Analysis of PSLQ, an integer relation finding algorithm

  • 4.84 MB
  • 4256 Downloads
  • English

National Aeronautics and Space Administration, National Technical Information Service, distributor , [Washington, D.C, Springfield, Va
Algorithms., Inte
Other titlesInteger relation finding algorithm.
StatementHelaman R.P. Ferguson, David H. Bailey, and Steve Arno.
SeriesNASA technical memorandum -- 112040.
ContributionsBailey, David H., Arno, Steve., United States. National Aeronautics and Space Administration.
The Physical Object
FormatMicroform
Pagination1 v.
ID Numbers
Open LibraryOL15498417M

That PSLQ(τ) is ‘polynomial time’ in the dimension and the number of bits of a smallest integer relation. Different τ or γ choices lead to different time and space requirements for the algorithm.

For dimension n = 2 we prove that PSLQ(τ) will construct a relation of smallest norm M x. We give examples in dimension n = 3, for some τ,for.

Download Analysis of PSLQ, an integer relation finding algorithm PDF

CiteSeerX — Analysis Of PSLQ, An Integer Relation Finding Algorithm. CiteSeerX - Document Details (Isaac Councill, Lee Giles, Pradeep Teregowda): Let K be either the real, complex, or quaternion number system and let O(K) be the corresponding integers.

Let x = (x 1 ; ; xn) be a vector in K n. The vector x has an integer relation if there exists a vector m = (m 1 ; ; mn) 2 O(K) n, m 6= 0. In this paper we define the parameterized integer relation construction algorithm PSLQ(τ), where the parameter τ can be freely chosen in a certain interval. Beginning with an arbitrary vector x =(x1,xn) ∈ K n, iterations of PSLQ(τ) will produce lower bounds on the norm of any possible relation for x.

In this paper we define the parameterized integer relation construction algorithm PSLQ(r), where the parameter rcan be freely chosen in a certain interval.

Beginning with an arbitrary vector X = (Xl, Xn) _ K n, iterations of PSLQ(r) will produce lower bounds on the norm of any possible relation for X.

PSLQ: An Algorithm to Discover Integer Relations David H. Bailey and J.M. Borweiny 1. Introduction. Let x= (x 1;x 2; ;x n) be a vector of real or complex numbers.

x is said to possess an integer relation if there exist integers a i, not all zero, such that a 1x 1 + a 2x 2 + + a nx n = 0. The pslq algorithm computes integer relations for real numbers and Gaussian integer relations for complex numbers.

We endeavour to extend pslq to find integer relations consisting of algebraic integers from some quadratic extension fields (in both the real and complex cases). Find. Previous. Next. Page: Presentation Mode Open Print Current View.

Tools. Zoom Out. Zoom In. More Information Less Information Close Enter the password to open this PDF file: Cancel OK. File name: File size: Title: Author:. PSLQ, an efficient, numerically stable algorithm guaranteed to find relations among integers in a limited number of iterations, is a shining example of modern computer technology applied to the discovery of new facts of mathematics and physics -- truly an algorithm of the century.

Thus PSLQ(τ) can be used to prove that there are no relations for x of norm less than a given size. Let Mx be the smallest norm of Analysis of PSLQ relation for x.

For the real and complex case and each fixed parameter τ in a certain interval, we prove that PSLQ(τ) constructs a relation. The PSLQ Integer Relation Algorithm In a new algorithm, known as ``PSLQ'' algorithm, was developed by Ferguson [].It appears to combine some of the best features separately possessed by previous algorithms, including fast run times, numerical stability, numerical efficiency (i.e.

successfully recovering a relation when the input is known to only limited precision), and a guaranteed. As we shall see, integer relation algorithms have a variety of interesting applications, including the recognition of a numeric constant in terms of the mathematical formula that it satisfies At the present time, the most effective scheme for integer relation detection is Ferguson’s “PSLQ” : Manju Devi.

Get this from a library. Analysis of PSLQ, an integer relation finding algorithm. [Helaman Ferguson; David H Bailey; Steve Arno; United States.

National Aeronautics and Space Administration.]. The PSLQ algorithm () can be used to find integer relations between real numbers. A sample is included that demonstrates PSLQ finding the polynomial that has a specific surd as one of it's roots. Analysis of PSLQ, An Integer Relation Finding Algorithm number system and let O(K) be the corresponding integers.

Let × = (Xl, • • •, ×n) be a vector in K n. The vector × has an integer relation if there exists a vector m = (ml, mn) E O(K) n, m = _ O, such that mlx I + m2x 2 + + mnXn = O.

In this paper we define the. 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.

At the present time the “PSLQ” algorithm of mathematician-sculptor Helaman Ferguson is the most widely used integer relation algorithm. It was named one of ten “algorithms of the century” by Computing in Science and Engineering. Ferguson, DHB and S. Arno, “Analysis of PSLQ, an integer relation finding algorithm,”File Size: 1MB.

