Abstract

Quotients for eigenvalue problems (generalized or not) are considered. To have a quotient optimally approximating an eigenvalue, conditions are formulated to maximize the one-dimensional projection of the eigenvalue problem. Respective optimal quotient iterations are derived under the assumption that applying the inverse is affordable. Inexact methods are also considered if applying the inverse is not affordable. Then, to approximate an eigenvector, optimality conditions are formulated to minimize linear independency over a subspace. Equivalence transformations are performed for preconditioning iterations and steering the convergence. These ideas extend to subspaces in a natural way. For the standard eigenvalue problem, a new Arnoldi method arises as an alternative to the classical Arnoldi method.