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

2015
*
Mathematics of Computation
*

However, no rigorous

doi:10.1090/s0025-5718-2015-02931-7
fatcat:hflxltkbojcmrbtth4pjy44uru
*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 ...##
###
An asymptotic expansion for the error term in the Brent-McMillan algorithm for Euler's constant
[article]

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*. ...

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

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*. ...

##
###
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]

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 ...

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

1980
*
Mathematics of Computation
*

We describe several new

doi:10.1090/s0025-5718-1980-0551307-4
fatcat:6jir7iaa7vfgdmsr7t3xsuylui
*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 ...##
###
Arbitrary-precision computation of the gamma function
[article]

2021
*
arXiv
*
pre-print

We discuss

arXiv:2109.08392v1
fatcat:jq7vdq6xijf5de5gaqdehvy5uu
*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. ...##
###
Forecasting volatility in oil prices with a class of nonlinear volatility models: smooth transition RBF and MLP neural networks augmented GARCH approach

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. ...

##
###
Computing hypergeometric functions rigorously
[article]

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)! ...

##
###
Modern Computer Arithmetic (version 0.5.1)
[article]

2010
*
arXiv
*
pre-print

This is

arXiv:1004.4710v1
fatcat:p3qiz53pxvbl3bprkmkz4lgady
*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 ...##
###
On the Complexity of Familiar Functions and Numbers

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. ...

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

2009
*
Journal of Geophysical Research
*

Direct and

doi:10.1029/2008jd010270
fatcat:l35eyem6cjfengaxa5dhsonewu
*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. ...##
###
MP users guide
[article]

2010
*
arXiv
*
pre-print

MP is

arXiv:1004.3173v2
fatcat:6brfey2gczgjdmthybdlwhj5ji
*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. ...##
###
Fraction-Free Computation of Matrix Rational Interpolants and Matrix GCDs

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 ...

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

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

doi:10.1007/978-3-319-32377-0_20
fatcat:ardscakdifh53mkyu3vlvamrji
*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 ...
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