Three-part splittings and Varga's type extensions of the successive overrelaxation (SOR) theory

Varga, in his excellent book [4] and in a later paper of his [5], extended the SOR theory in various directions by having considered the well known Ostrowski-Reich theorem as a starting point. In this paper we extend the theory by considering three-part splittings of Varga's type, where one of the basic parts is negative definite instead of being positive definite. Thus we are able to construct SOR-type schemes which converge for all the values of the overrelaxation parameter ω which do not belong to the familiar interval [0,2]. Then by following a similar but more complicated analysis, than that in [5], we are able to obtain the corresponding optimum schemes in the various possible cases.