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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# [ Log From Exponential 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 := exp
(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,&-%$expG6#%\"xG\"\"\"%\"cG!\"\"" }}{PARA 11 "" 
1 "" {XPPMATH 20 "6#>-%\"AG6$\"\"!F'*$,&-%$expG6#%\"xG\"\"\"%\"cG!\"\"
F0" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>-%\"BG6$\"\"!F'*&-%$expG6#%\"xG
\"\"\",&F)F-%\"cG!\"\"F0" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%,Maximum
TermG\"\"%" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"wG\"\"\"" }}{PARA 
11 "" 1 "" {XPPMATH 20 "6#>-%\"AG6$\"\"!\"\"\"*&-%$expG6#%\"xGF(,&F*F(
%\"cG!\"\"!\"#" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>-%\"BG6$\"\"!\"\"\"
*(-%$expG6#%\"xGF(%\"cGF(,&F*F(F.!\"\"!\"#" }}{PARA 11 "" 1 "" 
{XPPMATH 20 "6#>-%\"AG6$%\"lG\"\"!*&)%\"xGF'\"\"\",&-%$expG6#F+F,%\"cG
!\"\"F2" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>-%\"BG6$%\"lG\"\"!*()%\"xG
F'\"\"\"-%$expG6#F+F,,&F-F,%\"cG!\"\"F2" }}{PARA 11 "" 1 "" {XPPMATH 
20 "6#>-%\"AG6$%\"lG\"\"\",$*&,(*&)%\"xGF'F(-%$expG6#F.F(!\"\"*(F'F()F
.,&F'F(F2F(F(F/F(F(*(F'F(F4F(%\"cGF(F2F(,&F/F(F7F2!\"#F2" }}{PARA 11 "
" 1 "" {XPPMATH 20 "6#>-%\"BG6$%\"lG\"\"\",$*(-%$expG6#%\"xGF(,(*&)F.F
'F(%\"cGF(!\"\"*(F'F()F.,&F'F(F3F(F(F+F(F(*(F'F(F5F(F2F(F3F(,&F+F(F2F3
!\"#F3" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%(xprimeAG,&%\"xG\"\"\"*(F&
F',&-%$expG6#F&F'%\"cG!\"\"F',(*&F*F'F&F'F'*&%\"lGF'F*F'F.*&F2F'F-F'F'
F.F." }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%(xprimeBG,&%\"xG\"\"\"*(F&F'
,&-%$expG6#F&F'%\"cG!\"\"F',(*&F-F'F&F'F'*&%\"lGF'F*F'F.*&F2F'F-F'F'F.
F." }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"wG\"\"#" }}{PARA 11 "" 1 "" 
{XPPMATH 20 "6#>-%\"AG6$\"\"!\"\"#,$*(-%$expG6#%\"xG\"\"\",&F+F/%\"cGF
/F/,&F+F/F1!\"\"!\"$#F/F(" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>-%\"BG6$
\"\"!\"\"#,$**-%$expG6#%\"xG\"\"\"%\"cGF/,&F+F/F0F/F/,&F+F/F0!\"\"!\"$
#F/F(" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>-%\"AG6$%\"lG\"\"!*&)%\"xGF'
\"\"\",&-%$expG6#F+F,%\"cG!\"\"F2" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>
-%\"BG6$%\"lG\"\"!*()%\"xGF'\"\"\"-%$expG6#F+F,,&F-F,%\"cG!\"\"F2" }}
{PARA 11 "" 1 "" {XPPMATH 20 "6#>-%\"AG6$%\"lG\"\"#,$*&,6*&)%\"xGF'\"
\"\"-%$expG6#,$F.F(F/!\"\"*(F-F/-F16#F.F/%\"cGF/F4*(F'F/)F.,&F'F/F4F/F
/F0F/F(**F'F/F:F/F6F/F8F/!\"#*(F'F()F.,&F'F/F=F/F/F0F/F4**F'F(F?F/F6F/
F8F/F(*(F'F(F?F/F8F(F4*(F'F/F?F/F0F/F/**F'F/F?F/F6F/F8F/F=*(F'F/F?F/F8
F(F/F/,&F6F/F8F4!\"$#F4F(" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>-%\"BG6$
%\"lG\"\"#,$*(-%$expG6#%\"xG\"\"\",6*()F.F'F/F+F/%\"cGF/!\"\"*&F2F/F3F
(F4**F'F/)F.,&F'F/F4F/F/F+F/F3F/F(*(F'F/F7F/F3F(!\"#*(F'F()F.,&F'F/F:F
/F/-F,6#,$F.F(F/F4**F'F(F<F/F+F/F3F/F(*(F'F(F<F/F3F(F4*(F'F/F<F/F>F/F/
**F'F/F<F/F+F/F3F/F:*(F'F/F<F/F3F(F/F/,&F+F/F3F4!\"$#F4F(" }}{PARA 11 
"" 1 "" {XPPMATH 20 "6#>%(xprimeAG,&%\"xG\"\"\"**,&-%$expG6#F&F'%\"cG!
