{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 "Hyperlink" -1 17 "" 0 1 0 128 128 1 2 0 1 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 "Purple Emphasis" -1 259 "Times" 1 12 102 0 230 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 "Dark Red Emphasis" -1 261 "Tim es" 1 12 128 0 0 1 0 1 0 0 0 0 0 0 0 0 }{CSTYLE "" 260 262 "" 1 18 0 0 0 0 0 0 0 0 0 0 0 0 0 0 }{CSTYLE "" 260 263 "" 1 18 0 0 0 0 0 0 0 0 0 0 0 0 0 0 }{CSTYLE "" 260 264 "" 1 18 0 0 0 0 0 0 0 0 0 0 0 0 0 0 } {CSTYLE "" -1 265 "" 0 1 0 0 0 0 0 1 0 0 0 0 0 0 0 1 }{CSTYLE "" -1 266 "" 0 1 0 0 0 0 0 1 0 0 0 0 0 0 0 1 }{CSTYLE "" -1 267 "" 0 1 0 0 0 0 0 1 0 0 0 0 0 0 0 1 }{CSTYLE "" -1 268 "" 0 1 0 0 0 0 0 1 0 0 0 0 0 0 0 1 }{CSTYLE "" -1 269 "" 0 1 0 0 0 0 0 1 0 0 0 0 0 0 0 1 } {CSTYLE "" -1 270 "" 0 1 0 0 0 0 0 1 0 0 0 0 0 0 0 1 }{CSTYLE "" -1 271 "" 0 1 0 0 0 0 0 1 0 0 0 0 0 0 0 1 }{CSTYLE "" -1 272 "" 0 1 0 0 0 0 1 0 0 0 0 0 0 0 0 1 }{CSTYLE "" -1 273 "" 0 1 0 0 0 0 1 0 0 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 }{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 "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 "Bulle t 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 1 2 2 2 2 1 1 1 1 }1 1 0 0 0 0 1 0 1 0 2 2 0 1 }{PSTYLE "Normal" -1 257 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 " -1 258 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 -1 56 "Evaluating the cosine function fr om its Maclaurin series" }}{PARA 0 "" 0 "" {TEXT -1 37 "by Peter Stone , Nanaimo, B.C., Canada" }}{PARA 0 "" 0 "" {TEXT -1 19 "Version: 25.3 .2007" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 8 "restart;" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 " ;" }}}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 50 "Graphing partial sums of th e Maclaurin series for " }{XPPEDIT 18 0 "cos(x)" "6#-%$cosG6#%\"xG" } {TEXT -1 1 " " }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}{PARA 0 "" 0 "" {TEXT -1 25 "The Maclaurin series for " }{XPPEDIT 18 0 "cos( x);" "6#-%$cosG6#%\"xG" }{TEXT -1 5 " is " }{XPPEDIT 18 0 "Sum((-1)^n *x^(2*n)/(2*n)!,n = 0 .. infinity)=1-x^2/2!+x^4/4!-x^6/6!+x^8/8!+` . . . `" "6#/-%$SumG6$*(),$\"\"\"!\"\"%\"nGF*)%\"xG*&\"\"#F*F,F*F*-%*fact orialG6#*&F0F*F,F*F+/F,;\"\"!%)infinityG,.F*F**&F.F0-F26#F0F+F+*&F.\" \"%-F26#F>F+F**&F.\"\"'-F26#FBF+F+*&F.\"\")-F26#FFF+F*%(~.~.~.~GF*" } {TEXT -1 2 ". " }}{PARA 0 "" 0 "" {TEXT -1 55 "We can make a compariso n of finite sums approximating " }{XPPEDIT 18 0 "cos(x);" "6#-%$cosG6 #%\"xG" }{TEXT -1 24 " graphically as follows." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 27 "First we define a functio n " }{TEXT 0 2 "MC" }{TEXT -1 87 " with two arguments or input paramet ers. The degree of the polynomial approximation to " }{XPPEDIT 18 0 "c os(x);" "6#-%$cosG6#%\"xG" }{TEXT -1 4 " is " }{XPPEDIT 18 0 "2*n;" "6 #*&\"\"#\"\"\"%\"nGF%" }{TEXT -1 2 ".\n" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 106 "x := 'x': i := 'i': n := 'n':\nMC := (n,x) -> sum((- 1)^i*x^(2*i)/(2*i)!,i=0..n);\nMC(1,x);\nMC(2,x);\nMC(3,x);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%#MCGf*6$%\"nG%\"xG6\"6$%)operatorG%&arrowG F)-%$sumG6$*&*&)!\"\"%\"iG\"\"\")9%,$F4\"\"#F5F5-%*factorialG6#F8F3/F4 ;\"\"!9$F)F)F)" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#,&\"\"\"F$*&#F$\"\"# F$*$)%\"xGF'F$F$!\"\"" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#,(\"\"\"F$*&# F$\"\"#F$*$)%\"xGF'F$F$!\"\"*&#F$\"#CF$)F*\"\"%F$F$" }}{PARA 11 "" 1 " " {XPPMATH 20 "6#,*\"\"\"F$*&#F$\"\"#F$*$)%\"xGF'F$F$!\"\"*&#F$\"#CF$) F*\"\"%F$F$*&#F$\"$?(F$*$)F*\"\"'F$F$F+" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 36 "evalf(MC(3,Pi/4));\ne valf(sqrt(2)/2);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#$\"+Z@.rq!#5" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#$\"+5y1rq!#5" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 39 "The degree 2, 4, and 6 ap proximations " }{XPPEDIT 18 0 "y = 1-x^2/2;" "6#/%\"yG,&\"\"\"F&*&%\" xG\"\"#F)!\"\"F*" }{TEXT -1 3 " , " }{XPPEDIT 18 0 "y = 1-x^2/2!-x^4/4 !;" "6#/%\"yG,(\"\"\"F&*&%\"xG\"\"#-%*factorialG6#F)!\"\"F-*&F(\"\"%-F +6#F/F-F-" }{TEXT -1 6 " and " }{XPPEDIT 18 0 "y = 1-x^2/2!+x^4/4!-x^ 6/6!;" "6#/%\"yG,*\"\"\"F&*&%\"xG\"\"#-%*factorialG6#F)!