{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 "" -1 23 "Courier" 1 10 0 0 0 0 0 0 0 0 0 0 3 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 "" -1 260 "" 1 14 0 0 72 0 0 1 1 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 "" -1 263 "" 0 1 0 0 0 0 0 1 0 0 0 0 0 0 0 1 }{CSTYLE "" -1 264 "" 0 10 0 0 0 0 0 0 0 0 0 0 3 0 0 1 } {CSTYLE "" -1 265 "" 0 1 0 0 0 0 0 1 0 0 0 0 0 0 0 1 }{CSTYLE "" -1 266 "" 0 10 0 0 0 0 0 0 0 0 0 0 3 0 0 1 }{CSTYLE "Purple Emphasis" -1 267 "Times" 1 12 102 0 230 1 0 1 0 0 0 0 0 0 0 0 }{CSTYLE "Red Emphasi s" -1 268 "Times" 1 12 255 0 0 1 0 1 0 0 0 0 0 0 0 0 }{CSTYLE "Dark Re d Emphasis" -1 269 "Times" 1 12 128 0 0 1 0 1 0 0 0 0 0 0 0 0 } {CSTYLE "" -1 270 "" 0 1 0 0 0 0 1 0 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 271 "" 0 1 0 0 0 0 1 0 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 272 "" 0 1 0 0 0 0 1 0 0 0 0 0 0 0 0 0 }{CSTYLE "Grey Emphasis" -1 273 "Times" 1 12 96 52 84 1 0 1 0 0 0 0 0 0 0 0 }{PSTYLE "Normal" -1 0 1 {CSTYLE "" -1 -1 "Times" 1 12 0 0 0 1 2 2 2 2 2 2 1 1 1 1 }1 1 0 0 0 0 1 0 1 0 2 2 0 1 }{PSTYLE "Heading 1" -1 3 1 {CSTYLE "" -1 -1 "Times" 1 18 0 0 128 1 2 1 2 2 2 2 1 1 1 1 }1 1 0 0 8 4 3 0 3 0 2 2 0 1 }{PSTYLE "Heading 2 " -1 4 1 {CSTYLE "" -1 -1 "Times" 1 14 128 0 0 1 2 1 2 2 2 2 1 1 1 1 } 1 1 0 0 8 2 1 0 1 0 2 2 0 1 }{PSTYLE "Maple Output" -1 11 1 {CSTYLE " " -1 -1 "Times" 1 12 0 0 0 1 2 2 2 2 2 2 1 1 1 1 }3 3 0 0 0 0 1 0 1 0 2 2 0 1 }{PSTYLE "Maple Output" -1 12 1 {CSTYLE "" -1 -1 "Times" 1 12 0 0 0 1 2 2 2 2 2 2 1 1 1 1 }1 3 0 0 0 0 1 0 1 0 2 2 0 1 }{PSTYLE "Map le Plot" -1 13 1 {CSTYLE "" -1 -1 "Times" 1 12 0 0 0 1 2 2 2 2 2 2 1 1 1 1 }3 1 0 0 0 0 1 0 1 0 2 2 0 1 }{PSTYLE "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 }{PSTYLE "Norma l" -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 }} {SECT 0 {PARA 3 "" 0 "" {TEXT 259 0 "" }{TEXT -1 37 "Approximating Fun ctions using Moments" }}{PARA 0 "" 0 "" {TEXT -1 37 "by Peter Stone, N anaimo, B.C., Canada" }}{PARA 0 "" 0 "" {TEXT -1 20 "Version: 25.3.20 07\n" }}{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 approximation procedures" }} {PARA 0 "" 0 "" {TEXT -1 17 "The Maple m-file " }{TEXT 273 10 "fcnappr x.m" }{TEXT -1 37 " contains the code for the procedure " }{TEXT 0 10 "momentpoly" }{TEXT -1 25 " used in this worksheet. " }}{PARA 0 "" 0 " " {TEXT -1 123 "It can be read into a Maple session by a command simil ar to the one that follows, where the file path gives its location. \+ " }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 36 "read \"K:\\\\Maple/procd rs/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 77 "Introduction to the moment scheme for constructing