A dual adaptive procedure for the automatic determination of iteration parameters for Chebyshev acceleration

Chebyshev acceleration for a symmetrizable basic iterative method u ( n +1) = Gu ( n ) + k ; requires estimates of the extreme eigenvalues m ( G ) and M ( G ) of the iteration matrix G . Adaptive procedures are often used in order to obtain good estimates for m ( G ) and M ( G ). Some existing adaptive procedures are able to give an estimate of either m ( G ) or M ( G ) but not both on any given iteration. In this paper we present an adaptive procedure which can estimate both m ( G ) and M ( G ) at the same time and which has other useful properties. Numerical results are given which show the new procedure usually requires fewer iterations than previous procedures.