Jacobi Method

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, and strictly upper triangular parts of , respectively.

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.