19 September 2007

A quick thought on Newton's method

John Armstrong talks about fractals in Newton's method. (If you're not familiar with Newton's method, read that post first; what follows is in the nature of a comment to it.) This is something you wouldn't expect; when you're taught Newton's method the impression is that it'll always find the closest root to your initial estimate, or at least that the region which eventually leads to any given root is "nice-looking". That is very far from being true. Basically, if you're trying to find a zero of f(z) using Newton's method, and f'(z) is small, then you can suddenly find yourself making arbitrarily large steps. The self-similar nature of the picture you get if you color the initial values which lead to one zero red, another zero green, another zero blue, and so on isn't too surprising if you think about it this way; the picture has to be its own image under the transformation which takes c to c-(f'(c)/f(c)), which is analytic when f(c) isn't zero. An example of such a picture accompanies the post.

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