appr oximating polynomials " }}{PARA 0 "" 0 "" {TEXT -1 16 "Acknowledgement :" }}{PARA 0 "" 0 "" {TEXT -1 112 "The ideas in this section are adapt ed from a Maple worksheet written by Tim Howard of Columbus State Univ ersity." }}{PARA 0 "" 0 "" {TEXT 260 0 "" }}{PARA 0 "" 0 "" {TEXT -1 60 "We consider a scheme for determining a quadratic polynomial " } {XPPEDIT 18 0 "p(x) = a*x^2+b*x+c;" "6#/-%\"pG6#%\"xG,(*&%\"aG\"\"\"*$ F'\"\"#F+F+*&%\"bGF+F'F+F+%\"cGF+" }{TEXT -1 22 " using the so-called " }{TEXT 267 7 "moments" }{TEXT -1 135 " of a function. The moment of a function is needed when determining the center of mass of an object or the centroid of a planar region." }}{PARA 0 "" 0 "" {TEXT -1 39 "I f we have a planar region bounded by " }{XPPEDIT 18 0 "y = f(x)" "6#/ %\"yG-%\"fG6#%\"xG" }{TEXT -1 6 ", the " }{TEXT 270 1 "x" }{TEXT -1 7 " axis, " }{XPPEDIT 18 0 "x = a" "6#/%\"xG%\"aG" }{TEXT -1 7 ", and \+ " }{XPPEDIT 18 0 "x = b" "6#/%\"xG%\"bG" }{TEXT -1 49 ", then the cent roid of the region has coordinates" }{XPPEDIT 18 0 "``(x[c],y[c]);" "6 #-%!G6$&%\"xG6#%\"cG&%\"yG6#F)" }{TEXT -1 7 ", where" }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "x[c] = int(x*f(x),x = a .. b)/int( f(x),x = a .. b);" "6#/&%\"xG6#%\"cG*&-%$intG6$*&F%\"\"\"-%\"fG6#F%F-/ F%;%\"aG%\"bGF--F*6$-F/6#F%/F%;F3F4!\"\"" }{TEXT -1 7 " and " } {XPPEDIT 18 0 "y[c] = int(f(x)^2/2,x = a .. b)/int(f(x),x = a .. b);" "6#/&%\"yG6#%\"cG*&-%$intG6$*&-%\"fG6#%\"xG\"\"#F1!\"\"/F0;%\"aG%\"bG \"\"\"-F*6$-F.6#F0/F0;F5F6F2" }{TEXT -1 2 " ." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 34 "The integral in the numer ator of " }{XPPEDIT 18 0 "x[c]" "6#&%\"xG6#%\"cG" }{TEXT -1 35 " is k nown as the \"first moment of " }{XPPEDIT 18 0 "f(x)" "6#-%\"fG6#%\"x G" }{TEXT -1 19 " over the interval " }{XPPEDIT 18 0 "[a,b]" "6#7$%\"a G%\"bG" }{TEXT -1 3 "\". " }}{PARA 0 "" 0 "" {TEXT -1 41 "This idea ca n be extended to define the \"" }{TEXT 271 1 "n" }{TEXT -1 15 "-th mom ent of " }{XPPEDIT 18 0 "f(x) " "6#-%\"fG6#%\"xG" }{TEXT -1 19 " over the interval " }{XPPEDIT 18 0 "[a,b]" "6#7$%\"aG%\"bG" }{TEXT -1 2 " \":" }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "int( x^n*f(x), x=a..b)" "6#-%$intG6$*&)%\"xG%\"nG\"\"\"-%\"fG6#F(F*/F(;%\"aG%\"bG" } {TEXT -1 2 " ." }}{PARA 0 "" 0 "" {TEXT -1 234 "It turns out that, for certain kinds of functions, a function can be completely determined i f we know only its moments. If we don't know all of the moments, we s ometimes can still obtain a reasonable approximation using some of the m." