Darboux integral: Difference between revisions

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en>Ozob
No, it's really meant to be the other way around, because that avoids having to use improper integrals
en>Vinícius Machado Vogt
A reference for the part about their equivalence.
 
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'''Richard Peirce Brent''' (born 20 April 1946, [[Melbourne]]) is an [[Australia]]n [[mathematician]] and [[computer scientist]]. He is an emeritus professor at the [[Australian National University]] and a conjoint professor at the [[University of Newcastle (Australia)|University of Newcastle]] (Australia). From March 2005 to March 2010 he was a [[Federation Fellowship|Federation Fellow]]<ref>[http://www.arc.gov.au/pdf/2004_ff/2004_FedFellows_Selection_Report.pdf Federation Fellowships Funding Outcomes 2004]. Australian Research Council</ref> at the [[Australian National University]]. His research interests include [[number theory]] (in particular [[factorisation]]), [[random number generators]], [[computer architecture]], and [[analysis of algorithms]].
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In 1973, he published a [[root-finding algorithm]] (an algorithm for solving equations numerically) which is now known as [[Brent's method]].<ref>Brent (1973). ''Algorithms for Minimization without Derivatives.'' Prentice-Hall, Englewood Cliffs, NJ. Reprinted by  Dover Publications, Mineola, New York, January 2002. ISBN 0-486-41998-3.</ref>
 
In 1975 he and [[Eugene Salamin (mathematician)|Eugene Salamin]] independently conceived the [[Salamin–Brent algorithm]], used in high-precision calculation of <math>\pi</math>. At the same time, he showed that all the [[elementary functions]] (such as log(''x''), sin(''x'') etc.) can be evaluated to high precision in the same time as <math>\pi</math> (apart from a small constant factor) using the [[arithmetic-geometric mean]] of [[Carl Friedrich Gauss]].<ref>{{cite journal | last = Brent | first = R.P. | year = 1975 | id = {{citeseerx|10.1.1.119.3317}} | title = Multiple-Precision Zero-Finding Methods and the Complexity of Elementary Function Evaluation | journal = Analytic Computational Complexity | editor-last = Traub | editor-first = J.F. | publisher = Academic Press | location = New York }}</ref>
 
In 1979 he showed that the first 75 million complex zeros of the [[Riemann zeta function]] lie on the critical line, providing some experimental evidence for the [[Riemann Hypothesis]].<ref>{{cite journal | last = Brent | first = R.P. | year = 1979 | jstor = 2006473 | title = On the Zeros of the Riemann Zeta Function in the Critical Strip | journal = Mathematics of Computation | volume = 33 | issue = 148 | pages = 1361–1372 }}</ref>
 
In 1980 he and Nobel laureate [[Edwin McMillan]] found a new algorithm for high-precision computation of the [[Euler-Mascheroni constant]] <math>\gamma</math> using [[Bessel functions]], and showed that <math>\gamma</math> can not have a simple rational form ''p''/''q'' (where ''p'' and ''q'' are integers) unless ''q'' is extremely large (greater than 10<sup>15000</sup>).<ref>Brent, R.P. and MacMillan, E.M. (1980). "[http://www.ams.org/journals/mcom/1980-34-149/S0025-5718-1980-0551307-4/S0025-5718-1980-0551307-4.pdf Some New Algorithms for High-Precision Computation of Euler's Constant]". ''Mathematics of Computation'' '''34''' (149) 305-312.</ref>
 
In 1980 he and [[John Pollard (mathematician)|John Pollard]] factored the eighth [[Fermat number]] using a variant of the [[Pollard rho]] algorithm.<ref>{{cite journal | last1 = Brent | first1 = R.P. | last2 = Pollard | first2 = J.M. | year = 1981 | jstor = 2007666 | title = Factorization of the Eighth Fermat Number | journal = Mathematics of Computation | volume = 36 | issue = 154 | pages = 627–630 }}</ref> He later factored the tenth<ref>{{cite journal | last = Brent | first = R.P. | year = 1999 | jstor = 2585124 | title = Factorization of the Tenth Fermat Number | journal = Mathematics of Computation | volume = 68 | issue = 225 | pages = 429–451 }}</ref> and eleventh Fermat numbers using Lenstra's [[elliptic curve factorisation]] algorithm.
 
In 2002, Brent, Samuli Larvala and [[Paul Zimmermann]] discovered a very large [[primitive polynomial (field theory)|primitive trinomials]] over GF(2):
:<math> x^{6972593} + x^{3037958} + 1. </math>
The degree 6972593 is the exponent of a [[Mersenne prime]].<ref>Brent, R.P. and Larvala, S. and Zimmerman, P. (2005). "[http://www.ams.org/journals/mcom/2005-74-250/S0025-5718-04-01673-4/S0025-5718-04-01673-4.pdf A primitive trinomial of degree 6972593]". ''Mathematics of Computation'' '''74''' (250) 1001-1002.</ref>
 
In 2009, Brent and Paul Zimmermann discovered some even larger primitive trinomials, for example:
:<math> x^{43112609} + x^{3569337} + 1. </math>
The degree 43112609 is again the exponent of a Mersenne prime.<ref>Brent, R.P. and Zimmerman, P. (2011). "[http://www.ams.org/notices/201102/rtx110200233p.pdf The great trinomial hunt]". ''Notices of the American Mathematical Society'' '''58''' 233-239.</ref>
 
In 2010, Brent and Paul Zimmermann published "Modern Computer Arithmetic", (Cambridge University Press, 2010), a book about algorithms for performing arithmetic, and their implementation on modern computers.
 
Brent is a Fellow of the [[Association for Computing Machinery]], the [[IEEE]], [[Society for Industrial and Applied Mathematics|SIAM]] and the [[Australian Academy of Science]]. In 2005, he was awarded the [[Hannan Medal]] by the [[Australian Academy of Science]].
 
==References==
<references/>
 
==External links==
* [http://wwwmaths.anu.edu.au/~brent/ Richard Brent's home page]
{{Use dmy dates|date=September 2011}}
 
{{Authority control|VIAF=223659213|LCCN=n/86/851707}}
{{Persondata <!-- Metadata: see [[Wikipedia:Persondata]]. -->
| NAME              = Brent, Richard
| ALTERNATIVE NAMES =
| SHORT DESCRIPTION = Australian academic
| DATE OF BIRTH    = 20 April 1946
| PLACE OF BIRTH    =
| DATE OF DEATH    =
| PLACE OF DEATH    =
}}
{{DEFAULTSORT:Brent, Richard}}
[[Category:1946 births]]
[[Category:Australian computer scientists]]
[[Category:Australian mathematicians]]
[[Category:Australian academics]]
[[Category:Australian National University faculty]]
[[Category:Complex systems scientists]]
[[Category:Fellows of the Association for Computing Machinery]]
[[Category:Living people]]
[[Category:People from the Australian Capital Territory]]
[[Category:Fellows of the Australian Academy of Science]]
[[Category:Fellows of Society for Industrial and Applied Mathematics]]
[[Category:University of Newcastle (Australia) faculty]]

Latest revision as of 22:02, 22 December 2014

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