{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 "Grey Emphasis " -1 262 "Times" 1 12 96 52 84 1 0 1 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 }{CSTYLE "" 261 265 "" 0 1 0 0 0 0 0 0 1 0 0 0 0 0 0 0 }{CSTYLE "" 261 266 "" 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 1 0 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 269 "" 0 1 0 0 0 0 1 0 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 270 "" 0 1 0 0 0 0 1 0 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 271 "" 0 1 0 0 0 0 1 0 0 0 0 0 0 0 0 0 } {CSTYLE "" -1 272 "" 0 1 0 0 0 0 1 0 0 0 0 0 0 0 0 0 }{CSTYLE "" 258 273 "" 0 1 0 0 0 0 0 0 1 0 0 0 0 0 0 0 }{CSTYLE "" -1 274 "System" 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 275 "System" 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 276 "System" 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 277 "System" 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 278 "" 0 1 0 0 0 0 1 0 0 0 0 0 0 0 0 0 }{CSTYLE "Grey E mphasis " -1 279 "Times" 1 12 96 52 84 1 0 1 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 280 "" 0 1 0 0 0 0 1 0 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 281 "" 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 "Times" 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 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 "Bullet Item" -1 15 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 3 3 1 0 1 0 2 2 15 2 }{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 "Normal" -1 257 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 }} {SECT 0 {PARA 3 "" 0 "" {TEXT -1 38 "Examples of interpolating polynom ials " }}{PARA 0 "" 0 "" {TEXT -1 37 "by Peter Stone, Nanaimo, B.C., C anada" }}{PARA 0 "" 0 "" {TEXT -1 21 "Version: 24.3.2007 " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 8 "res tart;" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 56 "load interpolation and function approxima tion procedures" }}{PARA 0 "" 0 "" {TEXT -1 17 "The Maple m-file " } {TEXT 262 10 "fcnapprx.m" }{TEXT -1 37 " contains the code for the pro cedure " }{TEXT 0 13 "newton_interp" }{TEXT -1 25 " used in this works heet. " }}{PARA 0 "" 0 "" {TEXT -1 123 "It can be read into a Maple se ssion by a command similar to the one that follows, where the file pat h gives its location. " }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 36 "r ead \"K:\\\\Maple/procdrs/fcnapprx.m\";" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}} {SECT 1 {PARA 4 "" 0 "" {TEXT -1 20 "The Maple procedure " }{TEXT 0 6 "interp" }{TEXT -1 38 " for finding interpolating polynomials" }} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}{PARA 0 "" 0 "" {TEXT -1 20 "The Maple procedure " }{TEXT 0 6 "interp" }{TEXT -1 105 " const ructs the interpolating polynomial for a given data set consisting of \+ a list of independent values " }{XPPEDIT 18 0 "[x[0],` . . . `,x[n]]" "6#7%&%\"xG6#\"\"!%(~.~.~.~G&F%6#%\"nG" }{TEXT -1 37 " and correspond ing dependent values " }{XPPEDIT 18 0 "[y[0],` . . . `,y[n]]" "6#7%&% \"yG6#\"\"!%(~.~.~.~G&F%6#%\"nG" }{TEXT -1 56 " together with a variab le to be used for the polynomial." }}{PARA 0 "" 0 "" {TEXT -1 25 "We c an use the procedure " }{TEXT 0 6 "interp" }{TEXT -1 57 " to find the \+ parabola which passes through the 3 points: " }{XPPEDIT 18 0 "``(1,2), ``(2,3)" "6$-%!G6$\"\"\"\"\"#-F$6$F'\"\"$" }{TEXT -1 5 " and " } {XPPEDIT 18 0 "``(4,1)" "6#-%!G6$\"\"%\"\"\"" }{TEXT -1 1 "." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 36 "x \+ := 'x':\ninterp([1,2,4],[2,3,1],x);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6 #,(*$)%\"xG\"\"#\"\"\"#!\"#\"\"$F&F+#!\"\"F+F(" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 18 "The Maple command " } {TEXT 0 7 "unapply" }{TEXT -1 92 " can be used to construct an explici t interpolating polynomial function in conjunction with " }{TEXT 0 6 " interp" }{TEXT -1 2 ". " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 57 "x :='x':\np := unapply(interp([1,2, 4],[2,3,1],x),x);\np(3);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"pGf*6# %\"xG6\"6$%)operatorG%&arrowGF(,&*&,&9$#!\"#\"\"$F2\"\"\"F3F/F3F3#!\" \"F2F3F(F(F(" }}{PARA 11 "" 1 "" {XPPMATH 20 "6##\"\")\"\"$" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 14 "The procedure \+ " }{TEXT 0 6 "interp" }{TEXT -1 145 " essentially uses the Lagrange in terpolation formula, but the algebraic form of the output differs from the standard form given by the procedure " }{TEXT 0 8 "lagrange" } {TEXT -1 83 " given in the worksheet on Lagrange interpolation and als o in the previous section." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 165 "n := 2:\nx := 'x': y := 'y':\nxval s := [seq(x[i],i=1..n)];\nyvals := [seq(y[i],i=1..n)];\npol1 := interp (xvals,yvals,z);\npol2 := lagrange(xvals,yvals,z);\nis(pol1=pol2);\n" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%&xvalsG7$&%\"xG6#\"\"\"&F'6#\"\"# " }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%&yvalsG7$&%\"yG6#\"\"\"&F'6#\"\" #" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%%pol1G,&*(,&&%\"yG6#\"\"#!\"\"& F)6#\"\"\"F/F/,&&%\"xGF*F,&F2F.F/F,%\"zGF/F/*&,&*&F(F/F3F/F/*&F-F/F1F/ F,F/F0F,F/" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%%pol2G,&*(,&%\"zG\"\" \"&%\"xG6#\"\"#!\"\"F),&F*F.&F+6#F)F)F.&%\"yGF1F)F)*(,&F(F)F0F.F),&F*F )F0F.F.&F3F,F)F)" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%%trueG" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 64 "The expressions becomes rather complicated for larger values of " }{TEXT 263 1 "n" } {TEXT -1 45 ", so we just look at the leading coefficient." