{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 1 0 0 0 0 0 0 0 0 1 }{CSTYLE "" -1 260 "" 0 1 0 0 0 0 1 0 0 0 0 0 0 0 0 1 }{CSTYLE "" -1 261 "" 0 1 0 0 0 0 0 1 0 0 0 0 0 0 0 1 }{CSTYLE "" -1 262 "" 0 1 0 0 0 0 0 1 0 0 0 0 0 0 0 1 }{CSTYLE "Purple Emphasis" -1 263 "Times" 1 12 102 0 230 1 0 1 0 0 0 0 0 0 0 0 }{CSTYLE "Red Emphasis" -1 264 "Tim es" 1 12 255 0 0 1 0 1 0 0 0 0 0 0 0 0 }{CSTYLE "Dark Red Emphasis" -1 265 "Times" 1 12 128 0 0 1 0 1 0 0 0 0 0 0 0 0 }{CSTYLE "" 264 266 "" 1 18 0 0 0 0 0 0 0 0 0 0 0 0 0 0 }{CSTYLE "" 265 267 "" 0 1 0 0 0 0 0 0 1 0 0 0 0 0 0 0 }{CSTYLE "" -1 268 "" 0 1 0 0 0 0 0 0 1 0 0 0 0 0 0 0 }{CSTYLE "" -1 269 "" 0 1 0 0 0 0 1 0 0 0 0 0 0 0 0 0 }{CSTYLE " " -1 270 "" 0 1 0 0 0 0 1 0 0 0 0 0 0 0 0 0 }{CSTYLE "" 258 271 "" 0 1 0 0 0 0 0 0 1 0 0 0 0 0 0 0 }{CSTYLE "" 258 272 "" 0 1 0 0 0 0 0 0 1 0 0 0 0 0 0 0 }{CSTYLE "" 258 273 "" 0 1 0 0 0 0 0 0 1 0 0 0 0 0 0 0 }{CSTYLE "" -1 274 "" 0 1 0 0 0 0 1 0 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 275 "" 0 1 0 0 0 0 1 0 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 276 "" 0 1 0 0 0 0 1 0 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 277 "" 0 1 0 0 0 0 1 0 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 278 "" 0 1 0 0 0 0 1 0 0 0 0 0 0 0 0 0 } {CSTYLE "" -1 279 "" 0 1 0 0 0 0 1 0 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 280 "" 0 1 0 0 0 0 1 0 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 281 "" 0 1 0 0 0 0 1 0 0 0 0 0 0 0 0 0 }{PSTYLE "Normal" -1 0 1 {CSTYLE "" -1 -1 "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 "Bullet Item" -1 15 1 {CSTYLE "" -1 -1 "Times" 1 12 0 0 0 1 2 2 2 2 2 2 1 1 1 1 }1 1 0 0 3 3 1 0 1 0 2 2 15 2 }{PSTYLE "Normal" -1 256 1 {CSTYLE "" -1 -1 "Times " 1 12 0 0 0 1 2 2 2 2 2 2 1 1 1 1 }3 1 0 0 0 0 1 0 1 0 2 2 0 1 } {PSTYLE "Normal" -1 257 1 {CSTYLE "" -1 -1 "Times" 1 18 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" -1 258 1 {CSTYLE "" -1 -1 "Times" 1 12 0 0 0 1 2 2 2 2 2 2 1 1 1 1 }1 1 0 0 0 0 1 0 1 0 2 2 0 1 }{PSTYLE "Normal" -1 259 1 {CSTYLE "" -1 -1 "Times" 1 12 0 0 0 1 2 2 2 2 2 2 1 1 1 1 }1 1 0 0 0 0 1 0 1 0 2 2 0 1 }} {SECT 0 {PARA 3 "" 0 "" {TEXT -1 28 "Gaussian-Legendre Quadrature" }} {PARA 0 "" 0 "" {TEXT -1 37 "by Peter Stone, Nanaimo, B.C., Canada" }} {PARA 0 "" 0 "" {TEXT -1 19 "Version: 24.3.2007" }}{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 40 "Basic idea of Gauss integration formulas" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}{PARA 0 "" 0 "" {TEXT -1 103 " The Newton-Cotes integration formulas, which include Simpson's Rule, i nvolve computing function values " }{XPPEDIT 18 0 "f(x[i])" "6#-%\"fG6 #&%\"xG6#%\"iG" }{TEXT -1 25 " at equally spaced nodes " }{XPPEDIT 18 0 "x[i]" "6#&%\"xG6#%\"iG" }{TEXT -1 75 ". The function values at the \+ nodes are then multiplied by suitable weights " }{XPPEDIT 18 0 "w[i]" "6#&%\"wG6#%\"iG" }{TEXT -1 38 " to obtain a numerical apoproximation \+ " }{XPPEDIT 18 0 "Sum(w[i]*f(x[i]),i = 1 .. n);" "6#-%$SumG6$*&&%\"wG6 #%\"iG\"\"\"-%\"fG6#&%\"xG6#F*F+/F*;F+%\"nG" }{TEXT -1 18 " for the in tegral " }{XPPEDIT 18 0 "Int(f(x),x = a .. b);" "6#-%$IntG6$-%\"fG6#% \"xG/F);%\"aG%\"bG" }{TEXT -1 1 "." }}{PARA 0 "" 0 "" {TEXT 263 11 "Ga uss rules" }{TEXT -1 9 " involve " }{TEXT 263 57 "choosing the locatio n of the nodes as well as the weights" }{TEXT -1 117 " in an effort to reduce the number of function evaluations required to achieve a parti cular precision for the result." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }} {PARA 0 "" 0 "" {TEXT -1 202 "Simpson's rule is exact on cubics, and r equires 3 function evaluations. The Gauss 2-point rule is exact on cub ics, but only requires 2 function evaluations at 2 special nodes. More generally, the Gauss " }{TEXT 269 1 "n" }{TEXT -1 46 "-point rule is \+ exact on polynomials of degree " }{XPPEDIT 18 0 "2*n -1" "6#,&*&\"\"# \"\"\"%\"nGF&F&F&!\"\"" }{TEXT -1 15 ", and requires " }{TEXT 270 1 "n " }{TEXT -1 107 " function evaluations. It is sufficient to locate the nodes required to evaluate integrals on the interval " }{XPPEDIT 18 0 "[-1,1]" "6#7$,$\"\"\"!\"\"F%" }{TEXT -1 61 ". General integrals ca n be handled by a change of variables." }}{PARA 0 "" 0 "" {TEXT -1 0 " " }}{PARA 0 "" 0 "" {TEXT -1 29 "The problem is to find nodes " } {XPPEDIT 18 0 "x[1],x[2],` . . . `,x[n];" "6&&%\"xG6#\"\"\"&F$6#\"\"#% (~.~.~.~G&F$6#%\"nG" }{TEXT -1 14 ", and weights " }{XPPEDIT 18 0 "w[1 ],w[2],` . . . `,w[n];" "6&&%\"wG6#\"\"\"&F$6#\"\"#%(~.~.~.~G&F$6#%\"n G" }{TEXT -1 28 " so that for any polynomial" }}{PARA 256 "" 0 "" {TEXT -1 2 " " }{XPPEDIT 18 0 "p(x) = sum(a[i]*x^i,i = 0 .. 2*n-1);" "6#/-%\"pG6#%\"xG-%$sumG6$*&&%\"aG6#%\"iG\"\"\")F'F/F0/F/;\"\"!,&*&\" \"#F0%\"nGF0F0F0!\"\"" }{TEXT -1 12 " ------- (i)" }}{PARA 0 "" 0 "" {TEXT -1 9 "of degree" }{XPPEDIT 18 0 "`` <= 2*n-1;" "6#1%!G,&*&\"\"# \"\"\"%\"nGF(F(F(!\"\"" }{TEXT -1 9 ", we have" }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "int(p(x),x = -1 .. 1) = sum(w[i]*p(x[i] ),i = 1 .. n);" "6#/-%$intG6$-%\"pG6#%\"xG/F*;,$\"\"\"!\"\"F.-%$sumG6$ *&&%\"wG6#%\"iGF.-F(6#&F*6#F7F./F7;F.%\"nG" }{TEXT -1 13 " ------- (ii )" }}{PARA 0 "" 0 "" {TEXT -1 4 "The " }{TEXT 260 1 "n" }{TEXT -1 20 " nodes together the " }{TEXT 259 1 "n" }{TEXT -1 20 " weights constitu te " }{XPPEDIT 18 0 "2*n" "6#*&\"\"#\"\"\"%\"nGF%" }{TEXT -1 10 " unkn owns." }}{PARA 0 "" 0 "" {TEXT -1 150 "We can substitute the general e xpression for the polynomial from (i) in both sides of (ii), and the i ntegral on the left can be found in terms of the " }{XPPEDIT 18 0 "2*n " "6#*&\"\"#\"\"\"%\"nGF%" }{TEXT -1 14 " coefficients " }{XPPEDIT 18 0 "a[i]" "6#&%\"aG6#%\"iG" }{TEXT -1 1 "." }}{PARA 0 "" 0 "" {TEXT -1 71 "Since we want equation (ii) to hold for any choice of the coeffici ents " }{XPPEDIT 18 0 "a[i]" "6#&%\"aG6#%\"iG" }{TEXT -1 63 ", it woul d have to hold in the particular cases when a certain " }{XPPEDIT 18 0 "a[i]" "6#&%\"aG6#%\"iG" }{TEXT -1 24 " is 1 and the remaining " } {XPPEDIT 18 0 "a[j]" "6#&%\"aG6#%\"jG" }{TEXT -1 6 " with " }{XPPEDIT 18 0 "j<>i" "6#0%\"jG%\"iG" }{TEXT -1 14 " are all zero." }}{PARA 0 " " 0 "" {TEXT -1 39 "Thus we can equate the coefficients of " } {XPPEDIT 18 0 "a[i]" "6#&%\"aG6#%\"iG" }{TEXT -1 47 " on the left and \+ right sides of (ii) to obtain " }{XPPEDIT 18 0 "2*n" "6#*&\"\"#\"\"\"% \"nGF%" }{TEXT -1 19 " equations for the " }{XPPEDIT 18 0 "2*n" "6#*& \"\"#\"\"\"%\"nGF%" }{TEXT -1 10 " unknowns " }{XPPEDIT 18 0 "x[1], `. . `, x[n],w[1],` . . `,w[n]" "6(&%\"xG6#\"\"\"%%.~.~G&F$6#%\"nG&%\"wG 6#F&%&~.~.~G&F,6#F*" }{TEXT -1 38 ", which can hopefully then be solve d. " }}{PARA 0 "" 0 "" {TEXT -1 46 "We attempt to follow this program \+ in the case " }{XPPEDIT 18 0 "n = 3" "6#/%\"nG\"\"$" }{TEXT -1 3 ". \+ " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 18 "Gauss 3-point rule" }}{PARA 0 "" 0 "" {TEXT -1 1 " " }}{PARA 0 "" 0 "" {TEXT 271 6 "Step 1" }{TEXT -1 39 ": Set up a general degree 5 po lynomial " }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "p(x)=a[0 ]+a[1]*t+a[2]*t^2+a[3]*t^3+a[4]*t^4+a[5]*t^5" "6#/-%\"pG6#%\"xG,.&%\"a G6#\"\"!\"\"\"*&&F*6#F-F-%\"tGF-F-*&&F*6#\"\"#F-*$F1F5F-F-*&&F*6#\"\"$ F-*$F1F:F-F-*&&F*6#\"\"%F-*$F1F?F-F-*&&F*6#\"\"&F-*$F1FDF-F-" }{TEXT -1 2 ", " }}{PARA 0 "" 0 "" {TEXT -1 18 "with coefficients " } {XPPEDIT 18 0 "a[0],a[1],a[2],a[3],a[4]" "6'&%\"aG6#\"\"!&F$6#\"\"\"&F $6#\"\"#&F$6#\"\"$&F$6#\"\"%" }{TEXT -1 5 " and " }{XPPEDIT 18 0 "a[5] " "6#&%\"aG6#\"\"&" }{TEXT -1 29 ", and find an expression for " } {XPPEDIT 18 0 "Int(p(t),t=-1..1)" "6#-%$IntG6$-%\"pG6#%\"tG/F);,$\"\" \"!