Arithmétique et algorithmique en algèbre linéaire exacte pour la bibliothèque LinBox. (Arithmetic and algorithmic in exact linear algebra for the LinBox library)

L'algebre lineaire numerique a connu depuis quelques decennies des developpements intensifs autant au niveau mathematique qu'informatique qui ont permis d'aboutir a de veritable standard logiciel comme BLAS ou LAPACK. Dans le cadre du calcul exact ou formel, la situation n'est pas aussi avancee, en particulier a cause de la diversite des problematiques et de la jeunesse des progres theoriques. Cette these s'inscrit dans une tendance recente qui vise a federer des codes performants provenant de bibliotheques specialisees au sein d'une unique plateforme de calcul. En particulier, l'emergence de bibliotheques robustes et portables comme GMP ou NTL pour le calcul exact s'avere etre un reel atout pour le developpement d'applications en algebre lineaire exacte. Dans cette these, nous etudions la faisabilite et la pertinence de la reutilisation de codes specialises pour developper une bibliotheque d'algebre lineaire exacte performante, a savoir la bibliotheque LinBox. Nous nous appuyons sur les mecanismes C++ de programmation generique (classes abtraites, classes templates) pour fournir une abstraction des composantes mathematiques et ainsi permettre le plugin de composants externes. Notre objectif est alors de concevoir et de valider des boites a outils generiques haut niveau dans LinBox pour l'implantation d'algorithmes en algebre lineaire exacte. En particulier, nous proposons des routines de calcul hybride "exact/numerique" pour des matrices denses sur un corps finis permettant d'approcher les performances obtenues par des bibliotheques numeriques comme LAPACK. A un plus haut niveau, ces routines nous permettent de valider la reutilisation de codes specifiques sur un probleme classique du calcul formel: la resolution de systemes lineaires diophantiens. La bibliotheque LinBox est disponible a www. Linalg. Org.

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