M. Madalena Martins

The AOR iterative method for new preconditioned linear systems: 461

Abstract In this paper we apply the AOR method to preconditioned linear systems different from those considered in Evans and Martins (Internat. J. Comput. Math. 5 (1995) 69–76), Gunawardena et al. (Linear Algebra Appl. 154–156 (1991) 123–143) and Li and Evans (Technical Report No. 901, Department of Computer Studies, University of Loughborough, 1994). Our results show that some improvements in the convergence rate of this iterative method can be obtained.

On an accelerated overrelaxation iterative method for linear systems with strictly diagonally dominant matrix

We consider a linear system Ax = b of n simultaneous equations, where A is a strictly diagonally dominant matrix. We get bounds for the spectral radius of the matrix L rsl,which is accociated with the Accelerated Overrelaxation iterative method (AOR). Sufficient conditions for the convergence of that method will be given, which improve the results of Theorem 3, Section 4 of [2], applied to this type of matrices.

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

Considering the linear systems Ax = b, where the matrix A is irreducible and diagonally dominant, we obtain bounds for the spectral radius of the 4 ,, 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 X of the SOR method, and we improve the results of Theorems 5, 6 of [3] for the AOR method.

On the convergence of the modified overrelaxation method

Abstract The modified overrelaxation (MSOR) method is applied to a linear system Ax=b, where A has property A. We get bounds for the spectral radius of the iteration matrix of this method, and we achieve convergence conditions for the MSOR method when A is strictly diagonally dominant. We extend our conclusions to another kind of matrices—H,L,M or Stieltjes. In the last section we use the vectorial norms, getting convergence conditions for the MSOR method, when A is a block-H matrix. We also generalize a theorem of Roberts for this kind of matrices.

A variant of the AOR method for augmented systems

In this paper we present a variant of the Accelerated Overrelaxation iterative method (AOR), denoted by modified AOR-like method (MAOR-like method) for solving the augmented systems, i.e. the AOR-like method with three real parameters , r and . For special values of r ,  and  we get the MSOR-like method, the AOR-like method and the SOR-like method. An equation relating the involved parameters and the eigenvalues of the iteration matrix of the MAOR-like method is obtained. Furthermore, some convergence conditions for the MAOR-like method are derived. This paper generalizes the main results of Li, Li, Nie, and Evans 2004 and Shao, Li, and Li (2007). Numerical examples are presented to show that, for a suitable choice of the involved parameters, the MAOR-like method is superior when compared to the above iterative methods and to the SSOR-like method presented by Zheng, Wang, and Wu (2009).

Extended convergence regions for the AOR method

Abstract 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.

Generalized diagonal dominance in connection with the accelerated overrelaxation (AOR) method

With the definition of generalized diagonal dominant matrices we improve the known results about the intervals of convergence of the (AOR) method for linear systems. We consider this problem for different kinds of matrices and we get some important results forH-matrices.

A note on the convergence of the MSOR method

Abstract We enlarge the intervals of convergence of the MSOR method when the matrix A of Ax = b is an H -matrix, and we also correct one of the intervals given in an earlier paper.

Further results on the preconditioned sor method

Several methods have been developed on the preconditioned iterative methodsi. e iterative methods applied to preconditioned linear systems. Usui, Kohno and Niki [4] have proposed the adaptive Gauss-Seidel (GS) method, and the same authors [5] have presented the pre-conditioned SOR method. They have shown, with the aid of numerical examples, that the two methods have a better rate of convergence in comparison with the classical SOR method. In this paper we will prove theoretically the improvement in the rate of convergence.

An error bound for the SSOR and USSOR methods

Abstract We consider the SSOR and USSOR methods in order to approximate the solution of the linear system Ax = b . We establish a bound for the norm of e n = x − x n in terms of the norms of δ n = x n − x n − 1 , δ n + 1 , and their inner product, where x n is the n th iteration vector obtained using the USSOR method. Similar results are given for the SSOR method.