The transport of images method: computing all zeros of harmonic mappings by continuation

Abstract: We present a continuation method to compute all zeros of a harmonic mapping $f$ in the complex plane. Our method works without any prior knowledge of the number of zeros or their approximate location. We start by computing all solution of $f(z) = \eta$ with $|\eta|$ sufficiently large and then track all solutions as $\eta$ tends to $0$ to finally obtain all zeros of $f$. Using theoretical results on harmonic mappings we analyze where and how the number of solutions of $f(z) = \eta$ changes and incorporate this into the method. We prove that our method is guaranteed to compute all zeros, as long as none of them is singular. In our numerical example the method always terminates with the correct number of zeros, is very fast compared to general purpose root finders and is highly accurate in terms of the residual. An easy-to-use MATLAB implementation is freely available online.