Abstract
We investigate successive matrix squaring (SMS) algorithms for computing the generalized inverse of a given matrix .
1. Introduction
Throughout this paper, the symbol denotes a set of all complex matrices. Let , and the symbols , and stand for the range, the null space, the spectrum of matrix , and the matrix norm, respectively.
A matrix is called a inverse of matrix if holds. The symbols , , and denote, respectively, the MoorePenrose inverse, the index, and the Drazin inverse of , and, obviously, (see [1] for details). Let , , and and , and there exists and unique matrix such that then is called inverse of with the prescribed range and null space of , denoted by .
In [1], it is well known that the generalized inverse of a given matrix with the prescribed range and null space is very important in applications of many mathematics branches such as stable approximations of illposed problems, linear and nonlinear problems involving rankdeficient generalized, and the applications to statistics [2]. In particular, the generalized inverse plays an important role for the iterative methods for solving nonlinear equations [1, 2].
In recent years, successive matrix squaring algorithms are investigated for computing the generalized inverse of a given matrix in [3–7]. In [3], the authors exhibit a deterministic iterative algorithm for linear system solution and matrix inversion based on a repeated matrix squaring scheme. Wei derives a successive matrix squaring (SMS) algorithm to approximate the Drazin inverse in [4]. Wei et al. in [5] derive a successive matrix squaring (SMS) algorithm to approximate the weighted generalized inverse , which can be expressed in the form of successive squaring of a composite matrix . Stanimirović and CvetkovićIlić derive a successive matrix squaring (SMS) algorithm to approximate an outer generalized inverse with prescribed range and null space of a given matrix in [6]. In [7], authors introduce a new algorithm based on the successive matrix squaring (SMS) method and this algorithm uses the strategy of displacement rank in order to find various outer inverses with prescribed ranges and null spaces of a square Toeplitz matrix.
In this paper, based on [3–5], we investigate successive matrix squaring algorithms for computing the generalized inverse of a matrix in Section 2 and also give a numerical example for illustrating our results in Section 3.
The following given lemma suggests that the generalized inverse is unique.
Lemma 1.1 (see [1, Theorem 2.14]). Let with rank , let be a subspace of of dimension , and let be a subspace of of dimension . Then, has a inverse such that and if and only if in which case is unique.
The following nations are stated in Banach space but they are true in the finite dimension space. Throughout this paper, let denote the Banach space and let stand for the set of all bounded linear operators from to , in particular .
In the following, we state two lemmas which are given for Banach space but it can be used also for the finite dimension space.
Lemma 1.2 (see [8, Section 4]). Let and and , respectively, closed subspaces of and . Then the following statements are equivalent:(i)has a inverse such that and ,(ii) is a complemented subspace of , is invertible and .
Lemma 1.3 (see [9, Section 3]). Suppose that the conditions of Lemma 1.2 are satisfied. If we take , then holds and has the following matrix form: where is invertible. Moreover, has the matrix following form:
From (1.5), we obtain the following projections (see [9]):
2. Main Result
In this section, we consider successive matrix squaring (SMS) algorithms for computing the generalized inverse .
Let and the sequence in , and we can define the iterative form as follows ([10, Theorem 2.2] for computing the generalized inverse in the infinite space case): From [10], the authors have proved that the iteration (2.1) converges to the generalized inverse if and only if , where and (for the proof see [11] and [10, Theorem 2.1] when ).
In the following, we give the algorithm for computing the generalized inverse of a matrix .
Let and . It is not difficult to see that the above fact can be written as follows: From (2.2) and letting , we have
By (2.3), we prove that the iterative (2.1) is equal to the right upper block in the matrix . Note that we defined the new iterative form as follows:
From the new iterative form (2.4), we arrive at
Assume that , and by (2.5), we have
By (2.4)–(2.6), we have Algorithm 1.

From (2.4)–(2.6) and Algorithm 1, we obtain the following result.
Theorem 2.1. Let , and the sequence converges to the generalized inverse if and only if . In this case where and
Proof. From the proof in [11] and [10, Theorem 2.1] when and according to (2.4), (2.5) and (2.6), we easily finish the proof of the former of the theorem. In the following, we only prove the last section, that is, prove that the inequality (2.7) holds.
