Backward Errors and Small Sample Condition Estimation for $\star$-Sylveter Equations

TL;DR

This paper develops a componentwise perturbation analysis and condition estimation algorithms for $oldsymbol{ ext{ extsterling}}$-Sylvester equations, providing reliable condition numbers and backward error bounds that enhance numerical stability assessment.

Contribution

It introduces new algorithms for estimating normwise, mixed, and componentwise condition numbers and defines a sharp, computable componentwise backward error bound for $oldsymbol{ ext{ extsterling}}$-Sylvester equations.

Findings

Algorithms produce reliable condition estimates under componentwise perturbations.

The backward error bound is sharp and reliable for well-conditioned and moderately ill-conditioned problems.

Numerical examples confirm the effectiveness of the proposed methods.

Abstract

In this paper, we adopt a componentwise perturbation analysis for $\star$-Sylvester equations. Based on the small condition estimation (SCE), we devise the algorithms to estimate normwise, mixed and componentwise condition numbers for $\star$-Sylvester equations. We also define a componentwise backward error with a sharp and easily computable bound. Numerical examples illustrate that our algorithm under componentwise perturbations produces reliable estimates, and the new derived computable bound for the componentwise backward error is sharp and reliable for well conditioned and moderate ill-conditioned $\star$-Sylvester equations under large or small perturbations.