Generalized AOR Method Research Articles

On the Some New Preconditioned Generalized AOR Methods for Solving Weighted Linear Least Squares Problems

Recently, in the paper [Z.G. Huang, L.G. Wang, Z. Xu, J.J. Cui, Some new preconditioned generalized AOR methods for solving weighted linear least squares problems, Computational and Applied Mathematics, 37(2018) 415-438.], Huang et al, by using the generalized accelerated over-relaxation (GAOR) methods, proposed some new preconditioners for solving weighted linear least squares problems and discuss their comparison results. In this paper, we present a new model of GAOR methods to solve the weighted linear least squares problems. We prove that the new model is superior to the existing mentioned methods. Numerical examples are also reported to confirm our theoretical analysis.

Generalized AOR method for solving a class of generalized saddle point problems

In this paper, by constructing appropriate splitting, a generalized AOR (GAOR) method for solving a class of generalized saddle point problems has been presented, and the convergence of the method is studied. The numerical experiments results show the efficiency and robustness of the new method.

Some new preconditioned generalized AOR methods for solving weighted linear least squares problems

In this paper, we propose some new preconditioned GAOR methods for solving weighted linear least squares problems and discuss their comparison results. Comparison results show that the convergence rates of the new preconditioned GAOR methods are better than those of the preconditioned GAOR methods presented by Shen et al. (Appl Math Mech Engl Ed 33(3):375–384, 2012) and Wang et al. (J Appl Math, doi: 10.1155/2012/563586 , 2012) whenever these methods are convergent. Finally, numerical experiments are provided to confirm the theoretical results obtained in this paper.

On the iterative methods for weighted linear least squares problem

Purpose The Weighted Linear Least Squares (WLLS) problem has many different applications in sciences and engineering. The purpose of this paper is to introduce an iterative scheme for solving the WLLS problem. Design/methodology/approach By considering the splitting techniques in conjunction with Generalized Accelerated Over-relaxation (GAOR) method the authors design a new iterative method to solve the weighted linear least squares problem. Furthermore, within the computational framework, some models of iterative schemes candidates are investigated and evaluated. Findings In this paper, the authors propose an efficient iterative scheme for solving the WLLS problem. The proposed scheme presented promising results from the aspects of both convergence behavior and performance. Moreover, comparative results for the proposed schemes are also presented. Research limitations/implications Comparison between the new methods and other similar methods for the studied problem shows a remarkable agreement and reveals that the new model is much better in comparison with the existing methods in point of view rate of convergence and computing efficiency, as illustrated by the theoretical analysis and numerical results presented. Originality/value For solving WLLS more attention has recently been paid on a special class of splitting techniques with the preconditioned GAOR method. In this paper, the authors use a different splitting for the GAOR method and present a promising class of methods. The convergence results of the iterative algorithm are also proposed. Several examples are given to show the efficiency of the presented methods.

Some new preconditioned generalized AOR methods for generalized least-squares problems

In this paper, we present some new preconditioned GAOR methods for solving generalized least-squares problems and their comparison results. Comparison results show that the convergence rates of the new preconditioned GAOR methods are better than those of the preconditioned GAOR methods presented by Zhou et al. (2009) [19] and Wang et al. (2013) [18] whenever these methods are convergent. Lastly, numerical experiments are provided to confirm the theoretical results.

The matrix analogue of the scalar AOR iterative method

The Accelerated Overrelaxation (AOR) and the Generalized AOR (GAOR) iterative methods for the solution of linear systems of algebraic equations (Ax=b,A∈Cn×n,det(A)≠0,b∈Cn) have been around for about four decades and a plethora of variations of them have been proposed. In this work a novel algorithm is introduced, the Matrix Analogue of the AOR (MAAOR) iterative method, which is analysed and studied. The MAAOR method generalizes both the AOR and the GAOR. Sufficient convergence conditions for the GAOR method are determined when the coefficient matrix A of the linear system to be solved is a Hermitian matrix with positive diagonal elements. Similarly, sufficient convergence conditions for the MAAOR method are determined when A is a nonsingular H-matrix. The new convergence conditions are the most general ones so far. Numerical examples are presented in support of the theory developed.

Preconditioned generalized mixed-type splitting iterative method for solving weighted least-squares problems

Recently, Shen et al. [Preconditioned iterative methods for solving weighted linear least squares problems, Appl. Math. Mech.-Engl. Ed. 33(3) (2012), pp. 375–384] have considered four kinds of preconditioned generalized accelerated overrelaxation (GAOR) methods for solving the linear systems based on a class of weighted least-squares problems and examined their convergence rates. More recently, Yun [Comparison results on the preconditioned GAOR method for generalized least squares problems, Int. J. Comput. Math. 89 (2012), pp. 2094–2105] has focused on the same problem and suggested three different types of preconditioned GAOR methods and studied their convergence properties. In this paper, we first introduce the generalized mixed-type splitting (GMTS) iterative method for solving the linear systems corresponding to the weighted least-squares problems. The GMTS iterative method exploits auxiliary matrices L1 and D1 which gratify certain conditions. In order to improve the convergence rate of the GMTS method, different types of preconditioners are applied. In addition, the convergence of the (preconditioned) GMTS iterative methods is discussed. It is both theoretically and experimentally demonstrated that by appropriate choices of auxiliary matrices L1 and D1, the (preconditioned) GMTS method outperforms the (preconditioned) GAOR method.

Modified Preconditioned GAOR Methods for Systems of Linear Equations

Three kinds of preconditioners are proposed to accelerate the generalized AOR (GAOR) method for the linear system from the generalized least squares problem. The convergence and comparison results are obtained. The comparison results show that the convergence rate of the preconditioned generalized AOR (PGAOR) methods is better than that of the original GAOR methods. Finally, some numerical results are reported to confirm the validity of the proposed methods.

