{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 0 0 1 0 0 0 0 0 0 0 }{CSTYLE "" -1 267 "" 0 1 0 0 0 0 0 0 1 0 0 0 0 0 0 0 }{CSTYLE "" -1 268 "" 0 1 0 0 0 0 0 0 1 0 0 0 0 0 0 0 }{CSTYLE "" -1 269 "" 0 1 0 0 0 0 0 0 1 0 0 0 0 0 0 0 } {CSTYLE "" -1 270 "" 0 1 0 0 0 0 0 0 1 0 0 0 0 0 0 0 }{CSTYLE "" -1 271 "" 0 1 0 0 0 0 0 0 1 0 0 0 0 0 0 0 }{CSTYLE "" -1 272 "" 0 1 0 0 0 0 0 0 1 0 0 0 0 0 0 0 }{CSTYLE "" -1 273 "" 0 1 0 0 0 0 0 0 1 0 0 0 0 0 0 0 }{CSTYLE "Cyan Emphasis" -1 274 "Times" 1 12 0 160 160 1 0 1 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 275 "" 0 1 0 0 0 0 0 0 1 0 0 0 0 0 0 0 }{CSTYLE "" -1 276 "" 0 1 0 0 0 0 0 0 1 0 0 0 0 0 0 0 }{CSTYLE "" -1 277 "" 0 1 0 0 0 0 0 0 1 0 0 0 0 0 0 0 }{CSTYLE "" -1 278 "" 0 1 0 0 0 0 0 0 1 0 0 0 0 0 0 0 }{CSTYLE "" 261 279 "" 1 18 0 0 0 0 0 0 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 280 "" 1 12 0 0 0 0 0 0 0 0 0 0 0 0 0 0 } {CSTYLE "" -1 281 "" 0 1 0 0 0 0 1 0 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 282 "" 0 1 0 0 0 0 1 0 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 283 "" 0 1 0 0 0 0 1 0 0 0 0 0 0 0 0 0 }{PSTYLE "Normal" -1 0 1 {CSTYLE "" -1 -1 "Tim es" 1 12 0 0 0 1 2 2 2 2 2 2 1 1 1 1 }1 1 0 0 0 0 1 0 1 0 2 2 0 1 } {PSTYLE "Heading 1" -1 3 1 {CSTYLE "" -1 -1 "Times" 1 18 0 0 128 1 2 1 2 2 2 2 1 1 1 1 }1 1 0 0 8 4 3 0 3 0 2 2 0 1 }{PSTYLE "Heading 2" -1 4 1 {CSTYLE "" -1 -1 "Times" 1 14 128 0 0 1 2 1 2 2 2 2 1 1 1 1 }1 1 0 0 8 2 1 0 1 0 2 2 0 1 }{PSTYLE "Heading 3" -1 5 1 {CSTYLE "" -1 -1 "Times" 1 12 128 0 0 1 1 1 2 2 2 2 1 1 1 1 }1 1 0 0 0 0 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 Output" -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 }{PSTYLE "Maple Output" -1 259 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 }} {SECT 0 {PARA 3 "" 0 "" {TEXT -1 102 "Derivation of 6 stage, order 5 R unge-Kutta schemes, each with a 7 stage, order 4 FSAL embedded scheme \n" }{TEXT 280 24 "(stage-order 2 examples)" }}{PARA 0 "" 0 "" {TEXT -1 45 "by Peter Stone, Gabriola Island, B.C., Canada" }}{PARA 0 "" 0 " " {TEXT -1 19 "Version: 16.12.2012" }}{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 "lo ad procedures for constructing Runge-Kutta schemes etc. " }}{PARA 0 " " 0 "" {TEXT -1 18 "The Maple m-files " }{TEXT 262 9 "butcher.m" } {TEXT -1 2 ", " }{TEXT 262 7 "roots.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 0 102 "read \" C:\\\\Maple/procdrs/butcher.m\";\nread \"C:\\\\Maple/procdrs/roots.m\" ;\nread \"C:\\\\Maple/procdrs/intg.m\";" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}} {PARA 0 "" 0 "" {TEXT -1 40 "#======================================= " }}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 41 "General Papakostas-Papageorgi ou solution " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 175 "The following subsection contains the equations that comprise \+ the Papakostas-Papageorgiou algorithm to obtain a general solution for the Runge-Kutta coefficients in terms of " }{XPPEDIT 18 0 "c[2]" "6# &%\"cG6#\"\"#" }{TEXT -1 3 ", " }{XPPEDIT 18 0 "c[3]" "6#&%\"cG6#\"\" $" }{TEXT -1 3 ", " }{XPPEDIT 18 0 "c[4]" "6#&%\"cG6#\"\"%" }{TEXT -1 3 ", " }{XPPEDIT 18 0 "c[5]" "6#&%\"cG6#\"\"&" }{TEXT -1 7 " and \+ " }{XPPEDIT 18 0 "`b*`[7]" "6#&%#b*G6#\"\"(" }{TEXT -1 1 "." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 127 "See: A Family \+ of Fifth-order Runge-Kutta Pairs, Mathematics of Computation, Volume 6 5, Number 215, July 1996, Pages 1165-1181. " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 51 "This algorithm was introduced i n another worksheet." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}{SECT 1 {PARA 4 "" 0 "" {TEXT 0 8 "P P_eqs_F" }{TEXT -1 42 ": equations of Papakostas and Papageorgiou" }} {PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 2525 "PP_eqs_F := [b[6]=(5*c[3]*(2*c[4]*(3*c[5]-2)-4*c[5]+3)-5*c[4]*(4 *c[5]-3)+3*(5*c[5]-4))/\n (60*c[6]*(c[3]-c[6])*(c[4] -c[6])*(c[5]-c[6])),\n\nb[5]=-(12*b[6]*c[6]*(c[4]-c[6])*(c[3]-c[6])-2* c[3]*(3*c[4]-2)+4*c[4]-3)/\n (12*c[5]*(c[3]-c[5])*(c[4]-c[5])), \n\nb[4]=-(6*b[5]*c[5]*(c[3]-c[5])+6*b[6]*c[6]*(c[3]-c[6])-3*c[3]+2)/( 6*c[4]*(c[3]-c[4])),\n\nb[3]=-(2*b[4]*c[4]+2*b[5]*c[5]+2*b[6]*c[6]-1)/ (2*c[3]),\n\nk[1]=1440*b[4]*b[5]*`b*`[7]*c[4]*c[5]*(c[5]-c[6])*(c[4]-c [5])*(c[3]^2-c[3]*(c[4]+c[5])+c[4]*c[5]),\n\nk[2]=1440*b[5]^2*`b*`[7]* c[5]^2*(c[5]-c[6])*(c[4]-c[5])*(c[3]^2-2*c[3]*c[5]+c[5]^2),\n\nk[3]=12 0*b[5]*c[5]*(c[5]-c[6])*(c[4]-c[5])*(c[3]-c[5])*(12*b[6]*`b*`[7]*c[6]* (c[3]-c[6])-2*c[3]+1),\n\nk[4]=(5*c[3]*(4*c[6]-3)-2*(5*c[6]-4))*(12*`b *`[7]*(c[4]-1)*(c[3]-1)-2*c[3]*(3*c[4]-2)+4*c[4]-3),\n\nm[1]=b[5]*c[5] *(c[4]-c[5])*(5*c[3]^2*(4*c[5]-3)-c[3]*(20*c[5]^2-5*c[5]-8)+2*c[5]*(5* c[5]-4)),\n\nm[2]=b[6]*c[6]*(c[4]-c[6])*(5*c[3]^2*(4*c[6]-3)-c[3]*(20* c[6]^2-5*c[6]-8)+2*c[6]*(5*c[6]-4)),\n\n`b*`[6]=-b[6]*(k[1]+k[2]+k[3]+ k[4])/(12*(m[1]+m[2])),\n\n`b*`[5]=-(12*`b*`[6]*c[6]*(c[4]-c[6])*(c[3] -c[6])+12*`b*`[7]*(c[4]-1)*(c[3]-1)-2*c[3]*(3*c[4]-2)+4*c[4]-3)/\n \+ (12*c[5]*(c[3]-c[5])*(c[4]-c[5])),\n\n`b*`[4]=-(6*`b*`[5]*c[5]*( c[3]-c[5])+6*`b*`[6]*c[6]*(c[3]-c[6])+6*`b*`[7]*(c[3]-1)-3*c[3]+2)/(6* c[4]*(c[3]-c[4])),\n\n`b*`[3]=-(2*`b*`[4]*c[4]+2*`b*`[5]*c[5]+2*`b*`[6 ]*c[6]+2*`b*`[7]-1)/(2*c[3]),\n\na[3,2]=c[3]^2/(2*c[2]),\n\na[6,2]=-(a [3,2]*(b[3]*(b[4]*`b*`[5]*(c[3]-c[4])+b[5]*`b*`[4]*(c[5]-c[3]))+b[4]*b [5]*`b*`[3]*(c[4]-c[5])))/\n (b[4]*(b[5]*`b*`[6]*(c[4]-c[5])+b [6]*`b*`[5]*(c[6]-c[4]))+b[5]*b[6]*`b*`[4]*(c[5]-c[6])),\n\na[5,2]=(a[ 3,2]*b[3]*(c[3]-c[4])+a[6,2]*b[6]*(c[6]-c[4]))/(b[5]*(c[4]-c[5])),\n\n a[4,2]=-(a[3,2]*b[3]+a[5,2]*b[5]+a[6,2]*b[6])/b[4],\n\na[6,5]=(5*c[3]* (2*c[4]-1)-5*c[4]+3)/(60*b[6]*c[5]*(c[3]-c[5])*(c[4]-c[5])),\n\na[6,4] =-(60*a[5,4]*b[5]*c[4]*(c[4]-c[5])*(c[3]-c[4])+5*c[3]*(2*c[5]-1)-5*c[5 ]+3)/\n (60*b[6]*c[4]*(c[3]-c[4])*(c[4]-c[5])),\n\na[5 ,4]=-(5*c[3]*(4*c[6]-3)-2*(5*c[6]-4))/(120*b[5]*c[4]*(c[3]-c[4])*(c[5] -c[6])),\n\na[4,3]=-(2*a[4,2]*c[2]-c[4]^2)/(2*c[3]),\n\na[5,3]=-(2*a[5 ,2]*c[2]+2*a[5,4]*c[4]-c[5]^2)/(2*c[3]),\n\na[6,3]=-(2*a[6,2]*c[2]+2*a [6,4]*c[4]+2*a[6,5]*c[5]-c[6]^2)/(2*c[3]),\n\nb[1]=1-b[3]-b[4]-b[5]-b[ 6],\n\n`b*`[1]=1-`b*`[3]-`b*`[4]-`b*`[5]-`b*`[6]-`b*`[7],\n\na[2,1]=c[ 2],\n\na[3,1]=c[3]-a[3,2],\n\na[4,1]=c[4]-a[4,2]-a[4,3],\n\na[5,1]=c[5 ]-a[5,2]-a[5,3]-a[5,4],\n\na[6,1]=c[6]-a[6,2]-a[6,3]-a[6,4]-a[6,5],\n \na[7,1]=b[1], a[7,2]=b[2], a[7,3]=b[3], a[7,4]=b[4], a[7,5]=b[5], a[7 ,6]=b[6]]:" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 " " {MPLTEXT 1 0 1 ";" }}}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 141 "e1 := \{c[6]=1,c[7]=1,b[2]=0,`b*`[2]=0\} :\neqns := subs(e1,PP_eqs_F):\nnops(%);\nvars := indets(eqns)minus \{c [2],c[3],c[4],c[5],`b*`[7]\};\nnops(%);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#\"#P" }}{PARA 12 "" 1 "" {XPPMATH 20 "6#>%%varsG " 0 " " {MPLTEXT 1 0 22 "infolevel[solve] := 4:" }}}{EXCHG {PARA 0 "> " 0 " " {MPLTEXT 1 0 94 "sol := solve(\{op(eqns)\},vars):\ne2 := `union`(e1, remove(has,sol,\{k,m\})):\ninfolevel[solve] := 0:" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 2 "e2" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 10256 "e2 := \{c[7] = 1, a[6,2] \+ = 1/2*(-15*c[4]*c[3]^2+30*c[5]*c[4]*c[3]^2-30*c[3]*c[4]*c[5]+16*c[4]*c [3]-5*c[3]*c[5]+3*c[3]+10*c[4]*c[5]-6*c[4])/(-12+15*c[3]+15*c[4]+15*c[ 5]-20*c[4]*c[3]-20*c[4]*c[5]+30*c[3]*c[4]*c[5]-20*c[3]*c[5])/c[3]/c[2] , c[6] = 1, a[2,1] = c[2], a[5,4] = 1/2*(c[4]*c[3]-c[3]*c[5]-c[4]*c[5] +c[5]^2)*c[5]*(5*c[3]-2)/(c[3]-c[4])/(3-5*c[3]+10*c[4]*c[3]-5*c[4])/c[ 4], `b*`[2] = 0, b[2] = 0, b[4] = 1/60*(10*c[3]*c[5]-5*c[3]-5*c[5]+3)/ (-c[4]^2+c[4]+c[4]*c[3]-c[3])/c[4]/(c[4]-c[5]), a[7,1] = 1/60*(5*c[4]- 10*c[4]*c[5]+30*c[3]*c[4]*c[5]-10*c[4]*c[3]-10*c[3]*c[5]-3+5*c[3]+5*c[ 5])/c[3]/c[4]/c[5], a[6,4] = 1/2*(6-30*c[5]^2*c[3]+20*c[5]^2*c[3]^2-10 *c[5]^2*c[4]-16*c[3]-8*c[4]-14*c[5]-7*c[4]^2*c[3]+30*c[5]^2*c[3]*c[4]+ 25*c[5]*c[4]*c[3]^2-20*c[5]^2*c[4]*c[3]^2+23*c[4]*c[3]+14*c[4]*c[5]-39 *c[3]*c[4]*c[5]+39*c[3]*c[5]-15*c[4]*c[3]^2+10*c[3]^2+2*c[4]^2+10*c[5] ^2-25*c[3]^2*c[5]+5*c[4]^2*c[3]^2)/c[4]/(12*c[4]^2-12*c[4]*c[3]-15*c[4 ]^3-12*c[4]*c[5]+12*c[3]*c[5]-20*c[4]^2*c[5]^2+20*c[4]^3*c[5]+20*c[4]^ 3*c[3]-20*c[4]^2*c[3]^2+20*c[5]^2*c[3]^2-20*c[4]^2*c[5]*c[3]+15*c[4]*c [3]^2+15*c[5]^2*c[4]+30*c[5]^2*c[4]^2*c[3]+30*c[5]*c[4]^2*c[3]^2-30*c[ 5]^2*c[4]*c[3]^2-30*c[5]*c[4]^3*c[3]+15*c[3]*c[4]*c[5]-15*c[3]^2*c[5]- 15*c[5]^2*c[3]), `b*`[1] = 1/60*(-460*c[5]*c[4]*c[3]*`b*`[7]+760*c[5]* c[4]*c[3]^2*`b*`[7]+3*c[3]-6*c[4]-20*c[4]^2*c[5]-60*c[4]^2*c[3]-1800*` b*`[7]*c[4]^2*c[5]*c[3]^2+100*c[5]*c[4]*c[3]^2+100*c[3]^2*c[5]*`b*`[7] -100*c[3]^3*c[5]*c[4]+300*c[3]^3*c[4]*`b*`[7]-600*c[3]^3*`b*`[7]*c[4]^ 2+940*`b*`[7]*c[4]^2*c[3]^2-60*c[5]*c[3]*`b*`[7]-560*c[4]^2*c[3]*`b*`[ 7]+300*c[5]*c[4]^2*c[3]^3-100*c[4]^2*c[3]^3+140*c[4]^2*c[5]*c[3]+120*` b*`[7]*c[4]^2+120*c[5]*c[4]*`b*`[7]-340*c[5]*c[4]^2*c[3]^2-200*`b*`[7] *c[4]^2*c[5]+1000*c[5]*c[4]^2*c[3]*`b*`[7]-600*c[3]^3*c[5]*c[4]*`b*`[7 ]+29*c[4]*c[3]-380*`b*`[7]*c[4]*c[3]^2+252*`b*`[7]*c[4]*c[3]+10*c[4]*c [5]-50*c[3]*c[4]*c[5]+1200*c[3]^3*`b*`[7]*c[4]^2*c[5]+36*`b*`[7]*c[3]- 72*`b*`[7]*c[4]-60*`b*`[7]*c[3]^2-5*c[3]*c[5]-60*c[4]*c[3]^2-5*c[3]^2+ 10*c[4]^2+10*c[3]^2*c[5]+50*c[4]*c[3]^3+130*c[4]^2*c[3]^2)/(10*c[4]*c[ 3]^2-8*c[4]*c[3]-c[3]+2*c[4])/c[3]/c[4]/c[5], a[6,1] = 1/2*(10*c[3]*c[ 2]*c[5]^2-14*c[3]*c[2]*c[5]-30*c[3]^2*c[2]*c[5]^2-25*c[3]^3*c[2]*c[5]+ 10*c[3]^3*c[2]-108*c[3]^2*c[2]*c[4]*c[5]-3*c[5]*c[4]*c[3]^2+5*c[5]^2*c [4]*c[3]^2-30*c[5]^2*c[4]^2*c[3]^3+30*c[5]^2*c[4]^2*c[3]^2-10*c[5]^2*c [4]^2*c[3]+15*c[5]*c[4]^2*c[3]^3+6*c[4]^2*c[5]*c[3]-30*c[3]^2*c[2]*c[4 ]^2-30*c[3]^3*c[4]*c[2]+10*c[3]*c[2]*c[4]^2-16*c[3]*c[2]*c[4]-16*c[5]* c[4]^2*c[3]^2+39*c[3]*c[2]*c[4]*c[5]-30*c[3]*c[2]*c[5]^2*c[4]-9*c[3]*c [2]*c[5]*c[4]^2+6*c[3]*c[2]+10*c[2]*c[4]^2*c[5]^2-10*c[3]*c[2]*c[4]^2* c[5]^2-60*c[3]^3*c[2]*c[5]*c[4]^2-40*c[3]^2*c[2]*c[4]^2*c[5]^2+46*c[3] ^2*c[2]*c[4]-16*c[2]*c[3]^2-6*c[2]*c[5]*c[4]^2+39*c[3]^2*c[2]*c[5]+60* c[2]*c[4]^2*c[5]^2*c[3]^3+20*c[3]^3*c[2]*c[4]^2+20*c[3]^3*c[2]*c[5]^2- 60*c[3]^3*c[2]*c[5]^2*c[4]+75*c[3]^3*c[2]*c[4]*c[5]+80*c[3]^2*c[2]*c[5 ]^2*c[4]+65*c[3]^2*c[2]*c[5]*c[4]^2)/(-12+15*c[3]+15*c[4]+15*c[5]-20*c [4]*c[3]-20*c[4]*c[5]+30*c[3]*c[4]*c[5]-20*c[3]*c[5])/c[3]^2/c[2]/c[4] /c[5], a[7,2] = 0, b[3] = -1/60*(10*c[4]*c[5]-5*c[4]-5*c[5]+3)/(c[3]-1 )/c[3]/(c[3]-c[5])/(c[3]-c[4]), a[6,3] = -1/2*(10*c[5]^2*c[3]-10*c[5]^ 2*c[3]^2+6*c[3]-6*c[4]^2*c[5]+16*c[4]^2*c[3]-40*c[5]^2*c[3]*c[4]-45*c[ 5]*c[4]*c[3]^2+60*c[5]^2*c[4]*c[3]^2-20*c[3]^3*c[5]*c[4]-10*c[5]^2*c[4 ]^2*c[3]-20*c[5]*c[4]^2*c[3]^3+15*c[4]^2*c[3]^3-19*c[4]^2*c[5]*c[3]+10 *c[4]^2*c[5]^2+45*c[5]*c[4]^2*c[3]^2-10*c[3]^4*c[4]-20*c[4]*c[5]^2*c[3 ]^3-16*c[4]*c[3]+45*c[3]*c[4]*c[5]+20*c[3]^4*c[5]*c[4]-14*c[3]*c[5]+17 *c[4]*c[3]^2-8*c[3]^2+2*c[3]^3+14*c[3]^2*c[5]+9*c[4]*c[3]^3-31*c[4]^2* c[3]^2)/(-30*c[5]^2*c[3]*c[4]+20*c[5]^2*c[4]-15*c[4]*c[5]+15*c[4]*c[3] +30*c[5]*c[4]*c[3]^2-20*c[4]*c[3]^2+15*c[3]^2+12*c[5]-12*c[3]+20*c[5]^ 2*c[3]-15*c[5]^2-20*c[3]^2*c[5])/(c[3]-c[4])/c[3]^2, a[4,1] = 1/2*c[4] *(2*c[2]*c[3]^2-3*c[4]*c[3]^2+2*c[4]^2*c[3]+2*c[3]*c[2]*c[4]-2*c[2]*c[ 4]^2)/c[2]/c[3]^2, a[5,3] = -1/2*c[5]*(-2*c[5]^2*c[3]+6*c[4]^2*c[5]-6* c[4]^2*c[3]+15*c[5]^2*c[3]*c[4]-5*c[5]*c[4]*c[3]^2-20*c[5]^2*c[4]*c[3] ^2+20*c[3]^3*c[5]*c[4]+20*c[5]^2*c[4]^2*c[3]-5*c[4]^2*c[5]*c[3]-10*c[4 ]^2*c[5]^2-20*c[5]*c[4]^2*c[3]^2-4*c[3]*c[4]*c[5]+4*c[4]*c[3]^2+2*c[3] ^2*c[5]-10*c[4]*c[3]^3+15*c[4]^2*c[3]^2)/(3-5*c[3]+10*c[4]*c[3]-5*c[4] )/c[3]^2/(c[3]-c[4]), a[4,3] = -c[4]^2*(c[3]-c[4])/c[3]^2, a[5,2] = 1/ 2*c[5]*(-15*c[4]*c[3]^2+30*c[5]*c[4]*c[3]^2-20*c[5]^2*c[3]*c[4]+6*c[4] *c[3]-5*c[5]^2*c[3]+3*c[3]*c[5]-6*c[4]*c[5]+10*c[5]^2*c[4])/c[2]/(3-5* c[3]+10*c[4]*c[3]-5*c[4])/c[3], `b*`[5] = 1/60*(-460*c[5]*c[4]*c[3]*`b *`[7]+760*c[5]*c[4]*c[3]^2*`b*`[7]+3*c[3]-6*c[4]-60*c[4]^2*c[3]-1800*` b*`[7]*c[4]^2*c[5]*c[3]^2+100*c[3]^2*c[5]*`b*`[7]+300*c[3]^3*c[4]*`b*` [7]-600*c[3]^3*`b*`[7]*c[4]^2+940*`b*`[7]*c[4]^2*c[3]^2-60*c[5]*c[3]*` b*`[7]-560*c[4]^2*c[3]*`b*`[7]-100*c[4]^2*c[3]^3+120*`b*`[7]*c[4]^2+12 0*c[5]*c[4]*`b*`[7]-200*`b*`[7]*c[4]^2*c[5]+1000*c[5]*c[4]^2*c[3]*`b*` [7]-600*c[3]^3*c[5]*c[4]*`b*`[7]+29*c[4]*c[3]-380*`b*`[7]*c[4]*c[3]^2+ 252*`b*`[7]*c[4]*c[3]+1200*c[3]^3*`b*`[7]*c[4]^2*c[5]+36*`b*`[7]*c[3]- 72*`b*`[7]*c[4]-60*`b*`[7]*c[3]^2-60*c[4]*c[3]^2-5*c[3]^2+10*c[4]^2+50 *c[4]*c[3]^3+130*c[4]^2*c[3]^2)/c[5]/(c[5]^2*c[3]-c[5]^3*c[3]+c[5]^2*c [3]^2-2*c[5]^2*c[4]+2*c[5]^3*c[4]+2*c[4]^2*c[5]-2*c[4]^2*c[3]+7*c[5]^2 *c[3]*c[4]-8*c[5]^3*c[4]*c[3]-9*c[5]*c[4]*c[3]^2-2*c[5]^2*c[4]*c[3]^2+ 10*c[5]^3*c[4]*c[3]^2+10*c[3]^3*c[5]*c[4]-10*c[5]^2*c[4]^2*c[3]^2+8*c[ 5]^2*c[4]^2*c[3]+10*c[5]*c[4]^2*c[3]^3-10*c[4]^2*c[3]^3-6*c[4]^2*c[5]* c[3]-2*c[4]^2*c[5]^2+2*c[5]*c[4]^2*c[3]^2-10*c[4]*c[5]^2*c[3]^3+c[3]*c [4]*c[5]+c[4]*c[3]^2-c[3]^2*c[5]+8*c[4]^2*c[3]^2), a[4,2] = 1/2*(3*c[3 ]-2*c[4])*c[4]^2/c[3]/c[2], `b*`[6] = -1/60*(-1000*c[5]*c[4]*c[3]*`b*` [7]+1780*c[5]*c[4]*c[3]^2*`b*`[7]-12*c[3]+24*c[4]+40*c[4]^2*c[5]+160*c [4]^2*c[3]-2880*`b*`[7]*c[4]^2*c[5]*c[3]^2-280*c[5]*c[4]*c[3]^2+160*c[ 3]^2*c[5]*`b*`[7]+200*c[3]^3*c[5]*c[4]+900*c[3]^3*c[4]*`b*`[7]-1200*c[ 3]^3*`b*`[7]*c[4]^2+2020*`b*`[7]*c[4]^2*c[3]^2-120*c[5]*c[3]*`b*`[7]-1 160*c[4]^2*c[3]*`b*`[7]-300*c[5]*c[4]^2*c[3]^3+200*c[4]^2*c[3]^3-220*c [4]^2*c[5]*c[3]+240*`b*`[7]*c[4]^2+240*c[5]*c[4]*`b*`[7]+440*c[5]*c[4] ^2*c[3]^2-320*`b*`[7]*c[4]^2*c[5]+1600*c[5]*c[4]^2*c[3]*`b*`[7]-1200*c [3]^3*c[5]*c[4]*`b*`[7]-111*c[4]*c[3]-1400*`b*`[7]*c[4]*c[3]^2+792*`b* `[7]*c[4]*c[3]-30*c[4]*c[5]+140*c[3]*c[4]*c[5]+1800*c[3]^3*`b*`[7]*c[4 ]^2*c[5]+96*`b*`[7]*c[3]-192*`b*`[7]*c[4]-120*`b*`[7]*c[3]^2+15*c[3]*c [5]+220*c[4]*c[3]^2+15*c[3]^2-30*c[4]^2-20*c[3]^2*c[5]-150*c[4]*c[3]^3 -310*c[4]^2*c[3]^2)/(-10*c[4]*c[3]^2+10*c[5]*c[4]*c[3]^2+8*c[4]*c[3]-8 *c[3]*c[4]*c[5]+c[3]-c[3]*c[5]-2*c[4]+2*c[4]*c[5])/(c[4]-1)/(c[3]-1), \+ `b*`[3] = 1/60*(-460*c[5]*c[4]*c[3]*`b*`[7]+760*c[5]*c[4]*c[3]^2*`b*`[ 7]+3*c[3]-6*c[4]-20*c[4]^2*c[5]-40*c[4]^2*c[3]-1800*`b*`[7]*c[4]^2*c[5 ]*c[3]^2+50*c[5]*c[4]*c[3]^2+100*c[3]^2*c[5]*`b*`[7]+300*c[3]^3*c[4]*` b*`[7]-600*c[3]^3*`b*`[7]*c[4]^2+940*`b*`[7]*c[4]^2*c[3]^2-60*c[5]*c[3 ]*`b*`[7]-560*c[4]^2*c[3]*`b*`[7]+80*c[4]^2*c[5]*c[3]+120*`b*`[7]*c[4] ^2+120*c[5]*c[4]*`b*`[7]-100*c[5]*c[4]^2*c[3]^2-200*`b*`[7]*c[4]^2*c[5 ]+1000*c[5]*c[4]^2*c[3]*`b*`[7]-600*c[3]^3*c[5]*c[4]*`b*`[7]+19*c[4]*c [3]-380*`b*`[7]*c[4]*c[3]^2+252*`b*`[7]*c[4]*c[3]+10*c[4]*c[5]-30*c[3] *c[4]*c[5]+1200*c[3]^3*`b*`[7]*c[4]^2*c[5]+36*`b*`[7]*c[3]-72*`b*`[7]* c[4]-60*`b*`[7]*c[3]^2-5*c[3]*c[5]-30*c[4]*c[3]^2+10*c[4]^2+50*c[4]^2* c[3]^2)/(c[3]-1)/c[3]/(-10*c[5]*c[4]*c[3]^2+8*c[3]*c[4]*c[5]-2*c[4]*c[ 5]+2*c[4]*c[3]+10*c[4]*c[3]^3-8*c[4]*c[3]^2+c[3]*c[5]-c[3]^2)/(c[3]-c[ 4]), a[3,2] = 1/2*c[3]^2/c[2], a[7,6] = 1/60*(-12+15*c[3]+15*c[4]+15*c [5]-20*c[4]*c[3]-20*c[4]*c[5]+30*c[3]*c[4]*c[5]-20*c[3]*c[5])/(c[5]-1) /(c[4]-1)/(c[3]-1), a[5,1] = 1/2*(2*c[3]*c[2]*c[5]^2-5*c[3]^2*c[2]*c[5 ]^2+5*c[3]^3*c[2]*c[5]+10*c[3]^2*c[2]*c[4]*c[5]-3*c[5]*c[4]*c[3]^2+5*c [5]^2*c[4]*c[3]^2+20*c[5]^2*c[4]^2*c[3]^2-10*c[5]^2*c[4]^2*c[3]-30*c[5 ]*c[4]^2*c[3]^3+15*c[4]^2*c[3]^3+6*c[4]^2*c[5]*c[3]-25*c[3]^2*c[2]*c[4 ]^2-15*c[3]^3*c[4]*c[2]+6*c[3]*c[2]*c[4]^2-2*c[3]*c[2]*c[4]*c[5]-5*c[3 ]*c[2]*c[5]^2*c[4]+5*c[3]*c[2]*c[5]*c[4]^2+10*c[2]*c[4]^2*c[5]^2-20*c[ 3]*c[2]*c[4]^2*c[5]^2+8*c[3]^2*c[2]*c[4]-6*c[2]*c[5]*c[4]^2-2*c[3]^2*c [2]*c[5]+20*c[3]^3*c[2]*c[4]^2+20*c[3]^2*c[2]*c[5]*c[4]^2-6*c[4]^2*c[3 ]^2)*c[5]/c[4]/c[3]^2/(3-5*c[3]+10*c[4]*c[3]-5*c[4])/c[2], a[6,5] = (- 3+8*c[3]+8*c[4]+3*c[5]+5*c[4]^2*c[5]+15*c[4]^2*c[3]-15*c[5]*c[4]*c[3]^ 2-15*c[4]^2*c[5]*c[3]+10*c[5]*c[4]^2*c[3]^2-23*c[4]*c[3]-8*c[4]*c[5]+2 3*c[3]*c[4]*c[5]-8*c[3]*c[5]+15*c[4]*c[3]^2-5*c[3]^2-5*c[4]^2+5*c[3]^2 *c[5]-10*c[4]^2*c[3]^2)/c[5]/(-20*c[5]^3*c[3]+20*c[5]^2*c[3]^2-20*c[5] ^3*c[4]-15*c[4]^2*c[5]+15*c[4]^2*c[3]+20*c[5]^2*c[3]*c[4]+30*c[5]^3*c[ 4]*c[3]-30*c[5]^2*c[4]*c[3]^2-30*c[5]^2*c[4]^2*c[3]+20*c[4]^2*c[5]^2+3 0*c[5]*c[4]^2*c[3]^2-12*c[4]*c[3]+12*c[4]*c[5]-15*c[3]*c[4]*c[5]+15*c[ 5]^3+12*c[3]*c[5]+15*c[4]*c[3]^2-12*c[5]^2-15*c[3]^2*c[5]-20*c[4]^2*c[ 3]^2), a[7,3] = -1/60*(10*c[4]*c[5]-5*c[4]-5*c[5]+3)/(c[3]-1)/c[3]/(c[ 3]-c[5])/(c[3]-c[4]), b[5] = -1/60*(3-5*c[3]+10*c[4]*c[3]-5*c[4])/c[5] /(-c[5]^2-c[5]^2*c[3]+c[5]^3+c[3]*c[5]-c[5]^2*c[4]+c[3]*c[4]*c[5]-c[4] *c[3]+c[4]*c[5]), b[1] = 1/60*(5*c[4]-10*c[4]*c[5]+30*c[3]*c[4]*c[5]-1 0*c[4]*c[3]-10*c[3]*c[5]-3+5*c[3]+5*c[5])/c[3]/c[4]/c[5], b[6] = 1/60* (-12+15*c[3]+15*c[4]+15*c[5]-20*c[4]*c[3]-20*c[4]*c[5]+30*c[3]*c[4]*c[ 5]-20*c[3]*c[5])/(c[5]-1)/(c[4]-1)/(c[3]-1), a[3,1] = 1/2*c[3]*(2*c[2] -c[3])/c[2], `b*`[4] = -1/60*(-460*c[5]*c[4]*c[3]*`b*`[7]+760*c[5]*c[4 ]*c[3]^2*`b*`[7]+3*c[3]-6*c[4]-1800*`b*`[7]*c[4]^2*c[5]*c[3]^2+130*c[5 ]*c[4]*c[3]^2+100*c[3]^2*c[5]*`b*`[7]-100*c[3]^3*c[5]*c[4]+300*c[3]^3* c[4]*`b*`[7]-600*c[3]^3*`b*`[7]*c[4]^2+940*`b*`[7]*c[4]^2*c[3]^2-60*c[ 5]*c[3]*`b*`[7]-560*c[4]^2*c[3]*`b*`[7]+120*`b*`[7]*c[4]^2+120*c[5]*c[ 4]*`b*`[7]-200*`b*`[7]*c[4]^2*c[5]+1000*c[5]*c[4]^2*c[3]*`b*`[7]-600*c [3]^3*c[5]*c[4]*`b*`[7]+34*c[4]*c[3]-380*`b*`[7]*c[4]*c[3]^2+252*`b*`[ 7]*c[4]*c[3]+10*c[4]*c[5]-60*c[3]*c[4]*c[5]+1200*c[3]^3*`b*`[7]*c[4]^2 *c[5]+36*`b*`[7]*c[3]-72*`b*`[7]*c[4]-60*`b*`[7]*c[3]^2-5*c[3]*c[5]-70 *c[4]*c[3]^2-5*c[3]^2+10*c[3]^2*c[5]+50*c[4]*c[3]^3)/(-c[4]^2+c[4]+c[4 ]*c[3]-c[3])/c[4]/(-10*c[5]*c[4]*c[3]^2+10*c[4]^2*c[3]^2+8*c[3]*c[4]*c [5]+c[3]*c[5]-8*c[4]^2*c[3]-c[4]*c[3]-2*c[4]*c[5]+2*c[4]^2), a[7,4] = \+ 1/60*(10*c[3]*c[5]-5*c[3]-5*c[5]+3)/(-c[4]^2+c[4]+c[4]*c[3]-c[3])/c[4] /(c[4]-c[5]), a[7,5] = -1/60*(3-5*c[3]+10*c[4]*c[3]-5*c[4])/c[5]/(-c[5 ]^2-c[5]^2*c[3]+c[5]^3+c[3]*c[5]-c[5]^2*c[4]+c[3]*c[4]*c[5]-c[4]*c[3]+ c[4]*c[5])\}:" }{TEXT -1 0 "" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 34 "general coefficients in terms of " } {XPPEDIT 18 0 "c[2]" "6#&%\"cG6#\"\"#" }{TEXT -1 3 ", " }{XPPEDIT 18 0 "c[3]" "6#&%\"cG6#\"\"$" }{TEXT -1 3 ", " }{XPPEDIT 18 0 "c[4]" "6# &%\"cG6#\"\"%" }{TEXT -1 3 ", " }{XPPEDIT 18 0 "c[5]" "6#&%\"cG6#\"\" &" }{TEXT -1 7 " and " }{XPPEDIT 18 0 "`b*`[7];" "6#&%#b*G6#\"\"(" } {TEXT -1 1 " " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 575 "for ii from 2 to 7 do\n print(`============== ========================================`);\n if 2<=ii and ii<=5 the n\n print(c[ii]*` is a parameter`)\n else\n print(c[ii]=su bs(e2,c[ii]));\n end if:\n print(``); \n for jj to ii-1 do\n \+ print(a[ii,jj]=subs(e2,a[ii,jj]));print(``);\n end do:\nend do: \n`######################################################`;\n``;\nfor \+ ii to 6 do\n print(b[ii]=subs(e2,b[ii]));print(``);\nend do:\n`##### #################################################`;\n``;\nfor ii to 6 \+ do\n print(`b*`[ii]=subs(e2,`b*`[ii]));print(``);\nend do:" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%W========================================== ============G" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#*&&%\"cG6#\"\"#\"\"\" %0~is~a~parameterGF(" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%!G" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/&%\"aG6$\"\"#\"\"\"&%\"cG6#F'" }}{PARA 11 " " 1 "" {XPPMATH 20 "6#%!G" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%W======= ===============================================G" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#*&&%\"cG6#\"\"$\"\"\"%0~is~a~parameterGF(" }}{PARA 11 " " 1 "" {XPPMATH 20 "6#%!G" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/&%\"aG6$ \"\"$\"\"\",$*&#F(\"\"#F(*(&%\"cG6#F'F(,&*&F,F(&F/6#F,F(F(F.!\"\"F(F3F 5F(F(" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%!G" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/&%\"aG6$\"\"$\"\"#,$*&#\"\"\"F(F,*&&%\"cG6#F'F(&F/6#F( !\"\"F,F," }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%!G" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%W===================================================== =G" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#*&&%\"cG6#\"\"%\"\"\"%0~is~a~par ameterGF(" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%!G" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/&%\"aG6$\"\"%\"\"\",$*&#F(\"\"#F(**&%\"cG6#F'F(,,*(F,F (&F/6#F,F()&F/6#\"\"$F,F(F(*(F8F(F.F(F5F(!\"\"*(F,F()F.F,F(F6F(F(**F,F (F6F(F3F(F.F(F(*(F,F(F3F(F F,F6F,FF,F7F,F ,**\"#SF,F2F,F7F,F?F,F;**\"#XF,F3F,F?F,F:F,F;**\"#gF,F2F,F?F,F:F,F,** \"#?F,)F7F(F,F3F,F?F,F;**F1F,F2F,F>F,F7F,F;**FKF,F3F,F>F,FLF,F;*(\"#:F ,F>F,FLF,F,**\"#>F,F>F,F3F,F7F,F;*(F1F,F>F,F2F,F,**FGF,F3F,F>F,F:F,F,* (F1F,)F7FAF,F?F,F;**FKF,F?F,F2F,FLF,F;*(FCF,F?F,F7F,F;**FGF,F7F,F?F,F3 F,F,**FKF,FVF,F3F,F?F,F,*(\"#9F,F7F,F3F,F;*(\"#F,F:F,F ;F,,:**\"#IF,F2F,F7F,F?F,F;*(FKF,F2F,F?F,F,*(FPF,F?F,F3F,F;*(FPF,F?F,F 7F,F,**FcoF,F3F,F?F,F:F,F,*(FKF,F?F,F:F,F;*&FPF,F:F,F,*&\"#7F,F3F,F,*& F[pF,F7F,F;*(FKF,F2F,F7F,F,*&FPF,F2F,F;*(FKF,F:F,F3F,F;F;,&F7F,F?F;F;F 7!\"#F,F;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%!G" }}{PARA 12 "" 1 "" {XPPMATH 20 "6#/&%\"aG6$\"\"'\"\"%,$*&#\"\"\"\"\"#F,*(,LF'F,*(\"#IF,)& %\"c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inF/F4F/F8F/F/**F >F/F0F/F4F/F8F/F,*&F>F/FQF/F/*(FinF/F0F/F8F/F/*(F>F/F4F/F@F/F/*&FinF/F SF/F,*(F>F/F@F/F8F/F,*(FPF/F;F/F@F/F,F," }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%!G" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%W======================= ===============================G" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/& %\"cG6#\"\"(\"\"\"" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%!G" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/&%\"aG6$\"\"(\"\"\",$*&#F(\"#gF(**,2*&\"\"&F(& %\"cG6#\"\"%F(F(*(\"#5F(F1F(&F26#F0F(!\"\"**\"#IF(&F26#\"\"$F(F1F(F7F( F(*(F6F(F1F(FF9*&F0F(FF+*&F>F+F1F+F+F1F9F9F>F9,&F>F+F5F9F9F+F +" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%!G" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/&%\"bG6#\"\"&,$*&#\"\"\"\"#gF+*(,*\"\"$F+*&F'F+&%\"cG6#F/F+!\" \"*(\"#5F+&F26#\"\"%F+F1F+F+*&F'F+F7F+F4F+&F2F&F4,2*$)F;\"\"#F+F4*&F>F +F1F+F4*$)F;F/F+F+*&F1F+F;F+F+*&F>F+F7F+F4*(F1F+F7F+F;F+F+*&F7F+F1F+F4 *&F7F+F;F+F+F4F+F4" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%!G" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/&%\"bG6#\"\"',$*&#\"\"\"\"#gF+**,2\"#7!\"\"*& \"#:F+&%\"cG6#\"\"$F+F+*&F2F+&F46#\"\"%F+F+*&F2F+&F46#\"\"&F+F+*(\"#?F +F8F+F3F+F0*(F@F+F8F+FF+F5F+F8F+F+**\"$!QF+F9F+F5 F+F?F+F<**\"$_#F+F9F+F5F+F8F+F+*(\"#5F+F5F+F1F+F+**\"#IF+F8F+F5F+F1F+F <*,\"%+7F+FQF+F9F+FFF+F1F+F+*(\"#OF+F9F+F8F+F+*(\"#sF+F9F+F5F+F<*(F,F+ F9F+F?F+F<*(F4F+F8F+F1F+F<*(FgoF+F5F+F?F+F<*&FeoF+FFF+F+*(FLF+FFF+F?F+ F+F+,&F8F+F+F*&\"#CF+F5F+F+*(\"#SF+) F5FBF+F1F+F+*(\"$g\"F+FIF+F8F+F+*,\"%!)GF+F;F+FIF+F1F+FAF+F>**\"$!GF+F 1F+F5F+FAF+F>**FKF+FAF+F1F+F;F+F+**\"$+#F+)F8F:F+F1F+F5F+F+**\"$+*F+FS F+F5F+F;F+F+**\"%+7F+FSF+F;F+FIF+F>**\"%??F+F;F+FIF+FAF+F+**\"$?\"F+F1 F+F8F+F;F+F>**\"%g6F+FIF+F8F+F;F+F>**\"$+$F+F1F+FIF+FSF+F>*(FRF+FIF+FS F+F+**\"$?#F+FIF+F1F+F8F+F>*(\"$S#F+F;F+FIF+F+**F^oF+F1F+F5F+F;F+F+** \"$S%F+F1F+FIF+FAF+F+**\"$?$F+F;F+FIF+F1F+F>*,\"%+;F+F1F+FIF+F8F+F;F+F +*,FWF+FSF+F1F+F5F+F;F+F>*(\"$6\"F+F5F+F8F+F>**\"%+9F+F;F+F5F+FAF+F>** \"$#zF+F;F+F5F+F8F+F+*(\"#IF+F5F+F1F+F>**\"$S\"F+F8F+F5F+F1F+F+*,\"%+= F+FSF+F;F+FIF+F1F+F+*(\"#'*F+F;F+F8F+F+*(\"$#>F+F;F+F5F+F>*(FenF+F;F+F AF+F>*(\"#:F+F8F+F1F+F+*(F\\oF+F5F+FAF+F+*&FipF+FAF+F+*&F^pF+FIF+F>*( \"#?F+FAF+F1F+F>*(\"$]\"F+F5F+FSF+F>*(\"$5$F+FIF+FAF+F>F+,2*(\"#5F+F5F +FAF+F>**FeqF+F1F+F5F+FAF+F+*(\"\")F+F5F+F8F+F+**FhqF+F8F+F5F+F1F+F>F8 F+*&F8F+F1F+F>*&FBF+F5F+F>*(FBF+F5F+F1F+F+F>,&F5F+F+F>F>,&F8F+F+F>F>F+ F>" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%!G" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 94 "We can obtain an exp ression for the square of the principal error norm in terms of the nod es " }{XPPEDIT 18 0 "c[2]" "6#&%\"cG6#\"\"#" }{TEXT -1 3 ", " } {XPPEDIT 18 0 "c[3]" "6#&%\"cG6#\"\"$" }{TEXT -1 3 ", " }{XPPEDIT 18 0 "c[4]" "6#&%\"cG6#\"\"%" }{TEXT -1 7 " and " }{XPPEDIT 18 0 "c[5] " "6#&%\"cG6#\"\"&" }{TEXT -1 2 ". " }}{PARA 0 "" 0 "" {TEXT -1 0 "" } }{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 171 "errterms5_6 := PrincipalEr rorTerms(5,6,'expanded'):\nsm := 0:\nfor ct to nops(errterms5_6) do\n \+ sm := sm+(simplify(subs(e2,errterms5_6[ct])))^2;\nend do:\nsm := sim plify(sm):" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 " " {MPLTEXT 1 0 1 ";" }}}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 49 "sm: squar e of principal error norm in terms of " }{XPPEDIT 18 0 "c[2]" "6#&% \"cG6#\"\"#" }{TEXT -1 3 ", " }{XPPEDIT 18 0 "c[3]" "6#&%\"cG6#\"\"$ " }{TEXT -1 3 ", " }{XPPEDIT 18 0 "c[4]" "6#&%\"cG6#\"\"%" }{TEXT -1 7 " and " }{XPPEDIT 18 0 "c[5]" "6#&%\"cG6#\"\"&" }{TEXT -1 1 " " }} {PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1665 "sm := 1/25920000*(12042*c[5]^2*c[3]^2+297000*c[3]*c[2]^2*c[4]^2* c[5]-13500*c[3]^2*c[2]*c[5]^2+28044*c[3]^3*c[5]-270000*c[3]*c[2]^2*c[4 ]^2*c[5]^2-22140*c[3]^4*c[5]-148500*c[3]^2*c[2]^2*c[4]*c[5]-270000*c[3 ]^2*c[2]^2*c[4]^2*c[5]+135000*c[3]^2*c[2]^2*c[4]*c[5]^2-22140*c[5]^2*c [3]^3+270000*c[3]^2*c[2]^2*c[4]^2*c[5]^2-18000*c[3]^2*c[2]*c[4]*c[5]+8 1000*c[3]*c[2]^2*c[5]*c[4]-67500*c[3]*c[2]^2*c[5]^2*c[4]+96444*c[5]*c[ 4]*c[3]^2-76140*c[5]^2*c[4]*c[3]^2-613800*c[5]^2*c[4]^2*c[3]^3-221940* c[3]^3*c[5]*c[4]+153450*c[5]^2*c[4]^2*c[3]^2+675180*c[5]*c[4]^2*c[3]^3 -240140*c[4]^2*c[3]^3+198000*c[3]^2*c[2]*c[4]^2-49500*c[3]^3*c[4]*c[2] -55800*c[3]*c[2]*c[4]^2+12000*c[3]*c[2]*c[4]-184140*c[5]*c[4]^2*c[3]^2 -34200*c[3]*c[2]*c[4]*c[5]+27000*c[3]*c[2]*c[5]^2*c[4]+162000*c[3]*c[2 ]*c[5]*c[4]^2+613800*c[4]^2*c[3]^4*c[5]^2-613800*c[4]^2*c[3]^4*c[5]-73 800*c[4]*c[3]^4*c[5]^2-42140*c[3]^4*c[4]+223450*c[4]^2*c[3]^4+189180*c [4]*c[5]^2*c[3]^3+18450*c[5]^2*c[3]^4+10642*c[3]^4-135000*c[3]*c[2]*c[ 4]^2*c[5]^2+27900*c[2]^2*c[4]^2+6975*c[3]^2*c[2]^2+540000*c[3]^3*c[2]* c[5]*c[4]^2+81180*c[3]^4*c[5]*c[4]+540000*c[3]^2*c[2]*c[4]^2*c[5]^2+69 00*c[3]^2*c[2]*c[4]-6000*c[2]*c[3]^2-36856*c[4]*c[3]^2+17100*c[3]^2*c[ 2]*c[5]+6552*c[3]^2-12856*c[3]^3-27900*c[3]*c[2]^2*c[4]-99000*c[3]*c[2 ]^2*c[4]^2+49500*c[3]^2*c[2]^2*c[4]+90000*c[3]^2*c[2]^2*c[4]^2-20250*c [3]^2*c[2]^2*c[5]+16875*c[3]^2*c[2]^2*c[5]^2-81000*c[5]*c[2]^2*c[4]^2+ 67500*c[5]^2*c[2]^2*c[4]^2-540000*c[2]*c[4]^2*c[5]^2*c[3]^3-180000*c[3 ]^3*c[2]*c[4]^2-135000*c[3]^3*c[2]*c[5]^2*c[4]+148500*c[3]^3*c[2]*c[4] *c[5]+13500*c[3]^2*c[2]*c[5]^2*c[4]-594000*c[3]^2*c[2]*c[5]*c[4]^2-160 56*c[3]^2*c[5]+88044*c[4]*c[3]^3+66442*c[4]^2*c[3]^2)/c[3]^2:" }}} {PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 67 "The stability function for the order 5 scheme is only affected by " }{XPPEDIT 18 0 "c[3]" "6#&%\"cG6#\"\"$" }{TEXT -1 7 " and " } {XPPEDIT 18 0 "c[4]" "6#&%\"cG6#\"\"%" }{TEXT -1 2 ". " }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 34 "StabilityFunction(5,6,'expanded'); " }}{PARA 11 "" 1 "" {XPPMATH 20 "6#,0\"\"\"F$%\"zGF$*&\"\"#!\"\"F%F'F $*&\"\"'F(F%\"\"$F$*&\"#CF(F%\"\"%F$*&\"$?\"F(F%\"\"&F$*0&%\"bG6#F*F$& %\"aG6$F*F1F$&F76$F1F.F$&F76$F.F+F$&F76$F+F'F$&%\"cG6#F'F$)F%F*F$F$" } }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 57 "simplify(subs(e2,b[6]*a[6,5]*a[5,4]*a[4,3]*a[3,2]*c[2]));" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#,$*&#\"\"\"\"$S#F&*&,&*&\"\"&F&&%\"cG6 #\"\"$F&F&\"\"#!\"\"F&&F-6#\"\"%F&F&F1" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}}{PARA 0 "" 0 "" {TEXT 279 30 "______________________________" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 87 "#==================================================================== ==================" }}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 22 "The stabili ty radius " }{XPPEDIT 18 0 "rho" "6#%$rhoG" }{TEXT -1 32 " and imagi nary axis inclusion " }{XPPEDIT 18 0 "rho[2]" "6#&%$rhoG6#\"\"#" } {TEXT -1 42 " for a 6 stage, order 5 Runge-Kutta scheme" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 13 "The (linear) " } {TEXT 260 16 "stability radius" }{TEXT -1 1 " " }{XPPEDIT 18 0 "rho" " 6#%$rhoG" }{TEXT -1 145 " associated with a stability region of an exp licit Runge-Kutta method is the radius of the largest disc that can fi t inside the stability region." }}{PARA 0 "" 0 "" {TEXT -1 99 "The int erval given by the intersection of the stability region with the imagi nary axis has radius " }{XPPEDIT 18 0 "rho[2]" "6#&%$rhoG6#\"\"#" } {TEXT -1 8 " where " }{XPPEDIT 18 0 "rho[2]" "6#&%$rhoG6#\"\"#" } {TEXT -1 27 " is the (linear stability)" }{TEXT 260 25 " imaginary ax is inclusion" }{TEXT -1 1 "." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 138 "Suppose we are given a 6 stage, order 5 Runge -Kutta scheme constructed via the Papakostas-Papageorgiou algorithm wi th stability function " }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "R(z)= 1+z+1/2" "6#/-%\"RG6#%\"zG,(\"\"\"F)F'F)*&F)F)\"\"#!\"\"F) " }{TEXT -1 1 " " }{XPPEDIT 18 0 "z^2+1/6" "6#,&*$%\"zG\"\"#\"\"\"*&F' F'\"\"'!\"\"F'" }{TEXT -1 1 " " }{XPPEDIT 18 0 "z^3+1/24" "6#,&*$%\"zG \"\"$\"\"\"*&F'F'\"#C!\"\"F'" }{TEXT -1 1 " " }{XPPEDIT 18 0 "z^4+1/12 0" "6#,&*$%\"zG\"\"%\"\"\"*&F'F'\"$?\"!\"\"F'" }{TEXT -1 1 " " } {XPPEDIT 18 0 "z^5+t[6]*z^6" "6#,&*$%\"zG\"\"&\"\"\"*&&%\"tG6#\"\"'F'* $F%F,F'F'" }{TEXT -1 2 ", " }}{PARA 0 "" 0 "" {TEXT -1 7 "where " } {XPPEDIT 18 0 "t[6]=(2-5*c[3])*c[4]/240" "6#/&%\"tG6#\"\"'*(,&\"\"#\" \"\"*&\"\"&F+&%\"cG6#\"\"$F+!\"\"F+&F/6#\"\"%F+\"$S#F2" }{TEXT -1 1 ". " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 71 "For e xample, the \"classic\" Prince-Dormand 6 stage, order 5 scheme has " }{XPPEDIT 18 0 "c[3] = 3/10" "6#/&%\"cG6#\"\"$*&F'\"\"\"\"#5!\"\"" } {TEXT -1 7 " and " }{XPPEDIT 18 0 "c[4]=4/5" "6#/&%\"cG6#\"\"%*&F'\" \"\"\"\"&!\"\"" }{TEXT -1 6 ", so " }{XPPEDIT 18 0 "t[6]=1/600" "6#/& %\"tG6#\"\"'*&\"\"\"F)\"$+'!\"\"" }{TEXT -1 32 " and the stability fu nction is " }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "R(z) = \+ 1+z+z^2/2+z^3/6+z^4/24+z^5/120+z^6/600" "6#/-%\"RG6#%\"zG,0\"\"\"F)F'F )*&F'\"\"#F+!\"\"F)*&F'\"\"$\"\"'F,F)*&F'\"\"%\"#CF,F)*&F'\"\"&\"$?\"F ,F)*&F'F/\"$+'F,F)" }{TEXT -1 2 ". " }}{PARA 0 "" 0 "" {TEXT -1 0 "" } }{PARA 0 "" 0 "" {TEXT -1 25 "For this scheme we have " }{XPPEDIT 18 0 "rho" "6#%$rhoG" }{TEXT -1 1 " " }{TEXT 271 1 "~" }{TEXT -1 18 " 1.6 53283946 and " }{XPPEDIT 18 0 "rho[2]" "6#&%$rhoG6#\"\"#" }{TEXT -1 1 " " }{TEXT 270 1 "~" }{TEXT -1 14 " 0.9971890086." }}{PARA 0 "" 0 " " {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 258 "R := z -> 1+z+1/2*z^2+1/6*z^3+1/24*z^4+1/120*z^5+1/600*z^6:\nrho=-newton(R(z)=1 ,z=-2.8)/2;\nreal_part := proc(u)\n Re(newton(R(z)=exp(u*Pi*I),z=1.0 *I))\nend proc:\nu0 := bisect('real_part'(u),u=0.2..0.5):\nrho[2]=Im(e valf[12](newton(R(z)=exp(u0*Pi*I),z=1.0*I)));" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/%$rhoG$\"+YRG`;!\"*" }}{PARA 11 "" 1 "" {XPPMATH 20 "6 #/&%$rhoG6#\"\"#$\"-#f3!*=(**!#7" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }} {PARA 0 "" 0 "" {TEXT -1 54 "The following picture shows how the stabi lity radius " }{XPPEDIT 18 0 "rho" "6#%$rhoG" }{TEXT -1 53 " and the linear stability imaginary axis inclusion " }{XPPEDIT 18 0 "rho[2]" "6#&%$rhoG6#\"\"#" }{TEXT -1 13 " depend on " }{XPPEDIT 18 0 "t[6]" "6#&%\"tG6#\"\"'" }{TEXT -1 12 " for 0.0014" }{XPPEDIT 18 0 "`` <= t[ 6];" "6#1%!G&%\"tG6#\"\"'" }{XPPEDIT 18 0 "``<=0" "6#1%!G\"\"!" } {TEXT -1 6 ".0078." }}{PARA 0 "" 0 "" {TEXT -1 4 "The " }{TEXT 260 6 " purple" }{TEXT -1 25 " curve is the graph for " }{XPPEDIT 18 0 "rho" "6#%$rhoG" }{TEXT -1 12 ", while the " }{TEXT 257 5 "green" }{TEXT -1 25 " curve is the graph for " }{XPPEDIT 18 0 "rho[2]" "6#&%$rhoG6#\" \"#" }{TEXT -1 1 "." }}{PARA 0 "" 0 "" {TEXT -1 41 "The curves interse ct at the point where " }{XPPEDIT 18 0 "t[6]" "6#&%\"tG6#\"\"'" } {TEXT -1 2 " " }{TEXT 265 1 "~" }{TEXT -1 24 " 0.002946139890 where \+ " }{XPPEDIT 18 0 "rho=rho[2]" "6#/%$rhoG&F$6#\"\"#" }{TEXT -1 2 " " }{TEXT 266 1 "~" }{TEXT -1 15 " 1.398032276. " }}{PARA 0 "" 0 "" {TEXT -1 20 "The points coloured " }{TEXT 274 4 "cyan" }{TEXT -1 27 " \+ correspond to the values " }{XPPEDIT 18 0 "rho" "6#%$rhoG" }{TEXT -1 1 " " }{TEXT 273 1 "~" }{TEXT -1 18 " 1.653283946 and " }{XPPEDIT 18 0 "rho[2]" "6#&%$rhoG6#\"\"#" }{TEXT -1 1 " " }{TEXT 272 1 "~" }{TEXT -1 21 " 0.9971890086 where " }{XPPEDIT 18 0 "t[6]=1/600" "6#/&%\"tG6# \"\"'*&\"\"\"F)\"$+'!\"\"" }{TEXT -1 43 " associated with the Prince-D ormand scheme." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1156 "pts := []: pts2 := []:\nfor i from 0 to 320 do \n t6 := 0.14e-2+i*2e-5;\n R := z -> 1+z+1/2*z^2+1/6*z^3+1/24*z^4+ 1/120*z^5+t6*z^6:\n z0 := -newton(R(z)=1,z=-4)/2;\n pts := [op(pts ),[t6,z0]];\n real_part := proc(u)\n Re(newton(R(z)=exp(u*Pi*I) ,z=1.*I))\n end proc:\n u0 := bisect('real_part'(u),u=7.5e-2..0.5) ;\n rr := Im(newton(R(z)=exp(u0*Pi*I),z=1.*I));\n pts2 := [op(pts2 ),[t6,rr]];\nend do:\np1 := plot([pts,pts2],color=[COLOR(RGB,.5,0,1),C OLOR(RGB,0,.75,0)]):\np2 := plot([[.2946139890e-2,0],[.2946139890e-2,1 .398032276]],color=brown,linestyle=2):\np3 := plot([[[.2946139890e-2,1 .398032276]]$4],style=point,symbol=[diamond,cross,circle$2],\n \+ symbolsize=[10$3,15],color=[coral$3,black]):\np4 := plot([[.13888 88889e-2,0],[.1388888889e-2,1.77]],color=black,linestyle=3):\np5 := pl ot([[.1666666667e-2,0],[.1666666667e-2,1.653283946]],color=COLOR(RGB,. 7,0,.7),linestyle=2):\np6 := plot([[[.1666666667e-2,.997189008592],[.1 666666667e-2,1.653283946]]$4],style=point,\n symbol=[diamond,cross, circle$2],symbolsize=[10$3,15],color=[cyan$3,black]):\nplots[display]( [p||(1..6)],font=[HELVETICA,9],labels=[`t[6]`,``],view=[0.0012..0.0078 ,0..1.77]);" }}{PARA 13 "" 1 "" {GLPLOT2D 614 614 614 {PLOTDATA 2 "62- %'CURVESG6$7]_l7$$\"3++++++++9!#?$\"3)******>gg0x\"!#<7$$\"3/++++++?9F *$\"35+++P$)zf5%G%p\"F-7$$\"3-++++++!e\"F*$\"3%******fy%4'o\"F-7$$\"33+++++++;F *$\"3/+++92:y;F-7$$\"3#*************>;F*$\"3!******R2R/n\"F-7$$\"3(*** **********R;F*$\"3/+++yz%Hm\"F-7$$\"3-++++++g;F*$\"35+++%Qmcl\"F-7$$\" 33++++++!o\"F*$\"35+++ASe[;F-7$$\"3#**************p\"F*$\"3$******4K\" pT;F-7$$\"3'*************>F*$\"3++++)Q([\"e\"F-7$$\"30++++++?>F*$\"3#******R/+id \"F-7$$\"35++++++S>F*$\"3)******zCD5d\"F-7$$\"3%*************f>F*$\"3% ******\\tefc\"F-7$$\"3+++++++!)>F*$\"3#******zY'*4c\"F-7$$\"31+++++++? 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\"#:Fb_u-F$6%7$7$$\"3%*******)))))))Q\"F*F_`q7$F[au$\"3-++++++qf3!*=(**Fi`qFi au-Fi]u6&F[`qF_`qFi^uFi^uF]_uFb_u-F$6&FbbuFfbuFh_uFb_u-F$6&FbbuFfbuF]` uFb_u-F$6&FbbuFb`uFd`uFb_u-%+AXESLABELSG6%%%t[6]G%!G-%%FONTG6#%(DEFAUL TG-Fdcu6$%*HELVETICAG\"\"*-%%VIEWG6$;$\"#7!\"%$\"#yFadu;F_`q$\"$x\"F^] u" 1 2 0 1 10 0 2 9 1 4 2 1.000000 45.000000 45.000000 0 0 "Curve 1" " Curve 2" "Curve 3" "Curve 4" "Curve 5" "Curve 6" "Curve 7" "Curve 8" " Curve 9" "Curve 10" "Curve 11" "Curve 12" "Curve 13" }}}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 64 "The following procedur e can be used to calculate the value of " }{XPPEDIT 18 0 "t[6]" "6#& %\"tG6#\"\"'" }{TEXT -1 10 " where " }{XPPEDIT 18 0 "rho=rho[2]" "6 #/%$rhoG&F$6#\"\"#" }{TEXT -1 1 "." }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 361 "rad_inc := proc(t_6)\n local R,t6,lft,rgt,real_par t,u0;\n lft := ll_; rgt := rr_;\n R := z -> 1+z+1/2*z^2+1/6*z^3+1/ 24*z^4+1/120*z^5+t_6*z^6:\n real_part := proc(u)\n Re(newton(R( z)=exp(u*Pi*I),z=1.4*I))\n end proc:\n u0 := bisect('real_part'(u) ,u=lft..rgt);\n Im(evalf[Digits+4](newton(R(z)=exp(u0*Pi*I),z=1.4*I) ))+newton(R(z)=1,z=-2.8)/2;\nend proc:" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 86 "ll_ := .44: rr_ := .47: \nevalf[20](bisect('rad_inc'(t_6),t_6=0.0028..0.003,info=true));" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#$\"5?_l7,!*)Rh%H!#A" }}}{PARA 0 "" 0 " " {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 11 "Thus when " }{XPPEDIT 18 0 "t[6]" "6#&%\"tG6#\"\"'" }{TEXT -1 2 " " }{TEXT 268 1 "~" } {TEXT -1 64 " 0.0029461398900112655220 and the associated common val ue of " }{XPPEDIT 18 0 "rho" "6#%$rhoG" }{TEXT -1 7 " and " } {XPPEDIT 18 0 "rho[2]" "6#&%$rhoG6#\"\"#" }{TEXT -1 6 " is " } {XPPEDIT 18 0 "rho=rho[2]" "6#/%$rhoG&F$6#\"\"#" }{TEXT -1 2 " " } {TEXT 267 1 "~" }{TEXT -1 20 " 1.39803227622108. " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 123 "t6 := 0.002 9461398900112655220:\nR := z -> 1+z+1/2*z^2+1/6*z^3+1/24*z^4+1/120*z^5 +t6*z^6:\nevalf[15](-newton(R(z)=1,z=-4)/2);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#$\"03@iFK!)R\"!#9" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }} {PARA 0 "" 0 "" {TEXT -1 30 "The imaginary axis inclusion " } {XPPEDIT 18 0 "rho[2]" "6#&%$rhoG6#\"\"#" }{TEXT -1 16 " is zero when " }{XPPEDIT 18 0 "t[6]<=1/720" "6#1&%\"tG6#\"\"'*&\"\"\"F)\"$?(!\"\" " }{TEXT -1 2 " " }{TEXT 269 1 "~" }{TEXT -1 13 " 0.00138889." }} {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}}{PARA 0 "" 0 "" {TEXT -1 87 "#============================================================== ========================" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 " " 0 "" {TEXT -1 53 "#================================================= ===" }}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 15 "A scheme with " } {XPPEDIT 18 0 "rho=rho[2]" "6#/%$rhoG&F$6#\"\"#" }{TEXT -1 33 " and mi nimum principal error norm" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}{PARA 0 "" 0 "" {TEXT -1 39 "#--------------------------------- -----" }}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 28 "checking the combined sc heme" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 35 "coefficient s of the combined scheme" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1674 "ee := \{c[2]=17/150,\nc[3]=230/13 79,\nc[4]=1420370/2342387,\nc[5]=133/208,\nc[6]=1,\nc[7]=1,\n\na[2,1]= 17/150,\na[3,1]=1424390/32327897,\na[3,2]=3967500/32327897,\na[4,1]=11 725892832745653435650/6798789880492106984987,\na[4,2]=-204812814453768 1132500/295599560021395955869,\na[4,3]=39503685458474301158220/6798789 880492106984987,\na[5,1]=12161603119957121528213173/257321525068563161 50988800,\na[5,2]=-1141151413305973125/551372154861039616,\na[5,3]=368 3267576399581353585064593/1765398936832403041633484800,\na[5,4]=508180 6128006459902107790523/33857948797038081396772249600,\na[6,1]=-7781666 32716677780702867/2973121469222107001182300,\na[6,2]=104590105895625/6 8427659422081,\na[6,3]=-6240811551117087470432064137/66004141676210428 66395511800,\na[6,4]=-963956247541399247752708263264927/63142814465243 6311985040262094600,\na[6,5]=4149774798413464954572800/188163962429323 9507076793,\na[7,1]=10715793331/868982366000,\na[7,2]=0,\na[7,3]=38075 3067446365335491/1017424549568194596000,\na[7,4]=-33988229203965941757 637160569/199599379574494873119383412000,\na[7,5]=73328266119626752/10 8862061490007875,\na[7,6]=175006801591/1589096299500,\n\nb[1]=10715793 331/868982366000,\nb[2]=0,\nb[3]=380753067446365335491/101742454956819 4596000,\nb[4]=-33988229203965941757637160569/199599379574494873119383 412000,\nb[5]=73328266119626752/108862061490007875,\nb[6]=175006801591 /1589096299500,\n\n`b*`[1]=36166679554284109/3035692917188954400,\n`b* `[2]=0,\n`b*`[3]=1333442845934451852070878169/355425911933675601060908 6400,\n`b*`[4]=-372495557697385192340734287120958393/20918344717582119 76844177479899662400,\n`b*`[5]=129508101849060942811648/19014873140464 0113216825,\n`b*`[6]=58914464694395831/693916322394528225,\n`b*`[7]=1/ 40\}:" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 390 "subs(ee,matrix([[c[2],a[2,1],``$3],\n \+ [c[3],a[3,1],a[3,2],``$2],[c[4],a[4,1],a[4,2],a[4,3],``],\n [c[5 ],a[5,1],a[5,2],a[5,3],a[5,4]],\n [c[6],a[6,1],a[6,2],a[6,3],a[6, 4]],[``$4,a[6,5]],\n [c[7],a[7,1],a[7,2],a[7,3],a[7,4]],[``$3,a[7 ,5],a[7,6]],\n [`b`,b[1],b[2],b[3],b[4]],[``$3,b[5],b[6]],\n \+ [`b*`,`b*`[1],`b*`[2],`b*`[3],`b*`[4]],[``$2,`b*`[5],`b*`[6],`b*`[7]] ]));" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#K%'matrixG6#7.7'#\"#<\"$]\"F(% !GF+F+7'#\"$I#\"%z8#\"(!RC9\")(*yKK#\"(+v'RF2F+F+7'#\"(q.U\"\"((QUB#\" 8]cV`cuKG*es6\"7()\\)p5#\\!))*y)z'#!7+D8\"oPX9G\"[?\"6pe&fR@+c*f&H#\"8 ?#e6Iu%eao.&RF;F+7'#\"$L\"\"$3##\";tJ@G:7d*>Jgh@\"\";+)))4:;j&o]_@tD#! 4DJ(fI89:T6\"3;'R5'[:s8b#\"=$fk]e`8e*RwvE$o$\"=+[[L;/.C$o$*)Rl<#\"=B0z 2@!*fk+Gh!=3&\">+'\\Ax'R\"3Qqz[z&Q$7'\"\"\"#!9nGq!yxmrKm;y(\":+B=,q5A# p97tH#\"0Dc*e5!f/\"\"/\"3A%fwUo#!=PT1K/Z(3<6b63C'\"=+=^&RmG/@w;9/g'#!B F\\Ej#3FvZ#*RTvCcR'*\"B+Y4i-/&)>JOClW\"G9j7'F+F+F+F+#\":+Gda\\Y8%)zu( \\T\":$zwq]RKHC'R;)=7'FR#\",JLz:2\"\"-+gO#)*o)\"\"!#\"6\"\\N`OYu1`2Q\" 7+gf%>o&\\XU<5#!>p0;PwvTf'R?H#))R$\"?+?T$Q>J([\\u&z$*f*>7'F+F+F+#\"2_n i>hEGL(\"3vy+!\\h?')3\"#\"-\"f,o+v\"\".+&*H'4*e\"7'%\"bGF^oFaoFboFeoFh o7'%#b*G#\"24TGa&zm;O\"4+W&*)=\"fUb$#!E$Re47(GM2M#>&Q(pdb\\s$\"F+Cm**)zu@e#\"2JeR%pkW\"*e\"3D#GXRAj\"Rp#FR\"#SQ(ppri nt16\"" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 132 "subs(ee,matrix([seq([c[i],seq(a[i,j],j=1..i-1),``$(8 -i)],i=2..7),\n[`b`,seq(b[i],i=1..6),``],[`b*`,seq(`b*`[i],i=1..7)]])) : evalf(%);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#K%'matrixG6#7*7*$\"+LLL L6!#5F(%!GF+F+F+F+F+7*$\"+s_(ym\"F*$\"+2221W!#6$\"+,#osA\"F*F+F+F+F+F+ 7*$\"++4]\"F*F+F+F+7*$\"\"\"\"\"!$ !+])Qth#F*$\"+MqZG:F9$!+S<=b%*F*$!+t&Gm_\"F9$\"+)f.a?#F9F+F+7*FJ$\"+$y UJB\"F1$FLFL$\"+#RABu$F*$!+(QAGq\"F*$\"+p!))et'F*$\"+ZwH,6F*F+7*%\"bGF XFZFenFgnFinF[oF+7*%#b*G$\"+g8Q\">\"F1FZ$\"+]fn^PF*$!+(R72y\"F*$\"+[V) 3\"oF*$\"+J'R,\\)F1$\"+++++DF1Q(pprint26\"" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 146 "RK5_6eqs := [op(R owSumConditions(6,'expanded')),op(OrderConditions(5,6,'expanded'))]:\n `RK4_7eqs*` := subs(b=`b*`,OrderConditions(4,7,'expanded')):" }}} {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 7 "Check: " } }{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 148 "simplify(subs(ee,RK5_6eqs) ):\nmap(u->`if`(lhs(u)=rhs(u),0,1),%);\nnops(%);\nsimplify(subs(ee,`RK 4_7eqs*`)):\nmap(u->`if`(lhs(u)=rhs(u),0,1),%);\nnops(%);" }{TEXT -1 0 "" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#78\"\"!F$F$F$F$F$F$F$F$F$F$F$F$ F$F$F$F$F$F$F$F$F$" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#\"#A" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#7*\"\"!F$F$F$F$F$F$F$" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#\"\")" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 92 "We can calculate the principal error norm, that is, th e 2-norm of the principal error terms." }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 123 "errterms5_6 := PrincipalErrorTerms(5,6,'expanded'): \nsqrt(add(subs(ee,errterms5_6[i])^2,i=1.. nops(errterms5_6))):\nevalf (%);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#$\"+Lr[&H#!#7" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 87 "We can also calculat e the 2 norm of the principal error of the order 4 embedded scheme." } }{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 145 "`errterms4_7*` := subs(b=` b*`,PrincipalErrorTerms(4,7,'expanded')):\nsqrt(add(subs(ee,`errterms4 _7*`[i])^2,i=1.. nops(`errterms4_7*`))):\nevalf(%);" }}{PARA 11 "" 1 " " {XPPMATH 20 "6#$\"+(Q:X8\"!#7" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }} {PARA 0 "" 0 "" {TEXT -1 86 "Next we set-up stage-order condtions to c heck for stage-orders from 2 to 5 inclusive. " }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 85 "for ct from 2 to 4 do\n so||ct||_6 := StageOrd erConditions(ct,6,'expanded');\nend do:" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 58 "Stages 3 to 6 have the following r espective stage-orders. " }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 173 "[seq([seq(expand(subs(ee,so||i||_6[j])),i=2..4)],j=1..4)]:\nmap(proc( L) local i; for i to nops(L) do if not evalb(L[i]) then break end if e nd do; i end proc,%):\nsimplify(%); " }}{PARA 11 "" 1 "" {XPPMATH 20 " 6#7&\"\"#F$F$F$" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 27 "The simplifying conditions:" }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "Sum(b[i]*a[i,j],i = j+1 .. 6) = b[j]*(1-c[j]) " "6#/-%$SumG6$*&&%\"bG6#%\"iG\"\"\"&%\"aG6$F+%\"jGF,/F+;,&F0F,F,F,\" \"'*&&F)6#F0F,,&F,F,&%\"cG6#F0!\"\"F," }{TEXT -1 3 ", " }{XPPEDIT 18 0 "j=1" "6#/%\"jG\"\"\"" }{TEXT -1 7 " . . 5 " }}{PARA 0 "" 0 "" {TEXT -1 8 "(where " }{XPPEDIT 18 0 "c[1]=0" "6#/&%\"cG6#\"\"\"\"\"! " }{TEXT -1 17 " ) are satisfied." }}{PARA 256 "" 0 "" {TEXT -1 1 " " }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 154 "[Sum(b[i]*a[i,1],i=1+1..6 )=b[1],seq(Sum(b[i]*a[i,j],i=j+1..6)=b[j]*(1-c[j]),j=2..5)];\neval(sub s(Sum=add,%)):\nsubs(ee,%);\nmap(u->lhs(u)-rhs(u),%);\nnops(%);" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#7'/-%$SumG6$*&&%\"bG6#%\"iG\"\"\"&%\"a G6$F,F-F-/F,;\"\"#\"\"'&F*6#F-/-F&6$*&F)F-&F/6$F,F3F-/F,;\"\"$F4*&&F*6 #F3F-,&F-F-&%\"cGFB!\"\"F-/-F&6$*&F)F-&F/6$F,F?F-/F,;\"\"%F4*&&F*6#F?F -,&F-F-&FEFRFFF-/-F&6$*&F)F-&F/6$F,FOF-/F,;\"\"&F4*&&F*6#FOF-,&F-F-&FE FjnFFF-/-F&6$*&F)F-&F/6$F,FgnF-/F,;F4F4*&&F*6#FgnF-,&F-F-&FEFgoFFF-" } }{PARA 12 "" 1 "" {XPPMATH 20 "6#7'/#\",JLz:2\"\"-+gO#)*o)F%/\"\"!F)/# \"3HXr?#)43hF\"3+S+5$))p[&))F+/#!8(yY(zah]F35X\"\"9+gV_Dp%yN7[;#F//#\" 0W(f4uRDN\"10,?`T\\^9F3" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#7'\"\"!F$F$ F$F$" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#\"\"&" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 27 "The simplifying condition : " }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "Sum(b[i]*a[i,2] ,i = 3 .. 6)=0" "6#/-%$SumG6$*&&%\"bG6#%\"iG\"\"\"&%\"aG6$F+\"\"#F,/F+ ;\"\"$\"\"'\"\"!" }{TEXT -1 1 " " }}{PARA 0 "" 0 "" {TEXT -1 13 "is sa tisfied." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 59 "Sum(b[i]*a[i,2],i=3..6);\neval(subs(Sum=add,%));\nsub s(ee,%);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#-%$SumG6$*&&%\"bG6#%\"iG\" \"\"&%\"aG6$F*\"\"#F+/F*;\"\"$\"\"'" }}{PARA 11 "" 1 "" {XPPMATH 20 "6 #,**&&%\"bG6#\"\"$\"\"\"&%\"aG6$F(\"\"#F)F)*&&F&6#\"\"%F)&F+6$F1F-F)F) *&&F&6#\"\"&F)&F+6$F7F-F)F)*&&F&6#\"\"'F)&F+6$F=F-F)F)" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#\"\"!" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 27 "The simplifying condition: " }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "Sum(b[i]*Sum(a[i,j]*a[j,2],j = 3 .. i-1 ),i = 2 .. 6)=0" "6#/-%$SumG6$*&&%\"bG6#%\"iG\"\"\"-F%6$*&&%\"aG6$F+% \"jGF,&F16$F3\"\"#F,/F3;\"\"$,&F+F,F,!\"\"F,/F+;F6\"\"'\"\"!" }{TEXT -1 1 " " }}{PARA 0 "" 0 "" {TEXT -1 13 "is satisfied." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 82 "Sum(b[i]* Sum(a[i,j]*a[j,2],j=3..i-1),i =2 ..6);\neval(subs(Sum=add,%));\nsubs(e e,%);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#-%$SumG6$*&&%\"bG6#%\"iG\"\" \"-F$6$*&&%\"aG6$F*%\"jGF+&F06$F2\"\"#F+/F2;\"\"$,&F*F+F+!\"\"F+/F*;F5 \"\"'" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#,(*(&%\"bG6#\"\"%\"\"\"&%\"aG 6$F(\"\"$F)&F+6$F-\"\"#F)F)*&&F&6#\"\"&F),&*&&F+6$F4F-F)F.F)F)*&&F+6$F 4F(F)&F+6$F(F0F)F)F)F)*&&F&6#\"\"'F),(*&&F+6$FAF-F)F.F)F)*&&F+6$FAF(F) F " 0 "" {MPLTEXT 1 0 64 "Su m(b[i]*c[i]*a[i,2],i=3..6);\neval(subs(Sum=add,%));\nsubs(ee,%);" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#-%$SumG6$*(&%\"bG6#%\"iG\"\"\"&%\"cGF) F+&%\"aG6$F*\"\"#F+/F*;\"\"$\"\"'" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#, **(&%\"bG6#\"\"$\"\"\"&%\"cGF'F)&%\"aG6$F(\"\"#F)F)*(&F&6#\"\"%F)&F+F2 F)&F-6$F3F/F)F)*(&F&6#\"\"&F)&F+F9F)&F-6$F:F/F)F)*(&F&6#\"\"'F)&F+F@F) &F-6$FAF/F)F)" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#\"\"!" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 56 "None of the 20 princ ipal error conditions are satisfied." }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 123 "RK5_6err_eqs := PrincipalErrorConditions(5,6,'expand ed'):\nexpand(subs(ee,RK5_6err_eqs)):\nmap(u->`if`(lhs(u)=rhs(u),0,1), %);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#76\"\"\"F$F$F$F$F$F$F$F$F$F$F$F $F$F$F$F$F$F$F$" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> \+ " 0 "" {MPLTEXT 1 0 1 ";" }}}}{PARA 0 "" 0 "" {TEXT -1 39 "#---------- ----------------------------" }}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 26 "d etermination of the nodes" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 2 "e2" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 10256 "e2 := \{c[7] = 1, a[6,2] = 1/2*(-15*c[4]*c[3]^2+30 *c[5]*c[4]*c[3]^2-30*c[3]*c[4]*c[5]+16*c[4]*c[3]-5*c[3]*c[5]+3*c[3]+10 *c[4]*c[5]-6*c[4])/(-12+15*c[3]+15*c[4]+15*c[5]-20*c[4]*c[3]-20*c[4]*c [5]+30*c[3]*c[4]*c[5]-20*c[3]*c[5])/c[3]/c[2], c[6] = 1, a[2,1] = c[2] , a[5,4] = 1/2*(c[4]*c[3]-c[3]*c[5]-c[4]*c[5]+c[5]^2)*c[5]*(5*c[3]-2)/ (c[3]-c[4])/(3-5*c[3]+10*c[4]*c[3]-5*c[4])/c[4], `b*`[2] = 0, b[2] = 0 , b[4] = 1/60*(10*c[3]*c[5]-5*c[3]-5*c[5]+3)/(-c[4]^2+c[4]+c[4]*c[3]-c [3])/c[4]/(c[4]-c[5]), a[7,1] = 1/60*(5*c[4]-10*c[4]*c[5]+30*c[3]*c[4] *c[5]-10*c[4]*c[3]-10*c[3]*c[5]-3+5*c[3]+5*c[5])/c[3]/c[4]/c[5], a[6,4 ] = 1/2*(6-30*c[5]^2*c[3]+20*c[5]^2*c[3]^2-10*c[5]^2*c[4]-16*c[3]-8*c[ 4]-14*c[5]-7*c[4]^2*c[3]+30*c[5]^2*c[3]*c[4]+25*c[5]*c[4]*c[3]^2-20*c[ 5]^2*c[4]*c[3]^2+23*c[4]*c[3]+14*c[4]*c[5]-39*c[3]*c[4]*c[5]+39*c[3]*c [5]-15*c[4]*c[3]^2+10*c[3]^2+2*c[4]^2+10*c[5]^2-25*c[3]^2*c[5]+5*c[4]^ 2*c[3]^2)/c[4]/(12*c[4]^2-12*c[4]*c[3]-15*c[4]^3-12*c[4]*c[5]+12*c[3]* c[5]-20*c[4]^2*c[5]^2+20*c[4]^3*c[5]+20*c[4]^3*c[3]-20*c[4]^2*c[3]^2+2 0*c[5]^2*c[3]^2-20*c[4]^2*c[5]*c[3]+15*c[4]*c[3]^2+15*c[5]^2*c[4]+30*c [5]^2*c[4]^2*c[3]+30*c[5]*c[4]^2*c[3]^2-30*c[5]^2*c[4]*c[3]^2-30*c[5]* c[4]^3*c[3]+15*c[3]*c[4]*c[5]-15*c[3]^2*c[5]-15*c[5]^2*c[3]), `b*`[1] \+ = 1/60*(-460*c[5]*c[4]*c[3]*`b*`[7]+760*c[5]*c[4]*c[3]^2*`b*`[7]+3*c[3 ]-6*c[4]-20*c[4]^2*c[5]-60*c[4]^2*c[3]-1800*`b*`[7]*c[4]^2*c[5]*c[3]^2 +100*c[5]*c[4]*c[3]^2+100*c[3]^2*c[5]*`b*`[7]-100*c[3]^3*c[5]*c[4]+300 *c[3]^3*c[4]*`b*`[7]-600*c[3]^3*`b*`[7]*c[4]^2+940*`b*`[7]*c[4]^2*c[3] ^2-60*c[5]*c[3]*`b*`[7]-560*c[4]^2*c[3]*`b*`[7]+300*c[5]*c[4]^2*c[3]^3 -100*c[4]^2*c[3]^3+140*c[4]^2*c[5]*c[3]+120*`b*`[7]*c[4]^2+120*c[5]*c[ 4]*`b*`[7]-340*c[5]*c[4]^2*c[3]^2-200*`b*`[7]*c[4]^2*c[5]+1000*c[5]*c[ 4]^2*c[3]*`b*`[7]-600*c[3]^3*c[5]*c[4]*`b*`[7]+29*c[4]*c[3]-380*`b*`[7 ]*c[4]*c[3]^2+252*`b*`[7]*c[4]*c[3]+10*c[4]*c[5]-50*c[3]*c[4]*c[5]+120 0*c[3]^3*`b*`[7]*c[4]^2*c[5]+36*`b*`[7]*c[3]-72*`b*`[7]*c[4]-60*`b*`[7 ]*c[3]^2-5*c[3]*c[5]-60*c[4]*c[3]^2-5*c[3]^2+10*c[4]^2+10*c[3]^2*c[5]+ 50*c[4]*c[3]^3+130*c[4]^2*c[3]^2)/(10*c[4]*c[3]^2-8*c[4]*c[3]-c[3]+2*c [4])/c[3]/c[4]/c[5], a[6,1] = 1/2*(10*c[3]*c[2]*c[5]^2-14*c[3]*c[2]*c[ 5]-30*c[3]^2*c[2]*c[5]^2-25*c[3]^3*c[2]*c[5]+10*c[3]^3*c[2]-108*c[3]^2 *c[2]*c[4]*c[5]-3*c[5]*c[4]*c[3]^2+5*c[5]^2*c[4]*c[3]^2-30*c[5]^2*c[4] ^2*c[3]^3+30*c[5]^2*c[4]^2*c[3]^2-10*c[5]^2*c[4]^2*c[3]+15*c[5]*c[4]^2 *c[3]^3+6*c[4]^2*c[5]*c[3]-30*c[3]^2*c[2]*c[4]^2-30*c[3]^3*c[4]*c[2]+1 0*c[3]*c[2]*c[4]^2-16*c[3]*c[2]*c[4]-16*c[5]*c[4]^2*c[3]^2+39*c[3]*c[2 ]*c[4]*c[5]-30*c[3]*c[2]*c[5]^2*c[4]-9*c[3]*c[2]*c[5]*c[4]^2+6*c[3]*c[ 2]+10*c[2]*c[4]^2*c[5]^2-10*c[3]*c[2]*c[4]^2*c[5]^2-60*c[3]^3*c[2]*c[5 ]*c[4]^2-40*c[3]^2*c[2]*c[4]^2*c[5]^2+46*c[3]^2*c[2]*c[4]-16*c[2]*c[3] ^2-6*c[2]*c[5]*c[4]^2+39*c[3]^2*c[2]*c[5]+60*c[2]*c[4]^2*c[5]^2*c[3]^3 +20*c[3]^3*c[2]*c[4]^2+20*c[3]^3*c[2]*c[5]^2-60*c[3]^3*c[2]*c[5]^2*c[4 ]+75*c[3]^3*c[2]*c[4]*c[5]+80*c[3]^2*c[2]*c[5]^2*c[4]+65*c[3]^2*c[2]*c [5]*c[4]^2)/(-12+15*c[3]+15*c[4]+15*c[5]-20*c[4]*c[3]-20*c[4]*c[5]+30* c[3]*c[4]*c[5]-20*c[3]*c[5])/c[3]^2/c[2]/c[4]/c[5], a[7,2] = 0, b[3] = -1/60*(10*c[4]*c[5]-5*c[4]-5*c[5]+3)/(c[3]-1)/c[3]/(c[3]-c[5])/(c[3]- c[4]), a[6,3] = -1/2*(10*c[5]^2*c[3]-10*c[5]^2*c[3]^2+6*c[3]-6*c[4]^2* c[5]+16*c[4]^2*c[3]-40*c[5]^2*c[3]*c[4]-45*c[5]*c[4]*c[3]^2+60*c[5]^2* c[4]*c[3]^2-20*c[3]^3*c[5]*c[4]-10*c[5]^2*c[4]^2*c[3]-20*c[5]*c[4]^2*c [3]^3+15*c[4]^2*c[3]^3-19*c[4]^2*c[5]*c[3]+10*c[4]^2*c[5]^2+45*c[5]*c[ 4]^2*c[3]^2-10*c[3]^4*c[4]-20*c[4]*c[5]^2*c[3]^3-16*c[4]*c[3]+45*c[3]* c[4]*c[5]+20*c[3]^4*c[5]*c[4]-14*c[3]*c[5]+17*c[4]*c[3]^2-8*c[3]^2+2*c [3]^3+14*c[3]^2*c[5]+9*c[4]*c[3]^3-31*c[4]^2*c[3]^2)/(-30*c[5]^2*c[3]* c[4]+20*c[5]^2*c[4]-15*c[4]*c[5]+15*c[4]*c[3]+30*c[5]*c[4]*c[3]^2-20*c [4]*c[3]^2+15*c[3]^2+12*c[5]-12*c[3]+20*c[5]^2*c[3]-15*c[5]^2-20*c[3]^ 2*c[5])/(c[3]-c[4])/c[3]^2, a[4,1] = 1/2*c[4]*(2*c[2]*c[3]^2-3*c[4]*c[ 3]^2+2*c[4]^2*c[3]+2*c[3]*c[2]*c[4]-2*c[2]*c[4]^2)/c[2]/c[3]^2, a[5,3] = -1/2*c[5]*(-2*c[5]^2*c[3]+6*c[4]^2*c[5]-6*c[4]^2*c[3]+15*c[5]^2*c[3 ]*c[4]-5*c[5]*c[4]*c[3]^2-20*c[5]^2*c[4]*c[3]^2+20*c[3]^3*c[5]*c[4]+20 *c[5]^2*c[4]^2*c[3]-5*c[4]^2*c[5]*c[3]-10*c[4]^2*c[5]^2-20*c[5]*c[4]^2 *c[3]^2-4*c[3]*c[4]*c[5]+4*c[4]*c[3]^2+2*c[3]^2*c[5]-10*c[4]*c[3]^3+15 *c[4]^2*c[3]^2)/(3-5*c[3]+10*c[4]*c[3]-5*c[4])/c[3]^2/(c[3]-c[4]), a[4 ,3] = -c[4]^2*(c[3]-c[4])/c[3]^2, a[5,2] = 1/2*c[5]*(-15*c[4]*c[3]^2+3 0*c[5]*c[4]*c[3]^2-20*c[5]^2*c[3]*c[4]+6*c[4]*c[3]-5*c[5]^2*c[3]+3*c[3 ]*c[5]-6*c[4]*c[5]+10*c[5]^2*c[4])/c[2]/(3-5*c[3]+10*c[4]*c[3]-5*c[4]) /c[3], `b*`[5] = 1/60*(-460*c[5]*c[4]*c[3]*`b*`[7]+760*c[5]*c[4]*c[3]^ 2*`b*`[7]+3*c[3]-6*c[4]-60*c[4]^2*c[3]-1800*`b*`[7]*c[4]^2*c[5]*c[3]^2 +100*c[3]^2*c[5]*`b*`[7]+300*c[3]^3*c[4]*`b*`[7]-600*c[3]^3*`b*`[7]*c[ 4]^2+940*`b*`[7]*c[4]^2*c[3]^2-60*c[5]*c[3]*`b*`[7]-560*c[4]^2*c[3]*`b *`[7]-100*c[4]^2*c[3]^3+120*`b*`[7]*c[4]^2+120*c[5]*c[4]*`b*`[7]-200*` b*`[7]*c[4]^2*c[5]+1000*c[5]*c[4]^2*c[3]*`b*`[7]-600*c[3]^3*c[5]*c[4]* `b*`[7]+29*c[4]*c[3]-380*`b*`[7]*c[4]*c[3]^2+252*`b*`[7]*c[4]*c[3]+120 0*c[3]^3*`b*`[7]*c[4]^2*c[5]+36*`b*`[7]*c[3]-72*`b*`[7]*c[4]-60*`b*`[7 ]*c[3]^2-60*c[4]*c[3]^2-5*c[3]^2+10*c[4]^2+50*c[4]*c[3]^3+130*c[4]^2*c [3]^2)/c[5]/(c[5]^2*c[3]-c[5]^3*c[3]+c[5]^2*c[3]^2-2*c[5]^2*c[4]+2*c[5 ]^3*c[4]+2*c[4]^2*c[5]-2*c[4]^2*c[3]+7*c[5]^2*c[3]*c[4]-8*c[5]^3*c[4]* c[3]-9*c[5]*c[4]*c[3]^2-2*c[5]^2*c[4]*c[3]^2+10*c[5]^3*c[4]*c[3]^2+10* c[3]^3*c[5]*c[4]-10*c[5]^2*c[4]^2*c[3]^2+8*c[5]^2*c[4]^2*c[3]+10*c[5]* c[4]^2*c[3]^3-10*c[4]^2*c[3]^3-6*c[4]^2*c[5]*c[3]-2*c[4]^2*c[5]^2+2*c[ 5]*c[4]^2*c[3]^2-10*c[4]*c[5]^2*c[3]^3+c[3]*c[4]*c[5]+c[4]*c[3]^2-c[3] ^2*c[5]+8*c[4]^2*c[3]^2), a[4,2] = 1/2*(3*c[3]-2*c[4])*c[4]^2/c[3]/c[2 ], `b*`[6] = -1/60*(-1000*c[5]*c[4]*c[3]*`b*`[7]+1780*c[5]*c[4]*c[3]^2 *`b*`[7]-12*c[3]+24*c[4]+40*c[4]^2*c[5]+160*c[4]^2*c[3]-2880*`b*`[7]*c [4]^2*c[5]*c[3]^2-280*c[5]*c[4]*c[3]^2+160*c[3]^2*c[5]*`b*`[7]+200*c[3 ]^3*c[5]*c[4]+900*c[3]^3*c[4]*`b*`[7]-1200*c[3]^3*`b*`[7]*c[4]^2+2020* `b*`[7]*c[4]^2*c[3]^2-120*c[5]*c[3]*`b*`[7]-1160*c[4]^2*c[3]*`b*`[7]-3 00*c[5]*c[4]^2*c[3]^3+200*c[4]^2*c[3]^3-220*c[4]^2*c[5]*c[3]+240*`b*`[ 7]*c[4]^2+240*c[5]*c[4]*`b*`[7]+440*c[5]*c[4]^2*c[3]^2-320*`b*`[7]*c[4 ]^2*c[5]+1600*c[5]*c[4]^2*c[3]*`b*`[7]-1200*c[3]^3*c[5]*c[4]*`b*`[7]-1 11*c[4]*c[3]-1400*`b*`[7]*c[4]*c[3]^2+792*`b*`[7]*c[4]*c[3]-30*c[4]*c[ 5]+140*c[3]*c[4]*c[5]+1800*c[3]^3*`b*`[7]*c[4]^2*c[5]+96*`b*`[7]*c[3]- 192*`b*`[7]*c[4]-120*`b*`[7]*c[3]^2+15*c[3]*c[5]+220*c[4]*c[3]^2+15*c[ 3]^2-30*c[4]^2-20*c[3]^2*c[5]-150*c[4]*c[3]^3-310*c[4]^2*c[3]^2)/(-10* c[4]*c[3]^2+10*c[5]*c[4]*c[3]^2+8*c[4]*c[3]-8*c[3]*c[4]*c[5]+c[3]-c[3] *c[5]-2*c[4]+2*c[4]*c[5])/(c[4]-1)/(c[3]-1), `b*`[3] = 1/60*(-460*c[5] *c[4]*c[3]*`b*`[7]+760*c[5]*c[4]*c[3]^2*`b*`[7]+3*c[3]-6*c[4]-20*c[4]^ 2*c[5]-40*c[4]^2*c[3]-1800*`b*`[7]*c[4]^2*c[5]*c[3]^2+50*c[5]*c[4]*c[3 ]^2+100*c[3]^2*c[5]*`b*`[7]+300*c[3]^3*c[4]*`b*`[7]-600*c[3]^3*`b*`[7] *c[4]^2+940*`b*`[7]*c[4]^2*c[3]^2-60*c[5]*c[3]*`b*`[7]-560*c[4]^2*c[3] *`b*`[7]+80*c[4]^2*c[5]*c[3]+120*`b*`[7]*c[4]^2+120*c[5]*c[4]*`b*`[7]- 100*c[5]*c[4]^2*c[3]^2-200*`b*`[7]*c[4]^2*c[5]+1000*c[5]*c[4]^2*c[3]*` b*`[7]-600*c[3]^3*c[5]*c[4]*`b*`[7]+19*c[4]*c[3]-380*`b*`[7]*c[4]*c[3] ^2+252*`b*`[7]*c[4]*c[3]+10*c[4]*c[5]-30*c[3]*c[4]*c[5]+1200*c[3]^3*`b *`[7]*c[4]^2*c[5]+36*`b*`[7]*c[3]-72*`b*`[7]*c[4]-60*`b*`[7]*c[3]^2-5* c[3]*c[5]-30*c[4]*c[3]^2+10*c[4]^2+50*c[4]^2*c[3]^2)/(c[3]-1)/c[3]/(-1 0*c[5]*c[4]*c[3]^2+8*c[3]*c[4]*c[5]-2*c[4]*c[5]+2*c[4]*c[3]+10*c[4]*c[ 3]^3-8*c[4]*c[3]^2+c[3]*c[5]-c[3]^2)/(c[3]-c[4]), a[3,2] = 1/2*c[3]^2/ c[2], a[7,6] = 1/60*(-12+15*c[3]+15*c[4]+15*c[5]-20*c[4]*c[3]-20*c[4]* c[5]+30*c[3]*c[4]*c[5]-20*c[3]*c[5])/(c[5]-1)/(c[4]-1)/(c[3]-1), a[5,1 ] = 1/2*(2*c[3]*c[2]*c[5]^2-5*c[3]^2*c[2]*c[5]^2+5*c[3]^3*c[2]*c[5]+10 *c[3]^2*c[2]*c[4]*c[5]-3*c[5]*c[4]*c[3]^2+5*c[5]^2*c[4]*c[3]^2+20*c[5] ^2*c[4]^2*c[3]^2-10*c[5]^2*c[4]^2*c[3]-30*c[5]*c[4]^2*c[3]^3+15*c[4]^2 *c[3]^3+6*c[4]^2*c[5]*c[3]-25*c[3]^2*c[2]*c[4]^2-15*c[3]^3*c[4]*c[2]+6 *c[3]*c[2]*c[4]^2-2*c[3]*c[2]*c[4]*c[5]-5*c[3]*c[2]*c[5]^2*c[4]+5*c[3] *c[2]*c[5]*c[4]^2+10*c[2]*c[4]^2*c[5]^2-20*c[3]*c[2]*c[4]^2*c[5]^2+8*c [3]^2*c[2]*c[4]-6*c[2]*c[5]*c[4]^2-2*c[3]^2*c[2]*c[5]+20*c[3]^3*c[2]*c [4]^2+20*c[3]^2*c[2]*c[5]*c[4]^2-6*c[4]^2*c[3]^2)*c[5]/c[4]/c[3]^2/(3- 5*c[3]+10*c[4]*c[3]-5*c[4])/c[2], a[6,5] = (-3+8*c[3]+8*c[4]+3*c[5]+5* c[4]^2*c[5]+15*c[4]^2*c[3]-15*c[5]*c[4]*c[3]^2-15*c[4]^2*c[5]*c[3]+10* c[5]*c[4]^2*c[3]^2-23*c[4]*c[3]-8*c[4]*c[5]+23*c[3]*c[4]*c[5]-8*c[3]*c [5]+15*c[4]*c[3]^2-5*c[3]^2-5*c[4]^2+5*c[3]^2*c[5]-10*c[4]^2*c[3]^2)/c [5]/(-20*c[5]^3*c[3]+20*c[5]^2*c[3]^2-20*c[5]^3*c[4]-15*c[4]^2*c[5]+15 *c[4]^2*c[3]+20*c[5]^2*c[3]*c[4]+30*c[5]^3*c[4]*c[3]-30*c[5]^2*c[4]*c[ 3]^2-30*c[5]^2*c[4]^2*c[3]+20*c[4]^2*c[5]^2+30*c[5]*c[4]^2*c[3]^2-12*c [4]*c[3]+12*c[4]*c[5]-15*c[3]*c[4]*c[5]+15*c[5]^3+12*c[3]*c[5]+15*c[4] *c[3]^2-12*c[5]^2-15*c[3]^2*c[5]-20*c[4]^2*c[3]^2), a[7,3] = -1/60*(10 *c[4]*c[5]-5*c[4]-5*c[5]+3)/(c[3]-1)/c[3]/(c[3]-c[5])/(c[3]-c[4]), b[5 ] = -1/60*(3-5*c[3]+10*c[4]*c[3]-5*c[4])/c[5]/(-c[5]^2-c[5]^2*c[3]+c[5 ]^3+c[3]*c[5]-c[5]^2*c[4]+c[3]*c[4]*c[5]-c[4]*c[3]+c[4]*c[5]), b[1] = \+ 1/60*(5*c[4]-10*c[4]*c[5]+30*c[3]*c[4]*c[5]-10*c[4]*c[3]-10*c[3]*c[5]- 3+5*c[3]+5*c[5])/c[3]/c[4]/c[5], b[6] = 1/60*(-12+15*c[3]+15*c[4]+15*c [5]-20*c[4]*c[3]-20*c[4]*c[5]+30*c[3]*c[4]*c[5]-20*c[3]*c[5])/(c[5]-1) /(c[4]-1)/(c[3]-1), a[3,1] = 1/2*c[3]*(2*c[2]-c[3])/c[2], `b*`[4] = -1 /60*(-460*c[5]*c[4]*c[3]*`b*`[7]+760*c[5]*c[4]*c[3]^2*`b*`[7]+3*c[3]-6 *c[4]-1800*`b*`[7]*c[4]^2*c[5]*c[3]^2+130*c[5]*c[4]*c[3]^2+100*c[3]^2* c[5]*`b*`[7]-100*c[3]^3*c[5]*c[4]+300*c[3]^3*c[4]*`b*`[7]-600*c[3]^3*` b*`[7]*c[4]^2+940*`b*`[7]*c[4]^2*c[3]^2-60*c[5]*c[3]*`b*`[7]-560*c[4]^ 2*c[3]*`b*`[7]+120*`b*`[7]*c[4]^2+120*c[5]*c[4]*`b*`[7]-200*`b*`[7]*c[ 4]^2*c[5]+1000*c[5]*c[4]^2*c[3]*`b*`[7]-600*c[3]^3*c[5]*c[4]*`b*`[7]+3 4*c[4]*c[3]-380*`b*`[7]*c[4]*c[3]^2+252*`b*`[7]*c[4]*c[3]+10*c[4]*c[5] -60*c[3]*c[4]*c[5]+1200*c[3]^3*`b*`[7]*c[4]^2*c[5]+36*`b*`[7]*c[3]-72* `b*`[7]*c[4]-60*`b*`[7]*c[3]^2-5*c[3]*c[5]-70*c[4]*c[3]^2-5*c[3]^2+10* c[3]^2*c[5]+50*c[4]*c[3]^3)/(-c[4]^2+c[4]+c[4]*c[3]-c[3])/c[4]/(-10*c[ 5]*c[4]*c[3]^2+10*c[4]^2*c[3]^2+8*c[3]*c[4]*c[5]+c[3]*c[5]-8*c[4]^2*c[ 3]-c[4]*c[3]-2*c[4]*c[5]+2*c[4]^2), a[7,4] = 1/60*(10*c[3]*c[5]-5*c[3] -5*c[5]+3)/(-c[4]^2+c[4]+c[4]*c[3]-c[3])/c[4]/(c[4]-c[5]), a[7,5] = -1 /60*(3-5*c[3]+10*c[4]*c[3]-5*c[4])/c[5]/(-c[5]^2-c[5]^2*c[3]+c[5]^3+c[ 3]*c[5]-c[5]^2*c[4]+c[3]*c[4]*c[5]-c[4]*c[3]+c[4]*c[5])\}:" }{TEXT -1 0 "" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 49 "s m: square of principal error norm in terms of " }{XPPEDIT 18 0 "c[2] " "6#&%\"cG6#\"\"#" }{TEXT -1 3 ", " }{XPPEDIT 18 0 "c[3]" "6#&%\"cG6 #\"\"$" }{TEXT -1 3 ", " }{XPPEDIT 18 0 "c[4]" "6#&%\"cG6#\"\"%" } {TEXT -1 7 " and " }{XPPEDIT 18 0 "c[5]" "6#&%\"cG6#\"\"&" }{TEXT -1 1 " " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1665 "sm := 1/25920000*(12042*c[5]^2*c[3]^2+297000*c[3]*c [2]^2*c[4]^2*c[5]-13500*c[3]^2*c[2]*c[5]^2+28044*c[3]^3*c[5]-270000*c[ 3]*c[2]^2*c[4]^2*c[5]^2-22140*c[3]^4*c[5]-148500*c[3]^2*c[2]^2*c[4]*c[ 5]-270000*c[3]^2*c[2]^2*c[4]^2*c[5]+135000*c[3]^2*c[2]^2*c[4]*c[5]^2-2 2140*c[5]^2*c[3]^3+270000*c[3]^2*c[2]^2*c[4]^2*c[5]^2-18000*c[3]^2*c[2 ]*c[4]*c[5]+81000*c[3]*c[2]^2*c[5]*c[4]-67500*c[3]*c[2]^2*c[5]^2*c[4]+ 96444*c[5]*c[4]*c[3]^2-76140*c[5]^2*c[4]*c[3]^2-613800*c[5]^2*c[4]^2*c [3]^3-221940*c[3]^3*c[5]*c[4]+153450*c[5]^2*c[4]^2*c[3]^2+675180*c[5]* c[4]^2*c[3]^3-240140*c[4]^2*c[3]^3+198000*c[3]^2*c[2]*c[4]^2-49500*c[3 ]^3*c[4]*c[2]-55800*c[3]*c[2]*c[4]^2+12000*c[3]*c[2]*c[4]-184140*c[5]* c[4]^2*c[3]^2-34200*c[3]*c[2]*c[4]*c[5]+27000*c[3]*c[2]*c[5]^2*c[4]+16 2000*c[3]*c[2]*c[5]*c[4]^2+613800*c[4]^2*c[3]^4*c[5]^2-613800*c[4]^2*c [3]^4*c[5]-73800*c[4]*c[3]^4*c[5]^2-42140*c[3]^4*c[4]+223450*c[4]^2*c[ 3]^4+189180*c[4]*c[5]^2*c[3]^3+18450*c[5]^2*c[3]^4+10642*c[3]^4-135000 *c[3]*c[2]*c[4]^2*c[5]^2+27900*c[2]^2*c[4]^2+6975*c[3]^2*c[2]^2+540000 *c[3]^3*c[2]*c[5]*c[4]^2+81180*c[3]^4*c[5]*c[4]+540000*c[3]^2*c[2]*c[4 ]^2*c[5]^2+6900*c[3]^2*c[2]*c[4]-6000*c[2]*c[3]^2-36856*c[4]*c[3]^2+17 100*c[3]^2*c[2]*c[5]+6552*c[3]^2-12856*c[3]^3-27900*c[3]*c[2]^2*c[4]-9 9000*c[3]*c[2]^2*c[4]^2+49500*c[3]^2*c[2]^2*c[4]+90000*c[3]^2*c[2]^2*c [4]^2-20250*c[3]^2*c[2]^2*c[5]+16875*c[3]^2*c[2]^2*c[5]^2-81000*c[5]*c [2]^2*c[4]^2+67500*c[5]^2*c[2]^2*c[4]^2-540000*c[2]*c[4]^2*c[5]^2*c[3] ^3-180000*c[3]^3*c[2]*c[4]^2-135000*c[3]^3*c[2]*c[5]^2*c[4]+148500*c[3 ]^3*c[2]*c[4]*c[5]+13500*c[3]^2*c[2]*c[5]^2*c[4]-594000*c[3]^2*c[2]*c[ 5]*c[4]^2-16056*c[3]^2*c[5]+88044*c[4]*c[3]^3+66442*c[4]^2*c[3]^2)/c[3 ]^2:" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 " " {TEXT -1 23 "The stability function " }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "R(z)= 1+z+1/2" "6#/-%\"RG6#%\"zG,(\"\"\"F)F'F)*& F)F)\"\"#!