Research Article | Open Access
Iterative Algorithm for Solving a System of Nonlinear Matrix Equations
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.
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 [1–3]. In the recent years, some special case of the system (1.2) has been studied in many papers [4–14]. For example, Costa and Marques  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  Mukaidani et al. proposed a numerical algorithm for finding solution of cross-coupled algebraic Riccati equations. In  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  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
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 .
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 .
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 .
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 .