\"\"F-*&F(\"\" %-F+6#F/F-F&*&F(\"\"'-F+6#F3F-F-" }{TEXT -1 28 " can be plotted as fo llows." }}{PARA 0 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "y = cos(x);" "6#/%\"yG-%$cosG6#%\"xG" }{TEXT -1 33 " is also plotted for comparison . " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 182 "MC := (n,x) -> sum((-1)^i*x ^(2*i)/(2*i)!,i = 0..n):\nplot([cos(x),MC(1,x),MC(2,x),MC(3,x)],x=-10. .10,-1.5..1.5,\n color=[black,red,green,blue],linestyle=[2,1$3] ,\nnumpoints=100);" }}{PARA 13 "" 1 "" {GLPLOT2D 486 260 260 {PLOTDATA 2 "6(-%'CURVESG6%7\\v7$$!#5\"\"!$!3QCXw!H:2R)!#=7$$!3)fllllL ')y*!#<$!3G/'z13;`M*F-7$$!3I$RRRHzmp*F1$!3(*eL&H4?Ej*F-7$$!3lIJJJ\\s/' *F1$!3@MqX_=`Q)*F-7$$!3wJKK#y=Ib*F1$!3X*y%*))3%)y\"**F-7$$!3(GLLLj78]* F1$!3Cm8#H^E2(**F-7$$!3)RVVV[1'\\%*F1$!3aGIU!)y\"p***F-7$$!33NNNN.!zR* F1$!3e`7q$=)Q'***F-7$$!3QXWWW9&eM*F1$!3+gxk!>p)o**F-7$$!3!RNNNb-QH*F1$ !3[/%)ef(\\V\"**F-7$$!3TiiiiOvT#*F1$!3B/n1bv(H$)*F-7$$!3#4<<V*F-7$$!39EDDD()\\#)*)F1$! 3&4Q4l(\\yP!*F-7$$!3'efff>#R!z)F1$!3KWB$3&*eV0)F-7$$!3%=@@@,x9f)F1$!32 y![79iVs'F-7$$!3iVWWW'edQ)F1$!3SF%zTfd12&F-7$$!3AIJJJ$HKG)F1$!35o?vr'o =;%F-7$$!3e====+q!=)F1$!3i4[j;uO4KF-7$$!3!orrr,O_2)F1$!3Q\"eGV+OX>#F-7 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\"Q!6\"-%%VIEWG6$;F(Fbbn;$Fifo!\"\"$\"#:F\\_r" 1 2 0 1 10 0 2 9 1 4 2 1.000000 45.000000 45.000000 0 0 "Curve 1" "Curve 2" "Curve 3" "Curve \+ 4" }}}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" } }{PARA 0 "" 0 "" {TEXT -1 216 "Successively higher degree Taylor polyn omials produce progressively better approximations over a fixed interv al centred at the origin.\nThe next picture shows the graphs of the de gree 12, 14 and 16 Taylor polynomials." }}{PARA 0 "" 0 "" {TEXT -1 0 " " }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 186 "MC := (n,x) -> sum((-1) ^i*x^(2*i)/(2*i)!,i = 0..n):\nplot([cos(x),MC(6,x),MC(7,x),MC(8,x)],x= -10..10,-1.5..1.5,\n color=[black,magenta,coral,cyan],linestyle =[2,1$3],\nnumpoints=100);" }}{PARA 13 "" 1 "" {GLPLOT2D 529 299 299 {PLOTDATA 2 "6(-%'CURVESG6%7\\v7$$!#5\"\"!$!3QCXw!H:2R)!#=7$$!3)fllllL ')y*!#<$!3G/'z13;`M*F-7$$!3I$RRRHzmp*F1$!3(*eL&H4?Ej*F-7$$!3lIJJJ\\s/' *F1$!3@MqX_=`Q)*F-7$$!3wJKK#y=Ib*F1$!3X*y%*))3%)y\"**F-7$$!3(GLLLj78]* 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3$[B2&o93H\\F17$Fg]n$\"3H)G_;Lcvh(F17$F\\^n$\"3Im7(*R9a+7Fffn7$Fa^n$\" 3\\/%=h'4*G!=Fffn7$Ff^n$\"3'QzKbI8W>#Fffn7$F[_n$\"3[u_0Wt]jEFffn7$Fe_n $\"3o$zRO$QSmKFffn7$F_`n$\"332X!>+)\\$*RFffn7$Fi`n$\"3QO'e3C5f'[Fffn7$ Fcan$\"3i8T%pL7@\"fFffn7$$\"3GJJJ\"3u*['*F1$\"3Il;g[QLYkFffn7$Fhan$\"3 %*f%>(eE0DqFffn7$$\"3E\"**)*)R=0U(*F1$\"3%>U!QpLp^wFffn7$F]bn$\"3O#oWn +E*H$)Fffn7$F\\ho$\"3=,.CBysn\"*Fffn7$Faho$\"3x?G:KXL35Fbcn7$Ffho$\"3k <%)o:uM36Fbcn7$FbbnFd^q-Febn6&FgbnF`ioF]ioF]ioFaio-%+AXESLABELSG6$Q\"x 6\"Q!6\"-%%VIEWG6$;F(Fbbn;$Fbcn!\"\"$\"#:Fbdr" 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" }}}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 38 "An adaptiv e graph plotting procedure: " }{TEXT 0 5 "graph" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 79 "The procedure in this sec tion is an alternative to the standard Maple function " }{TEXT 0 4 "pl ot" }{TEXT -1 195 ", for plotting the graph of a single function. It a llows control over how the plotting is performed, in a manner which is a bit different from the control one has over the standard Maple rout ine " }{TEXT 0 4 "plot" }{TEXT -1 1 "." }}{PARA 0 "" 0 "" {TEXT -1 0 " " }}{PARA 0 "" 0 "" {TEXT -1 120 "In particular, the precision of the \+ floating point calculations is not changed internally, whereas, the Ma ple procedure " }{TEXT 0 4 "plot" }{TEXT -1 137 " often uses hardware \+ floating point arithmetic whereby the effective precision is increased to the equivalent of about 15 decimal digits." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 12 "graph: usage" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }{TEXT 269 18 "Calling Sequenc e:\n" }}{PARA 0 "" 0 "" {TEXT 270 4 " " }{TEXT -1 17 "graph( f, xrn g )\n" }{TEXT 271 1 "\n" }{TEXT -1 26 " graph( f, xrng, yrng )" }} {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 256 "" 0 "" {TEXT 267 11 "Parame ters:" }}{PARA 0 "" 0 "" {TEXT -1 2 " " }}{PARA 0 "" 0 "" {TEXT -1 3 " " }{TEXT 23 4 "f - " }{TEXT -1 83 " an expression involving a s ingle variable, say x, or a function x -> f(x)\n\n " }{TEXT 23 7 "x rng - " }{TEXT -1 79 " horizontal plotting range in the form x=a.. b, when x is an expression in x" }}{PARA 0 "" 0 "" {TEXT -1 19 " \+ " }{TEXT 256 2 "OR" }{TEXT -1 48 " in the form a..b when f is afunction x -> f(x)" }}{PARA 0 "" 0 "" {TEXT -1 12 " \+ " }}{PARA 0 "" 0 "" {TEXT -1 4 " " }{TEXT 23 7 "yrng - " }{TEXT -1 92 " vertical range (optional), which can be given in the form c.. d, or in the form y=c..d.