## Euclidean dynamics

Venue: | Discrete and Continuous Dynamical Systems |

Citations: | 2 - 1 self |

### BibTeX

@ARTICLE{Vallée_euclideandynamics,

author = {Brigitte Vallée},

title = {Euclidean dynamics},

journal = {Discrete and Continuous Dynamical Systems},

year = {},

volume = {15},

pages = {2006}

}

### OpenURL

### Abstract

Abstract. We study a general class of Euclidean algorithms which compute the greatest common divisor [gcd], and we perform probabilistic analyses of their main parameters. We view an algorithm as a dynamical system restricted to rational inputs, and combine tools imported from dynamics, such as transfer operators, with various tools of analytic combinatorics: generating functions, Dirichlet series, Tauberian theorems, Perron’s formula and quasi-powers theorems. Such dynamical analyses can be used to perform the average-case analysis of algorithms, but also (dynamical) analysis in distribution. 1. Introduction. Computing the Greatest Common Divisor [Gcd

### Citations

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Citation Context ...s Theorem] are fulfilled for all the variables Rn whose moment generating function is “closely” related to the n-th power G n w, applied to some function F of F. Proof. Elementary perturbation theory =-=[53]-=- implies that Gw inherits the dominant eigenvalue property and the spectral gap when w is near 0. This proves that the n–th iterate G n w of the operator behaves as a uniform quasi–power where the mai... |

760 | Factoring polynomials with rational coefficients
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Citation Context ...solves the problem exactly using what resembles a lifting of the Euclidean algorithm to 2–dimensional lattices. In recent times, an important discovery was made by Lenstra, Lenstra and Lovász in 1982 =-=[64]-=-; their algorithm, called the LLL algorithm, is able to find reduced bases in all dimensions d ≥ 3. The LLL algorithm itself proceeds by stages based on the Gaussian algorithm as the main reduction st... |

293 | An Introduction To The Analysis of Algorithms
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Citation Context ...e case in analysis of algorithms, generating functions. The crucial rôle of generating functions in the analysis of data structures and algorithms is well described in books by Flajolet and Sedgewick =-=[33, 34]-=-. i=1sEUCLIDEAN DYNAMICS 289 We consider a general parameter R defined on Ω, � Ω, and we wish to study its distribution on ΩN, when endowed with the uniform probability. Our final probabilistic tool [... |

256 |
Thermodynamic Formalism
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Citation Context ...e density transformer is a special case of a transfer operator, and the general notion of transfer operators was introduced by Ruelle, in connection with his thermodynamic formalism (see for instance =-=[85, 86]-=-). Then Mayer has applied such operators to the classic continued fraction transformation. After works of Chernov [21], Dolgopyat [29] was interested in the decay of correlations for hyperbolic flows ... |

175 |
Produits tensoriels topologiques et espaces nucléaires
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(Show Context)
Citation Context ... ∈ M}, A∞(M) is a Banach space. Under previous conditions, the transfer operator operator Hs acts on A∞(M) for ℜ(s) > σ and is compact. It is moreover nuclear of order 0 (in the sense of Grothendieck =-=[40]-=-, [41]). Property (i) is essential here, and it is necessary that the closure M of disk M is mapped inside M. Mayer [72, 73, 71] deeply studied the transfer operator Hs relative to the Classical Eucli... |

154 |
operators and classical function theory
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Citation Context ...t a functional space where the density transformer acts and is compact or quasi–compact. 8.1. Compacity and analytic spaces. There exist results due to Schwartz [87], Shapiro and Taylor [92], Shapiro =-=[91]-=- which exhibit sufficient conditions under which transfer operators are compact on a functional space of analytic functions defined on some disk M. The operator Hs is the sum of the component operator... |

152 |
Fundamental Problems of Algorithmic Algebra
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(Show Context)
Citation Context ...puting the Greatest Common Divisor [Gcd] –on integers or polynomials– is a central problem in computer algebra, and all the gcd algorithms are based on main principles due to Euclid. See for instance =-=[113]-=- or [110] for a description of use of Gcd Algorithms in Computer Algebra. According to Knuth [56], “we might call Euclid’s method the granddaddy of all algorithms, because it is the oldest nontrivial ... |

