I was invited to give an undergraduate talk at Seattle University later this month about Newton iteration and Smale’s Alpha Theory — a small but important component of my research code, abelfunctions. Here’s a cute little picture I created in preparation for the talk.
The complex function $f(z) = z^9 - 1$ has nine roots, shown as red dots above. The coloring indicates which root Newton iteration will converge to if the initial root guess is given in that color region. For example, any initial guess in a black region will converge to the “black root” on the lower right-hand side of the image.
The point of Smale’s Alpha Theory is to make sure your initial guess converges to the root you wanted.