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Download Algorithmes paralleles pour le calcul formel: algebre by Dumas J.-G. PDF

By Dumas J.-G.

Summary: In each fi eld of scientifi с and business learn, the extension of using laptop technology has led to an expanding desire for computing energy. it really is therefore very important to exploit those computing assets in parallel. during this thesis we search to compute the canonical type of very huge sparse matrices with integer coeffi cients, specifically the integer Smith basic shape. through 'Very large'', we suggest one million indeterminates and 1000000 equations, i.e. thousand billion of coeffi cients. these days, such platforms will not be even storable. although, we're drawn to platforms for which lots of those coeffi cients are exact; as a consequence we discuss sparse structures. we wish to remedy those structures in an actual manner, i.e. we paintings with integers or in smaller algebraic constructions the place all of the easy mathematics operations are nonetheless legitimate, specifically fi nitefi elds. The rebuilding of the total answer from the smaller suggestions is then particularly effortless.

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Tant donné qu’il y a ✁ ✏✂ ☎✏ ✡ racines primitives dans p pour ✏ premier, la probabilité d’en trou✝ ☎✓ ver une est de ✓ et donc l’espérance du nombre de tirages pour tomber sur ✓ une racine primitive est de ☎ ✓ ✝ . 1) ✕ ✢ ✗ ✁ ✗✙✘✙✕✔✗✙✒✦✕ En outre, comme n’est pas premier, nous pouvons ✍ utiliser cette inégalité pour toutes les racines primitives de nombres premiers. Il est de plus conjecturé que ☎ ☎☞ ✡✡ ☎✌ ✝ pour un nombre infini de ☞ , cette ✌ borne semble donc très bonne. Ainsi, pour ✄ ☞ ✄ , elle donne une valeur maximale d’environ ✞ .

R : r + } } ✌ ✟✌ ✗ ☛; ✌ ✟✠ ✠ GFQ_AXPY(r, , ,j) { // ✌ ✟ ✠ if (j==0) GFQ_MUL(r, , ); else if (( ==0) || ( ==0)) r=j; else { r = + - j - ; r = (r<0)? r + : r; r = t_plus1[(r>0)? r : r + ]; if (r) { r = r + j r = (r>0)? r : r + ; } } } ✎ ✎ ✎ ✑ ☛ ✎✑ ✑ ]; ☛ ☛ ✓✟✟ ☎✞ ✌ ✟✌ ✟✟ ✠ ✌ ✟✒ ☎✞ ✟ GFQ_NEG(r,i) { // ✞ ✌ ✟ if ( i==0 ) r=0; else { r = i ; r = (r>0)? 1 Implémentations 63 ✞☛ . Ainsi, on gagne une opération pour le calcul de GFQ_ADD et GFQ_AXPY. Dans ce cas, la taille du corps n’est plus limitée par la taille du mot machine, mais par l’espace mémoire disponible.

Ces synchronisations sont, soit explicites, comme par exemple pour les langages à base de processus légers ou pour Cilk, soit implicites et déterminées à partir de la déclaration par les tâches des accès à la mémoire qu’elles effectuent, comme dans Jade ou Athapascan-1. On parle alors de langages effectuant une analyse du flot de données. 1 – Comparaison de différents langages de programmation parallèle de « haut niveau » [65 - Galilée (1999)] Enfin, le parallélisme peut être de type « série-parallèle » si les synchronisations entre les tâches sont effectuées par fratrie : la tâche mère est seule capable de synchroniser ses filles, et cette synchronisation est globale sur l’ensemble des filles créées.

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