116 |
Products of random matrices with applications to Schrödinger operators
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Citation Context ...p2(n) pd(n) ��� , , . . . , = 0, q(n) q(n) lim n→∞ ||q(n)(x1, x2, . . . , xd) − (p1(n), p2(n), . . . , pd(n))|| = 0. These convergence properties are closely related with the Lyapounov exponents (see =-=[12]-=-) of the set of matrices used by the algorithm. In particular, if the second Lyapounov exponent is strictly negative, then the algorithm is strongly convergent. As in the lattice reduction problem, th... |

98 |
Positive transfer operators and decay of correlations
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(Show Context)
Citation Context ...mall” so that there is a spectral gap. Property UDE is true under quite general “positivity” hypotheses (for instance, theorems due to Krasnoselski [58], or cone methods as described in Baladi’s book =-=[4]-=-), and very often true for systems with complete branches [as soon as they are topologically mixing]. In contrast, Property SG is both central and not so easy to obtain in a quite general framework. [... |

72 | On convergence rates in the central limit theorems for combinatorial structures
- Hwang
- 1998
(Show Context)
Citation Context ..., and for each parameter Rn = Cn or Rn = ℓn, a quasi-powers approximation (with uniform remainder term) will be be obtained for E[exp(wRn)]. In this case, the following Quasi-Powers Theorems of Hwang =-=[47, 48, 49]-=- will be used and this will lead to the asymptotic gaussian law. This theorem is a compact and versatile statement, which encapsulates the consequences of the Lévy continuity theorem and the Berry-Ess... |

71 |
On decay of correlations in Anosov flows
- Dolgopyat
- 1998
(Show Context)
Citation Context ... 7.4. UNI Conditions imply US Property. First, we have previously seen that a system which satisfies Condition UNI is not C 2 conjugated to a dynamical system with affine branches. In fact, Dolgopyat =-=[29]-=- proved that the UNI Condition is sufficient to also imply that the quasi-inverse of the transfer operator satisfies US(s) –at least, in the case of one–variable transfer operators Hs which are relate... |

66 |
Schnelle Berechnung von Kettenbruchentwicklungen. (Fast computation of continued fraction expansions
- Schönhage
- 1971
(Show Context)
Citation Context ...e for instance Gouezel’s work [39] for occurrences of these laws in continued fraction related matters.s344 BRIGITTE VALL ÉE Figure 22. Japanese dynamical systems (iv) There exist fast gcd algorithms =-=[93]-=- which are based on a Divide and Conquer method, and use the principles of the Lehmer method [61]. For a readable treatment of such algorithms, see the book of Yap [113], for instance. Dynamical Analy... |

55 |
Dynamical zeta functions for piecewise monotone maps of the interval
- Ruelle
- 1994
(Show Context)
Citation Context ...e density transformer is a special case of a transfer operator, and the general notion of transfer operators was introduced by Ruelle, in connection with his thermodynamic formalism (see for instance =-=[85, 86]-=-). Then Mayer has applied such operators to the classic continued fraction transformation. After works of Chernov [21], Dolgopyat [29] was interested in the decay of correlations for hyperbolic flows ... |

48 |
Théorèmes limites pour les structures combinatoires et les fonctions arithmétiques, Thèse, Ecole Polytechnique
- Hwang
- 1994
(Show Context)
Citation Context ..., and for each parameter Rn = Cn or Rn = ℓn, a quasi-powers approximation (with uniform remainder term) will be be obtained for E[exp(wRn)]. In this case, the following Quasi-Powers Theorems of Hwang =-=[47, 48, 49]-=- will be used and this will lead to the asymptotic gaussian law. This theorem is a compact and versatile statement, which encapsulates the consequences of the Lévy continuity theorem and the Berry-Ess... |

