AOR Method Research Articles

Matrix multi-splitting multi-parameter relaxation methods

In this paper, we propose a class of matrix multisplitting multiparameter relaxation methods, for solving large nonsingular systems of equations. This new class of method includes the well known matrix multisplitting relaxation methods such as the matrix multisplitting SOR, methods as well as the extrapolated matrix multisplitting AOR method as its special cases, as well as the matrix multisplitting SSOR and SAOR methods. It therefore forms a series of relaxation methods in the sense of matrix multisplitting which affords more flexible choices for practical application and also makes the parallel computation of serial relaxation methods become possible. The convergence theory of this new class of methods is established under the condition that the coefficient matrix of the system of equations is an H-matrix.

An error bound on the AOR method for solving linear systems

We investigate the numerical stability, for the symmetric positive definite and consistently ordered matrix A, of the AOR iterative method to obtain the solution of the linear system Ax = b. We give an error bound ∥ε n∥ 2⩽ 1 D 2 (ω−1) 2+μ 2∣ω(r−ω)∣ 2∥δ n∥ 2−2(ω−1) 2〈δ n,δ n+1〉+2μ 2∣ω(r−ω)∣〈∣δ n∣,∣δ n+1∣〉+∥δ n+1∥ 2 for the error estimation.

Comparisons of nonnegative splittings of matrices

The comparison of the asymptotic rates of convergence of two iteration matrices induced by two splitting of the same matrix has arisen in the work of many authors. In this article, some new comparison theorems for two nonnegative splittings (a splitting A= M− N is nonnegative if M −1 exist and M −1 N is nonnegative) are derived. They extend the known results in the literature. In addition, we also point out three incorrect conditions in a paper by Beauwens. Furthermore, we give some reasonable conditions ensuring the strict inequality between the asymptotic convergence rates. This also answers the open question which additional and appropriate conditions should be imposed on Miller-Neumann splittings to obtain strict inequality. Finally, some applications to a class of generalized AOR, SOR, and JOR iterative methods whose special cases imply block (also point) AOR, SOR, and JOR iterative methods for solving linear systems are discussed.

Some recent results on the modified SOR theory

For the solution of the linear system Ax=( I− T) x= c (1), where T is a weakly cyclic of index p⩾2 matrix of a special structure, the block SOR method with different relaxation factors associated with the row blocks of A is considered. First, a well-known relationship connecting the eigenvalue spectra of the block Jacobi matrix T of A and its associated modified SOR (MSOR) matrix is proved for all p⩾3, via an approach due to Varga, Niethammer, and Cai. Next, it is shown that the matrix analogue of the eigenvalue relationship holds at least for p=3. This, together with the facts that the matrix analogue holds true for p=2 and also for any p⩾3, provided all relaxation factors coincide, suggests a more general validity of the aforementioned matrix identity. Then, based on the matrix relationship, an equivalence is established between the MSOR method and a stationary p-parameter p-step iterative one for the solution of (1). So one can study convergence properties and determine optimal parameters of both methods by studying the simpler of the two, that is, the p-step one. Finally, some applications of the theory developed when p=2 are presented, and a brief discussion concerning comparisons of the optimum MSOR and AOR methods in a specific case is given.

Convergence analysis of the modified sor (MSOR) method

In this paper we present an analysis of the convergence of the MSOR method with fixed parameters, for solving a linear system of equations Ax=b, where A belongs to a subclass of consistently ordered matrices. The optimum parameters and the optimum virtual spectral radius of the MSOR method are also obtained. These results extend those of Young [5, pp. 277-281]. Finally, a comparison of the optimum MSOR with the optimum SOR and AOR methods is made, showing the superiority of MSOR.

The AOR method for solving linear interval equations

This paper is motivated by the paper [7], where the SOR method for solving linear interval equations was considered. It is known that sometimes the AOR method for systems of linear (“point”) equations converges faster than the SOR method. We give some sufficient conditions for the convergence of the interval AOR method for the same class of interval matrices which are considered in [7].

Optimum extrapolated method in a special case with application to sor method

Given a linear system Ax = b and an iterative method x (m + 1) = Gx (m)+k, m = 0,1,2,…(1) to solve it, we determine analytically the optimum extrapolation factor of the extrapolated method of (1), when all the eigenvalues of G have the same modulus (Section 2). Then using the SOR theory in the case of consistently ordered matrices A and applying the results of Section 2 to the extrapolated SOR (ESOR) method, we show (Section 3), that the globally optimum parameters of it (and also of the AOR method) [2, 11, 14] are recovered.

Extended convergence regions for the AOR method

In this paper, we give sufficient conditions for the convergence of the (AOR) method, when the matrix A for Ax = b is a strictly diagonally dominant matrix. These results improve the conclusions obtained in the Theorem 4 [10]. With the notion of generalized diagonal dominant matrix, we enlarge the convergence regions given in Theorem 9 [10], when A is a nonsingular H-matrix. In the last section we generalize theorem 6 of Robert [11] and we present some results which extend the convergence regions for the (AOR) method.

A survey of AOR and TOR methods

This paper makes a survey on the AOR and TOR methods. We present the recent results about convergence and determination of the optimum parameters, mainly those obtained by Chinese authors.

New Parallel Iterative Schemes And The Aor Method

On presente une famille de schemas iteratifs paralleles pour la solution des systemes d'equations lineaires. On etablit les conditions de consistance et de convergence

Note on irreducible diagonally dominant matrices and the convergence of the AOR iterative method

Considering the linear systems A x = b Ax = b , where the matrix A is irreducible and diagonally dominant, we obtain bounds for the spectral radius of the L r , ω {L_{r,\omega }} matrix of the AOR method and we achieve the convergence conditions given in [2] by a different method. If A is strictly diagonally dominant, we get larger intervals for the parameter ω \omega of the SOR method, and we improve the results of Theorems 5, 6 of [3] for the AOR method.