Efficient algorithms and implementation in exact linear algebra. (Algorithmes et implantations efficaces en algèbre linéaire exacte)

This "habilitation a diriger des recherches" manuscript concerns the efficiency in exact linear algebra which is the main research area we have been investigating since the beginning of our PhD thesis in 2001. The goal of this document is not to provide an exhaustive list of all our results. We chose to focus only on few major results for which we provide a concise presentation where we emphasize our salient ideas and techniques. To further improve the readability and the understandability of these work, we provide for most of them some introduction of the context giving the main objectives and the existing results of the literature.The first chapter of this manuscript intends to give a larger view on our research activity, our projects and our student supervision. For this, we provide a brief description of our results obtained mainly while being a member of the LIRMM laboratory in the teams ARITH and ECO. We then describe seven articles that have been published in internationally renowned conferences and journals, either through collaborative works or as the only one investi- gator. We would like to take this occasion to thank all our co-authors for these fruitful collaborations. Even if all these articles fit the large context of computer algebra and they embrace the same motivations for improving efficiency, we decided to provide their description within three different chapters according to the main object of study. The chapters 2 and 3 focus on the efficiency of arithmetic for the main objects arising in computer algebra: the univariate polynomials for the first one, and the integers for the second one. Since the major result of Karatsuba in 1962 who showed a subquadratic complexity algorithm for the multiplication, this operation occupies a central role to improve efficiency in computer algebra. Since then many other results have improved the complexity upper bound for this problem while not being able to completely established its complexity yet. We shall mention the very recent result in March 2019 which provides a new upper bound of O(nlog(n)). This finally proves a conjecture which have been proposed almost fifty years ago. Even if all these theoretical results are fundamental, their scope in practice remain nowadays limited to unreachable sizes of instances. Our work is more concerned by the latter point where we do want to improve efficiency of algorithms in practice. For polynomials, we study the multiplication problem for non standard basis (Chebyshev basis) for which we propose both theoretical and practical equivalences with the algorithms available for the classical monomial basis. We also study the question regarding the fast probabilistic verification of polynomial products (full, short and middle) that is highly relevant to delegating computation. Finally, we also consider these operations within the framework of space complexity. In particular, we provide algorithmic reductions that enable a space complexity of O(1) while still offering the best possible asymptotic time complexity. On the integer side, we provide practical improvements for modular multiplication on multi-core machines. In particular, we design a novel approach that aims to minimize the number of synchronizations inherent to fine grained parallel computing. Lastly, we study the conversions between classical binary representation and the multi-modular one. More precisely, we focus on the case where several integers have to be converted simultaneously. There, we provide an algorithm that reduces the complexity to matrix multiplication and that allows to outperform in practice the best asymptotic algorithms on a wide range of values. The last chapter is about the continuation of our PhD studies on efficient linear algebra. In particular, we focus on two aspects of our work. The first one concerns efficient dense linear algebra over prime fields. In particular, we recall our approach that revealed highly efficient in practice that is to combine: delayed modular arithmetic, sub-cubic matrix multiplication a la Strassen and optimized numerical BLAS libraries. We further detail how re-using this approach and our fast conversions with multi-modular integer representation yields one of the fastest code for integer matrix multiplication to date for integers of moderate bit length. The last part of this manuscript is devoted to linear algebra over univariate polynomial rings. We first recall our PhD’s result that demonstrates the first efficient reduction to matrix multiplication for the minimal approximant basis problem. We emphasize the fact that this result has considerably helped to design faster algorithms for linear algebra with univariate polynomials. We also show that extending this result to the model of online computing brings improvement to block Wiedemann type of algorithms where finely integrated early termination has been made possible. Finally, we demonstrate how we’ve been able to provide an optimal certificate that allows an efficient Monte-Carlo verification procedure for minimal approximant basis.