Two Efficient Inexact Algorithms for a Class of Large Sparse Complex Linear Systems

Abstract

Recently Salkuyeh et al. (Int J Comput Math 92:802-815, 2015) studied the generalized SOR (GSOR) iterative method for a class of complex symmetric linear system of equations. In this paper, we present an inexact variant of the GSOR method in which the conjugate gradient and the preconditioned conjugate gradient methods are regarded as its inner iteration processes at each step of the GSOR outer iteration. Moreover, we construct a new method called shifted GSOR iteration method which is obtained from combination of a shiftsplitting iteration scheme and the GSOR iteration method. The convergence analysis of the proposed methods are presented. Some numerical experiments are given to show the performance of the methods and are compared with those of the inexact MHSS method.Mathematics Subject Classification. 65F10, 93C10.

A new double-step splitting iteration method for certain block two-by-two linear systems

“…Next, by using the proof techniques of Theorem 3.1 of Bai et al (2003) and Theorem 2 of Edalatpour et al (2016), we proceed to analyze the convergence conditions of the INDSS iteration method. To this end, an useful result will be used as follows.…”

“…Lemma 3.1 (Edalatpour et al 2016) For every B ∈ C n×n and ε > 0, there exists a norm • on C n such that for the corresponding induced norm, B ≤ ρ(B) + ε.…”

Robust iteration methods for complex systems with an indefinite matrix term

“…where M and K are two square matrices, typically inertia and stiffness matrices, respectively. When the matrices M and K are symmetric positive semidefinite with at least one of them being positive definite, there are several iteration methods for solving the system [9,10,11,12,13,14,15,16,21,22,23,25]. Some of them are directly applied to the main system (see [21,22,23,25,11,12]) and some to its real form [9,10,13,14,15].…”

A preconditioned SSOR iteration method for solving complex symmetric system of linear equations

“…Splitting iteration methods such as generalized successive overrelaxation (GSOR) [18], preconditioned GSOR (PGSOR) [16], scale splitting (SCSP) [15], two-parameter two-step SCSP (TTSCSP) in [19] and parameterized splitting (PS) in [20] proposed for the linear system (1). Edalatpour et al in [13] solved inexactly the linear system (1) by two efficient methods. Using the Hermitian and skew-Hermitian splitting [5] of the coefficient matrix A, Bai et al proposed the modified HSS (MHSS) iteration method in [2] and preconditioned MHSS (PMHSS) iteration method in [6].…”