Weighted Average Iterative Algorithm for Solving a Nonlinear Matrix Equation

1. Introduction

The nonlinear matrix equation

$$X^p-{\displaystyle \sum _{i=1}^m{A}_i^T{X}^{-1}{A}_i=Q},$$

(1.1)

Where p is a positive integer, $A_1,A_2,\dots A_m$ are non-singular real matrices, and $Q$ is a positive definite matrix, has been widely used in many fields, such as control theory [1,2], ladder network [3], random filtering [4], statistics [5], etc.

For the nonlinear matrix equation (1.1) with p=1, Duan et al. [6] proved it has a unique positive definite solution and gave the existence interval of the unique Hermitian positive definite solution. They also gave an iterative algorithm to compute the unique Hermitian positive definite solution and a error estimation formula of the iterative approximate solution; Yin et al. [7] considered the perturbation analysis of the positive definite solution, obtained two perturbation bounds of the unique positive definite solution, and derived the expression of the condition number of the unique solution; Li et al. [8] proposed a fixed point accelerated iteration method, and based on the basic characteristics of the Thompson distance, proved the convergence and error estimation of the proposed algorithm. For the general cases of the nonlinear matrix equation (1.1), Lim et al. [9] proved the nonlinear matrix equation $X-{\displaystyle \sum _{i=1}^m{M}_i{X}^{{\delta }_i}}{M_i}^{*}=Q$ with $\left|{\delta }_i\right|<1$ has an unique positive definite solution and gave the existence interval of the unique Hermitian positive definite solution; Jin et al. [10] discussed the solution of the matrix equation $X^p-{\displaystyle \sum _{i=1}^m{A}_i^T{X}^{{\delta }_i}{A}_i=Q}$ through the properties of Thompson metric and two fixed point theorems in ordered Banach spaces, and estimated the bounds of the positive definite solution, and, furthermore, investigated perturbation analysis of the solution with respect to small perturbations on the coefficient matrices. For the similar cases of the nonlinear matrix equation (1.1), Erfanifar et al. [11] first introduced the necessary and sufficient conditions for the existence of the solution of nonlinear matrix equation $X^p+{\displaystyle \sum _{i=1}^m{A}_i^{*}{X}^{-{\delta }_i}{A}_i=Q}$ and then proposed the inversion-free iteration method to find this solution of the nonlinear matrix equation; Liu et al. [12] proposed an iterative algorithm for solving the maximum Hermitian positive definite solution of a nonlinear matrix equation $X^s+{\displaystyle \sum _{i=1}^m{A_i}^{*}}{X}^{-t_i}{A}_i=Q$ and proved the convergence of proposed algorithm; Li et al. [13,14] derived necessary conditions and sufficient conditions for the existence and uniqueness of the Hermitian positive definite solution for the matrix equation $X+{\displaystyle \sum _{i=1}^m{A}_i^{*}{X}^{{\delta }_i}}{A}_i=I$ with $q\in (0,1)$; Meng et al. [15] proposed a fixed point iteration method with step parameters to solve the positive definite solution of the matrix equation $X^s-A^{*}{X}^{-t}A=Q$; Meng et al. [16] and Mouhadjer et al. [17] studied the nonlinear matrix equation $X^p=A+M^{*}{(X^{-1}+B)}^{-1}M$ with $p\ge 1$; Sayevand et al. [18] studied the nonlinear matrix equation $X+A^{*}{X}^{-1}A+B^{*}{X}^{-1}B=I$; Pakhira et al. [19] studied the nonlinear matrix equation $X^s+A^{*}{X}^{-t}A+B^{*}{X}^{-p}B=I$ with $s,t,p\ge 1$, etc.

In this paper, we propose a weighted average iterative algorithm to solve the nonlinear matrix equation (1.1). Based on the basic characteristics of the Thompson metric space, we will prove the convergence and error estimation formula of the proposed algorithm. Meanwhile, we will give some numerical experiments to illustrate the feasibility and effectiveness of the proposed algorithm.

