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{SECT 0 {EXCHG {PARA 256 "" 0 "" {TEXT 257 27 "Some Interpolation Exam
ples" }}{PARA 0 "" 0 "" {TEXT 256 18 "Mth 351 Aug 7 2001" }}{PARA 0 "
" 0 "" {TEXT 258 16 "Bent E. Petersen" }}{PARA 0 "" 0 "" {TEXT 259 38 
"Filename: 351u2001_interp_examples.mws" }}{PARA 0 "" 0 "" {TEXT -1 0 
"" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 
73 "In this worksheet we illustrate some of Maple's interpolation faci
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{MPLTEXT 1 0 8 "restart;" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 0 "" }}
{PARA 0 "" 0 "" {TEXT 260 57 "Example: Cubic Interpolation Spline for \+
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{EXCHG {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 376 "I
n the above examples we see the interpolation polynomial has smaller e
rror than the interpolation spline. The next example will demonstrate \+
this is not always the case. One problem with interpolation polynomial
s is that they are of high degree if there are many nodes. Evaluating \+
a polynomial of high degree runs a serious risk of loss of significanc
e errors due to roundoff." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}}{EXCHG 
{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT 262 61 "Example.
 A function pathological for equispaced interpolation" }}{PARA 0 "" 0 
"" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 
0 "" {TEXT -1 68 "We use a do-loop to build a list of  21  equispaced \+
nodes in  [-1,1]" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 ">
 " 0 "" {MPLTEXT 1 0 67 "a:=-1: XX2:=[]:for k from 1 to 21 do XX2:=[op
(XX2),a];a:=a+1/10;od:" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 0 "" }}
{PARA 0 "" 0 "" {TEXT -1 31 "Let's see what  XX2  looks like" }}{PARA 
0 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 4 "XX
2;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#77!\"\"#!\"*\"#5#!\"%\"\"&#!\"(F
'#!\"$F*#F$\"\"##!\"#F*#F.F'#F$F*#F$F'\"\"!#\"\"\"F'#F8F*#\"\"$F'#F0F*
#F8F0#F;F*#\"\"(F'#\"\"%F*#\"\"*F'F8" }}}{EXCHG {PARA 0 "" 0 "" {TEXT 
-1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 128 "Now here's the standard exampl
e of a function whose interpolation polynomials with equispaced nodes \+
yield very bad approximation" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}}
{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 19 "f:=x->1/(1+20*x^2);" }}
{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"fGR6#%\"xG6\"6$%)operatorG%&arrowG
F(*&\"\"\"F-,&F-F-*&\"#?F-)9$\"\"#F-F-!\"\"F(F(F(" }}}{EXCHG {PARA 0 "
> " 0 "" {MPLTEXT 1 0 16 "YY3:=map(f,XX2);" }}{PARA 11 "" 1 "" 
{XPPMATH 20 "6#>%$YY3G77#\"\"\"\"#@#\"\"&\"#')#F*\"#p#F*\"#a#F*\"#T#F'
\"\"'#F*F(#F*\"#9#F*\"\"*#F*F3F'F9F7F5F4F2F0F.F,F)F&" }}}{EXCHG {PARA 
0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 59 "Let's look at t
he natural cubic spline interpolation first." }}{PARA 0 "" 0 "" {TEXT 
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Y3,x,cubic):" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 83 "plot(ex3,x=
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 expect the interpolation polynomial with Chebyshev nodes to much bett
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0 "> " 0 "" {MPLTEXT 1 0 81 "n:=20: XX3:=[]: for k from 0 to n do XX3:
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{PARA 0 "" 0 "" {TEXT -1 292 "Now if we ask Maple to compute the inter
polation polynomial for the nodes determined by  XX3  and  YY4  we wil
l have a very long wait. Maple will try to do the calculation exactly.
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3gMWC,*yD\\\"F-7$$\"3Mm\"z>w'Q\"z)F1$\"3kJq!fK]RU#F-7$$\"36+]i!*z>W))F
1$\"3WTYr?b7LKF-7$$\"3yK3F>#4q*))F1$\"3=1+O=aXoQF-7$$\"3amm\"zW?)\\*)F
1$\"37_@YbN1$G%F-7$$\"3'GeRA1Ei(*)F1$\"3G!y)[_\"3^R%F-7$$\"3I+Dcw;j-!*
F1$\"3R)HouZV$QWF-7$$\"3,fRs$[Me,*F1$\"3b))*R#Q!3KV%F-7$$\"3i;a)3HP!H!
