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

ABSTRACTWe present an iterative method based on the Hermitian and skew-Hermitian splitting (HSS) for solving the continuous Sylvester equation. By using the HSS of the coefficient matrices A and B, we establish a method which is practically inner/outer iterations, by employing a conjugate gradient on the normal equations (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.KEYWORDS: CGNRSylvester equationHermitian and skew-Hermitian splittingpreconditioningnested iterations2010 AMS SUBJECT CLASSIFICATIONS: 15A2415A3015A6965F1065F30 AcknowledgmentsThe authors are very much indebted to the referees for their constructive comments and suggestions which greatly improved the original manuscript of this paper.Disclosure statementNo potential conflict of interest was reported by the authors.