# Simple-iteration method

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A method for approximately solving a system of linear algebraic equations that can be transformed to the form and whose solution is looked for as the limit of a sequence , where is an initial approximation. In order that the simple-iteration method converges for any initial approximation it is necessary and sufficient that all eigenvalues of are less than one in modulus; it is sufficient that some norm of is less than one. If in some norm, compatible with the norm of a vector , satisfies , then the simple-iteration method converges at the rate of a geometric series and the estimate holds for its error.

In the case of a cubic, octahedral or spherical vector norm, the condition is fulfilled if

1) , ;

2) , ;

3) .

The simplest version of the method corresponds to the case when one takes , where is the identity matrix, for . If all diagonal entries of are non-zero, then, choosing and , where is the diagonal matrix with as diagonal entries those of , one obtains the Jacobi method or the method of simultaneous displacement.

A particular case of the simple-iteration method is the method with and , where is an iteration parameter, chosen from the condition that the norm of is minimal with respect to . If and are the minimal and maximal eigenvalues of a symmetric positive-definite matrix and , then one has for the matrix in the spherical norm the estimate , with .

For a system of non-linear algebraic equations the simple-iteration method has the form The problem of choosing the iteration parameter is solved in dependence on the differentiability properties of the . Often it is subjected to the requirement that the method converges locally in a neighbourhood of a solution.

How to Cite This Entry:
Simple-iteration method. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Simple-iteration_method&oldid=16747
This article was adapted from an original article by E.S. Nikolaev (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article