\"\"F'F&F',(*&F*F'F&F'F'*&%\"lGF'F*F'F.*&F2F'F-F'F'F',6*&-F+6#,$F&\"\"
#F'F&F9F'*(F*F'F-F'F&F9F'*(F2F'F6F'F&F'!\"#**F2F'F*F'F-F'F&F'F9*&F2F9F
6F'F'*(F2F9F*F'F-F'F<*&F2F9F-F9F'*&F2F'F6F'F.*(F2F'F*F'F-F'F9*&F2F'F-F
9F.F.F<" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%(xprimeBG,&%\"xG\"\"\"**,
&-%$expG6#F&F'%\"cG!\"\"F'F&F',(*&F-F'F&F'F'*&%\"lGF'F*F'F.*&F2F'F-F'F
'F',6*(F*F'F-F'F&\"\"#F'*&F&F6F-F6F'**F2F'F*F'F-F'F&F'!\"#*(F&F'F2F'F-
F6F6*&F2F6-F+6#,$F&F6F'F'*(F2F6F*F'F-F'F9*&F2F6F-F6F'*&F2F'F<F'F.*(F2F
'F*F'F-F'F6*&F2F'F-F6F.F.F9" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"wG
\"\"$" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>-%\"AG6$\"\"!\"\"$,$*(-%$exp
G6#%\"xG\"\"\",(-F,6#,$F.\"\"#F/*&F+F/%\"cGF/\"\"%*$F6F4F/F/,&F+F/F6!
\"\"!\"%#F/\"\"'" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>-%\"BG6$\"\"!\"\"
$,$**-%$expG6#%\"xG\"\"\"%\"cGF/,(-F,6#,$F.\"\"#F/*&F+F/F0F/\"\"%*$F0F
5F/F/,&F+F/F0!\"\"!\"%#F/\"\"'" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>-%
\"AG6$%\"lG\"\"!*&)%\"xGF'\"\"\",&-%$expG6#F+F,%\"cG!\"\"F2" }}{PARA 
11 "" 1 "" {XPPMATH 20 "6#>-%\"BG6$%\"lG\"\"!*()%\"xGF'\"\"\"-%$expG6#
F+F,,&F-F,%\"cG!\"\"F2" }}{PARA 12 "" 1 "" {XPPMATH 20 "6#>-%\"AG6$%\"
lG\"\"$,$*&,P*&)%\"xGF'\"\"\"-%$expG6#,$F.F(F/!\"\"*(F-F/-F16#,$F.\"\"
#F/%\"cGF/!\"%*(F-F/-F16#F.F/F:F9F4*(F'F/)F.,&F'F/F4F/F/F0F/F(**F'F/F@
F/F=F/F:F9!\"$*(F'F9)F.,&F'F/!\"#F/F/F0F/FC**F'F9FEF/F6F/F:F/\"\"'**F'
F9FEF/F=F/F:F9FC*(F'F/FEF/F0F/F(**F'F/FEF/F6F/F:F/!\"'**F'F/FEF/F=F/F:
F9F(*(F'F()F.,&F'F/FCF/F/F0F/F/**F'F(FPF/F6F/F:F/FC**F'F(FPF/F=F/F:F9F
(*(F'F(FPF/F:F(F4*(F'F9FPF/F0F/FC**F'F9FPF/F6F/F:F/\"\"***F'F9FPF/F=F/
F:F9!\"**(F'F9FPF/F:F(F(*(F'F/FPF/F0F/F9**F'F/FPF/F6F/F:F/FM**F'F/FPF/
F=F/F:F9FI*(F'F/FPF/F:F(FGF/,&F=F/F:F4F;#F4FI" }}{PARA 12 "" 1 "" 
{XPPMATH 20 "6#>-%\"BG6$%\"lG\"\"$,$*(-%$expG6#%\"xG\"\"\",P*()F.F'F/-
F,6#,$F.\"\"#F/%\"cGF/!\"\"*(F2F/F+F/F7F6!\"%*&F2F/F7F(F8**F7F/F'F/)F.