\"\"F)" }{TEXT -1 1 " " }{XPPEDIT 18 0 "z^2+1/6" "6#,&*$%\"z G\"\"#\"\"\"*&F'F'\"\"'!\"\"F'" }{TEXT -1 1 " " }{XPPEDIT 18 0 "z^3+1/ 24" "6#,&*$%\"zG\"\"$\"\"\"*&F'F'\"#C!\"\"F'" }{TEXT -1 1 " " } {XPPEDIT 18 0 "z^4+1/120" "6#,&*$%\"zG\"\"%\"\"\"*&F'F'\"$?\"!\"\"F'" }{TEXT -1 1 " " }{XPPEDIT 18 0 "z^5+t[6]*z^6" "6#,&*$%\"zG\"\"&\"\"\"* &&%\"tG6#\"\"'F'*$F%F,F'F'" }{TEXT -1 2 ", " }}{PARA 0 "" 0 "" {TEXT -1 47 "of a 6 stage, order 5 Runge-Kutta scheme with " }{XPPEDIT 18 0 "rho=rho[2]" "6#/%$rhoG&F$6#\"\"#" }{TEXT -1 7 " has " }{XPPEDIT 18 0 "t[6]" "6#&%\"tG6#\"\"'" }{TEXT -1 2 " " }{TEXT 275 1 "~" } {TEXT -1 1 " " }{XPPEDIT 18 0 "103/34961" "6#*&\"$.\"\"\"\"\"&h\\$!\" \"" }{TEXT -1 2 ". " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 50 "0.00294613989001;\nconvert(%,rational,7);\n evalf(%);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#$\"-,!*)Rh%H!#9" }}{PARA 11 "" 1 "" {XPPMATH 20 "6##\"$.\"\"&h\\$" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#$\"+&)*Rh%H!#7" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 39 "Using this rational approximation for " }{XPPEDIT 18 0 "t[6]" "6#&%\"tG6#\"\"'" }{TEXT -1 8 " gives." }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "t[6] = ``;" "6#/&%\"tG6#\"\"'%!G" } {XPPEDIT 18 0 "(2-5*c[3])*c[4]/240=103/34961" "6#/*(,&\"\"#\"\"\"*&\" \"&F'&%\"cG6#\"\"$F'!\"\"F'&F+6#\"\"%F'\"$S#F.*&\"$.\"F'\"&h\\$F." } {TEXT -1 1 "." }}{PARA 0 "" 0 "" {TEXT -1 8 "so that " }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "(2-5*c[3])*c[4] = 24720/34961" "6# /*&,&\"\"#\"\"\"*&\"\"&F'&%\"cG6#\"\"$F'!\"\"F'&F+6#\"\"%F'*&\"&?Z#F' \"&h\\$F." }{TEXT -1 1 " " }}{PARA 0 "" 0 "" {TEXT -1 4 "and " }} {PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "c[4]=24720/(69922-174 805*c[3])" "6#/&%\"cG6#\"\"%*&\"&?Z#\"\"\",&\"&A*pF**&\"'0[ " 0 "" {MPLTEXT 1 0 84 "(2-5*c[3])*c[4]=240*103/3496 1;\nc[4]=solve(%,c[4]);\nc[4]=numer(rhs(%))/denom(rhs(%));" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/*&,&\"\"#\"\"\"*&\"\"&F'&%\"cG6#\"\"$F'!\" \"F'&F+6#\"\"%F'#\"&?Z#\"&h\\$" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/&% \"cG6#\"\"%,$*&#\"&?Z#\"&h\\$\"\"\"*&F-F-,&*&\"\"&F-&F%6#\"\"$F-F-\"\" #!\"\"F6F-F6" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/&%\"cG6#\"\"%,$*&\"&? Z#\"\"\",&*&\"'0[ " 0 "" {MPLTEXT 1 0 101 "plot(-24720/34961*1/(-2+5*c[3]),c[3]=0..0.27,c[4]=0..1,color=COLO R(RGB,.9,.1,.2),font=[HELVETICA,9]);" }}{PARA 13 "" 1 "" {GLPLOT2D 485 401 401 {PLOTDATA 2 "6'-%'CURVESG6#7S7$$\"\"!F)$\"3Eh]Y\")zONN!#=7 $$\"3)3++]iJ_)e!#?$\"3))\\W7w3;)e$F,7$$\"34+]P>Sf+6!#>$\"3`okrS`RNOF,7 $$\"3/++v`7Zw;F6$\"3cxf!pVB+p$F,7$$\"3I++Dwd;cAF6$\"3oPhDLspYPF,7$$\"3 C+](=60J$GF6$\"3cV\">^(p&[!QF,7$$\"3K+]PH<+oLF6$\"3wv\"o(*4:/'QF,7$$\" 33+]P%>c=#RF6$\"3^))G4_$y'>RF,7$$\"3M+]PuKl%\\%F6$\"37\"o\\SM8H)RF,7$$ \"3;+](oQ8c1&F6$\"3!=h(RC#4![SF,7$$\"3%4++D/;Hl&F6$\"324p;ajAe8H\"F,$\"3!zdn&>>(3A&F,7$$\"3-+]73tm[8F,$\"3JqShd@sL`F,7$$ \"3,+vVn`>39F,$\"3eu]kRgAcaF,7$$\"3/+D1=Z,g9F,$\"3;*Q:M*3anbF,7$$\"3P+ ]PO5)f^\"F,$\"3)4jlFm!)Hp&F,7$$\"3/++]>(*zt:F,$\"3N()RCg)\\'GeF,7$$\"3 /+](3-k.j\"F,$\"3o7y'[T#ynfF,7$$\"39+v$4*G4&o\"F,$\"3dc9U/B()3hF,7$$\" 3A+]7q*feu\"F,$\"3!3aK)3_btiF,7$$\"3(*****\\$*=Y+=F,$\"3;4p[C;HHkF,7$$ \"3A++D,6we=F,$\"3x#=F,$\"3;c(p&GASrnF,7$$\"3 K++vHNMp>F,$\"3cv@Z#G!*R'pF,7$$\"30+v$R='oB?F,$\"3-!*QJe*ya:(F,7$$\"3M +vo:**[!3#F,$\"3c`GJ;)HsO(F,7$$\"3.++DYf-O@F,$\"3?(>:W#F,$\"3`Vf_9m*4$yF,7$$\"3_+++\"zj,D#F,$\"3oPZX'\\(f\"3)F,7$$\"3E +]it&GuI#F,$\"3'y!4G\\A-b$)F,7$$\"3]+v$\\<>UO#F,$\"3=47yx'*3X')F,7$$\" 3P++]RQS;CF,$\"3kV^L7O(*H*)F,7$$\"3K+]7aL@wCF,$\"3/qy<&G![!G*F,7$$\"3U +++(p2(HDF,$\"3G?;E0k8='*F,7$$\"37+v$R#Qu'e#F,$\"3t&QQ1ZI1+\"!#<7$$\"3 _+D1vYLTEF,$\"34'ptgcN3/\"F_z7$$\"3;+++++++FF,$\"3e![eF`0y3\"F_z-%%FON TG6$%*HELVETICAG\"\"*-%+AXESLABELSG6$Q%c[3]6\"Q%c[4]Fc[l-%&COLORG6&%$R GBG$F^[l!\"\"$\"\"\"Fj[l$\"\"#Fj[l-%%VIEWG6$;F($\"#F!\"#;F($F\\\\lF)" 1 2 0 1 10 0 2 6 1 4 2 1.000000 45.000000 45.000000 0 0 "Curve 1" }}}} {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 130 "We can u se the preceding relation to obtain an expression for the square of th e principal error norm in terms of the three nodes " }{XPPEDIT 18 0 " c[2]" "6#&%\"cG6#\"\"#" }{TEXT -1 3 ", " }{XPPEDIT 18 0 "c[3]" "6#&% \"cG6#\"\"$" }{TEXT -1 7 " and " }{XPPEDIT 18 0 "c[5]" "6#&%\"cG6#\" \"&" }{TEXT -1 1 "." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 57 "sm2 := simplify(subs(c[4]=24720/(69922-1748 05*c[3]),sm)):" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> \+ " 0 "" {MPLTEXT 1 0 1 ";" }}}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 50 "sm2: square of principal error norm in terms of " }{XPPEDIT 18 0 "c[2]" "6#&%\"cG6#\"\"#" }{TEXT -1 3 ", " }{XPPEDIT 18 0 "c[3]" "6#&%\"cG6# \"\"$" }{TEXT -1 7 " and " }{XPPEDIT 18 0 "c[5]" "6#&%\"cG6#\"\"&" } {TEXT -1 1 " " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1303 "sm2 := 1/31681277824320000*(1047363032943900*c [3]^5*c[5]+30557662050000*c[3]^4*c[2]+325185338162050*c[3]^6+170490873 60000*c[2]^2+563772739061250*c[5]^2*c[3]^6-676527286873500*c[3]^6*c[5] -870688777041000*c[3]^2*c[2]^2*c[5]+772517817067500*c[3]^2*c[2]^2*c[5] ^2+295218698355900*c[3]^2*c[2]^2+8930004940128*c[3]^2+246424776858000* c[3]^3*c[2]*c[5]+170642194506000*c[3]^2*c[2]*c[5]^2+39880963872000*c[3 ]*c[2]*c[5]-35826844320000*c[3]*c[2]*c[5]^2-108721125936000*c[3]*c[2]^ 2-291648648330000*c[3]^3*c[2]*c[5]^2-162705348363600*c[3]^2*c[2]*c[5]+ 249761911165248*c[3]^4+429428252545650*c[5]^2*c[3]^4-58321947765184*c[ 3]^3-560131782652440*c[3]^4*c[5]-121689243502200*c[5]^2*c[3]^3+1418243 89563216*c[3]^3*c[5]+21038509205928*c[5]^2*c[3]^2+41247792000000*c[5]^ 2*c[2]^2-49497350400000*c[5]*c[2]^2-24322404603744*c[3]^2*c[5]+3214965 03840000*c[3]*c[2]^2*c[5]-281663017200000*c[3]*c[2]^2*c[5]^2-786970248 00000*c[3]^3*c[2]+51731307192000*c[2]*c[3]^2-13356512640000*c[3]*c[2]- 470891829035040*c[3]^5+515645797921875*c[3]^4*c[2]^2*c[5]^2-3844052673 79500*c[3]^3*c[2]^2+170842607662500*c[3]^4*c[2]*c[5]^2+213133596474375 *c[3]^4*c[2]^2-808642423642500*c[5]^2*c[3]^5-995875884337500*c[3]^3*c[ 2]^2*c[5]^2-618774957506250*c[3]^4*c[2]^2*c[5]+1136715136605000*c[3]^3 *c[2]^2*c[5]-119174095372500*c[3]^4*c[2]*c[5])/c[3]^2/(5*c[3]-2)^2:" } }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 8 "Taking " }{XPPEDIT 18 0 "c[5] = 7/10;" "6#/&%\"cG6#\"\"&*&\"\"(\" \"\"\"#5!\"\"" }{TEXT -1 103 ", we obtain the following surface for th e graph of the square of the principal error norm in terms of " } {XPPEDIT 18 0 "c[2]" "6#&%\"cG6#\"\"#" }{TEXT -1 7 " and " } {XPPEDIT 18 0 "c[3]" "6#&%\"cG6#\"\"$" }{TEXT -1 2 ". 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8E1E282F7B3EDED3956A8E30273EDC2A59F7DCADE33EDA46F1515FA94E3ED8FA482650 22E53ED822BB78A0DD293ED7A835D3FAA33C3ED7799FA2F6FCD93ED78B2FFAFD4D753E D7D558789E06583ED85422D79AF7BC3ED906E98D0F29A03ED9F062672AEA8F3EDB16FE 2BC51BF93EDC85ACF16AC2D13EDE4D27575081B23EE042FA6BBB94633EE1A9C6EB3553 F63EE3749C57C98EE93EE5C8D91F7024413EE8DDC04FD00A813EED06AAB0EA83743EEF D883352A9FFB3EEA0DE281626C713EE5DC57E8C78AD83EE2CCBEE07722053EE08EEDF9 803E023EDDD77B7C3979CE3EDB790CA51FB83D3ED9CB93F0D38F733ED8A7908A055172 3ED7F080E5DA6F993ED791EAFD366B913ED77D648AA064F23ED7A94B403B78333ED80F FCBE5877E13ED8AF72FBDB05D83ED98938F991C3143EDAA2B86ACF52FC3EDC05EE3973 E8B53EDDC2A461A308F23EDFF0679CCEC5843EE158CAA3CB8B4A3EE31C0EC87BF3773E E56682B4E83BCE3EE86E3F362761D23EEC84F57023FA133EF1F02B5ABD51A73EED1E48 1986376B3EE8321C09D906493EE493C8C24BA53A3EE1E79403CFFD213EDFDC73D42870 783EDCF6866B766BE83EDADDE8F2686E593ED964CE779480903ED869F59995DDB03ED7 D54183F62BAB3ED79574A7DD63973ED79EB1C8F8996D3ED7E98BD07173BD3ED8728436 52BF743ED939E87E7A43613EDA440D04217EEB3EDB99F110A553423EDD4A6907155CE5 3EDF6C0356940F283EE110013746049F3EE2CB82535773793EE50C3057B29D8D3EE807 024537837F3EEC0C0F4CC6B0B63EF4290DEAFA311B3EF042E8450FFEC03EEAD0587F5E 25553EE697ABC3A5B9073EE373BCD436558A3EE11CA17FD4C9063EDEBF5EA463C4553E DC31472B0EBF4D3EDA5A75414E6A6E3ED91493EF2CED733ED843A337363D7F3ED7D360 52B448F63ED7B58C12D5697E3ED7E0D00DE5EB7F3ED8501D3C76568B3ED90270F5E46B 1E3ED9FAFBF7BBA0563EDB41B576FF02843EDCE47547A77DB93EDEF8C804C7070D3EE0 CF6AA5A4BF4D3EE282F6F85C0E623EE4B9E207CF49643EE7A8097D006DEE3EEB9BF846 D2A7663EF696E94B4BEE7E3EF2223DE997DF6A3EEDB70D4956E84C3EE8D867E4855D6F 3EE533686AB347503EE276F47F5E6F413EE069CAA7F3E0AE3EDDC5AE9AC682913EDB88 84E7330F2B3ED9F05BE69F9F003ED8DD1016F6A1483ED837278B25154A3ED7EDDA1DD1 E8E43ED7F5C976B5DF063ED8483E0E45CAB53ED8E2D25FD03B833ED9C785459DB6EF3E DAFD3B6C80F62F3EDC90C923596B673EDE96B5A767AC653EE09706EEE7BC263EE2426C B789C50D3EE46F97C53E3FC53EE75154DD8221AA3EEB34B05E47DE7B3EF939BD7BB289 B13EF42D24FA5ABD903EF0731D33E1A7853EEB55FD24EA92783EE72696C746D25F3EE3 FD32E8B12AD33EE19995370131C13EDF9B1F418FB7C03EDCEEFD69426EF23EDAFD4D7F EDF1F23ED9A188233756483ED8C0CA512FC89D3ED8479BE9EE179E3ED828780AE14DC6 3ED85AE6ABC11C563ED8DB0CBC3DB2C83ED9A9A8EDC7C2E63EDACC82F12B2EED3EDC4F 649A2B25EB3EDE45CC3E75FF983EE066D6130EFA783EE209E390E096ED3EE42D518FFF 7FE73EE702E466BC9E043EEAD637932655B23EFC118A7C2E02C73EF6639D7758995A3E F22EEFED51ACB63EEE106B84D5581A3EE94D47E9F0F6AE3EE5AF5CBBCCFBD03EE2EF0E FF59D4F93EE0D8CC8FB52FB53EDE8DDEC77C89A83EDC3B68BB17E6AF3EDA910B5BF85D 4D3ED97048A4D462F43ED8C2D17729F57B3ED878DBCA6838693ED8881714E84ADE3ED8 EB200B2CD2243ED9A166F039C4383EDAAF8C04FDAC193EDC2047AC1CAD913EDE060BC9 F200613EE03ED8121A7A5E3EE1D95B846084003EE3F30F68130A403EE6BCB818AFE457 3EEA808DE56E0C793EFF1E504CBE59BC3EF8C5A7609172B63EF40EFED0FB83E63EF083 D98222D72A3EEBA77BD2B1B4663EE78D71F8B1E2443EE46A3800FDCABF3EE2048E1A2B 3C7D3EE0329480F0AF7E3EDDAAAD981D7D4D3EDBAB99C139B6453EDA45A28612E43E3E D95F7AC58582673ED8E6F4B54A9EE13ED8CFCF49BB576F3ED9130C4C9D996C3ED9AEBF 4CF3BA913EDAA656A7F86DE73EDC0372592E01C83EDDD77449DBAF683EE01F0CEC0A3C 0C3EE1B0D492098C393EE3C0D14D78DF213EE67ECFF35BF39C3EEA33B3551F02B03F01 300776B1C7443EFB5342B60549A03EF61349DEDF2CE63EF21DE9D19DCA963EEE353281 890B523EE997729F5FDE123EE60B103BED12D23EE350D4402A027F3EE13A6E0C3877A3 3EDF4B1C16FEB5463EDCF13352FB608C3EDB40D7F4EB4CB23EDA1D97D500BED03ED972 C2CB88812E3ED9320F4A3A40923ED952D1809008E73ED9D1B203F5A6433EDAB0E2DA1B 74503EDBF8E4A15F22D83EDDBA05BE330B1C3EE00774A0DE3F9A3EE1904EB9DBB0433E E396974030FD4C3EE6492BF6C0CC313EE9EFA7E2393AF1-%&COLORG6&%$RGBG$\"*+++ +\"!\")F'F'-%+AXESLABELSG6%%%c[2]G%%c[3]G%!G-%+LIGHTMODELG6#%(LIGHT_2G -%*AXESSTYLEG6#%$BOXG" 1 2 0 1 10 0 2 1 4 2 2 1.000000 45.000000 45.000000 0 0 "Curve 1" }}}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 " " 0 "" {TEXT -1 58 "The following calculations give (approximate) valu es for " }{XPPEDIT 18 0 "c[2]" "6#&%\"cG6#\"\"#" }{TEXT -1 7 " and \+ " }{XPPEDIT 18 0 "c[3]" "6#&%\"cG6#\"\"$" }{TEXT -1 82 " to minimize \+ the square of the principal error norm for three fixed choices for " }{XPPEDIT 18 0 "c[5]" "6#&%\"cG6#\"\"&" }{TEXT -1 10 ", namely " } {XPPEDIT 18 0 "c[5]=3/5" "6#/&%\"cG6#\"\"&*&\"\"$\"\"\"F'!\"\"" } {TEXT -1 3 ", " }{XPPEDIT 18 0 "c[5]=13/20" "6#/&%\"cG6#\"\"&*&\"#8\" \"\"\"#?!\"\"" }{TEXT -1 7 " and " }{XPPEDIT 18 0 "c[5]=7/10" "6#/&% \"cG6#\"\"&*&\"\"(\"\"\"\"#5!\"\"" }{TEXT -1 1 "." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 247 "uu := \{c[5 ]=3/5.\}:\nmn := evalf[12](minimize(eval(sm2,uu),c[2]=0.1..0.15,c[3]=0 .15..0.25,'location'));\nsubs(op(1,op(1,op(2,[mn]))),[c[2],c[3]]):\nrr := map(convert,%,rational,5):\nc[2]=op(1,rr),c[3]=op(2,rr);\n`princip al error norm:`,sqrt(op(1,[mn]));" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#> %#mnG6$$\"-^\"*o#3S&!#<<#7$<$/&%\"cG6#\"\"#$\"-BE7u35!#7/&F.6#\"\"$$\" -d\\N)*e:F3F&" }}{PARA 11 "" 1 "" {XPPMATH 20 "6$/&%\"cG6#\"\"##\"#B\" $G#/&F%6#\"\"$#\"#T\"$j#" }}{PARA 11 "" 1 "" {XPPMATH 20 "6$%6principa l~error~norm:G$\"+?z'RK#!#7" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 249 "uu := \{c[5]=13/20.\}:\nmn \+ := evalf[12](minimize(eval(sm2,uu),c[2]=0.1..0.15,c[3]=0.15..0.25,'loc ation'));\nsubs(op(1,op(1,op(2,[mn]))),[c[2],c[3]]):\nrr := map(conver t,%,rational,5):\nc[2]=op(1,rr),c[3]=op(2,rr);\n`principal error norm: `,sqrt(op(1,[mn]));" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%#mnG6$$\"-(>. >*y_!#<<#7$<$/&%\"cG6#\"\"#$\"-='es&p6!#7/&F.6#\"\"$$\"-&*))=b*p\"F3F& " }}{PARA 11 "" 1 "" {XPPMATH 20 "6$/&%\"cG6#\"\"##\"#?\"$r\"/&F%6#\" \"$#\"#V\"$`#" }}{PARA 11 "" 1 "" {XPPMATH 20 "6$%6principal~error~nor m:G$\"+J)*e(H#!#7" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 " > " 0 "" {MPLTEXT 1 0 247 "uu := \{c[5]=7/10.\}:\nmn := evalf[12](mini mize(eval(sm2,uu),c[2]=0.1..0.2,c[3]=0.15..0.25,'location'));\nsubs(op (1,op(1,op(2,[mn]))),[c[2],c[3]]):\nrr := map(convert,%,rational,5):\n c[2]=op(1,rr),c[3]=op(2,rr);\n`principal error norm:`,sqrt(op(1,[mn])) ;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%#mnG6$$\"-H9v(\\f&!#<<#7$<$/&% \"cG6#\"\"#$\"-vBs-d8!#7/&F.6#\"\"$$\"-8Sr'['=F3F&" }}{PARA 11 "" 1 " " {XPPMATH 20 "6$/&%\"cG6#\"\"##\"#>\"$S\"/&F%6#\"\"$#\"#e\"$6$" }} {PARA 11 "" 1 "" {XPPMATH 20 "6$%6principal~error~norm:G$\"+\"[q`O#!#7 " }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 14 "The \+ procedure " }{TEXT 0 8 "minimize" }{TEXT -1 75 " has difficulty handli ng minimization with respect to the three variables " }{XPPEDIT 18 0 "c[2]" "6#&%\"cG6#\"\"#" }{TEXT -1 3 ", " }{XPPEDIT 18 0 "c[3]" "6#&% \"cG6#\"\"$" }{TEXT -1 7 " and " }{XPPEDIT 18 0 "c[5]" "6#&%\"cG6#\" \"&" }{TEXT -1 86 ", so we set up a procedure to perform minimizations with repect to the two variables " }{XPPEDIT 18 0 "c[2]" "6#&%\"cG6# \"\"#" }{TEXT -1 7 " and " }{XPPEDIT 18 0 "c[3]" "6#&%\"cG6#\"\"$" } {TEXT -1 85 " as in the previous examples and then use the one dimens ional minmization procedure " }{TEXT 0 7 "findmin" }{TEXT -1 29 " to m inimize with repect to " }{XPPEDIT 18 0 "c[5]" "6#&%\"cG6#\"\"&" } {TEXT -1 1 "." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 143 "prin_err := proc(c_5)\n local mn;\n mn := e valf[Digits+2](minimize(eval(sm2,c[5]=c_5),c[2]=0.1..0.2,c[3]=0.15..0. 25));\n sqrt(mn);\nend proc:" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }} {PARA 0 "" 0 "" {TEXT -1 21 "Using the procedure " }{TEXT 0 8 "prin_e rr" }{TEXT -1 55 " three previous examples can be performed as follow s. " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 40 "prin_err(.6),prin_err(.65),prin_err(.7);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6%$\"+>z'RK#!#7$\"+J)*e(H#F%$\"+\"[q`O#F%" }}} {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 55 "These thr ee values can be used as starting values for " }{TEXT 0 7 "findmin" } {TEXT -1 1 "." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 110 "evalf[12](findmin('prin_err'(c5),c5=\{0.6,0.65, 0.7\},convergence='location'));\nc[5]=convert(op(1,%),rational,7);" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#7$$\"-8boA%R'!#7$\"-EFr[&H#!#9" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#/&%\"cG6#\"\"&#\"$L\"\"$3#" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 250 "u u := \{c[5]=133/208.\}:\nmn := evalf[12](minimize(eval(sm2,uu),c[2]=0. 1..0.2,c[3]=0.15..0.25,'location'));\nsubs(op(1,op(1,op(2,[mn]))),[c[2 ],c[3]]):\nrr := map(convert,%,rational,5):\nc[2]=op(1,rr),c[3]=op(2,r r);\n`principal error norm:`,sqrt(op(1,[mn]));" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%#mnG6$$\"-)4:h#p_!#<<#7$<$/&%\"cG6#\"\"#$\"-a#[^K8\"! #7/&F.6#\"\"$$\"-CsO(ym\"F3F&" }}{PARA 11 "" 1 "" {XPPMATH 20 "6$/&%\" cG6#\"\"##\"#<\"$]\"/&F%6#\"\"$#\"$I#\"%z8" }}{PARA 11 "" 1 "" {XPPMATH 20 "6$%6principal~error~norm:G$\"+Fr[&H#!#7" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 14 "The value of " } {XPPEDIT 18 0 "c[4]" "6#&%\"cG6#\"\"%" }{TEXT -1 20 " corresponding t o " }{XPPEDIT 18 0 "c[3]=230/1379" "6#/&%\"cG6#\"\"$*&\"$I#\"\"\"\"%z 8!\"\"" }{TEXT -1 10 " is . . ." }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 54 "c[4]=24720/(69922-174805*c[3]);\nsubs(c[3]=230/1379,%);" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#/&%\"cG6#\"\"%,$*&\"&?Z#\"\"\",&\"&A*p F+*&\"'0[ " 0 "" {MPLTEXT 1 0 112 "e3 := \{c[2]=17/150,c[3]=23 0/1379,c[4]=1420370/2342387,c[5]=133/208,`b*`[7]=1/40\}:\ne4 := `union `(subs(e3,e2),e3):" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 2 "e4 " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1751 "e4 := \{a[6,4] = -963956247541399247752708263264927/63142814 4652436311985040262094600, `b*`[3] = 1333442845934451852070878169/3554 259119336756010609086400, a[7,2] = 0, `b*`[6] = 58914464694395831/6939 16322394528225, a[4,1] = 11725892832745653435650/679878988049210698498 7, a[6,3] = -6240811551117087470432064137/6600414167621042866395511800 , a[7,5] = 73328266119626752/108862061490007875, a[7,4] = -33988229203 965941757637160569/199599379574494873119383412000, a[3,2] = 3967500/32 327897, a[4,2] = -2048128144537681132500/295599560021395955869, a[7,6] = 175006801591/1589096299500, `b*`[1] = 36166679554284109/30356929171 88954400, c[2] = 17/150, c[3] = 230/1379, c[7] = 1, a[5,1] = 121616031 19957121528213173/25732152506856316150988800, a[5,3] = 368326757639958 1353585064593/1765398936832403041633484800, a[6,2] = 104590105895625/6 8427659422081, a[4,3] = 39503685458474301158220/6798789880492106984987 , a[6,1] = -778166632716677780702867/2973121469222107001182300, a[5,2] = -1141151413305973125/551372154861039616, c[6] = 1, a[6,5] = 4149774 798413464954572800/1881639624293239507076793, `b*`[7] = 1/40, `b*`[2] \+ = 0, b[2] = 0, c[5] = 133/208, c[4] = 1420370/2342387, a[7,3] = 380753 067446365335491/1017424549568194596000, b[5] = 73328266119626752/10886 2061490007875, b[1] = 10715793331/868982366000, b[6] = 175006801591/15 89096299500, a[3,1] = 1424390/32327897, b[3] = 380753067446365335491/1 017424549568194596000, `b*`[4] = -372495557697385192340734287120958393 /2091834471758211976844177479899662400, `b*`[5] = 12950810184906094281 1648/190148731404640113216825, a[2,1] = 17/150, a[5,4] = 5081806128006 459902107790523/33857948797038081396772249600, b[4] = -339882292039659 41757637160569/199599379574494873119383412000, a[7,1] = 10715793331/86 8982366000\}:" }{TEXT -1 0 "" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 366 "subs(e4,matrix([[ c[2],a[2,1],``$3],\n [c[3],a[3,1],a[3,2],``$2],[c[4],a[4,1],a[4,2],a[ 4,3],``],\n [c[5],a[5,1],a[5,2],a[5,3],a[5,4]],\n [c[6],a[6,1],a[6,2 ],a[6,3],a[6,4]],[``$4,a[6,5]],\n [c[7],a[7,1],a[7,2],a[7,3],a[7,4]], [``$3,a[7,5],a[7,6]],\n [`b`,b[1],b[2],b[3],b[4]],[``$3,b[5],b[6]],\n [`b*`,`b*`[1],`b*`[2],`b*`[3],`b*`[4]],[``$2,`b*`[5],`b*`[6],`b*`[7] ]]));" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#K%'matrixG6#7.7'#\"#<\"$]\"F( %!GF+F+7'#\"$I#\"%z8#\"(!RC9\")(*yKK#\"(+v'RF2F+F+7'#\"(q.U\"\"((QUB# \"8]cV`cuKG*es6\"7()\\)p5#\\!))*y)z'#!7+D8\"oPX9G\"[?\"6pe&fR@+c*f&H# \"8?#e6Iu%eao.&RF;F+7'#\"$L\"\"$3##\";tJ@G:7d*>Jgh@\"\";+)))4:;j&o]_@t D#!4DJ(fI89:T6\"3;'R5'[:s8b#\"=$fk]e`8e*RwvE$o$\"=+[[L;/.C$o$*)Rl<#\"= B0z2@!*fk+Gh!=3&\">+'\\Ax'R\"3Qqz[z&Q$7'\"\"\"#!9nGq!yxmrKm;y(\":+B=,q 5A#p97tH#\"0Dc*e5!f/\"\"/\"3A%fwUo#!=PT1K/Z(3<6b63C'\"=+=^&RmG/@w;9/g' #!BF\\Ej#3FvZ#*RTvCcR'*\"B+Y4i-/&)>JOClW\"G9j7'F+F+F+F+#\":+Gda\\Y8%)z u(\\T\":$zwq]RKHC'R;)=7'FR#\",JLz:2\"\"-+gO#)*o)\"\"!#\"6\"\\N`OYu1`2Q \"7+gf%>o&\\XU<5#!>p0;PwvTf'R?H#))R$\"?+?T$Q>J([\\u&z$*f*>7'F+F+F+#\"2 _ni>hEGL(\"3vy+!\\h?')3\"#\"-\"f,o+v\"\".+&*H'4*e\"7'%\"bGF^oFaoFboFeo Fho7'%#b*G#\"24TGa&zm;O\"4+W&*)=\"fUb$#!E$Re47(GM2M#>&Q(pdb\\s$\"F+Cm**)zu@e#\"2JeR%pkW\"*e\"3D#GXRAj\"Rp#FR\"#SQ)pp rint116\"" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 " " {MPLTEXT 1 0 132 "subs(e4,matrix([seq([c[i],seq(a[i,j],j=1..i-1),``$ (8-i)],i=2..7),\n[`b`,seq(b[i],i=1..6),``],[`b*`,seq(`b*`[i],i=1..7)]] )): evalf(%);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#K%'matrixG6#7*7*$\"+L LLL6!#5F(%!GF+F+F+F+F+7*$\"+s_(ym\"F*$\"+2221W!#6$\"+,#osA\"F*F+F+F+F+ F+7*$\"++4]\"F*F+F+F+7*$\"\"\"\"\" !$!+])Qth#F*$\"+MqZG:F9$!+S<=b%*F*$!+t&Gm_\"F9$\"+)f.a?#F9F+F+7*FJ$\"+ $yUJB\"F1$FLFL$\"+#RABu$F*$!+(QAGq\"F*$\"+p!))et'F*$\"+ZwH,6F*F+7*%\"b GFXFZFenFgnFinF[oF+7*%#b*G$\"+g8Q\">\"F1FZ$\"+]fn^PF*$!+(R72y\"F*$\"+[ V)3\"oF*$\"+J'R,\\)F1$\"+++++DF1Q)pprint126\"" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}}{PARA 0 "" 0 "" {TEXT -1 39 "#--------------------------------------" }} {SECT 1 {PARA 4 "" 0 "" {TEXT -1 39 "characteristics of the embedded s cheme " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 35 "coefficient s of the combined scheme" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1674 "ee := \{c[2]=17/150,\nc[3]=230/13 79,\nc[4]=1420370/2342387,\nc[5]=133/208,\nc[6]=1,\nc[7]=1,\n\na[2,1]= 17/150,\na[3,1]=1424390/32327897,\na[3,2]=3967500/32327897,\na[4,1]=11 725892832745653435650/6798789880492106984987,\na[4,2]=-204812814453768 1132500/295599560021395955869,\na[4,3]=39503685458474301158220/6798789 880492106984987,\na[5,1]=12161603119957121528213173/257321525068563161 50988800,\na[5,2]=-1141151413305973125/551372154861039616,\na[5,3]=368 3267576399581353585064593/1765398936832403041633484800,\na[5,4]=508180 6128006459902107790523/33857948797038081396772249600,\na[6,1]=-7781666 32716677780702867/2973121469222107001182300,\na[6,2]=104590105895625/6 8427659422081,\na[6,3]=-6240811551117087470432064137/66004141676210428 66395511800,\na[6,4]=-963956247541399247752708263264927/63142814465243 6311985040262094600,\na[6,5]=4149774798413464954572800/188163962429323 9507076793,\na[7,1]=10715793331/868982366000,\na[7,2]=0,\na[7,3]=38075 3067446365335491/1017424549568194596000,\na[7,4]=-33988229203965941757 637160569/199599379574494873119383412000,\na[7,5]=73328266119626752/10 8862061490007875,\na[7,6]=175006801591/1589096299500,\n\nb[1]=10715793 331/868982366000,\nb[2]=0,\nb[3]=380753067446365335491/101742454956819 4596000,\nb[4]=-33988229203965941757637160569/199599379574494873119383 412000,\nb[5]=73328266119626752/108862061490007875,\nb[6]=175006801591 /1589096299500,\n\n`b*`[1]=36166679554284109/3035692917188954400,\n`b* `[2]=0,\n`b*`[3]=1333442845934451852070878169/355425911933675601060908 6400,\n`b*`[4]=-372495557697385192340734287120958393/20918344717582119 76844177479899662400,\n`b*`[5]=129508101849060942811648/19014873140464 0113216825,\n`b*`[6]=58914464694395831/693916322394528225,\n`b*`[7]=1/ 40\}:" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 " " {TEXT -1 5 "Let " }{XPPEDIT 18 0 "T[5,6];" "6#&%\"TG6$\"\"&\"\"'" } {TEXT -1 127 " denote the vector whose components are the principal e rror terms of the 6 stage, order 5 scheme (the error terms of order 6) ." }}{PARA 0 "" 0 "" {TEXT -1 5 "Let " }{XPPEDIT 18 0 "`T*`[4,7];" "6 #&%#T*G6$\"\"%\"\"(" }{TEXT -1 145 " denote the vector whose componen ts are the principal error terms of the embedded 7 stage, order 4 sche me (the error terms of order 5) and let " }{XPPEDIT 18 0 "`T*`[5,7]; " "6#&%#T*G6$\"\"&\"\"(" }{TEXT -1 99 " denote the vector whose compo nents are the error terms of order 6 of the embedded order 4 scheme." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 39 "Denote \+ the 2-norm of these vectors by " }{XPPEDIT 18 0 "abs(abs(T[5,6]));" " 6#-%$absG6#-F$6#&%\"TG6$\"\"&\"\"'" }{TEXT -1 3 ", " }{XPPEDIT 18 0 " abs(abs(`T*`[4,7]));" "6#-%$absG6#-F$6#&%#T*G6$\"\"%\"\"(" }{TEXT -1 7 " and " }{XPPEDIT 18 0 "abs(abs(`T*`[5,7]));" "6#-%$absG6#-F$6#&%# T*G6$\"\"&\"\"(" }{TEXT -1 15 " respectively." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 8 "Define: " }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "A[6] = abs(abs(T[5,6]));" "6#/&%\" AG6#\"\"'-%$absG6#-F)6#&%\"TG6$\"\"&F'" }{TEXT -1 6 ", " } {XPPEDIT 18 0 "B[6] = abs(abs(`T*`[5,7]))/abs(abs(`T*`[4,7]));" "6#/&% \"BG6#\"\"'*&-%$absG6#-F*6#&%#T*G6$\"\"&\"\"(\"\"\"-F*6#-F*6#&F/6$\"\" %F2!\"\"" }{TEXT -1 6 ", " }{XPPEDIT 18 0 "C[6] = abs(abs(`T*`[5,7 ]-T[5,6]))/abs(abs(`T*`[4,7]));" "6#/&%\"CG6#\"\"'*&-%$absG6#-F*6#,&&% #T*G6$\"\"&\"\"(\"\"\"&%\"TG6$F2F'!\"\"F4-F*6#-F*6#&F06$\"\"%F3F8" } {TEXT -1 1 "." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 77 "Dormand and Prince have suggested that as well as attempt ing to ensure that " }{XPPEDIT 18 0 "A[6];" "6#&%\"AG6#\"\"'" }{TEXT -1 73 " is a minimum, if the embedded scheme is to be used for error \+ control, " }{XPPEDIT 18 0 "B[6];" "6#&%\"BG6#\"\"'" }{TEXT -1 7 " an d " }{XPPEDIT 18 0 "C[6];" "6#&%\"CG6#\"\"'" }{TEXT -1 27 " should b e chosen so that " }}{PARA 0 "" 0 "" {TEXT -1 68 "they are \"similar i n magnitude\" and also not differ too much from 1." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 186 "errterms5_6 := PrincipalErrorTerms(5,6,'expanded'):\n`errterms5_7*` :=subs(b=`b*` ,PrincipalErrorTerms(5,7,'expanded')):\n`errterms4_7*` := subs(b=`b*`, PrincipalErrorTerms(4,7,'expanded')):" }}}{PARA 0 "" 0 "" {TEXT -1 0 " " }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 326 "snmB := sqrt(add(evalf( subs(ee,`errterms5_7*`[i]))^2,i=1..nops(`errterms5_7*`))):\nsdnB := sq rt(add(evalf(subs(ee,`errterms4_7*`[i]))^2,i=1..nops(`errterms4_7*`))) :\nsnmC := sqrt(add((evalf(subs(ee,`errterms5_7*`[i])-subs(ee,errterms 5_6[i])))^2,i=1..nops(errterms5_6))):\n'B'[6]= evalf[8](snmB/sdnB);\n' C'[6]= evalf[8](snmC/sdnB);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/&%\"BG 6#\"\"'$\")#3kF\"!\"(" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/&%\"CG6#\"\" '$\")[8&G\"!\"(" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> \+ " 0 "" {MPLTEXT 1 0 1 ";" }}}}{PARA 0 "" 0 "" {TEXT -1 39 "#---------- ----------------------------" }}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 26 "a bsolute stability regions" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 35 "coefficients of the combined scheme" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1674 "ee := \{c[2]=17/ 150,\nc[3]=230/1379,\nc[4]=1420370/2342387,\nc[5]=133/208,\nc[6]=1,\nc [7]=1,\n\na[2,1]=17/150,\na[3,1]=1424390/32327897,\na[3,2]=3967500/323 27897,\na[4,1]=11725892832745653435650/6798789880492106984987,\na[4,2] =-2048128144537681132500/295599560021395955869,\na[4,3]=39503685458474 301158220/6798789880492106984987,\na[5,1]=12161603119957121528213173/2 5732152506856316150988800,\na[5,2]=-1141151413305973125/55137215486103 9616,\na[5,3]=3683267576399581353585064593/176539893683240304163348480 0,\na[5,4]=5081806128006459902107790523/33857948797038081396772249600, \na[6,1]=-778166632716677780702867/2973121469222107001182300,\na[6,2]= 104590105895625/68427659422081,\na[6,3]=-6240811551117087470432064137/ 6600414167621042866395511800,\na[6,4]=-9639562475413992477527082632649 27/631428144652436311985040262094600,\na[6,5]=414977479841346495457280 0/1881639624293239507076793,\na[7,1]=10715793331/868982366000,\na[7,2] =0,\na[7,3]=380753067446365335491/1017424549568194596000,\na[7,4]=-339 88229203965941757637160569/199599379574494873119383412000,\na[7,5]=733 28266119626752/108862061490007875,\na[7,6]=175006801591/1589096299500, \n\nb[1]=10715793331/868982366000,\nb[2]=0,\nb[3]=38075306744636533549 1/1017424549568194596000,\nb[4]=-33988229203965941757637160569/1995993 79574494873119383412000,\nb[5]=73328266119626752/108862061490007875,\n b[6]=175006801591/1589096299500,\n\n`b*`[1]=36166679554284109/30356929 17188954400,\n`b*`[2]=0,\n`b*`[3]=1333442845934451852070878169/3554259 119336756010609086400,\n`b*`[4]=-372495557697385192340734287120958393/ 2091834471758211976844177479899662400,\n`b*`[5]=1295081018490609428116 48/190148731404640113216825,\n`b*`[6]=58914464694395831/69391632239452 8225,\n`b*`[7]=1/40\}:" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}}{PARA 0 "" 0 "" {TEXT -1 0 " " }}{PARA 0 "" 0 "" {TEXT -1 79 "The stability function R for the 6 \+ stage, order 5 scheme is given as follows." }}{EXCHG {PARA 0 "> " 0 " " {MPLTEXT 1 0 75 "subs(ee,StabilityFunction(5,6,'expanded')):\nR := u napply(%,z):\n'R(z)'=R(z);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/-%\"RG6 #%\"zG,0\"\"\"F)F'F)*&#F)\"\"#F)*$)F'F,F)F)F)*&#F)\"\"'F)*$)F'\"\"$F)F )F)*&#F)\"#CF)*$)F'\"\"%F)F)F)*&#F)\"$?\"F)*$)F'\"\"&F)F)F)*&#\"$.\"\" &h\\$F)*$)F'F1F)F)F)" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 124 "We can find the point where the boundary of the stabi lity region intersects the negative real axis by solving the equation: " }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "R(z) = 1;" "6#/- %\"RG6#%\"zG\"\"\"" }{TEXT -1 2 ". " }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 26 "z0 := newton(R(z)=1,z=-3);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%#z0G$!+IX1'z#!\"*" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" } }{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 301 "z0 := newton(R(z)=1,z=-3): \np1 := plot([R(z),1],z=-3.19..0.49,color=[red,blue]):\np2 := plot([[[ z0,1]]$3],style=point,symbol=[circle,cross,diamond],color=black):\np3 \+ := plot([[z0,0],[z0,1]],linestyle=3,color=COLOR(RGB,0,.5,0)):\nplots[d isplay]([p1,p2,p3],view=[-3.19..0.49,-0.0..1.47],font=[HELVETICA,9]); " }}{PARA 13 "" 1 "" {GLPLOT2D 401 211 211 {PLOTDATA 2 "6+-%'CURVESG6$ 7X7$$!3%**************=$!#<$\"3!Rg$3pHAa@F*7$$!3'HLLjB$*)\\JF*$\"3iy5! 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" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1426 "R := z -> 1+z+1/2*z^2+1/6*z^3+1/24*z^4+1/120*z^5+103/34961*z^6 :\npts := []: z0 := 0: tt := 0: \nwhile tt<=161/20 do\n zz := newton (R(z)=exp(tt*Pi*I),z=z0):\n z0 := zz:\n if (1/2<=tt and tt<=1) or \+ (139/20<=tt and tt<=149/20) then\n hh := 1/40\n else \n hh := 1/20\n end if;\n tt := tt+hh;\n pts := [op(pts),[Re(zz),Im(z z)]]:\nend do:\np1 := plot(pts,color=COLOR(RGB,.5,.1,.13)):\np2 := plo ts[polygonplot]([seq([pts[i-1],pts[i],[-1.4,0]],i=2..nops(pts))],\n \+ style=patchnogrid,color=COLOR(RGB,1,.2,.25)):\npts := []: z0 := 1.5+3.5*I:\nfor ct from 0 to 60 do\n zz := newton(R(z)=exp(ct*Pi/30 *I),z=z0):\n z0 := zz:\n pts := [op(pts),[Re(zz),Im(zz)]]:\nend do :\np3 := plot(pts,color=COLOR(RGB,.5,.1,.13)):\np4 := plots[polygonplo t]([seq([pts[i-1],pts[i],[1.44,3.34]],i=2..nops(pts))],\n sty le=patchnogrid,color=COLOR(RGB,1,.2,.25)):\npts := []: z0 := 1.5-3.5*I :\nfor ct from 0 to 60 do\n zz := newton(R(z)=exp(ct*Pi/30*I),z=z0): \n z0 := zz:\n pts := [op(pts),[Re(zz),Im(zz)]]:\nend do:\np5 := p lot(pts,color=COLOR(RGB,.5,.1,.13)):\np6 := plots[polygonplot]([seq([p ts[i-1],pts[i],[1.44,-3.34]],i=2..nops(pts))],\n style=patchn ogrid,color=COLOR(RGB,1,.2,.25)):\np7 := plot([[[-3.29,0],[1.99,0]],[[ 0,-3.89],[0,3.89]]],color=black,linestyle=3):\nplots[display]([p||(1.. 7)],view=[-3.29..1.99,-3.89..3.89],font=[HELVETICA,9],\n color=COLOR (RGB,.4,0,1),labels=[`Re(z)`,`Im(z)`],axes=boxed,scaling=constrained); " }}{PARA 13 "" 1 "" {GLPLOT2D 408 566 566 {PLOTDATA 2 "60-%'CURVESG6$ 7bv7$$\"\"!F)F(7$$\"3))*****4FUSI#!#D$\"3%******fG'zq:!#=7$$\"32+++qNc 29!#B$\"3<+++]ueTJF07$$\"3-+++!4L\"y9!#A$\"3,+++w@I7ZF07$$\"3h+++YL'GM (F:$\"3=+++\"zcDG'F07$$\"30+++;'yJM#!#@$\"3_+++_$=6&yF07$$\"3S+++dcY%R &FE$\"3A+++S96:%*F07$$\"3L+++P]OB\"*FE$\"3#*******yL#p4\"!#<7$$\"3v+++ 0v:5**FE$\"3++++9([0D\"FR7$$!39+++')*z,W%F:$\"3)******\\X)Q,9FR7$$!3*) *****z#))eUN!#?$\"35+++$\"3#******4bw0p\"FR7$$!3\"******pAJFt\"Feo$\"3#******fkj &f*o%Feo$\"35+++eD\\c>FR7$$!3x*****p/(e3hFeo$ \"34+++Q*Q#=?FR7$$!3%******\\\\l&exFeo$\"3'******H-9x2#FR7$$!3W*****4s Asj*Feo$\"3\")*****fI6Z8#FR7$$!3)*******GDat6F0$\"3?+++%pN!*=#FR7$$!30 +++]$RPS\"F0$\"39+++C4_SAFR7$$!33+++9F?_;F0$\"3#)*****\\xW!*G#FR7$$!36 +++%Rvk\">F0$\"3++++Q(QXL#FR7$$!3!******\\![&R>#F0$\"3;+++^/*pP#FR7$$! 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\"\"\"\"!" }{TEXT -1 2 ". " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 289 "We can distort the boundary curve horizontally b y taking the 11th root of the real part of points along the curve. In \+ this way we see that the largest interval on the nonnegative imaginary axis that contains the origin and lies inside the stability region is [ 0, 0.34 ] approximately. " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 313 "R := z -> 1+z+1/2*z^2+1/6*z ^3+1/24*z^4+1/120*z^5+103/34961*z^6:\nDigits := 20:\npts := []: z0 := \+ 0:\nfor ct from 0 to 50 do\n zz := newton(R(z)=exp(ct*Pi/100*I),z=z0 ):\n z0 := zz:\n pts := [op(pts),[surd(Re(zz),11),Im(zz)]]:\nend d o:\nplot(pts,color=COLOR(RGB,1,0,.15),thickness=2,font=[HELVETICA,9]); \nDigits := 10:" }}{PARA 13 "" 1 "" {GLPLOT2D 261 276 276 {PLOTDATA 2 "6(-%'CURVESG6#7U7$$\"\"!F)F(7$$\"5\"\\VNsa)Q%RT)!#@$\"5Ei\"\\%e`EfTJF -7$$\"5K9\\zx C%*F-7$$\"5OkwZ(=_f2z\"F3$\"5vswVx_qjc7F37$$\"5DXKh#4`L:-#F3$\"57>cPc& G'zq:F37$$\"5a'*ywtgDaJAF3$\"5g:&fAZWb\\)=F37$$\"5Qj%e]U2FbU#F3$\"5&f% z:&[U9\"*>#F37$$\"5XxxiviXe1EF3$\"5r'\\XEU-tK^#F37$$\"5u8(e#z3q'ox#F3$ \"5-%o$[h!*3VFGF37$$\"5Q3,lciu$z$HF3$\"5oAT_[]ueTJF37$$\"5GV0]')y'p44$ F3$\"5,@lc$*>=ubMF37$$\"5%fX+fOwnoB$F3$\"5(fGNXfp#*)pPF37$$\"5Q`)GU:xT jP$F3$\"5u_e_'oAQS3%F37$$\"5Qny))p:v&*4NF3$\"5>e\"o79'e<)R%F37$$\"5SL/ D530J$=P,khXl&F 37$$\"5*\\,DI)Q8S.TF3$\"5(44Je.u$eofF37$$\"5C'*R*\\R[4*3UF3$\"5:(eNt6z cDG'F37$$\"54f8:*o\"pQ5VF3$\"5t@N6\\ftY'f'F37$$\"5aka)>UY:zS%F3$\"5jg \"[+^R*H5pF37$$\"5TzDggbba,XF3$\"5w7\\&*)[$R.CsF37$$\"5K/RiQYtH\"f%F3$ \"5Onf6a3)[w`(F37$$\"5;+G_4\")3;xYF3$\"5_-e8<_$=6&yF37$$\"5gOr[o#H&4fZ F3$\"56*H!Q?PJTk\")F37$$\"5[zMcVap-P[F3$\"5_B(Q6Ds*\\x%)F37$$\"5C=Zu/5 k%3\"\\F3$\"5p#>SZ%Q/M!z)F37$$\"50ro')y(y./)\\F3$\"5Da>.w6K*G5*F37$$\" 5+5]8$eB-b/&F3$\"5nUb-YS96:%*F37$$\"5%*fl1;k'))e5&F3$\"5>'\\H+u#R%ps*F 37$$\"5J>4_2'oS7;&F3$\"5S\"p`d!)[LQ+\"!#>7$$\"5:?!f3l,[6@&F3$\"5?\"p9' o)RA\\.\"Ffu7$$\"5ZG=Zu8b3b_F3$\"5\"fT!yScY&f1\"Ffu7$$\"5U!R\"\\1sJP#H &F3$\"5.7Q\\QzL#p4\"Ffu7$$\"5-'zxGz\"\\6A`F3$\"5q+3.[.8#y7\"Ffu7$$\"5/ K[s3c85V`F3$\"5ff0iA@3ke6Ffu7$$\"5bXPCFDxk``F3$\"5A2\">!pFSP*=\"Ffu7$$ \"5lG8OT)>28N&F3$\"5,Ks[5sF,?7Ffu7$$\"51&*y)y)zDKK`F3$\"5seU;w8([0D\"F fu7$$\"5:<[G2t=Y!H&F3$\"5g]*\\Q.Qt4G\"Ffu7$$\"5Ey\\w?Dp99_F3$\"5RvtgcE #y7J\"Ffu7$$\"5Rn:TwA@]x]F3$\"5Ci$\\d4pa9M\"Ffu7$$\"5^!=g\"4O0j'z%F3$ \"5==NbD[U\\r8Ffu7$$!5+Y,`-)=p2-%F3$\"5nBOkHb%)Q,9Ffu7$$!58&4Or`tWj+&F 3$\"5vqaq:))*G6V\"Ffu7$$!5H]:^<'z*Gj`F3$\"50,A]towqg9Ffu7$$!5NK'=V=7=F h&F3$\"5HuHhHzk6!\\\"Ffu7$$!5BNw=y-k)Q\"eF3$\"5z)Rk))RcZ$>:Ffu7$$!5i)= E7vpBq)fF3$\"5yK5b%o@$R[:Ffu-%%FONTG6$%*HELVETICAG\"\"*-%*THICKNESSG6# \"\"#-%+AXESLABELSG6$Q!6\"F]\\l-%&COLORG6&%$RGBG$\"\"\"F)F($\"#:!\"#-% %VIEWG6$%(DEFAULTGF[]l" 1 2 0 1 10 2 2 6 1 4 2 1.000000 45.000000 45.000000 0 0 "Curve 1" }}}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 " " 0 "" {TEXT -1 123 "The relevant intersection point of the boundary c urve with the imaginary axis can be determined more accurately as foll ows." }}{PARA 0 "" 0 "" {TEXT -1 86 "First we look for points on the b oundary curve either side of the intersection point. " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 111 "Digits : = 15:\nz0 := 1.4*I:\nfor ct from 43 to 46 do\n newton(R(z)=exp(ct*Pi /100*I),z=z0);\nend do;\nDigits := 10:" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#^$$\"0Tww8uHy&!#=$\"0\\d4pa9M\"!#9" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#^$$\"0n`DPBC4$!#=$\"0ab#[U\\r8!#9" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#^$$!0UEf)*z,W%!#>$\"0W'Hb%)Q,9!#9" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#^$$!0y68#HS^\\!#=$\"00d\"))*G6V\"!#9" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 107 "Then we apply the bisection me thod to calculate the parameter value associated with the intersection point." }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 176 "real_part := pro c(u)\n Re(newton(R(z)=exp(u*Pi*I),z=1.4*I))\nend proc:\nDigits := 15 :\nu0 := bisect('real_part'(u),u=0.43..0.46);\nnewton(R(z)=exp(u0*Pi*I ),z=1.4*I);\nDigits := 10:" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%#u0G$ \"0e`v,[()[%!#:" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#^#$\"0,BVGK!)R\"!#9 " }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 4 "The \+ " }{TEXT 260 112 "largest interval on the nonnegative imaginary axis t hat contains the origin and lies inside the stability region" }{TEXT -1 36 " is [ 0, 1.3980 ] (approximately)." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 23 "-newton(R(z)=1,z=- 3)/2;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#$\"+lA.)R\"!\"*" }}}{PARA 0 " " 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT 260 4 "Note" }{TEXT -1 27 ": The stability radius is " }{XPPEDIT 18 0 "rho" "6#%$rhoG" } {TEXT -1 2 " " }{TEXT 276 1 "~" }{TEXT -1 54 " 1.398032265 while th e imaginary axis inclusion is " }{XPPEDIT 18 0 "rho[2]" "6#&%$rhoG6# \"\"#" }{TEXT -1 2 " " }{TEXT 277 1 "~" }{TEXT -1 19 " 1.398032284, \+ so " }{XPPEDIT 18 0 "rho" "6#%$rhoG" }{TEXT -1 2 " " }{TEXT 278 1 "~ " }{TEXT -1 2 " " }{XPPEDIT 18 0 "rho[2]" "6#&%$rhoG6#\"\"#" }{TEXT -1 14 " as required." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 37 "#------------------------------------" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 80 "The stability function R* for the 7 stage, order 4 scheme is given as follows." }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 97 "subs(ee,subs(b=`b*`,StabilityFuncti on(4,7,'expanded'))):\n`R*` := unapply(%,z):\n'`R*`(z)'=`R*`(z);" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#/-%#R*G6#%\"zG,2\"\"\"F)F'F)*&#F)\"\"# F)*$)F'F,F)F)F)*&#F)\"\"'F)*$)F'\"\"$F)F)F)*&#F)\"#CF)*$)F'\"\"%F)F)F) *&#\"-[\")eq(*F)*$)F'\"\"&F)F)F)*&#\".xepX8@\"\"0+'4uSH&)[ F)*$)F'F1F)F)F)*&#\"$.\"\"(S%)R\"F)*$)F'\"\"(F)F)F)" }}}{PARA 0 "" 0 " " {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 124 "We can find the point \+ where the boundary of the stability region intersects the negative rea l axis by solving the equation: " }}{PARA 256 "" 0 "" {TEXT -1 1 " " } {XPPEDIT 18 0 "`R*`(z) = 1;" "6#/-%#R*G6#%\"zG\"\"\"" }{TEXT -1 2 ". \+ " }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 32 "z_0 := newton(`R*`(z)=1, z=-3.2);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%$z_0G$!+$y:G;$!\"*" }}} {PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 320 "z_0 := newton(`R*`(z)=1,z=-3.2):\np_1 := plot([`R*`(z),1],z=-3.69 ..0.49,color=[red,blue]):\np_2 := plot([[[z_0,1]]$3],style=point,symbo l=[circle,cross,diamond],color=black):\np_3 := plot([[z_0,0],[z_0,1]], linestyle=3,color=COLOR(RGB,0,.5,0)):\nplots[display]([p_1,p_2,p_3],vi ew=[-3.69..0.49,-0.07..1.47],font=[HELVETICA,9]);" }}{PARA 13 "" 1 "" {GLPLOT2D 411 218 218 {PLOTDATA 2 "6+-%'CURVESG6$7X7$$!3%************* *o$!#<$\"3%R]U(z[2#f#F*7$$!3Om;zZRWWOF*$\"35NF *$\"3'**Hp`kF)H>F*7$$!3')\\i+xX.vMF*$\"3S$e-P\"yu#y\"F*7$$!3UL$3LTd/V$ F*$\"3Q+O\"*GaYX;F*7$$!37LLBjZe&Q$F*$\"3?JF`N=l;:F*7$$!3FL$eJ672M$F*$ \"3U\"p=7a\\nR\"F*7$$!31]i&\\t_gH$F*$\"3'GTv]!)*y&G\"F*7$$!3'o;anN$R^K F*$\"3w8*\\7)*GF=\"F*7$$!3eL3(fV$eoJF*$\"3MDZC$pn5,\"F*7$$!33+v`f&QG3$ F*$\"3wfIODf4w&)!#=7$$!3;L3n&3hT*HF*$\"3p5)=]&)o5A(Fco7$$!3/+ve,!od!HF *$\"3+M&\\Y!)R)ygFco7$$!3rmmr\\[%[\"GF*$\"3u()pvSxg&4&Fco7$$!3OL$e!)of 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" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 " " {MPLTEXT 1 0 1531 "`R*` := z -> 1+z+1/2*z^2+1/6*z^3+1/24*z^4 +\n \+ 921239267717/97705881481920*z^5+1211345695877/488529407409600*z^6+1 03/1398440*z^7:\npts := []: z0 := 0: tt := 0: \nwhile tt<=161/20 do\n \+ zz := newton(`R*`(z)=exp(tt*Pi*I),z=z0):\n z0 := zz:\n if (9/20< =tt and tt<=21/20) \n or (137/20<=tt and tt<=15/2) then\n hh \+ := 1/40\n else \n hh := 1/20\n end if;\n tt := tt+hh;\n p ts := [op(pts),[Re(zz),Im(zz)]]:\nend do:\np_1 := plot(pts,color=COLOR (RGB,.38,.07,.1)):\np_2 := plots[polygonplot]([seq([pts[i-1],pts[i],[- 1.6,0]],i=2..nops(pts))],\n style=patchnogrid,color=COLOR(RGB ,.75,.14,.2)):\npts := []: z0 := 1.4+3.6*I:\nfor ct from 0 to 60 do\n \+ zz := newton(`R*`(z)=exp(ct*Pi/30*I),z=z0):\n z0 := zz:\n pts := [op(pts),[Re(zz),Im(zz)]]:\nend do:\np_3 := plot(pts,color=COLOR(RGB, .38,.07,.1)):\np_4 := plots[polygonplot]([seq([pts[i-1],pts[i],[1.37,3 .39]],i=2..nops(pts))],\n style=patchnogrid,color=COLOR(RGB,. 75,.14,.2)):\npts := []: z0 := 1.4-3.6*I:\nfor ct from 0 to 60 do\n \+ zz := newton(`R*`(z)=exp(ct*Pi/30*I),z=z0):\n z0 := zz:\n pts := [ op(pts),[Re(zz),Im(zz)]]:\nend do:\np_5 := plot(pts,color=COLOR(RGB,.3 8,.07,.1)):\np_6 := plots[polygonplot]([seq([pts[i-1],pts[i],[1.37,-3. 39]],i=2..nops(pts))],\n style=patchnogrid,color=COLOR(RGB,.7 5,.14,.2)):\np_7 := plot([[[-3.79,0],[1.99,0]],[[0,-3.89],[0,3.89]]],c olor=black,linestyle=3):\nplots[display]([p_||(1..7)],view=[-3.79..1.9 9,-3.89..3.89],font=[HELVETICA,9],\n color=COLOR(RGB,.4,0,1),labels= [`Re(z)`,`Im(z)`],axes=boxed,scaling=constrained);" }}{PARA 13 "" 1 " " {GLPLOT2D 486 581 581 {PLOTDATA 2 "60-%'CURVESG6$7gv7$$\"\"!F)F(7$$! 3x*****4MeNT#!#F$\"3-+++k_zq:!#=7$$!3y*****H.[jz%!#D$\"34+++rrbTJF07$$ !3'******4&*H2:\"!#B$\"31+++l?57ZF07$$!3'******4`\"*f5\"!#A$\"3M+++*G# )=G'F07$$!3%*******Q@!*fjF@$\"3#*******)o$o\\yF07$$!3F+++4$QUi#!#@$\"3 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"Curve 9" "Curve 10" "Curve 11" "Curve 12" "Curve 13" "Cur ve 14" }}}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}}{PARA 0 "" 0 "" {TEXT -1 39 "#----------------- ---------------------" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" } }}}{PARA 0 "" 0 "" {TEXT -1 55 "#===================================== =================" }}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 15 "A scheme wit h " }{XPPEDIT 18 0 "t[6] = 1/500;" "6#/&%\"tG6#\"\"'*&\"\"\"F)\"$+&! \"\"" }{TEXT -1 34 " and minimum principal error norm" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}{PARA 0 "" 0 "" {TEXT -1 39 "# --------------------------------------" }}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 28 "checking the combined scheme" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 35 "coefficients of the combined scheme" }}{PARA 0 " " 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1111 "ee \+ := \{c[2]=34/195,\nc[3]=137/525,\nc[4]=252/365,\nc[5]=31/40,\nc[6]=1, \nc[7]=1,\n\na[2,1]=34/195,\na[3,1]=82063/1249500,\na[3,2]=243997/1249 500,\na[4,1]=101345382936/124124821241,\na[4,2]=-14189381388/453010296 5,\na[4,3]=21949459056/7301460073,\na[5,1]=-2095505049916883/707267524 239360,\na[5,2]=498515020731/40972513280,\na[5,3]=-2529970133596857/27 1730069595136,\na[5,4]=6431020141871/7296710270976,\na[6,1]=7897134155 27611/1013168913843996,\na[6,2]=-743202495/291283372,\na[6,3]=62545595 7178350900/271098296009435219,\na[6,4]=21511957375165/104104742407029, \na[6,5]=1735464775680/6719560297073,\na[7,1]=990209/12842928,\na[7,2] =0,\na[7,3]=3027623484375/7555589800144,\na[7,4]=469280932525/13891867 38576,\na[7,5]=43980800/446349501,\na[7,6]=406471/4735152,\n\nb[1]=990 209/12842928,\nb[2]=0,\nb[3]=3027623484375/7555589800144,\nb[4]=469280 932525/1389186738576,\nb[5]=43980800/446349501,\nb[6]=406471/4735152, \n\n`b*`[1]=2124247802213/27446171926320,\n`b*`[2]=0,\n`b*`[3]=6147225 8224069875/153778919202330832,\n`b*`[4]=203730456418295501/59375647149 4983888,\n`b*`[5]=17760294550016/190775579270913,\n`b*`[6]=53945083294 7/10119327608880,\n`b*`[7]=1/30\}:" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" } }{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 137 "subs(ee,mat rix([seq([c[i],seq(a[i,j],j=1..i-1),``$(8-i)],i=2..7),\n[`b`,seq(b[i], i=1..6),``],[`b*`,seq(`b*`[i],i=1..7)]]));\n``;\nevalf(%%);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#K%'matrixG6#7*7*#\"#M\"$&>F(%!GF+F+F+F+F+7*# \"$P\"\"$D&#\"&j?)\"(+&\\7#\"'(*RCF2F+F+F+F+F+7*#\"$_#\"$l$#\"-OHQX85 \"-T7#[7C\"#!,)Q\"Q*=9\"+lH5IX#\",c!f%\\>#\"+t+Y,tF+F+F+F+7*#\"#J\"#S# !1$)o\"*\\]]&4#\"0g$RU_nsq#\"-J2-:&)\\\",!G8D(4%#!1dofL,(*HD\"0O^fp+tr ##\".r=9?5V'\".w4F5nH(F+F+F+7*\"\"\"#\"06w_:Mr*y\"1'*R%Q\"*oJ,\"#!*&\\ -Ku\"*sLG\"H#\"3+4Nyr&fXD'\"3>_V4gH)4r##\"/l^Pd>^@\"0HqSUZ5/\"#\".!ovZ YN<\".tqHg&>nF+F+7*FS#\"'4-**\")GH%G\"\"\"!#\".vV[Bw-$\".W,!)*ebv#\"-D D$4Gp%\".w&Qn=*Q\"#\")+3)R%\"*,&\\jW#\"'rkS\"(_^t%F+7*%\"bGF^oFaoFboFe oFhoF[pF+7*%#b*G#\".8A!yCC@\"/?j#>*yP:#\" 3,bH=kXIP?\"3))Q)\\\\rkv$f#\"/;+b%Hgx\"\"084Fzbx!>#\"-ZH$3XR&\"/!))3wK >,\"#FS\"#IQ)pprint146\"" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%!G" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#K%'matrixG6#7*7*$\"+W(*eVF*F+F+F+F+F+7*$\"+*e4T!pF*$\"+ #)ezk\")F*$!+yDCKJ!\"*$\"+[R<1IF;F+F+F+F+7*$\"++++]xF*$!+R\"=G'HF;$\"+ %)fq;7!\")$!+V0g5$*F;$\"+d%)e8))F*F+F+F+7*$\"\"\"\"\"!$\"+*>*[%z(F*$!+ qeZ^DF;$\"+L(=rI#F;$\"+8jPm?F*$\"+geq#e#F*F+F+7*FK$\"+(*)\\,r(F1$FMFM$ \"+o082SF*$\"+v#)4yLF*$\"+?WW`)*F1$\"+ds6%e)F1F+7*%\"bGFYFenFfnFhnFjnF \\oF+7*%#b*G$\"+1qoRxF1Fen$\"+oPW(*RF*$\"+%R77V$F*$\"+9?_4$*F1$\"+Ag*3 L&F1$\"+LLLLLF1Q)pprint156\"" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 146 "RK5_6eqs := [op(RowSumCondi tions(6,'expanded')),op(OrderConditions(5,6,'expanded'))]:\n`RK4_7eqs* ` := subs(b=`b*`,OrderConditions(4,7,'expanded')):" }}}{PARA 0 "" 0 " " {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 7 "Check: " }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 148 "simplify(subs(ee,RK5_6eqs)):\nmap(u->`if `(lhs(u)=rhs(u),0,1),%);\nnops(%);\nsimplify(subs(ee,`RK4_7eqs*`)):\nm ap(u->`if`(lhs(u)=rhs(u),0,1),%);\nnops(%);" }{TEXT -1 0 "" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#78\"\"!F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F $F$F$" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#\"#A" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#7*\"\"!F$F$F$F$F$F$F$" }}{PARA 11 "" 1 "" {XPPMATH 20 " 6#\"\")" }}}{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." }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 123 "errterms5_6 := PrincipalErrorTerms(5,6,'expanded'):\nsqrt(add(sub s(ee,errterms5_6[i])^2,i=1.. nops(errterms5_6))):\nevalf(%);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#$\"+@Bfw')!#8" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 87 "We can also calculate the 2 norm o f the principal error of the order 4 embedded scheme." }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 145 "`errterms4_7*` := subs(b=`b*`,PrincipalE rrorTerms(4,7,'expanded')):\nsqrt(add(subs(ee,`errterms4_7*`[i])^2,i=1 .. nops(`errterms4_7*`))):\nevalf(%);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#$\"+4C()y6!#7" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 86 "Next we set-up stage-order condtions to check for stage-o rders from 2 to 5 inclusive. " }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 85 "for ct from 2 to 4 do\n so||ct||_6 := StageOrderConditions(ct, 6,'expanded');\nend do:" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 " " 0 "" {TEXT -1 58 "Stages 3 to 6 have the following respective stage- orders. " }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 173 "[seq([seq(expan d(subs(ee,so||i||_6[j])),i=2..4)],j=1..4)]:\nmap(proc(L) local i; for \+ i to nops(L) do if not evalb(L[i]) then break end if end do; i end pro c,%):\nsimplify(%); " }}{PARA 11 "" 1 "" {XPPMATH 20 "6#7&\"\"#F$F$F$ " }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 27 "The \+ simplifying conditions:" }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "Sum(b[i]*a[i,j],i = j+1 .. 6) = b[j]*(1-c[j])" "6#/-%$SumG6$*&&% \"bG6#%\"iG\"\"\"&%\"aG6$F+%\"jGF,/F+;,&F0F,F,F,\"\"'*&&F)6#F0F,,&F,F, &%\"cG6#F0!\"\"F," }{TEXT -1 3 ", " }{XPPEDIT 18 0 "j=1" "6#/%\"jG\" \"\"" }{TEXT -1 7 " . . 5 " }}{PARA 0 "" 0 "" {TEXT -1 8 "(where " } {XPPEDIT 18 0 "c[1]=0" "6#/&%\"cG6#\"\"\"\"\"!" }{TEXT -1 17 " ) are s atisfied." }}{PARA 256 "" 0 "" {TEXT -1 1 " " }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 154 "[Sum(b[i]*a[i,1],i=1+1..6)=b[1],seq(Sum(b[i]*a[i, j],i=j+1..6)=b[j]*(1-c[j]),j=2..5)];\neval(subs(Sum=add,%)):\nsubs(ee, %);\nmap(u->lhs(u)-rhs(u),%);\nnops(%);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#7'/-%$SumG6$*&&%\"bG6#%\"iG\"\"\"&%\"aG6$F,F-F-/F,;\"\"#\"\"'&F* 6#F-/-F&6$*&F)F-&F/6$F,F3F-/F,;\"\"$F4*&&F*6#F3F-,&F-F-&%\"cGFB!\"\"F- /-F&6$*&F)F-&F/6$F,F?F-/F,;\"\"%F4*&&F*6#F?F-,&F-F-&FEFRFFF-/-F&6$*&F) F-&F/6$F,FOF-/F,;\"\"&F4*&&F*6#FOF-,&F-F-&FEFjnFFF-/-F&6$*&F)F-&F/6$F, FgnF-/F,;F4F4*&&F*6#FgnF-,&F-F-&FEFgoFFF-" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#7'/#\"'4-**\")GH%G\"F%/\"\"!F)/#\"+v=!pw&\",)epJZ>F+/# \"+&=,dG\"\",_zo$H7F//#\"(?&*4\"\")*Q%f\\F3" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#7'\"\"!F$F$F$F$" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#\"\" &" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 27 "The simplifying condition: " }}{PARA 256 "" 0 "" {TEXT -1 1 " " } {XPPEDIT 18 0 "Sum(b[i]*a[i,2],i = 3 .. 6)=0" "6#/-%$SumG6$*&&%\"bG6#% \"iG\"\"\"&%\"aG6$F+\"\"#F,/F+;\"\"$\"\"'\"\"!" }{TEXT -1 1 " " }} {PARA 0 "" 0 "" {TEXT -1 13 "is satisfied." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 59 "Sum(b[i]*a[i,2],i= 3..6);\neval(subs(Sum=add,%));\nsubs(ee,%);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#-%$SumG6$*&&%\"bG6#%\"iG\"\"\"&%\"aG6$F*\"\"#F+/F*;\"\" $\"\"'" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#,**&&%\"bG6#\"\"$\"\"\"&%\"a G6$F(\"\"#F)F)*&&F&6#\"\"%F)&F+6$F1F-F)F)*&&F&6#\"\"&F)&F+6$F7F-F)F)*& &F&6#\"\"'F)&F+6$F=F-F)F)" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#\"\"!" }} }{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 27 "The simp lifying condition: " }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "Sum(b[i]*Sum(a[i,j]*a[j,2],j = 3 .. i-1),i = 2 .. 