Hermitian Positive Definite Solution Research Articles

Solving system of nonlinear matrix equations over Hermitian positive definite matrices

In this article, we first introduce the necessary and sufficient conditions for the existence of the Hermitian positive definite solution group of a system of nonlinear matrix equations Then we propose an inversion-free iteration method to find the Hermitian positive definite solution group of the system of nonlinear matrix equations. Finally, the accuracy and effectiveness of the proposed method in compare to some existing algorithms are demonstrated by various numerical examples.

Elegant Iterative Methods for Solving a Nonlinear Matrix Equation X-A* eX A=I

The nonlinear matrix equation was solved by Gao (2016) via standard fixed point method. In this paper, three more elegant iterative methods are proposed to find the approximate solution of the nonlinear matrix equation namely: Newton’s method; modified fixed point method and a combination of Newton’s method and fixed point method. The convergence of Newton’s method and modified fixed point method are derived. Comparative numerical experimental results indicate that the new developed algorithms have both less computational time and good convergence properties when compared to their respective standard algorithms.
 Keywords: Hermitian positive definite solution; nonlinear matrix equation; modified fixed point method; iterative method

On Hermitian positive definite solutions of a nonlinear matrix equation

In this paper, the Hermitian positive definite solutions of the matrix equation $$X^s +A^* X^{ - t}A = Q$$ , where A is an $$n \times n$$ nonsingular complex matrix, Q is an $$n \times n$$ Hermitian positive definite matrix and $$s, t> 0$$ , are discussed. Some conditions for the existence of Hermitian positive definite solutions of this equation are derived. In addition, two iterative methods to obtaining the maximum or minimum Hermitian positive definite solutions of this equation are proposed. In addition, a necessary and sufficient condition for the existence of these solutions is presented. Theoretical results are illustrated by some numerical examples.

Investigation of positive definite solution of nonlinear matrix equation $$X^{p}=Q +\sum \nolimits _{i=1}^m A_i^*X^{\delta }A_i$$

In this paper, we consider the nonlinear matrix equation $$X^{p}=Q +\sum \nolimits _{i=1}^m A_i^*X^{\delta }A_i,$$ where $$A_i (i=1,2,\ldots ,m)$$ are $$n\times n$$ nonsingular complex matrices, Q is a $$n\times n$$ Hermitian positive definite (HPD) matrix, $$p\ge 1, m\ge 1$$ are positive integers, and $$\delta \in (0,1)$$ . We discuss the solution of this equation via properties of Thompson metric and two fixed point theorems in ordered Banach spaces and estimate the bounds of the HPD solution. Furthermore, perturbation analysis is investigated.

On the nonlinear matrix equation X p = A + ∑ i = 1 m M i ∗ ( B + X − 1 ) − 1 M i

We establish several necessary and sufficient conditions for the existence and uniqueness of Hermitian positive definite (HPD) solutions to the general matrix equation X p = A + ∑ i = 1 m M i ∗ ( B + X − 1 ) − 1 M i , where p, m are positive integers, M i (i = 1, 2, …, m) are n × n nonsingular complex matrices, A and B are HPD matrices, and then give three algorithms to get the unique solution. Moreover, we give two numerical examples to illustrate the effectiveness of the theoretical results and the behavior of the considered methods.

Verified computation for the geometric mean of two matrices

An algorithm for numerically computing an interval matrix containing the geometric mean of two Hermitian positive definite (HPD) matrices is proposed. We consider a special continuous-time algebraic Riccati equation (CARE) where the geometric mean is the unique HPD solution, and compute an interval matrix containing a solution to the equation. We invent a change of variables designed specifically for the special CARE. By the aid of this special change of variables, the proposed algorithm gives smaller radii, and is more successful than previous approaches. Solutions to the equation are not necessarily Hermitian. We thus establish a theory for verifying that the contained solution is Hermitian. Finally, the positive definiteness of the solution is verified. Numerical results show effectiveness, efficiency, and robustness of the algorithm.

