Abstract and Applied Analysis

Volume 2013 (2013), Article ID 843785, 7 pages

http://dx.doi.org/10.1155/2013/843785

Research Article

The Inversion-Free Iterative Methods for Solving the Nonlinear Matrix Equation

Na Huang and Changfeng Ma

School of Mathematics and Computer Science, Fujian Normal University, Fuzhou 350007, China

Received 1 August 2013; Accepted 1 November 2013

Academic Editor: Józef Banas

Copyright © 2013 Na Huang and Changfeng Ma. 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 present two inversion-free iterative methods for computing the maximal positive definite solution of the equation . We prove that the sequences generated by the two iterative schemes are monotonically increasing and bounded above. We also present some numerical results to compare our proposed methods with some previously developed inversion-free techniques for solving the same matrix equation.

1. Introduction

In this paper, we consider the nonlinear matrix equation:

where , is the identity matrix, and a Hermitian positive definite solution is required.

Specifically, if , the nonlinear matrix equation (1) reduces to

The nonlinear matrix equation (2) has many applications in nano research, control theory, dynamic programming, statistics, ladder networks, stochastic filtering, and so forth (see [1–7]). The special case (2) has been widely studied by some authors (see [8–27]). Different iterative methods for computing the positive definite solutions of (2) have been proposed, for example, the fixed-point iteration (see [15]), structure-preserving doubling algorithm (see [7, 16]), and some inversion-free iterations (see [17, 20, 23, 27]). Among them, structure-preserving doubling algorithm has been seen as one of the most efficient algorithms as it has quadratic convergence rate.

However, very little research has been done on the solutions to (1) in the case . In [28], Long et al. stated the application background of (1) and presented some conditions for the existence of the positive definite solution of this equation. Moreover, they proposed some iterative algorithms to find the positive definite solution. Popchev et al. made a complete perturbation analysis of (1) (see [29]). In [30], Liu and Chen considered the nonlinear matrix equation . They studied the existence of the positive definite solution of this equation.

Motivated and inspired by the works mentioned above, in this paper, we propose two new inversion-free iterative methods for obtaining the maximal positive definite solution of (1). We prove that the sequences generated by the two iterative schemes are monotonically increasing and bounded above. In addition, we also provide some numerical results to illustrate the effectiveness of the proposed algorithm.

Throughout this paper, we use the following notations: for , we write , and to denote the conjugate transpose, the inverse, and the Frobenius norm of the matrix , respectively. For any , , we write (or ) if is a Hermitian positive semidefinite matrix. And we write (or ) if is a Hermitian positive definite matrix. We use to denote the zero matrix of size implied by context and to denote the identity matrix of size implied by context.

This paper is organized as follows. In Section 2, we present two new iterative methods to solve the nonlinear matrix equation (1). In Section 3, the convergence analysis of the proposed methods is given. Some numerical experiments are reported in Section 4. Finally, we conclude this paper in Section 5.

2. New Inversion-Free Iterative Methods

In this section, we present two new inversion free iterative algorithms for solving problem (1). Let ; then the nonlinear matrix equation (1) is equivalent to

Obviously, if is a Hermitian positive definite solution of (3), we have . That is, .

Premultiplying and postmultiplying by on (3) simultaneously, we get

Adding to both sides of the above equation, we have

Obviously, solves (5) if is a solution of (3). Conversely, if is a nonsingular solution of (5), solves (3) as well.

Thus, we just need to solve the matrix equation (5) if we want to get a Hermitian positive definite solution of (3). By this point, we present the following iterative scheme.

Algorithm 1 (an inversion-free iterative algorithm for (1)).

Step 1. Input the matrix . Take initial matrix and tolerance error . Set .

Step 2. Obtain by the following iterative scheme:

where .

Step 3. Stop if . Otherwise, , go to Step 2.

Remark 2. If , Algorithm 1 reduces to Algorithm 2.1 in [23] with and . Moreover, as is a Hermitian matrix, from (6) we know that is also a Hermitian matrix, for all .

On the other hand, premultiplying and postmultiplying by on (3), respectively, we have

Adding the above two equations, we can get

By some simple calculating, we obtain

By the above analysis, we know that if is a solution of (3), is a solution of (9). Now we prove that a Hermitian positive definite solution of (9) is also a solution of (3).

