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Precise error estimate of the Brent-McMillan algorithm for the computation of Euler's constant
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<span title="2017-12-11">2017</span>
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arXiv
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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
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... ave 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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