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-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 "Error" -1 8 1 {CSTYLE "" -1 -1 "Courier" 1 10 255 0 255 1 2 2 2 2 2 1 1 1 3 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 "Normal " -1 258 1 {CSTYLE "" -1 -1 "Times" 1 12 0 0 0 1 2 2 2 2 2 2 1 1 1 1 } 2 1 0 0 0 0 1 0 1 0 2 2 0 1 }{PSTYLE "Normal" -1 259 1 {CSTYLE "" -1 -1 "Times" 1 12 0 0 0 1 2 1 2 2 2 2 1 1 1 1 }1 1 0 0 0 0 1 0 1 0 2 2 0 1 }} {SECT 0 {PARA 3 "" 0 "" {TEXT -1 63 "The absolute stability region of \+ an explicit Runge-Kutta method" }}{PARA 0 "" 0 "" {TEXT -1 46 "by Pete r Stone, Gabriola Island, B.C., Canada " }}{PARA 0 "" 0 "" {TEXT -1 18 "Version: 5.12.2011" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 8 "restart;" }}}{EXCHG {PARA 0 "> " 0 " " {MPLTEXT 1 0 1 ";" }}}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 82 "load proc edures for constructing Runge-Kutta schemes and for root-finding metho ds " }}{PARA 0 "" 0 "" {TEXT -1 18 "The Maple m-files " }{TEXT 262 9 " butcher.m" }{TEXT -1 5 " and " }{TEXT 262 7 "roots.m" }{TEXT -1 33 " a re required by this worksheet. " }}{PARA 0 "" 0 "" {TEXT -1 134 "They \+ can be read into a Maple session by commands similar to those that fol low, where each file path gives the location of the m-file." }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 69 "read \"C:\\\\Maple/procdrs/butcher. m\";\nread \"C:\\\\Maple/procdrs/roots.m\";" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}{PARA 0 "" 0 "" {TEXT -1 86 "#------------------------------------- ------------------------------------------------" }}{SECT 1 {PARA 4 " " 0 "" {TEXT -1 63 "The absolute stability region of an explicit Runge -Kutta method" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 18 "Consider an order " }{TEXT 265 1 "p" }{TEXT -1 50 " expli cit Runge-Kutta method with s stages where " }{XPPEDIT 18 0 "p<=s" "6 #1%\"pG%\"sG" }{TEXT -1 2 ". " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }} {PARA 0 "" 0 "" {TEXT -1 4 "Let " }}{PARA 256 "" 0 "" {TEXT -1 1 " " } {XPPEDIT 18 0 "[A] = matrix([[a[1,1], a[1,2], a[1,3], `.`, `.`, a[1,s- 1], a[1,s]], [a[2,1], a[2,2], a[2,3], `.`, `.`, a[2,s-1], a[2,s]], [a[ 3,1], a[3,2], a[3,3], `.`, `.`, a[3,s-1], a[3,s]], [a[4,1], a[4,2], a[ 4,3], `.`, `.`, a[4,s-1], a[4,s]], [`.`, `.`, `.`, `.`, `.`, `.`, `.`] , [`.`, `.`, `.`, ` . `, `.`, `.`, `.`], [a[s,1], a[s,2], a[s,3], ` . `, ` . `, a[s,s-1], a[s,s]]]);" "6#/7#%\"AG-%'matrixG6#7)7)&%\"aG6 $\"\"\"F.&F,6$F.\"\"#&F,6$F.\"\"$%\".GF5&F,6$F.,&%\"sGF.F.!\"\"&F,6$F. F97)&F,6$F1F.&F,6$F1F1&F,6$F1F4F5F5&F,6$F1,&F9F.F.F:&F,6$F1F97)&F,6$F4 F.&F,6$F4F1&F,6$F4F4F5F5&F,6$F4,&F9F.F.F:&F,6$F4F97)&F,6$\"\"%F.&F,6$F XF1&F,6$FXF4F5F5&F,6$FX,&F9F.F.F:&F,6$FXF97)F5F5F5F5F5F5F57)F5F5F5%$~. ~GF5F5F57)&F,6$F9F.&F,6$F9F1&F,6$F9F4%&~~.~~GFfo&F,6$F9,&F9F.F.F:&F,6$ F9F9" }{TEXT -1 1 " " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 " " {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 78 "be the square matrix wh ose entries are the linking coefficients of the method." }}{PARA 0 "" 0 "" {TEXT -1 20 "Since the method is " }{TEXT 260 8 "explicit" } {TEXT -1 3 ", " }{XPPEDIT 18 0 "a[i,j]=0" "6#/&%\"aG6$%\"iG%\"jG\"\"! " }{TEXT -1 9 ", where " }{XPPEDIT 18 0 "i<=j" "6#1%\"iG%\"jG" } {TEXT -1 11 ", that is, " }}{PARA 256 "" 0 "" {TEXT -1 1 " " } {XPPEDIT 18 0 "[A] = matrix([[0, 0, 0, `.`, `.`, 0, 0], [a[2,1], 0, 0, `.`, `.`, 0, 0], [a[3,1], a[3,2], 0, `.`, `.`, 0, 0], [a[4,1], a[4,2] , a[4,3], `.`, `.`, 0, 0], [`.`, `.`, `.`, `.`, `.`, `.`, `.`], [`.`, \+ `.`, `.`, ` . `, `.`, `.`, `.`], [a[s,1], a[s,2], a[s,3], ` . `, ` \+ . `, a[s,s-1], 0]]);" "6#/7#%\"AG-%'matrixG6#7)7)\"\"!F+F+%\".GF,F+F+ 7)&%\"aG6$\"\"#\"\"\"F+F+F,F,F+F+7)&F/6$\"\"$F2&F/6$F6F1F+F,F,F+F+7)&F /6$\"\"%F2&F/6$F " }{TEXT 289 1 "s" }{TEXT -1 10 ", so that " }} {PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "y[n] = y[n-1]*(1+Sum( z^i*[b]^T*[A]^(i-1)*[1],i = 1 .. s));" "6#/&%\"yG6#%\"nG*&&F%6#,&F'\" \"\"F,!\"\"F,,&F,F,-%$SumG6$**)%\"zG%\"iGF,)7#%\"bG%\"TGF,)7#%\"AG,&F5 F,F,F-F,7#F,F,/F5;F,%\"sGF,F," }{TEXT -1 20 " ------- (iii). " }} {PARA 256 "" 0 "" {TEXT -1 1 " " }{TEXT 291 26 "______________________ ____" }{TEXT -1 23 " " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 23 "The row sum conditions " }} {PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "Sum(a[i,j],j=1..s)=c[ i]" "6#/-%$SumG6$&%\"aG6$%\"iG%\"jG/F+;\"\"\"%\"sG&%\"cG6#F*" }{TEXT -1 3 ", " }{XPPEDIT 18 0 "i=1" "6#/%\"iG\"\"\"" }{TEXT -1 12 ", 2 . . . , " }{TEXT 276 1 "s" }{TEXT -1 1 "," }}{PARA 0 "" 0 "" {TEXT -1 33 "can be written in the matrix form" }}{PARA 256 "" 0 "" {TEXT -1 1 " \+ " }{XPPEDIT 18 0 "[A]*[1]= [c]" "6#/*&7#%\"AG\"\"\"7#F'F'7#%\"cG" } {TEXT -1 1 "." }}{PARA 0 "" 0 "" {TEXT -1 34 "The (abreviated) order c onditions " }}{PARA 256 "" 0 "" {TEXT -1 2 " " }{XPPEDIT 18 0 "b*e = \+ 1" "6#/*&%\"bG\"\"\"%\"eGF&F&" }{TEXT -1 3 ", " }{XPPEDIT 18 0 "b*c=1 /2" "6#/*&%\"bG\"\"\"%\"cGF&*&F&F&\"\"#!\"\"" }{TEXT -1 3 ", " } {XPPEDIT 18 0 "b*``(a*c) = 1/6" "6#/*&%\"bG\"\"\"-%!G6#*&%\"aGF&%\"cGF &F&*&F&F&\"\"'!\"\"" }{TEXT -1 3 ", " }{XPPEDIT 18 0 "b*``(a*``(a*c)) = 1/24" "6#/*&%\"bG\"\"\"-%!G6#*&%\"aGF&-F(6#*&F+F&%\"cGF&F&F&*&F&F& \"#C!\"\"" }{TEXT -1 3 ", " }{XPPEDIT 18 0 "b*``(a*``(a*``(a*c))) = 1 /120" "6#/*&%\"bG\"\"\"-%!G6#*&%\"aGF&-F(6#*&F+F&-F(6#*&F+F&%\"cGF&F&F &F&*&F&F&\"$?\"!\"\"" }{TEXT -1 10 ", . . . , " }{XPPEDIT 18 0 "b*``(a *``(a*``*` . . .`*(a*c))) = 1/p!" "6#/*&%\"bG\"\"\"-%!G6#*&%\"aGF&-F(6 #**F+F&F(F&%'~.~.~.GF&*&F+F&%\"cGF&F&F&F&*&F&F&-%*factorialG6#%\"pG!\" \"" }{TEXT -1 1 " " }}{PARA 0 "" 0 "" {TEXT -1 31 "then give rise to t he relations" }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "[b]^T *[A]^(k-1)*[1]=1/k!" "6#/*()7#%\"bG%\"TG\"\"\")7#%\"AG,&%\"kGF)F)!\"\" F)7#F)F)*&F)F)-%*factorialG6#F.F/" }{TEXT -1 3 ", " }{XPPEDIT 18 0 "k =1" "6#/%\"kG\"\"\"" }{TEXT -1 13 ", 2, . . . , " }{TEXT 277 1 "p" } {TEXT -1 1 "." }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{TEXT 282 15 "_____ __________" }{TEXT -1 26 " " }}{PARA 0 "" 0 " " {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 45 "These relations enable \+ (iii) to be written as" }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "y[n] = y[n-1]*(1+z+z^2/2!+` . . . `+z^p/p!+Sum(z^i*[b]^T*[A]^(i- 1)*[1],i = p+1 .. infinity));" "6#/&%\"yG6#%\"nG*&&F%6#,&F'\"\"\"F,!\" \"F,,.F,F,%\"zGF,*&F/\"\"#-%*factorialG6#F1F-F,%(~.~.~.~GF,*&)F/%\"pGF ,-F36#F8F-F,-%$SumG6$**)F/%\"iGF,)7#%\"bG%\"TGF,)7#%\"AG,&F@F,F,F-F,7# F,F,/F@;,&F8F,F,F,%)infinityGF,F," }{TEXT -1 1 "." }}{PARA 0 "" 0 "" {TEXT -1 7 "Since [" }{TEXT 278 1 "A" }{TEXT -1 23 "] is lower triangu lar, " }{XPPEDIT 18 0 "[A]^(j-1)=0 " "6#/)7#%\"AG,&%\"jG\"\"\"F)!\"\" \"\"!" }{TEXT -1 7 " when " }{TEXT 280 1 "j" }{TEXT -1 3 " > " } {TEXT 279 1 "s" }{TEXT -1 10 ", so that " }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "y[n] = y[n-1]*(1+z+z^2/2!+` . . . `+z^p/p!+Su m(z^i*[b]^T*[A]^(i-1)*[1],i = p+1 .. s));" "6#/&%\"yG6#%\"nG*&&F%6#,&F '\"\"\"F,!\"\"F,,.F,F,%\"zGF,*&F/\"\"#-%*factorialG6#F1F-F,%(~.~.~.~GF ,*&)F/%\"pGF,-F36#F8F-F,-%$SumG6$**)F/%\"iGF,)7#%\"bG%\"TGF,)7#%\"AG,& F@F,F,F-F,7#F,F,/F@;,&F8F,F,F,%\"sGF,F," }{TEXT -1 1 "." }}{PARA 0 "" 0 "" {TEXT -1 5 "Thus " }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "y[n]=R(z)*y[n-1]" "6#/&%\"yG6#%\"nG*&-%\"RG6#%\"zG\"\"\"&F%6#,&F 'F-F-!\"\"F-" }{TEXT -1 2 ", " }}{PARA 0 "" 0 "" {TEXT -1 7 "where " }{XPPEDIT 18 0 "z=h*lambda" "6#/%\"zG*&%\"hG\"\"\"%'lambdaGF'" }{TEXT -1 6 " and " }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "R(z) \+ = 1+z+z^2/2!+` . . . `+z^p/p!+``;" "6#/-%\"RG6#%\"zG,.\"\"\"F)F'F)*&F' \"\"#-%*factorialG6#F+!\"\"F)%(~.~.~.~GF)*&)F'%\"pGF)-F-6#F3F/F)%!GF) " }{XPPEDIT 18 0 "Sum(z^i*[b]^T*[A]^(i-1)*[1],i = p+1 .. s);" "6#-%$Su mG6$**)%\"zG%\"iG\"\"\")7#%\"bG%\"TGF*)7#%\"AG,&F)F*F*!\"\"F*7#F*F*/F) ;,&%\"pGF*F*F*%\"sG" }{TEXT -1 17 " ------- (iv). " }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{TEXT 281 34 "__________________________________ " }{TEXT -1 22 " " }}{PARA 0 "" 0 "" {TEXT -1 0 " " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 4 "The " }{TEXT 260 28 "region of absolute stability" }{TEXT -1 103 " (or absol ute stability region) of the Runge-Kutta method is the region in the c omplex plane for which " }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "abs(R(z))<=1" "6#1-%$absG6#-%\"RG6#%\"zG\"\"\"" }{TEXT -1 1 "." }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{TEXT 283 7 "_______" }{TEXT -1 1 " " }}{PARA 0 "" 0 "" {TEXT -1 16 "R is called the " }{TEXT 260 17 "st abilty function" }{TEXT -1 27 " of the Runge-Kutta method." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 67 "Points on the bou ndary region can be found by solving the equation " }}{PARA 256 "" 0 " " {TEXT -1 1 " " }{XPPEDIT 18 0 "R(z)=exp(theta*i)" "6#/-%\"RG6#%\"zG- %$expG6#*&%&thetaG\"\"\"%\"iGF-" }{TEXT -1 1 " " }}{PARA 0 "" 0 "" {TEXT -1 22 "for various values of " }{XPPEDIT 18 0 "theta" "6#%&theta G" }{TEXT -1 1 "." }}{PARA 0 "" 0 "" {TEXT -1 79 "This can be done by \+ applying Newton's method. Starting from a particular point " } {XPPEDIT 18 0 "z[0]" "6#&%\"zG6#\"\"!" }{TEXT -1 60 " on the boundary \+ curve associated with the parameter value " }{XPPEDIT 18 0 "theta[0] " "6#&%&thetaG6#\"\"!" }{TEXT -1 13 ", the value " }{XPPEDIT 18 0 "z[ 1]" "6#&%\"zG6#\"\"\"" }{TEXT -1 51 " for a nearby point on the curve corresponding to " }{XPPEDIT 18 0 "theta[1]" "6#&%&thetaG6#\"\"\"" } {TEXT -1 23 " may be found by using " }{XPPEDIT 18 0 "z[0]" "6#&%\"zG6 #\"\"!" }{TEXT -1 41 " as the starting approximation to solve " } {XPPEDIT 18 0 "R(z) = exp(theta[1]*i)" "6#/-%\"RG6#%\"zG-%$expG6#*&&%& thetaG6#\"\"\"F/%\"iGF/" }{TEXT -1 5 " for " }{XPPEDIT 18 0 "z[1]" "6# &%\"zG6#\"\"\"" }{TEXT -1 1 "." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }} {PARA 0 "" 0 "" {TEXT -1 21 "For methods of order " }{XPPEDIT 18 0 "p< =4" "6#1%\"pG\"\"%" }{TEXT -1 7 " where " }{XPPEDIT 18 0 "s=p" "6#/%\" sG%\"pG" }{TEXT -1 113 " the region of absolute stability does not dep end on the coefficients of the method and is the region for which " } }{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "abs(1+z+z^2/2!+` . . . `+z^p/p!) <= 1;" "6#1-%$absG6#,,\"\"\"F(%\"zGF(*&F)\"\"#-%*factoria lG6#F+!\"\"F(%(~.~.~.~GF(*&)F)%\"pGF(-F-6#F3F/F(F(" }{TEXT -1 1 "." }} {PARA 256 "" 0 "" {TEXT -1 1 " " }{TEXT 284 15 "_______________" } {TEXT -1 3 " " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 88 "The following picture shows the outer boundary of the reg ions of absolute stability for " }{TEXT 285 1 "p" }{TEXT -1 37 " stage Runge-Kutta methods of order " }{XPPEDIT 18 0 "p=1" "6#/%\"pG\"\"\" " }{TEXT -1 10 ", 2, 3, 4." }}{PARA 0 "" 0 "" {TEXT -1 43 "The regions become progressively larger as " }{TEXT 286 1 "p" }{TEXT -1 11 " incr eases." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 988 "pts := []:\nz0 := 0:\nfor ct from 0 to 80 do\n zz \+ := newton(1+z=exp(ct*Pi/40*I),z=z0):\n z0 := zz:\n pts := [op(pts) ,[Re(zz),Im(zz)]]:\nend do:\np1 := plot(pts,thickness=2,style=line,col or=COLOR(RGB,0,.2,1)):\npts := []:\nz0 := 0:\nfor ct from 0 to 80 do\n zz := newton(1+z+z^2/2=exp(ct*Pi/20*I),z=z0):\n z0 := zz:\n pts := [op(pts),[Re(zz),Im(zz)]]:\nend do:\np2 := plot(pts,thickness=2,st yle=line,color=COLOR(RGB,0,.65,0)):\npts := []:\nz0 := 0:\nfor ct from 0 to 120 do\n zz := newton(1+z+z^2/2+z^3/6=exp(ct*Pi/20*I),z=z0):\n z0 := zz:\n pts := [op(pts),[Re(zz),Im(zz)]]:\nend do:\np3 := plo t(pts,thickness=2,style=line):\npts := []:\nz0 := 0:\nfor ct from 0 to 160 do\n zz := newton(1+z+z^2/2+z^3/6+z^4/24=exp(ct*Pi/20*I),z=z0): \n z0 := zz:\n pts := [op(pts),[Re(zz),Im(zz)]]:\nend do:\np4 := p lot(pts,thickness=2,color=COLOR(RGB,0,.6,.8),style=line):\nplots[displ ay]([p1,p2,p3,p4],font=[HELVETICA,9],view=[-3.19..0.59,-3.19..3.19],\n labels=[`Re(z)`,`Im(z)`],axes=boxed,scaling=constrained);" }}{PARA 13 "" 1 "" {GLPLOT2D 614 614 614 {PLOTDATA 2 "6+-%'CURVESG6&7]p7$$\"\" !F)F(7$$!3A+++nim#3$!#?$\"3w*****Hd4f%y!#>7$$!3$*******Rf;J7F0$\"3'*** ****\\YMk:!#=7$$!3%*******fz+jFF0$\"33+++ROXMBF67$$!3%)******p$[V*[F0$ \"3A+++W*p,4$F67$$!3h******[n/7wF0$\"3B+++CV$o#QF67$$!3-+++eZ$**3\"F6$ \"3%******p*\\!*RXF67$$!3))*****fN)ft9F6$\"3,+++Zc)\\A&F67$$!30+++c+$) 4>F6$\"31+++BD&y(eF67$$!3))*****RMSfR#F6$\"3C+++$[![%\\'F67$$!3@+++)=K *GHF6$\"3M+++7y1rqF67$$!3v*****p^>b]$F6$\"3c*****flfSg(F67$$!3&******p ZZ@7%F6$\"3m*****R%*p,4)F67$$!3)******HN9]x%F6$\"36+++W;SE&)F67$$!3]** ***H+&4gaF6$\"3Q+++U_15*)F67$$!3y*****fnlJ<'F6$\"3*)*****\\K&zQ#*F67$$ !3L+++c+$)4pF6$\"3))*****H;l0^*F67$$!3k*****4OYbm(F6$\"3K+++/#*pB(*F67 $$!3x******\\`lN%)F6$\"3#******fS$)o()*F67$$!3>+++V!4a@*F6$\"3?+++PL

