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" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 55 "We can make a comparison of finite sums approximating " }{XPPEDIT 18 0 "ln(1+x);" "6#-%#lnG6#,&\"\"\"F'%\"xGF'" }{TEXT -1 24 " graphically as follows." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 " " 0 "" {TEXT -1 27 "First we define a function " }{TEXT 0 2 "LN" } {TEXT -1 84 " with two arguments or input parameters. The degree of th e polynomial approximation " }{TEXT 262 7 "LN(n,x)" }{TEXT -1 4 " to \+ " }{XPPEDIT 18 0 "ln(1+x);" "6#-%#lnG6#,&\"\"\"F'%\"xGF'" }{TEXT -1 4 " is " }{XPPEDIT 18 0 "n;" "6#%\"nG" }{TEXT -1 2 ".\n" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 101 "x := 'x': i := 'i': n := 'n':\nLN := (n, x) -> sum((-1)^(i+1)*x^i/i,i=1..n);\nLN(2,x);\nLN(3,x);\nLN(4,x);" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#>%#LNGf*6$%\"nG%\"xG6\"6$%)operatorG%& arrowGF)-%$sumG6$*()!\"\",&%\"iG\"\"\"F5F5F5)9%F4F5F4F2/F4;F59$F)F)F) " }}{PARA 11 "" 1 "" {XPPMATH 20 "6#,&%\"xG\"\"\"*&#F%\"\"#F%*$)F$F(F% F%!\"\"" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#,(%\"xG\"\"\"*&#F%\"\"#F%*$ )F$F(F%F%!\"\"*&#F%\"\"$F%)F$F.F%F%" }}{PARA 11 "" 1 "" {XPPMATH 20 "6 #,*%\"xG\"\"\"*&#F%\"\"#F%*$)F$F(F%F%!\"\"*&#F%\"\"$F%)F$F.F%F%*&#F%\" \"%F%*$)F$F2F%F%F+" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 34 "evalf(LN(10,0.2));\nevalf(ln(1.2));" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#$\"+_b@B=!#5" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#$\"+ob@B=!#5" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 38 "The degree 1, 2 and 3 approximations " } {XPPEDIT 18 0 "y = x;" "6#/%\"yG%\"xG" }{TEXT -1 3 " , " }{XPPEDIT 18 0 "y = x-x^2/2;" "6#/%\"yG,&%\"xG\"\"\"*&F&\"\"#F)!\"\"F*" }{TEXT -1 6 " and " }{XPPEDIT 18 0 "y = x-x^2/2+x^3/3;" "6#/%\"yG,(%\"xG\"\"\"* &F&\"\"#F)!\"\"F**&F&\"\"$F,F*F'" }{TEXT -1 28 " can be plotted as fo llows." }}{PARA 0 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "y = ln(1+x); " "6#/%\"yG-%#lnG6#,&\"\"\"F)%\"xGF)" }{TEXT -1 33 " is also plotted f or comparison. " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 175 "LN := (n,x) -> sum((-1)^(i+1)*x^i/i,i=1..n):\nplot([ln(1+x),LN(1,x),LN(2,x),LN(3, x)],x=-1..1.5,-.5..1.5,\n color=[black,red,green,blue],linestyl e=[2,1$3],\nnumpoints=100);" }}{PARA 13 "" 1 "" {GLPLOT2D 526 299 299 {PLOTDATA 2 "6(-%'CURVESG6%7iq7$$!31+++-Nu\"***!#=$!3U/()GX)R$*4(!#<7$ $!37+++/q[$)**F*$!3.f(GZm#>1kF-7$$!3;+++10Bv**F*$!3K&f?mbF2+'F-7$$!3I+ ++4S(p'**F*$!38)*zW9b/8dF-7$$!3U+++85Y]**F*$!3.snC'R!e2`F-7$$!3\\***** z,[R$**F*$!3'e.))QL)*)>]F-7$$!3z*****p-A4!**F*$!3Zx)zdALWh%F-7$$!35+++ Og*y')*F*$!3M*4GL:^nK%F-7$$!3e*****R0W=!)*F*$!32K*>_/'G@RF-7$$!35+++r? zN(*F*$!3xbK)*oRgLOF-7$$!3)******z6\\3i*F*$!3!\\%edShSsKF-7$$!3x*****R ;1f]*F*$!32U(49#\\h2IF-7$$!3;+++>aPZ#*F*$!3Gz)4H$Qx'e#F-7$$!3w*****\\' 48()*)F*$!3O#[*Qg\")z*G#F-7$$!3A+++2M7G()F*$!3#\\Vk3l\"4i?F-7$$!3)**** **\\C!*z[)F*$!3/Sz`j`9*)=F-7$$!3X+++liMR#)F*$!3Ci*pD\"***ot\"F-7$$!37+ 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"" {MPLTEXT 1 0 1 ";" }}}}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 38 "An adaptive graph plotting procedure: " }{TEXT 0 5 "gr aph" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 79 "Th e procedure in this section is an alternative to the standard Maple fu nction " }{TEXT 0 4 "plot" }{TEXT -1 195 ", for plotting the graph of \+ a single function. It allows control over how the plotting is performe d, in a manner which is a bit different 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 parti cular, the precision of the floating point calculations is not changed internally, whereas, the Maple procedure " }{TEXT 0 4 "plot" }{TEXT -1 137 " often uses hardware floating point arithmetic whereby the eff ective precision is increased to the equivalent of about 15 decimal di gits." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 12 "graph: usag e" }}{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 variable, say x, or a function x -> \+ f(x)\n\n " }{TEXT 23 7 "xrng - " }{TEXT -1 79 " horizontal plot ting 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), whi ch can be given in the form c..d, or 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 method." }}{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 ar e plotted along sections of the curve where the graph is bending." }} {PARA 15 "" 0 "" {TEXT -1 107 "Any maximum or minimum points are locat ed approximately by parabolic interpolation and added to the graph.\n " }}{PARA 256 "" 0 "" {TEXT 273 8 "Options:" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 19 "adaptive=true/false" }}{PARA 0 "" 0 "" {TEXT -1 144 "Adaptive plotting will, where necessary, sub-div ide the plotting interval in an attempt to get a good graphical repres entation 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 "numpoints=n\nFor non-adaptive plotti ng the interval for the plot is subdivided into a fixed number of sub- intervals of equal width by means \"numpoints\" points. \nFor adaptive plotting \"numpoints\" controls the spacing of points along the curve , that is, in the direction of the curve rather than just horizontally . " }}{PARA 0 "" 0 "" {TEXT -1 57 "The spacing between points is gener ally no greater than " }{XPPEDIT 18 0 "plotwidth/numpoints;" "6#*&%*p lotwidthG\"\"\"%*numpointsG!\"\"" }{TEXT -1 97 ". Note that, in genera l, the number of points plotted could be vastly different from \"numpo ints\"." }}{PARA 0 "" 0 "" {TEXT -1 36 "The default value is \"numpoin ts=33\"." }}{PARA 0 "" 0 "" {TEXT -1 12 "\nlinearity=n" }}{PARA 0 "" 0 "" {TEXT -1 82 "\"linearity\" controls the 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 dev iate no more than " }{TEXT 277 16 "tol * plotheight" }{TEXT -1 62 " fr om 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 def ault value is \"linearity=10\"." }}{PARA 0 "" 0 "" {TEXT -1 12 "\nmaxp oints=n" }}{PARA 0 "" 0 "" {TEXT -1 108 "This option provides a cut-of f for the adaptive subdivision by specifying a minimum horizontal dist ance 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\" 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 standard plot options" }{TEXT -1 184 ": color, linestyle, thickness, scaling, xtickmarks, yt ickmarks, tickmarks, labels, style, symbol, symbolsize, title, axes, f ont, labelfont, titlefont, axesfont, view, labeldirections." }}{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, place the cursor any where after the prompt [ > and press [Enter].\nYou can then close up \+ the subsection." }}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 21 "graph: impleme ntation" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "grap h" {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,xrang e,yrange,Options,startoptions,rs,\n mxpts,lnrty,nmpts,delta,pdat, adptv,curve,dev,m,addmaxmin,\n fn,n1,n2,proctype,vars,y1,y2,testv als,aa,bb,i,opt;\n\n if nargs<2 then\n error \"at least 2 argum ents are required; the basic syntax is: 'graph(f(x),x=a..b)'.