(* 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[ 102187354, 2505671] NotebookOptionsPosition[ 102183315, 2505555] NotebookOutlinePosition[ 102184011, 2505580] CellTagsIndexPosition[ 102183968, 2505577] WindowFrame->Normal ContainsDynamic->False*) (* Beginning of Notebook Content *) Notebook[{ Cell[CellGroupData[{ Cell["The Mandelbrot Set", "Title"], Cell[TextData[{ "The Mandelbrot set, the topic of this notebook, became famous as a simple \ model which produces extraordinarily complicated (and \"beautiful\") fractal \ structures. It is defined as the set of all points in the complex plane, (", Cell[BoxData[ FormBox[ SubscriptBox["c", "x"], TraditionalForm]]], ", ", Cell[BoxData[ FormBox[ SubscriptBox["c", "y"], TraditionalForm]]], ") such that the complex map " }], "Text"], Cell[BoxData[ RowBox[{"z", "\[Rule]", FormBox[ RowBox[{ SuperscriptBox["z", "2"], " ", "+", " ", "c"}], TraditionalForm]}]], "DisplayFormula"], Cell[TextData[{ "i.e. ", Cell[BoxData[ FormBox[ SubscriptBox["z", RowBox[{"n", "+", "1"}]], TraditionalForm]]], "= ", Cell[BoxData[ FormBox[ SuperscriptBox[ SubscriptBox["z", "n"], "2"], TraditionalForm]]], "+ c, does ", StyleBox["not", FontSlant->"Italic"], " escape to infinity starting from z = 0. As we shall see it is related to \ the logistic map, and can be thought of as a generalization of the logistic \ map to the complex plane.\n\nThe code below is used to generate a Mandelbrot \ set. It takes as arguments the real and imaginary parts (", StyleBox["cx", FontWeight->"Bold"], ", ", StyleBox["cy", FontWeight->"Bold"], ") of the complex number ", StyleBox["c", FontWeight->"Bold"], ",", " and returns (minus) the number of iterations that the map takes to escape \ (in practice to get to |z| > 2, because one can show that a point will escape \ to infinity if it once gets to a value where |z| > 2) up to a maximum of ", StyleBox["lim", FontWeight->"Bold"], ". Note the use of the W", StyleBox["hile", FontWeight->"Bold"], " command (a similar programming style to that used in C)." }], "Text"], Cell[BoxData[ RowBox[{"Clear", "[", "\"\\"", "]"}]], "Input", CellChangeTimes->{{3.421114772110346*^9, 3.421114772619689*^9}}], Cell[BoxData[ RowBox[{ RowBox[{ RowBox[{"mandelbrot", "[", RowBox[{"cx_", ",", " ", "cy_", ",", " ", "lim_"}], "]"}], " ", ":=", RowBox[{"(", "\n", "\t\t ", RowBox[{ RowBox[{"z", " ", "=", " ", "0"}], ";", " ", RowBox[{"ct", "=", "0"}], ";", "\n", "\t\t\t ", RowBox[{"While", " ", "[", " ", RowBox[{ RowBox[{ RowBox[{ RowBox[{"Abs", "[", "z", "]"}], " ", "<", " ", "2.0"}], " ", "&&", " ", RowBox[{"ct", " ", "<=", " ", "lim"}]}], ",", "\n", "\t\t\t\t ", RowBox[{ RowBox[{"z", " ", "=", " ", RowBox[{ RowBox[{"z", "^", "2"}], " ", "+", " ", "cx", " ", "+", " ", RowBox[{"I", " ", "cy"}]}]}], ";", " ", "\n", "\t\t\t\t ", RowBox[{"++", "ct"}]}]}], " ", "\n", "\t\t\t ", "]"}], " ", ";", "\n", "\t\t\t ", RowBox[{"-", "ct"}]}], "\n", "\t\t ", ")"}]}], ";"}]], "Input"], Cell["For example,", "Text"], Cell[CellGroupData[{ Cell[BoxData[ RowBox[{"mandelbrot", "[", RowBox[{"0.5", ",", " ", "0.5", ",", " ", "50"}], "]"}]], "Input"], Cell[BoxData[ RowBox[{"-", "5"}]], "Output", CellChangeTimes->{3.4211025703409233`*^9, 3.421106357068555*^9, 3.421114780645597*^9, 3.4831923468449306`*^9}] }, Open ]], Cell[TextData[{ "so starting from c = 0.5 + 0.5 I, it takes 5 iterations for the point z = 0 \ to escape to |z| > 2.