By applying (2.5) and (2.6), we obtain
By the iteration (2.4) and (2.9), we arrive at
The following corollary given the result is the same as theorem in [6, Theorem 2.3]. It also presents an explicit representation of the the generalized inverse and the sequence (2.4) converges to a inverse of a given matrix by its fullrank decomposition.
Corollary 2.2. Let be full rank decomposition, and the sequence converges to the inverse if and only if . In this case where and
Proof. From Theorem 2.5 and by [6, Theorem 2.3], we have the result.
In the following, we consider the improvement of the iterative form (2.1) (see [11] for computing the MoorePenrose inverse and the Drazin inverse of the matrix case and [10, Theorem 2.2] for computing the generalized inverse in the infinite space case): Let be a block matrix and then
By induction if has the following form: then
Similarly to the iterative form (2.4), we also define the new iterative scheme Note that from (2.18)
Let , and by (2.18), and (2.19), we arrive at
From (2.14) to (2.20), we find that if one wants to compute the generalized inverse then we only compute the element () of the matrix . Similarly to Algorithm 1, we also obtain Algorithm 2.

Analogous to Theorem 2.5 by Algorithm 2 and sequence (2.18), we also have the following theorem.
Theorem 2.3. Let , and the sequence converges to the generalized inverse if and only if . In this case where and
Proof. Similarly the proof in [10, Theorem 2.1], we can prove the former of this theorem. Analogous to the proof of Theorem 2.5, we finish the proof of the theorem.
In the following, we extend the sequence (2.4) to By (2.26) and by induction, we have
Assume that , we easily have
Similarly, from (2.23) and (2.25), we obtain the following result.
Theorem 2.4. Let , and the sequence converges to the generalized inverse if and only if . In this case where and
Proof. From (2.25) and only using instead of in Theorem 2.1, we easily have that converges to the generalized inverse if and only if . Similarly to the formula (2.29), we obtain that where , , and are the same as Theorem 2.5.
In the following, we consider the dually iterative form.
Let and the sequence in , and we can define the iterative form as follows (see [11] and [10, Theorem 2.3]):
Let and . It is not difficult to see that the above fact can be written as follows:
From iterative forms (2.26) and (2.29), we have the following theorem.
Theorem 2.5. Let , and the sequence converges to the generalized inverse if and only if . In this case where and
Similarly to Corollary 2.2, we have the result as follows.
Corollary 2.6. Let , full rank decomposition, and the sequence converges to the inverse if and only if . In this case where and
In the following, we consider the improvement of the iterative form (2.29) (see [11] for computing the MoorePenrose inverse and the Drazin inverse of the matrix case and [10, Theorem 2.3] for computing the generalized inverse in the infinite space case):
It is similar to (2.14), and we have
Analogous to Theorem 2.5 by Algorithm 2 and from (2.36), we obtain the theorem in the following.
Theorem 2.7. Let , and the sequence converges to the generalized inverse if and only if . In this case where and
Dually, we give the SMS algorithm for computing the generalized inverse which are analogous to the iterative form (2.23) as follows and omit their proofs:
Similarly Theorem 2.4, from (2.35) and (2.39), we obtain the following result.
Theorem 2.8. Let , and the sequence converges to the generalized inverse if and only if . In this case where and
3. Example
Here is an example to verify the effectiveness of the SMS method.
Example 3.1. Let
Let ; .
Take
By (2.2), we have
From [10, 12], we easily have the generalized inverse in
Then, from Algorithm 1, we obtain
But by the iteration (2.1), we get
From the data in (3.5) and (3.6), we obtain Table 1.
From the above in (3.5), (3.6), and Table 1, we know that we only need two steps by Algorithm 1, but five steps by using iterative form (2.1).
Acknowledgments
X. Liu is supported by the National Natural Science Foundation of China (11061005), College of Mathematics and Computer Science, Guangxi University for Nationalities, Nanning, China, and Y. Qin is supported by the Innovation Project of Guangxi University for Nationalities (gxunchx2011075), College of Mathematics and Computer Science, Guangxi University for Nationalities, Nanning, China.