Abstract
We investigate two iterative methods for computing the DMP inverse. The necessary and sufficient conditions for convergence of our schemes are considered and the error estimate is also derived. Numerical examples are given to test the accuracy and effectiveness of our methods.
1. Introduction
Let be the set of all complex matrices with rank . For any given a matrix , let , , and be the range space, the null space, and the Frobenius norm of matrix , respectively. For a nonnegative integer , if , then is called the index of and . In recent years, the generalized inverse has been applied in many fields of engineering and technology, such as control[1], the least squares problem [2, 3], matrix decomposition [4], image restoration, statistics (see [5]), and preconditioning [6–8]. In particular, -inverse plays an important role in stable approximations of ill-posed problems (see [1, 9]) and in linear and nonlinear problems [6, 10]. In [11], Baksalary and Trenkle investigate the core inverse.
For a given matrix , there exists matrix satisfying (see [8])(1),(2),(3),(4),
where is called the Moore-Penrose inverse of , denoted by , and it is unique. For a given matrix , there exists a matrix satisfying(1),(2),(3),
where is called the Drain inverse of , denoted by , and it is unique. Based on the Drazin inverse and the Moore-Penrose (MP) inverse, a new generalized inverse is defined in [12] as (see also [13, 14]): for a matrix , there is matrix satisfying (see [12])(1),(2),(3),
where is called DMP inverse, denoted by , and it is unique. It is shown that in [12]. In [15], Yu and Deng get some characterizations of DMP inverse in a Hilbert space. By using idempotent element, some new properties of DMP inverse are given in [16].
So far, there are few results on computation of the DMP inverse by the iterative methods given in [17–22]. Recently, a family of higher-order convergent iterative methods are developed in [23] and applied to compute the Moore-Penrose inverse; the method is extended to compute the generalized inverse in [20]. In this paper, we develop two iterative methods to compute the DMP inverse of a given matrix . The proposed method (I) is higher-order and the proposed method (II) can be implemented easily.
The paper is organized as follows. The proposed iterative methods for computing DMP inverse are given and some lemmas used for its convergence analysis are given in Section 2. The stability and convergence analysis of our scheme (1) and (4) are given, and numerical examples are given to test the corresponding theoretical results in Sections 3 and 4, respectively.
2. Preliminaries and Iterative Scheme
Lemma 1 (see [23]). If and , then(i),(ii).
Lemma 2 (see [12]). Let , . If exist, then(i),(ii).
As in [20], we develop a iterative scheme to compute the DMP inverse as follows.
Scheme I:where , , and . Following the line [21], we develop the iterative following scheme as Let and, by , we have Thus, an efficient high-order iterative method can be written as follows.
Scheme II:The iterative method given in (4) is applied to compute the Drazin inverse by [21]. Here, we use the sequence of iterative to compute the DMP inverse.
3. Scheme I for the DMP Inverse
In this section, we consider the numerical analysis of Scheme I (1) and present a numerical example to test our numerical theoretical results.
3.1. Stability and Convergence Analysis
Theorem 3. Let with . For an arbitrary positive integer , the sequence (1) converges if only if . Moreover, we have
Proof. Let for convenience. If , , then . From Lemma 2, we attain .
Now, we test by using mathematical induction. Assume that for any positive integer . By Lemma 2, we have Let ; thenSimilarly, by Lemma 2 and (7), we derive and Thus, we have Next, we investigate the necessary and sufficient condition for convergent property of Scheme I (1). Assume that the sequence converges to . Thus, , while . Therefore, we have .
Conversely, let for some scalar . Then . Thus, we have and
As in [24], we show that Scheme I (1) is asymptotically stable as the following result.
Theorem 4. Let and let the sequence be generated by (1) with an initial If , , and , then Scheme I (1) is asymptotically stable.
Proof. Let be the numerical perturbation of in Scheme I (1). Thus, it can be written into as . Here, we perform a first-order error analysis; that is, we formally neglect quadratic or higher terms. The manipulation is meaningful, while is sufficiently small. Further, we have Let ; we have Similarly, we have andBy (13) and , we derive By (7), we obtain Thus, we derive We can conclude that the perturbation at the iterate is bounded. Therefore, the sequence generated by (1) is asymptotically stable.
3.2. Numerical Example
Here is an example for computing DMP inverse in the iterative method (1).
Example 1. Let where and . Thus, . To test the high accuracy and efficiency of Scheme I (1), the DMP inverse of is given as Here, we apply Scheme I given in (1) to compute the DMP inverse with . The errors and are given in Table 1. From the numerical results note that (1) converges to and it has high-order accuracy.
4. Scheme II for the DMP Inverse
Here, the numerical analysis of Scheme II is derived and a numerical example is given to test our numerical theoretical results. Note that it is difficult to construct a projection given in Theorems 3 and 4 with satisfying .
4.1. Stability and Convergence Analysis
Theorem 2. Let with ; the sequence defined by Scheme II. If initial approximation satisfies and , then Scheme II converges to and
Proof. Similar to the proof of Theorem 9 in [21], let for ; thenThus, we have Since , we obtain that Similarly, . So we can get the conclusion , .
Since , , we attain . By Lemma 2, we have and So, we have
If the condition is satisfied, then when . In other words, , the scheme satisfies the following error estimate:
Theorem 3. Let be a singular square matrix with and the sequence defined by Scheme II. If the initial approximation is chosen such that and , then the order of convergence of the sequence is nine.
Proof. Let and ; we have .
Now, let ; we have Since DMP inverse is a special of -inverse, by citing the proof of Theorem 3.1 in [25], we have and and ).
Since when , we have .
It would be easy to find the error inequality of the high-order iterative as follows: Thus, the sequence converges to DMP inverse and the convergence order is nine.
It is easy to find with satisfying . So the method to compute DMP inverse is more convenient than another.
In what follows, we investigate the stability of the iterative method (4).
Theorem 4. Let ; the sequence defined by Scheme II with the initial approximation satisfies that is asymptotically stable for computing DMP inverse.
Proof. Let be the numerical perturbation introduced in Scheme II. Next, the modified valueappears instead of the exact value . Here, we formally neglect quadratic or higher terms such as . This formal manipulation is meaningful if is sufficiently small and further has , then, , when , .
Simplifying (28) we attain Using the matrix identity, we have , then We can conclude that the perturbation at the iterate is bounded. Therefore, the sequence generated by Scheme II is asymptotically stable.
4.2. Numerical Example
The numerical examples are worked out by using high level language Matlab R2013a on an Intel(R) core running on Windows 10 Professional Version.
Example 1. Let Take where . Thus, . To test the efficiency and accuracy of our scheme, we present the DMP inverse of as In Table 2, we give the errors , . The results show that the proposed method (4) converges to and has high-order accuracy.
5. Conclusions
We have developed two iterative methods for computing the DMP inverse. The proposed scheme has high-order accuracy and Scheme II can be implemented without constructing the projection . The stability, convergence analysis, and the error estimate of our schemes are given. Numerical examples show that our schemes have high-order accuracy and effectiveness. It is more interesting that we shall extend these methods to compute other generalized inverse, such as -generalized inverse [18, 20, 22].
Conflicts of Interest
No potential conflicts of interest were reported by the authors.
Acknowledgments
This work is supported partially by the National Natural Science Foundation of China (Grant no. 11361009), the Special Fund for Scientific and Technological Bases and Talents of Guangxi (Grant no. 2016AD05050), and the Special Fund for Bagui Scholars of Guangxi.