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 25 "The coefficient g iven by " }{TEXT 0 8 "lagrange" }{TEXT -1 36 " looks different from th at given by " }{TEXT 0 6 "interp" }{TEXT -1 47 ", but if we apply the \+ simplification procedure " }{TEXT 0 6 "normal" }{TEXT -1 8 " to the " }{TEXT 0 8 "lagrange" }{TEXT -1 74 " coefficient, we end up with exact ly the same expression as that given by " }{TEXT 0 6 "interp" }{TEXT -1 1 "." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 212 "n := 3:\nx := 'x': y := 'y': z := 'z':\nxvals := [se q(x[i],i=1..n)];\nyvals := [seq(y[i],i=1..n)];\nterm1 := coeff(interp( xvals,yvals,z),z,2);\ncoeff(lagrange(xvals,yvals,z),z,2);\nterm2 := no rmal(%);\nis(term1=term2);\n" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%&xva lsG7%&%\"xG6#\"\"\"&F'6#\"\"#&F'6#\"\"$" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%&yvalsG7%&%\"yG6#\"\"\"&F'6#\"\"#&F'6#\"\"$" }}{PARA 11 "" 1 " " {XPPMATH 20 "6#>%&term1G,$**,.*&&%\"yG6#\"\"$\"\"\"&%\"xG6#\"\"#F-F- *&F)F-&F/6#F-F-!\"\"*&&F*F0F-&F/F+F-F5*&F7F-F3F-F-*&&F*F4F-F8F-F-*&F;F -F.F-F5F-,&F.F5F3F-F5,&F8F-F3F5F5,&F8F-F.F5F5F5" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#,(*(,&&%\"xG6#\"\"#!\"\"&F'6#\"\"\"F-F*,&F+F-&F'6#\"\"$ F*F*&%\"yGF,F-F-*(,&F&F-F+F*F*,&F&F-F/F*F*&F3F(F-F-*(,&F/F-F+F*F*,&F/F -F&F*F*&F3F0F-F-" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%&term2G,$**,.*&& %\"yG6#\"\"$\"\"\"&%\"xG6#\"\"#F-F-*&F)F-&F/6#F-F-!\"\"*&&F*F0F-&F/F+F -F5*&F7F-F3F-F-*&&F*F4F-F8F-F-*&F;F-F.F-F5F-,&F.F5F3F-F5,&F8F-F3F5F5,& F8F-F.F5F5F5" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%%trueG" }}}{PARA 0 " " 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 211 "n := 4:\nx := 'x': y := 'y': z := 'z':\nxvals := [seq(x[i],i=1..n)];\nyval s := [seq(y[i],i=1..n)];\nterm1 := coeff(interp(xvals,yvals,z),z,2);\n coeff(lagrange(xvals,yvals,z),z,2);\nterm2 := normal(%);\nis(term1=ter m2);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%&xvalsG7&&%\"xG6#\"\"\"&F'6# \"\"#&F'6#\"\"$&F'6#\"\"%" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%&yvalsG 7&&%\"yG6#\"\"\"&F'6#\"\"#&F'6#\"\"$&F'6#\"\"%" }}{PARA 12 "" 1 "" {XPPMATH 20 "6#>%&term1G*0,R*(&%\"yG6#\"\"%\"\"\"&%\"xG6#\"\"#F,)&F.6# \"\"$F4F,F,*(&F)F/F,&F.6#F,F,)&F.F*F4F,!\"\"*(F(F,F7F,F1F,F;*(&F)F8F,F -F,F9F,F,*(F>F,F1F,F:F,F,*(F6F,F1F,F7F,F,*(F6F,F1F,F:F,F;*(F>F,)F-F4F, F2F,F,*(F>F,F-F,F1F,F;*(&F)F3F,)F7F4F,F-F,F,*(FFF,FCF,F:F,F,*(F(F,FCF, F2F,F;*(FFF,FGF,F:F,F;*(F(F,FCF,F7F,F,*(FFF,F-F,F9F,F;*(F6F,FGF,F:F,F, *(FFF,F7F,F9F,F,*(F6F,FGF,F2F,F;*(F(F,FGF,F2F,F,*(F>F,FCF,F:F,F;*(F6F, F2F,F9F,F,*(F(F,FGF,F-F,F;*(FFF,FCF,F7F,F;*(F>F,F2F,F9F,F;F,,&F-F;F7F, F;,&F2F,F7F;F;,&F2F,F-F;F;,&F:F,F7F;F;,&F:F,F-F;F;,&F:F,F2F;F;" }} {PARA 12 "" 1 "" {XPPMATH 20 "6#,**(,&*&,&*&&%\"xG6#\"\"#\"\"\",&F)!\" \"&F*6#F-F-F/F/*&F.F/&F*6#\"\"$F-F/F-,&F0F-F3F/F/F-*(F.F/F6F/&F*6#\"\" %F-F/F-,&F0F-F8F/F/&%\"yGF1F-F-*(,&*&,&*&F0F-,&F)F-F0F/F/F/*&FCF/F3F-F /F-,&F)F-F3F/F/F-*(FCF/FEF/F8F-F/F-,&F)F-F8F/F/&F=F+F-F-*(,&*&,&*&F0F- ,&F3F-F0F/F/F/*&FNF/F)F-F/F-,&F3F-F)F/F/F-*(FNF/FPF/F8F-F/F-,&F3F-F8F/ F/&F=F4F-F-*(,&*&,&*&F0F-,&F8F-F0F/F/F/*&FYF/F)F-F/F-,&F8F-F)F/F/F-*(F YF/FenF/F3F-F/F-,&F8F-F3F/F/&F=F9F-F-" }}{PARA 12 "" 1 "" {XPPMATH 20 "6#>%&term2G,$*0,R*(&%\"yG6#\"\"%\"\"\"&%\"xG6#\"\"#F-)&F/6#\"\"$F5F-! \"\"*(&F*F0F-&F/6#F-F-)&F/F+F5F-F-*(F)F-F9F-F2F-F-*(&F*F:F-F.F-F;F-F6* (F?F-F2F-F " 0 "" {MPLTEXT 1 0 57 "CurveFitti ng:-PolynomialInterpolation([1,2,4],[2,3,1],x);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#,(*$)%\"xG\"\"#\"\"\"#!\"#\"\"$*&F+F(F&F(F(#F(F+!\"\"" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 23 "The Maple commands \+ . . " }{TEXT 0 7 "map,zip" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 " " 0 "" {TEXT -1 18 "The Maple command " }{TEXT 0 3 "map" }{TEXT -1 79 " can be used to apply a function of a single variable to all members \+ of a list." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 " " {MPLTEXT 1 0 26 "map(x->x^2,[1,2,3,4,5,6]);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#7(\"\"\"\"\"%\"\"*\"#;\"#D\"#O" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 36 "[$1..10];\nm ap(sqrt,%);\nmap(evalf,%);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#7,\"\"\" \"\"#\"\"$\"\"%\"\"&\"\"'\"\"(\"\")\"\"*\"#5" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#7,\"\"\"*$\"\"##F$F&*$\"\"$F'F&*$\"\"&F'*$\"\"'F'*$\"\" (F',$*&F&F$F&F'F$F)*$\"#5F'" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#7,$\"\" \"\"\"!$\"+iN@99!\"*$\"+330K " 0 "" {MPLTEXT 1 0 44 "zip((x,y)->x+y,[1,2,3,4,5,6] ,[6,5,4,3,2,1]);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#7(\"\"(F$F$F$F$F$ " }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 13 "The \+ function " }{XPPEDIT 18 0 "proc (x, y) options operator, arrow; [x, y] end;" "6#f*6$%\"xG%\"yG7\"6$%)operatorG%&arrowG6\"7$F%F&F+F+F+" } {TEXT -1 21 " in conjunction with " }{TEXT 0 3 "zip" }{TEXT -1 43 " ca n be used to convert a pair of lists of " }{TEXT 270 1 "x" }{TEXT -1 5 " and " }{TEXT 271 1 "y" }{TEXT -1 44 " values into a list of points to be plotted." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> \+ " 0 "" {MPLTEXT 1 0 56 "points := zip((x,y)->[x,y],[1,2,3,4,5,6],[6,5, 4,3,2,1]);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%'pointsG7(7$\"\"\"\"\" '7$\"\"#\"\"&7$\"\"$\"\"%7$F.F-7$F+F*7$F(F'" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 39 "A collection of points along th e curve " }{XPPEDIT 18 0 "y=x^2" "6#/%\"yG*$%\"xG\"\"#" }{TEXT -1 46 " can be constructed (and plotted) as follows. " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 174 "xvals := [s eq(i,i=-3..3)];\nyvals := map(x->x^2,xvals);\npts := zip((x,y)->[x,y], xvals,yvals);\nplot([pts,pts],style=point,symbol=[circle,cross],symbol size=[15,15],color=brown);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%&xvals G7)!