44 | An average–case analysis of the Gaussian algorithm for lattice reduction
- Flajolet, Vallée
- 1997
(Show Context)
Citation Context ...lgorithm (K). 9.7. Generalizations of the Euclid Algorithm in higher dimensions. There are many various points of view for a generalization of the Euclidean algorithms. The lattice reduction problem. =-=[26]-=-[109]. The lattice reduction problem consists in finding a short basis of a lattice of Euclidean space given an initially skew basis. This reduction problem is well–known to be central to many areas o... |

43 |
Multi-Dimensional Continued Fraction Algorithms
- Brentjes
- 1981
(Show Context)
Citation Context ... n + O(n), n → ∞, α where α is the entropy of the CFE dynamical system and the constant which appears in the term O(n) is a variant of the Porter constant [see Section 7. 10]. Other generalizations. (=-=[95, 16, 10, 18]-=-). The Jacobi–Perron algorithm, or its variants, were introduced to generalize the continued fraction algorithm, from the point of view of the construction of a rational approximation of a vector of R... |

42 |
La théorie de Fredholm
- Grothendieck
- 1956
(Show Context)
Citation Context ... A∞(M) is a Banach space. Under previous conditions, the transfer operator operator Hs acts on A∞(M) for ℜ(s) > σ and is compact. It is moreover nuclear of order 0 (in the sense of Grothendieck [40], =-=[41]-=-). Property (i) is essential here, and it is necessary that the closure M of disk M is mapped inside M. Mayer [72, 73, 71] deeply studied the transfer operator Hs relative to the Classical Euclidean D... |

38 |
Some limit theorems for stationary Markov chains, Theory Prob
- Nagaev
- 1957
(Show Context)
Citation Context ...eorem for costs [stated here as Theorem 1] is quite well-known. See for instance [22] or [17] for interval maps, and [1] for a more abstract framework and references to the pioneering paper of Nagaev =-=[78]-=-. The situation is less clear for continuants: There are previous works due to Philipp [80] which have been generalized by Vallée [101]. These results are extended to our general framework for the fir... |

36 |
Continued fractions and related transformations
- Mayer
(Show Context)
Citation Context ...and is compact. It is moreover nuclear of order 0 (in the sense of Grothendieck [40], [41]). Property (i) is essential here, and it is necessary that the closure M of disk M is mapped inside M. Mayer =-=[72, 73, 71]-=- deeply studied the transfer operator Hs relative to the Classical Euclidean Dynamical System [often called the Ruelle-Mayer operator] and proved that Hs satisfies properties (i), (ii), (iii) for some... |

35 |
Local limit theorems for partial sums of stationary sequences generated by Gibbs–Markov maps. Stochastic Dynamics 1(2):193–237
- Aaronson, Denker
- 2001
(Show Context)
Citation Context ...uotients, the triples di = (mi, ǫi, ai, bi) are the digits. The continuants are defined when one “splits”s0 ≤ r < u 2 EUCLIDEAN DYNAMICS 285 Alg., X. η Division Set of LFT’s Conditions on J or F. (G) =-=[0, 1]-=-, 0 v = mu + r 0 ≤ r < u 1 G = { m + x , m ≥ 1} F = G T {m ≥ 2} (M) [0, 1], 1 v = mu − r 0 ≤ r < u 1 M = { m − x , m ≥ 2} F = M T {m ≥ 3} v = mu + ǫr ǫ = ±1, (K) [0, 1/2], 0 (m, ǫ) ≥ (2,+1) 1 K = { , ... |

34 |
Transformations dilatantes de l’intervalle et théorèmes limites
- Broise
- 1996
(Show Context)
Citation Context ...extensions of the MSBgcd’s lead to dynamical systems where X is a real compact interval endowed with the usual absolute value [see Figure 10]. For surveys on dynamical systems on a real interval, see =-=[17]-=-, [22]. For Type 1, there are many different possible generalizations for the integer part function which can be defined from the binary expansion of a real number. Such possible generalizations are d... |

