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

R. B. Paris

doi:10.5539/jmr.v11n3p60

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

R. B. Paris^*

^*Corresponding author for this work

Research output: Contribution to journal › Article › peer-review

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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 language	English
Pages (from-to)	60-66
Number of pages	7
Journal	Journal of Mathematics Research
Volume	11
Issue number	3
Early online date	22 May 2019
DOIs	https://doi.org/10.5539/jmr.v11n3p60
Publication status	Published - 22 May 2019

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abstract = "The Brent-McMillan algorithm is the fastest known procedure for the high-precision computation of Euler{\textquoteright}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.",

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AB - 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.

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