# Positive Definite Solutions of the Matrix EquationXr-∑i=1mAi∗X-δiAi=I

Authors: Asmaa M. Al-Dubiban
Journal: Abstract and Applied Analysis
 (2015)

---

## Abstract

We investigate the nonlinear matrix equationXr-∑i=1mAi∗X-δiAi=I, where r is a positive integer and δi∈(0,1],fori=1,2,…,m. We establish necessary and sufficient conditions for the existence of positive definite solutions of this equation. A sufficient condition for the equation to have a unique positive definite solution is established. An iterative algorithm is provided to compute the positive definite solutions for the equation and error estimate. Finally, some numerical examples are given to show the effectiveness and convergence of this algorithm.


---

## Body

## 1. Introduction

We consider the nonlinear matrix equation:(1)Xr-∑i=1mAi∗X-δiAi=I,where I is an n×n identity matrix, r is a positive integer, Ai are n×n complex matrices, and δi∈(0,1], for i=1,2,…,m. This type of equations arises in solving a large-scale system of linear equations. In many physical applications, we must solve the system of linear equation(2)Mx=f,where M arises from a finite difference approximation to an elliptic partial differential equation [1]. For example, let(3)M=I0⋯0A10I⋯0A2⋮⋮⋱⋮⋮00⋯IAmA1∗A2∗⋯Am∗-I.We can rewrite M as M=M~+D, where(4)M~=Xδ10⋯0A10Xδ2⋯0A2⋮⋮⋱⋮⋮00⋯XδmAmA1∗A2∗⋯Am∗-I,D=I-Xδ10⋯000I-Xδ2⋯00⋮⋮⋱⋮⋮00⋯I-Xδm000⋯00.We can decompose the matrix M~ to the LU decomposition(5)M~=-I0⋯000-I⋯00⋮⋮⋱⋮⋮00⋯-I0-A1∗X-δ1-A2∗X-δ2⋯-Am∗X-δm-I-Xδ10⋯0-A10-Xδ2⋯0-A2⋮⋮⋱⋮⋮00⋯-Xδm-Am00⋯0Xrif and only if X is a solution of (1). In addition, similar types of (1) have many applications in various areas, including control systems, ladder networks, dynamic programming, control theory, stochastic filtering, statistics, and optimal interpolation problem [2–5].During the last few years, many researchers worked to develop the theory and numerical approaches for positive definite solutions to the nonlinear matrix equations of the form (1) [6–9]. Recently, Duan et al. [10] showed that the equation X-∑i=1m‍Ai∗XδiAi=Q(0<|δi|<1) always has a unique positive definite solution by using the fixed point theorem of mixed monotone operators. They proposed an iterative method for computing the unique positive definite solution. Lim [11] presented a new proof for the existence and uniqueness of the positive definite solution for the last equation. Yin and Liu [12] derived necessary conditions and sufficient conditions for the existence of positive definite solutions for the matrix equations X±A∗X-qA=Q with q≥1; they also proposed iterative methods for obtaining positive definite solutions of these equations. He and Long [13] obtained some conditions for the existence of positive definite solution of the equation X+∑i=1m‍Ai∗X-1Ai=I. They also presented two iterative algorithms to find the maximal positive definite solution of this equation. Duan et al. [14] used the Thompson metric to prove that the matrix equation X-∑i=1m‍Ni∗X-1Ni=I always has a unique positive definite solution and they gave a precise perturbation bound for the unique positive definite solution. Similar kinds of equations have been investigated by many authors [15–20].In this paper, we study the positive definite solutions of (1). The paper is organized as follows. In Section 2, we derive necessary and sufficient conditions for the existence of positive definite solutions of  (1). We give a sufficient condition for the existence of a unique positive definite solution of this equation. In Section 3, we propose an iterative algorithm to obtain the positive definite solutions of (1). Moreover, we investigate convergence rate of the proposed algorithm. Finally, we give some numerical examples in Section 4 to ensure the performance and the effectiveness of the suggested algorithm.We start with some notations which will be used in this paper. The symbolCn×n denotes the set of n×n complex matrices. A⋆ denotes the complex conjugate transpose of A. We write A>0(A≥0), if matrix A is positive definite (positive semidefinite). If A-B is positive definite (positive semidefinite), then we write A>B(A≥B). Moreover, we denote ρ(A) by the spectral radius of A. ·, ·∞, and ·F denote the spectral, infinity, and Frobenius norm. For A=(a1,a2,…,an)=(aij) and a matrix B, A⊗B=(aijB) is a Kronecker product. vec(A) is a vector defined by vec(A)=(a1T,a2T,…,anT)T.


