Relaxed algorithms for p-adic numbers - Archive ouverte HAL Access content directly
Preprints, Working Papers, ... Year : 2010

Relaxed algorithms for p-adic numbers


Current implementations of p-adic numbers usually rely on so called zealous algorithms, which compute with truncated p-adic expansions at a precision that can be specified by the user. In combination with Newton-Hensel type lifting techniques, zealous algorithms can be made very efficient from an asymptotic point of view. In the similar context of formal power series, another so called lazy technique is also frequently implemented. In this context, a power series is essentially a stream of coefficients, with an effective promise to obtain the next coefficient at every stage. This technique makes it easier to solve implicit equations and also removes the burden of determining appropriate precisions from the user. Unfortunately, naive lazy algorithms are not competitive from the asymptotic complexity point of view. For this reason, a new relaxed approach was proposed by van der Hoeven in the 90's, which combines the advantages of the lazy approach with the asymptotic efficiency of the zealous approach. In this paper, we show how to adapt the lazy and relaxed approaches to the context of p-adic numbers. We report on our implementation in the C++ library algebramix of Mathemagix, and show significant speedups in the resolution of p-adic functional equations when compared to the classical Newton iteration.
Fichier principal
Vignette du fichier
padic-2010-05-22-1.pdf (364.73 Ko) Télécharger le fichier
Origin : Files produced by the author(s)

Dates and versions

hal-00486680 , version 1 (26-05-2010)


  • HAL Id : hal-00486680 , version 1


Jérémy Berthomieu, Joris van der Hoeven, Grégoire Lecerf. Relaxed algorithms for p-adic numbers. 2010. ⟨hal-00486680⟩
673 View
333 Download


Gmail Facebook X LinkedIn More