Stationary iterative methods are methods for solving a linear
system of equations

where
is a given matrix and is a given vector. Stationary iterative methods can be expressed
in the simple form

where neither nor depends upon the iteration count . The four main stationary methods are the Jacobi
method , Gauss-Seidel method , successive
overrelaxation method (SOR), and symmetric
successive overrelaxation method (SSOR).

The Jacobi method is based on solving for every variable locally with respect to the other variables; one iteration corresponds to
solving for every variable once. It is easy to understand and implement, but convergence
is slow.

The Gauss-Seidel method is similar to the Jacobi method except that it uses updated values as soon as they are available. It
generally converges faster than the Jacobi method ,
although still relatively slowly.

The successive overrelaxation method can be derived from the Gauss-Seidel method
by introducing an extrapolation parameter . This method can converge faster than Gauss-Seidel by
an order of magnitude.

Finally, the symmetric successive overrelaxation method is useful as a preconditioner for nonstationary methods.
However, it has no advantage over the successive
overrelaxation method as a stand-alone iterative method.

See also Gauss-Seidel Method ,

Jacobi Method ,

Linear System of Equations ,

Nonstationary Iterative Method ,

Successive Overrelaxation Method ,

Symmetric Successive Overrelaxation
Method
This entry contributed by Noel Black and Shirley Moore, adapted from Barrett et al. (1994) (author's link )

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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. Varga, R.
Matrix
Iterative Analysis. Englewood Cliffs, NJ: Prentice-Hall, 1962. Young,
D. Iterative
Solutions of Large Linear Systems. New York: Academic Press, 1971. Referenced
on Wolfram|Alpha Stationary Iterative Method
Cite this as:
Black, Noel and Moore, Shirley . "Stationary Iterative Method." From MathWorld --A
Wolfram Web Resource, created by Eric W. Weisstein .
https://mathworld.wolfram.com/StationaryIterativeMethod.html

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