Most of the successful methods for solving large eigenvalue problems consist of combining projection-type techniques with a few other strategies such as deflation and preconditioning. The standard Petrov-Galerkin conditions used to define projection methods give rise to a variety of different methods, some of which have been considered in recent years. These methods involve two subspaces: a subspace K from which the approximate eigenvector is extracted (called the "right" subspace) and a subspace L of the same dimension used to define constraints for computing these eigenvectors (the "left" subspace). One common choice is to take the left and right space to be the same and this leads to a standard orthogonal projection method exemplified by the Lanczos algorithm in the Hermitian case and the Arnoldi algorithm in the non-Hermitian case. A somewhat intriguing variation, derived by comparison with the GMRES algorithm for solving linear systems is to take L=AK. An (incorrect) motivation for this approach is that it acts as a projection operator for the inverse of the matrix. We will consider these options and a few others, which involve inverting the operator. We will compare some of these variants both experimentally and theoretically.

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