A preconditioned Jacobi-Davidson method for solving large generalized memory machines

In this paper we apply the recently proposed Jacobi-Davidson method for calculating extreme eigenvalues of large matrices to a generalized eigenproblem. This leads to an algorithm that computes the extreme eigensolutions of a matrix pencil (A;B), where A and B are general matrices. Factorization of either of them is avoided. Instead we need to solve two linear systems with sucient, but modest accuracy. If both linear systems are solved accurately enough, an asymptotically quadratic speed of convergence can be achieved. Interior eigenvalues in the vicinity of a given complex number can be computed without factorization as well. We illustrate the procedure with a few numerical examples, one of them being an application in magnetohydrodynamics.