Investigation of the existence and uniqueness of extremal and positive definite solutions of nonlinear matrix equations
- Abdel-Shakoor M Sarhan^{1} and
- Naglaa M El-Shazly^{1}Email author
https://doi.org/10.1186/s13660-016-1083-3
© Sarhan and El-Shazly 2016
Received: 28 December 2015
Accepted: 9 May 2016
Published: 31 May 2016
Abstract
We consider two nonlinear matrix equations \(X^{r} \pm \sum_{i = 1}^{m} A_{i}^{*}X^{\delta_{i}}A_{i} = I\), where \(- 1 < \delta_{i} < 0\), and r, m are positive integers. For the first equation (plus case), we prove the existence of positive definite solutions and extremal solutions. Two algorithms and proofs of their convergence to the extremal positive definite solutions are constructed. For the second equation (negative case), we prove the existence and the uniqueness of a positive definite solution. Moreover, the algorithm given in (Duan et al. in Linear Algebra Appl. 429:110-121, 2008) (actually, in (Shi et al. in Linear Multilinear Algebra 52:1-15, 2004)) for \(r = 1\) is proved to be valid for any r. Numerical examples are given to illustrate the performance and effectiveness of all the constructed algorithms. In Appendix, we analyze the ordering on the positive cone \(\overline{P(n)}\).
Keywords
1 Introduction
Dehghan and Hajarian [18] constructed an iterative method to solve the general coupled matrix equations \(\sum_{j = 1}^{p} A_{ij}X_{j}B_{ij} = M_{i}\), \(i = 1,2, \ldots,p\) (including the generalized (coupled) Lyapunov and Sylvester matrix equations as particular cases) over generalized bisymmetric matrix group \((X_{1},X_{2}, \ldots,X_{p})\) by extending the idea of conjugate gradient (CG) method. They determined the solvability of the general coupled matrix equations over generalized bisymmetric matrix group in the absence of roundoff errors. In addition, they obtained the optimal approximation generalized bisymmetric solution group to a given matrix group $({\stackrel{\u2322}{X}}_{1},{\stackrel{\u2322}{X}}_{2},\dots ,{\stackrel{\u2322}{X}}_{p})$ in Frobenius norm by finding the least Frobenius norm of the generalized bisymmetric solution group of new general coupled matrix equations.
Hajarian [19] derived a simple and efficient matrix algorithm to solve the general coupled matrix equations \(\sum_{j = 1}^{p} A_{ij}X_{j}B_{ij} = C_{i}\), \(i = 1,2, \ldots,p\) (including several linear matrix equations as particular cases) based on the conjugate gradients squared (CGS) method.
Hajarian [20] developed the conjugate gradient squared (CGS) and biconjugate gradient stabilized (Bi-CGSTAB) methods for obtaining matrix iterative methods for solving the Sylvester-transpose matrix equation \(\sum_{i = 1}^{k} (A_{i}XB_{i} + C_{i}X^{T}D_{i}) = E\) and the periodic Sylvester matrix equation $({\stackrel{\u2322}{A}}_{j}{\stackrel{\u2322}{X}}_{j}{\stackrel{\u2322}{B}}_{j}+{\stackrel{\u2322}{C}}_{j}{\stackrel{\u2322}{X}}_{j+1}{\stackrel{\u2322}{D}}_{j})={\stackrel{\u2322}{E}}_{j}$ for \(j = 1,2, \ldots,\lambda\).
- (a1)
The proof of the uniqueness of a positive definite solution of Equation (1.3).
- (a2)
An algorithm for obtaining that solution with an original proof of convergence.
- (b1)
- (b2)
The same algorithm given by [2] is used with the same slightly changed proof.
INVERSE-FREE iterative Algorithm 4.1
α | k | ||
---|---|---|---|
tol = 10^{−4} | tol = 10^{−6} | tol = 10^{−8} | |
1.2 | 7 | 14 | 21 |
1.4 | 8 | 15 | 22 |
1.6 | 9 | 16 | 23 |
1.8 | 10 | 17 | 24 |
2 | 10 | 17 | 24 |
INVERSE-FREE iterative Algorithm 4.3
α | k | ||
---|---|---|---|
tol = 10^{−4} | tol = 10^{−6} | tol = 10^{−8} | |
0.5 | 12 | 19 | 26 |
0.7 | 6 | 13 | 20 |
0.9 | 3 | 10 | 17 |
2 Preliminaries
- 1.
For \(A,B \in C^{n \times n}\), we write \(A > 0\) (≥0) if the matrix A is Hermitian positive definite (HPD) (semidefinite). If \(A - B > 0\) (\(A - B \ge 0\)), then we write \(A > B\) (\(A \ge B\)).
- 2.
If a Hermitian positive definite matrix X satisfies \(A \le X \le B\), then we write \(X \in [A,B]\).
- 3.
By a solution we mean a Hermitian positive definite solution.
- 4.
If Equation (1.1) has the maximal solution \(X_{L}\) (minimal solution \(X_{S}\)), then for any solution X, \(X_{S} \le X \le X_{L}\).
- 5.
\(P(n)\) denotes the set of all \(n \times n\) positive definite matrices.
- 6.Let E be a real Banach space. A nonempty convex closed set \(P \subset E\) is called a cone if:
- (i)
\(x \in P\), \(\lambda \ge 0\) implies \(\lambda x \in P\).
