Section: New Results

Tools for numerically guaranteed computations

Participant : Sylvain Chevillard.

The overall and long-term goal is to enhance the quality of numerical computations. The progress made during year 2013 is the following:

• Publication of a work with Marc Mezzarobba (who was with Aric project-team at that time, and who is now with LIP6) about the efficient evaluation of the Airy $\text{Ai}\left(x\right)$ function when $x$ is moderately large [22] . The Taylor series of the Airy Ai function (as many others such as, e.g., Bessel functions or erf) is ill-conditioned when $x$ is not small. To overcome this difficulty, we extend a method by Gawronski, Müller and Reinhard, known to solve the issue in the case of the error function erf. We rewrite $\text{Ai}\left(x\right)$ as $G\left(x\right)/F\left(x\right)$ where $F$ and $G$ are two functions with well-conditioned series. However, the coefficients of $G$ turn out to obey a three-terms ill-conditioned recurrence. We evaluate this recurrence using Miller's backward algorithm with a rigorous error analysis. Function $\text{Ai}$ is an example, but ideally the process could be automated to handle some appropriate class of functions in a future work.

• A more general endeavor is to develop a tool that helps developers of libms in their task. This is performed by the software Sollya (http://sollya.gforge.inria.fr/ ), developed in collaboration with C. Lauter (Université Pierre et Marie Curie) and M. Joldeş (LAAS). In 2013, we released version version 4.0 (in May) and 4.1 (in November) of Sollya. Among other things these releases make available to the user all features of Sollya as a C library. They also introduce the possibility of computing Chebyshev models, and a generalization of Remez algorithm allowing the user to compute a $L^{\infty }$ best approximation of a real-valued function on a bounded real interval by any linear combination of given functions.