Perturbation Analysis for the Matrix Equation 𝑋−∑^𝑚_𝑖=1𝐴^∗_𝑖𝑋𝐴_𝑖+∑^𝑛_𝑗=1𝐵^∗_𝑗𝑋𝐵_𝑗=𝐼

Abstract

We consider the perturbation analysis of the matrix equation 𝑋−∑^𝑚_𝑖=1𝐴^∗_𝑖𝑋𝐴_𝑖+∑^𝑛_𝑗=1𝐵^∗_𝑗𝑋𝐵_𝑗=𝐼. Based on the matrix differentiation, we first give a precise perturbation bound for the positive definite solution. A numerical example is presented to illustrate the sharpness of the perturbation bound.

1. Introduction

In this paper, we consider the matrix equation 𝑋−^𝑚_𝑖=1𝐴^∗_𝑖𝑋𝐴_𝑖+^𝑛_𝑗=1𝐵^∗_𝑗𝑋𝐵_𝑗=𝐼, (1.1) where 𝐴₁,𝐴₂,…,𝐴_𝑚,𝐵₁,𝐵₂,…,𝐵_𝑛 are 𝑛×𝑛 complex matrices, 𝐼 is an 𝑛×𝑛 identity matrix, 𝑚,𝑛 are nonnegative integers and the positive definite solution 𝑋 is practical interest. Here, 𝐴^∗_𝑖 and 𝐵^∗_𝑖 denote the conjugate transpose of the matrices 𝐴_𝑖 and 𝐵_𝑖, respectively. Equation (1.1) arises in solving some nonlinear matrix equations with Newton method. See, for example, the nonlinear matrix equation which appears in Sakhnovich [1]. Solving these nonlinear matrix equations gives rise to (1.1). On the other hand, (1.1) is the general case of the generalized Lyapunov equation 𝑀𝑌𝑆^∗+𝑆𝑌𝑀^∗+^𝑡_𝑘=1𝑁_𝑘𝑌𝑁^∗_𝑘+𝐶𝐶^∗=0, (1.2) whose positive definite solution is the controllability Gramian of the bilinear control system (see [2, 3] for more details) 𝑀𝑥̇(𝑡)=𝑆𝑥(𝑡)+^𝑡_𝑘=1𝑁_𝑘𝑥(𝑡)𝑢_𝑘(𝑡)+𝐶𝑢(𝑡). (1.3) Set 1𝐸=√2()1𝑀−𝑆+𝐼,𝐹=√2(𝑀+𝑆+𝐼),𝐺=𝑆−𝐼,𝑄=𝐶𝐶^∗,𝑋=𝑄^−1/2𝑌𝑄^−1/2,𝐴_𝑖=𝑄^−1/2𝑁_𝑖𝑄^1/2𝐴,𝑖=1,2,…,𝑡,_𝑡+1=𝑄^−1/2𝐹𝑄^1/2,𝐴_𝑡+2=𝑄^−1/2𝐺𝑄^1/2,𝐵₁=𝑄^−1/2𝐸𝑄^1/2,𝐵₂=𝑄^−1/2𝐶𝑄^1/2, (1.4) then (1.2) can be equivalently written as (1.1) with 𝑚=𝑡+2 and 𝑛=2.

Some special cases of (1.1) have been studied. Based on the kronecker product and fixed point theorem in partially ordered sets, Reurings [4] and Ran and Reurings [5, 6] gave some sufficient conditions for the existence of a unique positive definite solution of the linear matrix equations 𝑋−∑^𝑚_𝑖=1𝐴^∗_𝑖𝑋𝐴_𝑖=𝐼 and 𝑋+∑^𝑛_𝑗=1𝐵^∗_𝑖𝑋𝐵_𝑖=𝐼. And the expressions for these unique positive definite solutions were also derived under some constraint conditions. For the general linear matrix equation (1.1), Reurings [[4], Page 61] pointed out that it is hard to find sufficient conditions for the existence of a positive definite solution, because the map 𝐺(𝑋)=𝐼+^𝑚_𝑖=1𝐴^∗_𝑖𝑋𝐴_𝑖−^𝑛_𝑗=1𝐵^∗_𝑗𝑋𝐵_𝑗 (1.5) is not monotone and does not map the set of 𝑛×𝑛 positive definite matrices into itself. Recently, Berzig [7] overcame these difficulties by making use of Bhaskar-Lakshmikantham coupled fixed point theorem and gave a sufficient condition for (1.1) existing a unique positive definite solution. An iterative method was constructed to compute the unique positive definite solution, and the error estimation was given too.

