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Non-Linear Rational Riemann-Hilbert-Problems with Circular Target Curves

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Abstract

This article may be considered as a continuation of [3], where we studied non-linear Riemann-Hilbert problems with circular target curves ¦w − c¦ = r and Hölder continuous coefficients c and r. Here we assume that c and r 2 are rational functions and emphasize algorithmic and numerical aspects. It is shown that all solutions of the problem are rational and can be obtained by solving an interpolation problem of (generalized) Nevanlinna-Pick type. This problem is in turn reduced to a linear system, which leads to efficient (and stable) numerical methods. Special emphasis is on the Laurent case, which is of importance in applications. We propose an a- posteriori estimate which allows one to verify the accuracy of the approximate solution and report on some test calculations.

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Authors and Affiliations

  1. Department of Mathematics, Åbo Akademi University, FIN-20500, Åbo, Finland

    Christer Glader

  2. Institute of Applied Analysis, Tech Univ Bergakademie Freiberg, D-09596, Freiberg, Germany

    Elias Wegert

Authors
  1. Christer Glader
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  2. Elias Wegert
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Correspondence to Christer Glader.

Additional information

The first author was supported by the Magnus Ehrnrooth Foundation. The second author was supported by grant We 1704-8 of the Deutsche Forschungsgemeinschaft.

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Glader, C., Wegert, E. Non-Linear Rational Riemann-Hilbert-Problems with Circular Target Curves. Comput. Methods Funct. Theory 9, 653–678 (2009). https://doi.org/10.1007/BF03321750

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