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" }}{PARA 0 "" 0 "" {TEXT -1 123 "It can be read into a Maple session by a command similar to the one that follows, where the file path gives its location. " } }{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 36 "read \"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 35 "Definition of Chebyshev polynomials" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 179 "simplify(ex pand(cos(2*arccos(x))));\nsimplify(expand(cos(3*arccos(x))));\nsimplif y(expand(cos(4*arccos(x))));\nsimplify(expand(cos(5*arccos(x))));\nsim plify(expand(cos(6*arccos(x))));" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }} {PARA 0 "" 0 "" {TEXT -1 4 "The " }{TEXT 267 21 "Chebyshev polynomials " }{TEXT -1 79 " of the 1st kind are based on the following family of \+ trigonometric expansions." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 134 "cos(2*u)=expand(cos(2*u));\ncos(3* u)=expand(cos(3*u));\ncos(4*u)=expand(cos(4*u));\ncos(5*u)=expand(cos( 5*u));\ncos(6*u)=expand(cos(6*u));" }}{PARA 11 "" 1 "" {XPPMATH 20 "6# /-%$cosG6#,$*&\"\"#\"\"\"%\"uGF*F*,&*&F)F*)-F%6#F+F)F*F*F*!\"\"" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#/-%$cosG6#,$*&\"\"$\"\"\"%\"uGF*F*,&*& \"\"%F*)-F%6#F+F)F*F**&F)F*F0F*!\"\"" }}{PARA 11 "" 1 "" {XPPMATH 20 " 6#/-%$cosG6#,$*&\"\"%\"\"\"%\"uGF*F*,(*&\"\")F*)-F%6#F+F)F*F**&F.F*)F0 \"\"#F*!\"\"F*F*" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/-%$cosG6#,$*&\"\" &\"\"\"%\"uGF*F*,(*&\"#;F*)-F%6#F+F)F*F**&\"#?F*)F0\"\"$F*!\"\"*&F)F*F 0F*F*" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/-%$cosG6#,$*&\"\"'\"\"\"%\"u GF*F*,**&\"#KF*)-F%6#F+F)F*F**&\"#[F*)F0\"\"%F*!\"\"*&\"#=F*)F0\"\"#F* F*F*F6" }}}{PARA 0 "" 0 "" {TEXT -1 34 "\nIf we replace each occurrenc e of " }{XPPEDIT 18 0 "cos(u)" "6#-%$cosG6#%\"uG" }{TEXT -1 4 " by " } {TEXT 279 1 "x" }{TEXT -1 105 ", we get the family of Chebyshev polyno mials of the 1st kind.\nThis can be done automatically as follows.\n" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 129 "expand(cos(2*arccos(x))); \nexpand(cos(3*arccos(x)));\nexpand(cos(4*arccos(x)));\nexpand(cos(5*a rccos(x)));\nexpand(cos(6*arccos(x)));" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#,&*$)%\"xG\"\"#\"\"\"F'F(!\"\"" }}{PARA 11 "" 1 "" {XPPMATH 20 " 6#,&*$)%\"xG\"\"$\"\"\"\"\"%*&F'F(F&F(!\"\"" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#,(*$)%\"xG\"\"%\"\"\"\"\")*&F)F()F&\"\"#F(!\"\"F(F(" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#,(*$)%\"xG\"\"&\"\"\"\"#;*&\"#?F()F&\" \"$F(!\"\"*&F'F(F&F(F(" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#,**$)%\"xG\" \"'\"\"\"\"#K*&\"#[F()F&\"\"%F(!\"\"*&\"#=F()F&\"\"#F(F(F(F." }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 9 "Thus the " }{TEXT 278 1 "n" } {TEXT -1 57 " th Chebyshev polynomial of the first kind is defined by \+ " }{XPPEDIT 18 0 "T[n](x) = cos(n*arccos*x);" "6#/-&%\"TG6#%\"nG6#%\"x G-%$cosG6#*(F(\"\"\"%'arccosGF/F*F/" }{TEXT -1 1 "." }}{PARA 0 "" 0 " " {TEXT -1 56 "These polynomials are available via the Maple procedure " }{TEXT 0 10 "ChebyshevT" }{TEXT -1 2 ". " }}{PARA 0 "" 0 "" {TEXT -1 64 "To see the polynomials in the same form as above we need to use " }{TEXT 0 6 "expand" }{TEXT -1 1 "." }}{PARA 0 "" 0 "" {TEXT -1 0 " " }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 27 "ChebyshevT(6,x);\nexpand (%);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#-%+ChebyshevTG6$\"\"'%\"xG" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#,**$)%\"xG\"\"'\"\"\"\"#K*&\"#[F()F&\" \"%F(!\"\"*&\"#=F()F&\"\"#F(F(F(F." }}}{PARA 0 "" 0 "" {TEXT -1 0 "" } }{PARA 0 "" 0 "" {TEXT -1 67 "They are generally used with the domain \+ restricted to the interval " }{XPPEDIT 18 0 "[-1,1]" "6#7$,$\"\"\"!\" \"F%" }{TEXT -1 26 " in line with the formula " }{XPPEDIT 18 0 "T[n](x ) = cos(n*arccos*x);" "6#/-&%\"TG6#%\"nG6#%\"xG-%$cosG6#*(F(\"\"\"%'ar ccosGF/F*F/" }{TEXT -1 1 "." }}{PARA 0 "" 0 "" {TEXT -1 47 "The values will then also lie in the interval. 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" }} {PARA 0 "" 0 "" {TEXT -1 13 "Substituting " }{XPPEDIT 18 0 "theta = ar ccos*x;" "6#/%&thetaG*&%'arccosG\"\"\"%\"xGF'" }{TEXT -1 14 " implies \+ that " }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "cos(n*arccos *x) = 2*x*cos((n-1)*arccos*x)-cos((n-2)*arccos*x);" "6#/-%$cosG6#*(%\" nG\"\"\"%'arccosGF)%\"xGF),&*(\"\"#F)F+F)-F%6#*(,&F(F)F)!\"\"F)F*F)F+F )F)F)-F%6#*(,&F(F)F.F3F)F*F)F+F)F3" }{TEXT -1 1 " " }}{PARA 0 "" 0 "" {TEXT -1 4 "for " }{XPPEDIT 18 0 "n>=2" "6#1\"\"#%\"nG" }{TEXT -1 6 " \+ when " }{XPPEDIT 18 0 "-1<=x" "6#1,$\"\"\"!\"\"%\"xG" }{XPPEDIT 18 0 " ``<=1" "6#1%!G\"\"\"" }{TEXT -1 2 ". " }}{PARA 0 "" 0 "" {TEXT -1 0 " " }}{PARA 0 "" 0 "" {TEXT -1 82 "It follows that the Chebyshev polynom ials of the first kind satisfy the following " }{TEXT 267 19 "recurren ce relation" }{TEXT -1 2 ": " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "T[n](x) = 2*x*T[n-1](x)-T[n -2](x);" "6#/-&%\"TG6#%\"nG6#%\"xG,&*(\"\"#\"\"\"F*F.-&F&6#,&F(F.F.!\" \"6#F*F.F.-&F&6#,&F(F.F-F36#F*F3" }{TEXT -1 1 " " }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{TEXT 280 18 "__________________" }{TEXT -1 1 " " }} {PARA 0 "" 0 "" {TEXT -1 5 "when " }{XPPEDIT 18 0 "n>=2" "6#1\"\"#%\"n G" }{TEXT -1 2 ". " }}{PARA 0 "" 0 "" {TEXT -1 68 "They are therefore \+ determined completely by the inductive formulas: " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "PIECE WISE([T[0](x) = 1, ``],[T[1](x) = x, ``],[T[n](x) = 2*x*T[n-1](x)-T[n- 2](x), 1 < n]);" "6#-%*PIECEWISEG6%7$/-&%\"TG6#\"\"!6#%\"xG\"\"\"%!G7$ /-&F*6#F/6#F.F.F07$/-&F*6#%\"nG6#F.,&*(\"\"#F/F.F/-&F*6#,&F " 0 "" {MPLTEXT 1 0 272 "ChebT := proc(n::nonnegint,x)\n local \+ i,t,prevt,newt;\n prevt := 1;\n if n=0 then return prevt end if;\n t := x;\n if n=1 then return t end if;\n for i from 1 to n-1 do \n newt := 2*x*t - prevt;\n prevt := t;\n t := newt;\n \+ end do;\n return t;\nend proc:" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" } }{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 24 "ChebT(6,x);\nsimplify(%);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#,**(\"\"#\"\"\"%\"xGF&,(*(F%F&F'F&,( *(F%F&F'F&,&*(F%F&F'F&,&*&F%F&)F'F%F&F&F&!\"\"F&F&F'F1F&F&*&F%F&F0F&F1 F&F&F&F&*(F%F&F'F&F.F&F1F'F&F&F&*(F%F&F'F&F,F&F1*&F%F&F0F&F&F&F1" }} 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t7$Feo$!3O&>@iHpW>(F_t7$Fho$!3s+HJk];n>F07$F[p$!3%3[0FBL%oIF07$F^p$!3- E*G:>][>%F07$Fap$!3_u;nCg6%G&F07$Fdp$!3)oqI^!\\!f:'F07$Fgp$!3ai*p![CH4 qF07$Fjp$!3qi'p#[.R(y(F07$F]q$!3x!y:Rm/3W)F07$F`q$!37nHGB%>hF07$Fbv$\"3+?TzoRk \"*oF07$Fev$\"3!RB178F$ovF07$Fhv$\"3P\\dv\"=@*>#)F07$F[w$\"3#RRwKKEkv) F07$F^w$\"3G3(**o)*>[A*F07$Faw$\"3Ym]Rwi=e&*F07$Fdw$\"3&QX>%e8)4#)*F07 $$\"3)>555S,#GGF0$\"3\\L5RyB71**F07$Fgw$\"3AS1S!=PU'**F07$$\"3!yooo=%> HIF0$\"3Oi(e_$G([***F07$Fjw$\"3I_4mh?&y***F07$$\"3d%HHH44WB$F0$\"3'Qr/ (3u6r**F07$F]x$\"3#oLuxmGO\"**F07$$\"3P?>>>C\"eV$F0$\"3.!*G86%4L$)*F07 $F`x$\"3!\\#>`PIlE(*F07$Fcx$\"3%GiQ@e0fU*F07$Ffx$\"3;/nQ#)R2m*)F07$Fix $\"3?+tX+X)eV)F07$F\\y$\"3g`l#3'e%[y(F07$F_y$\"3N*z))4s@W,(F07$Fby$\"3 WB$)yIx3:iF07$Fey$\"3gJSB>Vqt_F07$Fhy$\"3kQe\"yb2nD%F07$F[z$\"3s(*zR\" Rv))3$F07$F^z$\"3P[z4]xp+?F07$Faz$\"3sx8:\"HJ@$pF_t7$Fdz$!3;yP%)p$=8>& F_t7$Fgz$!3`3&oqo:Gt\"F07$Fjz$!3Xl)>$\\i%R.$F07$F][l$!3I51W)[&48VF07$F `[l$!33iKQ#*\\HJaF07$Fc[l$!3u!)GF>bZYlF07$Ff[l$!33PK!*G*4p`(F07$Fi[l$! 3]E*He]Y'o%)F07$F\\\\l$!3rhrhfS*p7*F07$F_\\l$!3O`W*H.Qwn*F07$$\"3C@??? cv!)yF0$!3%R+mD>!**[)*F07$Fb\\l$!3/Y*[D)H(z&**F07$$\"3TA@@rxjI!)F0$!3s \\)z(Q`N()**F07$$\"3sRQQQdF!3)F0$!3m!pl9jY'****F07$$\"3/dbb0P\"*H\")F0 $!3y?zRERuBI!>F_t7$Fj]l$\"3 Adt+RpYb:F07$F]cp$\"3Yj*R+X4hF$F07$F]^l$\"3OwI2lxj[^F07$$\"3GURRRIWT)* F0$\"3)p#>MX)[>G'F07$Fecp$\"33&o,P9vuY(F07$$\"3w!)zzzw9Z**F0$\"3w%zvcg >kq)F0F_^l-Fa^l6&Fc^lFf^lFd^lFd^l-%*THICKNESSG6#\"\"#-%+AXESLABELSG6$Q \"x6\"Q!F_`r-%%VIEWG6$;F(F+%(DEFAULTG" 1 2 0 1 10 2 2 9 1 4 2 1.000000 45.000000 45.000000 0 0 "Curve 1" "Curve 2" "Curve 3" "Curve \+ 4" "Curve 5" "Curve 6" }}}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 90 "Chebyshev polynomials of the first kind are also in \+ the package of orthogonal polynomials " }{TEXT 0 9 "orthopoly" }{TEXT -1 4 " as " }{TEXT 0 12 "orthopoly[T]" }{TEXT -1 2 ". " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}} {SECT 1 {PARA 4 "" 0 "" {TEXT -1 63 "Properties of Chebyshev polynomia ls .. orthogonality relations " }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}{PARA 0 "" 0 "" {TEXT -1 15 "The polynomial " }{XPPEDIT 18 0 "T[n](x);" "6#-&%\"TG6#%\"nG6#%\"xG" }{TEXT -1 5 " has " }{TEXT 281 1 "n" }{TEXT -1 23 " zeros in the interval " }{XPPEDIT 18 0 "[-1,1]" " 6#7$,$\"\"\"!\"\"F%" }{TEXT -1 24 " located at the points: " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 256 "" 0 "" {TEXT -1 2 " " }{XPPEDIT 18 0 "x = cos((k-1/2)*Pi/n);" "6#/%\"xG-%$cosG6#*(,&%\"kG\"\"\"*&F+F+ \"\"#!\"\"F.F+%#PiGF+%\"nGF." }{TEXT -1 4 ", " }{XPPEDIT 18 0 "k = 1 , 2,` . . . `, n" "6&/%\"kG\"\"\"\"\"#%(~.~.~.~G%\"nG" }{TEXT -1 1 ". " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 51 "The l ocal maximum and minimum points are located at" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "x = c os(k*Pi/n);" "6#/%\"xG-%$cosG6#*(%\"kG\"\"\"%#PiGF*%\"nG!\"\"" }{TEXT -1 4 ", " }{XPPEDIT 18 0 "k = 1, 2,` . . . `, n" "6&/%\"kG\"\"\"\"\" #%(~.~.~.~G%\"nG" }{TEXT -1 1 "." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }} {PARA 0 "" 0 "" {TEXT -1 35 "At all of the local maximum points " } {XPPEDIT 18 0 "T[n](x) = 1;" "6#/-&%\"TG6#%\"nG6#%\"xG\"\"\"" }{TEXT -1 40 " and at all of the local minimum points " }{XPPEDIT 18 0 "T[n]( x) = -1;" "6#/-&%\"TG6#%\"nG6#%\"xG,$\"\"\"!\"\"" }{TEXT -1 1 "." }} {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 38 "The Cheby shev polynomials satisfy the " }{TEXT 267 22 "orthogonality relation" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 256 "" 0 "" {TEXT -1 2 " " } {XPPEDIT 18 0 "Int(T[i](x)*T[j](x)/sqrt(1-x^2),x = -1 .. 