Recently, the matrix equations of the form (1.1) have been studied by many authors (see [8–14]). Some numerical methods for solving the well-known Lyapunov equation 𝑋+𝐴^𝑇𝑋𝐴=𝑄, such as Bartels-Stewart method and Hessenberg-Schur method, have been proposed in [12]. Based on the fixed point theorem, the sufficient and necessary conditions for the existence of a positive definite solution of the matrix equation 𝑋^𝑠±𝐴^∗𝑋^−𝑡𝐴=𝑄,𝑠,𝑡∈𝑁 have been given in [8, 9]. The fixed point iterative method and inversion-free iterative method were developed for solving the matrix equations 𝑋±𝐴^∗𝑋^−𝛼𝐴=𝑄,𝛼>0 in [13, 14]. By making use of the fixed point theorem of mixed monotone operator, the matrix equation 𝑋−∑^𝑚_𝑖=1𝐴^∗_𝑖𝑋^𝛿𝑖𝐴_𝑖=𝑄,0<|𝛿_𝑖|<1 was studied in [10], and derived a sufficient condition for the existence of a positive definite solution. Assume that 𝐹 maps positive definite matrices either into positive definite matrices or into negative definite matrices, the general nonlinear matrix equation 𝑋+𝐴^∗𝐹(𝑋)𝐴=𝑄 was studied in [11], and the fixed point iterative method was constructed to compute the positive definite solution under some additional conditions.

Motivated by the works and applications in [2–6], we continue to study the matrix equation (1.1). Based on a new mathematical tool (i.e., the matrix differentiation), we firstly give a differential bound for the unique positive definite solution of (1.1), and then use it to derive a precise perturbation bound for the unique positive definite solution. A numerical example is used to show that the perturbation bound is very sharp.

Throughout this paper, we write 𝐵>0(𝐵≥0) if the matrix 𝐵 is positive definite (semidefinite). If 𝐵−𝐶 is positive definite (semidefinite), then we write 𝐵>𝐶(𝐵≥𝐶). If a positive definite matrix 𝑋 satisfies 𝐵≤𝑋≤𝐶, we denote that 𝑋∈[𝐵,𝐶]. The symbols 𝜆₁(𝐵) and 𝜆_𝑛(𝐵) denote the maximal and minimal eigenvalues of an 𝑛×𝑛 Hermitian matrix 𝐵, respectively. The symbol 𝐻^𝑛×𝑛 stands for the set of 𝑛×𝑛 Hermitian matrices. The symbol ‖𝐵‖ denotes the spectral norm of the matrix 𝐵.

2. Perturbation Analysis for the Matrix Equation (1.1)

Based on the matrix differentiation, we firstly give a differential bound for the unique positive definite solution 𝑋_𝑈 of (1.1), and then use it to derive a precise perturbation bound for 𝑋_𝑈 in this section.

Definition 2.1 ([15], Definition 3.6). Let 𝐹=(𝑓_𝑖𝑗)_𝑚×𝑛, then the matrix differentiation of 𝐹 is 𝑑𝐹=(𝑑𝑓_𝑖𝑗)_𝑚×𝑛. For example, let 𝐹=𝑠+𝑡𝑠²−2𝑡2𝑠+𝑡³𝑡². (2.1) Then 𝑑𝐹=𝑑𝑠+𝑑𝑡2𝑠𝑑𝑠−2𝑑𝑡2𝑑𝑠+3𝑡².𝑑𝑡2𝑡𝑑𝑡 (2.2)

Lemma 2.2 ([15], Theorem 3.2). The matrix differentiation has the following properties:(1)𝑑(𝐹₁±𝐹₂)=𝑑𝐹₁±𝑑𝐹₂; (2)𝑑(𝑘𝐹)=𝑘(𝑑𝐹), where 𝑘 is a complex number;(3)𝑑(𝐹^∗)=(𝑑𝐹)^∗; (4)𝑑(𝐹₁𝐹₂𝐹₃)=(𝑑𝐹₁)𝐹₂𝐹₃+𝐹₁(𝑑𝐹₂)𝐹₃+𝐹₁𝐹₂(𝑑𝐹₃); (5)𝑑𝐹⁻¹=−𝐹⁻¹(𝑑𝐹)𝐹⁻¹; (6)𝑑𝐹=0, where 𝐹 is a constant matrix.

