**Category: Attractor Fractal**

This is a visualization of Newton's method to calculate the complex solution of
z^{n}-1 = 0 by using the iteration formula

with fn(z) := z^{n} - 1, and n > 2 the polynomial degree of fn (also real values for n are allowed).

> solve(z^3-1=0, z);

The graph shows the starting points from which the method does or does not converge to one of the above roots. The documentation of FRACTINT explains the result: "The coloring of the plot shows the "basins of attraction" for each root of the polynomial -- i.e., an initial guess within any area of a given color would lead you to one of the roots. As you can see, things get [fractal] along certain radial lines or "spokes," those being the lines between actual roots." The fractal properties of Netwon's method to calculate complex roots were first discovered by John Hubbard of Orsay, France.

> restart: > newton := proc(x, y) > local z, m; > z := evalf(x+y*I); > for m from 0 to 50 while abs(z^3-1) >= 0.001 do > z := z - (z^3-1)/(3*z^2) > od; > m > end: > plot3d(0, -2 .. 2, -1.5 .. 1.5, orientation=[-90,0], > grid=[250, 250], style=patchnogrid, scaling=constrained, > color=newton);

The following statements have been taken from the book *Maple V Release 4* written by Michael
Kofler and published by Addison-Wesley. It uses the **complexplot3d** function which is
new with Release 4. However, with large values for 'grid', this command uses a very large
amount of memory so Windows systems missing the proper resources are likely to crash with
**complexplot3d**. The following images are the views of the Maple plot from different
orientations. The computation consumed around 11 Mbytes of memory.

> p := (z-1)*(z^2+z+5/4);

> f := unapply(z-p/(diff(p,z)-0.5*I), z);

> complexplot3d(f@@4, -4-4*I .. 4+4*I, view=-1 .. 2, > style=patchnogrid, orientation=[-143, 65], > grid=[150, 150]);

*MAPLE V FRACTALS NEWTON #1.01 current as of May 23, 1999Author: Alexander F. Walz, alexander.f.walz@t-online.de
Original file location: http://www.math.utsa.edu/mirrors/maple/mfrnewt.htm*