research-article

Integer Relation Detection

Author:

David H. BaileyAuthors Info & Claims

Computing in Science and Engineering, Volume 2, Issue 1

Pages 24 - 28

https://doi.org/10.1109/5992.814653

Published: 01 January 2000 Publication History

Abstract

Practical algorithms for integer relation detection have become a staple in the emerging discipline of "experimental mathematics"-using modern computer technology to explore mathematical research. After briefly discussing the problem of integer relation detection, the author describes several recent, remarkable applications of these techniques in both mathematics and physics.

References

[1]

H.R.P. Ferguson and R.W. Forcade, "Generalization of the Euclidean Algorithm for Real Numbers to All Dimensions Higher Than Two," Bulletin Am. Mathematical Soc., Vol. 1, No. 6, Nov. 1979, pp. 912-914.

[2]

J. Hastad, et al., "Polynomial Time Algorithms for Finding Integer Relations Among Real Numbers," SIAM J. Computing, Vol. 18, No. 5, Oct. 1989, pp. 859-881.

Digital Library

[3]

H.R.P. Ferguson D.H. Bailey and S. Arno, "Analysis of PSLQ, An Integer Relation Finding Algorithm," Mathematics of Computation, Vol. 68, No. 225, Jan. 1999, pp. 351-369.

Digital Library

[4]

D.H. Bailey and D. Broadhurst, "Parallel Integer Relation Detection: Techniques and Applications," submitted for publication, 1999; also available at www.nersc.gov/~dhbailey, online paper 34 (current Nov. 1999).

[5]

D.H. Bailey, "Multiprecision Translation and Execution of Fortran Programs," ACM Trans. Mathematical Software, Vol. 19, No. 3, 1993, pp. 288-319.

Digital Library

[6]

D.H. Bailey, "A Fortran-90 Based Multiprecision System," ACM Trans. Mathematical Software, Vol. 21, No. 4, 1995, pp. 379-387.

Digital Library

[7]

S. Chatterjee and H. Harjono, "MPFUN++: A Multiple Precision Floating Point Computation Package in C++," University of North Carolina, Sept. 1998; www.cs.unc.edu/Research/HARPOON/mpfun++ (current Nov. 1999).

[8]

D.H. Bailey P.B. Borwein and S. Plouffe, "On The Rapid Computation of Various Polylogarithmic Constants," Mathematics of Computation, Vol. 66, No. 218, 1997, pp. 903-913.

Digital Library

[9]

D.J. Broadhurst, "Polylogarithmic Ladders, Hypergeometric Series and the Ten Millionth Digits of ς(3) and ς(5)," preprint, Mar. 1998; xxx.lanl.gov/abs/math/9803067 (current Nov. 1999).

[10]

D.J. Broadhurst, "Massive 3-loop Feynman Diagrams Reducible to SC* Primitives of Algebras of the Sixth Root of Unity," preprint, March 1998, to appear in European Physical J. Computing; xxx.lanl.gov/abs/hep-th/9803091 (current Nov. 1999).

[11]

D.H. Bailey J.M. Borwein and R. Girgensohn, "Experimental Evaluation of Euler Sums," Experimental Mathematics, Vol. 4, No. 1, 1994, pp. 17-30.

[12]

D. Borwein and J.M. Borwein, "On an Intriguing Integral and Some Series Related to ς(4)," Proc. American Mathematical Soc., Vol. 123, 1995, pp. 111-118.

[13]

D. Borwein J.M. Borwein and R. Girgensohn, "Explicit Evaluation of Euler Sums," Proc. Edinburgh Mathematical Soc., Vol. 38, 1995, pp. 277-294.

[14]

J.M. Borwein D.M. Bradley and D.J. Broadhurst, "Evaluations of k-fold Euler/Zagier Sums: A Compendium of Results for Arbitrary k," Electronic J. Combinatorics, Vol. 4, No. 2, #R5, 1997.

[15]

J.M. Borwein D.J. Broadhurst and J. Kamnitzer, "Central Binomial Sums and Multiple Clausen Values (with Connections to Zeta Values)," No. 137, 1999; www.cecm.sfu.ca/preprints/1999pp.html (current Nov. 1999).

[16]

D.J. Broadhurst J.A. Gracey and D. Kreimer, "Beyond the Triangle and Uniqueness Relations: Non-Zeta Counterterms at Large N from Positive Knots," Zeitschrift für Physik, Vol. C75, 1997, pp. 559-574.

[17]

D.J. Broadhurst and D. Kreimer, "Association of Multiple Zeta Values with Positive Knots via Feynman Diagrams up to 9 Loops," Physics Letters, Vol. B383, 1997, pp. 403-412.

[18]

J.M. Borwein, et al., "Combinatorial Aspects of Multiple Zeta Values," Electronic J. Combinatorics, Vol. 5, No. 1, #R38, 1998.

Cited By

Gamarnik DKizildağ E(2019)High-Dimensional Linear Regression and Phase Retrieval via PSLQ Integer Relation Algorithm2019 IEEE International Symposium on Information Theory (ISIT)10.1109/ISIT.2019.8849681(1437-1441)Online publication date: 7-Jul-2019
https://dl.acm.org/doi/10.1109/ISIT.2019.8849681
Lu MHe BLuo Q(2010)Supporting extended precision on graphics processorsProceedings of the Sixth International Workshop on Data Management on New Hardware10.1145/1869389.1869392(19-26)Online publication date: 7-Jun-2010
https://dl.acm.org/doi/10.1145/1869389.1869392
Bailey D(2005)High-Precision Floating-Point Arithmetic in Scientific ComputationComputing in Science and Engineering10.1109/MCSE.2005.527:3(54-61)Online publication date: 1-May-2005
https://dl.acm.org/doi/10.1109/MCSE.2005.52
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cover image Computing in Science and Engineering

Computing in Science and Engineering Volume 2, Issue 1

January 2000

90 pages

ISSN:1521-9615

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Copyright © 2000.

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IEEE Educational Activities Department

United States

Publication History

Published: 01 January 2000

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Cited By

Gamarnik DKizildağ E(2019)High-Dimensional Linear Regression and Phase Retrieval via PSLQ Integer Relation Algorithm2019 IEEE International Symposium on Information Theory (ISIT)10.1109/ISIT.2019.8849681(1437-1441)Online publication date: 7-Jul-2019
https://dl.acm.org/doi/10.1109/ISIT.2019.8849681
Lu MHe BLuo Q(2010)Supporting extended precision on graphics processorsProceedings of the Sixth International Workshop on Data Management on New Hardware10.1145/1869389.1869392(19-26)Online publication date: 7-Jun-2010
https://dl.acm.org/doi/10.1145/1869389.1869392
Bailey D(2005)High-Precision Floating-Point Arithmetic in Scientific ComputationComputing in Science and Engineering10.1109/MCSE.2005.527:3(54-61)Online publication date: 1-May-2005
https://dl.acm.org/doi/10.1109/MCSE.2005.52
Bailey DBroadhurst DHida YLi XThompson BGiles RReed DKelley K(2002)High performance computing meets experimental mathematicsProceedings of the 2002 ACM/IEEE conference on Supercomputing10.5555/762761.762768(1-12)Online publication date: 16-Nov-2002
https://dl.acm.org/doi/10.5555/762761.762768

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