SOR-like Method Research Articles

Generalized and novel iterative scheme for best approximate solution of large and sparse augmented linear systems

This article presents a generalized novel iterative technique for obtaining the approximate solution of systems of linear equations(ABBT0)(xy)=(pq). Parameters involved in the scheme play a vital role to obtain the rapid and best approximate solution. The technique is applied to some important problems and the results are presented in detail. The article compares the effectiveness of the new presented iterative scheme with modified SOR-like method, generalized SOR and SOR-like method for solving large systems of linear equations. The convergence criteria for the iterative scheme are also analysed and it can be observed in the article that the theoretical derivation is supported by numerical calculations and graphical exhibition. Overall, the article provides a new and effective method with a detailed analysis of its performance and convergence criteria for solving systems of linear equations of the form presented. The newly derived technique has potential applications in various fields of science and engineering where systems of linear equations arise and need to be solved.

A modified generalized SOR-like method for solving an absolute value equation

In this paper, we propose a modified generalized SOR-like (MGSOR) method for solving an absolute value equation (AVE), which is obtained by reformulating equivalently AVE as a two-by-two block nonlinear equation and by introducing the transformation with a general nonsingular matrix P. The convergence results of the MGSOR method are obtained under certain assumptions imposed on the involved parameters. Furthermore, the optimal parameters minimizing the convergence rate of the MGSOR method for solving AVE are studied in detail. Numerical experiments further illustrate that the MGSOR method is efficient and has better performance than some existing iteration methods in aspects of the number of iteration steps and CPU time.

Generalized SOR-like iteration method for solving weakly nonlinear systems

Generalized successive overrelaxation method is an efficient tool for solving a class of two-by-two augmented systems. In this paper, we transform the weakly nonlinear equations into the form of such two-by-two augmented systems. By applying the generalized successive overrelaxation method to this augmented system, we develop a class of generalized successive overrelaxation like iteration methods for solving the weakly nonlinear equations. Under some reasonable assumptions on the matrix splitting and two relaxation parameters, we are able to establish the global convergence results of the proposed methods. As a by-product, we improve the existing convergence results of the successive overrelaxation like iteration methods in terms of giving a weaker convergence condition. Finally, we present numerical results to demonstrate the usefulness and efficiency of the proposed methods.

A modified SOR-like method for absolute value equations associated with second order cones

In this paper, we propose a modified SOR-like method for solving absolute value equations associated with second order cones (SOCAVE in short), which is obtained by reformulating the SOCAVE as a two-by-two block nonlinear equation. The convergence analysis and error estimation of this method are established under mild assumptions on system matrix and iteration parameters. And, the optimal iteration parameters and the corresponding optimal convergence factor are studied. In particular, we present the approximate optimal iteration parameters which are independent of the number of iterations. Numerical results are given to show the efficiency of the proposed iteration method with suitable parameters.

Kaczmarz method for saddle point systems

The Kaczmarz method is presented for solving saddle point systems. The convergence is analyzed. Numerical examples, compared with classical Krylov subspace methods, SOR-like method (2001) and recent modified SOR-like method (2014), show that the Kaczmarz algorithm is efficient in convergence rate and CPU time.

THE MODIFIED ASSOR-LIKE METHOD FOR SADDLE POINT PROBLEMS

<abstract> In this paper, we established the modified accelerated symmetric SOR-like (MASSOR) method for solving the large sparse saddle point systems of linear equations. The convergence of the MASSOR method for solving saddle point problems is analyzed. Numerical examples are presented to show the effectiveness of the proposed method. </abstract>

Modified SOR-Like Method for Absolute Value Equations

In this paper, based on the work of Ke and Ma, a modified SOR-like method is presented to solve the absolute value equations (AVE), which is gained by equivalently expressing the implicit fixed-point equation form of the AVE as a two-by-two block nonlinear equation. Under certain conditions, the convergence conditions for the modified SOR-like method are presented. The computational efficiency of the modified SOR-like method is better than that of the SOR-like method by some numerical experiments.

SOR-Like New Iterative Method for Solving the Epidemic Model and the Prey and Predator Problem

Our aim in this paper is to propose an SOR-like new iterative method by introducing a relaxation parameter ω to improve the new iterative method proposed by Daftardar-Gejji and Jafari (NIM) [J. Math. Anal. Appl. 316 (2006) 753–763] in order to solve two problems. The first one is the problem of the spread of a nonfatal disease in a population which is assumed to have constant size over the period of the epidemic, and the other one is the problem of prey and predator. The proposed method is not limited to these two problems but can be applicable to a wide range of systems of nonlinear functional problem. The results, for different values of ω, show that we found some known methods and our method compared to methods using the calculation of special polynomials and derivatives like the Adomian decomposition method (ADM), the calculation of the Lagrange multiplier as in the variational iterative method (VIM), or the construction of a homotopy as in the homotopy perturbation method (HPM) has several advantages, such as very effective and very simple to implement. Unfortunately, these methods do not guarantee a valid approximation in large time interval. To overcome this, we applied our method for approximating the solution of the problems in a sequence of time intervals as a multistage approach. Some numerical results are presented with plots according to the parameter ω.

Generalized SOR-Like Iteration Method for Linear Complementarity Problem

In this paper, we present a generalized SOR-like iteration method to solve the non-Hermitian positive definite linear complementarity problem (LCP), which is obtained by reformulating equivalently the implicit fixed-point equation of the LCP as a two-by-two block nonlinear equation. The convergence properties of the generalized SOR-like iteration method are discussed under certain conditions. Numerical experiments show that the generalized SOR-like method is efficient, compared with the SOR-like method and the modulus-based SOR method.

