PSLQ: An Algorithm to Discover Integer Relations
Abstract
Let x = (x{sub 1}, x{sub 2} {hor_ellipsis}, x{sub n}) be a vector of real or complex numbers. x is said to possess an integer relation if there exist integers a{sub i}, not all zero, such that a{sub 1}x{sub 1} + a{sub 2}x{sub 2} + {hor_ellipsis} + a{sub n}x{sub n} = 0. By an integer relation algorithm, we mean a practical computational scheme that can recover the vector of integers ai, if it exists, or can produce bounds within which no integer relation exists. As we will see in the examples below, an integer relation algorithm can be used to recognize a computed constant in terms of a formula involving known constants, or to discover an underlying relation between quantities that can be computed to high precision. At the present time, the most effective algorithm for integer relation detection is the 'PSLQ' algorithm of mathematiciansculptor Helaman Ferguson [10, 4]. Some efficient 'multilevel' implementations of PSLQ, as well as a variant of PSLQ that is wellsuited for highly parallel computer systems, are given in [4]. PSLQ constructs a sequence of integervalued matrices B{sub n} that reduces the vector y = xB{sub n}, until either the relation is found (as one ofmore »
 Authors:
 Publication Date:
 Research Org.:
 Lawrence Berkeley National Lab. (LBNL), Berkeley, CA (United States)
 Sponsoring Org.:
 Computational Research Division
 OSTI Identifier:
 963658
 Report Number(s):
 LBNL2144E
TRN: US200918%%277
 DOE Contract Number:
 DEAC0205CH11231
 Resource Type:
 Technical Report
 Country of Publication:
 United States
 Language:
 English
 Subject:
 97; ACCURACY; ALGORITHMS; COMPUTERS; DETECTION; IMPLEMENTATION; MAPLES; MATRICES; VECTORS
Citation Formats
Bailey, David H., and Borwein, J. M. PSLQ: An Algorithm to Discover Integer Relations. United States: N. p., 2009.
Web. doi:10.2172/963658.
Bailey, David H., & Borwein, J. M. PSLQ: An Algorithm to Discover Integer Relations. United States. doi:10.2172/963658.
Bailey, David H., and Borwein, J. M. Fri .
"PSLQ: An Algorithm to Discover Integer Relations". United States.
doi:10.2172/963658. https://www.osti.gov/servlets/purl/963658.
@article{osti_963658,
title = {PSLQ: An Algorithm to Discover Integer Relations},
author = {Bailey, David H. and Borwein, J. M.},
abstractNote = {Let x = (x{sub 1}, x{sub 2} {hor_ellipsis}, x{sub n}) be a vector of real or complex numbers. x is said to possess an integer relation if there exist integers a{sub i}, not all zero, such that a{sub 1}x{sub 1} + a{sub 2}x{sub 2} + {hor_ellipsis} + a{sub n}x{sub n} = 0. By an integer relation algorithm, we mean a practical computational scheme that can recover the vector of integers ai, if it exists, or can produce bounds within which no integer relation exists. As we will see in the examples below, an integer relation algorithm can be used to recognize a computed constant in terms of a formula involving known constants, or to discover an underlying relation between quantities that can be computed to high precision. At the present time, the most effective algorithm for integer relation detection is the 'PSLQ' algorithm of mathematiciansculptor Helaman Ferguson [10, 4]. Some efficient 'multilevel' implementations of PSLQ, as well as a variant of PSLQ that is wellsuited for highly parallel computer systems, are given in [4]. PSLQ constructs a sequence of integervalued matrices B{sub n} that reduces the vector y = xB{sub n}, until either the relation is found (as one of the columns of B{sub n}), or else precision is exhausted. At the same time, PSLQ generates a steadily growing bound on the size of any possible relation. When a relation is found, the size of smallest entry of the vector y abruptly drops to roughly 'epsilon' (i.e. 10{sup p}, where p is the number of digits of precision). The size of this drop can be viewed as a 'confidence level' that the relation is real and not merely a numerical artifact  a drop of 20 or more orders of magnitude almost always indicates a real relation. Very high precision arithmetic must be used in PSLQ. If one wishes to recover a relation of length n, with coefficients of maximum size d digits, then the input vector x must be specified to at least nd digits, and one must employ nddigit floatingpoint arithmetic. Maple and Mathematica include multiple precision arithmetic facilities and Maple ships with a full implementation of PSLQ. One may also use any of several freeware multiprecision software packages, for example the ARPREC package by the first author and colleagues at LBNL [7]. In the remaining sections we describe various representative applications of PSLQ. More detail about these examples is given in [8] and the references therein.},
doi = {10.2172/963658},
journal = {},
number = ,
volume = ,
place = {United States},
year = {Fri Apr 03 00:00:00 EDT 2009},
month = {Fri Apr 03 00:00:00 EDT 2009}
}

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This Annual Report describes, in general, the progress during the year 1976 on a program under the direction of The Metal Properties Council, Inc. designed to screen materials for use in such plants with respect to the various unique corrosive environments obtaining therein; and, finally to provide useful corrosion data as well as reliable information on other properties needed for the design, construction, and operation of such plants. Corrosion tests were conducted in the gasifier gas composition at 1800/sup 0/F and 1000 psi; results are reported. Evaluation of metals and refractories continued in the HYGAS, Consol and Synthane pilot plants.more » 
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