The number of steps in the Euclidean algorithm
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It is shown that given any ϵ > 0 there exists c0 > 0 such that | L(u, v) − (12π −2 log 2) log v | log v) 1 2 +ϵ for all except at most x 2 exp {−c 0 ( log x) ϵ 2 } of the pairs u, v with 1 ≤ u ≤ v ≤ x.About:
This article is published in Journal of Number Theory.The article was published on 1970-11-01 and is currently open access. It has received 92 citations till now. The article focuses on the topics: Bound graph & Euclidean algorithm.read more
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Book
Analytic Combinatorics
TL;DR: This text can be used as the basis for an advanced undergraduate or a graduate course on the subject, or for self-study, and is certain to become the definitive reference on the topic.
Journal ArticleDOI
Euclidean algorithms are Gaussian
Viviane Baladi,Brigitte Vallée +1 more
TL;DR: A central limit theorem is obtained for a general class of additive parameters (costs, observables) associated to three standard Euclidean algorithms, with optimal speed of convergence, and very precise asymptotic estimates and error terms for the mean and variance of such parameters are provided.
The euclidean algorithm in algebraic number fields
TL;DR: The authors survey what is known about Euclidean number fields from a number theoretical (and number geometrical) point of view and put some emphasis on the open problems in this field.
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An upper bound on the average number of iterations of the LLL algorithm
Hervé Daudé,Brigitte Vallée +1 more
TL;DR: An upper bound is established regarding the average number of iterations of the lattice reduction algorithm of Lenstra, Lenstra and Lovasz (the LLL algorithm), which is essentially independent of the length of the input vectors, so that, in any fixed dimension, the L LL algorithm turns out to be of complexity O(1) on average.
Journal ArticleDOI
An Average-Case Analysis of the Gaussian Algorithm for Lattice Reduction
TL;DR: The Gaussian algorithm for lattice reduction in dimension 2 is analysed and it is found that, when applied to random inputs in a continuous model, the complexity is constant on average, its probability distribution decays geometrically, and the dynamics are characterized by a conditional invariant measure.
References
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Book ChapterDOI
On the Average Length of a Class of Finite Continued Fractions
TL;DR: In this article, a finite continued fraction (FFLF) representation of a natural integer with respect to a given natural integer N and a non-natural integer N can be found.