Subproper splitting for rectangular matrices

AbstractIn this paper iterative schemes for approximating a solution to a rectangular but consistent linear system Ax = b are studied. Let A ϵ Cm × nr. The splitting A = M − N is called subproper if R(A) ⊆ R(M) and R(A∗) ⊆R(M∗). Consider the iteration xi = M†Nxi−1 + M†b. We characterize the convergence of this scheme to a solution of the linear system. When A ϵ Rm×nr, monotonicity and the concept of subproper regular splitting are used to determine a necessary and a sufficient condition for the scheme to converge to a solution