2.5 Newton domains of attraction
(type=newtbasin)
The Newton formula is an algorithm used to find the roots of polynomial
equations by successive "guesses" that converge on the correct value as
you feed the results of each approximation back into the formula. It
works very well -- unless you are unlucky enough to pick a value that is
on a line BETWEEN two actual roots. In that case, the sequence explodes
into chaos, with results that diverge more and more wildly as you
continue the iteration.
This fractal type shows the results for the polynomial Z^n - 1, which
has n roots in the complex plane. Use the <T>ype command and enter
"newtbasin" in response to the prompt. You will be asked for a
parameter, the "order" of the equation (an integer from 3 through 10 --
3 for x^3-1, 7 for x^7-1, etc.). A second parameter is a flag to turn on
alternating shades showing changes in the number of iterations needed to
attract an orbit. Some people like stripes and some don't, as always,
Fractint gives you a choice!
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 a
bit weird along certain radial lines or "spokes," those being the lines
between actual roots. By "weird," we mean infinitely complex in the good
old fractal sense. Zoom in and see for yourself.
This fractal type is symmetric about the origin, with the number of
"spokes" depending on the order you select. It uses floating-point math
if you have an FPU, or a somewhat slower integer algorithm if you don't
have one.