6)=0" "6#/-%$SumG 6$*&&%\"bG6#%\"iG\"\"\"-F%6$*&&%\"aG6$F+%\"jGF,&F16$F3\"\"#F,/F3;\"\"$ ,&F+F,F,!\"\"F,/F+;F6\"\"'\"\"!" }{TEXT -1 1 " " }}{PARA 0 "" 0 "" {TEXT -1 13 "is satisfied." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 82 "Sum(b[i]*Sum(a[i,j]*a[j,2],j=3..i-1 ),i =2 ..6);\neval(subs(Sum=add,%));\nsubs(ee,%);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#-%$SumG6$*&&%\"bG6#%\"iG\"\"\"-F$6$*&&%\"aG6$F*%\"jGF+& F06$F2\"\"#F+/F2;\"\"$,&F*F+F+!\"\"F+/F*;F5\"\"'" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#,(*(&%\"bG6#\"\"%\"\"\"&%\"aG6$F(\"\"$F)&F+6$F-\"\"#F)F )*&&F&6#\"\"&F),&*&&F+6$F4F-F)F.F)F)*&&F+6$F4F(F)&F+6$F(F0F)F)F)F)*&&F &6#\"\"'F),(*&&F+6$FAF-F)F.F)F)*&&F+6$FAF(F)F " 0 "" {MPLTEXT 1 0 64 "Sum(b[i]*c[i]*a[i,2],i=3..6) ;\neval(subs(Sum=add,%));\nsubs(ee,%);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#-%$SumG6$*(&%\"bG6#%\"iG\"\"\"&%\"cGF)F+&%\"aG6$F*\"\"#F+/F*;\" \"$\"\"'" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#,**(&%\"bG6#\"\"$\"\"\"&% \"cGF'F)&%\"aG6$F(\"\"#F)F)*(&F&6#\"\"%F)&F+F2F)&F-6$F3F/F)F)*(&F&6#\" \"&F)&F+F9F)&F-6$F:F/F)F)*(&F&6#\"\"'F)&F+F@F)&F-6$FAF/F)F)" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#\"\"!" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }} {PARA 0 "" 0 "" {TEXT -1 78 "None of the 20 principal error conditions are satisfied by the order 5 scheme." }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 123 "RK5_6err_eqs := PrincipalErrorConditions(5,6,'expand ed'):\nexpand(subs(ee,RK5_6err_eqs)):\nmap(u->`if`(lhs(u)=rhs(u),0,1), %);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#76\"\"\"F$F$F$F$F$F$F$F$F$F$F$F $F$F$F$F$F$F$F$" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 94 "None of the 9 principal error conditions are satisfied by the 7 stage embedded order 4 scheme." }}{PARA 0 "" 0 "" {TEXT -1 98 " The largest magnitude of an error term is about 645.5 times the lowest magnitude of an error term." }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 147 "`errterms4_7*` := subs(b=`b*`,PrincipalErrorTerms(4,7,'expanded') ):\nsubs(ee,`errterms4_7*`):\nmap(abs,evalf[8](%));\nevalf[4](max(op(% ))/min(op(%)));" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#7+$\")kv36!#5$\")`7 6C!#6$\")z&>*HF)$\")i61y!#7$\")qBb^!#8$\")(f5c(F1$\")u/J5F.$\")W3P!)F. $\")BT= " 0 "" {MPLTEXT 1 0 1 ";" }} }}{PARA 0 "" 0 "" {TEXT -1 39 "#-------------------------------------- " }}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 26 "determination of the nodes" } }{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 2 "e2" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 10256 "e2 := \{c [7] = 1, a[6,2] = 1/2*(-15*c[4]*c[3]^2+30*c[5]*c[4]*c[3]^2-30*c[3]*c[4 ]*c[5]+16*c[4]*c[3]-5*c[3]*c[5]+3*c[3]+10*c[4]*c[5]-6*c[4])/(-12+15*c[ 3]+15*c[4]+15*c[5]-20*c[4]*c[3]-20*c[4]*c[5]+30*c[3]*c[4]*c[5]-20*c[3] *c[5])/c[3]/c[2], c[6] = 1, a[2,1] = c[2], a[5,4] = 1/2*(c[4]*c[3]-c[3 ]*c[5]-c[4]*c[5]+c[5]^2)*c[5]*(5*c[3]-2)/(c[3]-c[4])/(3-5*c[3]+10*c[4] *c[3]-5*c[4])/c[4], `b*`[2] = 0, b[2] = 0, b[4] = 1/60*(10*c[3]*c[5]-5 *c[3]-5*c[5]+3)/(-c[4]^2+c[4]+c[4]*c[3]-c[3])/c[4]/(c[4]-c[5]), a[7,1] = 1/60*(5*c[4]-10*c[4]*c[5]+30*c[3]*c[4]*c[5]-10*c[4]*c[3]-10*c[3]*c[ 5]-3+5*c[3]+5*c[5])/c[3]/c[4]/c[5], a[6,4] = 1/2*(6-30*c[5]^2*c[3]+20* c[5]^2*c[3]^2-10*c[5]^2*c[4]-16*c[3]-8*c[4]-14*c[5]-7*c[4]^2*c[3]+30*c [5]^2*c[3]*c[4]+25*c[5]*c[4]*c[3]^2-20*c[5]^2*c[4]*c[3]^2+23*c[4]*c[3] +14*c[4]*c[5]-39*c[3]*c[4]*c[5]+39*c[3]*c[5]-15*c[4]*c[3]^2+10*c[3]^2+ 2*c[4]^2+10*c[5]^2-25*c[3]^2*c[5]+5*c[4]^2*c[3]^2)/c[4]/(12*c[4]^2-12* c[4]*c[3]-15*c[4]^3-12*c[4]*c[5]+12*c[3]*c[5]-20*c[4]^2*c[5]^2+20*c[4] ^3*c[5]+20*c[4]^3*c[3]-20*c[4]^2*c[3]^2+20*c[5]^2*c[3]^2-20*c[4]^2*c[5 ]*c[3]+15*c[4]*c[3]^2+15*c[5]^2*c[4]+30*c[5]^2*c[4]^2*c[3]+30*c[5]*c[4 ]^2*c[3]^2-30*c[5]^2*c[4]*c[3]^2-30*c[5]*c[4]^3*c[3]+15*c[3]*c[4]*c[5] -15*c[3]^2*c[5]-15*c[5]^2*c[3]), `b*`[1] = 1/60*(-460*c[5]*c[4]*c[3]*` b*`[7]+760*c[5]*c[4]*c[3]^2*`b*`[7]+3*c[3]-6*c[4]-20*c[4]^2*c[5]-60*c[ 4]^2*c[3]-1800*`b*`[7]*c[4]^2*c[5]*c[3]^2+100*c[5]*c[4]*c[3]^2+100*c[3 ]^2*c[5]*`b*`[7]-100*c[3]^3*c[5]*c[4]+300*c[3]^3*c[4]*`b*`[7]-600*c[3] ^3*`b*`[7]*c[4]^2+940*`b*`[7]*c[4]^2*c[3]^2-60*c[5]*c[3]*`b*`[7]-560*c [4]^2*c[3]*`b*`[7]+300*c[5]*c[4]^2*c[3]^3-100*c[4]^2*c[3]^3+140*c[4]^2 *c[5]*c[3]+120*`b*`[7]*c[4]^2+120*c[5]*c[4]*`b*`[7]-340*c[5]*c[4]^2*c[ 3]^2-200*`b*`[7]*c[4]^2*c[5]+1000*c[5]*c[4]^2*c[3]*`b*`[7]-600*c[3]^3* c[5]*c[4]*`b*`[7]+29*c[4]*c[3]-380*`b*`[7]*c[4]*c[3]^2+252*`b*`[7]*c[4 ]*c[3]+10*c[4]*c[5]-50*c[3]*c[4]*c[5]+1200*c[3]^3*`b*`[7]*c[4]^2*c[5]+ 36*`b*`[7]*c[3]-72*`b*`[7]*c[4]-60*`b*`[7]*c[3]^2-5*c[3]*c[5]-60*c[4]* c[3]^2-5*c[3]^2+10*c[4]^2+10*c[3]^2*c[5]+50*c[4]*c[3]^3+130*c[4]^2*c[3 ]^2)/(10*c[4]*c[3]^2-8*c[4]*c[3]-c[3]+2*c[4])/c[3]/c[4]/c[5], a[6,1] = 1/2*(10*c[3]*c[2]*c[5]^2-14*c[3]*c[2]*c[5]-30*c[3]^2*c[2]*c[5]^2-25*c [3]^3*c[2]*c[5]+10*c[3]^3*c[2]-108*c[3]^2*c[2]*c[4]*c[5]-3*c[5]*c[4]*c [3]^2+5*c[5]^2*c[4]*c[3]^2-30*c[5]^2*c[4]^2*c[3]^3+30*c[5]^2*c[4]^2*c[ 3]^2-10*c[5]^2*c[4]^2*c[3]+15*c[5]*c[4]^2*c[3]^3+6*c[4]^2*c[5]*c[3]-30 *c[3]^2*c[2]*c[4]^2-30*c[3]^3*c[4]*c[2]+10*c[3]*c[2]*c[4]^2-16*c[3]*c[ 2]*c[4]-16*c[5]*c[4]^2*c[3]^2+39*c[3]*c[2]*c[4]*c[5]-30*c[3]*c[2]*c[5] ^2*c[4]-9*c[3]*c[2]*c[5]*c[4]^2+6*c[3]*c[2]+10*c[2]*c[4]^2*c[5]^2-10*c [3]*c[2]*c[4]^2*c[5]^2-60*c[3]^3*c[2]*c[5]*c[4]^2-40*c[3]^2*c[2]*c[4]^ 2*c[5]^2+46*c[3]^2*c[2]*c[4]-16*c[2]*c[3]^2-6*c[2]*c[5]*c[4]^2+39*c[3] ^2*c[2]*c[5]+60*c[2]*c[4]^2*c[5]^2*c[3]^3+20*c[3]^3*c[2]*c[4]^2+20*c[3 ]^3*c[2]*c[5]^2-60*c[3]^3*c[2]*c[5]^2*c[4]+75*c[3]^3*c[2]*c[4]*c[5]+80 *c[3]^2*c[2]*c[5]^2*c[4]+65*c[3]^2*c[2]*c[5]*c[4]^2)/(-12+15*c[3]+15*c [4]+15*c[5]-20*c[4]*c[3]-20*c[4]*c[5]+30*c[3]*c[4]*c[5]-20*c[3]*c[5])/ c[3]^2/c[2]/c[4]/c[5], a[7,2] = 0, b[3] = -1/60*(10*c[4]*c[5]-5*c[4]-5 *c[5]+3)/(c[3]-1)/c[3]/(c[3]-c[5])/(c[3]-c[4]), a[6,3] = -1/2*(10*c[5] ^2*c[3]-10*c[5]^2*c[3]^2+6*c[3]-6*c[4]^2*c[5]+16*c[4]^2*c[3]-40*c[5]^2 *c[3]*c[4]-45*c[5]*c[4]*c[3]^2+60*c[5]^2*c[4]*c[3]^2-20*c[3]^3*c[5]*c[ 4]-10*c[5]^2*c[4]^2*c[3]-20*c[5]*c[4]^2*c[3]^3+15*c[4]^2*c[3]^3-19*c[4 ]^2*c[5]*c[3]+10*c[4]^2*c[5]^2+45*c[5]*c[4]^2*c[3]^2-10*c[3]^4*c[4]-20 *c[4]*c[5]^2*c[3]^3-16*c[4]*c[3]+45*c[3]*c[4]*c[5]+20*c[3]^4*c[5]*c[4] -14*c[3]*c[5]+17*c[4]*c[3]^2-8*c[3]^2+2*c[3]^3+14*c[3]^2*c[5]+9*c[4]*c [3]^3-31*c[4]^2*c[3]^2)/(-30*c[5]^2*c[3]*c[4]+20*c[5]^2*c[4]-15*c[4]*c [5]+15*c[4]*c[3]+30*c[5]*c[4]*c[3]^2-20*c[4]*c[3]^2+15*c[3]^2+12*c[5]- 12*c[3]+20*c[5]^2*c[3]-15*c[5]^2-20*c[3]^2*c[5])/(c[3]-c[4])/c[3]^2, a [4,1] = 1/2*c[4]*(2*c[2]*c[3]^2-3*c[4]*c[3]^2+2*c[4]^2*c[3]+2*c[3]*c[2 ]*c[4]-2*c[2]*c[4]^2)/c[2]/c[3]^2, a[5,3] = -1/2*c[5]*(-2*c[5]^2*c[3]+ 6*c[4]^2*c[5]-6*c[4]^2*c[3]+15*c[5]^2*c[3]*c[4]-5*c[5]*c[4]*c[3]^2-20* c[5]^2*c[4]*c[3]^2+20*c[3]^3*c[5]*c[4]+20*c[5]^2*c[4]^2*c[3]-5*c[4]^2* c[5]*c[3]-10*c[4]^2*c[5]^2-20*c[5]*c[4]^2*c[3]^2-4*c[3]*c[4]*c[5]+4*c[ 4]*c[3]^2+2*c[3]^2*c[5]-10*c[4]*c[3]^3+15*c[4]^2*c[3]^2)/(3-5*c[3]+10* c[4]*c[3]-5*c[4])/c[3]^2/(c[3]-c[4]), a[4,3] = -c[4]^2*(c[3]-c[4])/c[3 ]^2, a[5,2] = 1/2*c[5]*(-15*c[4]*c[3]^2+30*c[5]*c[4]*c[3]^2-20*c[5]^2* c[3]*c[4]+6*c[4]*c[3]-5*c[5]^2*c[3]+3*c[3]*c[5]-6*c[4]*c[5]+10*c[5]^2* c[4])/c[2]/(3-5*c[3]+10*c[4]*c[3]-5*c[4])/c[3], `b*`[5] = 1/60*(-460*c [5]*c[4]*c[3]*`b*`[7]+760*c[5]*c[4]*c[3]^2*`b*`[7]+3*c[3]-6*c[4]-60*c[ 4]^2*c[3]-1800*`b*`[7]*c[4]^2*c[5]*c[3]^2+100*c[3]^2*c[5]*`b*`[7]+300* c[3]^3*c[4]*`b*`[7]-600*c[3]^3*`b*`[7]*c[4]^2+940*`b*`[7]*c[4]^2*c[3]^ 2-60*c[5]*c[3]*`b*`[7]-560*c[4]^2*c[3]*`b*`[7]-100*c[4]^2*c[3]^3+120*` b*`[7]*c[4]^2+120*c[5]*c[4]*`b*`[7]-200*`b*`[7]*c[4]^2*c[5]+1000*c[5]* c[4]^2*c[3]*`b*`[7]-600*c[3]^3*c[5]*c[4]*`b*`[7]+29*c[4]*c[3]-380*`b*` [7]*c[4]*c[3]^2+252*`b*`[7]*c[4]*c[3]+1200*c[3]^3*`b*`[7]*c[4]^2*c[5]+ 36*`b*`[7]*c[3]-72*`b*`[7]*c[4]-60*`b*`[7]*c[3]^2-60*c[4]*c[3]^2-5*c[3 ]^2+10*c[4]^2+50*c[4]*c[3]^3+130*c[4]^2*c[3]^2)/c[5]/(c[5]^2*c[3]-c[5] ^3*c[3]+c[5]^2*c[3]^2-2*c[5]^2*c[4]+2*c[5]^3*c[4]+2*c[4]^2*c[5]-2*c[4] ^2*c[3]+7*c[5]^2*c[3]*c[4]-8*c[5]^3*c[4]*c[3]-9*c[5]*c[4]*c[3]^2-2*c[5 ]^2*c[4]*c[3]^2+10*c[5]^3*c[4]*c[3]^2+10*c[3]^3*c[5]*c[4]-10*c[5]^2*c[ 4]^2*c[3]^2+8*c[5]^2*c[4]^2*c[3]+10*c[5]*c[4]^2*c[3]^3-10*c[4]^2*c[3]^ 3-6*c[4]^2*c[5]*c[3]-2*c[4]^2*c[5]^2+2*c[5]*c[4]^2*c[3]^2-10*c[4]*c[5] ^2*c[3]^3+c[3]*c[4]*c[5]+c[4]*c[3]^2-c[3]^2*c[5]+8*c[4]^2*c[3]^2), a[4 ,2] = 1/2*(3*c[3]-2*c[4])*c[4]^2/c[3]/c[2], `b*`[6] = -1/60*(-1000*c[5 ]*c[4]*c[3]*`b*`[7]+1780*c[5]*c[4]*c[3]^2*`b*`[7]-12*c[3]+24*c[4]+40*c [4]^2*c[5]+160*c[4]^2*c[3]-2880*`b*`[7]*c[4]^2*c[5]*c[3]^2-280*c[5]*c[ 4]*c[3]^2+160*c[3]^2*c[5]*`b*`[7]+200*c[3]^3*c[5]*c[4]+900*c[3]^3*c[4] *`b*`[7]-1200*c[3]^3*`b*`[7]*c[4]^2+2020*`b*`[7]*c[4]^2*c[3]^2-120*c[5 ]*c[3]*`b*`[7]-1160*c[4]^2*c[3]*`b*`[7]-300*c[5]*c[4]^2*c[3]^3+200*c[4 ]^2*c[3]^3-220*c[4]^2*c[5]*c[3]+240*`b*`[7]*c[4]^2+240*c[5]*c[4]*`b*`[ 7]+440*c[5]*c[4]^2*c[3]^2-320*`b*`[7]*c[4]^2*c[5]+1600*c[5]*c[4]^2*c[3 ]*`b*`[7]-1200*c[3]^3*c[5]*c[4]*`b*`[7]-111*c[4]*c[3]-1400*`b*`[7]*c[4 ]*c[3]^2+792*`b*`[7]*c[4]*c[3]-30*c[4]*c[5]+140*c[3]*c[4]*c[5]+1800*c[ 3]^3*`b*`[7]*c[4]^2*c[5]+96*`b*`[7]*c[3]-192*`b*`[7]*c[4]-120*`b*`[7]* c[3]^2+15*c[3]*c[5]+220*c[4]*c[3]^2+15*c[3]^2-30*c[4]^2-20*c[3]^2*c[5] -150*c[4]*c[3]^3-310*c[4]^2*c[3]^2)/(-10*c[4]*c[3]^2+10*c[5]*c[4]*c[3] ^2+8*c[4]*c[3]-8*c[3]*c[4]*c[5]+c[3]-c[3]*c[5]-2*c[4]+2*c[4]*c[5])/(c[ 4]-1)/(c[3]-1), `b*`[3] = 1/60*(-460*c[5]*c[4]*c[3]*`b*`[7]+760*c[5]*c [4]*c[3]^2*`b*`[7]+3*c[3]-6*c[4]-20*c[4]^2*c[5]-40*c[4]^2*c[3]-1800*`b *`[7]*c[4]^2*c[5]*c[3]^2+50*c[5]*c[4]*c[3]^2+100*c[3]^2*c[5]*`b*`[7]+3 00*c[3]^3*c[4]*`b*`[7]-600*c[3]^3*`b*`[7]*c[4]^2+940*`b*`[7]*c[4]^2*c[ 3]^2-60*c[5]*c[3]*`b*`[7]-560*c[4]^2*c[3]*`b*`[7]+80*c[4]^2*c[5]*c[3]+ 120*`b*`[7]*c[4]^2+120*c[5]*c[4]*`b*`[7]-100*c[5]*c[4]^2*c[3]^2-200*`b *`[7]*c[4]^2*c[5]+1000*c[5]*c[4]^2*c[3]*`b*`[7]-600*c[3]^3*c[5]*c[4]*` b*`[7]+19*c[4]*c[3]-380*`b*`[7]*c[4]*c[3]^2+252*`b*`[7]*c[4]*c[3]+10*c [4]*c[5]-30*c[3]*c[4]*c[5]+1200*c[3]^3*`b*`[7]*c[4]^2*c[5]+36*`b*`[7]* c[3]-72*`b*`[7]*c[4]-60*`b*`[7]*c[3]^2-5*c[3]*c[5]-30*c[4]*c[3]^2+10*c [4]^2+50*c[4]^2*c[3]^2)/(c[3]-1)/c[3]/(-10*c[5]*c[4]*c[3]^2+8*c[3]*c[4 ]*c[5]-2*c[4]*c[5]+2*c[4]*c[3]+10*c[4]*c[3]^3-8*c[4]*c[3]^2+c[3]*c[5]- c[3]^2)/(c[3]-c[4]), a[3,2] = 1/2*c[3]^2/c[2], a[7,6] = 1/60*(-12+15*c [3]+15*c[4]+15*c[5]-20*c[4]*c[3]-20*c[4]*c[5]+30*c[3]*c[4]*c[5]-20*c[3 ]*c[5])/(c[5]-1)/(c[4]-1)/(c[3]-1), a[5,1] = 1/2*(2*c[3]*c[2]*c[5]^2-5 *c[3]^2*c[2]*c[5]^2+5*c[3]^3*c[2]*c[5]+10*c[3]^2*c[2]*c[4]*c[5]-3*c[5] *c[4]*c[3]^2+5*c[5]^2*c[4]*c[3]^2+20*c[5]^2*c[4]^2*c[3]^2-10*c[5]^2*c[ 4]^2*c[3]-30*c[5]*c[4]^2*c[3]^3+15*c[4]^2*c[3]^3+6*c[4]^2*c[5]*c[3]-25 *c[3]^2*c[2]*c[4]^2-15*c[3]^3*c[4]*c[2]+6*c[3]*c[2]*c[4]^2-2*c[3]*c[2] *c[4]*c[5]-5*c[3]*c[2]*c[5]^2*c[4]+5*c[3]*c[2]*c[5]*c[4]^2+10*c[2]*c[4 ]^2*c[5]^2-20*c[3]*c[2]*c[4]^2*c[5]^2+8*c[3]^2*c[2]*c[4]-6*c[2]*c[5]*c [4]^2-2*c[3]^2*c[2]*c[5]+20*c[3]^3*c[2]*c[4]^2+20*c[3]^2*c[2]*c[5]*c[4 ]^2-6*c[4]^2*c[3]^2)*c[5]/c[4]/c[3]^2/(3-5*c[3]+10*c[4]*c[3]-5*c[4])/c [2], a[6,5] = (-3+8*c[3]+8*c[4]+3*c[5]+5*c[4]^2*c[5]+15*c[4]^2*c[3]-15 *c[5]*c[4]*c[3]^2-15*c[4]^2*c[5]*c[3]+10*c[5]*c[4]^2*c[3]^2-23*c[4]*c[ 3]-8*c[4]*c[5]+23*c[3]*c[4]*c[5]-8*c[3]*c[5]+15*c[4]*c[3]^2-5*c[3]^2-5 *c[4]^2+5*c[3]^2*c[5]-10*c[4]^2*c[3]^2)/c[5]/(-20*c[5]^3*c[3]+20*c[5]^ 2*c[3]^2-20*c[5]^3*c[4]-15*c[4]^2*c[5]+15*c[4]^2*c[3]+20*c[5]^2*c[3]*c [4]+30*c[5]^3*c[4]*c[3]-30*c[5]^2*c[4]*c[3]^2-30*c[5]^2*c[4]^2*c[3]+20 *c[4]^2*c[5]^2+30*c[5]*c[4]^2*c[3]^2-12*c[4]*c[3]+12*c[4]*c[5]-15*c[3] *c[4]*c[5]+15*c[5]^3+12*c[3]*c[5]+15*c[4]*c[3]^2-12*c[5]^2-15*c[3]^2*c [5]-20*c[4]^2*c[3]^2), a[7,3] = -1/60*(10*c[4]*c[5]-5*c[4]-5*c[5]+3)/( c[3]-1)/c[3]/(c[3]-c[5])/(c[3]-c[4]), b[5] = -1/60*(3-5*c[3]+10*c[4]*c [3]-5*c[4])/c[5]/(-c[5]^2-c[5]^2*c[3]+c[5]^3+c[3]*c[5]-c[5]^2*c[4]+c[3 ]*c[4]*c[5]-c[4]*c[3]+c[4]*c[5]), b[1] = 1/60*(5*c[4]-10*c[4]*c[5]+30* c[3]*c[4]*c[5]-10*c[4]*c[3]-10*c[3]*c[5]-3+5*c[3]+5*c[5])/c[3]/c[4]/c[ 5], b[6] = 1/60*(-12+15*c[3]+15*c[4]+15*c[5]-20*c[4]*c[3]-20*c[4]*c[5] +30*c[3]*c[4]*c[5]-20*c[3]*c[5])/(c[5]-1)/(c[4]-1)/(c[3]-1), a[3,1] = \+ 1/2*c[3]*(2*c[2]-c[3])/c[2], `b*`[4] = -1/60*(-460*c[5]*c[4]*c[3]*`b*` [7]+760*c[5]*c[4]*c[3]^2*`b*`[7]+3*c[3]-6*c[4]-1800*`b*`[7]*c[4]^2*c[5 ]*c[3]^2+130*c[5]*c[4]*c[3]^2+100*c[3]^2*c[5]*`b*`[7]-100*c[3]^3*c[5]* c[4]+300*c[3]^3*c[4]*`b*`[7]-600*c[3]^3*`b*`[7]*c[4]^2+940*`b*`[7]*c[4 ]^2*c[3]^2-60*c[5]*c[3]*`b*`[7]-560*c[4]^2*c[3]*`b*`[7]+120*`b*`[7]*c[ 4]^2+120*c[5]*c[4]*`b*`[7]-200*`b*`[7]*c[4]^2*c[5]+1000*c[5]*c[4]^2*c[ 3]*`b*`[7]-600*c[3]^3*c[5]*c[4]*`b*`[7]+34*c[4]*c[3]-380*`b*`[7]*c[4]* c[3]^2+252*`b*`[7]*c[4]*c[3]+10*c[4]*c[5]-60*c[3]*c[4]*c[5]+1200*c[3]^ 3*`b*`[7]*c[4]^2*c[5]+36*`b*`[7]*c[3]-72*`b*`[7]*c[4]-60*`b*`[7]*c[3]^ 2-5*c[3]*c[5]-70*c[4]*c[3]^2-5*c[3]^2+10*c[3]^2*c[5]+50*c[4]*c[3]^3)/( -c[4]^2+c[4]+c[4]*c[3]-c[3])/c[4]/(-10*c[5]*c[4]*c[3]^2+10*c[4]^2*c[3] ^2+8*c[3]*c[4]*c[5]+c[3]*c[5]-8*c[4]^2*c[3]-c[4]*c[3]-2*c[4]*c[5]+2*c[ 4]^2), a[7,4] = 1/60*(10*c[3]*c[5]-5*c[3]-5*c[5]+3)/(-c[4]^2+c[4]+c[4] *c[3]-c[3])/c[4]/(c[4]-c[5]), a[7,5] = -1/60*(3-5*c[3]+10*c[4]*c[3]-5* c[4])/c[5]/(-c[5]^2-c[5]^2*c[3]+c[5]^3+c[3]*c[5]-c[5]^2*c[4]+c[3]*c[4] *c[5]-c[4]*c[3]+c[4]*c[5])\}:" }{TEXT -1 0 "" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 49 "sm: square of principal error \+ norm in terms of " }{XPPEDIT 18 0 "c[2]" "6#&%\"cG6#\"\"#" }{TEXT -1 3 ", " }{XPPEDIT 18 0 "c[3]" "6#&%\"cG6#\"\"$" }{TEXT -1 3 ", " } {XPPEDIT 18 0 "c[4]" "6#&%\"cG6#\"\"%" }{TEXT -1 7 " and " } {XPPEDIT 18 0 "c[5]" "6#&%\"cG6#\"\"&" }{TEXT -1 1 " " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1665 "sm := 1 /25920000*(12042*c[5]^2*c[3]^2+297000*c[3]*c[2]^2*c[4]^2*c[5]-13500*c[ 3]^2*c[2]*c[5]^2+28044*c[3]^3*c[5]-270000*c[3]*c[2]^2*c[4]^2*c[5]^2-22 140*c[3]^4*c[5]-148500*c[3]^2*c[2]^2*c[4]*c[5]-270000*c[3]^2*c[2]^2*c[ 4]^2*c[5]+135000*c[3]^2*c[2]^2*c[4]*c[5]^2-22140*c[5]^2*c[3]^3+270000* c[3]^2*c[2]^2*c[4]^2*c[5]^2-18000*c[3]^2*c[2]*c[4]*c[5]+81000*c[3]*c[2 ]^2*c[5]*c[4]-67500*c[3]*c[2]^2*c[5]^2*c[4]+96444*c[5]*c[4]*c[3]^2-761 40*c[5]^2*c[4]*c[3]^2-613800*c[5]^2*c[4]^2*c[3]^3-221940*c[3]^3*c[5]*c [4]+153450*c[5]^2*c[4]^2*c[3]^2+675180*c[5]*c[4]^2*c[3]^3-240140*c[4]^ 2*c[3]^3+198000*c[3]^2*c[2]*c[4]^2-49500*c[3]^3*c[4]*c[2]-55800*c[3]*c [2]*c[4]^2+12000*c[3]*c[2]*c[4]-184140*c[5]*c[4]^2*c[3]^2-34200*c[3]*c [2]*c[4]*c[5]+27000*c[3]*c[2]*c[5]^2*c[4]+162000*c[3]*c[2]*c[5]*c[4]^2 +613800*c[4]^2*c[3]^4*c[5]^2-613800*c[4]^2*c[3]^4*c[5]-73800*c[4]*c[3] ^4*c[5]^2-42140*c[3]^4*c[4]+223450*c[4]^2*c[3]^4+189180*c[4]*c[5]^2*c[ 3]^3+18450*c[5]^2*c[3]^4+10642*c[3]^4-135000*c[3]*c[2]*c[4]^2*c[5]^2+2 7900*c[2]^2*c[4]^2+6975*c[3]^2*c[2]^2+540000*c[3]^3*c[2]*c[5]*c[4]^2+8 1180*c[3]^4*c[5]*c[4]+540000*c[3]^2*c[2]*c[4]^2*c[5]^2+6900*c[3]^2*c[2 ]*c[4]-6000*c[2]*c[3]^2-36856*c[4]*c[3]^2+17100*c[3]^2*c[2]*c[5]+6552* c[3]^2-12856*c[3]^3-27900*c[3]*c[2]^2*c[4]-99000*c[3]*c[2]^2*c[4]^2+49 500*c[3]^2*c[2]^2*c[4]+90000*c[3]^2*c[2]^2*c[4]^2-20250*c[3]^2*c[2]^2* c[5]+16875*c[3]^2*c[2]^2*c[5]^2-81000*c[5]*c[2]^2*c[4]^2+67500*c[5]^2* c[2]^2*c[4]^2-540000*c[2]*c[4]^2*c[5]^2*c[3]^3-180000*c[3]^3*c[2]*c[4] ^2-135000*c[3]^3*c[2]*c[5]^2*c[4]+148500*c[3]^3*c[2]*c[4]*c[5]+13500*c [3]^2*c[2]*c[5]^2*c[4]-594000*c[3]^2*c[2]*c[5]*c[4]^2-16056*c[3]^2*c[5 ]+88044*c[4]*c[3]^3+66442*c[4]^2*c[3]^2)/c[3]^2:" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 64 " We shall const ruct a 6 stage, order 5 Runge-Kutta scheme with " }{XPPEDIT 18 0 "t[6 ] = 1/500;" "6#/&%\"tG6#\"\"'*&\"\"\"F)\"$+&!\"\"" }{TEXT -1 36 " so \+ that the stability function is " }}{PARA 256 "" 0 "" {TEXT -1 1 " " } {XPPEDIT 18 0 "R(z)= 1+z+1/2" "6#/-%\"RG6#%\"zG,(\"\"\"F)F'F)*&F)F)\" \"#!\"\"F)" }{TEXT -1 1 " " }{XPPEDIT 18 0 "z^2+1/6" "6#,&*$%\"zG\"\"# \"\"\"*&F'F'\"\"'!\"\"F'" }{TEXT -1 1 " " }{XPPEDIT 18 0 "z^3+1/24" "6 #,&*$%\"zG\"\"$\"\"\"*&F'F'\"#C!\"\"F'" }{TEXT -1 1 " " }{XPPEDIT 18 0 "z^4+1/120" "6#,&*$%\"zG\"\"%\"\"\"*&F'F'\"$?\"!\"\"F'" }{TEXT -1 1 " " }{XPPEDIT 18 0 "z^5+1/500;" "6#,&*$%\"zG\"\"&\"\"\"*&F'F'\"$+&!\" \"F'" }{TEXT -1 1 " " }{XPPEDIT 18 0 "z^6" "6#*$%\"zG\"\"'" }{TEXT -1 1 "." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 96 "A 6 stage, order 5 Runge-Kutta scheme will have precisely this stabilit y function provided that " }}{PARA 256 "" 0 "" {TEXT -1 1 " " } {XPPEDIT 18 0 "(2-5*c[3])*c[4]/240 = 1/500;" "6#/*(,&\"\"#\"\"\"*&\"\" &F'&%\"cG6#\"\"$F'!\"\"F'&F+6#\"\"%F'\"$S#F.*&F'F'\"$+&F." }{TEXT -1 2 ", " }}{PARA 0 "" 0 "" {TEXT -1 8 "so that " }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "(2-5*c[3])*c[4] = 12/25" "6#/*&,&\"\"# \"\"\"*&\"\"&F'&%\"cG6#\"\"$F'!\"\"F'&F+6#\"\"%F'*&\"#7F'\"#DF." } {TEXT -1 2 ", " }}{PARA 0 "" 0 "" {TEXT -1 9 "that is, " }}{PARA 256 " " 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "c[4] = 12/(50-125*c[3]);" "6#/&% \"cG6#\"\"%*&\"#7\"\"\",&\"#]F**&\"$D\"F*&F%6#\"\"$F*!\"\"F2" }{TEXT -1 2 ". " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 78 "(2-5*c[3])*c[4]=240/500;\nc[4]=solve(%,c[4]);\nc[4]=n umer(rhs(%))/denom(rhs(%));" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/*&,&\" \"#\"\"\"*&\"\"&F'&%\"cG6#\"\"$F'!\"\"F'&F+6#\"\"%F'#\"#7\"#D" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#/&%\"cG6#\"\"%,$*&#\"#7\"#D\"\"\"*&F-F -,&*&\"\"&F-&F%6#\"\"$F-F-\"\"#!\"\"F6F-F6" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/&%\"cG6#\"\"%,$*&\"#7\"\"\",&*&\"$D\"F+&F%6#\"\"$F+F+ \"#]!\"\"F3F3" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 52 "The following picture is the graph of this equation." }} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 92 "plot(12/(50-125*c[3]),c[3]=0 ..0.32,c[4]=0..1,color=COLOR(RGB,.1,.75,.2),font=[HELVETICA,9]);" }} {PARA 13 "" 1 "" {GLPLOT2D 485 401 401 {PLOTDATA 2 "6'-%'CURVESG6#7U7$ $\"\"!F)$\"3!**************R#!#=7$$\"3YLLLL$*3vp!#?$\"3:m5elKfUCF,7$$ \"3tmmmExS/8!#>$\"3O;L#or-4[#F,7$$\"3KLLL$*)Gp)>F6$\"3-\\J/&4Za_#F,7$$ \"3iLLL`U(Rn#F6$\"3YWBk)*>$>d#F,7$$\"3smmm1UvdLF6$\"3z3/!=vE*>EF,7$$\" 3YLLL`dq\"*RF6$\"3C7Ya>C0mEF,7$$\"3')******fe7[YF6$\"3@LBkrhb:FF,7$$\" 3YLLLtk*pK&F6$\"3Yb#oGmC(oFF,7$$\"3J+++S**o.gF6$\"3M6!Gg(e$Q#GF,7$$\"3 3ommYBv*p'F6$\"3Cfb$HWhG)GF,7$$\"3?MLL$\\WGJ(F6$\"3Qo0)zVLp$HF,7$$\"3b ,++S5/.!)F6$\"3v.hfC^G+IF,7$$\"3u++++;2'p)F6$\"3a4%Rav2n1$F,7$$\"3[+++ g\\$RO*F6$\"38!\\J%)phN8$F,7$$\"3yLLLt)=/(**F6$\"3=;;-)yZo>$F,7$$\"3om mm!pe\"p5F,$\"3)=v`T**4bF$F,7$$\"3ummm#3^-8\"F,$\"3-zGQ%RS_M$F,7$$\"3* ******z0;8?\"F,$\"3XLw&fo$=IMF,7$$\"3kmmm-1Ak7F,$\"3Ll&=O'[04NF,7$$\"3 3+++eoBL8F,$\"3aY<,j(p)*f$F,7$$\"33+++am&*)R\"F,$\"3'3,`=hE3p$F,7$$\"3 #pmmmOGvY\"F,$\"3I'*\\K#4j2z$F,7$$\"3(ommYg)\\I:F,$\"3w,u23WU()QF,7$$ \"3YLLL80U)f\"F,$\"33.wc`#pt*RF,7$$\"3[LLL(Gs*o;F,$\"3a&=Ej:b$=TF,7$$ \"33+++9yQI6W_'=F,$\"3MHF,$\"3b%)=POB!Gk%F, 7$$\"39+++Ir9(*>F,$\"3G]!HZ'G;$z%F,7$$\"31+++st;p?F,$\"3;+Ix)H[>(\\F,7 $$\"3rmmmi2)Q8#F,$\"3(ylq*3eOW^F,7$$\"3K+++?h(H?#F,$\"3baYrqg;U`F,7$$ \"3YLLL$y'elAF,$\"3;\"3M?<8]`&F,7$$\"3E+++wn.MBF,$\"3ML?PC@VidF,7$$\"3 mmmm%)GW)R#F,$\"3U\"R]Y\\mT*fF,7$$\"35+++EewlCF,$\"3+A\"\\TV$>diF,7$$ \"3immm1jeJDF,$\"3p!=%[trmPlF,7$$\"3,+++]U\\+EF,$\"3UIeSGXcfoF,7$$\"3[ LLL*ygom#F,$\"3#\\%3/:%[5?(F,7$$\"3eLLLh,tMFF,$\"3gIR?ZVJ(e(F,7$$\"3*p mm1aP?!GF,$\"34!zhzq1O,)F,7$$\"3O+++Kg)Q'GF,$\"3W:WU-l&)\\%)F,7$$\"3:n mm#QrZ$HF,$\"3,eZU@*\\@,*F,7$$\"3yLLL&3s\")*HF,$\"3!*ze)o.%[#e*F,7$$\" 3/+++e/xlIF,$\"3dTS]f?\")4$F,$\"3yg#y='QWk5F_z7$ $\"3K+++E2ZIJF,$\"3/WfhNc//6F_z7$$\"3=+++j`BlJF,$\"3=FeTeZ-]6F_z7$$\"3 1+++++++KF,$\"3;+++++++7F_z-%%FONTG6$%*HELVETICAG\"\"*-%+AXESLABELSG6$ Q%c[3]6\"Q%c[4]F]\\l-%&COLORG6&%$RGBG$\"\"\"!\"\"$\"#v!\"#$\"\"#Fe\\l- %%VIEWG6$;F($\"#KFh\\l;F($Fd\\lF)" 1 2 0 1 10 0 2 6 1 4 2 1.000000 45.000000 45.000000 0 0 "Curve 1" }}}}{PARA 0 "" 0 "" {TEXT -1 0 "" }} {PARA 0 "" 0 "" {TEXT -1 130 "We can use the preceding relation to obt ain an expression for the square of the principal error norm in terms \+ of the three nodes " }{XPPEDIT 18 0 "c[2]" "6#&%\"cG6#\"\"#" }{TEXT -1 3 ", " }{XPPEDIT 18 0 "c[3]" "6#&%\"cG6#\"\"$" }{TEXT -1 7 " and \+ " }{XPPEDIT 18 0 "c[5]" "6#&%\"cG6#\"\"&" }{TEXT -1 1 "." }}{PARA 0 " " 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 48 "sm2 : = simplify(subs(c[4]=12/(50-125*c[3]),sm)):" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 50 "sm2: square of principal error norm in t erms of " }{XPPEDIT 18 0 "c[2]" "6#&%\"cG6#\"\"#" }{TEXT -1 3 ", " } {XPPEDIT 18 0 "c[3]" "6#&%\"cG6#\"\"$" }{TEXT -1 7 " and " } {XPPEDIT 18 0 "c[5]" "6#&%\"cG6#\"\"&" }{TEXT -1 1 " " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1020 "sm2 := \+ 1/16200000000*(-270690000*c[3]^5+166281250*c[3]^6+164506800*c[3]^4+401 7600*c[2]^2+3834048*c[3]^2-11664000*c[5]*c[2]^2+9030000*c[3]^3*c[2]-83 5200*c[3]*c[2]-161437500*c[3]^3*c[2]^2-8789760*c[3]^2*c[5]+593167500*c [3]^5*c[5]-19500000*c[3]^4*c[2]-300105000*c[3]^2*c[2]^2*c[5]-465862500 *c[5]^2*c[3]^5-40509760*c[3]^3-345937500*c[3]^6*c[5]+11610000*c[3]^2*c [2]*c[5]^2+263317500*c[3]^2*c[2]^2*c[5]^2+108984375*c[3]^4*c[2]^2-3240 000*c[3]*c[2]*c[5]^2+2808000*c[3]*c[2]*c[5]-348000*c[2]*c[3]^2-1026000 0*c[3]^3*c[2]*c[5]^2+91368000*c[3]*c[2]^2*c[5]-2286000*c[3]^2*c[2]*c[5 ]-79380000*c[3]*c[2]^2*c[5]^2+263671875*c[3]^4*c[2]^2*c[5]^2+44437500* c[3]^4*c[2]*c[5]+475875000*c[3]^3*c[2]^2*c[5]-316406250*c[3]^4*c[2]^2* c[5]-413437500*c[3]^3*c[2]^2*c[5]^2-8437500*c[3]^4*c[2]*c[5]^2+9720000 *c[5]^2*c[2]^2+101947500*c[3]^2*c[2]^2-30996000*c[3]*c[2]^2+271368450* c[5]^2*c[3]^4-66544200*c[5]^2*c[3]^3+90205920*c[3]^3*c[5]+6517800*c[5] ^2*c[3]^2-19890000*c[3]^3*c[2]*c[5]+288281250*c[5]^2*c[3]^6-363544200* c[3]^4*c[5])/c[3]^2/(5*c[3]-2)^2:" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 8 "Taking " }{XPPEDIT 18 0 "c[5] = 7/8;" "6#/&%\"cG6#\"\"&*&\"\"(\"\"\"\"\")!\"\"" }{TEXT -1 103 ", we o btain the following surface for the graph of the square of the princip al error norm in terms of " }{XPPEDIT 18 0 "c[2]" "6#&%\"cG6#\"\"#" } {TEXT -1 7 " and " }{XPPEDIT 18 0 "c[3]" "6#&%\"cG6#\"\"$" }{TEXT -1 2 ". 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vF/$\"\"#F)-%+AXESLABELS G6%%%c[2]G%%c[3]G%!G-%*AXESSTYLEG6#%$BOXG-%+LIGHTMODELG6#%(LIGHT_4G" 1 2 0 1 10 0 2 1 6 2 2 1.000000 45.000000 45.000000 0 0 "Curve 1" }}}} {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 58 "The follo wing calculations give (approximate) values for " }{XPPEDIT 18 0 "c[2 ]" "6#&%\"cG6#\"\"#" }{TEXT -1 7 " and " }{XPPEDIT 18 0 "c[3]" "6#&% \"cG6#\"\"$" }{TEXT -1 82 " to minimize the square of the principal e rror norm for three fixed choices for " }{XPPEDIT 18 0 "c[5]" "6#&%\" cG6#\"\"&" }{TEXT -1 10 ", namely " }{XPPEDIT 18 0 "c[5] = 3/4;" "6#/ &%\"cG6#\"\"&*&\"\"$\"\"\"\"\"%!\"\"" }{TEXT -1 3 ", " }{XPPEDIT 18 0 "c[5] = 7/9;" "6#/&%\"cG6#\"\"&*&\"\"(\"\"\"\"\"*!\"\"" }{TEXT -1 7 " and " }{XPPEDIT 18 0 "c[5] = 4/5;" "6#/&%\"cG6#\"\"&*&\"\"%\"\"\"F '!\"\"" }{TEXT -1 1 "." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 244 "uu := \{c[5]=3/4.\}:\nmn := evalf[ 12](minimize(eval(sm2,uu),c[2]=0.1..0.2,c[3]=0.2..0.3,'location'));\ns ubs(op(1,op(1,op(2,[mn]))),[c[2],c[3]]):\nrr := map(convert,%,rational ,5):\nc[2]=op(1,rr),c[3]=op(2,rr);\n`principal error norm:`,sqrt(op(1, [mn]));" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%#mnG6$$\"-)pLjVo(!#=<#7$< $/&%\"cG6#\"\"$$\"-4<>?[D!#7/&F.6#\"\"#$\"-&\\sZXk\"F3F&" }}{PARA 11 " " 1 "" {XPPMATH 20 "6$/&%\"cG6#\"\"##\"#P\"$D#/&F%6#\"\"$#\"#`\"$3#" } }{PARA 11 "" 1 "" {XPPMATH 20 "6$%6principal~error~norm:G$\"+W+0m()!#8 " }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 244 "uu := \{c[5]=7/9.\}:\nmn := evalf[12](minimize(eval( sm2,uu),c[2]=0.1..0.2,c[3]=0.2..0.3,'location'));\nsubs(op(1,op(1,op(2 ,[mn]))),[c[2],c[3]]):\nrr := map(convert,%,rational,5):\nc[2]=op(1,rr ),c[3]=op(2,rr);\n`principal error norm:`,sqrt(op(1,[mn]));" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%#mnG6$$\"-/nyLIv!#=<#7$<$/&%\"cG6#\"\"#$\" -myQ7bvx')!#8" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 244 "uu := \{ c[5]=4/5.\}:\nmn := evalf[12](minimize(eval(sm2,uu),c[2]=0.1..0.2,c[3] =0.2..0.3,'location'));\nsubs(op(1,op(1,op(2,[mn]))),[c[2],c[3]]):\nrr := map(convert,%,rational,5):\nc[2]=op(1,rr),c[3]=op(2,rr);\n`princip al error norm:`,sqrt(op(1,[mn]));" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#> %#mnG6$$\"-S=%*\\)p(!#=<#7$<$/&%\"cG6#\"\"#$\"-LqE<^=!#7/&F.6#\"\"$$\" -X`DhuEF3F&" }}{PARA 11 "" 1 "" {XPPMATH 20 "6$/&%\"cG6#\"\"##\"#(*\"$ C&/&F%6#\"\"$#\"#B\"#')" }}{PARA 11 "" 1 "" {XPPMATH 20 "6$%6principal ~error~norm:G$\"+5$4Tx)!#8" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 14 "The procedure \+ " }{TEXT 0 8 "minimize" }{TEXT -1 75 " has difficulty handling minimiz ation with respect to the three variables " }{XPPEDIT 18 0 "c[2]" "6# &%\"cG6#\"\"#" }{TEXT -1 3 ", " }{XPPEDIT 18 0 "c[3]" "6#&%\"cG6#\"\" $" }{TEXT -1 7 " and " }{XPPEDIT 18 0 "c[5]" "6#&%\"cG6#\"\"&" } {TEXT -1 86 ", so we set up a procedure to perform minimizations with \+ repect to the two variables " }{XPPEDIT 18 0 "c[2]" "6#&%\"cG6#\"\"# " }{TEXT -1 7 " and " }{XPPEDIT 18 0 "c[3]" "6#&%\"cG6#\"\"$" } {TEXT -1 84 " as in the previous examples and the use the one dimensi onal minmization procedure " }{TEXT 0 7 "findmin" }{TEXT -1 29 " to mi nimize with repect to " }{XPPEDIT 18 0 "c[5]" "6#&%\"cG6#\"\"&" } {TEXT -1 1 "." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 141 "prin_err := proc(c_5)\n local mn;\n mn := e valf[Digits+2](minimize(eval(sm2,c[5]=c_5),c[2]=0.1..0.5,c[3]=0.2..0.3 ));\n sqrt(mn);\nend proc:" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }} {PARA 0 "" 0 "" {TEXT -1 21 "Using the procedure " }{TEXT 0 8 "prin_e rr" }{TEXT -1 55 " three previous examples can be performed as follow s. " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 45 "prin_err(3/4.),prin_err(7/9.),prin_err(4/5.);" }} {PARA 11 "" 1 "" {XPPMATH 20 "6%$\"+Q+0m()!#8$\"+S>vx')F%$\"+5$4Tx)F% " }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 55 "Thes e three values can be used as starting values for " }{TEXT 0 7 "findm in" }{TEXT -1 1 "." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 " > " 0 "" {MPLTEXT 1 0 122 "evalf[12](findmin('prin_err'(c5),c5=\{3/4., 7/9.,4/5.\},convergence='location'));\nc[5]=convert(op(1,%),rational,4 );\nevalf(%);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#7$$\"-n;T**\\x!#7$\"- +(H#fw')!#:" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/&%\"cG6#\"\"&#\"#J\"#S " }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/&%\"cG6#\"\"&$\"++++]x!#5" }}} {PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 246 "uu := \{c[5]=31/40.\}:\nmn := evalf[12](minimize(eval(sm2,uu),c[2 ]=0.1..0.2,c[3]=0.2..0.3,'location'));\nsubs(op(1,op(1,op(2,[mn]))),[c [2],c[3]]):\nrr := map(convert,%,rational,5):\nc[2]=op(1,rr),c[3]=op(2 ,rr);\n`principal error norm:`,sqrt(op(1,[mn]));" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%#mnG6$$\"-OT`KGv!#=<#7$<$/&%\"cG6#\"\"#$\"-kIQgV/&F%6#\"\"$#\"$P\"\"$D&" }}{PARA 11 "" 1 "" {XPPMATH 20 "6$%6principal~error~norm:G$\"+pAfw')!#8" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 14 "The value of " }{XPPEDIT 18 0 "c[4]" "6#&%\"cG6#\"\"%" }{TEXT -1 20 " corresponding to " } {XPPEDIT 18 0 "c[3] = 137/525;" "6#/&%\"cG6#\"\"$*&\"$P\"\"\"\"\"$D&! \"\"" }{TEXT -1 10 " is . . ." }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 44 "c[4]=12/(50-125*c[3]);\nsubs(c[3]=137/525,%);" }}{PARA 11 "" 1 " " {XPPMATH 20 "6#/&%\"cG6#\"\"%,$*&\"#7\"\"\",&\"#]F+*&\"$D\"F+&F%6#\" \"$F+!\"\"F3F+" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/&%\"cG6#\"\"%#\"$_# \"$l$" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 7 " Taking " }{XPPEDIT 18 0 "`b*`[7] = 1/30;" "6#/&%#b*G6#\"\"(*&\"\"\"F) \"#I!\"\"" }{TEXT -1 74 " gives suitable characteristics for the embe dded 7 stage, order 4 scheme." }}{PARA 0 "" 0 "" {TEXT -1 80 " Using t he general solution we can find the coefficients of the combined schem e." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 101 "e3 := \{c[2]=34/195,c[3]=137/525,c[4]=252/365,c[5]=3 1/40,`b*`[7]=1/30\}:\ne4 := `union`(subs(e3,e2),e3):" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}} {SECT 1 {PARA 4 "" 0 "" {TEXT -1 2 "e4" }}{PARA 0 "" 0 "" {TEXT -1 0 " " }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1188 "e4 := \{c[4] = 252/365 , c[5] = 31/40, `b*`[7] = 1/30, `b*`[1] = 2124247802213/27446171926320 , a[2,1] = 34/195, b[3] = 3027623484375/7555589800144, a[3,2] = 243997 /1249500, a[7,6] = 406471/4735152, c[6] = 1, c[7] = 1, b[2] = 0, `b*`[ 2] = 0, a[6,3] = 625455957178350900/271098296009435219, a[5,1] = -2095 505049916883/707267524239360, a[4,2] = -14189381388/4530102965, a[7,2] = 0, `b*`[6] = 539450832947/10119327608880, a[6,5] = 1735464775680/67 19560297073, b[1] = 990209/12842928, b[5] = 43980800/446349501, a[4,1] = 101345382936/124124821241, a[7,3] = 3027623484375/7555589800144, a[ 5,2] = 498515020731/40972513280, a[5,3] = -2529970133596857/2717300695 95136, b[6] = 406471/4735152, a[4,3] = 21949459056/7301460073, `b*`[4] = 203730456418295501/593756471494983888, a[3,1] = 82063/1249500, a[6, 1] = 789713415527611/1013168913843996, `b*`[5] = 17760294550016/190775 579270913, a[5,4] = 6431020141871/7296710270976, a[7,1] = 990209/12842 928, a[6,2] = -743202495/291283372, b[4] = 469280932525/1389186738576, a[6,4] = 21511957375165/104104742407029, `b*`[3] = 61472258224069875/ 153778919202330832, a[7,4] = 469280932525/1389186738576, a[7,5] = 4398 0800/446349501, c[2] = 34/195, c[3] = 137/525\}:" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 137 "s ubs(e4,matrix([seq([c[i],seq(a[i,j],j=1..i-1),``$(8-i)],i=2..7),\n[`b` ,seq(b[i],i=1..6),``],[`b*`,seq(`b*`[i],i=1..7)]]));\n``;\nevalf(%%); " }}{PARA 11 "" 1 "" {XPPMATH 20 "6#K%'matrixG6#7*7*#\"#M\"$&>F(%!GF+F +F+F+F+7*#\"$P\"\"$D&#\"&j?)\"(+&\\7#\"'(*RCF2F+F+F+F+F+7*#\"$_#\"$l$# \"-OHQX85\"-T7#[7C\"#!,)Q\"Q*=9\"+lH5IX#\",c!f%\\>#\"+t+Y,tF+F+F+F+7*# \"#J\"#S#!1$)o\"*\\]]&4#\"0g$RU_nsq#\"-J2-:&)\\\",!G8D(4%#!1dofL,(*HD \"0O^fp+tr##\".r=9?5V'\".w4F5nH(F+F+F+7*\"\"\"#\"06w_:Mr*y\"1'*R%Q\"*o J,\"#!*&\\-Ku\"*sLG\"H#\"3+4Nyr&fXD'\"3>_V4gH)4r##\"/l^Pd>^@\"0HqSUZ5/ \"#\".!ovZYN<\".tqHg&>nF+F+7*FS#\"'4-**\")GH%G\"\"\"!#\".vV[Bw-$\".W,! )*ebv#\"-DD$4Gp%\".w&Qn=*Q\"#\")+3)R%\"*,&\\jW#\"'rkS\"(_^t%F+7*%\"bGF ^oFaoFboFeoFhoF[pF+7*%#b*G#\".8A!yCC@\"/?j#>*yP:#\"3,bH=kXIP?\"3))Q)\\\\rkv$f#\"/;+b%Hgx\"\"084Fzbx!>#\"-ZH$3XR &\"/!))3wK>,\"#FS\"#IQ)pprint126\"" }}{PARA 11 "" 1 "" {XPPMATH 20 "6# %!G" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#K%'matrixG6#7*7*$\"+W(*eVF*F+F+F+F+F+7*$\"+*e4T! pF*$\"+#)ezk\")F*$!+yDCKJ!\"*$\"+[R<1IF;F+F+F+F+7*$\"++++]xF*$!+R\"=G' HF;$\"+%)fq;7!\")$!+V0g5$*F;$\"+d%)e8))F*F+F+F+7*$\"\"\"\"\"!$\"+*>*[% z(F*$!+qeZ^DF;$\"+L(=rI#F;$\"+8jPm?F*$\"+geq#e#F*F+F+7*FK$\"+(*)\\,r(F 1$FMFM$\"+o082SF*$\"+v#)4yLF*$\"+?WW`)*F1$\"+ds6%e)F1F+7*%\"bGFYFenFfn FhnFjnF\\oF+7*%#b*G$\"+1qoRxF1Fen$\"+oPW(*RF*$\"+%R77V$F*$\"+9?_4$*F1$ \"+Ag*3L&F1$\"+LLLLLF1Q)pprint136\"" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 57 "#-------------------------------------- ------------------" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 " > " 0 "" {MPLTEXT 1 0 1 ";" }}}}{PARA 0 "" 0 "" {TEXT -1 39 "#-------- ------------------------------" }}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 39 "characteristics of the embedded scheme " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 35 "coefficients of the combined scheme" }}{PARA 0 " " 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1111 "ee \+ := \{c[2]=34/195,\nc[3]=137/525,\nc[4]=252/365,\nc[5]=31/40,\nc[6]=1, \nc[7]=1,\n\na[2,1]=34/195,\na[3,1]=82063/1249500,\na[3,2]=243997/1249 500,\na[4,1]=101345382936/124124821241,\na[4,2]=-14189381388/453010296 5,\na[4,3]=21949459056/7301460073,\na[5,1]=-2095505049916883/707267524 239360,\na[5,2]=498515020731/40972513280,\na[5,3]=-2529970133596857/27 1730069595136,\na[5,4]=6431020141871/7296710270976,\na[6,1]=7897134155 27611/1013168913843996,\na[6,2]=-743202495/291283372,\na[6,3]=62545595 7178350900/271098296009435219,\na[6,4]=21511957375165/104104742407029, \na[6,5]=1735464775680/6719560297073,\na[7,1]=990209/12842928,\na[7,2] =0,\na[7,3]=3027623484375/7555589800144,\na[7,4]=469280932525/13891867 38576,\na[7,5]=43980800/446349501,\na[7,6]=406471/4735152,\n\nb[1]=990 209/12842928,\nb[2]=0,\nb[3]=3027623484375/7555589800144,\nb[4]=469280 932525/1389186738576,\nb[5]=43980800/446349501,\nb[6]=406471/4735152, \n\n`b*`[1]=2124247802213/27446171926320,\n`b*`[2]=0,\n`b*`[3]=6147225 8224069875/153778919202330832,\n`b*`[4]=203730456418295501/59375647149 4983888,\n`b*`[5]=17760294550016/190775579270913,\n`b*`[6]=53945083294 7/10119327608880,\n`b*`[7]=1/30\}:" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" } }{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 5 "Let " }{XPPEDIT 18 0 "T[5 ,6];" "6#&%\"TG6$\"\"&\"\"'" }{TEXT -1 127 " denote the vector whose \+ components are the principal error terms of the 6 stage, order 5 schem e (the error terms of order 6)." }}{PARA 0 "" 0 "" {TEXT -1 5 "Let " }{XPPEDIT 18 0 "`T*`[4,7];" "6#&%#T*G6$\"\"%\"\"(" }{TEXT -1 145 " de note the vector whose components are the principal error terms of the \+ embedded 7 stage, order 4 scheme (the error terms of order 5) and let \+ " }{XPPEDIT 18 0 "`T*`[5,7];" "6#&%#T*G6$\"\"&\"\"(" }{TEXT -1 99 " \+ denote the vector whose components are the error terms of order 6 of t he embedded order 4 scheme." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 39 "Denote the 2-norm of these vectors by " } {XPPEDIT 18 0 "abs(abs(T[5,6]));" "6#-%$absG6#-F$6#&%\"TG6$\"\"&\"\"' " }{TEXT -1 3 ", " }{XPPEDIT 18 0 "abs(abs(`T*`[4,7]));" "6#-%$absG6# -F$6#&%#T*G6$\"\"%\"\"(" }{TEXT -1 7 " and " }{XPPEDIT 18 0 "abs(abs (`T*`[5,7]));" "6#-%$absG6#-F$6#&%#T*G6$\"\"&\"\"(" }{TEXT -1 15 " re spectively." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 8 "Define: " }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "A[6 ] = abs(abs(T[5,6]));" "6#/&%\"AG6#\"\"'-%$absG6#-F)6#&%\"TG6$\"\"&F' " }{TEXT -1 6 ", " }{XPPEDIT 18 0 "B[6] = abs(abs(`T*`[5,7]))/abs( abs(`T*`[4,7]));" "6#/&%\"BG6#\"\"'*&-%$absG6#-F*6#&%#T*G6$\"\"&\"\"( \"\"\"-F*6#-F*6#&F/6$\"\"%F2!\"\"" }{TEXT -1 6 ", " }{XPPEDIT 18 0 "C[6] = abs(abs(`T*`[5,7]-T[5,6]))/abs(abs(`T*`[4,7]));" "6#/&%\"CG6 #\"\"'*&-%$absG6#-F*6#,&&%#T*G6$\"\"&\"\"(\"\"\"&%\"TG6$F2F'!\"\"F4-F* 6#-F*6#&F06$\"\"%F3F8" }{TEXT -1 1 "." }}{PARA 0 "" 0 "" {TEXT -1 0 " " }}{PARA 0 "" 0 "" {TEXT -1 77 "Dormand and Prince have suggested tha t as well as attempting to ensure that " }{XPPEDIT 18 0 "A[6];" "6#&% \"AG6#\"\"'" }{TEXT -1 73 " is a minimum, if the embedded scheme is t o be used for error control, " }{XPPEDIT 18 0 "B[6];" "6#&%\"BG6#\"\" '" }{TEXT -1 7 " and " }{XPPEDIT 18 0 "C[6];" "6#&%\"CG6#\"\"'" } {TEXT -1 27 " should be chosen so that " }}{PARA 0 "" 0 "" {TEXT -1 68 "they are \"similar in magnitude\" and also not differ too much fro m 1." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 186 "errterms5_6 := PrincipalErrorTerms(5,6,'expanded'): \n`errterms5_7*` :=subs(b=`b*`,PrincipalErrorTerms(5,7,'expanded')):\n `errterms4_7*` := subs(b=`b*`,PrincipalErrorTerms(4,7,'expanded')):" } }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 326 "snmB := sqrt(add(evalf(subs(ee,`errterms5_7*`[i]))^2,i=1..nops( `errterms5_7*`))):\nsdnB := sqrt(add(evalf(subs(ee,`errterms4_7*`[i])) ^2,i=1..nops(`errterms4_7*`))):\nsnmC := sqrt(add((evalf(subs(ee,`errt erms5_7*`[i])-subs(ee,errterms5_6[i])))^2,i=1..nops(errterms5_6))):\n' B[6]'= evalf[8](snmB/sdnB);\n'C[6]'= evalf[8](snmC/sdnB);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/&%\"BG6#\"\"'$\")Vt$4\"!\"(" }}{PARA 11 "" 1 " " {XPPMATH 20 "6#/&%\"CG6#\"\"'$\")f!)y9!\"(" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}}{PARA 0 "" 0 "" {TEXT -1 39 "#--------------------------------------" }} {SECT 1 {PARA 4 "" 0 "" {TEXT -1 26 "absolute stability regions" }} {PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 35 "coefficients of the comb ined scheme" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 " " {MPLTEXT 1 0 1111 "ee := \{c[2]=34/195,\nc[3]=137/525,\nc[4]=252/365 ,\nc[5]=31/40,\nc[6]=1,\nc[7]=1,\n\na[2,1]=34/195,\na[3,1]=82063/12495 00,\na[3,2]=243997/1249500,\na[4,1]=101345382936/124124821241,\na[4,2] =-14189381388/4530102965,\na[4,3]=21949459056/7301460073,\na[5,1]=-209 5505049916883/707267524239360,\na[5,2]=498515020731/40972513280,\na[5, 3]=-2529970133596857/271730069595136,\na[5,4]=6431020141871/7296710270 976,\na[6,1]=789713415527611/1013168913843996,\na[6,2]=-743202495/2912 83372,\na[6,3]=625455957178350900/271098296009435219,\na[6,4]=21511957 375165/104104742407029,\na[6,5]=1735464775680/6719560297073,\na[7,1]=9 90209/12842928,\na[7,2]=0,\na[7,3]=3027623484375/7555589800144,\na[7,4 ]=469280932525/1389186738576,\na[7,5]=43980800/446349501,\na[7,6]=4064 71/4735152,\n\nb[1]=990209/12842928,\nb[2]=0,\nb[3]=3027623484375/7555 589800144,\nb[4]=469280932525/1389186738576,\nb[5]=43980800/446349501, \nb[6]=406471/4735152,\n\n`b*`[1]=2124247802213/27446171926320,\n`b*`[ 2]=0,\n`b*`[3]=61472258224069875/153778919202330832,\n`b*`[4]=20373045 6418295501/593756471494983888,\n`b*`[5]=17760294550016/190775579270913 ,\n`b*`[6]=539450832947/10119327608880,\n`b*`[7]=1/30\}:" }}}{PARA 0 " " 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}} }{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 79 "The stab ility function R for the 6 stage, order 5 scheme is given as follows ." }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 75 "subs(ee,StabilityFuncti on(5,6,'expanded')):\nR := unapply(%,z):\n'R(z)'=R(z);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/-%\"RG6#%\"zG,0\"\"\"F)F'F)*&#F)\"\"#F)*$)F'F,F)F )F)*&#F)\"\"'F)*$)F'\"\"$F)F)F)*&#F)\"#CF)*$)F'\"\"%F)F)F)*&#F)\"$?\"F )*$)F'\"\"&F)F)F)*&#F)\"$+&F)*$)F'F1F)F)F)" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 124 "We can find the point where th e boundary of the stability region intersects the negative real axis b y solving the equation: " }}{PARA 256 "" 0 "" {TEXT -1 1 " " } {XPPEDIT 18 0 "R(z) = 1;" "6#/-%\"RG6#%\"zG\"\"\"" }{TEXT -1 2 ". " }} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 26 "z0 := newton(R(z)=1,z=-3);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%#z0G$!+X#pA6$!\"*" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 301 "z0 := \+ newton(R(z)=1,z=-3):\np1 := plot([R(z),1],z=-3.59..0.49,color=[red,blu e]):\np2 := plot([[[z0,1]]$3],style=point,symbol=[circle,cross,diamond ],color=black):\np3 := plot([[z0,0],[z0,1]],linestyle=3,color=COLOR(RG B,0,.5,0)):\nplots[display]([p1,p2,p3],view=[-3.59..0.49,-0.0..1.47],f ont=[HELVETICA,9]);" }}{PARA 13 "" 1 "" {GLPLOT2D 401 211 211 {PLOTDATA 2 "6+-%'CURVESG6$7X7$$!3')*************e$!#<$\"3u[y#)G\"*)eP #F*7$$!3')****\\0Q`XNF*$\"3;mr-&pj)*>#F*7$$!3')*****4hn5]$F*$\"3!z:-fd %HN?F*7$$!3!)**\\#H\"yPiMF*$\"3&)*frT`d4!>F*7$$!3u***\\[,)oBMF*$\"3-yy 5;.Yu\"F*7$$!3-++lQj)=;$ F*$\"3IKe0\"p))y4\"F*7$$!3e***\\k^d53$F*$\"3=qsu9imE%*!#=7$$!3y***\\G& RO(*HF*$\"3#=$=0fxZR!)F^o7$$!3n***\\'\\z!3\"HF*$\"3#*)Q3d7?Z\"oF^o7$$! 3t***\\E`HX#GF*$\"3E%)y\">j\"=#y&F^o7$$!3#*****zv:yNFF*$\"37z%)e0**3$* [F^o7$$!3%)****4FBhdEF*$\"3]IG>(z?$QUF^o7$$!3q****RnAhpDF*$\"36!\\psvI yi$F^o7$$!3')*****4(3D\"[#F*$\"38e5ik>SLJF^o7$$!3i****f#H)4'R#F*$\"3;[ &o&zQ8bFF^o7$$!3u***\\Ofr(=BF*$\"37Od)\\El[[#F^o7$$!3t****RpE#oA#F*$\" 3;cOd\"GfGC#F^o7$$!3))****fp)H*[@F*$\"3'pIfeA%3(4#F^o7$$!3#****\\g-A$e ?F*$\"3OReT\\x=()>F^o7$$!3'*****fJ(=\"y>F*$\"3r^F\\![kq$>F^o7$$!3g*** \\g+B,*=F*$\"30Ua+>>!p#>F^o7$$!3f***\\hEIj!=F*$\"3=>CS_w)f&>F^o7$$!3a* ***\\K8!*=*y%G6#F^o7$$ !3))****\\XQ,_:F*$\"3^KK%oV[&RAF^o7$$!3w***\\OLg?Y\"F*$\"3&[#p*)4Q;,CF ^o7$$!3))***\\r`bPQ\"F*$\"3u'QE*QtklDF^o7$$!3k****p6U=*H\"F*$\"3w8`ofx cnFF^o7$$!3%)****>dP\"=@\"F*$\"3,_fJz1_-IF^o7$$!3#)*****G\")Qj7\"F*$\" 3yQ`\\o:leKF^o7$$!3f***\\#4ujV5F*$\"3oFtqHH(>`$F^o7$$!30(****p+;\"=&*F ^o$\"3kO!GQ\\2i'QF^o7$$!3g)****fF?Ip)F^o$\"3=,CS<&[d>%F^o7$$!34'*****p W07yF^o$\"3=_BM2tC!e%F^o7$$!3W++]7&oP,(F^o$\"3_rSA#>+*f\\F^o7$$!3Z%*** *f5J59'F^o$\"3(*[*=f][;T&F^o7$$!34(***\\?K&)>`F^o$\"3s>!o1`OX(eF^o7$$! 3)y***\\=d[hWF^o$\"3H)*e9j9%4S'F^o7$$!3T'******euAi$F^o$\"3!oK?&eaDhpF ^o7$$!3H++]7$)pVFF^o$\"3nQ; F)F^o7$$!3'R****HW!>K5F^o$\"3\\%RU6!RH>!*F^o7$$!3IP***\\c8-u\"!#>$\"3[ c?$*pS[F)*F^o7$$\"3al++!3pa9'Faz$\"3qevY9BQj5F*7$$\"3y0++H^L=:F^o$\"3C yY,Uk'R;\"F*7$$\"3K0++)3%pEBF^o$\"3MbR]0V'>E\"F*7$$\"3)[++&RLd)=$F^o$ \"3Ujb!*Qcbv8F*7$$\"3S1+]c<]8SF^o$\"3!pROt[UQ\\\"F*7$$\"3!************ ***[F^o$\"3zo&)pXKKK;F*-%'COLOURG6&%$RGBG$\"*++++\"!\")$\"\"!Fj\\lFi\\ l-F$6$7S7$F($\"\"\"Fj\\l7$F3F_]l7$F=F_]l7$FGF_]l7$FQF_]l7$FenF_]l7$Fjn F_]l7$F`oF_]l7$FeoF_]l7$FjoF_]l7$F_pF_]l7$FdpF_]l7$FipF_]l7$F^qF_]l7$F cqF_]l7$FhqF_]l7$F]rF_]l7$FbrF_]l7$FgrF_]l7$F\\sF_]l7$FasF_]l7$FfsF_]l 7$F[tF_]l7$F`tF_]l7$FetF_]l7$FjtF_]l7$F_uF_]l7$FduF_]l7$FiuF_]l7$F^vF_ ]l7$FcvF_]l7$FhvF_]l7$F]wF_]l7$FbwF_]l7$FgwF_]l7$F\\xF_]l7$FaxF_]l7$Ff xF_]l7$F[yF_]l7$F`yF_]l7$FeyF_]l7$FjyF_]l7$F_zF_]l7$FezF_]l7$FjzF_]l7$ F_[lF_]l7$Fd[lF_]l7$Fi[lF_]l7$F^\\lF_]l-Fc\\l6&Fe\\lFi\\lFi\\lFf\\l-F$ 6&7#7$$!3!)*****\\CpA6$F*F_]l-%'SYMBOLG6#%'CIRCLEG-Fc\\l6&Fe\\lFj\\lFj \\lFj\\l-%&STYLEG6#%&POINTG-F$6&Fe`l-Fj`l6#%&CROSSGF]alF_al-F$6&Fe`l-F j`l6#%(DIAMONDGF]alF_al-F$6%7$7$Fg`lFi\\lFf`l-%&COLORG6&Fe\\lFi\\l$\" \"&!\"\"Fi\\l-%*LINESTYLEG6#\"\"$-%%FONTG6$%*HELVETICAG\"\"*-%+AXESLAB ELSG6%Q\"z6\"Q!Fdcl-F\\cl6#%(DEFAULTG-%%VIEWG6$;$!$f$!\"#$\"#\\F_dl;$! \"!Fj\\l$\"$Z\"F_dl" 1 2 0 1 10 0 2 9 1 4 2 1.000000 45.000000 45.000000 0 0 "Curve 1" "Curve 2" "Curve 3" "Curve 4" "Curve 5" "Curve 6" }}}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 50 "T he following picture shows the stability region. " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1658 "R := z -> \+ 1+z+1/2*z^2+1/6*z^3+1/24*z^4+1/120*z^5+1/500*z^6:\npts := []: z0 := 0: tt := 0: \nwhile tt<=161/20 do\n zz := newton(R(z)=exp(tt*Pi*I),z=z 0):\n z0 := zz:\n if (1/2<=tt and tt<=1) or (139/20<=tt and tt<=14 9/20) then\n hh := 1/60\n else \n hh := 1/20\n end if;\n tt := tt+hh;\n pts := [op(pts),[Re(zz),Im(zz)]]:\nend do:\np1 := \+ plot(pts,color=COLOR(RGB,.13,.38,.18)):\np2 := plots[polygonplot]([seq ([pts[i-1],pts[i],[-1.55,0]],i=2..nops(pts))],\n style=patchn ogrid,color=COLOR(RGB,.25,.75,.35)):\npts := []: z0 := 1.2+3.75*I: tt \+ := 0: \nwhile tt<=51/25 do\n zz := newton(R(z)=exp(tt*Pi*I),z=z0):\n z0 := zz:\n if (11/25<=tt and tt<=33/25) then\n hh := 1/50\n else \n hh := 1/25\n end if;\n tt := tt+hh;\n pts := [op (pts),[Re(zz),Im(zz)]]:\nend do:\np3 := plot(pts,color=COLOR(RGB,.13,. 38,.18)):\np4 := plots[polygonplot]([seq([pts[i-1],pts[i],[1.14,3.57]] ,i=2..nops(pts))],\n style=patchnogrid,color=COLOR(RGB,.25,.7 5,.35)):\npts := []: z0 := 1.2-3.75*I: tt := 0: \nwhile tt<=51/25 do\n zz := newton(R(z)=exp(tt*Pi*I),z=z0):\n z0 := zz:\n if (18/25<= tt and tt<=8/5) then\n hh := 1/50\n else \n hh := 1/25\n \+ end if;\n tt := tt+hh;\n pts := [op(pts),[Re(zz),Im(zz)]]:\nend d o:\np5 := plot(pts,color=COLOR(RGB,.13,.38,.18)):\np6 := plots[polygon plot]([seq([pts[i-1],pts[i],[1.14,-3.57]],i=2..nops(pts))],\n \+ style=patchnogrid,color=COLOR(RGB,.25,.75,.35)):\np7 := plot([[[-3.69 ,0],[1.69,0]],[[0,-4.09],[0,4.09]]],color=black,linestyle=3):\nplots[d isplay]([p||(1..7)],view=[-3.69..1.69,-4.09..4.09],font=[HELVETICA,9], \n color=COLOR(RGB,.4,0,1),labels=[`Re(z)`,`Im(z)`],axes=boxed,scali ng=constrained);" }}{PARA 13 "" 1 "" {GLPLOT2D 469 558 558 {PLOTDATA 2 "60-%'CURVESG6$7fw7$$\"\"!F)F(7$$\"3K******QKx-!*!#E$\"33+++3jzq:!#= 7$$\"3_+++i8)\\U&!#C$\"3C+++i-fTJF07$$\"37+++GgQ_b!#B$\"3>+++H#\\Br%F0 7$$\"3.+++)fd1k#!#A$\"3A+++v!**GG'F07$$\"3L+++1TY:yF@$\"3q*****Hs'o_yF 07$$\"37+++e0[_:!#@$\"33+++p+W?%*F07$$\"3-+++l>-\\$\"3*******f\"zk668F[p$\"3'*******GEOhFS7$$!3/+++jKb2KF[p$\"3'* *****H;Qt&>FS7$$!3G+++h*)f,RF[p$\"3>+++De]0?FS7$$!3#******HE*f-ZF[p$\" 3')*****\\$[F`?FS7$$!31+++3$z-i&F[p$\"37+++q5f+@FS7$$!3Q*****4tt&H%yF[p$\"3%)*****\\]nN>#FS7$$!3M+++R%)\\k\" *F[p$\"3@+++o?.RAFS7$$!3'******>woM1\"F0$\"3A+++16l$G#FS7$$!32+++UYqD7 F0$\"3=+++f*zsK#FS7$$!36+++MD?.9F0$\"3))*****fKh(pBFS7$$!3'******f?]cf \"F0$\"3@+++/G$4T#FS7$$!34+++M`N-=F0$\"37+++@mj]CFS7$$!3))*****H'*fA-# F0$\"33+++w-t)[#FS7$$!3%******z#[*RD#F0$\"3!*******oj4DDFS7$$!35+++#pm f\\#F0$\"3!)******o)\\'fDFS7$$!3))*****H!RYYFF0$\"3-+++!pTBf#FS7$$!3D+ ++SAw.IF0$\"3*******fKeJi#FS7$$!37+++2M0 v#FS7$$!3z*****4KM)4YF0$\"3'*******p]/%GFS7$$!3)******\\LJ%[kF0$ \"3)******\\#*Q/)GFS7$$!3))*****4Vr8?(F0$\"30+++(=I?\"HFS7$$!3o*****R? 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***3%FSFd\\yFg\\y-%*AXESSTYLEG6#%$BOXG-%(SCALINGG6#%,CONSTRAINEDG-%%FO NTG6$%*HELVETICAG\"\"*-%+AXESLABELSG6%%&Re(z)G%&Im(z)G-F]^y6#%(DEFAULT G-Fagn6&Fcgn$\"\"%!\"\"F($\"\"\"F)-%%VIEWG6$;$!$p$Ffgn$\"$p\"Ffgn;$!$4 %Ffgn$\"$4%Ffgn" 1 2 0 1 10 0 2 6 1 2 1 1.000000 45.000000 45.000000 0 0 "Curve 1" "Curve 2" "Curve 3" "Curve 4" "Curve 5" "Curve 6" "Curve 7" "Curve 8" }}}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 4 "The " }{TEXT 260 30 "interval of absolute stability" } {TEXT -1 89 " (or stability interval) is the intersection of the stab ility region with the real line." }}{PARA 0 "" 0 "" {TEXT -1 59 "For t his scheme the stability interval is (approximately) " }{XPPEDIT 18 0 "[-3.1123, 0];" "6#7$,$-%&FloatG6$\"&B6$!\"%!\"\"\"\"!" }{TEXT -1 2 ". " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 260 "W e can distort the boundary curve horizontally by taking the 11th root \+ of the real part of points along the curve. In this way we see that th e largest interval on the nonnegative imaginary axis that contains the origin and lies inside the stability region is " }{XPPEDIT 18 0 "[0, 1.2];" "6#7$\"\"!-%&FloatG6$\"#7!\"\"" }{TEXT -1 18 " approximately. " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 310 "R := z -> 1+z+1/2*z^2+1/6*z^3+1/24*z^4+1/120*z^5+1/5 00*z^6:\nDigits := 15:\npts := []: z0 := 0:\nfor ct from 0 to 43 do\n \+ zz := newton(R(z)=exp(ct*Pi/100*I),z=z0):\n z0 := zz:\n pts := [ op(pts),[surd(Re(zz),11),Im(zz)]]:\nend do:\nplot(pts,color=COLOR(RGB, 0,.7,.15),thickness=2,font=[HELVETICA,9]);\nDigits := 10:" }}{PARA 13 "" 1 "" {GLPLOT2D 261 276 276 {PLOTDATA 2 "6(-%'CURVESG6#7N7$$\"\"!F)F (7$$\"0#\\R15\"zs(!#;$\"0N(e`EfTJF-7$$\"0()zv([jF6!#:$\"0!p'oI&=$G'F-7 $$\"0BODydiS\"F3$\"0mKa&zxC%*F-7$$\"0u`u%QOW;F3$\"0+Wu0PmD\"F37$$\"0#3 ?Ms+c=F3$\"0Q(z2jzq:F37$$\"0-i7Gf%[?F3$\"0+YU_b\\)=F37$$\"0G(R[&egA#F3 $\"0!HIeY6*>#F37$$\"0pW'yUj\"R#F3$\"0$*\\shtK^#F37$$\"0'=6\\m9ZDF3$\"0 \\e$RAVFGF37$$\"0!GZaR,%p#F3$\"0O0;E!fTJF37$$\"02N]k&HLGF3$\"0U%=zsubM F37$$\"0))eM(=!e'HF3$\"0Nhip-*pPF37$$\"0867ai@4$F3$\"0p!enc0%3%F37$$\" 0i+mfsG@$F3$\"0=82.0#)R%F37$$\"0D!3pEKGLF3$\"0,F*G#\\Br%F37$$\"0$\\-J- #)QMF3$\"0hx[A'[E]F37$$\"0m#3xVgWNF3$\"0UQbS81M&F37$$\"0B5]6cek$F3$\"0 dQ!zusacF37$$\"0=<]M0Fu$F3$\"0tJVOC)ofF37$$\"0(Q+]_BNQF3$\"0t#*[2**GG' F37$$\"0wU)3X[BRF3$\"0aAzfXpf'F37$$\"0#Gl4vW2SF3$\"0_M3xc4\"pF37$$\"0( >RLD2(3%F3$\"0'34dT#\\A(F37$$\"0Hu:mdA;%F3$\"0l^P#z$)QvF37$$\"0\"Re$*Q %GB%F3$\"07\"QBno_yF37$$\"0n_/e/')H%F3$\"0R;Kedk;)F37$$\"0GEtpH#fVF3$ \"09IZ!e8![)F37$$\"0s#Q8DI9WF3$\"0-&*o([q$z)F37$$\"00YBakKY%F3$\"0bnlS Yr5*F37$$\"0^%G'eh`]%F3$\"0!RVp+W?%*F37$$\"0M?2td&RXF3$\"0uz[ejNt*F37$ $\"0-#)[\"=RkXF3$\"0N%zr#\\Y+\"!#97$$\"0#>(*GKsxXF3$\"0x;68?f.\"Ffu7$$ \"0dC8DXid%F3$\"0bm[8mr1\"Ffu7$$\"03f9%*\\Wb%F3$\"0c0aI%Q)4\"Ffu7$$\"0 n]i6p?]%F3$\"0B4B]r&H6Ffu7$$\"0t'\\z!egR%F3$\"0l#3RWsg6Ffu7$$\"0hKL_r4 ;%F3$\"0H+KqR=>\"Ffu7$$!04s**)[CaNF3$\"0E9#zP\"HA\"Ffu7$$!0@Rm!o?$Q%F3 $\"0XUX2VRD\"Ffu7$$!0I,J)>j0ZF3$\"0(*4u%R#\\G\"Ffu7$$!0cZq&3#p$\\F3$\" 0I!GOF&eJ\"Ffu7$$!0XEhWQs7&F3$\"0.#)**yDnM\"Ffu-%%FONTG6$%*HELVETICAG \"\"*-%*THICKNESSG6#\"\"#-%+AXESLABELSG6$Q!6\"Fjy-%&COLORG6&%$RGBGF($ \"\"(!\"\"$\"#:!\"#-%%VIEWG6$%(DEFAULTGFiz" 1 2 0 1 10 2 2 6 1 4 2 1.000000 45.000000 44.000000 0 0 "Curve 1" }}}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 123 "The relevant intersection poin t of the boundary curve with the imaginary axis can be determined more accurately as follows." }}{PARA 0 "" 0 "" {TEXT -1 86 "First we look \+ for points on the boundary curve either side of the intersection point . " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 111 "Digits := 15:\nz0 := 1.2*I:\nfor ct from 37 to 40 do \n newton(R(z)=exp(ct*Pi/100*I),z=z0);\nend do;\nDigits := 10:" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#^$$\"0kDV[V\\=\"!#=$\"0l#3RWsg6!#9" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#^$$\"0^-O7XNZ'!#>$\"0H+KqR=>\"!#9" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#^$$!0f&4*ffM9\"!#>$\"0E9#zP\"HA\"!#9" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#^$$!0\"[5t8RZ6!#=$\"0XUX2VRD\"!#9" } }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 107 "Then w e apply the bisection method to calculate the parameter value associat ed with the intersection point." }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 176 "real_part := proc(u)\n Re(newton(R(z)=exp(u*Pi*I),z=1.2*I)) \nend proc:\nDigits := 15:\nu0 := bisect('real_part'(u),u=0.37..0.40); \nnewton(R(z)=exp(u0*Pi*I),z=1.2*I);\nDigits := 10:" }}{PARA 11 "" 1 " " {XPPMATH 20 "6#>%#u0G$\"0jAGuxo)Q!#:" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#^#$\"0[zPjQ)=7!#9" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 " " 0 "" {TEXT -1 4 "The " }{TEXT 260 112 "largest interval on the nonne gative imaginary axis that contains the origin and lies inside the sta bility region" }{TEXT -1 5 " is " }{XPPEDIT 18 0 "[ 0, 1.2188 ]" "6#7 $\"\"!-%&FloatG6$\"&)=7!\"%" }{TEXT -1 18 " (approximately)." }} {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 37 "#-------- ----------------------------" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 80 "The stability function R* for the 7 stage, or der 4 scheme is given as follows." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 97 "subs(ee,subs(b=`b*`,Stabilit yFunction(4,7,'expanded'))):\n`R*` := unapply(%,z):\n'`R*`(z)'=`R*`(z) ;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/-%#R*G6#%\"zG,2\"\"\"F)F'F)*&#F) \"\"#F)*$)F'F,F)F)F)*&#F)\"\"'F)*$)F'\"\"$F)F)F)*&#F)\"#CF)*$)F'\"\"%F )F)F)*&#\")>nNS\"+++8uUF)*$)F'\"\"&F)F)F)*&#\")fw[>\",++RAG\"F)*$)F'F1 F)F)F)*&#F)\"&+]\"F)*$)F'\"\"(F)F)F)" }}}{PARA 0 "" 0 "" {TEXT -1 0 " " }}{PARA 0 "" 0 "" {TEXT -1 124 "We can find the point where the boun dary of the stability region intersects the negative real axis by solv ing the equation: " }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "`R*`(z) = 1;" "6#/-%#R*G6#%\"zG\"\"\"" }{TEXT -1 2 ". " }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 32 "z_0 := newton(`R*`(z)=1,z=-4.4);" } }{PARA 11 "" 1 "" {XPPMATH 20 "6#>%$z_0G$!+.1%\\Q%!\"*" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 320 "z_0 := newton(`R*`(z)=1,z=-4.4):\np_1 := plot([`R*`(z),1],z=-4.99..0.49,colo r=[red,blue]):\np_2 := plot([[[z_0,1]]$3],style=point,symbol=[circle,c ross,diamond],color=black):\np_3 := plot([[z_0,0],[z_0,1]],linestyle=3 ,color=COLOR(RGB,0,.5,0)):\nplots[display]([p_1,p_2,p_3],view=[-4.99.. 0.49,-0.07..1.47],font=[HELVETICA,9]);" }}{PARA 13 "" 1 "" {GLPLOT2D 436 269 269 {PLOTDATA 2 "6+-%'CURVESG6$7X7$$!3A++++++!*\\!#<$\"3)\\0U- q\"\\+FF*7$$!3cLLe(zv-$\\F*$\"3W^eCBm`qCF*7$$!3#pmm^f^0([F*$\"3Kq=RV0> cAF*7$$!3#3]()f)ee=[F*$\"3!R&R&=uw<3#F*7$$!3%QL3o F*7$$!3y*\\7>Iz\"3ZF*$\"3iae=Rn>YD[wV\"F*7$$!33nm64>3KXF*$\"3O&RBQ+b)*H\"F *7$$!3I+Dh]K`tWF*$\"3'e:IY*z;t6F*7$$!3\\L$3@f%)\\T%F*$\"3OGkb,@Kc5F*7$ $!3jm;u*Q?kI%F*$\"3)\\fx9oc!Q')!#=7$$!3E+]ZY%3S>%F*$\"3)eYz2&o)[%pF^o7 $$!3%omTR&=vxSF*$\"3!*QVvf(Q&yaF^o7$$!3/+]x(4o='RF*$\"3c*>!*G9]-F%F^o7 $$!3OF*$\"3m/J)*zJ`#)=F^o7$$!3?++]Qxz+NF*$\"3;e%zz-ozP\"F^o7$ $!3/++5QhU'Q$F*$\"3eg8<.)HA,\"F^o7$$!3mm;%zwlDG$F*$\"3skgU([?7$$ !3[LLBUd1fJF*$\"3G3OBzV6mcF\\r7$$!3eLL$4-XW0$F*$\"31\"*=NK'ehh%F\\r7$$ !3;+]n]iuKHF*$\"3oS([$HBh9SF\\r7$$!3ILL$z@A]#GF*$\"3lTC$G'=,WRF\\r7$$! 3!)**\\n!)=$oq#F*$\"3uf\"G$>_bwUF\\r7$$!3')**\\-InG%f#F*$\"3H)R^b_&*G# \\F\\r7$$!3ILL3sw&oZ#F*$\"3e9wqj55\")eF\\r7$$!3#HL3&R6-pBF*$\"3]U%*eQX uwpF\\r7$$!3#om;4([q_AF*$\"3-5)=(3\"H*e$)F\\r7$$!3xm;%z&\\)=8#F*$\"3iG Kn]uY#***F\\r7$$!3$***\\_o3rE?F*$\"3Yp0r(o4p:\"F^o7$$!3hmmhq*>J\">F*$ \"32uo!4gCKM\"F^o7$$!35++?e%pdz\"F*$\"3%zexf(>*Qb\"F^o7$$!3+++:w['4o\" F*$\"3[dALzkMzHdL\"F*$\"3=X%Q3Jlxf#F^o7$$!3z****\\%R. u@\"F*$\"36t#f8qf(QHF^o7$$!3pm;aLE=56F*$\"3_^X8%F^o7$$!3%H** \\(H-wtwF^o$\"3;A\"*[V>zRYF^o7$$!3cOLL)\\%eYlF^o$\"3mA[r.]-&>&F^o7$$!3 g.+vof`m`F^o$\"3)[XVoKpl%eF^o7$$!3Dkmm#)*3+B%F^o$\"3c[asS2k]lF^o7$$!3Y jm;()funIF^o$\"3!)f\"*zXc8etF^o7$$!3HBL3;r5:>F^o$\"3+6G+C]5d#)F^o7$$!3 _^****>q^f&)F\\r$\"3WE%>(3xlz\"*F^o7$$\"3OMnm\"G*fzNF\\r$\"3\\7y)GQWk. \"F*7$$\"3\"RLL8'ppV9F^o$\"3Cw:>97Jb6F*7$$\"3@0+D$4>8g#F^o$\"3u;fBDD5( H\"F*7$$\"3Q,+v#=6$4PF^o$\"3E%oZ[C\"4\\9F*7$$\"3!***************[F^o$ \"3,q54r$[Bj\"F*-%'COLOURG6&%$RGBG$\"*++++\"!\")$\"\"!Fj\\lFi\\l-F$6$7 S7$F($\"\"\"Fj\\l7$F3F_]l7$F=F_]l7$FGF_]l7$FQF_]l7$FenF_]l7$FjnF_]l7$F `oF_]l7$FeoF_]l7$FjoF_]l7$F_pF_]l7$FdpF_]l7$FipF_]l7$F^qF_]l7$FcqF_]l7 $FhqF_]l7$F^rF_]l7$FcrF_]l7$FhrF_]l7$F]sF_]l7$FbsF_]l7$FgsF_]l7$F\\tF_ ]l7$FatF_]l7$FftF_]l7$F[uF_]l7$F`uF_]l7$FeuF_]l7$FjuF_]l7$F_vF_]l7$Fdv F_]l7$FivF_]l7$F^wF_]l7$FcwF_]l7$FhwF_]l7$F]xF_]l7$FbxF_]l7$FgxF_]l7$F \\yF_]l7$FayF_]l7$FfyF_]l7$F[zF_]l7$F`zF_]l7$FezF_]l7$FjzF_]l7$F_[lF_] l7$Fd[lF_]l7$Fi[lF_]l7$F^\\lF_]l-Fc\\l6&Fe\\lFi\\lFi\\lFf\\l-F$6&7#7$$ !3W+++.1%\\Q%F*F_]l-%'SYMBOLG6#%'CIRCLEG-Fc\\l6&Fe\\lFj\\lFj\\lFj\\l-% &STYLEG6#%&POINTG-F$6&Fe`l-Fj`l6#%&CROSSGF]alF_al-F$6&Fe`l-Fj`l6#%(DIA MONDGF]alF_al-F$6%7$7$Fg`lFi\\lFf`l-%&COLORG6&Fe\\lFi\\l$\"\"&!\"\"Fi \\l-%*LINESTYLEG6#\"\"$-%%FONTG6$%*HELVETICAG\"\"*-%+AXESLABELSG6%Q\"z 6\"Q!Fdcl-F\\cl6#%(DEFAULTG-%%VIEWG6$;$!$*\\!\"#$\"#\\F_dl;$!\"(F_dl$ \"$Z\"F_dl" 1 2 0 1 10 0 2 9 1 4 2 1.000000 45.000000 45.000000 0 0 "C urve 1" "Curve 2" "Curve 3" "Curve 4" "Curve 5" "Curve 6" }}}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 82 "The following pic ture shows the stability region for the 7 stage, order 4 scheme. " }} {PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1823 "`R*` := z -> 1+z+1/2*z^2+1/6*z^3+1/24*z^4+40356719/4274130000*z^ 5+19487659/12822390000*z^6+1/15000*z^7:\npts := []: z0 := 0: tt := 0: \+ \nwhile tt<=241/30 do\n zz := newton(`R*`(z)=exp(tt*Pi*I),z=z0):\n \+ z0 := zz:\n if (1/3<=tt and tt<=29/30) \n or (211/30<=tt and tt <=23/3) then\n hh := 1/60\n else \n hh := 1/30\n end if; \n tt := tt+hh;\n pts := [op(pts),[Re(zz),Im(zz)]]:\nend do:\np_1 \+ := plot(pts,color=COLOR(RGB,.1,.33,.15)):\np_2 := plots[polygonplot]([ seq([pts[i-1],pts[i],[-2.2,0]],i=2..nops(pts))],\n style=patc hnogrid,color=COLOR(RGB,.2,.65,.3)):\npts := []: z0 := 1+3.8*I: tt := \+ 0:\nwhile tt<=61/30 do\n zz := newton(`R*`(z)=exp(tt*Pi*I),z=z0):\n \+ z0 := zz:\n if (17/30<=tt and tt<=29/30) then\n hh := 1/120\n elif (13/30<=tt and tt<=19/15) then\n hh := 1/60\n else \n \+ hh := 1/30\n end if;\n tt := tt+hh;\n pts := [op(pts),[Re(zz ),Im(zz)]]:\nend do:\np_3 := plot(pts,color=COLOR(RGB,.1,.33,.15)):\np _4 := plots[polygonplot]([seq([pts[i-1],pts[i],[0.94,3.61]],i=2..nops( pts))],\n style=patchnogrid,color=COLOR(RGB,.2,.65,.3)):\npts := []: z0 := 1-3.8*I: tt := 0:\nwhile tt<=61/30 do\n zz := newton(` R*`(z)=exp(tt*Pi*I),z=z0):\n z0 := zz:\n if (16/15<=tt and tt<=22/ 15) then\n hh := 1/120\n elif (23/30<=tt and tt<=8/5) then\n \+ hh := 1/60\n else \n hh := 1/30\n end if;\n tt := tt+hh; \n pts := [op(pts),[Re(zz),Im(zz)]]:\nend do:\np_5 := plot(pts,color =COLOR(RGB,.1,.33,.15)):\np_6 := plots[polygonplot]([seq([pts[i-1],pts [i],[0.94,-3.61]],i=2..nops(pts))],\n style=patchnogrid,color =COLOR(RGB,.2,.65,.3)):\np_7 := plot([[[-5.09,0],[1.29,0]],[[0,-4.29], [0,4.29]]],color=black,linestyle=3):\nplots[display]([p_||(1..7)],view =[-5.09..1.29,-4.29..4.29],font=[HELVETICA,9],\n color=COLOR(RGB,.4, 0,1),labels=[`Re(z)`,`Im(z)`],axes=boxed,scaling=constrained);" }} {PARA 13 "" 1 "" {GLPLOT2D 486 581 581 {PLOTDATA 2 "60-%'CURVESG6$7d\\ 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6" "Curve 7" "Curve 8" }}}}{PARA 0 "" 0 " " {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 74 "The stability interval \+ of the order 4 embedded scheme is (approximately) " }{XPPEDIT 18 0 "[ -4.3849, 0];" "6#7$,$-%&FloatG6$\"&\\Q%!