# High-Order Iterative Methods for the DMP Inverse

Authors: Xiaoji Liu; Naping Cai
Journal: Journal of Mathematics
 (2018)

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## 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.


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## Body

## 1. Introduction

LetCrm×n be the set of all m×n complex matrices with rank r. For any given a matrix A∈Cm×n, let R(A), N(A), and A be the range space, the null space, and the Frobenius norm of matrix A, respectively. For a nonnegative integer l, if rank(Al+1)=rank(Al), then l is called the index of A and ind(A)=l. 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, {2}-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 matrixA∈Cm×n, there exists matrix X∈Cn×m satisfying (see [8])(1)
AXA=A,(2)
XAX=X,(3)
(AX)∗=AX,(4)
(XA)∗=XA,whereX is called the Moore-Penrose inverse of A, denoted by X=A†, and it is unique. For a given matrix A∈Cn×n, there exists a matrix X∈Cn×n satisfying(1)
XAX=X,(2)
Al+1X=Al,(3)
AX=XA,whereX is called the Drain inverse of A, denoted by X=AD, and it is unique. Based on the Drazin (D) inverse and the Moore-Penrose (MP) inverse, a new generalized inverse is defined in [12] as (see also [13, 14]): for a matrix A∈Crn×n, there is matrix X∈Cn×n satisfying (see [12])(1)
XAX=X,(2)
XA=ADA,(3)
AlX=AlA†,whereX is called DMP inverse, denoted by X=AD,†, and it is unique. It is shown that X=ADAA† 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 A∈Cn×n. 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 Section2. 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]).
IfA∈Cn×n and Cn=L⨁M, then(i)
PL,MA=A⇔R(A)⊆L,(ii)
APL,M=A⇔N(A)⊇M.Lemma 2 (see [12]).
LetA∈Cn×n, ind(A)=l. If AD,† exist, then(i)
AAD,†=PR(Al),N(AlA†),R(AD,†)⊆R(Al),(ii)
AD,†A=PR(Al),N(Al),N(AD,†)⊇N(AlA†).As in [20], we develop a iterative scheme to compute the DMP inverse as follows.Scheme I:(1)Xn=Xn-1Ct1I-Ct2AXn-1+⋯+-1t-1CttAXn-1t-1,where α∈C\{0}, R(Y)⊆R(Al), and X0=αY. Following the line [21], we develop the iterative following scheme as (2)Yn=Vn-F′Vn-1FVn,Zn=Vn-2-1F′VnF′Vn-1+F′Yn-1,Vn+1=Zn-2-1FVnF′Yn-12+F′′WnFWnF′Wn-2.Let ψn=AVn and, by V-1-A=O, we have (3)Vn+1=-18Vn-7I+9AVn-5AVn2+AVn3×12I-42AVn+103AVn2-156AVn3+157AVn4-104AVn5+43AVn6-10AVn7+AVn8=18Vn84I-402ψn+1159ψn2-2241φn3+3060ψn4-3024ψn5+2178ψn6-1134φn7+416ψn8-102nψn9+15ψn10-ψn11=18VnϕVn.Thus, an efficient high-order iterative method can be written as follows.Scheme II:(4)ψn=AVn,Xn=-7I+ψn9I+ψn5I-ψn,θn=ψnXn,Vn+1=-18VnXn12I+θn6I+θn.The iterative method given in (4) is applied to compute the Drazin inverse by [21]. Here, we use the sequence of iterative {Vn}n=0n=∞ 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.
LetA∈Cm×m with ind(A)=l. For an arbitrary positive integer t≥2, the sequence (1) converges if only if ρ(PR(Al),N(AlA†)-AX0)<1. Moreover, we have (5)AD,†-Xn≤AD,†PRAl,NAlA†-AX0tn.Proof.
