Smooth and adaptive gradient method with retards

The gradient method with retards (GMR) is a nonmonotone iterative method recently developed to solve large, sparse, symmetric, and positive definite linear systems of equations. Its performance depends on the retard parameter m. The larger the m, the faster the convergence, but also the faster the loss of precision is observed in the intermediate computations of the algorithm. This loss of precision is mainly produced by the nonmonotone behavior of the norm of the gradient which also increases with m. In this work, we first use a recently developed inexpensive technique to smooth down the nonmonotone behavior of the method. Then we show that it is possible to choose m adaptively during the process to avoid loss of precision. Our adaptive choice of m can be viewed as a compromise between numerical stability and speed of convergence. Numerical results on some classical test problems are presented to illustrate the good numerical properties.