\"%!\"\"\"\"!" }{TEXT -1 2 ". \+ " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 79 "The s tability regions for the two schemes can be displayed in the same pict ure." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 241 "pp := plot([[[-5.09,0],[1.69,0]],[[0,-4.29],[0,4.29] ]],color=black,linestyle=3):\nplots[display]([pp,p||(1..6),p_||(1..6)] ,view=[-5.09..1.69,-4.29..4.29],font=[HELVETICA,9],\n labels =[`Re(z)`,`Im(z)`],axes=boxed,scaling=constrained);" }}{PARA 13 "" 1 " " {GLPLOT2D 445 617 617 {PLOTDATA 2 "65-%'CURVESG6%7$7$$!3')********** ***3&!#<$\"\"!F,7$$\"3%**************o\"F*F+-%'COLOURG6&%$RGBGF,F,F,-% *LINESTYLEG6#\"\"$-F$6%7$7$F+$!3/++++++!H%F*7$F+$\"3/++++++!H%F*F0F4-F 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/QHQFi\\oFhhilF^^clFdbs-%*AXESSTYLEG6#%$BOXG-%(SCALINGG6#%,CONSTRAINED G-%%FONTG6$%*HELVETICAG\"\"*-%+AXESLABELSG6%%&Re(z)G%&Im(z)G-F`\\\\m6# %(DEFAULTG-%%VIEWG6$;$!$4&F_in$\"$p\"F_in;$!$H%F_in$\"$H%F_in" 1 2 0 1 10 0 2 9 1 2 1 1.000000 45.000000 45.000000 0 0 "Curve 1" "Curve 2" "Curve 3" "Curve 4" "Curve 5" "Curve 6" "Curve 7" "Curve 8" "Curve 9" "Curve 10" "Curve 11" "Curve 12" "Curve 13" "Curve 14" }}}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}} {PARA 0 "" 0 "" {TEXT -1 39 "#--------------------------------------" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}}{PARA 0 "" 0 "" {TEXT -1 55 "#======================================================" }}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 15 "A scheme with " }{XPPEDIT 18 0 "t[6]=1/600" "6#/&%\"tG6#\"\"'*&\"\"\"F)\"$+'!\"\"" }{TEXT -1 35 " \+ and minimum principal error norm " }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}{PARA 0 "" 0 "" {TEXT -1 39 "#------------------ --------------------" }}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 28 "checking \+ the combined scheme" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 35 "c oefficients of the combined scheme" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1192 "ee := \{c[2]=64/315,\nc[3] =115/381,\nc[4]=762/935,\nc[5]=25/28,\nc[6]=1,\nc[7]=1,\n\na[2,1]=64/3 15,\na[3,1]=480815/6193536,\na[3,2]=462875/2064512,\na[4,1]=3449048252 19069/345923838700000,\na[4,2]=-2360077908867/601606676000,\na[4,3]=40 439332863108/10810119959375,\na[5,1]=12078745127989699/500969915778662 4,\na[5,2]=-791781731775/81669668864,\na[5,3]=39297175833216951/469446 1413969824,\na[5,4]=-10508413393960625/54097233987826176,\na[6,1]=2251 421737440863/828701767536000,\na[6,2]=-39895842357/3782730880,\na[6,3] =34564628685305112534/3916944972468643375,\na[6,4]=12051135495733565/3 6492943960723917,\na[6,5]=-26808346215168/82592376030125,\na[7,1]=2405 713/26289000,\na[7,2]=0,\na[7,3]=63896466577779/141024193000600,\na[7, 4]=454128848141375/589615117674696,\na[7,5]=-1359311744/2892576375,\na [7,6]=256979/1656648,\n\nb[1]=2405713/26289000,\nb[2]=0,\nb[3]=6389646 6577779/141024193000600,\nb[4]=454128848141375/589615117674696,\nb[5]= -1359311744/2892576375,\nb[6]=256979/1656648,\n\n`b*`[1]=1818563883019 /20194131951000,\n`b*`[2]=0,\n`b*`[3]=5513862498202899713/120365558967 94210600,\n`b*`[4]=324806515311046773125/452918159177876804664,\n`b*`[ 5]=-126112324722496/317422653663375,\n`b*`[6]=137695258717/12725690710 32,\n`b*`[7]=1/42\}:" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}}{PARA 0 "" 0 "" {TEXT -1 0 "" }} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 137 "subs(ee,matrix([seq([c[i],s eq(a[i,j],j=1..i-1),``$(8-i)],i=2..7),\n[`b`,seq(b[i],i=1..6),``],[`b* `,seq(`b*`[i],i=1..7)]]));\n``;\nevalf(%%);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#K%'matrixG6#7*7*#\"#k\"$:$F(%!GF+F+F+F+F+7*#\"$:\"\"$\" Q#\"':3[\"(ON>'#\"'vGY\"(7X1#F+F+F+F+F+7*#\"$i(\"$N*#\"0p!>_#[!\\M\"0+ +qQQ#fM#!.n)3z2gB\"-+gn1;g#\"/3J'GLR/%\"/v$f*>,\"3\"F+F+F+F+7*#\"#D\"# G#\"2*p*)z7X(y?\"\"1Cmyd\"*p4]#!-vC59#\"0v89[)GTX\"0'puw6:'*e#!+WI)Qc==\"/+5&>8%>?Fbo#\"48(**G? )\\iQ^&\"5+1@%z'*ebO?\"#\"6DJxY5J:l![K\"6kY!o(y26CF;$!+5,$\\p*F;$\"+wd'4P)F;$!+\\Z]U>F*F+F+F+7*$\" \"\"\"\"!$\"+id!or#F;$!+/Ooa5!\")$\"+ZXQC))F;$\"+\\$>BI$F*$!+)3ieC$F*F +F+7*FJ$\"+W^-^\"*F1$FLFL$\"+!)o)3`%F*$\"+ZN7-xF*$!+1:J*p%F*$\"+l&)>^: F*F+7*%\"bGFYFenFfnFhnFjnF\\oF+7*%#b*G$\"+[vS0!*F1Fen$\"+9/$4e%F*$\"+o tTrrF*$!+!f4I(RF*$\"+;e-#3\"F*$\"+\"Q_4Q#F1Q)pprint216\"" }}}{PARA 0 " " 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 146 "RK5_ 6eqs := [op(RowSumConditions(6,'expanded')),op(OrderConditions(5,6,'ex panded'))]:\n`RK4_7eqs*` := subs(b=`b*`,OrderConditions(4,7,'expanded' )):" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 7 "Ch eck: " }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 148 "simplify(subs(ee,R K5_6eqs)):\nmap(u->`if`(lhs(u)=rhs(u),0,1),%);\nnops(%);\nsimplify(sub s(ee,`RK4_7eqs*`)):\nmap(u->`if`(lhs(u)=rhs(u),0,1),%);\nnops(%);" } {TEXT -1 0 "" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#78\"\"!F$F$F$F$F$F$F$F $F$F$F$F$F$F$F$F$F$F$F$F$F$" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#\"#A" } }{PARA 11 "" 1 "" {XPPMATH 20 "6#7*\"\"!F$F$F$F$F$F$F$" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#\"\")" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 92 "We can calculate the principal error norm, that i s, the 2-norm of the principal error terms." }}{EXCHG {PARA 0 "> " 0 " " {MPLTEXT 1 0 123 "errterms5_6 := PrincipalErrorTerms(5,6,'expanded') :\nsqrt(add(subs(ee,errterms5_6[i])^2,i=1.. nops(errterms5_6))):\neval f(%);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#$\"+Mb:$)R!#8" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 87 "We can also calculat e the 2 norm of the principal error of the order 4 embedded scheme." } }{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 145 "`errterms4_7*` := subs(b=` b*`,PrincipalErrorTerms(4,7,'expanded')):\nsqrt(add(subs(ee,`errterms4 _7*`[i])^2,i=1.. nops(`errterms4_7*`))):\nevalf(%);" }}{PARA 11 "" 1 " " {XPPMATH 20 "6#$\"+*QAe;\"!#7" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }} {PARA 0 "" 0 "" {TEXT -1 86 "Next we set-up stage-order condtions to c heck for stage-orders from 2 to 5 inclusive. " }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 85 "for ct from 2 to 4 do\n so||ct||_6 := StageOrd erConditions(ct,6,'expanded');\nend do:" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 58 "Stages 3 to 6 have the following r espective stage-orders. " }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 173 "[seq([seq(expand(subs(ee,so||i||_6[j])),i=2..4)],j=1..4)]:\nmap(proc( L) local i; for i to nops(L) do if not evalb(L[i]) then break end if e nd do; i end proc,%):\nsimplify(%); " }}{PARA 11 "" 1 "" {XPPMATH 20 " 6#7&\"\"#F$F$F$" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 27 "The simplifying conditions:" }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "Sum(b[i]*a[i,j],i = j+1 .. 6) = b[j]*(1-c[j]) " "6#/-%$SumG6$*&&%\"bG6#%\"iG\"\"\"&%\"aG6$F+%\"jGF,/F+;,&F0F,F,F,\" \"'*&&F)6#F0F,,&F,F,&%\"cG6#F0!\"\"F," }{TEXT -1 3 ", " }{XPPEDIT 18 0 "j=1" "6#/%\"jG\"\"\"" }{TEXT -1 7 " . . 5 " }}{PARA 0 "" 0 "" {TEXT -1 8 "(where " }{XPPEDIT 18 0 "c[1]=0" "6#/&%\"cG6#\"\"\"\"\"! " }{TEXT -1 17 " ) are satisfied." }}{PARA 256 "" 0 "" {TEXT -1 1 " " }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 154 "[Sum(b[i]*a[i,1],i=1+1..6 )=b[1],seq(Sum(b[i]*a[i,j],i=j+1..6)=b[j]*(1-c[j]),j=2..5)];\neval(sub s(Sum=add,%)):\nsubs(ee,%);\nmap(u->lhs(u)-rhs(u),%);\nnops(%);" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#7'/-%$SumG6$*&&%\"bG6#%\"iG\"\"\"&%\"a G6$F,F-F-/F,;\"\"#\"\"'&F*6#F-/-F&6$*&F)F-&F/6$F,F3F-/F,;\"\"$F4*&&F*6 #F3F-,&F-F-&%\"cGFB!\"\"F-/-F&6$*&F)F-&F/6$F,F?F-/F,;\"\"%F4*&&F*6#F?F -,&F-F-&FEFRFFF-/-F&6$*&F)F-&F/6$F,FOF-/F,;\"\"&F4*&&F*6#FOF-,&F-F-&FE FjnFFF-/-F&6$*&F)F-&F/6$F,FgnF-/F,;F4F4*&&F*6#FgnF-,&F-F-&FEFgoFFF-" } }{PARA 11 "" 1 "" {XPPMATH 20 "6#7'/#\"(8dS#\")+!*GEF%/\"\"!F)/#\"-f8E 2x;\"-+\"Rh;I&F+/#\"-DGI*p&[\"._4()z\"3MF//#!)[oa[\"*D@>k*F3" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#7'\"\"!F$F$F$F$" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#\"\"&" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 27 "The simplifying condition: " }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "Sum(b[i]*a[i,2],i = 3 .. 6)=0" "6#/-%$S umG6$*&&%\"bG6#%\"iG\"\"\"&%\"aG6$F+\"\"#F,/F+;\"\"$\"\"'\"\"!" } {TEXT -1 1 " " }}{PARA 0 "" 0 "" {TEXT -1 13 "is satisfied." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 59 "Su m(b[i]*a[i,2],i=3..6);\neval(subs(Sum=add,%));\nsubs(ee,%);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#-%$SumG6$*&&%\"bG6#%\"iG\"\"\"&%\"aG6$F*\"\" #F+/F*;\"\"$\"\"'" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#,**&&%\"bG6#\"\"$ \"\"\"&%\"aG6$F(\"\"#F)F)*&&F&6#\"\"%F)&F+6$F1F-F)F)*&&F&6#\"\"&F)&F+6 $F7F-F)F)*&&F&6#\"\"'F)&F+6$F=F-F)F)" }}{PARA 11 "" 1 "" {XPPMATH 20 " 6#\"\"!" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 27 "The simplifying condition: " }}{PARA 256 "" 0 "" {TEXT -1 1 " " } {XPPEDIT 18 0 "Sum(b[i]*Sum(a[i,j]*a[j,2],j = 3 .. i-1),i = 2 .. 6)=0 " "6#/-%$SumG6$*&&%\"bG6#%\"iG\"\"\"-F%6$*&&%\"aG6$F+%\"jGF,&F16$F3\" \"#F,/F3;\"\"$,&F+F,F,!\"\"F,/F+;F6\"\"'\"\"!" }{TEXT -1 1 " " }} {PARA 0 "" 0 "" {TEXT -1 13 "is satisfied." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 82 "Sum(b[i]*Sum(a[i,j ]*a[j,2],j=3..i-1),i =2 ..6);\neval(subs(Sum=add,%));\nsubs(ee,%);" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#-%$SumG6$*&&%\"bG6#%\"iG\"\"\"-F$6$*&& %\"aG6$F*%\"jGF+&F06$F2\"\"#F+/F2;\"\"$,&F*F+F+!\"\"F+/F*;F5\"\"'" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#,(*(&%\"bG6#\"\"%\"\"\"&%\"aG6$F(\"\"$ F)&F+6$F-\"\"#F)F)*&&F&6#\"\"&F),&*&&F+6$F4F-F)F.F)F)*&&F+6$F4F(F)&F+6 $F(F0F)F)F)F)*&&F&6#\"\"'F),(*&&F+6$FAF-F)F.F)F)*&&F+6$FAF(F)F " 0 "" {MPLTEXT 1 0 64 "Sum(b[i]*c[i ]*a[i,2],i=3..6);\neval(subs(Sum=add,%));\nsubs(ee,%);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#-%$SumG6$*(&%\"bG6#%\"iG\"\"\"&%\"cGF)F+&%\"aG6$F* \"\"#F+/F*;\"\"$\"\"'" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#,**(&%\"bG6# \"\"$\"\"\"&%\"cGF'F)&%\"aG6$F(\"\"#F)F)*(&F&6#\"\"%F)&F+F2F)&F-6$F3F/ F)F)*(&F&6#\"\"&F)&F+F9F)&F-6$F:F/F)F)*(&F&6#\"\"'F)&F+F@F)&F-6$FAF/F) F)" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#\"\"!" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 78 "None of the 20 principal error \+ conditions are satisfied by the order 5 scheme." }}{EXCHG {PARA 0 "> \+ " 0 "" {MPLTEXT 1 0 123 "RK5_6err_eqs := PrincipalErrorConditions(5,6, 'expanded'):\nexpand(subs(ee,RK5_6err_eqs)):\nmap(u->`if`(lhs(u)=rhs(u ),0,1),%);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#76\"\"\"F$F$F$F$F$F$F$F$ F$F$F$F$F$F$F$F$F$F$F$" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 94 "None of the 9 principal error conditions are satisfi ed by the 7 stage embedded order 4 scheme." }}{PARA 0 "" 0 "" {TEXT -1 95 "The largest magnitude of an error term is about 59 times the lo west magnitude of an error term." }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 147 "`errterms4_7*` := subs(b=`b*`,PrincipalErrorTerms(4,7,'expand ed')):\nsubs(ee,`errterms4_7*`):\nmap(abs,evalf[8](%));\nevalf[4](max( op(%))/min(op(%)));" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#7+$\")i&Hs(!#6$ \");n:KF&$\")vf?zF&$\")aW25F&$\")$4y*R!#7$\")&*f9YF/$\")&=c*zF/$\")0*= 2\"F&$\")JgK8F/" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#$\"%Uf!\"#" }}} {PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}}{PARA 0 "" 0 "" {TEXT -1 39 "#------------------------------ --------" }}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 26 "determination of the \+ nodes" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 2 "e2" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 10256 "e2 := \{c[7] = 1, a[6,2] = 1/2*(-15*c[4]*c[3]^2+30*c[5]*c[4]*c[3]^2-3 0*c[3]*c[4]*c[5]+16*c[4]*c[3]-5*c[3]*c[5]+3*c[3]+10*c[4]*c[5]-6*c[4])/ (-12+15*c[3]+15*c[4]+15*c[5]-20*c[4]*c[3]-20*c[4]*c[5]+30*c[3]*c[4]*c[ 5]-20*c[3]*c[5])/c[3]/c[2], c[6] = 1, a[2,1] = c[2], a[5,4] = 1/2*(c[4 ]*c[3]-c[3]*c[5]-c[4]*c[5]+c[5]^2)*c[5]*(5*c[3]-2)/(c[3]-c[4])/(3-5*c[ 3]+10*c[4]*c[3]-5*c[4])/c[4], `b*`[2] = 0, b[2] = 0, b[4] = 1/60*(10*c [3]*c[5]-5*c[3]-5*c[5]+3)/(-c[4]^2+c[4]+c[4]*c[3]-c[3])/c[4]/(c[4]-c[5 ]), a[7,1] = 1/60*(5*c[4]-10*c[4]*c[5]+30*c[3]*c[4]*c[5]-10*c[4]*c[3]- 10*c[3]*c[5]-3+5*c[3]+5*c[5])/c[3]/c[4]/c[5], a[6,4] = 1/2*(6-30*c[5]^ 2*c[3]+20*c[5]^2*c[3]^2-10*c[5]^2*c[4]-16*c[3]-8*c[4]-14*c[5]-7*c[4]^2 *c[3]+30*c[5]^2*c[3]*c[4]+25*c[5]*c[4]*c[3]^2-20*c[5]^2*c[4]*c[3]^2+23 *c[4]*c[3]+14*c[4]*c[5]-39*c[3]*c[4]*c[5]+39*c[3]*c[5]-15*c[4]*c[3]^2+ 10*c[3]^2+2*c[4]^2+10*c[5]^2-25*c[3]^2*c[5]+5*c[4]^2*c[3]^2)/c[4]/(12* c[4]^2-12*c[4]*c[3]-15*c[4]^3-12*c[4]*c[5]+12*c[3]*c[5]-20*c[4]^2*c[5] ^2+20*c[4]^3*c[5]+20*c[4]^3*c[3]-20*c[4]^2*c[3]^2+20*c[5]^2*c[3]^2-20* c[4]^2*c[5]*c[3]+15*c[4]*c[3]^2+15*c[5]^2*c[4]+30*c[5]^2*c[4]^2*c[3]+3 0*c[5]*c[4]^2*c[3]^2-30*c[5]^2*c[4]*c[3]^2-30*c[5]*c[4]^3*c[3]+15*c[3] *c[4]*c[5]-15*c[3]^2*c[5]-15*c[5]^2*c[3]), `b*`[1] = 1/60*(-460*c[5]*c [4]*c[3]*`b*`[7]+760*c[5]*c[4]*c[3]^2*`b*`[7]+3*c[3]-6*c[4]-20*c[4]^2* c[5]-60*c[4]^2*c[3]-1800*`b*`[7]*c[4]^2*c[5]*c[3]^2+100*c[5]*c[4]*c[3] ^2+100*c[3]^2*c[5]*`b*`[7]-100*c[3]^3*c[5]*c[4]+300*c[3]^3*c[4]*`b*`[7 ]-600*c[3]^3*`b*`[7]*c[4]^2+940*`b*`[7]*c[4]^2*c[3]^2-60*c[5]*c[3]*`b* `[7]-560*c[4]^2*c[3]*`b*`[7]+300*c[5]*c[4]^2*c[3]^3-100*c[4]^2*c[3]^3+ 140*c[4]^2*c[5]*c[3]+120*`b*`[7]*c[4]^2+120*c[5]*c[4]*`b*`[7]-340*c[5] *c[4]^2*c[3]^2-200*`b*`[7]*c[4]^2*c[5]+1000*c[5]*c[4]^2*c[3]*`b*`[7]-6 00*c[3]^3*c[5]*c[4]*`b*`[7]+29*c[4]*c[3]-380*`b*`[7]*c[4]*c[3]^2+252*` b*`[7]*c[4]*c[3]+10*c[4]*c[5]-50*c[3]*c[4]*c[5]+1200*c[3]^3*`b*`[7]*c[ 4]^2*c[5]+36*`b*`[7]*c[3]-72*`b*`[7]*c[4]-60*`b*`[7]*c[3]^2-5*c[3]*c[5 ]-60*c[4]*c[3]^2-5*c[3]^2+10*c[4]^2+10*c[3]^2*c[5]+50*c[4]*c[3]^3+130* c[4]^2*c[3]^2)/(10*c[4]*c[3]^2-8*c[4]*c[3]-c[3]+2*c[4])/c[3]/c[4]/c[5] , a[6,1] = 1/2*(10*c[3]*c[2]*c[5]^2-14*c[3]*c[2]*c[5]-30*c[3]^2*c[2]*c [5]^2-25*c[3]^3*c[2]*c[5]+10*c[3]^3*c[2]-108*c[3]^2*c[2]*c[4]*c[5]-3*c [5]*c[4]*c[3]^2+5*c[5]^2*c[4]*c[3]^2-30*c[5]^2*c[4]^2*c[3]^3+30*c[5]^2 *c[4]^2*c[3]^2-10*c[5]^2*c[4]^2*c[3]+15*c[5]*c[4]^2*c[3]^3+6*c[4]^2*c[ 5]*c[3]-30*c[3]^2*c[2]*c[4]^2-30*c[3]^3*c[4]*c[2]+10*c[3]*c[2]*c[4]^2- 16*c[3]*c[2]*c[4]-16*c[5]*c[4]^2*c[3]^2+39*c[3]*c[2]*c[4]*c[5]-30*c[3] *c[2]*c[5]^2*c[4]-9*c[3]*c[2]*c[5]*c[4]^2+6*c[3]*c[2]+10*c[2]*c[4]^2*c [5]^2-10*c[3]*c[2]*c[4]^2*c[5]^2-60*c[3]^3*c[2]*c[5]*c[4]^2-40*c[3]^2* c[2]*c[4]^2*c[5]^2+46*c[3]^2*c[2]*c[4]-16*c[2]*c[3]^2-6*c[2]*c[5]*c[4] ^2+39*c[3]^2*c[2]*c[5]+60*c[2]*c[4]^2*c[5]^2*c[3]^3+20*c[3]^3*c[2]*c[4 ]^2+20*c[3]^3*c[2]*c[5]^2-60*c[3]^3*c[2]*c[5]^2*c[4]+75*c[3]^3*c[2]*c[ 4]*c[5]+80*c[3]^2*c[2]*c[5]^2*c[4]+65*c[3]^2*c[2]*c[5]*c[4]^2)/(-12+15 *c[3]+15*c[4]+15*c[5]-20*c[4]*c[3]-20*c[4]*c[5]+30*c[3]*c[4]*c[5]-20*c [3]*c[5])/c[3]^2/c[2]/c[4]/c[5], a[7,2] = 0, b[3] = -1/60*(10*c[4]*c[5 ]-5*c[4]-5*c[5]+3)/(c[3]-1)/c[3]/(c[3]-c[5])/(c[3]-c[4]), a[6,3] = -1/ 2*(10*c[5]^2*c[3]-10*c[5]^2*c[3]^2+6*c[3]-6*c[4]^2*c[5]+16*c[4]^2*c[3] -40*c[5]^2*c[3]*c[4]-45*c[5]*c[4]*c[3]^2+60*c[5]^2*c[4]*c[3]^2-20*c[3] ^3*c[5]*c[4]-10*c[5]^2*c[4]^2*c[3]-20*c[5]*c[4]^2*c[3]^3+15*c[4]^2*c[3 ]^3-19*c[4]^2*c[5]*c[3]+10*c[4]^2*c[5]^2+45*c[5]*c[4]^2*c[3]^2-10*c[3] ^4*c[4]-20*c[4]*c[5]^2*c[3]^3-16*c[4]*c[3]+45*c[3]*c[4]*c[5]+20*c[3]^4 *c[5]*c[4]-14*c[3]*c[5]+17*c[4]*c[3]^2-8*c[3]^2+2*c[3]^3+14*c[3]^2*c[5 ]+9*c[4]*c[3]^3-31*c[4]^2*c[3]^2)/(-30*c[5]^2*c[3]*c[4]+20*c[5]^2*c[4] -15*c[4]*c[5]+15*c[4]*c[3]+30*c[5]*c[4]*c[3]^2-20*c[4]*c[3]^2+15*c[3]^ 2+12*c[5]-12*c[3]+20*c[5]^2*c[3]-15*c[5]^2-20*c[3]^2*c[5])/(c[3]-c[4]) /c[3]^2, a[4,1] = 1/2*c[4]*(2*c[2]*c[3]^2-3*c[4]*c[3]^2+2*c[4]^2*c[3]+ 2*c[3]*c[2]*c[4]-2*c[2]*c[4]^2)/c[2]/c[3]^2, a[5,3] = -1/2*c[5]*(-2*c[ 5]^2*c[3]+6*c[4]^2*c[5]-6*c[4]^2*c[3]+15*c[5]^2*c[3]*c[4]-5*c[5]*c[4]* c[3]^2-20*c[5]^2*c[4]*c[3]^2+20*c[3]^3*c[5]*c[4]+20*c[5]^2*c[4]^2*c[3] -5*c[4]^2*c[5]*c[3]-10*c[4]^2*c[5]^2-20*c[5]*c[4]^2*c[3]^2-4*c[3]*c[4] *c[5]+4*c[4]*c[3]^2+2*c[3]^2*c[5]-10*c[4]*c[3]^3+15*c[4]^2*c[3]^2)/(3- 5*c[3]+10*c[4]*c[3]-5*c[4])/c[3]^2/(c[3]-c[4]), a[4,3] = -c[4]^2*(c[3] -c[4])/c[3]^2, a[5,2] = 1/2*c[5]*(-15*c[4]*c[3]^2+30*c[5]*c[4]*c[3]^2- 20*c[5]^2*c[3]*c[4]+6*c[4]*c[3]-5*c[5]^2*c[3]+3*c[3]*c[5]-6*c[4]*c[5]+ 10*c[5]^2*c[4])/c[2]/(3-5*c[3]+10*c[4]*c[3]-5*c[4])/c[3], `b*`[5] = 1/ 60*(-460*c[5]*c[4]*c[3]*`b*`[7]+760*c[5]*c[4]*c[3]^2*`b*`[7]+3*c[3]-6* c[4]-60*c[4]^2*c[3]-1800*`b*`[7]*c[4]^2*c[5]*c[3]^2+100*c[3]^2*c[5]*`b *`[7]+300*c[3]^3*c[4]*`b*`[7]-600*c[3]^3*`b*`[7]*c[4]^2+940*`b*`[7]*c[ 4]^2*c[3]^2-60*c[5]*c[3]*`b*`[7]-560*c[4]^2*c[3]*`b*`[7]-100*c[4]^2*c[ 3]^3+120*`b*`[7]*c[4]^2+120*c[5]*c[4]*`b*`[7]-200*`b*`[7]*c[4]^2*c[5]+ 1000*c[5]*c[4]^2*c[3]*`b*`[7]-600*c[3]^3*c[5]*c[4]*`b*`[7]+29*c[4]*c[3 ]-380*`b*`[7]*c[4]*c[3]^2+252*`b*`[7]*c[4]*c[3]+1200*c[3]^3*`b*`[7]*c[ 4]^2*c[5]+36*`b*`[7]*c[3]-72*`b*`[7]*c[4]-60*`b*`[7]*c[3]^2-60*c[4]*c[ 3]^2-5*c[3]^2+10*c[4]^2+50*c[4]*c[3]^3+130*c[4]^2*c[3]^2)/c[5]/(c[5]^2 *c[3]-c[5]^3*c[3]+c[5]^2*c[3]^2-2*c[5]^2*c[4]+2*c[5]^3*c[4]+2*c[4]^2*c [5]-2*c[4]^2*c[3]+7*c[5]^2*c[3]*c[4]-8*c[5]^3*c[4]*c[3]-9*c[5]*c[4]*c[ 3]^2-2*c[5]^2*c[4]*c[3]^2+10*c[5]^3*c[4]*c[3]^2+10*c[3]^3*c[5]*c[4]-10 *c[5]^2*c[4]^2*c[3]^2+8*c[5]^2*c[4]^2*c[3]+10*c[5]*c[4]^2*c[3]^3-10*c[ 4]^2*c[3]^3-6*c[4]^2*c[5]*c[3]-2*c[4]^2*c[5]^2+2*c[5]*c[4]^2*c[3]^2-10 *c[4]*c[5]^2*c[3]^3+c[3]*c[4]*c[5]+c[4]*c[3]^2-c[3]^2*c[5]+8*c[4]^2*c[ 3]^2), a[4,2] = 1/2*(3*c[3]-2*c[4])*c[4]^2/c[3]/c[2], `b*`[6] = -1/60* (-1000*c[5]*c[4]*c[3]*`b*`[7]+1780*c[5]*c[4]*c[3]^2*`b*`[7]-12*c[3]+24 *c[4]+40*c[4]^2*c[5]+160*c[4]^2*c[3]-2880*`b*`[7]*c[4]^2*c[5]*c[3]^2-2 80*c[5]*c[4]*c[3]^2+160*c[3]^2*c[5]*`b*`[7]+200*c[3]^3*c[5]*c[4]+900*c [3]^3*c[4]*`b*`[7]-1200*c[3]^3*`b*`[7]*c[4]^2+2020*`b*`[7]*c[4]^2*c[3] ^2-120*c[5]*c[3]*`b*`[7]-1160*c[4]^2*c[3]*`b*`[7]-300*c[5]*c[4]^2*c[3] ^3+200*c[4]^2*c[3]^3-220*c[4]^2*c[5]*c[3]+240*`b*`[7]*c[4]^2+240*c[5]* c[4]*`b*`[7]+440*c[5]*c[4]^2*c[3]^2-320*`b*`[7]*c[4]^2*c[5]+1600*c[5]* c[4]^2*c[3]*`b*`[7]-1200*c[3]^3*c[5]*c[4]*`b*`[7]-111*c[4]*c[3]-1400*` b*`[7]*c[4]*c[3]^2+792*`b*`[7]*c[4]*c[3]-30*c[4]*c[5]+140*c[3]*c[4]*c[ 5]+1800*c[3]^3*`b*`[7]*c[4]^2*c[5]+96*`b*`[7]*c[3]-192*`b*`[7]*c[4]-12 0*`b*`[7]*c[3]^2+15*c[3]*c[5]+220*c[4]*c[3]^2+15*c[3]^2-30*c[4]^2-20*c [3]^2*c[5]-150*c[4]*c[3]^3-310*c[4]^2*c[3]^2)/(-10*c[4]*c[3]^2+10*c[5] *c[4]*c[3]^2+8*c[4]*c[3]-8*c[3]*c[4]*c[5]+c[3]-c[3]*c[5]-2*c[4]+2*c[4] *c[5])/(c[4]-1)/(c[3]-1), `b*`[3] = 1/60*(-460*c[5]*c[4]*c[3]*`b*`[7]+ 760*c[5]*c[4]*c[3]^2*`b*`[7]+3*c[3]-6*c[4]-20*c[4]^2*c[5]-40*c[4]^2*c[ 3]-1800*`b*`[7]*c[4]^2*c[5]*c[3]^2+50*c[5]*c[4]*c[3]^2+100*c[3]^2*c[5] *`b*`[7]+300*c[3]^3*c[4]*`b*`[7]-600*c[3]^3*`b*`[7]*c[4]^2+940*`b*`[7] *c[4]^2*c[3]^2-60*c[5]*c[3]*`b*`[7]-560*c[4]^2*c[3]*`b*`[7]+80*c[4]^2* c[5]*c[3]+120*`b*`[7]*c[4]^2+120*c[5]*c[4]*`b*`[7]-100*c[5]*c[4]^2*c[3 ]^2-200*`b*`[7]*c[4]^2*c[5]+1000*c[5]*c[4]^2*c[3]*`b*`[7]-600*c[3]^3*c [5]*c[4]*`b*`[7]+19*c[4]*c[3]-380*`b*`[7]*c[4]*c[3]^2+252*`b*`[7]*c[4] *c[3]+10*c[4]*c[5]-30*c[3]*c[4]*c[5]+1200*c[3]^3*`b*`[7]*c[4]^2*c[5]+3 6*`b*`[7]*c[3]-72*`b*`[7]*c[4]-60*`b*`[7]*c[3]^2-5*c[3]*c[5]-30*c[4]*c [3]^2+10*c[4]^2+50*c[4]^2*c[3]^2)/(c[3]-1)/c[3]/(-10*c[5]*c[4]*c[3]^2+ 8*c[3]*c[4]*c[5]-2*c[4]*c[5]+2*c[4]*c[3]+10*c[4]*c[3]^3-8*c[4]*c[3]^2+ c[3]*c[5]-c[3]^2)/(c[3]-c[4]), a[3,2] = 1/2*c[3]^2/c[2], a[7,6] = 1/60 *(-12+15*c[3]+15*c[4]+15*c[5]-20*c[4]*c[3]-20*c[4]*c[5]+30*c[3]*c[4]*c [5]-20*c[3]*c[5])/(c[5]-1)/(c[4]-1)/(c[3]-1), a[5,1] = 1/2*(2*c[3]*c[2 ]*c[5]^2-5*c[3]^2*c[2]*c[5]^2+5*c[3]^3*c[2]*c[5]+10*c[3]^2*c[2]*c[4]*c [5]-3*c[5]*c[4]*c[3]^2+5*c[5]^2*c[4]*c[3]^2+20*c[5]^2*c[4]^2*c[3]^2-10 *c[5]^2*c[4]^2*c[3]-30*c[5]*c[4]^2*c[3]^3+15*c[4]^2*c[3]^3+6*c[4]^2*c[ 5]*c[3]-25*c[3]^2*c[2]*c[4]^2-15*c[3]^3*c[4]*c[2]+6*c[3]*c[2]*c[4]^2-2 *c[3]*c[2]*c[4]*c[5]-5*c[3]*c[2]*c[5]^2*c[4]+5*c[3]*c[2]*c[5]*c[4]^2+1 0*c[2]*c[4]^2*c[5]^2-20*c[3]*c[2]*c[4]^2*c[5]^2+8*c[3]^2*c[2]*c[4]-6*c [2]*c[5]*c[4]^2-2*c[3]^2*c[2]*c[5]+20*c[3]^3*c[2]*c[4]^2+20*c[3]^2*c[2 ]*c[5]*c[4]^2-6*c[4]^2*c[3]^2)*c[5]/c[4]/c[3]^2/(3-5*c[3]+10*c[4]*c[3] -5*c[4])/c[2], a[6,5] = (-3+8*c[3]+8*c[4]+3*c[5]+5*c[4]^2*c[5]+15*c[4] ^2*c[3]-15*c[5]*c[4]*c[3]^2-15*c[4]^2*c[5]*c[3]+10*c[5]*c[4]^2*c[3]^2- 23*c[4]*c[3]-8*c[4]*c[5]+23*c[3]*c[4]*c[5]-8*c[3]*c[5]+15*c[4]*c[3]^2- 5*c[3]^2-5*c[4]^2+5*c[3]^2*c[5]-10*c[4]^2*c[3]^2)/c[5]/(-20*c[5]^3*c[3 ]+20*c[5]^2*c[3]^2-20*c[5]^3*c[4]-15*c[4]^2*c[5]+15*c[4]^2*c[3]+20*c[5 ]^2*c[3]*c[4]+30*c[5]^3*c[4]*c[3]-30*c[5]^2*c[4]*c[3]^2-30*c[5]^2*c[4] ^2*c[3]+20*c[4]^2*c[5]^2+30*c[5]*c[4]^2*c[3]^2-12*c[4]*c[3]+12*c[4]*c[ 5]-15*c[3]*c[4]*c[5]+15*c[5]^3+12*c[3]*c[5]+15*c[4]*c[3]^2-12*c[5]^2-1 5*c[3]^2*c[5]-20*c[4]^2*c[3]^2), a[7,3] = -1/60*(10*c[4]*c[5]-5*c[4]-5 *c[5]+3)/(c[3]-1)/c[3]/(c[3]-c[5])/(c[3]-c[4]), b[5] = -1/60*(3-5*c[3] +10*c[4]*c[3]-5*c[4])/c[5]/(-c[5]^2-c[5]^2*c[3]+c[5]^3+c[3]*c[5]-c[5]^ 2*c[4]+c[3]*c[4]*c[5]-c[4]*c[3]+c[4]*c[5]), b[1] = 1/60*(5*c[4]-10*c[4 ]*c[5]+30*c[3]*c[4]*c[5]-10*c[4]*c[3]-10*c[3]*c[5]-3+5*c[3]+5*c[5])/c[ 3]/c[4]/c[5], b[6] = 1/60*(-12+15*c[3]+15*c[4]+15*c[5]-20*c[4]*c[3]-20 *c[4]*c[5]+30*c[3]*c[4]*c[5]-20*c[3]*c[5])/(c[5]-1)/(c[4]-1)/(c[3]-1), a[3,1] = 1/2*c[3]*(2*c[2]-c[3])/c[2], `b*`[4] = -1/60*(-460*c[5]*c[4] *c[3]*`b*`[7]+760*c[5]*c[4]*c[3]^2*`b*`[7]+3*c[3]-6*c[4]-1800*`b*`[7]* c[4]^2*c[5]*c[3]^2+130*c[5]*c[4]*c[3]^2+100*c[3]^2*c[5]*`b*`[7]-100*c[ 3]^3*c[5]*c[4]+300*c[3]^3*c[4]*`b*`[7]-600*c[3]^3*`b*`[7]*c[4]^2+940*` b*`[7]*c[4]^2*c[3]^2-60*c[5]*c[3]*`b*`[7]-560*c[4]^2*c[3]*`b*`[7]+120* `b*`[7]*c[4]^2+120*c[5]*c[4]*`b*`[7]-200*`b*`[7]*c[4]^2*c[5]+1000*c[5] *c[4]^2*c[3]*`b*`[7]-600*c[3]^3*c[5]*c[4]*`b*`[7]+34*c[4]*c[3]-380*`b* `[7]*c[4]*c[3]^2+252*`b*`[7]*c[4]*c[3]+10*c[4]*c[5]-60*c[3]*c[4]*c[5]+ 1200*c[3]^3*`b*`[7]*c[4]^2*c[5]+36*`b*`[7]*c[3]-72*`b*`[7]*c[4]-60*`b* `[7]*c[3]^2-5*c[3]*c[5]-70*c[4]*c[3]^2-5*c[3]^2+10*c[3]^2*c[5]+50*c[4] *c[3]^3)/(-c[4]^2+c[4]+c[4]*c[3]-c[3])/c[4]/(-10*c[5]*c[4]*c[3]^2+10*c [4]^2*c[3]^2+8*c[3]*c[4]*c[5]+c[3]*c[5]-8*c[4]^2*c[3]-c[4]*c[3]-2*c[4] *c[5]+2*c[4]^2), a[7,4] = 1/60*(10*c[3]*c[5]-5*c[3]-5*c[5]+3)/(-c[4]^2 +c[4]+c[4]*c[3]-c[3])/c[4]/(c[4]-c[5]), a[7,5] = -1/60*(3-5*c[3]+10*c[ 4]*c[3]-5*c[4])/c[5]/(-c[5]^2-c[5]^2*c[3]+c[5]^3+c[3]*c[5]-c[5]^2*c[4] +c[3]*c[4]*c[5]-c[4]*c[3]+c[4]*c[5])\}:" }{TEXT -1 0 "" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}} {PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 49 "sm: square of principal \+ error norm in terms of " }{XPPEDIT 18 0 "c[2]" "6#&%\"cG6#\"\"#" } {TEXT -1 3 ", " }{XPPEDIT 18 0 "c[3]" "6#&%\"cG6#\"\"$" }{TEXT -1 3 " , " }{XPPEDIT 18 0 "c[4]" "6#&%\"cG6#\"\"%" }{TEXT -1 7 " and " } {XPPEDIT 18 0 "c[5]" "6#&%\"cG6#\"\"&" }{TEXT -1 1 " " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1665 "sm := 1 /25920000*(12042*c[5]^2*c[3]^2+297000*c[3]*c[2]^2*c[4]^2*c[5]-13500*c[ 3]^2*c[2]*c[5]^2+28044*c[3]^3*c[5]-270000*c[3]*c[2]^2*c[4]^2*c[5]^2-22 140*c[3]^4*c[5]-148500*c[3]^2*c[2]^2*c[4]*c[5]-270000*c[3]^2*c[2]^2*c[ 4]^2*c[5]+135000*c[3]^2*c[2]^2*c[4]*c[5]^2-22140*c[5]^2*c[3]^3+270000* c[3]^2*c[2]^2*c[4]^2*c[5]^2-18000*c[3]^2*c[2]*c[4]*c[5]+81000*c[3]*c[2 ]^2*c[5]*c[4]-67500*c[3]*c[2]^2*c[5]^2*c[4]+96444*c[5]*c[4]*c[3]^2-761 40*c[5]^2*c[4]*c[3]^2-613800*c[5]^2*c[4]^2*c[3]^3-221940*c[3]^3*c[5]*c [4]+153450*c[5]^2*c[4]^2*c[3]^2+675180*c[5]*c[4]^2*c[3]^3-240140*c[4]^ 2*c[3]^3+198000*c[3]^2*c[2]*c[4]^2-49500*c[3]^3*c[4]*c[2]-55800*c[3]*c [2]*c[4]^2+12000*c[3]*c[2]*c[4]-184140*c[5]*c[4]^2*c[3]^2-34200*c[3]*c [2]*c[4]*c[5]+27000*c[3]*c[2]*c[5]^2*c[4]+162000*c[3]*c[2]*c[5]*c[4]^2 +613800*c[4]^2*c[3]^4*c[5]^2-613800*c[4]^2*c[3]^4*c[5]-73800*c[4]*c[3] ^4*c[5]^2-42140*c[3]^4*c[4]+223450*c[4]^2*c[3]^4+189180*c[4]*c[5]^2*c[ 3]^3+18450*c[5]^2*c[3]^4+10642*c[3]^4-135000*c[3]*c[2]*c[4]^2*c[5]^2+2 7900*c[2]^2*c[4]^2+6975*c[3]^2*c[2]^2+540000*c[3]^3*c[2]*c[5]*c[4]^2+8 1180*c[3]^4*c[5]*c[4]+540000*c[3]^2*c[2]*c[4]^2*c[5]^2+6900*c[3]^2*c[2 ]*c[4]-6000*c[2]*c[3]^2-36856*c[4]*c[3]^2+17100*c[3]^2*c[2]*c[5]+6552* c[3]^2-12856*c[3]^3-27900*c[3]*c[2]^2*c[4]-99000*c[3]*c[2]^2*c[4]^2+49 500*c[3]^2*c[2]^2*c[4]+90000*c[3]^2*c[2]^2*c[4]^2-20250*c[3]^2*c[2]^2* c[5]+16875*c[3]^2*c[2]^2*c[5]^2-81000*c[5]*c[2]^2*c[4]^2+67500*c[5]^2* c[2]^2*c[4]^2-540000*c[2]*c[4]^2*c[5]^2*c[3]^3-180000*c[3]^3*c[2]*c[4] ^2-135000*c[3]^3*c[2]*c[5]^2*c[4]+148500*c[3]^3*c[2]*c[4]*c[5]+13500*c [3]^2*c[2]*c[5]^2*c[4]-594000*c[3]^2*c[2]*c[5]*c[4]^2-16056*c[3]^2*c[5 ]+88044*c[4]*c[3]^3+66442*c[4]^2*c[3]^2)/c[3]^2:" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 31 "The Prince-Dorm and scheme has " }{XPPEDIT 18 0 "c[3] = 3/10" "6#/&%\"cG6#\"\"$*&F'\" \"\"\"#5!\"\"" }{TEXT -1 7 " and " }{XPPEDIT 18 0 "c[4]=4/5" "6#/&% \"cG6#\"\"%*&F'\"\"\"\"\"&!\"\"" }{TEXT -1 6 ", so " }{XPPEDIT 18 0 " t[6]=1/600" "6#/&%\"tG6#\"\"'*&\"\"\"F)\"$+'!\"\"" }{TEXT -1 32 " and the stability function is " }}{PARA 256 "" 0 "" {TEXT -1 1 " " } {XPPEDIT 18 0 "R(z)= 1+z+1/2" "6#/-%\"RG6#%\"zG,(\"\"\"F)F'F)*&F)F)\" \"#!\"\"F)" }{TEXT -1 1 " " }{XPPEDIT 18 0 "z^2+1/6" "6#,&*$%\"zG\"\"# \"\"\"*&F'F'\"\"'!\"\"F'" }{TEXT -1 1 " " }{XPPEDIT 18 0 "z^3+1/24" "6 #,&*$%\"zG\"\"$\"\"\"*&F'F'\"#C!\"\"F'" }{TEXT -1 1 " " }{XPPEDIT 18 0 "z^4+1/120" "6#,&*$%\"zG\"\"%\"\"\"*&F'F'\"$?\"!\"\"F'" }{TEXT -1 1 " " }{XPPEDIT 18 0 "z^5+1/600;" "6#,&*$%\"zG\"\"&\"\"\"*&F'F'\"$+'!\" \"F'" }{TEXT -1 1 " " }{XPPEDIT 18 0 "z^6" "6#*$%\"zG\"\"'" }{TEXT -1 2 ", " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 67 "t[6]=1/240*(2-5*c[3])*c[4];\nsubs(\{c[3]=3/10,c[4]=4/ 5\},%);\nevalf(%);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/&%\"tG6#\"\"',$ *&#\"\"\"\"$S#F+*&,&\"\"#F+*&\"\"&F+&%\"cG6#\"\"$F+!\"\"F+&F36#\"\"%F+ F+F+" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/&%\"tG6#\"\"'#\"\"\"\"$+'" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#/&%\"tG6#\"\"'$\"+nmmm;!#7" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 119 "A 6 stage, ord er 5 Runge-Kutta scheme will have the same stability function as the P rince-Dormand scheme provided that " }}{PARA 256 "" 0 "" {TEXT -1 1 " \+ " }{XPPEDIT 18 0 "(2-5*c[3])*c[4]/240 = 1/600;" "6#/*(,&\"\"#\"\"\"*& \"\"&F'&%\"cG6#\"\"$F'!\"\"F'&F+6#\"\"%F'\"$S#F.*&F'F'\"$+'F." }{TEXT -1 2 ", " }}{PARA 0 "" 0 "" {TEXT -1 9 "that is, " }}{PARA 256 "" 0 " " {TEXT -1 1 " " }{XPPEDIT 18 0 "c[4]=2/(10-25*c[3])" "6#/&%\"cG6#\"\" %*&\"\"#\"\"\",&\"#5F**&\"#DF*&F%6#\"\"$F*!\"\"F2" }{TEXT -1 2 ". " }} {PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 78 "(2-5*c[3])*c[4]=240/600;\nc[4]=solve(%,c[4]);\nc[4]=numer(rhs(%))/ denom(rhs(%));" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/*&,&\"\"#\"\"\"*&\" \"&F'&%\"cG6#\"\"$F'!\"\"F'&F+6#\"\"%F'#F&F)" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/&%\"cG6#\"\"%,$*&#\"\"#\"\"&\"\"\"*&F-F-,&*&F,F-&F%6# \"\"$F-F-F+!\"\"F4F-F4" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/&%\"cG6#\" \"%,$*&\"\"#\"\"\",&*&\"#DF+&F%6#\"\"$F+F+\"#5!\"\"F3F3" }}}{PARA 0 " " 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 52 "The following pict ure is the graph of this equation." }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 88 "plot(2/(10-25*c[3]),c[3]=0..0.32,c[4]=0..1,color=COLO R(RGB,.5,0,.9),font=[HELVETICA,9]);" }}{PARA 13 "" 1 "" {GLPLOT2D 485 401 401 {PLOTDATA 2 "6'-%'CURVESG6#7U7$$\"\"!F)$\"35+++++++?!#=7$$\"3Y LLLL$*3vp!#?$\"3L@U)zQ%\\N?F,7$$\"3tmmmExS/8!#>$\"37(4'oI*=u1#F,7$$\"3 KLLL$*)Gp)>F6$\"3.\"HpeCRX5#F,7$$\"3iLLL`U(Rn#F6$\"3&pGNbmwK9#F,7$$\"3 smmm1UvdLF6$\"3i2q;$HsK=#F,7$$\"3YLLL`dq\"*RF6$\"3750i\\.r@AF,7$$\"3') ******fe7[YF6$\"3+6'ojZjHE#F,7$$\"3YLLLtk*pK&F6$\"3h7-R_0F2BF,7$$\"3J+ ++S**o.gF6$\"3AwmNjl>`BF,7$$\"33ommYBv*p'F6$\"3zK'zd`%Q-CF,7$$\"3?MLL$ \\WGJ(F6$\"3%RZ];`WuW#F,7$$\"3b,++S5/.!)F6$\"34'3IQgP-]#F,7$$\"3u++++; 2'p)F6$\"3K3&*>'z*ebDF,7$$\"3[+++g\\$RO*F6$\"3Qv&f`T,8h#F,7$$\"3yLLLt) =/(**F6$\"3#oM^L#)RSm#F,7$$\"3ommm!pe\"p5F,$\"3'Hz%zh;fHFF,7$$\"3ummm# 3^-8\"F,$\"3O*R_'G.q(y#F,7$$\"3*******z0;8?\"F,$\"3k%pk\\S'[eGF,7$$\"3 kmmm-1Ak7F,$\"3)R!)[jQ7U#HF,7$$\"33+++eoBL8F,$\"3abkV$F,7$$\"33+++9yQI6W_'=F,$\"3#[xf>3,vu$F,7$$\"3 5+++%G$GK>F,$\"3=ql(p%>+pQF,7$$\"39+++Ir9(*>F,$\"3%*3UF(Q-V*RF,7$$\"31 +++st;p?F,$\"3w*\\x*[-HVTF,7$$\"3rmmmi2)Q8#F,$\"3A[bZ2:(pG%F,7$$\"3K++ +?h(H?#F,$\"3Cy)G*e]!=X%F,7$$\"3YLLL$y'elAF,$\"3#y1&pw4^7YF,7$$\"3E+++ wn.MBF,$\"3yF+Jqn--[F,7$$\"3mmmm%)GW)R#F,$\"3A$*>aX(Q^*\\F,7$$\"35+++E ewlCF,$\"3*\\$4zhyK9_F,7$$\"3immm1jeJDF,$\"3=$[.z(f0[aF,7$$\"3,+++]U\\ +EF,$\"34E:ntPI;dF,7$$\"3[LLL*ygom#F,$\"3D-2?zO(3+'F,7$$\"3eLLLh,tMFF, $\"3^vK+c>wAjF,7$$\"3*pmm1aP?!GF,$\"3'y:o**e0!ymF,7$$\"3O+++Kg)Q'GF,$ \"3oxON&3Z:/(F,7$$\"3:nmm#QrZ$HF,$\"37J1_M\\75vF,7$$\"3yLLL&3s\")*HF,$ \"3Op:2kLS&)zF,7$$\"3/+++e/xlIF,$\"3kX.?zf?j&)F,7$$\"3!******>f?\")4$F ,$\"3ox@l:bOq))F,7$$\"3K+++E2ZIJF,$\"3 " 0 "" {MPLTEXT 1 0 46 "sm2 : = simplify(subs(c[4]=2/(10-25*c[3]),sm)):" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 50 "sm2: square of principal error norm in t erms of " }{XPPEDIT 18 0 "c[2]" "6#&%\"cG6#\"\"#" }{TEXT -1 3 ", " } {XPPEDIT 18 0 "c[3]" "6#&%\"cG6#\"\"$" }{TEXT -1 7 " and " } {XPPEDIT 18 0 "c[5]" "6#&%\"cG6#\"\"&" }{TEXT -1 1 " " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 944 "sm2 := 1 /648000000*(-11249000*c[3]^5+6651250*c[3]^6+7236000*c[3]^4+111600*c[2] ^2+183848*c[3]^2-324000*c[5]*c[2]^2+945000*c[3]^3*c[2]+16800*c[3]*c[2] -5962500*c[3]^3*c[2]^2-413280*c[3]^2*c[5]+24538500*c[3]^5*c[5]-1275000 *c[3]^4*c[2]-10125000*c[3]^2*c[2]^2*c[5]-19372500*c[5]^2*c[3]^5-191848 0*c[3]^3-13837500*c[3]^6*c[5]-270000*c[3]^2*c[2]*c[5]^2+8842500*c[3]^2 *c[2]^2*c[5]^2+4359375*c[3]^4*c[2]^2-36000*c[3]*c[2]*c[5]-270000*c[2]* c[3]^2+1215000*c[3]^3*c[2]*c[5]^2+2808000*c[3]*c[2]^2*c[5]+684000*c[3] ^2*c[2]*c[5]-2430000*c[3]*c[2]^2*c[5]^2+10546875*c[3]^4*c[2]^2*c[5]^2+ 3262500*c[3]^4*c[2]*c[5]+17550000*c[3]^3*c[2]^2*c[5]-12656250*c[3]^4*c [2]^2*c[5]-15187500*c[3]^3*c[2]^2*c[5]^2-1687500*c[3]^4*c[2]*c[5]^2+27 0000*c[5]^2*c[2]^2+3442500*c[3]^2*c[2]^2-954000*c[3]*c[2]^2+11961450*c [5]^2*c[3]^4-3099600*c[5]^2*c[3]^3+4272120*c[3]^3*c[5]+295200*c[5]^2*c [3]^2-2520000*c[3]^3*c[2]*c[5]+11531250*c[5]^2*c[3]^6-16005600*c[3]^4* c[5])/c[3]^2/(5*c[3]-2)^2:" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}}{PARA 0 "" 0 "" {TEXT -1 0 " " }}{PARA 0 "" 0 "" {TEXT -1 8 "Taking " }{XPPEDIT 18 0 "c[5] = 7/8; " "6#/&%\"cG6#\"\"&*&\"\"(\"\"\"\"\")!\"\"" }{TEXT -1 103 ", we obtain the following surface for the graph of the square of the principal er ror norm in terms of " }{XPPEDIT 18 0 "c[2]" "6#&%\"cG6#\"\"#" } {TEXT -1 7 " and " }{XPPEDIT 18 0 "c[3]" "6#&%\"cG6#\"\"$" }{TEXT -1 2 ". 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03D4E7BD2873E9E68F11AE25DEA3E9C43CBF7D344FD3E9A50751DA3E67A3E988E69A65 844223E96FD3AD05232263E959C8F1E290EC83E946C23A79835A83E936BCDA1E119B03 E929B7C26FDB9D63E91FB3A443D5CBD3E918B315A514FDD3E914BABDA7AEB973E913D1 86E9311EC3E91600D9BF8CF533E91B54DF44E8C7E3E923DCCEA47666E3E92FAB464CE4 31A3E93ED6B2FACAB373E95179C6EAC67353E967B403F3C9E603E981AA5BD3BFB9C3E9 9F87EA9F56E683EA4560F0A71B8703EA2DF32817ED8093EA183A1CA3137793EA042F74 547CA553E9E39AD2B1484B23E9C21D9E623174A3E9A3DE3207146543E988D4C247EEA6 63E970FAE3FF29C883E95C4BA128972CE3E94AC3914F43E4F3E93C60F5F0639F63E931 23DB5D1EA1A3E9290E3EBBF83DE3E9242439E554E143E9226C35F6FD5F43E923EF25A0 296563E928B8C854F79E23E930D7F7CE247753E93C5F017A7991C3E94B640DD507C343 E95E0197F0CACB13E97456F7FB8FE173E98E8904030FAC83E9ACC2CAEE7815E3EA5BA9 AC498F70D3EA42AFE9F0051FF3EA2B826C16394EA3EA161A4E33450943EA027146AE62 0DD3E9E10357F20E1213E9C08CF58FB9B183E9A376BB16973A33E989B96A9AA3F703E9 734F48418029C3E9603432A9AD50A3E95065BF515AED23E943E35B87325723E93AAE72 867188B3E934CA997AD77623E9323DC248FB9693E93310761DE29F73E9374E18F5D318 43E93F053780E21743E94A47E10C019213E9592C0F640D6B53E96BCC1F05329C43E982 475A5B621943E99CC29B6184C7A3E9BB690795291283EA7325A5BE0E1E73EA588ECD41 2D62E3EA3FDCC3413092E3EA290816736E8443EA1409F7BFD61603EA00DC3F44EB9F93 E9DEF2E4FF77B6C3E9BFB997711CDDA3E9A40485B4FEF093E98BCD2FC43DC6F3E9770E BE2D4DDE33E965C61E92C2A6C3E957F224C20C5203E94D93B1047FB0D3E946ADDC6837 1243E943462BDF2C6293E94364CB38B6BE33E94714D1278065A3E94E648DBCAB4903E9 5965E501797713E9682EB7A7DD1303E97AD95C2800CAA3E991852B1342C8A3E9AC5721 EF22DFF3E9CB7AA0936E9A5-%&COLORG6&%$RGBGF*$\"\"!F;$\"\")F)-%+AXESLABEL SG6%%%c[2]G%%c[3]G%!G-%*AXESSTYLEG6#%$BOXG-%+LIGHTMODELG6#%(LIGHT_4G" 1 2 0 1 10 0 2 1 1 2 2 1.000000 39.000000 37.000000 0 0 "Curve 1" }}}} {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 58 "The follo wing calculations give (approximate) values for " }{XPPEDIT 18 0 "c[2 ]" "6#&%\"cG6#\"\"#" }{TEXT -1 7 " and " }{XPPEDIT 18 0 "c[3]" "6#&% \"cG6#\"\"$" }{TEXT -1 82 " to minimize the square of the principal e rror norm for three fixed choices for " }{XPPEDIT 18 0 "c[5]" "6#&%\" cG6#\"\"&" }{TEXT -1 10 ", namely " }{XPPEDIT 18 0 "c[5] = 7/8;" "6#/ &%\"cG6#\"\"&*&\"\"(\"\"\"\"\")!\"\"" }{TEXT -1 3 ", " }{XPPEDIT 18 0 "c[5] = 8/9;" "6#/&%\"cG6#\"\"&*&\"\")\"\"\"\"\"*!\"\"" }{TEXT -1 7 " and " }{XPPEDIT 18 0 "c[5] = 9/10;" "6#/&%\"cG6#\"\"&*&\"\"*\"\"\" \"#5!\"\"" }{TEXT -1 1 "." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 248 "uu := \{c[5]=7/8.\}:\nmn := evalf[ 12](minimize(eval(sm2,uu),c[2]=0.15..0.25,c[3]=0.25..0.35,'location')) ;\nsubs(op(1,op(1,op(2,[mn]))),[c[2],c[3]]):\nrr := map(convert,%,rati onal,5):\nc[2]=op(1,rr),c[3]=op(2,rr);\n`principal error norm:`,sqrt(o p(1,[mn]));" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%#mnG6$$\"-;3-Gi;!#=<# 7$<$/&%\"cG6#\"\"#$\"-A!RYh%>!#7/&F.6#\"\"$$\"-Ie>*f(HF3F&" }}{PARA 11 "" 1 "" {XPPMATH 20 "6$/&%\"cG6#\"\"##\"#H\"$\\\"/&F%6#\"\"$#\"#D\" #%)" }}{PARA 11 "" 1 "" {XPPMATH 20 "6$%6principal~error~norm:G$\"+sq5 xS!#8" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 248 "uu := \{c[5]=8/9.\}:\nmn := evalf[12](minimize(eval( sm2,uu),c[2]=0.15..0.25,c[3]=0.25..0.35,'location'));\nsubs(op(1,op(1, op(2,[mn]))),[c[2],c[3]]):\nrr := map(convert,%,rational,5):\nc[2]=op( 1,rr),c[3]=op(2,rr);\n`principal error norm:`,sqrt(op(1,[mn]));" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#>%#mnG6$$\"-XYeS!f\"!#=<#7$<$/&%\"cG6# \"\"$$\"-P`&3#4I!#7/&F.6#\"\"#$\"-]R=a6?F3F&" }}{PARA 11 "" 1 "" {XPPMATH 20 "6$/&%\"cG6#\"\"##\"#N\"$u\"/&F%6#\"\"$#\"#l\"$;#" }} {PARA 11 "" 1 "" {XPPMATH 20 "6$%6principal~error~norm:G$\"+v#*)z)R!#8 " }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 249 "uu := \{c[5]=9/10.\}:\nmn := evalf[12](minimize(eval (sm2,uu),c[2]=0.15..0.25,c[3]=0.25..0.35,'location'));\nsubs(op(1,op(1 ,op(2,[mn]))),[c[2],c[3]]):\nrr := map(convert,%,rational,5):\nc[2]=op (1,rr),c[3]=op(2,rr);\n`principal error norm:`,sqrt(op(1,[mn]));" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#>%#mnG6$$\"-k\"\"$v&/&F%6#\"\"$#\"#W\"$X\"" }} {PARA 11 "" 1 "" {XPPMATH 20 "6$%6principal~error~norm:G$\"+5Hq,S!#8" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 14 "The pr ocedure " }{TEXT 0 8 "minimize" }{TEXT -1 75 " has difficulty handling minimization with respect to the three variables " }{XPPEDIT 18 0 "c [2]" "6#&%\"cG6#\"\"#" }{TEXT -1 3 ", " }{XPPEDIT 18 0 "c[3]" "6#&%\" cG6#\"\"$" }{TEXT -1 7 " and " }{XPPEDIT 18 0 "c[5]" "6#&%\"cG6#\"\" &" }{TEXT -1 86 ", so we set up a procedure to perform minimizations w ith repect to the two variables " }{XPPEDIT 18 0 "c[2]" "6#&%\"cG6#\" \"#" }{TEXT -1 7 " and " }{XPPEDIT 18 0 "c[3]" "6#&%\"cG6#\"\"$" } {TEXT -1 84 " as in the previous examples and the use the one dimensi onal minmization procedure " }{TEXT 0 7 "findmin" }{TEXT -1 29 " to mi nimize with repect to " }{XPPEDIT 18 0 "c[5]" "6#&%\"cG6#\"\"&" } {TEXT -1 1 "." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 145 "prin_err := proc(c_5)\n local mn;\n mn := e valf[Digits+2](minimize(eval(sm2,c[5]=c_5),c[2]=0.15..0.25,c[3]=0.25.. 0.35));\n sqrt(mn);\nend proc:" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }} {PARA 0 "" 0 "" {TEXT -1 21 "Using the procedure " }{TEXT 0 8 "prin_e rr" }{TEXT -1 55 " three previous examples can be performed as follow s. " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 46 "prin_err(7/8.),prin_err(8/9.),prin_err(9/10.);" }} {PARA 11 "" 1 "" {XPPMATH 20 "6%$\"+1r5xS!#8$\"+N#*)z)RF%$\"+\")Hq,SF% " }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 55 "Thes e three values can be used as starting values for " }{TEXT 0 7 "findm in" }{TEXT -1 1 "." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 " > " 0 "" {MPLTEXT 1 0 123 "evalf[12](findmin('prin_err'(c5),c5=\{7/8., 8/9.,9/10.\},convergence='location'));\nc[5]=convert(op(1,%),rational, 4);\nevalf(%);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#7$$\"-b\")\\FF*)!#7$ \"-2i)\\J)R!#:" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/&%\"cG6#\"\"&#\"#D \"#G" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/&%\"cG6#\"\"&$\"+H9dG*)!#5" } }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 250 "uu := \{c[5]=25/28.\}:\nmn := evalf[12](minimize(eval(sm2,uu),c [2]=0.15..0.25,c[3]=0.25..0.35,'location'));\nsubs(op(1,op(1,op(2,[mn] ))),[c[2],c[3]]):\nrr := map(convert,%,rational,5):\nc[2]=op(1,rr),c[3 ]=op(2,rr);\n`principal error norm:`,sqrt(op(1,[mn]));" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%#mnG6$$\"-.+Ib'e\"!#=<#7$<$/&%\"cG6#\"\"#$\"-U=c sJ?!#7/&F.6#\"\"$$\"-%\\*>R=IF3F&" }}{PARA 11 "" 1 "" {XPPMATH 20 "6$/ &%\"cG6#\"\"##\"#k\"$:$/&F%6#\"\"$#\"$:\"\"$\"Q" }}{PARA 11 "" 1 "" {XPPMATH 20 "6$%6principal~error~norm:G$\"+%ybJ)R!#8" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 14 "The value of " } {XPPEDIT 18 0 "c[4]" "6#&%\"cG6#\"\"%" }{TEXT -1 20 " corresponding t o " }{XPPEDIT 18 0 "c[3] = 115/381;" "6#/&%\"cG6#\"\"$*&\"$:\"\"\"\" \"$\"Q!\"\"" }{TEXT -1 10 " is . . ." }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 42 "c[4]=2/(10-25*c[3]);\nsubs(c[3]=115/381,%);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/&%\"cG6#\"\"%,$*&\"\"#\"\"\",&\"#5F+*&\"#DF +&F%6#\"\"$F+!\"\"F3F+" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/&%\"cG6#\" \"%#\"$i(\"$N*" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 7 "Taking " }{XPPEDIT 18 0 "`b*`[7] = 1/42;" "6#/&%#b*G6#\"\" (*&\"\"\"F)\"#U!\"\"" }{TEXT -1 74 " gives suitable characteristics f or the embedded 7 stage, order 4 scheme." }}{PARA 0 "" 0 "" {TEXT -1 80 " Using the general solution we can find the coefficients of the co mbined scheme." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 101 "e3 := \{c[2]=64/315,c[3]=115/381,c[4]=762/935,c [5]=25/28,`b*`[7]=1/42\}:\ne4 := `union`(subs(e3,e2),e3):" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }} }{SECT 1 {PARA 4 "" 0 "" {TEXT -1 2 "e4" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1269 "e4 := \{a[6,5] = -268 08346215168/82592376030125, a[4,1] = 344904825219069/345923838700000, \+ a[7,3] = 63896466577779/141024193000600, b[5] = -1359311744/2892576375 , b[1] = 2405713/26289000, a[4,3] = 40439332863108/10810119959375, a[5 ,3] = 39297175833216951/4694461413969824, b[6] = 256979/1656648, a[5,2 ] = -791781731775/81669668864, a[3,1] = 480815/6193536, b[3] = 6389646 6577779/141024193000600, `b*`[4] = 324806515311046773125/4529181591778 76804664, a[7,6] = 256979/1656648, c[6] = 1, c[7] = 1, b[2] = 0, `b*`[ 2] = 0, a[3,2] = 462875/2064512, `b*`[5] = -126112324722496/3174226536 63375, a[7,2] = 0, `b*`[1] = 1818563883019/20194131951000, a[6,1] = 22 51421737440863/828701767536000, a[6,2] = -39895842357/3782730880, a[5, 4] = -10508413393960625/54097233987826176, a[7,1] = 2405713/26289000, \+ a[6,4] = 12051135495733565/36492943960723917, b[4] = 454128848141375/5 89615117674696, a[7,5] = -1359311744/2892576375, `b*`[3] = 55138624982 02899713/12036555896794210600, a[7,4] = 454128848141375/58961511767469 6, a[4,2] = -2360077908867/601606676000, a[5,1] = 12078745127989699/50 09699157786624, a[6,3] = 34564628685305112534/3916944972468643375, `b* `[6] = 137695258717/1272569071032, c[4] = 762/935, c[5] = 25/28, c[3] \+ = 115/381, c[2] = 64/315, `b*`[7] = 1/42, a[2,1] = 64/315\}:" }}} {PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 137 "subs(e4,matrix([seq([c[i],seq(a[i,j],j=1..i-1),``$(8 -i)],i=2..7),\n[`b`,seq(b[i],i=1..6),``],[`b*`,seq(`b*`[i],i=1..7)]])) ;\n``;\nevalf(%%);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#K%'matrixG6#7*7* #\"#k\"$:$F(%!GF+F+F+F+F+7*#\"$:\"\"$\"Q#\"':3[\"(ON>'#\"'vGY\"(7X1#F+ F+F+F+F+7*#\"$i(\"$N*#\"0p!>_#[!\\M\"0++qQQ#fM#!.n)3z2gB\"-+gn1;g#\"/3 J'GLR/%\"/v$f*>,\"3\"F+F+F+F+7*#\"#D\"#G#\"2*p*)z7X(y?\"\"1Cmyd\"*p4]# !-vC59#\"0v89[)GTX\"0'puw 6:'*e#!+WI)Qc==\"/+5&>8%>?Fbo#\"48(**G?)\\iQ^&\"5+1@%z'*ebO?\"#\"6DJxY 5J:l![K\"6kY!o(y26CF;$!+5,$ \\p*F;$\"+wd'4P)F;$!+\\Z]U>F*F+F+F+7*$\"\"\"\"\"!$\"+id!or#F;$!+/Ooa5! \")$\"+ZXQC))F;$\"+\\$>BI$F*$!+)3ieC$F*F+F+7*FJ$\"+W^-^\"*F1$FLFL$\"+! )o)3`%F*$\"+ZN7-xF*$!+1:J*p%F*$\"+l&)>^:F*F+7*%\"bGFYFenFfnFhnFjnF\\oF +7*%#b*G$\"+[vS0!*F1Fen$\"+9/$4e%F*$\"+otTrrF*$!+!f4I(RF*$\"+;e-#3\"F* $\"+\"Q_4Q#F1Q)pprint196\"" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 57 "#---------------------------------------------- ----------" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 " " {MPLTEXT 1 0 1 ";" }}}}{PARA 0 "" 0 "" {TEXT -1 39 "#--------------- -----------------------" }}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 39 "charac teristics of the embedded scheme " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}{SECT 1 {PARA 4 "" 0 " " {TEXT -1 35 "coefficients of the combined scheme" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1192 "ee := \{c[ 2]=64/315,\nc[3]=115/381,\nc[4]=762/935,\nc[5]=25/28,\nc[6]=1,\nc[7]=1 ,\n\na[2,1]=64/315,\na[3,1]=480815/6193536,\na[3,2]=462875/2064512,\na [4,1]=344904825219069/345923838700000,\na[4,2]=-2360077908867/60160667 6000,\na[4,3]=40439332863108/10810119959375,\na[5,1]=12078745127989699 /5009699157786624,\na[5,2]=-791781731775/81669668864,\na[5,3]=39297175 833216951/4694461413969824,\na[5,4]=-10508413393960625/540972339878261 76,\na[6,1]=2251421737440863/828701767536000,\na[6,2]=-39895842357/378 2730880,\na[6,3]=34564628685305112534/3916944972468643375,\na[6,4]=120 51135495733565/36492943960723917,\na[6,5]=-26808346215168/825923760301 25,\na[7,1]=2405713/26289000,\na[7,2]=0,\na[7,3]=63896466577779/141024 193000600,\na[7,4]=454128848141375/589615117674696,\na[7,5]=-135931174 4/2892576375,\na[7,6]=256979/1656648,\n\nb[1]=2405713/26289000,\nb[2]= 0,\nb[3]=63896466577779/141024193000600,\nb[4]=454128848141375/5896151 17674696,\nb[5]=-1359311744/2892576375,\nb[6]=256979/1656648,\n\n`b*`[ 1]=1818563883019/20194131951000,\n`b*`[2]=0,\n`b*`[3]=5513862498202899 713/12036555896794210600,\n`b*`[4]=324806515311046773125/4529181591778 76804664,\n`b*`[5]=-126112324722496/317422653663375,\n`b*`[6]=13769525 8717/1272569071032,\n`b*`[7]=1/42\}:" }}}{PARA 0 "" 0 "" {TEXT -1 0 " " }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 5 "Let " }{XPPEDIT 18 0 "T[5 ,6];" "6#&%\"TG6$\"\"&\"\"'" }{TEXT -1 127 " denote the vector whose \+ components are the principal error terms of the 6 stage, order 5 schem e (the error terms of order 6)." }}{PARA 0 "" 0 "" {TEXT -1 5 "Let " }{XPPEDIT 18 0 "`T*`[4,7];" "6#&%#T*G6$\"\"%\"\"(" }{TEXT -1 145 " de note the vector whose components are the principal error terms of the \+ embedded 7 stage, order 4 scheme (the error terms of order 5) and let \+ " }{XPPEDIT 18 0 "`T*`[5,7];" "6#&%#T*G6$\"\"&\"\"(" }{TEXT -1 99 " \+ denote the vector whose components are the error terms of order 6 of t he embedded order 4 scheme." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 39 "Denote the 2-norm of these vectors by " } {XPPEDIT 18 0 "abs(abs(T[5,6]));" "6#-%$absG6#-F$6#&%\"TG6$\"\"&\"\"' " }{TEXT -1 3 ", " }{XPPEDIT 18 0 "abs(abs(`T*`[4,7]));" "6#-%$absG6# -F$6#&%#T*G6$\"\"%\"\"(" }{TEXT -1 7 " and " }{XPPEDIT 18 0 "abs(abs (`T*`[5,7]));" "6#-%$absG6#-F$6#&%#T*G6$\"\"&\"\"(" }{TEXT -1 15 " re spectively." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 8 "Define: " }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "A[6 ] = abs(abs(T[5,6]));" "6#/&%\"AG6#\"\"'-%$absG6#-F)6#&%\"TG6$\"\"&F' " }{TEXT -1 6 ", " }{XPPEDIT 18 0 "B[6] = abs(abs(`T*`[5,7]))/abs( abs(`T*`[4,7]));" "6#/&%\"BG6#\"\"'*&-%$absG6#-F*6#&%#T*G6$\"\"&\"\"( \"\"\"-F*6#-F*6#&F/6$\"\"%F2!\"\"" }{TEXT -1 6 ", " }{XPPEDIT 18 0 "C[6] = abs(abs(`T*`[5,7]-T[5,6]))/abs(abs(`T*`[4,7]));" "6#/&%\"CG6 #\"\"'*&-%$absG6#-F*6#,&&%#T*G6$\"\"&\"\"(\"\"\"&%\"TG6$F2F'!\"\"F4-F* 6#-F*6#&F06$\"\"%F3F8" }{TEXT -1 1 "." }}{PARA 0 "" 0 "" {TEXT -1 0 " " }}{PARA 0 "" 0 "" {TEXT -1 77 "Dormand and Prince have suggested tha t as well as attempting to ensure that " }{XPPEDIT 18 0 "A[6];" "6#&% \"AG6#\"\"'" }{TEXT -1 73 " is a minimum, if the embedded scheme is t o be used for error control, " }{XPPEDIT 18 0 "B[6];" "6#&%\"BG6#\"\" '" }{TEXT -1 7 " and " }{XPPEDIT 18 0 "C[6];" "6#&%\"CG6#\"\"'" } {TEXT -1 27 " should be chosen so that " }}{PARA 0 "" 0 "" {TEXT -1 68 "they are \"similar in magnitude\" and also not differ too much fro m 1." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 186 "errterms5_6 := PrincipalErrorTerms(5,6,'expanded'): \n`errterms5_7*` :=subs(b=`b*`,PrincipalErrorTerms(5,7,'expanded')):\n `errterms4_7*` := subs(b=`b*`,PrincipalErrorTerms(4,7,'expanded')):" } }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 326 "snmB := sqrt(add(evalf(subs(ee,`errterms5_7*`[i]))^2,i=1..nops( `errterms5_7*`))):\nsdnB := sqrt(add(evalf(subs(ee,`errterms4_7*`[i])) ^2,i=1..nops(`errterms4_7*`))):\nsnmC := sqrt(add((evalf(subs(ee,`errt erms5_7*`[i])-subs(ee,errterms5_6[i])))^2,i=1..nops(errterms5_6))):\n' B[6]'= evalf[8](snmB/sdnB);\n'C[6]'= evalf[8](snmC/sdnB);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/&%\"BG6#\"\"'$\")LE`:!\"(" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/&%\"CG6#\"\"'$\")M0\"o\"!\"(" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}}{PARA 0 "" 0 "" {TEXT -1 39 "#--------------------------------------" }} {SECT 1 {PARA 4 "" 0 "" {TEXT -1 26 "absolute stability regions" }} {PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 35 "coefficients of the comb ined scheme" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 " " {MPLTEXT 1 0 1192 "ee := \{c[2]=64/315,\nc[3]=115/381,\nc[4]=762/935 ,\nc[5]=25/28,\nc[6]=1,\nc[7]=1,\n\na[2,1]=64/315,\na[3,1]=480815/6193 536,\na[3,2]=462875/2064512,\na[4,1]=344904825219069/345923838700000, \na[4,2]=-2360077908867/601606676000,\na[4,3]=40439332863108/108101199 59375,\na[5,1]=12078745127989699/5009699157786624,\na[5,2]=-7917817317 75/81669668864,\na[5,3]=39297175833216951/4694461413969824,\na[5,4]=-1 0508413393960625/54097233987826176,\na[6,1]=2251421737440863/828701767 536000,\na[6,2]=-39895842357/3782730880,\na[6,3]=34564628685305112534/ 3916944972468643375,\na[6,4]=12051135495733565/36492943960723917,\na[6 ,5]=-26808346215168/82592376030125,\na[7,1]=2405713/26289000,\na[7,2]= 0,\na[7,3]=63896466577779/141024193000600,\na[7,4]=454128848141375/589 615117674696,\na[7,5]=-1359311744/2892576375,\na[7,6]=256979/1656648, \n\nb[1]=2405713/26289000,\nb[2]=0,\nb[3]=63896466577779/1410241930006 00,\nb[4]=454128848141375/589615117674696,\nb[5]=-1359311744/289257637 5,\nb[6]=256979/1656648,\n\n`b*`[1]=1818563883019/20194131951000,\n`b* `[2]=0,\n`b*`[3]=5513862498202899713/12036555896794210600,\n`b*`[4]=32 4806515311046773125/452918159177876804664,\n`b*`[5]=-126112324722496/3 17422653663375,\n`b*`[6]=137695258717/1272569071032,\n`b*`[7]=1/42\}: " }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 " " {TEXT -1 79 "The stability function R for the 6 stage, order 5 sch eme is given as follows." }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 75 " subs(ee,StabilityFunction(5,6,'expanded')):\nR := unapply(%,z):\n'R(z) '=R(z);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/-%\"RG6#%\"zG,0\"\"\"F)F'F )*&#F)\"\"#F)*$)F'F,F)F)F)*&#F)\"\"'F)*$)F'\"\"$F)F)F)*&#F)\"#CF)*$)F' \"\"%F)F)F)*&#F)\"$?\"F)*$)F'\"\"&F)F)F)*&#F)\"$+'F)*$)F'F1F)F)F)" }}} {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 124 "We can f ind the point where the boundary of the stability region intersects th e negative real axis by solving the equation: " }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "R(z) = 1;" "6#/-%\"RG6#%\"zG\"\"\"" } {TEXT -1 2 ". " }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 26 "z0 := newt on(R(z)=1,z=-3);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%#z0G$!+$*yc1L!\" *" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 301 "z0 := newton(R(z)=1,z=-3):\np1 := plot([R(z),1],z=-3 .69..0.49,color=[red,blue]):\np2 := plot([[[z0,1]]$3],style=point,symb ol=[circle,cross,diamond],color=black):\np3 := plot([[z0,0],[z0,1]],li nestyle=3,color=COLOR(RGB,0,.5,0)):\nplots[display]([p1,p2,p3],view=[- 3.69..0.49,-0.0..1.47],font=[HELVETICA,9]);" }}{PARA 13 "" 1 "" {GLPLOT2D 401 211 211 {PLOTDATA 2 "6+-%'CURVESG6$7X7$$!3%************* *o$!#<$\"3gOY%H?7a(>F*7$$!3Om;zZRWWOF*$\"3u@wyNF*$\"3M)eHZ!e#*o9F*7$$!3')\\i+xX.vMF*$\"3yK\\Hd*GqN\"F*7$$!3UL$3LTd/ V$F*$\"3)35:cS/GD\"F*7$$!37LLBjZe&Q$F*$\"3)=/[i59_:\"F*7$$!3FL$eJ672M$ F*$\"3sSEwGeck5F*7$$!31]i&\\t_gH$F*$\"3wk7bWNa3)*!#=7$$!3'o;anN$R^KF*$ \"3.Tu7)RcE.*FZ7$$!3eL3(fV$eoJF*$\"3O$))*fKd>WxFZ7$$!33+v`f&QG3$F*$\"3 ')=!*4L(*4(f'FZ7$$!3;L3n&3hT*HF*$\"3S0Tkm%*=SFZ7$$!3OL$e!)ofZt#F*$\"3% =>Ugqo$*[$FZ7$$!31++:kFgWEF*$\"3q&fPNB\\'**HFZ7$$!3#****\\skvSb#F*$\"3 gw!\\v$QW2EFZ7$$!3#****\\e*f$oY#F*$\"33yk^z6\"=J#FZ7$$!3WL3nMSh(Q#F*$ \"3\\3gZ+Wq/@FZ7$$!3UmmJg9T$H#F*$\"38<\")=^n$\\#>FZ7$$!3QmmEB&4O@#F*$ \"3<%RuccMB#=FZ7$$!3\"**\\P#**3y?@F*$\"3'fUlY3dJv\"FZ7$$!3mmmwP=hQ?F*$ \"3'y2mjmAF*$\"3sMO)[$4FY'y!p\"F*$ \"3'eh@Y(=6t>FZ7$$!3F#FZ7$$!32+D'z\"4oH9F*$\"3=HiVy%p!QCFZ7$$!3KL$3fwOIM\"F*$ \"3eAnYPi)R.d&FZ$\"30lDq)yr\"HdFZ7$$!3Q+](y*Q$4p%FZ$ \"3Ow2][0tbiFZ7$$!3+om;pQ:JQFZ$\"3oOOI#R`t\"oFZ7$$!36-]PfQ/JHFZ$\"3AcU OA[WfuFZ7$$!3qILLR48k?FZ$\"3\"[=xx!)p\\8)FZ7$$!3ALLensex6FZ$\"3=T-jy\\ 5*)))FZ7$$!3+LmT]i%Q)H!#>$\"378I_&4Bgq*FZ7$$\"3)3+++$H6&4&Faz$\"3!**4) ym9F_5F*7$$\"36ML3'=^aV\"FZ$\"3]O`([*)eV:\"F*7$$\"3%ymmY'GijAFZ$\"3-Ad ?K*HSD\"F*7$$\"3!\\+DwgEm9$FZ$\"3#37UKN(zp8F*7$$\"3k+]PeQx\"*RFZ$\"3#e A)y#z)f!\\\"F*7$$\"3!***************[FZ$\"3Woh$>j=Bj\"F*-%'COLOURG6&%$ RGBG$\"*++++\"!\")$\"\"!Fj\\lFi\\l-F$6$7S7$F($\"\"\"Fj\\l7$F3F_]l7$F=F _]l7$FGF_]l7$FQF_]l7$FfnF_]l7$F[oF_]l7$F`oF_]l7$FeoF_]l7$FjoF_]l7$F_pF _]l7$FdpF_]l7$FipF_]l7$F^qF_]l7$FcqF_]l7$FhqF_]l7$F]rF_]l7$FbrF_]l7$Fg rF_]l7$F\\sF_]l7$FasF_]l7$FfsF_]l7$F[tF_]l7$F`tF_]l7$FetF_]l7$FjtF_]l7 $F_uF_]l7$FduF_]l7$FiuF_]l7$F^vF_]l7$FcvF_]l7$FhvF_]l7$F]wF_]l7$FbwF_] l7$FgwF_]l7$F\\xF_]l7$FaxF_]l7$FfxF_]l7$F[yF_]l7$F`yF_]l7$FeyF_]l7$Fjy F_]l7$F_zF_]l7$FezF_]l7$FjzF_]l7$F_[lF_]l7$Fd[lF_]l7$Fi[lF_]l7$F^\\lF_ ]l-Fc\\l6&Fe\\lFi\\lFi\\lFf\\l-F$6&7#7$$!3)******H*yc1LF*F_]l-%'SYMBOL G6#%'CIRCLEG-Fc\\l6&Fe\\lFj\\lFj\\lFj\\l-%&STYLEG6#%&POINTG-F$6&Fe`l-F j`l6#%&CROSSGF]alF_al-F$6&Fe`l-Fj`l6#%(DIAMONDGF]alF_al-F$6%7$7$Fg`lFi \\lFf`l-%&COLORG6&Fe\\lFi\\l$\"\"&!\"\"Fi\\l-%*LINESTYLEG6#\"\"$-%+AXE SLABELSG6%Q\"z6\"Q!F_cl-%%FONTG6#%(DEFAULTG-Fbcl6$%*HELVETICAG\"\"*-%% VIEWG6$;$!$p$!\"#$\"#\\F_dl;$!\"!Fj\\l$\"$Z\"F_dl" 1 2 0 1 10 0 2 9 1 4 2 1.000000 46.000000 44.000000 0 0 "Curve 1" "Curve 2" "Curve 3" "Cu rve 4" "Curve 5" "Curve 6" }}}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 50 "The following picture shows the stability regio n. " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1662 "R := z -> 1+z+1/2*z^2+1/6*z^3+1/24*z^4+1/120*z^5+1/ 600*z^6:\npts := []: z0 := 0: tt := 0: \nwhile tt<=161/20 do\n zz := newton(R(z)=exp(tt*Pi*I),z=z0):\n z0 := zz:\n if (9/20<=tt and tt <=21/20) \n or (140/20<=tt and tt<=152/20) then\n hh := 1/40 \n else \n hh := 1/20\n end if;\n tt := tt+hh;\n pts := [ op(pts),[Re(zz),Im(zz)]]:\nend do:\np1 := plot(pts,color=COLOR(RGB,.28 ,.05,.48)):\np2 := plots[polygonplot]([seq([pts[i-1],pts[i],[-1.65,0]] ,i=2..nops(pts))],\n style=patchnogrid,color=COLOR(RGB,.55,.1 ,.95)):\npts := []: z0 := 1+3.8*I: tt := 0: \nwhile tt<=61/30 do\n z z := newton(R(z)=exp(tt*Pi*I),z=z0):\n z0 := zz:\n if (17/30<=tt a nd tt<=39/30) then\n hh := 1/60\n else \n hh := 1/30\n e nd if;\n tt := tt+hh;\n pts := [op(pts),[Re(zz),Im(zz)]]:\nend do: \np3 := plot(pts,color=COLOR(RGB,.28,.05,.48)):\np4 := plots[polygonpl ot]([seq([pts[i-1],pts[i],[0.95,3.63]],i=2..nops(pts))],\n st yle=patchnogrid,color=COLOR(RGB,.55,.1,.95)):\npts := []: z0 := 1-3.8* I: tt := 0: \nwhile tt<=61/30 do\n zz := newton(R(z)=exp(tt*Pi*I),z= z0):\n z0 := zz:\n if (11/15<=tt and tt<=22/15) then\n hh := \+ 1/60\n else \n hh := 1/30\n end if;\n tt := tt+hh;\n pts \+ := [op(pts),[Re(zz),Im(zz)]]:\nend do:\np5 := plot(pts,color=COLOR(RGB ,.28,.05,.48)):\np6 := plots[polygonplot]([seq([pts[i-1],pts[i],[0.95, -3.63]],i=2..nops(pts))],\n style=patchnogrid,color=COLOR(RGB ,.55,.1,.95)):\np7 := plot([[[-3.99,0],[1.69,0]],[[0,-4.29],[0,4.29]]] ,color=black,linestyle=3):\nplots[display]([p||(1..7)],view=[-3.99..1. 69,-4.29..4.29],font=[HELVETICA,9],\n color=COLOR(RGB,.4,0,1),labels =[`Re(z)`,`Im(z)`],axes=boxed,scaling=constrained);" }}{PARA 13 "" 1 " " {GLPLOT2D 469 558 558 {PLOTDATA 2 "60-%'CURVESG6$7fv7$$\"\"!F)F(7$$ \"3?+++Ri=dS!#E$\"3#******fJ'zq:!#=7$$\"35+++l[BxB!#C$\"3++++_7fTJF07$ $\"3=+++anS+B!#B$\"3&)*****H!eO7ZF07$$\"3!3++gc,a#)*F:$\"3K+++D'>IG'F0 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(approximately) " }{XPPEDIT 18 0 "[-3.3066, 0];" "6#7$,$-%&FloatG6$ \"&mI$!\"%!\"\"\"\"!" }{TEXT -1 2 ". " }}{PARA 0 "" 0 "" {TEXT -1 0 " " }}{PARA 0 "" 0 "" {TEXT -1 260 "We can distort the boundary curve ho rizontally by taking the 11th root of the real part of points along th e curve. In this way we see that the largest interval on the nonnegati ve imaginary axis that contains the origin and lies inside the stabili ty region is " }{XPPEDIT 18 0 "[0, 1.0];" "6#7$\"\"!