{VERSION 2 3 "IBM INTEL NT" "2.3" } {USTYLETAB {CSTYLE "Maple Input" -1 0 "Courier" 0 1 255 0 0 1 0 1 0 0 1 0 0 0 0 }{CSTYLE "2D Math" -1 2 "Times" 0 1 0 0 0 0 0 0 2 0 0 0 0 0 0 }{CSTYLE "2D Output" 2 20 "" 0 1 0 0 255 1 0 0 0 0 0 0 0 0 0 } {PSTYLE "Normal" -1 0 1 {CSTYLE "" -1 -1 "" 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 }0 0 0 -1 -1 -1 0 0 0 0 0 0 -1 0 }{PSTYLE "Maple Output" 0 11 1 {CSTYLE "" -1 -1 "" 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 }3 3 0 -1 -1 -1 0 0 0 0 0 0 -1 0 }{PSTYLE "" 11 12 1 {CSTYLE "" -1 -1 "" 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 }1 0 0 -1 -1 -1 0 0 0 0 0 0 -1 0 }} {SECT 0 {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 2783 "################## ###################\n# Maple V, Release 4 script for the #\n# calculat ion of Schroder's methods #\n# #\n# \+ Copyright (C) 1996 by Dann Corbit #\n# [ Cube root Version ] \+ #\n# \"MaximumTerm\" controls the number #\n# of algorithms generat ed. #\n#####################################\n#>>>>>>>>>>>>>> >>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>\n# Reference:\n# \"On In finitely Many Algorithms For Solving Equations\"\n# by Ernst Schrod er, Translated by G. W. Stuart.\n# Synopsis:\n# Local convergence rat es can be made arbitrarily large.\n# Ernst Schroder showed, over 100 \+ years ago, how a family\n# of explicit single-point formulas could be designed to\n# have arbitrarily high orders of (local) convergence. \n# You can obtain a translation of Schroder's paper at:\n# ftp:// thales.cs.umd.edu/pub/reports/imase.ps\n# Web surfers may find it eas ier to pick it up via\n# Mr. Stuart's home page. The URL is:\n# h ttp://www.cs.umd.edu/~stewart/\n#<<<<<<<<<<<<<<<<<<<<<<<<<<<<<<<<<<<<< <<<<<<<<<<<<<<<<<<<<\n\n# Definition of y as a function of x\ny := x^3 - c;\n\n# Definition of the first term for Schroder's Series 'A'\n# E quation 55, Page 30.\nA(0, 0) := 1/y;\n\n# Definition of the first ter m for Schroder's Series 'B'\n# Equation 56, Page 31.\nB(0, 0) := diff( y, x)/y;\n\n# We can change MaximumTerm to process as many terms as we like\nMaximumTerm := 4;\n# Iterate to create terms from 1 to MaximumT erm\nfor i from 1 to MaximumTerm do w := i;\n\n # Definition of the \+ Lamda=0 terms for Schroder's Series 'A'\n # Equation 55, Page 30.