The focus of this paper is on numerical methods for finding a few eigenvalues and eigenvectors of a large sparse matrix. New preconditioning schemes are proposed for improving the effectiveness of a few methods for computing eigenvalues and eigenvectors. The basic framework of the preconditioned eigenvalue methods we consider is that of the Arnoldi method and the related Davidson method. Within this framework, it is possible to unravel new and more effective alternatives by varying the right-hand side and the matrix of the preconditioning equation. This paper first studies the effects of selecting various such right-hand sides. These comparisons indicate that a scheme based on the inexact-Newton method outperforms the others. We further study a number of Newton schemes for eigenvalue problems and test their potential as preconditioners. The experiments reveal that two schemes related to the Newton preconditioning can constitute good alternatives to other commonly used schemes.