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15
An Averagecase Analysis of the Gaussian Algorithm for Lattice Reduction
, 1996
"... .The Gaussian algorithm for lattice reduction in dimension 2 is analysed under its standard version. It is found that, when applied to random inputs in a continuous model, the complexity is constant on average, the probability distribution decays geometrically, and the dynamics is characterized by a ..."
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Cited by 40 (7 self)
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.The Gaussian algorithm for lattice reduction in dimension 2 is analysed under its standard version. It is found that, when applied to random inputs in a continuous model, the complexity is constant on average, the probability distribution decays geometrically, and the dynamics is characterized by a conditional invariant measure. The proofs make use of connections between lattice reduction, continued fractions, continuants, and functional operators. Analysis in the discrete model and detailed numerical data are also presented. Une analyse en moyenne de l'algorithme de Gauss de r'eduction des r'eseaux R'esum'e. L'algorithme de r'eduction des r'eseaux en dimension 2 qui est du `a Gauss est analys'e sous sa forme dite standard. Il est 'etabli ici que, sous un mod`ele continu, sa complexit'e est constante en moyenne et que la distribution de probabilit'es associ'ee decroit g'eom'etriquement tandis que la dynamique est caract'eris'ee par une densit'e conditionnelle invariante. Les preuves f...
Continued Fraction Algorithms, Functional Operators, and Structure Constants
, 1996
"... Continued fractions lie at the heart of a number of classical algorithms like Euclid's greatest common divisor algorithm or the lattice reduction algorithm of Gauss that constitutes a 2dimensional generalization. This paper surveys the main properties of functional operators,  transfer operat ..."
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Cited by 28 (4 self)
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Continued fractions lie at the heart of a number of classical algorithms like Euclid's greatest common divisor algorithm or the lattice reduction algorithm of Gauss that constitutes a 2dimensional generalization. This paper surveys the main properties of functional operators,  transfer operators  due to Ruelle and Mayer (also following Lévy, Kuzmin, Wirsing, Hensley, and others) that describe precisely the dynamics of the continued fraction transformation. Spectral characteristics of transfer operators are shown to have many consequences, like the normal law for logarithms of continuants associated to the basic continued fraction algorithm and a purely analytic estimation of the average number of steps of the Euclidean algorithm. Transfer operators also lead to a complete analysis of the "Hakmem" algorithm for comparing two rational numbers via partial continued fraction expansions and of the "digital tree" algorithm for completely sorting n real numbers by means of ...
Euclidean algorithms are Gaussian
, 2003
"... Abstract. We prove a Central Limit Theorem for a general class of costparameters associated to the three standard Euclidean algorithms, with optimal speed of convergence, and error terms for the mean and variance. For the most basic parameter of the algorithms, the number of steps, we go further an ..."
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Cited by 22 (10 self)
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Abstract. We prove a Central Limit Theorem for a general class of costparameters associated to the three standard Euclidean algorithms, with optimal speed of convergence, and error terms for the mean and variance. For the most basic parameter of the algorithms, the number of steps, we go further and prove a Local Limit Theorem (LLT), with speed of convergence O((log N) −1/4+ǫ). This extends and improves the LLT obtained by Hensley [27] in the case of the standard Euclidean algorithm. We use a “dynamical analysis ” methodology, viewing an algorithm as a dynamical system (restricted to rational inputs), and combining tools imported from dynamics, such as the crucial transfer operators, with various other techniques: Dirichlet series, Perron’s formula, quasipowers theorems, the saddle point method. Dynamical analysis had previously been used to perform averagecase analysis of algorithms. For the present (dynamical) analysis in distribution, we require precise estimates on the transfer operators, when a parameter varies along vertical lines in the complex plane. Such estimates build on results obtained only recently by Dolgopyat in the context of continuoustime dynamics [20]. 1.
Dynamical Analysis of a Class of Euclidean Algorithms
"... We develop a general framework for the analysis of algorithms of a broad Euclidean type. The averagecase complexity of an algorithm is seen to be related to the analytic behaviour in the complex plane of the set of elementary transformations determined by the algorithm. The methods rely on properti ..."
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Cited by 17 (4 self)
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We develop a general framework for the analysis of algorithms of a broad Euclidean type. The averagecase complexity of an algorithm is seen to be related to the analytic behaviour in the complex plane of the set of elementary transformations determined by the algorithm. The methods rely on properties of transfer operators suitably adapted from dynamical systems theory. As a consequence, we obtain precise averagecase analyses of algorithms for evaluating the Jacobi symbol of computational number theory fame, thereby solving conjectures of Bach and Shallit. These methods also provide a unifying framework for the analysis of an entire class of gcdlike algorithms together with new results regarding the probable behaviour of their cost functions. 1
Digits and Continuants in Euclidean Algorithms. Ergodic versus Tauberian Theorems
, 2000
"... We obtain new results regarding the precise average case analysis of the main quantities that intervene in algorithms of a broad Euclidean type. We develop a general framework for the analysis of such algorithms, where the averagecase complexity of an algorithm is related to the analytic behaviou ..."
