About the ''accurate mode'' of the IEEE 1788-2015 standard for interval arithmetic

The IEEE 1788-2015 standard for interval arithmetic defines three accuracy modes for the so-called set-based flavor: tightest, accurate and valid. This work in progress focuses on the accurate mode.First, an introduction to interval arithmetic and to the IEEE 1788-2015 standard is given, then the accurate mode is defined. How can this accurate mode be tested, when a library implementing interval arithmetic claims to provide this mode? The chosen approach is unit testing, and the elaboration of testing pairs for this approach is developed.A discussion closes this paper: how can the tester be tested? And if we go to the roots of the subject, is the accurate mode really relevant or should it be dropped off in the next version of the standard

Numerical reproducibility in HPC: issues in interval arithmetic

International audienceThe problem of numerical reproducibility is the problem of getting the same result when a numerical computation is run several times, whether on the same machine or on different machines. The accuracy of the result is a different issue. As far as interval arithmetic is concerned, the relevant issue is the inclusion property, that is, the guarantee that the exact result belongs to the computed resulting interval

Motivations for an arbitrary precision interval arithmetic and the MPFI library

This paper justifies why an arbitrary precision interval arithmetic is needed: to provide accurate results, interval computations require small input intervals; this explains why bisection is so often employed in interval algorithms. The MPFI library has been built in order to fulfill this need: indeed, no existing library met the required specifications. The main features of this library are briefly given and a comparison with a fixed-preci- sion interval arithmetic, on a specific problem, is presented: it shows that the overhead due to the multiple precision is completely admissible. Eventually, some applications based on MPFI are given: robotics, isolation of polynomial real roots (by an algorithm combining symbolic and numerical computations) and approximation of real roots with arbitrary accuracy.Cet article justifie le besoin d’une arithmétique par intervalles en précision arbitraire : pour fournir des résultats précis, un calcul par intervalles requiert des intervalles en entrée qui soient fins ; c’est pour cette raison que la bissection est un procédé si souvent employé dans les algorithmes par intervalles. La bibliothèque MPFI a été développée pour répondre à ce besoin : en effet, aucune bibliothèque existante n’offrait de spécifications satisfaisantes. Les caractéristiques de cette bibliothèque sont rapidement données puis une comparaison avec une bibliothèque d’arithmétique par intervalles en précision fixée est menée sur un problème spécifique : elle met en évidence le fait que le surcoût lié à la gestion de la précision multiple est tout à fait acceptable. Pour terminer, quelques applications basées sur MPFI sont présentées : robotique, isolation des racines réelles de polynômes (par un algorithme combinant calcul symbolique et calcul numérique) et approximation avec une précision arbitraire de zéros réel

Parallel Implementation of Interval Matrix Multiplication

International audienceTwo main and not necessarily compatible objectives when implementing the product of two dense matrices with interval coefficients are accuracy and efficiency. In this work, we focus on an implementation on multicore architectures. One direction successfully explored to gain performance in execution time is the representation of intervals by their midpoints and radii rather than the classical representation by endpoints. Computing with the midpoint-radius representation enables the use of optimized floating-point BLAS and consequently the performances benefit from the performances of the BLAS routines. Several variants of interval matrix multiplication have been proposed, that correspond to various trade-offs between accuracy and efficiency, including some efficient ones proposed by Rump in 2012. However, in order to guarantee that the computed result encloses the exact one, these efficient algorithms rely on an assumption on the order of execution of floating-point operations which is not verified by most implementations of BLAS. In this paper, an algorithm for interval matrix product is proposed that verifies this assumption. Furthermore, several optimizations are proposed and the implementation on a multicore architecture compares reasonably well with a non-guaranteed implementation based on MKL, the optimized BLAS of Intel: the overhead is most of the time less than 2 and never exceeds 3. This implementation also exhibits a good scalability

Solving and Certifying the Solution of a Linear System

The Reliable Computing journal has no more paper publication, only free, electronic publication.International audienceUsing floating-point arithmetic to solve a linear system yields a computed result, which is an approximation of the exact solution because of roundoff errors. In this paper, we present an approach to certify the computed solution. Here, "certify" means computing a guaranteed enclosure of the error. Our method is an iterative refinement method and thus it also improves the computed result. The method we present is inspired from the verifylss function of the IntLab library, with a first step, using floating-point arithmetic, to solve the linear system, followed by interval computations to get and refine an enclosure of the error. The specificity of our method is to relax the requirement of tightness of the error, in order to gain in performance. Indeed, only the order of magnitude of the error is needed. Experiments show a gain in accuracy and in performance, for various condition number of the matrix of the linear system

Certification of a Numerical Result: Use of Interval Arithmetic and Multiple Precision

