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" }{TEXT 257 0 "" }} {PARA 263 "" 0 "" {TEXT 256 7 "\251 2003 " }{TEXT -1 127 "Nathan Colli er, Autar Kaw, Jai Paul , University of South Florida , kaw@eng.usf.ed u , http://numericalmethods.eng.usf.edu/mws ." }}{PARA 259 "" 0 "" {TEXT -1 0 "" }}{PARA 258 "" 0 "" {TEXT -1 153 "NOTE: This worksheet d emonstrates the use of Maple to show how the choice of points for inte rpolation affects the polynomial interpolation of a function." }} {SECT 0 {PARA 260 "" 0 "" {TEXT 258 12 "Introduction" }{TEXT -1 0 "" } }{PARA 259 "" 0 "" {TEXT 259 105 "The following example illustrates th e difference in interpolation curves due to the selection of points. \+ " }{TEXT -1 116 "In 1901, Carl Runge published his work on dangers of \+ higher order interpolation. He took a simple looking function, " } {OLE 1 4102 1 "[xm]Br=WfoRrB:::wk;nyyI;G:;:j::>:B>N:F:nyyyyy]::yyyyyy: :::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::: fyyyyya:nYf::G:jy;:::::::::::::::::::::::::::::::::::::::::::::::::::: :::::::::::::::::::::::::::JcvGYMt>^:fBWMtNHm=;:::::::n:;`:Z@[::JRYLxiOMiY[Aj;J:M:<:=ja^GE=;:::::::::N;?R:yyyyyyA:yayA:<::: :::JDJ:j\\FHemj^HMmqnG;KaFFJufF>::::::;K;HYLkNG>::::::::NZ:nF>:nY>;V:;Jyk:[B::>tk^AFXDJ;a\\< njwr:WJ:@J;\\rRF:wyyAbR<:TnEj``pkDqqHqqTPt:yayQZ:J:jxi:SF;ny:EJ:F[^>>vDNryyYkwyyyy;g<>:oi:;JBB:Jf:;jCDZ<>lD^=l;J:Z:>:::::::::J?B:yay=J:B::::::nyyyYB::::::::: ::::yay=J:B:::::::::::::::::::jysy:>:<::::::::@J;DRNNZBH@?b[QK;\\RSNZB h@_rZVZIj?@k:DJ>E]:bj:@j:B:yI:xI;jy=jp@PqVdb`xq_eb`xqG:;J:> @CJ:f?=J>>:_c:MSGK:_;KT:=J:>?sJ:VY;syB:>L:[K<<:U K;^:>x;F:K:_KjEjk`j:Zo^=VY;><:[n:>:Cb::::Jhj:J:>@<:Uk:^:>X?B:K:_cJS>f<^v;F:;Jv:_;?F:=:GUGs:qAB:>L;Z<>Z<>ZJVdscRYEUXQZB:;T@CJ:f??JJS>f<>h=FZ:jXPM?JMJ?vYyyy=:;JHjw? :C:;j:<:J:>@C:Uk:^Z:JrOZ:V: \+ " 0 "" {MPLTEXT 1 0 21 "restart;with(linalg):" }}{PARA 7 "" 1 "" {TEXT -1 80 "Warning, the protected names norm and trace have been red efined 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 -1 168 "Let us first chose points which are equidistantly placed in [-1,1], and interpolate it. We will chose 11 points, [-1, - 0.8, -0.6, -0.4, -0.2, 0, 0.2, 0.4, 0.6, 0.8, 1]." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 29 "Runge's Function is given by:" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 24 "fRunge:=x->1/(1+25* x^2);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%'fRungeGf*6#%\"xG6\"6$%)ope ratorG%&arrowGF(*&\"\"\"F-,&F-F-*&\"#DF-)9$\"\"#F-F-!\"\"F(F(F(" }}}} {PARA 3 "" 0 "" {TEXT 263 18 "Plotting the data:" }}{EXCHG {PARA 0 "> \+ " 0 "" {MPLTEXT 1 0 62 "plot(fRunge,-1..1,-1..1,thickness=2,title=\"Ru nge's function\");" }}{PARA 13 "" 1 "" {GLPLOT2D 620 385 385 {PLOTDATA 2 "6'-%'CURVESG6$7ao7$$!