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Cited by 14 (5 self)
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We obtain new results regarding the precise average case analysis of the main quantities that intervene in algorithms of a broad Euclidean type. We develop a general framework for the analysis of such algorithms, where the averagecase complexity of an algorithm is related to the analytic behaviour in the complex plane of the set of elementary transformations determined by the algorithms. The methods rely on properties of transfer operators suitably adapted from dynamical systems theory and provide a unifying framework for the analysis of the main parameters digits and continuants that intervene in an entire class of gcdlike algorithms. We operate a general transfer from the continuous case (Continued Fraction Algorithms) to the discrete case (Euclidean Algorithms), where Ergodic Theorems are replaced by Tauberian Theorems.
The arithmeticgeometric scaling spectrum for continued fractions
 Arkivför Matematik
"... Abstract. To compare the continued fraction digits with the denominators of the corresponding approximants we introduce the arithmeticgeometric scaling. To determine its multifractal spectrum completely we impose a number theoretical free energy function and show that the Hausdor dimension of sets ..."
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Cited by 3 (1 self)
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Abstract. To compare the continued fraction digits with the denominators of the corresponding approximants we introduce the arithmeticgeometric scaling. To determine its multifractal spectrum completely we impose a number theoretical free energy function and show that the Hausdor dimension of sets consisting of irrationals with the same scaling exponent coincides with the Legendre transform of this free energy function. Furthermore, we determine the asymptotic of the local behaviour of the spectrum at the right boundary point and discuss a connection to the set of irrationals with continued fraction digits exceeding a given number which tends to in nity.
On The GaussKuzminWirsing Constant
, 1995
"... . We give strong supporting evidence for the fact that the GaussKuzminWirsing constant is j 2 j = 0:30366 30028 98732 6585974481 21901 55623 \Sigma 10 \Gamma31 : If x 0 is uniformly distributed in [0; 1], then the nth iterate x n = T (n) (x 0 ) of the continued fraction transformation T (x) ..."
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Cited by 2 (0 self)
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. We give strong supporting evidence for the fact that the GaussKuzminWirsing constant is j 2 j = 0:30366 30028 98732 6585974481 21901 55623 \Sigma 10 \Gamma31 : If x 0 is uniformly distributed in [0; 1], then the nth iterate x n = T (n) (x 0 ) of the continued fraction transformation T (x) = f 1 x g (where fxg = x \Gamma bxc denotes the fractional part) has a distribution F n (x) = Pr(x n ! x) that converges geometrically to the invariant density F1 (x) found by Gauß, namely, F1 (x) = log 2 (1 + x) : The rate of convergence is exponential being of the form O( n 2 ), as established by Wirsing [8] using operator methods (the PerronFrobenius theory). Babenko [1] and Mayer (see [7] for a survey) later provided a complete spectral decomposition of the form F n (x) = F1 (x) + 1 X k=2 n k \Psi k (x); for some eigenvalues k and some analytic eigenfunctions \Psi k (x). In the course of a joint work with H. Daud'e relative to an averagecase analysis of lattice reduction...
Euclidean dynamics
 Discrete and Continuous Dynamical Systems
"... 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 tran ..."
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Cited by 2 (1 self)
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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 quasipowers theorems. Such dynamical analyses can be used to perform the averagecase analysis of algorithms, but also (dynamical) analysis in distribution. 1. Introduction. Computing the Greatest Common Divisor [Gcd
The Asymptotic Growth Rate of Random Fibonacci Type Sequences.” The Fibonacci Quarterly 43
, 2005
"... In this paper, we use ergodic theory to compute the aysmptotic growth rate of a family of random Fibonacci type sequences. This extends the result in [2]. We also prove some Lochstype results regarding the effectiveness of various number theoretic expansions. 1. ..."
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In this paper, we use ergodic theory to compute the aysmptotic growth rate of a family of random Fibonacci type sequences. This extends the result in [2]. We also prove some Lochstype results regarding the effectiveness of various number theoretic expansions. 1.
Preface
"... The main justification for this book is that there have been significant advances in continued fractions over the past decade, but these remain for the most part scattered across the literature, and under the heading of topics from algebraic number theory to theoretical plasma physics. We now have a ..."
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The main justification for this book is that there have been significant advances in continued fractions over the past decade, but these remain for the most part scattered across the literature, and under the heading of topics from algebraic number theory to theoretical plasma physics. We now have a better understanding of the rate at which assorted continued fraction or greatest common denominator (gcd) algorithms complete their tasks. The number of steps required to complete a gcd calculation, for instance, has a Gaussian normal distribution. We know a lot more about badly approximable numbers. There are several related threads here. A badly approximable number is a number x such that {qp − qx: p, q ∈ Z and q ̸ = 0} is bounded below by a positive constant; badly approximable numbers have continued fraction expansions with bounded partial quotients, and so we are led to consider a kind of Cantor set EM consisting of all x ∈ [0, 1] such that the partial quotients of x are bounded above by M. The notion of a badly approximable rational number has the ring of crank mathematics, but it is quite natural to study the set of rationals r with partial quotients bounded by M. The number of such rationals with denominators up to n, say, turns out to be closely related to the Hausdorff dimension of EM, (comparable to n2dimEM) which is in turn related to the spectral radius of linear operators LM,s, acting on some suitably chosen space of functions f, and given by LM,sf(t) = ∑m k=1 (k + t) −sf(1/(k + t)). Similar operators have been studied by, among others, David Ruelle, in connection with theoretical onedimensional plasmas, and they are related to entropy. Alongside these developments there has been a dramatic increase in the computational power available to investigators. This has been helpful on the theoretical side, as one is more likely to seek a proof for a result when,