Superfast algorithms for Cauchy-like matrix computations and extensions

Abstract

An effective algorithm of [M. Morf, Ph.D. Thesis, Department of Electrical Engineering, Stanford University, Stanford, CA, 1974; in: Proceedings of the IEEE International Conference on ASSP, IEEE Computer Society Press, Silver Spring, MD, 1980, pp. 954–959; R.R. Bitmead and B.D.O. Anderson, Linear Algebra Appl. 34 (1980) 103–116] computes the solution x → =T −1 b → to a strongly nonsingular Toeplitz or Toeplitz-like linear system T x → = b → , a short displacement generator for the inverse T −1 of T, and det T. We extend this algorithm to the similar computations with n×n Cauchy and Cauchy-like matrices. Recursive triangular factorization of such a matrix can be computed by our algorithm at the cost of executing O(nr 2 log 3 n) arithmetic operations, where r is the scaling rank of the input Cauchy-like matrix C (r=1 if C is a Cauchy matrix). Consequently, the same cost bound applies to the computation of the determinant of C, a short scaling generator of C −1 , and the solution to a nonsingular linear system of n equations with such a matrix C. (Our algorithm does not use the reduction to Toeplitz-like computations.) We also relax the assumptions of strong nonsingularity and even nonsingularity of the input not only for the computations in the field of complex or real numbers, but even, where the algorithm runs in an arbitrary field. We achieve this by using randomization, and we also show a certain improvement of the respective algorithm by Kaltofen for Toeplitz-like computations in an arbitrary field. Our subject has close correlation to rational tangential (matrix) interpolation under passivity condition (e.g., to Nevanlinna–Pick tangential interpolation problems) and has further impact on the decoding of algebraic codes.