{VERSION 6 0 "IBM INTEL NT" "6.0" } {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 Comment" 2 18 "" 0 1 0 0 0 0 0 0 0 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 "Blue Emphasis" -1 256 "Times" 0 0 0 0 255 1 0 1 0 0 0 0 0 0 0 1 }{CSTYLE "Green Emphasis" -1 257 "Times" 1 12 0 128 0 1 0 1 0 0 0 0 0 0 0 1 }{CSTYLE "Maroon Emphasis" -1 258 "Times" 1 12 128 0 128 1 0 1 0 0 0 0 0 0 0 1 }{CSTYLE "Purple Emphasis" -1 259 "Times" 1 12 102 0 230 1 0 1 0 0 0 0 0 0 0 0 }{CSTYLE "Red Emphasis" -1 260 "Times " 1 12 255 0 0 1 0 1 0 0 0 0 0 0 0 0 }{CSTYLE "Dark Red Emphasis" -1 261 "Times" 1 12 128 0 0 1 0 1 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 262 "" 0 1 0 0 0 0 1 0 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 263 "" 0 1 0 0 0 0 1 0 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 264 "" 0 1 0 0 0 0 1 0 0 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 0 128 1 2 1 2 2 2 2 1 1 1 1 }1 1 0 0 8 4 3 0 3 0 2 2 0 1 }{PSTYLE "Heading 2" -1 4 1 {CSTYLE "" -1 -1 "Ti mes" 1 14 128 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 "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 Outpu t" -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 "Maple 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 "Normal" -1 256 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 "" 0 257 1 {CSTYLE "" -1 -1 "" 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 }3 0 0 -1 -1 -1 0 0 0 0 0 0 -1 0 }} {SECT 0 {PARA 3 "" 0 "" {TEXT -1 57 "The accuracy of an approximating \+ interpolating polynomial" }}{PARA 0 "" 0 "" {TEXT -1 37 "by Peter Ston e, Nanaimo, B.C., Canada" }}{PARA 0 "" 0 "" {TEXT -1 19 "Version: 24. 3.2007" }}{PARA 257 "" 0 "" {TEXT -1 1 " " }{GLPLOT2D 512 256 256 {PLOTDATA 2 "6*-%'CURVESG6$7ao7$$!3%**************>\"!#<$\"3&Gq-Fq-Fq# !#>7$$!3/+++IooZ6F*$\"3MULeqnFZHF-7$$!3*)****\\?%p@5\"F*$\"3gZdnSj\"y= $F-7$$!3/+++L.)40\"F*$\"3#z\\mez$z%\\$F-7$$!3g******4$>X***!#=$\"3#Qe? 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Tolme Runge (1856-1927) was the first holder of the chair in applied mathematics at the University of Gottingen, after ha ving been a professor at the technical university in Hanover from 1886 to 1904. The creation of this position was a result of the efforts of Felix Klein, who thereby helped establish applied mathematics as a re cognized part of mathematics. Runge studied in Munich and in Berlin, a nd was especially influenced by Weierstrass. After working on differen tial geometry, algebra and function theory, his extensive interests le d him to problems in physics, geodesy and astronomy, and hence to the \+ problem of applying numerical methods to practical situations." }} {PARA 0 "" 0 "" {TEXT -1 861 "Runge significantly influenced the devel opment of applied mathematics. One of the three theses which he defend ed as part of his dissertation in 1880 in Berlin was entitled \"The va lue of a mathematical discipline depends on its applicability to the n atural sciences.