33 | Large deviations of combinatorial distributions. II. Local limit theorems
- Hwang
- 1998
(Show Context)
Citation Context ..., and for each parameter Rn = Cn or Rn = ℓn, a quasi-powers approximation (with uniform remainder term) will be be obtained for E[exp(wRn)]. In this case, the following Quasi-Powers Theorems of Hwang =-=[47, 48, 49]-=- will be used and this will lead to the asymptotic gaussian law. This theorem is a compact and versatile statement, which encapsulates the consequences of the Lévy continuity theorem and the Berry-Ess... |

32 | Central limit theorem and stable laws for intermittent maps
- Gouëzel
(Show Context)
Citation Context ...of Section 7.4 to underlined operators H s,w . (iii) On another register, the extension to “large” costs or Bad Class is likely to lead us to the realm of stable laws: see for instance Gouezel’s work =-=[39]-=- for occurrences of these laws in continued fraction related matters.s344 BRIGITTE VALL ÉE Figure 22. Japanese dynamical systems (iv) There exist fast gcd algorithms [93] which are based on a Divide a... |

32 |
Euclid’s algorithm for large numbers
- Lehmer
- 1938
(Show Context)
Citation Context ...s of this algorithm. Lehmer-Euclid algorithm, Interrupted Algorithm. The Lehmer-Euclid algorithm is an improvement of the Euclid algorithm when applied for large integers. It was introduced by Lehmer =-=[62]-=- and first analyzed in the worst–case by Sorenson [97]. It uses what Daireaux et Vallée have called the Interrupted Euclidean algorithm [24]. This interrupted algorithm depends on some parameter α ∈ [... |

32 |
Metrical theory for a class of continued fraction transformations and their natural extensions
- Nakada
- 1981
(Show Context)
Citation Context ... 8.2) can be applied, and the transfer operator Hs is quasi-compact on BV . When parameter α belongs to [1/2, 1], this dynamical system Sα has been first extensively studied by Ito, Tanaka and Nakada =-=[51, 79]-=-. This is why the α–Euclidean algorithms are often nicknamed as “Japanese algorithms”. Later, Moussa, Cassa, Marmi [77] provided an extension of these results to the range α ∈ [ √ 2 − 1, 1/2]. Paper [... |

31 | Analysis of the binary Euclidean algorithm
- Brent
- 1976
(Show Context)
Citation Context ...n above [See Figure 17]. The density transformer H of the Binary Euclidean System, defined as H[f](x) := � 1 ( 1 + 2bx )2 1 f( 1 + 2b � 1 ) + ( x x + 2b)2 x f( ), (3.4) x + 2b was introduced by Brent =-=[14]-=- in his study of the binary gcd, but it is not easy to deal with. This is why the “induced version” of the transfer operator has been further introduced by Vallée [105]. Between two exchanges, there i... |

31 |
Maps of the interval with indifferent fixed points: thermodynamic formalism and phase transitions
- Prellberg
- 1991
(Show Context)
Citation Context ... (P1, P2, P3) of Figure 2, and the induced system is good. Consider the operators Qs,Ps relative to the sets {p}, Q, Qs := � h∈Q H s,[h], Ps := H s,[p] which satisfy Ps + Qs = Hs, and consider, as in =-=[82]-=-, the transfer operator � Hs of this induced dynamical system. It involves the operators Qs,Ps under the form �Hs = � k≥0 Qs ◦P k s = Qs ◦(I −Ps) −1 , so that � Hs ◦(I − Ps) = Qs. (2.17) Since the ope... |

30 |
Computational Problems Associated with Racah Algebra
- Stein
- 1967
(Show Context)
Citation Context ... numbers, and they are well–adapted to computing the Jacobi symbol [52] [61], for instance [the Quadratic Reciprocity law being only true for a pair of odd integers]. For the Binary division of Stein =-=[96]-=- described in [56] and the Plus-Minus division, of Brent and Kung [15], the main decision is made by the LSB’s; the MSB’s playsEUCLIDEAN DYNAMICS 283 only an auxilliary rôle, and only decide when the ... |

