Regular splitting with arbitarily small in spectral norm iteration matrix
by
I. Faragó
Eötvös Loránd University, Budapest

In order to solve the system of linear algebraic equations

A x=b, (1)

usually we construct the one step iteration of the form

Mx(j+1)=Nx(j)+b,     j=0, 1, ... (2)

where M and N defines a splitting of A

A=M-N. (3)

If the splitting (3) is regular (or weak regular) then the iteretive process (2) has a lot of good qualitative advantages. On the other hand, we make an effort to construct a regular splitting such that the rate of the convergence of the iteration (2) to the solution of (1) was convergent as fast as possible. Clearly, the iteration may have at most linear convergence. To the ratio of the possible "best" regular splitting is addressed the following

Theorem 1 Assume that A is symmetric positive definite irreducible matrix. Then for any number p > 0 there exists a regular splitting (3) such that s(M^-1N) < p where s(M^-1N) denotes the spectral radius of the itertion matrix M^-1N.

Date received: January 28, 2000