\"\n en d 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 type(rng,range) then\n \+ rs := rng;\n else\n error \"the 2nd argument, %1, is inv alid .. it should have the form 'a..b' to provide a horizontal range o ver which to plot the graph of %1\",rng,ff;\n end if;\n elif ty pe(ff,algebraic) then \n vars := indets(ff,name) minus indets(ff, realcons);\n if nops(vars)<>1 then \n if not type(ff,real cons) and not has(indets(ff),\{Int,Sum\}) then\n error \"th e 1st argument, %1, is invalid .. it should be an expression which dep ends 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,realcons) then\n erro r \"the 1st argument, %1, is invalid .. it should be an expression whi ch depends only on the variable %2\",ff,x;\n end if;\n \+ rs := op(2,rng);\n else\n error \"the 2nd argument, %1, \+ is invalid .. it should have the form '%2=a..b' to provide a horizonta l 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 value d procedure with a single argument\",ff; \n end if;\n \n xL := evalf(op(1,rs));\n xR := evalf(op(2,rs));\n if not type(xL,num eric) or not type(xR,numeric) then\n error \"each end point of th e horizontal range %1 must evaluate to a numeric\",rs;\n end if;\n \+ if xL>=xR then\n error \"2nd argument horizontal range is invali d\";\n end if;\n if proctype then\n xrange := xL..xR;\n els e\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) th en\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 n umeric\",rs;\n end if;\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 := false;\n adptv := true;\n\n if nargs>=s tartoptions then\n Options := [args[startoptions..nargs]];\n \+ if not type(Options,list(equation)) then\n error \"each optio nal argument after the %-1 argument must be an equation\",startoptions -1;\n end if;\n if hasoption(Options,'adaptive','adptv','Opt ions') then\n if not adptv=true then 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 integer greater than 1\"\n \+ end if;\n end if;\n if hasoption(Options,'linearity','ln rty','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 if not type(mxpts,posint) then\n \+ error \"\\\"maxpoints\\\" must be a positive integer\"\n end \+ if;\n end if;\n if hasoption(Options,'plotdata','pdat','Opti ons') or\n hasoption(Options,'plot_data','pdat','Options') the n\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','lines tyle','line_style','thickness',\n 'scaling','xtickmarks','y tickmarks','tickmarks','labels',\n 'style','symbol','symbol size','title','axes','font',\n 'labelfont','label_font','ti tlefont','title_font',\n 'axesfont','axes_font','view','lab eldirections'\}) 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 the n\n # Update estimate 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 height.\n if y3>ymax then ymax : = y3; height := ymax-ymin end if;\n if y34 then\n error \"1st argument %1 does not evaluat e 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 ymi n := 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 de lta := evalf(1/mxpts);\n nmpts := iquo(nmpts,2);\n h := eval f(1/nmpts)^2;\n curve := addmaxmin(adaptdiv([xL,yL],[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 else\n nmpts := nmpts-1;\n h := width/nmpts;\n n1 := iquo(nmp ts,2);\n n2 := nmpts-n1-1;\n curve := [[xL,yL],seq([xL+i*h,e valf(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,o p(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 79 "A procedure for evaluating the natural logarith m function from a Taylor series " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }} {PARA 0 "" 0 "" {TEXT -1 24 "By a change of variable " }{XPPEDIT 18 0 "u=x-1" "6#/%\"uG,&%\"xG\"\"\"F'!