- About this Journal ·
- Abstracting and Indexing ·
- Advance Access ·
- Aims and Scope ·
- Annual Issues ·
- Article Processing Charges ·
- Articles in Press ·
- Author Guidelines ·
- Bibliographic Information ·
- Citations to this Journal ·
- Contact Information ·
- Editorial Board ·
- Editorial Workflow ·
- Free eTOC Alerts ·
- Publication Ethics ·
- Reviewers Acknowledgment ·
- Submit a Manuscript ·
- Subscription Information ·
- Table of Contents

Abstract and Applied Analysis

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

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

## Perturbation Analysis of the Nonlinear Matrix Equation

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.

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.

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.

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.

#### 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.

#### References

- B. L. Buzbee, G. H. Golub, and C. W. Nielson, “On direct methods for solving Poisson's equations,”
*SIAM Journal on Numerical Analysis*, vol. 7, pp. 627–656, 1970. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet - W. N. Anderson Jr., G. B. Kleindorfer, P. R. Kleindorfer, and M. B. Woodroofe, “Consistent estimates of the parameters of a linear system,”
*Annals of Mathematical Statistics*, vol. 40, pp. 2064–2075, 1969. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet - W. N. Anderson, T. D. Morley, and G. E. Trapp, “The cascade limit, the shorted operator and quadratic optimalcontrol,” in
*Linear Circuits, Systems and Signal Processsing: Theory and Application*, C. I. Byrnes, F. C. Martin, and R. E. Saeks, Eds., pp. 3–7, North-Holland, New York, NY, USA, 1988. View at Google Scholar - R. S. Bucy, “A priori bounds for the Riccati equation,” in
*Proceedings of the 6th Berkeley Symposium on Mathematical Statistics and Probability*, vol. 3 of*Probability Theory*, pp. 645–656, University of California, Berkeley, Calif, USA, 1972. View at MathSciNet - D. V. Ouellette, “Schur complements and statistics,”
*Linear Algebra and its Applications*, vol. 36, pp. 187–295, 1981. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet - W. Pusz and S. L. Woronowicz, “Functional calculus for sesquilinear forms and the purification map,”
*Reports on Mathematical Physics*, vol. 8, no. 2, pp. 159–170, 1975. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet - J. Zabczyk, “Remarks on the control of discrete-time distributed parameter systems,”
*SIAM Journal on Control and Optimization*, vol. 12, pp. 721–735, 1974. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet - M. Berzig, X. F. Duan, and B. Samet, “Positive definite solution of the matrix equation $X=Q-{A}^{*}{X}^{-1}A+{B}^{*}{X}^{-1}B$ via Bhaskar—Lakshmikantham fixed point theorem,”
*Mathematical Sciences*, vol. 6, article 27, 2012. View at Google Scholar - J. Cai and G. L. Chen, “On the Hermitian positive definite solution of nonlinear matrix equation ${X}^{s}+{A}^{*}{X}^{-t}A=Q$,”
*Applied Mathematics and Computation*, vol. 217, pp. 2448–2456, 2010. View at Google Scholar - X. Duan and A. Liao, “On the existence of Hermitian positive definite solutions of the matrix equation ${X}^{s}+{A}^{*}{X}^{-t}A=Q$,”
*Linear Algebra and its Applications*, vol. 429, no. 4, pp. 673–687, 2008. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet - X. Duan, C. Li, and A. Liao, “Solutions and perturbation analsis for the nonlinear matrix equation $X+{\sum}_{i=1}^{m}{A}_{i}^{*}{X}^{-1}{A}_{i}=I$,”
*Applied Mathematics and Computation*, vol. 218, no. 8, pp. 4458–4466, 2011. View at Publisher · View at Google Scholar · View at MathSciNet - J. C. Engwerda, “On the existence of a positive definite solution of the matrix equation $X+{A}^{T}{X}^{-1}A=I$,”
*Linear Algebra and its Applications*, vol. 194, pp. 91–108, 1993. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet - A. Ferrante and B. C. Levy, “Hermitian solutions of the equation $X=Q+N{X}^{-1}{N}^{*}$,”
*Linear Algebra and its Applications*, vol. 247, pp. 359–373, 1996. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet - C.-H. Guo and P. Lancaster, “Iterative solution of two matrix equations,”
*Mathematics of Computation*, vol. 68, no. 228, pp. 1589–1603, 1999. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet - V. I. Hasanov, “Positive definite solutions of the matrix equations $X\pm {A}^{*}{X}^{-q}A=Q$,”
*Linear Algebra and its Applications*, vol. 404, pp. 166–182, 2005. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet - V. I. Hasanov and I. G. Ivanov, “On two perturbation estimates of the extreme solutions to the equations $X\pm {A}^{*}{X}^{-1}A=Q$,”
