Some notes on multisplltting methods and m-step preconditioners for linear systems Original Research Article

To solve the real nonsingular linear system Ax = b on parallel and vector machines, multisplitting methods and m-step preconditioners are considered. In this work, in particular: (1) We extend Adams and Ongʹs idea and theory of the m-step additive preconditioners and Ruggiero and Galliganiʹs method of the arithmetic mean, and we generalize the corresponding convergence results. (2) We determine suitable relaxed m-step preconditioners and solve the problem of minimizing the related condition number, with respect to the relaxation (extrapolation) parameter involved. (3) Finally, we complete the theoretical solution of the problem of determining the optimum SOR additive iterative method for 2-cyclic positive definite matrices.