scholarly journals The Feasibility of Homotopy Continuation Method for a Nonlinear Matrix Equation

2020 ◽  
Vol 2020 ◽  
pp. 1-13
Author(s):  
Mengran Wang ◽  
Jing Li
Keyword(s):  
Iterative Method ◽  
Matrix Equation ◽  
Continuation Method ◽  
Initial Approximation ◽  
Homotopy Continuation ◽  
Matrix Equations ◽  
Nonlinear Matrix Equation ◽  
Homotopy Continuation Method ◽  
Nonlinear Matrix

In this paper, we discuss the feasibility of homotopy continuation method for the nonlinear matrix equations X+∑i=1sBi∗X−1Bi+∑i=s+1mBi∗XtiBi=I with 0<ti<1. This iterative method does not depend on a good initial approximation to the solution of matrix equation.

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

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 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.

scholarly journals New collocation path-following approach for the optimal shape parameter using Kernel method

2021 ◽  
Vol 3 (2) ◽  
Author(s):  
Zouhair Saffah ◽  
Abdelaziz Timesli ◽  
Hassane Lahmam ◽  
Abderrahim Azouani ◽  
Mohamed Amdi
Keyword(s):  
Shape Parameter ◽  
Kernel Method ◽  
Continuation Method ◽  
Path Following ◽  
High Order ◽  
Homotopy Continuation ◽  
Homotopy Continuation Method ◽  
Optimal Value ◽  
Rbf Kernel ◽  
Order Algorithm

AbstractThe goal of this work is to develop a numerical method combining Radial Basic Functions (RBF) kernel and a high order algorithm based on Taylor series and homotopy continuation method. The local RBF approximation applied in strong form allows us to overcome the difficulties of numerical integration and to treat problems of large deformations. Furthermore, the high order algorithm enables to transform the nonlinear problem to a set of linear problems. Determining the optimal value of the shape parameter in RBF kernel is still an outstanding research topic. This optimal value depends on density and distribution of points and the considered problem for e.g. boundary value problems, integral equations, delay-differential equations etc. These have been extensively attempts in literature which end up choosing this optimal value by tests and error or some other ad-hoc means. Our contribution in this paper is to suggest a new strategy using radial basis functions kernel with an automatic reasonable choice of the shape parameter in the nonlinear case which depends on the accuracy and stability of the results. The computational experiments tested on some examples in structural analysis are performed and the comparison with respect to the state of art algorithms from the literature is given.

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