\n\nNext we write the function as a module, rather than \ just a function, so that local variables, ct, and z, can be defined in the \ routine independentally of whether any variables of the same name exist in \ the rest of the ", StyleBox["Mathematica", FontSlant->"Italic"], " session. Note that the local variables can be initialized at the same \ time as they are declared, and we do this here:" }], "Text"], Cell[BoxData[{ RowBox[{ RowBox[{"Clear", "[", "mandelbrot", "]"}], ";"}], "\[IndentingNewLine]", RowBox[{" ", RowBox[{ RowBox[{ RowBox[{"mandelbrot", "[", RowBox[{"cx_", ",", " ", "cy_", ",", " ", "lim_"}], "]"}], " ", ":=", RowBox[{"Module", " ", "[", " ", RowBox[{ RowBox[{"{", " ", RowBox[{ RowBox[{"z", " ", "=", " ", "0"}], ",", " ", RowBox[{"ct", "=", "0"}]}], "}"}], ",", "\n", "\t\t\t ", RowBox[{ RowBox[{"While", " ", "[", " ", RowBox[{ RowBox[{ RowBox[{ RowBox[{"Abs", "[", "z", "]"}], " ", "<", " ", "2.0"}], " ", "&&", " ", RowBox[{"ct", " ", "<=", " ", "lim"}]}], ",", "\n", "\t\t\t\t ", RowBox[{ RowBox[{"z", " ", "=", " ", RowBox[{ RowBox[{"z", "^", "2"}], " ", "+", " ", "cx", " ", "+", " ", RowBox[{"I", " ", "cy"}]}]}], ";", " ", "\n", "\t\t\t\t ", RowBox[{"++", "ct"}]}]}], " ", "\n", "\t\t\t ", "]"}], " ", ";", "\n", "\t\t\t ", RowBox[{"-", "ct"}]}]}], "\n", "\t\t ", "]"}]}], ";"}]}]}], "Input"], Cell[TextData[{ "This routine is quite slow. Next we compile it so that it runs faster. To \ compile a function ", StyleBox["f[x_] = x Log[x] ", "Input"], "and call the compiled version ", StyleBox["fc", "Input"], ", we write ", StyleBox["fc = Compile[{x}, x Log[x] ]", "Input"], ", where the quantity(ies) in curly brackets is (are) the argument(s). (The \ notation is similar to that for a pure function, for which we could write ", StyleBox["f = Function[{x}, x log[x] ]", "Input"], StyleBox[")", "Input", FontWeight->"Plain"], ". The following function is the compiled version of the above Module:" }], "Text"], Cell[BoxData[ RowBox[{" ", RowBox[{ RowBox[{"mandelbrotC", " ", "=", " ", RowBox[{"Compile", " ", "[", " ", RowBox[{ RowBox[{"{", " ", RowBox[{"cx", ",", " ", "cy", ",", " ", "lim"}], " ", "}"}], ",", "\n", "\t\t ", RowBox[{"Module", " ", "\n", "\t\t ", "[", " ", RowBox[{ RowBox[{"{", " ", RowBox[{ RowBox[{"z", " ", "=", " ", RowBox[{"0.", " ", "+", " ", RowBox[{"I", " ", "0."}]}]}], ",", " ", RowBox[{"ct", "=", "0"}]}], "}"}], ",", "\n", "\t\t\t ", RowBox[{ RowBox[{"While", " ", "[", " ", RowBox[{ RowBox[{ RowBox[{ RowBox[{"Abs", "[", "z", "]"}], " ", "<", " ", "2.0"}], " ", "&&", " ", RowBox[{"ct", " ", "<=", " ", "lim"}]}], ",", "\n", "\t\t\t\t ", RowBox[{ RowBox[{"z", " ", "=", " ", RowBox[{ RowBox[{"z", "^", "2"}], " ", "+", " ", "cx", " ", "+", " ", RowBox[{"I", " ", "cy"}]}]}], ";", " ", "\n", "\t\t\t\t ", RowBox[{"++", "ct"}]}]}], " ", "\n", "\t\t\t ", "]"}], " ", ";", "\n", "\t\t\t ", RowBox[{"-", "ct"}]}]}], "\n", "\t\t ", "]"}]}], "\n", "]"}]}], ";"}]}]], "Input"], Cell[TextData[{ "Note that the arguments appear on the RHS of the function definition, \ rather than on the LHS as in the uncompiled version.\n\nBelow we use show the \ whole of the Mandelbrot set with a coarse resolution and compare the timing \ of the compiled and uncompiled version. The value of the function being \ plotted is characterized by the degree of blackness in a ", StyleBox["DensityPlot", FontWeight->"Bold"], ". The smaller the number the blacker it is. Since we compute the ", StyleBox["negative", FontSlant->"Italic"], " of the number of iterations, the blackest region is where the number of \ iterations is greatest. 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