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Mathematics of Computation

Published by the American Mathematical Society, the Mathematics of Computation (MCOM) is devoted to research articles of the highest quality in all areas of pure and applied mathematics.

ISSN 1088-6842 (online) ISSN 0025-5718 (print)

The 2020 MCQ for Mathematics of Computation is 1.98.

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A bound for the error term in the Brent-McMillan algorithm
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by Richard P. Brent and Fredrik Johansson PDF
Math. Comp. 84 (2015), 2351-2359 Request permission

Abstract:

The Brent-McMillan algorithm B3 (1980), when implemented with binary splitting, is the fastest known algorithm for high-precision computation of Euler’s constant. However, no rigorous error bound for the algorithm has ever been published. We provide such a bound and justify the empirical observations of Brent and McMillan. We also give bounds on the error in the asymptotic expansions of functions related to the Bessel functions $I_0(x)$ and $K_0(x)$ for positive real $x$.
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Additional Information
  • Richard P. Brent
  • Affiliation: Mathematical Sciences Institute, Australian National University, Canberra, ACT 0200, Australia
  • Email: richard.brent@anu.edu.au
  • Fredrik Johansson
  • Affiliation: RISC, Johannes Kepler University, 4040 Linz, Austria
  • MR Author ID: 999321
  • Email: fredrik.johansson@risc.jku.at
  • Received by editor(s): November 29, 2013
  • Received by editor(s) in revised form: January 1, 2014
  • Published electronically: March 4, 2015
  • Additional Notes: The first author was supported by Australian Research Council grant DP140101417.
    The second author was supported by the Austrian Science Fund (FWF) grant Y464-N18.
  • © Copyright 2015 American Mathematical Society
  • Journal: Math. Comp. 84 (2015), 2351-2359
  • MSC (2010): Primary 33C10, 11Y60, 65G99, 65Y20, 68Q25, 68W40, 68W99
  • DOI: https://doi.org/10.1090/S0025-5718-2015-02931-7
  • MathSciNet review: 3356029