Accurate computations for eigenvalues of products of Cauchy-polynomial-Vandermonde matrices

In this paper, we consider the product eigenvalue problem for the class of Cauchy-polynomial-Vandermonde (CPV) matrices arising in a rational interpolation problem. We present the explicit expressions of minors of CPV matrices. An algorithm is designed to accurately compute the bidiagonal decomposition for strictly totally positive CPV matrices and their additive inverses. We then illustrate the sign regularity of the bidiagonal decomposition to show that all the eigenvalues of a product involving such matrices are computed to high relative accuracy. Numerical experiments are given to confirm the claimed high relative accuracy.