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{SECT 0 {EXCHG {PARA 256 "" 0 "" {TEXT 257 24 "Polynomial Interpolatio
n" }}{PARA 0 "" 0 "" {TEXT 256 30 "Mth 351 August 5 2002  Maple 6" }}
{PARA 0 "" 0 "" {TEXT 258 16 "Bent E. Petersen" }}{PARA 0 "" 0 "" 
{TEXT 259 42 "Filename: 351u2002_interpolation_polys.mws" }}{PARA 0 "
" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 268 "In this w
orksheet we will construct interpolation polynomials for a function  f
  in an interval for the cases of equispaced, Chebyshev and \"random\"
 nodes. I do a specific example for a simple function  f. I have appen
ded instructions on how to do the same examples in " }{TEXT 272 6 "Mat
lab" }{TEXT 274 1 " " }{TEXT -1 28 "for those of you who prefer " }
{TEXT 273 6 "Matlab" }{TEXT -1 1 "." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }
}{PARA 0 "" 0 "" {TEXT -1 45 "Everything you need for assignment 3 is \+
here." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT 275 32 
"Assignment 3. Due August 12 2002" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}
{PARA 258 "" 0 "" {TEXT 281 192 "Construct interpolation polynomials w
ith 13 (n=12) and with 21 (n=20) nodes for the function  f  given by  \+
f(x) = 1/(1+50*x^2)  on the interval [-1,1]. Try equispaced nodes and \+
Chebyshev nodes" }}{PARA 258 "" 0 "" {TEXT 282 0 "" }}{PARA 259 "" 0 "
" {TEXT 283 40 "- cos( (2*k+1)*Pi/(2*n+2) )   , k = 0..n" }}{PARA 0 "
" 0 "" {TEXT -1 0 "" }}{PARA 258 "" 0 "" {TEXT 284 93 "Try other nodes
 if you like. Look at the error (perhaps plot it) and comment on what \+
you see." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "" 0 "" 
{TEXT -1 160 "We first develop three simple routines to construct the \+
interpolation nodes. We compute the nodes as floating point numbers to
 avoid large symbolic expressions." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}
{PARA 0 "" 0 "" {TEXT 260 6 "Enodes" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }
}{PARA 0 "" 0 "" {TEXT -1 181 "Enodes(n,a,b,d) returns n+1 equispaced \+
nodes for the interval [a,b]. The nodes are evaluated to d decimal dig
its. If d is omitted then d is set equal to the current value of Digit
s." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "> " 0 "" 
{MPLTEXT 1 0 19 "Enodes:=proc(n,a,b)" }}{PARA 0 "> " 0 "" {MPLTEXT 1 
0 16 "  local L, k, d;" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 17 "if nargs \+
< 4 then" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 12 "  d:=Digits;" }}{PARA 
0 "> " 0 "" {MPLTEXT 1 0 4 "else" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 34 
"  d:=args[4]; # optional parameter" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 
3 "fi;" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 39 "L:=[seq(evalf(a+(b-a)*k/n
,d), k=0..n)];" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 4 "end:" }}}{EXCHG 
{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 25 "Here is a
 test of Enodes:" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 ">
 " 0 "" {MPLTEXT 1 0 17 "Enodes(10,3,6,4);" }}{PARA 11 "" 1 "" 
{XPPMATH 20 "6#7-$\"\"$\"\"!$\"%+L!\"$$\"%+OF)$\"%+RF)$\"%+UF)$\"%+XF)
$\"%+[F)$\"%+^F)$\"%+aF)$\"%+dF)$\"\"'F&" }}}{EXCHG {PARA 0 "" 0 "" 
{TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT 261 6 "Cnodes" }}{PARA 0 "" 0 "
" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 171 "Cnodes(n,a,b,d) return
s the n+1 Chebyshev nodes for the interval [a,b] evaluated to d digits
