Posts tagged ‘polynomial’

Newton fractals

I the moment I’m taking a course in Numerical Analysis, and here we were of course introduced to the Newton-Raphson method for finding roots. That is the method where, geometrical speaking, we take at tangent line to a function and use that to approximate the root, and keep on doing that until we reach satisfying precision.

Depending on where you start the iteration you get a a different root, and requiring a different amount of iterations. That’s self-evident. The “funny” part is that when searching for complex roots there are areas in the complex plane, where a very small difference in starting point gives different roots, and with remarkable difference in the number of iterations.

Newton for x^4+2

This is a plot where each root in the polynomial x^4+2 have been assigned a different color. And each point is colored according to which root a iteration starting here finds.

Now thats the mathematical part of it. Now to the beautiful part or at least more fascinating part. The pattern reapeats it self, as a fractal, when you zoom in.

So i modified my code to zoom in, and i used a diffenrent coloring scheme where i only colored according to the number of iterations taken. This made it possible to generate a video zooming in og showing the rapeating patterns in “nice” colors.

Here a some videos :

4 roots (14.6 mb)
7 roots (50 mb)

Yes I know that its very large files, and maybe i will upload them to google video at a later time for easy viewing and less quality.