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" }}{PARA 0 " " 0 "" {TEXT -1 55 "We can make a comparison of finite sums approximat ing " }{XPPEDIT 18 0 "exp(x);" "6#-%$expG6#%\"xG" }{TEXT -1 24 " grap hically as follows." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 " " {TEXT -1 27 "First we define a function " }{TEXT 0 3 "EXP" }{TEXT -1 84 " with two arguments or input parameters. The degree of the poly nomial approximation " }{TEXT 260 8 "EXP(n,x)" }{TEXT -1 4 " to " } {XPPEDIT 18 0 "exp(x);" "6#-%$expG6#%\"xG" }{TEXT -1 4 " is " } {XPPEDIT 18 0 "n;" "6#%\"nG" }{TEXT -1 1 "." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 95 "x := 'x': i := 'i' : n := 'n':\nEXP := (n,x) -> sum(x^i/i!,i=0..n);\nEXP(1,x);\nEXP(2,x); \nEXP(3,x);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%$EXPGf*6$%\"nG%\"xG6 \"6$%)operatorG%&arrowGF)-%$sumG6$*&)9%%\"iG\"\"\"-%*factorialG6#F3!\" \"/F3;\"\"!9$F)F)F)" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#,&\"\"\"F$%\"xG F$" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#,(\"\"\"F$%\"xGF$*&#F$\"\"#F$)F% F(F$F$" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#,*\"\"\"F$%\"xGF$*&#F$\"\"#F $)F%F(F$F$*&#F$\"\"'F$)F%\"\"$F$F$" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" } }{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 32 "evalf(EXP(10,1));\nevalf(ex p(1));" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#$\"+,=G=F!\"*" }}{PARA 11 " " 1 "" {XPPMATH 20 "6#$\"+G=G=F!\"*" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 38 "The degree 1, 2 and 3 approximations " }{XPPEDIT 18 0 "y = 1+x;" "6#/%\"yG,&\"\"\"F&%\"xGF&" }{TEXT -1 3 " , " }{XPPEDIT 18 0 "y = 1+x+x^2/2;" "6#/%\"yG,(\"\"\"F&%\"xGF&*&F'\"\"#F)!\"\"F&" }{TEXT -1 6 " and " }{XPPEDIT 18 0 "y = 1+x+x^2/2+x^3/6;" "6#/%\"yG,*\"\"\"F &%\"xGF&*&F'\"\"#F)!\"\"F&*&F'\"\"$\"\"'F*F&" }{TEXT -1 28 " can be p lotted as follows." }}{PARA 0 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "y = exp(x);" "6#/%\"yG-%$expG6#%\"xG" }{TEXT -1 33 " is also plotted fo r comparison. 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3u&f%>9@rliF17$F[el$\"3!HrFdM_(emF17$F`el$\"3)o@Dg6?n2(F17$Feel$\"3[sB Ik.MyuF17$Fjel$\"3')G'Q4x\"=GzF17$F_fl$\"3;RDA\"fc9R)F17$Fdfl$\"3fAu^ \\A0A*)F17$Fifl$\"3'*4(\\,/@hS*F17$F^gl$\"3U'G,cr6?***F17$Fdgl$\"3)y5O !>#Gt0\"Fbgl7$Figl$\"3?+AD7:%z6\"Fbgl7$F^hl$\"3S(o+jju<=\"Fbgl7$Fchl$ \"3[^7e[IH]7Fbgl7$Fhhl$\"3()>.H#y_oK\"Fbgl7$F]il$\"3(pn&4HJw+9Fbgl7$Fb il$\"3/&4,#Feo7$FI$\"3D5wTG+NH?Feo7$FN$\"3EQG(38([&)=Feo7$FS$\"3e,= \\=C,k\"o;Feo7$Fgn$\"3/x@Y$)4Q$f\"Feo7$F\\o$\"3/l =dEu!fa\"Feo7$Fao$\"3F*Hkr_*H6:Feo7$Fgo$\"3w93+:)y\\\\\"Feo7$F\\p$\"3E (R4xP2`\\\"Feo7$Fap$\"3I,Feo7$Fir$\"3Mrl!= &f\\Z?Feo7$F^s$\"3S?IU7!>U;#Feo7$Fcs$\"3UJ$*Rg'>kH#Feo7$Fhs$\"3L@'y<9( p?CFeo7$F]t$\"3Sm\"HD6BZc#Feo7$Fbt$\"31e*z]j\"\\CFFeo7$Fgt$\"3CfN*>u`> *GFeo7$F\\u$\"3'elJH3,\\1$Feo7$Fau$\"3QL'>EKy+F$Feo7$Ffu$\"3_I?#zd9oY$ Feo7$F[v$\"3%QJ4nAWo`1WFeo7$F_w$\"3kn+BC!RWo%Feo7$Fdw$\"3ql'\\8YnJ(\\Feo7$F iw$\"3G@,MY!*e%H&Feo7$F^x$\"3\\-F.v%=Qi&Feo7$Fcx$\"3%p74!R$4<)fFeo7$Fh x$\"3_+mN$oR\"fjFeo7$F]y$\"3-n'>I-Eps'Feo7$Fby$\"3)=82&=uourFeo7$Fgy$ \"3)RJ3$fPO+wFeo7$F\\z$\"3W2%\\J2z?3)Feo7$Faz$\"3alpm\"z7-:*Feo7$F[[l$\"3QHT]RHh#o*Feo7$F`[l$\"3CZ(obL^C.\"F17$Fe[l$ \"3y\"Qr\"))Hl%4\"F17$Fj[l$\"3]P>>U7*p;\"F17$F_\\l$\"3!='))RpsBL7F17$F d\\l$\"3,h)f1)eF88F17$Fi\\l$\"3!)yw;4z^&R\"F17$F^]l$\"3#zG>MD^G[\"F17$ Fc]l$\"3%4af*>zHv:F17$Fh]l$\"3#4qxNPL&p;F17$F]^l$\"35!>^TEaxx\"F17$Fb^ l$\"3A/4TorT()=F17$Fg^l$\"3**=f0$)H95?F17$F\\_l$\"3uw#)RxE5G@F17$Fa_l$ \"3!pK$Hb=`mAF17$Ff_l$\"3-,Q)pwZvS#F17$F[`l$\"3;D/Y3r%pb#F17$F``l$\"3; %>Gt&>)Gs#F17$Fe`l$\"3o;:[y')>&)GF17$Fj`l$\"3O\"HT!QJVhIF17$F_al$\"3]% )4AN]\\oKF17$Fdal$\"3%flsr!o'zY$F17$Fial$\"3'*Q`$\\R%[%o$F17$F^bl$\"3V VhWmpW=RF17$Fcbl$\"3a:`F\\xZYTF17$Fhbl$\"3ZIEc]://WF17$F]cl$\"3-;!H+&f GvYF17$Fbcl$\"3%=,)3Z2s$)\\F17$Fgcl$\"35.(oXfADF&F17$F\\dl$\"3ADi,EhEE cF17$Fadl$\"31^$zb)[YlfF17$Ffdl$\"3KD]eu!=3K'F17$F[el$\"3dbr2ms3EnF17$ F`el$\"35ln2#fH&erF17$Feel$\"3Q*HA.W([vvF17$Fjel$\"3odRug\"oT/)F17$F_f l$\"3*G6E7nq'G&)F17$Fdfl$\"3_THGF\"Re3*F17$Fifl$\"37h0%oLPif*F17$F^gl$ \"36*\\dy$Hm@5Fbgl7$Fdgl$\"3;u8:4V\\$3\"Fbgl7$Figl$\"3ev%GI]k#[6Fbgl7$ F^hl$\"3ouu@E[z;7Fbgl7$Fchl$\"3UoEH&e82H\"Fbgl7$Fhhl$\"3m.ikmztt8Fbgl7 $F]il$\"3+qs]V:Ja9Fbgl7$Fbil$\"3=Y'*QNtBQ:Fbgl7$Fgil$\"3yw>.<%>Ij\"Fbg l7$F\\jl$\"3q,ad'f!3LG(*H_2zF=Fbgl7$Ffjl$\"3z********* \\7%>Fbgl-F[[m6&F][mFf^nFc^nFc^nFg^n-%+AXESLABELSG6$Q\"x6\"Q!Fbep-%%VI EWG6$;F(Ffjl;$!\"&!