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Burbulla and C.T.J. Dodson Notebook to support the book Self-Tutor for Computer Calculus ;[s] 3:0,0;75,1;104,2;143,-1; 3:1,0,-9 -*-times-bold-r-normal-*-*-120-75-75-*-*-*-*,times,1,12,0,0,0;1,0,-9 -*-times-bold-i-normal-*-*-170-75-75-*-*-*-*,times,3,17,0,0,0;1,0,-9 -*-times-bold-r-normal-*-*-240-75-75-*-*-*-*,times,1,24,0,0,0; :[font = subtitle; inactive; preserveAspect] Using Mathematica 2 ;[s] 2:0,0;6,1;20,-1; 2:1,0,-9 -*-times-bold-r-normal-*-*-230-75-75-*-*-*-*,times,1,23,0,0,0;1,0,-9 -*-times-medium-i-normal-*-*-230-75-75-*-*-*-*,times,2,23,0,0,0; :[font = subsubtitle; inactive; preserveAspect; fontName = "Times"] by D. C. M. Burbulla and C. T. J. Dodson, University of Toronto Published by: Prentice-Hall Canada (English) ISBN 0130152803 and Prentice-Hall Japan (Japanese) ISBN4-8101-8558-3 C3055 P2880E ;[s] 5:0,0;128,1;129,2;145,3;192,4;222,-1; 5:1,0,-9 -*-times-medium-i-normal-*-*-120-75-75-*-*-*-*,times,2,12,0,0,0;1,0,-9 -*-times-medium-i-normal-*-*-110-75-75-*-*-*-*,times,2,11,0,0,0;1,0,-9 -*-courier-medium-i-normal-*-*-110-75-75-*-*-*-*,courier,2,11,0,0,0;1,0,-9 -*-times-medium-i-normal-*-*-120-75-75-*-*-*-*,times,2,12,0,0,0;1,0,-9 -*-courier-medium-i-normal-*-*-110-75-75-*-*-*-*,courier,2,11,0,0,0; :[font = section; inactive; Cclosed; preserveAspect; startGroup] Methods of Approximation :[font = subsection; inactive; Cclosed; preserveAspect; fontName = "Times"; startGroup] Chap4.m :[font = input; Cclosed; preserveAspect; fontName = "Courier"; startGroup] bisect/: bisect[f_][{a_, b_}] := If[N[f[a]*f[b]] > 0, {a, b}, If[N[f[a]*f[(a + b)/2]] <= 0, N[{a, (a + b)/2}], N[{(a + b)/2, b}]]] bisect::usage = "bisect[f][{a, b}] determines which half of the interval {a, b} contains a solution to the equation f[x] = 0. It is assumed that f is continuous on {a, b}. If f[a] and f[b] have the same sign, then the output is the original interval." tangent/: tangent[f_, a_][x_] := f'[a]*(x - a) + f[a] normal/: normal[f_, a_][x_] := (a - x)/f'[a] + f[a] tangentplot/: tangentplot[f_, a_, {b_, c_}] := Plot[{f[x], tangent[f, a][x]}, {x, b, c}] normalplot/: normalplot[f_, a_, {b_, c_}] := Plot[{f[x], normal[f, a][x]}, {x, b, c}, AspectRatio -> Automatic] errorplot/: errorplot[f_, a_, {b_, c_}] := Plot[Abs[f[x] - tangent[f, a][x]], {x, b, c}, PlotRange -> All] errorplot::usage = "errorplot[f, a, {b, c}] plots the absolute value of the difference of f and the tangent to f at x = a, for each x in the interval {b, c}." iteration/: iteration[f_][x_] := x - f[x]/f'[x] point/: point[f_, a_, n_] := Nest[iteration[f], N[a], n] newtonsmethod/: newtonsmethod[f_, a_, {b_, c_}][n_] := (one = Block[{x, i}, Plot[{f[x], tangent[f, point[f, a, n]][x]}, {x, First[Union[{b}, Table[point[f, a, i], {i, 0, n + 1}]]], Last[Union[{c}, Table[point[f, a, i], {i, 0, n + 1}]]]}, DisplayFunction -> Identity]]; two = Block[{i}, ListPlot[Table[{point[f, a, i], 0}, {i, 0, n + 1}], DisplayFunction -> Identity]]; three = Block[{i}, ListPlot[Table[{point[f, a, i], f[point[f, a, i]]}, {i, 0, n}], DisplayFunction -> Identity]]; Show[one, two, three, PlotLabel -> If[n == 0, If[a == 0, "0 = initial guess", If[a == 1, "1 = initial guess", a " = initial guess"]], {(n + 1) " iterations completed", N[Nest[iteration[f], a, n + 1]] " = approximation"} ], DisplayFunction -> $DisplayFunction]) newtonsmethod::usage = "newtonsmethod[f, a, {b, c}][n] shows Newton's method being used to generate the (n + 1)st approximation to the solution of the equation f[x] = 0 from the nth approximation, starting with initial guess x0 = a. All previous approximations are shown along the x-axis, and the value of the current approximation is indicated in the plot label. The interval {b, c} is the initial x-interval used for the plot; however the x-interval will adjust automatically, if necessary, to include all approximations x0, x1, ... , xn, x(n + 1)." fixedpoint/: fixedpoint[g_, a_, {b_, c_}][n_] := (one = Block[{x, i}, Plot[{g[x], x}, {x, b, c}, AspectRatio -> Automatic, PlotRange -> {{b, c}, {b, c}}, DisplayFunction -> Identity]]; two = Block[{i}, ListPlot[Table[{Nest[g, N[a], i], 0}, {i, 0, n + 1}], DisplayFunction -> Identity]]; three = Block[{i}, ListPlot[Table[{Nest[g, N[a], i], Nest[g, N[a], i]}, {i, 0, n + 1}], DisplayFunction -> Identity]]; four = Block[{i}, ListPlot[Table[{Nest[g, N[a], i], Nest[g, N[a], i + 1]}, {i, 0, n + 1}], DisplayFunction -> Identity]]; five = Block[{i}, ListPlot[Table[{Nest[g, N[a], Floor[i/2]], Mod[i + 1, 2]*Nest[g, N[a], Floor[i/2]] + Mod[i, 2]*g[Nest[g, N[a], Floor[i/2]]]}, {i, 0, 2*n + 3}], PlotJoined -> True, DisplayFunction -> Identity]]; Show[one, two, three, four, five, PlotLabel -> If[n == 0, {"beginning fixed point iteration", If[a == 0, "0 = initial guess", If[a == 1, "1 = initial guess", a " = initial guess"]]}, {(n + 1) " iterations completed", N[Nest[g, a, n + 1]] " = approximation"}], DisplayFunction -> $DisplayFunction]) fixedpoint::usage = "fixedpoint[f, a, {b, c}][n] performs (n + 1) iterations of the Fixed Point iteration method for approximating solutions to the equation f[x] = x, starting with x0 = a. The output plots f and the identity function, y = x, on the interval {b, c}; shows all successive approximations x0, x1, ... , xn, x(n + 1) along the x-axis; indicates the value of the current approximation in the PlotLabel; and shows the `cobweb' of points homing in on the fixed point." :[font = output; output; inactive; preserveAspect] "bisect[f][{a, b}] determines which half of the\ interval {a, b} contains a solution to the\ equation f[x] = 0. It is assumed that f is\ continuous on {a, b}. If f[a] and f[b] have the\ same sign, then the output is the original\ interval." ;[o] bisect[f][{a, b}] determines which half of the\ interval {a, b} contains a solution to the\ equation f[x] = 0. It is assumed that f is\ continuous on {a, b}. If f[a] and f[b] have the\ same sign, then the output is the original interval. :[font = output; output; inactive; preserveAspect] "errorplot[f, a, {b, c}] plots the absolute value of\ the difference of f and the tangent to f at x = a,\ for each x in the interval {b, c}." ;[o] errorplot[f, a, {b, c}] plots the absolute value of\ the difference of f and the tangent to f at x = a,\ for each x in the interval {b, c}. :[font = message; inactive; preserveAspect] TagSetDelayed::tagnf: Tag point not found in -Graphics-[f_, a_, n_]. :[font = output; output; inactive; preserveAspect; endGroup; endGroup] $Failed ;[o] $Failed :[font = subsection; inactive; Cclosed; preserveAspect; fontName = "Times"; startGroup] Linear Approximation :[font = input; Cclosed; preserveAspect; fontName = "Courier"; startGroup] tangent[Sqrt, 4][x] :[font = output; output; inactive; preserveAspect; endGroup] 2 + (-4 + x)/4 ;[o] -4 + x 2 + ------ 4 :[font = input; Cclosed; preserveAspect; fontName = "Courier"; startGroup] Simplify[%] :[font = output; output; inactive; preserveAspect; endGroup] 1 + x/4 ;[o] x 1 + - 4 :[font = input; Cclosed; preserveAspect; fontName = "Courier"; startGroup] tangent[Sqrt, 4][{7.2, 2.5, 4.8, 3.95}] // N :[font = output; output; inactive; preserveAspect; endGroup] {2.8, 1.625, 2.2, 1.9875} ;[o] {2.8, 1.625, 2.2, 1.9875} :[font = input; Cclosed; preserveAspect; fontName = "Courier"; startGroup] Sqrt[{7.2, 2.5, 4.8, 3.95}] // N :[font = output; output; inactive; preserveAspect; endGroup] {2.683281572999748, 1.581138830084189, 2.190890230020664, 1.987460691435179} ;[o] {2.68328, 1.58114, 2.19089, 1.98746} :[font = input; preserveAspect; fontName = "Courier"] We can see the error in tangent approximations using the new function errorplot: ;[s] 1:0,0;80,-1; 1:1,0,-9 -*-courier-medium-r-normal-*-*-120-75-75-*-*-*-*,courier,0,12,0,0,0; :[font = input; Cclosed; preserveAspect; fontName = "Courier"; startGroup] errorplot[Sqrt, 4, {0, 8}]; :[font = postscript; PostScript; 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s P P % End of Graphics MathPictureEnd :[font = input; Cclosed; preserveAspect; fontName = "Courier"; startGroup] Plot[f[x], {x, -2, -1.5}]; :[font = postscript; PostScript; formatAsPostScript; output; inactive; preserveAspect; pictureLeft = 34; pictureWidth = 282; pictureHeight = 174; endGroup] %! %%Creator: Mathematica %%AspectRatio: .61803 MathPictureStart %% Graphics /Courier findfont 10 scalefont setfont % Scaling calculations 3.833333 1.904762 0.107653 0.247833 [ [(-1.9)] .21429 .10765 0 2 Msboxa [(-1.8)] .40476 .10765 0 2 Msboxa [(-1.7)] .59524 .10765 0 2 Msboxa [(-1.6)] .78571 .10765 0 2 Msboxa [(-1.5)] .97619 .10765 0 2 Msboxa [(0.5)] .01131 .23157 1 0 Msboxa [(1)] .01131 .35549 1 0 Msboxa [(1.5)] .01131 .4794 1 0 Msboxa [(2)] .01131 .60332 1 0 Msboxa [ -0.001 -0.001 0 0 ] [ 1.001 .61903 0 0 ] ] MathScale % Start of Graphics 1 setlinecap 1 setlinejoin newpath [ ] 0 setdash 0 g p p .002 w .21429 .10765 m .21429 .1139 L s P [(-1.9)] .21429 .10765 0 2 Mshowa p .002 w .40476 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startGroup] Plot[f[x], {x, -1.75, -1.5}]; :[font = postscript; PostScript; formatAsPostScript; output; inactive; preserveAspect; pictureLeft = 34; pictureWidth = 282; pictureHeight = 174; endGroup] %! %%Creator: Mathematica %%AspectRatio: .61803 MathPictureStart %% Graphics /Courier findfont 10 scalefont setfont % Scaling calculations 6.690476 3.809524 0.189116 0.46507 [ [(-1.75)] .02381 .18912 0 2 Msboxa [(-1.65)] .40476 .18912 0 2 Msboxa [(-1.6)] .59524 .18912 0 2 Msboxa [(-1.55)] .78571 .18912 0 2 Msboxa [(-1.5)] .97619 .18912 0 2 Msboxa [(-0.4)] .20179 .00309 1 0 Msboxa [(-0.2)] .20179 .0961 1 0 Msboxa [(0.2)] .20179 .28213 1 0 Msboxa [(0.4)] .20179 .37514 1 0 Msboxa [(0.6)] .20179 .46816 1 0 Msboxa [(0.8)] .20179 .56117 1 0 Msboxa [ -0.001 -0.001 0 0 ] [ 1.001 .61903 0 0 ] ] MathScale % Start of Graphics 1 setlinecap 1 setlinejoin newpath [ ] 0 setdash 0 g p p .002 w .02381 .18912 m .02381 .19537 L s P [(-1.75)] .02381 .18912 0 2 Mshowa p .002 w .40476 .18912 m .40476 .19537 L s P 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.17051 L s P p .001 w .21429 .20772 m .21804 .20772 L s P p .001 w .21429 .22632 m .21804 .22632 L s P p .001 w .21429 .24492 m .21804 .24492 L s P p .001 w .21429 .26353 m .21804 .26353 L s P p .001 w .21429 .30073 m .21804 .30073 L s P p .001 w .21429 .31934 m .21804 .31934 L s P p .001 w .21429 .33794 m .21804 .33794 L s P p .001 w .21429 .35654 m .21804 .35654 L s P p .001 w .21429 .39375 m .21804 .39375 L s P p .001 w .21429 .41235 m .21804 .41235 L s P p .001 w .21429 .43095 m .21804 .43095 L s P p .001 w .21429 .44956 m .21804 .44956 L s P p .001 w .21429 .48676 m .21804 .48676 L s P p .001 w .21429 .50536 m .21804 .50536 L s P p .001 w .21429 .52397 m .21804 .52397 L s P p .001 w .21429 .54257 m .21804 .54257 L s P p .001 w .21429 .57977 m .21804 .57977 L s P p .001 w .21429 .59838 m .21804 .59838 L s P p .001 w .21429 .61698 m .21804 .61698 L s P p .002 w .21429 0 m .21429 .61803 L s P P 0 0 m 1 0 L 1 .61803 L 0 .61803 L closepath clip newpath p p .004 w .02381 .60332 m .06349 .57994 L .10317 .55644 L .14286 .53282 L .18254 .50909 L .22222 .48524 L .2619 .46128 L .30159 .43721 L .34127 .41304 L .38095 .38877 L .42063 .3644 L .46032 .33993 L .5 .31538 L .53968 .29073 L .57937 .266 L .61905 .24119 L .65873 .2163 L .69841 .19133 L .7381 .16629 L .77778 .14119 L .81746 .11601 L .85714 .09077 L .89683 .06548 L .93651 .04012 L .97619 .01472 L s P P % End of Graphics MathPictureEnd :[font = input; Cclosed; preserveAspect; fontName = "Courier"; startGroup] Plot[f[x], {x, -1.625, -1.5}]; :[font = postscript; PostScript; formatAsPostScript; output; inactive; preserveAspect; pictureLeft = 34; pictureWidth = 282; pictureHeight = 174; endGroup] %! %%Creator: Mathematica %%AspectRatio: .61803 MathPictureStart %% Graphics /Courier findfont 10 scalefont setfont % Scaling calculations 12.404762 7.619048 0.356141 0.910469 [ [(-1.62)] .0619 .35614 0 2 Msboxa [(-1.58)] .36667 .35614 0 2 Msboxa [(-1.56)] .51905 .35614 0 2 Msboxa [(-1.54)] .67143 .35614 0 2 Msboxa [(-1.52)] .82381 .35614 0 2 Msboxa [(-1.5)] .97619 .35614 0 2 Msboxa [(-0.3)] .20179 .083 1 0 Msboxa [(-0.2)] .20179 .17405 1 0 Msboxa [(-0.1)] .20179 .26509 1 0 Msboxa [(0.1)] .20179 .44719 1 0 Msboxa [(0.2)] .20179 .53823 1 0 Msboxa [ -0.001 -0.001 0 0 ] [ 1.001 .61903 0 0 ] ] MathScale % Start of Graphics 1 setlinecap 1 setlinejoin newpath [ ] 0 setdash 0 g p p .002 w .0619 .35614 m .0619 .36239 L s P [(-1.62)] .0619 .35614 0 2 Mshowa p .002 w .36667 .35614 m .36667 .36239 L s P [(-1.58)] .36667 .35614 0 2 Mshowa p .002 w .51905 .35614 m .51905 .36239 L s P [(-1.56)] .51905 .35614 0 2 Mshowa p .002 w .67143 .35614 m .67143 .36239 L s P [(-1.54)] .67143 .35614 0 2 Mshowa p .002 w .82381 .35614 m .82381 .36239 L s P [(-1.52)] .82381 .35614 0 2 Mshowa p .002 w .97619 .35614 m .97619 .36239 L s P [(-1.5)] .97619 .35614 0 2 Mshowa p .001 w .09238 .35614 m .09238 .35989 L s P p .001 w .12286 .35614 m .12286 .35989 L s P p .001 w .15333 .35614 m .15333 .35989 L s P p .001 w .18381 .35614 m .18381 .35989 L s P p .001 w .24476 .35614 m .24476 .35989 L s P p .001 w .27524 .35614 m .27524 .35989 L s P p .001 w .30571 .35614 m .30571 .35989 L s P p .001 w .33619 .35614 m .33619 .35989 L s P p .001 w .39714 .35614 m .39714 .35989 L s P p .001 w .42762 .35614 m .42762 .35989 L s P p .001 w .4581 .35614 m .4581 .35989 L s P p .001 w .48857 .35614 m .48857 .35989 L s P p .001 w .54952 .35614 m .54952 .35989 L s P p .001 w .58 .35614 m .58 .35989 L s P p .001 w .61048 .35614 m .61048 .35989 L s P p .001 w .64095 .35614 m .64095 .35989 L s P p .001 w .7019 .35614 m .7019 .35989 L s P p .001 w .73238 .35614 m .73238 .35989 L s P p .001 w .76286 .35614 m .76286 .35989 L s P p .001 w .79333 .35614 m .79333 .35989 L s P p .001 w .85429 .35614 m .85429 .35989 L s P p .001 w .88476 .35614 m .88476 .35989 L s P p .001 w .91524 .35614 m .91524 .35989 L s P p .001 w .94571 .35614 m .94571 .35989 L s P p .001 w .03143 .35614 m .03143 .35989 L s P p .001 w .00095 .35614 m .00095 .35989 L s P p .002 w 0 .35614 m 1 .35614 L s P p .002 w .21429 .083 m .22054 .083 L s P [(-0.3)] .20179 .083 1 0 Mshowa p .002 w .21429 .17405 m .22054 .17405 L s P [(-0.2)] .20179 .17405 1 0 Mshowa p .002 w .21429 .26509 m .22054 .26509 L s P [(-0.1)] .20179 .26509 1 0 Mshowa p .002 w .21429 .44719 m .22054 .44719 L s P [(0.1)] .20179 .44719 1 0 Mshowa p .002 w .21429 .53823 m .22054 .53823 L s P [(0.2)] .20179 .53823 1 0 Mshowa p .001 w .21429 .10121 m .21804 .10121 L s P p .001 w .21429 .11942 m .21804 .11942 L s P p .001 w .21429 .13763 m .21804 .13763 L s P p .001 w .21429 .15584 m .21804 .15584 L s P p .001 w .21429 .19226 m .21804 .19226 L s P p .001 w .21429 .21047 m .21804 .21047 L s P p .001 w .21429 .22868 m .21804 .22868 L s P p .001 w .21429 .24688 m .21804 .24688 L s P p .001 w .21429 .2833 m .21804 .2833 L s P p .001 w .21429 .30151 m .21804 .30151 L s P p .001 w .21429 .31972 m .21804 .31972 L s P p .001 w .21429 .33793 m .21804 .33793 L s P p .001 w .21429 .37435 m .21804 .37435 L s P p .001 w .21429 .39256 m .21804 .39256 L s P p .001 w .21429 .41077 m .21804 .41077 L s P p .001 w .21429 .42898 m .21804 .42898 L s P p .001 w .21429 .4654 m .21804 .4654 L s P p .001 w .21429 .48361 m .21804 .48361 L s P p .001 w .21429 .50182 m .21804 .50182 L s P p .001 w .21429 .52003 m .21804 .52003 L s P p .001 w .21429 .06479 m .21804 .06479 L s P p .001 w .21429 .04658 m .21804 .04658 L s P p .001 w .21429 .02837 m .21804 .02837 L s P p .001 w .21429 .01016 m .21804 .01016 L s P p .001 w .21429 .55644 m .21804 .55644 L s P p .001 w .21429 .57465 m .21804 .57465 L s P p .001 w .21429 .59286 m .21804 .59286 L s P p .001 w .21429 .61107 m .21804 .61107 L s P p .002 w .21429 0 m .21429 .61803 L s P P 0 0 m 1 0 L 1 .61803 L 0 .61803 L closepath clip newpath p p .004 w .02381 .60332 m .06349 .57922 L .10317 .55507 L .14286 .53089 L .18254 .50666 L .22222 .48239 L .2619 .45809 L .30159 .43374 L .34127 .40936 L .38095 .38494 L .42063 .36048 L .46032 .33599 L .5 .31146 L .53968 .2869 L .57937 .26231 L .61905 .23768 L .65873 .21302 L .69841 .18834 L .7381 .16362 L .77778 .13887 L .81746 .11409 L .85714 .08929 L .89683 .06446 L .93651 .0396 L .97619 .01472 L s P P % End of Graphics MathPictureEnd :[font = input; preserveAspect; fontName = "Courier"] Now try the new bisect function: ;[s] 1:0,0;32,-1; 1:1,0,-9 -*-courier-medium-r-normal-*-*-120-75-75-*-*-*-*,courier,0,12,0,0,0; :[font = input; Cclosed; preserveAspect; fontName = "Courier"; startGroup] ?bisect :[font = print; inactive; preserveAspect; endGroup] bisect[f][{a, b}] determines which half of the interval {a, b} contains a solution to the equation f[x] = 0. It is assumed that f is continuous on {a, b}. If f[a] and f[b] have the same sign, then the output is the original interval. :[font = input; Cclosed; preserveAspect; fontName = "Courier"; startGroup] bisect[f][{-2, -1}] (* f as before *) :[font = output; output; inactive; preserveAspect; endGroup] {-2., -1.5} ;[o] {-2., -1.5} :[font = input; Cclosed; preserveAspect; fontName = "Courier"; startGroup] bisect[f][%] :[font = output; output; inactive; preserveAspect; endGroup] {-1.75, -1.5} ;[o] {-1.75, -1.5} :[font = input; Cclosed; preserveAspect; fontName = "Courier"; startGroup] bisect[f][%] :[font = output; output; inactive; preserveAspect; endGroup] {-1.625, -1.5} ;[o] {-1.625, -1.5} :[font = input; Cclosed; preserveAspect; fontName = "Courier"; startGroup] bisect[f][%] :[font = output; output; inactive; preserveAspect; endGroup] {-1.625, -1.5625} ;[o] {-1.625, -1.5625} :[font = input; preserveAspect; fontName = "Courier"] More neatly, we can use Nest: ;[s] 1:0,0;29,-1; 1:1,0,-9 -*-courier-medium-r-normal-*-*-120-75-75-*-*-*-*,courier,0,12,0,0,0; :[font = input; Cclosed; preserveAspect; fontName = "Courier"; startGroup] ?Nest :[font = print; inactive; preserveAspect; endGroup] Nest[f, expr, n] gives an expression with f applied n times to expr. :[font = input; Cclosed; preserveAspect; fontName = "Courier"; startGroup] Nest[bisect[f], {-2, -1}, 4] :[font = output; output; inactive; preserveAspect; endGroup] {-1.625, -1.5625} ;[o] {-1.625, -1.5625} :[font = input; Cclosed; preserveAspect; fontName = "Courier"; startGroup] Nest[bisect[f], {-2, -1}, 18] :[font = output; output; inactive; preserveAspect; endGroup] {-1.571994781494141, -1.571990966796875} ;[o] {-1.57199, -1.57199} :[font = input; Cclosed; preserveAspect; fontName = "Courier"; startGroup] Nest[bisect[f], {-4, -3}, 19] :[font = output; output; inactive; preserveAspect; endGroup] {-3.514137268066406, -3.514135360717773} ;[o] {-3.51414, -3.51414} :[font = input; Cclosed; preserveAspect; fontName = "Courier"; startGroup] Nest[bisect[f], {1, 2}, 18] :[font = output; output; inactive; preserveAspect; endGroup] {1.086128234863281, 1.086132049560547} ;[o] {1.08613, 1.08613} :[font = input; preserveAspect; fontName = "Courier"] You can try also NRoots: ;[s] 1:0,0;24,-1; 1:1,0,-9 -*-courier-medium-r-normal-*-*-120-75-75-*-*-*-*,courier,0,12,0,0,0; :[font = input; Cclosed; preserveAspect; fontName = "Courier"; startGroup] ?NRoots :[font = print; inactive; preserveAspect; endGroup] NRoots[lhs==rhs, var] gives a list of numerical approximations to the roots of a polynomial equation. :[font = input; Cclosed; preserveAspect; fontName = "Courier"; startGroup] NRoots[x^3 + 4 x^2 - 6 == 0, x] :[font = output; output; inactive; preserveAspect; endGroup; endGroup] x == -3.514136929335291 || x == -1.571993268316203 || x == 1.086130197651494 ;[o] x == -3.51414 || x == -1.57199 || x == 1.08613 :[font = subsection; inactive; Cclosed; preserveAspect; fontName = "Times"; startGroup] Newton's Method :[font = input; preserveAspect; fontName = "Courier"] Newton's method requires an initial choice, and the sequence may not converge. However, when Newton's method does work, it works so well that the convergence is usually much quicker than that of the bisection method. ;[s] 1:0,0;217,-1; 1:1,0,-9 -*-courier-medium-r-normal-*-*-120-75-75-*-*-*-*,courier,0,12,0,0,0; :[font = text; inactive; preserveAspect] Example 1 :[font = input; preserveAspect; fontName = "Courier"] Clear[f]; f[x_] := x^3 + 4 x^2 - 6 ;[s] 2:0,0;1,1;35,-1; 2:1,0,-9 -*-courier-bold-r-normal-*-*-100-75-75-*-*-*-*,courier,1,10,0,0,0;1,0,-9 -*-courier-bold-r-normal-*-*-120-75-75-*-*-*-*,courier,1,12,0,0,0; :[font = input; Cclosed; preserveAspect; fontName = "Courier"; startGroup] Plot[f[x], {x, -4, 3}]; ;[s] 2:0,0;1,1;24,-1; 2:1,0,-9 -*-courier-bold-r-normal-*-*-100-75-75-*-*-*-*,courier,1,10,0,0,0;1,0,-9 -*-courier-bold-r-normal-*-*-120-75-75-*-*-*-*,courier,1,12,0,0,0; :[font = postscript; PostScript; formatAsPostScript; output; inactive; preserveAspect; pictureLeft = 34; pictureWidth = 282; pictureHeight = 174; endGroup] %! %%Creator: Mathematica 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m .64966 .16772 L s P p .001 w .67687 .16397 m .67687 .16772 L s P p .001 w .73129 .16397 m .73129 .16772 L s P p .001 w .7585 .16397 m .7585 .16772 L s P p .001 w .78571 .16397 m .78571 .16772 L s P p .001 w .81293 .16397 m .81293 .16772 L s P p .001 w .86735 .16397 m .86735 .16772 L s P p .001 w .89456 .16397 m .89456 .16772 L s P p .001 w .92177 .16397 m .92177 .16772 L s P p .001 w .94898 .16397 m .94898 .16772 L s P p .002 w 0 .16397 m 1 .16397 L s P p .002 w .56803 .03959 m .57428 .03959 L s P [(-5)] .55553 .03959 1 0 Mshowa p .002 w .56803 .28834 m .57428 .28834 L s P [(5)] .55553 .28834 1 0 Mshowa p .002 