Transformations of Cauchy matrices, Trummer's problem and a Cauchy-like linear solver

Abstract

Computations with dense structured matrices are ubiquitous in sciences, communication and engineering. Exploitation of structure simplifies the computations dramatically but sometimes leads to irregularity, e.g. to numerical stability problems. We will follow the line of [P90], [PLST93], [PZHY97] by exploiting the transformations among the most celebrated classes of structured matrices (that is, among the matrices of Toeplitz, Hankel, Cauchy and Vandermonde types), as a general tool for avoiding irregularity problems and improving computations with the structured matrices of the above classes. In this paper we apply the transformation techniques to Trummer's problem of multiplication of a Cauchy matrix by a vector and to multipoint polynomial evaluation, which should demonstrate the power of this general approach. In both cases (of Trummer's problem and polynomial evaluation) the resulting algorithms use only a few arithmetic operations, allow their effective parallelization and are stable numerically. The new transformations are described in sections 2–6. Subsequent sections reduce Cauchy-like matrix computations to recursive solution of Trummer's problem.