Block AOR iteration for nonsymmetric matrices

Abstract:

We consider a class of matrices that are of interest to numerical applications and are large, sparse, but not symmetric or diagonally dominant. We give a criterion for the existence of (and we actually construct) the inverse matrix in terms of powers of a "small" matrix. We use this criterion to find that the spectral radius of the Jacobi iteration matrix, corresponding to a block tridiagonal partition, is in general $\geqslant 1$. We also derive conditions (that are satisfied in cases of interest to applications) for the Jacobi matrix to have spectral radius = 1. We introduce convergent "block AOR" iterative schemes such as extrapolated Jacobi and extrapolated Gauss-Seidel schemes with optimum (under) relaxation parameter $\omega = .5$. A numerical example pertaining to the solution of Poisson’s equation is given, as a demonstration of some of our hypotheses and results. A comparison with SOR, applied to the 5-point finite difference method, is also included.