Integer relation algorithm: Difference between revisions

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{{Short description|Mathematical procedure}}

An '''integer relation''' between a set of real numbers ''x''₁, ''x''₂, ..., ''x''_''n'' is a set of integers ''a''₁, ''a''₂, ..., ''a''_''n'', not all 0, such that

:<math>a_1x_1 + a_2x_2 + \cdots + a_nx_n = 0.\,</math>

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An '''integer relation algorithm''' is an [[algorithm]] for finding integer relations. Specifically, given a set of real numbers known to a given precision, an integer relation algorithm will either find an integer relation between them, or will determine that no integer relation exists with coefficients whose magnitudes are less than a certain [[upper bound]].<ref>Since the set of real numbers can only be specified up to a finite precision, an algorithm that did not place limits on the size of its coefficients would always find an integer relation for sufficiently large coefficients. Results of interest occur when the size of the coefficients in an integer relation is small compared to the precision with which the real numbers are specified.</ref>

== History ==

For the case ''n'' = 2, an extension of the [[Euclidean algorithm]] can determinefind whetherany thereinteger isrelation anthat integer relationexists between any two real numbers ''x''₁ and ''x''₂. The algorithm generates successive terms of the [[continued fraction]] expansion of ''x''₁/''x''₂; if there is an integer relation between the numbers, then their ratio is rational and the algorithm eventually terminates.

*The first generalFerguson–Forcade algorithm that was proved to work for all values of ''n'' was the '''Ferguson-Forcade algorithm''', published in 1979 by [[Helaman Ferguson]] and [[R.W. Forcade]].<ref>{{MathWorld|urlname=IntegerRelation|title=Integer Relation}}</ref> SubsequentAlthough developmentsthe paper treats general ''n'', focussingit onis improvingnot bothclear efficiencyif andthe numericalpaper stability,fully producedsolves the followingproblem algorithms:because it lacks the detailed steps, proofs, and a precision bound that are crucial for a reliable implementation.

*The first algorithm with complete proofs was the '''[[Lenstra–Lenstra–Lovász lattice basis reduction algorithm|'''LLL algorithm]]''']], developed by [[Arjen Lenstra]], [[Hendrik Lenstra]] and [[László Lovász]] in 1982.<ref>{{MathWorld|urlname=LLLAlgorithm|title=LLL Algorithm}}</ref>

*The '''HJLS algorithm''', developed by [[Johan Håstad]], Bettina Just, [[Jeffrey Lagarias]], and [[Claus P. Schnorr|Claus-Peter Schnorr]] in 1986.<ref>{{MathWorld|urlname=HJLSAlgorithm|title=HJLS Algorithm}}</ref><ref>Johan Håstad, Bettina Just, Jeffrey Lagarias, Claus-Peter Schnorr: ''Polynomial time algorithms for finding integer relations among real numbers.'' Preliminary version: STACS 1986 (''Symposium Theoret. Aspects Computer Science'') Lecture Notes Computer Science 210 (1986), p. 105–118. ''SIAM J. Comput.'', Vol. 18 (1989), pp. 859–881</ref>

*The [[Lenstra–Lenstra–Lovász lattice basis reduction algorithm|'''LLL algorithm''']], developed by [[Arjen Lenstra]], [[Hendrik Lenstra]] and [[László Lovász]] in 1982.<ref>{{MathWorld|urlname=LLLAlgorithm|title=LLL Algorithm}}</ref>

*The '''PSOS algorithm''', developed by Ferguson in 1988.<ref>{{MathWorld|urlname=PSOSAlgorithm|title=PSOS Algorithm}}</ref>

