Abstract
The main aim of this paper is to compute the generalized inverse over Banach spaces by using semi-iterative method and to present the error bounds of the semi-iterative method for approximating .
1. Introduction
The semi-iterative method was originally inspired by the classical theory of summation and had been discussed by numerous authors [1, 2], which played an important role in many applications (see [3]). In 1962, Varga [3] defined the : for the vectors . The weights form an infinite lower triangular matrix This semi-iterative formula is used to solve the Drazin inverse solution of linear equations in Banach spaces in [1] and the solution of singular linear systems of algebraic equations in [4]. In 1996, Chen [5] defined the iterative formula: In 2010, Liu et al. [2] extended the iterative method to compute the generalized inverse over Banach spaces.
As we know, the iterative method in Liu et al. [2] can be used to compute the generalized inverse when . However, we can not apply it to compute the generalized inverse when . Therefor, it is necessary to study the semi-iterative method for computing the generalized inverse when . In this paper, we use the semi-iterative method to compute the generalized inverse over Banach spaces and present the error bounds of the semi-iterative method for approximating .
Now we list some notations used in this paper.
Let and be arbitrary Banach spaces. The symbol denotes the set of all bounded linear operators from to . Let . For any , we denote its range, null space, and norm by , , and , respectively. If , then we denote its spectrum and spectral radius by and . If and , then the restriction of on is defined by .
Let with . 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 on a Banach space, and we also give some conditions for the existence of semi-iterative convergence to the generalized inverse and its existence and estimate the error bounds of the semi-iterative method for approximating . In Section 3, we give an example for computing the generalized inverse when in our semi-iterative method.
The following lemmas are needed in what follows.
Lemma 1 (see [6, Section 4]). Let and be Banach spaces, , and and closed subspaces of and , respectively. Then the following statements are equivalent:(i)has a -inverse such that and .(ii) is a complemented subspace of , is closed, is invertible, and .In the case when (i) or (ii) holds, is unique and one denotes it by .
Lemma 2 (see [7, Section 3]). Suppose that the conditions of Lemma 2 are satisfied. If one takes , then holds and has the following matrix form: where is invertible. Moreover, has the following matrix form: Consequently,
Lemma 3 (see [2, Section 2]). Let and . Define the sequence in in the following way:
where and with . Then the iteration (7) converges if and only if ; equivalently, .
In this case, assume that and are closed subspaces of and , respectively. If and and , then exists and converges to , and when ,
2. Semi-Iterative Method for Computing Generalized Inverses
In the section, we will discuss semi-iterative method for computing generalized inverses . First, we deduce convergent conditions and error bounds of our semi-iterative method for computing generalized inverse .
Theorem 4. Let and . Define the sequence in in the following way: where and with . Then the semi-iteration (9) converges if and only if In this case, assume that and are closed subspaces of and , respectively. If and and , then exists and converges to , and when ,
Proof. From (7),
Let , , . Then we have
hence,
Let (). We have
From (15), we have
thus,
hence,
Therefore, we have
Denote , and , , . Then from (19), we have
So
Since and , then . On the other hand,
so . Since
then . As
and , then ; hence . Furthermore,
The proof is complete.
Similarly, we can obtain the following theorem.
Theorem 5. Let and . Define the sequence in in the following way: where and with . Then the semi-iteration (26) converges if and only if In this case, assume that and are closed subspaces of and , respectively. If and and , then exists and converges to , and when ,
3. Examples
We give an example for computing by the semi-iterative (9). Let the symbol denote the Frobenius norm.
Example 1. Consider the matrix
, , and . Obviously, and . Moreover, we choose
Therefore, for any , we have . If the scalar , it is easily verified that
The error estimate for diverse scalar and parameter of (25) and of (18) is given in Table 1.
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 .
Conflict of Interests
The authors declare that there is no conflict of interests regarding the publication of this paper.
Acknowledgments
This work was supported by the National Natural Science Foundation of China (11061005) and the Ministry of Education Science and Technology Key Project (210164).