\"$!\"#!\"\"\"\"!\"\"\"\"\"#\"\"$" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%&yvalsG7)\"\"*\"\"%\"\"\"\"\"!F(F'F&" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%$ptsG7)7$!\"$\"\"*7$!\"#\"\"%7$!\"\"\"\"\"7$\"\"!F07$ F.F.7$\"\"#F+7$\"\"$F(" }}{PARA 13 "" 1 "" {GLPLOT2D 376 291 291 {PLOTDATA 2 "6(-%'CURVESG6$7)7$$!\"$\"\"!$\"\"*F*7$$!\"#F*$\"\"%F*7$$! \"\"F*$\"\"\"F*7$$F*F*F87$F5F57$$\"\"#F*F07$$\"\"$F*F+-%'SYMBOLG6$%'CI RCLEG\"#:-F$6$F&-FA6$%&CROSSGFD-%'COLOURG6&%$RGBG$\")#)eqk!\")$\"))eqk \"FPFQ-%+AXESLABELSG6$Q!6\"FV-%&STYLEG6#%&POINTG-%%VIEWG6$%(DEFAULTGFi n" 1 5 0 1 10 0 2 6 1 4 2 1.000000 45.000000 45.000000 0 0 "Curve 1" " Curve 2" }}}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 66 "The same collection of points can also be constructed as follows. \+ " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 49 "xvals := [seq(i,i=-3..3)];\nmap(x->[x,x^2],xvals);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%&xvalsG7)!\"$!\"#!\"\"\"\"!\"\"\"\"\"#\"\" $" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#7)7$!\"$\"\"*7$!\"#\"\"%7$!\"\"\" \"\"7$\"\"!F.7$F,F,7$\"\"#F)7$\"\"$F&" }}}{PARA 0 "" 0 "" {TEXT -1 0 " " }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}}{SECT 1 {PARA 4 " " 0 "" {TEXT -1 37 "Nested evaluation of polynomials . . " }{TEXT 0 18 "convert(..,horner)" }}{PARA 0 "" 0 "" {TEXT -1 87 "This section co ntains a discussion of a different methods for evaluating polynomials. " }}{PARA 0 "" 0 "" {TEXT -1 48 "A standard approach to evaluating a polynomial " }{XPPEDIT 18 0 "p(x) = a[1]+a[2]*x+a[3]*x^2;" "6#/-%\"p G6#%\"xG,(&%\"aG6#\"\"\"F,*&&F*6#\"\"#F,F'F,F,*&&F*6#\"\"$F,*$F'F0F,F, " }{TEXT -1 12 " + ..... + " }{XPPEDIT 18 0 "a[n]*x^(n-1);" "6#*&&%\" aG6#%\"nG\"\"\")%\"xG,&F'F(F(!\"\"F(" }{TEXT -1 40 " is illustrated \+ by the following loop." }}{PARA 0 "" 0 "" {TEXT -1 87 "The number of a dditions and multiplications performed is accumulated in the variables \"" }{TEXT 262 7 "num_add" }{TEXT -1 7 "\" and \"" }{TEXT 262 8 "num_ mult" }{TEXT -1 2 "\"." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 292 "n := 12:\nx := 'x': a := 'a':\nnum _add := 0: num_mult := 0:\nxpower := 1:\npval := a[1]:\nfor i from 2 t o n do\n xpower := x*xpower;\n pval := pval + a[i]*xpower;\n num _add := num_add + 1;\n num_mult := num_mult + 2;\nend do:\npval;\npr int(num_add,`additions`);\nprint(num_mult,`multiplications`);" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#,:&%\"aG6#\"\"\"F'*&&F%6#\"\"#F'%\"xGF 'F'*&&F%6#\"\"$F')F,F+F'F'*&&F%6#\"\"%F')F,F0F'F'*&&F%6#\"\"&F')F,F5F' F'*&&F%6#\"\"'F')F,F:F'F'*&&F%6#\"\"(F')F,F?F'F'*&&F%6#\"\")F')F,FDF'F '*&&F%6#\"\"*F')F,FIF'F'*&&F%6#\"#5F')F,FNF'F'*&&F%6#\"#6F')F,FSF'F'*& &F%6#\"#7F')F,FXF'F'" }}{PARA 11 "" 1 "" {XPPMATH 20 "6$\"#6%*addition sG" }}{PARA 11 "" 1 "" {XPPMATH 20 "6$\"#A%0multiplicationsG" }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 67 "A more efficient algorithm for \+ evaluatng polynomials is based on a " }{TEXT 259 6 "nested" }{TEXT -1 66 " organization of the polynomial, which we illustrate for the case \+ " }{XPPEDIT 18 0 "n=4" "6#/%\"nG\"\"%" }{TEXT -1 1 "." }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "a[1]+a[2]*x+a[3]*x^2+a[4]*x^3 = x* (x*(x*a[4]+a[3])+a[2])+a[1];" "6#/,*&%\"aG6#\"\"\"F(*&&F&6#\"\"#F(%\"x GF(F(*&&F&6#\"\"$F(*$F-F,F(F(*&&F&6#\"\"%F(*$F-F1F(F(,&*&F-F(,&*&F-F(, &*&F-F(&F&6#F6F(F(&F&6#F1F(F(F(&F&6#F,F(F(F(&F&6#F(F(" }{TEXT -1 1 " \+ " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 124 "The \+ total number of multiplications and additions required to evaluate the nested form is less than for the previous method." }}{PARA 0 "" 0 "" {TEXT -1 100 "A loop which performs this nested multiplication is give n below. This evaluation method is known as " }{TEXT 259 15 "Horner's \+ method" }{TEXT -1 1 "." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 259 "n := 1 2:\nx := 'x': a := 'a':\nnum_add := 0: num_mult := 0:\npval := a[n]:\n for i from n-1 to 1 by -1 do\n pval := x*pval + a[i];\n num_add := num_add + 1;\n num_mult := num_mult + 1;\nend do:\npval;\nprint(num _add,`additions`);\nprint(num_mult,`multiplications`);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#,&*&%\"xG\"\"\",&*&F%F&,&*&F%F&,&*&F%F&,&*&F%F&,&* &F%F&,&*&F%F&,&*&F%F&,&*&F%F&,&*&F%F&,&*&F%F&&%\"aG6#\"#7F&F&&F<6#\"#6 F&F&F&&F<6#\"#5F&F&F&&F<6#\"\"*F&F&F&&F<6#\"\")F&F&F&&F<6#\"\"(F&F&F&& F<6#\"\"'F&F&F&&F<6#\"\"&F&F&F&&F<6#\"\"%F&F&F&&F<6#\"\"$F&F&F&&F<6#\" \"#F&F&F&&F<6#F&F&" }}{PARA 11 "" 1 "" {XPPMATH 20 "6$\"#6%*additionsG " }}{PARA 11 "" 1 "" {XPPMATH 20 "6$\"#6%0multiplicationsG" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 10 "The Maple " } {TEXT 0 7 "convert" }{TEXT -1 66 " function can be used to achieve thi s rearrangement algebraically." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 34 "convert(7+5*x+2*x^2+3*x^3,ho rner);" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 59 "Examples o f finding and graphing interpolating polynomials " }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 9 "Exam ple 1" }}{PARA 0 "" 0 "" {TEXT -1 4 "Let " }{XPPEDIT 18 0 "f(x)=exp(-x ^2)" "6#/-%\"fG6#%\"xG-%$expG6#,$*$F'\"\"#!\"\"" }{TEXT -1 3 ". " }} {PARA 0 "" 0 "" {TEXT -1 32 "We take a set of equally spaced " }{TEXT 269 1 "x" }{TEXT -1 71 " values to find a degree 10 interpolating poly nomial approximation for " }{XPPEDIT 18 0 "f(x);" "6#-%\"fG6#%\"xG" } {TEXT -1 17 " on the interval " }{XPPEDIT 18 0 "[0, 2];" "6#7$\"\"!\" \"#" }{TEXT -1 2 ".\n" }}{PARA 0 "" 0 "" {TEXT -1 3 "We " }{TEXT 259 22 "increase the precision" }{TEXT -1 48 " for the calculations to red uce rounding errors." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 " " {TEXT -1 73 "The polynomial is converted into a function by using th e Maple procedure " }{TEXT 0 7 "unapply" }{TEXT -1 1 "." