Chebyshev Interpolation Polynomial-based Tools for Rigorous Computing

17 pagesInternational audiencePerforming numerical computations, yet being able to provide rigorous mathematical statements about the obtained result, is required in many domains like global optimization, ODE solving or integration. Taylor models, which associate to a function a pair made of a Taylor approximation polynomial and a rigorous remainder bound, are a widely used rigorous computation tool. This approach benefits from the advantages of numerical methods, but also gives the ability to make reliable statements about the approximated function. Despite the fact that approximation polynomials based on interpolation at Chebyshev nodes offer a quasi-optimal approximation to a function, together with several other useful features, an analogous to Taylor models, based on such polynomials, has not been yet well-established in the field of validated numerics. This paper presents a preliminary work for obtaining such interpolation polynomials together with validated interval bounds for approximating univariate functions. We propose two methods that make practical the use of this: one is based on a representation in Newton basis and the other uses Chebyshev polynomial basis. We compare the quality of the obtained remainders and the performance of the approaches to the ones provided by Taylor models

Special Section on Emerging and Impacting Trends on Computer Arithmetic

The papers in this special section focus on emerging and impacting trends on computer arithmetic. The computer arithmetic. field encompasses the definition and standardization of arithmetic systems for computers. It also deals with issues pertaining to hardware and software implementations, testing, and verification. Researchers and practitioners of this field also work on challenges associated with using Computer Arithmetic to perform scientific and engineering calculations. As such, Computer Arithmetic can be regarded as a truly multi-disciplinary field, which builds upon mathematics, computer science and electrical engineering. Thus, the range of topics addressed by Computer Arithmetic is generally very broad, spanning from highly theoretical to extremely practical contributions. Computer Arithmetic has been an active research field since the advent of computers, and it is progressively evolving following continuously advancements in technology

Certified and fast computation of supremum norms of approximation errors

The version available on HAL corresponds to the version initially submitted to the conference and slightly differs from the published version since it does not account for remarks made by the referees.International audienceIn many numerical programs there is a need for a high-quality floating-point approximation of useful functions f, such as exp, sin, erf. In the actual implementation, the function is replaced by a polynomial p, leading to an approximation error (absolute or relative) epsilon = p-f or epsilon = p/f-1. The tight yet certain bounding of this error is an important step towards safe implementations. The main difficulty of this problem is due to the fact that this approximation error is very small and the difference p-f is highly cancellating. In consequence, previous approaches for computing the supremum norm in this degenerate case, have proven to be either unsafe, not sufficiently tight or too tedious in manual work. We present a safe and fast algorithm that computes a tight lower and upper bound for the supremum norms of approximation errors. The algorithm is based on a combination of several techniques, including enhanced interval arithmetic, automatic differentiation and isolation of the roots of a polynomial. We have implemented our algorithm and timings on several examples are given

Automatic generation of polynomial-based hardware architectures for function evaluation

International audienceMany applications require the evaluation of some function through polynomial approximation. This article details an architecture generator for this class of problems that improves upon the literature in two aspects. Firstly, it benefits from recent advances related to constrained-coefficient polynomial approximation. Secondly, it refines the error analysis of polynomial evaluation to reduce the size of the multipliers used. As a result, architectures for evaluating arbitrary functions with precisions up to 64 bits, making efficient use of the resources of recent FPGAs, can be obtained in seconds. An open-source implementation is provided in the FloPoCo project

Efficient and accurate computation of upper bounds of approximation errors

International audienceFor purposes of actual evaluation, mathematical functions f are commonly replaced by approximation polynomials p. Examples include floating-point implementations of elementary functions, quadrature or more theoretical proof work involving transcendental functions. Replacing f by p induces a relative error epsilon = p/f - 1. In order to ensure the validity of the use of p instead of f, the maximum error, i.e. the supremum norm of epsilon must be safely bounded above. Numerical algorithms for supremum norms are efficient but cannot offer the required safety. Previous validated approaches often require tedious manual intervention. If they are automated, they have several drawbacks, such as the lack of quality guarantees. In this article a novel, automated supremum norm algorithm with a priori quality is proposed. It focuses on the validation step and paves the way for formally certified supremum norms. Key elements are the use of intermediate approximation polynomials with bounded approximation error and a non-negativity test based on a sum-of-squares expression of polynomials. The new algorithm was implemented in the Sollya tool. The article includes experimental results on real-life examples

Racines carrées multiplicatives sur FPGA

10 pagesLes implantations actuelles de la racine carrée dans des bibliothèques d'opérateurs pour FPGA utilisent presque toutes une récurrence à base d'additions. Ce choix est particulièrement bien adapté à la structure des blocs logiques élémentaires d'un FPGA. Toutefois, il peut être remis en question à présent que la plupart des FPGA haute-performance incluent un grand nombre de blocs multiplieurs et de blocs mémoires. Cet article discute l'implantation d'une racine carrée compatible IEEE-754 en utilisant ces nouvelles ressources, et compare les performances obtenues avec l'approche classique

Implementation and performance evaluation of an extended precision floating-point arithmetic library for high-accuracy semidefinite programming

