# Solutions and Improved Perturbation Analysis for the Matrix EquationX-A*X-pA=Q(p>0)

Authors: Jing Li
Journal: Abstract and Applied Analysis
 (2013)

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## Abstract

The nonlinear matrix equationX-A*X-pA=Q with p>0 is investigated. We consider two cases of this equation: the case p≥1 and the case 0<p<1. In the case p≥1, a new sufficient condition for the existence of a unique positive definite solution for the matrix equation is obtained. A perturbation estimate for the positive definite solution is derived. Explicit expressions of the condition number for the positive definite solution are given. In the case 0<p<1, a new sharper perturbation bound for the unique positive definite solution is derived. A new backward error of an approximate solution to the unique positive definite solution is obtained. The theoretical results are illustrated by numerical examples.


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## Body

## 1. Introduction

In this paper, we consider the Hermitian positive definite solution of the nonlinear matrix equation(1)X-A*X-pA=Q,
where A, Q, and X are n×n complex matrices, Q is a positive definite matrix, and p>0. This type of nonlinear matrix equations arises in the analysis of ladder networks, the dynamic programming, control theory, stochastic filtering, statistics, and many applications [1–7].In the last few years, (1) was investigated in some special cases. For the nonlinear matrix equations X-A*X-1A=Q [8–12], X-A*X-2A=Q [13, 14], X-A*X-nA=Q [15, 16], and Xs-A*X-tA=Q [17], there were many contributions in the literature to the solvability, numerical solutions, and perturbation analysis. In addition, the related equations X+A*X-1A=Q [9–11, 18–23], X+A*X-2A=Q [13, 24, 25], X+A*X-nA=Q [16, 26], Xs+A*X-tA=Q [17, 27–30], X+A*X-qA=Q [31–33], and X±∑i=1mAi*X-1Ai=Q [34–36] were studied by many scholars.In [31], a sufficient condition for the equation X-A*X-pA=Q(0<p≤1) to have a unique positive definite solution was provided. When the coefficient matrix A is nonsingular, several sufficient conditions for the equation X-A*X-qA=Q(q≥1) to have a unique positive definite solution were given in [37]. When the coefficient matrix A is an arbitrary complex matrix, necessary conditions and sufficient conditions for the existence of positive definite solutions for the equation X-A*X-qA=Q(q≥1) were derived in [38]. Li and Zhang in [39] proved that there always exists a unique positive definite solution to the equation X-A*X-pA=Q(0<p<1). They also obtained a perturbation bound and a backward error of an approximate solution for the unique solution of the equation X-A*X-pA=Q(0<p<1).As a continuation of the previous results, the rest of the paper is organized as follows. Section2 gives some preliminary lemmas that will be needed to develop this work. In Section 3, a new sufficient condition for (1) with p≥1 having a unique positive definite solution is derived. In Section 4, a perturbation bound for the positive definite solution to (1) with p≥1 is given. In Section 5, applying the integral representation of matrix function, we also discuss the explicit expressions of condition number for the positive definite solution to (1) with p≥1. Furthermore, in Section 6, a new sharper perturbation bound for the unique positive definite solution to (1) with 0<p<1 is given. In Section 7, a new backward error of an approximate solution to (1) with 0<p<1 is obtained. Finally, several numerical examples are presented in Section 8.We denote by𝒞n×n the set of n×n complex matrices, by ℋn×n the set of n×n Hermitian matrices, by I the identity matrix, 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=(aij)∈𝒞n×n with columns ai and a matrix B, A⊗B=(aijB) is a Kronecker product, and vecA is a vector defined by vecA=(a1T,…,anT)T. For X,Y∈ℋn×n, we write X≥Y (resp., X>Y) if X-Y is Hermitian positive semidefinite (resp., definite). Let κ¯=λmax(A*A), κ_=λmin(A*A).


## 2. Preliminaries

In this section we quote some preliminary lemmas that we use later.Lemma 1 (see [39, Lemma 3.2]).
For every positive definite matrixX∈ℋn×n and 0<p<1, then (i)
X-p=(sinpπ/π)∫0∞‍(λI+X)-1λ-pdλ.(ii)
X-p=(sinpπ/pπ)∫0∞‍(λI+X)-1X(λI+X)-1λ-pdλ.Lemma 2 (see [39, Theorem 2.5]).
There exists a unique positive definite solutionX of X-A*X-pA=Q(0<p<1) and the iteration
(2)X0>0,Xn=Q+A*Xn-1-pA,n=1,2,…
converges to X.Lemma 3 (see [32, Lemma 2]).
(i) IfX∈ℋn×n, then ∥e-X∥=e-λmin(X).
(ii) IfX∈ℋn×n and r>0, then X-r=(1/Γ(r))∫0∞‍e-sXsr-1ds.
(iii) IfA,B∈𝒞n×n, then eA+B-eA=∫01‍e(1-t)ABet(A+B)dt.


