High-Order Iterative Methods for the DMP Inverse

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.

For a given matrix  ∈ C 푚×푛 , there exists matrix  ∈ C 푛×푚 satisfying (see [8]) where  is called the Drain inverse of , denoted by  =  퐷 , 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  ∈ C 푛×푛 푟 , there is matrix  ∈ C 푛×푛 satisfying (see [12]) where  is called DMP inverse, denoted by  =  퐷, † , and it is unique.It is shown that  =  퐷  † 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][18][19][20][21][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  ∈ C 푛×푛 .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.
Scheme II: The iterative method given in ( 4) is applied to compute the Drazin inverse by [21].Here, we use the sequence of iterative { 푛 } 푛=∞ 푛=0 to compute the DMP inverse.

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.

Numerical Example.
Here is an example for computing DMP inverse in the iterative method (1).

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 ( 푅(퐴 푙 ),푁(퐴 푙 퐴 † ) −  0 ) < 1.
It is easy to find  0 with satisfying ‖ −  0 ‖ < 1.So the method to compute DMP inverse is more convenient than another.
Proof.Let △ be the numerical perturbation introduced in Scheme II.Next, the modified value Simplifying (28) we attain Using the matrix identity, we have We can conclude that the perturbation at the iterate  + 1 is bounded.Therefore, the sequence { 푛 } 푛=∞ 푛=0 generated by Scheme II is asymptotically stable.
To test the efficiency and accuracy of our scheme, we present the DMP inverse of  as In Table 2, we give the errors ‖ 퐷, † −  푛 ‖, ‖ 푛 −  푛−1 ‖.The results show that the proposed method (4) converges to  퐷, † and has high-order accuracy.

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  푅(퐴 푙 ),푁(퐴 푙 퐴 † ) .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 {2}-generalized inverse [18,20,22].

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.

Table 2 :
Numerical results of Scheme II (28) in Example 1 in Section 4.2.