Jacobi Method (original) (raw)
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The Jacobi method is a method of solving a matrix equation on a matrix that has no zeros along its main diagonal (Bronshtein and Semendyayev 1997, p. 892). Each diagonal element is solved for, and an approximate value plugged in. The process is then iterated until it converges. This algorithm is a stripped-down version of the Jacobi transformation method of matrix diagonalization.
The Jacobi method is easily derived by examining each of the 
equations in the linear system of equations 
in isolation. If, in the 
th equation
 |
(1) |
|---|
solve for the value of 
while assuming the other entries of 
remain fixed. This gives
 |
(2) |
|---|
which is the Jacobi method.
In this method, the order in which the equations are examined is irrelevant, since the Jacobi method treats them independently. The definition of the Jacobi method can be expressed with matrices as
 |
(3) |
|---|
where the matrices 
,
, and 
represent thediagonal,strictly lower triangular, andstrictly upper triangular parts of 
, respectively.
See also
Gauss-Seidel Method, Stationary Iterative Method, Successive Overrelaxation Method, Symmetric Successive Overrelaxation Method
Portions of this entry contributed by Noel Black and Shirley Moore, adapted from Barrett et al. (1994) (author's link)
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References
Acton, F. S. Numerical Methods That Work, 2nd printing. Washington, DC: Math. Assoc. Amer., pp. 161-163, 1990.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.Bronshtein, I. N. and Semendyayev, K. A. Handbook of Mathematics, 3rd ed. New York: Springer-Verlag, p. 892, 1997.Hageman, L. and Young, D. Applied Iterative Methods. New York: Academic Press, 1981.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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Cite this as:
Black, Noel; Moore, Shirley; and Weisstein, Eric W. "Jacobi Method." From MathWorld--A Wolfram Resource. https://mathworld.wolfram.com/JacobiMethod.html