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


  •  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.



This work is licensed under a Creative Commons Attribution 4.0 License.
  • Issn(Print): 1916-9795
  • Issn(Onlne): 1916-9809
  • Started: 2009
  • Frequency: bimonthly

Journal Metrics

Google-based Impact Factor (2019): 2.75

  • h-index (February 2019): 17
  • i10-index (February 2019): 39
  • h5-index (February 2019): 9
  • h5-median (February 2019): 9

( The data was calculated based on Google Scholar Citations. Click Here to Learn More. )

Contact