- (ii)
\(x \in P\), \(- x \in P\) implies \(x = \theta\), where θ denotes the zero element.
- (i)
In this paper, we consider P to be the cone of \(n \times n\) positive semidefinite matrices, denoted \(\overline{P(n)}\); its interior is the set of \(n \times n\) positive definite matrices \(P(n)\).
Definition 2.1
([1])
A cone \(P \subset E\) is said to be normal if there exists a constant \(M > 0\) such that \(\theta \le x \le y\) implies \(\Vert x \Vert \le M\Vert y \Vert \).
Definition 2.2
([1])
Let P be a solid cone of a real Banach space E, and \(\Gamma:P^{o} \to P^{o}\). Let \(0 \le a < 1\). Then Γ is said to be a-concave if \(\Gamma (tx) \ge t^{a}\Gamma (x)\) \(\forall x \in P^{o}\), \(0 < t < 1\).
Similarly, Γ is said to be \(( - a)\)-convex if \(\Gamma (tx) \le t^{ - a}\Gamma (x)\) \(\forall x \in P^{o}\), \(0 < t < 1\).
Lemma 2.3
([1])
Let P be a normal cone in a real Banach space E, and let \(\Gamma:P^{o} \to P^{o}\) be a-concave and increasing (or \(( - a)\)-convex and decreasing) for an \(a \in [0,1)\). Then Γ has exactly one fixed point x in \(P^{o}\).
Lemma 2.4
([4])
If \(A \ge B > 0\) (or \(A > B > 0\)), then \(A^{\gamma} \ge B^{\gamma} > 0\) (or \(A^{\gamma} > B^{\gamma} > 0\)) for all \(\gamma \in (0,1]\), and \(B^{\gamma} \ge A^{\gamma} > 0\) (or \(B^{\gamma} > A^{\gamma} > 0\)) for all \(\gamma \in [ - 1,0)\).
Definition 2.5
([22])
A function f is said to be matrix monotone of order n if it is monotone with respect to this order on \(n \times n\) Hermitian matrices, that is, if \(A \le B\) implies \(f(A) \le f(B)\). If f is matrix monotone of order n for all n, then we say that f is matrix monotone or operator monotone.
Theorem 2.6
([22])
Every operator monotone function f on an interval I is continuously differentiable.
Definition 2.7
([1])
Let \(D \subset E\). An operator \(f:D \to E\) is said to be an increasing operator if \(y_{1} \ge y_{2}\) implies \(f(y_{1}) \ge f(y_{2})\), where \(y_{1},y_{2} \in D\). Similarly, f is said to be a decreasing operator if \(y_{1} \ge y_{2}\) implies \(f(y_{1}) \le f(y_{2})\), where \(y_{1},y_{2} \in D\).
Theorem 2.8
(Brouwer’s Fixed Point, [23])
Every continuous map of a closed bounded convex set in \(R^{n}\) into itself has a fixed point.
3 On the existence of positive definite solutions of \(X^{r}+\sum_{i=1}^{m}A_{i}^{*}X^{\delta_{i}}A_{i}=I\)
Theorem 3.1
The mapping F defined by (3.1) is operator monotone.
Proof
Suppose \(X_{1} \ge X_{2} > 0\). Then, \(F(X_{1}) = (I - \sum_{i = 1}^{m} A_{i}^{*}X_{1}^{\delta_{i}}A_{i} )^{\frac{1}{r}}\) and \(F(X_{2}) = (I - \sum_{i = 1}^{m} A_{i}^{*} X_{2}^{\delta_{i}}A_{i} )^{\frac{1}{r}}\).
Since \(X_{1} \ge X_{2}\), we have \(X_{1}^{\delta_{i}} \le X_{2}^{\delta_{i}}\) for all \(i = 1,2,\ldots,m\).
Then \(A_{i}^{*}X_{1}^{\delta_{i}}A_{i} \le A_{i}^{*}X_{2}^{\delta_{i}}A_{i}\) and \(\sum_{i = 1}^{m} A_{i}^{*}X_{1}^{\delta_{i}}A_{i} \le \sum_{i = 1}^{m} A_{i}^{*}X_{2}^{\delta_{i}}A_{i}\).
Therefore, \(I - \sum_{i = 1}^{m} A_{i}^{*}X_{1}^{\delta_{i}}A_{i} \ge I - \sum_{i = 1}^{m} A_{i}^{*}X_{2}^{\delta_{i}}A_{i}\), and since r is a positive integer, \(0 < \frac{1}{r} \le 1\).
Hence, \((I - \sum_{i = 1}^{m} A_{i}^{*}X_{1}^{\delta_{i}}A_{i} )^{\frac{1}{r}} \ge (I - \sum_{i = 1}^{m} A_{i}^{*}X_{2}^{\delta_{i}}A_{i} )^{\frac{1}{r}}\), that is, \(F(X_{1}) \ge F(X_{2})\). Thus, \(F(X)\) is operator monotone. □
The following theorem proves the existence of positive definite solutions for Equation (1.1), based on the Brouwer fixed point theorem.
Theorem 3.2
If a real number \(\beta < 1\) satisfies \((1 - \beta^{r})I \ge \sum_{i = 1}^{m} \beta^{\delta_{i}}A_{i}^{*}A_{i}\), then Equation (1.1) has positive definite solutions.