1) = PIECEWIS E([0, i <> j],[Pi/2, `i =`*j <> 0],[Pi, `i =`*j = 0]);" "6#/-%$IntG6$* (-&%\"TG6#%\"iG6#%\"xG\"\"\"-&F*6#%\"jG6#F.F/-%%sqrtG6#,&F/F/*$F.\"\"# !\"\"F;/F.;,$F/F;F/-%*PIECEWISEG6%7$\"\"!0F,F37$*&%#PiGF/F:F;0*&%$i~=G F/F3F/FC7$FG/*&FJF/F3F/FC" }{TEXT -1 4 " . " }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{TEXT 282 21 "_____________________" }{TEXT -1 1 " " } }{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 57 "These re sults can be verified by making the subsitution " }{XPPEDIT 18 0 "x=c os*theta" "6#/%\"xG*&%$cosG\"\"\"%&thetaGF'" }{TEXT -1 22 ", or, more \+ precisely, " }{XPPEDIT 18 0 "theta=arccos*x" "6#/%&thetaG*&%'arccosG\" \"\"%\"xGF'" }{TEXT -1 18 " in the integral. " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT 267 4 "Note" }{TEXT -1 80 ": The following trigonometric formulas are useful in the subsequent discuss ion. " }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "sin*alpha+si n*beta = 2*sin((alpha+beta)/2)*cos((alpha-beta)/2);" "6#/,&*&%$sinG\" \"\"%&alphaGF'F'*&F&F'%%betaGF'F'*(\"\"#F'-F&6#*&,&F(F'F*F'F'F,!\"\"F' -%$cosG6#*&,&F(F'F*F1F'F,F1F'" }{TEXT -1 9 ", " }{XPPEDIT 18 0 "sin*alpha-sin*beta = 2*cos((alpha+beta)/2)*sin((alpha-beta)/2);" "6#/ ,&*&%$sinG\"\"\"%&alphaGF'F'*&F&F'%%betaGF'!\"\"*(\"\"#F'-%$cosG6#*&,& F(F'F*F'F'F-F+F'-F&6#*&,&F(F'F*F+F'F-F+F'" }{TEXT -1 2 ", " }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "cos*alpha+cos*beta = 2*cos( (alpha+beta)/2)*cos((alpha-beta)/2);" "6#/,&*&%$cosG\"\"\"%&alphaGF'F' *&F&F'%%betaGF'F'*(\"\"#F'-F&6#*&,&F(F'F*F'F'F,!\"\"F'-F&6#*&,&F(F'F*F 1F'F,F1F'" }{TEXT -1 7 ", " }{XPPEDIT 18 0 "cos*alpha-cos*beta = \+ -2*sin((alpha+beta)/2)*sin((alpha-beta)/2);" "6#/,&*&%$cosG\"\"\"%&alp haGF'F'*&F&F'%%betaGF'!\"\",$*(\"\"#F'-%$sinG6#*&,&F(F'F*F'F'F.F+F'-F0 6#*&,&F(F'F*F+F'F.F+F'F+" }{TEXT -1 2 ". " }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 37 "Proof of th e orthogonality relations " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 256 "" 0 "" {TEXT -1 2 " " }{XPPEDIT 18 0 "Int(T[i](x)*T[j](x)/sqrt(1 -x^2),x = -1 .. 1)" "6#-%$IntG6$*(-&%\"TG6#%\"iG6#%\"xG\"\"\"-&F)6#%\" jG6#F-F.-%%sqrtG6#,&F.F.*$F-\"\"#!\"\"F:/F-;,$F.F:F." }{TEXT -1 9 " \+ ... " }{XPPEDIT 18 0 "PIECEWISE([x=cos*theta, x =-1*` implies `* the ta*` = `*Pi],[dx=-sin*theta*d*theta, x =1*` implies `* theta*` = `*0], [-dx=sin*theta*d*theta,``])" "6#-%*PIECEWISEG6%7$/%\"xG*&%$cosG\"\"\"% &thetaGF+/F(,$*,F+F+%*~implies~GF+F,F+%$~=~GF+%#PiGF+!\"\"7$/%#dxG,$** %$sinGF+F,F+%\"dGF+F,F+F3/F(*,F+F+F0F+F,F+F1F+\"\"!F+7$/,$F6F3**F9F+F, F+F:F+F,F+%!G" }{TEXT -1 1 " " }}{PARA 256 "" 0 "" {TEXT -1 1 " " } {XPPEDIT 18 0 "`` = Int(cos*i*theta*cos*j*theta*sin*theta/sqrt(1-cos^2 *theta),theta = 0 .. Pi);" "6#/%!G-%$IntG6$*4%$cosG\"\"\"%\"iGF*%&thet aGF*F)F*%\"jGF*F,F*%$sinGF*F,F*-%%sqrtG6#,&F*F**&F)\"\"#F,F*!\"\"F5/F, ;\"\"!%#PiG" }{TEXT -1 1 " " }}{PARA 256 "" 0 "" {TEXT -1 1 " " } {XPPEDIT 18 0 "`` = Int(cos*i*theta*cos*j*theta,theta = 0 .. Pi);" "6# /%!G-%$IntG6$*.%$cosG\"\"\"%\"iGF*%&thetaGF*F)F*%\"jGF*F,F*/F,;\"\"!%# PiG" }{TEXT -1 15 " ------- (i). " }}{PARA 0 "" 0 "" {TEXT -1 3 "If \+ " }{XPPEDIT 18 0 "i<>j" "6#0%\"iG%\"jG" }{TEXT -1 31 ", the integral ( i) is equal to " }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "`` =1/2" "6#/%!G*&\"\"\"F&\"\"#!\"\"" }{TEXT -1 1 " " }{XPPEDIT 18 0 "Int (cos((i+j)*x)+cos((i-j)*x),x = 0 .. Pi);" "6#-%$IntG6$,&-%$cosG6#*&,&% \"iG\"\"\"%\"jGF-F-%\"xGF-F--F(6#*&,&F,F-F.!\"\"F-F/F-F-/F/;\"\"!%#PiG " }{TEXT -1 1 " " }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "` ` = ``(sin(i+j)/(i+j)+sin(i-j)/(i-j));" "6#/%!G-F$6#,&*&-%$sinG6#,&%\" iG\"\"\"%\"jGF.F.,&F-F.F/F.!\"\"F.*&-F*6#,&F-F.F/F1F.,&F-F.F/F1F1F." } {TEXT -1 2 " " }{XPPEDIT 18 0 "PIECEWISE([Pi, ``],[``, ``],[0, ``]); " "6#-%*PIECEWISEG6%7$%#PiG%!G7$F(F(7$\"\"!F(" }{XPPEDIT 18 0 "`` = 0 " "6#/%!G\"\"!" }{TEXT -1 2 ". " }}{PARA 0 "" 0 "" {TEXT -1 3 "If " } {XPPEDIT 18 0 "i=j" "6#/%\"iG%\"jG" }{TEXT -1 6 ", but " }{XPPEDIT 18 0 "i<>0" "6#0%\"iG\"\"!" }{TEXT -1 5 " and " }{XPPEDIT 18 0 "j<>0" "6# 0%\"jG\"\"!" }{TEXT -1 32 ", then the integral (i) becomes " }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "Int(cos^2*i*theta,theta=0.. Pi)" "6#-%$IntG6$*(%$cosG\"\"#%\"iG\"\"\"%&thetaGF*/F+;\"\"!%#PiG" } {TEXT -1 1 " " }}{PARA 256 "" 0 "" {TEXT -1 2 " " }{XPPEDIT 18 0 "`` \+ = Int((1+cos*2*i*x)/2,x = 0 .. Pi);" "6#/%!G-%$IntG6$*&,&\"\"\"F***%$c osGF*\"\"#F*%\"iGF*%\"xGF*F*F*F-!\"\"/F/;\"\"!%#PiG" }{TEXT -1 2 " " }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "`` = x/2+sin*2*i*x/ 4;" "6#/%!G,&*&%\"xG\"\"\"\"\"#!\"\"F(*,%$sinGF(F)F(%\"iGF(F'F(\"\"%F* F(" }{TEXT -1 2 " " }{XPPEDIT 18 0 "PIECEWISE([Pi, ``],[``, ``],[0, ` `]);" "6#-%*PIECEWISEG6%7$%#PiG%!G7$F(F(7$\"\"!F(" }{XPPEDIT 18 0 "`` \+ = Pi/2;" "6#/%!G*&%#PiG\"\"\"\"\"#!\"\"" }{TEXT -1 1 "." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 4 " If " }{XPPEDIT 18 0 "i=j" "6#/%\"iG%\"jG" }{XPPEDIT 18 0 "``=0" "6#/%!G\"\"!" }{TEXT -1 27 ", the integral (i) becomes " }{XPPEDIT 18 0 "Int(1,theta=0..Pi)=Pi " "6#/-%$IntG6$\"\"\"/%&thetaG;\"\"!%#PiGF," }{TEXT -1 2 ". " }}} {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 41 "The Cheby shev polynomials also satisfy a " }{TEXT 267 31 "discrete orthogonalit y relation" }{TEXT -1 1 "." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 3 "If " }{XPPEDIT 18 0 "x[k] = cos(k*Pi/n);" "6#/&%\" xG6#%\"kG-%$cosG6#*(F'\"\"\"%#PiGF,%\"nG!\"\"" }{TEXT -1 5 " for " } {XPPEDIT 18 0 "k = 0,1,` . . . `,n" "6&/%\"kG\"\"!\"\"\"%(~.~.~.~G%\"n G" }{TEXT -1 6 ", and " }{XPPEDIT 18 0 "i<=n, j<=n" "6$1%\"iG%\"nG1%\" jGF%" }{TEXT -1 7 ", then " }}{PARA 256 "" 0 "" {TEXT -1 3 " " } {XPPEDIT 18 0 "Sum(``^`#`*T[i](x[k])*T[j](x[k]),k = 0 .. n) = PIECEWIS E([0, i <> j],[n/2, `i =`*j <> 0*` or n`],[n, `i =`*j = 0*` or n`]); " "6#/-%$SumG6$*()%!G%\"#G\"\"\"-&%\"TG6#%\"iG6#&%\"xG6#%\"kGF+-&F.6#% \"jG6#&F36#F5F+/F5;\"\"!%\"nG-%*PIECEWISEG6%7$F?0F0F97$*&F@F+\"\"#!\" \"0*&%$i~=GF+F9F+*&F?F+%'~or~~nGF+7$F@/*&FLF+F9F+*&F?F+FNF+" }{TEXT -1 3 " , " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 6 "where " }{XPPEDIT 18 0 "Sum(``^`#`);" "6#-%$SumG6#)%!G%\"#G" } {TEXT -1 64 " denote a finite sum whose first and last terms are halv ed, so " }}{PARA 256 "" 0 "" {TEXT -1 2 " " }{XPPEDIT 18 0 "Sum(``^`# `*u[k],k = 0 .. n) = u[0]/2+u[1]+u[2]+` . . . `+u[n-1]+u[n]/2;" "6#/-% $SumG6$*&)%!G%\"#G\"\"\"&%\"uG6#%\"kGF+/F/;\"\"!%\"nG,.*&&F-6#F2F+\"\" #!\"\"F+&F-6#F+F+&F-6#F8F+%(~.~.~.~GF+&F-6#,&F3F+F+F9F+*&&F-6#F3F+F8F9 F+" }{TEXT -1 2 ". " }}{PARA 0 "" 0 "" {TEXT -1 27 "Such a sum may be \+ called a " }{TEXT 267 15 "trapezoidal sum" }{TEXT -1 129 ", since it h as the form of the sum which occurs in the description of the trapezoi dal rule for approximate numerical integration." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 46 "Proof of the disc rete orthogonality relations " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }} {PARA 0 "" 0 "" {TEXT -1 52 "First we establish the following summatio n formula. " }}{PARA 0 "" 0 "" {TEXT -1 3 "If " }{XPPEDIT 18 0 "phi<>2 *m*Pi" "6#0%$phiG*(\"\"#\"\"\"%\"mGF'%#PiGF'" }{TEXT -1 8 ", where " } {TEXT 283 1 "m" }{TEXT -1 24 " is an integer, so that " }{XPPEDIT 18 0 "sin(phi/2)<>0" "6#0-%$sinG6#*&%$phiG\"\"\"\"\"#!\"\"\"\"!" }{TEXT -1 5 " and " }{XPPEDIT 18 0 "cos*k*phi<>1" "6#0*(%$cosG\"\"\"%\"kGF&%$ phiGF&F&" }{TEXT -1 8 ", then " }}{PARA 256 "" 0 "" {TEXT -1 1 " " } {XPPEDIT 18 0 "Sum(cos*k*phi,k=0..(n-1))=1+cos*phi+cos*2*phi+` . . . ` +cos*(n-1)*phi" "6#/-%$SumG6$*(%$cosG\"\"\"%\"kGF)%$phiGF)/F*;\"\"!,&% \"nGF)F)!\"\",,F)F)*&F(F)F+F)F)*(F(F)\"\"#F)F+F)F)%(~.~.~.~GF)*(F(F),& F0F)F)F1F)F+F)F)" }{TEXT -1 1 " " }}{PARA 256 "" 0 "" {TEXT -1 2 " " }{XPPEDIT 18 0 "``= 1/2 + 1/2" "6#/%!G,&*&\"\"\"F'\"\"#!\"\"F'*&F'F'F( F)F'" }{TEXT -1 1 " " }{XPPEDIT 18 0 "sin*n*phi*cot(phi/2)-1/2;" "6#,& **%$sinG\"\"\"%\"nGF&%$phiGF&-%$cotG6#*&F(F&\"\"#!\"\"F&F&*&F&F&F-F.F. " }{TEXT -1 1 " " }{XPPEDIT 18 0 "cos*n*phi" "6#*(%$cosG\"\"\"%\"nGF%% $phiGF%" }{TEXT -1 15 " ------- (i). " }}{PARA 258 "" 0 "" {TEXT -1 1 " " }}{PARA 0 "" 0 "" {TEXT -1 53 "The \"telescoping\" method can be used via the formula:" }}{PARA 256 "" 0 "" {TEXT -1 2 " " }{XPPEDIT 18 0 "2*cos*k*phi*sin(phi/2) = sin((k+1/2)*phi)-sin((k-1/2)*phi);" "6# /*,\"\"#\"\"\"%$cosGF&%\"kGF&%$phiGF&-%$sinG6#*&F)F&F%!\"\"F&,&-F+6#*& ,&F(F&*&F&F&F%F.F&F&F)F&F&-F+6#*&,&F(F&*&F&F&F%F.F.F&F)F&F." }{TEXT -1 2 ", " }}{PARA 0 "" 0 "" {TEXT -1 8 "We have " }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "2*sin(phi/2)*Sum(cos*k*phi,k = 0 .. n-1 ) = sin(phi/2)+cos*phi*sin(phi/2)+cos*2*phi*sin(phi/2)+` . . . `+cos*( n-1)*phi*sin(phi/2);" "6#/*(\"\"#\"\"\"-%$sinG6#*&%$phiGF&F%!\"\"F&-%$ SumG6$*(%$cosGF&%\"kGF&F+F&/F2;\"\"!,&%\"nGF&F&F,F&,,-F(6#*&F+F&F%F,F& *(F1F&F+F&-F(6#*&F+F&F%F,F&F&**F1F&F%F&F+F&-F(6#*&F+F&F%F,F&F&%(~.~.~. ~GF&**F1F&,&F7F&F&F,F&F+F&-F(6#*&F+F&F%F,F&F&" }{TEXT -1 1 " " }} {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 256 "" 0 "" {TEXT -1 1 " " } {XPPEDIT 18 0 "`` = sin(phi/2)+sin(3*phi/2)-sin(phi/2)+sin(5*phi/2)-si n(3*phi/2)+` . . . `+sin((n-1/2)*phi)-sin((n-3/2)*phi);" "6#/%!G,2-%$s inG6#*&%$phiG\"\"\"\"\"#!\"\"F+-F'6#*(\"\"$F+F*F+F,F-F+-F'6#*&F*F+F,F- F--F'6#*(\"\"&F+F*F+F,F-F+-F'6#*(F1F+F*F+F,F-F-%(~.~.~.~GF+-F'6#*&,&% \"nGF+*&F+F+F,F-F-F+F*F+F+-F'6#*&,&FAF+*&F1F+F,F-F-F+F*F+F-" }{TEXT -1 1 " " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "`` = sin(phi/2)+sin((n-1/2)*phi);" "6#/%!G,&-%$s inG6#*&%$phiG\"\"\"\"\"#!\"\"F+-F'6#*&,&%\"nGF+*&F+F+F,F-F-F+F*F+F+" } {XPPEDIT 18 0 "``= sin(phi/2)+sin(n*phi-phi/2)" "6#/%!G,&-%$sinG6#*&%$ phiG\"\"\"\"\"#!\"\"F+-F'6#,&*&%\"nGF+F*F+F+*&F*F+F,F-F-F+" }{TEXT -1 1 " " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "``=sin(phi/2)+sin*n*phi*cos(phi/2)-cos*n*phi*sin(ph i/2)" "6#/%!G,(-%$sinG6#*&%$phiG\"\"\"\"\"#!