-%&FloatG6$\"#5! \"\"" }{TEXT -1 18 " approximately. " }}{PARA 0 "" 0 "" {TEXT -1 0 " " }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 311 "R := z -> 1+z+1/2*z^2+1 /6*z^3+1/24*z^4+1/120*z^5+1/600*z^6:\nDigits := 15:\npts := []: z0 := \+ 0:\nfor ct from 0 to 35 do\n zz := newton(R(z)=exp(ct*Pi/100*I),z=z0 ):\n z0 := zz:\n pts := [op(pts),[surd(Re(zz),11),Im(zz)]]:\nend d o:\nplot(pts,color=COLOR(RGB,.55,0,.95),thickness=2,font=[HELVETICA,9] );\nDigits := 10:" }}{PARA 13 "" 1 "" {GLPLOT2D 261 276 276 {PLOTDATA 2 "6(-%'CURVESG6#7F7$$\"\"!F)F(7$$\"0<611cJ>(!#;$\"0O)e`EfTJF-7$$\"0o@ \"\\#3&\\5!#:$\"0o&*pI&=$G'F-7$$\"0`0#p[i38F3$\"0?Jw&zxC%*F-7$$\"0_l%) [l)H:F3$\"0:*3fqjc7F37$$\"0\"\\\"\\Z!GE F3$\"0`iAbb\\)=F37$$\"0-rY%=&)o?F3$\"0][0u9\"*>#F37$$\"0%*H\"zWh@AF3$ \"0krh#QF8DF37$$\"0G_H'GokBF3$\"0%H^9FVFGF37$$\"0>$oAlL*\\#F3$\"0lF>D \"fTJF37$$\"0Q#\\Ge^EEF3$\"0ijB?\\dX$F37$$\"0lu,cApu#F3$\"0(Q1>i!*pPF3 7$$\"0'yHbC3hGF3$\"0Q17\"=1%3%F37$$\"0Z-H?&QpHF3$\"0*[X0`@)R%F37$$\"0i -@;5@2$F3$\"0=DD!eO7ZF37$$\"0rDtQU%pJF3$\"0V5!H@^E]F37$$\"03P?Z\"[hKF3 $\"0_d[y_1M&F37$$\"0T)RHWC[LF3$\"0@Tq)eyacF37$$\"0o$pqGmHMF3$\"0pW,64* ofF37$$\"0V:u$Qd0NF3$\"00@Zi>IG'F37$$\"0L+x_-dd$F3$\"0QzW.9rf'F37$$\"0 GH^YL'ROF3$\"0W3NJ)=6pF37$$\"0!QC:Pw'p$F3$\"0X<\\uP_A(F37$$\"0Ebi8Eiu$ F3$\"0cLC'oDRvF37$$\"0&)y<,fny$F3$\"0rWmRRK&yF37$$\"0fCM6sk\"QF3$\"0y3 xAyr;)F37$$\"0_K-A$RKQF3$\"0EgkMl5[)F37$$\"0KWf4*[HQF3$\"0iDf#=*[z)F37 $$\"0FH[-$>)z$F3$\"0:-`!yk3\"*F37$$\"0=3=&\\P;PF3$\"0L`%)[ABU*F37$$\"0 W3*fQ71NF3$\"0V>*eT!ft*F37$$!0!z&\\SlgA$F3$\"0uK8-Q\\+\"!#97$$!0&\\ea) HH!QF3$\"0>!\\@PFO5Ffu7$$!02IME4,3%F3$\"0\\_o4'fn5Ffu7$$!0I.o!>G)G%F3$ \"02Y=m.*)4\"Ffu-%%FONTG6$%*HELVETICAG\"\"*-%*THICKNESSG6#\"\"#-%+AXES LABELSG6$Q!6\"Fbw-%&COLORG6&%$RGBG$\"#b!\"#F($\"#&*Fjw-%%VIEWG6$%(DEFA ULTGF`x" 1 2 0 1 10 2 2 6 1 4 2 1.000000 45.000000 45.000000 0 0 "Curv e 1" }}}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 123 "The relevant intersection point of the boundary curve with the imagin ary axis can be determined more accurately as follows." }}{PARA 0 "" 0 "" {TEXT -1 86 "First we look for points on the boundary curve eithe r side of the intersection point. " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 111 "Digits := 15:\nz0 := 1.0*I: \nfor ct from 30 to 33 do\n newton(R(z)=exp(ct*Pi/100*I),z=z0);\nend do;\nDigits := 10:" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#^$$\"0iXD*Gtn=! #>$\"0L`%)[ABU*!#:" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#^$$\"0JyxZsB%)*! #?$\"0V>*eT!ft*!#:" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#^$$!04Ze7r\"RR!# ?$\"0uK8-Q\\+\"!#9" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#^$$!0XUxNKgS#!#> $\"0>!\\@PFO5!#9" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 107 "Then we apply the bisection method to calculate the para meter value associated with the intersection point." }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 176 "real_part := proc(u)\n Re(newton(R(z)=ex p(u*Pi*I),z=1.0*I))\nend proc:\nDigits := 15:\nu0 := bisect('real_part '(u),u=0.30..0.33);\nnewton(R(z)=exp(u0*Pi*I),z=1.0*I);\nDigits := 10: " }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%#u0G$\"0TKT+x_<$!#:" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#^#$\"0IDj3!*=(**!#:" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 4 "The " }{TEXT 260 112 "largest in terval on the nonnegative imaginary axis that contains the origin and \+ lies inside the stability region" }{TEXT -1 36 " is [ 0, 0.9972 ] (a pproximately)." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 37 "#------------------------------------" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 80 "The stability function R * for the 7 stage, order 4 scheme is given as follows." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 97 "subs(ee ,subs(b=`b*`,StabilityFunction(4,7,'expanded'))):\n`R*` := unapply(%,z ):\n'`R*`(z)'=`R*`(z);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/-%#R*G6#%\" zG,2\"\"\"F)F'F)*&#F)\"\"#F)*$)F'F,F)F)F)*&#F)\"\"'F)*$)F'\"\"$F)F)F)* &#F)\"#CF)*$)F'\"\"%F)F)F)*&#\"((py;\"*g\"eV=F)*$)F'\"\"&F)F)F)*&#\"(T XD\"\"*+3z@*F)*$)F'F1F)F)F)*&#F)\"&+_#F)*$)F'\"\"(F)F)F)" }}}{PARA 0 " " 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 124 "We can find the p oint where the boundary of the stability region intersects the negativ e real axis by solving the equation: " }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "`R*`(z) = 1;" "6#/-%#R*G6#%\"zG\"\"\"" }{TEXT -1 2 ". " }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 30 "z_0 := newton(`R*`( z)=1,z=-4);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%$z_0G$!+t#3*fU!\"*" } }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 318 "z_0 := newton(`R*`(z)=1,z=-4):\np_1 := plot([`R*`(z),1],z=-4.79 ..0.49,color=[red,blue]):\np_2 := plot([[[z_0,1]]$3],style=point,symbo l=[circle,cross,diamond],color=black):\np_3 := plot([[z_0,0],[z_0,1]], linestyle=3,color=COLOR(RGB,0,.5,0)):\nplots[display]([p_1,p_2,p_3],vi ew=[-4.79..0.49,-0.07..1.47],font=[HELVETICA,9]);" }}{PARA 13 "" 1 "" {GLPLOT2D 411 220 220 {PLOTDATA 2 "6+-%'CURVESG6$7X7$$!3/++++++!z%!#<$ \"31*4e!zHf\"[#F*7$$!3n*****H^bCt%F*$\"3]:r+KC-lAF*7$$!3=+++E5\"\\n%F* $\"3Wg)\\nNvR1#F*7$$!3[++bv=%[i%F*$\"3n%)fRi;+,>F*7$$!3))****4DFxuXF*$ \"3!HehRCy&[*zWI\"F*7$$!3%*****>[Uz[VF *$\"3UfWb)Q+\"y6F*7$$!3K++0-CQ#H%F*$\"3U)yS:FUB1\"F*7$$!3$)*****ebqfB% F*$\"39W(*z'Q8(f&*!#=7$$!3U++q]&o88%F*$\"3qKzO)Gih\"yFin7$$!3p****4L#f 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{MPLTEXT 1 0 1831 "`R*` := \+ z -> 1+z+1/2*z^2+1/6*z^3+1/24*z^4+1678697/184358160*z^5+1254541/921790 800*z^6+1/25200*z^7:\npts := []: z0 := 0: tt := 0: \nwhile tt<=161/20 \+ do\n zz := newton(`R*`(z)=exp(tt*Pi*I),z=z0):\n z0 := zz:\n if ( 9/20<=tt and tt<=11/10) \n or (137/20<=tt and tt<=151/20) then\n \+ hh := 1/60\n else \n hh := 1/20\n end if;\n tt := tt+hh ;\n pts := [op(pts),[Re(zz),Im(zz)]]:\nend do:\np_1 := plot(pts,colo r=COLOR(RGB,.19,.04,.39)):\np_2 := plots[polygonplot]([seq([pts[i-1],p ts[i],[-2.1,0]],i=2..nops(pts))],\n style=patchnogrid,color=C OLOR(RGB,.37,.07,.78)):\npts := []: z0 := 0.9+3.9*I: tt := 0:\nwhile t t<=61/30 do\n zz := newton(`R*`(z)=exp(tt*Pi*I),z=z0):\n z0 := zz: \n if (21/30<=tt and tt<=29/30) then\n hh := 1/120\n elif (13 /30<=tt and tt<=19/15) then\n hh := 1/60\n else \n hh := 1 /30\n end if;\n tt := tt+hh;\n pts := [op(pts),[Re(zz),Im(zz)]]: \nend do:\np_3 := plot(pts,color=COLOR(RGB,.19,.04,.39)):\np_4 := plot s[polygonplot]([seq([pts[i-1],pts[i],[0.78,3.60]],i=2..nops(pts))],\n \+ style=patchnogrid,color=COLOR(RGB,.37,.07,.78)):\npts := []: \+ z0 := 0.9-3.9*I: tt := 0:\nwhile tt<=61/30 do\n zz := newton(`R*`(z) =exp(tt*Pi*I),z=z0):\n z0 := zz:\n if (16/15<=tt and tt<=4/3) then \n hh := 1/120\n elif (23/30<=tt and tt<=8/5) then\n hh := 1/60\n else \n hh := 1/30\n end if;\n tt := tt+hh;\n pts := [op(pts),[Re(zz),Im(zz)]]:\nend do:\np_5 := plot(pts,color=COLOR(R GB,.19,.04,.39)):\np_6 := plots[polygonplot]([seq([pts[i-1],pts[i],[0. 78,-3.6]],i=2..nops(pts))],\n style=patchnogrid,color=COLOR(R GB,.37,.07,.78)):\np_7 := plot([[[-4.89,0],[1.49,0]],[[0,-4.39],[0,4.3 9]]],color=black,linestyle=3):\nplots[display]([p_||(1..7)],view=[-4.8 9..1.49,-4.39..4.39],font=[HELVETICA,9],\n color=COLOR(RGB,.4,0,1),l abels=[`Re(z)`,`Im(z)`],axes=boxed,scaling=constrained);" }}{PARA 13 " " 1 "" {GLPLOT2D 486 581 581 {PLOTDATA 2 "60-%'CURVESG6$7dx7$$\"\"!F)F 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0 "" 0 "" {TEXT -1 39 "#------------------ --------------------" }}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 28 "checking \+ the combined scheme" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 35 "c oefficients of the combined scheme" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 968 "ee := \{c[2]=35/159,\nc[3]= 42/131,\nc[4]=131/143,\nc[5]=21/22,\nc[6]=1,\nc[7]=1,\n\na[2,1]=35/159 ,\na[3,1]=7476/85805,\na[3,2]=20034/85805,\na[4,1]=2438549411/19839619 80,\na[4,2]=-3707256508/716430715,\na[4,3]=25077455105/5158301148,\na[ 5,1]=105337889067/64388030080,\na[5,2]=-1698584121/245755840,\na[5,3]= 6869523776931/1096562558080,\na[5,4]=-927215289/26981535520,\na[6,1]=6 7512025387/32454592380,\na[6,2]=-20051384/2293935,\na[6,3]=10587214001 321/1373901639516,\na[6,4]=731293420/8209319229,\na[6,5]=-144610048/10 77690663,\na[7,1]=669707/6932520,\na[7,2]=0,\na[7,3]=2215522905683/457 0867891800,\na[7,4]=349043981/116904400,\na[7,5]=-2234144/575505,\na[7 ,6]=9363/7120,\n\nb[1]=669707/6932520,\nb[2]=0,\nb[3]=2215522905683/45 70867891800,\nb[4]=349043981/116904400,\nb[5]=-2234144/575505,\nb[6]=9 363/7120,\n\n`b*`[1]=2243660497/23535905400,\n`b*`[2]=0,\n`b*`[3]=7589 131232781673/15518096492661000,\n`b*`[4]=1104461697911/396890438000,\n `b*`[5]=-6925033984/1953839475,\n`b*`[6]=3529851/3021550,\n`b*`[7]=1/1 12\}:" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 137 "subs(ee,matrix([seq([c[i],seq(a[i,j],j=1.. i-1),``$(8-i)],i=2..7),\n[`b`,seq(b[i],i=1..6),``],[`b*`,seq(`b*`[i],i =1..7)]]));\n``;\nevalf(%%);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#K%'mat rixG6#7*7*#\"#N\"$f\"F(%!GF+F+F+F+F+7*#\"#U\"$J\"#\"%wu\"&0e)#\"&M+#F2 F+F+F+F+F+7*#F/\"$V\"#\"+6%\\&QC\"+!)>'R)>#!+3lD2P\"*:2V;(#\",0^Xx]#\" +[6Ie^F+F+F+F+7*#\"#@\"#A#\"-n!*)yL0\"\",!3I!)Qk#!+@Te)p\"\"*SevX##\". JpxB&po\".!3eDc'4\"#!**G:s#*\",?b`\")p#F+F+F+7*\"\"\"#\",(QD?^n\",!Q#f aC$#!)%Q^+#\"(NRH##\"/@8+9se5\".;&R;!RP\"#\"*?MHJ(\"+H#>$4#)#!*[+hW\" \"+j1px5F+F+7*FR#\"'2(p'\"(?D$p\"\"!#\".$o0H_:A\".+=*y'3d%#\"*\")R/\\$ \"*+W!p6#!(WTB#\"'0bd#\"%j$*\"%?rF+7*%\"bGF]oF`oFaoFdoFgoFjoF+7*%#b*G# \"+(\\gOC#\",+a!f`BF`o#\"1t;yK78*e(\"2+5m#\\'4=b\"#\".6zphW5\"\"-+!Q/* oR#!+%)R.Dp\"+v%RQ&>#\"(^)HN\"(]:-$#FR\"$7\"Q)pprint266\"" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%!G" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#K%'matri xG6#7*7*$\"+iyD,A!#5F(%!GF+F+F+F+F+7*$\"+qo51KF*$\"+w*yFr)!#6$\"+t*G[L #F*F+F+F+F+F+7*$\"+h\"R3;*F*$\"+?68H7!\"*$!+B(>Y<&F9$\"+>Ddh[F9F+F+F+F +7*$\"+XXXX&*F*$\"+&e&)fj\"F9$!+wMn6pF9$\"+a\")fkiF9$!+03[OMF1F+F+F+7* $\"\"\"\"\"!$\"+b%*>!3#F9$!+$zU5u)F9$\"+rv%fq(F9$\"+u%)33*)F1$!+t6&=M \"F*F+F+7*FJ$\"+7)o.m*F1$FLFL$\"+T30Z[F*$\"+/:s&)HF9$!+m$e?)QF9$\"+!4G ]J\"F9F+7*%\"bGFXFZFenFgnFinF[oF+7*%#b*G$\"+?i#H`*F1FZ$\"+fR]!*[F*$\"+ pty#y#F9$!+[3KWNF9$\"+\\_Ao6F9$\"+H9dG*)!#7Q)pprint276\"" }}}{PARA 0 " " 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 146 "RK5_ 6eqs := [op(RowSumConditions(6,'expanded')),op(OrderConditions(5,6,'ex panded'))]:\n`RK4_7eqs*` := subs(b=`b*`,OrderConditions(4,7,'expanded' )):" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 7 "Ch eck: " }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 148 "simplify(subs(ee,R K5_6eqs)):\nmap(u->`if`(lhs(u)=rhs(u),0,1),%);\nnops(%);\nsimplify(sub s(ee,`RK4_7eqs*`)):\nmap(u->`if`(lhs(u)=rhs(u),0,1),%);\nnops(%);" } {TEXT -1 0 "" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#78\"\"!F$F$F$F$F$F$F$F $F$F$F$F$F$F$F$F$F$F$F$F$F$" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#\"#A" } }{PARA 11 "" 1 "" {XPPMATH 20 "6#7*\"\"!F$F$F$F$F$F$F$" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#\"\")" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 92 "We can calculate the principal error norm, that i s, the 2-norm of the principal error terms." }}{EXCHG {PARA 0 "> " 0 " " {MPLTEXT 1 0 123 "errterms5_6 := PrincipalErrorTerms(5,6,'expanded') :\nsqrt(add(subs(ee,errterms5_6[i])^2,i=1.. nops(errterms5_6))):\neval f(%);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#$\"+7bsf=!#8" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 87 "We can also calculat e the 2 norm of the principal error of the order 4 embedded scheme." } }{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 145 "`errterms4_7*` := subs(b=` b*`,PrincipalErrorTerms(4,7,'expanded')):\nsqrt(add(subs(ee,`errterms4 _7*`[i])^2,i=1.. nops(`errterms4_7*`))):\nevalf(%);" }}{PARA 11 "" 1 " " {XPPMATH 20 "6#$\"+8-gWd!#8" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }} {PARA 0 "" 0 "" {TEXT -1 86 "Next we set-up stage-order condtions to c heck for stage-orders from 2 to 5 inclusive. " }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 85 "for ct from 2 to 4 do\n so||ct||_6 := StageOrd erConditions(ct,6,'expanded');\nend do:" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 58 "Stages 3 to 6 have the following r espective stage-orders. " }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 173 "[seq([seq(expand(subs(ee,so||i||_6[j])),i=2..4)],j=1..4)]:\nmap(proc( L) local i; for i to nops(L) do if not evalb(L[i]) then break end if e nd do; i end proc,%):\nsimplify(%); " }}{PARA 11 "" 1 "" {XPPMATH 20 " 6#7&\"\"#F$F$F$" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 27 "The simplifying conditions:" }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "Sum(b[i]*a[i,j],i = j+1 .. 6) = b[j]*(1-c[j]) " "6#/-%$SumG6$*&&%\"bG6#%\"iG\"\"\"&%\"aG6$F+%\"jGF,/F+;,&F0F,F,F,\" \"'*&&F)6#F0F,,&F,F,&%\"cG6#F0!\"\"F," }{TEXT -1 3 ", " }{XPPEDIT 18 0 "j=1" "6#/%\"jG\"\"\"" }{TEXT -1 7 " . . 5 " }}{PARA 0 "" 0 "" {TEXT -1 8 "(where " }{XPPEDIT 18 0 "c[1]=0" "6#/&%\"cG6#\"\"\"\"\"! " }{TEXT -1 17 " ) are satisfied." }}{PARA 256 "" 0 "" {TEXT -1 1 " " }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 154 "[Sum(b[i]*a[i,1],i=1+1..6 )=b[1],seq(Sum(b[i]*a[i,j],i=j+1..6)=b[j]*(1-c[j]),j=2..5)];\neval(sub s(Sum=add,%)):\nsubs(ee,%);\nmap(u->lhs(u)-rhs(u),%);\nnops(%);" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#7'/-%$SumG6$*&&%\"bG6#%\"iG\"\"\"&%\"a G6$F,F-F-/F,;\"\"#\"\"'&F*6#F-/-F&6$*&F)F-&F/6$F,F3F-/F,;\"\"$F4*&&F*6 #F3F-,&F-F-&%\"cGFB!\"\"F-/-F&6$*&F)F-&F/6$F,F?F-/F,;\"\"%F4*&&F*6#F?F -,&F-F-&FEFRFFF-/-F&6$*&F)F-&F/6$F,FOF-/F,;\"\"&F4*&&F*6#FOF-,&F-F-&FE FjnFFF-/-F&6$*&F)F-&F/6$F,FgnF-/F,;F4F4*&&F*6#FgnF-,&F-F-&FEFgoFFF-" } }{PARA 11 "" 1 "" {XPPMATH 20 "6#7'/#\"'2(p'\"(?D$pF%/\"\"!F)/#\",$f)Q 7p\"\",+i1e8&F+/#\"(,EK(\")+hAHF//#!'_:5\"'0bdF3" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#7'\"\"!F$F$F$F$" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#\"\" &" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 27 "The simplifying condition: " }}{PARA 256 "" 0 "" {TEXT -1 1 " " } {XPPEDIT 18 0 "Sum(b[i]*a[i,2],i = 3 .. 6)=0" "6#/-%$SumG6$*&&%\"bG6#% \"iG\"\"\"&%\"aG6$F+\"\"#F,/F+;\"\"$\"\"'\"\"!" }{TEXT -1 1 " " }} {PARA 0 "" 0 "" {TEXT -1 13 "is satisfied." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 59 "Sum(b[i]*a[i,2],i= 3..6);\neval(subs(Sum=add,%));\nsubs(ee,%);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#-%$SumG6$*&&%\"bG6#%\"iG\"\"\"&%\"aG6$F*\"\"#F+/F*;\"\" $\"\"'" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#,**&&%\"aG6$\"\"%\"\"#\"\"\" &%\"bG6#F(F*F**&&F&6$\"\"$F)F*&F,6#F1F*F**&&F&6$\"\"&F)F*&F,6#F7F*F**& &F&6$\"\"'F)F*&F,6#F=F*F*" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#\"\"!" }} }{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 27 "The simp lifying condition: " }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "Sum(b[i]*Sum(a[i,j]*a[j,2],j = 3 .. i-1),i = 2 .. 6)=0" "6#/-%$SumG 6$*&&%\"bG6#%\"iG\"\"\"-F%6$*&&%\"aG6$F+%\"jGF,&F16$F3\"\"#F,/F3;\"\"$ ,&F+F,F,!\"\"F,/F+;F6\"\"'\"\"!" }{TEXT -1 1 " " }}{PARA 0 "" 0 "" {TEXT -1 13 "is satisfied." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 82 "Sum(b[i]*Sum(a[i,j]*a[j,2],j=3..i-1 ),i =2 ..6);\neval(subs(Sum=add,%));\nsubs(ee,%);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#-%$SumG6$*&&%\"bG6#%\"iG\"\"\"-F$6$*&&%\"aG6$F*%\"jGF+& F06$F2\"\"#F+/F2;\"\"$,&F*F+F+!\"\"F+/F*;F5\"\"'" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#,(*(&%\"bG6#\"\"%\"\"\"&%\"aG6$F(\"\"$F)&F+6$F-\"\"#F)F )*&&F&6#\"\"&F),&*&&F+6$F4F-F)F.F)F)*&&F+6$F4F(F)&F+6$F(F0F)F)F)F)*&&F &6#\"\"'F),(*&&F+6$FAF-F)F.F)F)*&&F+6$FAF(F)F " 0 "" {MPLTEXT 1 0 64 "Sum(b[i]*c[i]*a[i,2],i=3..6) ;\neval(subs(Sum=add,%));\nsubs(ee,%);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#-%$SumG6$*(&%\"bG6#%\"iG\"\"\"&%\"cGF)F+&%\"aG6$F*\"\"#F+/F*;\" \"$\"\"'" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#,**(&%\"bG6#\"\"$\"\"\"&% \"cGF'F)&%\"aG6$F(\"\"#F)F)*(&F&6#\"\"%F)&F+F2F)&F-6$F3F/F)F)*(&F&6#\" \"&F)&F+F9F)&F-6$F:F/F)F)*(&F&6#\"\"'F)&F+F@F)&F-6$FAF/F)F)" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#\"\"!" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }} {PARA 0 "" 0 "" {TEXT -1 78 "None of the 20 principal error conditions are satisfied by the order 5 scheme." }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 123 "RK5_6err_eqs := PrincipalErrorConditions(5,6,'expand ed'):\nexpand(subs(ee,RK5_6err_eqs)):\nmap(u->`if`(lhs(u)=rhs(u),0,1), %);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#76\"\"\"F$F$F$F$F$F$F$F$F$F$F$F $F$F$F$F$F$F$F$" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 94 "None of the 9 principal error conditions are satisfied by the 7 stage embedded order 4 scheme." }}{PARA 0 "" 0 "" {TEXT -1 95 " The largest magnitude of an error term is about 31 times the lowest ma gnitude of an error term." }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 147 "`errterms4_7*` := subs(b=`b*`,PrincipalErrorTerms(4,7,'expanded') ):\nsubs(ee,`errterms4_7*`):\nmap(abs,evalf[8](%));\nevalf[4](max(op(% ))/min(op(%)));" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#7+$\")71HE!#6$\")Bz 0;F&$\")TJkYF&$\")b8FZ!#7$\")@;mWF-$\")INRZF-$\")TKK*)F-$\")4k_`F-$\") 2s)[\"F-" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#$\"%KJ!\"#" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}} {PARA 0 "" 0 "" {TEXT -1 39 "#--------------------------------------" }}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 26 "determination of the nodes" }} {PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 2 "e2" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 10256 "e2 := \{c [7] = 1, a[6,2] = 1/2*(-15*c[4]*c[3]^2+30*c[5]*c[4]*c[3]^2-30*c[3]*c[4 ]*c[5]+16*c[4]*c[3]-5*c[3]*c[5]+3*c[3]+10*c[4]*c[5]-6*c[4])/(-12+15*c[ 3]+15*c[4]+15*c[5]-20*c[4]*c[3]-20*c[4]*c[5]+30*c[3]*c[4]*c[5]-20*c[3] *c[5])/c[3]/c[2], c[6] = 1, a[2,1] = c[2], a[5,4] = 1/2*(c[4]*c[3]-c[3 ]*c[5]-c[4]*c[5]+c[5]^2)*c[5]*(5*c[3]-2)/(c[3]-c[4])/(3-5*c[3]+10*c[4] *c[3]-5*c[4])/c[4], `b*`[2] = 0, b[2] = 0, b[4] = 1/60*(10*c[3]*c[5]-5 *c[3]-5*c[5]+3)/(-c[4]^2+c[4]+c[4]*c[3]-c[3])/c[4]/(c[4]-c[5]), a[7,1] = 1/60*(5*c[4]-10*c[4]*c[5]+30*c[3]*c[4]*c[5]-10*c[4]*c[3]-10*c[3]*c[ 5]-3+5*c[3]+5*c[5])/c[3]/c[4]/c[5], a[6,4] = 1/2*(6-30*c[5]^2*c[3]+20* c[5]^2*c[3]^2-10*c[5]^2*c[4]-16*c[3]-8*c[4]-14*c[5]-7*c[4]^2*c[3]+30*c [5]^2*c[3]*c[4]+25*c[5]*c[4]*c[3]^2-20*c[5]^2*c[4]*c[3]^2+23*c[4]*c[3] +14*c[4]*c[5]-39*c[3]*c[4]*c[5]+39*c[3]*c[5]-15*c[4]*c[3]^2+10*c[3]^2+ 2*c[4]^2+10*c[5]^2-25*c[3]^2*c[5]+5*c[4]^2*c[3]^2)/c[4]/(12*c[4]^2-12* c[4]*c[3]-15*c[4]^3-12*c[4]*c[5]+12*c[3]*c[5]-20*c[4]^2*c[5]^2+20*c[4] ^3*c[5]+20*c[4]^3*c[3]-20*c[4]^2*c[3]^2+20*c[5]^2*c[3]^2-20*c[4]^2*c[5 ]*c[3]+15*c[4]*c[3]^2+15*c[5]^2*c[4]+30*c[5]^2*c[4]^2*c[3]+30*c[5]*c[4 ]^2*c[3]^2-30*c[5]^2*c[4]*c[3]^2-30*c[5]*c[4]^3*c[3]+15*c[3]*c[4]*c[5] -15*c[3]^2*c[5]-15*c[5]^2*c[3]), `b*`[1] = 1/60*(-460*c[5]*c[4]*c[3]*` b*`[7]+760*c[5]*c[4]*c[3]^2*`b*`[7]+3*c[3]-6*c[4]-20*c[4]^2*c[5]-60*c[ 4]^2*c[3]-1800*`b*`[7]*c[4]^2*c[5]*c[3]^2+100*c[5]*c[4]*c[3]^2+100*c[3 ]^2*c[5]*`b*`[7]-100*c[3]^3*c[5]*c[4]+300*c[3]^3*c[4]*`b*`[7]-600*c[3] ^3*`b*`[7]*c[4]^2+940*`b*`[7]*c[4]^2*c[3]^2-60*c[5]*c[3]*`b*`[7]-560*c [4]^2*c[3]*`b*`[7]+300*c[5]*c[4]^2*c[3]^3-100*c[4]^2*c[3]^3+140*c[4]^2 *c[5]*c[3]+120*`b*`[7]*c[4]^2+120*c[5]*c[4]*`b*`[7]-340*c[5]*c[4]^2*c[ 3]^2-200*`b*`[7]*c[4]^2*c[5]+1000*c[5]*c[4]^2*c[3]*`b*`[7]-600*c[3]^3* c[5]*c[4]*`b*`[7]+29*c[4]*c[3]-380*`b*`[7]*c[4]*c[3]^2+252*`b*`[7]*c[4 ]*c[3]+10*c[4]*c[5]-50*c[3]*c[4]*c[5]+1200*c[3]^3*`b*`[7]*c[4]^2*c[5]+ 36*`b*`[7]*c[3]-72*`b*`[7]*c[4]-60*`b*`[7]*c[3]^2-5*c[3]*c[5]-60*c[4]* c[3]^2-5*c[3]^2+10*c[4]^2+10*c[3]^2*c[5]+50*c[4]*c[3]^3+130*c[4]^2*c[3 ]^2)/(10*c[4]*c[3]^2-8*c[4]*c[3]-c[3]+2*c[4])/c[3]/c[4]/c[5], a[6,1] = 1/2*(10*c[3]*c[2]*c[5]^2-14*c[3]*c[2]*c[5]-30*c[3]^2*c[2]*c[5]^2-25*c [3]^3*c[2]*c[5]+10*c[3]^3*c[2]-108*c[3]^2*c[2]*c[4]*c[5]-3*c[5]*c[4]*c [3]^2+5*c[5]^2*c[4]*c[3]^2-30*c[5]^2*c[4]^2*c[3]^3+30*c[5]^2*c[4]^2*c[ 3]^2-10*c[5]^2*c[4]^2*c[3]+15*c[5]*c[4]^2*c[3]^3+6*c[4]^2*c[5]*c[3]-30 *c[3]^2*c[2]*c[4]^2-30*c[3]^3*c[4]*c[2]+10*c[3]*c[2]*c[4]^2-16*c[3]*c[ 2]*c[4]-16*c[5]*c[4]^2*c[3]^2+39*c[3]*c[2]*c[4]*c[5]-30*c[3]*c[2]*c[5] ^2*c[4]-9*c[3]*c[2]*c[5]*c[4]^2+6*c[3]*c[2]+10*c[2]*c[4]^2*c[5]^2-10*c [3]*c[2]*c[4]^2*c[5]^2-60*c[3]^3*c[2]*c[5]*c[4]^2-40*c[3]^2*c[2]*c[4]^ 2*c[5]^2+46*c[3]^2*c[2]*c[4]-16*c[2]*c[3]^2-6*c[2]*c[5]*c[4]^2+39*c[3] ^2*c[2]*c[5]+60*c[2]*c[4]^2*c[5]^2*c[3]^3+20*c[3]^3*c[2]*c[4]^2+20*c[3 ]^3*c[2]*c[5]^2-60*c[3]^3*c[2]*c[5]^2*c[4]+75*c[3]^3*c[2]*c[4]*c[5]+80 *c[3]^2*c[2]*c[5]^2*c[4]+65*c[3]^2*c[2]*c[5]*c[4]^2)/(-12+15*c[3]+15*c [4]+15*c[5]-20*c[4]*c[3]-20*c[4]*c[5]+30*c[3]*c[4]*c[5]-20*c[3]*c[5])/ c[3]^2/c[2]/c[4]/c[5], a[7,2] = 0, b[3] = -1/60*(10*c[4]*c[5]-5*c[4]-5 *c[5]+3)/(c[3]-1)/c[3]/(c[3]-c[5])/(c[3]-c[4]), a[6,3] = -1/2*(10*c[5] ^2*c[3]-10*c[5]^2*c[3]^2+6*c[3]-6*c[4]^2*c[5]+16*c[4]^2*c[3]-40*c[5]^2 *c[3]*c[4]-45*c[5]*c[4]*c[3]^2+60*c[5]^2*c[4]*c[3]^2-20*c[3]^3*c[5]*c[ 4]-10*c[5]^2*c[4]^2*c[3]-20*c[5]*c[4]^2*c[3]^3+15*c[4]^2*c[3]^3-19*c[4 ]^2*c[5]*c[3]+10*c[4]^2*c[5]^2+45*c[5]*c[4]^2*c[3]^2-10*c[3]^4*c[4]-20 *c[4]*c[5]^2*c[3]^3-16*c[4]*c[3]+45*c[3]*c[4]*c[5]+20*c[3]^4*c[5]*c[4] -14*c[3]*c[5]+17*c[4]*c[3]^2-8*c[3]^2+2*c[3]^3+14*c[3]^2*c[5]+9*c[4]*c [3]^3-31*c[4]^2*c[3]^2)/(-30*c[5]^2*c[3]*c[4]+20*c[5]^2*c[4]-15*c[4]*c [5]+15*c[4]*c[3]+30*c[5]*c[4]*c[3]^2-20*c[4]*c[3]^2+15*c[3]^2+12*c[5]- 12*c[3]+20*c[5]^2*c[3]-15*c[5]^2-20*c[3]^2*c[5])/(c[3]-c[4])/c[3]^2, a [4,1] = 1/2*c[4]*(2*c[2]*c[3]^2-3*c[4]*c[3]^2+2*c[4]^2*c[3]+2*c[3]*c[2 ]*c[4]-2*c[2]*c[4]^2)/c[2]/c[3]^2, a[5,3] = -1/2*c[5]*(-2*c[5]^2*c[3]+ 6*c[4]^2*c[5]-6*c[4]^2*c[3]+15*c[5]^2*c[3]*c[4]-5*c[5]*c[4]*c[3]^2-20* c[5]^2*c[4]*c[3]^2+20*c[3]^3*c[5]*c[4]+20*c[5]^2*c[4]^2*c[3]-5*c[4]^2* c[5]*c[3]-10*c[4]^2*c[5]^2-20*c[5]*c[4]^2*c[3]^2-4*c[3]*c[4]*c[5]+4*c[ 4]*c[3]^2+2*c[3]^2*c[5]-10*c[4]*c[3]^3+15*c[4]^2*c[3]^2)/(3-5*c[3]+10* c[4]*c[3]-5*c[4])/c[3]^2/(c[3]-c[4]), a[4,3] = -c[4]^2*(c[3]-c[4])/c[3 ]^2, a[5,2] = 1/2*c[5]*(-15*c[4]*c[3]^2+30*c[5]*c[4]*c[3]^2-20*c[5]^2* c[3]*c[4]+6*c[4]*c[3]-5*c[5]^2*c[3]+3*c[3]*c[5]-6*c[4]*c[5]+10*c[5]^2* c[4])/c[2]/(3-5*c[3]+10*c[4]*c[3]-5*c[4])/c[3], `b*`[5] = 1/60*(-460*c [5]*c[4]*c[3]*`b*`[7]+760*c[5]*c[4]*c[3]^2*`b*`[7]+3*c[3]-6*c[4]-60*c[ 4]^2*c[3]-1800*`b*`[7]*c[4]^2*c[5]*c[3]^2+100*c[3]^2*c[5]*`b*`[7]+300* c[3]^3*c[4]*`b*`[7]-600*c[3]^3*`b*`[7]*c[4]^2+940*`b*`[7]*c[4]^2*c[3]^ 2-60*c[5]*c[3]*`b*`[7]-560*c[4]^2*c[3]*`b*`[7]-100*c[4]^2*c[3]^3+120*` b*`[7]*c[4]^2+120*c[5]*c[4]*`b*`[7]-200*`b*`[7]*c[4]^2*c[5]+1000*c[5]* c[4]^2*c[3]*`b*`[7]-600*c[3]^3*c[5]*c[4]*`b*`[7]+29*c[4]*c[3]-380*`b*` [7]*c[4]*c[3]^2+252*`b*`[7]*c[4]*c[3]+1200*c[3]^3*`b*`[7]*c[4]^2*c[5]+ 36*`b*`[7]*c[3]-72*`b*`[7]*c[4]-60*`b*`[7]*c[3]^2-60*c[4]*c[3]^2-5*c[3 ]^2+10*c[4]^2+50*c[4]*c[3]^3+130*c[4]^2*c[3]^2)/c[5]/(c[5]^2*c[3]-c[5] ^3*c[3]+c[5]^2*c[3]^2-2*c[5]^2*c[4]+2*c[5]^3*c[4]+2*c[4]^2*c[5]-2*c[4] ^2*c[3]+7*c[5]^2*c[3]*c[4]-8*c[5]^3*c[4]*c[3]-9*c[5]*c[4]*c[3]^2-2*c[5 ]^2*c[4]*c[3]^2+10*c[5]^3*c[4]*c[3]^2+10*c[3]^3*c[5]*c[4]-10*c[5]^2*c[ 4]^2*c[3]^2+8*c[5]^2*c[4]^2*c[3]+10*c[5]*c[4]^2*c[3]^3-10*c[4]^2*c[3]^ 3-6*c[4]^2*c[5]*c[3]-2*c[4]^2*c[5]^2+2*c[5]*c[4]^2*c[3]^2-10*c[4]*c[5] ^2*c[3]^3+c[3]*c[4]*c[5]+c[4]*c[3]^2-c[3]^2*c[5]+8*c[4]^2*c[3]^2), a[4 ,2] = 1/2*(3*c[3]-2*c[4])*c[4]^2/c[3]/c[2], `b*`[6] = -1/60*(-1000*c[5 ]*c[4]*c[3]*`b*`[7]+1780*c[5]*c[4]*c[3]^2*`b*`[7]-12*c[3]+24*c[4]+40*c [4]^2*c[5]+160*c[4]^2*c[3]-2880*`b*`[7]*c[4]^2*c[5]*c[3]^2-280*c[5]*c[ 4]*c[3]^2+160*c[3]^2*c[5]*`b*`[7]+200*c[3]^3*c[5]*c[4]+900*c[3]^3*c[4] *`b*`[7]-1200*c[3]^3*`b*`[7]*c[4]^2+2020*`b*`[7]*c[4]^2*c[3]^2-120*c[5 ]*c[3]*`b*`[7]-1160*c[4]^2*c[3]*`b*`[7]-300*c[5]*c[4]^2*c[3]^3+200*c[4 ]^2*c[3]^3-220*c[4]^2*c[5]*c[3]+240*`b*`[7]*c[4]^2+240*c[5]*c[4]*`b*`[ 7]+440*c[5]*c[4]^2*c[3]^2-320*`b*`[7]*c[4]^2*c[5]+1600*c[5]*c[4]^2*c[3 ]*`b*`[7]-1200*c[3]^3*c[5]*c[4]*`b*`[7]-111*c[4]*c[3]-1400*`b*`[7]*c[4 ]*c[3]^2+792*`b*`[7]*c[4]*c[3]-30*c[4]*c[5]+140*c[3]*c[4]*c[5]+1800*c[ 3]^3*`b*`[7]*c[4]^2*c[5]+96*`b*`[7]*c[3]-192*`b*`[7]*c[4]-120*`b*`[7]* c[3]^2+15*c[3]*c[5]+220*c[4]*c[3]^2+15*c[3]^2-30*c[4]^2-20*c[3]^2*c[5] -150*c[4]*c[3]^3-310*c[4]^2*c[3]^2)/(-10*c[4]*c[3]^2+10*c[5]*c[4]*c[3] ^2+8*c[4]*c[3]-8*c[3]*c[4]*c[5]+c[3]-c[3]*c[5]-2*c[4]+2*c[4]*c[5])/(c[ 4]-1)/(c[3]-1), `b*`[3] = 1/60*(-460*c[5]*c[4]*c[3]*`b*`[7]+760*c[5]*c [4]*c[3]^2*`b*`[7]+3*c[3]-6*c[4]-20*c[4]^2*c[5]-40*c[4]^2*c[3]-1800*`b *`[7]*c[4]^2*c[5]*c[3]^2+50*c[5]*c[4]*c[3]^2+100*c[3]^2*c[5]*`b*`[7]+3 00*c[3]^3*c[4]*`b*`[7]-600*c[3]^3*`b*`[7]*c[4]^2+940*`b*`[7]*c[4]^2*c[ 3]^2-60*c[5]*c[3]*`b*`[7]-560*c[4]^2*c[3]*`b*`[7]+80*c[4]^2*c[5]*c[3]+ 120*`b*`[7]*c[4]^2+120*c[5]*c[4]*`b*`[7]-100*c[5]*c[4]^2*c[3]^2-200*`b *`[7]*c[4]^2*c[5]+1000*c[5]*c[4]^2*c[3]*`b*`[7]-600*c[3]^3*c[5]*c[4]*` b*`[7]+19*c[4]*c[3]-380*`b*`[7]*c[4]*c[3]^2+252*`b*`[7]*c[4]*c[3]+10*c [4]*c[5]-30*c[3]*c[4]*c[5]+1200*c[3]^3*`b*`[7]*c[4]^2*c[5]+36*`b*`[7]* c[3]-72*`b*`[7]*c[4]-60*`b*`[7]*c[3]^2-5*c[3]*c[5]-30*c[4]*c[3]^2+10*c [4]^2+50*c[4]^2*c[3]^2)/(c[3]-1)/c[3]/(-10*c[5]*c[4]*c[3]^2+8*c[3]*c[4 ]*c[5]-2*c[4]*c[5]+2*c[4]*c[3]+10*c[4]*c[3]^3-8*c[4]*c[3]^2+c[3]*c[5]- c[3]^2)/(c[3]-c[4]), a[3,2] = 1/2*c[3]^2/c[2], a[7,6] = 1/60*(-12+15*c [3]+15*c[4]+15*c[5]-20*c[4]*c[3]-20*c[4]*c[5]+30*c[3]*c[4]*c[5]-20*c[3 ]*c[5])/(c[5]-1)/(c[4]-1)/(c[3]-1), a[5,1] = 1/2*(2*c[3]*c[2]*c[5]^2-5 *c[3]^2*c[2]*c[5]^2+5*c[3]^3*c[2]*c[5]+10*c[3]^2*c[2]*c[4]*c[5]-3*c[5] *c[4]*c[3]^2+5*c[5]^2*c[4]*c[3]^2+20*c[5]^2*c[4]^2*c[3]^2-10*c[5]^2*c[ 4]^2*c[3]-30*c[5]*c[4]^2*c[3]^3+15*c[4]^2*c[3]^3+6*c[4]^2*c[5]*c[3]-25 *c[3]^2*c[2]*c[4]^2-15*c[3]^3*c[4]*c[2]+6*c[3]*c[2]*c[4]^2-2*c[3]*c[2] *c[4]*c[5]-5*c[3]*c[2]*c[5]^2*c[4]+5*c[3]*c[2]*c[5]*c[4]^2+10*c[2]*c[4 ]^2*c[5]^2-20*c[3]*c[2]*c[4]^2*c[5]^2+8*c[3]^2*c[2]*c[4]-6*c[2]*c[5]*c [4]^2-2*c[3]^2*c[2]*c[5]+20*c[3]^3*c[2]*c[4]^2+20*c[3]^2*c[2]*c[5]*c[4 ]^2-6*c[4]^2*c[3]^2)*c[5]/c[4]/c[3]^2/(3-5*c[3]+10*c[4]*c[3]-5*c[4])/c [2], a[6,5] = (-3+8*c[3]+8*c[4]+3*c[5]+5*c[4]^2*c[5]+15*c[4]^2*c[3]-15 *c[5]*c[4]*c[3]^2-15*c[4]^2*c[5]*c[3]+10*c[5]*c[4]^2*c[3]^2-23*c[4]*c[ 3]-8*c[4]*c[5]+23*c[3]*c[4]*c[5]-8*c[3]*c[5]+15*c[4]*c[3]^2-5*c[3]^2-5 *c[4]^2+5*c[3]^2*c[5]-10*c[4]^2*c[3]^2)/c[5]/(-20*c[5]^3*c[3]+20*c[5]^ 2*c[3]^2-20*c[5]^3*c[4]-15*c[4]^2*c[5]+15*c[4]^2*c[3]+20*c[5]^2*c[3]*c [4]+30*c[5]^3*c[4]*c[3]-30*c[5]^2*c[4]*c[3]^2-30*c[5]^2*c[4]^2*c[3]+20 *c[4]^2*c[5]^2+30*c[5]*c[4]^2*c[3]^2-12*c[4]*c[3]+12*c[4]*c[5]-15*c[3] *c[4]*c[5]+15*c[5]^3+12*c[3]*c[5]+15*c[4]*c[3]^2-12*c[5]^2-15*c[3]^2*c [5]-20*c[4]^2*c[3]^2), a[7,3] = -1/60*(10*c[4]*c[5]-5*c[4]-5*c[5]+3)/( c[3]-1)/c[3]/(c[3]-c[5])/(c[3]-c[4]), b[5] = -1/60*(3-5*c[3]+10*c[4]*c [3]-5*c[4])/c[5]/(-c[5]^2-c[5]^2*c[3]+c[5]^3+c[3]*c[5]-c[5]^2*c[4]+c[3 ]*c[4]*c[5]-c[4]*c[3]+c[4]*c[5]), b[1] = 1/60*(5*c[4]-10*c[4]*c[5]+30* c[3]*c[4]*c[5]-10*c[4]*c[3]-10*c[3]*c[5]-3+5*c[3]+5*c[5])/c[3]/c[4]/c[ 5], b[6] = 1/60*(-12+15*c[3]+15*c[4]+15*c[5]-20*c[4]*c[3]-20*c[4]*c[5] +30*c[3]*c[4]*c[5]-20*c[3]*c[5])/(c[5]-1)/(c[4]-1)/(c[3]-1), a[3,1] = \+ 1/2*c[3]*(2*c[2]-c[3])/c[2], `b*`[4] = -1/60*(-460*c[5]*c[4]*c[3]*`b*` [7]+760*c[5]*c[4]*c[3]^2*`b*`[7]+3*c[3]-6*c[4]-1800*`b*`[7]*c[4]^2*c[5 ]*c[3]^2+130*c[5]*c[4]*c[3]^2+100*c[3]^2*c[5]*`b*`[7]-100*c[3]^3*c[5]* c[4]+300*c[3]^3*c[4]*`b*`[7]-600*c[3]^3*`b*`[7]*c[4]^2+940*`b*`[7]*c[4 ]^2*c[3]^2-60*c[5]*c[3]*`b*`[7]-560*c[4]^2*c[3]*`b*`[7]+120*`b*`[7]*c[ 4]^2+120*c[5]*c[4]*`b*`[7]-200*`b*`[7]*c[4]^2*c[5]+1000*c[5]*c[4]^2*c[ 3]*`b*`[7]-600*c[3]^3*c[5]*c[4]*`b*`[7]+34*c[4]*c[3]-380*`b*`[7]*c[4]* c[3]^2+252*`b*`[7]*c[4]*c[3]+10*c[4]*c[5]-60*c[3]*c[4]*c[5]+1200*c[3]^ 3*`b*`[7]*c[4]^2*c[5]+36*`b*`[7]*c[3]-72*`b*`[7]*c[4]-60*`b*`[7]*c[3]^ 2-5*c[3]*c[5]-70*c[4]*c[3]^2-5*c[3]^2+10*c[3]^2*c[5]+50*c[4]*c[3]^3)/( -c[4]^2+c[4]+c[4]*c[3]-c[3])/c[4]/(-10*c[5]*c[4]*c[3]^2+10*c[4]^2*c[3] ^2+8*c[3]*c[4]*c[5]+c[3]*c[5]-8*c[4]^2*c[3]-c[4]*c[3]-2*c[4]*c[5]+2*c[ 4]^2), a[7,4] = 1/60*(10*c[3]*c[5]-5*c[3]-5*c[5]+3)/(-c[4]^2+c[4]+c[4] *c[3]-c[3])/c[4]/(c[4]-c[5]), a[7,5] = -1/60*(3-5*c[3]+10*c[4]*c[3]-5* c[4])/c[5]/(-c[5]^2-c[5]^2*c[3]+c[5]^3+c[3]*c[5]-c[5]^2*c[4]+c[3]*c[4] *c[5]-c[4]*c[3]+c[4]*c[5])\}:" }{TEXT -1 0 "" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 49 "sm: square of principal error \+ norm in terms of " }{XPPEDIT 18 0 "c[2]" "6#&%\"cG6#\"\"#" }{TEXT -1 3 ", " }{XPPEDIT 18 0 "c[3]" "6#&%\"cG6#\"\"$" }{TEXT -1 3 ", " } {XPPEDIT 18 0 "c[4]" "6#&%\"cG6#\"\"%" }{TEXT -1 7 " and " } {XPPEDIT 18 0 "c[5]" "6#&%\"cG6#\"\"&" }{TEXT -1 1 " " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1665 "sm := 1 /25920000*(12042*c[5]^2*c[3]^2+297000*c[3]*c[2]^2*c[4]^2*c[5]-13500*c[ 3]^2*c[2]*c[5]^2+28044*c[3]^3*c[5]-270000*c[3]*c[2]^2*c[4]^2*c[5]^2-22 140*c[3]^4*c[5]-148500*c[3]^2*c[2]^2*c[4]*c[5]-270000*c[3]^2*c[2]^2*c[ 4]^2*c[5]+135000*c[3]^2*c[2]^2*c[4]*c[5]^2-22140*c[5]^2*c[3]^3+270000* c[3]^2*c[2]^2*c[4]^2*c[5]^2-18000*c[3]^2*c[2]*c[4]*c[5]+81000*c[3]*c[2 ]^2*c[5]*c[4]-67500*c[3]*c[2]^2*c[5]^2*c[4]+96444*c[5]*c[4]*c[3]^2-761 40*c[5]^2*c[4]*c[3]^2-613800*c[5]^2*c[4]^2*c[3]^3-221940*c[3]^3*c[5]*c [4]+153450*c[5]^2*c[4]^2*c[3]^2+675180*c[5]*c[4]^2*c[3]^3-240140*c[4]^ 2*c[3]^3+198000*c[3]^2*c[2]*c[4]^2-49500*c[3]^3*c[4]*c[2]-55800*c[3]*c [2]*c[4]^2+12000*c[3]*c[2]*c[4]-184140*c[5]*c[4]^2*c[3]^2-34200*c[3]*c [2]*c[4]*c[5]+27000*c[3]*c[2]*c[5]^2*c[4]+162000*c[3]*c[2]*c[5]*c[4]^2 +613800*c[4]^2*c[3]^4*c[5]^2-613800*c[4]^2*c[3]^4*c[5]-73800*c[4]*c[3] ^4*c[5]^2-42140*c[3]^4*c[4]+223450*c[4]^2*c[3]^4+189180*c[4]*c[5]^2*c[ 3]^3+18450*c[5]^2*c[3]^4+10642*c[3]^4-135000*c[3]*c[2]*c[4]^2*c[5]^2+2 7900*c[2]^2*c[4]^2+6975*c[3]^2*c[2]^2+540000*c[3]^3*c[2]*c[5]*c[4]^2+8 1180*c[3]^4*c[5]*c[4]+540000*c[3]^2*c[2]*c[4]^2*c[5]^2+6900*c[3]^2*c[2 ]*c[4]-6000*c[2]*c[3]^2-36856*c[4]*c[3]^2+17100*c[3]^2*c[2]*c[5]+6552* c[3]^2-12856*c[3]^3-27900*c[3]*c[2]^2*c[4]-99000*c[3]*c[2]^2*c[4]^2+49 500*c[3]^2*c[2]^2*c[4]+90000*c[3]^2*c[2]^2*c[4]^2-20250*c[3]^2*c[2]^2* c[5]+16875*c[3]^2*c[2]^2*c[5]^2-81000*c[5]*c[2]^2*c[4]^2+67500*c[5]^2* c[2]^2*c[4]^2-540000*c[2]*c[4]^2*c[5]^2*c[3]^3-180000*c[3]^3*c[2]*c[4] ^2-135000*c[3]^3*c[2]*c[5]^2*c[4]+148500*c[3]^3*c[2]*c[4]*c[5]+13500*c [3]^2*c[2]*c[5]^2*c[4]-594000*c[3]^2*c[2]*c[5]*c[4]^2-16056*c[3]^2*c[5 ]+88044*c[4]*c[3]^3+66442*c[4]^2*c[3]^2)/c[3]^2:" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 64 " We shall const ruct a 6 stage, order 5 Runge-Kutta scheme with " }{XPPEDIT 18 0 "t[6 ] = 1/660;" "6#/&%\"tG6#\"\"'*&\"\"\"F)\"$g'!\"\"" }{TEXT -1 36 " so \+ that the stability function is " }}{PARA 256 "" 0 "" {TEXT -1 1 " " } {XPPEDIT 18 0 "R(z)= 1+z+1/2" "6#/-%\"RG6#%\"zG,(\"\"\"F)F'F)*&F)F)\" \"#!\"\"F)" }{TEXT -1 1 " " }{XPPEDIT 18 0 "z^2+1/6" "6#,&*$%\"zG\"\"# \"\"\"*&F'F'\"\"'!\"\"F'" }{TEXT -1 1 " " }{XPPEDIT 18 0 "z^3+1/24" "6 #,&*$%\"zG\"\"$\"\"\"*&F'F'\"#C!\"\"F'" }{TEXT -1 1 " " }{XPPEDIT 18 0 "z^4+1/120" "6#,&*$%\"zG\"\"%\"\"\"*&F'F'\"$?\"!\"\"F'" }{TEXT -1 1 " " }{XPPEDIT 18 0 "z^5+1/660;" "6#,&*$%\"zG\"\"&\"\"\"*&F'F'\"$g'!\" \"F'" }{TEXT -1 1 " " }{XPPEDIT 18 0 "z^6" "6#*$%\"zG\"\"'" }{TEXT -1 1 "." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 119 "A 6 stage, order 5 Runge-Kutta scheme \+ will have the same stability function as the Prince-Dormand scheme pro vided that " }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "(2-5*c [3])*c[4]/240 = 1/660;" "6#/*(,&\"\"#\"\"\"*&\"\"&F'&%\"cG6#\"\"$F'!\" \"F'&F+6#\"\"%F'\"$S#F.*&F'F'\"$g'F." }{TEXT -1 2 ", " }}{PARA 0 "" 0 "" {TEXT -1 8 "so that " }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "(2-5*c[3])*c[4] = 4/11;" "6#/*&,&\"\"#\"\"\"*&\"\"&F'&%\"cG6#\" \"$F'!\"\"F'&F+6#\"\"%F'*&F1F'\"#6F." }{TEXT -1 2 ", " }}{PARA 0 "" 0 "" {TEXT -1 9 "that is, " }}{PARA 256 "" 0 "" {TEXT -1 1 " " } {XPPEDIT 18 0 "c[4] = 4/(22-55*c[3]);" "6#/&%\"cG6#\"\"%*&F'\"\"\",&\" #AF)*&\"#bF)&F%6#\"\"$F)!\"\"F1" }{TEXT -1 2 ". " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 78 "(2-5*c[3])*c [4]=240/660;\nc[4]=solve(%,c[4]);\nc[4]=numer(rhs(%))/denom(rhs(%));" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/*&,&\"\"#\"\"\"*&\"\"&F'&%\"cG6#\" \"$F'!\"\"F'&F+6#\"\"%F'#F1\"#6" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/&% \"cG6#\"\"%,$*&#F'\"#6\"\"\"*&F,F,,&*&\"\"&F,&F%6#\"\"$F,F,\"\"#!\"\"F 5F,F5" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/&%\"cG6#\"\"%,$*&F'\"\"\",&* &\"#bF*&F%6#\"\"$F*F*\"#A!\"\"F2F2" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" } }{PARA 0 "" 0 "" {TEXT -1 52 "The following picture is the graph of th is equation." }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 88 "plot(4/(22-5 5*c[3]),c[3]=0..0.34,c[4]=0..1,color=COLOR(RGB,.9,.5,0),font=[HELVETIC A,9]);" }}{PARA 13 "" 1 "" {GLPLOT2D 485 401 401 {PLOTDATA 2 "6'-%'CUR VESG6#7U7$$\"\"!F)$\"3B========!#=7$$\"3+nmm;C.6u!#?$\"3)R7I1C/D&=F,7$ $\"3YLLe4K$fQ\"!#>$\"3)Gw'fk)RM)=F,7$$\"3omm;\\>66@F6$\"3K[O&*y\")[>>F ,7$$\"3&pmmTk(4TGF6$\"3o'Q!evk>d>F,7$$\"3=LLeWQhnNF6$\"3W^U%o*eA'*>F,7 $$\"3gmm\"HO(=TUF6$\"3!3.cN&z#Q.#F,7$$\"3L++DEPjQ\\F6$\"394d%y=&Gu?F,7 $$\"3%om;HvL*fcF6$\"3\"3.W!fZ&y6#F,7$$\"3o***\\7c?*yjF6$\"3/N[`:LV$RDF,7$$\"3TLLLv<*3?\"F,$\"3o=P!36I#)f#F ,7$$\"3%***\\i6$)Rw7F,$\"3Wx)=]'4EqEF,7$$\"3DLLL!RMKM\"F,$\"3#QI*\\XlV PFF,7$$\"3@+]ihTc;9F,$\"3:WW(RcP^\"GF,7$$\"39+]([W\"R'[\"F,$\"3#pYK;BT L*GF,7$$\"3qLLe*))[#f:F,$\"3ar;N\"o3(zHF,7$$\"3\\L$euwahi\"F,$\"39Vq1y 0pjIF,7$$\"3#om;az@$)p\"F,$\"37-HN6)\\(fJF,7$$\"3um;HbIGtF,$\"3K\\NQK: 8yMF,7$$\"3++++p=#=)>F,$\"3d&[Mt=5Og$F,7$$\"37++D*)40`?F,$\"3(fR4)[#[a t$F,7$$\"3A+]iD)o>7#F,$\"3mqt$*[u_sQF,7$$\"3A++vK.\\)>#F,$\"3_z()*p<=q .%F,7$$\"3CLLL5$[sE#F,$\"3%Gb4yi7s>%F,7$$\"3I++]F@mSBF,$\"3CrNQF#4HQ%F ,7$$\"3zm;Hdf=2CF,$\"33gHD(Hhfc%F,7$$\"3M++]uS\"*zCF,$\"3!HW]B`=Wy%F,7 $$\"33L$e\\cX$[DF,$\"3u)fr6nd*4]F,7$$\"31+]7!>w)>EF,$\"3W85e]%>'p_F,7$ $\"39LL$3X5)*o#F,$\"3G(R\"eEq*3b&F,7$$\"3/+]iS^-jFF,$\"3a::<5jWzeF,7$$ \"3+nmm)eRN$GF,$\"3fz$\\z\\o[B'F,7$$\"3ommT'z]c!HF,$\"3)=D)*=%)4dk'F,7 $$\"3#RLep)[;xHF,$\"3-+:9n8O5rF,7$$\"3C+++4*yG/$F,$\"3gF/0uXa)f(F,7$$ \"3)QL$3%f%>=JF,$\"3ktG#> " 0 "" {MPLTEXT 1 0 46 "sm2 : = simplify(subs(c[4]=4/(22-55*c[3]),sm)):" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 50 "sm2: square of principal error norm in t erms of " }{XPPEDIT 18 0 "c[2]" "6#&%\"cG6#\"\"#" }{TEXT -1 3 ", " } {XPPEDIT 18 0 "c[3]" "6#&%\"cG6#\"\"$" }{TEXT -1 7 " and " } {XPPEDIT 18 0 "c[5]" "6#&%\"cG6#\"\"&" }{TEXT -1 1 " " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 995 "sm2 := 1 /3136320000*(-55372240*c[3]^5+32192050*c[3]^6+36579248*c[3]^4+446400*c [2]^2+990912*c[3]^2-1296000*c[5]*c[2]^2+5766000*c[3]^3*c[2]+163200*c[3 ]*c[2]-27769500*c[3]^3*c[2]^2-2230272*c[3]^2*c[5]+120552300*c[3]^5*c[5 ]-7260000*c[3]^4*c[2]-45009000*c[3]^2*c[2]^2*c[5]-95386500*c[5]^2*c[3] ^5-10064192*c[3]^3-66973500*c[3]^6*c[5]-2646000*c[3]^2*c[2]*c[5]^2+392 17500*c[3]^2*c[2]^2*c[5]^2+21099375*c[3]^4*c[2]^2+216000*c[3]*c[2]*c[5 ]^2-417600*c[3]*c[2]*c[5]-1768800*c[2]*c[3]^2+9180000*c[3]^3*c[2]*c[5] ^2+11880000*c[3]*c[2]^2*c[5]+4712400*c[3]^2*c[2]*c[5]-10260000*c[3]*c[ 2]^2*c[5]^2+51046875*c[3]^4*c[2]^2*c[5]^2+19057500*c[3]^4*c[2]*c[5]+81 675000*c[3]^3*c[2]^2*c[5]-61256250*c[3]^4*c[2]^2*c[5]-70537500*c[3]^3* c[2]^2*c[5]^2-11137500*c[3]^4*c[2]*c[5]^2+1080000*c[5]^2*c[2]^2+153099 00*c[3]^2*c[2]^2-4039200*c[3]*c[2]^2+60642450*c[5]^2*c[3]^4-16279560*c [5]^2*c[3]^3+22483296*c[3]^3*c[5]+1583208*c[5]^2*c[3]^2-15714000*c[3]^ 3*c[2]*c[5]+55811250*c[5]^2*c[3]^6-81001800*c[3]^4*c[5])/c[3]^2/(5*c[3 ]-2)^2:" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 " " {TEXT -1 8 "Taking " }{XPPEDIT 18 0 "c[5] = 20/21;" "6#/&%\"cG6#\" \"&*&\"#?\"\"\"\"#@!\"\"" }{TEXT -1 103 ", we obtain the following sur face for the graph of the square of the principal error norm in terms \+ of " }{XPPEDIT 18 0 "c[2]" "6#&%\"cG6#\"\"#" }{TEXT -1 7 " and " } {XPPEDIT 18 0 "c[3]" "6#&%\"cG6#\"\"$" }{TEXT -1 2 ". " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 143 "plot3d (eval(sm2,c[5]=20/21),c[2]=0.1..0.4,c[3]=0.31..0.33,color=COLOR(RGB,.9 ,.4,0),lightmodel=light4,\n axes=boxed,labels=[`c[2]`,`c[3]`,``]);" }}{PARA 13 "" 1 "" {GLPLOT3D 476 394 394 {PLOTDATA 3 "6&-%%GRIDG6&;$\" \"\"!\"\"$\"\"%F);$\"#J!\"#$\"#LF/X,%)anythingG6\"6\"[gl'!%\"!!#\\bm\" :\":3E70C325C3EF9C173E6FBF03196B11EE3E6E238F52BA09BD3E6CB3F899F395403E 6B704A3D9150123E6A5892AE9CD67B3E696CE3BFEE7F483E68AD52EAC4BB213E6819F9 997458143E67B2F57852C1C73E677868CDF8BAAF3E676A7ADB00B1763E678958425A1B 713E67D5337AD74B903E684E454ACB638E3E68F4CD4EE5F9B43E69C9128D17F9833E6A CB6414EE6AD43E6BFC19AEAD00943E6D5B949A7A10EF3E6EEA4061798DCD3E705449DD 30F2F63E714B88C1F7D89B3E725B24EB7DE39E3E73836D999A683F3E7053D5229F89BA 3E6EDBFC222AA7C23E6D3C96F8DFDF2F3E6BC97C0431597C3E6A82AF4254A9223E6968 37ABD2CA073E687A1F6F36CBBC3E67B87431E36C023E67234756B671993E66BAAE4AB0 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c[2]G%%c[3]G%!G-%*AXESS TYLEG6#%$BOXG-%+LIGHTMODELG6#%(LIGHT_4G" 1 2 0 1 10 0 2 1 6 2 2 1.000000 45.000000 45.000000 0 0 "Curve 1" }}}}{PARA 0 "" 0 "" {TEXT -1 58 "The following calculations give (approximate) values for " } {XPPEDIT 18 0 "c[2]" "6#&%\"cG6#\"\"#" }{TEXT -1 7 " and " } {XPPEDIT 18 0 "c[3]" "6#&%\"cG6#\"\"$" }{TEXT -1 82 " to minimize the square of the principal error norm for three fixed choices for " } {XPPEDIT 18 0 "c[5]" "6#&%\"cG6#\"\"&" }{TEXT -1 10 ", namely " } {XPPEDIT 18 0 "c[5] = 20/21;" "6#/&%\"cG6#\"\"&*&\"#?\"\"\"\"#@!\"\"" }{TEXT -1 3 ", " }{XPPEDIT 18 0 "c[5] = 21/22;" "6#/&%\"cG6#\"\"&*&\" #@\"\"\"\"#A!\"\"" }{TEXT -1 7 " and " }{XPPEDIT 18 0 "c[5] = 22/23; " "6#/&%\"cG6#\"\"&*&\"#A\"\"\"\"#B!\"\"" }{TEXT -1 1 "." }}{PARA 0 " " 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 249 "uu : = \{c[5]=20/21.\}:\nmn := evalf[12](minimize(eval(sm2,uu),c[2]=0.2..0. 25,c[3]=0.25..0.35,'location'));\nsubs(op(1,op(1,op(2,[mn]))),[c[2],c[ 3]]):\nrr := map(convert,%,rational,5):\nc[2]=op(1,rr),c[3]=op(2,rr); \n`principal error norm:`,sqrt(op(1,[mn]));" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%#mnG6$$\"-Gj)pIZ$!#><#7$<$/&%\"cG6#\"\"$$\"-*[dqN?$!# 7/&F.6#\"\"#$\"- " 0 "" {MPLTEXT 1 0 249 "uu := \{c[5]=21/2 2.\}:\nmn := evalf[12](minimize(eval(sm2,uu),c[2]=0.2..0.25,c[3]=0.25. .0.35,'location'));\nsubs(op(1,op(1,op(2,[mn]))),[c[2],c[3]]):\nrr := \+ map(convert,%,rational,5):\nc[2]=op(1,rr),c[3]=op(2,rr);\n`principal e rror norm:`,sqrt(op(1,[mn]));" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%#mn G6$$\"-,4kceM!#><#7$<$/&%\"cG6#\"\"$$\"-\"\\`De?$!#7/&F.6#\"\"#$\"-G?& f5?#F3F&" }}{PARA 11 "" 1 "" {XPPMATH 20 "6$/&%\"cG6#\"\"##\"#N\"$f\"/ &F%6#\"\"$#\"#U\"$J\"" }}{PARA 11 "" 1 "" {XPPMATH 20 "6$%6principal~e rror~norm:G$\"+K@sf=!#8" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 249 "uu := \{c[5]=22/23.\}:\nmn := eval f[12](minimize(eval(sm2,uu),c[2]=0.2..0.25,c[3]=0.25..0.35,'location') );\nsubs(op(1,op(1,op(2,[mn]))),[c[2],c[3]]):\nrr := map(convert,%,rat ional,5):\nc[2]=op(1,rr),c[3]=op(2,rr);\n`principal error norm:`,sqrt( op(1,[mn]));" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%#mnG6$$\"-[Xb <#7$<$/&%\"cG6#\"\"#$\"-3`T!>?#!#7/&F.6#\"\"$$\"-3S@k2KF3F&" }}{PARA 11 "" 1 "" {XPPMATH 20 "6$/&%\"cG6#\"\"##\"#C\"$4\"/&F%6#\"\"$#\"#<\"# `" }}{PARA 11 "" 1 "" {XPPMATH 20 "6$%6principal~error~norm:G$\"+hG_l= !#8" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 14 "T he procedure " }{TEXT 0 8 "minimize" }{TEXT -1 75 " has difficulty han dling minimization with respect to the three variables " }{XPPEDIT 18 0 "c[2]" "6#&%\"cG6#\"\"#" }{TEXT -1 3 ", " }{XPPEDIT 18 0 "c[3]" "6#&%\"cG6#\"\"$" }{TEXT -1 7 " and " }{XPPEDIT 18 0 "c[5]" "6#&%\"c G6#\"\"&" }{TEXT -1 86 ", so we set up a procedure to perform minimiza tions with repect to the two variables " }{XPPEDIT 18 0 "c[2]" "6#&% \"cG6#\"\"#" }{TEXT -1 7 " and " }{XPPEDIT 18 0 "c[3]" "6#&%\"cG6#\" \"$" }{TEXT -1 84 " as in the previous examples and the use the one d imensional minmization procedure " }{TEXT 0 7 "findmin" }{TEXT -1 29 " to minimize with repect to " }{XPPEDIT 18 0 "c[5]" "6#&%\"cG6#\"\"& " }{TEXT -1 1 "." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> \+ " 0 "" {MPLTEXT 1 0 144 "prin_err := proc(c_5)\n local mn;\n mn := evalf[Digits+2](minimize(eval(sm2,c[5]=c_5),c[2]=0.2..0.25,c[3]=0.25. .0.35));\n sqrt(mn);\nend proc:" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }} {PARA 0 "" 0 "" {TEXT -1 21 "Using the procedure " }{TEXT 0 8 "prin_e rr" }{TEXT -1 55 " three previous examples can be performed as follow s. " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 51 "prin_err(20/21.),prin_err(21/22.),prin_err(22/23.);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6%$\"+8uhj=!#8$\"+b?sf=F%$\"+^J_l=F%" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 55 "These \+ three values can be used as starting values for " }{TEXT 0 7 "findmin " }{TEXT -1 1 "." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> \+ " 0 "" {MPLTEXT 1 0 128 "evalf[12](findmin('prin_err'(c5),c5=\{20/21., 21/22.,22/23.\},convergence='location'));\nc[5]=convert(op(1,%),ration al,4);\nevalf(%);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#7$$\"-6#3(eU&*!#7 $\"-r?eif=!#:" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/&%\"cG6#\"\"&#\"#@\" #A" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/&%\"cG6#\"\"&$\"+XXXX&*!#5" }}} {PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 249 "uu := \{c[5]=21/22.\}:\nmn := evalf[12](minimize(eval(sm2,uu),c[2 ]=0.2..0.25,c[3]=0.25..0.35,'location'));\nsubs(op(1,op(1,op(2,[mn]))) ,[c[2],c[3]]):\nrr := map(convert,%,rational,5):\nc[2]=op(1,rr),c[3]=o p(2,rr);\n`principal error norm:`,sqrt(op(1,[mn]));" }}{PARA 11 "" 1 " " {XPPMATH 20 "6#>%#mnG6$$\"-)>7m&eM!#><#7$<$/&%\"cG6#\"\"$$\"-=^b#e?$ !#7/&F.6#\"\"#$\"-[_'f5?#F3F&" }}{PARA 11 "" 1 "" {XPPMATH 20 "6$/&%\" cG6#\"\"##\"#N\"$f\"/&F%6#\"\"$#\"#U\"$J\"" }}{PARA 11 "" 1 "" {XPPMATH 20 "6$%6principal~error~norm:G$\"+b?sf=!#8" }}}{PARA 0 "" 0 " " {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 14 "The value of " } {XPPEDIT 18 0 "c[4]" "6#&%\"cG6#\"\"%" }{TEXT -1 20 " corresponding t o " }{XPPEDIT 18 0 "c[3] = 115/381;" "6#/&%\"cG6#\"\"$*&\"$:\"\"\"\" \"$\"Q!\"\"" }{TEXT -1 10 " is . . ." }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 41 "c[4]=4/(22-55*c[3]);\nsubs(c[3]=42/131,%);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/&%\"cG6#\"\"%,$*&F'\"\"\",&\"#AF**&\"#bF*&F %6#\"\"$F*!\"\"F2F*" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/&%\"cG6#\"\"%# \"$J\"\"$V\"" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 7 "Taking " }{XPPEDIT 18 0 "`b*`[7] = 1/112;" "6#/&%#b*G6#\" \"(*&\"\"\"F)\"$7\"!\"\"" }{TEXT -1 74 " gives suitable characteristi cs for the embedded 7 stage, order 4 scheme." }}{PARA 0 "" 0 "" {TEXT -1 80 " Using the general solution we can find the coefficients of the combined scheme." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 101 "e3 := \{c[2]=35/159,c[3]=42/131,c[4]=131/143 ,c[5]=21/22,`b*`[7]=1/112\}:\ne4 := `union`(subs(e3,e2),e3):" }}} {PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 2 "e4" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1045 "e4 := \{a[ 6,5] = -144610048/1077690663, a[4,1] = 2438549411/1983961980, a[7,3] = 2215522905683/4570867891800, a[3,2] = 20034/85805, b[5] = -2234144/57 5505, `b*`[7] = 1/112, b[1] = 669707/6932520, a[4,3] = 25077455105/515 8301148, a[5,3] = 6869523776931/1096562558080, b[6] = 9363/7120, a[5,2 ] = -1698584121/245755840, `b*`[1] = 2243660497/23535905400, a[3,1] = \+ 7476/85805, c[6] = 1, c[7] = 1, b[2] = 0, `b*`[2] = 0, `b*`[4] = 11044 61697911/396890438000, a[7,2] = 0, `b*`[5] = -6925033984/1953839475, ` b*`[6] = 3529851/3021550, a[2,1] = 35/159, a[6,1] = 67512025387/324545 92380, a[6,3] = 10587214001321/1373901639516, a[6,2] = -20051384/22939 35, a[5,4] = -927215289/26981535520, a[7,1] = 669707/6932520, a[7,6] = 9363/7120, a[4,2] = -3707256508/716430715, a[6,4] = 731293420/8209319 229, b[4] = 349043981/116904400, a[5,1] = 105337889067/64388030080, `b *`[3] = 7589131232781673/15518096492661000, a[7,4] = 349043981/1169044 00, a[7,5] = -2234144/575505, b[3] = 2215522905683/4570867891800, c[2] = 35/159, c[3] = 42/131, c[4] = 131/143, c[5] = 21/22\}:" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }} }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 137 "subs(e4,matrix([seq([c[i],seq(a[i,j],j=1..i-1),``$(8-i)],i=2..7 ),\n[`b`,seq(b[i],i=1..6),``],[`b*`,seq(`b*`[i],i=1..7)]]));\n``;\neva lf(%%);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#K%'matrixG6#7*7*#\"#N\"$f\" F(%!GF+F+F+F+F+7*#\"#U\"$J\"#\"%wu\"&0e)#\"&M+#F2F+F+F+F+F+7*#F/\"$V\" #\"+6%\\&QC\"+!)>'R)>#!+3lD2P\"*:2V;(#\",0^Xx]#\"+[6Ie^F+F+F+F+7*#\"#@ \"#A#\"-n!*)yL0\"\",!3I!)Qk#!+@Te)p\"\"*SevX##\".JpxB&po\".!3eDc'4\"#! **G:s#*\",?b`\")p#F+F+F+7*\"\"\"#\",(QD?^n\",!Q#faC$#!)%Q^+#\"(NRH##\" /@8+9se5\".;&R;!RP\"#\"*?MHJ(\"+H#>$4#)#!*[+hW\"\"+j1px5F+F+7*FR#\"'2( p'\"(?D$p\"\"!#\".$o0H_:A\".+=*y'3d%#\"*\")R/\\$\"*+W!p6#!(WTB#\"'0bd# \"%j$*\"%?rF+7*%\"bGF]oF`oFaoFdoFgoFjoF+7*%#b*G#\"+(\\gOC#\",+a!f`BF`o #\"1t;yK78*e(\"2+5m#\\'4=b\"#\".6zphW5\"\"-+!Q/*oR#!+%)R.Dp\"+v%RQ&># \"(^)HN\"(]:-$#FR\"$7\"Q)pprint246\"" }}{PARA 11 "" 1 "" {XPPMATH 20 " 6#%!G" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#K%'matrixG6#7*7*$\"+iyD,A!#5F (%!GF+F+F+F+F+7*$\"+qo51KF*$\"+w*yFr)!#6$\"+t*G[L#F*F+F+F+F+F+7*$\"+h \"R3;*F*$\"+?68H7!\"*$!+B(>Y<&F9$\"+>Ddh[F9F+F+F+F+7*$\"+XXXX&*F*$\"+& e&)fj\"F9$!+wMn6pF9$\"+a\")fkiF9$!+03[OMF1F+F+F+7*$\"\"\"\"\"!$\"+b%*> !3#F9$!+$zU5u)F9$\"+rv%fq(F9$\"+u%)33*)F1$!+t6&=M\"F*F+F+7*FJ$\"+7)o.m *F1$FLFL$\"+T30Z[F*$\"+/:s&)HF9$!+m$e?)QF9$\"+!4G]J\"F9F+7*%\"bGFXFZFe nFgnFinF[oF+7*%#b*G$\"+?i#H`*F1FZ$\"+fR]!*[F*$\"+pty#y#F9$!+[3KWNF9$\" +\\_Ao6F9$\"+H9dG*)!#7Q)pprint256\"" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 57 "#-------------------------------------- ------------------" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 " > " 0 "" {MPLTEXT 1 0 1 ";" }}}}{PARA 0 "" 0 "" {TEXT -1 39 "#-------- ------------------------------" }}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 39 "characteristics of the embedded scheme " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 35 "coefficients of the combined scheme" }}{PARA 0 " " 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 968 "ee : = \{c[2]=35/159,\nc[3]=42/131,\nc[4]=131/143,\nc[5]=21/22,\nc[6]=1,\nc [7]=1,\n\na[2,1]=35/159,\na[3,1]=7476/85805,\na[3,2]=20034/85805,\na[4 ,1]=2438549411/1983961980,\na[4,2]=-3707256508/716430715,\na[4,3]=2507 7455105/5158301148,\na[5,1]=105337889067/64388030080,\na[5,2]=-1698584 121/245755840,\na[5,3]=6869523776931/1096562558080,\na[5,4]=-927215289 /26981535520,\na[6,1]=67512025387/32454592380,\na[6,2]=-20051384/22939 35,\na[6,3]=10587214001321/1373901639516,\na[6,4]=731293420/8209319229 ,\na[6,5]=-144610048/1077690663,\na[7,1]=669707/6932520,\na[7,2]=0,\na [7,3]=2215522905683/4570867891800,\na[7,4]=349043981/116904400,\na[7,5 ]=-2234144/575505,\na[7,6]=9363/7120,\n\nb[1]=669707/6932520,\nb[2]=0, \nb[3]=2215522905683/4570867891800,\nb[4]=349043981/116904400,\nb[5]=- 2234144/575505,\nb[6]=9363/7120,\n\n`b*`[1]=2243660497/23535905400,\n` b*`[2]=0,\n`b*`[3]=7589131232781673/15518096492661000,\n`b*`[4]=110446 1697911/396890438000,\n`b*`[5]=-6925033984/1953839475,\n`b*`[6]=352985 1/3021550,\n`b*`[7]=1/112\}:" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 5 "Let " }{XPPEDIT 18 0 "T[5,6];" "6#&%\"TG6$\"\"&\"\"'" }{TEXT -1 127 " denote the vector whose compon ents are the principal error terms of the 6 stage, order 5 scheme (the error terms of order 6)." }}{PARA 0 "" 0 "" {TEXT -1 5 "Let " } {XPPEDIT 18 0 "`T*`[4,7];" "6#&%#T*G6$\"\"%\"\"(" }{TEXT -1 145 " den ote the vector whose components are the principal error terms of the e mbedded 7 stage, order 4 scheme (the error terms of order 5) and let \+ " }{XPPEDIT 18 0 "`T*`[5,7];" "6#&%#T*G6$\"\"&\"\"(" }{TEXT -1 99 " d enote the vector whose components are the error terms of order 6 of th e embedded order 4 scheme." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 39 "Denote the 2-norm of these vectors by " } {XPPEDIT 18 0 "abs(abs(T[5,6]));" "6#-%$absG6#-F$6#&%\"TG6$\"\"&\"\"' " }{TEXT -1 3 ", " }{XPPEDIT 18 0 "abs(abs(`T*`[4,7]));" "6#-%$absG6# -F$6#&%#T*G6$\"\"%\"\"(" }{TEXT -1 7 " and " }{XPPEDIT 18 0 "abs(abs (`T*`[5,7]));" "6#-%$absG6#-F$6#&%#T*G6$\"\"&\"\"(" }{TEXT -1 15 " re spectively." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 8 "Define: " }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "A[6 ] = abs(abs(T[5,6]));" "6#/&%\"AG6#\"\"'-%$absG6#-F)6#&%\"TG6$\"\"&F' " }{TEXT -1 6 ", " }{XPPEDIT 18 0 "B[6] = abs(abs(`T*`[5,7]))/abs( abs(`T*`[4,7]));" "6#/&%\"BG6#\"\"'*&-%$absG6#-F*6#&%#T*G6$\"\"&\"\"( \"\"\"-F*6#-F*6#&F/6$\"\"%F2!\"\"" }{TEXT -1 6 ", " }{XPPEDIT 18 0 "C[6] = abs(abs(`T*`[5,7]-T[5,6]))/abs(abs(`T*`[4,7]));" "6#/&%\"CG6 #\"\"'*&-%$absG6#-F*6#,&&%#T*G6$\"\"&\"\"(\"\"\"&%\"TG6$F2F'!\"\"F4-F* 6#-F*6#&F06$\"\"%F3F8" }{TEXT -1 1 "." }}{PARA 0 "" 0 "" {TEXT -1 0 " " }}{PARA 0 "" 0 "" {TEXT -1 77 "Dormand and Prince have suggested tha t as well as attempting to ensure that " }{XPPEDIT 18 0 "A[6];" "6#&% \"AG6#\"\"'" }{TEXT -1 73 " is a minimum, if the embedded scheme is t o be used for error control, " }{XPPEDIT 18 0 "B[6];" "6#&%\"BG6#\"\" '" }{TEXT -1 7 " and " }{XPPEDIT 18 0 "C[6];" "6#&%\"CG6#\"\"'" } {TEXT -1 27 " should be chosen so that " }}{PARA 0 "" 0 "" {TEXT -1 68 "they are \"similar in magnitude\" and also not differ too much fro m 1." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 186 "errterms5_6 := PrincipalErrorTerms(5,6,'expanded'): \n`errterms5_7*` :=subs(b=`b*`,PrincipalErrorTerms(5,7,'expanded')):\n `errterms4_7*` := subs(b=`b*`,PrincipalErrorTerms(4,7,'expanded')):" } }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 326 "snmB := sqrt(add(evalf(subs(ee,`errterms5_7*`[i]))^2,i=1..nops( `errterms5_7*`))):\nsdnB := sqrt(add(evalf(subs(ee,`errterms4_7*`[i])) ^2,i=1..nops(`errterms4_7*`))):\nsnmC := sqrt(add((evalf(subs(ee,`errt erms5_7*`[i])-subs(ee,errterms5_6[i])))^2,i=1..nops(errterms5_6))):\n' B[6]'= evalf[8](snmB/sdnB);\n'C[6]'= evalf[8](snmC/sdnB);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/&%\"BG6#\"\"'$\")7x7 " 0 "" {MPLTEXT 1 0 1 ";" }}}}{PARA 0 "" 0 "" {TEXT -1 39 "#--------------------------------------" }}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 26 "absolute stability regions" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}} {SECT 1 {PARA 4 "" 0 "" {TEXT -1 35 "coefficients of the combined sche me" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 968 "ee := \{c[2]=35/159,\nc[3]=42/131,\nc[4]=131/143,\nc [5]=21/22,\nc[6]=1,\nc[7]=1,\n\na[2,1]=35/159,\na[3,1]=7476/85805,\na[ 3,2]=20034/85805,\na[4,1]=2438549411/1983961980,\na[4,2]=-3707256508/7 16430715,\na[4,3]=25077455105/5158301148,\na[5,1]=105337889067/6438803 0080,\na[5,2]=-1698584121/245755840,\na[5,3]=6869523776931/10965625580 80,\na[5,4]=-927215289/26981535520,\na[6,1]=67512025387/32454592380,\n a[6,2]=-20051384/2293935,\na[6,3]=10587214001321/1373901639516,\na[6,4 ]=731293420/8209319229,\na[6,5]=-144610048/1077690663,\na[7,1]=669707/ 6932520,\na[7,2]=0,\na[7,3]=2215522905683/4570867891800,\na[7,4]=34904 3981/116904400,\na[7,5]=-2234144/575505,\na[7,6]=9363/7120,\n\nb[1]=66 9707/6932520,\nb[2]=0,\nb[3]=2215522905683/4570867891800,\nb[4]=349043 981/116904400,\nb[5]=-2234144/575505,\nb[6]=9363/7120,\n\n`b*`[1]=2243 660497/23535905400,\n`b*`[2]=0,\n`b*`[3]=7589131232781673/155180964926 61000,\n`b*`[4]=1104461697911/396890438000,\n`b*`[5]=-6925033984/19538 39475,\n`b*`[6]=3529851/3021550,\n`b*`[7]=1/112\}:" }}}{PARA 0 "" 0 " " {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}} {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 79 "The stabi lity function R for the 6 stage, order 5 scheme is given as follows. " }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 75 "subs(ee,StabilityFunctio n(5,6,'expanded')):\nR := unapply(%,z):\n'R(z)'=R(z);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/-%\"RG6#%\"zG,0\"\"\"F)F'F)*&#F)\"\"#F)*$)F'F,F)F )F)*&#F)\"\"'F)*$)F'\"\"$F)F)F)*&#F)\"#CF)*$)F'\"\"%F)F)F)*&#F)\"$?\"F )*$)F'\"\"&F)F)F)*&#F)\"$g'F)*$)F'F1F)F)F)" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 124 "We can find the point where th e boundary of the stability region intersects the negative real axis b y solving the equation: " }}{PARA 256 "" 0 "" {TEXT -1 1 " " } {XPPEDIT 18 0 "R(z) = 1;" "6#/-%\"RG6#%\"zG\"\"\"" }{TEXT -1 2 ". " }} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 28 "z0 := newton(R(z)=1,z=-3.4); " }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%#z0G$!+j^AFM!\"*" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 303 "z0 := \+ newton(R(z)=1,z=-3.4):\np1 := plot([R(z),1],z=-3.89..0.49,color=[red,b lue]):\np2 := plot([[[z0,1]]$3],style=point,symbol=[circle,cross,diamo nd],color=black):\np3 := plot([[z0,0],[z0,1]],linestyle=3,color=COLOR( RGB,0,.5,0)):\nplots[display]([p1,p2,p3],view=[-3.89..0.49,-0.0..1.47] ,font=[HELVETICA,9]);" }}{PARA 13 "" 1 "" {GLPLOT2D 414 205 205 {PLOTDATA 2 "6+-%'CURVESG6$7X7$$!37++++++!*Q!#<$\"3r@1tiiSLAF*7$$!3D+] PKUEUQF*$\"3)QOtYY)4j?F*7$$!3%****\\ZYGXz$F*$\"3Ine.o\"oT!>F*7$$!3')\\ 7eGQ*Hv$F*$\"3)ph$f56muf9r$F*$\"3%47!=9I\"Hl\"F*7$$!3!* \\(o+:\\Zm$F*$\"3Pe+_e!*zC:F*7$$!3/+]s2\"R!=OF*$\"3$)[5d$F*$\"3)RU,*=Q'QH\"F*7$$!3)***\\2u(**R_$F*$\"3S4R+(=*=!>\"F* 7$$!3/](=Ne.sZ$F*$\"3kIO'\\OwW4\"F*7$$!33+D'HR2/V$F*$\"3X=pv!R'y05F*7$ $!3C+D,v_jVLF*$\"3U@2#RaOae)!#=7$$!3A+D\"Hx(y`KF*$\"3MJl5qNvusF^o7$$!3 ;+Drdt'3;$F*$\"3*R`7-k9b^F^o7$$!3G+ +b(RrH(HF*$\"34-GN9g2EVF^o7$$!3S+](*4W0*)GF*$\"3*p,8Okr#=PF^o7$$!3))** *\\wv$e%z#F*$\"3Ae0/0G@aJF^o7$$!3Y++v*>D(*p#F*$\"3QOxRV2z*p#F^o7$$!3.+ +N-9J3EF*$\"3U_#)4u^4aBF^o7$$!3&**\\7n\"*)HDDF*$\"3$4d(=1%)e3@F^o7$$!3 #)***\\@/*eECF*$\"3+:e2\"pG/*=F^o7$$!3F++gI)oHM#F*$\"3Qfher.UgF*$\"3MgP*GL'4[;F^o7$$!3; +]7[0K\")=F*$\"3([$GrrVG8+]PA>;-F^o7$$!37+Dr\"=%f0;F*$\"3AV\\x9\"RY2#F^o7$$! 3Y+vez;`@:F*$\"3hF,C9W\"3B#F^o7$$!3A+]Ku=uI9F*$\"3rw#y*fZuACF^o7$$!3=+ +q@r%pL\"F*$\"3S%G0vJ@mk#F^o7$$!3G+]-bs=X7F*$\"3+nekf!G:*GF^o7$$!37+D \"e'[Sc6F*$\"3qph'[)p&R:$F^o7$$!3S+]dfr#y0\"F*$\"3O:*[ZkckZ$F^o7$$!39/ ++hq]#p*F^o$\"3'pca/-Rhz$F^o7$$!3#4++vNknu)F^o$\"3=BWQ+WBrTF^o7$$!3C.] 7.4y*)yF^o$\"3+8dvRhtVXF^o7$$!3S,+]G;(G&pF^o$\"3m$H+BY(f*)\\F^o7$$!3=1 ]7;IJrgF^o$\"3=GxV[uB\\aF^o7$$!3%4+DmDI)\\^F^o$\"3y(\\O2#)[^(fF^o7$$!3 I-+]FC\"*[UF^o$\"3*y_R+0@%QlF^o7$$!3w0]7`\\t0LF^o$\"3a\\=*>*z6&=(F^o7$ $!3U)****f%HM(R#F^o$\"3Y2#G%G$p$oyF^o7$$!3#G+]n\"4Qo9F^o$\"33!=lwkPVj) F^o7$$!3'*z*\\7i66Z&!#>$\"3![F#p;hen%*F^o7$$\"3y/++I1S%*HFaz$\"3%\\#pg OoRI5F*7$$\"3?&**\\-I$op7F^o$\"3:iS9g5QN6F*7$$\"3x,++=/[P@F^o$\"3!Qfd3 i5$Q7F*7$$\"31'*\\(Q9LF1$F^o$\"3YPgWlNNe8F*7$$\"3-)*\\7i!=$[RF^o$\"3I( >eRrMT[\"F*7$$\"3!***************[F^o$\"3,T9xMlJK;F*-%'COLOURG6&%$RGBG $\"*++++\"!