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

Richard P. Brent, Fredrik Johansson
2015 Mathematics of Computation  
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.  ...  The Brent-McMillan algorithm B3 (1980), when implemented with binary splitting, is the fastest known algorithm for high-precision computation of Euler's constant.  ...  To bound the error in the Brent-McMillan algorithm we must bound the errors in evaluating the transcendental functions I 0 (2n), K 0 (2n) and S 0 (2n) occurring in (1.1) (we ignore the error in evaluating  ... 
doi:10.1090/s0025-5718-2015-02931-7 fatcat:hflxltkbojcmrbtth4pjy44uru

An asymptotic expansion for the error term in the Brent-McMillan algorithm for Euler's constant [article]

R B Paris
2019 arXiv   pre-print
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 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  ...  4 (16x) In [3] , Brent and Johansson obtained a bound for the remainder term in the optimally truncated expansion (1.2) given by 24e −8x , thereby providing rigour to the algorithm.  ... 
arXiv:1809.04342v2 fatcat:pa6kkxcmrzh3zpuyvukt46457a

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 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  ...  Brent and Johansson (2015) obtained a bound for the remainder term in the optimally truncated expansion (1.2) given by 24e −8x , thereby providing rigour to the algorithm.  ... 
doi:10.5539/jmr.v11n3p60 fatcat:yqy6uvwze5ayllqa4lcnym5rwe

Page 5476 of Mathematical Reviews Vol. , Issue 86m [page]

1986 Mathematical Reviews  
The authors give a method for calculating ¢(s) by expanding it as 1 s-1 s—1)(s—2 saath ge te They give bounds for the truncation error: for |s| = 20, and truncation error 10~°, over 200 terms are needed  ...  , the most economical being those of R. P. Brent and E. M. McMillan [Math. Comp. 34 (1980), no. 149, 305-312; MR 82g:10002].  ... 

Precise error estimate of the Brent-McMillan algorithm for the computation of Euler's constant [article]

Jean-Pierre Demailly
2017 arXiv   pre-print
Brent and McMillan introduced in 1980 a new algorithm for the computation of Euler's constant γ, based on the use of the Bessel functions I_0(x) and K_0(x).  ...  Brent-McMillan conjectured in 1980 that the error is of the same magnitude as the last computed term, and Brent-Johansson partially proved it in 2015.  ...  (2.11) +∞ 1 Precise error estimate of the Brent-McMillan algorithm for Euler's constant  ... 
arXiv:1610.01893v3 fatcat:2nxauh3xnffariacei4emnpbsy

Some new algorithms for high-precision computation of Euler's constant

Richard P. Brent, Edwin M. McMillan
1980 Mathematics of Computation  
We describe several new algorithms for the high-precision computation of Euler's constant y = 0.577 ....  ...  Using one of the algorithms, which is based on an identity involving Bessel functions, 7 has been computed to 30,100 decimal places.  ...  This work was initiated while the first author was visiting  ... 
doi:10.1090/s0025-5718-1980-0551307-4 fatcat:6jir7iaa7vfgdmsr7t3xsuylui

Arbitrary-precision computation of the gamma function [article]

Fredrik Johansson
2021 arXiv   pre-print
We discuss the best methods available for computing the gamma function Γ(z) in arbitrary-precision arithmetic with rigorous error bounds.  ...  Besides attempting to summarize the existing state of the art, we present some new formulas, estimates, bounds and algorithmic improvements and discuss implementation results.  ...  The best available method for Euler's constant −ψ(1) = γ is the Brent-McMillan algorithm [BM80] based on Bessel functions.  ... 
arXiv:2109.08392v1 fatcat:jq7vdq6xijf5de5gaqdehvy5uu

Forecasting volatility in oil prices with a class of nonlinear volatility models: smooth transition RBF and MLP neural networks augmented GARCH approach

Melike Bildirici, Özgür Ersin
2015 Petroleum Science  
The models are compared in terms of MSE, RMSE, and MAE criteria for in-sample and out-of-sample forecast capabilities.  ...  In this study, the forecasting capabilities of a new class of nonlinear econometric models, namely, the LSTAR-LST-GARCH-RBF and MLP models are evaluated.  ...  , and reproduction in any medium, provided you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons license, and indicate if changes were made.  ... 
doi:10.1007/s12182-015-0035-8 fatcat:4qa5d4wlqbhxrm6np6xu3xu74m

Computing hypergeometric functions rigorously [article]

