By Pascal G.
Summary: For a number of a long time, numerical linear algebra has obvious extensive advancements in either mathematical and desktop technological know-how conception that have ended in actual usual software program like BLAS or lapack. In laptop algebra the location has now not complex as a lot, specifically due to the variety of the issues and due to a lot of the theoretical development were performed lately. This thesis falls right into a contemporary category of labor which goals at uniforming high-performance codes from many really expert libraries right into a unmarried platform of computation. specifically, the emergence of strong and transportable libraries like GMP or ntl for designated computation has became out to be a true asset for the improvement of purposes in detailed linear algebra. during this thesis, we examine the feasibility and the relevance of the re-use of specialised codes to enhance a excessive functionality special linear algebra library, specifically the LinBox library. We use the standard programming mechanisms of C++ (abstract type, template classification) to supply an abstraction of the mathematical items and hence to permit the plugin of exterior parts. Our goal is then to layout and validate, in LinBox. excessive point everyday toolboxes for the implementation of algorithms in distinct linear algebra. specifically, we advise ''exact/numeric'' hybrid computation workouts for dense matrices over finite fields which just about fit with the functionality acquired via numerical libraries like LAPACK. On the next point, we reuse those hybrid workouts to unravel very successfully a classical challenge of laptop algebra : fixing diophantine linear structures. therefore, this allowed us to validate the main of code reuse in LinBox library and extra as a rule in desktop algebra. The LinBox library is on the market at www.linalg.org.
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Extra resources for Arithmetique et algorithmique en algebre lineaire exacte pour la bibliotheque LinBox
Example text
Le calcul de la solution modulo des nombres premiers pi repose sur des calculs dans les corps finis Z/pi Z. Les performances des op´erations arithm´etiques de ces corps finis sont donc un crit`ere important de l’efficacit´e de cette m´ethode. Dans certains cas, le calcul de solutions probabilistes permet d’obtenir des gains en complexit´e non n´egligeables. Sur un corps fini, les probabilit´es de r´eussite de ces m´ethodes sont directement reli´ees `a la taille du corps finis. Afin d’augmenter les probabilit´es de r´eussite, une approche classique consiste `a plonger les corps finis dans une extension alg´ebrique.
C’est lors de la construction du g´en´erateur que l’it´erateur est positionn´e `a partir d’une graine d’al´ea et d’un corps fini, ou d’un sous-ensemble. Le parcours de la structure du g´en´erateur se fait au travers de la fonction random(Element &a) qui affecte la valeur de l’it´erateur `a a et incr´emente l’it´erateur. 1 pr´esente le mod`ele de base des corps finis avec ses diff´erentes encapsulations de types. Toutes les implantations de corps finis de la biblioth`eque LinBox doivent int´egrer ces trois types de donn´ees en respectant leurs arch´etypes respectifs.
LambdaSparseMatrix (***) : pr´econditionneur creux. MatrixBlackbox : adaptateur g´en´erique de matrices conformes avec MatrixArchetype. MoorePenrose : matrice boˆıte noire pour l’inverse g´en´eralis´ee de Moore-Penrose. NAGSparse : adaptateur g´en´erique pour les matrices creuses (format NAG). Hankel : matrice boˆıte noire de type Hankel (bas´ee sur les polynˆomes de NTL). Sylvester : matrice boˆıte noire de type Sylvester (`a partir de deux polynˆomes). Toeplitz : matrice boˆıte noire de type Toeplitz (bas´ee sur les polynˆomes de NTL).