Nested splitting conjugate gradient method for matrix equation<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si12.gif" display="inline" overflow="scroll"><mml:mi>A</mml:mi><mml:mi>X</mml:mi><mml:mi>B</mml:mi><mml:mo>=</mml:mo><mml:mi>C</mml:mi></mml:math>and preconditioning

Zak, Mohammad Khorsand; Toutounian, Faezeh

doi:10.1016/j.camwa.2013.05.004

Cited by 25 publications

(6 citation statements)

References 14 publications

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“…In this paper, we present the nested splitting conjugate gradient (NSCG) iterative method for solving the generalized Sylvester equation AXB + CXD = E with large sparse coefficient matrices, which is an extension of [16][17][18]. Based on the NSCG method, we propose the preconditioned nested splitting conjugate gradient (PNSCG) iterative method, which extends the scope of the NSCG iteration method in applications.…”

Section: Discussionmentioning

confidence: 98%

“…In this section, we will give an implementation of the NSCG method for the generalized Sylvester equation (1.1) and present sufficient conditions for the convergent splitting when the coefficient matrices are nonsymmetric, which extend those results in [16][17][18].…”

Section: The Nscg Methodsmentioning

confidence: 96%

“…Our purpose in the present paper is twofold. Firstly, we present the nested splitting conjugate gradient (NSCG) iterative method for solving the generalized Sylvester equation (1.1) by using the symmetric and skew-symmetric splitting of the matrices A, B, C and D. This method was first proposed in [16] for solving system of linear equations Ax = b, then was extended to solve the matrix equation AXB = C and AX + XB = C , respectively, see [17,18]. For more detail on this method, we refer to [28,29] and references therein.…”

Section: Introductionmentioning

confidence: 99%

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A preconditioned nested splitting conjugate gradient iterative method for the large sparse generalized Sylvester equation

2014

Computers & Mathematics with Applications

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a b s t r a c tA nested splitting conjugate gradient (NSCG) iterative method and a preconditioned NSCG (PNSCG) iterative method are presented for solving the generalized Sylvester equation with large sparse coefficient matrices, respectively. Both methods are actually inner/outer iterations, which employ the CG-like method as inner iteration to approximate each outer iteration, while each outer iteration is induced by a convergent and symmetric positive definite splitting of the coefficient matrices. Convergence conditions of both methods are studied in depth and numerical experiments demonstrate the efficiency of the proposed methods. Moreover, experimental results show that the PNSCG method is more accurate, robust and effective than the NSCG method.

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Section: Discussionmentioning

confidence: 98%

Section: The Nscg Methodsmentioning

confidence: 96%

Section: Introductionmentioning

confidence: 99%

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A preconditioned nested splitting conjugate gradient iterative method for the large sparse generalized Sylvester equation

2014

Computers & Mathematics with Applications

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“…For Example 1, we compare ADMM with the Hermitian and skew‐Hermitian splitting (HSS) iteration method , the gradient‐based iteration (GI) method and the nested splitting conjugate gradient (NSCG) method . For the HSS method, we choose ε k =0.01, η k =0.01 when γ =0, and ε k =0.1, η k =0.1 when γ =1.…”

Section: Numerical Experimentsmentioning

confidence: 99%

“…presented gradient based iteration methods as well as least squares based iteration methods for the linear matrix equation , which were extensions of the Jacobi and Gauss‐Seidel iterations for A x = b . In , a class of nested splitting conjugate gradient (NSCG) methods were proposed, which were extensions of the NSCG method for A x = b .…”

Section: Introductionmentioning

confidence: 99%

The Unified Frame of Alternating Direction Method of Multipliers for Three Classes of Matrix Equations Arising in Control Theory

2017

Asian Journal of Control

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In this paper, the unified frame of alternating direction method of multipliers (ADMM) is proposed for solving three classes of matrix equations arising in control theory including the linear matrix equation, the generalized Sylvester matrix equation and the quadratic matrix equation. The convergence properties of ADMM and numerical results are presented. The numerical results show that ADMM tends to deliver higher quality solutions with less computing time on the tested problems.

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On the Krylov subspace methods based on tensor format for positive definite Sylvester tensor equations

Beik

Saberi-Movahed

Ahmadi‐Asl

2016

Numerical Linear Algebra App

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This paper deals with studying some of well-known iterative methods in their tensor forms to solve a Sylvester tensor equation. More precisely, the tensor form of the Arnoldi process and full orthogonalization method are derived by using a product between two tensors. Then tensor forms of the conjugate gradient and nested conjugate gradient algorithms are also presented. Rough estimation of the required number of operations for the tensor form of the Arnoldi process is obtained, which reveals the advantage of handling the algorithms based on tensor format over their classical forms in general. Some numerical experiments are examined, which confirm the feasibility and applicability of the proposed algorithms in practice. Copyright

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Nested splitting conjugate gradient method for matrix equationAXB=Cand preconditioning

Cited by 25 publications

References 14 publications

A preconditioned nested splitting conjugate gradient iterative method for the large sparse generalized Sylvester equation

A preconditioned nested splitting conjugate gradient iterative method for the large sparse generalized Sylvester equation

The Unified Frame of Alternating Direction Method of Multipliers for Three Classes of Matrix Equations Arising in Control Theory

On the Krylov subspace methods based on tensor format for positive definite Sylvester tensor equations

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