Section: New Results

Reliability and Accuracy

Standardization of interval arithmetic

We contributed to the creation in 2008 and N. Revol chairs the IEEE 1788 working group on the standardization of interval arithmetic http://grouper.ieee.org/groups/1788/ . More than 140 persons from over 20 countries take part in the discussions, around 1500 messages were exchanged in 2012. We are currently voting on portions of the text of the standard and have good hope that the group will reach a final version of the standard within the allotted time. An extension has been granted for 2 more years, until December 2014.

The annual in-person meeting, chaired by N. Revol, took place at the end of the SCAN 2012 conference in Novosibirsk, Russia, the 28th of September. It was broadcasted via the Web and feedback was possible through e-mails. More than 20 persons attended the meeting.

V. Lefèvre participated in various discussions, either in the mailing-list or in small subgroups (he sent around 390 mail messages in 2012). He proposed a motion, which passed, on properties needed by number formats for operations between intervals and numbers (constructors, midpoint, etc.).

The latest discussions dealt with:

• flavors: even if there continues to be a give-and-take between proponents of a “small” standard involving just basic interval arithmetic and those who also want to also include the less common “modal arithmetic”, this motion about “flavors” intends to allow inclusion of modal interval arithmetic consistently and simply, possibly at a later stage or revision of the standard;

• expressions: what is regarded as an expression by P1788, the relation with the programming languages, what this implies concerning the allowed optimizations, etc.;

• decorations: what are the properties of functions we want to track along a computation, how the empty interval is handled, etc.;

• reproducibility: across several runs of a translated (e.g., compiled) program or across platforms, representation-independent behavior, reproducibility for parallel programs, etc.

A personal view of the current status of the work of the IEEE P1788 group and of directions for future work has been presented in [46] , [45] .

Interval matrix multiplication

Several formulas exist for the product of two intervals using the midpoint-radius representation: they trade off accuracy for efficiency. The use of these formulas for the product of matrices with interval coefficients allows to use BLAS3 routines and to benefit from their performances in terms of execution time [48] . The accuracy of these methods are studied in [42] . As it can be difficult to ensure that a prescribed rounding mode is actually in use, formulas that are oblivious to the rounding mode are developed [22] . The implementations of these variants on multicores are compared in [47] .

Rigorous polynomial approximation using Taylor models in Coq

One of the most common and practical ways of representing a real function on machines is by using a polynomial approximation. It is then important to properly handle the error introduced by such an approximation. N. Brisebarre, M. Joldes (Uppsala Univ., Sweden), E. Martin-Dorel, M. Mayero, J.-M. Muller, I. Pasca, L. Rideau (Marelle), and L. Théry (Marelle) have worked on the problem of offering guaranteed error bounds for a specific kind of rigorous polynomial approximation called Taylor model [26] . They carry out this work in the Coq proof assistant, with a special focus on genericity and efficiency for our implementation. They give an abstract interface for rigorous polynomial approximations, parameterized by the type of coefficients and the implementation of polynomials, and they instantiate this interface to the case of Taylor models with interval coefficients, while providing all the machinery for computing them.