\n" }}{PARA 256 "" 0 "" {TEXT 266 12 "Descrip tion:" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 15 "" 0 "" {TEXT -1 14 "The procedure " }{TEXT 0 5 "graph" }{TEXT -1 22 " plots the graph of \+ a " }{TEXT 258 15 "single function" }{TEXT -1 26 " using an adaptive m ethod." }}{PARA 15 "" 0 "" {TEXT -1 88 "An even spacing is aimed for a long sections of the curve which are approximately linear." }}{PARA 15 "" 0 "" {TEXT -1 79 "More points are plotted along sections of the \+ curve where the graph is bending." }}{PARA 15 "" 0 "" {TEXT -1 107 "An y maximum or minimum points are located approximately by parabolic int erpolation and added to the graph.\n" }}{PARA 256 "" 0 "" {TEXT 268 8 "Options:" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 19 "adaptive=true/false" }}{PARA 0 "" 0 "" {TEXT -1 144 "Adaptive plot ting will, where necessary, sub-divide the plotting interval in an att empt to get a good graphical representation of the function. " }} {PARA 0 "" 0 "" {TEXT -1 111 "By default, this option is set to \"true \", but it can be turned off by setting the \"adaptive\" option to \"f alse\"." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 312 "numpoints=n\nFor non-adaptive plotting the interval for the plot \+ is subdivided into a fixed number of sub-intervals of equal width by m eans \"numpoints\" points. \nFor adaptive plotting \"numpoints\" contr ols the spacing of points along the curve, that is, in the direction o f the curve rather than just horizontally. " }}{PARA 0 "" 0 "" {TEXT -1 57 "The spacing between points is generally no greater than " } {XPPEDIT 18 0 "plotwidth/numpoints;" "6#*&%*plotwidthG\"\"\"%*numpoint sG!\"\"" }{TEXT -1 97 ". Note that, in general, the number of points p lotted could be vastly different from \"numpoints\"." }}{PARA 0 "" 0 " " {TEXT -1 36 "The default value is \"numpoints=33\"." }}{PARA 0 "" 0 "" {TEXT -1 12 "\nlinearity=n" }}{PARA 0 "" 0 "" {TEXT -1 82 "\"linear ity\" controls the tolerance for the allowed deviation from a straight line." }}{PARA 0 "" 0 "" {TEXT -1 93 "Along any arc between two point s on the graph, the curve will generally deviate no more than " } {TEXT 272 16 "tol * plotheight" }{TEXT -1 62 " from the straight line \+ segment joining the two points, where " }{XPPEDIT 18 0 "tol = 10^(-4)/ linearity;" "6#/%$tolG*&)\"#5,$\"\"%!\"\"\"\"\"%*linearityGF*" }{TEXT -1 2 " ." }}{PARA 0 "" 0 "" {TEXT -1 98 "Thus increasing \"linearity\" reduces the tolerance, and so gives more points at bends in the graph ." }}{PARA 0 "" 0 "" {TEXT -1 36 "The default value is \"linearity=10 \"." }}{PARA 0 "" 0 "" {TEXT -1 12 "\nmaxpoints=n" }}{PARA 0 "" 0 "" {TEXT -1 108 "This option provides a cut-off for the adaptive subdivis ion by specifying a minimum horizontal distance of " }{XPPEDIT 18 0 " plotwidth/maxpoints" "6#*&%*plotwidthG\"\"\"%*maxpointsG!\"\"" }{TEXT -1 65 " between points for the plot. \nThe default is \"maxpoints=1000 \". " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 119 "plotdata=true/false\nSetting the option \"plotdata\" to \"true\" caus es the data points to be returned instead of the graph." }}{PARA 0 "" 0 "" {TEXT -1 61 "This option can also be specified using the word \"p lot_data\"." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT 265 31 "Available standard plot options" }{TEXT -1 184 ": color, line style, thickness, scaling, xtickmarks, ytickmarks, tickmarks, labels, \+ style, symbol, symbolsize, title, axes, font, labelfont, titlefont, ax esfont, view, labeldirections." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}} {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT 259 16 "How to a ctivate:" }{TEXT 256 1 "\n" }{TEXT -1 154 "To make the procedure activ e 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 21 "graph: implementation" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "graph" {MPLTEXT 1 0 9031 "graph := proc(ff,rng)\n local fx,x,y,t1,t2,eps,xL,yL,xR,yR,h,ymin,ymax,yM in,yMax,\n width,height,adaptdiv,xrange,yrange,Options,startoptio ns,rs,\n mxpts,lnrty,nmpts,delta,pdat,adptv,curve,dev,m,addmaxmin ,\n fn,n1,n2,proctype,vars,y1,y2,testvals,aa,bb,i,opt;\n\n if n args<2 then\n error \"at least 2 arguments are required; the basi c syntax is: 'graph(f(x),x=a..b)'.