An iterative method for computing the approximate inverse of a square matrix and the Moore–Penrose inverse of a non-square matrix

In this paper, an iterative scheme is proposed to find the roots of a nonlinear equation. It is shown that this iterative method has fourth order convergence in the neighborhood of the root. Based on this iterative scheme, we propose the main contribution of this paper as a new high-order computational algorithm for finding an approximate inverse of a square matrix. The analytical discussions show that this algorithm has fourth-order convergence as well. Next, the iterative method will be extended by theoretical analysis to find the pseudo-inverse (also known as the Moore–Penrose inverse) of a singular or rectangular matrix. Numerical examples are also made on some practical problems to reveal the efficiency of the new algorithm for computing a robust approximate inverse of a real (or complex) matrix.