A. Hadjidimos

Accelerated Overrelaxation Method

This paper describes a method for the numerical solution of linear systems of equations. The method is a two-parameter generalization of the Successive Over- relaxation (SOR) method such that when the two parameters involved are equal it co- incides with the SOR method. Finally, a numerical example is given to show the su- periority of the new method. 1. Introduction. For the numerical solution of linear systems, numerous direct as well as indirect methods exist. Among the indirect or iterative methods the Succes- sive Overrelaxation (SOR) and related methods play a very important role and are the most popular ones. These methods are fully covered in the excellent books by Varga (1), by Wachspress (2) and in the most recent one by Young (3). The purpose of this paper is to present a two-parameter generalization of the SOR method and also the first basic results concerning this method which has been called Accelerated Overrelaxation (AOR) method. As will be seen, the well-known methods of Jacobi, of Gauss-Seidel, of Simultaneous Overrelaxation and of Successive Overrelaxation can be derived, as special cases, from the AOR method. Finally a characteristic numerical example, which we give in a special case, shows the superiority of the AOR method. 2. Derivation of the AOR Method. We consider a system of N linear equations with N unknowns written in matrix form

The principle of extrapolation in connection with the accelerated overrelaxation method

Abstract This paper makes use of the extrapolation principle to improve and extend most of the theoretical results concerning the accelerated overrelaxation (AOR) method, as well as to discover some new ones.

More on modifications and improvements of classical iterative schemes for M-matrices

In the last four decades many articles have been devoted to the modifications and improvements of classes of preconditioners for linear systems whose matrix coefficient is an M-matrix in order to improve on the convergence rates of the classical iterative schemes (Jacobi, Gauss–Seidel, etc.). The present work is a contribution towards the generalization of the most common preconditioners used so far.

Successive overrelaxation (SOR) and related methods

Abstract Covering the last half of the 20th century, we present some of the basic and well-known results for the SOR theory and related methods as well as some that are not as well known. Most of the earlier results can be found in the excellent books by Varga (Matrix Iterative Analysis, Prentice-Hall, Englewood Cliffs, NJ, 1962) Young (Iterative Solution of Large Linear systems, Academic Press, New York, 1971) and Berman and Plemmons (Nonnegative Matrices in the Mathematical Sciences, SIAM, Philadelphia, PA, 1994) while some of the most recent ones are given in the bibliography of this paper. In this survey, both the point and the block SOR methods are considered for the solution of a linear system of the form Ax=b, where A∈ C n,n and b∈ C n ⧹{0}. Some general results concerning the SOR and related methods are given and also some more specific ones in cases where A happens to possess some further property, e.g., positive definiteness, L-, M-, H-matrix property, p-cyclic consistently ordered property etc.

Optimum accelerated overrelaxation method in a special case

Absbact. In this paper we give the optimum parameters for the Accelerated Overrelaxation (AOR) method in the special case where the matrix coefficient of the linear system, which is solved, is consistently ordered with nonvanishing diagonal elements. Under certain assumptions, concerning the eigenvalues of the corresponding Jacobi matrix, it is shown that the optimum AOR method gives better convergence rates than the optimum SOR does, while in the remaining cases the optimum AOR method coincides with the optimum SOR one.

On Iterative Solution for Linear Complementarity Problem with an

The numerous applications of the linear complementarity problem (LCP) in, e.g., the solution of linear and convex quadratic programming, free boundary value problems of fluid mechanics, and moving boundary value problems of economics make its efficient numerical solution a very imperative and interesting area of research. For the solution of the LCP, many iterative methods have been proposed, especially, when the matrix of the problem is a real positive definite or an

H_{+}

H_{+}

-Matrix

-matrix. In this work we assume that the real matrix of the LCP is an

The optimal solution of the extrapolation problem of a first order scheme

H_{+}

On the equivalence to the k -step iterative Euler methods and successive overrelaxation (SOR) methods for k -cyclic matrices

-matrix and solve it by using a new method, the scaled extrapolated block modulus algorithm, as well as an improved version of the very recently introduced modulus-based matrix splitting modified AOR iteration method. As is shown by numerical examples, the two new methods are very effective and competitive with each other. (A corrected PDF is attached to this article.)