A Fast Bjorck-Pereyra-type Algorithm for Solving Cauchy Linear Equations

In this paper we propose a new fast O(n) algorithm for solving Cauchy linear systems of equations. An a priori rounding error analysis, which provides bounds for the forward, backward and residual errors associated with the computed solution, indicates that for the class of totally positive Cauchy matrices the proposed 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, actually can be rapidly solved with a favorably high precision. The results indicate to a close resemblance between the numerical properties of Cauchy matrices and the much-studied Vandermonde matrices. In particular, our proposed Cauchy solver is an analog of the the well-known Bjorck-Pereyra algorithm for Vandermonde systems, whose often favorable numerical properties motivated a number of authors, who extended this algorithm to the more general Vandermonde-related structures, and obtained favorable forward error bounds. Here we obtain also nice backward and residual error bounds for the Bj orck-Pereyra algorithm, and show that all these forward and backward stability results hold not just for Vandermonde and Vandermonde-related structures, but that they have counterparts for other classes of structured matrices, in particular for Cauchy matrices. The proposed algorithm is used to illustrate via computed examples the connection of high relative accuracy to the concepts of e ective well-conditioning and of total positivity.