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Algèbre linéaire exacte efficace : le calcul du polynôme caractéristique

Abstract : Linear algebra is a building block in scientific computation. Initially dominated by the numerical computation, it has been the scene of major breakthrough in exact computation during the last decade. These algorithmic progresses making the exact computation approach feasible, it became necessary to consider these algorithms from the viewpoint of practicability. We present the building of a set of basic exact linear algebra subroutines. Their efficiency over a finite field near the numerical BLAS. Beyond the applications in exact computation, we show that they offer an
alternative to the multiprecision numerical methods for the resolution of ill-conditioned problems.

The computation of the characteristic polynomial is part of the classic problem in linear algebra. Its exact computation, e.g. helps determine the similarity equivalence between two matrices, using the Frobenius normal form, or the cospectrality of two graphs. The improvement of its theoretical complexity remains an open problem, for both dense or black-box methods. We address this problem from the viewpoint of efficiency in practice: adaptive algorithms for dense or black-box matrices are derived from the best existing algorithms, to ensure high efficiency in practice.
It makes it possible to handle problems whose dimensions was up to now unreachable.
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Submitted on : Sunday, November 5, 2006 - 5:32:26 PM
Last modification on : Wednesday, July 28, 2021 - 3:56:54 AM
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  • HAL Id : tel-00111346, version 1



Clément Pernet. Algèbre linéaire exacte efficace : le calcul du polynôme caractéristique. Génie logiciel [cs.SE]. Université Joseph-Fourier - Grenoble I, 2006. Français. ⟨tel-00111346⟩



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