Dynamical analysis of α-Euclidean algorithms

Bourdon, Jérémie; Daireaux, Benoît; Vallée, Brigitte

doi:10.1016/s0196-6774(02)00218-3

Cited by 17 publications

(16 citation statements)

References 9 publications

Supporting

Mentioning

Contrasting

Order By: Relevance

“…This mirror operation appears in [42] where Schweiger relates it to the natural extension, and in [7], where the authors use the geometric notion of "folded" and "unfolded." Clearly, the mirror map is an involution satisfying the morphism property (h • k) * = k * • h * .…”

Section: Uni Condition and Euclidean Dynamical Systemsmentioning

confidence: 99%

“…Proof. We refer to [4,9,48] except for claim (7) (which follows from λ ′ (1) = 0 and the implicit function theorem) and for claim ( 6): (6.a) Taking the derivatives at (1, 0) of H s,w [f s,w ] = λ(s, w)f s,w (with respect to s or w), integrating on I with respect to µ 1,0 (equal to the Lebesgue measure), and using that H * 1 preserves µ 1 , gives the expressions as integrals. To finish, apply Rohlin's formula.…”

Section: W)]mentioning

confidence: 99%

See 1 more Smart Citation

Euclidean algorithms are Gaussian

Baladi

Vallée

2005

Journal of Number Theory

Self Cite

307

View full text Add to dashboard Cite

We obtain a central limit theorem for a general class of additive parameters (costs, observables) associated to three standard Euclidean algorithms, with optimal speed of convergence. We also provide very precise asymptotic estimates and error terms for the mean and variance of such parameters. For costs that are lattice (including the number of steps), we go further and establish a local limit theorem, with optimal speed of convergence. We view an algorithm as a dynamical system restricted to rational inputs, and combine tools imported from dynamics, such as transfer operators, with various other techniques: Dirichlet series, Perron's formula, quasi-powers theorems, and the saddle-point method. Such dynamical analyses had previously been used to perform the average-case analysis of algorithms. For the present (dynamical) analysis in distribution, we require estimates on transfer operators when a parameter varies along vertical lines in the complex plane. To prove them, we adapt techniques introduced recently by Dolgopyat in the context of continuous-time dynamics (Ann. Math. 147 (1998) 357).

show abstract

Section: Uni Condition and Euclidean Dynamical Systemsmentioning

confidence: 99%

Section: W)]mentioning

confidence: 99%

Euclidean algorithms are Gaussian

Baladi

Vallée

2005

Journal of Number Theory

Self Cite

307

View full text Add to dashboard Cite

show abstract

“…We prove that an irrational number x is a Brjuno number if and only if B 0 (x) < ∞ and B 0 (−x) < ∞. Moreover the difference between B 1 and the even part of B 0 is bounded and numerical simulations suggest that it is 1 2 -Hölder-continuous.…”

Section: Introductionmentioning

confidence: 84%

“…It is known [1,4] that the maps A α admit a unique absolutely continuous invariant density dµ α (x) dx which is bounded from above and below by constants depending on α. (The density is known explicitly when α ∈ [γ , 1], see [9,8].)…”

Section: The (α U)-brjuno Functionsmentioning

confidence: 99%

Generalized Brjuno functions associated to α-continued fractions

Luzzi

Marmi

Nakada

et al. 2010

Journal of Approximation Theory

View full text Add to dashboard Cite

For 0 ≤ α ≤ 1 given, we consider the one-parameter family of α-continued fraction maps, which include the Gauss map (α = 1), the nearest integer (α = 1/2) and by-excess (α = 0) continued fraction maps. To each of these expansions and to each choice of a positive function u on the interval I α we associate a generalized Brjuno function B (α,u) (x). When α = 1/2 or α = 1, and u(x) = − log(x), these functions were introduced by Yoccoz in his work on linearization of holomorphic maps.We compare the functions obtained with different values of α and we prove that the set of (α, u)-Brjuno numbers does not depend on the choice of α provided that α = 0. We then consider the case α = 0, u(x) = − log(x) and we prove that x is a Brjuno number (for α = 0) if and only if both x and −x are Brjuno numbers for α = 0.

