Preconditioned Eigensolvers - an Oxymoron?

Abstract

>1.1 Preconditioning for linear systems In recent decades, the study of preconditioners for iterative methods for solving large linear systems of equations, arising from discretizations of stationary boundary value problems of mathematical physics, has become a major focus of numerical analysts and engineers. In each iteration step of such a method, a linear system with a special matrix, the preconditioner, has to be solved. The given system matrix can be available only in terms of a matrix-vector multiplication. The basic theory of convergence of these methods is well developed for the symmetric, positive definite case. It is known that the preconditioner should approximate the inverse of the matrix of the original system well in order to obtain rapid convergence. For finite element/difference problems, it is desirable that the rate of convergence is independent of the mesh size. There are several methods of constructing preconditioners, which allow the use of efficient methods,