Consider the nonlinear matrix equation with 0 < p_i < 1. Two perturbation bounds and the backward error of an approximate solution to the equation are derived. Explicit expressions of the condition number for the equation are obtained. The theoretical results are illustrated by numerical examples.

1. Introduction

In this paper we consider the Hermitian positive definite solution of the nonlinear matrix equation

()

where 0 < p_i < 1 (i = 1,2, …, m), A₁, A₂, …, A_m are n × n complex matrices, m is a positive integer, and Q is a positive definite matrix. Here,

denotes the conjugate transpose of the matrix A_i.

When m > 1, (1) is recognized as playing an important role in solving a system of linear equations. For example, in many physical calculations, one must solve the system of linear equation

()

where

()

arises in a finite difference approximation to an elliptic partial differential equation (for more information, refer to [1]). We can rewrite M as

, where

()

can be factored as

()

if and only if X is a solution of equation

. When m = 1, this type of nonlinear matrix equations arises in ladder networks, dynamic programming, control theory, stochastic filtering, statistics, and so forth [2–7].

For the similar equations X ± A^*X^−pA = Q, X^s ± A^*X^−tA = Q, , and X = Q − A^*X⁻¹A + B^*X⁻¹B, there were many contributions in the literature to the theory, numerical solutions, and perturbation analysis [8–32]. Jia and Gao [33] derived two perturbation estimates for the solution of the equation X − A^*X^qA = Q with 0 < q < 1. In addition, Duan et al. [34] proved that the equation has a unique positive definite solution. They also proposed an iterative method for obtaining the unique positive definite solution. However, to our best knowledge, there has been no perturbation analysis for (1) with m > 1 in the known literature.

As a continuation of the previous results, the rest of the paper is organized as follows. In Section 2, some preliminary lemmas are given. In Section 3, two perturbation bounds for the unique solution to (1) are derived. Furthermore, in Section 4, we obtain the backward error of an approximate solution to (1). In Section 5, we also discuss the condition number of the unique solution to (1). Finally, several numerical examples are presented in Section 6.

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 i the imaginary unit, 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 = (a₁, …, a_n) = (a_ij) ∈ 𝒞^n×n and a matrix B, A ⊗ B = (a_ijB) is a Kronecker product, and vec A is a vector defined by . For X, Y ∈ ℋ^n×n, we write X ≥ Y (X > Y, resp.) if X − Y is Hermitian positive semidefinite (definite, resp.).

2. Preliminaries

Lemma 1 (see [35].)If A ≥ B > 0 and 0 ≤ γ ≤ 1, then A^γ ≥ B^γ.

Lemma 2 (see [33].)For any Hermitian positive definite matrix X and Hermitian matrix ΔX, one has

(i)
, 0 < q < 1;
(ii)
.

In addition, if X + ΔX ≥ (1/ν)X > 0 and 0 < q < 1, then

(iii)
.

Lemma 3 (see [34].)The matrix equation always has a unique positive definite solution X. The matrix sequence X_k:

()

converges to the unique positive definite solution X for arbitrary initial positive definite matrices X₁, X₂, …, X_m.

3. Perturbation Bounds

Here the perturbed equation

()

is considered, where 0 < p_i < 1 and

and

are small perturbations of A_i and Q in (1), respectively. We assume that X and

are the solutions of (1) and (7), respectively. Let

and

By Lemma 3, we know that (1) always has a unique positive definite solution X; then in this section two perturbation bounds for the unique positive definite solution of (1) are developed. The relative perturbation bound in Theorem 5 does not depend on any knowledge of the actual solution X of (1). Furthermore, a sharper perturbation bound in Theorem 8 is derived.

To prove the next theorem, we first verify the following lemma.

Lemma 4. If X is a solution of (1), then

()

Proof. By Lemma 3, (1) with 0 < p_i < 1 always has a unique positive definite solution X. Then X > 0, and it follows that . Therefore X ≥ Q. By Lemma 1 and (1), we have .

The next theorem generalizes [33, Theorem 4] with m = 1, ∥ΔQ∥ = 0 to arbitrary integer m ≥ 1, ∥ΔQ∥ > 0.

Theorem 5. Let . If

()

then

()

where

()

Proof. Let

()

Obviously, Ω is a nonempty bounded convex closed set. Let

()

Evidently, f : Ω ↦ ℋ^n×n is continuous. We will prove that f(Ω)⊆Ω.