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0 0 "Curve 1" "Curve 2" "Curve 3" "Curve 4" }}}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}} {PARA 0 "" 0 "" {TEXT -1 86 "#---------------------------------------- ---------------------------------------------" }}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 56 "The construction of stability functions using tall tre es" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 257 "" 0 "" {TEXT -1 19 "U sing the relations" }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "[b]^T*[1] = 1;" "6#/*&)7#%\"bG%\"TG\"\"\"7#F)F)F)" }{TEXT -1 1 " " }} {PARA 0 "" 0 "" {TEXT -1 4 "and " }}{PARA 256 "" 0 "" {TEXT -1 1 " " } {XPPEDIT 18 0 "[A]*[1]=[c]" "6#/*&7#%\"AG\"\"\"7#F'F'7#%\"cG" }{TEXT -1 2 ", " }}{PARA 0 "" 0 "" {TEXT -1 12 "the equation" }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "y[n] = y[n-1]*(1+Sum(z^i*[b]^T*[A] ^(i-1)*[1],i = 1 .. s));" "6#/&%\"yG6#%\"nG*&&F%6#,&F'\"\"\"F,!\"\"F,, &F,F,-%$SumG6$**)%\"zG%\"iGF,)7#%\"bG%\"TGF,)7#%\"AG,&F5F,F,F-F,7#F,F, /F5;F,%\"sGF,F," }{TEXT -1 19 " ------- (iii). " }}{PARA 257 "" 0 " " {TEXT -1 51 "of the previous section can be written in the form:" }} {PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "y[n] = y[n-1]*(1+z+Su m(z^i*[b]^T*[A]^(i-2)*[c],i = 2 .. s));" "6#/&%\"yG6#%\"nG*&&F%6#,&F' \"\"\"F,!\"\"F,,(F,F,%\"zGF,-%$SumG6$**)F/%\"iGF,)7#%\"bG%\"TGF,)7#%\" AG,&F5F,\"\"#F-F,7#%\"cGF,/F5;F>%\"sGF,F," }{TEXT -1 1 "." }}{PARA 0 " " 0 "" {TEXT -1 33 "Similarly, for a method of order " }{TEXT 292 1 "p " }{TEXT -1 10 ", equation" }}{PARA 256 "" 0 "" {TEXT -1 1 " " } {XPPEDIT 18 0 "R(z) = 1+z+z^2/2!+` . . . `+z^p/p!+``;" "6#/-%\"RG6#%\" zG,.\"\"\"F)F'F)*&F'\"\"#-%*factorialG6#F+!\"\"F)%(~.~.~.~GF)*&)F'%\"p GF)-F-6#F3F/F)%!GF)" }{XPPEDIT 18 0 "Sum(z^i*[b]^T*[A]^(i-1)*[1],i = p +1 .. s);" "6#-%$SumG6$**)%\"zG%\"iG\"\"\")7#%\"bG%\"TGF*)7#%\"AG,&F)F *F*!\"\"F*7#F*F*/F);,&%\"pGF*F*F*%\"sG" }{TEXT -1 17 " ------- (iv). \+ " }}{PARA 0 "" 0 "" {TEXT -1 27 "can be written in the form:" }} {PARA 256 "" 0 "" {TEXT -1 2 " " }{XPPEDIT 18 0 "R(z) = 1+z+z^2/2!+` \+ . . . `+z^p/p!+``;" "6#/-%\"RG6#%\"zG,.\"\"\"F)F'F)*&F'\"\"#-%*factori alG6#F+!\"\"F)%(~.~.~.~GF)*&)F'%\"pGF)-F-6#F3F/F)%!GF)" }{XPPEDIT 18 0 "Sum(z^i*[b]^T*[A]^(i-2)*[c],i = p+1 .. s);" "6#-%$SumG6$**)%\"zG%\" iG\"\"\")7#%\"bG%\"TGF*)7#%\"AG,&F)F*\"\"#!\"\"F*7#%\"cGF*/F);,&%\"pGF *F*F*%\"sG" }{TEXT -1 2 ". " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 14 "The expression" }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "[b]^T*[A]^(k-2)*[c];" "6#*()7#%\"bG%\"TG\"\"\")7#% \"AG,&%\"kGF(\"\"#!\"\"F(7#%\"cGF(" }{TEXT -1 1 " " }}{PARA 0 "" 0 "" {TEXT -1 72 "can be identified with the left hand side of the simple o rder condition " }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "b* ``(a*``(a*``*` . . .`*a*c)) = 1/k!" "6#/*&%\"bG\"\"\"-%!G6#*&%\"aGF&-F (6#*,F+F&F(F&%'~.~.~.GF&F+F&%\"cGF&F&F&*&F&F&-%*factorialG6#%\"kG!\"\" " }}{PARA 0 "" 0 "" {TEXT -1 19 "in which there are " }{XPPEDIT 18 0 " ``(k-2)" "6#-%!G6#,&%\"kG\"\"\"\"\"#!\"\"" }{TEXT -1 2 " " }{TEXT 293 1 "a" }{TEXT -1 4 "' s." }}{PARA 0 "" 0 "" {TEXT -1 58 "This order condition is associated with a tree which has " }{TEXT 294 1 "k" } {TEXT -1 12 " nodes and " }{XPPEDIT 18 0 "``(k-1);" "6#-%!G6#,&%\"kG \"\"\"F(!\"\"" }{TEXT -1 37 " vertical segments with no branching." }} {PARA 0 "" 0 "" {TEXT -1 22 "Such trees are called " }{TEXT 260 10 "ta ll trees" }{TEXT -1 2 ". " }}{PARA 0 "" 0 "" {TEXT -1 31 "For example, the tall tree tree" }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{GLPLOT2D 92 239 239 {PLOTDATA 2 "6,-%'CURVESG6&7&7$$\"\"!F)F(7$F($\"\"\"F)7$F($\" \"#F)7$F($\"\"$F)-%'SYMBOLG6#%'CIRCLEG-%&COLORG6&%$RGBGF(F(F(-%&STYLEG 6#%&POINTG-F$6&F&-F46#%(DIAMONDGF7F;-F$6&F&-F46#%&CROSSGF7F;-F$6%7$F'F *-F86&F:F+F(F(-%*THICKNESSG6#F/-F$6%7$F*F-FLFN-F$6%7$F-F0FLFN-F$6%7#F0 -%'COLOURG6&F:$\"#5!\"\"F(F(-F<6#%%LINEG-%+AXESLABELSG6%Q!6\"F`o-%%FON TG6#%(DEFAULTG-%*AXESSTYLEG6#%%NONEG-%%VIEWG6$FeoFeo" 1 2 0 1 10 0 2 9 1 1 2 1.000000 45.000000 45.000000 0 0 "Curve 1" "Curve 2" "Curve 3 " "Curve 4" "Curve 5" "Curve 6" "Curve 7" }}}{PARA 0 "" 0 "" {TEXT -1 19 "is associated with " }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "b*``(a*``(a*c)) = 1/4!" "6#/*&%\"bG\"\"\"-%!G6#*&%\"aGF&-F(6#*&F +F&%\"cGF&F&F&*&F&F&-%*factorialG6#\"\"%!\"\"" }{TEXT -1 2 ". " }} {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 101 "This gives another method of constructing the \+ stability function of a Runge-Kutta method using Maple." }}{PARA 0 "" 0 "" {TEXT -1 125 "We use this method of construction to obtain the st ability function of the Prince-Dormand 6 stage order 5 Runge-Kutta met hod." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 60 "coefficient s for the Prince-Dormand 6 stage, order 5 scheme" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 396 "# coefficie nts by Prince and Dormand\nee := \{c[2]=1/5,c[3]=3/10,c[4]=4/5,c[5]=8/ 9,c[6]=1,\na[2,1]=1/5,\na[3,1]=3/40,a[3,2]=9/40,\na[4,1]=44/45,a[4,2]= -56/15,a[4,3]=32/9,\na[5,1]=19372/6561,a[5,2]=-25360/2187,a[5,3]=64448 /6561,a[5,4]=-212/729,\na[6,1]=9017/3168,a[6,2]=-355/33,a[6,3]=46732/5 247,a[6,4]=49/176,a[6,5]=-5103/18656,\nb[1]=35/384,b[2]=0,b[3]=500/111 3,b[4]=125/192,b[5]=-2187/6784,b[6]=11/84\}:" }}}{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 0 303 "s tages_ := 6:\norder_ := 5:\ntt[1] :=[]:\nphi := u_->[u_]:\nfor ct to s tages_ do tt[ct+1] := phi(tt[ct]) end do:\ncffs := [seq(subs(_S=stages _,TreeOrderCondition(tt[ii])),ii=1..stages_)]:\ncffs := map(u_->lhs(u_ ),%):\n1+add(cffs[ii]*z^ii,ii=1..stages_);\nsubs('Sum'='add',%):\nsubs (c[1]=0,value(%));\nsubs(ee,%);\n" }}{PARA 12 "" 1 "" {XPPMATH 20 "6#, 0\"\"\"F$*&%\"zGF$-%$SumG6$&%\"bG6#%\"iG/F-;F$\"\"'F$F$*&)F&\"\"#F$-F( 6$*&F*F$&%\"cGF,F$F.F$F$*&)F&\"\"$F$-F(6$*&F*F$-F(6$*&&%\"aG6$F-%\"jGF $&F86#FEF$/FE;F$,&F-F$F$!\"\"F$F.F$F$*&)F&\"\"%F$-F(6$*&F*F$-F(6$*&FBF $-F(6$*&&FC6$FE%\"kGF$&F86#FZF$/FZ;F$,&FEF$F$FKF$FHF$F.F$F$*&)F&\"\"&F $-F(6$*&F*F$-F(6$*&FBF$-F(6$*&FXF$-F(6$*&&FC6$FZ%\"lGF$&F86#F[pF$/F[p; F$,&FZF$F$FKF$FgnF$FHF$F.F$F$*&-F(6$*&F*F$-F(6$*&FBF$-F(6$*&FXF$-F(6$* &FioF$-F(6$*&&FC6$F[p%\"mGF$&F86#FcqF$/Fcq;F$,&F[pF$F$FKF$F^pF$FgnF$FH F$F.F$)F&F0F$F$" }}{PARA 12 "" 1 "" {XPPMATH 20 "6#,0\"\"\"F$*&,.&%\"b G6#\"\"'F$&F(6#\"\"&F$&F(6#\"\"#F$&F(6#\"\"$F$&F(6#\"\"%F$&F(6#F$F$F$% \"zGF$F$*&)F9F0F$,,*&F.F$&%\"cGF/F$F$*&F1F$&F?F2F$F$*&F4F$&F?F5F$F$*&F +F$&F?F,F$F$*&F'F$&F?F)F$F$F$F$*&)F9F3F$,**(F1F$&%\"aG6$F3F0F$F>F$F$*& F4F$,&*&&FM6$F6F0F$F>F$F$*&&FM6$F6F3F$FAF$F$F$F$*&F+F$,(*&&FM6$F-F0F$F >F$F$*&&FM6$F-F3F$FAF$F$*&&FM6$F-F6F$FCF$F$F$F$*&F'F$,**&&FM6$F*F0F$F> F$F$*&&FM6$F*F3F$FAF$F$*&&FM6$F*F6F$FCF$F$*&&FM6$F*F-F$FEF$F$F$F$F$F$* &)F9F6F$,(**F4F$FUF$FLF$F>F$F$*&F+F$,&*(FgnF$FLF$F>F$F$*&FjnF$FPF$F$F$ F$*&F'F$,(*(FboF$FLF$F>F$F$*&FeoF$FPF$F$*&FhoF$FXF$F$F$F$F$F$*&)F9F-F$ ,&*,F+F$FjnF$FUF$FLF$F>F$F$*&F'F$,&**FeoF$FUF$FLF$F>F$F$*&FhoF$F_pF$F$ F$F$F$F$*0F'F$FhoF$FjnF$FUF$FLF$F>F$)F9F*F$F$" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#,0*&#\"\"\"\"$+'F&*$)%\"zG\"\"'F&F&F&*&#F&\"$?\"F&*$)F* \"\"&F&F&F&*&#F&\"#CF&*$)F*\"\"%F&F&F&*&#F&F+F&*$)F*\"\"$F&F&F&*&#F&\" \"#F&*$)F*F?F&F&F&F*F&F&F&" }}}{PARA 0 "" 0 "" {TEXT -1 1 " " }}{PARA 0 "" 0 "" {TEXT -1 164 "In the following modification of the previous \+ computation we make use of the simplification of the left side of the \+ order condition to its right side for terms in " }{XPPEDIT 18 0 "z^i " "6#)%\"zG%\"iG" }{TEXT -1 9 ", where " }{TEXT 287 1 "i" }{TEXT -1 50 " is less than or equal to the order of the method." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 334 "stages _ := 6:\norder_ := 5:\ntt[1] :=[]:\nphi := u_->[u_]:\nfor ct to stages do tt[ct+1] := phi(tt[ct]) end do:\ncffs := [seq(subs(_S=stages_,Tree OrderCondition(tt[ii])),ii=1..stages_)]:\ncffs := map(u_->`if`(rhs(u_) >=1/order_!,rhs(u_),lhs(u_)),%):\n1+add(cffs[ii]*z^ii,ii=1..stages_); \nsubs('Sum'='add',%):\nsubs(c[1]=0,value(%));\nsubs(ee,%);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#,0\"\"\"F$%\"zGF$*&\"\"#!\"\"F%F'F$*&\"\"'F( F%\"\"$F$*&\"#CF(F%\"\"%F$*&\"$?\"F(F%\"\"&F$*&-%$SumG6$*&&%\"bG6#%\"i GF$-F46$*&&%\"aG6$F:%\"jGF$-F46$*&&F?6$FA%\"kGF$-F46$*&&F?6$FG%\"lGF$- F46$*&&F?6$FM%\"mGF$&%\"cG6#FSF$/FS;F$,&FMF$F$F(F$/FM;F$,&FGF$F$F(F$/F G;F$,&FAF$F$F(F$/FA;F$,&F:F$F$F(F$/F:;F$F*F$)F%F*F$F$" }}{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$&F7 6$F1F.F$&F76$F.F+F$&F76$F+F'F$&%\"cG6#F'F$)F%F*F$F$" }}{PARA 11 "" 1 " " {XPPMATH 20 "6#,0*&#\"\"\"\"$+'F&*$)%\"zG\"\"'F&F&F&*&#F&\"$?\"F&*$) F*\"\"&F&F&F&*&#F&\"#CF&*$)F*\"\"%F&F&F&*&#F&F+F&*$)F*\"\"$F&F&F&*&#F& \"\"#F&*$)F*F?F&F&F&F*F&F&F&" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }} {PARA 0 "" 0 "" {TEXT -1 22 "######################" }}{PARA 0 "" 0 " " {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 118 "The following computat ion obtains the stability function by a simple \"bootstraps\" method b ased on direct substitution." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 155 "X[1] := 1:\nfor ii from 2 t o 6 do\n X[ii] := 1+z*add(a[ii,jj]*X[jj],jj=1..ii-1);\nend do:\n1+z* add(b[i]*X[i],i=1..6):\ncollect(expand(%),z):\nsort(subs(ee,%));" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#,0*&#\"\"\"\"$+'F&*$)%\"zG\"\"'F&F&F&* &#F&\"$?\"F&*$)F*\"\"&F&F&F&*&#F&\"#CF&*$)F*\"\"%F&F&F&*&#F&F+F&*$)F* \"\"$F&F&F&*&#F&\"\"#F&*$)F*F?F&F&F&F*F&F&F&" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}}{PARA 0 "" 0 "" {TEXT -1 86 "#---------------------------------------------- ---------------------------------------" }}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 33 "Graphing a stability region with " }{TEXT 0 11 "contourpl ot" }{TEXT -1 1 " " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT 259 9 "Example 1" }{TEXT -1 1 ":" }}{PARA 0 "" 0 "" {TEXT -1 0 " " }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 69 "1+z+z^2/2+z^3/3!+z^4/4!; \nlambda := unapply(abs(subs(z=x+I*y,%)),x,y);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#,,\"\"\"F$%\"zGF$*&#F$\"\"#F$*$)F%F(F$F$F$*&#F$\"\"'F$* $)F%\"\"$F$F$F$*&#F$\"#CF$*$)F%\"\"%F$F$F$" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%'lambdaGf*6$%\"xG%\"yG6\"6$%)operatorG%&arrowGF)-%$ab sG6#,.\"\"\"F19$F1*&9%F1^#F1F1F1*&#F1\"\"#F1*$),&F2F1F3F1F8F1F1F1*&#F1 \"\"'F1*$)F;\"\"$F1F1F1*&#F1\"#CF1*$)F;\"\"%F1F1F1F)F)F)" }}}{PARA 0 " " 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 358 "p1 : = plots[contourplot](lambda(x,y),x=-3..0.3,y=-3..3,contours=[1],\n \+ grid=[50,50],filled=true,coloring=[COLOR(RGB,.45,.5,.9),white]): \np2 := plot([[[-3.19,0],[0.59,0]],[[0,-3.19],[0,3.19]]],color=black,l inestyle=3):\nplots[display]([p1,p2],font=[HELVETICA,9],view=[-3.19..0 .59,-3.19..3.19],labels=[`Re(z)`,`Im(z)`],\n axes=boxed,scaling=cons trained);" }}{PARA 13 "" 1 "" {GLPLOT2D 408 499 499 {PLOTDATA 2 "6,-%' CURVESG6^`l7$7$$!3iez*[C71t#!#<$!3a%[JI0+.a(!#=7$$!3'4,2%fs$[u#F*$!3c9 5bxQpMnF-7$F.7$$!3e#[)e3qo\\FF*$!39g(oA**zyQ'F-7$7$$!3s5eQHxZgFF*$!33[ Ej\"3/-^&F-F47$7$$!3E5eQHxZgFF*F=7$$!31=*)*G!p&ow#F*$!3+Gqz*H57&[F-7$7 $$!3e>S#4hx8x#F*$!3[!G9dG9dG%F-FC7$7$FJ$!3/\"G9dG9dG%F-7$$!3)epx%)**Hm x#F*$!3g(y\"*p&y-\\MF-7$7$$!3[2n4Sn[yFF*$!3)G\"fz*[C71$F-FR7$FX7$$!3[) =TpW<>y#F*$!3#QYkWj-%G@F-7$7$$!3[rq8%[lEy#F*$!3dXv(QpMn$=F-Fhn7$7$$!3/ rq8%[lEy#F*Fao7$$!3q<()=peI%y#F*$!3#*H6+&zp[g)!#>7$7$$!3=U'4Z]uXy#F*$! 3Z#y\"fz*[C7'F\\pFgo7$F^p7$$!3onK;/Xd%y#F*$\"3O/\"Go)3())o$F\\p7$7$$!3 ]CR?/Xd%y#F*$\"3s!*=fz*[C7'F\\pFdp7$Fjp7$$!3=-4$pPdHy#F*$\"3))GaGrX(Rc \"F-7$7$$!3[6:*Q[lEy#F*$\"3Rcv(QpMn$=F-F`q7$7$Fgq$\"36cv(QpMn$=F-7$$!3 kQ3['oO$zFF*$\"3\"45[SsV'Hp#F*$!3'3v(Q pMn$=*F-7$F^v7$$!3=H.NMrEw(QpMn$=*F-F`x7$Ffx7$$!31pO3!3se#zb0EF*$!3[[C71`Ej6F*7$F[z7$$!3b$*3(=Q/Bj#F *$!3;$*)yg\"*3r4\"F*7$7$$!3c*>/H[YZl#F*$!3&=hIlK;3/\"F*Faz7$FgzFiu7$Fb y7$$!3I)z%ee.udEF*$\"3u\\7Pn4xI5F*7$7$$!3GHV'zZYZl#F*$\"3'HhIlK;3/\"F* F^[l7$Fd[l7$$!3QbE9I#y#HEF*$\"38EG_A1Z,6F*7$7$$!3-;#G3#zb0EF*$\"3g\\C7 1`Ej6F*Fj[l7$F`\\l7$$!3%f8)R)4M4g#F*$\"3/GY7<[Qs6F*7$7$Fgy$\"3[l.yG'eI =\"F*Ff\\l7$7$$!3eF9dG9dGDF*$!33<$[]-z5J\"F*7$$!3$HnkO\"yxVDF*$!39&G9d G9dG\"F*7$7$$!3PtYm8yxVDF*Fh]lFfy7$F\\]l7$$!3wvZ**3!G$oDF*$\"3ij7!y** \\bB\"F*7$7$$!35uju1yxVDF*$\"3E'G9dG9dG\"F*F_^l7$Fe^l7$$!3aX;Q&)psNDF* $\"3]\"f0!*QC()H\"F*7$7$Fa]l$\"3iz5j7!z5J\"F*F[_l7$7$$!3oqb$R9F*7$7$$!3)oS?5bxQR#F*$!3]\"[tll\\X]\"F*Fbbl7$ 7$$!3!o\"fz*[C7Y#F*F\\bl7$$!38hK1\\kMCCF*$\"35wwuh7cj9F*7$7$Fibl$\"3:h ?2W'\\X]\"F*Facl7$7$$!3b'*[C71`EBF*$!3ob^r:G(4e\"F*7$$!3q640>&=EP#F*$! 3Wez*[C71`\"F*7$F`dlFhbl7$Fgcl7$$!3Q#pT&>1?&Q#F*$\"3^^d%e+O[^\"F*7$7$$ !3IcHU3&=EP#F*$\"3xfz*[C71`\"F*Fgdl7$7$F^el$\"3bfz*[C71`\"F*7$$!3fu+*> ,\\NM#F*$\"3#G>*))*ya:c\"F*7$7$F\\dl$\"3!\\2gG!G(4e\"F*Ffel7$7$F\\dl$! 