show abstract

“…Here we have to work with characteristic functions of some intervals, and we are led to work with a larger space, the space of functions with bounded variation on the unit interval I. This functional space was used previously in dynamical analysis [4], and the main properties of the transfer operator, when acting on this functional space, can be found there. For (s) > 1, the operator H s acts on BV (I) and the map s → H s is analytic.…”

Section: Spectral Properties Of the Transfer Operatormentioning

confidence: 99%

Dynamical Analysis of the Parametrized Lehmer–Euclid Algorithm

Daireaux¹,

Vallée²

2004

Combinator. Probab. Comp.

Self Cite

View full text Add to dashboard Cite

The Lehmer-Euclid Algorithm is an improvement of the Euclid Algorithm when applied to large integers. The original Lehmer-Euclid Algorithm replaces divisions on multi-precision integers by divisions on single-precision integers. Here we study a slightly different algorithm that replaces computations on n-bit integers by computations on µn-bit integers. This algorithm depends on the truncation degree µ ∈]0, 1[ and is denoted as the LE µ algorithm. The original Lehmer-Euclid Algorithm can be viewed as the limit of the LE µ algorithms for µ → 0. We provide here a precise analysis of the LE µ algorithm. For this purpose, we are led to study what we call the Interrupted Euclid Algorithm. This algorithm depends on some parameter α ∈ [0, 1] and is denoted by E α . When running with an input (a, b), it performs the same steps as the usual Euclid Algorithm, but it stops as soon as the current integer is smaller than a α , so that E 0 is the classical Euclid Algorithm. We obtain a very precise analysis of the algorithm E α , and describe the behaviour of main parameters (number of iterations, bit complexity) as a function of parameter α. Since the Lehmer-Euclid Algorithm LE µ when running on n-bit integers can be viewed as a sequence of executions of the Interrupted Euclid Algorithm E 1/2 on µn-bit integers, we then come back to the analysis of the LE µ algorithm and obtain our results. B. Daireaux and B. Valléedivision algorithm which is about twice as slow as Karatsuba multiplication -in most of the cases, since there is a small probability of failure.A significant improvement in the speed of the Euclid Algorithm when high-precision numbers are involved can be achieved with the so-called Lehmer-Euclid Algorithm, which uses a method due to Lehmer [18]. Working only with the leading digits of large integers, it is possible to simulate most of the multiple-precision divisions by single-precision divisions, which leads to a significant speed-up of the algorithm. The first version of this algorithm appeared in [18]; then some variants were described in Knuth [16], and finally Collins [7] and Jebelean [14] provided various improvements to the algorithm. Nowadays, most of computer algebra systems and multi-precision libraries use many of these variants. However, there exist very few analyses of the Lehmer-Euclid Algorithm. Sorenson [23] obtained a worst-case analysis of this algorithm, but, to the best of our knowledge, there does not exist any precise average-case analysis of the Lehmer-Euclid Algorithm. It is the purpose of this paper to provide such an analysis. Main resultsThe original Lehmer-Euclid Algorithm replaces divisions on multi-precision integers by divisions on single-precision integers (sometimes double precision is used). Here, we study a slightly different algorithm that replaces computations on n-bit integers by computations on µn-bit integers. This algorithm depends on the truncation degree µ ∈]0, 1[ and is denoted as the LE µ algorithm. This Lehmer-Euclid Algorithm can be viewed as a sequence of executions o...

show abstract

Dynamical analysis of α-Euclidean algorithms

Cited by 17 publications

References 9 publications

Euclidean algorithms are Gaussian

Euclidean algorithms are Gaussian

Generalized Brjuno functions associated to α-continued fractions

Dynamical Analysis of the Parametrized Lehmer–Euclid Algorithm

Contact Info

Product

Resources

About