\"\"" }{TEXT -1 26 " the Maclaurin se ries for " }{XPPEDIT 18 0 "ln(1+u);" "6#-%#lnG6#,&\"\"\"F'%\"uGF'" } {TEXT -1 46 " gives rise to the Taylor series expansion of " } {XPPEDIT 18 0 "ln(x)" "6#-%#lnG6#%\"xG" }{TEXT -1 7 " about " } {XPPEDIT 18 0 "x=1" "6#/%\"xG\"\"\"" }{TEXT -1 1 "." }}{PARA 257 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "Sum((-1)^(n+1)*(x-1)^n/n,n = 1 .. in finity) = x-1-(x-1)^2/2+(x-1)^3/3-(x-1)^4/4+(x-1)^5/5-(x-1)^6/6+` . . \+ . `;" "6#/-%$SumG6$*(),$\"\"\"!\"\",&%\"nGF*F*F*F*),&%\"xGF*F*F+F-F*F- F+/F-;F*%)infinityG,2F0F*F*F+*&,&F0F*F*F+\"\"#F7F+F+*&,&F0F*F*F+\"\"$F :F+F**&,&F0F*F*F+\"\"%F=F+F+*&,&F0F*F*F+\"\"&F@F+F**&,&F0F*F*F+\"\"'FC F+F+%(~.~.~.~GF*" }{TEXT -1 1 " " }}{PARA 0 "" 0 "" {TEXT -1 41 "Altho ugh this Taylor series converges to " }{XPPEDIT 18 0 "ln(x);" "6#-%#ln G6#%\"xG" }{TEXT -1 15 " for values of " }{TEXT 268 1 "x" }{TEXT -1 11 " such that " }{XPPEDIT 18 0 "0 " 0 "" {MPLTEXT 1 0 648 "ln_series := proc(xx::realcons)\n local x,pow,sum,term,eps,k,ma xit,even;\n\n maxit := Digits*20;\n eps := Float(2,-Digits);\n\n \+ x := evalf(1-xx);\n if x>1 or x<=-1 then\n error \"Taylor seri es only converges in the interval (0,2]\"\n end if;\n\n pow := x; \n sum := pow;\n even := false; \n for k from 2 to maxit do\n \+ pow := pow*x;\n term := pow/k;\n sum := sum + term;\n \+ if abs(term)<=eps*abs(sum) then break end if;\n end do;\n\n if k >maxit then\n print(`sum of`,k-1,`terms of series is`,sum);\n \+ error \"reached the maximum number, %1, of iterations without conver gence\",maxit;\n end if;\n evalf(-sum);\nend proc:" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 95 "When using the defau lt precision of 10 digits, we can get about 10 digit accuracy in compu ting " }{XPPEDIT 18 0 "ln(x)" "6#-%#lnG6#%\"xG" }{TEXT -1 27 " using t his procedure with " }{XPPEDIT 18 0 "1/10 < x;" "6#2*&\"\"\"F%\"#5!\" \"%\"xG" }{XPPEDIT 18 0 "``<19/10" "6#2%!G*&\"#>\"\"\"\"#5!\"\"" } {TEXT -1 2 ". " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 55 "xx := 1.333521432;\nevalf(ln_series(xx));\nevalf (ln(xx));" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%#xxG$\"+K9_L8!\"*" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#$\"+l8ByG!#5" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#$\"+l8ByG!#5" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 43 "We can check the accuracy of the procedure " } {TEXT 0 9 "ln_series" }{TEXT -1 12 " as follows." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 59 "plot(ln(x)-' evalf[10]@ln_series'(x),x=0.1..1.9,color=blue);" }}{PARA 13 "" 1 "" {GLPLOT2D 422 258 258 {PLOTDATA 2 "6&-%'CURVESG6#7]]n7$$\"\"\"!\"\"$F* !\"*7$$\"+**3E75!#5$!\"&F,7$$\"+)z@X-\"F0$!\"'F,7$$\"+)p#yO5F0F+7$$\"+ (fV!\\5F0$\"\"(F,7$$\"+'\\/81\"F0$!\"(F,7$$\"+'RlN2\"F0$!\"#F,7$$\"+&H Ee3\"F0$\"\"!FN7$$\"+%>(3)4\"F0$\"\"$F,7$$\"+$4[.6\"F0FR7$$\"+#**3E7\" F0$!\"$F,7$$\"+#*)p[8\"F0$\"\"#F,7$$\"+\"zIr9\"F0FH7$$\"+!p\"Rf6F0FH7$ 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{TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }} }}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 18 "Argument reduction" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}{PARA 0 "" 0 "" {TEXT -1 19 "T he Maple function " }{TEXT 0 6 "ilog10" }{TEXT -1 20 " gives the expon ent " }{XPPEDIT 18 0 "k = ilog10(x);" "6#/%\"kG-%'ilog10G6#%\"xG" } {TEXT -1 11 " such that " }{XPPEDIT 18 0 "10^k <= x;" "6#1)\"#5%\"kG% \"xG" }{XPPEDIT 18 0 "`` < 10^(k+1);" "6#2%!G)\"#5,&%\"kG\"\"\"F)F)" } {TEXT -1 1 "." }}{PARA 0 "" 0 "" {TEXT -1 10 "Then, if " }{XPPEDIT 18 0 "u = x*`.`*10^(-k);" "6#/%\"uG*(%\"xG\"\"\"%\".GF')\"#5,$%\"kG!\" \"F'" }{TEXT -1 10 ", we have " }{XPPEDIT 18 0 "1<=u" "6#1\"\"\"%\"uG " }{XPPEDIT 18 0 "``<10" "6#2%!G\"#5" }{TEXT -1 1 "." }}{PARA 0 "" 0 " " {TEXT -1 80 "Since Maple's software floating point arithmetic base i s 10, the computation of " }{TEXT 262 9 "ilog10(x)" }{TEXT -1 75 " is \+ very efficient, and does not require the computation of any logarithms ." