Algorithmes hybrides et adaptatifs en algèbre linéaire exacte

Abstract : In recent years, considerable progress has been made in exact linear algebra which offered us a significant number of different algorithms for each problem. This dissertation explores adaptive algorithms as a means to get effective solutions. As a result, we obtain very fast solutions in practice, the average complexity of which is comparable to the state of the art. In this thesis we consider the modeling and the use of adaptive algorithms on the example of the computation of the integer determinant, of rational algorithms and of the Smith normal form. As a measure of the performance of adaptive algorithms, we propose the expected complexity and validate our conclusions with an experimental evaluation. To build an adaptive algorithm, we use discovered characteristics of matrices and the explicit comparison of partial computation times. Our algorithms are based on the Chinese Remaindering algorithm with full and rational reconstruction and preconditioning. Our goal is to exploit the early termination condition faster than other algorithms. To improve the algorithms, we can also use heuristics. In this thesis we propose a heuristic scheme for calculating the Smith normal form of an exact sequence of matrices. Instead of treating each matrix separately, we show that the elimination of a row or column of matrix causes reductions in the neighboring matrices. Thus, we succeeded in computing some Smith forms for whole sequences, despite the fact that it was not possible for any of the larger matrices taken separately.
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Anna Urbanska-Marszałek. Algorithmes hybrides et adaptatifs en algèbre linéaire exacte. Symbolic Computation [cs.SC]. Grenoble University, 2010. English. ⟨tel-02143044⟩

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