Alternating Direction Iteration Methods For n Space Variables

By J. Douglas, Jr., R. B. Kellogg, and R. S. Varga We consider the iterative solution of the system of linear equations (1 ) (Xi + X-, + • ■ ■ + Xn)z = /, n 2; 2, where each Xj, 1 | j | n, is a Hermitian and positive definite N X N matrix. If n = 2, the iterative methods of Peaceman-Rachford [1, Chapter 7], or D'yakonov [2] and Kellogg [3], may be used to solve (1). In this paper these methods are generalized to n ^ 2, and are shown, in a sense, to be dual to one another. Let p > 0 be fixed, and define z¡ = (pi + Xj)z. From (1) we get the compound nN X nN matrix equation