Lemma 2.3 ([7], Theorem 3.1). If ^𝑚_𝑖=1𝐴^∗_𝑖𝐴_𝑖<12𝐼,^𝑛_𝑗=1𝐵^∗_𝑗𝐵_𝑗<12𝐼, (2.3) then (1.1) has a unique positive definite solution 𝑋_𝑈 and 𝑋_𝑈∈𝐼−2^𝑛_𝑗=1𝐵^∗_𝑗𝐵_𝑗,𝐼+2^𝑚_𝑖=1𝐴^∗_𝑖𝐴_𝑖. (2.4)

Theorem 2.4. If ^𝑚_𝑖=1𝐴_𝑖²<12,^𝑛_𝑗=1𝐵_𝑗²<12, (2.5) then (1.1) has a unique positive definite solution 𝑋_𝑈, and it satisfies 𝑑𝑋_𝑈≤21+2∑^𝑚_𝑖=1𝐴_𝑖²∑^𝑚_𝑖=1𝐴_𝑖𝑑𝐴_𝑖+∑^𝑛_𝑗=1𝐵_𝑗𝑑𝐵_𝑗1−∑^𝑚_𝑖=1𝐴_𝑖²−∑^𝑛_𝑗=1𝐵_𝑗². (2.6)

Proof. Since 𝜆₁𝐴^∗_𝑖𝐴_𝑖≤𝐴^∗_𝑖𝐴_𝑖≤𝐴_𝑖²𝜆,𝑖=1,2,…,𝑚,₁𝐵^∗_𝑗𝐵_𝑗≤𝐵^∗_𝑗𝐵_𝑗≤𝐵_𝑗²,𝑗=1,2,…,𝑛, (2.7) then 𝐴^∗_𝑖𝐴_𝑖≤𝜆₁𝐴^∗_𝑖𝐴_𝑖𝐼≤𝐴^∗_𝑖𝐴_𝑖𝐼≤𝐴_𝑖²𝐵𝐼,𝑖=1,2,…,𝑚,_∗𝑗𝐵_𝑗≤𝜆₁𝐵^∗_𝑗𝐵_𝑗𝐼≤𝐵^∗_𝑗𝐵_𝑗≤𝐵_𝑗²𝐼,𝑗=1,2,…,𝑛, (2.8) consequently, ^𝑚_𝑖=1𝐴^∗_𝑖𝐴_𝑖≤^𝑚_𝑖=1𝐴_𝑖²𝐼,_𝑛𝑗=1𝐵^∗_𝑗𝐵_𝑗≤^𝑛_𝑗=1𝐵_𝑗²𝐼. (2.9) Combining (2.5)–(2.9) we have ^𝑚_𝑖=1𝐴^∗_𝑖𝐴_𝑖≤^𝑚_𝑖=1𝐴_𝑖²1𝐼<2𝐼,_𝑛𝑗=1𝐵^∗_𝑗𝐵_𝑗≤^𝑛_𝑗=1𝐵_𝑗²1𝐼<2𝐼. (2.10) Then by Lemma 2.3 we obtain that (1.1) has a unique positive definite solution 𝑋_𝑈, which satisfies 𝑋_𝑈∈𝐼−2^𝑛_𝑗=1𝐵^∗_𝑗𝐵_𝑗,𝐼+2^𝑚_𝑖=1𝐴^∗_𝑖𝐴_𝑖. (2.11) Noting that 𝑋_𝑈 is the unique positive definite solution of (1.1), then 𝑋_𝑈−^𝑚_𝑖=1𝐴^∗_𝑖𝑋_𝑈𝐴_𝑖+^𝑛_𝑗=1𝐵^∗_𝑗𝑋_𝑈𝐵_𝑗=𝐼. (2.12) It is known that the elements of 𝑋_𝑈 are differentiable functions of the elements of 𝐴_𝑖 and 𝐵_𝑖. Differentiating (2.12), and by Lemma 2.2, we have 𝑑𝑋_𝑈−^𝑚_𝑖=1𝑑𝐴^∗_𝑖𝑋_𝑈𝐴_𝑖+𝐴^∗_𝑖𝑑𝑋_𝑈𝐴_𝑖+𝐴^∗_𝑖𝑋_𝑈𝑑𝐴_𝑖+^𝑛_𝑗=1𝑑𝐵^∗_𝑗𝑋_𝑈𝐵_𝑗+𝐵^∗_𝑗𝑑𝑋_𝑈𝐵_𝑗+𝐵^∗_𝑗𝑋_𝑈𝑑𝐵_𝑗=0, (2.13) which implies that 