TAIWANESE JOURNAL OF MATHEMATICS
Vol. 18, No. 5, pp. 1345-1364, October 2014
DOI: 10.11650/tjm.18.2014.3747
This paper is available online at http://journal.taiwanmathsoc.org.tw
ON THE EXISTENCE OF POSITIVE DEFINITE SOLUTIONS OF A
NONLINEAR MATRIX EQUATION
Jing Li* and Yuhai Zhang
Abstract. In this paper the nonlinear matrix equation X −
m
i=1
A∗i X −pi Ai = Q
with pi > 0 is investigated. Necessary and sufficient conditions for the existence of Hermitian positive definite solutions are obtained. An effective iterative
method to obtain the unique solution is established. A perturbation bound and the
backward error of an approximate solution to this solution is evaluated. Moreover,
an explicit expression of the condition number for the positive definite solution is
given. The theoretical results are illustrated by numerical examples.
1. INTRODUCTION
In this paper we consider the sensitivity analysis of the nonlinear matrix equation
(1.1)
X−
m
i=1
A∗i X −pi Ai = Q, pi > 0,
where A1 , A2 , · · · , Am are n × n complex matrices, Q is an n × n Hermitian positive
definite matrix, m is a positive integer and the Hermitian positive definite solution X
is required. Here, A∗i denotes the conjugate transpose of the matrix Ai .
This type of nonlinear matrix equations arises in many practical problems, such as
ladder networks, dynamic programming, control theory, stochastic filtering, statistics
and so forth [1, 2, 3, 4, 5, 6]. When p1 = p2 = · · · = pm = 1, the matrix equation
(1.1) reduces to a special case of the nonlinear matrix equation
(1.2)
− C)−1 A,
X = Q + A∗ (X
Received September 5, 2013, accepted January 16, 2014.
Communicated by Wen-Wei Lin.
2010 Mathematics Subject Classification: 15A24, 65F10, 65H05.
Key words and phrases: Nonlinear matrix equation, Positive definite solution, Iteration, Perturbation
estimate, Backward error, Condition number, Fixed point theorem.
*Corresponding author.
1345
1346
Jing Li and Yuhai Zhang
where Q is an n × n positive definite matrix, C is an mn × mn positive semi-definite
is the m × m block diagonal matrix
matrix, A is an arbitrary mn × n matrix and X
with on each diagonal entry the n × n matrix X. In [7], the matrix equation (1.2)
is recognized as playing an important role in modelling certain optimal interpolation
problems. Let C = 0, A = (AT1 , AT2 , · · · , ATm )T , where Ai , i = 1, 2, · · · , m, are n×n
m
− C)−1 A can be written as X − A∗ X −1 Ai = Q.
matrices. Then X = Q + A∗ (X
The general nonlinear matrix equation X −
m
i=1
i=1
i
A∗i X −pi Ai = Q (pi > 0) comes from
solving a system of linear equations in many physical calculations [8]. When solving
m
A∗i X −pi Ai = Q, we often do not know Ai and
the nonlinear matrix equation X −
i=1
available. Then we can solve the
i and Q
Q exactly, but have only approximations A
m
∗ −p
exactly which gives a different solution X.
We would
X
iA
i = Q
− A
equation X
i=1
i
influence the error in X.
Motivated by this,
i and Q
like to know how the errors of A
m
(pi > 0).
∗ X
−pi A
i = Q
−
A
we consider in this paper the sensitivity analysis of X
i=1
i
For the case m > 1, the solvability and numerical solutions to the matrix equation
m
with 0 < pi ≤ 1 have been studied in [9, 11, 10]. Duan et
∗ X
−pi A
i = Q
A
X−
i
i=1
al. [9] proved that the equation X −
m
i=1
A∗i X δi Ai = Q (0 < |δi | < 1) always has
a unique Hermitian positive definite solution. They also proposed an iterative method
for obtaining the unique Hermitian positive definite solution. Duan and Liao [10]
∗ r
showed that equation X − m
i=1 Ai X Ai = Q with −1 ≤ r < 0 or 0 < r < 1 has
aunique Hermitian positive definite solution. Lim [11] showed that the equation X −
m
∗ δi
i=1 Ai X Ai = Q (0 < |δi | < 1) has a unique Hermitian positive definite solution.
However, these papers have not examined the sensitivity analysis about the equation
(1.1) and limited the range of pi in (0, 1]. Yin and Fang [12] obtained an explicit
expression of the condition number and also gave two perturbation estimates for the
∗ −1 A = Q. Duan et al.
unique Hermitian positive definite solution of X − m
i
i=1 Ai X
[13] gave two
perturbation estimates for the Hermitian positive definite solution of the
∗ δi
equation X − m
i=1 Ai X Ai = Q with 0 < |δi | < 1. Whereas, to our best knowledge,
there
have been no literatures paying attention to the sensitivity analysis
the equation
m for
∗ X −pi A = Q with p > 0. The reason is that X −
∗ X −pi A = Q
A
A
X− m
i
i
i
i=1 i
i=1 i
does not always have unique Hermitian positive definite solution in the case pi > 0.
