A note on conjugate gradient convergence – Part III

Summary. In this paper we again consider the rate of convergence of the conjugate gradient method. We start with a general analysis of the conjugate gradient method for uniformly bounded solutions vectors and matrices whose eigenvalues are uniformly bounded and positive. We show that in such cases a fixed finite number of iterations of the method gives some fixed amount of improvement 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 $g_\beta$ we see this same effect. For other functions we show that the gain made by performing a fixed number of iterations of the method tends to zero as the size of the matrix tends to infinity.