Euclidean Norm Minimization of the SOR Operators
APOSTOLOS HADJIDIMOS, Michael Neumann · SIAM Journal on Matrix Analysis and Applications · 1998
Because the spectral radius is only an asymptotic measure of therate of convergence of a linear iterative method, Golub and dePillis [ Toward an effective two-parameter method, in Iterative Methods for Large Linear Systems, Academic Press, New York, 1990] have raised in a recent paper the question of determining, for each $k\geq 1$, a relaxation parameter $\omega\in (0,2)$ and a pair of relaxation parameters $\omega_1$ and $\omega_2$ which minimize the Euclidean norm of the kth power of the SOR and MSOR iteration matrices, respectively, associated with a real symmetric positive definite matrix with "Property A." Here we use a reduction of these operators which they derived from the SVD 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.