It is hard to find sufficient conditions for the existence
Hermitian positive
mof a ∗unique
−pi
definite solution, because the map L(X) = Q + i=1 Ai X Ai with pi > 0 is
not monotonic. There are two difficulties for considering the sensitivity analysis for
the equation (1.1). One is how to find some reasonable restrictions on the coefficient
On the Existence of Positive Definite Solutions of a Nonlinear Matrix Equation
1347
matrices ensuring this equation has a unique Hermitian positive definite solution. The
other one is how to find a reasonable expression of the matrix X −pi in the case pi > 0
which is easy to handle. By using the integral representation of matrix function, the
fixed point theorem and the technique developed in [14], we derive a sufficient condition
for the existence of a unique Hermitian positive definite solution to the matrix equation
(1.1) and consider the sensitivity analysis of this equation.
The rest of the paper is organized as follows. In Section 2, we give some preliminary knowledge that will be used to develop this work. In Section 3, we derive
necessary and sufficient conditions for the existence of Hermitian positive definite solutions to the equation (1.1). In Section 4, we give a perturbation bound for the unique
solution to the equation (1.1), which is independent of the exact solution of the equation
(1.1). In Section 5, the backward error estimates of an approximate solution for the
unique solution to the equation (1.1) are discussed. In Section 6, applying the integral
representation of matrix function, we also discuss the explicit expression of the condition number for the Hermitian positive definite solution to the equation (1.1). Finally,
several numerical examples are presented in Section 7.
We denote by C n×n the set of n × n complex matrices, by Hn×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 λmax (M ) and λmin (M ) the maximal and
minimal eigenvalues of M , respectively. For A = (a1 , . . . , an ) = (aij ) ∈ C n×n and a
matrix B, A ⊗ B = (aij B) is a Kronecker product, and vecA is a vector defined by
vecA = (aT1 , . . ., aTn )T . For X, Y ∈ Hn×n , we write X ≥ Y (resp. X > Y ) if X − Y
is Hermitian positive semi-definite (resp. definite).
2. PRELIMINARIES
In this section we quote some preliminary lemmas that we will use later.
p
Lemma 2.1. [15, Lemma 1]. If X, Y ≥ βI > 0 and p > 0, then X −p − Y −p ≤
− Y .
β −(p+1) X
Lemma 2.2. [15, Lemma 2].
(i) If X ∈ Hn×n , then e−X = e−λmin(X) .
∞ −sX r−1
1
s ds.
(ii) If X ∈ Hn×n and r > 0, then X −r = Γ(r)
0 e
1
(iii) If A, B ∈ C n×n , then eA+B − eA = 0 e(1−t)A Bet(A+B) dt.
3. NECESSARY AND SUFFICIENT CONDITIONS
In this section, we derive the properties of the Hermitian positive definite solutions
of (1.1), including uniqueness and estimates of the solutions.
1348
Jing Li and Yuhai Zhang
Theorem 3.1. The nonlinear matrix equation (1.1) always has Hermitian positive
definite solutions.
−pi
∗
Proof. Let Ω = {X : Q ≤ X ≤ Q + m
i=1 λmin (Q)Ai Ai }. Obviously, Ω is a
m
A∗i X −pi Ai . Evidently, F
nonempty bounded convex closed set. Let F (X) = Q +
i=1
is continuous. For X ∈ Ω, we have X ≥ Q > 0, which implies that F (X) ≥ Q.
i
It follows from X ≥ Q ≥ λmin (Q)I that X −pi ≤ λ−p
min (Q)I, which implies that
m
i
∗
F (X) ≤ Q+ λ−p
min (Q)Ai Ai . Therefore F (Ω) ⊆ Ω. By Brouwer fixed point theorem,
i=1
there exists X ∈ Ω satisfies F (X) = X, which means X is a solution of Eq.(1.1).
Theorem 3.2. If X is an Hermitian positive definite solution of (1.1), then Q ≤
m
A∗i Ai
.
X ≤Q+
pi
i=1 λmin (Q)
Proof. That X is an Hermitian positive definite solution of (1.1) implies X >
0. Then X −pi > 0 and A∗i X −pi Ai ≥ 0, i = 1, 2, · · · , m. Hence X = Q +
m
i
A∗i X −pi Ai ≥ Q ≥ λmin (Q)I. Consequently, X −pi ≤ λ−p
min (Q)I and X ≤ Q +
i=1
m
i=1
−p
λmini (Q)A∗i Ai .
Theorem 3.3. If
q=
m
i=1
i −1
pi Ai 2 λ−p
min (Q) < 1,
then
(i) The matrix equation (1.1) has a unique Hermitian positive definite solution X.
(ii) The iteration
(3.1) X0 ∈ Q, Q+
m
i=1
∗
i
λ−p
min (Q)Ai Ai
, Xn = Q+
m
i=1
−pi
A∗i Xn−1
Ai , n = 1, 2, · · ·
converges to X. Moreover,
qn
X1 − X0 .
1−q
−pi
∗
Proof. Let Ω = {X : Q ≤ X ≤ Q + m
i=1 λmin (Q)Ai Ai }. Obviously, Ω is a
m
A∗i X −pi Ai . Evidently, F
nonempty bounded convex closed set. Let F (X) = Q +
Xn − X ≤
i=1
On the Existence of Positive Definite Solutions of a Nonlinear Matrix Equation
1349
is continuous. The proof of Theorem 3.1 implies that F (Ω) ⊆ Ω. According to Lemma
2.1, we obtain ∀X1 , X2 ∈ Ω,
m
m ∗ −pi
−pi
−pi
−pi
∗
Ai (X1 − X2 )Ai ≤
(X
−
X
)A
F (X1 ) − F (X2 ) = Ai 1
i
2
i=1
≤
m
i=1
i=1
−(p +1)
pi λmini
(Q)A∗i Ai X1 − X2 = qX1 − X2 .
