Jacobi-Davidson Methods
  G. Sleijpen and H. van der Vorst

The Lanczos method (see §4.4) is quite effective in computing eigenvalues in the ends of the spectrum of $A$ if these eigenvalues are well separated from the remaining spectrum, or if it is applied to a shifted and inverted matrix operator $(A - \sigma I)^{-1}$, for some reasonable shift $\sigma$ close to the interesting eigenvalues.

If none of these conditions is fulfilled, for instance, if the computation of a vector $(A-\sigma I)^{-1}y$ for given $y$ is not feasible with a direct solver, then variants of the Jacobi-Davidson method [411] offer an attractive alternative.