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An Asymptotic Expansion for the Error Term in the Brent-McMillan Algorithm for Euler’s Constant

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  • R. B. Paris

Abstract

The Brent-McMillan algorithm is the fastest known procedure for the high-precision computation of Euler’s constant γ and is based on the modified Bessel functions I_0(2x) and K_0(2x). An error estimate for this algorithm relies on the optimally truncated asymptotic expansion for the product I_0(2x)K_0(2x) when x assumes large positive integer values. An asymptotic expansion for this optimal error term is derived by exploiting the techniques developed in hyperasymptotics, thereby enabling more precise information on the error term than recently obtained bounds and estimates.

Suggested Citation

  • R. B. Paris, 2019. "An Asymptotic Expansion for the Error Term in the Brent-McMillan Algorithm for Euler’s Constant," Journal of Mathematics Research, Canadian Center of Science and Education, vol. 11(3), pages 60-66, June.
    • Handle: RePEc:ibn:jmrjnl:v:11:y:2019:i:3:p:60
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    More about this item

    Keywords

    Euler's constant; Brent-McMillan algorithm; asymptotic expansion; optimal truncation; exponentially improved expansion;
    All these keywords.

    JEL classification:

    • R00 - Urban, Rural, Regional, Real Estate, and Transportation Economics - - General - - - General
    • Z0 - Other Special Topics - - General

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