{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 "" -1 259 "" 0 1 0 0 0 0 0 1 0 0 0 0 0 0 0 1 }{CSTYLE "Purple Emphasis" -1 260 "Times" 1 12 102 0 230 1 0 1 0 0 0 0 0 0 0 0 }{CSTYLE "Red Emphasis" -1 261 "Times" 1 12 255 0 0 1 0 1 0 0 0 0 0 0 0 0 }{CSTYLE "Dark Red Emphasis" -1 262 "Times" 1 12 128 0 0 1 0 1 0 0 0 0 0 0 0 0 }{CSTYLE "Blue emphasis" -1 263 "Times" 0 0 0 0 255 1 0 1 0 0 0 0 0 0 0 1 }{CSTYLE "Grey Emphasis" -1 264 "Tim es" 1 12 96 52 84 1 0 1 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 265 "" 0 1 0 0 0 0 1 0 0 0 0 0 0 0 0 0 }{PSTYLE "Normal" -1 0 1 {CSTYLE "" -1 -1 "Tim es" 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 "Maple Plot" -1 13 1 {CSTYLE "" -1 -1 "Times" 1 12 0 0 0 1 2 2 2 2 2 2 1 1 1 1 }3 1 0 0 0 0 1 0 1 0 2 2 0 1 }{PSTYLE "Normal" -1 256 1 {CSTYLE " " -1 -1 "Times" 1 12 0 0 0 1 2 2 2 2 2 2 1 1 1 1 }3 1 0 0 0 0 1 0 1 0 2 2 0 1 }{PSTYLE "Normal" -1 257 1 {CSTYLE "" -1 -1 "Times" 1 12 0 0 0 1 2 1 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 259 48 "Jacobi polynomials and interpola ting polynomials" }}{PARA 0 "" 0 "" {TEXT -1 37 "by Peter Stone, Nanai mo, B.C., Canada" }}{PARA 0 "" 0 "" {TEXT -1 19 "Version: 25.3.2007" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 8 "restart;" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}} {SECT 1 {PARA 4 "" 0 "" {TEXT -1 56 "load interpolation and function a pproximation procedures" }}{PARA 0 "" 0 "" {TEXT -1 17 "The Maple m-fi le " }{TEXT 264 10 "fcnapprx.m" }{TEXT -1 37 " contains the code for t he procedure " }{TEXT 0 9 "interpoly" }{TEXT -1 25 " used in this work sheet. " }}{PARA 0 "" 0 "" {TEXT -1 123 "It can be read into a Maple s ession by a command similar to the one that follows, where the file pa th 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 18 "Jacobi polynomials" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}{PARA 0 "" 0 "" {TEXT -1 4 "Th e " }{TEXT 260 18 "Jacobi polynomials" }{TEXT -1 1 " " }{XPPEDIT 18 0 "P(n,alpha,beta,x)" "6#-%\"PG6&%\"nG%&alphaG%%betaG%\"xG" }{TEXT -1 36 " satisfy the orthogonality relation:" }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "Int(w(x)*P(n,alpha,beta,x)*P(m,alpha,beta,x), x = -1 .. 1) = PIECEWISE([0, n <> m],[2^(alpha+beta+1)*GAMMA(n+alpha+1 )*GAMMA(n+beta+1)/((2*n+alpha+beta+1)*GAMMA(n+alpha+beta+1)), n = m]); " "6#/-%$IntG6$*(-%\"wG6#%\"xG\"\"\"-%\"PG6&%\"nG%&alphaG%%betaGF+F,-F .6&%\"mGF1F2F+F,/F+;,$F,!