Theorem 3. Let be a Hermitian positive definite solution of the nonlinear matrix equation (9). Then is also a positive definite solution of the nonlinear matrix equation (3).

Proof. By the nonlinear matrix equation (9), we have

This implies that is a solution of the matrix equation , where is unknown. Since is positive definite, and have no common eigenvalue and the matrix equation has a unique solution (see [31]). As solves the equation , we must have . Namely, is a positive definite solution of (3).

Thus, we just need to solve the matrix equation (9) if we want to get a Hermitian positive definite solution of (3). By this point, we present the following iterative scheme.

Algorithm 4 (another inversion free iterative algorithm).

Step 1. Input the matrix . Take initial matrix and tolerance error . Set .

Step 2. Obtain by the following iterative scheme:

Step 3. Stop if . Otherwise, go to Step 2.

Obviously, for all , sequence generated by Algorithm 4 with the initial matrix are all Hermitian matrices.

3. Convergence Analysis

In this section, we will prove that the sequences generated by Algorithms 1 and 4 with the initial matrix converge to the minimal positive definite solution of (3). In the first place, we introduce the following lemmas.

Lemma 5 (see [28]). If (1) has a positive definite solution , then .

Lemma 6 (see [23]). If and are Hermitian matrices of the same order with , then .

Lemma 7. If are both Hermitian positive semidefinite, then is a Hermitian positive semidefinite matrix as well.

Proof. Obviously, is a Hermitian matrix. If one of and is positive definite, without loss of generality, suppose that . Then by the assumption of the lemma, we know that , are well defined and , . Then we get

Assume that neither of and is positive definite. Then for , we have . Thus by the above analysis, we have . Let ; we get . This completes the proof.

Now we are in a position to prove that generated by Algorithm 1 with the initial matrix converges to the minimal positive definite solution of (3).

Theorem 8. The nonlinear matrix equation (3) has a positive definite solution and the sequence is generated by Algorithm 1 with the initial matrix . Let be the minimal positive definite solution of (3). Then the sequence is well defined, , and

Proof. Let be any positive definite solution of (3). Firstly, we will prove for all by mathematical induction.

For , we have . By (6) we have

On the other hand, by Lemma 5, we have . This together with Lemma 6 and (3) yields

where the second inequality follows from the fact that . Hence holds for .

Assume that holds for . Since

by (6), Lemma 6, and the fact that , we get

Moreover, it follows from (6) that

As , this together with (6) and Lemma 6 follows that

This implies that . Thus, by (18) and the fact that , we obtain . Therefore, holds for .

By the principle of mathematical induction, is true for all . The sequence is now well defined, monotonically increasing, and bounded above. Let . Then is a positive definite solution of the nonlinear matrix equation (3) by (6). Since for any positive definite solution of (3), is the minimal positive definite solution of (3). This completes the proof.

Remark 9. Let be the maximal positive definite solution of the nonlinear matrix equation (1). By the relationship between (1) and (3), we get that is the minimal positive definite solution of (3). So from Theorem 8, we know that the sequence generated by Algorithm 1 with the initial matrix converges to the inverse of the maximal positive definite solution of (1).

Now we consider the convergence theorem of Algorithm 4.

Theorem 10. The nonlinear matrix equation (3) has a positive definite solution and the sequence is generated by Algorithm 4 with the initial matrix . Let be the minimal positive definite solution of (3). Then the sequence is well defined, , and .

Proof. Let be any positive definite solution of (3). Firstly, we will prove for all by induction.

For , we have ; then . On the other hand, by the fact that , (3), and Lemma 7, we get

Hence is true for .

Assume that holds for . Then by Lemma 7 we have

Similarly, we get . This together with (11) yields

On the other hand, by Lemma 7 we obtain

Similarly, we get

Using the above inequalities, we can deduce that

where the first equality follows from (11) and the third equality follows from (3). Hence, holds for .

Thus, by the principle of induction, is true for all . The sequence is now well defined, monotonically increasing, and bounded above. Let . Then is a positive definite solution of the nonlinear matrix equation (3) by (11) and Theorem 3. Since for any positive definite solution of (3), is the minimal positive definite solution of (3). This completes the proof.