\n \+ A(0, w) := simplify(sum('(-1)^(a-1)*y^(a-1)*diff(y, x$a)*y^(w-a+1)*A (0, w-a)/a!', 'a'=1..w)/y^(w+1));\n\n # Definition of the Lamda=0 te rms for Schroder's Series 'B'\n # Equation 55, Page 30.\n B(0, w) \+ := simplify(sum('(-1)^(a)*y^(a)*diff(y, x$(a+1))*y^(w-a+1)*A(0, w-a)/a !', 'a'=0..w)/y^(w+1));\n\n # Definition of the Omega=0 terms for Sc hroder's Series 'A'\n # Equation 70, Page 40.\n A(l, 0) := simplif y(x^l*y*A(0, 0)/y);\n\n # Definition of the Omega=0 terms for Schrod er's Series 'B'\n # Equation 70, Page 40.\n B(l, 0) := simplify(x^ l*y*B(0, 0)/y);\n\n # Definition of the Lamda, Omega terms for Schro der's Series 'A'\n # Equation 70, Page 40.\n A(l, w) := simplify(s um('(-1)^(a)*l!/(a!*(l-a)!)*x^(l-a)*y^a*y^(w-a+1)*A(0, w-a)', 'a'=0..w )/y^(w+1));\n\n # Definition of the Lamda, Omega terms for Schroder' s Series 'B'\n # Equation 70, Page 40.\n B(l, w) := simplify(sum(' (-1)^(a)*l!/(a!*(l-a)!)*x^(l-a)*y^a*y^(w-a+1)*B(0, w-a)', 'a'=0..w)/y^ (w+1));\n\n # Definition of the Algorithm based on A(Lamda, Omega) P age 39.\n xprimeA := x - simplify(A(l, w-1)/A(l, w));\n\n # Defini tion of the Algorithm based on B(Lamda, Omega) Page 39.\n xprimeB := x - simplify(B(l, w-1)/B(l, w));\nod;\n\n" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"yG,&*$%\"xG\"\"$\"\"\"%\"cG!\"\"" }}{PARA 11 "" 1 " " {XPPMATH 20 "6#>-%\"AG6$\"\"!F'*$,&*$%\"xG\"\"$\"\"\"%\"cG!\"\"F/" } }{PARA 11 "" 1 "" {XPPMATH 20 "6#>-%\"BG6$\"\"!F',$*&%\"xG\"\"#,&*$F* \"\"$\"\"\"%\"cG!\"\"F1F." }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%,Maximu mTermG\"\"%" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"wG\"\"\"" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>-%\"AG6$\"\"!\"\"\",$*&%\"xG\"\"#,&*$F+\"\" $F(%\"cG!\"\"!\"#F/" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>-%\"BG6$\"\"! \"\"\",$*(%\"xGF(,&*$F+\"\"$F(%\"cG\"\"#F(,&F-F(F/!\"\"!\"#F." }} {PARA 11 "" 1 "" {XPPMATH 20 "6#>-%\"AG6$%\"lG\"\"!*&)%\"xGF'\"\"\",&* $F+\"\"$F,%\"cG!\"\"F1" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>-%\"BG6$%\" lG\"\"!,$*&)%\"xG,&F'\"\"\"\"\"#F.F.,&*$F,\"\"$F.%\"cG!\"\"F4F2" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#>-%\"AG6$%\"lG\"\"\"*&,()%\"xG,&F'F(\" \"#F(\"\"$*&F'F(F+F(!\"\"*(F'F()F,,&F'F(F1F(F(%\"cGF(F(F(,&*$F,F/F(F5F 1!\"#" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>-%\"BG6$%\"lG\"\"\",$*(%\"xG F(,*)F+,&F'F(\"\"$F(F(*&)F+F'F(%\"cGF(\"\"#*&F'F(F-F(!