Computation of Smallest Eigenvalues using Spectral Schur Complements

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 engineering. This technique is based on exploiting a high level of dimensional reduction via domain decomposition 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 subspaces 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.