(* Content-type: application/mathematica *) (*** Wolfram Notebook File ***) (* http://www.wolfram.com/nb *) (* CreatedBy='Mathematica 6.0' *) (*CacheID: 234*) (* Internal cache information: NotebookFileLineBreakTest NotebookFileLineBreakTest NotebookDataPosition[ 145, 7] NotebookDataLength[ 23488146, 679847] NotebookOptionsPosition[ 23485554, 679767] NotebookOutlinePosition[ 23486246, 679792] CellTagsIndexPosition[ 23486203, 679789] WindowFrame->Normal ContainsDynamic->False*) (* Beginning of Notebook Content *) Notebook[{ Cell[CellGroupData[{ Cell["Fractals from Newton's Method", "Title"], Cell[TextData[{ "Newton's method for solving the equation ", StyleBox["f(z) = 0", "InlineFormula"], " generates succesive estimates for the root from the iteration" }], "Text"], Cell[BoxData[ RowBox[{ RowBox[{"z", "'"}], " ", "=", " ", RowBox[{"z", " ", "-", " ", FractionBox[ RowBox[{"f", RowBox[{"(", "z", ")"}]}], RowBox[{ RowBox[{"f", "'"}], RowBox[{"(", "z", ")"}]}]]}]}]], "DisplayFormula"], Cell[TextData[{ "Here we take f(z) = ", Cell[BoxData[ FormBox[ RowBox[{ SuperscriptBox["z", "n"], " ", "-", " ", "1", " "}], TraditionalForm]]], ", whose roots are at z = ", Cell[BoxData[ FormBox[ SuperscriptBox["e", "i\[Theta]"], TraditionalForm]]], "where \[Theta] = 2 \[Pi] k / n, where k takes integer values from 0 to \ n-1. \n\nIt turns out that, for n > 2, the boundaries of the regions of the \ complex plane which converge to each of the roots have a ", StyleBox["complicated fractal geometry", FontWeight->"Bold"], ". \n\nTo see this we generate a function, ", StyleBox["newt(n, z)", "Input"], ", which finds which root of ", Cell[BoxData[ FormBox[ RowBox[{ SuperscriptBox["z", "n"], " ", "=", " ", "1"}], TraditionalForm]]], " one converges too starting from the given value of z, and returns the \ value of \[Theta]/(2 \[Pi]). ", StyleBox["newt", FontWeight->"Bold"], " uses the ", StyleBox["Mathematica", FontSlant->"Italic"], " function ", StyleBox["FixedPoint[fun, z, k] ", FontWeight->"Bold"], "(where the third argument is the maximum number of iterations) to determine \ which root the iterations of the Newton Raphson method converge to. The \ argument of z then gives \[Theta]." }], "Text", CellChangeTimes->{3.482973634324123*^9}], Cell[BoxData[ RowBox[{"Clear", "[", "\"\\"", "]"}]], "Input", CellChangeTimes->{{3.4211152635280447`*^9, 3.421115267023038*^9}}], Cell[BoxData[ RowBox[{ RowBox[{ RowBox[{"newt", "[", RowBox[{"n_", ",", " ", "z_"}], "]"}], " ", ":=", " ", RowBox[{ RowBox[{"Arg", "[", RowBox[{"FixedPoint", "[", RowBox[{ RowBox[{ RowBox[{"#", "-", RowBox[{ RowBox[{"(", RowBox[{ RowBox[{"#", "^", "n"}], " ", "-", " ", "1"}], ")"}], "/", RowBox[{"(", RowBox[{"n", " ", RowBox[{"#", "^", RowBox[{"(", RowBox[{"n", "-", "1"}], ")"}]}]}], ")"}]}]}], "&"}], ",", RowBox[{"N", "[", "z", "]"}], ",", "50"}], "]"}], "]"}], " ", "/", " ", RowBox[{"(", RowBox[{"2", " ", "Pi"}], ")"}]}]}], ";"}]], "Input"], Cell[TextData[{ "The function f(z) = ", Cell[BoxData[ FormBox[ SuperscriptBox["z", "n"], TraditionalForm]]], "- 1 is represented as a pure function, ", StyleBox["#^n - 1", FontWeight->"Bold"], " (where ", StyleBox["#", FontWeight->"Bold"], " is the value of z) which is terminated by a ", StyleBox["&", FontWeight->"Bold"], ". Hence