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.
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.
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, 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.
. 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.
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example the method always
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Matlab implementation of our method
MATLAB implementation