{VERSION 5 0 "IBM INTEL NT" "5.0" } {USTYLETAB {CSTYLE "Maple Input" -1 0 "Courier" 0 1 255 0 0 1 0 1 0 0 1 0 0 0 0 1 }{CSTYLE "2D Math" -1 2 "Times" 0 1 0 0 0 0 0 0 2 0 0 0 0 0 0 1 }{CSTYLE "2D Output" 2 20 "" 0 1 0 0 255 1 0 0 0 0 0 0 0 0 0 1 } {CSTYLE "" -1 256 "" 0 10 0 0 0 0 1 0 0 0 0 0 0 0 0 1 }{CSTYLE "" -1 257 "" 1 20 0 0 0 0 0 0 0 0 0 0 0 0 0 1 }{CSTYLE "" -1 258 "" 1 16 0 0 0 0 0 0 1 0 0 0 0 0 0 1 }{CSTYLE "" -1 259 "" 0 1 0 0 0 0 0 2 0 0 0 0 0 0 0 1 }{CSTYLE "" -1 260 "" 0 1 0 0 0 0 1 1 0 0 0 0 0 0 0 1 } {CSTYLE "" -1 261 "" 0 1 0 0 0 0 0 0 2 0 0 0 0 0 0 0 }{CSTYLE "" -1 262 "" 1 16 0 0 0 0 0 0 1 0 0 0 0 0 0 0 }{CSTYLE "" -1 263 "" 1 14 0 0 0 0 0 0 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 264 "" 0 1 0 0 0 0 0 0 1 0 0 0 0 0 0 0 }{CSTYLE "" -1 265 "" 1 16 0 0 0 0 0 0 1 0 0 0 0 0 0 0 } {CSTYLE "" -1 266 "" 0 1 0 0 0 0 0 0 1 0 0 0 0 0 0 0 }{CSTYLE "" -1 267 "" 0 1 0 0 0 0 1 0 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 268 "" 0 1 0 0 0 0 0 2 0 0 0 0 0 0 0 1 }{CSTYLE "" -1 269 "" 1 16 0 0 0 0 0 0 1 0 0 0 0 0 0 0 }{CSTYLE "" -1 270 "" 1 16 0 0 0 0 0 0 1 0 0 0 0 0 0 0 } {CSTYLE "" -1 271 "" 1 16 0 0 0 0 0 0 1 0 0 0 0 0 0 0 }{PSTYLE "Normal " -1 0 1 {CSTYLE "" -1 -1 "Times" 1 12 0 0 0 1 2 2 2 2 2 2 1 1 1 1 }1 1 0 0 0 0 1 0 1 0 2 2 0 1 }{PSTYLE "Heading 1" -1 3 1 {CSTYLE "" -1 -1 "Times" 1 18 0 0 0 1 2 1 2 2 2 2 1 1 1 1 }1 1 0 0 8 4 1 0 1 0 2 2 0 1 }{PSTYLE "Heading 2" -1 4 1 {CSTYLE "" -1 -1 "Times" 1 14 0 0 0 1 2 1 2 2 2 2 1 1 1 1 }1 1 0 0 8 2 1 0 1 0 2 2 0 1 }{PSTYLE "Warning" -1 7 1 {CSTYLE "" -1 -1 "Courier" 1 10 0 0 255 1 2 2 2 2 2 1 1 1 3 1 } 1 1 0 0 0 0 1 0 1 0 2 2 0 1 }{PSTYLE "Maple Output" -1 11 1 {CSTYLE " " -1 -1 "Times" 1 12 0 0 0 1 2 2 2 2 2 2 1 1 1 1 }3 3 0 0 0 0 1 0 1 0 2 2 0 1 }{PSTYLE "Maple Output" -1 12 1 {CSTYLE "" -1 -1 "Times" 1 12 0 0 0 1 2 2 2 2 2 2 1 1 1 1 }1 3 0 0 0 0 1 0 1 0 2 2 0 1 }{PSTYLE "Map le Plot" -1 13 1 {CSTYLE "" -1 -1 "Times" 1 12 0 0 0 1 2 2 2 2 2 2 1 1 1 1 }3 1 0 0 0 0 1 0 1 0 2 2 0 1 }{PSTYLE "AC - Title" -1 256 1 {CSTYLE "" -1 -1 "Times" 1 18 0 0 0 1 2 1 2 2 2 2 1 1 1 1 }3 1 0 0 12 12 1 0 1 0 1 2 258 1 }{PSTYLE "AC - Author" -1 257 1 {CSTYLE "" -1 -1 "Times" 1 12 0 0 0 1 1 2 2 2 2 2 1 1 1 