## 3. A Sufficient Condition for the Existence of a Unique Solution ofX-A*X-pA=Q(p≥1)

In this section, we derive a new sufficient condition for the existence of a unique solution ofX-A*X-pA=Q(p≥1) beginning with the lemma.Lemma 4 (see [38, Theorem 5, Remark 4]).
If(3)β>(pκ¯)1/(p+1),
then (1) has a unique positive definite solution X∈[βI,αI], where α and β are, respectively, positive solutions of the following equations:
(4)(x-λmax(Q))(λmin(Q)+κ_xp)p=κ¯,(5)(x-λmin(Q))(λmax(Q)+κ¯xp)p=κ_.
Furthermore,(6)λmin(Q)≤β≤α.Theorem 5.
If(7)((pκ¯)1/(p+1)-λmin(Q))(λmax(Q)+κ¯(pκ¯)p/(p+1))p<κ_,(8)(λmin(Q)p)p/(p+1)λmax1/(p+1)(Q)(κ¯)1/(p+1)+1-p>0,
then (1) has a unique positive definite solution.Proof.
We first proveβ>(pκ¯)1/(p+1), where β is the positive solution to (5). Let
(9)f(x)=(x-λmin(Q))(λmax(Q)+κ¯xp)p-κ_.
By computation, we obtain
(10)f′(x)=κ¯xp(λmax(Q)+κ¯xp)p-1×(λmax(Q)κ¯xp+p2λmin(Q)x-1+1-p2).
Define
(11)g(x)=λmax(Q)κ¯xp+p2λmin(Q)x-1+1-p2.
Then g(x) is decreasing on [0,λmin(Q)pκ¯/(λmax(Q))1/(p+1)] and increasing on [λmin(Q)pκ¯/(λmax(Q))1/(p+1),+∞), which implies that
(12)gmin=g((λmin(Q)pκ¯λmax(Q))1/(p+1))=(1+p)((λmin(Q)p)p/(p+1)λmax1/(p+1)(Q)(κ¯)1/(p+1)+1-p).
According to the condition (8), it follows that gmin>0. Note that
(13)f′(x)=κ¯xp(λmax(Q)+κ¯xp)p-1g(x),
which implies that f(x) is increasing on (0,+∞). Considering the condition (7), one sees that f((pκ¯)1/(p+1))<0. Combining that and the definition of β in Lemma 4, we obtain β>(pκ¯)1/(p+1). By Lemma 4, (1) has a unique positive definite solution.


## 4. Perturbation Bound forX-A*X-pA=Q(p≥1)

Li and Zhang in [39] proved that there always exists a unique positive definite solution to the equation X-A*X-pA=Q(0<p<1). They also obtained a perturbation bound for the unique solution. But their approach becomes invalid for the case of p≥1. Since the equation X-A*X-pA=Q(p≥1) does not always have a unique positive definite solution, there are two difficulties for a perturbation analysis of the equation X-A*X-pA=Q(p≥1). One difficulty is how to find some reasonable restrictions on the coefficient matrices of perturbed equation ensuring that this equation has a unique positive definite solution. The other difficulty is how to find an expression of ΔX which is easy to handle.Assume that the coefficient matrixA is perturbed to A~=ΔA+A. Let X~=ΔX+X with ΔX∈ℋn×n satisfying the perturbed equation
(14)X~-A~*X~-pA~=Q,p≥1.In the following, we derive a perturbation estimate for the positive definite solution to the matrix equationX-A*X-pA=Q(p≥1) beginning with the lemma.Lemma 6 (see [38, Corollary 1. Remark 4]).
If(15)p∥A∥2<λminp+1(Q),
then (1) has a unique positive definite solution X. Moreover, X≥λmin(Q)I.Theorem 7.
If(16)∥A∥<λminp+1(Q)p,∥ΔA∥<λminp+1(Q)p-∥A∥,
then
(17)X-A*X-pA=Q,X~-A~*X~-pA~=Q
have unique positive definite solutions X and X~, respectively. Furthermore,
(18)∥X~-X∥∥X∥≤(2∥A∥+∥ΔA∥)λminp+1(Q)-p∥A∥2∥ΔA∥≡ϱ.Proof.
By (16), it follows that ∥A~∥≤∥A∥+∥ΔA∥≤λmin(p+1)(Q)/p. According to Lemma 6, the condition (16) ensures that (1) and (14) have unique positive definite solutions X and X~, respectively. Furthermore, we obtain that
(19)X≥λmin(Q)I,X~≥λmin(Q)I.
Subtracting (14) from (1) gives
(20)ΔX=A~*X~-pA~-A*X-pA=A*(X~-p-X-p)A+ΔA*X~-pA+A~*X~-pΔA.
By Lemma 3 and inequalities in (19), we have
(21)∥ΔX+A*X-pA-A*X~-pA∥=∥ΔX+A*1Γ(p)∫0∞‍(e-sX-e-sX~)sp-1dsA∥=∥ΔX+A*1Γ(p)∫0∞‍∫01‍e-(1-t)sX~ΔXe-tsXdtspdsA∥≥∥ΔX∥-∥A∥2∥ΔX∥Γ(p)∫0∞‍∫01‍∥e-(1-t)sX~∥∥e-tsX∥dtspds≥∥ΔX∥-∥A∥2∥ΔX∥Γ(p)∫0∞‍∫01‍e-(1-t)sλmin(X~)e-tsλmin(X)dtspds≥∥ΔX∥-∥A∥2∥ΔX∥Γ(p)∫0∞‍∫01‍e-(1-t)sλmin(Q)e-tsλmin(Q)dtspds=∥ΔX∥-∥A∥2∥ΔX∥Γ(p)∫0∞‍∫01‍e-sλmin(Q)dtspds=∥ΔX∥-Γ(p+1)Γ(p)·∥A∥2∥ΔX∥λminp+1(Q)=λminp+1(Q)-p∥A∥2λminp+1(Q)∥ΔX∥.
Noting (16), we have
(22)λminp+1(Q)-p∥A∥2>0.
Combining (20) and (21), one sees that
(23)λminp+1(Q)-p∥A∥2λminp+1(Q)∥ΔX∥≤∥ΔA*X~-pA+A~*X~-pΔA∥≤(∥ΔA∥+2∥A∥)∥ΔA∥∥X~-p∥≤(∥ΔA∥+2∥A∥)∥ΔA∥λmin-p(Q),
which implies that
(24)∥ΔX∥∥X∥≤(∥ΔA∥+2∥A∥)λminp+1(Q)-p∥A∥2∥ΔA∥.