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 208 "Si nce the quadratic polynomial we seek contains three undetermined coeff icients, we need three equations to determine the polynomial. The mome nts scheme calls for us to match the 0th, 1st, and 2nd moments of " } {XPPEDIT 18 0 "p(x)" "6#-%\"pG6#%\"xG" }{TEXT -1 6 " over " }{XPPEDIT 18 0 "[a,b]" "6#7$%\"aG%\"bG" }{TEXT -1 40 " with the 0th, 1st, and 2n d moments of " }{XPPEDIT 18 0 "f(x)" "6#-%\"fG6#%\"xG" }{TEXT -1 16 " , respectively." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 3 "If " }{XPPEDIT 18 0 "f(x) = exp(x);" "6#/-%\"fG6#%\"xG-%$e xpG6#F'" }{TEXT -1 22 " , and the inteval is " }{XPPEDIT 18 0 "[-1,1] " "6#7$,$\"\"\"!\"\"F%" }{TEXT -1 42 ", the three determining equation s become: " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "int(a*x^2+b*x+c,x = -1 .. 1) = int(exp(x),x = -1 .. 1);" "6#/-%$intG6$,(*&%\"aG\"\"\"*$%\"xG\"\"#F*F**&%\"bGF*F,F*F *%\"cGF*/F,;,$F*!\"\"F*-F%6$-%$expG6#F,/F,;,$F*F4F*" }{TEXT -1 2 ", " }}{PARA 256 "" 0 "" {TEXT -1 0 "" }}{PARA 256 "" 0 "" {TEXT -1 1 " " } {XPPEDIT 18 0 "int(x*(a*x^2+b*x+c),x = -1 .. 1) = int(x*exp(x),x = -1 \+ .. 1);" "6#/-%$intG6$*&%\"xG\"\"\",(*&%\"aGF)*$F(\"\"#F)F)*&%\"bGF)F(F )F)%\"cGF)F)/F(;,$F)!\"\"F)-F%6$*&F(F)-%$expG6#F(F)/F(;,$F)F5F)" } {TEXT -1 2 ", " }}{PARA 258 "" 0 "" {TEXT -1 4 "and " }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "int(x^2*(a*x^2+b*x+c),x = -1 .. 1) = int(x^2*exp(x),x = -1 .. 1);" "6#/-%$intG6$*&%\"xG\"\"#,(*&%\"aG\" \"\"*$F(F)F-F-*&%\"bGF-F(F-F-%\"cGF-F-/F(;,$F-!\"\"F--F%6$*&F(F)-%$exp G6#F(F-/F(;,$F-F5F-" }{TEXT -1 2 ". " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 286 "a := 'a': b := 'b': c := \+ 'c':\nInt(a*x^2+b*x+c,x=-1..1)=Int(exp(x),x=-1..1);\neq1 := value(%); \nInt(x*(a*x^2+b*x+c),x=-1..1)=Int(x*exp(x),x=-1..1);\neq2 := value(%) ;\nInt(x^2*(a*x^2+b*x+c),x=-1..1)=Int(x^2*exp(x),x=-1..1);\neq3 := val ue(%);\nsols := solve(\{eq1,eq2,eq3\},\{a,b,c\});\nassign(sols);" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#/-%$IntG6$,(*&%\"aG\"\"\")%\"xG\"\"#F* F**&%\"bGF*F,F*F*%\"cGF*/F,;!\"\"F*-F%6$-%$expG6#F,F1" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%$eq1G/,&*(\"\"#\"\"\"\"\"$!\"\"%\"aGF)F)*&F(F)% \"cGF)F),&-%$expG6#F)F)-F16#F+F+" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/- %$IntG6$*&%\"xG\"\"\",(*&%\"aGF))F(\"\"#F)F)*&%\"bGF)F(F)F)%\"cGF)F)/F (;!\"\"F)-F%6$*&F(F)-%$expG6#F(F)F2" }}{PARA 11 "" 1 "" {XPPMATH 20 "6 #>%$eq2G/,$*(\"\"#\"\"\"\"\"$!\"\"%\"bGF)F),$*&F(F)-%$expG6#F+F)F)" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#/-%$IntG6$*&)%\"xG\"\"#\"\"\",(*&%\"aG F+F(F+F+*&%\"bGF+F)F+F+%\"cGF+F+/F);!\"\"F+-F%6$*&F(F+-%$expG6#F)F+F2 " }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%$eq3G/,&*(\"\"#\"\"\"\"\"&!