Comparison results on the preconditioned GAOR method for generalized least squares problems

Recently, Zhou et al. [Preconditioned GAOR methods for solving weighted linear least squares problems, J. Comput. Appl. Math. 224 (2009), pp. 242–249] have proposed the preconditioned generalized accelerated over relaxation (GAOR) methods for solving generalized least squares problems and studied their convergence rates. In this paper, we propose a new type of preconditioners and study the convergence rates of the new preconditioned GAOR methods for solving generalized least squares problems. Comparison results show that the convergence rates of the new preconditioned GAOR methods are better than those of the preconditioned GAOR methods presented by Zhou et al. whenever these methods are convergent. Lastly, numerical experiments are provided in order to confirm the theoretical results studied in this paper.

Generalized AOR Method for Solving Absolute Complementarity Problems

We introduce and consider a new class of complementarity problems, which is called the absolute value complementarity problem. We establish the equivalence between the absolute complementarity problems and the fixed point problem using the projection operator. This alternative equivalent formulation is used to discuss the existence of a solution of the absolute value complementarity problem. A generalized AOR method is suggested and analyzed for solving the absolute the complementarity problems. We discuss the convergence of generalized AOR method for the L‐matrix. Several examples are given to illustrate the implementation and efficiency of the method. Results are very encouraging and may stimulate further research in this direction.

Convergence of GAOR method for doubly diagonally dominant matrices

In this paper, we obtain bounds for the spectral radius of the matrix l ω, r which is the iterative matrix of the generalized accelerated overrelaxation (GAOR) iterative method. Moreover, we present one convergence theorem of the GAOR method. Finally, we present two numerical examples.

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.

Convergence of generalized AOR iterative method for linear systems with strictly diagonally dominant matrices

In this paper, some improvements on Darvishi and Hessari [On convergence of the generalized AOR method for linear systems with diagonally dominant coefficient matrices, Appl. Math. Comput. 176 (2006) 128–133] are presented for bounds of the spectral radius of l ω , r , which is the iterative matrix of the generalized AOR (GAOR) method. Subsequently, some new sufficient conditions for convergence of GAOR method will be given, which improve some results of Darvishi and Hessari [On convergence of the generalized AOR method for linear systems with diagonally dominant coefficient matrices, Appl. Math. Comput. 176 (2006) 128–133].

Generalized AOR methods for linear complementarity problem

In this paper, we firstly establish a class of generalized AOR (GAOR) methods for solving a linear complementarity problem LCP( M, q), whose special case reduces to generalized SOR (GSOR) method. Then, some sufficient conditions for convergence of the GAOR and GSOR methods are presented, when the system matrix M is an H-matrix, M-matrix and a strictly or irreducible diagonally dominant matrix. Moreover, when M is an L-matrix, we discuss the monotone convergence of the new methods. Lastly, we report some computational results with the proposed methods.

On convergence of the generalized AOR method for linear systems with diagonally dominant coefficient matrices

In this paper, we obtain bounds for the spectral radius of the matrix ℓ ω, r , which is the iterative matrix of generalized accelerated overrelaxation (GAOR) iterative method. We consider a linear system Hy = f, where H is an n × n diagonally dominant real matrix, f is an n-real vector of known values and y is an n-real vector of unknown variables. Sufficient conditions for the convergence of GAOR method will be presented.

Generalized AOR method for the augmented system

In this article, the accelerated over-relaxation (AOR) method is generalized for solving the saddle point problem or the augmented system. The successive over-relaxation (SOR)-like method is the special case of the generalized AOR (GAOR) method. The connection between the parameters and the eigenvalues of the iteration matrix of the GAOR method is given. Therefore, a necessary and sufficient condition for the convergence of the GAOR method is derived. Numerical examples are also given to show that the GAOR method is better than the SOR-like method in certain cases.

Convergence of the generalized AOR method

A generalized AOR (GAOR) method is proposed here to solve the linear system with some special structure. The convergence of the GAOR method is studied. The convergence result obtained here is an extension of the classical convergence result of the regular AOR method with weaker condition. In comparing with the generalized SOR (GSOR) method, we show that the GAOR method is better than the GSOR method under certain conditions. In some cases, the GAOR method is convergent even if the GSOR method does not converge. Finally, the GAOR method is applied to solve the generalized least-squares (LS) problems. Three numerical algorithms for the generalized LS problems are given.

On the convergence of the generalized AOR method

We consider the generalized AOR method presented by K. R. James. We give some convergence theorems for various kinds of matrices, namely when the coefficient matrix of the linear system is positive definite, an H-, L-, or M-matrix, or strictly or irreducibly diagonally dominant.

Convergence of parallel multisplitting methods for h-matrices

This paper investigates the convergence of parallel multisplitting methods for solving linear systems Ax = b, where A is an H-matrix. Sufficient conditions of convergence for general parallel methods are derived. A class of parallel generalized AOR methods, parallel block AOR methods and parallel AOR methods are proposed, and many convergence conditions are obtained. In case A is an M-matrix, we compare the asymptotic convergence rates of the parallel methods.

On the convergence of some generalized iterative methods

This paper is concerned with the generalized accelerated overrelaxation (GAOR) method, which constitutes a generalization of the basic iterative methods for the solution of linear systems. A number of new theoretical results are presented concerning the convergence theory of the GAOR method and special cases of it. Much attention is given to linear systems with positive definite coefficient matrices. Although the problem of determining the various optimum parameters of the GAOR method in the general case is very difficult to solve theoretically and is still an open one, the numerical examples which are presented in this paper show that a suitable exploitation of it may give much better results than the other basic iterative methods.