The method of steepest descent, or the optimum gradient method, has been known to mathematicians since the time of Cauchy [l]. Others who have discussed this method include Curry 12], Forsythe and Motzkin [3], Householder [4], Kantorovitch [5], and Temple [6]. Its infrequent application in computational work is no doubt due to the slowness with which it converges. This slowness of convergence is unfortunately generally true of gradient methods. However, with the advent of large-scale computing S it has become feasible to seriously consider em in practical numerical analysis. In a forthcoming paper, Hestenes and Stein [7] discuss a large class of gradient procedures for solving systems of linear equations. These procedures contain the optimum gradient method as a special case. The present note is mainly a report on some numerical experiments with them that were carried out on the IBM Card-Programmed Electronic Calculator at the Institute for Numerical Analysis of the National Bureau of Standards. Some attention is also given I to an experiment in which the problem of solving a system of linear equations was changed to an equivalent eigenvalue problem and then solved by a modification of one of the gradient methods discussed by Hestenes and Karush [8]. The most striking result of the experiments indicates that there is a large class of gradient methods that is self-accelerating, that is, which irregularly shows a large increase in the rate of convergence without the introduction of any modification in the computational routine. This behavior is in sharp contrast to that of the method of steepest descent, which in the light of the present results can no longer be considered as optimum from an over-all point of view unless modified by some special accelerating routine [9j.