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Fast Computation of the Circular Map

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Abstract

This paper presents a new numerical implementation of Koebe’s iterative method for computing the circular map of bounded and unbounded multiply connected regions of connectivity \(m\). The computational cost of the presented method is \(O(m^2n+mn\log n)\) where \(n\) is the number of nodes in the discretization of each boundary component. The accuracy and efficiency of the method presented are demonstrated by several numerical examples. These examples include regions with high connectivity, a region with close-to-touching boundaries, and a region with piecewise smooth boundaries.

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Acknowledgments

The author would like to thank the editor-in-charge and the anonymous referees for their valuable comments and suggestions which improved the presentation of this paper. Further, the author thanks Prof. Leslie Greengard and Dr. Zydrunas Gimbutas for making the MATLAB toolbox FMMLIB2D [15] publicly available.

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  1. Department of Mathematics, Faculty of Science, King Khalid University, P. O. Box 9004, Abha, 61413, Saudi Arabia

    Mohamed M. S. Nasser

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  1. Mohamed M. S. Nasser
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Correspondence to Mohamed M. S. Nasser.

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Communicated by Darren Crowdy.

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Nasser, M.M.S. Fast Computation of the Circular Map. Comput. Methods Funct. Theory 15, 187–223 (2015). https://doi.org/10.1007/s40315-014-0098-3

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