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{SECT 0 {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 2858 "##################
###################\n# Maple V, Release 4 script for the #\n# calculat
ion of Schroder's methods #\n#                                   #\n# \+
Copyright (C) 1996 by Dann Corbit #\n# [ Generic 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# Generic definition of y as some function of \+
x\ny := 2^x-c;\nreadlib(C):\n# Definition of the first term for Schrod
er's Series 'A'\n# Equation 55, Page 30.\nA(0, 0) := 1/y;\n\n# Definit
ion of the first term for Schroder's Series 'B'\n# Equation 56, Page 3
1.\nB(0, 0) := diff(y, x)/y;\n\n# We can change MaximumTerm to process
 as many terms as we like\nMaximumTerm := 10;\n# Iterate to create ter
ms from 1 to MaximumTerm\nfor i from 1 to MaximumTerm do w := i;\n\n  \+
 # Definition of the Lamda=0 terms for Schroder's Series 'A'\n   # Equ
ation 55, Page 30.\n   A(0, w) := simplify(sum('(-1)^(a-1)*y^(a-1)*dif
f(y, x$a)*y^(w-a+1)*A(0, w-a)/a!', 'a'=1..w)/y^(w+1));\n\n   # Definit
ion of the Lamda=0 terms for Schroder's Series 'B'\n   # Equation 55, \+
Page 30.\n   B(0, w) := simplify(sum('(-1)^(a)*y^(a)*diff(y, x$(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) := simplify(x^l*y*A(0, 0)/y);\n\n   # Definition of the Ome
ga=0 terms for Schroder'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 Schroder's Series 'A'\n   # Equation 70, Page 40.\n   \+
A(l, w) := simplify(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, Omeg
a 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) Page 39.\n   xprimeA := x - simplify(A(l, w-1)/A(l
, w));\n\n   # Definition of the Algorithm based on B(Lamda, Omega) Pa
ge 39.\n   xprimeB := x - simplify(B(l, w-1)/B(l, w)); C(xprimeB, opti
mized);\nod;\nl:= 0; \n C(xprimeB, optimized);\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$\"\"!F'*()\"\"#%\"xG\"\"\"-%#
lnG6#F*F,,&F)F,%\"cG!\"\"F2" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%,Maxi
mumTermG\"#5" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"wG\"\"\"" }}{PARA 
11 "" 1 "" {XPPMATH 20 "6#>-%\"AG6$\"\"!\"\"\"*()\"\"#%\"xGF(-%#lnG6#F
+F(,&F*!\"\"%\"cGF(!\"#" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>-%\"BG6$\"
\"!\"\"\"*(-%#lnG6#\"\"#F-,0)\"#;%\"xGF(*&)\"\")F1F(%\"cGF(!\"#*&)\"\"
%F1F()F-,$F1F-F(!\"\"*()F-,&F(F(F1F(F(F8F(F5F(F(*&)F-,$F1\"\"$F(F5F(F(
*&)F-,&F(F(F1F-F(F5F-F<*&)F-F1F(F5FCF(F(,&FHF<F5F(!\"%" }}{PARA 11 "" 
1 "" {XPPMATH 20 "6#>-%\"AG6$\"\"!F',$*$,&)\"\"#%\"xG!\"\"%\"cG\"\"\"F
.F." }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>-%\"BG6$\"\"!F',$*()\"\"#%\"xG
\"\"\"-%#lnG6#F+F-,&F*!\"\"%\"cGF-F2F2" }}{PARA 8 "" 1 "" {TEXT -1 64 
"Error, (in sum) the argument to factorial should be non-negative" }}
{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"lG\"\"!" }}{PARA 6 "" 1 "" {TEXT 
-1 704 "      t1 = log(2.0);\n      t2 = x*t1;\n      t3 = pow(4.0,2.0
*x);\n      t4 = c*c;\n      t5 = t4*t4;\n      t8 = pow(8.0,x);\n    \+
  t9 = t5*c;\n      t12 = 1.0+x;\n      t13 = pow(2.0,t12);\n      t15
 = t4*c;\n      t16 = t5*t15;\n      t19 = pow(128.0,x);\n      t23 = \+
pow(8.0,1.0+2.0*x);\n      t27 = pow(4.0,x);\n      t29 = t5*t4;\n    \+
  t32 = pow(32.0,x);\n      t36 = pow(64.0,x);\n      t41 = t1*t1;\n  \+
    t43 = x*x;\n      t52 = pow(2.0,x);\n      t59 = pow(4.0,t12);\n  \+
    t66 = x+x*(-40.0*t2*t3*t5+30.0*t2*t8*t9+t13*x*t1*t16+2.0*t2*c*t19+
t23*x*\nt1*t4-12.0*t27*x*t1*t29+30.0*t32*x*t1*t15-20.0*t2*t4*t36)/(-5.
0*t41*t32*t15*t43\n+5.0*t41*t9*t8*t43+4.0*t41*t36*t4*t43+t41*t52*t16*t
43-t41*c*t19*t43-t59*t41*t29\n*t43);" }}}{EXCHG {PARA 0 "> " 0 "" 
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