\"\"F+**F'F+%\"nGF+F*F+-%$ cosG6#*&F*F+F,F-F+F+**F1F+F/F+F*F+-F'6#*&F*F+F,F-F+F-" }{TEXT -1 2 ". \+ " }}{PARA 0 "" 0 "" {TEXT -1 6 "Hence " }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "Sum(cos*k*phi,k = 0 .. n-1)=1/2+1/2" "6#/-%$SumG 6$*(%$cosG\"\"\"%\"kGF)%$phiGF)/F*;\"\"!,&%\"nGF)F)!\"\",&*&F)F)\"\"#F 1F)*&F)F)F4F1F)" }{TEXT -1 1 " " }{XPPEDIT 18 0 "sin*n*phi*cot(phi/2) \+ -1/2" "6#,&**%$sinG\"\"\"%\"nGF&%$phiGF&-%$cotG6#*&F(F&\"\"#!\"\"F&F&* &F&F&F-F.F." }{TEXT -1 1 " " }{XPPEDIT 18 0 "cos*n*phi" "6#*(%$cosG\" \"\"%\"nGF%%$phiGF%" }{TEXT -1 2 ". " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 29 "The formula (i) now follows. " }}{PARA 0 "" 0 "" {TEXT -1 21 "It follows also that " }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "Sum(cos*k*phi,k = 1 .. n-1)=1/2" "6#/-% $SumG6$*(%$cosG\"\"\"%\"kGF)%$phiGF)/F*;F),&%\"nGF)F)!\"\"*&F)F)\"\"#F 0" }{TEXT -1 1 " " }{XPPEDIT 18 0 "sin*n*phi*cot(phi/2)-1/2;" "6#,&**% $sinG\"\"\"%\"nGF&%$phiGF&-%$cotG6#*&F(F&\"\"#!\"\"F&F&*&F&F&F-F.F." } {TEXT -1 1 " " }{XPPEDIT 18 0 "cos*n*phi-1/2" "6#,&*(%$cosG\"\"\"%\"nG F&%$phiGF&F&*&F&F&\"\"#!\"\"F+" }{TEXT -1 16 " ------- (ii). " }} {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 22 "As a spec ial case, if " }{TEXT 286 1 "m" }{TEXT -1 19 " is an integer and " } {TEXT 285 1 "m" }{TEXT -1 28 " is not an even multiple of " }{TEXT 284 1 "n" }{TEXT -1 14 ", then taking " }{XPPEDIT 18 0 "phi=m*Pi/n" "6 #/%$phiG*(%\"mG\"\"\"%#PiGF'%\"nG!\"\"" }{TEXT -1 8 " gives " }} {PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "Sum(cos(k*m*Pi/n),k = 1 .. n-1) = 1/2" "6#/-%$SumG6$-%$cosG6#**%\"kG\"\"\"%\"mGF,%#PiGF,%\" nG!\"\"/F+;F,,&F/F,F,F0*&F,F,\"\"#F0" }{TEXT -1 1 " " }{XPPEDIT 18 0 " sin*m*Pi*cot(k*m*Pi/(2*n))-1/2" "6#,&**%$sinG\"\"\"%\"mGF&%#PiGF&-%$co tG6#**%\"kGF&F'F&F(F&*&\"\"#F&%\"nGF&!\"\"F&F&*&F&F&F/F1F1" }{TEXT -1 1 " " }{XPPEDIT 18 0 "cos*m*Pi -1/2" "6#,&*(%$cosG\"\"\"%\"mGF&%#PiGF& F&*&F&F&\"\"#!\"\"F+" }{TEXT -1 2 ". " }}{PARA 0 "" 0 "" {TEXT -1 6 "H ence " }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "Sum(cos(k*m* Pi/n),k = 1 .. n-1)=1/2" "6#/-%$SumG6$-%$cosG6#**%\"kG\"\"\"%\"mGF,%#P iGF,%\"nG!\"\"/F+;F,,&F/F,F,F0*&F,F,\"\"#F0" }{XPPEDIT 18 0 "``((-1)^m +1);" "6#-%!G6#,&),$\"\"\"!\"\"%\"mGF)F)F)" }{TEXT -1 16 " ------- (i ii)." }}{PARA 0 "" 0 "" {TEXT -1 4 "Thus" }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "Sum(cos(k*m*Pi/n),k = 1 .. n-1)= PIECEWISE([- 1, ` m even`],[0, ` m odd`])" "6#/-%$SumG6$-%$cosG6#**%\"kG\"\"\"%\"m GF,%#PiGF,%\"nG!\"\"/F+;F,,&F/F,F,F0-%*PIECEWISEG6$7$,$F,F0%)~m~~evenG 7$\"\"!%'~m~oddG" }{TEXT -1 2 ". " }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 99 "m := 'm':\nassume(m_,integer):\nSum(cos(m*k*Pi/n),k=1 ..n-1);\ncombine(subs(m_=m,value(subs(m=m_,%))));" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#-%$SumG6$-%$cosG6#**%\"mG\"\"\"%\"kGF+%#PiGF+%\"nG!\"\" /F,;F+,&F.F+F+F/" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#,&*&\"\"#!\"\")F&% \"mG\"\"\"F&#F)F%F&" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }} }{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 3 "If " } {XPPEDIT 18 0 "x[k] = cos(k*Pi/n);" "6#/&%\"xG6#%\"kG-%$cosG6#*(F'\"\" \"%#PiGF,%\"nG!\"\"" }{TEXT -1 5 " for " }{XPPEDIT 18 0 "k = 0,1,` . . . `,n" "6&/%\"kG\"\"!\"\"\"%(~.~.~.~G%\"nG" }{TEXT -1 6 ", and " } {XPPEDIT 18 0 "0 <= i;" "6#1\"\"!%\"iG" }{XPPEDIT 18 0 "`` <= n;" "6#1 %!G%\"nG" }{TEXT -1 3 ", " }{XPPEDIT 18 0 "0 <= j;" "6#1\"\"!%\"jG" } {XPPEDIT 18 0 "`` <= n;" "6#1%!G%\"nG" }{TEXT -1 9 ", then " }} {PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "Sum(``^`#`*T[i](x[k]) *T[j](x[k]),k = 0 .. n) = Sum(``^`#`*cos(i*k*Pi/n)*cos(j*k*Pi/n),k = 0 .. n)" "6#/-%$SumG6$*()%!G%\"#G\"\"\"-&%\"TG6#%\"iG6#&%\"xG6#%\"kGF+- &F.6#%\"jG6#&F36#F5F+/F5;\"\"!%\"nG-F%6$*()F)F*F+-%$cosG6#**F0F+F5F+%# PiGF+F@!\"\"F+-FF6#**F9F+F5F+FIF+F@FJF+/F5;F?F@" }{TEXT -1 1 " " }} {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 256 "" 0 "" {TEXT -1 1 " " } {XPPEDIT 18 0 "``=1/2+ ``" "6#/%!G,&*&\"\"\"F'\"\"#!\"\"F'F$F'" } {XPPEDIT 18 0 "Sum(cos(i*k*Pi/n)*cos(j*k*Pi/n),k = 1 .. n-1)" "6#-%$Su mG6$*&-%$cosG6#**%\"iG\"\"\"%\"kGF,%#PiGF,%\"nG!\"\"F,-F(6#**%\"jGF,F- F,F.F,F/F0F,/F-;F,,&F/F,F,F0" }{XPPEDIT 18 0 "``+1/2" "6#,&%!G\"\"\"*& F%F%\"\"#!\"\"F%" }{TEXT -1 1 " " }{XPPEDIT 18 0 "(-1)^(i+j)" "6#),$\" \"\"!\"\",&%\"iGF%%\"jGF%" }{TEXT -1 2 ". " }}{PARA 0 "" 0 "" {TEXT -1 6 "Hence " }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "Sum(` `^`#`*T[i](x[k])*T[j](x[k]),k = 0 .. n)=1/2+1/2" "6#/-%$SumG6$*()%!G% \"#G\"\"\"-&%\"TG6#%\"iG6#&%\"xG6#%\"kGF+-&F.6#%\"jG6#&F36#F5F+/F5;\" \"!%\"nG,&*&F+F+\"\"#!\"\"F+*&F+F+FCFDF+" }{TEXT -1 1 " " }{XPPEDIT 18 0 "Sum(cos((i+j)*k*Pi/n)+cos((i-j)*k*Pi/n),k = 1 .. n-1)" "6#-%$Sum G6$,&-%$cosG6#**,&%\"iG\"\"\"%\"jGF-F-%\"kGF-%#PiGF-%\"nG!\"\"F--F(6#* *,&F,F-F.F2F-F/F-F0F-F1F2F-/F/;F-,&F1F-F-F2" }{XPPEDIT 18 0 "``+1/2" " 6#,&%!G\"\"\"*&F%F%\"\"#!\"\"F%" }{TEXT -1 1 " " }{XPPEDIT 18 0 "(-1)^ (i+j)" "6#),$\"\"\"!\"\",&%\"iGF%%\"jGF%" }{TEXT -1 15 " ------- (iv). " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 3 "If " }{XPPEDIT 18 0 "i<>j" "6#0%\"iG%\"jG" }{TEXT -1 55 " the formula (iii) can be applied to (iv) to show that " }}{PARA 0 "" 0 "" {TEXT -1 0 " " }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "Sum(``^`#`*T[i](x [k])*T[j](x[k]),k = 0 .. n) = 1/2-1/4;" "6#/-%$SumG6$*()%!G%\"#G\"\"\" -&%\"TG6#%\"iG6#&%\"xG6#%\"kGF+-&F.6#%\"jG6#&F36#F5F+/F5;\"\"!%\"nG,&* &F+F+\"\"#!\"\"F+*&F+F+\"\"%FDFD" }{XPPEDIT 18 0 "``( (-1)^(i+j)+1+(-1 )^(i-j)+1) +1/2" "6#,&-%!G6#,*),$\"\"\"!\"\",&%\"iGF*%\"jGF*F*F*F*),$F *F+,&F-F*F.F+F*F*F*F**&F*F*\"\"#F+F*" }{XPPEDIT 18 0 " (-1)^(i+j)" "6# ),$\"\"\"!\"\",&%\"iGF%%\"jGF%" }{XPPEDIT 18 0 " ``=0" "6#/%!G\"\"!" } {TEXT -1 2 ". " }}{PARA 0 "" 0 "" {TEXT -1 3 "If " }{XPPEDIT 18 0 "i=j " "6#/%\"iG%\"jG" }{XPPEDIT 18 0 "``<> 0" "6#0%!G\"\"!" }{TEXT -1 4 " \+ or " }{TEXT 287 1 "n" }{TEXT -1 14 " (iv) becomes " }}{PARA 256 "" 0 " " {TEXT -1 1 " " }{XPPEDIT 18 0 "Sum(``^`#`*T[i](x[k])^2,k = 0 .. n) = 1/2+1/2;" "6#/-%$SumG6$*&)%!G%\"#G\"\"\"*$-&%\"TG6#%\"iG6#&%\"xG6#%\" kG\"\"#F+/F6;\"\"!%\"nG,&*&F+F+F7!\"\"F+*&F+F+F7F>F+" }{TEXT -1 1 " " }{XPPEDIT 18 0 "Sum(cos(2*i*k*Pi/n)+1,k = 1 .. n-1);" "6#-%$SumG6$,&-% $cosG6#*,\"\"#\"\"\"%\"iGF,%\"kGF,%#PiGF,%\"nG!\"\"F,F,F,/F.;F,,&F0F,F ,F1" }{XPPEDIT 18 0 "``+1/2 = 1/2+1/2;" "6#/,&%!G\"\"\"*&F&F&\"\"#!\" \"F&,&*&F&F&F(F)F&*&F&F&F(F)F&" }{XPPEDIT 18 0 "``(-1+n-1)+1/2 =n/2" " 6#/,&-%!G6#,(\"\"\"!\"\"%\"nGF)F)F*F)*&F)F)\"\"#F*F)*&F+F)F-F*" } {TEXT -1 2 ". " }}{PARA 0 "" 0 "" {TEXT -1 3 "If " }{XPPEDIT 18 0 "i=j " "6#/%\"iG%\"jG" }{XPPEDIT 18 0 "`` = 0;" "6#/%!G\"\"!" }{TEXT -1 4 " or " }{TEXT 288 1 "n" }{TEXT -1 14 " (iv) becomes " }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "Sum(``^`#`*T[i](x[k])^2,k = 0 .. n) \+ = 1/2+1/2;" "6#/-%$SumG6$*&)%!G%\"#G\"\"\"*$-&%\"TG6#%\"iG6#&%\"xG6#% \"kG\"\"#F+/F6;\"\"!%\"nG,&*&F+F+F7!\"\"F+*&F+F+F7F>F+" }{TEXT -1 1 " \+ " }{XPPEDIT 18 0 "Sum(2,k = 1 .. n-1);" "6#-%$SumG6$\"\"#/%\"kG;\"\"\" ,&%\"nGF*F*!\"\"" }{XPPEDIT 18 0 "``+1/2 = 1/2+``(n-1)+1/2;" "6#/,&%!G \"\"\"*&F&F&\"\"#!\"\"F&,(*&F&F&F(F)F&-F%6#,&%\"nGF&F&F)F&*&F&F&F(F)F& " }{XPPEDIT 18 0 "`` = n;" "6#/%!G%\"nG" }{TEXT -1 2 ". " }}{PARA 0 " " 0 "" {TEXT -1 0 "" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 " " {TEXT -1 62 "Here are some examples of the discrete orthogonality re lation." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 161 "alias(T=ChebyshevT):\nSM := (i,j,n) -> T(i,1)*T(j,1) /2+\n Sum(T(i,cos(Pi*k/n))*T(j,cos(Pi*k/n)),k=1..n-1)+\n T (i,-1)*T(j,-1)/2:\n'SM(i,j,n)'=SM(i,j,n);\n\n" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/-%#SMG6%%\"iG%\"jG%\"nG,(*&#\"\"\"\"\"#F-*&-%\"TG6$F'F -F--F16$F(F-F-F-F--%$SumG6$*&-F16$F'-%$cosG6#*(%#PiGF-%\"kGF-F)!\"\"F- -F16$F(F;F-/F@;F-,&F)F-F-FAF-*&F,F-*&-F16$F'FAF--F16$F(FAF-F-F-" }}} {PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 55 "i=3,j=5,n=20;\nSM(op(map(rhs,[%])));\nsimplify(value(%));" }} {PARA 11 "" 1 "" {XPPMATH 20 "6%/%\"iG\"\"$/%\"jG\"\"&/%\"nG\"#?" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#,(*&#\"\"\"\"\"#F&*&-%\"TG6$\"\"$F&F&- F*6$\"\"&F&F&F&F&-%$SumG6$*&-F*6$F,-%$cosG6#,$*(\"#?!\"\"%#PiGF&%\"kGF &F&F&-F*6$F/F6F&/F>;F&\"#>F&*&F%F&*&-F*6$F,F " 0 "" {MPLTEXT 1 0 55 "i=5,j=5,n=20;\nSM(op(map (rhs,[%])));\nsimplify(value(%));" }}{PARA 11 "" 1 "" {XPPMATH 20 "6%/ %\"iG\"\"&/%\"jGF%/%\"nG\"#?" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#,(*&# \"\"\"\"\"#F&*$)-%\"TG6$\"\"&F&F'F&F&F&-%$SumG6$*$)-F+6$F--%$cosG6#,$* (\"#?!\"\"%#PiGF&%\"kGF&F&F'F&/F=;F&\"#>F&*&F%F&*$)-F+6$F-F;F'F&F&F&" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#\"#5" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 57 "i=20,j=20,n=20;\nSM(op( map(rhs,[%])));\nsimplify(value(%));" }}{PARA 11 "" 1 "" {XPPMATH 20 " 6%/%\"iG\"#?/%\"jGF%/%\"nGF%" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#,(*&# \"\"\"\"\"#F&*$)-%\"TG6$\"#?F&F'F&F&F&-%$SumG6$*$)-F+6$F--%$cosG6#,$*( F-!\"\"%#PiGF&%\"kGF&F&F'F&/F<;F&\"#>F&*&F%F&*$)-F+6$F-F:F'F&F&F&" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#\"#?" }}}{PARA 0 "" 0 "" {TEXT -1 0 " " }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}}{SECT 1 {PARA 4 " " 0 "" {TEXT -1 42 "Expressing a polynomial as a Chebyshev sum" }} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}{PARA 0 "" 0 "" {TEXT -1 13 "A polynomial " }{XPPEDIT 18 0 "p(x);" "6#-%\"pG6#%\"xG" }{TEXT -1 11 " of degree " }{TEXT 274 1 "n" }{TEXT -1 27 " can be expressed a s a sum " }{XPPEDIT 18 0 "Sum(``^`*`*c[k]*T[k](x),k = 0 .. n);" "6#-%$ SumG6$*()%!G%\"*G\"\"\"&%\"cG6#%\"kGF*-&%\"TG6#F.6#%\"xGF*/F.;\"\"!%\" nG" }{TEXT -1 7 ", for " }{XPPEDIT 18 0 "-1<=x " "6#1,$\"\"\"!\"\"%\" xG" }{XPPEDIT 18 0 "``<=1" "6#1%!G\"\"\"" }{TEXT -1 1 "," }}{PARA 0 " " 0 "" {TEXT -1 6 "where " }{XPPEDIT 18 0 "Sum(``^`*`)" "6#-%$SumG6#)% !G%\"*G" }{TEXT -1 51 " denotes a sum whose first term is halved, so t hat " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "Sum(``^`*`*c[k]*T[k](x),k = 0 .. n) = ``(c[0]/2)*T[ 0](x)+c[1]*T[1](x)+c[2]*T[2](x)+` . . . `+c[n-1]*T[n-1](x)+c[n]*T[n](x );" "6#/-%$SumG6$*()%!G%\"*G\"\"\"&%\"cG6#%\"kGF+-&%\"TG6#F/6#%\"xGF+/ F/;\"\"!