Gradient method with retards and generalizations

A generalization of the steepest descent and other methods for solving a la rge 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)))(Ax(k) - b), where lambda(x(nu(k))) is the steepest descent step at a previous iteration nu(k) is an element of {k; k - 1 ,..., max {0, k - m} }. The global convergence to the solution of the problem is established und er a more general framework, and numerical experiments are performed that s uggest that some strategies for the choice of nu(k) give rise to efficient methods for obtaining approximate solutions of the system.