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High Performance and Reliable Algebraic Computing

Clément Pernet 1, 2
1 MOAIS - PrograMming and scheduling design fOr Applications in Interactive Simulation
Inria Grenoble - Rhône-Alpes, LIG - Laboratoire d'Informatique de Grenoble
2 ARIC - Arithmetic and Computing
Inria Grenoble - Rhône-Alpes, LIP - Laboratoire de l'Informatique du Parallélisme
Abstract : This manuscript presents contributions on high performance algebraic computating, lying at the interface between computer algebra, coding theory and parallel computing.Exact linear algebra, over a finite field or the field of rationals is a core component of many applications using intensive algebraic computations: from cryptanalysis based on algebraic sieves or polynomial system solving, to testing conjectures in computational number theory or list decoding... Development of both algorithmic and efficient implementations in exact linear algebra has now reached a great level of maturity. We survey the milestones in these recent progresses, illustrated by our contributions, trying to exhibit a few general guidelines, as for example the importance of making theoretical algorithmic reductions effective. Matrix multiplication, combining conversions between modular, integral andfloating point arithmetic, the use of numerical BLAS, and of sub-cubic algorithms, with low memory footprint, is used as a building block. Gaussian elimination, harnesses its efficiency through many recursive reductions. Rank deficiency and rank profile computations is a specificity of exact computations that we explore in details. Lastly the computation of the characteristic polynomial and of the Frobenius normal form illustrates how an asymptotic improvement of the complexity can be made practical to achieve the best performance. We propose a paralellization of these algorithms based on recursive tasks maintaining the best asymptotical performances. Efficient implementations of work-stealing schedulers handling recursive tasks with dataflow dependencies allow to reach performances similar to state of the art numerical libraries with good scaling ability.In the context of large scale distributed computing, the reliability of remote resources is in question. In order to support errors, either due to physical alterations or malicious corruption, we propose an algorithm based fault tolerance(ABFT) based on evaluation-interpolation schemes, naturally arising in computer algebra. Some classic error correcting codes (Reed-Solomon and CRT codes) based on interpolation are studied and generalized to the case of rational fractionsand their interleaving. We also propose sparse interpolation codes, for which the difficult correction capacity analysis hardly reflects their good behavior in practice. These codes open many perspectives, in particular on symbolic-numeric decodingtechniques jointly supporting errors and numerical noise.
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Submitted on : Thursday, December 11, 2014 - 6:42:34 PM
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Clément Pernet. High Performance and Reliable Algebraic Computing. Symbolic Computation [cs.SC]. Université Joseph Fourier, Grenoble 1, 2014. ⟨tel-01094212⟩

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