\"\n end if;\n\n # Collect all t he input data.\n if type(ff,procedure) or \n (op(0,ff)=`@@` an d nops(ff)=2 and type(op(1,ff),procedure)) then\n proctype := tru e;\n if type(rng,range) then\n rs := rng;\n else\n \+ error \"the 2nd argument, %1, is invalid .. it should have the \+ form 'a..b' to provide a horizontal range over which to plot the graph of %1\",rng,ff;\n end if;\n elif type(ff,algebraic) then \n \+ vars := indets(ff,name) minus indets(ff,realcons);\n if nops(v ars)<>1 then \n if not type(ff,realcons) and not has(indets(ff ),\{Int,Sum\}) then\n error \"the 1st argument, %1, is inva lid .. it should be an expression which depends only on a single varia ble\",ff;\n end if;\n else\n x := op(1,vars);\n \+ end if;\n if type(rng,name=range) then\n proctype := \+ false;\n x := op(1,rng);\n if not member(x,vars) and n ot type(ff,realcons) then\n error \"the 1st argument, %1, i s invalid .. it should be an expression which depends only on the vari able %2\",ff,x;\n end if;\n rs := op(2,rng);\n el se\n error \"the 2nd argument, %1, is invalid .. it should hav e the form '%2=a..b' to provide a horizontal range over which to plot \+ the graph of %3\",rng,x,ff;\n end if;\n else\n error \"the 1st argument, %1, is invalid .. it should be an algebraic expression \+ in a single variable, or a numerical valued procedure with a single ar gument\",ff; \n end if;\n \n xL := evalf(op(1,rs));\n xR \+ := evalf(op(2,rs));\n if not type(xL,numeric) or not type(xR,numeric ) then\n error \"each end point of the horizontal range %1 must e valuate to a numeric\",rs;\n end if;\n if xL>=xR then\n error \"2nd argument horizontal range is invalid\";\n end if;\n if proc type then\n xrange := xL..xR;\n else\n xrange := x=xL..xR; \n end if;\n\n startoptions := 3; \n yrange := NULL;\n yMax : = infinity;\n yMin := -infinity;\n if nargs>2 then\n if type( args[3],range) or type(args[3],name=range) then\n startoptions := 4;\n if type(args[3],range) then\n rs := args[3 ]; \n else\n rs := op(2,args[3]);\n y := op(1,args[3]);\n end if;\n yMin := evalf(op(1,rs));\n yMax := evalf(op(2,rs));\n if not type(yMin,numeric) \+ or not type(yMax,numeric) then\n error \"each end point of \+ the vertical range %1 must evaluate to a numeric\",rs;\n end i f;\n if yMin>=yMax then\n error \"the 3rd argument \+ vertical range is invalid\";\n end if;\n if type(args[ 3],range) then\n yrange := yMin..yMax; \n else\n \+ yrange := y=yMin..yMax;\n end if;\n end if;\n \+ end if;\n\n # Get the options, but first set default\n nmpts := 33 ;\n lnrty := 10;\n Options := [];\n mxpts := 1000;\n pdat := f alse;\n adptv := true;\n\n if nargs>=startoptions then\n Opti ons := [args[startoptions..nargs]];\n if not type(Options,list(eq uation)) then\n error \"each optional argument after the %-1 a rgument must be an equation\",startoptions-1;\n end if;\n if hasoption(Options,'adaptive','adptv','Options') then\n if not adptv=true then adptv := false end if;\n end if;\n if hasop tion(Options,'numpoints','nmpts','Options') then\n if not type (nmpts,posint) or nmpts<2 then\n error \"\\\"numpoints\\\" \+ must be an integer greater than 1\"\n end if;\n end if;\n if hasoption(Options,'linearity','lnrty','Options') then\n \+ if not type(lnrty,posint) then\n error \"\\\"linearity\\ \" must be a positive integer\"\n end if;\n end if;\n \+ if hasoption(Options,'maxpoints','mxpts','Options') then\n i f not type(mxpts,posint) then\n error \"\\\"maxpoints\\\" m ust be a positive integer\"\n end if;\n end if;\n if hasoption(Options,'plotdata','pdat','Options') or\n hasoption (Options,'plot_data','pdat','Options') then\n if not pdat=true then pdat := false end if;\n end if;\n for i to nops(Option s) do\n opt := op(i,Options); \n if not member(op(1,op t),\n \{'color','colour','linestyle','line_style','thicknes s',\n 'scaling','xtickmarks','ytickmarks','tickmarks','labe ls',\n 'style','symbol','symbolsize','title','axes','font', \n 'labelfont','label_font','titlefont','title_font',\n \+ 'axesfont','axes_font','view','labeldirections'\}) then\n \+ error \"unknown or invalid option: %1\",opt;\n end if; \n end do; \n end if; \n\n # Recursively define d procedure to construct plotting data.