## 2. Conditions for the Existence of the Solutions

In this section, we will derive the necessary and sufficient conditions for existence of the positive definite solutions of (1). We give a sufficient condition for the existence of a unique positive definite solution of (1). We begin with some lemmas, which will play a key role to establish our results.Lemma 1 (see [21]).
IfA>B>0  (or A≥B>0), then Aα>Bα>0  (or Aα≥Bα>0), for all α∈(0,1], and 0<Aα<Bα  (or 0<Aα≤Bα), for all α∈[-1,0).Lemma 2 (see [22]).
IfA and B are n×n matrices, then ABF≤AFB and ABF≤ABF.Lemma 3 (see [23]).
LetA and B be positive operators such that A≥aI≥0 and B≥bI≥0; then, f(A)-f(B)≤C(a,b)A-B, for every nonnegative operator monotone function f on (0,∞), where C(a,b)=(f(a)-f(b))/(a-b) if a≠b and C(a,b)=f′(a) if a=b. Here, · stands for a unitarily invariant norm on matrices.Theorem 4.
If (1) has a positive definite solution X, then(6)I≤X≤I+∑i=1mAi∗Ai1/r.Proof.
SinceX is a positive definite solution of (1), then(7)Xr=I+∑i=1mAi∗X-δiAi≥I;by Lemma 1, we have X≥I. Also, from inequality X≥I and Lemma 1, we have Ai∗X-δiAi≤Ai∗Ai. Therefore,(8)Xr=I+∑i=1mAi∗X-δiAi≤I+∑i=1mAi∗Ai.Hence, X≤I+∑i=1m‍Ai∗Ai1/r implies that I≤X≤I+∑i=1m‍Ai∗Ai1/r.Theorem 5.
Equation (1) has a positive definite solution X if and only if the coefficient matrices Ai, i=1,2,…,m, can be factored as(9)Ai=W∗Wδi/2YiW∗Wr/2,where W is a nonsingular matrix, Z=(W∗W)-r/2, and ZY1⋮Ym is column-orthonormal. In this case, X=W∗W is a solution of (1).Proof.
LetX be a positive definite solution of (1); then, X=W∗W, where W is a nonsingular matrix. Furthermore, (1) can be rewritten as(10)W∗Wr-∑i=1mAi∗W∗W-δiAi=I,which implies that(11)W∗W-r/2W∗W-r/2+∑i=1mW∗W-r/2·Ai∗W∗W-δi/2W∗W-δi/2AiW∗W-r/2=I.Let Z=(W∗W)-r/2,Yi=(W∗W)-δi/2Ai(W∗W)-r/2; then, Ai=(W∗W)δi/2Yi(W∗W)r/2 and (11) turns into(12)Z∗Z+∑i=1mYi∗Yi=I;that is,(13)ZY1⋮Ym∗ZY1⋮Ym=I,which means that ZY1⋮Ym is column-orthonormal.
Conversely, suppose thatAi has the decomposition (9). Let X=W∗W; then, X is a positive definite matrix, and we have(14)
X
r
-
∑
i
=
1
m
A
i
∗
X
-
δ
i
A
i
=
W
∗
W
r
-
∑
i
=
1
m
A
i
∗
W
∗
W
-
δ
i
A
i
=
W
∗
W
r
-
∑
i
=
1
m
W
∗
W
r
/
2
Y
i
∗
W
∗
W
δ
i
/
2
·
W
∗
W
-
δ
i
W
∗
W
δ
i
/
2
Y
i
W
∗
W
r
/
2
=
W
∗
W
r
-
∑
i
=
1
m
W
∗
W
r
/
2
Y
i
∗
Y
i
W
∗
W
r
/
2
=
W
∗
W
r
/
2
I
-
∑
i
=
1
m
Y
i
∗
Y
i