\")$\"\"!Fj\\lFi\\l-F$6$7S7$F($\"\"\"Fj\\l7$F3F_]l7$F=F_]l7 $FGF_]l7$FQF_]l7$FenF_]l7$FjnF_]l7$F`oF_]l7$FeoF_]l7$FjoF_]l7$F_pF_]l7 $FdpF_]l7$FipF_]l7$F^qF_]l7$FcqF_]l7$FhqF_]l7$F]rF_]l7$FbrF_]l7$FgrF_] l7$F\\sF_]l7$FasF_]l7$FfsF_]l7$F[tF_]l7$F`tF_]l7$FetF_]l7$FjtF_]l7$F_u F_]l7$FduF_]l7$FiuF_]l7$F^vF_]l7$FcvF_]l7$FhvF_]l7$F]wF_]l7$FbwF_]l7$F gwF_]l7$F\\xF_]l7$FaxF_]l7$FfxF_]l7$F[yF_]l7$F`yF_]l7$FeyF_]l7$FjyF_]l 7$F_zF_]l7$FezF_]l7$FjzF_]l7$F_[lF_]l7$Fd[lF_]l7$Fi[lF_]l7$F^\\lF_]l-F c\\l6&Fe\\lFi\\lFi\\lFf\\l-F$6&7#7$$!38+++j^AFMF*F_]l-%'SYMBOLG6#%'CIR CLEG-Fc\\l6&Fe\\lFj\\lFj\\lFj\\l-%&STYLEG6#%&POINTG-F$6&Fe`l-Fj`l6#%&C ROSSGF]alF_al-F$6&Fe`l-Fj`l6#%(DIAMONDGF]alF_al-F$6%7$7$Fg`lFi\\lFf`l- %&COLORG6&Fe\\lFi\\l$\"\"&!\"\"Fi\\l-%*LINESTYLEG6#\"\"$-%%FONTG6$%*HE LVETICAG\"\"*-%+AXESLABELSG6%Q\"z6\"Q!Fdcl-F\\cl6#%(DEFAULTG-%%VIEWG6$ ;$!$*Q!\"#$\"#\\F_dl;$!\"!Fj\\l$\"$Z\"F_dl" 1 2 0 1 10 0 2 9 1 4 2 1.000000 45.000000 45.000000 0 0 "Curve 1" "Curve 2" "Curve 3" "Curve \+ 4" "Curve 5" "Curve 6" }}}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 50 "The following picture shows the stability region. " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1650 "R := z -> 1+z+1/2*z^2+1/6*z^3+1/24*z^4+1/120*z^5+1/660*z^6:\np ts := []: z0 := 0: tt := 0: \nwhile tt<=161/20 do\n zz := newton(R(z )=exp(tt*Pi*I),z=z0):\n z0 := zz:\n if (1/2<=tt and tt<=1) or (139 /20<=tt and tt<=149/20) then\n hh := 1/40\n else \n hh := \+ 1/20\n end if;\n tt := tt+hh;\n pts := [op(pts),[Re(zz),Im(zz)]] :\nend do:\np1 := plot(pts,color=COLOR(RGB,.48,.28,.05)):\np2 := plots [polygonplot]([seq([pts[i-1],pts[i],[-1.7,0]],i=2..nops(pts))],\n \+ style=patchnogrid,color=COLOR(RGB,.95,.55,.1)):\npts := []: z0 := 1+3.9*I: tt := 0: \nwhile tt<=61/30 do\n zz := newton(R(z)=exp(tt*P i*I),z=z0):\n z0 := zz:\n if (17/30<=tt and tt<=13/10) then\n \+ hh := 1/60\n else \n hh := 1/30\n end if;\n tt := tt+hh;\n pts := [op(pts),[Re(zz),Im(zz)]]:\nend do:\np3 := plot(pts,color=CO LOR(RGB,.48,.28,.05)):\np4 := plots[polygonplot]([seq([pts[i-1],pts[i] ,[0.85,3.64]],i=2..nops(pts))],\n style=patchnogrid,color=COL OR(RGB,.95,.55,.1)):\npts := []: z0 := 1-3.9*I: tt := 0: \nwhile tt<=6 1/30 do\n zz := newton(R(z)=exp(tt*Pi*I),z=z0):\n z0 := zz:\n if (11/15<=tt and tt<=22/15) then\n hh := 1/60\n else \n hh \+ := 1/30\n end if;\n tt := tt+hh;\n pts := [op(pts),[Re(zz),Im(zz )]]:\nend do:\np5 := plot(pts,color=COLOR(RGB,.48,.28,.05)):\np6 := pl ots[polygonplot]([seq([pts[i-1],pts[i],[0.85,-3.64]],i=2..nops(pts))], \n style=patchnogrid,color=COLOR(RGB,.95,.55,.1)):\np7 := plo t([[[-4.09,0],[1.49,0]],[[0,-4.29],[0,4.29]]],color=black,linestyle=3) :\nplots[display]([p||(1..7)],view=[-4.09..1.49,-4.29..4.29],font=[HEL VETICA,9],\n color=COLOR(RGB,.4,0,1),labels=[`Re(z)`,`Im(z)`],axes=b oxed,scaling=constrained);" }}{PARA 13 "" 1 "" {GLPLOT2D 469 558 558 {PLOTDATA 2 "60-%'CURVESG6$7bv7$$\"\"!F)F(7$$\"3(******R%>>4=!#E$\"3)) ******=jzq:!#=7$$\"3V*****f^w)=**!#D$\"3/+++-j!\\`yF07$$!3?+++ld9hV!#A$\"3(******z+zJU*F07$$!3******* p7`g6#!#@$\"3%******RaS\"*4\"!#<7$$!3'******H?+*eqFQ$\"35+++KTsb7FT7$$ !3!*******)4#y,>!#?$\"3#*******o1*>T\"FT7$$!3*******Hnv:U%Fgn$\"3'**** ***R!Rzc\"FT7$$!3K+++PJGekFgn$\"3'******zBMek\"FT7$$!3w+++PmN5#*Fgn$\" 3*)********[sB$\"3/+++:qm,=FT7$$!3')*****f()H Ow\"F\\p$\"3!*******)3L(z=FT7$$!33+++D9$)zBF\\p$\"3)******HM5!e>FT7$$! 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\"\"\"\"!" }{TEXT -1 2 ". " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 289 "We can distort the boundary curve horizontally b y taking the 11th root of the real part of points along the curve. In \+ this way we see that the largest interval on the nonnegative imaginary axis that contains the origin and lies inside the stability region is [ 0, 0.76 ] approximately. " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 309 "R := z -> 1+z+1/2*z^2+1/6*z ^3+1/24*z^4+1/120*z^5+1/660*z^6:\nDigits := 15:\npts := []: z0 := 0:\n for ct from 0 to 56 do\n zz := newton(R(z)=exp(ct*Pi/200*I),z=z0):\n z0 := zz:\n pts := [op(pts),[surd(Re(zz),11),Im(zz)]]:\nend do:\n plot(pts,color=COLOR(RGB,.9,.5,0),thickness=2,font=[HELVETICA,9]);\nDi gits := 10:" }}{PARA 13 "" 1 "" {GLPLOT2D 261 276 276 {PLOTDATA 2 "6(- %'CURVESG6#7en7$$\"\"!F)F(7$$\"0ry#o\"Qze%!#;$\"0*[zEjzq:F-7$$\"0qY&=) e^p'F-$\"0\"))e`EfTJF-7$$\"0=5o:F1N)F-$\"0&zO!)*)Q7ZF-7$$\"0P?$e2]m(*F -$\"0AaqI&=$G'F-7$$\"0=2fqQE5\"!#:$\"03xLj\")R&yF-7$$\"02d7aht@\"FB$\" 0`I'ezxC%*F-7$$\"0]k$))*4MK\"FB$\"0\"y7Gub*4\"FB7$$\"0[?nUzCU\"FB$\"0% p$)fqjc7FB7$$\"0M]rl\\d^\"FB$\"0Ft/p;PT\"FB7$$\"0(*[hXzSg\"FB$\"0$4>>j zq:FB7$$\"0U5$[b6)o\"FB$\"0szY%f(ys\"FB7$$\"0$[qhzNoFB$\"0jKzx9\"*>#FB7$$\" 0FB$\"0A'[hV>cBFB7$$\"0gE&Q&>#e?FB$\"04N6#RF8DFB7$$\"0cAS]+T 7#FB$\"0\"z$zW`.n#FB7$$\"0M=^)\\o(=#FB$\"0&))[IHVFGFB7$$\"0c^[/#4\\AFB $\"0&[XaB^%)HFB7$$\"0d:kp>%3BFB$\"0&Q1-fkuCFB$\"01*HX!HGh$FB7 $$\"0`Ea>,j_#FB$\"0fp-#y!*pPFB7$$\"0hq[0Hhd#FB$\"0nCqN')p#RFB7$$\"0*G( [#)R%FB7$$\"0&fPO#)=dFFB$\"0I16&pHbXFB7$$\"0KF$H$yxz#FB$\"0Z b`LtBr%FB7$$\"0eoKZ+k$GFB$\"0^XZ-\\%p[FB7$$\"0!z?YJ(H(GFB$\"0@vC!R_E]F B7$$\"07GY$4R2HFB$\"0xJt$yf$=&FB7$$\"0Lw+I>&RHFB$\"0$>e#oq1M&FB7$$\"0$ )z9M(=pHFB$\"0mV[FUx\\&FB7$$\"0!f7Zw<'*HFB$\"0.WHV7[l&FB7$$\"0**p/z6-- $FB$\"0]pn&4)=\"eFB7$$\"0,M]3H4/$FB$\"0(\\6Ew%*ofFB7$$\"0,#y1X&y0$FB$ \"0rQ&*>7g7'FB7$$\"0FL^U[.2$FB$\"0vIKTuIG'FB7$$\"0?OJ8Cv2$FB$\"0k-*zR8 SkFB7$$\"0*HqvX5yIFB$\"0-2xe!>(f'FB7$$\"0B/)4X:qIFB$\"0*f>**QCanFB7$$ \"0d@6hG00$FB$\"0cF.b$H6pFB7$$\"0nl'3%oN,$FB$\"0,I&\\\"R$oqFB7$$\"0F) \\*3 " 0 "" {MPLTEXT 1 0 112 "Digits := 15:\nz0 := 0.76*I:\nfor ct from 23 to 26 d o\n newton(R(z)=exp(ct*Pi/100*I),z=z0);\nend do;\nDigits := 10:" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#^$$\"0Um+u\"zd9!#?$\"0AMoF!QDs!#:" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#^$$\"0ZM`4p,@\"!#@$\"0kR#*GZ%Rv!#:" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#^$$!0wKE6;9;#!#?$\"0w&=K1\\`y!#:" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#^$$!06\"fLf$[u&!#?$\"0Ffd&e]n\")!#:" } }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 107 "Then w e apply the bisection method to calculate the parameter value associat ed with the intersection point." }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 178 "real_part := proc(u)\n Re(newton(R(z)=exp(u*Pi*I),z=0.76*I) )\nend proc:\nDigits := 15:\nu0 := bisect('real_part'(u),u=0.23..0.26) ;\nnewton(R(z)=exp(u0*Pi*I),z=0.76*I);\nDigits := 10:" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%#u0G$\"0z)H3Tw1C!#:" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#^#$\"0N`YJ!pgv!#:" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }} {PARA 0 "" 0 "" {TEXT -1 4 "The " }{TEXT 260 112 "largest interval on \+ the nonnegative imaginary axis that contains the origin and lies insid e the stability region" }{TEXT -1 36 " is [ 0, 0.7561 ] (approximate ly)." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 37 "# ------------------------------------" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 80 "The stability function R* for the 7 s tage, order 4 scheme is given as follows." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 97 "subs(ee,subs(b=`b*`,S tabilityFunction(4,7,'expanded'))):\n`R*` := unapply(%,z):\n'`R*`(z)'= `R*`(z);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/-%#R*G6#%\"zG,2\"\"\"F)F' F)*&#F)\"\"#F)*$)F'F,F)F)F)*&#F)\"\"'F)*$)F'\"\"$F)F)F)*&#F)\"#CF)*$)F '\"\"%F)F)F)*&#\"&^$H\"(+WT$F)*$)F'\"\"&F)F)F)*&#\"&\\R$\")+3!R#F)*$)F 'F1F)F)F)*&#F)\"&?R(F)*$)F'\"\"(F)F)F)" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 124 "We can find the point where the bou ndary of the stability region intersects the negative real axis by sol ving the equation: " }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "`R*`(z) = 1;" "6#/-%#R*G6#%\"zG\"\"\"" }{TEXT -1 2 ". " }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 32 "z_0 := newton(`R*`(z)=1,z=-3.7);" } }{PARA 11 "" 1 "" {XPPMATH 20 "6#>%$z_0G$!+H4(Rm$!\"*" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 320 "z_0 := newton(`R*`(z)=1,z=-3.7):\np_1 := plot([`R*`(z),1],z=-4.19..0.49,colo r=[red,blue]):\np_2 := plot([[[z_0,1]]$3],style=point,symbol=[circle,c ross,diamond],color=black):\np_3 := plot([[z_0,0],[z_0,1]],linestyle=3 ,color=COLOR(RGB,0,.5,0)):\nplots[display]([p_1,p_2,p_3],view=[-4.19.. 0.49,-0.07..1.47],font=[HELVETICA,9]);" }}{PARA 13 "" 1 "" {GLPLOT2D 410 236 236 {PLOTDATA 2 "6+-%'CURVESG6$7X7$$!3Q++++++!>%!#<$\"3)*G#ynX K#[CF*7$$!3l++DfY**QTF*$\"3mZuB0\\?bAF*7$$!3-++]=$*)z3%F*$\"3;B8\"pbA` 2#F*7$$!3O+vBW)4O/%F*$\"39]9By,%*G>F*7$$!3E+](*p.B**RF*$\"3=uK4%f1:z\" F*7$$!3W+Dm45K\\RF*$\"3c,30jL4Z;F*7$$!3;++N\\;T**QF*$\"3Tcb8GS%G^\"F*7 $$!3A++Sd9<\\QF*$\"3![`NU&**Q(Q\"F*7$$!3E++Xl7$*)z$F*$\"3U>gxXc4r7F*7$ $!3U+DOc)H*[PF*$\"3cOiI.)>R;\"F*7$$!37+]FZ%G*)p$F*$\"3_&yz\\l'yk5F*7$$ 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%&STYLEG6#%&POINTG-F$6&Ff`l-F[al6#%&CROSSGF^alF`al-F$6&Ff`l-F[al6#%(DI AMONDGF^alF`al-F$6%7$7$Fh`lFj\\lFg`l-%&COLORG6&Ff\\lFj\\l$\"\"&!\"\"Fj \\l-%*LINESTYLEG6#\"\"$-%%FONTG6$%*HELVETICAG\"\"*-%+AXESLABELSG6%Q\"z 6\"Q!Fecl-F]cl6#%(DEFAULTG-%%VIEWG6$;$!$>%!\"#$\"#\\F`dl;$!\"(F`dl$\"$ Z\"F`dl" 1 2 0 1 10 0 2 9 1 4 2 1.000000 45.000000 45.000000 0 0 "Curv e 1" "Curve 2" "Curve 3" "Curve 4" "Curve 5" "Curve 6" }}}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 81 "The following pictur e shows the stability region for the 7 stage, order 4 scheme." }} {PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1813 "`R*` := z -> 1+z+1/2*z^2+1/6*z^3+1/24*z^4+29351/3414400*z^5+3394 9/23900800*z^6+1/73920*z^7:\npts := []: z0 := 0: tt := 0: \nwhile tt<= 161/20 do\n zz := newton(`R*`(z)=exp(tt*Pi*I),z=z0):\n z0 := zz:\n if (9/20<=tt and tt<=11/10) \n or (137/20<=tt and tt<=151/20) t hen\n hh := 1/60\n else \n hh := 1/20\n end if;\n tt : = tt+hh;\n pts := [op(pts),[Re(zz),Im(zz)]]:\nend do:\np_1 := plot(p ts,color=COLOR(RGB,.38,.18,0)):\np_2 := plots[polygonplot]([seq([pts[i -1],pts[i],[-1.8,0]],i=2..nops(pts))],\n style=patchnogrid,co lor=COLOR(RGB,.75,.35,0)):\npts := []: z0 := 0.9+3.9*I: tt := 0:\nwhil e tt<=61/30 do\n zz := newton(`R*`(z)=exp(tt*Pi*I),z=z0):\n z0 := \+ zz:\n if (21/30<=tt and tt<=29/30) then\n hh := 1/120\n elif \+ (13/30<=tt and tt<=19/15) then\n hh := 1/60\n else \n hh : = 1/30\n end if;\n tt := tt+hh;\n pts := [op(pts),[Re(zz),Im(zz) ]]:\nend do:\np_3 := plot(pts,color=COLOR(RGB,.38,.18,0)):\np_4 := plo ts[polygonplot]([seq([pts[i-1],pts[i],[0.80,3.62]],i=2..nops(pts))],\n style=patchnogrid,color=COLOR(RGB,.75,.35,0)):\npts := []: z 0 := 0.9-3.9*I: tt := 0:\nwhile tt<=61/30 do\n zz := newton(`R*`(z)= exp(tt*Pi*I),z=z0):\n z0 := zz:\n if (16/15<=tt and tt<=4/3) then \n hh := 1/120\n elif (23/30<=tt and tt<=8/5) then\n hh := 1/60\n else \n hh := 1/30\n end if;\n tt := tt+hh;\n pts := [op(pts),[Re(zz),Im(zz)]]:\nend do:\np_5 := plot(pts,color=COLOR(R GB,.38,.18,0)):\np_6 := plots[polygonplot]([seq([pts[i-1],pts[i],[0.80 ,-3.62]],i=2..nops(pts))],\n style=patchnogrid,color=COLOR(RG B,.75,.35,0)):\np_7 := plot([[[-4.29,0],[1.39,0]],[[0,-4.29],[0,4.29]] ],color=black,linestyle=3):\nplots[display]([p_||(1..7)],view=[-4.29.. 1.39,-4.29..4.29],font=[HELVETICA,9],\n color=COLOR(RGB,.4,0,1),labe ls=[`Re(z)`,`Im(z)`],axes=boxed,scaling=constrained);" }}{PARA 13 "" 1 "" {GLPLOT2D 486 581 581 {PLOTDATA 2 "60-%'CURVESG6$7dx7$$\"\"!F)F(7 $$!39+++.<]CN!#E$\"33+++tgzq:!#=7$$!3/+++5j#yM#!#C$\"3?+++rVeTJF07$$!3 2+++r$eP%G!#B$\"3(******\\BCBr%F07$$!32+++)yQYs\"!#A$\"3c******[Y!HG'F 07$$!3K+++ywAfrF@$\"3%)*****f')4J&yF07$$!3(******f(*=3L#!#@$\"3n*****f 9+++#3$GNLFW$\"3*)*****4Y\"\\79FS7$$!3:+++&)>*[B%FW$ \"3#********z>ZY\"FS7$$!3%)*****z&)Q2L&FW$\"3\"******4#*op^\"FS7$$!3G+ ++D*4gl'FW$\"33+++eDDp:FS7$$!3/+++TD8[#)FW$\"3#******>o(e@;FS7$$!3(*** ***z_J\\,\"!#>$\"33+++g[*Rn\"FS7$$!30+++vNsS7F`p$\"3#******He)\\E$=FS7$$! 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" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 723 "SO5_6 := SimpleOrderConditions(5,6,'expanded'): \n`SO4_7*` := subs(b=`b*`,SimpleOrderConditions(4,7,'expanded')): \ncd ns := [seq(SO5_6[i],i=[1,2,4,8,12,13,15,16]),seq(`SO4_7*`[i],i=[1,2,4, 7,8])]:\nSO_eqs := [op(RowSumConditions(6,'expanded')),op(StageOrderCo nditions(2,6,'expanded'))]:\nsimp_eqns := [add(b[i]*a[i,2],i=3..6)=0,a dd(b[i]*add(a[i,j]*a[j,2],j=3..i-1),i=2..6)=0,\n add(b[i]*c[i] *a[i,2],i=3..6)=0,add(`b*`[i]*a[i,2],i=3..6)=0]:\nwt_eqns := [seq(b[i] =a[7,i],i=1..6)]:\ncdns := [op(wt_eqns),op(cdns),op(simp_eqns),op(SO_e qs)]:\n\nerrterms5_6 := PrincipalErrorTerms(5,6,'expanded'):\n`errterm s5_7*` :=subs(b=`b*`,PrincipalErrorTerms(5,7,'expanded')):\n`errterms4 _7*` := subs(b=`b*`,PrincipalErrorTerms(4,7,'expanded')):" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 9 "OR . . . " }} {PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 2678 "cdns := [b[6]=(5*c[3]*(2*c[4]*(3*c[5]-2)-4*c[5]+3)-5*c[4]*(4*c[5 ]-3)+3*(5*c[5]-4))/\n (60*c[6]*(c[3]-c[6])*(c[4]-c[6 ])*(c[5]-c[6])),\nb[5]=-(12*b[6]*c[6]*(c[4]-c[6])*(c[3]-c[6])-2*c[3]*( 3*c[4]-2)+4*c[4]-3)/\n (12*c[5]*(c[3]-c[5])*(c[4]-c[5])),\nb[4]= -(6*b[5]*c[5]*(c[3]-c[5])+6*b[6]*c[6]*(c[3]-c[6])-3*c[3]+2)/(6*c[4]*(c [3]-c[4])),\nb[3]=-(2*b[4]*c[4]+2*b[5]*c[5]+2*b[6]*c[6]-1)/(2*c[3]),\n k[1]=1440*b[4]*b[5]*`b*`[7]*c[4]*c[5]*(c[5]-c[6])*(c[4]-c[5])*(c[3]^2- c[3]*(c[4]+c[5])+c[4]*c[5]),\nk[2]=1440*b[5]^2*`b*`[7]*c[5]^2*(c[5]-c[ 6])*(c[4]-c[5])*(c[3]^2-2*c[3]*c[5]+c[5]^2),\nk[3]=120*b[5]*c[5]*(c[5] -c[6])*(c[4]-c[5])*(c[3]-c[5])*(12*b[6]*`b*`[7]*c[6]*(c[3]-c[6])-2*c[3 ]+1),\nk[4]=(5*c[3]*(4*c[6]-3)-2*(5*c[6]-4))*(12*`b*`[7]*(c[4]-1)*(c[3 ]-1)-2*c[3]*(3*c[4]-2)+4*c[4]-3),\nm[1]=b[5]*c[5]*(c[4]-c[5])*(5*c[3]^ 2*(4*c[5]-3)-c[3]*(20*c[5]^2-5*c[5]-8)+2*c[5]*(5*c[5]-4)),\nm[2]=b[6]* c[6]*(c[4]-c[6])*(5*c[3]^2*(4*c[6]-3)-c[3]*(20*c[6]^2-5*c[6]-8)+2*c[6] *(5*c[6]-4)),\n`b*`[6]=-b[6]*(k[1]+k[2]+k[3]+k[4])/(12*(m[1]+m[2])),\n `b*`[5]=-(12*`b*`[6]*c[6]*(c[4]-c[6])*(c[3]-c[6])+12*`b*`[7]*(c[4]-1)* (c[3]-1)-2*c[3]*(3*c[4]-2)+4*c[4]-3)/\n (12*c[5]*(c[3]-c[5])* (c[4]-c[5])),\n`b*`[4]=-(6*`b*`[5]*c[5]*(c[3]-c[5])+6*`b*`[6]*c[6]*(c[ 3]-c[6])+6*`b*`[7]*(c[3]-1)-3*c[3]+2)/(6*c[4]*(c[3]-c[4])),\n`b*`[3]=- (2*`b*`[4]*c[4]+2*`b*`[5]*c[5]+2*`b*`[6]*c[6]+2*`b*`[7]-1)/(2*c[3]),\n a[3,2]=c[3]^2/(2*c[2]),\na[6,2]=-(a[3,2]*(b[3]*(b[4]*`b*`[5]*(c[3]-c[4 ])+b[5]*`b*`[4]*(c[5]-c[3]))+b[4]*b[5]*`b*`[3]*(c[4]-c[5])))/\n \+ (b[4]*(b[5]*`b*`[6]*(c[4]-c[5])+b[6]*`b*`[5]*(c[6]-c[4]))+b[5]*b[6]* `b*`[4]*(c[5]-c[6])),\na[5,2]=(a[3,2]*b[3]*(c[3]-c[4])+a[6,2]*b[6]*(c[ 6]-c[4]))/(b[5]*(c[4]-c[5])),\na[4,2]=-(a[3,2]*b[3]+a[5,2]*b[5]+a[6,2] *b[6])/b[4],\na[6,5]=(5*c[3]*(2*c[4]-1)-5*c[4]+3)/(60*b[6]*c[5]*(c[3]- c[5])*(c[4]-c[5])),\na[6,4]=-(60*a[5,4]*b[5]*c[4]*(c[4]-c[5])*(c[3]-c[ 4])+5*c[3]*(2*c[5]-1)-5*c[5]+3)/\n (60*b[6]*c[4]*(c[3] -c[4])*(c[4]-c[5])),\na[5,4]=-(5*c[3]*(4*c[6]-3)-2*(5*c[6]-4))/(120*b[ 5]*c[4]*(c[3]-c[4])*(c[5]-c[6])),\na[4,3]=-(2*a[4,2]*c[2]-c[4]^2)/(2*c [3]),\na[5,3]=-(2*a[5,2]*c[2]+2*a[5,4]*c[4]-c[5]^2)/(2*c[3]),\na[6,3]= -(2*a[6,2]*c[2]+2*a[6,4]*c[4]+2*a[6,5]*c[5]-c[6]^2)/(2*c[3]),\nb[1]=1- b[3]-b[4]-b[5]-b[6],\n`b*`[1]=1-`b*`[3]-`b*`[4]-`b*`[5]-`b*`[6]-`b*`[7 ],\na[2,1]=c[2], a[3,1]=c[3]-a[3,2], a[4,1]=c[4]-a[4,2]-a[4,3],\na[5,1 ]=c[5]-a[5,2]-a[5,3]-a[5,4], a[6,1]=c[6]-a[6,2]-a[6,3]-a[6,4]-a[6,5], \na[7,1]=b[1], a[7,2]=b[2], a[7,3]=b[3], a[7,4]=b[4], a[7,5]=b[5], a[7 ,6]=b[6]]:\n\nerrterms5_6 := PrincipalErrorTerms(5,6,'expanded'):\n`er rterms5_7*` :=subs(b=`b*`,PrincipalErrorTerms(5,7,'expanded')):\n`errt erms4_7*` := subs(b=`b*`,PrincipalErrorTerms(4,7,'expanded')):" }}} {PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}{SECT 1 {PARA 4 "" 0 "" {TEXT 0 13 "calc_RKcoeff s" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 2121 "calc_RKcoeffs := proc()\n local eqns,sol,sm,ct,Rz ,stb5,stb4,nmB,snmB,dnB,abserrs,errtrm,sdnB,nmC,snmC,B_6,C_6,nrm;\n \+ global e1,e2;\n\n e1 := \{c[2]=c_2,c[3]=c_3,c[4]=c_4,c[5]=c_5,c[6]=1 ,c[7]=1,b[2]=0,`b*`[2]=0,`b*`[7]=bs_7\};\n eqns := subs(e1,[op(cdns) ]);\n sol := solve(\{op(eqns)\});\n e2 := `union`(e1,remove(has,so l,\{k,m\})): ## combined scheme\n Digits := 14;\n sm := 0;\n for ct to nops(errterms5_6) do\n sm := sm+(evalf(subs(e2,errterms5_6 [ct])))^2;\n end do;\n Rz := subs(e2,StabilityFunction(5,6,'expand ed'));\n stb5 := max(fsolve(Rz=1,z=-8..-1e-7),fsolve(Rz=-1,z=-8..-1e -7));\n stb5 := evalf[8](stb5);\n Rz := subs(e2,subs(b=`b*`,Stabil ityFunction(4,7,'expanded')));\n stb4 := max(fsolve(Rz=1,z=-8..-1e-7 ),fsolve(Rz=-1,z=-8..-1e-7));\n stb4 := evalf[8](stb4);\n nmB := 0 ;\n for ct to nops(`errterms5_7*`) do\n nmB := nmB+evalf(subs(e 2,`errterms5_7*`[ct]))^2;\n end do:\n snmB := sqrt(nmB);\n dnB : = 0;\n abserrs := [];\n for ct to nops(`errterms4_7*`) do\n e rrtrm := evalf(subs(e2,`errterms4_7*`[ct]));\n dnB := dnB+errtrm^ 2;\n abserrs := [op(abserrs),abs(errtrm)];\n end do;\n sdnB : = sqrt(dnB);\n nmC := 0;\n for ct to nops(errterms5_6) do\n n mC := nmC+(evalf(subs(e2,`errterms5_7*`[ct]))-evalf(subs(e2,errterms5_ 6[ct])))^2;\n end do;\n snmC := sqrt(nmC);\n B_6 := evalf[8](snm B/sdnB);\n C_6 := evalf[8](snmC/sdnB);\n print(`nodes:`,c[2]=c_2,c [3]=c_3,c[4]=c_4,c[5]=c_5);\n print(`order 5 weights:`,seq(b[i]=eval f[6](subs(e2,b[i])),i=[1,$3..6]));\n print(`order 4 weights:`,seq(`b *`[i]=evalf[6](subs(e2,`b*`[i])),i=[1,$3..7]));\n nrm := max(seq(seq (subs(e2,abs(a[i,j])),j=1..i-1),i=2..7));\n print(infinity*`-norm of linking coeffs`=evalf[10](nrm));\n print(`2-norm of principal error of order 5 scheme` = evalf[10](sqrt(sm)));\n print(`2-norm of princ ipal error of order 4 scheme` = evalf[10](sdnB));\n print(`order 5 s tability interval` = [stb5,0]);\n print(`order 4 stability interval` = [stb4,0]);\n print(`max-min principal error ratio of order 4 sche me` = evalf[4](max(op(abserrs))/min(op(abserrs))));\n print('B[6]'=B _6,'C[6]'=C_6);\nend proc:" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}}{PARA 0 "" 0 "" {TEXT -1 0 " " }}{PARA 0 "" 0 "" {TEXT -1 57 "#------------------------------------ --------------------" }}{SECT 1 {PARA 5 "" 0 "" {TEXT -1 21 "Prince-Do rmand scheme" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 79 "c_2 := 1/5: c_3 := 3/10: c_4 := 4/5: c_5 := 8/9: b s_7 := 1/40:\ncalc_RKcoeffs();" }}{PARA 11 "" 1 "" {XPPMATH 20 "6'%'no des:G/&%\"cG6#\"\"##\"\"\"\"\"&/&F&6#\"\"$#F/\"#5/&F&6#\"\"%#F5F+/&F&6 #F+#\"\")\"\"*" }}{PARA 11 "" 1 "" {XPPMATH 20 "6(%1order~5~weights:G/ &%\"bG6#\"\"\"$\"'e9\"*!\"(/&F&6#\"\"$$\"'O#\\%!\"'/&F&6#\"\"%$\"'U5lF 2/&F&6#\"\"&$!'wBKF2/&F&6#\"\"'$\"'_48F2" }}{PARA 11 "" 1 "" {XPPMATH 20 "6)%1order~4~weights:G/&%#b*G6#\"\"\"$\"'K\"**)!\"(/&F&6#\"\"$$\"'* [`%!\"'/&F&6#\"\"%$\"'iShF2/&F&6#\"\"&$!'7:FF2/&F&6#\"\"'$\"'w/*)F+/&F &6#\"\"($\"'++DF+" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/*&%)infinityG\" \"\"%8-norm~of~linking~coeffsGF&$\"+K$z&f6!\")" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/%L2-norm~of~principal~error~of~order~5~schemeG$\"+5;!3 *R!#8" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/%L2-norm~of~principal~error~ of~order~4~schemeG$\"+^r&H=\"!#7" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/% ;order~5~stability~intervalG7$$!)zc1L!\"(\"\"!" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/%;order~4~stability~intervalG7$$!)j)\\Q%!\"(\"\"!" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#/%Pmax-min~principal~error~ratio~of~or der~4~schemeG$\"%vt!\"#" }}{PARA 11 "" 1 "" {XPPMATH 20 "6$/&%\"BG6#\" \"'$\")6pT:!\"(/&%\"CGF&$\")YLl;F*" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" } }{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}{SECT 1 {PARA 4 "" 0 " " {TEXT -1 2 "e2" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> \+ " 0 "" {MPLTEXT 1 0 680 "e2 := \{b[4] = 125/192, c[7] = 1, c[6] = 1, ` b*`[2] = 0, b[2] = 0, a[7,2] = 0, b[6] = 11/84, a[3,2] = 9/40, a[4,1] \+ = 44/45, a[2,1] = 1/5, a[6,1] = 9017/3168, a[6,3] = 46732/5247, a[5,3] = 64448/6561, a[4,2] = -56/15, a[5,2] = -25360/2187, `b*`[1] = 5179/5 7600, a[6,2] = -355/33, a[4,3] = 32/9, b[5] = -2187/6784, a[6,5] = -51 03/18656, `b*`[3] = 7571/16695, a[7,6] = 11/84, a[3,1] = 3/40, c[4] = \+ 4/5, c[5] = 8/9, c[2] = 1/5, a[6,4] = 49/176, c[3] = 3/10, b[3] = 500/ 1113, a[5,1] = 19372/6561, a[7,5] = -2187/6784, `b*`[7] = 1/40, a[7,3] = 500/1113, a[7,1] = 35/384, `b*`[6] = 187/2100, b[1] = 35/384, a[7,4 ] = 125/192, `b*`[4] = 393/640, a[5,4] = -212/729, `b*`[5] = -92097/33 9200\}:" }{TEXT -1 0 "" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}}{PARA 0 "" 0 "" {TEXT -1 0 " " }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 137 "subs(e2,matrix([seq([c[ i],seq(a[i,j],j=1..i-1),``$(8-i)],i=2..7),\n[`b`,seq(b[i],i=1..6),``], [`b*`,seq(`b*`[i],i=1..7)]]));\n``;\nevalf(%%);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#K%'matrixG6#7*7*#\"\"\"\"\"&F(%!GF+F+F+F+F+7*#\"\"$\"#5 #F.\"#S#\"\"*F1F+F+F+F+F+7*#\"\"%F*#\"#W\"#X#!#c\"#:#\"#KF3F+F+F+F+7*# \"\")F3#\"&s$>\"%hl#!&g`#\"%(=##\"&[W'FD#!$7#\"$H(F+F+F+7*F)#\"%#!%(=#\"%%y'#\"#6\"#%)F+7*%\"bGFhnF[oF\\oF_oFboF eoF+7*%#b*G#\"%z^\"&+w&F[o#\"%rv\"&&p;#\"$$R\"$S'#!&(4#*\"'+#R$#\"$(= \"%+@#F)F1Q(pprint06\"" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%!G" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#K%'matrixG6#7*7*$\"+++++?!#5F(%!GF+F+F +F+F+7*$\"+++++IF*$\"+++++v!#6$\"++++]AF*F+F+F+F+F+7*$\"+++++!)F*$\"+y xxx(*F*$!+LLLLP!\"*$\"+cbbbNF;F+F+F+F+7*$\"+*)))))))))F*$\"+*o)f_HF;$! +K$z&f6!\")$\"+_G*G#)*F;$!+yK43HF*F+F+F+7*$\"\"\"\"\"!$\"+`_FYGF;$!+wv vv5FE$\"+=FU1*)F;$\"+444%y#F*$!+OIJNFF*F+F+7*FK$\"+LLe9\"*F1$FMFM$\"+$ )HO#\\%F*$\"+nmT5lF*$!+#zhPA$F*$\"+5Q_48F*F+7*%\"bGFYFenFfnFhnFjnF\\oF +7*%#b*G$\"+W%>8**)F1Fen$\"+'o!*[`%F*$\"++]iShF*$!+@Q7:FF*$\"+0>w/*)F1 $\"+++++DF1Q(pprint16\"" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}}{PARA 0 "" 0 "" {TEXT -1 57 " #--------------------------------------------------------" }}{SECT 1 {PARA 5 "" 0 "" {TEXT -1 13 "scheme with " }{XPPEDIT 18 0 "rho=rho[2] " "6#/%$rhoG&F$6#\"\"#" }{TEXT -1 33 " and minimum principal error nor m" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 102 "c_2 := 17/150: c_3 := 230/1379: c_4 := 1420370/23423 87: c_5 := 133/208: bs_7 := 1/40:\ncalc_RKcoeffs();" }}{PARA 11 "" 1 " " {XPPMATH 20 "6'%'nodes:G/&%\"cG6#\"\"##\"#<\"$]\"/&F&6#\"\"$#\"$I#\" %z8/&F&6#\"\"%#\"(q.U\"\"((QUB/&F&6#\"\"&#\"$L\"\"$3#" }}{PARA 11 "" 1 "" {XPPMATH 20 "6(%1order~5~weights:G/&%\"bG6#\"\"\"$\"'9L7!\"(/&F&6 #\"\"$$\"'KUP!\"'/&F&6#\"\"%$!'#Gq\"F2/&F&6#\"\"&$\"'*et'F2/&F&6#\"\"' $\"'I,6F2" }}{PARA 11 "" 1 "" {XPPMATH 20 "6)%1order~4~weights:G/&%#b* G6#\"\"\"$\"'Q\">\"!\"(/&F&6#\"\"$$\"'o^P!\"'/&F&6#\"\"%$!'r!y\"F2/&F& 6#\"\"&$\"')3\"oF2/&F&6#\"\"'$\"'9!\\)F+/&F&6#\"\"($\"'++DF+" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/*&%)infinityG\"\"\"%8-norm~of~linking~coeff sGF&$\"+\"GD(Gp!\"*" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/%L2-norm~of~pr incipal~error~of~order~5~schemeG$\"+Kr[&H#!#7" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/%L2-norm~of~principal~error~of~order~4~schemeG$\"+)Q:X 8\"!#7" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/%;order~5~stability~interva lG7$$!)X1'z#!\"(\"\"!" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/%;order~4~st ability~intervalG7$$!)e\"G;$!\"(\"\"!" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/%Pmax-min~principal~error~ratio~of~order~4~schemeG$\"%L(*!\"\"" }} {PARA 11 "" 1 "" {XPPMATH 20 "6$/&%\"BG6#\"\"'$\")#3kF\"!\"(/&%\"CGF&$ \")[8&G\"F*" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 2 "e2" }} {PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1751 "e2 := \{c[7] = 1, c[6] = 1, `b*`[2] = 0, b[2] = 0, a[7,2] = 0, ` b*`[6] = 58914464694395831/693916322394528225, a[4,2] = -2048128144537 681132500/295599560021395955869, a[5,2] = -1141151413305973125/5513721 54861039616, a[6,2] = 104590105895625/68427659422081, a[7,6] = 1750068 01591/1589096299500, a[3,1] = 1424390/32327897, b[4] = -33988229203965 941757637160569/199599379574494873119383412000, a[6,1] = -778166632716 677780702867/2973121469222107001182300, a[4,3] = 395036854584743011582 20/6798789880492106984987, a[2,1] = 17/150, a[4,1] = 11725892832745653 435650/6798789880492106984987, b[3] = 380753067446365335491/1017424549 568194596000, a[6,3] = -6240811551117087470432064137/66004141676210428 66395511800, a[6,5] = 4149774798413464954572800/1881639624293239507076 793, `b*`[5] = 129508101849060942811648/190148731404640113216825, a[5, 1] = 12161603119957121528213173/25732152506856316150988800, a[5,4] = 5 081806128006459902107790523/33857948797038081396772249600, `b*`[4] = - 372495557697385192340734287120958393/209183447175821197684417747989966 2400, b[5] = 73328266119626752/108862061490007875, b[1] = 10715793331/ 868982366000, a[5,3] = 3683267576399581353585064593/176539893683240304 1633484800, a[6,4] = -963956247541399247752708263264927/63142814465243 6311985040262094600, `b*`[3] = 1333442845934451852070878169/3554259119 336756010609086400, `b*`[7] = 1/40, c[3] = 230/1379, c[4] = 1420370/23 42387, c[5] = 133/208, c[2] = 17/150, b[6] = 175006801591/158909629950 0, a[7,4] = -33988229203965941757637160569/199599379574494873119383412 000, a[7,5] = 73328266119626752/108862061490007875, `b*`[1] = 36166679 554284109/3035692917188954400, a[7,3] = 380753067446365335491/10174245 49568194596000, a[3,2] = 3967500/32327897, a[7,1] = 10715793331/868982 366000\}:" }{TEXT -1 0 "" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}}{PARA 0 "" 0 "" {TEXT -1 0 " " }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 132 "subs(e2,matrix([seq([c[ i],seq(a[i,j],j=1..i-1),``$(8-i)],i=2..7),\n[`b`,seq(b[i],i=1..6),``], [`b*`,seq(`b*`[i],i=1..7)]])):\nevalf(%);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#K%'matrixG6#7*7*$\"+LLLL6!#5F(%!GF+F+F+F+F+7*$\"+s_(ym \"F*$\"+2221W!#6$\"+,#osA\"F*F+F+F+F+F+7*$\"++4]\"F*F+F+F+7*$\"\"\"\"\"!$!+])Qth#F*$\"+MqZG:F9$!+S<=b%*F *$!+t&Gm_\"F9$\"+)f.a?#F9F+F+7*FJ$\"+$yUJB\"F1$FLFL$\"+#RABu$F*$!+(QAG q\"F*$\"+p!))et'F*$\"+ZwH,6F*F+7*%\"bGFXFZFenFgnFinF[oF+7*%#b*G$\"+g8Q \">\"F1FZ$\"+]fn^PF*$!+(R72y\"F*$\"+[V)3\"oF*$\"+J'R,\\)F1$\"+++++DF1Q (pprint76\"" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}}{PARA 0 "" 0 "" {TEXT -1 57 "#-------------- ------------------------------------------" }}{SECT 1 {PARA 5 "" 0 "" {TEXT -1 13 "scheme with " }{XPPEDIT 18 0 "t[6] = 1/500;" "6#/&%\"tG6 #\"\"'*&\"\"\"F)\"$+&!\"\"" }{TEXT -1 35 " and minimum principal erro r norm " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 91 "c_2 := 34/195: c_3 := 137/525: c_4 := 252/365: c_5 := 31/40: bs_7 := 1/30:\ncalc_RKcoeffs():" }}{PARA 11 "" 1 "" {XPPMATH 20 "6'%'nodes:G/&%\"cG6#\"\"##\"#M\"$&>/&F&6#\"\"$#\"$P\"\"$D&/&F&6#\" \"%#\"$_#\"$l$/&F&6#\"\"&#\"#J\"#S" }}{PARA 11 "" 1 "" {XPPMATH 20 "6( %1order~5~weights:G/&%\"bG6#\"\"\"$\"':5x!\"(/&F&6#\"\"$$\"'82S!\"'/&F &6#\"\"%$\"'5yLF2/&F&6#\"\"&$\"'W`)*F+/&F&6#\"\"'$\"'7%e)F+" }}{PARA 11 "" 1 "" {XPPMATH 20 "6)%1order~4~weights:G/&%#b*G6#\"\"\"$\"'pRx!\" (/&F&6#\"\"$$\"'W(*R!\"'/&F&6#\"\"%$\"'@JMF2/&F&6#\"\"&$\"'_4$*F+/&F&6 #\"\"'$\"'!4L&F+/&F&6#\"\"($\"'LLLF+" }}{PARA 11 "" 1 "" {XPPMATH 20 " 6#/*&%)infinityG\"\"\"%8-norm~of~linking~coeffsGF&$\"+%)fq;7!\")" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#/%L2-norm~of~principal~error~of~order~ 5~schemeG$\"+ " 0 "" {MPLTEXT 1 0 1 ";" }}}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 2 "e2" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1188 "e2 := \{b[ 4] = 469280932525/1389186738576, a[3,2] = 243997/1249500, `b*`[7] = 1/ 30, a[3,1] = 82063/1249500, `b*`[3] = 61472258224069875/15377891920233 0832, a[7,3] = 3027623484375/7555589800144, a[7,1] = 990209/12842928, \+ a[5,2] = 498515020731/40972513280, b[5] = 43980800/446349501, c[6] = 1 , c[7] = 1, b[2] = 0, `b*`[2] = 0, a[4,2] = -14189381388/4530102965, a [6,2] = -743202495/291283372, a[7,6] = 406471/4735152, a[7,2] = 0, c[5 ] = 31/40, c[4] = 252/365, c[3] = 137/525, c[2] = 34/195, a[7,4] = 469 280932525/1389186738576, `b*`[4] = 203730456418295501/5937564714949838 88, a[6,3] = 625455957178350900/271098296009435219, `b*`[6] = 53945083 2947/10119327608880, a[6,5] = 1735464775680/6719560297073, a[2,1] = 34 /195, a[6,4] = 21511957375165/104104742407029, a[4,1] = 101345382936/1 24124821241, `b*`[1] = 2124247802213/27446171926320, a[6,1] = 78971341 5527611/1013168913843996, a[5,1] = -2095505049916883/707267524239360, \+ a[5,3] = -2529970133596857/271730069595136, b[6] = 406471/4735152, a[7 ,5] = 43980800/446349501, b[3] = 3027623484375/7555589800144, b[1] = 9 90209/12842928, `b*`[5] = 17760294550016/190775579270913, a[5,4] = 643 1020141871/7296710270976, a[4,3] = 21949459056/7301460073\}:" }}} {PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 137 "subs(e2,matrix([seq([c[i],seq(a[i,j],j=1..i-1),``$(8 -i)],i=2..7),\n[`b`,seq(b[i],i=1..6),``],[`b*`,seq(`b*`[i],i=1..7)]])) ;\n``;\nevalf(%%);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#K%'matrixG6#7*7* #\"#M\"$&>F(%!GF+F+F+F+F+7*#\"$P\"\"$D&#\"&j?)\"(+&\\7#\"'(*RCF2F+F+F+ F+F+7*#\"$_#\"$l$#\"-OHQX85\"-T7#[7C\"#!,)Q\"Q*=9\"+lH5IX#\",c!f%\\># \"+t+Y,tF+F+F+F+7*#\"#J\"#S#!1$)o\"*\\]]&4#\"0g$RU_nsq#\"-J2-:&)\\\",! G8D(4%#!1dofL,(*HD\"0O^fp+tr##\".r=9?5V'\".w4F5nH(F+F+F+7*\"\"\"#\"06w _:Mr*y\"1'*R%Q\"*oJ,\"#!*&\\-Ku\"*sLG\"H#\"3+4Nyr&fXD'\"3>_V4gH)4r##\" /l^Pd>^@\"0HqSUZ5/\"#\".!ovZYN<\".tqHg&>nF+F+7*FS#\"'4-**\")GH%G\"\"\" !#\".vV[Bw-$\".W,!)*ebv#\"-DD$4Gp%\".w&Qn=*Q\"#\")+3)R%\"*,&\\jW#\"'rk S\"(_^t%F+7*%\"bGF^oFaoFboFeoFhoF[pF+7*%#b*G#\".8A!yCC@\"/?j#>*yP:#\"3,bH=kXIP?\"3))Q)\\\\rkv$f#\"/;+b%Hgx\"\"0 84Fzbx!>#\"-ZH$3XR&\"/!))3wK>,\"#FS\"#IQ(pprint46\"" }}{PARA 11 "" 1 " " {XPPMATH 20 "6#%!G" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#K%'matrixG6#7* 7*$\"+W(*eVF*F+F+ F+F+F+7*$\"+*e4T!pF*$\"+#)ezk\")F*$!+yDCKJ!\"*$\"+[R<1IF;F+F+F+F+7*$\" ++++]xF*$!+R\"=G'HF;$\"+%)fq;7!\")$!+V0g5$*F;$\"+d%)e8))F*F+F+F+7*$\" \"\"\"\"!$\"+*>*[%z(F*$!+qeZ^DF;$\"+L(=rI#F;$\"+8jPm?F*$\"+geq#e#F*F+F +7*FK$\"+(*)\\,r(F1$FMFM$\"+o082SF*$\"+v#)4yLF*$\"+?WW`)*F1$\"+ds6%e)F 1F+7*%\"bGFYFenFfnFhnFjnF\\oF+7*%#b*G$\"+1qoRxF1Fen$\"+oPW(*RF*$\"+%R7 7V$F*$\"+9?_4$*F1$\"+Ag*3L&F1$\"+LLLLLF1Q(pprint56\"" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}} {PARA 0 "" 0 "" {TEXT -1 57 "#---------------------------------------- ----------------" }}{SECT 1 {PARA 5 "" 0 "" {TEXT -1 13 "scheme with \+ " }{XPPEDIT 18 0 "t[6] = 1/600;" "6#/&%\"tG6#\"\"'*&\"\"\"F)\"$+'!\"\" " }{TEXT -1 35 " and minimum principal error norm " }}{PARA 0 "" 0 " " {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 91 "c_2 := 64/ 315: c_3 := 115/381: c_4 := 762/935: c_5 := 25/28: bs_7 := 1/42:\ncalc _RKcoeffs();" }}{PARA 11 "" 1 "" {XPPMATH 20 "6'%'nodes:G/&%\"cG6#\"\" ##\"#k\"$:$/&F&6#\"\"$#\"$:\"\"$\"Q/&F&6#\"\"%#\"$i(\"$N*/&F&6#\"\"&# \"#D\"#G" }}{PARA 11 "" 1 "" {XPPMATH 20 "6(%1order~5~weights:G/&%\"bG 6#\"\"\"$\"'.^\"*!\"(/&F&6#\"\"$$\"'*3`%!\"'/&F&6#\"\"%$\"'7-xF2/&F&6# \"\"&$!'J*p%F2/&F&6#\"\"'$\"'?^:F2" }}{PARA 11 "" 1 "" {XPPMATH 20 "6) %1order~4~weights:G/&%#b*G6#\"\"\"$\"'T0!*!\"(/&F&6#\"\"$$\"'$4e%!\"'/ &F&6#\"\"%$\"'UrrF2/&F&6#\"\"&$!',tRF2/&F&6#\"\"'$\"'.#3\"F2/&F&6#\"\" ($\"'&4Q#F+" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/*&%)infinityG\"\"\"%8- norm~of~linking~coeffsGF&$\"+/Ooa5!\")" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/%L2-norm~of~principal~error~of~order~5~schemeG$\"+Ob:$)R!#8" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#/%L2-norm~of~principal~error~of~order~ 4~schemeG$\"+*QAe;\"!#7" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/%;order~5~ stability~intervalG7$$!)zc1L!\"(\"\"!" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/%;order~4~stability~intervalG7$$!)$3*fU!\"(\"\"!" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/%Pmax-min~principal~error~ratio~of~order~4~schemeG$ \"%Uf!\"#" }}{PARA 11 "" 1 "" {XPPMATH 20 "6$/&%\"BG6#\"\"'$\")LE`:!\" (/&%\"CGF&$\")M0\"o\"F*" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 2 "e2" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1269 "e2 := \{a[5,1] = 12078745127989699/5009699157786624 , a[4,1] = 344904825219069/345923838700000, a[4,3] = 40439332863108/10 810119959375, a[3,1] = 480815/6193536, `b*`[4] = 324806515311046773125 /452918159177876804664, `b*`[7] = 1/42, a[6,1] = 2251421737440863/8287 01767536000, b[1] = 2405713/26289000, c[6] = 1, c[7] = 1, b[2] = 0, `b *`[2] = 0, `b*`[3] = 5513862498202899713/12036555896794210600, a[7,2] \+ = 0, a[3,2] = 462875/2064512, a[5,3] = 39297175833216951/4694461413969 824, `b*`[1] = 1818563883019/20194131951000, a[6,5] = -26808346215168/ 82592376030125, a[7,5] = -1359311744/2892576375, a[5,4] = -10508413393 960625/54097233987826176, c[3] = 115/381, c[5] = 25/28, c[4] = 762/935 , a[7,4] = 454128848141375/589615117674696, a[7,6] = 256979/1656648, b [6] = 256979/1656648, b[4] = 454128848141375/589615117674696, `b*`[6] \+ = 137695258717/1272569071032, a[7,1] = 2405713/26289000, c[2] = 64/315 , a[5,2] = -791781731775/81669668864, a[4,2] = -2360077908867/60160667 6000, `b*`[5] = -126112324722496/317422653663375, a[2,1] = 64/315, a[7 ,3] = 63896466577779/141024193000600, a[6,4] = 12051135495733565/36492 943960723917, a[6,2] = -39895842357/3782730880, a[6,3] = 3456462868530 5112534/3916944972468643375, b[3] = 63896466577779/141024193000600, b[ 5] = -1359311744/2892576375\}:" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 137 "subs(e2,matrix([s eq([c[i],seq(a[i,j],j=1..i-1),``$(8-i)],i=2..7),\n[`b`,seq(b[i],i=1..6 ),``],[`b*`,seq(`b*`[i],i=1..7)]]));\n``;\nevalf(%%);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#K%'matrixG6#7*7*#\"#k\"$:$F(%!GF+F+F+F+F+7*#\"$:\" \"$\"Q#\"':3[\"(ON>'#\"'vGY\"(7X1#F+F+F+F+F+7*#\"$i(\"$N*#\"0p!>_#[!\\ M\"0++qQQ#fM#!.n)3z2gB\"-+gn1;g#\"/3J'GLR/%\"/v$f*>,\"3\"F+F+F+F+7*#\" #D\"#G#\"2*p*)z7X(y?\"\"1Cmyd\"*p4]#!-vC59#\"0v89[)GTX\"0'puw6:'*e#!+WI)Qc==\"/+5&>8%>?Fbo#\"48 (**G?)\\iQ^&\"5+1@%z'*ebO?\"#\"6DJxY5J:l![K\"6kY!o(y26CF;$!+5,$\\p*F;$\"+wd'4P)F;$!+\\Z]U>F*F+F+F+7*$ \"\"\"\"\"!$\"+id!or#F;$!+/Ooa5!\")$\"+ZXQC))F;$\"+\\$>BI$F*$!+)3ieC$F *F+F+7*FJ$\"+W^-^\"*F1$FLFL$\"+!)o)3`%F*$\"+ZN7-xF*$!+1:J*p%F*$\"+l&)> ^:F*F+7*%\"bGFYFenFfnFhnFjnF\\oF+7*%#b*G$\"+[vS0!*F1Fen$\"+9/$4e%F*$\" +otTrrF*$!+!f4I(RF*$\"+;e-#3\"F*$\"+\"Q_4Q#F1Q(pprint76\"" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }} }}{PARA 0 "" 0 "" {TEXT -1 57 "#-------------------------------------- ------------------" }}{SECT 1 {PARA 5 "" 0 "" {TEXT -1 13 "scheme with " }{XPPEDIT 18 0 "t[6] = 1/660;" "6#/&%\"tG6#\"\"'*&\"\"\"F)\"$g'!\" \"" }{TEXT -1 35 " and minimum principal error norm " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 91 "c_2 := 35 /159: c_3 := 42/131: c_4 := 131/143: c_5 := 21/22: bs_7 := 1/112:\ncal c_RKcoeffs();" }}{PARA 11 "" 1 "" {XPPMATH 20 "6'%'nodes:G/&%\"cG6#\" \"##\"#N\"$f\"/&F&6#\"\"$#\"#U\"$J\"/&F&6#\"\"%#F2\"$V\"/&F&6#\"\"&#\" #@\"#A" }}{PARA 11 "" 1 "" {XPPMATH 20 "6(%1order~5~weights:G/&%\"bG6# \"\"\"$\"'Pg'*!\"(/&F&6#\"\"$$\"'0Z[!\"'/&F&6#\"\"%$\"'s&)H!\"&/&F&6# \"\"&$!'1#)QF9/&F&6#\"\"'$\"'.:8F9" }}{PARA 11 "" 1 "" {XPPMATH 20 "6) %1order~4~weights:G/&%#b*G6#\"\"\"$\"'$H`*!\"(/&F&6#\"\"$$\"']!*[!\"'/ &F&6#\"\"%$\"'z#y#!\"&/&F&6#\"\"&$!'KWNF9/&F&6#\"\"'$\"'Bo6F9/&F&6#\" \"($\"'dG*)!\")" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/*&%)infinityG\"\" \"%8-norm~of~linking~coeffsGF&$\"+$zU5u)!\"*" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/%L2-norm~of~principal~error~of~order~5~schemeG$\"+8bsf =!#8" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/%L2-norm~of~principal~error~o f~order~4~schemeG$\"+6-gWd!#8" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/%;or der~5~stability~intervalG7$$!)_AFM!\"(\"\"!" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/%;order~4~stability~intervalG7$$!)4(Rm$!\"(\"\"!" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#/%Pmax-min~principal~error~ratio~of~or der~4~schemeG$\"%KJ!\"#" }}{PARA 11 "" 1 "" {XPPMATH 20 "6$/&%\"BG6#\" \"'$\")7x7 " 0 "" {MPLTEXT 1 0 1 ";" }}}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 2 "e2" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1045 "e2 := \{`b*`[6] = 3529851/3021550, a[3,1] = 7476/85805, `b*`[3] = 7589131232781673/15518096492661000, `b*`[1] = 2243660497/23535905400, a[3,2] = 20034/85805, `b*`[7] = 1/112, b[5] = -2234144/575505, c[6] = 1, c[7] = 1, b[2] = 0, `b*`[2] = 0, `b*`[5] = -6925033984/1953839475, a[6,5] = -144610048/1077690663, a[7,4] = 3490 43981/116904400, a[7,2] = 0, b[1] = 669707/6932520, b[3] = 22155229056 83/4570867891800, a[6,3] = 10587214001321/1373901639516, b[6] = 9363/7 120, `b*`[4] = 1104461697911/396890438000, a[7,3] = 2215522905683/4570 867891800, a[5,2] = -1698584121/245755840, a[4,1] = 2438549411/1983961 980, a[2,1] = 35/159, a[7,5] = -2234144/575505, a[4,2] = -3707256508/7 16430715, a[4,3] = 25077455105/5158301148, a[6,1] = 67512025387/324545 92380, c[2] = 35/159, c[3] = 42/131, c[4] = 131/143, c[5] = 21/22, a[6 ,4] = 731293420/8209319229, a[6,2] = -20051384/2293935, a[7,6] = 9363/ 7120, a[7,1] = 669707/6932520, a[5,4] = -927215289/26981535520, b[4] = 349043981/116904400, a[5,1] = 105337889067/64388030080, a[5,3] = 6869 523776931/1096562558080\}:" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}}{PARA 0 "" 0 "" {TEXT -1 0 " " }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 137 "subs(e2,matrix([seq([c[ i],seq(a[i,j],j=1..i-1),``$(8-i)],i=2..7),\n[`b`,seq(b[i],i=1..6),``], [`b*`,seq(`b*`[i],i=1..7)]]));\n``;\nevalf(%%);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#K%'matrixG6#7*7*#\"#N\"$f\"F(%!GF+F+F+F+F+7*#\"#U\"$J\" #\"%wu\"&0e)#\"&M+#F2F+F+F+F+F+7*#F/\"$V\"#\"+6%\\&QC\"+!)>'R)>#!+3lD2 P\"*:2V;(#\",0^Xx]#\"+[6Ie^F+F+F+F+7*#\"#@\"#A#\"-n!*)yL0\"\",!3I!)Qk# !+@Te)p\"\"*SevX##\".JpxB&po\".!3eDc'4\"#!**G:s#*\",?b`\")p#F+F+F+7*\" \"\"#\",(QD?^n\",!Q#faC$#!)%Q^+#\"(NRH##\"/@8+9se5\".;&R;!RP\"#\"*?MHJ (\"+H#>$4#)#!*[+hW\"\"+j1px5F+F+7*FR#\"'2(p'\"(?D$p\"\"!#\".$o0H_:A\". +=*y'3d%#\"*\")R/\\$\"*+W!p6#!(WTB#\"'0bd#\"%j$*\"%?rF+7*%\"bGF]oF`oFa oFdoFgoFjoF+7*%#b*G#\"+(\\gOC#\",+a!f`BF`o#\"1t;yK78*e(\"2+5m#\\'4=b\" #\".6zphW5\"\"-+!Q/*oR#!+%)R.Dp\"+v%RQ&>#\"(^)HN\"(]:-$#FR\"$7\"Q(ppri nt86\"" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%!G" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#K%'matrixG6#7*7*$\"+iyD,A!#5F(%!GF+F+F+F+F+7*$\"+qo51KF *$\"+w*yFr)!#6$\"+t*G[L#F*F+F+F+F+F+7*$\"+h\"R3;*F*$\"+?68H7!\"*$!+B(> Y<&F9$\"+>Ddh[F9F+F+F+F+7*$\"+XXXX&*F*$\"+&e&)fj\"F9$!+wMn6pF9$\"+a\") fkiF9$!+03[OMF1F+F+F+7*$\"\"\"\"\"!$\"+b%*>!3#F9$!+$zU5u)F9$\"+rv%fq(F 9$\"+u%)33*)F1$!+t6&=M\"F*F+F+7*FJ$\"+7)o.m*F1$FLFL$\"+T30Z[F*$\"+/:s& )HF9$!+m$e?)QF9$\"+!4G]J\"F9F+7*%\"bGFXFZFenFgnFinF[oF+7*%#b*G$\"+?i#H `*F1FZ$\"+fR]!*[F*$\"+pty#y#F9$!+[3KWNF9$\"+\\_Ao6F9$\"+H9dG*)!#7Q(ppr int96\"" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}}{PARA 0 "" 0 "" {TEXT -1 57 "#----------------- ---------------------------------------" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}}{PARA 0 "" 0 "" {TEXT -1 58 "#================= ========================================" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 31 "#==============================" } }{SECT 1 {PARA 4 "" 0 "" {TEXT -1 32 "Test-bed procedures for examples " }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 7 "RK5step" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 2432 "rk5step := proc(x_rk5step::realco ns)\n local c2,c3,c4,c5,c6,a21,a31,a32,a41,a42,a43,a51,a52,a53,a54, \n a61,a62,a63,a64,a65,f1,f2,f3,f4,f5,f6,b1,b2,b3,b4,b5,b6,\n xk,y k,t,jF,jM,jS,n,h,data,fn,xx,ys,saveDigits;\n options `Copyright 2004 by Peter Stone`;\n \n data := SOLN_;\n\n saveDigits := Digits; \n Digits := max(trunc(evalhf(Digits)),Digits+5);\n\n # procedure \+ to evaluate the slope field\n fn := proc(X_,Y_)\n local val; \n val := traperror(evalf(FXY_));\n if val=lasterror or not ty pe(val,numeric) then\n error \"evaluation of slope field faile d at %1\",evalf([X_,Y_],saveDigits);\n end if;\n val;\n \+ end proc;\n\n xx := evalf(x_rk5step);\n n := nops(data);\n\n if (data[1,1]data[n,1] or xxdata[1,1])) then\n err or \"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_; c5 := c5_; c6 := c6_; \n a21 := c2; a31 := a31_; a3 2 := a32_; \n a41 := a41_; a42 := a42_; a43 := a43_;\n a51 := a51_ ; a52 := a52_; a53 := a53_; a54 := a54_;\n a61 := a61_; a62 := a62_; a63 := a63_; a64 := a64_; a65 := a65_;\n b1 := b1_; b2 := b2_; b3 : = b3_; b4 := b4_; b5 := b5_; b6 := b6_; \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 := tr unc((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 i f;\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(x k + c2*h,yk + t*h);\n t := a31*f1 + a32*f2;\n f3 := fn(xk + c3*h,y k + t*h);\n t := a41*f1 + a42*f2 + a43*f3;\n f4 := fn(xk + c4*h,yk + t*h);\n t := a51*f1 + a52*f2 + a53*f3 + a54*f4;\n f5 := fn(xk + c5*h,yk + t*h);\n t := a61*f1 + a62*f2 + a63*f3 + a64*f4 + a65*f5; \n f6 := fn(xk + c6*h,yk + t*h);\n \n t := b1*f1 + b2*f2 + b3*f3 + b4*f4 + b5*f5 + b6*f6; \n ys := yk + t*h;\n\n evalf[saveDigi ts](ys);\nend proc: # of rk5step" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}}{SECT 1 {PARA 4 "" 0 " " {TEXT -1 29 "RK5_1 Prince-Dormand scheme " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 2304 "RK5_1 := proc(fx y,x,y,xx,yy,h,stps,bb)\n local c2,c3,c4,c5,c6,a21,a31,a32,a41,a42,a4 3,a51,a52,a53,a54,\n a61,a62,a63,a64,a65,f1,f2,f3,f4,f5,f6,b1,b2,b3, b4,b5,b6,\n 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([[1/5,1/5,0,0,0,0,0],\n \+ [3/10,3/40,9/40,0,0,0,0],\n [4/5,44/45,-56 /15,32/9,0,0,0],\n [8/9,19372/6561,-25360/2187,64448/65 61,-212/729,0,0],\n [1,9017/3168,-355/33,46732/5247,49/ 176,-5103/18656,0],\n [0,35/384,0,500/1113,125/192,-218 7/6784,11/84]]);\n \n c2 := evalf(A[1,1]);\n c3 := evalf(A[2,1]); \n c4 := evalf(A[3,1]);\n c5 := evalf(A[4,1]);\n c6 := evalf(A[5 ,1]);\n 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 := eval f(A[3,4]);\n a51 := evalf(A[4,2]);\n a52 := evalf(A[4,3]);\n a53 := evalf(A[4,4]);\n a54 := evalf(A[4,5]);\n a61 := evalf(A[5,2]); \n a62 := evalf(A[5,3]);\n a63 := evalf(A[5,4]);\n a64 := evalf( A[5,5]);\n a65 := evalf(A[5,6]);\n b1 := evalf(A[6,2]);\n b2 := \+ evalf(A[6,3]);\n b3 := evalf(A[6,4]);\n b4 := evalf(A[6,5]);\n b 5 := evalf(A[6,6]);\n b6 := evalf(A[6,7]);\n xk := evalf(xx);\n \+ yk := evalf(yy);\n soln := [xk,yk]; \n for k from 1 to stps do\n \+ f1 := fn(xk,yk);\n t := a21*f1;\n f2 := fn(xk + c2*h,yk \+ + t*h);\n t := a31*f1 + a32*f2;\n f3 := fn(xk + c3*h,yk + t* h);\n t := a41*f1 + a42*f2 + a43*f3;\n f4 := fn(xk + c4*h,yk + t*h);\n t := a51*f1 + a52*f2 + a53*f3 + a54*f4;\n f5 := f n(xk + c5*h,yk + t*h);\n t := a61*f1 + a62*f2 + a63*f3 + a64*f4 + a65*f5;\n f6 := fn(xk + c6*h,yk + t*h);\n \n yk := yk + ( b1*f1 + b2*f2 + b3*f3 + b4*f4 + b5*f5 + b6*f6)*h;\n xk := xk + h: \n soln := soln,[xk,yk];\n end do;\n if bb=true then\n e qns := \{SOLN_=[soln],FXY_=fxy,X_=x,Y_=y,c2_=c2,c3_=c3,\n c4_= c4,c5_=c5,c6_=c6,a31_=a31,a32_=a32,a41_=a41,\n a42_=a42,a43_=a 43,a51_=a51,a52_=a52,a53_=a53,a54_=a54,\n a61_=a61,a62_=a62,a6 3_=a63,a64_=a64,a65_=a65,\n b1_=b1,b2_=b2,b3_=b3,b4_=b4,b5_=b5 ,b6_=b6\};\n return subs(eqns,eval(rk5step)); \n else\n re turn evalf[saveDigits]([soln]);\n end if;\nend proc:" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}} {SECT 1 {PARA 4 "" 0 "" {TEXT -1 22 "RK5_2 A scheme with " } {XPPEDIT 18 0 "rho=rho[2]" "6#/%$rhoG&F$6#\"\"#" }{TEXT -1 1 " " }} {PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 3069 "RK5_2 := proc(fxy,x,y,xx,yy,h,stps,bb)\n local c2,c3,c4,c5,c6, a21,a31,a32,a41,a42,a43,a51,a52,a53,a54,\n a61,a62,a63,a64,a65,f1,f2 ,f3,f4,f5,f6,b1,b2,b3,b4,b5,b6,\n t,k,fn,xk,yk,soln,eqns,A,saveDigit s;\n\n saveDigits := Digits;\n Digits := max(trunc(evalhf(Digits)) ,Digits+5);\n\n fn := unapply(fxy,x,y);\n \n A := matrix([[17 /150,17/150,0,0,0,0,0],\n [230/1379,1424390/32327897,3967500/ 32327897,0,0,0,0],\n [1420370/2342387,11725892832745653435650/679 8789880492106984987,\n -2048128144537681132500/295599560021395 955869,39503685458474301158220/6798789880492106984987,0,0,0],\n [ 133/208, 12161603119957121528213173/25732152506856316150988800,\n \+ -1141151413305973125/551372154861039616,3683267576399581353585064 593/1765398936832403041633484800,\n 508180612800645990210779 0523/33857948797038081396772249600,0,0],\n [1,-778166632716677780 702867/2973121469222107001182300,104590105895625/68427659422081,\n \+ -6240811551117087470432064137/6600414167621042866395511800,\n \+ -963956247541399247752708263264927/631428144652436311985040 262094600,\n 4149774798413464954572800/1881639624293239507 076793,0],\n [0,10715793331/868982366000,0,380753067446365335491/ 1017424549568194596000,\n -33988229203965941757637160569/19 9599379574494873119383412000,\n 73328266119626752/10886206 1490007875,175006801591/1589096299500]]);\n \n c2 := evalf(A[1,1]); \n c3 := evalf(A[2,1]);\n c4 := evalf(A[3,1]);\n c5 := evalf(A[4 ,1]);\n c6 := evalf(A[5,1]);\n a21 := c2;\n a31 := evalf(A[2,2]) ;\n a32 := evalf(A[2,3]);\n a41 := evalf(A[3,2]);\n a42 := evalf (A[3,3]);\n a43 := evalf(A[3,4]);\n a51 := evalf(A[4,2]);\n a52 \+ := evalf(A[4,3]);\n a53 := evalf(A[4,4]);\n a54 := evalf(A[4,5]); \n a61 := evalf(A[5,2]);\n a62 := evalf(A[5,3]);\n a63 := evalf( A[5,4]);\n a64 := evalf(A[5,5]);\n a65 := evalf(A[5,6]);\n b1 := evalf(A[6,2]);\n b2 := evalf(A[6,3]);\n b3 := evalf(A[6,4]);\n \+ b4 := evalf(A[6,5]);\n b5 := evalf(A[6,6]);\n b6 := evalf(A[6,7]); \n xk := evalf(xx);\n yk := evalf(yy);\n soln := [xk,yk]; \n f or k from 1 to stps do\n f1 := fn(xk,yk);\n t := a21*f1;\n \+ f2 := fn(xk + c2*h,yk + t*h);\n t := a31*f1 + a32*f2;\n \+ f3 := fn(xk + c3*h,yk + t*h);\n t := a41*f1 + a42*f2 + a43*f3;\n \+ f4 := fn(xk + c4*h,yk + t*h);\n t := a51*f1 + a52*f2 + a53*f 3 + a54*f4;\n f5 := fn(xk + c5*h,yk + t*h);\n t := a61*f1 + \+ a62*f2 + a63*f3 + a64*f4 + a65*f5;\n f6 := fn(xk + c6*h,yk + t*h) ;\n \n yk := yk + (b1*f1 + b2*f2 + b3*f3 + b4*f4 + b5*f5 + b6*f 6)*h;\n xk := xk + h:\n soln := soln,[xk,yk];\n end do;\n \+ if bb=true then\n eqns := \{SOLN_=[soln],FXY_=fxy,X_=x,Y_=y,c2_ =c2,c3_=c3,\n c4_=c4,c5_=c5,c6_=c6,a31_=a31,a32_=a32,a41_=a41, \n a42_=a42,a43_=a43,a51_=a51,a52_=a52,a53_=a53,a54_=a54,\n \+ a61_=a61,a62_=a62,a63_=a63,a64_=a64,a65_=a65,\n b1_=b1,b 2_=b2,b3_=b3,b4_=b4,b5_=b5,b6_=b6\};\n return subs(eqns,eval(rk5s tep)); \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 22 "RK5_3 \+ A scheme with " }{XPPEDIT 18 0 "t[6] = 1/500;" "6#/&%\"tG6#\"\"'*&\" \"\"F)\"$+&!\"\"" }{TEXT -1 2 " " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 2639 "RK5_3 := proc(fxy,x,y,xx,y y,h,stps,bb)\n local c2,c3,c4,c5,c6,a21,a31,a32,a41,a42,a43,a51,a52, a53,a54,\n a61,a62,a63,a64,a65,f1,f2,f3,f4,f5,f6,b1,b2,b3,b4,b5,b6, \n t,k,fn,xk,yk,soln,eqns,A,saveDigits;\n\n saveDigits := Digits; \n Digits := max(trunc(evalhf(Digits)),Digits+5);\n\n fn := unappl y(fxy,x,y);\n \n A := matrix([[34/195,34/195,0,0,0,0,0],\n \+ [137/525,82063/1249500,243997/1249500,0,0,0,0],\n [252/365,1 01345382936/124124821241,-14189381388/4530102965,21949459056/730146007 3,0,0,0],\n [31/40,-2095505049916883/707267524239360,4985150207 31/40972513280,\n -2529970133596857/271730069595136,643102 0141871/7296710270976,0,0],\n [1, 789713415527611/1013168913843 996,-743202495/291283372,\n 625455957178350900/2710982960094 35219,21511957375165/104104742407029,1735464775680/6719560297073,0],\n [0, 990209/12842928,0,3027623484375/7555589800144,469280932525 /1389186738576,\n 43980800/446349501,406471/4735152]]);\n \+ \n c2 := evalf(A[1,1]);\n c3 := evalf(A[2,1]);\n c4 := evalf(A[3 ,1]);\n c5 := evalf(A[4,1]);\n c6 := evalf(A[5,1]);\n a21 := c2; \n a31 := evalf(A[2,2]);\n a32 := evalf(A[2,3]);\n a41 := evalf( A[3,2]);\n a42 := evalf(A[3,3]);\n a43 := evalf(A[3,4]);\n a51 : = evalf(A[4,2]);\n a52 := evalf(A[4,3]);\n a53 := evalf(A[4,4]);\n a54 := evalf(A[4,5]);\n a61 := evalf(A[5,2]);\n a62 := evalf(A[ 5,3]);\n a63 := evalf(A[5,4]);\n a64 := evalf(A[5,5]);\n a65 := \+ evalf(A[5,6]);\n b1 := evalf(A[6,2]);\n b2 := evalf(A[6,3]);\n b 3 := evalf(A[6,4]);\n b4 := evalf(A[6,5]);\n b5 := evalf(A[6,6]); \n b6 := evalf(A[6,7]);\n xk := evalf(xx);\n yk := evalf(yy);\n \+ soln := [xk,yk]; \n for k from 1 to stps do\n f1 := fn(xk,yk) ;\n t := a21*f1;\n f2 := fn(xk + c2*h,yk + t*h);\n t := a31*f1 + a32*f2;\n f3 := fn(xk + c3*h,yk + t*h);\n t := a41 *f1 + a42*f2 + a43*f3;\n f4 := fn(xk + c4*h,yk + t*h);\n t : = a51*f1 + a52*f2 + a53*f3 + a54*f4;\n f5 := fn(xk + c5*h,yk + t* h);\n t := a61*f1 + a62*f2 + a63*f3 + a64*f4 + a65*f5;\n f6 \+ := fn(xk + c6*h,yk + t*h);\n \n yk := yk + (b1*f1 + b2*f2 + b3* f3 + b4*f4 + b5*f5 + b6*f6)*h;\n xk := xk + h:\n soln := sol n,[xk,yk];\n end do;\n if bb=true then\n eqns := \{SOLN_=[sol n],FXY_=fxy,X_=x,Y_=y,c2_=c2,c3_=c3,\n c4_=c4,c5_=c5,c6_=c6,a3 1_=a31,a32_=a32,a41_=a41,\n a42_=a42,a43_=a43,a51_=a51,a52_=a5 2,a53_=a53,a54_=a54,\n a61_=a61,a62_=a62,a63_=a63,a64_=a64,a65 _=a65,\n b1_=b1,b2_=b2,b3_=b3,b4_=b4,b5_=b5,b6_=b6\};\n r eturn subs(eqns,eval(rk5step)); \n else\n return evalf[saveDigi ts]([soln]);\n end if;\nend proc:" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 22 "RK5_4 A scheme with " }{XPPEDIT 18 0 "t[6] = 1/600 ;" "6#/&%\"tG6#\"\"'*&\"\"\"F)\"$+'!\"\"" }{TEXT -1 2 " " }}{PARA 0 " " 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 2765 "RK5 _4 := proc(fxy,x,y,xx,yy,h,stps,bb)\n local c2,c3,c4,c5,c6,a21,a31,a 32,a41,a42,a43,a51,a52,a53,a54,\n a61,a62,a63,a64,a65,f1,f2,f3,f4,f5 ,f6,b1,b2,b3,b4,b5,b6,\n 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([[64/315,64/3 15,0,0,0,0,0],\n [115/381,480815/6193536,462875/2064512,0,0,0 ,0],\n [762/935,344904825219069/345923838700000,-236007790886 7/601606676000,\n 40439332863108/10810119959375,0,0,0] ,\n [25/28,12078745127989699/5009699157786624,-791781731775/8 1669668864,\n 39297175833216951/4694461413969824,-1050 8413393960625/54097233987826176,0,0],\n [1,2251421737440863/8 28701767536000,-39895842357/3782730880,\n 345646286853 05112534/3916944972468643375,12051135495733565/36492943960723917,\n \+ -26808346215168/82592376030125,0],\n [ 0, 2405713/26289000,0,63896466577779/141024193000600,\n \+ 454128848141375/589615117674696,-1359311744/2892576375,256979/165664 8]]);\n \n c2 := evalf(A[1,1]);\n c3 := evalf(A[2,1]);\n c4 := e valf(A[3,1]);\n c5 := evalf(A[4,1]);\n c6 := evalf(A[5,1]);\n a2 1 := c2;\n a31 := evalf(A[2,2]);\n a32 := evalf(A[2,3]);\n a41 : = evalf(A[3,2]);\n a42 := evalf(A[3,3]);\n a43 := evalf(A[3,4]);\n a51 := evalf(A[4,2]);\n a52 := evalf(A[4,3]);\n a53 := evalf(A[ 4,4]);\n a54 := evalf(A[4,5]);\n a61 := evalf(A[5,2]);\n a62 := \+ evalf(A[5,3]);\n a63 := evalf(A[5,4]);\n a64 := evalf(A[5,5]);\n \+ a65 := evalf(A[5,6]);\n b1 := evalf(A[6,2]);\n b2 := evalf(A[6,3] );\n b3 := evalf(A[6,4]);\n b4 := evalf(A[6,5]);\n b5 := evalf(A [6,6]);\n b6 := evalf(A[6,7]);\n xk := evalf(xx);\n yk := evalf( yy);\n soln := [xk,yk]; \n for k from 1 to stps do\n f1 := fn (xk,yk);\n t := a21*f1;\n f2 := fn(xk + c2*h,yk + t*h);\n \+ t := a31*f1 + a32*f2;\n f3 := fn(xk + c3*h,yk + t*h);\n t := a41*f1 + a42*f2 + a43*f3;\n f4 := fn(xk + c4*h,yk + t*h);\n \+ t := a51*f1 + a52*f2 + a53*f3 + a54*f4;\n f5 := fn(xk + c5*h, yk + t*h);\n t := a61*f1 + a62*f2 + a63*f3 + a64*f4 + a65*f5;\n \+ f6 := fn(xk + c6*h,yk + t*h);\n \n yk := yk + (b1*f1 + b2*f 2 + b3*f3 + b4*f4 + b5*f5 + b6*f6)*h;\n xk := xk + h:\n soln := soln,[xk,yk];\n end do;\n if bb=true then\n eqns := \{SOL N_=[soln],FXY_=fxy,X_=x,Y_=y,c2_=c2,c3_=c3,\n c4_=c4,c5_=c5,c6 _=c6,a31_=a31,a32_=a32,a41_=a41,\n a42_=a42,a43_=a43,a51_=a51, a52_=a52,a53_=a53,a54_=a54,\n a61_=a61,a62_=a62,a63_=a63,a64_= a64,a65_=a65,\n b1_=b1,b2_=b2,b3_=b3,b4_=b4,b5_=b5,b6_=b6\};\n return subs(eqns,eval(rk5step)); \n else\n return evalf[s aveDigits]([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 22 "RK5_5 A scheme with " }{XPPEDIT 18 0 "t [6] = 1/660;" "6#/&%\"tG6#\"\"'*&\"\"\"F)\"$g'!\"\"" }{TEXT -1 2 " " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 2559 "RK5_5 := proc(fxy,x,y,xx,yy,h,stps,bb)\n local c2,c3,c4,c5,c 6,a21,a31,a32,a41,a42,a43,a51,a52,a53,a54,\n a61,a62,a63,a64,a65,f1, f2,f3,f4,f5,f6,b1,b2,b3,b4,b5,b6,\n t,k,fn,xk,yk,soln,eqns,A,saveDig its;\n\n saveDigits := Digits;\n Digits := max(trunc(evalhf(Digits )),Digits+5);\n\n fn := unapply(fxy,x,y);\n \n A := matrix([[ 35/159,35/159,0,0,0,0,0],\n [42/131,7476/85805,20034/85805,0, 0,0,0],\n [131/143,2438549411/1983961980,-3707256508/71643071 5,25077455105/5158301148,0,0,0],\n [21/22,105337889067/643880 30080,-1698584121/245755840,\n 6869523776931/1096562 558080,-927215289/26981535520,0,0],\n [1,67512025387/32454592 380,-20051384/2293935,10587214001321/1373901639516,\n \+ 731293420/8209319229,-144610048/1077690663,0],\n [0,669707/ 6932520,0,2215522905683/4570867891800,349043981/116904400,-2234144/575 505,9363/7120]]);\n \n c2 := evalf(A[1,1]);\n c3 := evalf(A[2,1]); \n c4 := evalf(A[3,1]);\n c5 := evalf(A[4,1]);\n c6 := evalf(A[5 ,1]);\n 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 := eval f(A[3,4]);\n a51 := evalf(A[4,2]);\n a52 := evalf(A[4,3]);\n a53 := evalf(A[4,4]);\n a54 := evalf(A[4,5]);\n a61 := evalf(A[5,2]); \n a62 := evalf(A[5,3]);\n a63 := evalf(A[5,4]);\n a64 := evalf( A[5,5]);\n a65 := evalf(A[5,6]);\n b1 := evalf(A[6,2]);\n b2 := \+ evalf(A[6,3]);\n b3 := evalf(A[6,4]);\n b4 := evalf(A[6,5]);\n b 5 := evalf(A[6,6]);\n b6 := evalf(A[6,7]);\n xk := evalf(xx);\n \+ yk := evalf(yy);\n soln := [xk,yk]; \n for k from 1 to stps do\n \+ f1 := fn(xk,yk);\n t := a21*f1;\n f2 := fn(xk + c2*h,yk \+ + t*h);\n t := a31*f1 + a32*f2;\n f3 := fn(xk + c3*h,yk + t* h);\n t := a41*f1 + a42*f2 + a43*f3;\n f4 := fn(xk + c4*h,yk + t*h);\n t := a51*f1 + a52*f2 + a53*f3 + a54*f4;\n f5 := f n(xk + c5*h,yk + t*h);\n t := a61*f1 + a62*f2 + a63*f3 + a64*f4 + a65*f5;\n f6 := fn(xk + c6*h,yk + t*h);\n \n yk := yk + ( b1*f1 + b2*f2 + b3*f3 + b4*f4 + b5*f5 + b6*f6)*h;\n xk := xk + h: \n soln := soln,[xk,yk];\n end do;\n if bb=true then\n e qns := \{SOLN_=[soln],FXY_=fxy,X_=x,Y_=y,c2_=c2,c3_=c3,\n c4_= c4,c5_=c5,c6_=c6,a31_=a31,a32_=a32,a41_=a41,\n a42_=a42,a43_=a 43,a51_=a51,a52_=a52,a53_=a53,a54_=a54,\n a61_=a61,a62_=a62,a6 3_=a63,a64_=a64,a65_=a65,\n b1_=b1,b2_=b2,b3_=b3,b4_=b4,b5_=b5 ,b6_=b6\};\n return subs(eqns,eval(rk5step)); \n else\n re turn evalf[saveDigits]([soln]);\n end if;\nend proc:" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}}{SECT 1 {PARA 4 "" 0 " " {TEXT -1 20 "Testing the examples" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}{PARA 0 "" 0 "" {TEXT -1 80 "These tests do not \+ make use of the embedded order 4 method for error correction." }} {SECT 1 {PARA 4 "" 0 "" {TEXT -1 47 "Test 1 of 6 stage, order 5 Runge- Kutta methods " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "dy/dx=12*x*cos(4*x)*exp(-x)*y" "6#/*&%# dyG\"\"\"%#dxG!\"\"*,\"#7F&%\"xGF&-%$cosG6#*&\"\"%F&F+F&F&-%$expG6#,$F +F(F&%\"yGF&" }{TEXT -1 6 ", " }{XPPEDIT 18 0 "y(0)=1" "6#/-%\"yG6 #\"\"!