\"\"F,-%*PIECEWISEG6$7$\"\"!0F0F57$**)\"\"#,( F1F,F2F,F,F,F,-%&GAMMAG6#,(F0F,F1F,F,F,F,-FF6#,(F0F,F2F,F,F,F,*&,**&FC F,F0F,F,F1F,F2F,F,F,F,-FF6#,*F0F,F1F,F2F,F,F,F,F9/F0F5" }{TEXT -1 2 ", " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 36 "with respect to the weight function " }{XPPEDIT 18 0 "w(x) = (1-x)^alpha*( 1+x)^beta;" "6#/-%\"wG6#%\"xG*&),&\"\"\"F+F'!\"\"%&alphaGF+),&F+F+F'F+ %%betaGF+" }{TEXT -1 8 ", where " }{XPPEDIT 18 0 "alpha" "6#%&alphaG" }{TEXT -1 5 " and " }{XPPEDIT 18 0 "beta" "6#%%betaG" }{TEXT -1 18 " a re greater than " }{XPPEDIT 18 0 "-1" "6#,$\"\"\"!\"\"" }{TEXT -1 1 ". " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 48 "The L egendre polynomials are obtained by taking " }{XPPEDIT 18 0 "alpha=bet a" "6#/%&alphaG%%betaG" }{XPPEDIT 18 0 "``=0" "6#/%!G\"\"!" }{TEXT -1 1 "." }}{PARA 0 "" 0 "" {TEXT -1 26 "The Chebyshev polynomials " } {XPPEDIT 18 0 "T(n,x)" "6#-%\"TG6$%\"nG%\"xG" }{TEXT -1 48 " are scala r multiples of the Jacobi polynomials " }{XPPEDIT 18 0 "P(n,-1/2,-1/2, x)" "6#-%\"PG6&%\"nG,$*&\"\"\"F)\"\"#!\"\"F+,$*&F)F)F*F+F+%\"xG" } {TEXT -1 2 ". " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 179 "alpha := 1;\nbeta := 3/4;\nn := 13; m := 14;\na lias(P=orthopoly[P]):\nw := x -> (1-x)^alpha*(1+x)^beta:\n'w(x)'=w(x); \nInt('w(x)*P(n,alpha,beta,x)*P(m,alpha,beta,x)',x=-1..1);\nvalue(%); " }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%&alphaG\"\"\"" }}{PARA 11 "" 1 " " {XPPMATH 20 "6#>%%betaG#\"\"$\"\"%" }}{PARA 11 "" 1 "" {XPPMATH 20 " 6#>%\"nG\"#8" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"mG\"#9" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/-%\"wG6#%\"xG*&,&\"\"\"F*F'!\"\"F*),&F'F*F* F*#\"\"$\"\"%F*" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#-%$IntG6$*(-%\"wG6# %\"xG\"\"\"-%\"PG6&%\"nG%&alphaG%%betaGF*F+-F-6&%\"mGF0F1F*F+/F*;!\"\" F+" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#\"\"!" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 302 "alpha := 1;\nbeta := 3/4;\nn := 13; m := 14;\nalias(P=orthopoly[P]):\nw := x -> (1-x)^a lpha*(1+x)^beta:\n'w(x)'=w(x);\nInt('w(x)*P(n,alpha,beta,x)*P(n,alpha, beta,x)',x=-1..1);\nvalue(%);\n2^(alpha+beta+1)*GAMMA(n+alpha+1)*GAMMA (n+beta+1)/( (2*n+alpha+beta+1)*n!*GAMMA(n+alpha+beta+1));\nn := \+ 'n': m := 'm':" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%&alphaG\"\"\"" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#>%%betaG#\"\"$\"\"%" }}{PARA 11 "" 1 " " {XPPMATH 20 "6#>%\"nG\"#8" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"mG \"#9" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/-%\"wG6#%\"xG*&,&\"\"\"F*F'! \"\"F*),&F'F*F*F*#\"\"$\"\"%F*" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#-%$I ntG6$*&-%\"wG6#%\"xG\"\"\")-%\"PG6&%\"nG%&alphaG%%betaGF*\"\"#F+/F*;! \"\"F+" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#,$*(\"$'*)\"\"\"\"%&y'!