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{SECT 0 {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 2786 "##################
###################\n# Maple V, Release 4 script for the #\n# calculat
ion of Schroder's methods #\n#                                   #\n# \+
Copyright (C) 1996 by Dann Corbit #\n# [ Exponential from Log 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 := log
(x) - c;\n\n# Definition of the first term for Schroder's Series 'A'\n
# Equation 55, Page 30.\nA(0, 0) := 1/y;\n\n# Definition of the first \+
term for Schroder's Series 'B'\n# Equation 56, Page 31.\nB(0, 0) := di
ff(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 Maxim
umTerm\nfor i from 1 to MaximumTerm do w := i;\n\n   # Definition of t
he 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$a)*y^(w-a+1
)*A(0, w-a)/a!', 'a'=1..w)/y^(w+1));\n\n   # Definition of the Lamda=0
 terms for Schroder's Series 'B'\n   # Equation 55, Page 30.\n   B(0, \+
w) := simplify(sum('(-1)^(a)*y^(a)*diff(y, x$(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
 Schroder's Series 'A'\n   # Equation 70, Page 40.\n   A(l, 0) := simp
lify(x^l*y*A(0, 0)/y);\n\n   # Definition of the Omega=0 terms for Sch
roder'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 Sc
hroder's Series 'A'\n   # Equation 70, Page 40.\n   A(l, w) := simplif
y(sum('(-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 Schrod
er's Series 'B'\n   # Equation 70, Page 40.\n   B(l, w) := simplify(su
m('(-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
) Page 39.\n   xprimeA := x - simplify(A(l, w-1)/A(l, w));\n\n   # Def
inition 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,&-%#lnG6#%\"xG\"\"\"%\"cG!\"\"" }}{PARA 11 "" 1 
"" {XPPMATH 20 "6#>-%\"AG6$\"\"!F'*$,&-%#lnG6#%\"xG\"\"\"%\"cG!\"\"F0
" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>-%\"BG6$\"\"!F'*&%\"xG!\"\",&-%#l
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mG\"\"%" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"wG\"\"\"" }}{PARA 11 "
" 1 "" {XPPMATH 20 "6#>-%\"AG6$\"\"!\"\"\"*&%\"xG!\"\",&-%#lnG6#F*F+%
\"cGF(!\"#" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>-%\"BG6$\"\"!\"\"\",$*(
,(!\"\"F(-%#lnG6#%\"xGF,%\"cGF(F(F0!\"#,&F-F,F1F(F2F," }}{PARA 11 "" 
1 "" {XPPMATH 20 "6#>-%\"AG6$%\"lG\"\"!,$*&)%\"xGF'\"\"\",&-%#lnG6#F,!
\"\"%\"cGF-F2F2" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>-%\"BG6$%\"lG\"\"!
