{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 "Dark Red Emphasis" -1 259 "Times" 1 12 128 0 0 1 0 1 0 0 0 0 0 0 0 0 }{CSTYLE "Red Emphasis" -1 260 "Times" 1 12 255 0 0 1 0 1 0 0 0 0 0 0 0 0 }{CSTYLE "Purple Emphasis" -1 261 " Times" 1 12 103 0 230 1 0 1 0 0 0 0 0 0 0 0 }{CSTYLE "Grey Emphasis" -1 262 "Times" 1 12 96 52 84 1 0 1 0 0 0 0 0 0 0 0 }{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 1 0 1 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 "Warning" -1 7 1 {CSTYLE "" -1 -1 "Courier" 1 10 0 0 255 1 2 2 2 2 2 1 1 1 3 1 } 1 1 0 0 0 0 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 }} {SECT 0 {PARA 3 "" 0 "" {TEXT -1 59 "Numerical integration of function s of varying \"smoothness\"." }}{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 46 "load numerical integrati on procedures and data" }}{PARA 0 "" 0 "" {TEXT -1 18 "The Maple m-fil es " }{TEXT 262 6 "intg.m" }{TEXT -1 5 " and " }{TEXT 262 8 "gkdata.m " }{TEXT -1 100 " contain the code and data for the special numerical \+ integration procedures used in this worksheet. 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20 "6#%.adaptive~modeG" }}{PARA 7 "" 1 "" {TEXT -1 39 "Warning, reached max subdivision depth\n" }}{PARA 11 "" 1 "" {XPPMATH 20 "6$%Dnumber~of~function~evaluations~-->~G\"%0D" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#$\"+&\\*)G3%!#5" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 109 "Of course, it is much more eff icient to split the interval of integration into two subintervals at t he point " }{XPPEDIT 18 0 "x=Pi/4" "6#/%\"xG*&%#PiG\"\"\"\"\"%!\"\"" } {TEXT -1 62 ", which is the point at which the derivative is discontin uous." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 64 "quad(Int(f(x),x=0..1),method=Gauss_Kronrod,split=[Pi/ 4],info=1);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%SSplitting~interval~of ~integration~using~the~pointsG" }}{PARA 11 "" 1 "" {XPPMATH 20 "6%$\" \"!F$$\"+N;)R&y!#5$\"\"\"F$" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%!G" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#%Madaptive~10-21~node~Gauss-Kronrod~qu adratureG" }}{PARA 11 "" 1 "" {XPPMATH 20 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"quad(Int(g(x),x=0..1) ,method=Clenshaw_Curtis,info=2);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%E iterative~Clenshaw-Curtis~quadratureG" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%ioinserting~2~new~evaluation~points~between~each~previous~pair~at~ each~iterationG" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%coand~calculating~ the~Chebyshev~coefficients~using~a~fast~cosine~transformG" }}{PARA 11 "" 1 "" {XPPMATH 20 "6$%Fstep~1:~approx~value~of~integral~-->~G$\".s%G M!3<$!#8" }}{PARA 11 "" 1 "" {XPPMATH 20 "6$%Fstep~2:~approx~value~of~ integral~-->~G$\".y;.qd;$!#8" }}{PARA 11 "" 1 "" {XPPMATH 20 "6$%Fstep ~3:~approx~value~of~integral~-->~G$\".'\\%Q?e;$!#8" }}{PARA 7 "" 1 "" {TEXT -1 105 "Warning, integrand may have a singularity near the inter val of integration - using simple error estimate\n" }}{PARA 11 "" 1 " " {XPPMATH 20 "6$%Fstep~4:~approx~value~of~integral~-->~G$\".!eZ[#e;$! #8" }}{PARA 11 "" 1 "" {XPPMATH 20 "6$%Fstep~5:~approx~value~of~integr al~-->~G$\"./(*)[#e;$!#8" }}{PARA 11 "" 1 "" {XPPMATH 20 "6$%Fstep~6:~ approx~value~of~integral~-->~G$\".]S*[#e;$!#8" }}{PARA 7 "" 1 "" {TEXT -1 46 "Warning, reached maximum number of iterations\n" }}{PARA 11 "" 1 "" {XPPMATH 20 "6$%Dnumber~of~function~evaluations~-->~G\"%f9 " }}{PARA 11 "" 1 "" {XPPMATH 20 "6#$\"+%*[#e;$!