\"\"F-" }{TEXT -1 30 " in terms of the coefficients." }}{PARA 0 "" 0 "" {TEXT -1 54 "Note that only the coefficients of the even powers o f " }{TEXT 274 1 "t" }{TEXT -1 15 " are involved. " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 75 "sum(a[i]*t^i ,i=0..5);\np := unapply(%,t);\nInt(p(t),t=-1..1);\nls := value(%);" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#,.&%\"aG6#\"\"!\"\"\"*&&F%6#F(F(%\"tGF (F(*&&F%6#\"\"#F()F,F0F(F(*&&F%6#\"\"$F()F,F5F(F(*&&F%6#\"\"%F()F,F:F( F(*&&F%6#\"\"&F()F,F?F(F(" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"pGf*6 #%\"tG6\"6$%)operatorG%&arrowGF(,.&%\"aG6#\"\"!\"\"\"*&&F.6#F1F19$F1F1 *&&F.6#\"\"#F1)F5F9F1F1*&&F.6#\"\"$F1)F5F>F1F1*&&F.6#\"\"%F1)F5FCF1F1* &&F.6#\"\"&F1)F5FHF1F1F(F(F(" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#-%$Int G6$,.&%\"aG6#\"\"!\"\"\"*&&F(6#F+F+%\"tGF+F+*&&F(6#\"\"#F+)F/F3F+F+*&& F(6#\"\"$F+)F/F8F+F+*&&F(6#\"\"%F+)F/F=F+F+*&&F(6#\"\"&F+)F/FBF+F+/F/; !\"\"F+" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%#lsG,(*&\"\"#\"\"\"&%\"aG 6#\"\"!F(F(*&#F'\"\"$F(&F*6#F'F(F(*&#F'\"\"&F(&F*6#\"\"%F(F(" }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT 272 6 "Step 2" }{TEXT -1 24 ": We want to find nodes " }{XPPEDIT 18 0 "x[1],x[2]" "6$&%\"xG6#\"\"\"&F$6#\"\" #" }{TEXT -1 5 " and " }{XPPEDIT 18 0 "x[3]" "6#&%\"xG6#\"\"$" }{TEXT -1 24 ", together with weights " }{XPPEDIT 18 0 "w[1],w[2]" "6$&%\"wG6 #\"\"\"&F$6#\"\"#" }{TEXT -1 5 " and " }{XPPEDIT 18 0 "w[3]" "6#&%\"wG 6#\"\"$" }{TEXT -1 8 " so that" }}{PARA 256 "" 0 "" {TEXT -1 1 " " } {XPPEDIT 18 0 "Int(p(t),t = -1 .. 1)=Sum(p(x[i])*w[i],i=1..3)" "6#/-%$ IntG6$-%\"pG6#%\"tG/F*;,$\"\"\"!\"\"F.-%$SumG6$*&-F(6#&%\"xG6#%\"iGF.& %\"wG6#F9F./F9;F.\"\"$" }{TEXT -1 1 "," }}{PARA 0 "" 0 "" {TEXT -1 8 " that is," }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "2*a[0]+2/ 3*a[2]+2/5*a[4] = p(x[1])*w[1]+p(x[2])*w[2]+p(x[2])*w[1]" "6#/,(*&\"\" #\"\"\"&%\"aG6#\"\"!F'F'*(F&F'\"\"$!\"\"&F)6#F&F'F'*(F&F'\"\"&F.&F)6# \"\"%F'F',(*&-%\"pG6#&%\"xG6#F'F'&%\"wG6#F'F'F'*&-F96#&F<6#F&F'&F?6#F& F'F'*&-F96#&F<6#F&F'&F?6#F'F'F'" }{TEXT -1 1 "." }}{PARA 0 "" 0 "" {TEXT -1 21 "When each expression " }{XPPEDIT 18 0 "p(x[i])*w[i]" "6#* &-%\"pG6#&%\"xG6#%\"iG\"\"\"&%\"wG6#F*F+" }{TEXT -1 85 " is expanded, \+ the right side is a complicated expression involving the coefficients \+ " }{XPPEDIT 18 0 "a[0],a[1],a[2],a[3],a[4],a[5];" "6(&%\"aG6#\"\"!&F$ 6#\"\"\"&F$6#\"\"#&F$6#\"\"$&F$6#\"\"%&F$6#\"\"&" }{TEXT -1 12 ", the \+ nodes " }{XPPEDIT 18 0 "x[1],x[2],x[3]" "6%&%\"xG6#\"\"\"&F$6#\"\"#&F$ 6#\"\"$" }{TEXT -1 17 " and the weights " }{XPPEDIT 18 0 "w[1],w[2],w[ 3]" "6%&%\"wG6#\"\"\"&F$6#\"\"#&F$6#\"\"$" }{TEXT -1 2 ". " }}{PARA 0 "" 0 "" {TEXT -1 57 "We then form a system of 6 equations in the six u nknowns " }{XPPEDIT 18 0 "x[1], x[2], x[3],w[1], w[2], w[3]" "6(&%\"xG 6#\"\"\"&F$6#\"\"#&F$6#\"\"$&%\"wG6#F&&F.6#F)&F.6#F," }{TEXT -1 34 ", \+ by equating the coefficients of " }{XPPEDIT 18 0 "a[0], a[1], a[2], a[ 3], a[4], a[5]" "6(&%\"aG6#\"\"!&F$6#\"\"\"&F$6#\"\"#&F$6#\"\"$&F$6#\" \"%&F$6#\"\"&" }{TEXT -1 29 " on the left and right sides." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 104 "rs \+ := sum(p(x[i])*w[i],i=1..3);\ncollect(ls-rs,[seq(a[i],i=0..5)]);\neqns := coeffs(%,[seq(a[i],i=0..5)]);" }}{PARA 12 "" 1 "" {XPPMATH 20 "6#> %#rsG,(*&,.&%\"aG6#\"\"!\"\"\"*&&F)6#F,F,&%\"xGF/F,F,*&&F)6#\"\"#F,)F0 F5F,F,*&&F)6#\"\"$F,)F0F:F,F,*&&F)6#\"\"%F,)F0F?F,F,*&&F)6#\"\"&F,)F0F DF,F,F,&%\"wGF/F,F,*&,.F(F,*&F.F,&F1F4F,F,*&F3F,)FKF5F,F,*&F8F,)FKF:F, F,*&F=F,)FKF?F,F,*&FBF,)FKFDF,F,F,&FGF4F,F,*&,.F(F,*&F.F,&F1F9F,F,*&F3 F,)FXF5F,F,*&F8F,)FXF:F,F,*&F=F,)FXF?F,F,*&FBF,)FXFDF,F,F,&FGF9F,F," } }{PARA 12 "" 1 "" {XPPMATH 20 "6#,.*&,*\"\"#\"\"\"&%\"wG6#F'!\"\"&F)6# F&F+&F)6#\"\"$F+F'&%\"aG6#\"\"!F'F'*&,(*&&%\"xGF*F'F(F'F+*&&F9F-F'F,F' F+*&&F9F/F'F.F'F+F'&F2F*F'F'*&,**&)F8F&F'F(F'F+#F&F0F'*&)F;F&F'F,F'F+* &)F=F&F'F.F'F+F'&F2F-F'F'*&,(*&)F8F0F'F(F'F+*&)F;F0F'F,F'F+*&)F=F0F'F. F'F+F'&F2F/F'F'*&,**&)F=\"\"%F'F.F'F+#F&\"\"&F'*&)F;FVF'F,F'F+*&)F8FVF 'F(F'F+F'&F26#FVF'F'*&,(*&)F;FXF'F,F'F+*&)F8FXF'F(F'F+*&)F=FXF'F.F'F+F '&F26#FXF'F'" }}{PARA 12 "" 1 "" {XPPMATH 20 "6#>%%eqnsG6(,*\"\"#\"\" \"&%\"wG6#F(!\"\"&F*6#F'F,&F*6#\"\"$F,,(*&&%\"xGF+F(F)F(F,*&&F5F.F(F-F (F,*&&F5F0F(F/F(F,,**&)F4F'F(F)F(F,#F'F1F(*&)F7F'F(F-F(F,*&)F9F'F(F/F( F,,**&)F9\"\"%F(F/F(F,#F'\"\"&F(*&)F7FEF(F-F(F,*&)F4FEF(F)F(F,,(*&)F7F GF(F-F(F,*&)F4FGF(F)F(F,*&)F9FGF(F/F(F,,(*&)F4F1F(F)F(F,*&)F7F1F(F-F(F ,*&)F9F1F(F/F(F," }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT 273 6 "Step 3" }{TEXT -1 36 ": Solve the system of equations for " }{XPPEDIT 18 0 "x[1], x[2], x[3], w[1], w[2], w[3]" "6(&%\"xG6#\"\" \"&F$6#\"\"#&F$6#\"\"$&%\"wG6#F&&F.6#F)&F.6#F," }{TEXT -1 1 "." }} {PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 81 "soln := solve(\{eqns\},\{seq(w[i],i=1..3),seq(x[j],j=1..3)\}):\nma p(allvalues,[soln]);" }}{PARA 12 "" 1 "" {XPPMATH 20 "6#7(<(/&%\"xG6# \"\"$,$*&\"\"&!\"\"\"#:#\"\"\"\"\"#F0/&%\"wG6#F1#\"\")\"\"*/&F'6#F0,$* &F,F-F.F/F-/&F'F5\"\"!/&F4F;#F,F8/&F4F(FC<(/F&FFAFD<(/FEF6/F ?F*F9/F&F@FA/F3FC<(FJ/F?F " 0 "" {MPLTEXT 1 0 1 ";" }}}}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 20 "Legendre p olynomials" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}{PARA 0 " " 0 "" {TEXT -1 152 "Maple starts to have problems if the pattern of t he previous section is used in an attempt to find Gauss integration ru les involving more than 3 points." }}{PARA 0 "" 0 "" {TEXT -1 61 "Fort unately, there is another approach to this problem using " }{TEXT 263 20 "Legendre polynomials" }{TEXT -1 1 "." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 52 "For each non-negative integer defi ne a polynomial by" }}{PARA 256 "" 0 "" {TEXT -1 3 " " }{XPPEDIT 18 0 "P[n](x)=1/(2^n*n!)" "6#/-&%\"PG6#%\"nG6#%\"xG*&\"\"\"F,*&)\"\"#F(F, -%*factorialG6#F(F,!\"\"" }{TEXT -1 2 " " }{XPPEDIT 18 0 "d^n/(d*x^n) ;" "6#*&)%\"dG%\"nG\"\"\"*&F%F')%\"xGF&F'!\"\"" }{TEXT -1 2 " " } {XPPEDIT 18 0 "(x^2-1)^n" "6#),&*$%\"xG\"\"#\"\"\"F(!\"\"%\"nG" } {TEXT -1 5 ". " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 8 "We have " }{XPPEDIT 18 0 "P(0,x) = 1" "6#/-%\"PG6$\"\"!%\" xG\"\"\"" }{TEXT -1 29 ", and the next few values of " }{TEXT 275 1 "n " }{TEXT -1 11 " give . . ." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 136 "P := (n,x) -> expand(diff((x^2-1)^ n,x$n)/(2^n*n!));\n'P(1,x)'=P(1,x);\n'P(2,x)'=P(2,x);\n'P(3,x)'=P(3,x) ;\n'P(4,x)'=P(4,x);\n'P(5,x)'=P(5,x);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"PGf*6$%\"nG%\"xG6\"6$%)operatorG%&arrowGF)-%'expandG6#*&-%%diff G6$),&*$)9%\"\"#\"\"\"F:F:!\"\"9$-%\"$G6$F8F " 0 "" {MPLTEXT 1 0 1 ";" }} }{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 56 "The Lege ndre polynomials satisfy the recurrence relation" }}{PARA 256 "" 0 "" {TEXT -1 3 " " }{XPPEDIT 18 0 "P(n,x) = [(2*n-1)*P(n-1,x)*x-(n-1)*P( n-2,x)];" "6#/-%\"PG6$%\"nG%\"xG7#,&*(,&*&\"\"#\"\"\"F'F/F/F/!\"\"F/-F %6$,&F'F/F/F0F(F/F(F/F/*&,&F'F/F/F0F/-F%6$,&F'F/F.F0F(F/F0" }{TEXT -1 1 " " }{XPPEDIT 18 0 "1/n;" "6#*&\"\"\"F$%\"nG!\"\"" }{TEXT -1 14 " - ------ (i)," }}{PARA 0 "" 0 "" {TEXT -1 16 "and the relation" }}{PARA 256 "" 0 "" {TEXT -1 3 " " }{XPPEDIT 18 0 "(x^2-1)*`P '`(n,x) = n*(x *P(n,x)-P(n-1,x));" "6#/*&,&*$%\"xG\"\"#\"\"\"F)!\"\"F)-%$P~'G6$%\"nGF 'F)*&F.F),&*&F'F)-%\"PG6$F.F'F)F)-F36$,&F.F)F)F*F'F*F)" }{TEXT -1 2 ". " }}{PARA 256 "" 0 "" {TEXT -1 1 " " }}{PARA 0 "" 0 "" {TEXT -1 73 "T he recurrence relation (i) provides a convenient way to construct them . " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 274 "P := 'P':\nP := proc(n::nonnegint,x::algebraic)\n \+ local px;\n option remember;\n if n=0 then px := 1\n elif n=1 th en px := x else\n px := ((2*n-1)*P(n-1,x)*x-(n-1)*P(n-2,x))/n\n \+ end if;\n normal(px);\nend proc:\nfor n from 0 to 10 do\n print(' P'(n,x)=P(n,x))\nend do;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/-%\"PG6$ \"\"!