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A Newton method for best uniform rational approximation

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Abstract

We present a novel algorithm, inspired by the recent BRASIL algorithm, for best uniform rational approximation of real continuous functions on real intervals based on a formulation of the problem as a nonlinear system of equations and barycentric interpolation. We derive a closed form for the Jacobian of the system of equations and formulate a Newton’s method for its solution. The resulting method for best uniform rational approximation can handle singularities and arbitrary degrees for numerator and denominator. We give some numerical experiments which indicate that it typically converges globally and exhibits superlinear convergence in a neighborhood of the solution. A software implementation of the algorithm is provided. Interesting auxiliary results include formulae for the derivatives of barycentric rational interpolants with respect to the interpolation nodes, and for the derivative of the nullspace of a full-rank matrix.

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Algorithm 1
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Data Availability

Data sharing not applicable to this article as no datasets were generated or analyzed during the current study.

Notes

  1. https://github.com/c-f-h/baryrat

  2. https://pypi.org/project/gmpy2/

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Funding

This work was supported by the bilateral project KP-06-Austria/8/2019 (WTZ BG 03/2019), funded by Bulgarian National Science Fund and OeAD (Austria). The second author received support from the Austrian Science Fund (FWF) grant P 33956-NBL.

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Authors and Affiliations

  1. Bulgarian Academy of Sciences, Institute of Mathematics and Informatics, Acad. G. Bonchev, Bl. 8, 1113, Sofia, Bulgaria

    Irina Georgieva

  2. Johann Radon Institute for Computational and Applied Mathematics (RICAM), Altenberger Str. 69, 4040, Linz, Austria

    Clemens Hofreither

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  1. Irina Georgieva
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  2. Clemens Hofreither
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Correspondence to Clemens Hofreither.

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Georgieva, I., Hofreither, C. A Newton method for best uniform rational approximation. Numer Algor 93, 1741–1758 (2023). https://doi.org/10.1007/s11075-022-01487-5

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