The last equality is due to the fact that ||A∗i Ai || = ||Ai ||2 (refer to [19, Problem 11.
Page 312]). From q < 1, it follows that F is a contractive mapping. By Banach
contraction mapping principle, the theorem is proved.
4. PERTURBATION BOUNDS
In this section we develop a relative perturbation bound for the unique solution of
(1.1), which does not need any knowledge of the exact solution X of (1.1) and is easy
to calculate.
Li and Zhang [16] proved the existence of a unique positive definite solution to
the equation X − A∗ X −p A = Q (0 < p < 1) and also obtained a perturbation
bound for the unique solution. However, their approach becomes invalid for X −
m
∗ −pi A = Q (p > 0). Since the latter equation does not always have a
i
i
i=1 Ai X
unique positive definite
solution,
there are two difficulties for a perturbation analysis
∗ X −pi A = Q (p > 0). One is how to find some
A
of the equation X − m
i
i
i=1 i
reasonable restrictions on the coefficient matrices of perturbed equation ensuring this
perturbed equation has a unique positive definite solution. The other one is how to
find an expression of X −pi (pi > 0) which is easy to handle.
Consider the perturbed matrix equation
−
X
(4.1)
m
pi > 0,
i ∗ X
−pi A
i = Q,
A
i=1
is an Hermitian positive definite matrix.
where Q
m
2 −pi −1
2 −pi −1 If m
i=1 pi Ai λmin (Q) < 1 and
i=1 pi Ai λmin (Q) < 1. According to
Theorem 3.3, equations (1.1) and (4.1) have unique Hermitian positive definite solutions
i − Ai , i = 1, 2, · · · , m, ΔQ = Q
− Q and
respectively. Let ΔAi = A
X and X,
ΔX = X − X, then we have the following theorem.
Theorem 4.1. If
m
(4.2)
i=1
pi Ai 2 Λ−pi −1 < 1 and
m
i=1
with Λ = min{λmin (Q), λmin(Q)},
i 2 Λ−pi −1 < 1
pi A
1350
Jing Li and Yuhai Zhang
then
X−
m
i=1
A∗i X −pi Ai
−
= Q and X
m
i ∗ X
−pi A
i = Q
A
i=1
respectively. Furthermore,
have unique Hermitian positive definite solutions X and X,
− X
X
≤
X
m
+ ΔAi )ΔAiΛ−pi + ||ΔQ||
ξ.
2 −pi
Λ− m
i=1 pi Ai Λ
i=1 (2Ai
Proof. According to Theorem 3.3, the condition (4.2) ensures that (1.1) and (4.1)
respectively. Furthermore, we obtain
have unique positive definite solution X and X,
that
≥ λmin (Q)I
≥ ΛI.
X ≥ λmin (Q)I ≥ ΛI, X
(4.3)
Subtracting (4.1) from (1.1) gives
ΔX =
(4.4)
m −pi A
i − A∗i X −pi Ai + ΔQ
i ∗ X
A
i=1
m −pi − X −pi )Ai + ΔA∗ X
i ∗ X
−pi Ai + A
−pi ΔAi + ΔQ.
A∗i (X
=
i
i=1
By Lemma 2.2 and the inequalities in (4.3), we have
m −pi Ai A∗i X −pi Ai − Ai ∗ X
ΔX +
i=1
∞
m
pi −1
∗ 1
−sX
−sX
Ai
(e
−e
)s
dsAi = ΔX +
Γ(pi ) 0
i=1
∞ 1
m
1
A∗i
e−(1−t)sX ΔXe−tsX dtspi dsAi = ΔX +
Γ(pi ) 0
0
i=1
(4.5)
m
Ai 2 ΔX ∞ 1 −(1−t)sX ≥ ΔX −
e
e−tsX dtspi ds
Γ(pi )
0
0
i=1
m
Ai 2 ΔX ∞ 1
e−(1−t)sλmin(X) e−tsλmin(X) dtspi ds
= ΔX −
Γ(pi )
0
0
i=1
m
Ai 2 ΔX ∞ 1 −(1−t)sΛ −tsΛ pi
e
e
dts ds
≥ ΔX −
Γ(pi )
0
0
i=1
On the Existence of Positive Definite Solutions of a Nonlinear Matrix Equation
m
Ai 2 ΔX
1351
∞ 1
e−sΛ dtspi ds
i=1
m
m
Γ(pi + 1) Ai2 ΔX
pi Ai 2
·
= 1−
ΔX.
= ΔX −
Γ(pi )
Λpi+1
Λpi+1
= ΔX −
Γ(pi )
0
0
i=1
i=1
Noticing the conditions in (4.2), we have
1−
m
piAi 2
i=1
Λpi+1
> 0.
