiYzH7F/$\"/u#z9&RmCF27$$\":nmmmmTN@+d&)G\"F/$\"/#[_ *fxhHF27$$\":LLLLL3xJ@PIM\"F/$\"/Dq$Q!=CMF27$$\":+++++v$4;$[%*R\"F/$\" /Jia9c!)QF27$$\":LLLLL3F%f()yd9F/$\"/zfc:-6VF27$$\":+++++v=U5Uf^\"F/$ \"/q*o+!pTZF27$$\":nmmmmm\"Hn*fdd\"F/$\"/?Vf2([<&F27$$\":LLLLL$ektvWG; 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