4

I'm quite a beginner using Mathematica. I'd like to plot the fractals of higher degree polynomials.

I have an example for z^3 - 1 which looks like this:

F[z_] := z^3 - 1;
NewtonRaphsonStep[z_] := z - F[z]/F'[z]
NoOfIterations = 6; 
GridSize = .01
MatrixPlot[
  Table[
    θ = Arg[Nest[NewtonRaphsonStep, i + j I, NoOfIterations] /. 
      {Indeterminate -> 10}]; 
    If[θ > 1.8, 1, If[θ < -1.8, 2, 0]], 
    {j, -2., 2., GridSize}, {i, -2., 2., GridSize}], 
  Mesh -> None, Mesh -> False, ImageSize -> {500, 500}, 
  ColorRules -> {2 -> Red, 0 -> Orange, 1 -> Pink}]

And this works just fine. But for z^4 - 1 and higher degrees, I don't know how to plot the fractal, I'm not sure how to handle more roots.

Could you help me please?

m_goldberg
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Beginneruser
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1 Answers1

6

Here is a simpler ListDensityPlot version using FixedPoint and avoiding indeterminate values by choosing a step size which skips 0 + 0 I. It is also much faster by mapping over precomputed coordinates.

ListDensityPlot[Arg @ FixedPoint[NewtonRaphsonStep, #] & /@ 
   Table[i + j I, {j, -2., 2., 0.011}, {i, -2., 2., 0.011}], 
     ColorFunction -> "Rainbow"]

enter image description here

eldo
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