Solutions and improved perturbation analysis for the matrix equation X-A^{*}X^{-p}A=Q (p>0)

In this paper the nonlinear matrix equation X-A^{*}X^{-p}A=Q with p>0 is investigated. We consider two cases of this equation: the case p>1 and the case 0<p<1. In the case p>1, a new sufficient condition for the existence of a unique positive definite solution for the matrix equation is obtained. A perturbation estimate for the positive definite solution is derived. Explicit expressions of the condition number for the positive definite solution are given. In the case 0<p<1, a new sharper perturbation bound for the unique positive definite solution is evaluated. A new backward error of an approximate solution to the unique positive definite solution is obtained. The theoretical results are illustrated by numerical examples.


Introduction
In this paper we consider the Hermitian positive definite solution of the nonlinear matrix equation where A, Q and X are n × n complex matrices, Q is a positive definite matrix and p > 0. This type of nonlinear matrix equations arises in the analysis of ladder networks, the dynamic programming, control theory, stochastic filtering, statistics and many applications [1-4, 26, 27, 36].
In [13], a sufficient condition for the equation X − A * X −p A = Q (0 < p ≤ 1) to have a unique positive definite solution was provided. When the coefficient matrix A is nonsingular, several sufficient conditions for the equation X − A * X −q A = Q (q ≥ 1) to have a unique positive definite solution were given in [33]. When the coefficient matrix A is an arbitrary complex matrix, necessary conditions and sufficient conditions for the existence of positive definite solutions for the equation X−A * X −q A = Q (q ≥ 1) were derived in [31]. Li and Zhang in [24] proved that there always exists a unique positive definite solution to the equation X − A * X −p A = Q (0 < p < 1). They also obtained a perturbation bound and a backward error of an approximate solution for the unique solution of the equation X − A * X −p A = Q (0 < p < 1).
As a continuation of the previous results, the rest of the paper is organized as follows. Section 2 gives some preliminary lemmas that will be needed to develop this work. In Section 3, a new sufficient condition for Eq.(1.1) with p > 1 existing a unique positive definite solution is derived. In Section 4, a perturbation bound for the positive definite solution to Eq.(1.1) with p > 1 is given. In Section 5, applying the integral representation of matrix function, we also discuss the explicit expressions of condition number for the positive definite solution to Eq.(1.1) with p > 1. Furthermore, in Section 6, a new sharper perturbation bound for the unique positive definite solution to Eq.(1.1) with 0 < p < 1 is evaluated. In Section 7, a new backward error of an approximate solution to Eq.(1.1) with 0 < p < 1 is obtained. Finally, several numerical examples are presented in Section 8.
We denote by C n×n the set of n × n complex matrices, by H n×n the set of n × n Hermitian matrices, by I the identity matrix, by · the spectral norm, by · F the Frobenius norm and by λ max (M) and λ min (M) the maximal and minimal eigenvalues of M, respectively. For A = (a 1 , . . . , a n ) = (a i j ) ∈ C n×n and a matrix B, A ⊗ B = (a i j B) is a Kronecker product, and vecA is a vector defined by vecA

Priliminaries
In this section, we will give some preliminary lemmas that will be needed to develop this work.
Lemma 2.1. [24] For every positive definite matrix X ∈ H n×n , if 0 < p < 1, then Lemma 2.2. [24] There exists a unique positive definite solution X of X − A * X −p A = Q (0 < p < 1) and the iteration converges to X.
(ii) If X ∈ H n×n and r > 0, then

A sufficient condition for the existence of a unique solution of
In this section, we derive a new sufficient condition for the existence of a unique solution of X − A * X −p A = Q (p > 1) beginning with the lemma.
then Eq.(1.1) has a unique positive definite solution.

Perturbation bound for
Li and Zhang in [24] proved that there always exists a unique positive definite solution to the equation X − A * X −p A = Q (0 < p < 1). They also obtained a perturbation bound for the unique solution. But their approaches will become invalid for the case of p > 1. Since the equation does not always have a unique positive definite solution, there are two difficulties for perturbation analysis to the equation X − A * X −p A = Q (p > 1). One difficulty is how to find some reasonable restrictions on the coefficient matrices of perturbed equation ensuring this equation has a unique positive definite solution. The other difficulty is how to find an expression of ∆X which is easy to handle.
Assume that the coefficient matrix A is perturbed to A = ∆A + A. Let X = ∆X + X with ∆X ∈ H n×n satisfying the perturbed equation In the following, we derive a perturbation estimate for the positive definite solution to the matrix equation