W
∗
W
r
/
2
=
W
∗
W
r
/
2
·
Z
∗
Z
W
∗
W
r
/
2
=
I
.Hence, X=W∗W is a positive definite solution of (1).Theorem 6.
IfA1,A2,…,Am are invertible, then (1) has a positive definite solution X if and only if Ai, i=1,2,…,m, can be factored as(15)Ai=P∗Γδi/2ViΘP,where P is unitary, Γ>0,Θ>0 are diagonal matrices such that Γr-Θ2=I, and V1V2⋮Vm is column-orthonormal. In this case, X=P∗ΓP is a solution of (1).Proof.
LetX be a positive definite solution of (1); then, by spectral decomposition of X, we have X=P∗ΓP, where P is unitary and Γ>0 is diagonal. Therefore, (1) can be rewritten as(16)P∗ΓPr-∑i=1mAi∗P∗ΓP-δiAi=I,which implies that(17)Γr-I=∑i=1mPAi∗P∗Γ-δiPAiP∗,(18)∑i=1mΓr-I-1/2PAi∗P∗Γ-δiPAiP∗Γr-I-1/2=I.Let Vi=Γ-δi/2PAiP∗(Γr-I)-1/2 and Θ=(Γr-I)1/2. Then, Ai=P∗Γδi/2ViΘP, Θ>0, is a diagonal matrix and Γr-Θ2=I. Moreover, (18) turns into ∑i=1m‍Vi∗Vi=I, which means that V1V2⋮Vm is column-orthonormal.
Conversely, suppose thatAi has the decomposition (15). Let X=P∗ΓP; then X is a positive definite matrix, and we have (19)Xr-∑i=1mAi∗X-δiAi=P∗ΓPr-∑i=1mAi∗P∗ΓP-δiAi=P∗ΓrP-∑i=1mAi∗P∗Γ-δiPAi=P∗ΓrP-∑i=1mP∗ΘVi∗ViΘP=P∗Γr-Θ2P=I.Hence, X=P∗ΓP is a positive definite solution of (1).Theorem 7.
Ifq=∑i=1m‍δiAi2<1, then (1) has a unique positive definite solution.Proof.
We consider the mappingf(X)=I+∑i=1m‍Ai∗X-δiAi1/r.
LetΩ=X=X∗:I≤X≤[I+∑i=1m‍Ai∗Ai]1/r. Obviously, Ω is a bounded closed convex set and f is continuous on Ω.
IfX∈Ω, then by Lemma 1, X-δi≤I and(20)I≤I+∑i=1mAi∗X-δiAi1/r≤I+∑i=1mAi∗Ai1/r;that is,(21)I≤fX≤I+∑i=1mAi∗Ai1/r,which means that f(Ω)⊆Ω. By Brouwer’s fixed point theorem, f has a fixed point in Ω; thus, (1) has a solution in Ω.
Now, we prove thatf is a contraction mapping on Ω. For arbitrary X,Y∈Ω, we get(22)fXr-fYrF=∑k=0r-1fXkfX-fYfYr-1-kF=vec∑k=0r-1fXkfX-fYfYr-1-k=∑k=0r-1vecfXkfX-fYfYr-1-k=∑k=0r-1fYr-1-k⊗fXkvecfX-fY≥rvecfX-fY=rfX-fYF.According to the definition of the mapping f, we have(23)fXr-fYr=∑i=1mAi∗X-δi-Y-δiAi.From (22) and (23), and by Lemmas 2 and 3, we have(24)fX-fYF≤1rfXr-fYrF=1r∑i=1mAi∗X-δi-Y-δiAiF≤1r∑i=1mAi2X-δi-Y-δiF≤1r∑i=1mAi2X-δiY-δiYδi-XδiF≤1r∑i=1mδiAi2X-YF=qrX-YF.Since q=∑i=1m‍δiAi2<1, then f is a contraction mapping on Ω. By Banach’s fixed point theorem, the mapping f has a unique fixed point in Ω. This means that (1) has a unique positive definite solution in Ω.


