Journal of Mathematics

Volume 2018, Article ID 8175935, 6 pages

https://doi.org/10.1155/2018/8175935

## High-Order Iterative Methods for the DMP Inverse

^{1}School of Science, Guangxi University for Nationalities, Guangxi Key Laboratory of Hybrid Computation and IC Design Analysis, Nanning 530006, China^{2}School of Science, Guangxi University for Nationalities, Guangxi, China

Correspondence should be addressed to Xiaoji Liu; moc.621@27uilijoaix

Received 1 December 2017; Revised 4 March 2018; Accepted 26 March 2018; Published 7 May 2018

Academic Editor: Shwetabh Srivastava

Copyright © 2018 Xiaoji Liu and Naping Cai. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

#### 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*

*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*

*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*

*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.