6. Krylov Subspace Methods

This chapter will examine one of the most important classes of methods available for computing eigenvalues and eigenvectors of large matrices. These techniques are based on projections methods, both orthogonal and oblique, onto Krylov subpaces, i.e., subspaces spanned by the iterates of the simple power method. What may appear to be a trivial extension of a very slow algorithm turns out to be one of the most successful methods for extracting eigenvalues of large matrices, especially in the Hermitian case.