Solving two generalized nonlinear matrix equations

In this paper, we consider the numerical solutions of two generalized nonlinear matrix equations. Newton’s method is applied to compute one of the generalized nonlinear matrix equations and a generalized Stein equation is obtained, then we adapt the generalized Smith method to find the maximal Hermitian positive definite solution. Furthermore, we consider the properties of the solution for the generalized nonlinear matrix equation. Newton’s method is also applied to the other generalized nonlinear matrix equation to find the minimal Hermitian positive definite solution. Finally, two numerical examples are presented to illustrate the effectiveness of the theoretical results and the convergence behaviour of the considered methods for two generalized nonlinear matrix equations, respectively.

Solution of a Class of Nonlinear Matrix Equations

In this article, we present several necessary and sufficient conditions for the existence of Hermitian positive definite solutions of nonlinear matrix equations of the form $$X^s+A^*X^{-t}A+B^*X^{-p}B=Q$$ , where $$s,t,p \ge 1$$ , A, B are nonsingular matrices and Q is a Hermitian positive definite matrix. We derive some iterations to compute the solutions followed by some examples. In this context, we also discuss about the maximal and the minimal Hermitian positive definite solution of this particular nonlinear matrix equation.

Two Structure-Preserving-Doubling Like Algorithms to Solve the Positive Definite Solution of the Equation $$X-{A^{\rm{H}}}{\overline{X}}^{-1}A=Q$$

In this paper, we study the nonlinear matrix equation $$X-{A^{\rm{H}}}{\overline{X}}^{-1}A=Q$$ , where $$A,Q \in {{\mathbb {C}}}^{n\times n}$$ , Q is a Hermitian positive definite matrix and $$X \in {{\mathbb {C}}}^{n\times n}$$ is an unknown matrix. We prove that the equation always has a unique Hermitian positive definite solution. We present two structure-preserving-doubling like algorithms to find the Hermitian positive definite solution of the equation, and the convergence theories are established. Finally, we show the effectiveness of the algorithms by numerical experiments.

On the positive definite solution of a class of pair of nonlinear matrix equations

We find some necessary and sufficient conditions for the existence of Hermitian positive definite solution of a pair of nonlinear matrix equations of the form: $$\begin{aligned} X^{s_1}+A^*X^{-t_1}A+B^*Y^{-p_1}B=Q_1\\ Y^{s_2}+A^*Y^{-t_2}A+B^*X^{-p_2}B=Q_2, \end{aligned}$$and provide some algorithms for finding solutions. Finally, we give some numerical examples and study the convergence history of the iterations.

On the Hermitian positive definite solution and Newton's method for a nonlinear matrix equation

ABSTRACT In this paper, necessary and sufficient conditions for the existence of the Hermitian positive definite solution for a nonlinear matrix equation are derived. Then, a sufficient condition for the existence of the unique Hermitian positive definite solution is presented. Further, we propose to use Newton's method to solve this nonlinear matrix equation with some constraints. In addition, we prove the convergence of Newton's method. Finally, we present some numerical examples to illustrate the effectiveness of the derived results.

Inequalities for the eigenvalues of the positive definite solutions of the nonlinear matrix equation

In this paper, the Hermitian positive definite solutions of the matrix equation Xs + A*X-tA = Q are considered, where Q is a Hermitian positive definite matrix, s and t are positive integers. Bounds for the sum of eigenvalues of the solutions to the equation are given. The equivalent conditions for solutions of the equation are obtained. The eigenvalues of the solutions of the equation with the case AQ = QA are investigated.

Verified computation for the Hermitian positive definite solution of the conjugate discrete-time algebraic Riccati equation

A fast iterative algorithm is proposed for numerically computing an interval matrix containing the Hermitian positive definite solution to the conjugate discrete-time algebraic Riccati equation. This algorithm requires only cubic complexity per iteration, and verifies the uniqueness of the contained solution. Numerical results show effectiveness and robustness of the algorithm.

