A Newton method for harmonic mappings in the plane

Abstract

We present an iterative root finding method for harmonic mappings in the complex plane, which is a generalization of Newton's method for analytic functions. The complex formulation of the method allows an analysis in a complex variables spirit. For zeros close to poles of $f=h+\bar{g}$ we construct initial points for which the harmonic Newton iteration is guaranteed to converge. Moreover, we study the number of solutions of $f(z)=\eta$ close to the critical set of $f$ for certain $\eta \in \mathbb{C}$. We provide a Matlab implementation of the method, and illustrate our results with several examples and numerical experiments, including phase plots and plots of the basins of attraction.

arXiv comment26 pages, 10 figures. Improved visualization of the dynamics of the harmonic Newton map. Some minor further improvements

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Cited by in corpus (2)

• On the zeros of polyanalytic polynomials
• The transport of images method: computing all zeros of harmonic mappings by continuation