## 5. Condition Number forX-A*X-pA=Q(p≥1)

A condition number is a measurement of the sensitivity of the positive definite stabilizing solutions to small changes in the coefficient matrices. In this section, we apply the theory of condition number developed by Rice [40] to derive explicit expressions of the condition number for the matrix equation X-A*X-pA=Q(p≥1).Here we consider the perturbed equation(25)X~-A~*X~-pA~=Q~,p≥1,
where A~ and Q~ are small perturbations of A and Q in (1), respectively.Suppose thatp∥A∥2<λminp+1(Q) and p∥A~∥2<λminp+1(Q~). According to Lemma 6, (1) and (25) have unique positive definite solutions X and X~, respectively. Let ΔX=X~-X, ΔQ=Q~-Q, and ΔA=A~-A.Subtracting (25) from (1) gives
(26)ΔX=A~*X~-pA~-A*X-pA+ΔQ=A*(X~-p-X-p)A+ΔA*X~-pA+A~*X~-pΔA+ΔQ=-A*1Γ(p)∫0∞‍(e-sX-e-sX~)sp-1dsA+ΔA*X~-pA+A~*X~-pΔA+ΔQ=-A*1Γ(p)∫0∞‍∫01‍e-(1-t)sX~(X~-X)e-tsXdtspdsA+ΔA*X~-pA+A~*X~-pΔA+ΔQ=-A*1Γ(p)∫0∞‍∫01‍(e-(1-t)sX~-e-(1-t)sX)×ΔXe-tsXdtspdsA+ΔQ-A*1Γ(p)∫0∞‍∫01‍e-(1-t)sXΔXe-tsXdtspdsA-(A~*X-pΔA-A~*(X+ΔX)-pΔA)+A~*X-pΔA-(ΔA*X-pA-ΔA*(X+ΔX)-pA)+ΔA*X-pA=A*1Γ(p)∫0∞‍∫01‍∫01‍e-(1-m)(1-t)sXΔXe-m(1-t)sX~×ΔXe-tsXdm(1-t)dtsp+1dsA+ΔQ-A*1Γ(p)∫0∞‍∫01‍e-(1-t)sXΔXe-tsXdtspdsA+ΔA*X-pΔA+A*X-pΔA+ΔA*X-pA-A~*1Γ(p)∫0∞‍∫01‍e-(1-t)s(X+ΔX)ΔXe-tsXdtspdsΔA-ΔA*1Γ(p)∫0∞‍∫01‍e-(1-t)s(X+ΔX)ΔXe-tsXdtspdsA.
Therefore,
(27)ΔX+A*1Γ(p)∫0∞‍∫01‍e-(1-t)sXΔXe-tsXdtspdsA=E+h(ΔX),
where
(28)B=X-pA,E=ΔQ+(B*ΔA+ΔA*B)+ΔA*X-pΔA,h(ΔX)=A*1Γ(p)∫0∞‍∫01‍∫01‍e-(1-m)(1-t)sX×ΔXe-m(1-t)sX~ΔXe-tsX×dm(1-t)dtsp+1dsA-A~*1Γ(p)∫0∞‍∫01‍e-(1-t)s(X+ΔX)ΔXe-tsXdtspdsΔA-ΔA*1Γ(p)∫0∞‍∫01‍e-(1-t)s(X+ΔX)ΔXe-tsXdtspdsA.Lemma 8.
If(29)p∥A∥2<λminp+1(Q),
then the linear operator V:ℋn×n→ℋn×n defined by
(30)VW=W+1Γ(p)∫0∞‍∫01‍A*e-(1-t)sXWe-tsXAdtspds,1Γ(p)∫0∞‍∫01‍A*e-(1-t)sXWe-tsXAdtspdsW∈ℋn×n
is invertible.Proof.
Define the operatorR:ℋn×n→ℋn×n by
(31)RZ=1Γ(p)∫0∞‍∫01‍A*e-(1-t)sXZe-tsXAdtspds,∫0∞‍∫01‍A*e-(1-t)sXZe-tsXAdtspdsZ∈ℋn×n,
it follows that(32)VW=W+RW.
Then,V is invertible if and only if I+R is invertible.
According to Lemma3 and the condition (29), we have