Combining (4.4) with (4.5), one sees that
m
m
∗
pi Ai 2
∗ −pi
−pi
(ΔA
A
+
A
ΔA
)
+
ΔQ
X
X
ΔX
≤
1−
i
i
i
i
p
+1
i
Λ
≤
m
i=1
i=1
−pi + ||ΔQ||
(ΔAi + 2Ai) ΔAi X
i=1
m
(ΔAi + 2Ai )ΔAiΛ−pi + ||ΔQ||,
≤
i=1
which implies that
m
ΔX
≤
X
(2Ai + ΔAi )ΔAiΛ−pi + ||ΔQ||
i=1
Λ−
m
.
pi Ai 2 Λ−pi
i=1
5. BACKWARD ERROR
In this section, we obtain some estimates for the backward error of the approximate
solution of (1.1).
> 0 be an approximation to the unique solution X to the equation (1.1),
Let X
and let ΔAi ∈ C n×n (i = 1, 2, · · · , m) and ΔQ ∈ Hn×n be the corresponding perturbations of the coefficient matrices Ai (i = 1, 2, · · · , m) and Q in the equation (1.1).
can be defined by
A backward error of the approximate solution X
ΔA1 ΔA2
ΔAm ΔQ :
,
,··· ,
,
η(X) = min α1
α2
αm
ρ F
m
(5.1)
∗
−p
i (Ai + ΔAi ) = Q + ΔQ ,
−
(Ai + ΔAi ) X
X
i=1
1352
Jing Li and Yuhai Zhang
where α1 , α2 , · · · , αm and ρ are positive parameters. Taking αi = ||Ai||F , i =
and
1, 2, · · · , m and ρ = ||Q||F in (5.1) gives the relative backward error ηrel (X),
taking αi = 1, i = 1, 2, · · · , m and ρ = 1 in (5.1) gives the absolute backward error
ηabs(X).
Let
+
R = Q−X
(5.2)
m
−pi Ai .
A∗i X
i=1
Note that
−
Q=X
m
−pi (Ai + ΔAi ) − ΔQ.
(Ai + ΔAi )∗ X
i=1
It follows from (5.2) that
(5.3)
−
m
−pi Ai + A∗ X
−pi ΔAi ) − ΔQ = R +
(ΔA∗i X
i
i=1
Let
m
−pi ΔAi .
ΔA∗i X
i=1
−pi Ai )T ⊗ I Π = Ui2 + iΩi2 ,
−pi Ai )∗ = Ui1 + iΩi1 , (X
I ⊗ (X
vecΔAi = xi + iyi , vecΔQ = q1 + iq2 , vecR = r1 + ir2 ,
−pi ΔAi ) = ai + ibi, i = 1, 2, · · · , m,
vec(ΔA∗i X
g=
xT y T q T q T
xT1 y1T
,
,··· , m, m, 1 , 2
α1 α1
αm αm ρ ρ
Ui =
Ui1 + Ui2 Ωi2 − Ωi1
Ωi1 + Ωi2 Ui1 − Ui2
T
,
,
T = [−α1 U1 , −α2 U2 , · · · , −αm Um , −ρI2n2 ],
where Π is the vec-permutation. Then (5.3) can be rewritten as
m ai
r1
+
.
(5.4)
Tg =
r2
bi
i=1
It follows from ρ > 0 that 2n2 × 2(m + 1)n2 matrix T is full row rank. Hence,
T T † = I2n2 , which implies that every solution to the equation
m ai r1
†
†
+T
(5.5)
g=T
r2
bi
i=1
On the Existence of Positive Definite Solutions of a Nonlinear Matrix Equation
1353
must be a solution to the equation (5.4). Consequently, for any solution g to (5.5) we
have
≤ ||g||.
η(X)
(5.6)
Then we can state the estimates of the backward error as follows.
be given matrices. Where Ai ∈
Theorem 5.1. Let A1 , A2 , · · · , Am, Q and X
be the backward
X and Q are Hermitian positive definite matrices. Let η(X)
error defined by (5.1). If
C n×n ,
s
r< m
,
2
4
ti αi
(5.7)
i=1
then we have that
≤ B(r),
U (r) ≤ η(X)
(5.8)
where
† r1 −pi , s = T † −1 , ti = X
(5.9) r = T
,
r2 2
s2 − 4rs( m
i=1 ti αi )
2rs
(5.10) B(r) =
, U (r) =
.
2
2
s + s2 − 4rs( m
s + s2 − 4rs( m
i=1 ti αi )
i=1 ti αi )
2r
Proof.
Let
L(g) = T
†
2(m+1)n2×1
→C
Obviously, L : C
that the quadratic equation
r1
r2
m ai +T
.
bi
†
i=1
2(m+1)n2 ×1
is continuous. The condition (5.7) ensures
m
(5.11)
1 2 2
ti αi )x
x=r+ (
s
i=1
in x has two positive real roots. The smaller one is
B(r) =
2rs
.