%\"xG\"\"\"" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/-%\"PG6$\"\"\"%\" xGF(" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/-%\"PG6$\"\"#%\"xG,&*(\"\"$\" \"\"F'!\"\"F(F'F,#F,F'F-" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/-%\"PG6$ \"\"$%\"xG,&*&#\"\"&\"\"#\"\"\"*$)F(F'F.F.F.*&#F'F-F.F(F.!\"\"" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#/-%\"PG6$\"\"%%\"xG,(*&#\"#N\"\")\"\" \"*$)F(F'F.F.F.*&#\"#:F'F.*$)F(\"\"#F.F.!\"\"#\"\"$F-F." }}{PARA 11 " " 1 "" {XPPMATH 20 "6#/-%\"PG6$\"\"&%\"xG,(*&#\"#j\"\")\"\"\"*$)F(F'F. F.F.*&#\"#N\"\"%F.*$)F(\"\"$F.F.!\"\"*&#\"#:F-F.F(F.F." }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/-%\"PG6$\"\"'%\"xG,**&#\"$J#\"#;\"\"\"*$)F(F'F.F. F.*&#\"$:$F-F.*$)F(\"\"%F.F.!\"\"*&#\"$0\"F-F.*$)F(\"\"#F.F.F.#\"\"&F- F7" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/-%\"PG6$\"\"(%\"xG,**&#\"$H%\"# ;\"\"\"*$)F(F'F.F.F.*&#\"$$pF-F.*$)F(\"\"&F.F.!\"\"*&#\"$:$F-F.*$)F(\" \"$F.F.F.*&#\"#NF-F.F(F.F7" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/-%\"PG6 $\"\")%\"xG,,*&#\"%Nk\"$G\"\"\"\"*$)F(F'F.F.F.*&#\"%.I\"#KF.*$)F(\"\"' F.F.!\"\"*&#\"%lM\"#kF.*$)F(\"\"%F.F.F.*&#\"$:$F4F.*$)F(\"\"#F.F.F8#\" #NF-F." }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/-%\"PG6$\"\"*%\"xG,,*&#\"&b @\"\"$G\"\"\"\"*$)F(F'F.F.F.*&#\"%Nk\"#KF.*$)F(\"\"(F.F.!\"\"*&#\"%4!* \"#kF.*$)F(\"\"&F.F.F.*&#\"%b6F4F.*$)F(\"\"$F.F.F8*&#\"$:$F-F.F(F.F." }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/-%\"PG6$\"#5%\"xG,.*&#\"&*=Y\"$c#\" \"\"*$)F(F'F.F.F.*&#\"'&R4\"F-F.*$)F(\"\")F.F.!\"\"*&#\"&X]%\"$G\"F.*$ )F(\"\"'F.F.F.*&#\"&:]\"F;F.*$)F(\"\"%F.F.F7*&#\"%lMF-F.*$)F(\"\"#F.F. F.#\"#jF-F7" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 35 "plot([seq(P(n,x),n=0..7)],x=-1..1);" }}{PARA 13 " " 1 "" {GLPLOT2D 558 355 355 {PLOTDATA 2 "6,-%'CURVESG6$7S7$$!\"\"\"\" !$\"\"\"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$$ !3]++]PYx\"\\#F0F+7$$!3EMLLL7i)4#F0F+7$$!3c****\\P'psm\"F0F+7$$!3')*** *\\74_c7F0F+7$$!3)3LLL3x%z#)!#>F+7$$!3KMLL3s$QM%FjoF+7$$!3]^omm;zr)*!# @F+7$$\"3%pJL$ezw5VFjoF+7$$\"3s*)***\\PQ#\\\")FjoF+7$$\"3GKLLe\"*[H7F0 F+7$$\"3I*******pvxl\"F0F+7$$\"3#z****\\_qn2#F0F+7$$\"3U)***\\i&p@[#F0 F+7$$\"3B)****\\2'HKHF0F+7$$\"3ElmmmZvOLF0F+7$$\"3i******\\2goPF0F+7$$ \"3UKL$eR<*fTF0F+7$$\"3m******\\)Hxe%F0F+7$$\"3ckm;H!o-*\\F0F+7$$\"3y) ***\\7k.6aF0F+7$$\"3#emmmT9C#eF0F+7$$\"33****\\i!*3`iF0F+7$$\"3%QLLL$* zym'F0F+7$$\"3wKLL3N1#4(F0F+7$$\"3Nmm;HYt7vF0F+7$$\"3Y*******p(G**yF0F 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M$f.#[RF07$F\\[m$!32->(Henr#QF07$$\"3'HL3_!HWb!*F0$!3g6%oBF)osMF07$Fbt $!3uSYAH(y8&HF07$Fd_n$!3e#fXa\"H=wAF07$Fd[m$!3?&y%H\"4#*zS\"F07$F\\`n$ !3KT_)RX'3^KFjo7$Fet$\"3#RmV\"=F07$Fd`n$ \"351z/[f43FF07$Fhjp$\"3SNn*)e-3)o$F07$F\\\\m$\"3;zUJ#4\">cZF07$F`[q$ \"3(yJWi7Go\"fF07$F\\an$\"3IdSEd.`urF07$Fh[q$\"3kST))H*yR`)F0FgtFbx-%+ AXESLABELSG6$Q\"x6\"Q!Fjdr-%%VIEWG6$;F(F+%(DEFAULTG" 1 2 0 1 10 0 2 9 1 4 2 1.000000 45.000000 45.000000 0 0 "Curve 1" "Curve 2" "Curve 3" " Curve 4" "Curve 5" "Curve 6" "Curve 7" "Curve 8" }}}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT 263 5 "Notes" }{TEXT -1 2 ": " }}{PARA 15 "" 0 "" {TEXT -1 5 " The " }{TEXT 276 1 "n" }{TEXT -1 24 " th Legendre polynomial " }{XPPEDIT 18 0 "P(n,x)" "6#-%\"PG6$%\"nG%\"xG" }{TEXT -1 13 " has exac tly " }{TEXT 277 1 "n" }{TEXT -1 42 " distinct roots between between - 1 and 1. " }}{PARA 15 "" 0 "" {TEXT -1 13 "The roots of " }{XPPEDIT 18 0 "P(n-1,x)" "6#-%\"PG6$,&%\"nG\"\"\"F(!\"\"%\"xG" }{TEXT -1 1 " " }{TEXT 263 10 "interleave" }{TEXT -1 14 " the roots of " }{XPPEDIT 18 0 "P(n,x)" "6#-%\"PG6$%\"nG%\"xG" }{TEXT -1 40 ", that is, there is ex actly one root of " }{XPPEDIT 18 0 "P(n-1,x)" "6#-%\"PG6$,&%\"nG\"\"\" F(!\"\"%\"xG" }{TEXT -1 40 " between each adjacent pair of roots of " }{XPPEDIT 18 0 "P(n,x)" "6#-%\"PG6$%\"nG%\"xG" }{TEXT -1 1 "." }} {PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 35 "fsolve(P(3,x),x);\nfsolve(P(4,x),x);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6%$!+#pmfu(!#5\"\"!$\"+#pmfu(F%" }}{PARA 11 "" 1 "" {XPPMATH 20 "6 &$!+;JO6')!#5$!+O/\")*R$F%$\"+O/\")*R$F%$\"+;JO6')F%" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }} {PARA 0 "" 0 "" {TEXT -1 56 "The Legendre polynomials form an example \+ of a family of " }{TEXT 263 22 "orthogonal polynomials" }{TEXT -1 1 ". " }}{PARA 0 "" 0 "" {TEXT -1 48 "The orthogonality is expressed by the fact that " }}{PARA 256 "" 0 "" {TEXT -1 2 " " }{XPPEDIT 18 0 "Int(P (n,x)*P(m,x),x = -1 .. 1) = PIECEWISE([0, m <> n],[2/(2*n+1), m = n]); " "6#/-%$IntG6$*&-%\"PG6$%\"nG%\"xG\"\"\"-F)6$%\"mGF,F-/F,;,$F-!\"\"F- -%*PIECEWISEG6$7$\"\"!0F0F+7$*&\"\"#F-,&*&F=F-F+F-F-F-F-F4/F0F+" } {TEXT -1 3 " ." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> \+ " 0 "" {MPLTEXT 1 0 41 "Int('P(5,x)'*'P(3,x)',x=-1..1);\nvalue(%);" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#-%$IntG6$*&-%\"PG6$\"\"&%\"xG\"\"\"-F( 6$\"\"$F+F,/F+;!\"\"F," }}{PARA 11 "" 1 "" {XPPMATH 20 "6#\"\"!" }}} {PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 43 "Int('P(10,x)'*'P(10,x)',x=-1..1);\nvalue(%);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#-%$IntG6$*$)-%\"PG6$\"#5%\"xG\"\"#\"\"\"/F,;!\"\"F." }} {PARA 11 "" 1 "" {XPPMATH 20 "6##\"\"#\"#@" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 59 "The Legendre polynomials are av ailable through the package " }{TEXT 0 9 "orthopoly" }{TEXT -1 1 "." } }{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 77 "P := 'P':\nwith(orthopoly);\nP(5,x);\nInt('P(5,x)'*'P(5,x)',x=-1 ..1);\nvalue(%);\n" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#7(%\"GG%\"HG%\"L G%\"PG%\"TG%\"UG" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#,(*$)%\"xG\"\"&\" \"\"#\"#j\"\")*&#\"#N\"\"%F(*$)F&\"\"$F(F(!\"\"*&#\"#:F+F(F&F(F(" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#-%$IntG6$*$)-%\"PG6$\"\"&%\"xG\"\"#\" \"\"/F,;!\"\"F." }}{PARA 11 "" 1 "" {XPPMATH 20 "6##\"\"#\"#6" }}} {PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 61 "Legendre polynomials as a \"coordinate system\" for polynomials" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}{PARA 0 "" 0 "" {TEXT -1 20 "The use of the term " }{TEXT 263 10 "orthogonal" }{TEXT -1 91 " in connection with Legend re polynomials can be linked to the notion of orthogonal vectors." }} {PARA 0 "" 0 "" {TEXT -1 62 "We can think of a general polynomials def ined on the interval " }{XPPEDIT 18 0 "[-1,1]" "6#7$,$\"\"\"!\"\"F%" } {TEXT -1 43 " as a vectors, with the integration formula" }}{PARA 256 "" 0 "" {TEXT -1 3 " p " }{TEXT 261 1 "." }{TEXT -1 7 " q = " } {XPPEDIT 18 0 "Int(p(x)*q(x),x = -1 .. 1);" "6#-%$IntG6$*&-%\"pG6#%\"x G\"\"\"-%\"qG6#F*F+/F*;,$F+!\"\"F+" }{TEXT -1 1 " " }}{PARA 0 "" 0 "" {TEXT -1 12 "providing a " }{TEXT 263 11 "dot product" }{TEXT -1 1 ". " }}{PARA 0 "" 0 "" {TEXT -1 48 "Thus we can form the \"length of the \+ projection\" " }{XPPEDIT 18 0 "a[j]" "6#&%\"aG6#%\"jG" }{TEXT -1 25 " \+ of a general polynomial " }{XPPEDIT 18 0 "q(x)" "6#-%\"qG6#%\"xG" } {TEXT -1 43 " in the direction of a Legendre polynomial " }{XPPEDIT 18 0 "P(j,x)" "6#-%\"PG6$%\"jG%\"xG" }{TEXT -1 24 " by means of the fo rmula" }}{PARA 256 "" 0 "" {TEXT -1 2 " " }{XPPEDIT 18 0 "a[j] = Int( q(x)*P(j,x),x = -1 .. 1)/Int(P(j,x)*P(j,x),x = -1 .. 1);" "6#/&%\"aG6# %\"jG*&-%$IntG6$*&-%\"qG6#%\"xG\"\"\"-%\"PG6$F'F0F1/F0;,$F1!\"\"F1F1-F *6$*&-F36$F'F0F1-F36$F'F0F1/F0;,$F1F8F1F8" }{TEXT -1 4 " . " }}{PARA 0 "" 0 "" {TEXT -1 60 "The denominator is the \"square of the length\" of the vector " }{XPPEDIT 18 0 "P(j,x)" "6#-%\"PG6$%\"jG%\"xG" } {TEXT -1 1 "." }}{PARA 0 "" 0 "" {TEXT -1 17 "The coefficients " } {XPPEDIT 18 0 "a[j]" "6#&%\"aG6#%\"jG" }{TEXT -1 9 " are the " }{TEXT 263 11 "coordinates" }{TEXT -1 4 " of " }{XPPEDIT 18 0 "q(x)" "6#-%\"q G6#%\"xG" }{TEXT -1 76 " in the \"orthogonal\" coordinate system provi ded by the Legendre polynomials." }}{PARA 0 "" 0 "" {TEXT -1 44 "For e xample, we can find the coordinates of " }{XPPEDIT 18 0 "q(x)=5*x^5-3* x^4+2*x^3-x^2+4*x-7" "6#/-%\"qG6#%\"xG,.