The Feasibility of Homotopy Continuation Method for a Nonlinear Matrix Equation

In this paper, we discuss the feasibility of homotopy continuation method for the nonlinear matrix equations X + 􏽐 si � 1 B ∗ i X − 1 B i + 􏽐 mi � s + 1 B ∗ i X t i B i � I with 0 < t i < 1. This iterative method does not depend on a good initial approximation to


Introduction
In this paper, we consider the Hermitian positive definite (HPD) solutions of the nonlinear matrix equation: where B i (i � 1, 2, . . . , m) are n × n complex nonsingular matrices, I is an n × n identity matrix, and s ≥ 1, m ≥ 2 are positive integers. Here, B * i denotes the conjugate transpose of the matrix B i . e nonlinear matrix equations of (1) or some special cases are applicable to many fields such as nanoresearch, ladder networks, dynamic programming, control theory, stochastic filtering, and statistics [1][2][3][4][5][6][7][8].
According to our knowledge, the matrix equation (1) has not been treated explicitly in the literatures. e reason is that (1) does not always have a unique Hermitian positive definite solution. It is hard to find sufficient conditions for the existence of a unique Hermitian positive definite solution because the map ere are two difficulties for discussing the solvability and iterative method for the matrix equation (1). One is how to find a suitable set and some reasonable restrictions on the coefficient matrices ensuring this equation has a unique Hermitian positive definite solution in this set. e other one is how to find a reasonable expression of F(X) − F(Y), which was important for discussing the feasibility of the homotopy continuation iterative method (see [34], for more details), which was not dependent on a good initial approximation to the solution of matrix equation.
In this paper, firstly, we derive necessary and sufficient conditions for the existence of Hermitian positive definite solutions to equation (1) in Section 3. en, discuss the homotopy continuation methods for obtaining the unique Hermitian positive definite solution in Section 4. e following notations are used throughout this paper. We denote by C n×n , H n×n , and U n×n the set of all n × n complex matrices, Hermitian matrices, and unitary matrices, respectively. For A � (a 1 , . . . , a n ) � (a ij ) ∈ C n×n and a matrix B, A ⊗ B � (a ij B) is a Kronecker product, and vec A is a vector defined by vecA � (a T 1 , . . . , a T n ) T . e symbol ‖ · ‖ stands for the spectral norm. We denote by λ i (M) the eigenvalues of M, by λ 1 (M) and λ n (M) the maximal and minimal eigenvalues of M, respectively. For X,

Preliminaries
In this section, we present some lemmas that will be needed to develop this paper.

Hermitian Positive Definite Solutions
In this section, some sufficient and necessary conditions for the existence and uniqueness of HPD solution of (1) are derived.

Theorem 1. (1) has a HPD solution if and only if there exist
where Λ 2 + Γ � I and m i�1 Q * i Q i � I. In this case, Y � P * ΓP is a HPD solution of (1).
Proof. Assume Y is an HPD solution of (1). According to the spectral decomposition theorem, we have that there exists P ∈ U n×n and a diagonal matrix Γ > 0 such that Y � P * ΓP.
en, (1) can be expressed as Multiplying the left side of (8) by P and the right side by P * , we obtain Recall that B i (i � 1, 2, . . . , m) are nonsingular matrices. en, erefore, we can rewrite (10) as Let It is easy to verify that Γ + Λ 2 � I and which means Y is an HPD solution of (1). Proof.

□
Step 1. We will prove that (1) has a HPD solution on Obviously, Ω is a bounded convex closed set and F is continuous on Ω.
Step 2. We will prove that if (1) has a HPD solution on [(m/(m + 1))I, I], then the HPD solution is unique.

The Homotopy Continuation Iterative Method
In this section, by means of the homotopy continuation iterative method (see [34], for more details), we derive a numerical iterative process for solving the matrix equation (1). Define the nonlinear map F: P n×n ⊂ C n×n ⟶ C n×n by Consider the homotopy H: [0, 1] × P n×n ⟶ P n×n : en, at t � 0, the solution of H(t, X) � 0 is a known matrix I, while at t � 1, the solution X of H(t, X) � 0 also solves F(X) � X. To discuss the numerical method for solving the homotopy equation H(t, X) � 0, we rewrite the homotopy equation H(t, X) � 0 as the following fixed point form.
Assume that G: [0, 1] × P n×n ⊂ [0, 1] × C n×n ⟶ P n×n is a map such that where X: [0, 1] ⟶ P n×n denotes the solution of H(t, X) � 0. en, for each t, we can consider the iterative process: Since for a fixed t, this process will converge to X(t) only for starting values near that point; to overcome the local convergence of iterative process, we consider the following numerical continuation process.
A partition of J � [0, 1]: and a sequence of integers j k , k � 1, . . . , N − 1, is chosen with the property that the points are well-defined and such that converges to X(1) as j ⟶ ∞. e main idea is to choose partition (44) so that X(t k ) lies in some domain of attraction D t (k+1) , for each k, 1 ≤ k ≤ N.
en, if X k,0 ∈ D t (k+1) , the sequence generated by (43) for t � t k must produce an iterate X k,j k ∈ D t (k+1) , which in turn can be taken as the starting point X (k+1),0 � X k,j k for the next iteration involving t k+1 . us, with X 1,0 � X 0 as initial point, the entire process can be carried out until finally t k � t N � 1 is reached. For t � 1, X N,0 � X N,j (N− 1) is then in D 1 which ensures that (45) converges to X(1) as j ⟶ ∞.
To discuss the feasibility of the abovementioned numerical continuation process, we will use the following definition and lemmas which can be found in [34].
Definition 2 (see [34]). Let G: D ⊂ R n ⟶ R n be a given mapping. en, any nonempty set D 0 ⊂ D is a domain of attraction of the iterative process: with respect to the point x * , if for any x 0 ∈ D 0 , we have x n ⊂ D and lim n⟶∞ x n � x * . If x * ∈ int(D 0 ) for some domain of attraction D 0 , then x * is a point of attraction of (47).
Lemma 7 (see [34]). Let G: D ⊂ R n ⟶ R n be Fréchet differentiable at the fixed point x * ∈ int(D) of G. If ρ(G ′ (x * )) < 1, then x * is a point of attraction of (47) and, more precisely, there is an open ball S(x * , r) with center x * and radius r > 0 which is a domain of attraction of (47) with respect to x * . Here, ρ(·) denotes the spectral radius of G ′ (x * ).
In what follows, we derive a sufficient condition for the existence of a unique HPD solution of the homotopy equation H(t, X) � 0 for all t ∈ [0, 1]. Proof. Since t > 0, then the homotopy equation H(t, X) � 0 can be rewritten as By the hypothesis of the theorem, we have It follows from eorem 2 that the homotopy equation H(t, X) � 0 has a unique HPD solution on [(m/(m+ 1))I, I].
In the next theorem, the local convergence of the iterative process (43) is obtained. □ Theorem 4. If F(X) � X has a unique HPD solution X * on ((m/(m + 1))I, I), then there exists an open ball N(X * , δ) with center X * and radius δ > 0 such that, for any starting value X 0 ∈ N(X * , δ), X n � F(X n− 1 ) converges to X * as n ⟶ ∞.