Symmetric block-SOR methods for rank-deficient least squares problems

Darvishi, M. T.; Khani, F.; Hamedi-Nezhad, S.; Zheng, Bing

doi:10.1016/j.cam.2007.03.023

Cited by 9 publications

(5 citation statements)

References 12 publications

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“…Liu [15] applied the minimization technique in Equation (6) to search for the optimal value of the relaxation parameter used in the SOR at each iteration step. For dealing with the different extensions of the SOR and AOR for different kinds of linear systems, there are many papers such as the symmetric SOR method for the augmented systems [16][17][18][19][20], the generalized AOR method for the linear complementarity problem [21], the generalized least-square problems [22], and a class of generalized saddle point problems [23], as well as the symmetric successive over-relaxation method for the rank-deficient linear systems [24], the symmetric block SOR methods for the rank-deficient least-squares problems [25].…”

Section: Introductionmentioning

confidence: 99%

Dynamical Optimal Values of Parameters in the SSOR, AOR, and SAOR Testing Using Poisson Linear Equations

Liu,

El-Zahar,

Chang

2023

Mathematics

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This paper proposes a dynamical approach to determine the optimal values of the parameters used in each iteration of the symmetric successive over-relaxation (SSOR), accelerated over-relaxation (AOR), and symmetric accelerated over-relaxation (SAOR) methods for solving linear equation systems. When the optimal values of the parameters in the SSOR, AOR, and SAOR are hard to determine as some fixed values, they are obtained by minimizing the merit functions, which are based on the maximal projection technique between the left- and right-hand-side vectors, which involves the input vector, the previous step values of the variables, and the parameters. The novelty is a new concept of the dynamical optimal values of the parameters, instead of the fixed values and the maximal projection technique. In a preferred range, the optimal values of the parameters can be quickly determined by using the golden section search algorithm with a loose convergence criterion. Without knowing and having the theoretical optimal values in general, the new methods might provide an alternative and proper choice of the values of the parameters for accelerating the convergence speed. Numerical testings of the linear Poisson equation discretized to a matrix–vector form and a Lyapunov equation form were used to assess the performance of the DOSSOR, DOAOR, and DOSAOR dynamical optimal methods.

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Section: Introductionmentioning

confidence: 99%

Dynamical Optimal Values of Parameters in the SSOR, AOR, and SAOR Testing Using Poisson Linear Equations

Liu,

El-Zahar,

Chang

2023

Mathematics

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“…Also, Zhang and Lu [12] presented the generalized symmetric SOR (GSSOR) method for solving large sparse augmented systems. One may find details of symmetric iterative methods to solve different kinds of linear systems in [13][14][15][16][17].…”

Section: Introductionmentioning

confidence: 99%

A modified symmetric successive overrelaxation method for augmented systems

Darvishi

Hessari

2011

Computers & Mathematics with Applications

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a b s t r a c tIn this paper, we establish a modified symmetric successive overrelaxation (MSSOR) method, to solve augmented systems of linear equations, which uses two relaxation parameters. This method is an extension of the symmetric SOR (SSOR) iterative method. The convergence of the MSSOR method for augmented systems is studied. Numerical examples show that the new method is an efficient method.

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“…Tian et al [7,11] studied the AOR method. For rank deficient linear least squares problems, the symmetric SOR(SSOR) method is also studied, see, e.g, [3][4][5]15]. Recently, Yun et al [6,13] proposed the unsymmetric SOR(USSOR) method to solve saddle point problems.…”

Section: Introductionmentioning

confidence: 99%

A note on the optimal parameters of USSOR method for solving linear least squares problems

Zhang

2019

Filomat

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For solving rank deficient linear least squares problems, unsymmetric successive overrelaxation (USSOR) type methods are investigated by some researchers recently. In this note, we continue to study the USSOR method for solving rank deficient linear least squares problems and obtain the optimal iteration parameters and the corresponding optimal convergence factors. Numerical experiments are given to examine the feasibility and effectiveness of the USSOR method with optimal parameters.

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Symmetric block-SOR methods for rank-deficient least squares problems

Cited by 9 publications

References 12 publications

Dynamical Optimal Values of Parameters in the SSOR, AOR, and SAOR Testing Using Poisson Linear Equations

Dynamical Optimal Values of Parameters in the SSOR, AOR, and SAOR Testing Using Poisson Linear Equations

A modified symmetric successive overrelaxation method for augmented systems

A note on the optimal parameters of USSOR method for solving linear least squares problems

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