𝑑𝑋_𝑈−^𝑚_𝑖=1𝐴^∗_𝑖𝑑𝑋_𝑈𝐴_𝑖+^𝑛_𝑗=1𝐵^∗_𝑗𝑑𝑋_𝑈𝐵_𝑗=^𝑚_𝑖=1𝑑𝐴^∗_𝑖𝑋_𝑈𝐴_𝑖+^𝑚_𝑖=1𝐴^∗_𝑖𝑋_𝑈𝑑𝐴_𝑖−^𝑛_𝑗=1𝑑𝐵^∗_𝑗𝑋_𝑈𝐵_𝑗−^𝑛_𝑗=1𝐵^∗_𝑗𝑋_𝑈𝑑𝐵_𝑗. (2.14) By taking spectral norm for both sides of (2.14), we obtain that 𝑑𝑋_𝑈−^𝑚_𝑖=1𝐴^∗_𝑖𝑑𝑋_𝑈𝐴_𝑖+^𝑛_𝑗=1𝐵^∗_𝑗𝑑𝑋_𝑈𝐵_𝑗=^𝑚_𝑖=1𝑑𝐴^∗_𝑖𝑋_𝑈𝐴_𝑖+^𝑚_𝑖=1𝐴^∗_𝑖𝑋_𝑈𝑑𝐴_𝑖−^𝑛_𝑗=1𝑑𝐵^∗_𝑗𝑋_𝑈𝐵_𝑗−^𝑛_𝑗=1𝐵^∗_𝑗𝑋_𝑈𝑑𝐵_𝑗≤^𝑚_𝑖=1𝑑𝐴^∗_𝑖𝑋_𝑈𝐴_𝑖+^𝑚_𝑖=1𝐴^∗_𝑖𝑋_𝑈𝑑𝐴_𝑖+^𝑛_𝑗=1𝑑𝐵^∗_𝑗𝑋_𝑈𝐵_𝑗+^𝑛_𝑗=1𝐵^∗_𝑗𝑋_𝑈𝑑𝐵_𝑗≤^𝑚_𝑖=1𝑑𝐴^∗_𝑖𝑋_𝑈𝐴_𝑖+^𝑚_𝑖=1𝐴^∗_𝑖𝑋_𝑈𝑑𝐴_𝑖+^𝑛_𝑗=1𝑑𝐵^∗_𝑗𝑋_𝑈𝐵_𝑗+^𝑛_𝑗=1𝐵^∗_𝑗𝑋_𝑈𝑑𝐵_𝑗≤^𝑚_𝑖=1𝑑𝐴^∗_𝑖𝑋_𝑈𝐴_𝑖+^𝑚_𝑖=1𝐴^∗_𝑖𝑋_𝑈𝑑𝐴_𝑖+^𝑛_𝑗=1𝑑𝐵^∗_𝑗𝑋_𝑈𝐵_𝑗+^𝑛_𝑗=1𝐵^∗_𝑗𝑋_𝑈𝑑𝐵_𝑗=2^𝑚_𝑖=1𝐴_𝑖𝑋_𝑈𝑑𝐴_𝑖+2^𝑛_𝑗=1𝐵_𝑗𝑋_𝑈𝑑𝐵_𝑗=2^𝑚_𝑖=1𝐴_𝑖𝑑𝐴_𝑖+^𝑛_𝑗=1𝐵_𝑗𝑑𝐵_𝑗𝑋_𝑈, (2.15) and noting (2.11) we obtain that 𝑋_𝑈≤𝐼+2^𝑚_𝑖=1𝐴^∗_𝑖𝐴_𝑖≤1+2^𝑚_𝑖=1𝐴_𝑖². (2.16) Then 𝑑𝑋_𝑈−^𝑚_𝑖=1𝐴^∗_𝑖𝑑𝑋_𝑈𝐴_𝑖+^𝑛_𝑗=1𝐵^∗_𝑗𝑑𝑋_𝑈𝐵_𝑗≤2^𝑚_𝑖=1𝐴_𝑖𝑑𝐴_𝑖+^𝑛_𝑗=1𝐵_𝑗𝑑𝐵_𝑗𝑋_𝑈≤21+2^𝑚_𝑖=1𝐴_𝑖^2𝑚_𝑖=1𝐴_𝑖𝑑𝐴_𝑖+^𝑛_𝑗=1𝐵_𝑗𝑑𝐵_𝑗,𝑑𝑋_𝑈−^𝑚_𝑖=1𝐴^∗_𝑖𝑑𝑋_𝑈𝐴_𝑖+^𝑛_𝑗=1𝐵^∗_𝑗𝑑𝑋_𝑈𝐵_𝑗≥𝑑𝑋_𝑈−^𝑚_𝑖=1𝐴^∗_𝑖𝑑𝑋_𝑈𝐴_𝑖−^𝑛_𝑗=1𝐵^∗_𝑗𝑑𝑋_𝑈𝐵_𝑗≥𝑑𝑋_𝑈−^𝑚_𝑖=1𝐴^∗_𝑖𝑑𝑋_𝑈𝐴_𝑖−^𝑛_𝑗=1𝐵^∗_𝑗𝑑𝑋_𝑈𝐵_𝑗≥𝑑𝑋_𝑈−^𝑚_𝑖=1𝐴^∗_𝑖𝑑𝑋_𝑈𝐴_𝑖−^𝑛_𝑗=1𝐵^∗_𝑗𝑑𝑋_𝑈𝐵_𝑗=1−^𝑚_𝑖=1𝐴_𝑖²−^𝑛_𝑗=1𝐵_𝑗²𝑑𝑋_𝑈. (2.17)(2.18) Due to (2.5) we have 1−^𝑚_𝑖=1𝐴_𝑖²−^𝑛_𝑗=1𝐵_𝑗²>0. (2.19) Combining (2.17), (2.18) and noting (2.19), we have 1−^𝑚_𝑖=1𝐴_𝑖²−^𝑛_𝑗=1𝐵_𝑗²𝑑𝑋_𝑈≤𝑑𝑋_𝑈−^𝑚_𝑖=1𝐴^∗_𝑖𝑑𝑋_𝑈𝐴_𝑖+^𝑛_𝑗=1𝐵^∗_𝑗𝑑𝑋_𝑈𝐵_𝑗≤21+2^𝑚_𝑖=1𝐴_𝑖^2𝑚_𝑖=1𝐴_𝑖𝑑𝐴_𝑖+^𝑛_𝑗=1𝐵_𝑗𝑑𝐵_𝑗, (2.20) which implies that 𝑑𝑋_𝑈≤21+2∑^𝑚_𝑖=1𝐴_𝑖²∑^𝑚_𝑖=1𝐴_𝑖𝑑𝐴_𝑖+∑^𝑛_𝑗=1𝐵_𝑗𝑑𝐵_𝑗1−∑^𝑚_𝑖=1𝐴_𝑖²−∑^𝑛_𝑗=1𝐵_𝑗². (2.21)

