From: rusin@washington.math.niu.edu (Dave Rusin)
Newsgroups: sci.math
Subject: Re: Fracltes and Newtons Method
Date: 7 Dec 1994 22:09:32 GMT

In article <3c50gb$urv@library.erc.clarkson.edu>,
James E. Kogler <koglerje@craft.camp.clarkson.edu> wrote:
> I have recently been told that fracltes are very closely related to
>finding roots using newtons method. How is this so? could someone shed 
>some light on the subject for me? 

Example: Use Newton's method to find the roots of  z^3-1=0  in the
complex plane. Of course, you know the three roots are  1, w, and w^2
where  w=(-1/2)+i sqrt(3)/2. But the question is, which starting places
for Newton's method will lead to which root? It is true that the
set of places which converges to  1,  say, is an open set, and includes
1, and is disjoint from the other two open sets, but it's not what
you might think -- it's not just the points closer to  1  than to 
either of the other two roots. Moreover, elementary topology then
shows that there is a closed, non-empty set of points which when used
as the starting point for Newton's method will not converge to any
of the three roots. That set is a fractal.

dave