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PDj$*F1$!3i6eAiFMZ8F-7$$\"3X+]P%y+QT*F1$!3Ox8otDR2DF-7$$\"3Y++voyMk%*F
1$!3#)[!y_+zQX$F-7$$\"3W+]7`\\*[^*F1$!3S7>L!)QWdSF-7$$\"3W++]P?Wl&*F1$
!3'G+1>P(Q(>%F-7$$\"3Q+D\"Gyh(>'*F1$!3vmJDRfbDPF-7$$\"3K+]7G:3u'*F1$!3
u:s?*pVic#F-7$$\"3'3D\"y+9C,(*F1$!3'e!=S@47Y<F-7$$\"3G+vVt7SG(*F1$!3'o
8&pRMnazFao7$$\"3q\\P4Y6cb(*F1$\"3yV2Q?p^kCFao7$$\"3A++v=5s#y*F1$\"3!z
WW`%*)\\B8F-7$$\"3v]iS\"*3))4)*F1$\"3Qo>Zd\\YfBF-7$$\"3;+D1k2/P)*F1$\"
3+CS^)=hWD$F-7$$\"3e\\(=nj+U')*F1$\"3]$G=j$p'3)QF-7$$\"35+]P40O\"*)*F1
$\"3k%[3:(*)yySF-7$$\"3k]7.#Q?&=**F1$\"3[%[#eZ$))4l$F-7$$\"31+voa-oX**
F1$\"3o%o<6SEsN#F-7$$\"3ACc,\">g#f**F1$\"39k2%zOWsH\"F-7$$\"3[\\PMF,%G
(**F1$!3(y*=>,4>-#*F97$$\"3uu=nj+U')**F1$!3iOsm4#3c&=F-7$$\"\"\"F*$!3S
T^P$)HaUSF--%&TITLEG6#QSerror~in~Chebyshev~interpolation~polynomial~fo
r~~f6\"-%'COLOURG6&%$RGBG$F*F*Fbdq$\"*++++\"!\")-%+AXESLABELSG6$Q\"xF]
dqQ!6\"-%*THICKNESSG6#\"\"$-%%VIEWG6$;F(Fecq%(DEFAULTG" 1 2 0 1 10 3 
2 6 1 4 2 1.000000 45.000000 45.000000 0 0 "Curve 1" }}}}{EXCHG {PARA 
0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 149 "Even for this \+
example the cubic spline does a little better than the Chebyshev inter
polation polynomial. Of course, that will not always be the case." }}
{PARA 0 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 0 "" 
}}{PARA 0 "" 0 "" {TEXT 263 38 "Example. A sample natural cubic spline
" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 
0 "" }}{PARA 0 "" 0 "" {TEXT -1 168 "In the examples above I suppresse
d the formulae for the cubic splines because they a very large. To see
 what a cubic spline \"looks like\" let's look at a simple example." }
}{PARA 0 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 
0 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 41 "XX6:=[0,1,2,3,4,5]
: YY6:=[-1,-1,1,0,2,2]:" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 29 "
ex6:=spline(XX6,YY6,x,cubic);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%$ex
6G-%*PIECEWISEG6'7$,(!\"\"\"\"\"*&#\"\"*\"#6F+%\"xGF+F**&#F.F/F+)F0\"
\"$F+F+2F0F+7$,*#\"#@F/F+*&#\"$0\"F/F+F0F+F**&#\"#'*F/F+)F0\"\"#F+F+*&
#\"#BF/F+*$F3F+F+F*2F0FA7$,*#!$(QF/F+*&#\"$2&F/F+F0F+F+*&#\"$5#F/F+*$F
@F+F+F**&#\"#GF/F+F3F+F+2F0F47$,*\"#!*F+*&#\"$q)F/F+F0F+F**&#\"$\\#F/F
+F@F+F+*&#FDF/F+FEF+F*2F0\"\"%7$,*#!%e5F/F+*&#\"$m'F/F+F0F+F+*&#\"$N\"
F/F+FQF+F**&F2F+F3F+F+%*otherwiseG" }}}{EXCHG {PARA 0 "" 0 "" {TEXT 
-1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 104 "To see how it behaves we will \+
plot it together with a piecewise linear function joining the data poi
nts." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "> " 0 "" 
{MPLTEXT 1 0 33 "ex6lin:=spline(XX6,YY6,x,linear);" }}{PARA 11 "" 1 "
" {XPPMATH 20 "6#>%'ex6linG-%*PIECEWISEG6'7$!\"\"2%\"xG\"\"\"7$,&!\"$F
,*&\"\"#F,F+F,F,2F+F17$,&\"\"$F,F+F)2F+F57$,&!\"'F,*&F1F,F+F,F,2F+\"\"
%7$F1%*otherwiseG" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 130 "plot(
[ex6,ex6lin],x=0..5,thickness=3,color=[blue,red], title=\"Natural cubi
c spline compared with piecewise linear interpolation\");" }}{PARA 13 
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FO3=F27$Fj[l$\"3\"HLL$3s?6>F27$$\"3Y\\7`p2_#)RF2$\"3!*)\\i!R:/l>F27$$
\"3*G$e*[$zV4SF2$\"\"#F)7$$\"3L;/E+^NOSF2F[]m7$F_\\lF[]m7$$\"3ym\"zW7@
^6%F2F[]m7$Fd\\lF[]m7$F^]lF[]m7$Fh]lF[]m7$Fb^lF[]m7$F\\_lF[]m7$Fa_lF[]
m7$Ff_lF[]m7$F[`lF[]m7$F``lF[]m-Fe`l6&Fg`lFh`lF(F(-%&TITLEG6#Q\\oNatur
al~cubic~spline~compared~with~piecewise~linear~interpolation6\"-%+AXES
LABELSG6$Q\"xFc^mQ!6\"-%*THICKNESSG6#\"\"$-%%VIEWG6$;F(F``l%(DEFAULTG
" 1 2 0 1 10 3 2 9 1 4 2 1.000000 45.000000 45.000000 0 0 "Curve 1" "C
urve 2" }}}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" 
{TEXT -1 138 "As you can see the spline does not oscillate much betwee
n the data points. Let's try the same comparison for the interpolation
 polynomial." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "> " 
0 "" {MPLTEXT 1 0 23 "ex7:=interp(XX6,YY6,x);" }}{PARA 11 "" 1 "" 
{XPPMATH 20 "6#>%$ex7G,.*$)%\"xG\"\"&\"\"\"#!#6\"#g*&#\"#b\"#CF*)F(\"
\"%F*F**&\"#5F*)F(\"\"$F*!\"\"*&#\"$D%F1F*)F(\"\"#F*F**&#\"$*eF-F*F(F*
F8F*F8" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 133 "plot([ex7,ex6lin
],x=0..5,thickness=3,color=[blue,red],title=\"Interpolation polynomial
 compared with piecewise linear interpolation\");" }}{PARA 13 "" 1 "" 
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