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 264 "f := x -> exp(-x^2); \nDigits := 20:\nleft := 0;\nright := 2;\ndeg := 10;\nh := abs(right-left)/deg;\nxvals := [seq(left+h*i,i=0..deg)];\nxvals := evalf(xvals):\nyvals := map(f,evalf(xvals)); \nx := 'x':\npx := inter p(xvals,yvals,x):\nDigits := 10:\np := unapply(evalf(px),x);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"fGf*6#%\"xG6\"6$%)operatorG%&arrowGF(-%$ expG6#,$*$)9$\"\"#\"\"\"!\"\"F(F(F(" }}{PARA 11 "" 1 "" {XPPMATH 20 "6 #>%%leftG\"\"!" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%&rightG\"\"#" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#>%$degG\"#5" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"hG#\"\"\"\"\"&" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#> %&xvalsG7-\"\"!#\"\"\"\"\"&#\"\"#F)#\"\"$F)#\"\"%F)F(#\"\"'F)#\"\"(F)# \"\")F)#\"\"*F)F+" }}{PARA 12 "" 1 "" {XPPMATH 20 "6#>%&yvalsG7-$\"\" \"\"\"!$\"5W4KK_\"R%*yg*!#?$\"5YQ8@m*)yV@&)F+$\"5@d5.rgKwwpF+$\"5Cd&[I /CCHF&F+$\"5g@BWr6WzyOF+$\"5sc<7#oex#pBF+$\"5:'*\\/@4Ue39F+$\"5!*fu*HV /u/t(!#@$\"5SP2()*)4&*Q;RF:$\"5%H!=M()))QcJ=F:" }}{PARA 12 "" 1 "" {XPPMATH 20 "6#>%\"pGf*6#%\"xG6\"6$%)operatorG%&arrowGF(,8*&$\"+1\"\\' =6!#7\"\"\")9$\"#5F1F1*&$\"+c?![;%F0F1)F3\"\"*F1!\"\"*&$\"+g[CA_!#8F1F 3F1F1*&$\"+xB`1G!#6F1)F3\"\")F1F:*&$\"+W.]25!\"*F1)F3\"\"#F1F:*&$\"+3O J!4#!#5F1)F3\"\"(F1F1*&$\"+gQ`lVFBF1)F3\"\"$F1F1*&$\"+'\\FSo%FNF1)F3\" \"'F1F:*&$\"+K*yli$FNF1)F3\"\"%F1F1*&$\"+S>F!f#FNF1)F3\"\"&F1F1$F1\"\" !F1F(F(F(" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 94 "The graph of the interpolating polynomial can be plotted along \+ with the interpolation points.\n" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 154 "pts := evalf(zip((x,y)->[x,y],xvals,yvals));\nplot([p(x),pts] ,x=left..right,style=[line,point],symbol=circle,\n \+ color=[red,black]);" }}{PARA 12 "" 1 "" {XPPMATH 20 "6#>%$ptsG7- 7$$\"\"!F($\"\"\"F(7$$\"+++++?!#5$\"+#R%*yg*F.7$$\"+++++SF.$\"+!*yV@&) F.7$$\"+++++gF.$\"+hKwwpF.7$$\"+++++!)F.$\"+SU#HF&F.7$F)$\"+7WzyOF.7$$ \"+++++7!\"*$\"+(ex#pBF.7$$\"+++++9FF$\"+4Ue39F.7$$\"+++++;FF$\"+WSZIx !#67$$\"+++++=FF$\"+5&*Q;RFS7$$\"\"#F($\"+*)QcJ=FS" }}{PARA 13 "" 1 " " {GLPLOT2D 400 300 300 {PLOTDATA 2 "6'-%'CURVESG6%7S7$$\"\"!F)$\"\"\" F)7$$\"39LLLL3VfV!#>$\"3uc\")[C,8\")**!#=7$$\"3'pmm;H[D:)F/$\"3n-_(>&z 'Q$**F27$$\"3LLLLe0$=C\"F2$\"37C\"ykIOq%)*F27$$\"3ILLL3RBr;F2$\"3`QVdD 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1.000000 45.000000 45.000000 0 0 "Curve 1" }}}}{PARA 0 "" 0 "" {TEXT -1 0 "" }} {PARA 0 "" 0 "" {TEXT -1 4 "The " }{TEXT 259 22 "maximum absolute erro r" }{TEXT -1 25 " in using the polynomial " }{XPPEDIT 18 0 "p(x)" "6#- %\"pG6#%\"xG" }{TEXT -1 16 " to approximate " }{XPPEDIT 18 0 "f(x)=exp (-x^2)" "6#/-%\"fG6#%\"xG-%$expG6#,$*$F'\"\"#!\"\"" }{TEXT -1 17 " on \+ the interval " }{XPPEDIT 18 0 "[0,2]" "6#7$\"\"!\"\"#" }{TEXT -1 16 " \+ is about 1.2e-5" }{XPPEDIT 18 0 "`` = 1.2;" "6#/%!G-%&FloatG6$\"#7!\" \"" }{TEXT -1 1 " " }{TEXT 277 1 "x" }{TEXT -1 1 " " }{XPPEDIT 18 0 "1 0^(-5)" "6#)\"#5,$\"\"&!\"\"" }{TEXT -1 1 "." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 44 "numapprox[infnorm] (p(x)-f(x),x=left..right);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#$\"+\"\\ Q!=7!#9" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 9 "Example 2" }}{PARA 0 "" 0 "" {TEXT 273 8 "Question" }{TEXT -1 2 ": " }}{PARA 0 " " 0 "" {TEXT -1 46 "(a) Find a degree 16 polynomial approximation " } {XPPEDIT 18 0 "p(x)" "6#-%\"pG6#%\"xG" }{TEXT -1 5 " for " }{XPPEDIT 18 0 "f(x) = sin(2*x^2);" "6#/-%\"fG6#%\"xG-%$sinG6#*&\"\"#\"\"\"*$F'F ,F-" }{TEXT -1 17 " on the interval " }{XPPEDIT 18 0 "0<=x" "6#1\"\"!% \"xG" }{XPPEDIT 18 0 "``<=1" "6#1%!G\"\"\"" }{TEXT -1 33 " which agree s with the values of " }{XPPEDIT 18 0 "f(x)" "6#-%\"fG6#%\"xG" }{TEXT -1 7 " at 17 " }{TEXT 262 14 "equally spaced" }{TEXT -1 11 " values of " }{TEXT 272 1 "x" }{TEXT -1 15 " between 0 and " }{XPPEDIT 18 0 "1; " "6#\"\"\"" }{TEXT -1 71 " inclusive. Calculate the coefficients of t he interpolating polynomial " }{XPPEDIT 18 0 "p(x)" "6#-%\"pG6#%\"xG" }{TEXT -1 1 " " }{TEXT 262 20 "correct to 15 digits" }{TEXT -1 2 ". " }}{PARA 0 "" 0 "" {TEXT -1 49 "(b) Plot a graph of the interpolating p olynomial " }{XPPEDIT 18 0 "p(x)" "6#-%\"pG6#%\"xG" }{TEXT -1 32 " fou nd in (a) over the interval " }{XPPEDIT 18 0 "0<=x" "6#1\"\"!%\"xG" } {XPPEDIT 18 0 "``<=1" "6#1%!G\"\"\"" }{TEXT -1 64 " along with the int erpolation points used for its construction. " }}{PARA 0 "" 0 "" {TEXT -1 17 "(c) Estimate the " }{TEXT 262 22 "maximum absolute error " }{TEXT -1 25 " in using the polynomial " }{XPPEDIT 18 0 "p(x)" "6#-% \"pG6#%\"xG" }{TEXT -1 16 " to approximate " }{XPPEDIT 18 0 "f(x)" "6# -%\"fG6#%\"xG" }{TEXT -1 18 " in the interval " }{XPPEDIT 18 0 "0<=x " "6#1\"\"!%\"xG" }{XPPEDIT 18 0 "``<=1" "6#1%!G\"\"\"" }{TEXT -1 1 ". " }}{PARA 0 "" 0 "" {TEXT -1 56 "(d) Compare the values of your interp olating polynomial " }{XPPEDIT 18 0 "p(x)" "6#-%\"pG6#%\"xG" }{TEXT -1 18 " and the function " }{XPPEDIT 18 0 "f(x)" "6#-%\"fG6#%\"xG" } {TEXT -1 5 " for " }{XPPEDIT 18 0 "x=2*sqrt(6)/5" "6#/%\"xG*(\"\"#\"\" \"-%%sqrtG6#\"\"'F'\"\"&!\"\"" }{TEXT -1 87 ". \n What is the rela tive error in using the first value to approximate the second? " }} {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT 265 8 "Solution " }{TEXT -1 2 ": " }}{PARA 0 "" 0 "" {TEXT -1 4 "(a) " }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 279 "f := x -> sin(2*x^2); \nDigits := 20:\nl eft := 0;\nright := 1;\ndeg := 16;\nh := abs(right-left)/deg;\nxvals : = [seq(left+h*i,i=0..deg)];\nxvals := evalf(xvals):\nyvals := map(f,xv als); \nx := 'x':\npx := interp(evalf(xvals),yvals,x):\nDigits := 15: \np := unapply(evalf(px),x);\nDigits := 10:" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"fGf*6#%\"xG6\"6$%)operatorG%&arrowGF(-%$sinG6#,$*& \"\"#\"\"\")9$F1F2F2F(F(F(" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%%leftG \"\"!" