Backward Errors and Small Sample Condition Estimation for $\star$-Sylveter Equations
Provides practical error estimation tools for researchers solving $\\star$-Sylvester equations, though the contribution is incremental.
The paper develops algorithms for estimating normwise, mixed, and componentwise condition numbers for $\\star$-Sylvester equations, and defines a sharp, computable bound for componentwise backward error. Numerical examples show the estimates are reliable for well-conditioned and moderately ill-conditioned 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.