A new version of successive approximations method for solving Sylvester matrix equations

Kaabi, Amer; Kerayechian, Asghar; Toutounian, Faezeh

doi:10.1016/j.amc.2006.08.007

Cited by 11 publications

(4 citation statements)

References 15 publications

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“…An iteration based on block Arnoldi was recently proposed in [149], where a series of Sylvester equations is solved by means of the block method above; however, the full rank of the right-hand side cannot be ensured after the first iteration of the process.…”

Section: Sylvester Equation Largementioning

confidence: 99%

Computational Methods for Linear Matrix Equations

Simoncini¹

2016

SIAM Rev.

376

428

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Given the matrices A, B, C, D and E of conforming dimensions, we consider the linear matrix equation AXE + DXB = C in the unknown matrix X. Our aim is to provide an overview of the major algorithmic developments that have taken place in the past few decades in the numerical solution of this and of related problems, which are becoming a reliable tool in the numerical formulation of advanced application models.

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Section: Sylvester Equation Largementioning

confidence: 99%

Computational Methods for Linear Matrix Equations

Simoncini¹

2016

SIAM Rev.

376

428

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“…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.

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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.

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“…In practical applications, we solve the linear matrix equations of large dimensions by effective iterative methods. There are several ideas to formulate an iterative procedure, namely, one can use matrix sign function [5], block recursion [6,7], Krylov subspace [8,9], Hermitian and skew-Hermitian splitting [10,11], and other related research works; see, e.g., [12][13][14][15]. In the recent decade, the ideas of gradients, hierarchical identification and minimization of associated norm-error functions have encouraged and brought about many researches; see, e.g., [16][17][18][19][20][21][22][23][24][25][26][27][28].…”

Section: Introductionmentioning

confidence: 99%

Gradient-descent iterative algorithm for solving a class of linear matrix equations with applications to heat and Poisson equations

Kittisopaporn

Chansangiam

2020

Adv Differ Equ

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In this paper, we introduce a new iterative algorithm for solving a generalized Sylvester matrix equation of the form p t=1 A t XB t = C which includes a class of linear matrix equations. The objective of the algorithm is to minimize an error at each iteration by the idea of gradient-descent. We show that the proposed algorithm is widely applied to any problems with any initial matrices as long as such problem has a unique solution. The convergence rate and error estimates are given in terms of the condition number of the associated iteration matrix. Furthermore, we apply the proposed algorithm to sparse systems arising from discretizations of the one-dimensional heat equation and the two-dimensional Poisson's equation. Numerical simulations illustrate the capability and effectiveness of the proposed algorithm comparing to well-known methods and recent methods.

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A new version of successive approximations method for solving Sylvester matrix equations

Cited by 11 publications

References 15 publications

Computational Methods for Linear Matrix Equations

Computational Methods for Linear Matrix Equations

On RGI Algorithms for Solving Sylvester Tensor Equations

Gradient-descent iterative algorithm for solving a class of linear matrix equations with applications to heat and Poisson equations

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