We give a formal definition of preconditioned eigensolvers as polynomial methods for generalized symmetric eigenvalue problems.
We present a survey of some theoretical convergence rate estimates for
preconditioned iterative methods for symmetric eigenvalue problems. We
consider preconditioned analogs of the power method, the steepest
descent/ascent method, the Lanczos-type methods by Scott and Davidson, the
conjugate gradient methods, as well as their block variants.
We suggest a new preconditioned conjugate gradient method, and argue that this is a genuine conjugate gradient method by comparing it with the preconditioned conjugate gradient method for linear systems of equations.