\"\" \"\"##\"\"$\"\"%F&" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#,$*(\"$'*)\"\"\" \"%&y'!\"\"\"\"##\"\"$\"\"%F&" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 56 "The Jacobi polynomials satisfy the recurrence relations:" }}{PARA 256 "" 0 "" {TEXT -1 2 " " }{XPPEDIT 18 0 "P(0,alpha,beta,x) = 1;" "6 #/-%\"PG6&\"\"!%&alphaG%%betaG%\"xG\"\"\"" }{TEXT -1 1 " " }}{PARA 256 "" 0 "" {TEXT -1 2 " " }{XPPEDIT 18 0 "P(1,alpha,beta,x)=(alpha-b eta)/2+(1+(alpha+beta)/2)*x" "6#/-%\"PG6&\"\"\"%&alphaG%%betaG%\"xG,&* &,&F(F'F)!\"\"F'\"\"#F.F'*&,&F'F'*&,&F(F'F)F'F'F/F.F'F'F*F'F'" }{TEXT -1 2 " " }}{PARA 0 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "P(n+1,alpha ,beta,x)" "6#-%\"PG6&,&%\"nG\"\"\"F(F(%&alphaG%%betaG%\"xG" }{TEXT -1 4 " = " }}{PARA 0 "" 0 "" {TEXT -1 2 " " }{XPPEDIT 18 0 "((2*n+alpha +beta-1)*(alpha^2-beta^2+(2*n+alpha+beta-2)*(2*n+alpha+beta)*x)*P(n-1, alpha,beta,x)-2*(n+alpha-1)*(n+beta-1)*(2*n+alpha+beta)*P(n-2,alpha,be ta,x))/(2*n*(n+alpha+beta)*(2*n+alpha+beta-2));" "6#*&,&*(,**&\"\"#\" \"\"%\"nGF)F)%&alphaGF)%%betaGF)F)!\"\"F),(*$F+F(F)*$F,F(F-*(,**&F(F)F *F)F)F+F)F,F)F(F-F),(*&F(F)F*F)F)F+F)F,F)F)%\"xGF)F)F)-%\"PG6&,&F*F)F) F-F+F,F6F)F)*,F(F),(F*F)F+F)F)F-F),(F*F)F,F)F)F-F),(*&F(F)F*F)F)F+F)F, F)F)-F86&,&F*F)F(F-F+F,F6F)F-F)**F(F)F*F),(F*F)F+F)F,F)F),**&F(F)F*F)F )F+F)F,F)F(F-F)F-" }{TEXT -1 1 " " }}{PARA 0 "" 0 "" {TEXT -1 4 "for \+ " }{XPPEDIT 18 0 "n>1" "6#2\"\"\"%\"nG" }{TEXT -1 1 "." }}{PARA 256 " " 0 "" {TEXT -1 1 " " }}{PARA 0 "" 0 "" {TEXT -1 70 "The recurrence re lation provides a convenient way to construct them.\n" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 649 "alias(P=P):\nP := proc(n::nonnegin t,a::algebraic,b::algebraic,x::algebraic)\n local px,c,d,e,f,g,h,k; \n option remember;\n if n=0 then 1\n elif n=1 then (a-b)/2+(1+( a+b)/2)*x\n else\n h := 2*n+a+b;\n g := h-1;\n k := h -2;\n c := 2*(n*(n+a+b)*k);\n d := a^2-b^2;\n e := k*h; \n f := 2*(n+a-1)*(n+b-1)*h;\n px := (g*(d+e*x)*P(n-1,a,b,x) -f*P(n-2,a,b,x))/c;\n if type(a,'numeric') and type(b,'numeric') \+ then expand(px)\n else convert(map(normal,series(px,x,n+1)),polyn om)\n end if\n end if\nend proc:\nalpha := -1/8;\nbeta := -7/8; \nfor n from 0 to 7 do\n print('P'(n,alpha,beta,x)=P(n,alpha,beta,x) )\nend do;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%&alphaG#!\"\"\"\")" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#>%%betaG#!\"(\"\")" }}{PARA 11 "" 1 " " {XPPMATH 20 "6#/-%\"PG6&\"\"!#!\"\"\"\")#!