A fast parallel Björck–Pereyra-type algorithm for solving Cauchy linear equations

Abstract

We propose a new fast O(n2) parallel algorithm for solving Cauchy systems of linear equations. We perform an a priori rounding error analysis and obtain componentwise bounds for the forward, backward and residual errors. These bounds indicate that for the class of totally positive Cauchy matrices the new algorithm is forward and backward stable, producing a remarkably high relative accuracy. In particular, Hilbert linear systems, often considered to be too ill-conditioned to be attacked, can be rapidly solved with high precision. The results indicate a close resemblance between the numerical properties of Cauchy matrices and the much-studied Vandermonde matrices. In fact, our proposed Cauchy solver is an analog of the well-known Björck–Pereyra algorithm for Vandermonde systems. As a by-product we obtain favorably backward error bounds for the original Björck–Pereyra algorithm. Several computed examples illustrate a connection of high relative accuracy to the concepts of effective well-conditioning and total positivity.