\"\"$\"#?F*" 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 adaptive graph plott ing procedure: " }{TEXT 0 5 "graph" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }} {PARA 0 "" 0 "" {TEXT -1 79 "The procedure in this section is an alter native to the standard Maple function " }{TEXT 0 4 "plot" }{TEXT -1 195 ", for plotting the graph of a single function. It allows control \+ over how the plotting is performed, in a manner which is a bit differe nt from the control one has over the standard Maple routine " }{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 poin t calculations is not changed internally, whereas, the Maple procedure " }{TEXT 0 4 "plot" }{TEXT -1 137 " often uses hardware floating poin t arithmetic whereby the effective precision is increased to the equiv alent 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 274 18 "Calling Sequence:\n" }}{PARA 0 "" 0 "" {TEXT 275 4 " " }{TEXT -1 17 "graph( f, xrng )\n" }{TEXT 276 1 "\n" }{TEXT -1 26 " graph( f, xrng, yrng )" }}{PARA 0 "" 0 " " {TEXT -1 0 "" }}{PARA 256 "" 0 "" {TEXT 272 11 "Parameters:" }} {PARA 0 "" 0 "" {TEXT -1 2 " " }}{PARA 0 "" 0 "" {TEXT -1 3 " " } {TEXT 23 4 "f - " }{TEXT -1 83 " an expression involving a single v ariable, say x, or a function x -> f(x)\n\n " }{TEXT 23 7 "xrng - \+ " }{TEXT -1 79 " horizontal plotting range in the form x=a..b, whe n 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, o r in the form y=c..d.\n" }}{PARA 256 "" 0 "" {TEXT 271 12 "Description :" }}{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 metho d." }}{PARA 15 "" 0 "" {TEXT -1 88 "An even spacing is aimed for along 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 "Any maxi mum or minimum points are located approximately by parabolic interpola tion and added to the graph.\n" }}{PARA 256 "" 0 "" {TEXT 273 8 "Optio ns:" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 19 "ad aptive=true/false" }}{PARA 0 "" 0 "" {TEXT -1 144 "Adaptive plotting w ill, where necessary, sub-divide the plotting interval in an attempt t o 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 \"false\"." }} {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 312 "numpoint s=n\nFor non-adaptive plotting the interval for the plot is subdivided into a fixed number of sub-intervals of equal width by means \"numpoi nts\" points. \nFor adaptive plotting \"numpoints\" controls the spaci ng of points along the curve, that is, in the direction of the curve r ather than just horizontally. " }}{PARA 0 "" 0 "" {TEXT -1 57 "The spa cing between points is generally no greater than " }{XPPEDIT 18 0 "pl otwidth/numpoints;" "6#*&%*plotwidthG\"\"\"%*numpointsG!\"\"" }{TEXT -1 97 ". Note that, in general, the number of points plotted could be \+ vastly different from \"numpoints\"." }}{PARA 0 "" 0 "" {TEXT -1 36 "T he default value is \"numpoints=33\"." }}{PARA 0 "" 0 "" {TEXT -1 12 " \nlinearity=n" }}{PARA 0 "" 0 "" {TEXT -1 82 "\"linearity\" controls t he tolerance for the allowed deviation from a straight line." }}{PARA 0 "" 0 "" {TEXT -1 93 "Along any arc between two points on the graph, \+ the curve will generally deviate no more than " }{TEXT 277 16 "tol * p lotheight" }{TEXT -1 62 " from the straight line segment joining the t wo points, where " }{XPPEDIT 18 0 "tol = 10^(-4)/linearity;" "6#/%$tol G*&)\"#5,$\"\"%!\"\"\"\"\"%*linearityGF*" }{TEXT -1 2 " ." }}{PARA 0 " " 0 "" {TEXT -1 98 "Thus increasing \"linearity\" reduces the toleranc e, 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 opt ion provides a cut-off for the adaptive subdivision by specifying a mi nimum 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\n Setting the option \"plotdata\" to \"true\" causes 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 \"plot_data\"." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT 270 31 "Available stan dard plot options" }{TEXT -1 184 ": color, linestyle, thickness, scal ing, xtickmarks, ytickmarks, tickmarks, labels, style, symbol, symbols ize, title, axes, font, labelfont, titlefont, axesfont, view, labeldir ections." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT 259 16 "How to activate:" }{TEXT 256 1 " \n" }{TEXT -1 154 "To make the procedure active open the subsection, p lace 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,yMin,yMax,\n \+ width,height,adaptdiv,xrange,yrange,Options,startoptions,rs,\n mx pts,lnrty,nmpts,delta,pdat,adptv,curve,dev,m,addmaxmin,\n fn,n1,n 2,proctype,vars,y1,y2,testvals,aa,bb,i,opt;\n\n if nargs<2 then\n \+ error \"at least 2 arguments are required; the basic syntax is: 'gr aph(f(x),x=a..b)'.\"\n end if;\n\n # Collect all the input data.\n if type(ff,procedure) or \n (op(0,ff)=`@@` and nops(ff)=2 and type(op(1,ff),procedure)) then\n proctype := true;\n if typ e(rng,range) then\n rs := rng;\n else\n error \"t he 2nd argument, %1, is invalid .. it should have the form 'a..b' to p rovide a horizontal range over which to plot the graph of %1\",rng,ff; \n end if;\n elif type(ff,algebraic) then \n vars := indet s(ff,name) minus indets(ff,realcons);\n if nops(vars)<>1 then \n \+ if not type(ff,realcons) and not has(indets(ff),\{Int,Sum\}) t hen\n error \"the 1st argument, %1, is invalid .. it should be an expression which depends only on a single variable\",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 not type(ff,realc ons) then\n error \"the 1st argument, %1, is invalid .. it \+ should be an expression