Incorporating the external zeros and poles of the integrand into Gauss-type quadrature rules (Q6546962)

This paper concerns the construction of Gaussian quadrature formulas for integrals \N\[\NI_\omega(g)=\int_a^b g(x)\omega(x)dx \]\Nwhere \(\omega\) is a positive weight and the function \(g\) that has some known zeros and poles in \(\mathbb{R}\setminus[a,b]\). These zeros and poles are used to construct a rational function \(R\), whose sign is chosen such that \(R\) is positive in \([a,b]\). Then \(I_\omega(g) = \int_a^b f(x) \tilde\omega(x)dx\) with \(\tilde\omega=R\omega\) and a Gaussian quadrature for the modified weight \(\tilde\omega\) has to be constructed. A Stieltjes procedure is used to compute the modified Jacobi matrix from which nodes and weights are recovered by a Golub-Welsch procedure. Error estimates are obtained numerically by adding additional nodes and weights obtaining (generalized) averaged Gaussian quadrature formulas as proposed by \textit{D. P. Laurie} [Math. Comput. 65, No. 214, 739--747 (1996; Zbl 0843.41020)] or \textit{M. M. Spalević} [Math. Comput. 86, No. 306, 1877--1885 (2017; Zbl 1361.65012)], and adaptations for the case where \(R\) is only approximately known. The method given is illustrated by several numerical examples.