The Investigation on Two Kinds of Nonlinear Matrix Equations

In this paper, we consider two kinds of nonlinear matrix equations $$X+ \sum _{i=1}^{m}B_{i}^*X^{t_{i}}B_{i}=I\;(0<t_{i}<1)$$ and $$X^{s}-\sum _{i=1}^{m}A_{i}^*X^{p_{i}}A_{i}=I\;(p_{i}>1,\;s\ge 1)$$ . By means of the integral representation of matrix functions, properties of Kronecker product and the monotonic p-concave operator fixed point theorem, we derive necessary conditions and sufficient conditions for the existence and uniqueness of the Hermitian positive definite solution for the matrix equations. We also obtain some properties of the Hermitian positive definite solutions, the bounds of the determinant’s sum for $$A_{i}^{*}A_{i}$$ and the spectral radius of $$A_{i}$$ .

Solvability for a Nonlinear Matrix Equation

In this paper, the matrix equation $$X+\sum _{i=1}^{m}A_{i}^*X^{-q_{i}}A_{i}=I$$ with $$0<q_{i}\le 1$$ is investigated. Based on the integral representation of matrix functions and the properties of Kronecker product, we discuss the uniqueness of the Hermitian positive definite (HPD) solution of the above equation. Some properties of the HPD solution are obtained.

On the matrix equation

In this paper we investigate the matrix equation . Some sufficient and necessary conditions for the existence of Hermitian positive definite solutions as well as for the existence of the smallest Hermitian positive definite solution of the considered equation are derived. Two iterative methods for computing the smallest positive definite solution are proposed. Some numerical examples are presented to illustrate the theoretical results.

The Hermitian Positive Definite Solution of the Nonlinear Matrix Equation

Abstract: In this paper, we study the nonlinear matrix equation X s ± ∑ i = 1 m A i T X δ i A i = Q $X^{s}\pm\sum^{m}_{i=1}A^{T}_{i}X^{\delta_{i}}A_{i}=Q$ , where A i ( i = 1 , 2 , … , m ) $A_{i}\;(i=1,2,\ldots,m)$ is n × n $n\times n$ nonsingular real matrix and Q $Q$ is n × n $n\times n$ Hermitian positive definite matrix. It is shown that the equation has an unique Hermitian positive definite solution under some conditions. Iterative algorithms for obtaining the Hermitian positive definite solution of the equation are proposed. Finally, numerical examples are reported to illustrate the effectiveness of algorithms.

The positive definite solution of the nonlinear matrix equation [formula omitted

We consider the nonlinear matrix equation Xp=A+M(B+X−1)−1M∗, where p≥1 is a positive integer, M is an arbitrary n×n matrix, and A and B are Hermitian positive semidefinite matrices. An elegant estimate of the Hermitian positive definite (HPD) solution is derived. A fixed-point iteration and an inversion-free variant iteration for obtaining the HPD solution are proposed. Some numerical examples are presented to show the efficiency of the proposed two iterative methods.

Convergence analysis of some iterative methods for a nonlinear matrix equation

We study iterative methods for finding the largest Hermitian positive definite solution of the matrix equation X+∑i=1mAi∗X−1Ai=Q. Convergence rates of the basic fixed point iteration, inversion free variants of the basic fixed point iteration, and Stein iteration are considered. Some numerical examples are presented to illustrate the convergence behaviour of the various algorithms.

Computing the Hermitian Positive Definite Solutions of a Nonlinear Matrix Equation

In this paper, we consider a nonlinear matrix equation. We propose necessary and sufficient conditions for the existence of Hermitian positive definite solutions. Some necessary conditions and sufficient conditions for the existence of Hermitian positive definite solutions of this equation is also derived. Based on the Banach fixed point theorem, the existence and the uniqueness of the Hermitian positive definite solution are studied. An iterative method for obtaining the Hermitian positive definite solution of this equation is proposed. Finally, some numerical examples are presented to illustrate the performance and efficiency of the proposed algorithm.