Project Extra Credit

Basins of Attraction

This project is for extra credit. It can be done at any time to replace an assignment that you did not do so well on. It is due the last day of class at the latest.

For this project you will be computing the basins of attraction for the cube root of -1. We will be iterating over the points of the complex plane from (-1, -1) to (1,1). For each point we will be iterating using Newton’s method to find which of the three roots of the cube root of -1 Newton’s method converges to after 1000 iterations. You might think that the point will converge to the closest root, but this is not always the case! To see this we will color each point with a separate color depending on the root it converges to. As you can see from the graph, most points do converge to the closest attractor. But on the boundaries between the regions (the roots are z = (-1, 0), z = (.5, sin /3), z = (.5, -sin) points do NOT converge to the closest root. Your program should illustrate it.

Newton’s method for the cube roots of -1 says:

Iterate this 1000 times for each point in the plane and color based on the root it is converging to and you should get something like:

or this:

Note: This time use the built in complex class.