Subproper and Regular Splittings for a Restricted Rectangular System

Let Ax = b be a rectricted rectangular and consistent linear system, where A is an m by n matrix and x is in a subspace T of C^n. The concept of subproper splitting A=M-N introduced by Neumann is generalized. A necessary and sufficient condition on the subproper splitting such that the iterative sequence converges to a solution of Ax = b is given. Monotonicity and the concept of regular subproper splitting are used to study convergence. Numerical examples are given to verify our conclusions