1 }3 1 0 0 8 8 1 0 1 0 2 2 259 1 }{PSTYLE "AC - Note" -1 258 1 {CSTYLE "" -1 -1 "Times" 1 12 0 0 0 1 2 2 2 2 2 2 1 1 1 1 }3 1 0 0 0 0 1 0 1 0 2 2 0 1 }{PSTYLE "AC - Normal Text" -1 259 1 {CSTYLE "" -1 -1 "Times" 1 12 0 0 0 1 2 2 2 2 2 2 1 1 1 1 }1 1 0 0 0 0 1 0 1 0 2 2 0 1 }{PSTYLE "AC - Section Heading" -1 260 1 {CSTYLE "" -1 -1 "Times" 1 16 0 0 0 1 2 1 2 2 2 2 1 1 1 1 }1 1 0 0 12 0 1 0 1 0 2 2 260 1 }{PSTYLE "AC - Disclaimer" -1 261 1 {CSTYLE "" -1 -1 "Times" 1 9 0 0 0 1 2 2 2 2 2 2 1 1 1 1 }1 1 0 0 12 0 1 0 1 0 2 2 0 1 }{PSTYLE "Normal" -1 262 1 {CSTYLE "" -1 -1 "Times" 1 12 0 0 0 1 2 2 2 2 2 2 1 1 1 1 }3 1 0 0 0 0 1 0 1 0 2 2 0 1 } {PSTYLE "AC - Author" -1 263 1 {CSTYLE "" -1 -1 "Times" 1 12 0 0 0 1 1 2 2 2 2 2 1 1 1 1 }3 1 0 0 4 4 1 0 1 0 2 2 259 1 }} {SECT 0 {PARA 256 "" 0 "" {TEXT 261 35 "The Need for Spline Interpolat ion. " }{TEXT 257 0 "" }}{PARA 263 "" 0 "" {TEXT 256 7 "\251 2003 " } {TEXT -1 127 "Nathan Collier, Autar Kaw, Jai Paul , University of Sout h Florida , kaw@eng.usf.edu , http://numericalmethods.eng.usf.edu/mws \+ ." }}{PARA 259 "" 0 "" {TEXT -1 0 "" }}{PARA 258 "" 0 "" {TEXT -1 224 "NOTE: This worksheet demonstrates the use of Maple to show why we nee d to understand spline interpolation. It illustrates how interpolation using splines can be more accurate when compared to interpolation usi ng polynomials." }}{SECT 0 {PARA 260 "" 0 "" {TEXT 258 12 "Introductio n" }{TEXT -1 0 "" }}{PARA 259 "" 0 "" {TEXT 259 34 "The following exam ple illustrates " }{TEXT -1 81 "the need for spline interpolation as o pposed to using polynomial interpolation. " }{TEXT 268 114 "In the ye ar 1901, Runge tried to explain that higher order interpolation is a b ad idea. He took a simple function " }{OLE 1 4102 1 "[xm]Br=WfoRrB:::w k;nyyI;G:;:j::>:B>N:F:nyyyyy]::yyyyyy::::::::::::::::::::::::::::::::: ::::::::::::::::::::::::::::::::::::::fyyyyya:nYf::G:jy;:::::::::::::: :::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::JcvGY Mt>^:fBWMtNHm=;:::::::n:;`:Z@[::Jj ::::::;K;HYLkNG;J::::::::JCN:ry:>:<:::: ::?J:j;>:wAE:GJ:v:JyK?j?J@j@>:W:YJ:>\\:B:]:wAyA::::::::::: :::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::j:b:B:::g KRn@mKV>@kKT>AUKUF@]kTvEALpFFamsfFaMHJ;NwtnD^fEi<>:AR:=r:Ob<=b >AfSDJ;]\\LVjrc:?VDDJ=q]=n;;R:?B\\]k:nyyYZDjysy?bm?:;JZC:bKi:UTTAeVYuV YeScEBETVeURcUTYeU;sFWCF;B=BKaDBETV:;rZ::jqnFcmuFF_ml ^EAJyE:=j>r:yU:VZ:^=>:EJ:F[^>>vDN\\:B:;xyyQVy yyyYJeZ:>u=j?