About this Journal Submit a Manuscript Table of Contents
Journal of Applied Mathematics
Volume 2012 (2012), Article ID 461407, 15 pages
http://dx.doi.org/10.1155/2012/461407
Research Article

Iterative Algorithm for Solving a System of Nonlinear Matrix Equations

Faculty of Science and Arts, Qassim University, P.O. Box 1162, Buraydah 51431, Saudi Arabia

Received 15 August 2012; Accepted 7 November 2012

Academic Editor: Abdel-Maksoud A. Soliman

Copyright © 2012 Asmaa M. Al-Dubiban. 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 discuss the positive definite solutions for the system of nonlinear matrix equations and , where , are two positive integers. Some properties of solutions are studied, and the necessary and sufficient conditions for the existence of positive definite solutions are given. An iterative algorithm for obtaining positive definite solutions of the system is proposed. Moreover, the error estimations are found. Finally, some numerical examples are given to show the efficiency of the proposed iterative algorithm.

1. Introduction

In this paper, we consider the system of nonlinear matrix equations that can be expressed in the form where are two positive integers, are unknown matrices, is the identity matrix, and are given nonsingular matrices. All matrices are defined over the complex field.

System of nonlinear matrix equations of the form of (1.1) is a special case of the system of algebraic discrete-type Riccati equations of the form where [1, 2], when . It is well known that the algebraic Riccati equations often arise in control theory, stability theory, communication system, dynamic programming, signal processing, statistics, and so forth [13]. In the recent years, some special case of the system (1.2) has been studied in many papers [414]. For example, Costa and Marques [5] have studied the maximal and stabilizing hermitian solutions for discrete-time-coupled algebraic Riccati equations. Czornik and Świerniak [7, 8] have studied the lower and the upper bounds on the solution of coupled algebraic Riccati equation. In [13] Mukaidani et al. proposed a numerical algorithm for finding solution of cross-coupled algebraic Riccati equations. In [4] Aldubiban has studied the properties of special case of Sys. (1.2) and obtained the sufficient conditions for existence of a positive definite solution and proposed an iterative algorithm to calculate the solutions. In [10] Ivanov proposed a method to solve the discrete-time-coupled algebraic Riccati equations.

This paper is organized as following: in Section 2, we derive the necessary and sufficient conditions of existence solutions for the Sys. (1.1). In Section 3, we introduce an iterative algorithm to obtain the positive definite solutions of Sys. (1.1). We discuss the convergence of the proposed iterative algorithm and study the convergence of the algorithm. Some numerical examples are given to illustrate the efficiency for suggested algorithm in Section 4.

The following notations are used throughout the rest of the paper. The notation means that is a positive semidefinite (positive definite), denotes the complex conjugate transpose of , and is the identity matrix. Moreover, is used as a different notation for . We denote by the spectral radius of , , and mean the eigenvalues of and , respectively. The norm used in this paper is the spectral norm of the matrix , that is, unless otherwise noted.

2. Conditions for Existence of the Solutions

In this section, we will discuss some properties of the solutions for the Sys. (1.1), and we obtain the necessary and sufficient conditions for the existence of the solutions of Sys. (1.1).

Theorem 2.1. If are the smallest and the largest eigenvalues of a solution X of Sys. (1.1), respectively, and are the smallest and the largest eigenvalues of a solution Y of Sys. (1.1), respectively, are eigenvalues of , then

Proof. Let be an eigenvector corresponding to an eigenvalue of the matrix and , and let be an eigenvector corresponding to an eigenvalue of the matrix and . Since the solution of Sys. (1.1) is a positive definite solution then and .
From the Sys. (1.1), we have that is Hence Also, from the Sys. (1.1), we have that is Hence

Theorem 2.2. If Sys. (1.1) has a positive definite solution , then

Proof. Since is a positive definite solution of Sys. (1.1), then From the inequality , we have that is Hence From the inequality , we have that is Hence

Theorem 2.3. Sys. (1.1) has a positive definite solution if and only if the matrices have the factorization where are nonsingular matrices satisfying the following system: In this case the solution is .

Proof. Let Sys. (1.1) have a positive definite solution ; then , where are nonsingular matrices. Then Sys. (1.1) can be rewritten as Letting , , then , , then the Sys. (1.1) is an equivalent to Sys. (2.17).
Conversely, if have the factorization (2.16) and satisfy Sys. (2.17), let , then are positive definite matrices, and we have Hence Sys. (1.1) has a positive definite solution.

3. Iterative Algorithm for Solving the System

In this section, we will investigate the iterative solution of the Sys. (1.1). From this section to the end of the paper we will consider that the matrices are normal satisyfing and .

Let us consider the following iterative algorithm.

Algorithm 3.1. Take .
For compute

Lemma 3.2. For the Sys. (1.1), we have where , are determined by Algorithm 3.1.

Proof. Since , then Using the conditions , we obtain Also, we have Using the conditions , we obtain By the same manner, we get Further, assume that for each it is satisfied that Now, by induction, we will prove Since the two matrices are normal and using the equalities (3.8), therefor Similarly By using the conditions and the equalities (3.8), we have Also, we can prove Therefore, the equalities (3.2) are true for all .

Corollary 3.3. From Lemma 3.2, we have where , are determined by Algorithm 3.1.

Lemma 3.4. For the Sys. (1.1), we have where , are determined by Algorithm 3.1.

Proof. Since then .
By using the equalities (3.14), we have Similarly we get Further, assume that for each it is satisfied that Now, we will prove From the equalities (3.18), we have By using the equalities (3.14) and (3.20), we have By the same manner, we can prove Therefore, the equalities (3.15) are true for all .

