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Abstract and Applied Analysis
Volume 2013 (2013), Article ID 575964, 12 pages
http://dx.doi.org/10.1155/2013/575964
Research Article

Solutions and Improved Perturbation Analysis for the Matrix Equation

School of Mathematics and Statistics, Shandong University, Weihai, Weihai 264209, China

Received 16 November 2012; Revised 2 May 2013; Accepted 7 May 2013

Academic Editor: Carlos Vazquez

Copyright © 2013 Jing Li. 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

The nonlinear matrix equation with is investigated. We consider two cases of this equation: the case and the case In the case , a new sufficient condition for the existence of a unique positive definite solution for the matrix equation is obtained. A perturbation estimate for the positive definite solution is derived. Explicit expressions of the condition number for the positive definite solution are given. In the case , a new sharper perturbation bound for the unique positive definite solution is derived. A new backward error of an approximate solution to the unique positive definite solution is obtained. The theoretical results are illustrated by numerical examples.

1. Introduction

In this paper, we consider the Hermitian positive definite solution of the nonlinear matrix equation where , , and are complex matrices, is a positive definite matrix, and . This type of nonlinear matrix equations arises in the analysis of ladder networks, the dynamic programming, control theory, stochastic filtering, statistics, and many applications [17].

In the last few years, (1) was investigated in some special cases. For the nonlinear matrix equations [812], [13, 14], [15, 16], and [17], there were many contributions in the literature to the solvability, numerical solutions, and perturbation analysis. In addition, the related equations [911, 1823], [13, 24, 25], [16, 26], [17, 2730], [3133], and [3436] were studied by many scholars.

In [31], a sufficient condition for the equation to have a unique positive definite solution was provided. When the coefficient matrix is nonsingular, several sufficient conditions for the equation to have a unique positive definite solution were given in [37]. When the coefficient matrix is an arbitrary complex matrix, necessary conditions and sufficient conditions for the existence of positive definite solutions for the equation were derived in [38]. Li and Zhang in [39] proved that there always exists a unique positive definite solution to the equation . They also obtained a perturbation bound and a backward error of an approximate solution for the unique solution of the equation .

As a continuation of the previous results, the rest of the paper is organized as follows. Section 2 gives some preliminary lemmas that will be needed to develop this work. In Section 3, a new sufficient condition for (1) with having a unique positive definite solution is derived. In Section 4, a perturbation bound for the positive definite solution to (1) with is given. In Section 5, applying the integral representation of matrix function, we also discuss the explicit expressions of condition number for the positive definite solution to (1) with . Furthermore, in Section 6, a new sharper perturbation bound for the unique positive definite solution to (1) with is given. In Section 7, a new backward error of an approximate solution to (1) with is obtained. Finally, several numerical examples are presented in Section 8.

We denote by the set of complex matrices, by the set of Hermitian matrices, by the identity matrix, by the spectral norm, by the Frobenius norm and by and the maximal and minimal eigenvalues of , respectively. For with columns and a matrix , is a Kronecker product, and is a vector defined by . For , we write (resp., if is Hermitian positive semidefinite (resp., definite). Let , .

2. Preliminaries

In this section we quote some preliminary lemmas that we use later.

Lemma 1 (see [39, Lemma 3.2]). For every positive definite matrix and , then (i). (ii).

Lemma 2 (see [39, Theorem 2.5]). There exists a unique positive definite solution of and the iteration converges to .

Lemma 3 (see [32, Lemma 2]). (i) If , then .
(ii) If and , then .
(iii) If , then .

3. A Sufficient Condition for the Existence of a Unique Solution of

In this section, we derive a new sufficient condition for the existence of a unique solution of beginning with the lemma.

Lemma 4 (see [38, Theorem 5, Remark 4]). If then (1) has a unique positive definite solution , where and are, respectively, positive solutions of the following equations:
Furthermore,

Theorem 5. If then (1) has a unique positive definite solution.

Proof. We first prove , where is the positive solution to (5). Let By computation, we obtain Define Then is decreasing on and increasing on , which implies that According to the condition (8), it follows that . Note that which implies that is increasing on . Considering the condition (7), one sees that . Combining that and the definition of in Lemma 4, we obtain . By Lemma 4, (1) has a unique positive definite solution.

