Newton Fractals

The coolest problem in numerical analysis, ever.

Newton's Method is a well known procedure for finding roots of functions.  Specifically, we start with a function f(x) and an initial value of x0.  We then define values xn by the equation

xn+1 = xn - f(xn)/f'(xn)

In most cases, xn will converge to a root of f(x) rapidly.

While most people learn this using real functions, there is no reason why Newton's Method cannot be used with complex functions.  So, given a function f(z) with multiple roots, we can color each point z in the complex plane by which root Newton's Method will converge to if z0 = z.  We can also alter the color depending on how quickly we converge to the root, and we will color the point black if we don't converge to anything in a predetermined number of steps.

Since this is an iterative complex system, it is not too surprising that we end up with fractals.  What is wonderful about this scheme is the great variety of images one can make with this simple concept.  The students in my numerical analysis class were given a program that would create these fractals.  They picked the function, the colors, and the error tolerance.  Some of the results are listed below.

If you would like to try the program: newton2.nb.  You need Mathematica to run it.  If you create a picture that you are particularly proud of, send me the code and I'll include it here.

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