^;UTRcETcTX[US:oi:;JBB:Jf:;j CDZ<>lD^=l;J:@[C:>Z:: ::::::kJ;@:NZ:>Z:vYxY:>Z::::::J<<:;t:B:FZ:vCJ:=b:KfF@CJ:f?=J>JSdJ@IJyaj:jN`Q>JSJAXJ;>:PM>ZDJEPj:jRPM>JSJUMj:JNPM> JSng;FaAF:;jR:_KjDJFEj:>:MSGK:_;KT:=J:>?sJ:VY;sy>Z:JBC:JS>v>:<:Uk:^:>X?B:K:_c JS>f<^v;F:;Jv:_;?F:=:GUGs:qAB:>L;Z<>Z<>ZJVdscRYEUXQZB:IbJS>f<>h=FZ:>:uSGO:G;Ojysy=:;JHjw ?:Uk:^Z:JrOZ:V:< J::::nc;ocC[:::::::::::::::::::::::::::::::::::::::::::::::::::::::::: ::::::::3:" }{TEXT -1 207 " and chose equidistantly spaced data points to interpolate this function. The same function is used in the follow ing example and is interpolated using polynomial interpolation and cub ic spline interpolation." }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 8 "r estart;" }}}{PARA 7 "" 1 "" {TEXT -1 80 "Warning, the protected names \+ norm and trace have been redefined and unprotected\n" }}}{PARA 260 "" 0 "" {TEXT -1 0 "" }}{SECT 0 {PARA 3 "" 0 "" {TEXT 262 17 "Section I : Data." }}{EXCHG {PARA 0 "" 0 "" {TEXT -1 127 "The points chosen in th is example are 9 equidistant points in [-1,1]. They are [-1, -0.75, -0 .5, -0.25, 0, 0.25, 0.5, 0.75, 1]." }}}}{PARA 3 "" 0 "" {TEXT 263 26 " Plotting Runge's Function:" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 24 "fRunge:=x->1/(1+25*x^2);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%'fRu ngeGf*6#%\"xG6\"6$%)operatorG%&arrowGF(*&\"\"\"F-,&F-F-*&\"#DF-)9$\"\" #F-F-!\"\"F(F(F(" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 64 "plot(fR unge,-1..1,-0.5..1,thickness=4,title=\"Runge's function\");" }}{PARA 13 "" 1 "" {GLPLOT2D 428 307 307 {PLOTDATA 2 "6'-%'CURVESG6$7ao7$$!\" \"\"\"!$\"3QYQ:YQ:YQ!#>7$$!3ommm;p0k&*!#=$\"3G\"\\24&eu*=%F-7$$!3wKL$3 X&F-7$$!3\"QLL3i.9!zF1$\"3DPXyk$y6-'F-7$$!3\"omm T!R=0vF1$\"3oThr]RWImF-7$$!3u****\\P8#\\4(F1$\"3%RPydGV8O(F-7$$!3+nm;/ siqmF1$\"3=)o;ES()yC)F-7$$!3[++](y$pZiF1$\"3/368)oX]H*F-7$$!33LLL$yaE \"eF1$\"3alN'zNm&e5F17$$!3hmmm\">s%HaF1$\"3c8976Gx%>\"F17$$!3Q+++]$*4) *\\F1$\"3C'*>*3w9-Q\"F17$$!39+++]_&\\c%F1$\"3lyZ%*)o#Q5;F17$$!31+++]1a