3Yb^r:G(4e\"F*7$$!3?6Wr6\\FiAF*$!3QAqJ\"zSuk\"F*7$7$$!3?'QpMn$=fAF*$!3 UdKrBod];F*Fcfl7$7$F\\dl$\"3ou+'G!G(4e\"F*7$$!3:Vy;kN&3I#F*$\"3(p^JDov jg\"F*7$7$$!3k'QpMn$=fAF*$\"37r4D6od];F*Fbgl7$7$$!3%e(QpMn$=>#F*$!3PI) )HaWz43mv\">ecAF*$!35&z*[C71`;F*7$7$Fdhl$!3K&z*[C71`;F*Fifl7 $7$FjflF[hl7$$!3ab_*oNt#eAF*$\"3UAfW.hS^;F*7$7$$!3#zljW!>ecAF*$\"3?'z* [C71`;F*F^il7$7$Feil$\"3U'z*[C71`;F*7$$!35Z!z!o%pC@#F*$\"3-2,>&Gv0p\"F *7$7$F_hl$\"3g4DLSWz4e2$p(ew\"F*Fgjl7$Fcjl7$$!37\"z4'oWcm@F*$\"3-(H$QH1 cH6#F*$!3uJ;3/-^v6#F*$\"3'Gj\"3/-^v;.=F*7$7$F^\\m$\"3)z0)32V@:=F*Fe]m7$F]\\m7$$!3iQ4_VfI@? 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fm7&Fe^bnFe^bnFi]fmFd[fm7&Fc[fmFc[fmFd[fmFe^bn7&Fe^bnFe^bnFi]fmFj_an7& Fc_fmFc_fmFb_gmFc]^n7&Fbf_nFbf_nFahfmF\\ifm7&F\\`gnF\\`gnFgbhnFjihn7&F i_fmFi_fmF]\\bnFhhbn7&Fh]hnFh]hnFfcgnFdcgn7&F[_[nF[_[nF^eenF_hjm7&F[_[ nF[_[nF^eenFaeen7&Ff\\gmFf\\gm7$$\"3pD:NvB*G$zF-$!3#e)*G?%f6oPF*Fhas7& Ff\\gmF[ajnF[_sFcbs7&F[ajnF[ajnFf\\gmF[\\jm7&F[\\jmF[ajnFbj[oFjbs7&Fa^ \\nFa^\\nFchhmF[j[n7&Fa^\\nFa^\\nFchhmFehhm7&F^g^nF^g^nFf^hmFihhm7&Fce fmFcefmFihhmF^g^n7&FcefmFcefmFihhmFbefm7&F[ahnF[ahnFejgnF]bhn7&FdabnFd abnFhcimF`abn7&FdabnFdabnFhcimF\\dim7&FhabnFhabnF\\dimFdabn7&FhabnFhab nF\\dimF`dim7&7$Fh]mF]^lF^h_oF`dimFhabn7&F^h_oF^h_oF`dimF_e`n7&F]d[oF] d[oF_e`nF^h_o7&F]d[oF]d[oF_e`nFce`n7&F]bhnF]bhnF_hgnFejgn7&FghhmFghhmF e_jnFb_in7&Ffb[oFfb[oF_fcnF`[[o7&F]bhnF]bhnF_hgnFi]hn7&F\\egnF\\egnFji hnF\\\\jn7&F\\]^nF\\]^nFa]fnFdbfm7&F`]^nF`]^nFdbfmF\\]^n7&F`]^nF`]^nFd bfmFbbfm7&Fec^nFec^nFbbfmF`]^n7&F[g`nF[g`nFeigmF^bhm7&FidgmFidgmF^bhmF [g`n7&FidgmFidgmF^bhmFhdgm7&Fj\\anFj\\anFjdimFdjem7&FcjemFcjemFdjemFj \\an7&Fi]hnFi]hnFdcgnF_hgn7&F\\egnF\\egnFjihnF\\`gn7&FjghmFjghmF^a]nF] jgm7&FjghmFjghmF^a]nFdjgm7&Fi]hnFi]hnFdcgnFh]hn7&F]canF]canF\\ifmFbf_n 7&F[ahnF[ahnFejgnF]^hn7&FidhnFidhnF]^hnF[ahn7&F[gcnF[gcnFifcnFhfgn7&Fh fgnF[gcnFdednFael7&FhfgnF[ein7$$!30K3`qygIHF*Fg\\sFael7&F[einF[ein7$$! 3)3poQ_!y**GF*$\"3hL " 0 "" {MPLTEXT 1 0 1 ";" }}}}{PARA 0 "" 0 "" {TEXT -1 86 "#------------ ---------------------------------------------------------------------- ---" }}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 73 "The absolute stability fun ction for a 6 stage, order 5 Runge-Kutta method" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 26 "The stabilty 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)-%*factorialG6#\"\"#!\"\"F)" } {TEXT -1 1 " " }{XPPEDIT 18 0 "z^2+1/3!" "6#,&*$%\"zG\"\"#\"\"\"*&F'F' -%*factorialG6#\"\"$!\"\"F'" }{TEXT -1 1 " " }{XPPEDIT 18 0 "z^3+1/4! " "6#,&*$%\"zG\"\"$\"\"\"*&F'F'-%*factorialG6#\"\"%!\"\"F'" }{TEXT -1 1 " " }{XPPEDIT 18 0 "z^4+1/5!" "6#,&*$%\"zG\"\"%\"\"\"*&F'F'-%*factor ialG6#\"\"&!\"\"F'" }{TEXT -1 2 " " }{XPPEDIT 18 0 "z^5+[b]^T*[A]^5*z ^6" "6#,&*$%\"zG\"\"&\"\"\"*()7#%\"bG%\"TGF'*$7#%\"AGF&F'F%\"\"'F'" } {TEXT -1 1 " " }}{PARA 0 "" 0 "" {TEXT -1 6 "where " }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "[A]= matrix([[0, 0, 0, 0, 0, 0], [a[ 2,1], 0, 0, 0, 0, 0], [a[3,1], a[3,2], 0, 0, 0, 0], [a[4,1], a[4,2], a [4,3], 0, 0, 0], [a[5,1], a[5,2], a[5,3], a[5,4], 0, 0], [a[6,1], a[6, 2], a[6,3], a[6,4], a[6,5], 0]])" "6#/7#%\"AG-%'matrixG6#7(7(\"\"!F+F+ F+F+F+7(&%\"aG6$\"\"#\"\"\"F+F+F+F+F+7(&F.6$\"\"$F1&F.6$F5F0F+F+F+F+7( &F.6$\"\"%F1&F.6$F;F0&F.6$F;F5F+F+F+7(&F.6$\"\"&F1&F.6$FCF0&F.6$FCF5&F .6$FCF;F+F+7(&F.6$\"\"'F1&F.6$FMF0&F.6$FMF5&F.6$FMF;&F.6$FMFCF+" } {TEXT -1 1 " " }}{PARA 0 "" 0 "" {TEXT -1 4 "and " }}{PARA 256 "" 0 " " {TEXT -1 1 " " }{XPPEDIT 18 0 "[b]^T = [b[1], b[2], b[3], b[4], b[5] , b[6]];" "6#/)7#%\"bG%\"TG7(&F&6#\"\"\"&F&6#\"\"#&F&6#\"\"$&F&6#\"\"% &F&6#\"\"&&F&6#\"\"'" }{TEXT -1 2 ". " }}{PARA 0 "" 0 "" {TEXT -1 4 "N ow " }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "[A]^5 = matrix ([[0, 0, 0, 0, 0, 0], [0, 0, 0, 0, 0, 0], [0, 0, 0, 0, 0, 0], [0, 0, 0 , 0, 0, 0], [0, 0, 0, 0, 0, 0], [a[6,5]*a[5,4]*a[4,3]*a[3,2]*a[2,1], 0 , 0, 0, 0, 0]])" "6#/*$7#%\"AG\"\"&-%'matrixG6#7(7(\"\"!F-F-F-F-F-7(F- F-F-F-F-F-7(F-F-F-F-F-F-7(F-F-F-F-F-F-7(F-F-F-F-F-F-7(*,&%\"aG6$\"\"'F '\"\"\"&F56$F'\"\"%F8&F56$F;\"\"$F8&F56$F>\"\"#F8&F56$FAF8F8F-F-F-F-F- " }{TEXT -1 2 ", " }}{PARA 0 "" 0 "" {TEXT -1 28 "so that the coeffici ent of " }{XPPEDIT 18 0 "z^6" "6#*$%\"zG\"\"'" }{TEXT -1 5 " is:" }} {PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "[b]^T*[A]^5=b[6]*a[6, 5]*a[5,4]*a[4,3]*a[3,2]*a[2,1]" "6#/*&)7#%\"bG%\"TG\"\"\"*$7#%\"AG\"\" &F)*.&F'6#\"\"'F)&%\"aG6$F1F-F)&F36$F-\"\"%F)&F36$F7\"\"$F)&F36$F:\"\" #F)&F36$F=F)F)" }{TEXT -1 2 ". " }}{PARA 256 "" 0 "" {TEXT -1 1 " " } {TEXT 321 23 "_______________________" }{TEXT -1 1 " " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> \+ " 0 "" {MPLTEXT 1 0 145 "A := matrix([seq([seq(a[i,j],j=1..i-1),seq(0, j=i..6)],i=1..6)]);\n``;\nB := matrix([[seq(b[i],i=1..6)]]);\n``;\neva lm(A^5);\n``;\nevalm(B &* A^5)[1,1];" }}{PARA 11 "" 1 "" {XPPMATH 20 " 6#>%\"AGK%'matrixG6#7(7(\"\"!F*F*F*F*F*7(&%\"aG6$\"\"#\"\"\"F*F*F*F*F* 7(&F-6$\"\"$F0&F-6$F4F/F*F*F*F*7(&F-6$\"\"%F0&F-6$F:F/&F-6$F:F4F*F*F*7 (&F-6$\"\"&F0&F-6$FBF/&F-6$FBF4&F-6$FBF:F*F*7(&F-6$\"\"'F0&F-6$FLF/&F- 6$FLF4&F-6$FLF:&F-6$FLFBF*Q(pprint36\"" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%!G" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"BGK%'matrixG6#7#7(&% \"bG6#\"\"\"&F+6#\"\"#&F+6#\"\"$&F+6#\"\"%&F+6#\"\"&&F+6#\"\"'Q(pprint 46\"" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%!G" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#K%'matrixG6#7(7(\"\"!F(F(F(F(F(F'F'F'F'7(*,&%\"aG6$\"\" '\"\"&\"\"\"&F,6$F/\"\"%F0&F,6$F3\"\"$F0&F,6$F6\"\"#F0&F,6$F9F0F0F(F(F (F(F(Q(pprint56\"" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%!G" }}{PARA 11 " " 1 "" {XPPMATH 20 "6#*.&%\"bG6#\"\"'\"\"\"&%\"aG6$F'\"\"&F(&F*6$F,\" \"%F(&F*6$F/\"\"$F(&F*6$F2\"\"#F(&F*6$F5F(F(" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 22 "The special procedure " } {TEXT 0 17 "StabilityFunction" }{TEXT -1 18 " may also be used." }} {PARA 0 "" 0 "" {TEXT -1 0 "" }}{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 1 ";" }}}}{PARA 0 "" 0 "" {TEXT -1 86 "#-------------------------------------------------------- -----------------------------" }}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 4 "T he " }{TEXT 319 16 "stability radius" }{TEXT -1 6 " and " }{TEXT 316 24 "imaginary axis inclusion" }{TEXT -1 42 " of a 6 stage, order 5 Run ge-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 -1 146 " associated wi th a stability region of an explicit Runge-Kutta method is the radius \+ of the largest disc that can fit inside the stability region. " }} {PARA 0 "" 0 "" {TEXT -1 50 "This largest disc is always centred at th e point " }{XPPEDIT 18 0 "-rho;" "6#,$%$rhoG!\"\"" }{TEXT -1 19 " on the real axis." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 86 "The stability function of a general 6 stage, order 5 Rung e-Kutta scheme has the form: " }}{PARA 256 "" 0 "" {TEXT -1 1 " " } {XPPEDIT 18 0 "1+z+z^2/2+z^3/6+z^4/24+z^5/120+t[6]*z^6;" "6#,0\"\"\"F$ %\"zGF$*&F%\"\"#F'!\"\"F$*&F%\"\"$\"\"'F(F$*&F%\"\"%\"#CF(F$*&F%\"\"& \"$?\"F(F$*&&%\"tG6#F+F$*$F%F+F$F$" }{TEXT -1 2 ", " }}{PARA 0 "" 0 " " {TEXT -1 7 "where " }{XPPEDIT 18 0 "t[6]=b[6]*a[6,5]*a[5,4]*a[4,3]* a[3,2]*c[2]" "6#/&%\"tG6#\"\"'*.&%\"bG6#F'\"\"\"&%\"aG6$F'\"\"&F,&F.6$ F0\"\"%F,&F.6$F3\"\"$F,&F.6$F6\"\"#F,&%\"cG6#F9F," }{TEXT -1 1 "." }} {PARA 0 "" 0 "" {TEXT -1 0 "" }}{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 "" }}{PARA 0 "" 0 "" {TEXT -1 76 "The following is a sample calculatio n of the value of the stability radius " }{XPPEDIT 18 0 "rho" "6#%$rh oG" }{TEXT -1 7 " for " }{XPPEDIT 18 0 "t[6]=1/400" "6#/&%\"tG6#\"\" '*&\"\"\"F)\"$+%!\"\"" }{TEXT -1 11 " = 0.0025. " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 59 "R := z -> 1+ z+1/2*z^2+1/6*z^3+1/24*z^4+1/120*z^5+1/400*z^6;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"RGf*6#%\"zG6\"6$%)operatorG%&arrowGF(,0\"\"\"F-9$F- *&#F-\"\"#F-*$)F.F1F-F-F-*&#F-\"\"'F-*$)F.\"\"$F-F-F-*&#F-\"#CF-*$)F. \"\"%F-F-F-*&#F-\"$?\"F-*$)F.\"\"&F-F-F-*&#F-\"$+%F-*$)F.F6F-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 interse cts 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 := n ewton(R(z)=1,z=-3);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%#z0G$!+0nA=H! \"*" }}}{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,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=[- 3.19..0.49,-0.0..1.47],font=[HELVETICA,9]);" }}{PARA 13 "" 1 "" {GLPLOT2D 386 232 232 {PLOTDATA 2 "6+-%'CURVESG6$7X7$$!3%************* *=$!#<$\"3wuuXpGAa@F*7$$!3'HLLjB$*)\\JF*$\"3iR!e/Cpv*>F*7$$!3Tmmmsky4J F*$\"3[]#QT9c5&=F*7$$!3e***\\?z*)[2$F*$\"3)zAv78?9t\"F*7$$!3<2Jqj!*Q\"F*7$$!3[mm1p!)*>#HF*$\"3u2w=uD3&G\"F*7$$!3smm'3'H\\# )GF*$\"3ML()=$)>L)=\"F*7$$!3q***\\NhvJ%GF*$\"3E2xU\\J$))4\"F*7$$!37LLB m#eQ!GF*$\"3.$f5k1Td,\"F*7$$!3smmOQQ&4t#F*$\"3gDE#48YVx)!#=7$$!3)***** 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This situation is typical but does not always occur." }} {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 46 "The assoc iated stability radius is therefore " }{XPPEDIT 18 0 "rho" "6#%$rhoG " }{TEXT -1 2 " " }{TEXT 320 1 "~" }{TEXT -1 15 " 1.459113353. 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DEFAULTG-%(SCALINGG6#%,CONSTRAINEDG-%%VIEWG6$FghoFgho" 1 2 0 1 10 0 2 9 1 4 1 1.000000 45.000000 45.000000 0 0 "Curve 1" "Curve 2" }}}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 148 "Apart from the stabiltity radius, another important characterstic of the stability region is its intersection of the region with the im aginary axis." }}{PARA 0 "" 0 "" {TEXT -1 56 "This intersection can be a closed interval of the form " }{XPPEDIT 18 0 "[-rho[2],rho[2]]" "6 #7$,$&%$rhoG6#\"\"#!\"\"&F&6#F(" }{TEXT -1 33 ", or consist of the ori gin alone." }}{PARA 0 "" 0 "" {TEXT -1 46 "In the first case the radiu s of the interval " }{XPPEDIT 18 0 "rho[2]" "6#&%$rhoG6#\"\"#" } {TEXT -1 29 " gives the linear stability " }{TEXT 260 24 "imaginary a xis inclusion" }{TEXT -1 29 ". In the scond case we have " }{XPPEDIT 18 0 "rho[2]=0" "6#/&%$rhoG6#\"\"#\"\"!" }{TEXT -1 2 ". " }}{PARA 0 " " 0 "" {TEXT -1 235 "In trying to achieve good stability properties we can attempt to maximize the stability radius. However, it may be pre ferable to try to compromise between stretching the stability region i n the real and imaginary directions and have " }{XPPEDIT 18 0 "rho" " 6#%$rhoG" }{TEXT -1 6 " and " }{XPPEDIT 18 0 "rho[2]" "6#&%$rhoG6#\" \"#" }{TEXT -1 37 " not too different from each other. " }}{PARA 0 " " 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 373 "The imaginary axi s inclusion cannot be identified or estimated from a picture such as t he one above. However, if we take a suitable negative integer power of the real part of a boundary point this will move the point away from \+ the imaginary axis without changing sides. Taking the 11th root is usu ally sufficient as in the following picture shows that a rough estimat e of " }{XPPEDIT 18 0 "rho[2]" "6#&%$rhoG6#\"\"#" }{TEXT -1 6 " is \+ " }{XPPEDIT 18 0 "rho[2]" "6#&%$rhoG6#\"\"#" }{TEXT -1 2 " " }{TEXT 317 1 "~" }{TEXT -1 7 " 1.35 " }}{PARA 0 "" 0 "" {TEXT -1 156 "Anothe r way of saying this is that the largest interval on the nonnegative i maginary axis that contains the origin and lies inside the stability r egion is " }{XPPEDIT 18 0 "[ 0, 1.35 ]" "6#7$\"\"!-%&FloatG6$\"$N\"! \"#" }{TEXT -1 15 " 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+1/400*z^6:\nDigits := 20:\npts := []: z0 := 0:\n for ct from 0 to 52 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:\n plot(pts,color=COLOR(RGB,.8,0,.8),thickness=2,font=[HELVETICA,9]);\nDi gits := 10:" }}{PARA 13 "" 1 "" {GLPLOT2D 261 276 276 {PLOTDATA 2 "6(- %'CURVESG6#7W7$$\"\"!F)F(7$$\"5#)Q2&R1UI'f\")!#@$\"5X()Qee`EfTJF-7$$\" 5BnK^tT7n!>\"!#?$\"5(p!GPn1`=$G'F-7$$\"5#)z@?:j`&\\[\"F3$\"5X*p&[8_zxC %*F-7$$\"5/hC!*4#)y_OF3$\"5*R@ vZgH'zq:F37$$\"5li%fO2%Qyj@F3$\"5MR^4A#[b\\)=F37$$\"5yr`udW,v^BF3$\"5= F4K$\\`9\"*>#F37$$\"5'=ep6)f%Rr_#F3$\"5o*p(='QItK^#F37$$\"5T%4&)fm]M?p #F3$\"55%>h2m_Ju#GF37$$\"52)=-ZKpYz%GF3$\"5rvgN1w()eTJF37$$\"5c\"Gm=8l 4g*HF3$\"5L*[ydURWdX$F37$$\"5k-[rq(>&4PJF3$\"5(H],r;T(*)pPF37$$\"5=,$> )3sl)=F$F3$\"5ZeuFF^k/%3%F37$$\"5vm(4([cm#4S$F3$\"5;U3qy;'*=)R%F37$$\" 56(>Ea(*y]Y_$F3$\"5#Qh'\\*\\OCBr%F37$$\"5))[cs[%o4Mk$F3$\"5VoZ*4AOZk-& F37$$\"5Rw0N6Vd[dPF3$\"5le0mB'RE%F3$\"5oy f6T,%41\"pF37$$\"50,t$z8\")zJN%F3$\"5E(**e:3t`WA(F37$$\"54!Hv)*emT%QWF 3$\"5Dvbs-=&4#QvF37$$\"50`xCvR.r>XF3$\"5*\\I!e%>$y&=&yF37$$\"5OJ\")eO( )=\"pf%F3$\"5&*e*G#\\upPl\")F37$$\"5(z&3:0gA$*pYF3$\"5!oTptsHU(y%)F37$ $\"5,,/u1X;hQZF3$\"5X,o!p*Rg#>z)F37$$\"5Csi%*=sit-[F3$\"5Z\\-Q^'>(*[5* F37$$\"5ykH2K;n-i[F3$\"5C$*>lA:9i<%*F37$$\"5o(fxW4;@h\"\\F3$\"5&>zmo)Q 41I(*F37$$\"5%Qd:uq9^X'\\F3$\"5XM.0-fu@/5!#>7$$\"5jb.PV'p0n+&F3$\"5?'