On the Faber Transform and Efficient Numerical Rational Approximation

Algorithms are presented for numerical type (m, n) rational approximation (m ≧ n - 1) which can be effectively applied either in general simply connected regions of the complex plane or, as a special case, on intervals of the real line. In the former case particularly, the methods are much more effi...

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Bibliographic Details
Published in:SIAM journal on numerical analysis 1983-10, Vol.20 (5), p.989-1000
Main Author: Ellacott, S. W.
Format: Article
Language:English
Subjects:
Algorithms
Approximation
Chebyshev approximation
Coefficients
Fourier transforms
Geometric planes
Mathematical functions
Mathematical theorems
Polynomials
Rational functions
Real lines
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Summary:Algorithms are presented for numerical type (m, n) rational approximation (m ≧ n - 1) which can be effectively applied either in general simply connected regions of the complex plane or, as a special case, on intervals of the real line. In the former case particularly, the methods are much more efficient than those previously proposed; but they are also competitive even for intervals. The approximations are based on the observation that the Faber transform of a rational function is itself rational, and are of two types, a class of Padé-like approximants and approximations of rational Carathéodory-Fejér type based on the singular value decomposition of a Hankel matrix of Faber coefficients which generalise those recently introduced for the unit disc by Trefethen (Numer. Math., 37 (1981), pp. 297-320). These latter, particularly, give rise to approximations which are sufficiently near to best for practical purposes.
ISSN:0036-1429
1095-7170
DOI:10.1137/0720069