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT 263 8 "Exam ples" }{TEXT -1 2 ": " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 98 "ilog10(0.012345);\nilog10(0.12345);\nilog 10(1.2345);\nilog10(12.345);\nilog10(123.45);\nilog10(1234.5);" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#!\"#" }}{PARA 11 "" 1 "" {XPPMATH 20 " 6#!\"\"" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#\"\"!" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#\"\"\"" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#\"\"#" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#\"\"$" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 14 "Note that, if " }{XPPEDIT 18 0 "x = u*`.`*10^k;" "6#/%\"x G*(%\"uG\"\"\"%\".GF')\"#5%\"kGF'" }{TEXT -1 7 ", then " }}{PARA 257 " " 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "ln(x) = ln(u)+k*ln(10);" "6#/-%# lnG6#%\"xG,&-F%6#%\"uG\"\"\"*&%\"kGF,-F%6#\"#5F,F," }{TEXT -1 13 " --- ---- (i)." }}{PARA 257 "" 0 "" {TEXT -1 1 " " }{TEXT 264 23 "_________ ___ " }{TEXT -1 1 " " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }} {PARA 0 "" 0 "" {TEXT -1 104 "This formula can be used to reduce the a rgument range of the natural logarithm function to the interval " } {XPPEDIT 18 0 "[1,10]" "6#7$\"\"\"\"#5" }{TEXT -1 44 " at the cost of \+ storing the constant ln(10)." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 58 "The argument range can be further reduced to th e interval " }{XPPEDIT 18 0 "[1, R];" "6#7$\"\"\"%\"RG" }{TEXT -1 9 " \+ , where " }{XPPEDIT 18 0 "R = 10^(1/8);" "6#/%\"RG)\"#5*&\"\"\"F(\"\") !\"\"" }{TEXT -1 13 ", as follows." }}{PARA 0 "" 0 "" {TEXT -1 1 " " } }{PARA 0 "" 0 "" {TEXT -1 7 "Divide " }{TEXT 278 1 "u" }{TEXT -1 17 " \+ successively by " }{TEXT 279 1 "R" }{TEXT -1 11 " to obtain " }{TEXT 265 1 "v" }{TEXT -1 6 " with " }{XPPEDIT 18 0 "1<=v" "6#1\"\"\"%\"vG" }{XPPEDIT 18 0 "`` " 0 "" {MPLTEXT 1 0 106 "ln1 := x -> if x >=1 and x<=1.333521432 then ln(x) else FAIL end if;\nplot('ln1'(x),x=0 .9..1.4,thickness=2);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%$ln1Gf*6#% \"xG6\"6$%)operatorG%&arrowGF(@%31\"\"\"9$1F0$\"+K9_L8!\"*-%#lnG6#F0%% FAILGF(F(F(" }}{PARA 13 "" 1 "" {GLPLOT2D 400 300 300 {PLOTDATA 2 "6&- %'CURVESG6$7N7$$\"3!******\\El-+\"!#<$\"35+RgM#)H_E!#@7$$\"3#*******Q^ g+5F*$\"3_NoFs(f&\\gF-7$$\"3/+++))[G,5F*$\"3/cG#yBjSG\"!#?7$$\"33+++OY '>+\"F*$\"3)pJsmi3F'>F87$$\"3,+++LTK.5F*$\"3AT->$G?'=LF87$$\"3%******* HOo/5F*$\"3u)eqC$fpsYF87$$\"3'******>Tt%45F*$\"3#3vBHj?)G%*F87$$\"31++ +&>jU,\"F*$\"3_))4GjKC;9!#>7$$\"35+++;v/D5F*$\"3AU'3*eo*QZ#FR7$$\"35++ +>h(e.\"F*$\"3f#[lK/cZ_$FR7$$\"33+++%[6j/\"F*$\"3bQ&=kr5r_%FR7$$\"33++ +&z(yb5F*$\"3VDDgQ8sGaFR7$$\"3/+++Yg0n5F*$\"37+S0l(\\.\\'FR7$$\"33+++K \"F* $\"3eKdBnMm^>?\"F*$\"3`Q*=(\\m>R=Fjp7$$\"3)******* QU077F*$\"3*[ifHQmJ#>Fjp7$$\"35+++-uIB7F*$\"3J&*\\)*e]@W7F*$\"3mAMOU#[]=#Fjp7$$\"3!* *****\\$z*RD\"F*$\"3-0\"*yazOjAFjp7$$\"3++++YKpk7F*$\"3)f,E\\*fH[BFjp7 $$\"31+++,nvu7F*$\"3j\\JcwLbFCFjp7$$\"3'*******4fF&G\"F*$\"37$>BD6M(4D Fjp7$$\"33+++g.c&H\"F*$\"3EO0%)>JV*e#Fjp7$$\"3#******pAFjI\"F*$\"3cJsJ hb>sEFjp7$$\"31+++)*pp;8F*$\"33?GNkKE^FFjp7$$\"3'******HH**>K\"F*$\"3% \\(eLm?X\"z#Fjp7$$\"34+++)e,tK\"F*$\"3Ai2j++[JGFjp7$$\"3#******>yI*H8F *$\"3)>C.F(*o7&GFjp7$$\"31+++x*fDL\"F*$\"33WqEp)=5(GFjp7$$\"3\"******f i))GL\"F*$\"3SH6=j[[tGFjp7$$\"31+++ws@L8F*$\"3#=>LmD]f(GFjp7$%*undefin edGF^y-%'COLOURG6&%$RGBG$\"#5!