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.
Highlights
The Brent-McMillan algorithm (Brent & McMillan, 1980), when implemented with binary splitting, is the fastest known method of high-precision computation of Euler’s constant γ
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
The error resulting from K0(2x)/I0(2x) in the Brent-McMillan algorithm in (1.1) at optimal truncation has the expansion
Summary
An Asymptotic Expansion for the Error Term in the Brent-McMillan Algorithm for Euler’s Constant. B. Paris, Division of Computing and Mathematics, Abertay University, Dundee DD1 1HG, UK. Received: April 11, 2019 Accepted: May 16, 2019 Online Published: May 22, 2019 doi:10.5539/jmr.v11n3p60
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