This paper develops an accelerated version of the steepest descent method by a two-step iteration. The new algorithm uses information with delay to define the iterations. Specifically, in the first step, a prediction of the new test point is calculated by using the gradient method with the exact minimal gradient steplength and then, a correction is computed by a weighted sum between the prediction and the iterate predecessor to the previous point. A convergence result is studied. Some numerical experiments are performed, in order to compare the efficiency and effectiveness of the proposed method with similar methods existing in the literature. The numerical results show that the new algorithm, presents a competitive performance to the classical conjugate gradient method, which makes this procedure a good alternative to solve large-scale problems.
Centro de Investigación en Matemáticas CIMAT A.C., march 2019