* [/z!y\"RN5Ffu7$$\"5!3MGOirw\"4&F3$\"5IGgYrH$='f6Ffu7$$\"5K:84WzTI\"3&F3$\"5+Gi/o%pH0>\"Ffu7$ $\"5#Q\\N#H`1u]]F3$\"5$RH3P'p5P@7Ffu7$$\"5zh`Ci;Qg*)\\F3$\"5[gw[F3$\"5`^YO&*p'=GG\"Ffu7$$\"5$=46pa>w&[YF3$\"5y !*HJ;Z?T88Ffu7$$\"5WTF'))fN!)of$F3$\"5AS,O\"Q#)4RM\"Ffu7$$!5&e?x9;*of: ZF3$\"5BJRO73bIu8Ffu7$$!5!)*))*>6L%\\S4&F3$\"5nU@V4^Ef/9Ffu7$$!5tbP16h #[&\\`F3$\"53y?]!z([wM9Ffu7$$!5EXe'H2HLPb&F3$\"5n\"*)Q-@\"f\"[Y\"Ffu7$ $!5!4>z(Rs_4HdF3$\"5h:d0.\"fRZ\\\"Ffu7$$!5$Hz\"Q[T>q&)eF3$\"5Tc'**\\6( )HX_\"Ffu7$$!5[JY*)4(GZ*GgF3$\"5@T(*o9A3=a:Ffu7$$!5lzI!\\jsi?;'F3$\"5! H)3J.4mo$e\"Ffu7$$!5\"*[5)yvARrG'F3$\"5OEtmwh9/8;Ffu-%%FONTG6$%*HELVET ICAG\"\"*-%+AXESLABELSG6$Q!6\"Fc\\l-%&COLORG6&%$RGBG$\"\")!\"\"F(Fi\\l -%*THICKNESSG6#\"\"#-%%VIEWG6$%(DEFAULTGFc]l" 1 2 0 1 10 2 2 6 1 4 2 1.000000 45.000000 45.000000 0 0 "Curve 1" }}}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 123 "The relevant intersection poin t of the boundary curve with the imaginary axis can be determined more accurately as follows." }}{PARA 0 "" 0 "" {TEXT -1 86 "First we look \+ for points on the boundary curve either side of the intersection point . " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 112 "Digits := 15:\nz0 := 1.36*I:\nfor ct from 42 to 45 d o\n newton(R(z)=exp(ct*Pi/100*I),z=z0);\nend do;\nDigits := 10:" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#^$$\"0*y)49!Q!>#!#=$\"08jr/7MJ\"!#9" } }{PARA 11 "" 1 "" {XPPMATH 20 "6#^$$\"0\"GEbZs.8!#>$\"0g8Q#)4RM\"!#9" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#^$$!0\\ITo8Rc#!#=$\"0kB\"3bIu8!#9" } }{PARA 11 "" 1 "" {XPPMATH 20 "6#^$$!02*y=fp$*f!#=$\"0K%4^Ef/9!#9" }}} {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 107 "Then we \+ apply the bisection method to calculate the parameter value associated with the intersection point." }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 178 "real_part := proc(u)\n Re(newton(R(z)=exp(u*Pi*I),z=1.36*I)) \nend proc:\nDigits := 15:\nu0 := bisect('real_part'(u),u=0.42..0.45); \nnewton(R(z)=exp(u0*Pi*I),z=1.36*I);\nDigits := 10:" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%#u0G$\"0&=E?;[0V!#:" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#^#$\"0#)pNqybM\"!#9" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 4 "The " }{TEXT 260 112 "largest interval on the nonn egative imaginary axis that contains the origin and lies inside the st ability region" }{TEXT -1 5 " is " }{XPPEDIT 18 0 "[0, 1.3456];" "6#7 $\"\"!-%&FloatG6$\"&cM\"!\"%" }{TEXT -1 18 " (approximately)." }} {PARA 0 "" 0 "" {TEXT -1 6 "Thus " }{XPPEDIT 18 0 "rho[2]" "6#&%$rhoG 6#\"\"#" }{TEXT -1 2 " " }{TEXT 318 1 "~" }{TEXT -1 9 " 1.3456." }} {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 71 "The follo wing picture shows the main part of the stability region (the " } {TEXT 261 3 "red" }{TEXT -1 98 " curve) together with the boundary of \+ the largest disc that fits inside the stability region (the " }{TEXT 256 4 "blue" }{TEXT -1 54 " curve) and has radius equal to the stabili ty radius " }{XPPEDIT 18 0 "rho" "6#%$rhoG" }{TEXT -1 2 ". " }}{PARA 0 "" 0 "" {TEXT -1 27 "The line segment (drawn in " }{TEXT 257 5 "gree n" }{TEXT -1 27 ") represents the interval " }{XPPEDIT 18 0 "[-rho[2] , rho[2]]" "6#7$,$&%$rhoG6#\"\"#!\"\"&F&6#F(" }{TEXT -1 74 " on the i maginary axis giving the intersection with the stability region." }} {PARA 0 "" 0 "" {TEXT -1 27 "The line segment (drawn in " }{TEXT 264 6 "orange" }{TEXT -1 27 ") represents the interval " }{XPPEDIT 18 0 " [-2*rho, 0];" "6#7$,$*&\"\"#\"\"\"%$rhoGF'!\"\"\"\"!" }{TEXT -1 69 " \+ on the real axis giving the intersection with the stability region." } }{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 627 "R := z -> 1+z+1/2*z^2+1/6*z^3+1/24*z^4+1/120*z^5+1/400*z^6:\npt s := []:z0 := 0:\nfor ct from 0 to 360 do\n zz := newton(R(z)=exp(ct *Pi/30*I),z=z0):\n z0 := zz:\n pts := [op(pts),[Re(zz),Im(zz)]]:\n end do:\np1 := plot(pts,color=red):\nrd := 1.45911335264: rd2 := 1.345 57870357: \np2 := plot([-rd+rd*cos(t),rd*sin(t),t=0..2*Pi],color=blue ):\np3 := plot([[-2*rd,0],[0,0]],thickness=4,color=COLOR(RGB,1,.6,0)): \np4 := plot([[0,-rd2],[0,rd2]],thickness=4,color=COLOR(RGB,0,.75,0)): \np5 := plot([[[-.07,-rd2],[.07,-rd2]],[[-.07,rd2],[.07,rd2]]],color=C OLOR(RGB,0,.5,0)):\nplots[display]([p||(1..5)],scaling=constrained,fon t=[HELVETICA,9]);" }}{PARA 13 "" 1 "" {GLPLOT2D 353 456 456 {PLOTDATA 2 "6,-%'CURVESG6$7eal7$$\"\"!F)F(7$$\"3++++'puZX\"!#E$\"3/+++\\v>Z5!#= 7$$\"3P+++#\\L\"4\"*!#D$\"3/+++t[R%4#F07$$\"3!******REcl***!#C$\"3(*** ***fx)eTJF07$$\"3j*****HiQ=K&!#B$\"3A+++.;w)=%F07$$\"3++++k'zw)=!#A$\" 3z*****R4aeB&F07$$\"3()******>$yj7&FF$\"3;+++6#=FG'F07$$\"3(******\\:R S9\"!#@$\"3p*****4r\\!HtF07$$\"31+++bASx@FQ$\"3#*******GfIu$)F07$$\"37 +++,H9*e$FQ$\"3Q+++:9i<%*F07$$\"3%********[/M4&FQ$\"3++++sTxX5!#<7$$\" 3S+++\\)>L&fFQ$\"3'******z/+$\\6F]o7$$\"3C+++f&)GsZFQ$\"35+++!>O@D\"F] o7$$!3.+++W*HT#pFF$\"3-+++#Q`SN\"F]o7$$!31+++$[;WO\"!#?$\"3-+++\\F\"[X \"F]o7$$!3#******pV>_#QF`p$\"3'******>#3=a:F]o7$$!3=*****f0-Q&zF`p$\"3 !*******Rl$>l\"F]o7$$!3%******RY,;V\"!#>$\"3'******4Hkyu\"F]o7$$!3!*** ***fA#=^BF`q$\"31+++'*3tT=F]o7$$!3>+++@cu8OF`q$\"3(******\\t\\K$>F]o7$ $!39+++Omet_F`q$\"3)******R4T?-#F]o7$$!3k+++*H\"=ttF`q$\"3.+++pTg2@F]o 7$$!3N*****f()*\\N**F`q$\"3#******H#4K*=#F]o7$$!31+++_op&H\"F0$\"3))** ****>)3lE#F]o7$$!3#******HiK/k\"F0$\"3#)*****z$4_QBF]o7$$!3#******\\@; =-#F0$\"3!)*****zPi[S#F]o7$$!3++++RcGKCF0$\"37+++8VFlCF]o7$$!3)******f l6R'GF0$\"3&)*****zd_(>DF]o7$$!3;+++NMW4LF0$\"3#******R+0&oDF]o7$$!3A+ ++qH*Gw$F0$\"3<+++o$y=h#F]o7$$!3\")*****H'*z'>UF0$\"35+++5kG]EF]o7$$!3 !******4]*\\wYF0$\"3++++cd:%o#F]o7$$!3[+++M^2J^F0$\"3<+++DE*Qr#F]o7$$! 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" }{TEXT -1 5 ".004." }}{PARA 0 "" 0 "" {TEXT -1 4 "The " }{TEXT 260 6 "purple" }{TEXT -1 25 " curve is the graph for " }{XPPEDIT 18 0 "rh o" "6#%$rhoG" }{TEXT -1 12 ", while the " }{TEXT 257 5 "green" }{TEXT -1 25 " curve is the graph for " }{XPPEDIT 18 0 "rho[2]" "6#&%$rhoG6# \"\"#" }{TEXT -1 1 "." }}{PARA 0 "" 0 "" {TEXT -1 41 "The curves inter sect at the point where " }{XPPEDIT 18 0 "t[6]" "6#&%\"tG6#\"\"'" } {TEXT -1 2 " " }{TEXT 322 1 "~" }{TEXT -1 24 " 0.002946139890 where \+ " }{XPPEDIT 18 0 "rho=rho[2]" "6#/%$rhoG&F$6#\"\"#" }{TEXT -1 2 " " }{TEXT 323 1 "~" }{TEXT -1 15 " 1.398032276. " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 814 "pts := []: \+ pts2 := []:\nfor i from 0 to 50 do\n t6 := 0.2e-2+i*4e-5;\n R := z -> 1+z+1/2*z^2+1/6*z^3+1/24*z^4+1/120*z^5+t6*z^6:\n z0 := -newton(R (z)=1,z=-4)/2;\n pts := [op(pts),[t6,z0]];\n real_part := proc(u) \n Re(newton(R(z)=exp(u*Pi*I),z=1.4*I))\n end proc:\n u0 := b isect('real_part'(u),u=0.38..0.5);\n rr := Im(newton(R(z)=exp(u0*Pi* I),z=1.4*I));\n pts2 := [op(pts2),[t6,rr]];\nend do:\np1 := plot([pt s,pts2],color=[COLOR(RGB,.5,0,1),COLOR(RGB,0,.75,0)]):\np2 := plot([[. 2946139890e-2,1.21],[.2946139890e-2,1.398032276]],color=black,linestyl e=3):\np3 := plot([[[.2946139890e-2,1.398032276]]$4],style=point,symbo l=[diamond,cross,circle$2],\n symbolsize=[10$3,15],color=[ coral$3,black]):\nplots[display]([p1,p2,p3],font=[HELVETICA,9],labels= [`t[6]`,``],view=[0.00198..0.00402,1.21..1.563]);\n\n" }}{PARA 13 "" 1 "" {GLPLOT2D 586 372 372 {PLOTDATA 2 "6,-%'CURVESG6$7U7$$\"31+++++++ ?!#?$\"33+++AY8c:!#<7$$\"3:++++++S?F*$\"3)*******zypY:F-7$$\"3#)****** ******z?F*$\"3,+++[;iP:F-7$$\"3$*************>@F*$\"3(******fm\"))G:F- 7$$\"3/++++++g@F*$\"3++++afX?:F-7$$\"39+++++++AF*$\"3-+++PXK7:F-7$$\"3 \")************RAF*$\"3*******>ApW]\"F-7$$\"3#*************zAF*$\"3#** ****>Rto\\\"F-7$$\"3-++++++?BF*$\"3%*******H=_*[\"F-7$$\"37++++++gBF*$ \"37+++z0S#[\"F-7$$\"3!)*************R#F*$\"3++++)z'\\v9F-7$$\"3!***** ********RCF*$\"35+++e')zo9F-7$$\"3+++++++![#F*$\"3%******RB&Hi9F-7$$\" 36++++++?DF*$\"3!******4UwfX\"F-7$$\"3A++++++gDF*$\"3++++cG$)\\9F-7$$ \"3))*************f#F*$\"3++++We&QW\"F-7$$\"3**************REF*$\"3'** ******HP!Q9F-7$$\"35++++++!o#F*$\"36+++*ppBV\"F-7$$\"3?++++++?FF*$\"3 \"******p,YoU\"F-7$$\"3))************fFF*$\"3-+++&pf9U\"F-7$$\"3)***** *********z#F*$\"3)******zf/iT\"F-7$$\"33++++++SGF*$\"3)*******z\\269F- 7$$\"3?++++++!)GF*$\"3*******>XlgS\"F-7$$\"3&)************>HF*$\"3\"** ****R&4<,9F-7$$\"3'*************fHF*$\"3/+++HnQ'R\"F-7$$\"31+++++++IF* $\"3%******zH3JF*$\"33+++unEy8F-7$ $\"31++++++gJF*$\"3%******zkrRP\"F-7$$\"3;+++++++KF*$\"3-+++`Mwp8F-7$$ \"3$)************RKF*$\"3'******p+RcO\"F-7$$\"3%*************zKF*$\"3+ +++%H&fh8F-7$$\"3/++++++?LF*$\"3'******fXHwN\"F-7$$\"39++++++gLF*$\"3+ +++x(QPN\"F-7$$\"3#)*************R$F*$\"3)******fn?*\\8F-7$$\"3#****** *******RMF*$\"3)******4qshM\"F-7$$\"3-++++++![$F*$\"33+++;D\\U8F-7$$\" 38++++++?NF*$\"31+++'*y()Q8F-7$$\"3!)************fNF*$\"3/+++DnKN8F-7$ $\"3!**************f$F*$\"3(*******yp$=L\"F-7$$\"3,++++++SOF*$\"3)**** ***HnSG8F-7$$\"37++++++!o$F*$\"3(******f8M]K\"F-7$$\"3A++++++?PF*$\"3( ******RV<RF*$\"3)*******pe\"fI\"F-7$$\"3) *************fRF*$\"3++++%e,HI\"F-7$$\"35+++++++SF*$\"35++++G$**H\"F-- %&COLORG6&%$RGBG$\"\"&!\"\"$\"\"!Fj[l$\"\"\"Fj[l-F$6$7U7$F($\"3#****** RjQ)=7F-7$F/$\"3#******fznUB\"F-7$F4$\"3)******4\"pO[7F-7$F9$\"3)***** *4z)Hh7F-7$F>$\"3/+++<0?t7F-7$FC$\"3-+++)Q)=%G\"F-7$FH$\"31+++V^p[8F-7$Fjo$\"3'*******HUla8F-7$F_p$\"35+++eKFg8F-7$Fdp$\"3++++'Gz bO\"F-7$Fip$\"3'*******)f'fq8F-7$F^q$\"3'******f.Z`P\"F-7$Fcq$\"33+++) G])z8F-7$Fhq$\"33+++jT7%Q\"F-7$F]r$\"33+++(z%=)Q\"F-7$Fbr$\"3%******R& o/#R\"F-7$Fgr$\"3-+++%oBdR\"F-7$F\\s$\"3'******>ZF#*R\"F-7$Fas$\"3**** ****f$pDS\"F-7$Ffs$\"33+++a&fdS\"F-7$F[t$\"3++++Du!)39F-7$F`t$\"30+++ \"f@9F-7$Fdu$\"35+++!4\">A9F-7$Fiu$\"3/+++dVaC9F-7$F^v$\"33+++6M!oU\"F- 7$Fcv$\"30+++jK(*G9F-7$Fhv$\"3-+++y&e5V\"F-7$F]w$\"3%********pjIV\"F-7 $Fbw$\"3-+++%o#*\\V\"F-7$Fgw$\"3,+++>$\\oV\"F-7$F\\x$\"3*)*****p9P'Q9F -7$Fax$\"3-+++%[f.W\"F-7$Ffx$\"3!******pV>?W\"F-7$F[y$\"3\"******\\#*> OW\"F-7$F`y$\"3%******Roj^W\"F-7$Fey$\"31+++\"H`mW\"F-7$Fjy$\"3/+++k64 [9F-7$F_z$\"3)******zez%\\9F-7$Fdz$\"31+++72#3X\"F-7$Fiz$\"33+++ll6_9F -7$F^[l$\"3-+++j!pLX\"F--Fc[l6&Fe[lFi[l$\"#v!\"#Fi[l-F$6%7$7$$\"3))*** ****))Rh%HF*$\"3'*************47F-7$Fbfl$\"3!******fFK!)R\"F--%'COLOUR G6&Fe[lFj[lFj[lFj[l-%*LINESTYLEG6#\"\"$-F$6&7#Fffl-Fjfl6&Fe[l$\"*++++ \"!\")$\")AR!)\\FgglFi[l-%'SYMBOLG6$%(DIAMONDG\"#5-%&STYLEG6#%&POINTG- F$6&FbglFcgl-F[hl6$%&CROSSGF^hlF_hl-F$6&FbglFcgl-F[hl6$%'CIRCLEGF^hlF_ hl-F$6&FbglFifl-F[hl6$F\\il\"#:F_hl-%%FONTG6$%*HELVETICAG\"\"*-%+AXESL ABELSG6%%%t[6]G%!G-Fcil6#%(DEFAULTG-%%VIEWG6$;$\"$)>!\"&$\"$-%Fejl;$\" $@\"F]fl$\"%j:!\"$" 1 2 0 1 10 0 2 9 1 4 2 1.000000 45.000000 45.000000 0 0 "Curve 1" "Curve 2" "Curve 3" "Curve 4" "Curve 5" "Curve 6" "Curve 7" }}}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 64 "The following procedure can be used to calculate the valu e of " }{XPPEDIT 18 0 "t[6]" "6#&%\"tG6#\"\"'" }{TEXT -1 10 " wher e " }{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_part,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](newt on(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=0.0 028..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 325 1 "~" }{TEXT -1 64 " 0.0029461398900112655220 and the associated common value 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 324 1 "~" }{TEXT -1 20 " 1.39803227622108. " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 123 "t6 \+ := 0.0029461398900112655220:\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 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}}{PARA 0 "" 0 "" {TEXT -1 78 "#-------------------------------------------------------- ---------------------" }}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 27 "The stab ility region when " }{XPPEDIT 18 0 "rho = rho[2]" "6#/%$rhoG&F$6#\"\" #" }{TEXT -1 1 " " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 8 "We set " }{XPPEDIT 18 0 "t[6] = 346370519/117567573819" " 6#/&%\"tG6#\"\"'*&\"*>0PY$\"\"\"\"->Qdnv6!\"\"" }{TEXT -1 54 ", which \+ is a rational approximation for the value of " }{XPPEDIT 18 0 "t[6]" "6#&%\"tG6#\"\"'" }{TEXT -1 51 " obtained in the previous subsection \+ which gives " }{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 60 "convert(.29461398900112655220e-2,rational,21);\n evalf[20](%);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6##\"*>0PY$\"->Qdnv6" } }{PARA 11 "" 1 "" {XPPMATH 20 "6#$\"5?_l7,!*)Rh%H!#A" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 90 "R := z -> 1+z+1/2*z^2+1/6*z^3+1/24*z^4+1/120*z^5+346370519/117567573819*z^6:\n' R(z)'=R(z);\n" }}{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)\"#C F)*$)F'\"\"%F)F)F)*&#F)\"$?