On subsequences of convergents to a quadratic irrational given by some numerical schemes
- [1] Laboratoire Arithmétique, Géométrie et Applications Université Paris-13, Institut Galilée 99 avenue Jean-Baptiste Clément F - 93 430 Villetaneuse.
Journal de Théorie des Nombres de Bordeaux (2010)
- Volume: 22, Issue: 2, page 449-474
- ISSN: 1246-7405
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Abstract
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topRittaud, Benoît. "On subsequences of convergents to a quadratic irrational given by some numerical schemes." Journal de Théorie des Nombres de Bordeaux 22.2 (2010): 449-474. <http://eudml.org/doc/116414>.
@article{Rittaud2010,
abstract = {Given a quadratic irrational $\alpha $, we are interested in how some numerical schemes applied to a convenient function $f$ provide subsequences of convergents to $\alpha $. 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.},
affiliation = {Laboratoire Arithmétique, Géométrie et Applications Université Paris-13, Institut Galilée 99 avenue Jean-Baptiste Clément F - 93 430 Villetaneuse.},
author = {Rittaud, Benoît},
journal = {Journal de Théorie des Nombres de Bordeaux},
keywords = {quadratic irrational; continued fraction expansion; subsequences of convergents},
language = {eng},
number = {2},
pages = {449-474},
publisher = {Université Bordeaux 1},
title = {On subsequences of convergents to a quadratic irrational given by some numerical schemes},
url = {http://eudml.org/doc/116414},
volume = {22},
year = {2010},
}
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
PB - Université Bordeaux 1
VL - 22
IS - 2
SP - 449
EP - 474
AB - Given a quadratic irrational $\alpha $, we are interested in how some numerical schemes applied to a convenient function $f$ provide subsequences of convergents to $\alpha $. 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.
LA - eng
KW - quadratic irrational; continued fraction expansion; subsequences of convergents
UR - http://eudml.org/doc/116414
ER -
References
top- E. Burger, On Newton’s method and rational approximations to quadratic irrationals. Canad. Math. Bull. 47 (2004), 12–16. Zbl1080.11007MR2032263
- G. Hardy and E. Wright, An Introduction to the Theory of Numbers. Oxford University Press, 1965. Zbl0020.29201
- T. Komatsu, Continued fractions and Newton’s approximations, II. Fibonacci Quart. 39 (2001), 336–338. Zbl0992.11008MR1851533
- G. Rieger, The golden section and Newton approximation. Fibonacci Quart. 37 (1999), 178–179. Zbl0943.11004MR1690469
- D. Rosen, A class of continued fractions associated with certain properly discontinuous groups. Duke Math. J. 21 (1954), 549–563. Zbl0056.30703MR65632
- 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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