Further results on the convergence behavior of conjugate-gradients and Ritz values

Abstract

The usual estimates for conjugate gradients (CG) specify a non trivial rate of convergence right from the beginning. We investigate situations where the same can be said for Ritz values (considered as approximations to eigenvalues). We investigate the effect on the convergence behavior of Ritz values of multiplying the weight functions by certain functions of polynomial growth. This is shown not to change the convergence behavior but only to cause a delay of at most a certain number of steps. We give competitive alternatives for the usual Ritz error estimates. We show that, in a sense, CG errors can be considered as Ritz errors and vice versa. Hence, properties for CG errors should be expected to have parallels for Ritz values and vice versa. Finally, some smaller results are given.