\n adaptdiv := proc(pL,pR)\n \+ local x0,x1,x2,y0,y1,y2,p1,dx,dy,divL,divR;\n \n x0 := pL[1] ;\n x2 := pR[1];\n\n x1 := (x0+x2)/2;\n y1 := evalf(fn( x1));\n\n if y1<=yMax and y1>=yMin then\n # Update estima te of the height.\n if y1>ymax then ymax := y1; height := ymax -ymin end if;\n if y1 0 then p := -p end if;\n q := abs(q);\n \+ x3 := x0+p/q;\n y3 := evalf(fn(x3));\n if y3 <=yMax and y3>=yMin then\n # Update estimate of the heig ht.\n if y3>ymax then ymax := y3; height := ymax-ymin en d if;\n if y34 then\n \+ error \"1st argument %1 does not evaluate to a numeric at some point , or points, in the plotting interval\",ff;\n end if;\n\n width := xR-xL;\n\n if adptv then \n ymin := max(yMin,min(yL,yR));\n ymax := min(yMax,max(yL,yR));\n\n height := ymax-ymin;\n \+ eps := evalf(1/lnrty*0.0001);\n delta := evalf(1/mxpts);\n \+ nmpts := iquo(nmpts,2);\n h := evalf(1/nmpts)^2;\n curve : = addmaxmin(adaptdiv([xL,yL],[xR,yR]));\n if pdat=true then\n \+ return curve;\n else\n return plot(curve,xrange,yran ge,op(Options));\n end if;\n else\n nmpts := nmpts-1;\n \+ h := width/nmpts;\n n1 := iquo(nmpts,2);\n n2 := nmpts-n1 -1;\n curve := [[xL,yL],seq([xL+i*h,evalf(fn(xL+i*h))],i=1..n1), \n seq([xR+(i-n2)*h,evalf(fn(xR+(i-n2)*h))],i=0..n2-1),[ xR,yR]];\n if pdat=true then\n return curve;\n else \n return plot(curve,xrange,yrange,op(Options));\n end if ;\n end if;\nend proc: # of graph" }}}{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 section. " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> \+ " 0 "" {MPLTEXT 1 0 1 ";" }}}}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 72 "A p rocedure for evaluating the cosine function from its Maclaurin series " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 43 "Altho ugh the Maclaurin series converges to " }{XPPEDIT 18 0 "cos(x);" "6#-% $cosG6#%\"xG" }{TEXT -1 19 " for all values of " }{TEXT 274 1 "x" } {TEXT -1 85 ", it is only of practical use in evaluating the cosine f unction for small values of " }{TEXT 273 1 "x" }{TEXT -1 2 ".\n" }} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 665 "cos_series := proc(xx::real cons)\n local x,z,sum,term,eps,k,even,maxit;\n\n maxit := Digits*1 0;\n eps := Float(1,-Digits);\n\n x := evalf(xx);\n\n term := 1. 0;\n sum := term;\n z := x*x;\n even := false; \n for k from 1 to maxit by 2 do\n term := term*z/(k*(k+1));\n if even then \n sum := sum+term;\n else\n sum := sum-term;\n \+ end if;\n if abs(term)<=eps*abs(sum) then break end if;\n \+ even := not even;\n end do;\n\n if k > maxit then\n print(` sum of`,(k+1)/2,`terms of cosine series is`,sum);\n error \"reach ed the maximum number, %1, of iterations without convergence\",maxit; \n end if;\n evalf(sum);\nend proc:" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 95 "When using the default precision o f 10 digits, we can get about 10 digit accuracy in computing " } {XPPEDIT 18 0 "cos(x)" "6#-%$cosG6#%\"xG" }{TEXT -1 27 " using this pr ocedure with " }{XPPEDIT 18 0 "abs(x) <= 2;" "6#1-%$absG6#%\"xG\"\"#" }{TEXT -1 1 "." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 55 "xx := 1.9;\nevalf(cos_series(xx),10);\nevalf(cos (xx),10);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%#xxG$\"#>!\"\"" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#$!+oc*GB$!#5" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#$!+pc*GB$!#5" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 43 "We can check the accuracy of the procedure " } {TEXT 0 10 "cos_series" }{TEXT -1 12 " as follows." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 71 "plot(cos(x)- 'evalf[10]@cos_series'(x),x=-2..2,color=blue,numpoints=80);" }}{PARA 13 "" 1 "" {GLPLOT2D 467 257 257 {PLOTDATA 2 "6&-%'CURVESG6#7jcr7$$!\" #\"\"!$!\"\"!#57$$!+?XM)*>!\"*$F)F-7$$!+S!*o'*>F1F27$$!+gN.&*>F1$F*F*7 $$!+!3yL*>F1F+7$$!++Es\"*>F1$\"\"\"F-7$$!+?r1!*>F1F97$$!+S;T))>F1F97$$ !+ghv')>F1F@7$$!+!o+^)>F1$\"\"#F-7$$!++_W$)>F1$!\"'F-7$$!+?(*y\")>F1F+ 7$$!+SU8!)>F1F27$$!+g(y%y>F1F97$$!+!GBo(>F1$!\"$F-7$$!++y;v>F1F+7$$!+> B^t>F1F[o7$$!+Qo&=(>F1FN7$$!+e8?q>F1$\"\"$F-7$$!+yeao>F1F@7$$!+)R!*o'> F1F97$$!+=\\Bl>F1Fio7$$!+Q%zN'>F1$!\"%F-7$$!+eR#>'>F1$!\"&F-7$$!+y%o-' >F1FN7$$!+)*Hhe>F1F+7$$!+=v&p&>F1Fio7$$!+Q?Ib>F1F97$$!+elk`>F1F+7$$!+y 5*>&>F1F[o7$$!+)fN.&>F1Fgp7$$!+=,o[>F1F97$$!+QY-Z>F1F[o7$$!+9UeX>F1F97 $$!+*yVT%>F1F97$$!+kLqU>F1F27$$!+SHET>F1F@7$$!+;D#)R>F1F97$$!+#4#QQ>F1 $\"\"&F-7$$!+o;%p$>F1FS7$$!+V7]N>F1F+7$$!+=31M>F1F@7$$!+%R?E$>F1FN7$$! 