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

R. B. Paris
2019 Journal of Mathematics Research  
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
more » ... ling more precise information on the error term than recently obtained bounds and estimates.
doi:10.5539/jmr.v11n3p60 fatcat:yqy6uvwze5ayllqa4lcnym5rwe