## 3. The Iterative Algorithm

In this section, we consider the fixed point iteration method and its speed of convergence. We present the following iterative algorithm to compute the positive definite solution of (1).Algorithm 8.
Consider(25)X0=I,Xk+1=I+∑i=1mAi∗Xk-δiAi1/r,fork=0,1,2,….Theorem 9.
Ifq=∑i=1m‍δiAi2<1, then (1) has a unique positive definite solution X and satisfies(26)X2k≤X≤X2k+1,(27)X2k+1-X2k≤q2kI+∑i=1mAi∗Ai1/r-I,where the sequence {Xk}, k=0,1,2,…, is determined by Algorithm 8.Proof.
According to Theorem7, (1) has a unique positive definite solution. Consider the sequence {Xk} generated from Algorithm 8 and using Lemma 1.
SinceAi∗Ai≥0, then we have(28)X1=I+∑i=1mAi∗Ai1/r≥I=X0,which implies that(29)∑i=1mAi∗X1-δiAi≤∑i=1mAi∗Ai;then,(30)I≤X2=I+∑i=1mAi∗X1-δiAi1/r≤I+∑i=1mAi∗Ai1/r=X1;that is,(31)X0≤X2≤X1.We find the relation between X2, X3, X4, and X5. Using X0≤X2≤X1, we obtain(32)X3=I+∑i=1mAi∗X2-δiAi1/r≤I+∑i=1mAi∗Ai1/r=X1,X3=I+∑i=1mAi∗X2-δiAi1/r≥I+∑i=1mAi∗X1-δiAi1/r=X2.Hence, X2≤X3≤X1. By the same way, we can prove that(33)X0≤X2≤X4≤X5≤X3≤X1.So, assume that the inequalities(34)X0≤X2k≤X2k+2≤X2k+3≤X2k+1≤X1hold for k.
Now, fork+1 and using inequalities (34), we have(35)X2k+4=I+∑i=1mAi∗X2k+3-δiAi1/r≤I+∑i=1mAi∗X2k+2-δiAi1/r=X2k+3,X2k+4=I+∑i=1mAi∗X2k+3-δiAi1/r≥I+∑i=1mAi∗X2k+1-δiAi1/r=X2k+2.Similarly(36)X2k+5=I+∑i=1mAi∗X2k+4-δiAi1/r≤I+∑i=1mAi∗X2k+2-δiAi1/r=X2k+3,X2k+5=I+∑i=1mAi∗X2k+4-δiAi1/r≥I+∑i=1mAi∗X2k+3-δiAi1/r=X2k+4.Therefore, inequalities (34) are true, for all k=0,1,2,…; that is, the subsequences {X2k} and {X2k+1} are monotonic and bounded. Hence, these are convergent to positive definite matrices.