Theorem 3.5. If are satisfying the following conditions: (i), (ii), then the Sys. (1.1) has a positive definite solution , which satisfy where , , are determined by Algorithm 3.1.

Proof. For we have and .
Since then and , hence , , that is, We find the relation between , and and the relation between , and .
Since , then , , , .
Since , then , , , .
Also since , then , , , .
Thus we get So, assume that for each it is satisfied that Now, we will prove and .
By using the inequalities (3.26) we have Also we have Similarly Also we have Therefore, the inequalities (3.26) are true for all ; consequently the subsequences , and are monotonic and bounded. therefore they are convergent to positive definite matrices. To prove that the sequences have a common limit, we have Since , then we have Consequently Also, to prove that the sequences have a common limit, we have Since , then we have Consequently By using (3.36) in (3.33) and (3.33) in (3.36), we have Therefore Consequently the subsequences are convergent and have a common positive definite limit . Also, the subsequences are convergent and have a common positive definite limit . Therefore is a positive definite solution of Sys. (1.1).

4. Numerical Examples

In this section the numerical examples are given to display the flexibility of the method. The solutions are computed for some different matrices with different orders. In the following examples we denote by the solutions which are obtained by iterative Algorithm 3.1, and , , and .

Example 4.1. Consider Sys. (1.1) with , and matrices By computation, we get The results are given in the Table 1.

tab1
Table 1

Example 4.2. Consider Sys. (1.1) with , and matrices By computation, we get The results are given in the Table 2.

tab2
Table 2

5. Conclusion

In this paper we considered the system of nonlinear matrix equations (1.1) where are two positive integers. We achieved the general conditions for the existence of a positive definite solution. Moreover, we discussed an iterative algorithm from which a solution can always be calculated numerically, whenever the system is solvable. The numerical examples included in this paper showed the efficiency of the iterative algorithm which is described.

References

  1. P. Lancaster and L. Rodman, Algebraic Riccati Equations, Oxford Science, 1995.
  2. B. Meini, “Matrix equations and structures: effcient solution of special discrete algebraic Riccati equations,” in Proceedings of the WLSSCOO, 2000.
  3. W. N. Anderson Jr., T. D. Morley, and G. E. Trapp, “Positive solutions to X=ABX1B*,” Linear Algebra and its Applications, vol. 134, pp. 53–62, 1990. View at Publisher · View at Google Scholar · View at Zentralblatt MATH
  4. A. M. Aldubiban, Iterative algorithms for computing the positive definite solutions for nonlinear matrix equations [Ph.D. thesis], Riyadh University for Girls, Riyadh, Saudi Arabia, 2008.
  5. O. L. V. Costa and R. P. Marques, “Maximal and stabilizing Hermitian solutions for discrete-time coupled algebraic Riccati equations,” Mathematics of Control, Signals, and Systems, vol. 12, no. 2, pp. 167–195, 1999. View at Publisher · View at Google Scholar · View at Zentralblatt MATH
  6. O. L. V. Costa and J. C. C. Aya, “Temporal difference methods for the maximal solution of discrete-time coupled algebraic Riccati equations,” Journal of Optimization Theory and Applications, vol. 109, no. 2, pp. 289–309, 2001. View at Publisher · View at Google Scholar · View at Zentralblatt MATH
  7. A. Czornik and A. Swierniak, “Lower bounds on the solution of coupled algebraic Riccati equation,” Automatica, vol. 37, no. 4, pp. 619–624, 2001. View at Publisher · View at Google Scholar · View at Zentralblatt MATH
  8. A. Czornik and A. Świerniak, “Upper bounds on the solution of coupled algebraic Riccati equation,” Journal of Inequalities and Applications, vol. 6, no. 4, pp. 373–385, 2001. View at Publisher · View at Google Scholar · View at Zentralblatt MATH
  9. R. Davies, P. Shi, and R. Wiltshire, “Upper solution bounds of the continuous and discrete coupled algebraic Riccati equations,” Automatica, vol. 44, no. 4, pp. 1088–1096, 2008. View at Publisher · View at Google Scholar
  10. I. G. Ivanov, “A method to solve the discrete-time coupled algebraic Riccati equations,” Applied Mathematics and Computation, vol. 206, no. 1, pp. 34–41, 2008. View at Publisher · View at Google Scholar · View at Zentralblatt MATH
  11. I. G. Ivanov, “Stein iterations for the coupled discrete-time Riccati equations,” Nonlinear Analysis. Theory, Methods & Applications A, vol. 71, no. 12, pp. 6244–6253, 2009. View at Publisher · View at Google Scholar · View at Zentralblatt MATH
  12. H. Mukaidani, “Newton's method for solving cross-coupled sign-indefinite algebraic Riccati equations for weakly coupled large-scale systems,” Applied Mathematics and Computation, vol. 188, no. 1, pp. 103–115, 2007. View at Publisher · View at Google Scholar · View at Zentralblatt MATH
  13. H. Mukaidani, S. Yamamoto, and T. Yamamoto, “A Numerical algorithm for finding solution of cross-coupled algebraic Riccati equations,” IEICE Transactions, vol. E91-A, pp. 682–685, 2008. View at Publisher · View at Google Scholar
  14. H. Mukaidani, “Numerical computation of cross-coupled algebraic Riccati equations related to H-constrained LQG control problem,” Applied Mathematics and Computation, vol. 199, no. 2, pp. 663–676, 2008. View at Publisher · View at Google Scholar