A note on an SOR-like method for augmented systems

Li, Changjun; Zheng, Lianxi; Evans, D.; Zhang, Tie

doi:10.1093/imanum/23.4.581

Cited by 14 publications

(10 citation statements)

References 6 publications

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“…Theorem 4.1 shows that the GSOR method, when the optimal parameters are employed, has the same asymptotic convergence rate as the PCG method studied in [22] (here, we have assumed that x PCG := A −1 (b −By PCG ) is computed in exact arithmetic, see Method 3.1 in Section 3). However, we notice that this asymptotic convergence rate is global for the optimal GSOR method in the sense that it is an estimate with respect to the whole variable, but is partial for the PCG method in the sense that it is only an estimate with respect to the second block of the whole variable.…”

Section: Theorem 41 Consider the Gsor Method Letmentioning

confidence: 97%

“…Although the above PCG method is interesting and valuable the claims for its convergence rate in Theorems 1 and 2 in [22] are flawed because the error measures on the {x (k) PCG } and {y (k) PCG } components are different. In addition, the authors fail to mention that, compared with SOR-like and GSOR, PCG requires two extra inner products per iteration step and needs to compute the initial residual vector r (0) := B T A −1 (By (0) PCG − b) + q at the starting as well as to recover the final approximate vector x PCG := A −1 (b − By PCG ) at the terminating.…”

Section: Methods 31 (The Pcg Method)mentioning

confidence: 99%

“…Here we briefly discuss the results in [22] in which the classic conjugate gradient method of Hestenes and Stiefel (see [19,16]) is adapted, with preconditioning, to the solution of the augmented linear system (1.1).…”

Section: Related Workmentioning

confidence: 99%

See 2 more Smart Citations

On generalized successive overrelaxation methods for augmented linear systems

2005

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For the augmented system of linear equations, Golub, Wu and Yuan recently studied an SOR-like method (BIT 41(2001)71-85). By further accelerating it with another parameter, in this paper we present a generalized SOR (GSOR) method for the augmented linear system. We prove its convergence under suitable restrictions on the iteration parameters, and determine its optimal iteration parameters and the corresponding optimal convergence factor. Theoretical analyses show that the GSOR method has faster asymptotic convergence rate than the SOR-like method. Also numerical results show that the GSOR method is more effective than the SOR-like method when they are applied to solve the augmented linear system. This GSOR method is further generalized to obtain a framework of the relaxed splitting iterative methods for solving both symmetric and nonsymmetric augmented linear systems by using the techniques of vector extrapolation, matrix relaxation and inexact iteration. Besides, we also demonstrate a complete version about the convergence theory of the SOR-like method.

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Section: Theorem 41 Consider the Gsor Method Letmentioning

confidence: 97%

Section: Methods 31 (The Pcg Method)mentioning

confidence: 99%

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On generalized successive overrelaxation methods for augmented linear systems

2005

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“…The above assumptions are necessary to ensure the existence of a unique solution in the above augmented linear system (1.1). The augmented linear system (1.1) arises in many different applications of scientific computing such as weighted least-squares problems [18,22,27], finite element discretization of the Navier-Stokes equations [13][14][15], constrained optimization [25], equilibrium system and saddle point problems [7,19], etc.…”

Section: Introductionmentioning

confidence: 99%

Accelerated SOR-like method for augmented linear systems

Njeru

Guo

2015

Bit Numer Math

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By accelerating the SOR-like method by two parameters alpha and omega, we propose an accelerated SOR-like (ASOR) method for solving the augmented linear systems. The convergence of the ASOR method is discussed under suitable restrictions on the iteration parameters alpha and omega, and the experimental optimal values of the iteration parameters are determined. Numerical results are presented to show the efficiency of the ASOR method when the iteration parameters alpha and omega are suitably chosen.

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“…Recently, the augmented linear system has attracted more and more researchers and various kinds of iteration methods have been established and discussed. For example, the Uzawa-type methods [13,15], the preconditioned Krylov subspace methods [7,9,18], the relaxation methods [8,10,14,17], and the Hermitian and skew-Hermitian splitting methods [3][4][5][6]12], etc. Moreover, the singular augmented linear system has been specially studied in [11,19].…”

Section: Introductionmentioning

confidence: 99%

On Contraction and Semi-Contraction Factors of GSOR Method for Augmented Linear Systems

Chen¹,

Jiang

Zheng

2010

JCM

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The generalized successive overrelaxation (GSOR) method was presented and studied by Bai, Parlett and Wang [Numer. Math. 102(2005), pp.1-38] for solving the augmented system of linear equations, and the optimal iteration parameters and the corresponding optimal convergence factor were exactly obtained. In this paper, we further estimate the contraction and the semi-contraction factors of the GSOR method. The motivation of the study is that the convergence speed of an iteration method is actually decided by the contraction factor but not by the spectral radius in finite-step iteration computations. For the nonsingular augmented linear system, under some restrictions we obtain the contraction domain of the parameters involved, which guarantees that the contraction factor of the GSOR method is less than one. For the singular but consistent augmented linear system, we also obtain the semi-contraction domain of the parameters in a similar fashion. Finally, we use two numerical examples to verify the theoretical results and the effectiveness of the GSOR method.Mathematics subject classification: 65F10, 65F50; CR: G1.3.

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A note on an SOR-like method for augmented systems

Cited by 14 publications

References 6 publications

On generalized successive overrelaxation methods for augmented linear systems

On generalized successive overrelaxation methods for augmented linear systems

Accelerated SOR-like method for augmented linear systems

On Contraction and Semi-Contraction Factors of GSOR Method for Augmented Linear Systems

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