Title
Fast and stable rational interpolation in roots of unity and chebyshev points Fast and stable rational interpolation in roots of unity and chebyshev points
Author
Faculty/Department
Faculty of Sciences. Mathematics and Computer Science
Publication type
article
Publication
Philadelphia, Pa ,
Subject
Mathematics
Source (journal)
SIAM journal on numerical analysis. - Philadelphia, Pa
Volume/pages
50(2012) :3 , p. 1713-1734
ISSN
0036-1429
ISI
000310210700032
Carrier
E
Target language
English (eng)
Full text (Publishers DOI)
Affiliation
University of Antwerp
Abstract
A new method for interpolation by rational functions of prescribed numerator and denominator degrees is presented. When the interpolation nodes are roots of unity or Chebyshev points, the algorithm is particularly simple and relies on discrete Fourier transform matrices, which results in a fast implementation using the fast Fourier transform. The method is generalized for arbitrary grids, which requires the construction of polynomials orthogonal on the set of interpolation nodes. The appearance of common factors in the numerator and denominator due to finite-precision arithmetic is explained by the behavior of the singular values of the linear system associated with the rational interpolation problem. The new algorithm has connections with other methods, particularly the work of Jacobi and Kronecker, Berrut and Mittelmann, and Egecioglu and Koc. Short MATLAB codes and numerical experiments are included.
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