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{SECT 0 {EXCHG {PARA 258 "" 0 "" {TEXT 258 24 "Polynomial Approximatio
n" }}{PARA 258 "" 0 "" {TEXT 259 8 "in Maple" }}{PARA 258 "" 0 "" 
{TEXT 256 30 "polynomial_approximationR4.mws" }}{PARA 259 "" 0 "" 
{TEXT 260 14 "Brian E. Blank" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}}
{EXCHG {PARA 257 "" 0 "" {TEXT -1 84 "Click on a [+]  sign to expand a
 section. Click on a [-] sign to collapse a section." }}}{SECT 1 
{PARA 3 "" 0 "" {TEXT -1 13 " Introduction" }}{PARA 0 "" 0 "" {TEXT 
-1 9 "In this  " }{TEXT 257 5 "MAPLE" }{TEXT -1 3 "   " }{HYPERLNK 17 
"worksheet" 2 "worksheet" "" }{TEXT -1 77 "  we will learn several way
s to approximate the solutions of equations using " }{TEXT 261 6 " MAP
LE" }{TEXT -1 3 ".  " }}}{SECT 0 {PARA 260 "" 0 "" {TEXT -1 11 " Code \+
for  " }{TEXT 306 16 "BernsteinApprox " }}{PARA 0 "" 0 "" {TEXT -1 0 "
" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1173 "BernsteinApprox := pro
c()\noption `Copyright (c) 2000 by Brian E. Blank. \nAll rights to the
 code are reserved. \nHowever, all use of the code including distribut
ion is permitted.`:\nlocal jj, aa, bb, ff, NN, xx:\nif nargs <> 4   th
en \n   ERROR(`BernsteinApprox expects 4 arguments.`);\nelif not type(
args[1],\{name,procedure\})\n   then\n   ERROR(`BernsteinApprox expect
s its first argument to be a name.`);\nelif not type(args[3],range)\n \+
  then\n   ERROR(`BernsteinApprox expects its third argument to be a r
ange.`);\nelif not type(args[2],posint) \nthen\n   ERROR(`BernsteinApp
rox expects its second argument to be a positive integer.`);\nelif not
 type(args[4],algebraic)\nthen\n   ERROR(`BernsteinApprox expects its \+
fourth argument to be an algebraic expression.`);\nelif not type(evalf
(op(1,args[3])),realcons) or not type(evalf(op(2,args[3])),realcons) o
r (evalf(op(1,args[3])) >= evalf(op(2,args[3]))) then \nERROR(`Bernste
inApprox received an improper range\nas its third argument.`);\n else
\nff := args[1]:\nNN := args[2]: \naa := op(1,args[3]):\nbb := op(2,ar
gs[3]):\nxx := (args[4]-aa)/(bb-aa):\nRETURN( sum( ff(aa+(bb-aa)*jj/NN
)*binomial(NN,jj)*xx^jj*(1-xx)^(NN-jj) ,jj=0..NN ));\nfi:\nend: " }}}
{PARA 0 "" 0 "" {TEXT -1 0 "" }}}{SECT 0 {PARA 3 "" 0 "" {TEXT -1 17 "
 Description of  " }{TEXT 305 15 "BernsteinApprox" }}{SECT 1 {PARA 0 "
" 0 "" {TEXT 26 10 "Function: " }{TEXT 307 1 " " }{TEXT 347 15 "Bernst
einApprox" }{TEXT -1 57 " - returns the Bernstein uniform approximatin
g polynomial" }}{PARA 0 "" 0 "usage" {TEXT 26 17 "Calling Sequence:" }
{TEXT -1 1 "\n" }{TEXT 317 33 "      BernsteinApprox(f, a, b, N)" }}
{PARA 0 "" 0 "" {TEXT 26 11 "Parameters:" }{TEXT -1 3 "\n  " }{TEXT 
308 1 " " }{TEXT 310 1 "f" }{TEXT 309 3 " - " }{TEXT -1 48 "a function
 name or procedure, or an equation\n   " }{TEXT 312 1 "a" }{TEXT 311 
3 " - " }{TEXT -1 17 "a real number\n   " }{TEXT 314 1 "b" }{TEXT 313 
3 " - " }{TEXT -1 17 "a real number\n   " }{TEXT 315 1 "N" }{TEXT 316 
3 " - " }{TEXT -1 18 "a positive integer" }}}{SECT 0 {PARA 4 "" 0 "" 
{TEXT 26 15 "Description of " }{TEXT 318 16 " BernsteinApprox" }}