LetP=PR(Al),N(AlA†) for convenience. If X0=αY, R(Y)⊆R(Al), then R(AX0)=R(αAY)⊆R(AAl)⊆R(Al). From Lemma 2, we attain PAX0=AX0.
Now, we testPAXn=AXn by using mathematical induction. Assume that PAXn=AXn for any positive integer n>0. By Lemma 2, we have (6)PAXn+1=PAXnCt1I-Ct2AXn+⋯+-1t-1CttAXnt-1=Ct1AXn-Ct2AXn2+⋯+-1t-1AXnt=AXn+1.Let En=P-AXn; then(7)En+1=P-Ct1AXn+Ct2AXn2+⋯+-1tAXnt=Ent.Similarly, by Lemma 2 and (7), we derive AD,†AXn=PR(Al),N(Al)Xn=Xn and (8)AD,†-Xn=AD,†AAD,†-AD,†AXn=AD,†P-AXn=AD,†En.Thus, we have (9)AD,†-Xn≤AD,†E0tn.
Next, we investigate the necessary and sufficient condition for convergent property of Scheme I (1). Assume that the sequence {Xn} converges to AD,†. Thus, En=E0tn=P-AXn→0, while n→∞. Therefore, we have ρ(E0)<1.
Conversely, letρ(PR(Al),N(AlA†)-AX0)<1 for some scalar α∈C\{0}. Then AD,†-Xn=AD,†En=AD,†E0tn. Thus, we have AD,†-Xn→0 and (10)AD,†-Xn≤AD,†PRAl,NAlA†-AX0tn.As in [24], we show that Scheme I (1) is asymptotically stable as the following result.Theorem 4.
LetA∈Cm×m and let the sequence {Xn}n=0n=∞ be generated by (1) with an initial X0=αY. If α∈C\{0}, R(Y)⊆R(Al), and ρ(PR(Al),N(AlA†)-AX0)<1, then Scheme I (1) is asymptotically stable.Proof.
Let△Xn be the numerical perturbation of Xn in Scheme I (1). Thus, it can be written into as Xn=Xn~+△Xn. Here, we perform a first-order error analysis; that is, we formally neglect quadratic or higher terms. The manipulation is meaningful, while △Xn is sufficiently small. Further, we have (11)XnA△Xn≈△XnAXn.Let ϕ(Xn)=∑i=1t(-1)i-1Cti(AXn)i-1; we have (12)Xn+1=XnϕXn,AXn+A△Xnt≈AXnt+tAXnt-1A△Xn.
Similarly, we haveX~n+1=X~nϕ(X~n) and(13)X~n+1=Xn+△Xn∑i=1t-1i-1CtiAXn+A△Xni-1=tXn+t△Xn+∑i=2t-1i-1CtiXn+△XnAXni-1+∑i=2t-1i-1CtiXn+△Xni-1AXni-2A·△Xn≈Xn+1+t△Xn+∑i=2t-1i-1Cti·△XnAXni-1+∑i=2t-1i-1Ctii-1XnAXni-2A△Xn.By (13) and PAXn=AXn, we derive (14)X~n+1=Xn+1+t△Xn+∑i=2t-1i-1iCti△XnAXni-1=Xn+1+t△XnP-AXnt-1.By (7), we obtain (15)△Xn=t△XnP-AXnt-1=t△XnEnt-1.Thus, we derive (16)△Xn≤t△XnEnt-1≤tn+1△X0E0tn-1.We can conclude that the perturbation at the iterate n+1 is bounded. Therefore, the sequence {Xn}n=0n=∞ 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(17)A=1100000000010000,Y=AlA∗=2000000000000000,where l=ind(A)=2 and X0=αY. Thus, R(Y)⊆R(Al). To test the high accuracy and efficiency of Scheme I (1), the DMP inverse of A is given as (18)AD,†=1.0000000000000000000.Here, we apply Scheme I given in (1) to compute the DMP inverse with α=0.2. The errors AD,†-Xn and Xn-Xn-1 are given in Table 1. From the numerical results note that (1) converges to AD,† and it has high-order accuracy.Table 1
Numerical results of Scheme I withα=0.2 in Example 1.
Order Steps A D , † - X n X n - X n - 1 cpu time t = 5 4 0 3.3307 e - 16 0.000380 t = 6 4 6.6613 e - 16 1.7764 e - 15 0.001192 t = 7 4 2.4425 e - 15 7.1054 e - 15 0.000473 t = 8 3 9.7700 e - 15 1.9429 e - 14 0.000606 t = 9 3 3.9968 e - 15 1.0658 e - 14 0.000516