A new SOR-like method for solving absolute value equations

In this paper, by constructing equivalently the absolute value equation (AVE) as the new block equation of two-by-two, we present a new SOR-like (NSOR) method for solving the AVE. The convergence and optimal parameters of NSOR method for solving the AVE are studied in detail. Numerical experiments are presented to show that the NSOR method is implementable and efficient.

Optimal accelerated SOR-like (ASOR) method for singular symmetric saddle point problems

In a recent paper a new iterative method for the solution of the nonsingular symmetric saddle point problem was proposed (Njeru and Guo, 2016). The ASOR method belongs to the family of the SOR-like methods and uses two parameters α and ω. Convergence intervals for the parameters involved were found. In the present work we analyze and study an extension of the above problem to the singular case, and determine optimal values for the two parameters as well as for the semi-convergence factor of the ASOR method. Numerical results are presented to show the efficiency of the optimal singular ASOR method.

A modified ASOR-like method for augmented linear systems

For solving an augmented linear system, Njeru and Guo presented an accelerated SOR-like (ASOR) method in (P. N. Njeru and X.-P. Guo. Accelerated SOR-like method for augmented linear systems, BIT Numerical Mathematics. 56, 557–571 2016). By further accelerating the ASOR method, we introduce another parameter and propose a modified ASOR method. We first characterize properties about eigenpairs of the iteration matrix, then discuss the convergence of the modified ASOR method. Numerical experiments are given to show the effectiveness of the modified ASOR method.

An accelerated symmetric SOR-like method for augmented systems

Recently, Njeru and Guo presented an accelerated SOR-like (ASOR) method for solving the large and sparse augmented systems. In this paper, we establish an accelerated symmetric SOR-like (ASSOR) method, which is an extension of the ASOR method. Furthermore, the convergence properties of the ASSOR method for augmented systems are studied under suitable restrictions, and the functional equation between the iteration parameters and the eigenvalues of the relevant iteration matrix is established in detail. Finally, numerical examples show that the ASSOR is an efficient iteration method.

Some generalizations of the new SOR-like method for solving symmetric saddle-point problems

Saddle-point problems arise in many areas of scientific computing and engineering applications. Research on the efficient numerical methods of these problems has become a hot topic in recent years. In this paper, we propose some generalizations of the new SOR-like method based on the original method. Convergence of these methods is discussed under suitable restrictions on iteration parameters. Numerical experiments are given to show that these methods are effective and efficient.

An Accelerated SOR-Like Method for Generalised Saddle Point Problems

An Accelerated SOR-Like Method for Generalised Saddle Point Problems

On equivalence of optimal relaxed block iterative methods for the singular nonsymmetric saddle point problem

There exist many classes of relaxed block iterative methods for the solution of the nonsingular and singular saddle point problems. Recently, the singular nonsymmetric saddle point problem has been optimally solved by means of a stationary linear second-order iterative method using the Manteuffel algorithm [Hadjidimos (2016) [19]]. The main purpose of this work is to extend, analyze and study a number of classes of stationary iterative methods based on generalizations of SOR-like methods, determine their optimal parameters, via the optimal parameters in the aforementioned work, and show the equivalence of the optimal methods studied. Finally, a computational comparison of the performances of the above optimal methods and their nonstationary counterparts shows the superiority of the latter methods.

SOR-like Methods with Optimization Model for Augmented Linear Systems

AbstractThere has been a lot of study on the SOR-like methods for solving the augmented system of linear equations since the outstanding work of Golub, Wu and Yuan (BIT 41(2001)71-85) was presented fifteen years ago. Based on the SOR-like methods, we establish a class of accelerated SOR-like methods for large sparse augmented linear systems by making use of optimization technique, which will find the optimal relaxation parameter ω by optimization models. We demonstrate the convergence theory of the new methods under suitable restrictions. The numerical examples show these methods are effective.

A generalized modified SOR-like method for the singular saddle point problems

Recently, Guo et al. proposed a modified SOR-like (MSOR-like) iteration method for solving the nonsingular saddle point problem. In this paper, we further prove the semi-convergence of this method when it is applied to solve the singular saddle point problems under suitable conditions on the involved iteration parameters. Moreover, the optimal iteration parameters and the corresponding optimal semi-convergence factor for the MSOR-like method are determined. In addition, numerical experiments are used to show the feasibility and effectiveness of the MSOR-like method for solving singular saddle point problems, arising from the incompressible flow problems.

Generalized ASOR and Modified ASOR Methods for Saddle Point Problems

Recently, the accelerated successive overrelaxation- (SOR-) like (ASOR) method was proposed for saddle point problems. In this paper, we establish a generalized accelerated SOR-like (GASOR) method and a modified accelerated SOR-like (MASOR) method, which are extension of the ASOR method, for solving both nonsingular and singular saddle point problems. The sufficient conditions of the convergence (semiconvergence) for solving nonsingular (singular) saddle point problems are derived. Finally, numerical examples are carried out, which show that the GASOR and MASOR methods have faster convergence rates than the SOR-like, generalized SOR (GSOR), modified SOR-like (MSOR-like), modified symmetric SOR (MSSOR), generalized symmetric SOR (GSSOR), generalized modified symmetric SOR (GMSSOR), and ASOR methods with optimal or experimentally found optimal parameters when the iteration parameters are suitably chosen.

Accelerated SOR-like method for augmented linear systems

By accelerating the SOR-like method by two parameters alpha and omega, we propose an accelerated SOR-like (ASOR) method for solving the augmented linear systems. The convergence of the ASOR method is discussed under suitable restrictions on the iteration parameters alpha and omega, and the experimental optimal values of the iteration parameters are determined. Numerical results are presented to show the efficiency of the ASOR method when the iteration parameters alpha and omega are suitably chosen.