The following figure shows the basins of attraction for *z* → *z*^{5}−1
using Newton’s method in the complex plane:

Here’s the code (we find the fixed point of

which should
land on a solution to *z*^{5} = 1, for a 125-by-125 array of complex
starting values between −1.5 − 1.5 *i* and
1.5 + 1.5 *i*;
we color each point on the array according to the solution that Newton’s
method ends with, or rather the phase of the
ending point, which amounts to the same thing):

theFunction[z_] := z^5; iterateThis[z_] = z - (theFunction[z] - 1)/theFunction'[z]; (* at the moment, the graph is backwards, but I can live with that *) ContourPlot[ Arg[ FixedPoint[ iterateThis, x + I y, 30 ]], {x, -1.5, 1.5}, {y, -1.5, 1.5}, ContourLines -> False, PlotPoints -> 125, ColorFunction -> Hue, Axes -> False, Frame -> False ];

Using the enhanced `Compile`

function in *Mathematica* version 3.0,
we can make this roughly a zillion times faster:

(* works just for z^n = 1, integer n > 2; should be a snap to modify it to work with any poly. function, but I have to get back to work now *) newt = Compile[{{n, _Integer}, {z, _Complex}}, Arg[ FixedPoint[# - (#^n - 1)/(n #^(n - 1))&, z, 35] ]]; Show[ GraphicsArray[ Partition[ Table[ ListDensityPlot[Table[newt[n, b + a I], {a, -1.1, 1.1, 0.03}, {b, 1.1, -1.1, -0.03}], Mesh -> False, ColorFunction -> Hue, Frame -> False, DisplayFunction -> Identity], {n, 4, 12}], 3], GraphicsSpacing -> 0.05]]

More of the same:

Designed and rendered too long ago using *Mathematica* 3.0 for the Apple Macintosh.

Fame, esoteric and imagined: The previous image was used on the cover of The Mathematica Journal, issue 7, volume 1.

© 1996–2021 Robert Dickau.

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