On the effectiveness of adaptive Chebyshev acceleration for solving systems of linear equations

Abstract

A symmetrizable basic iterative method u ( n + 1) = Gu ( n) + k can be greatly accelerated by Chebyshev acceleration. This method requires estimates of the extreme eigenvalues m( G) and M( G) of the iteration matrix G. An adaptive procedure for finding the eigenvalues was introduced by Hageman and Young (1981). We describe a scheme of using contours to test the effectiveness of this adaptive Chebyshev acceleration procedure. We conclude that the adaptive process is not sensitive to the starting estimate unless it is very close to M( G). Moreover, the adaptive procedure takes at most 35% more work than the optimal nonadaptive procedure.