,$*&)%\"xG,&F'\"\"\"!\"\"F.F.,&-%#lnG6#F,F/%\"cGF.F/F/" }}{PARA 11 "" 
1 "" {XPPMATH 20 "6#>-%\"AG6$%\"lG\"\"\"*()%\"xG,&F'F(!\"\"F(F(,(F(F(*
&F'F(-%#lnG6#F+F(F-*&F'F(%\"cGF(F(F(,&F0F-F4F(!\"#" }}{PARA 11 "" 1 "
" {XPPMATH 20 "6#>-%\"BG6$%\"lG\"\"\"*()%\"xG,&F'F(!\"#F(F(,,F(F(-%#ln
G6#F+F(%\"cG!\"\"*&F'F(F/F(F3*&F'F(F2F(F(F(,&F/F3F2F(F-" }}{PARA 11 "
" 1 "" {XPPMATH 20 "6#>%(xprimeAG,&%\"xG\"\"\"*(F&F',&-%#lnG6#F&!\"\"%
\"cGF'F',(F'F'*&%\"lGF'F*F'F-*&F1F'F.F'F'F-F'" }}{PARA 11 "" 1 "" 
{XPPMATH 20 "6#>%(xprimeBG,&%\"xG\"\"\"*(F&F',&-%#lnG6#F&!\"\"%\"cGF'F
',,F'F'F*F'F.F-*&%\"lGF'F*F'F-*&F1F'F.F'F'F-F'" }}{PARA 11 "" 1 "" 
{XPPMATH 20 "6#>%\"wG\"\"#" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>-%\"AG6
$\"\"!\"\"#,$*(,(!\"#\"\"\"-%#lnG6#%\"xG!\"\"%\"cGF-F-,&F.F2F3F-!\"$F1
F,#F-F(" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>-%\"BG6$\"\"!\"\"#,$*(,.F(
\"\"\"-%#lnG6#%\"xG\"\"$%\"cG!\"$*$F-F(F(*&F-F,F2F,!\"%*$F2F(F(F,F0F3,
&F-!\"\"F2F,F3#F9F(" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>-%\"AG6$%\"lG
\"\"!,$*&)%\"xGF'\"\"\",&-%#lnG6#F,!\"\"%\"cGF-F2F2" }}{PARA 11 "" 1 "
" {XPPMATH 20 "6#>-%\"BG6$%\"lG\"\"!,$*&)%\"xG,&F'\"\"\"!\"\"F.F.,&-%#
lnG6#F,F/%\"cGF.F/F/" }}{PARA 12 "" 1 "" {XPPMATH 20 "6#>-%\"AG6$%\"lG
\"\"#,$*()%\"xG,&F'\"\"\"!\"#F.F.,8F(F.-%#lnG6#F,F.%\"cG!\"\"*&F'F.F1F
.F/*&F'F.F4F.F(*&F'F(F1F(F.*(F'F(F1F.F4F.F/*&F'F(F4F(F.*&F'F.F1F(F5*(F
'F.F1F.F4F.F(*&F'F.F4F(F5F.,&F1F5F4F.!\"$#F5F(" }}{PARA 12 "" 1 "" 
{XPPMATH 20 "6#>-%\"BG6$%\"lG\"\"#,$*()%\"xG,&F'\"\"\"!\"$F.F.,>F(F.-%
#lnG6#F,\"\"$%\"cGF/*$F1F(F(*&F1F.F5F.!\"%*$F5F(F(*&F'F.F1F.!\"#*&F'F.
F1F(F/*(F'F.F1F.F5F.\"\"'*&F'F.F5F.F(*&F'F.F5F(F/*&F'F(F1F(F.*(F'F(F1F
.F5F.F;*&F'F(F5F(F.F.,&F1!\"\"F5F.F/#FEF(" }}{PARA 11 "" 1 "" 
{XPPMATH 20 "6#>%(xprimeAG,&%\"xG\"\"\"**F&F',(F'F'*&%\"lGF'-%#lnG6#F&
F'!\"\"*&F+F'%\"cGF'F'F',&F,F/F1F'F',8\"\"#F'F,F'F1F/F*!\"#F0F4*&F+F4F
,F4F'*(F+F4F,F'F1F'F5*&F+F4F1F4F'*&F+F'F,F4F/*(F+F'F,F'F1F'F4*&F+F'F1F
4F/F/F4" }}{PARA 12 "" 1 "" {XPPMATH 20 "6#>%(xprimeBG,&%\"xG\"\"\"**F
&F',,F'F'-%#lnG6#F&F'%\"cG!\"\"*&%\"lGF'F*F'F.*&F0F'F-F'F'F',&F*F.F-F'