Proof
From (3.2) and (3.3) we get that \(F(X) \in D_{1}\); therefore, \(F:D_{1} \to D_{1}\). F is continuous since it is operator monotone. Therefore, F has a fixed point in \(D_{1}\), which is a solution of Equation (1.1).
The following remark, in addition to Examples 5.1 and 5.2 in Section 5, assures the validity of this theorem. □
Remark
Let us consider the simple case \(r = m = n = 1\), \(\delta = - \frac{1}{2}\). It is clear that β does not exist for \(a^{2} > \frac{2\sqrt{3}}{9}\) (where \(a^{2}\) is \(A^{*}A\) in this simple case) and also that no solution exists. For \(a^{2} \le \frac{2\sqrt{3}}{9}\), there exist β and solutions (i.e., the theorem holds). For instance, for \(a^{2} = \frac{2\sqrt{3}}{9}\), there exist \(\beta = \frac{1}{3}\) and the solution, namely \(x = \frac{1}{3}\).
Theorem 3.3
The mapping F has the maximal and the minimal elements in \(D_{1} = [\beta I,(I - \sum_{i = 1}^{m} A_{i}^{*}A_{i})^{\frac{1}{r}} ]\), where β is given in Theorem 3.2.
Proof
By Theorems 2.6 and 3.1 the mapping F is continuous and bounded above since \(F(X) < I\). Let \(\sup_{X \in D_{1}}F(X) = Y\). So, there exists an X̂ in \(D_{1}\) satisfying \(Y - \varepsilon I < F(\hat{X}) \le Y\). We can choose a sequence \(\{ X_{n} \}\) in \(D_{1}\) satisfying \(Y - (\frac{1}{n})I < F(X_{n}) \le Y\). Since \(D_{1}\) is compact, the sequence \(\{ X_{n} \}\) has a subsequence \(\{ X_{n_{k}} \}\) convergent to \(\overline{X} \in D_{1}\). So, \(Y - (\frac{1}{n})I < F(X_{n_{k}}) \le Y\). Taking the limit as \(n \to \infty\), by the continuity of F we get \(\lim_{n \to \infty} F(X_{n_{k}}) = F(\overline{X}) = Y = \max \{ F(X):X \in D_{1} \}\). Hence, F has a maximal element \(\overline{X} \in D_{1}\).
Similarly, we can prove that F has a minimal element in \(D_{1}\), noting that F is bounded below by the zero matrix. □
4 Two algorithms for obtaining extremal positive definite solution of \(X^{r}+\sum_{i=1}^{m}A_{i}^{*}X^{\delta_{i}}A_{i}=I\)
In this section, we present two algorithms for obtaining the extremal positive definite solutions of Equation (1.1). The main idea of the algorithms is to avoid computing the inverses of matrices.
Algorithm 4.1
(INVERSE-FREE Algorithm)
Theorem 4.2
Suppose that Equation (1.1) has a positive definite solution. Then the iterative Algorithm 4.1 generates subsequences \(\{ X_{2k} \}\) and \(\{ X_{2k + 1} \}\) that are decreasing and converge to the maximal solution \(X_{L}\).
Proof
Suppose that Equation (1.1) has a solution. We first prove that the subsequences \(\{ X_{2k} \}\) and \(\{ X_{2k + 1} \}\) are decreasing and the subsequences \(\{ Y_{2k} \}\) and \(\{ Y_{2k + 1} \}\) are increasing. Consider the sequence of matrices generated by (4.1).
So we get \(X_{1} < X_{0}\). Also, \(Y_{1} = Y_{0}[2I - X_{0}Y_{0}^{ - \frac{1}{\delta}} ] = \alpha^{\delta} [2I - \alpha \alpha^{ - 1}I] = \alpha^{\delta} I = Y_{0}\).
From (4.2) we get \(2\alpha^{\delta} I - \alpha^{\delta - 1}(I - \sum_{i = 1}^{m} \alpha^{\delta_{i}}A_{i}^{*}A_{i} )^{\frac{1}{r}} > \alpha^{\delta} I\). Thus, \(Y_{2} > Y_{0}\).
Since \(Y_{2} > Y_{0}\), we have \(Y_{2}^{\frac{\delta_{i}}{\delta}} > Y_{0}^{\frac{\delta_{i}}{\delta}}\), so we get \((I - \sum_{i = 1}^{m} A_{i}^{*} Y_{2}^{\frac{\delta_{i}}{\delta}} A_{i})^{\frac{1}{r}} < (I - \sum_{i = 1}^{m} A_{i}^{*}Y_{0}^{\frac{\delta_{i}}{\delta}} A_{i})^{\frac{1}{r}}\).
Thus, \(X_{3} < X_{1}\). Also, \(Y_{3} = Y_{2}[2I - X_{2}Y_{2}^{ - \frac{1}{\delta}} ]\). Since \(Y_{2} > Y_{0}\) and \(X_{2} < X_{0}\), we have \(Y_{2}^{ - \frac{1}{\delta}} > Y_{0}^{ - \frac{1}{\delta}}\) and \(- X_{2} > - X_{0}\), so that \(- X_{2}Y_{2}^{ - \frac{1}{\delta}} > - X_{0}Y_{0}^{ - \frac{1}{\delta}}\) and \(2I - X_{2}Y_{2}^{ - \frac{1}{\delta}} > 2I - X_{0}Y_{0}^{ - \frac{1}{\delta}}\), and we get \(Y_{2}[2I - X_{2}Y_{2}^{ - \frac{1}{\delta}} ] > Y_{0}[2I - X_{0}Y_{0}^{ - \frac{1}{\delta}} ]\). Thus, \(Y_{3} > Y_{1}\).
