scholarly journals An Inversion-Free Method for Finding Positive Definite Solution of a Rational Matrix Equation

2014 ◽  
Vol 2014 ◽  
pp. 1-5 ◽  
Author(s):  
Fazlollah Soleymani ◽  
Mahdi Sharifi ◽  
Solat Karimi Vanani ◽  
Farhad Khaksar Haghani ◽  
Adem Kılıçman
Keyword(s):  
Matrix Equation ◽  
Numerical Experiments ◽  
Iterative Scheme ◽  
Positive Definite ◽  
Minimal Solution ◽  
New Method ◽  
Rational Matrix ◽  
Positive Definite Solution ◽  
Definite Solution

A new iterative scheme has been constructed for finding minimal solution of a rational matrix equation of the formX+A*X-1A=I. The new method is inversion-free per computing step. The convergence of the method has been studied and tested via numerical experiments.

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.

scholarly journals Notes on the Hermitian Positive Definite Solutions of a Matrix Equation

2014 ◽  
Vol 2014 ◽  
pp. 1-8 ◽  
Author(s):  
Jing Li ◽  
Yuhai Zhang
Keyword(s):  
Fixed Point ◽  
Fixed Point Theorem ◽  
Point Theorem ◽  
Matrix Equation ◽  
Positive Definite ◽  
Nonlinear Matrix Equation ◽  
Positive Definite Solution ◽  
Hermitian Positive Definite Solution ◽  
The Matrix ◽  
Definite Solution

The nonlinear matrix equation,X-∑i=1mAi*XδiAi=Q,with-1≤δi<0is investigated. A fixed point theorem in partially ordered sets is proved. And then, by means of this fixed point theorem, the existence of a unique Hermitian positive definite solution for the matrix equation is derived. Some properties of the unique Hermitian positive definite solution are obtained. A residual bound of an approximate solution to the equation is evaluated. The theoretical results are illustrated by numerical examples.

scholarly journals Latest Inversion-Free Iterative Scheme for Solving a Pair of Nonlinear Matrix Equations

2021 ◽  
Vol 2021 ◽  
pp. 1-22
Author(s):  
Sourav Shil ◽  
Hemant Kumar Nashine
Keyword(s):  
Convergence Rate ◽  
Convergence Analysis ◽  
Iterative Scheme ◽  
Positive Definite ◽  
Matrix Equations ◽  
Identity Matrix ◽  
Positive Definite Solution ◽  
Nonlinear Matrix ◽  
Definite Solution

In this work, the following system of nonlinear matrix equations is considered, X 1 + A ∗ X 1 − 1 A + B ∗ X 2 − 1 B = I  and  X 2 + C ∗ X 2 − 1 C + D ∗ X 1 − 1 D = I , where A , B , C ,  and  D are arbitrary n × n matrices and I is the identity matrix of order n . Some conditions for the existence of a positive-definite solution as well as the convergence analysis of the newly developed algorithm for finding the maximal positive-definite solution and its convergence rate are discussed. Four examples are also provided herein to support our results.

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