Computation of Smallest Eigenvalues using Spectral Schur Complements Constantine Bekas Yousef Saad January 23, 2004 Abstract The Automated Multilevel Substructing method (AMLS ) was recently presented as an alternative to well-established methods for computing eigenvalues of large matrices in the context of structural engi- neering. This technique is based on exploiting a high level of dimensional reduction via domain decom- position and projection methods. This paper takes a purely algebraic look at the method and explains that it can be viewed as a technique based on a first order approximation to a nonlinear eigenvalue problem. A ‘corrective projection’ viewpoint leads us to explore variants of the method which use Krylov sub- spaces instead of eigenbasis to construct subspaces of approximants. The nonlinear eigenvalue problem viewpoint yields a second order approximation as an enhancement to the first order technique inherent to AMLS . Numerical experiments are presented to validate the approaches presented. 1 Introduction The numerical solution of large sparse symmetric eigenvalue problems continues to be at the forefront of current research in scientifi c computing. In the last few decades, projection methods such as the Lanczos algorithm and its variants, have dominated the scene. For example, a block version of this method combined with shift-and-invert [7], whereby the problem is replaced by , is used in major commercial structural engineering packages such as MCS.NASTRAN [10]. ARPACK [12], a package based on an implicitly restarted Arnoldi/Lanczos process, is currently the best known public- domain eigenvalue package for large eigenvalue problems. The Lanczos process scales poorly with the number of eigenvalues to be computed, because of the need to orthogonalize large Krylov bases. In recent years, an alternative approach has emerged in structural engineering which has been reported to be superior to the standard shift-and-invert Lanczos approach. The algorithm, called Automated Multilevel Substructuring method (AMLS ) is rooted in a domain decompo- sition framework. It has been reported as being capable of computing thousands of the smallest normal modes of dynamic structures on commodity workstations and of being orders of magnitude faster than the standard approach [11]. The paper [3] presents a theoretical framework for the algorithm from the point of domain decom- position, using adequate functional spaces and operators on them. The goal of our paper is to present a different, yet complementary viewpoint, which is entirely algebraic. AMLS is essentially a Schur comple- ment method. Schur complement techniques are well understood for solving linear systems and play a major role in Domain Decomposition techniques [15, 16]. Relatively speaking, the formulation of this method for eigenvalue problems has been essentially neglected so far. One could of course extend the approach used for linear systems in order to compute eigenvalues, by formulating a Schur complement problem for each different eigenvalue, (e.g., by solving the eigenvalue problem as a sequence of linear systems through shift- and-invert). A more complete framework was suggested in early work by Abramov [1, 2] and Chichov [5]. Work supported by NSF under grants NSF/ITR-0082094 and NSF ACI-0305120, and by the Minnesota Supercomputer Institute Department of Computer Science & Engineering, University of Minnesota, Twin cities, Minneapolis, MN. Email: bekas,saad cs.umn.edu 1