Computing eigenvalues of quasi-rational Bernstein-Vandermonde matrices to high relative accuracy

In this article, we consider how to accurately solve the eigenvalue problem for a class of quasi-rational Bernstein-Vandermonde (q-RBV) matrices. This class of matrices belongs to generalized sign regular matrices with signature (1, horizontal ellipsis ,1,-1). An algorithm is developed to accurately compute the parameter matrix for q-RBV matrices. Based on the parameter matrix, all the eigenvalues of q-RBV matrices have been computed to high relative accuracy. The perturbation theory for the eigenvalues of q-RBV matrices and the error analysis of our proposed algorithm are provided. Numerical experiments are performed to confirm the claimed high relative accuracy.