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A bound for the error term in the Brent-McMillan algorithm

Authors: Richard P. Brent and Fredrik Johansson
Journal: Math. Comp. 84 (2015), 2351-2359
MSC (2010): Primary 33C10, 11Y60, 65G99, 65Y20, 68Q25, 68W40, 68W99
Published electronically: March 4, 2015
MathSciNet review: 3356029
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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

Fredrik Johansson
Affiliation: RISC, Johannes Kepler University, 4040 Linz, Austria

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.
Article copyright: © Copyright 2015 American Mathematical Society