\"(F*%\"xG\"\"\"" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/-%\"PG6&\"\"\"#!\"\"\"\")#!\"(F*%\"xG,&#\" \"$F*F'*&\"\"#F)F-F'F'" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/-%\"PG6&\" \"##!\"\"\"\")#!\"(F*%\"xG,(#\"#R\"$G\"F)*&#\"\"$F*\"\"\"F-F5F5*&#F4\" \"%F5*$)F-F'F5F5F5" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/-%\"PG6&\"\"$#! \"\"\"\")#!\"(F*%\"xG,*#\"$*>\"%C5F)*&#\"$8#\"$c#\"\"\"F-F6F)*&#\"\"* \"#;F6*$)F-\"\"#F6F6F6*&#\"\"&\"\"%F6*$)F-F'F6F6F6" }}{PARA 11 "" 1 " " {XPPMATH 20 "6#/-%\"PG6&\"\"%#!\"\"\"\")#!\"(F*%\"xG,,#\"%bt\"&oF$\" \"\"*&#\"$&H\"$7&F2F-F2F)*&#\"$:&\"$c#F2*$)F-\"\"#F2F2F)*&#\"#:\"#;F2* $)F-\"\"$F2F2F2*&#\"#NFAF2*$)F-F'F2F2F2" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/-%\"PG6&\"\"&#!\"\"\"\")#!\"(F*%\"xG,.#\"&(RQ\"'W@E\"\"\"*&#\"& N(p\"&Ob'F2F-F2F2*&#\"%le\"%'4%F2*$)F-\"\"#F2F2F)*&#\"%DZ\"%C5F2*$)F- \"\"$F2F2F)*&#\"$0\"\"#kF2*$)F-\"\"%F2F2F2*&#\"#j\"#;F2*$)F-F'F2F2F2" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/-%\"PG6&\"\"'#!\"\"\"\")#!\"(F*%\"x G,0#\"'F)z(\"(/V>%F)*&#\"'j->\"'W@E\"\"\"F-F6F6*&#\"$J#\"#KF6*$)F-F'F6 F6F6*&#\"&@0\"\"%C5F6*$)F-\"\"%F6F6F)*&#\"%4MF@F6*$)F-\"\"$F6F6F)*&#\" $*=\"#kF6*$)F-\"\"&F6F6F6*&#\"'FVZ\"'s58F6*$)F-\"\"#F6F6F6" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/-%\"PG6&\"\"(#!\"\"\"\")#!\"(F*%\"xG,2#\"(J K5%\")KWbLF)*&#\")x#=0\"\"(3')Q)\"\"\"F-F6F)*&#\"$H%\"#KF6*$)F-F'F6F6F 6*&#\"$$p\"$G\"F6*$)F-\"\"'F6F6F6*&#\"&:7'\"%#>)F6*$)F-\"\"%F6F6F)*&# \"'v()p\"&Ob'F6*$)F-\"\"$F6F6F6*&#\"&pf%\"%[?F6*$)F-\"\"&F6F6F)*&#\"'B @m\"'W@EF6*$)F-\"\"#F6F6F6" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 85 "alpha := -1/8;\nbeta := -7/8;\nplot ([seq(P(n,alpha,beta,x),n=0..7)],x=-1..1);\np := 'P':" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%&alphaG#!\"\"\"\")" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%%betaG#!\"(\"\")" }}{PARA 13 "" 1 "" {GLPLOT2D 522 309 309 {PLOTDATA 2 "6,-%'CURVESG6$7S7$$!\"\"\"\"!$\"\"\"F*7$$!3ommm;p0k&*!#=F +7$$!3wKL$3s%HaF0F+7$$!3Q++ +]$*4)*\\F0F+7$$!39+++]_&\\c%F0F+7$$!31+++]1aZTF0F+7$$!3umm;/#)[oPF0F+ 7$$!3hLLL$=exJ$F0F+7$$!3*RLLLtIf$HF0F+7$$!3]++]PYx\"\\#F0F+7$$!3EMLLL7 i)4#F0F+7$$!3c****\\P'psm\"F0F+7$$!3')****\\74_c7F0F+7$$!3)3LLL3x%z#)! 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obtain i nitial estimate for root \n ns := n*n;\n for i from 1 to n d o\n if irem(n,2)=1 and i=iquo(n,2)+1 and signum(0,alf-bet,0)=0 th en\n z := 0;\n goto(1111);\n elif i=1 then\n \+ an := alf/n;\n bn := bet/n;\n r1 := (1+alf)*(2.78/ (4+ns)+0.768*an/n);\n r2 := 1+(1.48+0.452*an)*an+(0.96+0.83*an )*bn;\n z := 1-r1/r2;\n elif i=2 then\n r1 := (4. 