Convergence domains of the SSOR method for a class of generalized consistently ordered matrices

Abstract

The functional equation relating the eigenvalues of the block Symmetric Successive Overrelaxation (SSOR) iteration matrix with those of the block Jacobi iteration matrix found by Chong and Cai (1985) is used in order to obtain precise domains of convergence of the block SSOR iteration method associated with a class of generalized consistently ordered (GCO) ( k, p− k)-matrices A( p⩾2, k=1, …, p−1). We show that the domain of convergence depends only on the relaxation parameter ω, the spectral radius of the block Jacobi iteration matrix and the value of l = k p . Unlike the case ( k, p− k) = (1, p−1), p⩾3, which was studied in an earlier paper by the authors (1989), beyond certain critical values of l the domain of convergence does not grow monotonically with l and manifests various sorts of behavior. However, as in the case ( k, p− k) = (1, p−1), it is shown that the intersection of the convergence domains taken over all pairs ( p, k) coincides with the exact domain of convergence of the point SSOR iteration method associated with nonsingular H-matrices A, that is, matrices which can be diagonally scaled to being strictly diagonally dominant.