\"\"\"" }{TEXT -1 1 " " }}{PARA 0 "" 0 "" {TEXT -1 10 "Solution: " }}{PARA 256 "" 0 "" {TEXT -1 2 " " }{XPPEDIT 18 0 "y=exp(-12/17*x* cos(4*x)*exp(-x)+180/289*exp(-x)*cos(4*x)+48/17*exp(-x)*sin(4*x)*x+96/ 289*exp(-x)*sin(4*x)-180/289)" "6#/%\"yG-%$expG6#,,*,\"#7\"\"\"\"# " 0 "" {MPLTEXT 1 0 229 "de := diff(y(x),x)=12*x*cos(4*x)*exp(-x)*y(x); \nic := y(0)=1;\ndsolve(\{de,ic\},y(x)):\ny(x)=simplify(numer(rhs(%))/ convert(denom(rhs(%)),exp));\nf := unapply(rhs(%),x):\nplot(f(x),x=0.. 5,0..1.45,font=[HELVETICA,9],labels=[`x`,`y(x)`]);" }}{PARA 11 "" 1 " " {XPPMATH 20 "6#>%#deG/-%%diffG6$-%\"yG6#%\"xGF,,$*,\"#7\"\"\"F,F0-%$ cosG6#,$*&\"\"%F0F,F0F0F0-%$expG6#,$F,!\"\"F0F)F0F0" }}{PARA 11 "" 1 " " {XPPMATH 20 "6#>%#icG/-%\"yG6#\"\"!\"\"\"" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/-%\"yG6#%\"xG-%$expG6#,,*&#\"#7\"#<\"\"\"*(F'F0-%$cosG 6#,$*&\"\"%F0F'F0F0F0-F)6#,$F'!\"\"F0F0F;*&#\"$!=\"$*GF0*&F2F0F8F0F0F0 *&#\"#[F/F0*(F8F0-%$sinGF4F0F'F0F0F0*&#\"#'*F?F0*&F8F0FEF0F0F0#F>F?F; " }}{PARA 13 "" 1 "" {GLPLOT2D 577 286 286 {PLOTDATA 2 "6&-%'CURVESG6$ 7er7$$\"\"!F)$\"\"\"F)7$$\"3gmmTN@Ki8!#>$\"3Fk>e\"G.6+\"!#<7$$\"3ALL$3 FWYs#F/$\"3!H*fm:2P/5F27$$\"3%)***\\iSmp3%F/$\"3Qn()\\Dat45F27$$\"3Wmm mT&)G\\aF/$\"34$Q7t`Dr,\"F27$$\"3m****\\7G$R<)F/$\"3S2-*\\9jw.\"F27$$ \"3GLLL3x&)*3\"!#=$\"3U([#>C\\El5F27$$\"3))**\\i!R(*Rc\"FJ$\"3>&=^@[0u 7\"F27$$\"3umm\"H2P\"Q?FJ$\"3k\\#o#G?)=?\"F27$$\"3!***\\PMnNrDFJ$\"3s_ j<)f!R*G\"F27$$\"3MLL$eRwX5$FJ$\"37'\\4u:c`O\"F27$$\"3_LLe*[`HP$FJ$\"3 [!\\'y0#yNR\"F27$$\"3rLLL$eI8k$FJ$\"3N\"Ha_9o@T\"F27$$\"3_L$3-8>bx$FJ$ \"3@))>@pAD<9F27$$\"3*QL$3xwq4RFJ$\"3a@g!fsi#>9F27$$\"3EM$eRA'*Q/%FJ$ \"3^DvP/8/=9F27$$\"33ML$3x%3yTFJ$\"3bF0p:\"oMT\"F27$$\"3h+]PfyG7ZFJ$\" 3e=U+Y19h8F27$$\"3emm\"z%4\\Y_FJ$\"3Yii#4W6uD\"F27$$\"3'QLL3FGT\\&FJ$ 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PMfE5F27$$\"3?+](=xpe=#F2$\"3ES3-I\\16#*FJ7$$\"3mLeRA9WRAF2$\"3IMhv&[? ^3)FJ7$$\"37nm\"H28IH#F2$\"3H\\m$Q)R4@qFJ7$$\"3$p;a8d3AM#F2$\"39j2HRJ+ ZhFJ7$$\"3um;zpSS\"R#F2$\"3#>07(=j$QR&FJ7$$\"3-+v$41oWW#F2$\"3QVRl9U0B ZFJ7$$\"3GLL3_?`(\\#F2$\"3/\\cKWs=$>%FJ7$$\"3AL3_D1l_DF2$\"3o6E$fFc$yP FJ7$$\"3fL$e*)>pxg#F2$\"3ym)*p(*f`&[$FJ7$$\"3%omm\"z+vbEF2$\"3jG&[,$f< =LFJ7$$\"33+]Pf4t.FF2$\"3%R>3YHT'HFF2$\"3/C%)f*f*e+K FJ7$$\"3om\"zWi^bv#F2$\"3-Gu$[oUh>$FJ7$$\"3)*\\7.d>Y\"y#F2$\"3#p*R$)o? n4KFJ7$$\"3uLLe*Gst!GF2$\"3>.X!=mk1C$FJ7$$\"3)om\"H2\"34'GF2$\"3'[>IF2$\"3a-&\\&*p%H, TFJ7$$\"3F+]i!RU07$F2$\"3'fkDHe#=P[FJ7$$\"3+++v=S2LKF2$\"3K%\\5FaXpw&F J7$$\"3Jmmm\"p)=MLF2$\"3))zmB`6`OlFJ7$$\"3GLLeR%p\")Q$F2$\"3#o,C;(=8fo FJ7$$\"3B++](=]@W$F2$\"3#G%=QV$\\;4(FJ7$$\"3C$ekyZ2mY$F2$\"3u,muc\"4C(FJ7$$\"3hTgx.2vFNF2$\"3/^M\"Q[;l C(FJ7$$\"35L$e*[$z*RNF2$\"3=wJ%fi2nC(FJ7$$\"3)*\\PMFwrmNF2$\"3R[i&\\xl (GsFJ7$$\"3%o;Hd!fX$f$F2$\"3IEKi0hy'=(FJ7$$\"3r$e9T=%>?OF2$\"3(>gS`&3d ArFJ7$$\"3e++]iC$pk$F2$\"3ma\\oRiHQqFJ7$$\"3ILe*[t\\sp$F2$\"3'e9/wG(3M oFJ7$$\"3[m;H2qcZPF2$\"3CYQ8S*3be'FJ7$$\"3O+]7.\"fF&QF2$\"3**Q8E[N&3+' FJ7$$\"3Ymm;/OgbRF2$\"3kN#z0%oN^aFJ7$$\"3w**\\ilAFjSF2$\"3[i8#)*p//*\\ FJ7$$\"3ym\"zW7@^6%F2$\"3>C%QCunR#[FJ7$$\"3yLLL$)*pp;%F2$\"3g*yCm#3E'p %FJ7$$\"3)QL3-$H**>UF2$\"3$*o:W?mr0YFJ7$$\"3)RL$3xe,tUF2$\"3!\\Bp&*))o Xb%FJ7$$\"3h+v=n(*fDVF2$\"3kIpK$)H$3a%FJ7$$\"3Cn;HdO=yVF2$\"3u&G6!oNOh XFJ7$$\"3MMe9\"z-lU%F2$\"3kC\">#=Lu2YFJ7$$\"3a+++D>#[Z%F2$\"3w_(eqj7vn %FJ7$$\"3SnmT&G!e&e%F2$\"3W>T$>g**p!\\FJ7$$\"3#RLLL)Qk%o%F2$\"3'yDBP_q :;&FJ7$$\"37+]iSjE!z%F2$\"3J;fP@m(pV&FJ7$$\"3a+]P40O\"*[F2$\"3!>+$=fU- gcFJ7$$\"\"&F)$\"3h(Q0fOqh\"eFJ-%'COLOURG6&%$RGBG$\"#5!\"\"F(F(-%+AXES LABELSG6$%\"xG%%y(x)G-%%FONTG6$%*HELVETICAG\"\"*-%%VIEWG6$;F(F]am;F($ \"$X\"!\"#" 1 2 0 1 10 0 2 9 1 4 2 1.000000 45.000000 45.000000 0 0 "C urve 1" }}}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 32 "The following code constructs a " }{TEXT 260 17 "discrete solution " }{TEXT -1 44 " based on each of the methods and gives the " }{TEXT 260 22 "root mean square error" }{TEXT -1 18 " of each solution." }} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 690 "F := (x,y) -> 12*x*cos(4*x) *exp(-x)*y: hh := 0.01: numsteps := 500: x0 := 0: y0 := 1:\nmatrix([[` slope field: `,F(x,y)],[`initial point: `,``(x0,y0)],[`step width: \+ `,hh],\n[`no. of steps: `,numsteps]]);``;\nmthds := [`Prince-Dorman d scheme`,`scheme with equal radii`,`scheme with `*t[6]=1/500,`scheme \+ with `*t[6]=1/600,\n`scheme with `*t[6]=1/660]: errs := []:\nDigits := 25:\nfor ct to 5 do\n Fn_RK5_||ct := RK5_||ct(F(x,y),x,y,x0,y0,hh,n umsteps,false);\n sm := 0: numpts := nops(Fn_RK5_||ct):\n for ii t o numpts do\n sm := sm+(Fn_RK5_||ct[ii,2]-f(Fn_RK5_||ct[ii,1]))^2 ;\n end do:\n errs := [op(errs),sqrt(sm/numpts)];\nend do:\nDigits := 10:\nlinalg[transpose]([mthds,evalf(errs)]);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#K%'matrixG6#7&7$%0slope~field:~~~G,$*,\"#7\"\"\"%\"xGF, -%$cosG6#,$*&\"\"%F,F-F,F,F,-%$expG6#,$F-!\"\"F,%\"yGF,F,7$%0initial~p oint:~G-%!G6$\"\"!F,7$%/step~width:~~~G$F,!\"#7$%1no.~of~steps:~~~G\"$ +&Q*pprint1116\"" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%!G" }}{PARA 11 " " 1 "" {XPPMATH 20 "6#K%'matrixG6#7'7$%6Prince-Dormand~schemeG$\"+&4X5 q\"!#?7$%8scheme~with~equal~radiiG$\"+d'zwM\"!#>7$/*&%-scheme~with~G\" \"\"&%\"tG6#\"\"'F5#F5\"$+&$\"+],dQ^F+7$/F3#F5\"$+'$\"+>VC.:F+7$/F3#F5 \"$g'$\"+npkCM!#@Q*pprint1126\"" }}}{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 0 621 " F := (x,y) -> 12*x*cos(4*x)*exp(-x)*y: hh := 0.01: numsteps := 500: x0 := 0: y0 := 1:\nmatrix([[`slope field: `,F(x,y)],[`initial point: ` ,``(x0,y0)],[`step width: `,hh],\n[`no. of steps: `,numsteps]]);`` ;\nmthds := [`Prince-Dormand scheme`,`scheme with equal radii`,`scheme with `*t[6]=1/500,`scheme with `*t[6]=1/600,\n`scheme with `*t[6]=1/6 60]: errs := []:\nDigits := 25:\nfor ct to 5 do\n fn_RK5_||ct := RK5 _||ct(F(x,y),x,y,x0,y0,hh,numsteps,true);\nend do:\nxx := 4.999: fxx : = evalf(f(xx)):\nfor ct to 5 do\n errs := [op(errs),abs(fn_RK5_||ct( xx)-fxx)];\nend do:\nDigits := 10:\nlinalg[transpose]([mthds,evalf(err s)]);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#K%'matrixG6#7&7$%0slope~field :~~~G,$*,\"#7\"\"\"%\"xGF,-%$cosG6#,$*&\"\"%F,F-F,F,F,-%$expG6#,$F-!\" \"F,%\"yGF,F,7$%0initial~point:~G-%!G6$\"\"!F,7$%/step~width:~~~G$F,! \"#7$%1no.~of~steps:~~~G\"$+&Q*pprint1136\"" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%!G" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#K%'matrixG6#7'7$ %6Prince-Dormand~schemeG$\"+jFY\"y\"!#?7$%8scheme~with~equal~radiiG$\" +&z`8X\"!#>7$/*&%-scheme~with~G\"\"\"&%\"tG6#\"\"'F5#F5\"$+&$\"+m7!\\[ &F+7$/F3#F5\"$+'$\"+,bgr:F+7$/F3#F5\"$g'$\"+]0#HQ$!#@Q*pprint1146\"" } }}{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 0 365 "mthds := [`Prince-Dormand s cheme`,`scheme with equal radii`,`scheme with `*t[6]=1/500,`scheme wit h `*t[6]=1/600,\n`scheme with `*t[6]=1/660]: errs := []:\nDigits := 20 :\nfor ct to 5 do\n sm := NCint((f(x)-'fn_RK5_||ct'(x))^2,x=0..5,ada ptive=false,numpoints=7,factor=200);\n errs := [op(errs),sqrt(sm/5)] ;\nend do:\nDigits := 10:\nlinalg[transpose]([mthds,evalf(errs)]);" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#K%'matrixG6#7'7$%6Prince-Dormand~schem eG$\"+R<7,7$/*&%-scheme~wi th~G\"\"\"&%\"tG6#\"\"'F5#F5\"$+&$\"+yk8S^F+7$/F3#F5\"$+'$\"+1w;.:F+7$ /F3#F5\"$g'$\"+DxH@M!#@Q*pprint1026\"" }}}{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 0 411 "evalf[20](plot(['fn_RK5_1'(x)-f(x) ,'fn_RK5_2'(x)-f(x),'fn_RK5_3'(x)-f(x),'fn_RK5_4'(x)-f(x),\n'fn_RK5_5' (x)-f(x)],x=0..5,color=[brown,red,COLOR(RGB,0,.7,0),COLOR(RGB,.5,0,1), COLOR(RGB,1,.35,0)],\nlegend=[`Prince-Dormand scheme`,`scheme with equ al radii`,`scheme with t[6]=1/500`,`scheme with t[6]=1/600`,\n`scheme \+ with t[6]=1/660`],font=[HELVETICA,9],title=`error curves for 6 stage o rder 5 Runge-Kutta methods`));" }}{PARA 13 "" 1 "" {GLPLOT2D 809 639 639 {PLOTDATA 2 "6+-%'CURVESG6%7er7$$\"\"!F)F(7$$\"5NLLL$3FWYs#!#@$\"( [$p9!#>7$$\"5qmmmmT&)G\\aF-$\"(rpK#F07$$\"5SLLL3x1h6oF-$\"(5[@#F07$$\" 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" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 32 "The following code constructs a " }{TEXT 260 17 "discrete solution" }{TEXT -1 44 " based on each of the methods and gives the \+ " }{TEXT 260 22 "root mean square error" }{TEXT -1 18 " of each soluti on." }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 693 "G := (x,y) -> x/y: h h := 0.05: numsteps := 200: x0 := 0: y0 := 1:\nmatrix([[`slope field: \+ `,G(x,y)],[`initial point: `,``(x0,y0)],[`step width: `,hh],\n[`no . of steps: `,numsteps]]);``;\nmthds := [`Prince-Dormand scheme`,`sc heme with equal radii`,`scheme with `*t[6]=1/500,`scheme with `*t[6]=1 /600,\n`scheme with `*t[6]=1/660]: errs := []:\nDigits := 20:\ng := x \+ -> sqrt(1+x^2):\nfor ct to 5 do\n Gn_RK5_||ct := RK5_||ct(G(x,y),x,y ,x0,y0,hh,numsteps,false);\n sm := 0: numpts := nops(Gn_RK5_||ct):\n for ii to numpts do\n sm := sm+(Gn_RK5_||ct[ii,2]-g(Gn_RK5_||c t[ii,1]))^2;\n end do:\n errs := [op(errs),sqrt(sm/numpts)];\nend \+ do:\nDigits := 10:\nlinalg[transpose]([mthds,evalf(errs)]);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#K%'matrixG6#7&7$%0slope~field:~~~G*&%\"xG\" \"\"%\"yG!\"\"7$%0initial~point:~G-%!G6$\"\"!F+7$%/step~width:~~~G$\" \"&!\"#7$%1no.~of~steps:~~~G\"$+#Q)pprint936\"" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%!G" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#K%'matrixG6#7'7$ %6Prince-Dormand~schemeG$\"+Y)>F!f!#A7$%8scheme~with~equal~radiiG$\"+g #)Gp()!#@7$/*&%-scheme~with~G\"\"\"&%\"tG6#\"\"'F5#F5\"$+&$\"+P2giTF07 $/F3#F5\"$+'$\"+2[u.sF+7$/F3#F5\"$g'$\"+)=nLr(F+Q)pprint946\"" }}} {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 "9.99;" "6#-%&FloatG6$\"$ ***!\"#" }{TEXT -1 16 " is also given." }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 623 "G := (x,y) -> x/y: hh := 0.05: numsteps := 200: x0 : = 0: y0 := 1:\nmatrix([[`slope field: `,G(x,y)],[`initial point: `,` `(x0,y0)],[`step width: `,hh],\n[`no. of steps: `,numsteps]]);``; \nmthds := [`Prince-Dormand scheme`,`scheme with equal radii`,`scheme \+ with `*t[6]=1/500,`scheme with `*t[6]=1/600,\n`scheme with `*t[6]=1/66 0]: errs := []:\nDigits := 20:\nfor ct to 5 do\n gn_RK5_||ct := RK5_ ||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 5 do\n errs := [op( errs),abs(gn_RK5_||ct(xx)-gxx)];\nend do:\nDigits := 10:\nlinalg[trans pose]([mthds,evalf(errs)]);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#K%'matr ixG6#7&7$%0slope~field:~~~G*&%\"xG\"\"\"%\"yG!\"\"7$%0initial~point:~G -%!G6$\"\"!F+7$%/step~width:~~~G$\"\"&!\"#7$%1no.~of~steps:~~~G\"$+#Q) pprint956\"" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%!G" }}{PARA 11 "" 1 " " {XPPMATH 20 "6#K%'matrixG6#7'7$%6Prince-Dormand~schemeG$\"'TF5!#=7$% 8scheme~with~equal~radiiG$\"(,%)e#F+7$/*&%-scheme~with~G\"\"\"&%\"tG6# \"\"'F4#F4\"$+&$\"('H69F+7$/F2#F4\"$+'$\"&^k&F+7$/F2#F4\"$g'$\"&=E)F+Q )pprint966\"" }}}{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$\"\"!\"#5" } {TEXT -1 82 " of each Runge-Kutta method is estimated as follows usin g the special procedure " }{TEXT 0 5 "NCint" }{TEXT -1 98 " to perfo rm numerical integration by the 7 point Newton-Cotes method over 100 e qual subintervals." }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 390 "mthds := [`Prince-Dormand scheme`,`scheme with equal radii`,`scheme with `* t[6]=1/500,`scheme with `*t[6]=1/600,\n`scheme with `*t[6]=1/660]: err s := []:\nDigits := 20:\ng := x -> sqrt(1+x^2):\nfor ct to 5 do\n sm := NCint((g(x)-'gn_RK5_||ct'(x))^2,x=0..10,adaptive=false,numpoints=7 ,factor=100);\n errs := [op(errs),sqrt(sm/10)];\nend do:\nDigits := \+ 10:\nlinalg[transpose]([mthds,evalf(errs)]);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#K%'matrixG6#7'7$%6Prince-Dormand~schemeG$\"+3d8*)e!#A7$ %8scheme~with~equal~radiiG$\"+Y+!eu)!#@7$/*&%-scheme~with~G\"\"\"&%\"t G6#\"\"'F5#F5\"$+&$\"+[UX^TF07$/F3#F5\"$+'$\"+%H[p=(F+7$/F3#F5\"$g'$\" +\"QaQp(F+Q)pprint976\"" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 " " 0 "" {TEXT -1 92 "The following error graphs are constructed using t he numerical procedures for the solutions." }}{EXCHG {PARA 0 "> " 0 " " {MPLTEXT 1 0 412 "evalf[20](plot([g(x)-'gn_RK5_1'(x),g(x)-'gn_RK5_2' (x),g(x)-'gn_RK5_3'(x),g(x)-'gn_RK5_4'(x),\ng(x)-'gn_RK5_5'(x)],x=0..1 0,color=[brown,red,COLOR(RGB,0,.7,0),COLOR(RGB,.5,0,1),COLOR(RGB,1,.35 ,0)],\nlegend=[`Prince-Dormand scheme`,`scheme with equal radii`,`sche me with t[6]=1/500`,`scheme with t[6]=1/600`,\n`scheme with t[6]=1/660 `],font=[HELVETICA,9],title=`error curves for 6 stage order 5 Runge-Ku tta methods`));" }}{PARA 13 "" 1 "" {GLPLOT2D 920 663 663 {PLOTDATA 2 "6+-%'CURVESG6%7\\q7$$\"\"!F)F(7$$\"5ILLL3x1h6o!#A$\"#Y!#>7$$\"5mmmmTN @Ki8!#@$\"%dHF07$$\"5******\\7.K[V?F4$\"&RO$F07$$\"5KLLL$3FWYs#F4$\"'& p)=F07$$\"5lmm;aQ`!eS$F4$\"'\"G=(F07$$\"5)******\\iSmp3%F4$\"(,#R@F07$ $\"5KLL$eRZF\"oZF4$\"((px`F07$$\"5lmmmmT&)G\\aF4$\"(g/9(F07$$\"5+++]P4 '\\/8'F4$\"(o&QrF07$$F,F4$\"($)*\\rF07$$\"5lmm;zWF07$$\"5mmm\"HdXqm c\"F_p$\")U>E>F07$$\"5++++]ilyM;F_p$\")>KC>F07$$\"5LLLLe*)4D2>F_p$\")! 3L/#F07$$\"5mmmmm;arz@F_p$\")[DNBF07$$\"5LLL$e*)4bQl#F_p$\")s9&f#F07$$ \"5++++D\"y%*z7$F_p$\")1F2FF07$$\"5mm;H#=qHlC$F_p$\")a%zp#F07$$\"5LLLe RAY1lLF_p$\")uY*o#F07$$\"5++](oHa*f$[$F_p$\")&*p$p#F07$$\"5mmm;ajW8-OF _p$\")\\p)o#F07$$\"5+++vo/V?RQF_p$\")*>rl#F07$$\"5LLLL$e9ui2%F_p$\")]p kDF07$$\"5mmm;H2Q\\4YF_p$\")?IeBF07$$\"5++++voMrU^F_p$\"),i-@F07$$\"5N LL$3-8Lfn&F_p$\")1'=#=F07$$\"5mmmmm\"z_\"4iF_p$\")l4N:F07$$\"5lmmm;zp! fu'F_p$\")9_b7F07$$\"5lmmmmm6m#G(F_p$\"(zE#**F07$$\"5lmmm;a`T>yF_p$\"( _E](F07$$\"5lmmmmT&phN)F_p$\"(+lJ&F07$$\"5ILL$3-js.*))F_p$\"(()>Q$F07$ $\"5++++v=ddC%*F_p$\"(yHo\"F07$$\"5lmm;H2)y(e**F_p$\"'A`?F07$$\"5LLLLe *=)H\\5F0$!(C$o5F07$$\"5++++v=JN[6F0$!(<\"HCF07$$\"5mmmm\"z/3uC\"F0$!( AjL$F07$$\"5++++vo3p)H\"F0$!(>!>PF07$$\"5LLLLe*ot*\\8F0$!(]k,%F07$$\"5 mmmmT5lD,9F0$!(DM?%F07$$\"5++++DJ$RDX\"F0$!(?.M%F07$$\"5LLLLekGhe:F0$! 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'Yh')F07$F]dl$!'y/%)F07$Fbdl$!'!o9)F07$Fgdl$!'?EzF07$F\\el$!';)p(F07$F ael$!':&\\(F07$Ffel$!')RH(F07$F[fl$!'R2rF07$F`fl$!'$>#pF07$Fefl$!'<_nF 07$Fjfl$!'!pe'F07$F_gl$!'vIkF07$Fdgl$!'h$H'F07$Figl$!'UVhF07$F^hl$!') \\,'F07$Fchl$!'!Q)eF07$Fhhl$!'[jdF07$F]il$!&&RcFail-F]hn6&Feil$\"\"&Fa hnF($\"\"\"F)-F\\jl6#%7scheme~with~t[6]=1/600G-F$6%7aqF'7$F+$\"#ZF07$F 2$\"%!)HF07$F8$\"&%)Q$F07$F=$\"'c**=F07$FB$\"'MDsF07$FG$\"(s)\\@F07$FL $\"(&f)R&F07$FQ$\"(`_;(F07$FV$\"(rL;(F07$Fen$\"(B^<(F07$Fin$\"(\"yisF0 7$F^o$\"(6`g(F07$Fco$\"(F07$Fer$\")#f2*>F07$Fjr$\")e%))) >F07$Fa]o$\")\\!)))>F07$Ff]o$\")IW(*>F07$F[^o$\")'\\A.#F07$F_s$\")C'*G @F07$Fc^o$\")M#3N#F07$Fh^o$\"):,yCF07$F]_o$\")LquCF07$Fds$\"):3sCF07$F e^m$\")a?!f#F07$Fis$\")kmHGF07$F]_m$\")X?uGF07$F^t$\")r8eIF07$Fct$\")b !*\\IF07$Fht$\")>UgIF07$F]u$\")[D]JF07$Fbu$\")^elJF07$$\"5LL$e9TQp1s$F _p$\"))yQ:$F07$Fgu$\")IFYJF07$$\"5mm;/ED#Rx&RF_p$\")*f&fJF07$F\\v$\")% [$oJF07$Fav$\")(*R!3$F07$Ffv$\")M[IHF07$F[w$\")-&*RFF07$F`w$\")V.FDF07 $Few$\")x_0BF07$Fjw$\")LA'3#F07$F_x$\"),'f(=F07$Fdx$\")Hwy;F07$Fix$\") q$y\\\"F07$F^y$\")I4L8F07$Fcy$\")>D%=\"F07$Fhy$\")w[]5F07$F]z$\"(6C()) F07$Fbz$\"(#4JwF07$Ff[l$\"(1J'eF07$F`\\l$\"(Aa&[F07$Fj\\l$\"(!p.UF07$F _]l$\"((*ys$F07$Fd]l$\"(cTT$F07$Fi]l$\"(W`7$F07$F^^l$\"(e:*GF07$Fc^l$ \"(N()p#F07$Fh^l$\"(l^a#F07$F]_l$\"(gZQ#F07$Fb_l$\"(%*QE#F07$Fg_l$\"(# oP@F07$F\\`l$\"(or.#F07$Fa`l$\"(Qr$>F07$Ff`l$\"(b/&=F07$F[al$\"(!zn\"F07$Fgdl$\"(\"4g6F07$F\\el $\"(-n7\"F07$Fael$\"(!)p4\"F07$Ffel$\"(Iv1\"F07$F[fl$\"(:-/\"F07$F`fl$ \"(mI,\"F07$Fefl$\"'<#))*F07$Fjfl$\"'ES'*F07$F_gl$\"'r6%*F07$Fdgl$\"') 4@*F07$Figl$\"'9\"**)F07$F^hl$\"'<.))F07$Fchl$\"'=6')F07$Fhhl$\"'1N%)F 07$F]il$\"&OD)Fail-F]hn6&FeilF[]p$\"#N!\"#F(-F\\jl6#%7scheme~with~t[6] =1/660G-%&TITLEG6#%Uerror~curves~for~6~stage~order~5~Runge-Kutta~metho dsG-%%FONTG6$%*HELVETICAG\"\"*-%+AXESLABELSG6$Q\"x6\"Q!Fhaq-%%VIEWG6$; F(F]il%(DEFAULTG" 1 2 0 1 10 0 2 9 1 4 2 1.000000 45.000000 45.000000 0 1 "Prince-Dormand scheme" "scheme with equal radii" "scheme with t[6 ]=1/500" "scheme with t[6]=1/600" "scheme with t[6]=1/660" }}}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 46 "Test 3 of 6 stage, order 5 Ru nge-Kutta methods" }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 " dy/dx = -x*y;" "6#/*&%#dyG\"\"\"%#dxG!\"\",$*&%\"xGF&%\"yGF&F(" } {TEXT -1 2 ", " }{XPPEDIT 18 0 "y(0)=1" "6#/-%\"yG6#\"\"!\"\"\"" } {TEXT -1 1 " " }}{PARA 0 "" 0 "" {TEXT -1 10 "Solution: " }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "y = exp(-x^2/2);" "6#/%\"yG-%$e xpG6#,$*&%\"xG\"\"#F+!\"\"F," }{TEXT -1 2 ". " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 32 "The following code constr ucts a " }{TEXT 260 17 "discrete solution" }{TEXT -1 44 " based on eac h of the methods and gives the " }{TEXT 260 22 "root mean square error " }{TEXT -1 18 " of each solution." }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 693 "H := (x,y) -> -x*y: hh := 0.1: numsteps := 100: x0 : = 0: y0 := 1:\nmatrix([[`slope field: `,H(x,y)],[`initial point: `,` `(x0,y0)],[`step width: `,hh],\n[`no. of steps: `,numsteps]]);``; \nmthds := [`Prince-Dormand scheme`,`scheme with equal radii`,`scheme \+ with `*t[6]=1/500,`scheme with `*t[6]=1/600,\n`scheme with `*t[6]=1/66 0]: errs := []:\nDigits := 20:\nh := x -> exp(-x^2/2):\nfor ct to 5 do \n Hn_RK5_||ct := RK5_||ct(H(x,y),x,y,x0,y0,hh,numsteps,false);\n \+ sm := 0: numpts := nops(Hn_RK5_||ct):\n for ii to numpts do\n s m := sm+(Hn_RK5_||ct[ii,2]-h(Hn_RK5_||ct[ii,1]))^2;\n end do:\n er rs := [op(errs),sqrt(sm/numpts)];\nend do:\nDigits := 10:\nlinalg[tran spose]([mthds,evalf(errs)]);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#K%'mat rixG6#7&7$%0slope~field:~~~G,$*&%\"xG\"\"\"%\"yGF,!\"\"7$%0initial~poi nt:~G-%!G6$\"\"!F,7$%/step~width:~~~G$F,F.7$%1no.~of~steps:~~~G\"$+\"Q )pprint886\"" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%!G" }}{PARA 11 "" 1 " " {XPPMATH 20 "6#K%'matrixG6#7'7$%6Prince-Dormand~schemeG$\"+dZw!Q&!#= 7$%8scheme~with~equal~radiiG$\"+'=2hQ#!#<7$/*&%-scheme~with~G\"\"\"&% \"tG6#\"\"'F5#F5\"$+&$\"+&[-D.\"F07$/F3#F5\"$+'$\"+5uJ\\`F+7$/F3#F5\"$ g'$\"+U'R[.$F+Q)pprint896\"" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 30 "The following code constructs " }{TEXT 260 20 " numerical procedures" }{TEXT -1 56 " for solutions based on each of th e Runge-Kutta schemes." }}{PARA 0 "" 0 "" {TEXT -1 75 "The error in th e value obtained by each of the methods at the point where " } {XPPEDIT 18 0 "9.99;" "6#-%&FloatG6$\"$***!\"#" }{TEXT -1 16 " is als o given." }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 623 "H := (x,y) -> - x*y: hh := 0.1: numsteps := 100: x0 := 0: y0 := 1:\nmatrix([[`slope fi eld: `,H(x,y)],[`initial point: `,``(x0,y0)],[`step width: `,hh], \n[`no. of steps: `,numsteps]]);``;\nmthds := [`Prince-Dormand schem e`,`scheme with equal radii`,`scheme with `*t[6]=1/500,`scheme with `* t[6]=1/600,\n`scheme with `*t[6]=1/660]: errs := []:\nDigits := 20:\nf or ct to 5 do\n hn_RK5_||ct := RK5_||ct(H(x,y),x,y,x0,y0,hh,numsteps ,true);\nend do:\nh := x -> exp(-x^2/2):\nxx := 9.99: hxx := evalf(h(x x)):\nfor ct to 5 do\n errs := [op(errs),abs(hn_RK5_||ct(xx)-hxx)]; \nend do:\nDigits := 10:\nlinalg[transpose]([mthds,evalf(errs)]);" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#K%'matrixG6#7&7$%0slope~field:~~~G,$*& %\"xG\"\"\"%\"yGF,!\"\"7$%0initial~point:~G-%!G6$\"\"!F,7$%/step~width :~~~G$F,F.7$%1no.~of~steps:~~~G\"$+\"Q)pprint146\"" }}{PARA 11 "" 1 " " {XPPMATH 20 "6#%!G" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#K%'matrixG6#7' 7$%6Prince-Dormand~schemeG$\"+nc[3J!#L7$%8scheme~with~equal~radiiG$\"+ gb'zA\"!#K7$/*&%-scheme~with~G\"\"\"&%\"tG6#\"\"'F5#F5\"$+&$\"+D.3haF+ 7$/F3#F5\"$+'$\"+)>w.6$F+7$/F3#F5\"$g'$\"+B'\\60#F+Q)pprint156\"" }}} {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 4 "The " } {TEXT 260 22 "root mean square error" }{TEXT -1 110 " over the interva l [0, 0.5] of each Runge-Kutta method is estimated as follows using \+ the special procedure " }{TEXT 0 5 "NCint" }{TEXT -1 97 " to perform numerical integration by the 7 point Newton-Cotes method over 50 equa l subintervals." }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 389 "mthds := [`Prince-Dormand scheme`,`scheme with equal radii`,`scheme with `*t[6 ]=1/500,`scheme with `*t[6]=1/600,\n`scheme with `*t[6]=1/660]: errs : = []:\nDigits := 20:\nh := x -> exp(-x^2/2):\nfor ct to 5 do\n sm := NCint((h(x)-'hn_RK5_||ct'(x))^2,x=0..10,adaptive=false,numpoints=7,fa ctor=50);\n errs := [op(errs),sqrt(sm/10)];\nend do:\nDigits := 10: \nlinalg[transpose]([mthds,evalf(errs)]);" }}{PARA 11 "" 1 "" {XPPMATH 20 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y$\"2;vO%=M%*4WF-7$Fey$\"2>#\\&F-7$Fh[l$\"2b\"=9@&=D+'F-7$F]\\l$\"2K&zFvtF.gF-7 $Fb\\l$\"2r4u&pq$p,'F-7$Fg\\l$\"2nqHB<)\\GgF-7$F\\]l$\"2sUvw)>[ZgF-7$F a]l$\"2iackj#pxgF-7$Ff]l$\"2(47/tD[ChF-7$F[^l$\"2!Hd$Q*y1+jF-7$F`^l$\" 2#eIV@vGblF-7$Fe^l$\"2*Rr9tVIblF-7$Fj^l$\"2jo-zu)QclF-7$F__l$\"2:$yO9k $*flF-7$Fd_l$\"22qC9S74d'F-7$Fi_l$\"2qm?)*Qs@e'F-7$F^`l$\"2+pEhU')**f' F-7$Fc`l$\"2pR/cM?ui'F-7$Fh`l$\"2VdHb6&oomF-7$F]al$\"2\")fXZ%3e\\oF-7$ Fbal$\"28E_u\\jL:(F-7$Fgal$\"2*>CvzTO`rF-7$F\\bl$\"2&f4U]yR`rF-7$Fabl$ \"2v*GSfix`rF-7$Ffbl$\"2&)Q6B$[)f:(F-7$F[cl$\"23Z>8`I\\;(F-7$F`cl$\"2C L@u&[l$>(F-7$Fecl$\"2([P3F\"*[.yF-7$Fjcl$\"2ekDp%32*\\)F-7$F_dl$\"2RLP Y%\\U`#*F-7$Fddl$\"3hn!RiC&y15F--F`jn6&F[elFbjo$\"#N!\"#FchmFael-Ffel6 #%7scheme~with~t[6]=1/660G-%&TITLEG6#%hnrelative~error~curves~for~6~st age~order~5~Runge-Kutta~methodsG-%%FONTG6$%*HELVETICAG\"\"*-%+AXESLABE LSG6$Q\"x6\"Q!Fd[q-%%VIEWG6$;F(Fddl%(DEFAULTG" 1 2 0 1 10 0 2 9 1 4 2 1.000000 45.000000 45.000000 0 1 "Prince-Dormand scheme" "scheme with \+ equal radii" "scheme with t[6]=1/500" "scheme with t[6]=1/600" "scheme with t[6]=1/660" }}}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 " > " 0 "" {MPLTEXT 1 0 1 ";" }}}}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 47 "T est 4 of 6 stage, order 5 Runge-Kutta methods " }}{PARA 0 "" 0 "" {TEXT -1 81 "F. G. Lether: Mathematics of Computation, Vol. 20, no. 95 , (July 1966) page 381. " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "dy/dx = -32*x*y*ln(2);" "6#/*&% #dyG\"\"\"%#dxG!\"\",$**\"#KF&%\"xGF&%\"yGF&-%#lnG6#\"\"#F&F(" }{TEXT -1 6 ", " }{XPPEDIT 18 0 "y(-1) = 1/8;" "6#/-%\"yG6#,$\"\"\"!\"\"* &F(F(\"\")F)" }{TEXT -1 1 " " }}{PARA 0 "" 0 "" {TEXT -1 10 "Solution: " }}{PARA 256 "" 0 "" {TEXT -1 2 " " }{XPPEDIT 18 0 "y = 2^(13-6*x^2 );" "6#/%\"yG)\"\"#,&\"#8\"\"\"*&\"\"'F)*$%\"xGF&F)!\"\"" }{TEXT -1 2 ". " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 191 "de := diff(y(x),x)=-32*x*y(x)*ln(2);\nic := y(-1)=1/ 8;\ndsolve(\{de,ic\},y(x)):\ny(x)=2^simplify(log[2](rhs(%)));\nk := un apply(rhs(%),x):\nplot(k(x),x=-1..1,font=[HELVETICA,9],labels=[`x`,`y( x)`]);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%#deG/-%%diffG6$-%\"yG6#%\" xGF,,$**\"#K\"\"\"F,F0F)F0-%#lnG6#\"\"#F0!\"\"" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%#icG/-%\"yG6#!\"\"#\"\"\"\"\")" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/-%\"yG6#%\"xG)\"\"#,&\"#8\"\"\"*&\"#;F,)F'F)F,!\"\"" } }{PARA 13 "" 1 "" {GLPLOT2D 577 286 286 {PLOTDATA 2 "6&-%'CURVESG6$7eo 7$$!\"\"\"\"!$\"3+++++++]7!#=7$$!3ommm;p0k&*F-$\"3!Hg[[\"[$)=KF-7$$!3w KL$3$3(F-7$$!3mmmmT%p\"e()F-$\"3!=E-TWD`l\"!#<7$$!3 :mmm\"4m(G$)F-$\"3M\"fONp()[t$F=7$$!3\"QLL3i.9!zF-$\"3A!e'4A%*Rg!)F=7$ $!3\"ommT!R=0vF-$\"3%z2Mbncie\"!#;7$$!3u****\\P8#\\4(F-$\"3C>dT>$)H#3$ FM7$$!3+nm;/siqmF-$\"3gp%*z`g)4*eFM7$$!3[++](y$pZiF-$\"3%R6L-Y$zz5!#:7 $$!33LLL$yaE\"eF-$\"3xvp\"p)==K>Fgn7$$!3hmmm\">s%HaF-$\"3dBW_P%Gb6$Fgn 7$$!3Q+++]$*4)*\\F-$\"3e;N4:OF ap7$$!3]++]PYx\"\\#F-$\"37]-4,Tp9TFap7$$!3QnmTNz>&H#F-$\"3y(*QMk^JnXFa p7$$!3EMLLL7i)4#F-$\"3yCPsPtXE]Fap7$$!3#pm;aVXH)=F-$\"3_cYryDpGbFap7$$ !3c****\\P'psm\"F-$\"38i9x*[!p=gFap7$$!3s*****\\F&*=Y\"F-$\"3K`b3X@Jjk Fap7$$!3')****\\74_c7F-$\"3co2Qfx9woFap7$$!3ZmmT5VBU5F-$\"3E!>K?nVAE(F ap7$$!3)3LLL3x%z#)!#>$\"3C'Q/NU&H#f(Fap7$$!3gKL$e9d;J'Fft$\"3@erx_1&z$ yFap7$$!3KMLL3s$QM%Fft$\"3<%pUH&HNA!)Fap7$$!3'ym;aQdDG$Fft$\"3%eWuwq(o %4)Fap7$$!3T,+]ivF@AFft$\"3[vW[$G&HZ\")Fap7$$!3=o;/^wj!p\"Fft$\"3]3j^O K2m\")Fap7$$!3'\\L$eRx**f6Fft$\"355#oX4%yz\")Fap7$$!3S<+D\"GyNH'!#?$\" 3#QE)R9AS)=)Fap7$$!3]^omm;zr)*!#@$\"3;#)*eHY6>>)Fap7$$\"3o'H$3x\"yY_%F jv$\"3Q>*>/AS,>)Fap7$$\"3&yK$3_Nl.5Fft$\"3eqLS$Q`G=)Fap7$$\"3/E$ekGR[b \"Fft$\"3-l$f@nl+<)Fap7$$\"3@CL$3-Dg5#Fft$\"3kJX?)*G!=:)Fap7$$\"3e?Le* ['R3KFft$\"3E'yGoI5!*4)Fap7$$\"3%pJL$ezw5VFft$\"3U_-I6(**[-)Fap7$$\"3L `mmmJ+IiFft$\"3%pB(\\hv&o%yFap7$$\"3s*)***\\PQ#\\\")Fft$\"3!QM&=wHL5wF ap7$$\"3ilm\"z\\1A-\"F-$\"3#*[#H(\\2i&H(Fap7$$\"3GKLLe\"*[H7F-$\"3))\\ \\;@heFpFap7$$\"3ylm;HCjV9F-$\"3)e+$\\9-Y,lFap7$$\"3I*******pvxl\"F-$ \"3S%z:5s)zRgFap7$$\"3g)***\\7JFn=F-$\"31))p(30[[c&Fap7$$\"3#z****\\_q n2#F-$\"3ae5F\"zuv2&Fap7$$\"3=)**\\P/q%zAF-$\"3ZUhzOe!Rg%Fap7$$\"3U)** *\\i&p@[#F-$\"3r&f%4uLbOTFap7$$\"3L)**\\(=GB2FF-$\"3WV]5@%**Rj$Fap7$$ \"3B)****\\2'HKHF-$\"3ul]=$GLo:$Fap7$$\"3uJL$3UDX8$F-$\"3sKZjodBbFFap7 $$\"3ElmmmZvOLF-$\"3!>\\-t_7IQ#Fap7$$\"3i******\\2goPF-$\"3Q>G9F7l&p\" Fap7$$\"3UKL$eR<*fTF-$\"3?\"Fap7$$\"3m******\\)Hxe%F-$\"3V-? C_;$p$zFgn7$$\"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$$\"3fKLLLb dQ()F-$\"3TxwT%Qu%>ei< " 0 "" {MPLTEXT 1 0 700 "K := (x,y) -> -32*x*y(x)*ln (2): hh := 0.01: numsteps := 200: x0 := -1: y0 := 1/8:\nmatrix([[`slop e field: `,K(x,y)],[`initial point: `,``(x0,y0)],[`step width: `,h h],\n[`no. of steps: `,numsteps]]);``;\nmthds := [`Prince-Dormand sc heme`,`scheme with equal radii`,`scheme with `*t[6]=1/500,`scheme with `*t[6]=1/600,\n`scheme with `*t[6]=1/660]: errs := []:\nDigits := 20: \nfor ct to 5 do\n Kn_RK5_||ct := RK5_||ct(evalf(K(x,y)),x,y,x0,eval f(y0),hh,numsteps,false);\n sm := 0: numpts := nops(Kn_RK5_||ct):\n \+ for ii to numpts do\n sm := sm+(Kn_RK5_||ct[ii,2]-k(Kn_RK5_||ct [ii,1]))^2;\n end do:\n errs := [op(errs),sqrt(sm/numpts)];\nend d o:\nDigits := 10:\nlinalg[transpose]([mthds,evalf(errs)]);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#K%'matrixG6#7&7$%0slope~field:~~~G,$**\"#K\" \"\"%\"xGF,-%\"yG6#F-F,-%#lnG6#\"\"#F,!\"\"7$%0initial~point:~G-%!G6$F 5#F,\"\")7$%/step~width:~~~G$F,!\"#7$%1no.~of~steps:~~~G\"$+#Q*pprint1 156\"" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%!G" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#K%'matrixG6#7'7$%6Prince-Dormand~schemeG$\"+!Q$4m6!#77$ %8scheme~with~equal~radiiG$\"+/\\+@vF+7$/*&%-scheme~with~G\"\"\"&%\"tG 6#\"\"'F4#F4\"$+&$\"+Qn/QGF+7$/F2#F4\"$+'$\"+<_3f6F+7$/F2#F4\"$g'$\"+u EaER!#8Q*pprint1166\"" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 30 "The following code constructs " }{TEXT 260 20 "numer ical procedures" }{TEXT -1 56 " for solutions based on each of the Run ge-Kutta schemes." }}{PARA 0 "" 0 "" {TEXT -1 75 "The error in the val ue obtained by each of the methods at the point where " }{XPPEDIT 18 0 "x = 0;" "6#/%\"xG\"\"!" }{TEXT -1 20 ".995 is also given." }} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 631 "K := (x,y) -> -32*x*y(x)*ln (2): hh := 0.01: numsteps := 200: x0 := -1: y0 := 1/8:\nmatrix([[`slop e field: `,K(x,y)],[`initial point: `,``(x0,y0)],[`step width: `,h h],\n[`no. of steps: `,numsteps]]);``;\nmthds := [`Prince-Dormand sc heme`,`scheme with equal radii`,`scheme with `*t[6]=1/500,`scheme with `*t[6]=1/600,\n`scheme with `*t[6]=1/660]: errs := []:\nDigits := 20: \nfor ct to 5 do\n kn_RK5_||ct := RK5_||ct(evalf(K(x,y)),x,y,x0,eval f(y0),hh,numsteps,true);\nend do:\nxx := 0.995: kxx := evalf(k(xx)):\n for ct to 5 do\n errs := [op(errs),abs(kn_RK5_||ct(xx)-kxx)];\nend d o:\nDigits := 10:\nlinalg[transpose]([mthds,evalf(errs)]);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#K%'matrixG6#7&7$%0slope~field:~~~G,$**\"#K\" \"\"%\"xGF,-%\"yG6#F-F,-%#lnG6#\"\"#F,!\"\"7$%0initial~point:~G-%!G6$F 5#F,\"\")7$%/step~width:~~~G$F,!\"#7$%1no.~of~steps:~~~G\"$+#Q*pprint1 176\"" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%!G" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#K%'matrixG6#7'7$%6Prince-Dormand~schemeG$\"+kbI48!#;7$% 8scheme~with~equal~radiiG$\"+Pm3QrF+7$/*&%-scheme~with~G\"\"\"&%\"tG6# \"\"'F4#F4\"$+&$\"+CQcRGF+7$/F2#F4\"$+'$\"+m1=08F+7$/F2#F4\"$g'$\"+s*3 !fg!# " 0 "" {MPLTEXT 1 0 366 "mt hds := [`Prince-Dormand scheme`,`scheme with equal radii`,`scheme with `*t[6]=1/500,`scheme with `*t[6]=1/600,\n`scheme with `*t[6]=1/660]: \+ errs := []:\nDigits := 20:\nfor ct to 5 do\n sm := NCint((k(x)-'kn_R K5_||ct'(x))^2,x=-1..1,adaptive=false,numpoints=7,factor=100);\n err s := [op(errs),sqrt(sm/2)];\nend do:\nDigits := 10:\nlinalg[transpose] ([mthds,evalf(errs)]);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#K%'matrixG6# 7'7$%6Prince-Dormand~schemeG$\"+d2+p6!#77$%8scheme~with~equal~radiiG$ \"+N\\wRvF+7$/*&%-scheme~with~G\"\"\"&%\"tG6#\"\"'F4#F4\"$+&$\"+pZ7XGF +7$/F2#F4\"$+'$\"+T^(>;\"F+7$/F2#F4\"$g'$\"+ZrKOR!#8Q)pprint856\"" }}} {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 92 "The follo wing error graphs are constructed using the numerical procedures for t he solutions." }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 433 "evalf[20]( plot(['kn_RK5_1'(x)-k(x),'kn_RK5_2'(x)-k(x),'kn_RK5_3'(x)-k(x),'kn_RK5 _4'(x)-k(x),\n'kn_RK5_5'(x)-k(x)],x=-1..1,\ncolor=[brown,red,COLOR(RGB ,0,.7,0),COLOR(RGB,.5,0,1),COLOR(RGB,1,.35,0)],thickness=[1$3,2,1],\nl egend=[`Prince-Dormand scheme`,`scheme with equal radii`,`scheme with \+ t[6]=1/500`,`scheme with t[6]=1/600`,\n`scheme with t[6]=1/660`],font= [HELVETICA,9],title=`error curves for 6 stage order 5 Runge-Kutta meth ods`));" }}{PARA 13 "" 1 "" {GLPLOT2D 818 571 571 {PLOTDATA 2 "6+-%'CU RVESG6&7eo7$$!\"\"\"\"!$F*F*7$$!5nmmmm;p0k&*!#?$\".*=t9XBDF/7$$!5LLLL$ 37$$!5nmmmm\"4 m(G$)F/$\".#*[`QR])F<7$$!5LLLL$3i.9!zF/$\"/)etO=>5#F<7$$!5mmmm;/R=0vF/ $\".kj=2PW%!#=7$$!5++++]P8#\\4(F/$\".'e$\\Vn7*FL7$$!5mmmm;/siqmF/$\"/l )eQ%\\2=FL7$$!5++++](y$pZiF/$\".IDyebR$!#<7$$!5LLLLL$yaE\"eF/$\".>Iqmj ='Ffn7$$!5mmmmm\">s%HaF/$\"/EG\"Q5u+\"Ffn7$$!5+++++]$*4)*\\F/$\"/X5'\\ 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" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 165 "d e := diff(y(x),x)=16/((16*x+1)*y(x));\nic := y(0)=1;\ndsolve(\{de,ic\} ,y(x));\ns := unapply(rhs(%),x):\nplot(s(x),x=0..0.5,0..2.6,font=[HELV ETICA,9],labels=[`x`,`y(x)`]);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%#d eG/-%%diffG6$-%\"yG6#%\"xGF,,$*(\"#;\"\"\",&*&F/F0F,F0F0F0F0!\"\"F)F3F 0" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%#icG/-%\"yG6#\"\"!\"\"\"" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#/-%\"yG6#%\"xG*$,&*&\"\"#\"\"\"-%#lnG6 #,&*&\"#;F,F'F,F,F,F,F,F,F,F,#F,F+" }}{PARA 13 "" 1 "" {GLPLOT2D 503 318 318 {PLOTDATA 2 "6&-%'CURVESG6$7U7$$\"\"!F)$\"\"\"F)7$$\"3WmmmT&)G \\a!#?$\"3EP,(Qyl.3\"!#<7$$\"3ILLL3x&)*3\"!#>$\"3?25A!pa&\\6F27$$\"3-+ ]i!R(*Rc\"F6$\"3oz*p77wF?\"F27$$\"3umm\"H2P\"Q?F6$\"3]_vibZz]7F27$$\"3 MLL$eRwX5$F6$\"3!Qb+fY'3W8F27$$\"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$)F6$\"3K!\\`od36k\"F 27$$\"3!))**\\7`l2Q*F6$\"3#HUv\"fmC$o\"F27$$\"3smm;/j$o/\"!#=$\"3:'H!f >cuAjU6Fco$\"3K$o8QC!za=F27$$\"3)*****\\P[6j9 Fco$\"39iuo+OIZ=F27$$\"3KL$e*[z(yb\"Fco$\"3Q:]fA\\>F27$$\"3))**\\iSj0 x=Fco$\"3-5Hbh&QF%>F27$$\"3Wmmm\"pW`(>Fco$\"3So#znsrC'>F27$$\"35+]i!f# =$3#Fco$\"3w)>Y)R!pI)>F27$$\"3/+](=xpe=#Fco$\"3?*eB@.[<+#F27$$\"3smm\" H28IH#Fco$\"3/Fyh^(\\.-#F27$$\"3km;zpSS\"R#Fco$\"3)4US+%ypO?F27$$\"3GL L3_?`(\\#Fco$\"3#4Cj+a0O0#F27$$\"3#HLe*)>pxg#Fco$\"3ab\\mG7Vq?F27$$\"3 u**\\Pf4t.FFco$\"3Cx7m@=^%3#F27$$\"32LLe*Gst!GFco$\"3Q>IFco$\"3&ocGC'[]F@ F27$$\"3h**\\i!RU07$Fco$\"3HCH$Q\")f.9#F27$$\"3b***\\(=S2LKFco$\"3C`wr Wc9a@F27$$\"3Kmmm\"p)=MLFco$\"3;=S,IA7m@F27$$\"3!*****\\(=]@W$Fco$\"3w 4%eC\"p]y@F27$$\"35L$e*[$z*RNFco$\"3UyOr,.R*=#F27$$\"3#*****\\iC$pk$Fc o$\"3wIdFs1%4?#F27$$\"39m;H2qcZPFco$\"3Qbx\"QY%\\6AF27$$\"3q**\\7.\"fF &QFco$\"3f+!e(oz@AAF27$$\"3Ymm;/OgbRFco$\"36qG(yA8CB#F27$$\"3y**\\ilAF jSFco$\"3v.zLgjzUAF27$$\"3YLLL$)*pp;%Fco$\"3IImU*yHDD#F27$$\"3?LL3xe,t UFco$\"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;L LL$)Qk%o%Fco$\"35e8d5)>wH#F27$$\"37+]iSjE!z%Fco$\"3e%4h.zwhI#F27$$\"35 +]P40O\"*[Fco$\"3Nwd2,K=9BF27$$\"3++++++++]Fco$\"3m'>())[`fABF2-%'COLO URG6&%$RGBG$\"#5!\"\"F(F(-%+AXESLABELSG6$%\"xG%%y(x)G-%%FONTG6$%*HELVE TICAG\"\"*-%%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 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 0 683 "S := (x,y) -> 16/((16*x+1)*y): hh := 0.005: numsteps := 100: \+ x0 := 0: y0 := 1:\nmatrix([[`slope field: `,S(x,y)],[`initial point: `,``(x0,y0)],[`step width: `,hh],\n[`no. of steps: `,numsteps]]); ``;\nmthds := [`Prince-Dormand scheme`,`scheme with equal radii`,`sche me with `*t[6]=1/500,`scheme with `*t[6]=1/600,\n`scheme with `*t[6]=1 /660]: errs := []:\nDigits := 20:\nfor ct to 5 do\n Sn_RK5_||ct := R K5_||ct(S(x,y),x,y,x0,y0,hh,numsteps,false);\n sm := 0: numpts := no ps(Sn_RK5_||ct):\n for ii to numpts do\n sm := sm+(Sn_RK5_||ct[ ii,2]-s(Sn_RK5_||ct[ii,1]))^2;\n end do:\n errs := [op(errs),sqrt( sm/numpts)];\nend do:\nDigits := 10:\nlinalg[transpose]([mthds,evalf(e rrs)]);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#K%'matrixG6#7&7$%0slope~fie ld:~~~G,$*(\"#;\"\"\",&*&F+F,%\"xGF,F,F,F,!\"\"%\"yGF0F,7$%0initial~po int:~G-%!G6$\"\"!F,7$%/step~width:~~~G$\"\"&!\"$7$%1no.~of~steps:~~~G \"$+\"Q)pprint766\"" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%!G" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#K%'matrixG6#7'7$%6Prince-Dormand~schemeG$\"+ $z#)*z>!#>7$%8scheme~with~equal~radiiG$\"+F:Y$)=!#=7$/*&%-scheme~with~ G\"\"\"&%\"tG6#\"\"'F5#F5\"$+&$\"+9y4LKF+7$/F3#F5\"$+'$\"+k#>d7$F+7$/F 3#F5\"$g'$\"+jS\"3+%F+Q)pprint776\"" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 30 "The following code constructs " }{TEXT 260 20 "numerical procedures" }{TEXT -1 56 " for solutions based on ea ch of the Runge-Kutta schemes." }}{PARA 0 "" 0 "" {TEXT -1 75 "The err or in the value obtained by each of the methods at the point where " }{XPPEDIT 18 0 "x = 0;" "6#/%\"xG\"\"!" }{TEXT -1 21 ".4995 is also g iven." }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 615 "S := (x,y) -> 16/( (16*x+1)*y): hh := 0.005: numsteps := 100: x0 := 0: y0 := 1:\nmatrix([ [`slope field: `,S(x,y)],[`initial point: `,``(x0,y0)],[`step width: `,hh],\n[`no. of steps: `,numsteps]]);``;\nmthds := [`Prince-Dorm and scheme`,`scheme with equal radii`,`scheme with `*t[6]=1/500,`schem e with `*t[6]=1/600,\n`scheme with `*t[6]=1/660]: errs := []:\nDigits \+ := 20:\nfor ct to 5 do\n sn_RK5_||ct := RK5_||ct(S(x,y),x,y,x0,y0,hh ,numsteps,true);\nend do:\nxx := 0.4995: sxx := evalf(s(xx)):\nfor ct \+ to 5 do\n errs := [op(errs),abs(sn_RK5_||ct(xx)-sxx)];\nend do:\nDig its := 10:\nlinalg[transpose]([mthds,evalf(errs)]);" }}{PARA 11 "" 1 " " {XPPMATH 20 "6#K%'matrixG6#7&7$%0slope~field:~~~G,$*(\"#;\"\"\",&*&F +F,%\"xGF,F,F,F,!\"\"%\"yGF0F,7$%0initial~point:~G-%!G6$\"\"!F,7$%/ste p~width:~~~G$\"\"&!\"$7$%1no.~of~steps:~~~G\"$+\"Q)pprint786\"" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#%!G" }}{PARA 11 "" 1 "" {XPPMATH 20 "6 #K%'matrixG6#7'7$%6Prince-Dormand~schemeG$\"++9%**o\"!#>7$%8scheme~wit h~equal~radiiG$\"+aG>#e\"!#=7$/*&%-scheme~with~G\"\"\"&%\"tG6#\"\"'F5# F5\"$+&$\"+HHe9EF+7$/F3#F5\"$+'$\"+nE,NEF+7$/F3#F5\"$g'$\"+N(=&RLF+Q)p print796\"" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 4 "The " }{TEXT 260 22 "root mean square error" }{TEXT -1 110 " ove r the interval [0, 0.5] of each Runge-Kutta method is estimated as f ollows 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 0 368 "mthds := [`Prince-Dormand scheme`,`scheme with equal radii`,`sche me with `*t[6]=1/500,`scheme with `*t[6]=1/600,\n`scheme with `*t[6]=1 /660]: errs := []:\nDigits := 20:\nfor ct to 5 do\n sm := NCint((s(x )-'sn_RK5_||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]([mthds,evalf(errs)]);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#K% 'matrixG6#7'7$%6Prince-Dormand~schemeG$\"+,'R(z>!#>7$%8scheme~with~equ al~radiiG$\"+jv-$)=!#=7$/*&%-scheme~with~G\"\"\"&%\"tG6#\"\"'F5#F5\"$+ &$\"+nY-KKF+7$/F3#F5\"$+'$\"+#)y4DJF+7$/F3#F5\"$g'$\"+OBy**RF+Q)pprint 806\"" }}}{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 0 413 " evalf[20](plot(['sn_RK5_1'(x)-s(x),'sn_RK5_2'(x)-s(x),'sn_RK5_3'(x)-s( x),'sn_RK5_4'(x)-s(x),\n'sn_RK5_5'(x)-s(x)],x=0..0.5,color=[brown,red, COLOR(RGB,0,.7,0),COLOR(RGB,.5,0,1),COLOR(RGB,1,.35,0)],\nlegend=[`Pri nce-Dormand scheme`,`scheme with equal radii`,`scheme with t[6]=1/500` ,`scheme with t[6]=1/600`,\n`scheme with t[6]=1/660`],font=[HELVETICA, 9],title=`error curves for 6 stage order 5 Runge-Kutta methods`));" }} {PARA 13 "" 1 "" {GLPLOT2D 875 539 539 {PLOTDATA 2 "6+-%'CURVESG6%7dq7 $$\"\"!F)F(7$$\"5SLLL3x1h6o!#B$\"&F.&!#>7$$\"5ommmTN@Ki8!#A$\"(yF07$$\"5omm;aQ`!eS$F4$\"*0Cc-$F07$$\"5NLLeRseQYPF4$ \"*,p\\R%F07$$\"5-+++D1k'p3%F4$\"*U'H>eF07$$\"5pmmT5SpaFWF4$\"*5%=+pF0 7$$\"5OLL$eRZF\"oZF4$\"*$p)y%pF07$$\"50++D\"y+3(3^F4$\"*O!eYeF07$$\"5q mmmmT&)G\\aF4$\"*\\s8#eF07$$\"50++]P4'\\/8'F4$\"*a!)fx&F07$$F,F4$\"*HE ,y&F07$$\"5qmm;zW*F4$\"+(3nu+\"F07$$\"5qmmm\"z%\\DO&*F4$\"+f[6*=\"F07$$\"5NLL3x\" [No()*F4$\"+tt'GS\"F07$$\"5+++Dc,;u@5!#@$\"+tDQ#[\"F07$$\"5nm;z%\\l*zb 5Fbr$\"+7q4x9F07$$\"5MLLLL3x&)*3\"Fbr$\"+56$>Z\"F07$$\"5nm\"z>'o^7\\6F 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O\"FI7$$\"3TM$3_5,-`%FI$\"3CFB-Gn\\(\\\"FI7$$\"3SnmT&G!e&e%FI$\"3t\\(p 9r/Xi\"FI7$$\"3m+]P%37^j%FI$\"3_eaMDR_K " 0 "" {MPLTEXT 1 0 695 "U := (x,y) -> (1+2*(x+1)*si n(3*x))*exp(-y): hh := 0.01: numsteps := 500: x0 := 0: y0 := 0:\nmatri x([[`slope field: `,U(x,y)],[`initial point: `,``(x0,y0)],[`step wid th: `,hh],\n[`no. of steps: `,numsteps]]);``;\nmthds := [`Prince-D ormand scheme`,`scheme with equal radii`,`scheme with `*t[6]=1/500,`sc heme with `*t[6]=1/600,\n`scheme with `*t[6]=1/660]: errs := []:\nDigi ts := 25:\nfor ct to 5 do\n Un_RK5_||ct := RK5_||ct(U(x,y),x,y,x0,y0 ,hh,numsteps,false);\n sm := 0: numpts := nops(Un_RK5_||ct):\n for ii to numpts do\n sm := sm+(Un_RK5_||ct[ii,2]-u(Un_RK5_||ct[ii,1 ]))^2;\n end do:\n errs := [op(errs),sqrt(sm/numpts)];\nend do:\nD igits := 10:\nlinalg[transpose]([mthds,evalf(errs)]);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#K%'matrixG6#7&7$%0slope~field:~~~G*&,&\"\"\"F+*(\" \"#F+,&%\"xGF+F+F+F+-%$sinG6#,$*&\"\"$F+F/F+F+F+F+F+-%$expG6#,$%\"yG! \"\"F+7$%0initial~point:~G-%!G6$\"\"!FA7$%/step~width:~~~G$F+!\"#7$%1n o.~of~steps:~~~G\"$+&Q)pprint646\"" }}{PARA 11 "" 1 "" {XPPMATH 20 "6# %!G" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#K%'matrixG6#7'7$%6Prince-Dorman d~schemeG$\"+HV!pc)!#A7$%8scheme~with~equal~radiiG$\"+%R][E$!#@7$/*&%- scheme~with~G\"\"\"&%\"tG6#\"\"'F5#F5\"$+&$\"+]9ve?F07$/F3#F5\"$+'$\"+ L#*yVnF+7$/F3#F5\"$g'$\"+8ap[CF+Q)pprint656\"" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 30 "The following code constr ucts " }{TEXT 260 20 "numerical procedures" }{TEXT -1 56 " for solutio ns based on each of the Runge-Kutta schemes." }}{PARA 0 "" 0 "" {TEXT -1 75 "The error in the value obtained by each of the methods at the p oint where " }{XPPEDIT 18 0 "x = 4.999;" "6#/%\"xG-%&FloatG6$\"%**\\! \"$" }{TEXT -1 16 " is also given." }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 626 "U := (x,y) -> (1+2*(x+1)*sin(3*x))*exp(-y): hh := 0. 01: numsteps := 500: x0 := 0: y0 := 0:\nmatrix([[`slope field: `,U(x ,y)],[`initial point: `,``(x0,y0)],[`step width: `,hh],\n[`no. of st eps: `,numsteps]]);``;\nmthds := [`Prince-Dormand scheme`,`scheme wi th equal radii`,`scheme with `*t[6]=1/500,`scheme with `*t[6]=1/600,\n `scheme with `*t[6]=1/660]: errs := []:\nDigits := 25:\nfor ct to 5 do \n un_RK5_||ct := RK5_||ct(U(x,y),x,y,x0,y0,hh,numsteps,true);\nend \+ do:\nxx := 4.999: uxx := evalf(u(xx)):\nfor ct to 5 do\n errs := [op (errs),abs(un_RK5_||ct(xx)-uxx)];\nend do:\nDigits := 10:\nlinalg[tran spose]([mthds,evalf(errs)]);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#K%'mat rixG6#7&7$%0slope~field:~~~G*&,&\"\"\"F+*(\"\"#F+,&%\"xGF+F+F+F+-%$sin G6#,$*&\"\"$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)ppri nt696\"" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%!G" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#K%'matrixG6#7'7$%6Prince-Dormand~schemeG$\"+R)yws(!#A7$ %8scheme~with~equal~radiiG$\"+qjFNE!#@7$/*&%-scheme~with~G\"\"\"&%\"tG 6#\"\"'F5#F5\"$+&$\"+n#pD\">F07$/F3#F5\"$+'$\"+a,SogF+7$/F3#F5\"$g'$\" +<>1d6F+Q)pprint706\"" }}}{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 u sing the special procedure " }{TEXT 0 5 "NCint" }{TEXT -1 98 " to pe rform numerical integration by the 7 point Newton-Cotes method over 20 0 equal subintervals." }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 365 "mt hds := [`Prince-Dormand scheme`,`scheme with equal radii`,`scheme with `*t[6]=1/500,`scheme with `*t[6]=1/600,\n`scheme with `*t[6]=1/660]: \+ errs := []:\nDigits := 20:\nfor ct to 5 do\n sm := NCint((u(x)-'un_R K5_||ct'(x))^2,x=0..5,adaptive=false,numpoints=7,factor=200);\n errs := [op(errs),sqrt(sm/5)];\nend do:\nDigits := 10:\nlinalg[transpose]( [mthds,evalf(errs)]);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#K%'matrixG6#7 '7$%6Prince-Dormand~schemeG$\"+>s8f&)!#A7$%8scheme~with~equal~radiiG$ \"+N\"ygC$!#@7$/*&%-scheme~with~G\"\"\"&%\"tG6#\"\"'F5#F5\"$+&$\"+TX%* \\?F07$/F3#F5\"$+'$\"+CfXXnF+7$/F3#F5\"$g'$\"+VP0dCF+Q)pprint716\"" }} }{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 92 "The foll owing error graphs are constructed using the numerical procedures for \+ the solutions." }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 411 "evalf[20] (plot(['un_RK5_1'(x)-u(x),'un_RK5_2'(x)-u(x),'un_RK5_3'(x)-u(x),'un_RK 5_4'(x)-u(x),\n'un_RK5_5'(x)-u(x)],x=0..5,color=[brown,red,COLOR(RGB,0 ,.7,0),COLOR(RGB,.5,0,1),COLOR(RGB,1,.35,0)],\nlegend=[`Prince-Dormand scheme`,`scheme with equal radii`,`scheme with t[6]=1/500`,`scheme wi th t[6]=1/600`,\n`scheme with t[6]=1/660`],font=[HELVETICA,9],title=`e rror curves for 6 stage order 5 Runge-Kutta methods`));" }}{PARA 13 " " 1 "" {GLPLOT2D 933 657 657 {PLOTDATA 2 "6+-%'CURVESG6%7_r7$$\"\"!F)F (7$$\"5NLLL$3FWYs#!#@$!(]x]'F-7$$\"5qmmmmT&)G\\aF-$!)w`v6F-7$$\"5SLLL3 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45.000000 0 0 "Curve 1" }}}}{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 0 690 "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 := [`Prince-Dormand scheme`,`scheme with eq ual radii`,`scheme with `*t[6]=1/500,`scheme with `*t[6]=1/600,\n`sche me with `*t[6]=1/660]: errs := []:\nDigits := 30:\nfor ct to 5 do\n \+ Vn_RK5_||ct := RK5_||ct(V(x,y),x,y,x0,y0,hh,numsteps,false);\n sm := 0: numpts := nops(Vn_RK5_||ct):\n for ii to numpts do\n sm := \+ sm+(Vn_RK5_||ct[ii,2]-v(Vn_RK5_||ct[ii,1]))^2;\n end do:\n errs := [op(errs),sqrt(sm/numpts)];\nend do:\nDigits := 10:\nlinalg[transpose ]([mthds,evalf(errs)]);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#K%'matrixG6 #7&7$%0slope~field:~~~G,$*&,&\"\"\"F,*&\"\"%F,-%$cosG6#,$*&\"\"$F,%\"x GF,F,F,F,F,,&%\"yGF,#F,F4!\"\"F,F97$%0initial~point:~G-%!G6$\"\"!F,7$% /step~width:~~~G$\"\"#!\"#7$%1no.~of~steps:~~~G\"$]#Q)pprint596\"" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#%!G" }}{PARA 11 "" 1 "" {XPPMATH 20 "6 #K%'matrixG6#7'7$%6Prince-Dormand~schemeG$\"+(z\"fnY!#>7$%8scheme~with ~equal~radiiG$\"+bDOgI!#=7$/*&%-scheme~with~G\"\"\"&%\"tG6#\"\"'F5#F5 \"$+&$\"+vR6r6F07$/F3#F5\"$+'$\"+Aw$4`%F+7$/F3#F5\"$g'$\"+\">2ph\"F+Q) pprint606\"" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 30 "The following code constructs " }{TEXT 260 20 "numerical proced ures" }{TEXT -1 56 " for solutions based on each of the Runge-Kutta sc hemes." }}{PARA 0 "" 0 "" {TEXT -1 75 "The error in the value obtained by each of the methods at the point where " }{XPPEDIT 18 0 "x = 4.99 9;" "6#/%\"xG-%&FloatG6$\"%**\\!\"$" }{TEXT -1 16 " is also given." } }{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 621 "V := (x,y) -> -(1+4*cos(3* x))*(y-1/3): hh := 0.02: numsteps := 250: x0 := 0: y0 := 1:\nmatrix([[ `slope field: `,V(x,y)],[`initial point: `,``(x0,y0)],[`step width: \+ `,hh],\n[`no. of steps: `,numsteps]]);``;\nmthds := [`Prince-Dorma nd scheme`,`scheme with equal radii`,`scheme with `*t[6]=1/500,`scheme with `*t[6]=1/600,\n`scheme with `*t[6]=1/660]: errs := []:\nDigits : = 30:\nfor ct to 5 do\n vn_RK5_||ct := RK5_||ct(V(x,y),x,y,x0,y0,hh, numsteps,true);\nend do:\nxx := 4.999: vxx := evalf(v(xx)):\nfor ct to 5 do\n errs := [op(errs),abs(vn_RK5_||ct(xx)-vxx)];\nend do:\nDigit s := 10:\nlinalg[transpose]([mthds,evalf(errs)]);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#K%'matrixG6#7&7$%0slope~field:~~~G,$*&,&\"\"\"F,*&\"\"% F,-%$cosG6#,$*&\"\"$F,%\"xGF,F,F,F,F,,&%\"yGF,#F,F4!\"\"F,F97$%0initia l~point:~G-%!G6$\"\"!F,7$%/step~width:~~~G$\"\"#!\"#7$%1no.~of~steps:~ ~~G\"$]#Q)pprint616\"" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%!G" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#K%'matrixG6#7'7$%6Prince-Dormand~schemeG$\"+ &)4s_@!#?7$%8scheme~with~equal~radiiG$\"+^&*4+8!#>7$/*&%-scheme~with~G \"\"\"&%\"tG6#\"\"'F5#F5\"$+&$\"+)=n'R^F+7$/F3#F5\"$+'$\"+E+a&3#F+7$/F 3#F5\"$g'$\"+3#))Q%z!#@Q)pprint626\"" }}}{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 integration by the 7 point Newton-Cotes \+ method over 100 equal subintervals." }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 365 "mthds := [`Prince-Dormand scheme`,`scheme with equal radii`,`scheme with `*t[6]=1/500,`scheme with `*t[6]=1/600,\n`scheme \+ with `*t[6]=1/660]: errs := []:\nDigits := 20:\nfor ct to 5 do\n sm \+ := NCint((v(x)-'vn_RK5_||ct'(x))^2,x=0..5,adaptive=false,numpoints=7,f actor=100);\n errs := [op(errs),sqrt(sm/5)];\nend do:\nDigits := 10: \nlinalg[transpose]([mthds,evalf(errs)]);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#K%'matrixG6#7'7$%6Prince-Dormand~schemeG$\"+Y75gY!#>7$% 8scheme~with~equal~radiiG$\"+;lTdI!#=7$/*&%-scheme~with~G\"\"\"&%\"tG6 #\"\"'F5#F5\"$+&$\"+N.$)p6F07$/F3#F5\"$+'$\"+.p>BXF+7$/F3#F5\"$g'$\"+* =AEh\"F+Q)pprint636\"" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 92 "The following error graphs are constructed using the numerical procedures for the solutions." }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 411 "evalf[20](plot(['vn_RK5_1'(x)-v(x),'vn_RK5_2'(x)-v(x ),'vn_RK5_3'(x)-v(x),'vn_RK5_4'(x)-v(x),\n'vn_RK5_5'(x)-v(x)],x=0..5,c olor=[brown,red,COLOR(RGB,0,.7,0),COLOR(RGB,.5,0,1),COLOR(RGB,1,.35,0) ],\nlegend=[`Prince-Dormand scheme`,`scheme with equal radii`,`scheme \+ with t[6]=1/500`,`scheme with t[6]=1/600`,\n`scheme with t[6]=1/660`], font=[HELVETICA,9],title=`error curves for 6 stage order 5 Runge-Kutta methods`));" }}{PARA 13 "" 1 "" {GLPLOT2D 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`^l$\",C#)[*)[\"F07$F\\fl$\"-xXAOW9F97$$\"5,+++D\"=EX8#F`^l$\"-?p-M!H \"F97$Fafl$\"-,-LVe6F97$Fffl$\",<+eM.)F97$F[gl$\",,Sw6a&F97$F`gl$\",bd P&4YF97$Fegl$\",rF))R&RF97$Fjgl$\",r)3F&\\$F97$F_hl$\",ZdLL@$F97$Fdhl$ \",D-aR3$F97$Fihl$\",_jlT/$F97$F^il$\",4CfH4$F97$Fcil$\",([vaMKF97$Fhi l$\",4y5J[$F97$F]jl$\",DsF)\\QF97$Fbjl$\",uX%Gv\\F97$Fgjl$\",4\"z\\-nF 97$F\\[m$\",h&HT\"R*F97$Fa[m$\"-.@9/E7F97$Ff[m$\"-@N'\\S^\"F97$F[\\m$ \"-uQVs6;F97$F`\\m$\"-S#>eqn\"F97$Fe\\m$\"-$)R)*4'o\"F97$Fj\\m$\"-xog* Qp\"F97$F_]m$\"-29-_)p\"F97$Fd]m$\"-!R79sp\"F97$Fi]m$\"-.G%HXp\"F97$F^ ^m$\"-S?#z$)o\"F97$Fc^m$\"-sx6Uw;F97$Fh^m$\"-pxhyi;F97$F]_m$\"-wDE#zd \"F97$Fb_m$\"-zO9<\\9F97$Fg_m$\"-NV?k&4\"F97$F\\`m$\",RKG*)f(F97$Fa`m$ \",#Q3IQ_F97$Ff`m$\",`pu6v$F97$F[am$\"-)>s9Jg#F-7$F`am$\"-Qr,XT\"F-7$Fjam$\",s!ym*R)F-7$F_bm$\",DK@3!pF-7$Fdbm$\",Z?C7S'F -7$Fibm$\",([cCFnF-7$F^cm$\",vM\"ffzF--Fgbp6&FecmFh[r$\"#N!\"#F(-F\\dm 6#%7scheme~with~t[6]=1/660G-%&TITLEG6#%Uerror~curves~for~6~stage~order ~5~Runge-Kutta~methodsG-%%FONTG6$%*HELVETICAG\"\"*-%+AXESLABELSG6$Q\"x 6\"Q!Fjis-%%VIEWG6$;F(F^cm%(DEFAULTG" 1 2 0 1 10 0 2 9 1 4 2 1.000000 45.000000 45.000000 0 1 "Prince-Dormand scheme" "scheme with equal rad ii" "scheme with t[6]=1/500" "scheme with t[6]=1/600" "scheme with t[6 ]=1/660" }}}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 46 "Test 8 of \+ 6 stage, order 5 Runge-Kutta methods" }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "dy/dx=x*(9-x^2)/(1+y^2)" "6#/*&%#dyG\"\"\"%#dxG!\" \"*(%\"xGF&,&\"\"*F&*$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+sqr t(64+9*x^8-324*x^6+2916*x^4))^(1/3);" "6#/-%$rhoG6#%\"xG),(*&\"#a\"\" \"*$F'\"\"#F,F,*&\"\"$F,*$F'\"\"%F,!\"\"-%%sqrtG6#,*\"#kF,*&\"\"*F,*$F '\"\")F,F,*&\"$C$F,*$F'\"\"'F,F3*&\"%;HF,*$F'F2F,F,F,*&F,F,F0F3" } {TEXT -1 2 ". " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 178 "de := diff(y(x),x)=x*(9-x^2)/(1+y(x)^2);\nic := y(0)=0;\ndsolve(\{de,ic\},y(x));\nw := unapply(rhs(%),x):\nplot(w(x), x=0..4,0..3.7,numpoints=75,font=[HELVETICA,9],labels=[`x`,`y(x)`]);" } }{PARA 11 "" 1 "" {XPPMATH 20 "6#>%#deG/-%%diffG6$-%\"yG6#%\"xGF,*(F, \"\"\",&\"\"*F.*$)F,\"\"#F.!\"\"F.,&F.F.*$)F)F3F.F.F4" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%#icG/-%\"yG6#\"\"!F)" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/-%\"yG6#%\"xG,&*&\"\"#!\"\",(*&\"\"$\"\"\")F'\"\"%F/F+ *&\"#aF/)F'F*F/F/*$,*\"#kF/*&\"\"*F/)F'\"\")F/F/*&\"$C$F/)F'\"\"'F/F+* &\"%;HF/F0F/F/#F/F*F/#F/F.F/*&F*F/F,#F+F.F+" }}{PARA 13 "" 1 "" {GLPLOT2D 503 318 318 {PLOTDATA 2 "6&-%'CURVESG6$7io7$$\"\"!F)F(7$$\"3 ()=*=*=*Qx#G!#>$\"3_LLtbH2)f$!#?7$$\"3uPy$y$yZbcF-$\"3ZF^'eEW*Q9F-7$$ \"3;_8N^$ye6)F-$\"3C$\"3aT8Yqv-h6F>7$$\"3oKCVKs3o@F>$\"3c?q**e5wz?F>7$$\"3$4\" 3\"3T.Ds#F>$\"3+#H`Y\")*G6KF>7$$\"3jy$y$y\"=lB$F>$\"3L\\!fpl0?S%F>7$$ \"3G(H(H(p](oPF>$\"3XzNO%Rmzr&F>7$$\"3/I(H(Hj=>VF>$\"3n)f#*4g%))4rF>7$ $\"373\"3\"3n&y'[F>$\"3[oK(4I9f[)F>7$$\"3oKCVK;BKaF>$\"3y!44&)e3J')*F> 7$$\"3etH(HPL$HfF>$\"3xCnB_*)['F>$\"3g[#>`Yr+B\" Fho7$$\"3Ul['[')o30(F>$\"3?yA^q$e=N\"Fho7$$\"3)yH(H(pzBf(F>$\"3)GF.C0: VY\"Fho7$$\"3m53\"3TBT3)F>$\"3ipKusL^i:Fho7$$\"3u\\'['[U&)o')F>$\"3_Io w_)\\Zn\"Fho7$$\"3ynvcnz>k\"*F>$\"31*=-!RxBm$\"3 +Ep.yT!)o=Fho7$$\"3\">*=*=tV]-\"Fho$\"3eT(\\)>LNc>Fho7$$\"3!z$y$y&G+\" 3\"Fho$\"3!4zI&*RE\"\\?Fho7$$\"37>*=*y\"*GM6Fho$\"3tVizMbXM@Fho7$$\"3' \\'['['y))*=\"Fho$\"3fa-R]1_?AFho7$$\"3s%f%f9[%4C\"Fho$\"3#y2[?x6qH#Fh o7$$\"3[KCVKm,'H\"Fho$\"3)*e,ywZ!pP#Fho7$$\"3c8N^t2A`8Fho$\"3WUCG,38dC Fho7$$\"3Y'['[Yr,.9Fho$\"3@#)=I[fvCDFho7$$\"3q8N^$f)zc9Fho$\"3%3DX;Fho$\"3iG:8:1'ez#Fho7$$\"3c%f%fuKqx;Fho$\"3 [Z4\"[C.I'GFho7$$\"3=^8N\"ft,t\"Fho$\"3!\\dJru-7#HFho7$$\"3EaS0ao>'y\" Fho$\"3!*)Rs,)f8\")HFho7$$\"3tcnvcA'p$=Fho$\"3`s'\\F\\'[LIFho7$$\"3363 \"3DiC*=Fho$\"3wC-.o]f)3$Fho7$$\"3$)******>MoW>Fho$\"3Mca,!\\@%QJFho7$ $\"3!*['[')ep#**>Fho$\"3Or<<>%)R)=$Fho7$$\"31Yf%faPE0#Fho$\"3!)Qt2wy;N KFho7$$\"3C^8N^)3&3@Fho$\"3Ik#>&=s*=G$Fho7$$\"3E>*=*e&>B;#Fho$\"3HYm&4 R?ZK$Fho7$$\"3_f%f%z([t@#Fho$\"3-!>y6]piO$Fho7$$\"3oq-FIB#>F#Fho$\"3%> b4%)\\*>0MFho7$$\"3vvcnb(p?K#Fho$\"35nW;n-%*QMFho7$$\"3eCVKkVazBFho$\" 3))))QO[-#QDFho$\"3]n/[Ud8hNFho7$$\"3AdnvO#pkf #Fho$\"3[Y'[5$oF(e$Fho7$$\"3'>;i@A4pk#Fho$\"3e\"*)HR3Wug$Fho7$$\"3$ovc n0fTq#Fho$\"3E1Ex&>gui$Fho7$$\"3;Yf%f1Ojv#Fho$\"3)*3G:eW%Hk$Fho7$$\"3[ aS09&4M\"GFho$\"3#G5xO+Znl$Fho7$$\"3OnvcZUliGFho$\"3?xe_?\\\"fm$Fho7$$ \"3163\"3DQ(=HFho$\"3I;`e!RAJn$Fho7$$\"3k*=*=HE\"H(HFho$\"3IP\"\\xLemn $Fho7$$\"3%pvcnh^q-$Fho$\"3e^O4S8lwOFho7$$\"3-dnvO9*43$Fho$\"3y,I^j&GH n$Fho7$$\"3m(H(H<2\"G8$Fho$\"3]\\w?2VllOFho7$$\"3'>*=*=tG))=$Fho$\"3#R \\Gi]bMl$Fho7$$\"3-A;ihz@UKFho$\"3q^o+4DNPOFho7$$\"3!>*=*=h2%)H$Fho$\" 3%>QOHhJ`h$Fho7$$\"3A;i@wFF\\LFho$\"3YxQ[8&f0f$Fho7$$\"3)\\'['[I)[0MFh o$\"3q>yn%)HPdNFho7$$\"3c0aS&)HLfMFho$\"3a59(yGT$>NFho7$$\"3['['[1m/8N Fho$\"3uI!*3pTpuMFho7$$\"3E#*=*=p]\"pNFho$\"3s<,a='3,U$Fho7$$\"37.Fq-Y #3i$Fho$\"3w<#3Q&zxhLFho7$$\"31wcnbdutOFho$\"3?^'pWBqHH$Fho7$$\"32dnv' *p'o+i*\\/FFFho7$$\"\"%F)$\"3CxC=rRoRDFho-%'COLOU RG6&%$RGBG$\"#5!\"\"F(F(-%+AXESLABELSG6$%\"xG%%y(x)G-%%FONTG6$%*HELVET ICAG\"\"*-%%VIEWG6$;F(F`cl;F($\"#PFjcl" 1 2 0 1 10 0 2 9 1 4 2 1.000000 45.000000 45.000000 0 0 "Curve 1" }}}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 32 "The following code constructs a " }{TEXT 260 17 "discrete solution" }{TEXT -1 44 " based on each of t he methods and gives the " }{TEXT 260 22 "root mean square error" } {TEXT -1 18 " of each solution." }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 684 "W := (x,y) -> x*(9-x^2)/(1+y^2): hh := 0.01: numsteps := 400: x0 := 0: y0 := 0:\nmatrix([[`slope field: `,W(x,y)],[`initial point : `,``(x0,y0)],[`step width: `,hh],\n[`no. of steps: `,numsteps]]) ;``;\nmthds := [`Prince-Dormand scheme`,`scheme with equal radii`,`sch eme with `*t[6]=1/500,`scheme with `*t[6]=1/600,\n`scheme with `*t[6]= 1/660]: errs := []:\nDigits := 30:\nfor ct to 5 do\n Wn_RK5_||ct := \+ RK5_||ct(W(x,y),x,y,x0,y0,hh,numsteps,false);\n sm := 0: numpts := n ops(Wn_RK5_||ct):\n for ii to numpts do\n sm := sm+(Wn_RK5_||ct [ii,2]-w(Wn_RK5_||ct[ii,1]))^2;\n end do:\n errs := [op(errs),sqrt (sm/numpts)];\nend do:\nDigits := 10:\nlinalg[transpose]([mthds,evalf( errs)]);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#K%'matrixG6#7&7$%0slope~fi eld:~~~G*(%\"xG\"\"\",&\"\"*F+*$)F*\"\"#F+!\"\"F+,&F+F+*$)%\"yGF0F+F+F 17$%0initial~point:~G-%!G6$\"\"!F;7$%/step~width:~~~G$F+!\"#7$%1no.~of ~steps:~~~G\"$+%Q)pprint136\"" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%!G" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#K%'matrixG6#7'7$%6Prince-Dormand~sch emeG$\"+$z.,*=!#A7$%8scheme~with~equal~radiiG$\"+R$G]@\"!#@7$/*&%-sche me~with~G\"\"\"&%\"tG6#\"\"'F5#F5\"$+&$\"+/WBJIF+7$/F3#F5\"$+'$\"+_LQg >F+7$/F3#F5\"$g'$\"+oc1J8F+Q)pprint146\"" }}}{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 = 3.499;" "6#/%\"xG-%&FloatG6$\"%*\\$!\"$" } {TEXT -1 16 " is also given." }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 615 "W := (x,y) -> x*(9-x^2)/(1+y^2): hh := 0.01: numsteps := 400: x 0 := 0: y0 := 0:\nmatrix([[`slope field: `,W(x,y)],[`initial point: \+ `,``(x0,y0)],[`step width: `,hh],\n[`no. of steps: `,numsteps]]);` `;\nmthds := [`Prince-Dormand scheme`,`scheme with equal radii`,`schem e with `*t[6]=1/500,`scheme with `*t[6]=1/600,\n`scheme with `*t[6]=1/ 660]: errs := []:\nDigits := 30:\nfor ct to 5 do\n wn_RK5_||ct := RK 5_||ct(W(x,y),x,y,x0,y0,hh,numsteps,true);\nend do:\nxx := 3.499: wxx \+ := evalf(w(xx)):\nfor ct to 5 do\n errs := [op(errs),abs(wn_RK5_||ct (xx)-wxx)];\nend do:\nDigits := 10:\nlinalg[transpose]([mthds,evalf(er rs)]);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#K%'matrixG6#7&7$%0slope~fiel d:~~~G*(%\"xG\"\"\",&\"\"*F+*$)F*\"\"#F+!\"\"F+,&F+F+*$)%\"yGF0F+F+F17 $%0initial~point:~G-%!G6$\"\"!F;7$%/step~width:~~~G$F+!\"#7$%1no.~of~s teps:~~~G\"$+%Q)pprint196\"" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%!G" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#K%'matrixG6#7'7$%6Prince-Dormand~schem eG$\"+0333#)!#C7$%8scheme~with~equal~radiiG$\"+A]^Z]!#B7$/*&%-scheme~w ith~G\"\"\"&%\"tG6#\"\"'F5#F5\"$+&$\"+5]8iSF+7$/F3#F5\"$+'$\"+p!HcI\"F 07$/F3#F5\"$g'$\"+5$zN0(F+Q)pprint206\"" }}}{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 "[0,4]" " 6#7$\"\"!