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

Filomat ◽  
2019 ◽  
Vol 33 (9) ◽  
pp. 2667-2671
Author(s):  
Guoxing Wu ◽  
Ting Xing ◽  
Duanmei Zhou
Keyword(s):  
Matrix Equation ◽  
Positive Definite Matrix ◽  
Positive Definite ◽  
Nonlinear Matrix Equation ◽  
The Matrix ◽  
Hermitian Positive Definite Matrix ◽  
Nonlinear Matrix ◽  
Positive Integers ◽  
Equivalent Conditions

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.

Iteration with Stepsize Parameter and Condition Numbers for a Nonlinear Matrix Equation

2018 ◽  
Vol 34 ◽  
pp. 217-230
Author(s):  
Syed M. Raza Shah Naqvi ◽  
Jie Meng ◽  
Hyun-Min Kim
Keyword(s):  
Matrix Equation ◽  
Positive Definite Matrix ◽  
Positive Definite ◽  
Condition Numbers ◽  
Symmetric Positive Definite Matrix ◽  
Nonlinear Matrix Equation ◽  
Symmetric Positive Definite ◽  
The Matrix ◽  
Nonlinear Matrix ◽  
Definite Solution

In this paper, the nonlinear matrix equation $X^p+A^TXA=Q$, where $p$ is a positive integer, $A$ is an arbitrary $n\times n$ matrix, and $Q$ is a symmetric positive definite matrix, is considered. A fixed-point iteration with stepsize parameter for obtaining the symmetric positive definite solution of the matrix equation is proposed. The explicit expressions of the normwise, mixed and componentwise condition numbers are derived. Several numerical examples are presented to show the efficiency of the proposed iterative method with proper stepsize parameter and the sharpness of the three kinds of condition numbers.

The Hermitian Positive Definite Solution of the Nonlinear Matrix Equation

2017 ◽  
Vol 18 (5) ◽  
pp. 293-301
Author(s):  
Xindong Zhang ◽  
Xinlong Feng
Keyword(s):  
Matrix Equation ◽  
Iterative Algorithms ◽  
Positive Definite Matrix ◽  
Positive Definite ◽  
Nonlinear Matrix Equation ◽  
Positive Definite Solution ◽  
Hermitian Positive Definite Solution ◽  
Hermitian Positive Definite Matrix ◽  
Nonlinear Matrix ◽  
Definite Solution

Abstract:In this paper, we study the nonlinear matrix equation $X^{s}\pm\sum^{m}_{i=1}A^{T}_{i}X^{\delta_{i}}A_{i}=Q$, where $A_{i}\;(i=1,2,\ldots,m)$ is $n\times n$ nonsingular real matrix and $Q$ is $n\times n$ 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.

scholarly journals On the Hermitian Positive Definite Solutions of Nonlinear Matrix EquationXs+A∗X−t1A+B∗X−t2B=Q

2011 ◽  
Vol 2011 ◽  
pp. 1-18 ◽  
Author(s):  
Aijing Liu ◽  
Guoliang Chen
Keyword(s):  
Iterative Method ◽  
Matrix Equation ◽  
Sufficient Conditions ◽  
Positive Definite Matrix ◽  
Positive Definite ◽  
Nonlinear Matrix Equation ◽  
Positive Definite Solution ◽  
Hermitian Positive Definite Solution ◽  
Nonlinear Matrix ◽  
Definite Solution

Nonlinear matrix equationXs+A∗X−t1A+B∗X−t2B=Qhas many applications in engineering; control theory; dynamic programming; ladder networks; stochastic filtering; statistics and so forth. In this paper, the Hermitian positive definite solutions of nonlinear matrix equationXs+A∗X−t1A+B∗X−t2B=Qare considered, whereQis a Hermitian positive definite matrix,A,Bare nonsingular complex matrices,sis a positive number, and0<ti≤1,i=1,2. Necessary and sufficient conditions for the existence of Hermitian positive definite solutions are derived. A sufficient condition for the existence of a unique Hermitian positive definite solution is given. In addition, some necessary conditions and sufficient conditions for the existence of Hermitian positive definite solutions are presented. Finally, an iterative method is proposed to compute the maximal Hermitian positive definite solution, and numerical example is given to show the efficiency of the proposed iterative method.