\"F)*$)F'\"\"&F)F)F)*&#\"*>0PY$\"->Qdnv6F)* $)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 stabilit y region intersects the negative real axis by solving the equation: " }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "R(z) = 1;" "6#/-%\" RG6#%\"zG\"\"\"" }{TEXT -1 2 ". " }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 26 "z0 := newton(R(z)=1,z=-3);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6# >%#z0G$!+_X1'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],styl e=point,symbol=[circle,cross,diamond],color=black):\np3 := plot([[z0,0 ],[z0,1]],linestyle=3,color=COLOR(RGB,0,.5,0)):\nplots[display]([p1,p2 ,p3],view=[-3.19..0.49,-0.0..1.47],font=[HELVETICA,9]);" }}{PARA 13 " " 1 "" {GLPLOT2D 386 232 232 {PLOTDATA 2 "6+-%'CURVESG6$7X7$$!3%****** ********=$!#<$\"3wuuXpGAa@F*7$$!3'HLLjB$*)\\JF*$\"3iR!e/Cpv*>F*7$$!3Tm mmsky4JF*$\"3[]#QT9c5&=F*7$$!3e***\\?z*)[2$F*$\"3)zAv78?9t\"F*7$$!3<2Jqj!*Q\"F*7$$!3[mm1p!)*>#HF*$\"3u2w=uD3&G\"F*7$$!3smm '3'H\\#)GF*$\"3ML()=$)>L)=\"F*7$$!3q***\\NhvJ%GF*$\"3E2xU\\J$))4\"F*7$ $!37LLBm#eQ!GF*$\"3.$f5k1Td,\"F*7$$!3smmOQQ&4t#F*$\"3gDE#48YVx)!#=7$$! 3)*****4EbYbEF*$\"3,:%z.!zIRvF^o7$$!3emmc0aRxDF*$\"3A8RPMjU]kF^o7$$!3G ++!plv&*\\#F*$\"3p0sd%42X`&F^o7$$!3@LL8![G&>CF*$\"3/WUzP!G&[ZF^o7$$!3k mmE$)G-\\BF*$\"3E,n!*e%HC<%F^o7$$!31++S!G]'pAF*$\"3#)*Hps2$))QOF^o7$$! 33+++m<&**=#F*$\"3*)*y$p.@?6KF^o7$$!3F++gzu98@F*$\"3%)\\;*)))GF^o7$ $!3smmcH=SV?F*$\"3KH6F*$\"3GzX;7FhoCF^o7$$!3SLL $\\D6-*=F*$\"3)Gao$>oUeBF^o7$$!3)*****HLl[3=F*$\"31#f4B:9eG#F^o7$$!3hL L$pIYht\"F*$\"3m`L@(3gcE#F^o7$$!31++I8wxc;F*$\"3UBfVVur&G#F^o7$$!3+++! z%)*>\"e\"F*$\"3oi5%>-m9M#F^o7$$!3RLLLyBM-:F*$\"3W[YMk&**QV#F^o7$$!3@L Lj/m#*H9F*$\"3'R]%[$*Q!pa#F^o7$$!3smmm4k\"=N\"F*$\"3iE*yE^'p'p#F^o7$$! 3%omm&p=oq7F*$\"3G`Ejy44\")GF^o7$$!3A++!R,a+?\"F*$\"3EL$*))\\IYkIF^o7$ $!3umm'[*RxB6F*$\"3gC?)[9-eG$F^o7$$!3F++?r#p\\/\"F*$\"3m5)\\48'oRNF^o7 $$!3+/++MAuy'*F^o$\"3C#G+wm0J\"QF^o7$$!3h.++0!3G$*)F^o$\"3MsbWBz#=5%F^ o7$$!38,++A_d/\")F^o$\"3'4PQnCs8X%F^o7$$!3)yLL$H7PgtF^o$\"3A9;D3Ex#z%F ^o7$$!3++++?YxllF^o$\"3;O+)32xv=&F^o7$$!3#ymm;*>vXeF^o$\"3!pf;_E!4ubF^ o7$$!3],++wqde]F^o$\"3>K([WSe,.'F^o7$$!3bPLLEo!zJ%F^o$\"3qKXW3!fN\\'F^ o7$$!3<,++,IpVNF^o$\"36cP&))H!>;qF^o7$$!3oOLLtuv'y#F^o$\"3sF^o$\"3G\"[e3Uj>>)F^o7$$!3/ommA45J7F^o$\"3V&=P$GKmT))F ^o7$$!3l!ommWJg]%!#>$\"35Q!o5B)Rf&*F^o7$$\"3&ymmm<$))R\"F*7$$\"3K)*****[LT+TF^o$\"3o9:pkq)o]\"F*7$$\"3!************* **[F^o$\"3s=D\\TjLK;F*-%'COLOURG6&%$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$Fc qF_]l7$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$F_uF_]l7$FduF_]l7$FiuF_]l7$F^vF_] l7$FcvF_]l7$FhvF_]l7$F]wF_]l7$FbwF_]l7$FgwF_]l7$F\\xF_]l7$FaxF_]l7$Ffx F_]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#)*****>bkgz#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#%(DIAMONDGF]alF_al-F$6%7$7$Fg`lFi\\lFf`l-%&COLORG6&Fe\\lFi\\l$\"\"& !\"\"Fi\\l-%*LINESTYLEG6#\"\"$-%%FONTG6$%*HELVETICAG\"\"*-%+AXESLABELS G6%Q\"z6\"Q!Fdcl-F\\cl6#%(DEFAULTG-%%VIEWG6$;$!$>$!\"#$\"#\\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 49 "The follo wing picture shows the stability region." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1304 "R := z -> 1+z+1/2*z ^2+1/6*z^3+1/24*z^4+1/120*z^5+346370519/117567573819*z^6:\npts := []: \+ z0 := 0:\nfor ct from 0 to 200 do\n zz := newton(R(z)=exp(ct*Pi/20*I ),z=z0):\n z0 := zz:\n pts := [op(pts),[Re(zz),Im(zz)]]:\nend do: \np1 := plot(pts,color=COLOR(RGB,.4,.2,.05)):\np2 := plots[polygonplot ]([seq([pts[i-1],pts[i],[-1.4,0]],i=2..nops(pts))],\n style=p atchnogrid,color=COLOR(RGB,.8,.4,.2)):\npts := []: z0 := 1.5+3.5*I:\nf or ct from 0 to 60 do\n zz := newton(R(z)=exp(ct*Pi/30*I),z=z0):\n \+ z0 := zz:\n pts := [op(pts),[Re(zz),Im(zz)]]:\nend do:\np3 := plot( pts,color=COLOR(RGB,.4,.2,.05)):\np4 := plots[polygonplot]([seq([pts[i -1],pts[i],[1.44,3.33]],i=2..nops(pts))],\n style=patchnogrid ,color=COLOR(RGB,.8,.4,.2)):\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 := plot(pts,color= COLOR(RGB,.4,.2,.05)):\np6 := plots[polygonplot]([seq([pts[i-1],pts[i] ,[1.44,-3.33]],i=2..nops(pts))],\n style=patchnogrid,color=CO LOR(RGB,.8,.4,.2)):\np7 := plot([[[-3.39,0],[1.89,0]],[[0,-3.69],[0,3. 69]]],color=black,linestyle=3):\nplots[display]([p||(1..7)],view=[-3.3 9..1.89,-3.69..3.69],font=[HELVETICA,9],\n color=COLOR(RGB,.4,0,1),l abels=[`Re(z)`,`Im(z)`],axes=boxed,scaling=constrained);" }}{PARA 13 " " 1 "" {GLPLOT2D 614 614 614 {PLOTDATA 2 "60-%'CURVESG6$7ew7$$\"\"!F)F (7$$\"3;+++J@//B!#D$\"3%******fG'zq:!#=7$$\"33+++$[jvS\"!#B$\"3<+++]ue TJF07$$\"3%******z*H8y9!#A$\"3,+++w@I7ZF07$$\"3P+++xG'GM(F:$\"3=+++\"z cDG'F07$$\"38+++h%yJM#!#@$\"3_+++_$=6&yF07$$\"3?+++v_Y%R&FE$\"3I+++T96 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" }}{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#^$$\"0f!pfH(Hy&!#=$\"0@s9pa9M\"!#9" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#^$$\"0Ft\\>AC4$!#=$\"07W)[U\\r8!#9" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#^$$!0*GBK:>SW!#>$\"0KmfX)Q,9!#9" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#^$$!0'yVFSS^\\!#=$\"0/;*))*G6V\"!#9" } }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 107 "Then w e apply the bisection method to calculate the parameter value associat ed with the intersection point." }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 176 "real_part := proc(u)\n Re(newton(R(z)=exp(u*Pi*I),z=1.4*I)) \nend proc:\nDigits := 15:\nu0 := bisect('real_part'(u),u=0.43..0.46); \nnewton(R(z)=exp(u0*Pi*I),z=1.4*I);\nDigits := 10:" }}{PARA 11 "" 1 " " {XPPMATH 20 "6#>%#u0G$\"0r*y()zu)[%!#:" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#^#$\"0-.hFK!)R\"!#9" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 4 "The " }{TEXT 260 112 "largest interval on the nonn egative imaginary axis that contains the origin and lies inside the st ability region" }{TEXT -1 5 " is " }{XPPEDIT 18 0 "[ 0, 1.3980 ]" "6# 7$\"\"!-%&FloatG6$\"&!)R\"!\"%" }{TEXT -1 18 " (approximately)." }} {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 71 "The follo wing picture shows the main part of the stability region (the " } {TEXT 261 3 "red" }{TEXT -1 98 " curve) together with the boundary of \+ the largest disc that fits inside the stability region (the " }{TEXT 256 4 "blue" }{TEXT -1 54 " curve) and has radius equal to the stabili ty radius " }{XPPEDIT 18 0 "rho" "6#%$rhoG" }{TEXT -1 2 ". " }}{PARA 0 "" 0 "" {TEXT -1 27 "The line segment (drawn in " }{TEXT 257 5 "gree n" }{TEXT -1 27 ") represents the interval " }{XPPEDIT 18 0 "[-rho[2] , rho[2]]" "6#7$,$&%$rhoG6#\"\"#!\"\"&F&6#F(" }{TEXT -1 74 " on the i maginary axis giving the intersection with the stability region." }} {PARA 0 "" 0 "" {TEXT -1 27 "The line segment (drawn in " }{TEXT 264 6 "orange" }{TEXT -1 27 ") represents the interval " }{XPPEDIT 18 0 " [-2*rho, 0];" "6#7$,$*&\"\"#\"\"\"%$rhoGF'!\"\"\"\"!" }{TEXT -1 69 " \+ on the real axis giving the intersection with the stability region." } }{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 614 "R := z -> 1+z+1/2*z^2+1/6*z^3+1/24*z^4+1/120*z^5+346370519/1175 67573819*z^6:\npts := []: z0 := 0:\nfor ct from 0 to 360 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:\np1 := plot(pts,color=red):\nrd := 1.3980322 76: \np2 := plot([-rd+rd*cos(t),rd*sin(t),t=0..2*Pi],color=blue):\np3 \+ := plot([[-2*rd,0],[0,0]],thickness=4,color=COLOR(RGB,1,.6,0)):\np4 := plot([[0,-rd],[0,rd]],thickness=4,color=COLOR(RGB,0,.75,0)):\np5 := p lot([[[-.07,-rd],[.07,-rd]],[[-.07,rd],[.07,rd]]],color=COLOR(RGB,0,.5 ,0)):\nplots[display]([p||(1..5)],scaling=constrained,font=[HELVETICA, 9]);" }}{PARA 13 "" 1 "" {GLPLOT2D 353 456 456 {PLOTDATA 2 "6,-%'CURVE SG6$7eal7$$\"\"!F)F(7$$\"3>+++k5\"*R?!#E$\"3/+++\\v>Z5!#=7$$\"3'****** zuL#z7!#C$\"3'******\\z%R%4#F07$$\"33+++$[jvS\"!#B$\"3<+++]ueTJF07$$\" 35+++%41I_(F:$\"3)******zz^()=%F07$$\"33+++IrG$o#!#A$\"3Y*****f53eB&F0 7$$\"3P+++xG'GM(FE$\"3=+++\"zcDG'F07$$\"3!******4bigl\"!#@$\"3Z*****\\ r'eGtF07$$\"3A+++\"HF&*>$FP$\"3]+++(*G;t$)F07$$\"3?+++v_Y%R&FP$\"3I+++ T96:%*F07$$\"3C+++1)H*[zFP$\"3)******46t_/\"!#<7$$\"3/+++,<(4+\"!#?$\" 35+++VoP[6F\\o7$$\"3r+++Ik:5**FP$\"3++++9([0D\"F\\o7$$\"35+++vbqs\\FP$ \"31+++3N[^8F\\o7$$!3-+++=zvg&)FP$\"3/+++\"fm3X\"F\\o7$$!34+++))))eUNF `o$\"3'******z@$R[:F\\o7$$!3X+++Ix[1\")F`o$\"3(******H.wPk\"F\\o7$$!3. +++45K7:!#>$\"3\"******f'ftOF\\o7$$!3G+++O#\\*4cF_q$\"3! 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" }}{PARA 0 "" 0 "" {TEXT -1 73 "The slope of the tangent (derivative) to the curve at a p articular point " }{TEXT 303 1 "z" }{TEXT -1 100 " can be estimated fr om the slope of the line segment joining two neighbouring points eithe r side of " }{TEXT 304 1 "z" }{TEXT -1 14 " on the curve." }}{PARA 0 " " 0 "" {TEXT -1 96 "The graph shows the slope of the line perpendicula r to the tangent (the normal line) at a point " }{TEXT 305 1 "z" } {TEXT -1 26 " on the curve (plotted in " }{TEXT 256 4 "blue" }{TEXT -1 82 ") and the slope of the radius of the appropriate circle passing through the point " }{TEXT 306 1 "z" }{TEXT -1 26 " on the curve (plo tted in " }{TEXT 261 3 "red" }{TEXT -1 3 "). " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1233 "Digits := \+ 14:\nt6 := 7/8284:\nct_start := 47234: ct_end := 47280:\ntheta_div := \+ 10000:\nz_approx := -4+1.77*I:\n\nR := z -> 1+z+1/2*z^2+1/6*z^3+1/24*z ^4+1/120*z^5+t6*z^6:\npts := []: pts2 := []: pts3 := []:\nz0 := z_appr ox: zp := z0:\nfor ct from ct_start to ct_end do\n theta := evalf(ct *Pi/theta_div);\n zz := newton(R(z)=exp(theta*I),z=z0);\n normal_s lope := -Re(zz-zp)/Im(zz-zp);\n radius_slope := Im(z0)/(Re(z0)-abs(z 0)/(2*cos(argument(z0))));\n pts := [op(pts),[theta,radius_slope]]; \n pts2 := [op(pts2),[theta,normal_slope]];\n pts3 := [op(pts3),[t heta,normal_slope-radius_slope,zz]];\n zp := z0; z0 := zz;\nend do: \npts := [op(3..nops(pts),pts)]:\npts2 := [op(3..nops(pts2),pts2)]:\np ts3 := [op(3..nops(pts3),pts3)]:\ntheta0 := 0:\nfor ii from 2 to nops( pts3) do\n y1 := pts3[ii-1,2]; y2 := pts3[ii,2];\n if pts3[ii-1,2] *pts3[ii,2]<0 then\n theta1 := pts3[ii-1,1]; theta2 := pts3[ii,1] ;\n theta0 := (theta2*y1-theta1*y2)/(y1-y2);\n z0 := pts3[ii ,3];\n break;\n end if;\nend do: \nplot([pts,pts2],color=[red ,blue],style=line,font=[HELVETICA,9]);\nif theta0<>0 then\n zz := ne wton(R(z)=exp(theta0*I),z=z0);\n rho := -abs(zz)/(2*cos(argument(zz) ));\nend if:\nDigits := 10:\n[evalf(theta0),evalf(zz)];\n[t6,evalf(rho )];\n" }}{PARA 13 "" 1 "" {GLPLOT2D 494 342 342 {PLOTDATA 2 "6(-%'CURV ESG6$7O7$$\"3%**p\\eqiR[\"!#;$!31+$)e!\\@i0\"!#<7$$\"3-+ix@T*R[\"F*$!3 ++.<9r.c5F-7$$\"33+FqPb-%[\"F*$!33+%3By_e0\"F-7$$\"3)**HHO&p0%[\"F*$!3 3+[)\\\\oc0\"F-7$$\"3/+ebp$)3%[\"F*$!3-+7=_U[b5F-7$$\"3%**R#[&y>T[\"F* $!31+:)Q0+`0\"F-7$$\"3++*39?^T[\"F*$!3++o1+f6b5F-7$$\"33+aLELS@%[\"F*$!3!**\\EesZZ0\"F-7$$\"3/+&)=\\aC% [\"F*$!3)**fk`qjX0\"F-7$$\"3#***\\6loF%[\"F*$!3%**R>$H(zV0\"F-7$$\"3++ ;/\"G3V[\"F*$!3%***Gn(z&>a5F-7$$\"33+\"oppRV[\"F*$!3/+qS5>,a5F-7$$\"3' **f%*G6rV[\"F*$!31+b]n!GQ0\"F-7$$\"3/+7#)GDS%[\"F*$!3)**R]*oUk`5F-7$$ \"3#**pZZ%RV%[\"F*$!3-+Rs90Y`5F-7$$\"3++Vng`Y%[\"F*$!3'**f4[!oF`5F-7$$ 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hy-%&STYLEG6#%%LINEG-%+AXESLABELSG6$Q!6\"F`cl-%%FONTG6$%*HELVETICAG\" \"*-%%VIEWG6$%(DEFAULTGFjcl" 1 6 0 1 10 0 2 9 1 4 2 1.000000 45.000000 45.000000 0 0 "Curve 1" "Curve 2" }}}{PARA 11 "" 1 "" {XPPMATH 20 "6#7$$\"+TKp%[\"!\")^$$!+7*GU,%!\"*$\"+%otBs\"F*" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#7$#\"\"(\"%%G)$\"+C9iwB!\"*" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 117 "The point of inters ection of the two graphs can be read off visually and used to obtain a n estimate of the value of " }{XPPEDIT 18 0 "rho" "6#%$rhoG" }{TEXT -1 43 " corresponding to the particlar value of " }{XPPEDIT 18 0 "t[ 6]" "6#&%\"tG6#\"\"'" }{TEXT -1 1 "." }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 132 "Digits := 20:\nzz := newton(R(z)=exp(14.84692*I),z=- 4+1.72*I);\nrho := -abs(zz)/(2*cos(argument(zz))):\nDigits := 10:\n[t6 ,evalf(rho)];\n" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%#zzG^$$!5D'y!)y1* HA9S!#>$\"5k,9>z7$zBs\"F(" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#7$#\"\"( \"%%G)$\"+B9iwB!\"*" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 " " {TEXT -1 46 "The following table gives some results where " } {XPPEDIT 18 0 "rho" "6#%$rhoG" }{TEXT -1 32 " is close to its maximum value." }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "matrix([[t [6], rho], [___________, _______________], [.844990e-3, 2.376621421], \+ [.845000e-3, 2.376621422], [.845005e-3, 2.376621423], [.845010e-3, 2.3 76621423], [.845020e-3, 2.376621420]]);" "6#-%'matrixG6#7)7$&%\"tG6#\" \"'%$rhoG7$%,___________G%0_______________G7$-%&FloatG6$\"'!*\\%)!\"*- F26$\"+@9iwBF57$-F26$\"'+]%)F5-F26$\"+A9iwBF57$-F26$\"'0]%)F5-F26$\"+B 9iwBF57$-F26$\"'5]%)F5-F26$\"+B9iwBF57$-F26$\"'?]