Spectral Variants of Krylov Subspace Methods

Krylov iterative methods usually solve an optimization problem, per iteration, to obtain a vector whose components are the step lengths associated with the previous search directions. This vector can be viewed as the solution of a multiparameter optimization problem. In that sense, Krylov methods can be combined with the spectral choice of step length that has recently been developed to accelerate descent methods in optimization. In this work, we discuss different spectral variants of Krylov methods and present encouraging preliminary numerical experiments, with and without preconditioning.