Accurate computations of singular values and linear systems for Polynomial-Vandermonde-type matrices

In this paper, we consider how to accurately solve the singular value problem and the linear system for a class of Polynomial-Vandermonde-type (PVT) matrices, which belongs to the class of negative matrices introduced by Huang and Xue (Adv Comput Math 47:73, 2021), and these negative matrices arise in some applications such as interpolation problems. In order to parameterize PVT matrices, we present the explicit expressions of initial minors of such matrices. An algorithm is designed to accurately compute the parametrization matrix for PVT matrices. Based on the accurate parametrization algorithm, all the singular values, both large and small, of PVT matrices are computed, and the linear systems associated with PVT matrices are solved to high relative accuracy. Numerical experiments are performed to confirm the claimed high relative accuracy.