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By Pascal G.
Summary: For a number of a long time, numerical linear algebra has obvious in depth advancements in either mathematical and computing device technological know-how conception that have ended in actual common software program like BLAS or lapack. In computing device algebra the location has now not complex as a lot, specifically as a result variety of the issues and thanks to a lot of the theoretical development were performed lately. This thesis falls right into a fresh type of labor which goals at uniforming high-performance codes from many really good libraries right into a unmarried platform of computation. particularly, the emergence of sturdy and conveyable libraries like GMP or ntl for special computation has became out to be a true asset for the advance of functions in unique linear algebra. during this thesis, we examine the feasibility and the relevance of the re-use of specialised codes to increase a excessive functionality unique linear algebra library, particularly the LinBox library. We use the conventional programming mechanisms of C++ (abstract category, template type) to supply an abstraction of the mathematical items and hence to permit the plugin of exterior elements. Our target is then to layout and validate, in LinBox. excessive point familiar toolboxes for the implementation of algorithms in specified linear algebra. specifically, we suggest ''exact/numeric'' hybrid computation exercises 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 exercises to resolve very successfully a classical challenge of laptop algebra : fixing diophantine linear structures. accordingly, this allowed us to validate the primary of code reuse in LinBox library and extra more often than not in computing device algebra. The LinBox library is obtainable at www.linalg.org.
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Extra resources for Arithmetique et algorithmique en algebre lineaire exacte pour la bibliotheque LinBox
Example text
Nous nous int´eressons maintenant aux d´etails d’implantation de l’arch´etype de donn´ees des corps finis. 3 d´ecrit les diff´erents m´ecanismes C++ utilis´es pour obtenir une interface de donn´ees compilable. La classe FieldArchetype int`egre la classe FieldAbstract au travers du pointeur de donn´ees _ptr. Afin de conserver le polymorphisme de l’interface abstraite, le constructeur est g´en´erique sur le type de corps fini. La construction d’un arch´etype sur un corps fini concret entraˆıne l’allocation du pointeur _ptr au travers de l’enveloppe.
28 29 31 34 38 45 45 48 48 50 51 53 59 59 60 61 63 65 28 Arithm´etique des corps finis L’algorithmique en calcul exact et plus particuli`erement en alg`ebre lin´eaire n´ecessite des calculs sur de tr`es grands entiers. Les donn´ees calcul´ees ont g´en´eralement un grossissement de l’ordre de la taille du probl`eme `a r´esoudre. 4]. Le calcul de la solution modulo des nombres premiers pi repose sur des calculs dans les corps finis Z/pi Z.
1. Arch´etype de donn´ees 31 Soit le type Field qui d´efinit une implantation de corps premiers de type Z/pZ en accord avec le mod`ele de base de la biblioth`eque LinBox. On d´efinit alors l’exemple suivant : 1 2 3 4 5 6 Field F (19); Field :: Element a ,b , c ; F . init (a ,13); F . init (b ,3); F . init ( c ); F . mul (c ,a , b ); Ce premier exemple construit tout d’abord `a la ligne 1 le corps premier F de caract´eristique 19. La ligne 2 d´eclare trois ´el´ements a,b,c du type d’´el´ement encapsul´e par le type Field.