(33)∥RW∥≤∥A∥2∥W∥1Γ(p)∫0∞‍∫01‍∥e-(1-t)sX∥∥e-tsX∥dtspds=∥A∥2∥W∥1Γ(p)∫0∞‍∫01‍e-(1-t)sλmin(X)e-tsλmin(X)dtspds≤∥A∥2∥W∥1Γ(p)∫0∞‍∫01‍e-(1-t)sλmin(Q)e-tsλmin(Q)dtspds=∥A∥2∥W∥1Γ(p)∫0∞‍e-sλmin(Q)spds=p∥A∥2λminp+1(Q)∥W∥<∥W∥,
which implies that ∥R∥<1 and I+R is invertible. Therefore, the operator V is invertible.Thus, we can rewrite (27) as
(34)ΔX=V-1ΔQ+V-1(B*ΔA+ΔA*B)+V-1(ΔA*X-pΔA)+V-1(h(ΔX)),(35)
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.By the theory of condition number developed by Rice [40], we define the condition number of the Hermitian positive definite solution X to the matrix equation X-A*X-pA=Q(p≥1) by
(36)c(X)=limδ→0sup∥(ΔA/η,ΔQ/ρ)∥F≤δ∥ΔX∥Fξδ,
where ξ, η, and ρ are positive parameters. Taking ξ=η=ρ=1 in (36) gives the absolute condition number cabs(X), and taking ξ=∥X∥F, η=∥A∥F, and ρ=∥Q∥F in (36) gives the relative condition number crel(X).Substituting (35) into (36), we get
(37)c(X)=1ξmax(ΔA/η,ΔQ/ρ)≠0ΔA∈𝒞n×n,ΔQ∈ℋn×n∥V-1(ΔQ+B*ΔA+ΔA*B)∥F∥(ΔA/η,ΔQ/ρ)∥F=1ξmax(E,H)≠0E∈𝒞n×n,H∈ℋn×n∥V-1(ρH+η(B*E+E*B))∥F∥(E,H)∥F.
Let V be the matrix representation of the linear operator V. It follows from Lemma 4.3.2. in [41] that
(38)vec(VW)=V·vecW.
By Lemma 4.3.1. in [41], we have
(39)vec(VW)=(I⊗I+1Γ(p)∫0∞‍∫01‍(e-tsXA)T⊗(A*e-(1-t)sX)dtspds)I+1Γ(p)∫0∞‍∫01‍(e-tsXA)T⊗(A*e-(1-t)sX)dtspds.·vecW.Then,(40)V=I⊗I+1Γ(p)∫0∞‍∫01‍(e-tsXA)T⊗(A*e-(1-t)sX)dtspds.Let(41)V-1=S+iΣ,V-1(I⊗B*)=V-1(I⊗(X-pA)*)=U1+iΩ1,V-1(BT⊗I)Π=V-1((X-pA)T⊗I)Π=U2+iΩ2,Sc=[S-ΣΣS],Uc=[U1+U2Ω2-Ω1Ω1+Ω2U1-U2],(42)vecH=x+iy,vecE=a+ib,g=(xT,yT,aT,bT)T,wherex,y,a,b∈ℛn2, S,Σ,U1,U2,Ω1,Ω2∈ℛn2×n2, M=(E,H),  i=-1,   Π is the vec-permutation matrix, that is,
(43)vecET=ΠvecE.Furthermore, we obtain that(44)c(X)=1ξmaxM≠0∥V-1(ρH+η(B*E+E*B))∥F∥(E,H)∥F(45)=1ξmaxM≠0∥ρV-1vecH+ηV-1×((I⊗B*)vecE+(BT⊗I)vecE*)∥)×(∥vec(E,H)∥)-1=1ξmaxM≠0∥ρ(S+iΣ)(x+iy)+η[(U1+iΩ1)(a+ib)+(U2+iΩ2)(a-ib)]∥×(∥vec(E,H)∥)-1=1ξmaxg≠0∥(ρSc,ηUc)g∥∥g∥=1ξ∥(ρSc,ηUc)∥,E∈𝒞n×n,H∈ℋn×n.Then, we have the following theorem.Theorem 9.
Ifp∥A∥2<λminp+1(Q), then the condition number c(X) defined by (36) has the explicit expression
(46)c(X)=1ξ∥(ρSc,ηUc)∥,
where the matrices Sc and Uc are defined by (40)-(41).Remark 10.
From (46) we have the relative condition number
(47)crel(X)=∥(∥Q∥FSc,∥A∥FUc)∥∥X∥F.


