EXISTENCE AND UNIQUENESS OF SPLITTINGS FOR STATIONARY ITERATIVE METHODS WITH APPLICATIONS TO ALTERNATING METHODS

Given a nonsingular matrix A, and a matrix T of the same order, under certain very mild conditions, there is a unique splitting A = B - C, s uch that T = B-1C. Moreover, all properties of the splitting are deriv ed directly from the iteration matrix T. These results do not hold whe n the matrix A is singular. In this case, given a matrix T and a split ting A = B - C such that T = B-1C, there are infinitely many other spl ittings corresponding to the same matrices A and T, and different spli ttings can have different properties. For instance, when T is nonnegat ive, some of these splittings can be regular splittings, while others can be only weak splittings. Analogous results hold in the symmetric p ositive semidefinite case. Given a singular matrix A, not for all iter ation matrices T there is a splitting corresponding to them. Necessary and sufficient conditions for the existence of such splittings are ex amined. As an illustration of the theory developed, the convergence of certain alternating iterations is analyzed. Different cases where the matrix is monotone, singular, and positive (semi)definite are studied .