2)
s + s2 − 4rs( m
t
α
i=1 i i
1354
Jing Li and Yuhai Zhang
2
Define Ω = g ∈ C 2(m+1)n ×1 : ||g|| ≤ B(r) . Then for any g ∈ Ω, we have
||L(g)||
m m
m
1
1 1 ai ∗ −pi
ti ΔAi2F
≤ r+
ΔAi X ΔAi ≤ r +
bi = r + s
s
s
F
i=1
i=1
i=1
m
m
2
1
ΔA
ΔA
1
ΔA
1
2
m
≤r+
≤ r+
ti α2i ti α2i g2
α , α ,··· , α
s
s
1
2
m
F
i=1
i=1
m
1 2
≤ r+
ti αi B 2 (r) = B(r).
s
i=1
The last equality is due to the fact that B(r) is a solution to the quadratic equation
(5.11). Thus we have proved that L(Ω) ⊂ Ω. By the Schauder fixed-point theorem,
there exists a g∗ ∈ Ω such that L(g∗ ) = g∗ , which means that g∗ is a solution to (5.5),
and hence it follows from (5.6) that
≤ ||g∗ || ≤ B(r).
η(X)
Suppose that ( ΔA1 min , · · · , ΔAm min , ΔQmin )
Next we derive a lower bound for η(X).
α1
αm
ρ
satisfies
ΔA1 min
ΔAm min ΔQmin .
,··· ,
,
(5.12)
η(X) = α
α
ρ
1
Then we have
(5.13)
m
T gmin =
r1
r2
F
m ai∗
+
,
bi∗
i=1
where
−pi ΔAi min ) = ai∗ + ibi∗ ,
vec(ΔATimin X
vec(ΔAi min ) = xi∗ + iyi∗ ,
vec(ΔQmin ) = q1∗ + iq2∗ ,
T
T
T
T
T
T
q2∗
x1∗ y1∗
xTm∗ ym∗
q1∗
,
,
,··· ,
,
,
.
gmin =
α1 α1
αm αm ρ ρ
Let a singular value decomposition of T be T = W (E, 0)Z T , where W and Z are orthogonal matrices, E = diag(e1, e2 , · · · , e2n2 ) with e1 ≥ · · · ≥ e2n2 > 0. Substituting
this decomposition into (5.13) and letting
2
v
T
, v ∈ R2n ,
Z gmin =
∗
On the Existence of Positive Definite Solutions of a Nonlinear Matrix Equation
we get
v = E −1 W T
r1
r2
1355
m ai∗
.
bi∗
+ E −1 W T
i=1
It follows from (5.12) that
v η(X) = gmin = ∗ ≥ v
m −1 T
r1 −1 T
ai∗ W
W
−
E
≥
E
r2 bi∗ i=1
m † r1 a
i∗
†
− T ·
≥
T
r2
bi∗ i=1
m
m
1
1 ∗
−pi
ΔA
≥
r
−
ti ΔAi min 2F
X
≥ r−
ΔAi min
i min s
F
s
i=1
i=1
m
2
1
ΔAm min 2 ΔA1 min
≥ r−
ti αi ,··· ,
s
α1
αm
F
i=1
m
1 2
≥ r−
ti αi B 2 (r).
s
i=1
Here we have used the fact that
ΔA1 min
ΔAm min ,··· ,
α1
αm
F
ΔA1 min
ΔAm min ΔQmin = η(X)
≤ B(r).
≤
,··· ,
,
α1
αm
ρ
F
Let
1
U (r) = r −
s
m
i=1
ti α2i
B 2 (r).
Since B(r) is a solution to the equation (5.11), we have
m
1 2 2
ti αi )B (r),
B(r) = r + (
s
i=1
which implies that
U (r) = r −
m
1 s
i=1
ti α2i
2
s2 − 4rs( m
i=1 ti αi )
> 0.
B 2 (r) = 2r − B(r) =
2)
s + s2 − 4rs( m
t
α
i=1 i i
2r
1356
Jing Li and Yuhai Zhang
≥ U (r).
Then η(X)
6. CONDITION NUMBER
A condition number is a measurement of the sensitivity of the Hermitian positive
definite stabilizing solutions to small changes in the coefficient matrices. In this section,
we apply the theory of condition number developed by Rice [17] to study condition
numbers of the unique solution to (1.1). The difficulty for obtaining explicit expressions
of the condition number for (1.1) is how to find expressions of ΔX and X −pi (pi > 0)
which are easy to handle. Here we consider the perturbed equation
−
X
(6.1)
m
i=1
Suppose that
m
i=1
∗ X
−pi A
i = Q.
A
i
i −1
piAi 2 λ−p
min (Q) < 1 and
m
i=1
i 2 λ−pi −1 (Q)
< 1. According to
pi A
min
Theorem 3.3, equations (1.1) and (6.1) have unique Hermitian positive definite solutions
i − Ai , ΔQ = Q
− Q and ΔX = X
− X.