2. Preliminaries

Let $S{R}^{n\times n}$ denote the set of $n\times n$ real symmetric matrices, $S{R}_0^{n\times n}$ denote the set of $n\times n$ real symmetric positive semi-definite matrices, and $S{R}_{+}^{n\times n}$ denote the set of $n\times n$ real symmetric positive definite matrices. The notations $A^{*}$, $A^T$ and ${‖A‖}_F$ represent the conjugate transpose, transpose and Frobenius norm of the matrix $A$, respectively. The notations $B\ge C$ means $B-C\in S{R}_0^{n\times n}$, $B>C$ means $B-C\in S{R}_{+}^{n\times n}$, and $X\in \left[B,C\right]$ means $B\le X\le C$. The notations ${\lambda }_1\left(X\right)$ and ${\lambda }_n\left(X\right)$ indicate the maximum and minimum eigenvalues of the real symmetric positive definite matrix X, respectively, and $I$ indicate the identity matrix of size implied by context.

Definition 2.1.

[6] Let $A,B\in S{R}_{+}^{n\times n}$ , the Thompson distance between the matrices $A$ and $B$ is defined as follows

$$d\left(A,B\right)=\max \left\{\log M\left(A/B\right),\log M\left(B/A\right)\right\},$$

where

$$M\left(A/B\right)=\inf \left\{\delta >0:A\le \delta B\right\}={\lambda }_1\left(B^{-{\displaystyle \frac{1}{2}}}A{B}^{-{\displaystyle \frac{1}{2}}}\right).$$

Thompson distance has the following basic properties:

Lemma 2.1.

[6] For any non-singular matrix $M$ , the following equations and inequalities

$$d\left(X,Y\right)=d\left(X^{-1},Y^{-1}\right)=d\left(M^{*}XM,M^{*}YM\right),$$

$$d\left(X^r,Y^r\right)\le \left|r\right|d\left(X,Y\right),\left|r\right|\le 1,$$

hold for any $X,Y\in S{R}_{+}^{n\times n}$.

Lemma 2.2.

[6] For any $A,B,C,D\in S{R}_{+}^{n\times n}$ , there are

$$d\left(A+B,C+D\right)\le \max \left\{d\left(A,C\right),d\left(B,D\right)\right\}.$$

Lemma 2.3.

[6] For any $A\in S{R}_0^{n\times n}$ , $X,Y\in S{R}_{+}^{n\times n}$

$$d\left(A+X,A+Y\right)\le {\displaystyle \frac{\alpha }{\alpha +\beta }}d\left(X,Y\right),$$

where$\alpha =\max \left\{{\lambda }_1\left(X\right),{\lambda }_1\left(Y\right)\right\}$,$\beta ={\lambda }_n\left(A\right)$ .

In addition, the following lemma 2.4 can be obtained by direct calculation according to Thompson’s distance definition:

Lemma 2.4.

For the constant $\alpha \ne 0$ , $A,B\in S{R}_{+}^{n\times n}$ . We have $d\left(\alpha A,\alpha B\right)=d\left(A,B\right)$.

The follow Lemma 2.5 and Lemma 2.6 from [11] can be used to prove the fact that the nonlinear matrix equation (1.1) has a unique positive definite solution in given metric space.

Lemma 2.5.

If $\varphi$ is compression mapping on the metric space $\Omega$ and has a compression coefficient $\alpha \in \left(0，1\right)$ , then the mapping $\varphi$ has a unique fixed point on $\Omega$.

Lemma 2.6.

For any $X,Y\in S{R}_{+}^{n\times n}$ . If $0<X\le Y$ , then $Y^{\alpha }\le X^{\alpha }$ holds for any $\alpha \in [-1,0)$ , and $X^{\alpha }\le Y^{\alpha }$ holds for any $\alpha \in \left(0，1\right]$.

Lemma 2.7.

Assume that X is a positive definite solution of the nonlinear matrix equation (1.1), then the following inequality holds

$$Q^{\frac{1}{p}}\le X\le {\left(Q+{\displaystyle \sum _{i=1}^m{A_i}^T{Q}^{-\frac{1}{p}}{A}_i}\right)}^{\frac{1}{p}}.$$

(2.1)

Proof. 

Since X is a positive definite solution of the nonlinear matrix equation (1.1), then

$$X={\left(Q+{\displaystyle \sum _{i=1}^m{A_i}^T{X}^{-1}{A}_i}\right)}^{\frac{1}{p}}.$$

Hence, we have $X\ge Q^{\frac{1}{p}}$. And so we have by Lemma 2.6 that

$X^{-1}\le Q^{-\frac{1}{p}}$, $X={\left(Q+{\displaystyle \sum _{i=1}^m{A_i}^T{X}^{-1}{A}_i}\right)}^{\frac{1}{p}}\le {\left(Q+{\displaystyle \sum _{i=1}^m{A_i}^T{Q}^{-\frac{1}{p}}{A}_i}\right)}^{\frac{1}{p}}$.