F',>\"\"#F'F*\"\"$F-!\"$*$F*F4F4*&F*F'F-F'!\"%*$F-F4F4F/!\"#*&F0F'F*F4
F6*(F0F'F*F'F-F'\"\"'F1F4*&F0F'F-F4F6*&F0F4F*F4F'*(F0F4F*F'F-F'F;*&F0F
4F-F4F'F.F4" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"wG\"\"$" }}{PARA 
11 "" 1 "" {XPPMATH 20 "6#>-%\"AG6$\"\"!\"\"$,$*(,.F(\"\"\"-%#lnG6#%\"
xGF(%\"cG!\"$*$F-\"\"#F,*&F-F,F1F,!\"#*$F1F4F,F,,&F-!\"\"F1F,!\"%F0F2#
F,F(" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>-%\"BG6$\"\"!\"\"$,$*(,6!\"'
\"\"\"-%#lnG6#%\"xG!#7%\"cG\"#7*$F.\"\"#!#6*&F.F-F3F-\"#A*$F3F6F7*$F.F
(F,*&F.F6F3F-\"#=*&F.F-F3F6!#=*$F3F(\"\"'F-F1!\"%,&F.!\"\"F3F-FB#FDFA
" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>-%\"AG6$%\"lG\"\"!,$*&)%\"xGF'\"
\"\",&-%#lnG6#F,!\"\"%\"cGF-F2F2" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>-
%\"BG6$%\"lG\"\"!,$*&)%\"xG,&F'\"\"\"!\"\"F.F.,&-%#lnG6#F,F/%\"cGF.F/F
/" }}{PARA 12 "" 1 "" {XPPMATH 20 "6#>-%\"AG6$%\"lG\"\"$,$*()%\"xG,&F'
\"\"\"!\"$F.F.,V\"\"'F.*&F'F.%\"cGF.F1*&F'F.-%#lnG6#F,F.!\"'F3F8F5F1*&
F'F(F3F(F.*&F'F.F3F(\"\"#*&F'F;F3F(F/*&F'F.F3F;F8*$F3F;F;*&F'F.F5F(!\"
#*&F'F;F5F(F(*&F'F(F5F(!\"\"*$F5F;F;*&F5F.F3F.!\"%*&F'F;F3F;F(*(F'F(F5
F.F3F;F/*(F'F(F5F;F3F.F(*(F'F.F5F.F3F;F8*(F'F.F5F;F3F.F1*(F'F;F5F.F3F;
\"\"**(F'F;F5F;F3F.!\"**(F'F;F5F.F3F.F8*(F'F.F5F.F3F.\"#7*&F'F;F5F;F(*
&F'F.F5F;F8F.,&F5FCF3F.FF#F.F1" }}{PARA 12 "" 1 "" {XPPMATH 20 "6#>-%
\"BG6$%\"lG\"\"$,$*()%\"xG,&F'\"\"\"!\"%F.F.,hn\"\"'F.*&F'F.%\"cGF.F1*
&F'F.-%#lnG6#F,F.!\"'F3!#7F5\"#7*&F'F(F3F(F.*&F'F.F3F(\"#6*&F'\"\"#F3F
(F8*$F3F(F8*&F'F.F3F?F9*$F3F?F=*&F'F.F5F(!#6*&F'F?F5F(F1*&F'F(F5F(!\"
\"*$F5F(F1*$F5F?F=*&F5F.F3F.!#A*&F5F?F3F.!#=*&F5F.F3F?\"#=*&F'F?F3F?F(
*(F'F(F5F.F3F?!\"$*(F'F(F5F?F3F.F(*(F'F.F5F.F3F?!#L*(F'F.F5F?F3F.\"#L*
(F'F?F5F.F3F?FO*(F'F?F5F?F3F.FM*(F'F?F5F.F3F.F8*(F'F.F5F.F3F.\"#C*&F'F
?F5F?F(*&F'F.F5F?F9F.,&F5FGF3F.F/#F.F1" }}{PARA 12 "" 1 "" {XPPMATH 
20 "6#>%(xprimeAG,&%\"xG\"\"\"**F&F',8\"\"#F'-%#lnG6#F&F'%\"cG!\"\"*&%
\"lGF'F+F'!\"#*&F1F'F.F'F**&F1F*F+F*F'*(F1F*F+F'F.F'F2*&F1F*F.F*F'*&F1
F'F+F*F/*(F1F'F+F'F.F'F**&F1F'F.F*F/F',&F+F/F.F'F',V\"\"'F'F3F<F0!\"'F
.F=F+F<*&F1\"\"$F.F?F'*&F1F'F.F?F**&F1F*F.F?!\"$F9F=*$F.F*F**&F1F'F+F?