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%&rightG\"\"\"" }}{PARA 11 " " 1 "" {XPPMATH 20 "6#>%$degG\"#;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#> %\"hG#\"\"\"\"#;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%&xvalsG73\"\"!# \"\"\"\"#;#F(\"\")#\"\"$F)#F(\"\"%#\"\"&F)#F-F+#\"\"(F)#F(\"\"##\"\"*F )#F1F+#\"#6F)#F-F/#\"#8F)#F4F+#\"#:F)F(" }}{PARA 12 "" 1 "" {XPPMATH 20 "6#>%&yvalsG73$\"\"!F'$\"5s/JGQF0U7y!#A$\"5S(ygK&)R\"\\CJ!#@$\"5WA/ g[gyXDqF-$\"5'**oF_QLZnC\"!#?$\"5;\"z4H*G5tS>F2$\"5#fKOjk^nbx#F2$\"5UY HpQ#o3`t$F2$\"5F+I?/'QbUz%F2$\"54:jJ,A+:9fF2$\"5ysO`a9^nTqF2$\"5!HPy%f )Q*z1\")F2$\"5#H;&4*4%fnA!*F2$\"5kIF5Xwg#zo*F2$\"5@KgRAf9=#***F2$\"5)p c#e9]QjD)*F2$\"5S&p\"oDoU(H4*F2" }}{PARA 12 "" 1 "" {XPPMATH 20 "6#>% \"pGf*6#%\"xG6\"6$%)operatorG%&arrowGF(,B*&$\"0De$))))****>!#9\"\"\")9 $\"\"#F1F1*&$\"0*=$)=U![R$!#=F1)F3\"\"%F1!\"\"*&$\"0=P&e-U^H!#%\\D^8F0F1)F3\"\"'F1F;*&$\"0*)**Rl4Tb#!#>F1)F3\"\"$F1F 1*&$\"0c.zXN`7*!#:F1)F3\"#5F1F;*&$\"0$pNlelM;F0F1)F3\"#6F1F1*&$\"0m-:L HDm\"F0F1)F3\"#7F1F;*&$\"0vh142P>\"F0F1)F3\"#8F1F1*&$\"0s%e7[!***y!#;F 1)F3\"\"(F1F1*&$\"05?Dbqie#FPF1)F3\"\")F1F;*&$\"0w2RJwEO'FPF1)F3\"\"*F 1F1*&$\"0usH7F/'eFPF1)F3\"#9F1F;*&$\"0^g4M>MW\"FPF1)F3\"#:F1F1*&$\"0x& [6YHK5F_oF1)F3\"#;F1F;*&$\".$=X8tw?!#?F1F3F1F1F(F(F(" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 4 "(b) " }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 135 "pts := 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"The maximum absolute er ror in using the polynomial " }{XPPEDIT 18 0 "p(x)" "6#-%\"pG6#%\"xG" }{TEXT -1 16 " to approximate " }{XPPEDIT 18 0 "f(x)" "6#-%\"fG6#%\"xG " }{TEXT -1 17 " in the interval " }{XPPEDIT 18 0 "0<=x" "6#1\"\"!%\"x G" }{XPPEDIT 18 0 "``<=1" "6#1%!G\"\"\"" }{TEXT -1 17 " is about 2.2e- 10" }{XPPEDIT 18 0 "`` = 2.2;" "6#/%!G-%&FloatG6$\"#A!\"\"" }{TEXT -1 1 " " }{TEXT 275 1 "x" }{TEXT -1 1 " " }{XPPEDIT 18 0 "10^(-10)" "6#) \"#5,$F$!\"\"" }{TEXT -1 2 ". " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }} {PARA 0 "" 0 "" {TEXT -1 3 "(d)" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 177 "Digits := 15:\nxx := evalf(2*sqrt(6)/5);\napprox_val := p(xx) ;\naccurate_val := f(xx);\nabserr := abs(approx_val-accurate_val);\nre lerr := evalf(abserr/accurate_val,3);\nDigits := 10:" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%#xxG$\"0sK6(*ezz*!#:" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%+approx_valG$\"/_pMZX'R*!#9" }}{PARA 11 "" 1 "" {XPPMATH 20 "6 #>%-accurate_valG$\"0D`otakR*!#:" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>% 'abserrG$\"'0e@!#:" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%'relerrG$\"$I# !#7" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 28 "T he relative error in using " }{XPPEDIT 18 0 "p(2*sqrt(6)/5)" "6#-%\"pG 6#*(\"\"#\"\"\"-%%sqrtG6#\"\"'F(\"\"&!\"\"" }{TEXT -1 16 " to approxim ate " }{XPPEDIT 18 0 "f(2*sqrt(6)/5)" "6#-%\"fG6#*(\"\"#\"\"\"-%%sqrtG 6#\"\"'F(\"\"&!\"\"" }{TEXT -1 17 " is about 2.3e-10" }{XPPEDIT 18 0 " `` = 2.3;" "6#/%!G-%&FloatG6$\"#B!\"\"" }{TEXT -1 1 " " }{TEXT 276 1 " x" }{TEXT -1 1 " " }{XPPEDIT 18 0 "10^(-10)" "6#)\"#5,$F$!\"\"" } {TEXT -1 2 ". " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 9 "Exampl e 3" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{PARA 0 "" 0 "" {TEXT -1 22 "Consider the function " }{XPPEDIT 18 0 "f(x)" "6#-%\"fG6# %\"xG" }{TEXT -1 12 " defined by " }{XPPEDIT 18 0 "f(x)=PIECEWISE([sin (x)/x,x<>0],[1,x=0])" "6#/-%\"fG6#%\"xG-%*PIECEWISEG6$7$*&-%$sinG6#F' \"\"\"F'!\"\"0F'\"\"!7$F0/F'F3" }{TEXT -1 1 "." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 97 "To work with this functio n in the Maple environment we need to define it by means of a procedur e." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 81 "f := proc(x) if x<>0 then sin(x)/x else 1 end if end \+ proc; \nplot(f(x),x=-12..12);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"f Gf*6#%\"xG6\"F(F(@%09$\"\"!*&-%$sinG6#F+\"\"\"F+!\"\"F1F(F(F(" }} {PARA 13 "" 1 "" {GLPLOT2D 541 210 210 {PLOTDATA 2 "6%-%'CURVESG6$7ar7 $$!#7\"\"!$!3%y&pL$)4WrW!#>7$$!3)******\\TVQ<\"!#;$!3_e<$RpxXF'F-7$$!3 '*******HooZ6F1$!3\\pTi:>LBxF-7$$!3&****\\_7G\\7\"F1$!3(zK9-I$*[g)F-7$ $!3$*****\\?%p@5\"F1$!3[bfEH/#*p!*F-7$$!3'***\\i[@P*3\"F1$!3K&zk#pg-K \"*F-7$$!3+++vw[dw5F1$!33OVF>:[W!*F-7$$!3-+]([gxP1\"F1$!3T++4!eJ^!))F- 7$$!31+++L.)40\"F1$!3sDe#G<%>9%)F-7$$!3\"******>8;_-\"F1$!3seB1hLc!=(F -7$$!3Q******4$>X***!#<$!3sE(y(yFin$\"3u)pa&zf,p7Fap7$$!3g*****\\f #*4v(Fin$\"3IfL'R?@LG\"Fap7$$!3U******pD6CwFin$\"3Q(Hl\"4t7x7Fap7$$!39 +++XDB(\\(Fin$\"3#=0H%pR%)\\7Fap7$$!3#*****\\U\"4iB(Fin$\"3jpXD?i^E6Fa p7$$!3q******Rd=vpFin$\"3Q$*Q/)fQy9*F-7$$!3%*******HmO:lFin$\"3C:RwPSm JNF-7$$!3e******>#>x*fFin$!3q`m&Rc'>&p%F-7$$!3;++++j%zZ&Fin$!3*[SO$**) zhJ\"Fap7$$!3*)******Rv\\F_Fin$!3!QO(=@Rtk;Fap7$$!3i******z([q(\\Fin$! 3)=+1>zk#R>Fap7$$!3=++DYILj[Fin$!3=n`)G1BG.#Fap7$$!3')****\\7th\\ZFin$ !3osLKTR(R5#Fap7$$!3o**\\iX%fFp%Fin$!3:g'\\6&=`I@Fap7$$!3`***\\(y:!fj% Fin$!3;)GyO*4x]@Fap7$$!3E+](=rV!zXFin$!3qQ4Kl_Zk@Fap7$$!34+++Xe=AXFin$ !3MqO)[sS9<#Fap7$$!3i+](=MwXX%Fin$!3yWQl;!*oq@Fap7$$!3E++vQo'pQ%Fin$!3 cO#R;uU)f@Fap7$$!3\"***\\iNtN>VFin$!3=*f8iYL'Q@Fap7$$!3c****\\Kyu^UFin $!3e3/=6A#o5#Fap7$$!3s***\\i#)Gl6%Fin$!3HM`q%=#f5?Fap7$$!3!*******>)48 )RFin$!3%G#p=QG()p=Fap7$$!3!*******\\L@_PFin$!3w5D\\855G:Fap7$$!3!**** ***zo6BNFin$!3'G+2rfNo0\"Fap7$$!3-++]AKicKFin$!3)R@srB>W_$F-7$$!3;+++l &H,*HFin$\"3up*474!4Y]F-7$$!3U++]AvBaFFin$\"31%Gkyk*[r8Fap7$$!3p+++![X $=DFin$\"39>Cw?