### 5.1. The Real Case

In this subsection, we consider the real case, that is, where all the coefficients matricesA, Q of the matrix equation X-A*X-pA=Q(p≥1) are real. In such a case the corresponding solution X is also real. Similar arguments as in Theorem 9 give the following theorem.Theorem 11.
LetA, Q be real, c(X) the condition number defined by (36). If p∥A∥2<λminp+1(Q), then c(X) has the explicit expression
(48)c(X)=1ξ∥(ρSr,ηUr)∥,
where
(49)Sr=(I⊗I+1Γ(p)∫0∞‍∫01‍(e-tsXA)T⊗(ATe-(1-t)sX)dtspds1ξ),-1Ur=Sr[I⊗(ATX-p)+((ATX-p)⊗I)Π].Proof.
Let(50)Sr=V-1,Ur=V-1((I⊗BT)+(BT⊗I)Π)Ur=Sr((I⊗(X-pA)T)+((X-pA)T⊗I)Π)vecH=x,vecE=a,g=(xT,aT)T,
where x,a,∈ℛn2, M=(E,H),   i=-1,    Π is the vec-permutation matrix, that is,
(51)vecET=ΠvecE.
It follows from (44) that
(52)c(X)=1ξmaxM≠0∥V-1(ρH+η(BTE+ETB))∥F∥(E,H)∥F=1ξmaxM≠0∥ρV-1vecH+ηV-1((I⊗BT)vecE+(BT⊗I)vecET)∥×(∥vec(E,H)∥)-1=1ξmaxg≠0∥(ρSr,ηUr)g∥∥g∥=1ξ∥(ρSr,ηUr)∥.Remark 12.
In the real case the relative condition number is given by(53)crel(X)=∥(∥Q∥FSr,∥A∥FUr)∥∥X∥F.


## 5.1. The Real Case

In this subsection, we consider the real case, that is, where all the coefficients matricesA, Q of the matrix equation X-A*X-pA=Q(p≥1) are real. In such a case the corresponding solution X is also real. Similar arguments as in Theorem 9 give the following theorem.Theorem 11.
LetA, Q be real, c(X) the condition number defined by (36). If p∥A∥2<λminp+1(Q), then c(X) has the explicit expression
(48)c(X)=1ξ∥(ρSr,ηUr)∥,
where
(49)Sr=(I⊗I+1Γ(p)∫0∞‍∫01‍(e-tsXA)T⊗(ATe-(1-t)sX)dtspds1ξ),-1Ur=Sr[I⊗(ATX-p)+((ATX-p)⊗I)Π].Proof.
Let(50)Sr=V-1,Ur=V-1((I⊗BT)+(BT⊗I)Π)Ur=Sr((I⊗(X-pA)T)+((X-pA)T⊗I)Π)vecH=x,vecE=a,g=(xT,aT)T,
where x,a,∈ℛn2, M=(E,H),   i=-1,    Π is the vec-permutation matrix, that is,
(51)vecET=ΠvecE.
It follows from (44) that
(52)c(X)=1ξmaxM≠0∥V-1(ρH+η(BTE+ETB))∥F∥(E,H)∥F=1ξmaxM≠0∥ρV-1vecH+ηV-1((I⊗BT)vecE+(BT⊗I)vecET)∥×(∥vec(E,H)∥)-1=1ξmaxg≠0∥(ρSr,ηUr)g∥∥g∥=1ξ∥(ρSr,ηUr)∥.Remark 12.
In the real case the relative condition number is given by(53)crel(X)=∥(∥Q∥FSr,∥A∥FUr)∥∥X∥F.