z - f(z) / f'(z) is represented by the hieroglyphics # -(#^n - \ 1)/(n #^(n-1))& . The FixedPoint function will either converge to a fixed \ point or stop after the specified number of iterations (50 here). \n\nIt will \ be more efficient to compile this code, specifying that n is integer and z is \ complex:" }], "Text", CellChangeTimes->{{3.421120068084948*^9, 3.4211200814752073`*^9}}], Cell[BoxData[ RowBox[{ RowBox[{"newtC", " ", "=", " ", RowBox[{"Compile", "[", RowBox[{ RowBox[{"{", RowBox[{ RowBox[{"{", RowBox[{"n", ",", " ", "_Integer"}], "}"}], ",", " ", RowBox[{"{", RowBox[{"z", ",", " ", "_Complex"}], "}"}]}], "}"}], ",", RowBox[{ RowBox[{"Arg", "[", RowBox[{"FixedPoint", "[", RowBox[{ RowBox[{ RowBox[{"#", "-", RowBox[{ RowBox[{"(", RowBox[{ RowBox[{"#", "^", "n"}], "-", "1"}], ")"}], "/", RowBox[{"(", RowBox[{"n", " ", RowBox[{"#", "^", RowBox[{"(", RowBox[{"n", "-", "1"}], ")"}]}]}], ")"}]}]}], "&"}], ",", RowBox[{"N", "[", "z", "]"}], ",", "50"}], "]"}], "]"}], " ", "/", " ", RowBox[{"(", RowBox[{"2", " ", "Pi"}], ")"}]}]}], "]"}]}], ";"}]], "Input"], Cell[TextData[{ "Note, as discussed before, that the arguments of ", StyleBox["newtC", FontWeight->"Bold"], " appear as the first argument of ", StyleBox["Compile", FontWeight->"Bold"], " on the right hand side, where we have the option, used here, of specifying \ whether they are real, integer, or complex (the default is real)." }], "Text", CellChangeTimes->{3.421145144040444*^9}], Cell[TextData[{ "For example, for n = 3, starting from 2, -2 + 2 I, and -2 - 2 I, ", StyleBox["newt", FontWeight->"Bold"], " converges to the three cube roots of -1 (which have argument 0, 2\[Pi]/3, \ and -2\[Pi]/3, so the output from ", StyleBox["newt", FontWeight->"Bold"], " is 0, 1/3, and -1/3 since I divide by 2 \[Pi]). " }], "Text", CellChangeTimes->{3.4211159720848513`*^9, 3.4211451526019897`*^9}], Cell[CellGroupData[{ Cell[BoxData[ RowBox[{"{", RowBox[{ RowBox[{"newtC", "[", RowBox[{"3", ",", " ", "2"}], "]"}], ",", " ", RowBox[{"newtC", "[", RowBox[{"3", ",", " ", RowBox[{ RowBox[{"-", "2"}], " ", "+", " ", RowBox[{"2", "I"}]}]}], "]"}], ",", " ", RowBox[{"newtC", "[", RowBox[{"3", ",", " ", RowBox[{ RowBox[{"-", "2"}], " ", "-", " ", RowBox[{"2", " ", "I"}]}]}], "]"}]}], "}"}]], "Input", CellChangeTimes->{{3.421117286076272*^9, 3.4211172921951113`*^9}}], Cell[BoxData[ RowBox[{"{", RowBox[{"0.`", ",", "0.33333333333333337`", ",", RowBox[{"-", "0.33333333333333337`"}]}], "}"}]], "Output", CellChangeTimes->{3.421115304310625*^9, 3.421115582870071*^9, 3.4211172328455143`*^9, 3.421117293108358*^9, 3.4211173578982677`*^9, 3.421119509897586*^9, 3.482973785162219*^9, 3.483755789589739*^9, 3.483756162312478*^9, 3.514837291221777*^9, 3.515350684225337*^9}] }, Open ]], Cell[TextData[{ "In previous versions of ", StyleBox["Mathematica", FontSlant->"Italic"], " I constructed a routine ", StyleBox["newtonplot", FontWeight->"Bold"], " which produces a density plot of the values produced by ", StyleBox["newtC", FontWeight->"Bold"], ", such that each color corresponds to a particular root. The color is \ specified by the option ", StyleBox["ColorFunction \[Rule] (Hue[ #]&) ", FontWeight->"Bold"], " of the ", StyleBox["DensityPlot", FontWeight->"Bold"], " command. The argument of the function ", StyleBox["Hue[h] ", FontWeight->"Bold"], "has to lie", StyleBox[" ", FontWeight->"Bold"], "between 0 and 1. (If an argument outside this range is given, it is shifted \ by an appropriate integer to put it into this range, e.g. -0.3 \[Rule] 0.7.) \ As ", StyleBox["h", FontWeight->"Bold"], " ranges between 0 and 1 the corresponding color goes through red, yellow, \ green, cyan, blue, magenta, and back to red again. The output from ", StyleBox["newt", FontWeight->"Bold"], " is therefore an appropriate argument for ", StyleBox["Hue", FontWeight->"Bold"], ". Note that ", StyleBox["Hue", FontWeight->"Bold"], " is indicated as a \"pure function\", i.e. its argument is specified by ", StyleBox["#", FontWeight->"Bold"], " and the end of the function is indicated by ", StyleBox["&", FontWeight->"Bold"], ". The option ", StyleBox["ColorFunctionScaling \[Rule] False", FontWeight->"Bold"], " prevents ", StyleBox["Mathematica", FontSlant->"Italic"], " from trying to scale the range of values fed to the ", StyleBox["ColorFunction", FontWeight->"Bold"], " to lie between 0 and 1, which leads to less satisfactory colors. (It's \ better to do the scaling oneself, which I have done by dividing the argument \ of the root by 2 \[Pi].)" }], "Text", CellChangeTimes->{{3.421145144040444*^9, 3.421145239405717*^9}, 3.482973695521969*^9}], Cell[BoxData[ RowBox[{ RowBox[{"newtonplot", "[", RowBox[{ "n_", ",", " ", "npoint_", ",", " ", "xmin_", ",", "xmax_", ",", " ", "ymin_", ",", " ", "ymax_"}], "]"}], " ", ":=", RowBox[{"DensityPlot", "[", "\[IndentingNewLine]", RowBox[{ RowBox[{"newtC", "[", RowBox[{"n", ",", " ", RowBox[{"x", " ", "+", " ", RowBox[{"I", " ", "y"}]}]}], "]"}], ",", "\[IndentingNewLine]", RowBox[{"{", RowBox[{"x", ",", "xmin", ",", " ", "xmax"}], "}"}], ",", RowBox[{"{", RowBox[{"y", ",", "ymin", ",", "ymax"}], "}"}], ",", RowBox[{"Mesh", "\[Rule]", "False"}], ",", RowBox[{"PlotPoints", "\[Rule]", "npoint"}], ",", " ", RowBox[{"Frame", " ", "\[Rule]", " ", "False"}], ",", RowBox[{"ColorFunctionScaling", " ", "\[Rule]", " ", "False"}], ",", " ", RowBox[{"ColorFunction", "\[Rule]", RowBox[{"(", " ", RowBox[{ RowBox[{"Hue", "[", "#", "]"}], "&"}], ")"}]}]}], "]"}]}]], "Input", Evaluatable->False, CellChangeTimes->{ 3.4211152954369802`*^9, {3.4211172191741667`*^9, 3.421117224043105*^9}, { 3.421117336072164*^9, 3.421117354020336*^9}}, FontSlant->"Italic"], Cell[TextData[{ StyleBox["However DensityPlot is impossibly slow with version 6. \n\n", FontWeight->"Bold"], "I therefore, instead, explicitly create a list of values to be plotted, and \ then plot them using ", StyleBox["ListDensityPlot", FontWeight->"Bold"], ":" }], "Text", CellChangeTimes->{{3.4211452466405993`*^9, 3.4211453234633217`*^9}, { 3.4211455669810677`*^9, 3.421145576825377*^9}}], Cell[BoxData[ RowBox[{ RowBox[{ RowBox[{"array", "[", RowBox[{ "n_", ",", " ", "npoint_", ",", " ", "xmin_", ",", "xmax_", ",", " ", "ymin_", ",", " ", "ymax_"}], "]"}], " ", ":=", RowBox[{"Flatten", "[", RowBox[{ RowBox[{"Table", "[", RowBox[{ RowBox[{"{", RowBox[{"x", ",", " ", "y", ",", " ", RowBox[{"newtC", "[", RowBox[{"n", ",", " ", RowBox[{"x", " ", "+", " ", RowBox[{"I", " ", "y"}]}]}], "]"}]}], "}"}], ",", " ", RowBox[{"{", RowBox[{"x", ",", " ", "xmin", ",", " ", "xmax", ",", " ", RowBox[{ RowBox[{"(", RowBox[{"xmax", "-", "xmin"}], ")"}], "/", "npoint"}]}], "}"}], ",", " ", RowBox[{"{", RowBox[{"y", ",", " ", "ymin", ",", " ", "ymax", ",", " ", RowBox[{ RowBox[{"(", RowBox[{"ymax", "-", "ymin"}], ")"}], "/", "npoint"}]}], "}"}]}], "]"}], ",", " ", "1"}], "]"}]}], ";"}]], "Input", CellChangeTimes->{{3.421117428242423*^9, 3.421117503524171*^9}, { 3.421117555972892*^9, 3.4211175884424267`*^9}, {3.42111761921064*^9, 3.4211176225460567`*^9}, {3.421117705389296*^9, 3.421117717065837*^9}, { 3.421117759736829*^9, 3.421117773250139*^9}, {3.421117907748217*^9, 3.421117983395575*^9}, {3.4211180295366898`*^9, 3.42111803871218*^9}, { 3.4211180793545513`*^9, 3.421118110690854*^9}, {3.4211181471594*^9, 3.421118148583745*^9}, {3.4211182018121147`*^9, 3.42111821483044*^9}, { 3.4211189562767277`*^9, 3.421118959979432*^9}, {3.421119047497843*^9, 3.421119129123863*^9}}], Cell[TextData[{ "This will show us the regions in the complex plane which converge to each \ of the roots. We need to remove one set of brackets with ", StyleBox["Flatten", FontWeight->"Bold"], " ( or ", StyleBox["[[1]] ", FontWeight->"Bold"], "). If we don't do that" }], "Text", CellChangeTimes->{{3.4211453666527243`*^9, 3.4211453717001534`*^9}, { 3.515351043084795*^9, 3.5153510993802357`*^9}}], Cell[BoxData[ RowBox[{ RowBox[{ RowBox[{"array2", "[", RowBox[{ "n_", ",", " ", "npoint_", ",", " ", "xmin_", ",", "xmax_", ",", " ", "ymin_", ",", " ", "ymax_"}], "]"}], " ", ":=", RowBox[{"Table", "[", RowBox[{ RowBox[{"{", RowBox[{"x", ",", " ", "y", ",", " ", RowBox[{"newtC", "[", RowBox[{"n", ",", " ", RowBox[{"x", " ", "+", " ", RowBox[{"I", " ", "y"}]}]}], "]"}]}], "}"}], ",", " ", RowBox[{"{", RowBox[{"x", ",", " ", "xmin", ",", " ", "xmax", ",", " ", RowBox[{ RowBox[{"(", RowBox[{"xmax", "-", "xmin"}], ")"}], "/", "npoint"}]}], "}"}], ",", " ", RowBox[{"{", RowBox[{"y", ",", " ", "ymin", ",", " ", "ymax", ",", " ", RowBox[{ RowBox[{"(", RowBox[{"ymax", "-", "ymin"}], ")"}], "/", "npoint"}]}], "}"}]}], "]"}]}], ";"}]], "Input", CellChangeTimes->{{3.421117428242423*^9, 3.421117503524171*^9}, { 3.421117555972892*^9, 3.4211175884424267`*^9}, {3.42111761921064*^9, 3.4211176225460567`*^9}, {3.421117705389296*^9, 3.421117717065837*^9}, { 3.421117759736829*^9, 3.421117773250139*^9}, {3.421117907748217*^9, 3.421117983395575*^9}, {3.4211180295366898`*^9, 3.42111803871218*^9}, { 3.4211180793545513`*^9, 3.421118110690854*^9}, {3.4211181471594*^9, 3.421118148583745*^9}, {3.4211182018121147`*^9, 3.42111821483044*^9}, { 3.4211189562767277`*^9, 3.421118959979432*^9}, {3.421119047497843*^9, 3.421119129123863*^9}, {3.5153509375141153`*^9, 3.5153509489449863`*^9}}], Cell[CellGroupData[{ Cell[BoxData[ RowBox[{"array2", "[", RowBox[{ "3", ",", " ", "4", ",", " ", "0.1", ",", " ", "1", ",", " ", "0.1", ",", " ", "1"}], "]"}]], "Input", CellChangeTimes->{{3.5153508061885347`*^9, 3.515350820774502*^9}, { 3.5153508753164053`*^9, 3.515350878152218*^9}, 3.51535097853939*^9}], Cell[BoxData[ RowBox[{"{", RowBox[{ RowBox[{"{", RowBox[{ RowBox[{"{", RowBox[{"0.1`", ",", "0.1`", ",", RowBox[{"-", "0.33333333333333337`"}]}], "}"}], ",", RowBox[{"{", RowBox[{"0.1`", ",", "0.325`", ",", RowBox[{"-", "0.3333333333333333`"}]}], "}"}], ",", RowBox[{"{", RowBox[{"0.1`", ",", "0.55`", ",", "0.