30 | Dynamical sources in information theory: fundamental intervals and word prefixes, Algorithmica 29
- Vallée
- 2001
(Show Context)
Citation Context ...te the main works of our group which are closely related to the present survey. Note that the Caen group introduced dynamical methods in another algorithmic domain, the Information Theory Domain. See =-=[104]-=- for an instance of such a work. Average-case dynamical analysis. A precise description of Euclidean analyses can be found in the following papers. Paper [35] is itself a survey paper where transfer o... |

29 | Continued fraction Algorithms, Functional operators and Structure constants, Theoretical Computer Science 194
- Flajolet, Vallee
- 1998
(Show Context)
Citation Context ...main, the Information Theory Domain. See [104] for an instance of such a work. Average-case dynamical analysis. A precise description of Euclidean analyses can be found in the following papers. Paper =-=[35]-=- is itself a survey paper where transfer operators are used for analysing the Euclid Algorithm together some of its generalization on higher dimensions. Paper [101] introduces for the first time the u... |

29 |
The number of steps in the Euclidean algorithm
- Hensley
- 1994
(Show Context)
Citation Context ...e polynomial gcd can be found in [38, 56].sEUCLIDEAN DYNAMICS 343 Distributional analysis. Concerning the standard Euclidean algorithm and the number of steps (i.e., the constant cost c ≡ 1), Hensley =-=[44]-=- has obtained a Central Limit Theorem, and a Local Limit Theorem with speed of convergence O((log N) −1/24 ). Hensley has used a transfer operator Hs,0, to obtain distributional results on rational tr... |

28 |
A zata function related to the continued fraction transformation
- Mayer
- 1976
(Show Context)
Citation Context ...and is compact. It is moreover nuclear of order 0 (in the sense of Grothendieck [40], [41]). Property (i) is essential here, and it is necessary that the closure M of disk M is mapped inside M. Mayer =-=[72, 73, 71]-=- deeply studied the transfer operator Hs relative to the Classical Euclidean Dynamical System [often called the Ruelle-Mayer operator] and proved that Hs satisfies properties (i), (ii), (iii) for some... |

28 |
Real numbers with bounded partial quotients: a survey, Enseign
- Shallit
- 1992
(Show Context)
Citation Context ...tance, the reals whose all digits in continued fraction expansion are at most M are deeply studied since they are badly approximable by rationals, and intervene in many contexts of number theory (see =-=[90]-=-). Hensley precisely studied this set EM, its Hausdorff dimension tM, and exhibits the asymptotic behaviour of |tM − 1| when the constraint bound M tends to ∞ [45]. These results have been generalized... |

27 |
Worst-case complexity bounds for algorithms in the theory of integral quadratic forms
- Lagarias
- 1980
(Show Context)
Citation Context ... the Gaussian algorithm, both in the average case and in probability. Like its one–dimensional counterpart, the algorithm is known to be of worst–case logarithmic complexity, a result due to Lagarias =-=[60]-=-, with best possible bounds being provided by Vallée [100]. The analysis provided in [26] is another instance of a dynamical analysis; it deals with the transfer operator Hs relative to the Classical ... |

26 |
Continued Fractions. The University of Chicago
- Khinchin
- 1964
(Show Context)
Citation Context ...as first studied by Gauss himself. The density transformer, also known as the Perron-Frobenius operator, was introduced early in the study of continued fractions (see for instance Lévy [63], Khinchin =-=[54]-=-, Kuzmin [59], Wirsing [111] and Babenko [3]). It was more recently deeply studied by Mayer, in a sequence of papers [72, 73, 71, 74, 75, 76]. The Centered system was studied by Rieger [83, 84], the E... |