F2*&F1F*F+F?F?*&F1F?F+F?F/*$F+F*F**&F+F'F.F'!\"%F6F?*(F1F?F+F'F.F*FB*(
F1F?F+F*F.F'F?*(F1F'F+F'F.F*F=*(F1F'F+F*F.F'F<*(F1F*F+F'F.F*\"\"**(F1F
*F+F*F.F'!\"*F5F=F8\"#7F4F?F7F=F/F?" }}{PARA 12 "" 1 "" {XPPMATH 20 "6
#>%(xprimeBG,&%\"xG\"\"\"**F&F',>\"\"#F'-%#lnG6#F&\"\"$%\"cG!\"$*$F+F*
F**&F+F'F/F'!\"%*$F/F*F**&%\"lGF'F+F'!\"#*&F6F'F+F*F0*(F6F'F+F'F/F'\"
\"'*&F6F'F/F'F**&F6F'F/F*F0*&F6F*F+F*F'*(F6F*F+F'F/F'F7*&F6F*F/F*F'F',
&F+!\"\"F/F'F',hnF:F'F;F:F5!\"'F/!#7F+\"#7*&F6F.F/F.F'*&F6F'F/F.\"#6*&
F6F*F/F.FC*$F/F.FCF<FDF4FH*&F6F'F+F.!#6*&F6F*F+F.F:*&F6F.F+F.FA*$F+F.F
:F1FHF2!#A*&F+F*F/F'!#=*&F+F'F/F*\"#=F?F.*(F6F.F+F'F/F*F0*(F6F.F+F*F/F
'F.*(F6F'F+F'F/F*!#L*(F6F'F+F*F/F'\"#L*(F6F*F+F'F/F*FT*(F6F*F+F*F/F'FR
F>FCF9\"#CF=F.F8FDFAF." }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"wG\"\"%
" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>-%\"AG6$\"\"!\"\"%,$*(,6!#7\"\"\"
-%#lnG6#%\"xG!#=%\"cG\"#=*$F.\"\"#!#6*&F.F-F3F-\"#A*$F3F6F7*$F.\"\"$!
\"$*&F.F6F3F-\"\"**&F.F-F3F6!\"**$F3F<F<F-,&F.!\"\"F3F-!\"&F1!\"%#F-\"
#7" }}{PARA 12 "" 1 "" {XPPMATH 20 "6#>-%\"BG6$\"\"!\"\"%,$*(,@\"#7\"
\"\"%\"cG!#I*$F.\"\"$!#D*$F.F(F,*$-%#lnG6#%\"xGF1\"#D*$F.\"\"#\"#N*$F5
F;F<*&F5F-F.F-!#q*&F5F1F.F-!#[*&F5F;F.F;\"#s*&F5F-F.F1FAF5\"#I*&F5F;F.
F-!#v*&F5F-F.F;\"#v*$F5F(F,F-F8!\"&,&F5!\"\"F.F-FK#FMF," }}{PARA 11 "
" 1 "" {XPPMATH 20 "6#>-%\"AG6$%\"lG\"\"!,$*&)%\"xGF'\"\"\",&-%#lnG6#F
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"" 1 "" {XPPMATH 20 "6#>-%\"AG6$%\"lG\"\"%,$*()%\"xG,&F'\"\"\"!\"%F.F.
