A bound for the error term in the Brent-McMillan algorithm
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- by Richard P. Brent and Fredrik Johansson PDF
- Math. Comp. 84 (2015), 2351-2359 Request permission
Abstract:
The Brent-McMillan algorithm B3 (1980), when implemented with binary splitting, is the fastest known algorithm for high-precision computation of Euler’s constant. However, no rigorous error bound for the algorithm has ever been published. We provide such a bound and justify the empirical observations of Brent and McMillan. We also give bounds on the error in the asymptotic expansions of functions related to the Bessel functions $I_0(x)$ and $K_0(x)$ for positive real $x$.References
- Milton Abramowitz and Irene A. Stegun (eds.), Handbook of mathematical functions with formulas, graphs, and mathematical tables, Dover Publications, Inc., New York, 1992. Reprint of the 1972 edition. MR 1225604
- R. P. Brent, Ramanujan and Euler’s Constant, Presented at the CARMA Workshop on Exploratory Experimentation and Computation in Number Theory, Newcastle, Australia, July 2010, http://maths-people.anu.edu.au/~brent/pd/Euler_CARMA_10.pdf.
- Richard P. Brent and Edwin M. McMillan, Some new algorithms for high-precision computation of Euler’s constant, Math. Comp. 34 (1980), no. 149, 305–312. MR 551307, DOI 10.1090/S0025-5718-1980-0551307-4
- Richard P. Brent and Paul Zimmermann, Modern computer arithmetic, Cambridge Monographs on Applied and Computational Mathematics, vol. 18, Cambridge University Press, Cambridge, 2011. MR 2760886
- J. D. Jackson and W. K. H. Panofsky, Edwin Mattison McMillan 1907–1991, Biographical Memoirs, Nat. Acad. Sci. (USA), 69:213–237, 1996.
- C. Koutschan, HolonomicFunctions (User’s Guide), Technical Report 10-01, RISC Report Series, University of Linz, Austria, 2010.
- National Institute of Standards and Technology, Digital Library of Mathematical Functions, http://dlmf.nist.gov/, 2013.
- Frank W. J. Olver, Asymptotics and special functions, AKP Classics, A K Peters, Ltd., Wellesley, MA, 1997. Reprint of the 1974 original [Academic Press, New York; MR0435697 (55 #8655)]. MR 1429619
- A. J. Yee, Euler-Mascheroni Constant, http://www.numberworld.org/digits/EulerGamma/, 2011.
Additional Information
- Richard P. Brent
- Affiliation: Mathematical Sciences Institute, Australian National University, Canberra, ACT 0200, Australia
- Email: richard.brent@anu.edu.au
- Fredrik Johansson
- Affiliation: RISC, Johannes Kepler University, 4040 Linz, Austria
- MR Author ID: 999321
- Email: fredrik.johansson@risc.jku.at
- Received by editor(s): November 29, 2013
- Received by editor(s) in revised form: January 1, 2014
- Published electronically: March 4, 2015
- Additional Notes: The first author was supported by Australian Research Council grant DP140101417.
The second author was supported by the Austrian Science Fund (FWF) grant Y464-N18. - © Copyright 2015 American Mathematical Society
- Journal: Math. Comp. 84 (2015), 2351-2359
- MSC (2010): Primary 33C10, 11Y60, 65G99, 65Y20, 68Q25, 68W40, 68W99
- DOI: https://doi.org/10.1090/S0025-5718-2015-02931-7
- MathSciNet review: 3356029