\"\"% \"aGF)F)*(F(F)\"\"$F+%\"cGF)F),&-%$expG6#F)F)*&F*F)-F26#F+F)F+" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#>%%solsG<%/%\"cG,&*&#\"\"$\"\"%\"\"\"- %$expG6#F-F-!\"\"*&#\"#LF,F--F/6#F1F-F-/%\"bG,$*&F+F-F5F-F-/%\"aG,&*&# \"#:F,F-F.F-F-*&#\"$0\"F,F-F5F-F1" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }} {PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 81 "p := unapply(evalf(a*x^2+b*x+c),x);\nplot([exp(x),p(x)],x=-1..1,co lor=[red,blue]);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"pGf*6#%\"xG6\" 6$%)operatorG%&arrowGF(,(*&$\"*G:sO&!\"*\"\"\")9$\"\"#F1F1*&$\"+C$QO5 \"F0F1F3F1F1$\"*>SH'**F0F1F(F(F(" }}{PARA 13 "" 1 "" {GLPLOT2D 379 304 304 {PLOTDATA 2 "6&-%'CURVESG6$7S7$$!\"\"\"\"!$\"3MBWr6WzyO!#=7$$! 3ommm;p0k&*F-$\"3.W^QtfrUQF-7$$!3wKL$3f&F-7$$!3hmmm\" >s%HaF-$\"3R/uvRHL5eF-7$$!3Q+++]$*4)*\\F-$\"3Eh`=4&fk1'F-7$$!39+++]_& 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Ur+@JDF-7$$\"3ElmmmZvOLF1$\"3GTqs^o#>#GF-7$$\"3i******\\2goPF1$\"3kIx> w#*ztIF-7$$\"3UKL$eR<*fTF1$\"3J.SiF!H/C$F-7$$\"3m******\\)Hxe%F1$\"35' *3)*3wrWLF-7$$\"37KLeR*)**)y%F1$\"3gS;`$eABO$F-7$$\"3ckm;H!o-*\\F1$\"3 U0oZVd,eLF-7$$\"3nJL$3A_1?&F1$\"3Asv5C!['GLF-7$$\"3y)***\\7k.6aF1$\"3M MpoIeLsKF-7$$\"3#emmmT9C#eF1$\"3uZMp`*4$yIF-7$$\"33****\\i!*3`iF1$\"3X !GqGn()\\u#F-7$$\"3%QLLL$*zym'F1$\"3![AUN!ph%G#F-7$$\"3wKLL3N1#4(F1$\" 3]mLE`7F-7$$\"3a****\\P$[/a)F1 $!3k/:sYs\")z=F-7$$\"3fKLLLbdQ()F1$!3#*z!*\\J/XcDF-7$$\"3amm\"zW?)\\*) F1$!3cm3X]K7NLF-7$$\"3[++]i`1h\"*F1$!3k'RuxK2^<%F-7$$\"3Y++++PDj$*F1$! 3\\?$**R9c'Q]F-7$$\"3W++]P?Wl&*F1$!3K)Gz;`%fifF-7$$\"3A++v=5s#y*F1$!3g Y'=.EX_-(F-7$$\"\"\"F*$!3I0X!fu&zi\")F--%+AXESLABELSG6$Q\"x6\"Q!Fg]l-% 'COLOURG6&%$RGBG$F*F*F]^l$\"*++++\"!\")-%%VIEWG6$;F(F_]l%(DEFAULTG" 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 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 25 "The general moment sche me" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}{PARA 0 "" 0 "" {TEXT -1 70 "The general moment scheme for constructing a polynomial a pproximation " }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "p(x) = c[0]+c[1]*x+c[2]*x^2+` . . . `+a[n]*x^n;" "6#/-%\"pG6#%\"xG,,&%\"cG 6#\"\"!\"\"\"*&&F*6#F-F-F'F-F-*&&F*6#\"\"#F-*$F'F4F-F-%(~.~.~.~GF-*&&% \"aG6#%\"nGF-)F'F;F-F-" }{TEXT -1 1 " " }}{PARA 0 "" 0 "" {TEXT -1 10 "of degree " }{TEXT 272 1 "n" }{TEXT -1 16 " for a function " } {XPPEDIT 18 0 "f(x)" "6#-%\"fG6#%\"xG" }{TEXT -1 24 " defined on an in terval " }{XPPEDIT 18 0 "[a,b]" "6#7$%\"aG%\"bG" }{TEXT -1 31 " is to \+ construct a sequence of " }{XPPEDIT 18 0 "n+1" "6#,&%\"nG\"\"\"F%F%" } {TEXT -1 39 " linear equations for the coefficients " }{XPPEDIT 18 0 " c[0],c[1],c[2],` . . . `,c[n];" "6'&%\"cG6#\"\"!&F$6#\"\"\"&F$6#\"\"#% (~.~.~.