w .56803 .41272 m .57428 .41272 L s P [(10)] .55553 .41272 1 0 Mshowa p .002 w .56803 .53709 m .57428 .53709 L s P [(15)] .55553 .53709 1 0 Mshowa p .001 w .56803 .06447 m .57178 .06447 L s P p .001 w .56803 .08934 m .57178 .08934 L s P p .001 w .56803 .11422 m .57178 .11422 L s P p .001 w .56803 .13909 m .57178 .13909 L s P p .001 w .56803 .18884 m .57178 .18884 L s P p .001 w 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.19494 .24999 L .19742 .25024 L .19866 .25033 L .1999 .25041 L .20114 .25048 L .20238 .25052 L .20362 .25055 L .20486 .25057 L .2061 .25056 L .20734 .25054 L .20858 .25051 L .20982 .25045 L .21106 .25039 L .2123 .2503 L .21726 .2498 L .21974 .24946 L .22222 .24906 L .23214 .24686 L .24206 .24376 L .2619 .23509 L .28175 .22351 L .30159 .20948 L .34127 .17594 L .38095 .13817 L .42063 .09986 L .46032 .06473 L .48016 .04952 L .5 .03648 L .50992 .03093 L .51984 .02609 L .52976 .02203 L .53968 .01881 L .54464 .01753 L .5496 .01648 L .55208 .01604 L .55456 .01567 L Mistroke .55704 .01535 L .55952 .0151 L .56076 .01499 L .562 .01491 L .56324 .01484 L .56448 .01478 L .56572 .01474 L .56696 .01472 L .5682 .01472 L .56944 .01473 L .57068 .01475 L .57192 .0148 L .57316 .01486 L .5744 .01494 L .57688 .01514 L .57937 .01542 L .58433 .01619 L .58929 .01724 L .59425 .01859 L .59921 .02024 L .60913 .02448 L .61905 .03002 L .63889 .04522 L .65873 .06631 L .67857 .09374 L .69841 .12799 L .7381 .21877 L .77778 .34235 L .81746 .50243 L Mfstroke .84037 .61803 m .81746 .50243 L s s s s s P P % End of Graphics MathPictureEnd :[font = input; Cclosed; preserveAspect; fontName = "Courier"; startGroup] ?newtonsmethod ;[s] 2:0,0;1,1;15,-1; 2:1,0,-9 -*-courier-bold-r-normal-*-*-100-75-75-*-*-*-*,courier,1,10,0,0,0;1,0,-9 -*-courier-bold-r-normal-*-*-120-75-75-*-*-*-*,courier,1,12,0,0,0; :[font = print; inactive; preserveAspect; endGroup] newtonsmethod[f, a, {b, c}][n] shows Newton's method being used to generate the (n + 1)st approximation to the solution of the equation f[x] = 0 from the nth approximation, starting with initial guess x0 = a. All previous approximations are shown along the x-axis, and the value of the current approximation is indicated in the plot label. The interval {b, c} is the initial x-interval used for the plot; however the x-interval will adjust automatically, if necessary, to include all approximations x0, x1, ... , xn, x(n + 1). :[font = input; Cclosed; preserveAspect; fontName = "Courier"; startGroup] newtonsmethod[f, 0.4, {0, 3}][0]; :[font = postscript; PostScript; formatAsPostScript; output; inactive; preserveAspect; pictureLeft = 34; pictureWidth = 282; pictureHeight = 174; endGroup] %! %%Creator: Mathematica %%AspectRatio: .61803 MathPictureStart %% Graphics /Courier findfont 10 scalefont setfont % Scaling calculations 0.02381 0.31746 0.119886 0.015539 [ [(0.5)] .18254 .11989 0 2 Msboxa [(1)] .34127 .11989 0 2 Msboxa [(1.5)] .5 .11989 0 2 Msboxa [(2)] .65873 .11989 0 2 Msboxa [(2.5)] .81746 .11989 0 2 Msboxa [(3)] .97619 .11989 0 2 Msboxa [(0.4 = initial guess)] .5 .61803 0 -2 Msboxa [(-5)] .01131 .04219 1 0 Msboxa [(5)] .01131 .19758 1 0 Msboxa [(10)] .01131 .27528 1 0 Msboxa [(15)] .01131 .35298 1 0 Msboxa [(20)] .01131 .43068 1 0 Msboxa [(25)] .01131 .50837 1 0 Msboxa [(30)] .01131 .58607 1 0 Msboxa [ -0.001 -0.001 0 0 ] [ 1.001 .61903 0 0 ] ] MathScale % Start of Graphics 1 setlinecap 1 setlinejoin newpath [ ] 0 setdash 0 g p p .002 w .18254 .11989 m .18254 .12614 L s P [(0.5)] .18254 .11989 0 2 Mshowa p .002 w .34127 .11989 m .34127 .12614 L s P [(1)] .34127 .11989 0 2 Mshowa p .002 w .5 .11989 m .5 .12614 L s P [(1.5)] .5 .11989 0 2 Mshowa p .002 w .65873 .11989 m .65873 .12614 L s P [(2)] .65873 .11989 0 2 Mshowa p .002 w .81746 .11989 m .81746 .12614 L s P [(2.5)] .81746 .11989 0 2 Mshowa p .002 w .97619 .11989 m .97619 .12614 L s P [(3)] .97619 .11989 0 2 Mshowa p .001 w .05556 .11989 m .05556 .12364 L s P p .001 w .0873 .11989 m .0873 .12364 L s P p .001 w .11905 .11989 m .11905 .12364 L s P p .001 w .15079 .11989 m .15079 .12364 L s P p .001 w .21429 .11989 m .21429 .12364 L s P p .001 w .24603 .11989 m .24603 .12364 L s P p .001 w .27778 .11989 m .27778 .12364 L s P p .001 w .30952 .11989 m .30952 .12364 L s P p .001 w .37302 .11989 m .37302 .12364 L s P p .001 w .40476 .11989 m .40476 .12364 L s P p .001 w .43651 .11989 m .43651 .12364 L s P p .001 w .46825 .11989 m .46825 .12364 L s P p .001 w .53175 .11989 m .53175 .12364 L s P p .001 w .56349 .11989 m .56349 .12364 L s P p .001 w .59524 .11989 m .59524 .12364 L s P p .001 w .62698 .11989 m .62698 .12364 L s P p .001 w .69048 .11989 m .69048 .12364 L s P p .001 w .72222 .11989 m .72222 .12364 L s P p .001 w .75397 .11989 m .75397 .12364 L s P p .001 w .78571 .11989 m .78571 .12364 L s P p .001 w .84921 .11989 m .84921 .12364 L s P p .001 w .88095 .11989 m .88095 .12364 L s P p .001 w .9127 .11989 m .9127 .12364 L s P p .001 w .94444 .11989 m .94444 .12364 L s P p .002 w 0 .11989 m 1 .11989 L s P [(0.4 = initial guess)] .5 .61803 0 -2 Mshowa p .002 w .02381 .04219 m .03006 .04219 L s P [(-5)] .01131 .04219 1 0 Mshowa p .002 w .02381 .19758 m .03006 .19758 L s P [(5)] .01131 .19758 1 0 Mshowa p .002 w .02381 .27528 m .03006 .27528 L s P [(10)] .01131 .27528 1 0 Mshowa p .002 w .02381 .35298 m .03006 .35298 L s P [(15)] .01131 .35298 1 0 Mshowa p .002 w .02381 .43068 m .03006 .43068 L s P [(20)] .01131 .43068 1 0 Mshowa p .002 w .02381 .50837 m .03006 .50837 L s P [(25)] .01131 .50837 1 0 Mshowa p .002 w .02381 .58607 m .03006 .58607 L s P [(30)] .01131 .58607 1 0 Mshowa p .001 w .02381 .05773 m .02756 .05773 L s P p .001 w .02381 .07327 m .02756 .07327 L s P p .001 w .02381 .08881 m .02756 .08881 L s P p .001 w .02381 .10435 m .02756 .10435 L s P p .001 w .02381 .13543 m .02756 .13543 L s P p .001 w .02381 .15097 m .02756 .15097 L s P p .001 w .02381 .1665 m .02756 .1665 L s P p .001 w .02381 .18204 m .02756 .18204 L s P p .001 w .02381 .21312 m .02756 .21312 L s P p .001 w .02381 .22866 m .02756 .22866 L s P p .001 w .02381 .2442 m .02756 .2442 L s P p .001 w .02381 .25974 m .02756 .25974 L s P p .001 w .02381 .29082 m .02756 .29082 L s P p .001 w .02381 .30636 m .02756 .30636 L s P p .001 w .02381 .3219 m .02756 .3219 L s P p .001 w .02381 .33744 m .02756 .33744 L s P p .001 w .02381 .36852 m .02756 .36852 L s P p .001 w .02381 .38406 m .02756 .38406 L s P p .001 w .02381 .3996 m .02756 .3996 L s P p .001 w .02381 .41514 m .02756 .41514 L s P p .001 w .02381 .44622 m .02756 .44622 L s P p .001 w .02381 .46175 m .02756 .46175 L s P p .001 w .02381 .47729 m .02756 .47729 L s P p .001 w .02381 .49283 m .02756 .49283 L s P p .001 w .02381 .52391 m .02756 .52391 L s P p .001 w .02381 .53945 m .02756 .53945 L s P p .001 w .02381 .55499 m .02756 .55499 L s P p .001 w .02381 .57053 m .02756 .57053 L s P p .001 w .02381 .02665 m .02756 .02665 L s P p .001 w .02381 .01111 m .02756 .01111 L s P p .001 w .02381 .60161 m .02756 .60161 L s P p .001 w .02381 .61715 m .02756 .61715 L s P p .002 w .02381 0 m .02381 .61803 L s P P 0 0 m 1 0 L 1 .61803 L 0 .61803 L closepath clip newpath p p p p .004 w .02381 .02665 m .02505 .02665 L .02629 .02665 L .02753 .02666 L .02877 .02666 L 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End of Graphics MathPictureEnd :[font = input; Cclosed; preserveAspect; fontName = "Courier"; startGroup] newtonsmethod[f, 0.4, {0, 3}][1]; :[font = postscript; PostScript; formatAsPostScript; output; inactive; preserveAspect; pictureLeft = 34; pictureWidth = 282; pictureHeight = 174; endGroup] %! %%Creator: Mathematica %%AspectRatio: .61803 MathPictureStart %% Graphics /Courier findfont 10 scalefont setfont % Scaling calculations 0.02381 0.31746 0.226225 0.006616 [ [(0.5)] .18254 .22622 0 2 Msboxa [(1)] .34127 .22622 0 2 Msboxa [(1.5)] .5 .22622 0 2 Msboxa [(2)] .65873 .22622 0 2 Msboxa [(2.5)] .81746 .22622 0 2 Msboxa [(3)] .97619 .22622 0 2 Msboxa [({2 iterations completed, 1.28603 = approximation})] .5 .61803 0 -2 Msboxa [(-20)] .01131 .09391 1 0 Msboxa [(20)] .01131 .35854 1 0 Msboxa [(40)] .01131 .49085 1 0 Msboxa [ -0.001 -0.001 0 0 ] [ 1.001 .61903 0 0 ] ] MathScale % Start of Graphics 1 setlinecap 1 setlinejoin newpath [ ] 0 setdash 0 g p p .002 w .18254 .22622 m .18254 .23247 L s P [(0.5)] .18254 .22622 0 2 Mshowa p .002 w .34127 .22622 m .34127 .23247 L s P [(1)] .34127 .22622 0 2 Mshowa p .002 w .5 .22622 m .5 .23247 L s P [(1.5)] .5 .22622 0 2 Mshowa p .002 w .65873 .22622 m .65873 .23247 L s P [(2)] .65873 .22622 0 2 Mshowa p .002 w .81746 .22622 m .81746 .23247 L s P [(2.5)] .81746 .22622 0 2 Mshowa p .002 w .97619 .22622 m .97619 .23247 L s P [(3)] .97619 .22622 0 2 Mshowa p .001 w .05556 .22622 m .05556 .22997 L s P p .001 w .0873 .22622 m .0873 .22997 L s P p .001 w .11905 .22622 m .11905 .22997 L s P p .001 w .15079 .22622 m .15079 .22997 L s P p .001 w .21429 .22622 m .21429 .22997 L s P p .001 w .24603 .22622 m .24603 .22997 L s P p .001 w .27778 .22622 m .27778 .22997 L s P p .001 w .30952 .22622 m .30952 .22997 L s P p .001 w .37302 .22622 m .37302 .22997 L s P p .001 w .40476 .22622 m .40476 .22997 L s P p .001 w .43651 .22622 m .43651 .22997 L s P p .001 w .46825 .22622 m .46825 .22997 L s P p .001 w .53175 .22622 m .53175 .22997 L s P p .001 w .56349 .22622 m .56349 .22997 L s P p .001 w .59524 .22622 m .59524 .22997 L s P p .001 w .62698 .22622 m .62698 .22997 L s P p .001 w .69048 .22622 m .69048 .22997 L s P p .001 w .72222 .22622 m .72222 .22997 L s P p .001 w .75397 .22622 m .75397 .22997 L s P p .001 w .78571 .22622 m .78571 .22997 L s P p .001 w .84921 .22622 m .84921 .22997 L s P p .001 w .88095 .22622 m .88095 .22997 L s P p .001 w .9127 .22622 m .9127 .22997 L s P p .001 w .94444 .22622 m .94444 .22997 L s P p .002 w 0 .22622 m 1 .22622 L s P [({2 iterations completed, 1.28603 = approximation})] .5 .61803 0 -2 Mshowa p .002 w .02381 .09391 m .03006 .09391 L s P [(-20)] .01131 .09391 1 0 Mshowa p .002 w .02381 .35854 m .03006 .35854 L s P [(20)] .01131 .35854 1 0 Mshowa p .002 w .02381 .49085 m .03006 .49085 L s P [(40)] .01131 .49085 1 0 Mshowa p .001 w .02381 .12037 m .02756 .12037 L s P p .001 w .02381 .14684 m .02756 .14684 L s P p .001 w .02381 .1733 m .02756 .1733 L s P p .001 w .02381 .19976 m .02756 .19976 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preserveAspect; fontName = "Courier"; startGroup] newtonsmethod[f, 0.4, {0, 3}][2]; :[font = postscript; PostScript; formatAsPostScript; output; inactive; preserveAspect; pictureLeft = 34; pictureWidth = 282; pictureHeight = 174; endGroup] %! %%Creator: Mathematica %%AspectRatio: .61803 MathPictureStart %% Graphics /Courier findfont 10 scalefont setfont % Scaling calculations 0.02381 0.31746 0.149133 0.007968 [ [(0.5)] .18254 .14913 0 2 Msboxa [(1)] .34127 .14913 0 2 Msboxa [(1.5)] .5 .14913 0 2 Msboxa [(2)] .65873 .14913 0 2 Msboxa [(2.5)] .81746 .14913 0 2 Msboxa [(3)] .97619 .14913 0 2 Msboxa [({3 iterations completed, 1.1062 = approximation})] .5 .61803 0 -2 Msboxa [(-10)] .01131 .06945 1 0 Msboxa [(10)] .01131 .22881 1 0 Msboxa [(20)] .01131 .3085 1 0 Msboxa [(30)] .01131 .38818 1 0 Msboxa [(40)] .01131 .46786 1 0 Msboxa [(50)] .01131 .54754 1 0 Msboxa [ -0.001 -0.001 0 0 ] [ 1.001 .61903 0 0 ] ] MathScale % Start of Graphics 1 setlinecap 1 setlinejoin newpath [ ] 0 setdash 0 g p p .002 w .18254 .14913 m .18254 .15538 L s P [(0.5)] .18254 .14913 0 2 Mshowa p .002 w .34127 .14913 m .34127 .15538 L s P [(1)] .34127 .14913 0 2 Mshowa p .002 w .5 .14913 m .5 .15538 L s P [(1.5)] .5 .14913 0 2 Mshowa p .002 w .65873 .14913 m .65873 .15538 L s P [(2)] .65873 .14913 0 2 Mshowa p .002 w .81746 .14913 m .81746 .15538 L s P [(2.5)] .81746 .14913 0 2 Mshowa p .002 w .97619 .14913 m .97619 .15538 L s P [(3)] .97619 .14913 0 2 Mshowa p .001 w .05556 .14913 m .05556 .15288 L s P p .001 w .0873 .14913 m .0873 .15288 L s P p .001 w .11905 .14913 m .11905 .15288 L s P p .001 w .15079 .14913 m .15079 .15288 L s P p .001 w .21429 .14913 m .21429 .15288 L s P p .001 w .24603 .14913 m .24603 .15288 L s P p .001 w .27778 .14913 m .27778 .15288 L s P p .001 w .30952 .14913 m .30952 .15288 L s P p .001 w .37302 .14913 m .37302 .15288 L s P p .001 w .40476 .14913 m .40476 .15288 L s P p .001 w .43651 .14913 m .43651 .15288 L s P p .001 w .46825 .14913 m .46825 .15288 L s P p .001 w .53175 .14913 m .53175 .15288 L s P p .001 w .56349 .14913 m .56349 .15288 L s P p .001 w .59524 .14913 m .59524 .15288 L s P p .001 w .62698 .14913 m .62698 .15288 L s P p .001 w .69048 .14913 m .69048 .15288 L s P p .001 w .72222 .14913 m .72222 .15288 L s P p .001 w .75397 .14913 m .75397 .15288 L s P p .001 w .78571 .14913 m .78571 .15288 L s P p .001 w .84921 .14913 m .84921 .15288 L s P p .001 w .88095 .14913 m .88095 .15288 L s P p .001 w .9127 .14913 m .9127 .15288 L s P p .001 w .94444 .14913 m .94444 .15288 L s P p .002 w 0 .14913 m 1 .14913 L s P [({3 iterations completed, 1.1062 = approximation})] .5 .61803 0 -2 Mshowa p .002 w .02381 .06945 m .03006 .06945 L s P [(-10)] .01131 .06945 1 0 Mshowa p .002 w .02381 .22881 m .03006 .22881 L s P [(10)] .01131 .22881 1 0 Mshowa p .002 w .02381 .3085 m .03006 .3085 L s P [(20)] .01131 .3085 1 0 Mshowa p .002 w .02381 .38818 m .03006 .38818 L s P [(30)] .01131 .38818 1 0 Mshowa p .002 w .02381 .46786 m .03006 .46786 L s P [(40)] 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L .34127 .14116 L .38095 .15301 L .42063 .16669 L .46032 .1823 L .5 .19993 L .53968 .21968 L .57937 .24164 L .61905 .2659 L .65873 .29256 L .69841 .32171 L .7381 .35344 L .77778 .38785 L .81746 .42503 L .85714 .46507 L .89683 .50807 L .93651 .55412 L .97619 .60332 L s P P p p .004 w .02381 .01472 m .06349 .0299 L .10317 .04509 L .14286 .06028 L .18254 .07547 L .22222 .09066 L .2619 .10585 L .30159 .12104 L .34127 .13623 L .38095 .15142 L .42063 .16661 L .46032 .1818 L .5 .19699 L .53968 .21217 L .57937 .22736 L .61905 .24255 L .65873 .25774 L .69841 .27293 L .7381 .28812 L .77778 .30331 L .81746 .3185 L .85714 .33369 L .89683 .34888 L .93651 .36407 L .97619 .37925 L s P P P p .008 w .15079 .14913 Mdot .60766 .14913 Mdot .43207 .14913 Mdot .37498 .14913 Mdot P p .008 w .15079 .10693 Mdot .60766 .2587 Mdot .43207 .17098 Mdot P P % End of Graphics MathPictureEnd :[font = input; Cclosed; preserveAspect; fontName = "Courier"; startGroup] newtonsmethod[f, 0.4, {0, 3}][3]; :[font = postscript; PostScript; formatAsPostScript; output; inactive; preserveAspect; pictureLeft = 34; pictureWidth = 282; pictureHeight = 174; endGroup] %! %%Creator: Mathematica %%AspectRatio: .61803 MathPictureStart %% Graphics /Courier findfont 10 scalefont setfont % Scaling calculations 0.02381 0.31746 0.128114 0.008337 [ [(0.5)] .18254 .12811 0 2 Msboxa [(1)] .34127 .12811 0 2 Msboxa [(1.5)] .5 .12811 0 2 Msboxa [(2)] .65873 .12811 0 2 Msboxa [(2.5)] .81746 .12811 0 2 Msboxa [(3)] .97619 .12811 0 2 Msboxa [({4 iterations completed, 1.08636 = approximation})] .5 .61803 0 -2 Msboxa [(-10)] .01131 .04474 1 0 Msboxa [(10)] .01131 .21148 1 0 Msboxa [(20)] .01131 .29485 1 0 Msboxa [(30)] .01131 .37822 1 0 Msboxa [(40)] .01131 .46159 1 0 Msboxa [(50)] .01131 .54496 1 0 Msboxa [ -0.001 -0.001 0 0 ] [ 1.001 .61903 0 0 ] ] MathScale % Start of Graphics 1 setlinecap 1 setlinejoin newpath [ ] 0 setdash 0 g p p .002 w .18254 .12811 m .18254 .13436 L s P [(0.5)] .18254 .12811 0 2 Mshowa p .002 w .34127 .12811 m .34127 .13436 L s P [(1)] .34127 .12811 0 2 Mshowa p .002 w .5 .12811 m .5 .13436 L s P [(1.5)] .5 .12811 0 2 Mshowa p .002 w .65873 .12811 m .65873 .13436 L s P [(2)] .65873 .12811 0 2 Mshowa p .002 w .81746 .12811 m .81746 .13436 L s P [(2.5)] .81746 .12811 0 2 Mshowa p .002 w .97619 .12811 m .97619 .13436 L s P [(3)] .97619 .12811 0 2 Mshowa p .001 w .05556 .12811 m .05556 .13186 L s P p .001 w .0873 .12811 m .0873 .13186 L s P p .001 w .11905 .12811 m .11905 .13186 L s P p .001 w .15079 .12811 m .15079 .13186 L s P p .001 w .21429 .12811 m .21429 .13186 L s P p .001 w .24603 .12811 m .24603 .13186 L s P p .001 w .27778 .12811 m .27778 .13186 L s P p .001 w .30952 .12811 m .30952 .13186 L s P p .001 w .37302 .12811 m .37302 .13186 L s P p .001 w .40476 .12811 m .40476 .13186 L s P p .001 w .43651 .12811 m .43651 .13186 L s P p .001 w .46825 .12811 m .46825 .13186 L s P p .001 w .53175 .12811 m .53175 .13186 L s P p .001 w .56349 .12811 m .56349 .13186 L s P p .001 w .59524 .12811 m .59524 .13186 L s P p .001 w .62698 .12811 m .62698 .13186 L s P p .001 w .69048 .12811 m .69048 .13186 L s P p .001 w .72222 .12811 m .72222 .13186 L s P p .001 w .75397 .12811 m .75397 .13186 L s P p .001 w .78571 .12811 m .78571 .13186 L s P p .001 w .84921 .12811 m .84921 .13186 L s P p .001 w .88095 .12811 m .88095 .13186 L s P p .001 w .9127 .12811 m .9127 .13186 L s P p .001 w .94444 .12811 m .94444 .13186 L s P p .002 w 0 .12811 m 1 .12811 L s P [({4 iterations completed, 1.08636 = approximation})] .5 .61803 0 -2 Mshowa p .002 w .02381 .04474 m .03006 .04474 L s P [(-10)] .01131 .04474 1 0 Mshowa p .002 w .02381 .21148 m .03006 .21148 L s P [(10)] .01131 .21148 1 0 Mshowa p .002 w .02381 .29485 m .03006 .29485 L s P [(20)] .01131 .29485 1 0 Mshowa p .002 w .02381 .37822 m .03006 .37822 L s P [(30)] .01131 .37822 1 0 Mshowa p .002 w .02381 .46159 m .03006 .46159 L s P [(40)] .01131 .46159 1 0 Mshowa p .002 w .02381 .54496 m .03006 .54496 L s P [(50)] .01131 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%%Creator: Mathematica %%AspectRatio: .61803 MathPictureStart %% Graphics /Courier findfont 10 scalefont setfont % Scaling calculations 0.02381 0.31746 0.125971 0.008375 [ [(0.5)] .18254 .12597 0 2 Msboxa [(1)] .34127 .12597 0 2 Msboxa [(1.5)] .5 .12597 0 2 Msboxa [(2)] .65873 .12597 0 2 Msboxa [(2.5)] .81746 .12597 0 2 Msboxa [(3)] .97619 .12597 0 2 Msboxa [({5 iterations completed, 1.08613 = approximation})] .5 .61803 0 -2 Msboxa [(-10)] .01131 .04223 1 0 Msboxa [(10)] .01131 .20972 1 0 Msboxa [(20)] .01131 .29346 1 0 Msboxa [(30)] .01131 .37721 1 0 Msboxa [(40)] .01131 .46095 1 0 Msboxa [(50)] .01131 .5447 1 0 Msboxa [ -0.001 -0.001 0 0 ] [ 1.001 .61903 0 0 ] ] MathScale % Start of Graphics 1 setlinecap 1 setlinejoin newpath [ ] 0 setdash 0 g p p .002 w .18254 .12597 m .18254 .13222 L s P [(0.5)] .18254 .12597 0 2 Mshowa p .002 w .34127 .12597 m .34127 .13222 L s P [(1)] .34127 .12597 0 2 Mshowa p .002 w .5 .12597 m .5 .13222 L s P [(1.5)] .5 .12597 0 2 Mshowa p .002 w .65873 .12597 m .65873 .13222 L s P [(2)] .65873 .12597 0 2 Mshowa p .002 w .81746 .12597 m .81746 .13222 L s P [(2.5)] .81746 .12597 0 2 Mshowa p .002 w .97619 .12597 m .97619 .13222 L s P [(3)] .97619 .12597 0 2 Mshowa p .001 w .05556 .12597 m .05556 .12972 L s P p .001 w .0873 .12597 m .0873 .12972 L s P p .001 w .11905 .12597 m .11905 .12972 L s P p .001 w .15079 .12597 m .15079 .12972 L s P p .001 w .21429 .12597 m .21429 .12972 L s P p .001 w .24603 .12597 m .24603 .12972 L s P p .001 w .27778 .12597 m .27778 .12972 L s P p .001 w .30952 .12597 m .30952 .12972 L s P p .001 w .37302 .12597 m .37302 .12972 L s P p .001 w .40476 .12597 m .40476 .12972 L s P p .001 w .43651 .12597 m .43651 .12972 L s P p .001 w .46825 .12597 m .46825 .12972 L s P p .001 w .53175 .12597 m .53175 .12972 L s P p .001 w .56349 .12597 m .56349 .12972 L s P p .001 w .59524 .12597 m .59524 .12972 L s P p .001 w .62698 .12597 m .62698 .12972 L s P p .001 w .69048 .12597 m .69048 .12972 L s P p .001 w .72222 .12597 m .72222 .12972 L s P p .001 w .75397 .12597 m .75397 .12972 L s P p .001 w .78571 .12597 m .78571 .12972 L s P p .001 w .84921 .12597 m .84921 .12972 L s P p .001 w .88095 .12597 m .88095 .12972 L s P p .001 w .9127 .12597 m .9127 .12972 L s P p .001 w .94444 .12597 m .94444 .12972 L s P p .002 w 0 .12597 m 1 .12597 L s P [({5 iterations completed, 1.08613 = approximation})] .5 .61803 0 -2 Mshowa p .002 w .02381 .04223 m .03006 .04223 L s P [(-10)] .01131 .04223 1 0 Mshowa p .002 w .02381 .20972 m .03006 .20972 L s P [(10)] .01131 .20972 1 0 Mshowa p .002 w .02381 .29346 m .03006 .29346 L s P [(20)] .01131 .29346 1 0 Mshowa p .002 w .02381 .37721 m .03006 .37721 L s P [(30)] .01131 .37721 1 0 Mshowa p .002 w .02381 .46095 m .03006 .46095 L s P [(40)] .01131 .46095 1 0 Mshowa p .002 w .02381 .5447 m .03006 .5447 L s P [(50)] .01131 .5447 1 0 Mshowa p .001 w .02381 .05897 m .02756 .05897 L s P p .001 w .02381 .07572 m .02756 .07572 L s P p .001 w .02381 .09247 m .02756 .09247 L s P p .001 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.02756 .00873 L s P p .001 w .02381 .56145 m .02756 .56145 L s P p .001 w .02381 .5782 m .02756 .5782 L s P p .001 w .02381 .59494 m .02756 .59494 L s P p .001 w .02381 .61169 m .02756 .61169 L s P p .002 w .02381 0 m .02381 .61803 L s P P 0 0 m 1 0 L 1 .61803 L 0 .61803 L closepath clip newpath p p p p .004 w .02381 .07572 m .02505 .07572 L .02629 .07573 L .02753 .07573 L .02877 .07573 L .03125 .07574 L .03373 .07576 L .03621 .07578 L .03869 .0758 L .04365 .07586 L .04861 .07593 L .05357 .07602 L .06349 .07626 L .07341 .07657 L .08333 .07696 L .10317 .07795 L .12302 .07925 L .14286 .08088 L .18254 .08514 L .22222 .09085 L .2619 .0981 L .30159 .10698 L .34127 .1176 L .38095 .13004 L .42063 .14442 L .46032 .16083 L .5 .17936 L .53968 .20011 L .57937 .22319 L .61905 .24869 L .65873 .27671 L .69841 .30735 L .7381 .3407 L .77778 .37686 L .81746 .41594 L .85714 .45802 L .89683 .50322 L .93651 .55162 L .97619 .60332 L s P P p p .004 w .02381 .01472 m .06349 .02752 L .10317 .04032 L .14286 .05313 L .18254 .06593 L .22222 .07874 L .2619 .09154 L .30159 .10434 L .34127 .11715 L .38095 .12995 L .42063 .14276 L .46032 .15556 L .5 .16836 L .53968 .18117 L .57937 .19397 L .61905 .20678 L .65873 .21958 L .69841 .23239 L .7381 .24519 L .77778 .25799 L .81746 .2708 L .85714 .2836 L .89683 .29641 L .93651 .30921 L .97619 .32201 L s P P P p .008 w .15079 .12597 Mdot .60766 .12597 Mdot .43207 .12597 Mdot .37498 .12597 Mdot .36869 .12597 Mdot .36861 .12597 Mdot P p .008 w .15079 .08162 Mdot .60766 .24112 Mdot .43207 .14894 Mdot .37498 .12805 Mdot .36869 .12599 Mdot P P % End of Graphics MathPictureEnd :[font = input; preserveAspect; fontName = "Courier"] Note that each graph produced with newtonsmethod keeps track of all the previous approximations, and gives you the value of the current approximation. Of course, if the initial guess were closer to the root then the convergence would be quicker. ;[s] 1:0,0;248,-1; 1:1,0,-9 -*-courier-medium-r-normal-*-*-120-75-75-*-*-*-*,courier,0,12,0,0,0; :[font = input; preserveAspect; fontName = "Courier"] Example 2 ;[s] 1:0,0;9,-1; 1:1,0,-9 -*-courier-medium-r-normal-*-*-120-75-75-*-*-*-*,courier,0,12,0,0,0; :[font = input; Cclosed; preserveAspect; fontName = "Courier"; startGroup] ?iteration :[font = print; inactive; preserveAspect; endGroup] Global`iteration iteration[f_][x_] := x - f[x]/Derivative[1][f][x] :[font = input; Cclosed; preserveAspect; fontName = "Courier"; startGroup] Nest[iteration[f], 0.4, 6] (* f as before *) :[font = output; output; inactive; preserveAspect; endGroup] 1.086130197651495 ;[o] 1.08613 :[font = input; Cclosed; preserveAspect; fontName = "Courier"; startGroup] Nest[iteration[f], -1.5, 4] :[font = output; output; inactive; preserveAspect; endGroup] -1.571993268316203 ;[o] -1.57199 :[font = input; Cclosed; preserveAspect; fontName = "Courier"; startGroup] Nest[iteration[f], -3.5, 5] :[font = output; output; inactive; preserveAspect; endGroup] -3.514136929335291 ;[o] -3.51414 :[font = input; Cclosed; preserveAspect; fontName = "Courier"; startGroup] ?FindRoot :[font = print; inactive; preserveAspect; endGroup] FindRoot[lhs == rhs, {x, x0}] searches for a numerical solution to the equation lhs == rhs, starting with x == x0. :[font = input; Cclosed; preserveAspect; fontName = "Courier"; startGroup] FindRoot[f[x] == 0, {x, 1.5}] :[font = output; output; inactive; preserveAspect; endGroup] {x -> 1.086130197651494} ;[o] {x -> 1.08613} :[font = input; Cclosed; preserveAspect; fontName = "Courier"; startGroup] FindRoot[f[x] == 0, {x, -1.5}] :[font = output; output; inactive; preserveAspect; endGroup] {x -> -1.571993268316203} ;[o] {x -> -1.57199} :[font = input; Cclosed; preserveAspect; fontName = "Courier"; startGroup] FindRoot[f[x] == 0, {x, -3.5}] :[font = output; output; inactive; preserveAspect; endGroup] {x -> -3.514136929335291} ;[o] {x -> -3.51414} :[font = input; Cclosed; preserveAspect; fontName = "Courier"; startGroup] FindRoot[Sin[x] == x, {x, 12}] :[font = message; inactive; preserveAspect] FindRoot::cvnwt: Newton's method failed to converge to the prescribed accuracy after 15 iterations. :[font = output; output; inactive; preserveAspect; endGroup; endGroup] {x -> -0.02199972487539759} ;[o] {x -> -0.0219997} :[font = subsection; inactive; Cclosed; preserveAspect; fontName = "Times"; startGroup] Functional Iteration and Fixed Points :[font = input; preserveAspect; fontName = "Courier"] Finding a fixed point of a function g(x) is equivalent to solving the equation g(x) = x. The Brouwer Fixed Point Theorem says that every continuous function from a closed n-ball to itself (in particular, from a a closed interval to itself) has a fixed point. Finding it is an interesting exercise in successive approximations. The Mathematica function FixedPoint provides one method, for example, we estimate a fixed point of the cosine function starting form an initial guess of 0.0. ;[s] 6:0,0;244,1;247,2;335,3;346,4;489,5;490,-1; 6:1,0,-9 -*-courier-medium-r-normal-*-*-120-75-75-*-*-*-*,courier,0,12,0,0,0;1,0,-9 -*-courier-medium-i-normal-*-*-120-75-75-*-*-*-*,courier,2,12,0,0,0;1,0,-9 -*-courier-medium-r-normal-*-*-120-75-75-*-*-*-*,courier,0,12,0,0,0;1,0,-9 -*-courier-medium-i-normal-*-*-120-75-75-*-*-*-*,courier,2,12,0,0,0;1,0,-9 -*-courier-medium-r-normal-*-*-120-75-75-*-*-*-*,courier,0,12,0,0,0;1,0,-9 -*-courier-bold-r-normal-*-*-100-75-75-*-*-*-*,courier,1,10,0,0,0; :[font = input; Cclosed; preserveAspect; fontName = "Courier"; startGroup] FixedPoint[Cos,0.0,20] :[font = output; output; inactive; preserveAspect; endGroup] 0.7389377567153445 ;[o] 0.738938 :[font = input; Cclosed; preserveAspect; fontName = "Courier"; startGroup] Cos[%] :[font = output; output; inactive; preserveAspect; endGroup] 0.7391843997714937 ;[o] 0.739184 :[font = input; preserveAspect; fontName = "Courier"] After a few more iterations the procedure homes in on the fixed point at 0.739085. In our book we defined a function, fixedpoint, to illustrate graphically the progress of functional iteration to find a fixed point; we look at this and some other functions. ;[s] 4:0,0;120,1;130,2;259,3;260,-1; 4:1,0,-9 -*-courier-medium-r-normal-*-*-120-75-75-*-*-*-*,courier,0,12,0,0,0;1,0,-9 -*-courier-bold-r-normal-*-*-100-75-75-*-*-*-*,courier,1,10,0,0,0;1,0,-9 -*-courier-medium-r-normal-*-*-120-75-75-*-*-*-*,courier,0,12,0,0,0;1,0,-9 -*-courier-bold-r-normal-*-*-120-75-75-*-*-*-*,courier,1,12,0,0,0; :[font = input; preserveAspect; fontName = "Courier"] Example 1 ;[s] 1:0,0;9,-1; 1:1,0,-9 -*-courier-medium-r-normal-*-*-120-75-75-*-*-*-*,courier,0,12,0,0,0; :[font = input; Cclosed; preserveAspect; fontName = "Courier"; startGroup] FindRoot[Cos[x] == x, {x, 0.0}] :[font = output; output; inactive; preserveAspect; endGroup] {x -> 0.7390851332151608} ;[o] {x -> 0.739085} :[font = input; Cclosed; preserveAspect; fontName = "Courier"; startGroup] Nest[Cos, 0.0, 35] :[font = output; output; inactive; preserveAspect; endGroup] 0.7390855263619245 ;[o] 0.739086 :[font = input; Cclosed; preserveAspect; fontName = "Courier"; startGroup] ?fixedpoint :[font = print; inactive; preserveAspect; endGroup] fixedpoint[f, a, {b, c}][n] performs (n + 1) iterations of the Fixed Point iteration method for approximating solutions to the equation f[x] = x, starting with x0 = a. The output plots f and the identity function, y = x, on the interval {b, c}; shows all successive approximations x0, x1, ... , xn, x(n + 1) along the x-axis; indicates the value of the current approximation in the PlotLabel; and shows the `cobweb' of points homing in on the fixed point. :[font = input; Cclosed; preserveAspect; fontName = "Courier"; startGroup] fixedpoint[Cos, 0.0, {-0.5, 2}][0]; :[font = postscript; PostScript; formatAsPostScript; output; inactive; preserveAspect; pictureLeft = 34; pictureWidth = 282; pictureHeight = 282; endGroup] %! %%Creator: Mathematica %%AspectRatio: 1 MathPictureStart %% Graphics /Courier findfont 10 scalefont setfont % Scaling calculations 0.2 0.4 0.2 0.4 [ [(-0.5)] 0 .2 0 2 Msboxa [(0.5)] .4 .2 0 2 Msboxa [(1)] .6 .2 0 2 Msboxa [(1.5)] .8 .2 0 2 Msboxa [(2)] 1 .2 0 2 Msboxa [({beginning fixed point iteration, 0 = initial guess})] .5 1 0 -2 Msboxa [(-0.5)] .1875 0 1 0 Msboxa [(0.5)] .1875 .4 1 0 Msboxa [(1)] .1875 .6 1 0 Msboxa [(1.5)] .1875 .8 1 0 Msboxa [(2)] .1875 1 1 0 Msboxa [ -0.001 -0.001 0 0 ] [ 1.001 1.001 0 0 ] ] MathScale % Start of Graphics 1 setlinecap 1 setlinejoin newpath [ ] 0 setdash 0 g p p .002 w 0 .2 m 0 .20625 L s P [(-0.5)] 0 .2 0 2 Mshowa p .002 w .4 .2 m .4 .20625 L s P [(0.5)] .4 .2 0 2 Mshowa p .002 w .6 .2 m .6 .20625 L s P [(1)] .6 .2 0 2 Mshowa p .002 w .8 .2 m .8 .20625 L s P [(1.5)] .8 .2 0 2 Mshowa p .002 w 1 .2 m 1 .20625 L s P [(2)] 1 .2 0 2 Mshowa p .001 w .04 .2 m .04 .20375 L s P p .001 w .08 .2 m .08 .20375 L s P p .001 w .12 .2 m .12 .20375 L s P p .001 w .16 .2 m .16 .20375 L s P p .001 w .24 .2 m .24 .20375 L s P p .001 w .28 .2 m .28 .20375 L s P p .001 w .32 .2 m .32 .20375 L s P p .001 w .36 .2 m .36 .20375 L s P p .001 w .44 .2 m .44 .20375 L s P p .001 w .48 .2 m .48 .20375 L s P p .001 w .52 .2 m .52 .20375 L s P p .001 w .56 .2 m .56 .20375 L s P p .001 w .64 .2 m .64 .20375 L s P p .001 w .68 .2 m .68 .20375 L s P p .001 w .72 .2 m .72 .20375 L s P p .001 w .76 .2 m .76 .20375 L s P p .001 w .84 .2 m .84 .20375 L s P p .001 w .88 .2 m .88 .20375 L s P p .001 w .92 .2 m .92 .20375 L s P p .001 w .96 .2 m .96 .20375 L s P p .002 w 0 .2 m 1 .2 L s P [({beginning fixed point iteration, 0 = initial guess})] .5 1 0 -2 Mshowa p .002 w .2 0 m .20625 0 L s P [(-0.5)] .1875 0 1 0 Mshowa p .002 w .2 .4 m .20625 .4 L s P [(0.5)] .1875 .4 1 0 Mshowa p .002 w .2 .6 m .20625 .6 L s P [(1)] .1875 .6 1 0 Mshowa p .002 w .2 .8 m .20625 .8 L s P [(1.5)] .1875 .8 1 0 Mshowa p .002 w .2 1 m .20625 1 L s P [(2)] .1875 1 1 0 Mshowa p .001 w .2 .04 m .20375 .04 L s P p .001 w .2 .08 m .20375 .08 L s P p .001 w .2 .12 m .20375 .12 L s P p .001 w .2 .16 m .20375 .16 L s P p .001 w .2 .24 m .20375 .24 L s P p .001 w .2 .28 m .20375 .28 L s P p .001 w .2 .32 m .20375 .32 L s P p .001 w .2 .36 m .20375 .36 L s P p .001 w .2 .44 m .20375 .44 L s P p .001 w .2 .48 m .20375 .48 L s P p .001 w .2 .52 m .20375 .52 L s P p .001 w .2 .56 m .20375 .56 L s P p .001 w .2 .64 m .20375 .64 L s P p .001 w .2 .68 m .20375 .68 L s P p .001 w .2 .72 m .20375 .72 L s P p .001 w .2 .76 m .20375 .76 L s P p .001 w .2 .84 m .20375 .84 L s P p .001 w .2 .88 m .20375 .88 L s P p .001 w .2 .92 m .20375 .92 L s P p .001 w .2 .96 m .20375 .96 L s P p .002 w .2 0 m .2 1 L s P P 0 0 m 1 0 L 1 1 L 0 1 L closepath clip newpath p p p p .004 w 0 .55103 m .04167 .56907 L .08333 .58311 L .10417 .58857 L .125 .59299 L .13542 .5948 L .14583 .59634 L .15625 .59761 L .16667 .59861 L .17188 .59901 L .17708 .59934 L .18229 .59961 L .1849 .59971 L .1875 .5998 L .1901 .59988 L .19141 .59991 L .19271 .59993 L .19401 .59996 L .19531 .59997 L .19661 .59999 L .19792 .59999 L .19922 .6 L .20052 .6 L .20182 .6 L .20313 .59999 L .20443 .59998 L .20573 .59996 L .20703 .59994 L .20833 .59991 L .21354 .59977 L .21615 .59967 L .21875 .59956 L .22917 .59894 L .23438 .59852 L .23958 .59804 L .25 .59688 L .27083 .59374 L .29167 .58954 L .33333 .57798 L .375 .56233 L .41667 .54274 L .45833 .51944 L .5 .49268 L .54167 .46274 L .58333 .42995 L .625 .39468 L .66667 .35729 L .70833 .31819 L .75 .27782 L .79167 .2366 L Mistroke .83333 .19499 L .875 .15342 L .91667 .11237 L .95833 .07226 L 1 .03354 L Mfstroke P P p p .004 w 0 0 m .04167 .04167 L .08333 .08333 L .125 .125 L .16667 .16667 L .20833 .20833 L .25 .25 L .29167 .29167 L .33333 .33333 L .375 .375 L .41667 .41667 L .45833 .45833 L .5 .5 L .54167 .54167 L .58333 .58333 L .625 .625 L .66667 .66667 L .70833 .70833 L .75 .75 L .79167 .79167 L .83333 .83333 L .875 .875 L .91667 .91667 L .95833 .95833 L 1 1 L s P P P p .008 w .2 .2 Mdot .6 .2 Mdot P p .008 w .2 .2 Mdot .6 .6 Mdot P p .008 w .2 .6 Mdot .6 .41612 Mdot P .004 w .2 .2 m .2 .6 L .6 .6 L .6 .41612 L s P % End of Graphics MathPictureEnd :[font = input; Cclosed; preserveAspect; fontName = "Courier"; startGroup] fixedpoint[Cos, 0.0, {-0.5, 2}][1]; :[font = postscript; PostScript; formatAsPostScript; output; inactive; preserveAspect; pictureLeft = 34; pictureWidth = 282; pictureHeight = 282; endGroup] %! %%Creator: Mathematica %%AspectRatio: 1 MathPictureStart %% Graphics /Courier findfont 10 scalefont setfont % Scaling calculations 0.2 0.4 0.2 0.4 [ [(-0.5)] 0 .2 0 2 Msboxa [(0.5)] .4 .2 0 2 Msboxa [(1)] .6 .2 0 2 Msboxa [(1.5)] .8 .2 0 2 Msboxa [(2)] 1 .2 0 2 Msboxa [({2 iterations completed, 0.540302 = approximation})] .5 1 0 -2 Msboxa [(-0.5)] .1875 0 1 0 Msboxa [(0.5)] .1875 .4 1 0 Msboxa [(1)] .1875 .6 1 0 Msboxa [(1.5)] .1875 .8 1 0 Msboxa [(2)] .1875 1 1 0 Msboxa [ -0.001 -0.001 0 0 ] [ 1.001 1.001 0 0 ] ] MathScale % Start of Graphics 1 setlinecap 1 setlinejoin newpath [ ] 0 setdash 0 g p p .002 w 0 .2 m 0 .20625 L s P [(-0.5)] 0 .2 0 2 Mshowa p .002 w .4 .2 m .4 .20625 L s P [(0.5)] .4 .2 0 2 Mshowa p .002 w .6 .2 m .6 .20625 L s P [(1)] .6 .2 0 2 Mshowa p .002 w .8 .2 m .8 .20625 L s P [(1.5)] .8 .2 0 2 Mshowa p .002 w 1 .2 m 1 .20625 L s P [(2)] 1 .2 0 2 Mshowa p .001 w .04 .2 m .04 .20375 L s P p .001 w .08 .2 m .08 .20375 L s P p .001 w .12 .2 m .12 .20375 L s P p .001 w .16 .2 m .16 .20375 L s P p .001 w .24 .2 m .24 .20375 L s P p .001 w .28 .2 m .28 .20375 L s P p .001 w .32 .2 m .32 .20375 L s P p .001 w .36 .2 m .36 .20375 L s P p .001 w .44 .2 m .44 .20375 L s P p .001 w .48 .2 m .48 .20375 L s P p .001 w .52 .2 m .52 .20375 L s P p .001 w .56 .2 m .56 .20375 L s P p .001 w .64 .2 m .64 .20375 L s P p .001 w .68 .2 m .68 .20375 L s P p .001 w .72 .2 m .72 .20375 L s P p .001 w .76 .2 m .76 .20375 L s P p .001 w .84 .2 m .84 .20375 L s P p .001 w .88 .2 m .88 .20375 L s P p .001 w .92 .2 m .92 .20375 L s P p .001 w .96 .2 m .96 .20375 L s P p .002 w 0 .2 m 1 .2 L s P [({2 iterations completed, 0.540302 = approximation})] .5 1 0 -2 Mshowa p .002 w .2 0 m .20625 0 L s P [(-0.5)] .1875 0 1 0 Mshowa p .002 w .2 .4 m .20625 .4 L s P [(0.5)] .1875 .4 1 0 Mshowa p .002 w .2 .6 m .20625 .6 L s P [(1)] .1875 .6 1 0 Mshowa p .002 w .2 .8 m .20625 .8 L s P [(1.5)] .1875 .8 1 0 Mshowa p .002 w .2 1 m .20625 1 L s P [(2)] .1875 1 1 0 Mshowa p .001 w .2 .04 m .20375 .04 L s P p .001 w .2 .08 m .20375 .08 L s P p .001 w .2 .12 m .20375 .12 L s P p .001 w .2 .16 m .20375 .16 L s P p .001 w .2 .24 m .20375 .24 L s P p .001 w .2 .28 m .20375 .28 L s P p .001 w .2 .32 m .20375 .32 L s P p .001 w .2 .36 m .20375 .36 L s P p .001 w .2 .44 m .20375 .44 L s P p .001 w .2 .48 m .20375 .48 L s P p .001 w .2 .52 m .20375 .52 L s P p .001 w .2 .56 m .20375 .56 L s P p .001 w .2 .64 m .20375 .64 L s P p .001 w .2 .68 m .20375 .68 L s P p .001 w .2 .72 m .20375 .72 L s P p .001 w .2 .76 m .20375 .76 L s P p .001 w .2 .84 m .20375 .84 L s P p .001 w .2 .88 m .20375 .88 L s P p .001 w .2 .92 m .20375 .92 L s P p .001 w .2 .96 m .20375 .96 L s P p .002 w .2 0 m .2 1 L s P P 0 0 m 1 0 L 1 1 L 0 1 L closepath clip newpath p p p p .004 w 0 .55103 m .04167 .56907 L .08333 .58311 L .10417 .58857 L .125 .59299 L .13542 .5948 L .14583 .59634 L .15625 .59761 L .16667 .59861 L .17188 .59901 L .17708 .59934 L .18229 .59961 L .1849 .59971 L .1875 .5998 L .1901 .59988 L .19141 .59991 L .19271 .59993 L .19401 .59996 L .19531 .59997 L .19661 .59999 L .19792 .59999 L .19922 .6 L .20052 .6 L .20182 .6 L .20313 .59999 L .20443 .59998 L .20573 .59996 L .20703 .59994 L .20833 .59991 L .21354 .59977 L .21615 .59967 L .21875 .59956 L .22917 .59894 L .23438 .59852 L .23958 .59804 L .25 .59688 L .27083 .59374 L .29167 .58954 L .33333 .57798 L .375 .56233 L .41667 .54274 L .45833 .51944 L .5 .49268 L .54167 .46274 L .58333 .42995 L .625 .39468 L .66667 .35729 L .70833 .31819 L .75 .27782 L .79167 .2366 L Mistroke .83333 .19499 L .875 .15342 L .91667 .11237 L .95833 .07226 L 1 .03354 L Mfstroke P P p p .004 w 0 0 m .04167 .04167 L .08333 .08333 L .125 .125 L .16667 .16667 L .20833 .20833 L .25 .25 L .29167 .29167 L .33333 .33333 L .375 .375 L .41667 .41667 L .45833 .45833 L .5 .5 L .54167 .54167 L .58333 .58333 L .625 .625 L .66667 .66667 L .70833 .70833 L .75 .75 L .79167 .79167 L .83333 .83333 L .875 .875 L .91667 .91667 L .95833 .95833 L 1 1 L s P P P p .008 w .2 .2 Mdot .6 .2 Mdot .41612 .2 Mdot P p .008 w .2 .2 Mdot .6 .6 Mdot .41612 .41612 Mdot P p .008 w .2 .6 Mdot .6 .41612 Mdot .41612 .54302 Mdot P .004 w .2 .2 m .2 .6 L .6 .6 L .6 .41612 L .41612 .41612 L .41612 .54302 L s P % End of Graphics MathPictureEnd :[font = input; Cclosed; preserveAspect; fontName = "Courier"; startGroup] fixedpoint[Cos, 0.0, {-0.5, 2}][2]; :[font = postscript; PostScript; formatAsPostScript; output; inactive; preserveAspect; pictureLeft = 34; pictureWidth = 282; pictureHeight = 282; endGroup] %! %%Creator: Mathematica %%AspectRatio: 1 MathPictureStart %% Graphics /Courier findfont 10 scalefont setfont % Scaling calculations 0.2 0.4 0.2 0.4 [ [(-0.5)] 0 .2 0 2 Msboxa [(0.5)] .4 .2 0 2 Msboxa [(1)] .6 .2 0 2 Msboxa [(1.5)] .8 .2 0 2 Msboxa [(2)] 1 .2 0 2 Msboxa [({3 iterations completed, 0.857553 = approximation})] .5 1 0 -2 Msboxa [(-0.5)] .1875 0 1 0 Msboxa [(0.5)] .1875 .4 1 0 Msboxa [(1)] .1875 .6 1 0 Msboxa [(1.5)] .1875 .8 1 0 Msboxa [(2)] .1875 1 1 0 Msboxa [ -0.001 -0.001 0 0 ] [ 1.001 1.001 0 0 ] ] MathScale % Start of Graphics 1 setlinecap 1 setlinejoin newpath [ ] 0 setdash 0 g p p .002 w 0 .2 m 0 .20625 L s P [(-0.5)] 0 .2 0 2 Mshowa p .002 w .4 .2 m .4 .20625 L s P [(0.5)] .4 .2 0 2 Mshowa p .002 w .6 .2 m .6 .20625 L s P [(1)] .6 .2 0 2 Mshowa p .002 w .8 .2 m .8 .20625 L s P [(1.5)] .8 .2 0 2 Mshowa p .002 w 1 .2 m 1 .20625 L s P [(2)] 1 .2 0 2 Mshowa p .001 w .04 .2 m .04 .20375 L s P p .001 w .08 .2 m .08 .20375 L s P p .001 w .12 .2 m .12 .20375 L s P p .001 w .16 .2 m .16 .20375 L s P p .001 w .24 .2 m .24 .20375 L s P p .001 w .28 .2 m .28 .20375 L s P p .001 w .32 .2 m .32 .20375 L s P p .001 w .36 .2 m .36 .20375 L s P p .001 w .44 .2 m .44 .20375 L s P p .001 w .48 .2 m .48 .20375 L s P p .001 w .52 .2 m .52 .20375 L s P p .001 w .56 .2 m .56 .20375 L s P p .001 w .64 .2 m .64 .20375 L s P p .001 w .68 .2 m .68 .20375 L s P p .001 w .72 .2 m .72 .20375 L s P p .001 w .76 .2 m .76 .20375 L s P p .001 w .84 .2 m .84 .20375 L s P p .001 w .88 .2 m .88 .20375 L s P p .001 w .92 .2 m .92 .20375 L s P p .001 w .96 .2 m .96 .20375 L s P p .002 w 0 .2 m 1 .2 L s P [({3 iterations completed, 0.857553 = approximation})] .5 1 0 -2 Mshowa p .002 w .2 0 m .20625 0 L s P [(-0.5)] .1875 0 1 0 Mshowa p .002 w .2 .4 m .20625 .4 L s P [(0.5)] .1875 .4 1 0 Mshowa p .002 w .2 .6 m .20625 .6 L s P [(1)] .1875 .6 1 0 Mshowa p .002 w .2 .8 m .20625 .8 L s P [(1.5)] .1875 .8 1 0 Mshowa p .002 w .2 1 m .20625 1 L s P [(2)] .1875 1 1 0 Mshowa p .001 w .2 .04 m .20375 .04 L s P p .001 w .2 .08 m .20375 .08 L s P p .001 w .2 .12 m .20375 .12 L s P p .001 w .2 .16 m .20375 .16 L s P p .001 w .2 .24 m .20375 .24 L s P p .001 w .2 .28 m .20375 .28 L s P p .001 w .2 .32 m .20375 .32 L s P p .001 w .2 .36 m .20375 .36 L s P p .001 w .2 .44 m .20375 .44 L s P p .001 w .2 .48 m .20375 .48 L s P p .001 w .2 .52 m .20375 .52 L s P p .001 w .2 .56 m .20375 .56 L s P p .001 w .2 .64 m .20375 .64 L s P p .001 w .2 .68 m .20375 .68 L s P p .001 w .2 .72 m .20375 .72 L s P p .001 w .2 .76 m .20375 .76 L s P p .001 w .2 .84 m .20375 .84 L s P p .001 w .2 .88 m .20375 .88 L s P p .001 w .2 .92 m .20375 .92 L s P p .001 w .2 .96 m .20375 .96 L s P p .002 w .2 0 m .2 1 L s P P 0 0 m 1 0 L 1 1 L 0 1 L closepath clip newpath p p p p .004 w 0 .55103 m .04167 .56907 L .08333 .58311 L .10417 .58857 L .125 .59299 L .13542 .5948 L .14583 .59634 L .15625 .59761 L .16667 .59861 L .17188 .59901 L .17708 .59934 L .18229 .59961 L .1849 .59971 L .1875 .5998 L .1901 .59988 L .19141 .59991 L .19271 .59993 L .19401 .59996 L .19531 .59997 L .19661 .59999 L .19792 .59999 L .19922 .6 L .20052 .6 L .20182 .6 L .20313 .59999 L .20443 .59998 L .20573 .59996 L .20703 .59994 L .20833 .59991 L .21354 .59977 L .21615 .59967 L .21875 .59956 L .22917 .59894 L .23438 .59852 L .23958 .59804 L .25 .59688 L .27083 .59374 L .29167 .58954 L .33333 .57798 L .375 .56233 L .41667 .54274 L .45833 .51944 L .5 .49268 L .54167 .46274 L .58333 .42995 L .625 .39468 L .66667 .35729 L .70833 .31819 L .75 .27782 L .79167 .2366 L Mistroke .83333 .19499 L .875 .15342 L .91667 .11237 L .95833 .07226 L 1 .03354 L Mfstroke P P p p .004 w 0 0 m .04167 .04167 L .08333 .08333 L .125 .125 L .16667 .16667 L .20833 .20833 L .25 .25 L .29167 .29167 L .33333 .33333 L .375 .375 L .41667 .41667 L .45833 .45833 L .5 .5 L .54167 .54167 L .58333 .58333 L .625 .625 L .66667 .66667 L .70833 .70833 L .75 .75 L .79167 .79167 L .83333 .83333 L .875 .875 L .91667 .91667 L .95833 .95833 L 1 1 L s P P P p .008 w .2 .2 Mdot .6 .2 Mdot .41612 .2 Mdot .54302 .2 Mdot P p .008 w .2 .2 Mdot .6 .6 Mdot .41612 .41612 Mdot .54302 .54302 Mdot P p .008 w .2 .6 Mdot .6 .41612 Mdot .41612 .54302 Mdot .54302 .46172 Mdot P .004 w .2 .2 m .2 .6 L .6 .6 L .6 .41612 L .41612 .41612 L .41612 .54302 L .54302 .54302 L .54302 .46172 L s P % End of Graphics MathPictureEnd :[font = input; Cclosed; preserveAspect; fontName = "Courier"; startGroup] fixedpoint[Cos, 0.0, {-0.5, 2}][6]; :[font = postscript; PostScript; formatAsPostScript; output; inactive; preserveAspect; pictureLeft = 34; pictureWidth = 282; pictureHeight = 282; endGroup] %! %%Creator: Mathematica %%AspectRatio: 1 MathPictureStart %% Graphics /Courier findfont 10 scalefont setfont % Scaling calculations 0.2 0.4 0.2 0.4 [ [(-0.5)] 0 .2 0 2 Msboxa [(0.5)] .4 .2 0 2 Msboxa [(1)] .6 .2 0 2 Msboxa [(1.5)] .8 .2 0 2 Msboxa [(2)] 1 .2 0 2 Msboxa [({7 iterations completed, 0.76396 = approximation})] .5 1 0 -2 Msboxa [(-0.5)] .1875 0 1 0 Msboxa [(0.5)] .1875 .4 1 0 Msboxa [(1)] .1875 .6 1 0 Msboxa [(1.5)] .1875 .8 1 0 Msboxa [(2)] .1875 1 1 0 Msboxa [ -0.001 -0.001 0 0 ] [ 1.001 1.001 0 0 ] ] MathScale % Start of Graphics 1 setlinecap 1 setlinejoin newpath [ ] 0 setdash 0 g p p .002 w 0 .2 m 0 .20625 L s P [(-0.5)] 0 .2 0 2 Mshowa p .002 w .4 .2 m .4 .20625 L s P [(0.5)] .4 .2 0 2 Mshowa p .002 w .6 .2 m .6 .20625 L s P [(1)] .6 .2 0 2 Mshowa p .002 w .8 .2 m .8 .20625 L s P [(1.5)] .8 .2 0 2 Mshowa p .002 w 1 .2 m 1 .20625 L s P [(2)] 1 .2 0 2 Mshowa p .001 w .04 .2 m .04 .20375 L s P p .001 w .08 .2 m .08 .20375 L s P p .001 w .12 .2 m .12 .20375 L s P p .001 w .16 .2 m .16 .20375 L s P p .001 w .24 .2 m .24 .20375 L s P p .001 w .28 .2 m .28 .20375 L s P p .001 w .32 .2 m .32 .20375 L s P p .001 w .36 .2 m .36 .20375 L s P p .001 w .44 .2 m .44 .20375 L s P p .001 w .48 .2 m .48 .20375 L s P p .001 w .52 .2 m .52 .20375 L s P p .001 w .56 .2 m .56 .20375 L s P p .001 w .64 .2 m .64 .20375 L s P p .001 w .68 .2 m .68 .20375 L s P p .001 w .72 .2 m .72 .20375 L s P p .001 w .76 .2 m .76 .20375 L s P p .001 w .84 .2 m .84 .20375 L s P p .001 w .88 .2 m .88 .20375 L s P p .001 w .92 .2 m .92 .20375 L s P p .001 w .96 .2 m .96 .20375 L s P p .002 w 0 .2 m 1 .2 L s P [({7 iterations completed, 0.76396 = approximation})] .5 1 0 -2 Mshowa p .002 w .2 0 m .20625 0 L s P [(-0.5)] .1875 0 1 0 Mshowa p .002 w .2 .4 m .20625 .4 L s P [(0.5)] .1875 .4 1 0 Mshowa p .002 w .2 .6 m .20625 .6 L s P [(1)] .1875 .6 1 0 Mshowa p .002 w .2 .8 m .20625 .8 L s P [(1.5)] .1875 .8 1 0 Mshowa p .002 w .2 1 m .20625 1 L s P [(2)] .1875 1 1 0 Mshowa p .001 w .2 .04 m .20375 .04 L s P p .001 w .2 .08 m .20375 .08 L s P p .001 w .2 .12 m .20375 .12 L s P p .001 w .2 .16 m .20375 .16 L s P p .001 w .2 .24 m .20375 .24 L s P p .001 w .2 .28 m .20375 .28 L s P p .001 w .2 .32 m .20375 .32 L s P p .001 w .2 .36 m .20375 .36 L s P p .001 w .2 .44 m .20375 .44 L s P p .001 w .2 .48 m .20375 .48 L s P p .001 w .2 .52 m .20375 .52 L s P p .001 w .2 .56 m .20375 .56 L s P p .001 w .2 .64 m .20375 .64 L s P p .001 w .2 .68 m .20375 .68 L s P p .001 w .2 .72 m .20375 .72 L s P p .001 w .2 .76 m .20375 .76 L s P p .001 w .2 .84 m .20375 .84 L s P p .001 w .2 .88 m .20375 .88 L s P p .001 w .2 .92 m .20375 .92 L s P p .001 w .2 .96 m .20375 .96 L s P p .002 w .2 0 m .2 1 L s P P 0 0 m 1 0 L 1 1 L 0 1 L closepath clip newpath p p p p .004 w 0 .55103 m .04167 .56907 L .08333 .58311 L .10417 .58857 L .125 .59299 L .13542 .5948 L .14583 .59634 L .15625 .59761 L .16667 .59861 L .17188 .59901 L .17708 .59934 L .18229 .59961 L .1849 .59971 L .1875 .5998 L .1901 .59988 L .19141 .59991 L .19271 .59993 L .19401 .59996 L .19531 .59997 L .19661 .59999 L .19792 .59999 L .19922 .6 L .20052 .6 L .20182 .6 L .20313 .59999 L .20443 .59998 L .20573 .59996 L .20703 .59994 L .20833 .59991 L .21354 .59977 L .21615 .59967 L .21875 .59956 L .22917 .59894 L .23438 .59852 L .23958 .59804 L .25 .59688 L .27083 .59374 L .29167 .58954 L .33333 .57798 L .375 .56233 L .41667 .54274 L .45833 .51944 L .5 .49268 L .54167 .46274 L .58333 .42995 L .625 .39468 L .66667 .35729 L .70833 .31819 L .75 .27782 L .79167 .2366 L Mistroke .83333 .19499 L .875 .15342 L .91667 .11237 L .95833 .07226 L 1 .03354 L Mfstroke P P p p .004 w 0 0 m .04167 .04167 L .08333 .08333 L .125 .125 L .16667 .16667 L .20833 .20833 L .25 .25 L .29167 .29167 L .33333 .33333 L .375 .375 L .41667 .41667 L .45833 .45833 L .5 .5 L .54167 .54167 L .58333 .58333 L .625 .625 L .66667 .66667 L .70833 .70833 L .75 .75 L .79167 .79167 L .83333 .83333 L .875 .875 L .91667 .91667 L .95833 .95833 L 1 1 L s P P P p .008 w .2 .2 Mdot .6 .2 Mdot .41612 .2 Mdot .54302 .2 Mdot .46172 .2 Mdot .51739 .2 Mdot .48055 .2 Mdot .50558 .2 Mdot P p .008 w .2 .2 Mdot .6 .6 Mdot .41612 .41612 Mdot .54302 .54302 Mdot .46172 .46172 Mdot .51739 .51739 Mdot .48055 .48055 Mdot .50558 .50558 Mdot P p .008 w .2 .6 Mdot .6 .41612 Mdot .41612 .54302 Mdot .54302 .46172 Mdot .46172 .51739 Mdot .51739 .48055 Mdot .48055 .50558 Mdot .50558 .48884 Mdot P .004 w .2 .2 m .2 .6 L .6 .6 L .6 .41612 L .41612 .41612 L .41612 .54302 L .54302 .54302 L .54302 .46172 L .46172 .46172 L .46172 .51739 L .51739 .51739 L .51739 .48055 L .48055 .48055 L .48055 .50558 L .50558 .50558 L .50558 .48884 L s P % End of Graphics MathPictureEnd :[font = input; Cclosed; preserveAspect; fontName = "Courier"; startGroup] ?FixedPoint :[font = print; inactive; preserveAspect; endGroup] FixedPoint[f, expr] starts with expr, then applies f repeatedly until the result no longer changes. FixedPoint[f, expr, n] stops after at most n steps. :[font = input; Cclosed; preserveAspect; fontName = "Courier"; startGroup] FixedPoint[Cos, 0.0] :[font = output; output; inactive; preserveAspect; endGroup] 0.7390851332151606 ;[o] 0.739085 :[font = input; Cclosed; preserveAspect; fontName = "Courier"; startGroup] ?FixedPointList :[font = print; inactive; preserveAspect; endGroup] FixedPointList[f, expr] generates a list giving the results of applying f repeatedly, starting with expr, until the results no longer change. FixedPointList[f, expr, n] stops after at most n steps. :[font = input; Cclosed; preserveAspect; fontName = "Courier"; startGroup] FixedPointList[Cos, 0.0, 40] :[font = output; output; inactive; preserveAspect; endGroup] {0., 1., 0.5403023058681398, 0.857553215846393, 0.6542897904977792, 0.7934803587425657, 0.7013687736227566, 0.7639596829006542, 0.7221024250267078, 0.7504177617637606, 0.7314040424225098, 0.7442373549005569, 0.7356047404363474, 0.7414250866101093, 0.7375068905132429, 0.7401473355678757, 0.7383692041223232, 0.7395672022122561, 0.7387603198742113, 0.7393038923969059, 0.7389377567153445, 0.7391843997714937, 0.7390182624274122, 0.7391301765296711, 0.7390547907469174, 0.7391055719265363, 0.7390713652989449, 0.7390944073790913, 0.7390788859949921, 0.7390893414033927, 0.7390822985224024, 0.7390870426953322, 0.7390838469650002, 0.7390859996481299, 0.7390845495752127, 0.7390855263619245, 0.7390848683867142, 0.739085311606762, 0.7390850130484204, 0.7390852141609172, 0.739085078689123} ;[o] {0., 1., 0.540302, 0.857553, 0.65429, 0.79348, 0.701369, 0.76396, 0.722102, 0.750418, 0.731404, 0.744237, 0.735605, 0.741425, 0.737507, 0.740147, 0.738369, 0.739567, 0.73876, 0.739304, 0.738938, 0.739184, 0.739018, 0.73913, 0.739055, 0.739106, 0.739071, 0.739094, 0.739079, 0.739089, 0.739082, 0.739087, 0.739084, 0.739086, 0.739085, 0.739086, 0.739085, 0.739085, 0.739085, 0.739085, 0.739085} :[font = input; Cclosed; preserveAspect; fontName = "Courier"; startGroup] ListPlot[%]; :[font = postscript; PostScript; formatAsPostScript; output; inactive; preserveAspect; pictureLeft = 34; pictureWidth = 282; pictureHeight = 174; endGroup] %! %%Creator: Mathematica %%AspectRatio: .61803 MathPictureStart %% Graphics /Courier findfont 10 scalefont setfont % Scaling calculations 0.02381 0.023229 -40.865523 55.729601 [ [(10)] .2561 .37438 0 2 Msboxa [(20)] .48839 .37438 0 2 Msboxa [(30)] .72067 .37438 0 2 Msboxa [(40)] .95296 .37438 0 2 Msboxa [(0.734)] .01131 .04 1 0 Msboxa [(0.736)] .01131 .15146 1 0 Msboxa [(0.738)] .01131 .26292 1 0 Msboxa [(0.742)] .01131 .48584 1 0 Msboxa [(0.744)] .01131 .5973 1 0 Msboxa [ -0.001 -0.001 0 0 ] [ 1.001 .61903 0 0 ] ] MathScale % Start of Graphics 1 setlinecap 1 setlinejoin newpath [ ] 0 setdash 0 g p p .002 w .2561 .37438 m .2561 .38063 L s P [(10)] .2561 .37438 0 2 Mshowa p .002 w .48839 .37438 m .48839 .38063 L s P [(20)] .48839 .37438 0 2 Mshowa p .002 w .72067 .37438 m .72067 .38063 L s P [(30)] .72067 .37438 0 2 Mshowa p .002 w .95296 .37438 m .95296 .38063 L s P [(40)] .95296 .37438 0 2 Mshowa p .001 w .07027 .37438 m .07027 .37813 L s P p .001 w .11672 .37438 m .11672 .37813 L s P p .001 w .16318 .37438 m .16318 .37813 L s P p .001 w .20964 .37438 m .20964 .37813 L s P p .001 w .30256 .37438 m .30256 .37813 L s P p .001 w .34901 .37438 m .34901 .37813 L s P p .001 w .39547 .37438 m .39547 .37813 L s P p .001 w .44193 .37438 m .44193 .37813 L s P p .001 w .53484 .37438 m .53484 .37813 L s P p .001 w .5813 .37438 m .5813 .37813 L s P p .001 w .62776 .37438 m .62776 .37813 L s P p .001 w .67422 .37438 m .67422 .37813 L s P p .001 w .76713 .37438 m .76713 .37813 L s P p .001 w .81359 .37438 m .81359 .37813 L s P p .001 w .86005 .37438 m .86005 .37813 L s P p .001 w .9065 .37438 m .9065 .37813 L s P p .001 w .99942 .37438 m .99942 .37813 L s P p .002 w 0 .37438 m 1 .37438 L s P p .002 w .02381 .04 m .03006 .04 L s P [(0.734)] .01131 .04 1 0 Mshowa p .002 w .02381 .15146 m .03006 .15146 L s P [(0.736)] .01131 .15146 1 0 Mshowa p .002 w .02381 .26292 m .03006 .26292 L s P [(0.738)] .01131 .26292 1 0 Mshowa p .002 w .02381 .48584 m .03006 .48584 L s P [(0.742)] .01131 .48584 1 0 Mshowa p .002 w .02381 .5973 m .03006 .5973 L s P [(0.744)] .01131 .5973 1 0 Mshowa p .001 w .02381 .0623 m .02756 .0623 L s P p .001 w .02381 .08459 m .02756 .08459 L s P p .001 w .02381 .10688 m .02756 .10688 L s P p .001 w .02381 .12917 m .02756 .12917 L s P p .001 w .02381 .17375 m .02756 .17375 L s P p .001 w .02381 .19605 m .02756 .19605 L s P p .001 w .02381 .21834 m .02756 .21834 L s P p .001 w .02381 .24063 m .02756 .24063 L s P p .001 w .02381 .28521 m .02756 .28521 L s P p .001 w .02381 .30751 m .02756 .30751 L s P p .001 w .02381 .3298 m .02756 .3298 L s P p .001 w .02381 .35209 m .02756 .35209 L s P p .001 w .02381 .39667 m .02756 .39667 L s P p .001 w .02381 .41897 m .02756 .41897 L s P p .001 w .02381 .44126 m .02756 .44126 L s P p .001 w .02381 .46355 m .02756 .46355 L s P p .001 w .02381 .50813 m .02756 .50813 L s P p .001 w .02381 .53042 m .02756 .53042 L s P p .001 w .02381 .55272 m .02756 .55272 L s P p .001 w .02381 .57501 m .02756 .57501 L s P p .001 w .02381 .01771 m .02756 .01771 L s P p .002 w .02381 0 m .02381 .61803 L s P P 0 0 m 1 0 L 1 .61803 L 0 .61803 L closepath clip newpath p .008 w .04704 -40.86552 Mdot .07027 14.86408 Mdot .0935 -10.75469 Mdot .11672 6.92558 Mdot .13995 -4.40221 Mdot .16318 3.35482 Mdot .18641 -1.77852 Mdot .20964 1.70965 Mdot .23287 -0.62304 Mdot .2561 .95496 Mdot .27933 -0.10467 Mdot .30256 .61053 Mdot .32578 .12944 Mdot .34901 .4538 Mdot .37224 .23544 Mdot .39547 .38259 Mdot .4187 .2835 Mdot .44193 .35026 Mdot .46516 .30529 Mdot .48839 .33559 Mdot .51161 .31518 Mdot .53484 .32893 Mdot .55807 .31967 Mdot .5813 .32591 Mdot .60453 .32171 Mdot .62776 .32454 Mdot .65099 .32263 Mdot .67422 .32391 Mdot .69744 .32305 Mdot .72067 .32363 Mdot .7439 .32324 Mdot .76713 .3235 Mdot .79036 .32332 Mdot .81359 .32344 Mdot .83682 .32336 Mdot .86005 .32342 Mdot .88328 .32338 Mdot .9065 .32341 Mdot .92973 .32339 Mdot .95296 .3234 Mdot .97619 .32339 Mdot P % End of Graphics MathPictureEnd :[font = input; preserveAspect; fontName = "Courier"] Example 2 Consider next the family of parabolic functions for k>0: ;[s] 1:0,0;68,-1; 1:1,0,-9 -*-courier-medium-r-normal-*-*-120-75-75-*-*-*-*,courier,0,12,0,0,0; :[font = input; preserveAspect; fontName = "Courier"] Clear[g]; g[k_][x_] := k x (1 - x) :[font = input; Cclosed; preserveAspect; fontName = "Courier"; startGroup] FixedPoint[g[2.6], 0.1] :[font = output; output; inactive; preserveAspect; endGroup] 0.6153846153846154 ;[o] 0.615385 :[font = input; Cclosed; preserveAspect; fontName = "Courier"; startGroup] fixedpoint[g[2.6], 0.1, {0, 1}][20]; :[font = postscript; PostScript; formatAsPostScript; output; inactive; preserveAspect; pictureLeft = 34; pictureWidth = 282; pictureHeight = 282; endGroup] %! %%Creator: Mathematica %%AspectRatio: 1 MathPictureStart %% Graphics /Courier findfont 10 scalefont setfont % Scaling calculations 0 1 0 1 [ [(0.2)] .2 0 0 2 Msboxa [(0.4)] .4 0 0 2 Msboxa [(0.6)] .6 0 0 2 Msboxa [(0.8)] .8 0 0 2 Msboxa [(1)] 1 0 0 2 Msboxa [({21 iterations completed, 0.615388 = approximation})] .5 1 0 -2 Msboxa [(0)] -0.0125 0 1 0 Msboxa [(0.2)] -0.0125 .2 1 0 Msboxa [(0.4)] -0.0125 .4 1 0 Msboxa [(0.6)] -0.0125 .6 1 0 Msboxa [(0.8)] -0.0125 .8 1 0 Msboxa [(1)] -0.0125 1 1 0 Msboxa [ -0.001 -0.001 0 0 ] [ 1.001 1.001 0 0 ] ] MathScale % Start of Graphics 1 setlinecap 1 setlinejoin newpath [ ] 0 setdash 0 g p p .002 w .2 0 m .2 .00625 L s P [(0.2)] .2 0 0 2 Mshowa p .002 w .4 0 m .4 .00625 L s P [(0.4)] .4 0 0 2 Mshowa p .002 w .6 0 m .6 .00625 L s P [(0.6)] .6 0 0 2 Mshowa p .002 w .8 0 m .8 .00625 L s P [(0.8)] .8 0 0 2 Mshowa p .002 w 1 0 m 1 .00625 L s P [(1)] 1 0 0 2 Mshowa p .001 w .04 0 m .04 .00375 L s P p .001 w .08 0 m .08 .00375 L s P p .001 w .12 0 m .12 .00375 L s P p .001 w .16 0 m .16 .00375 L s P p .001 w .24 0 m .24 .00375 L s P p .001 w .28 0 m .28 .00375 L s P p .001 w .32 0 m .32 .00375 L s P p .001 w .36 0 m .36 .00375 L s P p .001 w .44 0 m .44 .00375 L s P p .001 w .48 0 m .48 .00375 L s P p .001 w .52 0 m .52 .00375 L s P p .001 w .56 0 m .56 .00375 L s P p .001 w .64 0 m .64 .00375 L s P p .001 w .68 0 m .68 .00375 L s P p .001 w .72 0 m .72 .00375 L s P p .001 w .76 0 m .76 .00375 L s P p .001 w .84 0 m .84 .00375 L s P p .001 w .88 0 m .88 .00375 L s P p .001 w .92 0 m .92 .00375 L s P p .001 w .96 0 m .96 .00375 L s P p .002 w 0 0 m 1 0 L s P [({21 iterations completed, 0.615388 = approximation})] .5 1 0 -2 Mshowa p .002 w 0 0 m .00625 0 L s P [(0)] -0.0125 0 1 0 Mshowa p .002 w 0 .2 m .00625 .2 L s P [(0.2)] -0.0125 .2 1 0 Mshowa p .002 w 0 .4 m .00625 .4 L s P [(0.4)] -0.0125 .4 1 0 Mshowa p .002 w 0 .6 m .00625 .6 L s P [(0.6)] -0.0125 .6 1 0 Mshowa p .002 w 0 .8 m .00625 .8 L s P [(0.8)] -0.0125 .8 1 0 Mshowa p .002 w 0 1 m .00625 1 L s P [(1)] -0.0125 1 1 0 Mshowa p .001 w 0 .04 m .00375 .04 L s P p .001 w 0 .08 m .00375 .08 L s P p .001 w 0 .12 m .00375 .12 L s P p .001 w 0 .16 m .00375 .16 L s P p .001 w 0 .24 m .00375 .24 L s P p .001 w 0 .28 m .00375 .28 L s P p .001 w 0 .32 m .00375 .32 L s P p .001 w 0 .36 m .00375 .36 L s P p .001 w 0 .44 m .00375 .44 L s P p .001 w 0 .48 m .00375 .48 L s P p .001 w 0 .52 m .00375 .52 L s P p .001 w 0 .56 m .00375 .56 L s P p .001 w 0 .64 m .00375 .64 L s P p .001 w 0 .68 m .00375 .68 L s P p .001 w 0 .72 m .00375 .72 L s P p .001 w 0 .76 m .00375 .76 L s P p .001 w 0 .84 m .00375 .84 L s P p .001 w 0 .88 m .00375 .88 L s P p .001 w 0 .92 m .00375 .92 L s P p .001 w 0 .96 m .00375 .96 L s P p .002 w 0 0 m 0 1 L s P P 0 0 m 1 0 L 1 1 L 0 1 L closepath clip newpath p p p p .004 w 0 0 m .04167 .10382 L .08333 .19861 L .125 .28438 L .16667 .36111 L .20833 .42882 L .25 .4875 L .29167 .53715 L .33333 .57778 L .375 .60938 L .39583 .62179 L .41667 .63194 L .4375 .63984 L .44792 .64295 L .45833 .64549 L .46875 .64746 L .47396 .64824 L .47917 .64887 L .48438 .64937 L .48698 .64956 L .48958 .64972 L .49219 .64984 L .49349 .64989 L .49479 .64993 L .49609 .64996 L .4974 .64998 L .4987 .65 L .5 .65 L .5013 .65 L .5026 .64998 L .50391 .64996 L .50521 .64993 L .50651 .64989 L .50781 .64984 L .51042 .64972 L .51302 .64956 L .51563 .64937 L .52083 .64887 L .52604 .64824 L .53125 .64746 L .54167 .64549 L .55208 .64295 L .5625 .63984 L .58333 .63194 L .60417 .62179 L .625 .60938 L .66667 .57778 L .70833 .53715 L .75 .4875 L .79167 .42882 L Mistroke .83333 .36111 L .875 .28438 L .91667 .19861 L .95833 .10382 L 1 0 L Mfstroke P P p p .004 w 0 0 m .04167 .04167 L .08333 .08333 L .125 .125 L .16667 .16667 L .20833 .20833 L .25 .25 L .29167 .29167 L .33333 .33333 L .375 .375 L .41667 .41667 L .45833 .45833 L .5 .5 L .54167 .54167 L .58333 .58333 L .625 .625 L .66667 .66667 L .70833 .70833 L .75 .75 L .79167 .79167 L .83333 .83333 L .875 .875 L .91667 .91667 L .95833 .95833 L 1 1 L s P P P p .008 w .1 0 Mdot .234 0 Mdot .46603 0 Mdot .647 0 Mdot .59382 0 Mdot .62712 0 Mdot .60799 0 Mdot .61968 0 Mdot .61276 0 Mdot .61694 0 Mdot .61444 0 Mdot .61595 0 Mdot .61505 0 Mdot .61559 0 Mdot .61526 0 Mdot .61546 0 Mdot .61534 0 Mdot .61541 0 Mdot .61537 0 Mdot .61539 0 Mdot .61538 0 Mdot .61539 0 Mdot P p .008 w .1 .1 Mdot .234 .234 Mdot .46603 .46603 Mdot .647 .647 Mdot .59382 .59382 Mdot .62712 .62712 Mdot .60799 .60799 Mdot .61968 .61968 Mdot .61276 .61276 Mdot .61694 .61694 Mdot .61444 .61444 Mdot .61595 .61595 Mdot .61505 .61505 Mdot .61559 .61559 Mdot .61526 .61526 Mdot .61546 .61546 Mdot .61534 .61534 Mdot .61541 .61541 Mdot .61537 .61537 Mdot .61539 .61539 Mdot .61538 .61538 Mdot .61539 .61539 Mdot P p .008 w .1 .234 Mdot .234 .46603 Mdot .46603 .647 Mdot .647 .59382 Mdot .59382 .62712 Mdot .62712 .60799 Mdot .60799 .61968 Mdot .61968 .61276 Mdot .61276 .61694 Mdot .61694 .61444 Mdot .61444 .61595 Mdot .61595 .61505 Mdot .61505 .61559 Mdot .61559 .61526 Mdot .61526 .61546 Mdot .61546 .61534 Mdot .61534 .61541 Mdot .61541 .61537 Mdot .61537 .61539 Mdot .61539 .61538 Mdot .61538 .61539 Mdot .61539 .61538 Mdot P .004 w .1 .1 m .1 .234 L .234 .234 L .234 .46603 L .46603 .46603 L .46603 .647 L .647 .647 L .647 .59382 L .59382 .59382 L .59382 .62712 L .62712 .62712 L .62712 .60799 L .60799 .60799 L .60799 .61968 L .61968 .61968 L .61968 .61276 L .61276 .61276 L .61276 .61694 L .61694 .61694 L .61694 .61444 L .61444 .61444 L .61444 .61595 L .61595 .61595 L .61595 .61505 L .61505 .61505 L .61505 .61559 L .61559 .61559 L .61559 .61526 L .61526 .61526 L .61526 .61546 L .61546 .61546 L .61546 .61534 L .61534 .61534 L .61534 .61541 L .61541 .61541 L .61541 .61537 L .61537 .61537 L .61537 .61539 L .61539 .61539 L .61539 .61538 L .61538 .61538 L .61538 .61539 L .61539 .61539 L .61539 .61538 L s P % End of Graphics MathPictureEnd :[font = input; Cclosed; preserveAspect; fontName = "Courier"; startGroup] ListPlot[ FixedPointList[g[2.6], 0.1, 20] ]; :[font = postscript; PostScript; formatAsPostScript; output; inactive; preserveAspect; pictureLeft = 34; pictureWidth = 282; pictureHeight = 174; endGroup] %! %%Creator: Mathematica %%AspectRatio: .61803 MathPictureStart %% Graphics /Courier findfont 10 scalefont setfont % Scaling calculations 0.02381 0.045351 -3.730522 6.698358 [ [(5)] .25057 .28849 0 2 Msboxa [(10)] .47732 .28849 0 2 Msboxa [(15)] .70408 .28849 0 2 Msboxa [(20)] .93084 .28849 0 2 Msboxa [(0.56)] .01131 .02056 1 0 Msboxa [(0.58)] .01131 .15453 1 0 Msboxa [(0.62)] .01131 .42246 1 0 Msboxa [(0.64)] .01131 .55643 1 0 Msboxa [ -0.001 -0.001 0 0 ] [ 1.001 .61903 0 0 ] ] MathScale % Start of Graphics 1 setlinecap 1 setlinejoin newpath [ ] 0 setdash 0 g p p .002 w .25057 .28849 m .25057 .29474 L s P [(5)] .25057 .28849 0 2 Mshowa p .002 w .47732 .28849 m .47732 .29474 L s P [(10)] .47732 .28849 0 2 Mshowa p .002 w .70408 .28849 m .70408 .29474 L s P [(15)] .70408 .28849 0 2 Mshowa p .002 w .93084 .28849 m .93084 .29474 L s P [(20)] .93084 .28849 0 2 Mshowa p .001 w .06916 .28849 m .06916 .29224 L s P p .001 w .11451 .28849 m .11451 .29224 L s P p .001 w .15986 .28849 m .15986 .29224 L s P p .001 w .20522 .28849 m .20522 .29224 L s P p .001 w .29592 .28849 m .29592 .29224 L s P p .001 w .34127 .28849 m .34127 .29224 L s P p .001 w .38662 .28849 m .38662 .29224 L s P p .001 w .43197 .28849 m .43197 .29224 L s P p .001 w .52268 .28849 m .52268 .29224 L s P p .001 w .56803 .28849 m .56803 .29224 L s P p .001 w .61338 .28849 m .61338 .29224 L s P p .001 w .65873 .28849 m .65873 .29224 L s P p .001 w .74943 .28849 m .74943 .29224 L s P p .001 w .79478 .28849 m .79478 .29224 L s P p .001 w .84014 .28849 m .84014 .29224 L s P p .001 w .88549 .28849 m .88549 .29224 L s P p .001 w .97619 .28849 m .97619 .29224 L s P p .002 w 0 .28849 m 1 .28849 L s P p .002 w .02381 .02056 m .03006 .02056 L s P [(0.56)] .01131 .02056 1 0 Mshowa p .002 w .02381 .15453 m .03006 .15453 L s P [(0.58)] .01131 .15453 1 0 Mshowa p .002 w .02381 .42246 m .03006 .42246 L s P [(0.62)] .01131 .42246 1 0 Mshowa p .002 w .02381 .55643 m .03006 .55643 L s P [(0.64)] .01131 .55643 1 0 Mshowa p .001 w .02381 .04735 m .02756 .04735 L s P p .001 w .02381 .07415 m .02756 .07415 L s P p .001 w .02381 .10094 m .02756 .10094 L s P p .001 w .02381 .12773 m .02756 .12773 L s P p .001 w .02381 .18132 m .02756 .18132 L s P p .001 w .02381 .20811 m .02756 .20811 L s P p .001 w .02381 .23491 m .02756 .23491 L s P p .001 w .02381 .2617 m .02756 .2617 L s P p .001 w .02381 .31529 m .02756 .31529 L s P p .001 w .02381 .34208 m .02756 .34208 L s P p .001 w .02381 .36887 m .02756 .36887 L s P p .001 w .02381 .39567 m .02756 .39567 L s P p .001 w .02381 .44925 m .02756 .44925 L s P p .001 w .02381 .47605 m .02756 .47605 L s P p .001 w .02381 .50284 m .02756 .50284 L s P p .001 w .02381 .52963 m .02756 .52963 L s P p .001 w .02381 .58322 m .02756 .58322 L s P p .001 w .02381 .61001 m .02756 .61001 L s P p .002 w .02381 0 m .02381 .61803 L s P P 0 0 m 1 0 L 1 .61803 L 0 .61803 L closepath clip newpath p .008 w .06916 -3.06069 Mdot .11451 -2.16311 Mdot .15986 -0.60886 Mdot .20522 .60332 Mdot .25057 .24707 Mdot .29592 .47013 Mdot .34127 .342 Mdot .38662 .42032 Mdot .43197 .37396 Mdot .47732 .40198 Mdot .52268 .38524 Mdot .56803 .39531 Mdot .61338 .38928 Mdot .65873 .3929 Mdot .70408 .39073 Mdot .74943 .39203 Mdot .79478 .39125 Mdot .84014 .39172 Mdot .88549 .39144 Mdot .93084 .39161 Mdot .97619 .39151 Mdot P % End of Graphics MathPictureEnd :[font = input; preserveAspect; fontName = "Courier"] Example 3 ;[s] 1:0,0;9,-1; 1:1,0,-9 -*-courier-medium-r-normal-*-*-120-75-75-*-*-*-*,courier,0,12,0,0,0; :[font = input; Cclosed; preserveAspect; fontName = "Courier"; startGroup] fixedpoint[g[3.2], 0.1, {0, 1}][50]; :[font = postscript; PostScript; formatAsPostScript; output; inactive; preserveAspect; pictureLeft = 34; pictureWidth = 282; pictureHeight = 282; endGroup] %! %%Creator: Mathematica %%AspectRatio: 1 MathPictureStart %% Graphics /Courier findfont 10 scalefont setfont % Scaling calculations 0 1 0 1 [ [(0.2)] .2 0 0 2 Msboxa [(0.4)] .4 0 0 2 Msboxa [(0.6)] .6 0 0 2 Msboxa [(0.8)] .8 0 0 2 Msboxa [(1)] 1 0 0 2 Msboxa [({51 iterations completed, 0.799455 = approximation})] .5 1 0 -2 Msboxa [(0)] -0.0125 0 1 0 Msboxa [(0.2)] -0.0125 .2 1 0 Msboxa [(0.4)] -0.0125 .4 1 0 Msboxa [(0.6)] -0.0125 .6 1 0 Msboxa [(0.8)] -0.0125 .8 1 0 Msboxa [(1)] -0.0125 1 1 0 Msboxa [ -0.001 -0.001 0 0 ] [ 1.001 1.001 0 0 ] ] MathScale % Start of Graphics 1 setlinecap 1 setlinejoin newpath [ ] 0 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