Preconditioned GSOR iterative method for a class of complex symmetric system of linear equations

Abstract

In this paper, we present a preconditioned variant of the generalized successive overrelaxation (GSOR) iterative method for solving a broad class of complex symmetric linear systems. We study conditions under which the spectral radius of the iteration matrix of the preconditioned GSOR method is smaller than that of the GSOR method and determine the optimal values of iteration parameters. Numerical experiments are given to verify the validity of the presented theoretical results and the effectiveness of the preconditioned GSOR method.

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

“…To accelerate the convergence rate of the GSOR method, they then proposed the preconditioned GSOR (PGSOR) iteration method [17], for the linear system (2) and established the conditions under which the PGSOR iteration method is more effective than the GSOR iteration method. In this paper, to improve the computing efficiency of the GSOR iteration method we employ a Krylov subspace iteration scheme such as the conjugate gradient (CG) method or preconditioned conjugate gradient (PCG) method (see [21]) to solve inexactly the two linear subsystems at each step of the GSOR iteration.…”

“…Moreover, similar to IGSOR, we establish the inexact variant of SGSOR (ISGSOR). To speed up the convergence of the GSOR method one may precondition the system (2) by the preconditioner proposed in [17].…”

“…In [17], it has been proved that for ω = 1, if we set the iteration parameter of the method to be β = 0.828, then the convergence factor would be ρ(G β ) = 0.172, where G β is the iteration matrix of the method with the iteration parameter β. In the section of numerical experiments we apply the IGSOR and the ISGSOR for solving the equivalent system (26) with ω = 1 instead of solving the system (2), and they are called inexact preconditioned GSOR (IPGSOR) and inexact shifted preconditioned GSOR (ISPGSOR), respectively.…”

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

“…Based on the successive overrelaxation (SOR) method, recently Salkuyeh et al [19] applied the generalized SOR (GSOR) to efficiently solve the block two-by-two linear system (2). After that, they proposed a preconditioned variant of the GSOR (PGSOR) method [24] and established conditions such that the PGSOR iteration method to be more effective than the GSOR iteration method.…”

On SSOR iteration method for a class of block two-by-two linear systems

“…A large variety of effective iteration methods have been proposed in the literature for solving the linear system (1), such as C-to-R iteration methods [1,2,5,12,17], preconditioned Krylov subspace methods [14,16,26], splitting iteration methods [11,14,22,23,27,31], Hermitian and skew-Hermitian splitting (HSS) method and its variants [4, 6-9, 18, 21, 24, 30].…”