Abstract and Applied Analysis

Volume 2013 (2013), Article ID 979832, 11 pages

http://dx.doi.org/10.1155/2013/979832

Research Article

Perturbation Analysis of the Nonlinear Matrix Equation

Jing Li

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

Received 15 March 2013; Accepted 7 May 2013

Academic Editor: Vejdi I. Hasanov

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.

Consider the nonlinear matrix equation with . Two perturbation bounds and the backward error of an approximate solution to the equation are derived. Explicit expressions of the condition number for the equation are 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 , are complex matrices, is a positive integer, and is a positive definite matrix. Here, denotes the conjugate transpose of the matrix .

When , (1) is recognized as playing an important role in solving a system of linear equations. For example, in many physical calculations, one must solve the system of linear equation

where

arises in a finite difference approximation to an elliptic partial differential equation (for more information, refer to [1]). We can rewrite as , where

can be factored as

if and only if is a solution of equation . When   , this type of nonlinear matrix equations arises in ladder networks, dynamic programming, control theory, stochastic filtering, statistics, and so forth [2–7].

For the similar equations , , , and , there were many contributions in the literature to the theory, numerical solutions, and perturbation analysis [8–32]. Jia and Gao [33] derived two perturbation estimates for the solution of the equation with . In addition, Duan et al. [34] proved that the equation has a unique positive definite solution. They also proposed an iterative method for obtaining the unique positive definite solution. However, to our best knowledge, there has been no perturbation analysis for (1) with in the known literature.

As a continuation of the previous results, the rest of the paper is organized as follows. In Section 2, some preliminary lemmas are given. In Section 3, two perturbation bounds for the unique solution to (1) are derived. Furthermore, in Section 4, we obtain the backward error of an approximate solution to (1). In Section 5, we also discuss the condition number of the unique solution to (1). Finally, several numerical examples are presented in Section 6.

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

2. Preliminaries

Lemma 1 (see [35]). If and , then .

Lemma 2 (see [33]). For any Hermitian positive definite matrix and Hermitian matrix , one has

(i),  ;(ii).

In addition, if and , then

(iii).

Lemma 3 (see [34]). The matrix equation always has a unique positive definite solution . The matrix sequence :

converges to the unique positive definite solution for arbitrary initial positive definite matrices .

3. Perturbation Bounds

Here the perturbed equation

is considered, where   and   and are small perturbations of and in (1), respectively. We assume that and are the solutions of (1) and (7), respectively. Let , and .

By Lemma 3, we know that (1) always has a unique positive definite solution ; then in this section two perturbation bounds for the unique positive definite solution of (1) are developed. The relative perturbation bound in Theorem 5 does not depend on any knowledge of the actual solution of (1). Furthermore, a sharper perturbation bound in Theorem 8 is derived.

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

Lemma 4. If is a solution of (1), then

Proof. By Lemma 3, (1) with always has a unique positive definite solution . Then , and it follows that . Therefore . By Lemma 1 and (1), we have   .

The next theorem generalizes [33, Theorem 4] with ,   to arbitrary integer ,  .

Theorem 5. Let . If

then

where

Proof. Let

Obviously, is a nonempty bounded convex closed set. Let

Evidently, is continuous. We will prove that .

For every , it follows that Thus

According to (9), we have

Therefore

From Lemmas 2 and 4, it follows that

Therefore

That is, . By Brouwer's fixed point theorem, there exists a such that . Moreover, by Lemma 3, we know that and are the unique solutions to (1) and (7), respectively. Then

Remark 6. According to

we get for and . Therefore (1) is well posed.

Next, a sharper perturbation estimate is derived.

Subtracting (1) from (7), we have

where

Lemma 7. If , then the linear operator defined by

is invertible.

Proof. It suffices to show that the following equation:

has a unique solution for every  . Define the operator by

Let . Thus (21) is equivalent to

According to Lemma 2, we have

which implies that and is invertible. Therefore, the operator is invertible.

