Uniformly fast evaluation of holonomic functions

Joris van der Hoeven (1)

1 LIX - Laboratoire d'informatique de l'École polytechnique [Palaiseau]

In a series of previous articles, we have given ecient algorithms for the evaluation of holonomic functions over the algebraic numbers and for the computation of their limits at singularities. The focus of these articles was mainly on the ecient evaluation at a xed point. In the present note, we will show that there exist uniformly ecient algorithms for evaluating holonomic functions. The main technical diculty is to maintain uniform eciency near irregular singularities. We will introduce a variant of accelerato-summation for this purpose that we call expedito-summation.

Mots clés: holonomic function; special function; fast evaluation; accelero-summation; expedito-summation

Domaines: Logiciel mathématique [cs.MS]

Joris van der Hoeven. Uniformly fast evaluation of holonomic functions. 2016. ⟨hal-01374898v2⟩