scholarly journals Iterative methods for the extremal positive definite solution of the matrix equation X+A*X-αA=Q

2007 ◽  
Vol 200 (2) ◽  
pp. 520-527 ◽  
Author(s):  
Zhen-yun Peng ◽  
Salah M. El-Sayed ◽  
Xiang-lin Zhang
Keyword(s):  
Iterative Methods ◽  
Matrix Equation ◽  
Positive Definite ◽  
Positive Definite Solution ◽  
The Matrix ◽  
Definite Solution

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 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 Necessary and Sufficient Conditions for the Existence of a Positive Definite Solution for the Matrix Equation

2013 ◽  
Vol 2013 ◽  
pp. 1-7 ◽  
Author(s):  
Naglaa M. El-Shazly
Keyword(s):  
Matrix Equation ◽  
Sufficient Conditions ◽  
Positive Definite ◽  
Necessary And Sufficient Conditions ◽  
Identity Matrix ◽  
Positive Definite Solution ◽  
The Matrix ◽  
Necessary And Sufficient ◽  
Real Matrices ◽  
Definite Solution

In this paper necessary and sufficient conditions for the matrix equation to have a positive definite solution are derived, where , is an identity matrix, are nonsingular real matrices, and is an odd positive integer. These conditions are used to propose some properties on the matrices , . Moreover, relations between the solution and the matrices are derived.

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