On the convergence rate of the conjugate gradients in presence of rounding errors *

among which Axelsson [2], Andersson [1], Jennings [8] and Axelsson and Lindskog [4], where explicit bounds on the number of iterations are derived under simple assumptions on the eigenvalue distribution, are particularly relevant for the discussion of preconditioning techniques. All of them however make the assumption of exact arithmetic computation and, since rounding errors imply some loss of orthogonality (see for instance [7]), their validity in the context of finite precision arithmetic remains subject to further examination. Greenbaum made recently [5] a thorough stability analysis of the conjugate gradient algorithm, mentioning already as corollary some conclusions about rounding error effects on the convergence rate. The convergence behaviour was further investigated by Strakos [10] and by Geenbaum and Strakos [6], who found