An asymptotic expansion for the error term in the Brent-McMillan algorithm for Euler’s constant

R. B. Paris*

*Corresponding author for this work

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    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.
    Original languageEnglish
    Pages (from-to)60-66
    Number of pages7
    JournalJournal of Mathematics Research
    Volume11
    Issue number3
    Early online date22 May 2019
    DOIs
    Publication statusPublished - 22 May 2019

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