## 3.1. Stability and Convergence Analysis

Theorem 3.
LetA∈Cm×m with ind(A)=l. For an arbitrary positive integer t≥2, the sequence (1) converges if only if ρ(PR(Al),N(AlA†)-AX0)<1. Moreover, we have (5)AD,†-Xn≤AD,†PRAl,NAlA†-AX0tn.Proof.
LetP=PR(Al),N(AlA†) for convenience. If X0=αY, R(Y)⊆R(Al), then R(AX0)=R(αAY)⊆R(AAl)⊆R(Al). From Lemma 2, we attain PAX0=AX0.
Now, we testPAXn=AXn by using mathematical induction. Assume that PAXn=AXn for any positive integer n>0. By Lemma 2, we have (6)PAXn+1=PAXnCt1I-Ct2AXn+⋯+-1t-1CttAXnt-1=Ct1AXn-Ct2AXn2+⋯+-1t-1AXnt=AXn+1.Let En=P-AXn; then(7)En+1=P-Ct1AXn+Ct2AXn2+⋯+-1tAXnt=Ent.Similarly, by Lemma 2 and (7), we derive AD,†AXn=PR(Al),N(Al)Xn=Xn and (8)AD,†-Xn=AD,†AAD,†-AD,†AXn=AD,†P-AXn=AD,†En.Thus, we have (9)AD,†-Xn≤AD,†E0tn.
Next, we investigate the necessary and sufficient condition for convergent property of Scheme I (1). Assume that the sequence {Xn} converges to AD,†. Thus, En=E0tn=P-AXn→0, while n→∞. Therefore, we have ρ(E0)<1.
Conversely, letρ(PR(Al),N(AlA†)-AX0)<1 for some scalar α∈C\{0}. Then AD,†-Xn=AD,†En=AD,†E0tn. Thus, we have AD,†-Xn→0 and (10)AD,†-Xn≤AD,†PRAl,NAlA†-AX0tn.As in [24], we show that Scheme I (1) is asymptotically stable as the following result.Theorem 4.
LetA∈Cm×m and let the sequence {Xn}n=0n=∞ be generated by (1) with an initial X0=αY. If α∈C\{0}, R(Y)⊆R(Al), and ρ(PR(Al),N(AlA†)-AX0)<1, then Scheme I (1) is asymptotically stable.Proof.
Let△Xn be the numerical perturbation of Xn in Scheme I (1). Thus, it can be written into as Xn=Xn~+△Xn. Here, we perform a first-order error analysis; that is, we formally neglect quadratic or higher terms. The manipulation is meaningful, while △Xn is sufficiently small. Further, we have (11)XnA△Xn≈△XnAXn.Let ϕ(Xn)=∑i=1t(-1)i-1Cti(AXn)i-1; we have (12)Xn+1=XnϕXn,AXn+A△Xnt≈AXnt+tAXnt-1A△Xn.
Similarly, we haveX~n+1=X~nϕ(X~n) and(13)X~n+1=Xn+△Xn∑i=1t-1i-1CtiAXn+A△Xni-1=tXn+t△Xn+∑i=2t-1i-1CtiXn+△XnAXni-1+∑i=2t-1i-1CtiXn+△Xni-1AXni-2A·△Xn≈Xn+1+t△Xn+∑i=2t-1i-1Cti·△XnAXni-1+∑i=2t-1i-1Ctii-1XnAXni-2A△Xn.By (13) and PAXn=AXn, we derive (14)X~n+1=Xn+1+t△Xn+∑i=2t-1i-1iCti△XnAXni-1=Xn+1+t△XnP-AXnt-1.By (7), we obtain (15)△Xn=t△XnP-AXnt-1=t△XnEnt-1.Thus, we derive (16)△Xn≤t△XnEnt-1≤tn+1△X0E0tn-1.We can conclude that the perturbation at the iterate n+1 is bounded. Therefore, the sequence {Xn}n=0n=∞ 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(17)A=1100000000010000,Y=AlA∗=2000000000000000,where l=ind(A)=2 and X0=αY. Thus, R(Y)⊆R(Al). To test the high accuracy and efficiency of Scheme I (1), the DMP inverse of A is given as (18)AD,†=1.0000000000000000000.Here, we apply Scheme I given in (1) to compute the DMP inverse with α=0.2. The errors AD,†-Xn and Xn-Xn-1 are given in Table 1. From the numerical results note that (1) converges to AD,† and it has high-order accuracy.Table 1
Numerical results of Scheme I withα=0.2 in Example 1.
Order Steps A D , † - X n X n - X n - 1 cpu time t = 5 4 0 3.3307 e - 16 0.000380 t = 6 4 6.6613 e - 16 1.7764 e - 15 0.001192 t = 7 4 2.4425 e - 15 7.1054 e - 15 0.000473 t = 8 3 9.7700 e - 15 1.9429 e - 14 0.000606 t = 9 3 3.9968 e - 15 1.0658 e - 14 0.000516