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%=\\F1Fcp7$$\"3+++++](=#\\F1F^p7$$\"3++++D1kJ\\F1Fio7$$\"37+++:3JK\\F1 Fdo7$$\"3++++]iST\\F1F_o7$$\"3********))f&R%\\F1Fg_m7$$\"3++++v=<^\\F1 Fen7$$\"3+++++v$4'\\F1FV7$$\"3e******GKRh\\F1$\"3'**************z%F-7$ $\"3++++DJqq\\F1FF7$$\"3W+++Z\"*>!)\\F1FA7$$\"3++++](o/)\\F1F<7$$\"3++ ++vVB!*\\F1F77$$\"\"&F*F+-%'COLOURG6&%$RGBGFfqFfq$\"*++++\"!\")-%+AXES LABELSG6$Q\"x6\"Q!Fgew-%%VIEWG6$;F(Fjdw%(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 35 "Argument reduction to the interval " } {XPPEDIT 18 0 "[0, Pi/2];" "6#7$\"\"!*&%#PiG\"\"\"\"\"#!\"\"" }{TEXT -1 1 " " }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}{PARA 0 "" 0 "" {TEXT -1 181 "The first point to observe in the design of a proce dure to evaluate the cosine function is that the problem can be reduce d to that of evaluating the cosine function on the interval " } {XPPEDIT 18 0 "[0, Pi/2];" "6#7$\"\"!*&%#PiG\"\"\"\"\"#!\"\"" }{TEXT -1 9 ". " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 56 "We can reduce the interval over which we need to define \+ " }{XPPEDIT 18 0 "cos(x)" "6#-%$cosG6#%\"xG" }{TEXT -1 17 " to the int erval " }{XPPEDIT 18 0 "[-Pi, Pi];" "6#7$,$%#PiG!\"\"F%" }{TEXT -1 17 " by periodicity: " }}{PARA 257 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "cos(x+2*Pi) = cos(x);" "6#/-%$cosG6#,&%\"xG\"\"\"*&\"\"#F)%#PiGF)F)-F %6#F(" }{TEXT -1 1 "." }}{PARA 257 "" 0 "" {TEXT -1 1 " " }{TEXT 262 12 "____________" }{TEXT -1 2 " " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }} {PARA 0 "" 0 "" {TEXT -1 49 "Thus we can subtract a suitable integer m ultiple " }{XPPEDIT 18 0 "2*k*Pi;" "6#*(\"\"#\"\"\"%\"kGF%%#PiGF%" } {TEXT -1 4 " of " }{XPPEDIT 18 0 "2*Pi" "6#*&\"\"#\"\"\"%#PiGF%" } {TEXT -1 6 " from " }{TEXT 276 1 "x" }{TEXT -1 37 ", or add a suitable integer multiple " }{XPPEDIT 18 0 "2*k*Pi;" "6#*(\"\"#\"\"\"%\"kGF%%# PiGF%" }{TEXT -1 4 " to " }{TEXT 275 1 "x" }{TEXT -1 57 ", to ensure t hat the resulting number is in the interval " }{XPPEDIT 18 0 "[-Pi, Pi ];" "6#7$,$%#PiG!\"\"F%" }{TEXT -1 1 "." }}{PARA 0 "" 0 "" {TEXT -1 24 "The required integer is " }}{PARA 257 "" 0 "" {TEXT -1 1 " " } {XPPEDIT 18 0 "k = floor((x+Pi)/(2*Pi));" "6#/%\"kG-%&floorG6#*&,&%\"x G\"\"\"%#PiGF+F+*&\"\"#F+F,F+!\"\"" }{TEXT -1 2 ", " }}{PARA 0 "" 0 " " {TEXT -1 8 "that is," }}{PARA 257 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "k = floor(x/(2*Pi)+1/2);" "6#/%\"kG-%&floorG6#,&*&%\"xG\"\"\"*& \"\"#F+%#PiGF+!\"\"F+*&F+F+F-F/F+" }{TEXT -1 14 " ------- (i), " }} {PARA 0 "" 0 "" {TEXT -1 6 "where " }{XPPEDIT 18 0 "floor(z)" "6#-%&fl oorG6#%\"zG" }{TEXT -1 23 " is the largest integer" }{XPPEDIT 18 0 "`` <= z;" "6#1%!G%\"zG" }{TEXT -1 1 "." }}{PARA 0 "" 0 "" {TEXT -1 41 "T he cosine function is an even function: " }}{PARA 257 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "cos(-x) = cos(x);" "6#/-%$cosG6#,$%\"xG!\"\"- F%6#F(" }{TEXT -1 14 " ------- (ii)," }}{PARA 257 "" 0 "" {TEXT -1 1 " " }{TEXT 263 9 "_________" }{TEXT -1 17 " " }}{PARA 0 "" 0 "" {TEXT -1 23 "so we can replace x by " }{XPPEDIT 18 0 "abs(x) " "6#-%$absG6#%\"xG" }{TEXT -1 2 ". " }}{PARA 0 "" 0 "" {TEXT -1 44 "T his reduces the interval for evaluation to " }{XPPEDIT 18 0 "[0,Pi]" " 6#7$\"\"!%#PiG" }{TEXT -1 1 "." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }} {PARA 0 "" 0 "" {TEXT -1 12 "The formula:" }}{PARA 257 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "cos(Pi-x) = -cos(x);" "6#/-%$cosG6#,&%#PiG\" \"\"%\"xG!\"\",$-F%6#F*F+" }{TEXT -1 15 " ------- (iii)," }}{PARA 257 "" 0 "" {TEXT -1 1 " " }{TEXT 264 11 "___________" }{TEXT -1 18 " \+ " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 42 "can now be used to reduce the interval to " }{XPPEDIT 18 0 "[0,Pi/2]" "6#7$\"\"!*&%#PiG\"\"\"\"\"#!\"\"" }{TEXT -1 1 "." }} {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 46 "To illust rate this idea, we define a function " }{TEXT 0 4 "cos1" }{TEXT -1 47 " using Maple's cosine function on the interval " }{XPPEDIT 18 0 "[0, \+ Pi/2];" "6#7$\"\"!*&%#PiG\"\"\"\"\"#!\"\"" }{TEXT -1 58 ", but ensure \+ that it gives no value outside this interval." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 112 "cos1 := x - > if x>0 and x<=1.5707963267949 then cos(x) else FAIL end if;\nplot('c os1'(x),x=-0.2..2,thickness=2); " }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>% %cos1Gf*6#%\"xG6\"6$%)operatorG%&arrowGF(@%32\"\"!9$1F0$\"/\\zEjzq:!