Gradient Method with Retards and Generalizations

A generalization of the steepest descent and other methods for solving a large scale symmetric positive definitive system Ax = b is presented. Given a positive integer m, the new iteration is given by xk+1 = xk −λ(xν(k))(Axk − b), where λ(xν(k)) is the steepest descent step at a previous iteration ν(k) ∈ {k, k−1, . . . , max{0, k−m}}. The global convergence to the solution of the problem is established under a more general framework, and numerical experiments are performed that suggest that some strategies for the choice of ν(k) give rise to efficient methods for obtaining approximate solutions of the system.