## 6. New Perturbation Bound forX-A*X-pA=Q(0<p<1)

Here, we consider the perturbed equation(54)X~-A~*X~-pA~=Q~,0<p<1,
where A~ and Q~ are small perturbations of A and Q in (1), respectively. We assume that X and X~ are the solutions of (1) and (54), respectively. Let ΔX=X~-X, ΔQ=Q~-Q, and ΔA=A~-A.In this section, we develop a new perturbation bound for the solution of (1) which is sharper than that in [39, Theorem 3.1].Subtracting (1) from (54), using Lemma 1, we have
(55)ΔX+sinpππ∫0∞‍[(λI+X)-1A]*×ΔX[(λI+X)-1A]λ-pdλ=E+h(ΔX),
where
(56)B=X-pA,E=ΔQ+(B*ΔA+ΔA*B)+ΔA*X-pΔA,h(ΔX)=sinpππA*∫0∞‍(λI+X)-1×ΔX(λI+X+ΔX)-1×ΔX(λI+X)-1λ-pdλA-sinpππA~*∫0∞‍(λI+X)-1ΔX(λI+X+ΔX)-1λ-pdλΔA-sinpππΔA*∫0∞‍(λI+X)-1ΔX(λI+X+ΔX)-1λ-pdλA.
By Lemma  5.1 in [39], the linear operator L:ℋn×n→ℋn×n defined by
(57)LW=W+sinpππ∫0∞‍[(λI+X)-1A]*×W[(λI+X)-1A]λ-pdλ,W∈ℋn×n
is invertible.We also define operatorP:𝒞n×n→ℋn×n by
(58)PZ=L-1(B*Z+Z*B),Z∈𝒞n×n,i=1,2,…,m.
Thus, we can rewrite (55) as
(59)ΔX=L-1ΔQ+PΔA+L-1(ΔA*X-pΔA)+L-1(h(ΔX)).
Define
(60)∥L-1∥=maxW∈ℋn×n∥W∥=1∥L-1W∥,∥P∥=maxZ∈𝒞n×n∥Z∥=1∥PZ∥.
Now we denote
(61)l=∥L-1∥-1,ζ=∥X-1∥,ξ=∥X-p∥,n=∥P∥,η=pξ∥A∥2,ϵ=1l∥ΔQ∥+n∥ΔA∥+ξl∥ΔA∥2,σ=plζξ(2∥A∥+∥ΔA∥)∥ΔA∥.Theorem 13.
If(62)σ<1,ϵ<l(1-σ)2ζ(l+lσ+2η+2(lσ+η)(η+l)),
then
(63)∥X~-X∥≤2lϵl(1+ζϵ-σ)+l2(1+ζϵ-σ)2-4lζϵ(l+η)≡μ*Proof.
Let(64)f(ΔX)=L-1ΔQ+PΔA+L-1(ΔA*X-pΔA)+L-1(h(ΔX)).
Obviously, f:ℋn×n→ℋn×n is continuous. The condition (62) ensures that the quadratic equation
(65)ζ(l+η)x2-l(1+ζϵ-σ)x+lϵ=0
in x has two positive real roots. The smaller one is
(66)μ*=2lϵl(1+ζϵ-σ)+l2(1+ζϵ-σ)2-4lζϵ(l+η).
Define Ω={ΔX∈ℋn×n:∥ΔX∥≤μ*}. Then for any ΔX∈Ω, by (62), we have
(67)∥X-1ΔX∥≤∥X-1∥∥ΔX∥≤ζμ*≤ζ·2ϵ1+ζϵ-σ=1+ζϵ+σ-11+ζϵ-σ≤1+(ζ·l(1-σ)2/ζ(l+lσ+2η+2(lσ+η)(η+l))+σ-1(lσ+η)(η+l))×(1+ζϵ-σ)-1≤1+l(1-σ)2+(σ-1)(l+lσ+2η)(l+lσ+2η)(1+ζϵ-σ)=1+-2(1-σ)(lσ+η)(lσ+l+2η)(1+ζϵ-σ)<1.It follows that I-X-1ΔX is nonsingular and
(68)∥I-X-1ΔX∥≤11-∥X-1ΔX∥≤11-ζ∥ΔX∥.
Therefore
(69)∥f(ΔX)∥≤1l∥ΔQ∥+n∥ΔA∥+ξl∥ΔAi∥2+plζξ∥A∥2∥ΔX∥21-ζ∥ΔX∥+plζξ(2∥A∥+∥ΔA∥)∥ΔA∥·∥ΔX∥1-ζ∥ΔX∥≤ϵ+σ∥ΔX∥1-ζ∥ΔX∥+ηζ∥ΔX∥2l(1-ζ∥ΔX∥)≤ϵ+σμ*1-ζμ*+ηζμ*2l(1-ζμ*)=μ*,
for ΔX∈Ω, in which the last equality is due to the fact that μ* is a solution to (65). That is f(Ω)⊆Ω. According to Schauder fixed point theorem, there exists ΔX*∈Ω such that f(ΔX*)=ΔX*. It follows that X+ΔX* is a Hermitian solution of (54). By Lemma 2, we know that the solution of (54) is unique. Then ΔX*=X~-X and ∥X~-X∥≤μ*.


