A new iterative method for solving a class of complex symmetric system of linear equations

Abstract

We present a new stationary iterative method, called Scale-Splitting (SCSP) method, and investigate its convergence properties. The SCSP method naturally results in a simple matrix splitting preconditioner, called SCSP-preconditioner, for the original linear system. Some numerical comparisons are presented between the SCSP-preconditioner and several available block preconditioners, such as PGSOR (Hezari et al. Numer. Linear Algebra Appl. 22, 761-776, 2015) and rotate block triangular preconditioners (Bai Sci. China Math. 56, 2523-2538, 2013, when they are applied to expedite the convergence rate of Krylov subspace iteration methods for solving the original complex system and its block real formulation, respectively. Numerical experiments show that the SCSP-preconditioner can compete with PGSOR-preconditioner and even more effective than the rotate block triangular preconditioners.

On symmetric block triangular splitting iteration method for a class of complex symmetric system of linear equations

“…is increasing and decreasing about µ 2 min and µ 2 max , respectively, then we hope to choose a proper parameter ω which minimizes the µ 2 max . Based on the Theorem 1 of [15], the latter result of (3.7) is easy to obtain.…”

On Euler preconditioned SHSS iterative method for a class of complex symmetric linear systems

“…Moreover, Hezari et al [30] presented a preconditioned variant of the GSOR (PGSOR) iterative method, Chen and Ma [21] established the AOR-Uzawa iterative method. For more iterative methods and Krylov subspace methods in real arithmetic for solving non-singular linear system (1.4), see [10,15,16,31,34] and references therein. When W, T are both symmetric positive semi-definite satisfying null(W ) ∩ null(T ) = {0}, then the coefficient matrix A of (1.1) is singular.…”

Two-parameter TSCSP method for solving complex symmetric system of linear equations

“…Hezari et al in [14] presented the Scale-Splitting (SCSP) iteration method for solving (1) which can be described as follows.…”

“…At each iteration of the SCSP iteration method, it is required to solve a linear system with coefficient matrix αW + T . In [14] it was proved that if W and T are symmetric positive semidefinite matrices satisfying null(W ) ∩ null(T ) = {0}, then the SCSP iteration method is convergent provided that…”