>pjM`eKb*Fap7$$!3x%***\\i)o!RRFap $\"3e)pW#*\\%RV(*Fap7$$!3E(****\\2Lbm#Fap$\"31V%**e>-?))*Fap7$$!3])** \\7=l(G?Fap$\"3Ti@&o*GaJ**Fap7$$!3u****\\(G(*>R\"Fap$\"3+%fL!**ptn**Fa p7$$!3y4+]PRH_vF-$\"3yL>/?l\\!***Fap7$$!3=A+++]h%=\"F-$\"37j8>;hw****F ap7$$\"3@y**\\7QhHaF-$\"3g7P=ts3&***Fap7$$\"3'y***\\iUQ/7Fap$\"3)[\"Q+ P=%e(**Fap7$$\"3!z**\\P92e'=Fap$\"3eS()3..3U**Fap7$$\"3%z****\\-Is_#Fa p$\"3+NbtE7*Q*)*Fap7$$\"3-)***\\(yv+&QFap$\"3iS=\"**=tZv*Fap7$$\"33)** ***\\:#H<&Fap$\"3/OkJ?R%*f&*Fap7$$\"3K(******z.gZ(Fap$\"3'Qj/&yq<%4*Fa p7$$\"3a'*****\\g3z(*Fap$\"3Wm!y*>bl![)Fap7$$\"33++](zZmA\"Fin$\"3'4FD ,ctUn(Fap7$$\"3_+++!*pQv9Fin$\"39&zl'*ycqu'Fap7$$\"3%)*****\\\"*eBt\"F in$\"3+A-@W;I(p&Fap7$$\"3;******R3L*)>Fin$\"31<:*G)*GHf%Fap7$$\"3A**** *\\tF2C#Fin$\"3Q(=\"p\"\\d#)\\$Fap7$$\"3G******HY7#\\#Fin$\"3v5k,SdpEC Fap7$$\"3f****\\_SONFFin$\"3XSpLm*)eW9Fap7$$\"3!******\\Z.'yHFin$\"3o) fX3L/yW&F-7$$\"3y****\\#Qz'[KFin$!3oC*QE?>+H$F-7$$\"3l*******Gb(=NFin$ !3;:c^A:jY5Fap7$$\"3)*)****\\]I9w$Fin$!3EIEk:xOW:Fap7$$\"3J)*****>d5/S Fin$!3ev<>$e_n*=Fap7$$\"3%))****\\^fO8%Fin$!3PYh)=1B_-#Fap7$$\"3Q***** *4L@jUFin$!33n#*3TF(H6#Fap7$$\"3>****\\2-*zK%Fin$!3#3bqVNF>9#Fap7$$\"3 ))*****\\5nFR%Fin$!3W*e^h-w6;#Fap7$$\"3g++]-SadWFin$!3c'pknwJ4<#Fap7$$ \"3U++++4KAXFin$!3?09a*\\Fin$!3`L+2,%*\\D>Fap7$$\"39++]Z$)e[_Fin$!36V [)>S\")yj\"Fap7$$\"3P,++?eF0bFin$!3Ecr%Q7iZF\"Fap7$$\"3Q)****\\j@$))fF in$!3!)=xfqe$H&[F-7$$\"3K+++&pVK\\'Fin$\"3)QF\"pM=H6KF-7$$\"3)*)****** H(*o)pFin$\"3AJ!R#*[R4E*F-7$$\"3%*)***\\(3-`C(Fin$\"3v!QR%4^KK6Fap7$$ \"3*))****\\(oq.vFin$\"3-QDY0Ev^7Fap7$$\"3a***\\i8W\"GwFin$\"3k&yA- " 0 "" {MPLTEXT 1 0 1 ";" }}}{PARA 0 " " 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 22 "Note that by defin ing " }{XPPEDIT 18 0 "f(x)" "6#-%\"fG6#%\"xG" }{TEXT -1 26 " to have t he value 1 when " }{XPPEDIT 18 0 "x=0" "6#/%\"xG\"\"!" }{TEXT -1 17 ", we ensure that " }{XPPEDIT 18 0 "f(x)" "6#-%\"fG6#%\"xG" }{TEXT -1 18 " is continuous at " }{XPPEDIT 18 0 "x=0" "6#/%\"xG\"\"!" }{TEXT -1 1 "." }}{PARA 0 "" 0 "" {TEXT -1 32 "We take a set of equally space d " }{TEXT 268 1 "x" }{TEXT -1 71 " values to find a degree 12 interpo lating polynomial approximation for " }{XPPEDIT 18 0 "f(x);" "6#-%\"fG 6#%\"xG" }{TEXT -1 17 " on the interval " }{XPPEDIT 18 0 "[0, pi];" "6 #7$\"\"!%#piG" }{TEXT -1 168 ".\nWe increase the precision to 20 digit s for the calculations to reduce rounding errors and retain 15 digit v alues for the coefficients of the interpolating polynomial " } {XPPEDIT 18 0 "p(x)" "6#-%\"pG6#%\"xG" }{TEXT -1 2 ". " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 317 "f := p roc(x) if x<>0 then sin(x)/x else 1 end if end proc; \nDigits := 20:\n left := 0;\nright := Pi;\ndeg := 12;\nh := abs(right-left)/deg;\nxvals := [seq(left+h*i,i=0..deg)];\nxvals := evalf(xvals);\nyvals := map(f, evalf(xvals)); \nx := 'x':\npx := interp(xvals,yvals,x):\nDigits := 15 :\np := unapply(evalf(px),x);\nDigits := 10:" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"fGf*6#%\"xG6\"F(F(@%09$\"\"!*&-%$sinG6#F+\"\"\"F+! \"\"F1F(F(F(" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%%leftG\"\"!" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#>%&rightG%#PiG" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%$degG\"#7" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"hG,$ *&\"#7!\"\"%#PiG\"\"\"F*" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%&xvalsG7 /\"\"!,$*&\"#7!\"\"%#PiG\"\"\"F,,$*&\"\"'F*F+F,F,,$*&\"\"%F*F+F,F,,$*& \"\"$F*F+F,F,,$*(\"\"&F,F)F*F+F,F,,$*&\"\"#F*F+F,F,,$*(\"\"(F,F)F*F+F, F,,$*(F;F,F5F*F+F,F,,$*(F5F,F2F*F+F,F,,$*(F8F,F/F*F+F,F,,$*(\"#6F,F)F* F+F,F,F+" }}{PARA 12 "" 1 "" {XPPMATH 20 "6#>%&xvalsG7/$\"\"!F'$\"5aO% \\\"*z(Q*zh#!#?$\"54t))H)fv()fB&F*$\"5i4$[uRj\")R&yF*$\"5iu(f'>^v>Z5!# >$\"5F=Zd**Qp**38F1$\"5#>m*[zEjzq:F1$\"5e0YSf9dfK=F1$\"5B\\&>$R-^R%4#F 1$\"5*G\\M#>!\\%>cBF1$F)F1$\"5?!Qk!zlKzzGF1$\"5&QKz*e`EfTJF1" }}{PARA 12 "" 1 "" {XPPMATH 20 "6#>%&yvalsG7/\"\"\"$\"5Q>#p`YHfh))*!#?$\"5h9?P ^&e'H\\&*F)$\"5bpg5dhJ;.!*F)$\"5Eu!)oKJM$*p#)F)$\"5h&[PtevH\"ztF)$\"54 V8enBx>mjF)$\"5*R?)4[op!3F&F)$\"59PSMm:n'\\8%F)$\"5\\c`.>(Qa5+$F)$\"5# HSu-rJf)4>F)$\"5\"GcE%fSvT()*)!#@$!5o%GH^;/)4*=\"!#R" }}{PARA 12 "" 1 "" {XPPMATH 20 "6#>%\"pGf*6#%\"xG6\"6$%)operatorG%&arrowGF(,<*&$\"0RKe %RfPrNFBF1)F3\"\"*F1F1*&$\"0#*)pO%\\B:$FBF1)F3\"\"$F1F1*& $\"0AMJ)3GnE!#?F1)F3\"\")F1F1*&$\"0Ce)yaCL$)!#F1)F3\"\"(F1F1*&$\"0@G3&z\\_:F>F1)F3\"\"&F1F1*&$\"0X76Fuf)>!# =F1)F3\"\"'F1FE$F1\"\"!F1F(F(F(" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }} {PARA 0 "" 0 "" {TEXT -1 94 "The graph of the interpolating polynomial can be plotted along with the interpolation points.\n" 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-Fjo6#%%LINEGF]p-%+AXESLABELSG6$Q\"x6\"Q!F[al-%%VIEWG6$;F($\"+aEfTJ!\" *%(DEFAULTG" 1 2 0 1 10 0 2 9 1 4 2 1.000000 45.000000 45.000000 0 0 " Curve 1" "Curve 2" "Curve 3" "Curve 4" }}}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 33 "The absolute error curve is . . ." } }{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 41 "plot(p(x)-f(x),x=left..right,color=blue);" }}{PARA 13 "" 1 "" {GLPLOT2D 491 331 331 {PLOTDATA 2 "6&-%'CURVESG6#7jp7$$\"3:+++/1$*R@!# ?$\"3vqNfbWKA6!#H7$$\"3I+++37')zUF*$\"3c$=O:e\\#)=#F-7$$\"3e******4=z> kF*$\"39,(ppvi'*>$F-7$$\"3;+++:Csf&)F*$\"3wtO_\\i*y:%F-7$$\"35+++i$eRG \"!#>$\"30N/c-PPAfF-7$$\"3.+++$[W>r\"F@$\"3Ip>%ph/]\\(F-7$$\"3%)*****R s;zc#F@$\"3hF_ o$\"3F,0lr=(*=YF-7$$\"37+++![:zG#F_o$\"3!**oqAS>(y=F-7$$\"32+++#3o^i#F _o$!3(R&HTrnk>K!#J7$$\"3E+++\"y<3'HF_o$!33E3VQ?2g6F-7$$\"3!*******zuY' H$F_o$!3w$Gq.-&yd;F-7$$\"3;+++S]EuLF_o$!3\"\\^Sv!y_,E(!