Open Access

On nonlinear matrix equations \(X\pm\sum_{i=1}^{m}A_{i}^{*}X^{-n_{i}}A_{i}=I\)

Journal of Inequalities and Applications20152015:147

https://doi.org/10.1186/s13660-015-0667-7

Received: 15 February 2015

Accepted: 15 April 2015

Published: 28 April 2015

Abstract

We study the nonlinear matrix equations \(X+\sum_{i=1}^{m}A_{i}^{*}X^{-n_{i}}A_{i}=I\) and \(X-\sum_{i=1}^{m}A_{i}^{*}X^{-n_{i}}A_{i}=I\), where \(n_{i}\) are positive integers for \(i=1,2,\ldots,m\). The iterative algorithms for obtaining positive definite solutions for these equations are proposed. The necessary and sufficient conditions for the existence of positive definite solutions of these equations are derived. Moreover, the rate of convergence of the sequences generated from the algorithms is studied. The efficiency of proposed algorithms is illustrated by numerical examples.

Keywords

matrix equation positive definite solution iterative algorithm

1 Introduction

Consider the nonlinear matrix equations:
$$ X+\sum_{i=1}^{m}A_{i}^{*}X^{-n_{i}}A_{i}=I $$
(1.1)
and
$$ X-\sum_{i=1}^{m}A_{i}^{*}X^{-n_{i}}A_{i}=I, $$
(1.2)
where X is an unknown square matrix, I is the identity matrix, \(A_{i}\) are square complex matrices and \(n_{i}\) are positive integers for \(i=1,2,\ldots,m\).

Nonlinear matrix equations of type (1.1) and (1.2) have many applications in engineering, control theory, dynamic programming, stochastic filtering, ladder networks, statistics, etc.; see [14] and the references therein. When \(m=1\) and \(n_{1}=1\), (1.2) arises in the analysis of stationary Gaussian reciprocal processes over a finite interval [3]. When \(m>1\) and \(n_{i}=1\) for \(i=1,2,\ldots,m\), (1.1) arises in solving a large-scale system of linear equations in many physical calculations [5] and (1.2) is recognized as playing an important role in modeling certain optimal interpolation problems [6, 7].

In the last few years, many authors have been greatly interested in developing the theory and numerical approaches for positive definite solutions to the nonlinear matrix equations of the form (1.1) and (1.2). Similar types of (1.1) and (1.2) have been investigated [812]. The matrix equations \(X\pm A^{*}X^{-1}A=Q\) have been studied by several authors [14, 13, 14] and different iterative algorithms for computing the positive definite solutions with linear and quadratic rate of convergence are proposed. Ivanov et al. [15] derived sufficient conditions for the existence of positive definite solutions for the matrix equations \(X\pm A^{*}X^{-2}A=I\) and they proposed iterative algorithms for obtaining positive definite solutions of these equations. El-Sayed [16] presented two iterative methods for calculating the positive definite solutions of the matrix equation \(X-A^{*}X^{-n}A=Q\), for the integer \(n\geq1\), the first method is derived for a normal matrix A and for the second method a sufficient condition for convergence is given for \(n = 2^{k}\). El-Sayed and Ran [17] studied the general matrix equation \(X+ A^{*}F(X)A=Q\) where F maps positive definite matrices either into positive definite matrices or into negative definite matrices and satisfies some monotonicity property. Hasanov and Ivanov [18] considered the matrix equations \(X\pm A^{*}X^{-n}A=Q\), they studied the solutions and perturbation analysis of these solutions. They also derived a sufficient condition for the existence of a unique positive definite solution of the equation \(X- A^{*}X^{-n}A=Q\). Hasanov [19] established and proved theorems for the necessary and sufficient conditions of the existence of positive definite solutions for the matrix equations \(X\pm A^{*}X^{-q}A=Q\) with \(0< q\leq1\), he showed that the equation \(X- A^{*}X^{-q}A=Q\) has a unique positive definite solution by using the properties of matrix sequence in Banach space. Also, in [5] some conditions for the existence of positive definite solution of the equation \(X+\sum_{i=1}^{m}A_{i}^{*}X^{-1}A_{i}=I\) have been obtained and two iterative algorithms to find the maximal positive definite solution of this equation have been presented. Duan et al. [20] gave two perturbation estimates for the positive definite solution of the equation \(X-\sum_{i=1}^{m}A_{i}^{*}X^{\delta_{i}}A_{i}=Q\) with \(0 < |\delta_{i}|< 1\). Duan et al. [6] studied the equation \(X-\sum_{i=1}^{m}N_{i}^{*}X^{-1}N_{i}=I\), they used the Thompson metric to prove that the matrix equation always has a unique positive definite solution and they derived a precise perturbation bound for the unique positive definite solution. In addition, other nonlinear matrix equations such as \(X^{s}\pm A^{T}X^{-t}A=I_{n}\) [21], \(AX^{2} + BX+C = 0\) [22], and \(X=Q +A^{H}(I\otimes X-C)^{-\delta}A^{*}\) [23] have been investigated.

In this paper, we study the positive definite solutions of (1.1) and (1.2). We derive the necessary and sufficient conditions for the existence of positive definite solutions. We suggest iterative algorithms for obtaining positive definite solutions of these equations. Moreover, under some conditions we obtain the rates of convergence of the iterative sequences of approximate solutions and the stopping criterions. Finally, we give some numerical examples to ensure the performance and the effectiveness of the suggested iterative algorithms.

The following notations will be used in this paper. \(A^{\star}\) denotes the complex conjugate transpose of A. We write \(A > 0\) (\(A\geq0\)), if matrix A is positive definite (positive semidefinite). If \(A-B\) is positive definite (positive semidefinite), then we write \(A > B\) (\(A\geq B\)). Moreover, we denote \(\rho(A)\) by the spectral radius of A. We use \(\lambda_{\mathrm{max}}(A)\) and \(\lambda _{\mathrm{min}}(A)\) to denote the maximal and minimal eigenvalues of A. \(\| \cdot \|\) and \(\| \cdot \|_{\infty}\) denote the spectral and infinity norm, respectively.

Lemma 1.1

[24]

If \(A\geq B> 0\), then \(A^{-1}\leq B^{-1}\).

Lemma 1.2

[24]

If A and B are positive definite matrices for which \(A-B>0\) and \(AB=BA\) are satisfied, then \(A^{n}-B^{n}>0\).

Lemma 1.3

[25]

If \(A>B>0\) (or \(A\geq B>0\)), then \(A^{\alpha}>B^{\alpha}>0\) (or \(A^{\alpha}\geq B^{\alpha}>0\)), for all \(\alpha\in(0,1]\), and \(0< A^{\alpha}<B^{\alpha}\) (or \(0< A^{\alpha }\leq B^{\alpha}\)), for all \(\alpha\in[-1,0)\).

2 The matrix equation \(X+\sum_{i=1}^{m}A_{i}^{*}X^{-n_{i}}A_{i}=I\)

In this section, we give some necessary and sufficient conditions for the existence of positive definite solutions of (1.1). We present the following iterative algorithm to compute the positive definite solution of (1.1).

Algorithm 2.1

$$\left \{ \begin{array}{l} X_{0} =I, \\ X_{s+1} = I-\sum_{i=1}^{m}A_{i}^{*}X^{-n_{i}}_{s}A_{i},\quad \mbox{for } s=0,1,2,\ldots. \end{array} \right . $$

Remark 2.1

Letting \(m=1\) in Algorithm 2.1, we get Algorithm (2.2) in [10] which is proposed for obtaining the positive definite solutions of the matrix equation \(X+A^{*}X^{-n}A=I\). Also, letting \(n_{i}=1\), \(\forall i=1,2,\ldots,m\), in Algorithm 2.1, we get Algorithm 2.7 in [5], which is proposed for obtaining the positive definite solutions of the matrix equation \(X+\sum_{i=1}^{m}A_{i}^{*}X^{-1}A_{i}=I\).

The following theorem provides the necessary condition for the existence of positive definite solutions of (1.1).

