The asymptotically fastest algorithms for many linear algebra problems on integer matrices, including solving a system of linear equations and computing the determinant, use high-order lifting. For a square nonsingular integer matrix A, high-order lifting computes B congruent to A^{-1} mod X^k and matrix R with AB = I + RX^k for non-negative integers X and k. Here, we present a deterministic method -- "double-plus-one" lifting -- to compute the high-order residue R as well as a succinct representation of B. As an application, we give a fully deterministic algorithm to certify the unimodularity of A. The cost of the algorithm is O((log n) n^{omega} M(log n + log ||A||)) bit operations, where ||A|| denotes the largest entry in absolute value, M(t) the cost of multiplying two integers bounded in bit length by t, and omega the exponent of matrix multiplication. Unimodularity certification is then applied to give a heuristic, but certified, algorithm for computing the determinant and Hermite normal form of a square, nonsingular integer matrix. Though most effective on random matrices, a highly optimized implementation of the latter algorithm demonstrates the techniques' effectiveness across a variety of inputs: empirical running times grow as O(n^3log n). A comparison against the fastest known Hermite normal algorithms -- those available in Sage and Magma -- show our implementation is, in all cases, highly competitive, and often surpasses existing, state-of-the-art implementations.

The asymptotically fastest algorithms for many linear algebra problems on integer matrices, including solving a system of linear equations and computing the determinant, use high-order lifting. For a square nonsingular integer matrix A, high-order lifting computes B congruent to A^{-1} mod X^k and matrix R with AB = I + RX^k for non-negative integers X and k. Here, we present a deterministic method -- "double-plus-one" lifting -- to compute the high-order residue R as well as a succinct representation of B. As an application, we give a fully deterministic algorithm to certify the unimodularity of A. The cost of the algorithm is O((log n) n^{omega} M(log n + log ||A||)) bit operations, where ||A|| denotes the largest entry in absolute value, M(t) the cost of multiplying two integers bounded in bit length by t, and omega the exponent of matrix multiplication. Unimodularity certification is then applied to give a heuristic, but certified, algorithm for computing the determinant and Hermite normal form of a square, nonsingular integer matrix. Though most effective on random matrices, a highly optimized implementation of the latter algorithm demonstrates the techniques' effectiveness across a variety of inputs: empirical running times grow as O(n^3log n). A comparison against the fastest known Hermite normal algorithms -- those available in Sage and Magma -- show our implementation is, in all cases, highly competitive, and often surpasses existing, state-of-the-art implementations.

Dans ce mémoire de thèse, nous développons d'abord des multiplications matricielles efficaces. Nous créons de nouveaux ordonnancements qui permettent de réduire la taille de la mémoire supplémentaire nécessaire lors d'une multiplication du type Winograd tout en gardant une bonne complexité, grâce au développement d'outils externes ad hoc (jeu de galets), à des calculs fins de complexité et à de nouveaux algorithmes hybrides. Nous utilisons ensuite des technologies parallèles (multicœurs et GPU) pour accélérer efficacement la multiplication entre matrice creuse et vecteur dense (SpMV), essentielles aux algorithmes dits /boîte noire/, et créons de nouveaux formats hybrides adéquats. Enfin, nous établissons des méthodes de /design/ générique orientées vers l'efficacité, notamment par conception par briques de base, et via des auto-optimisations. Nous proposons aussi des méthodes pour améliorer et standardiser la qualité du code de manière à pérenniser et rendre plus robuste le code produit. Cela permet de pérenniser de rendre plus robuste le code produit. Ces méthodes sont appliquées en particulier à la bibliothèque de calcul exact LinBox. / We first expose in this memoir efficient matrix multiplication techniques. We set up new schedules that allow us to minimize the extra memory requirements during a Winograd-style matrix multiplication, while keeping the complexity competitive. In order to get them, we develop external tools (pebble game), tight complexity computations and new hybrid algorithms. Then we use parallel technologies (multicore CPU and GPU) in order to accelerate efficiently the sparse matrix--dense vector multiplication (SpMV), crucial to /blackbox/ algorithms and we set up new hybrid formats to store them. Finally, we establish generic design methods focusing on efficiency, especially via building block conceptions or self-optimization. We also propose tools for improving and standardizing code quality in order to make it more sustainable and more robust. This is in particular applied to the LinBox computer algebra library.