#I7$$\"35+++?n_'*[F_o$!3E(3=c&F_o$\"3%*4y 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)***\\/l#fTJF`x$\"3yX1s\\K%**o$F^u-%'COLOURG6&%$RGBG$\"\"!FdilFcil$\"* ++++\"!\")-%+AXESLABELSG6$Q\"x6\"Q!F\\jl-%%VIEWG6$;Fcil$\"+aEfTJ!\"*%( DEFAULTG" 1 2 0 1 10 0 2 6 1 4 2 1.000000 45.000000 45.000000 0 0 "Cur ve 1" }}}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 4 " The " }{TEXT 259 22 "maximum absolute error" }{TEXT -1 25 " in using t he polynomial " }{XPPEDIT 18 0 "p(x)" "6#-%\"pG6#%\"xG" }{TEXT -1 16 " to approximate " }{XPPEDIT 18 0 "f(x) = PIECEWISE([sin(x)/x, x <> 0], [1, x = 0])" "6#/-%\"fG6#%\"xG-%*PIECEWISEG6$7$*&-%$sinG6#F'\"\"\"F'! \"\"0F'\"\"!7$F0/F'F3" }{TEXT -1 17 " on the interval " }{XPPEDIT 18 0 "[0, Pi];" "6#7$\"\"!%#PiG" }{TEXT -1 14 " is about 1.6 " }{TEXT 274 1 "x" }{TEXT -1 1 " " }{XPPEDIT 18 0 "10^(-11);" "6#)\"#5,$\"#6!\" \"" }{TEXT -1 1 "." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 " > " 0 "" {MPLTEXT 1 0 38 "numapprox[infnorm](p(x)-f(x),x=0..Pi);" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#$\"+].'=d\"!#?" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 5 "Tasks" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 2 "Q1 " }}{PARA 0 "" 0 "" {TEXT -1 46 "(a) Find a degree 10 polynomial appro ximation " }{XPPEDIT 18 0 "p(x)" "6#-%\"pG6#%\"xG" }{TEXT -1 5 " for \+ " }{XPPEDIT 18 0 "arctan(x);" "6#-%'arctanG6#%\"xG" }{TEXT -1 17 " on \+ the interval " }{XPPEDIT 18 0 "[0, 1];" "6#7$\"\"!\"\"\"" }{TEXT -1 34 ", which agrees with the values of " }{XPPEDIT 18 0 "arctan(x)" "6# -%'arctanG6#%\"xG" }{TEXT -1 32 " at 11 equally spaced values of " } {TEXT 267 1 "x" }{TEXT -1 55 " between 0 and 1 inclusive. You may use the procedure " }{TEXT 0 6 "interp" }{TEXT -1 1 "." }}{PARA 0 "" 0 " " {TEXT 259 4 "Note" }{TEXT -1 2 ": " }{TEXT 261 1 "I" }{TEXT 266 21 " ncrease the precision" }{TEXT -1 67 " from 10 digits to 15 or 20 digit s when performing the calculation." }}{PARA 0 "" 0 "" {TEXT -1 44 "(b) Illustrate the interpolating polynomial " }{XPPEDIT 18 0 "p(x)" "6#-% \"pG6#%\"xG" }{TEXT -1 137 " from part (a) and the interpolation point s in a graph.\n(c) Plot absolute and relative error curves arising in \+ the use of the polynomial " }{XPPEDIT 18 0 "p(x)" "6#-%\"pG6#%\"xG" } {TEXT -1 16 " to approximate " }{XPPEDIT 18 0 "arctan(x)" "6#-%'arctan G6#%\"xG" }{TEXT -1 17 " in the interval " }{XPPEDIT 18 0 "[0, 1];" "6 #7$\"\"!\"\"\"" }{TEXT -1 72 ", and estimate the maximum absolute and \+ relative errors on the interval " }{XPPEDIT 18 0 "[0,1]" "6#7$\"\"!\" \"\"" }{TEXT -1 1 "." }}{PARA 0 "" 0 "" {TEXT -1 39 "_________________ ______________________" }}{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 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" } }}{PARA 0 "" 0 "" {TEXT -1 39 "_______________________________________ " }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}}{SECT 1 {PARA 4 " " 0 "" {TEXT -1 2 "Q2" }}{PARA 0 "" 0 "" {TEXT -1 47 "(a) Find a degr ee 10 polynomial approximation " }{XPPEDIT 18 0 "p(x)" "6#-%\"pG6#%\"x G" }{TEXT -1 5 " for " }{XPPEDIT 18 0 "f(x) = exp(-x^3);" "6#/-%\"fG6# %\"xG-%$expG6#,$*$F'\"\"$!\"\"" }{TEXT -1 17 " on the interval " } {XPPEDIT 18 0 "[0, 1];" "6#7$\"\"!\"\"\"" }{TEXT -1 33 " which agrees \+ with the values of " }{XPPEDIT 18 0 "f(x)" "6#-%\"fG6#%\"xG" }{TEXT -1 7 " at 11 " }{TEXT 262 14 "equally spaced" }{TEXT -1 11 " values of " }{TEXT 264 1 "x" }{TEXT -1 15 " between 0 and " }{XPPEDIT 18 0 "1; " "6#\"\"\"" }{TEXT -1 11 " inclusive." }}{PARA 0 "" 0 "" {TEXT -1 17 "(b) Estimate the " }{TEXT 262 22 "maximum absolute error" }{TEXT -1 25 " in using the polynomial " }{XPPEDIT 18 0 "p(x)" "6#-%\"pG6#%\"xG " }{TEXT -1 16 " to approximate " }{XPPEDIT 18 0 "f(x)" "6#-%\"fG6#%\" xG" }{TEXT -1 17 " in the interval " }{XPPEDIT 18 0 "[0, 1];" "6#7$\" \"!\"\"\"" }{TEXT -1 1 "." }}{PARA 0 "" 0 "" {TEXT -1 56 "(c) Compare \+ the values of your interpolating polynomial " }{XPPEDIT 18 0 "p(x)" "6 #-%\"pG6#%\"xG" }{TEXT -1 18 " and the function " }{XPPEDIT 18 0 "f(x) " "6#-%\"fG6#%\"xG" }{TEXT -1 5 " for " }{XPPEDIT 18 0 "x = Pi/4;" "6# /%\"xG*&%#PiG\"\"\"\"\"%!\"\"" }{TEXT -1 85 ".\n What is the relat ive error in using the first value to approximate the second?" }} {PARA 0 "" 0 "" {TEXT -1 39 "_______________________________________" }}{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 0 "" } }{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{PARA 0 "" 0 "" {TEXT -1 39 "_______________________________________" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 2 "Q3" }} {PARA 0 "" 0 "" {TEXT -1 4 "Let " }{TEXT 262 2 "Sn" }{TEXT -1 94 " be \+ the numerical \"black box\" function defined by the procedure in the \+ following subsection. " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 40 "code for \"black box\" numerical function " }{TEXT 0 2 "Sn" } {TEXT -1 2 " " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1470 "Sn := proc(x::realcons)\n local xx,eps,saveD igits,doS,val,p,q,maxit;\n \n if x=0 then return 0. end if;\n\n \+ doS := proc(x,eps,maxit)\n local t,p,s,u,v,h,i; \n # set up \+ a starting approximation\n if x<2.6 and x>-2.6 then \n s \+ := .9741807232*x-12.95848365/(x+32.34978388/\n (x+.702777931 1/(x-20.37123456/(x+35.08632448/(x-5.546018485/x)))));\n elif x>0 then \n s := x-1.570796327+(.2346004180+.4696458221*x)/\n \+ (1.+(-.6231393515+.4753495862*x)*x);\n else\n s := x+1.570796327+(-.2346004180+.4696458221*x)/\n (1.