\"\"$\"\"!FgyFfy-%*THICKNESSG6#\"\"#-%+A XESLABELSG6$Q\"x6\"Q!F`z-%%VIEWG6$;$\"\"*Fey$\"#9Fey%(DEFAULTG" 1 2 0 1 10 2 2 9 1 4 2 1.000000 45.000000 45.000000 0 0 "Curve 1" }}}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 406 "ln2 := proc(xx)\n local x,u,R,L, k,r;\n\n x := evalf(xx);\n if x=0 then return Float(-infinity) end if;\n\n k := ilog10(x);\n u := Scale10(x,-k); # u=x*10^(-k)\n r := 0; \n if u>1.333521432 then\n R := evalf(10^(1/8));\n \+ while u>1.333521432 do\n u := u/R;\n r := r+1;\n \+ end do;\n end if;\n if k<>0 or r<> 0 then L := evalf(ln(10)/8) end if;\n ln1(u)+(8*k+r)*L;\nend proc; " }}{PARA 12 "" 1 "" {XPPMATH 20 "6#>%$ln2Gf*6#%#xxG6(%\"xG%\"uG%\"RG%\"LG%\"kG%\"rG6\"F/C*>8$-%&eva lfG6#9$@$/F2\"\"!O$!\"\"%)infinityG>8(-%'ilog10G6#F2>8%-%(Scale10G6$F2 ,$F?F<>8)F9@$2$\"+K9_L8!\"*FDC$>8&-F46#*$)\"#5#\"\"\"\"\")FY?(F/FYFYF/ FLC$>FD*&FDFYFRF<>FJ,&FJFYFYFY@$50F?F90FJF9>8'-F46#,$*&FXFY-%#lnG6#FWF YFY,&-%$ln1G6#FDFY*&,&*&FZFYF?FYFYFJFYFYF`oFYFYF/F/F/" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 26 "plot('ln2'(x),x=0.01..10);" }}{PARA 13 "" 1 " " {GLPLOT2D 400 300 300 {PLOTDATA 2 "6%-%'CURVESG6$7gn7$$\"\"\"!\"#$!+ '=q^g%!\"*7$$\"+1*z/o\"!#6$!+u24'3%F-7$$\"+7)f4O#F1$!+[>5YPF-7$$\"+=(R 9/$F1$!+\">RG\\$F-7$$\"+D'>>s$F1$!+A1$4H$F-7$$\"+P%zG3&F1$!+mAHzHF-7$$ \"+]#RQW'F1$!+mc/UFF-7$$\"+v)ed;*F1$!+0bp*Q#F-7$$\"+]yw)=\"!#5$!+UxmH@ F-7$$\"+v<:LsTFU$!+v7UT()FU7$$\"+RO%HI'FU$!+SKo:YFU7$$\"+sL\"yW)FU$!+ kXx'o\"FU7$$\"+\"*)[#e5F-$\"+Y_bhcF17$$\"+S1;c7F-$\"+!e*f!G#FU7$$\"+#z '3h9F-$\"+s`!=z$FU7$$\"+7<-t;F-$\"+)*RJY^FU7$$\"+`pF%)=F-$\"+#oTaL'FU7 $$\"+O*y:5#F-$\"+m$*)oU(FU7$$\"+S'yHH#F-$\"+%)o^)H)FU7$$\"+v$\\%3DF-$ \"+Qyk'>*FU7$$\"+`[![s#F-$\"+7oR-5F-7$$\"+XMILHF-$\"+W#Hh2\"F-7$$\"+U, kAJF-$\"+W)y'Q6F-7$$\"+(yzxM$F-$\"+wtH37F-7$$\"+*f-&QNF-$\"+WOqj7F-7$$ \"+p&e.w$F-$\"+sTF-$\"+\">V% G9F-7$$\"+/yOxVF-$\"+'eZkZ\"F-7$$\"+?,W\"f%F-$\"+FP>C:F-7$$\"+J`-)y%F- $\"+z!=hc\"F-7$$\"+S!p++&F-$\"+?cF-$\"+=oF-7$$\"+R( yG3(F-$\"+F/od>F-7$$\"+g5d'H(F-$\"+@XS()>F-7$$\"+\"))Qw\\(F-$\"+c\")e9 ?F-7$$\"+)o7yq(F-$\"+\\WBU?F-7$$\"+,gH8zF-$\"+%QW&o?F-7$$\"+()zTG\")F- $\"+:jO&4#F-7$$\"+FggN$)F-$\"+Bi`?@F-7$$\"+Ad[Z&)F-$\"+srjX@F-7$$\"+Z4 hd()F-$\"+XJ#*p@F-7$$\"+1Up]*)F-$\"+%4J<>#F-7$$\"+5%*)><*F-$\"+6U:;AF- 7$$\"+z%=*p$*F-$\"+'R/vB#F-7$$\"+\\@&4e*F-$\"+xpxfAF-7$$\"+)HQHy*F-$\" +z)R1G#F-7$$\"#5\"\"!$\"+$4&e-BF--%'COLOURG6&%$RGBG$Fi]l!\"\"$Fj]lFj]l Fc^l-%+AXESLABELSG6$Q\"x6\"Q!Fh^l-%%VIEWG6$;F(Fh]l%(DEFAULTG" 1 2 0 1 10 0 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 52 "The following exa mple illustrates how the procedure " }{TEXT 0 3 "ln2" }{TEXT -1 8 " wo rks. " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 52 "xx := 9;\nprintlevel := 10:\nln2(xx);\nprintlevel := \+ 1:" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%#xxG\"\"*" }}{PARA 9 "" 1 "" {TEXT -1 24 "\{--> enter ln2, args = 9" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"xG$\"\"*\"\"!" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"kG\"\"! " }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"uG$\"\"*\"\"!" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"rG\"\"!" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>% \"RG$\"+K9_L8!\"*" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"uG$\"+&)y/\\n !\"*" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"rG\"\"\"" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"uG$\"+G>2h]!\"*" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"rG\"\"#" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"uG$\"+K&o_z$!\"* " }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"rG\"\"$" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"uG$\"+&*)\\g%G!\"*" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"rG\"\"%" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"uG$\"+OjBM@!