A family of iterative methods for computing Moore–Penrose inverse of a matrix

Abstract

This paper improves on generalized properties of a family of iterative methods to compute the approximate inverses of square matrices originally proposed in [1]. And while the methods of [1] can be used to compute the inner inverses of any matrix, it has not been proved that these sequences converge (in norm) to a fixed inner inverse of the matrix. In this paper, it is proved that the sequences indeed are convergent to a fixed inner inverse of the matrix which is the Moore–Penrose inverse of the matrix. The convergence proof of these sequences is given by fundamental matrix calculus, and numerical experiments show that the third-order iterations are as good as the second-order iterations.