SOME NOTES ON MULTISPLITTING METHODS AND M-STEP PRECONDITIONERS FOR LINEAR-SYSTEMS

To solve the real nonsingular linear system Ax = b on parallel and vec tor machines, multisplitting methods and m-step preconditioners are co nsidered. In this work, in particular: (1) We extend Adams and Ong's i dea and theory of the m-step additive preconditioners and Ruggiero and Galligani's method of the arithmetic mean, and we generalize the corr esponding convergence results. (2) We determine suitable relaxed m-ste p preconditioners and solve the problem of minimizing the related cond ition number, with respect to the relaxation (extrapolation) parameter involved. (3) Finally, we complete the theoretical solution of the pr oblem of determining the optimum SOR additive iterative method for 2-c yclic positive definite matrices.