Accurate computations with some Catalan-Stieltjes matrices

Catalan-Stieltjes matrices and their corresponding Catalan-like numbers are very important in combinatorics. In this paper, we give a sufficient condition for the Catalan-Stieltjes matrix to be triangular strictly totally positive (Delta\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\Delta $$\end{document}STP), that is, all its minors are positive except those which are zero due to the triangular structure. On this basis, we show that it is possible to compute the bidiagonal factorization of a class of Catalan-Stieltjes matrices with high relative accuracy (HRA). We also demonstrate that this class of matrices includes many well-known combinatorial triangles, such as the Catalan triangle of Aigner and the Bell triangle. Moreover, we construct an accurate algorithm to compute the bidiagonal factorization of some Catalan-Stieltjes matrices. The algorithm guarantees HRA for several linear algebraic computations associated with these matrices, like computing their singular values, their inverses, as well as the solutions of some linear systems. We show the accuracy and effectiveness of the proposed algorithm through numerical experiments.