Simple-iteration method

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.

References

[1]	D.K. Faddeev, V.N. Faddeeva, "Computational methods of linear algebra" , Freeman (1963) (Translated from Russian)
[2]	I.S. Berezin, N.P. Zhidkov, "Computing methods" , Pergamon (1973) (Translated from Russian)
[3]	J.M. Ortega, W.C. Rheinboldt, "Iterative solution of non-linear equations in several variables" , Acad. Press (1970)
[4]	A.A. Samarskii, E.S. Nikolaev, "Numerical methods for grid equations" , 1–2 , Birkhäuser (1989) (Translated from Russian)