Results 1  10
of
34
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 ..."
Abstract

Cited by 28 (12 self)
 Add to MetaCart
(Show Context)
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.
Dynamics of the Binary Euclidean Algorithm: Functional Analysis and Operators
 ALGORITHMICA
, 1998
"... We provide here a complete averagecase analysis of the binary continued fraction representation of a random rational whose numerator and denominator are odd and less than N. We analyze the three main parameters of the binary continued fraction expansion, namely, the height, the number of steps of ..."
Abstract

Cited by 24 (3 self)
 Add to MetaCart
We provide here a complete averagecase analysis of the binary continued fraction representation of a random rational whose numerator and denominator are odd and less than N. We analyze the three main parameters of the binary continued fraction expansion, namely, the height, the number of steps of the binary Euclidean algorithm, and finally the sum of the exponents of powers of 2 contained in the numerators of the binary continued fraction. The average values of these parameters are shown to be asymptotic to Ai log N, and the three constants Ai are related to the invariant measure of the Perron–Frobenius operator linked to this dynamical system. The binary Euclidean algorithm was previously studied in 1976 by Brent who provided a partial analysis of the number of steps, based on a heuristic model and some unproven conjecture. Our methods are quite different, not relying on unproven assumptions, and more general, since they allow us to study all the parameters of the binary continued fraction expansion.
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 ..."
Abstract

Cited by 19 (5 self)
 Add to MetaCart
(Show Context)
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
Average BitComplexity of Euclidean Algorithms
 Proceedings ICALP’00, Lecture Notes Comp. Science 1853, 373–387
, 2000
"... We obtain new results regarding the precise average bitcomplexity of five algorithms of a broad Euclidean type. We develop a general framework for analysis of algorithms, where the averagecase complexity of an algorithm is seen to be related to the analytic behaviour in the complex plane of the set ..."
Abstract

Cited by 18 (7 self)
 Add to MetaCart
(Show Context)
We obtain new results regarding the precise average bitcomplexity of five algorithms of a broad Euclidean type. We develop a general framework for analysis of algorithms, where 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 algorithms. The methods rely on properties of transfer operators suitably adapted from dynamical systems theory and provide a unifying framework for the analysis of an entire class of gcdlike algorithms. Keywords: Averagecase Analysis of algorithms, BitComplexity, Euclidean Algorithms, Dynamical Systems, Ruelle operators, Generating Functions, Dirichlet Series, Tauberian Theorems. 1 Introduction Motivations. Euclid's algorithm was analysed first in the worst case in 1733 by de Lagny, then in the averagecase around 1969 independently by Heilbronn [12] and Dixon [6], and finally in distribution by Hensley [13] who proved in 1994 that the Eu...
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 ..."
Abstract

Cited by 16 (6 self)
 Add to MetaCart
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.
Some LinearTime Algorithms for Systolic Arrays
, 2000
"... We survey some recent results on lineartime algorithms for systolic arrays. In particular, we show how the greatest common divisor (GCD) of two polynomials of degree n over a finite field can be computed in time O(n) on a linear systolic array of O(n) cells; similarly for the GCD of two nbit binar ..."
Abstract

Cited by 12 (7 self)
 Add to MetaCart
We survey some recent results on lineartime algorithms for systolic arrays. In particular, we show how the greatest common divisor (GCD) of two polynomials of degree n over a finite field can be computed in time O(n) on a linear systolic array of O(n) cells; similarly for the GCD of two nbit binary numbers. We show how n by n Toeplitz systems of linear equations can be solved in time O(n) on a linear array of O(n) cells, each of which has constant memory size (independent of n). Finally, we outline how a twodimensional square array of O(n) by O(n) cells can be used to solve (to working accuracy) the eigenvalue problem for a symmetric real n by n matrix in time O(nS(n)). Here S(n) is a slowly growing function of n; for practical purposes S(n) can be regarded as a constant. In addition to their theoretical interest, these results have potential applications in the areas of errorcorrecting codes, symbolic and algebraic computation, signal processing and image processing. For example, systolic GCD arrays for error correction have been implemented with the microprogrammable “PSC” chip.
An Analysis of Lehmer's Euclidean GCD Algorithm
 Proceedings Of The 1995 International Symposium On Symbolic And Algebraic Computation
, 1995
"... Let u and v be positive integers. We show that a slightly modified version of D. H. Lehmer's greatest common divisor algorithm will compute gcd(u; v) (with u ? v) using at most Of(log u log v)=k + k log v + log u + k 2 g bit operations and O(log u + k2 2k ) space, where k is the number of ..."
Abstract

Cited by 7 (3 self)
 Add to MetaCart
(Show Context)
Let u and v be positive integers. We show that a slightly modified version of D. H. Lehmer's greatest common divisor algorithm will compute gcd(u; v) (with u ? v) using at most Of(log u log v)=k + k log v + log u + k 2 g bit operations and O(log u + k2 2k ) space, where k is the number of bits in the multiprecision base of the algorithm. This is faster than Euclid's algorithm by a factor that is roughly proportional to k. Letting n be the number of bits in u and v, and setting k = b(log n)=4c, we obtain a subquadratic running time of O(n 2 = log n) in linear space. 1 Introduction Let u and v be positive integers. The greatest common divisor (GCD) of u and v is the largest integer d such that d divides both u and v. The most wellknown algorithm for computing GCDs is the Euclidean Algorithm. Much is known about this algorithm: the number of iterations required is \Theta(log v), and the worstcase running time is \Theta(log u log v), where time is measured in bit operation...
A Binary Algorithm for the Jacobi Symbol
 ACM SIGSAM Bulletin
, 1993
"... We present a new algorithm to compute the Jacobi symbol, based on Stein's binary algorithm for the greatest common divisor, and we determine the worstcase behavior of this algorithm. Our implementation of the algorithm runs approximately 725% faster than traditional methods on inputs of size ..."
Abstract

Cited by 6 (1 self)
 Add to MetaCart
(Show Context)
We present a new algorithm to compute the Jacobi symbol, based on Stein's binary algorithm for the greatest common divisor, and we determine the worstcase behavior of this algorithm. Our implementation of the algorithm runs approximately 725% faster than traditional methods on inputs of size 1001000 decimal digits. 1 Introduction Efficient computation of the Jacobi symbol \Gamma a n \Delta is an important component of the Monte Carlo primality test of Solovay and Strassen [9]. Algorithms for computing the Jacobi symbol can also be found on symbolic algebra systems such as Mathematica and Maple. Several efficient algorithms modeled on Euclid's algorithm for computing the greatest common divisor (gcd) have been proposed and analyzed; see, for example, [12, 3, 8]. Indeed, it is possible to compute \Gamma a n \Delta in O((log a)(log n)) bit operations using the "naive arithmetic" model. Using Schonhage's result [7], it is possible (see [1]) to compute \Gamma a n \Delta (...
Euclidean dynamics
 DISCRETE AND CONTINUOUS DYNAMICAL SYSTEMS
, 2006
"... 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 ope ..."
Abstract

Cited by 4 (2 self)
 Add to MetaCart
(Show Context)
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.