*Linear Algebra and its Applications*, vol. 413, no. 1, pp. 81–92, 2006. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet - V. I. Hasanov, I. G. Ivanov, and F. Uhlig, “Improved perturbation estimates for the matrix equations $X\pm {A}^{*}{X}^{-1}A=Q$,”
*Linear Algebra and its Applications*, vol. 379, pp. 113–135, 2004, Tenth Conference of the International Linear Algebra Society. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet - Y.-M. He and J.-H. Long, “On the Hermitian positive definite solution of the nonlinear matrix equation $X+{\sum}_{i=1}^{m}{A}_{i}^{*}{X}^{-1}{A}_{i}$,”
*Applied Mathematics and Computation*, vol. 216, no. 12, pp. 3480–3485, 2010. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet - I. G. Ivanov and S. M. El-sayed, “Properties of positive definite solutions of the equation $X+{A}^{*}{X}^{-2}A=I$,”
*Linear Algebra and its Applications*, vol. 279, no. 1–3, pp. 303–316, 1998. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet - I. G. Ivanov, V. I. Hasanov, and B. V. Minchev, “On matrix equations $X\pm {A}^{*}{X}^{-2}A=I$,”
*Linear Algebra and its Applications*, vol. 326, no. 1–3, pp. 27–44, 2001. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet - J. Li and Y. H. Zhang, “The Hermitian positive definite solution and its perturbation analysis for the matrix equation $X-{A}^{*}{X}^{-1}A=Q$,”
*Mathematica Numerica Sinica*, vol. 30, no. 2, pp. 129–142, 2008 (Chinese). View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet - J. Li and Y. Zhang, “Perturbation analysis of the matrix equation $X-{A}^{*}{X}^{-q}A=Q$,”
*Linear Algebra and its Applications*, vol. 431, no. 9, pp. 1489–1501, 2009. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet - X.-G. Liu and H. Gao, “On the positive definite solutions of the matrix equations ${X}^{s}\pm {A}^{T}{X}^{-t}A={I}_{n}$,”
*Linear Algebra and its Applications*, vol. 368, pp. 83–97, 2003. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet - J.-G. Sun and S.-F. Xu, “Perturbation analysis of maximal solution of the matrix equation $X+{A}^{*}{X}^{-1}A=P$. II,”
*Linear Algebra and its Applications*, vol. 362, pp. 211–228, 2003. View at Publisher · View at Google Scholar · View at MathSciNet - H. Xiao and J. T. Wang, “On the matrix equation $X-{A}^{*}{X}^{-p}A=Q\hspace{0.17em}\hspace{0.17em}(p>1)$,”
*Chinese Journal of Engineering Mathematics*, vol. 26, no. 2, pp. 305–309, 2009. View at Google Scholar · View at MathSciNet - S.-F. Xu, “Perturbation analysis of the maximal solution of the matrix equation $X+{A}^{*}{X}^{-1}A=P$,”
*Linear Algebra and its Applications*, vol. 336, pp. 61–70, 2001. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet - X. Yin, S. Liu, and L. Fang, “Solutions and perturbation estimates for the matrix equation ${X}^{s}+{A}^{*}{X}^{-t}A=Q$,”
*Linear Algebra and its Applications*, vol. 431, no. 9, pp. 1409–1421, 2009. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet - X. Yin, S. Liu, and T. Li, “On positive definite solutions of the matrix equation $X+{A}^{*}{X}^{-q}A=Q\hspace{0.17em}\hspace{0.17em}(0<q\le 1)$,”
*Taiwanese Journal of Mathematics*, vol. 16, no. 4, pp. 1391–1407, 2012. View at Google Scholar · View at MathSciNet - X. Zhan, “Computing the extremal positive definite solutions of a matrix equation,”
*SIAM Journal on Scientific Computing*, vol. 17, no. 5, pp. 1167–1174, 1996. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet - X. Zhan and J. Xie, “On the matrix equation $X+{A}^{T}{X}^{-1}A=I$,”
*Linear Algebra and its Applications*, vol. 247, pp. 337–345, 1996. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet - Y. Zhang, “On Hermitian positive definite solutions of matrix equation $X+{A}^{*}{X}^{-2}A=I$,”
*Linear Algebra and its Applications*, vol. 372, pp. 295–304, 2003. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet - Y.-H. Zhang, “On Hermitian positive definite solutions of matrix equation $X-{A}^{*}{X}^{-2}A=I$,”
*Journal of Computational Mathematics*, vol. 23, no. 4, pp. 408–418, 2005. View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet - G. Jia and D. Gao, “Perturbation estimates for the nonlinear matrix equation $X-{A}^{*}{X}^{q}A=Q(0<q<1)$,”
*Journal of Applied Mathematics and Computing*, vol. 35, no. 1-2, pp. 295–304, 2011. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet - X. Duan, A. Liao, and B. Tang, “On the nonlinear matrix equation $X-{\sum}_{i=1}^{m}{A}_{i}^{*}{X}^{{\delta}_{i}}{A}_{i}=Q$,”
*Linear Algebra and its Applications*, vol. 429, no. 1, pp. 110–121, 2008. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet - X. Zhan,
*Matrix Inequalities*, vol. 1790 of*Lecture Notes in Mathematics*, Springer, Berlin, Germany, 2002. View at Publisher · View at Google Scholar · View at MathSciNet - J. R. Rice, “A theory of condition,”
*SIAM Journal on Numerical Analysis*, vol. 3, no. 2, pp. 287–310, 1966. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet