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Rational interpolation through the optimal attachment of poles to the interpolating polynomial

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Abstract

After recalling some pitfalls of polynomial interpolation (in particular, slopes limited by Markov's inequality) and rational interpolation (e.g., unattainable points, poles in the interpolation interval, erratic behavior of the error for small numbers of nodes), we suggest an alternative for the case when the function to be interpolated is known everywhere, not just at the nodes. The method consists in replacing the interpolating polynomial with a rational interpolant whose poles are all prescribed, written in its barycentric form as in [4], and optimizing the placement of the poles in such a way as to minimize a chosen norm of the error.

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Authors and Affiliations

  1. Département de Mathématiques, Université de Fribourg, CH-1700, Fribourg/Pérolles, Switzerland

    Jean-Paul Berrut

  2. Department of Mathematics, Arizona State University, Tempe, AZ, 85287-1804, USA

    Hans D. Mittelmann

Authors
  1. Jean-Paul Berrut
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  2. Hans D. Mittelmann
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Berrut, JP., Mittelmann, H.D. Rational interpolation through the optimal attachment of poles to the interpolating polynomial. Numerical Algorithms 23, 315–328 (2000). https://doi.org/10.1023/A:1019168504808

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