4. Numerical Experiments

In this section, we will give some numerical examples to support our Algorithms 1 and 4. All experiments were run on a PC with Pentium(R) Dual-Core CPU E5800 @2.40 GHz. All the programming is implemented in MATLAB R2011b (7.13). We report the number of required iterations (Iter.), the norm of the residual (Res.) when the process is stopped, the required CPU time (CPU), and the number of matrix-matrix (MM) products required. In our implementation, we stop all considered algorithms when with . We compare our Algorithm 1 (A1) and Algorithm 4 (A2) with the following inverse-free methods for solving (1).

(i)In [28], for finding positive definite solution of (1), Long et al. proposed the following iteration:

(ii)In [30], for solving positive definite solution of (1), Liu and Chen proposed the following iteration:

Example 11. For the first experiment, we consider (1) when and are given as in Example 4.1 from [28]:

We could obtain the maximal solution (the first 4 digits) by the iterative schemes A1-A2 and B1–B3. The maximal solution is

Our numerical results are reported in Table 1.

Table 1: Numerical results of Example 11.

Scheme
Iter.
CPU
Res.
MM

A1
14
0.0006

84

A2
20
0.0008

100

B1
26
0.0011

156

B2
15
0.0007

90

Example 12. In this test, the matrices and are given as in Example 4.2 from [28]:

We will obtain the maximal solution (the first 4 digits) by the iterative schemes A1-A2 and B1–B3. The maximal solution is

Our numerical results are reported in Table 2.

Table 2: Numerical results of Example 12.

Scheme
Iter.
CPU
Res.
MM

A1
48
0.0027

288

A2
83
0.0040

415

B1
92
0.0049

552

B2
49
0.0028

294

Example 13. In this experiment we solve (1) with the matrices and as follows:

We could obtain the maximal solution (the first 4 digits) by the iterative schemes A1-A2 and B1–B3. The unique positive definite solution is

The numerical results are listed in Table 3.

Table 3: Numerical results of Example 13.

Scheme
Iter.
CPU
Res.
MM

A1
28
0.0016

168

A2
47
0.0025

235

B1
54
0.0032

324

B2
29
0.0018

174

Example 14. In this test, the matrices and are provided with the following forms:

where is identity matrix and and are randomly generated with entries and . For different matrix dimension (DIM) , the numerical results are reported in Table 4.

Table 4: Numerical results of Example 14.

DIM
Scheme
Iter.
CPU
Res.
MM

64
A1
16
0.0050

96

A2
24
0.0070

120

B1
32
0.0419

192

B2
17
0.0064

102

128
A1
17
0.0324

102

A2
24
0.0620

120

B1
32
0.1036

192

B2
18
0.0454

108

256
A1
17
0.3721

102

A2
24
0.4954

120

B1
32
0.6902

192

B2
18
0.4155

108

512
A1
17
2.5991

168

A2
24
2.9816

235

B1
32
4.4817

324

B2
18
2.7676

174

1024
A1
17
20.2298

168

A2
24
26.0464

235

B1
32
36.6167

324

B2
18
21.8226

174

From the above experiments, we find that Algorithm 1 has an advantage in the number of iterations and CPU time. Although Algorithm 4 performs worse than B2, it performs better than B1. In Example 14, we can see that the iterations of Algorithms 1 and 4 are invariant with the dimension, which are the same as the performance of B1 and B2. In a word, our algorithms are promising.

5. Conclusion

In this paper, we propose two inversion-free iterative algorithms for obtaining the maximal positive definite solution of the equation . We prove that the sequences generated by the proposed iterative schemes are monotonically increasing and bounded above. Some numerical results are also reported in the paper, which confirm the good theoretical properties of our approach. Although we prove that the sequence generated by Algorithms 1 and 4 with the initial matrix converges to the minimal positive definite solution of (3) (or the maximal positive definite solution of (1)), we do not yet give the analysis on the convergence rate of the two methods, which is our work in the future.

Conflict of Interests

The authors declare that there is no conflict of interests regarding the publication of this paper.

Authors’ Contribution

All authors contributed equally and significantly in writing this paper. All authors read and approved the final paper.

Acknowledgments

The project is supported by the National Natural Science Foundation of China (Grant nos. 11071041 and 11201074), Fujian Natural Science Foundation (Grant No. 2013J01006) and The University Special Fund Project of Fujian (Grant no. JK2013060).

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