Hence, the inequality (2.1) holds. ■

Let

$$G\left(X\right)={\left(Q+{\displaystyle \sum _{i=1}^m{A}_i^T}{X}^{-1}{A}_i\right)}^{{\displaystyle \frac{1}{p}}}.$$

(2.2)

and

$$\Omega =\left\{X\left|Q^{{\displaystyle \frac{1}{p}}}\le X\le {\left(Q+{\displaystyle \sum _{i=1}^m{A_i}^T{Q}^{-\frac{1}{p}}{A}_i}\right)}^{{\displaystyle \frac{1}{p}}}\right.\right\},$$

(2.3)

then, we have the following Lemma 2.8.

Lemma 2.8.

The matrix function$G(X)$satisfies the Lipschitz condition on Thompson metric space$\Omega$, that is, for any$X,Y\in \Omega$, we have

$$d\left(G\left(X\right),G\left(Y\right)\right)<Ld\left(X,Y\right),$$

(2.4)

where$0<L=\frac{1}{p}\frac{{\lambda }_1\left({\displaystyle \sum _{i=1}^m{A}_i^T{Q}^{-\frac{1}{p}}{A}_i}\right)}{{\lambda }_1\left({\displaystyle \sum _{i=1}^m{A}_i^T{Q}^{-\frac{1}{p}}{A}_i}\right)+{\lambda }_n(Q)}<1$

Proof. 

It can known from Lemma 2.1, Lemma 2.2, Lemma 2.3 and Lemma 2.6 that

$$d\left(G\left(X\right),G\left(Y\right)\right)=d\left({\left(Q+{\displaystyle \sum _{i-1}^m{A}_i^T{X}^{-1}{A}_i}\right)}^{\frac{1}{p}},{\left(Q+{\displaystyle \sum _{i-1}^m{A}_i^T{Y}^{-1}{A}_i}\right)}^{\frac{1}{p}}.\right)$$

$\le \frac{1}{p}d\left(Q+{\displaystyle {\displaystyle \sum _{i-1}^m}{A}_i^T{X}^{-1}{A}_i},Q+{\displaystyle {\displaystyle \sum _{i-1}^m}{A}_i^T{Y}^{-1}{A}_i}\right)$

$\le \frac{1}{p}\frac{{\lambda }_1\left({\displaystyle \sum _{i=1}^m{A}_i^T{Q}^{-\frac{1}{p}}{A}_i}\right)}{{\lambda }_1\left({\displaystyle \sum _{i=1}^m{A}_i^T{Q}^{-\frac{1}{p}}{A}_i}\right)+{\lambda }_n(Q)}d\left({\displaystyle {\displaystyle \sum _{i-1}^m}{A}_i^T{X}^{-1}{A}_i},{\displaystyle {\displaystyle \sum _{i-1}^m}{A}_i^T{Y}^{-1}{A}_i}\right)$

$=\frac{1}{p}\frac{{\lambda }_1\left({\displaystyle \sum _{i=1}^m{A}_i^T{Q}^{-\frac{1}{p}}{A}_i}\right)}{{\lambda }_1\left({\displaystyle \sum _{i=1}^m{A}_i^T{Q}^{-\frac{1}{p}}{A}_i}\right)+{\lambda }_n(Q)}\max \left\{d\left(A_1^T{X}^{-1}{A}_1,A_1^T{Y}^{-1}{A}_1\right),\dots ,d\left(A_m^T{X}^{-1}{A}_m,A_m^T{Y}^{-1}{A}_m\right)\right\}$$=\frac{1}{p}\frac{{\lambda }_1\left({\displaystyle \sum _{i=1}^m{A}_i^T{Q}^{-\frac{1}{p}}{A}_i}\right)}{{\lambda }_1\left({\displaystyle \sum _{i=1}^m{A}_i^T{Q}^{-\frac{1}{p}}{A}_i}\right)+{\lambda }_n(Q)}d(X,Y)$.

Hence, the inequality (2.4) holds. ■

According to Lemma 2.5, Lemma 2.7 and Lemma 2.8, the following lemma can be immediately obtained.

Lemma 2.9.

The nonlinear matrix equation (1.1) has a unique positive definite solution, and, furthermore, its unique positive definite solution X* satisfies $X\in \Omega$ given as (2.3).

