(*********************************************************************** Mathematica-Compatible Notebook This notebook can be used on any computer system with Mathematica 3.0, MathReader 3.0, or any compatible application. The data for the notebook starts with the line of stars above. To get the notebook into a Mathematica-compatible application, do one of the following: * Save the data starting with the line of stars above into a file with a name ending in .nb, then open the file inside the application; * Copy the data starting with the line of stars above to the clipboard, then use the Paste menu command inside the application. Data for notebooks contains only printable 7-bit ASCII and can be sent directly in email or through ftp in text mode. Newlines can be CR, LF or CRLF (Unix, Macintosh or MS-DOS style). NOTE: If you modify the data for this notebook not in a Mathematica- compatible application, you must delete the line below containing the word CacheID, otherwise Mathematica-compatible applications may try to use invalid cache data. For more information on notebooks and Mathematica-compatible applications, contact Wolfram Research: web: http://www.wolfram.com email: info@wolfram.com phone: +1-217-398-0700 (U.S.) Notebook reader applications are available free of charge from Wolfram Research. ***********************************************************************) (*CacheID: 232*) (*NotebookFileLineBreakTest NotebookFileLineBreakTest*) (*NotebookOptionsPosition[ 605345, 13135]*) (*NotebookOutlinePosition[ 606399, 13171]*) (* CellTagsIndexPosition[ 606355, 13167]*) (*WindowFrame->Normal*) Notebook[{ Cell[CellGroupData[{Cell[TextData[{ StyleBox[ " (C) Copyright 1994 D.C.M. Burbulla and C.T.J. Dodson\n \ ", Evaluatable->False, AspectRatioFixed->True, FontSize->12, PrivateFontOptions->{"FontPostScriptName"->Automatic}], StyleBox[" Notebook to support the book", Evaluatable->False, AspectRatioFixed->True, FontSize->17, FontSlant->"Italic", PrivateFontOptions->{"FontPostScriptName"->Automatic}], StyleBox["\n Self-Tutor for Computer Calculus", Evaluatable->False, AspectRatioFixed->True, PrivateFontOptions->{"FontPostScriptName"->Automatic}] }], "Title", Evaluatable->False, AspectRatioFixed->True], Cell[TextData[{ StyleBox["Using ", Evaluatable->False, AspectRatioFixed->True, FontSize->23, PrivateFontOptions->{"FontPostScriptName"->Automatic}], StyleBox["Mathematica 2 ", Evaluatable->False, AspectRatioFixed->True, FontSize->23, FontWeight->"Plain", FontSlant->"Italic", PrivateFontOptions->{"FontPostScriptName"->Automatic}] }], "Subtitle", Evaluatable->False, AspectRatioFixed->True], Cell[TextData[{ StyleBox[ " by\n D. C. M. Burbulla and C. T. J. Dodson, University of Toronto\n\ Published by: \n Prentice-Hall Canada (English) ", Evaluatable->False, AspectRatioFixed->True, FontFamily->"times", FontSize->12, PrivateFontOptions->{"FontPostScriptName"->Automatic}], StyleBox[" ", Evaluatable->False, AspectRatioFixed->True, FontFamily->"times", FontSize->11, PrivateFontOptions->{"FontPostScriptName"->Automatic}], StyleBox[" ISBN 0130152803", Evaluatable->False, AspectRatioFixed->True, FontFamily->"courier", FontSize->11, PrivateFontOptions->{"FontPostScriptName"->Automatic}], StyleBox[" and \n Prentice-Hall Japan (Japanese) ", Evaluatable->False, AspectRatioFixed->True, FontFamily->"times", FontSize->12, PrivateFontOptions->{"FontPostScriptName"->Automatic}], StyleBox["ISBN4-8101-8558-3 C3055 P2880E", Evaluatable->False, AspectRatioFixed->True, FontFamily->"courier", FontSize->11, PrivateFontOptions->{"FontPostScriptName"->Automatic}] }], "Subsubtitle", Evaluatable->False, AspectRatioFixed->True, FontFamily->"Times"], Cell[CellGroupData[{Cell[TextData["Calculus Projects"], "Section", Evaluatable->False, AspectRatioFixed->True], Cell[TextData[StyleBox[ "Here we suggest some project directions that may be interesting to pursue.