which depends only on the variable %2\",ff,x; \n end if;\n rs := op(2,rng);\n else\n er ror \"the 2nd argument, %1, is invalid .. it should have 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 var iable, or a numerical valued procedure with a single argument\",ff; \+ \n end if;\n \n xL := evalf(op(1,rs));\n xR := evalf(op(2,r s));\n if not type(xL,numeric) or not type(xR,numeric) then\n e rror \"each end point of the horizontal range %1 must evaluate to a nu meric\",rs;\n end if;\n if xL>=xR then\n error \"2nd argument horizontal range is invalid\";\n end if;\n if proctype 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(yM ax,numeric) then\n error \"each end point of the vertical r ange %1 must evaluate to a numeric\",rs;\n end if;\n i f 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 yrang e := 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 := false;\n adpt v := true;\n\n if nargs>=startoptions then\n Options := [args[s tartoptions..nargs]];\n if not type(Options,list(equation)) then \n error \"each optional argument after the %-1 argument must \+ be an equation\",startoptions-1;\n end if;\n if hasoption(Op tions,'adaptive','adptv','Options') then\n if not adptv=true t hen adptv := false end if;\n end if;\n if hasoption(Options, 'numpoints','nmpts','Options') then\n if not type(nmpts,posint ) or nmpts<2 then\n error \"\\\"numpoints\\\" must be an in teger greater than 1\"\n end if;\n end if;\n if haso ption(Options,'linearity','lnrty','Options') then\n if not typ e(lnrty,posint) then\n error \"\\\"linearity\\\" must be a \+ positive integer\"\n end if;\n end if;\n if hasoptio n(Options,'maxpoints','mxpts','Options') then\n if not type(mx pts,posint) then\n error \"\\\"maxpoints\\\" must be a posi tive integer\"\n end if;\n end if;\n if hasoption(Op tions,'plotdata','pdat','Options') or\n hasoption(Options,'plo t_data','pdat','Options') then\n if not pdat=true then pdat := false end if;\n end if;\n for i to nops(Options) do\n \+ opt := op(i,Options); \n if not member(op(1,opt),\n \+ \{'color','colour','linestyle','line_style','thickness',\n \+ 'scaling','xtickmarks','ytickmarks','tickmarks','labels',\n \+ 'style','symbol','symbolsize','title','axes','font',\n \+ 'labelfont','label_font','titlefont','title_font',\n 'axes font','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 defined 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 estimate of the heig ht.\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 height.\n \+ if y3>ymax then ymax := y3; height := ymax-ymin end if;\n \+ if y34 then\n error \"1 st 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 := addmaxmi n(adaptdiv([xL,yL],[xR,yR]));\n if pdat=true then\n retur n curve;\n else\n return plot(curve,xrange,yrange,op(Opti ons));\n end if;\n else\n nmpts := nmpts-1;\n h := wi dth/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 r eturn 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 sectio n. " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 78 "A procedur e for evaluating the exponential function from its Maclaurin series " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 43 "Althoug h the Maclaurin series converges to " }{XPPEDIT 18 0 "exp(x);" "6#-%$e xpG6#%\"xG" }{TEXT -1 19 " for all values of " }{TEXT 278 1 "x" } {TEXT -1 89 ", it is only of practical use in evaluating the exponenti al function for small values of " }{TEXT 279 1 "x" }{TEXT -1 2 ".\n" } }{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 513 "exp_series := proc(xx::rea lcons)\n local x,sum,term,eps,k,maxit;\n\n maxit := Digits*10;\n \+ eps := Float(2,-Digits);\n\n x := evalf(xx);\n term := 1.0;\n s um := term;\n for k from 1 to maxit do\n term := term*x/k;\n \+ sum := sum + term;\n if abs(term)<=eps*abs(sum) then break end if;\n end do;\n if k>maxit then\n print(`sum of`,k,`terms of exponential series is`,sum);\n error \"reached the maximum numbe r, %1, of iterations without convergence\",maxit;\n end if;\n eval f(sum);\nend proc:" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 " " {TEXT -1 95 "When using the default precision of 10 digits, we can g et about 10 digit accuracy in computing " }{XPPEDIT 18 0 "exp(x)" "6#- %$expG6#%\"xG" }{TEXT -1 27 " using this procedure with " }{XPPEDIT 18 0 "abs(x) <= 2;" "6#1-%$absG6#%\"xG\"\"#" }{TEXT -1 1 "." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 50 "xx := 0.75;\nevalf(exp_series(xx));\nevalf(exp(xx));" }}{PARA 11 "" 1 " " {XPPMATH 20 "6#>%#xxG$\"#v!\"#" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#$ \"+<++<@!\"*" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#$\"+<++<@!\"*" }}} {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 43 "We can ch eck the accuracy of the procedure " }{TEXT 0 13 "arcsin_series" } {TEXT -1 12 " as follows." