On PMHSS Iteration Methods for Continuous Sylvester Equations (Journal of Computational Mathematics, 2018)
The modified Hermitian and skew-Hermitian splitting (MHSS) iteration method and preconditioned MHSS (PMHSS) iteration method were introduced respectively. In the paper, on the basis of the MHSS iteration method, we present a PMHSS iteration method for solving large sparse continuous Sylvester equations with non-Hermitian and complex symmetric positive definite ⁄ semi-definite matrices. Under suitable conditions, we prove the convergence of the PMHSS iteration method and discuss the spectral properties of the preconditioned matrix. Moreover, to reduce the computing cost, we establish an inexact variant of the PMHSS iteration method and analyze its convergence property in detail. Numerical results show that the PMHSS iteration method and its inexact variant are efficient and robust solvers for this class of continuous Sylvester equations.
Keywords: Continuous Sylvester equation; PMHSS iteration; Inexact PMHSS iteration; Preconditioning; Convergence.