Hence, \(\{ X_{2k} \}\) and \(\{ X_{2k + 1} \}\), \(k = 0,1,2,\ldots\) , are decreasing, whereas \(\{ Y_{2k} \}\) and \(\{ Y_{2k + 1} \}\), \(k = 0,1,2,\ldots\) , are increasing.
Now, we show that \(\{ X_{2k} \}\) and \(\{ X_{2k + 1} \}\) are bounded from below by \(X_{L}\) (\(X_{k} > X_{L}\)) and that \(\{ Y_{2k} \}\) and \(\{ Y_{2k + 1} \}\) are bounded from above by \(X_{L}^{\delta}\).
Since \(X_{L}\) is a solution of (1.1), we have that \((I - \sum_{i = 1}^{m} A_{i}^{*}X_{L}^{\delta_{i}}A_{i} )^{\frac{1}{r}} < I\) and \(\alpha I - (I - \sum_{i = 1}^{m} A_{i}^{*}X_{L}^{\delta_{i}}A_{i} )^{\frac{1}{r}} > \alpha I - I = (\alpha - 1)I > 0\), \(\alpha > 1\), that is, \(X_{0} > X_{L}\). Also, \(X_{1} - X_{L} = (I - \sum_{i = 1}^{m} A_{i}^{*}X_{0}^{\delta_{i}}A_{i} )^{\frac{1}{r}} - (I - \sum_{i = 1}^{m} A_{i}^{*}X_{L}^{\delta_{i}}A_{i})^{\frac{1}{r}}\).
Since \(X_{0} > X_{L}\), we have \(X_{0}^{\delta_{i}} < X_{L}^{\delta_{i}}\), and therefore \((I - \sum_{i = 1}^{m} A_{i}^{*}X_{0}^{\delta_{i}}A_{i} )^{\frac{1}{r}} > (I - \sum_{i = 1}^{m} A_{i}^{*}X_{L}^{\delta_{i}}A_{i})^{\frac{1}{r}}\). So we get \(X_{1} > X_{L}\). Also, \(X_{L}^{\delta} - Y_{0} = (I - \sum_{i - 1}^{m} A_{i}^{*}X_{L}^{\delta_{i}}A_{i} )^{\frac{\delta}{r}} - \alpha^{\delta} I\).
From (4.3) and (4.4) we get \((I - \sum_{i = 1}^{m} A_{i}^{*}X_{L}^{\delta_{i}}A_{i} )^{\frac{\delta}{r}} > \alpha^{\delta} I\), so we have \(X_{L}^{\delta} > Y_{0}\) and \(X_{L}^{\delta} > Y_{1}\).
Assume that \(X_{2k} > X_{L}\), \(X_{2k + 1} > X_{L}\) at \(k = t\) that is, \(X_{2t} > X_{L}\), \(X_{2t + 1} > X_{L}\). Also, \(Y_{2t} < X_{L}^{\delta}\) and \(Y_{2t + 1} < X_{L}^{\delta}\).
Since \(Y_{2t + 1} < X_{L}^{\delta}\), we have \(Y_{2t + 1}^{\frac{\delta_{i}}{\delta}} < X_{L}^{\delta_{i}}\) and thus \(\sum_{i = 1}^{m} A_{i}^{*} Y_{2t + 1}^{\frac{\delta_{i}}{\delta}} A_{i} < \sum_{i = 1}^{m} A_{i}^{*}X_{L}^{\delta_{i}} A_{i}\). Therefore, \((I - \sum_{i = 1}^{m} A_{i}^{*} Y_{2t + 1}^{\frac{\delta_{i}}{\delta}} A_{i})^{\frac{1}{r}} > (I - \sum_{i = 1}^{m} A_{i}^{*}X_{L}^{\delta_{i}} A_{i})^{\frac{1}{r}}\).
Hence, \(X_{2t + 2} - X_{L} > 0\), that is, \(X_{2t + 2} > X_{L}\). Also, \(X_{L}^{\delta} - Y_{2t + 2} = X_{L}^{\delta} - Y_{2t + 1}[2I - X_{2t + 1}Y_{2t + 1}^{ - \frac{1}{\delta}} ]\).
Since, \(Y_{2t + 2} < X_{L}^{\delta}\), we have \(Y_{2t + 2}^{\frac{\delta_{i}}{\delta}} < X_{L}^{\delta_{i}}\) and \(\sum_{i = 1}^{m} A_{i}^{*} Y_{2t + 2}^{\frac{\delta_{i}}{\delta}} A_{i} < \sum_{i = 1}^{m} A_{i}^{*}X_{L}^{\delta_{i}} A_{i}\). Therefore, \((I - \sum_{i = 1}^{m} A_{i}^{*} Y_{2t + 2}^{\frac{\delta_{i}}{\delta}} A_{i})^{\frac{1}{r}} > (I - \sum_{i = 1}^{m} A_{i}^{*}X_{L}^{\delta_{i}} A_{i})^{\frac{1}{r}}\).