1+alf)/((1+alf)*(1+0.156*alf));\n r2 := 1+0.06*(n-8)*(1+0.12*a lf)/n;\n r3 := 1+0.012*bet*(1+0.25*abs(alf))/n;\n z := z - (1-z)*r1*r2*r3; \n elif i=3 then\n r1 := (1.67+0.28* alf)/(1+0.37*alf);\n r2 := 1+0.22*(n-8)/n;\n r3 := 1+8 *bet/((6.28+bet)*ns);\n z := z-(x[1]-z)*r1*r2*r3;\n elif \+ i=n-1 then\n r1 := (1+0.235*bet)/(0.766+0.119*bet);\n \+ r2 := 1/(1+0.639*(n-4)/(1+0.71*(n-4)));\n r3 := 1/(1+20*alf/(( 7.5*alf)*ns));\n z := z+(z-x[n-3])*r1*r2*r3;\n elif i=n t hen\n r1 := (1+0.37*bet)/(1.67+0.28*bet);\n r2 := 1/(1 +0.22*(n-8)/n);\n r3 := 1/(1+8*alf/((6.28+alf)*ns));\n \+ z := z+(z-x[n-2])*r1*r2*r3;\n else\n z := 3*x[i-1]-3*x[i -2]+x[i-3];\n end if;\n \n eps := Float(5,-Digits);\n \+ ab := alf+bet;\n ds := alf*alf-bet*bet;\n for k from 1 t o Digits*5 do\n # evaluate Jacobi poly at z .. p1\n te mp := 2+ab;\n p1 := (alf-bet+temp*z)/2;\n p2 := 1;\n \+ for j from 2 to n do\n p3 := p2;\n p2 := \+ p1;\n temp := 2*j+ab;\n a := 2*j*(j+ab)*(temp-2) ;\n b := (temp-1)*(ds+temp*(temp-2)*z);\n c := 2 *(j-1+alf)*(j-1+bet)*temp;\n p1 := (b*p2-c*p3)/a;\n \+ end do;\n\n # get the derivative and apply the Newton formula \n h := p1*(temp*(1-z*z))/(n*(alf-bet-temp*z)*p1+2*(n+alf)*(n+ bet)*p2);\n z := z-h;\n if abs(h)<=eps then break end \+ if;\n end do;\n 1111:\n x := [op(x),z];\n end do;\n \+ [seq(x[n-i],i=0..n-1)];\nend proc: # of jacobizeros" }}}{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 14 "The proce dure " }{TEXT 0 11 "jacobizeros" }{TEXT -1 31 " is somewhat faster tha n using " }{TEXT 0 6 "fsolve" }{TEXT -1 18 ", especially when " } {TEXT 264 1 "n" }{TEXT -1 10 " is large." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 181 "n := 30;\nalf := -3/ 8;\nbet := -3/4;\nst := time():\nevalf(evalf(jacobizeros(n,alf,bet),12 ));\ntime()-st;\ntime():\nevalf(evalf(fsolve(orthopoly[P](n,alf,bet,x) ),12));\ntime()-st;\nn := 'n':" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\" nG\"#I" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%$alfG#!\"$\"\")" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%$betG#!\"$\"\"%" }}{PARA 12 "" 1 "" {XPPMATH 20 "6#7@$!+y*\\P***!#5$!+@oy(*)*F&$!+wQu#p*F&$!+i$f5Q*F&$!+?U \+ " 0 "" {MPLTEXT 1 0 1 ";" }}}{PARA 0 "" 0 "" {TEXT -1 30 "When approxi mating a function " }{XPPEDIT 18 0 "f(x)" "6#-%\"fG6#%\"xG" }{TEXT -1 144 " by an interpolating polynomial with equally spaced nodes, the ma ximum points on the absolute error curve often have quite different ma gnitudes." }}{PARA 0 "" 0 "" {TEXT -1 51 "Suppose that it is possible \+ to expand the function " }{XPPEDIT 18 0 "f(x)" "6#-%\"fG6#%\"xG" } {TEXT -1 57 " in a series of orthogonal (Jacobi) polynomials so that: \+ " }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "f(x) = a[0]*P(0,a lpha,beta,x)+a[1]*P(1,alpha,beta,x)+a[2]*P(2,alpha,beta,x)+` . . . `" "6#/-%\"fG6#%\"xG,**&&%\"aG6#\"\"!