Theorem 2.1

If (1.1) has a positive definite solution X, then
$$ \bigl(A_{i}A^{*}_{i} \bigr)^{\frac{1}{n_{i}}}< X \leq I-\sum_{i=1}^{m}A_{i}^{*}A_{i}, \quad i=1,2,\ldots,m. $$
(2.1)

Proof

Since X is a positive definite solution of (1.1), then \(X\leq I\) and \(\sum_{i=1}^{m}A_{i}^{*}X^{-n_{i}}A_{i}< I\). Using the inequality \(X\leq I\) and Lemmas 1.1, 1.2, we have
$$X=I-\sum_{i=1}^{m}A_{i}^{*}X^{-n_{i}}A_{i} \leq I-\sum_{i=1}^{m}A_{i}^{*}A_{i}. $$
Also, from the inequality \(\sum_{i=1}^{m}A_{i}^{*}X^{-n_{i}}A_{i}< I\), we have \(A_{i}^{*}X^{-n_{i}}A_{i}< I\). Then
$$A_{i}^{*}X^{-n_{i}/2}X^{-n_{i}/2}A_{i} < I, $$
which implies that
$$X^{-n_{i}/2}A_{i} A_{i}^{*}X^{-n_{i}/2}< I. $$
Using Lemma 1.3, we obtain
$$\bigl(A_{i} A_{i}^{*}\bigr)^{1/n_{i}}< X. $$
This completes the proof. □

Remark 2.2

Letting \(n_{i}=1\), \(\forall i=1,2,\ldots,m\), in (2.1) we get the condition \(A_{i}A^{*}_{i}< X \leq I-\sum_{i=1}^{m}A_{i}^{*}A_{i}\), which is necessary for the existence of positive definite solutions of the matrix equation \(X+\sum_{i=1}^{m}A_{i}^{*}X^{-1}A_{i}=I\) ([5], Theorem 2.1).

Lemma 2.1

If \(A_{i}\), \(i=1,2,\ldots,m\), are hermitian matrices and \(A_{i}A_{j}=A_{j}A_{i}\), for all \(i,j=1,2,\ldots,m\), then
$$ A_{j}X_{s}=X_{s}A_{j}, \quad j=1,2,3,\ldots,m, $$
(2.2)
where the sequence \(\{X_{s}\}\), \(s=0,1,2,\ldots\) , is determined by Algorithm 2.1.

Proof

Since \(X_{0}=I\), \(A_{j}X_{0}=X_{0}A_{j}\). By using the condition \(A_{i}A_{j}=A_{j}A_{i}\), we have
$$A_{j}X_{1} = A_{j} \Biggl(I-\sum _{i=1}^{m}A^{2}_{i} \Biggr) = A_{j}-\sum_{i=1}^{m}A_{j}A^{2}_{i} = A_{j}-\sum_{i=1}^{m}A^{2}_{i}A_{j} = \Biggl(I-\sum_{i=1}^{m}A^{2}_{i} \Biggr)A_{j} =X_{1}A_{j}. $$
We suppose that \(A_{j}X_{s}=X_{s}A_{j}\). Then for \(X_{s+1}\), we have
$$\begin{aligned} A_{j}X_{s+1} =& A_{j} \Biggl(I-\sum _{i=1}^{m}A_{i}X^{-n_{i}}_{s}A_{i} \Biggr) \\ =& A_{j}-\sum_{i=1}^{m}A_{j}A_{i}X^{-n_{i}}_{s}A_{i} \\ =& A_{j}-\sum_{i=1}^{m}A_{i}A_{j}X^{-n_{i}}_{s}A_{i} \\ =& A_{j}-\sum_{i=1}^{m}A_{i}X^{-n_{i}}_{s}A_{j}A_{i} \\ =& A_{j}-\sum_{i=1}^{m}A_{i}X^{-n_{i}}_{s}A_{i}A_{j} \\ =& \Biggl( I-\sum_{i=1}^{m}A_{i}X^{-n_{i}}_{s}A_{i} \Biggr)A_{j} \\ =&X_{s+1}A_{j}. \end{aligned}$$
Hence, the equalities (2.2) are true, for all \(s=0,1,2,\ldots \) . □

Lemma 2.2

If \(A_{i}\), \(i=1,2,\ldots,m\), are hermitian matrices and \(A_{i}A_{j}=A_{j}A_{i}\), for all \(i,j=1,2,\ldots,m\), then
$$ X_{s}X_{r}=X_{r}X_{s}. $$
(2.3)
Here the sequences \(\{X_{s}\}\), \(\{X_{r}\}\), \(s,r=0,1,2,\ldots\) , are determined by Algorithm 2.1.

Proof

Since \(X_{0}=I\), \(X_{0}X_{r}=X_{r}X_{0}\), \(\forall r=0,1,2,\ldots\) . According to Lemma 2.1, we have
$$\begin{aligned} X_{1}X_{r} =& \Biggl(I-\sum _{i=1}^{m}A^{2}_{i} \Biggr) \Biggl(I-\sum_{i=1}^{m}A_{i}X^{-n_{i}}_{r-1}A_{i} \Biggr) \\ =& I-\sum_{i=1}^{m}A^{2}_{i}- \sum_{i=1}^{m}A_{i}X^{-n_{i}}_{r-1}A_{i}+ \sum_{j=1}^{m}\sum _{i=1}^{m}A^{2}_{j}A_{i}X^{-n_{i}}_{r-1}A_{i} \\ =& I-\sum_{i=1}^{m}A^{2}_{i}- \sum_{i=1}^{m}A_{i}X^{-n_{i}}_{r-1}A_{i}+ \sum_{j=1}^{m}\sum _{i=1}^{m}A_{i}X^{-n_{i}}_{r-1}A_{i}A^{2}_{j} \\ =& I-\sum_{i=1}^{m}A^{2}_{i}- \sum_{i=1}^{m}A_{i}X^{-n_{i}}_{r-1}A_{i}+ \sum_{i=1}^{m}\sum _{j=1}^{m}A_{i}X^{-n_{i}}_{r-1}A_{i}A^{2}_{j} \\ =& \Biggl(I-\sum_{i=1}^{m}A_{i}X^{-n_{i}}_{r-1}A_{i} \Biggr) \Biggl(I-\sum_{i=1}^{m}A^{2}_{i} \Biggr) \\ =& X_{r}X_{1}. \end{aligned}$$
That is, \(X_{1}X_{r}=X_{r}X_{1}\), \(\forall r=0,1,2,\ldots\) . We suppose that \(X_{s}X_{r}=X_{r}X_{s}\), \(\forall r=0,1,2,\ldots\) . Then for \(X_{s+1}\), we have
$$\begin{aligned} X_{s+1} X_{r} =& \Biggl(I-\sum _{i=1}^{m}A_{i}X^{-n_{i}}_{s}A_{i} \Biggr) \Biggl(I-\sum_{i=1}^{m}A_{i}X^{-n_{i}}_{r-1}A_{i} \Biggr) \\ =& I-\sum_{i=1}^{m}A_{i}X^{-n_{i}}_{s}A_{i}- \sum_{i=1}^{m}A_{i}X^{-n_{i}}_{r-1}A_{i}+ \sum_{j=1}^{m}A_{j}X^{-n_{j}}_{s}A_{j} \sum_{i=1}^{m}A_{i}X^{-n_{i}}_{r-1}A_{i} \\ =& I-\sum_{i=1}^{m}A_{i}X^{-n_{i}}_{s}A_{i}- \sum_{i=1}^{m}A_{i}X^{-n_{i}}_{r-1}A_{i}+ \sum_{j=1}^{m}\sum _{i=1}^{m}A_{j}X^{-n_{j}}_{s}A_{j}A_{i}X^{-n_{i}}_{r-1}A_{i} \\ =& I-\sum_{i=1}^{m}A_{i}X^{-n_{i}}_{s}A_{i}- \sum_{i=1}^{m}A_{i}X^{-n_{i}}_{r-1}A_{i}+ \sum_{j=1}^{m}\sum _{i=1}^{m}A_{i}X^{-n_{j}}_{s}A_{i}A_{j}X^{-n_{i}}_{r-1}A_{j} \\ =& I-\sum_{i=1}^{m}A_{i}X^{-n_{i}}_{s}A_{i}- \sum_{i=1}^{m}A_{i}X^{-n_{i}}_{r-1}A_{i}+ \sum_{j=1}^{m}\sum _{i=1}^{m}A_{i}A_{i}X^{-n_{j}}_{s}X^{-n_{i}}_{r-1}A_{j}A_{j} \\ =& I-\sum_{i=1}^{m}A_{i}X^{-n_{i}}_{s}A_{i}- \sum_{i=1}^{m}A_{i}X^{-n_{i}}_{r-1}A_{i}+ \sum_{j=1}^{m}\sum _{i=1}^{m}A_{i}A_{i}X^{-n_{i}}_{r-1}X^{-n_{j}}_{s}A_{j}A_{j} \\ =& I-\sum_{i=1}^{m}A_{i}X^{-n_{i}}_{s}A_{i}- \sum_{i=1}^{m}A_{i}X^{-n_{i}}_{r-1}A_{i}+ \sum_{j=1}^{m}\sum _{i=1}^{m}A_{i}X^{-n_{i}}_{r-1}A_{i}A_{j}X^{-n_{j}}_{s}A_{j} \\ =& I-\sum_{i=1}^{m}A_{i}X^{-n_{i}}_{s}A_{i}- \sum_{i=1}^{m}A_{i}X^{-n_{i}}_{r-1}A_{i}+ \sum_{i=1}^{m}A_{i}X^{-n_{i}}_{r-1}A_{i} \sum_{j=1}^{m}A_{j}X^{-n_{j}}_{s}A_{j} \\ =& \Biggl(I-\sum_{i=1}^{m}A_{i}X^{-n_{i}}_{r-1}A_{i} \Biggr) \Biggl(I-\sum_{i=1}^{m}A_{i}X^{-n_{i}}_{s}A_{i} \Biggr) \\ =& X_{r}X_{s+1}. \end{aligned}$$
Therefore, the equality (2.3) is true, for all \(s,r=0,1,2,\ldots \) . □

Remark 2.3

When we compare Lemmas 2.1 and 2.2 by Lemmas 4 and 5 in [10], we note that the sequence \(\{X_{s}\}\), \(s=0,1,2,\ldots\) (which is defined by Algorithm 2.1) satisfies the same properties of the sequence \(\{X_{s}\}\), \(s=0,1,2,\ldots \) (which is defined by Algorithm (2.2) in [10]).