*The '''PSLQ algorithm''', developed by Ferguson and [[David H. Bailey (mathematician)|Bailey]] in 1992 and substantially simplified by Ferguson, Bailey, and Arno in 1999.<ref>Helaman R. P. Ferguson, David H. Bailey, and Steve Arno: "Analysis of PSLQ, an integer relation finding algorithm", Math. Comp., vol.68, no.225 (Jan. 1999), pp.351-369.</ref><ref>[https://www.davidhbailey.com/dhbpapers/pslq-comp-alg.pdf David H. Bailey and J.M. Borwein: "PSLQ: An Algorithm to Discover Integer Relations" (May 14, 2020)]</ref><ref>{{MathWorld|urlname=PSLQAlgorithm|title=PSLQ Algorithm}}</ref><ref>[http://crd.lbl.gov/~dhbailey/dhbpapers/pslq.pdf ''A Polynomial Time, Numerically Stable Integer Relation Algorithm''] {{Webarchive|url=https://web.archive.org/web/20070717073907/http://crd.lbl.gov/~dhbailey/dhbpapers/pslq.pdf |date=2007-07-17 }} by Helaman R. P. Ferguson and David H. Bailey; RNR Technical Report RNR-91-032; July 14, 1992</ref> In 2000 the PSLQ algorithm was selected as one of the "Top Ten Algorithms of the Century" by [[Jack Dongarra]] and Francis Sullivan<ref>{{cite journal |author-first=Barry Arthur |author-last=Cipra |author-link=Barry Arthur Cipra |url=http://www.uta.edu/faculty/rcli/TopTen/topten.pdf |title=The Best of the 20th Century: Editors Name Top 10 Algorithms |journal=SIAM News |volume=33 |issue=4 |access-date=2012-08-17 |archive-date=2021-04-24 |archive-url=https://web.archive.org/web/20210424004030/https://www.uta.edu/faculty/rcli/TopTen/topten.pdf |url-status=dead }}</ref> even though it is considered essentially equivalent to HJLS.<ref>Jingwei Chen, Damien Stehlé, Gilles Villard: [http://perso.ens-lyon.fr/damien.stehle/downloads/PSLQHJLS.pdf ''A New View on HJLS and PSLQ: Sums and Projections of Lattices.''], [http://www.issac-conference.org/2013/ ISSAC'13]</ref><ref>Helaman R. P. Ferguson, David H. Bailey and Steve Arno, ANALYSIS OF PSLQ, AN INTEGER RELATION FINDING ALGORITHM: [http://crd-legacy.lbl.gov/~dhbailey/dhbpapers/cpslq.pdf]</ref>

*The '''HJLS algorithm''', developed by Ferguson and [[David H. Bailey|David Bailey]] in 1992.<ref>{{MathWorld|urlname=HJLSAlgorithm|title=HJLS Algorithm}}</ref>

*The LLL algorithm has been improved by numerous authors. Modern LLL implementations can solve integer relation problems with ''n'' above 500.

*The '''PSLQ algorithm''', also developed by Ferguson and Bailey in 1992 and substantially simplified by Ferguson, Bailey, and Arno in 1999.<ref>{{MathWorld|urlname=PSLQAlgorithm|title=PSLQ Algorithm}}</ref><ref>[http://crd.lbl.gov/~dhbailey/dhbpapers/pslq.pdf ''A Polynomial Time, Numerically Stable Integer Relation Algorithm''] by [[Helaman Ferguson|Helaman R. P. Ferguson]] and [[David H. Bailey]]; RNR Technical Report RNR-91-032; July 14, 1992</ref>

In 2000 the PSLQ algorithm was selected as one of the "Top Ten Algorithms of the Century" by [[Jack Dongarra]] and Francis Sullivan.<ref>[http://amath.colorado.edu/resources/archive/topten.pdf ''The Best of the 20th Century: Editors Name Top 10 Algorithms''] by Barry A. Cipra; SIAM News, Volume 33, Number 4</ref>

==Applications==

Integer relation algorithms have two mainnumerous applications. The first application is to determine whether a given real number ''x'' is likely to be [[algebraic number|algebraic]], by searching for an integer relation between a set of powers of ''x'' {1, ''x'', ''x''², ..., ''x''^''n''}. The second application is to search for an integer relation between a real number ''x'' and a set of mathematical constants such as ''e'', π{{pi}} and ln(2), which will lead to an expression for ''x'' as a linear combination of these constants.

