Conjugate gradient algorithm for consistent generalized Sylvester-transpose matrix equations

<abstract><p>We develop an effective algorithm to find a well-approximate solution of a generalized Sylvester-transpose matrix equation where all coefficient matrices and an unknown matrix are rectangular. The algorithm aims to construct a finite sequence of approximated solutions from any given initial matrix. It turns out that the associated residual matrices are orthogonal, and thus, the desire solution comes out in the final step with a satisfactory error. We provide numerical experiments to show the capability and performance of the algorithm.</p></abstract>

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An iterative method to solve a nonlinear matrix equation

Electronic Journal of Linear Algebra ◽

10.13001/1081-3810.2951 ◽

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.

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Two Iterative Methods for Solving Linear Interval Systems

Applied Computational Intelligence and Soft Computing ◽

10.1155/2018/2797038 ◽

2018 ◽

Vol 2018 ◽

pp. 1-13 ◽

Cited By ~ 2

Author(s):

Esmaeil Siahlooei ◽

Seyed Abolfazl Shahzadeh Fazeli

Keyword(s):

Iterative Method ◽

Conjugate Gradient ◽

Numerical Experiments ◽

Gradient Algorithm ◽

Interval Matrix ◽

Interval Numbers ◽

Linear Interval ◽

Linear Interval Systems ◽

Diagonally Dominant ◽

Interval Systems

Conjugate gradient is an iterative method that solves a linear system Ax=b, where A is a positive definite matrix. We present this new iterative method for solving linear interval systems Ãx̃=b̃, where Ã is a diagonally dominant interval matrix, as defined in this paper. Our method is based on conjugate gradient algorithm in the context view of interval numbers. Numerical experiments show that the new interval modified conjugate gradient method minimizes the norm of the difference of Ãx̃ and b̃ at every step while the norm is sufficiently small. In addition, we present another iterative method that solves Ãx̃=b̃, where Ã is a diagonally dominant interval matrix. This method, using the idea of steepest descent, finds exact solution x̃ for linear interval systems, where Ãx̃=b̃; we present a proof that indicates that this iterative method is convergent. Also, our numerical experiments illustrate the efficiency of the proposed methods.

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Inconsistent LR Fuzzy Matrix Equation

Mathematical Problems in Engineering ◽

10.1155/2020/4065809 ◽

2020 ◽

Vol 2020 ◽

pp. 1-9

Author(s):

Xiaobin Guo ◽

Lijuan Wu

Keyword(s):

Approximate Solution ◽

Matrix Equation ◽

Generalized Inverse ◽

Matrix Theory ◽

Sufficient Conditions ◽

Mean Value ◽

Matrix Equations ◽

Fuzzy Matrix ◽

Left And Right ◽

The Mean

In this paper, the inconsistent LR fuzzy matrix equation A X ˜ = B ˜ is proposed and discussed. Firstly, the LR fuzzy matrix equation is transformed into two crisp matrix equations in which one determines the mean value and the other determines the left and right extends of fuzzy approximate solution. Secondly, the approximate solution of the LR fuzzy matrix equation is obtained by solving two crisp matrix equations according to the generalized inverse of crisp matrix theory. Then, sufficient conditions for the existence of strong LR fuzzy approximate solution are given. Finally, some numerical examples are given to illustrate our proposed method.

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