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

2020 ◽  
Author(s):  
Xiao-Xia Guo ◽  
Hong-Xiao Wu
Keyword(s):  
Positive Definite ◽  
Structure Preserving ◽  
Positive Definite Solution ◽  
Definite Solution

scholarly journals Best Proximity Point Results for Generalized Θ-Contractions and Application to Matrix Equations

Symmetry ◽  
2019 ◽  
Vol 11 (1) ◽  
pp. 93
Author(s):  
Zhenhua Ma ◽  
Azhar Hussain ◽  
Muhammad Adeel ◽  
Nawab Hussain ◽  
Ekrem Savas
Keyword(s):  
Metric Spaces ◽  
Best Proximity Point ◽  
Positive Definite ◽  
Matrix Equations ◽  
Nonlinear Matrix Equation ◽  
Complete Metric Spaces ◽  
Partially Ordered ◽  
Positive Definite Solution ◽  
Nonlinear Matrix ◽  
Definite Solution

In this paper, we introduce the notion of C ´ iri c ´ type α - ψ - Θ -contraction and prove best proximity point results in the context of complete metric spaces. Moreover, we prove some best proximity point results in partially ordered complete metric spaces through our main results. As a consequence, we obtain some fixed point results for such contraction in complete metric and partially ordered complete metric spaces. Examples are given to illustrate the results obtained. Moreover, we present the existence of a positive definite solution of nonlinear matrix equation X = Q + ∑ i = 1 m A i * γ ( X ) A i and give a numerical example.

On the Nonlinear Matrix Equation Based on Engineering Calculation

2012 ◽  
Vol 450-451 ◽  
pp. 158-161
Author(s):  
Dong Jie Gao
Keyword(s):  
Matrix Equation ◽  
Iteration Method ◽  
Engineering Calculation ◽  
Positive Definite ◽  
Nonlinear Matrix Equation ◽  
Positive Definite Solution ◽  
Nonlinear Matrix ◽  
Definite Solution

We consider the positive definite solution of the nonlinear matrix equation . We prove that the equation always has a unique positive definite solution. The iteration method for the equation is given.

scholarly journals The Inversion-Free Iterative Methods for Solving the Nonlinear Matrix EquationX+AHX-1A+BHX-1B=I

2013 ◽  
Vol 2013 ◽  
pp. 1-7 ◽  
Author(s):  
Na Huang ◽  
Changfeng Ma
Keyword(s):  
Iterative Methods ◽  
Matrix Equation ◽  
Positive Definite ◽  
Numerical Results ◽  
Iterative Schemes ◽  
Positive Definite Solution ◽  
Nonlinear Matrix ◽  
Definite Solution

We present two inversion-free iterative methods for computing the maximal positive definite solution of the equationX+AHX-1A+BHX-1B=I. We prove that the sequences generated by the two iterative schemes are monotonically increasing and bounded above. We also present some numerical results to compare our proposed methods with some previously developed inversion-free techniques for solving the same matrix equation.

scholarly journals Necessary and Sufficient Conditions for the Existence of a Hermitian Positive Definite Solution of a Type of Nonlinear Matrix Equations

2009 ◽  
Vol 2009 ◽  
pp. 1-13 ◽  
Author(s):  
Wenling Zhao ◽  
Hongkui Li ◽  
Xueting Liu ◽  
Fuyi Xu
Keyword(s):  
Matrix Equation ◽  
Sufficient Conditions ◽  
Positive Definite ◽  
Necessary And Sufficient Conditions ◽  
Nonsingular Matrix ◽  
Positive Definite Solution ◽  
Hermitian Positive Definite Solution ◽  
Nonlinear Matrix ◽  
Necessary And Sufficient ◽  
Definite Solution

We study the Hermitian positive definite solutions of the nonlinear matrix equationX+A∗X−2A=I, whereAis ann×nnonsingular matrix. Some necessary and sufficient conditions for the existence of a Hermitian positive definite solution of this equation are given. However, based on the necessary and sufficient conditions, some properties and the equivalent equations ofX+A∗X−2A=Iare presented while the matrix equation has a Hermitian positive definite solution.

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