A generalized second order iterative algorithm for computing the Moore–Penrose inverse

Abstract

In this paper, we propose a generalized second order iterative algorithm for computing the Moore–Penrose inverse. The method arises from the second Penrose equation XAX = X and is further generalized by two real parameters. A detailed theoretical analysis is conducted to show that, under certain conditions, the new approach possesses linear, quadratic, and cubic convergence. As a result, various linear and quadratic convergence schemes can be extracted. Efficiency analysis of the method is considered to state its relation with respect to the condition number and number of iterations. We provide adequate examples to validate the new iterative scheme including matrices produced from real-life problems. Moreover, the applicability of method is also examine on one-dimensional heat problems. The convergence and error analysis, as well as the average CPU time analysis, are also given.