Some recent results on the modified SOR theory

Abstract

For the solution of the linear system Ax=( I− T) x= c (1), where T is a weakly cyclic of index p⩾2 matrix of a special structure, the block SOR method with different relaxation factors associated with the row blocks of A is considered. First, a well-known relationship connecting the eigenvalue spectra of the block Jacobi matrix T of A and its associated modified SOR (MSOR) matrix is proved for all p⩾3, via an approach due to Varga, Niethammer, and Cai. Next, it is shown that the matrix analogue of the eigenvalue relationship holds at least for p=3. This, together with the facts that the matrix analogue holds true for p=2 and also for any p⩾3, provided all relaxation factors coincide, suggests a more general validity of the aforementioned matrix identity. Then, based on the matrix relationship, an equivalence is established between the MSOR method and a stationary p-parameter p-step iterative one for the solution of (1). So one can study convergence properties and determine optimal parameters of both methods by studying the simpler of the two, that is, the p-step one. Finally, some applications of the theory developed when p=2 are presented, and a brief discussion concerning comparisons of the optimum MSOR and AOR methods in a specific case is given.