ON POSITIVE DEFINITE SOLUTIONS OF THE MATRIX EQUATION X

TAIWANESE JOURNAL OF MATHEMATICS Vol. 16, No. 4, pp. 1391-1407, August 2012 This paper is available online at http://journal.taiwanmathsoc.org.tw

ON POSITIVE DEFINITE SOLUTIONS OF THE MATRIX EQUATION X + A∗ X −q A = Q(0 < q ≤ 1) Xiaoyan Yin*, Sanyang Liu and Tiexiang Li Abstract. Consider the nonlinear matrix equation X + A∗ X −q A = Q where 0 < q ≤ 1. A new sufficient condition for this equation to have positive definite solution is provided and two iterative methods for the maximal positive definite solution are proposed. Applying the theory of condition number developed by Rice, an explicit expression of the condition number of the maximal positive definite solution is obtained. The theoretical results are illustrated by numerical examples.

1. INTRODUCTION In this paper, we investigate the nonlinear matrix equation (1.1)

X+A∗ X −q A = Q,

where 0 < q ≤ 1, A is nonsingular and Q is positive definite. Eq.(1.1) can be viewed as a natural extension of the scalar equation x + a2 /xq = 1. Following [1], this matrix equation can be appeared during solving systems of linear equations of the form: (1.2)

M x = f,

where the positive definite matrix M arises from a finite difference approximation to Q A in which we an elliptic differential equation. As an example, let M = A∗ Q q X A ˜ = ˜ + diag[Q − X q , 0] where M . We can can rewrite M as M = M A∗ Q ˜ , for solving Eq.(1.2), to the LU decomposition decompose M q q I 0 X A X A ˜ = . M= A∗ X −q I A∗ Q 0 X Received January 27, 2011, accepted August 29, 2011. Communicated by Wen-Wei Lin. 2010 Mathematics Subject Classification: 15A24, 65F10, 65H05. Key words and phrases: Nonlinear matrix equation, Positive definite solution, Condition number. *Corresponding author.

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Xiaoyan Yin, Sanyang Liu and Tiexiang Li

For such a decomposition to exist, the matrix X must be a solution of the matrix ˜ y = f is transformed to equation X + A∗ X −q A = Q. The solution of the system M the the solution of two linear systems that have lower triangular block coefficient matrix and upper triangular block coefficient matrix, respectively. The Woodbury formula can be applied to compute the solution of Eq.(1.2). Moreover, this type of nonlinear matrix equations have many applications in control theory, dynamic programming, statistics, stochastic filtering, nano research, see for example [1-5] and the references therein. The case that q = 1 has been studied extensively by many authors [1-7]. Also there are many contributions in the literature to the theory, applications and numerical solutions of Eq.(1.1) in the case that q is a positive integer, see for instance [8,9] and the references therein. The first attempt to solve Eq.(1.1) for q = 1 was made for q = 12 and Q = I by El-Sayed [10]. In [11], M.A.Ramadan investigated the equation 1 X + A∗ X − 2m A = I. Then V.I.Hasanov investigated Eq.(1.1) for q = 1/n, n ∈ N [12]. Eq.(1.1) with arbitrary 0 < q ≤ 1 and q ≥ 1 were considered in [13,14] and [15], respectively, where several necessary conditions and sufficient conditions for the existence of the positive definite solutions were derived. When Q = I, [16] treat iterative methods for X + A∗ X −q A = I. Zhenyun Peng and his coauthors [17,18] discussed Eq.(1.1) with two cases: q ≥ 1 and 0 < q < 1. They established iterative methods to obtain the positive definite solutions and considered the convergence rates of the considered methods. Based on these, in this paper, we continue to study the matrix equation X + ∗ −q A X A = Q with 0 < q ≤ 1 and Q positive definite. We derive a new sufficient condition for Eq.(1.1) to have positive definite solutions and also we propose two iterative methods for obtaining the maximal positive definite solution. Moreover, by the theory of condition number developed by Rice, we give an explicit expression of the condition number of the maximal positive definite solution to Eq.(1.1). The following notations are used throughout the rest of the paper. Denote the set of n × n Hermitian matrices by H n×n . The notation A ≥ 0(A > 0) means that A is Hermitian positive semidefinite (positive definite). For A, B ∈ H n×n , we write A ≥ B(A > B) if A − B ≥ 0(> 0). We denote by σ1 (A) and σm (A) the maximal and minimal singular values of A, respectively. Similarly, λ1 (A) and λm (A) means the maximal and the minimal eigenvalues of A, respectively. For n × n square matrix A = (a1 , ..., an) = (aij ) and a matrix B, A ⊗ B = (aij B) is the Kronecker product, and vec(A) = (aT1 , ..., aTn )T . Unless otherwise noted, the symbol · denotes the matrix spectral norm (i.e., A = ρ(AA∗ ) = σ1 (A)), and · F the Frobenius norm. The paper is organized as follows. In Section 2, we give a new sufficient condition for Eq.(1.1) to have positive definite solutions and we offer two iterative methods for the maximal positive definite solution. An explicit expression of the condition number for the maximal solution is obtained in Section 3. Finally, several numerical examples are given in Section 4.