## 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 projectionP given in Theorems 3 and 4 with satisfying ρ(PR(Al),N(AlA†)-AX0)<1.


### 4.1. Stability and Convergence Analysis

Theorem 2.
LetA∈Cm×m with ind(A)=l; the sequence {Vn}n=0n=∞ defined by Scheme II. If initial approximation V0 satisfies I-AV0<1 and R(V0)⊆R(Al), then Scheme II converges to AD,† and (19)AD,†-VnAD,†≤I-AV09n.Proof.
Similar to the proof of Theorem 9 in [21], let Fn=I-AVn for n≥0; then(20)Fn+1=I-AVn+1=I-A-18Vn-7I+9AVn-5AVn2+AVn3×12I-42AVn+103AVn2-156AVn3+157AVn4-104AVn5+43AVn6-10AVn7+AVn8=18I+Fn3Fn9=18Fn9+3Fn10+3Fn11+Fn12.Thus, we have (21)Fn+1≤18Fn9+3Fn10+3Fn11+Fn12.Since F0<1, we obtain that (22)F1≤18F09+3F010+3F011+F012≤F09<1.Similarly, F2<F19<1. So we can get the conclusion Fn≤1, Fn+1≤Fn9≤F09n+1.
SinceR(V0)⊆R(Al), R(Vn)⊆R(Vn-1), we attain R(Vn)⊆R(Al). By Lemma 2, we have AD,†AVn=Vn and (23)δn=AD,†-AD,†AVn=AD,†I-AVn=AD,†Fn.So, we have δn≤AD,†Fn≤AD,†F09n.
If the conditionI-AV0<1 is satisfied, then δn→0 when n→∞. In other words, Vn→AD,†, the scheme {Vn}n=0∞ satisfies the following error estimate: (24)AD,†-VnAD,†≤I-AV09n.Theorem 3.
LetA∈Cm×m be a singular square matrix with ind(A)=l and the sequence {Vn}n=0n=∞ defined by Scheme II. If the initial approximation V0 is chosen such that I-AV0<1 and R(V0)⊆R(Al), then the order of convergence of the sequence {Vn}n=0n=∞ is nine.Proof.
LetF0=I-AV0 and Fn=I-AVn; we have Fn=(1/8)(Fn9+3Fn10+3Fn11+Fn12).
Now, letδn=AD,†-Vn; we have (25)Aδn+1=AAD,†-I+I-AVn+1=18Fn9+3Fn10+3Fn11+Fn12+8AAD,†-I=18Fn9+AAD,†-I+3Fn10+AAD,†-I+3Fn11+AAD,†-I+Fn12+AAD,†-I. Since DMP inverse is a special of {2}-inverse, by citing the proof of Theorem 3.1 in [25], we have (I-AAD,†)t=0 and (Fn)t+AAD,†-I=(Aδn)t and Aδn+1=(1/8)((Aδn)9+3(Aδn)10+3(Aδn)11+(Aδn)12).
SinceAδn≤1 when n→∞, we have Aδn+1≤(1/8)(Aδn9+3Aδn10+3Aδn11+Aδn12)≤Aδn9.