\"\"\"-%\"PG6&F-%&alphaG%%betaGF'F.F .*&&F+6#F.F.-F06&F.F2F3F'F.F.*&&F+6#\"\"#F.-F06&F " 0 "" {MPLTEXT 1 0 1 ";" }}}}{SECT 1 {PARA 4 "" 0 " " {TEXT 0 9 "interpoly" }{TEXT -1 79 ": examples with nodes spaced in \+ the pattern of the zeros of Jacobi polynomials " }}{EXCHG {PARA 0 "> \+ " 0 "" {MPLTEXT 1 0 1 ";" }}}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 9 "Examp le 1" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 76 "C onsider the problem of finding a polynomial approximation for the func tion " }{XPPEDIT 18 0 "f(x)=arctan(x)/x" "6#/-%\"fG6#%\"xG*&-%'arctanG 6#F'\"\"\"F'!\"\"" }{TEXT -1 17 " on the interval " }{XPPEDIT 18 0 "[0 ,1]" "6#7$\"\"!\"\"\"" }{TEXT -1 2 ". " }}{PARA 0 "" 0 "" {TEXT -1 22 "The Jacobi polynomial " }{XPPEDIT 18 0 "P(n,-1/2,-1/2,x);" "6#-%\"PG6 &%\"nG,$*&\"\"\"F)\"\"#!\"\"F+,$*&F)F)F*F+F+%\"xG" }{TEXT -1 48 " has \+ the same zeros as the Chebyshev polynomial " }{XPPEDIT 18 0 "T(n,x)" " 6#-%\"TG6$%\"nG%\"xG" }{TEXT -1 1 "." }}{PARA 0 "" 0 "" {TEXT -1 56 "F irst we construct a degree 10 interpolating polynomial " }{XPPEDIT 18 0 "p(x)" "6#-%\"pG6#%\"xG" }{TEXT -1 60 " whose nodes are distributed \+ in the pattern of the zeros of " }{XPPEDIT 18 0 "T(10,x)" "6#-%\"TG6$ \"#5%\"xG" }{TEXT -1 1 "." }}{PARA 0 "" 0 "" {TEXT -1 90 "We get a sli ghtly different error curve constructing a degree 10 interpolating pol ynomial " }{XPPEDIT 18 0 "q(x)" "6#-%\"qG6#%\"xG" }{TEXT -1 61 " whose nodes are distributed in the pattern of the zeros of " }{XPPEDIT 18 0 "P(10,-1/2,-19/20,x);" "6#-%\"PG6&\"#5,$*&\"\"\"F)\"\"#!\"\"F+,$*&\" #>F)\"#?F+F+%\"xG" }{TEXT -1 2 ". " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 239 "f := x -> arctan(x)/x;\nint erpoly(f(x),x=0..1,10,spacing=Chebyshev,info=true):\np := unapply(%,x) ;\ninterpoly(f(x),x=0..1,10,params=[-1/2,-19/20],info=true):\nq := una pply(%,x);\nplot([f(x)-p(x),f(x)-q(x)],x=0..1,color=[red,COLOR(RGB,0,. 7,0)]);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"fGf*6#%\"xG6\"6$%)opera torG%&arrowGF(*&-%'arctanG6#9$\"\"\"F0!\"\"F(F(F(" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%aonodes~are~distributed~according~to~the~zeros~of~a~Ch ebyshev~polynomialG" }}{PARA 11 "" 1 "" {XPPMATH 20 "6$%2weight~functi on:~G*&\"\"\"F%*$,&F%F%*$)%\"xG\"\"#F%!