To prove the subsequences{X2k}, {X2k+1} have a common limit, we have(37)X2k+1r-X2kr=∑i=1mAi∗X2k-δi-X2k-1-δiAi≤∑i=1mAi2X2k-δi-X2k-1-δi=∑i=1mAi2X2k-δiX2k-1δi-X2kδiX2k-1-δi≤∑i=1mAi2X2k-δiX2k-1-δiX2k-1δi-X2kδi.Since I≤X2k,I≤X2k-1, for each k=1,2,3,…, and δi∈(0,1], then X2k-δi,X2k-1-δi≤1. By using Lemma 3, we have(38)X2k+1r-X2kr≤∑i=1mAi2X2kδi-X2k-1δi≤∑i=1mδiAi2X2k-X2k-1.Since(39)X2k+1r-X2kr-X2k-X2k+1≥X2kr-X2kr-X2k-X2k+1=X2k+1-X2k≥X2k-X2k=0,from (38) and (39), we have (40)X2k+1-X2k≤∑i=1mδiAi2X2k-X2k-1.Hence,(41)X2k+1-X2k≤qX2k-X2k-1≤⋯≤q2kI+∑i=1mAi∗Ai1/r-I;that is,(42)X2k+1-X2k⟶0,ask⟶∞.Consequently, the subsequences {X2k} and {X2k+1} are convergent and have a common limit X, which is a positive definite solution of (1).From Theorem9, we can deduce the following corollary.Corollary 10.
From inequality (27), one has the following upper bound:(43)max⁡X2k+1-X,X-X2k≤q2kI+∑i=1mAi∗Ai1/r-I.Theorem 11.
If (1) has a positive definite solution and after k iterative steps of Algorithm 8 one has I-Xk-1Xk-1<ɛ, then(44)Xkr-∑i=1mAi∗Xk-δiAi-I<ɛ∑i=1mδiAi2I+∑i=1mAi∗Ai1/r.Proof.
By using Algorithm8, we have (45)Xkr-∑i=1mAi∗Xk-δiAi-I=Xkr-∑i=1mAi∗Xk-δiAi-Xkr+∑i=1mAi∗Xk-1-δiAi=∑i=1mAi∗Xk-1-δi-Xk-δiAi.Taking norm of the above equation, we get (46)Xkr-∑i=1mAi∗Xk-δiAi-I=∑i=1mAi∗Xk-1-δi-Xk-δiAi≤∑i=1mAi2Xk-1-δi-Xk-δi≤∑i=1mAi2Xk-1-δiXk-δiXkδi-Xk-1δi.By Theorem 9, we have Xk-1-δi≤I, Xk-δi≤I, and Xk≤(I+∑i=1m‍Ai∗Ai)1/r. By using Lemma 3, we have(47)Xkr-∑i=1mAi∗Xk-δiAi-I≤∑i=1mAi2Xkδi-Xk-1δi≤∑i=1mδiAi2Xk-Xk-1≤∑i=1mδiAi2XkI-Xk-1Xk-1<ɛ∑i=1mδiAi2I+∑i=1mAi∗Ai1/r.