For every ΔX ∈ Ω, it follows that Thus

()

According to (9), we have

()

Therefore

()

From Lemmas 2 and 4, it follows that

()

Therefore

()

That is, f(Ω)⊆Ω. By Brouwer′s fixed point theorem, there exists a ΔX ∈ Ω such that f(ΔX) = ΔX. Moreover, by Lemma 3, we know that X and

are the unique solutions to (1) and (7), respectively. Then

()

Remark 6. According to

()

we get

for ∥ΔQ∥ → 0 and ∥ΔA_i∥ → 0 (i = 1,2, …, m). Therefore (1) is well posed.

Next, a sharper perturbation estimate is derived.

Subtracting (1) from (7), we have

()

where

()

Lemma 7. If , then the linear operator L : ℋ^n×n → ℋ^n×n defined by

()

is invertible.

Proof. It suffices to show that the following equation:

()

has a unique solution for every V ∈ ℋ^n×n. Define the operator M : ℋ^n×n → ℋ^n×n by

()

Let Y = X^−1/2WX^−1/2. Thus (21) is equivalent to

()

According to Lemma 2, we have

()

which implies that | | M|| < 1 and I + M is invertible. Therefore, the operator L is invertible.

Furthermore, we define operators P_i : 𝒞^n×n → ℋ^n×n by

()

Thus, we can rewrite (21) as

()

Define

()

Now we denote that

()

Theorem 8. Suppose that X and are the solutions of (1) and (7), respectively. If ,

()

then

()

Proof. Let

()

Obviously, f : ℋ^n×n → ℋ^n×n is continuous. The condition (32) ensures that the quadratic equation (ζ + θ)x² − (1 + ζϵ − σ)x + ϵ = 0 with respect to the variable x has two positive real roots. The smaller one is

()

Define Ω = {ΔX ∈ ℋ^n×n : ∥ΔX∥≤ν}. Then for any ΔX ∈ Ω, by (32), we have

()

It follows that I − X⁻¹ΔX is nonsingular and

()

Using (22) and Lemma 2, we have

()

Noting (31) and (34), it follows that

()

for ΔX ∈ Ω. That is, f(Ω)⊆Ω. According to Schauder fixed point theorem, there exists ΔX_* ∈ Ω such that f(ΔX_*) = X_*. It follows that X + ΔX_* is a Hermitian solution of (7). By Lemma 3, we know that the solution of (7) is unique. Then

Remark 9. From Theorem 8, we get the first order perturbation bound for the solution as follows:

()

as (ΔA₁, ΔA₂, …, ΔA_m, ΔQ) → 0.

Combining this with (29) gives

()

as (ΔA₁, ΔA₂, …, ΔA_m, ΔQ) → 0.

4. Backward Error

In this section, a backward error of an approximate solution for the unique solution to (1) is obtained.

Theorem 10. Let be an approximation to the solution X of (1). If and the residual satisfies

()

then

()

Proof. Let

()

where

. Obviously, Ψ is a nonempty bounded convex closed set. Let

()

Evidently g : Ψ ↦ ℋ^n×n is continuous. We will prove that g(Ψ)⊆Ψ. For every ΔX ∈ Ψ, we have

()

Hence

()

That is,

()

Using (43), one sees that

()

Therefore

According to (17), we obtain

()

for ΔX ∈ Ψ. That is, g(Ψ)⊆Ψ. By Brouwer’s fixed point theorem, there exists a ΔX ∈ Ψ such that g(ΔX) = ΔX. Hence

is a solution of (1). Moreover, by Lemma 3, we know that the solution X of (1) is unique. Then

()

5. Condition Number

In this section, we apply the theory of condition number developed by Rice [36] to study condition numbers of the unique solution to (1).

5.1. The Complex Case

Suppose that X and

are the solutions of the matrix equations (1) and (7), respectively. Let

and

. Using Theorem 8 and Remark 9, we have

()

as (ΔA₁, ΔA₂, …, ΔA_m, ΔQ) → 0.

By the theory of condition number developed by Rice [36], we define the condition number of the Hermitian positive definite solution X to (1) by

()

where ξ, ρ, and η_i, i = 1,2, …, m, are positive parameters. Taking ξ = η_i = ρ = 1 in (54) gives the absolute condition number c_abs(X), and taking ξ = ∥X∥_F,

, and ρ = ∥Q∥_F in (54) gives the relative condition number c_rel(X).

Substituting (53) into (54), we get

()

Let L be the matrix representation of the linear operator L. Then it is easy to see that

()

Let

()

vec H = x + iy, vec E_i = a_i + ib_i,

, M = (E₁, E₂, …, E_m, H), where

, i = 1,2, …, m, and Π is the vec-permutation matrix, such that

()

Then we obtain that

()

Then we have the following theorem.

Theorem 11. If , then the condition number c(X) defined by (54) has the explicit expression

()

where the matrices S_c and U_i are defined as in (57).

