Abstract

The nonlinear matrix equation, with is investigated. A fixed point theorem in partially ordered sets is proved. And then, by means of this fixed point theorem, the existence of a unique Hermitian positive definite solution for the matrix equation is derived. Some properties of the unique Hermitian positive definite solution are obtained. A residual bound of an approximate solution to the equation is evaluated. The theoretical results are illustrated by numerical examples.

1. Introduction

In this paper we consider the Hermitian positive definite solution of the following nonlinear matrix equation: where , are complex matrices, is an Hermitian positive definite matrix, and is a positive integer. Here, denotes the conjugate transpose of the matrix .

This type of nonlinear matrix equations arises in many practical applications. Equation (1) in the case comes from ladder networks, dynamic programming, control theory, stochastic filtering, statistics, and so forth [1–7]. When , (1) in the case is recognized as playing an important role in modeling certain optimal interpolation problems (see [8, 9] for more details). The Hermitian positive definite solutions of the general-type equation (1) play an important role in connection with a certain system of linear equations in many physical calculations (see [7, 10] for more details). When solving the nonlinear matrix equation (1), we often do not avoid some round-off errors. Then we only get an approximation . After we computed , we would like to know how good our computation was. Motivated by these, we consider in this paper the Hermitian positive definite solutions and the residual bound of (1).

In the last few years, (1) was investigated in some special cases. For the nonlinear matrix equations, [11–17], [18], [19, 20], [21], [22–25], and [9, 26, 27], there were many contributions in the literature to the solvability, numerical solutions, and perturbation analysis. In addition, the related general equations, [28–34], were studied by some scholars.

For the case and changes with , Duan et al. [35] proved that (1) with has a unique Hermitian positive definite solution by fixed point theorems for monotone and mixed monotone operators in a normal cone. Lim [36] showed that (1) with has a unique Hermitian positive definite solution by using a strict contraction for the Thompson metric on the open convex cone of positive definite matrices. Shi et al. [37] studied the existence and uniqueness of solutions of nonlinear matrix equations, and , with . Li [10] gave perturbation analysis for the positive definite solution of (1) with . Duan et al. [38] gave two perturbation estimates for the positive definite solution of (1) with . However, these papers have not considered (1) in the case for some and the approaches in these papers will become invalid in this case. Meanwhile, in some practical problems (e.g., in certain optimal interpolation problems), the case of for some is required. To our best knowledge, there has been no literature paying attention to the Hermitian positive definite solutions and the residual bound for (1) with . By using the integral representation of matrix function and the fixed point theorem, we prove the existence of a unique Hermitian positive definite solution to (1) and consider the residual bound of this equation. Note that the integral representation of matrix function in the case is different from the case . Furthermore, the monotonicity of in the former case differs from the latter. Based on the above arguments, we will consider (1) with in this paper.

The rest of the paper is organized as follows. In Section 2, we give some preliminary lemmas that will be needed to develop this work. In Section 3, a fixed point theorem in partially ordered sets is proved. And then, by means of this fixed point theorem, the existence of a unique Hermitian positive definite solution for the matrix equation (1) with is derived. We propose an iterative method to compute the Hermitian positive definite solution. We also obtain some properties of the unique Hermitian positive definite solution. Furthermore, in Section 4, a residual bound for the unique Hermitian positive definite solution to (1) with is given. Finally, several numerical examples are presented in Section 5.

We denote by the set of complex matrices, by the set of Hermitian matrices, by the identity matrix, by the spectral norm, and by and the maximal and minimal eigenvalues of , respectively. For , we write (resp., if is Hermitian positive semidefinite (resp., definite). Further, the sets and are defined by and , respectively.

2. Preliminaries

Lemma 1 (see [39]). If , then .

Lemma 2 (see [39]). If and , then .

For the sake of completeness we will provide the proof of the next lemma.

Lemma 3 (see [17]). For every positive definite matrix , if , with , then

Proof. Suppose that is a positive definite matrix. If , then

Lemma 4 (see [23]). For every Hermitian positive definite matrix and , it yields that(i); (ii), . In addition, if and , with , then

3. The Positive Definite Solutions

In this section, we use a new method, which is different from the approaches applied in [35, 36] to prove that (1) with has a unique Hermitian positive definite solution . Meanwhile, we give an iterative method to compute the unique Hermitian positive definite solution for arbitrary initial positive definite matrix. Moreover, we obtain some properties of the Hermitian positive definite solution to (1).

Theorem 5. Let be a partially ordered metric space, with the property that, for any two elements and in , there is a positive number such that .
Let be a continuous, order reversing map such that there is a with for all . Put . Then maps into itself.
Consider . Assume in addition that for there is a number such that, for all , Then has a unique fixed point in , and, for every , the iteration , started with , converges to the unique fixed point.

Proof. To show that maps into itself, we only need to show that for we have . In fact, since is order reversing, something much stronger holds; for we have . In particular, maps into . Moreover, maps into . It follows that if there is a fixed point of , then it is in .
The fact that is order reversing means that is order preserving and one can check that there are two matrices, and , such that these two form a periodic orbit which is the attractor of the iteration of for any starting value. In addition, we have and , so that .
It remains to show that, under the extra condition (5), and are equal. In fact, we will show that , which is enough. By the assumption on , there is a such that . Let . Suppose that . Then, using (5), which contradicts the definition of . So , and in particular .

The following results are immediate consequences of Theorem 5.

Theorem 6. There exists a unique Hermitian positive definite solution of (1) with , and the iteration, converges to .

Proof. Let . Then the set is partially ordered and for any two elements, and , in there is a positive number such that . A map associated with (1) is defined by Obviously, is continuous, and a solution of (1) is a fixed point of . Let and . By Lemmas 1 and 2, we obtain that is order reversing. So . Then and as also . That is, maps into itself and is also order reversing.
By Theorem 5, it remains to prove that, for , there exists a number such that, for all , In fact, choose . Note that For , a calculation gives which completes the proof.

Theorem 7. If is an Hermitian positive definite solution of (1), then , where and are, respectively, the solutions of the following equations: Moreover,

Proof. Define the sequences and as From (16), it follows that Supposing and , then Hence, for each , we have and , which imply that the sequences and are monotonic and bounded. Therefore, they are convergent to certain positive numbers. Let Taking limits in (16) yields which imply that Therefore and satisfy (13) and (14), respectively. We will prove that for any positive definite solution . According to Lemmas 1 and 2 and the sequences defined by (16), it follows that for each Hermitian positive definite solution . From , it follows that . Hence Since , it follows that and . Note that inequality (22) implies . By similar induction, it yields that Taking limits on both sides of inequality (23), we have .

Corollary 8. Every Hermitian positive definite solution of (1) is in , where and are defined as in Theorem 7.

Proof. We suppose that is the Hermitian positive definite solution of (1). By Theorem 7, it follows that Using , we obtain . Applying inequality (24) yields .

Remark 9. The above estimate of Hermitian positive definite solution of (1) is more precise than that in Theorem 7.

4. Residual Bound

The matrix equation (1) with includes the following cases: first, some in and others ; secondly, all in ; thirdly, all . Without loss of generality, let , , and , . Then (1) can be rewritten as In this section, a residual bound of an approximate solution for the unique solution to (25) is developed.

Theorem 10. Let be an approximation to the solution of (25). If and the residual satisfies then