Parallel Computation of Echelon Forms

We propose efficient parallel algorithms and implementations on shared memory architectures of LU factorization over a finite field. Compared to the corresponding numerical routines, we have identified three main difficulties specific to linear algebra over finite fields. First, the arithmetic complexity could be dominated by modular reductions. Therefore, it is mandatory to delay as much as possible these reductions while mixing fine-grain parallelizations of tiled iterative and recursive algorithms. Second, fast linear algebra variants, e.g., using Strassen-Winograd algorithm, never suffer from instability and can thus be widely used in cascade with the classical algorithms. There, trade-offs are to be made between size of blocks well suited to those fast variants or to load and communication balancing. Third, many applications over finite fields require the rank profile of the matrix (quite often rank deficient) rather than the solution to a linear system. It is thus important to design parallel algorithms that preserve and compute this rank profile. Moreover, as the rank profile is only discovered during the algorithm, block size has then to be dynamic. We propose and compare several block decomposition: tile iterative with left-looking, right-looking and Crout variants, slab and tile recursive. Experiments demonstrate that the tile recursive variant performs better and matches the performance of reference numerical software when no rank deficiency occur. Furthermore, even in the most heterogeneous case, namely when all pivot blocks are rank deficient, we show that it is possbile to maintain a high efficiency. This work is partly funded by the HPAC project of the French Agence Nationale de la Recherche (ANR 11 BS02 013). Université de Grenoble. Laboratoire LJK, umr CNRS, INRIA, UJF, UPMF, GINP. 51, av. des Mathématiques, F38041 Grenoble, France. INRIA. Laboratoire LIG, umr CNRS, INRIA, UJF, UPMF, GINP. 51, av. J. Kuntzmann, F38330 Montbonnot St-Martin, France. Université de Grenoble. Laboratoire de l’Informatique du Parallélisme umr CNRS, INRIA, UCBL, ÉNS de Lyon. 46 Allée d’Italie, F69364 LYON Cedex 07, France. [email protected], [email protected], [email protected], [email protected].