\n\ For subsequent use, we introduce some new functions.", AspectRatioFixed->True, FontFamily->"courier", FontWeight->"Plain", PrivateFontOptions->{"FontPostScriptName"->Automatic}]], "Input", AspectRatioFixed->True, FontFamily->"Courier"], Cell[CellGroupData[{Cell[TextData["Chap13.m"], "Subsection", Evaluatable->False, AspectRatioFixed->True, FontFamily->"Times"], Cell[CellGroupData[{Cell[TextData[ "points/: points[{a_, b_}, {c_, d_}] := Block[{x, y},\n Table[N[{x, \ y}], {x, a, b, (b - a)/10}, {y, c, d, (d - c)/10}]]\n\t \n\nf/: f[x_] := x^4 \ - 1\n\n\niteration/: iteration[f_][x_] := x - f[x]/Derivative[1][f][x]\n\n\n\ attractors/: attractors[f_, {a_, b_}, {c_, d_}, n_] := \n \ Block[{x, y}, Table[Nest[iteration[f], N[x + y*I], n],\n\t {x, a, b, (b - \ a)/10}, {y, c, d, (d - c)/10}]]\n\t \n\t \n\t \ncolour/: \ colour[i_, j_] := Which[\n attractor[[i, j]] == 1., RGBColor[1, 0, \ 0],\n\t attractor[[i, j]] == 1. I, RGBColor[0, 1, 0],\n\t attractor[[i, j]] \ == -1., RGBColor[0, 0, 1],\n\t attractor[[i, j]] == -1. I, RGBColor[1, 1, 0],\ \n\t True, RGBColor[1, 0, 1]]\n\n\npatch/: patch[f_, {a_, b_}, {c_, d_}, n_] \ := (\n lattice = points[{a, b}, {c, d}];\n\tattractor = attractors[f, \ {a, b}, {c, d}, n];\n\tShow[Table[Graphics[{colour[i, j], Point[lattice[[i, \ j]]]}],\n\t{i, 1, 11}, {j, 1, 11}]])"], "Input", AspectRatioFixed->True, FontFamily->"Courier"], Cell[TextData[ "General::spell1: \n Possible spelling error: new symbol name \n \ \"attractor\" is similar to existing symbol \n \"attractors\"."], "Message",\ Evaluatable->False, AspectRatioFixed->True]}, Open]]}, Open]], Cell[CellGroupData[{Cell[TextData["Newton's Method in the Complex Plane"], "Subsection", Evaluatable->False, AspectRatioFixed->True, FontFamily->"Times"], Cell[TextData[{ StyleBox[ "Recall that the formula for Newton's Method is defined as \niteration[f] \ in the package Chap4.m above. We need this package called in,\nand then \ proceed to investigate the convergence of Newton's method for the\nfunction \ f(x) = x^3 - 3 x^2 - x + 3 which has zeros 1, -1, 3 -3.", AspectRatioFixed->True, FontFamily->"courier", FontWeight->"Plain", PrivateFontOptions->{"FontPostScriptName"->Automatic}], StyleBox["\n", AspectRatioFixed->True, FontFamily->"courier", FontSize->10, PrivateFontOptions->{"FontPostScriptName"->Automatic}] }], "Input", AspectRatioFixed->True, FontFamily->"Courier"], Cell[TextData["Clear[f]; f[x_] := x^3 - 3 x^2 - x + 3\n"], "Input", AspectRatioFixed->True, FontFamily->"Courier"], Cell[CellGroupData[{Cell[TextData["Solve[f[x] == 0, x]\n"], "Input", AspectRatioFixed->True, FontFamily->"Courier"], Cell[OutputFormData["\<\ {{x -> -1}, {x -> 1}, {x -> 3}}\ \>", "\<\ {{x -> -1}, {x -> 1}, {x -> 3}}\ \>"], "Output", Evaluatable->False, AspectRatioFixed->True]}, Open]], Cell[TextData[{ StyleBox[ "By making use of iteration[f], define a function root, as\nfollows, to \ perform n iterations of Newton's method with initial choice x:", AspectRatioFixed->True, FontFamily->"courier", FontWeight->"Plain", PrivateFontOptions->{"FontPostScriptName"->Automatic}], StyleBox["\n", AspectRatioFixed->True, FontFamily->"courier", FontSize->10, PrivateFontOptions->{"FontPostScriptName"->Automatic}] }], "Input", AspectRatioFixed->True, FontFamily->"Courier"], Cell[TextData["root[f_, x_, n_] := Nest[iteration[f], N[x], n]"], "Input", AspectRatioFixed->True, FontFamily->"Courier"], Cell[CellGroupData[{Cell[TextData["root[f, 4.5, 20]"], "Input", AspectRatioFixed->True, FontFamily->"Courier"], Cell[OutputFormData["\<\ 3.\ \>", "\<\ 3.