\"* " }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"rG\"\"&" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"uG$\"+q9X+;!\"*" }}{PARA 11 "" 1 "" {XPPMATH 20 "6# >%\"rG\"\"'" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"uG$\"+!Hp,?\"!\"*" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"rG\"\"(" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"LG$\"+m8ByG!#5" }}{PARA 9 "" 1 "" {TEXT -1 34 "\{-- > enter ln1, args = 1.200169290" }}{PARA 9 "" 1 "" {TEXT -1 40 "<-- ex it ln1 (now in ln2) = .1824626218\}" }}{PARA 11 "" 1 "" {XPPMATH 20 "6 #$\"+yXA(>#!\"*" }}{PARA 9 "" 1 "" {TEXT -1 46 "<-- exit ln2 (now at t op level) = 2.197224578\}" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#$\"+yXA(> #!\"*" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 14 "evalf(ln(xx));" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#$\" +xXA(>#!\"*" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 77 "An arbi trary precision procedure to evaluate the natural logarithm function: \+ " }{TEXT 0 5 "logAP" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}} {SECT 1 {PARA 4 "" 0 "" {TEXT -1 21 "logAP: implementation" }}{PARA 0 "" 0 "" {TEXT 259 4 "Note" }{TEXT -1 16 ": The procedure " }{TEXT 0 5 "logAP" }{TEXT -1 20 " uses two constants " }{XPPEDIT 18 0 "ln(10);" " 6#-%#lnG6#\"#5" }{TEXT -1 5 " and " }{XPPEDIT 18 0 "10^(-1/8);" "6#)\" #5,$*&\"\"\"F'\"\")!\"\"F)" }{TEXT -1 2 ". " }}{PARA 0 "" 0 "" {TEXT -1 60 "This limits the precision of the procedure when calculating " } {XPPEDIT 18 0 "ln(x);" "6#-%#lnG6#%\"xG" }{TEXT -1 6 " with " } {XPPEDIT 18 0 "10^(1/8) < x;" "6#2)\"#5*&\"\"\"F'\"\")!\"\"%\"xG" } {TEXT -1 1 "." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 9660 "logAP := proc(xx::realcons)\n local x,term,e ps,k,i,maxit,saveDigits,\n extraDigits,sum,u,t,c,c2,c4,L,r,m,pow;\n \+ \n # Increase precision\n saveDigits := Digits;\n Digits := Di gits+length(Digits)+1;\n x := evalf(xx);\n\n if x=0 then return Fl oat(-infinity) end if;\n \n # Reduce the range to 1<=z<=10^(1 /8) approximately\n k := ilog10(x);\n u := Scale10(x,-k); # u=z*10 ^(-k)\n r := 0;\n if u > 1.3335214321633240257 then\n c := evalf(root8of10inv);\n # Multiply successively by 10^(-1/8)\n \+ # but the no. of multiplications needed can be reduced ..\n if u>3.1622776601683793320 then\n c2 := c*c;\n c4 := c2* c2;\n if u>5.6234132519034908039 then # u>c^6=10^(3/4) \n \+ if u>7.4989420933245582730 then # u>c^7=10^(7/8)\n \+ u := u*c4*c2*c;\n r := 7;\n else\n \+ u := u*c4*c2;\n r := 6;\n end if;\n \+ else\n if u>4.2169650342858224857 then # u>c^5=10^(5/8 )\n u := u*c4*c;\n r := 5;\n el se\n u := u*c4;\n r := 4;\n end if;\n end if;\n else\n if u>1.778279410038922801 2 then # u>c^2=10^(1/4)\n c2 := c*c;\n if u>2.37 13737056616552617 then\n u := u*c2*c;\n r \+ := 3;\n else\n u := u*c2;\n r : = 2;\n end if;\n else\n u := u*c;\n \+ r := 1;\n end if;\n end if;\n end if; \n\n \+ # Initialisation for Maclaurin series loop\n eps := Float(1,-saveDi gits);\n maxit := Digits*4;\n\n t := 1.0-u;\n pow := t;\n sum \+ := pow;\n for i from 2 to maxit do\n pow := pow*t;\n term \+ := pow/i;\n sum := sum+term;\n if abs(term)<=eps*abs(sum) th en break end if;\n end do;\n sum := -sum;\n\n if k<>0 or r<>0 th en\n L := evalf(Ln10)*0.125;\n sum := sum + (8*k+r)*L\n en d if;\n\n Digits := saveDigits;\n evalf(sum);\nend proc: # of logA P\n\n`evalf/constant/root8of10inv` := proc()\nlocal d,r,r2,r4;\nglobal _root8of10inv;\n if Digits<=55 then evalf(.749894209332455827302184 2756151364384418679181649710146)\n elif Digits<=length(op(1,_root8of 10inv)) then evalf(_root8of10inv)\n else\n d := length(op(1,_ro ot8of10inv));\n r := _root8of10inv;\n while d " 0 "" {MPLTEXT 1 0 1 ";" }}}}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 27 "Testing the procedure logAP" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 50 "plot('logAP'(x),x=0.01..10,color=red,thic kness=2);" }}{PARA 13 "" 1 "" {GLPLOT2D 450 266 266 {PLOTDATA 2 "6'-%' CURVESG6#7gn7$$\"\"\"!