ZTF1$\"37***3b:1m)=F17$$!3umm;/#)[oPF1$\"39#)GKr1i(>#F17$$!3hLLL$=exJ$ F1$\"3K;lyg0LlEF17$$!3*RLLLtIf$HF1$\"3])*zu],lpJF17$$!3]++]PYx\"\\#F1$ \"3sWl%Q&H#F1$\"3f&oE[qcfJ%F17$$!3EMLLL7i)4#F1$\"3 ]Td%Rs=&fZF17$$!3#pm;aVXH)=F1$\"3$*>y\"yY%=,`F17$$!3c****\\P'psm\"F1$ \"3+.wmc%*))**eF17$$!3s*****\\F&*=Y\"F1$\"3dt')>^ipcgb*F17$$\"3L`mmmJ+IiF-$\"3RP59#[,b6*F17$$ \"3s*)***\\PQ#\\\")F-$\"3yVTRoQ9w&)F17$$\"3ilm\"z\\1A-\"F1$\"3W(>+iv*y GzF17$$\"3GKLLe\"*[H7F1$\"3'H:E_JjtD(F17$$\"3ylm;HCjV9F1$\"3w9@ga8aulF 17$$\"3I*******pvxl\"F1$\"3EnqC)y)[FfF17$$\"3g)***\\7JFn=F1$\"3!4Jjzd, GM&F17$$\"3#z****\\_qn2#F1$\"3`hP(oka<\"[F17$$\"3=)**\\P/q%zAF1$\"3oS/ g()\\s\\VF17$$\"3U)***\\i&p@[#F1$\"38e\"[=?cl$RF17$$\"3B)****\\2'HKHF1 $\"3=hqHij,vJF17$$\"3ElmmmZvOLF1$\"3<)*Qq#=nIk#F17$$\"3i******\\2goPF1 $\"3%*Gocd#=v>#F17$$\"3UKL$eR<*fTF1$\"3Q#)=)eB,v(=F17$$\"3m******\\)Hx e%F1$\"3'Q))Qpt!)pf\"F17$$\"3ckm;H!o-*\\F1$\"37N^*o:]RQ\"F17$$\"3y)*** \\7k.6aF1$\"3aR4Y9y%>?\"F17$$\"3#emmmT9C#eF1$\"31\\`8zZRb5F17$$\"33*** *\\i!*3`iF1$\"3%*)y\"f(o+0G*F-7$$\"3%QLLL$*zym'F1$\"3/m'e9@CTD)F-7$$\" 3wKLL3N1#4(F1$\"3'zd`$>+%oO(F-7$$\"3Nmm;HYt7vF1$\"3D%oGOI/!=mF-7$$\"3Y *******p(G**yF1$\"3[z=$395U-'F-7$$\"3]mmmT6KU$)F1$\"3\"f%G`9>?NaF-7$$ \"3fKLLLbdQ()F1$\"3q&pK\\TLu(\\F-7$$\"3[++]i`1h\"*F1$\"3Aq5l\"oD$\\XF- 7$$\"3W++]P?Wl&*F1$\"3YW2DpLe)=%F-7$$\"\"\"F*F+-%'COLOURG6&%$RGBG$\"#5 F)$F*F*Fcal-%&TITLEG6#Q1Runge's~function6\"-%+AXESLABELSG6$Q!FhalF\\bl -%*THICKNESSG6#\"\"%-%%VIEWG6$;F(F[al;$!\"&F)F[al" 1 2 0 1 10 4 2 9 1 4 2 1.000000 45.000000 45.000000 0 0 "Curve 1" }}}}{SECT 0 {PARA 3 "" 0 "" {TEXT 269 37 "Section II: Polynomial Interpolation." }}{PARA 0 " " 0 "" {TEXT -1 387 "The function is interpolated using 9 equidistant \+ data points in [-1,1] to obtain an 8th order polynomial. By looking \+ at the plot of the original function and the 8th order polynomial, you can see that the polynomial interpolation does not accurately represe nt the function. One may think that choosing more points would help i n alleviating this problem, but in fact it makes it worse." }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 171 "poly_fn:=interp([-1,-0.75,-0.5,-0. 