%)F5-F26$\"+?9iwBF5" }{TEXT -1 1 " " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 8 "Taking " }{XPPEDIT 18 0 " t[6] = 7/8284;" "6#/&%\"tG6#\"\"'*&\"\"(\"\"\"\"%%G)!\"\"" }{TEXT -1 1 " " }{TEXT 308 1 "~" }{TEXT -1 21 " 0.000845002 gives " }{XPPEDIT 18 0 "rho" "6#%$rhoG" }{TEXT -1 1 " " }{TEXT 301 1 "~" }{TEXT -1 24 " \+ 2.3766214 and taking " }{XPPEDIT 18 0 "t[6] = 1/1183;" "6#/&%\"tG6# \"\"'*&\"\"\"F)\"%$=\"!\"\"" }{TEXT -1 1 " " }{TEXT 307 1 "~" }{TEXT -1 21 " 0.000845309 gives " }{XPPEDIT 18 0 "rho" "6#%$rhoG" }{TEXT -1 1 " " }{TEXT 302 1 "~" }{TEXT -1 72 " 2.3766206, which is close eno ugh to the maximum for practical purposes." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 354 "R := z -> 1+z+1/2 *z^2+1/6*z^3+1/24*z^4+1/120*z^5+1/1183*z^6:\npts := []:\nz0 := 0:\nfor ct from 0 to 360 do\n zz := newton(R(z)=exp(ct*Pi/30*I),z=z0):\n \+ z0 := zz:\n pts := [op(pts),[Re(zz),Im(zz)]]:\nend do:\np1 := plot(p ts,color=red):\nrd := 2.37662: \np2 := plot([-rd+rd*cos(t),rd*sin(t),t =0..2*Pi],color=blue):\nplots[display]([p1,p2],scaling=constrained);" }}{PARA 13 "" 1 "" {GLPLOT2D 391 461 461 {PLOTDATA 2 "6'-%'CURVESG6$7e al7$$\"\"!F)F(7$$!3')*4Dp*)=W:(!#F$\"3'**4t;b(>Z5!#=7$$!3K+7:y2c^X!#D$ \"3++PIj^R%4#F07$$!3\\**GBn6uK^!#C$\"33+E-#p$fTJF07$$!3,+Xr-5?VG!#B$\" 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$\"$H%Fe^^l" 1 2 0 1 10 0 2 9 1 2 1 1.000000 45.000000 45.000000 0 0 " Curve 1" "Curve 2" "Curve 3" "Curve 4" }}}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}}{PARA 0 "" 0 "" {TEXT -1 86 "#-------------------------------------------------------- -----------------------------" }}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 61 " The stability radius of a 7 stage, order 5 Runge-Kutta scheme" }} {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 13 "The (line ar) " }{TEXT 260 16 "stability radius" }{TEXT -1 2 " " }{XPPEDIT 18 0 "rho" "6#%$rhoG" }{TEXT -1 111 " associated with a stability region \+ is the radius of the largest disc that can fit inside the stability re gion." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 86 " The stability function of a general 7 stage, order 5 Runge-Kutta schem e has the form: " }}{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+t[6]*z^6+t[7]*z^7;" "6#/-%\"RG6#% \"zG,2\"\"\"F)F'F)*&F'\"\"#F+!\"\"F)*&F'\"\"$\"\"'F,F)*&F'\"\"%\"#CF,F )*&F'\"\"&\"$?\"F,F)*&&%\"tG6#F/F)*$F'F/F)F)*&&F86#\"\"(F)*$F'F>F)F)" }{TEXT -1 2 ", " }}{PARA 0 "" 0 "" {TEXT -1 24 "where the coefficients " }{XPPEDIT 18 0 "t[6]" "6#&%\"tG6#\"\"'" }{TEXT -1 7 " and " } {XPPEDIT 18 0 "t[7]" "6#&%\"tG6#\"\"(" }{TEXT -1 45 " are associated \+ with \"tall\" trees. In detail" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }} {PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "t[6]= b[6]*a[6,5]*a[5 ,4]*a[4,3]*a[3,2]*c[2]+b[7]" "6#/&%\"tG6#\"\"',&*.&%\"bG6#F'\"\"\"&%\" aG6$F'\"\"&F-&F/6$F1\"\"%F-&F/6$F4\"\"$F-&F/6$F7\"\"#F-&%\"cG6#F:F-F-& F+6#\"\"(F-" }{TEXT -1 2 " (" }{XPPEDIT 18 0 " a[7,5]*a[5,4]*a[4,3]*a[ 3,2]*c[2]+a[7,6]*a[6,4]*a[4,3]*a[3,2]*c[2]+``" "6#,(*,&%\"aG6$\"\"(\" \"&\"\"\"&F&6$F)\"\"%F*&F&6$F-\"\"$F*&F&6$F0\"\"#F*&%\"cG6#F3F*F**,&F& 6$F(\"\"'F*&F&6$F:F-F*&F&6$F-F0F*&F&6$F0F3F*&F56#F3F*F*%!GF*" }{TEXT -1 1 " " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "a[7,6]*a[6,5]*a[5,3]*a[3,2]*c[2]+a[7,6]*a[6,5]*a [5,4]*a[4,2]*c[2]+a[7,6]*a[6,5]*a[5,4]*a[4,3]*c[3]" "6#,(*,&%\"aG6$\" \"(\"\"'\"\"\"&F&6$F)\"\"&F*&F&6$F-\"\"$F*&F&6$F0\"\"#F*&%\"cG6#F3F*F* *,&F&6$F(F)F*&F&6$F)F-F*&F&6$F-\"\"%F*&F&6$F>F3F*&F56#F3F*F**,&F&6$F(F )F*&F&6$F)F-F*&F&6$F-F>F*&F&6$F>F0F*&F56#F0F*F*" }{TEXT -1 3 " ) " }} {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 3 "and" }} {PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "t[7]=b[7]*a[7,6]*a[6, 5]*a[5,4]*a[4,3]*a[3,2]*c[2]*z^7" "6#/&%\"tG6#\"\"(*2&%\"bG6#F'\"\"\"& %\"aG6$F'\"\"'F,&F.6$F0\"\"&F,&F.6$F3\"\"%F,&F.6$F6\"\"$F,&F.6$F9\"\"# F,&%\"cG6#F " 0 "" {MPLTEXT 1 0 23 "StabilityFunction(5,7);" }}{PARA 12 "" 1 "" {XPPMATH 20 "6#,2\"\"\"F$%\"zGF$*&\"\"#!\"\"F%F'F$*&\"\"'F(F%\"\"$F$*&\"#CF(F% \"\"%F$*&\"$?\"F(F%\"\"&F$*&,&*.&%\"bG6#F*F$&%\"aG6$F*F1F$&F96$F1F.F$& F96$F.F+F$&F96$F+F'F$&%\"cG6#F'F$F$*&&F66#\"\"(F$,&*,&F96$FGF1F$F;F$F= F$F?F$FAF$F$*&&F96$FGF*F$,&**&F96$F*F.F$F=F$F?F$FAF$F$*&F8F$,&*(&F96$F 1F+F$F?F$FAF$F$*&F;F$,&*&&F96$F.F'F$FAF$F$*&F=F$&FB6#F+F$F$F$F$F$F$F$F $F$F$F$)F%F*F$F$*2FEF$FMF$F8F$F;F$F=F$F?F$FAF$)F%FGF$F$" }}}{PARA 0 " " 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 81 "We can investigate stability regions that are obtained by varing the parameters " } {XPPEDIT 18 0 "t[6]" "6#&%\"tG6#\"\"'" }{TEXT -1 7 " and " } {XPPEDIT 18 0 "t[7]" "6#&%\"tG6#\"\"(" }{TEXT -1 115 " leaving aside \+ the question as to whether it is possible to construct a scheme having the chosen parameter values." }}{PARA 0 "" 0 "" {TEXT -1 1 " " }} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}{SECT 1 {PARA 4 "" 0 " " {TEXT -1 72 "Sample calculation of the stability radius for a 7 stag e, order 5 scheme" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 76 "The following is a sample calculation of the value of the stability radius " }{XPPEDIT 18 0 "rho" "6#%$rhoG" }{TEXT -1 8 " wh en " }{XPPEDIT 18 0 "t[6] = 0;" "6#/&%\"tG6#\"\"'\"\"!" }{TEXT -1 12 ".0018 and " }{XPPEDIT 18 0 "t[7] = 0;" "6#/&%\"tG6#\"\"(\"\"!" } {TEXT -1 6 ".0002." }}{PARA 0 "" 0 "" {TEXT -1 224 "The largest inscri bed circle for the stability region will have a radius line radius at \+ the point of contact with the boundary of the stability region that is perpendicular to the tangent to the boundary curve at that point." }} {PARA 0 "" 0 "" {TEXT -1 73 "The slope of the tangent (derivative) to \+ the curve at a particular point " }{TEXT 295 1 "z" }{TEXT -1 100 " can be estimated from the slope of the line segment joining two neighbour ing points either side of " }{TEXT 296 1 "z" }{TEXT -1 14 " on the cur ve." }}{PARA 0 "" 0 "" {TEXT -1 96 "The graph shows the slope of the l ine perpendicular to the tangent (the normal line) at a point " } {TEXT 297 1 "z" }{TEXT -1 26 " on the curve (plotted in " }{TEXT 256 4 "blue" }{TEXT -1 82 ") and the slope of the radius of the appropriat e circle passing through the point " }{TEXT 298 1 "z" }{TEXT -1 26 " o n the curve (plotted in " }{TEXT 261 3 "red" }{TEXT -1 3 "). " }} {PARA 0 "" 0 "" {TEXT -1 91 "The point of intersection of the two grap hs is used to obtain an estimate of the value of " }{XPPEDIT 18 0 "rh o" "6#%$rhoG" }{TEXT -1 45 " corresponding to the particular values o f " }{XPPEDIT 18 0 "t[6]" "6#&%\"tG6#\"\"'" }{TEXT -1 6 " and " } {XPPEDIT 18 0 "t[7]" "6#&%\"tG6#\"\"(" }{TEXT -1 1 "." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 141 "The section of the bo undary curve close to the point of tangency to consider and the finene ss of the subdivision to be used can be adjusted. " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 153 "It is necessary to provi de a suitable starting approximation for applying Newton's method to o btain the first point on the section of the boundary curve." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1248 " Digits := 14:\nt6 := 9/5000: t7 := 1/5000:\nct_start := 35950: ct_end \+ := 35980:\ntheta_div := 10000:\nz_approx := -3.8+1.38*I:\n\nR := z -> \+ 1+z+1/2*z^2+1/6*z^3+1/24*z^4+1/120*z^5+t6*z^6+t7*z^7:\npts := []: pts2 := []: pts3 := []:\nz0 := z_approx: zp := z0:\nfor ct from ct_start t o ct_end do\n theta := evalf(ct*Pi/theta_div);\n zz := newton(R(z) =exp(theta*I),z=z0);\n normal_slope := -Re(zz-zp)/Im(zz-zp);\n rad ius_slope := Im(z0)/(Re(z0)-abs(z0)/(2*cos(argument(z0))));\n pts := [op(pts),[theta,radius_slope]];\n pts2 := [op(pts2),[theta,normal_s lope]];\n pts3 := [op(pts3),[theta,normal_slope-radius_slope,zz]];\n zp := z0; z0 := zz;\nend do:\npts := [op(3..nops(pts),pts)]:\npts2 \+ := [op(3..nops(pts2),pts2)]:\npts3 := [op(3..nops(pts3),pts3)]:\ntheta 0 := 0:\nfor ii from 2 to nops(pts3) do\n y1 := pts3[ii-1,2]; y2 := \+ pts3[ii,2];\n if pts3[ii-1,2]*pts3[ii,2]<0 then\n theta1 := pts 3[ii-1,1]; theta2 := pts3[ii,1];\n theta0 := (theta2*y1-theta1*y2 )/(y1-y2);\n z0 := pts3[ii,3];\n break;\n end if;\nend do: \nplot([pts,pts2],color=[red,blue],font=[HELVETICA,9]);\nif theta0< >0 then\n zz := newton(R(z)=exp(theta0*I),z=z0);\n rho := -abs(zz) /(2*cos(argument(zz)));\nend if:\nDigits := 10:\n[evalf(theta0),evalf( zz)];\n[t6,t7,evalf(rho)];\n" }}{PARA 13 "" 1 "" {GLPLOT2D 442 324 324 {PLOTDATA 2 "6'-%'CURVESG6$7?7$$\"3'**f=3Rl%H6!#;$!3:+)fQ'f&H6F*$!39+o8mD'*>!)F-7$$\"31+Z_a5fH6F*$!3R+=o$R,$=!)F-7 $$\"3'**H^/ZA'H6F*$!3p**fz[/k;!)F-7$$\"3-+yP')QlH6F*$!3P+XRJ(z\\,)F-7$ $\"3#**R/BI&oH6F*$!3b*459C>L,)F-7$$\"3****3B=nrH6F*$!3b+Dvy*e;,)F-7$$ \"31+u:M\"[(H6F*$!3o*4aL%*)**4!)F-7$$\"3&***R3]&z(H6F*$!3++08N\"R$3!)F 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z0 := zz:\n pts := [op(pts),[Re(zz),Im(zz)]]: \nend do:\np1 := plot(pts,color=red):\nrd := 2.1531: \np2 := plot([-rd +rd*cos(t),rd*sin(t),t=0..2*Pi],color=blue):\nplots[display]([p1,p2],s caling=constrained,font=[HELVETICA,9]);" }}{PARA 13 "" 1 "" {GLPLOT2D 389 453 453 {PLOTDATA 2 "6(-%'CURVESG6$7eal7$$\"\"!F)F(7$$\"3s*****p&H u&R&!#F$\"31+++^v>Z5!#=7$$\"3v******40j.M!#D$\"3)*******H]R%4#F07$$\"3 /+++T\"3Iy$!#C$\"3u******H9fTJF07$$\"3++++as*>0#!#B$\"3;+++>6y)=%F07$$ \"3A+++]%=#puF@$\"3X+++;\\%fB&F07$$\"3')*****>w*R+@!#A$\"3a+++=R.$G'F0 7$$\"3')*****fJ9E\"\\FK$\"3&******f'[%*HtF07$$\"3[*****4fb$p**FK$\"3E+ ++$*o[w$)F07$$\"3))*****fuX$*z\"!#@$\"3^+++C5MA%*F07$$\"3.+++%RAl#HFen $\"33+++u:qY5!#<7$$\"3(******z([t-VFen$\"33+++H/)4:\"F]o7$$\"3s*****RC \\gm&Fen$\"3#******RC`#\\dFen$\"31+++.3gh9F]o7$$\"3/+++kut^?Fen$\"3#******4\\= Rc\"F]o7$$!3C+++))H8)o'Fen$\"3%*******3uLl;F]o7$$!35+++*e\\qJ#!#?$\"3# ******4*Gnl2&F_q$\"3#******p\">uk=F]o7$$!34+++hmd? 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"6#*$%\"zG\"\"(" }{TEXT -1 29 " in the stability \+ function: " }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "R(z) = \+ 1+z+z^2/2+z^3/6+z^4/24+z^5/120+t[6]*z^6+t[7]*z^7" "6#/-%\"RG6#%\"zG,2 \"\"\"F)F'F)*&F'\"\"#F+!\"\"F)*&F'\"\"$\"\"'F,F)*&F'\"\"%\"#CF,F)*&F' \"\"&\"$?\"F,F)*&&%\"tG6#F/F)*$F'F/F)F)*&&F86#\"\"(F)*$F'F>F)F)" } {TEXT -1 2 ", " }}{PARA 0 "" 0 "" {TEXT -1 22 "the stability radius \+ " }{XPPEDIT 18 0 "rho" "6#%$rhoG" }{TEXT -1 1 " " }{TEXT 300 1 "~" } {TEXT -1 12 " 3.04298260." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 " " 0 "" {TEXT -1 54 "Note that the associated real stability interval i s: " }{XPPEDIT 18 0 "[-2*rho,0]" "6#7$,$*&\"\"#\"\"\"%$rhoGF'!\"\"\" \"!" }{TEXT -1 1 "." }}{PARA 0 "" 0 "" {TEXT -1 1 " " }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1301 "Digits := 20:\nt6 := .117605e-2: t7 := \+ .7827904795e-4:\nct_start := 3759140: ct_end := 3759240:\ntheta_div := 1000000:\nz_approx := -5+2*I:\n\nR := z -> 1+z+1/2*z^2+1/6*z^3+1/24*z ^4+1/120*z^5+t6*z^6+t7*z^7:\npts := []: pts2 := []: pts3 := []:\nz0 := z_approx: zp := z0:\nfor ct from ct_start to ct_end do\n theta := e valf(ct*Pi/theta_div);\n zz := newton(R(z)=exp(theta*I),z=z0);\n n ormal_slope := -Re(zz-zp)/Im(zz-zp);\n radius_slope := Im(z0)/(Re(z0 )-abs(z0)/(2*cos(argument(z0))));\n pts := [op(pts),[theta,radius_sl ope]];\n pts2 := [op(pts2),[theta,normal_slope]];\n pts3 := [op(pt s3),[theta,normal_slope-radius_slope,zz]];\n zp := z0; z0 := zz;\nen d do:\npts := [op(3..nops(pts),pts)]:\npts2 := [op(3..nops(pts2),pts2) ]:\npts3 := [op(3..nops(pts3),pts3)]:\ntheta0 := 0:\nfor ii from 2 to \+ nops(pts3) do\n y1 := pts3[ii-1,2]; y2 := pts3[ii,2];\n if pts3[ii -1,2]*pts3[ii,2]<0 then\n theta1 := pts3[ii-1,1]; theta2 := pts3[ ii,1];\n theta0 := (theta2*y1-theta1*y2)/(y1-y2);\n z0 := pt s3[ii,3];\n break;\n end if;\nend do: \nplot([pts,pts2],color =[red,blue],font=[HELVETICA,9]);\nif theta0<>0 then\n zz := newton(R (z)=exp(theta0*I),z=z0);\n rho := -abs(zz)/(2*cos(argument(zz)));\ne nd if:\nrr := 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{TEXT -1 1 " " }{XPPEDIT 18 0 "-5.439066268" "6#,$-%&F loatG6$\"+oi1Ra!\"*!\"\"" }{TEXT -1 1 " " }{TEXT 314 1 "+" }{TEXT -1 1 " " }{XPPEDIT 18 0 "1.875773481*i" "6#*&-%&FloatG6$\"+\"[td(=!\"*\" \"\"%\"iGF)" }{TEXT -1 16 " approximately." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 158 "The following picture shows th e boundary of the stability region (excluding \"satellites\") along wi th the circle associated with the maximal stability radius.\n" }} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 447 "t6 := 0.117605e-2: t7 := 0. 