respectively. Let ΔAi = A
X and X,
Subtracting (6.1) from (1.1) gives
ΔX =
m −pi A
i − A∗ X −pi Ai + ΔQ
∗ X
A
i
i=1
=
i
m −pi − X −pi )Ai + ΔA∗i X
i ∗ X
−pi Ai + A
−pi ΔAi + ΔQ
A∗i (X
i=1
m
=−
=−
+
i=1
m
A∗i
Γ(pi )
A∗i
∞
m −sX
−sX
pi −1
∗ X
−pi Ai + A
−pi ΔAi +ΔQ
−e
dsAi +
e
s
ΔA∗i X
i
Γ(pi )
i=1
m 0
0
∞ 1
0
i=1
− X e−tsX dtspi dsAi
e−(1−t)sX X
∗i X
−pi Ai + A
−pi ΔAi + ΔQ
ΔA∗i X
i=1
∞ 1
m
A∗i
(e−(1−t)sX − e−(1−t)sX )ΔXe−tsX dtspi dsAi
=−
Γ(pi ) 0
0
i=1
+
m
i=1
∗ (X + ΔX)−pi ΔAi
A
i
∞ 1
m
m
A∗i
∗i X −pi ΔAi
−
e−(1−t)sX ΔXe−tsX dtspi dsAi −
A
Γ(pi ) 0
0
i=1
i=1
1357
On the Existence of Positive Definite Solutions of a Nonlinear Matrix Equation
+
m
i ∗ X −pi ΔAi −
A
i=1
+
m
i=1
m
ΔA∗i X −pi Ai − ΔA∗i (X + ΔX)−pi Ai
i=1
ΔA∗i X −pi Ai + ΔQ
∞ 1 1
m
A∗i
e−(1−m)(1−t)sX
=
Γ(pi ) 0
0
0
i=1
ΔXe−m(1−t)sX ΔXe−tsX dm(1 − t)dtspi +1 dsAi + ΔQ
m
m
A∗ ∞ 1
−(1−t)sX
−tsX
pi
i
e
ΔXe
dts dsAi +
ΔA∗i X −pi ΔAi
−
Γ(pi ) 0
0
i=1
i=1
∗ ∞ 1
m
A
i
−
e−(1−t)s(X+ΔX) ΔXe−tsX dtspi dsΔAi
Γ(pi ) 0
0
+
i=1
m
(A∗i X −pi ΔAi + ΔA∗i X −pi Ai )
i=1
∞ 1
m
ΔA∗i
e−(1−t)s(X+ΔX) ΔXe−tsX dtspi dsAi .
−
Γ(pi ) 0
0
i=1
Therefore
(6.2)
∞ 1
m
A∗i
e−(1−t)sX ΔXe−tsX dtspi dsAi = E + h(ΔX),
ΔX +
Γ(pi ) 0
0
i=1
where
Bi = X −pi Ai ,
m
m
∗
∗
(Bi ΔAi + ΔAi Bi ) +
ΔA∗i X −pi ΔAi ,
E = ΔQ +
i=1
i=1
∞ 1 1
m
A∗i
e−(1−m)(1−t)sX
h(ΔX) =
Γ(pi ) 0
0
0
i=1
ΔXe−m(1−t)sX ΔXe−tsX dm(1 − t)dtspi +1 dsAi
m ∗ ∞ 1
Ai
−
e−(1−t)s(X+ΔX) ΔXe−tsX dtspi dsΔAi
Γ(pi ) 0
0
i=1
∞ 1
m
ΔA∗i
−
e−(1−t)s(X+ΔX) ΔXe−tsX dtspi dsAi .
Γ(pi ) 0
0
i=1
1358
Jing Li and Yuhai Zhang
Lemma 6.3. If
m
(6.3)
i=1
i −1
pi Ai 2 λ−p
min (Q) < 1,
then the linear operator V : Hn×n → Hn×n defined by
∞ 1
m
1
A∗i e−(1−t)sX W e−tsX Ai dtspi ds, W ∈ Hn×n
(6.4) VW = W +
Γ(pi ) 0
0
i=1
is invertible.
Defining the operator R : Hn×n → Hn×n by
∞ 1
m
1
A∗i e−(1−t)sX Ze−tsX Ai dtspi ds, Z ∈ Hn×n,
RZ =
Γ(pi ) 0
0
Proof.
i=1
it follows that
VW = W + RW.
Then V is invertible if and only if I + R is invertible.
According to Lemma 2.2 and the condition (6.3), we have
∞ 1
m
1
−(1−t)sX ||Ai||2 ||W ||
||RW || ≤
e
e−tsX dtspi ds
Γ(pi) 0
0
i=1
m
∞
1
1
||Ai ||2 ||W ||
e−(1−t)sλmin (X)e−tsλmin (X)dtspi ds
=
Γ(pi) 0
0
i=1
m
∞
1
1
||Ai ||2 ||W ||
e−(1−t)sλmin (Q)e−tsλmin (Q)dtspi ds
≤
Γ(pi) 0
0
i=1
m
∞
1
||Ai||2 ||W ||
e−sλmin (Q)spi ds
=
Γ(pi) 0
i=1
=
m
pi ||Ai||2
i=1
p +1
i
λmin
(Q)
||W || < ||W ||,
which implies that ||R|| < 1 and I + R is invertible. Therefore, the operator V is
invertible.
Based on the arguments above, we can rewrite (6.2) as
ΔX=V −1 ΔQ+V −1
m
m
(Bi∗ ΔAi +ΔA∗i Bi )+V −1
(ΔA∗i X −pi ΔAi )+V −1 (h(ΔX)).
i=1
i=1
On the Existence of Positive Definite Solutions of a Nonlinear Matrix Equation
1359
Obviously,
(6.5)
ΔX = V −1 ΔQ + V −1
m
i=1
(Bi∗ ΔAi + ΔA∗i Bi )
+O(||(ΔA1 , ΔA2 , · · · , ΔAm , ΔQ)||2F )
as (ΔA1 , ΔA2 , · · · , ΔAm , ΔQ) → 0.
