(Basin of Attraction) Consider the complex polynomial z3 − 1, whose zeros are the three cube roots of unity. Generate a picture showing three basins of attraction in the complex plane in the square region defined by −1?Real(z)?1 and −1?Imaginary(z)?1. To do this, use a mesh of 1000 × 1000 pixels inside the square. The center point of each pixel is used to start the iteration of Newton’s method. Assign a particular basin color to each pixel if convergence to a root is obtained with nmax = 10 iterations. The large number of iterations suggested can be avoided by doing some analysis with the aid of Theorem 1, since the iterates get within a certain neighborhood of the root and the iteration can be stopped. The criterion for convergence is to check both |zn+1 − zn| < ε and |z3 n+1 − 1| < ε with a small value such as ε = 10−4 as well as a maximum number of iterations.
Hint: It is best to debug your program and get a crude picture with only a small number of pixels such as 10 × 10.
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