Some convergence results related to the JOR iterative method for symmetric, positive-definite matrices

摘要

The Jacobi method for iteratively solving a set of linear algebraic equations is well known. However, it suffers from the drawback that it does not converge for all linear algebraic systems. In science and engineering, one often encounters algebraic systems of the form Ax = b where the matrix A is symmetric and positive- definite. In this paper we show that a slight variant of the Jacobi method, namely the Jacobi overrelaxation (JOR) method, can be made to ensure convergence of the iterative scheme for such matrices. Thus we expand the applicability of JOR iterative methods to all symmetric, positive-definite matrices in a manner which is computationally convenient and simple. In particular, we show that this variant scheme can be explicitly determined without the need for any eigenvalue computations. We show that if eigenvalue computations are employed, one can obtain the maximum rate of convergence by a proper choice of parameters for the JOR scheme. This maximum rate of convergence is also explicitly obtained.