Fredrik Johansson
2016 arXiv   pre-print
The output can be used directly for interval computations or to generate provably correct floating-point approximations in any format.  ...  The functions _0F_1, _1F_1, _2F_1 and _2F_0 (or the Kummer U-function) are supported for unrestricted complex parameters and argument, and by extension, we cover exponential and trigonometric integrals  ...  Euler's constant γ = −ψ(1) is computed using the Brent-McMillan algorithm, for which rigorous error bounds were derived in [15] . 2F1 (a, b, −n, z) = (a) n+1 (b) n+1 z n+1 (n + 1)!  ... 
arXiv:1606.06977v2 fatcat:ulmwaxmmybdalhihlzbggwb7qm

Modern Computer Arithmetic (version 0.5.1) [article]

Richard P. Brent, Paul Zimmermann
2010 arXiv   pre-print
This is a draft of a book about algorithms for performing arithmetic, and their implementation on modern computers.  ...  At the same time, we provide a self-contained introduction for the reader who is not an expert in the field, and exercises at the end of each chapter.  ...  Acknowledgements We thank the French National Institute for Research in Computer Science and Control (INRIA), the Australian National University (ANU), and the Australian Research Council (ARC), for their  ... 
arXiv:1004.4710v1 fatcat:p3qiz53pxvbl3bprkmkz4lgady

On the Complexity of Familiar Functions and Numbers

J. M. Borwein, P. B. Borwein
1988 SIAM Review  
For most functions, provably optimal methods are not known; however the gap between what is known and what is possible is often small.  ...  The intent is to suggest that it is possible to base a taxonomy of such functions and numbers on their computational complexity.  ...  A variation of the above method for computing y has been used by Brent and McMillan [10] to compute over 29,000 partial quotients of the continued fraction of y.  ... 
doi:10.1137/1030134 fatcat:dyuvva7n6ffhrf7w6fr5pipvbi

Detection of a tropospheric ozone anomaly using a newly developed ozone retrieval algorithm for an up-looking infrared interferometer

K. J. Lightner, W. W. McMillan, K. J. McCann, R. M. Hoff, M. J. Newchurch, E. J. Hintsa, C. D. Barnet
2009 Journal of Geophysical Research  
Direct and the KLOBBER Retrieval Algorithm for Indirect a Date in 2003 UTC Time Ozonesonde (DU) KLOBBER (DU) Statistical % Error Final % Error 2 June 21:56 46.1 45.6 À28.0 À1.1  ...  Special thanks go to Brent Holben for supplying AERONET data from the COVE site and to Ken Rutledge for making the use of Chesapeake Light a possibility.  ... 
doi:10.1029/2008jd010270 fatcat:l35eyem6cjfengaxa5dhsonewu

MP users guide [article]

Richard P. Brent
2010 arXiv   pre-print
MP is a package of ANSI Standard Fortran (ANS X3.9-1966) subroutines for performing multiple-precision floating-point arithmetic and evaluating elementary and special functions.  ...  The User's Guide describes the routines and their calling sequences, example and test programs, use of the Augment precompiler, and gives installation instructions for the package.  ...  The method was discovered by Edwin McMillan and Richard Brent, and is faster than the method of Sweeney (used in earlier versions of MPEUL). See -R. P. Brent and E. M.  ... 
arXiv:1004.3173v2 fatcat:6brfey2gczgjdmthybdlwhj5ji

Fraction-Free Computation of Matrix Rational Interpolants and Matrix GCDs

Bernhard Beckermann, George Labahn
2000 SIAM Journal on Matrix Analysis and Applications  
The algorithms are suitable for computation in exact arithmetic domains where growth of coe cients in intermediate computations are a central concern.  ...  The algorithms are fast and compute all solutions to a given problem.  ...  For a proof of part (b), it remains to establish the (rough) bound for N .Notice that N j~ j, with the latter quantity being bounded above by N # , the McMillan degree of G (see the remark before Theorem  ... 
doi:10.1137/s0895479897326912 fatcat:jwa6tvzvrjdiva75mor64ryyai

The computation of previously inaccessible digits of π2 and Catalan's constant (2013) [chapter]

David H. Bailey, Jonathan M. Borwein, Andrew Mattingly, Glenn Wightwick
2016 Pi: The Next Generation  
Acknowledgements Thanks are due to many colleagues, but most explicitly to Prof Mary-Anne Williams of University Technology Sydney who conceived the idea of a π-related computation to conclude in conjunction  ...  We also wish to thank Matthew Tam who constructed the database version of [1] .  ...  If we set N = 10, 000, 001 in (24), since we know there are no errors in the first 10,000,000 elements, then we obtain an upper bound of 1.563 × 10 −9 which suggests it is truly unlikely that errors will  ... 
doi:10.1007/978-3-319-32377-0_20 fatcat:ardscakdifh53mkyu3vlvamrji
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