The following theorem provides the sufficient condition for the existence of positive definite solutions of (1.1).

Theorem 2.2

Let \(A_{i}\), \(i=1,2,\ldots,m\), be hermitian matrices and \(A_{i}A_{j}=A_{j}A_{i}\), for all \(i,j=1,2,\ldots,m\). If \(A^{2}_{i}\leq\frac{(\alpha-1)}{nm\alpha^{(2n_{i}+1)}}I\), where \(\alpha>1\) and \(n=\max_{1 \leq i \leq m}\{n_{i}\}\), then (1.1) has a positive definite solution.

Proof

We consider the sequence \(\{X_{s}\}\) generated from Algorithm 2.1. For \(X_{0}\), we have \(X_{0}=I>\frac{1}{\alpha}I\). For \(X_{1}\), we have
$$\begin{aligned}& X_{1}=I-\sum_{i=1}^{m}A^{2}_{i} \geq I-\sum_{i=1}^{m}\frac{(\alpha -1)}{nm\alpha^{(2n_{i}+1)}}I>I- \sum_{i=1}^{m}\frac{(\alpha -1)}{nm\alpha}I \geq I- \frac{(\alpha-1)}{\alpha}I=\frac{1}{\alpha}I, \\& X_{1}=I-\sum_{i=1}^{m}A^{2}_{i} \leq I=X_{0}. \end{aligned}$$
That is,
$$X_{0}\geq X_{1}>\frac{1}{\alpha}I. $$
We suppose that
$$ X_{s-1}\geq X_{s} > \frac{1}{\alpha}I. $$
(2.4)
Using the inequalities (2.4) and Lemmas 1.1, 1.2, and 2.2, we obtain
$$\begin{aligned} X_{s+1} =& I-\sum_{i=1}^{m}A_{i}X^{-n_{i}}_{s}A_{i} \\ \leq& I-\sum_{i=1}^{m}A_{i}X^{-n_{i}}_{s-1}A_{i} \\ =& X_{s}. \end{aligned}$$
Also
$$\begin{aligned} X_{s+1} =& I-\sum_{i=1}^{m}A_{i}X^{-n_{i}}_{s}A_{i} \\ >& I-\sum_{i=1}^{m}\alpha^{n_{i}} A^{2}_{i} \\ \geq& I-\sum_{i=1}^{m} \alpha^{n_{i}}\frac{(\alpha-1)}{nm\alpha ^{(2n_{i}+1)}}I \\ =& I-\sum_{i=1}^{m}\frac{1}{m} \frac{(\alpha-1)}{n\alpha ^{(n_{i}+1)}}I \\ >& I-\frac{(\alpha-1)}{\alpha}I \\ =&\frac{1}{\alpha}I. \end{aligned}$$
Therefore the inequalities (2.4) are true, for all \(s=1,2,\ldots \) . That is, the sequence \(\{X_{s}\}\) is monotonically decreasing and bounded below by \(\frac{1}{\alpha}I\). Hence, the sequence \(\{X_{s}\}\) converges to a positive definite solution X of (1.1). □

Theorem 2.3

Let \(A_{i}\), \(i=1,2,\ldots,m\), be hermitian matrices and \(A_{i}A_{j}=A_{j}A_{i}\), for all \(i,j=1,2,\ldots,m\). If \(A^{2}_{i}\leq\frac{(\alpha-1)}{nm\alpha^{(2n_{i}+1)}}I\), then (1.1) has a positive definite solution X which satisfies
$$ \|X_{s+1}-X\|< \biggl(\frac{\alpha-1}{\alpha}\biggr) \|X_{s}-X\|, $$
(2.5)
where \(\alpha>1\), \(n=\max_{1 \leq i \leq m}\{n_{i}\}\), and \(\{ X_{s}\}\), \(s=0,1,2,\ldots\) , is the sequence determined by Algorithm 2.1.

Proof

By Theorem 2.2, we know that the sequence \(\{X_{s}\}\), \(s=0,1,2,\ldots \) , is convergent to a positive definite solution X of (1.1). We consider the spectral norm of the matrix \(X_{s+1}-X\). We have
$$\begin{aligned} \Vert X_{s+1}-X\Vert =&\Biggl\Vert I-\sum _{i=1}^{m}A_{i}X^{-n_{i}}_{s}A_{i}-I+ \sum_{i=1}^{m}A_{i}X^{-n_{i}}A_{i} \Biggr\Vert \\ =&\Biggl\Vert \sum_{i=1}^{m}A_{i} \bigl(X^{-n_{i}}-X^{-n_{i}}_{s}\bigr)A_{i}\Biggr\Vert \\ \leq&\sum_{i=1}^{m}\bigl\Vert A_{i}\bigl(X^{-n_{i}}-X^{-n_{i}}_{s} \bigr)A_{i}\bigr\Vert \\ \leq&\sum_{i=1}^{m}\Vert A_{i}\Vert ^{2}\bigl\Vert X^{-n_{i}} \bigl(X^{n_{i}}_{s}-X^{n_{i}}\bigr)X^{-n_{i}}_{s} \bigr\Vert \\ \leq&\sum_{i=1}^{m}\Vert A_{i}\Vert ^{2}\bigl\Vert X^{-n_{i}}\bigr\Vert \bigl\Vert X^{-n_{i}}_{s}\bigr\Vert \bigl\Vert X^{n_{i}}_{s}-X^{n_{i}}\bigr\Vert \\ =&\sum_{i=1}^{m}\Vert A_{i} \Vert ^{2}\bigl\Vert X^{-n_{i}}\bigr\Vert \bigl\Vert X^{-n_{i}}_{s}\bigr\Vert \Biggl\Vert (X_{s}-X)\sum _{r=1}^{n_{i}}X^{n_{i}-r}_{s}X^{r-1} \Biggr\Vert \\ \leq&\sum_{i=1}^{m}\Vert A_{i}\Vert ^{2}\bigl\Vert X^{-n_{i}}\bigr\Vert \bigl\Vert X^{-n_{i}}_{s}\bigr\Vert \Vert X_{s}-X \Vert \Biggl(\sum_{r=1}^{n_{i}}\Vert X_{s}\Vert ^{n_{i}-r}\Vert X\Vert ^{r-1}\Biggr). \end{aligned}$$
From the proof of Theorem 2.2, we obtain \(X^{-n_{i}}_{s}<\alpha ^{n_{i}}I\), \(X^{-n_{i}} \leq\alpha^{n_{i}}I\), and \(X\leq X_{s}\leq I\). Then we have
$$\begin{aligned} \Vert X_{s+1}-X\Vert \leq& \sum _{i=1}^{m}\Vert A_{i}\Vert ^{2}\bigl\Vert X^{-n_{i}}\bigr\Vert \bigl\Vert X^{-n_{i}}_{s}\bigr\Vert \Vert X_{s}-X\Vert \Biggl(\sum_{r=1}^{n_{i}}\Vert X_{s}\Vert ^{n_{i}-r}\Vert X\Vert ^{r-1}\Biggr) \\ < & \sum_{i=1}^{m}n_{i} \alpha^{2n_{i}}\Vert A_{i}\Vert ^{2}\Vert X_{s}-X\Vert \\ \leq& \sum_{i=1}^{m}n_{i} \alpha^{2n_{i}}\frac{(\alpha -1)}{nm\alpha^{(2n_{i}+1)}}\Vert X_{s}-X\Vert \\ \leq& \sum_{i=1}^{m}n\frac{(\alpha-1)}{nm\alpha} \Vert X_{s}-X\Vert \\ =& \frac{(\alpha-1)}{\alpha} \Vert X_{s}-X\Vert . \end{aligned}$$
 □

