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Gauss-Seidel Method

The Gauss-Seidel method (called Seidel's method by Jeffreys and Jeffreys 1988, p. 305) is a technique for solving the n equations of the linear system of equations Ax=b one at a time in sequence, and uses previously computed results as soon as they are available,

x_i^((k))=(b_i-sum_(j<i)a_(ij)x_j^((k))-sum_(j>i)a_(ij)x_j^((k-1)))/(a_(ii)).

There are two important characteristics of the Gauss-Seidel method should be noted. Firstly, the computations appear to be serial. Since each component of the new iterate depends upon all previously computed components, the updates cannot be done simultaneously as in the Jacobi method. Secondly, the new iterate x^((k)) depends upon the order in which the equations are examined. If this ordering is changed, the components of the new iterates (and not just their order) will also change.

In terms of matrices, the definition of the Gauss-Seidel method can be expressed as

x^((k))=(D-L)^(-1)(U x^((k-1))+b),

where the matrices D, -L, and -U represent the diagonal, strictly lower triangular, and strictly upper triangular parts of A, respectively.

The Gauss-Seidel method is applicable to strictly diagonally dominant, or symmetric positive definite matrices A.

See also

Jacobi Method, Linear System of Equations, Nonstationary Iterative Method, Stationary 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.Jeffreys, H. and Jeffreys, B. S. Methods of Mathematical Physics, 3rd ed. Cambridge, England: Cambridge University Press, pp. 305-306, 1988.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. Numerical Recipes in FORTRAN: The Art of Scientific Computing, 2nd ed. Cambridge, England: Cambridge University Press, pp. 864-866, 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.

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Gauss-Seidel Method

Cite this as:

Black, Noel and Moore, Shirley. "Gauss-Seidel Method." From MathWorld--A Wolfram Web Resource, created by Eric W. Weisstein. https://mathworld.wolfram.com/Gauss-SeidelMethod.html

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