%\"nG,.*&-F)6#*&&F-6#F8F+\"\"#!\"\"F+-&F26#F86#F5F+F+*&&F-6#F+ F+-&F26#F+6#F5F+F+*&&F-6#FAF+-&F26#FA6#F5F+F+%(~.~.~.~GF+*&&F-6#,&F9F+ F+FBF+-&F26#,&F9F+F+FB6#F5F+F+*&&F-6#F9F+-&F26#F96#F5F+F+" }{TEXT -1 2 ". " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 17 " The coefficients " }{XPPEDIT 18 0 "c[k]" "6#&%\"cG6#%\"kG" }{TEXT -1 21 " can be computed from" }}{PARA 256 "" 0 "" {TEXT -1 3 " " } {XPPEDIT 18 0 "c[k]=2/Pi" "6#/&%\"cG6#%\"kG*&\"\"#\"\"\"%#PiG!\"\"" } {TEXT -1 1 " " }{XPPEDIT 18 0 "Int(p(x)*T[k](x)/sqrt(1-x^2),x = -1 .. \+ 1) = 2/Pi;" "6#/-%$IntG6$*(-%\"pG6#%\"xG\"\"\"-&%\"TG6#%\"kG6#F+F,-%%s qrtG6#,&F,F,*$F+\"\"#!\"\"F9/F+;,$F,F9F,*&F8F,%#PiGF9" }{TEXT -1 1 " \+ " }{XPPEDIT 18 0 "Int(f(cos*theta)*cos*k*theta,theta = 0 .. Pi);" "6#- %$IntG6$**-%\"fG6#*&%$cosG\"\"\"%&thetaGF,F,F+F,%\"kGF,F-F,/F-;\"\"!%# PiG" }{TEXT -1 3 " . " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 25 "We call the coefficients " }{XPPEDIT 18 0 "c[k];" "6#& %\"cG6#%\"kG" }{TEXT -1 5 " the " }{TEXT 267 22 "Chebyshev coefficient s" }{TEXT -1 4 " of " }{XPPEDIT 18 0 "p(x);" "6#-%\"pG6#%\"xG" }{TEXT -1 1 "." }}{PARA 0 "" 0 "" {TEXT -1 17 "For example, let " }{XPPEDIT 18 0 "p(x) = 2*x^6+3*x^5-5*x^4+7*x^3-4*x^2+8*x-3.;" "6#/-%\"pG6#%\"xG, 0*&\"\"#\"\"\"*$F'\"\"'F+F+*&\"\"$F+*$F'\"\"&F+F+*&F1F+*$F'\"\"%F+!\" \"*&\"\"(F+*$F'F/F+F+*&F4F+*$F'F*F+F5*&\"\")F+F'F+F+-%&FloatG6$F/\"\"! F5" }{TEXT -1 1 "\n" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 317 "alia s(T=ChebyshevT):\np := x -> 2*x^6+3*x^5-5*x^4+7*x^3-4*x^2+8*x-3:\n'p(x )'=p(x);\nn := degree(p(x));\nc := array(0..n):\nfor k from 0 to n do \n 'c'[k]=2/Pi*Int('p(cos(theta))'*cos(k*theta),theta=0..Pi);\n c[ k] := value(rhs(%));\nend do;\nk := 'k':\nq := unapply(c[0]/2+sum(c[k] *T(k,x),k=1..n),x):\nn := 'n':\n'q(x)'=q(x);\n\n" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/-%\"pG6#%\"xG,0*&\"\"#\"\"\")F'\"\"'F+F+*&\"\"$F+)F'\" \"&F+F+*&F1F+)F'\"\"%F+!\"\"*&\"\"(F+)F'F/F+F+*&F4F+)F'F*F+F5*&\"\")F+ F'F+F+F/F5" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"nG\"\"'" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/&%\"cG6#\"\"!,$,$-%$IntG6$-%\"pG6#-%$cosG6#%&t hetaG/F3;F'%#PiG*&\"\"#\"\"\"F6!\"\"F9" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>&%\"cG6#\"\"!#!#D\"\"#" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/&%\" cG6#\"\"\",$,$-%$IntG6$*&-%\"pG6#-%$cosG6#%&thetaGF'F1F'/F4;\"\"!%#PiG *&\"\"#F'F8!\"\"F'" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>&%\"cG6#\"\"\"# \"$@\"\"\")" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/&%\"cG6#\"\"#,$,$-%$In tG6$*&-%\"pG6#-%$cosG6#%&thetaG\"\"\"-F26#,$*&F'F5F4F5F5F5/F4;\"\"!%#P iG*&F'F5F=!\"\"F5" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>&%\"cG6#\"\"##!# d\"#;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/&%\"cG6#\"\"$,$,$-%$IntG6$*& -%\"pG6#-%$cosG6#%&thetaG\"\"\"-F26#,$*&F'F5F4F5F5F5/F4;\"\"!%#PiG*&\" \"#F5F=!\"\"F5" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>&%\"cG6#\"\"$#\"#V \"#;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/&%\"cG6#\"\"%,$,$-%$IntG6$*&- %\"pG6#-%$cosG6#%&thetaG\"\"\"-F26#,$*&F'F5F4F5F5F5/F4;\"\"!%#PiG*&\" \"#F5F=!\"\"F5" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>&%\"cG6#\"\"%#!\"\" F'" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/&%\"cG6#\"\"&,$,$-%$IntG6$*&-% \"pG6#-%$cosG6#%&thetaG\"\"\"-F26#,$*&F'F5F4F5F5F5/F4;\"\"!%#PiG*&\"\" #F5F=!\"\"F5" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>&%\"cG6#\"\"&#\"\"$\" #;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/&%\"cG6#\"\"',$,$-%$IntG6$*&-% \"pG6#-%$cosG6#%&thetaG\"\"\"-F26#,$*&F'F5F4F5F5F5/F4;\"\"!%#PiG*&\"\" #F5F=!\"\"F5" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>&%\"cG6#\"\"'#\"\"\" \"#;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/-%\"qG6#%\"xG,0#\"#D\"\"%!\" \"*&#\"$@\"\"\")\"\"\"-%\"TG6$F1F'F1F1*&#\"#d\"#;F1-F36$\"\"#F'F1F,*&# \"#VF8F1-F36$\"\"$F'F1F1*&#F1F+F1-F36$F+F'F1F,*&#FAF8F1-F36$\"\"&F'F1F 1*&#F1F8F1-F36$\"\"'F'F1F1" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 116 "If we form the Chebyshev sum with the coeffici ents as computed above, we obtain the original form of the polynomial \+ " }{XPPEDIT 18 0 "p(x);" "6#-%\"pG6#%\"xG" }{TEXT -1 1 "." }}{PARA 0 " " 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 31 "'q(x) '=q(x);\n``=expand(rhs(%));" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/-%\"qG 6#%\"xG,0#\"#D\"\"%!\"\"*&#\"$@\"\"\")\"\"\"-%\"TG6$F1F'F1F1*&#\"#d\"# ;F1-F36$\"\"#F'F1F,*&#\"#VF8F1-F36$\"\"$F'F1F1*&#F1F+F1-F36$F+F'F1F,*& #FAF8F1-F36$\"\"&F'F1F1*&#F1F8F1-F36$\"\"'F'F1F1" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/%!G,0*&\"\"#\"\"\")%\"xG\"\"'F(F(*&\"\"$F()F*\"\"&F(F( *&F/F()F*\"\"%F(!\"\"*&\"\"(F()F*F-F(F(*&F2F()F*F'F(F3*&\"\")F(F*F(F(F -F3" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 60 "An alterna tive method for calculating Chebyshev coefficients" }}{EXCHG {PARA 0 " > " 0 "" {MPLTEXT 1 0 1 ";" }}}{PARA 0 "" 0 "" {TEXT -1 19 "Given a po lynomial " }{XPPEDIT 18 0 "p(x);" "6#-%\"pG6#%\"xG" }{TEXT -1 11 " of \+ degree " }{TEXT 275 1 "n" }{TEXT -1 9 ", define " }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "d[j]=2/n" "6#/&%\"dG6#%\"jG*&\"\"#\"\" \"%\"nG!\"\"" }{TEXT -1 1 " " }{XPPEDIT 18 0 "Sum(``^`#`*p(x[k])*T[j]( x[k]),k = 0 .. n);" "6#-%$SumG6$*()%!G%\"#G\"\"\"-%\"pG6#&%\"xG6#%\"kG F*-&%\"TG6#%\"jG6#&F/6#F1F*/F1;\"\"!%\"nG" }{TEXT -1 8 ", for " } {XPPEDIT 18 0 "j = 0,1,` . . . `,n" "6&/%\"jG\"\"!\"\"\"%(~.~.~.~G%\"n G" }{TEXT -1 1 "," }}{PARA 256 "" 0 "" {TEXT -1 2 " " }{TEXT 270 27 " ___________________________" }{TEXT -1 2 " " }}{PARA 0 "" 0 "" {TEXT -1 6 "where " }{XPPEDIT 18 0 "x[k] = cos(k*Pi/n);" "6#/&%\"xG6#%\"kG-% $cosG6#*(F'\"\"\"%#PiGF,%\"nG!\"\"" }{TEXT -1 2 ", " }{XPPEDIT 18 0 "k = 0,1,` . . . `, n" "6&/%\"kG\"\"!\"\"\"%(~.~.~.~G%\"nG" }{TEXT -1 1 "." }}{PARA 0 "" 0 "" {TEXT -1 4 "Then" }}{PARA 256 "" 0 "" {TEXT -1 2 " " }{XPPEDIT 18 0 "d[j]=2/n" "6#/&%\"dG6#%\"jG*&\"\"#\"\"\"%\"nG! \"\"" }{TEXT -1 1 " " }{XPPEDIT 18 0 "Sum(``^`#`*p(cos(k*Pi/n))*cos(k* j*Pi/n),k = 0 .. n);" "6#-%$SumG6$*()%!G%\"#G\"\"\"-%\"pG6#-%$cosG6#*( %\"kGF*%#PiGF*%\"nG!\"\"F*-F/6#**F2F*%\"jGF*F3F*F4F5F*/F2;\"\"!F4" } {TEXT -1 7 ", for " }{XPPEDIT 18 0 "j = 0,1,` . . . `,n" "6&/%\"jG\" \"!\"\"\"%(~.~.~.~G%\"nG" }{TEXT -1 1 "," }}{PARA 0 "" 0 "" {TEXT -1 36 "The discrete orthogonality relation " }}{PARA 256 "" 0 "" {TEXT -1 3 " " }{XPPEDIT 18 0 "Sum(``^`#`*T[i](x[k])*T[j](x[k]),k = 0 .. n ) = PIECEWISE([0, i <> j],[n/2, `i =`*j <> 0*` or n`],[n, `i =`*j = 0 *` or n`]);" "6#/-%$SumG6$*()%!G%\"#G\"\"\"-&%\"TG6#%\"iG6#&%\"xG6#% \"kGF+-&F.6#%\"jG6#&F36#F5F+/F5;\"\"!%\"nG-%*PIECEWISEG6%7$F?0F0F97$*& F@F+\"\"#!\"\"0*&%$i~=GF+F9F+*&F?F+%'~or~~nGF+7$F@/*&FLF+F9F+*&F?F+FNF +" }{TEXT -1 3 " , " }}{PARA 0 "" 0 "" {TEXT -1 17 "implies that the \+ " }{XPPEDIT 18 0 "d[j]" "6#&%\"dG6#%\"jG" }{TEXT -1 35 " are the Cheby shev coefficients of " }{XPPEDIT 18 0 "p(x);" "6#-%\"pG6#%\"xG" } {TEXT -1 1 "." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 16 "Chebyshev series" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}{PARA 0 "" 0 "" {TEXT -1 26 "For an arbitrary fu nction " }{XPPEDIT 18 0 "f(x)" "6#-%\"fG6#%\"xG" }{TEXT -1 25 " define d on the interval " }{XPPEDIT 18 0 "[-1,1]" "6#7$,$\"\"\"!\"\"F%" } {TEXT -1 29 ", we can define the infinite " }{TEXT 267 16 "Chebyshev s eries" }{TEXT -1 2 ": " }}{PARA 256 "" 0 "" {TEXT -1 2 " " }{XPPEDIT 18 0 "Sum(``^`*`*c[k]*T[k](x),k = 0 .. infinity);" "6#-%$SumG6$*()%!G% \"*G\"\"\"&%\"cG6#%\"kGF*-&%\"TG6#F.6#%\"xGF*/F.;\"\"!%)infinityG" } {TEXT -1 1 " " }}{PARA 0 "" 0 "" {TEXT -1 23 "where the coefficients \+ " }{XPPEDIT 18 0 "c[k]" "6#&%\"cG6#%\"kG" }{TEXT -1 15 " are defined b y" }}{PARA 256 "" 0 "" {TEXT -1 3 " " }{XPPEDIT 18 0 "c[k]=2/Pi" "6# /&%\"cG6#%\"kG*&\"\"#\"\"\"%#PiG!\"\"" }{TEXT -1 1 " " }{XPPEDIT 18 0 "Int(f(x)*T[k](x)/sqrt(1-x^2),x = -1 .. 1) = 2/Pi;" "6#/-%$IntG6$*(-% \"fG6#%\"xG\"\"\"-&%\"TG6#%\"kG6#F+F,-%%sqrtG6#,&F,F,*$F+\"\"#!\"\"F9/ F+;,$F,F9F,*&F8F,%#PiGF9" }{TEXT -1 1 " " }{XPPEDIT 18 0 "Int(f(cos*th eta)*cos*k*theta,theta = 0 .. Pi);" "6#-%$IntG6$**-%\"fG6#*&%$cosG\"\" \"%&thetaGF,F,F+F,%\"kGF,F-F,/F-;\"\"!%#PiG" }{TEXT -1 2 ". " }}{PARA 0 "" 0 "" {TEXT -1 51 "Under suitable conditions this series converges to " }{XPPEDIT 18 0 "f(x)" "6#-%\"fG6#%\"xG" }{TEXT -1 1 "." }}{PARA 0 "" 0 "" {TEXT -1 97 "The alternative expression for the coefficients given in the previous section for the case where " }{XPPEDIT 18 0 "f( x)" "6#-%\"fG6#%\"xG" }{TEXT -1 26 " is a polynomial gives an " } {TEXT 267 34 "approximation for the coefficients" }{TEXT -1 25 " in th e general case, if " }{TEXT 276 1 "n" }{TEXT -1 11 " is chosen " } {TEXT 267 18 "sufficiently large" }{TEXT -1 1 "." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 35 "We may still call the coe fficients " }{XPPEDIT 18 0 "c[j]" "6#&%\"cG6#%\"jG" }{TEXT -1 27 " com puted using the formula" }}{PARA 256 "" 0 "" {TEXT -1 2 " " } {XPPEDIT 18 0 "c[j] = 2/n;" "6#/&%\"cG6#%\"jG*&\"\"#\"\"\"%\"nG!\"\"" }{TEXT -1 1 " " }{XPPEDIT 18 0 "Sum(``^`#`*f(x[k])*T[j](x[k]),k = 0 .. n);" "6#-%$SumG6$*()%!G%\"#G\"\"\"-%\"fG6#&%\"xG6#%\"kGF*-&%\"TG6#%\" jG6#&F/6#F1F*/F1;\"\"!%\"nG" }{TEXT -1 2 ", " }}{PARA 0 "" 0 "" {TEXT 267 22 "Chebyshev coefficients" }{TEXT -1 18 " for the function " } {XPPEDIT 18 0 "f(x)" "6#-%\"fG6#%\"xG" }{TEXT -1 1 "." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 34 "Example: the \+ Chebyshev series for " }{XPPEDIT 18 0 "exp(x)" "6#-%$expG6#%\"xG" }} {PARA 0 "" 0 "" {TEXT -1 22 "Consider the function " }{XPPEDIT 18 0 "f (x) = exp(x);" "6#/-%\"fG6#%\"xG-%$expG6#F'" }{TEXT -1 25 " defined on the interval " }{XPPEDIT 18 0 "[-1,1]" "6#7$,$\"\"\"!\"\"F%" }{TEXT -1 1 "." }}{PARA 0 "" 0 "" {TEXT -1 53 "We calculate the first few Che byshev coefficients of " }{XPPEDIT 18 0 "f(x)" "6#-%\"fG6#%\"xG" } {TEXT -1 1 "." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 284 "alias(T=ChebyshevT):\nf := x -> exp(x):\n'f(x)' =f(x);\nn := 8;\nc := array(0..n):\nk := 'k':for k from 0 to n do\n \+ 'c'[k]=2/Pi*Int('f(cos(theta))'*cos(k*theta),theta=0..Pi);\n c[k] := evalf(rhs(%));\nend do;\nk := 'k':\ng := unapply(c[0]/2+sum(c[k]*T(k, x),k=1..n),x):\nn := 'n':\n'g(x)'=g(x);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/-%\"fG6#%\"xG-%$expGF&" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"n G\"\")" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/&%\"cG6#\"\"!,$,$-%$IntG6$- %\"fG6#-%$cosG6#%&thetaG/F3;F'%#PiG*&\"\"#\"\"\"F6!\"\"F9" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>&%\"cG6#\"\"!$\"+c<8KD!\"*" }}{PARA 11 "" 1 " " {XPPMATH 20 "6#/&%\"cG6#\"\"\",$,$-%$IntG6$*&-%\"fG6#-%$cosG6#%&thet aGF'F1F'/F4;\"\"!%#PiG*&\"\"#F'F8!\"\"F'" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>&%\"cG6#\"\"\"$\"+3#=.8\"!\"*" }}{PARA 11 "" 1 "" {XPPMATH 20 " 6#/&%\"cG6#\"\"#,$,$-%$IntG6$*&-%\"fG6#-%$cosG6#%&thetaG\"\"\"-F26#,$* &F'F5F4F5F5F5/F4;\"\"!%#PiG*&F'F5F=!\"\"F5" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>&%\"cG6#\"\"#$\"+%R`\\r#!#5" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/&%\"cG6#\"\"$,$,$-%$IntG6$*&-%\"fG6#-%$cosG6#%&thetaG \"\"\"-F26#,$*&F'F5F4F5F5F5/F4;\"\"!%#PiG*&\"\"#F5F=!\"\"F5" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>&%\"cG6#\"\"$$\"+%)\\oLW!#6" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/&%\"cG6#\"\"%,$,$-%$IntG6$*&-%\"fG6#-%$cosG6#%&th etaG\"\"\"-F26#,$*&F'F5F4F5F5F5/F4;\"\"!%#PiG*&\"\"#F5F=!\"\"F5" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#>&%\"cG6#\"\"%$\"+U/Cua!#7" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/&%\"cG6#\"\"&,$,$-%$IntG6$*&-%\"fG6#-%$cosG 6#%&thetaG\"\"\"-F26#,$*&F'F5F4F5F5F5/F4;\"\"!%#PiG*&\"\"#F5F=!\"\"F5 " }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>&%\"cG6#\"\"&$\"+=JEHa!#8" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#/&%\"cG6#\"\"',$,$-%$IntG6$*&-%\"fG6#- %$cosG6#%&thetaG\"\"\"-F26#,$*&F'F5F4F5F5F5/F4;\"\"!%#PiG*&\"\"#F5F=! \"\"F5" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>&%\"cG6#\"\"'$\"+)HKx\\%!#9 " }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/&%\"cG6#\"\"(,$,$-%$IntG6$*&-%\"f G6#-%$cosG6#%&thetaG\"\"\"-F26#,$*&F'F5F4F5F5F5/F4;\"\"!%#PiG*&\"\"#F5 F=!\"\"F5" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>&%\"cG6#\"\"($\"+7kV)>$! #:" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/&%\"cG6#\"\"),$,$-%$IntG6$*&-% \"fG6#-%$cosG6#%&thetaG\"\"\"-F26#,$*&F'F5F4F5F5F5/F4;\"\"!%#PiG*&\"\" #F5F=!\"\"F5" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>&%\"cG6#\"\")$\"+*GD@ *>!#;" }}{PARA 12 "" 1 "" {XPPMATH 20 "6#/-%\"gG6#%\"xG,4$\"+ye1m7!\"* \"\"\"*&$\"+3#=.8\"F+F,-%\"TG6$F,F'F,F,*&$\"+%R`\\r#!#5F,-F16$\"\"#F'F ,F,*&$\"+%)\\oLW!#6F,-F16$\"\"$F'F,F,*&$\"+U/Cua!#7F,-F16$\"\"%F'F,F,* &$\"+=JEHa!#8F,-F16$\"\"&F'F,F,*&$\"+)HKx\\%!#9F,-F16$\"\"'F'F,F,*&$\" +7kV)>$!#:F,-F16$\"\"(F'F,F,*&$\"+*GD@*>!#;F,-F16$\"\")F'F,F," }}} {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 56 "We can ob tain the explicit polynomial approximation for " }{XPPEDIT 18 0 "f(x)= exp(x)" "6#/-%\"fG6#%\"xG-%$expG6#F'" }{TEXT -1 56 " by substituting f or the various Chebyshev polynomials. " }}{PARA 0 "" 0 "" {TEXT -1 0 " " }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 52 "g(x);\nq := unapply(expa nd(value(%)),x):\n'q(x)'=q(x);" }}{PARA 12 "" 1 "" {XPPMATH 20 "6#,4$ \"+ye1m7!\"*\"\"\"*&$\"+3#=.8\"F&F'-%\"TG6$F'%\"xGF'F'*&$\"+%R`\\r#!#5 F'-F,6$\"\"#F.F'F'*&$\"+%)\\oLW!#6F'-F,6$\"\"$F.F'F'*&$\"+U/Cua!#7F'-F ,6$\"\"%F.F'F'*&$\"+=JEHa!#8F'-F,6$\"\"&F.F'F'*&$\"+)HKx\\%!#9F'-F,6$ \"\"'F.F'F'*&$\"+7kV)>$!#:F'-F,6$\"\"(F.F'F'*&$\"+*GD@*>!#;F'-F,6$\"\" )F.F'F'" }}{PARA 12 "" 1 "" {XPPMATH 20 "6#/-%\"qG6#%\"xG,4*&$\"+4!*** ****!#5\"\"\"F'F-F-*&$\"+6hfG$)!#7F-)F'\"\"&F-F-*&$\"+/')omT!#6F-)F'\" \"%F-F-*&$\"+c)zmm\"F,F-)F'\"\"$F-F-*&$\"+B(*****\\F,F-)F'\"\"#F-F-$\" +,+++5!\"*F-*&$\"+/$**p/#!#8F-)F'\"\"(F-F-*&$\"+q.#*\\D!#9F-)F'\"\")F- F-*&$\"+GfF)Q\"F1F-)F'\"\"'F-F-" }}}{PARA 0 "" 0 "" {TEXT -1 1 " " }} {PARA 0 "" 0 "" {TEXT -1 24 "Here is the error curve." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 37 "plot(q(x) -exp(x),x=-1..1,color=blue);" }}{PARA 13 "" 1 "" {GLPLOT2D 441 243 243 {PLOTDATA 2 "6&-%'CURVESG6#7aw7$$!\"\"\"\"!$\"3a$zj$pd)e2\"!#D7$$! 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" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 518 "Digits := 10:\ne := x -> exp(x)-q(x):\n'e(x)'=exp(x) -'q(x)';\nx1 := -1.;\nx2 := evalf(evalf[16](fsolve(D(e)(x),x=-0.94))); \nx3 := evalf(evalf[16](fsolve(D(e)(x),x=-0.76)));\nx4 := evalf(evalf[ 16](fsolve(D(e)(x),x=-0.5)));\nx5 := evalf(evalf[16](fsolve(D(e)(x),x= -0.17)));\nx6 := evalf(evalf[16](fsolve(D(e)(x),x=0.18)));\nx7 := eval f(evalf[16](fsolve(D(e)(x),x=0.5)));\nx8 := evalf(evalf[16](fsolve(D(e )(x),x=0.77)));\nx9 := evalf(evalf[16](fsolve(D(e)(x),x=0.94)));\nx10 \+ := 1.;\nevalf[14](map(e,[x1,x2,x3,x4,x5,x6,x7,x8,x9,x10]));" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/-%\"eG6#%\"xG,&-%$expGF&\"\"\"-%\"qGF&!\"\" " }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%#x1G$!\"\"\"\"!" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%#x2G$!+AU`!R*!#5" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%#x3G$!+km/Mw!#5" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%#x4G$!+m& ej&\\!#5" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%#x5G$!+%*3qw;!#5" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#>%#x6G$\"+O!pzz\"!#5" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%#x7G$\"+m*)oW]!#5" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%#x8G$\"+ojP)o(!#5" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%#x9G$\"+ba x.%*!#5" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%$x10G$\"\"\"\"\"!" }} {PARA 12 "" 1 "" {XPPMATH 20 "6#7,$!(')e2\"!#9$\"(3l-\"F&$!(bm4\"F&$\" (8].\"F&$!(4&Q6F&$\"'1p5!#8$!'*y;\"F1$\"'^>6F1$!'Nx6F1$\"'fW6F1" }}} {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 59 "By this c alculation the maximum absolute error occurs when " }{TEXT 291 1 "x" } {TEXT -1 1 " " }{TEXT 290 1 "~" }{TEXT -1 15 " 0.9403775455. " }} {PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 35 "xx := x9;\nevalf[16](h(xx)-exp(xx));" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%#xxG$\"+bax.%*!#5" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#$\")#et< \"!#:" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 165 "We can see the reason for the characteristic \"equi-ripple\" prop erty of the error curve if we compare it with the graph of first omitt ed term in the Chebyshev series." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 137 "c9=2/Pi*Int(f(cos(theta))*c os(9*theta),theta=0..Pi);\nc9 := evalf[14](rhs(%));\nplot([exp(x)-h(x) ,c9*T(9,x)],x=-1..1,color=[blue,magenta]);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/%#c9G,$,$-%$IntG6$*&-%$expG6#-%$cosG6#%&thetaG\"\"\"-F /6#,$*&\"\"*F2F1F2F2F2/F1;\"\"!%#PiG*&\"\"#F2F;!\"\"F2" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%#c9G$\"/\"pY-xO5\"!#@" }}{PARA 13 "" 1 "" {GLPLOT2D 360 270 270 {PLOTDATA 2 "6&-%'CURVESG6$7aw7$$!\"\"\"\"!$!3Yd `Jrd)e2\"!#D7$$!3-n;HdNvs**!#=$!3%*4([tN#\\Z&)!#E7$$!3/MLe9r]X**F1$!3g ;n$=$Rz$\\'F47$$!3/,](=ng#=**F1$!3SU]MN=E\"f%F47$$!3%pmm\"HU,\"*)*F1$! 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" }}{PARA 0 "" 0 "" {TEXT -1 63 "Thus the err or curve is not much different from the graph of a " }{TEXT 267 38 "ve rtically scaled Chebyshev polynomial" }{TEXT -1 90 ". As a result the \+ maximum and minimum values occur with very uniform deviations from zer o." }}{PARA 0 "" 0 "" {TEXT -1 112 "This explains why a truncated seri es of Chebyshev polynomials provides such a good approximation for a f unction." }}{PARA 0 "" 0 "" {TEXT -1 108 "It is very not very differen t from the minimax polynomial approximation for the function of the sa me degree." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 4 "The " }{TEXT 267 18 "minimax polynomial" }{TEXT -1 25 " approxim ation of degree " }{TEXT 271 1 "n" }{TEXT -1 16 " for a function " } {XPPEDIT 18 0 "f(x)" "6#-%\"fG6#%\"xG" }{TEXT -1 16 " on an interval \+ " }{XPPEDIT 18 0 "[a,b]" "6#7$%\"aG%\"bG" }{TEXT -1 19 " is the polyno mial " }{XPPEDIT 18 0 "p(x)" "6#-%\"pG6#%\"xG" }{TEXT -1 60 " such tha t the maximum absolute error is a minimum, that is," }}{PARA 0 "" 0 " " {TEXT -1 1 "\n" }{TEXT 259 75 " \+ max " }{XPPEDIT 18 0 "abs( f(x) - p( x) )" "6#-%$absG6#,&-%\"fG6#%\"xG\"\"\"-%\"pG6#F*!\"\"" }{TEXT 260 6 " " }}{PARA 0 "" 0 "" {TEXT 261 37 " x i n [a,b]" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 34 "is minimized over all polynomials " }{XPPEDIT 18 0 "p(x)" "6#-%\"p G6#%\"xG" }{TEXT -1 10 " of degree" }{XPPEDIT 18 0 "``<= n " "6#1%!G% \"nG" }{TEXT -1 2 ". " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 62 "Such a minimax polynomial can be obtained using the pr ocedure " }{TEXT 0 7 "minimax" }{TEXT -1 8 " in the " }{TEXT 0 9 "numa pprox" }{TEXT -1 10 " package. " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 95 "evalf[20](numapprox[minimax] (exp(x),x=-1..1,8)):\np := unapply(evalf(expand(%)),x):\n'p(x)'=p(x); " }}{PARA 12 "" 1 "" {XPPMATH 20 "6#/-%\"pG6#%\"xG,4$\"+,+++5!\"*\"\" \"*&$\"+5!*******!#5F,F'F,F,*&$\"+\\&*****\\F0F,)F'\"\"#F,F,*&$\"+b)zm m\"F0F,)F'\"\"$F,F,*&$\"+JupmT!#6F,)F'\"\"%F,F,*&$\"+'Q'fG$)!#7F,)F'\" \"&F,F,*&$\"+)pM\")Q\"FDF,)F'\"\"'F,F,*&$\"+Qx*p/#!#8F,)F'\"\"(F,F,*&$ \"+&o\")pb#!#9F,)F'\"\")F,F," }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 511 "e := x -> exp(x)-p(x):\n'e( x)'=exp(x)-'p(x)';\nx1 := -1.;\nx2 := evalf(evalf[16](fsolve(D(e)(x),x =-0.94)));\nx3 := evalf(evalf[16](fsolve(D(e)(x),x=-0.76)));\nx4 := ev alf(evalf[16](fsolve(D(e)(x),x=-0.5)));\nx5 := evalf(evalf[16](fsolve( D(e)(x),x=-0.17)));\nx6 := evalf(evalf[16](fsolve(D(e)(x),x=0.18)));\n x7 := evalf(evalf[16](fsolve(D(e)(x),x=0.5)));\nx8 := evalf(evalf[16]( fsolve(D(e)(x),x=0.77)));\nx9 := evalf(evalf[16](fsolve(D(e)(x),x=0.94 )));\nx10 := 1.;\nevalf(evalf[16](map(e,[x1,x2,x3,x4,x5,x6,x7,x8,x9,x1 0])));" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/-%\"eG6#%\"xG,&-%$expGF&\" \"\"-%\"pGF&!