On Positive Definite Solutions of the Matrix Equation X +A∗ X −q A = Q(0 < q ≤ 1)

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2. POSITIVE DEFINITE SOLUTIONS In this section, we provide a sufficient condition for Eq.(1.1) to have positive definite solutions and also we propose two iterative methods for obtaining the maximal positive definite solution. To prove our main results, we need the following lemmas: Lemma 2.1. ([19]). Let A, B be positive definite. Then for any unitary invariant norm | · |, we have |B t At B t | ≤ |(BAB)t |, if 0 ≤ t ≤ 1; |(BAB)t | ≤ |B t At B t |, if t ≥ 1. Lemma 2.2. ([20]). If A > B > 0(or A ≥ B > 0), then A q > B q > 0(or Aq ≥ B > 0) for all q ∈ (0, 1], and 0 < A q < B q (or 0 < Aq ≤ B q ) for all q ∈ [−1, 0). q

Lemma 2.3. ([20]). For any Hermite matrices X, Y ≥ bI > 0, q > 0, we have X −q − Y −q ≤ qb−(q+1) X − Y . Lemma 2.4. ([3]). If Eq. (1.1) has a positive definite solution X, then 2 (Q−q/2AQ−1/2 ) ≤ σm

qq and X ≤ αQ, (q + 1)q+1

2 (Q−q/2 AQ−1/2 ) in where α is a solution of the scalar equation x q (1 − x) = σm q [ q+1 , 1].

Lemma 2.5. ([3]). If C and P are Hermitian matrices of the same order with P > 0, then CP C + P −1 ≥ 2C. Now consider the following scalar equations: (2.1)

2 (Q−q/2AQ−1/2 ), xq (1−x) = σm

(2.2)

xq (1−x) = σ12 (Q−q/2AQ−1/2 ).

Let

f (x) = xq (1 − x), x ∈ [0, 1].

q ], monotonically decreasing on Obviously, f (x) is monotonically increasing on [0, q+1 q [ q+1 , 1], and qq q )= . max f (x) = f ( q+1 (q + 1)q+1 x∈[0,1]

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Thus, if σ12 (Q−q/2AQ−1/2 )
0|α1 Q ≤ X ≤ β1 Q}, ϕ3 = {X > 0|β1 Q < X < β2 Q}, ϕ4 = {X > 0|β2 Q ≤ X ≤ α2 Q}, ϕ5 = {X > 0|α2 Q < X < Q}. Then we have the following result. Theorem 2.1. Suppose that σ 12 (Q−q/2AQ−1/2)
0 : q+1 Q for any X ∈ Ω. Combining Lemma 2.2 with the assumption A2 Q−1 q+1 < qq , we have (q+1)q+1 G(X) = Q − A∗ X −q A (1 + q)q −1/2 ∗ −q Q A Q AQ−1/2I]Q1/2 ≥ Q1/2[I − qq (1 + q)q 2 −q/2 ≥ Q1/2[I − σ1 (Q AQ−1/2)I]Q1/2 qq