~G&F$6#%\"nG" }{TEXT -1 28 " by equating the moments of " } {XPPEDIT 18 0 "p(x)" "6#-%\"pG6#%\"xG" }{TEXT -1 5 " and " }{XPPEDIT 18 0 "f(x)" "6#-%\"fG6#%\"xG" }{TEXT -1 7 ", thus:" }}{PARA 256 "" 0 " " {TEXT -1 1 " " }{XPPEDIT 18 0 "int(x^k*p(x),x = a .. b) = int(x^k*f( x),x = a .. b);" "6#/-%$intG6$*&)%\"xG%\"kG\"\"\"-%\"pG6#F)F+/F);%\"aG %\"bG-F%6$*&)F)F*F+-%\"fG6#F)F+/F);F1F2" }{TEXT -1 3 ", " }{XPPEDIT 18 0 "k = 0, 1,` . . . `, n" "6&/%\"kG\"\"!\"\"\"%(~.~.~.~G%\"nG" } {TEXT -1 1 "." }}{PARA 0 "" 0 "" {TEXT -1 98 "The following procedure \+ constructs the moment polynomial of specified degree for a given funct ion." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 38 "N ote that the moments of the function " }{XPPEDIT 18 0 "f(x)" "6#-%\"fG 6#%\"xG" }{TEXT -1 54 " are computed using numerical integration by ap plying " }{TEXT 0 9 "evalf/int" }{TEXT -1 1 "." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 76 "A more user friendly vers ion of this procedure is given in the next section." }}{PARA 0 "" 0 " " {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 900 "momentpol := proc(f::procedure,a::realcons,b::realcons,n::posint)\n local c,m ,A,ba,i,j,k,sys,eqn,saveDigits,pol,x;\n\n c := array(0..n);\n m := array(1..n+1);\n ba := array(1..2*n+1);\n \n # Calculate all the \+ moments of the function.\n for i from 0 to n do\n m[i+1] := eva lf(Int(x^i*f(x),x=a..b));\n end do;\n\n # Construct symbolic expre ssions for the moments of the\n # the polynomial involving the coeff icients as unknowns.\n for i from 1 to 2*n+1 do ba[i] := evalf(b^i-a ^i) end do;\n sys := [seq(add(c[k]/(j+k+1)*ba[j+k+1],k=0..n),j=0..n) ];\n A := array(1..n+1,1..n+1,sparse);\n for i to n+1 do\n eq n := op(i,sys);\n eqn := collect(eqn,c,'distributed');\n for j to n+1 do A[i,j] := coeff(eqn,c[j-1]) end do;\n end do;\n\n # S olve the system of equations for the coefficients.\n m := linalg[lin solve](A,m);\n pol := add(m[k]*x^(k-1),k=1..n+1);\n evalf(pol);\ne nd proc:" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 22 "momentpol(exp,-1,1,3);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#,*$\"+\">SH'**!#5\"\"\"*&$\"+,([&z**F&F'%\"xGF'F'*&$\"+Y_@n`F&F' )F+\"\"#F'F'*&$\"+#)3Rh " 0 "" {MPLTEXT 1 0 1 ";" }}}}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 46 "A procedure for computing moment polynomials: \+ " }{TEXT 0 10 "momentpoly" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 17 "momentpoly: usage" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 " " 0 "" {TEXT -1 0 "" }{TEXT 261 18 "Calling Sequence:\n" }}{PARA 0 "" 0 "" {TEXT 262 2 " " }{TEXT -1 28 " momentpoly( fx, rng deg ) " } {TEXT 263 1 "\n" }{TEXT -1 0 "" }}{PARA 257 "" 