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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 : 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.〔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.〕 == History == For the case ''n'' = 2, an extension of the Euclidean algorithm can determine whether there is an integer relation 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 Ferguson–Forcade algorithm was published in 1979 by Helaman Ferguson and R.W. Forcade. Although the paper treats general ''n'', it is not clear if the paper fully solves the problem because it lacks detailed steps, proofs, and a precision bound; crucial for a reliable implementation. *The first algorithm with complete proofs was the LLL algorithm, developed by Arjen Lenstra, Hendrik Lenstra and László Lovász in 1982. *The HJLS algorithm, developed by Johan Håstad, Bettina Just, Jeffrey Lagarias, and Claus-Peter Schnorr in 1986.〔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. Computing'', Vol. 18 (1989), p. 859–881〕 *The PSOS algorithm, developed by Ferguson in 1988. *The PSLQ algorithm, developed by Ferguson and Bailey in 1992 and substantially simplified by Ferguson, Bailey, and Arno in 1999.〔(''A Polynomial Time, Numerically Stable Integer Relation Algorithm'' ) by Helaman R. P. Ferguson and David H. Bailey; RNR Technical Report RNR-91-032; July 14, 1992〕 In 2000 the PSLQ algorithm was selected as one of the "Top Ten Algorithms of the Century" by Jack Dongarra and Francis Sullivan even though it is considered essentially equivalent to HJLS.〔Jingwei Chen, Damien Stehlé, Gilles Villard: ''(A New View on HJLS and PSLQ: Sums and Projections of Lattices.'' ), (ISSAC'13 )〕〔Helaman R. P. Ferguson, David H. Bailey and Steve Arno, ANALYSIS OF PSLQ, AN INTEGER RELATION FINDING ALGORITHM: ()〕 *The LLL algorithm has been improved by numerous authors. Modern LLL implementations can solve integer relation problems with ''n'' above 500. 抄文引用元・出典: フリー百科事典『 ウィキペディア（Wikipedia）』 ■ウィキペディアで「Integer relation algorithm」の詳細全文を読む スポンサード リンク
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