An efficient two-step iterative method for solving a class of complex symmetric linear systems

Abstract

In this paper, a new two-step iterative method called the two-step parameterized (TSP) iteration method for a class of complex symmetric linear systems is developed. We investigate its convergence conditions and derive the quasi-optimal parameters which minimize the upper bound of the spectral radius of the iteration matrix of the TSP iteration method. Meanwhile, some more practical ways to choose iteration parameters for the TSP iteration method are proposed. Furthermore, comparisons of the TSP iteration method with some existing ones are given, which show that the upper bound of the spectral radius of the TSP iteration method is smaller than those of the modified Hermitian and skew-Hermitian splitting (MHSS), the preconditioned MHSS (PMHSS), the combination method of real part and imaginary part (CRI) and the parameterized variant of the fixed-point iteration adding the asymmetric error (PFPAE) iteration methods proposed recently. Inexact version of the TSP iteration (ITSP) method and its convergence properties are also presented. Numerical experiments demonstrate that both TSP and ITSP are effective and robust when they are used either as linear solvers or as matrix splitting preconditioners for the Krylov subspace iteration methods and they have comparable advantages over some known ones for the complex symmetric linear systems.