\" He later explained \"It was not the aim of my thesi s to assert that every theorem must have a practical application. I on ly meant to observe that mathematics for its own sake is on the same l evel as chess or other games. They gain in value only through relation ships to natural science. In my opinion, before a mathematician choses to work in a given area, he should ask himself if it has applications before he devotes his time and effort to it. Men such as Gauss, Lagra nge, Jacobi etc. have undoubtedly done so.\" A hundred years later, th is Credo for an applied mathematician remains incisive." }}{PARA 0 "" 0 "" {TEXT -1 768 "In honouring Runge on his 70-th birthday, E. Trefft z characterized him as follows: \"If Runge has succeeeded in building \+ bridges between mathematics and technical science, then it rests on tw o characteristics of a true applied mathematician. First, his deep mat hematical knowledge, which is manifest in his early papers in pure mat hematics, and which always reappears in his application of pure mathem atics to problems in applied mathematics. And secondly, the untiring e nergy with which he developed his methods to the point where they had \+ real practicability, and not just to the point where the mathematician considers it \"simple\", but to the point where the practitioner lose s his distaste for the mathematical mechanics\". (Z. agnew. Math. Mech . 6, 423-424 (1926))." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 20 "The Runge phenomenon" }}{PARA 0 "" 0 "" {TEXT -1 137 "For a numerical example, we take the set of equally spac ed data points used previously to find a degree 10 polynomial approxim ation for " }{XPPEDIT 18 0 "f(x) = 1/(1+25*x^2);" "6#/-%\"fG6#%\"xG*& \"\"\"F),&F)F)*&\"#DF)*$F'\"\"#F)F)!\"\"" }{TEXT -1 18 " on the inter val " }{XPPEDIT 18 0 "[-1 ,1]" "6#7$,$\"\"\"!\"\"F%" }{TEXT -1 131 ". \+ This is a classic example to illustrate the problems associated with u sing an interpolating polynomial to approximate a function." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 189 "x := 'x':\nf := x->1/(1+25*x^2);\nleft := -1;\nright := 1;\ndeg := 10; \nh := abs(right-left)/deg;\nxvals := [seq(left+h*i,i=0..deg)]:\nyvals := map(f,xvals):\np := unapply(interp(xvals,yvals,x),x);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"fGf*6#%\"xG6\"6$%)operatorG%&arrowGF(*&\"\" \"F-,&F-F-*&\"#DF-)9$\"\"#F-F-!\"\"F(F(F(" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%%leftG!\"\"" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%&rig htG\"\"\"" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%$degG\"#5" }}{PARA 11 " " 1 "" {XPPMATH 20 "6#>%\"hG#\"\"\"\"\"&" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"pGf*6#%\"xG6\"6$%)operatorG%&arrowGF(,.*$)9$\"#5\"\"\"#!'D1R \"%o<*&#\"'v$4\"\"$@#F1)F/\"\")F1F1*&#\"&v=&\"$O\"F1*$)F/\"\"'F1F1!\" \"*&#\"&DX&\"$U%F1)F/\"\"%F1F1*&#\"%DPF8F1*$)F/\"\"#F1F1FBF1F1F(F(F(" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 4 "The " } {TEXT 0 3 "zip" }{TEXT -1 41 " command can be used to zip the lists of " }{TEXT 262 1 "x" }{TEXT -1 5 " and " }{TEXT 263 1 "y" }{TEXT -1 42 " values together to form a list of points." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 168 "points := evalf(z ip((x,y)->[x,y],xvals,yvals));\nplot([points,f(x),p(x)],x=left..right, symbol=circle,\ncolor=[blue,green,red],style=[point,line,line],\nlines tyle=[1,1,3]);" }}{PARA 12 "" 1 "" {XPPMATH 20 "6#>%'pointsG7-7$$!