Given a quadratic irrational , we are interested in how some numerical schemes applied to a convenient function provide subsequences of convergents to . 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 which provide arithmetical subsequences of convergents.
Un irrationnel quadratique étant donné, nous nous intéressons à la manière dont une fonction convenablement choisie produit des sous-suites de réduites de . 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 qui fournissent des sous-suites arithmétiques de réduites.
@article{JTNB_2010__22_2_449_0, author = {Rittaud, Beno{\^\i}t}, title = {On subsequences of convergents to a quadratic irrational given by some numerical schemes}, journal = {Journal de th\'eorie des nombres de Bordeaux}, pages = {449--474}, publisher = {Universit\'e Bordeaux 1}, volume = {22}, number = {2}, year = {2010}, doi = {10.5802/jtnb.726}, zbl = {1223.11087}, mrnumber = {2769073}, language = {en}, url = {http://archive.numdam.org/articles/10.5802/jtnb.726/} }
TY - JOUR AU - Rittaud, Benoît TI - On subsequences of convergents to a quadratic irrational given by some numerical schemes JO - Journal de théorie des nombres de Bordeaux PY - 2010 SP - 449 EP - 474 VL - 22 IS - 2 PB - Université Bordeaux 1 UR - http://archive.numdam.org/articles/10.5802/jtnb.726/ DO - 10.5802/jtnb.726 LA - en ID - JTNB_2010__22_2_449_0 ER -
%0 Journal Article %A Rittaud, Benoît %T On subsequences of convergents to a quadratic irrational given by some numerical schemes %J Journal de théorie des nombres de Bordeaux %D 2010 %P 449-474 %V 22 %N 2 %I Université Bordeaux 1 %U http://archive.numdam.org/articles/10.5802/jtnb.726/ %R 10.5802/jtnb.726 %G en %F JTNB_2010__22_2_449_0
Rittaud, Benoît. 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. http://archive.numdam.org/articles/10.5802/jtnb.726/
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