It would be easy to find the error inequality of the high-order iterative as follows:(26)δn+1=Vn+1-AD,†≤AD,†A9δn9. Thus, the sequence converges to DMP inverse and the convergence order is nine.It is easy to findV0 with satisfying I-AV0<1. 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.
LetA∈Cm×m; the sequence {Vn}n=0n=∞ defined by Scheme II with the initial approximation V0 satisfies I-AV0<1 that is asymptotically stable for computing DMP inverse.Proof.
Let△n be the numerical perturbation introduced in Scheme II. Next, the modified value(27)Vn~=Vn+△nappears instead of the exact value Vn. Here, we formally neglect quadratic or higher terms such as (△n)2. This formal manipulation is meaningful if △n is sufficiently small and further has Vn△n≈△nVn, then(28)Vn+1~=18Vn~ϕVn~=18Vn+△nϕVn+△n≈Vn+1+1884I-2×402ψ+3×1159ψ2-4×2241φ3+5×3060ψ4-6×3024ψ5+7×2178ψ6-8×1134φ7+9×416ψ8-10×102ψ9+11×15ψ10-12×ψ11△n=Vn+1+189×F8+10×3F9+11×3F10+12×F11·△n,Vn+(△n)t=Vnt+tVnt-1△n, △n+1=Vn+1~-Vn+1, when k→∞, Vn→AD,†.
Simplifying (28) we attain (29)△n+1=189×F8+10×3F9+11×3F10+12×F11△n≈189×I-AAD,†8+10×3I-AAD,†9+11×3I-AAD,†10+12×I-AAD,†11△n.Using the matrix identity, we have (I-AAD,†)t=I-AAD,†,t≥1,(30)△n+1≈189+30+33+12I-AAD,†△n≈848I-AAD,†△n, then (31)△n+1≤848n+1I-AAD,†△0.
We can conclude that the perturbation at the iteraten+1 is bounded. Therefore, the sequence {Vn}n=0n=∞ 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(32)A=110011001.Take (33)V0=0.2-0.50.100.5-0.1000.1,where ind(A)=l=1,R(V0)⊆R(Al). Thus, I-AV0<1. To test the efficiency and accuracy of our scheme, we present the DMP inverse of A as (34)AD,†=1-1101-1001.
In Table2, we give the errors AD,†-Xn, Xn-Xn-1. The results show that the proposed method (4) converges to AD,† and has high-order accuracy.Table 2
Numerical results of Scheme II (28) in Example 1 in Section 4.2.
Steps A D , † - X n X n - X n - 1 CPU time 5 2.6038 e - 14 2.5121 e - 14 0.002925


