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{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# [ Square 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*x
 - 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,&*$%\"xG\"\"#\"\"\"%\"cG!\"\"" }}{PARA 11 "" 1 "
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}{PARA 11 "" 1 "" {XPPMATH 20 "6#>-%\"BG6$\"\"!F',$*&%\"xG\"\"\",&*$F*
\"\"#F+%\"cG!\"\"F0F." }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%,MaximumTer
mG\"\"%" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"wG\"\"\"" }}{PARA 11 "
" 1 "" {XPPMATH 20 "6#>-%\"AG6$\"\"!\"\"\",$*&%\"xGF(,&*$F+\"\"#F(%\"c
G!\"\"!\"#F." }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>-%\"BG6$\"\"!\"\"\",$
*&,&*$%\"xG\"\"#F(%\"cGF(F(,&F,F(F/!\"\"!\"#F." }}{PARA 11 "" 1 "" 
{XPPMATH 20 "6#>-%\"AG6$%\"lG\"\"!*&)%\"xGF'\"\"\",&*$F+\"\"#F,%\"cG!
\"\"F1" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>-%\"BG6$%\"lG\"\"!,$*&)%\"x
G,&F'\"\"\"F.F.F.,&*$F,\"\"#F.%\"cG!\"\"F3F1" }}{PARA 11 "" 1 "" 
{XPPMATH 20 "6#>-%\"AG6$%\"lG\"\"\",$*&,()%\"xG,&F'F(F(F(!\"#*&F'F(F,F
(F(*(F'F()F-,&F'F(!\"\"F(F(%\"cGF(F4F(,&*$F-\"\"#F(F5F4F/F4" }}{PARA 
11 "" 1 "" {XPPMATH 20 "6#>-%\"BG6$%\"lG\"\"\",$*&,*)%\"xG,&F'F(\"\"#F
(!\"\"*&)F-F'F(%\"cGF(F0*&F'F(F,F(F(*(F'F(F2F(F3F(F0F(,&*$F-F/F(F3F0!
\"#F8" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%(xprimeAG,&%\"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'F0*&)F&F+F'F/F'F0*&F+F'F
2F'F'*(F+F'F5F'F/F'F0F0F'" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"wG\"
\"#" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>-%\"AG6$\"\"!\"\"#*&,&*$%\"xGF
(\"\"$%\"cG\"\"\"F/,&F+F/F.!\"\"!\"$" }}{PARA 11 "" 1 "" {XPPMATH 20 "
6#>-%\"BG6$\"\"!\"\"#,$*(%\"xG\"\"\",&*$F+F(F,%\"cG\"\"$F,,&F.F,F/!\"
\"!\"$F(" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>-%\"AG6$%\"lG\"\"!*&)%\"x
GF'\"\"\",&*$F+\"\"#F,%\"cG!\"\"F1" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#
>-%\"BG6$%\"lG\"\"!,$*&)%\"xG,&F'\"\"\"F.F.F.,&*$F,\"\"#F.%\"cG!\"\"F3
F1" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>-%\"AG6$%\"lG\"\"#,$*&,2)%\"xG,
&F'\"\"\"F(F/\"\"'*&)F-F'F/%\"cGF/F(*&F'F/F,F/!\"&*(F'F/F2F/F3F/F0*&F'
F(F,F/F/*(F'F(F2F/F3F/!\"#*(F'F()F-,&F'F/F9F/F/F3F(F/*(F'F/F;F/F3F(!\"
\"F/,&*$F-F(F/F3F>!\"$#F/F(" }}{PARA 12 "" 1 "" {XPPMATH 20 "6#>-%\"BG
6$%\"lG\"\"#*&,2)%\"xG,&F'\"\"\"\"\"$F.F(*&)F,,&F'F.F.F.F.%\"cGF.\"\"'
*&F'F.F+F.!\"$*(F'F.)F,,&F'F.!\"\"F.F.F3F(F.*&F'F(F+F.F.*(F'F(F1F.F3F.