Theorem 2.4

Let \(A_{i}\), \(i=1,2,\ldots,m\), be hermitian matrices and \(A_{i}A_{j}=A_{j}A_{i}\), for all \(i,j=1,2,\ldots,m\). If \(A^{2}_{i}\leq\frac{(\alpha-1)}{nm\alpha^{(2n_{i}+1)}}I\), where \(\alpha>1\), \(n=\max_{1 \leq i \leq m}\{n_{i}\}\), and after s iterative steps of Algorithm 2.1, we have \(\| I-X^{n_{i}}_{s}X^{-n_{i}}_{s-1}\|<\varepsilon\), then
$$ \Biggl\Vert X_{s}+\sum_{i=1}^{m}A_{i}X_{s}^{-n_{i}}A_{i}-I \Biggr\Vert < \frac{(\alpha-1)}{\alpha}\varepsilon. $$
(2.6)

Proof

From Algorithm 2.1, we have
$$\begin{aligned} X_{s}+\sum_{i=1}^{m}A_{i}X_{s}^{-n_{i}}A_{i}-I =& X_{s}+\sum_{i=1}^{m}A_{i}X_{s}^{-n_{i}}A_{i}-X_{s}- \sum_{i=1}^{m}A_{i}X_{s-1}^{-n_{i}}A_{i} \\ =& \sum_{i=1}^{m}A_{i} \bigl(X_{s}^{-n_{i}}-X_{s-1}^{-n_{i}} \bigr)A_{i}. \end{aligned}$$
By taking the norm on both sides of the above equation, we have
$$\begin{aligned} \Biggl\Vert X_{s}+\sum_{i=1}^{m}A_{i}X_{s}^{-n_{i}}A_{i}-I \Biggr\Vert =& \Biggl\Vert \sum_{i=1}^{m}A_{i} \bigl(X_{s}^{-n_{i}}-X_{s-1}^{-n_{i}} \bigr)A_{i}\Biggr\Vert \\ \leq& \sum_{i=1}^{m}\|A_{i} \|^{2}\bigl\Vert X_{s}^{-n_{i}}-X_{s-1}^{-n_{i}} \bigr\Vert \\ \leq& \sum_{i=1}^{m}\|A_{i} \|^{2}\bigl\Vert X_{s}^{-n_{i}}\bigr\Vert \bigl\Vert I-X_{s}^{n_{i}}X_{s-1}^{-n_{i}}\bigr\Vert . \end{aligned}$$
From the proof of Theorem 2.2, we have \(X_{s}^{-n_{i}}<\alpha ^{n_{i}}I\), then
$$\begin{aligned} \Biggl\Vert X_{s}+\sum_{i=1}^{m}A_{i}X_{s}^{-n_{i}}A_{i}-I \Biggr\Vert < & \sum_{i=1}^{m} \frac{(\alpha-1)}{nm\alpha^{(2n_{i}+1)}}\alpha ^{n_{i}}\bigl\Vert I-X_{s}^{n_{i}}X_{s-1}^{-n_{i}} \bigr\Vert \\ <& \sum_{i=1}^{m}\frac{(\alpha-1)}{m\alpha}\bigl\Vert I-X_{s}^{n_{i}}X_{s-1}^{-n_{i}}\bigr\Vert \\ <& \frac{(\alpha-1)}{\alpha}\varepsilon. \end{aligned}$$
 □

3 The matrix equation \(X-\sum_{i=1}^{m}A_{i}^{*}X^{-n_{i}}A_{i}=I\)

In this section, we give some necessary and sufficient conditions for the existence of positive definite solutions of (1.2). We present the following iterative algorithm to compute the positive definite solution of (1.2).

Algorithm 3.1

$$\left \{ \begin{array}{l} X_{0} =I, \\ X_{s+1} = I+\sum_{i=1}^{m}A_{i}^{*}X^{-n_{i}}_{s}A_{i},\quad \mbox {for } s=0,1,2,\ldots. \end{array} \right . $$

Remark 3.1

Letting \(m=1\) in Algorithm 3.1, we get Algorithm (2.1) in [16] which is proposed for obtaining the positive definite solutions of the matrix equation \(X-A^{*}X^{-n}A=I\). Also, letting \(n_{i}=1\), \(\forall i=1,2,\ldots,m\), in Algorithm 3.1, we get Algorithm (2.1) in [6] which is proposed for obtaining the positive definite solutions of the matrix equation \(X-\sum_{i=1}^{m}A_{i}^{*}X^{-1}A_{i}=I\).

The following theorem provides the necessary condition for the existence of positive definite solutions of (1.2).

Theorem 3.1

If (1.2) has a positive definite solution X, then
$$ I\leq X \leq I+\sum_{i=1}^{m}A_{i}^{*}A_{i}. $$
(3.1)

Proof

Since X is a positive definite solution of (1.2), then \(\sum_{i=1}^{m}A_{i}^{*}X^{-n_{i}}A_{i}\geq0\). Thus we get
$$X=I+\sum_{i=1}^{m}A_{i}^{*}X^{-n_{i}}A_{i} \geq I. $$
Also, from the inequality \(X\geq I\) and Lemmas 1.1 and 1.2, we have
$$X=I+\sum_{i=1}^{m}A_{i}^{*}X^{-n_{i}}A_{i} \leq I+\sum_{i=1}^{m}A_{i}^{*}A_{i}. $$
This completes the proof. □

Remark 3.2

The condition (3.1) is the same necessary condition for the existence of positive definite solutions of the matrix equation \(X-\sum_{i=1}^{m}A_{i}^{*}X^{-1}A_{i}=I\) ([6], Remark 2.1). Also, if A is an invertible matrix and \(m=1\) in condition (3.1), then we get the condition \(I< X < I+A^{*}A\), which is necessary for the existence of positive definite solutions of the matrix equation \(X-A^{*}X^{-n}A=I\) ([16], Corollary 2.1).

Lemma 3.1

If \(A_{i}\), \(i=1,2,\ldots,m\), are hermitian matrices and \(A_{i}A_{j}=A_{j}A_{i}\), for all \(i,j=1,2,\ldots,m\), then
$$ A_{j}X_{s}=X_{s}A_{j}, \quad j=1,2,3,\ldots,m, $$
(3.2)
where the sequence \(\{X_{s}\}\), \(s=0,1,2,\ldots\) , is determined by Algorithm 3.1.

Proof

The proof is similar to the proof of Lemma 2.1. □

Lemma 3.2

If \(A_{i}\), \(i=1,2,\ldots,m\), are hermitian matrices and \(A_{i}A_{j}=A_{j}A_{i}\), for all \(i,j=1,2,\ldots,m\), then
$$ X_{s}X_{r}=X_{r}X_{s}. $$
(3.3)
Here the sequences \(\{X_{s}\}\), \(\{X_{r}\}\), \(s,r=0,1,2,\ldots\) , are determined by Algorithm 3.1.

Proof

The proof is similar to the proof of Lemma 2.2. □

Remark 3.3

When we compare Lemmas 3.1 and 3.2 by Lemmas 2.3 and 2.4 in [16], we note that the sequence \(\{X_{s}\} \), \(s=0,1,2,\ldots\) (which is defined by Algorithm 3.1) satisfies the same properties of the sequence \(\{X_{s}\}\), \(s=0,1,2,\ldots\) (which is defined by Algorithm (2.1) in [16]).

Theorem 3.2

Let \(A_{i}\), \(i=1,2,\ldots,m\), be hermitian matrices and \(A_{i}A_{j}=A_{j}A_{i}\), for all \(i,j=1,2,\ldots,m\). If \(q=\sum_{i=1}^{m}n_{i}\|A_{i}\|^{2} (1+\sum_{i=1}^{m}\|A_{i}\| ^{2} )^{n_{i}-1}<1\), then (1.2) has a positive definite solution X which satisfies
$$ X_{2s}\leq X\leq X_{2s+1} $$
(3.4)
and
$$ \|X_{2s+1}-X_{2s}\|\leq q^{2s} \sum _{i=1}^{m}\|A_{i} \|^{2}, $$
(3.5)
where the sequence \(\{X_{s}\}\), \(s=0,1,2,\ldots\) , is determined by Algorithm 3.1.

