This paper improves on generalized properties of a family of iterative methods to compute the approximate inverses of square matrices originally proposed in . And while the methods of  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. © 2012 Elsevier Inc. All rights reserved.
Weiguo, L., Juan, L., & Tiantian, Q. (2013). A family of iterative methods for computing Moore-Penrose inverse of a matrix. Linear Algebra and Its Applications, 438(1), 47–56. https://doi.org/10.1016/j.laa.2012.08.004