On the Line Successive Overrelaxation Method

Youssef, I. K.

doi:10.11648/j.acm.20160503.12

Cited by 2 publications

(2 citation statements)

References 14 publications

Supporting

Mentioning

Contrasting

Order By: Relevance

“…If % > (( ), then b?B(%)@ (−1,0). This result means that % should be chosen from the interval (% , % ), where % is defined in (11) and % = K K = .…”

Section: Convergence Of the Methodsmentioning

confidence: 99%

See 1 more Smart Citation

On Optimal Parameter Not Only for the SOR Method

Grzegórski¹

2019

ACM

View full text Add to dashboard Cite

The Jacobi, Gauss-Seidel and SOR methods belong to the class of simple iterative methods for linear systems. Because of the parameter ω, the SOR method is more effective than the Gauss-Seidel method. Here, a new approach to the simple iterative methods is proposed. A new parameter q can be introduced to every simple iterative method. Then, if a matrix of a system is positive definite and the parameter q is sufficiently large, the method is convergent. The original Jacobi method is convergent only if the matrix is diagonally dominated, while the Jacobi method with the parameter q is convergent for every positive definite matrix. The optimality criterion for the choice of the parameter q is given, and thus, interesting results for the Jacobi, Richardson and Gauss-Seidel methods are obtained. The Gauss-Seidel method with the parameter q, in a sense, is equivalent to the SOR method. From the formula for the optimal value of q results the formula for optimal value of ω. Up to present, this formula was known only in special cases. Practical useful approximate formula for optimal value ω is also given. The influence of the parameter q on the speed of convergence of the simple iterative methods is shown in a numerical example. Numerical experiments confirm: for very large scale systems the speed of convergence of the SOR method with optimal or approximate parameter ω is near the same (in some cases better) as the speed of convergence of the conjugate gradients method.

show abstract

“…If % > (( ), then b?B(%)@ (−1,0). This result means that % should be chosen from the interval (% , % ), where % is defined in (11) and % = K K = .…”

Section: Convergence Of the Methodsmentioning

confidence: 99%

“…Here, we give such formula for every positive definite and symmetric matrix. In papers [11][12][13] the authors use an approximate value of the parameter . Nevanlinna [14] explains why the SOR and conjugate gradients methods are essentially equally fast for discretized Laplacians.…”

Section: Introductionmentioning

confidence: 99%