Restrictions on implicit filtering techniques for orthogonal projection methods

We consider the class of the Orthogonal Projection Methods (OPM) to solve i teratively large eigenvalue problems. An OPM is a method that projects a la rge eigenvalue problem on a smaller subspace. In this subspace, an approxim ation of the eigenvalue spectrum can be computed from a small eigenvalue pr oblem using a direct method. Examples of OPMs are the Arnoldi and the David son method. We show how an OPM can be restarted - implicitly and explicitly . This restart can be used to remove a specific subset of vectors from the approximation subspace. This is called explicit filtering. An implicit rest art can also be combined with an implicit filtering step, i.e. the applicat ion of a polynomial or rational function on the subspace, even if inaccurat e arithmetic is assumed. However, the condition for the implicit applicatio n of a tilter is that the rank of the residual matrix must be small. (C) 19 99 Elsevier Science Inc. All rights reserved.