scholarly journals Extended Branciari quasi-b-distance spaces, implicit relations and application to nonlinear matrix equations

2021 ◽  
Vol 2021 (1) ◽  
Author(s):  
Reena Jain ◽  
Hemant Kumar Nashine ◽  
Vahid Parvaneh
Keyword(s):  
Positive Definite Matrix ◽  
Weak Limit ◽  
Positive Definite ◽  
Contractive Condition ◽  
Sufficient Condition ◽  
Nonlinear Matrix Equation ◽  
Property A ◽  
Underlying Space ◽  
Hermitian Positive Definite Matrix ◽  
Nonlinear Matrix

AbstractThis study introduces extended Branciari quasi-b-distance spaces, a novel implicit contractive condition in the underlying space, and basic fixed-point results, a weak well-posed property, a weak limit shadowing property and generalized Ulam–Hyers stability. The given notions and results are exemplified by suitable models. We apply these results to obtain a sufficient condition ensuring the existence of a unique positive-definite solution of a nonlinear matrix equation (NME) $\mathcal{X}=\mathcal{Q} + \sum_{i=1}^{k}\mathcal{A}_{i}^{*} \mathcal{G(X)}\mathcal{A}_{i}$ X = Q + ∑ i = 1 k A i ∗ G ( X ) A i , where $\mathcal{Q}$ Q is an $n\times n$ n × n Hermitian positive-definite matrix, $\mathcal{A}_{1}$ A 1 , $\mathcal{A}_{2}$ A 2 , …, $\mathcal{A}_{m}$ A m are $n \times n$ n × n matrices, and $\mathcal{G}$ G is a nonlinear self-mapping of the set of all Hermitian matrices that are continuous in the trace norm. We demonstrate this sufficient condition for the NME $\mathcal{X}= \mathcal{Q} +\mathcal{A}_{1}^{*}\mathcal{X}^{1/3} \mathcal{A}_{1}+\mathcal{A}_{2}^{*}\mathcal{X}^{1/3} \mathcal{A}_{2}+ \mathcal{A}_{3}^{*}\mathcal{X}^{1/3}\mathcal{A}_{3}$ X = Q + A 1 ∗ X 1 / 3 A 1 + A 2 ∗ X 1 / 3 A 2 + A 3 ∗ X 1 / 3 A 3 , and visualize this through convergence analysis and a solution graph.

scholarly journals Solutions and Improved Perturbation Analysis for the Matrix EquationX-A*X-pA=Q  (p>0)

2013 ◽  
Vol 2013 ◽  
pp. 1-12 ◽  
Author(s):  
Jing Li
Keyword(s):  
Matrix Equation ◽  
Positive Definite ◽  
Nonlinear Matrix Equation ◽  
Backward Error ◽  
Perturbation Bound ◽  
Positive Definite Solution ◽  
The Matrix ◽  
Nonlinear Matrix ◽  
Theoretical Results ◽  
Definite Solution

The nonlinear matrix equationX-A*X-pA=Qwithp>0is investigated. We consider two cases of this equation: the casep≥1and the case0<p<1.In the casep≥1, a new sufficient condition for the existence of a unique positive definite solution for the matrix equation is obtained. A perturbation estimate for the positive definite solution is derived. Explicit expressions of the condition number for the positive definite solution are given. In the case0<p<1, a new sharper perturbation bound for the unique positive definite solution is derived. A new backward error of an approximate solution to the unique positive definite solution is obtained. The theoretical results are illustrated by numerical examples.

scholarly journals Hermitian Positive Definite Solution of the Matrix Equation X=Q+∑i=1mAi(B+X-1)-1Ai*

