An iterative method to solve a nonlinear matrix equation

2016 ◽  
Vol 31 ◽  
pp. 620-632
Author(s):  
Peng Jingjing ◽  
Liao Anping ◽  
Peng Zhenyun
Keyword(s):  
Approximate Solution ◽  
Iterative Method ◽  
Matrix Equation ◽  
Closed Ball ◽  
Open Ball ◽  
Nonlinear Matrix Equation ◽  
Initial Matrix ◽  
Convergence Results ◽  
The Matrix ◽  
Nonlinear Matrix

n this paper, an iterative method to solve one kind of nonlinear matrix equation is discussed. For each initial matrix with some conditions, the matrix sequences generated by the iterative method are shown to lie in a fixed open ball. The matrix sequences generated by the iterative method are shown to converge to the only solution of the nonlinear matrix equation in the fixed closed ball. In addition, the error estimate of the approximate solution in the fixed closed ball, and a numerical example to illustrate the convergence results are given.

scholarly journals Sensitivity Analysis of the Matrix Equation from Interpolation Problems

2013 ◽  
Vol 2013 ◽  
pp. 1-8 ◽  
Author(s):  
Jing Li ◽  
Yuhai Zhang
Keyword(s):  
Sensitivity Analysis ◽  
Approximate Solution ◽  
Matrix Equation ◽  
Nonlinear Matrix Equation ◽  
Backward Error ◽  
Perturbation Bound ◽  
Interpolation Problems ◽  
The Matrix ◽  
Nonlinear Matrix ◽  
Theoretical Results

This paper studies the sensitivity analysis of a nonlinear matrix equation connected to interpolation problems. The backward error estimates of an approximate solution to the equation are derived. A residual bound of an approximate solution to the equation is obtained. A perturbation bound for the unique solution to the equation is evaluated. This perturbation bound is independent of the exact solution of this equation. The theoretical results are illustrated by numerical examples.

A novel iterative method for the solution of a nonlinear matrix equation

2020 ◽  
Vol 153 ◽  
pp. 503-518 ◽  
Author(s):  
Raziyeh Erfanifar ◽  
Khosro Sayevand ◽  
Hamid Esmaeili
Keyword(s):  
Iterative Method ◽  
Matrix Equation ◽  
Nonlinear Matrix Equation ◽  
Nonlinear Matrix

Newton's iterative method to solve a nonlinear matrix equation

2018 ◽  
Vol 67 (9) ◽  
pp. 1867-1878
Author(s):  
Jingjing Peng ◽  
Anping Liao ◽  
Zhenyun Peng ◽  
Zhencheng Chen
Keyword(s):  
Iterative Method ◽  
Matrix Equation ◽  
Nonlinear Matrix Equation ◽  
Newton’S Iterative Method ◽  
Nonlinear Matrix

scholarly journals Inversion free Iterative Method for Finding Symmetric Solution of the Nonlinear Matrix Equation 𝑿 − 𝑨∗𝑿𝒒𝑨 = 𝑰 (𝒒 ≥ 𝟐)

2021 ◽  
Vol 47 (4) ◽  
pp. 1392-1401
Author(s):  
Chacha Stephen Chacha
Keyword(s):  
Positive Integer ◽  
Iterative Method ◽  
Matrix Equation ◽  
Symmetric Solution ◽  
Hermitian Matrix ◽  
Initial Guess ◽  
Nonlinear Matrix Equation ◽  
Numerical Examples ◽  
Nonlinear Matrix

In this paper, we propose the inversion free iterative method to find symmetric solution of thenonlinear matrix equation 𝑿 − 𝑨∗𝑿𝒒𝑨 = 𝑰 (𝒒 ≥ 𝟐), where 𝑋 is an unknown symmetricsolution, 𝐴 is a given Hermitian matrix and 𝑞 is a positive integer. The convergence of theproposed method is derived. Numerical examples demonstrate that the proposed iterative methodis quite efficient and converges well when the initial guess is sufficiently close to the approximatesolution. Keywords: Symmetric solution, nonlinear matrix equation, inversion free, iterative method

scholarly journals Comment on “Perturbation Analysis of the Nonlinear Matrix Equation ”

2013 ◽  
Vol 2013 ◽  
pp. 1-2 ◽  
Author(s):  
Maher Berzig ◽  
Erdal Karapınar
Keyword(s):  
Perturbation Analysis ◽  
Matrix Equation ◽  
Nonlinear Matrix Equation ◽  
The Matrix ◽  
Nonlinear Matrix

We show that the perturbation estimate for the matrix equation due to J. Li, is wrong. Our discussion is supported by a counterexample.

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.

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 A New Inversion-Free Iterative Scheme to Compute Maximal and Minimal Solutions of a Nonlinear Matrix Equation

Mathematics ◽  
2021 ◽  
Vol 9 (23) ◽  
pp. 2994
Author(s):  
Malik Zaka Ullah
Keyword(s):  
Iterative Method ◽  
Matrix Equation ◽  
Iterative Scheme ◽  
Positive Definite ◽  
Nonlinear Matrix Equation ◽  
Numerical Tests ◽  
Maximal Solutions ◽  
Minimal And Maximal Solutions ◽  
Nonlinear Matrix ◽  
Convergence Of The Scheme

The goal of this article is to investigate a new solver in the form of an iterative method to solve X+A∗X−1A=I as an important nonlinear matrix equation (NME), where A,X,I are appropriate matrices. The minimal and maximal solutions of this NME are discussed as Hermitian positive definite (HPD) matrices. The convergence of the scheme is given. Several numerical tests are also provided to support the theoretical discussions.

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