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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- Richard P. Brent
- Affiliation: Mathematical Sciences Institute, Australian National University, Canberra, ACT 0200, Australia
- Email: firstname.lastname@example.org
- Fredrik Johansson
- Affiliation: RISC, Johannes Kepler University, 4040 Linz, Austria
- MR Author ID: 999321
- Email: email@example.com
- 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