25,0,0.25,0.5,0.75,1],[fRunge(-1),fRunge(-0.75),fRunge(-0.5),fRunge(-0 .25),fRunge(0),fRunge(0.25),fRunge(0.5),fRunge(0.75),fRunge(1)],t);" } }{PARA 12 "" 1 "" {XPPMATH 20 "6#>%(poly_fnG,4*&$\"+V+$*o`!\")\"\"\")% \"tG\"\")F*F**&$\"#8F)F*)F,\"\"(F*!\"\"$\"+++++5!\"*F**&$\"+/,:G5!\"(F *)F,\"\"'F*F3*&$\"#;!#5F*F,F*F**&$\"#>F)F*)F,\"\"&F*F**&$\"+bMI?8F)F*) F,\"\"#F*F3*&$\"+61sOhF)F*)F,\"\"%F*F**&$\"\"$F)F*)F,FRF*F3" }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 174 "poly_fn:=t->interp([-1,-0.7 5,-0.5,-0.25,0,0.25,0.5,0.75,1],[fRunge(-1),fRunge(-0.75),fRunge(-0.5) ,fRunge(-0.25),fRunge(0),fRunge(0.25),fRunge(0.5),fRunge(0.75),fRunge( 1)],t):" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 116 "plot([fRunge,po ly_fn],-1..1,-1..1,thickness=4,color=[red,green],legend=[\"Runge's Fun ction\",\"8th order Polynomial\"]);" }}{PARA 13 "" 1 "" {GLPLOT2D 502 402 402 {PLOTDATA 2 "6'-%'CURVESG6%7ao7$$!\"\"\"\"!$\"3QYQ:YQ:YQ!#>7$$ !3ommm;p0k&*!#=$\"3G\"\\24&eu*=%F-7$$!3wKL$3X&F- 7$$!3\"QLL3i.9!zF1$\"3DPXyk$y6-'F-7$$!3\"ommT!R=0vF1$\"3oThr]RWImF-7$$ !3u****\\P8#\\4(F1$\"3%RPydGV8O(F-7$$!3+nm;/siqmF1$\"3=)o;ES()yC)F-7$$ !3[++](y$pZiF1$\"3/368)oX]H*F-7$$!33LLL$yaE\"eF1$\"3alN'zNm&e5F17$$!3h mmm\">s%HaF1$\"3c8976Gx%>\"F17$$!3Q+++]$*4)*\\F1$\"3C'*>*3w9-Q\"F17$$! 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" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 189 "Can you repeat the example by choosing 20 equidistant data points in [-1,1] and compare the results obtained from polynomia l interpolation and spline interpolation for a value of x = -0.45?" }} }{PARA 4 "" 0 "" {TEXT 266 11 "References:" }}{PARA 0 "" 0 "" {TEXT -1 4 "[1] " }{TEXT 267 133 "Autar Kaw, Holistic Numerical Methods Inst itute, See http://numericalmethods.eng.usf.edu/mws/ind/05inp/mws_ind_i np_spe_needspline.pdf" }}{PARA 261 "" 0 "" {TEXT -1 0 "" }{TEXT 260 11 "Disclaimer:" }{TEXT -1 248 " While every effort has been made to v alidate the solutions in this worksheet, University of South Florida a nd the contributors are not responsible for any errors contained and a re not liable for any damages resulting from the use of this material. " }}}{MARK "1 1" 88 }{VIEWOPTS 1 1 0 1 1 1803 1 1 1 1 }{PAGENUMBERS 0 1 2 33 1 1 }