On the Minimization of the l(2)-norms of the SOR and MSOR Operators

Michael Neumann t Department of Mathematics University of Connecticut . Storrs, CT 06269-3009. Because the spectral radius is only an asymptotic measure of the rate of convergence of a linear iterative method, Golub and dePillis have raised in a recent paper the question of determining, for each k ~ 1, a relaxation parameter w E (0,2) and a pair of relaxation parameters WI and W2 which minimize the £2-norm of the k-th power of the SOR and MSOR iteration matrices, respectively, associated with a real symmetric positive definite matrix with property "Au. Here we use a reduction of these operators which they derived from the singular value decomposition of the associated block Jacobi matrix to obtain the minimizing relaxation parameters for the case k = 1 for both operators. We conclude the paper with two brief sections in which we assess what our results imply. Subject Classification: AMS(MOS): 65FlO, CR Category: 5.14