On semi-convergence of the generalized shift-splitting iteration method for singular nonsymmetric saddle point problems

Cao, Yang; Miao, Shu-Xin

doi:10.1016/j.camwa.2016.02.027

Cited by 32 publications

(23 citation statements)

References 19 publications

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“…Additionally, if B is rank deficient and B T u = 0, then it holds that λ = 0 or v = 0, λ = 0 by virtue of the second equation of (24). Similar to the derivation of (19), we also deduce (19) as B is rank deficient, v = 0 and λ = 0.…”

Section: Spectral Analysis Of the Mgssp Preconditioned Matrixsupporting

confidence: 53%

“…If B is rank deficient and u = 0, then λ = 0. Besides, if B is rank deficient and B T u = 0, then λ = 0 or λ satisfies the Inequalities (19). If B T u = 0, then the eigenvalues of the preconditioned matrix P −1 M GSSP A satisfy…”

Section: Spectral Analysis Of the Mgssp Preconditioned Matrixmentioning

confidence: 99%

“…Bai [3] derived some necessary and sufficient conditions to assure the semi-convergence of the HSS method. Chen et al [21] and Cao et al [19] investigated the generalized shift-splitting iteration method for singular saddle point problems. Very recently, Dou et al [24] introduced the modifying the parameterized inexact Uzawa (PIU) for singular saddle point problems, and Zheng and Lu [44] proved the semi-convergence of the upper and lower triangular (ULT) splitting iteration method for singular saddle point problems.…”

Section: Introductionmentioning

confidence: 99%

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A modified generalized shift-splitting preconditioner for nonsymmetric saddle point problems

et al. 2017

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For the nonsymmetric saddle point problems with nonsymmetric positive definite (1,1) parts, the modified generalized shift-splitting (MGSSP) preconditioner as well as the MGSSP iteration method are derived in this paper, which generalize the MSSP preconditioner and the MSSP iteration method newly developed by Huang and Su (J. Comput. Appl. Math. 2017), respectively. The convergent and semi-convergent analysis of the MGSSP iteration method are presented, and we prove that this method is unconditionally convergent and semi-convergent. In addition, some spectral properties of the preconditioned matrix are carefully analyzed. Numerical results demonstrate the robustness and effectiveness of the MGSSP preconditioner and the MGSSP iteration method, and also illustrate that the MGSSP iteration method outperforms the GSS and GMSS iteration methods, and the MGSSP preconditioner is superior to the shift-splitting (SS), generalized SS (GSS), modified SS (MSS) and generalized MSS (GMSS) preconditioners for the GMRES method for solving the nonsymmetric saddle point problems.for the saddle point problem (1), where α is a positive constant and I is the identity matrix.where α ≥ 0, β > 0 and I is the identity matrix. It is easy to see that P SS is a special case of P GSS when α = β. Numerical results in [21,20] confirmed that the GSS preconditioner is superior to the SS preconditioner.

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Section: Spectral Analysis Of the Mgssp Preconditioned Matrixsupporting

confidence: 53%

Section: Spectral Analysis Of the Mgssp Preconditioned Matrixmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

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A modified generalized shift-splitting preconditioner for nonsymmetric saddle point problems

et al. 2017

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“…The reported numerical results show that, the SEPSS preconditioner outperforms the other methods from both the iteration counts and CPU time point of view. As the numerical results show the strategies presented in (17) are quite suitable to estimate the optimal values of α and β.…”

Section: Semi-convergence Of Epss Iteration Methodsmentioning

confidence: 92%

“…(b) When A is PD, the P GSS preconditioner was studied in [16,17,34,36,38] and P ESS preconditioner was studied in [35].…”

Section: Semi-convergence Of Epss Iteration Methodsmentioning

confidence: 99%

An extension of the positive-definite and skew-Hermitian splitting method for preconditioning of generalized saddle point problems

Masoudi

Salkuyeh

2020

Computers & Mathematics with Applications

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A parameterized extended shift‐splitting preconditioner for nonsymmetric saddle point problems

Vakili

Ebadi

Vuik

2022

Numerical Linear Algebra App

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In this article, a parameterized extended shift-splitting (PESS) method and its induced preconditioner are given for solving nonsingular and nonsymmetric saddle point problems with nonsymmetric positive definite (1,1) part. The convergence analysis of the PESS iteration method is discussed. The distribution of eigenvalues of the preconditioned matrix is provided. A number of experiments are given to verify the efficiency of the PESS method for solving nonsymmetric saddle-point problems.

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On semi-convergence of the generalized shift-splitting iteration method for singular nonsymmetric saddle point problems

Cited by 32 publications

References 19 publications

A modified generalized shift-splitting preconditioner for nonsymmetric saddle point problems

A modified generalized shift-splitting preconditioner for nonsymmetric saddle point problems

An extension of the positive-definite and skew-Hermitian splitting method for preconditioning of generalized saddle point problems

A parameterized extended shift‐splitting preconditioner for nonsymmetric saddle point problems

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