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| HAL : hal-00018223, version 4 |
| arXiv : cs/0601133 |
| Fiche détaillée |  Récupérer au format |
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| ACM Transactions on Mathematical Software 35, 3 (2009) article 19 |
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| Versions disponibles : | v1 (30-01-2006) | v2 (31-01-2006) | v3 (10-12-2007) | v4 (14-01-2009) |
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| Dense Linear Algebra over Finite Fields: the FFLAS and FFPACK packages |
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| Jean-Guillaume Dumas 1Pascal Giorgi 2 |
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| (2009) |
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| In the past two decades, some major efforts have been made to reduce exact (e.g. integer, rational, polynomial) linear algebra problems to matrix multiplication in order to provide algorithms with optimal asymptotic complexity. To provide efficient implementations of such algorithms one need to be careful with the underlying arithmetic. It is well known that modular techniques such as the Chinese remainder algorithm or the p-adic lifting allow very good practical performance, especially when word size arithmetic are used. Therefore, finite field arithmetic becomes an important core for efficient exact linear algebra libraries. In this paper, we study high performance implementations of basic linear algebra routines over word size prime fields: specially the matrix multiplication; our goal being to provide an exact alternate to the numerical BLAS library. We show that this is made possible by a carefull combination of numerical computations and asymptotically faster algorithms. Our kernel has several symbolic linear algebra applications enabled by diverse matrix multiplication reductions: symbolic triangularization, system solving, determinant and matrix inverse implementations are thus studied. |
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| 1 : | Laboratoire Jean Kuntzmann (LJK) |
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CNRS : UMR5224 – Université Joseph Fourier - Grenoble I – Université Pierre Mendès-France - Grenoble II – Institut Polytechnique de Grenoble |
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| 2 : | Laboratoire d'Informatique de Robotique et de Microélectronique de Montpellier (LIRMM) |
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CNRS : UMR5506 – Université Montpellier II - Sciences et Techniques du Languedoc |
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| 3 : | MOAIS (INRIA Grenoble Rhône-Alpes / LIG Laboratoire d'Informatique de Grenoble) |
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INRIA – Université Joseph Fourier - Grenoble I – Institut National Polytechnique de Grenoble (INPG) – Université Pierre Mendès-France - Grenoble II – CNRS : UMR5217 |
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| Domaine | : | Informatique/Calcul formel |
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| Word size Finite fields – BLAS level 1-2-3 – {Linear Algebra Package} – Winograd's symbolic Matrix Multiplication – Matrix Factorization – Exact Determinant – Exact Inverse |
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| hal-00018223, version 4 | |
| http://hal.archives-ouvertes.fr/hal-00018223 | |
| oai:hal.archives-ouvertes.fr:hal-00018223 | |
| Contributeur : Jean-Guillaume Dumas | |
| Soumis le : Mercredi 14 Janvier 2009, 17:18:19 | |
| Dernière modification le : Mercredi 14 Janvier 2009, 17:30:38 | |