\"\"*(F'F(F1F(F2 F(F(F(,&*$F+F/F(F2F5!\"#F/" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%(xprim eAG,&%\"xG\"\"\"*(F&F',&*$F&\"\"$F'%\"cG!\"\"F',(F*!\"$*&%\"lGF'F&F+F' *&F1F'F,F'F-F-F'" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%(xprimeBG,&%\"xG \"\"\"*()F&,&%\"lGF'F'F'F',&*$F&\"\"$F'%\"cG!\"\"F',*)F&,&F+F'F.F'F'*& )F&F+F'F/F'\"\"#*&F+F'F2F'F0*(F+F'F5F'F/F'F'F0F0" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"wG\"\"#" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>-%\"AG6 $\"\"!\"\"#,$*(%\"xG\"\"\",&*$F+\"\"$F(%\"cGF,F,,&F.F,F0!\"\"!\"$F/" } }{PARA 11 "" 1 "" {XPPMATH 20 "6#>-%\"BG6$\"\"!\"\"#,$*&,(*$%\"xG\"\"' \"\"\"*&F-\"\"$%\"cGF/\"\"(*$F2F(F/F/,&*$F-F1F/F2!\"\"!\"$F1" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>-%\"AG6$%\"lG\"\"!*&)%\"xGF'\"\"\",&*$F+\" \"$F,%\"cG!\"\"F1" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>-%\"BG6$%\"lG\" \"!,$*&)%\"xG,&F'\"\"\"\"\"#F.F.,&*$F,\"\"$F.%\"cG!\"\"F4F2" }}{PARA 12 "" 1 "" {XPPMATH 20 "6#>-%\"AG6$%\"lG\"\"#,$*&,2)%\"xG,&F'\"\"\"\" \"%F/\"#7*&)F-,&F'F/F/F/F/%\"cGF/\"\"'*&F'F/F,F/!\"(*(F'F/F3F/F5F/\"\" )*&F'F(F,F/F/*(F'F(F3F/F5F/!\"#*(F'F()F-,&F'F/F=F/F/F5F(F/*(F'F/F?F/F5 F(!\"\"F/,&*$F-\"\"$F/F5FB!\"$#F/F(" }}{PARA 11 "" 1 "" {XPPMATH 20 "6 #>-%\"BG6$%\"lG\"\"#,$*&,2)%\"xG,&F'\"\"\"\"\"'F/F(*&)F-,&F'F/\"\"$F/F /%\"cGF/\"#9*&)F-F'F/F5F(F(*&F'F/F,F/!\"$*(F'F/F8F/F5F(F4*&F'F(F,F/F/* (F'F(F2F/F5F/!\"#*(F'F(F8F/F5F(F/F/,&*$F-F4F/F5!\"\"F:#F4F(" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%(xprimeAG,&%\"xG\"\"\"**,&*$F&\"\"$F'%\"cG !\"\"F'F&F',(F*!\"$*&%\"lGF'F&F+F'*&F1F'F,F'F-F',2*$F&\"\"'\"#7*&F&F+F ,F'F5*&F1F'F&F5!\"(*(F1F'F&F+F,F'\"\")*&F1\"\"#F&F5F'*(F1F=F&F+F,F'!\" #*&F1F=F,F=F'*&F1F'F,F=F-F-F=" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%(xp rimeBG,&%\"xG\"\"\"**F&F',*)F&,&%\"lGF'\"\"$F'F'*&)F&F,F'%\"cGF'\"\"#* &F,F'F*F'!\"\"*(F,F'F/F'F0F'F'F',&*$F&F-F'F0F3F',2)F&,&F,F'\"\"'F'F1*& F*F'F0F'\"#9*&F/F'F0F1F1*&F,F'F8F'!\"$*(F,F'F/F'F0F1F-*&F,F1F8F'F'*(F, F1F*F'F0F'!\"#*(F,F1F/F'F0F1F'F3FC" }}{PARA 11 "" 1 "" {XPPMATH 20 "6# >%\"wG\"\"$" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>-%\"AG6$\"\"!\"\"$*&,( *$%\"xG\"\"'\"#5*&F,F(%\"cG\"\"\"\"#;*$F0\"\"#F1F1,&*$F,F(F1F0!\"\"!\" %" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>-%\"BG6$\"\"!