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Rational Interpolation through the Optimal Attachment of Poles to the Interpolating Polynomial

Jean-Paul Berrut
Département de Mathématiques
Université de Fribourg
CH-1700 Fribourg/Pérolles, Switzerland
Hans D. Mittelmann
Department of Mathematics
Arizona State University
Tempe, Arizona 85287-1804, USA

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. Keywords
Interpolation, rational interpolation, optimal interpolation

Classification: Primary 65D05, 41A05; Secondary 41A20


Hans Mittelmann