scholarly journals The optimal convergence factor of the gradient based iterative algorithm for linear matrix equations

Filomat ◽  
2012 ◽  
Vol 26 (3) ◽  
pp. 607-613 ◽  
Author(s):  
Xiang Wang ◽  
Dan Liao
Keyword(s):  
Iterative Algorithm ◽  
Numerical Experiments ◽  
Linear Matrix ◽  
Matrix Equations ◽  
Convergence Factor ◽  
Gradient Based ◽  
Gradient Type ◽  
And Mathematics ◽  
Computers And Mathematics ◽  
Linear Matrix Equations

A hierarchical gradient based iterative algorithm of [L. Xie et al., Computers and Mathematics with Applications 58 (2009) 1441-1448] has been presented for finding the numerical solution for general linear matrix equations, and the convergent factor has been discussed by numerical experiments. However, they pointed out that how to choose a best convergence factor is still a project to be studied. In this paper, we discussed the optimal convergent factor for the gradient based iterative algorithm and obtained the optimal convergent factor. Moreover, the theoretical results of this paper can be extended to other methods of gradient-type based. Results of numerical experiments are consistent with the theoretical findings.

Convergence of ADGI methods for solving systems of linear matrix equations

2014 ◽  
Vol 31 (4) ◽  
pp. 681-690 ◽  
Author(s):  
Masoud Hajarian
Keyword(s):  
Iterative Methods ◽  
Design Methodology ◽  
Linear Matrix ◽  
Matrix Equations ◽  
Hierarchical Identification ◽  
Content Type ◽  
Alternating Direction ◽  
Gradient Based ◽  
Engineering Physics ◽  
Linear Matrix Equations

Purpose – The linear matrix equations have wide applications in engineering, physics, economics and statistics. The purpose of this paper is to introduce iterative methods for solving the systems of linear matrix equations. Design/methodology/approach – According to the hierarchical identification principle, the authors construct alternating direction gradient-based iterative (ADGI) methods to solve systems of linear matrix equations. Findings – The authors propose efficient ADGI methods to solve the systems of linear matrix equations. It is proven that the ADGI methods consistently converge to the solution for any initial matrix. Moreover, the constructed methods are extended for finding the reflexive solution to the systems of linear matrix equations. Originality/value – This paper proposes efficient iterative methods without computing any matrix inverses, vec operator and Kronecker product for finding the solution of the systems of linear matrix equations.

On the numerical solution of a class of systems of linear matrix equations

2018 ◽  
Vol 40 (1) ◽  
pp. 207-225
Author(s):  
Valeria Simoncini
Keyword(s):  
Numerical Solution ◽  
Numerical Experiments ◽  
Original Problem ◽  
Linear Matrix ◽  
Matrix Equations ◽  
Structured Systems ◽  
New Methods ◽  
Computational Resources ◽  
Linear Matrix Equations

Abstract We consider the solution of systems of linear matrix equations in two or three unknown matrices. For dense problems we derive algorithms that determine the numerical solution by only involving matrices of the same size as those in the original problem, thus requiring low computational resources. For large and structured systems we show how the problem properties can be exploited to design effective algorithms with low memory and operation requirements. Numerical experiments illustrate the performance of the new methods.

scholarly journals Iterative Solutions of a Set of Matrix Equations by Using the Hierarchical Identification Principle

2014 ◽  
Vol 2014 ◽  
pp. 1-10 ◽  
Author(s):  
Huamin Zhang
Keyword(s):  
Unique Solution ◽  
Iterative Algorithm ◽  
Iterative Solution ◽  
Matrix Equations ◽  
Convergence Factor ◽  
Initial Value ◽  
Hierarchical Identification ◽  
The Real ◽  
Gradient Based ◽  
Iterative Solutions

This paper is concerned with iterative solution to a class of the real coupled matrix equations. By using the hierarchical identification principle, a gradient-based iterative algorithm is constructed to solve the real coupled matrix equationsA1XB1+A2XB2=F1andC1XD1+C2XD2=F2. The range of the convergence factor is derived to guarantee that the iterative algorithm is convergent for any initial value. The analysis indicates that if the coupled matrix equations have a unique solution, then the iterative solution converges fast to the exact one for any initial value under proper conditions. A numerical example is provided to illustrate the effectiveness of the proposed algorithm.

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