On the Splittings for Rectangular Systems

Abstract

Recently, M. Hanke and M. Neumann$^{[4]}$ have derived a necessary and sufficient condition on a splitting of $A=U-V$, which leads to a fixed point system, such that the iterative sequence converges to the least squares solution of minimum 2-norm of the system $Ax=b$. In this paper, we give a necessary and sufficient condition on the splitting such that the iterative sequence converges to the weighted Moore-Penrose solution of the system $Ax=b$ for every $x_0\in C^n$ and every $b\in C^m$. We also provide a necessary and sufficient condition such that the iterative sequence is convergent for every $x_0\in C^n$.