Successive Overrelaxation Method
المؤلف:
Barrett, R.; Berry, M.; Chan, T. F.; Demmel, J.; Donato, J.; Dongarra, J.; Eijkhout, V.; Pozo, R.; Romine, C.; and van der Vorst, H.
المصدر:
emplates for the Solution of Linear Systems: Building Blocks for Iterative Methods, 2nd ed. Philadelphia, PA: SIAM, 1994. http://www.netlib.org/linalg/html_templates/Templates.html.
الجزء والصفحة:
...
1-12-2021
1225
Successive Overrelaxation Method
The successive overrelaxation method (SOR) is a method of solving a linear system of equations 
derived by extrapolating the Gauss-Seidel method. This extrapolation takes the form of a weighted average between the previous iterate and the computed Gauss-Seidel iterate successively for each component,
where 
denotes a Gauss-Seidel iterate and 
is the extrapolation factor. The idea is to choose a value for 
that will accelerate the rate of convergence of the iterates to the solution.
In matrix terms, the SOR algorithm can be written as
where the matrices 
, 
, and 
represent the diagonal, strictly lower-triangular, and strictly upper-triangular parts of 
, respectively.
If 
, the SOR method simplifies to the Gauss-Seidel method. A theorem due to Kahan (1958) shows that SOR fails to converge if 
is outside the interval 
.
In general, it is not possible to compute in advance the value of 
that will maximize the rate of convergence of SOR. Frequently, some heuristic estimate is used, such as 
where 
is the mesh spacing of the discretization of the underlying physical domain.
REFERENCES:
Barrett, R.; Berry, M.; Chan, T. F.; Demmel, J.; Donato, J.; Dongarra, J.; Eijkhout, V.; Pozo, R.; Romine, C.; and van der Vorst, H. Templates for the Solution of Linear Systems: Building Blocks for Iterative Methods, 2nd ed. Philadelphia, PA: SIAM, 1994. http://www.netlib.org/linalg/html_templates/Templates.html.
Hageman, L. and Young, D. Applied Iterative Methods. New York: Academic Press, 1981.
Kahan, W. Gauss-Seidel Methods of Solving Large Systems of Linear Equations. Ph.D. thesis. Toronto, Canada, University of Toronto, 1958.
Press, W. H.; Flannery, B. P.; Teukolsky, S. A.; and Vetterling, W. T. "Successive Overrelaxation (SOR)." Numerical Recipes in FORTRAN: The Art of Scientific Computing, 2nd ed. Cambridge, England: Cambridge University Press, pp. 866-869, 1992.
Varga, R. Matrix Iterative Analysis. Englewood Cliffs, NJ: Prentice-Hall, 1962.
Young, D. Iterative Solutions of Large Linear Systems. New York: Academic Press, 1971.
الاكثر قراءة في الرياضيات التطبيقية
اخر الاخبار
اخبار العتبة العباسية المقدسة
