A fast solver for linear systems with displacement structure

TL;DR

This paper introduces a fast, memory-efficient solver for linear systems with Cauchy-like structure, leveraging the generalized Schur algorithm, and provides a robust software implementation with demonstrated effectiveness.

Contribution

It presents a novel fast solver for Cauchy-like structured systems using the generalized Schur algorithm, including new pivoting strategies and software tools.

Findings

Requires O(rn^2) operations and O(rn) memory

Includes robust Matlab and C-MEX software implementations

Demonstrates effectiveness through numerical experiments

Abstract

We describe a fast solver for linear systems with reconstructable Cauchy-like structure, which requires O(rn^2) floating point operations and O(rn) memory locations, where n is the size of the matrix and r its displacement rank. The solver is based on the application of the generalized Schur algorithm to a suitable augmented matrix, under some assumptions on the knots of the Cauchy-like matrix. It includes various pivoting strategies, already discussed in the literature, and a new algorithm, which only requires reconstructability. We have developed a software package, written in Matlab and C-MEX, which provides a robust implementation of the above method. Our package also includes solvers for Toeplitz(+Hankel)-like and Vandermonde-like linear systems, as these structures can be reduced to Cauchy-like by fast and stable transforms. Numerical experiments demonstrate the effectiveness of the software.