Condition number for
A condition number is a measurement of the sensitivity of the positive definite stabilizing solutions to small changes in the coefficient matrices. In this section, we apply the theory of condition number developed by Rice [28] to derive explicit expressions of the condition number for the matrix equation Here we consider the perturbed equation where A and Q are small perturbations of A and Q in Eq.(1.1), respectively. Suppose that p A 2 < λ p+1 min (Q) and p A 2 < λ p+1 min ( Q). According to Lemma 4.1, Eq.(1.1) and Eq.(5.1) have unique positive definite solutions X and X, respectively. Let ∆X = X − X, Therefore where Proof. Define the operator R : H n×n → H n×n by which implies that ||R|| < 1 and I + R is invertible. Therefore, the operator V is invertible.
Thus, we can rewrite (5.2) as Obviously, By the theory of condition number developed by Rice [6], we define the condition number of the Hermitian positive definite solution X to the matrix equation where ξ, η and ρ are positive parameters. Taking ξ = η = ρ = 1 in (5.6) gives the absolute condition number c abs (X), and taking ξ = ||X|| F , η = ||A|| F and ρ = ||Q|| F in (5.6) gives the relative condition number c rel (X). 7 Substituting (5.5) into (5.6), we get Let V be the matrix representation of the linear operator V. Then it is easy to see that where x, y, a, b ∈ R n 2 , S , Σ, U 1 , U 2 , Ω 1 , Ω 2 ∈ R n 2 ×n 2 , M = (E, H), i = √ −1, Π is the vecpermutation matrix, i.e., vec E T = Π vec E.
Furthermore, we obtain that Then we have the following theorem.

The real case
In this subsection we consider the real case, i.e., all the coefficient matrices A, Q of the matrix equation X − A * X −p A = Q (p > 1) are real. In such a case the corresponding solution X is also real. Completely similar arguments as in Theorem 5.2 give the following theorem. (5.6). If p A 2 < λ p+1 min (Q), then c(X) has the explicit expression

Remark 2.
In the real case the relative condition number is given by

New perturbation bound for
Here we consider the perturbed equation By Lemma 5.1 in [24], the linear operator L : H n×n → H n×n defined by is invertible. We also define operator P : C n×n → H n×n by Thus,we can rewrite (6.2) as

Now we denote
Theorem 6.1. If .
Obviously, f : H n×n → H n×n is continuous. The condition (6.5) ensures that the quadratic equation ζ(l + η)x 2 − l(1 + ζǫ − σ)x + lǫ = 0 in x has two positive real roots. The smaller one is .

New backward error for
In this section we evaluate a new backward error of an approximate solution to the unique solution, which is sharper than that in Theorem 4.1 [24]. Theorem 7.1. Let X > 0 be an approximation to the solution X of (1.1). If X − p 2 A 2 X −1 < 1 and the residual R( X) To prove the above theorem, we first verify the following lemma.

Numerical Examples
To illustrate the theoretical results of the previous sections, in this section four simple examples are given, which were carried out using MATLAB 7.1. For the stopping criterion we take Example 8.1. We consider the matrix equation

Suppose that the coefficient matrix A is perturbed to
and C is a random matrix generated by MATLAB function randn.
We compare our own result µ * X in Theorem 6.1with the perturbation bound ξ * proposed in Theorem 3.1 [24].
The condition in Theorem 3.1 [24] is The conditions in Theorem 6.1 are By computation, we list them in Table 1. The results listed in Table 1 show that the conditions in Theorem 3.1 [24] and Theorem 6.1 are satisfied.
By Theorem 3.1 in [24] and Theorem 6.1, we can compute the relative perturbation bounds ξ * , µ * X , respectively. These results averaged as the geometric mean of 10 randomly perturbed runs. Some results are listed in Table 2.
The results listed in Table 2 show that the perturbation bound µ * X given by Theorem 6.1 is fairly sharp, while the bound ξ * given by Theorem 3.1 in [24] is conservative.  [24]. The residual R( X k ) ≡ Q + A * X −p k A − X k satisfies the conditions in Theorem 4.1 [24] and in Theorem 7.1.
By Theorem 4.1 in [24] , we can compute the backward error bound By Theorem 7.1, we can compute the new backward error bound Some results are shown in Table3.
From the results listed in Table 3 we see that the new backward error bound θ||R( X k )|| is sharper than the backward error bound ν * ||R( X k )|| in [24]. Moreover, we see that the backward error θ||R( X)|| for an approximate solution X seems to be independent of the conditioning of the solution X.   The results listed in Table 4 show that the perturbation bound ̺ given by Theorem 4.2 is fairly sharp. By Remark 2, we can compute the relative condition number c rel (X). Some results are listed in