\"\"" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%#x1G$!\"\"\"\"! " }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%#x2G$!+.fU%Q*!#5" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%#x3G$!+9l/:w!#5" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%#x4G$!+l&z$=\\!#5" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%#x5G$!+^[- H;!#5" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%#x6G$\"+!z2R%=!#5" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%#x7G$\"+;%3[3&!#5" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%#x8G$\"+)[%#x9G$\"+&)*)R5%*!#5" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%$x10G$\"\" \"\"\"!" }}{PARA 12 "" 1 "" {XPPMATH 20 "6#7,$!*xg$*3\"!#;$\"*kuE7\"F& $!*_F@4\"F&$\"*F;(=6F&$!*dmc4\"F&$\")lx:6!#:$!)(>\"*4\"F1$\")e*46\"F1$ !)[Q/6F1$\")&Ru5\"F1" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 53 "numapprox [infnorm](p(x)-exp(x),x=-1..1,'xmax');\nxmax;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#$\"+kXnA6!#<" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#$!+o3H% Q*!#5" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 " " }}{PARA 0 "" 0 "" {TEXT -1 151 "The coefficients of this minimax pol ynomial bear more than a passing resemblance to those of the Chebyshev series, and the error curve is very similar." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 57 "plot([exp(x) -q(x),exp(x)-p(x)],x=-1..1,color=[blue,red]);" }}{PARA 13 "" 1 "" {GLPLOT2D 539 292 292 {PLOTDATA 2 "6&-%'CURVESG6$7aw7$$!\"\"\"\"!$!3Yd `Jrd)e2\"!#D7$$!3-n;HdNvs**!#=$!3%*4([tN#\\Z&)!#E7$$!3/MLe9r]X**F1$!3g ;n$=$Rz$\\'F47$$!3/,](=ng#=**F1$!3SU]MN=E\"f%F47$$!3%pmm\"HU,\"*)*F1$! 3yf908RdLGF47$$!3'GLekynP')*F1$!3f#)Hyd\\e97F47$$!3()***\\PM@l$)*F1$\" 3/;u5#R5or#!#F7$$!3)omT5!\\F4)*F1$\"3!G[BkbH5j\"F47$$!3!RLL$e%G?y*F1$ \"3KnVsz<5pGF47$$!3#om;HdNvs*F1$\"3\\q`YF)zJ+&F47$$!3u****\\(oUIn*F1$ \"3[B%\\Moler'F47$$!3xLL3-)\\&='*F1$\"39&=rn%QnY!)F47$$!3ommm;p0k&*F1$ \"3v.T\\!3wF.*F47$$!3D*\\P%[Hk;&*F1$\"3Q\\hcsMdP'*F47$$!3#HL3-)*G#p%*F 1$\"3-xh3Tc&H+\"F-7$$!3w\\P4'*>_X%*F1$\"3t,\"4&=!>_,\"F-7$$!3gm\"z>,:= U*F1$\"39'=f.B@H-\"F-7$$!3U$eky-3\")R*F1$\"3?*[\\=X+j-\"F-7$$!3E++vV5S 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(q(x)-exp(x),x=-1..1);\nnumapprox[infnorm](p(x)-exp(x),x=-1..1);" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#$\"+6#et<\"!#<" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#$\"+kXnA6!#<" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}}{SECT 1 {PARA 4 "" 0 " " {TEXT -1 56 "A procedure for constructing a finite Chebyshev series: " }{TEXT 0 10 "chebseries" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 17 "chebseries: usage" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 " " 0 "" {TEXT -1 0 "" }{TEXT 262 18 "Calling Sequence:\n" }}{PARA 0 "" 0 "" {TEXT 263 2 " " }{TEXT -1 28 " chebseries( fx, rng deg ) " } {TEXT 264 1 "\n" }{TEXT -1 0 "" }}{PARA 257 "" 0 "" {TEXT -1 15 "Param eters: " }}{PARA 0 "" 0 "" {TEXT 23 9 " f - " }{TEXT -1 61 " \+ an expression f(x) involving a single variable, say x, " }}{PARA 0 "" 0 "" {TEXT -1 88 " which evaluates to a real floating point number, or a procedure." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }{TEXT 23 12 " rng - " }{TEXT 272 68 "the range x=a . . b or a . . b for the function to be approxi mated." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 " " }{TEXT 23 12 " deg - " }{TEXT 266 53 "the degree of the resulti ng polynomial approximation." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT 267 4 "Note" }{TEXT 273 56 ": deg is an optional argum ent with a default value of 4." }{TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 10 " " }}{PARA 257 "" 0 "" {TEXT -1 12 "Description: " }}{PARA 0 "" 0 "" {TEXT -1 121 "The procedure attempts to find the \+ Chebyshev polynomial of specified degree which approximates fx on the \+ interval [a,b]." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT 265 8 "Options:" }{TEXT -1 1 "\n" }}{PARA 0 "" 0 "" {TEXT -1 46 "numpoints=n\nThe value of n used in the formula" }}{PARA 256 "" 0 "" {TEXT -1 2 " " }{XPPEDIT 18 0 "c[j] = 2/n;" "6#/&%\"cG6#%\"jG*&\"\"# \"\"\"%\"nG!\"\"" }{TEXT -1 1 " " }{XPPEDIT 18 0 "Sum(``^`#`*g(cos(k*P i/n))*cos(k*j*Pi/n),k = 0 .. n);" "6#-%$SumG6$*()%!G%\"#G\"\"\"-%\"gG6 #-%$cosG6#*(%\"kGF*%#PiGF*%\"nG!\"\"F*-F/6#**F2F*%\"jGF*F3F*F4F5F*/F2; \"\"!F4" }{TEXT -1 2 " " }}{PARA 0 "" 0 "" {TEXT -1 60 "for calculati ng the Chebyshev coefficients for the function " }{XPPEDIT 18 0 "g(x) \+ = f(``((a+b)/2)+``((b-a)/2)*x);" "6#/-%\"gG6#%\"xG-%\"fG6#,&-%!G6#*&,& %\"aG\"\"\"%\"bGF2F2\"\"#!\"\"F2*&-F-6#*&,&F3F2F1F5F2F4F5F2F'F2F2" } {TEXT -1 1 "." }}{PARA 0 "" 0 "" {TEXT -1 36 "The default is \"numpoin ts=2*Digits\"." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 29 "output=chebseries/poly/coeffs" }}{PARA 0 "" 0 "" {TEXT -1 48 "The output can be in one of the following forms:" }}{PARA 15 " " 0 "" {TEXT -1 25 "A finite Chebyshev series" }}{PARA 256 "" 0 "" {TEXT -1 2 " " }{XPPEDIT 18 0 "Sum(``^`*`*c[k]*T(k,(x-``((a+b)/2))/`` ((b-a)/2)),k = 0 .. n);" "6#-%$SumG6$*()%!G%\"*G\"\"\"&%\"cG6#%\"kGF*- %\"TG6$F.*&,&%\"xGF*-F(6#*&,&%\"aGF*%\"bGF*F*\"\"#!\"\"F and press [Enter].\nYou can \+ then close up the subsection." }}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 28 " chebseries: implementation " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 8695 "chebseries := proc(f,rng,a rg3)\n local proctype,a,b,d,m,ab,k,i,sum,x,Options,n2,j,delta,cs,val ,\n c,n,rs,saveDigits,pol,outpt,lnx,U,ChebT,fn,chp,u,t,vars,\n \+ startopts,makeproc,deg,evenfcn,oddfcn,rg,aa,bb,r,pi2,pi2p,pi2m;\n\n if nargs<2 then\n error \"at least 2 arguments are required; t he basic syntax is: 'chebseries(f(x),x=a..b)'.\"\n end if;\n \n i f type(f,procedure) then\n if nops([op(1,eval(f))])<>1 then\n \+ error \"the 1st argument, %1, is invalid .. it should be a proced ure with a single argument\",f;\n end if;\n proctype := true ;\n if type(rng,realcons..realcons) then\n rg := rng\n \+ else\n error \"the 2nd argument, %1, is invalid .. when the 1st argument is a procedure, the 2nd argument should have the form 'a ..b', where a and b are real constants, to provide the interval over w hich to construct the Chebyshev series\",rng;\n end if;\n elif \+ type(f,algebraic) then\n vars := indets(f,name) minus indets(f,re alcons);\n if nops(vars)<>1 then \n if not has(indets(f), \{Int,Sum\}) then\n error \"the 1st argument, %1, is invali d .. it should be an expression which depends only on a single variabl e\",f;\n end if;\n end if;\n if type(rng,name=realco ns..realcons) then\n proctype := false;\n x := op(1,rn g);\n if not member(x,vars) then\n error \"the 1st \+ argument, %1, is invalid .. it should be an expression which depends o nly on the variable %2\",f,x;\n end if;\n rg := op(2,r ng);\n else\n error \"the 2nd argument, %1, is invalid .. it should have the form 'x=a..b', where a and b are real constants, t o provide the interval over which to construct the Chebyshev series\", rng;\n end if;\n else\n error \"the 1st argument, %1, is i nvalid .. it should be an algebraic expression in a single variable, a n equation in a single variable, or a procedure with a single real arg ument\",f;\n end if;\n\n startopts := 3;\n deg := 4;\n if narg s>2 then\n if type(arg3,posint) then\n deg := arg3;\n \+ startopts := startopts + 1;\n elif not type(arg3,`=`) then\n error \"3rd argument must be a positive integer\"\n end \+ if;\n end if;\n\nChebT := proc(n::nonnegint,x)\n local i,t,prevt,n ewt;\n prevt := 1;\n if n=0 then return prevt end if;\n t := x; \n if n=1 then return t end if;\n for i from 1 to n-1 do\n ne wt := 2*x*t - prevt;\n prevt := t;\n t := newt;\n end do; \n return expand(t);\nend proc:\n\n # Get the options \"numpoints \".\n # Set the default values to start with.\n n := Digits*2;\n \+ outpt := 'chebseries';\n if startopts<=nargs then\n Options:=[ args[startopts..nargs]];\n if not type(Options,list(equation)) th en\n error \"each optional argument after the %-1 argument mus t be an equation\",startopts-1;\n end if;\n if hasoption(Opt ions,'output','outpt','Options') then \n if not member(outpt, \{'chebseries','poly','coeffs'\}) then\n error \"\\\"output \\\" must be 'chebseries','poly' or 'coeffs'\"\n end if;\n \+ end if;\n if hasoption(Options,'numpoints','n','Options') then \+ \n if not type(n,posint) then\n error \"\\\"numpoin ts\\\" must be a positive integer\"\n end if;\n end if;\n if nops(Options)>0 then\n error \"%1 is not a valid opti on for %2 .. the recognised options are \\\"numpoints\\\" and \\\"outp ut\\\"\",op(1,Options),procname;\n\n end if\n end if;\n n := \+ max(n,deg);\n\n # Increase precision for the computation by about 70 %.