7827904795e-4:\nR := z -> 1+z+1/2*z^2+1/6*z^3+1/24*z^4+1/120*z^5+t6*z^ 6+t7*z^7:\npts := []:\nz0 := 0:\nfor ct from 0 to 500 do\n zz := new ton(R(z)=exp(ct*Pi/50*I),z=z0):\n z0 := zz:\n pts := [op(pts),[Re( zz),Im(zz)]]:\nend do:\np1 := plot(pts,color=red):\nrd := 3.042982599: \np2 := plot([-rd+rd*cos(t),rd*sin(t),t=0..2*Pi],numpoints=100,color= blue):\nplots[display]([p1,p2],scaling=constrained,font=[HELVETICA,9], style=line);" }}{PARA 13 "" 1 "" 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:= 0.7827904795e-4:\nR := z -> 1+z+1/2*z^2+1/6*z^3+1/24*z^4+1 /120*z^5+t6*z^6+t7*z^7:\npts := []:\nz0 := 0:\nfor ct from 0 to 500 do \n zz := newton(R(z)=exp(ct*Pi/50*I),z=z0):\n z0 := zz:\n pts := [op(pts),[Re(zz),Im(zz)]]:\nend do:\np1 := plot(pts,color=COLOR(RGB,. 2,0,.1)):\np2 := plots[polygonplot]([seq([pts[i-1],pts[i],[-2.4,0]],i= 2..nops(pts))],\n style=patchnogrid,color=COLOR(RGB,.9,.1,.6) ):\npts := []:\nz0 := 0.8+4.2*I:\nfor ct from 0 to 100 do\n zz := ne wton(R(z)=exp(ct*Pi/50*I),z=z0):\n z0 := zz:\n pts := [op(pts),[Re (zz),Im(zz)]]:\nend do:\np3 := plot(pts,color=COLOR(RGB,.2,0,.1)):\np4 := plots[polygonplot]([seq([pts[i-1],pts[i],[0.52,3.85]],i=2..nops(pt s))],\n style=patchnogrid,color=COLOR(RGB,.9,.1,.6)):\npts := []:\nz0 := 0.8-4.2*I:\nfor ct from 0 to 100 do\n zz := newton(R(z)= exp(ct*Pi/50*I),z=z0):\n z0 := zz:\n pts := [op(pts),[Re(zz),Im(zz )]]:\nend do:\np5 := plot(pts,color=COLOR(RGB,.2,0,.1)):\np6 := plots[ polygonplot]([seq([pts[i-1],pts[i],[0.52,-3.85]],i=2..nops(pts))],\n \+ style=patchnogrid,color=COLOR(RGB,.9,.1,.6)):\np7 := plot([[[- 7.19,0],[1.19,0]],[[0,-4.39],[0,4.39]]],color=black,linestyle=3):\nplo ts[display]([p||(1..7)],view=[-7.19..1.19,-4.39..4.39],font=[HELVETICA ,9],\n labels=[`Re(z)`,`Im(z)`],axes=boxed,scaling=constr ained);" }}{PARA 13 "" 1 "" {GLPLOT2D 468 489 489 {PLOTDATA 2 "6/-%'CU RVESG6$7ajl7$$\"\"!F)F(7$$!3A+++jQtjS!#G$\"3)******zI&=$G'!#>7$$!3#*** ***fA$>t%)!#F$\"3-+++iqjc7!#=7$$!3W+++vX_a&*!#E$\"31++++c&\\)=F77$$!3M +++m68[`!#D$\"3')*****>=uK^#F77$$!3++++!=]a.#!#C$\"3#******\\$HfTJF77$ $!3y******e\"\\Q1'FG$\"31+++#>7*pPF77$$!3$*******Hd&[_\"!#B$\"39+++$oK #)R%F77$$!3))*****p5>mQ$FR$\"3z*****>\"ebE]F77$$!3f******\\M\")RoFR$\" 3a+++gS)[l&F77$$!3-+++_Pb\"G\"!#A$\"3')******3:A$G'F77$$!37+++o6\\fAF \\o$\"3*)*****z^u:\"pF77$$!3)******>g)3)y$F\\o$\"3)******Hf_*RvF77$$!3 )******\\0js3'F\\o$\"3u******G%p$o\")F77$$!3S*****zHhTV*F\\o$\"3k***** 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\"\"" }{TEXT -1 3 ",0]" }}{PARA 0 "" 0 "" {TEXT -1 1 " " }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1317 "Digits := 20:\nt6 := 0.1103845e-2 : t7 := 0.6537742564e-4:\nct_start := 2667550: ct_end := 2667650:\nthe ta_div := 1000000:\nz_approx := -4.3+3*I:\n\nR := z -> 1+z+1/2*z^2+1/6 *z^3+1/24*z^4+1/120*z^5+t6*z^6+t7*z^7:\npts := []: pts2 := []: pts3 := []:\nz0 := z_approx: zp := z0:\nfor ct from ct_start to ct_end do\n \+ theta := evalf(ct*Pi/theta_div);\n zz := newton(R(z)=exp(theta*I),z =z0);\n normal_slope := -Re(zz-zp)/Im(zz-zp);\n radius_slope := Im (z0)/(Re(z0)-abs(z0)/(2*cos(argument(z0))));\n pts := [op(pts),[thet a,radius_slope]];\n pts2 := [op(pts2),[theta,normal_slope]];\n pts 3 := [op(pts3),[theta,normal_slope-radius_slope,zz]];\n zp := z0; z0 := zz;\nend do:\npts := [op(3..nops(pts),pts)]:\npts2 := [op(3..nops( pts2),pts2)]:\npts3 := [op(3..nops(pts3),pts3)]:\ntheta0 := 0:\nfor ii from 2 to nops(pts3) do\n y1 := pts3[ii-1,2]; y2 := pts3[ii,2];\n \+ if pts3[ii-1,2]*pts3[ii,2]<0 then\n theta1 := pts3[ii-1,1]; thet a2 := pts3[ii,1];\n theta0 := (theta2*y1-theta1*y2)/(y1-y2);\n \+ z0 := pts3[ii,3];\n break;\n end if;\nend do: \nplot([pts, pts2],color=[red,blue],font=[HELVETICA,9],style=line);\nif theta0<>0 t hen\n zz := newton(R(z)=exp(theta0*I),z=z0);\n rho := -abs(zz)/(2* cos(argument(zz)));\nend if:\nrr := -newton(R(z)=-1,z=-6)/2:\nDigits : = 10:\n[evalf(theta0),evalf(zz)];\n[t6,t7,evalf(rho),evalf(rr)];" }} {PARA 13 "" 1 "" {GLPLOT2D 481 420 420 {PLOTDATA 2 "6(-%'CURVESG6$7_q7 $$\"5H8g(oiwh.Q)!#>$!5p\\uyT*\\<@a#F*7$$\"5Fs89'y!\\O!Q)F*$!5z#4(fBV?6 UDF*7$$\"5DJnSX\\!o.Q)F*$!5>u2(yqe1@a#F*7$$\"5B!4sY5>r.Q)F*$!5ix%3Y48, @a#F*7$$\"5@\\u$RELu.Q)F*$!5(o=5Q[n&4UDF*7$$\"5>3G?BuuP!Q)F*$!5#e)eZv= -4UDF*7$$\"5m 1$z?a#F*7$$\"58&)))*4!**oQ!Q)F*$!5)zvc_1&Q2UDF*7$$\"56WUEgS+R!Q)F*$!58 `#ynYRo?a#F*7$$\"54.'H&>#=$R!Q)F*$!5rcOwqQH1UDF*7$$\"52i\\zyBjR!Q)F*$! 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" }}{PARA 0 "" 0 "" {TEXT -1 1 " " }}{PARA 0 "" 0 "" {TEXT -1 24 "In \+ the following table " }{XPPEDIT 18 0 "r[1]" "6#&%\"rG6#\"\"\"" } {TEXT -1 43 " is the radius of the circle with centre " }{XPPEDIT 18 0 "-r[1]" "6#,$&%\"rG6#\"\"\"!\"\"" }{TEXT -1 89 " on the real axi s and tangential to the boundary of the stability region near the poin t " }{XPPEDIT 18 0 "-4.3+3*i" "6#,&-%&FloatG6$\"#V!\"\"F(*&\"\"$\"\"\" %\"iGF+F+" }{TEXT -1 56 " and r[2] is the radius of the circle with \+ centre at " }{XPPEDIT 18 0 "-r[2]" "6#,$&%\"rG6#\"\"#!\"\"" }{TEXT -1 88 " on the real axis and tangential to the boundary of the stabil ity region at the point " }{XPPEDIT 18 0 "-2*r[2]" "6#,$*&\"\"#\"\"\" &%\"rG6#F%F&!\"\"" }{TEXT -1 2 ". " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }} {PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "matrix([[t[6], t[7], \+ r[1], r[2]], [____________, ________________, ____________, __________ __], [.1103834e-2, .6537569275e-4, 3.173869831, 3.173869832], [.110383 5e-2, .6537585028e-4, 3.173869831, 3.173869832], [.1103836e-2, .653760 0782e-4, 3.173869832, 3.173869832], [.1103837e-2, .6537616535e-4, 3.17 3869832, 3.173869833], [.1103838e-2, .6537632289e-4, 3.173869832, 3.17 3869832], [.1103839e-2, .6537648042e-4, 3.173869832, 3.173869833], [.1 103840e-2, .6537663796e-4, 3.173869832, 3.173869833], [.1103841e-2, .6 537679550e-4, 3.173869832, 3.173869832], [.1103842e-2, .6537695304e-4, 3.173869832, 3.173869832], [.1103843e-2, .6537711057e-4, 3.173869832, 3.173869833], [.1103844e-2, .6537726811e-4, 3.173869832, 3.173869832] , [.1103845e-2, .6537742564e-4, 3.173869831, 3.173869833]]);" "6#-%'ma trixG6#707&&%\"tG6#\"\"'&F)6#\"\"(&%\"rG6#\"\"\"&F06#\"\"#7&%-________ ____G%1________________GF7F77&-%&FloatG6$\"(MQ5\"!\"*-F;6$\"+v#pv`'!#9 -F;6$\"+J)pQ<$F>-F;6$\"+K)pQ<$F>7&-F;6$\"(NQ5\"F>-F;6$\"+G]ePlFB-F;6$ \"+J)pQ<$F>-F;6$\"+K)pQ<$F>7&-F;6$\"(OQ5\"F>-F;6$\"+#y+w`'FB-F;6$\"+K) pQ<$F>-F;6$\"+K)pQ<$F>7&-F;6$\"(PQ5\"F>-F;6$\"+NlhPlFB-F;6$\"+K)pQ<$F> -F;6$\"+L)pQ<$F>7&-F;6$\"(QQ5\"F>-F;6$\"+*GKw`'FB-F;6$\"+K)pQ<$F>-F;6$ \"+K)pQ<$F>7&-F;6$\"(RQ5\"F>-F;6$\"+U![w`'FB-F;6$\"+K)pQ<$F>-F;6$\"+L) pQ<$F>7&-F;6$\"(SQ5\"F>-F;6$\"+'zjw`'FB-F;6$\"+K)pQ<$F>-F;6$\"+L)pQ<$F >7&-F;6$\"(TQ5\"F>-F;6$\"+]&zw`'FB-F;6$\"+K)pQ<$F>-F;6$\"+K)pQ<$F>7&-F ;6$\"(UQ5\"F>-F;6$\"+/`pPlFB-F;6$\"+K)pQ<$F>-F;6$\"+K)pQ<$F>7&-F;6$\"( VQ5\"F>-F;6$\"+d5rPlFB-F;6$\"+K)pQ<$F>-F;6$\"+L)pQ<$F>7&-F;6$\"(WQ5\"F >-F;6$\"+6osPlFB-F;6$\"+K)pQ<$F>-F;6$\"+K)pQ<$F>7&-F;6$\"(XQ5\"F>-F;6$ \"+kDuPlFB-F;6$\"+J)pQ<$F>-F;6$\"+L)pQ<$F>" }{TEXT -1 3 " " }}{PARA 0 "" 0 "" {TEXT -1 1 " " }}{PARA 0 "" 0 "" {TEXT -1 133 "The following picture shows the boundary of the stability region along with the cir cle associated with the maximal stability radius.\n" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 446 "t6 := .1103840e-2: t7 := .6537663796e-4:\n R := z -> 1+z+1/2*z^2+1/6*z^3+1/24*z^4+1/120*z^5+t6*z^6+t7*z^7:\npts : = []:\nz0 := 0:\nfor ct from 0 to 700 do\n zz := newton(R(z)=exp(ct* Pi/50*I),z=z0):\n z0 := zz:\n pts := [op(pts),[Re(zz),Im(zz)]]:\ne nd do:\np1 := plot(pts,color=red):\nrd := 3.173869832: \np2 := plot([- rd+rd*cos(t),rd*sin(t),t=0..2*Pi],numpoints=100,color=blue):\nplots[di splay]([p1,p2],scaling=constrained,font=[HELVETICA,9],style=line);" }} {PARA 13 "" 1 "" {GLPLOT2D 604 604 604 {PLOTDATA 2 "6)-%'CURVESG6$7ifm 7$$\"\"!F)F(7$$!3'******fequ]%!#G$\"3)******zI&=$G'!#>7$$!3++++'\\H-8 \"!#E$\"37+++jqjc7!#=7$$!3)******\\gacF\"!#D$\"3#******\\gb\\)=F77$$!3 #)******fq8JrF;$\"39+++>UF8DF77$$!3)******z\\Oxq#!#C$\"38+++7JfTJF77$$ !3b*****\\]LE/)FF$\"3C+++DG\"*pPF77$$!3#)*****R4F`,#!#B$\"3w*****p`M#) R%F77$$!3!)*****pXCxX%FQ$\"3^*****\\]gl-&F77$$!3A+++jhbh*)FQ$\"33+++&p 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calculations, the stability radius is simply the radius of the real stability interval." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 136 "t6 := .1139945e-2: t7 := .7 182784295e-4:\nR := z -> 1+z+1/2*z^2+1/6*z^3+1/24*z^4+1/120*z^5+t6*z^6 +t7*z^7:\n'rho' = newton(R(z)=-1,z=-6)/2;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/%$rhoG$!+Q^E$4$!\"*" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 306 "z0 := newton(R(z)=-1,z=-6 ):\np1 := plot([R(z),-1],z=-6.89..0.49,color=[red,blue]):\np2 := plot( [[[z0,-1]]$3],style=point,symbol=[circle,cross,diamond],color=black): \np3 := plot([[z0,0],[z0,-1]],linestyle=3,color=COLOR(RGB,0,.5,0)):\np lots[display]([p1,p2,p3],view=[-6.89..0.49,-1.47..1.47],font=[HELVETIC A,9]);" }}{PARA 13 "" 1 "" {GLPLOT2D 396 336 336 {PLOTDATA 2 "6+-%'CUR VESG6$7V7$$!3o*************)o!#<$!3e(R\\I(=N_JF*7$$!3y**\\7,&o&4oF*$!3 sn'pc9/[y#F*7$$!3*)***\\A+P\"HnF*$!3/jm7^mPbCF*7$$!3c\\P9&)R:fmF*$!3O> 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Stability Results for Explicit Runge-Kutta Methods, Bry njulf Owren and Kristian Seip, \n Division of Mathematical Scien ces, 7he University of Trondheim, N-7034 Trondheim-N711, Norway." }} {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 45 "In this p aper the following result is proved." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 32 "Given a stability region for an " } {TEXT 309 1 "s" }{TEXT -1 14 " stage, order " }{TEXT 310 1 "p" }{TEXT -1 44 " Runge-Kutta method, with stability radius " }{XPPEDIT 18 0 "r ho" "6#%$rhoG" }{TEXT -1 10 ", define: " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "H[s,p](r) = matr ix([[s], [s-p]])-(-1)^p;" "6#/-&%\"HG6$%\"sG%\"pG6#%\"rG,&-%'matrixG6# 7$7#F(7#,&F(\"\"\"F)!\"\"F4),$F4F5F)F5" }{TEXT -1 1 " " }{XPPEDIT 18 0 "Sum((-2*r)^k/k!,k = 0 .. p)" "6#-%$SumG6$*&),$*&\"\"#\"\"\"%\"rGF+! \"\"%\"kGF+-%*factorialG6#F.F-/F.;\"\"!%\"pG" }{TEXT -1 1 " " } {XPPEDIT 18 0 "matrix([[s-k], [s-p]])" "6#-%'matrixG6#7$7#,&%\"sG\"\" \"%\"kG!\"\"7#,&F)F*%\"pGF," }{TEXT -1 1 "." }}{PARA 0 "" 0 "" {TEXT -1 5 "Then " }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "rho < \+ r[0](s,p)" "6#2%$rhoG-&%\"rG6#\"\"!6$%\"sG%\"pG" }{TEXT -1 1 "," }} {PARA 0 "" 0 "" {TEXT -1 6 "where " }{XPPEDIT 18 0 "r[0](m, p)" "6#-&% \"rG6#\"\"!6$%\"mG%\"pG" }{TEXT -1 48 " is the unique positive root of the polynomial " }{XPPEDIT 18 0 "H[s,p](r)" "6#-&%\"HG6$%\"sG%\"pG6# %\"rG" }{TEXT -1 1 "." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT 259 9 "Example 1" }{TEXT -1 1 ":" }}{PARA 0 "" 0 "" {TEXT -1 88 "We find an upper bound for the stability radius of a 7 stage order 5 Runge Kutta scheme." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 166 "subs(\{s=6,p=5\},binomial(s,s-p)-( -1)^p*Sum((-2*r)^k/k!*binomial(s-k,s-p),k=0..p)):\nvalue(%): H := unap ply(%,r):\n'H(r)'=H(r);\nplot(H(r),r=-1.3..3.5,font=[HELVETICA,9]);" } }{PARA 11 "" 1 "" {XPPMATH 20 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RGBG$\"#5!\"\"$\"\"!Fc[lFb[l-%%FONTG6$%*HELVETICAG\"\"*-%+AXESLABELSG6 $Q\"r6\"Q!F]\\l-%%VIEWG6$;$!#8Fa[l$\"#NFa[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 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 20 "fsolve(H(r ),r=2..3);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#$\"+QU$z`#!\"*" }}} {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 98 "The maxim al stability radius of a 6 stage order 5 Runge Kutta scheme is approxi mately 2.37662142." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 " " {TEXT 259 9 "Example 2" }{TEXT -1 1 ":" }}{PARA 0 "" 0 "" {TEXT -1 88 "We find an upper bound for the stability radius of a 7 stage order 5 Runge Kutta scheme." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 164 "subs(\{s=7,p=5\},binomial(s,s-p)-( -1)^p*Sum((-2*r)^k/k!*binomial(s-k,s-p),k=0..p)):\nvalue(%): H := unap ply(%,r):\n'H(r)'=H(r);\nplot(H(r),r=-0.8..4,font=[HELVETICA,9]);" }} {PARA 11 "" 1 "" {XPPMATH 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*>\"Faq$\"3w.k$=TWgW#F-7$$\"3)******4)\\V)H\"Faq$\"3C2*36If`R#F-7$$\"3 v******\\DH,9Faq$\"3-/)o-_c\"\\BF-7$$\"3%******p!zu&\\\"Faq$\"3WK)4j;) f6BF-7$$\"3k******p2j(f\"Faq$\"3mshSf%3^F#F-7$$\"3$)*****4VeMq\"Faq$\" 3oaU+7%)GSAF-7$$\"3!)*****4s\"e&z\"Faq$\"3Rf$G\"*[09@#F-7$$\"3'******z Rx]*=Faq$\"3-2>66LZ!=#F-7$$\"3g*****z;my*>Faq$\"3-x4\\RtPZ@F-7$$\"3')* ****f#\\U)4#Faq$\"3@MTdHfI7@F-7$$\"3W*****\\p?d>#Faq$\"33i)e**49U2#F-7 $$\"3d*****z0^PI#Faq$\"3)=aLST/]-#F-7$$\"38*****R9@3S#Faq$\"3[ F-7$$\"3P******zTY/DFaq$\"3/&QF$yy:0>F-7$$\"31*****\\))*fqC8F-7$$\"3C*****\\PT25$Faq$\"3f2X\"Rg)e@6F-7$$\"37+ ++%=\"H+KFaq$\"3=,%=r<4`)))Faq7$$\"3/+++U_4-LFaq$\"3g7([IR:U1'Faq7$$\" 3G+++6j0.MFaq$\"3#f;(Gv0FmFFaq7$$\"3/+++[!He\\$Faq$!3Wa$3q/4_i(F*7$$\" 3'******R2d@g$Faq$!33oJv233raFaq7$$\"3k*****z7esp$Faq$!30l#*=_)*yN5F-7 $$\"35+++(ob')z$Faq$!3-B9(y+;dj\"F-7$$\"3#*******)31d*QFaq$!35rBK(QM\\ H#F-7$$\"\"%\"\"!