International audienceUsing floating-point arithmetic to solve a numerical problem yields a computed result, which is an approximation of the exact solution because of roundoff errors. In this paper, we present an approach to certify the computed solution. Here, "certify" means computing a guaranteed enclosure of the error between the computed, approximate, result and the exact, unknown result. We discuss an iterative refinement method: classically, such methods aim at computing an approximation of the error and they add it to the previous result to improve its accuracy. We add two ingredients: interval arithmetic is used to get an enclosure of the error instead of an approximation, and multiple precision is used to reach higher accuracy. We exemplify this approach on the certification of the solution of a linear system

Refining and verifying the solution of a linear system

International audienceThe problem considered here is to refine an approximate, numerical, solution of a linear system and simultaneously give an enclosure of the error between this approximate solution and the exact one: this is the verification step. Desirable properties for an algorithm solving this problem are accuracy of the results, complexity and performance of the actual implementation. A new algorithm is given, which has been designed with these desirable properties in mind. It is based on iterative refinement for accuracy, with well-chosen computing precisions, and uses interval arithmetic for verification

Arithmétique par intervalles

National audienceThis paper constitutes an introduction to interval arithmetic. This arithmetic allows on the one hand to take into account the measurement uncertainties on data and on the other hand to determine an enclosure of the computed result that is guaranteed to contain it: indeed, the main advantage of interval arithmetic is its reliability. The goal of this introduction is to emphasize the strong points of interval arithmetic and to explain how to alleviate its problems. The main advantage is to provide global information, such as for instance the range of a function over a whole set. This global information can serve to prove that an iteration is contractant and thus that it has a fixed point. It can also be used to detemine the global optimum of a function without being trapped by a local one.Cet article est une introduction à l'arithmétique par intervalles. Avec une telle arithmétique, il est possible à la fois de tenir compte des incertitudes sur les données et de retourner un encadrement contenant à coup sûr le résultat d'un calcul : la force de l'arithmétique par intervalles est en effet la fiabilité des résultats. L'objectif de cette introduction est de mettre en évidence les points forts d'une telle arithmétique et de montrer comment contourner ses faiblesses. Son avantage majeur est de fournir une information globale telle qu'un surencadrement de l'image d'un ensemble par une fonction. Cette information globale peut être utilisée pour déterminer le caractère contractant d'une itération et par conséquent pour prouver l'existence et l'unicité de la solution calculée. Elle peut aussi servir à optimiser globalement une fonction en évitant de se laisser piéger par un optimum local

Itérations affines et effet enveloppant: différentes approches.

International audienceAffine iterations of the form x(n+1) = Ax(n) + b converge, using real arithmetic, if the spectral radius of the matrix A is less than 1. However, substituting interval arithmetic to real arithmetic may lead to divergence of these iterations, in particular if the spectral radius of the absolute value of A is greater than 1. We will review different approaches to limit the overestimation of the iterates, when the components of the initial vector x(0) and b are intervals. We will compare, both theoretically and experimentally, the widths of the iterates computed by these different methods: the naive iteration, methods based on the QR-and SVD-factorization of A, and Lohner's QR-factorization method. The method based on the SVD-factorization is computationally less demanding and gives good results when the matrix is poorly scaled, it is superseded either by the naive iteration or by Lohner's method otherwise

Taylor models and floating-point arithmetic: proof that arithmetic operations are validated in COSY

The goal of this paper is to prove that the implementation of Taylor models in COSY, based on floating-point arithmetic, computes results satisfyin- g the «containment property», i.e. guaranteed results. First, Taylor models are defined and their implementation in the COSY software by Makino and Berz is detailed. Afterwards IEEE-754 floating-point arithmetic is introduced. Then the core of this paper is given: the algorithms implemented in COSY for multiplying a Taylor model by a scalar, for adding or multiplying two Taylor models are given and are proven to return Taylor models satisfying the containment property.L'objectif de ce travail est de démontrer que l'implantation des modèles de Taylor, telle qu'elle est réalisée dans le logiciel COSY, calcule des résultats qui sont garantis, c'est à dire qu''ils satisfont la propriété d'inclusion.Tout d'abord, les modèles de Taylor sont définis et leur implantation par Makino et Berz dans le logiciel COSY est détaillée. Ensuite l'arithmétique flottante, telle qu'elle est spécifiée par la norme IEEE-754, est présentée. Enfin on arrive au cœur du sujet : les algorithmes implantés dans COSY pour la multiplication d'un modèle de Taylor par un scalaire et pour la somme et le produit de deux modèles de Taylor sont donnés; il est démontré que ces algorithmes retournent de s modèles de Taylor qui satisfont la propriété d'inclusion