\n saveDigits := Digits;\n Digits := Digits + max(5,trunc(Digits *0.70));\n\n aa := op(1,rg);\n bb := op(2,rg); \n a := evalf(aa) ;\n b := evalf(bb);\n if not type([a,b],[numeric, numeric]) then\n error \"expecting a numeric range in the second argument\"\n e nd if;\n if a>=b then\n if proctype then\n error \"inva lid range of values\"\n else\n error \"invalid range of v alues for %1\",x;\n end if;\n end if;\n\n makeproc := proc(fx ,x,a,b)\n proc(_x)\n local y;\n y := traperror(ev alf(eval(subs(x=_x,fx))));\n if y=lasterror or not type(y,nume ric) then\n if evalf(_x-a)=0 then\n y := eval f(limit(fx,x=_x,'right'));\n elif evalf(_x-b)=0 then\n \+ y := evalf(limit(fx,x=_x,'left'));\n else\n \+ y := evalf(limit(fx,x=_x,'real'));\n end if;\n \+ end if;\n y;\n end proc;\n end proc;\n\n if proc type then\n if type(f,procedure) then fn := eval(f)\n else\n try fn := subs(_body=f(_t),proc(_t) evalf(_body) end proc)\n \+ catch:\n error \"expecting the 1st argument to be an o perator, but received %1\",f\n end try;\n fn := subs(_ t='t', eval(fn))\n end if\n else\n fn := makeproc(f,x,a,b) \n end if;\n \n if not type([fn(a),fn(.7101449275*a+.2898550725*b ),\n fn(.381966011*a+.618033989*b),fn(b)],list(numeric)) then\n \+ error \"function does not evaluate to a numeric\"\n end if;\n\n \+ d := evalf((b-a)*0.5);\n ab := evalf((a+b)*0.5);\n\n cs := array (0..n);\n val := array(0..n);\n \n # Set up cosines and funct ion values\n delta := evalf(Pi/n);\n pi2 := evalf(Pi/2);\n m := \+ iquo(n,2,'r');\n if r=0 then\n cs[m] := 0.0;\n val[m] := t raperror(fn(ab+d*cs[m]));\n if val[m]=lasterror then error val[m] end if;\n for k to m do\n cs[m+k] := cos(pi2+k*delta);\n val[m+k] := traperror(fn(ab+d*cs[m+k]));\n if val[m+k ]=lasterror then error val[m+k] end if;\n cs[m-k] := cos(pi2-k *delta);\n val[m-k] := traperror(fn(ab+d*cs[m-k]));\n \+ if val[m-k]=lasterror then error val[m-k] end if;\n end do;\n e lse\n pi2p := pi2+delta/2;\n pi2m := pi2-delta/2;\n for k from 0 to m do\n cs[m+k+1] := cos(pi2p+k*delta);\n \+ val[m+k+1] := traperror(fn(ab+d*cs[m+k+1]));\n if val[m+k+1]=l asterror then error val[m+k+1] end if;\n cs[m-k] := cos(pi2m-k *delta);\n val[m-k] := traperror(fn(ab+d*cs[m-k]));\n \+ if val[m-k]=lasterror then error val[m-k] end if;\n end do;\n e nd if;\n\n oddfcn := false;\n evenfcn := false;\n if not proctyp e and a+b=0 then\n oddfcn := traperror(type(f,oddfunc(x)));\n \+ if oddfcn<>true then oddfcn := false end if;\n if not oddfcn th en \n evenfcn := traperror(type(f,evenfunc(x)));\n if \+ evenfcn<>true then evenfcn := false end if;\n end if;\n end if; \n \n # Compute Chebyshev coefficients\n c := array(0..deg);\n \+ n2 := n+n;\n if oddfcn then\n for j from 0 to deg do\n \+ if irem(j,2)=1 then\n sum := val[0]*0.5;\n for k from 1 to n-1 do\n m := irem(j*k,n2);\n i f m>n then m := n2-m end if;\n sum := sum + cs[m]*val[k] ;\n end do;\n sum := sum - val[n]*0.5;\n \+ c[j] := 2*sum/n;\n else\n c[j] := 0.0;\n \+ end if; \n end do;\n elif evenfcn then\n sum := (val[0 ]+val[n])*0.5;\n for k from 1 to n-1 do\n sum := sum + va l[k];\n end do;\n c[0] := 2*sum/n;\n for j from 1 to de g do\n if irem(j,2)=0 then\n sum := val[0]*0.5;\n \+ for k from 1 to n-1 do\n m := irem(j*k,n2);\n \+ if m>n then m := n2-m end if;\n sum := sum + cs[m]*val[k];\n end do;\n sum := sum + val[n] *0.5;\n c[j] := 2*sum/n;\n else\n c[j] : = 0.0;\n end if; \n end do;\n else # neither even nor odd\n sum := (val[0]+val[n])*0.5;\n for k from 1 to n-1 do \n sum := sum + val[k];\n end do;\n c[0] := 2*sum/n; \n for j from 1 to deg do\n sum := val[0]*0.5;\n \+ for k from 1 to n-1 do\n m := irem(j*k,n2);\n if m>n then m := n2-m end if;\n sum := sum + cs[m]*val[k];\n \+ end do;\n if irem(j,2)=0 then\n sum := sum \+ + val[n]*0.5;\n else\n sum := sum - val[n]*0.5;\n \+ end if;\n c[j] := 2*sum/n; \n end do;\n end if ;\n \n if outpt='chebseries' then\n lnx := (2*x-aa-bb)/(bb-aa) ;\n Digits := saveDigits;\n pol := evalf(c[0]/2)+add(evalf(c [j])*U(j,lnx),j=1..deg);\n pol := subs(U=ChebyshevT,pol); \n \+ if proctype then\n return unapply(pol,x);\n else\n \+ return pol;\n end if;\n elif outpt='poly' then\n pol := c[0]/2+add(c[j]*ChebT(j,(x-ab)/d),j=1..deg);\n pol := sort(pol); \n Digits := saveDigits;\n if proctype then\n retur n unapply(evalf(pol),x);\n else\n return evalf(pol);\n \+ end if;\n else # outpt='coeffs'\n Digits := saveDigits;\n \+ return evalf(c);\n end if; \nend proc:" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 3 "; " }}}{PARA 0 "" 0 "" {TEXT -1 0 " " }}}{SECT 1 {PARA 4 "" 0 "" {TEXT 0 10 "chebseries" }{TEXT -1 11 ": e xamples " }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 9 "Example 1" }}{PARA 0 "" 0 "" {TEXT -1 50 " We construct a Chebyshev series approximation for " }{XPPEDIT 18 0 "f( x) = exp(x);" "6#/-%\"fG6#%\"xG-%$expG6#F'" }{TEXT -1 17 " on the inte rval " }{XPPEDIT 18 0 "[-1,1]" "6#7$,$\"\"\"!\"\"F%" }{TEXT -1 1 "." } }{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 185 "f := x -> exp(x):\n'f(x)'=f(x);\nchebseries(f(x),x=-1..1,8,outp ut='chebseries');\np := unapply(expand(%),x):\n'p(x)'=p(x);\nplot([p(x ),f(x)],x=-1..1,0..2.72,thickness=[1,2],tickmarks=[4,4]);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/-%\"fG6#%\"xG-%$expGF&" }}{PARA 12 "" 1 "" {XPPMATH 20 "6#,4$\"+ye1m7!\"*\"\"\"*&$\"+3#=.8\"F&F'-%+ChebyshevTG6$F '%\"xGF'F'*&$\"+&R`\\r#!#5F'-F,6$\"\"#F.F'F'*&$\"+&)\\oLW!#6F'-F,6$\" \"$F.F'F'*&$\"+U/Cua!#7F'-F,6$\"\"%F.F'F'*&$\"+>JEHa!#8F'-F,6$\"\"&F.F 'F'*&$\"+&HKx\\%!#9F'-F,6$\"\"'F.F'F'*&$\"+ikV)>$!#:F'-F,6$\"\"(F.F'F' *&$\"+1[7#*>!#;F'-F,6$\"\")F.F'F'" }}{PARA 12 "" 1 "" {XPPMATH 20 "6#/ -%\"pG6#%\"xG,4*&$\"+_(>*\\D!#9\"\"\")F'\"\")F-F-*&$\"+O$**p/#!#8F-)F' \"\"(F-F-*&$\"+RfF)Q\"!#7F-)F'\"\"'F-F-*&$\"+1hfG$)F9F-)F'\"\"&F-F-*&$ \"+/')omT!#6F-)F'\"\"%F-F-*&$\"+c)zmm\"!#5F-)F'\"\"$F-F-*&$\"+D(***** \\FJF-)F'\"\"#F-F-*&$\"+4!*******FJF-F'F-F-$\"+,+++5!\"*F-" }}{PARA 13 "" 1 "" {GLPLOT2D 502 334 334 {PLOTDATA 2 "6'-%'CURVESG6%7S7$$!\"\" \"\"!$\"3#3?p8_%zyO!#=7$$!3ommm;p0k&*F-$\"3[:V%\\)erUQF-7$$!3wKL$3Iu7*RF-7$$!3mmmmT%p\"e()F-$\"37(oB3/;_;%F-7$$!3:mmm\"4m(G$ )F-$\"3+r0Y7v'zM%F-7$$!3\"QLL3i.9!zF-$\"37>O'[26y`%F-7$$!3\"ommT!R=0vF -$\"3(fS4#ov@@ZF-7$$!3u****\\P8#\\4(F-$\"3dlFM)eR*=\\F-7$$!3+nm;/siqmF -$\"3?G%4RAQ@8&F-7$$!3[++](y$pZiF-$\"3!f)RZS)[QN&F-7$$!33LLL$yaE\"eF-$ 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18 0 "2046999336*x^7+0" "6#,&*&\"+O$**p/#\"\"\"*$%\"xG\"\" (F&F&\"\"!F&" }{TEXT -1 3 ".00" }{XPPEDIT 18 0 "1388275939*x^6+0" "6#, &*&\"+RfF)Q\"\"\"\"*$%\"xG\"\"'F&F&\"\"!F&" }{TEXT -1 3 ".00" } {XPPEDIT 18 0 "8328596106*x^5+0" "6#,&*&\"+1hfG$)\"\"\"*$%\"xG\"\"&F&F &\"\"!F&" }{TEXT -1 2 ".0" }{XPPEDIT 18 0 "4166688604*x^4+``;" "6#,&*& \"+/')omT\"\"\"*$%\"xG\"\"%F&F&%!GF&" }{TEXT -1 1 " " }}{PARA 256 "" 0 "" {TEXT -1 2 "0." }{XPPEDIT 18 0 "1666679856*x^3+0" "6#,&*&\"+c)zmm \"\"\"\"*$%\"xG\"\"$F&F&\"\"!F&" }{TEXT -1 1 "." }{XPPEDIT 18 0 "49999 99725*x^2+0" "6#,&*&\"+D(*****\\\"\"\"*$%\"xG\"\"#F&F&\"\"!F&" }{TEXT -1 1 "." }{XPPEDIT 18 0 "9999999009*x+1.000000001" "6#,&*&\"+4!******* \"\"\"%\"xGF&F&-%&FloatG6$\"+,+++5!\"*F&" }{TEXT -1 1 " " }}{PARA 0 " " 0 "" {TEXT -1 15 "to approximate " }{XPPEDIT 18 0 "f(x) = exp(x);" " 6#/-%\"fG6#%\"xG-%$expG6#F'" }{TEXT -1 17 " on the interval " } {XPPEDIT 18 0 "[-1,1]" "6#7$,$\"\"\"!\"\"F%" }{TEXT -1 14 " is about 1 .2 " }{TEXT 297 1 "x" }{TEXT -1 1 " " }{XPPEDIT 18 0 "10^(-8);" "6#)\" #5,$\"\")!\"\"" }{TEXT -1 2 ". 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" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 " " {MPLTEXT 1 0 51 "numapprox[infnorm](p(x)-f(x),x=-1..1):\nevalf[4](%) ;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#$\"%&>\"!#6" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 14 "The procedure " }{TEXT 0 8 "chebpade" }{TEXT -1 8 " in the " }{TEXT 0 9 "numapprox" }{TEXT -1 80 " package gives essentially the same result, except it seems to req uire that the " }{TEXT 277 1 "T" }{TEXT -1 44 " be defined to be the o rthogonal polynomial " }{TEXT 277 1 "T" }{TEXT -1 10 " from the " } {TEXT 0 9 "orthopoly" }{TEXT -1 9 " package." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 101 "with(orthopoly,T) :\nf := x -> exp(x):\n'f(x)'=f(x);\nnumapprox[chebpade](f(x),x=-1..1,8 );\nsort(eval(%));" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/-%\"fG6#%\"xG-% $expGF&" }}{PARA 12 "" 1 "" {XPPMATH 20 "6#,4*&$\"+ye1m7!\"*\"\"\"-%\" TG6$\"\"!%\"xGF(F(*&$\"+3#=.8\"F'F(-F*6$F(F-F(F(*&$\"+'R`\\r#!#5F(-F*6 $\"\"#F-F(F(*&$\"+&)\\oLW!#6F(-F*6$\"\"$F-F(F(*&$\"+U/Cua!#7F(-F*6$\" \"%F-F(F(*&$\"+>JEHa!#8F(-F*6$\"\"&F-F(F(*&$\"+'HKx\\%!#9F(-F*6$\"\"'F -F(F(*&$\"+ikV)>$!#:F(-F*6$\"\"(F-F(F(*&$\"+1[7#*>!#;F(-F*6$\"\")F-F(F (" }}{PARA 12 "" 1 "" {XPPMATH 20 "6#,4*&$\"+_(>*\\D!#9\"\"\")%\"xG\" \")F(F(*&$\"+O$**p/#!#8F()F*\"\"(F(F(*&$\"+SfF)Q\"!#7F()F*\"\"'F(F(*&$ \"+1hfG$)F5F()F*\"\"&F(F(*&$\"+/')omT!#6F()F*\"\"%F(F(*&$\"+c)zmm\"!#5 F()F*\"\"$F(F(*&$\"+F(*****\\FFF()F*\"\"#F(F(*&$\"+4!*******FFF(F*F(F( $\"+,+++5!\"*F(" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 80 "We can obtain a more accurate approximating polynomial by increasing the degree." }}{PARA 0 "" 0 "" {TEXT -1 35 "The precision \+ is increased for the " }{TEXT 0 4 "plot" }{TEXT -1 64 " command so tha t the accuracy of this polynomial can be checked." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 210 "Digits := 1 6:\nf := x -> exp(x):\n'f(x)'=f(x);\nchebseries(f(x),x=-1..1,12,output 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" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 51 "numapprox[infnorm](p(x)-f(x),x=-1..1):\nevalf[4](%); " }}{PARA 11 "" 1 "" {XPPMATH 20 "6#$\"%**>!#5" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 14 "The procedure " }{TEXT 0 8 "chebpade" }{TEXT -1 8 " in the " }{TEXT 0 9 "numapprox" }{TEXT -1 43 " package gives essentially the same result." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 102 "with(orthop oly,T):\nf := x -> cosh(x):\n'f(x)'=f(x);\nnumapprox[chebpade](f(x),x= -1..1,6);\nsort(eval(%));" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/-%\"fG6# %\"xG-%%coshGF&" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#,**&$\"+ye1m7!\"*\" \"\"-%\"TG6$\"\"!%\"xGF(F(*&$\"+'R`\\r#!#5F(-F*6$\"\"#F-F(F(*&$\"+U/Cu a!#7F(-F*6$\"\"%F-F(F(*&$\"+'HKx\\%!#9F(-F*6$\"\"'F-F(F(" }}{PARA 11 " " 1 "" {XPPMATH 20 "6#,**&$\"+NVFR9!#7\"\"\")%\"xG\"\"'F(F(*&$\"+/7]jT !#6F()F*\"\"%F(F(*&$\"+vM1+]!#5F()F*\"\"#F(F($\"+ " 0 "" {MPLTEXT 1 0 237 "Digits := 18:\nf := x -> co sh(x):\n'f(x)'=f(x);\nchebseries(f(x),x=-1..1,12,output='chebseries'); 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Fhel-%%VIEWG6$;F(Fgz%(DEFAULTG" 1 2 0 1 10 0 2 9 1 4 2 1.000000 45.000000 45.000000 0 0 "Curve 1" "Curve 2" }}}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 84 "The following (absolute) error \+ curve shows that the maximum absolute error in using " }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "p(x) =0 " "6#/-%\"pG6#%\"xG\"\"!" }{TEXT -1 1 "." }{XPPEDIT 18 0 "9999999014*x+0" "6#,&*&\"+9!