A typical approach in [[experimental mathematics]] is to use [[numerical method]]s and [[arbitrary precision arithmetic]] to find an approximate value for an [[series (mathematics)|infinite series]], [[infinite product]] or an [[integral]] to a high degree of precision (usually at least 100 significant figures), and then use an integer relation algorithm to search for an integer relation between this value and a set of mathematical constants. If an integer relation is found, this suggests a possible [[closed-form expression]] for the original series, product or integral. This conjecture can then be validated by formal algebraic methods. The higher the precision to which the inputs to the algorithm are known, the greater the level of confidence that any integer relation that is found is not just a [[Almost integer|numerical artifact]].

A notable success of this approach was the use of the PSLQ algorithm to find the integer relation that led to the [[Bailey–Borwein–Plouffe formula]] for the value of [[pi|{{pi}}]]. PSLQ has also helped find new identities involving [[multiple zeta function]]s and their appearance in [[quantum field theory]]; and in identifying bifurcation points of the [[logistic map]]. For example, where B₄ is the logistic map's fourth bifurcation point, the constant &alpha;&nbsp;=&nbsp;−''B''₄(''B''₄&nbsp;−&nbsp;2) is a root of a 120th-degree polynomial whose largest coefficient is 257³⁰.<ref>David H. Bailey and David J. Broadhurst, [http://crd.lbl.gov/~dhbailey/dhbpapers/ppslq.pdf "Parallel Integer Relation Detection: Techniques and Applications,"] {{Webarchive|url=https://web.archive.org/web/20110720013234/http://crd.lbl.gov/~dhbailey/dhbpapers/ppslq.pdf |date=2011-07-20 }} Mathematics of Computation, vol. 70, no. 236 (October 2000), pp. 1719–1736; LBNL-44481.</ref><ref>I. S. Kotsireas, and K. Karamanos, "Exact Computation of the bifurcation Point B4 of the logistic Map and the Bailey–Broadhurst Conjectures", I. J. Bifurcation and Chaos 14(7):2417–2423 (2004)</ref> Integer relation algorithms are combined with tables of high precision mathematical constants and heuristic search methods in applications such as the [[Inverse Symbolic Calculator]] or [[Plouffe's Inverter]].

A notable success of this approach was the use of the PSLQ algorithm to find the integer relation that led to the [[Bailey-Borwein-Plouffe formula]] for the value of [[pi|π]].

Integer relation finding can be used to [[Factorization of polynomials|factor polynomials]] of high degree.<ref>M. van Hoeij: ''Factoring polynomials and the knapsack problem.'' J. of Number Theory, 95, 167–189, (2002).</ref>

Integer relation algorithms are combined with tables of high precision mathematical constants and heuristic search methods in applications such as the [[Inverse Symbolic Calculator]] or [[Plouffe's Inverter]].

== References ==

{{reflist|colwidth=30em}}

== External links ==

* [https://web.archive.org/web/20080422084455/http://oldweb.cecm.sfu.ca/organics/papers/bailey/paper/html/paper.html ''Recognizing Numerical Constants''] by [[David H. Bailey (mathematician)|David H. Bailey]] and [[Simon Plouffe]]

* [http://crd.lbl.gov/~dhbailey/dhbpapers/tenproblems.pdf ''Ten Problems in Experimental Mathematics''] {{Webarchive|url=https://web.archive.org/web/20110610051846/http://crd.lbl.gov/~dhbailey/dhbpapers/tenproblems.pdf |date=2011-06-10 }} by [[David H. Bailey]], [[Jonathan Borwein|Jonathan M. Borwein]], Vishaal Kapoor, and [[Eric W. Weisstein]]

{{number theoretic algorithms}}