3. Iterative Algorithm and Its Convergence

This section we first propose a multi-steps weighted average iterative algorithm for solving the nonlinear matrix equation (1.1), and then, based on the basic characteristics of Thompson metric space, prove the convergence of the proposed algorithm and the error estimation formula of the iterative solution. Let $F(X)=G(X)-X$. we propose the multi-steps weighted average iterative algorithm to solve the nonlinear matrix equation (1.1) as the following Algorithm 3.1.

Algorithm 3.1 (Weighted average iteration algorithm to solve (1.1))：

Step 1. Given the initial iteration point $X_0\in \Omega$, error tolerance parametersε, and positive integer s;

Step 2. Calculate $F\left(X_0\right)=G\left(X_0\right)-X_0$, Let $k=0$;

Step 3. If ${‖F(X_k)‖}_F<\epsilon$, stop (in this case,$X_k$ is an approximate solution of the nonlinear matrix equation (1.1)), otherwise, go to Step 4;

Step 4. Let $i=0,X_{k+i}^{(k)}=X_k$

Step 5. Calculate $X_{k+i+1}^{(k)}=G(X_{k+i}^{(k)})$;

Step 6. Let $i\triangleq i+1$, and if $i<s$, go to Step 5;

Step 7. Find $c={(c_0,c_1,\dots ,c_s)}^T\in {ℝ}^{s+1}$ such that

$\underset{c}{\min }{‖{\displaystyle \sum _{i=0}^s{c}_i\left(X_{k+i+1}^{(k)}-X_{k+i}^{\left(k\right)}\right)}‖}_{F}s.t.\begin{array}{c}\end{array}{\displaystyle \sum _{i=0}^s{c}_i}=1$

Step 8. Calculate $X_{k+1}={\displaystyle \sum _{i=0}^s{c}_i}{X}_{k+i+1}^{\left(k\right)}$;

Step 9. Calculate $F\left(X_k\right)=G\left(X_k\right)-X_k$. Let $k\triangleq k+1$, and go to Step 3.

For Algorithm 3.1, the following convergence theorem exists.

Theorem 3.1.

The nonlinear matrix equation (1.1) has a unique positive definite solution, and for any initial matrix $X_0\in \Omega$, the matrix sequence $\left\{X_k\right\}$ generated by Algorithm 3.1 converges to a unique positive definite solution $X^{*}$, and has the following error estimation formula

$$d\left(X_k,X^{*}\right)\le L^{k-1}d\left(X_0,X^{*}\right).$$

(2.5)

Proof. 

The result that the nonlinear matrix equation (1.1) has a unique positive definite solution X* in Ω given as (2.3}) can be obtained directly by Lemma 2.9, and so we don’t prove it here. We prove the conclusion (2.5) by induction. When k=1and k=2, we have

$$X_1=c_0{X}_0^{(0)}+c_1{X}_1^{(0)}+\dots +c_s{X}_s^{(0)}$$

where ${\displaystyle \sum _{i=0}^s{c}_i}=1$, $X_0^{(0)}=X_0$, $X_i^{(0)}=G(X_{i-1}^{(0)})$, $i=1,2,\dots ,s$. So we have by Lemma 2.2, Lemma 2.4 and Lemma 2.8 that

$d\left(X_1,X^{*}\right)=d\left(c_0{X}_0^{(0)}+c_1{X}_1^{(0)}+\dots +c_s{X}_s^{(0)},X^{*}\right)$

$=d(c_0{X}_0^{(0)}+c_1{X}_1^{(0)}+\dots +c_s{X}_s^{(0)},c_0{X}^{*}+c_1{X}^{*}+\dots +c_s{X}^{*})$

$\le \max \{d(c_0{X}_0^{(0)},c_0{X}^{*}),d(c_1{X}_1^{(0)},c_1{X}^{*})，\dots ,d(c_s{X}_s^{(0)},c_s{X}^{*})\}$

$=\max \{d(X_0^{(0)},X^{*}),d(X_1^{(0)},X^{*})，\dots ,d(X_s^{(0)},X^{*})\}$

$=\max \{d(X_0^{(0)},X^{*}),d(G(X_0^{(0)}),G(X^{*}))，\dots ,d(G(X_{s-1}^{(0)}),G(X^{*}))\}$

$\le \max \{d(X_0,X^{*}),Ld(X_0,X^{*})，\dots ,L^{s-1}d(X_0,X^{*})\}=L^{1-1}d(X_0,X^{*}).$

$d\left(X_2,X^{*}\right)=d\left(c_0{X}_0^{(1)}+c_1{X}_1^{(1)}+\dots +c_s{X}_s^{(1)},X^{*}\right)$

