| Publications of Adhemar Bultheel | |
A generalized eigenvalue problem for quasi-orthogonal rational functions
Karl Deckers Adhemar Bultheel Joris Van Deun
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Rational Chebyshev example; $n \gt 1$ odd. weights: $w_1 = (1 - x^2)^{-1/2}$, $w_2 = (1 - x)^{1/2}(1 + x)^{-1/2}$, $w_3 = (1 - x^2)^{1/2}$ Parameter $\beta$ (qORF) relates to parameter $\tau$ (pORF). RGQ nodes real and simple if $\tau\in \mathbb{T}\setminus\{-1\}$. For $\beta\in [-1,1]$, the qORF nodes are in $[-1,1]$. $\rho^{[i\pm]}$ refers to RQF with pORF node in $\pm1$ for weight $w_i$. $\beta^{[i\pm]}$ refers to RQF with qORF node in $\pm1$ for weight $w_i$. |
Abstract:
In general, the zeros of an orthogonal rational function (ORF) on a subset of the real line, with poles among $\{\alpha_1\ldots,\alpha_n\}\subset(\mathbb{C}_0\cup\{\infty\}$), are not all real (unless $\alpha_n$ is real), and hence, they are not suitable to construct a rational Gaussian quadrature rule (RGQ). For this reason, the zeros of a so-called quasi-ORF (qORF) or a so-called para-ORF (pORF) are used instead. These zeros depend on one single parameter $\tau\in(\mathbb{C}\cup\{\infty\})$, which can always be chosen in such a way that the zeros are all real and simple. In this paper we provide a generalized eigenvalue problem to compute the zeros of a quasi-ORF and the corresponding weights in the RGQ. First, we study the connection between qORFs, pORFs and ORFs. Next, a condition is given for the parameter $\tau$ so that the zeros are all real and simple. Finally, some illustrative and numerical examples are given.
Status:
Accepted
BiBTeX entry:
@article{ArtDBVD10,
author = "K. Deckers and A. Bultheel and Van Deun, J.",
title = "A generalized eigenvalue problem for quasi-orthogonal rational functions",
year = "2010",
volume = "117",
number = "3",
pages = "463-506",
journal = "Numerische Mathematik",
url = "http://nalag.cs.kuleuven.be/papers/ade/GEP/index.html",
DOI = "10.1007/s00211-010-0356-x",
ZBL = "1213.65044",
MR = "2772416",
LIMO = "606941",
}
File(s):
preprint.pdf (352K)
See also Report TW 562