Finding Minimal Polynomials Up: Integer Relation Algorithms Previous: Rational Inputs The PSLQ Algorithm The first integer relation algorithm working reliably on vectors of non-trivial length was discovered by Ferguson and Forcade [].After a number of improvements, in a new algorithm, known as the PSLQ algorithm, was developed by Ferguson [].

Wolfram Language Revolutionary knowledge-based programming language.

Description Analysis of PSLQ, an integer relation finding algorithm PDF

Wolfram Cloud Central infrastructure for Wolfram's cloud products & services. Wolfram Science Technology-enabling science of the computational universe. So if the integer relation algorithm (where the output integers are indeed limitedit would be trivial to generate an arbitrarily large spurious solution) does return a relation, either there's a solution or a larger/no solution that, at least for the PSLQ algorithm, does come with a bound for other possible solutions.

$\endgroup. Nearly three decades ago the first integer relations algorithm was developed. Given a set of numbers {x1,xm}, an integer relations algorithm seeks integers {a1,am} such that a1x1++amxm = 0.

One of the most popular and efficient of these is the PSLQ algorithm, listed as one of the top ten algorithms of the 20th century[2]. At the present time, the most effective algorithm for integer relation detection is the 'PSLQ' algorithm of mathematician-sculptor Helaman Ferguson [10, 4].

Some efficient 'multi-level' implementations of PSLQ, as well as a variant of PSLQ that is well-suited for highly parallel computer systems, are given in.

PSLQ is an algorithm for finding integer relations. Namely, given n real numbers x = (x1, x2, x n) PSLQ tries to find integers m=(m1,m2,m n), not all zero, such that x m= m1x1 + + m nx n =0.

The vector m is called an integer relation for x. In case that no relation is found, PSLQ provides a lower bound for the norm of any potential integer relation.

In this paper we define the parameterized integer relation construction algorithm PSLQ(tau), where the parameter tau can be freely chosen in a certain interval Topics: Computer Programming and Software.

The search for integer relations and the PSLQ algorithm An important task in experimental mathematics is to search for inte- ger relations involving a finite set of computed numbers. An integer re- lation algorithm is a computational scheme that, for a given real vector X = {xi,X2.

It finds a relatively short vector in an integer lattice. In this chapter we give some examples of how LLL can be used to approach some of the central problems of the book.

Appendix B deals, in detail, with the LLL algorithm and the closely related PSLQ algorithm for finding integer : Peter Borwein. PSLQ Algorithm. GitHub Gist: instantly share code, notes, and snippets. Computational Excursions in Analysis and Number Theory, Appendix B * Java implementation of the PSLQ integer relation algorithm.

This is still a work in progress. Should be used only as an example of using MPFloat. MATHEMATICS OF COMPUTATION Vol NumberJanuaryPages { S (99) ANALYSIS OF PSLQ, AN INTEGER RELATION FINDING ALGORITHM HELAMAN R.

FERGUSON, DAVID H. BAILEY, AND STEVE ARNO Abstract.

Details Analysis of PSLQ, an integer relation finding algorithm PDF

Let K be either the real, complex, or quaternion number system and let O(K) be the corresponding integers. PSLQ: An Algorithm to Discover Integer Relations.

Author(s): Bailey, David H. et al. Main Content Metrics Author & Article Info. Main Content. Download PDF to View View Larger. Thumbnails Document Outline Attachments. Find.

Previous. Next. Presentation Mode Open Print Download Current View. Tools. Zoom by: 8. The rst integer relation algorithm with the required properties mentioned above was discovered in by Fergu-son and Forcade [19]. Since then, a number of other integer relation algorithms have been discovered, including the \HJLS" algorithm [21] (which is based on the LLL algorithm), and the \PSLQ" algorithm.

The PSLQ Algorithm. The celebrated integer relation finding algorithm PSLQ has been successfully used in many applications. However, the PSLQ was only analyzed theoretically for exact input. When the input data are irrational numbers, they must be approximate ones due to finite precision in : Yong Feng, Jingwei Chen and Wenyuan Wu.

PSLQ LLL and Integer Relations algorithms There are new algorithms that are widely used since the late 80’s but fast enough in the early ’s.

They are called LLL algorithms, PSLQ algorithms or more generally speaking: Integer Relations algorithms.

We focus on this version of the algorithm: Given a vector of real numbers.One application of PSLQ is to find the minimal polynomial of an algebraic number given a decimal approximation of the algebraic number.

The examples below illustrate this. Generally speaking, if the height of the minimal polynomial is m and its degree is n, then you need more than n ⁢ log 10 m correct decimal digits for the algebraic number.Chongqing Institute of Green and Intelligent Technology, CAS.

Chongqing Institute of Green and Intelligent Technology, CAS. View Profile.