International audienceSemidefinite programming (SDP) is widely used in optimization problems with many applications, however , certain SDP instances are ill-posed and need more precision than the standard double-precision available. Moreover, these problems are large-scale and could benefit from parallelization on specialized architectures such as GPUs. In this article, we implement and evaluate the performance of a floating-point expansion-based arithmetic library (newFPLib) in the context of such numerically highly accurate SDP solvers. We plugged-in the newFPLib with the state-of-the-art SDPA solver for both CPU and GPU-tuned implementations. We compare and contrast both the numerical accuracy and performance of SDPA-GMP,-QD and-DD, which employ other multiple-precision arithmetic libraries against SDPA-newFPLib. We show that our newFPLib is a very good trade-off for accuracy and speed when solving ill-conditioned SDP problems

A Power Series Expansion based Method to compute the Probability of Collision for Short-term Space Encounters

Rapport LAAS n° 15072This article provides a new method for computing the probability of collision between two spherical space objects involved in a short-term encounter under Gaussian-distributed uncertainty. In this model of conjunction, classical assumptions reduce the probability of collision to the integral of a two-dimensional Gaussian probability density function over a disk. The computational method presented here is based on an analytic expression for the integral, derived by use of Laplace transform and D-finite functions properties. The formula has the form of a product between an exponential term and a convergent power series with positive coefficients. Analytic bounds on the truncation error are also derived and are used to obtain a very accurate algorithm. Another contribution is the derivation of analytic bounds on the probability of collision itself, allowing for a very fast and - in most cases - very precise evaluation of the risk. The only other analytical method of the literature - based on an approximation - is shown to be a special case of the new formula. A numerical study illustrates the efficiency of the proposed algorithms on a broad variety of examples and favorably compares the approach to the other methods of the literature

Validated Numerics:Algorithms and Practical Applications in Aerospace

International audienceMy lecture will survey some classical and recent validated computing algorithms based on the theory of set-valued analysis, in suitable functional spaces, as well as by combining symbolic and numerical computations. These techniques are illustrated with some applications which appear in practical space mission analysis and design. This is only a short summary of the talk

Algorithmes symboliques-numeriques validés et applications au domaine spatial

National audienceWhen computing with finite precision, one strives to achieve accurate and/or guaranteed results without compromising efficiency. For this, we combine symbolic and numerical computation, which leads to the development of specific new computer arithmetic and approximation algorithms. Firstly, at the arithmetics level, we focus on high-precision arithmetic operations, using as basic building blocks the available operators for floating-point arithmetic. We are also interested in problems related to the efficient and reliable implementation and evaluation in fixed-precision of elementary and special functions. Secondly, at the symbolic-numeric level, we focus on effective polynomial approximations together with validated error bounds expressed in Taylor or Chebyshev basis. We exploit approximation algorithms mainly related to D-finite functions i.e., solutions of linear differential equations with polynomial coefficients. The theoretical tools developed above are then applied to problems coming from optimal control and aerospace. A first example consists of a new method to compute the orbital collision probability between two spherical objects involved in a short-term encounter, under Gaussian uncertainty. Another one discusses efficient and validated algorithms for impulsive spacecraft rendezvous. Finally, the obtained results are put in perspective: the goal is to bring more reliable computations in the field of optimal control theory and aerospace applications, by making further use of computer arithmetics, computer algebra and approximation theory tools.Le calcul scientifique en précision finie nécessite dans le même temps des résultats précis et/ou garantis et des algorithmes efficaces. Dans cette optique, le calcul symbolique et numérique sont conjointement exploités, conduisant au développement de nouveaux algorithmes spécifiques à l'arithmétique des ordinateurs et à l'approximation effective. Concernant l'arithmétique à virgule flottante, un premier objectif consiste à étendre la précision usuelle (binary32 ou binary64), en utilisant comme briques de base les opérateurs déjà disponibles au niveau matériel. Nous nous intéressons également aux problèmes liés à l'implantation et à l’évaluation en précision fixée, efficace et fiable, des fonctions élémentaires et spéciales. Au niveau symbolique-numérique, nous exploitons des approximations polynomiales effectives, exprimées en bases de Taylor ou de Chebyshev, avec bornes d'erreur validées ainsi que des algorithmes d’approximation principalement liés aux fonctions D-finies (solutions d’équations différentielles linéaires à coefficients polynomiaux). Les outils théoriques développés ci-dessus sont appliqués à de problèmes réels issus du contrôle optimal et de l'aérospatiale. Un premier exemple d'application réussie consiste en une nouvelle méthode permettant de calculer la probabilité de collision orbitale entre deux objets sphériques impliqués dans une rencontre à court terme, sous incertitude gaussienne. Dans un deuxième exemple, nous proposons des algorithmes efficaces et validés pour le problème du rendez-vous spatial sous hypothèses d'une propulsion impulsionnel et d'une modélisation képlerienne. Enfin, les résultats obtenus sont mis en perspective: l’objectif est d’apporter des méthodes de calcul plus fiables dans le domaine de la théorie du contrôle optimal et des applications aérospatiales, en utilisant plus systématiquement des outils spécifiques des champs de l’arithmétique des ordinateurs, du calcul formel et de la théorie de l’approximation