A bound for the error term in the Brent-McMillan algorithm

Richard P. Brent; Fredrik Johansson

doi:10.1090/S0025-5718-2015-02931-7

A bound for the error term in the Brent-McMillan algorithm

Richard P. Brent, Fredrik Johansson

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

4 Citations (Scopus)

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 I₀(x) and K₀(x) for positive real x.

Original language	English
Pages (from-to)	2351-2359
Number of pages	9
Journal	Mathematics of Computation
Volume	84
Issue number	295
DOIs	https://doi.org/10.1090/S0025-5718-2015-02931-7
Publication status	Published - 2015

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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 I0(x) and K0(x) for positive real x.",

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AB - 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 I0(x) and K0(x) for positive real x.

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A bound for the error term in the Brent-McMillan algorithm

Abstract

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