\ \>"], "Output", Evaluatable->False, AspectRatioFixed->True]}, Open]], Cell[CellGroupData[{Cell[TextData["root[f, -4.5, 20]"], "Input", AspectRatioFixed->True, FontFamily->"Courier"], Cell[OutputFormData["\<\ -1.\ \>", "\<\ -1.\ \>"], "Output", Evaluatable->False, AspectRatioFixed->True]}, Open]], Cell[CellGroupData[{Cell[TextData["root[f, 0.3, 20]"], "Input", AspectRatioFixed->True, FontFamily->"Courier"], Cell[OutputFormData["\<\ 1.\ \>", "\<\ 1.\ \>"], "Output", Evaluatable->False, AspectRatioFixed->True]}, Open]], Cell[TextData[{ StyleBox[ "Let us now consider the equation x^4 - 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However, things do not stay\nso \ predictable if we allow complex numbers as initial choices. We\nshall not \ cover any calculus of complex numbers here; we only point out\nthat all the \ calculus required in Newton's method works equally well for\ncomplex numbers.\ \n\nTo begin with, let us take an initial choice of 1 + 2i, where i is the\n\ square root of -1. Newton's method then converges to the root i:", AspectRatioFixed->True, FontFamily->"courier", FontWeight->"Plain", PrivateFontOptions->{"FontPostScriptName"->Automatic}]], "Input", AspectRatioFixed->True, FontFamily->"Courier"], Cell[CellGroupData[{Cell[TextData["root[f, 1 + 2 I, 20]"], "Input", AspectRatioFixed->True, FontFamily->"Courier"], Cell[OutputFormData["\<\ 0. + 1.*I\ \>", "\<\ 0. + 1. I\ \>"], "Output", Evaluatable->False, AspectRatioFixed->True]}, Open]], Cell[TextData[StyleBox[ "Instead of investigating things further with tables of calculations, let\nus \ use colours. To do this we shall represent the complex number x + y i\nby the \ point (x, y) in the plane. If the initial choice x + y i\nresults in \ Newton's method converging to the root 1 we shall colour in\nthe point (x, y) \ red; if the convergence is to the root i we shall\ncolour the point (x, y) \ green. Similarly we shall use blue to colour in\ninitial points which result \ in convergence to -1, and yellow for initial\npoints which result in \ convergence to -i. If there is no convergence, we\nshall colour in the \ initial point magenta.\n\nIn the package for this chapter, you will find all \ the necessary functions to do \nall of the above colouring. 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We shall need the function fixedpoint from \ Chap4.m:"], "Text", Evaluatable->False, AspectRatioFixed->True], Cell[TextData[ "fixedpoint/: fixedpoint[g_, a_, {b_, c_}][n_] := (one = \n \ Block[{x, i}, Plot[{g[x], x}, {x, b, c}, \n\t AspectRatio -> Automatic, \ PlotRange -> {{b, c}, {b, c}}, \n\t DisplayFunction -> Identity]];\n\t \ two = Block[{i}, ListPlot[Table[{Nest[g, N[a], i], 0}, \n\t {i, 0, n + \ 1}], DisplayFunction -> Identity]];\n\t three = Block[{i}, \ ListPlot[Table[{Nest[g, N[a], i], \n\t Nest[g, N[a], i]}, {i, 0, n + 1}], \ DisplayFunction -> Identity]];\n\t four = Block[{i}, \ ListPlot[Table[{Nest[g, N[a], i], \n\t Nest[g, N[a], i + 1]}, {i, 0, n + \ 1}],\n\t DisplayFunction -> Identity]];\n\t five = Block[{i}, \ ListPlot[Table[{Nest[g, N[a], Floor[i/2]],\n\t Mod[i + 1, 2]*Nest[g, \ N[a], Floor[i/2]] + \n\t Mod[i, 2]*g[Nest[g, N[a], Floor[i/2]]]}, {i, 0, \ 2*n + 3}],\n\t PlotJoined -> True, DisplayFunction -> Identity]];\n\t \ Show[one, two, three, four, five,\n 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67oo0P00ogoo0Goo000HOol00`00Oomoo`3oOol001Qoo`03001oogoo0?moo`00\ \>"], ImageRangeCache->{{{0, 281}, {173, 0}} -> {-24.8876, -0.288647, 1.03385, 0.0174943}}]}, Open]], Cell[TextData[{ StyleBox[ "This correctly reveals that there were frequency components at frequencies \ f=20, 40 and \n80 cycles per second. (Their ghosts appear again at \ frequencies 256-f where f=20,40,80.