\"\" \"\"!$\"+YQ:YQ!#67$$!+++++!)!#5$\"+THN#)eF,7$$!+++++gF0$\"+++++5F07$$! 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Q)[ew\"R#F17$F[al$\"3!fn4ts$zyAF17$$\"3IKLe9as;cF1$\"3[-o4xn)e&>F17$F` al$\"3!ze[[jBs\\\"F17$Feal$\"3%GjZK?6zj\"F-7$Fjal$!3vQ&QEF^xJ\"F17$$\" 3ILL$3sr*zoF1$!3)RW#)f**3F(>F17$F_bl$!3P4(>zd%yFCF17$$\"3>*\\P%)RZY9(F 1$!3G1'R$['Rl\\#F17$$\"3gl;a)GJs>(F1$!3*px$fi89WDF17$$\"39Leky^\")\\sF 1$!3I*y]>L^(oDF17$$\"3c***\\(o!*R-tF1$!3C224K,`oDF17$$\"3)f;a)eH)\\N(F 1$!3S5%R%=xkTDF17$$\"3^L$e*[oc2uF1$!3#>D&>UZH'[#F17$$\"3#**\\i!R2:guF1 $!3t.I83[p+CF17$Fdbl$!3E6KS;x6$G#F17$$\"3\"HL$ek6,1xF1$!3ArjstB#*\\:F1 7$Fibl$!3III'e\">4=HF-7$$\"3ymmTgg/5!)F1$\"3-RG?$\\Qm%oF-7$$\"3)HLL3U/ 37)F1$\"3E[R\"e6()=&=F17$$\"3=***\\7yi:B)F1$\"3**zB*pW4r?$F17$F^cl$\"3 Vwd%)*pE2u%F17$$\"3-LLeRZQT%)F1$\"3uz?ekD9\\iF17$$\"3a****\\P$[/a)F1$ \"3iS5%)z9qmyF17$$\"32mmTN>^R')F1$\"3)*Qs+QZqm&*F17$Fccl$\"32)33D:198 \"F[gl7$$\"36+]i!*z>W))F1$\"3IH@!)ep(yJ\"F[gl7$$\"3amm\"zW?)\\*)F1$\"3 C]unyB&y\\\"F[gl7$$\"3'HL3_!HWb!*F1$\"3c$)3n3H%Hm\"F[gl7$Fhcl$\"3+bO%z uSH!=F[gl7$$\"3Y++DJ&f@E*F1$\"3dg?J)oQ@!>F[gl7$$\"3Y++++PDj$*F1$\"3uY! =@%=Ta>F[gl7$$\"3+,v=Us_)Q*F1$\"3[J,+2bXe>F[gl7$$\"3X+]P%y+QT*F1$\"3KY E$>S$Re>F[gl7$$\"3!**\\ilKu!R%*F1$\"370dii5'R&>F[gl7$$\"3Y++voyMk%*F1$ \"3$)R$Q8?%)[%>F[gl7$$\"3W+]7`\\*[^*F1$\"3J9lcE\"\\;\">F[gl7$F]dl$\"31 O^$[q(Gc=F[gl7$$\"3Q+D\"Gyh(>'*F1$\"3gWs(QE%=pX#F 17$FEF[el-FS6&FUFVFDFDFddl-Fhn6#\"\"$-%'SYMBOLG6#%'CIRCLEG-%+AXESLABEL SG6$Q\"x6\"Q!6\"-%%VIEWG6$;F(FE%(DEFAULTG" 1 2 4 1 10 0 2 9 1 4 2 1.000000 45.000000 45.000000 0 0 "Curve 1" "Curve 2" "Curve 3" }}}} {PARA 0 "" 0 "" {TEXT -1 384 "\nNotice that towards the ends of the in terval the interpolating polynomial oscillates wildly away from the gr aph of the original function.\nAt the data points away from the centre of the interval the derivative of the interpolating polynomial differ s considerably from that of the function, There is nothing in the way \+ that the interpolating polynomial is constructed to prevent this. " }} {PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 51 "subs(x=0.8,diff(f(x),x));\nsubs(x=0.8,diff(p(x),x));" }}{PARA 11 " " 1 "" {XPPMATH 20 "6#$!+XI3%Q\"!#5" }}{PARA 11 "" 1 "" {XPPMATH 20 "6 #$\"*,i._*!\")" }}}{PARA 0 "" 0 "" {TEXT -1 67 "\nThe approximation is not too bad near the centre of the interval. " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 73 "The problem is that the a symptotic behaviour of this function is a very \"" }{TEXT 259 13 "un-p olynomial" }{TEXT -1 238 "\" type of behaviour. It is impossible for a polynomial to approximate a function which has a horizontal or obliqu e asymptote in an interval where the function is close to the asymptot e. Increasing the degree is no help in such situations." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 139 "However, it should \+ be possible to obtain a reasonable polynomial approximation for this f unction by means of interpolation in the interval " }{XPPEDIT 18 0 "[- 1/5,1/5]" "6#7$,$*&\"\"\"F&\"\"&!