#8 -%$cosG6#F0%%FAILGF(F(F(" }}{PARA 13 "" 1 "" {GLPLOT2D 334 248 248 {PLOTDATA 2 "6&-%'CURVESG6$7O7$$\"3%******\\#=Vk9!#?$\"2i1*>x#*)*****! #<7$$\"3G+++7)[DS%F*$\"3)GKwz3.*****!#=7$$\"3(********zl1M(F*$\"3Y4zLu 0t****F37$$\"38+++vLJ4>!#>$\"3G%H!f;t<)***F37$$\"3%*******p,c%3$F<$\"3 MH#e97V_***F37$$\"3u******4'oPE&F<$\"3o4DVo&\\h)**F37$$\"3g+++]q(HW(F< $\"3YDh:JQJs**F37$$\"31+++Hle&>\"F3$\"3UeV%zt8'G**F37$$\"3++++v+Ji;F3$ \"3Gx5Z@T:i)*F37$$\"3'******H$o`F@F3$\"39dW9[<`u(*F37$$\"3.+++Q(zgg#F3 $\"3'34(*H,NBm*F37$$\"3:+++*e!eFIF3$\"3a#zeX%zw/x&F3$\"3yKUEt(p2Q)F37$$\" 3s*****z*y/fiF3$\"3gXi(e(RL/\")F37$$\"3[*****Hk;:p'F3$\"3Yg&=N)=[VyF37 $$\"3U+++)R.g;(F3$\"3Gu<%3kI/a(F37$$\"3-+++'*p#yh(F3$\"3uK>O'>ngB(F37$ $\"3a+++?vD*3)F3$\"3!f6q+*4w-pF37$$\"3Z******4ziT'*)*F3$\"3I(**ylQ%*)*[&F37$$\"32+++2QCN5F-$\"34) e%HqF;.^F37$$\"3#******pKbB3\"F-$\"3e@:(pf(\\#p%F37$$\"36+++eZWG6F-$\" 3K8&y=re1G%F37$$\"3++++_'QI<\"F-$\"3q[Xbn/_tQF37$$\"3,+++oDbA7F-$\"3vh ICW^Z7MF37$$\"3!******R-VqE\"F-$\"3i!\\:#Qy.\"*HF37$$\"3*)*****>3YXJ\" F-$\"3'z_Rp_Tc\"F-$\"3lc;MAfeVmF*7$$\"3++++!e8rc\"F-$\"3ATRGq%QFo$F*7 $$\"31+++oW2q:F-$\"3%)*>b?K(e=s!#@7$%*undefinedGFcy-%'COLOURG6&%$RGBG$ \"#5!\"\"$\"\"!F\\zF[z-%*THICKNESSG6#\"\"#-%+AXESLABELSG6$Q\"x6\"Q!Fez -%%VIEWG6$;$!\"#Fjy$F`zF\\z%(DEFAULTG" 1 2 0 1 10 2 2 9 1 4 2 1.000000 45.000000 45.000000 0 0 "Curve 1" }}}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 25 "Now we define a function " } {TEXT 0 4 "cos2" }{TEXT -1 40 " using the formulas (i), (ii) and (iii) ." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 477 "cos2 := proc(xx)\n local x,pi,pi2,piBy2,k;\n\n p i := evalf(Pi);\n x := evalf(xx);\n \n # Reduce the range to -Pi <=x<=Pi\n if abs(x)>pi then\n pi2 := 2*pi;\n # Determine t he number of multiples of 2*Pi to add or subtract.\n k := floor(x /pi2+0.5);\n x := x-k*pi2;\n end if;\n\n # cos is an even fun ction\n x := abs(x);\n\n # Reduce the range to 0<=x<=Pi/2\n piBy 2 := pi*0.5;\n if x>piBy2 then\n -cos1(pi-x)\n else\n co s1(x);\n end if;\nend proc;" }}{PARA 12 "" 1 "" {XPPMATH 20 "6#>%%co s2Gf*6#%#xxG6'%\"xG%#piG%$pi2G%&piBy2G%\"kG6\"F.C(>8%-%&evalfG6#%#PiG> 8$-F36#9$@$2F1-%$absG6#F7C%>8&,$*&\"\"#\"\"\"F1FFFF>8(-%&floorG6#,&*&F 7FFFB!\"\"FF$\"\"&FNFF>F7,&F7FF*&FHFFFBFFFN>F7F=>8',$*&FOFFF1FFFF@%2FV F7,$-%%cos1G6#,&F1FFF7FNFN-FgnF?F.F.F." }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 52 "plot('cos2'(x),x=-3*Pi. .5*Pi,color=red,thickness=2);" }}{PARA 13 "" 1 "" {GLPLOT2D 510 139 139 {PLOTDATA 2 "6'-%'CURVESG6#7[x7$$!3^***H>%zxC%*!#<$!2y************ ***F*7$$!3Y`ymi,Ic$*F*$!3MeON;Jcw**!#=7$$!3U2dS$QAyG*F*$!3^Y'Q'[BO1**F 27$$!3QhN9/YM>#*F*$!3+P*=TvE(*y*F27$$!3M:9)[#o'3:*F*$!3Z#Qlu/.si*F27$$ !3DBrNm7\"R,*F*$!3SkwIf+vn\"*F27$$!3QHG$yqbp())F*$!3Auja,ygO&)F27$$!3a RO[4riQ')F*$!3=%z]A]Vd1(F27$$!3q\\W86&)H+%)F*$!3M.$eEQPa>&F27$$!3]Fwq$ z&Gm#)F*$!3OlgY3Y@2SF27$$!3a.3GwIFK\")F*$!3umWNtA8ZFF27$$!3czR&)e.E)*z F*$!3/Bq&>'oyP9F27$$!3gbrUTwCkyF*$!3d&okHr#eE5!#>7$$!3_&R=2gY$HxF*$\"3 ^!ez$fg7V7F27$$!3WN'4+cXWf(F*$\"3/d)=x!>KmDF27$$!3Ov3I>XafuF*$\"3U*)*> l'e)G%QF27$$!3H:@fyMkCtF*$\"3)Q$Q4j=i\\]F27$$!3/f<)*)f$Q!>(F*$\"3k5_wo HbfhF27$$!3#GSr$>P7cqF*$\"3M-`o4/ierF27$$!3eY5wRQ'=#pF*$\"34!GEjJU)G!) 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A`kF\"Fi\\n$\"3`YQG.:I/)*F27$$\"3ICjJ&=DDG\"Fi\\n$\"3YlbS])pnm*F27$$\" 3/^v;3\"pYH\"Fi\\n$\"3#GLX+?daG*F27$$\"3&zx=5.8oI\"Fi\\n$\"3S@gv/cPn() F27$$\"3ous/P8t?8Fi\\n$\"35hRn=RK:!)F27$$\"3Trd2V'\\YL\"Fi\\n$\"3!4@?I []#3rF27$$\"39oU5\\zc[8Fi\\n$\"3cc]W2()pjgF27$$\"3)[wK^D'[i8Fi\\n$\"3f *yb]#4(=!\\F27$$\"3?$)f'R&\\$\\P\"Fi\\n$\"3IugNg!)o\"y$F27$$\"3O,#*z_O Q(Q\"Fi\\n$\"3F5kA\")f(Hg#F27$$\"3]>Cj^B$)*R\"Fi\\n$\"3AiCMBw(RQ\"F27$ $\"3$yjl/0\"G79Fi\\n$\"3Xi'yTL%fN9Fdo7$$\"3&RMu)fRbD9Fi\\n$!3GWw*))4k4 =\"F27$$\"3C]IGpo#)Q9Fi\\n$!3%pDqw48Z[#F27$$\"3ac#G%)>cH<*F27$$\"32aOlTA\\V:Fi\\n$!3qk*e1^`&H'*F27$$\"3YSF :i#=.b\"Fi\\n$!3+%z)\\Th0\"z*F27$$\"3-F=l#GWrb\"Fi\\n$!3%G.WC\\bp!**F2 7$$\"3e84:..