## 7. New Backward Error forX-A*X-pA=Q(0<p<1)

In this section, we evaluate a new backward error estimate for an approximate solution to the unique solution, which is sharper than that in [39, Theorem 4.1].Theorem 14.
LetX~>0 be an approximation to the solution X of (1). If ∥X~-p/2A∥2∥X~-1∥<1 and the residual R(X~)≡Q+A*X~-pA-X~ satisfies
(70)∥R(X~)∥≤θ12min{1,θ12λmin(X~)},whereθ1=(1-∥X~-p/2A∥2∥X~-1∥)λmin(X~)+∥R(X~)∥>0,then
(71)∥X~-X∥≤θ∥R(X~)∥,whereθ=2λmin(X~)θ1+θ12-4λmin(X~)∥R(X~)∥.To prove the above theorem, we first verify the following lemma.Lemma 15.
For every positive definite matrixX∈ℋn×n, 0<p<1, if X+ΔX≥(1/ν)I>0, then
(72)∥A*((X+ΔX)-p-X-p)A∥≤p(∥ΔX∥+ν∥ΔX∥2)∥X-p/2A∥2∥X-1∥.Proof.
It follows from Lemma1 that
(73)∥A*((X+ΔX)-p-X-p)A∥=∥A*(sinpππ∫0∞‍((λI+X+ΔX)-1(λI+X)-1)×λ-pdλ)A∫0∞‍∥≤sinpππ(∥A*∫0∞‍(λI+X)-1ΔX(λI+X)-1λ-pdλA∥)+sinpππ(∥A*∫0∞‍(λI+X)-1ΔX(λI+X+ΔX)-1×ΔX(λI+X)-1λ-pdλA∫0∞‍∥).
Note thatΔX≤∥ΔX∥I, X+ΔX≥(1/ν)I>0, and X-p=(sinpπ/pπ)∫0∞‍(λI+X)-1X(λI+X)-1λ-pdλ, we have
(74)∥A*∫0∞‍(λI+X)-1ΔX(λI+X)-1λ-pdλA∥=∥A*∫0∞‍(λI+X)-1XX-1ΔX(λI+X)-1λ-pdλA∥≤∥A*∫0∞‍(λI+X)-1X(λI+X)-1λ-pdλA∥∥X-1∥∥ΔX∥=pπsin(pπ)·∥A*X-pA∥∥X-1∥∥ΔX∥,(75)∥A*∫0∞‍(λI+X)-1ΔX(λI+X+ΔX)-1×ΔX(λI+X)-1λ-pdλA∫0∞‍∥=∥A*∫0∞‍(λI+X)-1XX-1×ΔX(λI+X+ΔX)-1ΔX(λI+X)-1λ-pdλA∫0∞‍∥≤∥A*(∫0∞‍(λI+X)-1X(λI+X)-1λ-pdλA)∥·ν∥ΔX∥2∥X-1∥=pπsin(pπ)·∥A*X-pA∥·ν∥ΔX∥2∥X-1∥.
A combination of (73)–(75) gives
(76)∥A*((X+ΔX)-p-X-p)A∥≤p∥A*X-pA∥∥X-1∥∥ΔX∥+p∥A*X-pA∥ν∥ΔX∥2∥X-1∥=p(∥ΔX∥+ν∥ΔX∥2)∥X-p/2A∥2∥X-1∥.
Here, we have used the result∥A*X-pA∥=∥(X-p/2A)*(X-p/2A)∥=∥X-p/2A∥2 to derive the last equality (refer to [42, Problem 11. Page 312]).Proof of Theorem14.
Let(77)Ψ={ΔX∈ℋn×n:∥ΔX∥≤θ∥R(X~)∥}.
Obviously,Ψ is a nonempty bounded convex closed set. Let
(78)g(ΔX)=A*((X~+ΔX)-p-X~-p)A+R(X~).
Evidentlyg:Ψ↦ℋn×n is continuous. The condition (70) ensures that the equation
(79)y2-[(1-∥X~-p/2A∥2∥X~-1∥)λmin(X~)+∥R(X~)∥]y+λmin(X~)∥R(X~)∥=0
iny has two positive real roots. The smaller one is y*=2λmin(X~)∥R(X~)∥/θ1+θ12-4λmin(X~)∥R(X~)∥, where θ1=(1-∥X~-p/2A∥2∥X~-1∥)λmin(X~)+∥R(X~)∥.
We will prove thatg(Ψ)⊆Ψ. For every ΔX∈Ψ, we have
(80)ΔX≥-θ∥R(X~)∥I.
Hence,(81)X~+ΔX≥X~-θ∥R(X~)∥I≥(λmin(X~)-θ∥R(X~)∥)I.
Using (70) and (71), one sees that
(82)θ∥R(X~)∥=2λmin(X~)∥R(X~)∥θ1+θ12-4λmin(X~)∥R(X~)∥<2λmin(X~)∥R(X~)∥θ1<λmin(X~).
Therefore,(λmin(X~)-θ∥R(X~)∥)I>0.
According to (72), we obtain
(83)∥g(ΔX)∥≤p(∥ΔX∥+∥ΔX∥2λmin(X~)-θ∥R(X~)∥)×∥X~-p/2A∥2∥X~-1∥+∥R(X~)∥≤(θ∥R(X~)∥+(θ∥R(X~)∥)2λmin(X~)-θ∥R(X~)∥)×(p∥X~-p/2A∥2∥X~-1∥)+∥R(X~)∥=θ∥R(X~)∥,
forΔX∈Ψ, in which the last equality is due to the fact that θ∥R(X~)∥ is a solution to (79). That is g(Ψ)⊆Ψ. By Brouwer fixed point theorem, there exists a ΔX∈Ψ such that g(ΔX)=ΔX. Hence X~+ΔX is a solution of (1). Moreover, by Lemma 2, we know that the solution X of (1) is unique. Then
(84)∥X~-X∥=∥ΔX∥≤θ∥R(X~)∥.