`"}], "}"}], ",", RowBox[{"{", RowBox[{"0.1`", ",", "0.775`", ",", "0.3333333333333333`"}], "}"}], ",", RowBox[{"{", RowBox[{"0.1`", ",", "1.`", ",", "0.33333333333333337`"}], "}"}]}], "}"}], ",", RowBox[{"{", RowBox[{ RowBox[{"{", RowBox[{"0.325`", ",", "0.1`", ",", "0.`"}], "}"}], ",", RowBox[{"{", RowBox[{"0.325`", ",", "0.325`", ",", RowBox[{"-", "0.3333333333333333`"}]}], "}"}], ",", RowBox[{"{", RowBox[{"0.325`", ",", "0.55`", ",", RowBox[{"-", "0.33333333333333337`"}]}], "}"}], ",", RowBox[{"{", RowBox[{"0.325`", ",", "0.775`", ",", "0.3333333333333333`"}], "}"}], ",", RowBox[{"{", RowBox[{"0.325`", ",", "1.`", ",", "0.3333333333333333`"}], "}"}]}], "}"}], ",", RowBox[{"{", RowBox[{ RowBox[{"{", RowBox[{"0.55`", ",", "0.1`", ",", "0.`"}], "}"}], ",", RowBox[{"{", RowBox[{"0.55`", ",", "0.325`", ",", "0.`"}], "}"}], ",", RowBox[{"{", RowBox[{"0.55`", ",", "0.55`", ",", "0.`"}], "}"}], ",", RowBox[{"{", RowBox[{"0.55`", ",", "0.775`", ",", RowBox[{"-", "0.3333333333333333`"}]}], "}"}], ",", RowBox[{"{", RowBox[{"0.55`", ",", "1.`", ",", RowBox[{"-", "0.3333333333333333`"}]}], "}"}]}], "}"}], ",", RowBox[{"{", RowBox[{ RowBox[{"{", RowBox[{"0.775`", ",", "0.1`", ",", "0.`"}], "}"}], ",", RowBox[{"{", RowBox[{"0.775`", ",", "0.325`", ",", "0.`"}], "}"}], ",", RowBox[{"{", RowBox[{"0.775`", ",", "0.55`", ",", "0.`"}], "}"}], ",", RowBox[{"{", RowBox[{"0.775`", ",", "0.775`", ",", "0.`"}], "}"}], ",", RowBox[{"{", RowBox[{"0.775`", ",", "1.`", ",", "0.3333333333333333`"}], "}"}]}], "}"}], ",", RowBox[{"{", RowBox[{ RowBox[{"{", RowBox[{"1.`", ",", "0.1`", ",", "0.`"}], "}"}], ",", RowBox[{"{", RowBox[{"1.`", ",", "0.325`", ",", "0.`"}], "}"}], ",", RowBox[{"{", RowBox[{"1.`", ",", "0.55`", ",", "0.`"}], "}"}], ",", RowBox[{"{", RowBox[{"1.`", ",", "0.775`", ",", "0.`"}], "}"}], ",", RowBox[{"{", RowBox[{"1.`", ",", "1.`", ",", "0.`"}], "}"}]}], "}"}]}], "}"}]], "Output", CellChangeTimes->{{3.515350963913466*^9, 3.515350979123971*^9}}] }, Open ]], Cell[TextData[{ "We indeed have an unwanted extra set of brackets. However, using our \ function ", StyleBox["array", FontWeight->"Bold"], " we have a list of elements, each of which is a list of three quantities, \ x, y, and \[Theta] / 2\[Pi], as desired:" }], "Text", CellChangeTimes->{{3.515351109671163*^9, 3.515351240621779*^9}}], Cell[CellGroupData[{ Cell[BoxData[ RowBox[{"array", "[", RowBox[{ "3", ",", " ", "4", ",", " ", "0.1", ",", " ", "1", ",", " ", "0.1", ",", " ", "1"}], "]"}]], "Input", CellChangeTimes->{{3.5153508061885347`*^9, 3.515350820774502*^9}, { 3.5153508753164053`*^9, 3.515350878152218*^9}}], Cell[BoxData[ RowBox[{"{", RowBox[{ RowBox[{"{", RowBox[{"0.1`", ",", "0.1`", ",", RowBox[{"-", "0.33333333333333337`"}]}], "}"}], ",", RowBox[{"{", RowBox[{"0.1`", ",", "0.325`", ",", RowBox[{"-", "0.3333333333333333`"}]}], "}"}], ",", RowBox[{"{", RowBox[{"0.1`", ",", "0.55`", ",", "0.`"}], "}"}], ",", RowBox[{"{", RowBox[{"0.1`", ",", "0.775`", ",", "0.3333333333333333`"}], "}"}], ",", RowBox[{"{", RowBox[{"0.1`", ",", "1.`", ",", 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