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Journal of Applied Mathematics
Volume 2013, Article ID 767042, 10 pages
Research Article

Parallel RFSAI-BFGS Preconditioners for Large Symmetric Eigenproblems

1Department of Civil, Environmental, and Architectural Engineering, University of Padua, Via Trieste 63, 35100 Padova, Italy
2Department of Mathematics, University of Padua, Via Trieste 63, 35100 Padova, Italy

Received 9 April 2013; Revised 18 July 2013; Accepted 1 August 2013

Academic Editor: D. R. Sahu

Copyright © 2013 L. Bergamaschi and A. Martínez. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.


We propose a parallel preconditioner for the Newton method in the computation of the leftmost eigenpairs of large and sparse symmetric positive definite matrices. A sequence of preconditioners starting from an enhanced approximate inverse RFSAI (Bergamaschi and Martínez, 2012) and enriched by a BFGS-like update formula is proposed to accelerate the preconditioned conjugate gradient solution of the linearized Newton system to solve , being the Rayleigh quotient. In a previous work (Bergamaschi and Martínez, 2013) the sequence of preconditioned Jacobians is proven to remain close to the identity matrix if the initial preconditioned Jacobian is so. Numerical results onto matrices arising from various realistic problems with size up to 1.5 million unknowns account for the efficiency and the scalability of the proposed low rank update of the RFSAI preconditioner. The overall RFSAI-BFGS preconditioned Newton algorithm has shown comparable efficiencies with a well-established eigenvalue solver on all the test problems.