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 58 "plot(exp(x)-'evalf[10]@exp_series'( 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\"3'******poagq%F1Faiu7$$\"3++++]7.2ZF1Ffiu7$$\"3++++voz;ZF1F[ju7$$\"3 /+++QMxn)Qt%F1Ffiu7$$\"3++++D \"Gjt%F1Ffiu7$$\"3++++]P4YZF1F[ju7$$\"3++++v$fev%F1Ffiu7$$\"3+++++]ilZ F1Ffiu7$$\"3++++D1RvZF1Ffiu7$$\"3s*****fkGNy%F1Faiu7$$\"3++++]i:&y%F1F []p7$$\"3s*****4F%H$z%F1F[]p7$$\"3++++v=#\\z%F1Faiu7$$\"3+++++vo/[F1F[ ju7$$\"3++++DJX9[F1F[ju7$$\"3++++](=U#[F1$!3))**************HF[fu7$$\" 3W+++`%Ra#[F1F[]p7$$\"3++++vV)R$[F1F[]p7$$\"3++++++vV[F1Faiu7$$\"3++++ Dc^`[F1F[]p7$$\"3++++]7Gj[F1Fihu7$$\"3e*****fw%Qp[F1Fh^v7$$\"3++++vo/t [F1F[ju7$$\"3+++++D\"G)[F1$FjitF[fu7$$\"3Q+++wT9%)[F1F[]p7$$\"3++++D\" yD*[F1Ffiu7$$\"3u*****H'RC$*[F1F[]p7$$\"3++++]PM-\\F1F[]p7$$\"3++++v$4 @\"\\F1F[]p7$$\"3s*****fRZ-#\\F1F[ju7$$\"3+++++](=#\\F1Ffiu7$$\"3++++D 1kJ\\F1Fihu7$$\"3++++]iST\\F1Fihu7$$\"3++++v=<^\\F1F[]p7$$\"3+++++v$4' \\F1Ffiu7$$\"3++++DJqq\\F1F[]p7$$\"3g*****\\?r(z\\F1F[ju7$$\"3++++](o/ )\\F1$\"3))**************HF[fu7$$\"3++++vVB!*\\F1Ffiu7$$\"\"&F*F[]p-%' COLOURG6&%$RGBGF[]pF[]p$\"*++++\"!\")-%+AXESLABELSG6$Q\"x6\"Q!F`dv-%%V IEWG6$;F(Fccv%(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 19 "Argument reduction " }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 35 "Argument reduction t o the interval " }{XPPEDIT 18 0 "[-ln(10)/2, ln(10)/2]" "6#7$,$*&-%#ln G6#\"#5\"\"\"\"\"#!\"\"F,*&-F'6#F)F*F+F," }{TEXT -1 1 "." }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}{PARA 0 "" 0 "" {TEXT -1 44 "W e can subtract a suitable integer multiple " }{XPPEDIT 18 0 "k*ln(10); " "6#*&%\"kG\"\"\"-%#lnG6#\"#5F%" }{TEXT -1 4 " of " }{XPPEDIT 18 0 "l n(10);" "6#-%#lnG6#\"#5" }{TEXT -1 6 " from " }{TEXT 280 1 "x" }{TEXT -1 37 ", or add a suitable integer multiple " }{XPPEDIT 18 0 "k*ln(10) ;" "6#*&%\"kG\"\"\"-%#lnG6#\"#5F%" }{TEXT -1 4 " to " }{TEXT 281 1 "x " }{TEXT -1 57 ", to ensure that the resulting number is in the interv al " }{XPPEDIT 18 0 "[-ln(10)/2, ln(10)/2];" "6#7$,$*&-%#lnG6#\"#5\"\" \"\"\"#!\"\"F,*&-F'6#F)F*F+F," }{TEXT -1 1 "." }}{PARA 0 "" 0 "" {TEXT -1 4 "If " }{XPPEDIT 18 0 "abs(x)>ln(10)/2" "6#2*&-%#lnG6#\"#5 \"\"\"\"\"#!\"\"-%$absG6#%\"xG" }{TEXT -1 7 ", let " }}{PARA 257 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "k=floor(x/ln(10)+1/2)" "6#/%\"kG-% &floorG6#,&*&%\"xG\"\"\"-%#lnG6#\"#5!\"\"F+*&F+F+\"\"#F0F+" }{TEXT -1 14 " ------- (i), " }}{PARA 257 "" 0 "" {TEXT -1 1 " " }{TEXT 266 11 " ___________" }{TEXT -1 16 " " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 6 "where " }{XPPEDIT 18 0 "floor(x) " "6#-%&floorG6#%\"xG" }{TEXT 30 23 " is the largest integer" } {XPPEDIT 18 0 "``<= x" "6#1%!G%\"xG" }{TEXT 30 2 ". " }}{PARA 0 "" 0 " " {TEXT -1 4 "Set " }}{PARA 257 "" 0 "" {TEXT -1 2 " " }{XPPEDIT 18 0 "z = x-k*`.`*ln(10)" "6#/%\"zG,&%\"xG\"\"\"*(%\"kGF'%\".GF'-%#lnG6# \"#5F'!\"\"" }{TEXT -1 14 " ------- (ii)." }}{PARA 257 "" 0 "" {TEXT -1 1 " " }{TEXT 268 9 "_________" }{TEXT -1 17 " " }} {PARA 0 "" 0 "" {TEXT -1 5 "Then " }{TEXT 282 1 "z" }{TEXT -1 20 " is \+ in the interval " }{XPPEDIT 18 0 "[-ln(10)/2, ln(10)/2]" "6#7$,$*&-%#l nG6#\"#5\"\"\"\"\"#!\"\"F,*&-F'6#F)F*F+F," }{TEXT -1 4 " and" }}{PARA 257 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "exp(x) = exp(z)*`.`*10^k;" "6#/-%$expG6#%\"xG*(-F%6#%\"zG\"\"\"%\".GF,)\"#5%\"kGF," }{TEXT -1 16 " ------- (iii). " }}{PARA 257 "" 0 "" {TEXT -1 1 " " }{TEXT 267 7 "__ _____" }{TEXT -1 19 " " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 19 "This follows since " }{XPPEDIT 18 0 "k= (x-z)/ln(10)" "6#/%\"kG*&,&%\"xG\"\"\"%\"zG!\"\"F(-%#lnG6#\"#5F* " }{TEXT -1 10 ", so that " }}{PARA 257 "" 0 "" {TEXT -1 2 " " } {XPPEDIT 18 0 "exp(z)*`.`*10^k = exp(z)*`.`*10^((x-z)/ln(10));" "6#/*( -%$expG6#%\"zG\"\"\"%\".GF))\"#5%\"kGF)*(-F&6#F(F)F*F))F,*&,&%\"xGF)F( !\"\"F)-%#lnG6#F,F5F)" }{TEXT -1 1 " " }}{PARA 257 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "`` = exp(z)*`.`*10^((x-z)*log[10](exp(1)));" "6#/% !G*(-%$expG6#%\"zG\"\"\"%\".GF*)\"#5*&,&%\"xGF*F)!\"\"F*-&%$logG6#F-6# -F'6#F*F*F*" }{TEXT -1 1 " " }}{PARA 257 "" 0 "" {TEXT -1 1 " " } {XPPEDIT 18 0 "``= exp(z)*`.`*10^(log[10](exp(x-z))" "6#/%!G*(-%$expG6 #%\"zG\"\"\"%\".GF*)\"#5-&%$logG6#F-6#-F'6#,&%\"xGF*F)!\"\"F*" }{TEXT -1 1 " " }}{PARA 257 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "`` = exp(z )*`.`*exp(x-z);" "6#/%!G*(-%$expG6#%\"zG\"\"\"%\".GF*-F'6#,&%\"xGF*F)! \"\"F*" }{TEXT -1 1 " " }}{PARA 257 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "``= exp(x)" "6#/%!G-%$expG6#%\"xG" }{TEXT -1 1 " " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 46 "To illustrate this idea, we define a function " }{TEXT 0 4 "exp1" }{TEXT -1 52 " using Maple's exponential function on the inte rval " }{XPPEDIT 18 0 "[-ln(10)/2, ln(10)/2];" "6#7$,$*&-%#lnG6#\"#5\" \"\"\"\"#!\"\"F,*&-F'6#F)F*F+F," }{TEXT -1 58 ", but ensure that it gi ves no value outside this interval." }}{PARA 0 "" 0 "" {TEXT -1 0 "" } }{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 107 "exp1 := x -> if abs(x)<=1. 