Subproper and Regular Splitting for Restricted Rectangular Linear System

Iterative schemes for approximating a solution to restricted rectangular but consistent linear system of equations Ax=b, x∈T, are considered. The methods are based upon so-called subproper splitting A=M−N, which is a generalization of the concept of subproper splitting introduced by Neumann and studied further by Berman and Neumann and others. We give a necessary and sufficient condition on the splitting such that the iterative sequence converges to a solution of Ax=b in the case of b∈AT for every x0, where A∈Cm×n and T is a subspace of Cn. Monotonicity and the concept of regular subproper splitting are used to determine a necessary and a sufficient condition for the convergence of the iterative scheme. Finally, we present two numerical examples to verify our conclusions