!\"#*(F'F(F8F.F3F(F.*(F'F.F1F.F3F.F(F.,&*$F,F(F.F3F:F6" }}{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&\"\"%\"\"'*&F&F+F,F'F+
*&F1F'F&F5!\"&*(F1F'F&F+F,F'F6*&F1F+F&F5F'*(F1F+F&F+F,F'F/*&F1F+F,F+F'
*&F1F'F,F+F-F-F+" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%(xprimeBG,&%\"xG
\"\"\"**,&*$F&\"\"#F'%\"cG!\"\"F'F&F',*F*F-F,F-*&%\"lGF'F&F+F'*&F0F'F,
F'F-F',2*$F&\"\"%F+*&F&F+F,F'\"\"'*&F0F'F&F4!\"$*&F0F'F,F+F'*&F0F+F&F4
F'*(F0F+F&F+F,F'!\"#*&F0F+F,F+F'*(F0F'F&F+F,F'F+F-F+" }}{PARA 11 "" 1 
"" {XPPMATH 20 "6#>%\"wG\"\"$" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>-%\"
AG6$\"\"!\"\"$,$*(%\"xG\"\"\",&*$F+\"\"#F,%\"cGF,F,,&F.F,F0!\"\"!\"%\"
\"%" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>-%\"BG6$\"\"!\"\"$,$*&,(*$%\"x
G\"\"%\"\"\"*&F-\"\"#%\"cGF/\"\"'*$F2F1F/F/,&*$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,\"\"#F.%\"cG!\"\"F3F1" }}{PARA 
12 "" 1 "" {XPPMATH 20 "6#>-%\"AG6$%\"lG\"\"$,$*&,>)%\"xG,&F'\"\"\"F(F
/!#C*&)F-,&F'F/F/F/F/%\"cGF/F0*&F'F/F,F/\"#E*(F'F/F2F/F4F/!#I*(F'F/)F-
,&F'F/!\"\"F/F/F4\"\"#\"\"'*&F'F=F,F/!\"**(F'F=F2F/F4F/\"#@*(F'F=F:F/F
4F=!#:*(F'F/)F-,&F'F/!\"$F/F/F4F(!\"#*(F'F=FFF/F4F(F(*&F'F(F,F/F/*(F'F
(F2F/F4F/FH*(F'F(F:F/F4F=F(*(F'F(FFF/F4F(F<F/,&*$F-F=F/F4F<!\"%#F<F>" 
}}{PARA 12 "" 1 "" {XPPMATH 20 "6#>-%\"BG6$%\"lG\"\"$,$*&,>)%\"xG,&F'
\"\"\"\"\"%F/!\"'*&F'\"\"#F,F/F1*(F'F()F-,&F'F/!\"#F/F/%\"cGF(!\"\"*(F
'F()F-F'F/F8F3F(*(F'F()F-,&F'F/F3F/F/F8F/!\"$*&F'F(F,F/F/*(F'F/F=F/F8F
/F(*&F=F/F8F/!#O*(F'F/F;F/F8F3!#:*(F'F3F=F/F8F/\"#7*&F'F/F,F/\"#6*(F'F
3F;F/F8F3F1*(F'F/F5F/F8F(F/*&F;F/F8F3F1F/,&*$F-F3F/F8F9!\"%#F9F(" }}
{PARA 12 "" 1 "" {XPPMATH 20 "6#>%(xprimeAG,&%\"xG\"\"\"**,&*$F&\"\"#F
'%\"cG!\"\"F'F&F',2*$F&\"\"%\"\"'*&F&F+F,F'F+*&%\"lGF'F&F0!\"&*(F4F'F&
F+F,F'F1*&F4F+F&F0F'*(F4F+F&F+F,F'!\"#*&F4F+F,F+F'*&F4F'F,F+F-F',>*$F&
