Newton domains of attraction

Nice Image


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.

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Noel Giffin,