Perturbation analysis for the periodic generalized coupled Sylvester equation

TL;DR

This paper conducts a detailed perturbation analysis of the periodic generalized coupled Sylvester equation, deriving condition numbers and error bounds, and demonstrating their effectiveness through numerical examples.

Contribution

It introduces new perturbation bounds, condition numbers, and estimation methods specifically for the PGCS equation, advancing numerical stability analysis.

Findings

Derived normwise and componentwise perturbation bounds

Established effective condition numbers for PGCS equation

Validated results with numerical experiments

Abstract

In this paper, we consider the perturbation analysis for the periodic generalized coupled Sylvester (PGCS) equation. The normwise backward error for this equation is first obtained. Then, we present its normwise and componentwise perturbation bounds, from which the normwise and effective condition numbers are derived. Moreover, the mixed and componentwise condition numbers for the PGCS equation are also given. To estimate these condition numbers with high reliability, the probabilistic spectral norm estimator and the statistical condition estimation method are applied. The obtained results are illustrated by numerical examples.