Further results on iterative methods for computing generalized inverses

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