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

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.