$=d(c_0{X}_0^{(1)}+c_1{X}_1^{(1)}+\dots +c_s{X}_s^{(1)},c_0{X}^{*}+c_1{X}^{*}+\dots +c_s{X}^{*})$

$\le \max \{d(c_0{X}_0^{(1)},c_0{X}^{*}),d(c_1{X}_1^{(1)},c_1{X}^{*})，\dots ,d(c_s{X}_s^{(1)},c_s{X}^{*})\}$

$=\max \{d(X_0^{(1)},X^{*}),d(X_1^{(1)},X^{*})，\dots ,d(X_s^{(1)},X^{*})\}$

$=\max \{d(G(X_0^{(1)}),G(X^{*})),d(G(X_1^{(1)}),G(X^{*}))，\dots ,d(G(X_{s-1}^{(1)}),G(X^{*}))\}$

$\le \max \{Ld(X_0,X^{*}),L^{2}d(X_0,X^{*})，\dots ,L^{s}d(X_0,X^{*})\}=L^{2-1}d(X_0,X^{*}).$

Assumed that the the conclusion holds for $k=t(t>0)$, that is, $d\left(X_t,X^{*}\right)\le L^{t-1}d\left(X_0,X^{*}\right)$, then

$d\left(X_{t+1},X^{*}\right)=d\left(c_0{X}_0^{(t)}+c_1{X}_1^{(t)}+\dots +c_s{X}_s^{(t)},X^{*}\right)$

$=d(c_0{X}_0^{(t)}+c_1{X}_1^{(t)}+\dots +c_s{X}_s^{(t)},c_0{X}^{*}+c_1{X}^{*}+\dots +c_s{X}^{*})$

$\le \max \{d(c_0{X}_0^{(t)},c_0{X}^{*}),d(c_1{X}_1^{(t)},c_1{X}^{*})，\dots ,d(c_s{X}_s^{(t)},c_s{X}^{*})\}$

$=\max \{d(X_0^{(t)},X^{*}),d(X_1^{(t)},X^{*})，\dots ,d(X_s^{(t)},X^{*})\}$

$=\max \{d(G(X_0^{(t)}),G(X^{*})),d(G(X_1^{(t)}),G(X^{*}))，\dots ,d(G(X_{s-1}^{(t)}),G(X^{*}))\}$

$\le \max \{L^{t}d(X_0,X^{*}),L^{t+1}d(X_0,X^{*})，\dots ,L^{t+s}d(X_0,X^{*})\}=L^{(t+1)-1}d(X_0,X^{*}).$

Hence, by the principle of induction, the error estimation formula $d\left(X_k,X^{*}\right)\le L^{k-1}d\left(X_0,X^{*}\right)$ holds for all $k$

Notice that the compression coefficient $L\in (0,1)$, we have

$$\underset{k\to \infty }{\lim }d(X_k,X^{*})=0.$$

and this means

$\underset{k\to \infty }{\lim }{X}_k=X^{*}.$

Thus, the matrix sequence $\left\{X_k\right\}$ generated by Algorithm 3.1 converges to the unique positive definite solution $X^{*}$ of the nonlinear matrix equation (1.1). ■

4. Numerical Experiments

In this section, some numerical experiments will be tested to illustrate the feasibility of Algorithm 3.1. The experiments are divided into two parts. In the first part, the relationship between the fixed points iterations number s and the convergence of the Algorithm 3.1 is analyzed, and the selection range of the best fixed points iterations numbers are given. In the second part, the comparison of the convergence of Algorithm 3.1 with fixed point iterative algorithm [6,20] and Anderson acceleration algorithm [8,21] are given. In all the numerical experiments, the matrices $A_i(i=1,2,\dots ,m)$ are randomly generated by the MATLAB, that is $A_i=randn(n,n)$. To make sure the matrix equation (1.1) has a solution, the matrices Q is chosen as

$Q_0=randn(n,n)$, $Q=Q_0^T{Q}_0+0.1I$.

In all tests, the initial iteration matrix $X_0$ is chosen as $X_0={(Q+0.01I)}^{\frac{1}{p}}$, the termination condition is the residual $R(X_k)$ satisfied

${‖R(X_k)‖}_F={‖F(X)‖}_F={‖Q+{\displaystyle \sum _{i=1}^m{A}_i^T{X}_k^{-1}{A}_i-}{X}_k^p‖}_F<{10}^{-9}$. （4.1）

In all data results are obtained in the Window10, 64-bit, Matlab R2013a environment.