## 8. Numerical Examples

To illustrate the theoretical results of the previous sections, in this section four simple examples are given, which were carried out using MATLAB 7.1. For the stopping criterion we takeεk+1(X)=∥Xk-A*Xk-pA-Q∥<1.0e-10.Example 1.
We consider the matrix equation(85)X-A*X-1/3A=I,
where(86)A=A0∥A0∥,A0=(20.9501).
Suppose that the coefficient matrixA is perturbed to A~=A+ΔA, where
(87)ΔA=10-j∥CT+C∥(CT+C)
andC is a random matrix generated by MATLAB function randn.We compare our own resultμ*/∥X∥≜err2 in Theorem 13 with the perturbation bound ξ*≜err1 proposed in [39, Theorem 3.1].The assumption in [39, Theorem 3.1] is
(88)ass1=∥A∥2+ζ-∥A∥-∥ΔA∥>0.The assumptions in Theorem13 are
(89)ass2=1-σ>0,ass3=l(1-σ)2ζ(l+σl+2η+2(lσ+η)(η+l))-ϵ>0.By computation, we list them in Table1.Table 1
Assumptions check for Example1 with different values of j.
j
4
5
6
7
ass
1
0.0455
0.0456
0.0456
0.0456
ass
2
0.9999
1.0000
1.0000
1.0000
ass
3
0.3957
0.3959
0.3959
0.3959The results listed in Table1 show that the assumptions in Theorem 3.1 [39] and Theorem 13 are satisfied.By Theorem  3.1 in [39] and Theorem 13, we can compute the relative perturbation bounds err1,err2, 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 Example1 with different values of j.
j
4
5
6
7
∥
X
~
-
X
∥
∥
X
∥
6
.
8119
×
1
0
-
5
4
.
2332
×
1
0
-
6
4
.
3287
×
1
0
-
7
5
.
5767
×
1
0
-
8
err
1
2
.
6003
×
1
0
-
4
2
.
1375
×
1
0
-
5
1
.
9229
×
1
0
-
6
2
.
7300
×
1
0
-
7
err
2
8
.
8966
×
1
0
-
5
6
.
5825
×
1
0
-
6
7
.
2867
×
1
0
-
7
9
.
3455
×
1
0
-
8The results listed in Table2 show that the perturbation bound err2 given by Theorem 13 is fairly sharp, while the bound err1 given by Theorem  3.1 in [39] is conservative.Example 2.
Consider the equation(90)X-A*X-3/4A=Q,
for(91)A=(0.2-0.20.10.1),Q=(0.89390.29870.19910.6614).
Choose X~0=3Q. Let the approximate solution X~k be given with the iterative method (2), where k is the iteration number. Assume that the solution X of (1) is unknown.We compare our own result with the backward error proposed in Theorem 4.1 [39].The residualR(X~k)≡Q+A*X~k-pA-X~k satisfies the conditions in Theorem 4.1 [39] and in Theorem 14.By Theorem  4.1 in [39], we can compute the backward error bound
(92)∥X~k-X∥≤ν*∥R(X~k)∥,whereν*=2∥X~k∥∥X~k-1∥1-(3/4)∥X~k-3/8AX~k-1/2∥2.
By Theorem 14, we can compute the new backward error bound
(93)∥X~k-X∥≤θ∥R(X~k)∥,whereθ=2λmin(X~k)θ1+θ12-4λmin(X~k)∥R(X~k)∥,θ1=(1-∥X~k-3/8A∥2∥X~k-1∥)λmin(X~k)+∥R(X~k)∥.
Let
(94)κ1=ν*∥R(X~k)∥∥X~k-X∥,κ2=θ∥R(X~k)∥∥X~k-X∥.
Some results are shown in Table 3.Table 3
Results for Example2 with different values of k.
k
4
5
6
7
∥
X
~
k
-
X
∥
6
.
2131
×
1
0
-
6
1
.
5830
×
1
0
-
7
8
.
2486
×
1
0
-
9
6
.
0132
×
1
0
-
10
ν
*
∥
R
(
X
~
k
)
∥
2
.
5930
×
1
0
-
5
6
.
6257
×
1
0
-
7
3
.
5697
×
1
0
-
8
2
.
4646
×
1
0
-
9
κ
1
4.1734
4.1856
4.3277
4.0986
θ
∥
R
(
X
~
k
)
∥
7
.
0053
×
1
0
-
6
1
.
7900
×
1
0
-
7
9
.
6440
×
1
0
-
9
6
.
6583
×
1
0
-
10
κ
2
1.1275
1.1308
1.1692
1.1073From the results listed in Table3 we see that the new backward error bound θ∥R(X~k)∥ is sharper and closer to the actual error than the backward error bound ν*∥R(X~k)∥ in [39]. Moreover, we see that the backward error θ∥R(X~)∥ for an approximate solution X~ seems to be independent of the conditioning of the solution X.Example 3.
We consider the matrix equation(95)X-A*X-3A=5I,
where(96)A=A0∥A0∥,A0=(20.9501).
We now consider the perturbation bounds for the solutionX when the coefficient matrix A is perturbed to A~=A+ΔA, where
(97)ΔA=10-j∥CT+C∥(CT+C)
and C is a random matrix generated by MATLAB function randn.The conditions in Theorem7 are satisfied.By Theorem7, we can compute the relative perturbation bound ϱ with different values of j. These results averaged as the geometric mean of 10 randomly perturbed runs. Some results are listed in Table 4.Table 4
Results for Example3 with different values of j.
j
4
5
6
7
∥
X
~
-
X
∥
∥
X
∥
1
.
1892
×
1
0
-
7
2
.
1101
×
1
0
-
8
2
.
4085
×
1
0
-
9
1
.
6847
×
1
0
-
10
ϱ
2
.
0791
×
1
0
-
7
3
.
5353
×
1
0
-
8
3
.
9573
×
1
0
-
9
3
.
2580
×
1
0
-
10The results listed in Table4 show that the perturbation bound ϱ given by Theorem 7 is fairly sharp.Example 4.
Consider the matrix equationX-A*X-3A=Q, where
(98)A=(0.50.55-10-k11),Q=(5115).
By Remark12, we can compute the relative condition number crel(X). Some results are listed in Table 5.Table 5
Results for Example4 with different values of k.
k
1
3
5
7
9
c
rel
(
X
)
1.2510
1.0991
1.0009
1.0009
1.0009Table5 shows that the unique positive definite solution X is well conditioned.



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*Source: 575964-2013-06-02.xml*