Two-stage spectral preconditioners for iterative eigensolvers

In this paper, we present preconditioning techniques to accelerate the convergence of Krylov solvers at each step of an Inexact Newton's method for the computation of the leftmost eigenpairs of large and sparse symmetric positive definite matrices arising in large-scale scientific computations. We propose a two-stage spectral preconditioning strategy: The first stage produces a very rough approximation of a number of the leftmost eigenvectors. The second stage uses these approximations as starting vectors and also to construct the tuned preconditioner from an initial inverse approximation of the coefficient matrix, as proposed by Martinez. In the framework of the Implicitly Restarted Lanczos method. The action of this spectral preconditioner results in clustering a number of the eigenvalues of the preconditioned matrices close to one. We also study the combination of this approach with a BFGS-style updating of the proposed spectral preconditioner as described by Bergamaschi and Martinez. Extensive numerical testing on a set of representative large SPD matrices gives evidence of the acceleration provided by these spectral preconditioners.