## 4.1. Stability and Convergence Analysis

Theorem 2.
LetA∈Cm×m with ind(A)=l; the sequence {Vn}n=0n=∞ defined by Scheme II. If initial approximation V0 satisfies I-AV0<1 and R(V0)⊆R(Al), then Scheme II converges to AD,† and (19)AD,†-VnAD,†≤I-AV09n.Proof.
Similar to the proof of Theorem 9 in [21], let Fn=I-AVn for n≥0; then(20)Fn+1=I-AVn+1=I-A-18Vn-7I+9AVn-5AVn2+AVn3×12I-42AVn+103AVn2-156AVn3+157AVn4-104AVn5+43AVn6-10AVn7+AVn8=18I+Fn3Fn9=18Fn9+3Fn10+3Fn11+Fn12.Thus, we have (21)Fn+1≤18Fn9+3Fn10+3Fn11+Fn12.Since F0<1, we obtain that (22)F1≤18F09+3F010+3F011+F012≤F09<1.Similarly, F2<F19<1. So we can get the conclusion Fn≤1, Fn+1≤Fn9≤F09n+1.
SinceR(V0)⊆R(Al), R(Vn)⊆R(Vn-1), we attain R(Vn)⊆R(Al). By Lemma 2, we have AD,†AVn=Vn and (23)δn=AD,†-AD,†AVn=AD,†I-AVn=AD,†Fn.So, we have δn≤AD,†Fn≤AD,†F09n.
If the conditionI-AV0<1 is satisfied, then δn→0 when n→∞. In other words, Vn→AD,†, the scheme {Vn}n=0∞ satisfies the following error estimate: (24)AD,†-VnAD,†≤I-AV09n.Theorem 3.
LetA∈Cm×m be a singular square matrix with ind(A)=l and the sequence {Vn}n=0n=∞ defined by Scheme II. If the initial approximation V0 is chosen such that I-AV0<1 and R(V0)⊆R(Al), then the order of convergence of the sequence {Vn}n=0n=∞ is nine.Proof.
LetF0=I-AV0 and Fn=I-AVn; we have Fn=(1/8)(Fn9+3Fn10+3Fn11+Fn12).
Now, letδn=AD,†-Vn; we have (25)Aδn+1=AAD,†-I+I-AVn+1=18Fn9+3Fn10+3Fn11+Fn12+8AAD,†-I=18Fn9+AAD,†-I+3Fn10+AAD,†-I+3Fn11+AAD,†-I+Fn12+AAD,†-I. Since DMP inverse is a special of {2}-inverse, by citing the proof of Theorem 3.1 in [25], we have (I-AAD,†)t=0 and (Fn)t+AAD,†-I=(Aδn)t and Aδn+1=(1/8)((Aδn)9+3(Aδn)10+3(Aδn)11+(Aδn)12).
SinceAδn≤1 when n→∞, we have Aδn+1≤(1/8)(Aδn9+3Aδn10+3Aδn11+Aδn12)≤Aδn9.
It would be easy to find the error inequality of the high-order iterative as follows:(26)δn+1=Vn+1-AD,†≤AD,†A9δn9. Thus, the sequence converges to DMP inverse and the convergence order is nine.It is easy to findV0 with satisfying I-AV0<1. 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.
LetA∈Cm×m; the sequence {Vn}n=0n=∞ defined by Scheme II with the initial approximation V0 satisfies I-AV0<1 that is asymptotically stable for computing DMP inverse.Proof.
Let△n be the numerical perturbation introduced in Scheme II. Next, the modified value(27)Vn~=Vn+△nappears instead of the exact value Vn. Here, we formally neglect quadratic or higher terms such as (△n)2. This formal manipulation is meaningful if △n is sufficiently small and further has Vn△n≈△nVn, then(28)Vn+1~=18Vn~ϕVn~=18Vn+△nϕVn+△n≈Vn+1+1884I-2×402ψ+3×1159ψ2-4×2241φ3+5×3060ψ4-6×3024ψ5+7×2178ψ6-8×1134φ7+9×416ψ8-10×102ψ9+11×15ψ10-12×ψ11△n=Vn+1+189×F8+10×3F9+11×3F10+12×F11·△n,Vn+(△n)t=Vnt+tVnt-1△n, △n+1=Vn+1~-Vn+1, when k→∞, Vn→AD,†.
Simplifying (28) we attain (29)△n+1=189×F8+10×3F9+11×3F10+12×F11△n≈189×I-AAD,†8+10×3I-AAD,†9+11×3I-AAD,†10+12×I-AAD,†11△n.Using the matrix identity, we have (I-AAD,†)t=I-AAD,†,t≥1,(30)△n+1≈189+30+33+12I-AAD,†△n≈848I-AAD,†△n, then (31)△n+1≤848n+1I-AAD,†△0.
We can conclude that the perturbation at the iteraten+1 is bounded. Therefore, the sequence {Vn}n=0n=∞ 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(32)A=110011001.Take (33)V0=0.2-0.50.100.5-0.1000.1,where ind(A)=l=1,R(V0)⊆R(Al). Thus, I-AV0<1. To test the efficiency and accuracy of our scheme, we present the DMP inverse of A as (34)AD,†=1-1101-1001.
In Table2, we give the errors AD,†-Xn, Xn-Xn-1. The results show that the proposed method (4) converges to AD,† and has high-order accuracy.Table 2
Numerical results of Scheme II (28) in Example 1 in Section 4.2.
Steps A D , † - X n X n - X n - 1 CPU time 5 2.6038 e - 14 2.5121 e - 14 0.002925


## 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 projectionPR(Al),N(AlA†). 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].



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