A generalized eigenvalue problem for quasi-orthogonal rational functions

Abstract

In general, the zeros of an orthogonal rational function (ORF) on a subset of the real line, with poles among {α 1 , . . . , α n } ⊂ (C 0 ∪ {∞}), are not all real (unless α 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 or a so-called para-ORF are used instead. These zeros depend on one single parameter τ ∈ (C ∪ {∞}), 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 quasi-ORFs, para-ORFs and ORFs. Next, a condition is given for the parameter τ so that the zeros are all real and simple. Finally, some illustrative and numerical examples are given. Mathematics Subject Classification (2000)42C05 · 65D32 · 65F15

The existence and construction of rational Gauss-type quadrature rules

“…The case in which m = 1 has already been proved in [6,Thm. 5] under the less restrictive condition that ϕ…”

“…The existence of rational Gauss-Radau quadrature rules has been studied in [6,Sect. 5] for the case of i = 0.…”

“…, α n−1 , α}, G)-PRIQ (i.e., a rational Gaussian quadrature rule) too. Consequently, its nodes can be computed by solving a modified GEP of the form (2.7), with the coefficients a [ϕ] n−1 and b [ϕ] n−1 that appear in the last row of the matrices J n and S n replaced with a [P ] n−1,xn,1 and b [P ] n−1,xn,1 (depending on the fixed node; see [6,Sect. 3]), while the corresponding weights again can be found by means of the corresponding eigenvectors.…”

“…In [6,Thm. 7] it has been proved then that there exists polesα ∈ R 0 so that for every k n ∈ C 0 , the sequence of rational functions {ϕ 0 , .…”

Positive rational interpolatory quadrature formulas on the unit circle and the interval

“…A special case is obtained when one node in the n-point quadrature is fixed in advance, so that the weights are all positive and the quadrature is exact for every F ∈ R n−1,n−1 , which corresponds to the n-point rational Gauss-Radau quadrature formula. However, the existence of this n-point rational Gauss-Radau quadrature depends on the choice of the node (e.g., it surely does not exist whenever the node is a zero of ϕ n−1 ; see [10]). Whenever two nodes in an (n + 1)-point quadrature formula are fixed in advance, so that the weights are all positive and the quadrature is exact for every F ∈ R n,n−1 , we obtain the (n + 1)-point rational Gauss-Lobatto quadrature formula.…”