0 "" {TEXT -1 11 "Param eters:" }}{PARA 0 "" 0 "" {TEXT -1 4 " " }}{PARA 0 "" 0 "" {TEXT 23 10 " fx - " }{TEXT -1 55 " an expression involving a singl e variable, say x," }}{PARA 0 "" 0 "" {TEXT -1 85 " \+ where f(x) evaluates to a real floating point number." }} {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }{TEXT 23 12 " rng - " }{TEXT 264 62 "the range x=a..b for the definite \+ integral to be approximated." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }{TEXT 23 12 " deg - " }{TEXT 266 53 "the degree of the resulting polynomial approximation." }}{PARA 0 "" 0 "" {TEXT -1 10 " " }}{PARA 257 "" 0 "" {TEXT -1 12 "Description: " }}{PARA 0 "" 0 "" {TEXT -1 14 "The procedure " }{TEXT 0 10 "momentpo ly" }{TEXT -1 118 " attempts to find the polynomial px of specified de gree which approximates fx on the interval [a,b] to the extent that" } }{PARA 256 "" 0 "" {TEXT -1 2 " " }{XPPEDIT 18 0 "Int(x^k*px,x = a .. b) = Int(x^k*fx,x = a .. b);" "6#/-%$IntG6$*&)%\"xG%\"kG\"\"\"%#pxGF+ /F);%\"aG%\"bG-F%6$*&)F)F*F+%#fxGF+/F);F/F0" }{TEXT -1 26 ", k = 0, 1 , . . . , deg. " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT 265 8 "Options:" }{TEXT -1 1 "\n" }}{PARA 0 "" 0 "" {TEXT -1 15 "info=true/false" }}{PARA 0 "" 0 "" {TEXT -1 40 "With the option \"inf o=true\" the moments " }{XPPEDIT 18 0 "Int(x^k*fx,x=a..b)" "6#-%$IntG6 $*&)%\"xG%\"kG\"\"\"%#fxGF*/F(;%\"aG%\"bG" }{TEXT -1 59 ", k = 0, 1, . . . , deg are printed as they are computed.\n" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT 267 16 "How to activate:" } {TEXT 256 1 "\n" }{TEXT -1 154 "To make the procedure active open the \+ subsection, place the cursor anywhere after the prompt [ > and press \+ [Enter].\nYou can then close up the subsection." }}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 26 "momentpoly: implementation" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 3060 "momentpoly := pr oc(fx::algebraic,rng::equation,deg::posint)\n local a,b,fn,np,c,m,A, ba,i,j,k,sys,eq,saveDigits,pol,\n x,Options,rs,outpt,aa,bb,n,p rntflg;\n \n if nargs<3 then\n error \"at least 3 arguments ar e required; the basic syntax is: 'momentpoly(f(x),x=a..b,n)'.\"\n en d if;\n\n if not type(rng,name=realcons..realcons) then\n error \"the 2nd argument, %1, is invalid .. it should have the form 'name=a ..b', to provide the interval for the approximating polynomial\",rng; \n end if;\n \n x := op(1,rng);\n if not type(indets(fx,name) \+ minus \{x\},set(realcons)) then\n error \"the 1st argument, %1, i s invalid .. it should depend only on the variable %2\",fx,x;\n end \+ if;\n rs := op(2,rng);\n\n # Get the options \"output\" and \"info \".\n # Set the default values to start with.