4. Perturbation Bound for

Li and Zhang in [39] proved that there always exists a unique positive definite solution to the equation . They also obtained a perturbation bound for the unique solution. But their approach becomes invalid for the case of . Since the equation does not always have a unique positive definite solution, there are two difficulties for a perturbation analysis of the equation . One difficulty is how to find some reasonable restrictions on the coefficient matrices of perturbed equation ensuring that this equation has a unique positive definite solution. The other difficulty is how to find an expression of which is easy to handle.

Assume that the coefficient matrix is perturbed to . Let with satisfying the perturbed equation

In the following, we derive a perturbation estimate for the positive definite solution to the matrix equation beginning with the lemma.

Lemma 6 (see [38, Corollary 1. Remark 4]). If then (1) has a unique positive definite solution . Moreover, .

Theorem 7. If then have unique positive definite solutions and , respectively. Furthermore,

Proof. By (16), it follows that . According to Lemma 6, the condition (16) ensures that (1) and (14) have unique positive definite solutions and , respectively. Furthermore, we obtain that Subtracting (14) from (1) gives By Lemma 3 and inequalities in (19), we have Noting (16), we have Combining (20) and (21), one sees that which implies that

5. Condition Number for

A condition number is a measurement of the sensitivity of the positive definite stabilizing solutions to small changes in the coefficient matrices. In this section, we apply the theory of condition number developed by Rice [40] to derive explicit expressions of the condition number for the matrix equation .

Here we consider the perturbed equation where and are small perturbations of and in (1), respectively.

Suppose that and . According to Lemma 6, (1) and (25) have unique positive definite solutions and , respectively. Let , , and .

Subtracting (25) from (1) gives Therefore, where

Lemma 8. If then the linear operator defined by is invertible.

Proof. Define the operator by
it follows that
Then, is invertible if and only if is invertible.
According to Lemma 3 and the condition (29), we have which implies that and is invertible. Therefore, the operator is invertible.

Thus, we can rewrite (27) as

By the theory of condition number developed by Rice [40], we define the condition number of the Hermitian positive definite solution to the matrix equation by where , , and are positive parameters. Taking in (36) gives the absolute condition number , and taking , , and in (36) gives the relative condition number .

Substituting (35) into (36), we get Let be the matrix representation of the linear operator . It follows from Lemma . in [41] that By Lemma . in [41], we have

Then,

Let

where , , ,  ,   is the vec-permutation matrix, that is,

Furthermore, we obtain that

Then, we have the following theorem.

Theorem 9. If , then the condition number defined by (36) has the explicit expression where the matrices and are defined by (40)(41).

Remark 10. From (46) we have the relative condition number

5.1. The Real Case

In this subsection, we consider the real case, that is, where all the coefficients matrices , of the matrix equation are real. In such a case the corresponding solution is also real. Similar arguments as in Theorem 9 give the following theorem.

Theorem 11. Let , be real, the condition number defined by (36). If , then has the explicit expression where

Proof. Let where , ,   ,     is the vec-permutation matrix, that is, It follows from (44) that

Remark 12. In the real case the relative condition number is given by

6. New Perturbation Bound for

Here, we consider the perturbed equation where and are small perturbations of and in (1), respectively. We assume that and are the solutions of (1) and (54), respectively. Let , , and .

In this section, we develop a new perturbation bound for the solution of (1) which is sharper than that in [39, Theorem 3.1].

Subtracting (1) from (54), using Lemma 1, we have where By Lemma  5.1 in [39], the linear operator defined by is invertible.

We also define operator by Thus, we can rewrite (55) as Define Now we denote

Theorem 13. If then

Proof. Let Obviously, is continuous. The condition (62) ensures that the quadratic equation in has two positive real roots. The smaller one is Define . Then for any , by (62), we have It follows that is nonsingular and Therefore for , in which the last equality is due to the fact that is a solution to (65). That is . According to Schauder fixed point theorem, there exists such that . It follows that is a Hermitian solution of (54). By Lemma 2, we know that the solution of (54) is unique. Then and .