26 |
Analysis of the subtractive algorithm for greatest common divisors
- Yao, Knuth
- 1975
(Show Context)
Citation Context ...ational) density. Then, under the conjecture and the heuristic hypothesis, he obtains the average–case analysis of the Binary Algorithm (B). The Subtractive algorithm (T) was studied by Knuth and Yao =-=[112]-=-. Results on the average-case analysis of the polynomial gcd can be found in [38, 56].sEUCLIDEAN DYNAMICS 343 Distributional analysis. Concerning the standard Euclidean algorithm and the number of ste... |

25 | Exponential error terms for growth functions of negatively curved surfaces
- Pollicott, Sharp
- 1998
(Show Context)
Citation Context ...ater on, Pollicott and Sharp used Dolgopyat’s bounds together with Perron’s formula to find error terms in asymptotic estimates for geodesic flows on surfaces of variable negative curvature; see e.g. =-=[81]-=-, where only univariate Dirichlet series with positive cofficients appear. Truncated trajectories and metrical properties of continued fractions.. Central Limit Theorem for costs [stated here as Theor... |

24 |
The Jacobi-Perron algorithm; its theory and applications
- BERNSTEIN
- 1971
(Show Context)
Citation Context ... n + O(n), n → ∞, α where α is the entropy of the CFE dynamical system and the constant which appears in the term O(n) is a variant of the Porter constant [see Section 7. 10]. Other generalizations. (=-=[95, 16, 10, 18]-=-). The Jacobi–Perron algorithm, or its variants, were introduced to generalize the continued fraction algorithm, from the point of view of the construction of a rational approximation of a vector of R... |

24 |
The number of steps in the Euclidean algorithm
- Dixon
- 1970
(Show Context)
Citation Context ...al (de Lagny, Heilbronn) to probabilistic (Dixon). Average-case analysis. The standard Euclidean Algorithm was analysed first in the average-case around 1969 independently by Heilbronn [42] and Dixon =-=[28]-=-. The centered algorithm was studied by Rieger [83]. Brent [14] has analysed the Binary algorithm under some heuristic hypotheses. Brent’s work deals with the operator H relative to the plain [i.e., n... |

23 | Euclidean algorithms are Gaussian
- Baladi, Vallée
(Show Context)
Citation Context ...ee parameters of interest, the total cost relative to a digit– cost of moderate growth, the size of the ending continuant U at a fraction δ of the depth, the bit–complexity. These results appeared in =-=[7]-=-[5] for the total cost C and Good algorithms of Type 1. They are generalized here to all the Algorithms of the Good Class [even those of Type 2]. Results about continuants and bit–complexity can be fo... |

23 |
Markov approximations and decay of correlations for Anosov flows
- Chernov
- 1998
(Show Context)
Citation Context ...d by Ruelle, in connection with his thermodynamic formalism (see for instance [85, 86]). Then Mayer has applied such operators to the classic continued fraction transformation. After works of Chernov =-=[21]-=-, Dolgopyat [29] was interested in the decay of correlations for hyperbolic flows satisfying some uniform nonintegrability condition (UNI). Later on, Pollicott and Sharp used Dolgopyat’s bounds togeth... |

22 |
Dynamics of the Binary Euclidean Algorithm: Functional Analysis and Operators., Algorithmica
- Vallee
- 1998
(Show Context)
Citation Context ...) x + 2b was introduced by Brent [14] in his study of the binary gcd, but it is not easy to deal with. This is why the “induced version” of the transfer operator has been further introduced by Vallée =-=[105]-=-. Between two exchanges, there is a sequence of internal loops (composed with subtractions and binary shifts) that can be written as v = u+2 b1 v1, v1 = u + 2 b2 v2, v2 = u + 2 b3 v3, . . . vℓ−1 = u +... |

20 |
Metrical theory of continued fractions
- Iosifescu, Kraaikamp
- 2002
(Show Context)
Citation Context ...0] which have been generalized by Vallée [101]. These results are extended to our general framework for the first time in the present paper. A survey for metrical properties of continued fractions is =-=[50]-=-. 9.3. Euclidean Analysis. Inside the Euclidean framework, most of dynamical studies concern the continuous point of view [metric properties of continued fraction expansions for instance], and not the... |