*&\"\"&\"\"\"*$F'F*F+F+*&\"\"$ F+*$F'\"\"%F+!\"\"*&\"\"#F+*$F'F.F+F+*$F'F3F1*&F0F+F'F+F+\"\"(F1" } {TEXT -1 12 " as follows." }}{PARA 0 "" 0 "" {TEXT -1 16 "All coordina tes " }{XPPEDIT 18 0 "a[j]" "6#&%\"aG6#%\"jG" }{TEXT -1 61 " with j gr eater than the degree 5 of the polynomial are zero." }}{PARA 0 "" 0 " " {TEXT -1 1 " " }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 148 "with(ort hopoly):\nq := x -> 5*x^5-3*x^4+2*x^3-x^2+4*x-7;\nfor j from 0 to 5 do \n a[j] := int(P(j,x)*q(x),x=-1..1)/int(P(j,x)*P(j,x),x=-1..1);\nend do;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"qGf*6#%\"xG6\"6$%)operator G%&arrowGF(,.*&\"\"&\"\"\")9$F.F/F/*&\"\"$F/)F1\"\"%F/!\"\"*&\"\"#F/)F 1F3F/F/*$)F1F8F/F6*&F5F/F1F/F/\"\"(F6F(F(F(" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>&%\"aG6#\"\"!#!$>\"\"#:" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>&%\"aG6#\"\"\"#\"$d#\"#N" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>&% \"aG6#\"\"##!#]\"#@" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>&%\"aG6#\"\"$# \"$O\"\"#X" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>&%\"aG6#\"\"%#!#C\"#N" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>&%\"aG6#\"\"&#\"#S\"#j" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 19 "We can reconstruc t " }{XPPEDIT 18 0 "q(x)" "6#-%\"qG6#%\"xG" }{TEXT -1 11 " as the sum " }}{PARA 256 "" 0 "" {TEXT -1 3 " " }{XPPEDIT 18 0 "Sum(a[j]*P(j,x) ,j = 0 .. 5);" "6#-%$SumG6$*&&%\"aG6#%\"jG\"\"\"-%\"PG6$F*%\"xGF+/F*; \"\"!\"\"&" }{TEXT -1 1 "." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 44 "j := 'j':\nSum(a[j]*P(j,x),j=0..5); \nvalue(%);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#-%$SumG6$*&&%\"aG6#%\"j G\"\"\"-%\"PG6$F*%\"xGF+/F*;\"\"!\"\"&" }}{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-" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 33 "If we take a function defined on " }{XPPEDIT 18 0 "[-1,1] " "6#7$,$\"\"\"!\"\"F%" }{TEXT -1 209 " which is not a polynomial, the re will be infinitely many coordinates, but, under suitable conditions , the coordinates tend to zero. In this way we can obtain a polynomial approximation for the given function." }}{PARA 0 "" 0 "" {TEXT -1 45 "We try this out for the exponential function " }{XPPEDIT 18 0 "f(x)=e xp(x)" "6#/-%\"fG6#%\"xG-%$expG6#F'" }{TEXT -1 1 "." }}{PARA 0 "" 0 " " {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 152 "with(orth opoly):\nf := x -> exp(x):\n'f(x)'=f(x);\nfor j from 0 to 8 do\n a[j ] := evalf(int(P(j,x)*exp(x),x=-1..1)/int(P(j,x)*P(j,x),x=-1..1),15); \nend do;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/-%\"fG6#%\"xG-%$expGF&" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>&%\"aG6#\"\"!$\"0!Qk$>,_<\"!#9" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#>&%\"aG6#\"\"\"$\"0L9NKQO5\"!#9" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#>&%\"aG6#\"\"#$\"/QZ1N9yN!#9" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>&%\"aG6#\"\"$$\"-$oOjb/(!#8" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>&%\"aG6#\"\"%$\"+]\"G^'**!#7" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>&%\"aG6#\"\"&$\"*7'e*4\"!#6" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>&%\"aG6#\"\"'$\"'XX**!#5" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>&%\"aG6#\"\"($\"%=w!\"*" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>&% \"aG6#\"\")$\"\"'!\"(" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 123 "j := 'j':\nSum(a[j]*P(j,x),j=0..8) ;\ng := unapply(value(%),x);\nplot([g(x),exp(x)],x=-1..1,color=[red,gr een],thickness=[1,2]);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#-%$SumG6$*&& %\"aG6#%\"jG\"\"\"-%\"PG6$F*%\"xGF+/F*;\"\"!\"\")" }}{PARA 12 "" 1 "" {XPPMATH 20 "6#>%\"gGf*6#%\"xG6\"6$%)operatorG%&arrowGF(,4$\"+E+++5!\" *\"\"\"*&$\"+N$*******!#5F09$F0F0*&$\"+#f!****\\F4F0)F5\"\"#F0F0*&$\"+ ]onm;F4F0)F5\"\"$F0F0*&$\"+d4>nT!#6F0)F5\"\"%F0F0*&$\"+qgGH$)!#7F0)F5 \"\"&F0F0*&$\"+%4o&z8FIF0)F5\"\"'F0F0*&$\"+]idU?!#8F0)F5\"\"(F0F0*&$\" +]iS;I!#9F0)F5\"\")F0F0F(F(F(" }}{PARA 13 "" 1 "" {GLPLOT2D 400 300 300 {PLOTDATA 2 "6&-%'CURVESG6%7S7$$!\"\"\"\"!$\"3_)*\\JZczyO!#=7$$!3o mmm;p0k&*F-$\"3wwRbferUQF-7$$!3wKL$3f&F-7$$!3hmmm\" >s%HaF-$\"3C()*pY$HL5eF-7$$!3Q+++]$*4)*\\F-$\"3I5v^l$fk1'F-7$$!39+++]_ &\\c%F-$\"3axnF&>)*\\L'F-7$$!31+++]1aZTF-$\"3I8%*R7n-0mF-7$$!3umm;/#)[ oPF-$\"3apHysu>goF-7$$!3hLLL$=exJ$F-$\"3->&))Rh\"[wrF-7$$!3*RLLLtIf$HF -$\"3AC+[d!)zbuF-7$$!3]++]PYx\"\\#F-$\"3!o`c[G;Wz(F-7$$!3EMLLL7i)4#F-$ \"3`$*GwV,'p5)F-7$$!3c****\\P'psm\"F-$\"3E))F-7$$!3)3LLL3x%z#)!#>$\"33XFSU2S0#*F-7$$!3KMLL3s$QM %Fdr$\"3!)=0Kve\"\\d*F-7$$!3]^omm;zr)*!#@$\"3Ox6**RL8!***F-7$$\"3%pJL$ ezw5VFdr$\"3pUY=L.0W5!#<7$$\"3s*)***\\PQ#\\\")Fdr$\"3m#GV(p\\!\\3\"Fgs 7$$\"3GKLLe\"*[H7F-$\"3EaYoem#38\"Fgs7$$\"3I*******pvxl\"F-$\"3?&3$4L0 J!=\"Fgs7$$\"3#z****\\_qn2#F-$\"35B7#[g:3B\"Fgs7$$\"3U)***\\i&p@[#F-$ \"3/yY^kzt\"G\"Fgs7$$\"3B)****\\2'HKHF-$\"3emy;y0vS8Fgs7$$\"3ElmmmZvOL F-$\"3=>9#>)**3'R\"Fgs7$$\"3i******\\2goPF-$\"3)Qi[XI+xX\"Fgs7$$\"3UKL $eR<*fTF-$\"3uSI`IL(e^\"Fgs7$$\"3m******\\)Hxe%F-$\"3A8/E'[J@e\"Fgs7$$ \"3ckm;H!o-*\\F-$\"3Wh:g@v6Z;Fgs7$$\"3y)***\\7k.6aF-$\"3-&)fYu]d;Rqz%>Fgs7$$\"3wKLL3N1#4(F-$\"37@$) yUwPK?Fgs7$$\"3Nmm;HYt7vF-$\"3\\PS:kwp>@Fgs7$$\"3Y*******p(G**yF-$\"3f gn$4[RK?#Fgs7$$\"3]mmmT6KU$)F-$\"3X4/Tw[/.BFgs7$$\"3fKLLLbdQ()F-$\"3Ii S^Yi8'R#Fgs7$$\"3[++]i`1h\"*F-$\"3I@WRJ&R&*\\#Fgs7$$\"3W++]P?Wl&*F-$\" 3_b7@hlo-EFgs7$$\"\"\"F*$\"3:+:7*)=G=FFgs-%'COLOURG6&%$RGBG$\"*++++\"! \")$F*F*Fb[l-%*THICKNESSG6#Fhz-F$6%7S7$F($\"3MBWr6WzyOF-7$F/$\"3.W^Qtf rUQF-7$F4$\"3[$*)f+Ru7*RF-7$F9$\"3sbu^dg@lTF-7$F>$\"3o&GlMXnzM%F-7$FC$ \"3Pc)=:(4\"y`%F-7$FH$\"3G4;0fu@@ZF-7$FM$\"3tE.r.&R*=\\F-7$FR$\"30z!*= &=Q@8&F-7$FW$\"3+m09a)[QN&F-7$Ffn$\"3%*o7XaF!>f&F-7$F[o$\"3R/uvRHL5eF- 7$F`o$\"3Eh`=4&fk1'F-7$Feo$\"3en)*)HW)*\\L'F-7$Fjo$\"3QSdj9q-0mF-7$F_p $\"3Oc1T#y(>goF-7$Fdp$\"3;7<*Q)=[wrF-7$Fip$\"3)=y\\.E)zbuF-7$F^q$\"33w Fu%Q;Wz(F-7$Fcq$\"3v7-HV,'p5)F-7$Fhq$\"3k,F]goIk%)F-7$F]r$\"3//[k;i@>) )F-7$Fbr$\"3(RvD]\\+a?*F-7$Fhr$\"3Kh#G[g:\\d*F-7$F]s$\"3i$)QMzI8!***F- 7$Fcs$\"32Z0q6.0W5Fgs7$Fis$\"3cs^ka\\!\\3\"Fgs7$F^t$\"3;(zM@lE38\"Fgs7 $Fct$\"3W$e]g`5.=\"Fgs7$Fht$\"3TE;`;c\"3B\"Fgs7$F]u$\"3+&y'4$)zt\"G\"F gs7$Fbu$\"3H'>'[,1vS8Fgs7$Fgu$\"3k*>pi+!4'R\"Fgs7$F\\v$\"3oHAQE.qd9Fgs 7$Fav$\"3]!RtqMte^\"Fgs7$Ffv$\"3wD+V%\\J@e\"Fgs7$F[w$\"31]tZ?v6Z;Fgs7$ F`w$\"3'Qf'fjFgs7$Fdx$\"3PvSfBwPK?Fgs7$Fix$\"3[J]$\\l(p>@Fgs7$F^y$ \"3,i!=R[RK?#Fgs7$Fcy$\"3J=Ad$*[/.BFgs7$Fhy$\"3Q`]xri8'R#Fgs7$F]z$\"3q VqM`&R&*\\#Fgs7$Fbz$\"3`rQsflo-EFgs7$Fgz$\"34X!f%G=G=FFgs-F\\[l6&F^[lF b[lF_[lFb[l-Fd[l6#\"\"#-%+AXESLABELSG6$Q\"x6\"Q!Feel-%%VIEWG6$;F(Fgz%( DEFAULTG" 1 2 0 1 10 0 2 9 1 4 2 1.000000 45.000000 45.000000 0 0 "Cur ve 1" "Curve 2" }}}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 87 "The absolute error curve shows that the maximim absolute \+ error in using the polynomial " }{XPPEDIT 18 0 "g(x)" "6#-%\"gG6#%\"xG " }{TEXT -1 16 " to approximate " }{XPPEDIT 18 0 "f(x) = exp(x);" "6#/ -%\"fG6#%\"xG-%$expG6#F'" }{TEXT -1 40 " occurs at the left end of the interval." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 " " {MPLTEXT 1 0 37 "plot(g(x)-exp(x),x=-1..1,color=blue);" }}{PARA 13 " " 1 "" {GLPLOT2D 320 212 212 {PLOTDATA 2 "6&-%'CURVESG6#7cr7$$!\"\"\" \"!$\"3!4,*=v0gN7!#C7$$!3-n;HdNvs**!#=$\"3`,CETN$36\"F-7$$!3/MLe9r]X** F1$\"3$o?F'Q\">L#**!#D7$$!3/,](=ng#=**F1$\"3#4-g=QS*)z)F97$$!3%pmm\"HU ,\"*)*F1$\"3Hk`BewALxF97$$!3'GLekynP')*F1$\"3u[G=?TDCnF97$$!3()***\\PM @l$)*F1$\"31Eq\"=lO,x&F97$$!3)omT5!\\F4)*F1$\"3iRe'QEN!p[F97$$!3!RLL$e %G?y*F1$\"3YtndxB:>SF97$$!3#om;HdNvs*F1$\"3ahr1S$fgY#F97$$!3u****\\(oU In*F1$\"3GA&z5M)H(4\"F97$$!3xLL3-)\\&='*F1$!3&QV\\yp]0+\"!