Proof

We consider the matrix sequence \(\{X_{s}\}\) generated from Algorithm 3.1 and using Lemmas 1.1, 1.2 and 3.2. Since \(A_{i}^{2}\geq0\), then
$$X_{1}=I+\sum_{i=1}^{m}A^{2}_{i} \geq I=X_{0} $$
and
$$X_{2}=I+\sum_{i=1}^{m}A_{i}X_{1}^{-n_{i}}A_{i} \leq I+\sum_{i=1}^{m}A^{2}_{i}=X_{1}. $$
Consequently
$$X_{0}\leq X_{2}\leq X_{1}. $$
We find the relation between \(X_{2}\), \(X_{3}\), \(X_{4}\), \(X_{5}\). Using \(X_{0}\leq X_{2}\leq X_{1}\), we obtain
$$X_{3}=I+\sum_{i=1}^{m}A_{i}X_{2}^{-n_{i}}A_{i} \leq I+\sum_{i=1}^{m}A^{2}_{i}=X_{1} $$
and
$$X_{3}=I+\sum_{i=1}^{m}A_{i}X_{2}^{-n_{i}}A_{i} \geq I+\sum_{i=1}^{m}A_{i}X_{1}^{-n_{i}}A_{i}=X_{2}. $$
Hence \(X_{2}\leq X_{3}\leq X_{1}\). In the same way we can prove that
$$X_{0}\leq X_{2}\leq X_{4}\leq X_{5} \leq X_{3}\leq X_{1}. $$
We suppose that
$$ X_{0}\leq X_{2s} \leq X_{2s+2}\leq X_{2s+3}\leq X_{2s+1}\leq X_{1}. $$
(3.6)
Using the inequalities (3.6), we have
$$\begin{aligned}& X_{2s+4}=I+\sum_{i=1}^{m}A_{i}X_{2s+3}^{-n_{i}}A_{i} \leq I+\sum_{i=1}^{m}A_{i} X_{2s+2}^{-n_{i}}A_{i}=X_{2s+3}, \\& X_{2s+4}=I+\sum_{i=1}^{m}A_{i}X_{2s+3}^{-n_{i}}A_{i} \geq I+\sum_{i=1}^{m}A_{i} X_{2s+1}^{-n_{i}}A_{i}=X_{2s+2}. \end{aligned}$$
Similarly
$$\begin{aligned}& X_{2s+5}=I+\sum_{i=1}^{m}A_{i}X_{2s+4}^{-n_{i}}A_{i} \leq I+\sum_{i=1}^{m}A_{i} X_{2s+2}^{-n_{i}}A_{i} =X_{2s+3}, \\& X_{2s+5}=I+\sum_{i=1}^{m}A_{i}X_{2s+4}^{-n_{i}}A_{i} \geq I+\sum_{i=1}^{m}A_{i} X_{2s+3}^{-n_{i}}A_{i} =X_{2s+4}. \end{aligned}$$
Therefore, the inequalities (3.6) are true, for all \(s=0,1,2,\ldots \) . Consequently the subsequences \(\{X_{2s}\}\) and \(\{ X_{2s+1}\}\) are convergent to positive definite matrices. To prove that these sequences have a common limit, we consider
$$\begin{aligned} \Vert X_{2s+1}-X_{2s}\Vert =& \Biggl\Vert \sum_{i=1}^{m}A_{i} \bigl(X_{2s}^{-n_{i}}-X_{2s-1}^{-n_{i}} \bigr)A_{i}\Biggr\Vert \\ \leq& \sum_{i=1}^{m}\Vert A_{i}\Vert ^{2}\bigl\Vert X_{2s}^{-n_{i}}-X_{2s-1}^{-n_{i}} \bigr\Vert \\ =& \sum_{i=1}^{m}\Vert A_{i} \Vert ^{2}\bigl\Vert X_{2s}^{-n_{i}} \bigl(X_{2s-1}^{n_{i}}-X_{2s}^{n_{i}} \bigr)X_{2s-1}^{-n_{i}}\bigr\Vert \\ \leq& \sum_{i=1}^{m}\Vert A_{i}\Vert ^{2}\bigl\Vert X_{2s}^{-n_{i}} \bigr\Vert \bigl\Vert X_{2s-1}^{-n_{i}}\bigr\Vert \bigl\Vert X_{2s-1}^{n_{i}}-X_{2s}^{n_{i}}\bigr\Vert \\ =&\sum_{i=1}^{m}\Vert A_{i} \Vert ^{2}\bigl\Vert X^{-n_{i}}_{2s}\bigr\Vert \bigl\Vert X^{-n_{i}}_{2s-1}\bigr\Vert \Biggl\Vert (X_{2s-1}-X_{2s})\sum_{r=1}^{n_{i}}X^{n_{i}-r}_{2s-1}X^{r-1}_{2s} \Biggr\Vert \\ \leq&\sum_{i=1}^{m}\Vert A_{i}\Vert ^{2}\bigl\Vert X^{-n_{i}}_{2s} \bigr\Vert \bigl\Vert X^{-n_{i}}_{2s-1}\bigr\Vert \Vert X_{2s-1}-X_{2s}\Vert \\ &{}\times \Biggl(\sum _{r=1}^{n_{i}}\Vert X_{2s-1}\Vert ^{n_{i}-r}\Vert X_{2s}\Vert ^{r-1} \Biggr). \end{aligned}$$
From the inequalities (3.6), we have \(X_{2s}, X_{2s-1}\geq I\), and \(X_{2s}, X_{2s-1}\leq I+\sum_{i=1}^{m}A_{i}^{2}\), for all \(s=1,2,3,\ldots \) . Then we have
$$ \|X_{2s+1}-X_{2s}\|\leq \sum _{i=1}^{m}n_{i}\|A_{i} \|^{2}\| X_{2s-1}-X_{2s}\| \Biggl(1+\sum _{i=1}^{m}\|A_{i}\|^{2} \Biggr)^{n_{i}-1}. $$
Hence
$$\|X_{2s+1}-X_{2s}\| \leq q \|X_{2s}-X_{2s-1} \|\leq\cdots\leq q^{2s} \sum_{i=1}^{m} \|A_{i}\|^{2}, $$
that is,
$$\|X_{2s+1}-X_{2s}\|\rightarrow0, \quad \mbox{as } s\rightarrow \infty. $$
Hence, the two subsequences \(\{X_{2s}\}\) and \(\{X_{2s+1}\}\) have the same limit X, which is a positive definite solution of (1.2). □

From Theorem 3.2, we can deduce the following corollary.

Corollary 3.1

From inequality (3.5), we have the following upper bound:
$$ \max \bigl(\Vert X_{2s+1}-X \Vert ,\Vert X-X_{2s}\Vert \bigr)\leq q^{2s} \sum _{i=1}^{m}\|A_{i}\|^{2}. $$
(3.7)

Remark 3.4

Theorem 3.2 provides the sufficient condition \(q=\sum_{i=1}^{m}n_{i}\|A_{i}\|^{2} (1+ \sum_{i=1}^{m}\|A_{i}\| ^{2} )^{n_{i}-1}<1\) for the existence of positive definite solutions of (1.2), we note that when \(m=1\) we have the condition \(\|A\|^{2}(1+\|A\|^{2})^{n-1}<\frac{1}{n}\), which is sufficient for the existence of positive definite solutions of the matrix equation \(X-A^{*}X^{-n}A=I\) ([16], Theorem 2.1).

Theorem 3.3

Let \(A_{i}\), \(i=1,2,\ldots,m\), be hermitian matrices and \(A_{i}A_{j}=A_{j}A_{i}\), for all \(i,j=1,2,\ldots,m\). If \(q=\sum_{i=1}^{m}n_{i}\|A_{i}\|^{2} (1+\sum_{i=1}^{m}\|A_{i}\|^{2} )^{n_{i}-1}<1\), and after s iterative steps of Algorithm  3.1, we have \(\| I-X^{n_{i}}_{s-1}X^{-n_{i}}_{s}\|<\varepsilon\), then
$$ \Biggl\Vert X_{s}-\sum_{i=1}^{m}A_{i}X_{s}^{-n_{i}}A_{i}-I \Biggr\Vert < \varepsilon\sum_{i=1}^{m} \|A_{i}\|^{2}. $$
(3.8)