7. New Backward Error for

In this section, we evaluate a new backward error estimate for an approximate solution to the unique solution, which is sharper than that in [39, Theorem 4.1].

Theorem 14. Let be an approximation to the solution of (1). If and the residual satisfies then

To prove the above theorem, we first verify the following lemma.

Lemma 15. For every positive definite matrix , , if , then

Proof. It follows from Lemma 1 that
Note that , , and , we have
A combination of (73)–(75) gives
Here, we have used the result to derive the last equality (refer to [42, Problem 11. Page 312]).

Proof of Theorem 14. Let
Obviously, is a nonempty bounded convex closed set. Let
Evidently is continuous. The condition (70) ensures that the equation
in has two positive real roots. The smaller one is , where .
We will prove that . For every , we have
Hence,
Using (70) and (71), one sees that
Therefore, .
According to (72), we obtain
for , in which the last equality is due to the fact that is a solution to (79). That is . By Brouwer fixed point theorem, there exists a such that . Hence is a solution of (1). Moreover, by Lemma 2, we know that the solution of (1) is unique. Then

8. Numerical Examples

To illustrate the theoretical results of the previous sections, in this section four simple examples are given, which were carried out using MATLAB 7.1. For the stopping criterion we take .

Example 1. We consider the matrix equation
where
Suppose that the coefficient matrix is perturbed to , where
and is a random matrix generated by MATLAB function randn.

We compare our own result in Theorem 13 with the perturbation bound proposed in [39, Theorem 3.1].

The assumption in [39, Theorem 3.1] is

The assumptions in Theorem 13 are

By computation, we list them in Table 1.

tab1
Table 1: Assumptions check for Example 1 with different values of j.

The results listed in Table 1 show that the assumptions in Theorem 3.1 [39] and Theorem 13 are satisfied.

By Theorem  3.1 in [39] and Theorem 13, we can compute the relative perturbation bounds , respectively. These results averaged as the geometric mean of 10 randomly perturbed runs. Some results are listed in Table 2.

tab2
Table 2: Perturbation bounds for Example 1 with different values of j.

The results listed in Table 2 show that the perturbation bound given by Theorem 13 is fairly sharp, while the bound given by Theorem  3.1 in [39] is conservative.

Example 2. Consider the equation
for Choose . Let the approximate solution be given with the iterative method (2), where is the iteration number. Assume that the solution of (1) is unknown.

We compare our own result with the backward error proposed in Theorem 4.1 [39].

The residual satisfies the conditions in Theorem 4.1 [39] and in Theorem 14.

By Theorem  4.1 in [39], we can compute the backward error bound By Theorem 14, we can compute the new backward error bound Let Some results are shown in Table 3.

tab3
Table 3: Results for Example 2 with different values of k.

From the results listed in Table 3 we see that the new backward error bound is sharper and closer to the actual error than the backward error bound in [39]. Moreover, we see that the backward error for an approximate solution seems to be independent of the conditioning of the solution .

Example 3. We consider the matrix equation
where
We now consider the perturbation bounds for the solution when the coefficient matrix is perturbed to , where and is a random matrix generated by MATLAB function randn.

The conditions in Theorem 7 are satisfied.

By Theorem 7, we can compute the relative perturbation bound with different values of . These results averaged as the geometric mean of 10 randomly perturbed runs. Some results are listed in Table 4.

tab4
Table 4: Results for Example 3 with different values of j.

The results listed in Table 4 show that the perturbation bound given by Theorem 7 is fairly sharp.

Example 4. Consider the matrix equation , where
By Remark 12, we can compute the relative condition number . Some results are listed in Table 5.

tab5
Table 5: Results for Example 4 with different values of .

Table 5 shows that the unique positive definite solution is well conditioned.

Acknowledgments

The author would like to express her gratitude to the referees for their fruitful comments. This research project was funded by the National Nature Science Foundation of China (11201263), the Nature Science Foundation of Shandong Province (ZR2012AQ004), and Independent Innovation Foundation of Shandong University (IIFSDU), China.

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