scholarly journals An iterative method for the skew-symmetric solution and the optimal approximate solution of the matrix equation AXB=C

2008 ◽  
Vol 212 (2) ◽  
pp. 231-244 ◽  
Author(s):  
Guang-Xin Huang ◽  
Feng Yin ◽  
Ke Guo
Keyword(s):  
Approximate Solution ◽  
Iterative Method ◽  
Matrix Equation ◽  
Symmetric Solution ◽  
Optimal Approximate Solution ◽  
The Matrix

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.

Optimal Approximate Solution of a Class of Matrix Equation

2012 ◽  
Vol 220-223 ◽  
pp. 2180-2183
Author(s):  
Chun Rui Cheng ◽  
Jun Hui Zhu
Keyword(s):  
Approximate Solution ◽  
Singular Value Decomposition ◽  
Matrix Equation ◽  
Symmetric Solution ◽  
Singular Value ◽  
Solution Set ◽  
Optimal Approximate Solution ◽  
Generalized Singular Value Decomposition ◽  
Value Decomposition ◽  
Matrix Pair

The optimal approximate solution of a class of matrix equation is considered. Based on the generalized singular value decomposition, the expression of the optimal approximate solution to a given matrix pair is derived in the symmetric solution set.

scholarly journals An Iterative Algorithm for the Generalized Reflexive Solution of the Matrix EquationsAXB=E,CXD=F

2012 ◽  
Vol 2012 ◽  
pp. 1-20 ◽  
Author(s):  
Deqin Chen ◽  
Feng Yin ◽  
Guang-Xin Huang
Keyword(s):  
Approximate Solution ◽  
Iterative Algorithm ◽  
Matrix Equation ◽  
Iterative Solution ◽  
Linear Matrix ◽  
Optimal Approximate Solution ◽  
Linear Matrix Equation ◽  
Numerical Examples ◽  
Reflexive Matrix ◽  
The Matrix

An iterative algorithm is constructed to solve the linear matrix equation pairAXB=E, CXD=Fover generalized reflexive matrixX. When the matrix equation pairAXB=E, CXD=Fis consistent over generalized reflexive matrixX, for any generalized reflexive initial iterative matrixX1, the generalized reflexive solution can be obtained by the iterative algorithm within finite iterative steps in the absence of round-off errors. The unique least-norm generalized reflexive iterative solution of the matrix equation pair can be derived when an appropriate initial iterative matrix is chosen. Furthermore, the optimal approximate solution ofAXB=E, CXD=Ffor a given generalized reflexive matrixX0can be derived by finding the least-norm generalized reflexive solution of a new corresponding matrix equation pairAX̃B=Ẽ, CX̃D=F̃withẼ=E-AX0B, F̃=F-CX0D. Finally, several numerical examples are given to illustrate that our iterative algorithm is effective.

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.

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 New algorithms to compute the nearness symmetric solution of the matrix equation

SpringerPlus ◽  
2016 ◽  
Vol 5 (1) ◽  
Author(s):  
Zhen-yun Peng ◽  
Yang-zhi Fang ◽  
Xian-wei Xiao ◽  
Dan-dan Du
Keyword(s):  
Matrix Equation ◽  
Symmetric Solution ◽  
The Matrix ◽  
New Algorithms

scholarly journals An Iterative Algorithm for the Least Squares Generalized Reflexive Solutions of the Matrix Equations

2012 ◽  
Vol 2012 ◽  
pp. 1-18 ◽  
Author(s):  
Feng Yin ◽  
Guang-Xin Huang
Keyword(s):  
Approximate Solution ◽  
Iterative Algorithm ◽  
Applied Mathematics ◽  
Frobenius Norm ◽  
Matrix Equations ◽  
Optimal Approximate Solution ◽  
Numerical Examples ◽  
Reflexive Matrix ◽  
The Matrix ◽  
Residual Problem

The generalized coupled Sylvester systems play a fundamental role in wide applications in several areas, such as stability theory, control theory, perturbation analysis, and some other fields of pure and applied mathematics. The iterative method is an important way to solve the generalized coupled Sylvester systems. In this paper, an iterative algorithm is constructed to solve the minimum Frobenius norm residual problem: min over generalized reflexive matrix . For any initial generalized reflexive matrix , by the iterative algorithm, the generalized reflexive solution can be obtained within finite iterative steps in the absence of round-off errors, and the unique least-norm generalized reflexive solution can also be derived when an appropriate initial iterative matrix is chosen. Furthermore, the unique optimal approximate solution to a given matrix in Frobenius norm can be derived by finding the least-norm generalized reflexive solution of a new corresponding minimum Frobenius norm residual problem: with , . Finally, several numerical examples are given to illustrate that our iterative algorithm is effective.

scholarly journals The Structure of Symmetric Solutions of the Matrix Equation AX=B over a Principal Ideal Domain

2017 ◽  
Vol 2017 ◽  
pp. 1-7
Author(s):  
V. M. Prokip
Keyword(s):  
Matrix Equation ◽  
Principal Ideal ◽  
Symmetric Solution ◽  
Principal Ideal Domain ◽  
Ideal Domain ◽  
The Matrix

We investigate the structure of symmetric solutions of the matrix equation AX=B, where A and B are m-by-n matrices over a principal ideal domain R and X is unknown n-by-n matrix over R. We prove that matrix equation AX=B over R has a symmetric solution if and only if equation AX=B has a solution over R and the matrix ABT is symmetric. If symmetric solution exists we propose the method for its construction.

An iterative method for the least squares symmetric solution of matrix equation $AXB = C$

2006 ◽  
Vol 42 (2) ◽  
pp. 181-192 ◽  
Author(s):  
Jin-jun Hou ◽  
Zhen-yun Peng ◽  
Xiang-lin Zhang
Keyword(s):  
Iterative Method ◽  
Least Squares ◽  
Matrix Equation ◽  
Symmetric Solution
Sign in / Sign up

Export Citation Format

Share Document