Effective condition numbers and small sample statistical condition estimation for the generalized Sylvester equation

In this paper, we investigate the effective condition numbers for the generalized Sylvester equation (AX − YB, DX − YE) = (C, F), where A, D ∈ ℝm×m, B, E ∈ ℝn×n and C, F ∈ ℝm×n. We apply the small sample statistical method for the fast condition estimation of the generalized Sylvester equation, which requires O(m2n + mn2) flops, comparing with O(m3 + n3) flops for the generalized Schur and generalized Hessenberg-Schur methods for solving the generalized Sylvester equation. Numerical examples illustrate the sharpness of our perturbation bounds.

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