Theorem 2.5. Let 𝐴₁,𝐴₂,…,𝐴_𝑚,𝐵₁,𝐵₂,…,𝐵_𝑛 be perturbed matrices of 𝐴₁,𝐴₂,…,𝐴_𝑚,𝐵₁,𝐵₂,…,𝐵_𝑛 in (1.1) and Δ_𝑖=𝐴_𝑖−𝐴_𝑖,𝑖=1,2,…,𝑚,Δ_𝑗=𝐵_𝑗−𝐵_𝑗,𝑗=1,2,…,𝑛. If ^𝑚_𝑖=1𝐴_𝑖²<12,^𝑛_𝑗=1𝐵_𝑗²<12,2^𝑚_𝑖=1𝐴_𝑖Δ_𝑖+^𝑚_𝑖=1𝐴_𝑖²<12−^𝑚_𝑖=1𝐴_𝑖²,2^𝑛_𝑗=1𝐵_𝑗Δ_𝑗+^𝑛_𝑗=1Δ_𝑗²<12−^𝑛_𝑗=1𝐵_𝑗², (2.22)(2.23)(2.24) then (1.1) and its perturbed equation 𝑋−^𝑚_𝑖=1𝐴^∗_𝑖𝑋A_𝑖+^𝑛_𝑗=1𝐵^∗_𝑗𝑋𝐵_𝑗=𝐼 (2.25) have unique positive definite solutions 𝑋_𝑈 and 𝑋_𝑈, respectively, which satisfy 𝑋_𝑈−𝑋_𝑈≤𝑆_err, (2.26) where 𝑆_err=21+2∑^𝑚_𝑖=1𝐴_𝑖+Δ_𝑖²∑^𝑚_𝑖=1𝐴_𝑖+Δ_𝑖²Δ_𝑖+∑^𝑛_𝑗=1𝐵_𝑗+Δ_𝑗²Δ_𝑗1−∑^𝑚_𝑖=1𝐴_𝑖+Δ_𝑖²−∑^𝑛_𝑗=1𝐵_𝑗+Δ_𝑗². (2.27)