\"\"\"\"!$\"3QYQ:YQ:YQ!#>7$$!3ommm;p 0k&*!#=$\"3G\"\\24&eu*=%F-7$$!3wKL$3X&F-7$$!3\"Q LL3i.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$$!33LLL$yaE\"eF1$\"3alN'zNm&e5F17$$!3hmmm\">s %HaF1$\"3c8976Gx%>\"F17$$!3Q+++]$*4)*\\F1$\"3C'*>*3w9-Q\"F17$$!39+++]_ &\\c%F1$\"3lyZ%*)o#Q5;F17$$!31+++]1aZTF1$\"37***3b:1m)=F17$$!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$$\"3s*)***\\PQ#\\\")F-$\"3yVTRoQ9w&)F1 7$$\"3ilm\"z\\1A-\"F1$\"3W(>+iv*yGzF17$$\"3GKLLe\"*[H7F1$\"3'H:E_JjtD( F17$$\"3ylm;HCjV9F1$\"3w9@ga8aulF17$$\"3I*******pvxl\"F1$\"3EnqC)y)[Ff F17$$\"3g)***\\7JFn=F1$\"3!4Jjzd,GM&F17$$\"3#z****\\_qn2#F1$\"3`hP(oka <\"[F17$$\"3=)**\\P/q%zAF1$\"3oS/g()\\s\\VF17$$\"3U)***\\i&p@[#F1$\"38 e\"[=?cl$RF17$$\"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******\\)Hxe%F1$\"3'Q))Qpt!)pf\"F17$$\"3ckm;H!o -*\\F1$\"37N^*o:]RQ\"F17$$\"3y)***\\7k.6aF1$\"3aR4Y9y%>?\"F17$$\"3#emm mT9C#eF1$\"31\\`8zZRb5F17$$\"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$$\"3 Nmm;HYt7vF1$\"3D%oGOI/!=mF-7$$\"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$\"#5F)$F*F*Fcal-%+AXESLABELSG6$Q!6\"Fgal -%*THICKNESSG6#\"\"#-%&TITLEG6#Q1Runge's~functionFhal-%%VIEWG6$;F(F[al Fdbl" 1 2 0 1 10 2 2 9 1 4 2 1.000000 45.000000 45.000000 0 0 "Curve 1 " }}}}{PARA 3 "" 0 "" {TEXT -1 0 "" }}{SECT 0 {PARA 3 "" 0 "" {TEXT -1 0 "" }{TEXT 264 70 "Section II: Polynomial Interpolation with equid istantly spaced points." }}{PARA 0 "" 0 "" {TEXT -1 226 "Let us now in terpolate Runge's function using polynomial interpolation in [-1,1], c hoosing 11 equidistantly spaced data points, [-1, -0.8, -0.6, -0.4, -0 .2, 0, 0.2, 0.4, 0.6, 0.8, 1]. This will give us a 10th order polynomi al." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 207 "eq_poly:=interp([-1, -0.8, -0.6, -0.4, -0.2, 0, 0.2, 0.4, 0.6, 0.8, 1],[fRunge(-1),fRunge(-0.8),fRunge(-0.6),fRunge(-0.4), fRunge(-0.2),fRunge(0),fRunge(0.2),fRunge(0.4),fRunge(0.6),fRunge(0.8) ,fRunge(1)],t);" }}{PARA 12 "" 1 "" {XPPMATH 20 "6#>%(eq_polyG,8*&$\"+ RuT4A!\"(\"\"\")%\"tG\"#5F*!\"\"*&$\"#7F)F*)F,\"\"*F*F.$\"+++++5!\"*F* *&$\"+a]4\\\\F)F*)F,\"\")F*F**&$\"#?!#5F*F,F*F.*&$\"#AF)F*)F,\"\"(F*F* *&$\"+j._&o\"!\")F*)F,\"\"#F*F.*&$\"+Z#QV\"QF)F*)F,\"\"'F*F.*&$\"#6FHF *)F,\"\"$F*F**&$FDF)F*)F,\"\"&F*F.*&$\"+)G(fL7F)F*)F,\"\"%F*F*" }}} {EXCHG {PARA 0 "" 0 "" {TEXT -1 1 "." }}{PARA 0 "" 0 "" {TEXT -1 56 "W e will now plot this function against Runge's function." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 210 "eq_po ly:=t->interp([-1, -0.8, -0.6, -0.4, -0.2, 0, 0.2, 0.4, 0.6, 0.8, 1],[ fRunge(-1),fRunge(-0.8),fRunge(-0.6),fRunge(-0.4),fRunge(-0.2),fRunge( 0),fRunge(0.2),fRunge(0.4),fRunge(0.6),fRunge(0.8),fRunge(1)],t):" }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 235 "plot([fRunge,eq_poly],-1..1 ,-0.5..1,thickness=2,color=[red,green],title=\"Plot of Runge's Functio n and Polynomial interpolated with equidistantly spaced points\",legen d=[\"Runge's function\",\"Polynomial with equidistantly spaced points \"]);" }}{PARA 13 "" 1 "" {GLPLOT2D 747 423 423 {PLOTDATA 2 "6(-%'CURV ESG6%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%RPydGV8 O(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;F 17$$!31+++]1aZTF1$\"37***3b:1m)=F17$$!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$$!3EML LL7i)4#F1$\"3]Td%Rs=&fZF17$$!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$$\"3ilm\"z\\1A-\"F1$ \"3W(>+iv*yGzF17$$\"3GKLLe\"*[H7F1$\"3'H:E_JjtD(F17$$\"3ylm;HCjV9F1$\" 3w9@ga8aulF17$$\"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$$\"3m ******\\)Hxe%F1$\"3'Q))Qpt!)pf\"F17$$\"3ckm;H!o-*\\F1$\"37N^*o:]RQ\"F1 7$$\"3y)***\\7k.6aF1$\"3aR4Y9y%>?\"F17$$\"3#emmmT9C#eF1$\"31\\`8zZRb5F 17$$\"33****\\i!*3`iF1$\"3%*)y\"f(o+0G*F-7$$\"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&% $RGBG$\"*++++\"!\")$F*F*Fdal-%'LEGENDG6#Q1Runge's~function6\"-F$6%7ct7 $F)$\"+YQ:YQ!#67$$!+dNvs**!#5$\"+J$*HeCFdbl7$$!+9r]X**Fdbl$\"+ZrByVFdb l7$$!+r1E=**Fdbl$\"+v?H^hFdbl7$$!+HU,\"*)*Fdbl$\"+i%3Ty(Fdbl7$$!+'ynP' )*Fdbl$\"+Cs6$G*Fdbl7$$!+V8_O)*Fdbl$\"+hSXl5!\"*7$$!+,\\F4)*Fdbl$\"+X( R/>\"F`dl7$$!+e%G?y*Fdbl$\"+=3&QI\"F`dl7$$!+tb`F(*Fdbl$\"+y3=)\\\"F`dl 7$$!+(oUIn*Fdbl$\"+r;s_;F`dl7$$!+-)\\&='*Fdbl$\"+#3j9x\"F`dl7$$!+F`dl7$$!+!)*G#p%*Fdbl$\"+% GsD%>F`dl7$$!+'*>_X%*Fdbl$\"+by4_>F`dl7$$!+7]\"=U*Fdbl$\"+RRZd>F`dl7$$ !+G!3\")R*Fdbl$\"+w#H*e>F`dl7$$!+W5Su$*Fdbl$\"+D\\oc>F`dl7$$!+3Jdz#*Fd bl$\"+#zt[\">F`dl7$$!+r^u%=*Fdbl$\"+P-oH=F`dl7$$!+S75y!*Fdbl$\"+y[Z&p \"F`dl7$$!+2tXr*)Fdbl$\"+e$*=L:F`dl7$$!+vL\"['))Fdbl$\"+6Lv`8F`dl7$$!+ U%p\"e()Fdbl$\"+V'eh;\"F`dl7$$!+0'=3l)Fdbl$\"+/b\\k(*Fdbl7$$!+nxYV&)Fd bl$\"+T\\XA)Fdbl$\"+i?\">3$Fdbl7$$!+c[3:\")Fdbl$\"+O#epy\"Fdbl7$$! +QUC3!)Fdbl$\"+t.`smF`bl7$$!+@OS,zFdbl$!+,5K\\FF`bl7$$!+iPH.xFdbl$!+yg sj:Fdbl7$$!+/R=0vFdbl$!+\\:.-BFdbl7$$!+$3,RX(Fdbl$!+\"f4DT#Fdbl7$$!+i# =ES(Fdbl$!+Gfw#\\#Fdbl7$$!+TaL^tFdbl$!+J2TWDFdbl7$$!+?E0+tFdbl$!+&z%4p DFdbl7$$!+*zp([sFdbl$!+Gz\\oDFdbl7$$!+yp[(>(Fdbl$!+h\"=Va#Fdbl7$$!+dT? 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To do this, first the half interval from -1 to 1 is divided to give 'n' poi nts in [-1,1] ( 'n' needs to be odd). Then the number of divisions is (n-1), and the number of divisions are divided equally, that is (n-1) /2, between [-1,0] and [0,1]. For points from -1 to 0, starting with \+ -1 as the first point, and assuming the next points are -1+ " }{TEXT 271 1 "l" }{TEXT -1 6 ", -1+2" }{TEXT 275 3 "l, " }{TEXT -1 4 "-1+4" } {TEXT 276 10 "l, ......." }{TEXT -1 2 ",0" }{TEXT 277 2 ", " }{TEXT -1 