Section: New Results

Certified computing and computer algebra

Standardization of interval arithmetic

The IEEE 1788 working group is devoted to the standardization of interval arithmetic. V. Lefèvre and N. Revol are very active in this group. This year has been devoted to a ballot on the whole text of the standard [28] , and to editorial work to make it compliant with IEEE rules. The final, remaining step, is the so-called “Sponsor ballot” and it should be completed in 2015.

Interval linear algebra on multi-core processors

For the product of matrices with interval coefficients, fast approximate algorithms have been developed by Philippe Théveny: they compute an enclosure of the exact product. These algorithms rely on the representation of intervals by their midpoints and radii. This representation allows one to use optimized routines for the multiplication of matrices with floating-point coefficients. In [4] , the quality of the approximation of several algorithms is established, which accounts for roundoff errors and not only method's errors. A new algorithm is proposed, which requires even less (only 2) calls to a floating-point routine and still offers a good approximation quality, for a well specified type of input matrices. Three of the studied algorithms are implemented on a multi-core architecture. To avoid problems listed in [12] and to offer good performances, Philippe Théveny developed optimizations. The resulting implementations exhibit good performances: guaranteed results are obtained with an overhead less than 3, high numerical intensity and good scalability.

Numerical reproducibility

What is called numerical reproducibility is the problem of getting the same result when the scientific computation is run several times, either on the same machine or on different machines. In [12] , the focus is on interval computations using floating-point arithmetic: Nathalie Revol and Philippe Théveny identified implementation issues that may invalidate the inclusion property, and presented several ways to preserve this inclusion property. This work has also been replaced in the larger context of numerical validation [15] .

Faster multivariate interpolation with multiplicities

Muhammad Chowdhury (U. Western Ontario), Claude-Pierre Jeannerod, Vincent Neiger (ENS de Lyon), Éric Schost (U. Western Ontario), and Gilles Villard proposed in [38] a fast algorithm for interpolating multivariate polynomials with multiplicities. This algorithm relies on the reduction to a problem of simultaneous polynomial approximations, which is then solved using fast structured linear algebra techniques. This algorithm leads to the best known complexity bounds for the interpolation step of the list-decoding of Reed-Solomon codes, Parvaresh-Vardy codes or folded Reed-Solomon codes. In the special case of Reed-Solomon codes, it allows to accelerate the interpolation step of Guruswami and Sudan’s list-decoding by a factor (list size)/(multiplicity).

Polynomial system solving

M. Bardet (U. Rouen), J.-C. Faugère (PolSys team) and B. Salvy studied the complexity of Gröbner bases computation, in particular in the generic situation where the variables are in simultaneous Noether position with respect to the system. They gave a bound on the number of polynomials of each degree in a Gröbner basis computed by Faugère’s $F_5$ algorithm in this generic case for the grevlex ordering (which is also a bound on the number of polynomials for a reduced Gröbner basis, independently of the algorithm used) and used it to bound the complexity of the $F_5$ algorithm [5] .

Linear differential equations

In [6] , A. Bostan (SpecFun team), K. Raschel (U. Tours) and B. Salvy proved that the sequence ${\left(e_n^{𝔖}\right)}_{n\ge 0}$ of excursions in the quarter plane corresponding to a nonsingular step set $𝔖\subseteq {\{0,\pm 1\}}^2$ with infinite group does not satisfy any nontrivial linear recurrence with polynomial coefficients. Accordingly, in those cases, the trivariate generating function of the numbers of walks with given length and prescribed ending point is not D-finite. Moreover, they displayed the asymptotics of $e_n^{𝔖}$. This completes the classification of these walks.

Colleagues from the LAAS (Toulouse) and B. Salvy provided a new method for computing the probability of collision between two spherical space objects involved in a short-term encounter. In this specific framework of conjunction, classical assumptions reduce the probability of collision to the integral of a 2-D normal distribution over a disk shifted from the peak of the corresponding Gaussian function. Both integrand and domain of integration directly depend on the nature of the short-term encounter. Thus the inputs are the combined sphere radius, the mean relative position in the encounter plane at reference time as well as the relative position covariance matrix representing the uncertainties. The method they presented is based on an analytical expression for the integral. It has the form of a convergent power series whose coefficients verify a linear recurrence. It is derived using Laplace transform and properties of D-finite functions. The new method has been intensively tested on a series of test-cases and compares favorably to other existing works [29] .