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