4.1. Selection of Fixed Points Iteration Numbers

For each iteration of the Algorithm 3.1, fixed points iteration number (s) has an important relationship with convergence speed. Consider the nonlinear matrix equation

$$X-{A_1}^T{X}^{-1}{A}_1-{A_2}^T{X}^{-1}{A}_2-{A_3}^T{X}^{-1}{A}_3=Q.$$

(4.2)

We can be seen from Figure 1 that the convergence speed of Algorithm 3.1 is directly affected by the fixed points iteration number (s) . In general, when s chosen as 5 through 10, Algorithm 3.1 is relatively more efficient than other cases. But how to select the best fixed points iteration number (s) is an important problem which should be studied in future.

4.2. Comparison of Convergence of Different Algorithms

Notice that Algorithm 3.1 is most efficient when the fixed points iteration number s chosen as 5 to 10. Therefore, all the following tests, the fixed points iteration number s in Algorithm 3.1 is fixed at 5. In addition, for the convenience of the expression, we use the notations MPE, FIX and FIXA denote as Algorithm 3.1, fixed-point algorithm proposed in [6,20] and Anderson acceleration algorithm proposed in [8,21], respectively. In all tests, the maximum iteration numbers of each iteration algorithms are restricted to 10000, and the termination condition is the establishment of (4.1). Table 1 and Figure 2 report the comparison of MPE, FIXA and FIX to solve the nonlinear matrix equation (4.2) of the computational time (Time) with different matrix sizes.

Based on the tests reported in Table 1, Figure 2 and many other performed unreported tests which show similar patterns, we have that Algorithm 3.1 is more efficient than the fixed point accelerated iteration algorithm proposed in [8,21] and fixed point algorithm proposed in [6,20].

Table 4.1. Numerical comparison values of MPE, FIX and FIX with different matrix size.

n Time (seconds) Algor	MPE	FIXA	FIX
100	0.0461	0.0285	0.0393
200	0.0959	0.2193	0.1717
300	0.2334	0.5372	0.4636
400	0.6079	1.0863	1.2363
500	0.8371	1.6197	2.1035
600	1.5645	2.3805	3.2996
700	2.4385	3.5413	4.8826
800	3.3272	4.7722	6.8717
900	4.6206	6.4310	9.6169
1000	6.3372	8.2044	12.8925
1100	7.7366	10.4206	16.3306
1200	10.1329	12.3955	20.7740
1300	16.1424	23.8879	39.3865
1400	23.5979	28.5672	48.2884
1500	20.3946	20.8011	35.7082
1600	22.2414	24.5547	42.8654
1700	23.6987	29.6401	48.9275
1800	28.0851	34.3586	59.1632
1900	32.3917	38.3886	67.3332
2000	36.9877	40.8254	76.2251

Table 4.1. Numerical comparison values of MPE, FIX and FIX with different matrix size.

n Time (seconds) Algor	MPE	FIXA	FIX
100	0.0461	0.0285	0.0393
200	0.0959	0.2193	0.1717
300	0.2334	0.5372	0.4636
400	0.6079	1.0863	1.2363
500	0.8371	1.6197	2.1035
600	1.5645	2.3805	3.2996
700	2.4385	3.5413	4.8826
800	3.3272	4.7722	6.8717
900	4.6206	6.4310	9.6169
1000	6.3372	8.2044	12.8925
1100	7.7366	10.4206	16.3306
1200	10.1329	12.3955	20.7740
1300	16.1424	23.8879	39.3865
1400	23.5979	28.5672	48.2884
1500	20.3946	20.8011	35.7082
1600	22.2414	24.5547	42.8654
1700	23.6987	29.6401	48.9275
1800	28.0851	34.3586	59.1632
1900	32.3917	38.3886	67.3332
2000	36.9877	40.8254	76.2251

5. Conclusions

In this paper, a weighted average iterative algorithm (Algorithm 3.1) is proposed to solve the nonlinear matrix equation $X^p-{\displaystyle \sum _{i=1}^m{A}_i^T{X}^{-1}{A}_i=Q}$. The convergence and error estimation of the proposed algorithm are proved (Theorem 3.1). Some numerical comparison experiments with existing algorithms are given (Table 1, Figure 2). For Algorithm 3.1, the selection of the fixed point iteration number s is discussed. Problems which should be studied in near future are listed.