Hence, \(X_{2t + 3} - X_{L} > 0\), that is, \(X_{2t + 3} > X_{L}\), and thus \(X_{L}^{\delta} - Y_{2t + 3} = X_{L}^{\delta} - Y_{2t + 2}[2I - X_{2t + 2}Y_{2t + 2}^{ - \frac{1}{\delta}} ]\).
Then \(Y_{2t + 2}[2I - X_{2t + 2}Y_{2t + 2}^{ - \frac{1}{\delta}} ] < X_{L}^{\delta}\). Hence, \(Y_{2t + 3} < X_{L}^{\delta}\).
Since \(\{ X_{2k} \}\) and \(\{ X_{2k + 1} \}\) are decreasing and bounded from below by \(X_{L}\) and \(\{ Y_{2k} \}\) and \(\{ Y_{2k + 1} \}\) are increasing and bounded from above by \(X_{L}^{\delta}\), it follows that \(\lim_{k \to \infty} X_{k} = X\) and \(\lim_{k \to \infty} Y_{k} = Y\) exist.
Taking limits in (4.1) gives \(Y = X^{\delta}\) and \(X = (I - \sum_{i = 1}^{m} A_{i}^{*}X^{\delta_{i}} A_{i})^{\frac{1}{r}}\), that is, X is a solution. Hence, \(X = X_{L}\). □
Remark
We have proved that the maximal solution is unique; see the Appendix.
Now, we consider the case \(0 < \alpha < 1\).
Algorithm 4.3
(INVERSE-FREE Algorithm)
Theorem 4.4
Suppose that Equation (1.1) has a positive definite solution such that \(\sum_{i = 1}^{m} \alpha^{\delta_{i}}A_{i}^{*}A_{i} < (1 - \alpha^{r})I\). Then the iterative Algorithm 4.3 generates the subsequences \(\{ X_{2k} \}\) and \(\{ X_{2k + 1} \}\) that are increasing and converge to the minimal solution \(X_{S}\).
Proof
Suppose that Equation (1.1) has a solution. We first prove that the subsequences \(\{ X_{2k} \}\) and \(\{ X_{2k + 1} \}\) are increasing and the subsequences \(\{ Y_{2k} \}\) and \(\{ Y_{2k + 1} \}\) are decreasing. Consider the sequence of matrices generated by (4.5).
So we get \(X_{1} > X_{0}\). Also, \(Y_{1} = Y_{0}[2I - X_{0}Y_{0}^{ - \frac{1}{\delta}} ] = \alpha^{\delta} [2I - \alpha \alpha^{ - 1}I] = \alpha^{\delta} I = Y_{0}\).
From (4.7) we get \(2\alpha^{\delta} I - \alpha^{\delta - 1}(I - \sum_{i = 1}^{m} \alpha^{\delta_{i}}A_{i}^{*}A_{i} )^{\frac{1}{r}} < \alpha^{\delta} I\). Thus, \(Y_{2} < Y_{0}\).
Since \(Y_{2} < Y_{0}\), we have \(Y_{2}^{\frac{\delta_{i}}{\delta}} < Y_{0}^{\frac{\delta_{i}}{\delta}}\), so we get \((I - \sum_{i = 1}^{m} A_{i}^{*} Y_{2}^{\frac{\delta_{i}}{\delta}} A_{i})^{\frac{1}{r}} > (I - \sum_{i = 1}^{m} A_{i}^{*}Y_{0}^{\frac{\delta_{i}}{\delta}} A_{i})^{\frac{1}{r}}\). Thus, \(X_{3} > X_{1}\). Also, \(Y_{3} = Y_{2}[2I - X_{2}Y_{2}^{ - \frac{1}{\delta}} ]\).
Since \(Y_{2} < Y_{0}\) and \(X_{2} > X_{0}\), we have \(Y_{2}^{ - \frac{1}{\delta}} < Y_{0}^{ - \frac{1}{\delta}}\) and \(- X_{2} < - X_{0}\), so that \(- X_{2}Y_{2}^{ - \frac{1}{\delta}} < - X_{0}Y_{0}^{ - \frac{1}{\delta}}\) and \(2I - X_{2}Y_{2}^{ - \frac{1}{\delta}} < 2I - X_{0}Y_{0}^{ - \frac{1}{\delta}}\), and we get \(Y_{2}[2I - X_{2}Y_{2}^{ - \frac{1}{\delta}} ] < Y_{0}[2I - X_{0}Y_{0}^{ - \frac{1}{\delta}} ]\). Thus, \(Y_{3} < Y_{1}\).
Hence \(\{ X_{2k} \}\) and \(\{ X_{2k + 1} \}\), \(k = 0,1,2,\ldots\) , are increasing, whereas \(\{ Y_{2k} \}\) and \(\{ Y_{2k + 1} \}\), \(k = 0,1,2,\ldots\) , are decreasing.
Now, we show that \(\{ X_{2k} \}\) and \(\{ X_{2k + 1} \}\) are bounded from above by \(X_{S}\) (\(X_{S} > X_{k}\)), and \(\{ Y_{2k} \}\) and \(\{ Y_{2k + 1} \}\) are bounded from below by \(X_{S}^{\delta}\).
