Algorithmic design and analysis of symbolic or numerical algorithms.

On the symbolic side, we have so far obtained fast algorithms for basic operations on both polynomial matrices and structured matrices, but in a rather independent way. Both types turn out to have much in common, but this is sometimes not reflected by the complexities obtained, especially for applications in cryptology and coding theory. Our long term goal in this area is thus to explore these connections further, to provide a more unified treatment and bridge these complexity gaps, and to produce associated efficient implementations. A first step towards this goal will be the design and implementation of enhanced algorithms for various generalizations of Hermite-Padé approximation; in the context of list decoding, this should in particular make it possible to improve over the structured-matrix approach, which is so far the fastest known.

On the numerical side, we will continue to revisit and improve the classical error bounds of numerical analysis in the light of all the subtleties of IEEE floating-point arithmetic. These aspects will be developed jointly with the “symbolic floating-point” approach presented in the next paragraph. A complementary approach will also be studied, based on the estimation (possibly via automatic differentiation) of condition numbers in order to identify inputs leading to large backward errors. Finally, concerning interval arithmetic, a thorough analysis of the accuracy of several representations, such as mid-rad, is also to be done.

Symbolic floating-point arithmetic.

Our work on the analysis of algorithms in floating-point arithmetic leads us to manipulate floating-point data in their greatest generality, that is, as symbolic expressions in the base and the precision. A long-term goal here is to develop theorems as well as efficient data structures and algorithms for handling such quantities by computer rather than by hand as we do now. This is a completely new direction, whose main outcome will be a “symbolic floating-point toolbox” distributed in computer algebra systems like Sage and or Maple. In particular, such a toolbox will provide a way to check automatically the certificates of optimality we have obtained on the error bounds of various numerical algorithms. A PhD student has started on this subject in September 2014.

High-performance multiple precision arithmetic libraries.

Many numerical problems require higher precision than the conventional floating-point (single, double) formats. One solution is to use multiple precision libraries such as GNU MPFR, which allow the manipulation of very high precision numbers, but their generality (they are able to handle numbers with millions of digits), is a quite heavy alternative when high performance is needed. Our objective is to design a multiple precision arithmetic library that would allow to tackle problems where a precision of a few hundred bits is sufficient, but which have strong performance requirements. Applications include the process of long-term iteration of chaotic dynamical systems ranging from the classical Henon map to calculations of planetary orbits. The designed algorithms will be formally proved. We are in close contact with Warwick Tucker (Uppsala University, Sweden) and Mioara Joldes (LAAS, Toulouse) on this topic. A PhD student funded by a Région Rhône-Alpes grant has started on this topic in September 2014.

Interactions between arithmetics.

We will work on the interplay between floating-point and integer arithmetics, and especially on how to make the best use of both integer and floating-point basic operations when designing floating-point numerical kernels for embedded devices. This will be done in the context of the Metalibm ANR project and of our collaboration with STMicroelectronics. In addition, our work on the IEEE 1788 standard leads naturally to the development of associated reference libraries for interval arithmetic. A first direction will be to implement IEEE 1788 interval arithmetic using the fixed-precision hardware available for IEEE 754-2008 floating-point arithmetic. Another one will be to provide efficient support for multiple-precision intervals, in mid-rad representation and by developing MPFR-based code-generation tools aimed at handling families of functions.

Adequation algorithms/architectures.

So far, we have investigated how specific instructions like the fused multiply-add (FMA) impact the accuracy of computations, and have proposed several highly accurate FMA-based algorithms. The FMA being available on several recent architectures, we now want to understand its impact on such algorithms in terms of practical performances. This should be a medium term project, leading to FMA-based algorithms with best speed/accuracy/robustness tradeoff. On the other hand (and on the long term), a major issue is how to exploit the various levels of parallelism of recent and upcoming architectures to ensure simultaneously high performance and reliability. A first direction will be to focus on SIMD parallelism, offered by instruction sets via vector instructions. This kind of parallelism should be key for small numerical kernels like elementary functions, complex arithmetic, or low-dimensional matrix computations. A second direction will be at the multi-core processor level, especially for larger numerical or algebraic problems (and in conjunction with SIMD parallelism when handling sub-problems of small enough dimension). Finally, we will work on aspects of automatic adaptation (auto-tuning) to such architectural features, not only for speed, but also for accuracy. This could be done via the design and implementation of heuristics capable of inserting more accurate codes, based for example on error-free transforms, whenever needed.