 precision. If d is omitted then d is set equal to the current value o
f Digits." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "> " 0 "
" {MPLTEXT 1 0 19 "Cnodes:=proc(n,a,b)" }}{PARA 0 "> " 0 "" {MPLTEXT 
1 0 16 "  local L, k, d;" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 17 "if narg
s < 4 then" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 12 "  d:=Digits;" }}
{PARA 0 "> " 0 "" {MPLTEXT 1 0 4 "else" }}{PARA 0 "> " 0 "" {MPLTEXT 
1 0 34 "  d:=args[4]; # optional parameter" }}{PARA 0 "> " 0 "" 
{MPLTEXT 1 0 3 "fi;" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 66 "L:=[seq(eval
f(a+((b-a)/2)*(1-cos((2*k+1)*Pi/(2*n+2))),d),k=0..n)];" }}{PARA 0 "> \+
" 0 "" {MPLTEXT 1 0 4 "end:" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 0 "" }
}{PARA 0 "" 0 "" {TEXT -1 25 "Here is a test of Cnodes:" }}{PARA 0 "" 
0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 17 "Cnodes
(10,3,6,4);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#7-$\"%:I!\"$$\"%OJF&$\"
%nLF&$\"%*o$F&$\"%xSF&$\"%+XF&$\"%B\\F&$\"%6`F&$\"%LcF&$\"%keF&$\"%&)f
F&" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT 
262 6 "Rnodes" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" 
{TEXT -1 177 "Rnodes(n,a,b,d] returns n+1 random nodes for the interva
l [a,b]. The nodes are evaluated to d decimal digits. If d is omitted \+
then d is set equal to the current value of Digits." }}{PARA 0 "" 0 "
" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 19 "Rnodes:=p
roc(n,a,b)" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 16 "  local L, k, d;" }}
{PARA 0 "> " 0 "" {MPLTEXT 1 0 17 "if nargs < 4 then" }}{PARA 0 "> " 
0 "" {MPLTEXT 1 0 12 "  d:=Digits;" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 
4 "else" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 34 "  d:=args[4]; # optional
 parameter" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 3 "fi;" }}{PARA 0 "> " 0 
"" {MPLTEXT 1 0 6 "L:=\{\};" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 20 "# Av
oid coincidences" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 22 "while nops(L) <
 n+1 do" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 59 "  L:=\{seq(evalf(a+(b-a)
*(rand()/999999999999),d), k=0..n)\};" }}{PARA 0 "> " 0 "" {MPLTEXT 1 
0 3 "od;" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 17 "L:=sort([op(L)]);" }}
{PARA 0 "> " 0 "" {MPLTEXT 1 0 4 "end:" }}}{EXCHG {PARA 0 "" 0 "" 
{TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 109 "Here is test of Rnodes. \+
Note by setting the random seed we ensure we get the same \"random\" n
umbers each time!" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "
> " 0 "" {MPLTEXT 1 0 31 "randomize(1): Rnodes(10,3,6,4);" }}{PARA 11 
"" 1 "" {XPPMATH 20 "6#7-$\"%'4$!\"$$\"%zPF&$\"%jRF&$\"%JSF&$\"%#G%F&$
\"%BWF&$\"%vYF&$\"%8[F&$\"%p^F&$\"%P_F&$\"%S_F&" }}}{EXCHG {PARA 0 "" 
0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT 263 29 "Nodes for interpola
tion tests" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT 
-1 73 "Here are sets of nodes in the interval  [0,2]  for the experime
nts below." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "> " 0 "
" {MPLTEXT 1 0 21 "XE:=Enodes(12,0,2,6);" }}{PARA 12 "" 1 "" {XPPMATH 
20 "6#>%#XEG7/$\"\"!F'$\"'nm;!\"'$\"'LLLF*$\"'++]F*$\"'nmmF*$\"'LL$)F*
$\"\"\"F'$\"'nm6!\"&$\"'LL8F7$\"'++:F7$F)F7$\"'LL=F7$\"\"#F'" }}}
