Notes on the Hermitian Positive Definite Solutions of a Matrix Equation

i X δiA i = Q, with −1 ≤ δ i < 0 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.

This type of nonlinear matrix equations arises in many practical applications.Equation (1) in the case  = 1 comes from ladder networks, dynamic programming, control theory, stochastic filtering, statistics, and so forth [1][2][3][4][5][6][7].When  > 1, (1) in the case  1 =  2 = ⋅ ⋅ ⋅ =   = −1 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 generaltype 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 X.After we computed X, 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).
For the case  > 1 and   changes with , Duan et al. [35] proved that (1) with 0 < |  | < 1 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 0 < |  | < 1 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, with |  | < 1. Li [10] gave perturbation analysis for the positive definite solution of (1) with 0 <   < 1. Duan et al. [38] gave two perturbation estimates for the positive definite solution of (1) with 0 < |  | < 1.However, these papers have not considered (1) in the case   = −1 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   = −1 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 −1 ≤   < 0. 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 Furthermore, the monotonicity of    in the former case differs from the latter.Based on the above arguments, we will consider (1) with −1 ≤   < 0 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 −1 ≤   < 0 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 −1 ≤   < 0 is given.Finally, several numerical examples are presented in Section 5.
For the sake of completeness we will provide the proof of the next lemma.

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 −1 ≤   < 0 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 Θ be a partially ordered metric space, with the property that, for any two elements  and  in Θ, there is a positive number  such that  ≤ .Let F : Θ → Θ be a continuous, order reversing map such that there is a  ∈ Θ with F() ≥  for all  ∈ Θ.
Consider F 2 () = F(F()).Assume in addition that for 0 <  < 1 there is a number () > 0 such that, for all  ∈ , ( Then F has a unique fixed point in Θ, and, for every  0 ∈ Θ, the iteration  +1 = F(  ), started with  0 , converges to the unique fixed point.
Proof.To show that F maps  into itself, we only need to show that for  ∈  we have F() ≤ F().In fact, since F is order reversing, something much stronger holds; for  ≥  we have F() ≤ F().In particular, F maps  into .Moreover, F 2 maps Θ into .It follows that if there is a fixed point of F, then it is in .
The fact that F is order reversing means that F 2 is order preserving and one can check that there are two matrices,  (1)  and  (2) , such that these two form a periodic orbit which is the attractor of the iteration of F for any starting value.In addition, we have  (1) = lim  → ∞ F 2+1 () and  (2) = lim  → ∞ F 2 (), so that  (2) ≤  (1) .
The following results are immediate consequences of Theorem 5. Theorem 6.There exists a unique Hermitian positive definite solution  of (1) with −1 ≤   < 0, and the iteration, converges to .
Then the set X is partially ordered and for any two elements,  and , in X there is a positive number  such that  ≤ .A map  associated with ( 1) is defined by Obviously,  : X → X is continuous, and a solution of (1) is a fixed point of .Let Ω = [, ()] and L  () =    ( ∈ Ω, −1 ≤   < 0).By Lemmas 1 and 2, we obtain that L  is order reversing.So L  () ≤ L  ().Then and as L  () > 0 also () =  + ∑  =1  *  L  ()  ≥ .That is,  maps Ω into itself and  is also order reversing.
Corollary 8. Every Hermitian positive definite solution of (1) 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 Remark 9.The above estimate of Hermitian positive definite solution of ( 1) is more precise than that in Theorem 7.
Theorem 10.Let X > 0 be an approximation to the solution  of (25).

Numerical Examples
To illustrate the results of the previous sections, in this section, two simple examples are given, which were carried out using MATLAB 7.1.For the stopping criterion we take Example 1.In this example, we study the following matrix equation:  (41) Choose X0 = .Let the approximate solution X of  be given with the iterative method (7), where  is the iterative number.
(43) Some results are listed in Table 1.
The results listed in Table 1 show that the residual bound given by Theorem 10 is fairly sharp.

Table 1 :
Results for Example 2 with different values of .