Computational methods for generalized inverse matrices arising from proper splittings

AbstractA proper splitting of a rectangular matrix A is one of the form A = M − N, where A and M have the same range and null spaces. This concept was introduced by R. Plemmons as a means of generalizing to rectangular and singular matrices the concept of a regular splitting of a nonsingular matrix as introduced by R. Varga. In consideration of the linear system Ax=b, A. Berman and R. Plemmons used a proper splitting of A into M − N and showed that the iteration x(i+1)=M+Nx(i)+M+b converges to A+b, the best least-squares solution to the system, if and only if the spectral radius of M+N is less than one. The purpose of this paper is to further develop the characteristics of proper splittings and to extend these previous results by replacing the Moore-Penrose generalized inverse with a least-squares g-inverse, a minimum-norm g-inverse, or a g-inverse. Also, some criteria are given for comparing convergence rates of Mi−Ni, where A = M1−N1 = M2−N2, and a method is developed for constructing proper splittings of special types of matrices