Basis selection in LOBPCG

Authors:
U. Hetmaniuk;R. Lehoucq
Affiliations:
Sandia National Laboratories, Computational Mathematics and Algorithms Department, Albuquerque, NM;Sandia National Laboratories, Computational Mathematics and Algorithms Department, Albuquerque, NM
Venue:
Journal of Computational Physics
Year:
2006

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Abstract

The purpose of our paper is to discuss basis selection for Knyazev's locally optimal block preconditioned conjugate gradient (LOBPCG) method. An inappropriate choice of basis can lead to ill-conditioned Gram matrices in the Rayleigh-Ritz analysis that can delay convergence or produce inaccurate eigenpairs. We demonstrate that the choice of basis is not merely related to computing in finite precision arithmetic. We propose a representation that maintains orthogonality of the basis vectors and so has excellent numerical properties.