scholarly journals Solvability Theory and Iteration Method for One Self-Adjoint Polynomial Matrix Equation

2014 ◽  
Vol 2014 ◽  
pp. 1-7 ◽  
Author(s):  
Zhigang Jia ◽  
Meixiang Zhao ◽  
Minghui Wang ◽  
Sitao Ling
Keyword(s):  
Iterative Algorithm ◽  
Perturbation Analysis ◽  
Matrix Equation ◽  
Iteration Method ◽  
Numerical Experiments ◽  
Polynomial Matrix ◽  
Sufficient Conditions ◽  
Positive Definite ◽  
Solvability Theory ◽  
Adjoint Polynomial

The solvability theory of an important self-adjoint polynomial matrix equation is presented, including the boundary of its Hermitian positive definite (HPD) solution and some sufficient conditions under which the (unique or maximal) HPD solution exists. The algebraic perturbation analysis is also given with respect to the perturbation of coefficient matrices. An efficient general iterative algorithm for the maximal or unique HPD solution is designed and tested by 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 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 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.

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 The ULT-HSS hybrid iteration method for symmetric saddle point problems

2021 ◽  
pp. 128-128
Author(s):  
Jun-Feng Lu
Keyword(s):  
Saddle Point ◽  
Iteration Method ◽  
Numerical Experiments ◽  
Optimal Parameter ◽  
Sufficient Conditions ◽  
Saddle Point Problem ◽  
Point Problem ◽  
Saddle Point Problems ◽  
Necessary And Sufficient ◽  
Direction Iteration

This paper proposes a hybrid iteration method for solving symmetric saddle point problem arising in computational fluid dynamics. It is an implicit alternative direction iteration method and named as the ULT-HSS method. The convergence analysis is provided, and the necessary and sufficient conditions are given for the convergence of the method. Some practical approaches are formulated for setting the optimal parameter of the method. Numerical experiments are given to show its efficiency.

scholarly journals On preconditioned normal and skew-Hermitian splitting iteration method for continuous Sylvester equations AX + XB = C*

Filomat ◽  
2018 ◽  
Vol 32 (6) ◽  
pp. 2207-2217
Author(s):  
Xue-Qing Liang ◽  
Xiang Wang ◽  
Xiao-Bin Tang ◽  
Xiao-Yong Xiao
Keyword(s):  
Exact Solution ◽  
Theoretical Analysis ◽  
Iteration Method ◽  
Numerical Experiments ◽  
Convergence Property ◽  
Positive Definite ◽  
New Method ◽  
Splitting Iteration Method

In this paper, we present a preconditioned normal and skew-Hermitian splitting (PNSS) iteration method for continuous Sylvester equations AX + XB = C with positive definite/semi-definite matrices. Theoretical analysis shows that the PNSS methods will converge unconditionally to the exact solution of the continuous Sylvester equations. An inexact variant of the PNSS iteration method(IPNSS) and the analysis of its convergence property in detail have been established. Numerical experiments further show that this new method is more efficient and robust than the existing ones.

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.

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