$!3Hmmmmmm1JF--%'COLOURG6&%$RGBG$\"#5!\"\"$FhzFhzFb[l -%%FONTG6$%*HELVETICAG\"\"*-%+AXESLABELSG6$Q\"r6\"Q!F\\\\l-%%VIEWG6$;$ !\")Fa[lFfz%(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 20 "fsolve(H(r),r=3..4);" }}{PARA 11 " " 1 "" {XPPMATH 20 "6#$\"+#*z(oZ$!\"*" }}}{PARA 0 "" 0 "" {TEXT -1 0 " " }}{PARA 0 "" 0 "" {TEXT -1 98 "The maximal stability radius of a 7 s tage order 5 Runge Kutta scheme is approximately 3.17386983." }} {PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}}{PARA 0 "" 0 "" {TEXT -1 86 "#------------------------------ -------------------------------------------------------" }}{PARA 0 "" 0 "" {TEXT 327 27 "___________________________" }{TEXT -1 0 "" }} {PARA 0 "" 0 "" {TEXT -1 0 "" }}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 77 "A procedure for constructing the stability function of a Runge-Kutta sc heme: " }{TEXT 0 16 "StabiltyFunction" }{TEXT -1 1 " " }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 25 "Stability Function: usage" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }{TEXT 328 18 "Calling Sequence:\n" }}{PARA 0 " " 0 "" {TEXT -1 35 " StabilityFunction(order,stages)" }{TEXT 329 4 "\nor " }{TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 29 " StabilityFun ction(stages)" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 259 "" 0 "" {TEXT -1 11 "Parameters:" }}{PARA 0 "" 0 "" {TEXT -1 5 " " }} {PARA 0 "" 0 "" {TEXT 23 17 " order - " }{TEXT -1 69 " a positive integer giving the order of the Runge-Kutta scheme " }{TEXT 23 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT 23 17 " stages - " }{TEXT -1 79 " a positive integer giving the number of stages of the Runge-Kutta scheme" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 259 "" 0 "" {TEXT -1 12 "Description:" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 14 "The procedure " } {TEXT 0 17 "StabilityFunction" }{TEXT -1 85 " constructs the stability function of an explicit Runge-Kutta scheme of given order. " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT 330 7 "Option:" }} {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 120 "The opti onal argument \"expanded\" causes all the summations in the expression for the stability function to be expanded. " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT 260 16 "How to activate:" }}{PARA 0 "" 0 "" {TEXT -1 155 "To make the proc edure active, open the subsection, place the cursor anywhere after the prompt [ > and press [Enter].\nYou can then close up the subsection. " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 35 "Stability Function: implementation " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 8014 "StabilityFunction := \+ proc()\n local order,stages,s1,s2,tt,phi,ct,u_,trs,i_,cffs,stbfcn;\n \n if nargs=0 then\n error \"%1 uses a 1st argument, (of type p osint), which is missing\",'procname';\n end if;\n if nargs>0 then \n if not type(args[1],posint) then \n error \"the 1st ar gument, %1, is invalid .. it should be a positive integer\",args[1];\n end if;\n end if;\n if nargs=1 or nargs=2 and args[2]='expan ded' then\n stages := args[1];\n order := -1;\n else\n \+ order := args[1];\n end if; \n if nargs>1 then\n if args[ 2]<>'expanded' then\n if not type(args[2],posint) then \n \+ error \"the 2nd argument, %1, is invalid .. it should be a posi tive integer or the option 'expanded'\",args[2];\n end if;\n \+ if args[2]>27 then\n error \"the 2nd argument, %1, i s invalid .. it should be no greater than 27\",args[2];\n end \+ if;\n stages :=args[2];\n end if;\n end if;\n if narg s>2 and (args[3]<>'expanded' or nargs>3) then\n error \"the only \+ valid optional argument for %1 is 'expanded'\",procname;\n end if; \+ \n\n s1 := \{a,b,c\};\n unassign('a','b','c');\n s2 := \{a,b,c\} minus s1;\n if s2<>\{\} then \n WARNING(\"the variables, %1 ha ve been unassigned for use as coefficients\",s2);\n end if;\n s1 : = \{z\};\n unassign('z');\n s2 := \{z\} minus s1;\n if s2<>\{\} \+ then \n WARNING(\"the variable, %1 has been unassigned for use as a parameter\",op(s2));\n end if;\n if stages=1 or stages=2 then\n s1 := \{i\}; unassign('i'); s2 := \{i\} minus s1;\n elif stag es=3 then\n s1 := \{i,j\}; unassign('i','j'); s2 := \{i,j\} minu s s1;\n elif stages=4 then\n s1 := \{i,j,k\}; unassign('i','j' ,'k'); s2 := \{i,j,k\} minus s1;\n elif stages=5 then\n s1 := \+ \{i,j,k,l\}; unassign('i','j','k','l'); s2 := \{i,j,k,l\} minus s1;\n elif stages=6 then\n s1 := \{i,j,k,l,m\}; unassign('i','j','k ','l','m'); s2 := \{i,j,k,l,m\} minus s1;\n elif stages=7 then\n \+ s1 := \{i,j,k,l,m,n\}; unassign('i','j','k','l','m','n'); s2 := \{i, j,k,l,m,n\} minus s1;\n elif stages=8 then\n s1 := \{i,j,k,l,m, n,p\}; unassign('i','j','k','l','m','p');\n s2 := \{i,j,k,l,m,n,p \} minus s1;\n elif stages=9 then\n s1 := \{i,j,k,l,m,n,p,q\}; \+ unassign('i','j','k','l','m','p','q');\n s2 := \{i,j,k,l,m,n,p,q \} minus s1;\n elif stages=10 then\n s1 := \{i,j,k,l,m,n,p,q,r \}; unassign('i','j','k','l','m','p','q','r');\n s2 := \{i,j,k,l, m,n,p,q,r\} minus s1;\n elif stages=11 then\n s1 := \{i,j,k,l,m ,n,p,q,r,s\}; unassign('i','j','k','l','m','p','q','r','s');\n s2 := \{i,j,k,l,m,n,p,q,r,s\} minus s1;\n elif stages=12 then\n s 1 := \{i,j,k,l,m,n,p,q,r,s,t\};\n unassign('i','j','k','l','m','p ','q','r','s','t');\n s2 := \{i,j,k,l,m,n,p,q,r,s,t\} minus s1;\n elif stages=13 then\n s1 := \{i,j,k,l,m,n,p,q,r,s,t,u\};\n \+ unassign('i','j','k','l','m','p','q','r','s','t','u');\n s2 := \+ \{i,j,k,l,m,n,p,q,r,s,t,u\} minus s1;\n elif stages=14 then\n s 1 := \{i,j,k,l,m,n,p,q,r,s,t,u,v\};\n unassign('i','j','k','l','m ','p','q','r','s','t','u','v');\n s2 := \{i,j,k,l,m,n,p,q,r,s,t,u ,v\} minus s1;\n elif stages=15 then\n s1 := \{i,j,k,l,m,n,p,q, r,s,t,u,v,w\};\n unassign('i','j','k','l','m','p','q','r','s','t' ,'u','v','w');\n s2 := \{i,j,k,l,m,n,p,q,r,s,t,u,v,w\} minus s1; \n elif stages=16 then\n s1 := \{i,j,k,l,m,n,p,q,r,s,t,u,v,w,d \};\n unassign('i','j','k','l','m','p','q','r','s','t','u','v','w ','d');\n s2 := \{i,j,k,l,m,n,p,q,r,s,t,u,v,w,d\} minus s1;\n e lif stages=17 then\n s1 := \{i,j,k,l,m,n,p,q,r,s,t,u,v,w,d,h\};\n unassign('i','j','k','l','m','p','q','r','s','t','u','v','w','d' ,'h');\n s2 := \{i,j,k,l,m,n,p,q,r,s,t,u,v,w,d,h\} minus s1;\n \+ elif stages=18 then\n s1 := \{i,j,k,l,m,n,p,q,r,s,t,u,v,w,d,h,iot a\};\n unassign('i','j','k','l','m','p','q','r','s','t','u','v',' w','d','h','iota');\n s2 := \{i,j,k,l,m,n,p,q,r,s,t,u,v,w,d,h,iot a\} minus s1;\n elif stages=19 then\n s1 := \{i,j,k,l,m,n,p,q,r ,s,t,u,v,w,d,h,iota,kappa\};\n unassign('i','j','k','l','m','p',' q','r','s','t','u','v','w','d','h','iota','kappa');\n s2 := \{i,j ,k,l,m,n,p,q,r,s,t,u,v,w,d,h,iota,kappa\} minus s1;\n elif stages=20 then\n s1 := \{i,j,k,l,m,n,p,q,r,s,t,u,v,w,d,h,iota,kappa,lambda \};\n unassign('i','j','k','l','m','p','q','r','s','t','u','v','w ','d','h','iota',\n 'kappa','lambda');\n s2 : = \{i,j,k,l,m,n,p,q,r,s,t,u,v,w,d,h,iota,kappa,lambda\} minus s1;\n \+ elif stages=21 then\n s1 := \{i,j,k,l,m,n,p,q,r,s,t,u,v,w,d,h,iot a,kappa,lambda,mu\};\n unassign('i','j','k','l','m','p','q','r',' s','t','u','v','w','d','h','iota',\n 'kappa','lamb da','mu');\n s2 := \{i,j,k,l,m,n,p,q,r,s,t,u,v,w,d,h,iota,kappa,l ambda,mu\} minus s1;\n elif stages=22 then\n s1 := \{i,j,k,l,m, n,p,q,r,s,t,u,v,w,d,h,iota,kappa,lambda,mu,nu\};\n unassign('i',' j','k','l','m','p','q','r','s','t','u','v','w','d','h','iota',\n \+ 'kappa','lambda','mu','nu');\n s2 := \{i,j,k,l,m,n ,p,q,r,s,t,u,v,w,d,h,iota,kappa,lambda,mu,nu\} minus s1;\n elif stag es=23 then\n s1 := \{i,j,k,l,m,n,p,q,r,s,t,u,v,w,d,h,iota,kappa,l ambda,mu,nu,rho\};\n unassign('i','j','k','l','m','p','q','r','s' ,'t','u','v','w','d','h','iota',\n 'kappa','lambda ','mu','nu','rho');\n s2 := \{i,j,k,l,m,n,p,q,r,s,t,u,v,w,d,h,iot a,kappa,lambda,mu,nu,rho\} minus s1;\n elif stages=24 then\n s1 := \{i,j,k,l,m,n,p,q,r,s,t,u,v,w,d,h,iota,kappa,lambda,mu,nu,rho,sigm a\};\n unassign('i','j','k','l','m','p','q','r','s','t','u','v',' w','d','h','iota',\n 'kappa','lambda','mu','nu','r ho','sigma');\n s2 := \{i,j,k,l,m,n,p,q,r,s,t,u,v,w,d,h,iota,kapp a,lambda,mu,nu,rho,sigma\} minus s1;\n elif stages=25 then\n s1 := \{i,j,k,l,m,n,p,q,r,s,t,u,v,w,d,h,iota,kappa,lambda,mu,nu,rho,sigm a,tau\};\n unassign('i','j','k','l','m','p','q','r','s','t','u',' v','w','d','h','iota',\n 'kappa','lambda','mu','nu ','rho','sigma','tau');\n s2 := \{i,j,k,l,m,n,p,q,r,s,t,u,v,w,d,h ,iota,kappa,lambda,mu,nu,rho,sigma,tau\} minus s1;\n elif stages=26 \+ then\n s1 := \{i,j,k,l,m,n,p,q,r,s,t,u,v,w,d,h,iota,kappa,lambda, mu,nu,rho,sigma,tau,upsilon\};\n unassign('i','j','k','l','m','p' ,'q','r','s','t','u','v','w','d','h','iota',\n 'ka ppa','lambda','mu','nu','rho','sigma','tau','upsilon');\n s2 := \+ \{i,j,k,l,m,n,p,q,r,s,t,u,v,w,d,h,iota,kappa,lambda,mu,nu,rho,sigma,ta u,upsilon\} minus s1;\n elif stages=27 then\n s1 := \{i,j,k,l,m ,n,p,q,r,s,t,u,v,w,d,h,iota,kappa,lambda,mu,nu,rho,sigma,tau,upsilon,o mega\};\n unassign('i','j','k','l','m','p','q','r','s','t','u','v ','w','d','h','iota',\n 'kappa','lambda','mu','nu' ,'rho','sigma','tau','upsilon','omega');\n s2 := \{i,j,k,l,m,n,p, q,r,s,t,u,v,w,d,h,iota,kappa,lambda,mu,nu,rho,sigma,tau,upsilon,omega \} minus s1;\n end if;\n if s2<>\{\} then \n WARNING(\"the va riables, %1 have been unassigned for possible use as indices\",s2);\n \+ end if;\n if order<0 then\n tt[1] :=[];\n phi := u_->[u_ ];\n for ct to stages do tt[ct+1] := phi(tt[ct]) end do;\n c ffs := map(lhs,[seq(subs(_S=stages,TreeOrderCondition(tt[i_])),i_=1..s tages)]);\n stbfcn := 1+add(cffs[i_]*z^i_,i_=1..stages);\n i f member('expanded',[args]) then\n subs(c[1]=0,eval(subs('Sum' ='add',stbfcn)));\n else\n stbfcn;\n end if;\n els e\n if stages[u_];\n for ct to stages do tt[ct+1] := phi(tt[ct] ) end do;\n cffs := [seq(subs(_S=stages,TreeOrderCondition(tt[i_] )),i_=1..stages)];\n cffs := map(u_->`if`(rhs(u_)>=1/order!,rhs(u _),lhs(u_)),cffs);\n stbfcn := 1+add(cffs[i_]*z^i_,i_=1..stages); \n if nargs=3 and args[3]='expanded' then\n subs(c[1]=0,e val(subs('Sum'='add',stbfcn)));\n else\n stbfcn;\n e nd if;\n end if;\nend proc: # StabilityFunction" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 39 "Examples are gi ven in the next section." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 8 "Examples" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 21 "StabilityFunction(5);" }}{PARA 11 "" 1 " " {XPPMATH 20 "6#,.\"\"\"F$*&-%$SumG6$&%\"bG6#%\"iG/F,;F$\"\"&F$%\"zGF $F$*&-F'6$*&F)F$&%\"cGF+F$F-F$)F0\"\"#F$F$*&-F'6$*&F)F$-F'6$*&&%\"aG6$ F,%\"jGF$&F66#FCF$/FC;F$,&F,F$F$!\"\"F$F-F$)F0\"\"$F$F$*&-F'6$*&F)F$-F '6$*&F@F$-F'6$*&&FA6$FC%\"kGF$&F66#FXF$/FX;F$,&FCF$F$FIF$FFF$F-F$)F0\" \"%F$F$*&-F'6$*&F)F$-F'6$*&F@F$-F'6$*&FVF$-F'6$*&&FA6$FX%\"lGF$&F66#Fi oF$/Fio;F$,&FXF$F$FIF$FenF$FFF$F-F$)F0F/F$F$" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 32 "StabilityFun ction(5,'expanded');" }}{PARA 12 "" 1 "" {XPPMATH 20 "6#,.\"\"\"F$*&,, &%\"bG6#F$F$&F(6#\"\"#F$&F(6#\"\"$F$&F(6#\"\"%F$&F(6#\"\"&F$F$%\"zGF$F $*&,**&F*F$&%\"cGF+F$F$*&F-F$&F;F.F$F$*&F0F$&F;F1F$F$*&F3F$&F;F4F$F$F$ )F6F,F$F$*&,(*(F-F$&%\"aG6$F/F,F$F:F$F$*&F0F$,&*&&FG6$F2F,F$F:F$F$*&&F G6$F2F/F$F=F$F$F$F$*&F3F$,(*&&FG6$F5F,F$F:F$F$*&&FG6$F5F/F$F=F$F$*&&FG 6$F5F2F$F?F$F$F$F$F$)F6F/F$F$*&,&**F0F$FOF$FFF$F:F$F$*&F3F$,&*(FWF$FFF $F:F$F$*&FZF$FJF$F$F$F$F$)F6F2F$F$*.F3F$FZF$FOF$FFF$F:F$)F6F5F$F$" }}} {PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 23 "StabilityFunction(5,5);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#,.\"\" \"F$%\"zGF$*&#F$\"\"#F$*$)F%F(F$F$F$*&#F$\"\"'F$*$)F%\"\"$F$F$F$*&#F$ \"#CF$*$)F%\"\"%F$F$F$*&#F$\"$?\"F$*$)F%\"\"&F$F$F$" }}}{PARA 0 "" 0 " " {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 23 "StabilityF unction(5,6);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#,0\"\"\"F$%\"zGF$*&\" \"#!\"\"F%F'F$*&\"\"'F(F%\"\"$F$*&\"#CF(F%\"\"%F$*&\"$?\"F(F%\"\"&F$*& -%$SumG6$*&&%\"bG6#%\"iGF$-F46$*&&%\"aG6$F:%\"jGF$-F46$*&&F?6$FA%\"kGF $-F46$*&&F?6$FG%\"lGF$-F46$*&&F?6$FM%\"mGF$&%\"cG6#FSF$/FS;F$,&FMF$F$F (F$/FM;F$,&FGF$F$F(F$/FG;F$,&FAF$F$F(F$/FA;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 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$F 1F.F$&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 30 "StabilityFun ction(2,'expand');" }}{PARA 8 "" 1 "" {TEXT -1 127 "Error, (in Stabili tyFunction) the 2nd argument, expand, is invalid .. it should be a pos itive integer or the option 'expanded'\n" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}}{PARA 0 "" 0 " " {TEXT 331 27 "___________________________" }{TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}}{MARK "4 0 0" 0 }{VIEWOPTS 1 1 0 1 1 1803 1 1 1 1 }{PAGENUMBERS 0 1 2 33 1 1 }