\"\"#F%F+F," }}{PARA 11 "" 1 " " {XPPMATH 20 "6#%!G" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%'nodes:G" }} {PARA 12 "" 1 "" {XPPMATH 20 "6#7-$\"3\"QjL&f!z#*3&!#?$\"4I93uAB+%=XF& $\"57e3(G#G@D@7F&$\"5&*37?s7fz'H#F&$\"5:^^Gz:sL\"f$F&$\"\"&!\"\"$\"5&) [[r?%yi'3kF&$\"50\"z)zF(3/Kq(F&$\"5)=9Hr<(yuy()F&$\"5q&=fsn(*f\"[&*F&$ \"5>mjYS4s5\\**F&" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%!G" }}{PARA 11 " " 1 "" {XPPMATH 20 "6#%(values:G" }}{PARA 12 "" 1 "" {XPPMATH 20 "6#7- $\"5LO@0ZlO\"*****!#?$\"5=z\"QA0,.K***F&$\"5uYm#)4y\\s]**F&$\"5?vu&*zI*))F&$\"5E#QJW sxn3_)F&$\"5j[Ldl?#=o?)F&$\"5i06$yKP7O)zF&$\"5jd3d#yW:&oyF&" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%!G" }}{PARA 12 "" 1 "" {XPPMATH 20 "6#>%\"p Gf*6#%\"xG6\"6$%)operatorG%&arrowGF(,8$\"+4********!#5\"\"\"*&$\"+x\\' ))>#!#:F09$F0F0*&$\"+Ds?MLF/F0)F5\"\"#F0!\"\"*&$\"+RL?X8!#7F0)F5\"\"$F 0F0*&$\"+\\Y,&*=F/F0)F5\"\"%F0F0*&$\"+bBhzY!#6F0)F5\"\"&F0F0*&$\"+\")Q QuEF/F0)F5\"\"'F0F;*&$\"+7J>H>F/F0)F5\"\"(F0F0*&$\"+K\"[6-$FJF0)F5\"\" )F0F;*&$\"+!o(GEAFJF0)F5\"\"*F0F;*&$\"+/eMn\")F?F0)F5\"#5F0F0F(F(F(" } }{PARA 11 "" 1 "" {XPPMATH 20 "6#%^onodes~are~distributed~according~to ~the~zeros~of~a~Jacobi~polynomialG" }}{PARA 11 "" 1 "" {XPPMATH 20 "6$ %2weight~function:~G*&\"\"\"F%*&,&F%F%%\"xG!\"\"#F%\"\"#),&F(F%F%F%#\" #>\"#?F%F)" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%!G" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%'nodes:G" }}{PARA 12 "" 1 "" {XPPMATH 20 "6#7-$\"2Ar\" Qa/#[T%!#?$\"4P]!)R0'f_\"G$F&$\"5#>6\\B)\\?pZ5F&$\"5UU>nyX'f.5#F&$\"5, *>338e8sR$F&$\"5s))*>Kt&=xG[F&$\"5&Qr!HYVP>uiF&$\"55MS![0Kr9h(F&$\"5#p !\\'3s-Exs)F&$\"5akLH%)oftG&*F&$\"5!\\&>[8F)))o%**F&" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%!G" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%(values:G" } }{PARA 12 "" 1 "" {XPPMATH 20 "6#7-$\"5%Hy37.N*******!#?$\"5L!QH:l%GT' ***F&$\"5+%p]DlZ]O'**F&$\"5Y\\a.RVLsc)*F&$\"5(eqJ3!R8#*R'*F&$\"5/))oY* ))y_hJ*F&$\"51(REY*oR#3$*)F&$\"5?OZ6`$o(eZ&)F&$\"5rrqU/acp@#)F&$\"5I$p >$z\"48#*)zF&$\"5>=F92y$\\\"pyF&" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%! 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" }}{PARA 0 "" 0 "" {TEXT -1 63 "We construct a degree 7 inter polating polynomial approximation " }{XPPEDIT 18 0 "p(x)" "6#-%\"pG6#% \"xG" }{TEXT -1 5 " for " }{XPPEDIT 18 0 "f(x)" "6#-%\"fG6#%\"xG" } {TEXT -1 85 " based on interpolation points which are distributed in t he pattern of the zeros of " }{XPPEDIT 18 0 "P(7,-7/8,-7/8,x);" "6#-% \"PG6&\"\"(,$*&F&\"\"\"\"\")!\"\"F+,$*&F&F)F*F+F+%\"xG" }{TEXT -1 2 ". 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