By the condition number theory developed by Rice [17], we define the condition
number of the Hermitian positive definite solution X to (6.5) by
(6.6)
sup
c(X) = lim
δ→0
||(
ΔA1 ΔA2
m , ΔQ )|| ≤δ
, η ,··· , ΔA
F
η1
ηm
ρ
2
||ΔX||F
,
ξδ
where ξ, ρ and ηi , i = 1, 2, · · · , m are positive parameters. Taking ξ = ηi = ρ = 1 in
(6.6) gives the absolute condition number cabs (X), and taking ξ = ||X||F , ηi = ||Ai ||F
and ρ = ||Q||F in (6.6) gives the relative condition number crel (X).
Substituting (6.5) into (6.6), we get
m
−1 ∗
∗
V
(Bi ΔAi + ΔAi Bi ) + ΔQ 1
i=1
F
max
c(X) = ΔQ
ΔA
ΔA
ΔA
ξ ΔA1 ΔA2
1
2
m
ΔAm ΔQ
= 0 η1 , η2 , · · · , ηm , ρ F
η1 , η2 , · · · , ηm , ρ
ΔAi ∈ C n×n , ΔQ ∈ Hn×n
m
−1 ∗
∗
V
ηi (Bi Ei + Ei Bi + ρH) 1
i=1
F
max
=
.
ξ (E , E , · · · , E , H) = 0
(E1 , E2, · · · , Em, H) F
1
2
m
Ei ∈ C n×n , H ∈ Hn×n
Let V be the matrix representation of the linear operator V. It follows from Lemma
4.3.2 in [18] that
vec(VW ) = V · vecW.
By Lemma 4.3.1 in [18], we have
∞ 1
m
1
(e−tsX Ai )T ⊗(A∗i e−(1−t)sX )dtspi ds ·vecW.
vec(VW ) = I ⊗ I +
Γ(pi ) 0 0
i=1
Then
(6.7)
V = I ⊗I +
m
i=1
1
Γ(pi )
0
∞ 1
0
(e−tsX Ai )T ⊗ (A∗i e−(1−t)sX )dtspi ds.
1360
Jing Li and Yuhai Zhang
Let
(6.8) V −1 = S + iΣ, V −1 (I ⊗ Bi∗ ) = Ui1 + iΩi1 , V −1 (BiT ⊗ I)Π = Ui2 + iΩi2 ,
(6.9) Sc =
S −Σ
Σ S
,
Ui =
Ui1 + Ui2 Ωi2 − Ωi1
Ωi1 + Ωi2 Ui1 − Ui2
, i = 1, 2, · · · , m,
vecH = x + iy, vecEi = ai + ibi , g = (aT1 , bT1 , · · · , aTm, bTm, xT , y T )T ,
M = (E1 , E2, · · · , Em, H),
2
2 ×n2
where x, y, ai, bi ∈ Rn , S, Σ, Ui1, Ui2 , Ωi1 , Ωi2 ∈ Rn
√
−1, Π is the vec-permutation matrix, such that
, i = 1, 2, · · · , m, i =
vec AT = Π vec A.
Then we obtain that
c(X)
m
−1 ∗
∗
V
η
(ρH
+
B
E
+
E
B
)
i
i
i
i
i
1
i=1
F
max
ξ M = 0
||(E1, E2 , · · · , Em , H)||F
m
−1
−1
∗
T
∗ ρV vecH +
ηi V
(I ⊗ Bi )vecEi + (Bi ⊗ I)vecEi 1
i=1
=
max
ξ M = 0
vec (E1 , E2 , · · · , Em, H)
m
ρ(S + iΣ)(x + iy) +
ηi [(Ui1 + iΩi1 )(ai + ibi ) + (Ui2 + iΩi2 )(ai − ibi )]
1
i=1
=
max
ξ M = 0
vec (E1 , E2 , · · · , Em, H)
=
=
1
||(ρSc , η1 U1 , η2 U2 , · · · , ηm Um )g||
max
ξ g = 0
g
=
1
|| (ρSc , η1 U1 , η2 U2 , · · · , ηmUm )||, Ei ∈ C n×n .
ξ
Theorem 6.1. If
m
i=1
i −1
pi Ai 2 λ−p
min (Q) < 1 and
m
i=1
i −1
pi Ai + ΔAi 2 λ−p
min (Q +
ΔQ) < 1, the condition number c(X) defined by (6.6) has the explicit expression
c(X) =
(6.10)
1
|| (ρSc, η1 U1 , η2 U2 , · · · , ηmUm )||,
ξ
where the matrices Sc , Ui are defined as in (6.8).
Remark 6.1. From (6.10) we have the relative condition number
crel (X) =
|| (||Q||F Sc , ||A1 ||F U1 , ||A2 ||F U2 , · · · , ||Am||F Um )||
.
||X||F
1361
On the Existence of Positive Definite Solutions of a Nonlinear Matrix Equation
7. NUMERICAL EXAMPLES
To illustrate the theoretical results of the previous sections, in this section several
simple examples are given, which were carried out using MATLAB 7.1. For the
m
A∗i Xk−pi Ai − Q < 1.0e − 10.
stopping criterion we take εk+1 (X) = Xk −
i=1
Example 7.1. In this example, we study the matrix equation
X − A∗1 X −2 A1 − A∗2 X −3 A2 = I,
with
⎛
Ak =
1
k+2
⎜
⎜
+ 2 × 10−2
A, k = 1, 2, A = ⎜
⎜
||A||
⎝
2
1
0
0
0
1
2
1
0
0
0
1
2
1
0
0
0
1
2
1
0
0
0
1
2
⎞
⎟
⎟
⎟.