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≥ Q1/2[1 − q Q, > q+1

(1 + q)q A2 Q−1 q+1 ]Q1/2 qq

which gives G(Ω) ⊆ Ω. Moreover, for any X, Y ∈ Ω, X, Y ≥

q q λm (Q)I = I. q+1 (q + 1)Q−1

According to Lemma 2.3, we have G(X) − G(Y ) = A∗ (X −q − Y −q )A ≤ A2 X −q − Y −q ≤

(q + 1)q+1 −1 q+1 Q A2 X − Y < X − Y , qq

which means that G(X) is a contraction on Ω. By Banach’s fixed-point theorem, G(X) has a unique fixed point on Ω, i.e., matrix equation (1.1) has a unique positive definite solution X L ∈ Ω, and XL can be obtained by iteration (2.5). Combining the fact that ϕ4 ⊂ Ω, we obtain that the unique positive definite solution X L is in ϕ 4 . Next, we prove that XL is the maximal positive definite solution. Let X be an arbitrary positive definite solution to Eq.(1.1). Obviously, X ≤ Q. Since G(X) is monotonically increasing, then X = G(X) ≤ G(Q), X = Gk (X) ≤ Gk (Q) → XL , k → ∞. Consequently, X ≤ XL which means that X L is the maximal positive definite solution of Eq.(1.1). Corollary 2.1. If A 2 Q−1 q+1 < solution X L exists and satisfies

qq , (q+1)q+1

then the maximal positive definite

1 XL−1 < (1 + )Q−1 . q Moreover, for any other positive definite solution X of matrix Eq.(1.1), we have 1 X −1 > (1 + )Q−1 . q q

q Proof. Since A 2 Q−1 q+1 < (q+1) q+1 , then the unique maximal positive q Q according to Theorem definite solution X L exists and satisfies X L ≥ β2 Q > q+1 −1 −1 1 1 −1 2.1. Thus XL < (1 + q )Q which gives XL < (1 + q )Q−1 .

Let X = XL be any positive definite solution to Eq.(1.1). Then X < the proof of theorem 2.1 (3) and consequently X−1 > (1 + 1q )Q−1 .

q q+1 Q

from


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Definition 2.1. ([21]). Suppose that the sequence {X k } of matrices converges to the matrix B. We say that this sequence converges linearly to B, if there exists a number σ ∈ (0, 1) and p = 1 such that lim

Xk+1 − B = σ. Xk − Bp

The number σ is called the rate of convergence. If the above holds with σ = 0 and p = 1 or p > 1, then the sequence is said to converge superlinearly. In particular, if p = 2 and σ < ∞, the sequence is said to be quadratic rate of convergence. q

q Corollary 2.2. If A 2 Q−1 q+1 < (q+1) q+1 , then the sequence {X k } of positive definite matrices generated by (2.5) converges to the maximal solution X L with at least linear convergence rate.

Proof. By (2.5), we have Xk+1 = Q − A∗ Xk−q A and Xk+1 ≥ q −1 −1 q+1 Q I for each k = 1, 2, .... Using Lemma 2.3, we then obtain that Xk+1 − XL = A∗ (XL−q −Xk−q )A ≤ q( = Denote 0 < θ =

q q+1 Q

≥

q Q−1 −1 )−(q+1)A2 Xk −XL q+1

(q + 1)q+1 A2 Q−1 q+1 Xk − XL . qq

(q+1)q+1 A2 Q−1 q+1 qq

lim

< 1. Then we have

Xk+1 − XL ≤ θ, Xk − XL

which completes the proof. Next, we give an inversion-free iteration for XL .  Y = (γQ)−1, γ ∈ [α2 , 1]    0 Xk = Q − A∗ Ykq A, (2.6)    Yk+1 = 2Yk − Yk Xk Yk . where α2 is defined in (2.4). Theorem 2.2. If Eq. (1.1) has a positive definite solution, then the sequence {X k } generated by (2.6) is monotone decreasing and converges to the maximal positive definite solution X L .

