Further results on iterative methods for computing generalized inverses

In this paper, we reconsider the iterative method Xk=Xk−1+βY(I−AXk−1), k=1,2,…,β∈C∖{0} for computing the generalized inverse AT,S(2) over Banach spaces or the generalized Drazin inverse ad of a Banach algebra element a, reveal the intrinsic relationship between the convergence of such iterations and the existence of AT,S(2) or ad, and present the error bounds of the iterative methods for approximating AT,S(2) or ad. Moreover, we deduce some necessary and sufficient conditions for iterative convergence to AT,S(2) or ad.