#E7$$!3ommm; p0k&*F1$!37)G*ps;JQ6F97$$!3#HL3-)*G#p%*F1$!3u54h=nE+EF97$$!3E++vV5Su$* F1$!3G,hD%y)GtOF97$$!3_m;H2Jdz#*F1$!3V!>\")[V$45WF97$$!3wKL$3#=F97$$!3\"ommT!R=0vF1$\"3E(4xxo ZWj#F97$$!3tKL$3i_+I(F1$\"3dORMF+!)GKF97$$!3u****\\P8#\\4(F1$\"3CxhR(R 'QnNF97$$!3ILL$3d%)=/(F1$\"3[X&o%pOO9OF97$$!3%ommT!y%)))pF1$\"3%)=:7Lc CXOF97$$!3R++]P5\"e$pF1$\"3s8xi'[4/m$F97$$!3#RLL3FuF)oF1$\"3EF2(RHv-m$ F97$$!3Znm;/vtHoF1$\"3Smch/@IXOF97$$!3-,+]P2qwnF1$\"3CtTWXT)fh$F97$$!3 XLL$3(RmBnF1$\"3-![yp?ZGd$F97$$!3+nm;/siqmF1$\"3v!>sG6Xk^$F97$$!3uLL$e \\g\"fkF1$\"3MMYhvfXrJF97$$!3[++](y$pZiF1$\"3Qc(QBm+Ym#F97$$!33LLL$yaE \"eF1$\"3=H95`73#H\"F97$$!3hmmm\">s%HaF1$!3c!z9U:$3')>Qj+\"GF97$$!31+++]1aZTF1$!34$*fwqKOAIF97$$!3Ym;aQvx_SF1$!3K' o8'HQDyIF97$$!3SLL3FW,eRF1$!38?*RNVg*3JF97$$!3M+]i:8DjQF1$!3LA&=t`.[6$ F97$$!3umm;/#)[oPF1$!3'RxI#ooF'4$F97$$!3<++v$>BJa$F1$!3#HlJ'[:[eHF97$$ !3hLLL$=exJ$F1$!3iWeGJ>.*p#F97$$!3!QLL$eW%o7$F1$!3Cj[)Hg_fR#F97$$!3*RL LLtIf$HF1$!3<^[fwvpG?F97$$!3]++]PYx\"\\#F1$!3;m6)z#Ri))**Fao7$$!3EMLLL 7i)4#F1$\"3tUB.My!os%!#G7$$!3c****\\P'psm\"F1$\"3C\\r6cJ0d5F97$$!3')** **\\74_c7F1$\"3?(*Q.rVj()=F97$$!3ZmmT5VBU5F1$\"3'fMhexE9A#F97$$!3)3LLL 3x%z#)!#>$\"3')zw86*pPZ#F97$$!3gKL$e9d;J'Fg^l$\"3Jn/!)=&)*zi#F97$$!3KM LL3s$QM%Fg^l$\"3FhV\"[dA\\q#F97$$!3'ym;aQdDG$Fg^l$\"3#=n.(H#QRr#F97$$! 3T,+]ivF@AFg^l$\"3^E4*4#)3.q#F97$$!3'\\L$eRx**f6Fg^l$\"3i21H66LkEF97$$ !3]^omm;zr)*!#@$\"3(>1$[PHZ1EF97$$\"3@CL$3-Dg5#Fg^l$\"3V&*eRDUF>CF97$$ \"3%pJL$ezw5VFg^l$\"3?F97$$\"3UKL$eR<*fTF1$!3Q'3 ^c(\\.a;F97$$\"3m******\\)Hxe%F1$!37A%>?a7'p\")Fao7$$\"3ckm;H!o-*\\F1$ \"3mBik!R6U7\"Fao7$$\"3y)***\\7k.6aF1$\"3'3F!\\;\"Rp3\"F97$$\"3#emmmT9 C#eF1$\"3acm9rLq%)=F97$$\"3WKLeR>F97$$\"3Nmm;HYt7vF1$\"3Zch..g+>#*Fao7$$\"3Y*******p(G** yF1$!3Em6&zU,8)HFao7$$\"3]mmmT6KU$)F1$!3h&)3H')3=;@#F97$$\"3fKLLLbdQ()F1$!3v5#pw5*4EDF97$$\"3Mm\"z>w'Q\"z )F1$!33k3;+2*)oDF97$$\"36+]i!*z>W))F1$!3SncA9#y7f#F97$$\"3yK3F>#4q*))F 1$!3gVa5$e6:f#F97$$\"3amm\"zW?)\\*)F1$!3C*QN4X&ynDF97$$\"3'HL3_!HWb!*F 1$!3q>Ned%e4W#F97$$\"3[++]i`1h\"*F1$!3'o%ylQAE&>#F97$$\"3Y++++PDj$*F1$ !3^cH/x;/L8F97$$\"3W++]P?Wl&*F1$\"33Ezu$*RQ([\"Fao7$$\"3K+]7G:3u'*F1$ \"3I;]$*px-Z7F97$$\"3A++v=5s#y*F1$\"3i@XZbJK%e#F97$$\"3;+D1k2/P)*F1$\" 3q:U " 0 "" {MPLTEXT 1 0 40 "numapprox[infnorm](g(x)-exp(x),x=-1 ..1);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#$\"+v0gN7!#;" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}} {SECT 1 {PARA 4 "" 0 "" {TEXT -1 51 "Gauss integration formulas via Le gendre polynomials" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}} {PARA 0 "" 0 "" {TEXT -1 140 "We look for a direct method of construct ing a list of nodes or abscissas and associated or weights appearing i n Gauss integration formulas.\n" }}{PARA 0 "" 0 "" {TEXT -1 4 "Let " } {XPPEDIT 18 0 "x[1],x[2],` . . . `,x[n];" "6&&%\"xG6#\"\"\"&F$6#\"\"#% (~.~.~.~G&F$6#%\"nG" }{TEXT -1 8 " be the " }{TEXT 278 1 "n" }{TEXT -1 39 " real zeros of the Legendre polynomial " }{XPPEDIT 18 0 "P(n,x) " "6#-%\"PG6$%\"nG%\"xG" }{TEXT -1 1 "." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 103 "We use an approach which is similar to what we did to derive the coefficients in Newton-Cotes formulas." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 105 "Consid er the \"L\" polynomials which are used in the construction of the Lag range interpolating polynomial." }}{PARA 0 "" 0 "" {TEXT -1 18 "Specif ically, let " }{XPPEDIT 18 0 "L(i,x) = L[i](x);" "6#/-%\"LG6$%\"iG%\"x G-&F%6#F'6#F(" }{TEXT -1 59 " be the interpolating polynomial which ha s the value 1 at " }{XPPEDIT 18 0 "x[i];" "6#&%\"xG6#%\"iG" }{TEXT -1 37 " and the value 0 at the other zeros." }}{PARA 256 "" 0 "" {TEXT -1 2 " " }{XPPEDIT 18 0 "L[i](x) = product((x-x[k])/(x[i]-x[k]) ,k = 1 .. n);" "6#/-&%\"LG6#%\"iG6#%\"xG-%(productG6$*&,&F*\"\"\"&F*6# %\"kG!\"\"F0,&&F*6#F(F0&F*6#F3F4F4/F3;F0%\"nG" }}{PARA 256 "" 0 "" {TEXT -1 2 " " }{XPPEDIT 18 0 "k <> i;" "6#0%\"kG%\"iG" }{TEXT 262 0 "" }}{PARA 0 "" 0 "" {TEXT -1 26 "Define associated weights " } {XPPEDIT 18 0 "w[1],w[2],` . . . `,w[n];" "6&&%\"wG6#\"\"\"&F$6#\"\"#% (~.~.~.~G&F$6#%\"nG" }{TEXT -1 4 " by " }}{PARA 256 "" 0 "" {TEXT -1 2 " " }{XPPEDIT 18 0 "w[i] = Int(L[i](x),x = -1 .. 1);" "6#/&%\"wG6#% \"iG-%$IntG6$-&%\"LG6#F'6#%\"xG/F0;,$\"\"\"!\"\"F4" }{TEXT -1 1 "." }} {PARA 0 "" 0 "" {TEXT -1 4 "Let " }{XPPEDIT 18 0 "f(x)" "6#-%\"fG6#%\" xG" }{TEXT -1 25 " be any polynomial degree" }{XPPEDIT 18 0 "`` < n;" "6#2%!G%\"nG" }{TEXT -1 2 ". " }}{PARA 0 "" 0 "" {TEXT -1 5 "Then " } {XPPEDIT 18 0 "f(x)" "6#-%\"fG6#%\"xG" }{TEXT -1 62 " coincides with t he Lagrange interpolating polynomial for the " }{TEXT 279 1 "n" } {TEXT -1 12 " data points" }{XPPEDIT 18 0 "``(x[1], f(x[1])),``(x[2], \+ f(x[2])),` . . . `,``(x[n], f(x[n]))" "6&-%!G6$&%\"xG6#\"\"\"-%\"fG6#& F'6#F)-F$6$&F'6#\"\"#-F+6#&F'6#F3%(~.~.~.~G-F$6$&F'6#%\"nG-F+6#&F'6#F= " }{TEXT -1 5 ", so " }}{PARA 256 "" 0 "" {TEXT -1 2 " " }{XPPEDIT 18 0 "f(x)=Sum(L[i](x)*f(x[i]),i=1..n)" "6#/-%\"fG6#%\"xG-%$SumG6$*&-& %\"LG6#%\"iG6#F'\"\"\"-F%6#&F'6#F0F2/F0;F2%\"nG" }{TEXT -1 2 ", " }} {PARA 0 "" 0 "" {TEXT -1 4 "and " }}{PARA 256 "" 0 "" {TEXT -1 1 " " } {XPPEDIT 18 0 "Int(f(x),x = -1 .. 1)=Int(Sum(L[i](x)*f(x[i]),i = 1 .. \+ n),x = -1 .. 1)" "6#/-%$IntG6$-%\"fG6#%\"xG/F*;,$\"\"\"!\"\"F.-F%6$-%$ SumG6$*&-&%\"LG6#%\"iG6#F*F.-F(6#&F*6#F:F./F:;F.%\"nG/F*;,$F.F/F." } {TEXT -1 1 " " }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "``=S um(Int(L[i](x),x = -1 .. 1)*f(x[i]),i = 1 .. n)" "6#/%!G-%$SumG6$*&-%$ IntG6$-&%\"LG6#%\"iG6#%\"xG/F2;,$\"\"\"!\"\"F6F6-%\"fG6#&F26#F0F6/F0;F 6%\"nG" }{TEXT -1 1 " " }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "``=Sum(w[i]*f(x[i]),i = 1 .. n)" "6#/%!G-%$SumG6$*&&%\"wG6#%\"iG \"\"\"-%\"fG6#&%\"xG6#F,F-/F,;F-%\"nG" }{TEXT -1 3 ". " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 111 "However, it turns o ut that, because of the special choice of the nodes as the zeros of th e Legendre polynomial " }{XPPEDIT 18 0 "P(n,x)" "6#-%\"PG6$%\"nG%\"xG " }{TEXT -1 15 ", this formula " }{TEXT 263 25 "holds for all polynomi als" }{TEXT 256 1 " " }{XPPEDIT 18 0 "f(x)" "6#-%\"fG6#%\"xG" }{TEXT 256 1 " " }{TEXT 263 9 "of degree" }{XPPEDIT 18 0 "``<=2*n-1" "6#1%!G, &*&\"\"#\"\"\"%\"nGF(F(F(!\"\"" }{TEXT -1 1 "." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 77 "Let's try to see why this is true by considering as an example the polynomial" }}{PARA 256 "" 0 "" {TEXT -1 4 " " }{XPPEDIT 18 0 "f(x) = 3*x^4-x^7;" "6#/-%\"fG6# %\"xG,&*&\"\"$\"\"\"*$F'\"\"%F+F+*$F'\"\"(!\"\"" }{TEXT -1 1 "." }} {PARA 0 "" 0 "" {TEXT -1 58 "We want to show that it is possible to ob tain the integral" }}{PARA 256 "" 0 "" {TEXT -1 2 " " }{XPPEDIT 18 0 "Int(f(x),x = -1 .. 1);" "6#-%$IntG6$-%\"fG6#%\"xG/F);,$\"\"\"!\"\"F- " }{TEXT -1 1 " " }}{PARA 0 "" 0 "" {TEXT -1 82 "exactly by using a 4 \+ point Gauss formula constructed in the method outlined above." }} {PARA 0 "" 0 "" {TEXT -1 51 "We first find the zeros of the Legendre p olynomial " }{XPPEDIT 18 0 "P(4,x)" "6#-%\"PG6$\"\"%%\"xG" }{TEXT -1 1 "." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 50 "with(orthopoly):\nxg := [evalf(fsolve(P(4,x)),15)];" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%#xgG7&$!0`Sf6j8h)!#:$!0c[eV5)*R$F( $\"0c[eV5)*R$F($\"0`Sf6j8h)F(" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }} {PARA 0 "" 0 "" {TEXT -1 70 "Now define the Lagrange \"L\" polynomials using the following procedure." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 377 "L := proc(xvals::list,k::po sint,x)\nlocal v,xvalsdelete;\n if k > nops(xvals) then\n error \"2nd argument must no geater than the list length\"\n end if;\n \+ xvalsdelete := [op(1..k-1,xvals),op(k+1..nops(xvals),xvals)];\n if m ember(xvals[k], xvalsdelete) then\n error \"the same point has be en entered twice\"\n end if;\n mul((x-v)/(xvals[k]-v),v=xvalsdelet e);\nend proc:" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 37 "The graphs of the 4 \"L\" polynomials " }{XPPEDIT 18 0 " L(1,x) = L[1](x),L(2,x) = L[2](x),L(3,x) = L[3](x),L(4,x) = L[4](x);" "6&/-%\"LG6$\"\"\"%\"xG-&F%6#F'6#F(/-F%6$\"\"#F(-&F%6#F06#F(/-F%6$\"\" $F(-&F%6#F86#F(/-F%6$\"\"%F(-&F%6#F@6#F(" }{TEXT -1 62 " are plotted i n the following picture along with the graph of " }{XPPEDIT 18 0 "P(4, x)" "6#-%\"PG6$\"\"%%\"xG" }{TEXT -1 11 " (shown in " }{TEXT 264 3 "re d" }{TEXT -1 2 ")." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 " > " 0 "" {MPLTEXT 1 0 165 "plot([P(4,x),seq(L(xg,i,x),i=1..4)],x=-1..1 ,\n thickness=[2,1,1,1,1],color=[red,blue,green,magenta,coral],\n l egend=[`P(4,x)`,`L(1,x)`,`L(2,x)`,`L(3,x)`,`L(4,x)`]);" }}{PARA 13 "" 1 "" {GLPLOT2D 624 419 419 {PLOTDATA 2 "6)-%'CURVESG6&7ep7$$!\"\"\"\"! 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3cX89$*p5JBF07$F``l$!3cH^T)R4+G#F07$Fj`l$!3'*eGS!>QI>#F07$Fdal$!3q_,-= oip?F07$F^bl$!3-!G(fKvu6$\"3mw*Q_^Gx]#F07$FR$\"332:*)pE&oQ\"F07$Ff n$\"3*)zx3xb;5KFjn7$Fao$!3\"z$)*3j$GPa&Fjn7$F[p$!3t\"[?*4y))R7F07$Fep$ !3R3K*ySU&>#F07$Fcr$!33V3g) 45\"zAF07$F]s$!3UC'G$)4G8L#F07$Fgs$!3'4^ON4t*\\BF07$F\\t$!3Q'e%\\[w_NB F07$Fat$!3CJ3=:5U)G#F07$Fft$!3LAKz\\qTI@F07$F[u$!3#GR3WrRl%=F07$F`u$!3 wW.VIw]f9F07$Feu$!3H(p/@(\\_+5F07$Fju$!3+%pHo**>J>&Fjn7$F_v$\"3[g*\\)4 PM<7Fjn7$Fdv$\"3MfQr&)4XUrFjn7$Fiv$\"3#3yQ8[^&\\9F07$F^w$\"3AH\"y`=PB8 #F07$Fcw$\"3z'*z[]4M0HF07$Fhw$\"3-K,C\"Q4Yl$F07$F]x$\"3_G1b$em$RWF07$F gx$\"3czZ=6$GMCcjR*F07$F_\\l$\"3!QsilVZGv*F07$Fd\\l$\" 3+5z3Dg\\u**F07$Fi\\l$\"3=&Hx6_8(45Fdhl7$$\"3-mm\"H2fU'RF0$\"3H>^n7i46 5Fdhl7$F^]l$\"3#H[]B.G(45Fdhl7$$\"3/mm\"HiBQP%F0$\"3k_kt/+'\\+\"Fdhl7$ Fc]l$\"3!op]\\#*)Rm**F07$Fh]l$\"3+*=wA\\o'3(*F07$F]^l$\"3%*p'esP+'*G*F 07$Fb^l$\"3c%ye\\$z0A()F07$F\\_l$\"3wc**G-r$*\\zF07$F``l$\"3k`WXs\"F07$Ffdl$!3 G8f!>h\"3$)GF07$F`el$!3G=ZE[Me_SF07$Fjel$!3u.f7EOd#G&F07$Fdfl$!3%3HM&) )f7tmF07$F+$!3#Hlc!)\\Cj8)F0-F_gl6&FaglFbglFeglFbglF^bm-F[hl6#%'L(3,x) G-F$6&7U7$F($!3?E5\\j>$!3MSBMMzwhqFjn7$FR$!3YI$fjS;>(QFjn7$Ff n$!3W@(f3hB7())F[\\m7$Fao$\"3@+0T28N::Fjn7$F[p$\"3!Qj3jw(>]LFjn7$Fep$ \"3ocOLusb$f%Fjn7$F_q$\"3KT'>%*=!paaFjn7$Fcr$\"3u'RBL#*y%HfFjn7$Fgs$\" 3B*QbeU%yBgFjn7$Fat$\"3'*z\"*37@VqdFjn7$Fft$\"3G;AKgr2*G&Fjn7$F[u$\"3Q z!zh-T&)\\%Fjn7$F`u$\"37a4\"3DEH?Z:Fjn7$Fiv$!3!)>(o= &oqOIFjn7$F^w$!3#3#*pQ&*Q?K%Fjn7$Fcw$!3[!G;1m0Yl&Fjn7$Fhw$!3e_%f0:x$3o Fjn7$F]x$!3=A68)Hs+&yFjn7$Fgx$!3\"\\o>A)3#Qi)Fjn7$Fay$!3\"*RlYm!)*>A*F jn7$F\\z$!3f6uo+h(*Q&*Fjn7$Ffz$!34LFbX9sD&*Fjn7$F[[l$!3wiy#e@=5<*Fjn7$ F`[l$!3Y\\D4?RF#R)Fjn7$Fe[l$!3[/(R4'oR$=(Fjn7$Fj[l$!3!f8\\$GV2abFjn7$F _\\l$!3/ue^4C!)pJFjn7$Fd\\l$!38X\")o2ZF[\\m7$Fi\\l$\"3OMfd.$Fjn 7$F^]l$\"3k#y(3#yzq!oFjn7$Fc]l$\"3EYa.6%)oh6F07$Fh]l$\"3CQFsUYa$o\"F07 $F]^l$\"3lG.-T(p[I#F07$Fb^l$\"3+h\">@r(=\"*HF07$F\\_l$\"37j%H/%z[(z$F0 7$F``l$\"3%*>fbEJ.jYF07$Fdal$\"3MB%**\\ynEk&F07$F^bl$\"3F5(z+@2Fr'F07$ Fhbl$\"3C$z2/&3-'y(F07$Fbcl$\"3Z\"HOoM)\\E\"*F07$F\\dl$\"3N=byWh!H/\"F dhl7$Ffdl$\"3ckmIM%pH>\"Fdhl7$F`el$\"3-6$GI6()*o7Fdhl7$Fjel$\"37ZW3P\" )yZ8Fdhl7$Fdfl$\"3]5 " 0 "" {MPLTEXT 1 0 1 ";" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 61 "Note that each \"L\" polynomial has three zeros in common with " }{XPPEDIT 18 0 "P(4,x)" "6#-%\"PG6$\"\"%%\"xG" }{TEXT -1 1 ". " }}{PARA 0 "" 0 "" {TEXT -1 47 "The weights are now computed as the i ntegrals " }{XPPEDIT 18 0 "w[i] = Int(L[i](x),x = -1 .. 1);" "6#/&%\" wG6#%\"iG-%$IntG6$-&%\"LG6#F'6#%\"xG/F0;,$\"\"\"!\"\"F4" }{TEXT -1 1 " ." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 125 "wg := evalf([int(L(xg,1,x),x=-1..1),int(L(xg,2,x),x= -1..1),\n int(L(xg,3,x),x=-1..1),int(L(xg,4,x),x=-1..1)], 15);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%#wgG7&$\"0`u8X[&yM!#:$\"0\\D '[:X@lF(F)F&" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 18 " We check that " }{XPPEDIT 18 0 "Int(f(x),x = -1 .. 1) = Sum(w[i]*f(x[i]),i = 1 .. n);" "6#/-%$IntG6$-%\"fG6#%\"xG/F*;,$\"\" \"!\"\"F.-%$SumG6$*&&%\"wG6#%\"iGF.-F(6#&F*6#F7F./F7;F.%\"nG" }{TEXT -1 2 ". " }}{PARA 0 "" 0 "" {TEXT -1 29 "First evaluate the integral. \+ " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 49 "f := x -> 3*x^4-x^7;\nInt(f(x),x=-1..1);\nvalue(%);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"fGf*6#%\"xG6\"6$%)operatorG%&arrowGF(,&* &\"\"$\"\"\")9$\"\"%F/F/*$)F1\"\"(F/!\"\"F(F(F(" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#-%$IntG6$,&*&\"\"$\"\"\")%\"xG\"\"%F)F)*$)F+\"\"(F)!\" \"/F+;F0F)" }}{PARA 11 "" 1 "" {XPPMATH 20 "6##\"\"'\"\"&" }}}{PARA 0 "" 0 "" {TEXT -1 22 "Now calculate the sum " }{XPPEDIT 18 0 "Sum(w[i]* f(x[i]),i = 1 .. 4);" "6#-%$SumG6$*&&%\"wG6#%\"iG\"\"\"-%\"fG6#&%\"xG6 #F*F+/F*;F+\"\"%" }{TEXT -1 1 "." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 27 "sum(wg[i]*f(xg[i]),i=1..4); " }}{PARA 11 "" 1 "" {XPPMATH 20 "6#$\"+++++7!\"*" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 22 "Now let's investigate " } {TEXT 263 18 "why this all works" }{TEXT -1 1 "." }}{PARA 0 "" 0 "" {TEXT -1 16 "First we divide " }{XPPEDIT 18 0 "f(x) = 3*x^4-x^7;" "6#/ -%\"fG6#%\"xG,&*&\"\"$\"\"\"*$F'\"\"%F+F+*$F'\"\"(!\"\"" }{TEXT -1 4 " by " }{XPPEDIT 18 0 "P(4,x) = 35/8*x^4-15/4*x^2+3/8" "6#/-%\"PG6$\"\" %%\"xG,(*(\"#N\"\"\"\"\")!\"\"F(F'F,*(\"#:F,F'F.F(\"\"#F.*&\"\"$F,F-F. F," }{TEXT -1 19 " to get a quotient " }{XPPEDIT 18 0 "q(x)" "6#-%\"qG 6#%\"xG" }{TEXT -1 17 " and a remainder " }{XPPEDIT 18 0 "r(x)" "6#-% \"rG6#%\"xG" }{TEXT -1 2 ". " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 96 "qx := quo(f(x),P(4,x),x,'rx' ):\nq := unapply(qx,x):\n'q(x)'=q(x);\nr := unapply(rx,x):\n'r(x)'=r(x );" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/-%\"qG6#%\"xG,(*&#\"\")\"#N\"\" \"*$)F'\"\"$F-F-!\"\"*&#\"#[\"$X#F-F'F-F1#\"#CF,F-" }}{PARA 11 "" 1 " " {XPPMATH 20 "6#/-%\"rG6#%\"xG,**&#\"$f\"\"$X#\"\"\"*$)F'\"\"$F-F-!\" \"*&#\"#=F,F-F'F-F-#\"\"*\"#NF1*&#F4\"\"(F-*$)F'\"\"#F-F-F-" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 52 "We can check by multiplication that this is correct." }}{PARA 0 "" 0 "" {TEXT -1 0 " " }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 57 "'P(4,x)*q(x)+r(x)'=P(4,x )*q(x)+r(x);\n``=simplify(rhs(%));" }}{PARA 11 "" 1 "" {XPPMATH 20 "6# /,&*&-%\"PG6$\"\"%%\"xG\"\"\"-%\"qG6#F*F+F+-%\"rGF.F+,,*&,(#\"\"$\"\") F+*&#\"#NF6F+*$)F*F)F+F+F+*&#\"#:F)F+*$)F*\"\"#F+F+!\"\"F+,(*&#F6F9F+* $)F*F5F+F+FB*&#\"#[\"$X#F+F*F+FB#\"#CF9F+F+F+*(\"$f\"F+FKFBF*F5FB*(\"# =F+FKFBF*F+F+#\"\"*F9FB*(FQF+\"\"(FBF*FAF+" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/%!G,&*&\"\"$\"\"\")%\"xG\"\"%F(F(*$)F*\"\"(F(!\"\"" }} }{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 14 "The remainder " }{XPPEDIT 18 0 "r(x)" "6#-%\"rG6#%\"xG" }{TEXT -1 16 " has degree < 4." }}{PARA 0 "" 0 "" {TEXT -1 3 "Now" }}{PARA 256 "" 0 "" {TEXT -1 2 " " }{XPPEDIT 18 0 "Sum(w[i]*f(x[i]),i = 1 .. 4) = Sum(w[i]*(P(4,x[i])*q(x[i])+r(x[i ])),i = 1 .. 4);" "6#/-%$SumG6$*&&%\"wG6#%\"iG\"\"\"-%\"fG6#&%\"xG6#F+ F,/F+;F,\"\"%-F%6$*&&F)6#F+F,,&*&-%\"PG6$F5&F16#F+F,-%\"qG6#&F16#F+F,F ,-%\"rG6#&F16#F+F,F,/F+;F,F5" }{TEXT -1 1 " " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "`` = Sum(w[ i]*r(x[i]),i = 1 .. 4);" "6#/%!G-%$SumG6$*&&%\"wG6#%\"iG\"\"\"-%\"rG6# &%\"xG6#F,F-/F,;F-\"\"%" }{TEXT -1 11 ", because " }{XPPEDIT 18 0 "P( 4,x[i]) = 0;" "6#/-%\"PG6$\"\"%&%\"xG6#%\"iG\"\"!" }{TEXT -1 5 " for \+ " }{XPPEDIT 18 0 "i = 1,2,3,4" "6&/%\"iG\"\"\"\"\"#\"\"$\"\"%" }{TEXT -1 1 " " }}{PARA 256 "" 0 "" {TEXT -1 1 " " }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "`` = Int(r(x),x = -1 .. 1);" "6#/%!G-%$IntG6$ -%\"rG6#%\"xG/F+;,$\"\"\"!\"\"F/" }{TEXT -1 2 ", " }}{PARA 258 "" 0 " " {TEXT -1 70 "because the Gauss formula is certainly exact for polyno mials of degree" }{XPPEDIT 18 0 "`` < 4;" "6#2%!G\"\"%" }}{PARA 256 " " 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "`` = Int(P(4,x)*q(x)+r(x),x = -1 .. 