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Mathematics of Computation

ISSN 1088-6842(online) ISSN 0025-5718(print)



Parallel integer relation detection: Techniques and applications

Authors: David H. Bailey and David J. Broadhurst
Journal: Math. Comp. 70 (2001), 1719-1736
MSC (2000): Primary 11Y16; Secondary 11-04
Published electronically: July 3, 2000
MathSciNet review: 1836930
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Let $\{x_1, x_2, \cdots, x_n\}$ be a vector of real numbers. An integer relation algorithm is a computational scheme to find the $n$integers $a_k$, if they exist, such that $a_1 x_1 + a_2 x_2 + \cdots + a_n x_n= 0$. In the past few years, integer relation algorithms have been utilized to discover new results in mathematics and physics. Existing programs for this purpose require very large amounts of computer time, due in part to the requirement for multiprecision arithmetic, yet are poorly suited for parallel processing.

This paper presents a new integer relation algorithm designed for parallel computer systems, but as a bonus it also gives superior results on single processor systems. Single- and multi-level implementations of this algorithm are described, together with performance results on a parallel computer system. Several applications of these programs are discussed, including some new results in mathematical number theory, quantum field theory and chaos theory.

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Additional Information

David H. Bailey
Affiliation: Lawrence Berkeley Laboratory, MS 50B-2239, Berkeley, California 94720

David J. Broadhurst
Affiliation: Open University, Department of Physics, Milton Keynes MK7 6AA, United Kingdom

Received by editor(s): October 20, 1999
Published electronically: July 3, 2000
Additional Notes: The work of the first author was supported by the Director, Office of Computational and Technology Research, Division of Mathematical, Information, and Computational Sciences of the U.S. Department of Energy, under contract number DE-AC03-76SF00098.
Article copyright: © Copyright 2000 American Mathematical Society