Semi-Iterative Method for Computing the Generalized Inverse A ( 2 ) T , S

and Applied Analysis 3

In 2010, Liu et al. [2] extended the iterative method to compute the generalized inverse  (2)  , over Banach spaces.As we know, the iterative method in Liu et al. [2] can be used to compute the generalized inverse  (2)  , when ( − ) < 1.However, we can not apply it to compute the generalized inverse  (2)  , when ( − ) = 1.Therefor, it is necessary to study the semi-iterative method for computing the generalized inverse  (2)  , when ( − ) = 1.In this paper, we use the semi-iterative method to compute the generalized inverse  (2)  , over Banach spaces and present the error bounds of the semi-iterative method for approximating  (2)  , .Now we list some notations used in this paper.Let X and Y be arbitrary Banach spaces.The symbol B(X, Y) denotes the set of all bounded linear operators from X to Y. Let B(X) := B(X, X).For any  ∈ B(X, Y), we denote its range, null space, and norm by R(), N(), and ‖‖, respectively.If  ∈ B(X), then we denote its spectrum and spectral radius by () and ().If  ∈ B(X, Y) and  ⊂ X, then the restriction |  of  on  is defined by   → ,  ∈ .
Let ,  ⊂ X with  ⊕  = X.Denote  , by the projection from  onto .
The paper is organized as follows.Some lemmas will be presented in the remainder of this section.In Section 2, we reconsider the method to compute the generalized inverse  (2)  , on a Banach space, and we also give some conditions for the existence of semi-iterative convergence to the generalized inverse  (2)  , and its existence and estimate the error bounds of the semi-iterative method for approximating  (2)  , .In Section 3, we give an example for computing the generalized inverse  (2)  , when ( − ) = 1 in our semi-iterative method.
In the case when (i) or (ii) holds,  is unique and one denotes it by  (2)  , .

Semi-Iterative Method for Computing
Generalized Inverses  (2)   , In the section, we will discuss semi-iterative method for computing generalized inverses  (2)  , .First, we deduce convergent conditions and error bounds of our semi-iterative method for computing generalized inverse  (2)  , .
Similarly, we can obtain the following theorem.

Examples
We give an example for computing  (2)  , by the semi-iterative (9).Let the symbol ‖ ⋅ ‖ denote the Frobenius norm.
Table 1 shows that within limits the larger the parameters  and , the smaller the error bounds.But  can not be infinitely large, because when  is large enough, the error bounds are large as well.So we must choose the best  and .