Algebraic formulations for the solution of the nullspace‐free eigenvalue problem using the inexact Shift‐and‐Invert Lanczos method

Abstract

SUMMARYGiven the generalized symmetric eigenvalue problem Ax = Mx, with A semideÿnite and M deÿnite, we analyse some algebraic formulations for the approximation of the smallest non-zero eigenpairs, assuming that a sparse basis for the null space is available. In particular, we consider the inexact version of the Shift-and-Invert Lanczos method, and we show that apparently di erent algebraic formulations provide the same approximation iterates, under some natural hypotheses. Our results suggest that alternative strategies need to be explored to really take advantage of the special problem setting, other than reformulating the algebraic problem. Experiments on a real application problem corroborate our theoretical ÿndings. Copyright ? 2002 John Wiley & Sons, Ltd.KEY WORDS: generalized eigenvalue problem; Shift-and-Invert Lanczos; iterative system solvers 1. THE PROBLEM We are interested in the analysis of some algebraic formulations for the computation of some of the smallest non-zero positive eigenvalues and associated eigenvectors of the large n × n generalized eigenvalue problemwith M symmetric positive deÿnite and A symmetric positive semideÿnite. We assume that a sparse basis C for the null space of A is available, as is the case in several applications; see for instance References [1-3] and their references. A relevant feature of the considered setting is that the dimension of the null space is very high, compared with the problem dimension, reaching up to, say, one-third of the entire problem dimension. The fact that C is a basis for the null space of A plays a crucial role in our analysis, therefore we emphasize the constraint in terms of null space orthogonality. When only the smallest non-zero eigenvalue is requested,

Finite element approximation of eigenvalue problems

“…The two mixed formulations have essentially been introduced for the theoretical analysis of the finite element approximation. Some comments on the computational issues can be found, for instance, in Simoncini (2003) and Arbenz and Geus (1999). For multigrid solvers, the reader is referred to Hiptmair (1999b), Arnold, Falk and Winther (2000), Reitzinger and Schöberl (2002), and to the references therein.…”

SLEPc

“…Therefore, the zero eigenvalues must be avoided during the computation. This can be done in several ways (see Simoncini [2003] for a detailed discussion). One possible way is to solve the following modified problem instead of (8),…”

Modified edge finite elements for photonic crystals

“…To have an idea of the practical computation of the eigenvalues of this generalized eigensystem see [20]. If the matrix B has full rank, then system (13) has exactly N(h) = dim(E h ) real and positive eigenvalues:…”