151292546 then exp(x) else FAIL end if;\nplot('exp1'(x),x=-1.5..1.5,th ickness=2); " }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%%exp1Gf*6#%\"xG6\"6$ %)operatorG%&arrowGF(@%1-%$absG6#9$$\"+YDH^6!\"*-%$expGF0%%FAILGF(F(F( " }}{PARA 13 "" 1 "" {GLPLOT2D 334 248 248 {PLOTDATA 2 "6&-%'CURVESG6$ 7T7$$!3++++CA#*\\6!#<$\"31!Q\"yqRhmJ!#=7$$!3#******R%\\1[6F*$\"3Ir)fXs +D<$F-7$$!3++++%Q]V9\"F*$\"3qx$*)H32V=$F-7$$!3%******\\#ejS6F*$\"3u[A? vt:'>$F-7$$!3&******fq1K8\"F*$\"3j7POG/**>KF-7$$!36+++'exd7\"F*$\"37NO o-7+WKF-7$$!33+++%*y+&4\"F*$\"3+QP.h'p`M$F-7$$!3++++,#QU1\"F*$\"3?w%p: o0*\\MF-7$$!3-+++\"3%f+5F*$\"3)pa0*e&4mn$F-7$$!39+++\"oS:P*F-$\"3x8\\* )*=6u\"RF-7$$!3h*****\\<#)*=()F-$\"3gY<(RUq:=%F-7$$!3'******zG3U9)F-$ \"3m!*Qlbw&*GWF-7$$!3Y*****\\-\\r\\(F-$\"3eSAjCC,DZF-7$$!3?+++vGVZoF-$ \"3h$4%\\Cl>U]F-7$$!3_*****\\(4J@iF-$\"3#Q4#RiC*zO&F-7$$!35+++1Bt_cF-$ \"3qf9(eu[?o&F-7$$!3G+++!GPm(\\F-$\"3)HVE(oL\\zgF-7$$!3()*******4'*QS% F-$\"3;l$y#pa&yV'F-7$$!3&*******f>mPPF-$\"3y*H#)Q#yP\")oF-7$$!3w****** \\=$z9$F-$\"31>fj6#)R*H(F-7$$!3m******fW!4]#F-$\"3K\\;ToMI(y(F-7$$!3!* ******p8y%)=F-$\"3Ai=%e5'=#G)F-7$$!3)*******f:#>C\"F-$\"3@(GRX5,@$))F- 7$$!3V++++ev:l!#>$\"35V_KD$)>p$*F-7$$!31++++!p2[\"!#?$\"37ZZ(zE._)**F- 7$$\"31++++>:mkFir$\"3+TaXsyzm5F*7$$\"3-+++gdQA7F-$\"3'4U\\hmB+8\"F*7$ $\"3,+++SPBW=F-$\"3(Qe'3L[_-7F*7$$\"31+++]Nm'[#F-$\"3O>I*R79BG\"F*7$$ \"3()*******yb^6$F-$\"3#*3XFfI\\l8F*7$$\"36+++SVDBPF-$\"3EH9TV^5^9F*7$ $\"3!*******4TW)R%F-$\"3Qf39`cY_:F*7$$\"36+++]@80]F-$\"3ynANOwc\\;F*7$ $\"35+++?6!Hl&F-$\"31ago$He*f8>bNm=F*7$$\" 3!*******zZf\")oF-$\"3Ir%[2V\\+*>F*7$$\"3')******R?S&[(F-$\"3L`t-$)=\" R6#F*7$$\"3]+++?Yb;\")F-$\"3uja)[RK;D#F*7$$\"3`+++?;iL()F-$\"3yo\\gS& \\\\R#F*7$$\"3_+++!fL'z$*F-$\"3[%z#*3'HxaDF*7$$\"3(********)>=+5F*$\"3 btpetlx=FF*7$$\"3++++E&4Q1\"F*$\"3sz5VlwQ(*GF*7$$\"31+++g)f`4\"F*$\"3Y SaCk&e-*HF*7$$\"3#******R>5p7\"F*$\"3ipB&QG1h3$F*7$$\"3(******R1eT8\"F *$\"3!*z]FO_b3JF*7$$\"3,+++MfST6F*$\"3XHO&o[n68$F*7$$\"3.+++p)H]9\"F*$ \"3ZV=#)>_`UJF*7$$\"31+++/Ql[6F*$\"3#GO^MAWR:$F*7$$\"3++++sdY]6F*$\"3I $pX,Dk'fJF*7$%%FAILGF[[l-%'COLOURG6&%$RGBG$\"#5!\"\"$\"\"!Fd[lFc[l-%+A XESLABELSG6$Q\"x6\"Q!Fi[l-%*THICKNESSG6#\"\"#-%%VIEWG6$;$!#:Fb[l$\"#:F b[l%(DEFAULTG" 1 2 0 1 10 2 2 9 1 4 2 1.000000 45.000000 45.000000 0 0 "Curve 1" }}}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 25 "Now we define a function " }{TEXT 0 4 "exp2" }{TEXT -1 40 " usi ng the formulas (i), (ii) and (iii)." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 298 "exp2 := proc(xx)\n loca l x,ln10,k;\n\n x := evalf(xx);\n\n # Reduce the argument range to -ln(10)/2<=z<=ln(10)/2\n k := 0;\n if abs(x)>1.151292546 then\n \+ ln10 := evalf(ln(10)); \n k := floor(x/ln10+0.5);\n x : = x - k*ln10;\n end if;\n\n Scale10(exp1(x),k); # exp1(x)*10^k\nen d proc;" }}{PARA 12 "" 1 "" {XPPMATH 20 "6#>%%exp2Gf*6#%#xxG6%%\"xG%%l n10G%\"kG6\"F,C&>8$-%&evalfG6#9$>8&\"\"!@$2$\"+YDH^6!\"*-%$absG6#F/C%> 8%-F16#-%#lnG6#\"#5>F5-%&floorG6#,&*&F/\"\"\"FA!\"\"FN$\"\"&FOFN>F/,&F /FN*&F5FNFAFNFO-%(Scale10G6$-%%exp1GF>F5F,F,F," }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 22 "exp2(-1.5);\nexp(-1.5);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#$\"+,;IJA!#5" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#$\"+,;I JA!#5" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 46 "plot('exp2'(x),x=-3..3,color=red,thickness=2);" }} {PARA 13 "" 1 "" {GLPLOT2D 450 266 266 {PLOTDATA 2 "6'-%'CURVESG6#7V7$ $!\"$\"\"!$\"+Poqy\\!#67$$!+vq@pG!\"*$\"+(\\LVn&F-7$$!+^NUbFF1$\"+r$3# ejF-7$$!+K3XFEF1$\"+uOCEsF-7$$!+G)H')\\#F1$\"+GYv>#)F-7$$!+'3@/P#F1$\" +9r8W$*F-7$$!+r^b^AF1$\"+\"QaB0\"!#57$$!+,kZG@F1$\"+A\\=!>\"FM7$$!+h\" )=,?F1$\"+'yXFM7$$!+0)H%*\\\"F1$\"+TUdKAFM7$$!+vl[p8F1$\"+zXPU DFM7$$!+&>iUC\"F1$\"+3V`\")GFM7$$!+hkaI6F1$\"+'zn&GKFM7$$!*buK&**F1$\" +mR-'p$FM7$$!*?#z2))F1$\"+/$)fWTFM7$$!*\"RKvuF1$\"+!3O`t%FM7$$!*qjeH'F 1$\"+D97G`FM7$$!*\"*3=+&F1$\"+P&4U1'FM7$$!*ui&pPF1$\"+p1YfoFM7$$!*7VQ[ #F1$\"+#*4g+yFM7$$!*;^JI\"F1$\"+Ex=y()FM7$$!(Q:'HF1$\"+I%G/(**FM7$$\"* RIKH\"F1$\"+-x0Q6F17$$\"*^rZW#F1$\"+&[`pF\"F17$$\"*[n%)o$F1$\"+uf1Y9F1 7$$\"*5FL(\\F1$\"+8&HVk\"F17$$\"*e6.B'F1$\"+&HrX'=F17$$\"*p3lW(F1$\"+R hq0@F17$$\"*A))oz)F1$\"+/'\\,T#F17$$\"+Ik-,5F1$\"+(Qt5s#F17$$\"+D-eI6F 1$\"+*>`u4$F17$$\"+>_(zC\"F1$\"+FHG$[$F17$$\"+b*=jP\"F1$\"+LnHgRF17$$ \"+4/3(\\\"F1$\"+aMioWF17$$\"+C4JB;F1$\"+U%[)p]F17$$\"+DVsYn#f(=F1$\"+))['o_'F17$$\"+!)RO+?F1$\"+dgu\"R(F17$$\"+_!>w7#F 1$\"+%Gb[R)F17$$\"+*Q?QD#F1$\"+0?0C&*F17$$\"+5jypBF1$\"+g1^p5!\")7$$\" +Ujp-DF1$\"+Y'Q:A\"Fay7$$\"+gEd@EF1$\"+?Utv8Fay7$$\"+Mh%\\o#F1$\"+#=Td Y\"Fay7$$\"+4'>$[FF1$\"+Aojh:Fay7$$\"+5h(*3GF1$\"+>?Hf;Fay7$$\"+6EjpGF 1$\"+uR0j " 0 "" {MPLTEXT 1 0 1 ";" }}}}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 35 "Argument reduction to the interval " } {XPPEDIT 18 0 "[-ln(10)/8, ln(10)/8];" "6#7$,$*&-%#lnG6#\"#5\"\"\"\"\" )!\"\"F,*&-F'6#F)F*F+F," }{TEXT -1 1 "." }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}{PARA 0 "" 0 "" {TEXT -1 4 "Let " }{XPPEDIT 18 0 "L = ln(10)/4" "6#/%\"LG*&-%#lnG6#\"#5\"\"\"\"\"%!