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

by

Jean-Paul Berrut

Département de Mathématiques

Université de Fribourg

CH-1700 Fribourg/Pérolles, Switzerland

and

Hans D. Mittelmann

Department of Mathematics

Arizona State University

Tempe,
Arizona 85287-1804, USA

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.
Keywords

Interpolation, rational interpolation, optimal interpolation

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

- Introduction
- Attaching poles to the interpolating polynomial
- The optimization problem
- Numerical experiments
- Conclusion
- Bibliography
- About this document ...

2000-05-30