Generalized SOR iterative method for a class of complex symmetric linear system of equations

In this paper, to solve a broad class of complex symmetric linear systems, we recast the complex system in a real formulation and apply the generalized successive overrelaxation (GSOR) iterative method to the equivalent real system. We then investigate its convergence properties and determine its optimal iteration parameter as well as its corresponding optimal convergence factor. In addition, the resulting GSOR preconditioner is used to preconditioned Krylov subspace methods such as GMRES for solving the real equivalent formulation of the system. Finally, we give some numerical experiments to validate the theoretical results and compare the performance of the GSOR method with the modified Hermitian and skew-Hermitian splitting (MHSS) iteration.Comment: 14 page

A new relaxed HSS preconditioner for saddle point problems

We present a preconditioner for saddle point problems. The proposed preconditioner is extracted from a stationary iterative method which is convergent under a mild condition. Some properties of the preconditioner as well as the eigenvalues distribution of the preconditioned matrix are presented. The preconditioned system is solved by a Krylov subspace method like restarted GMRES. Finally, some numerical experiments on test problems arisen from finite element discretization of the Stokes problem are given to show the effectiveness of the preconditioner.Comment: 16 pages, two figures, Accepted for publication in Numerical Algorithms, 201

The WR-HSS iteration method for a system of linear differential equations and its applications to the unsteady discrete elliptic problem

We consider the numerical method for non-self-adjoint positive definite linear differential equations, and its application to the unsteady discrete elliptic problem, which is derived from spatial discretization of the unsteady elliptic problem with Dirichlet boundary condition. Based on the idea of the alternating direction implicit (ADI) iteration technique and the Hermitian/skew-Hermitian splitting (HSS), we establish a waveform relaxation (WR) iteration method for solving the non-self-adjoint positive definite linear differential equations, called the WR-HSS method. We analyze the convergence property of the WR-HSS method, and prove that the WR-HSS method is unconditionally convergent to the solution of the system of linear differential equations. In addition, we derive the upper bound of the contraction factor of the WR-HSS method in each iteration which is only dependent on the Hermitian part of the corresponding non-self-adjoint positive definite linear differential operator. Finally, the applications of the WR-HSS method to the unsteady discrete elliptic problem demonstrate its effectiveness and the correctness of the theoretical results.Comment: 30 pages, 5 figures, 13 table

Two-step scale-splitting method for solving complex symmetric system of linear equations

Based on the Scale-Splitting (SCSP) iteration method presented by Hezari et al. in (A new iterative method for solving a class of complex symmetric system linear of equations, Numerical Algorithms 73 (2016) 927-955), we present a new two-step iteration method, called TSCSP, for solving the complex symmetric system of linear equations $(W+iT)x=b$, where $W$ and $T$ are symmetric positive definite and symmetric positive semidefinite matrices, respectively. It is shown that if the matrices $W$ and $T$ are symmetric positive definite, then the method is unconditionally convergent. The optimal value of the parameter, which minimizes the spectral radius of the iteration matrix is also computed. Numerical {comparisons} of the TSCSP iteration method with the SCSP, the MHSS, the PMHSS and the GSOR methods are given to illustrate the effectiveness of the method.Comment: 13 pages. Current status: Unsubmitted. arXiv admin note: text overlap with arXiv:1403.5902, arXiv:1611.0370

On the generalized shift-splitting preconditioner for saddle point problems

In this paper, the generalized shift-splitting preconditioner is implemented for saddle point problems with symmetric positive definite (1,1)-block and symmetric positive semidefinite (2,2)-block. The proposed preconditioner is extracted form a stationary iterative method which is unconditionally convergent. Moreover, a relaxed version of the proposed preconditioner is presented and some properties of the eigenvalues distribution of the corresponding preconditioned matrix are studied. Finally, some numerical experiments on test problems arisen from finite element discretization of the Stokes problem are given to show the effectiveness of the preconditioners.Comment: 7 pages, 1 figure and 2 tables, Applied Mathematics Letters, 201

Multistep matrix splitting iteration preconditioning for singular linear systems

Multistep matrix splitting iterations serve as preconditioning for Krylov subspace methods for solving singular linear systems. The preconditioner is applied to the generalized minimal residual (GMRES) method and the flexible GMRES (FGMRES) method. We present theoretical and practical justifications for using this approach. Numerical experiments show that the multistep generalized shifted splitting (GSS) and Hermitian and skew-Hermitian splitting (HSS) iteration preconditioning are more robust and efficient compared to standard preconditioners for some test problems of large sparse singular linear systems.Comment: 16 page

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

We introduce a two-parameter version of the two-step scale-splitting iteration method, called TTSCSP, for solving a broad class of complex symmetric system of linear equations. We present some conditions for the convergence of the method. An upper bound for the spectral radius of the method is presented and optimal parameters which minimize this bound are given. Inexact version of the TTSCSP iteration method (ITTSCSP) is also presented. Some numerical experiments are reported to verify the effectiveness of the TTSCSP iteration method and the numerical results are compared with those of the TSCSP, the SCSP and the PMHSS iteration methods. Numerical comparison of the ITTSCSP method with the inexact version of TSCSP, SCSP and PMHSS are presented. We also compare the numerical results of the BiCGSTAB method in conjunction with the TTSCSP and the ILU preconditioners.Comment: 21 pages, Accepted for publication in CALCOLO, Feb 3, 2018. arXiv admin note: text overlap with arXiv:1611.0370

The nonlinear HSS-like iteration method for absolute value equations

Salkuyeh proposed the Picard-HSS iteration method to solve the absolute value equation (AVE), which is a class of non-differentiable NP-hard problem. To further improve its performance, a nonlinear HSS-like iteration method is proposed. Compared to that the Picard-HSS method is an inner-outer double-layer iteration scheme, the HSS-like iteration is only a monolayer and the iteration vector could be updated timely. Some numerical experiments are used to demonstrate that the nonlinear HSS-like method is feasible, robust and effective

A modification of the generalized shift-splitting method for singular saddle point problems

A modification of the generalized shift-splitting (GSS) method is presented for solving singular saddle point problems. In this kind of modification, the diagonal shift matrix is replaced by a block diagonal matrix which is symmetric positive definite. Semi-convergence of the proposed method is investigated. The induced preconditioner is applied to the saddle point problem and the preconditioned system is solved by the restarted generalized minimal residual method. Eigenvalue distribution of the preconditioned matrix is also discussed. Finally some numerical experiments are given to show the effectiveness and robustness of the new preconditioner. Numerical results show that the modified GSS method is superior to the classical GSS method.Comment: 21 pages, submitte

Semi-convergence of the EPSS method for singular generalized saddle point problems

Recently, in (M. Masoudi, D.K. Salkuyeh, An extension of positive-definite and skew-Hermitian splitting method for preconditioning of generalized saddle point problems, Computers \& Mathematics with Application, https://doi.org/10.1016/j.camwa.2019.10.030, 2019) an extension of the positive definite and skew-Hermitian splitting (EPSS) iteration method for nonsingular generalized saddle point problems has been presented. In this article, we study semi-convergence of the EPSS method for singular generalized saddle problems. Then a special case of EPSS (SEPSS) preconditioner is applied to the nonsingular generalized saddle point problems. Some numerical results are presented to show the effectiveness of the preconditioner.Comment: 14 pages, Submitte