{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 21 "XC:=Cnodes(12,0,2,6);" }}
{PARA 12 "" 1 "" {XPPMATH 20 "6#>%#XCG7/$\"%\"H(!\"'$\"&%)\\'F($\"';q<
F($\"'xoLF($\"'t_`F($\"'z1wF($\"\"\"\"\"!$\"'KR7!\"&$\"'tk9F8$\"'7j;F8
$\"')H#=F8$\"'-N>F8$\"'r#*>F8" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 
0 39 "randomize(77777): XR:=Rnodes(12,0,2,6);" }}{PARA 12 "" 1 "" 
{XPPMATH 20 "6#>%#XRG7/$\"':y_!\"($\"'pa:!\"'$\"'d,WF+$\"'g%e%F+$\"'tv
[F+$\"'d`aF+$\"'%[)yF+$\"'0#R)F+$\"'*oS\"!\"&$\"'L#[\"F:$\"':*\\\"F:$
\"'a;:F:$\"'9d>F:" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "
" 0 "" {TEXT 264 21 "Interpolation Samples" }}{PARA 0 "" 0 "" {TEXT 
-1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 33 "We will interpolate the functio
n " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "> " 0 "" 
{MPLTEXT 1 0 19 "f:=x->x+3*abs(x-1);" }}{PARA 11 "" 1 "" {XPPMATH 20 "
6#>%\"fGR6#%\"xG6\"6$%)operatorG%&arrowGF(,&9$\"\"\"*&\"\"$F.-%$absG6#
,&F-F.F.!\"\"F.F.F(F(F(" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 0 "" }}
{PARA 0 "" 0 "" {TEXT 265 24 "Equispaced interpolation" }}{PARA 0 "" 
0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 14 "YE:=ma
p(f,XE);" }}{PARA 12 "" 1 "" {XPPMATH 20 "6#>%#YEG7/$\"\"$\"\"!$\"(mmm
#!\"'$\"(MLL#F+$\"(+++#F+$\"(mmm\"F+$\"(MLL\"F+$\"\"\"F($\"'om;!\"&$\"
'KLBF8$\"'++IF8$\"'omOF8$\"'KLVF8$\"\"&F(" }}}{EXCHG {PARA 0 "> " 0 "
" {MPLTEXT 1 0 28 "pE:=sort(interp(XE,YE,x),x);" }}{PARA 12 "" 1 "" 
{XPPMATH 20 "6#>%#pEG,<*$)%\"xG\"#7\"\"\"$!+;T=X6!\"'*&$\"+#)3Au8!\"&F
*)F(\"#6F*F**&$\"+dohmsF1F*)F(\"#5F*!\"\"*&$\"+lq!yA#!\"%F*)F(\"\"*F*F
**&$\"+7<c$Q%F=F*)F(\"\")F*F9*&$\"+P2T(y&F=F*)F(\"\"(F*F**&$\"+)zG]?&F
=F*)F(\"\"'F*F9*&$\"+S7\"*pJF=F*)F(\"\"&F*F**&$\"+!yF=F\"F=F*)F(\"\"%F
*F9*&$\"+v4CsJF1F*)F(\"\"$F*F**&$\"+[Q-sVF-F*)F(\"\"#F*F9*&$\"+g\"GoX#
!\"(F*F(F*F*$Fgn\"\"!F*" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 63 "
plot(f(x)-pE,x=0..2,title=\"Error in equispaced interpolation\");" }}
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1 2 0 1 10 0 2 9 1 4 2 1.000000 45.000000 45.000000 0 0 "Curve 1" }}}}
{EXCHG {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 52 "No
te the errors are quite large in some cases above." }}{PARA 0 "" 0 "" 
{TEXT -1 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT 268 6 "Matlab" }}{PARA 
0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 60 "Here's one way \+
to do the first two examples above in Matlab:" }}{PARA 0 "" 0 "" 
{TEXT -1 1 "\n" }{TEXT 269 22 "Abscissae for plotting" }}{PARA 0 "" 0 
"" {TEXT -1 31 "\n>> Xeval = linspace(0,2,200);\n" }}{PARA 0 "" 0 "" 
{TEXT 280 12 "Common data\n" }{TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT 
-1 24 ">> a = 0, b = 2, n = 12\n" }}{PARA 0 "" 0 "" {TEXT 270 24 "Equi
spaced interpolation" }{TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }
}{PARA 0 "" 0 "" {TEXT -1 154 ">> XE = linspace(a,b,n+1)\n\n>> YE = XE
 + 3*abs(XE-1)\n\n>> pE = polyfit(XE,YE,n)\n\n>> ErrE = Xeval + 3*abs(
Xeval-1) - polyval(pE,Xeval);\n\n>> plot(Xeval,ErrE)\n" }}{PARA 0 "" 
0 "" {TEXT -1 1 "\n" }{TEXT 271 24 "Chebyshev interpolation\n" }{TEXT 
-1 257 "\n>> Xtmp = linspace(1/(2*n+2),(2*n+1)/(2*n+2),n+1)*pi\n\n>> X
C = 1 - cos(Xtmp)\n\n>> YC = XC + 3*abs(XC-1)\n\n>> pC = polyfit(XC,YC
,n)\n\n>> ErrC = Xeval + 3*abs(Xeval-1) - polyval(pC,Xeval);\n\n>> plo
t(Xeval,ErrC)\n\nI'll leave the random interpolation example to you." 
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