Proof. By (2.22) and Theorem 2.4, we know that (1.1) has a unique positive definite solution 𝑋_𝑈. And by (2.23) we have ^𝑚_𝑖=1𝐴_𝑖²=^𝑚_𝑖=1𝐴_𝑖+Δ_𝑖²≤^𝑚_𝑖=1𝐴_𝑖+Δ_𝑖²=^𝑚_𝑖=1𝐴_𝑖²+2𝐴_𝑖Δ_𝑖+Δ_𝑖²=^𝑚_𝑖=1𝐴_𝑖²+2^𝑚_𝑖=1𝐴_𝑖Δ_𝑖+^𝑚_𝑖=1Δ_𝑖²<^𝑚_𝑖=1𝐴_𝑖²+12−^𝑚_𝑖=1𝐴_𝑖²=12, (2.28) similarly, by (2.24) we have ^𝑛_𝑗=1𝐵_𝑗²<12. (2.29) By (2.28), (2.29), and Theorem 2.4 we obtain that the perturbed equation (2.25) has a unique positive definite solution 𝑋_𝑈.

Set 𝐴_𝑖(𝑡)=𝐴_𝑖+𝑡Δ_𝑖,𝐵_𝑗(𝑡)=𝐵_𝑗+𝑡Δ_𝑗,𝑡∈[0,1], (2.30) then by (2.23) we have ^𝑚_𝑖=1𝐴_𝑖(𝑡)²=^𝑚_𝑖=1𝐴_𝑖+𝑡Δ_𝑖²≤^𝑚_𝑖=1𝐴_𝑖+𝑡Δ_𝑖²=^𝑚_𝑖=1𝐴_𝑖²+2𝑡𝐴_𝑖Δ_𝑖+𝑡²Δ_𝑖²≤^𝑚_𝑖=1𝐴_𝑖²+2𝐴_𝑖Δ_𝑖+Δ_𝑖²=^𝑚_𝑖=1𝐴_𝑖²+2^𝑚_𝑖=1𝐴_𝑖Δ_𝑖+^𝑚_𝑖=1Δ_𝑖²<^𝑚_𝑖=1𝐴_𝑖²+12−^𝑚_𝑖=1𝐴_𝑖²=12, (2.31) similarly, by (2.24) we have ^𝑛_𝑗=1𝐵_𝑗(𝑡)²=^𝑚_𝑖=1𝐵_𝑗+𝑡Δ_𝑗²<12. (2.32)

Therefore, by (2.31), (2.32), and Theorem 2.4 we derive that for arbitrary 𝑡∈[0,1], the matrix equation 𝑋−^𝑚_𝑖=1𝐴^∗_𝑖(𝑡)𝑋𝐴_𝑖(𝑡)+^𝑛_𝑗=1𝐵^∗_𝑗(𝑡)𝑋𝐵_𝑗(𝑡)=𝐼 (2.33) has a unique positive definite solution 𝑋_𝑈(𝑡), especially, 𝑋_𝑈(0)=𝑋_𝑈,𝑋_𝑈(1)=𝑋_𝑈. (2.34)

