Parallel computation of the rank of large sparse matrices from algebraic K-theory

Jean-Guillaume Dumas (1) , Philippe Elbaz-Vincent (2) , Pascal Giorgi (3) , Anna Urbanska (1)

1 CASYS - Calculs Algébriques et Systèmes Dynamiques
2 I3M - Institut de Mathématiques et de Modélisation de Montpellier
3 LP2A - Laboratoire de Physique Appliquée et d'Automatique

This paper deals with the computation of the rank and of some integer Smith forms of a series of sparse matrices arising in algebraic K-theory. The number of non zero entries in the considered matrices ranges from 8 to 37 millions. The largest rank computation took more than 35 days on 50 processors. We report on the actual algorithms we used to build the matrices, their link to the motivic cohomology and the linear algebra and parallelizations required to perform such huge computations. In particular, these results are part of the first computation of the cohomology of the linear group GL_7(Z).

Mots clés: cohomology computation; sparse matrix; parallel rank and Smith form; block Wiedemann

Domaines: K-théorie et homologie [math.KT]; Calcul formel [cs.SC]; Calcul parallèle, distribué et partagé [cs.DC]; Théorie des nombres [math.NT]

Jean-Guillaume Dumas, Philippe Elbaz-Vincent, Pascal Giorgi, Anna Urbanska. Parallel computation of the rank of large sparse matrices from algebraic K-theory. PASCO '07 - International Workshop on Parallel Symbolic Computation, Jul 2007, London, Canada. pp.43-52, ⟨10.1145/1278177.1278186⟩. ⟨hal-00142141v2⟩