View in EDS

FAST AND STABLE RATIONAL INTERPOLATION IN ROOTS OF UNITY AND CHEBYSHEV POINTS.

Saved in:
Bibliographic Details
Title: FAST AND STABLE RATIONAL INTERPOLATION IN ROOTS OF UNITY AND CHEBYSHEV POINTS.
Authors: Pachón, Ricardo1 pachon@maths.ox.ac.uk, Gonnet, Pedro1 gonnet@maths.ox.ac.uk, Van Deun, Joris2 joris.vandeun@ua.ac.be
Source: SIAM Journal on Numerical Analysis. 2012, Vol. 50 Issue 3, p1713-1734. 22p.
Subjects: Interpolation, Numerical analysis, Chebyshev approximation, Fourier transforms, Linear differential equations
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 Eğgecioğlu and Koç. Short MATLAB codes and numerical experiments are included. [ABSTRACT FROM AUTHOR]
Copyright of SIAM Journal on Numerical Analysis is the property of Society for Industrial & Applied Mathematics and its content may not be copied or emailed to multiple sites without the copyright holder's express written permission. Additionally, content may not be used with any artificial intelligence tools or machine learning technologies. However, users may print, download, or email articles for individual use. This abstract may be abridged. No warranty is given about the accuracy of the copy. Users should refer to the original published version of the material for the full abstract. (Copyright applies to all Abstracts.)
Database: Engineering Source
Description
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 Eğgecioğlu and Koç. Short MATLAB codes and numerical experiments are included. [ABSTRACT FROM AUTHOR]
ISSN:00361429
DOI:10.1137/100797291