From Theorem 2.4 it follows that 𝑋_𝑈−𝑋_𝑈=𝑋_𝑈(1)−𝑋_𝑈(0)=¹₀𝑑𝑋_𝑈(𝑡)≤¹₀𝑑𝑋_𝑈(𝑡≤₁₀)21+2∑^𝑚_𝑖=1𝐴_𝑖(𝑡)²∑^𝑚_𝑖=1𝐴_𝑖(𝑡)𝑑𝐴_𝑖(𝑡)+∑^𝑛_𝑗=1𝐵_𝑗(𝑡)𝑑𝐵_𝑗(𝑡)1−∑^𝑚_𝑖=1𝐴_𝑖(𝑡)²−∑^𝑛_𝑗=1𝐵_𝑗(𝑡)²≤¹₀21+2∑^𝑚_𝑖=1𝐴_𝑖(𝑡)²∑^𝑚_𝑖=1𝐴_𝑖(𝑡)Δ_𝑖𝑑𝑡+∑^𝑛_𝑗=1𝐵_𝑗(𝑡)Δ_𝑗𝑑𝑡1−∑^𝑚_𝑖=1𝐴_𝑖(𝑡)²−∑^𝑛_𝑗=1𝐵_𝑗(𝑡)²=¹₀21+2∑^𝑚_𝑖=1𝐴_𝑖(𝑡)²∑^𝑚_𝑖=1𝐴_𝑖(𝑡)Δ_𝑖+∑^𝑛_𝑗=1𝐵_𝑗(𝑡)Δ_𝑗1−∑^𝑚_𝑖=1𝐴_i(𝑡)²−∑^𝑛_𝑗=1𝐵_𝑗(𝑡)²𝑑𝑡. (2.35) Noting that 𝐴_𝑖(𝑡)=𝐴_𝑖+𝑡Δ_𝑖≤𝐴_𝑖+𝑡Δ_𝑖𝐵,𝑖=1,2,…,𝑚,_𝑗(𝑡)=𝐵_𝑗+𝑡Δ_𝑖≤𝐵_𝑗+𝑡Δ_𝑖,𝑗=1,2,…,𝑛, (2.36) and combining Mean Value Theorem of Integration, we have𝑋_𝑈−𝑋_𝑈≤¹₀21+2∑^𝑚_𝑖=1𝐴_𝑖(𝑡)²∑^𝑚_𝑖=1𝐴_𝑖(𝑡)Δ_𝑖+∑^𝑛_𝑗=1𝐵_𝑗(𝑡)Δ_𝑗1−∑^𝑚_𝑖=1𝐴_𝑖(𝑡)²−∑^𝑛_𝑗=1𝐵_𝑗(𝑡)²≤𝑑𝑡₁₀21+2∑^𝑚_𝑖=1𝐴_𝑖+𝑡Δ_𝑖²∑^𝑚_𝑖=1𝐴_𝑖+𝑡Δ_𝑖²Δ_𝑖+∑^𝑛_𝑗=1𝐵_𝑗+𝑡Δ_𝑗²Δ_𝑗1−∑^𝑚_𝑖=1𝐴_𝑖+𝑡Δ_𝑖²−∑^𝑛_𝑗=1𝐵_𝑗+𝑡Δ_𝑗²=2∑𝑑𝑡1+2^𝑚_𝑖=1𝐴_𝑖+𝜉Δ_𝑖²∑^𝑚_𝑖=1𝐴_𝑖+𝜉Δ_𝑖²Δ_𝑖+∑^𝑛_𝑗=1𝐵_𝑗+𝜉Δ_𝑗²Δ_𝑗1−∑^𝑚_𝑖=1𝐴_𝑖+𝜉Δ_𝑖²−∑^𝑛_𝑗=1𝐵_𝑗+𝜉Δ_𝑗²×()[]≤21−0,𝜉∈0,11+2∑^𝑚_𝑖=1𝐴_𝑖+Δ_𝑖²∑^𝑚_𝑖=1𝐴_𝑖+Δ_𝑖²Δ_𝑖+∑^𝑛_𝑗=1𝐵_𝑗+Δ_𝑗²Δ_𝑗1−∑^𝑚_𝑖=1𝐴_𝑖+Δ_𝑖²−∑^𝑛_𝑗=1𝐵_𝑗+Δ_𝑗²=𝑆_err. (2.37)

3. Numerical Experiments

In this section, we use a numerical example to confirm the correctness of Theorem 2.5 and the precision of the perturbation bound for the unique positive definite solution 𝑋_𝑈 of (1.1).