Since \(X_{S} > X_{0}\), we have \(X_{S}^{\delta_{i}} < X_{0}^{\delta_{i}}\), and therefore \((I - \sum_{i = 1}^{m} A_{i}^{*}X_{S}^{\delta_{i}}A_{i} )^{\frac{1}{r}} > (I - \sum_{i = 1}^{m} A_{i}^{*}X_{0}^{\delta_{i}}A_{i})^{\frac{1}{r}}\). So we get \(X_{S} > X_{1}\). Also, \(Y_{0} - X_{S}^{\delta} = \alpha^{\delta} I - (I - \sum_{i - 1}^{m} A_{i}^{*}X_{S}^{\delta_{i}}A_{i} )^{\frac{\delta}{r}}\).
From (4.8) and (4.9) we get \(\alpha^{\delta} I > (I - \sum_{i = 1}^{m} A_{i}^{*}X_{S}^{\delta_{i}}A_{i} )^{\frac{\delta}{r}}\), and thus \(Y_{0} > X_{S}^{\delta}\) and \(Y_{1} > X_{S}^{\delta}\).
Assume that \(X_{2k} < X_{S}\) and \(X_{2k + 1} < X_{S}\) at \(k = t\), that is, \(X_{2t} < X_{S}\) and \(X_{2t + 1} < X_{S}\). Also, \(Y_{2t} > X_{S}^{\delta}\) and \(Y_{2t + 1} > X_{S}^{\delta}\).
Since \(Y_{2t + 1} > X_{S}^{\delta}\), we have \(Y_{2t + 1}^{\frac{\delta_{i}}{\delta}} > X_{S}^{\delta_{i}}\) and \(\sum_{i = 1}^{m} A_{i}^{*} Y_{2t + 1}^{\frac{\delta_{i}}{\delta}} A_{i} > \sum_{i = 1}^{m} A_{i}^{*}X_{S}^{\delta_{i}} A_{i}\). Therefore, \((I - \sum_{i = 1}^{m} A_{i}^{*} Y_{2t + 1}^{\frac{\delta_{i}}{\delta}} A_{i})^{\frac{1}{r}} < (I - \sum_{i = 1}^{m} A_{i}^{*}X_{S}^{\delta_{i}} A_{i})^{\frac{1}{r}}\).
Since \(Y_{2t + 2} > X_{S}^{\delta}\), we have \(Y_{2t + 2}^{\frac{\delta_{i}}{\delta}} > X_{S}^{\delta_{i}}\) and \(\sum_{i = 1}^{m} A_{i}^{*} Y_{2t + 2}^{\frac{\delta_{i}}{\delta}} A_{i} > \sum_{i = 1}^{m} A_{i}^{*}X_{S}^{\delta_{i}} A_{i}\). Therefore, \((I - \sum_{i = 1}^{m} A_{i}^{*} Y_{2t + 2}^{\frac{\delta_{i}}{\delta}} A_{i})^{\frac{1}{r}} < (I - \sum_{i = 1}^{m} A_{i}^{*}X_{S}^{\delta_{i}} A_{i})^{\frac{1}{r}}\).
Then \(Y_{2t + 2}[2I - X_{2t + 2}Y_{2t + 2}^{ - \frac{1}{\delta}} ] > X_{S}^{\delta}\), and hence \(Y_{2t + 3} > X_{S}^{\delta}\).
Since \(\{ X_{2k} \}\) and \(\{ X_{2k + 1} \}\) are increasing and bounded from above by \(X_{S}\) and \(\{ Y_{2k} \}\) and \(\{ Y_{2k + 1} \}\) are decreasing and bounded from below by \(X_{S}^{\delta}\), it follows that \(\lim_{k \to \infty} X_{k} = X\) and \(\lim_{k \to \infty} Y_{k} = Y\) exist. Taking the limits in (4.5) gives \(Y = X^{\delta}\) and \(X = (I - \sum_{i = 1}^{m} A_{i}^{*}X^{\delta_{i}} A_{i})^{\frac{1}{r}}\), that is, X is a solution of Equation (1.1). Hence, \(X = X_{S}\). □
Remark
We have proved that the minimal solution is unique; see the Appendix.
5 Numerical examples
In this section, we report a variety of numerical examples to illustrate the accuracy and efficiency of the two proposed Algorithms 4.1 and 4.3 to obtain the extremal positive definite solutions of Equation (1.1). The solutions are computed for different matrices \(A_{i}\), \(i = 1,2,\ldots,m\), and different values of α, r, δ, and \(\delta_{i}\), \(i = 1,2,\ldots,m\). All programs are written in MATLAB version 5.3. We denote \(\varepsilon (X_{k}) = \Vert X_{k}^{r} + \sum_{i = 1}^{m} A_{i}^{*}X_{k}^{\delta_{i}}A_{i} - I \Vert _{\infty} = \Vert X_{k + 1} - X_{k} \Vert _{\infty}\) for the stopping criterion, and we use \(\varepsilon (X_{k}) < \mathrm{tol}\) for different chosen tolerances.
Example 5.1
The eigenvalues of \(X_{L}\) are \((0.9556, 0.9984, 0.9695, 0.9969)\).
Note: Theorem 3.2 holds for \(\beta = \frac{1}{3}, \frac{1}{2},\ldots\) , etc.
Example 5.2
The eigenvalues of \(X_{S}\) are \((0.9180, 0.9909, 0.9733, 0.9966)\).
Also, Theorem 3.2 holds for \(\beta = \frac{1}{3}, \frac{1}{2},\ldots\) , etc.
Remark
From Table 1 we see that the number of iterations k increases as the value of α (\(\alpha > 1\)) increases, and from Table 2 we see that the number of iterations k decreases as the value of α (\(0 < \alpha < 1\)) increases. For details, see the Appendix in the end of this paper.