+(.6 231393515+.4753495862*x)*x);\n end if;\n # solve the equatio n x=s+arctan(s) for s by Halley's method \n for i to maxit do\n \+ t := s+arctan(s)-x;\n p := 1+s^2;\n u := 1+1/p; \n v := -2*s/p^2;\n h := t/(u-1/2*v*t/u);\n s \+ := s-h;\n if abs(h)<=eps*abs(s) then break end if;\n end \+ do;\n s;\n end proc;\n\n p := ilog10(Digits);\n q := Float( Digits,-p);\n maxit := trunc((p+(.02331061386+.1111111111*q))*2.0959 03274)+2;\n saveDigits := Digits;\n Digits := Digits+min(iquo(Digi ts,3),5);\n xx := evalf(x);\n eps := Float(3,-saveDigits-1);\n i f Digits<=trunc(evalhf(Digits)) and \n ilog10(xx)trunc(evalhf(DBL_MIN_10_EX P)) then\n val := evalhf(doS(xx,eps,maxit))\n else\n val : = doS(xx,eps,maxit)\n end if;\n evalf[saveDigits](val);\nend proc: " }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 " " {TEXT -1 46 "(a) Find a degree 12 polynomial approximation " } {XPPEDIT 18 0 "p(x)" "6#-%\"pG6#%\"xG" }{TEXT -1 5 " for " }{TEXT 262 5 "Sn(x)" }{TEXT -1 17 " on the interval " }{XPPEDIT 18 0 "[0, 3];" "6 #7$\"\"!\"\"$" }{TEXT -1 34 ", which agrees with the values of " } {TEXT 262 5 "Sn(x)" }{TEXT -1 7 " at 13 " }{TEXT 262 14 "equally space d" }{TEXT -1 11 " values of " }{TEXT 278 1 "x" }{TEXT -1 27 " between \+ 0 and 3 inclusive." }}{PARA 0 "" 0 "" {TEXT -1 84 "(b) Illustrate the \+ interpolating polynomial and the interpolation points in a graph." }} {PARA 0 "" 0 "" {TEXT -1 71 "(c) Plot the absolute error curve arising in the use of the polynomial " }{XPPEDIT 18 0 "p(x)" "6#-%\"pG6#%\"xG " }{TEXT -1 16 " to approximate " }{TEXT 262 5 "Sn(x)" }{TEXT -1 17 " \+ in the interval " }{XPPEDIT 18 0 "[0, 3];" "6#7$\"\"!\"\"$" }{TEXT -1 19 ", and estimate the " }{TEXT 262 22 "maximum absolute error" } {TEXT -1 17 " on the interval " }{XPPEDIT 18 0 "[0, 3];" "6#7$\"\"!\" \"$" }{TEXT -1 1 "." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 " " {TEXT 259 5 "Notes" }{TEXT -1 2 ": " }}{PARA 15 "" 0 "" {TEXT -1 11 "When using " }{TEXT 262 5 "Sn(x)" }{TEXT -1 30 " as (part of) an argu ment for " }{TEXT 0 4 "plot" }{TEXT -1 35 ", it must be enclosed in qu otes as " }{TEXT 0 7 "'Sn(x)'" }{TEXT -1 3 " or" }{TEXT 0 8 " 'Sn'(x) " }{TEXT -1 2 ". " }}{PARA 15 "" 0 "" {TEXT -1 13 "The function " } {TEXT 262 5 "Sn(x)" }{TEXT -1 47 " provides a numerical inverse for th e function " }{XPPEDIT 18 0 "g(x)=x+arctan(x)" "6#/-%\"gG6#%\"xG,&F'\" \"\"-%'arctanG6#F'F)" }{TEXT -1 2 ". " }}{PARA 0 "" 0 "" {TEXT -1 39 " _______________________________________" }}{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 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 39 "____________ ___________________________" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 2 "Q4" }}{PARA 257 "" 0 "" {TEXT -1 46 "(a) Find a degree 12 polynomial approximation " } {XPPEDIT 18 0 "p(x)" "6#-%\"pG6#%\"xG" }{TEXT -1 5 " for " }{XPPEDIT 18 0 "f(x)=ln(x+1)/(1+sin(x))" "6#/-%\"fG6#%\"xG*&-%#lnG6#,&F'\"\"\"F- F-F-,&F-F--%$sinG6#F'F-!\"\"" }{TEXT -1 17 " on the interval " } {XPPEDIT 18 0 "0<=x" "6#1\"\"!%\"xG" }{XPPEDIT 18 0 "`` <= 2;" "6#1%!G \"\"#" }{TEXT -1 33 " which agrees with the values of " }{XPPEDIT 18 0 "f(x)" "6#-%\"fG6#%\"xG" }{TEXT -1 7 " at 13 " }{TEXT 279 14 "equall y spaced" }{TEXT -1 11 " values of " }{TEXT 280 1 "x" }{TEXT -1 15 " b etween 0 and " }{XPPEDIT 18 0 "2;" "6#\"\"#" }{TEXT -1 11 " inclusive. " }}{PARA 0 "" 0 "" {TEXT -1 51 "(b) Plot the graph of the interpolati ng polynomial " }{XPPEDIT 18 0 "p(x)" "6#-%\"pG6#%\"xG" }{TEXT -1 83 " found in part (a) together with the interpolation points used in its \+ construction." }}{PARA 0 "" 0 "" {TEXT -1 17 "(c) Estimate the " } {TEXT 279 22 "maximum absolute error" }{TEXT -1 25 " in using the poly nomial " }{XPPEDIT 18 0 "p(x)" "6#-%\"pG6#%\"xG" }{TEXT -1 43 " found \+ in (a) to approximate the function " }{XPPEDIT 18 0 "f(x)=ln(x+1)/(1+ sin(x))" "6#/-%\"fG6#%\"xG*&-%#lnG6#,&F'\"\"\"F-F-F-,&F-F--%$sinG6#F'F -!\"\"" }{TEXT -1 17 " on the interval " }{XPPEDIT 18 0 "0<=x" "6#1\" \"!%\"xG" }{XPPEDIT 18 0 "`` <= 2;" "6#1%!G\"\"#" }{TEXT -1 1 "." }} {PARA 0 "" 0 "" {TEXT -1 39 "_______________________________________" }}{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 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 39 "_______________________________________" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 2 "Q5" }}{PARA 257 "" 0 "" {TEXT -1 46 "(a) Find a degree 10 \+ polynomial approximation " }{XPPEDIT 18 0 "p(x)" "6#-%\"pG6#%\"xG" } {TEXT -1 5 " for " }{XPPEDIT 18 0 "f(x) = arctan(x)/(x^2-3*x+4);" "6#/ -%\"fG6#%\"xG*&-%'arctanG6#F'\"\"\",(*$F'\"\"#F,*&\"\"$F,F'F,!\"\"\"\" %F,F2" }{TEXT -1 17 " on the interval " }{XPPEDIT 18 0 "0<=x" "6#1\"\" !%\"xG" }{XPPEDIT 18 0 "`` <= 2;" "6#1%!G\"\"#" }{TEXT -1 33 " which a grees with the values of " }{XPPEDIT 18 0 "f(x)" "6#-%\"fG6#%\"xG" } {TEXT -1 7 " at 11 " }{TEXT 279 14 "equally spaced" }{TEXT -1 11 " val ues of " }{TEXT 281 1 "x" }{TEXT -1 15 " between 0 and " }{XPPEDIT 18 0 "2;" "6#\"\"#" }{TEXT -1 11 " inclusive." }}{PARA 0 "" 0 "" {TEXT -1 51 "(b) Plot the graph of the interpolating polynomial " }{XPPEDIT 18 0 "p(x)" "6#-%\"pG6#%\"xG" }{TEXT -1 83 " found in part (a) togethe r with the interpolation points used in its construction." }}{PARA 0 " " 0 "" {TEXT -1 17 "(c) Estimate the " }{TEXT 279 22 "maximum absolute error" }{TEXT -1 25 " in using the polynomial " }{XPPEDIT 18 0 "p(x) " "6#-%\"pG6#%\"xG" }{TEXT -1 42 " found in (a) to approximate the fun ction " }{XPPEDIT 18 0 "f(x) = arctan(x)/(x^2-3*x+4);" "6#/-%\"fG6#%\" xG*&-%'arctanG6#F'\"\"\",(*$F'\"\"#F,*&\"\"$F,F'F,!\"\"\"\"%F,F2" } {TEXT -1 17 " on the interval " }{XPPEDIT 18 0 "0<=x" "6#1\"\"!%\"xG" }{XPPEDIT 18 0 "`` <= 2;" "6#1%!G\"\"#" }}{PARA 0 "" 0 "" {TEXT -1 39 "_______________________________________" }}{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 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 39 "____________ ___________________________" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}}}{MARK "4 0 \+ 0" 0 }{VIEWOPTS 1 1 0 1 1 1803 1 1 1 1 }{PAGENUMBERS 0 1 2 33 1 1 }