EUCLIDEAN NORM MINIMIZATION OF THE SOR OPERATORS

Because the spectral radius is only an asymptotic measure of the rate of convergence of a linear iterative method, Golub and dePillis [Towar d an effective two-parameter method, in Iterative Methods for Large Li near Systems, Academic Press, New York, 1990] have raised in a recent paper the question of determining, for each k greater than or equal to 1, a relaxation parameter omega epsilon (0,2) and a pair of relaxatio n parameters omega(1) and omega(2) which minimize the Euclidean norm o f the kth power of the SOR and MSOR iteration matrices, respectively, associated with a real symmetric positive definite matrix with ''Prope rty A''. Here we use a reduction of these operators which they derived from the SVD of the associated block Jacobi matrix to obtain the mini mizing relaxation parameters for the case k = 1 for both operators. We conclude the paper with two brief sections in which we assess what ou r results imply.