(Rc\"Fi\\n$!3mM.FX " 0 "" {MPLTEXT 1 0 1 ";" }}}}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 66 "An arbitrary precision procedure to evaluate the cosine f unction: " }{TEXT 0 8 "cosineAP" }{TEXT -1 2 " " }}{EXCHG {PARA 0 "> \+ " 0 "" {MPLTEXT 1 0 1 ";" }}}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 24 "cosi neAP: implementation" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1669 "cosineAP := proc(xx::realcons)\n loca l x,z,pi2,term,eps,k,even,maxit,saveDigits,\n extraDigits,sum,pi,ref lect;\n \n if xx=0 then return 1.0 end if;\n\n # increase precis ion \n saveDigits := Digits;\n Digits := Digits+length(Digits)+1; \n x := evalf(xx);\n # argument reduction involves loss of precisi on\n # increase precision further when x has large magnitude\n ext raDigits := max(ilog10(x),0);\n if extraDigits>0 then\n Digits \+ := Digits+extraDigits;\n x := evalf(xx)\n end if;\n\n pi := e valf(Pi); # Maple evaluates pi rapidly\n \n # Reduce the range \+ to -Pi<=x<=Pi approximately\n if x>3.1415926535897932385 or x<-3.141 5926535897932385 then\n pi2 := pi+pi;\n\n # Determine the nu mber of multiples of 2*Pi to add or subtract.\n k := floor(x/pi2+ 0.5);\n x := x - k*pi2;\n end if;\n\n # If x<0, replace x by \+ -x so that 0<=x<=Pi\n if x<0 then x := -x end if;\n\n # If x>Pi/2, replace x by Pi - x so that 0<=x<=Pi/2\n if x>1.5707963267948966192 then\n x := pi-x;\n reflect := true;\n else\n reflec t := false;\n end if;\n\n Digits := Digits-extraDigits;\n\n # In itialisation for Maclaurin series loop\n eps := Float(1,-saveDigits) ;\n maxit := Digits*4;\n\n # calculate cos(x)\n term := 1.0;\n \+ sum := term;\n z := x*x;\n even := false; \n for k from 1 to ma xit by 2 do\n term := term*z/(k*(k+1));\n if even then\n \+ sum := sum+term;\n else\n sum := sum-term;\n en d if;\n if abs(term)<=eps*abs(sum) then break end if;\n even := not even;\n end do;\n\n Digits := saveDigits;\n if reflect t hen\n -evalf(sum);\n else\n evalf(sum);\n end if;\nend p roc: # of cosineAP" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 30 " Testing the procedure cosineAP" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 56 "plot('cosineAP'(x),x=-2*Pi.. 5*Pi,color=red,thickness=2);" }}{PARA 13 "" 1 "" {GLPLOT2D 499 189 189 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\\h3;UXw7vfdn,f$*Giz)Rnt6x%pS^^h\"z:Z[0&oH)4>RP*>b#)eX#*yhb4xefhdjE0m<)[ ([hp5)\\h3;UXw7v " 0 "" {MPLTEXT 1 0 94 "xx := Pi/2-10^(-20):\nDigits := 50:\ncosineAP(xx);\nevalf(evalf(cos(xx),D igits+20));\nDigits := 10:" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#$\"Sdp%G 125uU++++++++++++++++\"!#p" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#$\"SLLLL $)****************************************!#q" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 18 "Speed comparisons " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 17 "Maple's function " } {TEXT 0 3 "cos" }{TEXT -1 16 " is faster than " }{TEXT 0 8 "cosineAP" }{TEXT -1 1 "." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 152 "st := time():\nfor i to 500 do cosineAP(rand()* Float(1,-11)) end do:\ntime()-st;\nst := time():\nfor i to 500 do cos( rand()*Float(1,-11)) end do:\ntime()-st;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#$\"$a)!\"$" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#$\"$V\"!\"$" }}} {PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 180 "st := time():\nDigits := 50:\nfor i to 100 do cosineAP(rand()*Flo at(1,-11)) end do:\ntime()-st;\nst := time():\nfor i to 100 do cos(ran d()*Float(1,-11)) end do:\ntime()-st;\nDigits := 10:" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#$\"%58!\"$" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#$\"$?% !\"$" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}} }{PARA 0 "" 0 "" {TEXT -1 0 "" }}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 4 "T ask" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 2 "Q1" }}{PARA 0 "" 0 "" {TEXT -1 35 "Argument reduct ion to the interval " }{XPPEDIT 18 0 "[0,Pi/4]" "6#7$\"\"!*&%#PiG\"\" \"\"\"%!\"\"" }{TEXT -1 53 " can be performed by using the trigonometr ic formula " }}{PARA 257 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "cos(x) =2*cos(x/2)^2-1" "6#/-%$cosG6#%\"xG,&*&\"\"#\"\"\"*$-F%6#*&F'F+F*!\"\" F*F+F+F+F0" }{TEXT -1 1 "." }}{PARA 0 "" 0 "" {TEXT -1 159 "Write an a rbitrary precision procedure procedure to evaluate the cosine function which encorporates this formula and uses the Maclaurin series on the \+ interval " }{XPPEDIT 18 0 "[0,Pi/4]" "6#7$\"\"!*&%#PiG\"\"\"\"\"%!\"\" " }{TEXT -1 1 "." }}{PARA 0 "" 0 "" {TEXT -1 40 "_____________________ ___________________" }}{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 40 "________________________________________" }} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}}{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 }