A generalized eigenvalue problem for quasi-orthogonal rational functions

In general, the zeros of an orthogonal rational function (ORF) on a subset of the real line, with poles among {α1, . . . , αn} ⊂ (C0 ∪ {∞}), are not all real (unless αn 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 paraORF 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.