2014 ◽  
Vol 2014 ◽  
pp. 1-8 ◽  
Author(s):  
Chun-Mei Li ◽  
Jing-Jing Peng
Keyword(s):  
Matrix Equation ◽  
Necessary Conditions ◽  
Sufficient Conditions ◽  
Positive Definite ◽  
Nonlinear Matrix Equation ◽  
Positive Definite Solution ◽  
Hermitian Positive Definite Solution ◽  
The Matrix ◽  
Nonlinear Matrix ◽  
Definite Solution

We consider the Hermitian positive definite solution of the nonlinear matrix equation X=Q+∑i=1mAi(B+X-1)-1Ai*. Some new sufficient conditions and necessary conditions for the existence of Hermitian positive definite solutions are derived. An iterative method is proposed to compute the Hermitian positive definite solution. In the end, an example is used to illustrate the correctness and application of our results.

scholarly journals An implicit relation, relational theoretic approach under w-distance and application to nonlinear matrix equations

2021 ◽  
Vol 2021 (1) ◽  
Author(s):  
Reena Jain ◽  
Hemant Kumar Nashine ◽  
Manuel De la Sen
Keyword(s):  
Metric Spaces ◽  
Sufficient Conditions ◽  
Positive Definite Matrix ◽  
Positive Definite ◽  
Contractive Condition ◽  
Theoretic Approach ◽  
Nonlinear Matrix Equation ◽  
Surface Plot ◽  
Hermitian Positive Definite Matrix ◽  
Nonlinear Matrix

AbstractWe propose a new class of implicit relations and an implicit type contractive condition based on it in the relational metric spaces under w-distance functional. Further we derive fixed points results based on them. Useful examples illustrate the applicability and effectiveness of the presented results. We apply these results to discuss sufficient conditions ensuring the existence of a unique positive definite solution of the nonlinear matrix equation (NME) of the form $\mathcal{U}=\mathcal{Q} + \sum_{i=1}^{k}\mathcal{A}_{i}^{*} \mathcal{G}\mathcal{(U)}\mathcal{A}_{i}$ U = Q + ∑ i = 1 k A i ∗ G ( U ) A i , where $\mathcal{Q}$ Q is an $n\times n$ n × n Hermitian positive definite matrix, $\mathcal{A}_{1}$ A 1 , $\mathcal{A}_{2}$ A 2 , …, $\mathcal{A}_{m}$ A m are $n \times n$ n × n matrices and $\mathcal{G}$ G is a nonlinear self-mapping of the set of all Hermitian matrices which are continuous in the trace norm. In order to demonstrate the obtained conditions, we consider an example together with convergence and error analysis and visualisation of solutions in a surface plot.

scholarly journals A two-sided iterative method for computing positive definite solutions of a nonlinear matrix equation

2003 ◽  
Vol 45 (1) ◽  
pp. 145-152 ◽  
Author(s):  
Salah M. El-Sayed
Keyword(s):  
Matrix Equation ◽  
Iterative Process ◽  
Additional Condition ◽  
Positive Definite ◽  
The Novel ◽  
Stopping Criterion ◽  
Nonlinear Matrix Equation ◽  
Solvability Conditions ◽  
The Matrix ◽  
Nonlinear Matrix

AbstractIn several recent papers, a one-sided iterative process for computing positive definite solutions of the nonlinear matrix equation X + A* X−1A = Q, where Q is positive definite, has been studied. In this paper, a two-sided iterative process for the same equation is investigated. The novel idea here is that the two sequences obtained by starting at two different values provide (a) an interval in which the solution is located, that is, Xk ≤ X ≤ Yk for all k and (b) a better stopping criterion. Some properties of solutions are discussed. Sufficient solvability conditions on a matrix A are derived. Moreover, when the matrix A is normal and satisfies an additional condition, the matrix equation has smallest and largest positive definite solutions. Finally, some numerical examples are given to illustrate the effectiveness of the algorithm.

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