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F1*(F'F.F<F5F3F.F1*(F'F9F<F5F3F.!#W*(F'F5F<F9F3F9!#O*(F'F(F<F9F3F9\"\"
'*&F'F.F3F(F6*(F'F9F<F9F3F9\"#m*(F'F9F<F.F3F5FD*&F'F.F<F.!#CF3FFF<\"#O
*&F'F(F<F(F.*&F'F.F<F(F6*&F'F5F<F(F6*&F'F9F<F(F:*&F'F5F3F5F(*&F'F.F3F5
\"#A*&F'F9F3F5!#=*$F3F5F6*(F'F.F<F.F3F5F1*(F'F.F<F9F3F9FF*&F'F.F3F9FF*
$F3F9FV*&F'F.F<F5!#A*&F'F9F<F5\"#=*&F'F5F<F5F/*$F<F5FH*$F<F9FV*&F<F.F3
F.FD*&F<F9F3F.FX*&F<F.F3F9F[o*&F'F9F3F9\"#7*(F'F5F<F.F3F9!#7*(F'F5F<F9
F3F.Fco*(F'F.F<F.F3F9!#m*(F'F.F<F9F3F.FK*(F'F9F<F.F3F9\"#a*(F'F9F<F9F3
F.!#a*(F'F9F<F.F3F.FN*(F'F.F<F.F3F.\"#s*&F'F9F<F9Fco*&F'F.F<F9FFF.,&F<
!\"\"F3F.!\"&#FdpF1" }}{PARA 12 "" 1 "" {XPPMATH 20 "6#>-%\"BG6$%\"lG
\"\"%,$*()%\"xG,&F'\"\"\"!\"&F.F.,jq\"#CF.*&F'F.%\"cGF.F1*&F'\"\"$F3F(
!#5*&F'F(F3F(F.*&F'\"\"#F3F(\"#N*(F'F(-%#lnG6#F,F5F3F.!\"%*(F'F5F<F5F3
F.\"#S*(F'F(F<F.F3F5F?*(F'F5F<F.F3F5FA*(F'F.F<F5F3F.\"$+#*(F'F9F<F5F3F
.!$S\"*(F'F5F<F9F3F9!#g*(F'F(F<F9F3F9\"\"'*&F'F.F3F(!#]*(F'F9F<F9F3F9
\"$5#*(F'F9F<F.F3F5FG*&F'F.F<F.!#CF3FIF<\"#g*&F'F(F<F(F.*&F'F.F<F(FM*&
F'F5F<F(F6*&F'F9F<F(F:*&F'F5F3F5F(*&F'F.F3F5\"#q*&F'F9F3F5!#I*$F3F5FM*
$F3F(F1*(F'F.F<F.F3F5FE*(F'F.F<F9F3F9!$+$*&F'F.F3F9FI*$F3F9FZ*&F'F.F<F
5!#q*&F'F9F<F5\"#I*&F'F5F<F5F?*$F<F5\"#]*$F<F9FZ*&F<F.F3F.FG*&F<F9F3F.
!$]\"*&F<F.F3F9\"$]\"*&F'F9F3F9\"#7*(F'F5F<F.F3F9!#7*(F'F5F<F9F3F.F\\p
*(F'F.F<F.F3F9!$5#*(F'F.F<F9F3F.FO*(F'F9F<F.F3F9\"#!**(F'F9F<F9F3F.!#!
**(F'F9F<F.F3F.FR*(F'F.F<F.F3F.\"$?\"*&F'F9F<F9F\\p*&F'F.F<F9FI*&F<F5F
3F.!#'**&F<F9F3F9\"$W\"*&F<F.F3F5F]q*$F<F(F1F.,&F<!\"\"F3F.F/#FcqF1" }
}{PARA 12 "" 1 "" {XPPMATH 20 "6#>%(xprimeAG,&%\"xG\"\"\"**F&F',V\"\"'
F'*&%\"lGF'%\"cGF'F**&F,F'-%#lnG6#F&F'!\"'F-F2F/F**&F,\"\"$F-F4F'*&F,F
'F-F4\"\"#*&F,F6F-F4!\"$*&F,F'F-F6F2*$F-F6F6*&F,F'F/F4!\"#*&F,F6F/F4F4
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