A note on conjugate gradient convergence - Part III

In this paper we again consider the rate of convergence of the conjugate gr adient method. We start with a general analysis of the conjugate gradient m ethod for uniformly bounded solutions vectors and matrices whose eigenvalue s are uniformly bounded and positive. We show that in such cases a fixed fi nite number of iterations of the method gives some fixed amount of improvem ent as the the size of the matrix tends to infinity. Then we specialize to the finite element (or finite difference) scheme for the problem y "(x) = g beta(x), y(0) = y(1) = 0. We show that for some classes of function gp we see this same effect. For other functions we show that the gain made by per forming a fixed number of iterations of the method tends to zero as the siz e of the matrix tends to infinity. Mathematics Subject Classification (1991 ): 65F10.