\"\"" }{TEXT -1 46 ". We can subtract a suitable integer multiple " }{XPPEDIT 18 0 "k* `.`*L;" "6#*(%\"kG\"\"\"%\".GF%%\"LGF%" }{TEXT -1 4 " of " }{TEXT 269 1 "L" }{TEXT -1 6 " from " }{TEXT 284 1 "x" }{TEXT -1 37 ", or add a s uitable integer multiple " }{XPPEDIT 18 0 "k*`.`*L;" "6#*(%\"kG\"\"\"% \".GF%%\"LGF%" }{TEXT -1 4 " of " }{TEXT 283 1 "L" }{TEXT -1 4 " to " }{TEXT 285 1 "x" }{TEXT -1 57 ", to ensure that the resulting number i s in the interval " }{XPPEDIT 18 0 "[-ln(10)/8, ln(10)/8];" "6#7$,$*&- %#lnG6#\"#5\"\"\"\"\")!\"\"F,*&-F'6#F)F*F+F," }{TEXT -1 1 "." }}{PARA 0 "" 0 "" {TEXT -1 4 "If " }{XPPEDIT 18 0 "ln(10)/8 < abs(x);" "6#2*& -%#lnG6#\"#5\"\"\"\"\")!\"\"-%$absG6#%\"xG" }{TEXT -1 7 ", let " }} {PARA 257 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "k = floor(x/L+1/2);" "6#/%\"kG-%&floorG6#,&*&%\"xG\"\"\"%\"LG!\"\"F+*&F+F+\"\"#F-F+" } {TEXT -1 14 " ------- (i), " }}{PARA 257 "" 0 "" {TEXT -1 1 " " } {TEXT 263 9 "_________" }{TEXT -1 18 " " }}{PARA 0 " " 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 6 "where " }{XPPEDIT 18 0 "floor(x)" "6#-%&floorG6#%\"xG" }{TEXT 30 23 " is the largest int eger" }{XPPEDIT 18 0 "``<= x" "6#1%!G%\"xG" }{TEXT 30 2 ". " }}{PARA 0 "" 0 "" {TEXT -1 4 "Set " }}{PARA 257 "" 0 "" {TEXT -1 2 " " } {XPPEDIT 18 0 "z = x-k*`.`*L;" "6#/%\"zG,&%\"xG\"\"\"*(%\"kGF'%\".GF'% \"LGF'!\"\"" }{TEXT -1 14 " ------- (ii)." }}{PARA 257 "" 0 "" {TEXT -1 1 " " }{TEXT 265 6 "______" }{TEXT -1 16 " " }} {PARA 0 "" 0 "" {TEXT -1 26 "Then z is in the interval " }{XPPEDIT 18 0 "[-ln(10)/8, ln(10)/8];" "6#7$,$*&-%#lnG6#\"#5\"\"\"\"\")!\"\"F,*&-F '6#F)F*F+F," }{TEXT -1 4 " and" }}{PARA 257 "" 0 "" {TEXT -1 1 " " } {XPPEDIT 18 0 "exp(x) = exp(z)*`.`*10^(k/4);" "6#/-%$expG6#%\"xG*(-F%6 #%\"zG\"\"\"%\".GF,)\"#5*&%\"kGF,\"\"%!\"\"F," }{TEXT -1 16 " ------- \+ (iii). " }}{PARA 257 "" 0 "" {TEXT -1 1 " " }{TEXT 264 8 "________" } {TEXT -1 19 " " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }} {PARA 0 "" 0 "" {TEXT -1 19 "This follows since " }{XPPEDIT 18 0 "k = \+ (x-z)/L;" "6#/%\"kG*&,&%\"xG\"\"\"%\"zG!\"\"F(%\"LGF*" }{TEXT -1 10 ", so that " }}{PARA 257 "" 0 "" {TEXT -1 2 " " }{XPPEDIT 18 0 "exp(z)* `.`*10^(k/4) = exp(z)*`.`*10^((x-z)/(4*L));" "6#/*(-%$expG6#%\"zG\"\" \"%\".GF))\"#5*&%\"kGF)\"\"%!\"\"F)*(-F&6#F(F)F*F))F,*&,&%\"xGF)F(F0F) *&F/F)%\"LGF)F0F)" }{TEXT -1 1 " " }}{PARA 257 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "`` = exp(z)*`.`*10^((x-z)/ln(10));" "6#/%!G*(-%$expG6# %\"zG\"\"\"%\".GF*)\"#5*&,&%\"xGF*F)!\"\"F*-%#lnG6#F-F1F*" }{TEXT -1 1 " " }}{PARA 257 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "``= exp(z)*` .`*10^((x-z)*log[10](exp(1)))" "6#/%!G*(-%$expG6#%\"zG\"\"\"%\".GF*)\" #5*&,&%\"xGF*F)!\"\"F*-&%$logG6#F-6#-F'6#F*F*F*" }{TEXT -1 1 " " }} {PARA 257 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "``= exp(z)*`.`*10^(lo g[10](exp(x-z))" "6#/%!G*(-%$expG6#%\"zG\"\"\"%\".GF*)\"#5-&%$logG6#F- 6#-F'6#,&%\"xGF*F)!\"\"F*" }{TEXT -1 1 " " }}{PARA 257 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "`` = exp(z)*`.`*exp(x-z);" "6#/%!G*(-%$expG6# %\"zG\"\"\"%\".GF*-F'6#,&%\"xGF*F)!\"\"F*" }{TEXT -1 1 " " }}{PARA 257 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "``= exp(x)" "6#/%!G-%$expG6 #%\"xG" }{TEXT -1 1 " " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT 259 4 "Note" }{TEXT -1 22 ": In order to compute " } {XPPEDIT 18 0 "10^(k/4)" "6#)\"#5*&%\"kG\"\"\"\"\"%!\"\"" }{TEXT -1 12 ", the value " }{XPPEDIT 18 0 "R=10^(1/4)" "6#/%\"RG)\"#5*&\"\"\"F( \"\"%!\"\"" }{TEXT -1 12 " is needed. " }}{PARA 0 "" 0 "" {TEXT -1 4 " If " }{XPPEDIT 18 0 "k=4*`.`*m+r" "6#/%\"kG,&*(\"\"%\"\"\"%\".GF(%\"m GF(F(%\"rGF(" }{TEXT -1 8 ", where " }{XPPEDIT 18 0 "0<=r" "6#1\"\"!% \"rG" }{XPPEDIT 18 0 "``<4" "6#2%!G\"\"%" }{TEXT -1 11 ", that is, " } {TEXT 286 1 "m" }{TEXT -1 5 " and " }{TEXT 287 1 "r" }{TEXT -1 9 " are the " }{TEXT 259 8 "quotient" }{TEXT -1 5 " and " }{TEXT 259 9 "remai nder" }{TEXT -1 20 " respectively, when " }{TEXT 288 1 "k" }{TEXT -1 23 " is divided by 4, then " }{XPPEDIT 18 0 "10^(k/4)=10^m*`.`*R^r" "6 #/)\"#5*&%\"kG\"\"\"\"\"%!\"\"*()F%%\"mGF(%\".GF()%\"RG%\"rGF(" } {TEXT -1 1 "." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 46 "To illustrate this idea, \+ we define a function " }{TEXT 0 4 "exp1" }{TEXT -1 52 " using Maple's \+ exponential function on the interval " }{XPPEDIT 18 0 "[-ln(10)/8, ln( 10)/8];" "6#7$,$*&-%#lnG6#\"#5\"\"\"\"\")!\"\"F,*&-F'6#F)F*F+F," } {TEXT -1 58 ", but ensure that it gives no value outside this interval ." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 106 "exp1 := x -> if abs(x)<=.2878231366 then exp(x) else FAIL end if;\nplot('exp1'(x),x=-0.5..0.5,thickness=2);" }}{PARA 11 " " 1 "" {XPPMATH 20 "6#>%%exp1Gf*6#%\"xG6\"6$%)operatorG%&arrowGF(@%1-% $absG6#9$$\"+m8ByG!#5-%$expGF0%%FAILGF(F(F(" }}{PARA 13 "" 1 "" {GLPLOT2D 334 248 248 {PLOTDATA 2 "6&-%'CURVESG6$7M7$$!37+++_7RwG!