## 4. Numerical Examples

In this section, we give some numerical examples to illustrate the convergence of the proposed algorithm. The solution is computed for different matricesAi, i=1,2,…,m, with different orders and different values of r and δi,  i=1,2,…,m. We denote X, the solution obtained by Algorithm 8 and ϵ1(Xk)=Xk+1-Xk∞, ϵ2(Xk)=X-Xk∞, and ϵ3(Xk)=Xkr-∑i=1m‍Ai∗Xk-δiAi-I∞.For computingZ=Y1/p with p an integer, we use the following iterative algorithm.Algorithm 12 (see [24]).
Consider(48)Z0=I,Zk+1=1pp-1Zk+Zk1-pY,fork=0,1,2,….Example 1.
Consider the matrix equation(49)X3-A1∗X-1/4A1-A2∗X-1/2A2-A3∗X-1/3A3=I,where A1, A2, and A3 are given by(50)A1=10.20.30.20.600.20.1-0.5,A2=0.30.20.50.10.3-0.40.50.60.1,A3=0.210.10.40.10.60.20.6-0.4.
By using Algorithms8 and 12, we have after 16 iterations(51)X≈X16=1.329520.1644940.09557210.1644941.425-0.02249660.0955721-0.02249661.28816.
The other results are listed in Table1.Table 1
Error analysis for Example1.
k
ϵ
1
(
X
k
)
ϵ
2
(
X
k
)
ϵ
3
(
X
k
)
1
3.96009
×
1
0
-
2
3.71552
×
1
0
-
2
2.78769
×
1
0
-
1
2
2.60028
×
1
0
-
3
2.44577
×
1
0
-
3
1.79779
×
1
0
-
2
3
1.64516
×
1
0
-
4
1.54515
×
1
0
-
4
1.14731
×
1
0
-
3
4
1.06330
×
1
0
-
5
1.00005
×
1
0
-
5
7.43950
×
1
0
-
5
5
6.72778
×
1
0
-
7
6.32454
×
1
0
-
7
4.71572
×
1
0
-
6
6
4.28759
×
1
0
-
8
4.03237
×
1
0
-
8
3.00787
×
1
0
-
7
7
2.71410
×
1
0
-
9
2.55216
×
1
0
-
9
1.90490
×
1
0
-
8Example 2.
Consider the matrix equation(52)X7-A1∗X-1/3A1-A2∗X-1/3A2-A3∗X-1/4A3=I,where A1, A2, and A3 are given by(53)A1=10.20.3-0.40.20.5-0.200.5,A2=0.5-0.2-0.50.40.400.30.6-0.1,A3=-11.50.4-1.51-0.90.200.4.
By using Algorithms8 and 12, we have after 13 iterations(54)X≈X13=1.26207-0.09560340.0322249-0.09560341.226290.002318670.03222490.002318671.15893.
The other results are listed in Table2.Table 2
Error analysis for Example2.
k
ϵ
1
(
X
k
)
ϵ
2
(
X
k
)
ϵ
3
(
X
k
)
1
7.87438
×
1
0
-
3
7.64002
×
1
0
-
3
2.44198
×
1
0
-
1
2
2.41622
×
1
0
-
4
2.34359
×
1
0
-
4
7.73525
×
1
0
-
3
3
7.47444
×
1
0
-
6
7.26322
×
1
0
-
6
2.43083
×
1
0
-
4
4
2.17530
×
1
0
-
7
2.11217
×
1
0
-
7
6.98136
×
1
0
-
6
5
6.50298
×
1
0
-
9
6.31369
×
1
0
-
9
2.08893
×
1
0
-
7
6
1.94921
×
1
0
-
10
1.89286
×
1
0
-
10
6.27981
×
1
0
-
9Example 3.
Consider the matrix equation(55)X4-A1∗X-1/4A1-A2∗X-1/5A2-A3∗X-1/6A3-A4∗X-1/3A4=I,where A1, A2, A3, and A4 are given by(56)A1=-0.1-0.10.4-1.30-1.70.5-20.2-1-0.50.40.50.10.8-0.4,A2=2.5-4.20.510.1-0.50-0.4-10.50.50.10.2-2.50.50,A3=1.520.100-0.200.61.20.12.6-0.4-0.62.4-0.20.1,A4=0.2-1.50.1-2-1.5-0.46.50.60.20.1-0.10.40.12.50.1-1.
