Accurate solutions of product linear systems associated with rank-structured matrices

In this paper, we consider how to accurately solve linear systems associated with a wide class of rank-structured matrices containing the well-known Vandermonde and Cauchy matrices, i.e., consecutive-rank-descending (CRD) matrices. We provide a mechanism to guarantee that the inverse of any product of CRD matrices is generated in a subtraction-free manner. With the mechanism, the solutions of linear systems associated with such products are accurately determined by the parameters of CRD factors, and we then accurately compute the solutions as warranted by these parameters. In particular, linear systems associated with products of Vandermonde and Cauchy matrices, whose nodes satisfy certain positive or negative properties, are solved to high relative accuracy. Error analysis and numerical experiments are provided to confirm the high accuracy.