Optimum Accelerated Overrelaxation (AOR) Method for Systems with Positive Definite Coefficient Matrix

An iterative method applied to nonsymmetric linear systems

“…We denote TG = --(D-L)-1U as the Gauss-Seidel iteration matrix for scheme (2) and Tf --89 -(D -L)-IU) for the iteration matrix of scheme (4). Then the following relation is obtained:…”

“…Varga [1] proposed a hybrid scheme that is nearly optimal. On the other hand, N. Gaitanos, J. Hadjidimos and A. Yeyios [2] reported an AOR method applied to the skew Hermitian splitting. These methods with an optimum extrapolation parameter are able to give a maximum asymptotic convergence rate.…”

A new generalized AOR iterative method for solving linear systems

“…This method is well-defined even when some elements on the diagonal of A are zero. Some very interesting results concerning the GAOR method were given in [2], [3], [5], [6], [7], [11], [15]. Authors showed that with spacial conditions, the GAOR iterative method converges when A is an M -matrix or is a Hermitian positive definite matrix.…”

“…For Hermitian positive definite coefficient matrix A, in [2] the author considered the splitting A = D − E − E H in which D is any Hermitian positive definite matrix and the generalized AOR method…”

Note To The Mixed-Type Splitting Iterative Method For The Positive Real Linear System

“…Therefore it is shown in this note the AOR method is convergent for the proper choice of the overrelaxation parameters o and t. This result about the AOR method seems new to our knowledge. For the basic results on consistently ordered matricies, M-Matrices, H-Matrices, and symmetric positive definite matrices in connection with AOR method, please see [1][2][3][4]7]. Before this section is ended, we note that for the above iterative method to be of practical value, D 1 must be chosen so that…”