Abstract
We explore iterative schemes for obtaining a solution to the linear system ( ∗) Ax = b, A ϵ C m × 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 − Q 1 − Q 2. 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.
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