Furthermore, we define operators by

Thus, we can rewrite (21) as

Define

Now we denote that

Theorem 8. Suppose that and are the solutions of (1) and (7), respectively. If ,

then

Proof. Let

Obviously, is continuous. The condition (32) ensures that the quadratic equation with respect to the variable has two positive real roots. The smaller one is

Define . Then for any , by (32), we have

It follows that is nonsingular and

Using (22) and Lemma 2, we have

Noting (31) and (34), it follows that

for . That is, . According to Schauder fixed point theorem, there exists such that . It follows that is a Hermitian solution of (7). By Lemma 3, we know that the solution of (7) is unique. Then .

Remark 9. From Theorem 8, we get the first order perturbation bound for the solution as follows:

as.

Combining this with (29) gives

as .

4. Backward Error

In this section, a backward error of an approximate solution for the unique solution to (1) is obtained.

Theorem 10. Let be an approximation to the solution of (1). If and the residual satisfies

then

Proof. Let

where . Obviously, is a nonempty bounded convex closed set. Let

Evidently is continuous. We will prove that . For every , we have

Hence

That is,

Using (43), one sees that

Therefore .

According to (17), we obtain

for . That is, . By Brouwer’s fixed point theorem, there exists a such that . Hence is a solution of (1). Moreover, by Lemma 3, we know that the solution of (1) is unique. Then

5. Condition Number

In this section, we apply the theory of condition number developed by Rice [36] to study condition numbers of the unique solution to (1).

5.1. The Complex Case

Suppose that and are the solutions of the matrix equations (1) and (7), respectively. Let , and . Using Theorem 8 and Remark 9, we have

as .

By the theory of condition number developed by Rice [36], we define the condition number of the Hermitian positive definite solution to (1) by

where , , and ,  , are positive parameters. Taking in (54) gives the absolute condition number , and taking , , and in (54) gives the relative condition number .

Substituting (53) into (54), we get

Let be the matrix representation of the linear operator . Then it is easy to see that

Let

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

Then we obtain that

Then we have the following theorem.

Theorem 11. If , then the condition number defined by (54) has the explicit expression

where the matrices and are defined as in (57).

Remark 12. From (60) we have the relative condition number

5.2. The Real Case

In this subsection we consider the real case. That is, all the coefficient matrices , of (1) are real. In such a case the corresponding solution is also real. Completely similar arguments as Theorem 11 give the following theorem.

Theorem 13. Let , be real and let be the condition number defined by (54). If , then has the explicit expression

where

Remark 14. In the real case the relative condition number is given by

6. Numerical Examples

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

Example 15. We consider the matrix equation

with

Suppose that the coefficient matrices and are perturbed to , where

and is a random matrix generated by MATLAB function randn.

We now consider the corresponding perturbation bounds for the solution in Theorems 5 and 8.

The assumptions in Theorem 5 are

The assumptions in Theorem 8 are

By computation, we list them in Table 1.

Table 1: Assumptions check for Example 15 with different values of .

4
5
6
7

The results listed in Table 1 show that the assumptions of Theorems 5 and 8 are satisfied.

By Theorems 5 and 8, we can compute the relative perturbation bounds and , respectively. These results averaged as the geometric mean of 10 randomly perturbed runs. Some results are listed in Table 2.

Table 2: Perturbation bounds for Example 15 with different values of .

4
5
6
7

The results listed in Table 2 show that the perturbation bound given by Theorem 8 is fairly sharp, while the bound given by Theorem 5 which does not depend on the exact solution is conservative.

Example 16. We consider the matrix equation

with

Choose , . Let the approximate solution of be given with the iterative method (6), where is the iteration number.

The residual satisfies the conditions in Theorem 10.

By Theorem 10, we can compute the backward error bound for as follows:

where

Some results are listed in Table 3.

Table 3: Backward error bound for Example 16 with different values of .

8
10
12
14

The results listed in Table 3 show that the error bound given by Theorem 10 is fairly sharp.

Example 17. We study the matrix equation

with

By Remark 14, we can compute the relative condition number . Some results are listed in Table 4.

Table 4 shows that the unique positive definite solution is well conditioned.

Table 4: Relative condition number for Example 17 with different values of .

1
3
5
7
9

1.0717
1.0228
1.0225
1.0225
1.0225

Acknowledgments

The author wishes to express her gratitude to the referees for their fruitful comments. The work was supported in part by National Nature Science Foundation of China (11201263), Natural Science Foundation of Shandong Province (ZR2012AQ004), and Independent Innovation Foundation of Shandong University (IIFSDU), China.

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