\nMore discussion of the Fourier transform \ function is given in the ", AspectRatioFixed->True, FontFamily->"courier", FontWeight->"Plain", PrivateFontOptions->{"FontPostScriptName"->Automatic}], StyleBox["Mathematica", AspectRatioFixed->True, FontFamily->"courier", FontWeight->"Plain", FontSlant->"Italic", PrivateFontOptions->{"FontPostScriptName"->Automatic}], StyleBox[" book.)", AspectRatioFixed->True, FontFamily->"courier", FontWeight->"Plain", PrivateFontOptions->{"FontPostScriptName"->Automatic}] }], "Input", AspectRatioFixed->True, FontFamily->"Courier"]}, Open]], Cell[CellGroupData[{Cell[TextData["Nesting Functions"], "Section", Evaluatable->False, AspectRatioFixed->True], Cell[TextData[StyleBox[ "The function Nest can quickly generate iterations of funtions. For example,\n\ if we take a quadratic and an initial value , say 0.1, then we can tabulate \ \nthe first three iterations as follows:", AspectRatioFixed->True, FontFamily->"courier", FontWeight->"Plain", PrivateFontOptions->{"FontPostScriptName"->Automatic}]], "Input", AspectRatioFixed->True, FontFamily->"Courier"], Cell[TextData["p[x_] := x^2"], "Input", AspectRatioFixed->True, FontFamily->"Courier"], Cell[CellGroupData[{Cell[TextData["ptab=Table[ Nest[p, 0.1, n], {n,3}]"], "Input", AspectRatioFixed->True, FontFamily->"Courier"], Cell[OutputFormData["\<\ {0.01, 0.0001, 1.000000000000001*10^-8}\ \>", "\<\ -8 {0.01, 0.0001, 1. 10 }\ \>"], "Output", Evaluatable->False, AspectRatioFixed->True]}, Open]], Cell[TextData[StyleBox[ "Which correctly gives p[0.1], p[p[0.1]], p[p[p[0.1]]].\n\nThe following \ function ", AspectRatioFixed->True, FontFamily->"courier", FontWeight->"Plain", PrivateFontOptions->{"FontPostScriptName"->Automatic}]], "Input", AspectRatioFixed->True, FontFamily->"Courier"], Cell[TextData["f[x_] := 1 - 2 x^2"], "Input", AspectRatioFixed->True, FontFamily->"Courier"], Cell[TextData[StyleBox[ "looks simple enough. It is bounded by +1 and -1 on the interval [-1,1] \nand \ achieves its single maximum, 1, at x=0; it has minima of -1 at the\n\ endpoints of the interval and it has two fixed points, at x=-1 and x=0.5. \n\ Now pick any number in the interval [-1,1] and Nest f. For example, \nwith \ initial value again 0.1 ", AspectRatioFixed->True, FontFamily->"courier", FontWeight->"Plain", PrivateFontOptions->{"FontPostScriptName"->Automatic}]], "Input", AspectRatioFixed->True, FontFamily->"Courier"], Cell[CellGroupData[{Cell[TextData["Table[ Nest[f, 0.1, n], {n,3}]"], "Input", AspectRatioFixed->True, FontFamily->"Courier"], Cell[OutputFormData["\<\ {0.98, -0.9208, -0.6957452799999995}\ \>", "\<\ {0.98, -0.9208, -0.695745}\ \>"], "Output", Evaluatable->False, AspectRatioFixed->True]}, Open]], Cell[TextData[ " which has no obvious pattern. Try a few more iterations"], "Text",\ Evaluatable->False, AspectRatioFixed->True], Cell[CellGroupData[{Cell[TextData["Table[ Nest[f, 0.1, n], {n,10}]"], "Input", AspectRatioFixed->True, FontFamily->"Courier"], Cell[OutputFormData["\<\ {0.98, -0.9208, -0.6957452799999995, 0.03187701071544469, 0.997967712375695, -0.991879109888755, -0.967648337267418, -0.872686609232799, -0.5231638358684814, 0.4525992016787534}\ \>", "\<\ {0.98, -0.9208, -0.695745, 0.031877, 0.997968, -0.991879, -0.967648, -0.872687, -0.523164, 0.452599}\ \>"], "Output", Evaluatable->False, AspectRatioFixed->True]}, Open]], Cell[TextData[StyleBox[ " In fact, try a large number and ListPlot them; so give the Table a name \ first", AspectRatioFixed->True, FontFamily->"courier", FontWeight->"Plain", PrivateFontOptions->{"FontPostScriptName"->Automatic}]], "Input", AspectRatioFixed->True, FontFamily->"Courier"], Cell[CellGroupData[{Cell[TextData["ftab=Table[ Nest[f, 0.1, n], {n,256}];"], "Input", AspectRatioFixed->True, 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