Proof

From Algorithm 3.1, we have
$$\begin{aligned} X_{s}-\sum_{i=1}^{m}A_{i}X_{s}^{-n_{i}}A_{i}-I =& X_{s}-\sum_{i=1}^{m}A_{i}X_{s}^{-n_{i}}A_{i}-X_{s}+ \sum_{i=1}^{m}A_{i}X_{s-1}^{-n_{i}}A_{i} \\ =& \sum_{i=1}^{m}A_{i} \bigl(X_{s-1}^{-n_{i}}-X_{s}^{-n_{i}} \bigr)A_{i}. \end{aligned}$$
By taking the norm on both sides of the above equation, we have
$$\begin{aligned} \Biggl\Vert X_{s}-\sum_{i=1}^{m}A_{i}X_{s}^{-n_{i}}A_{i}-I \Biggr\Vert =& \Biggl\Vert \sum_{i=1}^{m}A_{i} \bigl(X_{s-1}^{-n_{i}}-X_{s}^{-n_{i}} \bigr)A_{i}\Biggr\Vert \\ \leq& \sum_{i=1}^{m}\Vert A_{i}\Vert ^{2}\bigl\Vert X_{s-1}^{-n_{i}}-X_{s}^{-n_{i}} \bigr\Vert \\ \leq& \sum_{i=1}^{m}\Vert A_{i}\Vert ^{2}\bigl\Vert X_{s-1}^{-n_{i}} \bigr\Vert \bigl\Vert I-X_{s-1}^{n_{i}}X_{s}^{-n_{i}} \bigr\Vert . \end{aligned}$$
From the proof of Theorem 3.2, we have \(X_{s-1}^{-n_{i}}\leq I\), then
$$\begin{aligned} \Biggl\Vert X_{s}-\sum_{i=1}^{m}A_{i}X_{s}^{-n_{i}}A_{i}-I \Biggr\Vert \leq& \sum_{i=1}^{m}\Vert A_{i}\Vert ^{2}\bigl\Vert I-X_{s-1}^{n_{i}}X_{s}^{-n_{i}} \bigr\Vert \\ < &\varepsilon\sum_{i=1}^{m}\Vert A_{i}\Vert ^{2}. \end{aligned}$$
 □

4 Numerical examples

In this section, we use the iterative Algorithms 2.1 and 3.1 to compute the positive definite solutions of (1.1) and (1.2), respectively. The solutions are computed for different matrices \(A_{i}\), \(i=1,2,\ldots,m\), with different orders. We denote X, the solution obtained by Algorithms 2.1 and 3.1 and \(\epsilon(X_{s})=\|X-X_{s}\|_{\infty}\), \(R_{1}(X_{s})=\|X_{s}+\sum_{i=1}^{m}A_{i}^{*}X_{s}^{-n_{i}}A_{i}-I\|_{\infty}\), \(Y_{s}= I-\sum_{i=1}^{m}A_{i}^{*}A_{i}-X_{s}\), \(Z_{i,s}= X^{n_{i}}_{s}-A_{i}A^{*}_{i}\) (\(i=1,2,\ldots,m\)), \(R_{2}(X_{s})=\Vert X_{s}-\sum_{i=1}^{m}A_{i}^{*}X_{s}^{-n_{i}}A_{i}-I \Vert _{\infty}\).

Example 4.1

Consider the matrix equation
$$ X+A_{1}^{*}X^{-4}A_{1}+A_{2}^{*}X^{-5}A_{2}+A_{3}^{*}X^{-3}A_{3}=I, $$
(4.1)
where \(A_{1}\), \(A_{2}\), and \(A_{3} \) are given by
$$\begin{aligned}& A_{1}=\left ( \begin{array}{@{}c@{\quad}c@{\quad}c@{}} 0.091 & 0.015 & 0.004 \\ 0.014 & 0.029 & 0.045 \\ -0.043 & 0.071 &0.015 \end{array} \right ), \qquad A_{2}=\left ( \begin{array}{@{}c@{\quad}c@{\quad}c@{}} 0.098 & 0.014 & 0.05 \\ 0.034 & 0.025 & 0.07 \\ 0.051 & 0.04 & -0.001 \end{array} \right ), \\& A_{3}=\left ( \begin{array}{@{}c@{\quad}c@{\quad}c@{}} 0.09 & -0.008 & 0.025 \\ 0.034 & -0.087 & 0.015 \\ 0.02 & -0.02 & 0.044 \end{array} \right ). \end{aligned}$$
We use Algorithm 2.1 to solve (4.1). After 22 iterations, we get the positive definite solution
$$X\approx X_{22}=\left ( \begin{array}{@{}c@{\quad}c@{\quad}c@{}} 0.960979 & 0.0000874449 & -0.0134898 \\ 0.0000874449 & 0.982002 & -0.00301522 \\ -0.0134898 & -0.00301522 &0.986046 \end{array} \right ) $$
and \(R_{1}(X_{22})= 1.11022\times10^{-16}\), \(\lambda_{\mathrm{min}}(Y_{22})= 0.000488382\), \(\lambda_{\mathrm{min}}(Z_{1,22})=0.825427\), \(\lambda_{\mathrm{min}}(Z_{2,22})=0.778478\), \(\lambda _{\mathrm{min}}(Z_{3,22})=0.860326\).
The other results are listed in Table 1.
Table 1

Error analysis for Example 4.1

s

\(\boldsymbol{\epsilon(X_{s})}\)

\(\boldsymbol{R_{1}(X_{s})}\)

\(\boldsymbol{\lambda}_{\mathbf{min}}\boldsymbol{(Y_{s})}\)

\(\boldsymbol{\lambda}_{\mathbf{min}}\boldsymbol{(Z_{1,s})}\)

\(\boldsymbol{\lambda}_{\mathbf{min}}\boldsymbol{(Z_{2,s})}\)

\(\boldsymbol{\lambda}_{\mathbf{min}}\boldsymbol{(Z_{3,s})}\)

0

3.90209 × 10−2

3.33430 × 10−2

−3.82258 × 10−2

0.989301

0.980411

0.985409

1

5.67787 × 10−3

4.70245 × 10−3

1.11022 × 10−16

0.849272

0.807341

0.879195

2

9.75412 × 10−4

8.02922 × 10−4

4.48982 × 10−4

0.829614

0.783528

0.863646

3

1.72490 × 10−4

1.41756 × 10−4

4.83255 × 10−4

0.826170

0.779373

0.860915

4

3.07337 × 10−5

2.52483 × 10−5

4.87548 × 10−4

0.825560

0.778638

0.860431

5

5.48539 × 10−6

4.50591 × 10−6

4.88236 × 10−4

0.825451

0.778506

0.860345

6

9.79481 × 10−7

8.04563 × 10−7

4.88356 × 10−4

0.825431

0.778483

0.860329

7

1.74918 × 10−7

1.43680 × 10−7

4.88377 × 10−4

0.825428

0.778479

0.860327

8

3.12382 × 10−8

2.56594 × 10−8

4.88381 × 10−4

0.825427

0.778478

0.860326

Example 4.2

Consider the matrix equation
$$ X+A_{1}^{*}X^{-7}A_{1}+A_{2}^{*}X^{-2}A_{2}+A_{3}^{*}X^{-11}A_{3}+A_{4}^{*}X^{-4}A_{4}=I, $$
(4.2)
where \(A_{1}\), \(A_{2}\), \(A_{3}\), and \(A_{4} \) are given by
$$\begin{aligned}& A_{1}=\left ( \begin{array}{@{}c@{\quad}c@{\quad}c@{\quad}c@{}} 0.023 & 0.015 & 0.014 & 0.001 \\ 0.014 & 0.011 & 0.029 & 0.045 \\ 0.01 & -0.043 &0.071 & 0.015 \\ 0.011 & -0.043 & 0.071 & 0.015 \end{array} \right ), \\& A_{2}=\left ( \begin{array}{@{}c@{\quad}c@{\quad}c@{\quad}c@{}} 0.019 & 0.014 &0.085 & -0.01 \\ 0.034 & 0.062 & 0.035 & 0.07 \\ 0.029 &0.051 &0.04 & -0.001 \\ 0.026 & 0.01 & -0.043 & 0.015 \end{array} \right ), \\& A_{3}=\left ( \begin{array}{@{}c@{\quad}c@{\quad}c@{\quad}c@{}} 0.002 & 0.014 & 0.03 &0.025 \\ 0.034 & 0.087 &0.015 &0.067 \\ -0.022& 0.012 &-0.02 &0.044 \\ -0.041 & -0.034 & 0.071 &0.015 \end{array} \right ), \\& A_{4}=\left ( \begin{array}{@{}c@{\quad}c@{\quad}c@{\quad}c@{}} 0.017 & 0.045 & -0.002 &0.025 \\ -0.003 &0.034 &0.087 &0.012 \\ 0.04 & 0.027 &-0.02 & 0.044 \\ 0.051 & -0.083 &0.019 & 0.001 \end{array} \right ). \end{aligned}$$
We use Algorithm 2.1 to solve (4.2). After 29 iterations, we get the positive definite solution
$$X\approx X_{29}=\left ( \begin{array}{@{}c@{\quad}c@{\quad}c@{\quad}c@{}} 0.985316 & -0.00602454 &-0.00452105 &-0.00717771 \\ -0.00602454 & 0.963749 &0.00114964 &-0.0154917 \\ -0.00452105 & 0.00114964 &0.951857 & -0.0113565 \\ -0.00717771 & -0.0154917 &-0.0113565 &0.97492 \end{array} \right ) $$
and \(R_{1}(X_{29})= 1.11022\times10^{-16}\), \(\lambda_{\mathrm{min}}(Y_{29})= 0.00144519\), \(\lambda_{\mathrm{min}}(Z_{1,29})=0.649354\), \(\lambda_{\mathrm{min}}(Z_{2,29})=0.878163\), \(\lambda_{\mathrm{min}}(Z_{3,29})= 0.517277\), \(\lambda_{\mathrm{min}}(Z_{4,29})= 0.785219\).
The other results are listed in Table 2.
Table 2