#5" }}}{PARA 0 "" 0 " " {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 98 "It is much more efficie nt to split the interval of integration into two subintervals at the p oint " }{XPPEDIT 18 0 "x = 1/sqrt(2);" "6#/%\"xG*&\"\"\"F&-%%sqrtG6#\" \"#!\"\"" }{TEXT -1 69 ", which is the point at which the second deriv ative is discontinuous." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 77 "quad(In t(g(x),x=0..1),\n method=Clenshaw_Curtis,split=[1/sqrt(2)],info=1) ;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%SSplitting~interval~of~integrati on~using~the~pointsG" }}{PARA 11 "" 1 "" {XPPMATH 20 "6%$\"\"!F$$\"+7y 1rq!#5$\"\"\"F$" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%!G" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%Eiterative~Clenshaw-Curtis~quadratureG" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%ioinserting~2~new~evaluation~points~between ~each~previous~pair~at~each~iterationG" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%coand~calculating~the~Chebyshev~coefficients~using~a~fast~cosin e~transformG" }}{PARA 11 "" 1 "" {XPPMATH 20 "6$%Dnumber~of~function~e valuations~-->~G\"#>" }}{PARA 11 "" 1 "" {XPPMATH 20 "6&%.integral~ove rG;$\"\"!F&$\"+7y1rq!#5%$-->G$\"+-8^y6F)" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%Eiterative~Clenshaw-Curtis~quadratureG" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%ioinserting~2~new~evaluation~points~between~each~previ ous~pair~at~each~iterationG" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%coand~ calculating~the~Chebyshev~coefficients~using~a~fast~cosine~transformG " }}{PARA 11 "" 1 "" {XPPMATH 20 "6$%Dnumber~of~function~evaluations~- ->~G\"#>" }}{PARA 11 "" 1 "" {XPPMATH 20 "6&%.integral~overG;$\"+7y1rq !#5$\"\"\"\"\"!%$-->G$\"+#f8t)>F'" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#$ \"+%*[#e;$!#5" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 59 "Compl ex singularities close to the interval of integration " }}{PARA 0 "" 0 "" {TEXT -1 2 " " }}{PARA 0 "" 0 "" {TEXT -1 2 " " }{XPPEDIT 18 0 "Int(2/(2+sin(10*Pi*x)),x = 0 .. 1);" "6#-%$IntG6$*&\"\"#\"\"\",&F'F(- %$sinG6#*(\"#5F(%#PiGF(%\"xGF(F(!\"\"/F0;\"\"!F(" }{TEXT -1 1 " " }} {PARA 0 "" 0 "" {TEXT -1 14 "The function " }{XPPEDIT 18 0 "2/(2+sin( 10*Pi*x))" "6#*&\"\"#\"\"\",&F$F%-%$sinG6#*(\"#5F%%#PiGF%%\"xGF%F%!\" \"" }{TEXT -1 112 " has a number singularities in the complex plain j ust above and below the segment from 0 to 1 on the real axis." }} {PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 192 "sols := NULL:\nfor k from 0 to 5 do\n sols := sols,fsolve(2+sin (10*Pi*x),x=-0.05+k*0.2+0.04*I,complex);\n sols := sols,fsolve(2+sin (10*Pi*x),x=-0.05+k*0.2-0.04*I,complex);\nod:\nsols := [sols];" }} {PARA 12 "" 1 "" {XPPMATH 20 "6#>%%solsG7.^$$!+++++]!#6$\"+$=2?>%F)^$F '$!+$=2?>%F)^$$\"+++++:!#5F*^$F0F-^$$\"+++++NF2F*^$F5F-^$$\"+++++bF2F* ^$F9F-^$$\"+++++vF2F*^$F=F-^$$\"+++++&*F2F*^$FAF-" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 34 "The following addition to Maple's " }{TEXT 0 7 "convert" }{TEXT -1 125 " procedure can be used \+ to convert a list of complex numbers x+y*I to the correspond points [x ,y], which can then be plotted.\n" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 258 "`convert/point` := proc(a::\{complexcons,list(comple xcons)\})\n if type(a,complexcons) then [Re(a),Im(a)] else\n map( _z->[Re(_z),Im(_z)],a) fi\nend:\npts := convert(sols,point):\nplot(pts ,style=point,symbol=circle,color=red,\n view=[-0.1..1,-0.5..0. 5]);" }}{PARA 13 "" 1 "" {GLPLOT2D 281 187 187 {PLOTDATA 2 "6(-%'CURVE SG6#7.7$$!3G+++++++]!#>$\"3E+++$=2?>%F*7$F($!3E+++$=2?>%F*7$$\"3%***** *********\\\"!#=F+7$F1F.7$$\"3w*************\\$F3F+7$F6F.7$$\"3U++++++ 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