Example 5.3
INVERSE-FREE iterative Algorithm 4.1
α | k | ||
---|---|---|---|
tol = 10^{−6} | tol = 10^{−8} | tol = 10^{−10} | |
1.2 | 8 | 20 | 31 |
1.4 | 10 | 22 | 33 |
1.6 | 11 | 23 | 34 |
1.8 | 12 | 24 | 35 |
2 | 13 | 25 | 36 |
The eigenvalues of \(X_{L}\) are: \((0.99954, 0.99976, 0.99995, 0.99982, 0.99987, 0.99991)\).
Example 5.4
INVERSE-FREE iterative Algorithm 4.3
α | k | |||
---|---|---|---|---|
tol = 10^{−4} | tol = 10^{−6} | tol = 10^{−8} | tol = 10^{−10} | |
0.5 | 13 | 20 | 27 | 33 |
0.7 | 3 | 10 | 17 | 23 |
0.9 | 3 | 9 | 16 | 23 |
The eigenvalues of \(X_{S}\) are \((0.99181, 0.9944, 0.9992, 0.99882, 0.99596, 0.99676)\).
Remarks
- 1.
The obtained results for Examples 5.3 and 5.4 shown in Tables 3 and 4, respectively, indicate that increasing the dimension of the problem does not affect the efficiency of the proposed algorithms.
- 2.
From Tables 1, 2, 3, and 4, it is clear that we obtained a high accuracy for different values of α after a few numbers of steps; see the number of iterations, which indicate that our algorithms have high efficiency.
6 On the existence and the uniqueness of a positive definite solution of \(X^{r}-\sum_{i=1}^{m}A_{i}^{*}X^{\delta_{i}}A_{i}=I\)
Theorem 6.1
If Equation (1.2) has a positive definite solution X, then \(X \in [G^{2}(I),G(I)]\).
Proof
Then \((I + \sum_{i = 1}^{m} A_{i}^{*}A_{i} )^{\frac{\delta_{i}}{r}} \le X^{\delta_{i}} \le I\), and thus \(\sum_{i = 1}^{m} A_{i}^{*}(I + \sum_{i = 1}^{m} A_{i}^{*}A_{i} )^{\frac{\delta_{i}}{r}}A_{i} \le \sum_{i = 1}^{m} A_{i}^{*}X^{\delta_{i}}A_{i} \le \sum_{i = 1}^{m} A_{i}^{*}A_{i}\).
We have \(\sum_{i = 1}^{m} A_{i}^{*}X^{\delta_{i}}A_{i} = X^{r} - I\). Then \(\sum_{i = 1}^{m} A_{i}^{*}(I + \sum_{i = 1}^{m} A_{i}^{*}A_{i} )^{\frac{\delta_{i}}{r}}A_{i} \le X^{r} - I \le \sum_{i = 1}^{m} A_{i}^{*}A_{i}\).
Therefore, \((I + \sum_{i = 1}^{m} A_{i}^{*}(I + \sum_{i = 1}^{m} A_{i}^{*}A_{i} )^{\frac{\delta_{i}}{r}}A_{i})^{\frac{1}{r}} \le X \le (I + \sum_{i = 1}^{m} A_{i}^{*}A_{i} )^{\frac{1}{r}}\), that is, \(G^{2}(I) \le X \le G(I)\). Hence, \(X \in [G^{2}(I),G(I)]\).
We use the Brouwer fixed point theorem to prove the existence of positive definite solutions of Equation (1.2). Since \(P(n)\) is not complete, we consider the subset \(D_{2} = [G^{2}(I),G(I)] \subset P(n)\), which is compact. □
Theorem 6.2
Equation (1.2) has a Hermitian positive definite solution.
Proof
It is obvious that \(D_{2}\) is closed, bounded, and convex. To show that \(G:D_{2} \to D_{2}\), let \(X \in D_{2}\). Then \(G^{2}(I) \le X \le G(I)\), that is, \(X \le (I + \sum_{i = 1}^{m} A_{i}^{*}A_{i} )^{\frac{1}{r}}\), and thus \(X^{\delta_{i}} \ge (I + \sum_{i = 1}^{m} A_{i}^{*}A_{i} )^{\frac{\delta_{i}}{r}}\), \(- 1 < \delta_{i} < 0\). Therefore, \((I + \sum_{i = 1}^{m} A_{i}^{*}X^{\delta_{i}}A_{i})^{\frac{1}{r}} \ge (I + \sum_{i = 1}^{m} A_{i}^{*}(I + \sum_{i = 1}^{m} A_{i}^{*}A_{i} )^{\frac{\delta_{i}}{r}} A_{i} )^{\frac{1}{r}}\).
From (6.5) and (6.6) we get \(G(X) \in D_{2}\), that is, \(G:D_{2} \to D_{2}\).
According to Lemma 6.2.37 in [16], \(\sum_{i = 1}^{m} A_{i}^{*}X^{\delta_{i}}A_{i}\) is continuous on \(D_{2}\). Hence, \(G(X)\) is continuous on \(D_{2}\). By Brouwer’s fixed point theorem, G has a fixed point in \(D_{2}\), which is a solution of Equation (1.2). □
Remark
Applying Lemma 2.3, in [1], it is proved that Equation (1.2) has a unique positive definite solution for \(r = 1\). The following theorem shows that the uniqueness holds for any r, that is, for Equation (1.2).