\"\"$,$*(%\"xG\"\"# ,(*$F+\"\"'\"\"\"*&F+F(%\"cGF0\"#;*$F2F,\"#5F0,&*$F+F(F0F2!\"\"!\"%F( " }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>-%\"AG6$%\"lG\"\"!*&)%\"xGF'\"\" \",&*$F+\"\"$F,%\"cG!\"\"F1" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>-%\"BG 6$%\"lG\"\"!,$*&)%\"xG,&F'\"\"\"\"\"#F.F.,&*$F,\"\"$F.%\"cG!\"\"F4F2" }}{PARA 12 "" 1 "" {XPPMATH 20 "6#>-%\"AG6$%\"lG\"\"$,$*&,@)%\"xG,&F' \"\"\"\"\"'F/!#g*&)F-,&F'F/F(F/F/%\"cGF/!#'**&)F-F'F/F5\"\"#!\"'*&F'F/ F,F/\"#Z*(F'F/F3F/F5F/!#U*(F'F/F8F/F5F9!\"$*&F'F9F,F/!#7*(F'F9F3F/F5F/ \"#F*(F'F9F8F/F5F9!#=*(F'F/)F-,&F'F/F@F/F/F5F(!\"#*(F'F9FHF/F5F(F(*&F' F(F,F/F/*(F'F(F3F/F5F/F@*(F'F(F8F/F5F9F(*(F'F(FHF/F5F(!\"\"F/,&*$F-F(F /F5FP!\"%#FPF0" }}{PARA 12 "" 1 "" {XPPMATH 20 "6#>-%\"BG6$%\"lG\"\"$, $*&,>*&)%\"xG,&F'\"\"\"\"\"&F0F0%\"cGF0\"#'**&F'F()F.,&F'F0\"\")F0F0! \"\"*(F'F0)F.,&F'F0F8F0F0F2F(\"\"#*&)F.,&F'F0FF0F 2F<\"#R*(F'F0F-F0F2F0!#I*(F'F(F:F0F2F(F0F5\"\"'*&F'FF0F2F%(xprime AG,&%\"xG\"\"\"**,&*$F&\"\"$F'%\"cG!\"\"F'F&F',2*$F&\"\"'\"#7*&F&F+F,F 'F0*&%\"lGF'F&F0!\"(*(F4F'F&F+F,F'\"\")*&F4\"\"#F&F0F'*(F4F9F&F+F,F'! \"#*&F4F9F,F9F'*&F4F'F,F9F-F',@*$F&\"\"*!#g*&F&F0F,F'!#'**&F,F9F&F+!\" '*&F4F'F&F@\"#Z*(F4F'F&F0F,F'!#U*(F4F'F,F9F&F+!\"$*&F4F9F&F@!#7*(F4F9F &F0F,F'\"#F*(F4F9F,F9F&F+!#=*&F4F'F,F+F;*&F4F9F,F+F+*&F4F+F&F@F'*(F4F+ F&F0F,F'FK*(F4F+F,F9F&F+F+*&F4F+F,F+F-F-F+" }}{PARA 12 "" 1 "" {XPPMATH 20 "6#>%(xprimeBG,&%\"xG\"\"\"**,&*$F&\"\"$F'%\"cG!\"\"F'F&F' ,2*$F&\"\"'\"\"#*&F&F+F,F'\"#9*$F,F1F1*&%\"lGF'F&F0!\"$*&F6F'F,F1F+*&F 6F1F&F0F'*(F6F1F&F+F,F'!\"#*&F6F1F,F1F'F',>*&F&F0F,F'!#'**&F6F+F&\"\"* F'*&F6F'F,F+F;*&F,F1F&F+!#g*(F6F'F,F1F&F+!#R*(F6F'F&F0F,F'\"#I*&F6F+F, F+F-*$F&FA!\"'*&F6F1F&FAFK*&F6F'F&FA\"#6*(F6F1F&F0F,F'FA*(F6F+F,F1F&F+ F+*&F6F1F,F+F7*(F6F+F&F0F,F'F7F-F+" }}{PARA 11 "" 1 "" {XPPMATH 20 "6# >%\"wG\"\"%" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>-%\"AG6$\"\"!\"\"%,$*( %\"xG\"\"#,(*$F+\"\"'\"\"&*&F+\"\"$%\"cG\"\"\"\"#<*$F3F,F0F4,&*$F+F2F4 F3!\"\"!\"&F2" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>-%\"BG6$\"\"!\"\"%,$ *(%\"xG\"\"\",**$F+\"\"*F,*&F+\"\"'%\"cGF,\"#I*&F2\"\"#F+\"\"$\"#X*$F2 F6\"\"&F,,&*$F+F6F,F2!\"\"!\"&F6" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>- %\"AG6$%\"lG\"\"!*&)%\"xGF'\"\"\",&*$F+\"\"$F,%\"cG!