1 Introduction
Before we present our research, it is worth pointing out the real life experience (community volunteer activity) relevant to our research work. Besides the interest in our own problem, this class of equations appears, for instance, in the optimal control problem for infinite Markov jump linear systems [3] or linearquadratic twoperson zerosum differential games over an infinite horizon [11]
. An important recent development in the use of invariant subspacebased methods is the use of iterative refinement. As we know, this idea is closely related to the solution of the Riccati equation in Newton’s method. In particular, Newton’s method itself is an interesting yet useful technique, and now a fairly complete theory has been developed for this algorithm, including a computable estimate of the convergence area. We refer the interested reader to
[9, 6, 7, 4, 5] for details.The algebraic Riccati equations play fundamental roles in engineering, management, economic, finance, linearquadratic control and estimation systems as well as in many branches of applied mathematics. The purpose of this article is not to investigate the vast literature available for these equations, but to provide readers with references [9, 6, 7, 4, 5, 12, 8, 2, 10]
. Generally, algebraic Riccati equations that appear in game theory are nonsymmetric, so this makes the effective numerical methods developed for symmetric equations based on control theory inapplicable. Our problem is to find the solution of the coupled algebraic Riccati equations generated by twoperson linearquadratic nonzero sum deterministic differential games over an infinite horizon with closedloop Nash equilibria. These equations based on closedloop type Nash equilibria are symmetric, but they are coupled together so that the methods established to solve control problems cannot be directly applied. In this article, we develop an iteration approach to tackle these coupled algebraic Riccati equations, and propose an effective algorithm for finding positive definite solutions.
The article is organized as follows. Section 2 introduces some basic notions and presents formulation and algorithm. In Section 3, we present several numerical examples according to different dimensions to illustrate the proposed approach. Section 4 concludes this research work.
2 Formulation and Algorithm
Throughout this paper, we let and be the set of all (real) matrices and symmetric (real) matrices. To simplify the notation, we denote (for )
where the notation denotes transpose of matrix .
Definition 2.1.
The following is called a system of coupled algebraic Riccati equations (AREs):
(2.1) 
where , and satisfies
(2.2) 
and
Since we cannot solve the coupled algebraic Riccati equations (2.1)(2.2) directly, we separate the coupled equations into two decoupled equations (2.4) and (2.7) for finding positive definite solutions using iterations in the following algorithm.
[t]0.9 Algorithm:

Let be initial matrix solution of iteration and .

Substituting into (2.2) yields .
[t]0.9 Algorithm:

If , then set and go to Step 2. Otherwise, stop.
Since and for , we have . In addition, and . Therefore, applying the Schur method in [9], we can tackle equations (2.4) and (2.7
) directly. Although the Schurbased method has all the reliability and efficiency, it is relatively difficult to effectively parallelize and vectorize. Therefore, other methods have been reexamined in order to implement them on various advanced computing architectures. Due to its nonlinearity in
, it is difficult to study its numerical analysis. Numerical theory will be the subject of our future research. In addition, we also can tackle equations (2.4) and (2.7) using a computational approach via a semidefinite programming associated with linear matrix inequalities, whose feasibility is equivalent to the solvability of equations (2.4) and (2.7).3 Examples
In this section, we present some examples illustrating the algorithm in the previous section. We discuss about five cases: (1) , and ; (2) , and ; (3) , and ; (4) , and ; (5) , and . All computations are done on an iMac using Matlab software and double precision arithmetic.
Example 3.1.
Consider the following problem with , and . In this example,
Then the corresponding system of generalized AREs (2.1)(2.2) reads
(3.1) 
which is equivalent to
(3.2) 
This is an example with two scalar variables and two quadratic equations. Let be initial solution of iteration and . Using algorithm in Section 2, we can get
Then (2.4)(2.5) and (2.7)(2.8) reads
Several iterations leads to a pair of solution satisfying (3.1) and its equivalent equations (3.2).
Example 3.2.
Consider the following problem with , and . In this example,
Then the corresponding system of generalized AREs (2.1)(2.2) reads
(3.3) 
where , and . Let be initial matrix solution of iteration, where
(3.4) 
and . Using algorithm in Section 2, we can get
Then (2.4)(2.5) and (2.7)(2.8) reads
Several iterations leads to a pair of solution
(3.5) 
satisfying (3.3).
Example 3.3.
Consider the following problem with , and . In this example,
Then the corresponding system of generalized AREs (2.1)(2.2) reads
(3.6) 
where , and . Let be initial matrix solution of iteration, where
(3.7) 
and . Using algorithm in Section 2, we can get
where and . Then (2.4)(2.5) and (2.7)(2.8) reads
where . Several iterations leads to a pair of solution
(3.8) 
satisfying (3.6).
Example 3.4.
Consider the following problem with , and . In this example,
Then the corresponding system of generalized AREs (2.1)(2.2) reads
(3.9) 
where , and . Let be initial matrix solution of iteration, where
(3.10) 
and . Using algorithm in Section 2, we can get
where , and . Then (2.4)(2.5) and (2.7)(2.8) reads
where and . Several iterations leads to a pair of solution
(3.11) 
satisfying (3.9).
Example 3.5.
Consider the following problem with , and . In this example,
Then the corresponding system of generalized AREs (2.1)(2.2) reads
(3.12) 
where , and . Let be initial matrix solution of iteration, where
(3.13) 
and . Using algorithm in Section 2, we can get
where , and . Then (2.4)(2.5) and (2.7)(2.8) reads
where and . Several iterations leads to a pair of solution
(3.14) 
satisfying (3.12).
4 Conclusion
We have learned many methods of algebraic Riccati equations and their performance in computer finite arithmetic environments. However, many important questions remain unanswered and are the subject of ongoing research topics.
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