Remark 12. From (60) we have the relative condition number

()

5.2. The Real Case

In this subsection we consider the real case. That is, all the coefficient matrices A_i, Q of (1) are real. In such a case the corresponding solution X is also real. Completely similar arguments as Theorem 11 give the following theorem.

Theorem 13. Let A_i, Q be real and let c(X) be the condition number defined by (54). If , then c(X) has the explicit expression

()

where

()

Remark 14. In the real case the relative condition number is given by

()

6. Numerical Examples

To illustrate the results of the previous sections, in this section three simple examples are given, which were carried out using MATLAB 7.1. For the stopping criterion we take .

Example 15. We consider the matrix equation

()

with

()

Suppose that the coefficient matrices A₁ and A₂ are perturbed to

, where

()

and C is a random matrix generated by MATLAB function randn.

We now consider the corresponding perturbation bounds for the solution X in Theorems 5 and 8.

The assumptions in Theorem 5 are

()

The assumptions in Theorem 8 are

()

By computation, we list them in Table 1.

Table 1. Assumptions check for Example 15 with different values of j.

j	4	5	6	7
ass₁	1.0749	1.0753	1.0753	1.0753
ass₂	0.9248	0.9247	0.9247	0.9247
ass₃	0.8543	0.8550	0.8550	0.8550
ass₄	0.9999	1.0000	1.0000	1.0000
ass₅	0.7645	0.7648	0.7648	0.7648

The results listed in Table 1 show that the assumptions of Theorems 5 and 8 are satisfied.

By Theorems 5 and 8, we can compute the relative perturbation bounds ξ₁ and ξ₂ = ν/∥X∥, respectively. These results averaged as the geometric mean of 10 randomly perturbed runs. Some results are listed in Table 2.

Table 2. Perturbation bounds for Example 15 with different values of j.

j	4	5	6	7
	2.5627 × 10⁻⁵	3.8447 × 10⁻⁶	5.1681 × 10⁻⁷	2.1776 × 10⁻⁸
ξ₁	2.1885 × 10⁻⁴	1.9891 × 10⁻⁵	2.4026 × 10⁻⁶	1.8251 × 10⁻⁷
ξ₂	8.0828 × 10⁻⁵	7.4741 × 10⁻⁶	8.8011 × 10⁻⁷	6.9496 × 10⁻⁸

The results listed in Table 2 show that the perturbation bound ξ₂ given by Theorem 8 is fairly sharp, while the bound ξ₁ given by Theorem 5 which does not depend on the exact solution is conservative.

Example 16. We consider the matrix equation

()

with

()

Choose

. Let the approximate solution

of X be given with the iterative method (6), where k is the iteration number.

The residual satisfies the conditions in Theorem 10.

By Theorem 10, we can compute the backward error bound for

as follows:

()

where

()

Some results are listed in Table 3.

Table 3. Backward error bound for Example 16 with different values of k.

k	8	10	12	14
	6.1091 × 10⁻⁴	4.0865 × 10⁻⁵	2.6837 × 10⁻⁶	1.7372 × 10⁻⁷
	7.1435 × 10⁻⁴	4.7784 × 10⁻⁵	3.1381 × 10⁻⁶	2.0318 × 10⁻⁷

The results listed in Table 3 show that the error bound given by Theorem 10 is fairly sharp.

Example 17. We study the matrix equation

()

with

()

By Remark 14, we can compute the relative condition number c_rel(X). Some results are listed in Table 4.

Table 4 shows that the unique positive definite solution X is well conditioned.

Table 4. Relative condition number for Example 17 with different values of k.

k	1	3	5	7	9
c_rel(X)	1.0717	1.0228	1.0225	1.0225	1.0225

Acknowledgments

The author wishes to express her gratitude to the referees for their fruitful comments. The work was supported in part by National Nature Science Foundation of China (11201263), Natural Science Foundation of Shandong Province (ZR2012AQ004), and Independent Innovation Foundation of Shandong University (IIFSDU), China.

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Perturbation Analysis of the Nonlinear Matrix Equation

Abstract

1. Introduction

2. Preliminaries

3. Perturbation Bounds

4. Backward Error

5. Condition Number

5.1. The Complex Case

5.2. The Real Case

6. Numerical Examples

Acknowledgments

References

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References

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Perturbation Analysis of the Nonlinear Matrix Equation

This article relates to:

Comment on “Perturbation Analysis of the Nonlinear Matrix Equation X-∑i=1mAi*XpiAi=Q”

Abstract

1. Introduction

2. Preliminaries

3. Perturbation Bounds

4. Backward Error

5. Condition Number

5.1. The Complex Case

5.2. The Real Case

6. Numerical Examples

Acknowledgments

References

Citing Literature

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