Euclidean algorithm

Noun

Euclidean algorithm (plural Euclidean algorithms)

1. (historical) Any of certain algorithms first described in Euclid's Elements.
   • The Euclidean algorithms for finding a compound ratio also allowed a ratio and an inverse ratio, and more than two ratios to be compounded, since each compounded pair is equivalent to a single ratio between lines.1998, John J. Roche, The Mathematics of Measurement: A Critical History, The Athlone Press, page 44
2. Specifically, a method, based on a division algorithm, for finding the greatest common divisor (gcd) of two given integers; any of certain variations or generalisations of said method.
   • In a quadratic field Q (√), D a squarefree integer, with class number 1 any algebraic integer can be decomposed uniquely into primes but for only 21 domains Euclidean algorithms are known. We prove that for D<-19 even remainder sequences with possibly nondecreasing norms cannot determine the GCD of...1985, Erich Kaltofen, Heinrich Rolletschek, Arithmetic in Quadratic Fields with Unique Factorization, Bob F. Caviness (editor), EUROCAL '85, European Conference on Computer Algebra, Linz...
   • In this paper, we provide new analyses that characterize the precise average bit-complexity of a class of Euclidean algorithms.
     We consider here five algorithms that are all classical variations of the Euclidean algorithm and are called Classical (𝒢), By-Excess (ℒ), Centered (𝒦), Subtractive (𝒯)...2003, Ali Akhavi, Brigitte Vallée, Average Bit-Complexity of Euclidean Algorithms, Ugo Montanari, Jose D.P. Rolim, Emo Welzl (editors), Automata, Languages and Programming: 27th International...