Numerical methods for large-scale Lyapunov equations with symmetric banded data

TL;DR

This paper introduces two efficient numerical methods for solving large-scale Lyapunov equations with symmetric banded data, exploiting solution structure to save memory and computation, and demonstrates their effectiveness through numerical experiments.

Contribution

The paper presents novel solution methods tailored for large-scale Lyapunov equations with symmetric banded data, addressing both well-conditioned and ill-conditioned cases.

Findings

Methods effectively exploit solution structure for efficiency.

Numerical experiments confirm computational savings.

Approaches handle both well-conditioned and ill-conditioned matrices.

Abstract

The numerical solution of large-scale Lyapunov matrix equations with symmetric banded data has so far received little attention in the rich literature on Lyapunov equations. We aim to contribute to this open problem by introducing two efficient solution methods, which respectively address the cases of well conditioned and ill conditioned coefficient matrices. The proposed approaches conveniently exploit the possibly hidden structure of the solution matrix so as to deliver memory and computation saving approximate solutions. Numerical experiments are reported to illustrate the potential of the described methods.