On subsequences of convergents to a quadratic irrational given by some numerical schemes
Journal de Théorie des Nombres de Bordeaux, Volume 22 (2010) no. 2, pp. 449-474.

Given a quadratic irrational α, we are interested in how some numerical schemes applied to a convenient function f provide subsequences of convergents to α. We investigate three numerical schemes: secant-like methods and formal generalizations, which lead to linear recurring subsequences; the false position method, which leads to arithmetical subsequences of convergents and gives some interesting series expansions; Newton’s method, for which we complete a result of Edward Burger [1] about the existence of some functions f which provide arithmetical subsequences of convergents.

Un irrationnel quadratique α étant donné, nous nous intéressons à la manière dont une fonction f convenablement choisie produit des sous-suites de réduites de α. Nous étudions trois schémas numériques  : les méthodes type sécante et certaines généralisations formelles, qui conduisent à des sous-suites à récurrence linéaire  ; la méthode de la fausse position, qui conduit à des sous-suites arithmétiques de réduites et donne quelques intéressants développement en série  ; la méthode de Newton, pour laquelle nous complétons un résultat d’Edward Burger [1] sur l’existence de fonctions f qui fournissent des sous-suites arithmétiques de réduites.

Received:
Published online:
DOI: 10.5802/jtnb.726
Benoît Rittaud 1

1 Laboratoire Arithmétique, Géométrie et Applications Université Paris-13, Institut Galilée 99 avenue Jean-Baptiste Clément F - 93 430 Villetaneuse.
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Benoît Rittaud. On subsequences of convergents to a quadratic irrational given by some numerical schemes. Journal de Théorie des Nombres de Bordeaux, Volume 22 (2010) no. 2, pp. 449-474. doi : 10.5802/jtnb.726. https://jtnb.centre-mersenne.org/articles/10.5802/jtnb.726/

[1] E. Burger, On Newton’s method and rational approximations to quadratic irrationals. Canad. Math. Bull. 47 (2004), 12–16. | MR: 2032263 | Zbl: 1080.11007

[2] G. Hardy and E. Wright, An Introduction to the Theory of Numbers. Oxford University Press, 1965.

[3] T. Komatsu, Continued fractions and Newton’s approximations, II. Fibonacci Quart. 39 (2001), 336–338. | MR: 1851533 | Zbl: 0992.11008

[4] G. Rieger, The golden section and Newton approximation. Fibonacci Quart. 37 (1999), 178–179. | MR: 1690469 | Zbl: 0943.11004

[5] D. Rosen, A class of continued fractions associated with certain properly discontinuous groups. Duke Math. J. 21 (1954), 549–563. | MR: 65632 | Zbl: 0056.30703

[6] J.-A. Serret, Sur le développement en fraction continue de la racine carrée d’un nombre entier. J. Math. Pures Appl. XII (1836), 518–520.

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