\"\"F1" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>-%\"BG6$%\"lG\"\"!,$*&)%\"xG,&F'\"\"\"\"\"# F.F.,&*$F,\"\"$F.%\"cG!\"\"F4F2" }}{PARA 12 "" 1 "" {XPPMATH 20 "6#>-% \"AG6$%\"lG\"\"%,$*&,P*(F'\"\"#)%\"xG,&F'\"\"\"F-F1F1%\"cGF-\"$u\"*&)F /,&F'F1\"\"&F1F1F2F1\"%C7*(F'F(F5F1F2F1!\"%*&F'F()F/,&F'F1\"\")F1F1F1* &F'\"\"$F\"*&F'F1F-%\"BG6$%\"lG\"\"%,$*&,R* (F'\"\"$)%\"xG,&F'\"\"\"F1F1F1%\"cGF-F(*(F'F-)F/,&F'F1F(F1F1F2\"\"#!#C *(F'F-)F/,&F'F1!\"#F1F1F2F(F6*&F'F-)F/,&F'F1\"#5F1F1!#5*&F'F(F=F1F1*(F 'F()F/,&F'F1\"\"(F1F1F2F1!\"%*(F'F(F4F1F2F6\"\"'*(F'F(F.F1F2F-FF*(F'F6 F9F1F2F(!\"\"*&F.F1F2F-\"$?\"*&FCF1F2F1\"$?(F=\"#C*&F'F6F=F1\"#N*&F4F1 F2F6\"%!3\"*(F'F6F4F1F2F6!$9\"*(F'F6FCF1F2F1F(*(F'F-FCF1F2F1\"#G*(F'F1 F4F1F2F6\"$w#*(F'F1F.F1F2F-\"$k\"*(F'F1FCF1F2F1!$)Q*(F'F6F.F1F2F-\"#w* (F'F1F9F1F2F(F;*&F'F1F=F1!#]*(F'F(F9F1F2F(F1F1,&*$F/F-F1F2FK!\"&#F1\" \")" }}{PARA 12 "" 1 "" {XPPMATH 20 "6#>%(xprimeAG,&%\"xG\"\"\"**,&*$F &\"\"$F'%\"cG!\"\"F'F&F',@*$F&\"\"*!#g*&F&\"\"'F,F'!#'**&F,\"\"#F&F+! \"'*&%\"lGF'F&F0\"#Z*(F9F'F&F3F,F'!#U*(F9F'F,F6F&F+!\"$*&F9F6F&F0!#7*( F9F6F&F3F,F'\"#F*(F9F6F,F6F&F+!#=*&F9F'F,F+!\"#*&F9F6F,F+F+*&F9F+F&F0F '*(F9F+F&F3F,F'F>*(F9F+F,F6F&F+F+*&F9F+F,F+F-F',P*(F9F6F&F3F,F6\"$u\"* &F&F0F,F'\"%C7*(F9\"\"%F&F0F,F'!\"%*&F9FRF&\"#7F'*&F9F+F&FUFD*(F9F'F,F +F&F+\"#O*(F9FRF&F3F,F6F3*&F9F'F,FRF7*&F9F+F,FRF7*&F9FRF,FRF'*&F9F6F,F R\"#6*&F&F3F,F6\"$g$*(F9F'F&F3F,F6\"$_#*(F9F'F&F0F,F'\"#g*(F9F+F,F+F&F +FX*$F&FUFjn*&F9F6F&FU\"$>\"*&F9F'F&FU!$U$*(F9F6F&F0F,F'!$g#*(F9FRF,F+ F&F+FS*(F9F+F&F3F,F6!#s*(F9F6F,F+F&F+!#W*(F9F+F&F0F,F'F^oF-FR" }} {PARA 12 "" 1 "" {XPPMATH 20 "6#>%(xprimeBG,&%\"xG\"\"\"**,&*$F&\"\"$F '%\"cG!\"\"F'F&F',>*&F&\"\"'F,F'!#'**&%\"lGF+F&\"\"*F'*&F3F'F,F+!\"#*& F,\"\"#F&F+!#g*(F3F'F,F8F&F+!#R*(F3F'F&F0F,F'\"#I*&F3F+F,F+F-*$F&F4!\" '*&F3F8F&F4F@*&F3F'F&F4\"#6*(F3F8F&F0F,F'F4*(F3F+F,F8F&F+F+*&F3F8F,F+! \"$*(F3F+F&F0F,F'FGF',R*(F3F+F,F+F&F+\"\"%*(F3F+F&F0F,F8!#C*&F3F+F,FKF 8*&F3F+F&\"#7!#5*&F3FKF&FPF'*(F3FKF&F4F,F'!\"%*(F3FKF&F0F,F8F0*(F3FKF, F+F&F+FT*&F3F8F,FKF-*&F&F+F,F+\"$?\"*&F&F4F,F'\"$?(*$F&FP\"#C*&F3F8F&F P\"#N*&F&F0F,F8\"%!3\"*(F3F8F&F0F,F8!$9\"*(F3F8F&F4F,F'FK*(F3F+F&F4F,F '\"#G*(F3F'F&F0F,F8\"$w#*(F3F'F,F+F&F+\"$k\"*(F3F'F&F4F,F'!$)Q*(F3F8F, F+F&F+\"#w*&F3F'F,FKF6*&F3F'F&FP!#]*&F3FKF,FKF'F-FK" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}}{MARK "0 0 0" 225 }{VIEWOPTS 1 1 0 1 1 1803 }