Symmetric Successive Overrelaxation Method Research Articles

The Monomial Preconditioned SSOR Method for Linear Complementarity Problem

This paper aims to show that the existing preconditioned symmetric successive over-relaxation (SSOR) approach to solving the linear complementarity problem (LCP) is not valid. To overcome the flaws, we propose an efficient preconditioner called the monomial preconditioner. The convergence behavior of the proposed model is also established. Meanwhile, the efficiency of the new method is verified by numerical experiments.

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

We present a preconditioned version of the symmetric successive overrelaxation (SSOR) iteration method for a class of complex symmetric linear systems. The convergence results of the proposed method are established and conditions under which the spectral radius of the iteration matrix of the method is smaller than that of the SSOR method are analyzed. Numerical experiments illustrate the theoretical results and depict the efficiency of the new iteration method.

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

In this paper, the optimal iteration parameters of the symmetric successive overrelaxation (SSOR) method for a class of block two-by-two linear systems are obtained, which result in optimal convergence factor. An accelerated variant of the SSOR (ASSOR) method is presented, which significantly improves the convergence rate of the SSOR method. Furthermore, a more practical way to choose iteration parameters for the ASSOR method has also been proposed. Numerical experiments demonstrate the efficiency of the SSOR and ASSOR methods for solving a class of block two-by-two linear systems with the optimal parameters.

A comparison of the Extrapolated Successive Overrelaxation and the Preconditioned Simultaneous Displacement methods for augmented linear systems

In this paper we study the impact of two types of preconditioning on the numerical solution of large sparse augmented linear systems. The first preconditioning matrix is the lower triangular part whereas the second is the product of the lower triangular part with the upper triangular part of the augmented system's coefficient matrix. For the first preconditioning matrix we form the Generalized Modified Extrapolated Successive Overrelaxation (GMESOR) method, whereas the second preconditioning matrix yields the Generalized Modified Preconditioned Simultaneous Displacement (GMPSD) method, which is an extrapolated form of the Symmetric Successive Overrelaxation method. We find sufficient conditions for each aforementioned iterative method to converge. In addition, we develop a geometric approach, for determining the optimum values of their parameters and corresponding spectral radii. It is shown that both iterative methods studied (GMESOR and GMPSD) attain the same rate of convergence. Numerical results confirm our theoretical expectations.

Modified SSOR Modelling for Linear Complementarity Problems

In this paper, we present an efficient numerical method for solving the linear complementarity problems based on preconditioning strategy. Furthermore, the convergence properties of the proposed method have been analyzed and compared with symmetric successive over-relaxation (SSOR) method. Finally, some numerical experiments are illustrated to show the efficiency of the proposed method.

On the modified symmetric successive over-relaxation method for augmented systems

Recently, Darvishi and Hessari (Comput Math Appl 61:3128–3135, 2011) introduced a modified SSOR method for augmented systems. In this paper, we establish an improvement SSOR method for solving the large sparse augmented systems, which is the extension of the modified SSOR method. The convergence analysis of our method is studied under suitable restrictions on iteration parameters. Furthermore, the functional equation between the parameters and the eigenvalues of the iteration matrix of this method is obtained. Finally, numerical computations are presented based on a particular linear system, which clearly show the effectiveness of our algorithm.

A new modified SSOR iteration method for solving augmented linear systems

In this paper, we present a new symmetric successive over-relaxation method to find solution of the large sparse augmented linear systems, which is the extension of the symmetric successive overrelaxation iteration method. The convergence analysis of our method is also studied. Furthermore, the functional equation between the parameters and the eigenvalues of the relevant iteration matrix is obtained. Finally, numerical computations are presented to show the effectiveness of our algorithm.

Symmetric SOR Method for Absolute Complementarity Problems

We study symmetric successive overrelaxation (SSOR) method for absolute complementarity problems. Solving this problem is equivalent to solving the absolute value equations. Some examples are given to show the implementation and efficiency of the method.

A modified symmetric successive overrelaxation method for augmented systems

In this paper, we establish a modified symmetric successive overrelaxation (MSSOR) method, to solve augmented systems of linear equations, which uses two relaxation parameters. This method is an extension of the symmetric SOR (SSOR) iterative method. The convergence of the MSSOR method for augmented systems is studied. Numerical examples show that the new method is an efficient method.

Preface

Preface

Asynchronous Multisplitting GAOR Method and Asynchronous Multisplitting SSOR Method for Systems of Weakly Nonlinear Equations

In this article, we introduce two new asynchronous multisplitting methods for solving the system of weakly nonlinear equations Ax = G(x) in which A is an n × n real matrix and G(x) = (g1(x), g2(x), . . . , gn(x))T is a P-bounded mapping. First, by generalized accelerated overrelaxation (GAOR) technique, we introduce the asynchronous parallel multisplitting GAOR method (including the synchronous parallel multisplitting AOR method as a special case) for solving the system of weakly nonlinear equations. Second, asynchronous parallel multisplitting method based on symmetric successive overrelaxation (SSOR) multisplitting is introduced, which is called asynchronous parallel multisplitting SSOR method. Then under suitable conditions, we establish the convergence of the two introduced methods. The given results contain synchronous multisplitting iterations as a special case.

