Analysis of the SOR Iteration for the 9-Point Laplacian

Adams, Loyce M.; LeVeque, Randall J.; Young, David M.

doi:10.21236/ada212718

Cited by 5 publications

(8 citation statements)

References 2 publications

(3 reference statements)

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“…obviously increases with w, and differentiation confirms that the first line decreases with w. Therefore the optimal w is the crossover value w -1 = a2 = ('cosh) 2 , which reduces after a little work to 2 _of=tl 2 -2 h.…”

mentioning

confidence: 57%

Fourier Analysis of the Sor Iteration

LeVeque¹,

Trefethen²

1986

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The SOR iteration for solving linear systems of equations depends upon an overrelaxation factor w. We show that for the standard model problem of Poisson's equation on a rectangle, the optimal w and corresponding convergence rate can be rigorously obtained by Fourier analysis. The trick is to tilt the space-time grid so that the SOR stencil becomes symmetrical. The tilted grid also gives insight into the relation between convergence rates of several variants.

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confidence: 57%

Fourier Analysis of the Sor Iteration

LeVeque¹,

Trefethen²

1986

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“…We then show that PSOR is just the SOR method applied to a reordered linear system; hence, the SOR theory can be applied to the analysis of PSOR. In particular, we prove that the PSOR method has the same asymptotic rate of convergence as the red/black SOR (R/B SOR) method for the five-point stencil on strip and block partitions and as the R/B/G/O SOR method [3] for nine-point stencil strip partitions. We then demonstrate the parallel performance of the PSOR method on four different message-passing multiprocessors (a KSR1, the Intel Delta, a Paragon, and an IBM SP2) for solving a Poisson model problem.…”

Section: Introductionmentioning

confidence: 84%

“…Defined by using the multicolor ordering technique, the multicolor SOR method is a widely used parallel version of SOR and has been studied by many authors (such as Adams and Ortega [1], Adams and Jordan [2], and Adams, LeVeque, and Young [3]). Since the multicolor SOR method is parallel only within the same color, the two-color SOR method (usually referred to as the red/black SOR method) is preferred; but for some complicated problems more than two colors are required to define a multicolor ordering.…”

Section: Introductionmentioning

confidence: 99%

New Parallel SOR Method by Domain Partitioning

Xie¹,

Adams²

1999

SIAM J. Sci. Comput.

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In this paper we propose and analyze a new parallel SOR method, the PSOR method, formulated by using domain partitioning and interprocessor data communication techniques. We prove that the PSOR method has the same asymptotic rate of convergence as the Red/Black (R/B) SOR method for the five-point stencil on both strip and block partitions, and as the four-color (R/B/G/O) SOR method for the nine-point stencil on strip partitions. We also demonstrate the parallel performance of the PSOR method on four different MIMD multiprocessors (a KSR1, an Intel Delta, a Paragon, and an IBM SP2

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“…It is very convenient to control the convergence and convergence speed of the SOR algorithm by selecting the appropriate relaxation factor. Therefore, the highly efficient and controllable SOR method has a wide range of applications in the fields of computational science and engineering computing, see Cai et al (2008), Adams et al (1988), Bentrcia et al (2010) and Hwang and Lin (1992).…”

Section: Introductionmentioning

confidence: 99%

Modified Newton-GSOR method for solving complex nonlinear systems with symmetric Jacobian matrices

Xiao

2020

Comp. Appl. Math.

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In this paper, a new method for solving complex nonlinear systems with symmetric Jacobian matrices is proposed-modified Newton generalized successive overrelaxation (MN-GSOR) iteration method. The new MN-GSOR method helps us to solve nonlinear systems using the GSOR method as an inner iterative approximation for solving the Newton equations. Next, we use the Hölder continuous condition instead of the Lipschitz assumption to analyze and prove the convergence properties of the MN-GSOR method. Numerical experiments are conducted to show the superiority and effectiveness of MN-GSOR method, and comparisons are made to see the advantages over some other recently proposed methods. Furthermore, when the imaginary part of the Jacobian matrix is symmetric but non-definite, some recently proposed methods are not applicable, but the MN-GSOR method still works well. Keywords Complex systems of nonlinear equations • Convergence analysis • Modified Newton-PMHSS • Modified Newton-GSOR • Modified Newton-HSS Mathematics Subject Classification 60F10 • 60F50 with F : D ⊂ C n → C n being nonlinear and continuously differentiable. Communicated by Zhong-Zhi Bai.

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Analysis of the SOR Iteration for the 9-Point Laplacian

Cited by 5 publications

References 2 publications

Fourier Analysis of the Sor Iteration

Fourier Analysis of the Sor Iteration

New Parallel SOR Method by Domain Partitioning

Modified Newton-GSOR method for solving complex nonlinear systems with symmetric Jacobian matrices

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