⎟
⎠
By computation, q = 2A∗1 A1 + 3A∗2 A2 = 0.4673 < 1. Let X0 = 1.1I ∈ [I, I +
A∗1 A1 + A∗2 A2 ]. The conditions in Theorem 3.3 are satisfied. Algorithm (3.1) needs
17 iterations to obtain the unique positive definite solution
⎛
⎜
⎜
X=⎜
⎜
⎝
1.0578
0.0426
0.0426
1.0651
0.0073
0.0413
−0.0013 0.0074
0.0001 −0.0013
⎞
0.0073 −0.0013 0.0001
0.0413 0.0074 −0.0013 ⎟
⎟
1.0652 0.0413
0.0073 ⎟
⎟
0.0413 1.0651
0.0426 ⎠
0.0073 0.0426
1.0578
with the residual X − A∗1 X −2 A1 − A∗2 X −3 A2 − I = 5.6312e − 011.
Example 7.2. In this example, we consider the corresponding perturbation bound
for the solution X in Theorem 4.1.
We consider the matrix equation
X − A∗1 X −2 A1 − A∗2 X −3/2A2 = I,
with
⎛
A1 =
1
3
+ 2 × 10−2
A, A2 =
||A||
1
6
⎜
⎜
+ 3 × 10−2
A, A = ⎜
⎜
||A||
⎝
2
1
0
0
0
1
2
1
0
0
0
1
2
1
0
0
0
1
2
1
0
0
0
1
2
⎞
⎟
⎟
⎟.
⎟
⎠
1362
Jing Li and Yuhai Zhang
In this case, the matrix equation (1.1) has a unique positive definite solution X. Suppose
i = Ai +
that the coefficient matrices A1 , A2 and Q are respectively perturbed to A
ΔAi , i = 1, 2 and Q = I + ΔQ, where
ΔA1 =
10−j
3 × 10−j−1 T
T
(C
(C + C), ΔQ = 10−j × A
+
C),
ΔA
=
2
C T + C
C T + C
and C is a random matrix generated by MATLAB function randn.
By Theorem 4.1, we can compute the relative perturbation bound ξ. The results
averaged as the geometric mean of 20 randomly perturbed runs. Some results are listed
in Table 1.
Table 1: Perturbation bounds for Example 7.2 with different values of j
4
5
6
7
j
X−X
X
ξ
2.8216 × 10−4
7.4788 × 10−4
2.6097 × 10−5
6.9630 × 10−5
2.8774 × 10−6
7.4725 × 10−6
2.7732 × 10−7
7.5760 × 10−7
The results listed in Table 1 show that the perturbation bound ξ given by Theorem
4.1 is sharp.
Example 7.3. In this example, we consider the backward error of an approximate
solution for the unique solution X to the equation (1.1) in Theorem 5.1. We consider
X − A∗1 X −1/2 A1 − A∗2 X −3/2 A2 = Q,
with the coefficient matrices
⎛
⎞
√
1
0 1
1⎝
2
3
−1 1 1 ⎠ , A2 =
A1 , Q = X − A∗1 X −1/2A1 − A∗2 X −3/2A2 ,
A1 =
5
45
−1 −1 1
where X = diag(1, 2, 3), which ensures that there exists a unique positive solution in
equation (1.1).
Let
⎛
⎞
0.5 −0.1 0.2
= X + ⎝ −0.1 0.3
X
0.6 ⎠ × 10−j
0.2
0.6 −0.4
be an approximate solution to (1.1). Take α1 = ||A1 ||F , α2 = ||A2 ||F and ρ = ||Q||F
in Theorem 5.1. Some results on lower and upper bounds for the backward error η(X)
are displayed in Table 2.
decreases as the
The results listed in Table 2 show that the backward error of X
− X||F decreases.
error ||X
On the Existence of Positive Definite Solutions of a Nonlinear Matrix Equation
j
1
3
5
7
9
1363
Table 2: Backward error for Example 7.3 with different values of j
− X||F
||X
r
U (r)
B(r)
0.2298
2.3 × 10−3
2.2978 × 10−5
2.2978 × 10−7
2.2978 × 10−9
0.0636
6.3587 × 10−4
6.3587 × 10−6
6.3587 × 10−8
6.3587 × 10−10
0.0632
6.3583 × 10−4
6.3587 × 10−6
6.3587 × 10−8
6.3587 × 10−10
0.0639
6.3591 × 10−4
6.3587 × 10−6
6.3587 × 10−8
6.3587 × 10−10
ACKNOWLEDGMENTS
The authors wish to thank the anonymous referees for providing valuable comments
and suggestions which improved this paper. 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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Jing Li
School of Mathematics and Statistics
Shandong University
Weihai 264209
P. R. China
[email protected]
Yuhai Zhang
School of Mathematics
Shandong University
Jinan 250100
P. R. China