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Saving Flops in LU Based Shift-and-Invert Strategy

Laura Grigori 1 Desire W. Nuentsa 1 Hua Xiang 1
1 GRAND-LARGE - Global parallel and distributed computing
CNRS - Centre National de la Recherche Scientifique : UMR8623, Inria Saclay - Ile de France, UP11 - Université Paris-Sud - Paris 11, LIFL - Laboratoire d'Informatique Fondamentale de Lille, LRI - Laboratoire de Recherche en Informatique
Abstract : The shift-and-invert method is very efficient in eigenvalue computations, in particular when interior eigenvalues are sought. This method involves solving linear systems of the form $(A-\sigma I)z=b$. The shift $\sigma$ is variable, hence when a direct method is used to solve the linear system, the LU factorization of $(A-\sigma I)$ needs to be computed for every shift change. We present two strategies that reduce the number of floating point operations performed in the LU factorization when the shift changes. Both methods perform first a preprocessing step that aims at eliminating parts of the matrix that are not affected by the diagonal change. This leads to $43\%$ and $50\%$ flops savings respectively.
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Submitted on : Tuesday, June 10, 2008 - 9:52:06 AM
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Laura Grigori, Desire W. Nuentsa, Hua Xiang. Saving Flops in LU Based Shift-and-Invert Strategy. [Research Report] RR-6553, INRIA. 2008, pp.15. ⟨inria-00286417v2⟩

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