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 precondition-
ing 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 Martínez. 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 Martínez. Extensive numerical testing on a set of representative large SPD matrices gives evidence
of the acceleration provided by these spectral preconditioners.