,&F'F/F8F/F/F3F/F(*(F'F/F=F/F7F(!\"$**F'F6)F.,&F'F/!\"#F/F/F3F/F7F/F@*
*F'F6FBF/F+F/F7F6\"\"'*(F'F6FBF/F7F(F@**F'F/FBF/F3F/F7F/F(**F'F/FBF/F+
F/F7F6!\"'*(F'F/FBF/F7F(F(*(F'F()F.,&F'F/F@F/F/-F,6#,$F.F(F/F/**F'F(FM
F/F3F/F7F/F@**F'F(FMF/F+F/F7F6F(*(F'F(FMF/F7F(F8*(F'F6FMF/FOF/F@**F'F6
FMF/F3F/F7F/\"\"***F'F6FMF/F+F/F7F6!\"**(F'F6FMF/F7F(F(*(F'F/FMF/FOF/F
6**F'F/FMF/F3F/F7F/FJ**F'F/FMF/F+F/F7F6FF*(F'F/FMF/F7F(FDF/,&F+F/F7F8F
:#F8FF" }}{PARA 12 "" 1 "" {XPPMATH 20 "6#>%(xprimeAG,&%\"xG\"\"\"**,&
-%$expG6#F&F'%\"cG!\"\"F'F&F',6*&-F+6#,$F&\"\"#F'F&F4F'*(F*F'F-F'F&F4F
'*(%\"lGF'F1F'F&F'!\"#**F7F'F*F'F-F'F&F'F4*&F7F4F1F'F'*(F7F4F*F'F-F'F8
*&F7F4F-F4F'*&F7F'F1F'F.*(F7F'F*F'F-F'F4*&F7F'F-F4F.F',P*&-F+6#,$F&\"
\"$F'F&FEF'*(F1F'F-F'F&FE\"\"%*(F*F'F-F4F&FEF'*(F7F'FBF'F&F4!\"$**F7F'
F*F'F-F4F&F4FE*(F7F4FBF'F&F'FE**F7F4F1F'F-F'F&F'!\"'**F7F4F*F'F-F4F&F'
FE*(F7F'FBF'F&F'FJ**F7F'F1F'F-F'F&F'\"\"'**F7F'F*F'F-F4F&F'FJ*&F7FEFBF
'F.*(F7FEF1F'F-F'FE*(F7FEF*F'F-F4FJ*&F7FEF-FEF'*&F7F4FBF'FE*(F7F4F1F'F
-F'!\"**(F7F4F*F'F-F4\"\"**&F7F4F-FEFJ*&F7F'FBF'F8*(F7F'F1F'F-F'FR*(F7
F'F*F'F-F4FN*&F7F'F-FEF4F.FJ" }}{PARA 12 "" 1 "" {XPPMATH 20 "6#>%(xpr
imeBG,&%\"xG\"\"\"**,&-%$expG6#F&F'%\"cG!\"\"F'F&F',6*(F*F'F-F'F&\"\"#
F'*&F&F1F-F1F'**%\"lGF'F*F'F-F'F&F'!\"#*(F&F'F4F'F-F1F1*&F4F1-F+6#,$F&
F1F'F'*(F4F1F*F'F-F'F5*&F4F1F-F1F'*&F4F'F8F'F.*(F4F'F*F'F-F'F1*&F4F'F-
F1F.F',P*(F8F'F-F'F&\"\"$F'*(F*F'F-F1F&FB\"\"%*&F&FBF-FBF'**F-F'F4F'F8
F'F&F1!\"$*(F4F'F-FBF&F1FB**F4F1F8F'F-F'F&F'FB**F4F1F*F'F-F1F&F'!\"'*(
F4F1F-FBF&F'FB**F4F'F8F'F-F'F&F'FG**F4F'F*F'F-F1F&F'\"\"'*(F&F'F4F'F-F
BFG*&F4FB-F+6#,$F&FBF'F.*(F4FBF8F'F-F'FB*(F4FBF*F'F-F1FG*&F4FBF-FBF'*&
F4F1FRF'FB*(F4F1F8F'F-F'!\"**(F4F1F*F'F-F1\"\"**&F4F1F-FBFG*&F4F'FRF'F
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{XPPMATH 20 "6#>%\"wG\"\"%" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>-%\"AG6
$\"\"!\"\"%,$*(-%$expG6#%\"xG\"\"\",*-F,6#,$F.\"\"$F/*&-F,6#,$F.\"\"#F
/%\"cGF/\"#6*&F+F/F:F9F;*$F:F4F/F/,&F+F/F:!\"\"!\"&#F/\"#C" }}{PARA 
11 "" 1 "" {XPPMATH 20 "6#>-%\"BG6$\"\"!\"\"%,$**-%$expG6#%\"xG\"\"\"%