GAUSS-SEIDEL AND SUCCESSIVE OVERRELAXATION METHODS FOR RADIATIVE TRANSFER WITH PARTIAL FREQUENCY REDISTRIBUTION

The linearly-polarized solar limb spectrum that is produced by scattering processes contains a wealth of information on the physical conditions and magnetic fields of the solar outer atmosphere, but the modeling of many of its strongest spectral lines requires solving an involved non-LTE radiative transfer problem accounting for partial redistribution (PRD) effects. Fast radiative transfer methods for the numerical solution of PRD problems are also needed for a proper treatment of hydrogen lines when aiming at realistic time-dependent magnetohydrodynamic simulations of the solar chromosphere. Here we show how the two-level atom PRD problem with and without polarization can be solved accurately and efficiently via the application of highly convergent iterative schemes based on the Gauss-Seidel (GS) and Successive Overrelaxation (SOR) radiative transfer methods that had been previously developed for the complete redistribution (CRD) case. Of particular interest is the Symmetric SOR method, which allows us to reach the fully converged solution with an order of magnitude of improvement in the total computational time with respect to the Jacobi-based local ALI (Accelerated Lambda Iteration) method.

A modified SSOR iterative method for augmented systems

Golub, Wu and Yuan [G.H. Golub, X. Wu, J.Y. Yuan, SOR-like methods for augmented systems, BIT 41 (2001) 71–85] have presented the SOR-like algorithm to solve augmented systems. In this paper, we present the modified symmetric successive overrelaxation (MSSOR) method for solving augmented systems, which is based on Darvishi and Hessari’s work above. We derive its convergence under suitable restrictions on the iteration parameter, determine its optimal iteration parameter and the corresponding optimal convergence factor under certain conditions. Finally, we apply the MSSOR method to solve augmented systems.

Symmetric block-SOR methods for rank-deficient least squares problems

In this article, we develop symmetric block successive overrelaxation (S-block-SOR) methods for finding the solution of the rank-deficient least squares problems. We propose an S2-block-SOR and an S3-block-SOR method for solving such problems and the convergence of these two methods is studied. The comparisons between the S2-block and the S3-block methods are presented with some numerical examples.

Symmetric SOR method for augmented systems

In this paper, we develop symmetric successive overrelaxation (SSOR) method to find solution of augmented systems. These systems have appeared in many different applications of scientific computing, e.g., the finite element approximation to solve the Navier–Stokes equation and the constrained optimization. The convergence of SSOR method are studied. Finally, we apply the new method to solve augmented systems. The iteration number of this method to receive the solution is very better than SOR-like method.

Comments on “Symmetric successive overrelaxation methods for rank deficient linear systems”: [Applied Mathematics and Computation 173 (2006) 404–420

In this note, we present a critical review on the paper “Symmetric successive overrelaxation methods for rank deficient linear systems” by authors Darvishi and Khosro-Aghdam [M.T. Darvishi, R. Khosro-Aghdam, Symmetric successive overrelaxation methods for rank deficient linear systems, Appl. Math. Comput. 173 (2006) 404–420]. We point out that a sufficient condition for the semiconvergence of SSOR iteration matrix, one of the main results in their paper, is not true and its proof is unacceptable. In fact, this problem has been thoroughly solved and one nice set of necessary and sufficient conditions has been given out by us, see [B. Zheng, K. Wang,Symmetric successive overrelaxation methods for solving the rank deficient linear least squares problem, Appl. Math. Comput. 169 (2005) 1305–1323].

Modeling the 3D Terrain Effect on MT by the Boundary Element Method

ABSTRACT A numerical method is put forward in this paper, using the boundary element method (BEM) to model 3D terrain effects on magnetotelluric (MT) surveys. Using vector integral theory and electromagnetic field boundary conditions, the boundary problem of two electromagnetic fields in the upper half space (air) and lower half space (earth medium) was transformed into two vector integral equations just related to the topography: one magnetic equation for computing the magnetic field and the other electrical equation for computing the electrical field. The topography integral is decomposed into a series of integrals in a triangle element. For the integral in a triangle element, we suppose that the electromagnetic field in it is the stack of the electromagnetic field in the homogeneous earth and the topography response which is a constant; so the computation becomes simple, convenient and highly accurate. By decomposition and computation, each vector integral equation can be calculated by solving three linear equations that are related to the three Cartesian directions. The matrix of these linear equations is diagonally dominant and can be solved using the Symmetric Successive Over-Relaxation (SSOR) method. The apparent resistivity curve of MT on two 3D terrains calculated by BEM is shown in this paper.

Symmetric successive overrelaxation methods for fuzzy linear systems

Symmetric successive overrelaxation (SSOR) methods, for solving fuzzy linear systems, are discussed and followed by convergence theorems. Numerical examples are given to illustrate the procedure.

Symmetric successive overrelaxation methods for rank deficient linear systems

In this paper, we develop symmetric successive overrelaxation (symmetric SOR or SSOR) methods for finding the least square solution of minimal norm to the linear system Ax=b where A is an mxn matrix of rank r. The methods are obtained by first augmenting the system to a block 4x4 consistent system. The augmented coefficient matrix is then split by a subproper SSOR splitting. We state and prove some theorems and by some numerical examples we show the number of iterations for SSOR is less than SOR and accelerated overrelaxation methods for finding the least square solution of minimal norm to the linear system.

Symmetric successive overrelaxation methods for solving the rank deficient linear least squares problem

Symmetric successive overrelaxation (SSOR) methods are developed for finding the least squares solution of minimal norm to the linear system Ax = b , where A ∈ C m × n is a matrix of rank r and b ∈ C m . By first augmenting the system to a block 4 × 4 consistent system and then splitting the augmented coefficient matrix by subproper SSOR splittings, we obtain intervals for the relaxation parameter in which the subproper SSOR iteration matrix is semiconvergent and a solution for the augmented system. Subsequently, a method for transforming such a solution to the least squares solution of minimal 2-norm is presented.