Computing the Moore-Penrose inverse using its error bounds

Abstract

A new iterative scheme for the computation of the Moore-Penrose generalized inverse of an arbitrary rectangular or singular complex matrix is proposed. The method uses appropriate error bounds and is applicable without restrictions on the rank of the matrix. But, it requires that the rank of the matrix is known in advance or computed beforehand. The method computes a sequence of monotonic inclusion interval matrices which contain the Moore-Penrose generalized inverse and converge to it. Successive interval matrices are constructed by using previous approximations generated from the hyperpower iterative method of an arbitrary order and appropriate error bounds of the Moore-Penrose inverse. A convergence theorem of the introduced method is established. Numerical examples involving randomly generated matrices are presented to demonstrate the efficacy of the proposed approach. The main property of our method is that the successive interval matrices are not defined using principles of interval arithmetic, but using accurately defined error bounds of the Moore-Penrose inverse. © 2019 Elsevier Inc.