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

Richard P. Brent; Fredrik Johansson

doi:10.48550/arXiv.1312.0039

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

Richard P. Brent, Fredrik Johansson

Research Institute for Symbolic Computation

Research output: Working paper and reports › Preprint

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 modified Bessel functions.

Original language	English
Place of Publication	Hagenberg
Publisher	RISC
Number of pages	10
DOIs	https://doi.org/10.48550/arXiv.1312.0039
Publication status	Published - 2013

Publication series

Name	arXiv.org

Fields of science

101001 Algebra
101002 Analysis
101 Mathematics
102 Computer Sciences
102011 Formal languages
101009 Geometry
101013 Mathematical logic
101020 Technical mathematics
101025 Number theory
101012 Combinatorics
101005 Computer algebra
101006 Differential geometry
101003 Applied geometry
102025 Distributed systems

JKU Focus areas

Computation in Informatics and Mathematics

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10.48550/arXiv.1312.0039

Cite this

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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 modified Bessel functions.

UR - http://arxiv.org/abs/1312.0039

U2 - 10.48550/arXiv.1312.0039

DO - 10.48550/arXiv.1312.0039

M3 - Preprint

T3 - arXiv.org

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

PB - RISC

CY - Hagenberg

ER -

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

Abstract

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