F1!#C*&F&F0F,F'F>*&F4F'F&F1\"#E*(F4F'F&F0F,F'!#I*(F4F'F&F+F,F+F1*&F4F+
F&F1!\"**(F4F+F&F0F,F'\"#@*(F4F+F&F+F,F+!#:*&F4F'F,\"\"$F9*&F4F+F,FLFL
*&F4FLF&F1F'*(F4FLF&F0F,F'!\"$*(F4FLF,F+F&F+FL*&F4FLF,FLF-F-FL" }}
{PARA 12 "" 1 "" {XPPMATH 20 "6#>%(xprimeBG,&%\"xG\"\"\"**,&*$F&\"\"#F
'%\"cG!\"\"F'F&F',2*$F&\"\"%F+*&F&F+F,F'\"\"'*&%\"lGF'F&F0!\"$*&F4F'F,
F+F'*&F4F+F&F0F'*(F4F+F&F+F,F'!\"#*&F4F+F,F+F'*(F4F'F&F+F,F'F+F',>*$F&
F2!\"'*&F4F+F&F2F>*&F4\"\"$F,FAF-*(F4FAF,F+F&F+FA*(F4FAF&F0F,F'F5*&F4F
AF&F2F'*(F4F'F&F0F,F'FA*&F&F0F,F'!#O*(F4F'F&F+F,F+!#:*(F4F+F&F0F,F'\"#
7*&F4F'F&F2\"#6*(F4F+F&F+F,F+F>*&F4F'F,FAF'*&F&F+F,F+F>F-FA" }}{PARA 
11 "" 1 "" {XPPMATH 20 "6#>%\"wG\"\"%" }}{PARA 11 "" 1 "" {XPPMATH 20 
"6#>-%\"AG6$\"\"!\"\"%*&,(*$%\"xGF(\"\"&*&F,\"\"#%\"cG\"\"\"\"#5*$F0F/
F1F1,&*$F,F/F1F0!\"\"!\"&" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>-%\"BG6$
\"\"!\"\"%,$*(%\"xG\"\"\",(*$F+F(F,*&F+\"\"#%\"cGF,\"#5*$F1F0\"\"&F,,&
*$F+F0F,F1!\"\"!\"&F0" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>-%\"AG6$%\"l
G\"\"!*&)%\"xGF'\"\"\",&*$F+\"\"#F,%\"cG!\"\"F1" }}{PARA 11 "" 1 "" 
{XPPMATH 20 "6#>-%\"BG6$%\"lG\"\"!,$*&)%\"xG,&F'\"\"\"F.F.F.,&*$F,\"\"
#F.%\"cG!\"\"F3F1" }}{PARA 12 "" 1 "" {XPPMATH 20 "6#>-%\"AG6$%\"lG\"
\"%,$*&,N)%\"xG,&F'\"\"\"F(F/\"$?\"*&F'\"\"#F,F/\"#r*(F'\"\"$)F-,&F'F/
!\"#F/F/%\"cGF5\"#K*(F'F5)F-F'F/F9F2!#g*(F'F5)F-,&F'F/F2F/F/F9F/\"#[*&
F'F5F,F/!#9*(F'F/)F-,&F'F/!\"%F/F/F9F(!\"'*(F'F5FEF/F9F(FH*(F'F2F6F/F9
F5!#c*(F'F(FEF/F9F(F/*(F'F2FEF/F9F(\"#6*(F'F/F?F/F9F/\"$K\"*&F?F/F9F/
\"$S#*(F'F2F?F/F9F/!$w\"*&F'F/F,F/!$a\"*(F'F2F<F/F9F2\"$]\"*(F'F/F6F/F
9F5\"#G*&F<F/F9F2\"#C*(F'F(F?F/F9F/FG*(F'F(F<F/F9F2\"\"'*(F'F(F6F/F9F5
FG*&F'F(F,F/F/F/,&*$F-F2F/F9!\"\"!\"&#F/Ffn" }}{PARA 12 "" 1 "" 
{XPPMATH 20 "6#>-%\"BG6$%\"lG\"\"%,$*&,P*&F'F()%\"xG,&F'\"\"\"\"\"&F0F
0F0*(F'F0)F.,&F'F0!\"\"F0F0%\"cG\"\"$!\"%*(F'F0)F.,&F'F0!\"$F0F0F6F(\"