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Proof. Suppose X is a positive definite solution of Eq.(1.1), then X ≤ α 2 Q according to Lemma 2.4. Combining this with Lemma 2.2, we have Y0 = (γQ)−1 ≤ (α2 Q)−1 ≤ X −1 , X0 = Q − A∗ Y0q A ≥ Q − A∗ X −q A = X. Appling Lemma 2.5, we obtain that Y1 = 2Y0 − Y0 X0 Y0 ≤ X0−1 ≤ X −1 . Since X0 = Q −

2 (Q−q/2 AQ−1/2 ) A∗ Q−q A σm 1/2 ≤ Q [1 − ]Q1/2 ≤ γQ = Y0−1 γq γq

from the definition of γ, then Y1 − Y0 = Y0 − Y0 X0 Y0 = Y0 [Y0−1 − X0 ]Y0 ≥ 0. It follows that and

X1 = Q − A∗ Y1q A ≥ Q − A∗ X −q A = X X1 − X0 = −A∗ Y1q A + A∗ Y0q A = A∗ (Y0q − Y1q )A ≤ 0.

Thus X 0 ≥ X1 ≥ X and Y0 ≤ Y1 ≤ X −1 . Suppose that Xk−1 ≥ Xk ≥ X, Yk−1 ≤ Yk ≤ X −1 , then Yk+1 = 2Yk − Yk Xk Yk ≤ Xk−1 ≤ X −1 , Xk+1 = Q − A∗ Ykq A ≥ Q − A∗ X −q A = X, and

Yk+1 − Yk = Yk (Yk−1 − Xk )Yk ≥ Yk (Yk−1 − Xk−1 )Yk ≥ 0, Yk+1 = 2Yk − Yk Xk Yk ≤ Xk−1 ≤ X −1 .

According to Lemma 2.2, we get q )A ≤ 0. Xk+1 − Xk = A∗ (Ykq − Yk+1

Therefore, Xk ≥ Xk+1 ≥ X and Yk ≤ Yk+1 ≤ X −1 for each k = 0, 1, 2, .... Thus the sequences {Xk } and {Yk } are both convergent. Denote Xl = limk→∞ Xk and Y = limk→∞ Yk . Taking limit in iteration (2.6) leads to Y = X l−1 and Xl + A∗ Xl−q A = Q. Moreover, since Xk ≥ X for all k = 0, 1, 2, ..., then Xl = XL where XL is the maximal positive definite solution of Eq.(1.1). Remark 2.1. Let γ = 1 in (2.6), then we obtain Algorithm 2.1 in [18].


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3. CONDITION NUMBER OF THE MAXIMAL SOLUTION XL By the theory of condition number developed by Rice[22], we give in this section an explicit expression of the condition number of the maximal positive definite solution XL . 3.1. The complex case Lemma 3.1. ([23]). Let X be any n × n positive definite matrix, 0 < q ≤ 1. Then ∞ (i) X −q = sinπqπ 0 (λI + X)−1 λ−q dλ, ∞ (ii) X −q = sinqπqπ 0 (λI + X)−1 X(λI + X)−1 λ−q dλ. Denote the perturbed matrix equation of (1.1) by ˜ −q A˜ = Q, ˜ ˜ A˜∗ X X+

(3.1)

