Further results on the convergence behaviour of CG and Ritz values ∗

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 behaviour of Ritz values of multiplying the weight functions by certain functions of polynomial growth. This will be shown not change the convergence behaviour but only to cause a delay of at most a certain number of steps. Alternatives for the usual Ritz error estimates will be given which are sometimes or always competitive. We will 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.