\"\"F(*&F&F&F'F(" }{TEXT -1 49 ", for example. See the tasks in the next section." }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 5 "Tasks" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 2 "Q1" }}{PARA 0 "" 0 "" {TEXT -1 80 "Check that doubling the degree of the interpola ting polynomial approximation to " }{XPPEDIT 18 0 "f(x) = 1/(1+25*x^2) ;" "6#/-%\"fG6#%\"xG*&\"\"\"F),&F)F)*&\"#DF)*$F'\"\"#F)F)!\"\"" } {TEXT -1 18 " on the interval " }{XPPEDIT 18 0 "[-1,1]" "6#7$,$\"\"\" !\"\"F%" }{TEXT -1 114 " from 10 to 20 does not give any improvement i n the situation towards the edges of the interval. (Assume that the " }{TEXT 264 1 "x" }{TEXT -1 34 " values are again equally spaced.)" }} {PARA 0 "" 0 "" {TEXT -1 39 "Plot a graph to illustrate your answer." }}{PARA 0 "" 0 "" {TEXT -1 52 "_______________________________________ _____________" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" } }{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{PARA 0 "" 0 " " {TEXT -1 52 "____________________________________________________" } }{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 2 "Q2" }}{PARA 0 "" 0 "" {TEXT -1 34 "Obtain a polynomial \+ approximation " }{XPPEDIT 18 0 "p(x)" "6#-%\"pG6#%\"xG" }{TEXT -1 4 " \+ to " }{XPPEDIT 18 0 "f(x)" "6#-%\"fG6#%\"xG" }{TEXT -1 17 " on the int erval " }{XPPEDIT 18 0 "[-1/5, 1/5];" "6#7$,$*&\"\"\"F&\"\"&!\"\"F(*&F &F&F'F(" }{TEXT -1 47 ", which has a maximum absolute error less than \+ " }{XPPEDIT 18 0 "10^(-3);" "6#)\"#5,$\"\"$!\"\"" }{TEXT -1 2 ". " }} {PARA 0 "" 0 "" {TEXT -1 39 "Plot a graph to illustrate your answer." }}{PARA 0 "" 0 "" {TEXT -1 52 "_______________________________________ _____________" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" } }{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{PARA 0 "" 0 " " {TEXT -1 52 "____________________________________________________" } }{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 8 "Epilogue" }}{PARA 0 "" 0 "" {TEXT -1 58 "I f we increase the degree of the interpolating polynomial " }{XPPEDIT 18 0 "p(x)" "6#-%\"pG6#%\"xG" }{TEXT -1 5 " for " }{XPPEDIT 18 0 "f(x) = 1/(1+25*x^2)" "6#/-%\"fG6#%\"xG*&\"\"\"F),&F)F)*&\"#DF)*$F'\"\"#F)F )!\"\"" }{TEXT -1 46 " , with equally spaced points in the interval " }{XPPEDIT 18 0 "[-1,1]" "6#7$,$\"\"\"!\"\"F%" }{TEXT -1 91 ", we find \+ that the interpolating polynomial gives a progressively better approxi mation for " }{XPPEDIT 18 0 "f(x)" "6#-%\"fG6#%\"xG" }{TEXT -1 106 " i n an interval from approximately -0.7 to 0.7, but the approximation be comes worse outside this interval." }}{PARA 0 "" 0 "" {TEXT -1 24 "Her e are a few examples." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 129 "Note that the precision for the computations is progr essively increased, as the degree of the interpolating polynomial incr eases." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 11 "degree = 40" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 256 "x := 'x':\nf := x->1/(1+25*x^2);\n saveDigits := Digits: Digits := 25:\nleft := -1.;\nright := 1.;\ndeg : = 40;\nh := evalf(abs(right-left)/deg):\nxvals := [seq(left+h*i,i=0..d eg)]:\nyvals := map(f,xvals):\np := 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