\n outpt := 'poly';\n prntflg := false;\n if nargs>3 then\n Options:=[args[4..narg s]];\n if not type(Options,list(equation)) then\n error \+ \"each optional argument must be an equation\"\n end if;\n i f hasoption(Options,'output','outpt','Options') then \n if not member(outpt,\{'poly','coeffs'\}) then\n error \"\\\"outpu t\\\" must be 'poly' or 'coeffs'\"\n end if;\n end if;\n \+ if hasoption(Options,'info','prntflg','Options') then\n i f prntflg<>true then prntflg := false end if;\n end if;\n if nops(Options)>0 then\n error \"%1 is not a valid option for % 2\",op(1,Options),procname;\n end if;\n end if;\n n := deg;\n \n # Increase precision for the computation by about 30%.\n saveDi gits := Digits;\n Digits := Digits+max(5,trunc(Digits*0.3));\n fn \+ := unapply(fx,x);\n\n aa := op(1,rs);\n bb := op(2,rs);\n a := e valf(aa);\n b := evalf(bb);\n if a>=b then\n error \"invalid \+ range of values for %1\",x;\n end if;\n\n np := n + 1;\n c := ar ray(0..n);\n m := array(1..np);\n ba := array(1..np+n);\n \n # Calculate all the moments of the function.\n if prntflg then\n \+ print(`moments of function`);\n end if;\n for i from 0 to n do\n \+ m[i+1] := traperror(evalf(Int(x^i*fn(x),x=a..b)));\n if m[i+ 1]=lasterror then\n error \"failed to calculate %-1 moment\",i ;\n end if;\n if prntflg then print(m[i+1]) end if;\n end \+ do;\n\n # Construct symbolic expressions for the moments of the\n \+ # the polynomial involving the coefficients as unknowns.\n for i fro m 1 to 2*n+1 do ba[i] := b^i-a^i end do;\n sys := [seq(add(c[k]/(j+k +1)*ba[j+k+1],k=0..n),j=0..n)];\n A := array(1..np,1..np,sparse);\n \+ for i to np do\n eq := op(i,sys);\n eq := collect(eq,c,'di stributed');\n for j to np do A[i,j] := coeff(eq,c[j-1]) end do; \n end do;\n\n # Solve the system of equations for the coefficient s.\n m := traperror(linalg[linsolve](A,m));\n if m=lasterror then \n error \"failed to solve for the coefficients\"\n end if;\n\n if outpt='poly' then\n pol := add(m[k]*x^(k-1),k=1..np);\n \+ Digits := saveDigits;\n return evalf(pol);\n else # outpt=coe ffs\n Digits := saveDigits;\n return convert(evalf(m),list); \n end if; \nend proc:" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}}{PARA 0 "" 0 "" {TEXT -1 0 " " }}{PARA 0 "" 0 "" {TEXT -1 40 "Examples are given in the next sectio n. " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 3 "; " }}}}{SECT 1 {PARA 4 "" 0 "" {TEXT 0 10 "momentpol y" }{TEXT -1 10 ": examples" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 9 "Example 1" }}{PARA 0 "" 0 "" {TEXT -1 56 "We construct a polynomial approximation of degree 8 \+ for " }{XPPEDIT 18 0 "exp(x)" "6#-%$expG6#%\"xG" }{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 96 "momentpoly(exp(x),x=-1..1,8,info=true):\np := unapply (%,x);\nplot(p(x)-exp(x),x=-1..1,color=blue);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%4moments~of~functionG" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#$\"0g(G(Q-/N#!