*******\" \"\"%\"xGF&F&\"\"!F&" }{TEXT -1 1 "." }{XPPEDIT 18 0 "1666679856*x^3+0 " "6#,&*&\"+c)zmm\"\"\"\"*$%\"xG\"\"$F&F&\"\"!F&" }{TEXT -1 3 ".00" } {XPPEDIT 18 0 "8328596106*x^5+0" "6#,&*&\"+1hfG$)\"\"\"*$%\"xG\"\"&F&F &\"\"!F&" }{TEXT -1 4 ".000" }{XPPEDIT 18 0 "2046999336*x^7" "6#*&\"+O $**p/#\"\"\"*$%\"xG\"\"(F%" }{TEXT -1 1 " " }}{PARA 0 "" 0 "" {TEXT -1 15 "to approximate " }{XPPEDIT 18 0 "f(x) = cosh(x);" "6#/-%\"fG6#% \"xG-%%coshG6#F'" }{TEXT -1 17 " on the interval " }{XPPEDIT 18 0 "[-1 ,1]" "6#7$,$\"\"\"!\"\"F%" }{TEXT -1 14 " is about 1.1 " }{TEXT 294 1 "x" }{TEXT -1 1 " " }{XPPEDIT 18 0 "10^(-8);" "6#)\"#5,$\"\")!\"\"" } {TEXT -1 2 ". 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" }} {PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 51 "numapprox[infnorm](p(x)-f(x),x=-1..1):\nevalf[4](%);" }}{PARA 11 " " 1 "" {XPPMATH 20 "6#$\"%\\6!#6" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }} {PARA 0 "" 0 "" {TEXT -1 14 "The procedure " }{TEXT 0 8 "chebpade" } {TEXT -1 8 " in the " }{TEXT 0 9 "numapprox" }{TEXT -1 31 " package gi ves the same result." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 102 "with(orthopoly,T):\nf := x -> sinh(x):\n 'f(x)'=f(x);\nnumapprox[chebpade](f(x),x=-1..1,7);\nsort(eval(%));" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#/-%\"fG6#%\"xG-%%sinhGF&" }}{PARA 11 " " 1 "" {XPPMATH 20 "6#,**&$\"+3#=.8\"!\"*\"\"\"-%\"TG6$F(%\"xGF(F(*&$ \"+&)\\oLW!#6F(-F*6$\"\"$F,F(F(*&$\"+>JEHa!#8F(-F*6$\"\"&F,F(F(*&$\"+i kV)>$!#:F(-F*6$\"\"(F,F(F(" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#,**&$\"+ O$**p/#!#8\"\"\")%\"xG\"\"(F(F(*&$\"+1hfG$)!#7F()F*\"\"&F(F(*&$\"+c)zm m\"!#5F()F*\"\"$F(F(*&$\"+9!*******F5F(F*F(F(" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" 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_o7$$\"0S%[Y==1'*F-Feim7$$\"0]7Gyh(>'*F-F]im7$$\"0voal@pk*F-Feim7$$\"0 +D\"G:3u'*F-F^ho7$$\"05`WYhwo*F-F]im7$$\"0D\"y+9C,(*F-F]im7$$\"0S4rL@[ r*F-Feim7$$\"0]PMF,%G(*F-Feim7$$\"0gl(47)>u*F-Feim7$$\"0v$4Y6cb(*F-F]i m7$$\"0!>U#3T\"p(*F-F]im7$$\"0+](=5s#y*F-Feim7$$\"05y]&4I'z*F-Fd`o7$$ \"0D19*3))4)*F-Fd`o7$$\"0SMx#3YB)*F-Fd`o7$$\"0]iSwSq$)*F-Fiim7$$\"0g!R +2i])*F-Fd`o7$$\"0v=nj+U')*F-Ff[n7$$\"0!p/t0yx)*F-Fdhm7$$\"0+v$40O\"*) *F-Fiim7$$\"05.dWS\\!**F-Fdhm7$$\"0DJ?Q?&=**F-Fh_n7$$\"0Sf$=.5K**F-Fdh m7$$\"0](oa-oX**F-Fgjm7$$\"0g:5>g#f**F-Fgjm7$$\"0vVt7SG(**F-F]_n7$$\"0 !>nj+U')**F-F`\\n7$$FbdmF*F[hm-%+AXESLABELSG6$Q\"x6\"Q!Fght-%&COLORG6& %$RGBG$FPF)Fh_nFbht-%%VIEWG6$;F(Fbht;$FhsFF$Fg[nFF" 1 2 0 1 10 0 2 6 1 4 2 1.000000 45.000000 44.000000 0 0 "Curve 1" }}}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 42 "The following calculation shows that when " }{XPPEDIT 18 0 "x=0" "6#/%\"xG\"\"!" }{TEXT -1 47 " .02 the absolute error in the polynomial value " }{XPPEDIT 18 0 "q(x) " "6#-%\"qG6#%\"xG" }{TEXT -1 69 " is less than that obtained by evalu ating the analytical formula for " }{XPPEDIT 18 0 "f(x)" "6#-%\"fG6#% \"xG" }{TEXT -1 2 ". " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 251 "xx := 0.02;\nDigits := 15:\naccurate_val := evalf(evalf[25](f(xx)));\nfunction_val := evalf(f(xx));\nabs_error _in_function_val := abs(function_val-accurate_val);\npoly_val := evalf (q(xx));\nabs_error_in_poly_val := abs(poly_val-accurate_val);\nDigits := 10:\n" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%#xxG$\"\"#!\"#" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#>%-accurate_valG$\"0zP8+n+,\"!#9" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#>%-function_valG$\"0+Q8+n+,\"!#9" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#>%:abs_error_in_function_valG$\"#@!#9 " }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%)poly_valG$\"0rP8+n+,\"!#9" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#>%6abs_error_in_poly_valG$\"\")!#9" }} }{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 9 "Example 6" }}{PARA 0 " " 0 "" {TEXT -1 45 " We construct a polynomial approximation for " } {XPPEDIT 18 0 "f(x) = PIECEWISE([Int(1/(t+exp(t)),t = 0 .. x)/x, x <> \+ 0],[1, x = 0]);" "6#/-%\"fG6#%\"xG-%*PIECEWISEG6$7$*&-%$IntG6$*&\"\"\" F1,&%\"tGF1-%$expG6#F3F1!\"\"/F3;\"\"!F'F1F'F70F'F:7$F1/F'F:" }{TEXT -1 19 " on the interval " }{XPPEDIT 18 0 " [0,2]" "6#7$\"\"!\"\"#" } {TEXT -1 1 "." }}{PARA 0 "" 0 "" {TEXT 267 4 "Note" }{TEXT -1 3 ": " }{XPPEDIT 18 0 "Limit(Int(1/(t+exp(t)),t = 0 .. x)/x,x = 0) = 1;" "6#/ -%&LimitG6$*&-%$IntG6$*&\"\"\"F,,&%\"tGF,-%$expG6#F.F,!\"\"/F.;\"\"!% \"xGF,F6F2/F6F5F," }{TEXT -1 10 ", so that " }{XPPEDIT 18 0 "f(x)" "6# -%\"fG6#%\"xG" }{TEXT -1 18 " is continuous at " }{XPPEDIT 18 0 "x=0" "6#/%\"xG\"\"!" }{TEXT -1 2 ". " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 52 "Limit(Int(1/(t+exp(t)),t=0.. x)/x,x=0);\n``=value(%);\n" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#-%&Limit G6$,$-%$IntG6$*&\"\"\"F+,&%\"tGF+-%$expG6#F-F+!\"\"/F-;\"\"!%\"xG*$F5F 1/F5F4" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/%!G\"\"\"" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 78 "f := x -> Int(1/(t+exp(t)),t=0..x)/x:\n'f(x)'=f(x);\nplot(f(x),x=-.3..3,0..1.5) ;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/-%\"fG6#%\"xG,$-%$IntG6$*&\"\"\" F-,&%\"tGF--%$expG6#F/F-!\"\"/F/;\"\"!F'*$F'F3" }}{PARA 13 "" 1 "" {GLPLOT2D 437 251 251 {PLOTDATA 2 "6%-%'CURVESG6$7U7$$!\"$!\"\"$\"+eEj y9!\"*7$$!+cpMSE!#5$\"+\"R\"F-7$$!+7Rp!G#F1$\"+'\\`fJ\"F-7$$!+A F1$\"+AYIe7F-7$$!+K&H[l\"F1$\"+4\"zl?\"F-7$$!*z&z4&*F1$\"+h;=26F-7$$!* ^SYU#F1$\"+H]&\\-\"F-7$$\"*ESoi%F1$\"+%*y&3c*F17$$\"+elW;6F1$\"+-QU5!* F17$$\"+$zzLz\"F1$\"+TDv5&)F17$$\"+8^Y$\\#F1$\"+2(fX0)F17$$\"+^_I\">$F 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\nchebseries(f(x),x=0..2,20,output='chebseries');\nq := unapply(expand (%),x):\n'q(x)'=q(x);\nDigits := 10:" }}{PARA 12 "" 1 "" {XPPMATH 20 " 6#,L$\"3Ic\"H.'=;ze!#=\"\"\"*&$\"3OTj._e7OIF&F'-%+ChebyshevTG6$F',&%\" xGF'F'!\"\"F'F0*&$\"3$R'eBX+G\\x!#>F'-F,6$\"\"#F.F'F'*&$\"3$48z`'>>k@F 4F'-F,6$\"\"$F.F'F0*&$\"36!4v;!e*4W'!#?F'-F,6$\"\"%F.F'F'*&$\"39U;>'y[ #z>FAF'-F,6$\"\"&F.F'F0*&$\"3,j-FNF'-F,6$\"\"(F.F'F0*&$\"3\"Gq13M4bV'!#AF'-F,6$\"\")F.F'F'*&$\"3 /()z*oTZl5#FenF'-F,6$\"\"*F.F'F0*&$\"3dkU&RTE`&p!#BF'-F,6$\"#5F.F'F'*& $\"3(3\"[5kW78BFboF'-F,6$\"#6F.F'F0*&$\"3\"GA#yF.F'F0*&$\"3/Z!f1z8RS\"F \\sF'-F,6$\"#?F.F'F'" }}{PARA 12 "" 1 "" {XPPMATH 20 "6#/-%\"qG6#%\"xG ,L*&$\"3'>Q*Qc'*******!#=\"\"\"F'F-!\"\"*&$\"3))HFUH'e'4)*!#>F-)F'\"#< F-F.*&$\"3JVO`6iZ!e\"F2F-)F'\"#=F-F-*&$\"3X48Art0z:!#?F-)F'\"#>F-F.*&$ \"3b/0xM:bgt!#AF-)F'\"#?F-F-$\"3ls*f#**********F,F-*&$\"3CtGN4KeT:!# " 0 "" {MPLTEXT 1 0 58 "evalf[20](pl ot(q(x)-f(x),x=0..2,color=COLOR(RGB,.6,0,1)));" }}{PARA 13 "" 1 "" {GLPLOT2D 494 224 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0 "" }} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}}{SECT 1 {PARA 4 "" 0 " " {TEXT -1 9 "Example 7" }}{PARA 0 "" 0 "" {TEXT -1 50 "We construct a Chebyshev series approximation for " }{XPPEDIT 18 0 "sin(Pi/2*x)/x;" "6#*&-%$sinG6#*(%#PiG\"\"\"\"\"#!\"\"%\"xGF)F)F,F+" }{TEXT -1 17 " on \+ the interval " }{XPPEDIT 18 0 "[-1,1]" "6#7$,$\"\"\"!\"\"F%" }{TEXT -1 1 "." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 174 "f := x -> sin(Pi/2*x)/x:\n'f(x)'=f(x);\nchebseries(f (x),x=-1..1,8,output='chebseries');\np := unapply(expand(%),x):\n'p(x) '=p(x);\nplot(p(x)-f(x),x=-1..1,color=COLOR(RGB,.6,0,1));" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/-%\"fG6#%\"xG*&-%$sinG6#,$*(\"\"#!\"\"%#PiG\" \"\"F'F1F1F1F'F/" }}{PARA 12 "" 1 "" {XPPMATH 20 "6#,,$\"+i*yiF\"!\"* \"\"\"*&$\"+#p:E&G!#5F'-%+ChebyshevTG6$\"\"#%\"xGF'!\"\"*&$\"+2g,=\"*! #7F'-F-6$\"\"%F0F'F'*&$\"+N^(eO\"!#8F'-F-6$\"\"'F0F'F1*&$\"+e='\\=\"!# :F'-F-6$\"\")F0F'F'" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/-%\"pG6#%\"xG, 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$$\"3K+]7G:3u'*F1$!3UUflkoR)o&F-7$$\"3G+vVt7SG(*F1$!3u)>n<(R\">t%F-7$$ \"3A++v=5s#y*F1$!3mC)R<54)4MF-7$$\"3v]iS\"*3))4)*F1$!3iydHK1q'f#F-7$$ \"3;+D1k2/P)*F1$!3W*eh[*oxt;F-7$$\"3e\\(=nj+U')*F1$!3+')**\\dv)RM'FW7$ $\"35+]P40O\"*)*F1$\"3%H,\"))QgF$G&FW7$$\"3PDJqX/%\\!**F1$\"3'zXBD*3 DF-7$$\"31+voa-oX**F1$\"3Oz#o^m*p_KF-7$$\"3ACc,\">g#f**F1$\"3=\"z " 0 "" {MPLTEXT 1 0 64 "with(orthopoly,T):\nn umapprox[chebpade](f(x),x=-1..1,8);\neval(%);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#,,*&$\"+i*yiF\"!\"*\"\"\"-%\"TG6$\"\"!%\"xGF(F(*&$\"+#p :E&G!#5F(-F*6$\"\"#F-F(!\"\"*&$\"+1g,=\"*!#7F(-F*6$\"\"%F-F(F(*&$\"+O^ (eO\"!#8F(-F*6$\"\"'F-F(F5*&$\"+e='\\=\"!#:F(-F*6$\"\")F-F(F(" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#,,$\"+?jzq:!\"*\"\"\"*&$\"+/wjfk!#5F') %\"xG\"\"#F'!\"\"*&$\"+gA**oz!#6F')F-\"\"%F'F'*&$\"+r1:uY!#7F')F-\"\"' F'F/*&$\"+y6v;:!#8F')F-\"\")F'F'" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }} {PARA 0 "" 0 "" {TEXT -1 80 "We can obtain a more accurate approximati ng polynomial by increasing the degree." }}{PARA 0 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\"9++++]7.#Q?&=**F1$!&y8#F-7$$\"9++++v$f$=.5K**F1$!&Gh\"F-7$$\"9+++++v oa-oX**F1$!%u&*F-7$$\"9++++Dc,\">g#f**F1$!%x:F-7$$\"9++++]PMF,%G(**F1$ \"%2!)F-7$$\"9++++v=nj+U')**F1$\"&L$>F-7$$\"\"\"F*F+-%&COLORG6&%$RGBG$ \"\"'F)$F*F*Faiq-%+AXESLABELSG6$Q\"x6\"Q!F^jq-%%VIEWG6$%(DEFAULTG;$!\" %!#>$\"\"%Fgjq" 1 2 0 1 10 0 2 6 1 4 2 1.000000 45.000000 45.000000 0 0 "Curve 1" }}}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 15 "The polynomial " }{XPPEDIT 18 0 "r(x) = 2*x/Pi;" "6#/-%\"rG6#% \"xG*(\"\"#\"\"\"F'F*%#PiG!\"\"" }{TEXT -1 1 " " }{XPPEDIT 18 0 "q(2*x /Pi)" "6#-%\"qG6#*(\"\"#\"\"\"%\"xGF(%#PiG!\"\"" }{TEXT -1 31 " provid es an approximation for " }{XPPEDIT 18 0 "sin(x);" "6#-%$sinG6#%\"xG" }{TEXT -1 17 " on the interval " }{XPPEDIT 18 0 "[-Pi/2, Pi/2];" "6#7$ ,$*&%#PiG\"\"\"\"\"#!\"\"F)*&F&F'F(F)" }{TEXT -1 53 " which has good e rror characterstics with respect to " }{TEXT 267 14 "relative error" } {TEXT -1 2 ". 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