Example 3.1. Consider the symmetric linear matrix equation 𝑋−𝐴^∗₁𝑋𝐴₁−𝐴^∗₂𝑋𝐴₂+𝐵^∗₁𝑋𝐵₁+𝐵^∗₂𝑋𝐵₂=𝐼, (3.1) and its perturbed equation 𝑋−𝐴^∗₁𝑋𝐴₁−𝐴^∗₂𝑋𝐴₂+𝐵^∗₁𝑋𝐵₁+𝐵^∗₂𝑋𝐵₂=𝐼, (3.2) where 𝐴₁=0.02−0.10−0.020.08−0.100.02−0.06−0.120.14,𝐴₂=,𝐵0.08−0.10−0.020.08−0.100.02−0.06−0.120.14₁=0.470.020.04−0.100.36−0.02−0.040.010.47,𝐵₂=,𝐴0.100.100.050.150.2750.0750.050.050.175₁=𝐴₁+0.50.1−0.2−0.40.20.6−0.20.1−0.1×10^−𝑗,𝐴₂=𝐴₂+−0.40.1−0.20.50.7−1.31.10.90.6×10^−𝑗,𝐵₁=𝐵₁+0.80.20.05−0.20.120.14−0.25−0.20.26×10^−𝑗,𝐵₂=𝐵₂+0.20.20.1−0.30.15−0.150.1−0.10.25×10^−𝑗,𝑗∈𝑁. (3.3)
It is easy to verify that the conditions (2.22)–(2.24) are satisfied, then (3.1) and its perturbed equation (3.2) have unique positive definite solutions 𝑋_𝑈 and 𝑋_𝑈, respectively. From Berzig [7] it follows that the sequences {𝑋_𝑘} and {𝑌_𝑘} generated by the iterative method 𝑋₀=0,𝑌₀𝑋=2𝐼,_𝑘+1=𝐼+𝐴^∗₁𝑋_𝑘𝐴₁+𝐴^∗₂𝑋_𝑘𝐴₂−𝐵^∗₁𝑌_𝑘𝐵₁−𝐵^∗₂𝑌_𝑘𝐵₂,𝑌_𝑘+1=𝐼+𝐴^∗₁𝑌_𝑘𝐴₁+𝐴^∗₂𝑌_𝑘𝐴₂−𝐵^∗₁𝑋_𝑘𝐵₁−𝐵^∗₂𝑋_𝑘𝐵₂,𝑘=0,1,2,… (3.4) both converge to 𝑋_𝑈. Choose 𝜏=1.0×10⁻¹⁵ as the termination scalar, that is, 𝑅(𝑋)=𝑋−𝐴^∗₁𝑋𝐴₁−𝐴^∗₂𝑋𝐴₂+𝐵^∗₁𝑋𝐵₁+𝐵^∗₂𝑋𝐵₂−𝐼≤𝜏=1.0×10⁻¹⁵. (3.5) By using the iterative method (3.4) we can get the computed solution 𝑋_𝑘 of (3.1). Since 𝑅(𝑋_𝑘)<1.0×10⁻¹⁵, then the computed solution 𝑋_𝑘 has a very high precision. For simplicity, we write the computed solution as the unique positive definite solution 𝑋_𝑈. Similarly, we can also get the unique positive definite solution 𝑋_𝑈 of the perturbed equation (3.2).
Some numerical results on the perturbation bounds for the unique positive definite solution 𝑋_𝑈 are listed in Table 1.
From Table 1, we see that Theorem 2.5 gives a precise perturbation bound for the unique positive definite solution of (3.1).

Table 1 

Numerical results for the different values of 𝑗.

4. Conclusion

In this paper, we study the matrix equation (1.1) which arises in solving some nonlinear matrix equations and the bilinear control system. A new method of perturbation analysis is developed for the matrix equation (1.1). By making use of the matrix differentiation and its elegant properties, we derive a precise perturbation bound for the unique positive definite solution of (1.1). A numerical example is presented to illustrate the sharpness of the perturbation bound.

Acknowledgments

This research was supported by the National Natural Science Foundation of China (11101100; 11171205; 11261014; 11226323), the Natural Science Foundation of Guangxi Province (2012GXNSFBA053006; 2011GXNSFA018138), the Key Project of Scientific Research Innovation Foundation of Shanghai Municipal Education Commission (13ZZ080), the Natural Science Foundation of Shanghai (11ZR1412500), the Ph.D. Programs Foundation of Ministry of Education of China (20093108110001), the Discipline Project at the corresponding level of Shanghai (A. 13-0101-12-005), and Shanghai Leading Academic Discipline Project (J50101). The authors wish to thank Professor Y. Zhang and the anonymous referees for providing very useful suggestions for improving this paper. The authors also thank Dr. Xiaoshan Chen for discussing the properties of matrix differentiation.