19 |
Analytic Combinatorics, Book in preparation
- Flajolet, Sedgewick
- 1999
(Show Context)
Citation Context ...e case in analysis of algorithms, generating functions. The crucial rôle of generating functions in the analysis of data structures and algorithms is well described in books by Flajolet and Sedgewick =-=[33, 34]-=-. i=1sEUCLIDEAN DYNAMICS 289 We consider a general parameter R defined on Ω, � Ω, and we wish to study its distribution on ΩN, when endowed with the uniform probability. Our final probabilistic tool [... |

18 |
Sur les lois de probabilité dont dépendent les quotients complets et incomplets d’une fraction continue
- Lévy
- 1929
(Show Context)
Citation Context ...lidean system was first studied by Gauss himself. The density transformer, also known as the Perron-Frobenius operator, was introduced early in the study of continued fractions (see for instance Lévy =-=[63]-=-, Khinchin [54], Kuzmin [59], Wirsing [111] and Babenko [3]). It was more recently deeply studied by Mayer, in a sequence of papers [72, 73, 71, 74, 75, 76]. The Centered system was studied by Rieger ... |

18 |
Multidimensional continued fractions
- Schweiger
- 2000
(Show Context)
Citation Context ...in the worst case in 1733 by de Lagny, The Centered algorithm (K) has been considered by Rieger [83]. The Even Algorithm is introduced by Eisenstein [32]. The Even and Odd algorithms are described in =-=[94, 95]-=-. Pseudo–Euclidean Algorithms. Two of these Algorithms, the Pseudo-Classical ( ˘ G), the Pseudo-Centered ( ˘ K) have been studied by Shallit [88] who limited himself to a worst-case analysis and wrote... |

18 |
Opérateurs de Ruelle-Mayer généralisés et analyse en moyenne des algorithmes de Gauss et d’Euclide
- Vallée
- 1997
(Show Context)
Citation Context ...stract framework and references to the pioneering paper of Nagaev [78]. The situation is less clear for continuants: There are previous works due to Philipp [80] which have been generalized by Vallée =-=[101]-=-. These results are extended to our general framework for the first time in the present paper. A survey for metrical properties of continued fractions is [50]. 9.3. Euclidean Analysis. Inside the Eucl... |

17 | Average bit–complexity of Euclidean Algorithms
- Akhavi, Vallée
- 2000
(Show Context)
Citation Context ...th its weighted version H (c) s defined in Figure 4 via the relation S [1] C (s) = (I − Hs) −1 ◦ H (c) s ◦ (I − Hs) −1 [1](η). (5.7) For the second moment, with a supplementary derivation, one gets S =-=[2]-=- C (s) = (I − Hs) −1 ◦ H (c2 ) s ◦ (I − Hs) −1 [1](0)+ (5.8) +2(I − Hs) −1 ◦ H (c) s ◦ (I − Hs) −1 ◦ H (c) s ◦ (I − Hs) −1 [1](0). We observe that all these Dirichlet series have the same structure: q... |

17 |
Gauss’ algorithm revisited
- Vallée
- 1991
(Show Context)
Citation Context ...probability. Like its one–dimensional counterpart, the algorithm is known to be of worst–case logarithmic complexity, a result due to Lagarias [60], with best possible bounds being provided by Vallée =-=[100]-=-. The analysis provided in [26] is another instance of a dynamical analysis; it deals with the transfer operator Hs relative to the Classical Euclidean Algorithm with the special value s = 2, and all ... |

17 | Dynamical analysis of a class of Euclidean algorithms
- Vallee
- 2003
(Show Context)
Citation Context ... algorithms], and uses them for obtaining Theorem 2 for continuants in the case of the Classical Euclidean Algorithm. More general transfer operator with two variables are introduced in [104]. Papers =-=[106, 107]-=- provide an unifying point of view on Algorithms of Type 1 and 2. The analysis of the Binary Algorithm can be found in [105], while Daireaux explicits the Plus-Minus Dynamical System in [23]. A recent... |