Precise error estimate of the Brent-McMillan algorithm for the computation of Euler's constant - Archive ouverte HAL Accéder directement au contenu
Pré-Publication, Document De Travail Année : 2016

Precise error estimate of the Brent-McMillan algorithm for the computation of Euler's constant

Jean-Pierre Demailly

Résumé

Brent and McMillan introduced in 1980 a new algorithm for the computation of Euler's constant γ, based on the use of the Bessel functions I_0(x) and K_0(x). It is the fastest known algorithm for the computation of γ. The time complexity can still be improved by evaluating a certain divergent asymptotic expansion up to its minimal term. Brent-McMillan conjectured in 1980 that the error is of the same magnitude as the last computed term, and Brent-Johansson partially proved it in 2015. They also gave some numerical evidence for a more precise estimate of the error term. We find here an explicit expression of that optimal estimate, along with a complete self-contained formal proof and an even more precise error bound.
Fichier principal
Vignette du fichier
gamma_brent_mcmillan.pdf (214.74 Ko) Télécharger le fichier
Origine : Fichiers produits par l'(les) auteur(s)
Loading...

Dates et versions

hal-01377144 , version 1 (06-10-2016)
hal-01377144 , version 2 (08-10-2016)
hal-01377144 , version 3 (10-12-2017)

Identifiants

Citer

Jean-Pierre Demailly. Precise error estimate of the Brent-McMillan algorithm for the computation of Euler's constant. 2016. ⟨hal-01377144v3⟩

Collections

UGA CNRS FOURIER
129 Consultations
76 Téléchargements

Altmetric

Partager

Gmail Facebook X LinkedIn More