By using Algorithms8 and 12, we have after 18 iterations(57)X≈X18=1.85179-0.19784-0.08730590.0249042-0.197842.51638-0.0882272-0.0105432-0.0873059-0.08822722.503020.005414740.0249042-0.01054320.005414741.85343.
The other results are listed in Table3.Table 3
Error analysis for Example3.
k
ϵ
1
(
X
k
)
ϵ
2
(
X
k
)
ϵ
3
(
X
k
)
1
1.80539
×
1
0
-
1
1.73636
×
1
0
-
1
12.8192
2
7.29793
×
1
0
-
3
6.90272
×
1
0
-
3
4.63002
×
1
0
-
1
3
4.16292
×
1
0
-
4
3.95211
×
1
0
-
4
2.66295
×
1
0
-
2
4
2.22090
×
1
0
-
5
2.10808
×
1
0
-
5
1.41947
×
1
0
-
3
5
1.18818
×
1
0
-
6
1.12817
×
1
0
-
6
7.60338
×
1
0
-
5
6
6.32186
×
1
0
-
8
6.00133
×
1
0
-
8
3.98446
×
1
0
-
6
7
3.37643
×
1
0
-
9
3.20528
×
1
0
-
9
2.86995
×
1
0
-
7Example 4.
Consider the matrix equation(58)X5-A1∗X-1/2A1-A2∗X-1/3A2=I,where A1 and A2 are given by(59)A1=aij=inifi=j,aij=2j2n3+iifi≠j,A2=aij=12i+nifi=j,aij=j4n+iifi>j,aij=in3-jifi<j.
By using Algorithms8 and 12, we have solutions when n=10,50,100. The results are listed in Table 4.Table 4
Error analysis for Example4.
k
ϵ
1
(
X
k
)
ϵ
2
(
X
k
)
ϵ
3
(
X
k
)
n
=
10
1
7.81929
×
1
0
-
3
7.45968
×
1
0
-
3
6.72846
×
1
0
-
2
2
3.77022
×
1
0
-
4
3.59607
×
1
0
-
4
3.20257
×
1
0
-
3
3
1.82579
×
1
0
-
5
1.74147
×
1
0
-
5
1.55207
×
1
0
-
4
4
8.84049
×
1
0
-
7
8.43217
×
1
0
-
7
7.51571
×
1
0
-
6
5
4.28088
×
1
0
-
8
4.08315
×
1
0
-
8
3.63970
×
1
0
-
7
6
2.07307
×
1
0
-
9
1.97732
×
1
0
-
9
1.76271
×
1
0
-
8
n
=
50
1
7.79810
×
1
0
-
3
7.43990
×
1
0
-
3
6.69737
×
1
0
-
2
2
3.75515
×
1
0
-
4
3.58193
×
1
0
-
4
3.18340
×
1
0
-
3
3
1.81596
×
1
0
-
5
1.73221
×
1
0
-
5
1.54042
×
1
0
-
4
4
8.77998
×
1
0
-
7
8.37505
×
1
0
-
7
7.44758
×
1
0
-
6
5
4.24507
×
1
0
-
8
4.04929
×
1
0
-
8
3.60087
×
1
0
-
7
6
2.05247
×
1
0
-
9
1.95781
×
1
0
-
9
1.74100
×
1
0
-
8
n
=
100
1
7.86249
×
1
0
-
3
7.50582
×
1
0
-
3
9.92082
×
1
0
-
2
2
3.76538
×
1
0
-
4
3.59253
×
1
0
-
4
3.30164
×
1
0
-
3
3
1.81583
×
1
0
-
5
1.73209
×
1
0
-
5
1.55072
×
1
0
-
4
4
8.77928
×
1
0
-
7
8.37439
×
1
0
-
7
7.46052
×
1
0
-
6
5
4.24470
×
1
0
-
8
4.04894
×
1
0
-
8
3.60045
×
1
0
-
7
6
2.05227
×
1
0
-
9
1.95762
×
1
0
-
9
1.74078
×
1
0
-
8


## 5. Conclusion

We have presented existence theorems of positive definite solutions for a nonlinear matrix equation (1) under certain conditions. Uniqueness theorem of solutions for this matrix equation is proved. An iterative algorithm is introduced to obtain positive definite solutions for this matrix equation. Moreover, convergence and error analysis are studied. Finally, the numerical examples confirmed the efficient convergence and error reduction as predicted and this method is efficient and easy.



---

*Source: 473965-2015-07-12.xml*