\"\"%" }{TEXT -1 82 " of each Runge-Kutta method is estimate d as follows using the special procedure " }{TEXT 0 5 "NCint" }{TEXT -1 98 " to perform numerical integration by the 7 point Newton-Cotes \+ method over 200 equal subintervals." }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 365 "mthds := [`Prince-Dormand scheme`,`scheme with equal radii`,`scheme with `*t[6]=1/500,`scheme with `*t[6]=1/600,\n`scheme \+ with `*t[6]=1/660]: errs := []:\nDigits := 20:\nfor ct to 5 do\n sm \+ := NCint((w(x)-'wn_RK5_||ct'(x))^2,x=0..4,adaptive=false,numpoints=7,f actor=200);\n errs := [op(errs),sqrt(sm/4)];\nend do:\nDigits := 10: \nlinalg[transpose]([mthds,evalf(errs)]);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#K%'matrixG6#7'7$%6Prince-Dormand~schemeG$\"+m#[:r\"!#A7 $%8scheme~with~equal~radiiG$\"+OY)f>\"!#@7$/*&%-scheme~with~G\"\"\"&% \"tG6#\"\"'F5#F5\"$+&$\"+L'=t%HF+7$/F3#F5\"$+'$\"+n]gz " 0 " " {MPLTEXT 1 0 411 "evalf[20](plot([w(x)-'wn_RK5_1'(x),w(x)-'wn_RK5_2' (x),w(x)-'wn_RK5_3'(x),w(x)-'wn_RK5_4'(x),\nw(x)-'wn_RK5_5'(x)],x=0..4 ,color=[brown,red,COLOR(RGB,0,.7,0),COLOR(RGB,.5,0,1),COLOR(RGB,1,.35, 0)],\nlegend=[`Prince-Dormand scheme`,`scheme with equal radii`,`schem e with t[6]=1/500`,`scheme with t[6]=1/600`,\n`scheme with t[6]=1/660` ],font=[HELVETICA,9],title=`error curves for 6 stage order 5 Runge-Kut ta methods`));" }}{PARA 13 "" 1 "" {GLPLOT2D 972 759 759 {PLOTDATA 2 " 6+-%'CURVESG6%7jp7$$\"\"!F)F(7$$\"5LLLLLL3VfV!#@$!+^4^7F9$!*# Q@-NF-7$$\"5LLL$e*[e-Y8F9$!*+A]f$F-7$$\"5mmmmT&y`3W\"F9$!*v!=:OF-7$$\" 5LLLek`xE)[\"F9$!*I.pe$F-7$$\"5+++](=s\"oN:F9$!)M5^NF97$$\"5mmmT5!p&4$ e\"F9$!)Qr*\\$F97$$\"5LLLLLe'40j\"F9$!)y2$R$F97$$\"5mmmm\"H_(zV=F9$!)^ (>%F9$!(tS\"))F97$$\"5mmmmm;EI&R%F9$!)z\"*[9F97$$\"5+++++vCT$f% F9$!)KQ@=F97$$\"5mmmm;/uY#p%F9$!)fdS>F97$$\"5LLLLLLB_\"z%F9$!)dF??F97$ $\"5++++]isd!*[F9$!)n)\\1#F97$$\"5mmmmm\">K'*)\\F9$!)QPz?F97$$\"5NLLLL Ly>#4&F9$!)$y'p?F97$$\"5+++++vMw%>&F9$!)qHL?F97$$\"5qmmmm;\"HtH&F9$!)F 9v>F97$$\"5NLLLLeZ*)*R&F9$!)')p**=F97$$\"5qmmmmTg-0cF9$!(a(G7$$\"5 +++++Dt:5eF9$!(EMa\"F`w7$$\"5mmmmm\"fX(emF9$!'z`')F`w7$$\"5NLLLL3!z;3( F9$!':NhF`w7$$\"5+++++DCh/vF9$!'dJTF`w7$$\"5lmmm;H9lRzF9$!'WlGF`w7$$\" 5LLLLLL/pu$)F9$!'#>$>F`w7$$\"5mmmmm;c0T\"*F9$!&%Q&)F`w7$$\"5+++++I,Q+5 F`w$!&nL#F`w7$$\"5+++++]*3q3\"F`w$\"%RPF`w7$$\"5+++++q=\\q6F`w$\"&,x\" 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" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 172 "de := diff(y(x),x )=-(1+cos(2*x))*y(x)^3;\nic := y(0)=sqrt(2);\ndsolve(\{de,ic\},y(x)); \nm := unapply(rhs(%),x):\nplot(m(x),x=0..3,0..1.42,font=[HELVETICA,9] ,labels=[`x`,`y(x)`]);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%#deG/-%%di ffG6$-%\"yG6#%\"xGF,,$*&,&\"\"\"F0-%$cosG6#,$*&\"\"#F0F,F0F0F0F0)F)\" \"$F0!\"\"" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%#icG/-%\"yG6#\"\"!*$\" \"##\"\"\"F+" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/-%\"yG6#%\"xG*&\"\"\" F)*$,(*(\"\"#F)-%$cosGF&F)-%$sinGF&F)F)*&F-F)F'F)F)#F)F-F)#F)F-!\"\"" }}{PARA 13 "" 1 "" {GLPLOT2D 503 318 318 {PLOTDATA 2 "6&-%'CURVESG6$7Y 7$$\"\"!F)$\"3:&4tBc8UT\"!#<7$$\"3$*****\\ilyM;!#>$\"3ozW7@k#*H8F,7$$ \"3')*****\\7t&pKF0$\"3!G<)\\ef9f7F,7$$\"3z****\\(ofV!\\F0$\"3oe\"F,7$$\"3s******\\i9RlF0$\"3kESFh\"zh9\"F,7$$\"33++vVV)RQ*F0$\"3'f)* )e-w\\p5F,7$$\"3/++vVA)GA\"!#=$\"3V)o6<$fq15F,7$$\"3;+]iSS\"Ga\"FJ$\"3 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)**\\P4w)R7#F,$\"3!Qwx>a)*Q4&FJ7$$\"3;++]x%f\")=#F,$\"3q$pQbJ#)G/&FJ7$ $\"3!)**\\P/-a[AF,$\"3gJla\"HTu)\\FJ7$$\"3/+](=Yb;J#F,$\"3c:[>;?IA\\FJ 7$$\"3')****\\i@OtBF,$\"3m09))4iC_[FJ7$$\"3')**\\PfL'zV#F,$\"3%Gjf])o8 tZFJ7$$\"3>+++!*>=+DF,$\"3[G/4+_V#p%FJ7$$\"3-++DE&4Qc#F,$\"3!**R*=7x[1 YFJ7$$\"3=+]P%>5pi#F,$\"3f7E:iH**=XFJ7$$\"39+++bJ*[o#F,$\"3cgVvc$ovV%F J7$$\"33++Dr\"[8v#F,$\"3Ln\\jDQ5WVFJ7$$\"3++++Ijy5GF,$\"3OZ!)Q%zK7E%FJ 7$$\"31+]P/)fT(GF,$\"3)*4_&egIW<%FJ7$$\"31+]i0j\"[$HF,$\"3qns]&)H\\$4% FJ7$$\"\"$F)$\"3ntdq;jW4SFJ-%'COLOURG6&%$RGBG$\"#5!\"\"F(F(-%+AXESLABE LSG6$%\"xG%%y(x)G-%%FONTG6$%*HELVETICAG\"\"*-%%VIEWG6$;F(Fc\\l;F($\"$U \"!\"#" 1 2 0 1 10 0 2 9 1 4 2 1.000000 45.000000 45.000000 0 0 "Curve 1" }}}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 32 "T he following code constructs a " }{TEXT 260 17 "discrete solution" } {TEXT -1 44 " based on each of the methods and gives the " }{TEXT 260 22 "root mean square error" }{TEXT -1 18 " of each solution." }} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 697 "M := (x,y) -> -(1+cos(2*x)) *y^3: hh := 0.01: numsteps := 300: x0 := 0: y0 := sqrt(2):\nmatrix([[` slope field: `,M(x,y)],[`initial point: `,``(x0,y0)],[`step width: \+ `,hh],\n[`no. of steps: `,numsteps]]);``;\nmthds := [`Prince-Dorman d scheme`,`scheme with equal radii`,`scheme with `*t[6]=1/500,`scheme \+ with `*t[6]=1/600,\n`scheme with `*t[6]=1/660]: errs := []:\nDigits := 20:\nfor ct to 5 do\n Mn_RK5_||ct := RK5_||ct(M(x,y),x,y,x0,evalf(y 0),hh,numsteps,false);\n sm := 0: numpts := nops(Mn_RK5_||ct):\n f or ii to numpts do\n sm := sm+(Mn_RK5_||ct[ii,2]-m(Mn_RK5_||ct[ii ,1]))^2;\n end do:\n errs := [op(errs),sqrt(sm/numpts)];\nend do: \nDigits := 10:\nlinalg[transpose]([mthds,evalf(errs)]);" }}{PARA 11 " " 1 "" {XPPMATH 20 "6#K%'matrixG6#7&7$%0slope~field:~~~G,$*&,&\"\"\"F, -%$cosG6#,$*&\"\"#F,%\"xGF,F,F,F,)%\"yG\"\"$F,!\"\"7$%0initial~point:~ G-%!G6$\"\"!*$F2#F,F27$%/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$%6Prince-Dormand~schemeG$\"+NGtx B!#>7$%8scheme~with~equal~radiiG$\"+&y,l0#F+7$/*&%-scheme~with~G\"\"\" &%\"tG6#\"\"'F4#F4\"$+&$\"+$R'e)4%!#@7$/F2#F4\"$+'$\"+j)RB.#F+7$/F2#F4 \"$g'$\"+*Gl*[))!#?Q)pprint286\"" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }} {PARA 0 "" 0 "" {TEXT -1 30 "The following code constructs " }{TEXT 260 20 "numerical procedures" }{TEXT -1 56 " for solutions based on ea ch of the Runge-Kutta schemes." }}{PARA 0 "" 0 "" {TEXT -1 75 "The err or in the value obtained by each of the methods at the point where " }{XPPEDIT 18 0 "x = 2.999;" "6#/%\"xG-%&FloatG6$\"%**H!\"$" }{TEXT -1 16 " is also given." }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 628 "M : = (x,y) -> -(1+cos(2*x))*y^3: hh := 0.01: numsteps := 300: x0 := 0: y0 := sqrt(2):\nmatrix([[`slope field: `,M(x,y)],[`initial point: `,`` (x0,y0)],[`step width: `,hh],\n[`no. of steps: `,numsteps]]);``;\n mthds := [`Prince-Dormand scheme`,`scheme with equal radii`,`scheme wi th `*t[6]=1/500,`scheme with `*t[6]=1/600,\n`scheme with `*t[6]=1/660] : errs := []:\nDigits := 20:\nfor ct to 5 do\n mn_RK5_||ct := RK5_|| ct(M(x,y),x,y,x0,evalf(y0),hh,numsteps,true);\nend do:\nxx := 2.999: m xx := evalf(m(xx)):\nfor ct to 5 do\n errs := [op(errs),abs(mn_RK5_| |ct(xx)-mxx)];\nend do:\nDigits := 10:\nlinalg[transpose]([mthds,evalf (errs)]);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#K%'matrixG6#7&7$%0slope~f ield:~~~G,$*&,&\"\"\"F,-%$cosG6#,$*&\"\"#F,%\"xGF,F,F,F,)%\"yG\"\"$F,! \"\"7$%0initial~point:~G-%!G6$\"\"!*$F2#F,F27$%/step~width:~~~G$F,!\"# 7$%1no.~of~steps:~~~G\"$+$Q)pprint296\"" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%!G" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#K%'matrixG6#7'7$%6Prince- Dormand~schemeG$\"+&\\M5f%!#?7$%8scheme~with~equal~radiiG$\"+<>h1SF+7$ /*&%-scheme~with~G\"\"\"&%\"tG6#\"\"'F4#F4\"$+&$\")8#=<%F+7$/F2#F4\"$+ '$\"+.!o.$RF+7$/F2#F4\"$g'$\"+S0V< \+ " 0 "" {MPLTEXT 1 0 365 "mthds := [`Prince-Dormand scheme`,`scheme wit h equal radii`,`scheme with `*t[6]=1/500,`scheme with `*t[6]=1/600,\n` scheme with `*t[6]=1/660]: errs := []:\nDigits := 20:\nfor ct to 5 do \n sm := NCint((m(x)-'mn_RK5_||ct'(x))^2,x=0..3,adaptive=false,numpo ints=7,factor=150);\n errs := [op(errs),sqrt(sm/3)];\nend do:\nDigit s := 10:\nlinalg[transpose]([mthds,evalf(errs)]);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#K%'matrixG6#7'7$%6Prince-Dormand~schemeG$\"+1LlYB!#>7$% 8scheme~with~equal~radiiG$\"+e'f0.#F+7$/*&%-scheme~with~G\"\"\"&%\"tG6 #\"\"'F4#F4\"$+&$\"+z\"4.%R!#@7$/F2#F4\"$+'$\"+`B'e+#F+7$/F2#F4\"$g'$ \"+)\\&RM()!#?Q)pprint316\"" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 92 "The following error graphs are constructed usin g the numerical procedures for the solutions." }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 411 "evalf[20](plot(['mn_RK5_1'(x)-m(x),'mn_RK5_2'(x )-m(x),'mn_RK5_3'(x)-m(x),'mn_RK5_4'(x)-m(x),\n'mn_RK5_5'(x)-m(x)],x=0 ..3,color=[brown,red,COLOR(RGB,0,.7,0),COLOR(RGB,.5,0,1),COLOR(RGB,1,. 35,0)],\nlegend=[`Prince-Dormand scheme`,`scheme with equal radii`,`sc heme with t[6]=1/500`,`scheme with t[6]=1/600`,\n`scheme with t[6]=1/6 60`],font=[HELVETICA,9],title=`error curves for 6 stage order 5 Runge- Kutta methods`));" }}{PARA 13 "" 1 "" {GLPLOT2D 1026 602 602 {PLOTDATA 2 "6+-%'CURVESG6%7\\p7$$\"\"!F)F(7$$\"5+++]7.K[V?!#A$\"'Q'z \"!#>7$$\"5++++D1k'p3%F-$\")-dN:F07$$\"5+++]P4'\\/8'F-$\"*9#3$>#F07$$ \"5++++]7G$R<)F-$\"+5;_$[\"F07$$\"5+++Dc,;u@5!#@$\"+:+y;dF07$$\"5+++]( =#**3E7FC$\"+)[4)*e&F07$$\"5+++v=U#Q/V\"FC$\"+MUjzaF07$$\"5++++]ilyM;F C$\"+9I#)3bF07$$\"5+++D\"G)[8R=FC$\"+@yO`iF07$$F,FC$\"+`Yk,%)F07$$\"5+ ++vVB:$yC#FC$\"+.]#zA)F07$$\"5++++vV)z@X#FC$\"+-f;q!)F07$$\"5+++]P%[w3 'GFC$\"+atzj%)F07$$\"5+++++DJdpKFC$\"+DLde$*F07$$\"5+++DJX9#RZ$FC$\"+f iS)=*F07$$\"5+++]il(p#yOFC$\"+#=q*)4*F07$$\"5+++v$f3=E)QFC$\"+\"opxO*F 07$$F3FC$\"+D[y#))*F07$$\"5+++DcEZJ\"H%FC$\"+KYY.(*F07$$\"5+++](o/jc\\ %FC$\"+A\\pO&*F07$$\"5+++v=n8,+ZFC$\"+_#eIV*F07$$\"5++++](ofV!\\FC$\"+ 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"Curve 1" }}}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 " " 0 "" {TEXT -1 93 "The following code constructs a discrete solution \+ based on each of the methods and gives the " }{TEXT 260 22 "root mean \+ square error" }{TEXT -1 18 " of each solution." }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 721 "P := (x,y) -> -(2*sin(5*x)+3*cos(7*x))*sinh(y): hh := 0.01: numsteps := 500: x0 := 0: y0 := sqrt(5)/2:\nmatrix([[`slo pe field: `,P(x,y)],[`initial point: `,``(x0,y0)],[`step width: `, hh],\n[`no. of steps: `,numsteps]]);``;\nmthds := [`Prince-Dormand s cheme`,`scheme with equal radii`,`scheme with `*t[6]=1/500,`scheme wit h `*t[6]=1/600,\n`scheme with `*t[6]=1/660]: errs := []:\nDigits := 30 :\nfor ct to 5 do\n Pn_RK5_||ct := RK5_||ct(P(x,y),x,y,x0,evalf(y0), hh,numsteps,false);\n sm := 0: numpts := nops(Pn_RK5_||ct):\n for \+ ii to numpts do\n sm := sm+(Pn_RK5_||ct[ii,2]-evalf(p(Pn_RK5_||ct [ii,1])))^2;\n end do:\n errs := [op(errs),sqrt(sm/numpts)];\nend \+ do:\nDigits := 10:\nlinalg[transpose]([mthds,evalf(errs)]);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#K%'matrixG6#7&7$%0slope~field:~~~G,$*&,&*&\" \"#\"\"\"-%$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\"$+&Q)ppr int816\"" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%!G" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#K%'matrixG6#7'7$%6Prince-Dormand~schemeG$\"+I1(\\_%!#?7 $%8scheme~with~equal~radiiG$\"+\\Y'yh(!#>7$/*&%-scheme~with~G\"\"\"&% \"tG6#\"\"'F5#F5\"$+&$\"+*\\SZN#F07$/F3#F5\"$+'$\"+?nx7OF+7$/F3#F5\"$g '$\"+r-#3X#F+Q)pprint826\"" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 30 "The following code constructs " }{TEXT 260 20 " numerical procedures" }{TEXT -1 56 " for solutions based on each of th e Runge-Kutta schemes." }}{PARA 0 "" 0 "" {TEXT -1 75 "The error in th e value obtained by each of the methods at the point where " } {XPPEDIT 18 0 "x = 4.999;" "6#/%\"xG-%&FloatG6$\"%**\\!\"$" }{TEXT -1 16 " is also given." }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 645 "P : = (x,y) -> -(2*sin(5*x)+3*cos(7*x))*sinh(y): hh := 0.01: numsteps := 5 00: x0 := 0: y0 := sqrt(5)/2:\nmatrix([[`slope field: `,P(x,y)],[`in itial point: `,``(x0,y0)],[`step width: `,hh],\n[`no. of steps: `, numsteps]]);``;\nmthds := [`Prince-Dormand scheme`,`scheme with equal \+ radii`,`scheme with `*t[6]=1/500,`scheme with `*t[6]=1/600,\n`scheme w ith `*t[6]=1/660]: errs := []:\nDigits := 30:\nfor ct to 5 do\n pn_R K5_||ct := RK5_||ct(P(x,y),x,y,x0,evalf(y0),hh,numsteps,true);\nend do :\nxx := 4.999: pxx := evalf(p(xx)):\nfor ct to 5 do\n errs := [op(e rrs),abs(pn_RK5_||ct(xx)-pxx)];\nend do:\nDigits := 10:\nlinalg[transp ose]([mthds,evalf(errs)]);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#K%'matri xG6#7&7$%0slope~field:~~~G,$*&,&*&\"\"#\"\"\"-%$sinG6#,$*&\"\"&F.%\"xG F.F.F.F.*&\"\"$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)pprint406\"" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%!G" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#K%'matrixG6#7'7$ %6Prince-Dormand~schemeG$\"+jRyWC!#?7$%8scheme~with~equal~radiiG$\"+L) RGT#!#=7$/*&%-scheme~with~G\"\"\"&%\"tG6#\"\"'F5#F5\"$+&$\"+^>!QE'!#>7 $/F3#F5\"$+'$\"+L'paq$F+7$/F3#F5\"$g'$\"+-$))RH#F+Q)pprint416\"" }}} {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$\"\"!\"\"&" }{TEXT -1 82 " of each \+ Runge-Kutta method is estimated as follows using the special procedure " }{TEXT 0 5 "NCint" }{TEXT -1 98 " to perform numerical integratio n by the 7 point Newton-Cotes method over 200 equal subintervals." }} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 365 "mthds := [`Prince-Dormand s cheme`,`scheme with equal radii`,`scheme with `*t[6]=1/500,`scheme wit h `*t[6]=1/600,\n`scheme with `*t[6]=1/660]: errs := []:\nDigits := 20 :\nfor ct to 5 do\n sm := NCint((p(x)-'pn_RK5_||ct'(x))^2,x=0..5,ada ptive=false,numpoints=7,factor=200);\n errs := [op(errs),sqrt(sm/5)] ;\nend do:\nDigits := 10:\nlinalg[transpose]([mthds,evalf(errs)]);" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#K%'matrixG6#7'7$%6Prince-Dormand~schem eG$\"+Z[t([%!#?7$%8scheme~with~equal~radiiG$\"+47$/*&%-scheme~w ith~G\"\"\"&%\"tG6#\"\"'F5#F5\"$+&$\"+G![RP#F07$/F3#F5\"$+'$\"++%fTf$F +7$/F3#F5\"$g'$\"+o#eiY#F+Q)pprint366\"" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 92 "The following error graphs are con structed using the numerical procedures for the solutions." }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 364 "evalf[20](plot(['pn_RK5_1'(x)-p(x) ,'pn_RK5_3'(x)-p(x),'pn_RK5_4'(x)-p(x),\n'pn_RK5_5'(x)-p(x)],x=0..2.2, color=[brown,COLOR(RGB,0,.7,0),COLOR(RGB,.5,0,1),COLOR(RGB,1,.35,0)], \nlegend=[`Prince-Dormand scheme`,`scheme with t[6]=1/500`,`scheme wit h t[6]=1/600`,\n`scheme with t[6]=1/660`],font=[HELVETICA,9],title=`er ror curves for 6 stage order 5 Runge-Kutta methods`));" }}{PARA 13 "" 1 "" {GLPLOT2D 968 614 614 {PLOTDATA 2 "6*-%'CURVESG6%7it7$$\"\"!F)F(7 $$\"5mmmmm;RP&z%!#@$\"*H\"pd=!#?7$$\"5++++v=&)e\")oF-$\"*al\"3MF07$$\" 5LLLL$37.y'*)F-$\"*YfxL%F07$$\"5&****\\P%[ZMa&*F-$\"*#>a4VF07$$\"5mmmT gP')395F0$\"*2%fDWF07$$\"5LL$ek.!Gus5F0$\"*WZmJ%F07$$\"5+++]7jpRJ6F0$ \"*4RwF%F07$$\"5MLLek)G0([7F0$\"*f00+%F07$$\"5nmmm;9O,m8F0$\"*VA\"4OF0 7$$\"5nmmmmca=-;F0$\"*WQbO#F07$$\"5nmmm;*Hd$Q=F0$\"*Q&)QX\"F07$$\"5+++ ]7emSt?F0$\"))o[&pF07$$\"5LLLL3#F07$$\"5++++DhoPEDF0$!(&3*H#F07$$\"5LLLLL$GP`j#F0$!(9tc#F07$$\"5 mmmmT0xHWFF0$\"'-LfF07$$\"5LLLL3<@%*pHF0$\")I)fX\"F07$$\"5++++vGle&>$F 0$\")\"o(eOF07$$\"5LLLL3-$[*GMF0$\")u$z2&F07$$\"5mmmmTv+JiOF0$\")sXLbF 07$$\"5LLLLeaM#\\*QF0$\")\"o#pXF07$$\"5++++vLo`FTF0$\")c&pW#F07$$\"5mm m;/'G3oO%F0$!(h/U%F07$$\"5LLLLLQ(zgg%F0$!)6.\"Q&F07$$\"5mmmm;*e!eF]F0$ !*w=@E\"F07$$\"5NLLL3_c$[E&F0$!*K5ng\"F07$$\"5+++++:24-bF0$!*I*\\n>F07 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G. Lether: Mathematics of Computation, Vol. 20, no. 95, (July 1966) page 382. " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 256 "" 0 " " {TEXT -1 1 " " }{XPPEDIT 18 0 "dy/dx=exp(-x)/(x-1)^2" "6#/*&%#dyG\" \"\"%#dxG!\"\"*&-%$expG6#,$%\"xGF(F&*$,&F.F&F&F(\"\"#F(" }{TEXT -1 2 " " }{XPPEDIT 18 0 "cos(1/(x-1))-y" "6#,&-%$cosG6#*&\"\"\"F(,&%\"xGF(F (!\"\"F+F(%\"yGF+" }{TEXT -1 5 ", " }{XPPEDIT 18 0 "y(0)=sin*1" "6# /-%\"yG6#\"\"!*&%$sinG\"\"\"F*F*" }{TEXT -1 1 " " }}{PARA 0 "" 0 "" {TEXT -1 10 "Solution: " }}{PARA 256 "" 0 "" {TEXT -1 2 " " } {XPPEDIT 18 0 "y = -exp(-x)*sin(1/(x-1))" "6#/%\"yG,$*&-%$expG6#,$%\"x G!\"\"\"\"\"-%$sinG6#*&F-F-,&F+F-F-F,F,F-F," }{TEXT -1 1 " " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 185 "d e := diff(y(x),x)=exp(-x)/(x-1)^2*cos(1/(x-1))-y(x);\nic := y(0)=sin(1 );\ndsolve(\{de,ic\},y(x));\nq := unapply(rhs(%),x):\nplot(q(x),x=0..1 -1/(6*Pi),font=[HELVETICA,9],labels=[`x`,`y(x)`]);" }}{PARA 11 "" 1 " " {XPPMATH 20 "6#>%#deG/-%%diffG6$-%\"yG6#%\"xGF,,&*(-%$expG6#,$F,!\" \"\"\"\",&F,F4F4F3!\"#-%$cosG6#*&F4F4F5F3F4F4F)F3" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%#icG/-%\"yG6#\"\"!-%$sinG6#\"\"\"" }}{PARA 11 "" 1 " " {XPPMATH 20 "6#/-%\"yG6#%\"xG,$*&-%$expG6#,$F'!\"\"\"\"\"-%$sinG6#*& F/F/,&F'F/F/F.F.F/F." }}{PARA 13 "" 1 "" {GLPLOT2D 603 396 396 {PLOTDATA 2 "6&-%'CURVESG6$7[r7$$\"\"!F)$\"30l*y![)4ZT)!#=7$$\"3#>=\"* )>z2k?!#>$\"3m%p=Qu6DN)F,7$$\"3kMQU\"3@+'QF0$\"3m=O<,5$yO#)F,7$$\"38UQ!)p4'G\"zF0$\"3/Z^$zTlU<)F,7$$\" 3BY$*R0>JO**F0$\"36ty1)z*36\")F,7$$\"3wbXC%*4B\"=\"F,$\"3A;o(=P!Q^!)F, 7$$\"3M!)fxC(zaP\"F,$\"3E8^!ydw!))zF,7$$\"3kgswR?Pw:F,$\"3T8>lD8j?zF,7 $$\"3Qnlb!4?mx\"F,$\"3%p4\\s^P4&yF,7$$\"3OsvSC)*f#)>F,$\"3/$H(=wa6wxF, 7$$\"3)Rk+h)o-k@F,$\"3LR8d$>`qq(F,7$$\"3Q^Vo'yq#oBF,$\"3YB)Qc;#3DwF,7$ $\"3?0sMKLNtDF,$\"3,;%fG`C(F,7$$ \"3S+dSsVlWLF,$\"3&36sy[X09(F,7$$\"3EOur83&\\b$F,$\"37)QgTzpp+(F,7$$\" 3c#**Qwz)4TPF,$\"3M]xfz#>l(oF,7$$\"3wx#p)QELXRF,$\"3UR-VbS%zr'F,7$$\"3 \"yy;Gb6)RTF,$\"3%40quzD%\\lF,7$$\"3p2KM(*)HFM%F,$\"3W'4!o9@F_jF,7$$\" 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\\RTgHcmRGF,7$$\"3C\\b#*Q:tj#*F,$\"3M$o6)eV:lLF,7$$\"3oE3CP5fw#*F,$\"3 Ma)HpV]]I!H!G$*F,$\"3a%4t 07BP*GF,7$$\"3J=s\")G&))3M*F,$\"3f(3)H;[%o+#F,7$$\"3w&\\Kr-[PN*F,$\"33 n1kl[]%*F,$!3m (=[SoWqQ#F,7$$\"3%>saO,CmX*F,$!3GDS5eP#en\"F,7$$\"3AhB\"Gw`IY*F,$!3U$3 !Gg0_(o)F07$$\"3]++(>^$[p%*F,$!3V'=8$[D+C:!#C-%'COLOURG6&%$RGBG$\"#5! \"\"F(F(-%+AXESLABELSG6$%\"xG%%y(x)G-%%FONTG6$%*HELVETICAG\"\"*-%%VIEW G6$;F($\"+BN[p%*!#5%(DEFAULTG" 1 2 0 1 10 0 2 9 1 4 2 1.000000 45.000000 45.000000 0 0 "Curve 1" }}}}{PARA 0 "" 0 "" {TEXT -1 0 "" }} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 193 "Q := (x,y) -> exp(-x)/(x-1) ^2*cos(1/(x-1))-y;\nhh := evalf[28](1/500-1/(3000*Pi)):\nDigits := 25: \nfor ct to 5 do\n qn_RK5_||ct := RK5_||ct(Q(x,y),x,y,0,sin(1),hh,50 0,true);\nend do;\nDigits := 10:" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }} {PARA 0 "" 0 "" {TEXT -1 93 "The following code constructs a discrete \+ solution based on each of the methods and gives the " }{TEXT 260 22 "r oot mean square error" }{TEXT -1 18 " of each solution." }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 737 "Q := (x,y) -> exp(-x)/(x-1)^2*cos( 1/(x-1))-y: hh := 1/500-1/(3000*Pi): numsteps := 500: x0 := 0: y0 := s in(1):\nmatrix([[`slope field: `,Q(x,y)],[`initial point: `,``(x0,y0 )],[`step width: `,hh],\n[`no. of steps: `,numsteps]]);``;\nmthds \+ := [`Prince-Dormand scheme`,`scheme with equal radii`,`scheme with `*t [6]=1/500,`scheme with `*t[6]=1/600,\n`scheme with `*t[6]=1/660]: errs := []:\nDigits := 30:\nfor ct to 5 do\n Qn_RK5_||ct := RK5_||ct(Q(x ,y),x,y,x0,evalf[33](y0),evalf[33](hh),numsteps,false);\n sm := 0: n umpts := nops(Qn_RK5_||ct):\n for ii to numpts do\n sm := sm+(Q n_RK5_||ct[ii,2]-q(Qn_RK5_||ct[ii,1]))^2;\n end do:\n errs := [op( errs),sqrt(sm/numpts)];\nend do:\nDigits := 10:\nlinalg[transpose]([mt hds,evalf(errs)]);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#K%'matrixG6#7&7$ %0slope~field:~~~G,&*(-%$expG6#,$%\"xG!\"\"\"\"\",&F/F1F1F0!\"#-%$cosG 6#*&F1F1F2F0F1F1%\"yGF07$%0initial~point:~G-%!G6$\"\"!-%$sinG6#F17$%/s tep~width:~~~G,&#F1\"$+&F1*&F1F1*&\"%+IF1%#PiGF1F0F07$%1no.~of~steps:~ ~~GFFQ)pprint426\"" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%!G" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#K%'matrixG6#7'7$%6Prince-Dormand~schemeG$\"+44 \"*4J!#=7$%8scheme~with~equal~radiiG$\"+wn\\@J!#<7$/*&%-scheme~with~G \"\"\"&%\"tG6#\"\"'F5#F5\"$+&$\"+h:r*4&F+7$/F3#F5\"$+'$\"+[h14GF+7$/F3 #F5\"$g'$\"+amC9OF+Q)pprint436\"" }}}{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 = 0;" "6#/%\"xG\"\"!" }{TEXT -1 21 ".9469 is also g iven." }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 662 "Q := (x,y) -> exp( -x)/(x-1)^2*cos(1/(x-1))-y: hh := 1/500-1/(3000*Pi): numsteps := 500: \+ x0 := 0: y0 := sin(1):\nmatrix([[`slope field: `,Q(x,y)],[`initial p oint: `,``(x0,y0)],[`step width: `,hh],\n[`no. of steps: `,numstep s]]);``;\n\nmthds := [`Prince-Dormand scheme`,`scheme with equal radii `,`scheme with `*t[6]=1/500,`scheme with `*t[6]=1/600,\n`scheme with ` *t[6]=1/660]: errs := []:\nDigits := 30:\nfor ct to 5 do\n qn_RK5_|| ct := RK5_||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 5 do\n errs := [o p(errs),abs(qn_RK5_||ct(xx)-qxx)];\nend do:\nDigits := 10:\nlinalg[tra nspose]([mthds,evalf(errs)]);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#K%'ma trixG6#7&7$%0slope~field:~~~G,&*(-%$expG6#,$%\"xG!\"\"\"\"\",&F/F1F1F0 !\"#-%$cosG6#*&F1F1F2F0F1F1%\"yGF07$%0initial~point:~G-%!G6$\"\"!-%$si nG6#F17$%/step~width:~~~G,&#F1\"$+&F1*&F1F1*&\"%+IF1%#PiGF1F0F07$%1no. ~of~steps:~~~GFFQ)pprint446\"" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%!G" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#K%'matrixG6#7'7$%6Prince-Dormand~sch emeG$\"+N\"[)fD!#=7$%8scheme~with~equal~radiiG$\"+,_RiQ!#;7$/*&%-schem e~with~G\"\"\"&%\"tG6#\"\"'F5#F5\"$+&$\"+,Z1tx!#<7$/F3#F5\"$+'$\"+FNA' f'F+7$/F3#F5\"$g'$\"+Oo(>*[F>Q)pprint456\"" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 4 "The " }{TEXT 260 22 "root mean s quare error" }{TEXT -1 19 " over the interval " }{XPPEDIT 18 0 " [0, 1 -1/(6*Pi)] " "6#7$\"\"!,&\"\"\"F&*&F&F&*&\"\"'F&%#PiGF&!\"\"F+" } {TEXT -1 82 " of each Runge-Kutta method is estimated as follows 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 0 385 "mthds := [`Prince-Dormand scheme`,`scheme with equal radii`,`scheme with `* t[6]=1/500,`scheme with `*t[6]=1/600,\n`scheme with `*t[6]=1/660]: err s := []:\nDigits := 20:\nfor ct to 5 do\n sm := NCint((q(x)-'qn_RK5_ ||ct'(x))^2,x=0..1-1/(6*Pi),adaptive=false,numpoints=7,factor=200);\n \+ errs := [op(errs),sqrt(sm/(1-1/(6*Pi)))];\nend do:\nDigits := 10:\nl inalg[transpose]([mthds,evalf(errs)]);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#K%'matrixG6#7'7$%6Prince-Dormand~schemeG$\"+\"G,,7$!#=7$%8scheme ~with~equal~radiiG$\"+P031D!#<7$/*&%-scheme~with~G\"\"\"&%\"tG6#\"\"'F 5#F5\"$+&$\"+ " 0 "" {MPLTEXT 1 0 428 "evalf[20](plot(['qn_RK5_1'(x)-q(x),'qn_RK5_2'(x)-q(x ),'qn_RK5_3'(x)-q(x),'qn_RK5_4'(x)-q(x),\n'qn_RK5_5'(x)-q(x)],x=0..0.7 ,-2e-14..2e-14,\ncolor=[brown,red,COLOR(RGB,0,.7,0),COLOR(RGB,.5,0,1), COLOR(RGB,1,.35,0)],\nlegend=[`Prince-Dormand scheme`,`scheme with equ al radii`,`scheme with t[6]=1/500`,`scheme with t[6]=1/600`,\n`scheme \+ with t[6]=1/660`],font=[HELVETICA,9],title=`error curves for 6 stage o rder 5 Runge-Kutta methods`));" }}{PARA 13 "" 1 "" {GLPLOT2D 685 540 540 {PLOTDATA 2 "6+-%'CURVESG6%7fn7$$\"\"!F)F(7$$\"5mmmmm\"z+e_\"!#@$ \"#w!#?7$$\"5LLLL3->R`GF-$\"$a\"F07$$\"5mmmm;apSYVF-$\"$W#F07$$\"5lmmm ;z'=$\\eF-$\"$Z$F07$$\"5KLLL3Ft3XtF-$\"$g%F07$$\"5lmmmTNj&=t)F-$\"$'eF 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nom(rhs(%)),exp)));\nr := unapply(rhs(%),x):\nplot(r(x),x=0..5,font=[H ELVETICA,9],labels=[`x`,`y(x)`]);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#> %#deG/-%%diffG6$-%\"yG6#%\"xGF,,$*(\"\"&\"\"\"F)F0)-%$sinG6#,$*&\"\"(F 0F,F0F0F7F0F0" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%#icG/-%\"yG6#\"\"! \"\"\"" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/-%\"yG6#%\"xG-%$expG6#,,*&# \"#D\"#k\"\"\"-%$cosG6#,$*&\"\"(F0F'F0F0F0!\"\"*&#F0F/F0-F26#,$*&\"#NF 0F'F0F0F0F7*&#\"\"&\"%OJF0-F26#,$*&\"#\\F0F'F0F0F0F0*&#FAF/F0-F26#,$*& \"#@F0F'F0F0F0F0#\"#;FGF0" }}{PARA 13 "" 1 "" {GLPLOT2D 806 286 286 {PLOTDATA 2 "6&-%'CURVESG6$7_^l7$$\"\"!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$$ 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" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 681 "R := (x,y) -> 5*y*sin(7 *x)^7: hh := 0.01: numsteps := 500: x0 := 0: y0 := 1:\nmatrix([[`slope field: `,R(x,y)],[`initial point: `,``(x0,y0)],[`step width: `,hh ],\n[`no. of steps: `,numsteps]]);``;\nmthds := [`Prince-Dormand sch eme`,`scheme with equal radii`,`scheme with `*t[6]=1/500,`scheme with \+ `*t[6]=1/600,\n`scheme with `*t[6]=1/660]: errs := []:\nDigits := 25: \nfor ct to 5 do\n Rn_RK5_||ct := RK5_||ct(R(x,y),x,y,x0,y0,hh,numst eps,false);\n sm := 0: numpts := nops(Rn_RK5_||ct):\n for ii to nu mpts do\n sm := sm+(Rn_RK5_||ct[ii,2]-r(Rn_RK5_||ct[ii,1]))^2;\n \+ end do:\n errs := [op(errs),sqrt(sm/numpts)];\nend do:\nDigits := \+ 10:\nlinalg[transpose]([mthds,evalf(errs)]);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#K%'matrixG6#7&7$%0slope~field:~~~G,$*(\"\"&\"\"\"%\"yGF ,)-%$sinG6#,$*&\"\"(F,%\"xGF,F,F4F,F,7$%0initial~point:~G-%!G6$\"\"!F, 7$%/step~width:~~~G$F,!\"#7$%1no.~of~steps:~~~G\"$+&Q)pprint366\"" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#%!G" }}{PARA 11 "" 1 "" {XPPMATH 20 "6 #K%'matrixG6#7'7$%6Prince-Dormand~schemeG$\"+6!zy<\"!#=7$%8scheme~with ~equal~radiiG$\"+%>s$)e)F+7$/*&%-scheme~with~G\"\"\"&%\"tG6#\"\"'F4#F4 \"$+&$\"+-Et_\"*!#>7$/F2#F4\"$+'$\"+>2J>PF=7$/F2#F4\"$g'$\"+w3`$=%F=Q) pprint376\"" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 30 "The following code constructs " }{TEXT 260 20 "numerical proced ures" }{TEXT -1 56 " for solutions based on each of the Runge-Kutta sc hemes." }}{PARA 0 "" 0 "" {TEXT -1 75 "The error in the value obtained by each of the methods at the point where " }{XPPEDIT 18 0 "x = 4.99 9;" "6#/%\"xG-%&FloatG6$\"%**\\!\"$" }{TEXT -1 16 " is also given." } }{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 612 "R := (x,y) -> 5*y*sin(7*x) ^7: hh := 0.01: numsteps := 500: x0 := 0: y0 := 1:\nmatrix([[`slope fi eld: `,R(x,y)],[`initial point: `,``(x0,y0)],[`step width: `,hh], \n[`no. of steps: `,numsteps]]);``;\nmthds := [`Prince-Dormand schem e`,`scheme with equal radii`,`scheme with `*t[6]=1/500,`scheme with `* t[6]=1/600,\n`scheme with `*t[6]=1/660]: errs := []:\nDigits := 25:\nf or ct to 5 do\n rn_RK5_||ct := RK5_||ct(R(x,y),x,y,x0,y0,hh,numsteps ,true);\nend do:\nxx := 4.999: rxx := evalf(r(xx)):\nfor ct to 5 do\n \+ errs := [op(errs),abs(rn_RK5_||ct(xx)-rxx)];\nend do:\nDigits := 10: \nlinalg[transpose]([mthds,evalf(errs)]);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#K%'matrixG6#7&7$%0slope~field:~~~G,$*(\"\"&\"\"\"%\"yGF ,)-%$sinG6#,$*&\"\"(F,%\"xGF,F,F4F,F,7$%0initial~point:~G-%!G6$\"\"!F, 7$%/step~width:~~~G$F,!\"#7$%1no.~of~steps:~~~G\"$+&Q)pprint516\"" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#%!G" }}{PARA 11 "" 1 "" {XPPMATH 20 "6 #K%'matrixG6#7'7$%6Prince-Dormand~schemeG$\"+9/U#R#!#=7$%8scheme~with~ equal~radiiG$\"+=sw_7$/F3#F5\"$g'$\"+_A()G$)FDQ)p print526\"" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 4 "The " }{TEXT 260 22 "root mean square error" }{TEXT -1 20 " over the interval " }{XPPEDIT 18 0 "[0, 5];" "6#7$\"\"!\"\"&" }{TEXT -1 82 " of each Runge-Kutta method is estimated as follows using the spe cial procedure " }{TEXT 0 5 "NCint" }{TEXT -1 98 " to perform numeri cal integration by the 7 point Newton-Cotes method over 200 equal subi ntervals." }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 365 "mthds := [`Pri nce-Dormand scheme`,`scheme with equal radii`,`scheme with `*t[6]=1/50 0,`scheme with `*t[6]=1/600,\n`scheme with `*t[6]=1/660]: errs := []: \nDigits := 20:\nfor ct to 5 do\n sm := NCint((r(x)-'rn_RK5_||ct'(x) )^2,x=0..5,adaptive=false,numpoints=7,factor=200);\n errs := [op(err s),sqrt(sm/5)];\nend do:\nDigits := 10:\nlinalg[transpose]([mthds,eval f(errs)]);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#K%'matrixG6#7'7$%6Prince -Dormand~schemeG$\"+x/Vw6!#=7$%8scheme~with~equal~radiiG$\"+@Muw&)F+7$ /*&%-scheme~with~G\"\"\"&%\"tG6#\"\"'F4#F4\"$+&$\"+Io!z6*!#>7$/F2#F4\" $+'$\"+]A&=s$F=7$/F2#F4\"$g'$\"+T9)z;%F=Q)pprint536\"" }}}{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 0 411 "evalf[20](plot(['rn_R K5_1'(x)-r(x),'rn_RK5_2'(x)-r(x),'rn_RK5_3'(x)-r(x),'rn_RK5_4'(x)-r(x) ,\n'rn_RK5_5'(x)-r(x)],x=0..5,color=[brown,red,COLOR(RGB,0,.7,0),COLOR (RGB,.5,0,1),COLOR(RGB,1,.35,0)],\nlegend=[`Prince-Dormand scheme`,`sc heme with equal radii`,`scheme with t[6]=1/500`,`scheme with t[6]=1/60 0`,\n`scheme with t[6]=1/660`],font=[HELVETICA,9],title=`error curves \+ for 6 stage order 5 Runge-Kutta methods`));" }}{PARA 13 "" 1 "" {GLPLOT2D 806 597 597 {PLOTDATA 2 "6+-%'CURVESG6%7[fl7$$\"\"!F)F(7$$\" 5ommmTN@Ki8!#@$\"(AP*[!#>7$$\"5NLLL$3FWYs#F-$!)eJ58F07$$\"5-+++D1k'p3% F-$!*#zrA;F07$$\"5qmmmmT&)G\\aF-$!*Zzn0$F07$$\"5SLL3_v!p)*y&F-$!*culQ$ F07$$\"50++]P4'\\/8'F-$!*i$y)[%F07$$\"5qmm\"HK9I5Z'F-$!*&4X)\\%F07$$\" 5SLLL3x1h6oF-$!*>)obZF07$$\"5qmm;zWXu=#F07$$\"5MLe*) fI&*y/@Feo$!+D`v**QF07$$\"5++]7G))>Wr@Feo$!+onfFUF07$$\"5nmTN'fW%4QAFe 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scheme" "scheme with \+ t[6]=1/500" "scheme with t[6]=1/600" "scheme with t[6]=1/660" }}}} {PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 48 "Test 13 of 6 stage, ord er 5 Runge-Kutta methods " }}{PARA 256 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 72 "See: \"Mathematica in Action\" by Stan Wagon, \+ Springer-Verlag, page 302. " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 256 "" 0 "" {TEXT -1 3 " " }{XPPEDIT 18 0 "dy/dx = cos*x+2*y;" "6#/* &%#dyG\"\"\"%#dxG!\"\",&*&%$cosGF&%\"xGF&F&*&\"\"#F&%\"yGF&F&" }{TEXT -1 8 ", " }{XPPEDIT 18 0 "y(0) = -2/5;" "6#/-%\"yG6#\"\"!,$*&\" \"#\"\"\"\"\"&!\"\"F-" }{TEXT -1 1 " " }}{PARA 0 "" 0 "" {TEXT -1 10 " Solution: " }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "y = 1/5 ;" "6#/%\"yG*&\"\"\"F&\"\"&!\"\"" }{TEXT -1 1 " " }{XPPEDIT 18 0 "sin* x-2/5" "6#,&*&%$sinG\"\"\"%\"xGF&F&*&\"\"#F&\"\"&!\"\"F+" }{TEXT -1 1 " " }{XPPEDIT 18 0 "cos*x" "6#*&%$cosG\"\"\"%\"xGF%" }{TEXT -1 3 " . \+ " }}{PARA 0 "" 0 "" {TEXT 260 4 "Note" }{TEXT -1 2 ": " }}{PARA 0 "" 0 "" {TEXT -1 53 "The general solution of the differential equation \+ " }{XPPEDIT 18 0 "dy/dx = cos*x+2*y;" "6#/*&%#dyG\"\"\"%#dxG!\"\",&*& %$cosGF&%\"xGF&F&*&\"\"#F&%\"yGF&F&" }{TEXT -1 64 " contains an expon ential term, but with the initial condition " }{XPPEDIT 18 0 "y(0) = \+ -2/5" "6#/-%\"yG6#\"\"!,$*&\"\"#\"\"\"\"\"&!\"\"F-" }{TEXT -1 23 " th is term 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(d e,y(x));" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%#deG/-%%diffG6$-%\"yG6#% \"xGF,,&-%$cosGF+\"\"\"*&\"\"#F0F)F0F0" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/-%\"yG6#%\"xG,(*&#\"\"#\"\"&\"\"\"-%$cosGF&F-!\"\"*&#F-F,F--%$s inGF&F-F-*&-%$expG6#,$*&F+F-F'F-F-F-%$_C1GF-F-" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 105 "Any slight deviation of \+ a numerical solution from the correct solution tends to become rapidly magnified." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 " " {MPLTEXT 1 0 154 "de := diff(y(x),x)=cos(x)+2*y(x);\nic := y(0)=-2/5 ;\ndsolve(\{de,ic\},y(x));\ne := unapply(rhs(%),x):\nplot(e(x),x=0..8, font=[HELVETICA,9],labels=[`x`,`y(x)`]);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%#deG/-%%diffG6$-%\"yG6#%\"xGF,,&-%$cosGF+\"\"\"*&\"\"#F0F)F0F0 " }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%#icG/-%\"yG6#\"\"!#!\"#\"\"&" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#/-%\"yG6#%\"xG,&*&#\"\"#\"\"&\"\"\"-%$ 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 0 687 "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 := [`Prince-Dormand scheme`,`scheme with equal radii`,`scheme \+ with `*t[6]=1/500,`scheme with `*t[6]=1/600,\n`scheme with `*t[6]=1/66 0]: errs := []:\nDigits := 20:\nfor ct to 5 do\n En_RK5_||ct := RK5_ ||ct(E(x,y),x,y,x0,evalf(y0),hh,numsteps,false);\n sm := 0: numpts : = nops(En_RK5_||ct):\n for ii to numpts do\n sm := sm+(En_RK5_| |ct[ii,2]-e(En_RK5_||ct[ii,1]))^2;\n end do:\n errs := [op(errs),s qrt(sm/numpts)];\nend do:\nDigits := 10:\nlinalg[transpose]([mthds,eva lf(errs)]);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#K%'matrixG6#7&7$%0slope ~field:~~~G,&-%$cosG6#%\"xG\"\"\"*&\"\"#F.%\"yGF.F.7$%0initial~point:~ G-%!G6$\"\"!#!\"#\"\"&7$%/step~width:~~~G$F0F97$%1no.~of~steps:~~~G\"$ +%Q)pprint526\"" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%!G" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#K%'matrixG6#7'7$%6Prince-Dormand~schemeG$\"+8x.$o# !#:7$%8scheme~with~equal~radiiG$\"+$4tSY\"!#97$/*&%-scheme~with~G\"\" \"&%\"tG6#\"\"'F5#F5\"$+&$\"+$eO>&eF+7$/F3#F5\"$+'$\"+HQ*zl#F+7$/F3#F5 \"$g'$\"+d " 0 "" {MPLTEXT 1 0 618 "E : = (x,y) -> cos(x)+2*y: hh := 0.02: numsteps := 400: x0 := 0: y0 := -2/ 5:\nmatrix([[`slope field: `,E(x,y)],[`initial point: `,``(x0,y0)],[ `step width: `,hh],\n[`no. of steps: `,numsteps]]);``;\nmthds := [ `Prince-Dormand scheme`,`scheme with equal radii`,`scheme with `*t[6]= 1/500,`scheme with `*t[6]=1/600,\n`scheme with `*t[6]=1/660]: errs := \+ []:\nDigits := 20:\nfor ct to 5 do\n en_RK5_||ct := RK5_||ct(E(x,y), x,y,x0,evalf(y0),hh,numsteps,true);\nend do:\nxx := 7.999: exx := eval f(e(xx)):\nfor ct to 5 do\n errs := [op(errs),abs(en_RK5_||ct(xx)-ex x)];\nend do:\nDigits := 10:\nlinalg[transpose]([mthds,evalf(errs)]); " }}{PARA 11 "" 1 "" {XPPMATH 20 "6#K%'matrixG6#7&7$%0slope~field:~~~G ,&-%$cosG6#%\"xG\"\"\"*&\"\"#F.%\"yGF.F.7$%0initial~point:~G-%!G6$\"\" !#!\"#\"\"&7$%/step~width:~~~G$F0F97$%1no.~of~steps:~~~G\"$+%Q)pprint8 56\"" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%!G" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#K%'matrixG6#7'7$%6Prince-Dormand~schemeG$\"+kAy'[\"!#97 $%8scheme~with~equal~radiiG$\"+s].8\")F+7$/*&%-scheme~with~G\"\"\"&%\" tG6#\"\"'F4#F4\"$+&$\"+/.!GC$F+7$/F2#F4\"$+'$\"+7W!HZ\"F+7$/F2#F4\"$g' $\"+ajQqk!#:Q)pprint866\"" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 4 "The " }{TEXT 260 22 "root mean square error" } {TEXT -1 20 " over the interval " }{XPPEDIT 18 0 "[0, 8];" "6#7$\"\"! \"\")" }{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 0 365 "mthds := [`Prince-Dormand scheme`,`scheme with equal radii`,`sche me with `*t[6]=1/500,`scheme with `*t[6]=1/600,\n`scheme with `*t[6]=1 /660]: errs := []:\nDigits := 20:\nfor ct to 5 do\n sm := NCint((e(x )-'en_RK5_||ct'(x))^2,x=0..8,adaptive=false,numpoints=7,factor=200);\n errs := [op(errs),sqrt(sm/8)];\nend do:\nDigits := 10:\nlinalg[tran spose]([mthds,evalf(errs)]);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#K%'mat rixG6#7'7$%6Prince-Dormand~schemeG$\"+WkaLE!#:7$%8scheme~with~equal~ra diiG$\"+Co1P9!#97$/*&%-scheme~with~G\"\"\"&%\"tG6#\"\"'F5#F5\"$+&$\"+e =*Ru&F+7$/F3#F5\"$+'$\"+>X'*3EF+7$/F3#F5\"$g'$\"+cM5Y6F+Q*pprint1416\" " }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 92 "The \+ following error graphs are constructed using the numerical procedures \+ for the solutions." }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 412 "evalf [20](plot(['en_RK5_1'(x)-e(x),'en_RK5_2'(x)-e(x),'en_RK5_3'(x)-e(x),'e n_RK5_4'(x)-e(x),\n'en_RK5_5'(x)-e(x)],x=0..2,\ncolor=[brown,red,COLOR (RGB,0,.7,0),COLOR(RGB,.5,0,1),COLOR(RGB,1,.35,0)],\nlegend=[`Prince-D ormand scheme`,`scheme with equal radii`,`scheme with t[6]=1/500`,`sch eme with t[6]=1/600`,\n`scheme with t[6]=1/660`],font=[HELVETICA,9],ti tle=`error curves for 6 stage order 5 Runge-Kutta methods`));" }} {PARA 13 "" 1 "" {GLPLOT2D 604 604 604 {PLOTDATA 2 "6+-%'CURVESG6%7S7$ 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}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 426 "eval f[20](plot(['en_RK5_1'(x)-e(x),'en_RK5_2'(x)-e(x),'en_RK5_3'(x)-e(x),' en_RK5_4'(x)-e(x),\n'en_RK5_5'(x)-e(x)],x=2..8,-1e-6..4.3e-5,\ncolor=[ brown,red,COLOR(RGB,0,.7,0),COLOR(RGB,.5,0,1),COLOR(RGB,1,.35,0)],\nle gend=[`Prince-Dormand scheme`,`scheme with equal radii`,`scheme with t [6]=1/500`,`scheme with t[6]=1/600`,\n`scheme with t[6]=1/660`],font=[ HELVETICA,9],title=`error curves for 6 stage order 5 Runge-Kutta metho ds`));" }}{PARA 13 "" 1 "" {GLPLOT2D 813 648 648 {PLOTDATA 2 "6+-%'CUR VESG6%7Z7$$\"\"#\"\"!$\"+ua$4U*!#?7$$\"5+++++DHyI@!#>$\",a`jo@\"F-7$$ \"5++++v[kdWAF1$\",!)))p0_\"F-7$$\"5++++]n\"\\DP#F1$\",v!y8c>F-7$$\"5+ +++]s,P,DF1$\",Zm)\\@DF-7$$\"5++++v8*y&HEF1$\"-E9*)3]K!#@7$$\"5++++vG[ W[FF1$\"-]Q+'Q6%FH7$$\"5++++v)fB:(GF1$\"-y!RPND&FH7$$\"5++++vQ=\"))*HF 1$\"-1PXUnnFH7$$\"5++++vj=pDJF1$\".'[B,&Rr)!#A7$$\"5+++++lN?cKF1$\".M= ]008\"FH7$$\"5++++]U$e6P$F1$\".)RO71A9FH7$$\"5+++++&>q0]$F1$\".-?'QTT= FH7$$\"5+++++DM^IOF1$\".9=$>O(Q#FH7$$\"5+++++0ytbPF1$\"-X@HKmIF-7$$\"5 ++++vQNXpQF1$\"-!y67!\\QF-7$$\"5+++++XDn/SF1$\"-b\"e\\R/&F-7$$\"5+++++ !y?#>TF1$\"-@kqTUjF-7$$\"5++++v3wY_UF1$\"-\"='G=z#)F-7$$\"5+++++IOTqVF 1$\".05xy\"[5F-7$$\"5++++v3\">)*\\%F1$\".r\"=_\"yN\"F-7$$\"5++++DEP/BY F1$\".\\9<9tt\"F-7$$\"5++++](o:;v%F1$\".y_e$yYAF-7$$\"5++++v$)[op[F1$ \".cb]c_%GF-7$$\"5++++]i%Qq*\\F1$\".AS$[nqOF-7$$\"5++++vQIKH^F1$\".OWl xCy%F-7$$\"5++++D^rZW_F1$\".ooX[6-'F-7$$\"5++++]Zn%)o`F1$\".'fm!R;s(F- 7$$\"5+++++5FL(\\&F1$\"./3@JU)**F-7$$\"5++++]d6.BcF1$\"/MKf;!QG\"F-7$$ \"5++++vo3lWdF1$\"/mv\"[Ntj\"F-7$$\"5++++]A))ozeF1$\"0E$\\$)Q-X@FH7$$ \"5+++++Ik-,gF1$\"0M4$GHw7(F1$\"0$pS\\(3Cg#F-7$$\"5++ ++v)Q?QD(F1$\"0DFs\\(f\\LF-7$$\"5+++++5jyptF1$\"0?y/nbRA%F-7$$\"5++++D E8COuF1$\"0)**Q0$zV#[F-7$$\"5++++]Ujp-vF1$\"0^9\"o9:5bF-7$$\"5++++D,X8 ivF1$\"0!yWD7q0iF-7$$\"5+++++gEd@wF1$\"0FiHj]!*)pF-7$$\"5+++]PMh%\\o(F 1$\"0GR(*4zM$zF-7$$\"5++++v3'>$[xF1$\"0PX:\"z_0!*F-7$$\"5+++++5h(*3yF1 $\"1!4!)>[/n,\"F-7$$\"5++++D6EjpyF1$\"1[H25s$y9\"F-7$$\"5+++vVeWA-zF1$ \"1\")y-e)\\^A\"F-7$$\"5+++]i0j\"[$zF1$\"1R\"f%4*pwI\"F-7$$\"5+++D\"G: 3u'zF1$\"1[z;+\"[dR\"F-7$$\"\")F*$\"1mB8-)e(*[\"F--%'COLOURG6&%$RGBG$ \")#)eqk!\")$\"))eqk\"Fd]lFe]l-%'LEGENDG6#%6Prince-Dormand~schemeG-F$6 %7Z7$F($\",4Cg49&F-7$F/$\",k)>JSmF-7$F5$\",N=`vH)F-7$F:$\"--HQVn5F-7$F ?$\"-****f$fP\"F-7$FD$\".')fg4Nx\"FH7$FJ$\".I]#z%[C#FH7$FO$\".)e`@umGF H7$FT$\".'y*)R$Gp$FH7$FY$\"/'ox?1]v%Fgn7$Fin$\"./*H=\"*ohFH7$F^o$\".=t /j)fxFH7$Fco$\"/s%*4t\"[+\"FH7$Fho$\"/a#)H'GFI\"FH7$F]p$\".m*H@At;F-7$ Fbp$\".?Ev<.5#F-7$Fgp$\".[j'zO_FF-7$F\\q$\".WtA74Y$F-7$Faq$\".SbVhx^%F -7$Ffq$\".MO2t'>dF-7$F[r$\".'*>?*G4uF-7$F`r$\".w/DH,[*F-7$Fer$\"/[iu)= gA\"F-7$Fjr$\"/YJN?f_:F-7$F_s$\"/\\\"FH7$F\\v$ \"1e$>AhnK$>FH7$Fav$\"2ijKM(z*[W#Fgn7$Ffv$\"1!4i(=*z.;$FH7$F[w$\"17GqO tvBSFH7$F`w$\"1568q8Mz^FH7$Few$\"1g$>ITR$HmFH7$Fjw$\"0R+(H\")3%e)F-7$F _x$\"1=*ph9z45\"F-7$Fdx$\"1m^H)ov+U\"F-7$Fix$\"1!R`^r*zF=F-7$F^y$\"1o2 sDs\"\\I#F-7$Fcy$\"1T^jW[bKEF-7$Fhy$\"1U<#yAln+$F-7$F]z$\"1\"R*fB5J'Q$ F-7$Fbz$\"1r#)z0rw8QF-7$Fgz$\"1H!zJR?\"HVF-7$F\\[l$\"1Ex$4A8T\"\\F-7$F a[l$\"1iucon#za&F-7$Ff[l$\"1;#G;wF-7$Fj\\l$\"1\\]?ptFH\")F--F_]l6&Fa] l$\"*++++\"Fd]l$F*F*Fjhl-Fh]l6#%8scheme~with~equal~radiiG-F$6%7Z7$F($ \",.*)e_0#F-7$F/$\",h0OXl#F-7$F5$\",`d/pJ$F-7$F:$\",i;zoE%F-7$F?$\",mC <**\\&F-7$FD$\"-'=_()*)3(FH7$FJ$\"-S8'*)G(*)FH7$FO$\".[g`ce9\"FH7$FT$ \".1hZTgZ\"FH7$FY$\"/')z!y)e+>Fgn7$Fin$\".%42YslCFH7$F^o$\".y(zmi,JFH7 $Fco$\".#erTE;SFH7$Fho$\".a%Gy,2_FH7$F]p$\"-y^Y)yo'F-7$Fbp$\"-<([#*\\R )F-7$Fgp$\".fp#\\7+6F-7$F\\q$\".jD_JLQ\"F-7$Faq$\".mDVcd!=F-7$Ffq$\".n ()HjhG#F-7$F[r$\".WP(p]hHF-7$F`r$\".07zE#*y$F-7$Fer$\".a7SB/!\\F-7$Fjr $\".T&3Xu0iF-7$F_s$\".,c<`g+)F-7$Fds$\"/#RF7*4V5F-7$Fis$\"/thORE88F-7$ F^t$\"/gIIc:%o\"F-7$Fct$\"/2.Elkx@F-7$Fht$\"/yny<3+GF-7$F]u$\"/NYv&p6d $F-7$Fbu$\"0'>U9=[yYFH7$Fgu$\"0u9StmM'fFH7$F\\v$\"03N^m=ts(FH7$Fav$\"1 i+/+!=Bx*Fgn7$Ffv$\"1!)H9><@j7FH7$F[w$\"1i$[i#eI3;FH7$F`w$\"1+Z\\Wc>q? FH7$Few$\"1!*4EuMw\\EFH7$Fjw$\"0&=(RS\"3JMF-7$F_x$\"00vR1T1S%F-7$Fdx$ \"0ZBZ)y2wcF-7$Fix$\"0J%*f'4w0tF-7$F^y$\"0Vgm_6G@*F-7$Fcy$\"19D3C(QA0 \"F-7$Fhy$\"1Kj9X:\"=?\"F-7$F]z$\"1.C\"R%p^`8F-7$Fbz$\"1M&>L?sV_\"F-7$ Fgz$\"1FzH%>g.t\"F-7$F\\[l$\"1.Lx#G$=k>F-7$Fa[l$\"1`vPM,_>`.DF-7$F[\\l$\"1yaMqu:sEF-7$F`\\l$\"1)3$)p\"39_GF-7$Fe\\l$\"1( p6W%pCWIF-7$Fj\\l$\"1L#>.R#H\\KF--%&COLORG6&Fa]lFjhl$\"\"(!\"\"Fjhl-Fh ]l6#%7scheme~with~t[6]=1/500G-F$6%7Z7$F($\"+*oe7L*F-7$F/$\",yaQ`?\"F-7 $F5$\",g;Di]\"F-7$F:$\",Y;Tx$>F-7$F?$\",svZy\\#F-7$FD$\"-Eo\\l>KFH7$FJ $\"-]'f%QvSFH7$FO$\"-QfeW/_FH7$FT$\"-cmyA/nFH7$FY$\".'[A'3Ej)Fgn7$Fin$ \".u#=_&*>6FH7$F^o$\".Q'[B $G'F-7$Faq$\"-ak6#>?)F-7$Ffq$\".'RXnRQ5F-7$F[r$\".8-/V^M\"F-7$F`r$\". \"oO(*4@+D@FH7$Fgu$\"0%> *)*z^'3FFH7$F\\v$\"0obYj1)4NFH7$Fav$\"1iVtU2mQWFgn7$Ffv$\"0S;V5.wt&FH7 $F[w$\"0-q)yl/0tFH7$F`w$\"0?!G-])HS*FH7$Few$\"1SYr&eUN?\"FH7$Fjw$\"0I \\?4B%e:F-7$F_x$\"0T%G;W!))*>F-7$Fdx$\"0(ymfu6yDF-7$Fix$\"0OZN$=L=LF-7 $F^y$\"0r[ZhGX=%F-7$Fcy$\"0\"yR/yMzZF-7$Fhy$\"0ewh')=(eaF-7$F]z$\"0.Yp Dwx9'F-7$Fbz$\"05-Gw8Q#pF-7$Fgz$\"0EM;zE%fyF-7$F\\[l$\"0&y.Q*o9#*)F-7$ Fa[l$\"1HsB2W@25F-7$Ff[l$\"1#Q'oAJ7P6F-7$F[\\l$\"1e\"oF79P@\"F-7$F`\\l $\"1*[u&>RY&H\"F-7$Fe\\l$\"1?^?v*>FQ\"F-7$Fj\\l$\"1[G:nJ&eZ\"F--Fjcm6& Fa]l$\"\"&F^dmFjhl$\"\"\"F*-Fh]l6#%7scheme~with~t[6]=1/600G-F$6%7Z7$F( $\"+t#yp4%F-7$F/$\"+Ib)GH&F-7$F5$\"+8^&[h'F-7$F:$\"+^0l5&)F-7$F?$\",a( e9(4\"F-7$FD$\"-coRD99FH7$FJ$\"-!fw(>!z\"FH7$FO$\"-3d`@'G#FH7$FT$\"-c2 K4XHFH7$FY$\".'GI7D#z$Fgn7$Fin$\"-%3DD*>\\FH7$F^o$\"-=ljz)='FH7$Fco$\" -#oyNQ,)FH7$Fho$\".C/t!)*Q5FH7$F]p$\"-JyRYM8F-7$Fbp$\"-%Qc)3v;F-7$Fgp$ \"-Z[67&>#F-7$F\\q$\"--t&3-w#F-7$Faq$\"-$e()zIg$F-7$Ffq$\"-Cw;khXF-7$F [r$\"-`T9;4fF-7$F`r$\"-tW'=2c(F-7$Fer$\"-WP$**yx*F-7$Fjr$\".]+oT#Q7F-7 $F_s$\".Caleuf\"F-7$Fds$\".gnN383#F-7$Fis$\".Fg n7$Ffv$\"0]-q!o\\?DFH7$F[w$\"0AkW*e14KFH7$F`w$\"0SiFVy18%FH7$Few$\"0!y M\"*[4(G&FH7$Fjw$\"/,e,]1YoF-7$F_x$\"/Gh2aj!y)F-7$Fdx$\"0(y\\0DbK6F-7$ Fix$\"0QZ:_CxX\"F-7$F^y$\"0!GI-*R#Q=F-7$Fcy$\"0;#eJ1a*4#F-7$Fhy$\"0GYR _%)zR#F-7$F]z$\"0RXhc$o+FF-7$Fbz$\"0G$e%[#fTIF-7$Fgz$\"0WGT.-EX$F-7$F \\[l$\"0t$>&=^\">RF-7$Fa[l$\"0t)Q5jjCWF-7$Ff[l$\"04xF+=`*\\F-7$F[\\l$ \"0qXe'zxJ`F-7$F`\\l$\"0EBCB+4p&F-7$Fe\\l$\"0**)f&=6U2'F-7$Fj\\l$\"0/q (o+M$['F--Fjcm6&Fa]lFa_n$\"#N!\"#Fjhl-Fh]l6#%7scheme~with~t[6]=1/660G- %%FONTG6$%*HELVETICAG\"\"*-%&TITLEG6#%Uerror~curves~for~6~stage~order~ 5~Runge-Kutta~methodsG-%+AXESLABELSG6$Q\"x6\"Q!Ff[o-%%VIEWG6$;F(Fj\\l; $F^dm!\"'$\"#VF^\\o" 1 2 0 1 10 0 2 9 1 4 2 1.000000 45.000000 45.000000 0 1 "Prince-Dormand scheme" "scheme with equal radii" "schem e with t[6]=1/500" "scheme with t[6]=1/600" "scheme with t[6]=1/660" } }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 2 " ;" }}}}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 48 "Test 14 of 6 stage , order 5 Runge-Kutta methods " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }} {PARA 256 "" 0 "" {TEXT -1 3 " " }{XPPEDIT 18 0 "dy/dx = 10*x*cos*x- 10*y;" "6#/*&%#dyG\"\"\"%#dxG!\"\",&**\"#5F&%\"xGF&%$cosGF&F,F&F&*&F+F &%\"yGF&F(" }{TEXT -1 8 ", " }{XPPEDIT 18 0 "y(0) = sqrt(5);" "6 #/-%\"yG6#\"\"!-%%sqrtG6#\"\"&" }{TEXT -1 1 " " }}{PARA 0 "" 0 "" {TEXT -1 10 "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 0 703 "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 := [`Prince-Dormand schem e`,`scheme with equal radii`,`scheme with `*t[6]=1/500,`scheme with `* t[6]=1/600,\n`scheme with `*t[6]=1/660]: errs := []:\nDigits := 20:\nf or ct to 5 do\n Bn_RK5_||ct := RK5_||ct(B(x,y),x,y,x0,evalf(y0),hh,n umsteps,false);\n sm := 0: numpts := nops(Bn_RK5_||ct):\n for ii t o numpts do\n sm := sm+(Bn_RK5_||ct[ii,2]-evalf(b(Bn_RK5_||ct[ii, 1])))^2;\n end do:\n errs := [op(errs),sqrt(sm/numpts)];\nend do: \nDigits := 10:\nlinalg[transpose]([mthds,evalf(errs)]);" }}{PARA 11 " " 1 "" {XPPMATH 20 "6#K%'matrixG6#7&7$%0slope~field:~~~G,&*(\"#5\"\"\" %\"xGF,-%$cosG6#F-F,F,*&F+F,%\"yGF,!\"\"7$%0initial~point:~G-%!G6$\"\" !*$\"\"&#F,\"\"#7$%/step~width:~~~G$F,!\"#7$%1no.~of~steps:~~~G\"$+&Q) pprint836\"" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%!G" }}{PARA 11 "" 1 " " {XPPMATH 20 "6#K%'matrixG6#7'7$%6Prince-Dormand~schemeG$\"+nh=da!#>7 $%8scheme~with~equal~radiiG$\"+EP/$*G!#=7$/*&%-scheme~with~G\"\"\"&%\" tG6#\"\"'F5#F5\"$+&$\"+]P]d6F07$/F3#F5\"$+'$\"+i=HcaF+7$/F3#F5\"$g'$\" +Ya%\\n#F+Q)pprint846\"" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 " " 0 "" {TEXT -1 30 "The following code constructs " }{TEXT 260 20 "num erical procedures" }{TEXT -1 56 " for solutions based on each of the R unge-Kutta schemes." }}{PARA 0 "" 0 "" {TEXT -1 75 "The error in the v alue obtained by each of the methods at the point where " }{XPPEDIT 18 0 "x = 4.999;" "6#/%\"xG-%&FloatG6$\"%**\\!\"$" }{TEXT -1 16 " is \+ also given." }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 627 "B := (x,y) - > 10*x*cos(x)-10*y: hh := 0.01: numsteps := 500: x0 := 0: y0 := sqrt(5 ):\nmatrix([[`slope field: `,B(x,y)],[`initial point: `,``(x0,y0)],[ `step width: `,hh],\n[`no. of steps: `,numsteps]]);``;\nmthds := [ `Prince-Dormand scheme`,`scheme with equal radii`,`scheme with `*t[6]= 1/500,`scheme with `*t[6]=1/600,\n`scheme with `*t[6]=1/660]: errs := \+ []:\nDigits := 20:\nfor ct to 5 do\n bn_RK5_||ct := RK5_||ct(B(x,y), x,y,x0,evalf(y0),hh,numsteps,true);\nend do:\nxx := 4.999: bxx := eval f(b(xx)):\nfor ct to 5 do\n errs := [op(errs),abs(bn_RK5_||ct(xx)-bx x)];\nend do:\nDigits := 10:\nlinalg[transpose]([mthds,evalf(errs)]); " }}{PARA 11 "" 1 "" {XPPMATH 20 "6#K%'matrixG6#7&7$%0slope~field:~~~G ,&*(\"#5\"\"\"%\"xGF,-%$cosG6#F-F,F,*&F+F,%\"yGF,!\"\"7$%0initial~poin t:~G-%!G6$\"\"!*$\"\"&#F,\"\"#7$%/step~width:~~~G$F,!\"#7$%1no.~of~ste ps:~~~G\"$+&Q)pprint166\"" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%!G" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#K%'matrixG6#7'7$%6Prince-Dormand~schem eG$\"+wbs3Z!#?7$%8scheme~with~equal~radiiG$\"+?sicC!#>7$/*&%-scheme~wi th~G\"\"\"&%\"tG6#\"\"'F5#F5\"$+&$\"+Z=@-5F07$/F3#F5\"$+'$\"+286dYF+7$ /F3#F5\"$g'$\"+'GC,@#F+Q)pprint176\"" }}}{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 integration by the 7 point Newton-Cotes \+ method over 200 equal subintervals." }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 365 "mthds := [`Prince-Dormand scheme`,`scheme with equal radii`,`scheme with `*t[6]=1/500,`scheme with `*t[6]=1/600,\n`scheme \+ with `*t[6]=1/660]: errs := []:\nDigits := 20:\nfor ct to 5 do\n sm \+ := NCint((b(x)-'bn_RK5_||ct'(x))^2,x=0..5,adaptive=false,numpoints=7,f actor=200);\n errs := [op(errs),sqrt(sm/5)];\nend do:\nDigits := 10: \nlinalg[transpose]([mthds,evalf(errs)]);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#K%'matrixG6#7'7$%6Prince-Dormand~schemeG$\"+$=L;F&!#>7$ %8scheme~with~equal~radiiG$\"+]@$\\z#!#=7$/*&%-scheme~with~G\"\"\"&%\" tG6#\"\"'F5#F5\"$+&$\"+$\\8#=6F07$/F3#F5\"$+'$\"+L&p2F&F+7$/F3#F5\"$g' $\"+k.p$e#F+Q)pprint616\"" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 92 "The following error graphs are constructed using \+ the numerical procedures for the solutions." }}{EXCHG {PARA 0 "> " 0 " " {MPLTEXT 1 0 429 "evalf[20](plot(['bn_RK5_1'(x)-b(x),'bn_RK5_2'(x)-b (x),'bn_RK5_3'(x)-b(x),'bn_RK5_4'(x)-b(x),\n'bn_RK5_5'(x)-b(x)],x=0..0 .65,numpoints=100,font=[HELVETICA,9],\ncolor=[brown,red,COLOR(RGB,0,.7 ,0),COLOR(RGB,.5,0,1),COLOR(RGB,1,.35,0)],\nlegend=[`Prince-Dormand sc heme`,`scheme with equal radii`,`scheme with t[6]=1/500`,`scheme with \+ t[6]=1/600`,\n`scheme with t[6]=1/660`],title=`error curves for 6 stag e order 5 Runge-Kutta methods`));" }}{PARA 13 "" 1 "" {GLPLOT2D 874 550 550 {PLOTDATA 2 "6+-%'CURVESG6%7d\\l7$$\"\"!F)F(7$$\"5SSSSSS:N<7$$\"5\"333333.ZV$F-$\"):V\"4\"F07$$\"5+,,,,^)yLH%F-$\")7 ;)=%F07$$\"5?@@@@@Y0_^F-$\"*-KzD\"F07$$\"5STTTT\"RI2,'F-$\"*TQ0>$F07$$ \"5ihhhhhhSpoF-$\"*;f-:(F07$$\"5XXXX&zNNlh(F-$\"+vl5N8F07$$\"5IHHHHaXm j$)F-$\"+))GM_BF07$$\"55888j]Pz5\"*F-$\"+Nz'*\\RF07$$\"5&ppppp%H#z&)*F 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"" {TEXT -1 48 "Test 15 of 7 stage, order 5 Runge-Kutta m ethods " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 198 "See: Comparing Numerical Methods for Ordinary Differential Equati ons, Hull, Enright, Fellen and Sedgwick,\n Siam Journal on Nume rical Analysis, Vol. 9, No. 4 (Dec. 1972), page 617, Example A5." }} {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 256 "" 0 "" {TEXT -1 3 " " } {XPPEDIT 18 0 "dy/dx = (y-x)/(y+x);" "6#/*&%#dyG\"\"\"%#dxG!\"\"*&,&% \"yGF&%\"xGF(F&,&F+F&F,F&F(" }{TEXT -1 8 ", " }{XPPEDIT 18 0 "y( 1) = 1;" "6#/-%\"yG6#\"\"\"F'" }{TEXT -1 1 " " }}{PARA 0 "" 0 "" {TEXT -1 10 "Solution: " }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "2*ln((x^2+y^2)/(x^2))+4*arctan(y/x)+4*ln*x-2*ln*2-Pi = 0;" "6#/, ,*&\"\"#\"\"\"-%#lnG6#*&,&*$%\"xGF&F'*$%\"yGF&F'F'*$F.F&!\"\"F'F'*&\" \"%F'-%'arctanG6#*&F0F'F.F2F'F'*(F4F'F)F'F.F'F'*(F&F'F)F'F&F'F2%#PiGF2 \"\"!" }{TEXT -1 2 ". " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 73 "de := diff(y(x),x)=(y(x)-x)/(y(x)+x );\nic := y(1)=1;\ndsolve(\{de,ic\},y(x));" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%#deG/-%%diffG6$-%\"yG6#%\"xGF,*&,&F)\"\"\"F,!\"\"F/,& F)F/F,F/F0" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%#icG/-%\"yG6#\"\"\"F) " }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/-%\"yG6#%\"xG-%'RootOfG6#,,*&\"\" #\"\"\"-%#lnG6#*&,&*$)F'F-F.F.*$)%#_ZGF-F.F.F.F'!\"#F.!\"\"*&\"\"%F.-% 'arctanG6#*&F8F.F'F:F.F:*&F " 0 "" {MPLTEXT 1 0 1 ";" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 41 "The solution can be given more \+ simply as " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 " ln(x^2+y^2)+2*arctan(y/x)=ln*2+Pi/2" "6#/,&- %#lnG6#,&*$%\"xG\"\"#\"\"\"*$%\"yGF+F,F,*&F+F,-%'arctanG6#*&F.F,F*!\" \"F,F,,&*&F&F,F+F,F,*&%#PiGF,F+F4F," }{TEXT -1 2 ". " }}{PARA 0 "" 0 " " {TEXT -1 18 "The solution (for " }{TEXT 283 1 "x" }{TEXT -1 47 " inc reasing) is the section of the polar curve " }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "r=sqrt(2)*exp(Pi/4-theta)" "6#/%\"rG*&- %%sqrtG6#\"\"#\"\"\"-%$expG6#,&*&%#PiGF*\"\"%!\"\"F*%&thetaGF2F*" } {TEXT -1 5 ", " }{XPPEDIT 18 0 "-Pi/4<=theta" "6#1,$*&%#PiG\"\"\"\" \"%!\"\"F)%&thetaG" }{XPPEDIT 18 0 "``<=Pi/4" "6#1%!G*&%#PiG\"\"\"\"\" %!\"\"" }{TEXT -1 2 ". " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 7 "Check: " }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 60 " ln((x^2+y^2))+2*arctan(y/x)=ln(2)+Pi/2;\nimplicitdiff(%,y,x);" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#/,&-%#lnG6#,&*$)%\"xG\"\"#\"\"\"F-*$)% \"yGF,F-F-F-*&F,F--%'arctanG6#*&F0F-F+!\"\"F-F-,&-F&6#F,F-*&F,F6%#PiGF -F-" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#,$*&,&%\"xG\"\"\"%\"yG!\"\"F',& F(F'F&F'F)F)" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 471 "p1 := plot([sqrt(2)*exp(Pi/4-t),t,t=-Pi/4..Pi/4 ],coords=polar,thickness=2,color=red):\np2 := plot([sqrt(2)*exp(Pi/4-t ),t,t=Pi/4..2*Pi],coords=polar,color=black,linestyle=2):\np3 := plot([ sqrt(2)*exp(Pi/4-t),t,t=-Pi/3..-Pi/4],coords=polar,color=black,linesty le=2):\np4 := plot([[[1,1],[uu,-uu]]$4],style=point,symbol=[circle$2,d iamond,cross],\n symbolsize=[12,10$3],color=[black,gr een$3]):\nplots[display]([p1,p2,p3,p4],font=[HELVETICA,9],labels=[`x`, `y(x)`]);" }}{PARA 13 "" 1 "" {GLPLOT2D 567 520 520 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GF+F,F,*&F+F,-%'arctanG6#*&F.F,F*!\"\"F,F,,&*&F&F,F+F,F,*&%#PiGF,F+F4F ," }{TEXT -1 1 " " }}{PARA 0 "" 0 "" {TEXT -1 4 "for " }{TEXT 281 1 "y " }{TEXT -1 25 " numerically in terms of " }{TEXT 282 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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748 "C := \+ (x,y) -> (y-x)/(y+x): hh := 0.01: numsteps := 375: x0 := 1: y0 := 1:\n matrix([[`slope field: `,C(x,y)],[`initial point: `,``(x0,y0)],[`ste p width: `,hh],\n[`no. of steps: `,numsteps]]);``;\nmthds := [`Pri nce-Dormand scheme`,`scheme with equal radii`,`scheme with `*t[6]=1/50 0,`scheme with `*t[6]=1/600,\n`scheme with `*t[6]=1/660]:\nerrs := []: vals := []:\nDigits := 25:\nfor ct to 5 do\n Cn_RK5_||ct := RK5_|| ct(C(x,y),x,y,x0,y0,hh,numsteps,false);\n sm := 0: numpts := nops(Cn _RK5_||ct):\n for ii to numpts do\n if ct=1 then vals := [op(va ls),phi(Cn_RK5_||ct[ii,1])] end if;\n sm := sm+(Cn_RK5_||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~w idth:~~~G$F,!\"#7$%1no.~of~steps:~~~G\"$v$Q)pprint186\"" }}{PARA 11 " " 1 "" {XPPMATH 20 "6#%!G" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#K%'matrix G6#7'7$%6Prince-Dormand~schemeG$\"+(p=A>$!#?7$%8scheme~with~equal~radi iG$\"+txf@BF+7$/*&%-scheme~with~G\"\"\"&%\"tG6#\"\"'F4#F4\"$+&$\"+2m=' 4%!#@7$/F2#F4\"$+'$\"+gN]$*GF+7$/F2#F4\"$g'$\"+8@9`9F+Q)pprint196\"" } }}{PARA 0 "" 0 "" {TEXT -1 2 " " }}{PARA 0 "" 0 "" {TEXT -1 30 "The f ollowing code constructs " }{TEXT 260 20 "numerical procedures" } {TEXT -1 56 " for solutions based on each of the Runge-Kutta schemes. " }}{PARA 0 "" 0 "" {TEXT -1 75 "The error in the value obtained by ea ch of the methods at the point where " }{XPPEDIT 18 0 "x = 4.749;" "6 #/%\"xG-%&FloatG6$\"%\\Z!\"$" }{TEXT -1 16 " is also given." }} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 611 "C := (x,y) -> (y-x)/(y+x): \+ hh := 0.01: numsteps := 375: x0 := 1: y0 := 1:\nmatrix([[`slope field: `,C(x,y)],[`initial point: `,``(x0,y0)],[`step width: `,hh],\n[`n o. of steps: `,numsteps]]);``;\nmthds := [`Prince-Dormand scheme`,`s cheme with equal radii`,`scheme with `*t[6]=1/500,`scheme with `*t[6]= 1/600,\n`scheme with `*t[6]=1/660]: errs := []:\nDigits := 25:\nfor ct to 5 do\n cn_RK5_||ct := RK5_||ct(C(x,y),x,y,x0,y0,hh,numsteps,true );\nend do:\nxx := 4.749: cxx := evalf(phi(xx)):\nfor ct to 5 do\n e rrs := [op(errs),abs(cn_RK5_||ct(xx)-cxx)];\nend do:\nDigits := 10:\nl inalg[transpose]([mthds,evalf(errs)]);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#K%'matrixG6#7&7$%0slope~field:~~~G*&,&%\"yG\"\"\"%\"xG!\"\"F,,&F +F,F-F,F.7$%0initial~point:~G-%!G6$F,F,7$%/step~width:~~~G$F,!\"#7$%1n o.~of~steps:~~~G\"$v$Q)pprint316\"" }}{PARA 11 "" 1 "" {XPPMATH 20 "6# %!G" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#K%'matrixG6#7'7$%6Prince-Dorman d~schemeG$\"+))[*f/%!#>7$%8scheme~with~equal~radiiG$\"+r3paHF+7$/*&%-s cheme~with~G\"\"\"&%\"tG6#\"\"'F4#F4\"$+&$\"+tb6;a!#?7$/F2#F4\"$+'$\"+ *\\o.n$F+7$/F2#F4\"$g'$\"+!HG5%=F+Q)pprint326\"" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 4 "The " }{TEXT 260 22 "root \+ mean square error" }{TEXT -1 20 " over the interval " }{XPPEDIT 18 0 "[1, 4.75];" "6#7$\"\"\"-%&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 0 372 "mthds := [`Prince-Dormand s cheme`,`scheme with equal radii`,`scheme with `*t[6]=1/500,`scheme wit h `*t[6]=1/600,\n`scheme with `*t[6]=1/660]: errs := []:\nDigits := 20 :\nfor ct to 5 do\n sm := NCint(('phi'(x)-'cn_RK5_||ct'(x))^2,x=1..4 .75,adaptive=false,numpoints=7,factor=200);\n errs := [op(errs),sqrt (sm/5)];\nend do:\nDigits := 10:\nlinalg[transpose]([mthds,evalf(errs) ]);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#K%'matrixG6#7'7$%6Prince-Dorman d~schemeG$\"+3E\"ep\"!#?7$%8scheme~with~equal~radiiG$\"+L'>.Q\"F+7$/*& %-scheme~with~G\"\"\"&%\"tG6#\"\"'F4#F4\"$+&$\"+OC)*fE!#@7$/F2#F4\"$+' $\"+l\"e$Q:F+7$/F2#F4\"$g'$\"+'e " 0 "" {MPLTEXT 1 0 435 "evalf[20](plot(['phi'(x )-'cn_RK5_1'(x),'phi'(x)-'cn_RK5_2'(x),'phi'(x)-'cn_RK5_3'(x),\n'phi'( x)-'cn_RK5_4'(x),'phi'(x)-'cn_RK5_5'(x)],x=1..3.75,font=[HELVETICA,9], \ncolor=[brown,red,COLOR(RGB,0,.7,0),COLOR(RGB,.5,0,1),COLOR(RGB,1,.35 ,0)],\nlegend=[`Prince-Dormand scheme`,`scheme with equal radii`,`sche me with t[6]=1/500`,`scheme with t[6]=1/600`,\n`scheme with t[6]=1/660 `],title=`error curves for 6 stage order 5 Runge-Kutta methods`));" }} 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