#=$ \"3V0_I6@K+vF*7$$!31+++CSSqGF*$\"38p`upS\"[](F*7$$!3++++'z;W'GF*$\"3PW (G%=(3$4vF*7$$!3#******zcH%eGF*$\"35?l'*eg!Q^(F*7$$!3@+++c1[MGF*$\"3il #>>PA=`(F*7$$!3%******RuJ0\"GF*$\"3'e%*)o$)=))\\vF*7$$!3'*******>RjiFF *$\"35j#\\#=48'e(F*7$$!3*******f4OZr#F*$\"3ro\\F%*RbAwF*7$$!38+++')G*o g#F*$\"3dM#R'=H?0xF*7$$!3!******\\n\\!*\\#F*$\"3!4xO(zzu)y(F*7$$!33+++ DwZ#G#F*$\"3oA0KX.FfzF*7$$!3-+++D.xt?F*$\"3Y?#f$H<8F\")F*7$$!37+++.TC% )=F*$\"3aNQK#4JEG)F*7$$!3!******>4z)e;F*$\"3Yu\"4g&=Tr%)F*7$$!33+++n`' zY\"F*$\"3sD2GRkpM')F*7$$!3++++>t)eC\"F*$\"3,Eh_+#*fG))F*7$$!3%******p h5$\\5F*$\"3!fTxc\"f'Q+*F*7$$!3W*******=[jL)!#>$\"3A\"zq-xm,?*F*7$$!3' *******pXg#G'Fjp$\"3aK#4:@o5R*F*7$$!3;+++]&Q(RTFjp$\"3=Q'[a$yZ%f*F*7$$ !3%*******4'=><#Fjp$\"31lj!3x\\^y*F*7$$!3I++++q*e$\\!#@$\"3EiN&4Kl]*** F*7$$\"33+++qRQb@Fjp$\"3zv^g,yy@5!#<7$$\"3!)******z\">Y2%Fjp$\"3N%R#\\ 3xeT5Fgr7$$\"3++++!yXu9'Fjp$\"3Yh7_MLSj5Fgr7$$\"3%)*******[y))G)Fjp$\" 3M0>L \"Fgr7$$\"3%)******=0ksGF*$\"3\"e^v!*3wFL\"Fgr7$%*undefinedGFhx-%'COLO URG6&%$RGBG$\"#5!\"\"$\"\"!FayF`y-%*THICKNESSG6#\"\"#-%+AXESLABELSG6$Q \"x6\"Q!Fjy-%%VIEWG6$;$!\"&F_y$\"\"&F_y%(DEFAULTG" 1 2 0 1 10 2 2 9 1 4 2 1.000000 45.000000 45.000000 0 0 "Curve 1" }}}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 25 "Now we define a function \+ " }{TEXT 0 4 "exp2" }{TEXT -1 40 " using the formulas (i), (ii) and (i ii)." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 587 "exp2 := proc(xx)\n local x,L,R,Q,k,m,r;\n\n x := evalf(xx);\n\n # Reduce the argument range to -ln(10)/8<=z<=ln(10)/ 8\n k := 0;\n if abs(x)>.2878231366 then\n L := evalf(ln(10)/ 4);\n R := evalf(10^(1/4)); \n k := floor(x/L+0.5);\n \+ x := x-k*L;\n end if;\n\n # evaluate 10^(k/4) using 10^(1/4)\n m := iquo(k,4);\n r := irem(k,4);\n if r<0 then\n r := r+4;\n \+ m := m-1;\n end if;\n if r<2 then\n if r<1 then Q := 1 e lse Q := R end if;\n else\n if r<3 then Q := R^2 else Q := R^3 \+ end if;\n end if;\n\n Scale10(exp1(x)*Q,m); # exp1(x)*10^m\nend pr oc;" }}{PARA 12 "" 1 "" {XPPMATH 20 "6#>%%exp2Gf*6#%#xxG6)%\"xG%\"LG% \"RG%\"QG%\"kG%\"mG%\"rG6\"F0C*>8$-%&evalfG6#9$>8(\"\"!@$2$\"+m8ByG!#5 -%$absG6#F3C&>8%-F56#,$*&#\"\"\"\"\"%FK-%#lnG6#\"#5FKFK>8&-F56#*$)FPFJ FK>F9-%&floorG6#,&*&F3FKFE!\"\"FK$\"\"&FgnFK>F3,&F3FK*&F9FKFEFKFgn>8)- %%iquoG6$F9FL>8*-%%iremGFao@$2FcoF:C$>Fco,&FcoFKFLFK>F^o,&F^oFKFKFgn@% 2Fco\"\"#@%2FcoFK>8'FK>FcpFR@%2Fco\"\"$>Fcp*$)FRF_pFK>Fcp*$)FRFgpFK-%( Scale10G6$*&-%%exp1GFBFKFcpFKF^oF0F0F0" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 22 "exp2(-1.5);\nexp(-1.5);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#$\"+-;IJA!#5" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#$\"+,;IJA!#5" }} }{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 46 "plot('exp2'(x),x=-3..3,color=red,thickness=2);" }}{PARA 13 "" 1 "" {GLPLOT2D 450 266 266 {PLOTDATA 2 "6'-%'CURVESG6#7V7$$!\"$\"\"!$\"+ Noqy\\!#67$$!+vq@pG!\"*$\"+&\\LVn&F-7$$!+^NUbFF1$\"+r$3#ejF-7$$!+K3XFE F1$\"+qOCEsF-7$$!+G)H')\\#F1$\"+GYv>#)F-7$$!+'3@/P#F1$\"+9r8W$*F-7$$!+ r^b^AF1$\"+\"QaB0\"!#57$$!+,kZG@F1$\"+A\\=!>\"FM7$$!+h\")=,?F1$\"+'yX< N\"FM7$$!+O\"3V(=F1$\"+W5hM:FM7$$!+NkzVFM7$$!+0)H%*\\\"F1$\"+UUdKAFM7$$!+vl[p8F1$\"+yXPUDFM7$$!+&>iUC \"F1$\"+1V`\")GFM7$$!+hkaI6F1$\"+%zn&GKFM7$$!*buK&**F1$\"+jR-'p$FM7$$! *?#z2))F1$\"+-$)fWTFM7$$!*\"RKvuF1$\"+!3O`t%FM7$$!*qjeH'F1$\"+D97G`FM7 $$!*\"*3=+&F1$\"+Q&4U1'FM7$$!*ui&pPF1$\"+o1YfoFM7$$!*7VQ[#F1$\"+#*4g+y FM7$$!*;^JI\"F1$\"+Ex=y()FM7$$!(Q:'HF1$\"+I%G/(**FM7$$\"*RIKH\"F1$\"+- x0Q6F17$$\"*^rZW#F1$\"+&[`pF\"F17$$\"*[n%)o$F1$\"+uf1Y9F17$$\"*5FL(\\F 1$\"+8&HVk\"F17$$\"*e6.B'F1$\"+&HrX'=F17$$\"*p3lW(F1$\"+Shq0@F17$$\"*A ))oz)F1$\"+0'\\,T#F17$$\"+Ik-,5F1$\"+)Qt5s#F17$$\"+D-eI6F1$\"++KX(4$F1 7$$\"+>_(zC\"F1$\"+HHG$[$F17$$\"+b*=jP\"F1$\"+MnHgRF17$$\"+4/3(\\\"F1$ \"+`MioWF17$$\"+C4JB;F1$\"+T%[)p]F17$$\"+DVsYn#f( =F1$\"+()['o_'F17$$\"+!)RO+?F1$\"+`gu\"R(F17$$\"+_!>w7#F1$\"+%Gb[R)F17 $$\"+*Q?QD#F1$\"+0?0C&*F17$$\"+5jypBF1$\"+g1^p5!\")7$$\"+Ujp-DF1$\"+Y' Q:A\"Fay7$$\"+gEd@EF1$\"+@Utv8Fay7$$\"+Mh%\\o#F1$\"+$=TdY\"Fay7$$\"+4' >$[FF1$\"+Aojh:Fay7$$\"+5h(*3GF1$\"+??Hf;Fay7$$\"+6EjpGF1$\"+vR0j " 0 "" {MPLTEXT 1 0 1 ";" }}}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 71 "An arbi trary precision procedure to evaluate the exponential function: " } {TEXT 0 6 "expnAP" }{TEXT -1 1 " " }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 22 "expnAP: imp lementation" }}{PARA 0 "" 0 "" {TEXT 259 4 "Note" }{TEXT -1 16 ": The \+ procedure " }{TEXT 0 6 "expnAP" }{TEXT -1 20 " uses two constants " } {XPPEDIT 18 0 "L=ln(10)/4" "6#/%\"LG*&-%#lnG6#\"#5\"\"\"\"\"%!\"\"" } {TEXT -1 5 " and " }{XPPEDIT 18 0 "R=10^(1/4)" "6#/%\"RG)\"#5*&\"\"\"F (\"\"%!\"\"" }{TEXT -1 2 ". " }}{PARA 0 "" 0 "" {TEXT -1 60 "This limi ts the precision of the procedure when calculating " }{XPPEDIT 18 0 "e xp(x)" "6#-%$expG6#%\"xG" }{TEXT -1 6 " with " }{XPPEDIT 18 0 "abs(x)> ln(10)/8" "6#2*&-%#lnG6#\"#5\"\"\"\"\")!\"\"-%$absG6#%\"xG" }{TEXT -1 1 "." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 8991 "expnAP := proc(xx::realcons)\n local x,term,eps,k ,i,maxit,saveDigits,\n extraDigits,sum,L,R,Q,r,m;\n\n # increase p recision \n saveDigits := Digits;\n Digits := Digits+length(Digit s)+1;\n x := evalf(xx);\n # argument reduction involves loss of pr ecision\n # increase precision further when x has large magnitude\n \+ extraDigits := max(ilog10(x),0);\n if extraDigits>0 then\n Di gits := Digits+extraDigits;\n x := evalf(xx)\n end if;\n \+ \n # Reduce the range to -ln(10)/8<=z<=ln(10)/8 approximately\n k := 0;\n if abs(x)>0.28782313662425571050 then\n L := evalf(Ln1 0)*0.25;\n R := evalf(root4of10);\n # Determine the number o f multiples of L=ln(10)/4 to add or subtract.\n k := floor(x/L+0. 5);\n x := x - k*L;\n end if;\n\n Digits := Digits - extraDig its;\n\n # Initialisation for Maclaurin series loop\n eps := Float (1,-saveDigits);\n maxit := Digits;\n\n # calculate exp(x)\n ter m := 1.0;\n sum := term;\n for i from 1 to maxit do\n term := term*x/i;\n sum := sum+term;\n if abs(term)<=eps*abs(sum) t hen break end if;\n end do;\n\n m := iquo(k,4,'r');\n if r<0 the n\n r := r+4;\n m := m-1;\n end if;\n if r<2 then\n \+ if r<1 then Q := 1 else Q := R end if;\n else\n if r<3 then Q \+ := R^2 else Q := R^3 end if;\n end if;\n\n sum := Scale10(sum*Q,m) ;\n Digits := saveDigits;\n evalf(sum);\nend proc: # of expnAP\n\n `evalf/constant/root4of10` := proc()\nlocal d,r,r2,r4;\nglobal _root4o f10;\n if Digits<=55 then evalf(1.7782794100389228012254211951926848 44735790526402255358)\n elif Digits<=length(op(1,_root4of10)) then e valf(_root4of10)\n else\n d := length(op(1,_root4of10));\n \+ r := _root4of10;\n while d " 0 "" {MPLTEXT 1 0 1 ";" }}}}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 28 "Testing the procedure expnAP" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 48 "plot('expnAP'(x),x=-3..3,color=red,thickn ess=2);" }}{PARA 13 "" 1 "" {GLPLOT2D 450 266 266 {PLOTDATA 2 "6'-%'CU RVESG6#7V7$$!\"$\"\"!$\"+Poqy\\!#67$$!+vq@pG!\"*$\"+(\\LVn&F-7$$!+^NUb FF1$\"+r$3#ejF-7$$!+K3XFEF1$\"+uOCEsF-7$$!+G)H')\\#F1$\"+GYv>#)F-7$$!+ '3@/P#F1$\"+9r8W$*F-7$$!+r^b^AF1$\"+\"QaB0\"!#57$$!+,kZG@F1$\"+A\\=!> \"FM7$$!+h\")=,?F1$\"+'yXFM7$$!+0)H%*\\\"F1$\"+TUdKAFM7$$!+vl [p8F1$\"+zXPUDFM7$$!+&>iUC\"F1$\"+3V`\")GFM7$$!+hkaI6F1$\"+'zn&GKFM7$$ !*buK&**F1$\"+mR-'p$FM7$$!*?#z2))F1$\"+/$)fWTFM7$$!*\"RKvuF1$\"+!3O`t% FM7$$!*qjeH'F1$\"+D97G`FM7$$!*\"*3=+&F1$\"+P&4U1'FM7$$!*ui&pPF1$\"+p1Y foFM7$$!*7VQ[#F1$\"+#*4g+yFM7$$!*;^JI\"F1$\"+Ex=y()FM7$$!(Q:'HF1$\"+I% G/(**FM7$$\"*RIKH\"F1$\"+-x0Q6F17$$\"*^rZW#F1$\"+&[`pF\"F17$$\"*[n%)o$ F1$\"+uf1Y9F17$$\"*5FL(\\F1$\"+8&HVk\"F17$$\"*e6.B'F1$\"+&HrX'=F17$$\" *p3lW(F1$\"+Rhq0@F17$$\"*A))oz)F1$\"+/'\\,T#F17$$\"+Ik-,5F1$\"+(Qt5s#F 17$$\"+D-eI6F1$\"+*>`u4$F17$$\"+>_(zC\"F1$\"+FHG$[$F17$$\"+b*=jP\"F1$ \"+LnHgRF17$$\"+4/3(\\\"F1$\"+aMioWF17$$\"+C4JB;F1$\"+U%[)p]F17$$\"+DV sYn#f(=F1$\"+))['o_'F17$$\"+!)RO+?F1$\"+dgu\"R(F1 7$$\"+_!>w7#F1$\"+%Gb[R)F17$$\"+*Q?QD#F1$\"+0?0C&*F17$$\"+5jypBF1$\"+g 1^p5!\")7$$\"+Ujp-DF1$\"+Y'Q:A\"Fay7$$\"+gEd@EF1$\"+?Utv8Fay7$$\"+Mh% \\o#F1$\"+#=TdY\"Fay7$$\"+4'>$[FF1$\"+Aojh:Fay7$$\"+5h(*3GF1$\"+>?Hf;F ay7$$\"+6EjpGF1$\"+uR0j " 0 "" {MPLTEXT 1 0 66 "xx:= 1. 5;\nDigits := 100:\nexpnAP(xx);\nevalf(exp(xx));\nDigits := 10:" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#>%#xxG$\"#:!\"\"" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#$\"_q\"R`CQJQ0\")Hx8nYum$fy#3]Excqmp$o)\\d+>eF>,Yb?gA[1 Q.2*o\"[%!#**" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#$\"_q\"R`CQJQ0\")Hx8n Yum$fy#3]Excqmp$o)\\d+>eF>,Yb?gA[1Q.2*o\"[%!#**" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 67 "xx:= -1.5:\nDigits := 100:\nexpnAP(xx);\nevalf(exp (xx));\nDigits := 10:" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#$\"_q3fxDLIPc x6W98jm^Kg(=`$QuG$z5OH; " 0 " " {MPLTEXT 1 0 1 ";" }}}}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 18 "Speed co mparisons " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 17 "Maple's function " }{TEXT 0 3 "exp" }{TEXT -1 16 " is faster th an " }{TEXT 0 6 "expnAP" }{TEXT -1 1 "." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 150 "st := time():\nfor i t o 500 do expnAP(rand()*Float(1,-11)) end do:\ntime()-st;\nst := time() :\nfor i to 500 do exp(rand()*Float(1,-11)) end do:\ntime()-st;" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#$\"$C*!\"$" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#$\"$\\\"!\"$" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 178 "st := time():\nDigits := 50 :\nfor i to 100 do expnAP(rand()*Float(1,-11)) end do:\ntime()-st;\nst := time():\nfor i to 100 do exp(rand()*Float(1,-11)) end do:\ntime()- st;\nDigits := 10:" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#$\"%%>\"!\"$" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#$\"#*)!\"$" }}}{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 " " }}}{MARK "4 0 0" 0 }{VIEWOPTS 1 1 0 1 1 1803 1 1 1 1 }{PAGENUMBERS 0 1 2 33 1 1 }