51 "where the distance between -1 and the next point is" }{TEXT 279 4 " l, " }{TEXT -1 100 "and the distance between consecutive point s from -1 to 0 gets doubled after each point, the value of" }{TEXT 280 3 " l " }{TEXT -1 11 "is given by" }{TEXT 281 2 " l" }{TEXT -1 18 "=1/[2^\{(n-1)/2\}-1]" }{TEXT 282 1 "." }{TEXT -1 83 " \nThe points fr om 0 to 1 are mirror images about the y-axis of points from -1 to 0." }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 6 "n:=11:" }}{PARA 0 "> " 0 " " {MPLTEXT 1 0 236 "Xb:=matrix(n,1,0):\nd:=0:\nfor i from 2 to ((n-1)/ 2)+1 do\nd:=d+2^(i-2):\nend do:\nl:=1/d:\nXb[1,1]:=-1:\nfor i from 2 t o ((n-1)/2)+1 do\nXb[i,1]:=Xb[i-1,1]+2^(i-2)*l:\nend do:\nfor i from ( n-1)/2+2 to n do\nXb[i,1]:=Xb[i-1,1]+(2^(n-i))*l:\nend do:" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 73 "Yb:=matrix(n,1,0):\nfor i from 1 to n do\nYb[i,1]:=fRunge(Xb[i,1]):\nend do:" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 165 "When x and y data and order is given, the following cons tructs the matrix whose inverse is needed to find coefficients of the \+ polynomial which approximates the data." }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 98 "A:=matrix(n,n,0):\nfor i from 1 to n do\nfor j from 1 to n do\nA[i,j]:=Xb[i,1]^(j-1):\nend do:\nend do:" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 86 "This generates the coefficients for the polynomia l that approximates the x and y data." }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 27 "M:=evalm(inverse(A) &* Yb):" }}}{EXCHG {PARA 0 "" 0 " " {TEXT -1 136 "When given the polynomial order and specific x, this p rocedure uses the above calculated coefficients to calculate the appro ximated f(x)" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 99 "f2:=proc(x) \nlocal i,c,d;\nc:=0:\nfor i 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obtained by interpolation us ing more data points near the end points of -1 and 1." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 341 "plot([fR unge,eq_poly,f2],-1..1,-0.5..1,thickness=2,color=[RED,GREEN,BLUE],titl e=\"Runge's function and Interpolated polynomials with equidistant dat a points and with more data points near the ends of the interval [-1,1 ]\",legend=[\"Runge's function\",\"Polynomial with equidistantly space d data points\",\"Polynomial with more points at the end\"]);" }} {PARA 13 "" 1 "" {GLPLOT2D 789 525 525 {PLOTDATA 2 "6)-%'CURVESG6%7ao7 $$!\"\"\"\"!$\"3QYQ:YQ:YQ!#>7$$!3ommm;p0k&*!#=$\"3G\"\\24&eu*=%F-7$$!3 wKL$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$$!3hmmm\">s%HaF1$\"3c8976Gx%>\"F17$$!3Q+++] $*4)*\\F1$\"3C'*>*3w9-Q\"F17$$!39+++]_&\\c%F1$\"3lyZ%*)o#Q5;F17$$!31++ 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