3-part splittings for singular and rectangular linear systems

AbstractWe explore iterative schemes for obtaining a solution to the linear system (∗) Ax = b, A ϵ Cm × n, if the system is solvable, or for obtaining an approximate solution to (∗) if the system is not solvable. Our iterative schemes are obtained via a 3-part splitting of A into A = M − Q1 − Q2. The 3-part splitting of A is, in turn, a refinement of a (2-part) subproper splitting of A into A = M − Q. We indicate the possible usefulness of such refinements (of a 2-part splitting of A) to systems (∗) which arise from a discrete analog to the Neumann problem, where the conventional iterative schemes (i.e., iterative schemes induced by a 2-part splitting of A) are not necessarily convergent