Preconditioned Galerkin and minimal residual methods for solving Sylvester equations

Kaabi, Amer; Toutounian, Faezeh; Kerayechian, Asghar

doi:10.1016/j.amc.2006.02.021

Cited by 5 publications

(4 citation statements)

References 17 publications

Supporting

Mentioning

Contrasting

Order By: Relevance

“…All the numerical experiments were computed in double precision with MATLAB code. To compare the numerical results of the L-GLS algorithm with those of some existing methods, we chose some examples from Cai and Chen (2009), Chu (1989), Dehghan and Hajarian (2008, 2011a, 2012, Hajarian and Dehghan (2011), Kaabi et al (2006) and Khorsand Zak and Toutounian (2013). For more details about each example, one can refer to the related citation.…”

Section: Numerical Experimentsmentioning

confidence: 99%

See 1 more Smart Citation

A general iterative approach for solving the general constrained linear matrix equations system

Karimi

Dehghan

2015

Transactions of the Institute of Measurement and Control

View full text Add to dashboard Cite

Much of the research which has been done on special kinds of matrix equations has almost the same structure. In this paper, an iterative approach is proposed which is more general and includes all such matrix equations. This approach is based on the global least squares (GL-LSQR) method. To do so, a linear matrix operator, its adjoint and a proper inner product are defined. Then we demonstrate how to employ the new approach for solving the general linear matrix equations system. When the matrix equations are consistent, a least-norm solution can be obtained. Moreover, the optimal approximate solution to a given group of matrices can be derived. Also, some theoretical properties and the error analysis of the new method are discussed. Finally, some numerical experiments are given to compare the new iterative method with some existent methods in terms of their numerical results and illustrate the efficiency of the new method.

show abstract

Section: Numerical Experimentsmentioning

confidence: 99%

“…Consider the first test problem of Kaabi et al (2006). We compared the performance of the L-GLS, Galerkin (Gal) and the minimal residual (MR) by Kaabi et al (2006) algorithms. The numerical results are presented in Tables 3 and 4 with n = 0 and n = 10, respectively.…”

Section: Numerical Experimentsmentioning

confidence: 99%

A general iterative approach for solving the general constrained linear matrix equations system

Karimi

Dehghan

2015

Transactions of the Institute of Measurement and Control

View full text Add to dashboard Cite

show abstract

“…Zhou et al [50] introduced a modified version to improve the performance of HSS iteration. In [28,29,35], some Krylov subspace methods for obtaining an approximate solution of (1.2) have been proposed. From the hierarchical identification principle [14,16], some efficient gradient-based iterative methods for solving Sylvester matrix equation were proposed in [13,19].…”

Section: Introductionmentioning

confidence: 99%

On RGI Algorithms for Solving Sylvester Tensor Equations

Zhang

Wang

2022

Taiwanese J. Math.

View full text Add to dashboard Cite

This paper is concerned with studying the relaxed gradient-based iterative method based on tensor format to solve the Sylvester tensor equation. From the information given by the previous steps, we further develop a modified relaxed gradient-based iterative method which converges faster than the method above. Under some suitable conditions, we prove that the introduced methods are convergent to the unique solution for any initial tensor. At last, we provide some numerical examples to show that our methods perform much better than the GI algorithm proposed by Chen and Lu (Math. Probl. Eng. 2013) both in the number of iteration steps and the elapsed CPU time.

show abstract

“…An efficient iterative method based on Hermitian and skew Hermitian splitting has been proposed in [19]. Krylov subspace based methods have been presented in [20][21][22][23][24][25][26] for solving Sylvester equations and generalized Sylvester equations. Recently based on the idea of a hierarchical identification principle [27][28][29], some efficient gradient based iterative algorithms for solving generalized Sylvester equations and coupled (general coupled) Sylvester equations have been proposed in [27,[30][31][32].…”

Section: Mathematical Problems In Engineeringmentioning

confidence: 99%

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

Zhang

Sheng

2017

Mathematical Problems in Engineering

View full text Add to dashboard Cite

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 equation + = . 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) matrix 0 is taken. Finally, two numerical examples are given to illustrate that the introduced iterative algorithms are more efficient.

show abstract

Preconditioned Galerkin and minimal residual methods for solving Sylvester equations

Cited by 5 publications

References 17 publications

A general iterative approach for solving the general constrained linear matrix equations system

A general iterative approach for solving the general constrained linear matrix equations system

On RGI Algorithms for Solving Sylvester Tensor Equations

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

Contact Info

Product

Resources

About