1);" "6#/%!G-%$IntG6$,&*&-%\"PG6$\"\"%%\"xG\"\"\"-%\"qG6#F.F/F/-% \"rG6#F.F//F.;,$F/!\"\"F/" }{TEXT -1 2 ", " }}{PARA 259 "" 0 "" {TEXT -1 8 "because " }{XPPEDIT 18 0 "Int(P(4,x)*q(x),x = -1 .. 1) = 0;" "6# /-%$IntG6$*&-%\"PG6$\"\"%%\"xG\"\"\"-%\"qG6#F,F-/F,;,$F-!\"\"F-\"\"!" }{TEXT -1 18 ", since degree of " }{XPPEDIT 18 0 "q(x)" "6#-%\"qG6#%\" xG" }{TEXT -1 3 " is" }{XPPEDIT 18 0 "``< 4" "6#2%!G\"\"%" }{TEXT -1 12 " (see below)" }}{PARA 256 "" 0 "" {TEXT -1 2 " " }{XPPEDIT 18 0 " `` = Int(f(x),x = -1 .. 1);" "6#/%!G-%$IntG6$-%\"fG6#%\"xG/F+;,$\"\"\" !\"\"F/" }{TEXT -1 2 ". " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 " " 0 "" {TEXT -1 38 "Recall from the previous section that " }{XPPEDIT 18 0 "P(4,x)" "6#-%\"PG6$\"\"%%\"xG" }{TEXT -1 42 " is orthogonal to a ny polynomial of degree" }{XPPEDIT 18 0 "``<4" "6#2%!G\"\"%" }{TEXT -1 1 "." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 37 "Int('P(4,x)'*q(x),x=-1..1);\nvalue(%);" }}{PARA 11 " " 1 "" {XPPMATH 20 "6#-%$IntG6$*&-%\"PG6$\"\"%%\"xG\"\"\",(*$)F+\"\"$F ,#!\")\"#NF+#!#[\"$X##\"#CF3F,F,/F+;!\"\"F," }}{PARA 11 "" 1 "" {XPPMATH 20 "6#\"\"!" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 41 "A more convenient formula for the weights" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}{PARA 0 "" 0 "" {TEXT -1 56 "There is a mor e convenient way to calculate the weights " }{XPPEDIT 18 0 "w[j]" "6#& %\"wG6#%\"jG" }{TEXT -1 42 " in the Gauss-Legendre integration formula " }}{PARA 256 "" 0 "" {TEXT -1 2 " " }{XPPEDIT 18 0 "Int(f(x),x=-1..1 )=Sum(w[i]*f(x[j]),j=1..n)" "6#/-%$IntG6$-%\"fG6#%\"xG/F*;,$\"\"\"!\" \"F.-%$SumG6$*&&%\"wG6#%\"iGF.-F(6#&F*6#%\"jGF./F<;F.%\"nG" }{TEXT -1 2 ", " }}{PARA 0 "" 0 "" {TEXT -1 6 "where " }{XPPEDIT 18 0 "f(x)" "6# -%\"fG6#%\"xG" }{TEXT -1 26 " is a polynomial of degree" }{XPPEDIT 18 0 "``<=2*n-1" "6#1%!G,&*&\"\"#\"\"\"%\"nGF(F(F(!\"\"" }{TEXT -1 57 ", \+ although the derivation of the formula is a bit tricky." }}{PARA 0 "" 0 "" {TEXT -1 131 "If you are not interested in the details, just take a look at the formula at the end of this section, and skip to the nex t section." }}{PARA 0 "" 0 "" {TEXT -1 1 " " }}{PARA 0 "" 0 "" {TEXT -1 13 "To calculate " }{XPPEDIT 18 0 "w[j]" "6#&%\"wG6#%\"jG" }{TEXT -1 5 " let " }{XPPEDIT 18 0 "f(x) = P(n,x)*P(n-1,x)/(x-x[j]);" "6#/-% \"fG6#%\"xG*(-%\"PG6$%\"nGF'\"\"\"-F*6$,&F,F-F-!\"\"F'F-,&F'F-&F'6#%\" jGF1F1" }{TEXT -1 8 ", where " }{XPPEDIT 18 0 "x[j]" "6#&%\"xG6#%\"jG " }{TEXT -1 8 " is the " }{TEXT 280 1 "j" }{TEXT -1 12 " th zero of " }{XPPEDIT 18 0 "P(n,x)" "6#-%\"PG6$%\"nG%\"xG" }{TEXT -1 29 " correspo nding to the weight " }{XPPEDIT 18 0 "w[j]" "6#&%\"wG6#%\"jG" }{TEXT -1 1 "." }}{PARA 0 "" 0 "" {TEXT -1 6 "Since " }{XPPEDIT 18 0 "f(x)" " 6#-%\"fG6#%\"xG" }{TEXT -1 12 " has degree " }{XPPEDIT 18 0 "2*n-2" "6 #,&*&\"\"#\"\"\"%\"nGF&F&F%!\"\"" }{TEXT -1 27 ", the formula is exact for " }{XPPEDIT 18 0 "f(x)" "6#-%\"fG6#%\"xG" }{TEXT -1 1 "." }} {PARA 0 "" 0 "" {TEXT -1 16 "Moreover, since " }{XPPEDIT 18 0 "f(x[i]) =0" "6#/-%\"fG6#&%\"xG6#%\"iG\"\"!" }{TEXT -1 5 " for " }{XPPEDIT 18 0 "i<>j" "6#0%\"iG%\"jG" }{TEXT -1 8 " we have" }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "w[j]*f(x[j])=Int(f(x),x=-1..1)" "6#/*&& %\"wG6#%\"jG\"\"\"-%\"fG6#&%\"xG6#F(F)-%$IntG6$-F+6#F./F.;,$F)!\"\"F) " }{TEXT -1 1 "," }}{PARA 0 "" 0 "" {TEXT -1 2 "so" }}{PARA 256 "" 0 " " {TEXT -1 2 " " }{XPPEDIT 18 0 "w[j]=1/f(x[j])" "6#/&%\"wG6#%\"jG*& \"\"\"F)-%\"fG6#&%\"xG6#F'!\"\"" }{TEXT -1 1 " " }{XPPEDIT 18 0 "Int(f (x),x=-1..1)" "6#-%$IntG6$-%\"fG6#%\"xG/F);,$\"\"\"!\"\"F-" }{TEXT -1 2 ". " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 27 " Now, by L'Hospital's rule, " }{XPPEDIT 18 0 "Limit(P(n,x)/(x-x[j]) = ` P '`(n,x[j]),x = x[j]);" "6#-%&LimitG6$/*&-%\"PG6$%\"nG%\"xG\"\"\",&F, F-&F,6#%\"jG!\"\"F2-%$P~'G6$F+&F,6#F1/F,&F,6#F1" }{TEXT -1 48 ", so, s trictly speaking, we should have defined " }{XPPEDIT 18 0 "f(x)" "6#-% \"fG6#%\"xG" }{TEXT -1 15 " by the formula" }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "f(x) = PIECEWISE([P(n,x)*P(n-1,x)/(x-x[j]), x <> x[j]],[`P '`(n,x[j])*P(n-1,x[j]), x = x[j]]);" "6#/-%\"fG6#%\"xG-% *PIECEWISEG6$7$*(-%\"PG6$%\"nGF'\"\"\"-F.6$,&F0F1F1!\"\"F'F1,&F'F1&F'6 #%\"jGF5F50F'&F'6#F97$*&-%$P~'G6$F0&F'6#F9F1-F.6$,&F0F1F1F5&F'6#F9F1/F '&F'6#F9" }{TEXT -1 3 " , " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 13 "in order for " }{XPPEDIT 18 0 "f(x)" "6#-%\"fG6#% \"xG" }{TEXT -1 45 " to be continuous everywhere in the interval " } {XPPEDIT 18 0 "[-1,1]" "6#7$,$\"\"\"!\"\"F%" }{TEXT -1 1 "." }}{PARA 0 "" 0 "" {TEXT -1 5 "Thus " }}{PARA 256 "" 0 "" {TEXT -1 2 " " } {XPPEDIT 18 0 "w[j] = 1/(`P '`(n,x[j])*P(n-1,x[j]));" "6#/&%\"wG6#%\"j G*&\"\"\"F)*&-%$P~'G6$%\"nG&%\"xG6#F'F)-%\"PG6$,&F.F)F)!\"\"&F06#F'F)F 6" }{TEXT -1 1 " " }{XPPEDIT 18 0 "Int(P(n,x)*P(n-1,x)/(x-x[j]),x = -1 .. 1);" "6#-%$IntG6$*(-%\"PG6$%\"nG%\"xG\"\"\"-F(6$,&F*F,F,!\"\"F+F,, &F+F,&F+6#%\"jGF0F0/F+;,$F,F0F," }{TEXT -1 1 " " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 22 "Since the polynomials " } {XPPEDIT 18 0 "P(n,x)/(x-x[j])" "6#*&-%\"PG6$%\"nG%\"xG\"\"\",&F(F)&F( 6#%\"jG!\"\"F." }{TEXT -1 5 " and " }{XPPEDIT 18 0 "P(n-1,x)" "6#-%\"P G6$,&%\"nG\"\"\"F(!\"\"%\"xG" }{TEXT -1 18 " both have degree " } {XPPEDIT 18 0 "n-1" "6#,&%\"nG\"\"\"F%!\"\"" }{TEXT -1 16 ", the polyn omial" }}{PARA 256 "" 0 "" {TEXT -1 2 " " }{XPPEDIT 18 0 "Q(n-2,x) = \+ P(n,x)/(x-x[j])-a[n]/a[n-1];" "6#/-%\"QG6$,&%\"nG\"\"\"\"\"#!\"\"%\"xG ,&*&-%\"PG6$F(F,F),&F,F)&F,6#%\"jGF+F+F)*&&%\"aG6#F(F)&F86#,&F(F)F)F+F +F+" }{TEXT -1 1 " " }{XPPEDIT 18 0 "P(n-1,x)" "6#-%\"PG6$,&%\"nG\"\" \"F(!\"\"%\"xG" }{TEXT -1 1 "," }}{PARA 0 "" 0 "" {TEXT -1 6 "where " }{XPPEDIT 18 0 "a[n]" "6#&%\"aG6#%\"nG" }{TEXT -1 5 " and " }{XPPEDIT 18 0 "a[n-1]" "6#&%\"aG6#,&%\"nG\"\"\"F(!\"\"" }{TEXT -1 33 " are the \+ leading coefficients of " }{XPPEDIT 18 0 "P(n,x)" "6#-%\"PG6$%\"nG%\"x G" }{TEXT -1 5 " and " }{XPPEDIT 18 0 "P(n-1,x" "6#-%\"PG6$,&%\"nG\"\" \"F(!\"\"%\"xG" }{TEXT -1 14 "), has degree " }{XPPEDIT 18 0 "n-2" "6# ,&%\"nG\"\"\"\"\"#!\"\"" }{TEXT -1 1 "." }}{PARA 0 "" 0 "" {TEXT -1 1 " " }}{PARA 0 "" 0 "" {TEXT -1 12 "Then, since " }{XPPEDIT 18 0 "P(n-1 ,x)" "6#-%\"PG6$,&%\"nG\"\"\"F(!\"\"%\"xG" }{TEXT -1 18 " is orthogona l to " }{XPPEDIT 18 0 "Q(n-2,x)" "6#-%\"QG6$,&%\"nG\"\"\"\"\"#!\"\"%\" xG" }{TEXT -1 9 ", we have" }}{PARA 256 "" 0 "" {TEXT -1 3 " " } {XPPEDIT 18 0 "Int(P(n,x)*P(n-1,x)/(x-x[j]),x = -1 .. 1) = a[n]/a[n-1] ;" "6#/-%$IntG6$*(-%\"PG6$%\"nG%\"xG\"\"\"-F)6$,&F+F-F-!\"\"F,F-,&F,F- &F,6#%\"jGF1F1/F,;,$F-F1F-*&&%\"aG6#F+F-&F;6#,&F+F-F-F1F1" }{TEXT -1 1 " " }{XPPEDIT 18 0 "Int(P(n-1,x)^2,x=-1..1)" "6#-%$IntG6$*$-%\"PG6$, &%\"nG\"\"\"F,!\"\"%\"xG\"\"#/F.;,$F,F-F," }{TEXT -1 2 ". " }}{PARA 0 "" 0 "" {TEXT -1 110 "By examining the recurrence and orthogonalty rel ations for Legendre polynomials, we see that the right side is" }} {PARA 256 "" 0 "" {TEXT -1 2 " " }{XPPEDIT 18 0 "``((2*n-1)/n)*``(2/( 2*n-1)) = 2/n;" "6#/*&-%!G6#*&,&*&\"\"#\"\"\"%\"nGF,F,F,!\"\"F,F-F.F,- F&6#*&F+F,,&*&F+F,F-F,F,F,F.F.F,*&F+F,F-F." }{TEXT -1 1 "." }}{PARA 0 "" 0 "" {TEXT -1 4 "Thus" }}{PARA 256 "" 0 "" {TEXT -1 2 " " } {XPPEDIT 18 0 "w[j] = 2/(n*`P'`(n,x[j])*P(n-1,x[j]));" "6#/&%\"wG6#%\" jG*&\"\"#\"\"\"*(%\"nGF*-%#P'G6$F,&%\"xG6#F'F*-%\"PG6$,&F,F*F*!\"\"&F1 6#F'F*F7" }{TEXT -1 3 " . " }}{PARA 0 "" 0 "" {TEXT -1 27 "From the st andard relation " }}{PARA 256 "" 0 "" {TEXT -1 3 " " }{XPPEDIT 18 0 "(x^2-1)*`P '`(n,x) = n*(x*P(n,x)-P(n-1,x));" "6#/*&,&*$%\"xG\"\"#\"\" \"F)!