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arXiv · 1610.01893

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

Jean-Pierre Demailly
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Abstract

Brent and McMillan introduced in 1980 a new algorithm for the computation of Euler's constant $\gamma$, 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 $\gamma$. 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.

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Jean-Pierre Demailly. 2016-10-06. Precise error estimate of the Brent-McMillan algorithm for the computation of Euler's constant. https://arxiv.org/abs/1610.01893

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