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Electronic Transactions on Numerical Analysis.

Volume 38, pp. 146-167, 2011.

Copyright2011, Kent State University.

ISSN 1068-9613.

ETNA

Kent State University http://etna.math.kent.edu

ROBUST RATIONAL INTERPOLATION AND LEAST-SQUARES

PEDRO GONNETy, RICARDO PACH ´ONy,ANDLLOYD N. TREFETHENy

Abstract. An efficient and robust algorithm and a Matlab coderatdiskare presented for rational interpolation or linearized least-squares approximation of a function based on its values at points equally spaced on a circle. The use of the singular value decomposition enables the detection and elimination of spurious poles or Froissart doublets that commonly complicate such fits without contributing to the quality of the approximation. As an application, the algorithm leads to a method for the stable computation of certain radial basis function interpolants in the difficult case of smoothness parameter"close to zero.

Key words. Rational interpolation, spurious poles, Froissart doublets, Pad´e approximation, radial basis func- tions, ratdisk, singular value decomposition

AMS subject classifications. 41A20, 41A21, 65D05

Received February 10, 2011. Accepted for publication February 28, 2011. Published online May 18, 2011.

Recommended by L. Reichel. P. G. was supported by Swiss National Science Foundation Individual Support Fel- lowships Nr. PBEZP2-127959 and Nr. PA00P2-134146.

yOxford University Mathematical Institute, 25-29 St Giles, Oxford OX1 3LB, UK (Pedro.Gonnet,[email protected]).

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