The transport of images method: computing all zeros of harmonic mappings by continuation
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We present a novel continuation method to compute all zeros of a harmonic mapping f in the complex plane. Our method works without any prior knowledge about the number of zeros, or their approximate location, as long as the number of zeros is finite. We give a complete convergence analysis, which relies on results on the caustics of f, and on convergence results for Newton's method. In our numerical examples, the method 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 of our method is freely available online.
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