An iterative method for solving a system of linear algebraic equations . The solution is found as the limit of a sequence
the terms of which are computed from the formula
where the are the entries of the matrix , the components of the vector , and the diagonal entries of are assumed to be . The use of (*) differs from the simple-iteration method only in the fact that, at step , computation of the -th component utilizes the previously computed -th approximations of the first components.
The Seidel method can be expressed in matrix notation as follows. If , where
then formula (*) in matrix notation is . The Seidel method is equivalent to simple iteration applied to the system , which is equivalent to the initial system. The method is convergent if and only if all the eigenvalues of the matrix are less than 1 in absolute value, or, equivalently, if all roots of the equation are less than 1 in absolute value.
In practical work, the following sufficient conditions for convergence are more convenient. 1) Suppose that , , for all , ; then the Seidel method is convergent and the following estimate holds for the rate of convergence:
2) If is a positive-definite Hermitian matrix, then the Seidel method is convergent.
Among the available modifications of the Seidel method are methods that employ preliminary transformation of the system into an equivalent system (see ).
The method was proposed by L. Seidel in .
|||L. Seidel, Abh. Bayer. Akad. Wiss. Math.-Naturwiss. Kl. , 11 : 3 (1874) pp. 81–108|
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Seidel method. G.D. Kim (originator), Encyclopedia of Mathematics. URL: http://www.encyclopediaofmath.org/index.php?title=Seidel_method&oldid=18222