Accurate solution of structured linear systems via rank-revealing decompositions

Abstract

Linear systems of equations Ax = b, where the matrix A has some particular structure, arise frequently in applications. Very often structured matrices have huge condition numbers κ(A) = ‖A−1‖‖A‖ and, therefore, standard algorithms fail to compute accurate solutions of Ax = b. We say in this paper that a computed solution x is accurate if ‖x− x‖/‖x‖ = O(u), being u the unit roundoff. In this work, we introduce a framework that allows to solve accurately many classes of structured linear systems, independently of the condition number of A and efficiently, that is, with cost O(n3). For most of these classes no algorithms are known that are both accurate and efficient. The approach in this work relies on computing first an accurate rank-revealing decomposition of A, an idea that has been widely used in the last decades to compute singular value and eigenvalue decompositions of structured matrices with high relative accuracy. In particular, we illustrate the new method solving accurately Cauchy and Vandermonde linear systems with any distribution of nodes, i.e., without requiring A to be totally positive, for most right-hand sides b.