An iterative method for solving the continuous sylvester equation by emphasizing on the skew-hermitian parts of the coefficient matrices

We present an iterative method based on the Hermitian and skew-Hermitian splitting for solving the continuous Sylvester equation. By using the Hermitian and skew-Hermitian splitting of the coefficient matrices A and B, we establish a method which is practically inner/outer iterations, by employing a CGNR-like method as inner iteration to approximate each outer iterate, while each outer iteration is induced by a convergent splitting of the coefficient matrices. Via this method, a Sylvester equation with coefficient matrices SA and SB (which are the skew-Hermitian part of A and B, respectively) is solved iteratively by a CGNR-like method. Convergence conditions of this method are studied and numerical examples show the efficiency of this method. In addition, we show that the quasi-Hermitian splitting can induce accurate, robust and effective preconditioned Krylov subspace methods.