Error analysis for Example 4.2

s

\(\boldsymbol{\epsilon(X_{s})}\)

\(\boldsymbol{R_{1}(X_{s})}\)

\(\boldsymbol{\lambda}_{\mathbf{min}}\boldsymbol{(Y_{s})}\)

\(\boldsymbol{\lambda} _{\mathbf{min}}\boldsymbol{(Z_{1,s})}\)

\(\boldsymbol{\lambda}_{\mathbf{min}}\boldsymbol{(Z_{2,s})}\)

\(\boldsymbol{\lambda}_{\mathbf{min}}\boldsymbol{(Z_{3,s})}\)

\(\boldsymbol{\lambda}_{\mathbf{min}}\boldsymbol{(Z_{4,s})}\)

0

4.81432 × 10−2

3.79180 × 10−2

−4.33175 × 10−2

0.984520

0.982228

0.984476

0.987805

1

1.02252 × 10−2

7.49107 × 10−3

−1.78032 × 10−18

0.719649

0.902084

0.604645

0.831201

2

2.73413 × 10−3

2.08835 × 10−3

1.05313 × 10−3

0.664911

0.883407

0.535912

0.795502

3

6.45772 × 10−4

5.05650 × 10−4

1.33072 × 10−3

0.652517

0.879228

0.521034

0.787317

4

1.40122 × 10−4

1.08374 × 10−4

1.41687 × 10−3

0.650050

0.878406

0.518114

0.785684

5

3.17484 × 10−5

2.37848 × 10−5

1.43915 × 10−3

0.649533

0.878227

0.517495

0.785339

6

7.96360 × 10−6

5.93391 × 10−6

1.44386 × 10−3

0.649401

0.878180

0.517334

0.785250

7

2.02969 × 10−6

1.53853 × 10−6

1.44486 × 10−3

0.649366

0.878167

0.517291

0.785226

8

4.91154 × 10−7

3.76520 × 10−7

1.44510 × 10−3

0.649357

0.878164

0.517280

0.785220

9

1.14634 × 10−7

8.75007 × 10−8

1.44517 × 10−3

0.649355

0.878164

0.517278

0.785219

10

2.71333 × 10−8

2.05092 × 10−8

1.44518 × 10−3

0.649354

0.878164

0.517277

0.785219

Example 4.3

Consider the matrix equation
$$ X+A_{1}^{*}X^{-2}A_{1}+A_{2}^{*}X^{-2}A_{2}=I, $$
(4.3)
where \(A_{1}\) and \(A_{2}\) are given as in Example 3.1 from [5]:
$$\begin{aligned}& A_{1}=\left ( \begin{array}{@{}c@{\quad}c@{\quad}c@{}} 0.010 & -0.150 & -0.259 \\ 0.015 & 0.212 & -0.064 \\ 0.025 & -0.069 & 0.138 \end{array} \right ), \\& A_{2}=\left ( \begin{array}{@{}c@{\quad}c@{\quad}c@{}} 0.160 &-0.025 & 0.020 \\ -0.025 & -0.288 & -0.060 \\ 0.004 & -0.016 & -0.120 \end{array} \right ). \end{aligned}$$
We use Algorithm 2.1 to solve (4.3). After 78 iterations, we get the positive definite solution
$$X\approx X_{78}=\left ( \begin{array}{@{}c@{\quad}c@{\quad}c@{}} 0.970376 & -0.0101782 & -0.00533509 \\ -0.0101782 & 0.733948 & -0.0493223 \\ -0.00533509 & -0.0493223 & 0.869915 \end{array} \right ) $$
and \(R_{1}(X_{78})= 1.11022\times10^{-16}\), \(\lambda_{\mathrm{min}}(Y_{78})= 0.001965\), \(\lambda_{\mathrm{min}}(Z_{1,78})=0.479668\), \(\lambda_{\mathrm{min}}(Z_{2,78})=0.427158\).
The other results are listed in Table 3.
Table 3

Error analysis for Example 4.3

s

\(\boldsymbol{\epsilon(X_{s})}\)

\(\boldsymbol{R_{1}(X_{s})}\)

\(\boldsymbol{\lambda}_{\mathbf{min}}\boldsymbol{(Y_{s})}\)

\(\boldsymbol{\lambda}_{\mathbf{min}}\boldsymbol{(Z_{1,s})}\)

\(\boldsymbol{\lambda}_{\mathbf{min}}\boldsymbol{(Z_{2,s})}\)

0

2.66052 × 10−1

1.56030 × 10−1

−1.74299 × 10−1

0.900634

0.910937

4

1.95763 × 10−2

7.54210 × 10−3

1.94201 × 10−3

0.508120

0.455158

8

2.99945 × 10−3

1.09180 × 10−3

1.96189 × 10−3

0.483981

0.431403

12

4.96010 × 10−4

1.78894 × 10−4

1.96449 × 10−3

0.480380

0.427859

16

8.30170 × 10−5

2.98954 × 10−5

1.96492 × 10−3

0.479787

0.427276

20

1.39223 × 10−5

5.01230 × 10−6

1.96499 × 10−3

0.479688

0.427178

24

2.33562 × 10−6

8.40831 × 10−7

1.96500 × 10−3

0.479671

0.427162

28

3.91847 × 10−7

1.41065 × 10−7

1.96500 × 10−3

0.479669

0.427159

32

6.57408 × 10−8

2.36667 × 10−8

1.96500 × 10−3

0.479668

0.427158

Example 4.4

Consider the matrix equation
$$ X-A_{1}^{*}X^{-2}A_{1}-A_{2}^{*}X^{-6}A_{2}-A_{3}^{*}X^{-10}A_{3}-A_{4}^{*}X^{-3}A_{4}=I, $$
(4.4)
where \(A_{1}\), \(A_{2}\), \(A_{3}\), and \(A_{4}\) are given by
$$\begin{aligned}& A_{1}=\left ( \begin{array}{@{}c@{\quad}c@{\quad}c@{}} 0.1 & -0.1 & 0.4 \\ -0.1 & 0.2 & 0.5 \\ 0.4 & 0.2 & 0.1 \end{array} \right ), \qquad A_{2}=\left ( \begin{array}{@{}c@{\quad}c@{\quad}c@{}} 0.2 & -0.4 & 0.5 \\ 0.9 & 0.2 & 0 \\ 0.1 & -0.4 &0.3 \end{array} \right ), \\& A_{3}=\left ( \begin{array}{@{}c@{\quad}c@{\quad}c@{}} -0.2 & 0.2 & 0.1 \\ -0.8 & 0.1 & 0.5 \\ 0.8 & -0.2 &-0.4 \end{array} \right ), \qquad A_{4}=\left ( \begin{array}{@{}c@{\quad}c@{\quad}c@{}} 0 & 0.3 & 0.1 \\ 0.5 & 0.3 & 0.5 \\ 0.2 & -0.2 & -0.7 \end{array} \right ). \end{aligned}$$
We use Algorithm 3.1 to solve (4.4). After 133 iterations, we get the positive definite solution
$$X\approx X_{133}=\left ( \begin{array}{@{}c@{\quad}c@{\quad}c@{}} 1.85505 & 0.204339 & -0.139713 \\ 0.204339 & 1.21553 & 0.144921 \\ -0.139713 & 0.144921 & 1.6854 \end{array} \right ) $$
and \(R_{2}(X_{133})= 2.22045\times10^{-16}\), \(\lambda _{\mathrm{min}}(I+\sum_{i=1}^{4}A_{i}^{*}A_{i}-X_{133})= 0.490411\), \(\lambda_{\mathrm{min}}(X_{133}-I)=0.100459\).
The other results are listed in Table 4.
Table 4

Error analysis for Example 4.4

s

\(\boldsymbol{\epsilon(X_{s})}\)

\(\boldsymbol{R_{2}(X_{s})}\)

\(\boldsymbol{\lambda}_{\mathbf{min}}\boldsymbol{(I+\sum_{i=1}^{4}A_{i}^{*}A_{i}-X_{s})}\)

\(\boldsymbol{\lambda}_{\mathbf{min}}\boldsymbol{(X_{s}-I)}\)