\"\"F)-%$P~'G6$%\"nGF'F)*&F.F),&*&F'F)-%\"PG6$F.F'F)F)-F36$,&F.F) F)F*F'F*F)" }{TEXT -1 3 ", " }}{PARA 0 "" 0 "" {TEXT -1 33 "for Legen dre polynomials, we have" }}{PARA 256 "" 0 "" {TEXT -1 2 " " } {XPPEDIT 18 0 "n*P(n-1,x[j]) = (1-x[j]^2)*`P '`(n,x[j]);" "6#/*&%\"nG \"\"\"-%\"PG6$,&F%F&F&!\"\"&%\"xG6#%\"jGF&*&,&F&F&*$&F-6#F/\"\"#F+F&-% $P~'G6$F%&F-6#F/F&" }{TEXT -1 1 "," }}{PARA 0 "" 0 "" {TEXT -1 2 "so" }}{PARA 256 "" 0 "" {TEXT -1 2 " " }{XPPEDIT 18 0 "w[j] = 2/((1-x[j]^ 2)*`P '`(n,x[j])^2);" "6#/&%\"wG6#%\"jG*&\"\"#\"\"\"*&,&F*F**$&%\"xG6# F'F)!\"\"F**$-%$P~'G6$%\"nG&F/6#F'F)F*F1" }{TEXT -1 4 ". " }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{TEXT 266 14 "______________" }{TEXT -1 1 " " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 47 "Computation of Gauss-Legendre nodes and weights" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}{PARA 0 "" 0 "" {TEXT -1 57 "T he problem posed in the first section was to find nodes " }{XPPEDIT 18 0 "x[1], x[2],` . . . `,x[n]" "6&&%\"xG6#\"\"\"&F$6#\"\"#%(~.~.~.~G &F$6#%\"nG" }{TEXT -1 13 " and weights " }{XPPEDIT 18 0 "w[1],w[2],` . . . `,w[n];" "6&&%\"wG6#\"\"\"&F$6#\"\"#%(~.~.~.~G&F$6#%\"nG" }{TEXT -1 29 ", so that for any polynomial" }}{PARA 256 "" 0 "" {TEXT -1 2 " " }{XPPEDIT 18 0 "f(x) = sum(a[i]*x^i,i = 0 .. 2*n-1);" "6#/-%\"fG6# %\"xG-%$sumG6$*&&%\"aG6#%\"iG\"\"\")F'F/F0/F/;\"\"!,&*&\"\"#F0%\"nGF0F 0F0!\"\"" }{TEXT -1 1 " " }}{PARA 0 "" 0 "" {TEXT -1 10 "of degree " } {XPPEDIT 18 0 "2*n-1" "6#,&*&\"\"#\"\"\"%\"nGF&F&F&!\"\"" }{TEXT -1 9 ", we have" }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "int(f(x ),x = -1 .. 1) = sum(w[i]*f(x[i]),i = 1 .. n);" "6#/-%$intG6$-%\"fG6#% \"xG/F*;,$\"\"\"!\"\"F.-%$sumG6$*&&%\"wG6#%\"iGF.-F(6#&F*6#F7F./F7;F.% \"nG" }{TEXT -1 1 "." }}{PARA 0 "" 0 "" {TEXT -1 93 "The last three se ctions explain how the problem is solved for a general non-negative in teger " }{TEXT 281 1 "n" }{TEXT -1 34 " by means of Legendre polynomia ls." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 15 "" 0 "" {TEXT -1 15 "T ake the nodes " }{XPPEDIT 18 0 "x[1],x[2],` . . . `,x[n];" "6&&%\"xG6# \"\"\"&F$6#\"\"#%(~.~.~.~G&F$6#%\"nG" }{TEXT -1 11 " to be the " } {TEXT 263 5 "zeros" }{TEXT -1 28 " of the Legendre polynomial " } {XPPEDIT 18 0 "P(n,x)" "6#-%\"PG6$%\"nG%\"xG" }{TEXT -1 1 "." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 15 "" 0 "" {TEXT -1 36 "Calculate the \+ corresponding weights " }{XPPEDIT 18 0 "w[1],w[2],` . . . `,w[n];" "6& &%\"wG6#\"\"\"&F$6#\"\"#%(~.~.~.~G&F$6#%\"nG" }{TEXT -1 15 " by the fo rmula" }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "w[i] = 2/((1 -x[i]^2)*`P '`(n,x[i]));" "6#/&%\"wG6#%\"iG*&\"\"#\"\"\"*&,&F*F**$&%\" xG6#F'F)!\"\"F*-%$P~'G6$%\"nG&F/6#F'F*F1" }{TEXT -1 2 ", " }}{PARA 0 " " 0 "" {TEXT -1 7 " where " }{XPPEDIT 18 0 "`P '`(n,x);" "6#-%$P~'G6$% \"nG%\"xG" }{TEXT -1 27 " denotes the derivative of " }{XPPEDIT 18 0 " P(n,x)" "6#-%\"PG6$%\"nG%\"xG" }{TEXT -1 1 "." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 46 "The following subsection \+ contains a procedure " }{TEXT 0 6 "gauleg" }{TEXT -1 35 " for performi ng these calculations." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 22 "gauleg: implementation" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }} {PARA 0 "" 0 "" {TEXT -1 115 "This procedure is based on one given in \+ the book: Numerical Recipies in \"C\", Cambridege University Press, pa ge 152." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1246 "gauleg := proc(n::posint)\n local eps,m,xg,wg,z,h ,i,j,k,p1,p2,p3,pp,maxit;\n maxit := 2*Digits;\n Digits := Digits+ 3;\n eps := Float(5,4-Digits);\n m := trunc((n+1)/2);\n xg := [] ;\n wg := [];\n for i from 1 to m do\n z := evalf(cos(Pi*(i-0 .25)/(n+0.5))); # approximation for root\n for k from 1 to maxit \+ do\n # evaluate Legendre poly & derivative at z\n p1 : = 1;\n p2 := 0;\n for j from 1 to n do\n p3 := p2;\n p2 := p1;\n p1 := ((2*j-1)*z*p2-(j-1)* p3)/j;\n end do;\n pp := n*(z*p1-p2)/(z*z-1);# value o f derivative\n h := p1/pp;\n z := z-h;\n if ab s(h) " 0 "" {MPLTEXT 1 0 1 ";" }}}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}{PARA 0 "" 0 "" {TEXT -1 13 "For the case " }{XPPEDIT 18 0 "n = 3" "6#/%\"nG\"\"$" }{TEXT -1 86 ", we obtain the same nodes and weights as in the computation gi ven in the 2nd section." }}{PARA 0 "" 0 "" {TEXT -1 84 "The list of no des is given first, followed by the list of the corresponding weights. " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 10 "gauleg(3);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#7$7%$!+#pmfu(!#5 $\"\"!F)$\"+#pmfu(F'7%$\"+cbbbbF'$\"+*)))))))))F'F-" }}}{PARA 0 "" 0 " " {TEXT -1 1 " " }}{PARA 0 "" 0 "" {TEXT -1 97 "We can obtain the node s and weights for a 11 point Gauss-Legendre integration formula as fol lows." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 53 "xw := evalf(gauleg(11),15):\nxg := xw[1];\nwg := xw[2 ];" }}{PARA 12 "" 1 "" {XPPMATH 20 "6#>%#xgG7-$!0dg9e'G#y*!#:$!0&4o(*f iq))F($!0\\Sd0?:I(F($!07o?Hh4>&F($!0XB&f:V&p#F($\"\"!F2$\"0XB&f:V&p#F( $\"07o?Hh4>&F($\"0\\Sd0?:I(F($\"0&4o(*fiq))F($\"0dg9e'G#y*F(" }}{PARA 12 "" 1 "" {XPPMATH 20 "6#>%#wgG7-$\"0'ph6n&oc&!#;$\"00\\Yp.eD\"!#:$\" 0Mx#4@!H'=F+$\"0y>fkP>L#F+$\"0Z-^WX!GEF+$\"0,zx'3DHFF+F0F.F,F)F&" }}} {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT 267 7 "Example" }{TEXT -1 4 ": " }{XPPEDIT 18 0 "Int(cos(x^3),x = -1 .. 1);" "6#-%$I ntG6$-%$cosG6#*$%\"xG\"\"$/F*;,$\"\"\"!\"\"F/" }{TEXT -1 2 " " }} {PARA 0 "" 0 "" {TEXT -1 20 "We use the formula " }{XPPEDIT 18 0 "Int (f(x),x = -1 .. 1)" "6#-%$IntG6$-%\"fG6#%\"xG/F);,$\"\"\"!\"\"F-" } {TEXT -1 1 " " }{TEXT 268 1 "~" }{TEXT -1 1 " " }{XPPEDIT 18 0 "Sum(f( x[i])*w[i],i=1..11)" "6#-%$SumG6$*&-%\"fG6#&%\"xG6#%\"iG\"\"\"&%\"wG6# F-F./F-;F.\"#6" }{TEXT -1 8 ", where " }{XPPEDIT 18 0 "f(x)= cos(x^3) " "6#/-%\"fG6#%\"xG-%$cosG6#*$F'\"\"$" }{TEXT -1 15 " and the nodes " }{XPPEDIT 18 0 "x[i]" "6#&%\"xG6#%\"iG" }{TEXT -1 13 " and weights " } {XPPEDIT 18 0 "w[i]" "6#&%\"wG6#%\"iG" }{TEXT -1 44 " are obtained fro m the previous calculation." }}{PARA 0 "" 0 "" {TEXT -1 1 " " }} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 71 "f := x-> cos(x^3);\nSum('f(x g[i])'*'wg[i]',i=1..11);\nevalf(evalf(%,15));" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"fGf*6#%\"xG6\"6$%)operatorG%&arrowGF(-%$cosG6#*$)9$ \"\"$\"\"\"F(F(F(" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#-%$SumG6$*&-%\"fG 6#&%#xgG6#%\"iG\"\"\"&%#wgGF,F./F-;F.\"#6" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#$\"+\"))3M'=!\"*" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }} {PARA 0 "" 0 "" {TEXT -1 43 "This value agrees with the value given by \"" }{TEXT 0 9 "evalf/Int" }{TEXT -1 2 "\"." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 32 "Int(cos(x^3),x=-1. .1);\nevalf(%);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#-%$IntG6$-%$cosG6#* $)%\"xG\"\"$\"\"\"/F+;!\"\"F-" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#$\"+ \"))3M'=!\"*" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 4 "Task" }}{PARA 0 "" 0 "" {TEXT -1 73 "(a) O btain the nodes and weights for 19 point Gauss-Legendre integration." }}{PARA 0 "" 0 "" {TEXT -1 96 "(b) Use the nodes and weights obtained \+ in part (a) to find a 15 digit decimal approximation for " }{XPPEDIT 18 0 "Pi" "6#%#PiG" }{TEXT -1 15 " by evaluating " }}{PARA 256 "" 0 " " {TEXT -1 3 " " }{XPPEDIT 18 0 "Int(1/(1+x^2),x = -1 .. 1);" "6#-%$ IntG6$*&\"\"\"F',&F'F'*$%\"xG\"\"#F'!\"\"/F*;,$F'F,F'" }{TEXT -1 2 ". \+ " }}{PARA 0 "" 0 "" {TEXT -1 34 "__________________________________" } }{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" } }{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{PARA 0 "" 0 "" {TEXT -1 34 "__________________________________" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}}}{MARK "4 0 0" 0 }{VIEWOPTS 1 1 0 1 1 1803 1 1 1 1 }{PAGENUMBERS 0 1 2 33 1 1 }