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A single point's trajectory under \ the operation f:z\[Rule](1/z-nearest[1/z]) is computed using floating point \ arithmetic, and the computer keeps track of the number of times the \ trajectory lands within each pixel of an N by N grid partitioning the unit \ box. This count is then displayed in a false-color image. \ \>", "Subsubtitle"], Cell["\<\ It should be kept in mind that the accuracy with which our point is recorded \ is utterly inadequate to ensure that the computed trajectory agrees with the \ real trajectory an exact computation would reveal. We are trusting to our \ luck that the rounding errors inherent in floating point arithmetic will \ function as if they were small, independent random perturbations in the \ trajectory of our initially chosen point, and that a randomizing perturbation \ to the trajectory can only assist if not accelerate the convergence of the \ observed trajectory distribution to the invariant density. The density exists, and it is real-analytic in each of the 12 regions into \ which the unit box is dissected by the arcs cutting through the box of the \ circles |z-1|=1, |z+1|=1, |z-\[ImaginaryI]|=1, |z+\[ImaginaryI]|=1, |z-(1+\ \[ImaginaryI])|=1, |z-(1-\[ImaginaryI])|=1, |z+(1+\[ImaginaryI])|=1, and \ |z+(1-\[ImaginaryI])|=1. \ \>", "Text"], Cell[BoxData[ \(reim[z_] := {Re[z], Im[z]}\)], "Input", InitializationCell->True], Cell[BoxData[ \(nearest[z_] := Floor[Re[z] + 1/2] + I\ Floor[Im[z] + 1/2]\)], "Input", InitializationCell->True], Cell[BoxData[ \(Clear[hurwitzcount]; hurwitzcount[u_] := 0\)], "Input", InitializationCell->True], Cell[CellGroupData[{ Cell[BoxData[ \(n = 1000\)], "Input", InitializationCell->True], Cell[BoxData[ \(1000\)], "Output"] }, Open ]], Cell[CellGroupData[{ Cell[BoxData[ \(\(?Compile\)\)], "Input"], Cell[BoxData[ RowBox[{"\<\"Compile[{x1, x2, ... }, expr] creates a compiled function \ which evaluates expr assuming numerical values of the xi. Compile[{{x1, t1}, \ ... }, expr] assumes that xi is of a type which matches ti. Compile[{{x1, t1, \ n1}, ... }, expr] assumes that xi is a rank ni array of objects each of a \ type which matches ti. Compile[vars, expr, {{p1, pt1}, ... }] assumes that \ subexpressions in expr which match pi are of types which match pti.\"\>", " ", ButtonBox[ StyleBox["More\[Ellipsis]", "SR"], ButtonData:>"Compile", Active->True, ButtonStyle->"RefGuideLink"]}]], "Print", CellTags->"Info3311069516-8959663"] }, Open ]], Cell[CellGroupData[{ Cell[BoxData[ \(hurwitzcompy = Compile[{{z, _Real, 1}}, \[IndentingNewLine]Module[{u, v, w, x, y, j, k}, u = z[\([1]\)]; v = z[\([2]\)]; w = u^2 + v^2; x = u/w; y = \(-v\)/w; x -= Floor[x + 1/2]; y -= Floor[y + 1/2]; j = Floor[n\ x + n/2 + 1]; k = Floor[n\ y + n/2 + 1]; \(hurwitzcount[j, k]++\); {x, y}]]\)], "Input", InitializationCell->True], Cell[BoxData[ TagBox[\(CompiledFunction[{z}, Module[{u, v, w, x, y, j, k}, u = z\[LeftDoubleBracket]1\[RightDoubleBracket]; v = z\[LeftDoubleBracket]2\[RightDoubleBracket]; w = u\^2 + v\^2; x = u\/w; y = \(-\(v\/w\)\); x -= Floor[x + 1\/2]; y -= Floor[y + 1\/2]; j = Floor[n\ x + n\/2 + 1]; k = Floor[n\ y + n\/2 + 1]; \(hurwitzcount[j, k]++\); {x, y}], "-CompiledCode-"]\), False, Editable->False]], "Output"] }, Open ]], Cell[BoxData[ \(Do[hurwitzcount[j, k] = 0, {j, 1, n}, {k, 1, n}]\)], "Input", InitializationCell->True], Cell[CellGroupData[{ Cell[BoxData[ \(z0 = N[Log[4] - 1 + I \((Log[7] - 2)\)]\)], "Input", InitializationCell->True], Cell[BoxData[ \(\(\(0.3862943611198906`\)\(\[InvisibleSpace]\)\) - 0.05408985094468677`\ \[ImaginaryI]\)], "Output"] }, Open ]], Cell[CellGroupData[{ Cell[BoxData[ \(z0 = reim[z0]\)], "Input", InitializationCell->True], Cell[BoxData[ \({0.3862943611198906`, \(-0.05408985094468677`\)}\)], "Output"] }, Open ]], Cell[CellGroupData[{ Cell[BoxData[ \(huey = Function[z, RGBColor[Mod[3 z, 1], Mod[4 z, 1], Mod[2 z, 1]]]\)], "Input", InitializationCell->True], Cell[BoxData[ \(Function[z, RGBColor[Mod[3\ z, 1], Mod[4\ z, 1], Mod[2\ z, 1]]]\)], "Output"] }, Open ]], Cell[TextData[{ "The stage is set. Our function hurwitzcompy executes one instance of f, \ marks which pixel the moving point falls in, and updates the tally. Our \ function hurwitzcount keeps that tally. We now need to set a bound on the \ number of steps we'll make, and a name for the tally as we save it from time \ to time. This name, here \"tallyho.m\" for whimsical reasons, will be the \ filename of a data structure saved within ", StyleBox["Mathematica", FontSlant->"Italic"], "'s home folder by the code as it executes. If you have the patience to \ carry the computation to larger values of rbound, first setting n=1000, you \ should see something like what appears in the book. I removed the better part \ of the displays to hold down the size of this file. " }], "Text"], Cell["\<\ Here we use larger pixels. This accelerates convergence at the price of \ reducing resolution. We get our picture about 100 times faster than we would \ for the higher-resolution attempt. \ \>", "Text"], Cell[CellGroupData[{ Cell[BoxData[{ \(n = 100; rbound = 2^27; hurwitzcompy = Compile[{{z, _Real, 1}}, \[IndentingNewLine]Module[{u, v, w, x, y, j, k}, u = z[\([1]\)]; v = z[\([2]\)]; w = u^2 + v^2; x = u/w; y = \(-v\)/w; x -= Floor[x + 1/2]; y -= Floor[y + 1/2]; j = Floor[n\ x + n/2 + 1]; k = Floor[n\ y + n/2 + 1]; \(hurwitzcount[j, k]++\); {x, y}]];\), "\[IndentingNewLine]", \(r = 1; s = 1; z = z0; While[r \[LessEqual] rbound, z = hurwitzcompy[z]; If[r \[Equal] 2^s, \[IndentingNewLine]\(s++\); Print["\", r, "\<,z=\>", z]; \[IndentingNewLine]m = Table[hurwitzcount[a, b], {a, 1, n}, {b, 1, n}]; \[IndentingNewLine]m >> tallyho2.m; mm = m + Reverse[m]; mm = mm + Map[Reverse, mm]; \[IndentingNewLine]mm = mm + Transpose[mm]; If[r \[GreaterEqual] 2^22, \[IndentingNewLine]ListDensityPlot[mm, Mesh \[Rule] False, ColorFunction \[Rule] huey]]]; \[IndentingNewLine]\(r++\)]\)}], "Input"], 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