\"cGF/,*-F,6#,$F.\"\"$F/*&-F,6#,$F.\"\"#F/F0F/\"#6*&F+F/F0F:F;*$F0F5F/
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$%\"lG\"\"!*&)%\"xGF'\"\"\",&-%$expG6#F+F,%\"cG!\"\"F2" }}{PARA 11 "" 
1 "" {XPPMATH 20 "6#>-%\"BG6$%\"lG\"\"!*()%\"xGF'\"\"\"-%$expG6#F+F,,&
F-F,%\"cG!\"\"F2" }}{PARA 12 "" 1 "" {XPPMATH 20 "6#>-%\"AG6$%\"lG\"\"
%,$*&,\\q*(F'F()%\"xG,&F'\"\"\"!\"%F0F0-%$expG6#,$F.F(F0!\"\"**F'\"\"$
)F.,&F'F0!\"$F0F0-F36#,$F.\"\"#F0%\"cGF?\"#7*(F'F?F9F0F2F0!#7*(F'F8F-F
0F2F0\"\"'**F'F(F-F0-F36#,$F.F8F0F@F0F(*()F.F'F0FGF0F@F0!#6**F'F?F-F0F
GF0F@F0\"#W**F'F(F-F0F<F0F@F?!\"'*(F'F?F-F0F2F0FL**F'F?F-F0F<F0F@F?!#m
*(F'F0F-F0F2F0FE**F'F0)F.,&F'F0F6F0F0F<F0F@F?FC**F'F?F9F0FGF0F@F0\"#O*
(F'F0)F.,&F'F0!\"#F0F0F2F0FE**F'F?F9F0F<F0F@F?!#O**F'F8F-F0FGF0F@F0!#C
**F'F0F9F0FGF0F@F0F[o**F'F8F9F0FGF0F@F0FC*(F'F0F-F0F@F(FE*(F'F8F-F0F@F
(FE*(F'F(F-F0F@F(F6*(F'F?F-F0F@F(FL*(FKF0-F36#F.F0F@F8F6**F'F?FenF0Fco
F0F@F8FP**F'F0FenF0FcoF0F@F8FE**F'F8F-F0FcoF0F@F8F[o**F'F(F-F0FcoF0F@F
8F(**F'F8F9F0FcoF0F@F8F1**F'F?F9F0FcoF0F@F8FA**F'F0F9F0FcoF0F@F8!\")**
F'F0FVF0FcoF0F@F8F1**F'F?F-F0FcoF0F@F8FN**F'F0F-F0FcoF0F@F8F[o*(F'F0FV
F0F2F0F(**F'F0F-F0FGF0F@F0F[o**F'F8F-F0F<F0F@F?FY**F'F?FenF0FGF0F@F0FE
**F'F0F9F0F<F0F@F?\"#C*(FKF0F<F0F@F?FL*(F'F?FenF0F2F0FP**F'F0FenF0FGF0
F@F0FP*(F'F8F9F0F2F0F(*(F'F0F9F0F2F0\"\")**F'F?FenF0F<F0F@F?FE*&FKF0F2
F0F6**F'F0FenF0F<F0F@F?FP**F'F0FVF0FGF0F@F0FA**F'F0F-F0F<F0F@F?FYF0,&F
coF0F@F6!\"&#F6Fep" }}{PARA 12 "" 1 "" {XPPMATH 20 "6#>-%\"BG6$%\"lG\"
\"%,$*(-%$expG6#%\"xG\"\"\",\\q*(F'F()F.,&F'F/!\"%F/F/-F,6#,$F.F(F/!\"
\"**F'\"\"$)F.,&F'F/!\"$F/F/-F,6#,$F.\"\"#F/%\"cGFA!#7*(F'F:F2F/F5F/\"
\"'**F'F(F2F/-F,6#,$F.F:F/FBF/F(*()F.F'F/FGF/FBF/F8**F'FAF2F/FGF/FBF/
\"#W**F'F(F2F/F>F/FBFA!\"'*(F'FAF2F/F5F/!#6**F'FAF2F/F>F/FBFA!#m*(F'F/
F2F/F5F/FE**F'F/)F.,&F'F/F8F/F/F>F/FBFA\"#7**F'FAF;F/FGF/FBF/FC**F'FAF
;F/F>F/FBFA\"#O*(F'FA)F.,&F'F/!\"#F/F/FBF(FO**F'F:F2F/FGF/FBF/!#C*(F'F
/F;F/FBF(!\")*(F'FAF;F/FBF(FX*(F'F:F;F/FBF(F4**F'F/F;F/FGF/FBF/\"\")**
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