˜ are the slightly perturbed matrices of the matrices A and Q, respectively. where A˜ and Q From Theorem 2.1, we see that if ∆A and ∆Q are sufficiently small, then the ˜ L be the maximal positive solution to the perturbed matrix equation (3.1) exists. Let X ˜ ˜ −Q maximal positive definite solution to Eq.(3.1). Denote ∆A = A − A, ∆Q = Q ˜ L − XL . Subtracting (1.1) from (3.1) gives rise to and ∆X = X ∗ −q ˜ −q A−A ˜ XL A = ∆Q, i.e., ∆X + A˜∗ X L

(3.2)

˜ −q −X −q )A+A˜∗ (X ˜ −q −X −q )∆A+∆A∗ (X ˜ −q −X −q )A ∆X+A∗ (X L L L L L L = ∆Q − (∆A∗ XL−q A + ∆A∗ XL−q ∆A + A∗ XL−q ∆A).

Using Lemma 3.1, we have

(3.3)

˜ −q − X −q X L L sin qπ ∞ = [(λI + XL + ∆X)−1 − (λI + XL )−1 ]λ−q dλ π 0 sin qπ ∞ −(λI + XL )−1 ∆X(λI + XL + ∆X)−1 λ−q dλ = . π 0 ∞ sin qπ −(λI + XL )−1 ∆X(λI + XL )−1 λ−q dλ = π 0 ∞ sin qπ (λI +XL)−1 ∆X(λI +XL +∆X)−1 ∆X(λI +XL)−1 λ−q dλ + π 0

Combining (3.3) with (3.2), we obtain that ∞ (3.4) ∆X − sinπqπ 0 A∗ (λI +XL)−1 ∆X(λI +XL)−1 λ−q Adλ = E +h(∆X),

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where E = ∆Q − (∆A∗ XL−q A + ∆A∗ XL−q ∆A + A∗ XL−q ∆A), sin qπ ∗ ∞ A (λI + XL)−1 ∆X(λI + XL + ∆X)−1 h(∆X) = − π 0

∆X(λI + XL)−1 λ−q dλA sin qπ ˜∗ ∞ A (λI + XL)−1 ∆X(λI + XL + ∆X)−1 λ−q dλ∆A + π 0 ∞ sin qπ ∆A∗ (λI + XL )−1 ∆X(λI + XL + ∆X)−1 λ−q dλA. + π 0 q

q Then the linear operator L : Lemma 3.2. Let A2 Q−1 q+1 < (q+1) q+1 . n×n n×n →H defined by H sin qπ ∞ ∗ A (λI + XL )−1 W (λI + XL )−1 Aλ−q dλ (3.5) LW = W − π 0

is invertible. Proof.

It suffices to show that for any matrix V ∈ H n×n , the following equation

(3.6)

LW = V

has a unique solution. Define the operator M : H n×n → H n×n by sin qπ ∞ −1/2 ∗ XL A (λI + XL )−1 MZ = π 0 1/2

1/2

−1/2 −q

XL ZXL (λI + XL )−1 AXL

−1/2

Let Y = XL

−1/2

W XL

dλ, Z ∈ H n×n .

. Thus (3.6) is equivalent to −1/2

Y −MY = XL

(3.7) Notice that XL−1
0), Math. Num. Sin., 26 (2004), 61-72, (in Chinese). 21. A. M. Sarhan, N. M. El-Shazy and E. M. Shehata, On the existence of extremal positive ∗ δi definite solutions of the nonlinear matrix equation X r + Σm i=1 Ai X Ai = I, Mathematical and Computer Modelling, 51 (2010), 1107-1117. 22. J. R. Rice, A theory of condition, J. SIAM Numer. Anal., 3 (1966), 287-310. 23. J. Li and Y. H. Zhang, Perturbation analysis of the matrix equation X − A∗ X −p A = Q, Linear Algebra Appl., 431 (2009), 936-945. Xiaoyan Yin and Sanyang Liu Department of Mathematics Xidian University Xi’an 710071 P. R. China E-mail: [email protected] [email protected]


Tiexiang Li Department of Mathematics Southeast University Nanjing 211189 P. R. China E-mail: [email protected]

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