\"#*(F'F=F:F0F6F(F5*(F'F7F:F0F6F(!\"#*(F'F0)F.,&F'F0F0F0F0F6F=\"$W\"*&
F'F0F-F0!#]*(F'F=FBF0F6F=\"\"'*(F'F()F.,&F'F0F7F0F0F6F0F8*(F'F(FBF0F6F
=FHF-\"#C*&F'F7F-F0!#5*(F'F7FJF0F6F0\"#K*(F'F7F3F0F6F7\"#;*(F'F=FJF0F6
F0!#c*(F'F(F:F0F6F(F0*(F'F7FBF0F6F=!#O*(F'F(F3F0F6F7F8*&F'F=F-F0\"#N*&
FJF0F6F0\"$S#*&FBF0F6F=\"$?\"*(F'F=F3F0F6F7FS*(F'F0FJF0F6F0!##*F0,&*$F
.F=F0F6F5!\"&#F0\"#7" }}{PARA 12 "" 1 "" {XPPMATH 20 "6#>%(xprimeAG,&%
\"xG\"\"\"**,&*$F&\"\"#F'%\"cG!\"\"F'F&F',>*$F&\"\"'!#C*&F&\"\"%F,F'F1
*&%\"lGF'F&F0\"#E*(F5F'F&F3F,F'!#I*(F5F'F&F+F,F+F0*&F5F+F&F0!\"**(F5F+
F&F3F,F'\"#@*(F5F+F&F+F,F+!#:*&F5F'F,\"\"$!\"#*&F5F+F,FAFA*&F5FAF&F0F'
*(F5FAF&F3F,F'!\"$*(F5FAF,F+F&F+FA*&F5FAF,FAF-F',N*$F&\"\")\"$?\"*&F5F
+F&FK\"#r*(F5FAF,FAF&F+\"#K*(F5FAF,F+F&F3!#g*(F5FAF&F0F,F'\"#[*&F5FAF&
FK!#9*&F5F'F,F3!\"'*&F5FAF,F3FX*(F5F+F,FAF&F+!#c*&F5F3F,F3F'*&F5F+F,F3
\"#6*(F5F'F&F0F,F'\"$K\"*&F&F0F,F'\"$S#*(F5F+F&F0F,F'!$w\"*&F5F'F&FK!$
a\"*(F5F+F,F+F&F3\"$]\"*(F5F'F,FAF&F+\"#G*&F,F+F&F3\"#C*(F5F3F&F0F,F'!
\"%*(F5F3F,F+F&F3F0*(F5F3F,FAF&F+Fho*&F5F3F&FKF'F-F3" }}{PARA 12 "" 1 
"" {XPPMATH 20 "6#>%(xprimeBG,&%\"xG\"\"\"**,&*$F&\"\"#F'%\"cG!\"\"F'F
&F',>*$F&\"\"'!\"'*&%\"lGF+F&F0F1*&F3\"\"$F,F5F-*(F3F5F,F+F&F+F5*(F3F5
F&\"\"%F,F'!\"$*&F3F5F&F0F'*(F3F'F&F8F,F'F5*&F&F8F,F'!#O*(F3F'F&F+F,F+
!#:*(F3F+F&F8F,F'\"#7*&F3F'F&F0\"#6*(F3F+F&F+F,F+F1*&F3F'F,F5F'*&F&F+F
,F+F1F',P*&F3F8F&\"\")F'*(F3F'F,F5F&F+!\"%*&F3F'F,F8F+*&F3F+F,F8F-*&F3
F5F,F8!\"#*(F3F'F&F8F,F+\"$W\"*&F3F'F&FI!#]*(F3F+F,F+F&F8F0*(F3F8F&F0F
,F'FK*(F3F8F,F+F&F8F0*$F&FI\"#C*&F3F5F&FI!#5*(F3F5F&F0F,F'\"#K*(F3F5F,
F5F&F+\"#;*(F3F+F&F0F,F'!#c*&F3F8F,F8F'*(F3F5F,F+F&F8F=*(F3F8F,F5F&F+F
K*&F3F+F&FI\"#N*&F&F0F,F'\"$S#*&F,F+F&F8\"$?\"*(F3F+F,F5F&F+Fhn*(F3F'F
&F0F,F'!##*F-F8" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}}
{MARK "1 0 0" 0 }{VIEWOPTS 1 1 0 1 1 1803 }