{PARA 15 "" 0 "" {TEXT -1 8 "The call" }{TEXT 321 1 " " }{TEXT 323 30 
"BernsteinApprox(f, N, a..b, x)" }{TEXT -1 8 ", where " }{TEXT 319 1 "
f" }{TEXT -1 43 " is a real function of one variable, where " }{TEXT 
320 4 "a..b" }{TEXT -1 28 "  is an interval, and where " }{TEXT 322 1 
"N" }{TEXT -1 43 "  is a positive integer returns the degree " }{TEXT 
326 1 "N" }{TEXT -1 40 " Bernstein polynomial approximation of  " }
{TEXT 324 1 "f" }{TEXT -1 17 " on the interval " }{TEXT 325 4 "a..b" }
{TEXT -1 15 ".evaluated at  " }{TEXT 345 1 "x" }{TEXT -1 1 "." }}
{PARA 15 "" 0 "" {TEXT -1 5 "     " }{XPPEDIT 19 1 "Sum( f(a+(b-a)*k/N
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{SECT 0 {PARA 4 "" 0 "" {TEXT -1 8 "Examples" }}{PARA 0 "" 0 "" {TEXT 
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{MPLTEXT 1 0 133 "plot([sin(x),\nBernsteinApprox(sin, 10, 0..2*Pi, x),
\nBernsteinApprox(sin, 50, 0..2*Pi, x)], x = 0..2*Pi,\ncolor = [BLACK,
 PLUM, BLUE]);" }}{PARA 13 "" 1 "" {INLPLOT "6'-%'CURVESG6$7en7$\"\"!F
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0 "> " 0 "" {MPLTEXT 1 0 211 "plot([cos(2*x^2),\nBernsteinApprox(x -> \+
cos(2*x^2), 10, 0..3, x),\nBernsteinApprox(x -> cos(2*x^2), 100, 0..3,
 x),\nBernsteinApprox(x -> cos(2*x^2), 200, 0..3, x)], x = 0..3,\ncolo
r = [BLACK, BLUE, MAGENTA, WHEAT]);" }}{PARA 13 "" 1 "" {INLPLOT "6(-%
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}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}
{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 
0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}}{SECT 1 
{PARA 3 "" 0 "" {TEXT -1 33 " Copyright and Author Information" }}
{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 262 "" 0 "" {TEXT -1 63 "polynom
ial_approximationR4.mws     A Maple Release 4 worksheet." }}{PARA 263 
"" 0 "" {TEXT -1 0 "" }}{PARA 264 "" 0 "" {TEXT -1 24 "Author:  Brian \+
E. Blank " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 265 "" 0 "" {TEXT 
-1 40 "Date:  16  February 2001 (Last revision)" }}{PARA 0 "" 0 "" 
{TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 51 "This document may not be \+
distributed by any medium," }}{PARA 0 "" 0 "" {TEXT -1 55 "including p
rint, disk, and electronic transfer, without" }}{PARA 0 "" 0 "" {TEXT 
-1 39 "prior written permission of the author." }}{PARA 0 "" 0 "" 
{TEXT -1 0 "" }}{PARA 266 "" 0 "" {TEXT -1 49 "For more information,  \+
please contact the author:" }}{PARA 267 "" 0 "" {TEXT -1 4 "    " }}
{PARA 267 "" 0 "" {TEXT -1 32 "     Department of Mathematics, " }}
{PARA 0 "" 0 "" {TEXT -1 39 "     Washington University in St. Louis" 
}}{PARA 0 "" 0 "" {TEXT -1 26 "     St. Louis, MO   63130" }}{PARA 0 "
" 0 "" {TEXT -1 3 "   " }}{PARA 0 "" 0 "" {TEXT -1 33 "     Telephone:
    (314) 935-6763" }}{PARA 268 "" 0 "" {TEXT -1 44 "             e-ma
il:    brian@math.wustl.edu" }}{PARA 269 "" 0 "" {TEXT -1 0 "" }}
{PARA 0 "" 0 "" {TEXT -1 56 "Copyright:  \251 2001 Brian E. Blank,  Al
l Rights Reserved." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}}}{MARK "4 2" 0 
}{VIEWOPTS 1 1 0 1 1 1803 }