scholarly journals The Relaxed Gradient Based Iterative Algorithm for the Symmetric (Skew Symmetric) Solution of the Sylvester EquationAX+XB=C

2017 ◽  
Vol 2017 ◽  
pp. 1-8 ◽  
Author(s):  
Xiaodan Zhang ◽  
Xingping Sheng
Keyword(s):  
Iterative Methods ◽  
Iterative Algorithm ◽  
Matrix Equation ◽  
Symmetric Solution ◽  
Symmetric Matrix ◽  
Iterative Algorithms ◽  
Iterative Solution ◽  
Sylvester Matrix Equation ◽  
Numerical Examples ◽  
Gradient Based

In this paper, we present two different relaxed gradient based iterative (RGI) algorithms for solving the symmetric and skew symmetric solution of the Sylvester matrix equationAX+XB=C. By using these two iterative methods, it is proved that the iterative solution converges to its true symmetric (skew symmetric) solution under some appropriate assumptions when any initial symmetric (skew symmetric) matrixX0is taken. Finally, two numerical examples are given to illustrate that the introduced iterative algorithms are more efficient.

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 Modified Jacobi-Gradient Iterative Method for Generalized Sylvester Matrix Equation

Symmetry ◽  
2020 ◽  
Vol 12 (11) ◽  
pp. 1831
Author(s):  
Nopparut Sasaki ◽  
Pattrawut Chansangiam
Keyword(s):  
Iterative Method ◽  
Matrix Equation ◽  
Recursive Formula ◽  
Sylvester Matrix Equation ◽  
Hierarchical Identification ◽  
Sylvester Matrix ◽  
Generalized Sylvester Matrix Equation ◽  
Iteration Matrix ◽  
Diagonal Part ◽  
Gradient Based

We propose a new iterative method for solving a generalized Sylvester matrix equation A1XA2+A3XA4=E with given square matrices A1,A2,A3,A4 and an unknown rectangular matrix X. The method aims to construct a sequence of approximated solutions converging to the exact solution, no matter the initial value is. We decompose the coefficient matrices to be the sum of its diagonal part and others. The recursive formula for the iteration is derived from the gradients of quadratic norm-error functions, together with the hierarchical identification principle. We find equivalent conditions on a convergent factor, relied on eigenvalues of the associated iteration matrix, so that the method is applicable as desired. The convergence rate and error estimation of the method are governed by the spectral norm of the related iteration matrix. Furthermore, we illustrate numerical examples of the proposed method to show its capability and efficacy, compared to recent gradient-based iterative methods.

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 Trapezoidal Fully Fuzzy Sylvester Matrix Equation with Arbitrary Coefficients

2021 ◽  
Author(s):  
Ahmed Elsayed ◽  
Nazihah Ahmad ◽  
Ghassan Malkawi
Keyword(s):  
Linear Equation ◽  
Matrix Equation ◽  
Analytical Methods ◽  
Fuzzy Numbers ◽  
Coefficient Matrix ◽  
Sylvester Matrix Equation ◽  
Numerical Examples ◽  
Trapezoidal Fuzzy Numbers ◽  
Sylvester Matrix ◽  
Methods Numerical

Abstract Almost every existing method for solving trapezoidal fully fuzzy Sylvester matrix equation restricts the coefficient matrix and the solution to be positive fuzzy numbers only. In this paper, we develop new analytical methods to solve a trapezoidal fully fuzzy Sylvester matrix equation with restricted and unrestricted coefficients. The trapezoidal fully fuzzy Sylvester matrix equation is transferred to a system of crisp equations based on the sign of the coefficients by using Ahmd arithmetic multiplication operations between trapezoidal fuzzy numbers. The constructed method not only obtain a simple crisp system of linear equation that can be solved by any classical methods but also provide a widen the scope of the trapezoidal fully fuzzy Sylvester matrix equation in scientific applications. Furthermore, these methods have less steps and conceptually easy to understand when compared with existing methods. To illustrate the proposed methods numerical examples are solved.

scholarly journals A Gradient Based Iterative Solutions for Sylvester Tensor Equations

2013 ◽  
Vol 2013 ◽  
pp. 1-7 ◽  
Author(s):  
Zhen Chen ◽  
Linzhang Lu
Keyword(s):  
Exact Solution ◽  
Numerical Solution ◽  
Matrix Equation ◽  
Sylvester Matrix Equation ◽  
Tensor Equation ◽  
Initial Value ◽  
Hierarchical Identification ◽  
Gradient Based ◽  
Iterative Solutions ◽  
Special Case

This paper is concerned with the numerical solution of the Sylvester tensor equation, which includes the Sylvester matrix equation as special case. By applying hierarchical identification principle proposed by Ding and Chen, 2005, and by using tensor arithmetic concepts, an iterative algorithm and its modification are established to solve the Sylvester tensor equation. Convergence analysis indicates that the iterative solutions always converge to the exact solution for arbitrary initial value. Finally, some examples are provided to show that the proposed algorithms are effective.

scholarly journals Finite iterative algorithms for the generalized reflexive and anti-reflexive solutions of the linear matrix equation AXB = C

Filomat ◽  
2017 ◽  
Vol 31 (7) ◽  
pp. 2151-2162 ◽  
Author(s):  
Xiang Wang ◽  
Xiao-Bin Tang ◽  
Xin-Geng Gao ◽  
Wu-Hua Wu
Keyword(s):  
Iterative Method ◽  
Matrix Equation ◽  
Iterative Algorithms ◽  
Iterative Solution ◽  
Necessary Condition ◽  
Linear Matrix ◽  
Linear Matrix Equation ◽  
Reflexive Matrix ◽  
Roundoff Errors ◽  
Sufficient And Necessary Condition

In this paper, an iterative method is presented to solve the linear matrix equation AXB = C over the generalized reflexive (or anti-reflexive) matrix X (A ? Rpxn, B ? Rmxq, C ? Rpxq, X ? Rnxm). By the iterative method, the solvability of the equation AXB = C over the generalized reflexive (or anti-reflexive) matrix can be determined automatically. When the equation AXB = C is consistent over the generalized reflexive (or anti-reflexive) matrix X, for any generalized reflexive (or anti-reflexive) initial iterative matrix X1, the generalized reflexive (anti-reflexive) solution can be obtained within finite iterative steps in the absence of roundoff errors. The unique least-norm generalized reflexive (or anti-reflexive) iterative solution of AXB = C can be derived when an appropriate initial iterative matrix is chosen. A sufficient and necessary condition for whether the equation AXB = C is inconsistent is given. Furthermore, the optimal approximate solution of AXB = C for a given matrix X0 can be derived by finding the least-norm generalized reflexive (or anti-reflexive) solution of a new corresponding matrix equation AXB = C. Finally, several numerical examples are given to support the theoretical results of this paper.

scholarly journals New Iterative Methods for Solving Nonlinear Equations and Their Basins of Attraction

2022 ◽  
Vol 21 ◽  
pp. 9-16
Author(s):  
O. Ababneh
Keyword(s):  
Iterative Methods ◽  
Nonlinear Equations ◽  
Newton's Method ◽  
Order Of Convergence ◽  
Iterative Algorithms ◽  
Basins Of Attraction ◽  
Second Derivative ◽  
Numerical Examples ◽  
Convergence Results ◽  
Modified Newton’S Method

The purpose of this paper is to propose new modified Newton’s method for solving nonlinear equations and free from second derivative. Convergence results show that the order of convergence is four. Several numerical examples are given to illustrate that the new iterative algorithms are effective.In the end, we present the basins of attraction to observe the fractal behavior and dynamical aspects of the proposed algorithms.

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