Stationary Iterative Method

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


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


where neither B nor c depends upon the iteration count k. 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 omega. 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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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., 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.

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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.

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