#9" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#$\"0&)GM#))edt! #:" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#$\"0M=gAY))y)!#:" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#$\"0()\\#=S2&\\%!#:" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#$\"0cw)*zFP_&!#:" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#$\"01ApptHC$ !#:" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#$\"0kV8p\"=YS!#:" }}{PARA 11 " " 1 "" {XPPMATH 20 "6#$\"0S**o&3MQD!#:" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#$\"0(3o$!#:" }}{PARA 12 "" 1 "" {XPPMATH 20 "6#>%\"pGf*6#% \"xG6\"6$%)operatorG%&arrowGF(,4$\"+++++5!\"*\"\"\"*&$\"+r#*******!#5F 09$F0F0*&$\"+*y*****\\F4F0)F5\"\"#F0F0*&$\"+]tnm;F4F0)F5\"\"$F0F0*&$\" +2\\omT!#6F0)F5\"\"%F0F0*&$\"+RgokmgRi!#E7$$!3#om;HdNvs*F1$\"3E/,qO],n CFM7$$!3u****\\(oUIn*F1$!3HLokb1eBw!#F7$$!3xLL3-)\\&='*F1$!3'R0dA7UN\\ $FM7$$!3ommm;p0k&*F1$!3-\">Mvsi!pdFM7$$!3E++vV5Su$*F1$!3w2E'fLK(o5F-7$ $!3wKL$37+rp$et(FM7$$!3&emmmwnMa)F1$!3!y2z?PH?(RFM7$$! 3:mmm\"4m(G$)F1$!3@())fgBg^r\"FX7$$!3)****\\i&[3:\")F1$\"3a*3Hr^ac<$FM 7$$!3\"QLL3i.9!zF1$\"3:$)H7qox-eFM7$$!3'3++Dw$H.xF1$\"3A>8)=u%)*puFM7$ $!3\"ommT!R=0vF1$\"3'HLlhuTwO)FM7$$!37MLL$3,RX(F1$\"3e^#***H$[vZ)FM7$$ !3K++]i#=ES(F1$\"3!\\Y))Gj\")*Q&)FM7$$!3ammmTaL^tF1$\"3wIYZ!=HKb)FM7$$ !3tKL$3i_+I(F1$\"3G*\\\")R?Q<_)FM7$$!3Cmm;zp[(>(F1$\"3&e(RJ@)H!G$)FM7$ 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$!3w'z#yXD;%z#F-7$$\"\"\"F*$!3o_'[bca/<$F--%+AXESLABELSG6$Q\"x6\"Q!Feb m-%'COLOURG6&%$RGBG$F*F*F[cm$\"*++++\"!\")-%%VIEWG6$;F(F]bm%(DEFAULTG " 1 2 0 1 10 0 2 6 1 4 2 1.000000 45.000000 45.000000 0 0 "Curve 1" }} }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 9 "Example 2" }}{PARA 0 " " 0 "" {TEXT -1 56 "We construct a polynomial approximation of degree \+ 7 for " }{XPPEDIT 18 0 "ln(x);" "6#-%#lnG6#%\"xG" }{TEXT -1 17 " on th e interval " }{XPPEDIT 18 0 "[1,2]" "6#7$\"\"\"\"\"#" }{TEXT -1 1 "." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 82 "momentpoly(ln(x),x=1..2,7):\nq := unapply(%,x);\nplot(q(x)-ln(x) ,x=1..2,color=blue);" }}{PARA 12 "" 1 "" {XPPMATH 20 "6#>%\"qGf*6#%\"x G6\"6$%)operatorG%&arrowGF(,2$\"+yB([B#!\"*!\"\"*&$\"+6i3t[F/\"\"\"9$F 4F4*&$\"+LxHY]F/F4)F5\"\"#F4F0*&$\"+[-**QQF/F4)F5\"\"$F4F4*&$\"+mVNb>F /F4)F5\"\"%F4F0*&$\"+ru@Bj!#5F4)F5\"\"&F4F4*&$\"+u15u6FHF4)F5\"\"'F4F0 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