Research Topics of the Algorithms Project
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Computer algebra at large aims at making effective large portions of mathematics, paying due attention to complexity issues. For reasons mentioned above, our project specifically investigates the way mathematical objects originating in complex analysis can be dealt with in an algorithmic way by computer algebra systems. Our main contributions in this area concern the automation of asymptotic analysis and the handling of special functions. The mathematical foundations of our algorithms are deeply rooted in differential algebra (Hardy fields for asymptotic expansions and Ore algebras for special functions).
Over the years, in order to automate the average-case analysis of larger and ever larger classes of algorithms, we have developed algorithms and implementations for the following problems: the specification of formally specified combinatorial structures; the corresponding problems of enumeration and random generation; the automatic construction of asymptotic scales which is necessary for extracting the singular behaviour of generating functions; the automatic computation of asymptotic expansions in such scales; the automatic computation of asymptotic expansions satisfied by coefficients of generating series. An Encyclopedia of Combinatorial Structures, available on the web, gathers roughly one thousand structures for which generating series, recurrences, and asymptotic behaviour have been determined automatically using our libraries.
An important principle of computer algebra is that it is often easier to operate with equations defining a mathematical object implicitly rather than trying to obtain a ``closed-form'' expression of it. The class of linear differential and difference equations is particularly important in view of the large variety of functions and sequences they capture. In this area, we have developed the highly successful GFUN package (jointly with P. Zimmermann, from the Spaces project) dealing with the univariate case. In the multivariate case, we have developed the underlying theory based on Gröbner bases in Ore algebra, and an implementation in the MGFUN package. The algorithmic advances of the past few years have made it possible to start the implementation of a Dynamic Dictionary of Mathematical Functions, providing various information concerning classical functions (of wide use throughout sciences), including Bessel functions, Airy functions, ... The corresponding information is all automatically generated.