Theorem 6.3
Equation (1.2) has a unique positive definite solution.
Proof
Remarks
- 1.In [2], the authors considered the nonlinear matrix equation$$ X = \sum_{i = 1}^{m} A_{i}^{T}X^{\delta_{i}}A_{i},\quad \vert \delta_{i} \vert < 1, i = 1,2,\ldots,m. $$(6.7)They proved that the recursively defined matrix sequencewhere \(X_{1},X_{2},\ldots,X_{m}\) are arbitrary initial positive definite matrices, converge to the unique positive definite solution of (6.7).$$ X_{n + m + 1} = \sum_{i = 1}^{m} A_{i}^{T}X_{n + i}^{\delta_{i}}A_{i}, \quad n \ge 0, $$(6.8)In [1], the authors considered the nonlinear matrix equationwhere Q is a known positive definite matrix. They considered the same formula (6.8) as$$ X = Q + \sum_{i = 1}^{m} A_{i}^{*}X^{\delta_{i}}A_{i}, \quad 0 < \vert \delta_{i} \vert < 1, i = 1,2,\ldots,m, $$(6.9)$$ X_{n + m + 1} = Q + \sum_{i = 1}^{m} A_{i}^{*}X_{n + i}^{\delta_{i}}A_{i}, \quad n \ge 0. $$(6.10)
They used the algorithm given in [2] and used the main steps for the proof of convergence with slight changes to suit their problem (6.9).
- 2.
Since Q has no effect on the convergence of the algorithm and since in [2] it is proved that both \(X_{n + m + 1}\) and \(X_{n + i}\) converge to the unique solution X, we see that the proof given in [1] is redundant.
For our Equation (1.2), we use the recursive formulaaccording to the following proposition.$$ X_{n + m + 1} = \Biggl(I + \sum_{i = 1}^{m} A_{i}^{*}X_{n + i}^{\delta_{i}}A_{i} \Biggr)^{\frac{1}{r}} $$(6.11)
Proposition 6.4
The matrix sequence defined by (6.11) converges to the unique positive definite matrix solution X of Equation (1.2) for arbitrary initial positive definite matrices \(X_{1},X_{2},\ldots,X_{m}\), provided that it is valid for \(r = 1\).
Proof
We have that \(\sum_{i = 1}^{m} A_{i}^{*}X_{n + i}^{\delta_{i}}A_{i}\) is continuous and thus \(G_{n}(X) = I + \sum_{i = 1}^{m} A_{i}^{*}X_{n + i}^{\delta_{i}}A_{i}\) is continuous. Define \(F(G_{n}(X)) = (G_{n}(X))^{\frac{1}{r}}\), where r is a positive integer. Then F is also continuous. So, \(\lim_{n \to \infty} F(G_{n}(X)) = F\lim_{n \to \infty} G_{n}(X)\). Taking the limit of (6.11) as \(n \to \infty\) and using (6.9) and (6.10) with \(Q = I\), we obtain \(X = (I + \sum_{i = 1}^{m} A_{i}^{*}X^{\delta_{i}}A_{i} )^{\frac{1}{r}}\). □
7 Numerical examples
Example 7.1
We see that \(X_{9} \in [G^{2}(I),G(I)]\), where G is defined by (6.1).
Example 7.2
We see that \(X_{14} \in [G^{2}(I),G(I)]\), where G is defined by (6.1).
Remarks
- 1.
If we use \(X_{1}\) and \(X_{2}\) from Example 7.2, then we get the same solution \(X = X_{9}\), which proves the uniqueness of the solution of Equation (1.2).
- 2.If we putin Example 7.2, then we get the same solution$$\begin{aligned}& X_{1} = \left ( \textstyle\begin{array}{c@{\quad}c} 0.9355 & 0.4103 \\ 0.9169 & 0.8936 \end{array}\displaystyle \right ),\qquad X_{2} = \left ( \textstyle\begin{array}{c@{\quad}c} 0.3046 & 0.1934 \\ 0.1897 & 0.9822 \end{array}\displaystyle \right ),\quad \mbox{and} \\& X_{3} = \left ( \textstyle\begin{array}{c@{\quad}c} 0.3028 & 0.1509 \\ 0.5417 & 0.6979 \end{array}\displaystyle \right ) \end{aligned}$$which proves the uniqueness of the solution of Equation (1.2).$$X = X_{10} = \left ( \textstyle\begin{array}{c@{\quad}c} 1.0793 & 0.0693 \\ 0.0693 & 1.3536 \end{array}\displaystyle \right ), $$
The above examples show that the recursive formula defined by (6.11) is feasible and effective to compute the unique positive definite solution of Equation (1.2).
8 Conclusion
In this paper we considered two nonlinear matrix equations \(X^{r} \pm \sum_{i = 1}^{m} A_{i}^{*}X^{\delta_{i}}A_{i} = I\). For the first equation (plus case), the proofs of the existence of positive definite solutions beside the extremal solutions are given. Also two algorithms are suggested for computing the extremal solutions. For the second equation (negative case), the existence and uniqueness of a positive definite solution are proved. The algorithm in [1] is adapted for solving this equation. Numerical examples are introduced to illustrate the obtained theoretical results.
Declarations
Acknowledgements
The authors acknowledge the reviewers for reviewing the manuscript.
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
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