# On subsequences of convergents to a quadratic irrational given by some numerical schemes

Benoît Rittaud^{[1]}

- [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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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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