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 $x_{k+1} = x_k - \lambda (x_{\nu(k)}) (A x_k - b)$, where $\lambda (x_{\nu(k)})$ is the steepest descent step at a previous iteration $\nu(k) \in \{k, k-1, \ldots,$ 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 $\nu(k)$ give rise to efficient methods for obtaining approximate solutions of the system.