0

8.55046 × 10−1

2.65

0.715946

0

10

1.83959 × 10−1

3.26477 × 10−1

0.544118

0.076489

20

1.06950 × 10−2

1.87085 × 10−2

0.493691

0.098958

30

5.94976 × 10−4

1.04066 × 10−3

0.490594

0.100375

40

3.30916 × 10−5

5.78794 × 10−5

0.490422

0.100455

50

1.84048 × 10−6

3.21913 × 10−6

0.490412

0.100459

60

1.02364 × 10−7

1.79042 × 10−7

0.490411

0.100459

70

5.69327 × 10−9

9.95792 × 10−9

0.490411

0.100459

Example 4.5

Consider the matrix equation
$$ X-A_{1}^{*}X^{-5}A_{1}-A_{2}^{*}X^{-9}A_{2}-A_{3}^{*}X^{-14}A_{3}=I, $$
(4.5)
where \(A_{1}\), \(A_{2}\), and \(A_{3} \) are given by
$$\begin{aligned}& A_{1}=\left ( \begin{array}{@{}c@{\quad}c@{\quad}c@{\quad}c@{}} 0.01 & 0 & -0.01 & 0.04 \\ 0.11 & 0 & 0.12 & 0.05 \\ 0.06 & 0.05 & 0.12 &0.01 \\ 0.04 & -0.09 & 0.02 & 0.03 \end{array} \right ), \qquad A_{2}=\left ( \begin{array}{@{}c@{\quad}c@{\quad}c@{\quad}c@{}} 0 & -0.14 &0.05 &0.03 \\ 0.03 & 0.12 & 0 &0.01 \\ 0.01 & -0.04 & 0.09 &0.03 \\ 0.05 & 0.03 & 0 &0.32 \end{array} \right ), \\& A_{3}=\left ( \begin{array}{@{}c@{\quad}c@{\quad}c@{\quad}c@{}} -0.05 & 0.01 & 0.29 & 0.01 \\ -0.52 & 0 & 0.11 & -0.05 \\ 0 & -0.02 & 0 & -0.14 \\ 0.04 & -0.04 & 0.37 & 0.01 \end{array} \right ). \end{aligned}$$
We use Algorithm 3.1 to solve (4.5). After 78 iterations, we get the positive definite solution
$$X\approx X_{78}=\left ( \begin{array}{@{}c@{\quad}c@{\quad}c@{\quad}c@{}} 1.21033 & 0.0022825 & -0.00724383 & 0.036655 \\ 0.0022825 & 1.02578 & -0.00516744 & 0.00742036 \\ -0.00724383 & -0.00516744 & 1.07848 & 0.00429966 \\ 0.036655 & 0.00742036 & 0.00429966 & 1.069 \end{array} \right ) $$
and \(R_{2}(X_{78})= 2.22045\times10^{-16}\), \(\lambda _{\mathrm{min}}(I+\sum_{i=1}^{3}A_{i}^{*}A_{i}-X_{78})= 0.0223096\), \(\lambda_{\mathrm{min}}(X_{78}-I)=0.0237773\).
The other results are listed in Table 5.
Table 5

Error analysis for Example 4.5

s

\(\boldsymbol{\epsilon(X_{s})}\)

\(\boldsymbol{R_{2}(X_{s})}\)

\(\boldsymbol{\lambda}_{\mathbf{min}}\boldsymbol{(I+\sum_{i=1}^{3}A_{i}^{*}A_{i}-X_{s})}\)

\(\boldsymbol{\lambda} _{\mathbf{min}}\boldsymbol{(X_{s}-I)}\)

0

2.10332 × 10−1

2.95400 × 10−1

0.047059

0

3

3.11244 × 10−2

4.52684 × 10−2

0.017503

0.028153

6

5.17330 × 10−3

8.36352 × 10−3

0.023078

0.023109

9

1.10730 × 10−3

1.75717 × 10−3

0.022155

0.023907

12

2.28175 × 10−4

3.63481 × 10−4

0.022341

0.023751

15

4.73953 × 10−5

7.54424 × 10−5

0.022303

0.023783

18

9.83002 × 10−6

1.56497 × 10−5

0.022311

0.023776

21

2.03950 × 10−6

3.24685 × 10−6

0.022309

0.023778

24

4.23122 × 10−7

6.73607 × 10−7

0.022310

0.023777

27

8.77834 × 10−8

1.39750 × 10−7

0.022310

0.023777

30

1.82120 × 10−8

2.89934 × 10−8

0.022310

0.023777

Example 4.6

Consider the matrix equation
$$ X-A_{1}^{*}X^{-2}A_{1}-A_{2}^{*}X^{-2}A_{2}=I, $$
(4.6)
where \(A_{1}\) and \(A_{2} \) are given as in Example 4.1 from [6]:
$$\begin{aligned}& A_{1}=\left ( \begin{array}{@{}c@{\quad}c@{\quad}c@{}} 0.3060 & 0.6894 & 0.6093 \\ 0.2514 & 0.4285 &0.7642 \\ 0.0222 & 0.0987 & 0.8519 \end{array} \right ), \\& A_{2}=\left ( \begin{array}{@{}c@{\quad}c@{\quad}c@{}} 0.9529 & 0.6450 &0.4801 \\ 0.4410 & 0.1993 & 0.9823 \\ 0.9712 & 0.0052 & 0.9200 \end{array} \right ). \end{aligned}$$
We use Algorithm 3.1 to solve (4.6). After 792 iterations, we get the positive definite solution
$$X\approx X_{792}=\left ( \begin{array}{@{}c@{\quad}c@{\quad}c@{}} 1.49147 & 0.310408 & 0.378343 \\ 0.310408 & 1.44441 & 0.208234 \\ 0.378343 & 0.208234 & 1.8949 \end{array} \right ) $$
and \(R_{2}(X_{792})= 7.99361\times10^{-15}\), \(\lambda _{\mathrm{min}}(I+\sum_{i=1}^{2}A_{i}^{*}A_{i}-X_{792})= 0.183803\), \(\lambda_{\mathrm{min}}(X_{792}-I)=0.134276\).
The other results are listed in Table 6.
Table 6

Error analysis for Example 4.6

s

\(\boldsymbol{\epsilon(X_{s})}\)

\(\boldsymbol{R_{2}(X_{s})}\)

\(\boldsymbol{\lambda}_{\mathbf{min}}\boldsymbol{(I+\sum_{i=1}^{2}A_{i}^{*}A_{i}-X_{s})}\)

\(\boldsymbol{\lambda} _{\mathbf{min}}\boldsymbol{(X_{s}-I)}\)

0

8.94901 × 10−1

3.72279

0.501528

0

40

8.72900 × 10−2

1.79569 × 10−1

0.195273

0.120468

80

1.70413 × 10−2

3.36946 × 10−2

0.185914

0.131596

120

3.26922 × 10−3

6.41673 × 10−3

0.184203

0.133762

160

6.23654 × 10−4

1.22239 × 10−3

0.183879

0.134178

200

1.18834 × 10−4

2.32858 × 10−4

0.183817

0.134257

240

2.26383 × 10−5

4.43579 × 10−5

0.183805

0.134272

280

4.31248 × 10−6

8.44986 × 10−6

0.183803

0.134275

320

8.21498 × 10−7

1.60964 × 10−6

0.183803

0.134276

360

1.56490 × 10−7

3.06625 × 10−7

0.183803

0.134276

400

2.98101 × 10−8

5.84098 × 10−8

0.183803

0.134276

5 Conclusion

In this paper, we investigate the nonlinear matrix equations \(X\pm\sum_{i=1}^{m}A_{i}^{*}X^{-n_{i}}A_{i}=I\), where \(n_{i}\), \(i=1,2,\ldots,m\), are positive integers. Necessary and sufficient conditions for the existence of positive definite solutions are derived. Iterative algorithms are proposed to compute the positive definite solutions of these equations. Moreover, some numerical examples are given to illustrate the effectiveness and rapidly convergence rate (small run time) of the proposed iterative algorithms (see values of \(\epsilon(X_{s})\), \(R_{1}(X_{s})\), and \(R_{2}(X_{s})\)). Also, the values of \(\lambda _{\mathrm{min}}\) show that the solutions of the matrix equations satisfy the necessary conditions.

Declarations

Acknowledgements

The author is grateful to the editor and the reviewer for important comments and suggestions, which improved the quality of the paper.

Open Access This article is distributed under the terms of the Creative Commons Attribution 4.0 International License (http://creativecommons.org/licenses/by/4.0/), which permits unrestricted use, distribution, and reproduction in any medium, provided you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons license, and indicate if changes were made.

Authors’ Affiliations

(1)
Faculty of Science and Arts, Qassim University

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© Al-Dubiban; licensee Springer. 2015