\"#$!+'=q^g%!\"*7$$\"+1*z/o\"!#6$!+v24'3%F-7$$\" +7)f4O#F1$!+]>5YPF-7$$\"+=(R9/$F1$!+\">RG\\$F-7$$\"+D'>>s$F1$!+B1$4H$F -7$$\"+P%zG3&F1$!+nAHzHF-7$$\"+]#RQW'F1$!+nc/UFF-7$$\"+v)ed;*F1$!+2bp* Q#F-7$$\"+]yw)=\"!#5$!+UxmH@F-7$$\"+v<:LsTFU$!+%G@9u)FU7$$\"+RO%HI'FU$ !+WKo:YFU7$$\"+sL\"yW)FU$!+xXx'o\"FU7$$\"+\"*)[#e5F-$\"+Y_bhcF17$$\"+S 1;c7F-$\"+!e*f!G#FU7$$\"+#z'3h9F-$\"+p`!=z$FU7$$\"+7<-t;F-$\"+)*RJY^FU 7$$\"+`pF%)=F-$\"+!oTaL'FU7$$\"+O*y:5#F-$\"+l$*)oU(FU7$$\"+S'yHH#F-$\" +&)o^)H)FU7$$\"+v$\\%3DF-$\"+Myk'>*FU7$$\"+`[![s#F-$\"+7oR-5F-7$$\"+XM ILHF-$\"+W#Hh2\"F-7$$\"+U,kAJF-$\"+V)y'Q6F-7$$\"+(yzxM$F-$\"+wtH37F-7$ $\"+*f-&QNF-$\"+VOqj7F-7$$\"+p&e.w$F-$\"+sTF-$\"+\">V%G9F-7$$\"+/yOxVF-$\"+'eZkZ\"F-7$$\"+?,W\"f%F -$\"+EP>C:F-7$$\"+J`-)y%F-$\"+y!=hc\"F-7$$\"+S!p++&F-$\"+?cF-$\"+=oF-7$$\"+R(yG3(F-$\"+F/od>F-7$$\"+g5d'H(F-$\"+?XS()>F-7 $$\"+\"))Qw\\(F-$\"+b\")e9?F-7$$\"+)o7yq(F-$\"+\\WBU?F-7$$\"+,gH8zF-$ \"+$QW&o?F-7$$\"+()zTG\")F-$\"+:jO&4#F-7$$\"+FggN$)F-$\"+Ai`?@F-7$$\"+ Ad[Z&)F-$\"+srjX@F-7$$\"+Z4hd()F-$\"+XJ#*p@F-7$$\"+1Up]*)F-$\"+%4J<>#F -7$$\"+5%*)><*F-$\"+5U:;AF-7$$\"+z%=*p$*F-$\"+'R/vB#F-7$$\"+\\@&4e*F-$ \"+wpxfAF-7$$\"+)HQHy*F-$\"+y)R1G#F-7$$\"#5\"\"!$\"+$4&e-BF--%'COLOURG 6&%$RGBG$\"*++++\"!\")$Fj]lFj]lFd^l-%+AXESLABELSG6$Q\"x6\"Q!Fi^l-%*THI CKNESSG6#\"\"#-%%VIEWG6$;F(Fh]l%(DEFAULTG" 1 2 0 1 10 2 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 62 "xx:= 9;\nDigits := 100:\nlogAP(xx);\nevalf(ln(xx));\nDigits := 10:" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%#xxG\"\"*" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#$\"_qyTY F'4:BZP$z@Pke))\\Fn')QpM!*)\\Xc6\")\\H490XQZ!\\!z#Q>iLxXA(>#!#**" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#$\"_qyTYF'4:BZP$z@Pke))\\Fn')QpM!*)\\X c6\")\\H490XQZ!\\!z#Q>iLxXA(>#!#**" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" } }{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 70 "xx:= 0.0012345:\nDigits := 100:\nlogAP(xx);\nevalf(ln(xx));\nDig its := 10:" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#$!_qAUpT(4!RK2!z_C#yV#)= jZ![(>N;c(>ha&31\\'zX'>ZE1?-\"*R!z\"\\#*3(p'!#**" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#$!_qAUpT(4!RK2!z_C#yV#)=jZ![(>N;c(>ha&31\\'zX'>ZE1?-\"* R!z\"\\#*3(p'!#**" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 " > " 0 "" {MPLTEXT 1 0 1 ";" }}}}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 18 "S peed comparisons " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 17 "Maple's function " }{TEXT 0 2 "ln" }{TEXT -1 16 " is fast er than " }{TEXT 0 5 "logAP" }{TEXT -1 1 "." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 148 "st := time():\nfo r i to 500 do logAP(rand()*Float(1,-11)) end do:\ntime()-st;\nst := ti me():\nfor i to 500 do ln(rand()*Float(1,-11)) end do:\ntime()-st;" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#$\"%B8!\"$" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#$\"$0#!\"$" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 176 "st := time():\nDigits := 50:\nfor \+ i to 100 do logAP(rand()*Float(1,-11)) end do:\ntime()-st;\nst := time ():\nfor i to 100 do ln(rand()*Float(1,-11)) end do:\ntime()-st;\nDigi ts := 10:" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#$\"%eQ!\"$" }}{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 }