Computation of quasiseparable representations of Green matrices

TL;DR

This paper introduces algorithms for efficiently computing quasiseparable representations of Green matrices, specifically the inverse of band matrices, with linear complexity, supported by numerical experiments on stability and performance.

Contribution

It presents the first algorithms based on the Asplund theorem for quasiseparable representations of Green matrices, achieving linear complexity.

Findings

Algorithms confirmed linear complexity through numerical experiments.

Numerical results provide insights into stability properties.

First implementation of rank-structure based inversion algorithms for band matrices.

Abstract

The well-known Asplund theorem states that the inverse of a (possibly one-sided) band matrix $A$ is a Green matrix. In accordance with quasiseparable theory, such a matrix admits a quasiseparable representation in its rank-structured part. Based on this idea, we derive algorithms that compute a quasiseparable representation of $A^{-1}$ with linear complexity. Many inversion algorithms for band matrices exist in the literature. However, algorithms based on a computation of the rank structure performed theoretically via the Asplund theorem appear for the first time in this paper. Numerical experiments confirm complexity estimates and offer insight into stability properties.